LMFDB - Embedded newform 350.8.c.i.99.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,8,Mod(99,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 8, names="a")

//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);

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

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

Self dual:	no
Analytic conductor:	$109.334758919$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{8761})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4381x^{2} + 4796100 $ x^4 + 4381*x^2 + 4796100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.1
Root			$-47.3001i$ of defining polynomial
Character	$\chi$	$=$	350.99
Dual form			350.8.c.i.99.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000i q^{2} -49.3001i q^{3} -64.0000 q^{4} -394.401 q^{6} -343.000i q^{7} +512.000i q^{8} -243.501 q^{9} +O(q^{10})$	\(q-8.00000i q^{2} -49.3001i q^{3} -64.0000 q^{4} -394.401 q^{6} -343.000i q^{7} +512.000i q^{8} -243.501 q^{9} -3779.71 q^{11} +3155.21i q^{12} +6860.31i q^{13} -2744.00 q^{14} +4096.00 q^{16} -15256.5i q^{17} +1948.00i q^{18} -21008.6 q^{19} -16909.9 q^{21} +30237.7i q^{22} +92484.8i q^{23} +25241.7 q^{24} +54882.4 q^{26} -95814.7i q^{27} +21952.0i q^{28} -78621.5 q^{29} +152373. q^{31} -32768.0i q^{32} +186340. i q^{33} -122052. q^{34} +15584.0 q^{36} +445001. i q^{37} +168069. i q^{38} +338214. q^{39} +384839. q^{41} +135279. i q^{42} -264539. i q^{43} +241902. q^{44} +739878. q^{46} +225584. i q^{47} -201933. i q^{48} -117649. q^{49} -752147. q^{51} -439060. i q^{52} +263123. i q^{53} -766518. q^{54} +175616. q^{56} +1.03573e6i q^{57} +628972. i q^{58} -943718. q^{59} +2.40381e6 q^{61} -1.21899e6i q^{62} +83520.7i q^{63} -262144. q^{64} +1.49072e6 q^{66} +4.18860e6i q^{67} +976415. i q^{68} +4.55951e6 q^{69} +5.10301e6 q^{71} -124672. i q^{72} -3.16576e6i q^{73} +3.56001e6 q^{74} +1.34455e6 q^{76} +1.29644e6i q^{77} -2.70571e6i q^{78} +5.00648e6 q^{79} -5.25621e6 q^{81} -3.07871e6i q^{82} +549329. i q^{83} +1.08224e6 q^{84} -2.11631e6 q^{86} +3.87605e6i q^{87} -1.93521e6i q^{88} +3.34978e6 q^{89} +2.35308e6 q^{91} -5.91902e6i q^{92} -7.51202e6i q^{93} +1.80467e6 q^{94} -1.61547e6 q^{96} -1.54347e6i q^{97} +941192. i q^{98} +920362. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 256 q^{4} - 80 q^{6} - 38 q^{9}+O(q^{10}) $ 4 * q - 256 * q^4 - 80 * q^6 - 38 * q^9	$ 4 q - 256 q^{4} - 80 q^{6} - 38 q^{9} + 9030 q^{11} - 10976 q^{14} + 16384 q^{16} - 87404 q^{19} - 3430 q^{21} + 5120 q^{24} + 143152 q^{26} - 18522 q^{29} + 414056 q^{31} - 600528 q^{34} + 2432 q^{36} + 491546 q^{39} + 529596 q^{41} - 577920 q^{44} + 650208 q^{46} - 470596 q^{49} + 469410 q^{51} + 176240 q^{54} + 702464 q^{56} - 3810816 q^{59} - 238636 q^{61} - 1048576 q^{64} + 8860752 q^{66} + 13712652 q^{69} - 1225344 q^{71} + 2289152 q^{74} + 5593856 q^{76} + 25241518 q^{79} - 18995596 q^{81} + 219520 q^{84} - 8402336 q^{86} + 22356660 q^{89} + 6137642 q^{91} + 8121744 q^{94} - 327680 q^{96} + 5565060 q^{99}+O(q^{100}) $ 4 * q - 256 * q^4 - 80 * q^6 - 38 * q^9 + 9030 * q^11 - 10976 * q^14 + 16384 * q^16 - 87404 * q^19 - 3430 * q^21 + 5120 * q^24 + 143152 * q^26 - 18522 * q^29 + 414056 * q^31 - 600528 * q^34 + 2432 * q^36 + 491546 * q^39 + 529596 * q^41 - 577920 * q^44 + 650208 * q^46 - 470596 * q^49 + 469410 * q^51 + 176240 * q^54 + 702464 * q^56 - 3810816 * q^59 - 238636 * q^61 - 1048576 * q^64 + 8860752 * q^66 + 13712652 * q^69 - 1225344 * q^71 + 2289152 * q^74 + 5593856 * q^76 + 25241518 * q^79 - 18995596 * q^81 + 219520 * q^84 - 8402336 * q^86 + 22356660 * q^89 + 6137642 * q^91 + 8121744 * q^94 - 327680 * q^96 + 5565060 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	− 8.00000i	− 0.707107i
$2$	− 8.00000i	− 0.707107i
$3$	− 49.3001i	− 1.05420i	−0.849803	−	0.527101i	$-0.823279\pi$
$3$	− 49.3001i	− 1.05420i	0.849803	−	0.527101i	$-0.176721\pi$
$4$	−64.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−394.401	−0.745433
$7$	− 343.000i	− 0.377964i
$7$	− 343.000i	− 0.377964i
$8$	512.000i	0.353553i
$9$	−243.501	−0.111340
$10$	0	0
$11$	−3779.71	−0.856218	−0.428109	−	0.903727i	$-0.640820\pi$
$11$	−3779.71	−0.856218	−0.428109	+	0.903727i	$0.640820\pi$
$12$	3155.21i	0.527101i
$13$	6860.31i	0.866048i	0.901383	+	0.433024i	$0.142553\pi$
$13$	6860.31i	0.866048i	−0.901383	+	0.433024i	$0.857447\pi$
$14$	−2744.00	−0.267261
$15$	0	0
$16$	4096.00	0.250000
$17$	− 15256.5i	− 0.753153i	−0.926385	−	0.376577i	$-0.877101\pi$
$17$	− 15256.5i	− 0.753153i	0.926385	−	0.376577i	$-0.122899\pi$
$18$	1948.00i	0.0787293i
$19$	−21008.6	−0.702683	−0.351342	−	0.936247i	$-0.614274\pi$
$19$	−21008.6	−0.702683	−0.351342	+	0.936247i	$0.614274\pi$
$20$	0	0
$21$	−16909.9	−0.398451
$22$	30237.7i	0.605438i
$23$	92484.8i	1.58498i	0.609887	+	0.792488i	$0.291215\pi$
$23$	92484.8i	1.58498i	−0.609887	+	0.792488i	$0.708785\pi$
$24$	25241.7	0.372716
$25$	0	0
$26$	54882.4	0.612388
$27$	− 95814.7i	− 0.936826i
$28$	21952.0i	0.188982i
$29$	−78621.5	−0.598616	−0.299308	−	0.954157i	$-0.596756\pi$
$29$	−78621.5	−0.598616	−0.299308	+	0.954157i	$0.596756\pi$
$30$	0	0
$31$	152373.	0.918635	0.459317	−	0.888272i	$-0.348094\pi$
$31$	152373.	0.918635	0.459317	+	0.888272i	$0.348094\pi$
$32$	− 32768.0i	− 0.176777i
$33$	186340.i	0.902626i
$34$	−122052.	−0.532560
$35$	0	0
$36$	15584.0	0.0556700
$37$	445001.i	1.44429i	0.691741	+	0.722146i	$0.256844\pi$
$37$	445001.i	1.44429i	−0.691741	+	0.722146i	$0.743156\pi$
$38$	168069.i	0.496872i
$39$	338214.	0.912988
$40$	0	0
$41$	384839.	0.872038	0.436019	−	0.899938i	$-0.356388\pi$
$41$	384839.	0.872038	0.436019	+	0.899938i	$0.356388\pi$
$42$	135279.i	0.281747i
$43$	− 264539.i	− 0.507399i	−0.967283	−	0.253699i	$-0.918353\pi$
$43$	− 264539.i	− 0.507399i	0.967283	−	0.253699i	$-0.0816474\pi$
$44$	241902.	0.428109
$45$	0	0
$46$	739878.	1.12075
$47$	225584.i	0.316932i	0.987364	+	0.158466i	$0.0506548\pi$
$47$	225584.i	0.316932i	−0.987364	+	0.158466i	$0.949345\pi$
$48$	− 201933.i	− 0.263550i
$49$	−117649.	−0.142857
