LMFDB - Embedded newform 350.8.c.b.99.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,8,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, 8); 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, 8, names="a")

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	350.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-128,0,-1056] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	no
Analytic conductor:	$109.334758919$
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			350.8.c.b.99.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.00000i q^{2} +66.0000i q^{3} -64.0000 q^{4} -528.000 q^{6} -343.000i q^{7} -512.000i q^{8} -2169.00 q^{9} +40.0000 q^{11} -4224.00i q^{12} +4452.00i q^{13} +2744.00 q^{14} +4096.00 q^{16} +36502.0i q^{17} -17352.0i q^{18} +46222.0 q^{19} +22638.0 q^{21} +320.000i q^{22} +105200. i q^{23} +33792.0 q^{24} -35616.0 q^{26} +1188.00i q^{27} +21952.0i q^{28} +126334. q^{29} -170964. q^{31} +32768.0i q^{32} +2640.00i q^{33} -292016. q^{34} +138816. q^{36} +20954.0i q^{37} +369776. i q^{38} -293832. q^{39} +318486. q^{41} +181104. i q^{42} -77744.0i q^{43} -2560.00 q^{44} -841600. q^{46} +703716. i q^{47} +270336. i q^{48} -117649. q^{49} -2.40913e6 q^{51} -284928. i q^{52} -1.60328e6i q^{53} -9504.00 q^{54} -175616. q^{56} +3.05065e6i q^{57} +1.01067e6i q^{58} +1.17189e6 q^{59} -2.06887e6 q^{61} -1.36771e6i q^{62} +743967. i q^{63} -262144. q^{64} -21120.0 q^{66} -994268. i q^{67} -2.33613e6i q^{68} -6.94320e6 q^{69} +33280.0 q^{71} +1.11053e6i q^{72} +2.97145e6i q^{73} -167632. q^{74} -2.95821e6 q^{76} -13720.0i q^{77} -2.35066e6i q^{78} +2.37617e6 q^{79} -4.82201e6 q^{81} +2.54789e6i q^{82} +2.12236e6i q^{83} -1.44883e6 q^{84} +621952. q^{86} +8.33804e6i q^{87} -20480.0i q^{88} -6.92035e6 q^{89} +1.52704e6 q^{91} -6.73280e6i q^{92} -1.12836e7i q^{93} -5.62973e6 q^{94} -2.16269e6 q^{96} +4.95271e6i q^{97} -941192. i q^{98} -86760.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 128 q^{4} - 1056 q^{6} - 4338 q^{9} + 80 q^{11} + 5488 q^{14} + 8192 q^{16} + 92444 q^{19} + 45276 q^{21} + 67584 q^{24} - 71232 q^{26} + 252668 q^{29} - 341928 q^{31} - 584032 q^{34} + 277632 q^{36}+ \cdots - 173520 q^{99}+O(q^{100}) $ 2 * q - 128 * q^4 - 1056 * q^6 - 4338 * q^9 + 80 * q^11 + 5488 * q^14 + 8192 * q^16 + 92444 * q^19 + 45276 * q^21 + 67584 * q^24 - 71232 * q^26 + 252668 * q^29 - 341928 * q^31 - 584032 * q^34 + 277632 * q^36 - 587664 * q^39 + 636972 * q^41 - 5120 * q^44 - 1683200 * q^46 - 235298 * q^49 - 4818264 * q^51 - 19008 * q^54 - 351232 * q^56 + 2343788 * q^59 - 4137744 * q^61 - 524288 * q^64 - 42240 * q^66 - 13886400 * q^69 + 66560 * q^71 - 335264 * q^74 - 5916416 * q^76 + 4752336 * q^79 - 9644022 * q^81 - 2897664 * q^84 + 1243904 * q^86 - 13840692 * q^89 + 3054072 * q^91 - 11259456 * q^94 - 4325376 * q^96 - 173520 * 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 $

$2$	8.00000i	0.707107i
$2$	8.00000i	0.707107i
$3$	66.0000i	1.41130i	0.708560	+	0.705650i	$0.249345\pi$
$3$	66.0000i	1.41130i	−0.708560	+	0.705650i	$0.750655\pi$
$4$	−64.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−528.000	−0.997940
$7$	− 343.000i	− 0.377964i
$7$	− 343.000i	− 0.377964i
$8$	− 512.000i	− 0.353553i
$9$	−2169.00	−0.991770
$10$	0	0
$11$	40.0000	0.00906120	0.00453060	−	0.999990i	$-0.498558\pi$
$11$	40.0000	0.00906120	0.00453060	+	0.999990i	$0.498558\pi$
$12$	− 4224.00i	− 0.705650i
$13$	4452.00i	0.562022i	0.959705	+	0.281011i	$0.0906698\pi$
$13$	4452.00i	0.562022i	−0.959705	+	0.281011i	$0.909330\pi$
$14$	2744.00	0.267261
$15$	0	0
$16$	4096.00	0.250000
$17$	36502.0i	1.80196i	0.433859	+	0.900981i	$0.357151\pi$
$17$	36502.0i	1.80196i	−0.433859	+	0.900981i	$0.642849\pi$
$18$	− 17352.0i	− 0.701287i
$19$	46222.0	1.54601	0.773003	−	0.634402i	$-0.218754\pi$
$19$	46222.0	1.54601	0.773003	+	0.634402i	$0.218754\pi$
$20$	0	0
$21$	22638.0	0.533422
$22$	320.000i	0.00640723i
$23$	105200.i	1.80289i	0.432898	+	0.901443i	$0.357491\pi$
$23$	105200.i	1.80289i	−0.432898	+	0.901443i	$0.642509\pi$
$24$	33792.0	0.498970
$25$	0	0
$26$	−35616.0	−0.397410
$27$	1188.00i	0.0116156i
$28$	21952.0i	0.188982i
$29$	126334.	0.961894	0.480947	−	0.876750i	$-0.340293\pi$
$29$	126334.	0.961894	0.480947	+	0.876750i	$0.340293\pi$
$30$	0	0
$31$	−170964.	−1.03072	−0.515358	−	0.856975i	$-0.672341\pi$
$31$	−170964.	−1.03072	−0.515358	+	0.856975i	$0.672341\pi$
$32$	32768.0i	0.176777i
$33$	2640.00i	0.0127881i
$34$	−292016.	−1.27418
$35$	0	0
$36$	138816.	0.495885
$37$	20954.0i	0.0680081i	0.999422	+	0.0340041i	$0.0108259\pi$
$37$	20954.0i	0.0680081i	−0.999422	+	0.0340041i	$0.989174\pi$
$38$	369776.i	1.09319i
$39$	−293832.	−0.793182
$40$	0	0
$41$	318486.	0.721684	0.360842	−	0.932627i	$-0.382489\pi$
$41$	318486.	0.721684	0.360842	+	0.932627i	$0.382489\pi$
$42$	181104.i	0.377186i
$43$	− 77744.0i	− 0.149117i	−0.997217	−	0.0745585i	$-0.976245\pi$
$43$	− 77744.0i	− 0.149117i	0.997217	−	0.0745585i	$-0.0237548\pi$
$44$	−2560.00	−0.00453060
$45$	0	0
$46$	−841600.	−1.27483
$47$	703716.i	0.988678i	0.869269	+	0.494339i	$0.164590\pi$
$47$	703716.i	0.988678i	−0.869269	+	0.494339i	$0.835410\pi$
$48$	270336.i	0.352825i
$49$	−117649.	−0.142857
$50$	0	0
$51$	−2.40913e6	−2.54311
$52$	− 284928.i	− 0.281011i
$53$	− 1.60328e6i	− 1.47926i	−0.673016	−	0.739628i	$-0.735002\pi$
$53$	− 1.60328e6i	− 1.47926i	0.673016	−	0.739628i	$-0.264998\pi$
$54$	−9504.00	−0.00821350
$55$	0	0
$56$	−175616.	−0.133631
$57$	3.05065e6i	2.18188i
$58$	1.01067e6i	0.680162i
$59$	1.17189e6	0.742859	0.371429	−	0.928461i	$-0.378868\pi$
$59$	1.17189e6	0.742859	0.371429	+	0.928461i	$0.378868\pi$
$60$	0	0
$61$	−2.06887e6	−1.16702	−0.583511	−	0.812105i	$-0.698322\pi$
$61$	−2.06887e6	−1.16702	−0.583511	+	0.812105i	$0.698322\pi$
$62$	− 1.36771e6i	− 0.728826i
$63$	743967.i	0.374854i
$64$	−262144.	−0.125000
$65$	0	0
$66$	−21120.0	−0.00904253
