LMFDB - Embedded newform 175.9.d.c.76.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,9,Mod(76,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("175.76");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	175.d (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:	$71.2912567607$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			76.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	175.76
Dual form			175.9.d.c.76.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-127.000i q^{3} -256.000 q^{4} +2401.00i q^{7} -9568.00 q^{9} +O(q^{10})$	$q-127.000i q^{3} -256.000 q^{4} +2401.00i q^{7} -9568.00 q^{9} -23953.0 q^{11} +32512.0i q^{12} +56593.0i q^{13} +65536.0 q^{16} -97873.0i q^{17} +304927. q^{21} +381889. i q^{27} -614656. i q^{28} +85153.0 q^{29} +3.04203e6i q^{33} +2.44941e6 q^{36} +7.18731e6 q^{39} +6.13197e6 q^{44} +2.19149e6i q^{47} -8.32307e6i q^{48} -5.76480e6 q^{49} -1.24299e7 q^{51} -1.44878e7i q^{52} -2.29728e7i q^{63} -1.67772e7 q^{64} +2.50555e7i q^{68} +5.07427e7 q^{71} -3.34915e7i q^{73} -5.75112e7i q^{77} -7.01357e7 q^{79} -1.42757e7 q^{81} +5.41867e7i q^{83} -7.80613e7 q^{84} -1.08144e7i q^{87} -1.35880e8 q^{91} -1.65614e8i q^{97} +2.29182e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 512 q^{4} - 19136 q^{9}+O(q^{10}) $ 2 * q - 512 * q^4 - 19136 * q^9	$ 2 q - 512 q^{4} - 19136 q^{9} - 47906 q^{11} + 131072 q^{16} + 609854 q^{21} + 170306 q^{29} + 4898816 q^{36} + 14374622 q^{39} + 12263936 q^{44} - 11529602 q^{49} - 24859742 q^{51} - 33554432 q^{64} + 101485444 q^{71} - 140271454 q^{79} - 28551490 q^{81} - 156122624 q^{84} - 271759586 q^{91} + 458364608 q^{99}+O(q^{100}) $ 2 * q - 512 * q^4 - 19136 * q^9 - 47906 * q^11 + 131072 * q^16 + 609854 * q^21 + 170306 * q^29 + 4898816 * q^36 + 14374622 * q^39 + 12263936 * q^44 - 11529602 * q^49 - 24859742 * q^51 - 33554432 * q^64 + 101485444 * q^71 - 140271454 * q^79 - 28551490 * q^81 - 156122624 * q^84 - 271759586 * q^91 + 458364608 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$.

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

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 $

$2$	0	0		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	− 127.000i	− 1.56790i	−0.620823	−	0.783951i	$-0.713202\pi$
$3$	− 127.000i	− 1.56790i	0.620823	−	0.783951i	$-0.286798\pi$
$4$	−256.000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2401.00i	1.00000i
$7$	2401.00i	1.00000i
$8$	0	0
$9$	−9568.00	−1.45831
$10$	0	0
$11$	−23953.0	−1.63602	−0.818011	−	0.575202i	$-0.804923\pi$
$11$	−23953.0	−1.63602	−0.818011	+	0.575202i	$0.804923\pi$
$12$	32512.0i	1.56790i
$13$	56593.0i	1.98148i	0.135779	+	0.990739i	$0.456646\pi$
$13$	56593.0i	1.98148i	−0.135779	+	0.990739i	$0.543354\pi$
$14$	0	0
$15$	0	0
$16$	65536.0	1.00000
$17$	− 97873.0i	− 1.17184i	−0.810370	−	0.585919i	$-0.800734\pi$
$17$	− 97873.0i	− 1.17184i	0.810370	−	0.585919i	$-0.199266\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	304927.	1.56790
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	381889.i	0.718592i
$28$	− 614656.i	− 1.00000i
$29$	85153.0	0.120395	0.0601974	−	0.998186i	$-0.480827\pi$
$29$	85153.0	0.120395	0.0601974	+	0.998186i	$0.480827\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	3.04203e6i	2.56512i
$34$	0	0
$35$	0	0
$36$	2.44941e6	1.45831
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	7.18731e6	3.10676