$50$	0	0
$51$	−752147.	−0.793975
$52$	− 439060.i	− 0.433024i
$53$	263123.i	0.242769i	0.992606	+	0.121385i	$0.0387334\pi$
$53$	263123.i	0.242769i	−0.992606	+	0.121385i	$0.961267\pi$
$54$	−766518.	−0.662436
$55$	0	0
$56$	175616.	0.133631
$57$	1.03573e6i	0.740769i
$58$	628972.i	0.423285i
$59$	−943718.	−0.598219	−0.299110	−	0.954219i	$-0.596690\pi$
$59$	−943718.	−0.598219	−0.299110	+	0.954219i	$0.596690\pi$
$60$	0	0
$61$	2.40381e6	1.35595	0.677977	−	0.735083i	$-0.262857\pi$
$61$	2.40381e6	1.35595	0.677977	+	0.735083i	$0.262857\pi$
$62$	− 1.21899e6i	− 0.649573i
$63$	83520.7i	0.0420826i
$64$	−262144.	−0.125000
$65$	0	0
$66$	1.49072e6	0.638253
$67$	4.18860e6i	1.70140i	0.525651	+	0.850701i	$0.323822\pi$
$67$	4.18860e6i	1.70140i	−0.525651	+	0.850701i	$0.676178\pi$
$68$	976415.i	0.376577i
$69$	4.55951e6	1.67088
$70$	0	0
$71$	5.10301e6	1.69208	0.846042	−	0.533116i	$-0.178979\pi$
$71$	5.10301e6	1.69208	0.846042	+	0.533116i	$0.178979\pi$
$72$	− 124672.i	− 0.0393646i
$73$	− 3.16576e6i	− 0.952463i	−0.879320	−	0.476231i	$-0.842002\pi$
$73$	− 3.16576e6i	− 0.952463i	0.879320	−	0.476231i	$-0.157998\pi$
$74$	3.56001e6	1.02127
$75$	0	0
$76$	1.34455e6	0.351342
$77$	1.29644e6i	0.323620i
$78$	− 2.70571e6i	− 0.645580i
$79$	5.00648e6	1.14245	0.571226	−	0.820793i	$-0.306468\pi$
$79$	5.00648e6	1.14245	0.571226	+	0.820793i	$0.306468\pi$
$80$	0	0
$81$	−5.25621e6	−1.09894
$82$	− 3.07871e6i	− 0.616624i
$83$	549329.i	0.105453i	0.998609	+	0.0527265i	$0.0167912\pi$
$83$	549329.i	0.105453i	−0.998609	+	0.0527265i	$0.983209\pi$
$84$	1.08224e6	0.199225
$85$	0	0
$86$	−2.11631e6	−0.358785
$87$	3.87605e6i	0.631061i
$88$	− 1.93521e6i	− 0.302719i
$89$	3.34978e6	0.503676	0.251838	−	0.967769i	$-0.418965\pi$
$89$	3.34978e6	0.503676	0.251838	+	0.967769i	$0.418965\pi$
$90$	0	0
$91$	2.35308e6	0.327335
$92$	− 5.91902e6i	− 0.792488i
$93$	− 7.51202e6i	− 0.968426i
$94$	1.80467e6	0.224105
$95$	0	0
$96$	−1.61547e6	−0.186358
$97$	− 1.54347e6i	− 0.171710i	−0.996308	−	0.0858551i	$-0.972638\pi$
$97$	− 1.54347e6i	− 0.171710i	0.996308	−	0.0858551i	$-0.0273622\pi$
$98$	941192.i	0.101015i
$99$	920362.	0.0953313
$100$	0	0
$101$	−5.57655e6	−0.538568	−0.269284	−	0.963061i	$-0.586787\pi$
$101$	−5.57655e6	−0.538568	−0.269284	+	0.963061i	$0.586787\pi$
$102$	6.01717e6i	0.561425i
$103$	1.65280e7i	1.49036i	0.666866	+	0.745178i	$0.267635\pi$
$103$	1.65280e7i	1.49036i	−0.666866	+	0.745178i	$0.732365\pi$
$104$	−3.51248e6	−0.306194
$105$	0	0
$106$	2.10499e6	0.171664
$107$	− 1.57343e7i	− 1.24167i	−0.783943	−	0.620833i	$-0.786795\pi$
$107$	− 1.57343e7i	− 1.24167i	0.783943	−	0.620833i	$-0.213205\pi$
$108$	6.13214e6i	0.468413i
$109$	519023.	0.0383879	0.0191939	−	0.999816i	$-0.493890\pi$
$109$	519023.	0.0383879	0.0191939	+	0.999816i	$0.493890\pi$
$110$	0	0
$111$	2.19386e7	1.52257
$112$	− 1.40493e6i	− 0.0944911i
$113$	1.43175e6i	0.0933454i	0.998910	+	0.0466727i	$0.0148618\pi$
$113$	1.43175e6i	0.0933454i	−0.998910	+	0.0466727i	$0.985138\pi$
$114$	8.28581e6	0.523803
$115$	0	0
$116$	5.03177e6	0.299308
$117$	− 1.67049e6i	− 0.0964257i
$118$	7.54975e6i	0.423005i
$119$	−5.23298e6	−0.284665
$120$	0	0
$121$	−5.20093e6	−0.266890
$122$	− 1.92304e7i	− 0.958804i
$123$	− 1.89726e7i	− 0.919303i
$124$	−9.75189e6	−0.459317
$125$	0	0
$126$	668165.	0.0297569
$127$	− 2.53827e7i	− 1.09958i	−0.835304	−	0.549788i	$-0.814708\pi$
$127$	− 2.53827e7i	− 1.09958i	0.835304	−	0.549788i	$-0.185292\pi$
$128$	2.09715e6i	0.0883883i
$129$	−1.30418e7	−0.534900
$130$	0	0
$131$	4.24420e6	0.164948	0.0824738	−	0.996593i	$-0.473718\pi$
$131$	4.24420e6	0.164948	0.0824738	+	0.996593i	$0.473718\pi$
$132$	− 1.19258e7i	− 0.451313i
$133$	7.20595e6i	0.265589i
$134$	3.35088e7	1.20307
$135$	0	0
$136$	7.81132e6	0.266280
$137$	2.51894e7i	0.836944i	0.908230	+	0.418472i	$0.137434\pi$
$137$	2.51894e7i	0.836944i	−0.908230	+	0.418472i	$0.862566\pi$
$138$	− 3.64761e7i	− 1.18149i
$139$	−5.13160e7	−1.62069	−0.810347	−	0.585951i	$-0.800721\pi$
$139$	−5.13160e7	−1.62069	−0.810347	+	0.585951i	$0.800721\pi$
$140$	0	0
$141$	1.11213e7	0.334110
$142$	− 4.08241e7i	− 1.19648i
$143$	− 2.59300e7i	− 0.741526i
$144$	−997378.	−0.0278350
$145$	0	0
$146$	−2.53261e7	−0.673493
$147$	5.80011e6i	0.150600i
$148$	− 2.84801e7i	− 0.722146i
$149$	−6.56267e7	−1.62528	−0.812641	−	0.582765i	$-0.801971\pi$
$149$	−6.56267e7	−1.62528	−0.812641	+	0.582765i	$0.801971\pi$
$150$	0	0
$151$	6.46652e7	1.52845	0.764226	−	0.644949i	$-0.223121\pi$
$151$	6.46652e7	1.52845	0.764226	+	0.644949i	$0.223121\pi$
$152$	− 1.07564e7i	− 0.248436i
$153$	3.71496e6i	0.0838561i
$154$	1.03715e7	0.228834
$155$	0	0
$156$	−2.16457e7	−0.456494
$157$	6.31679e6i	0.130271i	0.997876	+	0.0651355i	$0.0207480\pi$
$157$	6.31679e6i	0.130271i	−0.997876	+	0.0651355i	$0.979252\pi$
$158$	− 4.00519e7i	− 0.807835i
$159$	1.29720e7	0.255928
$160$	0	0
$161$	3.17223e7	0.599065
$162$	4.20497e7i	0.777070i
$163$	− 9.71596e6i	− 0.175723i	−0.996133	−	0.0878616i	$-0.971997\pi$
$163$	− 9.71596e6i	− 0.175723i	0.996133	−	0.0878616i	$-0.0280033\pi$
$164$	−2.46297e7	−0.436019
$165$	0	0
$166$	4.39463e6	0.0745666
$167$	− 9.48652e7i	− 1.57616i	−0.615575	−	0.788079i	$-0.711076\pi$
$167$	− 9.48652e7i	− 1.57616i	0.615575	−	0.788079i	$-0.288924\pi$
$168$	− 8.65789e6i	− 0.140874i
$169$	1.56847e7	0.249962
$170$	0	0
$171$	5.11560e6	0.0782367
$172$	1.69305e7i	0.253699i
$173$	− 1.12549e8i	− 1.65265i	−0.563191	−	0.826327i	$-0.690426\pi$
$173$	− 1.12549e8i	− 1.65265i	0.563191	−	0.826327i	$-0.309574\pi$
$174$	3.10084e7	0.446228
$175$	0	0
$176$	−1.54817e7	−0.214055
$177$	4.65254e7i	0.630643i
$178$	− 2.67982e7i	− 0.356153i
$179$	3.58753e7	0.467530	0.233765	−	0.972293i	$-0.424895\pi$
$179$	3.58753e7	0.467530	0.233765	+	0.972293i	$0.424895\pi$
$180$	0	0
$181$	−606442.	−0.00760176	−0.00380088	−	0.999993i	$-0.501210\pi$
$181$	−606442.	−0.00760176	−0.00380088	+	0.999993i	$0.501210\pi$
$182$	− 1.88247e7i	− 0.231461i
$183$	− 1.18508e8i	− 1.42945i
$184$	−4.73522e7	−0.560374
$185$	0	0
$186$	−6.00962e7	−0.684781
$187$	5.76652e7i	0.644864i
$188$	− 1.44374e7i	− 0.158466i
