LMFDB - Embedded newform 1050.3.h.c.349.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,3,Mod(349,1050)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1050.349");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1050.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$28.6104277578$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.10
Character	$\chi$	$=$	1050.349
Dual form			1050.3.h.c.349.9

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -1.73205 q^{3} -2.00000 q^{4} -2.44949i q^{6} +(-1.46536 - 6.84490i) q^{7} -2.82843i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.41421i q^{2} -1.73205 q^{3} -2.00000 q^{4} -2.44949i q^{6} +(-1.46536 - 6.84490i) q^{7} -2.82843i q^{8} +3.00000 q^{9} -5.87800 q^{11} +3.46410 q^{12} +2.06416 q^{13} +(9.68016 - 2.07233i) q^{14} +4.00000 q^{16} -2.59754 q^{17} +4.24264i q^{18} -14.3990i q^{19} +(2.53808 + 11.8557i) q^{21} -8.31274i q^{22} -4.73565i q^{23} +4.89898i q^{24} +2.91916i q^{26} -5.19615 q^{27} +(2.93072 + 13.6898i) q^{28} +16.1571 q^{29} +48.3772i q^{31} +5.65685i q^{32} +10.1810 q^{33} -3.67347i q^{34} -6.00000 q^{36} -46.3013i q^{37} +20.3632 q^{38} -3.57522 q^{39} +74.7122i q^{41} +(-16.7665 + 3.58939i) q^{42} -10.4038i q^{43} +11.7560 q^{44} +6.69723 q^{46} -35.0672 q^{47} -6.92820 q^{48} +(-44.7054 + 20.0605i) q^{49} +4.49907 q^{51} -4.12831 q^{52} +39.9926i q^{53} -7.34847i q^{54} +(-19.3603 + 4.14467i) q^{56} +24.9397i q^{57} +22.8495i q^{58} +60.1573i q^{59} +73.1619i q^{61} -68.4156 q^{62} +(-4.39608 - 20.5347i) q^{63} -8.00000 q^{64} +14.3981i q^{66} -39.5941i q^{67} +5.19507 q^{68} +8.20239i q^{69} -39.3463 q^{71} -8.48528i q^{72} +11.8315 q^{73} +65.4800 q^{74} +28.7979i q^{76} +(8.61339 + 40.2343i) q^{77} -5.05613i q^{78} -41.2398 q^{79} +9.00000 q^{81} -105.659 q^{82} -126.222 q^{83} +(-5.07616 - 23.7114i) q^{84} +14.7133 q^{86} -27.9849 q^{87} +16.6255i q^{88} -92.1225i q^{89} +(-3.02473 - 14.1290i) q^{91} +9.47131i q^{92} -83.7917i q^{93} -49.5925i q^{94} -9.79796i q^{96} -4.26795 q^{97} +(-28.3698 - 63.2230i) q^{98} -17.6340 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 48 q^{4} + 72 q^{9}+O(q^{10}) $ 24 * q - 48 * q^4 + 72 * q^9	$ 24 q - 48 q^{4} + 72 q^{9} - 32 q^{11} - 16 q^{14} + 96 q^{16} - 12 q^{21} - 96 q^{29} - 144 q^{36} + 24 q^{39} + 64 q^{44} + 160 q^{46} - 236 q^{49} + 144 q^{51} + 32 q^{56} - 192 q^{64} + 496 q^{71} + 128 q^{74} + 416 q^{79} + 216 q^{81} + 24 q^{84} + 256 q^{86} - 316 q^{91} - 96 q^{99}+O(q^{100}) $ 24 * q - 48 * q^4 + 72 * q^9 - 32 * q^11 - 16 * q^14 + 96 * q^16 - 12 * q^21 - 96 * q^29 - 144 * q^36 + 24 * q^39 + 64 * q^44 + 160 * q^46 - 236 * q^49 + 144 * q^51 + 32 * q^56 - 192 * q^64 + 496 * q^71 + 128 * q^74 + 416 * q^79 + 216 * q^81 + 24 * q^84 + 256 * q^86 - 316 * q^91 - 96 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			1.41421i			0.707107i
$2$			1.41421i			0.707107i
$3$	−1.73205			−0.577350
$3$	−1.73205			−0.577350
$4$	−2.00000			−0.500000
$5$		0			0
$5$		0			0
$6$		−	2.44949i		−	0.408248i
$7$	−1.46536	−	6.84490i	−0.209337	−	0.977844i
$7$	−1.46536	−	6.84490i	−0.209337	−	0.977844i
$8$		−	2.82843i		−	0.353553i
$9$	3.00000			0.333333
$10$		0			0
$11$	−5.87800			−0.534363			−0.267182	−	0.963646i	$-0.586092\pi$
$11$	−5.87800			−0.534363			−0.267182	+	0.963646i	$0.586092\pi$
$12$	3.46410			0.288675
$13$	2.06416			0.158781			0.0793906	−	0.996844i	$-0.474703\pi$
$13$	2.06416			0.158781			0.0793906	+	0.996844i	$0.474703\pi$
$14$	9.68016	−	2.07233i	0.691440	−	0.148024i
$15$		0			0
$16$	4.00000			0.250000
$17$	−2.59754			−0.152796			−0.0763982	−	0.997077i	$-0.524342\pi$
$17$	−2.59754			−0.152796			−0.0763982	+	0.997077i	$0.524342\pi$
$18$			4.24264i			0.235702i
$19$		−	14.3990i		−	0.757840i	−0.925429	−	0.378920i	$-0.876296\pi$
$19$		−	14.3990i		−	0.757840i	0.925429	−	0.378920i	$-0.123704\pi$
$20$		0			0
$21$	2.53808	+	11.8557i	0.120861	+	0.564558i
$22$		−	8.31274i		−	0.377852i
$23$		−	4.73565i		−	0.205898i	−0.994687	−	0.102949i	$-0.967172\pi$
$23$		−	4.73565i		−	0.205898i	0.994687	−	0.102949i	$-0.0328278\pi$
$24$			4.89898i			0.204124i
$25$		0			0
$26$			2.91916i			0.112275i
$27$	−5.19615			−0.192450
$28$	2.93072	+	13.6898i	0.104669	+	0.488922i
$29$	16.1571			0.557140			0.278570	−	0.960416i	$-0.410140\pi$
$29$	16.1571			0.557140			0.278570	+	0.960416i	$0.410140\pi$
$30$		0			0
$31$			48.3772i			1.56055i	0.625434	+	0.780277i	$0.284922\pi$
$31$			48.3772i			1.56055i	−0.625434	+	0.780277i	$0.715078\pi$
$32$			5.65685i			0.176777i
$33$	10.1810			0.308515
$34$		−	3.67347i		−	0.108043i
$35$		0			0
$36$	−6.00000			−0.166667
$37$		−	46.3013i		−	1.25139i	−0.780069	−	0.625694i	$-0.784816\pi$
$37$		−	46.3013i		−	1.25139i	0.780069	−	0.625694i	$-0.215184\pi$
$38$	20.3632			0.535874
$39$	−3.57522			−0.0916724
$40$		0			0
$41$			74.7122i			1.82225i	0.412132	+	0.911124i	$0.364784\pi$
$41$			74.7122i			1.82225i	−0.412132	+	0.911124i	$0.635216\pi$
$42$	−16.7665	+	3.58939i	−0.399203	+	0.0854616i
$43$		−	10.4038i		−	0.241950i	−0.992656	−	0.120975i	$-0.961398\pi$
$43$		−	10.4038i		−	0.241950i	0.992656	−	0.120975i	$-0.0386021\pi$
$44$	11.7560			0.267182
$45$		0			0
$46$	6.69723			0.145592
$47$	−35.0672			−0.746111			−0.373055	−	0.927809i	$-0.621690\pi$
$47$	−35.0672			−0.746111			−0.373055	+	0.927809i	$0.621690\pi$
$48$	−6.92820			−0.144338
$49$	−44.7054	+	20.0605i	−0.912356	+	0.409398i
$50$		0			0
$51$	4.49907			0.0882170
$52$	−4.12831			−0.0793906
$53$			39.9926i			0.754578i	0.926096	+	0.377289i	$0.123144\pi$
$53$			39.9926i			0.754578i	−0.926096	+	0.377289i	$0.876856\pi$
$54$		−	7.34847i		−	0.136083i
$55$		0			0
$56$	−19.3603	+	4.14467i	−0.345720	+	0.0740119i
$57$			24.9397i			0.437539i
$58$			22.8495i			0.393958i
$59$			60.1573i			1.01962i	0.860288	+	0.509808i	$0.170284\pi$
$59$			60.1573i			1.01962i	−0.860288	+	0.509808i	$0.829716\pi$
$60$		0			0
$61$			73.1619i			1.19938i	0.800234	+	0.599688i	$0.204709\pi$
$61$			73.1619i			1.19938i	−0.800234	+	0.599688i	$0.795291\pi$
$62$	−68.4156			−1.10348
$63$	−4.39608	−	20.5347i	−0.0697791	−	0.325948i
$64$	−8.00000			−0.125000
$65$		0			0
$66$			14.3981i			0.218153i
$67$		−	39.5941i		−	0.590956i	−0.955350	−	0.295478i	$-0.904521\pi$
$67$		−	39.5941i		−	0.590956i	0.955350	−	0.295478i	$-0.0954789\pi$
$68$	5.19507			0.0763982
$69$			8.20239i			0.118875i
$70$		0			0