$67$	− 994268.i	− 0.403870i	−0.979399	−	0.201935i	$-0.935277\pi$
$67$	− 994268.i	− 0.403870i	0.979399	−	0.201935i	$-0.0647230\pi$
$68$	− 2.33613e6i	− 0.900981i
$69$	−6.94320e6	−2.54441
$70$	0	0
$71$	33280.0	0.0110352	0.00551759	−	0.999985i	$-0.498244\pi$
$71$	33280.0	0.0110352	0.00551759	+	0.999985i	$0.498244\pi$
$72$	1.11053e6i	0.350643i
$73$	2.97145e6i	0.894003i	0.894533	+	0.447002i	$0.147508\pi$
$73$	2.97145e6i	0.894003i	−0.894533	+	0.447002i	$0.852492\pi$
$74$	−167632.	−0.0480890
$75$	0	0
$76$	−2.95821e6	−0.773003
$77$	− 13720.0i	− 0.00342481i
$78$	− 2.35066e6i	− 0.560865i
$79$	2.37617e6	0.542228	0.271114	−	0.962547i	$-0.412608\pi$
$79$	2.37617e6	0.542228	0.271114	+	0.962547i	$0.412608\pi$
$80$	0	0
$81$	−4.82201e6	−1.00816
$82$	2.54789e6i	0.510307i
$83$	2.12236e6i	0.407423i	0.979031	+	0.203711i	$0.0653004\pi$
$83$	2.12236e6i	0.407423i	−0.979031	+	0.203711i	$0.934700\pi$
$84$	−1.44883e6	−0.266711
$85$	0	0
$86$	621952.	0.105442
$87$	8.33804e6i	1.35752i
$88$	− 20480.0i	− 0.00320362i
$89$	−6.92035e6	−1.04055	−0.520275	−	0.853999i	$-0.674170\pi$
$89$	−6.92035e6	−1.04055	−0.520275	+	0.853999i	$0.674170\pi$
$90$	0	0
$91$	1.52704e6	0.212424
$92$	− 6.73280e6i	− 0.901443i
$93$	− 1.12836e7i	− 1.45465i
$94$	−5.62973e6	−0.699101
$95$	0	0
$96$	−2.16269e6	−0.249485
$97$	4.95271e6i	0.550988i	0.961303	+	0.275494i	$0.0888414\pi$
$97$	4.95271e6i	0.550988i	−0.961303	+	0.275494i	$0.911159\pi$
$98$	− 941192.i	− 0.101015i
$99$	−86760.0	−0.00898662
$100$	0	0
$101$	3.23000e6	0.311945	0.155972	−	0.987761i	$-0.450149\pi$
$101$	3.23000e6	0.311945	0.155972	+	0.987761i	$0.450149\pi$
$102$	− 1.92731e7i	− 1.79825i
$103$	1.79909e6i	0.162227i	0.996705	+	0.0811135i	$0.0258476\pi$
$103$	1.79909e6i	0.162227i	−0.996705	+	0.0811135i	$0.974152\pi$
$104$	2.27942e6	0.198705
$105$	0	0
$106$	1.28262e7	1.04599
$107$	− 1.56429e7i	− 1.23445i	−0.786787	−	0.617225i	$-0.788257\pi$
$107$	− 1.56429e7i	− 1.23445i	0.786787	−	0.617225i	$-0.211743\pi$
$108$	− 76032.0i	− 0.00580782i
$109$	6.31890e6	0.467357	0.233679	−	0.972314i	$-0.424924\pi$
$109$	6.31890e6	0.467357	0.233679	+	0.972314i	$0.424924\pi$
$110$	0	0
$111$	−1.38296e6	−0.0959799
$112$	− 1.40493e6i	− 0.0944911i
$113$	1.02288e7i	0.666881i	0.942771	+	0.333441i	$0.108210\pi$
$113$	1.02288e7i	0.666881i	−0.942771	+	0.333441i	$0.891790\pi$
$114$	−2.44052e7	−1.54282
$115$	0	0
$116$	−8.08538e6	−0.480947
$117$	− 9.65639e6i	− 0.557396i
$118$	9.37515e6i	0.525281i
$119$	1.25202e7	0.681077
$120$	0	0
$121$	−1.94856e7	−0.999918
$122$	− 1.65510e7i	− 0.825209i
$123$	2.10201e7i	1.01851i
$124$	1.09417e7	0.515358
$125$	0	0
$126$	−5.95174e6	−0.265062
$127$	6.00725e6i	0.260233i	0.991499	+	0.130117i	$0.0415352\pi$
$127$	6.00725e6i	0.260233i	−0.991499	+	0.130117i	$0.958465\pi$
$128$	− 2.09715e6i	− 0.0883883i
$129$	5.13110e6	0.210449
$130$	0	0
$131$	2.06396e7	0.802144	0.401072	−	0.916047i	$-0.368638\pi$
$131$	2.06396e7	0.802144	0.401072	+	0.916047i	$0.368638\pi$
$132$	− 168960.i	− 0.00639404i
$133$	− 1.58541e7i	− 0.584335i
$134$	7.95414e6	0.285579
$135$	0	0
$136$	1.86890e7	0.637089
$137$	4.76199e6i	0.158222i	0.996866	+	0.0791109i	$0.0252081\pi$
$137$	4.76199e6i	0.158222i	−0.996866	+	0.0791109i	$0.974792\pi$
$138$	− 5.55456e7i	− 1.79917i
$139$	5.05723e6	0.159721	0.0798604	−	0.996806i	$-0.474553\pi$
$139$	5.05723e6	0.159721	0.0798604	+	0.996806i	$0.474553\pi$
$140$	0	0
$141$	−4.64453e7	−1.39532
$142$	266240.i	0.00780304i
$143$	178080.i	0.00509259i
$144$	−8.88422e6	−0.247942
$145$	0	0
$146$	−2.37716e7	−0.632156
$147$	− 7.76483e6i	− 0.201614i
$148$	− 1.34106e6i	− 0.0340041i
$149$	2.72736e7	0.675447	0.337723	−	0.941245i	$-0.390343\pi$
$149$	2.72736e7	0.675447	0.337723	+	0.941245i	$0.390343\pi$
$150$	0	0
$151$	6.48921e6	0.153381	0.0766906	−	0.997055i	$-0.475565\pi$
$151$	6.48921e6	0.153381	0.0766906	+	0.997055i	$0.475565\pi$
$152$	− 2.36657e7i	− 0.546596i
$153$	− 7.91728e7i	− 1.78713i
$154$	109760.	0.00242171
$155$	0	0
$156$	1.88052e7	0.396591
$157$	− 6.30810e7i	− 1.30092i	−0.759541	−	0.650459i	$-0.774577\pi$
$157$	− 6.30810e7i	− 1.30092i	0.759541	−	0.650459i	$-0.225423\pi$
$158$	1.90093e7i	0.383413i
$159$	1.05816e8	2.08767
$160$	0	0
$161$	3.60836e7	0.681427
$162$	− 3.85761e7i	− 0.712879i
$163$	− 8.32271e7i	− 1.50525i	−0.658450	−	0.752624i	$-0.728788\pi$
$163$	− 8.32271e7i	− 1.50525i	0.658450	−	0.752624i	$-0.271212\pi$
$164$	−2.03831e7	−0.360842
$165$	0	0
$166$	−1.69789e7	−0.288091
$167$	3.06916e7i	0.509931i	0.966950	+	0.254965i	$0.0820641\pi$
$167$	3.06916e7i	0.509931i	−0.966950	+	0.254965i	$0.917936\pi$
$168$	− 1.15907e7i	− 0.188593i
$169$	4.29282e7	0.684131
$170$	0	0
$171$	−1.00256e8	−1.53328
$172$	4.97562e6i	0.0745585i
$173$	5.27338e7i	0.774333i	0.922010	+	0.387167i	$0.126546\pi$
$173$	5.27338e7i	0.774333i	−0.922010	+	0.387167i	$0.873454\pi$
$174$	−6.67044e7	−0.959913
$175$	0	0
$176$	163840.	0.00226530
$177$	7.73450e7i	1.04840i
$178$	− 5.53628e7i	− 0.735780i
$179$	−8.42739e7	−1.09827	−0.549133	−	0.835735i	$-0.685042\pi$
$179$	−8.42739e7	−1.09827	−0.549133	+	0.835735i	$0.685042\pi$
$180$	0	0
$181$	−1.03956e8	−1.30309	−0.651547	−	0.758608i	$-0.725880\pi$
$181$	−1.03956e8	−1.30309	−0.651547	+	0.758608i	$0.725880\pi$
$182$	1.22163e7i	0.150207i
$183$	− 1.36546e8i	− 1.64702i
$184$	5.38624e7	0.637417
$185$	0	0
$186$	9.02690e7	1.02859
$187$	1.46008e6i	0.0163279i
$188$	− 4.50378e7i	− 0.494339i
$189$	407484.	0.00439030
$190$	0	0
$191$	−1.24775e8	−1.29572	−0.647861	−	0.761759i	$-0.724336\pi$
$191$	−1.24775e8	−1.29572	−0.647861	+	0.761759i	$0.724336\pi$
$192$	− 1.73015e7i	− 0.176413i
$193$	− 1.47589e8i	− 1.47776i	−0.673839	−	0.738878i	$-0.735356\pi$
$193$	− 1.47589e8i	− 1.47776i	0.673839	−	0.738878i	$-0.264644\pi$
$194$	−3.96217e7	−0.389607
$195$	0	0
$196$	7.52954e6	0.0714286
$197$	1.55812e8i	1.45200i	0.687692	+	0.726002i	$0.258624\pi$
$197$	1.55812e8i	1.45200i	−0.687692	+	0.726002i	$0.741376\pi$
$198$	− 694080.i	− 0.00635450i
