LMFDB - Embedded newform 320.5.b.d.191.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,5,Mod(191,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("320.191");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	320.b (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:	$33.0783881868$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.246034965625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 7x^{6} - 21x^{5} + 49x^{4} - 84x^{3} + 112x^{2} - 192x + 256 $ x^8 - 3x^7 + 7x^6 - 21x^5 + 49x^4 - 84x^3 + 112x^2 - 192*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{28}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.8
Root			$-1.02661 + 1.71641i$ of defining polynomial
Character	$\chi$	$=$	320.191
Dual form			320.5.b.d.191.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15.5779i q^{3} -11.1803 q^{5} -37.6230i q^{7} -161.671 q^{9} +O(q^{10})$	$q+15.5779i q^{3} -11.1803 q^{5} -37.6230i q^{7} -161.671 q^{9} -26.6928i q^{11} -58.0144 q^{13} -174.166i q^{15} +467.816 q^{17} -428.041i q^{19} +586.088 q^{21} -360.456i q^{23} +125.000 q^{25} -1256.68i q^{27} -964.509 q^{29} -417.993i q^{31} +415.818 q^{33} +420.638i q^{35} +1797.48 q^{37} -903.742i q^{39} -469.722 q^{41} -27.7492i q^{43} +1807.53 q^{45} +1538.96i q^{47} +985.506 q^{49} +7287.58i q^{51} +276.057 q^{53} +298.435i q^{55} +6667.98 q^{57} -3813.72i q^{59} +2051.87 q^{61} +6082.55i q^{63} +648.620 q^{65} -1165.73i q^{67} +5615.14 q^{69} +5689.40i q^{71} -2001.05 q^{73} +1947.24i q^{75} -1004.26 q^{77} -705.728i q^{79} +6481.10 q^{81} +1626.53i q^{83} -5230.34 q^{85} -15025.0i q^{87} +7156.62 q^{89} +2182.68i q^{91} +6511.45 q^{93} +4785.65i q^{95} -13005.4 q^{97} +4315.45i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 328 q^{9}+O(q^{10}) $ 8 * q - 328 * q^9	$ 8 q - 328 q^{9} - 352 q^{13} - 48 q^{17} - 16 q^{21} + 1000 q^{25} - 1200 q^{29} - 1120 q^{33} + 5728 q^{37} + 4896 q^{41} + 400 q^{45} - 5768 q^{49} - 2592 q^{53} + 3840 q^{57} - 7936 q^{61} - 1200 q^{65} + 2256 q^{69} - 14448 q^{73} - 2400 q^{77} - 936 q^{81} - 11200 q^{85} + 23760 q^{89} - 11360 q^{93} - 4368 q^{97}+O(q^{100}) $ 8 * q - 328 * q^9 - 352 * q^13 - 48 * q^17 - 16 * q^21 + 1000 * q^25 - 1200 * q^29 - 1120 * q^33 + 5728 * q^37 + 4896 * q^41 + 400 * q^45 - 5768 * q^49 - 2592 * q^53 + 3840 * q^57 - 7936 * q^61 - 1200 * q^65 + 2256 * q^69 - 14448 * q^73 - 2400 * q^77 - 936 * q^81 - 11200 * q^85 + 23760 * q^89 - 11360 * q^93 - 4368 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	15.5779i	1.73088i	0.501015	+	0.865438i	$0.332960\pi$
$3$	15.5779i	1.73088i	−0.501015	+	0.865438i	$0.667040\pi$
$4$	0	0
$5$	−11.1803	−0.447214
$5$	−11.1803	−0.447214
$6$	0	0
$7$	− 37.6230i	− 0.767817i	−0.923371	−	0.383909i	$-0.874578\pi$
$7$	− 37.6230i	− 0.767817i	0.923371	−	0.383909i	$-0.125422\pi$
$8$	0	0
$9$	−161.671	−1.99594
$10$	0	0
$11$	− 26.6928i	− 0.220602i	−0.993898	−	0.110301i	$-0.964819\pi$
$11$	− 26.6928i	− 0.220602i	0.993898	−	0.110301i	$-0.0351814\pi$
$12$	0	0
$13$	−58.0144	−0.343280	−0.171640	−	0.985160i	$-0.554907\pi$
$13$	−58.0144	−0.343280	−0.171640	+	0.985160i	$0.554907\pi$
$14$	0	0
$15$	− 174.166i	− 0.774072i
$16$	0	0
$17$	467.816	1.61874	0.809370	−	0.587300i	$-0.199809\pi$
$17$	467.816	1.61874	0.809370	+	0.587300i	$0.199809\pi$
$18$	0	0
$19$	− 428.041i	− 1.18571i	−0.805309	−	0.592855i	$-0.798001\pi$
$19$	− 428.041i	− 1.18571i	0.805309	−	0.592855i	$-0.201999\pi$
$20$	0	0
$21$	586.088	1.32900
$22$	0	0
$23$	− 360.456i	− 0.681391i	−0.940174	−	0.340695i	$-0.889338\pi$
$23$	− 360.456i	− 0.681391i	0.940174	−	0.340695i	$-0.110662\pi$
$24$	0	0
$25$	125.000	0.200000
$26$	0	0
$27$	− 1256.68i	− 1.72384i
$28$	0	0
$29$	−964.509	−1.14686	−0.573430	−	0.819255i	$-0.694388\pi$
$29$	−964.509	−1.14686	−0.573430	+	0.819255i	$0.694388\pi$
$30$	0	0
$31$	− 417.993i	− 0.434956i	−0.976065	−	0.217478i	$-0.930217\pi$
$31$	− 417.993i	− 0.434956i	0.976065	−	0.217478i	$-0.0697831\pi$
$32$	0	0
$33$	415.818	0.381834
$34$	0	0
$35$	420.638i	0.343378i
$36$	0	0
$37$	1797.48	1.31299	0.656495	−	0.754330i	$-0.272038\pi$
$37$	1797.48	1.31299	0.656495	+	0.754330i	$0.272038\pi$
$38$	0	0
$39$	− 903.742i	− 0.594176i
$40$	0	0
$41$	−469.722	−0.279430	−0.139715	−	0.990192i	$-0.544619\pi$
$41$	−469.722	−0.279430	−0.139715	+	0.990192i	$0.544619\pi$
$42$	0	0
$43$	− 27.7492i	− 0.0150077i	−0.999972	−	0.00750384i	$-0.997611\pi$
$43$	− 27.7492i	− 0.0150077i	0.999972	−	0.00750384i	$-0.00238857\pi$
$44$	0	0
$45$	1807.53	0.892609
$46$	0	0
$47$	1538.96i	0.696677i	0.937369	+	0.348339i	$0.113254\pi$
$47$	1538.96i	0.696677i	−0.937369	+	0.348339i	$0.886746\pi$
$48$	0	0
$49$	985.506	0.410457
$50$	0	0
$51$	7287.58i	2.80184i
$52$	0	0
$53$	276.057	0.0982758	0.0491379	−	0.998792i	$-0.484353\pi$
$53$	276.057	0.0982758	0.0491379	+	0.998792i	$0.484353\pi$
$54$	0	0
$55$	298.435i	0.0986561i
$56$	0	0
$57$	6667.98	2.05232
$58$	0	0
$59$	− 3813.72i	− 1.09558i	−0.836616	−	0.547791i	$-0.815469\pi$