$189$	−3.28645e7	−0.354087
$190$	0	0
$191$	9.62899e7	0.999917	0.499959	−	0.866049i	$-0.333349\pi$
$191$	9.62899e7	0.999917	0.499959	+	0.866049i	$0.333349\pi$
$192$	1.29237e7i	0.131775i
$193$	8.59534e6i	0.0860622i	0.999074	+	0.0430311i	$0.0137014\pi$
$193$	8.59534e6i	0.0860622i	−0.999074	+	0.0430311i	$0.986299\pi$
$194$	−1.23477e7	−0.121417
$195$	0	0
$196$	7.52954e6	0.0714286
$197$	1.29115e8i	1.20322i	0.798789	+	0.601611i	$0.205474\pi$
$197$	1.29115e8i	1.20322i	−0.798789	+	0.601611i	$0.794526\pi$
$198$	− 7.36290e6i	− 0.0674094i
$199$	6.91654e7	0.622162	0.311081	−	0.950383i	$-0.399309\pi$
$199$	6.91654e7	0.622162	0.311081	+	0.950383i	$0.399309\pi$
$200$	0	0
$201$	2.06498e8	1.79362
$202$	4.46124e7i	0.380825i
$203$	2.69672e7i	0.226255i
$204$	4.81374e7	0.396988
$205$	0	0
$206$	1.32224e8	1.05384
$207$	− 2.25201e7i	− 0.176471i
$208$	2.80998e7i	0.216512i
$209$	7.94065e7	0.601650
$210$	0	0
$211$	−1.05545e8	−0.773481	−0.386741	−	0.922188i	$-0.626399\pi$
$211$	−1.05545e8	−0.773481	−0.386741	+	0.922188i	$0.626399\pi$
$212$	− 1.68399e7i	− 0.121385i
$213$	− 2.51579e8i	− 1.78380i
$214$	−1.25875e8	−0.877990
$215$	0	0
$216$	4.90571e7	0.331218
$217$	− 5.22640e7i	− 0.347211i
$218$	− 4.15218e6i	− 0.0271443i
$219$	−1.56072e8	−1.00409
$220$	0	0
$221$	1.04664e8	0.652267
$222$	− 1.75509e8i	− 1.07662i
$223$	− 1.97859e8i	− 1.19478i	−0.801950	−	0.597392i	$-0.796204\pi$
$223$	− 1.97859e8i	− 1.19478i	0.801950	−	0.597392i	$-0.203796\pi$
$224$	−1.12394e7	−0.0668153
$225$	0	0
$226$	1.14540e7	0.0660052
$227$	1.27156e8i	0.721519i	0.932659	+	0.360759i	$0.117482\pi$
$227$	1.27156e8i	0.721519i	−0.932659	+	0.360759i	$0.882518\pi$
$228$	− 6.62865e7i	− 0.370385i
$229$	−5.90399e7	−0.324879	−0.162439	−	0.986719i	$-0.551936\pi$
$229$	−5.90399e7	−0.324879	−0.162439	+	0.986719i	$0.551936\pi$
$230$	0	0
$231$	6.39147e7	0.341161
$232$	− 4.02542e7i	− 0.211643i
$233$	1.10063e7i	0.0570026i	0.999594	+	0.0285013i	$0.00907347\pi$
$233$	1.10063e7i	0.0570026i	−0.999594	+	0.0285013i	$0.990927\pi$
$234$	−1.33639e7	−0.0681833
$235$	0	0
$236$	6.03980e7	0.299110
$237$	− 2.46820e8i	− 1.20437i
$238$	4.18638e7i	0.201289i
$239$	1.00012e8	0.473872	0.236936	−	0.971525i	$-0.423857\pi$
$239$	1.00012e8	0.473872	0.236936	+	0.971525i	$0.423857\pi$
$240$	0	0
$241$	−3.75862e7	−0.172969	−0.0864845	−	0.996253i	$-0.527563\pi$
$241$	−3.75862e7	−0.172969	−0.0864845	+	0.996253i	$0.527563\pi$
$242$	4.16075e7i	0.188720i
$243$	4.95850e7i	0.221681i
$244$	−1.53844e8	−0.677977
$245$	0	0
$246$	−1.51781e8	−0.650046
$247$	− 1.44125e8i	− 0.608557i
$248$	7.80151e7i	0.324787i
$249$	2.70820e7	0.111169
$250$	0	0
$251$	2.94280e8	1.17464	0.587318	−	0.809356i	$-0.300184\pi$
$251$	2.94280e8	1.17464	0.587318	+	0.809356i	$0.300184\pi$
$252$	− 5.34532e6i	− 0.0210413i
$253$	− 3.49566e8i	− 1.35709i
$254$	−2.03062e8	−0.777518
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	− 6.51717e6i	− 0.0239493i	−0.999928	−	0.0119747i	$-0.996188\pi$
$257$	− 6.51717e6i	− 0.0239493i	0.999928	−	0.0119747i	$-0.00381175\pi$
$258$	1.04334e8i	0.378232i
$259$	1.52635e8	0.545891
$260$	0	0
$261$	1.91444e7	0.0666499
$262$	− 3.39536e7i	− 0.116636i
$263$	3.11456e8i	1.05573i	0.849329	+	0.527863i	$0.177007\pi$
$263$	3.11456e8i	1.05573i	−0.849329	+	0.527863i	$0.822993\pi$
$264$	−9.54062e7	−0.319127
$265$	0	0
$266$	5.76476e7	0.187800
$267$	− 1.65145e8i	− 0.530976i
$268$	− 2.68070e8i	− 0.850701i
$269$	4.12347e8	1.29161	0.645803	−	0.763504i	$-0.276523\pi$
$269$	4.12347e8	1.29161	0.645803	+	0.763504i	$0.276523\pi$
$270$	0	0
$271$	5.01402e8	1.53036	0.765181	−	0.643816i	$-0.222650\pi$
$271$	5.01402e8	1.53036	0.765181	+	0.643816i	$0.222650\pi$
$272$	− 6.24906e7i	− 0.188288i
$273$	− 1.16007e8i	− 0.345077i
$274$	2.01515e8	0.591809
$275$	0	0
$276$	−2.91809e8	−0.835442
$277$	4.03541e8i	1.14080i	0.821368	+	0.570399i	$0.193211\pi$
$277$	4.03541e8i	1.14080i	−0.821368	+	0.570399i	$0.806789\pi$
$278$	4.10528e8i	1.14600i
$279$	−3.71030e7	−0.102281
$280$	0	0
$281$	−2.34291e8	−0.629916	−0.314958	−	0.949106i	$-0.601990\pi$
$281$	−2.34291e8	−0.629916	−0.314958	+	0.949106i	$0.601990\pi$
$282$	− 8.89705e7i	− 0.236251i
$283$	− 6.87212e8i	− 1.80235i	−0.433460	−	0.901173i	$-0.642707\pi$
$283$	− 6.87212e8i	− 1.80235i	0.433460	−	0.901173i	$-0.357293\pi$
$284$	−3.26592e8	−0.846042
$285$	0	0
$286$	−2.07440e8	−0.524338
$287$	− 1.32000e8i	− 0.329599i
$288$	7.97903e6i	0.0196823i
$289$	1.77578e8	0.432760
$290$	0	0
$291$	−7.60930e7	−0.181017
$292$	2.02609e8i	0.476231i
$293$	8.09889e8i	1.88100i	0.339792	+	0.940500i	$0.389643\pi$
$293$	8.09889e8i	1.88100i	−0.339792	+	0.940500i	$0.610357\pi$
$294$	4.64009e7	0.106490
$295$	0	0
$296$	−2.27840e8	−0.510634
$297$	3.62152e8i	0.802128i
$298$	5.25014e8i	1.14925i
$299$	−6.34474e8	−1.37266
$300$	0	0
$301$	−9.07367e7	−0.191779
$302$	− 5.17322e8i	− 1.08078i
$303$	2.74924e8i	0.567759i
$304$	−8.60512e7	−0.175671
$305$	0	0
$306$	2.97197e7	0.0592952
$307$	− 2.76546e8i	− 0.545486i	−0.962087	−	0.272743i	$-0.912069\pi$
$307$	− 2.76546e8i	− 0.545486i	0.962087	−	0.272743i	$-0.0879309\pi$
$308$	− 8.29723e7i	− 0.161810i
$309$	8.14832e8	1.57113
$310$	0	0
$311$	2.03805e8	0.384197	0.192099	−	0.981376i	$-0.438471\pi$
$311$	2.03805e8	0.384197	0.192099	+	0.981376i	$0.438471\pi$
$312$	1.73165e8i	0.322790i
$313$	2.65817e8i	0.489979i	0.969526	+	0.244990i	$0.0787845\pi$
$313$	2.65817e8i	0.489979i	−0.969526	+	0.244990i	$0.921215\pi$
$314$	5.05343e7	0.0921155
$315$	0	0
$316$	−3.20415e8	−0.571226
$317$	− 3.55891e8i	− 0.627494i	−0.949507	−	0.313747i	$-0.898416\pi$
$317$	− 3.55891e8i	− 0.627494i	0.949507	−	0.313747i	$-0.101584\pi$
$318$	− 1.03776e8i	− 0.180968i
$319$	2.97167e8	0.512546
$320$	0	0
$321$	−7.75703e8	−1.30897
$322$	− 2.53778e8i	− 0.423603i
$323$	3.20518e8i	0.529228i
$324$	3.36398e8	0.549472
$325$	0	0
$326$	−7.77277e7	−0.124255
$327$	− 2.55879e7i	− 0.0404685i
$328$	1.97037e8i	0.308312i
$329$	7.73753e7	0.119789
$330$	0	0
$331$	1.53871e8	0.233217	0.116608	−	0.993178i	$-0.462798\pi$
$331$	1.53871e8	0.233217	0.116608	+	0.993178i	$0.462798\pi$