$199$	1.33193e7	0.119810	0.0599052	−	0.998204i	$-0.480920\pi$
$199$	1.33193e7	0.119810	0.0599052	+	0.998204i	$0.480920\pi$
$200$	0	0
$201$	6.56217e7	0.569982
$202$	2.58400e7i	0.220578i
$203$	− 4.33326e7i	− 0.363562i
$204$	1.54184e8	1.27155
$205$	0	0
$206$	−1.43927e7	−0.114712
$207$	− 2.28179e8i	− 1.78805i
$208$	1.82354e7i	0.140506i
$209$	1.84888e6	0.0140087
$210$	0	0
$211$	−2.04940e8	−1.50189	−0.750945	−	0.660365i	$-0.770402\pi$
$211$	−2.04940e8	−1.50189	−0.750945	+	0.660365i	$0.770402\pi$
$212$	1.02610e8i	0.739628i
$213$	2.19648e6i	0.0155739i
$214$	1.25143e8	0.872888
$215$	0	0
$216$	608256.	0.00410675
$217$	5.86407e7i	0.389574i
$218$	5.05512e7i	0.330471i
$219$	−1.96116e8	−1.26171
$220$	0	0
$221$	−1.62507e8	−1.01274
$222$	− 1.10637e7i	− 0.0678681i
$223$	6.84858e7i	0.413555i	0.978388	+	0.206778i	$0.0662977\pi$
$223$	6.84858e7i	0.413555i	−0.978388	+	0.206778i	$0.933702\pi$
$224$	1.12394e7	0.0668153
$225$	0	0
$226$	−8.18301e7	−0.471556
$227$	− 1.93627e7i	− 0.109869i	−0.998490	−	0.0549344i	$-0.982505\pi$
$227$	− 1.93627e7i	− 0.109869i	0.998490	−	0.0549344i	$-0.0174950\pi$
$228$	− 1.95242e8i	− 1.09094i
$229$	−3.08157e8	−1.69569	−0.847847	−	0.530241i	$-0.822102\pi$
$229$	−3.08157e8	−1.69569	−0.847847	+	0.530241i	$0.822102\pi$
$230$	0	0
$231$	905520.	0.00483344
$232$	− 6.46830e7i	− 0.340081i
$233$	− 3.55797e7i	− 0.184271i	−0.995746	−	0.0921355i	$-0.970631\pi$
$233$	− 3.55797e7i	− 0.184271i	0.995746	−	0.0921355i	$-0.0293693\pi$
$234$	7.72511e7	0.394139
$235$	0	0
$236$	−7.50012e7	−0.371429
$237$	1.56827e8i	0.765247i
$238$	1.00161e8i	0.481594i
$239$	2.30056e8	1.09004	0.545018	−	0.838424i	$-0.316523\pi$
$239$	2.30056e8	1.09004	0.545018	+	0.838424i	$0.316523\pi$
$240$	0	0
$241$	5.03495e6	0.0231705	0.0115853	−	0.999933i	$-0.496312\pi$
$241$	5.03495e6	0.0231705	0.0115853	+	0.999933i	$0.496312\pi$
$242$	− 1.55885e8i	− 0.707049i
$243$	− 3.15655e8i	− 1.41121i
$244$	1.32408e8	0.583511
$245$	0	0
$246$	−1.68161e8	−0.720197
$247$	2.05780e8i	0.868890i
$248$	8.75336e7i	0.364413i
$249$	−1.40076e8	−0.574996
$250$	0	0
$251$	−1.03283e8	−0.412258	−0.206129	−	0.978525i	$-0.566087\pi$
$251$	−1.03283e8	−0.412258	−0.206129	+	0.978525i	$0.566087\pi$
$252$	− 4.76139e7i	− 0.187427i
$253$	4.20800e6i	0.0163363i
$254$	−4.80580e7	−0.184013
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	− 2.32282e8i	− 0.853592i	−0.904348	−	0.426796i	$-0.859642\pi$
$257$	− 2.32282e8i	− 0.853592i	0.904348	−	0.426796i	$-0.140358\pi$
$258$	4.10488e7i	0.148810i
$259$	7.18722e6	0.0257047
$260$	0	0
$261$	−2.74018e8	−0.953977
$262$	1.65117e8i	0.567201i
$263$	− 4.16749e8i	− 1.41263i	−0.707896	−	0.706317i	$-0.750356\pi$
$263$	− 4.16749e8i	− 1.41263i	0.707896	−	0.706317i	$-0.249644\pi$
$264$	1.35168e6	0.00452127
$265$	0	0
$266$	1.26833e8	0.413187
$267$	− 4.56743e8i	− 1.46853i
$268$	6.36332e7i	0.201935i
$269$	3.14679e8	0.985676	0.492838	−	0.870121i	$-0.335959\pi$
$269$	3.14679e8	0.985676	0.492838	+	0.870121i	$0.335959\pi$
$270$	0	0
$271$	1.92137e8	0.586433	0.293216	−	0.956046i	$-0.405274\pi$
$271$	1.92137e8	0.586433	0.293216	+	0.956046i	$0.405274\pi$
$272$	1.49512e8i	0.450490i
$273$	1.00784e8i	0.299795i
$274$	−3.80959e7	−0.111880
$275$	0	0
$276$	4.44365e8	1.27221
$277$	− 4.40393e8i	− 1.24498i	−0.782629	−	0.622489i	$-0.786122\pi$
$277$	− 4.40393e8i	− 1.24498i	0.782629	−	0.622489i	$-0.213878\pi$
$278$	4.04579e7i	0.112940i
$279$	3.70821e8	1.02223
$280$	0	0
$281$	3.59235e8	0.965842	0.482921	−	0.875664i	$-0.339576\pi$
$281$	3.59235e8	0.965842	0.482921	+	0.875664i	$0.339576\pi$
$282$	− 3.71562e8i	− 0.986642i
$283$	8.11467e7i	0.212823i	0.994322	+	0.106411i	$0.0339361\pi$
$283$	8.11467e7i	0.212823i	−0.994322	+	0.106411i	$0.966064\pi$
$284$	−2.12992e6	−0.00551759
$285$	0	0
$286$	−1.42464e6	−0.00360101
$287$	− 1.09241e8i	− 0.272771i
$288$	− 7.10738e7i	− 0.175322i
$289$	−9.22057e8	−2.24706
$290$	0	0
$291$	−3.26879e8	−0.777609
$292$	− 1.90173e8i	− 0.447002i
$293$	− 2.53416e8i	− 0.588569i	−0.955718	−	0.294285i	$-0.904919\pi$
$293$	− 2.53416e8i	− 0.588569i	0.955718	−	0.294285i	$-0.0950813\pi$
$294$	6.21187e7	0.142563
$295$	0	0
$296$	1.07284e7	0.0240445
$297$	47520.0i	0 0.000105252i
$298$	2.18189e8i	0.477613i
$299$	−4.68350e8	−1.01326
$300$	0	0
$301$	−2.66662e7	−0.0563609
$302$	5.19137e7i	0.108457i
$303$	2.13180e8i	0.440248i
$304$	1.89325e8	0.386501
$305$	0	0
$306$	6.33383e8	1.26369
$307$	8.72706e8i	1.72141i	0.509107	+	0.860703i	$0.329976\pi$
$307$	8.72706e8i	1.72141i	−0.509107	+	0.860703i	$0.670024\pi$
$308$	878080.i	0.00171241i
$309$	−1.18740e8	−0.228951
$310$	0	0
$311$	−6.71611e8	−1.26607	−0.633033	−	0.774124i	$-0.718190\pi$
$311$	−6.71611e8	−1.26607	−0.633033	+	0.774124i	$0.718190\pi$
$312$	1.50442e8i	0.280432i
$313$	1.92216e8i	0.354312i	0.984183	+	0.177156i	$0.0566896\pi$
$313$	1.92216e8i	0.354312i	−0.984183	+	0.177156i	$0.943310\pi$
$314$	5.04648e8	0.919888
$315$	0	0
$316$	−1.52075e8	−0.271114
$317$	− 1.33837e8i	− 0.235977i	−0.993015	−	0.117988i	$-0.962355\pi$
$317$	− 1.33837e8i	− 0.235977i	0.993015	−	0.117988i	$-0.0376446\pi$
$318$	8.46531e8i	1.47621i
$319$	5.05336e6	0.00871591
$320$	0	0
$321$	1.03243e9	1.74218
$322$	2.88669e8i	0.481842i
$323$	1.68720e9i	2.78584i
$324$	3.08609e8	0.504081
$325$	0	0
$326$	6.65817e8	1.06437
$327$	4.17048e8i	0.659581i
$328$	− 1.63065e8i	− 0.255154i
$329$	2.41375e8	0.373685
$330$	0	0
$331$	4.25298e8	0.644608	0.322304	−	0.946636i	$-0.395543\pi$
$331$	4.25298e8	0.644608	0.322304	+	0.946636i	$0.395543\pi$
$332$	− 1.35831e8i	− 0.203711i
$333$	− 4.54492e7i	− 0.0674484i
$334$	−2.45532e8	−0.360576
$335$	0	0
$336$	9.27252e7	0.133355
$337$	1.07703e9i	1.53293i	0.642288	+	0.766463i	$0.277985\pi$
$337$	1.07703e9i	1.53293i	−0.642288	+	0.766463i	$0.722015\pi$
$338$	3.43426e8i	0.483754i
$339$	−6.75098e8	−0.941170
$340$	0	0
$341$	−6.83856e6	−0.00933952
$342$	− 8.02044e8i	− 1.08419i
$343$	4.03536e7i	0.0539949i
$344$	−3.98049e7	−0.0527208
$345$	0	0
$346$	−4.21871e8	−0.547536