$59$	− 3813.72i	− 1.09558i	0.836616	−	0.547791i	$-0.184531\pi$
$60$	0	0
$61$	2051.87	0.551429	0.275714	−	0.961240i	$-0.411086\pi$
$61$	2051.87	0.551429	0.275714	+	0.961240i	$0.411086\pi$
$62$	0	0
$63$	6082.55i	1.53251i
$64$	0	0
$65$	648.620	0.153520
$66$	0	0
$67$	− 1165.73i	− 0.259686i	−0.991535	−	0.129843i	$-0.958553\pi$
$67$	− 1165.73i	− 0.259686i	0.991535	−	0.129843i	$-0.0414474\pi$
$68$	0	0
$69$	5615.14	1.17940
$70$	0	0
$71$	5689.40i	1.12863i	0.825561	+	0.564313i	$0.190859\pi$
$71$	5689.40i	1.12863i	−0.825561	+	0.564313i	$0.809141\pi$
$72$	0	0
$73$	−2001.05	−0.375501	−0.187751	−	0.982217i	$-0.560120\pi$
$73$	−2001.05	−0.375501	−0.187751	+	0.982217i	$0.560120\pi$
$74$	0	0
$75$	1947.24i	0.346175i
$76$	0	0
$77$	−1004.26	−0.169382
$78$	0	0
$79$	− 705.728i	− 0.113079i	−0.998400	−	0.0565396i	$-0.981993\pi$
$79$	− 705.728i	− 0.113079i	0.998400	−	0.0565396i	$-0.0180067\pi$
$80$	0	0
$81$	6481.10	0.987822
$82$	0	0
$83$	1626.53i	0.236106i	0.993007	+	0.118053i	$0.0376652\pi$
$83$	1626.53i	0.236106i	−0.993007	+	0.118053i	$0.962335\pi$
$84$	0	0
$85$	−5230.34	−0.723922
$86$	0	0
$87$	− 15025.0i	− 1.98507i
$88$	0	0
$89$	7156.62	0.903499	0.451750	−	0.892145i	$-0.350800\pi$
$89$	7156.62	0.903499	0.451750	+	0.892145i	$0.350800\pi$
$90$	0	0
$91$	2182.68i	0.263577i
$92$	0	0
$93$	6511.45	0.752856
$94$	0	0
$95$	4785.65i	0.530266i
$96$	0	0
$97$	−13005.4	−1.38223	−0.691113	−	0.722747i	$-0.742879\pi$
$97$	−13005.4	−1.38223	−0.691113	+	0.722747i	$0.742879\pi$
$98$	0	0
$99$	4315.45i	0.440307i
$100$	0	0
$101$	9618.86	0.942933	0.471467	−	0.881884i	$-0.343725\pi$
$101$	9618.86	0.942933	0.471467	+	0.881884i	$0.343725\pi$
$102$	0	0
$103$	− 20364.5i	− 1.91955i	−0.280767	−	0.959776i	$-0.590589\pi$
$103$	− 20364.5i	− 1.91955i	0.280767	−	0.959776i	$-0.409411\pi$
$104$	0	0
$105$	−6552.66	−0.594346
$106$	0	0
$107$	− 22152.3i	− 1.93486i	−0.253129	−	0.967432i	$-0.581460\pi$
$107$	− 22152.3i	− 1.93486i	0.253129	−	0.967432i	$-0.418540\pi$
$108$	0	0
$109$	4013.04	0.337770	0.168885	−	0.985636i	$-0.445983\pi$
$109$	4013.04	0.337770	0.168885	+	0.985636i	$0.445983\pi$
$110$	0	0
$111$	28001.0i	2.27262i
$112$	0	0
$113$	22690.2	1.77698	0.888489	−	0.458897i	$-0.151755\pi$
$113$	22690.2	1.77698	0.888489	+	0.458897i	$0.151755\pi$
$114$	0	0
$115$	4030.02i	0.304727i
$116$	0	0
$117$	9379.23	0.685165
$118$	0	0
$119$	− 17600.6i	− 1.24290i
$120$	0	0
$121$	13928.5	0.951335
$122$	0	0
$123$	− 7317.28i	− 0.483659i
$124$	0	0
$125$	−1397.54	−0.0894427
$126$	0	0
$127$	− 22038.9i	− 1.36641i	−0.730225	−	0.683206i	$-0.760585\pi$
$127$	− 22038.9i	− 1.36641i	0.730225	−	0.683206i	$-0.239415\pi$
$128$	0	0
$129$	432.274	0.0259764
$130$	0	0
$131$	− 4968.89i	− 0.289545i	−0.989465	−	0.144773i	$-0.953755\pi$
$131$	− 4968.89i	− 0.289545i	0.989465	−	0.144773i	$-0.0462451\pi$
$132$	0	0
$133$	−16104.2	−0.910409
$134$	0	0
$135$	14050.1i	0.770925i
$136$	0	0
$137$	−11945.1	−0.636426	−0.318213	−	0.948019i	$-0.603083\pi$
$137$	−11945.1	−0.636426	−0.318213	+	0.948019i	$0.603083\pi$
$138$	0	0
$139$	− 20123.7i	− 1.04154i	−0.853696	−	0.520772i	$-0.825644\pi$
$139$	− 20123.7i	− 1.04154i	0.853696	−	0.520772i	$-0.174356\pi$
$140$	0	0
$141$	−23973.8	−1.20586
$142$	0	0
$143$	1548.57i	0.0757282i
$144$	0	0
$145$	10783.5	0.512891
$146$	0	0
$147$	15352.1i	0.710450i
$148$	0	0
$149$	6127.67	0.276009	0.138004	−	0.990432i	$-0.455931\pi$
$149$	6127.67	0.276009	0.138004	+	0.990432i	$0.455931\pi$
$150$	0	0
$151$	15290.9i	0.670626i	0.942107	+	0.335313i	$0.108842\pi$
$151$	15290.9i	0.670626i	−0.942107	+	0.335313i	$0.891158\pi$
$152$	0	0
$153$	−75632.1	−3.23090
$154$	0	0
$155$	4673.30i	0.194518i
$156$	0	0
$157$	−26931.5	−1.09260	−0.546300	−	0.837589i	$-0.683964\pi$
$157$	−26931.5	−1.09260	−0.546300	+	0.837589i	$0.683964\pi$
$158$	0	0
$159$	4300.38i	0.170103i
$160$	0	0
$161$	−13561.4	−0.523183
$162$	0	0
$163$	− 19964.4i	− 0.751418i	−0.926738	−	0.375709i	$-0.877399\pi$
$163$	− 19964.4i	− 0.751418i	0.926738	−	0.375709i	$-0.122601\pi$
$164$	0	0
$165$	−4648.98	−0.170762
$166$	0	0
$167$	− 11615.2i	− 0.416481i	−0.978078	−	0.208240i	$-0.933226\pi$
$167$	− 11615.2i	− 0.416481i	0.978078	−	0.208240i	$-0.0667737\pi$
$168$	0	0
$169$	−25195.3	−0.882159
$170$	0	0
$171$	69201.8i	2.36660i
$172$	0	0
$173$	−27260.4	−0.910835	−0.455417	−	0.890278i	$-0.650510\pi$
$173$	−27260.4	−0.910835	−0.455417	+	0.890278i	$0.650510\pi$
$174$	0	0
$175$	− 4702.88i	− 0.153563i
$176$	0	0
$177$	59409.7	1.89632
$178$	0	0
$179$	36368.3i	1.13505i	0.823355	+	0.567527i	$0.192100\pi$
$179$	36368.3i	1.13505i	−0.823355	+	0.567527i	$0.807900\pi$
$180$	0	0
$181$	46165.5	1.40916	0.704580	−	0.709625i	$-0.251136\pi$
$181$	46165.5	1.40916	0.704580	+	0.709625i	$0.251136\pi$
$182$	0	0
$183$	31963.8i	0.954456i
$184$	0	0
$185$	−20096.5	−0.587187
$186$	0	0
$187$	− 12487.3i	− 0.357097i
$188$	0	0
$189$	−47280.1	−1.32360
$190$	0	0
$191$	− 23694.9i	− 0.649513i	−0.945798	−	0.324757i	$-0.894718\pi$
$191$	− 23694.9i	− 0.649513i	0.945798	−	0.324757i	$-0.105282\pi$
