LMFDB - Embedded newform 320.5.b.d.191.6

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			191.6
Root			$-0.641015 - 1.89449i$ of defining polynomial
Character	$\chi$	$=$	320.191
Dual form			320.5.b.d.191.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.14153i q^{3} +11.1803 q^{5} -63.6032i q^{7} +14.7155 q^{9} +O(q^{10})$	$q+8.14153i q^{3} +11.1803 q^{5} -63.6032i q^{7} +14.7155 q^{9} +33.3808i q^{11} -274.487 q^{13} +91.0251i q^{15} -284.950 q^{17} +5.17988i q^{19} +517.827 q^{21} +584.740i q^{23} +125.000 q^{25} +779.271i q^{27} -344.135 q^{29} +1466.69i q^{31} -271.771 q^{33} -711.105i q^{35} -1931.76 q^{37} -2234.74i q^{39} -976.795 q^{41} +2045.47i q^{43} +164.524 q^{45} -2561.74i q^{47} -1644.37 q^{49} -2319.93i q^{51} -121.922 q^{53} +373.209i q^{55} -42.1721 q^{57} -3456.33i q^{59} -4135.14 q^{61} -935.953i q^{63} -3068.86 q^{65} -1694.07i q^{67} -4760.68 q^{69} +7646.84i q^{71} -6008.30 q^{73} +1017.69i q^{75} +2123.12 q^{77} -4017.61i q^{79} -5152.50 q^{81} -3016.13i q^{83} -3185.84 q^{85} -2801.79i q^{87} +1190.40 q^{89} +17458.2i q^{91} -11941.1 q^{93} +57.9128i q^{95} -3021.43 q^{97} +491.215i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	8.14153i	0.904614i	0.891862	+	0.452307i	$0.149399\pi$
$3$	8.14153i	0.904614i	−0.891862	+	0.452307i	$0.850601\pi$
$4$	0	0
$5$	11.1803	0.447214
$5$	11.1803	0.447214
$6$	0	0
$7$	− 63.6032i	− 1.29802i	−0.760778	−	0.649012i	$-0.775182\pi$
$7$	− 63.6032i	− 1.29802i	0.760778	−	0.649012i	$-0.224818\pi$
$8$	0	0
$9$	14.7155	0.181673
$10$	0	0
$11$	33.3808i	0.275874i	0.990441	+	0.137937i	$0.0440472\pi$
$11$	33.3808i	0.275874i	−0.990441	+	0.137937i	$0.955953\pi$
$12$	0	0
$13$	−274.487	−1.62418	−0.812091	−	0.583531i	$-0.801671\pi$
$13$	−274.487	−1.62418	−0.812091	+	0.583531i	$0.801671\pi$
$14$	0	0
$15$	91.0251i	0.404556i
$16$	0	0
$17$	−284.950	−0.985986	−0.492993	−	0.870033i	$-0.664097\pi$
$17$	−284.950	−0.985986	−0.492993	+	0.870033i	$0.664097\pi$
$18$	0	0
$19$	5.17988i	0.0143487i	0.999974	+	0.00717435i	$0.00228368\pi$
$19$	5.17988i	0.0143487i	−0.999974	+	0.00717435i	$0.997716\pi$
$20$	0	0
$21$	517.827	1.17421
$22$	0	0
$23$	584.740i	1.10537i	0.833391	+	0.552684i	$0.186397\pi$
$23$	584.740i	1.10537i	−0.833391	+	0.552684i	$0.813603\pi$
$24$	0	0
$25$	125.000	0.200000
$26$	0	0
$27$	779.271i	1.06896i
$28$	0	0
$29$	−344.135	−0.409197	−0.204599	−	0.978846i	$-0.565589\pi$
$29$	−344.135	−0.409197	−0.204599	+	0.978846i	$0.565589\pi$
$30$	0	0
$31$	1466.69i	1.52621i	0.646275	+	0.763105i	$0.276326\pi$
$31$	1466.69i	1.52621i	−0.646275	+	0.763105i	$0.723674\pi$
$32$	0	0
$33$	−271.771	−0.249560
$34$	0	0
$35$	− 711.105i	− 0.580494i
$36$	0	0
$37$	−1931.76	−1.41107	−0.705537	−	0.708673i	$-0.749294\pi$
$37$	−1931.76	−1.41107	−0.705537	+	0.708673i	$0.749294\pi$
$38$	0	0
$39$	− 2234.74i	− 1.46926i
$40$	0	0
$41$	−976.795	−0.581080	−0.290540	−	0.956863i	$-0.593835\pi$
$41$	−976.795	−0.581080	−0.290540	+	0.956863i	$0.593835\pi$
$42$	0	0
$43$	2045.47i	1.10626i	0.833096	+	0.553128i	$0.186566\pi$
$43$	2045.47i	1.10626i	−0.833096	+	0.553128i	$0.813434\pi$
$44$	0	0
$45$	164.524	0.0812466
$46$	0	0
$47$	− 2561.74i	− 1.15969i	−0.814728	−	0.579843i	$-0.803114\pi$
$47$	− 2561.74i	− 1.15969i	0.814728	−	0.579843i	$-0.196886\pi$
$48$	0	0
$49$	−1644.37	−0.684868
$50$	0	0
$51$	− 2319.93i	− 0.891937i
$52$	0	0
$53$	−121.922	−0.0434041	−0.0217020	−	0.999764i	$-0.506909\pi$
$53$	−121.922	−0.0434041	−0.0217020	+	0.999764i	$0.506909\pi$
$54$	0	0
$55$	373.209i	0.123375i
$56$	0	0
$57$	−42.1721	−0.0129800
$58$	0	0
$59$	− 3456.33i	− 0.992913i	−0.868061	−	0.496457i	$-0.834634\pi$
$59$	− 3456.33i	− 0.992913i	0.868061	−	0.496457i	$-0.165366\pi$
$60$	0	0
$61$	−4135.14	−1.11130	−0.555649	−	0.831417i	$-0.687530\pi$
$61$	−4135.14	−1.11130	−0.555649	+	0.831417i	$0.687530\pi$
$62$	0	0
$63$	− 935.953i	− 0.235816i
$64$	0	0
$65$	−3068.86	−0.726356
$66$	0	0
$67$	− 1694.07i	− 0.377383i	−0.982036	−	0.188692i	$-0.939575\pi$
$67$	− 1694.07i	− 0.377383i	0.982036	−	0.188692i	$-0.0604247\pi$
$68$	0	0
$69$	−4760.68	−0.999932
$70$	0	0
$71$	7646.84i	1.51693i	0.651715	+	0.758464i	$0.274050\pi$
$71$	7646.84i	1.51693i	−0.651715	+	0.758464i	$0.725950\pi$
$72$	0	0
$73$	−6008.30	−1.12747	−0.563736	−	0.825955i	$-0.690636\pi$
$73$	−6008.30	−1.12747	−0.563736	+	0.825955i	$0.690636\pi$
$74$	0	0
$75$	1017.69i	0.180923i
$76$	0	0
$77$	2123.12	0.358092
$78$	0	0
$79$	− 4017.61i	− 0.643744i	−0.946783	−	0.321872i	$-0.895688\pi$
$79$	− 4017.61i	− 0.643744i	0.946783	−	0.321872i	$-0.104312\pi$
$80$	0	0
$81$	−5152.50	−0.785322
$82$	0	0
$83$	− 3016.13i	− 0.437818i	−0.975745	−	0.218909i	$-0.929750\pi$
$83$	− 3016.13i	− 0.437818i	0.975745	−	0.218909i	$-0.0702498\pi$
$84$	0	0