$347$	− 7.23764e8i	− 0.929916i	−0.885333	−	0.464958i	$-0.846069\pi$
$347$	− 7.23764e8i	− 0.929916i	0.885333	−	0.464958i	$-0.153931\pi$
$348$	− 5.33635e8i	− 0.678761i
$349$	4.48132e8	0.564310	0.282155	−	0.959369i	$-0.408951\pi$
$349$	4.48132e8	0.564310	0.282155	+	0.959369i	$0.408951\pi$
$350$	0	0
$351$	−5.28898e6	−0.00652825
$352$	1.31072e6i	0.00160181i
$353$	1.49946e9i	1.81435i	0.420749	+	0.907177i	$0.361767\pi$
$353$	1.49946e9i	1.81435i	−0.420749	+	0.907177i	$0.638233\pi$
$354$	−6.18760e8	−0.741329
$355$	0	0
$356$	4.42902e8	0.520275
$357$	8.26332e8i	0.961205i
$358$	− 6.74191e8i	− 0.776591i
$359$	2.56890e8	0.293033	0.146516	−	0.989208i	$-0.453194\pi$
$359$	2.56890e8	0.293033	0.146516	+	0.989208i	$0.453194\pi$
$360$	0	0
$361$	1.24260e9	1.39013
$362$	− 8.31650e8i	− 0.921427i
$363$	− 1.28605e9i	− 1.41118i
$364$	−9.77303e7	−0.106212
$365$	0	0
$366$	1.09236e9	1.16462
$367$	6.50424e8i	0.686856i	0.939179	+	0.343428i	$0.111588\pi$
$367$	6.50424e8i	0.686856i	−0.939179	+	0.343428i	$0.888412\pi$
$368$	4.30899e8i	0.450722i
$369$	−6.90796e8	−0.715744
$370$	0	0
$371$	−5.49924e8	−0.559106
$372$	7.22152e8i	0.727325i
$373$	4.66127e8i	0.465076i	0.972587	+	0.232538i	$0.0747030\pi$
$373$	4.66127e8i	0.465076i	−0.972587	+	0.232538i	$0.925297\pi$
$374$	−1.16806e7	−0.0115456
$375$	0	0
$376$	3.60303e8	0.349551
$377$	5.62439e8i	0.540606i
$378$	3.25987e6i	0.00310441i
$379$	−2.85860e8	−0.269721	−0.134861	−	0.990865i	$-0.543059\pi$
$379$	−2.85860e8	−0.269721	−0.134861	+	0.990865i	$0.543059\pi$
$380$	0	0
$381$	−3.96478e8	−0.367267
$382$	− 9.98202e8i	− 0.916213i
$383$	1.65075e9i	1.50136i	0.660665	+	0.750681i	$0.270275\pi$
$383$	1.65075e9i	1.50136i	−0.660665	+	0.750681i	$0.729725\pi$
$384$	1.38412e8	0.124743
$385$	0	0
$386$	1.18071e9	1.04493
$387$	1.68627e8i	0.147890i
$388$	− 3.16973e8i	− 0.275494i
$389$	−1.51304e9	−1.30325	−0.651624	−	0.758542i	$-0.725912\pi$
$389$	−1.51304e9	−1.30325	−0.651624	+	0.758542i	$0.725912\pi$
$390$	0	0
$391$	−3.84001e9	−3.24873
$392$	6.02363e7i	0.0505076i
$393$	1.36221e9i	1.13207i
$394$	−1.24649e9	−1.02672
$395$	0	0
$396$	5.55264e6	0.00449331
$397$	− 8.63794e8i	− 0.692857i	−0.938076	−	0.346428i	$-0.887394\pi$
$397$	− 8.63794e8i	− 0.692857i	0.938076	−	0.346428i	$-0.112606\pi$
$398$	1.06554e8i	0.0847187i
$399$	1.04637e9	0.824673
$400$	0	0
$401$	1.14042e8	0.0883199	0.0441599	−	0.999024i	$-0.485939\pi$
$401$	1.14042e8	0.0883199	0.0441599	+	0.999024i	$0.485939\pi$
$402$	5.24974e8i	0.403038i
$403$	− 7.61132e8i	− 0.579285i
$404$	−2.06720e8	−0.155972
$405$	0	0
$406$	3.46660e8	0.257077
$407$	838160.i	0 0.000616235i
$408$	1.23348e9i	0.899125i
$409$	1.18328e9	0.855176	0.427588	−	0.903974i	$-0.359363\pi$
$409$	1.18328e9	0.855176	0.427588	+	0.903974i	$0.359363\pi$
$410$	0	0
$411$	−3.14291e8	−0.223298
$412$	− 1.15142e8i	− 0.0811135i
$413$	− 4.01960e8i	− 0.280774i
$414$	1.82543e9	1.26434
$415$	0	0
$416$	−1.45883e8	−0.0993524
$417$	3.33777e8i	0.225414i
$418$	1.47910e7i	0.00990562i
$419$	−2.27959e8	−0.151394	−0.0756970	−	0.997131i	$-0.524118\pi$
$419$	−2.27959e8	−0.151394	−0.0756970	+	0.997131i	$0.524118\pi$
$420$	0	0
$421$	−3.90700e7	−0.0255186	−0.0127593	−	0.999919i	$-0.504062\pi$
$421$	−3.90700e7	−0.0255186	−0.0127593	+	0.999919i	$0.504062\pi$
$422$	− 1.63952e9i	− 1.06200i
$423$	− 1.52636e9i	− 0.980541i
$424$	−8.20878e8	−0.522996
$425$	0	0
$426$	−1.75718e7	−0.0110124
$427$	7.09623e8i	0.441093i
$428$	1.00114e9i	0.617225i
$429$	−1.17533e7	−0.00718718
$430$	0	0
$431$	−2.58620e9	−1.55594	−0.777968	−	0.628304i	$-0.783749\pi$
$431$	−2.58620e9	−1.55594	−0.777968	+	0.628304i	$0.783749\pi$
$432$	4.86605e6i	0.00290391i
$433$	1.78893e9i	1.05897i	0.848318	+	0.529486i	$0.177615\pi$
$433$	1.78893e9i	1.05897i	−0.848318	+	0.529486i	$0.822385\pi$
$434$	−4.69125e8	−0.275470
$435$	0	0
$436$	−4.04410e8	−0.233679
$437$	4.86255e9i	2.78727i
$438$	− 1.56893e9i	− 0.892162i
$439$	4.58905e8	0.258879	0.129440	−	0.991587i	$-0.458682\pi$
$439$	4.58905e8	0.258879	0.129440	+	0.991587i	$0.458682\pi$
$440$	0	0
$441$	2.55181e8	0.141681
$442$	− 1.30006e9i	− 0.716117i
$443$	− 1.38459e9i	− 0.756672i	−0.925668	−	0.378336i	$-0.876496\pi$
$443$	− 1.38459e9i	− 0.756672i	0.925668	−	0.378336i	$-0.123504\pi$
$444$	8.85097e7	0.0479900
$445$	0	0
$446$	−5.47887e8	−0.292428
$447$	1.80006e9i	0.953259i
$448$	8.99154e7i	0.0472456i
$449$	2.73611e9	1.42650	0.713248	−	0.700911i	$-0.247223\pi$
$449$	2.73611e9	1.42650	0.713248	+	0.700911i	$0.247223\pi$
$450$	0	0
$451$	1.27394e7	0.00653932
$452$	− 6.54641e8i	− 0.333441i
$453$	4.28288e8i	0.216467i
$454$	1.54901e8	0.0776890
$455$	0	0
$456$	1.56193e9	0.771411
$457$	2.43053e9i	1.19123i	0.803271	+	0.595614i	$0.203091\pi$
$457$	2.43053e9i	1.19123i	−0.803271	+	0.595614i	$0.796909\pi$
$458$	− 2.46525e9i	− 1.19904i
$459$	−4.33644e7	−0.0209309
$460$	0	0
$461$	3.94884e9	1.87723	0.938613	−	0.344971i	$-0.112111\pi$
$461$	3.94884e9	1.87723	0.938613	+	0.344971i	$0.112111\pi$
$462$	7.24416e6i	0.00341776i
$463$	− 2.57453e9i	− 1.20549i	−0.797933	−	0.602746i	$-0.794073\pi$
$463$	− 2.57453e9i	− 1.20549i	0.797933	−	0.602746i	$-0.205927\pi$
$464$	5.17464e8	0.240474
$465$	0	0
$466$	2.84638e8	0.130299
$467$	2.98482e8i	0.135616i	0.997698	+	0.0678078i	$0.0216005\pi$
$467$	2.98482e8i	0.135616i	−0.997698	+	0.0678078i	$0.978400\pi$
$468$	6.18009e8i	0.278698i
$469$	−3.41034e8	−0.152649
$470$	0	0
$471$	4.16335e9	1.83599
$472$	− 6.00010e8i	− 0.262640i
$473$	− 3.10976e6i	− 0.00135118i
$474$	−1.25462e9	−0.541112
$475$	0	0
$476$	−8.01292e8	−0.340539
$477$	3.47751e9i	1.46708i
$478$	1.84045e9i	0.770772i
$479$	−2.62000e9	−1.08925	−0.544625	−	0.838680i	$-0.683328\pi$
$479$	−2.62000e9	−1.08925	−0.544625	+	0.838680i	$0.683328\pi$
$480$	0	0
$481$	−9.32872e7	−0.0382221
$482$	4.02796e7i	0.0163840i
$483$	2.38152e9i	0.961698i
$484$	1.24708e9	0.499959
$485$	0	0
$486$	2.52524e9	0.997873
$487$	− 4.16662e9i	− 1.63468i	−0.576157	−	0.817339i	$-0.695448\pi$