$192$	0	0
$193$	−17101.4	−0.459111	−0.229556	−	0.973296i	$-0.573727\pi$
$193$	−17101.4	−0.459111	−0.229556	+	0.973296i	$0.573727\pi$
$194$	0	0
$195$	10104.1i	0.265724i
$196$	0	0
$197$	55597.2	1.43259	0.716293	−	0.697800i	$-0.245837\pi$
$197$	55597.2	1.43259	0.716293	+	0.697800i	$0.245837\pi$
$198$	0	0
$199$	58399.0i	1.47468i	0.675520	+	0.737342i	$0.263919\pi$
$199$	58399.0i	1.47468i	−0.675520	+	0.737342i	$0.736081\pi$
$200$	0	0
$201$	18159.7	0.449485
$202$	0	0
$203$	36287.8i	0.880579i
$204$	0	0
$205$	5251.65	0.124965
$206$	0	0
$207$	58275.1i	1.36001i
$208$	0	0
$209$	−11425.6	−0.261570
$210$	0	0
$211$	− 16960.3i	− 0.380951i	−0.981692	−	0.190475i	$-0.938997\pi$
$211$	− 16960.3i	− 0.380951i	0.981692	−	0.190475i	$-0.0610030\pi$
$212$	0	0
$213$	−88628.9	−1.95351
$214$	0	0
$215$	310.245i	0.00671163i
$216$	0	0
$217$	−15726.2	−0.333967
$218$	0	0
$219$	− 31172.1i	− 0.649947i
$220$	0	0
$221$	−27140.0	−0.555681
$222$	0	0
$223$	− 16817.9i	− 0.338191i	−0.985600	−	0.169095i	$-0.945915\pi$
$223$	− 16817.9i	− 0.338191i	0.985600	−	0.169095i	$-0.0540846\pi$
$224$	0	0
$225$	−20208.8	−0.399187
$226$	0	0
$227$	− 81040.8i	− 1.57272i	−0.617768	−	0.786361i	$-0.711963\pi$
$227$	− 81040.8i	− 1.57272i	0.617768	−	0.786361i	$-0.288037\pi$
$228$	0	0
$229$	−47720.6	−0.909987	−0.454994	−	0.890495i	$-0.650358\pi$
$229$	−47720.6	−0.909987	−0.454994	+	0.890495i	$0.650358\pi$
$230$	0	0
$231$	− 15644.3i	− 0.293179i
$232$	0	0
$233$	−15756.7	−0.290237	−0.145119	−	0.989414i	$-0.546356\pi$
$233$	−15756.7	−0.290237	−0.145119	+	0.989414i	$0.546356\pi$
$234$	0	0
$235$	− 17206.1i	− 0.311563i
$236$	0	0
$237$	10993.8	0.195726
$238$	0	0
$239$	− 110092.i	− 1.92735i	−0.267083	−	0.963673i	$-0.586060\pi$
$239$	− 110092.i	− 1.92735i	0.267083	−	0.963673i	$-0.413940\pi$
$240$	0	0
$241$	−26260.5	−0.452136	−0.226068	−	0.974112i	$-0.572587\pi$
$241$	−26260.5	−0.452136	−0.226068	+	0.974112i	$0.572587\pi$
$242$	0	0
$243$	− 829.219i	− 0.0140429i
$244$	0	0
$245$	−11018.3	−0.183562
$246$	0	0
$247$	24832.5i	0.407031i
$248$	0	0
$249$	−25338.0	−0.408670
$250$	0	0
$251$	− 24402.1i	− 0.387329i	−0.981068	−	0.193665i	$-0.937963\pi$
$251$	− 24402.1i	− 0.387329i	0.981068	−	0.193665i	$-0.0620374\pi$
$252$	0	0
$253$	−9621.57	−0.150316
$254$	0	0
$255$	− 81477.6i	− 1.25302i
$256$	0	0
$257$	30469.1	0.461311	0.230655	−	0.973036i	$-0.425913\pi$
$257$	30469.1	0.461311	0.230655	+	0.973036i	$0.425913\pi$
$258$	0	0
$259$	− 67626.8i	− 1.00814i
$260$	0	0
$261$	155933.	2.28906
$262$	0	0
$263$	− 45832.0i	− 0.662610i	−0.943524	−	0.331305i	$-0.892511\pi$
$263$	− 45832.0i	− 0.662610i	0.943524	−	0.331305i	$-0.107489\pi$
$264$	0	0
$265$	−3086.41	−0.0439503
$266$	0	0
$267$	111485.i	1.56385i
$268$	0	0
$269$	36915.5	0.510157	0.255078	−	0.966920i	$-0.417899\pi$
$269$	36915.5	0.510157	0.255078	+	0.966920i	$0.417899\pi$
$270$	0	0
$271$	− 123746.i	− 1.68497i	−0.538720	−	0.842485i	$-0.681092\pi$
$271$	− 123746.i	− 1.68497i	0.538720	−	0.842485i	$-0.318908\pi$
$272$	0	0
$273$	−34001.5	−0.456219
$274$	0	0
$275$	− 3336.60i	− 0.0441203i
$276$	0	0
$277$	−95245.1	−1.24132	−0.620659	−	0.784081i	$-0.713135\pi$
$277$	−95245.1	−1.24132	−0.620659	+	0.784081i	$0.713135\pi$
$278$	0	0
$279$	67577.2i	0.868144i
$280$	0	0
$281$	58728.9	0.743771	0.371885	−	0.928279i	$-0.378711\pi$
$281$	58728.9	0.743771	0.371885	+	0.928279i	$0.378711\pi$
$282$	0	0
$283$	71397.1i	0.891472i	0.895164	+	0.445736i	$0.147058\pi$
$283$	71397.1i	0.891472i	−0.895164	+	0.445736i	$0.852942\pi$
$284$	0	0
$285$	−74550.3	−0.917825
$286$	0	0
$287$	17672.4i	0.214551i
$288$	0	0
$289$	135330.	1.62032
$290$	0	0
$291$	− 202596.i	− 2.39246i
$292$	0	0
$293$	56264.5	0.655390	0.327695	−	0.944784i	$-0.393728\pi$
$293$	56264.5	0.655390	0.327695	+	0.944784i	$0.393728\pi$
$294$	0	0
$295$	42638.7i	0.489959i
$296$	0	0
$297$	−33544.3	−0.380282
$298$	0	0
$299$	20911.6i	0.233908i
$300$	0	0
$301$	−1044.01	−0.0115231
$302$	0	0
$303$	149842.i	1.63210i
$304$	0	0
$305$	−22940.6	−0.246607
$306$	0	0
$307$	175077.i	1.85760i	0.370577	+	0.928802i	$0.379160\pi$
$307$	175077.i	1.85760i	−0.370577	+	0.928802i	$0.620840\pi$
$308$	0	0
$309$	317236.	3.32251
$310$	0	0
$311$	− 23621.0i	− 0.244218i	−0.992517	−	0.122109i	$-0.961034\pi$
$311$	− 23621.0i	− 0.244218i	0.992517	−	0.122109i	$-0.0389657\pi$
$312$	0	0
$313$	−132585.	−1.35333	−0.676667	−	0.736289i	$-0.736576\pi$
$313$	−132585.	−1.35333	−0.676667	+	0.736289i	$0.736576\pi$
$314$	0	0
$315$	− 68004.9i	− 0.685361i
$316$	0	0
$317$	33855.0	0.336902	0.168451	−	0.985710i	$-0.446123\pi$
$317$	33855.0	0.336902	0.168451	+	0.985710i	$0.446123\pi$
$318$	0	0
$319$	25745.5i	0.252999i
$320$	0	0
$321$	345086.	3.34901
$322$	0	0
$323$	− 200244.i	− 1.91936i
$324$	0	0
$325$	−7251.80	−0.0686561
$326$	0	0
$327$	62514.8i	0.584638i
$328$	0	0
$329$	57900.4	0.534921
$330$	0	0
$331$	− 24338.3i	− 0.222144i	−0.993812	−	0.111072i	$-0.964572\pi$
$331$	− 24338.3i	− 0.222144i	0.993812	−	0.111072i	$-0.0354283\pi$
$332$	0	0
$333$	−290601.	−2.62064
