LMFDB - Embedded newform 350.6.c.a.99.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,6,Mod(99,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("350.99");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	350.c (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:	$56.1343369345$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	350.99
Dual form			350.6.c.a.99.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} -17.0000i q^{3} -16.0000 q^{4} -68.0000 q^{6} +49.0000i q^{7} +64.0000i q^{8} -46.0000 q^{9} +O(q^{10})$	\(q-4.00000i q^{2} -17.0000i q^{3} -16.0000 q^{4} -68.0000 q^{6} +49.0000i q^{7} +64.0000i q^{8} -46.0000 q^{9} -715.000 q^{11} +272.000i q^{12} +331.000i q^{13} +196.000 q^{14} +256.000 q^{16} +1699.00i q^{17} +184.000i q^{18} +1718.00 q^{19} +833.000 q^{21} +2860.00i q^{22} -3950.00i q^{23} +1088.00 q^{24} +1324.00 q^{26} -3349.00i q^{27} -784.000i q^{28} -4579.00 q^{29} +6756.00 q^{31} -1024.00i q^{32} +12155.0i q^{33} +6796.00 q^{34} +736.000 q^{36} +16518.0i q^{37} -6872.00i q^{38} +5627.00 q^{39} +18876.0 q^{41} -3332.00i q^{42} +2258.00i q^{43} +11440.0 q^{44} -15800.0 q^{46} +537.000i q^{47} -4352.00i q^{48} -2401.00 q^{49} +28883.0 q^{51} -5296.00i q^{52} -10984.0i q^{53} -13396.0 q^{54} -3136.00 q^{56} -29206.0i q^{57} +18316.0i q^{58} +25956.0 q^{59} +39188.0 q^{61} -27024.0i q^{62} -2254.00i q^{63} -4096.00 q^{64} +48620.0 q^{66} -4416.00i q^{67} -27184.0i q^{68} -67150.0 q^{69} -31880.0 q^{71} -2944.00i q^{72} -5018.00i q^{73} +66072.0 q^{74} -27488.0 q^{76} -35035.0i q^{77} -22508.0i q^{78} +27977.0 q^{79} -68111.0 q^{81} -75504.0i q^{82} +37644.0i q^{83} -13328.0 q^{84} +9032.00 q^{86} +77843.0i q^{87} -45760.0i q^{88} +17216.0 q^{89} -16219.0 q^{91} +63200.0i q^{92} -114852. i q^{93} +2148.00 q^{94} -17408.0 q^{96} +63175.0i q^{97} +9604.00i q^{98} +32890.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} - 136 q^{6} - 92 q^{9}+O(q^{10}) $ 2 * q - 32 * q^4 - 136 * q^6 - 92 * q^9	$ 2 q - 32 q^{4} - 136 q^{6} - 92 q^{9} - 1430 q^{11} + 392 q^{14} + 512 q^{16} + 3436 q^{19} + 1666 q^{21} + 2176 q^{24} + 2648 q^{26} - 9158 q^{29} + 13512 q^{31} + 13592 q^{34} + 1472 q^{36} + 11254 q^{39} + 37752 q^{41} + 22880 q^{44} - 31600 q^{46} - 4802 q^{49} + 57766 q^{51} - 26792 q^{54} - 6272 q^{56} + 51912 q^{59} + 78376 q^{61} - 8192 q^{64} + 97240 q^{66} - 134300 q^{69} - 63760 q^{71} + 132144 q^{74} - 54976 q^{76} + 55954 q^{79} - 136222 q^{81} - 26656 q^{84} + 18064 q^{86} + 34432 q^{89} - 32438 q^{91} + 4296 q^{94} - 34816 q^{96} + 65780 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 - 136 * q^6 - 92 * q^9 - 1430 * q^11 + 392 * q^14 + 512 * q^16 + 3436 * q^19 + 1666 * q^21 + 2176 * q^24 + 2648 * q^26 - 9158 * q^29 + 13512 * q^31 + 13592 * q^34 + 1472 * q^36 + 11254 * q^39 + 37752 * q^41 + 22880 * q^44 - 31600 * q^46 - 4802 * q^49 + 57766 * q^51 - 26792 * q^54 - 6272 * q^56 + 51912 * q^59 + 78376 * q^61 - 8192 * q^64 + 97240 * q^66 - 134300 * q^69 - 63760 * q^71 + 132144 * q^74 - 54976 * q^76 + 55954 * q^79 - 136222 * q^81 - 26656 * q^84 + 18064 * q^86 + 34432 * q^89 - 32438 * q^91 + 4296 * q^94 - 34816 * q^96 + 65780 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 4.00000i	− 0.707107i
$2$	− 4.00000i	− 0.707107i
$3$	− 17.0000i	− 1.09055i	−0.838257	−	0.545275i	$-0.816425\pi$
$3$	− 17.0000i	− 1.09055i	0.838257	−	0.545275i	$-0.183575\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−68.0000	−0.771136
$7$	49.0000i	0.377964i
$7$	49.0000i	0.377964i
$8$	64.0000i	0.353553i
$9$	−46.0000	−0.189300
$10$	0	0
$11$	−715.000	−1.78166	−0.890829	−	0.454339i	$-0.849876\pi$
$11$	−715.000	−1.78166	−0.890829	+	0.454339i	$0.849876\pi$
$12$	272.000i	0.545275i
$13$	331.000i	0.543212i	0.962408	+	0.271606i	$0.0875548\pi$
$13$	331.000i	0.543212i	−0.962408	+	0.271606i	$0.912445\pi$
$14$	196.000	0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	1699.00i	1.42584i	0.701245	+	0.712920i	$0.252628\pi$
$17$	1699.00i	1.42584i	−0.701245	+	0.712920i	$0.747372\pi$
$18$	184.000i	0.133856i
$19$	1718.00	1.09179	0.545895	−	0.837854i	$-0.316190\pi$
$19$	1718.00	1.09179	0.545895	+	0.837854i	$0.316190\pi$
$20$	0	0
$21$	833.000	0.412189
$22$	2860.00i	1.25982i
$23$	− 3950.00i	− 1.55696i	−0.627669	−	0.778480i	$-0.715991\pi$
$23$	− 3950.00i	− 1.55696i	0.627669	−	0.778480i	$-0.284009\pi$
$24$	1088.00	0.385568
$25$	0	0
$26$	1324.00	0.384109
$27$	− 3349.00i	− 0.884109i
$28$	− 784.000i	− 0.188982i
$29$	−4579.00	−1.01106	−0.505529	−	0.862810i	$-0.668702\pi$
$29$	−4579.00	−1.01106	−0.505529	+	0.862810i	$0.668702\pi$
$30$	0	0
$31$	6756.00	1.26266	0.631329	−	0.775516i	$-0.282510\pi$
$31$	6756.00	1.26266	0.631329	+	0.775516i	$0.282510\pi$
$32$	− 1024.00i	− 0.176777i
$33$	12155.0i	1.94299i
$34$	6796.00	1.00822
$35$	0	0
$36$	736.000	0.0946502
$37$	16518.0i	1.98360i	0.127816	+	0.991798i	$0.459203\pi$
$37$	16518.0i	1.98360i	−0.127816	+	0.991798i	$0.540797\pi$
$38$	− 6872.00i	− 0.772012i
$39$	5627.00	0.592400
$40$	0	0
$41$	18876.0	1.75368	0.876840	−	0.480782i	$-0.159647\pi$
$41$	18876.0	1.75368	0.876840	+	0.480782i	$0.159647\pi$
$42$	− 3332.00i	− 0.291462i
$43$	2258.00i	0.186231i	0.995655	+	0.0931157i	$0.0296826\pi$
$43$	2258.00i	0.186231i	−0.995655	+	0.0931157i	$0.970317\pi$
$44$	11440.0	0.890829
$45$	0	0
$46$	−15800.0	−1.10094
$47$	537.000i	0.0354593i	0.999843	+	0.0177296i	$0.00564381\pi$
$47$	537.000i	0.0354593i	−0.999843	+	0.0177296i	$0.994356\pi$
$48$	− 4352.00i	− 0.272638i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	28883.0	1.55495
$52$	− 5296.00i	− 0.271606i
$53$	− 10984.0i	− 0.537119i	−0.963263	−	0.268560i	$-0.913452\pi$
$53$	− 10984.0i	− 0.537119i	0.963263	−	0.268560i	$-0.0865477\pi$
$54$	−13396.0	−0.625159
$55$	0	0
$56$	−3136.00	−0.133631
$57$	− 29206.0i	− 1.19065i
$58$	18316.0i	0.714925i
$59$	25956.0	0.970751	0.485375	−	0.874306i	$-0.338683\pi$