$85$	−3185.84	−0.440946
$86$	0	0
$87$	− 2801.79i	− 0.370166i
$88$	0	0
$89$	1190.40	0.150284	0.0751421	−	0.997173i	$-0.476059\pi$
$89$	1190.40	0.150284	0.0751421	+	0.997173i	$0.476059\pi$
$90$	0	0
$91$	17458.2i	2.10823i
$92$	0	0
$93$	−11941.1	−1.38063
$94$	0	0
$95$	57.9128i	0.00641693i
$96$	0	0
$97$	−3021.43	−0.321122	−0.160561	−	0.987026i	$-0.551330\pi$
$97$	−3021.43	−0.321122	−0.160561	+	0.987026i	$0.551330\pi$
$98$	0	0
$99$	491.215i	0.0501189i
$100$	0	0
$101$	754.479	0.0739613	0.0369806	−	0.999316i	$-0.488226\pi$
$101$	754.479	0.0739613	0.0369806	+	0.999316i	$0.488226\pi$
$102$	0	0
$103$	2192.61i	0.206675i	0.994646	+	0.103337i	$0.0329521\pi$
$103$	2192.61i	0.206675i	−0.994646	+	0.103337i	$0.967048\pi$
$104$	0	0
$105$	5789.49	0.525123
$106$	0	0
$107$	− 13277.7i	− 1.15973i	−0.814713	−	0.579864i	$-0.803106\pi$
$107$	− 13277.7i	− 1.15973i	0.814713	−	0.579864i	$-0.196894\pi$
$108$	0	0
$109$	11298.9	0.951008	0.475504	−	0.879714i	$-0.342266\pi$
$109$	11298.9	0.951008	0.475504	+	0.879714i	$0.342266\pi$
$110$	0	0
$111$	− 15727.5i	− 1.27648i
$112$	0	0
$113$	−801.291	−0.0627528	−0.0313764	−	0.999508i	$-0.509989\pi$
$113$	−801.291	−0.0627528	−0.0313764	+	0.999508i	$0.509989\pi$
$114$	0	0
$115$	6537.59i	0.494336i
$116$	0	0
$117$	−4039.21	−0.295070
$118$	0	0
$119$	18123.7i	1.27983i
$120$	0	0
$121$	13526.7	0.923893
$122$	0	0
$123$	− 7952.60i	− 0.525653i
$124$	0	0
$125$	1397.54	0.0894427
$126$	0	0
$127$	12829.5i	0.795428i	0.917510	+	0.397714i	$0.130196\pi$
$127$	12829.5i	0.795428i	−0.917510	+	0.397714i	$0.869804\pi$
$128$	0	0
$129$	−16653.2	−1.00073
$130$	0	0
$131$	10711.7i	0.624189i	0.950051	+	0.312094i	$0.101031\pi$
$131$	10711.7i	0.624189i	−0.950051	+	0.312094i	$0.898969\pi$
$132$	0	0
$133$	329.457	0.0186250
$134$	0	0
$135$	8712.51i	0.478053i
$136$	0	0
$137$	−4859.57	−0.258915	−0.129457	−	0.991585i	$-0.541324\pi$
$137$	−4859.57	−0.258915	−0.129457	+	0.991585i	$0.541324\pi$
$138$	0	0
$139$	4641.90i	0.240252i	0.992759	+	0.120126i	$0.0383298\pi$
$139$	4641.90i	0.240252i	−0.992759	+	0.120126i	$0.961670\pi$
$140$	0	0
$141$	20856.5	1.04907
$142$	0	0
$143$	− 9162.58i	− 0.448070i
$144$	0	0
$145$	−3847.55	−0.182999
$146$	0	0
$147$	− 13387.7i	− 0.619541i
$148$	0	0
$149$	−22750.7	−1.02476	−0.512380	−	0.858759i	$-0.671236\pi$
$149$	−22750.7	−1.02476	−0.512380	+	0.858759i	$0.671236\pi$
$150$	0	0
$151$	18142.5i	0.795688i	0.917453	+	0.397844i	$0.130241\pi$
$151$	18142.5i	0.795688i	−0.917453	+	0.397844i	$0.869759\pi$
$152$	0	0
$153$	−4193.18	−0.179127
$154$	0	0
$155$	16398.1i	0.682542i
$156$	0	0
$157$	19382.1	0.786322	0.393161	−	0.919470i	$-0.371381\pi$
$157$	19382.1	0.786322	0.393161	+	0.919470i	$0.371381\pi$
$158$	0	0
$159$	− 992.632i	− 0.0392639i
$160$	0	0
$161$	37191.3	1.43480
$162$	0	0
$163$	− 6821.93i	− 0.256763i	−0.991725	−	0.128381i	$-0.959022\pi$
$163$	− 6821.93i	− 0.256763i	0.991725	−	0.128381i	$-0.0409782\pi$
$164$	0	0
$165$	−3038.49	−0.111607
$166$	0	0
$167$	− 43577.2i	− 1.56252i	−0.624203	−	0.781262i	$-0.714576\pi$
$167$	− 43577.2i	− 1.56252i	0.624203	−	0.781262i	$-0.285424\pi$
$168$	0	0
$169$	46782.0	1.63797
$170$	0	0
$171$	76.2245i	0.00260677i
$172$	0	0
$173$	14633.4	0.488938	0.244469	−	0.969657i	$-0.421386\pi$
$173$	14633.4	0.488938	0.244469	+	0.969657i	$0.421386\pi$
$174$	0	0
$175$	− 7950.40i	− 0.259605i
$176$	0	0
$177$	28139.8	0.898204
$178$	0	0
$179$	2927.65i	0.0913721i	0.998956	+	0.0456860i	$0.0145474\pi$
$179$	2927.65i	0.0913721i	−0.998956	+	0.0456860i	$0.985453\pi$
$180$	0	0
$181$	8172.78	0.249467	0.124733	−	0.992190i	$-0.460192\pi$
$181$	8172.78	0.249467	0.124733	+	0.992190i	$0.460192\pi$
$182$	0	0
$183$	− 33666.3i	− 1.00530i
$184$	0	0
$185$	−21597.8	−0.631052
$186$	0	0
$187$	− 9511.85i	− 0.272008i
$188$	0	0
$189$	49564.1	1.38753
$190$	0	0
$191$	21087.5i	0.578041i	0.957323	+	0.289020i	$0.0933295\pi$
$191$	21087.5i	0.578041i	−0.957323	+	0.289020i	$0.906670\pi$
$192$	0	0
$193$	−41681.9	−1.11901	−0.559504	−	0.828827i	$-0.689008\pi$
$193$	−41681.9	−1.11901	−0.559504	+	0.828827i	$0.689008\pi$
$194$	0	0
$195$	− 24985.2i	− 0.657072i
$196$	0	0
$197$	20183.4	0.520071	0.260035	−	0.965599i	$-0.416266\pi$
$197$	20183.4	0.520071	0.260035	+	0.965599i	$0.416266\pi$
$198$	0	0
$199$	− 43122.7i	− 1.08893i	−0.838784	−	0.544465i	$-0.816733\pi$
$199$	− 43122.7i	− 1.08893i	0.838784	−	0.544465i	$-0.183267\pi$
$200$	0	0
$201$	13792.4	0.341387
$202$	0	0
$203$	21888.1i	0.531148i
$204$	0	0
$205$	−10920.9	−0.259867
$206$	0	0
$207$	8604.74i	0.200816i
$208$	0	0
$209$	−172.908	−0.00395843
$210$	0	0
$211$	− 78286.9i	− 1.75843i	−0.476429	−	0.879213i	$-0.658069\pi$
$211$	− 78286.9i	− 1.75843i	0.476429	−	0.879213i	$-0.341931\pi$
$212$	0	0
$213$	−62256.9	−1.37224
$214$	0	0
$215$	22869.0i	0.494733i
