LMFDB - Embedded newform 320.5.b.d.191.4

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			191.4
Root			$1.21760 + 1.58665i$ of defining polynomial
Character	$\chi$	$=$	320.191
Dual form			320.5.b.d.191.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.20523i q^{3} -11.1803 q^{5} -30.6227i q^{7} +70.7265 q^{9} +O(q^{10})$	$q-3.20523i q^{3} -11.1803 q^{5} -30.6227i q^{7} +70.7265 q^{9} +168.004i q^{11} -3.15282 q^{13} +35.8356i q^{15} -229.376 q^{17} -151.168i q^{19} -98.1528 q^{21} +916.032i q^{23} +125.000 q^{25} -486.319i q^{27} +1004.39 q^{29} -1728.79i q^{31} +538.492 q^{33} +342.372i q^{35} +841.993 q^{37} +10.1055i q^{39} +2525.54 q^{41} +1653.22i q^{43} -790.746 q^{45} +2665.04i q^{47} +1463.25 q^{49} +735.204i q^{51} +3145.59 q^{53} -1878.34i q^{55} -484.527 q^{57} -3264.26i q^{59} +767.207 q^{61} -2165.83i q^{63} +35.2496 q^{65} -1859.46i q^{67} +2936.10 q^{69} +2774.43i q^{71} +2252.97 q^{73} -400.654i q^{75} +5144.73 q^{77} -906.591i q^{79} +4170.08 q^{81} -3833.20i q^{83} +2564.50 q^{85} -3219.31i q^{87} +1824.43 q^{89} +96.5478i q^{91} -5541.19 q^{93} +1690.10i q^{95} +15831.0 q^{97} +11882.3i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 3.20523i	− 0.356137i	−0.984018	−	0.178069i	$-0.943015\pi$
$3$	− 3.20523i	− 0.356137i	0.984018	−	0.178069i	$-0.0569849\pi$
$4$	0	0
$5$	−11.1803	−0.447214
$5$	−11.1803	−0.447214
$6$	0	0
$7$	− 30.6227i	− 0.624953i	−0.949926	−	0.312476i	$-0.898842\pi$
$7$	− 30.6227i	− 0.624953i	0.949926	−	0.312476i	$-0.101158\pi$
$8$	0	0
$9$	70.7265	0.873166
$10$	0	0
$11$	168.004i	1.38846i	0.719752	+	0.694231i	$0.244255\pi$
$11$	168.004i	1.38846i	−0.719752	+	0.694231i	$0.755745\pi$
$12$	0	0
$13$	−3.15282	−0.0186557	−0.00932787	−	0.999956i	$-0.502969\pi$
$13$	−3.15282	−0.0186557	−0.00932787	+	0.999956i	$0.502969\pi$
$14$	0	0
$15$	35.8356i	0.159269i
$16$	0	0
$17$	−229.376	−0.793689	−0.396844	−	0.917886i	$-0.629895\pi$
$17$	−229.376	−0.793689	−0.396844	+	0.917886i	$0.629895\pi$
$18$	0	0
$19$	− 151.168i	− 0.418747i	−0.977836	−	0.209373i	$-0.932858\pi$
$19$	− 151.168i	− 0.418747i	0.977836	−	0.209373i	$-0.0671424\pi$
$20$	0	0
$21$	−98.1528	−0.222569
$22$	0	0
$23$	916.032i	1.73163i	0.500365	+	0.865814i	$0.333199\pi$
$23$	916.032i	1.73163i	−0.500365	+	0.865814i	$0.666801\pi$
$24$	0	0
$25$	125.000	0.200000
$26$	0	0
$27$	− 486.319i	− 0.667104i
$28$	0	0
$29$	1004.39	1.19428	0.597141	−	0.802136i	$-0.296303\pi$
$29$	1004.39	1.19428	0.597141	+	0.802136i	$0.296303\pi$
$30$	0	0
$31$	− 1728.79i	− 1.79895i	−0.436969	−	0.899477i	$-0.643948\pi$
$31$	− 1728.79i	− 1.79895i	0.436969	−	0.899477i	$-0.356052\pi$
$32$	0	0
$33$	538.492	0.494483
$34$	0	0
$35$	342.372i	0.279487i
$36$	0	0
$37$	841.993	0.615042	0.307521	−	0.951541i	$-0.400501\pi$
$37$	841.993	0.615042	0.307521	+	0.951541i	$0.400501\pi$
$38$	0	0
$39$	10.1055i	0.00664400i
$40$	0	0
$41$	2525.54	1.50240	0.751202	−	0.660073i	$-0.229475\pi$
$41$	2525.54	1.50240	0.751202	+	0.660073i	$0.229475\pi$
$42$	0	0
$43$	1653.22i	0.894113i	0.894506	+	0.447057i	$0.147528\pi$
$43$	1653.22i	0.894113i	−0.894506	+	0.447057i	$0.852472\pi$
$44$	0	0
$45$	−790.746	−0.390492
$46$	0	0
$47$	2665.04i	1.20645i	0.797573	+	0.603223i	$0.206117\pi$
$47$	2665.04i	1.20645i	−0.797573	+	0.603223i	$0.793883\pi$
$48$	0	0
$49$	1463.25	0.609434
$50$	0	0
$51$	735.204i	0.282662i
$52$	0	0
$53$	3145.59	1.11982	0.559912	−	0.828552i	$-0.310835\pi$
$53$	3145.59	1.11982	0.559912	+	0.828552i	$0.310835\pi$
$54$	0	0
$55$	− 1878.34i	− 0.620939i
$56$	0	0
$57$	−484.527	−0.149131
$58$	0	0
$59$	− 3264.26i	− 0.937737i	−0.883268	−	0.468869i	$-0.844662\pi$
$59$	− 3264.26i	− 0.937737i	0.883268	−	0.468869i	$-0.155338\pi$
$60$	0	0
$61$	767.207	0.206183	0.103091	−	0.994672i	$-0.467127\pi$
$61$	767.207	0.206183	0.103091	+	0.994672i	$0.467127\pi$
$62$	0	0
$63$	− 2165.83i	− 0.545688i
$64$	0	0
$65$	35.2496	0.00834310
$66$	0	0
$67$	− 1859.46i	− 0.414227i	−0.978317	−	0.207113i	$-0.933593\pi$
$67$	− 1859.46i	− 0.414227i	0.978317	−	0.207113i	$-0.0664069\pi$
$68$	0	0
$69$	2936.10	0.616697
$70$	0	0
$71$	2774.43i	0.550374i	0.961391	+	0.275187i	$0.0887398\pi$
$71$	2774.43i	0.550374i	−0.961391	+	0.275187i	$0.911260\pi$
$72$	0	0
$73$	2252.97	0.422776	0.211388	−	0.977402i	$-0.432202\pi$
$73$	2252.97	0.422776	0.211388	+	0.977402i	$0.432202\pi$
$74$	0	0
$75$	− 400.654i	− 0.0712274i
$76$	0	0
$77$	5144.73	0.867723
$78$	0	0
$79$	− 906.591i	− 0.145264i	−0.997359	−	0.0726318i	$-0.976860\pi$
$79$	− 906.591i	− 0.145264i	0.997359	−	0.0726318i	$-0.0231398\pi$
$80$	0	0
$81$	4170.08	0.635586
$82$	0	0
$83$	− 3833.20i	− 0.556423i	−0.960520	−	0.278211i	$-0.910258\pi$
$83$	− 3833.20i	− 0.556423i	0.960520	−	0.278211i	$-0.0897416\pi$
$84$	0	0
$85$	2564.50	0.354948
