LMFDB - Embedded newform 32.17.c.a.31.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,17,Mod(31,32)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(32, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 17, names="a")

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

chi := DirichletCharacter("32.31");

S:= CuspForms(chi, 17);

N := Newforms(S);

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 17 $
Character orbit:	$[\chi]$	$=$	32.c (of order $2$, degree $1$, 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:	$51.9438540341$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4 x^{7} - 38020 x^{6} - 901434 x^{5} + 424670139 x^{4} + 18462385230 x^{3} + \cdots + 938226905832125 $ x^8 - 4x^7 - 38020x^6 - 901434x^5 + 424670139x^4 + 18462385230x^3 - 925550425910x^2 - 29930384649550*x + 938226905832125
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{90}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.8
Root			$-107.855 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	32.31
Dual form			32.17.c.a.31.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+12434.9i q^{3} -559134. q^{5} -8.71663e6i q^{7} -1.11580e8 q^{9} +O(q^{10})$	$q+12434.9i q^{3} -559134. q^{5} -8.71663e6i q^{7} -1.11580e8 q^{9} +1.47194e8i q^{11} +4.14624e8 q^{13} -6.95277e9i q^{15} -6.31695e8 q^{17} -2.19243e10i q^{19} +1.08390e11 q^{21} +7.67341e10i q^{23} +1.60043e11 q^{25} -8.52197e11i q^{27} +1.23361e11 q^{29} +1.90201e11i q^{31} -1.83033e12 q^{33} +4.87377e12i q^{35} -5.54587e11 q^{37} +5.15579e12i q^{39} -1.32545e13 q^{41} +4.26826e12i q^{43} +6.23880e13 q^{45} -8.72519e12i q^{47} -4.27467e13 q^{49} -7.85505e12i q^{51} +3.66747e13 q^{53} -8.23010e13i q^{55} +2.72626e14 q^{57} +1.78821e14i q^{59} +2.89428e14 q^{61} +9.72597e14i q^{63} -2.31830e14 q^{65} -1.70741e14i q^{67} -9.54179e14 q^{69} +2.91217e14i q^{71} +1.24552e15 q^{73} +1.99012e15i q^{75} +1.28303e15 q^{77} -1.72176e15i q^{79} +5.79384e15 q^{81} -4.17437e14i q^{83} +3.53202e14 q^{85} +1.53398e15i q^{87} -3.35006e15 q^{89} -3.61412e15i q^{91} -2.36512e15 q^{93} +1.22586e16i q^{95} -6.85255e15 q^{97} -1.64238e16i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 1121456 q^{5} - 189143672 q^{9}+O(q^{10}) $ 8 * q - 1121456 * q^5 - 189143672 * q^9	$ 8 q - 1121456 q^{5} - 189143672 q^{9} + 17627216 q^{13} - 7983031920 q^{17} + 131738539264 q^{21} + 159991390488 q^{25} - 191970074800 q^{29} - 295705712512 q^{33} + 2823854227536 q^{37} + 7729120566032 q^{41} + 108153928739408 q^{45} + 35978193324552 q^{49} + 185763363904080 q^{53} + 975883022595456 q^{57} + 15\!\cdots\!92 q^{61}+ \cdots - 16\!\cdots\!52 q^{97}+O(q^{100}) $ 8 * q - 1121456 * q^5 - 189143672 * q^9 + 17627216 * q^13 - 7983031920 * q^17 + 131738539264 * q^21 + 159991390488 * q^25 - 191970074800 * q^29 - 295705712512 * q^33 + 2823854227536 * q^37 + 7729120566032 * q^41 + 108153928739408 * q^45 + 35978193324552 * q^49 + 185763363904080 * q^53 + 975883022595456 * q^57 + 1588295187290192 * q^61 + 1407344840347936 * q^65 + 1721373686220544 * q^69 + 4735684644451728 * q^73 + 7884073041355520 * q^77 + 12832008243781256 * q^81 + 3759414296335776 * q^85 - 4267961369411184 * q^89 - 9651190567144448 * q^93 - 16022975903677552 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$.

$n$	$5$	$31$
$\chi(n)$	$1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	12434.9i	1.89527i	0.319352	+	0.947636i	$0.396535\pi$
$3$	12434.9i	1.89527i	−0.319352	+	0.947636i	$0.603465\pi$
$4$	0	0
$5$	−559134.	−1.43138	−0.715692	−	0.698416i	$-0.753889\pi$
$5$	−559134.	−1.43138	−0.715692	+	0.698416i	$0.753889\pi$
$6$	0	0
$7$	− 8.71663e6i	− 1.51204i	−0.654547	−	0.756022i	$-0.727140\pi$
$7$	− 8.71663e6i	− 1.51204i	0.654547	−	0.756022i	$-0.272860\pi$
$8$	0	0
$9$	−1.11580e8	−2.59206
$10$	0	0
$11$	1.47194e8i	0.686669i	0.939213	+	0.343334i	$0.111556\pi$
$11$	1.47194e8i	0.686669i	−0.939213	+	0.343334i	$0.888444\pi$
$12$	0	0
$13$	4.14624e8	0.508285	0.254142	−	0.967167i	$-0.418207\pi$
$13$	4.14624e8	0.508285	0.254142	+	0.967167i	$0.418207\pi$
$14$	0	0
$15$	− 6.95277e9i	− 2.71286i
$16$	0	0
$17$	−6.31695e8	−0.0905557	−0.0452779	−	0.998974i	$-0.514417\pi$
$17$	−6.31695e8	−0.0905557	−0.0452779	+	0.998974i	$0.514417\pi$
$18$	0	0
$19$	− 2.19243e10i	− 1.29091i	−0.763798	−	0.645455i	$-0.776668\pi$
$19$	− 2.19243e10i	− 1.29091i	0.763798	−	0.645455i	$-0.223332\pi$
$20$	0	0
$21$	1.08390e11	2.86573
$22$	0	0
$23$	7.67341e10i	0.979863i	0.871761	+	0.489932i	$0.162978\pi$
$23$	7.67341e10i	0.979863i	−0.871761	+	0.489932i	$0.837022\pi$
$24$	0	0
$25$	1.60043e11	1.04886
$26$	0	0
$27$	− 8.52197e11i	− 3.01738i
$28$	0	0
$29$	1.23361e11	0.246600	0.123300	−	0.992369i	$-0.460652\pi$
$29$	1.23361e11	0.246600	0.123300	+	0.992369i	$0.460652\pi$
$30$	0	0
$31$	1.90201e11i	0.223007i	0.993764	+	0.111503i	$0.0355666\pi$
$31$	1.90201e11i	0.223007i	−0.993764	+	0.111503i	$0.964433\pi$
$32$	0	0
$33$	−1.83033e12	−1.30142
$34$	0	0
$35$	4.87377e12i	2.16431i
$36$	0	0
$37$	−5.54587e11	−0.157890	−0.0789452	−	0.996879i	$-0.525155\pi$
$37$	−5.54587e11	−0.157890	−0.0789452	+	0.996879i	$0.525155\pi$
$38$	0	0
$39$	5.15579e12i	0.963338i
$40$	0	0
$41$	−1.32545e13	−1.65994	−0.829969	−	0.557810i	$-0.811642\pi$
$41$	−1.32545e13	−1.65994	−0.829969	+	0.557810i	$0.811642\pi$
$42$	0	0
