LMFDB - Embedded newform 350.6.c.d.99.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,6,Mod(99,350)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,0,-32,0,-80,0,0,286,0,-672] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000i q^{2} +10.0000i q^{3} -16.0000 q^{4} -40.0000 q^{6} -49.0000i q^{7} -64.0000i q^{8} +143.000 q^{9} -336.000 q^{11} -160.000i q^{12} +584.000i q^{13} +196.000 q^{14} +256.000 q^{16} +1458.00i q^{17} +572.000i q^{18} -470.000 q^{19} +490.000 q^{21} -1344.00i q^{22} -4200.00i q^{23} +640.000 q^{24} -2336.00 q^{26} +3860.00i q^{27} +784.000i q^{28} -4866.00 q^{29} -7372.00 q^{31} +1024.00i q^{32} -3360.00i q^{33} -5832.00 q^{34} -2288.00 q^{36} -14330.0i q^{37} -1880.00i q^{38} -5840.00 q^{39} +6222.00 q^{41} +1960.00i q^{42} +3704.00i q^{43} +5376.00 q^{44} +16800.0 q^{46} +1812.00i q^{47} +2560.00i q^{48} -2401.00 q^{49} -14580.0 q^{51} -9344.00i q^{52} -37242.0i q^{53} -15440.0 q^{54} -3136.00 q^{56} -4700.00i q^{57} -19464.0i q^{58} -34302.0 q^{59} +24476.0 q^{61} -29488.0i q^{62} -7007.00i q^{63} -4096.00 q^{64} +13440.0 q^{66} +17452.0i q^{67} -23328.0i q^{68} +42000.0 q^{69} +28224.0 q^{71} -9152.00i q^{72} +3602.00i q^{73} +57320.0 q^{74} +7520.00 q^{76} +16464.0i q^{77} -23360.0i q^{78} -42872.0 q^{79} -3851.00 q^{81} +24888.0i q^{82} -35202.0i q^{83} -7840.00 q^{84} -14816.0 q^{86} -48660.0i q^{87} +21504.0i q^{88} -26730.0 q^{89} +28616.0 q^{91} +67200.0i q^{92} -73720.0i q^{93} -7248.00 q^{94} -10240.0 q^{96} +16978.0i q^{97} -9604.00i q^{98} -48048.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} - 80 q^{6} + 286 q^{9} - 672 q^{11} + 392 q^{14} + 512 q^{16} - 940 q^{19} + 980 q^{21} + 1280 q^{24} - 4672 q^{26} - 9732 q^{29} - 14744 q^{31} - 11664 q^{34} - 4576 q^{36} - 11680 q^{39}+ \cdots - 96096 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 - 80 * q^6 + 286 * q^9 - 672 * q^11 + 392 * q^14 + 512 * q^16 - 940 * q^19 + 980 * q^21 + 1280 * q^24 - 4672 * q^26 - 9732 * q^29 - 14744 * q^31 - 11664 * q^34 - 4576 * q^36 - 11680 * q^39 + 12444 * q^41 + 10752 * q^44 + 33600 * q^46 - 4802 * q^49 - 29160 * q^51 - 30880 * q^54 - 6272 * q^56 - 68604 * q^59 + 48952 * q^61 - 8192 * q^64 + 26880 * q^66 + 84000 * q^69 + 56448 * q^71 + 114640 * q^74 + 15040 * q^76 - 85744 * q^79 - 7702 * q^81 - 15680 * q^84 - 29632 * q^86 - 53460 * q^89 + 57232 * q^91 - 14496 * q^94 - 20480 * q^96 - 96096 * q^99

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
$3$	10.0000i	0.641500i	0.947164	+	0.320750i	$0.103935\pi$
$3$	10.0000i	0.641500i	−0.947164	+	0.320750i	$0.896065\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−40.0000	−0.453609
$7$	− 49.0000i	− 0.377964i
$7$	− 49.0000i	− 0.377964i
$8$	− 64.0000i	− 0.353553i
$9$	143.000	0.588477
$10$	0	0
$11$	−336.000	−0.837255	−0.418627	−	0.908158i	$-0.637489\pi$
$11$	−336.000	−0.837255	−0.418627	+	0.908158i	$0.637489\pi$
$12$	− 160.000i	− 0.320750i
$13$	584.000i	0.958417i	0.877701	+	0.479208i	$0.159076\pi$
$13$	584.000i	0.958417i	−0.877701	+	0.479208i	$0.840924\pi$
$14$	196.000	0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	1458.00i	1.22359i	0.791017	+	0.611794i	$0.209552\pi$
$17$	1458.00i	1.22359i	−0.791017	+	0.611794i	$0.790448\pi$
$18$	572.000i	0.416116i
$19$	−470.000	−0.298685	−0.149343	−	0.988786i	$-0.547716\pi$
$19$	−470.000	−0.298685	−0.149343	+	0.988786i	$0.547716\pi$
$20$	0	0
$21$	490.000	0.242464
$22$	− 1344.00i	− 0.592028i
$23$	− 4200.00i	− 1.65550i	−0.561096	−	0.827751i	$-0.689620\pi$
$23$	− 4200.00i	− 1.65550i	0.561096	−	0.827751i	$-0.310380\pi$
$24$	640.000	0.226805
$25$	0	0
$26$	−2336.00	−0.677703
$27$	3860.00i	1.01901i
$28$	784.000i	0.188982i
$29$	−4866.00	−1.07443	−0.537214	−	0.843446i	$-0.680523\pi$
$29$	−4866.00	−1.07443	−0.537214	+	0.843446i	$0.680523\pi$
$30$	0	0
$31$	−7372.00	−1.37778	−0.688892	−	0.724864i	$-0.741903\pi$
$31$	−7372.00	−1.37778	−0.688892	+	0.724864i	$0.741903\pi$
$32$	1024.00i	0.176777i
$33$	− 3360.00i	− 0.537099i
$34$	−5832.00	−0.865207
$35$	0	0
$36$	−2288.00	−0.294239
$37$	− 14330.0i	− 1.72085i	−0.509581	−	0.860423i	$-0.670200\pi$
$37$	− 14330.0i	− 1.72085i	0.509581	−	0.860423i	$-0.329800\pi$
$38$	− 1880.00i	− 0.211202i
$39$	−5840.00	−0.614825
$40$	0	0
$41$	6222.00	0.578057	0.289028	−	0.957321i	$-0.406668\pi$
$41$	6222.00	0.578057	0.289028	+	0.957321i	$0.406668\pi$
$42$	1960.00i	0.171448i
$43$	3704.00i	0.305492i	0.988265	+	0.152746i	$0.0488116\pi$
$43$	3704.00i	0.305492i	−0.988265	+	0.152746i	$0.951188\pi$
$44$	5376.00	0.418627
$45$	0	0
$46$	16800.0	1.17062
$47$	1812.00i	0.119650i	0.998209	+	0.0598251i	$0.0190543\pi$
$47$	1812.00i	0.119650i	−0.998209	+	0.0598251i	$0.980946\pi$
$48$	2560.00i	0.160375i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	−14580.0	−0.784932
$52$	− 9344.00i	− 0.479208i
$53$	− 37242.0i	− 1.82114i	−0.413355	−	0.910570i	$-0.635643\pi$
$53$	− 37242.0i	− 1.82114i	0.413355	−	0.910570i	$-0.364357\pi$
$54$	−15440.0	−0.720548
$55$	0	0
$56$	−3136.00	−0.133631
$57$	− 4700.00i	− 0.191607i
$58$	− 19464.0i	− 0.759735i
$59$	−34302.0	−1.28289	−0.641445	−	0.767169i	$-0.721665\pi$
$59$	−34302.0	−1.28289	−0.641445	+	0.767169i	$0.721665\pi$
$60$	0	0
$61$	24476.0	0.842201	0.421101	−	0.907014i	$-0.361644\pi$
$61$	24476.0	0.842201	0.421101	+	0.907014i	$0.361644\pi$
$62$	− 29488.0i	− 0.974240i
$63$	− 7007.00i	− 0.222424i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	13440.0	0.379786
$67$	17452.0i	0.474961i	0.971392	+	0.237481i	$0.0763216\pi$
$67$	17452.0i	0.474961i	−0.971392	+	0.237481i	$0.923678\pi$
$68$	− 23328.0i	− 0.611794i
$69$	42000.0	1.06201
$70$	0	0
$71$	28224.0	0.664466	0.332233	−	0.943197i	$-0.392198\pi$
$71$	28224.0	0.664466	0.332233	+	0.943197i	$0.392198\pi$
$72$	− 9152.00i	− 0.208058i