$487$	− 4.16662e9i	− 1.63468i	0.576157	−	0.817339i	$-0.304552\pi$
$488$	1.05926e9i	0.412605i
$489$	5.49299e9	2.12436
$490$	0	0
$491$	2.41300e9	0.919967	0.459983	−	0.887928i	$-0.347855\pi$
$491$	2.41300e9	0.919967	0.459983	+	0.887928i	$0.347855\pi$
$492$	− 1.34528e9i	− 0.509256i
$493$	4.61144e9i	1.73330i
$494$	−1.64624e9	−0.614398
$495$	0	0
$496$	−7.00269e8	−0.257679
$497$	− 1.14150e7i	− 0.00417090i
$498$	− 1.12061e9i	− 0.406584i
$499$	1.04092e9	0.375029	0.187515	−	0.982262i	$-0.439957\pi$
$499$	1.04092e9	0.375029	0.187515	+	0.982262i	$0.439957\pi$
$500$	0	0
$501$	−2.02564e9	−0.719666
$502$	− 8.26261e8i	− 0.291510i
$503$	1.17273e9i	0.410876i	0.978670	+	0.205438i	$0.0658618\pi$
$503$	1.17273e9i	0.410876i	−0.978670	+	0.205438i	$0.934138\pi$
$504$	3.80911e8	0.132531
$505$	0	0
$506$	−3.36640e7	−0.0115515
$507$	2.83326e9i	0.965515i
$508$	− 3.84464e8i	− 0.130117i
$509$	8.13818e7	0.0273536	0.0136768	−	0.999906i	$-0.495646\pi$
$509$	8.13818e7	0.0273536	0.0136768	+	0.999906i	$0.495646\pi$
$510$	0	0
$511$	1.01921e9	0.337901
$512$	1.34218e8i	0.0441942i
$513$	5.49117e7i	0.0179579i
$514$	1.85826e9	0.603580
$515$	0	0
$516$	−3.28391e8	−0.105224
$517$	2.81486e7i	0.00895861i
$518$	5.74978e7i	0.0181759i
$519$	−3.48043e9	−1.09282
$520$	0	0
$521$	−2.77458e9	−0.859540	−0.429770	−	0.902939i	$-0.641405\pi$
$521$	−2.77458e9	−0.859540	−0.429770	+	0.902939i	$0.641405\pi$
$522$	− 2.19215e9i	− 0.674564i
$523$	− 4.99213e9i	− 1.52591i	−0.646449	−	0.762957i	$-0.723747\pi$
$523$	− 4.99213e9i	− 1.52591i	0.646449	−	0.762957i	$-0.276253\pi$
$524$	−1.32094e9	−0.401072
$525$	0	0
$526$	3.33399e9	0.998883
$527$	− 6.24053e9i	− 1.85731i
$528$	1.08134e7i	0.00319702i
$529$	−7.66221e9	−2.25040
$530$	0	0
$531$	−2.54184e9	−0.736745
$532$	1.01467e9i	0.292168i
$533$	1.41790e9i	0.405602i
$534$	3.65394e9	1.03841
$535$	0	0
$536$	−5.09065e8	−0.142790
$537$	− 5.56207e9i	− 1.54998i
$538$	2.51743e9i	0.696978i
$539$	−4.70596e6	−0.00129446
$540$	0	0
$541$	1.63095e9	0.442844	0.221422	−	0.975178i	$-0.428930\pi$
$541$	1.63095e9	0.442844	0.221422	+	0.975178i	$0.428930\pi$
$542$	1.53710e9i	0.414671i
$543$	− 6.86112e9i	− 1.83906i
$544$	−1.19610e9	−0.318545
$545$	0	0
$546$	−8.06275e8	−0.211987
$547$	2.00950e9i	0.524967i	0.964936	+	0.262484i	$0.0845416\pi$
$547$	2.00950e9i	0.524967i	−0.964936	+	0.262484i	$0.915458\pi$
$548$	− 3.04767e8i	− 0.0791109i
$549$	4.48738e9	1.15742
$550$	0	0
$551$	5.83941e9	1.48709
$552$	3.55492e9i	0.899586i
$553$	− 8.15026e8i	− 0.204943i
$554$	3.52315e9	0.880332
$555$	0	0
$556$	−3.23663e8	−0.0798604
$557$	− 4.47959e9i	− 1.09836i	−0.835704	−	0.549180i	$-0.814940\pi$
$557$	− 4.47959e9i	− 1.09836i	0.835704	−	0.549180i	$-0.185060\pi$
$558$	2.96657e9i	0.722827i
$559$	3.46116e8	0.0838071
$560$	0	0
$561$	−9.63653e7	−0.0230436
$562$	2.87388e9i	0.682954i
$563$	1.50730e9i	0.355976i	0.984033	+	0.177988i	$0.0569589\pi$
$563$	1.50730e9i	0.355976i	−0.984033	+	0.177988i	$0.943041\pi$
$564$	2.97250e9	0.697661
$565$	0	0
$566$	−6.49174e8	−0.150489
$567$	1.65395e9i	0.381050i
$568$	− 1.70394e7i	− 0.00390152i
$569$	−2.33088e9	−0.530428	−0.265214	−	0.964190i	$-0.585443\pi$
$569$	−2.33088e9	−0.530428	−0.265214	+	0.964190i	$0.585443\pi$
$570$	0	0
$571$	−2.91101e9	−0.654362	−0.327181	−	0.944962i	$-0.606099\pi$
$571$	−2.91101e9	−0.654362	−0.327181	+	0.944962i	$0.606099\pi$
$572$	− 1.13971e7i	− 0.00254630i
$573$	− 8.23517e9i	− 1.82865i
$574$	8.73926e8	0.192878
$575$	0	0
$576$	5.68590e8	0.123971
$577$	− 8.64805e9i	− 1.87414i	−0.349137	−	0.937072i	$-0.613525\pi$
$577$	− 8.64805e9i	− 1.87414i	0.349137	−	0.937072i	$-0.386475\pi$
$578$	− 7.37646e9i	− 1.58891i
$579$	9.74086e9	2.08556
$580$	0	0
$581$	7.27969e8	0.153991
$582$	− 2.61503e9i	− 0.549853i
$583$	− 6.41311e7i	− 0.0134038i
$584$	1.52138e9	0.316078
$585$	0	0
$586$	2.02733e9	0.416181
$587$	− 6.33513e9i	− 1.29277i	−0.763010	−	0.646387i	$-0.776279\pi$
$587$	− 6.33513e9i	− 1.29277i	0.763010	−	0.646387i	$-0.223721\pi$
$588$	4.96949e8i	0.100807i
$589$	−7.90230e9	−1.59349
$590$	0	0
$591$	−1.02836e10	−2.04921
$592$	8.58276e7i	0.0170020i
$593$	1.70162e9i	0.335098i	0.985864	+	0.167549i	$0.0535852\pi$
$593$	1.70162e9i	0.335098i	−0.985864	+	0.167549i	$0.946415\pi$
$594$	−380160.	−7.44241e−5 0
$595$	0	0
$596$	−1.74551e9	−0.337723
$597$	8.79071e8i	0.169088i
$598$	− 3.74680e9i	− 0.716484i
$599$	−3.01977e9	−0.574090	−0.287045	−	0.957917i	$-0.592673\pi$
$599$	−3.01977e9	−0.574090	−0.287045	+	0.957917i	$0.592673\pi$
$600$	0	0
$601$	−5.92708e9	−1.11373	−0.556865	−	0.830603i	$-0.687996\pi$
$601$	−5.92708e9	−1.11373	−0.556865	+	0.830603i	$0.687996\pi$
$602$	− 2.13330e8i	− 0.0398532i
$603$	2.15657e9i	0.400546i
$604$	−4.15309e8	−0.0766906
$605$	0	0
$606$	−1.70544e9	−0.311302
$607$	− 1.45649e9i	− 0.264331i	−0.991228	−	0.132165i	$-0.957807\pi$
$607$	− 1.45649e9i	− 0.264331i	0.991228	−	0.132165i	$-0.0421930\pi$
$608$	1.51460e9i	0.273298i
$609$	2.85995e9	0.513095
$610$	0	0
$611$	−3.13294e9	−0.555659
$612$	5.06706e9i	0.893565i
$613$	6.71607e9i	1.17762i	0.808273	+	0.588808i	$0.200403\pi$
$613$	6.71607e9i	1.17762i	−0.808273	+	0.588808i	$0.799597\pi$
$614$	−6.98164e9	−1.21722
$615$	0	0
$616$	−7.02464e6	−0.00121085
$617$	7.02027e9i	1.20325i	0.798779	+	0.601625i	$0.205480\pi$
$617$	7.02027e9i	1.20325i	−0.798779	+	0.601625i	$0.794520\pi$
$618$	− 9.49921e8i	− 0.161893i
$619$	−5.14352e9	−0.871652	−0.435826	−	0.900031i	$-0.643544\pi$
$619$	−5.14352e9	−0.871652	−0.435826	+	0.900031i	$0.643544\pi$
$620$	0	0
$621$	−1.24978e8	−0.0209417
$622$	− 5.37289e9i	− 0.895245i
$623$	2.37368e9i	0.393291i
$624$	−1.20354e9	−0.198296
$625$	0	0
$626$	−1.53773e9	−0.250536
$627$	1.22026e8i	0.0197704i
$628$	4.03719e9i	0.650459i
$629$	−7.64863e8	−0.122548
$630$	0	0
$631$	−4.41574e9	−0.699681	−0.349841	−	0.936809i	$-0.613764\pi$
$631$	−4.41574e9	−0.699681	−0.349841	+	0.936809i	$0.613764\pi$
$632$	− 1.21660e9i	− 0.191707i
$633$	− 1.35260e10i	− 2.11962i
$634$	1.07070e9	0.166861