$334$	0	0
$335$	13033.3i	0.116135i
$336$	0	0
$337$	5373.65	0.0473161	0.0236581	−	0.999720i	$-0.492469\pi$
$337$	5373.65	0.0473161	0.0236581	+	0.999720i	$0.492469\pi$
$338$	0	0
$339$	353466.i	3.07573i
$340$	0	0
$341$	−11157.4	−0.0959521
$342$	0	0
$343$	− 127411.i	− 1.08297i
$344$	0	0
$345$	−62779.2	−0.527445
$346$	0	0
$347$	78906.7i	0.655322i	0.944795	+	0.327661i	$0.106260\pi$
$347$	78906.7i	0.655322i	−0.944795	+	0.327661i	$0.893740\pi$
$348$	0	0
$349$	−139841.	−1.14811	−0.574053	−	0.818818i	$-0.694630\pi$
$349$	−139841.	−1.14811	−0.574053	+	0.818818i	$0.694630\pi$
$350$	0	0
$351$	72905.5i	0.591761i
$352$	0	0
$353$	−35542.6	−0.285233	−0.142616	−	0.989778i	$-0.545552\pi$
$353$	−35542.6	−0.285233	−0.142616	+	0.989778i	$0.545552\pi$
$354$	0	0
$355$	− 63609.4i	− 0.504737i
$356$	0	0
$357$	274181.	2.15130
$358$	0	0
$359$	179016.i	1.38900i	0.719491	+	0.694502i	$0.244375\pi$
$359$	179016.i	1.38900i	−0.719491	+	0.694502i	$0.755625\pi$
$360$	0	0
$361$	−52898.4	−0.405908
$362$	0	0
$363$	216977.i	1.64664i
$364$	0	0
$365$	22372.4	0.167929
$366$	0	0
$367$	32200.7i	0.239075i	0.992830	+	0.119537i	$0.0381411\pi$
$367$	32200.7i	0.239075i	−0.992830	+	0.119537i	$0.961859\pi$
$368$	0	0
$369$	75940.3	0.557725
$370$	0	0
$371$	− 10386.1i	− 0.0754578i
$372$	0	0
$373$	204296.	1.46839	0.734196	−	0.678937i	$-0.237559\pi$
$373$	204296.	1.46839	0.734196	+	0.678937i	$0.237559\pi$
$374$	0	0
$375$	− 21770.8i	− 0.154814i
$376$	0	0
$377$	55955.4	0.393694
$378$	0	0
$379$	− 135870.i	− 0.945903i	−0.881089	−	0.472951i	$-0.843189\pi$
$379$	− 135870.i	− 0.945903i	0.881089	−	0.472951i	$-0.156811\pi$
$380$	0	0
$381$	343319.	2.36509
$382$	0	0
$383$	155856.i	1.06249i	0.847218	+	0.531245i	$0.178276\pi$
$383$	155856.i	1.06249i	−0.847218	+	0.531245i	$0.821724\pi$
$384$	0	0
$385$	11228.0	0.0757498
$386$	0	0
$387$	4486.23i	0.0299543i
$388$	0	0
$389$	−91378.0	−0.603869	−0.301934	−	0.953329i	$-0.597632\pi$
$389$	−91378.0	−0.603869	−0.301934	+	0.953329i	$0.597632\pi$
$390$	0	0
$391$	− 168627.i	− 1.10299i
$392$	0	0
$393$	77404.8	0.501167
$394$	0	0
$395$	7890.28i	0.0505706i
$396$	0	0
$397$	177315.	1.12503	0.562515	−	0.826787i	$-0.309834\pi$
$397$	177315.	1.12503	0.562515	+	0.826787i	$0.309834\pi$
$398$	0	0
$399$	− 250870.i	− 1.57581i
$400$	0	0
$401$	198129.	1.23214	0.616070	−	0.787691i	$-0.288724\pi$
$401$	198129.	1.23214	0.616070	+	0.787691i	$0.288724\pi$
$402$	0	0
$403$	24249.6i	0.149312i
$404$	0	0
$405$	−72460.9	−0.441767
$406$	0	0
$407$	− 47979.9i	− 0.289648i
$408$	0	0
$409$	−176772.	−1.05674	−0.528370	−	0.849014i	$-0.677196\pi$
$409$	−176772.	−1.05674	−0.528370	+	0.849014i	$0.677196\pi$
$410$	0	0
$411$	− 186079.i	− 1.10158i
$412$	0	0
$413$	−143484.	−0.841206
$414$	0	0
$415$	− 18185.2i	− 0.105590i
$416$	0	0
$417$	313484.	1.80278
$418$	0	0
$419$	152807.i	0.870390i	0.900336	+	0.435195i	$0.143321\pi$
$419$	152807.i	0.870390i	−0.900336	+	0.435195i	$0.856679\pi$
$420$	0	0
$421$	−196597.	−1.10921	−0.554604	−	0.832115i	$-0.687130\pi$
$421$	−196597.	−1.10921	−0.554604	+	0.832115i	$0.687130\pi$
$422$	0	0
$423$	− 248805.i	− 1.39052i
$424$	0	0
$425$	58477.0	0.323748
$426$	0	0
$427$	− 77197.5i	− 0.423397i
$428$	0	0
$429$	−24123.4	−0.131076
$430$	0	0
$431$	230818.i	1.24255i	0.783592	+	0.621276i	$0.213385\pi$
$431$	230818.i	1.24255i	−0.783592	+	0.621276i	$0.786615\pi$
$432$	0	0
$433$	−284920.	−1.51966	−0.759832	−	0.650119i	$-0.774719\pi$
$433$	−284920.	−1.51966	−0.759832	+	0.650119i	$0.774719\pi$
$434$	0	0
$435$	167985.i	0.887752i
$436$	0	0
$437$	−154290.	−0.807932
$438$	0	0
$439$	327852.i	1.70118i	0.525832	+	0.850588i	$0.323754\pi$
$439$	327852.i	1.70118i	−0.525832	+	0.850588i	$0.676246\pi$
$440$	0	0
$441$	−159328.	−0.819245
$442$	0	0
$443$	− 179147.i	− 0.912854i	−0.889761	−	0.456427i	$-0.849129\pi$
$443$	− 179147.i	− 0.912854i	0.889761	−	0.456427i	$-0.150871\pi$
$444$	0	0
$445$	−80013.4	−0.404057
$446$	0	0
$447$	95456.2i	0.477737i
$448$	0	0
$449$	−308905.	−1.53226	−0.766129	−	0.642687i	$-0.777820\pi$
$449$	−308905.	−1.53226	−0.766129	+	0.642687i	$0.777820\pi$
$450$	0	0
$451$	12538.2i	0.0616428i
$452$	0	0
$453$	−238201.	−1.16077
$454$	0	0
$455$	− 24403.1i	− 0.117875i
$456$	0	0
$457$	56777.8	0.271861	0.135930	−	0.990718i	$-0.456598\pi$
$457$	56777.8	0.271861	0.135930	+	0.990718i	$0.456598\pi$
$458$	0	0
$459$	− 587895.i	− 2.79045i
$460$	0	0
$461$	259736.	1.22217	0.611084	−	0.791566i	$-0.290734\pi$
$461$	259736.	1.22217	0.611084	+	0.791566i	$0.290734\pi$
$462$	0	0
$463$	− 64677.1i	− 0.301709i	−0.988556	−	0.150855i	$-0.951797\pi$
$463$	− 64677.1i	− 0.301709i	0.988556	−	0.150855i	$-0.0482025\pi$
$464$	0	0
$465$	−72800.2	−0.336687
$466$	0	0
$467$	− 105097.i	− 0.481899i	−0.970538	−	0.240950i	$-0.922541\pi$
$467$	− 105097.i	− 0.481899i	0.970538	−	0.240950i	$-0.0774589\pi$
$468$	0	0
$469$	−43858.4	−0.199392
$470$	0	0
$471$	− 419536.i	− 1.89116i
$472$	0	0
$473$	−740.704	−0.00331072
$474$	0	0
$475$	− 53505.2i	− 0.237142i
$476$	0	0
$477$	−44630.3	−0.196152
$478$	0	0