$73$	3602.00i	0.0791109i	0.999217	+	0.0395555i	$0.0125942\pi$
$73$	3602.00i	0.0791109i	−0.999217	+	0.0395555i	$0.987406\pi$
$74$	57320.0	1.21682
$75$	0	0
$76$	7520.00	0.149343
$77$	16464.0i	0.316453i
$78$	− 23360.0i	− 0.434747i
$79$	−42872.0	−0.772869	−0.386435	−	0.922317i	$-0.626294\pi$
$79$	−42872.0	−0.772869	−0.386435	+	0.922317i	$0.626294\pi$
$80$	0	0
$81$	−3851.00	−0.0652170
$82$	24888.0i	0.408748i
$83$	− 35202.0i	− 0.560883i	−0.959871	−	0.280441i	$-0.909519\pi$
$83$	− 35202.0i	− 0.560883i	0.959871	−	0.280441i	$-0.0904808\pi$
$84$	−7840.00	−0.121232
$85$	0	0
$86$	−14816.0	−0.216015
$87$	− 48660.0i	− 0.689246i
$88$	21504.0i	0.296014i
$89$	−26730.0	−0.357704	−0.178852	−	0.983876i	$-0.557238\pi$
$89$	−26730.0	−0.357704	−0.178852	+	0.983876i	$0.557238\pi$
$90$	0	0
$91$	28616.0	0.362248
$92$	67200.0i	0.827751i
$93$	− 73720.0i	− 0.883849i
$94$	−7248.00	−0.0846055
$95$	0	0
$96$	−10240.0	−0.113402
$97$	16978.0i	0.183213i	0.995795	+	0.0916067i	$0.0292003\pi$
$97$	16978.0i	0.183213i	−0.995795	+	0.0916067i	$0.970800\pi$
$98$	− 9604.00i	− 0.101015i
$99$	−48048.0	−0.492705
$100$	0	0
$101$	99204.0	0.967667	0.483833	−	0.875160i	$-0.339244\pi$
$101$	99204.0	0.967667	0.483833	+	0.875160i	$0.339244\pi$
$102$	− 58320.0i	− 0.555031i
$103$	− 131644.i	− 1.22267i	−0.791373	−	0.611333i	$-0.790634\pi$
$103$	− 131644.i	− 1.22267i	0.791373	−	0.611333i	$-0.209366\pi$
$104$	37376.0	0.338852
$105$	0	0
$106$	148968.	1.28774
$107$	− 48852.0i	− 0.412499i	−0.978499	−	0.206250i	$-0.933874\pi$
$107$	− 48852.0i	− 0.412499i	0.978499	−	0.206250i	$-0.0661259\pi$
$108$	− 61760.0i	− 0.509504i
$109$	56374.0	0.454478	0.227239	−	0.973839i	$-0.427030\pi$
$109$	56374.0	0.454478	0.227239	+	0.973839i	$0.427030\pi$
$110$	0	0
$111$	143300.	1.10392
$112$	− 12544.0i	− 0.0944911i
$113$	8742.00i	0.0644043i	0.999481	+	0.0322021i	$0.0102520\pi$
$113$	8742.00i	0.0644043i	−0.999481	+	0.0322021i	$0.989748\pi$
$114$	18800.0	0.135486
$115$	0	0
$116$	77856.0	0.537214
$117$	83512.0i	0.564007i
$118$	− 137208.i	− 0.907140i
$119$	71442.0	0.462473
$120$	0	0
$121$	−48155.0	−0.299005
$122$	97904.0i	0.595526i
$123$	62220.0i	0.370823i
$124$	117952.	0.688892
$125$	0	0
$126$	28028.0	0.157277
$127$	− 315992.i	− 1.73847i	−0.494401	−	0.869234i	$-0.664612\pi$
$127$	− 315992.i	− 1.73847i	0.494401	−	0.869234i	$-0.335388\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	−37040.0	−0.195973
$130$	0	0
$131$	−24666.0	−0.125580	−0.0627900	−	0.998027i	$-0.520000\pi$
$131$	−24666.0	−0.125580	−0.0627900	+	0.998027i	$0.520000\pi$
$132$	53760.0i	0.268550i
$133$	23030.0i	0.112892i
$134$	−69808.0	−0.335848
$135$	0	0
$136$	93312.0	0.432604
$137$	− 303234.i	− 1.38031i	−0.723662	−	0.690155i	$-0.757542\pi$
$137$	− 303234.i	− 1.38031i	0.723662	−	0.690155i	$-0.242458\pi$
$138$	168000.i	0.750951i
$139$	−250586.	−1.10007	−0.550034	−	0.835142i	$-0.685385\pi$
$139$	−250586.	−1.10007	−0.550034	+	0.835142i	$0.685385\pi$
$140$	0	0
$141$	−18120.0	−0.0767557
$142$	112896.i	0.469848i
$143$	− 196224.i	− 0.802439i
$144$	36608.0	0.147119
$145$	0	0
$146$	−14408.0	−0.0559399
$147$	− 24010.0i	− 0.0916429i
$148$	229280.i	0.860423i
$149$	60594.0	0.223596	0.111798	−	0.993731i	$-0.464339\pi$
$149$	60594.0	0.223596	0.111798	+	0.993731i	$0.464339\pi$
$150$	0	0
$151$	124448.	0.444166	0.222083	−	0.975028i	$-0.428714\pi$
$151$	124448.	0.444166	0.222083	+	0.975028i	$0.428714\pi$
$152$	30080.0i	0.105601i
$153$	208494.i	0.720054i
$154$	−65856.0	−0.223766
$155$	0	0
$156$	93440.0	0.307412
$157$	− 76040.0i	− 0.246203i	−0.992394	−	0.123101i	$-0.960716\pi$
$157$	− 76040.0i	− 0.246203i	0.992394	−	0.123101i	$-0.0392840\pi$
$158$	− 171488.i	− 0.546501i
$159$	372420.	1.16826
$160$	0	0
$161$	−205800.	−0.625721
$162$	− 15404.0i	− 0.0461154i
$163$	124256.i	0.366310i	0.983084	+	0.183155i	$0.0586310\pi$
$163$	124256.i	0.366310i	−0.983084	+	0.183155i	$0.941369\pi$
$164$	−99552.0	−0.289028
$165$	0	0
$166$	140808.	0.396604
$167$	72420.0i	0.200940i	0.994940	+	0.100470i	$0.0320347\pi$
$167$	72420.0i	0.200940i	−0.994940	+	0.100470i	$0.967965\pi$
$168$	− 31360.0i	− 0.0857241i
$169$	30237.0	0.0814370
$170$	0	0
$171$	−67210.0	−0.175770
$172$	− 59264.0i	− 0.152746i
$173$	− 441552.i	− 1.12167i	−0.827926	−	0.560837i	$-0.810479\pi$
$173$	− 441552.i	− 1.12167i	0.827926	−	0.560837i	$-0.189521\pi$
$174$	194640.	0.487370
$175$	0	0
$176$	−86016.0	−0.209314
$177$	− 343020.i	− 0.822974i
$178$	− 106920.i	− 0.252935i
$179$	10692.0	0.0249417	0.0124709	−	0.999922i	$-0.496030\pi$
$179$	10692.0	0.0249417	0.0124709	+	0.999922i	$0.496030\pi$
$180$	0	0
$181$	−546064.	−1.23893	−0.619465	−	0.785024i	$-0.712651\pi$
$181$	−546064.	−1.23893	−0.619465	+	0.785024i	$0.712651\pi$
$182$	114464.i	0.256148i
$183$	244760.i	0.540272i
$184$	−268800.	−0.585308
$185$	0	0
$186$	294880.	0.624975
$187$	− 489888.i	− 1.02445i
$188$	− 28992.0i	− 0.0598251i
$189$	189140.	0.385149
$190$	0	0
$191$	−575976.	−1.14241	−0.571204	−	0.820808i	$-0.693523\pi$
$191$	−575976.	−1.14241	−0.571204	+	0.820808i	$0.693523\pi$
$192$	− 40960.0i	− 0.0801875i
$193$	− 413938.i	− 0.799912i	−0.916534	−	0.399956i	$-0.869025\pi$
$193$	− 413938.i	− 0.799912i	0.916534	−	0.399956i	$-0.130975\pi$
$194$	−67912.0	−0.129551
$195$	0	0
$196$	38416.0	0.0714286
$197$	494946.i	0.908641i	0.890838	+	0.454320i	$0.150118\pi$
$197$	494946.i	0.908641i	−0.890838	+	0.454320i	$0.849882\pi$
$198$	− 192192.i	− 0.348395i
$199$	−520364.	−0.931482	−0.465741	−	0.884921i	$-0.654212\pi$
$199$	−520364.	−0.931482	−0.465741	+	0.884921i	$0.654212\pi$
$200$	0	0
$201$	−174520.	−0.304688
$202$	396816.i	0.684244i
$203$	238434.i	0.406095i
$204$	233280.	0.392466
$205$	0	0
$206$	526576.	0.864556
$207$	− 600600.i	− 0.974225i
$208$	149504.i	0.239604i
$209$	157920.	0.250076
$210$	0	0
$211$	183284.	0.283412	0.141706	−	0.989909i	$-0.454741\pi$
