LMFDB - Embedded newform 160.6.d.a.81.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,6,Mod(81,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("160.81");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	160.d (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:	$25.6614111701$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 17 x^{18} + 78 x^{17} + 253 x^{16} - 884 x^{15} + 2396 x^{14} + 19376 x^{13} + \cdots + 1099511627776 $ x^20 - 2x^19 - 17x^18 + 78x^17 + 253x^16 - 884x^15 + 2396x^14 + 19376x^13 - 109104x^12 - 96128x^11 + 3580672x^10 - 1538048x^9 - 27930624x^8 + 79364096x^7 + 157024256x^6 - 926941184x^5 + 4244635648x^4 + 20937965568x^3 - 73014444032x^2 - 137438953472*x + 1099511627776
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{90}\cdot 3^{4}\cdot 5^{12} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			81.10
Root			$-2.63430 + 3.01006i$ of defining polynomial
Character	$\chi$	$=$	160.81
Dual form			160.6.d.a.81.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.67450i q^{3} -25.0000i q^{5} +38.2812 q^{7} +198.451 q^{9} +O(q^{10})$	$q-6.67450i q^{3} -25.0000i q^{5} +38.2812 q^{7} +198.451 q^{9} -491.486i q^{11} +956.299i q^{13} -166.863 q^{15} +339.725 q^{17} -1862.39i q^{19} -255.508i q^{21} +1991.10 q^{23} -625.000 q^{25} -2946.47i q^{27} -3573.66i q^{29} -7710.17 q^{31} -3280.42 q^{33} -957.029i q^{35} -2835.87i q^{37} +6382.82 q^{39} +10681.6 q^{41} -20581.0i q^{43} -4961.28i q^{45} +756.983 q^{47} -15341.6 q^{49} -2267.49i q^{51} -31628.2i q^{53} -12287.1 q^{55} -12430.5 q^{57} +4914.36i q^{59} -21422.9i q^{61} +7596.94 q^{63} +23907.5 q^{65} -6818.32i q^{67} -13289.6i q^{69} +11134.9 q^{71} +7305.98 q^{73} +4171.56i q^{75} -18814.6i q^{77} +23560.4 q^{79} +28557.4 q^{81} +23708.5i q^{83} -8493.12i q^{85} -23852.4 q^{87} -125652. q^{89} +36608.2i q^{91} +51461.6i q^{93} -46559.7 q^{95} +126994. q^{97} -97535.9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 196 q^{7} - 1620 q^{9}+O(q^{10}) $ 20 * q + 196 * q^7 - 1620 * q^9	$ 20 q + 196 q^{7} - 1620 q^{9} - 900 q^{15} + 4676 q^{23} - 12500 q^{25} - 7160 q^{31} + 5672 q^{33} + 44904 q^{39} + 11608 q^{41} - 44180 q^{47} + 18756 q^{49} + 24200 q^{55} + 5032 q^{57} - 240620 q^{63} + 200312 q^{71} - 105136 q^{73} - 282080 q^{79} + 65172 q^{81} + 332592 q^{87} - 3160 q^{89} - 144400 q^{95} + 147376 q^{97}+O(q^{100}) $ 20 * q + 196 * q^7 - 1620 * q^9 - 900 * q^15 + 4676 * q^23 - 12500 * q^25 - 7160 * q^31 + 5672 * q^33 + 44904 * q^39 + 11608 * q^41 - 44180 * q^47 + 18756 * q^49 + 24200 * q^55 + 5032 * q^57 - 240620 * q^63 + 200312 * q^71 - 105136 * q^73 - 282080 * q^79 + 65172 * q^81 + 332592 * q^87 - 3160 * q^89 - 144400 * q^95 + 147376 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 6.67450i	− 0.428169i	−0.976815	−	0.214085i	$-0.931323\pi$
$3$	− 6.67450i	− 0.428169i	0.976815	−	0.214085i	$-0.0686769\pi$