$71$	−39.3463			−0.554173			−0.277087	−	0.960845i	$-0.589369\pi$
$71$	−39.3463			−0.554173			−0.277087	+	0.960845i	$0.589369\pi$
$72$		−	8.48528i		−	0.117851i
$73$	11.8315			0.162075			0.0810377	−	0.996711i	$-0.474177\pi$
$73$	11.8315			0.162075			0.0810377	+	0.996711i	$0.474177\pi$
$74$	65.4800			0.884864
$75$		0			0
$76$			28.7979i			0.378920i
$77$	8.61339	+	40.2343i	0.111862	+	0.522524i
$78$		−	5.05613i		−	0.0648222i
$79$	−41.2398			−0.522023			−0.261012	−	0.965336i	$-0.584056\pi$
$79$	−41.2398			−0.522023			−0.261012	+	0.965336i	$0.584056\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$	−105.659			−1.28852
$83$	−126.222			−1.52075			−0.760374	−	0.649485i	$-0.774984\pi$
$83$	−126.222			−1.52075			−0.760374	+	0.649485i	$0.774984\pi$
$84$	−5.07616	−	23.7114i	−0.0604304	−	0.282279i
$85$		0			0
$86$	14.7133			0.171084
$87$	−27.9849			−0.321665
$88$			16.6255i			0.188926i
$89$		−	92.1225i		−	1.03508i	−0.855658	−	0.517542i	$-0.826847\pi$
$89$		−	92.1225i		−	1.03508i	0.855658	−	0.517542i	$-0.173153\pi$
$90$		0			0
$91$	−3.02473	−	14.1290i	−0.0332388	−	0.155263i
$92$			9.47131i			0.102949i
$93$		−	83.7917i		−	0.900986i
$94$		−	49.5925i		−	0.527580i
$95$		0			0
$96$		−	9.79796i		−	0.102062i
$97$	−4.26795			−0.0439995			−0.0219997	−	0.999758i	$-0.507003\pi$
$97$	−4.26795			−0.0439995			−0.0219997	+	0.999758i	$0.507003\pi$
$98$	−28.3698	−	63.2230i	−0.289488	−	0.645133i
$99$	−17.6340			−0.178121
$100$		0			0
$101$			90.7089i			0.898108i	0.893505	+	0.449054i	$0.148239\pi$
$101$			90.7089i			0.898108i	−0.893505	+	0.449054i	$0.851761\pi$
$102$			6.36264i			0.0623788i
$103$	−136.976			−1.32986			−0.664932	−	0.746904i	$-0.731540\pi$
$103$	−136.976			−1.32986			−0.664932	+	0.746904i	$0.731540\pi$
$104$		−	5.83832i		−	0.0561377i
$105$		0			0
$106$	−56.5581			−0.533567
$107$			23.3746i			0.218454i	0.994017	+	0.109227i	$0.0348375\pi$
$107$			23.3746i			0.218454i	−0.994017	+	0.109227i	$0.965162\pi$
$108$	10.3923			0.0962250
$109$	25.6873			0.235663			0.117832	−	0.993034i	$-0.462406\pi$
$109$	25.6873			0.235663			0.117832	+	0.993034i	$0.462406\pi$
$110$		0			0
$111$			80.1962i			0.722489i
$112$	−5.86144	−	27.3796i	−0.0523343	−	0.244461i
$113$		−	2.51092i		−	0.0222205i	−0.999938	−	0.0111103i	$-0.996463\pi$
$113$		−	2.51092i		−	0.0222205i	0.999938	−	0.0111103i	$-0.00353658\pi$
$114$	−35.2701			−0.309387
$115$		0			0
$116$	−32.3141			−0.278570
$117$	6.19247			0.0529271
$118$	−85.0753			−0.720977
$119$	3.80633	+	17.7799i	0.0319860	+	0.149411i
$120$		0			0
$121$	−86.4491			−0.714456
$122$	−103.467			−0.848087
$123$		−	129.405i		−	1.05208i
$124$		−	96.7543i		−	0.780277i
$125$		0			0
$126$	29.0405	−	6.21700i	0.230480	−	0.0493413i
$127$		−	19.5421i		−	0.153875i	−0.997036	−	0.0769375i	$-0.975486\pi$
$127$		−	19.5421i		−	0.153875i	0.997036	−	0.0769375i	$-0.0245142\pi$
$128$		−	11.3137i		−	0.0883883i
$129$			18.0200i			0.139690i
$130$		0			0
$131$			45.5826i			0.347959i	0.984749	+	0.173980i	$0.0556627\pi$
$131$			45.5826i			0.347959i	−0.984749	+	0.173980i	$0.944337\pi$
$132$	−20.3620			−0.154257
$133$	−98.5595	+	21.0997i	−0.741049	+	0.158644i
$134$	55.9945			0.417869
$135$		0			0
$136$			7.34694i			0.0540217i
$137$			177.239i			1.29372i	0.762610	+	0.646859i	$0.223918\pi$
$137$			177.239i			1.29372i	−0.762610	+	0.646859i	$0.776082\pi$
$138$	−11.5999			−0.0840575
$139$		−	33.1908i		−	0.238783i	−0.992847	−	0.119391i	$-0.961906\pi$
$139$		−	33.1908i		−	0.238783i	0.992847	−	0.119391i	$-0.0380944\pi$
$140$		0			0
$141$	60.7382			0.430767
$142$		−	55.6441i		−	0.391860i
$143$	−12.1331			−0.0848469
$144$	12.0000			0.0833333
$145$		0			0
$146$			16.7323i			0.114605i
$147$	77.4321	−	34.7458i	0.526749	−	0.236366i
$148$			92.6026i			0.625694i
$149$	43.9593			0.295029			0.147514	−	0.989060i	$-0.452873\pi$
$149$	43.9593			0.295029			0.147514	+	0.989060i	$0.452873\pi$
$150$		0			0
$151$	−182.752			−1.21028			−0.605139	−	0.796120i	$-0.706882\pi$
$151$	−182.752			−1.21028			−0.605139	+	0.796120i	$0.706882\pi$
$152$	−40.7264			−0.267937
$153$	−7.79261			−0.0509321
$154$	−56.8999	+	12.1812i	−0.369480	+	0.0790985i
$155$		0			0
$156$	7.15045			0.0458362
$157$	50.4042			0.321046			0.160523	−	0.987032i	$-0.448682\pi$
$157$	50.4042			0.321046			0.160523	+	0.987032i	$0.448682\pi$
$158$		−	58.3219i		−	0.369126i
$159$		−	69.2693i		−	0.435656i
$160$		0			0
$161$	−32.4151	+	6.93944i	−0.201336	+	0.0431021i
$162$			12.7279i			0.0785674i
$163$			101.758i			0.624281i	0.950036	+	0.312140i	$0.101046\pi$
$163$			101.758i			0.624281i	−0.950036	+	0.312140i	$0.898954\pi$
$164$		−	149.424i		−	0.911124i
$165$		0			0
$166$		−	178.505i		−	1.07533i
$167$	129.715			0.776738			0.388369	−	0.921504i	$-0.373039\pi$
$167$	129.715			0.776738			0.388369	+	0.921504i	$0.373039\pi$
$168$	33.5330	−	7.17877i	0.199601	−	0.0427308i
$169$	−164.739			−0.974789
$170$		0			0
$171$		−	43.1969i		−	0.252613i
$172$			20.8077i			0.120975i
$173$	−80.2634			−0.463950			−0.231975	−	0.972722i	$-0.574519\pi$
$173$	−80.2634			−0.463950			−0.231975	+	0.972722i	$0.574519\pi$
$174$		−	39.5766i		−	0.227452i
$175$		0			0
$176$	−23.5120			−0.133591
$177$		−	104.196i		−	0.588676i
$178$	130.281			0.731915
$179$	145.799			0.814519			0.407260	−	0.913312i	$-0.366484\pi$
$179$	145.799			0.814519			0.407260	+	0.913312i	$0.366484\pi$
$180$		0			0
$181$			323.062i			1.78487i	0.451175	+	0.892435i	$0.351005\pi$
$181$			323.062i			1.78487i	−0.451175	+	0.892435i	$0.648995\pi$
$182$	19.9814	−	4.27762i	0.109788	−	0.0235034i
$183$		−	126.720i		−	0.692460i
$184$	−13.3945			−0.0727959
$185$		0			0
$186$	118.499			0.637093
$187$	15.2683			0.0816488
$188$	70.1344			0.373055
$189$	7.61424	+	35.5672i	0.0402870	+	0.188186i
$190$		0			0
$191$	309.035			1.61799			0.808993	−	0.587818i	$-0.200013\pi$
$191$	309.035			1.61799			0.808993	+	0.587818i	$0.200013\pi$
$192$	13.8564			0.0721688
$193$			353.480i			1.83150i	0.401743	+	0.915752i	$0.368404\pi$
$193$			353.480i			1.83150i	−0.401743	+	0.915752i	$0.631596\pi$
$194$		−	6.03579i		−	0.0311123i
$195$		0			0
$196$	89.4109	−	40.1210i	0.456178	−	0.204699i
$197$			136.303i			0.691892i	0.938254	+	0.345946i	$0.112442\pi$
$197$			136.303i			0.691892i	−0.938254	+	0.345946i	$0.887558\pi$
$198$		−	24.9382i		−	0.125951i
$199$			62.4881i			0.314011i	0.987598	+	0.157005i	$0.0501840\pi$
$199$			62.4881i			0.314011i	−0.987598	+	0.157005i	$0.949816\pi$