$86$	0	0
$87$	− 3219.31i	− 0.425328i
$88$	0	0
$89$	1824.43	0.230329	0.115164	−	0.993346i	$-0.463261\pi$
$89$	1824.43	0.230329	0.115164	+	0.993346i	$0.463261\pi$
$90$	0	0
$91$	96.5478i	0.0116590i
$92$	0	0
$93$	−5541.19	−0.640674
$94$	0	0
$95$	1690.10i	0.187269i
$96$	0	0
$97$	15831.0	1.68253	0.841267	−	0.540620i	$-0.181810\pi$
$97$	15831.0	1.68253	0.841267	+	0.540620i	$0.181810\pi$
$98$	0	0
$99$	11882.3i	1.21236i
$100$	0	0
$101$	12246.1	1.20048	0.600240	−	0.799820i	$-0.295072\pi$
$101$	12246.1	1.20048	0.600240	+	0.799820i	$0.295072\pi$
$102$	0	0
$103$	1393.83i	0.131382i	0.997840	+	0.0656908i	$0.0209251\pi$
$103$	1393.83i	0.131382i	−0.997840	+	0.0656908i	$0.979075\pi$
$104$	0	0
$105$	1097.38	0.0995358
$106$	0	0
$107$	19221.2i	1.67886i	0.543470	+	0.839429i	$0.317110\pi$
$107$	19221.2i	1.67886i	−0.543470	+	0.839429i	$0.682890\pi$
$108$	0	0
$109$	−3429.78	−0.288678	−0.144339	−	0.989528i	$-0.546106\pi$
$109$	−3429.78	−0.288678	−0.144339	+	0.989528i	$0.546106\pi$
$110$	0	0
$111$	− 2698.78i	− 0.219039i
$112$	0	0
$113$	−6436.39	−0.504064	−0.252032	−	0.967719i	$-0.581099\pi$
$113$	−6436.39	−0.504064	−0.252032	+	0.967719i	$0.581099\pi$
$114$	0	0
$115$	− 10241.5i	− 0.774408i
$116$	0	0
$117$	−222.988	−0.0162896
$118$	0	0
$119$	7024.11i	0.496018i
$120$	0	0
$121$	−13584.3	−0.927827
$122$	0	0
$123$	− 8094.94i	− 0.535061i
$124$	0	0
$125$	−1397.54	−0.0894427
$126$	0	0
$127$	3262.31i	0.202264i	0.994873	+	0.101132i	$0.0322464\pi$
$127$	3262.31i	0.202264i	−0.994873	+	0.101132i	$0.967754\pi$
$128$	0	0
$129$	5298.94	0.318427
$130$	0	0
$131$	− 24866.0i	− 1.44899i	−0.689282	−	0.724493i	$-0.742074\pi$
$131$	− 24866.0i	− 1.44899i	0.689282	−	0.724493i	$-0.257926\pi$
$132$	0	0
$133$	−4629.16	−0.261697
$134$	0	0
$135$	5437.21i	0.298338i
$136$	0	0
$137$	−4464.13	−0.237846	−0.118923	−	0.992903i	$-0.537944\pi$
$137$	−4464.13	−0.237846	−0.118923	+	0.992903i	$0.537944\pi$
$138$	0	0
$139$	2234.68i	0.115661i	0.998326	+	0.0578304i	$0.0184183\pi$
$139$	2234.68i	0.115661i	−0.998326	+	0.0578304i	$0.981582\pi$
$140$	0	0
$141$	8542.07	0.429660
$142$	0	0
$143$	− 529.686i	− 0.0259028i
$144$	0	0
$145$	−11229.4	−0.534099
$146$	0	0
$147$	− 4690.06i	− 0.217042i
$148$	0	0
$149$	−5918.85	−0.266603	−0.133301	−	0.991076i	$-0.542558\pi$
$149$	−5918.85	−0.266603	−0.133301	+	0.991076i	$0.542558\pi$
$150$	0	0
$151$	2779.95i	0.121922i	0.998140	+	0.0609611i	$0.0194166\pi$
$151$	2779.95i	0.121922i	−0.998140	+	0.0609611i	$0.980583\pi$
$152$	0	0
$153$	−16223.0	−0.693022
$154$	0	0
$155$	19328.5i	0.804516i
$156$	0	0
$157$	−15602.7	−0.632995	−0.316497	−	0.948593i	$-0.602507\pi$
$157$	−15602.7	−0.632995	−0.316497	+	0.948593i	$0.602507\pi$
$158$	0	0
$159$	− 10082.3i	− 0.398811i
$160$	0	0
$161$	28051.3	1.08219
$162$	0	0
$163$	40312.2i	1.51726i	0.651520	+	0.758632i	$0.274132\pi$
$163$	40312.2i	1.51726i	−0.651520	+	0.758632i	$0.725868\pi$
$164$	0	0
$165$	−6020.52	−0.221139
$166$	0	0
$167$	− 21868.7i	− 0.784135i	−0.919936	−	0.392068i	$-0.871760\pi$
$167$	− 21868.7i	− 0.784135i	0.919936	−	0.392068i	$-0.128240\pi$
$168$	0	0
$169$	−28551.1	−0.999652
$170$	0	0
$171$	− 10691.6i	− 0.365636i
$172$	0	0
$173$	−43256.9	−1.44532	−0.722659	−	0.691205i	$-0.757080\pi$
$173$	−43256.9	−1.44532	−0.722659	+	0.691205i	$0.757080\pi$
$174$	0	0
$175$	− 3827.83i	− 0.124991i
$176$	0	0
$177$	−10462.7	−0.333963
$178$	0	0
$179$	− 9686.18i	− 0.302306i	−0.988510	−	0.151153i	$-0.951701\pi$
$179$	− 9686.18i	− 0.302306i	0.988510	−	0.151153i	$-0.0482986\pi$
$180$	0	0
$181$	−18040.2	−0.550661	−0.275330	−	0.961350i	$-0.588787\pi$
$181$	−18040.2	−0.550661	−0.275330	+	0.961350i	$0.588787\pi$
$182$	0	0
$183$	− 2459.08i	− 0.0734294i
$184$	0	0
$185$	−9413.77	−0.275055
$186$	0	0
$187$	− 38536.1i	− 1.10201i
$188$	0	0
$189$	−14892.4	−0.416908
$190$	0	0
$191$	− 25591.6i	− 0.701504i	−0.936468	−	0.350752i	$-0.885926\pi$
$191$	− 25591.6i	− 0.701504i	0.936468	−	0.350752i	$-0.114074\pi$
$192$	0	0
$193$	−37818.2	−1.01528	−0.507641	−	0.861569i	$-0.669482\pi$
$193$	−37818.2	−1.01528	−0.507641	+	0.861569i	$0.669482\pi$
$194$	0	0
$195$	− 112.983i	− 0.00297129i
$196$	0	0
$197$	13762.7	0.354625	0.177313	−	0.984155i	$-0.443260\pi$
$197$	13762.7	0.354625	0.177313	+	0.984155i	$0.443260\pi$
$198$	0	0
$199$	23429.5i	0.591640i	0.955244	+	0.295820i	$0.0955928\pi$
$199$	23429.5i	0.591640i	−0.955244	+	0.295820i	$0.904407\pi$
$200$	0	0
$201$	−5960.02	−0.147522
$202$	0	0
$203$	− 30757.2i	− 0.746370i
$204$	0	0
$205$	−28236.4	−0.671895
$206$	0	0
$207$	64787.7i	1.51200i
$208$	0	0
$209$	25396.7	0.581414
$210$	0	0
$211$	50926.3i	1.14387i	0.820299	+	0.571935i	$0.193807\pi$
$211$	50926.3i	1.14387i	−0.820299	+	0.571935i	$0.806193\pi$
$212$	0	0
$213$	8892.71	0.196009
$214$	0	0
$215$	− 18483.5i	− 0.399860i