$4$	0	0
$5$	− 25.0000i	− 0.447214i
$5$	− 25.0000i	− 0.447214i
$6$	0	0
$7$	38.2812	0.295284	0.147642	−	0.989041i	$-0.452832\pi$
$7$	38.2812	0.295284	0.147642	+	0.989041i	$0.452832\pi$
$8$	0	0
$9$	198.451	0.816671
$10$	0	0
$11$	− 491.486i	− 1.22470i	−0.790587	−	0.612349i	$-0.790225\pi$
$11$	− 491.486i	− 1.22470i	0.790587	−	0.612349i	$-0.209775\pi$
$12$	0	0
$13$	956.299i	1.56941i	0.619872	+	0.784703i	$0.287185\pi$
$13$	956.299i	1.56941i	−0.619872	+	0.784703i	$0.712815\pi$
$14$	0	0
$15$	−166.863	−0.191483
$16$	0	0
$17$	339.725	0.285105	0.142553	−	0.989787i	$-0.454469\pi$
$17$	339.725	0.285105	0.142553	+	0.989787i	$0.454469\pi$
$18$	0	0
$19$	− 1862.39i	− 1.18355i	−0.806104	−	0.591774i	$-0.798428\pi$
$19$	− 1862.39i	− 1.18355i	0.806104	−	0.591774i	$-0.201572\pi$
$20$	0	0
$21$	− 255.508i	− 0.126432i
$22$	0	0
$23$	1991.10	0.784825	0.392412	−	0.919789i	$-0.371641\pi$
$23$	1991.10	0.784825	0.392412	+	0.919789i	$0.371641\pi$
$24$	0	0
$25$	−625.000	−0.200000
$26$	0	0
$27$	− 2946.47i	− 0.777843i
$28$	0	0
$29$	− 3573.66i	− 0.789076i	−0.918880	−	0.394538i	$-0.870905\pi$
$29$	− 3573.66i	− 0.789076i	0.918880	−	0.394538i	$-0.129095\pi$
$30$	0	0
$31$	−7710.17	−1.44099	−0.720493	−	0.693462i	$-0.756085\pi$
$31$	−7710.17	−1.44099	−0.720493	+	0.693462i	$0.756085\pi$
$32$	0	0
$33$	−3280.42	−0.524379
$34$	0	0
$35$	− 957.029i	− 0.132055i
$36$	0	0
$37$	− 2835.87i	− 0.340551i	−0.985397	−	0.170275i	$-0.945534\pi$
$37$	− 2835.87i	− 0.340551i	0.985397	−	0.170275i	$-0.0544657\pi$
$38$	0	0
$39$	6382.82	0.671972
$40$	0	0
$41$	10681.6	0.992373	0.496187	−	0.868216i	$-0.334733\pi$
$41$	10681.6	0.992373	0.496187	+	0.868216i	$0.334733\pi$
$42$	0	0
$43$	− 20581.0i	− 1.69744i	−0.528841	−	0.848721i	$-0.677373\pi$
$43$	− 20581.0i	− 1.69744i	0.528841	−	0.848721i	$-0.322627\pi$
$44$	0	0
$45$	− 4961.28i	− 0.365226i
$46$	0	0
$47$	756.983	0.0499852	0.0249926	−	0.999688i	$-0.492044\pi$
$47$	756.983	0.0499852	0.0249926	+	0.999688i	$0.492044\pi$
$48$	0	0
$49$	−15341.6	−0.912807
$50$	0	0
$51$	− 2267.49i	− 0.122073i
$52$	0	0
$53$	− 31628.2i	− 1.54662i	−0.634025	−	0.773312i	$-0.718599\pi$
$53$	− 31628.2i	− 1.54662i	0.634025	−	0.773312i	$-0.281401\pi$
$54$	0	0
$55$	−12287.1	−0.547702
$56$	0	0
$57$	−12430.5	−0.506759
$58$	0	0
$59$	4914.36i	0.183796i	0.995768	+	0.0918981i	$0.0292934\pi$
$59$	4914.36i	0.183796i	−0.995768	+	0.0918981i	$0.970707\pi$
$60$	0	0
$61$	− 21422.9i	− 0.737147i	−0.929599	−	0.368573i	$-0.879846\pi$
$61$	− 21422.9i	− 0.737147i	0.929599	−	0.368573i	$-0.120154\pi$
$62$	0	0
$63$	7596.94	0.241150
$64$	0	0
$65$	23907.5	0.701860
$66$	0	0
$67$	− 6818.32i	− 0.185563i	−0.995687	−	0.0927813i	$-0.970424\pi$
$67$	− 6818.32i	− 0.185563i	0.995687	−	0.0927813i	$-0.0295758\pi$
$68$	0	0
$69$	− 13289.6i	− 0.336038i
$70$	0	0
$71$	11134.9	0.262143	0.131072	−	0.991373i	$-0.458158\pi$
$71$	11134.9	0.262143	0.131072	+	0.991373i	$0.458158\pi$
$72$	0	0