$200$		0			0
$201$			68.5789i			0.341189i
$202$	−128.282			−0.635058
$203$	−23.6759	−	110.594i	−0.116630	−	0.544796i
$204$	−8.99813			−0.0441085
$205$		0			0
$206$		−	193.713i		−	0.940356i
$207$		−	14.2070i		−	0.0686327i
$208$	8.25663			0.0396953
$209$			84.6371i			0.404962i
$210$		0			0
$211$	−188.967			−0.895579			−0.447790	−	0.894139i	$-0.647789\pi$
$211$	−188.967			−0.895579			−0.447790	+	0.894139i	$0.647789\pi$
$212$		−	79.9853i		−	0.377289i
$213$	68.1498			0.319952
$214$	−33.0566			−0.154470
$215$		0			0
$216$			14.6969i			0.0680414i
$217$	331.137	−	70.8900i	1.52598	−	0.326682i
$218$			36.3273i			0.166639i
$219$	−20.4928			−0.0935742
$220$		0			0
$221$	−5.36172			−0.0242612
$222$	−113.415			−0.510877
$223$	−264.391			−1.18561			−0.592805	−	0.805346i	$-0.701980\pi$
$223$	−264.391			−1.18561			−0.592805	+	0.805346i	$0.701980\pi$
$224$	38.7206	−	8.28933i	0.172860	−	0.0370059i
$225$		0			0
$226$	3.55098			0.0157123
$227$	24.7955			0.109231			0.0546157	−	0.998507i	$-0.482607\pi$
$227$	24.7955			0.109231			0.0546157	+	0.998507i	$0.482607\pi$
$228$		−	49.8795i		−	0.218770i
$229$		−	131.811i		−	0.575595i	−0.957691	−	0.287798i	$-0.907077\pi$
$229$		−	131.811i		−	0.575595i	0.957691	−	0.287798i	$-0.0929230\pi$
$230$		0			0
$231$	−14.9188	−	69.6879i	−0.0645836	−	0.301679i
$232$		−	45.6991i		−	0.196979i
$233$		−	145.909i		−	0.626217i	−0.949717	−	0.313109i	$-0.898630\pi$
$233$		−	145.909i		−	0.626217i	0.949717	−	0.313109i	$-0.101370\pi$
$234$			8.75747i			0.0374251i
$235$		0			0
$236$		−	120.315i		−	0.509808i
$237$	71.4295			0.301390
$238$	−25.1446	+	5.38296i	−0.105649	+	0.0226175i
$239$	411.485			1.72169			0.860847	−	0.508864i	$-0.169934\pi$
$239$	411.485			1.72169			0.860847	+	0.508864i	$0.169934\pi$
$240$		0			0
$241$		−	381.587i		−	1.58335i	−0.610943	−	0.791675i	$-0.709209\pi$
$241$		−	381.587i		−	1.58335i	0.610943	−	0.791675i	$-0.290791\pi$
$242$		−	122.258i		−	0.505196i
$243$	−15.5885			−0.0641500
$244$		−	146.324i		−	0.599688i
$245$		0			0
$246$	183.007			0.743930
$247$		−	29.7217i		−	0.120331i
$248$	136.831			0.551739
$249$	218.623			0.878004
$250$		0			0
$251$			236.750i			0.943229i	0.881805	+	0.471614i	$0.156329\pi$
$251$			236.750i			0.943229i	−0.881805	+	0.471614i	$0.843671\pi$
$252$	8.79216	+	41.0694i	0.0348895	+	0.162974i
$253$			27.8362i			0.110024i
$254$	27.6367			0.108806
$255$		0			0
$256$	16.0000			0.0625000
$257$	−187.550			−0.729768			−0.364884	−	0.931053i	$-0.618891\pi$
$257$	−187.550			−0.729768			−0.364884	+	0.931053i	$0.618891\pi$
$258$	−25.4841			−0.0987756
$259$	−316.928	+	67.8481i	−1.22366	+	0.261962i
$260$		0			0
$261$	48.4712			0.185713
$262$	−64.4636			−0.246044
$263$			167.502i			0.636890i	0.947941	+	0.318445i	$0.103161\pi$
$263$			167.502i			0.636890i	−0.947941	+	0.318445i	$0.896839\pi$
$264$		−	28.7962i		−	0.109076i
$265$		0			0
$266$	−29.8394	−	139.384i	−0.112178	−	0.524001i
$267$			159.561i			0.597606i
$268$			79.1881i			0.295478i
$269$			373.386i			1.38805i	0.719950	+	0.694026i	$0.244165\pi$
$269$			373.386i			1.38805i	−0.719950	+	0.694026i	$0.755835\pi$
$270$		0			0
$271$		−	395.308i		−	1.45870i	−0.684141	−	0.729350i	$-0.739823\pi$
$271$		−	395.308i		−	1.45870i	0.684141	−	0.729350i	$-0.260177\pi$
$272$	−10.3901			−0.0381991
$273$	5.23899	+	24.4721i	0.0191904	+	0.0896413i
$274$	−250.654			−0.914796
$275$		0			0
$276$		−	16.4048i		−	0.0594376i
$277$		−	50.0168i		−	0.180566i	−0.995916	−	0.0902830i	$-0.971223\pi$
$277$		−	50.0168i		−	0.180566i	0.995916	−	0.0902830i	$-0.0287772\pi$
$278$	46.9389			0.168845
$279$			145.132i			0.520185i
$280$		0			0
$281$	414.869			1.47640			0.738201	−	0.674580i	$-0.235675\pi$
$281$	414.869			1.47640			0.738201	+	0.674580i	$0.235675\pi$
$282$			85.8968i			0.304598i
$283$	40.4964			0.143097			0.0715484	−	0.997437i	$-0.477206\pi$
$283$	40.4964			0.143097			0.0715484	+	0.997437i	$0.477206\pi$
$284$	78.6926			0.277087
$285$		0			0
$286$		−	17.1588i		−	0.0599958i
$287$	511.398	−	109.480i	1.78187	−	0.381464i
$288$			16.9706i			0.0589256i
$289$	−282.253			−0.976653
$290$		0			0
$291$	7.39230			0.0254031
$292$	−23.6630			−0.0810377
$293$	23.5736			0.0804560			0.0402280	−	0.999191i	$-0.487192\pi$
$293$	23.5736			0.0804560			0.0402280	+	0.999191i	$0.487192\pi$
$294$	49.1380	+	109.506i	0.167136	+	0.372468i
$295$		0			0
$296$	−130.960			−0.442432
$297$	30.5430			0.102838
$298$			62.1679i			0.208617i
$299$		−	9.77513i		−	0.0326927i
$300$		0			0
$301$	−71.2133	+	15.2454i	−0.236589	+	0.0506491i
$302$		−	258.450i		−	0.855796i
$303$		−	157.112i		−	0.518523i
$304$		−	57.5958i		−	0.189460i
$305$		0			0
$306$		−	11.0204i		−	0.0360144i
$307$	313.711			1.02186			0.510930	−	0.859622i	$-0.329301\pi$
$307$	313.711			1.02186			0.510930	+	0.859622i	$0.329301\pi$
$308$	−17.2268	−	80.4687i	−0.0559311	−	0.261262i
$309$	237.249			0.767798
$310$		0			0
$311$		−	341.192i		−	1.09708i	−0.836125	−	0.548539i	$-0.815184\pi$
$311$		−	341.192i		−	1.09708i	0.836125	−	0.548539i	$-0.184816\pi$
$312$			10.1123i			0.0324111i
$313$	508.988			1.62616			0.813081	−	0.582151i	$-0.197789\pi$
$313$	508.988			1.62616			0.813081	+	0.582151i	$0.197789\pi$
$314$			71.2823i			0.227014i
$315$		0			0
$316$	82.4797			0.261012
$317$			355.980i			1.12297i	0.827488	+	0.561483i	$0.189769\pi$
$317$			355.980i			1.12297i	−0.827488	+	0.561483i	$0.810231\pi$
$318$	97.9616			0.308055
$319$	−94.9712			−0.297715
$320$		0			0
$321$		−	40.4860i		−	0.126124i
$322$	−9.81385	−	45.8419i	−0.0304778	−	0.142366i
$323$			37.4018i			0.115795i
$324$	−18.0000			−0.0555556
$325$		0			0
$326$	−143.907			−0.441433
$327$	−44.4917			−0.136060
$328$	211.318			0.644262
$329$	51.3861	+	240.032i	0.156189	+	0.729579i
$330$		0			0
$331$	−568.964			−1.71892			−0.859462	−	0.511199i	$-0.829201\pi$
$331$	−568.964			−1.71892			−0.859462	+	0.511199i	$0.829201\pi$
$332$	252.444			0.760374
$333$		−	138.904i		−	0.417129i
$334$			183.445i			0.549237i
$335$		0			0
$336$	10.1523	+	47.4229i	0.0302152	+	0.141140i
$337$		−	292.641i		−	0.868372i	−0.900823	−	0.434186i	$-0.857036\pi$
$337$		−	292.641i		−	0.868372i	0.900823	−	0.434186i	$-0.142964\pi$
$338$		−	232.976i		−	0.689280i
$339$			4.34904i			0.0128290i
$340$		0			0
$341$		−	284.361i		−	0.833903i
$342$	61.0896			0.178625
$343$	202.822	+	276.609i	0.591317	+	0.806439i
$344$	−29.4265			−0.0855422
$345$		0			0