$211$	183284.	0.283412	0.141706	+	0.989909i	$0.454741\pi$
$212$	595872.i	0.910570i
$213$	282240.i	0.426255i
$214$	195408.	0.291681
$215$	0	0
$216$	247040.	0.360274
$217$	361228.i	0.520753i
$218$	225496.i	0.321364i
$219$	−36020.0	−0.0507497
$220$	0	0
$221$	−851472.	−1.17271
$222$	573200.i	0.780591i
$223$	− 1.27746e6i	− 1.72023i	−0.510100	−	0.860115i	$-0.670392\pi$
$223$	− 1.27746e6i	− 1.72023i	0.510100	−	0.860115i	$-0.329608\pi$
$224$	50176.0	0.0668153
$225$	0	0
$226$	−34968.0	−0.0455407
$227$	1.28764e6i	1.65856i	0.558835	+	0.829279i	$0.311248\pi$
$227$	1.28764e6i	1.65856i	−0.558835	+	0.829279i	$0.688752\pi$
$228$	75200.0i	0.0958034i
$229$	−350936.	−0.442221	−0.221110	−	0.975249i	$-0.570968\pi$
$229$	−350936.	−0.442221	−0.221110	+	0.975249i	$0.570968\pi$
$230$	0	0
$231$	−164640.	−0.203004
$232$	311424.i	0.379867i
$233$	836154.i	1.00901i	0.863408	+	0.504506i	$0.168325\pi$
$233$	836154.i	1.00901i	−0.863408	+	0.504506i	$0.831675\pi$
$234$	−334048.	−0.398813
$235$	0	0
$236$	548832.	0.641445
$237$	− 428720.i	− 0.495796i
$238$	285768.i	0.327018i
$239$	−774336.	−0.876869	−0.438434	−	0.898763i	$-0.644467\pi$
$239$	−774336.	−0.876869	−0.438434	+	0.898763i	$0.644467\pi$
$240$	0	0
$241$	−1.15285e6	−1.27859	−0.639293	−	0.768963i	$-0.720773\pi$
$241$	−1.15285e6	−1.27859	−0.639293	+	0.768963i	$0.720773\pi$
$242$	− 192620.i	− 0.211428i
$243$	899470.i	0.977172i
$244$	−391616.	−0.421101
$245$	0	0
$246$	−248880.	−0.262212
$247$	− 274480.i	− 0.286265i
$248$	471808.i	0.487120i
$249$	352020.	0.359806
$250$	0	0
$251$	1.35801e6	1.36056	0.680282	−	0.732951i	$-0.261858\pi$
$251$	1.35801e6	1.36056	0.680282	+	0.732951i	$0.261858\pi$
$252$	112112.i	0.111212i
$253$	1.41120e6i	1.38608i
$254$	1.26397e6	1.22928
$255$	0	0
$256$	65536.0	0.0625000
$257$	317742.i	0.300083i	0.988680	+	0.150042i	$0.0479408\pi$
$257$	317742.i	0.300083i	−0.988680	+	0.150042i	$0.952059\pi$
$258$	− 148160.i	− 0.138574i
$259$	−702170.	−0.650418
$260$	0	0
$261$	−695838.	−0.632276
$262$	− 98664.0i	− 0.0887985i
$263$	1.05101e6i	0.936951i	0.883477	+	0.468475i	$0.155196\pi$
$263$	1.05101e6i	0.936951i	−0.883477	+	0.468475i	$0.844804\pi$
$264$	−215040.	−0.189893
$265$	0	0
$266$	−92120.0	−0.0798270
$267$	− 267300.i	− 0.229467i
$268$	− 279232.i	− 0.237481i
$269$	−1.18958e6	−1.00234	−0.501169	−	0.865349i	$-0.667097\pi$
$269$	−1.18958e6	−1.00234	−0.501169	+	0.865349i	$0.667097\pi$
$270$	0	0
$271$	−1.43008e6	−1.18287	−0.591435	−	0.806353i	$-0.701438\pi$
$271$	−1.43008e6	−1.18287	−0.591435	+	0.806353i	$0.701438\pi$
$272$	373248.i	0.305897i
$273$	286160.i	0.232382i
$274$	1.21294e6	0.976026
$275$	0	0
$276$	−672000.	−0.531003
$277$	− 63302.0i	− 0.0495699i	−0.999693	−	0.0247849i	$-0.992110\pi$
$277$	− 63302.0i	− 0.0495699i	0.999693	−	0.0247849i	$-0.00789010\pi$
$278$	− 1.00234e6i	− 0.777866i
$279$	−1.05420e6	−0.810795
$280$	0	0
$281$	−496614.	−0.375192	−0.187596	−	0.982246i	$-0.560070\pi$
$281$	−496614.	−0.375192	−0.187596	+	0.982246i	$0.560070\pi$
$282$	− 72480.0i	− 0.0542744i
$283$	− 1.15842e6i	− 0.859803i	−0.902876	−	0.429902i	$-0.858548\pi$
$283$	− 1.15842e6i	− 0.859803i	0.902876	−	0.429902i	$-0.141452\pi$
$284$	−451584.	−0.332233
$285$	0	0
$286$	784896.	0.567410
$287$	− 304878.i	− 0.218485i
$288$	146432.i	0.104029i
$289$	−705907.	−0.497168
$290$	0	0
$291$	−169780.	−0.117531
$292$	− 57632.0i	− 0.0395555i
$293$	1.43886e6i	0.979151i	0.871961	+	0.489575i	$0.162848\pi$
$293$	1.43886e6i	0.979151i	−0.871961	+	0.489575i	$0.837152\pi$
$294$	96040.0	0.0648013
$295$	0	0
$296$	−917120.	−0.608411
$297$	− 1.29696e6i	− 0.853170i
$298$	242376.i	0.158106i
$299$	2.45280e6	1.58666
$300$	0	0
$301$	181496.	0.115465
$302$	497792.i	0.314073i
$303$	992040.i	0.620758i
$304$	−120320.	−0.0746713
$305$	0	0
$306$	−833976.	−0.509155
$307$	989098.i	0.598954i	0.954104	+	0.299477i	$0.0968122\pi$
$307$	989098.i	0.598954i	−0.954104	+	0.299477i	$0.903188\pi$
$308$	− 263424.i	− 0.158226i
$309$	1.31644e6	0.784341
$310$	0	0
$311$	−2.22050e6	−1.30182	−0.650909	−	0.759155i	$-0.725612\pi$
$311$	−2.22050e6	−1.30182	−0.650909	+	0.759155i	$0.725612\pi$
$312$	373760.i	0.217373i
$313$	2.33008e6i	1.34434i	0.740396	+	0.672171i	$0.234638\pi$
$313$	2.33008e6i	1.34434i	−0.740396	+	0.672171i	$0.765362\pi$
$314$	304160.	0.174092
$315$	0	0
$316$	685952.	0.386435
$317$	− 427542.i	− 0.238963i	−0.992836	−	0.119481i	$-0.961877\pi$
$317$	− 427542.i	− 0.238963i	0.992836	−	0.119481i	$-0.0381232\pi$
$318$	1.48968e6i	0.826086i
$319$	1.63498e6	0.899569
$320$	0	0
$321$	488520.	0.264618
$322$	− 823200.i	− 0.442452i
$323$	− 685260.i	− 0.365468i
$324$	61616.0	0.0326085
$325$	0	0
$326$	−497024.	−0.259020
$327$	563740.i	0.291548i
$328$	− 398208.i	− 0.204374i
$329$	88788.0	0.0452235
$330$	0	0
$331$	−396616.	−0.198976	−0.0994879	−	0.995039i	$-0.531720\pi$
$331$	−396616.	−0.198976	−0.0994879	+	0.995039i	$0.531720\pi$
$332$	563232.i	0.280441i
$333$	− 2.04919e6i	− 1.01268i
$334$	−289680.	−0.142086
$335$	0	0
$336$	125440.	0.0606161
$337$	3.21819e6i	1.54361i	0.635860	+	0.771805i	$0.280646\pi$
$337$	3.21819e6i	1.54361i	−0.635860	+	0.771805i	$0.719354\pi$
$338$	120948.i	0.0575847i
$339$	−87420.0	−0.0413154
$340$	0	0
$341$	2.47699e6	1.15356
$342$	− 268840.i	− 0.124288i
$343$	117649.i	0.0539949i
$344$	237056.	0.108008
$345$	0	0
$346$	1.76621e6	0.793143
$347$	− 2.78018e6i	− 1.23951i	−0.784796	−	0.619755i	$-0.787232\pi$
$347$	− 2.78018e6i	− 1.23951i	0.784796	−	0.619755i	$-0.212768\pi$
$348$	778560.i	0.344623i
$349$	338800.	0.148895	0.0744475	−	0.997225i	$-0.476281\pi$
$349$	338800.	0.148895	0.0744475	+	0.997225i	$0.476281\pi$
$350$	0	0
$351$	−2.25424e6	−0.976635
$352$	− 344064.i	− 0.148007i
$353$	− 362046.i	− 0.154642i	−0.997006	−	0.0773209i	$-0.975363\pi$
$353$	− 362046.i	− 0.154642i	0.997006	−	0.0773209i	$-0.0246366\pi$