$59$	25956.0	0.970751	0.485375	+	0.874306i	$0.338683\pi$
$60$	0	0
$61$	39188.0	1.34843	0.674215	−	0.738535i	$-0.264482\pi$
$61$	39188.0	1.34843	0.674215	+	0.738535i	$0.264482\pi$
$62$	− 27024.0i	− 0.892833i
$63$	− 2254.00i	− 0.0715488i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	48620.0	1.37390
$67$	− 4416.00i	− 0.120183i	−0.998193	−	0.0600914i	$-0.980861\pi$
$67$	− 4416.00i	− 0.120183i	0.998193	−	0.0600914i	$-0.0191392\pi$
$68$	− 27184.0i	− 0.712920i
$69$	−67150.0	−1.69794
$70$	0	0
$71$	−31880.0	−0.750538	−0.375269	−	0.926916i	$-0.622450\pi$
$71$	−31880.0	−0.750538	−0.375269	+	0.926916i	$0.622450\pi$
$72$	− 2944.00i	− 0.0669278i
$73$	− 5018.00i	− 0.110211i	−0.998481	−	0.0551053i	$-0.982451\pi$
$73$	− 5018.00i	− 0.110211i	0.998481	−	0.0551053i	$-0.0175495\pi$
$74$	66072.0	1.40261
$75$	0	0
$76$	−27488.0	−0.545895
$77$	− 35035.0i	− 0.673403i
$78$	− 22508.0i	− 0.418890i
$79$	27977.0	0.504352	0.252176	−	0.967681i	$-0.418854\pi$
$79$	27977.0	0.504352	0.252176	+	0.967681i	$0.418854\pi$
$80$	0	0
$81$	−68111.0	−1.15347
$82$	− 75504.0i	− 1.24004i
$83$	37644.0i	0.599792i	0.953972	+	0.299896i	$0.0969520\pi$
$83$	37644.0i	0.599792i	−0.953972	+	0.299896i	$0.903048\pi$
$84$	−13328.0	−0.206095
$85$	0	0
$86$	9032.00	0.131685
$87$	77843.0i	1.10261i
$88$	− 45760.0i	− 0.629911i
$89$	17216.0	0.230387	0.115193	−	0.993343i	$-0.463251\pi$
$89$	17216.0	0.230387	0.115193	+	0.993343i	$0.463251\pi$
$90$	0	0
$91$	−16219.0	−0.205315
$92$	63200.0i	0.778480i
$93$	− 114852.i	− 1.37699i
$94$	2148.00	0.0250735
$95$	0	0
$96$	−17408.0	−0.192784
$97$	63175.0i	0.681736i	0.940111	+	0.340868i	$0.110721\pi$
$97$	63175.0i	0.681736i	−0.940111	+	0.340868i	$0.889279\pi$
$98$	9604.00i	0.101015i
$99$	32890.0	0.337269
$100$	0	0
$101$	−29250.0	−0.285314	−0.142657	−	0.989772i	$-0.545565\pi$
$101$	−29250.0	−0.285314	−0.142657	+	0.989772i	$0.545565\pi$
$102$	− 115532.i	− 1.09952i
$103$	− 149189.i	− 1.38562i	−0.721121	−	0.692809i	$-0.756373\pi$
$103$	− 149189.i	− 1.38562i	0.721121	−	0.692809i	$-0.243627\pi$
$104$	−21184.0	−0.192055
$105$	0	0
$106$	−43936.0	−0.379801
$107$	− 83742.0i	− 0.707105i	−0.935415	−	0.353552i	$-0.884974\pi$
$107$	− 83742.0i	− 0.707105i	0.935415	−	0.353552i	$-0.115026\pi$
$108$	53584.0i	0.442054i
$109$	−105377.	−0.849532	−0.424766	−	0.905303i	$-0.639644\pi$
$109$	−105377.	−0.849532	−0.424766	+	0.905303i	$0.639644\pi$
$110$	0	0
$111$	280806.	2.16321
$112$	12544.0i	0.0944911i
$113$	− 122754.i	− 0.904356i	−0.891928	−	0.452178i	$-0.850647\pi$
$113$	− 122754.i	− 0.904356i	0.891928	−	0.452178i	$-0.149353\pi$
$114$	−116824.	−0.841918
$115$	0	0
$116$	73264.0	0.505529
$117$	− 15226.0i	− 0.102830i
$118$	− 103824.i	− 0.686424i
$119$	−83251.0	−0.538917
$120$	0	0
$121$	350174.	2.17431
$122$	− 156752.i	− 0.953484i
$123$	− 320892.i	− 1.91248i
$124$	−108096.	−0.631329
$125$	0	0
$126$	−9016.00	−0.0505927
$127$	219196.i	1.20593i	0.797766	+	0.602967i	$0.206015\pi$
$127$	219196.i	1.20593i	−0.797766	+	0.602967i	$0.793985\pi$
$128$	16384.0i	0.0883883i
$129$	38386.0	0.203095
$130$	0	0
$131$	−96682.0	−0.492229	−0.246115	−	0.969241i	$-0.579154\pi$
$131$	−96682.0	−0.492229	−0.246115	+	0.969241i	$0.579154\pi$
$132$	− 194480.i	− 0.971494i
$133$	84182.0i	0.412658i
$134$	−17664.0	−0.0849820
$135$	0	0
$136$	−108736.	−0.504111
$137$	− 187288.i	− 0.852528i	−0.904599	−	0.426264i	$-0.859830\pi$
$137$	− 187288.i	− 0.852528i	0.904599	−	0.426264i	$-0.140170\pi$
$138$	268600.i	1.20063i
$139$	−176894.	−0.776562	−0.388281	−	0.921541i	$-0.626931\pi$
$139$	−176894.	−0.776562	−0.388281	+	0.921541i	$0.626931\pi$
$140$	0	0
$141$	9129.00	0.0386701
$142$	127520.i	0.530710i
$143$	− 236665.i	− 0.967819i
$144$	−11776.0	−0.0473251
$145$	0	0
$146$	−20072.0	−0.0779307
$147$	40817.0i	0.155793i
$148$	− 264288.i	− 0.991798i
$149$	199078.	0.734611	0.367306	−	0.930100i	$-0.380280\pi$
$149$	199078.	0.734611	0.367306	+	0.930100i	$0.380280\pi$
$150$	0	0
$151$	471583.	1.68312	0.841561	−	0.540162i	$-0.181637\pi$
$151$	471583.	1.68312	0.841561	+	0.540162i	$0.181637\pi$
$152$	109952.i	0.386006i
$153$	− 78154.0i	− 0.269912i
$154$	−140140.	−0.476168
$155$	0	0
$156$	−90032.0	−0.296200
$157$	72054.0i	0.233297i	0.993173	+	0.116648i	$0.0372151\pi$
$157$	72054.0i	0.233297i	−0.993173	+	0.116648i	$0.962785\pi$
$158$	− 111908.i	− 0.356630i
$159$	−186728.	−0.585756
$160$	0	0
$161$	193550.	0.588476
$162$	272444.i	0.815623i
$163$	385334.i	1.13597i	0.823038	+	0.567987i	$0.192278\pi$
$163$	385334.i	1.13597i	−0.823038	+	0.567987i	$0.807722\pi$
$164$	−302016.	−0.876840
$165$	0	0
$166$	150576.	0.424117
$167$	542957.i	1.50652i	0.657724	+	0.753259i	$0.271519\pi$
$167$	542957.i	1.50652i	−0.657724	+	0.753259i	$0.728481\pi$
$168$	53312.0i	0.145731i
$169$	261732.	0.704920
$170$	0	0
$171$	−79028.0	−0.206676
$172$	− 36128.0i	− 0.0931157i
$173$	370953.i	0.942331i	0.882045	+	0.471166i	$0.156167\pi$
$173$	370953.i	0.942331i	−0.882045	+	0.471166i	$0.843833\pi$
$174$	311372.	0.779662
$175$	0	0
$176$	−183040.	−0.445414
$177$	− 441252.i	− 1.05865i
$178$	− 68864.0i	− 0.162908i
$179$	754172.	1.75929	0.879646	−	0.475629i	$-0.157780\pi$
$179$	754172.	1.75929	0.879646	+	0.475629i	$0.157780\pi$
$180$	0	0
$181$	303840.	0.689364	0.344682	−	0.938720i	$-0.387987\pi$
$181$	303840.	0.689364	0.344682	+	0.938720i	$0.387987\pi$
$182$	64876.0i	0.145180i
$183$	− 666196.i	− 1.47053i
$184$	252800.	0.550469
$185$	0	0
$186$	−459408.	−0.973680
$187$	− 1.21478e6i	− 2.54036i
$188$	− 8592.00i	− 0.0177296i
$189$	164101.	0.334162
$190$	0	0
$191$	−186271.	−0.369455	−0.184728	−	0.982790i	$-0.559140\pi$
$191$	−186271.	−0.369455	−0.184728	+	0.982790i	$0.559140\pi$
$192$	69632.0i	0.136319i
$193$	92504.0i	0.178759i	0.995998	+	0.0893794i	$0.0284884\pi$
$193$	92504.0i	0.178759i	−0.995998	+	0.0893794i	$0.971512\pi$
$194$	252700.	0.482060
$195$	0	0
$196$	38416.0	0.0714286