$346$		−	113.510i		−	0.328063i
$347$			175.594i			0.506036i	0.967462	+	0.253018i	$0.0814232\pi$
$347$			175.594i			0.506036i	−0.967462	+	0.253018i	$0.918577\pi$
$348$	55.9697			0.160833
$349$			451.738i			1.29438i	0.762329	+	0.647190i	$0.224056\pi$
$349$			451.738i			1.29438i	−0.762329	+	0.647190i	$0.775944\pi$
$350$		0			0
$351$	−10.7257			−0.0305575
$352$		−	33.2510i		−	0.0944630i
$353$	539.172			1.52740			0.763700	−	0.645571i	$-0.223381\pi$
$353$	539.172			1.52740			0.763700	+	0.645571i	$0.223381\pi$
$354$	147.355			0.416256
$355$		0			0
$356$			184.245i			0.517542i
$357$	−6.59275	−	30.7957i	−0.0184671	−	0.0862624i
$358$			206.191i			0.575952i
$359$	−166.475			−0.463720			−0.231860	−	0.972749i	$-0.574481\pi$
$359$	−166.475			−0.463720			−0.231860	+	0.972749i	$0.574481\pi$
$360$		0			0
$361$	153.670			0.425678
$362$	−456.878			−1.26209
$363$	149.734			0.412491
$364$	6.04947	+	28.2579i	0.0166194	+	0.0776316i
$365$		0			0
$366$	179.209			0.489643
$367$	129.246			0.352169			0.176084	−	0.984375i	$-0.443657\pi$
$367$	129.246			0.352169			0.176084	+	0.984375i	$0.443657\pi$
$368$		−	18.9426i		−	0.0514745i
$369$			224.137i			0.607416i
$370$		0			0
$371$	273.746	−	58.6036i	0.737859	−	0.157961i
$372$			167.583i			0.450493i
$373$			423.280i			1.13480i	0.823443	+	0.567399i	$0.192050\pi$
$373$			423.280i			1.13480i	−0.823443	+	0.567399i	$0.807950\pi$
$374$			21.5927i			0.0577344i
$375$		0			0
$376$			99.1850i			0.263790i
$377$	33.3507			0.0884634
$378$	−50.2996	+	10.7682i	−0.133068	+	0.0284872i
$379$	−30.5074			−0.0804944			−0.0402472	−	0.999190i	$-0.512815\pi$
$379$	−30.5074			−0.0804944			−0.0402472	+	0.999190i	$0.512815\pi$
$380$		0			0
$381$			33.8479i			0.0888398i
$382$			437.042i			1.14409i
$383$	367.882			0.960529			0.480264	−	0.877124i	$-0.340541\pi$
$383$	367.882			0.960529			0.480264	+	0.877124i	$0.340541\pi$
$384$			19.5959i			0.0510310i
$385$		0			0
$386$	−499.897			−1.29507
$387$		−	31.2115i		−	0.0806500i
$388$	8.53590			0.0219997
$389$	−583.606			−1.50027			−0.750136	−	0.661283i	$-0.770012\pi$
$389$	−583.606			−1.50027			−0.750136	+	0.661283i	$0.770012\pi$
$390$		0			0
$391$			12.3010i			0.0314605i
$392$	56.7397	+	126.446i	0.144744	+	0.322567i
$393$		−	78.9515i		−	0.200894i
$394$	−192.761			−0.489241
$395$		0			0
$396$	35.2680			0.0890606
$397$	−618.783			−1.55865			−0.779323	−	0.626622i	$-0.784437\pi$
$397$	−618.783			−1.55865			−0.779323	+	0.626622i	$0.784437\pi$
$398$	−88.3715			−0.222039
$399$	170.710	−	36.5457i	0.427845	−	0.0915932i
$400$		0			0
$401$	577.546			1.44026			0.720132	−	0.693837i	$-0.244081\pi$
$401$	577.546			1.44026			0.720132	+	0.693837i	$0.244081\pi$
$402$	−96.9852			−0.241257
$403$			99.8580i			0.247787i
$404$		−	181.418i		−	0.449054i
$405$		0			0
$406$	156.403	−	33.4828i	0.385229	−	0.0824700i
$407$			272.159i			0.668696i
$408$		−	12.7253i		−	0.0311894i
$409$			488.227i			1.19371i	0.802349	+	0.596855i	$0.203583\pi$
$409$			488.227i			1.19371i	−0.802349	+	0.596855i	$0.796417\pi$
$410$		0			0
$411$		−	306.987i		−	0.746928i
$412$	273.952			0.664932
$413$	411.771	−	88.1522i	0.997025	−	0.213444i
$414$	20.0917			0.0485306
$415$		0			0
$416$			11.6766i			0.0280688i
$417$			57.4882i			0.137861i
$418$	−119.695			−0.286351
$419$			131.348i			0.313481i	0.987640	+	0.156740i	$0.0500986\pi$
$419$			131.348i			0.313481i	−0.987640	+	0.156740i	$0.949901\pi$
$420$		0			0
$421$	−556.789			−1.32254			−0.661270	−	0.750148i	$-0.729982\pi$
$421$	−556.789			−1.32254			−0.661270	+	0.750148i	$0.729982\pi$
$422$		−	267.240i		−	0.633270i
$423$	−105.202			−0.248704
$424$	113.116			0.266784
$425$		0			0
$426$			96.3784i			0.226240i
$427$	500.787	−	107.209i	1.17280	−	0.251074i
$428$		−	46.7492i		−	0.109227i
$429$	21.0152			0.0489864
$430$		0			0
$431$	−167.379			−0.388351			−0.194175	−	0.980967i	$-0.562203\pi$
$431$	−167.379			−0.388351			−0.194175	+	0.980967i	$0.562203\pi$
$432$	−20.7846			−0.0481125
$433$	−57.5429			−0.132894			−0.0664468	−	0.997790i	$-0.521166\pi$
$433$	−57.5429			−0.132894			−0.0664468	+	0.997790i	$0.521166\pi$
$434$	100.254	+	468.299i	0.230999	+	1.07903i
$435$		0			0
$436$	−51.3746			−0.117832
$437$	−68.1885			−0.156038
$438$		−	28.9811i		−	0.0661670i
$439$		−	838.459i		−	1.90993i	−0.296719	−	0.954965i	$-0.595893\pi$
$439$		−	838.459i		−	1.90993i	0.296719	−	0.954965i	$-0.404107\pi$
$440$		0			0
$441$	−134.116	+	60.1815i	−0.304119	+	0.136466i
$442$		−	7.58262i		−	0.0171553i
$443$		−	435.742i		−	0.983616i	−0.870704	−	0.491808i	$-0.836336\pi$
$443$		−	435.742i		−	0.983616i	0.870704	−	0.491808i	$-0.163664\pi$
$444$		−	160.392i		−	0.361244i
$445$		0			0
$446$		−	373.905i		−	0.838353i
$447$	−76.1398			−0.170335
$448$	11.7229	+	54.7592i	0.0261672	+	0.122230i
$449$	−612.776			−1.36476			−0.682378	−	0.730999i	$-0.739054\pi$
$449$	−612.776			−1.36476			−0.682378	+	0.730999i	$0.739054\pi$
$450$		0			0
$451$		−	439.158i		−	0.973743i
$452$			5.02184i			0.0111103i
$453$	316.536			0.698754
$454$			35.0662i			0.0772382i
$455$		0			0
$456$	70.5402			0.154693
$457$		−	3.25143i		−	0.00711473i	−0.999994	−	0.00355736i	$-0.998868\pi$
$457$		−	3.25143i		−	0.00711473i	0.999994	−	0.00355736i	$-0.00113235\pi$
$458$	186.409			0.407007
$459$	13.4972			0.0294057
$460$		0			0
$461$			662.733i			1.43760i	0.695218	+	0.718799i	$0.255308\pi$
$461$			662.733i			1.43760i	−0.695218	+	0.718799i	$0.744692\pi$
$462$	98.5536	−	21.0984i	0.213319	−	0.0456675i
$463$		−	193.306i		−	0.417507i	−0.977968	−	0.208754i	$-0.933059\pi$
$463$		−	193.306i		−	0.417507i	0.977968	−	0.208754i	$-0.0669407\pi$
$464$	64.6283			0.139285
$465$		0			0
$466$	206.346			0.442802
$467$	−353.116			−0.756138			−0.378069	−	0.925777i	$-0.623412\pi$
$467$	−353.116			−0.756138			−0.378069	+	0.925777i	$0.623412\pi$
$468$	−12.3849			−0.0264635
$469$	−271.018	+	58.0196i	−0.577863	+	0.123709i
$470$		0			0
$471$	−87.3026			−0.185356
$472$	170.151			0.360489
$473$			61.1538i			0.129289i
$474$			101.017i			0.213115i
$475$		0			0
$476$	−7.61266	−	35.5598i	−0.0159930	−	0.0747054i
$477$			119.978i			0.251526i
$478$			581.927i			1.21742i
$479$		−	608.762i		−	1.27090i	−0.772141	−	0.635451i	$-0.780814\pi$
$479$		−	608.762i		−	1.27090i	0.772141	−	0.635451i	$-0.219186\pi$
$480$		0			0
$481$		−	95.5732i		−	0.198697i
$482$	539.646			1.11960
$483$	56.1446	−	12.0195i	0.116241	−	0.0248850i
$484$	172.898			0.357228