$73$	7305.98	0.160462	0.0802308	−	0.996776i	$-0.474434\pi$
$73$	7305.98	0.160462	0.0802308	+	0.996776i	$0.474434\pi$
$74$	0	0
$75$	4171.56i	0.0856339i
$76$	0	0
$77$	− 18814.6i	− 0.361634i
$78$	0	0
$79$	23560.4	0.424732	0.212366	−	0.977190i	$-0.431883\pi$
$79$	23560.4	0.424732	0.212366	+	0.977190i	$0.431883\pi$
$80$	0	0
$81$	28557.4	0.483622
$82$	0	0
$83$	23708.5i	0.377754i	0.982001	+	0.188877i	$0.0604848\pi$
$83$	23708.5i	0.377754i	−0.982001	+	0.188877i	$0.939515\pi$
$84$	0	0
$85$	− 8493.12i	− 0.127503i
$86$	0	0
$87$	−23852.4	−0.337858
$88$	0	0
$89$	−125652.	−1.68149	−0.840746	−	0.541429i	$-0.817883\pi$
$89$	−125652.	−1.68149	−0.840746	+	0.541429i	$0.817883\pi$
$90$	0	0
$91$	36608.2i	0.463421i
$92$	0	0
$93$	51461.6i	0.616986i
$94$	0	0
$95$	−46559.7	−0.529299
$96$	0	0
$97$	126994.	1.37042	0.685212	−	0.728344i	$-0.259710\pi$
$97$	126994.	1.37042	0.685212	+	0.728344i	$0.259710\pi$
$98$	0	0
$99$	− 97535.9i	− 1.00018i
$100$	0	0
$101$	128800.i	1.25635i	0.778070	+	0.628177i	$0.216199\pi$
$101$	128800.i	1.25635i	−0.778070	+	0.628177i	$0.783801\pi$
$102$	0	0
$103$	70343.6	0.653328	0.326664	−	0.945140i	$-0.394075\pi$
$103$	70343.6	0.653328	0.326664	+	0.945140i	$0.394075\pi$
$104$	0	0
$105$	−6387.69	−0.0565419
$106$	0	0
$107$	90073.6i	0.760568i	0.924870	+	0.380284i	$0.124174\pi$
$107$	90073.6i	0.760568i	−0.924870	+	0.380284i	$0.875826\pi$
$108$	0	0
$109$	176201.i	1.42051i	0.703947	+	0.710253i	$0.251419\pi$
$109$	176201.i	1.42051i	−0.703947	+	0.710253i	$0.748581\pi$
$110$	0	0
$111$	−18928.0	−0.145813
$112$	0	0
$113$	−5466.04	−0.0402695	−0.0201348	−	0.999797i	$-0.506410\pi$
$113$	−5466.04	−0.0402695	−0.0201348	+	0.999797i	$0.506410\pi$
$114$	0	0
$115$	− 49777.4i	− 0.350984i
$116$	0	0
$117$	189779.i	1.28169i
$118$	0	0
$119$	13005.1	0.0841870
$120$	0	0
$121$	−80507.4	−0.499888
$122$	0	0
$123$	− 71294.1i	− 0.424904i
$124$	0	0
$125$	15625.0i	0.0894427i
$126$	0	0
$127$	−273870.	−1.50673	−0.753364	−	0.657603i	$-0.771570\pi$
$127$	−273870.	−1.50673	−0.753364	+	0.657603i	$0.771570\pi$
$128$	0	0
$129$	−137368.	−0.726793
$130$	0	0
$131$	131830.i	0.671175i	0.942009	+	0.335588i	$0.108935\pi$
$131$	131830.i	0.671175i	−0.942009	+	0.335588i	$0.891065\pi$
$132$	0	0
$133$	− 71294.3i	− 0.349483i
$134$	0	0
$135$	−73661.6	−0.347862
$136$	0	0
$137$	385733.	1.75584	0.877922	−	0.478804i	$-0.158929\pi$
$137$	385733.	1.75584	0.877922	+	0.478804i	$0.158929\pi$
$138$	0	0
$139$	445530.i	1.95587i	0.208910	+	0.977935i	$0.433008\pi$
$139$	445530.i	1.95587i	−0.208910	+	0.977935i	$0.566992\pi$
$140$	0	0
$141$	− 5052.48i	− 0.0214021i
$142$	0	0
$143$	470008.	1.92205
$144$	0	0
$145$	−89341.6	−0.352885
$146$	0	0
$147$	102397.i	0.390836i
$148$	0	0
$149$	− 125864.i	− 0.464448i	−0.972662	−	0.232224i	$-0.925400\pi$
$149$	− 125864.i	− 0.464448i	0.972662	−	0.232224i	$-0.0746003\pi$
$150$	0	0
$151$	449332.	1.60371	0.801853	−	0.597522i	$-0.203848\pi$