$332$	− 3.51570e7i	− 0.0527265i
$333$	− 1.08358e8i	− 0.160807i
$334$	−7.58922e8	−1.11451
$335$	0	0
$336$	−6.92631e7	−0.0996126
$337$	1.35700e9i	1.93141i	0.259644	+	0.965704i	$0.416395\pi$
$337$	1.35700e9i	1.93141i	−0.259644	+	0.965704i	$0.583605\pi$
$338$	− 1.25478e8i	− 0.176750i
$339$	7.05855e7	0.0984048
$340$	0	0
$341$	−5.75928e8	−0.786552
$342$	− 4.09248e7i	− 0.0553217i
$343$	4.03536e7i	0.0539949i
$344$	1.35444e8	0.179393
$345$	0	0
$346$	−9.00395e8	−1.16860
$347$	1.12588e9i	1.44656i	0.690553	+	0.723281i	$0.257367\pi$
$347$	1.12588e9i	1.44656i	−0.690553	+	0.723281i	$0.742633\pi$
$348$	− 2.48067e8i	− 0.315531i
$349$	1.52729e9	1.92324	0.961620	−	0.274386i	$-0.0884745\pi$
$349$	1.52729e9	1.92324	0.961620	+	0.274386i	$0.0884745\pi$
$350$	0	0
$351$	6.57318e8	0.811336
$352$	1.23854e8i	0.151359i
$353$	7.57091e8i	0.916087i	0.888930	+	0.458043i	$0.151450\pi$
$353$	7.57091e8i	0.916087i	−0.888930	+	0.458043i	$0.848550\pi$
$354$	3.72203e8	0.445932
$355$	0	0
$356$	−2.14386e8	−0.251838
$357$	2.57986e8i	0.300094i
$358$	− 2.87002e8i	− 0.330594i
$359$	−1.39223e9	−1.58811	−0.794055	−	0.607845i	$-0.792034\pi$
$359$	−1.39223e9	−1.58811	−0.794055	+	0.607845i	$0.792034\pi$
$360$	0	0
$361$	−4.52511e8	−0.506237
$362$	4.85154e6i	0.00537526i
$363$	2.56407e8i	0.281356i
$364$	−1.50597e8	−0.163668
$365$	0	0
$366$	−9.48063e8	−1.01077
$367$	− 3.97004e8i	− 0.419241i	−0.977783	−	0.209620i	$-0.932777\pi$
$367$	− 3.97004e8i	− 0.419241i	0.977783	−	0.209620i	$-0.0672228\pi$
$368$	3.78818e8i	0.396244i
$369$	−9.37084e7	−0.0970927
$370$	0	0
$371$	9.02513e7	0.0917581
$372$	4.80769e8i	0.484213i
$373$	1.14961e9i	1.14702i	0.819199	+	0.573509i	$0.194418\pi$
$373$	1.14961e9i	1.14702i	−0.819199	+	0.573509i	$0.805582\pi$
$374$	4.61321e8	0.455988
$375$	0	0
$376$	−1.15499e8	−0.112052
$377$	− 5.39367e8i	− 0.518430i
$378$	2.62916e8i	0.250377i
$379$	5.97062e8	0.563355	0.281677	−	0.959509i	$-0.409109\pi$
$379$	5.97062e8	0.563355	0.281677	+	0.959509i	$0.409109\pi$
$380$	0	0
$381$	−1.25137e9	−1.15917
$382$	− 7.70320e8i	− 0.707048i
$383$	1.56280e9i	1.42138i	0.703507	+	0.710688i	$0.251616\pi$
$383$	1.56280e9i	1.42138i	−0.703507	+	0.710688i	$0.748384\pi$
$384$	1.03390e8	0.0931791
$385$	0	0
$386$	6.87627e7	0.0608551
$387$	6.44153e7i	0.0564938i
$388$	9.87818e7i	0.0858551i
$389$	−1.45209e9	−1.25075	−0.625375	−	0.780324i	$-0.715054\pi$
$389$	−1.45209e9	−1.25075	−0.625375	+	0.780324i	$0.715054\pi$
$390$	0	0
$391$	1.41099e9	1.19373
$392$	− 6.02363e7i	− 0.0505076i
$393$	− 2.09239e8i	− 0.173888i
$394$	1.03292e9	0.850806
$395$	0	0
$396$	−5.89032e7	−0.0476657
$397$	2.41600e9i	1.93789i	0.247272	+	0.968946i	$0.420466\pi$
$397$	2.41600e9i	1.93789i	−0.247272	+	0.968946i	$0.579534\pi$
$398$	− 5.53324e8i	− 0.439935i
$399$	3.55254e8	0.279984
$400$	0	0
$401$	1.96890e9	1.52482	0.762410	−	0.647094i	$-0.224016\pi$
$401$	1.96890e9	1.52482	0.762410	+	0.647094i	$0.224016\pi$
$402$	− 1.65199e9i	− 1.26828i
$403$	1.04533e9i	0.795582i
$404$	3.56899e8	0.269284
$405$	0	0
$406$	2.15737e8	0.159987
$407$	− 1.68198e9i	− 1.23663i
$408$	− 3.85099e8i	− 0.280713i
$409$	−1.19251e9	−0.861851	−0.430925	−	0.902388i	$-0.641813\pi$
$409$	−1.19251e9	−0.861851	−0.430925	+	0.902388i	$0.641813\pi$
$410$	0	0
$411$	1.24184e9	0.882307
$412$	− 1.05779e9i	− 0.745178i
$413$	3.23695e8i	0.226106i
$414$	−1.80161e8	−0.124784
$415$	0	0
$416$	2.24798e8	0.153097
$417$	2.52988e9i	1.70854i
$418$	− 6.35252e8i	− 0.425431i
$419$	5.88605e8	0.390908	0.195454	−	0.980713i	$-0.437382\pi$
$419$	5.88605e8	0.390908	0.195454	+	0.980713i	$0.437382\pi$
$420$	0	0
$421$	1.95464e9	1.27667	0.638335	−	0.769759i	$-0.279624\pi$
$421$	1.95464e9	1.27667	0.638335	+	0.769759i	$0.279624\pi$
$422$	8.44362e8i	0.546934i
$423$	− 5.49298e7i	− 0.0352872i
$424$	−1.34719e8	−0.0858319
$425$	0	0
$426$	−2.01263e9	−1.26134
$427$	− 8.24505e8i	− 0.512502i
$428$	1.00700e9i	0.620833i
$429$	−1.27835e9	−0.781717
$430$	0	0
$431$	3.48899e8	0.209908	0.104954	−	0.994477i	$-0.466530\pi$
$431$	3.48899e8	0.209908	0.104954	+	0.994477i	$0.466530\pi$
$432$	− 3.92457e8i	− 0.234207i
$433$	2.14279e9i	1.26845i	0.773150	+	0.634223i	$0.218680\pi$
$433$	2.14279e9i	1.26845i	−0.773150	+	0.634223i	$0.781320\pi$
$434$	−4.18112e8	−0.245516
$435$	0	0
$436$	−3.32175e7	−0.0191939
$437$	− 1.94298e9i	− 1.11374i
$438$	1.24858e9i	0.709997i
$439$	1.17066e9	0.660394	0.330197	−	0.943912i	$-0.392885\pi$
$439$	1.17066e9	0.660394	0.330197	+	0.943912i	$0.392885\pi$
$440$	0	0
$441$	2.86476e7	0.0159057
$442$	− 8.37314e8i	− 0.461222i
$443$	− 2.47866e9i	− 1.35458i	−0.735717	−	0.677289i	$-0.763155\pi$
$443$	− 2.47866e9i	− 1.35458i	0.735717	−	0.677289i	$-0.236845\pi$
$444$	−1.40407e9	−0.761287
$445$	0	0
$446$	−1.58287e9	−0.844839
$447$	3.23540e9i	1.71337i
$448$	8.99154e7i	0.0472456i
$449$	2.26653e8	0.118168	0.0590840	−	0.998253i	$-0.481182\pi$
$449$	2.26653e8	0.118168	0.0590840	+	0.998253i	$0.481182\pi$
$450$	0	0
$451$	−1.45458e9	−0.746655
$452$	− 9.16321e7i	− 0.0466727i
$453$	− 3.18800e9i	− 1.61130i
$454$	1.01725e9	0.510191
$455$	0	0
$456$	−5.30292e8	−0.261901
$457$	2.71212e9i	1.32924i	0.747182	+	0.664619i	$0.231406\pi$
$457$	2.71212e9i	1.32924i	−0.747182	+	0.664619i	$0.768594\pi$
$458$	4.72319e8i	0.229724i
$459$	−1.46180e9	−0.705574
$460$	0	0
$461$	−1.33992e9	−0.636979	−0.318489	−	0.947926i	$-0.603176\pi$
$461$	−1.33992e9	−0.636979	−0.318489	+	0.947926i	$0.603176\pi$
$462$	− 5.11318e8i	− 0.241237i
$463$	− 2.25758e9i	− 1.05708i	−0.848908	−	0.528541i	$-0.822739\pi$
$463$	− 2.25758e9i	− 1.05708i	0.848908	−	0.528541i	$-0.177261\pi$
$464$	−3.22034e8	−0.149654
$465$	0	0
$466$	8.80501e7	0.0403069
$467$	3.61397e9i	1.64201i	0.570921	+	0.821005i	$0.306586\pi$
$467$	3.61397e9i	1.64201i	−0.570921	+	0.821005i	$0.693414\pi$
$468$	1.06911e8i	0.0482129i
$469$	1.43669e9	0.643069
$470$	0	0
$471$	3.11418e8	0.137332
$472$	− 4.83184e8i	− 0.211502i
$473$	9.99880e8i	0.434444i
$474$	−1.97456e9	−0.851621
$475$	0	0
$476$	3.34911e8	0.142333
$477$	− 6.40706e7i	− 0.0270299i
$478$	− 8.00098e8i	− 0.335078i
$479$	1.57398e9	0.654373	0.327186	−	0.944960i	$-0.393899\pi$