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	6.13197e6	1.63602
$45$	0	0
$46$	0	0
$47$	2.19149e6i	0.449105i	0.974462	+	0.224552i	$0.0720920\pi$
$47$	2.19149e6i	0.449105i	−0.974462	+	0.224552i	$0.927908\pi$
$48$	− 8.32307e6i	− 1.56790i
$49$	−5.76480e6	−1.00000
$50$	0	0
$51$	−1.24299e7	−1.83732
$52$	− 1.44878e7i	− 1.98148i
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	− 2.29728e7i	− 1.45831i
$64$	−1.67772e7	−1.00000
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	2.50555e7i	1.17184i
$69$	0	0
$70$	0	0
$71$	5.07427e7	1.99683	0.998413	−	0.0563101i	$-0.0179335\pi$
$71$	5.07427e7	1.99683	0.998413	+	0.0563101i	$0.0179335\pi$
$72$	0	0
$73$	− 3.34915e7i	− 1.17935i	−0.807640	−	0.589676i	$-0.799255\pi$
$73$	− 3.34915e7i	− 1.17935i	0.807640	−	0.589676i	$-0.200745\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.75112e7i	− 1.63602i
$78$	0	0
$79$	−7.01357e7	−1.80066	−0.900328	−	0.435211i	$-0.856674\pi$
$79$	−7.01357e7	−1.80066	−0.900328	+	0.435211i	$0.856674\pi$
$80$	0	0
$81$	−1.42757e7	−0.331634
$82$	0	0
$83$	5.41867e7i	1.14177i	0.821028	+	0.570887i	$0.193401\pi$
$83$	5.41867e7i	1.14177i	−0.821028	+	0.570887i	$0.806599\pi$
$84$	−7.80613e7	−1.56790
$85$	0	0
$86$	0	0
$87$	− 1.08144e7i	− 0.188767i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	−1.35880e8	−1.98148
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1.65614e8i	− 1.87072i	−0.353692	−	0.935362i	$-0.615074\pi$
$97$	− 1.65614e8i	− 1.87072i	0.353692	−	0.935362i	$-0.384926\pi$
$98$	0	0
$99$	2.29182e8	2.38583
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	− 1.12833e8i	− 1.00251i	−0.865301	−	0.501253i	$-0.832873\pi$
$103$	− 1.12833e8i	− 1.00251i	0.865301	−	0.501253i	$-0.167127\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	− 9.77636e7i	− 0.718592i
$109$	7.63077e7	0.540583	0.270292	−	0.962779i	$-0.412880\pi$
$109$	7.63077e7	0.540583	0.270292	+	0.962779i	$0.412880\pi$
$110$	0	0
$111$	0	0
$112$	1.57352e8i	1.00000i
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	−2.17992e7	−0.120395
$117$	− 5.41482e8i	− 2.88962i
$118$	0	0
$119$	2.34993e8	1.17184
$120$	0	0
$121$	3.59387e8	1.67657
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	− 7.78760e8i	− 2.56512i
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	2.78319e8	0.704152
$142$	0	0
$143$	− 1.35557e9i	− 3.24174i
$144$	−6.27048e8	−1.45831
$145$	0	0
$146$	0	0
$147$	7.32130e8i	1.56790i
$148$	0	0
$149$	3.98094e8	0.807683	0.403841	−	0.914829i	$-0.367675\pi$
$149$	3.98094e8	0.807683	0.403841	+	0.914829i	$0.367675\pi$
$150$	0	0
$151$	1.02439e9	1.97041	0.985204	−	0.171388i	$-0.0548253\pi$
$151$	1.02439e9	1.97041	0.985204	+	0.171388i	$0.0548253\pi$
$152$	0	0
$153$	9.36449e8i	1.70891i
$154$	0	0
$155$	0	0
$156$	−1.83995e9	−3.10676
$157$	− 2.32825e8i	− 0.383206i	−0.981473	−	0.191603i	$-0.938631\pi$
$157$	− 2.32825e8i	− 0.383206i	0.981473	−	0.191603i	$-0.0613685\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1.24361e9i	− 1.59889i	−0.600739	−	0.799445i	$-0.705127\pi$
$167$	− 1.24361e9i	− 1.59889i	0.600739	−	0.799445i	$-0.294873\pi$
$168$	0	0
$169$	−2.38704e9	−2.92626
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.67012e9i	1.86450i	0.361816	+	0.932250i	$0.382157\pi$
$173$	1.67012e9i	1.86450i	−0.361816	+	0.932250i	$0.617843\pi$
$174$	0	0
$175$	0	0
$176$	−1.56978e9	−1.63602
$177$	0	0
$178$	0	0
$179$	1.77908e9	1.73294	0.866472	−	0.499226i	$-0.166382\pi$