$354$	1.37208e6	0.581931
$355$	0	0
$356$	427680.	0.178852
$357$	714420.i	0.296676i
$358$	42768.0i	0.0176365i
$359$	−876528.	−0.358946	−0.179473	−	0.983763i	$-0.557439\pi$
$359$	−876528.	−0.358946	−0.179473	+	0.983763i	$0.557439\pi$
$360$	0	0
$361$	−2.25520e6	−0.910787
$362$	− 2.18426e6i	− 0.876056i
$363$	− 481550.i	− 0.191812i
$364$	−457856.	−0.181124
$365$	0	0
$366$	−979040.	−0.382030
$367$	− 2.98062e6i	− 1.15516i	−0.816335	−	0.577578i	$-0.803998\pi$
$367$	− 2.98062e6i	− 1.15516i	0.816335	−	0.577578i	$-0.196002\pi$
$368$	− 1.07520e6i	− 0.413875i
$369$	889746.	0.340173
$370$	0	0
$371$	−1.82486e6	−0.688326
$372$	1.17952e6i	0.441924i
$373$	3.91441e6i	1.45678i	0.685162	+	0.728391i	$0.259732\pi$
$373$	3.91441e6i	1.45678i	−0.685162	+	0.728391i	$0.740268\pi$
$374$	1.95955e6	0.724399
$375$	0	0
$376$	115968.	0.0423027
$377$	− 2.84174e6i	− 1.02975i
$378$	756560.i	0.272342i
$379$	−3.60661e6	−1.28974	−0.644868	−	0.764294i	$-0.723088\pi$
$379$	−3.60661e6	−1.28974	−0.644868	+	0.764294i	$0.723088\pi$
$380$	0	0
$381$	3.15992e6	1.11523
$382$	− 2.30390e6i	− 0.807804i
$383$	− 2.66644e6i	− 0.928826i	−0.885619	−	0.464413i	$-0.846265\pi$
$383$	− 2.66644e6i	− 0.928826i	0.885619	−	0.464413i	$-0.153735\pi$
$384$	163840.	0.0567012
$385$	0	0
$386$	1.65575e6	0.565623
$387$	529672.i	0.179775i
$388$	− 271648.i	− 0.0916067i
$389$	213366.	0.0714910	0.0357455	−	0.999361i	$-0.488619\pi$
$389$	213366.	0.0714910	0.0357455	+	0.999361i	$0.488619\pi$
$390$	0	0
$391$	6.12360e6	2.02565
$392$	153664.i	0.0505076i
$393$	− 246660.i	− 0.0805596i
$394$	−1.97978e6	−0.642506
$395$	0	0
$396$	768768.	0.246353
$397$	4.09408e6i	1.30371i	0.758345	+	0.651854i	$0.226008\pi$
$397$	4.09408e6i	1.30371i	−0.758345	+	0.651854i	$0.773992\pi$
$398$	− 2.08146e6i	− 0.658657i
$399$	−230300.	−0.0724205
$400$	0	0
$401$	942366.	0.292657	0.146328	−	0.989236i	$-0.453254\pi$
$401$	942366.	0.292657	0.146328	+	0.989236i	$0.453254\pi$
$402$	− 698080.i	− 0.215447i
$403$	− 4.30525e6i	− 1.32049i
$404$	−1.58726e6	−0.483833
$405$	0	0
$406$	−953736.	−0.287153
$407$	4.81488e6i	1.44079i
$408$	933120.i	0.277515i
$409$	4.84561e6	1.43232	0.716160	−	0.697936i	$-0.245898\pi$
$409$	4.84561e6	1.43232	0.716160	+	0.697936i	$0.245898\pi$
$410$	0	0
$411$	3.03234e6	0.885469
$412$	2.10630e6i	0.611333i
$413$	1.68080e6i	0.484887i
$414$	2.40240e6	0.688881
$415$	0	0
$416$	−598016.	−0.169426
$417$	− 2.50586e6i	− 0.705694i
$418$	631680.i	0.176830i
$419$	1.73485e6	0.482754	0.241377	−	0.970431i	$-0.422401\pi$
$419$	1.73485e6	0.482754	0.241377	+	0.970431i	$0.422401\pi$
$420$	0	0
$421$	−1.65145e6	−0.454109	−0.227055	−	0.973882i	$-0.572910\pi$
$421$	−1.65145e6	−0.454109	−0.227055	+	0.973882i	$0.572910\pi$
$422$	733136.i	0.200403i
$423$	259116.i	0.0704115i
$424$	−2.38349e6	−0.643870
$425$	0	0
$426$	−1.12896e6	−0.301408
$427$	− 1.19932e6i	− 0.318322i
$428$	781632.i	0.206250i
$429$	1.96224e6	0.514765
$430$	0	0
$431$	4.14360e6	1.07445	0.537223	−	0.843440i	$-0.319473\pi$
$431$	4.14360e6	1.07445	0.537223	+	0.843440i	$0.319473\pi$
$432$	988160.i	0.254752i
$433$	− 3.03966e6i	− 0.779121i	−0.921001	−	0.389561i	$-0.872627\pi$
$433$	− 3.03966e6i	− 0.779121i	0.921001	−	0.389561i	$-0.127373\pi$
$434$	−1.44491e6	−0.368228
$435$	0	0
$436$	−901984.	−0.227239
$437$	1.97400e6i	0.494474i
$438$	− 144080.i	− 0.0358855i
$439$	−2.54271e6	−0.629703	−0.314852	−	0.949141i	$-0.601955\pi$
$439$	−2.54271e6	−0.629703	−0.314852	+	0.949141i	$0.601955\pi$
$440$	0	0
$441$	−343343.	−0.0840682
$442$	− 3.40589e6i	− 0.829229i
$443$	− 2.43210e6i	− 0.588806i	−0.955681	−	0.294403i	$-0.904879\pi$
$443$	− 2.43210e6i	− 0.588806i	0.955681	−	0.294403i	$-0.0951208\pi$
$444$	−2.29280e6	−0.551961
$445$	0	0
$446$	5.10986e6	1.21639
$447$	605940.i	0.143437i
$448$	200704.i	0.0472456i
$449$	−1.82853e6	−0.428042	−0.214021	−	0.976829i	$-0.568656\pi$
$449$	−1.82853e6	−0.428042	−0.214021	+	0.976829i	$0.568656\pi$
$450$	0	0
$451$	−2.09059e6	−0.483981
$452$	− 139872.i	− 0.0322021i
$453$	1.24448e6i	0.284933i
$454$	−5.15057e6	−1.17278
$455$	0	0
$456$	−300800.	−0.0677432
$457$	− 1.58063e6i	− 0.354030i	−0.984208	−	0.177015i	$-0.943356\pi$
$457$	− 1.58063e6i	− 0.354030i	0.984208	−	0.177015i	$-0.0566441\pi$
$458$	− 1.40374e6i	− 0.312697i
$459$	−5.62788e6	−1.24685
$460$	0	0
$461$	5.09604e6	1.11681	0.558407	−	0.829567i	$-0.311413\pi$
$461$	5.09604e6	1.11681	0.558407	+	0.829567i	$0.311413\pi$
$462$	− 658560.i	− 0.143546i
$463$	− 7.02338e6i	− 1.52263i	−0.648384	−	0.761313i	$-0.724555\pi$
$463$	− 7.02338e6i	− 1.52263i	0.648384	−	0.761313i	$-0.275445\pi$
$464$	−1.24570e6	−0.268607
$465$	0	0
$466$	−3.34462e6	−0.713479
$467$	4.24845e6i	0.901443i	0.892665	+	0.450722i	$0.148833\pi$
$467$	4.24845e6i	0.901443i	−0.892665	+	0.450722i	$0.851167\pi$
$468$	− 1.33619e6i	− 0.282003i
$469$	855148.	0.179518
$470$	0	0
$471$	760400.	0.157939
$472$	2.19533e6i	0.453570i
$473$	− 1.24454e6i	− 0.255775i
$474$	1.71488e6	0.350581
$475$	0	0
$476$	−1.14307e6	−0.231236
$477$	− 5.32561e6i	− 1.07170i
$478$	− 3.09734e6i	− 0.620040i
$479$	−559284.	−0.111377	−0.0556883	−	0.998448i	$-0.517735\pi$
$479$	−559284.	−0.111377	−0.0556883	+	0.998448i	$0.517735\pi$
$480$	0	0
$481$	8.36872e6	1.64929
$482$	− 4.61140e6i	− 0.904097i
$483$	− 2.05800e6i	− 0.401400i
$484$	770480.	0.149502
$485$	0	0
$486$	−3.59788e6	−0.690965
$487$	1.32057e6i	0.252312i	0.992010	+	0.126156i	$0.0402640\pi$
$487$	1.32057e6i	0.252312i	−0.992010	+	0.126156i	$0.959736\pi$
$488$	− 1.56646e6i	− 0.297763i
$489$	−1.24256e6	−0.234988
$490$	0	0
$491$	6.27193e6	1.17408	0.587040	−	0.809558i	$-0.300293\pi$
$491$	6.27193e6	1.17408	0.587040	+	0.809558i	$0.300293\pi$
$492$	− 995520.i	− 0.185412i
$493$	− 7.09463e6i	− 1.31466i
$494$	1.09792e6	0.202420
$495$	0	0
$496$	−1.88723e6	−0.344446