$43$	4.26826e12i	0.365177i	0.983189	+	0.182588i	$0.0584476\pi$
$43$	4.26826e12i	0.365177i	−0.983189	+	0.182588i	$0.941552\pi$
$44$	0	0
$45$	6.23880e13	3.71023
$46$	0	0
$47$	− 8.72519e12i	− 0.366431i	−0.983073	−	0.183215i	$-0.941349\pi$
$47$	− 8.72519e12i	− 0.366431i	0.983073	−	0.183215i	$-0.0586506\pi$
$48$	0	0
$49$	−4.27467e13	−1.28627
$50$	0	0
$51$	− 7.85505e12i	− 0.171628i
$52$	0	0
$53$	3.66747e13	0.589060	0.294530	−	0.955642i	$-0.404837\pi$
$53$	3.66747e13	0.589060	0.294530	+	0.955642i	$0.404837\pi$
$54$	0	0
$55$	− 8.23010e13i	− 0.982887i
$56$	0	0
$57$	2.72626e14	2.44663
$58$	0	0
$59$	1.78821e14i	1.21788i	0.793218	+	0.608938i	$0.208404\pi$
$59$	1.78821e14i	1.21788i	−0.793218	+	0.608938i	$0.791596\pi$
$60$	0	0
$61$	2.89428e14	1.50974	0.754869	−	0.655876i	$-0.227700\pi$
$61$	2.89428e14	1.50974	0.754869	+	0.655876i	$0.227700\pi$
$62$	0	0
$63$	9.72597e14i	3.91930i
$64$	0	0
$65$	−2.31830e14	−0.727551
$66$	0	0
$67$	− 1.70741e14i	− 0.420473i	−0.977651	−	0.210237i	$-0.932577\pi$
$67$	− 1.70741e14i	− 0.420473i	0.977651	−	0.210237i	$-0.0674235\pi$
$68$	0	0
$69$	−9.54179e14	−1.85711
$70$	0	0
$71$	2.91217e14i	0.450973i	0.974246	+	0.225486i	$0.0723971\pi$
$71$	2.91217e14i	0.450973i	−0.974246	+	0.225486i	$0.927603\pi$
$72$	0	0
$73$	1.24552e15	1.54442	0.772212	−	0.635365i	$-0.219150\pi$
$73$	1.24552e15	1.54442	0.772212	+	0.635365i	$0.219150\pi$
$74$	0	0
$75$	1.99012e15i	1.98788i
$76$	0	0
$77$	1.28303e15	1.03827
$78$	0	0
$79$	− 1.72176e15i	− 1.13490i	−0.823409	−	0.567449i	$-0.807931\pi$
$79$	− 1.72176e15i	− 1.13490i	0.823409	−	0.567449i	$-0.192069\pi$
$80$	0	0
$81$	5.79384e15	3.12670
$82$	0	0
$83$	− 4.17437e14i	− 0.185339i	−0.995697	−	0.0926694i	$-0.970460\pi$
$83$	− 4.17437e14i	− 0.185339i	0.995697	−	0.0926694i	$-0.0295400\pi$
$84$	0	0
$85$	3.53202e14	0.129620
$86$	0	0
$87$	1.53398e15i	0.467375i
$88$	0	0
$89$	−3.35006e15	−0.851006	−0.425503	−	0.904957i	$-0.639903\pi$
$89$	−3.35006e15	−0.851006	−0.425503	+	0.904957i	$0.639903\pi$
$90$	0	0
$91$	− 3.61412e15i	− 0.768549i
$92$	0	0
$93$	−2.36512e15	−0.422659
$94$	0	0
$95$	1.22586e16i	1.84779i
$96$	0	0
$97$	−6.85255e15	−0.874336	−0.437168	−	0.899380i	$-0.644019\pi$
$97$	−6.85255e15	−0.874336	−0.437168	+	0.899380i	$0.644019\pi$
$98$	0	0
$99$	− 1.64238e16i	− 1.77988i
$100$	0	0
$101$	5.38789e15	0.497562	0.248781	−	0.968560i	$-0.419970\pi$
$101$	5.38789e15	0.497562	0.248781	+	0.968560i	$0.419970\pi$
$102$	0	0
$103$	− 8.37646e14i	− 0.0661246i	−0.999453	−	0.0330623i	$-0.989474\pi$
$103$	− 8.37646e14i	− 0.0661246i	0.999453	−	0.0330623i	$-0.0105260\pi$
$104$	0	0
$105$	−6.06047e16	−4.10197
$106$	0	0
$107$	9.09457e15i	0.529312i	0.964343	+	0.264656i	$0.0852585\pi$
$107$	9.09457e15i	0.529312i	−0.964343	+	0.264656i	$0.914742\pi$
$108$	0	0
$109$	1.62120e16	0.813625	0.406813	−	0.913512i	$-0.366640\pi$
$109$	1.62120e16	0.813625	0.406813	+	0.913512i	$0.366640\pi$
$110$	0	0
$111$	− 6.89622e15i	− 0.299245i
$112$	0	0
$113$	4.85879e16	1.82768	0.913841	−	0.406073i	$-0.133102\pi$
$113$	4.85879e16	1.82768	0.913841	+	0.406073i	$0.133102\pi$
$114$	0	0
$115$	− 4.29047e16i	− 1.40256i
$116$	0	0
$117$	−4.62635e16	−1.31750
$118$	0	0
$119$	5.50625e15i	0.136924i
$120$	0	0
$121$	2.42838e16	0.528486
$122$	0	0
$123$	− 1.64818e17i	− 3.14603i
$124$	0	0
$125$	−4.16867e15	−0.0699387
$126$	0	0
$127$	− 1.08250e17i	− 1.59955i	−0.600298	−	0.799777i	$-0.704951\pi$
$127$	− 1.08250e17i	− 1.59955i	0.600298	−	0.799777i	$-0.295049\pi$
$128$	0	0
$129$	−5.30753e16	−0.692110
$130$	0	0
$131$	3.67780e16i	0.424051i	0.977264	+	0.212026i	$0.0680060\pi$
$131$	3.67780e16i	0.424051i	−0.977264	+	0.212026i	$0.931994\pi$
$132$	0	0
$133$	−1.91106e17	−1.95191
$134$	0	0
$135$	4.76493e17i	4.31903i
$136$	0	0
$137$	1.85713e17	1.49650	0.748252	−	0.663415i	$-0.230893\pi$
$137$	1.85713e17	1.49650	0.748252	+	0.663415i	$0.230893\pi$
$138$	0	0
$139$	2.03178e17i	1.45800i	0.684511	+	0.729002i	$0.260016\pi$
$139$	2.03178e17i	1.45800i	−0.684511	+	0.729002i	$0.739984\pi$
$140$	0	0
$141$	1.08497e17	0.694486
$142$	0	0
$143$	6.10299e16i	0.349023i
$144$	0	0
$145$	−6.89754e16	−0.352980
$146$	0	0
$147$	− 5.31550e17i	− 2.43784i
$148$	0	0
$149$	8.57469e16	0.352962	0.176481	−	0.984304i	$-0.443529\pi$
$149$	8.57469e16	0.352962	0.176481	+	0.984304i	$0.443529\pi$
$150$	0	0
$151$	3.45376e17i	1.27784i	0.769274	+	0.638919i	$0.220618\pi$
$151$	3.45376e17i	1.27784i	−0.769274	+	0.638919i	$0.779382\pi$
$152$	0	0
$153$	7.04842e16	0.234726
$154$	0	0
$155$	− 1.06348e17i	− 0.319209i
$156$	0	0
$157$	3.11979e17	0.845139	0.422570	−	0.906330i	$-0.361128\pi$
$157$	3.11979e17	0.845139	0.422570	+	0.906330i	$0.361128\pi$
$158$	0	0
$159$	4.56045e17i	1.11643i
$160$	0	0
$161$	6.68862e17	1.48160
$162$	0	0
$163$	− 5.51492e17i	− 1.10672i	−0.832942	−	0.553360i	$-0.813345\pi$
$163$	− 5.51492e17i	− 1.10672i	0.832942	−	0.553360i	$-0.186655\pi$
$164$	0	0
$165$	1.02340e18	1.86284
$166$	0	0
$167$	− 3.24577e17i	− 0.536520i	−0.963346	−	0.268260i	$-0.913551\pi$
$167$	− 3.24577e17i	− 0.536520i	0.963346	−	0.268260i	$-0.0864487\pi$
$168$	0	0
$169$	−4.93504e17	−0.741646
$170$	0	0
$171$	2.44630e18i	3.34611i
$172$	0	0
$173$	1.38639e16	0.0172790	0.00863948	−	0.999963i	$-0.497250\pi$
$173$	1.38639e16	0.0172790	0.00863948	+	0.999963i	$0.497250\pi$
$174$	0	0
$175$	− 1.39504e18i	− 1.58592i
$176$	0	0
$177$	−2.22362e18	−2.30821
$178$	0	0