$151$	449332.	1.60371	0.801853	+	0.597522i	$0.203848\pi$
$152$	0	0
$153$	67418.8	0.232837
$154$	0	0
$155$	192754.i	0.644429i
$156$	0	0
$157$	− 475430.i	− 1.53935i	−0.638436	−	0.769675i	$-0.720418\pi$
$157$	− 475430.i	− 1.53935i	0.638436	−	0.769675i	$-0.279582\pi$
$158$	0	0
$159$	−211102.	−0.662217
$160$	0	0
$161$	76221.5	0.231746
$162$	0	0
$163$	210514.i	0.620602i	0.950638	+	0.310301i	$0.100430\pi$
$163$	210514.i	0.620602i	−0.950638	+	0.310301i	$0.899570\pi$
$164$	0	0
$165$	82010.6i	0.234509i
$166$	0	0
$167$	−366778.	−1.01768	−0.508842	−	0.860860i	$-0.669926\pi$
$167$	−366778.	−1.01768	−0.508842	+	0.860860i	$0.669926\pi$
$168$	0	0
$169$	−543215.	−1.46304
$170$	0	0
$171$	− 369593.i	− 0.966569i
$172$	0	0
$173$	242252.i	0.615393i	0.951485	+	0.307697i	$0.0995582\pi$
$173$	242252.i	0.615393i	−0.951485	+	0.307697i	$0.900442\pi$
$174$	0	0
$175$	−23925.7	−0.0590568
$176$	0	0
$177$	32800.9	0.0786959
$178$	0	0
$179$	− 587964.i	− 1.37157i	−0.727804	−	0.685785i	$-0.759459\pi$
$179$	− 587964.i	− 1.37157i	0.727804	−	0.685785i	$-0.240541\pi$
$180$	0	0
$181$	− 429244.i	− 0.973885i	−0.873434	−	0.486942i	$-0.838112\pi$
$181$	− 429244.i	− 0.973885i	0.873434	−	0.486942i	$-0.161888\pi$
$182$	0	0
$183$	−142987.	−0.315624
$184$	0	0
$185$	−70896.7	−0.152299
$186$	0	0
$187$	− 166970.i	− 0.349168i
$188$	0	0
$189$	− 112794.i	− 0.229685i
$190$	0	0
$191$	271725.	0.538947	0.269473	−	0.963008i	$-0.413150\pi$
$191$	271725.	0.538947	0.269473	+	0.963008i	$0.413150\pi$
$192$	0	0
$193$	871451.	1.68403	0.842015	−	0.539454i	$-0.181369\pi$
$193$	871451.	1.68403	0.842015	+	0.539454i	$0.181369\pi$
$194$	0	0
$195$	− 159570.i	− 0.300515i
$196$	0	0
$197$	207994.i	0.381844i	0.981605	+	0.190922i	$0.0611477\pi$
$197$	207994.i	0.381844i	−0.981605	+	0.190922i	$0.938852\pi$
$198$	0	0
$199$	−669523.	−1.19849	−0.599243	−	0.800567i	$-0.704532\pi$
$199$	−669523.	−1.19849	−0.599243	+	0.800567i	$0.704532\pi$
$200$	0	0
$201$	−45508.9	−0.0794523
$202$	0	0
$203$	− 136804.i	− 0.233001i
$204$	0	0
$205$	− 267039.i	− 0.443803i
$206$	0	0
$207$	395135.	0.640944
$208$	0	0
$209$	−915337.	−1.44949
$210$	0	0
$211$	− 617471.i	− 0.954796i	−0.878687	−	0.477398i	$-0.841580\pi$
$211$	− 617471.i	− 0.954796i	0.878687	−	0.477398i	$-0.158420\pi$
$212$	0	0
$213$	− 74319.6i	− 0.112242i
$214$	0	0
$215$	−514525.	−0.759119
$216$	0	0
$217$	−295154.	−0.425500
$218$	0	0
$219$	− 48763.7i	− 0.0687047i
$220$	0	0
$221$	324879.i	0.447446i
$222$	0	0
$223$	800666.	1.07817	0.539087	−	0.842250i	$-0.318769\pi$
$223$	800666.	1.07817	0.539087	+	0.842250i	$0.318769\pi$
$224$	0	0
$225$	−124032.	−0.163334
$226$	0	0
$227$	− 936259.i	− 1.20596i	−0.797758	−	0.602978i	$-0.793981\pi$
$227$	− 936259.i	− 1.20596i	0.797758	−	0.602978i	$-0.206019\pi$
$228$	0	0
$229$	− 541008.i	− 0.681735i	−0.940111	−	0.340867i	$-0.889279\pi$
$229$	− 541008.i	− 0.681735i	0.940111	−	0.340867i	$-0.110721\pi$
$230$	0	0
$231$	−125578.	−0.154841
$232$	0	0