$216$	0	0
$217$	93286.0	1.98106
$218$	0	0
$219$	− 48916.7i	− 1.01993i
$220$	0	0
$221$	78215.0	1.60142
$222$	0	0
$223$	80052.1i	1.60977i	0.593432	+	0.804884i	$0.297772\pi$
$223$	80052.1i	1.60977i	−0.593432	+	0.804884i	$0.702228\pi$
$224$	0	0
$225$	1839.44	0.0363346
$226$	0	0
$227$	96596.3i	1.87460i	0.348522	+	0.937301i	$0.386684\pi$
$227$	96596.3i	1.87460i	−0.348522	+	0.937301i	$0.613316\pi$
$228$	0	0
$229$	45904.6	0.875357	0.437679	−	0.899131i	$-0.355801\pi$
$229$	45904.6	0.875357	0.437679	+	0.899131i	$0.355801\pi$
$230$	0	0
$231$	17285.5i	0.323935i
$232$	0	0
$233$	48534.0	0.893993	0.446997	−	0.894536i	$-0.352494\pi$
$233$	48534.0	0.893993	0.446997	+	0.894536i	$0.352494\pi$
$234$	0	0
$235$	− 28641.2i	− 0.518627i
$236$	0	0
$237$	32709.5	0.582340
$238$	0	0
$239$	− 5241.47i	− 0.0917608i	−0.998947	−	0.0458804i	$-0.985391\pi$
$239$	− 5241.47i	− 0.0917608i	0.998947	−	0.0458804i	$-0.0146093\pi$
$240$	0	0
$241$	−46204.2	−0.795513	−0.397756	−	0.917491i	$-0.630211\pi$
$241$	−46204.2	−0.795513	−0.397756	+	0.917491i	$0.630211\pi$
$242$	0	0
$243$	21171.7i	0.358545i
$244$	0	0
$245$	−18384.6	−0.306282
$246$	0	0
$247$	− 1421.81i	− 0.0233049i
$248$	0	0
$249$	24555.9	0.396057
$250$	0	0
$251$	67180.8i	1.06635i	0.846006	+	0.533173i	$0.179000\pi$
$251$	67180.8i	1.06635i	−0.846006	+	0.533173i	$0.821000\pi$
$252$	0	0
$253$	−19519.1	−0.304943
$254$	0	0
$255$	− 25937.6i	− 0.398886i
$256$	0	0
$257$	−69998.4	−1.05979	−0.529897	−	0.848062i	$-0.677770\pi$
$257$	−69998.4	−1.05979	−0.529897	+	0.848062i	$0.677770\pi$
$258$	0	0
$259$	122866.i	1.83161i
$260$	0	0
$261$	−5064.12	−0.0743401
$262$	0	0
$263$	− 1033.64i	− 0.0149438i	−0.999972	−	0.00747188i	$-0.997622\pi$
$263$	− 1033.64i	− 0.0149438i	0.999972	−	0.00747188i	$-0.00237839\pi$
$264$	0	0
$265$	−1363.13	−0.0194109
$266$	0	0
$267$	9691.69i	0.135949i
$268$	0	0
$269$	−92501.0	−1.27833	−0.639163	−	0.769071i	$-0.720719\pi$
$269$	−92501.0	−1.27833	−0.639163	+	0.769071i	$0.720719\pi$
$270$	0	0
$271$	− 16540.1i	− 0.225216i	−0.993640	−	0.112608i	$-0.964080\pi$
$271$	− 16540.1i	− 0.225216i	0.993640	−	0.112608i	$-0.0359204\pi$
$272$	0	0
$273$	−142137.	−1.90713
$274$	0	0
$275$	4172.60i	0.0551748i
$276$	0	0
$277$	−28924.4	−0.376968	−0.188484	−	0.982076i	$-0.560357\pi$
$277$	−28924.4	−0.376968	−0.188484	+	0.982076i	$0.560357\pi$
$278$	0	0
$279$	21583.0i	0.277271i
$280$	0	0
$281$	141788.	1.79567	0.897837	−	0.440329i	$-0.145138\pi$
$281$	141788.	1.79567	0.897837	+	0.440329i	$0.145138\pi$
$282$	0	0
$283$	− 3926.64i	− 0.0490284i	−0.999699	−	0.0245142i	$-0.992196\pi$
$283$	− 3926.64i	− 0.0490284i	0.999699	−	0.0245142i	$-0.00780390\pi$
$284$	0	0
$285$	−471.499	−0.00580485
$286$	0	0
$287$	62127.3i	0.754256i
$288$	0	0
$289$	−2324.55	−0.0278319
$290$	0	0
$291$	− 24599.1i	− 0.290491i
$292$	0	0
$293$	−152172.	−1.77255	−0.886275	−	0.463160i	$-0.846715\pi$
$293$	−152172.	−1.77255	−0.886275	+	0.463160i	$0.846715\pi$
$294$	0	0
$295$	− 38643.0i	− 0.444044i
$296$	0	0
$297$	−26012.7	−0.294898
$298$	0	0
$299$	− 160503.i	− 1.79532i
$300$	0	0
$301$	130098.	1.43595
$302$	0	0
$303$	6142.61i	0.0669064i
$304$	0	0
$305$	−46232.2	−0.496987
$306$	0	0
$307$	− 16244.2i	− 0.172355i	−0.996280	−	0.0861773i	$-0.972535\pi$
$307$	− 16244.2i	− 0.172355i	0.996280	−	0.0861773i	$-0.0274651\pi$
$308$	0	0
$309$	−17851.2	−0.186961
$310$	0	0
$311$	− 76317.7i	− 0.789050i	−0.918885	−	0.394525i	$-0.870909\pi$
$311$	− 76317.7i	− 0.789050i	0.918885	−	0.394525i	$-0.129091\pi$
$312$	0	0
$313$	−90079.4	−0.919469	−0.459734	−	0.888057i	$-0.652055\pi$
$313$	−90079.4	−0.919469	−0.459734	+	0.888057i	$0.652055\pi$
$314$	0	0
$315$	− 10464.3i	− 0.105460i
$316$	0	0
$317$	63383.9	0.630755	0.315377	−	0.948966i	$-0.397869\pi$
$317$	63383.9	0.630755	0.315377	+	0.948966i	$0.397869\pi$
$318$	0	0
$319$	− 11487.5i	− 0.112887i
$320$	0	0
$321$	108101.	1.04911
$322$	0	0
$323$	− 1476.01i	− 0.0141476i
$324$	0	0
$325$	−34310.8	−0.324836
$326$	0	0
$327$	91990.5i	0.860295i
$328$	0	0
$329$	−162935.	−1.50530
$330$	0	0
$331$	− 70436.2i	− 0.642895i	−0.946927	−	0.321448i	$-0.895831\pi$
$331$	− 70436.2i	− 0.642895i	0.946927	−	0.321448i	$-0.104169\pi$
$332$	0	0
$333$	−28426.8	−0.256354
$334$	0	0
$335$	− 18940.3i	− 0.168771i
$336$	0	0
$337$	23577.5	0.207605	0.103802	−	0.994598i	$-0.466899\pi$
$337$	23577.5	0.207605	0.103802	+	0.994598i	$0.466899\pi$
$338$	0	0
$339$	− 6523.73i	− 0.0567671i
$340$	0	0
$341$	−48959.2	−0.421042
$342$	0	0
$343$	− 48124.3i	− 0.409049i
$344$	0	0
$345$	−53226.0	−0.447183
$346$	0	0
$347$	− 121680.i	− 1.01056i	−0.862956	−	0.505280i	$-0.831389\pi$
$347$	− 121680.i	− 1.01056i	0.862956	−	0.505280i	$-0.168611\pi$
$348$	0	0
$349$	14140.9	0.116098	0.0580491	−	0.998314i	$-0.481512\pi$