$179$	− 1.71129e18i	− 1.62367i	−0.583883	−	0.811837i	$-0.698468\pi$
$179$	− 1.71129e18i	− 1.62367i	0.583883	−	0.811837i	$-0.301532\pi$
$180$	0	0
$181$	2.27514e18	1.97505	0.987527	−	0.157447i	$-0.0503264\pi$
$181$	2.27514e18	1.97505	0.987527	+	0.157447i	$0.0503264\pi$
$182$	0	0
$183$	3.59900e18i	2.86136i
$184$	0	0
$185$	3.10089e17	0.226002
$186$	0	0
$187$	− 9.29814e16i	− 0.0621818i
$188$	0	0
$189$	−7.42829e18	−4.56241
$190$	0	0
$191$	− 8.77404e17i	− 0.495373i	−0.968840	−	0.247687i	$-0.920330\pi$
$191$	− 8.77404e17i	− 0.495373i	0.968840	−	0.247687i	$-0.0796704\pi$
$192$	0	0
$193$	1.09500e18	0.568796	0.284398	−	0.958706i	$-0.408206\pi$
$193$	1.09500e18	0.568796	0.284398	+	0.958706i	$0.408206\pi$
$194$	0	0
$195$	− 2.88278e18i	− 1.37891i
$196$	0	0
$197$	−4.03618e18	−1.77926	−0.889632	−	0.456678i	$-0.849039\pi$
$197$	−4.03618e18	−1.77926	−0.889632	+	0.456678i	$0.849039\pi$
$198$	0	0
$199$	− 2.79040e18i	− 1.13460i	−0.823511	−	0.567300i	$-0.807988\pi$
$199$	− 2.79040e18i	− 1.13460i	0.823511	−	0.567300i	$-0.192012\pi$
$200$	0	0
$201$	2.12314e18	0.796911
$202$	0	0
$203$	− 1.07529e18i	− 0.372870i
$204$	0	0
$205$	7.41104e18	2.37601
$206$	0	0
$207$	− 8.56195e18i	− 2.53986i
$208$	0	0
$209$	3.22711e18	0.886428
$210$	0	0
$211$	3.55424e18i	0.904663i	0.891850	+	0.452331i	$0.149408\pi$
$211$	3.55424e18i	0.904663i	−0.891850	+	0.452331i	$0.850592\pi$
$212$	0	0
$213$	−3.62125e18	−0.854716
$214$	0	0
$215$	− 2.38653e18i	− 0.522709i
$216$	0	0
$217$	1.65791e18	0.337196
$218$	0	0
$219$	1.54878e19i	2.92710i
$220$	0	0
$221$	−2.61916e17	−0.0460281
$222$	0	0
$223$	− 1.04679e19i	− 1.71168i	−0.517244	−	0.855838i	$-0.673042\pi$
$223$	− 1.04679e19i	− 1.71168i	0.517244	−	0.855838i	$-0.326958\pi$
$224$	0	0
$225$	−1.78576e19	−2.71871
$226$	0	0
$227$	− 9.84802e18i	− 1.39682i	−0.715696	−	0.698412i	$-0.753890\pi$
$227$	− 9.84802e18i	− 1.39682i	0.715696	−	0.698412i	$-0.246110\pi$
$228$	0	0
$229$	−6.59128e18	−0.871537	−0.435769	−	0.900059i	$-0.643523\pi$
$229$	−6.59128e18	−0.871537	−0.435769	+	0.900059i	$0.643523\pi$
$230$	0	0
$231$	1.59543e19i	1.96781i
$232$	0	0
$233$	−8.36447e18	−0.962922	−0.481461	−	0.876467i	$-0.659894\pi$
$233$	−8.36447e18	−0.962922	−0.481461	+	0.876467i	$0.659894\pi$
$234$	0	0
$235$	4.87855e18i	0.524503i
$236$	0	0
$237$	2.14099e19	2.15094
$238$	0	0
$239$	5.07789e18i	0.476980i	0.971145	+	0.238490i	$0.0766524\pi$
$239$	5.07789e18i	0.476980i	−0.971145	+	0.238490i	$0.923348\pi$
$240$	0	0
$241$	1.43437e19	1.26045	0.630223	−	0.776415i	$-0.282964\pi$
$241$	1.43437e19	1.26045	0.630223	+	0.776415i	$0.282964\pi$
$242$	0	0
$243$	3.53614e19i	2.90857i
$244$	0	0
$245$	2.39011e19	1.84115
$246$	0	0
$247$	− 9.09031e18i	− 0.656150i
$248$	0	0
$249$	5.19078e18	0.351267
$250$	0	0
$251$	− 1.21698e19i	− 0.772491i	−0.922396	−	0.386246i	$-0.873772\pi$
$251$	− 1.21698e19i	− 0.772491i	0.922396	−	0.386246i	$-0.126228\pi$
$252$	0	0
$253$	−1.12948e19	−0.672842
$254$	0	0
$255$	4.39203e18i	0.245665i
$256$	0	0
$257$	2.99621e19	1.57437	0.787187	−	0.616715i	$-0.211537\pi$
$257$	2.99621e19	1.57437	0.787187	+	0.616715i	$0.211537\pi$
$258$	0	0
$259$	4.83413e18i	0.238737i
$260$	0	0
$261$	−1.37646e19	−0.639202
$262$	0	0
$263$	− 1.63037e19i	− 0.712265i	−0.934436	−	0.356132i	$-0.884095\pi$
$263$	− 1.63037e19i	− 0.712265i	0.934436	−	0.356132i	$-0.115905\pi$
$264$	0	0
$265$	−2.05061e19	−0.843171
$266$	0	0
$267$	− 4.16576e19i	− 1.61289i
$268$	0	0
$269$	4.11698e19	1.50162	0.750812	−	0.660516i	$-0.229663\pi$
$269$	4.11698e19	1.50162	0.750812	+	0.660516i	$0.229663\pi$
$270$	0	0
$271$	1.26214e19i	0.433862i	0.976187	+	0.216931i	$0.0696047\pi$
$271$	1.26214e19i	0.433862i	−0.976187	+	0.216931i	$0.930395\pi$
$272$	0	0
$273$	4.49411e19	1.45661
$274$	0	0
$275$	2.35574e19i	0.720220i
$276$	0	0
$277$	−1.04220e19	−0.300685	−0.150343	−	0.988634i	$-0.548038\pi$
$277$	−1.04220e19	−0.300685	−0.150343	+	0.988634i	$0.548038\pi$
$278$	0	0
$279$	− 2.12225e19i	− 0.578047i
$280$	0	0
$281$	−3.18578e19	−0.819531	−0.409765	−	0.912191i	$-0.634389\pi$
$281$	−3.18578e19	−0.819531	−0.409765	+	0.912191i	$0.634389\pi$
$282$	0	0
$283$	3.86577e19i	0.939602i	0.882772	+	0.469801i	$0.155674\pi$
$283$	3.86577e19i	0.939602i	−0.882772	+	0.469801i	$0.844326\pi$
$284$	0	0
$285$	−1.52434e20	−3.50206
$286$	0	0
$287$	1.15534e20i	2.50990i
$288$	0	0
$289$	−4.82622e19	−0.991800
$290$	0	0
$291$	− 8.52107e19i	− 1.65711i
$292$	0	0
$293$	−3.81861e19	−0.703015	−0.351508	−	0.936185i	$-0.614331\pi$
$293$	−3.81861e19	−0.703015	−0.351508	+	0.936185i	$0.614331\pi$
$294$	0	0
$295$	− 9.99852e19i	− 1.74325i
$296$	0	0
$297$	1.25438e20	2.07194
$298$	0	0
$299$	3.18158e19i	0.498050i
$300$	0	0
$301$	3.72048e19	0.552163
$302$	0	0
$303$	6.69977e19i	0.943016i
$304$	0	0
$305$	−1.61829e20	−2.16101
$306$	0	0
$307$	− 3.65302e19i	− 0.462962i	−0.972839	−	0.231481i	$-0.925643\pi$
$307$	− 3.65302e19i	− 0.462962i	0.972839	−	0.231481i	$-0.0743571\pi$
$308$	0	0
$309$	1.04160e19	0.125324
$310$	0	0
$311$	− 4.96612e19i	− 0.567459i	−0.958904	−	0.283729i	$-0.908428\pi$
$311$	− 4.96612e19i	− 0.567459i	0.958904	−	0.283729i	$-0.0915718\pi$
$312$	0	0
$313$	−1.62800e19	−0.176726	−0.0883630	−	0.996088i	$-0.528164\pi$
$313$	−1.62800e19	−0.176726	−0.0883630	+	0.996088i	$0.528164\pi$
$314$	0	0
$315$	− 5.43813e20i	− 5.61003i
$316$	0	0
$317$	1.98478e20	1.94643	0.973215	−	0.229896i	$-0.0738387\pi$