$216$	0	0
$217$	−52940.3	−1.12426
$218$	0	0
$219$	− 7221.30i	− 0.150566i
$220$	0	0
$221$	723.182	0.0148069
$222$	0	0
$223$	− 518.539i	− 0.0104273i	−0.999986	−	0.00521365i	$-0.998340\pi$
$223$	− 518.539i	− 0.0104273i	0.999986	−	0.00521365i	$-0.00165956\pi$
$224$	0	0
$225$	8840.81	0.174633
$226$	0	0
$227$	27965.3i	0.542711i	0.962479	+	0.271355i	$0.0874719\pi$
$227$	27965.3i	0.542711i	−0.962479	+	0.271355i	$0.912528\pi$
$228$	0	0
$229$	69032.7	1.31639	0.658194	−	0.752848i	$-0.271321\pi$
$229$	69032.7	1.31639	0.658194	+	0.752848i	$0.271321\pi$
$230$	0	0
$231$	− 16490.1i	− 0.309028i
$232$	0	0
$233$	−73445.5	−1.35286	−0.676431	−	0.736506i	$-0.736474\pi$
$233$	−73445.5	−1.35286	−0.676431	+	0.736506i	$0.736474\pi$
$234$	0	0
$235$	− 29796.0i	− 0.539539i
$236$	0	0
$237$	−2905.84	−0.0517338
$238$	0	0
$239$	5261.50i	0.0921115i	0.998939	+	0.0460558i	$0.0146652\pi$
$239$	5261.50i	0.0921115i	−0.998939	+	0.0460558i	$0.985335\pi$
$240$	0	0
$241$	83930.3	1.44506	0.722528	−	0.691342i	$-0.242980\pi$
$241$	83930.3	1.44506	0.722528	+	0.691342i	$0.242980\pi$
$242$	0	0
$243$	− 52757.9i	− 0.893460i
$244$	0	0
$245$	−16359.7	−0.272547
$246$	0	0
$247$	476.604i	0.00781203i
$248$	0	0
$249$	−12286.3	−0.198163
$250$	0	0
$251$	96363.0i	1.52955i	0.644299	+	0.764774i	$0.277149\pi$
$251$	96363.0i	1.52955i	−0.644299	+	0.764774i	$0.722851\pi$
$252$	0	0
$253$	−153897.	−2.40430
$254$	0	0
$255$	− 8219.83i	− 0.126410i
$256$	0	0
$257$	60685.4	0.918794	0.459397	−	0.888231i	$-0.348066\pi$
$257$	60685.4	0.918794	0.459397	+	0.888231i	$0.348066\pi$
$258$	0	0
$259$	− 25784.1i	− 0.384372i
$260$	0	0
$261$	71037.1	1.04281
$262$	0	0
$263$	47710.6i	0.689769i	0.938645	+	0.344884i	$0.112082\pi$
$263$	47710.6i	0.689769i	−0.938645	+	0.344884i	$0.887918\pi$
$264$	0	0
$265$	−35168.7	−0.500801
$266$	0	0
$267$	− 5847.74i	− 0.0820286i
$268$	0	0
$269$	−65980.8	−0.911828	−0.455914	−	0.890024i	$-0.650687\pi$
$269$	−65980.8	−0.911828	−0.455914	+	0.890024i	$0.650687\pi$
$270$	0	0
$271$	15705.1i	0.213847i	0.994267	+	0.106924i	$0.0341000\pi$
$271$	15705.1i	0.213847i	−0.994267	+	0.106924i	$0.965900\pi$
$272$	0	0
$273$	309.458	0.00415219
$274$	0	0
$275$	21000.5i	0.277692i
$276$	0	0
$277$	90555.4	1.18020	0.590099	−	0.807331i	$-0.299089\pi$
$277$	90555.4	1.18020	0.590099	+	0.807331i	$0.299089\pi$
$278$	0	0
$279$	− 122272.i	− 1.57079i
$280$	0	0
$281$	24002.4	0.303978	0.151989	−	0.988382i	$-0.451432\pi$
$281$	24002.4	0.303978	0.151989	+	0.988382i	$0.451432\pi$
$282$	0	0
$283$	55978.1i	0.698949i	0.936946	+	0.349475i	$0.113640\pi$
$283$	55978.1i	0.698949i	−0.936946	+	0.349475i	$0.886360\pi$
$284$	0	0
$285$	5417.18	0.0666935
$286$	0	0
$287$	− 77338.8i	− 0.938931i
$288$	0	0
$289$	−30907.6	−0.370058
$290$	0	0
$291$	− 50741.9i	− 0.599213i
$292$	0	0
$293$	5265.22	0.0613312	0.0306656	−	0.999530i	$-0.490237\pi$
$293$	5265.22	0.0613312	0.0306656	+	0.999530i	$0.490237\pi$
$294$	0	0
$295$	36495.6i	0.419369i
$296$	0	0
$297$	81703.5	0.926249
$298$	0	0
$299$	− 2888.08i	− 0.0323048i
$300$	0	0
$301$	50625.9	0.558778
$302$	0	0
$303$	− 39251.6i	− 0.427536i
$304$	0	0
$305$	−8577.63	−0.0922078
$306$	0	0
$307$	6803.29i	0.0721842i	0.999348	+	0.0360921i	$0.0114910\pi$
$307$	6803.29i	0.0721842i	−0.999348	+	0.0360921i	$0.988509\pi$
$308$	0	0
$309$	4467.54	0.0467898
$310$	0	0
$311$	175182.i	1.81121i	0.424120	+	0.905606i	$0.360583\pi$
$311$	175182.i	1.81121i	−0.424120	+	0.905606i	$0.639417\pi$
$312$	0	0
$313$	−100372.	−1.02453	−0.512264	−	0.858828i	$-0.671193\pi$
$313$	−100372.	−1.02453	−0.512264	+	0.858828i	$0.671193\pi$
$314$	0	0
$315$	24214.8i	0.244039i
$316$	0	0
$317$	50852.5	0.506051	0.253025	−	0.967460i	$-0.418574\pi$
$317$	50852.5	0.506051	0.253025	+	0.967460i	$0.418574\pi$
$318$	0	0
$319$	168742.i	1.65822i
$320$	0	0
$321$	61608.6	0.597903
$322$	0	0
$323$	34674.2i	0.332355i
$324$	0	0
$325$	−394.103	−0.00373115
$326$	0	0
$327$	10993.3i	0.102809i
$328$	0	0
$329$	81610.6	0.753971
$330$	0	0
$331$	− 136931.i	− 1.24982i	−0.780698	−	0.624908i	$-0.785136\pi$
$331$	− 136931.i	− 1.24982i	0.780698	−	0.624908i	$-0.214864\pi$
$332$	0	0
$333$	59551.2	0.537034
$334$	0	0
$335$	20789.4i	0.185248i
$336$	0	0
$337$	196912.	1.73385	0.866926	−	0.498437i	$-0.166093\pi$
$337$	196912.	1.73385	0.866926	+	0.498437i	$0.166093\pi$
$338$	0	0
$339$	20630.1i	0.179516i
$340$	0	0
$341$	290444.	2.49778
$342$	0	0
$343$	− 118334.i	− 1.00582i
$344$	0	0
$345$	−32826.5	−0.275795
$346$	0	0
$347$	− 68004.5i	− 0.564779i	−0.959300	−	0.282390i	$-0.908873\pi$
$347$	− 68004.5i	− 0.564779i	0.959300	−	0.282390i	$-0.0911271\pi$
$348$	0	0
$349$	25273.0	0.207494	0.103747	−	0.994604i	$-0.466917\pi$
$349$	25273.0	0.207494	0.103747	+	0.994604i	$0.466917\pi$
$350$	0	0
$351$	1533.28i	0.0124453i