$197$	736368.i	1.35185i	0.736969	+	0.675926i	$0.236256\pi$
$197$	736368.i	1.35185i	−0.736969	+	0.675926i	$0.763744\pi$
$198$	− 131560.i	− 0.238485i
$199$	481620.	0.862128	0.431064	−	0.902321i	$-0.358138\pi$
$199$	481620.	0.862128	0.431064	+	0.902321i	$0.358138\pi$
$200$	0	0
$201$	−75072.0	−0.131065
$202$	117000.i	0.201747i
$203$	− 224371.i	− 0.382144i
$204$	−462128.	−0.777476
$205$	0	0
$206$	−596756.	−0.979780
$207$	181700.i	0.294733i
$208$	84736.0i	0.135803i
$209$	−1.22837e6	−1.94520
$210$	0	0
$211$	189531.	0.293072	0.146536	−	0.989205i	$-0.453188\pi$
$211$	189531.	0.293072	0.146536	+	0.989205i	$0.453188\pi$
$212$	175744.i	0.268560i
$213$	541960.i	0.818499i
$214$	−334968.	−0.499999
$215$	0	0
$216$	214336.	0.312580
$217$	331044.i	0.477240i
$218$	421508.i	0.600710i
$219$	−85306.0	−0.120190
$220$	0	0
$221$	−562369.	−0.774534
$222$	− 1.12322e6i	− 1.52962i
$223$	− 22597.0i	− 0.0304291i	−0.999884	−	0.0152145i	$-0.995157\pi$
$223$	− 22597.0i	− 0.0304291i	0.999884	−	0.0152145i	$-0.00484312\pi$
$224$	50176.0	0.0668153
$225$	0	0
$226$	−491016.	−0.639476
$227$	998117.i	1.28563i	0.766020	+	0.642816i	$0.222234\pi$
$227$	998117.i	1.28563i	−0.766020	+	0.642816i	$0.777766\pi$
$228$	467296.i	0.595326i
$229$	854644.	1.07695	0.538476	−	0.842641i	$-0.319000\pi$
$229$	854644.	1.07695	0.538476	+	0.842641i	$0.319000\pi$
$230$	0	0
$231$	−595595.	−0.734380
$232$	− 293056.i	− 0.357463i
$233$	1.25818e6i	1.51829i	0.650922	+	0.759144i	$0.274382\pi$
$233$	1.25818e6i	1.51829i	−0.650922	+	0.759144i	$0.725618\pi$
$234$	−60904.0	−0.0727120
$235$	0	0
$236$	−415296.	−0.485375
$237$	− 475609.i	− 0.550021i
$238$	333004.i	0.381072i
$239$	706581.	0.800142	0.400071	−	0.916484i	$-0.368985\pi$
$239$	706581.	0.800142	0.400071	+	0.916484i	$0.368985\pi$
$240$	0	0
$241$	616330.	0.683551	0.341775	−	0.939782i	$-0.388972\pi$
$241$	616330.	0.683551	0.341775	+	0.939782i	$0.388972\pi$
$242$	− 1.40070e6i	− 1.53747i
$243$	344080.i	0.373804i
$244$	−627008.	−0.674215
$245$	0	0
$246$	−1.28357e6	−1.35233
$247$	568658.i	0.593074i
$248$	432384.i	0.446417i
$249$	639948.	0.654103
$250$	0	0
$251$	190842.	0.191201	0.0956004	−	0.995420i	$-0.469523\pi$
$251$	190842.	0.191201	0.0956004	+	0.995420i	$0.469523\pi$
$252$	36064.0i	0.0357744i
$253$	2.82425e6i	2.77397i
$254$	876784.	0.852724
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.13094e6i	1.06809i	0.845456	+	0.534045i	$0.179329\pi$
$257$	1.13094e6i	1.06809i	−0.845456	+	0.534045i	$0.820671\pi$
$258$	− 153544.i	− 0.143610i
$259$	−809382.	−0.749729
$260$	0	0
$261$	210634.	0.191394
$262$	386728.i	0.348059i
$263$	− 1.67377e6i	− 1.49213i	−0.665874	−	0.746065i	$-0.731941\pi$
$263$	− 1.67377e6i	− 1.49213i	0.665874	−	0.746065i	$-0.268059\pi$
$264$	−777920.	−0.686950
$265$	0	0
$266$	336728.	0.291793
$267$	− 292672.i	− 0.251248i
$268$	70656.0i	0.0600914i
$269$	630942.	0.531629	0.265815	−	0.964024i	$-0.414359\pi$
$269$	630942.	0.531629	0.265815	+	0.964024i	$0.414359\pi$
$270$	0	0
$271$	−372476.	−0.308088	−0.154044	−	0.988064i	$-0.549230\pi$
$271$	−372476.	−0.308088	−0.154044	+	0.988064i	$0.549230\pi$
$272$	434944.i	0.356460i
$273$	275723.i	0.223906i
$274$	−749152.	−0.602828
$275$	0	0
$276$	1.07440e6	0.848972
$277$	− 867010.i	− 0.678930i	−0.940619	−	0.339465i	$-0.889754\pi$
$277$	− 867010.i	− 0.678930i	0.940619	−	0.339465i	$-0.110246\pi$
$278$	707576.i	0.549112i
$279$	−310776.	−0.239021
$280$	0	0
$281$	−1.94498e6	−1.46943	−0.734716	−	0.678375i	$-0.762685\pi$
$281$	−1.94498e6	−1.46943	−0.734716	+	0.678375i	$0.762685\pi$
$282$	− 36516.0i	− 0.0273439i
$283$	1.18501e6i	0.879543i	0.898110	+	0.439771i	$0.144941\pi$
$283$	1.18501e6i	0.879543i	−0.898110	+	0.439771i	$0.855059\pi$
$284$	510080.	0.375269
$285$	0	0
$286$	−946660.	−0.684351
$287$	924924.i	0.662829i
$288$	47104.0i	0.0334639i
$289$	−1.46674e6	−1.03302
$290$	0	0
$291$	1.07398e6	0.743467
$292$	80288.0i	0.0551053i
$293$	33669.0i	0.0229119i	0.999934	+	0.0114560i	$0.00364662\pi$
$293$	33669.0i	0.0229119i	−0.999934	+	0.0114560i	$0.996353\pi$
$294$	163268.	0.110162
$295$	0	0
$296$	−1.05715e6	−0.701307
$297$	2.39454e6i	1.57518i
$298$	− 796312.i	− 0.519449i
$299$	1.30745e6	0.845760
$300$	0	0
$301$	−110642.	−0.0703888
$302$	− 1.88633e6i	− 1.19015i
$303$	497250.i	0.311149i
$304$	439808.	0.272948
$305$	0	0
$306$	−312616.	−0.190857
$307$	27043.0i	0.0163760i	0.999966	+	0.00818802i	$0.00260636\pi$
$307$	27043.0i	0.0163760i	−0.999966	+	0.00818802i	$0.997394\pi$
$308$	560560.i	0.336702i
$309$	−2.53621e6	−1.51109
$310$	0	0
$311$	2.14919e6	1.26001	0.630004	−	0.776592i	$-0.283053\pi$
$311$	2.14919e6	1.26001	0.630004	+	0.776592i	$0.283053\pi$
$312$	360128.i	0.209445i
$313$	− 2.67052e6i	− 1.54076i	−0.637583	−	0.770381i	$-0.720066\pi$
$313$	− 2.67052e6i	− 1.54076i	0.637583	−	0.770381i	$-0.279934\pi$
$314$	288216.	0.164966
$315$	0	0
$316$	−447632.	−0.252176
$317$	250514.i	0.140018i	0.997546	+	0.0700090i	$0.0223028\pi$
$317$	250514.i	0.140018i	−0.997546	+	0.0700090i	$0.977697\pi$
$318$	746912.i	0.414192i
$319$	3.27398e6	1.80136
$320$	0	0
$321$	−1.42361e6	−0.771134
$322$	− 774200.i	− 0.416115i
$323$	2.91888e6i	1.55672i
$324$	1.08978e6	0.576733
$325$	0	0
$326$	1.54134e6	0.803255
$327$	1.79141e6i	0.926457i
$328$	1.20806e6i	0.620019i
$329$	−26313.0	−0.0134023
$330$	0	0
$331$	1.05899e6	0.531277	0.265639	−	0.964073i	$-0.414417\pi$
$331$	1.05899e6	0.531277	0.265639	+	0.964073i	$0.414417\pi$
$332$	− 602304.i	− 0.299896i
$333$	− 759828.i	− 0.375495i
$334$	2.17183e6	1.06527
$335$	0	0
$336$	213248.	0.103047
$337$	2.85025e6i	1.36712i	0.729893	+	0.683562i	$0.239570\pi$
$337$	2.85025e6i	1.36712i	−0.729893	+	0.683562i	$0.760430\pi$
$338$	− 1.04693e6i	− 0.498454i
$339$	−2.08682e6	−0.986246
$340$	0	0
$341$	−4.83054e6	−2.24962
$342$	316112.i	0.146142i
$343$	− 117649.i	− 0.0539949i
$344$	−144512.	−0.0658427
$345$	0	0
$346$	1.48381e6	0.666329
$347$	− 1.89141e6i	− 0.843259i	−0.906768	−	0.421630i	$-0.861458\pi$