$179$	1.77908e9	1.73294	0.866472	+	0.499226i	$0.166382\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.34435e9i	1.91715i
$188$	− 5.61021e8i	− 0.449105i
$189$	−9.16915e8	−0.718592
$190$	0	0
$191$	−4.10509e8	−0.308453	−0.154227	−	0.988036i	$-0.549289\pi$
$191$	−4.10509e8	−0.308453	−0.154227	+	0.988036i	$0.549289\pi$
$192$	2.13071e9i	1.56790i
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	1.47579e9	1.00000
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.04452e8i	0.120395i
$204$	3.18205e9	1.83732
$205$	0	0
$206$	0	0
$207$	0	0
$208$	3.70888e9i	1.98148i
$209$	0	0
$210$	0	0
$211$	2.05643e9	1.03749	0.518746	−	0.854928i	$-0.326399\pi$
$211$	2.05643e9	1.03749	0.518746	+	0.854928i	$0.326399\pi$
$212$	0	0
$213$	− 6.44433e9i	− 3.13083i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−4.25342e9	−1.84911
$220$	0	0
$221$	5.53893e9	2.32197
$222$	0	0
$223$	1.18298e9i	0.478365i	0.970975	+	0.239182i	$0.0768794\pi$
$223$	1.18298e9i	0.478365i	−0.970975	+	0.239182i	$0.923121\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2.82814e9i	− 1.06512i	−0.846393	−	0.532560i	$-0.821230\pi$
$227$	− 2.82814e9i	− 1.06512i	0.846393	−	0.532560i	$-0.178770\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	−7.30392e9	−2.56512
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.90724e9i	2.82325i
$238$	0	0
$239$	4.64492e9	1.42360	0.711798	−	0.702384i	$-0.247881\pi$
$239$	4.64492e9	1.42360	0.711798	+	0.702384i	$0.247881\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	4.31859e9i	1.23856i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	6.88171e9	1.79019
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	5.88103e9i	1.45831i
$253$	0	0
$254$	0	0
$255$	0	0
$256$	4.29497e9	1.00000
$257$	3.80518e9i	0.872252i	0.899886	+	0.436126i	$0.143650\pi$
$257$	3.80518e9i	0.872252i	−0.899886	+	0.436126i	$0.856350\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−8.14744e8	−0.175574
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	− 6.41420e9i	− 1.17184i
$273$	1.72567e10i	3.10676i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.88404e9	0.302179	0.151090	−	0.988520i	$-0.451722\pi$
$281$	1.88404e9	0.302179	0.151090	+	0.988520i	$0.451722\pi$
$282$	0	0
$283$	− 1.03677e10i	− 1.61636i	−0.588938	−	0.808178i	$-0.700454\pi$
$283$	− 1.03677e10i	− 1.61636i	0.588938	−	0.808178i	$-0.299546\pi$
$284$	−1.29901e10	−1.99683
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−2.60337e9	−0.373202
$290$	0	0
$291$	−2.10330e10	−2.93311
$292$	8.57383e9i	1.17935i
$293$	− 1.46163e10i	− 1.98320i	−0.129352	−	0.991599i	$-0.541290\pi$
$293$	− 1.46163e10i	− 1.98320i	0.129352	−	0.991599i	$-0.458710\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 9.14739e9i	− 1.17563i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5.30331e8i	0.0597027i	0.999554	+	0.0298513i	$0.00950338\pi$
$307$	5.30331e8i	0.0597027i	−0.999554	+	0.0298513i	$0.990497\pi$
$308$	1.47229e10i	1.63602i
$309$	−1.43298e10	−1.57183
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	− 2.44168e9i	− 0.254396i	−0.991877	−	0.127198i	$-0.959402\pi$
$313$	− 2.44168e9i	− 0.254396i	0.991877	−	0.127198i	$-0.0405984\pi$
$314$	0	0
$315$	0	0
$316$	1.79547e10	1.80066
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	−2.03967e9	−0.196969
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	3.65459e9	0.331634
$325$	0	0
$326$	0	0
$327$	− 9.69108e9i	− 0.847581i