$352$	0	0
$353$	−23471.6	−0.188362	−0.0941810	−	0.995555i	$-0.530023\pi$
$353$	−23471.6	−0.188362	−0.0941810	+	0.995555i	$0.530023\pi$
$354$	0	0
$355$	− 31019.1i	− 0.246135i
$356$	0	0
$357$	22513.9	0.176650
$358$	0	0
$359$	− 175609.i	− 1.36257i	−0.732018	−	0.681285i	$-0.761421\pi$
$359$	− 175609.i	− 1.36257i	0.732018	−	0.681285i	$-0.238579\pi$
$360$	0	0
$361$	107469.	0.824651
$362$	0	0
$363$	43540.9i	0.330434i
$364$	0	0
$365$	−25189.0	−0.189071
$366$	0	0
$367$	− 191326.i	− 1.42050i	−0.703950	−	0.710250i	$-0.748582\pi$
$367$	− 191326.i	− 1.42050i	0.703950	−	0.710250i	$-0.251418\pi$
$368$	0	0
$369$	178623.	1.31185
$370$	0	0
$371$	− 96326.3i	− 0.699837i
$372$	0	0
$373$	−227739.	−1.63689	−0.818446	−	0.574584i	$-0.805164\pi$
$373$	−227739.	−1.63689	−0.818446	+	0.574584i	$0.805164\pi$
$374$	0	0
$375$	4479.45i	0.0318539i
$376$	0	0
$377$	−3166.67	−0.0222802
$378$	0	0
$379$	115080.i	0.801167i	0.916260	+	0.400583i	$0.131192\pi$
$379$	115080.i	0.801167i	−0.916260	+	0.400583i	$0.868808\pi$
$380$	0	0
$381$	10456.5	0.0720336
$382$	0	0
$383$	112755.i	0.768670i	0.923194	+	0.384335i	$0.125569\pi$
$383$	112755.i	0.768670i	−0.923194	+	0.384335i	$0.874431\pi$
$384$	0	0
$385$	−57519.8	−0.388057
$386$	0	0
$387$	116926.i	0.780710i
$388$	0	0
$389$	183416.	1.21210	0.606049	−	0.795427i	$-0.292754\pi$
$389$	183416.	1.21210	0.606049	+	0.795427i	$0.292754\pi$
$390$	0	0
$391$	− 210116.i	− 1.37437i
$392$	0	0
$393$	−79701.5	−0.516038
$394$	0	0
$395$	10136.0i	0.0649639i
$396$	0	0
$397$	−64357.2	−0.408335	−0.204167	−	0.978936i	$-0.565449\pi$
$397$	−64357.2	−0.408335	−0.204167	+	0.978936i	$0.565449\pi$
$398$	0	0
$399$	14837.5i	0.0932000i
$400$	0	0
$401$	95942.1	0.596651	0.298326	−	0.954464i	$-0.403572\pi$
$401$	95942.1	0.596651	0.298326	+	0.954464i	$0.403572\pi$
$402$	0	0
$403$	5450.58i	0.0335608i
$404$	0	0
$405$	−46622.9	−0.284243
$406$	0	0
$407$	141458.i	0.853963i
$408$	0	0
$409$	−115605.	−0.691084	−0.345542	−	0.938403i	$-0.612305\pi$
$409$	−115605.	−0.691084	−0.345542	+	0.938403i	$0.612305\pi$
$410$	0	0
$411$	14308.6i	0.0847058i
$412$	0	0
$413$	−99960.5	−0.586041
$414$	0	0
$415$	42856.5i	0.248840i
$416$	0	0
$417$	7162.68	0.0411911
$418$	0	0
$419$	− 95151.9i	− 0.541987i	−0.962581	−	0.270994i	$-0.912648\pi$
$419$	− 95151.9i	− 0.541987i	0.962581	−	0.270994i	$-0.0873523\pi$
$420$	0	0
$421$	91153.3	0.514290	0.257145	−	0.966373i	$-0.417218\pi$
$421$	91153.3	0.514290	0.257145	+	0.966373i	$0.417218\pi$
$422$	0	0
$423$	188489.i	1.05343i
$424$	0	0
$425$	−28672.0	−0.158738
$426$	0	0
$427$	− 23493.9i	− 0.128855i
$428$	0	0
$429$	−1697.77	−0.00922495
$430$	0	0
$431$	45720.8i	0.246127i	0.992399	+	0.123063i	$0.0392719\pi$
$431$	45720.8i	0.246127i	−0.992399	+	0.123063i	$0.960728\pi$
$432$	0	0
$433$	−41171.0	−0.219592	−0.109796	−	0.993954i	$-0.535020\pi$
$433$	−41171.0	−0.219592	−0.109796	+	0.993954i	$0.535020\pi$
$434$	0	0
$435$	35993.0i	0.190213i
$436$	0	0
$437$	138474.	0.725114
$438$	0	0
$439$	305545.i	1.58543i	0.609595	+	0.792713i	$0.291332\pi$
$439$	305545.i	1.58543i	−0.609595	+	0.792713i	$0.708668\pi$
$440$	0	0
$441$	103491.	0.532138
$442$	0	0
$443$	− 301615.i	− 1.53690i	−0.639909	−	0.768451i	$-0.721028\pi$
$443$	− 301615.i	− 1.53690i	0.639909	−	0.768451i	$-0.278972\pi$
$444$	0	0
$445$	−20397.8	−0.103006
$446$	0	0
$447$	18971.3i	0.0949472i
$448$	0	0
$449$	44795.9	0.222201	0.111100	−	0.993809i	$-0.464563\pi$
$449$	44795.9	0.222201	0.111100	+	0.993809i	$0.464563\pi$
$450$	0	0
$451$	424301.i	2.08603i
$452$	0	0
$453$	8910.39	0.0434210
$454$	0	0
$455$	− 1079.44i	− 0.00521404i
$456$	0	0
$457$	−260232.	−1.24603	−0.623014	−	0.782211i	$-0.714092\pi$
$457$	−260232.	−1.24603	−0.623014	+	0.782211i	$0.714092\pi$
$458$	0	0
$459$	111550.i	0.529473i
$460$	0	0
$461$	−40597.5	−0.191028	−0.0955140	−	0.995428i	$-0.530449\pi$
$461$	−40597.5	−0.191028	−0.0955140	+	0.995428i	$0.530449\pi$
$462$	0	0
$463$	− 223112.i	− 1.04079i	−0.853927	−	0.520393i	$-0.825785\pi$
$463$	− 223112.i	− 1.04079i	0.853927	−	0.520393i	$-0.174215\pi$
$464$	0	0
$465$	61952.4	0.286518
$466$	0	0
$467$	− 149775.i	− 0.686760i	−0.939197	−	0.343380i	$-0.888428\pi$
$467$	− 149775.i	− 0.686760i	0.939197	−	0.343380i	$-0.111572\pi$
$468$	0	0
$469$	−56941.8	−0.258872
$470$	0	0
$471$	50010.3i	0.225433i
$472$	0	0
$473$	−277747.	−1.24144
$474$	0	0
$475$	− 18895.9i	− 0.0837494i
$476$	0	0
$477$	222476.	0.977793
$478$	0	0
$479$	− 255892.i	− 1.11528i	−0.830081	−	0.557642i	$-0.811706\pi$
$479$	− 255892.i	− 1.11528i	0.830081	−	0.557642i	$-0.188294\pi$
$480$	0	0
$481$	−2654.65	−0.0114741
$482$	0	0
$483$	− 89911.1i	− 0.385406i
$484$	0	0
$485$	−176995.	−0.752452
$486$	0	0
$487$	88785.3i	0.374354i	0.982326	+	0.187177i	$0.0599339\pi$