$317$	1.98478e20	1.94643	0.973215	+	0.229896i	$0.0738387\pi$
$318$	0	0
$319$	1.81579e19i	0.169333i
$320$	0	0
$321$	−1.13090e20	−1.00319
$322$	0	0
$323$	1.38494e19i	0.116899i
$324$	0	0
$325$	6.63578e19	0.533120
$326$	0	0
$327$	2.01594e20i	1.54204i
$328$	0	0
$329$	−7.60542e19	−0.554059
$330$	0	0
$331$	1.02834e20i	0.713695i	0.934163	+	0.356847i	$0.116148\pi$
$331$	1.02834e20i	0.713695i	−0.934163	+	0.356847i	$0.883852\pi$
$332$	0	0
$333$	6.18805e19	0.409261
$334$	0	0
$335$	9.54670e19i	0.601859i
$336$	0	0
$337$	−8.06137e19	−0.484585	−0.242292	−	0.970203i	$-0.577899\pi$
$337$	−8.06137e19	−0.484585	−0.242292	+	0.970203i	$0.577899\pi$
$338$	0	0
$339$	6.04185e20i	3.46395i
$340$	0	0
$341$	−2.79963e19	−0.153132
$342$	0	0
$343$	8.29278e19i	0.432860i
$344$	0	0
$345$	5.33514e20	2.65823
$346$	0	0
$347$	6.01724e19i	0.286261i	0.989704	+	0.143130i	$0.0457168\pi$
$347$	6.01724e19i	0.286261i	−0.989704	+	0.143130i	$0.954283\pi$
$348$	0	0
$349$	−1.54307e20	−0.701105	−0.350552	−	0.936543i	$-0.614006\pi$
$349$	−1.54307e20	−0.701105	−0.350552	+	0.936543i	$0.614006\pi$
$350$	0	0
$351$	− 3.53341e20i	− 1.53369i
$352$	0	0
$353$	−1.77971e20	−0.738160	−0.369080	−	0.929398i	$-0.620327\pi$
$353$	−1.77971e20	−0.738160	−0.369080	+	0.929398i	$0.620327\pi$
$354$	0	0
$355$	− 1.62830e20i	− 0.645516i
$356$	0	0
$357$	−6.84695e19	−0.259509
$358$	0	0
$359$	− 1.85755e19i	− 0.0673262i	−0.999433	−	0.0336631i	$-0.989283\pi$
$359$	− 1.85755e19i	− 0.0673262i	0.999433	−	0.0336631i	$-0.0107173\pi$
$360$	0	0
$361$	−1.92232e20	−0.666450
$362$	0	0
$363$	3.01966e20i	1.00162i
$364$	0	0
$365$	−6.96411e20	−2.21066
$366$	0	0
$367$	6.50326e20i	1.97607i	0.154230	+	0.988035i	$0.450710\pi$
$367$	6.50326e20i	1.97607i	−0.154230	+	0.988035i	$0.549290\pi$
$368$	0	0
$369$	1.47893e21	4.30265
$370$	0	0
$371$	− 3.19679e20i	− 0.890684i
$372$	0	0
$373$	2.68502e20	0.716600	0.358300	−	0.933607i	$-0.383357\pi$
$373$	2.68502e20	0.716600	0.358300	+	0.933607i	$0.383357\pi$
$374$	0	0
$375$	− 5.18369e19i	− 0.132553i
$376$	0	0
$377$	5.11484e19	0.125343
$378$	0	0
$379$	7.54994e19i	0.177349i	0.996061	+	0.0886747i	$0.0282632\pi$
$379$	7.54994e19i	0.177349i	−0.996061	+	0.0886747i	$0.971737\pi$
$380$	0	0
$381$	1.34608e21	3.03159
$382$	0	0
$383$	− 2.67239e20i	− 0.577178i	−0.957453	−	0.288589i	$-0.906814\pi$
$383$	− 2.67239e20i	− 0.577178i	0.957453	−	0.288589i	$-0.0931861\pi$
$384$	0	0
$385$	−7.17387e20	−1.48617
$386$	0	0
$387$	− 4.76251e20i	− 0.946559i
$388$	0	0
$389$	1.55323e20	0.296236	0.148118	−	0.988970i	$-0.452678\pi$
$389$	1.55323e20	0.296236	0.148118	+	0.988970i	$0.452678\pi$
$390$	0	0
$391$	− 4.84725e19i	− 0.0887322i
$392$	0	0
$393$	−4.57331e20	−0.803692
$394$	0	0
$395$	9.62697e20i	1.62447i
$396$	0	0
$397$	−1.13822e19	−0.0184459	−0.00922296	−	0.999957i	$-0.502936\pi$
$397$	−1.13822e19	−0.0184459	−0.00922296	+	0.999957i	$0.502936\pi$
$398$	0	0
$399$	− 2.37638e21i	− 3.69940i
$400$	0	0
$401$	1.60590e20	0.240195	0.120097	−	0.992762i	$-0.461679\pi$
$401$	1.60590e20	0.240195	0.120097	+	0.992762i	$0.461679\pi$
$402$	0	0
$403$	7.88617e19i	0.113351i
$404$	0	0
$405$	−3.23954e21	−4.47551
$406$	0	0
$407$	− 8.16316e19i	− 0.108418i
$408$	0	0
$409$	−6.36217e20	−0.812491	−0.406246	−	0.913764i	$-0.633162\pi$
$409$	−6.36217e20	−0.812491	−0.406246	+	0.913764i	$0.633162\pi$
$410$	0	0
$411$	2.30932e21i	2.83628i
$412$	0	0
$413$	1.55872e21	1.84148
$414$	0	0
$415$	2.33403e20i	0.265291i
$416$	0	0
$417$	−2.52650e21	−2.76332
$418$	0	0
$419$	− 3.54694e20i	− 0.373372i	−0.982420	−	0.186686i	$-0.940225\pi$
$419$	− 3.54694e20i	− 0.373372i	0.982420	−	0.186686i	$-0.0597748\pi$
$420$	0	0
$421$	6.49716e20	0.658365	0.329183	−	0.944266i	$-0.393227\pi$
$421$	6.49716e20	0.658365	0.329183	+	0.944266i	$0.393227\pi$
$422$	0	0
$423$	9.73552e20i	0.949809i
$424$	0	0
$425$	−1.01099e20	−0.0949803
$426$	0	0
$427$	− 2.52283e21i	− 2.28279i
$428$	0	0
$429$	−7.58900e20	−0.661494
$430$	0	0
$431$	− 1.52887e21i	− 1.28396i	−0.766720	−	0.641982i	$-0.778113\pi$
$431$	− 1.52887e21i	− 1.28396i	0.766720	−	0.641982i	$-0.221887\pi$
$432$	0	0
$433$	−1.58400e21	−1.28189	−0.640945	−	0.767587i	$-0.721457\pi$
$433$	−1.58400e21	−1.28189	−0.640945	+	0.767587i	$0.721457\pi$
$434$	0	0
$435$	− 8.57700e20i	− 0.668993i
$436$	0	0
$437$	1.68234e21	1.26492
$438$	0	0
$439$	− 2.15714e21i	− 1.56373i	−0.623446	−	0.781867i	$-0.714268\pi$
$439$	− 2.15714e21i	− 1.56373i	0.623446	−	0.781867i	$-0.285732\pi$
$440$	0	0
$441$	4.76965e21	3.33410
$442$	0	0
$443$	2.44868e21i	1.65083i	0.564527	+	0.825415i	$0.309059\pi$
$443$	2.44868e21i	1.65083i	−0.564527	+	0.825415i	$0.690941\pi$
$444$	0	0
$445$	1.87313e21	1.21812
$446$	0	0
$447$	1.06625e21i	0.668959i
$448$	0	0
$449$	2.39500e20	0.144989	0.0724945	−	0.997369i	$-0.476904\pi$
$449$	2.39500e20	0.144989	0.0724945	+	0.997369i	$0.476904\pi$
$450$	0	0
$451$	− 1.95097e21i	− 1.13983i
$452$	0	0
$453$	−4.29470e21	−2.42185
$454$	0	0
$455$	2.02078e21i	1.10009i
$456$	0	0
$457$	6.87433e20	0.361327	0.180664	−	0.983545i	$-0.442175\pi$
$457$	6.87433e20	0.361327	0.180664	+	0.983545i	$0.442175\pi$
$458$	0	0
$459$	5.38329e20i	0.273241i
$460$	0	0
$461$	1.28065e21	0.627803	0.313901	−	0.949456i	$-0.398364\pi$
$461$	1.28065e21	0.627803	0.313901	+	0.949456i	$0.398364\pi$
$462$	0	0
$463$	− 1.18085e21i	− 0.559172i	−0.960121	−	0.279586i	$-0.909803\pi$