$328$	0	0
$329$	−5.26176e9	−0.449105
$330$	0	0
$331$	−4.82996e9	−0.402376	−0.201188	−	0.979553i	$-0.564480\pi$
$331$	−4.82996e9	−0.402376	−0.201188	+	0.979553i	$0.564480\pi$
$332$	− 1.38718e10i	− 1.14177i
$333$	0	0
$334$	0	0
$335$	0	0
$336$	1.99837e10	1.56790
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 1.38413e10i	− 1.00000i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	2.76849e9i	0.188767i
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	−2.16122e10	−1.42387
$352$	0	0
$353$	− 2.17978e10i	− 1.40383i	−0.712261	−	0.701915i	$-0.752329\pi$
$353$	− 2.17978e10i	− 1.40383i	0.712261	−	0.701915i	$-0.247671\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 2.98441e10i	− 1.83732i
$358$	0	0
$359$	2.73956e10	1.64931	0.824656	−	0.565635i	$-0.191369\pi$
$359$	2.73956e10	1.64931	0.824656	+	0.565635i	$0.191369\pi$
$360$	0	0
$361$	1.69836e10	1.00000
$362$	0	0
$363$	− 4.56422e10i	− 2.62869i
$364$	3.47852e10	1.98148
$365$	0	0
$366$	0	0
$367$	− 2.27107e10i	− 1.25189i	−0.779867	−	0.625945i	$-0.784713\pi$
$367$	− 2.27107e10i	− 1.25189i	0.779867	−	0.625945i	$-0.215287\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.81906e9i	0.238560i
$378$	0	0
$379$	4.00209e10	1.93968	0.969841	−	0.243740i	$-0.0783742\pi$
$379$	4.00209e10	1.93968	0.969841	+	0.243740i	$0.0783742\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.21837e10i	1.96042i	0.197951	+	0.980212i	$0.436571\pi$
$383$	4.21837e10i	1.96042i	−0.197951	+	0.980212i	$0.563429\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	4.23972e10i	1.87072i
$389$	−1.64078e10	−0.716557	−0.358279	−	0.933615i	$-0.616636\pi$
$389$	−1.64078e10	−0.716557	−0.358279	+	0.933615i	$0.616636\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	−5.86707e10	−2.38583
$397$	− 2.28190e10i	− 0.918616i	−0.888277	−	0.459308i	$-0.848097\pi$
$397$	− 2.28190e10i	− 0.918616i	0.888277	−	0.459308i	$-0.151903\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.83147e10	1.86854	0.934268	−	0.356570i	$-0.116054\pi$
$401$	4.83147e10	1.86854	0.934268	+	0.356570i	$0.116054\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	2.88852e10i	1.00251i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−5.69377e10	−1.81247	−0.906237	−	0.422770i	$-0.861058\pi$
$421$	−5.69377e10	−1.81247	−0.906237	+	0.422770i	$0.861058\pi$
$422$	0	0
$423$	− 2.09681e10i	− 0.654936i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.72158e11	−5.08273
$430$	0	0
$431$	5.77837e10	1.67454	0.837271	−	0.546789i	$-0.184150\pi$
$431$	5.77837e10	1.67454	0.837271	+	0.546789i	$0.184150\pi$
$432$	2.50275e10i	0.718592i
$433$	3.28473e10i	0.934433i	0.884143	+	0.467216i	$0.154743\pi$
$433$	3.28473e10i	0.934433i	−0.884143	+	0.467216i	$0.845257\pi$
$434$	0	0
$435$	0	0
$436$	−1.95348e10	−0.540583
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	5.51576e10	1.45831
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 5.05580e10i	− 1.26637i
$448$	− 4.02821e10i	− 1.00000i
$449$	1.14383e10	0.281434	0.140717	−	0.990050i	$-0.455059\pi$
$449$	1.14383e10	0.281434	0.140717	+	0.990050i	$0.455059\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 1.30097e11i	− 3.08940i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	3.73766e10	0.842072
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	5.58059e9	0.120395
$465$	0	0
$466$	0	0
$467$	8.85676e9i	0.186212i	0.995656	+	0.0931060i	$0.0296795\pi$