$487$	88785.3i	0.374354i	−0.982326	+	0.187177i	$0.940066\pi$
$488$	0	0
$489$	129210.	0.540354
$490$	0	0
$491$	201316.i	0.835055i	0.908664	+	0.417528i	$0.137103\pi$
$491$	201316.i	0.835055i	−0.908664	+	0.417528i	$0.862897\pi$
$492$	0	0
$493$	−230383.	−0.947889
$494$	0	0
$495$	− 132848.i	− 0.542183i
$496$	0	0
$497$	84960.6	0.343958
$498$	0	0
$499$	− 357435.i	− 1.43548i	−0.696313	−	0.717739i	$-0.745177\pi$
$499$	− 357435.i	− 1.43548i	0.696313	−	0.717739i	$-0.254823\pi$
$500$	0	0
$501$	−70094.4	−0.279260
$502$	0	0
$503$	− 238591.i	− 0.943015i	−0.881862	−	0.471507i	$-0.843710\pi$
$503$	− 238591.i	− 0.943015i	0.881862	−	0.471507i	$-0.156290\pi$
$504$	0	0
$505$	−136916.	−0.536871
$506$	0	0
$507$	91512.8i	0.356013i
$508$	0	0
$509$	−108780.	−0.419870	−0.209935	−	0.977715i	$-0.567325\pi$
$509$	−108780.	−0.419870	−0.209935	+	0.977715i	$0.567325\pi$
$510$	0	0
$511$	− 68992.0i	− 0.264215i
$512$	0	0
$513$	−73515.6	−0.279348
$514$	0	0
$515$	− 15583.5i	− 0.0587556i
$516$	0	0
$517$	−447737.	−1.67510
$518$	0	0
$519$	138649.i	0.514731i
$520$	0	0
$521$	318609.	1.17377	0.586884	−	0.809671i	$-0.300354\pi$
$521$	318609.	1.17377	0.586884	+	0.809671i	$0.300354\pi$
$522$	0	0
$523$	265583.i	0.970948i	0.874251	+	0.485474i	$0.161353\pi$
$523$	265583.i	0.970948i	−0.874251	+	0.485474i	$0.838647\pi$
$524$	0	0
$525$	−12269.1	−0.0445138
$526$	0	0
$527$	396544.i	1.42781i
$528$	0	0
$529$	−559273.	−1.99854
$530$	0	0
$531$	− 230870.i	− 0.818801i
$532$	0	0
$533$	−7962.57	−0.0280284
$534$	0	0
$535$	− 214900.i	− 0.750808i
$536$	0	0
$537$	−31046.5	−0.107662
$538$	0	0
$539$	245832.i	0.846176i
$540$	0	0
$541$	−169497.	−0.579118	−0.289559	−	0.957160i	$-0.593509\pi$
$541$	−169497.	−0.579118	−0.289559	+	0.957160i	$0.593509\pi$
$542$	0	0
$543$	57823.1i	0.196111i
$544$	0	0
$545$	38346.1	0.129101
$546$	0	0
$547$	− 96446.8i	− 0.322339i	−0.986927	−	0.161170i	$-0.948473\pi$
$547$	− 96446.8i	− 0.322339i	0.986927	−	0.161170i	$-0.0515266\pi$
$548$	0	0
$549$	54261.8	0.180032
$550$	0	0
$551$	− 151831.i	− 0.500102i
$552$	0	0
$553$	−27762.2	−0.0907829
$554$	0	0
$555$	30173.3i	0.0979574i
$556$	0	0
$557$	200938.	0.647668	0.323834	−	0.946114i	$-0.395028\pi$
$557$	200938.	0.647668	0.323834	+	0.946114i	$0.395028\pi$
$558$	0	0
$559$	− 5212.29i	− 0.0166804i
$560$	0	0
$561$	−123517.	−0.392465
$562$	0	0
$563$	417434.i	1.31696i	0.752600	+	0.658478i	$0.228799\pi$
$563$	417434.i	1.31696i	−0.752600	+	0.658478i	$0.771201\pi$
$564$	0	0
$565$	71961.1	0.225424
$566$	0	0
$567$	− 127699.i	− 0.397211i
$568$	0	0
$569$	−424.191	−0.00131020	−0.000655100	−	1.00000i	$-0.500209\pi$
$569$	−424.191	−0.00131020	−0.000655100		1.00000i	$0.500209\pi$
$570$	0	0
$571$	− 606394.i	− 1.85987i	−0.367723	−	0.929935i	$-0.619862\pi$
$571$	− 606394.i	− 1.85987i	0.367723	−	0.929935i	$-0.380138\pi$
$572$	0	0
$573$	−82027.0	−0.249832
$574$	0	0
$575$	114504.i	0.346326i
$576$	0	0
$577$	−228321.	−0.685795	−0.342897	−	0.939373i	$-0.611408\pi$
$577$	−228321.	−0.685795	−0.342897	+	0.939373i	$0.611408\pi$
$578$	0	0
$579$	121216.i	0.361580i
$580$	0	0
$581$	−117383.	−0.347738
$582$	0	0
$583$	528471.i	1.55483i
$584$	0	0
$585$	2493.08	0.00728492
$586$	0	0
$587$	89650.3i	0.260181i	0.991502	+	0.130090i	$0.0415268\pi$
$587$	89650.3i	0.260181i	−0.991502	+	0.130090i	$0.958473\pi$
$588$	0	0
$589$	−261338.	−0.753306
$590$	0	0
$591$	− 44112.5i	− 0.126295i
$592$	0	0
$593$	−326334.	−0.928012	−0.464006	−	0.885832i	$-0.653588\pi$
$593$	−326334.	−0.928012	−0.464006	+	0.885832i	$0.653588\pi$
$594$	0	0
$595$	− 78531.9i	− 0.221826i
$596$	0	0
$597$	75097.1	0.210705
$598$	0	0
$599$	− 253846.i	− 0.707485i	−0.935343	−	0.353742i	$-0.884909\pi$
$599$	− 253846.i	− 0.707485i	0.935343	−	0.353742i	$-0.115091\pi$
$600$	0	0
$601$	−468553.	−1.29721	−0.648604	−	0.761126i	$-0.724647\pi$
$601$	−468553.	−1.29721	−0.648604	+	0.761126i	$0.724647\pi$
$602$	0	0
$603$	− 131513.i	− 0.361689i
$604$	0	0
$605$	151877.	0.414937
$606$	0	0
$607$	− 104709.i	− 0.284190i	−0.989853	−	0.142095i	$-0.954616\pi$
$607$	− 104709.i	− 0.284190i	0.989853	−	0.142095i	$-0.0453838\pi$
$608$	0	0
$609$	−98583.9	−0.265810
$610$	0	0
$611$	− 8402.39i	− 0.0225071i
$612$	0	0
$613$	−352323.	−0.937604	−0.468802	−	0.883303i	$-0.655314\pi$
$613$	−352323.	−0.937604	−0.468802	+	0.883303i	$0.655314\pi$
$614$	0	0
$615$	90504.2i	0.239287i
$616$	0	0
$617$	543388.	1.42738	0.713690	−	0.700461i	$-0.247022\pi$
$617$	543388.	1.42738	0.713690	+	0.700461i	$0.247022\pi$
$618$	0	0
$619$	− 656868.i	− 1.71434i	−0.515034	−	0.857169i	$-0.672221\pi$
$619$	− 656868.i	− 1.71434i	0.515034	−	0.857169i	$-0.327779\pi$
$620$	0	0
$621$	445483.	1.15518
$622$	0	0
$623$	− 55869.0i	− 0.143945i
$624$	0	0
$625$	15625.0	0.0400000
$626$	0	0
$627$	− 81402.5i	− 0.207063i
$628$	0	0
$629$	−193133.	−0.488152