$485$		0			0
$486$		−	22.0454i		−	0.0453609i
$487$			185.222i			0.380333i	0.981752	+	0.190167i	$0.0609028\pi$
$487$			185.222i			0.380333i	−0.981752	+	0.190167i	$0.939097\pi$
$488$	206.933			0.424044
$489$		−	176.250i		−	0.360429i
$490$		0			0
$491$	924.656			1.88321			0.941605	−	0.336720i	$-0.109317\pi$
$491$	924.656			1.88321			0.941605	+	0.336720i	$0.109317\pi$
$492$			258.811i			0.526038i
$493$	−41.9686			−0.0851290
$494$	42.0328			0.0850867
$495$		0			0
$496$			193.509i			0.390138i
$497$	57.6565	+	269.322i	0.116009	+	0.541895i
$498$			309.180i			0.620843i
$499$	644.190			1.29096			0.645481	−	0.763776i	$-0.276657\pi$
$499$	644.190			1.29096			0.645481	+	0.763776i	$0.276657\pi$
$500$		0			0
$501$	−224.674			−0.448450
$502$	−334.816			−0.666963
$503$	−774.419			−1.53960			−0.769800	−	0.638285i	$-0.779644\pi$
$503$	−774.419			−1.53960			−0.769800	+	0.638285i	$0.779644\pi$
$504$	−58.0809	+	12.4340i	−0.115240	+	0.0246706i
$505$		0			0
$506$	−39.3663			−0.0777990
$507$	285.337			0.562794
$508$			39.0842i			0.0769375i
$509$		−	675.381i		−	1.32688i	−0.748231	−	0.663439i	$-0.769096\pi$
$509$		−	675.381i		−	1.32688i	0.748231	−	0.663439i	$-0.230904\pi$
$510$		0			0
$511$	−17.3374	−	80.9855i	−0.0339284	−	0.158484i
$512$			22.6274i			0.0441942i
$513$			74.8192i			0.145846i
$514$		−	265.236i		−	0.516024i
$515$		0			0
$516$		−	36.0400i		−	0.0698449i
$517$	206.125			0.398694
$518$	−95.9517	−	448.204i	−0.185235	−	0.865259i
$519$	139.020			0.267862
$520$		0			0
$521$		−	572.162i		−	1.09820i	−0.835757	−	0.549100i	$-0.814971\pi$
$521$		−	572.162i		−	1.09820i	0.835757	−	0.549100i	$-0.185029\pi$
$522$			68.5486i			0.131319i
$523$	−85.2596			−0.163020			−0.0815102	−	0.996673i	$-0.525974\pi$
$523$	−85.2596			−0.163020			−0.0815102	+	0.996673i	$0.525974\pi$
$524$		−	91.1653i		−	0.173980i
$525$		0			0
$526$	−236.884			−0.450349
$527$		−	125.661i		−	0.238447i
$528$	40.7240			0.0771287
$529$	506.574			0.957606
$530$		0			0
$531$			180.472i			0.339872i
$532$	197.119	−	42.1993i	0.370524	−	0.0793221i
$533$			154.218i			0.289339i
$534$	−225.653			−0.422571
$535$		0			0
$536$	−111.989			−0.208935
$537$	−252.531			−0.470263
$538$	−528.048			−0.981501
$539$	262.778	−	117.916i	0.487530	−	0.218767i
$540$		0			0
$541$	−347.233			−0.641836			−0.320918	−	0.947107i	$-0.603991\pi$
$541$	−347.233			−0.641836			−0.320918	+	0.947107i	$0.603991\pi$
$542$	559.049			1.03146
$543$		−	559.559i		−	1.03050i
$544$		−	14.6939i		−	0.0270108i
$545$		0			0
$546$	−34.6087	+	7.40905i	−0.0633859	+	0.0135697i
$547$		−	593.025i		−	1.08414i	−0.840333	−	0.542070i	$-0.817641\pi$
$547$		−	593.025i		−	1.08414i	0.840333	−	0.542070i	$-0.182359\pi$
$548$		−	354.479i		−	0.646859i
$549$			219.486i			0.399792i
$550$		0			0
$551$		−	232.645i		−	0.422223i
$552$	23.1999			0.0420288
$553$	60.4312	+	282.283i	0.109279	+	0.510457i
$554$	70.7344			0.127679
$555$		0			0
$556$			66.3817i			0.119391i
$557$		−	836.159i		−	1.50118i	−0.660767	−	0.750591i	$-0.729769\pi$
$557$		−	836.159i		−	1.50118i	0.660767	−	0.750591i	$-0.270231\pi$
$558$	−205.247			−0.367826
$559$		−	21.4752i		−	0.0384171i
$560$		0			0
$561$	−26.4455			−0.0471399
$562$			586.714i			1.04397i
$563$	−894.630			−1.58904			−0.794520	−	0.607238i	$-0.792278\pi$
$563$	−894.630			−1.58904			−0.794520	+	0.607238i	$0.792278\pi$
$564$	−121.476			−0.215384
$565$		0			0
$566$			57.2706i			0.101185i
$567$	−13.1882	−	61.6041i	−0.0232597	−	0.108649i
$568$			111.288i			0.195930i
$569$	−559.583			−0.983450			−0.491725	−	0.870751i	$-0.663633\pi$
$569$	−559.583			−0.983450			−0.491725	+	0.870751i	$0.663633\pi$
$570$		0			0
$571$	439.360			0.769457			0.384729	−	0.923030i	$-0.374295\pi$
$571$	439.360			0.769457			0.384729	+	0.923030i	$0.374295\pi$
$572$	24.2662			0.0424235
$573$	−535.265			−0.934145
$574$	154.828	+	723.226i	0.269736	+	1.25998i
$575$		0			0
$576$	−24.0000			−0.0416667
$577$	−704.688			−1.22130			−0.610648	−	0.791902i	$-0.709091\pi$
$577$	−704.688			−1.22130			−0.610648	+	0.791902i	$0.709091\pi$
$578$		−	399.166i		−	0.690598i
$579$		−	612.246i		−	1.05742i
$580$		0			0
$581$	184.961	+	863.978i	0.318349	+	1.48705i
$582$			10.4543i			0.0179627i
$583$		−	235.077i		−	0.403219i
$584$		−	33.4645i		−	0.0573023i
$585$		0			0
$586$			33.3381i			0.0568910i
$587$	144.103			0.245491			0.122746	−	0.992438i	$-0.460830\pi$
$587$	144.103			0.245491			0.122746	+	0.992438i	$0.460830\pi$
$588$	−154.864	+	69.4916i	−0.263374	+	0.118183i
$589$	696.581			1.18265
$590$		0			0
$591$		−	236.083i		−	0.399464i
$592$		−	185.205i		−	0.312847i
$593$	612.101			1.03221			0.516105	−	0.856525i	$-0.327381\pi$
$593$	612.101			1.03221			0.516105	+	0.856525i	$0.327381\pi$
$594$			43.1943i			0.0727177i
$595$		0			0
$596$	−87.9186			−0.147514
$597$		−	108.233i		−	0.181294i
$598$	13.8241			0.0231173
$599$	−435.517			−0.727074			−0.363537	−	0.931580i	$-0.618431\pi$
$599$	−435.517			−0.727074			−0.363537	+	0.931580i	$0.618431\pi$
$600$		0			0
$601$			412.897i			0.687017i	0.939150	+	0.343508i	$0.111615\pi$
$601$			412.897i			0.687017i	−0.939150	+	0.343508i	$0.888385\pi$
$602$	−21.5602	−	100.711i	−0.0358143	−	0.167294i
$603$		−	118.782i		−	0.196985i
$604$	365.504			0.605139
$605$		0			0
$606$	222.190			0.366651
$607$	−868.480			−1.43077			−0.715387	−	0.698728i	$-0.753750\pi$
$607$	−868.480			−1.43077			−0.715387	+	0.698728i	$0.753750\pi$
$608$	81.4528			0.133968
$609$	41.0079	+	191.554i	0.0673365	+	0.314538i
$610$		0			0
$611$	−72.3842			−0.118468
$612$	15.5852			0.0254661
$613$			320.085i			0.522161i	0.965317	+	0.261080i	$0.0840788\pi$
$613$			320.085i			0.522161i	−0.965317	+	0.261080i	$0.915921\pi$
$614$			443.654i			0.722564i
$615$		0			0
$616$	113.800	−	24.3623i	0.184740	−	0.0395492i
$617$		−	287.002i		−	0.465157i	−0.972578	−	0.232578i	$-0.925284\pi$
$617$		−	287.002i		−	0.465157i	0.972578	−	0.232578i	$-0.0747162\pi$
$618$			335.521i			0.542915i
$619$		−	461.836i		−	0.746100i	−0.927811	−	0.373050i	$-0.878312\pi$
$619$		−	461.836i		−	0.746100i	0.927811	−	0.373050i	$-0.121688\pi$
$620$		0			0
$621$			24.6072i			0.0396251i
$622$	482.518			0.775752
$623$	−630.570	+	134.993i	−1.01215	+	0.216682i
$624$	−14.3009			−0.0229181
$625$		0			0
$626$			719.818i			1.14987i
$627$		−	146.596i		−	0.233805i
$628$	−100.808			−0.160523
$629$			120.269i			0.191207i
$630$		0			0
$631$	−13.3642			−0.0211794			−0.0105897	−	0.999944i	$-0.503371\pi$