$467$	8.85676e9i	0.186212i	−0.995656	+	0.0931060i	$0.970320\pi$
$468$	1.38619e11i	2.88962i
$469$	0	0
$470$	0	0
$471$	−2.95688e10	−0.600828
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−6.01582e10	−1.17184
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−9.20032e10	−1.67657
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.05331e11	1.81229	0.906147	−	0.422963i	$-0.139010\pi$
$491$	1.05331e11	1.81229	0.906147	+	0.422963i	$0.139010\pi$
$492$	0	0
$493$	− 8.33418e9i	− 0.141083i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.21833e11i	1.99683i
$498$	0	0
$499$	1.23010e11	1.98398	0.991992	−	0.126304i	$-0.0403115\pi$
$499$	1.23010e11	1.98398	0.991992	+	0.126304i	$0.0403115\pi$
$500$	0	0
$501$	−1.57939e11	−2.50690
$502$	0	0
$503$	2.98721e10i	0.466653i	0.972398	+	0.233326i	$0.0749610\pi$
$503$	2.98721e10i	0.466653i	−0.972398	+	0.233326i	$0.925039\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.03154e11i	4.58808i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	8.04131e10	1.17935
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 5.24927e10i	− 0.734745i
$518$	0	0
$519$	2.12105e11	2.92335
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	− 4.46525e10i	− 0.596815i	−0.954439	−	0.298407i	$-0.903545\pi$
$523$	− 4.46525e10i	− 0.596815i	0.954439	−	0.298407i	$-0.0964554\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	1.99363e11i	2.56512i
$529$	−7.83110e10	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 2.25944e11i	− 2.71708i
$538$	0	0
$539$	1.38084e11	1.63602
$540$	0	0
$541$	−1.71316e11	−1.99990	−0.999950	−	0.0100005i	$-0.996817\pi$
$541$	−1.71316e11	−1.99990	−0.999950	+	0.0100005i	$0.996817\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 1.68396e11i	− 1.80066i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	2.97733e11	3.00590
$562$	0	0
$563$	1.84055e11i	1.83195i	0.401235	+	0.915975i	$0.368581\pi$
$563$	1.84055e11i	1.83195i	−0.401235	+	0.915975i	$0.631419\pi$
$564$	−7.12496e10	−0.704152
$565$	0	0
$566$	0	0
$567$	− 3.42761e10i	− 0.331634i
$568$	0	0
$569$	−1.11713e11	−1.06575	−0.532873	−	0.846195i	$-0.678888\pi$
$569$	−1.11713e11	−1.06575	−0.532873	+	0.846195i	$0.678888\pi$
$570$	0	0
$571$	1.94640e11	1.83100	0.915499	−	0.402319i	$-0.131796\pi$
$571$	1.94640e11	1.83100	0.915499	+	0.402319i	$0.131796\pi$
$572$	3.47026e11i	3.24174i
$573$	5.21346e10i	0.483624i
$574$	0	0
$575$	0	0
$576$	1.60524e11	1.45831
$577$	2.20564e11i	1.98990i	0.100361	+	0.994951i	$0.468000\pi$
$577$	2.20564e11i	1.98990i	−0.100361	+	0.994951i	$0.532000\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.30102e11	−1.14177
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.03017e10i	0.170993i	0.996338	+	0.0854967i	$0.0272477\pi$
$587$	2.03017e10i	0.170993i	−0.996338	+	0.0854967i	$0.972752\pi$
$588$	− 1.87425e11i	− 1.56790i
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.44826e11i	1.17119i	0.810602	+	0.585597i	$0.199140\pi$
$593$	1.44826e11i	1.17119i	−0.810602	+	0.585597i	$0.800860\pi$
$594$	0	0
$595$	0	0
$596$	−1.01912e11	−0.807683
$597$	0	0
$598$	0	0
$599$	−1.18627e11	−0.921463	−0.460732	−	0.887540i	$-0.652413\pi$
$599$	−1.18627e11	−0.921463	−0.460732	+	0.887540i	$0.652413\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	−2.62243e11	−1.97041
$605$	0	0
$606$	0	0
$607$	− 2.64717e11i	− 1.94996i	−0.222283	−	0.974982i	$-0.571351\pi$
$607$	− 2.64717e11i	− 1.94996i	0.222283	−	0.974982i	$-0.428649\pi$