$233$	550428.	0.664218	0.332109	−	0.943241i	$-0.392240\pi$
$233$	550428.	0.664218	0.332109	+	0.943241i	$0.392240\pi$
$234$	0	0
$235$	− 18924.6i	− 0.0223541i
$236$	0	0
$237$	− 157254.i	− 0.181857i
$238$	0	0
$239$	−1.17782e6	−1.33378	−0.666889	−	0.745157i	$-0.732374\pi$
$239$	−1.17782e6	−1.33378	−0.666889	+	0.745157i	$0.732374\pi$
$240$	0	0
$241$	998860.	1.10780	0.553901	−	0.832583i	$-0.313139\pi$
$241$	998860.	1.10780	0.553901	+	0.832583i	$0.313139\pi$
$242$	0	0
$243$	− 906598.i	− 0.984915i
$244$	0	0
$245$	383539.i	0.408220i
$246$	0	0
$247$	1.78100e6	1.85747
$248$	0	0
$249$	158243.	0.161743
$250$	0	0
$251$	1.05232e6i	1.05430i	0.849773	+	0.527149i	$0.176739\pi$
$251$	1.05232e6i	1.05430i	−0.849773	+	0.527149i	$0.823261\pi$
$252$	0	0
$253$	− 978596.i	− 0.961174i
$254$	0	0
$255$	−56687.4	−0.0545929
$256$	0	0
$257$	−416435.	−0.393292	−0.196646	−	0.980475i	$-0.563005\pi$
$257$	−416435.	−0.393292	−0.196646	+	0.980475i	$0.563005\pi$
$258$	0	0
$259$	− 108560.i	− 0.100559i
$260$	0	0
$261$	− 709197.i	− 0.644415i
$262$	0	0
$263$	1.05502e6	0.940528	0.470264	−	0.882526i	$-0.344159\pi$
$263$	1.05502e6	0.940528	0.470264	+	0.882526i	$0.344159\pi$
$264$	0	0
$265$	−790705.	−0.691672
$266$	0	0
$267$	838665.i	0.719964i
$268$	0	0
$269$	1.73328e6i	1.46045i	0.683206	+	0.730226i	$0.260585\pi$
$269$	1.73328e6i	1.46045i	−0.683206	+	0.730226i	$0.739415\pi$
$270$	0	0
$271$	−1.32031e6	−1.09208	−0.546039	−	0.837760i	$-0.683865\pi$
$271$	−1.32031e6	−1.09208	−0.546039	+	0.837760i	$0.683865\pi$
$272$	0	0
$273$	244342.	0.198423
$274$	0	0
$275$	307179.i	0.244940i
$276$	0	0
$277$	− 5852.67i	− 0.00458305i	−0.999997	−	0.00229153i	$-0.999271\pi$
$277$	− 5852.67i	− 0.00458305i	0.999997	−	0.00229153i	$-0.000729416\pi$
$278$	0	0
$279$	−1.53009e6	−1.17681
$280$	0	0
$281$	1.46507e6	1.10686	0.553432	−	0.832895i	$-0.313318\pi$
$281$	1.46507e6	1.10686	0.553432	+	0.832895i	$0.313318\pi$
$282$	0	0
$283$	1.97541e6i	1.46619i	0.680124	+	0.733097i	$0.261926\pi$
$283$	1.97541e6i	1.46619i	−0.680124	+	0.733097i	$0.738074\pi$
$284$	0	0
$285$	310763.i	0.226630i
$286$	0	0
$287$	408903.	0.293032
$288$	0	0
$289$	−1.30444e6	−0.918715
$290$	0	0
$291$	− 847623.i	− 0.586774i
$292$	0	0
$293$	1.38276e6i	0.940976i	0.882406	+	0.470488i	$0.155922\pi$
$293$	1.38276e6i	0.940976i	−0.882406	+	0.470488i	$0.844078\pi$
$294$	0	0
$295$	122859.	0.0821962
$296$	0	0
$297$	−1.44815e6	−0.952623
$298$	0	0
$299$	1.90408e6i	1.23171i
$300$	0	0
$301$	− 787864.i	− 0.501228i
$302$	0	0
$303$	859675.	0.537933
$304$	0	0
$305$	−535573.	−0.329662
$306$	0	0
$307$	− 54523.5i	− 0.0330170i	−0.999864	−	0.0165085i	$-0.994745\pi$
$307$	− 54523.5i	− 0.0330170i	0.999864	−	0.0165085i	$-0.00525506\pi$
$308$	0	0
$309$	− 469508.i	− 0.279735i
$310$	0	0
$311$	1.07373e6	0.629499	0.314749	−	0.949175i	$-0.398079\pi$
$311$	1.07373e6	0.629499	0.314749	+	0.949175i	$0.398079\pi$
$312$	0	0
$313$	2.70724e6	1.56195	0.780973	−	0.624565i	$-0.214724\pi$