$497$	− 1.38298e6i	− 0.251144i
$498$	1.40808e6i	0.254422i
$499$	3.93785e6	0.707959	0.353979	−	0.935253i	$-0.384828\pi$
$499$	3.93785e6	0.707959	0.353979	+	0.935253i	$0.384828\pi$
$500$	0	0
$501$	−724200.	−0.128903
$502$	5.43204e6i	0.962063i
$503$	− 7.59830e6i	− 1.33905i	−0.742790	−	0.669525i	$-0.766498\pi$
$503$	− 7.59830e6i	− 1.33905i	0.742790	−	0.669525i	$-0.233502\pi$
$504$	−448448.	−0.0786386
$505$	0	0
$506$	−5.64480e6	−0.980104
$507$	302370.i	0.0522419i
$508$	5.05587e6i	0.869234i
$509$	7.82664e6	1.33900	0.669501	−	0.742812i	$-0.266508\pi$
$509$	7.82664e6	1.33900	0.669501	+	0.742812i	$0.266508\pi$
$510$	0	0
$511$	176498.	0.0299011
$512$	262144.i	0.0441942i
$513$	− 1.81420e6i	− 0.304363i
$514$	−1.27097e6	−0.212191
$515$	0	0
$516$	592640.	0.0979866
$517$	− 608832.i	− 0.100178i
$518$	− 2.80868e6i	− 0.459915i
$519$	4.41552e6	0.719554
$520$	0	0
$521$	8.94454e6	1.44366	0.721828	−	0.692072i	$-0.243302\pi$
$521$	8.94454e6	1.44366	0.721828	+	0.692072i	$0.243302\pi$
$522$	− 2.78335e6i	− 0.447087i
$523$	4.07481e6i	0.651407i	0.945472	+	0.325704i	$0.105601\pi$
$523$	4.07481e6i	0.651407i	−0.945472	+	0.325704i	$0.894399\pi$
$524$	394656.	0.0627900
$525$	0	0
$526$	−4.20403e6	−0.662524
$527$	− 1.07484e7i	− 1.68584i
$528$	− 860160.i	− 0.134275i
$529$	−1.12037e7	−1.74069
$530$	0	0
$531$	−4.90519e6	−0.754952
$532$	− 368480.i	− 0.0564462i
$533$	3.63365e6i	0.554019i
$534$	1.06920e6	0.162258
$535$	0	0
$536$	1.11693e6	0.167924
$537$	106920.i	0.0160001i
$538$	− 4.75834e6i	− 0.708760i
$539$	806736.	0.119608
$540$	0	0
$541$	−1.18676e7	−1.74329	−0.871644	−	0.490140i	$-0.836946\pi$
$541$	−1.18676e7	−1.74329	−0.871644	+	0.490140i	$0.836946\pi$
$542$	− 5.72032e6i	− 0.836416i
$543$	− 5.46064e6i	− 0.794775i
$544$	−1.49299e6	−0.216302
$545$	0	0
$546$	−1.14464e6	−0.164319
$547$	5.37801e6i	0.768516i	0.923226	+	0.384258i	$0.125543\pi$
$547$	5.37801e6i	0.768516i	−0.923226	+	0.384258i	$0.874457\pi$
$548$	4.85174e6i	0.690155i
$549$	3.50007e6	0.495616
$550$	0	0
$551$	2.28702e6	0.320916
$552$	− 2.68800e6i	− 0.375475i
$553$	2.10073e6i	0.292117i
$554$	253208.	0.0350512
$555$	0	0
$556$	4.00938e6	0.550034
$557$	5.64878e6i	0.771466i	0.922611	+	0.385733i	$0.126051\pi$
$557$	5.64878e6i	0.771466i	−0.922611	+	0.385733i	$0.873949\pi$
$558$	− 4.21678e6i	− 0.573318i
$559$	−2.16314e6	−0.292789
$560$	0	0
$561$	4.89888e6	0.657188
$562$	− 1.98646e6i	− 0.265301i
$563$	4.56407e6i	0.606850i	0.952855	+	0.303425i	$0.0981303\pi$
$563$	4.56407e6i	0.606850i	−0.952855	+	0.303425i	$0.901870\pi$
$564$	289920.	0.0383778
$565$	0	0
$566$	4.63367e6	0.607973
$567$	188699.i	0.0246497i
$568$	− 1.80634e6i	− 0.234924i
$569$	−8.00165e6	−1.03609	−0.518047	−	0.855352i	$-0.673341\pi$
$569$	−8.00165e6	−1.03609	−0.518047	+	0.855352i	$0.673341\pi$
$570$	0	0
$571$	−1.37164e7	−1.76055	−0.880275	−	0.474464i	$-0.842642\pi$
$571$	−1.37164e7	−1.76055	−0.880275	+	0.474464i	$0.842642\pi$
$572$	3.13958e6i	0.401220i
$573$	− 5.75976e6i	− 0.732855i
$574$	1.21951e6	0.154492
$575$	0	0
$576$	−585728.	−0.0735597
$577$	− 6.09797e6i	− 0.762510i	−0.924470	−	0.381255i	$-0.875492\pi$
$577$	− 6.09797e6i	− 0.762510i	0.924470	−	0.381255i	$-0.124508\pi$
$578$	− 2.82363e6i	− 0.351551i
$579$	4.13938e6	0.513144
$580$	0	0
$581$	−1.72490e6	−0.211994
$582$	− 679120.i	− 0.0831073i
$583$	1.25133e7i	1.52476i
$584$	230528.	0.0279699
$585$	0	0
$586$	−5.75544e6	−0.692364
$587$	8.08462e6i	0.968422i	0.874951	+	0.484211i	$0.160893\pi$
$587$	8.08462e6i	0.968422i	−0.874951	+	0.484211i	$0.839107\pi$
$588$	384160.i	0.0458214i
$589$	3.46484e6	0.411524
$590$	0	0
$591$	−4.94946e6	−0.582893
$592$	− 3.66848e6i	− 0.430211i
$593$	1.41575e6i	0.165330i	0.996577	+	0.0826649i	$0.0263431\pi$
$593$	1.41575e6i	0.165330i	−0.996577	+	0.0826649i	$0.973657\pi$
$594$	5.18784e6	0.603282
$595$	0	0
$596$	−969504.	−0.111798
$597$	− 5.20364e6i	− 0.597546i
$598$	9.81120e6i	1.12194i
$599$	−8.75460e6	−0.996941	−0.498470	−	0.866907i	$-0.666105\pi$
$599$	−8.75460e6	−0.996941	−0.498470	+	0.866907i	$0.666105\pi$
$600$	0	0
$601$	8.70276e6	0.982813	0.491407	−	0.870930i	$-0.336483\pi$
$601$	8.70276e6	0.982813	0.491407	+	0.870930i	$0.336483\pi$
$602$	725984.i	0.0816462i
$603$	2.49564e6i	0.279504i
$604$	−1.99117e6	−0.222083
$605$	0	0
$606$	−3.96816e6	−0.438942
$607$	1.69578e7i	1.86809i	0.357157	+	0.934045i	$0.383746\pi$
$607$	1.69578e7i	1.86809i	−0.357157	+	0.934045i	$0.616254\pi$
$608$	− 481280.i	− 0.0528006i
$609$	−2.38434e6	−0.260510
$610$	0	0
$611$	−1.05821e6	−0.114675
$612$	− 3.33590e6i	− 0.360027i
$613$	1.76743e7i	1.89973i	0.312658	+	0.949866i	$0.398780\pi$
$613$	1.76743e7i	1.89973i	−0.312658	+	0.949866i	$0.601220\pi$
$614$	−3.95639e6	−0.423524
$615$	0	0
$616$	1.05370e6	0.111883
$617$	9.70636e6i	1.02646i	0.858250	+	0.513232i	$0.171552\pi$
$617$	9.70636e6i	1.02646i	−0.858250	+	0.513232i	$0.828448\pi$
$618$	5.26576e6i	0.554613i
$619$	−1.48739e7	−1.56027	−0.780133	−	0.625613i	$-0.784849\pi$
$619$	−1.48739e7	−1.56027	−0.780133	+	0.625613i	$0.784849\pi$
$620$	0	0
$621$	1.62120e7	1.68697
$622$	− 8.88202e6i	− 0.920525i
$623$	1.30977e6i	0.135199i
$624$	−1.49504e6	−0.153706
$625$	0	0
$626$	−9.32031e6	−0.950593
$627$	1.57920e6i	0.160424i
$628$	1.21664e6i	0.123101i
$629$	2.08931e7	2.10561
$630$	0	0
$631$	1.26353e7	1.26331	0.631656	−	0.775248i	$-0.282375\pi$
$631$	1.26353e7	1.26331	0.631656	+	0.775248i	$0.282375\pi$
$632$	2.74381e6i	0.273251i
$633$	1.83284e6i	0.181809i
$634$	1.71017e6	0.168972
$635$	0	0
$636$	−5.95872e6	−0.584131
$637$	− 1.40218e6i	− 0.136917i
$638$	6.53990e6i	0.636092i
$639$	4.03603e6	0.391023
$640$	0	0
$641$	6.23398e6	0.599267	0.299634	−	0.954054i	$-0.403136\pi$
$641$	6.23398e6	0.599267	0.299634	+	0.954054i	$0.403136\pi$
$642$	1.95408e6i	0.187113i
$643$	1.06874e7i	1.01940i	0.860352	+	0.509701i	$0.170244\pi$