$631$	−13.3642			−0.0211794			−0.0105897	+	0.999944i	$0.503371\pi$
$632$			116.644i			0.184563i
$633$	327.301			0.517063
$634$	−503.432			−0.794057
$635$		0			0
$636$			138.539i			0.217828i
$637$	−92.2790	+	41.4080i	−0.144865	+	0.0650047i
$638$		−	134.310i		−	0.210517i
$639$	−118.039			−0.184724
$640$		0			0
$641$	−248.329			−0.387408			−0.193704	−	0.981060i	$-0.562050\pi$
$641$	−248.329			−0.387408			−0.193704	+	0.981060i	$0.562050\pi$
$642$	57.2558			0.0891835
$643$	−1084.03			−1.68590			−0.842950	−	0.537991i	$-0.819183\pi$
$643$	−1084.03			−1.68590			−0.842950	+	0.537991i	$0.819183\pi$
$644$	64.8302	−	13.8789i	0.100668	−	0.0215511i
$645$		0			0
$646$	−52.8942			−0.0818795
$647$	−1131.40			−1.74868			−0.874341	−	0.485312i	$-0.838706\pi$
$647$	−1131.40			−1.74868			−0.874341	+	0.485312i	$0.838706\pi$
$648$		−	25.4558i		−	0.0392837i
$649$		−	353.605i		−	0.544845i
$650$		0			0
$651$	−573.546	+	122.785i	−0.881023	+	0.188610i
$652$		−	203.515i		−	0.312140i
$653$		−	239.818i		−	0.367256i	−0.982996	−	0.183628i	$-0.941216\pi$
$653$		−	239.818i		−	0.367256i	0.982996	−	0.183628i	$-0.0587843\pi$
$654$		−	62.9207i		−	0.0962091i
$655$		0			0
$656$			298.849i			0.455562i
$657$	35.4945			0.0540251
$658$	−339.456	+	72.6709i	−0.515891	+	0.110442i
$659$	−340.662			−0.516938			−0.258469	−	0.966020i	$-0.583218\pi$
$659$	−340.662			−0.516938			−0.258469	+	0.966020i	$0.583218\pi$
$660$		0			0
$661$			88.1855i			0.133412i	0.997773	+	0.0667061i	$0.0212490\pi$
$661$			88.1855i			0.133412i	−0.997773	+	0.0667061i	$0.978751\pi$
$662$		−	804.637i		−	1.21546i
$663$	9.28678			0.0140072
$664$			357.010i			0.537666i
$665$		0			0
$666$	196.440			0.294955
$667$		−	76.5143i		−	0.114714i
$668$	−259.431			−0.388369
$669$	457.939			0.684512
$670$		0			0
$671$		−	430.046i		−	0.640903i
$672$	−67.0661	+	14.3575i	−0.0998007	+	0.0213654i
$673$		−	1258.19i		−	1.86952i	−0.355283	−	0.934759i	$-0.615616\pi$
$673$		−	1258.19i		−	1.86952i	0.355283	−	0.934759i	$-0.384384\pi$
$674$	413.857			0.614032
$675$		0			0
$676$	329.479			0.487394
$677$	−1194.41			−1.76427			−0.882134	−	0.470999i	$-0.843893\pi$
$677$	−1194.41			−1.76427			−0.882134	+	0.470999i	$0.843893\pi$
$678$	−6.15047			−0.00907149
$679$	6.25408	+	29.2137i	0.00921073	+	0.0430246i
$680$		0			0
$681$	−42.9471			−0.0630647
$682$	402.147			0.589658
$683$			890.365i			1.30361i	0.758387	+	0.651804i	$0.225988\pi$
$683$			890.365i			1.30361i	−0.758387	+	0.651804i	$0.774012\pi$
$684$			86.3938i			0.126307i
$685$		0			0
$686$	−391.184	+	286.833i	−0.570239	+	0.418124i
$687$			228.304i			0.332320i
$688$		−	41.6154i		−	0.0604875i
$689$			82.5511i			0.119813i
$690$		0			0
$691$		−	328.636i		−	0.475595i	−0.971315	−	0.237798i	$-0.923575\pi$
$691$		−	328.636i		−	0.475595i	0.971315	−	0.237798i	$-0.0764255\pi$
$692$	160.527			0.231975
$693$	25.8402	+	120.703i	0.0372874	+	0.174175i
$694$	−248.328			−0.357821
$695$		0			0
$696$			79.1531i			0.113726i
$697$		−	194.068i		−	0.278433i
$698$	−638.855			−0.915265
$699$			252.721i			0.361547i
$700$		0			0
$701$	144.549			0.206204			0.103102	−	0.994671i	$-0.467123\pi$
$701$	144.549			0.206204			0.103102	+	0.994671i	$0.467123\pi$
$702$		−	15.1684i		−	0.0216074i
$703$	−666.691			−0.948351
$704$	47.0240			0.0667954
$705$		0			0
$706$			762.505i			1.08004i
$707$	620.894	−	132.921i	0.878209	−	0.188007i
$708$			208.391i			0.294338i
$709$	1014.70			1.43117			0.715585	−	0.698526i	$-0.246160\pi$
$709$	1014.70			1.43117			0.715585	+	0.698526i	$0.246160\pi$
$710$		0			0
$711$	−123.719			−0.174008
$712$	−260.562			−0.365958
$713$	229.098			0.321315
$714$	43.5517	−	9.32356i	0.0609967	−	0.0130582i
$715$		0			0
$716$	−291.598			−0.407260
$717$	−712.713			−0.994020
$718$		−	235.432i		−	0.327899i
$719$		−	1364.28i		−	1.89747i	−0.316076	−	0.948734i	$-0.602365\pi$
$719$		−	1364.28i		−	1.89747i	0.316076	−	0.948734i	$-0.397635\pi$
$720$		0			0
$721$	200.719	+	937.588i	0.278390	+	1.30040i
$722$			217.322i			0.301000i
$723$			660.928i			0.914147i
$724$		−	646.123i		−	0.892435i
$725$		0			0
$726$			211.756i			0.291675i
$727$	−121.107			−0.166584			−0.0832922	−	0.996525i	$-0.526543\pi$
$727$	−121.107			−0.166584			−0.0832922	+	0.996525i	$0.526543\pi$
$728$	−39.9627	+	8.55524i	−0.0548938	+	0.0117517i
$729$	27.0000			0.0370370
$730$		0			0
$731$			27.0244i			0.0369690i
$732$			253.440i			0.346230i
$733$	1064.94			1.45285			0.726426	−	0.687245i	$-0.241180\pi$
$733$	1064.94			1.45285			0.726426	+	0.687245i	$0.241180\pi$
$734$			182.781i			0.249021i
$735$		0			0
$736$	26.7889			0.0363980
$737$			232.734i			0.315785i
$738$	−316.977			−0.429508
$739$	−658.565			−0.891156			−0.445578	−	0.895243i	$-0.647002\pi$
$739$	−658.565			−0.891156			−0.445578	+	0.895243i	$0.647002\pi$
$740$		0			0
$741$			51.4795i			0.0694730i
$742$	82.8781	+	387.135i	0.111695	+	0.521745i
$743$			844.054i			1.13601i	0.823026	+	0.568004i	$0.192284\pi$
$743$			844.054i			1.13601i	−0.823026	+	0.568004i	$0.807716\pi$
$744$	−236.999			−0.318547
$745$		0			0
$746$	−598.608			−0.802424
$747$	−378.666			−0.506916
$748$	−30.5366			−0.0408244
$749$	159.997	−	34.2522i	0.213614	−	0.0457305i
$750$		0			0
$751$	802.681			1.06882			0.534408	−	0.845227i	$-0.320535\pi$
$751$	802.681			1.06882			0.534408	+	0.845227i	$0.320535\pi$
$752$	−140.269			−0.186528
$753$		−	410.064i		−	0.544573i
$754$			47.1650i			0.0625531i
$755$		0			0
$756$	−15.2285	−	71.1343i	−0.0201435	−	0.0940930i
$757$			207.749i			0.274437i	0.990541	+	0.137219i	$0.0438163\pi$
$757$			207.749i			0.274437i	−0.990541	+	0.137219i	$0.956184\pi$
$758$		−	43.1440i		−	0.0569182i
$759$		−	48.2136i		−	0.0635226i
$760$		0			0
$761$			135.451i			0.177991i	0.996032	+	0.0889956i	$0.0283657\pi$
$761$			135.451i			0.177991i	−0.996032	+	0.0889956i	$0.971634\pi$
$762$	−47.8682			−0.0628192
$763$	−37.6411	−	175.827i	−0.0493331	−	0.230442i
$764$	−618.071			−0.808993
$765$		0			0
$766$			520.264i			0.679196i
$767$			124.174i			0.161896i
$768$	−27.7128			−0.0360844
$769$		−	12.5945i		−	0.0163778i	−0.999966	−	0.00818890i	$-0.997393\pi$
$769$		−	12.5945i		−	0.0163778i	0.999966	−	0.00818890i	$-0.00260664\pi$
$770$		0			0
$771$	324.847			0.421332
$772$		−	706.961i		−	0.915752i
$773$	−614.951			−0.795538			−0.397769	−	0.917486i	$-0.630215\pi$
$773$	−614.951			−0.795538			−0.397769	+	0.917486i	$0.630215\pi$