$479$	1.57398e9	0.654373	0.327186	+	0.944960i	$0.393899\pi$
$480$	0	0
$481$	−3.05284e9	−1.25082
$482$	3.00689e8i	0.122308i
$483$	− 1.56391e9i	− 0.631535i
$484$	3.32860e8	0.133445
$485$	0	0
$486$	3.96680e8	0.156752
$487$	4.55268e9i	1.78614i	0.449918	+	0.893070i	$0.351453\pi$
$487$	4.55268e9i	1.78614i	−0.449918	+	0.893070i	$0.648547\pi$
$488$	1.23075e9i	0.479402i
$489$	−4.78998e8	−0.185248
$490$	0	0
$491$	−1.27455e8	−0.0485929	−0.0242964	−	0.999705i	$-0.507735\pi$
$491$	−1.27455e8	−0.0485929	−0.0242964	+	0.999705i	$0.507735\pi$
$492$	1.21425e9i	0.459652i
$493$	1.19949e9i	0.450850i
$494$	−1.15300e9	−0.430315
$495$	0	0
$496$	6.24121e8	0.229659
$497$	− 1.75033e9i	− 0.639548i
$498$	− 2.16656e8i	− 0.0786081i
$499$	2.23201e9	0.804162	0.402081	−	0.915604i	$-0.368287\pi$
$499$	2.23201e9	0.804162	0.402081	+	0.915604i	$0.368287\pi$
$500$	0	0
$501$	−4.67687e9	−1.66159
$502$	− 2.35424e9i	− 0.830593i
$503$	2.83672e9i	0.993868i	0.867788	+	0.496934i	$0.165541\pi$
$503$	2.83672e9i	0.993868i	−0.867788	+	0.496934i	$0.834459\pi$
$504$	−4.27626e7	−0.0148784
$505$	0	0
$506$	−2.79653e9	−0.959605
$507$	− 7.73259e8i	− 0.263510i
$508$	1.62449e9i	0.549788i
$509$	2.96235e9	0.995689	0.497844	−	0.867266i	$-0.334125\pi$
$509$	2.96235e9	0.995689	0.497844	+	0.867266i	$0.334125\pi$
$510$	0	0
$511$	−1.08586e9	−0.359997
$512$	− 1.34218e8i	− 0.0441942i
$513$	2.01293e9i	0.658292i
$514$	−5.21374e7	−0.0169347
$515$	0	0
$516$	8.34674e8	0.267450
$517$	− 8.52643e8i	− 0.271363i
$518$	− 1.22108e9i	− 0.386003i
$519$	−5.54870e9	−1.74223
$520$	0	0
$521$	−2.72874e9	−0.845337	−0.422669	−	0.906284i	$-0.638907\pi$
$521$	−2.72874e9	−0.845337	−0.422669	+	0.906284i	$0.638907\pi$
$522$	− 1.53155e8i	− 0.0471286i
$523$	− 3.32258e9i	− 1.01559i	−0.861477	−	0.507796i	$-0.830460\pi$
$523$	− 3.32258e9i	− 1.01559i	0.861477	−	0.507796i	$-0.169540\pi$
$524$	−2.71629e8	−0.0824738
$525$	0	0
$526$	2.49165e9	0.746511
$527$	− 2.32468e9i	− 0.691873i
$528$	7.63250e8i	0.225657i
$529$	−5.14861e9	−1.51215
$530$	0	0
$531$	2.29796e8	0.0666057
$532$	− 4.61181e8i	− 0.132795i
$533$	2.64011e9i	0.755226i
$534$	−1.32116e9	−0.375457
$535$	0	0
$536$	−2.14456e9	−0.601536
$537$	− 1.76865e9i	− 0.492871i
$538$	− 3.29878e9i	− 0.913303i
$539$	4.44680e8	0.122317
$540$	0	0
$541$	5.68167e9	1.54271	0.771357	−	0.636403i	$-0.219578\pi$
$541$	5.68167e9	1.54271	0.771357	+	0.636403i	$0.219578\pi$
$542$	− 4.01122e9i	− 1.08213i
$543$	2.98977e7i	0.00801378i
$544$	−4.99925e8	−0.133140
$545$	0	0
$546$	−9.28059e8	−0.244006
$547$	− 5.82617e9i	− 1.52205i	−0.648725	−	0.761023i	$-0.724697\pi$
$547$	− 5.82617e9i	− 1.52205i	0.648725	−	0.761023i	$-0.275303\pi$
$548$	− 1.61212e9i	− 0.418472i
$549$	−5.85328e8	−0.150972
$550$	0	0
$551$	1.65173e9	0.420637
$552$	2.33447e9i	0.590747i
$553$	− 1.71722e9i	− 0.431806i
$554$	3.22833e9	0.806665
$555$	0	0
$556$	3.28422e9	0.810347
$557$	− 3.43671e9i	− 0.842655i	−0.906909	−	0.421327i	$-0.861564\pi$
$557$	− 3.43671e9i	− 0.842655i	0.906909	−	0.421327i	$-0.138436\pi$
$558$	2.96824e8i	0.0723234i
$559$	1.81482e9	0.439431
$560$	0	0
$561$	2.84290e9	0.679816
$562$	1.87433e9i	0.445418i
$563$	− 6.13771e9i	− 1.44953i	−0.688996	−	0.724765i	$-0.741948\pi$
$563$	− 6.13771e9i	− 1.44953i	0.688996	−	0.724765i	$-0.258052\pi$
$564$	−7.11764e8	−0.167055
$565$	0	0
$566$	−5.49770e9	−1.27445
$567$	1.80288e9i	0.415362i
$568$	2.61274e9i	0.598242i
$569$	3.61453e8	0.0822544	0.0411272	−	0.999154i	$-0.486905\pi$
$569$	3.61453e8	0.0822544	0.0411272	+	0.999154i	$0.486905\pi$
$570$	0	0
$571$	−1.97784e8	−0.0444596	−0.0222298	−	0.999753i	$-0.507077\pi$
$571$	−1.97784e8	−0.0444596	−0.0222298	+	0.999753i	$0.507077\pi$
$572$	1.65952e9i	0.370763i
$573$	− 4.74710e9i	− 1.05411i
$574$	−1.05600e9	−0.233062
$575$	0	0
$576$	6.38322e7	0.0139175
$577$	3.05290e9i	0.661603i	0.943700	+	0.330801i	$0.107319\pi$
$577$	3.05290e9i	0.661603i	−0.943700	+	0.330801i	$0.892681\pi$
$578$	− 1.42063e9i	− 0.306007i
$579$	4.23751e8	0.0907268
$580$	0	0
$581$	1.88420e8	0.0398575
$582$	6.08744e8i	0.127998i
$583$	− 9.94530e8i	− 0.207863i
$584$	1.62087e9	0.336746
$585$	0	0
$586$	6.47911e9	1.33007
$587$	− 3.97531e9i	− 0.811218i	−0.914047	−	0.405609i	$-0.867060\pi$
$587$	− 3.97531e9i	− 0.811218i	0.914047	−	0.405609i	$-0.132940\pi$
$588$	− 3.71207e8i	− 0.0753001i
$589$	−3.20115e9	−0.645509
$590$	0	0
$591$	6.36539e9	1.26844
$592$	1.82272e9i	0.361073i
$593$	5.28678e8i	0.104112i	0.998644	+	0.0520558i	$0.0165774\pi$
$593$	5.28678e8i	0.104112i	−0.998644	+	0.0520558i	$0.983423\pi$
$594$	2.89722e9	0.567190
$595$	0	0
$596$	4.20011e9	0.812641
$597$	− 3.40986e9i	− 0.655884i
$598$	5.07579e9i	0.970621i
$599$	5.74367e9	1.09193	0.545966	−	0.837807i	$-0.316163\pi$
$599$	5.74367e9	1.09193	0.545966	+	0.837807i	$0.316163\pi$
$600$	0	0
$601$	−4.45817e8	−0.0837715	−0.0418858	−	0.999122i	$-0.513337\pi$
$601$	−4.45817e8	−0.0837715	−0.0418858	+	0.999122i	$0.513337\pi$
$602$	7.25894e8i	0.135608i
$603$	− 1.01993e9i	− 0.189434i
$604$	−4.13858e9	−0.764226
$605$	0	0
$606$	2.19939e9	0.401466
$607$	− 3.19098e9i	− 0.579113i	−0.957161	−	0.289556i	$-0.906492\pi$
$607$	− 3.19098e9i	− 0.579113i	0.957161	−	0.289556i	$-0.0935078\pi$
$608$	6.88410e8i	0.124218i
$609$	1.32948e9	0.238519
$610$	0	0
$611$	−1.54758e9	−0.274478
$612$	− 2.37758e8i	− 0.0419280i
$613$	6.23010e9i	1.09240i	0.837653	+	0.546202i	$0.183927\pi$
$613$	6.23010e9i	1.09240i	−0.837653	+	0.546202i	$0.816073\pi$
$614$	−2.21237e9	−0.385717
$615$	0	0
$616$	−6.63778e8	−0.114417
$617$	− 9.29527e9i	− 1.59318i	−0.604522	−	0.796588i	$-0.706636\pi$
$617$	− 9.29527e9i	− 1.59318i	0.604522	−	0.796588i	$-0.293364\pi$
$618$	− 6.51865e9i	− 1.11096i
$619$	8.70241e9	1.47476	0.737381	−	0.675477i	$-0.236062\pi$
$619$	8.70241e9	1.47476	0.737381	+	0.675477i	$0.236062\pi$
$620$	0	0
$621$	8.86140e9	1.48485
$622$	− 1.63044e9i	− 0.271669i
$623$	− 1.14897e9i	− 0.190372i
$624$	1.38532e9	0.228247
$625$	0	0
$626$	2.12654e9	0.346468
$627$	− 3.91475e9i	− 0.634260i
$628$	− 4.04275e8i	− 0.0651355i
$629$	6.78915e9	1.08777
$630$	0	0