$643$	1.06874e7i	1.01940i	−0.860352	+	0.509701i	$0.829756\pi$
$644$	3.29280e6	0.312860
$645$	0	0
$646$	2.74104e6	0.258425
$647$	− 1.83258e7i	− 1.72109i	−0.509376	−	0.860544i	$-0.670124\pi$
$647$	− 1.83258e7i	− 1.72109i	0.509376	−	0.860544i	$-0.329876\pi$
$648$	246464.i	0.0230577i
$649$	1.15255e7	1.07411
$650$	0	0
$651$	−3.61228e6	−0.334063
$652$	− 1.98810e6i	− 0.183155i
$653$	− 7.28857e6i	− 0.668897i	−0.942414	−	0.334448i	$-0.891450\pi$
$653$	− 7.28857e6i	− 0.668897i	0.942414	−	0.334448i	$-0.108550\pi$
$654$	−2.25496e6	−0.206155
$655$	0	0
$656$	1.59283e6	0.144514
$657$	515086.i	0.0465550i
$658$	355152.i	0.0319779i
$659$	−4.54337e6	−0.407534	−0.203767	−	0.979019i	$-0.565319\pi$
$659$	−4.54337e6	−0.407534	−0.203767	+	0.979019i	$0.565319\pi$
$660$	0	0
$661$	−2.10021e7	−1.86964	−0.934821	−	0.355120i	$-0.884440\pi$
$661$	−2.10021e7	−1.86964	−0.934821	+	0.355120i	$0.884440\pi$
$662$	− 1.58646e6i	− 0.140697i
$663$	− 8.51472e6i	− 0.752292i
$664$	−2.25293e6	−0.198302
$665$	0	0
$666$	8.19676e6	0.716072
$667$	2.04372e7i	1.77872i
$668$	− 1.15872e6i	− 0.100470i
$669$	1.27746e7	1.10353
$670$	0	0
$671$	−8.22394e6	−0.705137
$672$	501760.i	0.0428620i
$673$	3.46923e6i	0.295253i	0.989043	+	0.147627i	$0.0471634\pi$
$673$	3.46923e6i	0.295253i	−0.989043	+	0.147627i	$0.952837\pi$
$674$	−1.28728e7	−1.09150
$675$	0	0
$676$	−483792.	−0.0407185
$677$	1.80916e7i	1.51707i	0.651631	+	0.758536i	$0.274085\pi$
$677$	1.80916e7i	1.51707i	−0.651631	+	0.758536i	$0.725915\pi$
$678$	− 349680.i	− 0.0292144i
$679$	831922.	0.0692481
$680$	0	0
$681$	−1.28764e7	−1.06397
$682$	9.90797e6i	0.815687i
$683$	4.67752e6i	0.383675i	0.981427	+	0.191838i	$0.0614447\pi$
$683$	4.67752e6i	0.383675i	−0.981427	+	0.191838i	$0.938555\pi$
$684$	1.07536e6	0.0878848
$685$	0	0
$686$	−470596.	−0.0381802
$687$	− 3.50936e6i	− 0.283685i
$688$	948224.i	0.0763730i
$689$	2.17493e7	1.74541
$690$	0	0
$691$	1.68960e7	1.34614	0.673069	−	0.739579i	$-0.264976\pi$
$691$	1.68960e7	1.34614	0.673069	+	0.739579i	$0.264976\pi$
$692$	7.06483e6i	0.560837i
$693$	2.35435e6i	0.186225i
$694$	1.11207e7	0.876466
$695$	0	0
$696$	−3.11424e6	−0.243685
$697$	9.07168e6i	0.707303i
$698$	1.35520e6i	0.105285i
$699$	−8.36154e6	−0.647282
$700$	0	0
$701$	2.40964e6	0.185207	0.0926035	−	0.995703i	$-0.470481\pi$
$701$	2.40964e6	0.185207	0.0926035	+	0.995703i	$0.470481\pi$
$702$	− 9.01696e6i	− 0.690585i
$703$	6.73510e6i	0.513991i
$704$	1.37626e6	0.104657
$705$	0	0
$706$	1.44818e6	0.109348
$707$	− 4.86100e6i	− 0.365744i
$708$	5.48832e6i	0.411487i
$709$	5.77010e6	0.431090	0.215545	−	0.976494i	$-0.430847\pi$
$709$	5.77010e6	0.431090	0.215545	+	0.976494i	$0.430847\pi$
$710$	0	0
$711$	−6.13070e6	−0.454816
$712$	1.71072e6i	0.126468i
$713$	3.09624e7i	2.28092i
$714$	−2.85768e6	−0.209782
$715$	0	0
$716$	−171072.	−0.0124709
$717$	− 7.74336e6i	− 0.562512i
$718$	− 3.50611e6i	− 0.253813i
$719$	1.43716e7	1.03677	0.518385	−	0.855147i	$-0.326533\pi$
$719$	1.43716e7	1.03677	0.518385	+	0.855147i	$0.326533\pi$
$720$	0	0
$721$	−6.45056e6	−0.462124
$722$	− 9.02080e6i	− 0.644024i
$723$	− 1.15285e7i	− 0.820214i
$724$	8.73702e6	0.619465
$725$	0	0
$726$	1.92620e6	0.135631
$727$	1.40705e7i	0.987353i	0.869646	+	0.493676i	$0.164347\pi$
$727$	1.40705e7i	0.987353i	−0.869646	+	0.493676i	$0.835653\pi$
$728$	− 1.83142e6i	− 0.128074i
$729$	−9.93049e6	−0.692073
$730$	0	0
$731$	−5.40043e6	−0.373796
$732$	− 3.91616e6i	− 0.270136i
$733$	− 3.75000e6i	− 0.257793i	−0.991658	−	0.128897i	$-0.958856\pi$
$733$	− 3.75000e6i	− 0.257793i	0.991658	−	0.128897i	$-0.0411436\pi$
$734$	1.19225e7	0.816819
$735$	0	0
$736$	4.30080e6	0.292654
$737$	− 5.86387e6i	− 0.397664i
$738$	3.55898e6i	0.240539i
$739$	−2.61318e7	−1.76019	−0.880093	−	0.474802i	$-0.842520\pi$
$739$	−2.61318e7	−1.76019	−0.880093	+	0.474802i	$0.842520\pi$
$740$	0	0
$741$	2.74480e6	0.183639
$742$	− 7.29943e6i	− 0.486720i
$743$	− 159072.i	− 0.0105711i	−0.999986	−	0.00528557i	$-0.998318\pi$
$743$	− 159072.i	− 0.0105711i	0.999986	−	0.00528557i	$-0.00168246\pi$
$744$	−4.71808e6	−0.312488
$745$	0	0
$746$	−1.56577e7	−1.03010
$747$	− 5.03389e6i	− 0.330067i
$748$	7.83821e6i	0.512227i
$749$	−2.39375e6	−0.155910
$750$	0	0
$751$	−2.65311e7	−1.71654	−0.858272	−	0.513196i	$-0.828461\pi$
$751$	−2.65311e7	−1.71654	−0.858272	+	0.513196i	$0.828461\pi$
$752$	463872.i	0.0299126i
$753$	1.35801e7i	0.872802i
$754$	1.13670e7	0.728143
$755$	0	0
$756$	−3.02624e6	−0.192575
$757$	1.52032e7i	0.964260i	0.876100	+	0.482130i	$0.160137\pi$
$757$	1.52032e7i	0.964260i	−0.876100	+	0.482130i	$0.839863\pi$
$758$	− 1.44264e7i	− 0.911981i
$759$	−1.41120e7	−0.889169
$760$	0	0
$761$	4.71380e6	0.295059	0.147530	−	0.989058i	$-0.452868\pi$
$761$	4.71380e6	0.295059	0.147530	+	0.989058i	$0.452868\pi$
$762$	1.26397e7i	0.788585i
$763$	− 2.76233e6i	− 0.171776i
$764$	9.21562e6	0.571204
$765$	0	0
$766$	1.06657e7	0.656779
$767$	− 2.00324e7i	− 1.22954i
$768$	655360.i	0.0400938i
$769$	1.58977e6	0.0969434	0.0484717	−	0.998825i	$-0.484565\pi$
$769$	1.58977e6	0.0969434	0.0484717	+	0.998825i	$0.484565\pi$
$770$	0	0
$771$	−3.17742e6	−0.192504
$772$	6.62301e6i	0.399956i
$773$	− 9.69095e6i	− 0.583334i	−0.956520	−	0.291667i	$-0.905790\pi$
$773$	− 9.69095e6i	− 0.583334i	0.956520	−	0.291667i	$-0.0942100\pi$
$774$	−2.11869e6	−0.127120
$775$	0	0
$776$	1.08659e6	0.0647757
$777$	− 7.02170e6i	− 0.417244i
$778$	853464.i	0.0505518i
$779$	−2.92434e6	−0.172657
$780$	0	0
$781$	−9.48326e6	−0.556327
$782$	2.44944e7i	1.43235i
$783$	− 1.87828e7i	− 1.09485i
$784$	−614656.	−0.0357143
$785$	0	0
$786$	986640.	0.0569642
$787$	1.57170e6i	0.0904549i	0.998977	+	0.0452275i	$0.0144013\pi$
$787$	1.57170e6i	0.0904549i	−0.998977	+	0.0452275i	$0.985599\pi$
$788$	− 7.91914e6i	− 0.454320i
$789$	−1.05101e7	−0.601054
$790$	0	0
$791$	428358.	0.0243425