$463$	− 1.18085e21i	− 0.559172i	0.960121	−	0.279586i	$-0.0901971\pi$
$464$	0	0
$465$	1.32242e21	0.604987
$466$	0	0
$467$	− 1.39742e21i	− 0.617722i	−0.951107	−	0.308861i	$-0.900052\pi$
$467$	− 1.39742e21i	− 0.617722i	0.951107	−	0.308861i	$-0.0999478\pi$
$468$	0	0
$469$	−1.48828e21	−0.635774
$470$	0	0
$471$	3.87942e21i	1.60177i
$472$	0	0
$473$	−6.28261e20	−0.250756
$474$	0	0
$475$	− 3.50883e21i	− 1.35399i
$476$	0	0
$477$	−4.09214e21	−1.52688
$478$	0	0
$479$	3.53067e21i	1.27401i	0.770860	+	0.637005i	$0.219827\pi$
$479$	3.53067e21i	1.27401i	−0.770860	+	0.637005i	$0.780173\pi$
$480$	0	0
$481$	−2.29945e20	−0.0802533
$482$	0	0
$483$	8.31722e21i	2.80803i
$484$	0	0
$485$	3.83150e21	1.25151
$486$	0	0
$487$	2.34571e21i	0.741382i	0.928756	+	0.370691i	$0.120879\pi$
$487$	2.34571e21i	0.741382i	−0.928756	+	0.370691i	$0.879121\pi$
$488$	0	0
$489$	6.85773e21	2.09754
$490$	0	0
$491$	− 1.72751e21i	− 0.511409i	−0.966755	−	0.255705i	$-0.917693\pi$
$491$	− 1.72751e21i	− 0.511409i	0.966755	−	0.255705i	$-0.0823074\pi$
$492$	0	0
$493$	−7.79265e19	−0.0223311
$494$	0	0
$495$	9.18311e21i	2.54770i
$496$	0	0
$497$	2.53843e21	0.681891
$498$	0	0
$499$	2.67312e21i	0.695367i	0.937612	+	0.347684i	$0.113032\pi$
$499$	2.67312e21i	0.695367i	−0.937612	+	0.347684i	$0.886968\pi$
$500$	0	0
$501$	4.03608e21	1.01685
$502$	0	0
$503$	− 3.36049e21i	− 0.820085i	−0.912066	−	0.410043i	$-0.865514\pi$
$503$	− 3.36049e21i	− 0.820085i	0.912066	−	0.410043i	$-0.134486\pi$
$504$	0	0
$505$	−3.01255e21	−0.712203
$506$	0	0
$507$	− 6.13666e21i	− 1.40562i
$508$	0	0
$509$	2.73914e21	0.607955	0.303978	−	0.952679i	$-0.401685\pi$
$509$	2.73914e21	0.607955	0.303978	+	0.952679i	$0.401685\pi$
$510$	0	0
$511$	− 1.08567e22i	− 2.33524i
$512$	0	0
$513$	−1.86838e22	−3.89517
$514$	0	0
$515$	4.68357e20i	0.0946497i
$516$	0	0
$517$	1.28429e21	0.251617
$518$	0	0
$519$	1.72396e20i	0.0327483i
$520$	0	0
$521$	−1.75204e21	−0.322733	−0.161366	−	0.986895i	$-0.551590\pi$
$521$	−1.75204e21	−0.322733	−0.161366	+	0.986895i	$0.551590\pi$
$522$	0	0
$523$	− 3.01690e21i	− 0.538948i	−0.963008	−	0.269474i	$-0.913150\pi$
$523$	− 3.01690e21i	− 0.538948i	0.963008	−	0.269474i	$-0.0868499\pi$
$524$	0	0
$525$	1.73471e22	3.00576
$526$	0	0
$527$	− 1.20149e20i	− 0.0201946i
$528$	0	0
$529$	2.44493e20	0.0398677
$530$	0	0
$531$	− 1.99528e22i	− 3.15680i
$532$	0	0
$533$	−5.49562e21	−0.843721
$534$	0	0
$535$	− 5.08509e21i	− 0.757649i
$536$	0	0
$537$	2.12797e22	3.07731
$538$	0	0
$539$	− 6.29203e21i	− 0.883245i
$540$	0	0
$541$	−6.64753e21	−0.905903	−0.452952	−	0.891535i	$-0.649629\pi$
$541$	−6.64753e21	−0.905903	−0.452952	+	0.891535i	$0.649629\pi$
$542$	0	0
$543$	2.82911e22i	3.74327i
$544$	0	0
$545$	−9.06468e21	−1.16461
$546$	0	0
$547$	1.19109e22i	1.48609i	0.669241	+	0.743046i	$0.266619\pi$
$547$	1.19109e22i	1.48609i	−0.669241	+	0.743046i	$0.733381\pi$
$548$	0	0
$549$	−3.22942e22	−3.91333
$550$	0	0
$551$	− 2.70460e21i	− 0.318339i
$552$	0	0
$553$	−1.50080e22	−1.71601
$554$	0	0
$555$	3.85591e21i	0.428335i
$556$	0	0
$557$	1.28394e22	1.38581	0.692906	−	0.721028i	$-0.256330\pi$
$557$	1.28394e22	1.38581	0.692906	+	0.721028i	$0.256330\pi$
$558$	0	0
$559$	1.76972e21i	0.185614i
$560$	0	0
$561$	1.15621e21	0.117851
$562$	0	0
$563$	9.75693e21i	0.966599i	0.875455	+	0.483299i	$0.160562\pi$
$563$	9.75693e21i	0.966599i	−0.875455	+	0.483299i	$0.839438\pi$
$564$	0	0
$565$	−2.71672e22	−2.61611
$566$	0	0
$567$	− 5.05027e22i	− 4.72771i
$568$	0	0
$569$	2.33547e21	0.212557	0.106279	−	0.994336i	$-0.466106\pi$
$569$	2.33547e21	0.212557	0.106279	+	0.994336i	$0.466106\pi$
$570$	0	0
$571$	− 1.04066e21i	− 0.0920918i	−0.998939	−	0.0460459i	$-0.985338\pi$
$571$	− 1.04066e21i	− 0.0920918i	0.998939	−	0.0460459i	$-0.0146621\pi$
$572$	0	0
$573$	1.09104e22	0.938868
$574$	0	0
$575$	1.22808e22i	1.02774i
$576$	0	0
$577$	−1.12210e20	−0.00913325	−0.00456663	−	0.999990i	$-0.501454\pi$
$577$	−1.12210e20	−0.00913325	−0.00456663	+	0.999990i	$0.501454\pi$
$578$	0	0
$579$	1.36162e22i	1.07802i
$580$	0	0
$581$	−3.63864e21	−0.280240
$582$	0	0
$583$	5.39827e21i	0.404489i
$584$	0	0
$585$	2.58675e22	1.88585
$586$	0	0
$587$	6.76129e21i	0.479651i	0.970816	+	0.239825i	$0.0770902\pi$
$587$	6.76129e21i	0.479651i	−0.970816	+	0.239825i	$0.922910\pi$
$588$	0	0
$589$	4.17001e21	0.287882
$590$	0	0
$591$	− 5.01894e22i	− 3.37219i
$592$	0	0
$593$	−2.34452e21	−0.153326	−0.0766629	−	0.997057i	$-0.524427\pi$
$593$	−2.34452e21	−0.153326	−0.0766629	+	0.997057i	$0.524427\pi$
$594$	0	0
$595$	− 3.07873e21i	− 0.195991i
$596$	0	0
$597$	3.46983e22	2.15037
$598$	0	0
$599$	1.33116e22i	0.803186i	0.915818	+	0.401593i	$0.131543\pi$
$599$	1.33116e22i	0.803186i	−0.915818	+	0.401593i	$0.868457\pi$
$600$	0	0
$601$	6.58865e20	0.0387080	0.0193540	−	0.999813i	$-0.493839\pi$
$601$	6.58865e20	0.0387080	0.0193540	+	0.999813i	$0.493839\pi$
$602$	0	0
$603$	1.90512e22i	1.08989i
$604$	0	0
$605$	−1.35779e22	−0.756467
$606$	0	0
$607$	− 2.83513e22i	− 1.53838i	−0.639022	−	0.769189i	$-0.720661\pi$
$607$	− 2.83513e22i	− 1.53838i	0.639022	−	0.769189i	$-0.279339\pi$
$608$	0	0
$609$	1.33711e22	0.706691
$610$	0	0
$611$	− 3.61767e21i	− 0.186251i
$612$	0	0
$613$	−1.12561e22	−0.564551	−0.282276	−	0.959333i	$-0.591089\pi$
$613$	−1.12561e22	−0.564551	−0.282276	+	0.959333i	$0.591089\pi$
$614$	0	0
$615$	9.21554e22i	4.50318i
$616$	0	0