$349$	14140.9	0.116098	0.0580491	+	0.998314i	$0.481512\pi$
$350$	0	0
$351$	− 213899.i	− 1.73618i
$352$	0	0
$353$	−226940.	−1.82122	−0.910608	−	0.413271i	$-0.864386\pi$
$353$	−226940.	−1.82122	−0.910608	+	0.413271i	$0.864386\pi$
$354$	0	0
$355$	85494.2i	0.678391i
$356$	0	0
$357$	−147555.	−1.15776
$358$	0	0
$359$	183457.i	1.42346i	0.702452	+	0.711731i	$0.252089\pi$
$359$	183457.i	1.42346i	−0.702452	+	0.711731i	$0.747911\pi$
$360$	0	0
$361$	130294.	0.999794
$362$	0	0
$363$	110128.i	0.835767i
$364$	0	0
$365$	−67174.8	−0.504221
$366$	0	0
$367$	94098.0i	0.698632i	0.937005	+	0.349316i	$0.113586\pi$
$367$	94098.0i	0.698632i	−0.937005	+	0.349316i	$0.886414\pi$
$368$	0	0
$369$	−14374.0	−0.105566
$370$	0	0
$371$	7754.63i	0.0563395i
$372$	0	0
$373$	−123826.	−0.890010	−0.445005	−	0.895528i	$-0.646798\pi$
$373$	−123826.	−0.890010	−0.445005	+	0.895528i	$0.646798\pi$
$374$	0	0
$375$	11378.1i	0.0809112i
$376$	0	0
$377$	94460.5	0.664611
$378$	0	0
$379$	282818.i	1.96892i	0.175609	+	0.984460i	$0.443811\pi$
$379$	282818.i	1.96892i	−0.175609	+	0.984460i	$0.556189\pi$
$380$	0	0
$381$	−104451.	−0.719555
$382$	0	0
$383$	− 69105.5i	− 0.471102i	−0.971862	−	0.235551i	$-0.924311\pi$
$383$	− 69105.5i	− 0.471102i	0.971862	−	0.235551i	$-0.0756895\pi$
$384$	0	0
$385$	23737.3	0.160143
$386$	0	0
$387$	30100.1i	0.200977i
$388$	0	0
$389$	28018.1	0.185157	0.0925783	−	0.995705i	$-0.470489\pi$
$389$	28018.1	0.185157	0.0925783	+	0.995705i	$0.470489\pi$
$390$	0	0
$391$	− 166622.i	− 1.08988i
$392$	0	0
$393$	−87209.6	−0.564650
$394$	0	0
$395$	− 44918.2i	− 0.287891i
$396$	0	0
$397$	261226.	1.65743	0.828715	−	0.559671i	$-0.189072\pi$
$397$	261226.	1.65743	0.828715	+	0.559671i	$0.189072\pi$
$398$	0	0
$399$	2682.28i	0.0168484i
$400$	0	0
$401$	−301385.	−1.87427	−0.937135	−	0.348967i	$-0.886533\pi$
$401$	−301385.	−1.87427	−0.937135	+	0.348967i	$0.886533\pi$
$402$	0	0
$403$	− 402586.i	− 2.47884i
$404$	0	0
$405$	−57606.7	−0.351207
$406$	0	0
$407$	− 64483.7i	− 0.389279i
$408$	0	0
$409$	306327.	1.83121	0.915605	−	0.402079i	$-0.131712\pi$
$409$	306327.	1.83121	0.915605	+	0.402079i	$0.131712\pi$
$410$	0	0
$411$	− 39564.4i	− 0.234218i
$412$	0	0
$413$	−219834.	−1.28883
$414$	0	0
$415$	− 33721.4i	− 0.195798i
$416$	0	0
$417$	−37792.2	−0.217335
$418$	0	0
$419$	13132.1i	0.0748007i	0.999300	+	0.0374003i	$0.0119077\pi$
$419$	13132.1i	0.0748007i	−0.999300	+	0.0374003i	$0.988092\pi$
$420$	0	0
$421$	189597.	1.06971	0.534856	−	0.844943i	$-0.320366\pi$
$421$	189597.	1.06971	0.534856	+	0.844943i	$0.320366\pi$
$422$	0	0
$423$	− 37697.4i	− 0.210683i
$424$	0	0
$425$	−35618.7	−0.197197
$426$	0	0
$427$	263008.i	1.44249i
$428$	0	0
$429$	74597.4	0.405331
$430$	0	0
$431$	312114.i	1.68019i	0.542440	+	0.840095i	$0.317501\pi$
$431$	312114.i	1.68019i	−0.542440	+	0.840095i	$0.682499\pi$
$432$	0	0
$433$	84490.5	0.450642	0.225321	−	0.974285i	$-0.427657\pi$
$433$	84490.5	0.450642	0.225321	+	0.974285i	$0.427657\pi$
$434$	0	0
$435$	− 31324.9i	− 0.165543i
$436$	0	0
$437$	−3028.88	−0.0158606
$438$	0	0
$439$	89602.9i	0.464936i	0.972604	+	0.232468i	$0.0746801\pi$
$439$	89602.9i	0.464936i	−0.972604	+	0.232468i	$0.925320\pi$
$440$	0	0
$441$	−24197.7	−0.124422
$442$	0	0
$443$	81699.7i	0.416306i	0.978096	+	0.208153i	$0.0667452\pi$
$443$	81699.7i	0.416306i	−0.978096	+	0.208153i	$0.933255\pi$
$444$	0	0
$445$	13309.1	0.0672091
$446$	0	0
$447$	− 185225.i	− 0.927013i
$448$	0	0
$449$	−153375.	−0.760787	−0.380393	−	0.924825i	$-0.624211\pi$
$449$	−153375.	−0.760787	−0.380393	+	0.924825i	$0.624211\pi$
$450$	0	0
$451$	− 32606.2i	− 0.160305i
$452$	0	0
$453$	−147708.	−0.719791
$454$	0	0
$455$	195189.i	0.942828i
$456$	0	0
$457$	−107730.	−0.515829	−0.257915	−	0.966168i	$-0.583035\pi$
$457$	−107730.	−0.515829	−0.257915	+	0.966168i	$0.583035\pi$
$458$	0	0
$459$	− 222053.i	− 1.05398i
$460$	0	0
$461$	130855.	0.615729	0.307865	−	0.951430i	$-0.400386\pi$
$461$	130855.	0.615729	0.307865	+	0.951430i	$0.400386\pi$
$462$	0	0
$463$	265844.i	1.24012i	0.784552	+	0.620062i	$0.212893\pi$
$463$	265844.i	1.24012i	−0.784552	+	0.620062i	$0.787107\pi$
$464$	0	0
$465$	−133505.	−0.617437
$466$	0	0
$467$	− 242239.i	− 1.11073i	−0.831605	−	0.555367i	$-0.812578\pi$
$467$	− 242239.i	− 1.11073i	0.831605	−	0.555367i	$-0.187422\pi$
$468$	0	0
$469$	−107749.	−0.489853
$470$	0	0
$471$	157800.i	0.711318i
$472$	0	0
$473$	−68279.3	−0.305187
$474$	0	0
$475$	647.485i	0.00286974i
$476$	0	0
$477$	−1794.14	−0.00788534
$478$	0	0
$479$	− 335632.i	− 1.46282i	−0.681935	−	0.731412i	$-0.738862\pi$
$479$	− 335632.i	− 1.46282i	0.681935	−	0.731412i	$-0.261138\pi$
$480$	0	0
$481$	530243.	2.29184
$482$	0	0
$483$	302794.i	1.29794i
$484$	0	0
$485$	−33780.7	−0.143610
$486$	0	0
$487$	17852.3i	0.0752725i	0.999292	+	0.0376363i	$0.0119828\pi$