$792$	3.07507e6i	0.174198i
$793$	1.42940e7i	0.807180i
$794$	−1.63763e7	−0.921860
$795$	0	0
$796$	8.32582e6	0.465741
$797$	2.25298e6i	0.125635i	0.998025	+	0.0628175i	$0.0200086\pi$
$797$	2.25298e6i	0.125635i	−0.998025	+	0.0628175i	$0.979991\pi$
$798$	− 921200.i	− 0.0512090i
$799$	−2.64190e6	−0.146403
$800$	0	0
$801$	−3.82239e6	−0.210501
$802$	3.76946e6i	0.206940i
$803$	− 1.21027e6i	− 0.0662360i
$804$	2.79232e6	0.152344
$805$	0	0
$806$	1.72210e7	0.933728
$807$	− 1.18958e7i	− 0.643000i
$808$	− 6.34906e6i	− 0.342122i
$809$	2.37938e7	1.27818	0.639090	−	0.769132i	$-0.279311\pi$
$809$	2.37938e7	1.27818	0.639090	+	0.769132i	$0.279311\pi$
$810$	0	0
$811$	5.32300e6	0.284187	0.142093	−	0.989853i	$-0.454617\pi$
$811$	5.32300e6	0.284187	0.142093	+	0.989853i	$0.454617\pi$
$812$	− 3.81494e6i	− 0.203048i
$813$	− 1.43008e7i	− 0.758812i
$814$	−1.92595e7	−1.01879
$815$	0	0
$816$	−3.73248e6	−0.196233
$817$	− 1.74088e6i	− 0.0912460i
$818$	1.93824e7i	1.01280i
$819$	4.09209e6	0.213174
$820$	0	0
$821$	1.48802e7	0.770464	0.385232	−	0.922820i	$-0.374121\pi$
$821$	1.48802e7	0.770464	0.385232	+	0.922820i	$0.374121\pi$
$822$	1.21294e7i	0.626121i
$823$	2.00601e7i	1.03236i	0.856479	+	0.516182i	$0.172647\pi$
$823$	2.00601e7i	1.03236i	−0.856479	+	0.516182i	$0.827353\pi$
$824$	−8.42522e6	−0.432278
$825$	0	0
$826$	−6.72319e6	−0.342867
$827$	− 1.21539e7i	− 0.617949i	−0.951070	−	0.308975i	$-0.900014\pi$
$827$	− 1.21539e7i	− 0.617949i	0.951070	−	0.308975i	$-0.0999858\pi$
$828$	9.60960e6i	0.487113i
$829$	−3.21197e7	−1.62325	−0.811625	−	0.584179i	$-0.801417\pi$
$829$	−3.21197e7	−1.62325	−0.811625	+	0.584179i	$0.801417\pi$
$830$	0	0
$831$	633020.	0.0317991
$832$	− 2.39206e6i	− 0.119802i
$833$	− 3.50066e6i	− 0.174798i
$834$	1.00234e7	0.499001
$835$	0	0
$836$	−2.52672e6	−0.125038
$837$	− 2.84559e7i	− 1.40397i
$838$	6.93938e6i	0.341359i
$839$	1.01320e6	0.0496922	0.0248461	−	0.999691i	$-0.492090\pi$
$839$	1.01320e6	0.0496922	0.0248461	+	0.999691i	$0.492090\pi$
$840$	0	0
$841$	3.16681e6	0.154394
$842$	− 6.60580e6i	− 0.321104i
$843$	− 4.96614e6i	− 0.240686i
$844$	−2.93254e6	−0.141706
$845$	0	0
$846$	−1.03646e6	−0.0497884
$847$	2.35960e6i	0.113013i
$848$	− 9.53395e6i	− 0.455285i
$849$	1.15842e7	0.551564
$850$	0	0
$851$	−6.01860e7	−2.84886
$852$	− 4.51584e6i	− 0.213128i
$853$	234824.i	0.0110502i	0.999985	+	0.00552510i	$0.00175870\pi$
$853$	234824.i	0.0110502i	−0.999985	+	0.00552510i	$0.998241\pi$
$854$	4.79730e6	0.225088
$855$	0	0
$856$	−3.12653e6	−0.145840
$857$	− 2.83802e7i	− 1.31997i	−0.751279	−	0.659985i	$-0.770563\pi$
$857$	− 2.83802e7i	− 1.31997i	0.751279	−	0.659985i	$-0.229437\pi$
$858$	7.84896e6i	0.363994i
$859$	−4.00081e7	−1.84997	−0.924986	−	0.380001i	$-0.875924\pi$
$859$	−4.00081e7	−1.84997	−0.924986	+	0.380001i	$0.875924\pi$
$860$	0	0
$861$	3.04878e6	0.140158
$862$	1.65744e7i	0.759748i
$863$	− 2.08030e7i	− 0.950823i	−0.879764	−	0.475411i	$-0.842299\pi$
$863$	− 2.08030e7i	− 0.950823i	0.879764	−	0.475411i	$-0.157701\pi$
$864$	−3.95264e6	−0.180137
$865$	0	0
$866$	1.21586e7	0.550922
$867$	− 7.05907e6i	− 0.318933i
$868$	− 5.77965e6i	− 0.260377i
$869$	1.44050e7	0.647088
$870$	0	0
$871$	−1.01920e7	−0.455211
$872$	− 3.60794e6i	− 0.160682i
$873$	2.42785e6i	0.107817i
$874$	−7.89600e6	−0.349646
$875$	0	0
$876$	576320.	0.0253748
$877$	− 3.03559e7i	− 1.33273i	−0.745624	−	0.666367i	$-0.767848\pi$
$877$	− 3.03559e7i	− 1.33273i	0.745624	−	0.666367i	$-0.232152\pi$
$878$	− 1.01708e7i	− 0.445267i
$879$	−1.43886e7	−0.628125
$880$	0	0
$881$	−2.58936e7	−1.12396	−0.561981	−	0.827150i	$-0.689961\pi$
$881$	−2.58936e7	−1.12396	−0.561981	+	0.827150i	$0.689961\pi$
$882$	− 1.37337e6i	− 0.0594452i
$883$	− 1.88813e7i	− 0.814950i	−0.913216	−	0.407475i	$-0.866409\pi$
$883$	− 1.88813e7i	− 0.814950i	0.913216	−	0.407475i	$-0.133591\pi$
$884$	1.36236e7	0.586354
$885$	0	0
$886$	9.72840e6	0.416349
$887$	2.34431e7i	1.00048i	0.865888	+	0.500238i	$0.166754\pi$
$887$	2.34431e7i	1.00048i	−0.865888	+	0.500238i	$0.833246\pi$
$888$	− 9.17120e6i	− 0.390296i
$889$	−1.54836e7	−0.657079
$890$	0	0
$891$	1.29394e6	0.0546033
$892$	2.04394e7i	0.860115i
$893$	− 851640.i	− 0.0357378i
$894$	−2.42376e6	−0.101425
$895$	0	0
$896$	−802816.	−0.0334077
$897$	2.45280e7i	1.01784i
$898$	− 7.31412e6i	− 0.302671i
$899$	3.58722e7	1.48033
$900$	0	0
$901$	5.42988e7	2.22833
$902$	− 8.36237e6i	− 0.342226i
$903$	1.81496e6i	0.0740709i
$904$	559488.	0.0227703
$905$	0	0
$906$	−4.97792e6	−0.201478
$907$	5.60873e6i	0.226384i	0.993573	+	0.113192i	$0.0361076\pi$
$907$	5.60873e6i	0.226384i	−0.993573	+	0.113192i	$0.963892\pi$
$908$	− 2.06023e7i	− 0.829279i
$909$	1.41862e7	0.569450
$910$	0	0
$911$	2.16215e7	0.863156	0.431578	−	0.902076i	$-0.357957\pi$
$911$	2.16215e7	0.863156	0.431578	+	0.902076i	$0.357957\pi$
$912$	− 1.20320e6i	− 0.0479017i
$913$	1.18279e7i	0.469602i
$914$	6.32252e6	0.250337
$915$	0	0
$916$	5.61498e6	0.221110
$917$	1.20863e6i	0.0474648i
$918$	− 2.25115e7i	− 0.881654i
$919$	−4.51695e7	−1.76424	−0.882119	−	0.471028i	$-0.843883\pi$
$919$	−4.51695e7	−1.76424	−0.882119	+	0.471028i	$0.843883\pi$
$920$	0	0
$921$	−9.89098e6	−0.384229
$922$	2.03842e7i	0.789706i
$923$	1.64828e7i	0.636835i
$924$	2.63424e6	0.101502
$925$	0	0
$926$	2.80935e7	1.07666
$927$	− 1.88251e7i	− 0.719512i
$928$	− 4.98278e6i	− 0.189934i
$929$	2.28729e7	0.869524	0.434762	−	0.900545i	$-0.356832\pi$
$929$	2.28729e7	0.869524	0.434762	+	0.900545i	$0.356832\pi$
$930$	0	0
$931$	1.12847e6	0.0426693
$932$	− 1.33785e7i	− 0.504506i
$933$	− 2.22050e7i	− 0.835117i
$934$	−1.69938e7	−0.637417
$935$	0	0
$936$	5.34477e6	0.199406
$937$	1.79616e7i	0.668336i	0.942514	+	0.334168i	$0.108455\pi$
$937$	1.79616e7i	0.668336i	−0.942514	+	0.334168i	$0.891545\pi$
$938$	3.42059e6i	0.126939i
$939$	−2.33008e7	−0.862395
$940$	0	0