$608$	0	0
$609$	2.59654e10	0.188767
$610$	0	0
$611$	−1.24023e11	−0.889891
$612$	− 2.39731e11i	− 1.70891i
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	4.71028e11	3.10676
$625$	0	0
$626$	0	0
$627$	0	0
$628$	5.96033e10i	0.383206i
$629$	0	0
$630$	0	0
$631$	−3.06967e11	−1.93631	−0.968153	−	0.250360i	$-0.919451\pi$
$631$	−3.06967e11	−1.93631	−0.968153	+	0.250360i	$0.919451\pi$
$632$	0	0
$633$	− 2.61167e11i	− 1.62669i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3.26247e11i	− 1.98148i
$638$	0	0
$639$	−4.85506e11	−2.91200
$640$	0	0
$641$	−3.28430e11	−1.94541	−0.972704	−	0.232049i	$-0.925457\pi$
$641$	−3.28430e11	−1.94541	−0.972704	+	0.232049i	$0.925457\pi$
$642$	0	0
$643$	− 1.66750e11i	− 0.975487i	−0.872987	−	0.487744i	$-0.837820\pi$
$643$	− 1.66750e11i	− 0.975487i	0.872987	−	0.487744i	$-0.162180\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 3.38044e11i	− 1.92911i	−0.263888	−	0.964553i	$-0.585005\pi$
$647$	− 3.38044e11i	− 1.92911i	0.263888	−	0.964553i	$-0.414995\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.20447e11i	1.71987i
$658$	0	0
$659$	−2.27348e11	−1.20545	−0.602726	−	0.797948i	$-0.705919\pi$
$659$	−2.27348e11	−1.20545	−0.602726	+	0.797948i	$0.705919\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	− 7.03444e11i	− 3.64062i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	3.18364e11i	1.59889i
$669$	1.50239e11	0.750029
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	6.11081e11	2.92626
$677$	1.42404e11i	0.677901i	0.940804	+	0.338950i	$0.110072\pi$
$677$	1.42404e11i	0.677901i	−0.940804	+	0.338950i	$0.889928\pi$
$678$	0	0
$679$	3.97639e11	1.87072
$680$	0	0
$681$	−3.59174e11	−1.67000
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	− 4.27550e11i	− 1.86450i
$693$	5.50267e11i	2.38583i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.99980e11	1.24228	0.621140	−	0.783700i	$-0.286670\pi$
$701$	2.99980e11	1.24228	0.621140	+	0.783700i	$0.286670\pi$
$702$	0	0
$703$	0	0
$704$	4.01865e11	1.63602
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−4.99684e11	−1.97747	−0.988736	−	0.149668i	$-0.952180\pi$
$709$	−4.99684e11	−1.97747	−0.988736	+	0.149668i	$0.952180\pi$
$710$	0	0
$711$	6.71059e11	2.62592
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−4.55446e11	−1.73294
$717$	− 5.89905e11i	− 2.23206i
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	2.70912e11	1.00251
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.94706e11i	1.41298i	0.707722	+	0.706491i	$0.249723\pi$
$727$	3.94706e11i	1.41298i	−0.707722	+	0.706491i	$0.750277\pi$
$728$	0	0
$729$	4.54798e11	1.61031
$730$	0	0
$731$	0	0
$732$	0	0
$733$	5.74743e11i	1.99094i	0.0950793	+	0.995470i	$0.469690\pi$
$733$	5.74743e11i	1.99094i	−0.0950793	+	0.995470i	$0.530310\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	3.51852e10	0.117973	0.0589864	−	0.998259i	$-0.481213\pi$
$739$	3.51852e10	0.117973	0.0589864	+	0.998259i	$0.481213\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 5.18459e11i	− 1.66507i
$748$	− 6.00154e11i	− 1.91715i
$749$	0	0
$750$	0	0
$751$	−2.15140e11	−0.676334	−0.338167	−	0.941086i	$-0.609807\pi$
$751$	−2.15140e11	−0.676334	−0.338167	+	0.941086i	$0.609807\pi$
$752$	1.43621e11i	0.449105i
$753$	0	0
$754$	0	0
$755$	0	0
$756$	2.34730e11	0.718592
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	1.83215e11i	0.540583i
$764$	1.05090e11	0.308453