$487$	17852.3i	0.0752725i	−0.999292	+	0.0376363i	$0.988017\pi$
$488$	0	0
$489$	55540.9	0.232271
$490$	0	0
$491$	138734.i	0.575466i	0.957711	+	0.287733i	$0.0929016\pi$
$491$	138734.i	0.575466i	−0.957711	+	0.287733i	$0.907098\pi$
$492$	0	0
$493$	98061.2	0.403463
$494$	0	0
$495$	5491.95i	0.0224138i
$496$	0	0
$497$	486363.	1.96901
$498$	0	0
$499$	175457.i	0.704644i	0.935879	+	0.352322i	$0.114608\pi$
$499$	175457.i	0.704644i	−0.935879	+	0.352322i	$0.885392\pi$
$500$	0	0
$501$	354785.	1.41348
$502$	0	0
$503$	− 244670.i	− 0.967040i	−0.875333	−	0.483520i	$-0.839358\pi$
$503$	− 244670.i	− 0.967040i	0.875333	−	0.483520i	$-0.160642\pi$
$504$	0	0
$505$	8435.33	0.0330765
$506$	0	0
$507$	380877.i	1.48173i
$508$	0	0
$509$	173309.	0.668937	0.334469	−	0.942407i	$-0.391443\pi$
$509$	173309.	0.668937	0.334469	+	0.942407i	$0.391443\pi$
$510$	0	0
$511$	382147.i	1.46349i
$512$	0	0
$513$	−4036.53	−0.0153382
$514$	0	0
$515$	24514.1i	0.0924277i
$516$	0	0
$517$	85513.1	0.319927
$518$	0	0
$519$	119139.i	0.442301i
$520$	0	0
$521$	218395.	0.804578	0.402289	−	0.915513i	$-0.368215\pi$
$521$	218395.	0.804578	0.402289	+	0.915513i	$0.368215\pi$
$522$	0	0
$523$	137830.i	0.503897i	0.967741	+	0.251949i	$0.0810714\pi$
$523$	137830.i	0.503897i	−0.967741	+	0.251949i	$0.918929\pi$
$524$	0	0
$525$	64728.4	0.234842
$526$	0	0
$527$	− 417932.i	− 1.50482i
$528$	0	0
$529$	−62080.0	−0.221840
$530$	0	0
$531$	− 50861.7i	− 0.180385i
$532$	0	0
$533$	268117.	0.943779
$534$	0	0
$535$	− 148450.i	− 0.518646i
$536$	0	0
$537$	−23835.6	−0.0826565
$538$	0	0
$539$	− 54890.3i	− 0.188937i
$540$	0	0
$541$	311998.	1.06600	0.533000	−	0.846115i	$-0.321065\pi$
$541$	311998.	1.06600	0.533000	+	0.846115i	$0.321065\pi$
$542$	0	0
$543$	66538.9i	0.225671i
$544$	0	0
$545$	126326.	0.425304
$546$	0	0
$547$	− 275725.i	− 0.921513i	−0.887527	−	0.460757i	$-0.847578\pi$
$547$	− 275725.i	− 0.921513i	0.887527	−	0.460757i	$-0.152422\pi$
$548$	0	0
$549$	−60850.6	−0.201893
$550$	0	0
$551$	− 1782.58i	− 0.00587145i
$552$	0	0
$553$	−255533.	−0.835596
$554$	0	0
$555$	− 175839.i	− 0.570859i
$556$	0	0
$557$	−301922.	−0.973161	−0.486580	−	0.873636i	$-0.661756\pi$
$557$	−301922.	−0.973161	−0.486580	+	0.873636i	$0.661756\pi$
$558$	0	0
$559$	− 561454.i	− 1.79676i
$560$	0	0
$561$	77441.0	0.246062
$562$	0	0
$563$	− 198587.i	− 0.626518i	−0.949668	−	0.313259i	$-0.898579\pi$
$563$	− 198587.i	− 0.626518i	0.949668	−	0.313259i	$-0.101421\pi$
$564$	0	0
$565$	−8958.70	−0.0280639
$566$	0	0
$567$	327715.i	1.01937i
$568$	0	0
$569$	363936.	1.12409	0.562044	−	0.827107i	$-0.310015\pi$
$569$	363936.	1.12409	0.562044	+	0.827107i	$0.310015\pi$
$570$	0	0
$571$	− 303041.i	− 0.929455i	−0.885454	−	0.464728i	$-0.846152\pi$
$571$	− 303041.i	− 0.929455i	0.885454	−	0.464728i	$-0.153848\pi$
$572$	0	0
$573$	−171685.	−0.522904
$574$	0	0
$575$	73092.5i	0.221074i
$576$	0	0
$577$	210571.	0.632479	0.316240	−	0.948679i	$-0.397580\pi$
$577$	210571.	0.632479	0.316240	+	0.948679i	$0.397580\pi$
$578$	0	0
$579$	− 339355.i	− 1.01227i
$580$	0	0
$581$	−191836.	−0.568299
$582$	0	0
$583$	− 4069.85i	− 0.0119741i
$584$	0	0
$585$	−45159.8	−0.131959
$586$	0	0
$587$	− 468352.i	− 1.35924i	−0.733564	−	0.679620i	$-0.762145\pi$
$587$	− 468352.i	− 1.35924i	0.733564	−	0.679620i	$-0.237855\pi$
$588$	0	0
$589$	−7597.26	−0.0218991
$590$	0	0
$591$	164324.i	0.470463i
$592$	0	0
$593$	−61460.0	−0.174777	−0.0873883	−	0.996174i	$-0.527852\pi$
$593$	−61460.0	−0.174777	−0.0873883	+	0.996174i	$0.527852\pi$
$594$	0	0
$595$	202629.i	0.572359i
$596$	0	0
$597$	351085.	0.985061
$598$	0	0
$599$	92590.3i	0.258055i	0.991641	+	0.129027i	$0.0411855\pi$
$599$	92590.3i	0.258055i	−0.991641	+	0.129027i	$0.958815\pi$
$600$	0	0
$601$	−58358.7	−0.161568	−0.0807842	−	0.996732i	$-0.525742\pi$
$601$	−58358.7	−0.161568	−0.0807842	+	0.996732i	$0.525742\pi$
$602$	0	0
$603$	− 24929.2i	− 0.0685603i
$604$	0	0
$605$	151233.	0.413178
$606$	0	0
$607$	− 72146.9i	− 0.195812i	−0.995196	−	0.0979062i	$-0.968785\pi$
$607$	− 72146.9i	− 0.195812i	0.995196	−	0.0979062i	$-0.0312145\pi$
$608$	0	0
$609$	−178203.	−0.480484
$610$	0	0
$611$	703165.i	1.88354i
$612$	0	0
$613$	481279.	1.28078	0.640392	−	0.768048i	$-0.278772\pi$
$613$	481279.	1.28078	0.640392	+	0.768048i	$0.278772\pi$
$614$	0	0
$615$	− 88912.8i	− 0.235079i
$616$	0	0
$617$	−471022.	−1.23729	−0.618644	−	0.785672i	$-0.712318\pi$
$617$	−471022.	−1.23729	−0.618644	+	0.785672i	$0.712318\pi$
$618$	0	0
$619$	− 668000.i	− 1.74339i	−0.490046	−	0.871697i	$-0.663020\pi$
$619$	− 668000.i	− 1.74339i	0.490046	−	0.871697i	$-0.336980\pi$
$620$	0	0
$621$	−455671.	−1.18159
$622$	0	0
$623$	− 75713.3i	− 0.195073i
$624$	0	0
$625$	15625.0	0.0400000
$626$	0	0
$627$	− 1407.74i	− 0.00358086i