$941$	−1.79697e7	−0.661558	−0.330779	−	0.943708i	$-0.607311\pi$
$941$	−1.79697e7	−0.661558	−0.330779	+	0.943708i	$0.607311\pi$
$942$	3.04160e6i	0.111680i
$943$	− 2.61324e7i	− 0.956974i
$944$	−8.78131e6	−0.320722
$945$	0	0
$946$	4.97818e6	0.180860
$947$	− 4.32115e7i	− 1.56576i	−0.622174	−	0.782879i	$-0.713750\pi$
$947$	− 4.32115e7i	− 1.56576i	0.622174	−	0.782879i	$-0.286250\pi$
$948$	6.85952e6i	0.247898i
$949$	−2.10357e6	−0.0758213
$950$	0	0
$951$	4.27542e6	0.153295
$952$	− 4.57229e6i	− 0.163509i
$953$	− 7.50965e6i	− 0.267848i	−0.990992	−	0.133924i	$-0.957242\pi$
$953$	− 7.50965e6i	− 0.267848i	0.990992	−	0.133924i	$-0.0427577\pi$
$954$	2.13024e7	0.757806
$955$	0	0
$956$	1.23894e7	0.438434
$957$	1.63498e7i	0.577074i
$958$	− 2.23714e6i	− 0.0787551i
$959$	−1.48585e7	−0.521708
$960$	0	0
$961$	2.57172e7	0.898288
$962$	3.34749e7i	1.16622i
$963$	− 6.98584e6i	− 0.242746i
$964$	1.84456e7	0.639293
$965$	0	0
$966$	8.23200e6	0.283833
$967$	1.69305e7i	0.582242i	0.956686	+	0.291121i	$0.0940283\pi$
$967$	1.69305e7i	0.582242i	−0.956686	+	0.291121i	$0.905972\pi$
$968$	3.08192e6i	0.105714i
$969$	6.85260e6	0.234448
$970$	0	0
$971$	2.86144e7	0.973949	0.486974	−	0.873416i	$-0.338101\pi$
$971$	2.86144e7	0.973949	0.486974	+	0.873416i	$0.338101\pi$
$972$	− 1.43915e7i	− 0.488586i
$973$	1.22787e7i	0.415787i
$974$	−5.28227e6	−0.178412
$975$	0	0
$976$	6.26586e6	0.210550
$977$	− 3.69445e7i	− 1.23826i	−0.785287	−	0.619132i	$-0.787485\pi$
$977$	− 3.69445e7i	− 1.23826i	0.785287	−	0.619132i	$-0.212515\pi$
$978$	− 4.97024e6i	− 0.166161i
$979$	8.98128e6	0.299489
$980$	0	0
$981$	8.06148e6	0.267450
$982$	2.50877e7i	0.830200i
$983$	− 3.88787e7i	− 1.28330i	−0.766998	−	0.641650i	$-0.778250\pi$
$983$	− 3.88787e7i	− 1.28330i	0.766998	−	0.641650i	$-0.221750\pi$
$984$	3.98208e6	0.131106
$985$	0	0
$986$	2.83785e7	0.929603
$987$	887880.i	0.0290109i
$988$	4.39168e6i	0.143133i
$989$	1.55568e7	0.505743
$990$	0	0
$991$	2.49212e7	0.806092	0.403046	−	0.915180i	$-0.367951\pi$
$991$	2.49212e7	0.806092	0.403046	+	0.915180i	$0.367951\pi$
$992$	− 7.54893e6i	− 0.243560i
$993$	− 3.96616e6i	− 0.127643i
$994$	5.53190e6	0.177586
$995$	0	0
$996$	−5.63232e6	−0.179903
$997$	− 1.01956e7i	− 0.324845i	−0.986721	−	0.162422i	$-0.948069\pi$
$997$	− 1.01956e7i	− 0.324845i	0.986721	−	0.162422i	$-0.0519307\pi$
$998$	1.57514e7i	0.500603i
$999$	5.53138e7	1.75356

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.c.d.99.2		2
5.2	odd	4		14.6.a.a.1.1	✓	1
5.3	odd	4		350.6.a.i.1.1		1
5.4	even	2	inner	350.6.c.d.99.1		2
15.2	even	4		126.6.a.f.1.1		1
20.7	even	4		112.6.a.c.1.1		1
35.2	odd	12		98.6.c.c.67.1		2
35.12	even	12		98.6.c.d.67.1		2
35.17	even	12		98.6.c.d.79.1		2
35.27	even	4		98.6.a.a.1.1		1
35.32	odd	12		98.6.c.c.79.1		2
40.27	even	4		448.6.a.l.1.1		1
40.37	odd	4		448.6.a.e.1.1		1
60.47	odd	4		1008.6.a.b.1.1		1
105.62	odd	4		882.6.a.x.1.1		1
140.27	odd	4		784.6.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.a.a.1.1	✓	1	5.2	odd	4
98.6.a.a.1.1		1	35.27	even	4
98.6.c.c.67.1		2	35.2	odd	12
98.6.c.c.79.1		2	35.32	odd	12
98.6.c.d.67.1		2	35.12	even	12
98.6.c.d.79.1		2	35.17	even	12
112.6.a.c.1.1		1	20.7	even	4
126.6.a.f.1.1		1	15.2	even	4
350.6.a.i.1.1		1	5.3	odd	4
350.6.c.d.99.1		2	5.4	even	2	inner
350.6.c.d.99.2		2	1.1	even	1	trivial
448.6.a.e.1.1		1	40.37	odd	4
448.6.a.l.1.1		1	40.27	even	4
784.6.a.i.1.1		1	140.27	odd	4
882.6.a.x.1.1		1	105.62	odd	4
1008.6.a.b.1.1		1	60.47	odd	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} +10.0000i q^{3} -16.0000 q^{4} -40.0000 q^{6} -49.0000i q^{7} -64.0000i q^{8} +143.000 q^{9} -336.000 q^{11} -160.000i q^{12} +584.000i q^{13} +196.000 q^{14} +256.000 q^{16} +1458.00i q^{17} +572.000i q^{18} -470.000 q^{19} +490.000 q^{21} -1344.00i q^{22} -4200.00i q^{23} +640.000 q^{24} -2336.00 q^{26} +3860.00i q^{27} +784.000i q^{28} -4866.00 q^{29} -7372.00 q^{31} +1024.00i q^{32} -3360.00i q^{33} -5832.00 q^{34} -2288.00 q^{36} -14330.0i q^{37} -1880.00i q^{38} -5840.00 q^{39} +6222.00 q^{41} +1960.00i q^{42} +3704.00i q^{43} +5376.00 q^{44} +16800.0 q^{46} +1812.00i q^{47} +2560.00i q^{48} -2401.00 q^{49} -14580.0 q^{51} -9344.00i q^{52} -37242.0i q^{53} -15440.0 q^{54} -3136.00 q^{56} -4700.00i q^{57} -19464.0i q^{58} -34302.0 q^{59} +24476.0 q^{61} -29488.0i q^{62} -7007.00i q^{63} -4096.00 q^{64} +13440.0 q^{66} +17452.0i q^{67} -23328.0i q^{68} +42000.0 q^{69} +28224.0 q^{71} -9152.00i q^{72} +3602.00i q^{73} +57320.0 q^{74} +7520.00 q^{76} +16464.0i q^{77} -23360.0i q^{78} -42872.0 q^{79} -3851.00 q^{81} +24888.0i q^{82} -35202.0i q^{83} -7840.00 q^{84} -14816.0 q^{86} -48660.0i q^{87} +21504.0i q^{88} -26730.0 q^{89} +28616.0 q^{91} +67200.0i q^{92} -73720.0i q^{93} -7248.00 q^{94} -10240.0 q^{96} +16978.0i q^{97} -9604.00i q^{98} -48048.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} - 80 q^{6} + 286 q^{9} - 672 q^{11} + 392 q^{14} + 512 q^{16} - 940 q^{19} + 980 q^{21} + 1280 q^{24} - 4672 q^{26} - 9732 q^{29} - 14744 q^{31} - 11664 q^{34} - 4576 q^{36} - 11680 q^{39}+ \cdots - 96096 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 - 80 * q^6 + 286 * q^9 - 672 * q^11 + 392 * q^14 + 512 * q^16 - 940 * q^19 + 980 * q^21 + 1280 * q^24 - 4672 * q^26 - 9732 * q^29 - 14744 * q^31 - 11664 * q^34 - 4576 * q^36 - 11680 * q^39 + 12444 * q^41 + 10752 * q^44 + 33600 * q^46 - 4802 * q^49 - 29160 * q^51 - 30880 * q^54 - 6272 * q^56 - 68604 * q^59 + 48952 * q^61 - 8192 * q^64 + 26880 * q^66 + 84000 * q^69 + 56448 * q^71 + 114640 * q^74 + 15040 * q^76 - 85744 * q^79 - 7702 * q^81 - 15680 * q^84 - 29632 * q^86 - 53460 * q^89 + 57232 * q^91 - 14496 * q^94 - 20480 * q^96 - 96096 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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