$479$	− 131187.i	− 0.571770i	−0.958264	−	0.285885i	$-0.907712\pi$
$479$	− 131187.i	− 0.571770i	0.958264	−	0.285885i	$-0.0922875\pi$
$480$	0	0
$481$	−104280.	−0.450724
$482$	0	0
$483$	− 211259.i	− 0.905566i
$484$	0	0
$485$	145404.	0.618150
$486$	0	0
$487$	− 34829.3i	− 0.146854i	−0.997301	−	0.0734271i	$-0.976606\pi$
$487$	− 34829.3i	− 0.146854i	0.997301	−	0.0734271i	$-0.0233936\pi$
$488$	0	0
$489$	311004.	1.30061
$490$	0	0
$491$	357740.i	1.48390i	0.670456	+	0.741950i	$0.266099\pi$
$491$	357740.i	1.48390i	−0.670456	+	0.741950i	$0.733901\pi$
$492$	0	0
$493$	−451213.	−1.85647
$494$	0	0
$495$	− 48248.2i	− 0.196911i
$496$	0	0
$497$	214053.	0.866578
$498$	0	0
$499$	235685.i	0.946523i	0.880922	+	0.473261i	$0.156923\pi$
$499$	235685.i	0.946523i	−0.880922	+	0.473261i	$0.843077\pi$
$500$	0	0
$501$	180941.	0.720877
$502$	0	0
$503$	− 94290.4i	− 0.372676i	−0.982486	−	0.186338i	$-0.940338\pi$
$503$	− 94290.4i	− 0.372676i	0.982486	−	0.186338i	$-0.0596619\pi$
$504$	0	0
$505$	−107542.	−0.421693
$506$	0	0
$507$	− 392490.i	− 1.52691i
$508$	0	0
$509$	145708.	0.562405	0.281202	−	0.959649i	$-0.409267\pi$
$509$	145708.	0.562405	0.281202	+	0.959649i	$0.409267\pi$
$510$	0	0
$511$	75285.5i	0.288316i
$512$	0	0
$513$	−537911.	−2.04398
$514$	0	0
$515$	227682.i	0.858450i
$516$	0	0
$517$	41079.2	0.153688
$518$	0	0
$519$	− 424659.i	− 1.57654i
$520$	0	0
$521$	115380.	0.425064	0.212532	−	0.977154i	$-0.431829\pi$
$521$	115380.	0.425064	0.212532	+	0.977154i	$0.431829\pi$
$522$	0	0
$523$	339555.i	1.24139i	0.784054	+	0.620693i	$0.213149\pi$
$523$	339555.i	1.24139i	−0.784054	+	0.620693i	$0.786851\pi$
$524$	0	0
$525$	73261.0	0.265799
$526$	0	0
$527$	− 195544.i	− 0.704081i
$528$	0	0
$529$	149913.	0.535707
$530$	0	0
$531$	616567.i	2.18671i
$532$	0	0
$533$	27250.6	0.0959229
$534$	0	0
$535$	247670.i	0.865298i
$536$	0	0
$537$	−566541.	−1.96464
$538$	0	0
$539$	− 26305.9i	− 0.0905474i
$540$	0	0
$541$	−274692.	−0.938539	−0.469269	−	0.883055i	$-0.655483\pi$
$541$	−274692.	−0.938539	−0.469269	+	0.883055i	$0.655483\pi$
$542$	0	0
$543$	719161.i	2.43908i
$544$	0	0
$545$	−44867.2	−0.151055
$546$	0	0
$547$	− 354189.i	− 1.18375i	−0.806029	−	0.591875i	$-0.798388\pi$
$547$	− 354189.i	− 1.18375i	0.806029	−	0.591875i	$-0.201612\pi$
$548$	0	0
$549$	−331727.	−1.10062
$550$	0	0
$551$	412850.i	1.35984i
$552$	0	0
$553$	−26551.6	−0.0868242
$554$	0	0
$555$	− 313061.i	− 1.01635i
$556$	0	0
$557$	2288.63	0.00737675	0.00368838	−	0.999993i	$-0.498826\pi$
$557$	2288.63	0.00737675	0.00368838	+	0.999993i	$0.498826\pi$
$558$	0	0
$559$	1609.85i	0.00515184i
$560$	0	0
$561$	194526.	0.618090
$562$	0	0
$563$	− 48825.1i	− 0.154037i	−0.997030	−	0.0770187i	$-0.975460\pi$
$563$	− 48825.1i	− 0.154037i	0.997030	−	0.0770187i	$-0.0245401\pi$
$564$	0	0
$565$	−253685.	−0.794689
$566$	0	0
$567$	− 243839.i	− 0.758467i
$568$	0	0
$569$	202790.	0.626356	0.313178	−	0.949695i	$-0.398606\pi$
$569$	202790.	0.626356	0.313178	+	0.949695i	$0.398606\pi$
$570$	0	0
$571$	412759.i	1.26597i	0.774163	+	0.632986i	$0.218171\pi$
$571$	412759.i	1.26597i	−0.774163	+	0.632986i	$0.781829\pi$
$572$	0	0
$573$	369116.	1.12423
$574$	0	0
$575$	− 45057.0i	− 0.136278i
$576$	0	0
$577$	−441255.	−1.32537	−0.662686	−	0.748897i	$-0.730584\pi$
$577$	−441255.	−1.32537	−0.662686	+	0.748897i	$0.730584\pi$
$578$	0	0
$579$	− 266404.i	− 0.794665i
$580$	0	0
$581$	61195.1	0.181286
$582$	0	0
$583$	− 7368.73i	− 0.0216798i
$584$	0	0
$585$	−104863.	−0.306415
$586$	0	0
$587$	416031.i	1.20739i	0.797214	+	0.603697i	$0.206306\pi$
$587$	416031.i	1.20739i	−0.797214	+	0.603697i	$0.793694\pi$
$588$	0	0
$589$	−178918.	−0.515732
$590$	0	0
$591$	866088.i	2.47963i
$592$	0	0
$593$	342542.	0.974101	0.487051	−	0.873374i	$-0.338073\pi$
$593$	342542.	0.974101	0.487051	+	0.873374i	$0.338073\pi$
$594$	0	0
$595$	196781.i	0.555840i
$596$	0	0
$597$	−909733.	−2.55250
$598$	0	0
$599$	472260.i	1.31622i	0.752923	+	0.658109i	$0.228643\pi$
$599$	472260.i	1.31622i	−0.752923	+	0.658109i	$0.771357\pi$
$600$	0	0
$601$	−211692.	−0.586078	−0.293039	−	0.956101i	$-0.594667\pi$
$601$	−211692.	−0.586078	−0.293039	+	0.956101i	$0.594667\pi$
$602$	0	0
$603$	188465.i	0.518317i
$604$	0	0
$605$	−155725.	−0.425450
$606$	0	0
$607$	9168.57i	0.0248842i	0.999923	+	0.0124421i	$0.00396055\pi$
$607$	9168.57i	0.0248842i	−0.999923	+	0.0124421i	$0.996039\pi$
$608$	0	0
$609$	−565287.	−1.52417
$610$	0	0
$611$	− 89281.8i	− 0.239156i
$612$	0	0
$613$	−164930.	−0.438915	−0.219457	−	0.975622i	$-0.570429\pi$
$613$	−164930.	−0.438915	−0.219457	+	0.975622i	$0.570429\pi$
$614$	0	0
$615$	81809.7i	0.216299i
$616$	0	0
$617$	32007.8	0.0840787	0.0420394	−	0.999116i	$-0.486615\pi$
$617$	32007.8	0.0840787	0.0420394	+	0.999116i	$0.486615\pi$
$618$	0	0
$619$	30448.1i	0.0794655i	0.999210	+	0.0397328i	$0.0126507\pi$
$619$	30448.1i	0.0794655i	−0.999210	+	0.0397328i	$0.987349\pi$
$620$	0	0
$621$	−452977.	−1.17461
$622$	0	0
$623$	− 269254.i	− 0.693722i
$624$	0	0
$625$	15625.0	0.0400000
$626$	0	0