$631$	7.77682e9	1.23225	0.616125	−	0.787648i	$-0.288701\pi$
$631$	7.77682e9	1.23225	0.616125	+	0.787648i	$0.288701\pi$
$632$	2.56332e9i	0.403918i
$633$	5.20339e9i	0.815405i
$634$	−2.84713e9	−0.443705
$635$	0	0
$636$	−8.30208e8	−0.127964
$637$	− 8.07108e8i	− 0.123721i
$638$	− 2.37733e9i	− 0.362425i
$639$	−1.24259e9	−0.188397
$640$	0	0
$641$	3.39730e9	0.509484	0.254742	−	0.967009i	$-0.418009\pi$
$641$	3.39730e9	0.509484	0.254742	+	0.967009i	$0.418009\pi$
$642$	6.20563e9i	0.925578i
$643$	4.82083e9i	0.715128i	0.933889	+	0.357564i	$0.116393\pi$
$643$	4.82083e9i	0.715128i	−0.933889	+	0.357564i	$0.883607\pi$
$644$	−2.03023e9	−0.299532
$645$	0	0
$646$	2.56414e9	0.374221
$647$	1.62271e9i	0.235546i	0.993041	+	0.117773i	$0.0375756\pi$
$647$	1.62271e9i	0.235546i	−0.993041	+	0.117773i	$0.962424\pi$
$648$	− 2.69118e9i	− 0.388535i
$649$	3.56699e9	0.512206
$650$	0	0
$651$	−2.57662e9	−0.366031
$652$	6.21821e8i	0.0878616i
$653$	− 5.88646e9i	− 0.827290i	−0.910438	−	0.413645i	$-0.864255\pi$
$653$	− 5.88646e9i	− 0.827290i	0.910438	−	0.413645i	$-0.135745\pi$
$654$	−2.04703e8	−0.0286156
$655$	0	0
$656$	1.57630e9	0.218009
$657$	7.70864e8i	0.106047i
$658$	− 6.19003e8i	− 0.0847036i
$659$	−4.54439e9	−0.618553	−0.309276	−	0.950972i	$-0.600087\pi$
$659$	−4.54439e9	−0.618553	−0.309276	+	0.950972i	$0.600087\pi$
$660$	0	0
$661$	−6.26700e9	−0.844024	−0.422012	−	0.906590i	$-0.638676\pi$
$661$	−6.26700e9	−0.844024	−0.422012	+	0.906590i	$0.638676\pi$
$662$	− 1.23097e9i	− 0.164909i
$663$	− 5.15996e9i	− 0.687620i
$664$	−2.81256e8	−0.0372833
$665$	0	0
$666$	−8.66864e8	−0.113708
$667$	− 7.27129e9i	− 0.948792i
$668$	6.07138e9i	0.788079i
$669$	−9.75448e9	−1.25954
$670$	0	0
$671$	−9.08569e9	−1.16099
$672$	5.54105e8i	0.0704368i
$673$	− 1.51044e10i	− 1.91007i	−0.296491	−	0.955036i	$-0.595817\pi$
$673$	− 1.51044e10i	− 1.91007i	0.296491	−	0.955036i	$-0.404183\pi$
$674$	1.08560e10	1.36571
$675$	0	0
$676$	−1.00382e9	−0.124981
$677$	1.37101e10i	1.69817i	0.528257	+	0.849084i	$0.322846\pi$
$677$	1.37101e10i	1.69817i	−0.528257	+	0.849084i	$0.677154\pi$
$678$	− 5.64684e8i	− 0.0695827i
$679$	−5.29409e8	−0.0649003
$680$	0	0
$681$	6.26882e9	0.760626
$682$	4.60742e9i	0.556176i
$683$	− 1.24954e9i	− 0.150064i	−0.997181	−	0.0750319i	$-0.976094\pi$
$683$	− 1.24954e9i	− 0.150064i	0.997181	−	0.0750319i	$-0.0239059\pi$
$684$	−3.27399e8	−0.0391184
$685$	0	0
$686$	3.22829e8	0.0381802
$687$	2.91067e9i	0.342487i
$688$	− 1.08355e9i	− 0.126850i
$689$	−1.80511e9	−0.210250
$690$	0	0
$691$	1.42757e10	1.64597	0.822987	−	0.568060i	$-0.192306\pi$
$691$	1.42757e10	1.64597	0.822987	+	0.568060i	$0.192306\pi$
$692$	7.20316e9i	0.826327i
$693$	− 3.15684e8i	− 0.0360319i
$694$	9.00701e9	1.02287
$695$	0	0
$696$	−1.98454e9	−0.223114
$697$	− 5.87129e9i	− 0.656778i
$698$	− 1.22183e10i	− 1.35994i
$699$	5.42610e8	0.0600922
$700$	0	0
$701$	−7.46099e7	−0.00818056	−0.00409028	−	0.999992i	$-0.501302\pi$
$701$	−7.46099e7	−0.00818056	−0.00409028	+	0.999992i	$0.501302\pi$
$702$	− 5.25855e9i	− 0.573701i
$703$	− 9.34884e9i	− 1.01488i
$704$	9.90829e8	0.107027
$705$	0	0
$706$	6.05673e9	0.647771
$707$	1.91276e9i	0.203560i
$708$	− 2.97763e9i	− 0.315322i
$709$	1.95350e9	0.205851	0.102925	−	0.994689i	$-0.467180\pi$
$709$	1.95350e9	0.205851	0.102925	+	0.994689i	$0.467180\pi$
$710$	0	0
$711$	−1.21908e9	−0.127201
$712$	1.71509e9i	0.178076i
$713$	1.40922e10i	1.45601i
$714$	2.06389e9	0.212199
$715$	0	0
$716$	−2.29602e9	−0.233765
$717$	− 4.93062e9i	− 0.499556i
$718$	1.11378e10i	1.12296i
$719$	1.09236e10	1.09602	0.548008	−	0.836473i	$-0.315387\pi$
$719$	1.09236e10	1.09602	0.548008	+	0.836473i	$0.315387\pi$
$720$	0	0
$721$	5.66910e9	0.563301
$722$	3.62008e9i	0.357963i
$723$	1.85300e9i	0.182344i
$724$	3.88123e7	0.00380088
$725$	0	0
$726$	2.05125e9	0.198949
$727$	7.23371e9i	0.698217i	0.937082	+	0.349109i	$0.113516\pi$
$727$	7.23371e9i	0.698217i	−0.937082	+	0.349109i	$0.886484\pi$
$728$	1.20478e9i	0.115730i
$729$	−9.05079e9	−0.865247
$730$	0	0
$731$	−4.03593e9	−0.382149
$732$	7.58450e9i	0.714724i
$733$	3.35715e9i	0.314852i	0.987531	+	0.157426i	$0.0503196\pi$
$733$	3.35715e9i	0.314852i	−0.987531	+	0.157426i	$0.949680\pi$
$734$	−3.17603e9	−0.296448
$735$	0	0
$736$	3.03054e9	0.280187
$737$	− 1.58317e10i	− 1.45677i
$738$	7.49668e8i	0.0686549i
$739$	1.94877e8	0.0177625	0.00888125	−	0.999961i	$-0.497173\pi$
$739$	1.94877e8	0.0177625	0.00888125	+	0.999961i	$0.497173\pi$
$740$	0	0
$741$	−7.10540e9	−0.641541
$742$	− 7.22010e8i	− 0.0648828i
$743$	− 1.29827e10i	− 1.16119i	−0.814193	−	0.580595i	$-0.802820\pi$
$743$	− 1.29827e10i	− 1.16119i	0.814193	−	0.580595i	$-0.197180\pi$
$744$	3.84615e9	0.342390
$745$	0	0
$746$	9.19689e9	0.811064
$747$	− 1.33762e8i	− 0.0117411i
$748$	− 3.69057e9i	− 0.322432i
$749$	−5.39687e9	−0.469305
$750$	0	0
$751$	−1.45261e10	−1.25144	−0.625718	−	0.780049i	$-0.715194\pi$
$751$	−1.45261e10	−1.25144	−0.625718	+	0.780049i	$0.715194\pi$
$752$	9.23992e8i	0.0792330i
$753$	− 1.45081e10i	− 1.23830i
$754$	−4.31494e9	−0.366585
$755$	0	0
$756$	2.10332e9	0.177044
$757$	2.35711e10i	1.97490i	0.157947	+	0.987448i	$0.449512\pi$
$757$	2.35711e10i	1.97490i	−0.157947	+	0.987448i	$0.550488\pi$
$758$	− 4.77649e9i	− 0.398352i
$759$	−1.72336e10	−1.43064
$760$	0	0
$761$	−5.21350e9	−0.428828	−0.214414	−	0.976743i	$-0.568784\pi$
$761$	−5.21350e9	−0.428828	−0.214414	+	0.976743i	$0.568784\pi$
$762$	1.00110e10i	0.819660i
$763$	− 1.78025e8i	− 0.0145092i
$764$	−6.16256e9	−0.499959
$765$	0	0
$766$	1.25024e10	1.00506
$767$	− 6.47420e9i	− 0.518086i
$768$	− 8.27119e8i	− 0.0658876i
$769$	−1.34080e10	−1.06321	−0.531607	−	0.846991i	$-0.678412\pi$
$769$	−1.34080e10	−1.06321	−0.531607	+	0.846991i	$0.678412\pi$
$770$	0	0
$771$	−3.21297e8	−0.0252474
$772$	− 5.50101e8i	− 0.0430311i
$773$	7.99869e9i	0.622861i	0.950269	+	0.311430i	$0.100808\pi$
$773$	7.99869e9i	0.622861i	−0.950269	+	0.311430i	$0.899192\pi$
$774$	5.15322e8	0.0399471
$775$	0	0
$776$	7.90255e8	0.0607087
$777$	− 7.52494e9i	− 0.575479i
$778$	1.16167e10i	0.884414i
$779$	−8.08492e9	−0.612766
$780$	0	0