$628$	0	0
$629$	550455.	1.39130
$630$	0	0
$631$	− 483839.i	− 1.21519i	−0.794249	−	0.607593i	$-0.792135\pi$
$631$	− 483839.i	− 1.21519i	0.794249	−	0.607593i	$-0.207865\pi$
$632$	0	0
$633$	637375.	1.59070
$634$	0	0
$635$	143438.i	0.355726i
$636$	0	0
$637$	451357.	1.11235
$638$	0	0
$639$	112527.i	0.275585i
$640$	0	0
$641$	−122259.	−0.297553	−0.148777	−	0.988871i	$-0.547534\pi$
$641$	−122259.	−0.297553	−0.148777	+	0.988871i	$0.547534\pi$
$642$	0	0
$643$	− 39691.4i	− 0.0960008i	−0.998847	−	0.0480004i	$-0.984715\pi$
$643$	− 39691.4i	− 0.0960008i	0.998847	−	0.0480004i	$-0.0152849\pi$
$644$	0	0
$645$	−186189.	−0.447542
$646$	0	0
$647$	− 293173.i	− 0.700351i	−0.936684	−	0.350175i	$-0.886122\pi$
$647$	− 293173.i	− 0.700351i	0.936684	−	0.350175i	$-0.113878\pi$
$648$	0	0
$649$	115375.	0.273919
$650$	0	0
$651$	759491.i	1.79209i
$652$	0	0
$653$	−751251.	−1.76181	−0.880904	−	0.473294i	$-0.843065\pi$
$653$	−751251.	−1.76181	−0.880904	+	0.473294i	$0.843065\pi$
$654$	0	0
$655$	119760.i	0.279146i
$656$	0	0
$657$	−88415.1	−0.204831
$658$	0	0
$659$	395047.i	0.909657i	0.890579	+	0.454828i	$0.150299\pi$
$659$	395047.i	0.909657i	−0.890579	+	0.454828i	$0.849701\pi$
$660$	0	0
$661$	7765.79	0.0177739	0.00888695	−	0.999961i	$-0.497171\pi$
$661$	7765.79	0.0177739	0.00888695	+	0.999961i	$0.497171\pi$
$662$	0	0
$663$	636790.i	1.44867i
$664$	0	0
$665$	3683.44	0.00832933
$666$	0	0
$667$	− 201230.i	− 0.452314i
$668$	0	0
$669$	−651747.	−1.45622
$670$	0	0
$671$	− 138034.i	− 0.306578i
$672$	0	0
$673$	602808.	1.33091	0.665456	−	0.746437i	$-0.268237\pi$
$673$	602808.	1.33091	0.665456	+	0.746437i	$0.268237\pi$
$674$	0	0
$675$	97408.8i	0.213792i
$676$	0	0
$677$	−131066.	−0.285964	−0.142982	−	0.989725i	$-0.545669\pi$
$677$	−131066.	−0.285964	−0.142982	+	0.989725i	$0.545669\pi$
$678$	0	0
$679$	192173.i	0.416824i
$680$	0	0
$681$	−786442.	−1.69579
$682$	0	0
$683$	326700.i	0.700338i	0.936687	+	0.350169i	$0.113876\pi$
$683$	326700.i	0.700338i	−0.936687	+	0.350169i	$0.886124\pi$
$684$	0	0
$685$	−54331.7	−0.115790
$686$	0	0
$687$	373734.i	0.791861i
$688$	0	0
$689$	33466.0	0.0704961
$690$	0	0
$691$	223027.i	0.467091i	0.972346	+	0.233545i	$0.0750327\pi$
$691$	223027.i	0.467091i	−0.972346	+	0.233545i	$0.924967\pi$
$692$	0	0
$693$	31242.8	0.0650555
$694$	0	0
$695$	51898.0i	0.107444i
$696$	0	0
$697$	278338.	0.572936
$698$	0	0
$699$	395141.i	0.808719i
$700$	0	0
$701$	−427680.	−0.870327	−0.435164	−	0.900351i	$-0.643309\pi$
$701$	−427680.	−0.870327	−0.435164	+	0.900351i	$0.643309\pi$
$702$	0	0
$703$	− 10006.3i	− 0.0202471i
$704$	0	0
$705$	233183.	0.469157
$706$	0	0
$707$	− 47987.3i	− 0.0960035i
$708$	0	0
$709$	642298.	1.27775	0.638873	−	0.769313i	$-0.279401\pi$
$709$	642298.	1.27775	0.638873	+	0.769313i	$0.279401\pi$
$710$	0	0
$711$	− 59121.1i	− 0.116951i
$712$	0	0
$713$	−857631.	−1.68702
$714$	0	0
$715$	− 102441.i	− 0.200383i
$716$	0	0
$717$	42673.6	0.0830081
$718$	0	0
$719$	465657.i	0.900759i	0.892837	+	0.450379i	$0.148711\pi$
$719$	465657.i	0.900759i	−0.892837	+	0.450379i	$0.851289\pi$
$720$	0	0
$721$	139457.	0.268269
$722$	0	0
$723$	− 376173.i	− 0.719632i
$724$	0	0
$725$	−43016.9	−0.0818395
$726$	0	0
$727$	− 4334.28i	− 0.00820064i	−0.999992	−	0.00410032i	$-0.998695\pi$
$727$	− 4334.28i	− 0.00820064i	0.999992	−	0.00410032i	$-0.00130518\pi$
$728$	0	0
$729$	−589722.	−1.10967
$730$	0	0
$731$	− 582856.i	− 1.09075i
$732$	0	0
$733$	580825.	1.08103	0.540515	−	0.841335i	$-0.318230\pi$
$733$	580825.	1.08103	0.540515	+	0.841335i	$0.318230\pi$
$734$	0	0
$735$	− 149679.i	− 0.277067i
$736$	0	0
$737$	56549.5	0.104110
$738$	0	0
$739$	373926.i	0.684695i	0.939573	+	0.342348i	$0.111222\pi$
$739$	373926.i	0.684695i	−0.939573	+	0.342348i	$0.888778\pi$
$740$	0	0
$741$	11575.7	0.0210819
$742$	0	0
$743$	− 733317.i	− 1.32835i	−0.747575	−	0.664177i	$-0.768782\pi$
$743$	− 733317.i	− 1.32835i	0.747575	−	0.664177i	$-0.231218\pi$
$744$	0	0
$745$	−254361.	−0.458287
$746$	0	0
$747$	− 44383.9i	− 0.0795397i
$748$	0	0
$749$	−844506.	−1.50536
$750$	0	0
$751$	88381.1i	0.156704i	0.996926	+	0.0783519i	$0.0249658\pi$
$751$	88381.1i	0.156704i	−0.996926	+	0.0783519i	$0.975034\pi$
$752$	0	0
$753$	−546955.	−0.964632
$754$	0	0
$755$	202839.i	0.355842i
$756$	0	0
$757$	−1.00702e6	−1.75729	−0.878647	−	0.477472i	$-0.841553\pi$
$757$	−1.00702e6	−1.75729	−0.878647	+	0.477472i	$0.841553\pi$
$758$	0	0
$759$	− 158915.i	− 0.275856i
$760$	0	0
$761$	−64709.6	−0.111738	−0.0558688	−	0.998438i	$-0.517793\pi$
$761$	−64709.6	−0.111738	−0.0558688	+	0.998438i	$0.517793\pi$
$762$	0	0
$763$	− 718648.i	− 1.23443i
$764$	0	0
$765$	−46881.2	−0.0801080
$766$	0	0
$767$	948717.i	1.61267i
$768$	0	0
$769$	156640.	0.264880	0.132440	−	0.991191i	$-0.457719\pi$