$635$	0	0
$636$	−6.77225e9	−1.04384
$637$	− 5.23773e8i	− 0.0802889i
$638$	4.04269e7i	0.00616308i
$639$	−7.21843e7	−0.0109443
$640$	0	0
$641$	6.94176e8	0.104104	0.0520519	−	0.998644i	$-0.483424\pi$
$641$	6.94176e8	0.104104	0.0520519	+	0.998644i	$0.483424\pi$
$642$	8.25944e9i	1.23191i
$643$	− 9.50809e9i	− 1.41044i	−0.708988	−	0.705220i	$-0.750848\pi$
$643$	− 9.50809e9i	− 1.41044i	0.708988	−	0.705220i	$-0.249152\pi$
$644$	−2.30935e9	−0.340713
$645$	0	0
$646$	−1.34976e10	−1.96989
$647$	− 7.73215e9i	− 1.12237i	−0.827691	−	0.561184i	$-0.810346\pi$
$647$	− 7.73215e9i	− 1.12237i	0.827691	−	0.561184i	$-0.189654\pi$
$648$	2.46887e9i	0.356439i
$649$	4.68758e7	0.00673119
$650$	0	0
$651$	−3.87028e9	−0.549806
$652$	5.32654e9i	0.752624i
$653$	5.06321e9i	0.711590i	0.934564	+	0.355795i	$0.115790\pi$
$653$	5.06321e9i	0.711590i	−0.934564	+	0.355795i	$0.884210\pi$
$654$	−3.33638e9	−0.466395
$655$	0	0
$656$	1.30452e9	0.180421
$657$	− 6.44508e9i	− 0.886645i
$658$	1.93100e9i	0.264235i
$659$	−8.08113e9	−1.09995	−0.549975	−	0.835181i	$-0.685363\pi$
$659$	−8.08113e9	−1.09995	−0.549975	+	0.835181i	$0.685363\pi$
$660$	0	0
$661$	6.30089e9	0.848588	0.424294	−	0.905524i	$-0.360522\pi$
$661$	6.30089e9	0.848588	0.424294	+	0.905524i	$0.360522\pi$
$662$	3.40239e9i	0.455806i
$663$	− 1.07255e10i	− 1.42928i
$664$	1.08665e9	0.144046
$665$	0	0
$666$	3.63594e8	0.0476932
$667$	1.32903e10i	1.73419i
$668$	− 1.96426e9i	− 0.254965i
$669$	−4.52006e9	−0.583651
$670$	0	0
$671$	−8.27549e7	−0.0105746
$672$	7.41802e8i	0.0942965i
$673$	9.62624e9i	1.21732i	0.793432	+	0.608659i	$0.208292\pi$
$673$	9.62624e9i	1.21732i	−0.793432	+	0.608659i	$0.791708\pi$
$674$	−8.61620e9	−1.08394
$675$	0	0
$676$	−2.74741e9	−0.342066
$677$	− 9.45429e9i	− 1.17103i	−0.810661	−	0.585516i	$-0.800892\pi$
$677$	− 9.45429e9i	− 1.17103i	0.810661	−	0.585516i	$-0.199108\pi$
$678$	− 5.40079e9i	− 0.665508i
$679$	1.69878e9	0.208254
$680$	0	0
$681$	1.27794e9	0.155058
$682$	− 5.47085e7i	− 0.00660403i
$683$	− 2.48879e8i	− 0.0298893i	−0.999888	−	0.0149447i	$-0.995243\pi$
$683$	− 2.48879e8i	− 0.0298893i	0.999888	−	0.0149447i	$-0.00475721\pi$
$684$	6.41635e9	0.766641
$685$	0	0
$686$	−3.22829e8	−0.0381802
$687$	− 2.03383e10i	− 2.39313i
$688$	− 3.18439e8i	− 0.0372793i
$689$	7.13779e9	0.831375
$690$	0	0
$691$	−3.46412e9	−0.399411	−0.199705	−	0.979856i	$-0.563998\pi$
$691$	−3.46412e9	−0.399411	−0.199705	+	0.979856i	$0.563998\pi$
$692$	− 3.37497e9i	− 0.387167i
$693$	2.97587e7i	0.00339662i
$694$	5.79011e9	0.657550
$695$	0	0
$696$	4.26908e9	0.479956
$697$	1.16254e10i	1.30045i
$698$	3.58506e9i	0.399027i
$699$	2.34826e9	0.260062
$700$	0	0
$701$	5.56322e9	0.609976	0.304988	−	0.952356i	$-0.401347\pi$
$701$	5.56322e9	0.609976	0.304988	+	0.952356i	$0.401347\pi$
$702$	− 4.23118e7i	− 0.00461617i
$703$	9.68536e8i	0.105141i
$704$	−1.04858e7	−0.00113265
$705$	0	0
$706$	−1.19956e10	−1.28294
$707$	− 1.10789e9i	− 0.117904i
$708$	− 4.95008e9i	− 0.524199i
$709$	−8.23697e9	−0.867971	−0.433986	−	0.900920i	$-0.642893\pi$
$709$	−8.23697e9	−0.867971	−0.433986	+	0.900920i	$0.642893\pi$
$710$	0	0
$711$	−5.15391e9	−0.537766
$712$	3.54322e9i	0.367890i
$713$	− 1.79854e10i	− 1.85826i
$714$	−6.61066e9	−0.679674
$715$	0	0
$716$	5.39353e9	0.549133
$717$	1.51837e10i	1.53837i
$718$	2.05512e9i	0.207206i
$719$	−5.85212e9	−0.587168	−0.293584	−	0.955933i	$-0.594848\pi$
$719$	−5.85212e9	−0.587168	−0.293584	+	0.955933i	$0.594848\pi$
$720$	0	0
$721$	6.17089e8	0.0613160
$722$	9.94081e9i	0.982973i
$723$	3.32307e8i	0.0327005i
$724$	6.65320e9	0.651547
$725$	0	0
$726$	1.02884e10	0.997858
$727$	1.51706e10i	1.46431i	0.681139	+	0.732154i	$0.261485\pi$
$727$	1.51706e10i	1.46431i	−0.681139	+	0.732154i	$0.738515\pi$
$728$	− 7.81842e8i	− 0.0751034i
$729$	1.02875e10	0.983472
$730$	0	0
$731$	2.83781e9	0.268703
$732$	8.73892e9i	0.823510i
$733$	1.55969e10i	1.46277i	0.681967	+	0.731383i	$0.261125\pi$
$733$	1.55969e10i	1.46277i	−0.681967	+	0.731383i	$0.738875\pi$
$734$	−5.20339e9	−0.485680
$735$	0	0
$736$	−3.44719e9	−0.318708
$737$	− 3.97707e7i	− 0.00365955i
$738$	− 5.52637e9i	− 0.506107i
$739$	6.95573e9	0.633997	0.316998	−	0.948426i	$-0.397325\pi$
$739$	6.95573e9	0.633997	0.316998	+	0.948426i	$0.397325\pi$
$740$	0	0
$741$	−1.35815e10	−1.22626
$742$	− 4.39939e9i	− 0.395348i
$743$	1.17803e10i	1.05365i	0.849975	+	0.526824i	$0.176617\pi$
$743$	1.17803e10i	1.05365i	−0.849975	+	0.526824i	$0.823383\pi$
$744$	−5.77722e9	−0.514296
$745$	0	0
$746$	−3.72902e9	−0.328858
$747$	− 4.60339e9i	− 0.404070i
$748$	− 9.34451e7i	− 0.00816396i
$749$	−5.36551e9	−0.466578
$750$	0	0
$751$	−6.96200e9	−0.599783	−0.299892	−	0.953973i	$-0.596951\pi$
$751$	−6.96200e9	−0.599783	−0.299892	+	0.953973i	$0.596951\pi$
$752$	2.88242e9i	0.247170i
$753$	− 6.81665e9i	− 0.581820i
$754$	−4.49951e9	−0.382266
$755$	0	0
$756$	−2.60790e7	−0.00219515
$757$	− 2.07114e10i	− 1.73530i	−0.497179	−	0.867648i	$-0.665631\pi$
$757$	− 2.07114e10i	− 1.73530i	0.497179	−	0.867648i	$-0.334369\pi$
$758$	− 2.28688e9i	− 0.190722i
$759$	−2.77728e8	−0.0230554
$760$	0	0
$761$	1.65392e10	1.36041	0.680204	−	0.733023i	$-0.261891\pi$
$761$	1.65392e10	1.36041	0.680204	+	0.733023i	$0.261891\pi$
$762$	− 3.17183e9i	− 0.259697i
$763$	− 2.16738e9i	− 0.176644i
$764$	7.98562e9	0.647861
$765$	0	0
$766$	−1.32060e10	−1.06162
$767$	5.21727e9i	0.417503i
$768$	1.10730e9i	0.0882063i
$769$	2.33650e10	1.85278	0.926391	−	0.376562i	$-0.122894\pi$
$769$	2.33650e10	1.85278	0.926391	+	0.376562i	$0.122894\pi$
$770$	0	0
$771$	1.53306e10	1.20467
$772$	9.44568e9i	0.738878i
$773$	− 7.09263e9i	− 0.552305i	−0.961114	−	0.276152i	$-0.910941\pi$
$773$	− 7.09263e9i	− 0.552305i	0.961114	−	0.276152i	$-0.0890595\pi$
$774$	−1.34901e9	−0.104574
$775$	0	0
$776$	2.53579e9	0.194804
$777$	4.74357e8i	0.0362770i
$778$	− 1.21043e10i	− 0.921536i
$779$	1.47211e10	1.11573
$780$	0	0
$781$	1.33120e6	9.99919e−5 0
$782$	− 3.07201e10i	− 2.29720i
$783$	1.50085e8i	0.0111730i