$781$	−1.92879e10	−1.44879
$782$	− 1.12879e10i	− 0.844095i
$783$	7.53309e9i	0.560799i
$784$	−4.81890e8	−0.0357143
$785$	0	0
$786$	−1.67392e9	−0.122957
$787$	7.88328e9i	0.576495i	0.957556	+	0.288248i	$0.0930726\pi$
$787$	7.88328e9i	0.576495i	−0.957556	+	0.288248i	$0.906927\pi$
$788$	− 8.26337e9i	− 0.601611i
$789$	1.53548e10	1.11295
$790$	0	0
$791$	4.91091e8	0.0352813
$792$	4.71226e8i	0.0337047i
$793$	1.64908e10i	1.17432i
$794$	1.93280e10	1.37030
$795$	0	0
$796$	−4.42659e9	−0.311081
$797$	4.95185e9i	0.346468i	0.984881	+	0.173234i	$0.0554217\pi$
$797$	4.95185e9i	0.346468i	−0.984881	+	0.173234i	$0.944578\pi$
$798$	− 2.84203e9i	− 0.197979i
$799$	3.44162e9	0.238698
$800$	0	0
$801$	−8.15673e8	−0.0560793
$802$	− 1.57512e10i	− 1.07821i
$803$	1.19657e10i	0.815516i
$804$	−1.32159e10	−0.896810
$805$	0	0
$806$	8.36262e9	0.562561
$807$	− 2.03288e10i	− 1.36161i
$808$	− 2.85519e9i	− 0.190412i
$809$	2.24582e9	0.149127	0.0745633	−	0.997216i	$-0.476244\pi$
$809$	2.24582e9	0.149127	0.0745633	+	0.997216i	$0.476244\pi$
$810$	0	0
$811$	−6.42965e9	−0.423267	−0.211634	−	0.977349i	$-0.567878\pi$
$811$	−6.42965e9	−0.423267	−0.211634	+	0.977349i	$0.567878\pi$
$812$	− 1.72590e9i	− 0.113128i
$813$	− 2.47192e10i	− 1.61331i
$814$	−1.34558e10	−0.874429
$815$	0	0
$816$	−3.08079e9	−0.198494
$817$	5.55759e9i	0.356541i
$818$	9.54012e9i	0.609421i
$819$	−5.72977e8	−0.0364455
$820$	0	0
$821$	−2.97337e10	−1.87520	−0.937601	−	0.347713i	$-0.886958\pi$
$821$	−2.97337e10	−1.87520	−0.937601	+	0.347713i	$0.886958\pi$
$822$	− 9.93473e9i	− 0.623885i
$823$	1.34850e9i	0.0843244i	0.999111	+	0.0421622i	$0.0134246\pi$
$823$	1.34850e9i	0.0843244i	−0.999111	+	0.0421622i	$0.986575\pi$
$824$	−8.46233e9	−0.526920
$825$	0	0
$826$	2.58956e9	0.159881
$827$	3.87853e9i	0.238450i	0.992867	+	0.119225i	$0.0380410\pi$
$827$	3.87853e9i	0.238450i	−0.992867	+	0.119225i	$0.961959\pi$
$828$	1.44129e9i	0.0882356i
$829$	−1.50571e10	−0.917910	−0.458955	−	0.888460i	$-0.651776\pi$
$829$	−1.50571e10	−0.917910	−0.458955	+	0.888460i	$0.651776\pi$
$830$	0	0
$831$	1.98946e10	1.20263
$832$	− 1.79839e9i	− 0.108256i
$833$	1.79491e9i	0.107593i
$834$	2.02391e10	1.20812
$835$	0	0
$836$	−5.08202e9	−0.300825
$837$	− 1.45996e10i	− 0.860601i
$838$	− 4.70884e9i	− 0.276414i
$839$	2.60321e9	0.152175	0.0760873	−	0.997101i	$-0.475757\pi$
$839$	2.60321e9	0.152175	0.0760873	+	0.997101i	$0.475757\pi$
$840$	0	0
$841$	−1.10685e10	−0.641659
$842$	− 1.56371e10i	− 0.902742i
$843$	1.15506e10i	0.664058i
$844$	6.75489e9	0.386741
$845$	0	0
$846$	−4.39439e8	−0.0249518
$847$	1.78392e9i	0.100875i
$848$	1.07775e9i	0.0606923i
$849$	−3.38796e10	−1.90004
$850$	0	0
$851$	−4.11558e10	−2.28917
$852$	1.61010e10i	0.891899i
$853$	8.00477e9i	0.441598i	0.975319	+	0.220799i	$0.0708665\pi$
$853$	8.00477e9i	0.441598i	−0.975319	+	0.220799i	$0.929133\pi$
$854$	−6.59604e9	−0.362394
$855$	0	0
$856$	8.05597e9	0.438995
$857$	− 1.19065e10i	− 0.646177i	−0.946369	−	0.323089i	$-0.895279\pi$
$857$	− 1.19065e10i	− 0.646177i	0.946369	−	0.323089i	$-0.104721\pi$
$858$	1.02268e10i	0.552758i
$859$	−5.62800e9	−0.302955	−0.151477	−	0.988461i	$-0.548403\pi$
$859$	−5.62800e9	−0.302955	−0.151477	+	0.988461i	$0.548403\pi$
$860$	0	0
$861$	−6.50760e9	−0.347464
$862$	− 2.79119e9i	− 0.148427i
$863$	8.80845e8i	0.0466511i	0.999728	+	0.0233255i	$0.00742543\pi$
$863$	8.80845e8i	0.0466511i	−0.999728	+	0.0233255i	$0.992575\pi$
$864$	−3.13966e9	−0.165609
$865$	0	0
$866$	1.71423e10	0.896926
$867$	− 8.75462e9i	− 0.456216i
$868$	3.34490e9i	0.173606i
$869$	−1.89231e10	−0.978188
$870$	0	0
$871$	−2.87351e10	−1.47349
$872$	2.65740e8i	0.0135722i
$873$	3.75835e8i	0.0191182i
$874$	−1.55438e10	−0.787530
$875$	0	0
$876$	9.98863e9	0.502044
$877$	2.47542e9i	0.123923i	0.998079	+	0.0619613i	$0.0197355\pi$
$877$	2.47542e9i	0.123923i	−0.998079	+	0.0619613i	$0.980264\pi$
$878$	− 9.36524e9i	− 0.466969i
$879$	3.99276e10	1.98295
$880$	0	0
$881$	2.56467e10	1.26362	0.631810	−	0.775123i	$-0.282312\pi$
$881$	2.56467e10	1.26362	0.631810	+	0.775123i	$0.282312\pi$
$882$	− 2.29181e8i	− 0.0112470i
$883$	2.59467e10i	1.26830i	0.773212	+	0.634148i	$0.218649\pi$
$883$	2.59467e10i	1.26830i	−0.773212	+	0.634148i	$0.781351\pi$
$884$	−6.69851e9	−0.326133
$885$	0	0
$886$	−1.98293e10	−0.957831
$887$	− 5.99555e9i	− 0.288467i	−0.989544	−	0.144234i	$-0.953928\pi$
$887$	− 5.99555e9i	− 0.288467i	0.989544	−	0.144234i	$-0.0460717\pi$
$888$	1.12326e10i	0.538311i
$889$	−8.70628e9	−0.415601
$890$	0	0
$891$	1.98670e10	0.940935
$892$	1.26630e10i	0.597392i
$893$	− 4.73920e9i	− 0.222703i
$894$	2.58832e10	1.21154
$895$	0	0
$896$	7.19323e8	0.0334077
$897$	3.12796e10i	1.44706i
$898$	− 1.81323e9i	− 0.0835574i
$899$	−1.19798e10	−0.549909
$900$	0	0
$901$	4.01434e9	0.182842
$902$	1.16366e10i	0.527965i
$903$	4.47333e9i	0.202173i
$904$	−7.33057e8	−0.0330026
$905$	0	0
$906$	−2.55040e10	−1.13936
$907$	3.49376e9i	0.155478i	0.996974	+	0.0777388i	$0.0247700\pi$
$907$	3.49376e9i	0.155478i	−0.996974	+	0.0777388i	$0.975230\pi$
$908$	− 8.13801e9i	− 0.360759i
$909$	1.35789e9	0.0599641
$910$	0	0
$911$	−2.09230e10	−0.916874	−0.458437	−	0.888727i	$-0.651590\pi$
$911$	−2.09230e10	−0.916874	−0.458437	+	0.888727i	$0.651590\pi$
$912$	4.24233e9i	0.185192i
$913$	− 2.07631e9i	− 0.0902908i
$914$	2.16970e10	0.939913
$915$	0	0
$916$	3.77855e9	0.162439
$917$	− 1.45576e9i	− 0.0623444i
$918$	1.16944e10i	0.498916i
$919$	3.59835e10	1.52932	0.764662	−	0.644432i	$-0.222906\pi$
$919$	3.59835e10	1.52932	0.764662	+	0.644432i	$0.222906\pi$
$920$	0	0
$921$	−1.36338e10	−0.575052
$922$	1.07193e10i	0.450412i
$923$	3.50082e10i	1.46543i
$924$	−4.09054e9	−0.170580
$925$	0	0
$926$	−1.80606e10	−0.747470
$927$	− 4.02457e9i	− 0.165936i
$928$	2.57627e9i	0.105821i
$929$	−2.05729e10	−0.841861	−0.420931	−	0.907093i	$-0.638296\pi$
$929$	−2.05729e10	−0.841861	−0.420931	+	0.907093i	$0.638296\pi$
$930$	0	0
$931$	2.47164e9	0.100383
$932$	− 7.04401e8i	− 0.0285013i
$933$	− 1.00476e10i	− 0.405021i
$934$	2.89118e10	1.16108
$935$	0	0
$936$	8.55290e8	0.0340916
$937$	1.62021e10i	0.643401i	0.946842	+	0.321700i	$0.104254\pi$