$347$	− 1.89141e6i	− 0.843259i	0.906768	−	0.421630i	$-0.138542\pi$
$348$	− 1.24549e6i	− 0.551304i
$349$	1.04232e6	0.458075	0.229038	−	0.973418i	$-0.426442\pi$
$349$	1.04232e6	0.458075	0.229038	+	0.973418i	$0.426442\pi$
$350$	0	0
$351$	1.10852e6	0.480259
$352$	732160.i	0.314956i
$353$	− 2.30309e6i	− 0.983725i	−0.870673	−	0.491862i	$-0.836316\pi$
$353$	− 2.30309e6i	− 0.983725i	0.870673	−	0.491862i	$-0.163684\pi$
$354$	−1.76501e6	−0.748581
$355$	0	0
$356$	−275456.	−0.115193
$357$	1.41527e6i	0.587716i
$358$	− 3.01669e6i	− 1.24401i
$359$	1.67594e6	0.686315	0.343157	−	0.939278i	$-0.388504\pi$
$359$	1.67594e6	0.686315	0.343157	+	0.939278i	$0.388504\pi$
$360$	0	0
$361$	475425.	0.192006
$362$	− 1.21536e6i	− 0.487454i
$363$	− 5.95296e6i	− 2.37119i
$364$	259504.	0.102657
$365$	0	0
$366$	−2.66478e6	−1.03982
$367$	− 94663.0i	− 0.0366872i	−0.999832	−	0.0183436i	$-0.994161\pi$
$367$	− 94663.0i	− 0.0366872i	0.999832	−	0.0183436i	$-0.00583928\pi$
$368$	− 1.01120e6i	− 0.389240i
$369$	−868296.	−0.331972
$370$	0	0
$371$	538216.	0.203012
$372$	1.83763e6i	0.688496i
$373$	− 953536.i	− 0.354867i	−0.984133	−	0.177433i	$-0.943221\pi$
$373$	− 953536.i	− 0.354867i	0.984133	−	0.177433i	$-0.0567794\pi$
$374$	−4.85914e6	−1.79631
$375$	0	0
$376$	−34368.0	−0.0125367
$377$	− 1.51565e6i	− 0.549219i
$378$	− 656404.i	− 0.236288i
$379$	−3.88824e6	−1.39045	−0.695225	−	0.718792i	$-0.744695\pi$
$379$	−3.88824e6	−1.39045	−0.695225	+	0.718792i	$0.744695\pi$
$380$	0	0
$381$	3.72633e6	1.31513
$382$	745084.i	0.261244i
$383$	2.93636e6i	1.02285i	0.859328	+	0.511425i	$0.170882\pi$
$383$	2.93636e6i	1.02285i	−0.859328	+	0.511425i	$0.829118\pi$
$384$	278528.	0.0963920
$385$	0	0
$386$	370016.	0.126402
$387$	− 103868.i	− 0.0352537i
$388$	− 1.01080e6i	− 0.340868i
$389$	−1.70377e6	−0.570871	−0.285435	−	0.958398i	$-0.592138\pi$
$389$	−1.70377e6	−0.570871	−0.285435	+	0.958398i	$0.592138\pi$
$390$	0	0
$391$	6.71105e6	2.21998
$392$	− 153664.i	− 0.0505076i
$393$	1.64359e6i	0.536801i
$394$	2.94547e6	0.955904
$395$	0	0
$396$	−526240.	−0.168634
$397$	− 1.19110e6i	− 0.379292i	−0.981853	−	0.189646i	$-0.939266\pi$
$397$	− 1.19110e6i	− 0.379292i	0.981853	−	0.189646i	$-0.0607340\pi$
$398$	− 1.92648e6i	− 0.609617i
$399$	1.43109e6	0.450024
$400$	0	0
$401$	−3.38330e6	−1.05070	−0.525351	−	0.850885i	$-0.676066\pi$
$401$	−3.38330e6	−1.05070	−0.525351	+	0.850885i	$0.676066\pi$
$402$	300288.i	0.0926772i
$403$	2.23624e6i	0.685891i
$404$	468000.	0.142657
$405$	0	0
$406$	−897484.	−0.270216
$407$	− 1.18104e7i	− 3.53409i
$408$	1.84851e6i	0.549758i
$409$	1.33185e6	0.393682	0.196841	−	0.980435i	$-0.436932\pi$
$409$	1.33185e6	0.393682	0.196841	+	0.980435i	$0.436932\pi$
$410$	0	0
$411$	−3.18390e6	−0.929725
$412$	2.38702e6i	0.692809i
$413$	1.27184e6i	0.366909i
$414$	726800.	0.208408
$415$	0	0
$416$	338944.	0.0960273
$417$	3.00720e6i	0.846880i
$418$	4.91348e6i	1.37546i
$419$	−5.82786e6	−1.62171	−0.810856	−	0.585246i	$-0.800998\pi$
$419$	−5.82786e6	−1.62171	−0.810856	+	0.585246i	$0.800998\pi$
$420$	0	0
$421$	2.47430e6	0.680374	0.340187	−	0.940358i	$-0.389510\pi$
$421$	2.47430e6	0.680374	0.340187	+	0.940358i	$0.389510\pi$
$422$	− 758124.i	− 0.207233i
$423$	− 24702.0i	− 0.00671245i
$424$	702976.	0.189900
$425$	0	0
$426$	2.16784e6	0.578766
$427$	1.92021e6i	0.509659i
$428$	1.33987e6i	0.353552i
$429$	−4.02331e6	−1.05546
$430$	0	0
$431$	4.61851e6	1.19759	0.598796	−	0.800902i	$-0.295646\pi$
$431$	4.61851e6	1.19759	0.598796	+	0.800902i	$0.295646\pi$
$432$	− 857344.i	− 0.221027i
$433$	58606.0i	0.0150218i	0.999972	+	0.00751091i	$0.00239082\pi$
$433$	58606.0i	0.0150218i	−0.999972	+	0.00751091i	$0.997609\pi$
$434$	1.32418e6	0.337459
$435$	0	0
$436$	1.68603e6	0.424766
$437$	− 6.78610e6i	− 1.69987i
$438$	341224.i	0.0849874i
$439$	−7.04298e6	−1.74419	−0.872097	−	0.489332i	$-0.837241\pi$
$439$	−7.04298e6	−1.74419	−0.872097	+	0.489332i	$0.837241\pi$
$440$	0	0
$441$	110446.	0.0270429
$442$	2.24948e6i	0.547678i
$443$	1.46894e6i	0.355627i	0.984064	+	0.177813i	$0.0569023\pi$
$443$	1.46894e6i	0.355627i	−0.984064	+	0.177813i	$0.943098\pi$
$444$	−4.49290e6	−1.08161
$445$	0	0
$446$	−90388.0	−0.0215166
$447$	− 3.38433e6i	− 0.801131i
$448$	− 200704.i	− 0.0472456i
$449$	7.48414e6	1.75197	0.875983	−	0.482341i	$-0.160213\pi$
$449$	7.48414e6	1.75197	0.875983	+	0.482341i	$0.160213\pi$
$450$	0	0
$451$	−1.34963e7	−3.12446
$452$	1.96406e6i	0.452178i
$453$	− 8.01691e6i	− 1.83553i
$454$	3.99247e6	0.909079
$455$	0	0
$456$	1.86918e6	0.420959
$457$	− 170320.i	− 0.0381483i	−0.999818	−	0.0190741i	$-0.993928\pi$
$457$	− 170320.i	− 0.0381483i	0.999818	−	0.0190741i	$-0.00607186\pi$
$458$	− 3.41858e6i	− 0.761520i
$459$	5.68995e6	1.26060
$460$	0	0
$461$	−4.28685e6	−0.939476	−0.469738	−	0.882806i	$-0.655652\pi$
$461$	−4.28685e6	−0.939476	−0.469738	+	0.882806i	$0.655652\pi$
$462$	2.38238e6i	0.519285i
$463$	3.38317e6i	0.733452i	0.930329	+	0.366726i	$0.119521\pi$
$463$	3.38317e6i	0.733452i	−0.930329	+	0.366726i	$0.880479\pi$
$464$	−1.17222e6	−0.252764
$465$	0	0
$466$	5.03274e6	1.07359
$467$	− 5.18029e6i	− 1.09916i	−0.835440	−	0.549581i	$-0.814787\pi$
$467$	− 5.18029e6i	− 1.09916i	0.835440	−	0.549581i	$-0.185213\pi$
$468$	243616.i	0.0514152i
$469$	216384.	0.0454248
$470$	0	0
$471$	1.22492e6	0.254422
$472$	1.66118e6i	0.343212i
$473$	− 1.61447e6i	− 0.331801i
$474$	−1.90244e6	−0.388924
$475$	0	0
$476$	1.33202e6	0.269459
$477$	505264.i	0.101677i
$478$	− 2.82632e6i	− 0.565786i
$479$	8.76779e6	1.74603	0.873014	−	0.487695i	$-0.162162\pi$
$479$	8.76779e6	1.74603	0.873014	+	0.487695i	$0.162162\pi$
$480$	0	0
$481$	−5.46746e6	−1.07751
$482$	− 2.46532e6i	− 0.483343i
$483$	− 3.29035e6i	− 0.641762i
$484$	−5.60278e6	−1.08715
$485$	0	0
$486$	1.37632e6	0.264319
$487$	− 270154.i	− 0.0516166i	−0.999667	−	0.0258083i	$-0.991784\pi$
$487$	− 270154.i	− 0.0516166i	0.999667	−	0.0258083i	$-0.00821594\pi$