$630$	0	0
$631$	110480.i	0.277476i	0.990329	+	0.138738i	$0.0443046\pi$
$631$	110480.i	0.277476i	−0.990329	+	0.138738i	$0.955695\pi$
$632$	0	0
$633$	163231.	0.407375
$634$	0	0
$635$	− 36473.8i	− 0.0904551i
$636$	0	0
$637$	−4613.37	−0.0113695
$638$	0	0
$639$	196226.i	0.480568i
$640$	0	0
$641$	−282590.	−0.687766	−0.343883	−	0.939013i	$-0.611742\pi$
$641$	−282590.	−0.687766	−0.343883	+	0.939013i	$0.611742\pi$
$642$	0	0
$643$	− 40275.4i	− 0.0974133i	−0.998813	−	0.0487066i	$-0.984490\pi$
$643$	− 40275.4i	− 0.0974133i	0.998813	−	0.0487066i	$-0.0155099\pi$
$644$	0	0
$645$	−59244.0	−0.142405
$646$	0	0
$647$	− 606942.i	− 1.44990i	−0.688801	−	0.724951i	$-0.741863\pi$
$647$	− 606942.i	− 1.44990i	0.688801	−	0.724951i	$-0.258137\pi$
$648$	0	0
$649$	548409.	1.30201
$650$	0	0
$651$	169686.i	0.400391i
$652$	0	0
$653$	95017.0	0.222831	0.111415	−	0.993774i	$-0.464462\pi$
$653$	95017.0	0.222831	0.111415	+	0.993774i	$0.464462\pi$
$654$	0	0
$655$	278011.i	0.648006i
$656$	0	0
$657$	159345.	0.369154
$658$	0	0
$659$	664518.i	1.53016i	0.643937	+	0.765079i	$0.277300\pi$
$659$	664518.i	1.53016i	−0.643937	+	0.765079i	$0.722700\pi$
$660$	0	0
$661$	−323882.	−0.741283	−0.370642	−	0.928776i	$-0.620862\pi$
$661$	−323882.	−0.741283	−0.370642	+	0.928776i	$0.620862\pi$
$662$	0	0
$663$	− 2317.97i	− 0.00527327i
$664$	0	0
$665$	51755.5	0.117034
$666$	0	0
$667$	920054.i	2.06805i
$668$	0	0
$669$	−1662.04	−0.00371355
$670$	0	0
$671$	128894.i	0.286277i
$672$	0	0
$673$	−301219.	−0.665047	−0.332524	−	0.943095i	$-0.607900\pi$
$673$	−301219.	−0.665047	−0.332524	+	0.943095i	$0.607900\pi$
$674$	0	0
$675$	− 60789.9i	− 0.133421i
$676$	0	0
$677$	−458020.	−0.999325	−0.499662	−	0.866220i	$-0.666543\pi$
$677$	−458020.	−0.999325	−0.499662	+	0.866220i	$0.666543\pi$
$678$	0	0
$679$	− 484786.i	− 1.05150i
$680$	0	0
$681$	89635.5	0.193279
$682$	0	0
$683$	− 209414.i	− 0.448914i	−0.974484	−	0.224457i	$-0.927939\pi$
$683$	− 209414.i	− 0.448914i	0.974484	−	0.224457i	$-0.0720609\pi$
$684$	0	0
$685$	49910.5	0.106368
$686$	0	0
$687$	− 221266.i	− 0.468815i
$688$	0	0
$689$	−9917.47	−0.0208912
$690$	0	0
$691$	42616.4i	0.0892525i	0.999004	+	0.0446263i	$0.0142097\pi$
$691$	42616.4i	0.0892525i	−0.999004	+	0.0446263i	$0.985790\pi$
$692$	0	0
$693$	363869.	0.757666
$694$	0	0
$695$	− 24984.5i	− 0.0517251i
$696$	0	0
$697$	−579298.	−1.19244
$698$	0	0
$699$	235410.i	0.481804i
$700$	0	0
$701$	496713.	1.01081	0.505405	−	0.862882i	$-0.331343\pi$
$701$	496713.	1.01081	0.505405	+	0.862882i	$0.331343\pi$
$702$	0	0
$703$	− 127282.i	− 0.257547i
$704$	0	0
$705$	−95503.3	−0.192150
$706$	0	0
$707$	− 375008.i	− 0.750243i
$708$	0	0
$709$	309547.	0.615792	0.307896	−	0.951420i	$-0.400375\pi$
$709$	309547.	0.615792	0.307896	+	0.951420i	$0.400375\pi$
$710$	0	0
$711$	− 64120.0i	− 0.126839i
$712$	0	0
$713$	1.58363e6	3.11512
$714$	0	0
$715$	5922.07i	0.0115841i
$716$	0	0
$717$	16864.3	0.0328043
$718$	0	0
$719$	410551.i	0.794163i	0.917783	+	0.397082i	$0.129977\pi$
$719$	410551.i	0.794163i	−0.917783	+	0.397082i	$0.870023\pi$
$720$	0	0
$721$	42682.7	0.0821072
$722$	0	0
$723$	− 269016.i	− 0.514638i
$724$	0	0
$725$	125549.	0.238857
$726$	0	0
$727$	714829.i	1.35249i	0.736678	+	0.676244i	$0.236393\pi$
$727$	714829.i	1.35249i	−0.736678	+	0.676244i	$0.763607\pi$
$728$	0	0
$729$	168675.	0.317392
$730$	0	0
$731$	− 379208.i	− 0.709648i
$732$	0	0
$733$	664653.	1.23705	0.618525	−	0.785765i	$-0.287731\pi$
$733$	664653.	1.23705	0.618525	+	0.785765i	$0.287731\pi$
$734$	0	0
$735$	52436.5i	0.0970642i
$736$	0	0
$737$	312397.	0.575138
$738$	0	0
$739$	132886.i	0.243326i	0.992571	+	0.121663i	$0.0388227\pi$
$739$	132886.i	0.243326i	−0.992571	+	0.121663i	$0.961177\pi$
$740$	0	0
$741$	1527.63	0.00278215
$742$	0	0
$743$	170672.i	0.309161i	0.987980	+	0.154581i	$0.0494026\pi$
$743$	170672.i	0.309161i	−0.987980	+	0.154581i	$0.950597\pi$
$744$	0	0
$745$	66174.8	0.119228
$746$	0	0
$747$	− 271109.i	− 0.485850i
$748$	0	0
$749$	588606.	1.04921
$750$	0	0
$751$	201923.i	0.358019i	0.983847	+	0.179010i	$0.0572893\pi$
$751$	201923.i	0.358019i	−0.983847	+	0.179010i	$0.942711\pi$
$752$	0	0
$753$	308866.	0.544728
$754$	0	0
$755$	− 31080.8i	− 0.0545253i
$756$	0	0
$757$	−363133.	−0.633686	−0.316843	−	0.948478i	$-0.602623\pi$
$757$	−363133.	−0.633686	−0.316843	+	0.948478i	$0.602623\pi$
$758$	0	0
$759$	493276.i	0.856261i
$760$	0	0
$761$	−106273.	−0.183508	−0.0917539	−	0.995782i	$-0.529247\pi$
$761$	−106273.	−0.183508	−0.0917539	+	0.995782i	$0.529247\pi$
$762$	0	0
$763$	105029.i	0.180410i
$764$	0	0
$765$	181378.	0.309929
$766$	0	0
$767$	10291.6i	0.0174942i
$768$	0	0
$769$	−573093.	−0.969109	−0.484554	−	0.874761i	$-0.661018\pi$
$769$	−573093.	−0.969109	−0.484554	+	0.874761i	$0.661018\pi$