$937$	1.62021e10i	0.643401i	−0.946842	+	0.321700i	$0.895746\pi$
$938$	− 1.14935e10i	− 0.454719i
$939$	1.31048e10	0.516537
$940$	0	0
$941$	−4.55390e10	−1.78164	−0.890820	−	0.454357i	$-0.849869\pi$
$941$	−4.55390e10	−1.78164	−0.890820	+	0.454357i	$0.849869\pi$
$942$	− 2.49135e9i	− 0.0971082i
$943$	3.55917e10i	1.38216i
$944$	−3.86547e9	−0.149555
$945$	0	0
$946$	7.99904e9	0.307198
$947$	3.11115e10i	1.19041i	0.803575	+	0.595204i	$0.202929\pi$
$947$	3.11115e10i	1.19041i	−0.803575	+	0.595204i	$0.797071\pi$
$948$	1.57965e10i	0.602187i
$949$	2.17181e10	0.824878
$950$	0	0
$951$	−1.75455e10	−0.661505
$952$	− 2.67928e9i	− 0.100644i
$953$	− 2.30174e10i	− 0.861452i	−0.902483	−	0.430726i	$-0.858258\pi$
$953$	− 2.30174e10i	− 0.861452i	0.902483	−	0.430726i	$-0.141742\pi$
$954$	−5.12565e8	−0.0191130
$955$	0	0
$956$	−6.40079e9	−0.236936
$957$	− 1.46503e10i	− 0.540326i
$958$	− 1.25919e10i	− 0.462712i
$959$	8.63997e9	0.316335
$960$	0	0
$961$	−4.29499e9	−0.156110
$962$	2.44227e10i	0.884467i
$963$	3.83131e9i	0.138247i
$964$	2.40551e9	0.0864845
$965$	0	0
$966$	−1.25113e10	−0.446562
$967$	4.54077e10i	1.61487i	0.589959	+	0.807433i	$0.299144\pi$
$967$	4.54077e10i	1.61487i	−0.589959	+	0.807433i	$0.700856\pi$
$968$	− 2.66288e9i	− 0.0943599i
$969$	1.58015e10	0.557913
$970$	0	0
$971$	5.50129e10	1.92840	0.964200	−	0.265176i	$-0.0854302\pi$
$971$	5.50129e10	1.92840	0.964200	+	0.265176i	$0.0854302\pi$
$972$	− 3.17344e9i	− 0.110840i
$973$	1.76014e10i	0.612564i
$974$	3.64214e10	1.26299
$975$	0	0
$976$	9.84599e9	0.338988
$977$	− 5.35876e10i	− 1.83837i	−0.393822	−	0.919187i	$-0.628847\pi$
$977$	− 5.35876e10i	− 1.83837i	0.393822	−	0.919187i	$-0.371153\pi$
$978$	3.83198e9i	0.130990i
$979$	−1.26612e10	−0.431257
$980$	0	0
$981$	−1.26382e8	−0.00427410
$982$	1.01964e9i	0.0343604i
$983$	5.14999e10i	1.72929i	0.502379	+	0.864647i	$0.332458\pi$
$983$	5.14999e10i	1.72929i	−0.502379	+	0.864647i	$0.667542\pi$
$984$	9.71397e9	0.325023
$985$	0	0
$986$	9.59590e9	0.318799
$987$	− 3.81461e9i	− 0.126282i
$988$	9.22403e9i	0.304278i
$989$	2.44658e10	0.804215
$990$	0	0
$991$	−3.15247e10	−1.02895	−0.514474	−	0.857506i	$-0.672013\pi$
$991$	−3.15247e10	−1.02895	−0.514474	+	0.857506i	$0.672013\pi$
$992$	− 4.99297e9i	− 0.162393i
$993$	− 7.58588e9i	− 0.245858i
$994$	−1.40027e10	−0.452229
$995$	0	0
$996$	−1.73325e9	−0.0555844
$997$	4.68548e10i	1.49734i	0.662942	+	0.748671i	$0.269308\pi$
$997$	4.68548e10i	1.49734i	−0.662942	+	0.748671i	$0.730692\pi$
$998$	− 1.78560e10i	− 0.568628i
$999$	4.26376e10	1.35305

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.8.c.i.99.1		4
5.2	odd	4		70.8.a.g.1.1	✓	2
5.3	odd	4		350.8.a.l.1.2		2
5.4	even	2	inner	350.8.c.i.99.4		4
20.7	even	4		560.8.a.h.1.2		2
35.27	even	4		490.8.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.8.a.g.1.1	✓	2	5.2	odd	4
350.8.a.l.1.2		2	5.3	odd	4
350.8.c.i.99.1		4	1.1	even	1	trivial
350.8.c.i.99.4		4	5.4	even	2	inner
490.8.a.k.1.2		2	35.27	even	4
560.8.a.h.1.2		2	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-8.00000i q^{2} -49.3001i q^{3} -64.0000 q^{4} -394.401 q^{6} -343.000i q^{7} +512.000i q^{8} -243.501 q^{9} +O(q^{10})\)	\(q-8.00000i q^{2} -49.3001i q^{3} -64.0000 q^{4} -394.401 q^{6} -343.000i q^{7} +512.000i q^{8} -243.501 q^{9} -3779.71 q^{11} +3155.21i q^{12} +6860.31i q^{13} -2744.00 q^{14} +4096.00 q^{16} -15256.5i q^{17} +1948.00i q^{18} -21008.6 q^{19} -16909.9 q^{21} +30237.7i q^{22} +92484.8i q^{23} +25241.7 q^{24} +54882.4 q^{26} -95814.7i q^{27} +21952.0i q^{28} -78621.5 q^{29} +152373. q^{31} -32768.0i q^{32} +186340. i q^{33} -122052. q^{34} +15584.0 q^{36} +445001. i q^{37} +168069. i q^{38} +338214. q^{39} +384839. q^{41} +135279. i q^{42} -264539. i q^{43} +241902. q^{44} +739878. q^{46} +225584. i q^{47} -201933. i q^{48} -117649. q^{49} -752147. q^{51} -439060. i q^{52} +263123. i q^{53} -766518. q^{54} +175616. q^{56} +1.03573e6i q^{57} +628972. i q^{58} -943718. q^{59} +2.40381e6 q^{61} -1.21899e6i q^{62} +83520.7i q^{63} -262144. q^{64} +1.49072e6 q^{66} +4.18860e6i q^{67} +976415. i q^{68} +4.55951e6 q^{69} +5.10301e6 q^{71} -124672. i q^{72} -3.16576e6i q^{73} +3.56001e6 q^{74} +1.34455e6 q^{76} +1.29644e6i q^{77} -2.70571e6i q^{78} +5.00648e6 q^{79} -5.25621e6 q^{81} -3.07871e6i q^{82} +549329. i q^{83} +1.08224e6 q^{84} -2.11631e6 q^{86} +3.87605e6i q^{87} -1.93521e6i q^{88} +3.34978e6 q^{89} +2.35308e6 q^{91} -5.91902e6i q^{92} -7.51202e6i q^{93} +1.80467e6 q^{94} -1.61547e6 q^{96} -1.54347e6i q^{97} +941192. i q^{98} +920362. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 256 q^{4} - 80 q^{6} - 38 q^{9}+O(q^{10}) \) 4 * q - 256 * q^4 - 80 * q^6 - 38 * q^9	\( 4 q - 256 q^{4} - 80 q^{6} - 38 q^{9} + 9030 q^{11} - 10976 q^{14} + 16384 q^{16} - 87404 q^{19} - 3430 q^{21} + 5120 q^{24} + 143152 q^{26} - 18522 q^{29} + 414056 q^{31} - 600528 q^{34} + 2432 q^{36} + 491546 q^{39} + 529596 q^{41} - 577920 q^{44} + 650208 q^{46} - 470596 q^{49} + 469410 q^{51} + 176240 q^{54} + 702464 q^{56} - 3810816 q^{59} - 238636 q^{61} - 1048576 q^{64} + 8860752 q^{66} + 13712652 q^{69} - 1225344 q^{71} + 2289152 q^{74} + 5593856 q^{76} + 25241518 q^{79} - 18995596 q^{81} + 219520 q^{84} - 8402336 q^{86} + 22356660 q^{89} + 6137642 q^{91} + 8121744 q^{94} - 327680 q^{96} + 5565060 q^{99}+O(q^{100}) \) 4 * q - 256 * q^4 - 80 * q^6 - 38 * q^9 + 9030 * q^11 - 10976 * q^14 + 16384 * q^16 - 87404 * q^19 - 3430 * q^21 + 5120 * q^24 + 143152 * q^26 - 18522 * q^29 + 414056 * q^31 - 600528 * q^34 + 2432 * q^36 + 491546 * q^39 + 529596 * q^41 - 577920 * q^44 + 650208 * q^46 - 470596 * q^49 + 469410 * q^51 + 176240 * q^54 + 702464 * q^56 - 3810816 * q^59 - 238636 * q^61 - 1048576 * q^64 + 8860752 * q^66 + 13712652 * q^69 - 1225344 * q^71 + 2289152 * q^74 + 5593856 * q^76 + 25241518 * q^79 - 18995596 * q^81 + 219520 * q^84 - 8402336 * q^86 + 22356660 * q^89 + 6137642 * q^91 + 8121744 * q^94 - 327680 * q^96 + 5565060 * q^99

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