$488$	2.50803e6i	0.476742i
$489$	6.55068e6	1.23884
$490$	0	0
$491$	4.85550e6	0.908930	0.454465	−	0.890765i	$-0.349830\pi$
$491$	4.85550e6	0.908930	0.454465	+	0.890765i	$0.349830\pi$
$492$	5.13427e6i	0.956238i
$493$	− 7.77972e6i	− 1.44161i
$494$	2.27463e6	0.419367
$495$	0	0
$496$	1.72954e6	0.315664
$497$	− 1.56212e6i	− 0.283677i
$498$	− 2.55979e6i	− 0.462521i
$499$	2.98576e6	0.536789	0.268394	−	0.963309i	$-0.413507\pi$
$499$	2.98576e6	0.536789	0.268394	+	0.963309i	$0.413507\pi$
$500$	0	0
$501$	9.23027e6	1.64293
$502$	− 763368.i	− 0.135199i
$503$	− 8.28783e6i	− 1.46057i	−0.683145	−	0.730283i	$-0.739388\pi$
$503$	− 8.28783e6i	− 1.46057i	0.683145	−	0.730283i	$-0.260612\pi$
$504$	144256.	0.0252963
$505$	0	0
$506$	1.12970e7	1.96149
$507$	− 4.44944e6i	− 0.768751i
$508$	− 3.50714e6i	− 0.602967i
$509$	−6.24307e6	−1.06808	−0.534040	−	0.845459i	$-0.679327\pi$
$509$	−6.24307e6	−1.06808	−0.534040	+	0.845459i	$0.679327\pi$
$510$	0	0
$511$	245882.	0.0416557
$512$	− 262144.i	− 0.0441942i
$513$	− 5.75358e6i	− 0.965261i
$514$	4.52377e6	0.755253
$515$	0	0
$516$	−614176.	−0.101547
$517$	− 383955.i	− 0.0631763i
$518$	3.23753e6i	0.530138i
$519$	6.30620e6	1.02766
$520$	0	0
$521$	7.49509e6	1.20971	0.604856	−	0.796335i	$-0.293230\pi$
$521$	7.49509e6	1.20971	0.604856	+	0.796335i	$0.293230\pi$
$522$	− 842536.i	− 0.135336i
$523$	3.80957e6i	0.609007i	0.952511	+	0.304503i	$0.0984905\pi$
$523$	3.80957e6i	0.609007i	−0.952511	+	0.304503i	$0.901510\pi$
$524$	1.54691e6	0.246115
$525$	0	0
$526$	−6.69508e6	−1.05509
$527$	1.14784e7i	1.80035i
$528$	3.11168e6i	0.485747i
$529$	−9.16616e6	−1.42413
$530$	0	0
$531$	−1.19398e6	−0.183764
$532$	− 1.34691e6i	− 0.206329i
$533$	6.24796e6i	0.952621i
$534$	−1.17069e6	−0.177659
$535$	0	0
$536$	282624.	0.0424910
$537$	− 1.28209e7i	− 1.91860i
$538$	− 2.52377e6i	− 0.375919i
$539$	1.71672e6	0.254523
$540$	0	0
$541$	7.67156e6	1.12691	0.563457	−	0.826145i	$-0.309471\pi$
$541$	7.67156e6	1.12691	0.563457	+	0.826145i	$0.309471\pi$
$542$	1.48990e6i	0.217851i
$543$	− 5.16528e6i	− 0.751786i
$544$	1.73978e6	0.252055
$545$	0	0
$546$	1.10289e6	0.158326
$547$	9.53845e6i	1.36304i	0.731798	+	0.681522i	$0.238681\pi$
$547$	9.53845e6i	1.36304i	−0.731798	+	0.681522i	$0.761319\pi$
$548$	2.99661e6i	0.426264i
$549$	−1.80265e6	−0.255258
$550$	0	0
$551$	−7.86672e6	−1.10386
$552$	− 4.29760e6i	− 0.600314i
$553$	1.37087e6i	0.190627i
$554$	−3.46804e6	−0.480076
$555$	0	0
$556$	2.83030e6	0.388281
$557$	7.45022e6i	1.01749i	0.860916	+	0.508746i	$0.169891\pi$
$557$	7.45022e6i	1.01749i	−0.860916	+	0.508746i	$0.830109\pi$
$558$	1.24310e6i	0.169014i
$559$	−747398.	−0.101163
$560$	0	0
$561$	−2.06513e7	−2.77039
$562$	7.77992e6i	1.03905i
$563$	3.36698e6i	0.447682i	0.974626	+	0.223841i	$0.0718596\pi$
$563$	3.36698e6i	0.447682i	−0.974626	+	0.223841i	$0.928140\pi$
$564$	−146064.	−0.0193351
$565$	0	0
$566$	4.74005e6	0.621931
$567$	− 3.33744e6i	− 0.435969i
$568$	− 2.04032e6i	− 0.265355i
$569$	4.05501e6	0.525063	0.262532	−	0.964923i	$-0.415443\pi$
$569$	4.05501e6	0.525063	0.262532	+	0.964923i	$0.415443\pi$
$570$	0	0
$571$	7.31585e6	0.939020	0.469510	−	0.882927i	$-0.344431\pi$
$571$	7.31585e6	0.939020	0.469510	+	0.882927i	$0.344431\pi$
$572$	3.78664e6i	0.483909i
$573$	3.16661e6i	0.402910i
$574$	3.69970e6	0.468691
$575$	0	0
$576$	188416.	0.0236626
$577$	9.76895e6i	1.22154i	0.791807	+	0.610771i	$0.209140\pi$
$577$	9.76895e6i	1.22154i	−0.791807	+	0.610771i	$0.790860\pi$
$578$	5.86698e6i	0.730457i
$579$	1.57257e6	0.194945
$580$	0	0
$581$	−1.84456e6	−0.226700
$582$	− 4.29590e6i	− 0.525711i
$583$	7.85356e6i	0.956963i
$584$	321152.	0.0389653
$585$	0	0
$586$	134676.	0.0162012
$587$	− 3.75689e6i	− 0.450021i	−0.974356	−	0.225011i	$-0.927758\pi$
$587$	− 3.75689e6i	− 0.450021i	0.974356	−	0.225011i	$-0.0722417\pi$
$588$	− 653072.i	− 0.0778965i
$589$	1.16068e7	1.37856
$590$	0	0
$591$	1.25183e7	1.47426
$592$	4.22861e6i	0.495899i
$593$	2.89048e6i	0.337546i	0.985655	+	0.168773i	$0.0539805\pi$
$593$	2.89048e6i	0.337546i	−0.985655	+	0.168773i	$0.946020\pi$
$594$	9.57814e6	1.11382
$595$	0	0
$596$	−3.18525e6	−0.367306
$597$	− 8.18754e6i	− 0.940194i
$598$	− 5.22980e6i	− 0.598043i
$599$	−1.32233e7	−1.50582	−0.752910	−	0.658124i	$-0.771350\pi$
$599$	−1.32233e7	−1.50582	−0.752910	+	0.658124i	$0.771350\pi$
$600$	0	0
$601$	3.47399e6	0.392321	0.196161	−	0.980572i	$-0.437153\pi$
$601$	3.47399e6	0.392321	0.196161	+	0.980572i	$0.437153\pi$
$602$	442568.i	0.0497724i
$603$	203136.i	0.0227506i
$604$	−7.54533e6	−0.841561
$605$	0	0
$606$	1.98900e6	0.220015
$607$	− 6.45088e6i	− 0.710636i	−0.934746	−	0.355318i	$-0.884373\pi$
$607$	− 6.45088e6i	− 0.710636i	0.934746	−	0.355318i	$-0.115627\pi$
$608$	− 1.75923e6i	− 0.193003i
$609$	−3.81431e6	−0.416747
$610$	0	0
$611$	−177747.	−0.0192619
$612$	1.25046e6i	0.134956i
$613$	8.43820e6i	0.906982i	0.891261	+	0.453491i	$0.149822\pi$
$613$	8.43820e6i	0.906982i	−0.891261	+	0.453491i	$0.850178\pi$
$614$	108172.	0.0115796
$615$	0	0
$616$	2.24224e6	0.238084
$617$	− 9.45501e6i	− 0.999882i	−0.866059	−	0.499941i	$-0.833355\pi$
$617$	− 9.45501e6i	− 0.999882i	0.866059	−	0.499941i	$-0.166645\pi$
$618$	1.01449e7i	1.06850i
$619$	−1.43145e6	−0.150158	−0.0750790	−	0.997178i	$-0.523921\pi$
$619$	−1.43145e6	−0.150158	−0.0750790	+	0.997178i	$0.523921\pi$
$620$	0	0
$621$	−1.32286e7	−1.37652
$622$	− 8.59674e6i	− 0.890960i
$623$	843584.i	0.0870780i
$624$	1.44051e6	0.148100
$625$	0	0
$626$	−1.06821e7	−1.08948
$627$	2.08823e7i	2.12134i
$628$	− 1.15286e6i	− 0.116648i
$629$	−2.80641e7	−2.82829
$630$	0	0
$631$	−1.01813e7	−1.01795	−0.508977	−	0.860780i	$-0.669976\pi$
$631$	−1.01813e7	−1.01795	−0.508977	+	0.860780i	$0.669976\pi$
$632$	1.79053e6i	0.178315i
$633$	− 3.22203e6i	− 0.319610i
$634$	1.00206e6	0.0990077
$635$	0	0
$636$	2.98765e6	0.292878
$637$	− 794731.i	− 0.0776018i
$638$	− 1.30959e7i	− 1.27375i
$639$	1.46648e6	0.142077