$627$	− 177987.i	− 0.452745i
$628$	0	0
$629$	840891.	2.12539
$630$	0	0
$631$	205747.i	0.516744i	0.966045	+	0.258372i	$0.0831861\pi$
$631$	205747.i	0.516744i	−0.966045	+	0.258372i	$0.916814\pi$
$632$	0	0
$633$	264206.	0.659379
$634$	0	0
$635$	246402.i	0.611078i
$636$	0	0
$637$	−57173.5	−0.140902
$638$	0	0
$639$	− 919809.i	− 2.25266i
$640$	0	0
$641$	−588976.	−1.43345	−0.716723	−	0.697358i	$-0.754359\pi$
$641$	−588976.	−1.43345	−0.716723	+	0.697358i	$0.754359\pi$
$642$	0	0
$643$	− 231907.i	− 0.560909i	−0.959867	−	0.280454i	$-0.909515\pi$
$643$	− 231907.i	− 0.560909i	0.959867	−	0.280454i	$-0.0904852\pi$
$644$	0	0
$645$	−4832.97	−0.0116170
$646$	0	0
$647$	− 427586.i	− 1.02144i	−0.859746	−	0.510722i	$-0.829378\pi$
$647$	− 427586.i	− 1.02144i	0.859746	−	0.510722i	$-0.170622\pi$
$648$	0	0
$649$	−101799.	−0.241687
$650$	0	0
$651$	− 244981.i	− 0.578056i
$652$	0	0
$653$	348146.	0.816461	0.408231	−	0.912879i	$-0.366146\pi$
$653$	348146.	0.816461	0.408231	+	0.912879i	$0.366146\pi$
$654$	0	0
$655$	55553.8i	0.129489i
$656$	0	0
$657$	323511.	0.749477
$658$	0	0
$659$	64952.7i	0.149564i	0.997200	+	0.0747818i	$0.0238260\pi$
$659$	64952.7i	0.149564i	−0.997200	+	0.0747818i	$0.976174\pi$
$660$	0	0
$661$	175663.	0.402048	0.201024	−	0.979586i	$-0.435573\pi$
$661$	175663.	0.402048	0.201024	+	0.979586i	$0.435573\pi$
$662$	0	0
$663$	− 422784.i	− 0.961816i
$664$	0	0
$665$	180051.	0.407147
$666$	0	0
$667$	347663.i	0.781460i
$668$	0	0
$669$	261987.	0.585367
$670$	0	0
$671$	− 54770.1i	− 0.121646i
$672$	0	0
$673$	166579.	0.367781	0.183891	−	0.982947i	$-0.441131\pi$
$673$	166579.	0.367781	0.183891	+	0.982947i	$0.441131\pi$
$674$	0	0
$675$	− 157085.i	− 0.344768i
$676$	0	0
$677$	528530.	1.15317	0.576584	−	0.817038i	$-0.304385\pi$
$677$	528530.	1.15317	0.576584	+	0.817038i	$0.304385\pi$
$678$	0	0
$679$	489301.i	1.06130i
$680$	0	0
$681$	1.26244e6	2.72219
$682$	0	0
$683$	144855.i	0.310521i	0.987874	+	0.155261i	$0.0496217\pi$
$683$	144855.i	0.310521i	−0.987874	+	0.155261i	$0.950378\pi$
$684$	0	0
$685$	133550.	0.284619
$686$	0	0
$687$	− 743387.i	− 1.57508i
$688$	0	0
$689$	−16015.3	−0.0337361
$690$	0	0
$691$	− 893708.i	− 1.87171i	−0.352380	−	0.935857i	$-0.614628\pi$
$691$	− 893708.i	− 1.87171i	0.352380	−	0.935857i	$-0.385372\pi$
$692$	0	0
$693$	162360.	0.338075
$694$	0	0
$695$	224989.i	0.465792i
$696$	0	0
$697$	−219743.	−0.452325
$698$	0	0
$699$	− 245456.i	− 0.502365i
$700$	0	0
$701$	24472.5	0.0498015	0.0249008	−	0.999690i	$-0.492073\pi$
$701$	24472.5	0.0498015	0.0249008	+	0.999690i	$0.492073\pi$
$702$	0	0
$703$	− 769397.i	− 1.55683i
$704$	0	0
$705$	268035.	0.539278
$706$	0	0
$707$	− 361891.i	− 0.724001i
$708$	0	0
$709$	−705573.	−1.40362	−0.701809	−	0.712365i	$-0.747624\pi$
$709$	−705573.	−1.40362	−0.701809	+	0.712365i	$0.747624\pi$
$710$	0	0
$711$	114096.i	0.225699i
$712$	0	0
$713$	−150668.	−0.296375
$714$	0	0
$715$	− 17313.5i	− 0.0338667i
$716$	0	0
$717$	1.71500e6	3.33600
$718$	0	0
$719$	− 430977.i	− 0.833675i	−0.908981	−	0.416837i	$-0.863138\pi$
$719$	− 430977.i	− 0.833675i	0.908981	−	0.416837i	$-0.136862\pi$
$720$	0	0
$721$	−766176.	−1.47387
$722$	0	0
$723$	− 409084.i	− 0.782592i
$724$	0	0
$725$	−120564.	−0.229372
$726$	0	0
$727$	709623.i	1.34264i	0.741168	+	0.671319i	$0.234272\pi$
$727$	709623.i	1.34264i	−0.741168	+	0.671319i	$0.765728\pi$
$728$	0	0
$729$	537887.	1.01213
$730$	0	0
$731$	− 12981.5i	− 0.0242935i
$732$	0	0
$733$	−99574.1	−0.185327	−0.0926634	−	0.995697i	$-0.529538\pi$
$733$	−99574.1	−0.185327	−0.0926634	+	0.995697i	$0.529538\pi$
$734$	0	0
$735$	− 171642.i	− 0.317723i
$736$	0	0
$737$	−31116.7	−0.0572873
$738$	0	0
$739$	436339.i	0.798979i	0.916738	+	0.399490i	$0.130813\pi$
$739$	436339.i	0.798979i	−0.916738	+	0.399490i	$0.869187\pi$
$740$	0	0
$741$	−386839.	−0.704520
$742$	0	0
$743$	220581.i	0.399568i	0.979840	+	0.199784i	$0.0640240\pi$
$743$	220581.i	0.399568i	−0.979840	+	0.199784i	$0.935976\pi$
$744$	0	0
$745$	−68509.5	−0.123435
$746$	0	0
$747$	− 262963.i	− 0.471252i
$748$	0	0
$749$	−833436.	−1.48562
$750$	0	0
$751$	− 201164.i	− 0.356673i	−0.983970	−	0.178336i	$-0.942929\pi$
$751$	− 201164.i	− 0.356673i	0.983970	−	0.178336i	$-0.0570715\pi$
$752$	0	0
$753$	380134.	0.670420
$754$	0	0
$755$	− 170958.i	− 0.299913i
$756$	0	0
$757$	−915352.	−1.59734	−0.798669	−	0.601771i	$-0.794462\pi$
$757$	−915352.	−1.59734	−0.798669	+	0.601771i	$0.794462\pi$
$758$	0	0
$759$	− 149884.i	− 0.260178i
$760$	0	0
$761$	363462.	0.627610	0.313805	−	0.949488i	$-0.398396\pi$
$761$	363462.	0.627610	0.313805	+	0.949488i	$0.398396\pi$
$762$	0	0
$763$	− 150983.i	− 0.259345i
$764$	0	0
$765$	845593.	1.44490
$766$	0	0
$767$	221250.i	0.376091i
$768$	0	0
$769$	−97671.8	−0.165164	−0.0825822	−	0.996584i	$-0.526317\pi$
$769$	−97671.8	−0.165164	−0.0825822	+	0.996584i	$0.526317\pi$
$770$	0	0
$771$	474645.i	0.798472i
$772$	0	0
$773$	−517767.	−0.866514	−0.433257	−	0.901270i	$-0.642636\pi$
$773$	−517767.	−0.866514	−0.433257	+	0.901270i	$0.642636\pi$
$774$	0	0