$617$	2.01816e22	0.960891	0.480446	−	0.877024i	$-0.340475\pi$
$617$	2.01816e22	0.960891	0.480446	+	0.877024i	$0.340475\pi$
$618$	0	0
$619$	− 3.10959e22i	− 1.44271i	−0.692567	−	0.721354i	$-0.743520\pi$
$619$	− 3.10959e22i	− 1.44271i	0.692567	−	0.721354i	$-0.256480\pi$
$620$	0	0
$621$	6.53926e22	2.95662
$622$	0	0
$623$	2.92012e22i	1.28676i
$624$	0	0
$625$	−2.20898e22	−0.948752
$626$	0	0
$627$	4.01287e22i	1.68002i
$628$	0	0
$629$	3.50330e20	0.0142979
$630$	0	0
$631$	− 3.95379e22i	− 1.57318i	−0.617475	−	0.786590i	$-0.711844\pi$
$631$	− 3.95379e22i	− 1.57318i	0.617475	−	0.786590i	$-0.288156\pi$
$632$	0	0
$633$	−4.41965e22	−1.71458
$634$	0	0
$635$	6.05264e22i	2.28958i
$636$	0	0
$637$	−1.77238e22	−0.653794
$638$	0	0
$639$	− 3.24939e22i	− 1.16895i
$640$	0	0
$641$	3.72172e22	1.30581	0.652905	−	0.757440i	$-0.273550\pi$
$641$	3.72172e22	1.30581	0.652905	+	0.757440i	$0.273550\pi$
$642$	0	0
$643$	− 2.08683e22i	− 0.714167i	−0.934072	−	0.357083i	$-0.883771\pi$
$643$	− 2.08683e22i	− 0.714167i	0.934072	−	0.357083i	$-0.116229\pi$
$644$	0	0
$645$	2.96762e22	0.990675
$646$	0	0
$647$	3.87080e21i	0.126057i	0.998012	+	0.0630284i	$0.0200759\pi$
$647$	3.87080e21i	0.126057i	−0.998012	+	0.0630284i	$0.979924\pi$
$648$	0	0
$649$	−2.63213e22	−0.836278
$650$	0	0
$651$	2.06159e22i	0.639078i
$652$	0	0
$653$	−6.50267e22	−1.96692	−0.983460	−	0.181127i	$-0.942025\pi$
$653$	−6.50267e22	−1.96692	−0.983460	+	0.181127i	$0.942025\pi$
$654$	0	0
$655$	− 2.05639e22i	− 0.606980i
$656$	0	0
$657$	−1.38974e23	−4.00323
$658$	0	0
$659$	7.54048e21i	0.211990i	0.994367	+	0.105995i	$0.0338028\pi$
$659$	7.54048e21i	0.211990i	−0.994367	+	0.105995i	$0.966197\pi$
$660$	0	0
$661$	−2.25647e22	−0.619182	−0.309591	−	0.950870i	$-0.600192\pi$
$661$	−2.25647e22	−0.619182	−0.309591	+	0.950870i	$0.600192\pi$
$662$	0	0
$663$	− 3.25689e21i	− 0.0872358i
$664$	0	0
$665$	1.06854e23	2.79394
$666$	0	0
$667$	9.46599e21i	0.241635i
$668$	0	0
$669$	1.30167e23	3.24409
$670$	0	0
$671$	4.26019e22i	1.03669i
$672$	0	0
$673$	2.85918e21	0.0679393	0.0339696	−	0.999423i	$-0.489185\pi$
$673$	2.85918e21	0.0679393	0.0339696	+	0.999423i	$0.489185\pi$
$674$	0	0
$675$	− 1.36389e23i	− 3.16481i
$676$	0	0
$677$	2.90803e22	0.659006	0.329503	−	0.944155i	$-0.393119\pi$
$677$	2.90803e22	0.659006	0.329503	+	0.944155i	$0.393119\pi$
$678$	0	0
$679$	5.97312e22i	1.32203i
$680$	0	0
$681$	1.22459e23	2.64736
$682$	0	0
$683$	4.59080e22i	0.969444i	0.874668	+	0.484722i	$0.161079\pi$
$683$	4.59080e22i	0.969444i	−0.874668	+	0.484722i	$0.838921\pi$
$684$	0	0
$685$	−1.03839e23	−2.14207
$686$	0	0
$687$	− 8.19618e22i	− 1.65180i
$688$	0	0
$689$	1.52062e22	0.299410
$690$	0	0
$691$	− 8.07657e22i	− 1.55383i	−0.629607	−	0.776914i	$-0.716784\pi$
$691$	− 8.07657e22i	− 1.55383i	0.629607	−	0.776914i	$-0.283216\pi$
$692$	0	0
$693$	−1.43160e23	−2.69126
$694$	0	0
$695$	− 1.13604e23i	− 2.08696i
$696$	0	0
$697$	8.37279e21	0.150317
$698$	0	0
$699$	− 1.04011e23i	− 1.82500i
$700$	0	0
$701$	−2.22888e21	−0.0382245	−0.0191122	−	0.999817i	$-0.506084\pi$
$701$	−2.22888e21	−0.0382245	−0.0191122	+	0.999817i	$0.506084\pi$
$702$	0	0
$703$	1.21589e22i	0.203822i
$704$	0	0
$705$	−6.06642e22	−0.994076
$706$	0	0
$707$	− 4.69642e22i	− 0.752336i
$708$	0	0
$709$	−9.18112e22	−1.43789	−0.718945	−	0.695067i	$-0.755375\pi$
$709$	−9.18112e22	−1.43789	−0.718945	+	0.695067i	$0.755375\pi$
$710$	0	0
$711$	1.92114e23i	2.94172i
$712$	0	0
$713$	−1.45949e22	−0.218516
$714$	0	0
$715$	− 3.41239e22i	− 0.499587i
$716$	0	0
$717$	−6.31430e22	−0.904007
$718$	0	0
$719$	− 5.04317e22i	− 0.706111i	−0.935602	−	0.353055i	$-0.885143\pi$
$719$	− 5.04317e22i	− 0.706111i	0.935602	−	0.353055i	$-0.114857\pi$
$720$	0	0
$721$	−7.30145e21	−0.0999832
$722$	0	0
$723$	1.78362e23i	2.38889i
$724$	0	0
$725$	1.97431e22	0.258649
$726$	0	0
$727$	1.12997e22i	0.144807i	0.997375	+	0.0724036i	$0.0230670\pi$
$727$	1.12997e22i	0.144807i	−0.997375	+	0.0724036i	$0.976933\pi$
$728$	0	0
$729$	−1.90309e23	−2.38583
$730$	0	0
$731$	− 2.69624e21i	− 0.0330689i
$732$	0	0
$733$	1.34229e23	1.61069	0.805347	−	0.592803i	$-0.201979\pi$
$733$	1.34229e23	1.61069	0.805347	+	0.592803i	$0.201979\pi$
$734$	0	0
$735$	2.97208e23i	3.48949i
$736$	0	0
$737$	2.51319e22	0.288726
$738$	0	0
$739$	− 1.18062e23i	− 1.32725i	−0.748064	−	0.663626i	$-0.769017\pi$
$739$	− 1.18062e23i	− 1.32725i	0.748064	−	0.663626i	$-0.230983\pi$
$740$	0	0
$741$	1.13037e23	1.24358
$742$	0	0
$743$	− 1.49057e22i	− 0.160487i	−0.996775	−	0.0802437i	$-0.974430\pi$
$743$	− 1.49057e22i	− 0.160487i	0.996775	−	0.0802437i	$-0.0255698\pi$
$744$	0	0
$745$	−4.79440e22	−0.505224
$746$	0	0
$747$	4.65774e22i	0.480408i
$748$	0	0
$749$	7.92740e22	0.800343
$750$	0	0
$751$	− 5.30335e22i	− 0.524120i	−0.965052	−	0.262060i	$-0.915598\pi$
$751$	− 5.30335e22i	− 0.524120i	0.965052	−	0.262060i	$-0.0844018\pi$
$752$	0	0
$753$	1.51330e23	1.46408
$754$	0	0
$755$	− 1.93111e23i	− 1.82908i
$756$	0	0
$757$	1.02184e23	0.947584	0.473792	−	0.880637i	$-0.342885\pi$
$757$	1.02184e23	0.947584	0.473792	+	0.880637i	$0.342885\pi$
$758$	0	0
$759$	− 1.40449e23i	− 1.27522i
$760$	0	0
$761$	−8.30921e22	−0.738725	−0.369362	−	0.929285i	$-0.620424\pi$
$761$	−8.30921e22	−0.738725	−0.369362	+	0.929285i	$0.620424\pi$
$762$	0	0
$763$	− 1.41314e23i	− 1.23024i
$764$	0	0
$765$	−3.94101e22	−0.335982
$766$	0	0
$767$	7.41435e22i	0.619028i
$768$	0	0