$640$	0	0
$641$	−1.76908e7	−1.70060	−0.850300	−	0.526298i	$-0.823580\pi$
$641$	−1.76908e7	−1.70060	−0.850300	+	0.526298i	$0.823580\pi$
$642$	5.69446e6i	0.545274i
$643$	1.82748e7i	1.74311i	0.490296	+	0.871556i	$0.336889\pi$
$643$	1.82748e7i	1.74311i	−0.490296	+	0.871556i	$0.663111\pi$
$644$	−3.09680e6	−0.294238
$645$	0	0
$646$	1.16755e7	1.10077
$647$	− 1.52897e6i	− 0.143594i	−0.997419	−	0.0717972i	$-0.977127\pi$
$647$	− 1.52897e6i	− 0.143594i	0.997419	−	0.0717972i	$-0.0228734\pi$
$648$	− 4.35910e6i	− 0.407812i
$649$	−1.85585e7	−1.72955
$650$	0	0
$651$	5.62775e6	0.520454
$652$	− 6.16534e6i	− 0.567987i
$653$	− 9.10088e6i	− 0.835219i	−0.908627	−	0.417610i	$-0.862868\pi$
$653$	− 9.10088e6i	− 0.835219i	0.908627	−	0.417610i	$-0.137132\pi$
$654$	7.16564e6	0.655104
$655$	0	0
$656$	4.83226e6	0.438420
$657$	230828.i	0.0208629i
$658$	105252.i	0.00947689i
$659$	430119.	0.0385811	0.0192906	−	0.999814i	$-0.493859\pi$
$659$	430119.	0.0385811	0.0192906	+	0.999814i	$0.493859\pi$
$660$	0	0
$661$	7.65248e6	0.681238	0.340619	−	0.940202i	$-0.389363\pi$
$661$	7.65248e6	0.681238	0.340619	+	0.940202i	$0.389363\pi$
$662$	− 4.23595e6i	− 0.375670i
$663$	9.56027e6i	0.844669i
$664$	−2.40922e6	−0.212058
$665$	0	0
$666$	−3.03931e6	−0.265515
$667$	1.80870e7i	1.57418i
$668$	− 8.68731e6i	− 0.753259i
$669$	−384149.	−0.0331844
$670$	0	0
$671$	−2.80194e7	−2.40244
$672$	− 852992.i	− 0.0728655i
$673$	− 2.18404e7i	− 1.85876i	−0.369128	−	0.929378i	$-0.620344\pi$
$673$	− 2.18404e7i	− 1.85876i	0.369128	−	0.929378i	$-0.379656\pi$
$674$	1.14010e7	0.966702
$675$	0	0
$676$	−4.18771e6	−0.352460
$677$	− 1.39504e7i	− 1.16981i	−0.811102	−	0.584905i	$-0.801132\pi$
$677$	− 1.39504e7i	− 1.16981i	0.811102	−	0.584905i	$-0.198868\pi$
$678$	8.34727e6i	0.697381i
$679$	−3.09558e6	−0.257672
$680$	0	0
$681$	1.69680e7	1.40205
$682$	1.93222e7i	1.59072i
$683$	2.29121e7i	1.87937i	0.342040	+	0.939686i	$0.388882\pi$
$683$	2.29121e7i	1.87937i	−0.342040	+	0.939686i	$0.611118\pi$
$684$	1.26445e6	0.103338
$685$	0	0
$686$	−470596.	−0.0381802
$687$	− 1.45289e7i	− 1.17447i
$688$	578048.i	0.0465578i
$689$	3.63570e6	0.291770
$690$	0	0
$691$	−1.69127e7	−1.34747	−0.673734	−	0.738974i	$-0.735310\pi$
$691$	−1.69127e7	−1.34747	−0.673734	+	0.738974i	$0.735310\pi$
$692$	− 5.93525e6i	− 0.471166i
$693$	1.61161e6i	0.127476i
$694$	−7.56562e6	−0.596274
$695$	0	0
$696$	−4.98195e6	−0.389831
$697$	3.20703e7i	2.50047i
$698$	− 4.16927e6i	− 0.323908i
$699$	2.13891e7	1.65577
$700$	0	0
$701$	−1.90087e7	−1.46102	−0.730510	−	0.682902i	$-0.760718\pi$
$701$	−1.90087e7	−1.46102	−0.730510	+	0.682902i	$0.760718\pi$
$702$	− 4.43408e6i	− 0.339594i
$703$	2.83779e7i	2.16567i
$704$	2.92864e6	0.222707
$705$	0	0
$706$	−9.21235e6	−0.695598
$707$	− 1.43325e6i	− 0.107838i
$708$	7.06003e6i	0.529326i
$709$	−1.66079e7	−1.24079	−0.620396	−	0.784289i	$-0.713028\pi$
$709$	−1.66079e7	−1.24079	−0.620396	+	0.784289i	$0.713028\pi$
$710$	0	0
$711$	−1.28694e6	−0.0954740
$712$	1.10182e6i	0.0814540i
$713$	− 2.66862e7i	− 1.96591i
$714$	5.66107e6	0.415578
$715$	0	0
$716$	−1.20668e7	−0.879646
$717$	− 1.20119e7i	− 0.872596i
$718$	− 6.70378e6i	− 0.485298i
$719$	−5.93610e6	−0.428232	−0.214116	−	0.976808i	$-0.568687\pi$
$719$	−5.93610e6	−0.428232	−0.214116	+	0.976808i	$0.568687\pi$
$720$	0	0
$721$	7.31026e6	0.523715
$722$	− 1.90170e6i	− 0.135768i
$723$	− 1.04776e7i	− 0.745447i
$724$	−4.86144e6	−0.344682
$725$	0	0
$726$	−2.38118e7	−1.67668
$727$	− 1.73276e7i	− 1.21591i	−0.793970	−	0.607957i	$-0.791989\pi$
$727$	− 1.73276e7i	− 1.21591i	0.793970	−	0.607957i	$-0.208011\pi$
$728$	− 1.03802e6i	− 0.0725898i
$729$	−1.07016e7	−0.745814
$730$	0	0
$731$	−3.83634e6	−0.265536
$732$	1.06591e7i	0.735266i
$733$	1.39829e7i	0.961255i	0.876925	+	0.480627i	$0.159591\pi$
$733$	1.39829e7i	0.961255i	−0.876925	+	0.480627i	$0.840409\pi$
$734$	−378652.	−0.0259418
$735$	0	0
$736$	−4.04480e6	−0.275234
$737$	3.15744e6i	0.214125i
$738$	3.47318e6i	0.234740i
$739$	1.14263e7	0.769649	0.384824	−	0.922990i	$-0.374262\pi$
$739$	1.14263e7	0.769649	0.384824	+	0.922990i	$0.374262\pi$
$740$	0	0
$741$	9.66719e6	0.646777
$742$	− 2.15286e6i	− 0.143551i
$743$	1.23126e7i	0.818236i	0.912481	+	0.409118i	$0.134164\pi$
$743$	1.23126e7i	0.818236i	−0.912481	+	0.409118i	$0.865836\pi$
$744$	7.35053e6	0.486840
$745$	0	0
$746$	−3.81414e6	−0.250929
$747$	− 1.73162e6i	− 0.113541i
$748$	1.94366e7i	1.27018i
$749$	4.10336e6	0.267261
$750$	0	0
$751$	−1.43093e7	−0.925806	−0.462903	−	0.886409i	$-0.653192\pi$
$751$	−1.43093e7	−0.925806	−0.462903	+	0.886409i	$0.653192\pi$
$752$	137472.i	0.00886481i
$753$	− 3.24431e6i	− 0.208514i
$754$	−6.06260e6	−0.388356
$755$	0	0
$756$	−2.62562e6	−0.167081
$757$	5.34505e6i	0.339010i	0.985529	+	0.169505i	$0.0542168\pi$
$757$	5.34505e6i	0.339010i	−0.985529	+	0.169505i	$0.945783\pi$
$758$	1.55530e7i	0.983197i
$759$	4.80122e7	3.02515
$760$	0	0
$761$	6.22568e6	0.389695	0.194848	−	0.980834i	$-0.437579\pi$
$761$	6.22568e6	0.389695	0.194848	+	0.980834i	$0.437579\pi$
$762$	− 1.49053e7i	− 0.929938i
$763$	− 5.16347e6i	− 0.321093i
$764$	2.98034e6	0.184728
$765$	0	0
$766$	1.17454e7	0.723264
$767$	8.59144e6i	0.527324i
$768$	− 1.11411e6i	− 0.0681594i
$769$	−1.57888e7	−0.962793	−0.481397	−	0.876503i	$-0.659870\pi$
$769$	−1.57888e7	−0.962793	−0.481397	+	0.876503i	$0.659870\pi$
$770$	0	0
$771$	1.92260e7	1.16481
$772$	− 1.48006e6i	− 0.0893794i
$773$	− 2.50453e7i	− 1.50757i	−0.657121	−	0.753785i	$-0.728226\pi$
$773$	− 2.50453e7i	− 1.50757i	0.657121	−	0.753785i	$-0.271774\pi$
$774$	−415472.	−0.0249281
$775$	0	0
$776$	−4.04320e6	−0.241030
$777$	1.37595e7i	0.817617i
$778$	6.81509e6i	0.403667i
$779$	3.24290e7	1.91465
$780$	0	0
$781$	2.27942e7	1.33720
$782$	− 2.68442e7i	− 1.56976i
$783$	1.53351e7i	0.893884i
$784$	−614656.	−0.0357143
$785$	0	0
$786$	6.57438e6	0.379576
$787$	1.28020e6i	0.0736784i	0.999321	+	0.0368392i	$0.0117289\pi$