$769$	156640.	0.264880	0.132440	+	0.991191i	$0.457719\pi$
$770$	0	0
$771$	− 569894.i	− 0.958706i
$772$	0	0
$773$	487717.	0.816224	0.408112	−	0.912932i	$-0.366187\pi$
$773$	487717.	0.816224	0.408112	+	0.912932i	$0.366187\pi$
$774$	0	0
$775$	183336.i	0.305242i
$776$	0	0
$777$	−1.00032e6	−1.65690
$778$	0	0
$779$	− 5059.68i	− 0.00833773i
$780$	0	0
$781$	−255257.	−0.418481
$782$	0	0
$783$	− 268174.i	− 0.437415i
$784$	0	0
$785$	216698.	0.351654
$786$	0	0
$787$	381516.i	0.615975i	0.951390	+	0.307988i	$0.0996556\pi$
$787$	381516.i	0.615975i	−0.951390	+	0.307988i	$0.900344\pi$
$788$	0	0
$789$	8415.45	0.0135183
$790$	0	0
$791$	50964.6i	0.0814547i
$792$	0	0
$793$	1.13504e6	1.80495
$794$	0	0
$795$	− 11098.0i	− 0.0175594i
$796$	0	0
$797$	−320530.	−0.504605	−0.252302	−	0.967648i	$-0.581188\pi$
$797$	−320530.	−0.504605	−0.252302	+	0.967648i	$0.581188\pi$
$798$	0	0
$799$	729969.i	1.14343i
$800$	0	0
$801$	17517.4	0.0273026
$802$	0	0
$803$	− 200562.i	− 0.311040i
$804$	0	0
$805$	415812.	0.641660
$806$	0	0
$807$	− 753100.i	− 1.15639i
$808$	0	0
$809$	−916837.	−1.40086	−0.700430	−	0.713721i	$-0.747009\pi$
$809$	−916837.	−1.40086	−0.700430	+	0.713721i	$0.747009\pi$
$810$	0	0
$811$	− 420414.i	− 0.639198i	−0.947553	−	0.319599i	$-0.896452\pi$
$811$	− 420414.i	− 0.639198i	0.947553	−	0.319599i	$-0.103548\pi$
$812$	0	0
$813$	134661.	0.203733
$814$	0	0
$815$	− 76271.5i	− 0.114828i
$816$	0	0
$817$	−10595.3	−0.0158733
$818$	0	0
$819$	256907.i	0.383008i
$820$	0	0
$821$	−522572.	−0.775283	−0.387641	−	0.921810i	$-0.626710\pi$
$821$	−522572.	−0.775283	−0.387641	+	0.921810i	$0.626710\pi$
$822$	0	0
$823$	− 1.09844e6i	− 1.62172i	−0.585241	−	0.810859i	$-0.699000\pi$
$823$	− 1.09844e6i	− 1.62172i	0.585241	−	0.810859i	$-0.301000\pi$
$824$	0	0
$825$	−33971.3	−0.0499120
$826$	0	0
$827$	831895.i	1.21635i	0.793804	+	0.608174i	$0.208098\pi$
$827$	831895.i	1.21635i	−0.793804	+	0.608174i	$0.791902\pi$
$828$	0	0
$829$	−877227.	−1.27645	−0.638224	−	0.769851i	$-0.720330\pi$
$829$	−877227.	−1.27645	−0.638224	+	0.769851i	$0.720330\pi$
$830$	0	0
$831$	− 235489.i	− 0.341011i
$832$	0	0
$833$	468562.	0.675270
$834$	0	0
$835$	− 487208.i	− 0.698782i
$836$	0	0
$837$	−1.14295e6	−1.63145
$838$	0	0
$839$	− 224242.i	− 0.318561i	−0.987233	−	0.159281i	$-0.949083\pi$
$839$	− 224242.i	− 0.318561i	0.987233	−	0.159281i	$-0.0509174\pi$
$840$	0	0
$841$	−588852.	−0.832557
$842$	0	0
$843$	1.15437e6i	1.62439i
$844$	0	0
$845$	523039.	0.732522
$846$	0	0
$847$	− 860343.i	− 1.19924i
$848$	0	0
$849$	31968.8	0.0443518
$850$	0	0
$851$	− 1.12958e6i	− 1.55976i
$852$	0	0
$853$	182588.	0.250943	0.125471	−	0.992097i	$-0.459956\pi$
$853$	182588.	0.250943	0.125471	+	0.992097i	$0.459956\pi$
$854$	0	0
$855$	852.216i	0.00116578i
$856$	0	0
$857$	−1.26573e6	−1.72337	−0.861687	−	0.507439i	$-0.830592\pi$
$857$	−1.26573e6	−1.72337	−0.861687	+	0.507439i	$0.830592\pi$
$858$	0	0
$859$	390821.i	0.529653i	0.964296	+	0.264826i	$0.0853147\pi$
$859$	390821.i	0.529653i	−0.964296	+	0.264826i	$0.914685\pi$
$860$	0	0
$861$	−505811.	−0.682310
$862$	0	0
$863$	1.15518e6i	1.55106i	0.631309	+	0.775531i	$0.282518\pi$
$863$	1.15518e6i	1.55106i	−0.631309	+	0.775531i	$0.717482\pi$
$864$	0	0
$865$	163607.	0.218660
$866$	0	0
$867$	− 18925.4i	− 0.0251772i
$868$	0	0
$869$	134111.	0.177592
$870$	0	0
$871$	465001.i	0.612940i
$872$	0	0
$873$	−44461.9	−0.0583391
$874$	0	0
$875$	− 88888.2i	− 0.116099i
$876$	0	0
$877$	286144.	0.372036	0.186018	−	0.982546i	$-0.440442\pi$
$877$	286144.	0.372036	0.186018	+	0.982546i	$0.440442\pi$
$878$	0	0
$879$	− 1.23891e6i	− 1.60347i
$880$	0	0
$881$	−1.13893e6	−1.46739	−0.733696	−	0.679478i	$-0.762206\pi$
$881$	−1.13893e6	−1.46739	−0.733696	+	0.679478i	$0.762206\pi$
$882$	0	0
$883$	1.25765e6i	1.61301i	0.591224	+	0.806507i	$0.298645\pi$
$883$	1.25765e6i	1.61301i	−0.591224	+	0.806507i	$0.701355\pi$
$884$	0	0
$885$	314613.	0.401689
$886$	0	0
$887$	126913.i	0.161309i	0.996742	+	0.0806544i	$0.0257010\pi$
$887$	126913.i	0.161309i	−0.996742	+	0.0806544i	$0.974299\pi$
$888$	0	0
$889$	815994.	1.03248
$890$	0	0
$891$	− 171994.i	− 0.216650i
$892$	0	0
$893$	13269.5	0.0166400
$894$	0	0
$895$	32732.2i	0.0408628i
$896$	0	0
$897$	1.30674e6	1.62407
$898$	0	0
$899$	− 504738.i	− 0.624521i
$900$	0	0
$901$	34741.7	0.0427958
$902$	0	0
$903$	1.05920e6i	1.29898i
$904$	0	0
$905$	91374.5	0.111565
$906$	0	0
$907$	318521.i	0.387190i	0.981082	+	0.193595i	$0.0620147\pi$
$907$	318521.i	0.387190i	−0.981082	+	0.193595i	$0.937985\pi$
$908$	0	0
$909$	11102.5	0.0134368
$910$	0	0
$911$	− 209882.i	− 0.252894i	−0.991973	−	0.126447i	$-0.959643\pi$
$911$	− 209882.i	− 0.252894i	0.991973	−	0.126447i	$-0.0403573\pi$
$912$	0	0
$913$	100681.	0.120783
$914$	0	0
$915$	− 376401.i	− 0.449582i