$769$	1.89213e23	1.54717	0.773587	−	0.633690i	$-0.218460\pi$
$769$	1.89213e23	1.54717	0.773587	+	0.633690i	$0.218460\pi$
$770$	0	0
$771$	3.72576e23i	2.98387i
$772$	0	0
$773$	1.98770e23	1.55925	0.779623	−	0.626249i	$-0.215411\pi$
$773$	1.98770e23	1.55925	0.779623	+	0.626249i	$0.215411\pi$
$774$	0	0
$775$	3.04404e22i	0.233903i
$776$	0	0
$777$	−6.01118e22	−0.452472
$778$	0	0
$779$	2.90595e23i	2.14283i
$780$	0	0
$781$	−4.28653e22	−0.309669
$782$	0	0
$783$	− 1.05128e23i	− 0.744087i
$784$	0	0
$785$	−1.74438e23	−1.20972
$786$	0	0
$787$	1.14525e23i	0.778221i	0.921191	+	0.389111i	$0.127218\pi$
$787$	1.14525e23i	0.778221i	−0.921191	+	0.389111i	$0.872782\pi$
$788$	0	0
$789$	2.02735e23	1.34994
$790$	0	0
$791$	− 4.23523e23i	− 2.76353i
$792$	0	0
$793$	1.20004e23	0.767377
$794$	0	0
$795$	− 2.54991e23i	− 1.59804i
$796$	0	0
$797$	−7.35302e22	−0.451647	−0.225823	−	0.974168i	$-0.572507\pi$
$797$	−7.35302e22	−0.451647	−0.225823	+	0.974168i	$0.572507\pi$
$798$	0	0
$799$	5.51166e21i	0.0331824i
$800$	0	0
$801$	3.73798e23	2.20586
$802$	0	0
$803$	1.83332e23i	1.06051i
$804$	0	0
$805$	−3.73984e23	−2.12073
$806$	0	0
$807$	5.11942e23i	2.84598i
$808$	0	0
$809$	6.12065e22	0.333587	0.166793	−	0.985992i	$-0.446659\pi$
$809$	6.12065e22	0.333587	0.166793	+	0.985992i	$0.446659\pi$
$810$	0	0
$811$	− 2.80811e23i	− 1.50054i	−0.661131	−	0.750270i	$-0.729923\pi$
$811$	− 2.80811e23i	− 1.50054i	0.661131	−	0.750270i	$-0.270077\pi$
$812$	0	0
$813$	−1.56945e23	−0.822287
$814$	0	0
$815$	3.08358e23i	1.58414i
$816$	0	0
$817$	9.35785e22	0.471411
$818$	0	0
$819$	4.03262e23i	1.99212i
$820$	0	0
$821$	−3.59769e23	−1.74292	−0.871462	−	0.490464i	$-0.836827\pi$
$821$	−3.59769e23	−1.74292	−0.871462	+	0.490464i	$0.836827\pi$
$822$	0	0
$823$	− 1.22099e23i	− 0.580116i	−0.957009	−	0.290058i	$-0.906325\pi$
$823$	− 1.22099e23i	− 0.580116i	0.957009	−	0.290058i	$-0.0936746\pi$
$824$	0	0
$825$	−2.92933e23	−1.36501
$826$	0	0
$827$	4.59734e21i	0.0210118i	0.999945	+	0.0105059i	$0.00334419\pi$
$827$	4.59734e21i	0.0210118i	−0.999945	+	0.0105059i	$0.996656\pi$
$828$	0	0
$829$	1.47823e23	0.662681	0.331341	−	0.943511i	$-0.392499\pi$
$829$	1.47823e23	0.662681	0.331341	+	0.943511i	$0.392499\pi$
$830$	0	0
$831$	− 1.29596e23i	− 0.569880i
$832$	0	0
$833$	2.70028e22	0.116480
$834$	0	0
$835$	1.81482e23i	0.767967i
$836$	0	0
$837$	1.62088e23	0.672897
$838$	0	0
$839$	7.31090e22i	0.297766i	0.988855	+	0.148883i	$0.0475678\pi$
$839$	7.31090e22i	0.297766i	−0.988855	+	0.148883i	$0.952432\pi$
$840$	0	0
$841$	−2.35029e23	−0.939188
$842$	0	0
$843$	− 3.96148e23i	− 1.55323i
$844$	0	0
$845$	2.75935e23	1.06158
$846$	0	0
$847$	− 2.11673e23i	− 0.799094i
$848$	0	0
$849$	−4.80703e23	−1.78080
$850$	0	0
$851$	− 4.25557e22i	− 0.154711i
$852$	0	0
$853$	−5.32042e20	−0.00189825	−0.000949125	−	1.00000i	$-0.500302\pi$
$853$	−5.32042e20	−0.00189825	−0.000949125		1.00000i	$0.500302\pi$
$854$	0	0
$855$	− 1.36781e24i	− 4.78957i
$856$	0	0
$857$	3.77246e23	1.29652	0.648258	−	0.761421i	$-0.275498\pi$
$857$	3.77246e23	1.29652	0.648258	+	0.761421i	$0.275498\pi$
$858$	0	0
$859$	4.82279e23i	1.62687i	0.581656	+	0.813435i	$0.302405\pi$
$859$	4.82279e23i	1.62687i	−0.581656	+	0.813435i	$0.697595\pi$
$860$	0	0
$861$	−1.43666e24	−4.75694
$862$	0	0
$863$	− 2.73382e23i	− 0.888554i	−0.895889	−	0.444277i	$-0.853461\pi$
$863$	− 2.73382e23i	− 0.888554i	0.895889	−	0.444277i	$-0.146539\pi$
$864$	0	0
$865$	−7.75180e21	−0.0247328
$866$	0	0
$867$	− 6.00134e23i	− 1.87973i
$868$	0	0
$869$	2.53432e23	0.779299
$870$	0	0
$871$	− 7.07931e22i	− 0.213720i
$872$	0	0
$873$	7.64605e23	2.26633
$874$	0	0
$875$	3.63367e22i	0.105750i
$876$	0	0
$877$	5.52719e23	1.57946	0.789728	−	0.613457i	$-0.210222\pi$
$877$	5.52719e23	1.57946	0.789728	+	0.613457i	$0.210222\pi$
$878$	0	0
$879$	− 4.74840e23i	− 1.33241i
$880$	0	0
$881$	1.85570e23	0.511330	0.255665	−	0.966765i	$-0.417706\pi$
$881$	1.85570e23	0.511330	0.255665	+	0.966765i	$0.417706\pi$
$882$	0	0
$883$	8.56281e22i	0.231703i	0.993267	+	0.115851i	$0.0369596\pi$
$883$	8.56281e22i	0.231703i	−0.993267	+	0.115851i	$0.963040\pi$
$884$	0	0
$885$	1.24330e24	3.30393
$886$	0	0
$887$	2.43926e23i	0.636605i	0.947989	+	0.318302i	$0.103113\pi$
$887$	2.43926e23i	0.636605i	−0.947989	+	0.318302i	$0.896887\pi$
$888$	0	0
$889$	−9.43576e23	−2.41859
$890$	0	0
$891$	8.52816e23i	2.14701i
$892$	0	0
$893$	−1.91293e23	−0.473029
$894$	0	0
$895$	9.56841e23i	2.32410i
$896$	0	0
$897$	−3.95625e23	−0.943940
$898$	0	0
$899$	2.34633e22i	0.0549936i
$900$	0	0
$901$	−2.31672e22	−0.0533427
$902$	0	0
$903$	4.62638e23i	1.04650i
$904$	0	0
$905$	−1.27211e24	−2.82706
$906$	0	0
$907$	− 2.02757e22i	− 0.0442708i	−0.999755	−	0.0221354i	$-0.992954\pi$
$907$	− 2.02757e22i	− 0.0442708i	0.999755	−	0.0221354i	$-0.00704649\pi$
$908$	0	0
$909$	−6.01178e23	−1.28971
$910$	0	0
$911$	4.63660e23i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$911$	4.63660e23i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$912$	0	0
$913$	6.14440e22	0.127266
$914$	0	0
$915$	− 2.01232e24i	− 4.09571i
$916$	0	0
$917$	3.20581e23	0.641184
$918$	0	0
$919$	5.88225e23i	1.15616i	0.815979	+	0.578082i	$0.196199\pi$
$919$	5.88225e23i	1.15616i	−0.815979	+	0.578082i	$0.803801\pi$
$920$	0	0
$921$	4.54249e23	0.877439
$922$	0	0
$923$	1.20746e23i	0.229223i
$924$	0	0
$925$	−8.87580e22	−0.165605
$926$	0	0
$927$	9.34642e22i	0.171399i