$770$	0	0
$771$	− 194511.i	− 0.327217i
$772$	0	0
$773$	890302.	1.48997	0.744986	−	0.667080i	$-0.232456\pi$
$773$	890302.	1.48997	0.744986	+	0.667080i	$0.232456\pi$
$774$	0	0
$775$	− 216099.i	− 0.359791i
$776$	0	0
$777$	−82644.0	−0.136889
$778$	0	0
$779$	− 381780.i	− 0.629126i
$780$	0	0
$781$	−466116.	−0.764173
$782$	0	0
$783$	− 488455.i	− 0.796711i
$784$	0	0
$785$	174443.	0.283084
$786$	0	0
$787$	413419.i	0.667484i	0.942664	+	0.333742i	$0.108311\pi$
$787$	413419.i	0.667484i	−0.942664	+	0.333742i	$0.891689\pi$
$788$	0	0
$789$	152924.	0.245652
$790$	0	0
$791$	197100.i	0.315016i
$792$	0	0
$793$	−2418.87	−0.00384650
$794$	0	0
$795$	112724.i	0.178354i
$796$	0	0
$797$	1.06112e6	1.67051	0.835254	−	0.549865i	$-0.185321\pi$
$797$	1.06112e6	1.67051	0.835254	+	0.549865i	$0.185321\pi$
$798$	0	0
$799$	− 611296.i	− 0.957542i
$800$	0	0
$801$	129036.	0.201115
$802$	0	0
$803$	378508.i	0.587008i
$804$	0	0
$805$	−313623.	−0.483968
$806$	0	0
$807$	211484.i	0.324736i
$808$	0	0
$809$	817968.	1.24980	0.624898	−	0.780706i	$-0.285140\pi$
$809$	817968.	1.24980	0.624898	+	0.780706i	$0.285140\pi$
$810$	0	0
$811$	590910.i	0.898420i	0.893426	+	0.449210i	$0.148295\pi$
$811$	590910.i	0.898420i	−0.893426	+	0.449210i	$0.851705\pi$
$812$	0	0
$813$	50338.7	0.0761589
$814$	0	0
$815$	− 450704.i	− 0.678541i
$816$	0	0
$817$	249913.	0.374407
$818$	0	0
$819$	6828.49i	0.0101802i
$820$	0	0
$821$	−646976.	−0.959846	−0.479923	−	0.877311i	$-0.659335\pi$
$821$	−646976.	−0.959846	−0.479923	+	0.877311i	$0.659335\pi$
$822$	0	0
$823$	− 998248.i	− 1.47380i	−0.676002	−	0.736900i	$-0.736289\pi$
$823$	− 998248.i	− 1.47380i	0.676002	−	0.736900i	$-0.263711\pi$
$824$	0	0
$825$	67311.5	0.0988966
$826$	0	0
$827$	− 992385.i	− 1.45101i	−0.688219	−	0.725503i	$-0.741607\pi$
$827$	− 992385.i	− 1.45101i	0.688219	−	0.725503i	$-0.258393\pi$
$828$	0	0
$829$	−725129.	−1.05513	−0.527565	−	0.849514i	$-0.676895\pi$
$829$	−725129.	−1.05513	−0.527565	+	0.849514i	$0.676895\pi$
$830$	0	0
$831$	− 290251.i	− 0.420312i
$832$	0	0
$833$	−335635.	−0.483701
$834$	0	0
$835$	244500.i	0.350676i
$836$	0	0
$837$	−840745.	−1.20009
$838$	0	0
$839$	954288.i	1.35568i	0.735212	+	0.677838i	$0.237083\pi$
$839$	954288.i	1.35568i	−0.735212	+	0.677838i	$0.762917\pi$
$840$	0	0
$841$	301521.	0.426311
$842$	0	0
$843$	− 76933.3i	− 0.108258i
$844$	0	0
$845$	319211.	0.447058
$846$	0	0
$847$	415988.i	0.579848i
$848$	0	0
$849$	179423.	0.248922
$850$	0	0
$851$	771292.i	1.06502i
$852$	0	0
$853$	327068.	0.449511	0.224756	−	0.974415i	$-0.427842\pi$
$853$	327068.	0.449511	0.224756	+	0.974415i	$0.427842\pi$
$854$	0	0
$855$	119535.i	0.163517i
$856$	0	0
$857$	−1.26713e6	−1.72527	−0.862637	−	0.505823i	$-0.831189\pi$
$857$	−1.26713e6	−1.72527	−0.862637	+	0.505823i	$0.831189\pi$
$858$	0	0
$859$	545595.i	0.739407i	0.929150	+	0.369704i	$0.120541\pi$
$859$	545595.i	0.739407i	−0.929150	+	0.369704i	$0.879459\pi$
$860$	0	0
$861$	−247889.	−0.334388
$862$	0	0
$863$	199428.i	0.267771i	0.990997	+	0.133885i	$0.0427454\pi$
$863$	199428.i	0.267771i	−0.990997	+	0.133885i	$0.957255\pi$
$864$	0	0
$865$	483627.	0.646366
$866$	0	0
$867$	99066.2i	0.131791i
$868$	0	0
$869$	152311.	0.201693
$870$	0	0
$871$	5862.56i	0.00772771i
$872$	0	0
$873$	1.11967e6	1.46913
$874$	0	0
$875$	42796.5i	0.0558975i
$876$	0	0
$877$	966378.	1.25646	0.628229	−	0.778028i	$-0.283780\pi$
$877$	966378.	1.25646	0.628229	+	0.778028i	$0.283780\pi$
$878$	0	0
$879$	− 16876.3i	− 0.0218423i
$880$	0	0
$881$	−410709.	−0.529154	−0.264577	−	0.964365i	$-0.585232\pi$
$881$	−410709.	−0.529154	−0.264577	+	0.964365i	$0.585232\pi$
$882$	0	0
$883$	− 1.15757e6i	− 1.48465i	−0.670039	−	0.742326i	$-0.733722\pi$
$883$	− 1.15757e6i	− 1.48465i	0.670039	−	0.742326i	$-0.266278\pi$
$884$	0	0
$885$	116977.	0.149353
$886$	0	0
$887$	− 908264.i	− 1.15442i	−0.816595	−	0.577211i	$-0.804141\pi$
$887$	− 908264.i	− 1.15442i	0.816595	−	0.577211i	$-0.195859\pi$
$888$	0	0
$889$	99900.7	0.126405
$890$	0	0
$891$	700590.i	0.882487i
$892$	0	0
$893$	402867.	0.505195
$894$	0	0
$895$	108295.i	0.135195i
$896$	0	0
$897$	−9256.98	−0.0115049
$898$	0	0
$899$	− 1.73639e6i	− 2.14846i
$900$	0	0
$901$	−721522.	−0.888792
$902$	0	0
$903$	− 162268.i	− 0.199002i
$904$	0	0
$905$	201696.	0.246263
$906$	0	0
$907$	− 1.16501e6i	− 1.41617i	−0.706127	−	0.708085i	$-0.749559\pi$
$907$	− 1.16501e6i	− 1.41617i	0.706127	−	0.708085i	$-0.250441\pi$
$908$	0	0
$909$	866124.	1.04822
$910$	0	0
$911$	− 446494.i	− 0.537996i	−0.963141	−	0.268998i	$-0.913307\pi$
$911$	− 446494.i	− 0.537996i	0.963141	−	0.268998i	$-0.0866925\pi$
$912$	0	0
$913$	643992.	0.772572
$914$	0	0
$915$	27493.3i	0.0328386i
$916$	0	0
$917$	−761465.	−0.905547
$918$	0	0
$919$	− 1.24513e6i	− 1.47429i	−0.675734	−	0.737146i	$-0.736173\pi$