$774$	44.1398			0.0570281
$775$		0			0
$776$			12.0716i			0.0155562i
$777$	548.936	−	117.516i	0.706481	−	0.151244i
$778$		−	825.343i		−	1.06085i
$779$	1075.78			1.38097
$780$		0			0
$781$	231.278			0.296130
$782$	−17.3963			−0.0222459
$783$	−83.9546			−0.107222
$784$	−178.822	+	80.2420i	−0.228089	+	0.102350i
$785$		0			0
$786$	111.654			0.142054
$787$	−783.038			−0.994966			−0.497483	−	0.867474i	$-0.665742\pi$
$787$	−783.038			−0.994966			−0.497483	+	0.867474i	$0.665742\pi$
$788$		−	272.605i		−	0.345946i
$789$		−	290.122i		−	0.367708i
$790$		0			0
$791$	−17.1870	+	3.67940i	−0.0217282	+	0.00465158i
$792$			49.8765i			0.0629753i
$793$			151.018i			0.190438i
$794$		−	875.091i		−	1.10213i
$795$		0			0
$796$		−	124.976i		−	0.157005i
$797$	50.5740			0.0634554			0.0317277	−	0.999497i	$-0.489899\pi$
$797$	50.5740			0.0634554			0.0317277	+	0.999497i	$0.489899\pi$
$798$	51.6834	+	241.421i	0.0647662	+	0.302532i
$799$	91.0884			0.114003
$800$		0			0
$801$		−	276.367i		−	0.345028i
$802$			816.773i			1.01842i
$803$	−69.5455			−0.0866071
$804$		−	137.158i		−	0.170594i
$805$		0			0
$806$	−141.221			−0.175212
$807$		−	646.724i		−	0.801393i
$808$	256.563			0.317529
$809$	−908.730			−1.12328			−0.561638	−	0.827383i	$-0.689828\pi$
$809$	−908.730			−1.12328			−0.561638	+	0.827383i	$0.689828\pi$
$810$		0			0
$811$			195.796i			0.241426i	0.992687	+	0.120713i	$0.0385181\pi$
$811$			195.796i			0.241426i	−0.992687	+	0.120713i	$0.961482\pi$
$812$	47.3518	+	221.187i	0.0583151	+	0.272398i
$813$			684.693i			0.842181i
$814$	−384.891			−0.472839
$815$		0			0
$816$	17.9963			0.0220542
$817$	−149.805			−0.183359
$818$	−690.458			−0.844080
$819$	−9.07420	−	42.3869i	−0.0110796	−	0.0517544i
$820$		0			0
$821$	−239.589			−0.291826			−0.145913	−	0.989297i	$-0.546612\pi$
$821$	−239.589			−0.291826			−0.145913	+	0.989297i	$0.546612\pi$
$822$	434.146			0.528158
$823$		−	1493.19i		−	1.81432i	−0.420781	−	0.907162i	$-0.638244\pi$
$823$		−	1493.19i		−	1.81432i	0.420781	−	0.907162i	$-0.361756\pi$
$824$			387.427i			0.470178i
$825$		0			0
$826$	124.666	+	582.332i	0.150927	+	0.705003i
$827$			1031.14i			1.24685i	0.781884	+	0.623424i	$0.214259\pi$
$827$			1031.14i			1.24685i	−0.781884	+	0.623424i	$0.785741\pi$
$828$			28.4139i			0.0343163i
$829$		−	113.418i		−	0.136814i	−0.997658	−	0.0684068i	$-0.978208\pi$
$829$		−	113.418i		−	0.136814i	0.997658	−	0.0684068i	$-0.0217916\pi$
$830$		0			0
$831$			86.6316i			0.104250i
$832$	−16.5133			−0.0198477
$833$	116.124	−	52.1079i	0.139405	−	0.0625545i
$834$	−81.3006			−0.0974827
$835$		0			0
$836$		−	169.274i		−	0.202481i
$837$		−	251.375i		−	0.300329i
$838$	−185.755			−0.221664
$839$			252.998i			0.301548i	0.988568	+	0.150774i	$0.0481765\pi$
$839$			252.998i			0.301548i	−0.988568	+	0.150774i	$0.951823\pi$
$840$		0			0
$841$	−579.949			−0.689595
$842$		−	787.419i		−	0.935177i
$843$	−718.575			−0.852402
$844$	377.934			0.447790
$845$		0			0
$846$		−	148.778i		−	0.175860i
$847$	126.679	+	591.736i	0.149562	+	0.698626i
$848$			159.971i			0.188645i
$849$	−70.1418			−0.0826170
$850$		0			0
$851$	−219.267			−0.257658
$852$	−136.300			−0.159976
$853$	911.188			1.06822			0.534108	−	0.845416i	$-0.320648\pi$
$853$	911.188			1.06822			0.534108	+	0.845416i	$0.320648\pi$
$854$	151.616	+	708.219i	0.177536	+	0.829296i
$855$		0			0
$856$	66.1133			0.0772352
$857$	−638.434			−0.744963			−0.372482	−	0.928040i	$-0.621493\pi$
$857$	−638.434			−0.744963			−0.372482	+	0.928040i	$0.621493\pi$
$858$			29.7199i			0.0346386i
$859$		−	644.249i		−	0.749999i	−0.927025	−	0.374999i	$-0.877643\pi$
$859$		−	644.249i		−	0.749999i	0.927025	−	0.374999i	$-0.122357\pi$
$860$		0			0
$861$	−885.767	+	189.625i	−1.02877	+	0.220239i
$862$		−	236.710i		−	0.274605i
$863$			623.500i			0.722480i	0.932473	+	0.361240i	$0.117647\pi$
$863$			623.500i			0.722480i	−0.932473	+	0.361240i	$0.882353\pi$
$864$		−	29.3939i		−	0.0340207i
$865$		0			0
$866$		−	81.3779i		−	0.0939699i
$867$	488.876			0.563871
$868$	−662.274	+	141.780i	−0.762989	+	0.163341i
$869$	242.408			0.278950
$870$		0			0
$871$		−	81.7283i		−	0.0938328i
$872$		−	72.6546i		−	0.0833195i
$873$	−12.8038			−0.0146665
$874$		−	96.4331i		−	0.110335i
$875$		0			0
$876$	40.9855			0.0467871
$877$			204.837i			0.233566i	0.993157	+	0.116783i	$0.0372582\pi$
$877$			204.837i			0.233566i	−0.993157	+	0.116783i	$0.962742\pi$
$878$	1185.76			1.35052
$879$	−40.8307			−0.0464513
$880$		0			0
$881$			1294.76i			1.46964i	0.678260	+	0.734822i	$0.262734\pi$
$881$			1294.76i			1.46964i	−0.678260	+	0.734822i	$0.737266\pi$
$882$	−85.1095	−	189.669i	−0.0964960	−	0.215044i
$883$		−	1600.29i		−	1.81233i	−0.422925	−	0.906165i	$-0.638997\pi$
$883$		−	1600.29i		−	1.81233i	0.422925	−	0.906165i	$-0.361003\pi$
$884$	10.7234			0.0121306
$885$		0			0
$886$	616.232			0.695521
$887$	−822.386			−0.927154			−0.463577	−	0.886057i	$-0.653434\pi$
$887$	−822.386			−0.927154			−0.463577	+	0.886057i	$0.653434\pi$
$888$	226.829			0.255438
$889$	−133.764	+	28.6362i	−0.150466	+	0.0322118i
$890$		0			0
$891$	−52.9020			−0.0593737
$892$	528.782			0.592805
$893$			504.931i			0.565433i
$894$		−	107.678i		−	0.120445i
$895$		0			0
$896$	−77.4413	+	16.5787i	−0.0864300	+	0.0185030i
$897$			16.9310i			0.0188752i
$898$		−	866.596i		−	0.965028i
$899$			781.633i			0.869447i
$900$		0			0
$901$		−	103.882i		−	0.115297i
$902$	621.063			0.688540
$903$	123.345	−	26.4058i	0.136595	−	0.0292423i
$904$	−7.10195			−0.00785614
$905$		0			0
$906$			447.649i			0.494094i
$907$			803.919i			0.886350i	0.896435	+	0.443175i	$0.146148\pi$
$907$			803.919i			0.886350i	−0.896435	+	0.443175i	$0.853852\pi$
$908$	−49.5910			−0.0546157
$909$			272.127i			0.299369i
$910$		0			0
$911$	−604.804			−0.663891			−0.331945	−	0.943299i	$-0.607705\pi$
$911$	−604.804			−0.663891			−0.331945	+	0.943299i	$0.607705\pi$
$912$			99.7589i			0.109385i
$913$	741.933			0.812632
$914$	4.59822			0.00503087
$915$		0			0
$916$			263.623i			0.287798i
$917$	312.009	−	66.7950i	0.340250	−	0.0728408i
$918$			19.0879i			0.0207929i
$919$	−152.377			−0.165807			−0.0829035	−	0.996558i	$-0.526419\pi$
$919$	−152.377			−0.165807			−0.0829035	+	0.996558i	$0.526419\pi$
$920$		0			0
$921$	−543.363			−0.589971
$922$	−937.246			−1.01654
$923$	−81.2169			−0.0879923
$924$	29.8376	+	139.376i	0.0322918	+	0.150840i
$925$		0			0
$926$	273.376			0.295222