$928$	0	0
$929$	−2.59102e23	−0.467030	−0.233515	−	0.972353i	$-0.575023\pi$
$929$	−2.59102e23	−0.467030	−0.233515	+	0.972353i	$0.575023\pi$
$930$	0	0
$931$	9.37189e23i	1.66047i
$932$	0	0
$933$	6.17531e23	1.07549
$934$	0	0
$935$	5.19891e22i	0.0890060i
$936$	0	0
$937$	−1.08006e24	−1.81773	−0.908866	−	0.417088i	$-0.863051\pi$
$937$	−1.08006e24	−1.81773	−0.908866	+	0.417088i	$0.863051\pi$
$938$	0	0
$939$	− 2.02440e23i	− 0.334944i
$940$	0	0
$941$	−7.78257e23	−1.26592	−0.632960	−	0.774185i	$-0.718160\pi$
$941$	−7.78257e23	−1.26592	−0.632960	+	0.774185i	$0.718160\pi$
$942$	0	0
$943$	− 1.01707e24i	− 1.62651i
$944$	0	0
$945$	4.15341e24	6.53056
$946$	0	0
$947$	5.47593e23i	0.846560i	0.905999	+	0.423280i	$0.139121\pi$
$947$	5.47593e23i	0.846560i	−0.905999	+	0.423280i	$0.860879\pi$
$948$	0	0
$949$	5.16420e23	0.785007
$950$	0	0
$951$	2.46805e24i	3.68902i
$952$	0	0
$953$	4.74081e23	0.696802	0.348401	−	0.937346i	$-0.386725\pi$
$953$	4.74081e23	0.696802	0.348401	+	0.937346i	$0.386725\pi$
$954$	0	0
$955$	4.90587e23i	0.709070i
$956$	0	0
$957$	−2.25792e23	−0.320932
$958$	0	0
$959$	− 1.61879e24i	− 2.26278i
$960$	0	0
$961$	6.91247e23	0.950268
$962$	0	0
$963$	− 1.01477e24i	− 1.37201i
$964$	0	0
$965$	−6.12254e23	−0.814166
$966$	0	0
$967$	8.54892e23i	1.11815i	0.829118	+	0.559074i	$0.188843\pi$
$967$	8.54892e23i	1.11815i	−0.829118	+	0.559074i	$0.811157\pi$
$968$	0	0
$969$	−1.72216e23	−0.221556
$970$	0	0
$971$	− 4.45296e23i	− 0.563501i	−0.959488	−	0.281751i	$-0.909085\pi$
$971$	− 4.45296e23i	− 0.563501i	0.959488	−	0.281751i	$-0.0909151\pi$
$972$	0	0
$973$	1.77103e24	2.20457
$974$	0	0
$975$	8.25151e23i	1.01041i
$976$	0	0
$977$	−1.19178e24	−1.43562	−0.717810	−	0.696239i	$-0.754856\pi$
$977$	−1.19178e24	−1.43562	−0.717810	+	0.696239i	$0.754856\pi$
$978$	0	0
$979$	− 4.93107e23i	− 0.584359i
$980$	0	0
$981$	−1.80893e24	−2.10896
$982$	0	0
$983$	− 1.28878e24i	− 1.47826i	−0.673561	−	0.739132i	$-0.735236\pi$
$983$	− 1.28878e24i	− 1.47826i	0.673561	−	0.739132i	$-0.264764\pi$
$984$	0	0
$985$	2.25677e24	2.54681
$986$	0	0
$987$	− 9.45725e23i	− 1.05009i
$988$	0	0
$989$	−3.27521e23	−0.357824
$990$	0	0
$991$	9.77998e23i	1.05135i	0.850684	+	0.525677i	$0.176188\pi$
$991$	9.77998e23i	1.05135i	−0.850684	+	0.525677i	$0.823812\pi$
$992$	0	0
$993$	−1.27873e24	−1.35265
$994$	0	0
$995$	1.56021e24i	1.62405i
$996$	0	0
$997$	7.46914e23	0.765085	0.382542	−	0.923938i	$-0.375049\pi$
$997$	7.46914e23	0.765085	0.382542	+	0.923938i	$0.375049\pi$
$998$	0	0
$999$	4.72617e23i	0.476415i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.17.c.a.31.8	yes	8
4.3	odd	2	inner	32.17.c.a.31.1	✓	8
8.3	odd	2		64.17.c.f.63.8		8
8.5	even	2		64.17.c.f.63.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.17.c.a.31.1	✓	8	4.3	odd	2	inner
32.17.c.a.31.8	yes	8	1.1	even	1	trivial
64.17.c.f.63.1		8	8.5	even	2
64.17.c.f.63.8		8	8.3	odd	2

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:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 17 \)
Character orbit:	\([\chi]\)	\(=\)	32.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+12434.9i q^{3} -559134. q^{5} -8.71663e6i q^{7} -1.11580e8 q^{9} +O(q^{10})\)	\(q+12434.9i q^{3} -559134. q^{5} -8.71663e6i q^{7} -1.11580e8 q^{9} +1.47194e8i q^{11} +4.14624e8 q^{13} -6.95277e9i q^{15} -6.31695e8 q^{17} -2.19243e10i q^{19} +1.08390e11 q^{21} +7.67341e10i q^{23} +1.60043e11 q^{25} -8.52197e11i q^{27} +1.23361e11 q^{29} +1.90201e11i q^{31} -1.83033e12 q^{33} +4.87377e12i q^{35} -5.54587e11 q^{37} +5.15579e12i q^{39} -1.32545e13 q^{41} +4.26826e12i q^{43} +6.23880e13 q^{45} -8.72519e12i q^{47} -4.27467e13 q^{49} -7.85505e12i q^{51} +3.66747e13 q^{53} -8.23010e13i q^{55} +2.72626e14 q^{57} +1.78821e14i q^{59} +2.89428e14 q^{61} +9.72597e14i q^{63} -2.31830e14 q^{65} -1.70741e14i q^{67} -9.54179e14 q^{69} +2.91217e14i q^{71} +1.24552e15 q^{73} +1.99012e15i q^{75} +1.28303e15 q^{77} -1.72176e15i q^{79} +5.79384e15 q^{81} -4.17437e14i q^{83} +3.53202e14 q^{85} +1.53398e15i q^{87} -3.35006e15 q^{89} -3.61412e15i q^{91} -2.36512e15 q^{93} +1.22586e16i q^{95} -6.85255e15 q^{97} -1.64238e16i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 1121456 q^{5} - 189143672 q^{9}+O(q^{10}) \) 8 * q - 1121456 * q^5 - 189143672 * q^9	\( 8 q - 1121456 q^{5} - 189143672 q^{9} + 17627216 q^{13} - 7983031920 q^{17} + 131738539264 q^{21} + 159991390488 q^{25} - 191970074800 q^{29} - 295705712512 q^{33} + 2823854227536 q^{37} + 7729120566032 q^{41} + 108153928739408 q^{45} + 35978193324552 q^{49} + 185763363904080 q^{53} + 975883022595456 q^{57} + 15\!\cdots\!92 q^{61}+ \cdots - 16\!\cdots\!52 q^{97}+O(q^{100}) \) 8 * q - 1121456 * q^5 - 189143672 * q^9 + 17627216 * q^13 - 7983031920 * q^17 + 131738539264 * q^21 + 159991390488 * q^25 - 191970074800 * q^29 - 295705712512 * q^33 + 2823854227536 * q^37 + 7729120566032 * q^41 + 108153928739408 * q^45 + 35978193324552 * q^49 + 185763363904080 * q^53 + 975883022595456 * q^57 + 1588295187290192 * q^61 + 1407344840347936 * q^65 + 1721373686220544 * q^69 + 4735684644451728 * q^73 + 7884073041355520 * q^77 + 12832008243781256 * q^81 + 3759414296335776 * q^85 - 4267961369411184 * q^89 - 9651190567144448 * q^93 - 16022975903677552 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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