$313$	2.70724e6	1.56195	0.780973	+	0.624565i	$0.214724\pi$
$314$	0	0
$315$	− 189923.i	− 0.107846i
$316$	0	0
$317$	1.52062e6i	0.849908i	0.905215	+	0.424954i	$0.139710\pi$
$317$	1.52062e6i	0.849908i	−0.905215	+	0.424954i	$0.860290\pi$
$318$	0	0
$319$	−1.75641e6	−0.966380
$320$	0	0
$321$	601196.	0.325652
$322$	0	0
$323$	− 632699.i	− 0.337436i
$324$	0	0
$325$	− 597687.i	− 0.313881i
$326$	0	0
$327$	1.17606e6	0.608217
$328$	0	0
$329$	28978.2	0.0147598
$330$	0	0
$331$	3.31163e6i	1.66139i	0.556729	+	0.830694i	$0.312056\pi$
$331$	3.31163e6i	1.66139i	−0.556729	+	0.830694i	$0.687944\pi$
$332$	0	0
$333$	− 562781.i	− 0.278118i
$334$	0	0
$335$	−170458.	−0.0829861
$336$	0	0
$337$	−2.65162e6	−1.27185	−0.635927	−	0.771749i	$-0.719382\pi$
$337$	−2.65162e6	−1.27185	−0.635927	+	0.771749i	$0.719382\pi$
$338$	0	0
$339$	36483.1i	0.0172422i
$340$	0	0
$341$	3.78944e6i	1.76477i
$342$	0	0
$343$	−1.23068e6	−0.564821
$344$	0	0
$345$	−332239.	−0.150281
$346$	0	0
$347$	2.89737e6i	1.29175i	0.763441	+	0.645877i	$0.223508\pi$
$347$	2.89737e6i	1.29175i	−0.763441	+	0.645877i	$0.776492\pi$
$348$	0	0
$349$	− 1.23263e6i	− 0.541713i	−0.962620	−	0.270856i	$-0.912693\pi$
$349$	− 1.23263e6i	− 0.541713i	0.962620	−	0.270856i	$-0.0873068\pi$
$350$	0	0
$351$	2.81770e6	1.22075
$352$	0	0
$353$	145202.	0.0620207	0.0310103	−	0.999519i	$-0.490128\pi$
$353$	145202.	0.0620207	0.0310103	+	0.999519i	$0.490128\pi$
$354$	0	0
$355$	− 278372.i	− 0.117234i
$356$	0	0
$357$	− 86802.3i	− 0.0360463i
$358$	0	0
$359$	−1.97212e6	−0.807600	−0.403800	−	0.914847i	$-0.632311\pi$
$359$	−1.97212e6	−0.807600	−0.403800	+	0.914847i	$0.632311\pi$
$360$	0	0
$361$	−992386.	−0.400786
$362$	0	0
$363$	537347.i	0.214037i
$364$	0	0
$365$	− 182649.i	− 0.0717606i
$366$	0	0
$367$	−371055.	−0.143805	−0.0719024	−	0.997412i	$-0.522907\pi$
$367$	−371055.	−0.143805	−0.0719024	+	0.997412i	$0.522907\pi$
$368$	0	0
$369$	2.11977e6	0.810443
$370$	0	0
$371$	− 1.21076e6i	− 0.456694i
$372$	0	0
$373$	− 586109.i	− 0.218125i	−0.994035	−	0.109063i	$-0.965215\pi$
$373$	− 586109.i	− 0.218125i	0.994035	−	0.109063i	$-0.0347849\pi$
$374$	0	0
$375$	104289.	0.0382966
$376$	0	0
$377$	3.41749e6	1.23838
$378$	0	0
$379$	− 1.93721e6i	− 0.692752i	−0.938096	−	0.346376i	$-0.887412\pi$
$379$	− 1.93721e6i	− 0.692752i	0.938096	−	0.346376i	$-0.112588\pi$
$380$	0	0
$381$	1.82794e6i	0.645135i
$382$	0	0
$383$	162691.	0.0566716	0.0283358	−	0.999598i	$-0.490979\pi$
$383$	162691.	0.0566716	0.0283358	+	0.999598i	$0.490979\pi$
$384$	0	0
$385$	−470366.	−0.161728
$386$	0	0
$387$	− 4.08432e6i	− 1.38625i
$388$	0	0
$389$	2.24326e6i	0.751633i	0.926694	+	0.375817i	$0.122638\pi$
$389$	2.24326e6i	0.751633i	−0.926694	+	0.375817i	$0.877362\pi$
$390$	0	0
$391$	676425.	0.223758
$392$	0	0
$393$	879899.	0.287377
$394$	0	0
$395$	− 589010.i	− 0.189946i
$396$	0	0
$397$	1.64851e6i	0.524947i	0.964939	+	0.262474i	$0.0845383\pi$