$787$	1.28020e6i	0.0736784i	−0.999321	+	0.0368392i	$0.988271\pi$
$788$	− 1.17819e7i	− 0.675926i
$789$	−2.84541e7	−1.62724
$790$	0	0
$791$	6.01495e6	0.341815
$792$	2.10496e6i	0.119242i
$793$	1.29712e7i	0.732484i
$794$	−4.76442e6	−0.268200
$795$	0	0
$796$	−7.70592e6	−0.431064
$797$	1.13798e7i	0.634584i	0.948328	+	0.317292i	$0.102774\pi$
$797$	1.13798e7i	0.634584i	−0.948328	+	0.317292i	$0.897226\pi$
$798$	− 5.72438e6i	− 0.318215i
$799$	−912363.	−0.0505593
$800$	0	0
$801$	−791936.	−0.0436123
$802$	1.35332e7i	0.742959i
$803$	3.58787e6i	0.196358i
$804$	1.20115e6	0.0655327
$805$	0	0
$806$	8.94494e6	0.484998
$807$	− 1.07260e7i	− 0.579768i
$808$	− 1.87200e6i	− 0.100874i
$809$	1.70542e7	0.916138	0.458069	−	0.888917i	$-0.348541\pi$
$809$	1.70542e7	0.916138	0.458069	+	0.888917i	$0.348541\pi$
$810$	0	0
$811$	−2.21494e7	−1.18252	−0.591262	−	0.806480i	$-0.701370\pi$
$811$	−2.21494e7	−1.18252	−0.591262	+	0.806480i	$0.701370\pi$
$812$	3.58994e6i	0.191072i
$813$	6.33209e6i	0.335986i
$814$	−4.72415e7	−2.49898
$815$	0	0
$816$	7.39405e6	0.388738
$817$	3.87924e6i	0.203326i
$818$	− 5.32738e6i	− 0.278375i
$819$	746074.	0.0388662
$820$	0	0
$821$	−1.01068e7	−0.523307	−0.261654	−	0.965162i	$-0.584268\pi$
$821$	−1.01068e7	−0.523307	−0.261654	+	0.965162i	$0.584268\pi$
$822$	1.27356e7i	0.657415i
$823$	− 1.83993e7i	− 0.946895i	−0.880822	−	0.473447i	$-0.843009\pi$
$823$	− 1.83993e7i	− 0.946895i	0.880822	−	0.473447i	$-0.156991\pi$
$824$	9.54810e6	0.489890
$825$	0	0
$826$	5.08738e6	0.259444
$827$	2.48056e7i	1.26121i	0.776105	+	0.630604i	$0.217193\pi$
$827$	2.48056e7i	1.26121i	−0.776105	+	0.630604i	$0.782807\pi$
$828$	− 2.90720e6i	− 0.147367i
$829$	1.19708e6	0.0604976	0.0302488	−	0.999542i	$-0.490370\pi$
$829$	1.19708e6	0.0604976	0.0302488	+	0.999542i	$0.490370\pi$
$830$	0	0
$831$	−1.47392e7	−0.740407
$832$	− 1.35578e6i	− 0.0679015i
$833$	− 4.07930e6i	− 0.203692i
$834$	1.20288e7	0.598835
$835$	0	0
$836$	1.96539e7	0.972598
$837$	− 2.26258e7i	− 1.11633i
$838$	2.33114e7i	1.14672i
$839$	3.17171e7	1.55557	0.777783	−	0.628533i	$-0.216344\pi$
$839$	3.17171e7	1.55557	0.777783	+	0.628533i	$0.216344\pi$
$840$	0	0
$841$	456092.	0.0222363
$842$	− 9.89722e6i	− 0.481097i
$843$	3.30647e7i	1.60249i
$844$	−3.03250e6	−0.146536
$845$	0	0
$846$	−98808.0	−0.00474642
$847$	1.71585e7i	0.821810i
$848$	− 2.81190e6i	− 0.134280i
$849$	2.01452e7	0.959186
$850$	0	0
$851$	6.52461e7	3.08838
$852$	− 8.67136e6i	− 0.409250i
$853$	3.18237e7i	1.49754i	0.662831	+	0.748769i	$0.269355\pi$
$853$	3.18237e7i	1.49754i	−0.662831	+	0.748769i	$0.730645\pi$
$854$	7.68085e6	0.360383
$855$	0	0
$856$	5.35949e6	0.249999
$857$	− 2.27853e7i	− 1.05975i	−0.848076	−	0.529874i	$-0.822239\pi$
$857$	− 2.27853e7i	− 1.05975i	0.848076	−	0.529874i	$-0.177761\pi$
$858$	1.60932e7i	0.746319i
$859$	1.85966e7	0.859907	0.429953	−	0.902851i	$-0.358530\pi$
$859$	1.85966e7	0.859907	0.429953	+	0.902851i	$0.358530\pi$
$860$	0	0
$861$	1.57237e7	0.722848
$862$	− 1.84740e7i	− 0.846825i
$863$	− 2.77046e7i	− 1.26627i	−0.774043	−	0.633133i	$-0.781769\pi$
$863$	− 2.77046e7i	− 1.26627i	0.774043	−	0.633133i	$-0.218231\pi$
$864$	−3.42938e6	−0.156290
$865$	0	0
$866$	234424.	0.0106220
$867$	2.49346e7i	1.12656i
$868$	− 5.29670e6i	− 0.238620i
$869$	−2.00036e7	−0.898582
$870$	0	0
$871$	1.46170e6	0.0652847
$872$	− 6.74413e6i	− 0.300355i
$873$	− 2.90605e6i	− 0.129053i
$874$	−2.71444e7	−1.20199
$875$	0	0
$876$	1.36490e6	0.0600951
$877$	2.41150e7i	1.05874i	0.848391	+	0.529370i	$0.177572\pi$
$877$	2.41150e7i	1.05874i	−0.848391	+	0.529370i	$0.822428\pi$
$878$	2.81719e7i	1.23333i
$879$	572373.	0.0249866
$880$	0	0
$881$	−1.26207e7	−0.547827	−0.273914	−	0.961754i	$-0.588318\pi$
$881$	−1.26207e7	−0.547827	−0.273914	+	0.961754i	$0.588318\pi$
$882$	− 441784.i	− 0.0191222i
$883$	− 6.01876e6i	− 0.259780i	−0.991528	−	0.129890i	$-0.958538\pi$
$883$	− 6.01876e6i	− 0.259780i	0.991528	−	0.129890i	$-0.0414624\pi$
$884$	8.99790e6	0.387267
$885$	0	0
$886$	5.87575e6	0.251466
$887$	− 2.36901e7i	− 1.01102i	−0.862821	−	0.505509i	$-0.831305\pi$
$887$	− 2.36901e7i	− 1.01102i	0.862821	−	0.505509i	$-0.168695\pi$
$888$	1.79716e7i	0.764811i
$889$	−1.07406e7	−0.455800
$890$	0	0
$891$	4.86994e7	2.05508
$892$	361552.i	0.0152145i
$893$	922566.i	0.0387141i
$894$	−1.35373e7	−0.566485
$895$	0	0
$896$	−802816.	−0.0334077
$897$	− 2.22266e7i	− 0.922344i
$898$	− 2.99365e7i	− 1.23883i
$899$	−3.09357e7	−1.27662
$900$	0	0
$901$	1.86618e7	0.765847
$902$	5.39854e7i	2.20933i
$903$	1.88091e6i	0.0767626i
$904$	7.85626e6	0.319738
$905$	0	0
$906$	−3.20676e7	−1.29792
$907$	1.10583e7i	0.446346i	0.974779	+	0.223173i	$0.0716414\pi$
$907$	1.10583e7i	0.446346i	−0.974779	+	0.223173i	$0.928359\pi$
$908$	− 1.59699e7i	− 0.642816i
$909$	1.34550e6	0.0540100
$910$	0	0
$911$	3.07573e6	0.122787	0.0613934	−	0.998114i	$-0.480446\pi$
$911$	3.07573e6	0.122787	0.0613934	+	0.998114i	$0.480446\pi$
$912$	− 7.47674e6i	− 0.297663i
$913$	− 2.69155e7i	− 1.06862i
$914$	−681280.	−0.0269749
$915$	0	0
$916$	−1.36743e7	−0.538476
$917$	− 4.73742e6i	− 0.186045i
$918$	− 2.27598e7i	− 0.891378i
$919$	1.89018e7	0.738270	0.369135	−	0.929376i	$-0.379654\pi$
$919$	1.89018e7	0.738270	0.369135	+	0.929376i	$0.379654\pi$
$920$	0	0
$921$	459731.	0.0178589
$922$	1.71474e7i	0.664310i
$923$	− 1.05523e7i	− 0.407701i
$924$	9.52952e6	0.367190
$925$	0	0
$926$	1.35327e7	0.518629
$927$	6.86269e6i	0.262298i
$928$	4.68890e6i	0.178731i
$929$	1.81458e7	0.689821	0.344911	−	0.938636i	$-0.387909\pi$
$929$	1.81458e7	0.689821	0.344911	+	0.938636i	$0.387909\pi$
$930$	0	0
$931$	−4.12492e6	−0.155970
$932$	− 2.01309e7i	− 0.759144i
$933$	− 3.65362e7i	− 1.37410i
$934$	−2.07212e7	−0.777225
$935$	0	0
$936$	974464.	0.0363560
$937$	2.17350e7i	0.808744i	0.914595	+	0.404372i	$0.132510\pi$
$937$	2.17350e7i	0.808744i	−0.914595	+	0.404372i	$0.867490\pi$
$938$	− 865536.i	− 0.0321202i
$939$	−4.53989e7	−1.68028
$940$	0	0