$765$	0	0
$766$	0	0
$767$	0	0
$768$	− 5.45461e11i	− 1.56790i
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	4.83257e11	1.36761
$772$	0	0
$773$	− 6.77429e11i	− 1.89734i	−0.316265	−	0.948671i	$-0.602429\pi$
$773$	− 6.77429e11i	− 1.89734i	0.316265	−	0.948671i	$-0.397571\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−1.21544e12	−3.26685
$782$	0	0
$783$	3.25190e10i	0.0865147i
$784$	−3.77802e11	−1.00000
$785$	0	0
$786$	0	0
$787$	3.08417e11i	0.803970i	0.915646	+	0.401985i	$0.131680\pi$
$787$	3.08417e11i	0.803970i	−0.915646	+	0.401985i	$0.868320\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.45514e11i	0.856312i	0.903705	+	0.428156i	$0.140837\pi$
$797$	3.45514e11i	0.856312i	−0.903705	+	0.428156i	$0.859163\pi$
$798$	0	0
$799$	2.14487e11	0.526277
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.02222e11i	1.92945i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−8.32147e11	−1.94270	−0.971351	−	0.237650i	$-0.923623\pi$
$809$	−8.32147e11	−1.94270	−0.971351	+	0.237650i	$0.923623\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	− 5.23398e10i	− 0.120395i
$813$	0	0
$814$	0	0
$815$	0	0
$816$	−8.14604e11	−1.83732
$817$	0	0
$818$	0	0
$819$	1.30010e12	2.88962
$820$	0	0
$821$	6.14999e11	1.35364	0.676818	−	0.736150i	$-0.263358\pi$
$821$	6.14999e11	1.35364	0.676818	+	0.736150i	$0.263358\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	− 9.49473e11i	− 1.98148i
$833$	5.64218e11i	1.17184i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−4.92995e11	−0.985505
$842$	0	0
$843$	− 2.39273e11i	− 0.473787i
$844$	−5.26447e11	−1.03749
$845$	0	0
$846$	0	0
$847$	8.62889e11i	1.67657i
$848$	0	0
$849$	−1.31670e12	−2.53429
$850$	0	0
$851$	0	0
$852$	1.64975e12i	3.13083i
$853$	− 1.03736e12i	− 1.95944i	−0.200361	−	0.979722i	$-0.564212\pi$
$853$	− 1.03736e12i	− 1.95944i	0.200361	−	0.979722i	$-0.435788\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.37034e11i	− 0.254042i	−0.991900	−	0.127021i	$-0.959458\pi$
$857$	− 1.37034e11i	− 0.254042i	0.991900	−	0.127021i	$-0.0405416\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.30628e11i	0.585144i
$868$	0	0
$869$	1.67996e12	2.94591
$870$	0	0
$871$	0	0
$872$	0	0
$873$	1.58459e12i	2.72810i
$874$	0	0
$875$	0	0
$876$	1.08888e12	1.84911
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	−1.85627e12	−3.10946
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	−1.41797e12	−2.32197
$885$	0	0
$886$	0	0
$887$	5.29565e11i	0.855509i	0.903895	+	0.427754i	$0.140695\pi$
$887$	5.29565e11i	0.855509i	−0.903895	+	0.427754i	$0.859305\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3.41947e11	0.542560
$892$	− 3.02844e11i	− 0.478365i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	7.24005e11i	1.06512i
$909$	0	0
$910$	0	0
$911$	−4.15670e10	−0.0603497	−0.0301749	−	0.999545i	$-0.509606\pi$
$911$	−4.15670e10	−0.0603497	−0.0301749	+	0.999545i	$0.509606\pi$
$912$	0	0
$913$	− 1.29793e12i	− 1.86797i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	9.43384e11	1.32259	0.661297	−	0.750124i	$-0.270006\pi$
$919$	9.43384e11	1.32259	0.661297	+	0.750124i	$0.270006\pi$
$920$	0	0
$921$	6.73521e10	0.0936079
$922$	0	0
$923$	2.87168e12i	3.95667i
$924$	1.86980e12	2.56512
$925$	0	0
$926$	0	0
$927$	1.07959e12i	1.46197i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	9.84225e11i	1.27684i	0.769689	+	0.638419i	$0.220411\pi$
$937$	9.84225e11i	1.27684i	−0.769689	+	0.638419i	$0.779589\pi$