$397$	1.64851e6i	0.524947i	−0.964939	+	0.262474i	$0.915462\pi$
$398$	0	0
$399$	−475854.	−0.149638
$400$	0	0
$401$	−2.96862e6	−0.921921	−0.460960	−	0.887421i	$-0.652495\pi$
$401$	−2.96862e6	−0.921921	−0.460960	+	0.887421i	$0.652495\pi$
$402$	0	0
$403$	− 7.37323e6i	− 2.26149i
$404$	0	0
$405$	− 713936.i	− 0.216283i
$406$	0	0
$407$	−1.39379e6	−0.417072
$408$	0	0
$409$	−2.21238e6	−0.653962	−0.326981	−	0.945031i	$-0.606031\pi$
$409$	−2.21238e6	−0.653962	−0.326981	+	0.945031i	$0.606031\pi$
$410$	0	0
$411$	− 2.57458e6i	− 0.751799i
$412$	0	0
$413$	188127.i	0.0542721i
$414$	0	0
$415$	592713.	0.168937
$416$	0	0
$417$	2.97369e6	0.837444
$418$	0	0
$419$	2.61133e6i	0.726653i	0.931662	+	0.363327i	$0.118359\pi$
$419$	2.61133e6i	0.726653i	−0.931662	+	0.363327i	$0.881641\pi$
$420$	0	0
$421$	− 3.16116e6i	− 0.869242i	−0.900614	−	0.434621i	$-0.856882\pi$
$421$	− 3.16116e6i	− 0.869242i	0.900614	−	0.434621i	$-0.143118\pi$
$422$	0	0
$423$	150224.	0.0408214
$424$	0	0
$425$	−212328.	−0.0570210
$426$	0	0
$427$	− 820094.i	− 0.217668i
$428$	0	0
$429$	− 3.13707e6i	− 0.822963i
$430$	0	0
$431$	−1.00949e6	−0.261764	−0.130882	−	0.991398i	$-0.541781\pi$
$431$	−1.00949e6	−0.261764	−0.130882	+	0.991398i	$0.541781\pi$
$432$	0	0
$433$	741789.	0.190134	0.0950672	−	0.995471i	$-0.469693\pi$
$433$	741789.	0.190134	0.0950672	+	0.995471i	$0.469693\pi$
$434$	0	0
$435$	596310.i	0.151095i
$436$	0	0
$437$	− 3.70819e6i	− 0.928878i
$438$	0	0
$439$	4.40195e6	1.09014	0.545071	−	0.838390i	$-0.316503\pi$
$439$	4.40195e6	1.09014	0.545071	+	0.838390i	$0.316503\pi$
$440$	0	0
$441$	−3.04455e6	−0.745463
$442$	0	0
$443$	7.17840e6i	1.73788i	0.494922	+	0.868938i	$0.335197\pi$
$443$	7.17840e6i	1.73788i	−0.494922	+	0.868938i	$0.664803\pi$
$444$	0	0
$445$	3.14130e6i	0.751986i
$446$	0	0
$447$	−840082.	−0.198863
$448$	0	0
$449$	1.08371e6	0.253686	0.126843	−	0.991923i	$-0.459516\pi$
$449$	1.08371e6	0.253686	0.126843	+	0.991923i	$0.459516\pi$
$450$	0	0
$451$	− 5.24984e6i	− 1.21536i
$452$	0	0
$453$	− 2.99906e6i	− 0.686658i
$454$	0	0
$455$	915206.	0.207248
$456$	0	0
$457$	1.76573e6	0.395487	0.197744	−	0.980254i	$-0.436639\pi$
$457$	1.76573e6	0.395487	0.197744	+	0.980254i	$0.436639\pi$
$458$	0	0
$459$	− 1.00099e6i	− 0.221767i
$460$	0	0
$461$	− 7.82078e6i	− 1.71395i	−0.515359	−	0.856975i	$-0.672341\pi$
$461$	− 7.82078e6i	− 1.71395i	0.515359	−	0.856975i	$-0.327659\pi$
$462$	0	0
$463$	−3.93389e6	−0.852844	−0.426422	−	0.904524i	$-0.640226\pi$
$463$	−3.93389e6	−0.852844	−0.426422	+	0.904524i	$0.640226\pi$
$464$	0	0
$465$	1.28654e6	0.275925
$466$	0	0
$467$	1.58189e6i	0.335648i	0.985817	+	0.167824i	$0.0536740\pi$
$467$	1.58189e6i	0.335648i	−0.985817	+	0.167824i	$0.946326\pi$
$468$	0	0
$469$	− 261013.i	− 0.0547937i
$470$	0	0
$471$	−3.17326e6	−0.659102
$472$	0	0
$473$	−1.01153e7	−2.07886
$474$	0	0
$475$	1.16399e6i	0.236710i
$476$	0	0
$477$	− 6.27665e6i	− 1.26308i
$478$	0	0