$916$	0	0
$917$	681299.	0.810212
$918$	0	0
$919$	330341.i	0.391139i	0.980690	+	0.195570i	$0.0626555\pi$
$919$	330341.i	0.391139i	−0.980690	+	0.195570i	$0.937344\pi$
$920$	0	0
$921$	132253.	0.155914
$922$	0	0
$923$	− 2.09896e6i	− 2.46377i
$924$	0	0
$925$	−241470.	−0.282215
$926$	0	0
$927$	32265.4i	0.0375472i
$928$	0	0
$929$	561048.	0.650083	0.325042	−	0.945700i	$-0.394622\pi$
$929$	561048.	0.650083	0.325042	+	0.945700i	$0.394622\pi$
$930$	0	0
$931$	− 8517.62i	− 0.00982696i
$932$	0	0
$933$	621343.	0.713786
$934$	0	0
$935$	− 106346.i	− 0.121646i
$936$	0	0
$937$	−677949.	−0.772179	−0.386090	−	0.922461i	$-0.626174\pi$
$937$	−677949.	−0.772179	−0.386090	+	0.922461i	$0.626174\pi$
$938$	0	0
$939$	− 733384.i	− 0.831765i
$940$	0	0
$941$	−49257.5	−0.0556280	−0.0278140	−	0.999613i	$-0.508855\pi$
$941$	−49257.5	−0.0556280	−0.0278140	+	0.999613i	$0.508855\pi$
$942$	0	0
$943$	− 571171.i	− 0.642307i
$944$	0	0
$945$	554144.	0.620524
$946$	0	0
$947$	928184.i	1.03499i	0.855688	+	0.517493i	$0.173134\pi$
$947$	928184.i	1.03499i	−0.855688	+	0.517493i	$0.826866\pi$
$948$	0	0
$949$	1.64920e6	1.83122
$950$	0	0
$951$	516042.i	0.570590i
$952$	0	0
$953$	1.38986e6	1.53033	0.765165	−	0.643834i	$-0.222657\pi$
$953$	1.38986e6	1.53033	0.765165	+	0.643834i	$0.222657\pi$
$954$	0	0
$955$	235765.i	0.258508i
$956$	0	0
$957$	93525.8	0.102119
$958$	0	0
$959$	309084.i	0.336078i
$960$	0	0
$961$	−1.22765e6	−1.32931
$962$	0	0
$963$	− 195388.i	− 0.210691i
$964$	0	0
$965$	−466018.	−0.500436
$966$	0	0
$967$	1.55110e6i	1.65877i	0.558679	+	0.829384i	$0.311308\pi$
$967$	1.55110e6i	1.65877i	−0.558679	+	0.829384i	$0.688692\pi$
$968$	0	0
$969$	12016.9	0.0127981
$970$	0	0
$971$	481613.i	0.510810i	0.966834	+	0.255405i	$0.0822089\pi$
$971$	481613.i	0.510810i	−0.966834	+	0.255405i	$0.917791\pi$
$972$	0	0
$973$	295240.	0.311853
$974$	0	0
$975$	− 279343.i	− 0.293852i
$976$	0	0
$977$	1.05588e6	1.10618	0.553089	−	0.833122i	$-0.313449\pi$
$977$	1.05588e6	1.10618	0.553089	+	0.833122i	$0.313449\pi$
$978$	0	0
$979$	39736.5i	0.0414595i
$980$	0	0
$981$	166269.	0.172772
$982$	0	0
$983$	844860.i	0.874335i	0.899380	+	0.437167i	$0.144018\pi$
$983$	844860.i	0.874335i	−0.899380	+	0.437167i	$0.855982\pi$
$984$	0	0
$985$	225658.	0.232583
$986$	0	0
$987$	− 1.32654e6i	− 1.36172i
$988$	0	0
$989$	−1.19607e6	−1.22282
$990$	0	0
$991$	− 1.91051e6i	− 1.94537i	−0.232132	−	0.972684i	$-0.574570\pi$
$991$	− 1.91051e6i	− 1.94537i	0.232132	−	0.972684i	$-0.425430\pi$
$992$	0	0
$993$	573459.	0.581572
$994$	0	0
$995$	− 482126.i	− 0.486984i
$996$	0	0
$997$	26126.1	0.0262836	0.0131418	−	0.999914i	$-0.495817\pi$
$997$	26126.1	0.0262836	0.0131418	+	0.999914i	$0.495817\pi$
$998$	0	0
$999$	− 1.50536e6i	− 1.50838i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.5.b.d.191.6		8
4.3	odd	2	inner	320.5.b.d.191.3		8
8.3	odd	2		20.5.b.a.11.4	yes	8
8.5	even	2		20.5.b.a.11.3	✓	8
24.5	odd	2		180.5.c.a.91.6		8
24.11	even	2		180.5.c.a.91.5		8
40.3	even	4		100.5.d.c.99.16		16
40.13	odd	4		100.5.d.c.99.2		16
40.19	odd	2		100.5.b.c.51.5		8
40.27	even	4		100.5.d.c.99.1		16
40.29	even	2		100.5.b.c.51.6		8
40.37	odd	4		100.5.d.c.99.15		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.5.b.a.11.3	✓	8	8.5	even	2
20.5.b.a.11.4	yes	8	8.3	odd	2
100.5.b.c.51.5		8	40.19	odd	2
100.5.b.c.51.6		8	40.29	even	2
100.5.d.c.99.1		16	40.27	even	4
100.5.d.c.99.2		16	40.13	odd	4
100.5.d.c.99.15		16	40.37	odd	4
100.5.d.c.99.16		16	40.3	even	4
180.5.c.a.91.5		8	24.11	even	2
180.5.c.a.91.6		8	24.5	odd	2
320.5.b.d.191.3		8	4.3	odd	2	inner
320.5.b.d.191.6		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)

\(f(q)\)	\(=\)	\(q+8.14153i q^{3} +11.1803 q^{5} -63.6032i q^{7} +14.7155 q^{9} +O(q^{10})\)	\(q+8.14153i q^{3} +11.1803 q^{5} -63.6032i q^{7} +14.7155 q^{9} +33.3808i q^{11} -274.487 q^{13} +91.0251i q^{15} -284.950 q^{17} +5.17988i q^{19} +517.827 q^{21} +584.740i q^{23} +125.000 q^{25} +779.271i q^{27} -344.135 q^{29} +1466.69i q^{31} -271.771 q^{33} -711.105i q^{35} -1931.76 q^{37} -2234.74i q^{39} -976.795 q^{41} +2045.47i q^{43} +164.524 q^{45} -2561.74i q^{47} -1644.37 q^{49} -2319.93i q^{51} -121.922 q^{53} +373.209i q^{55} -42.1721 q^{57} -3456.33i q^{59} -4135.14 q^{61} -935.953i q^{63} -3068.86 q^{65} -1694.07i q^{67} -4760.68 q^{69} +7646.84i q^{71} -6008.30 q^{73} +1017.69i q^{75} +2123.12 q^{77} -4017.61i q^{79} -5152.50 q^{81} -3016.13i q^{83} -3185.84 q^{85} -2801.79i q^{87} +1190.40 q^{89} +17458.2i q^{91} -11941.1 q^{93} +57.9128i q^{95} -3021.43 q^{97} +491.215i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 328 q^{9}+O(q^{10}) \) 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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