$927$	−410.928			−0.443288
$928$			91.3982i			0.0984894i
$929$		−	756.423i		−	0.814233i	−0.913376	−	0.407117i	$-0.866534\pi$
$929$		−	756.423i		−	0.814233i	0.913376	−	0.407117i	$-0.133466\pi$
$930$		0			0
$931$	288.850	+	643.712i	0.310258	+	0.691420i
$932$			291.817i			0.313109i
$933$			590.961i			0.633399i
$934$		−	499.382i		−	0.534670i
$935$		0			0
$936$		−	17.5149i		−	0.0187126i
$937$	62.9589			0.0671920			0.0335960	−	0.999435i	$-0.489304\pi$
$937$	62.9589			0.0671920			0.0335960	+	0.999435i	$0.489304\pi$
$938$	−82.0521	−	383.277i	−0.0874755	−	0.408611i
$939$	−881.594			−0.938865
$940$		0			0
$941$			513.455i			0.545648i	0.962064	+	0.272824i	$0.0879577\pi$
$941$			513.455i			0.545648i	−0.962064	+	0.272824i	$0.912042\pi$
$942$		−	123.465i		−	0.131066i
$943$	353.811			0.375197
$944$			240.629i			0.254904i
$945$		0			0
$946$	−86.4845			−0.0914212
$947$			1612.21i			1.70243i	0.524813	+	0.851217i	$0.324135\pi$
$947$			1612.21i			1.70243i	−0.524813	+	0.851217i	$0.675865\pi$
$948$	−142.859			−0.150695
$949$	24.4221			0.0257345
$950$		0			0
$951$		−	616.576i		−	0.648345i
$952$	50.2891	−	10.7659i	0.0528247	−	0.0113087i
$953$			1350.65i			1.41726i	0.705582	+	0.708629i	$0.250686\pi$
$953$			1350.65i			1.41726i	−0.705582	+	0.708629i	$0.749314\pi$
$954$	−169.674			−0.177856
$955$		0			0
$956$	−822.970			−0.860847
$957$	164.495			0.171886
$958$	860.920			0.898664
$959$	1213.19	−	259.719i	1.26505	−	0.270823i
$960$		0			0
$961$	−1379.35			−1.43533
$962$	135.161			0.140500
$963$			70.1237i			0.0728180i
$964$			763.174i			0.791675i
$965$		0			0
$966$	16.9981	+	79.4004i	0.0175964	+	0.0821951i
$967$		−	936.177i		−	0.968125i	−0.875033	−	0.484063i	$-0.839161\pi$
$967$		−	936.177i		−	0.968125i	0.875033	−	0.484063i	$-0.160839\pi$
$968$			244.515i			0.252598i
$969$		−	64.7819i		−	0.0668544i
$970$		0			0
$971$		−	135.578i		−	0.139627i	−0.997560	−	0.0698133i	$-0.977760\pi$
$971$		−	135.578i		−	0.139627i	0.997560	−	0.0698133i	$-0.0222404\pi$
$972$	31.1769			0.0320750
$973$	−227.188	+	48.6365i	−0.233492	+	0.0499862i
$974$	−261.944			−0.268936
$975$		0			0
$976$			292.648i			0.299844i
$977$		−	1726.37i		−	1.76701i	−0.468420	−	0.883506i	$-0.655177\pi$
$977$		−	1726.37i		−	1.76701i	0.468420	−	0.883506i	$-0.344823\pi$
$978$	249.255			0.254861
$979$			541.496i			0.553111i
$980$		0			0
$981$	77.0619			0.0785544
$982$			1307.66i			1.33163i
$983$	−73.5367			−0.0748085			−0.0374042	−	0.999300i	$-0.511909\pi$
$983$	−73.5367			−0.0748085			−0.0374042	+	0.999300i	$0.511909\pi$
$984$	−366.013			−0.371965
$985$		0			0
$986$		−	59.3525i		−	0.0601953i
$987$	−89.0033	−	415.747i	−0.0901756	−	0.421223i
$988$			59.4434i			0.0601654i
$989$	−49.2690			−0.0498170
$990$		0			0
$991$	1809.34			1.82578			0.912888	−	0.408209i	$-0.133847\pi$
$991$	1809.34			1.82578			0.912888	+	0.408209i	$0.133847\pi$
$992$	−273.663			−0.275870
$993$	985.474			0.992421
$994$	−380.878	+	81.5386i	−0.383178	+	0.0820308i
$995$		0			0
$996$	−437.246			−0.439002
$997$	−1094.62			−1.09792			−0.548958	−	0.835850i	$-0.684975\pi$
$997$	−1094.62			−1.09792			−0.548958	+	0.835850i	$0.684975\pi$
$998$			911.023i			0.912849i
$999$			240.589i			0.240830i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.3.h.c.349.10		24
5.2	odd	4		1050.3.f.c.601.6	yes	12
5.3	odd	4		1050.3.f.d.601.7	yes	12
5.4	even	2	inner	1050.3.h.c.349.17		24
7.6	odd	2	inner	1050.3.h.c.349.18		24
35.13	even	4		1050.3.f.d.601.10	yes	12
35.27	even	4		1050.3.f.c.601.3	✓	12
35.34	odd	2	inner	1050.3.h.c.349.9		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1050.3.f.c.601.3	✓	12	35.27	even	4
1050.3.f.c.601.6	yes	12	5.2	odd	4
1050.3.f.d.601.7	yes	12	5.3	odd	4
1050.3.f.d.601.10	yes	12	35.13	even	4
1050.3.h.c.349.9		24	35.34	odd	2	inner
1050.3.h.c.349.10		24	1.1	even	1	trivial
1050.3.h.c.349.17		24	5.4	even	2	inner
1050.3.h.c.349.18		24	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1050.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -1.73205 q^{3} -2.00000 q^{4} -2.44949i q^{6} +(-1.46536 - 6.84490i) q^{7} -2.82843i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} -1.73205 q^{3} -2.00000 q^{4} -2.44949i q^{6} +(-1.46536 - 6.84490i) q^{7} -2.82843i q^{8} +3.00000 q^{9} -5.87800 q^{11} +3.46410 q^{12} +2.06416 q^{13} +(9.68016 - 2.07233i) q^{14} +4.00000 q^{16} -2.59754 q^{17} +4.24264i q^{18} -14.3990i q^{19} +(2.53808 + 11.8557i) q^{21} -8.31274i q^{22} -4.73565i q^{23} +4.89898i q^{24} +2.91916i q^{26} -5.19615 q^{27} +(2.93072 + 13.6898i) q^{28} +16.1571 q^{29} +48.3772i q^{31} +5.65685i q^{32} +10.1810 q^{33} -3.67347i q^{34} -6.00000 q^{36} -46.3013i q^{37} +20.3632 q^{38} -3.57522 q^{39} +74.7122i q^{41} +(-16.7665 + 3.58939i) q^{42} -10.4038i q^{43} +11.7560 q^{44} +6.69723 q^{46} -35.0672 q^{47} -6.92820 q^{48} +(-44.7054 + 20.0605i) q^{49} +4.49907 q^{51} -4.12831 q^{52} +39.9926i q^{53} -7.34847i q^{54} +(-19.3603 + 4.14467i) q^{56} +24.9397i q^{57} +22.8495i q^{58} +60.1573i q^{59} +73.1619i q^{61} -68.4156 q^{62} +(-4.39608 - 20.5347i) q^{63} -8.00000 q^{64} +14.3981i q^{66} -39.5941i q^{67} +5.19507 q^{68} +8.20239i q^{69} -39.3463 q^{71} -8.48528i q^{72} +11.8315 q^{73} +65.4800 q^{74} +28.7979i q^{76} +(8.61339 + 40.2343i) q^{77} -5.05613i q^{78} -41.2398 q^{79} +9.00000 q^{81} -105.659 q^{82} -126.222 q^{83} +(-5.07616 - 23.7114i) q^{84} +14.7133 q^{86} -27.9849 q^{87} +16.6255i q^{88} -92.1225i q^{89} +(-3.02473 - 14.1290i) q^{91} +9.47131i q^{92} -83.7917i q^{93} -49.5925i q^{94} -9.79796i q^{96} -4.26795 q^{97} +(-28.3698 - 63.2230i) q^{98} -17.6340 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 48 q^{4} + 72 q^{9}+O(q^{10}) \) 24 * q - 48 * q^4 + 72 * q^9	\( 24 q - 48 q^{4} + 72 q^{9} - 32 q^{11} - 16 q^{14} + 96 q^{16} - 12 q^{21} - 96 q^{29} - 144 q^{36} + 24 q^{39} + 64 q^{44} + 160 q^{46} - 236 q^{49} + 144 q^{51} + 32 q^{56} - 192 q^{64} + 496 q^{71} + 128 q^{74} + 416 q^{79} + 216 q^{81} + 24 q^{84} + 256 q^{86} - 316 q^{91} - 96 q^{99}+O(q^{100}) \) 24 * q - 48 * q^4 + 72 * q^9 - 32 * q^11 - 16 * q^14 + 96 * q^16 - 12 * q^21 - 96 * q^29 - 144 * q^36 + 24 * q^39 + 64 * q^44 + 160 * q^46 - 236 * q^49 + 144 * q^51 + 32 * q^56 - 192 * q^64 + 496 * q^71 + 128 * q^74 + 416 * q^79 + 216 * q^81 + 24 * q^84 + 256 * q^86 - 316 * q^91 - 96 * q^99

Introduction

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