$941$	1.86808e7	0.687735	0.343868	−	0.939018i	$-0.388263\pi$
$941$	1.86808e7	0.687735	0.343868	+	0.939018i	$0.388263\pi$
$942$	− 4.89967e6i	− 0.179904i
$943$	− 7.45602e7i	− 2.73041i
$944$	6.64474e6	0.242688
$945$	0	0
$946$	−6.45788e6	−0.234618
$947$	2.24778e6i	0.0814476i	0.999170	+	0.0407238i	$0.0129664\pi$
$947$	2.24778e6i	0.0814476i	−0.999170	+	0.0407238i	$0.987034\pi$
$948$	7.60974e6i	0.275010i
$949$	1.66096e6	0.0598678
$950$	0	0
$951$	4.25874e6	0.152697
$952$	− 5.32806e6i	− 0.190536i
$953$	− 3.73293e7i	− 1.33143i	−0.746207	−	0.665714i	$-0.768127\pi$
$953$	− 3.73293e7i	− 1.33143i	0.746207	−	0.665714i	$-0.231873\pi$
$954$	2.02106e6	0.0718964
$955$	0	0
$956$	−1.13053e7	−0.400071
$957$	− 5.56577e7i	− 1.96447i
$958$	− 3.50711e7i	− 1.23463i
$959$	9.17711e6	0.322225
$960$	0	0
$961$	1.70144e7	0.594303
$962$	2.18698e7i	0.761917i
$963$	3.85213e6i	0.133855i
$964$	−9.86128e6	−0.341775
$965$	0	0
$966$	−1.31614e7	−0.453795
$967$	− 2.61870e7i	− 0.900573i	−0.892884	−	0.450287i	$-0.851322\pi$
$967$	− 2.61870e7i	− 0.900573i	0.892884	−	0.450287i	$-0.148678\pi$
$968$	2.24111e7i	0.768733i
$969$	4.96210e7	1.69768
$970$	0	0
$971$	−3.91957e7	−1.33410	−0.667052	−	0.745011i	$-0.732444\pi$
$971$	−3.91957e7	−1.33410	−0.667052	+	0.745011i	$0.732444\pi$
$972$	− 5.50528e6i	− 0.186902i
$973$	− 8.66781e6i	− 0.293513i
$974$	−1.08062e6	−0.0364984
$975$	0	0
$976$	1.00321e7	0.337108
$977$	− 3.03935e6i	− 0.101870i	−0.998702	−	0.0509348i	$-0.983780\pi$
$977$	− 3.03935e6i	− 0.101870i	0.998702	−	0.0509348i	$-0.0162201\pi$
$978$	− 2.62027e7i	− 0.875990i
$979$	−1.23094e7	−0.410470
$980$	0	0
$981$	4.84734e6	0.160817
$982$	− 1.94220e7i	− 0.642711i
$983$	− 1.59937e7i	− 0.527915i	−0.964534	−	0.263957i	$-0.914972\pi$
$983$	− 1.59937e7i	− 0.527915i	0.964534	−	0.263957i	$-0.0850278\pi$
$984$	2.05371e7	0.676163
$985$	0	0
$986$	−3.11189e7	−1.01937
$987$	447321.i	0.0146159i
$988$	− 9.09853e6i	− 0.296537i
$989$	8.91910e6	0.289955
$990$	0	0
$991$	3.63186e6	0.117475	0.0587375	−	0.998273i	$-0.481293\pi$
$991$	3.63186e6	0.117475	0.0587375	+	0.998273i	$0.481293\pi$
$992$	− 6.91814e6i	− 0.223208i
$993$	− 1.80028e7i	− 0.579384i
$994$	−6.24848e6	−0.200590
$995$	0	0
$996$	−1.02392e7	−0.327052
$997$	4.33287e7i	1.38051i	0.723568	+	0.690253i	$0.242501\pi$
$997$	4.33287e7i	1.38051i	−0.723568	+	0.690253i	$0.757499\pi$
$998$	− 1.19430e7i	− 0.379567i
$999$	5.53188e7	1.75371

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.c.a.99.1		2
5.2	odd	4		70.6.a.e.1.1	✓	1
5.3	odd	4		350.6.a.e.1.1		1
5.4	even	2	inner	350.6.c.a.99.2		2
15.2	even	4		630.6.a.b.1.1		1
20.7	even	4		560.6.a.h.1.1		1
35.27	even	4		490.6.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.6.a.e.1.1	✓	1	5.2	odd	4
350.6.a.e.1.1		1	5.3	odd	4
350.6.c.a.99.1		2	1.1	even	1	trivial
350.6.c.a.99.2		2	5.4	even	2	inner
490.6.a.m.1.1		1	35.27	even	4
560.6.a.h.1.1		1	20.7	even	4
630.6.a.b.1.1		1	15.2	even	4

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:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} -17.0000i q^{3} -16.0000 q^{4} -68.0000 q^{6} +49.0000i q^{7} +64.0000i q^{8} -46.0000 q^{9} +O(q^{10})\)	\(q-4.00000i q^{2} -17.0000i q^{3} -16.0000 q^{4} -68.0000 q^{6} +49.0000i q^{7} +64.0000i q^{8} -46.0000 q^{9} -715.000 q^{11} +272.000i q^{12} +331.000i q^{13} +196.000 q^{14} +256.000 q^{16} +1699.00i q^{17} +184.000i q^{18} +1718.00 q^{19} +833.000 q^{21} +2860.00i q^{22} -3950.00i q^{23} +1088.00 q^{24} +1324.00 q^{26} -3349.00i q^{27} -784.000i q^{28} -4579.00 q^{29} +6756.00 q^{31} -1024.00i q^{32} +12155.0i q^{33} +6796.00 q^{34} +736.000 q^{36} +16518.0i q^{37} -6872.00i q^{38} +5627.00 q^{39} +18876.0 q^{41} -3332.00i q^{42} +2258.00i q^{43} +11440.0 q^{44} -15800.0 q^{46} +537.000i q^{47} -4352.00i q^{48} -2401.00 q^{49} +28883.0 q^{51} -5296.00i q^{52} -10984.0i q^{53} -13396.0 q^{54} -3136.00 q^{56} -29206.0i q^{57} +18316.0i q^{58} +25956.0 q^{59} +39188.0 q^{61} -27024.0i q^{62} -2254.00i q^{63} -4096.00 q^{64} +48620.0 q^{66} -4416.00i q^{67} -27184.0i q^{68} -67150.0 q^{69} -31880.0 q^{71} -2944.00i q^{72} -5018.00i q^{73} +66072.0 q^{74} -27488.0 q^{76} -35035.0i q^{77} -22508.0i q^{78} +27977.0 q^{79} -68111.0 q^{81} -75504.0i q^{82} +37644.0i q^{83} -13328.0 q^{84} +9032.00 q^{86} +77843.0i q^{87} -45760.0i q^{88} +17216.0 q^{89} -16219.0 q^{91} +63200.0i q^{92} -114852. i q^{93} +2148.00 q^{94} -17408.0 q^{96} +63175.0i q^{97} +9604.00i q^{98} +32890.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} - 136 q^{6} - 92 q^{9}+O(q^{10}) \) 2 * q - 32 * q^4 - 136 * q^6 - 92 * q^9	\( 2 q - 32 q^{4} - 136 q^{6} - 92 q^{9} - 1430 q^{11} + 392 q^{14} + 512 q^{16} + 3436 q^{19} + 1666 q^{21} + 2176 q^{24} + 2648 q^{26} - 9158 q^{29} + 13512 q^{31} + 13592 q^{34} + 1472 q^{36} + 11254 q^{39} + 37752 q^{41} + 22880 q^{44} - 31600 q^{46} - 4802 q^{49} + 57766 q^{51} - 26792 q^{54} - 6272 q^{56} + 51912 q^{59} + 78376 q^{61} - 8192 q^{64} + 97240 q^{66} - 134300 q^{69} - 63760 q^{71} + 132144 q^{74} - 54976 q^{76} + 55954 q^{79} - 136222 q^{81} - 26656 q^{84} + 18064 q^{86} + 34432 q^{89} - 32438 q^{91} + 4296 q^{94} - 34816 q^{96} + 65780 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 - 136 * q^6 - 92 * q^9 - 1430 * q^11 + 392 * q^14 + 512 * q^16 + 3436 * q^19 + 1666 * q^21 + 2176 * q^24 + 2648 * q^26 - 9158 * q^29 + 13512 * q^31 + 13592 * q^34 + 1472 * q^36 + 11254 * q^39 + 37752 * q^41 + 22880 * q^44 - 31600 * q^46 - 4802 * q^49 + 57766 * q^51 - 26792 * q^54 - 6272 * q^56 + 51912 * q^59 + 78376 * q^61 - 8192 * q^64 + 97240 * q^66 - 134300 * q^69 - 63760 * q^71 + 132144 * q^74 - 54976 * q^76 + 55954 * q^79 - 136222 * q^81 - 26656 * q^84 + 18064 * q^86 + 34432 * q^89 - 32438 * q^91 + 4296 * q^94 - 34816 * q^96 + 65780 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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