$784$	−4.81890e8	−0.0357143
$785$	0	0
$786$	−1.08977e10	−0.800491
$787$	− 1.23030e10i	− 0.899703i	−0.893103	−	0.449851i	$-0.851477\pi$
$787$	− 1.23030e10i	− 0.899703i	0.893103	−	0.449851i	$-0.148523\pi$
$788$	− 9.97194e9i	− 0.726002i
$789$	2.75054e10	1.99365
$790$	0	0
$791$	3.50847e9	0.252057
$792$	4.44211e7i	0.00317725i
$793$	− 9.21062e9i	− 0.655892i
$794$	6.91036e9	0.489924
$795$	0	0
$796$	−8.52433e8	−0.0599052
$797$	− 3.66650e9i	− 0.256535i	−0.991740	−	0.128268i	$-0.959058\pi$
$797$	− 3.66650e9i	− 0.256535i	0.991740	−	0.128268i	$-0.0409417\pi$
$798$	8.37099e9i	0.583132i
$799$	−2.56870e10	−1.78156
$800$	0	0
$801$	1.50102e10	1.03199
$802$	9.12334e8i	0.0624516i
$803$	1.18858e8i	0.00810074i
$804$	−4.19979e9	−0.284991
$805$	0	0
$806$	6.08905e9	0.409616
$807$	2.07688e10i	1.39109i
$808$	− 1.65376e9i	− 0.110289i
$809$	2.96609e10	1.96954	0.984770	−	0.173861i	$-0.0556245\pi$
$809$	2.96609e10	1.96954	0.984770	+	0.173861i	$0.0556245\pi$
$810$	0	0
$811$	2.51278e10	1.65417	0.827087	−	0.562073i	$-0.189996\pi$
$811$	2.51278e10	1.65417	0.827087	+	0.562073i	$0.189996\pi$
$812$	2.77328e9i	0.181781i
$813$	1.26810e10i	0.827633i
$814$	−6.70528e6	−0.000435744 0
$815$	0	0
$816$	−9.86780e9	−0.635777
$817$	− 3.59348e9i	− 0.230536i
$818$	9.46623e9i	0.604700i
$819$	−3.31214e9	−0.210676
$820$	0	0
$821$	4.57772e9	0.288701	0.144350	−	0.989527i	$-0.453891\pi$
$821$	4.57772e9	0.288701	0.144350	+	0.989527i	$0.453891\pi$
$822$	− 2.51433e9i	− 0.157896i
$823$	− 1.93133e9i	− 0.120769i	−0.998175	−	0.0603846i	$-0.980767\pi$
$823$	− 1.93133e9i	− 0.120769i	0.998175	−	0.0603846i	$-0.0192327\pi$
$824$	9.21135e8	0.0573559
$825$	0	0
$826$	3.21568e9	0.198537
$827$	1.58094e10i	0.971958i	0.873971	+	0.485979i	$0.161537\pi$
$827$	1.58094e10i	0.971958i	−0.873971	+	0.485979i	$0.838463\pi$
$828$	1.46034e10i	0.894024i
$829$	2.46536e9	0.150293	0.0751465	−	0.997173i	$-0.476058\pi$
$829$	2.46536e9	0.150293	0.0751465	+	0.997173i	$0.476058\pi$
$830$	0	0
$831$	2.90660e10	1.75704
$832$	− 1.16707e9i	− 0.0702528i
$833$	− 4.29442e9i	− 0.257423i
$834$	−2.67022e9	−0.159392
$835$	0	0
$836$	−1.18328e8	−0.00700433
$837$	− 2.03105e8i	− 0.0119724i
$838$	− 1.82368e9i	− 0.107052i
$839$	−2.51861e10	−1.47229	−0.736147	−	0.676822i	$-0.763357\pi$
$839$	−2.51861e10	−1.47229	−0.736147	+	0.676822i	$0.763357\pi$
$840$	0	0
$841$	−1.28960e9	−0.0747598
$842$	− 3.12560e8i	− 0.0180444i
$843$	2.37095e10i	1.36309i
$844$	1.31162e10	0.750945
$845$	0	0
$846$	1.22109e10	0.693347
$847$	6.68355e9i	0.377933i
$848$	− 6.56703e9i	− 0.369814i
$849$	−5.35568e9	−0.300357
$850$	0	0
$851$	−2.20436e9	−0.122611
$852$	− 1.40575e8i	− 0.00778697i
$853$	− 1.07306e10i	− 0.591972i	−0.955192	−	0.295986i	$-0.904352\pi$
$853$	− 1.07306e10i	− 0.591972i	0.955192	−	0.295986i	$-0.0956481\pi$
$854$	−5.67698e9	−0.311900
$855$	0	0
$856$	−8.00915e9	−0.436444
$857$	− 2.79332e10i	− 1.51596i	−0.652279	−	0.757979i	$-0.726187\pi$
$857$	− 2.79332e10i	− 1.51596i	0.652279	−	0.757979i	$-0.273813\pi$
$858$	− 9.40262e7i	− 0.00508210i
$859$	1.94983e10	1.04959	0.524795	−	0.851229i	$-0.324142\pi$
$859$	1.94983e10	1.04959	0.524795	+	0.851229i	$0.324142\pi$
$860$	0	0
$861$	7.20989e9	0.384962
$862$	− 2.06896e10i	− 1.10021i
$863$	1.63551e10i	0.866193i	0.901348	+	0.433096i	$0.142579\pi$
$863$	1.63551e10i	0.866193i	−0.901348	+	0.433096i	$0.857421\pi$
$864$	−3.89284e7	−0.00205338
$865$	0	0
$866$	−1.43114e10	−0.748807
$867$	− 6.08558e10i	− 3.17128i
$868$	− 3.75300e9i	− 0.194787i
$869$	9.50467e7	0.00491324
$870$	0	0
$871$	4.42648e9	0.226984
$872$	− 3.23528e9i	− 0.165236i
$873$	− 1.07424e10i	− 0.546453i
$874$	−3.89004e10	−1.97090
$875$	0	0
$876$	1.25514e10	0.630854
$877$	2.68874e10i	1.34601i	0.739636	+	0.673007i	$0.234998\pi$
$877$	2.68874e10i	1.34601i	−0.739636	+	0.673007i	$0.765002\pi$
$878$	3.67124e9i	0.183055i
$879$	1.67255e10	0.830648
$880$	0	0
$881$	1.08918e10	0.536644	0.268322	−	0.963329i	$-0.413531\pi$
$881$	1.08918e10	0.536644	0.268322	+	0.963329i	$0.413531\pi$
$882$	2.04145e9i	0.100184i
$883$	3.99542e10i	1.95299i	0.215542	+	0.976495i	$0.430848\pi$
$883$	3.99542e10i	1.95299i	−0.215542	+	0.976495i	$0.569152\pi$
$884$	1.04004e10	0.506371
$885$	0	0
$886$	1.10767e10	0.535048
$887$	1.30306e10i	0.626946i	0.949597	+	0.313473i	$0.101493\pi$
$887$	1.30306e10i	0.626946i	−0.949597	+	0.313473i	$0.898507\pi$
$888$	7.08078e8i	0.0339340i
$889$	2.06049e9	0.0983589
$890$	0	0
$891$	−1.92880e8	−0.00913516
$892$	− 4.38309e9i	− 0.206778i
$893$	3.25272e10i	1.52850i
$894$	−1.44005e10	−0.674056
$895$	0	0
$896$	−7.19323e8	−0.0334077
$897$	− 3.09111e10i	− 1.43002i
$898$	2.18889e10i	1.00869i
$899$	−2.15986e10	−0.991439
$900$	0	0
$901$	5.85229e10	2.66556
$902$	1.01916e8i	0.00462400i
$903$	− 1.75997e9i	− 0.0795422i
$904$	5.23713e9	0.235778
$905$	0	0
$906$	−3.42630e9	−0.153065
$907$	4.04015e10i	1.79792i	0.438026	+	0.898962i	$0.355678\pi$
$907$	4.04015e10i	1.79792i	−0.438026	+	0.898962i	$0.644322\pi$
$908$	1.23921e9i	0.0549344i
$909$	−7.00587e9	−0.309377
$910$	0	0
$911$	1.98919e10	0.871690	0.435845	−	0.900022i	$-0.356450\pi$
$911$	1.98919e10	0.871690	0.435845	+	0.900022i	$0.356450\pi$
$912$	1.24955e10i	0.545470i
$913$	8.48943e7i	0.00369174i
$914$	−1.94443e10	−0.842325
$915$	0	0
$916$	1.97220e10	0.847847
$917$	− 7.07939e9i	− 0.303182i
$918$	− 3.46915e8i	− 0.0148004i
$919$	4.10990e10	1.74674	0.873368	−	0.487061i	$-0.161931\pi$
$919$	4.10990e10	1.74674	0.873368	+	0.487061i	$0.161931\pi$
$920$	0	0
$921$	−5.75986e10	−2.42942
$922$	3.15908e10i	1.32740i
$923$	1.48163e8i	0.00620201i
$924$	−5.79533e7	−0.00241672
$925$	0	0
$926$	2.05962e10	0.852412
$927$	− 3.90223e9i	− 0.160892i
$928$	4.13971e9i	0.170040i
$929$	1.90374e10	0.779027	0.389513	−	0.921021i	$-0.372643\pi$
$929$	1.90374e10	0.779027	0.389513	+	0.921021i	$0.372643\pi$
$930$	0	0
$931$	−5.43797e9	−0.220858
$932$	2.27710e9i	0.0921355i
$933$	− 4.43264e10i	− 1.78680i
$934$	−2.38786e9	−0.0958947
$935$	0	0