$919$	− 1.24513e6i	− 1.47429i	0.675734	−	0.737146i	$-0.263827\pi$
$920$	0	0
$921$	21806.1	0.0257075
$922$	0	0
$923$	− 8747.30i	− 0.0102676i
$924$	0	0
$925$	105249.	0.123008
$926$	0	0
$927$	98580.4i	0.114718i
$928$	0	0
$929$	−1.65942e6	−1.92276	−0.961378	−	0.275231i	$-0.911246\pi$
$929$	−1.65942e6	−1.92276	−0.961378	+	0.275231i	$0.911246\pi$
$930$	0	0
$931$	− 221196.i	− 0.255199i
$932$	0	0
$933$	561500.	0.645040
$934$	0	0
$935$	430846.i	0.492832i
$936$	0	0
$937$	44504.5	0.0506902	0.0253451	−	0.999679i	$-0.491932\pi$
$937$	44504.5	0.0506902	0.0253451	+	0.999679i	$0.491932\pi$
$938$	0	0
$939$	321716.i	0.364872i
$940$	0	0
$941$	−439028.	−0.495807	−0.247904	−	0.968785i	$-0.579742\pi$
$941$	−439028.	−0.495807	−0.247904	+	0.968785i	$0.579742\pi$
$942$	0	0
$943$	2.31347e6i	2.60160i
$944$	0	0
$945$	166502.	0.186447
$946$	0	0
$947$	983414.i	1.09657i	0.836291	+	0.548285i	$0.184719\pi$
$947$	983414.i	1.09657i	−0.836291	+	0.548285i	$0.815281\pi$
$948$	0	0
$949$	−7103.22	−0.00788720
$950$	0	0
$951$	− 162994.i	− 0.180223i
$952$	0	0
$953$	455101.	0.501097	0.250549	−	0.968104i	$-0.419389\pi$
$953$	455101.	0.501097	0.250549	+	0.968104i	$0.419389\pi$
$954$	0	0
$955$	286123.i	0.313722i
$956$	0	0
$957$	540857.	0.590552
$958$	0	0
$959$	136704.i	0.148642i
$960$	0	0
$961$	−2.06521e6	−2.23623
$962$	0	0
$963$	1.35945e6i	1.46592i
$964$	0	0
$965$	422821.	0.454048
$966$	0	0
$967$	249543.i	0.266865i	0.991058	+	0.133433i	$0.0426000\pi$
$967$	249543.i	0.266865i	−0.991058	+	0.133433i	$0.957400\pi$
$968$	0	0
$969$	111139.	0.118364
$970$	0	0
$971$	− 71856.1i	− 0.0762123i	−0.999274	−	0.0381062i	$-0.987867\pi$
$971$	− 71856.1i	− 0.0762123i	0.999274	−	0.0381062i	$-0.0121325\pi$
$972$	0	0
$973$	68431.9	0.0722825
$974$	0	0
$975$	1263.19i	0.00132880i
$976$	0	0
$977$	279175.	0.292474	0.146237	−	0.989250i	$-0.453284\pi$
$977$	279175.	0.292474	0.146237	+	0.989250i	$0.453284\pi$
$978$	0	0
$979$	306512.i	0.319803i
$980$	0	0
$981$	−242576.	−0.252064
$982$	0	0
$983$	1.47299e6i	1.52438i	0.647355	+	0.762188i	$0.275875\pi$
$983$	1.47299e6i	1.52438i	−0.647355	+	0.762188i	$0.724125\pi$
$984$	0	0
$985$	−153871.	−0.158593
$986$	0	0
$987$	− 261581.i	− 0.268517i
$988$	0	0
$989$	−1.51440e6	−1.54827
$990$	0	0
$991$	29557.4i	0.0300967i	0.999887	+	0.0150483i	$0.00479022\pi$
$991$	29557.4i	0.0300967i	−0.999887	+	0.0150483i	$0.995210\pi$
$992$	0	0
$993$	−438896.	−0.445106
$994$	0	0
$995$	− 261950.i	− 0.264589i
$996$	0	0
$997$	−1.37172e6	−1.37999	−0.689994	−	0.723815i	$-0.742387\pi$
$997$	−1.37172e6	−1.37999	−0.689994	+	0.723815i	$0.742387\pi$
$998$	0	0
$999$	− 409477.i	− 0.410297i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.5.b.d.191.4		8
4.3	odd	2	inner	320.5.b.d.191.5		8
8.3	odd	2		20.5.b.a.11.5	✓	8
8.5	even	2		20.5.b.a.11.6	yes	8
24.5	odd	2		180.5.c.a.91.3		8
24.11	even	2		180.5.c.a.91.4		8
40.3	even	4		100.5.d.c.99.5		16
40.13	odd	4		100.5.d.c.99.11		16
40.19	odd	2		100.5.b.c.51.4		8
40.27	even	4		100.5.d.c.99.12		16
40.29	even	2		100.5.b.c.51.3		8
40.37	odd	4		100.5.d.c.99.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.5.b.a.11.5	✓	8	8.3	odd	2
20.5.b.a.11.6	yes	8	8.5	even	2
100.5.b.c.51.3		8	40.29	even	2
100.5.b.c.51.4		8	40.19	odd	2
100.5.d.c.99.5		16	40.3	even	4
100.5.d.c.99.6		16	40.37	odd	4
100.5.d.c.99.11		16	40.13	odd	4
100.5.d.c.99.12		16	40.27	even	4
180.5.c.a.91.3		8	24.5	odd	2
180.5.c.a.91.4		8	24.11	even	2
320.5.b.d.191.4		8	1.1	even	1	trivial
320.5.b.d.191.5		8	4.3	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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	320.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.20523i q^{3} -11.1803 q^{5} -30.6227i q^{7} +70.7265 q^{9} +O(q^{10})\)	\(q-3.20523i q^{3} -11.1803 q^{5} -30.6227i q^{7} +70.7265 q^{9} +168.004i q^{11} -3.15282 q^{13} +35.8356i q^{15} -229.376 q^{17} -151.168i q^{19} -98.1528 q^{21} +916.032i q^{23} +125.000 q^{25} -486.319i q^{27} +1004.39 q^{29} -1728.79i q^{31} +538.492 q^{33} +342.372i q^{35} +841.993 q^{37} +10.1055i q^{39} +2525.54 q^{41} +1653.22i q^{43} -790.746 q^{45} +2665.04i q^{47} +1463.25 q^{49} +735.204i q^{51} +3145.59 q^{53} -1878.34i q^{55} -484.527 q^{57} -3264.26i q^{59} +767.207 q^{61} -2165.83i q^{63} +35.2496 q^{65} -1859.46i q^{67} +2936.10 q^{69} +2774.43i q^{71} +2252.97 q^{73} -400.654i q^{75} +5144.73 q^{77} -906.591i q^{79} +4170.08 q^{81} -3833.20i q^{83} +2564.50 q^{85} -3219.31i q^{87} +1824.43 q^{89} +96.5478i q^{91} -5541.19 q^{93} +1690.10i q^{95} +15831.0 q^{97} +11882.3i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 328 q^{9}+O(q^{10}) \) 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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