$775$	− 52249.1i	− 0.0869912i
$776$	0	0
$777$	1.05348e6	1.74496
$778$	0	0
$779$	201061.i	0.331323i
$780$	0	0
$781$	151866.	0.248977
$782$	0	0
$783$	1.21208e6i	1.97700i
$784$	0	0
$785$	301103.	0.488626
$786$	0	0
$787$	199421.i	0.321974i	0.986957	+	0.160987i	$0.0514677\pi$
$787$	199421.i	0.321974i	−0.986957	+	0.160987i	$0.948532\pi$
$788$	0	0
$789$	713967.	1.14690
$790$	0	0
$791$	− 853676.i	− 1.36440i
$792$	0	0
$793$	−119038.	−0.189295
$794$	0	0
$795$	− 48079.7i	− 0.0760725i
$796$	0	0
$797$	204919.	0.322600	0.161300	−	0.986905i	$-0.448431\pi$
$797$	204919.	0.322600	0.161300	+	0.986905i	$0.448431\pi$
$798$	0	0
$799$	719949.i	1.12774i
$800$	0	0
$801$	−1.15702e6	−1.80333
$802$	0	0
$803$	53413.6i	0.0828363i
$804$	0	0
$805$	151621.	0.233975
$806$	0	0
$807$	575065.i	0.883019i
$808$	0	0
$809$	658877.	1.00672	0.503359	−	0.864078i	$-0.332097\pi$
$809$	658877.	1.00672	0.503359	+	0.864078i	$0.332097\pi$
$810$	0	0
$811$	464925.i	0.706872i	0.935459	+	0.353436i	$0.114987\pi$
$811$	464925.i	0.706872i	−0.935459	+	0.353436i	$0.885013\pi$
$812$	0	0
$813$	1.92770e6	2.91648
$814$	0	0
$815$	223209.i	0.336044i
$816$	0	0
$817$	−11877.8	−0.0177947
$818$	0	0
$819$	− 352875.i	− 0.526082i
$820$	0	0
$821$	−202623.	−0.300610	−0.150305	−	0.988640i	$-0.548026\pi$
$821$	−202623.	−0.300610	−0.150305	+	0.988640i	$0.548026\pi$
$822$	0	0
$823$	316525.i	0.467313i	0.972319	+	0.233656i	$0.0750691\pi$
$823$	316525.i	0.467313i	−0.972319	+	0.233656i	$0.924931\pi$
$824$	0	0
$825$	51977.2	0.0763669
$826$	0	0
$827$	− 148466.i	− 0.217077i	−0.994092	−	0.108539i	$-0.965383\pi$
$827$	− 148466.i	− 0.217077i	0.994092	−	0.108539i	$-0.0346171\pi$
$828$	0	0
$829$	834966.	1.21495	0.607477	−	0.794337i	$-0.292182\pi$
$829$	834966.	1.21495	0.607477	+	0.794337i	$0.292182\pi$
$830$	0	0
$831$	− 1.48372e6i	− 2.14857i
$832$	0	0
$833$	461035.	0.664422
$834$	0	0
$835$	129862.i	0.186256i
$836$	0	0
$837$	−525283.	−0.749796
$838$	0	0
$839$	1.17926e6i	1.67527i	0.546229	+	0.837636i	$0.316063\pi$
$839$	1.17926e6i	1.67527i	−0.546229	+	0.837636i	$0.683937\pi$
$840$	0	0
$841$	222997.	0.315288
$842$	0	0
$843$	914872.i	1.28738i
$844$	0	0
$845$	281692.	0.394513
$846$	0	0
$847$	− 524032.i	− 0.730451i
$848$	0	0
$849$	−1.11222e6	−1.54303
$850$	0	0
$851$	− 647913.i	− 0.894659i
$852$	0	0
$853$	533912.	0.733790	0.366895	−	0.930262i	$-0.380421\pi$
$853$	533912.	0.733790	0.366895	+	0.930262i	$0.380421\pi$
$854$	0	0
$855$	− 773699.i	− 1.05838i
$856$	0	0
$857$	155720.	0.212023	0.106011	−	0.994365i	$-0.466192\pi$
$857$	155720.	0.212023	0.106011	+	0.994365i	$0.466192\pi$
$858$	0	0
$859$	1.28910e6i	1.74703i	0.486795	+	0.873516i	$0.338166\pi$
$859$	1.28910e6i	1.74703i	−0.486795	+	0.873516i	$0.661834\pi$
$860$	0	0
$861$	−275298.	−0.371362
$862$	0	0
$863$	− 573138.i	− 0.769551i	−0.923010	−	0.384776i	$-0.874279\pi$
$863$	− 573138.i	− 0.769551i	0.923010	−	0.384776i	$-0.125721\pi$
$864$	0	0
$865$	304780.	0.407338
$866$	0	0
$867$	2.10816e6i	2.80457i
$868$	0	0
$869$	−18837.9	−0.0249455
$870$	0	0
$871$	67629.2i	0.0891452i
$872$	0	0
$873$	2.10259e6	2.75883
$874$	0	0
$875$	52579.8i	0.0686757i
$876$	0	0
$877$	−1.08498e6	−1.41066	−0.705328	−	0.708881i	$-0.749200\pi$
$877$	−1.08498e6	−1.41066	−0.705328	+	0.708881i	$0.749200\pi$
$878$	0	0
$879$	876483.i	1.13440i
$880$	0	0
$881$	443843.	0.571844	0.285922	−	0.958253i	$-0.407700\pi$
$881$	443843.	0.571844	0.285922	+	0.958253i	$0.407700\pi$
$882$	0	0
$883$	− 1.21061e6i	− 1.55269i	−0.630310	−	0.776344i	$-0.717072\pi$
$883$	− 1.21061e6i	− 1.55269i	0.630310	−	0.776344i	$-0.282928\pi$
$884$	0	0
$885$	−664220.	−0.848058
$886$	0	0
$887$	1.17188e6i	1.48949i	0.667352	+	0.744743i	$0.267428\pi$
$887$	1.17188e6i	1.48949i	−0.667352	+	0.744743i	$0.732572\pi$
$888$	0	0
$889$	−829169.	−1.04916
$890$	0	0
$891$	− 172999.i	− 0.217915i
$892$	0	0
$893$	658738.	0.826057
$894$	0	0
$895$	− 406610.i	− 0.507612i
$896$	0	0
$897$	−325759.	−0.404866
$898$	0	0
$899$	403158.i	0.498834i
$900$	0	0
$901$	129144.	0.159083
$902$	0	0
$903$	− 16263.5i	− 0.0199452i
$904$	0	0
$905$	−516146.	−0.630195
$906$	0	0
$907$	− 201406.i	− 0.244827i	−0.992479	−	0.122413i	$-0.960937\pi$
$907$	− 201406.i	− 0.244827i	0.992479	−	0.122413i	$-0.0390634\pi$
$908$	0	0
$909$	−1.55509e6	−1.88203
$910$	0	0
$911$	− 606030.i	− 0.730227i	−0.930963	−	0.365113i	$-0.881030\pi$
$911$	− 606030.i	− 0.730227i	0.930963	−	0.365113i	$-0.118970\pi$
$912$	0	0
$913$	43416.7	0.0520853
$914$	0	0
$915$	− 357366.i	− 0.426846i
$916$	0	0
$917$	−186945.	−0.222318
$918$	0	0
$919$	− 1.13891e6i	− 1.34853i	−0.738491	−	0.674263i	$-0.764461\pi$
$919$	− 1.13891e6i	− 1.34853i	0.738491	−	0.674263i	$-0.235539\pi$
$920$	0	0
$921$	−2.72733e6	−3.21528
$922$	0	0
$923$	− 330067.i	− 0.387435i
$924$	0	0
$925$	224685.	0.262598
$926$	0	0
$927$	3.29235e6i	3.83130i
$928$	0	0
$929$	291783.	0.338087	0.169043	−	0.985609i	$-0.445932\pi$
$929$	291783.	0.338087	0.169043	+	0.985609i	$0.445932\pi$
$930$	0	0
$931$	− 421837.i	− 0.486683i
$932$	0	0