$938$	0	0
$939$	−3.10093e11	−0.398868
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	− 2.28025e12i	− 2.82325i
$949$	1.89539e12	2.33686
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	−1.18910e12	−1.42360
$957$	2.59038e11i	0.308827i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	8.52891e11	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	− 1.10556e12i	− 1.23856i
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−7.30112e11	−0.788340
$982$	0	0
$983$	1.41054e12i	1.51067i	0.655337	+	0.755337i	$0.272527\pi$
$983$	1.41054e12i	1.51067i	−0.655337	+	0.755337i	$0.727473\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6.68244e11i	0.704152i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	1.52931e11	0.158562	0.0792812	−	0.996852i	$-0.474738\pi$
$991$	1.52931e11	0.158562	0.0792812	+	0.996852i	$0.474738\pi$
$992$	0	0
$993$	6.13405e11i	0.630885i
$994$	0	0
$995$	0	0
$996$	−1.76172e12	−1.79019
$997$	− 1.43178e12i	− 1.44910i	−0.689225	−	0.724548i	$-0.742049\pi$
$997$	− 1.43178e12i	− 1.44910i	0.689225	−	0.724548i	$-0.257951\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.9.d.c.76.1		2
5.2	odd	4		35.9.c.a.34.1	✓	1
5.3	odd	4		35.9.c.b.34.1	yes	1
5.4	even	2	inner	175.9.d.c.76.2		2
7.6	odd	2	inner	175.9.d.c.76.2		2
35.13	even	4		35.9.c.a.34.1	✓	1
35.27	even	4		35.9.c.b.34.1	yes	1
35.34	odd	2	CM	175.9.d.c.76.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.9.c.a.34.1	✓	1	5.2	odd	4
35.9.c.a.34.1	✓	1	35.13	even	4
35.9.c.b.34.1	yes	1	5.3	odd	4
35.9.c.b.34.1	yes	1	35.27	even	4
175.9.d.c.76.1		2	1.1	even	1	trivial
175.9.d.c.76.1		2	35.34	odd	2	CM
175.9.d.c.76.2		2	5.4	even	2	inner
175.9.d.c.76.2		2	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	175.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-127.000i q^{3} -256.000 q^{4} +2401.00i q^{7} -9568.00 q^{9} +O(q^{10})\)	\(q-127.000i q^{3} -256.000 q^{4} +2401.00i q^{7} -9568.00 q^{9} -23953.0 q^{11} +32512.0i q^{12} +56593.0i q^{13} +65536.0 q^{16} -97873.0i q^{17} +304927. q^{21} +381889. i q^{27} -614656. i q^{28} +85153.0 q^{29} +3.04203e6i q^{33} +2.44941e6 q^{36} +7.18731e6 q^{39} +6.13197e6 q^{44} +2.19149e6i q^{47} -8.32307e6i q^{48} -5.76480e6 q^{49} -1.24299e7 q^{51} -1.44878e7i q^{52} -2.29728e7i q^{63} -1.67772e7 q^{64} +2.50555e7i q^{68} +5.07427e7 q^{71} -3.34915e7i q^{73} -5.75112e7i q^{77} -7.01357e7 q^{79} -1.42757e7 q^{81} +5.41867e7i q^{83} -7.80613e7 q^{84} -1.08144e7i q^{87} -1.35880e8 q^{91} -1.65614e8i q^{97} +2.29182e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 512 q^{4} - 19136 q^{9}+O(q^{10}) \) 2 * q - 512 * q^4 - 19136 * q^9	\( 2 q - 512 q^{4} - 19136 q^{9} - 47906 q^{11} + 131072 q^{16} + 609854 q^{21} + 170306 q^{29} + 4898816 q^{36} + 14374622 q^{39} + 12263936 q^{44} - 11529602 q^{49} - 24859742 q^{51} - 33554432 q^{64} + 101485444 q^{71} - 140271454 q^{79} - 28551490 q^{81} - 156122624 q^{84} - 271759586 q^{91} + 458364608 q^{99}+O(q^{100}) \) 2 * q - 512 * q^4 - 19136 * q^9 - 47906 * q^11 + 131072 * q^16 + 609854 * q^21 + 170306 * q^29 + 4898816 * q^36 + 14374622 * q^39 + 12263936 * q^44 - 11529602 * q^49 - 24859742 * q^51 - 33554432 * q^64 + 101485444 * q^71 - 140271454 * q^79 - 28551490 * q^81 - 156122624 * q^84 - 271759586 * q^91 + 458364608 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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