$936$	−4.94407e9	−0.197069
$937$	3.93830e10i	1.56394i	0.623315	+	0.781971i	$0.285786\pi$
$937$	3.93830e10i	1.56394i	−0.623315	+	0.781971i	$0.714214\pi$
$938$	− 2.72827e9i	− 0.107939i
$939$	−1.26863e10	−0.500040
$940$	0	0
$941$	−1.59186e10	−0.622791	−0.311395	−	0.950280i	$-0.600796\pi$
$941$	−1.59186e10	−0.622791	−0.311395	+	0.950280i	$0.600796\pi$
$942$	3.33068e10i	1.29824i
$943$	3.35047e10i	1.30111i
$944$	4.80008e9	0.185715
$945$	0	0
$946$	2.48781e7	0.000955428 0
$947$	3.57237e9i	0.136688i	0.997662	+	0.0683442i	$0.0217716\pi$
$947$	3.57237e9i	0.136688i	−0.997662	+	0.0683442i	$0.978228\pi$
$948$	− 1.00369e10i	− 0.382624i
$949$	−1.32289e10	−0.502450
$950$	0	0
$951$	8.83326e9	0.333034
$952$	− 6.41034e9i	− 0.240797i
$953$	− 4.05101e9i	− 0.151614i	−0.997123	−	0.0758068i	$-0.975847\pi$
$953$	− 4.05101e9i	− 0.151614i	0.997123	−	0.0758068i	$-0.0241532\pi$
$954$	−2.78201e10	−1.03738
$955$	0	0
$956$	−1.47236e10	−0.545018
$957$	3.33522e8i	0.0123008i
$958$	− 2.09600e10i	− 0.770216i
$959$	1.63336e9	0.0598022
$960$	0	0
$961$	1.71608e9	0.0623741
$962$	− 7.46298e8i	− 0.0270271i
$963$	3.39294e10i	1.22429i
$964$	−3.22237e8	−0.0115853
$965$	0	0
$966$	−1.90521e10	−0.680023
$967$	2.55791e10i	0.909689i	0.890571	+	0.454844i	$0.150305\pi$
$967$	2.55791e10i	0.909689i	−0.890571	+	0.454844i	$0.849695\pi$
$968$	9.97661e9i	0.353524i
$969$	−1.11355e11	−3.93166
$970$	0	0
$971$	4.10323e10	1.43833	0.719165	−	0.694840i	$-0.244525\pi$
$971$	4.10323e10	1.43833	0.719165	+	0.694840i	$0.244525\pi$
$972$	2.02019e10i	0.705603i
$973$	− 1.73463e9i	− 0.0603688i
$974$	3.33330e10	1.15589
$975$	0	0
$976$	−8.47410e9	−0.291756
$977$	4.87277e10i	1.67165i	0.548998	+	0.835824i	$0.315010\pi$
$977$	4.87277e10i	1.67165i	−0.548998	+	0.835824i	$0.684990\pi$
$978$	4.39439e10i	1.50215i
$979$	−2.76814e8	−0.00942863
$980$	0	0
$981$	−1.37057e10	−0.463511
$982$	1.93040e10i	0.650515i
$983$	6.94762e9i	0.233291i	0.993174	+	0.116646i	$0.0372142\pi$
$983$	6.94762e9i	0.233291i	−0.993174	+	0.116646i	$0.962786\pi$
$984$	1.07623e10	0.360099
$985$	0	0
$986$	−3.68915e10	−1.22563
$987$	1.59307e10i	0.527382i
$988$	− 1.31699e10i	− 0.434445i
$989$	8.17867e9	0.268841
$990$	0	0
$991$	−1.83565e10	−0.599144	−0.299572	−	0.954074i	$-0.596844\pi$
$991$	−1.83565e10	−0.599144	−0.299572	+	0.954074i	$0.596844\pi$
$992$	− 5.60215e9i	− 0.182206i
$993$	2.80697e10i	0.909735i
$994$	9.13203e7	0.00294927
$995$	0	0
$996$	8.96484e9	0.287498
$997$	3.44954e10i	1.10237i	0.834383	+	0.551185i	$0.185824\pi$
$997$	3.44954e10i	1.10237i	−0.834383	+	0.551185i	$0.814176\pi$
$998$	8.32735e9i	0.265186i
$999$	−2.48934e7	−0.000789958 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.8.c.b.99.2		2
5.2	odd	4		350.8.a.d.1.1		1
5.3	odd	4		14.8.a.b.1.1	✓	1
5.4	even	2	inner	350.8.c.b.99.1		2
15.8	even	4		126.8.a.c.1.1		1
20.3	even	4		112.8.a.d.1.1		1
35.3	even	12		98.8.c.a.79.1		2
35.13	even	4		98.8.a.c.1.1		1
35.18	odd	12		98.8.c.b.79.1		2
35.23	odd	12		98.8.c.b.67.1		2
35.33	even	12		98.8.c.a.67.1		2
40.3	even	4		448.8.a.b.1.1		1
40.13	odd	4		448.8.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.8.a.b.1.1	✓	1	5.3	odd	4
98.8.a.c.1.1		1	35.13	even	4
98.8.c.a.67.1		2	35.33	even	12
98.8.c.a.79.1		2	35.3	even	12
98.8.c.b.67.1		2	35.23	odd	12
98.8.c.b.79.1		2	35.18	odd	12
112.8.a.d.1.1		1	20.3	even	4
126.8.a.c.1.1		1	15.8	even	4
350.8.a.d.1.1		1	5.2	odd	4
350.8.c.b.99.1		2	5.4	even	2	inner
350.8.c.b.99.2		2	1.1	even	1	trivial
448.8.a.b.1.1		1	40.3	even	4
448.8.a.i.1.1		1	40.13	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.00000i q^{2} +66.0000i q^{3} -64.0000 q^{4} -528.000 q^{6} -343.000i q^{7} -512.000i q^{8} -2169.00 q^{9} +40.0000 q^{11} -4224.00i q^{12} +4452.00i q^{13} +2744.00 q^{14} +4096.00 q^{16} +36502.0i q^{17} -17352.0i q^{18} +46222.0 q^{19} +22638.0 q^{21} +320.000i q^{22} +105200. i q^{23} +33792.0 q^{24} -35616.0 q^{26} +1188.00i q^{27} +21952.0i q^{28} +126334. q^{29} -170964. q^{31} +32768.0i q^{32} +2640.00i q^{33} -292016. q^{34} +138816. q^{36} +20954.0i q^{37} +369776. i q^{38} -293832. q^{39} +318486. q^{41} +181104. i q^{42} -77744.0i q^{43} -2560.00 q^{44} -841600. q^{46} +703716. i q^{47} +270336. i q^{48} -117649. q^{49} -2.40913e6 q^{51} -284928. i q^{52} -1.60328e6i q^{53} -9504.00 q^{54} -175616. q^{56} +3.05065e6i q^{57} +1.01067e6i q^{58} +1.17189e6 q^{59} -2.06887e6 q^{61} -1.36771e6i q^{62} +743967. i q^{63} -262144. q^{64} -21120.0 q^{66} -994268. i q^{67} -2.33613e6i q^{68} -6.94320e6 q^{69} +33280.0 q^{71} +1.11053e6i q^{72} +2.97145e6i q^{73} -167632. q^{74} -2.95821e6 q^{76} -13720.0i q^{77} -2.35066e6i q^{78} +2.37617e6 q^{79} -4.82201e6 q^{81} +2.54789e6i q^{82} +2.12236e6i q^{83} -1.44883e6 q^{84} +621952. q^{86} +8.33804e6i q^{87} -20480.0i q^{88} -6.92035e6 q^{89} +1.52704e6 q^{91} -6.73280e6i q^{92} -1.12836e7i q^{93} -5.62973e6 q^{94} -2.16269e6 q^{96} +4.95271e6i q^{97} -941192. i q^{98} -86760.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 128 q^{4} - 1056 q^{6} - 4338 q^{9} + 80 q^{11} + 5488 q^{14} + 8192 q^{16} + 92444 q^{19} + 45276 q^{21} + 67584 q^{24} - 71232 q^{26} + 252668 q^{29} - 341928 q^{31} - 584032 q^{34} + 277632 q^{36}+ \cdots - 173520 q^{99}+O(q^{100}) \) 2 * q - 128 * q^4 - 1056 * q^6 - 4338 * q^9 + 80 * q^11 + 5488 * q^14 + 8192 * q^16 + 92444 * q^19 + 45276 * q^21 + 67584 * q^24 - 71232 * q^26 + 252668 * q^29 - 341928 * q^31 - 584032 * q^34 + 277632 * q^36 - 587664 * q^39 + 636972 * q^41 - 5120 * q^44 - 1683200 * q^46 - 235298 * q^49 - 4818264 * q^51 - 19008 * q^54 - 351232 * q^56 + 2343788 * q^59 - 4137744 * q^61 - 524288 * q^64 - 42240 * q^66 - 13886400 * q^69 + 66560 * q^71 - 335264 * q^74 - 5916416 * q^76 + 4752336 * q^79 - 9644022 * q^81 - 2897664 * q^84 + 1243904 * q^86 - 13840692 * q^89 + 3054072 * q^91 - 11259456 * q^94 - 4325376 * q^96 - 173520 * q^99

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