$933$	367965.	0.422711
$934$	0	0
$935$	139612.i	0.159698i
$936$	0	0
$937$	−1.11462e6	−1.26955	−0.634774	−	0.772698i	$-0.718907\pi$
$937$	−1.11462e6	−1.26955	−0.634774	+	0.772698i	$0.718907\pi$
$938$	0	0
$939$	− 2.06539e6i	− 2.34246i
$940$	0	0
$941$	254584.	0.287509	0.143755	−	0.989613i	$-0.454082\pi$
$941$	254584.	0.287509	0.143755	+	0.989613i	$0.454082\pi$
$942$	0	0
$943$	169314.i	0.190401i
$944$	0	0
$945$	528608.	0.591930
$946$	0	0
$947$	− 997643.i	− 1.11244i	−0.831036	−	0.556218i	$-0.812252\pi$
$947$	− 997643.i	− 1.11244i	0.831036	−	0.556218i	$-0.187748\pi$
$948$	0	0
$949$	116089.	0.128902
$950$	0	0
$951$	527389.i	0.583136i
$952$	0	0
$953$	−146915.	−0.161763	−0.0808815	−	0.996724i	$-0.525774\pi$
$953$	−146915.	−0.161763	−0.0808815	+	0.996724i	$0.525774\pi$
$954$	0	0
$955$	264917.i	0.290471i
$956$	0	0
$957$	−401060.	−0.437911
$958$	0	0
$959$	449411.i	0.488659i
$960$	0	0
$961$	748803.	0.810813
$962$	0	0
$963$	3.58137e6i	3.86186i
$964$	0	0
$965$	191200.	0.205321
$966$	0	0
$967$	− 1.47654e6i	− 1.57904i	−0.613727	−	0.789518i	$-0.710331\pi$
$967$	− 1.47654e6i	− 1.57904i	0.613727	−	0.789518i	$-0.289669\pi$
$968$	0	0
$969$	3.11939e6	3.32217
$970$	0	0
$971$	− 1.55163e6i	− 1.64569i	−0.568265	−	0.822846i	$-0.692385\pi$
$971$	− 1.55163e6i	− 1.64569i	0.568265	−	0.822846i	$-0.307615\pi$
$972$	0	0
$973$	−757113.	−0.799715
$974$	0	0
$975$	− 112968.i	− 0.118835i
$976$	0	0
$977$	13609.0	0.0142572	0.00712862	−	0.999975i	$-0.497731\pi$
$977$	13609.0	0.0142572	0.00712862	+	0.999975i	$0.497731\pi$
$978$	0	0
$979$	− 191030.i	− 0.199314i
$980$	0	0
$981$	−648792.	−0.674167
$982$	0	0
$983$	− 179673.i	− 0.185941i	−0.995669	−	0.0929706i	$-0.970364\pi$
$983$	− 179673.i	− 0.185941i	0.995669	−	0.0929706i	$-0.0296363\pi$
$984$	0	0
$985$	−621596.	−0.640672
$986$	0	0
$987$	901966.i	0.925882i
$988$	0	0
$989$	−10002.3	−0.0102261
$990$	0	0
$991$	− 1.04471e6i	− 1.06377i	−0.846816	−	0.531886i	$-0.821484\pi$
$991$	− 1.04471e6i	− 1.06377i	0.846816	−	0.531886i	$-0.178516\pi$
$992$	0	0
$993$	379139.	0.384503
$994$	0	0
$995$	− 652920.i	− 0.659499i
$996$	0	0
$997$	−100146.	−0.100750	−0.0503748	−	0.998730i	$-0.516042\pi$
$997$	−100146.	−0.100750	−0.0503748	+	0.998730i	$0.516042\pi$
$998$	0	0
$999$	− 2.25886e6i	− 2.26339i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.5.b.d.191.8		8
4.3	odd	2	inner	320.5.b.d.191.1		8
8.3	odd	2		20.5.b.a.11.1	✓	8
8.5	even	2		20.5.b.a.11.2	yes	8
24.5	odd	2		180.5.c.a.91.7		8
24.11	even	2		180.5.c.a.91.8		8
40.3	even	4		100.5.d.c.99.4		16
40.13	odd	4		100.5.d.c.99.14		16
40.19	odd	2		100.5.b.c.51.8		8
40.27	even	4		100.5.d.c.99.13		16
40.29	even	2		100.5.b.c.51.7		8
40.37	odd	4		100.5.d.c.99.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.5.b.a.11.1	✓	8	8.3	odd	2
20.5.b.a.11.2	yes	8	8.5	even	2
100.5.b.c.51.7		8	40.29	even	2
100.5.b.c.51.8		8	40.19	odd	2
100.5.d.c.99.3		16	40.37	odd	4
100.5.d.c.99.4		16	40.3	even	4
100.5.d.c.99.13		16	40.27	even	4
100.5.d.c.99.14		16	40.13	odd	4
180.5.c.a.91.7		8	24.5	odd	2
180.5.c.a.91.8		8	24.11	even	2
320.5.b.d.191.1		8	4.3	odd	2	inner
320.5.b.d.191.8		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	320.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+15.5779i q^{3} -11.1803 q^{5} -37.6230i q^{7} -161.671 q^{9} +O(q^{10})\)	\(q+15.5779i q^{3} -11.1803 q^{5} -37.6230i q^{7} -161.671 q^{9} -26.6928i q^{11} -58.0144 q^{13} -174.166i q^{15} +467.816 q^{17} -428.041i q^{19} +586.088 q^{21} -360.456i q^{23} +125.000 q^{25} -1256.68i q^{27} -964.509 q^{29} -417.993i q^{31} +415.818 q^{33} +420.638i q^{35} +1797.48 q^{37} -903.742i q^{39} -469.722 q^{41} -27.7492i q^{43} +1807.53 q^{45} +1538.96i q^{47} +985.506 q^{49} +7287.58i q^{51} +276.057 q^{53} +298.435i q^{55} +6667.98 q^{57} -3813.72i q^{59} +2051.87 q^{61} +6082.55i q^{63} +648.620 q^{65} -1165.73i q^{67} +5615.14 q^{69} +5689.40i q^{71} -2001.05 q^{73} +1947.24i q^{75} -1004.26 q^{77} -705.728i q^{79} +6481.10 q^{81} +1626.53i q^{83} -5230.34 q^{85} -15025.0i q^{87} +7156.62 q^{89} +2182.68i q^{91} +6511.45 q^{93} +4785.65i q^{95} -13005.4 q^{97} +4315.45i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 328 q^{9}+O(q^{10}) \) 8 * q - 328 * q^9	\( 8 q - 328 q^{9} - 352 q^{13} - 48 q^{17} - 16 q^{21} + 1000 q^{25} - 1200 q^{29} - 1120 q^{33} + 5728 q^{37} + 4896 q^{41} + 400 q^{45} - 5768 q^{49} - 2592 q^{53} + 3840 q^{57} - 7936 q^{61} - 1200 q^{65} + 2256 q^{69} - 14448 q^{73} - 2400 q^{77} - 936 q^{81} - 11200 q^{85} + 23760 q^{89} - 11360 q^{93} - 4368 q^{97}+O(q^{100}) \) 8 * q - 328 * q^9 - 352 * q^13 - 48 * q^17 - 16 * q^21 + 1000 * q^25 - 1200 * q^29 - 1120 * q^33 + 5728 * q^37 + 4896 * q^41 + 400 * q^45 - 5768 * q^49 - 2592 * q^53 + 3840 * q^57 - 7936 * q^61 - 1200 * q^65 + 2256 * q^69 - 14448 * q^73 - 2400 * q^77 - 936 * q^81 - 11200 * q^85 + 23760 * q^89 - 11360 * q^93 - 4368 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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