$479$	5.28209e6	1.05188	0.525941	−	0.850521i	$-0.323713\pi$
$479$	5.28209e6	1.05188	0.525941	+	0.850521i	$0.323713\pi$
$480$	0	0
$481$	2.71194e6	0.534463
$482$	0	0
$483$	− 508740.i	− 0.0992266i
$484$	0	0
$485$	− 3.17486e6i	− 0.612872i
$486$	0	0
$487$	−6.18631e6	−1.18198	−0.590989	−	0.806680i	$-0.701262\pi$
$487$	−6.18631e6	−1.18198	−0.590989	+	0.806680i	$0.701262\pi$
$488$	0	0
$489$	1.40508e6	0.265723
$490$	0	0
$491$	8.31222e6i	1.55601i	0.628256	+	0.778007i	$0.283769\pi$
$491$	8.31222e6i	1.55601i	−0.628256	+	0.778007i	$0.716231\pi$
$492$	0	0
$493$	− 1.21406e6i	− 0.224970i
$494$	0	0
$495$	−2.43840e6	−0.447292
$496$	0	0
$497$	426255.	0.0774068
$498$	0	0
$499$	− 9.46854e6i	− 1.70228i	−0.524936	−	0.851142i	$-0.675911\pi$
$499$	− 9.46854e6i	− 1.70228i	0.524936	−	0.851142i	$-0.324089\pi$
$500$	0	0
$501$	2.44806e6i	0.435741i
$502$	0	0
$503$	7.15769e6	1.26140	0.630700	−	0.776027i	$-0.282768\pi$
$503$	7.15769e6	1.26140	0.630700	+	0.776027i	$0.282768\pi$
$504$	0	0
$505$	3.22000e6	0.561859
$506$	0	0
$507$	3.62569e6i	0.626427i
$508$	0	0
$509$	1.08667e6i	0.185910i	0.995670	+	0.0929551i	$0.0296313\pi$
$509$	1.08667e6i	0.185910i	−0.995670	+	0.0929551i	$0.970369\pi$
$510$	0	0
$511$	279681.	0.0473817
$512$	0	0
$513$	−5.48746e6	−0.920614
$514$	0	0
$515$	− 1.75859e6i	− 0.292177i
$516$	0	0
$517$	− 372046.i	− 0.0612168i
$518$	0	0
$519$	1.61691e6	0.263493
$520$	0	0
$521$	4.39810e6	0.709856	0.354928	−	0.934894i	$-0.384505\pi$
$521$	4.39810e6	0.709856	0.354928	+	0.934894i	$0.384505\pi$
$522$	0	0
$523$	− 1.66013e6i	− 0.265392i	−0.991157	−	0.132696i	$-0.957637\pi$
$523$	− 1.66013e6i	− 0.265392i	0.991157	−	0.132696i	$-0.0423634\pi$
$524$	0	0
$525$	159692.i	0.0252863i
$526$	0	0
$527$	−2.61934e6	−0.410833
$528$	0	0
$529$	−2.47188e6	−0.384050
$530$	0	0
$531$	975259.i	0.150101i
$532$	0	0
$533$	1.02148e7i	1.55744i
$534$	0	0
$535$	2.25184e6	0.340136
$536$	0	0
$537$	−3.92436e6	−0.587264
$538$	0	0
$539$	7.54016e6i	1.11791i
$540$	0	0
$541$	6.71821e6i	0.986872i	0.869782	+	0.493436i	$0.164259\pi$
$541$	6.71821e6i	0.986872i	−0.869782	+	0.493436i	$0.835741\pi$
$542$	0	0
$543$	−2.86499e6	−0.416988
$544$	0	0
$545$	4.40503e6	0.635270
$546$	0	0
$547$	− 6.46880e6i	− 0.924391i	−0.886778	−	0.462195i	$-0.847062\pi$
$547$	− 6.46880e6i	− 0.924391i	0.886778	−	0.462195i	$-0.152938\pi$
$548$	0	0
$549$	− 4.25140e6i	− 0.602006i
$550$	0	0
$551$	−6.65554e6	−0.933909
$552$	0	0
$553$	901919.	0.125416
$554$	0	0
$555$	473200.i	0.0652097i
$556$	0	0
$557$	1.40656e7i	1.92097i	0.278325	+	0.960487i	$0.410221\pi$
$557$	1.40656e7i	1.92097i	−0.278325	+	0.960487i	$0.589779\pi$
$558$	0	0
$559$	1.96816e7	2.66398
$560$	0	0
$561$	−1.11444e6	−0.149503
$562$	0	0
$563$	6.70453e6i	0.891451i	0.895170	+	0.445726i	$0.147054\pi$
$563$	6.70453e6i	0.891451i	−0.895170	+	0.445726i	$0.852946\pi$
$564$	0	0
$565$	136651.i	0.0180091i
$566$	0	0
$567$	1.09321e6	0.142806
$568$	0	0
$569$	8.42220e6	1.09055	0.545274	−	0.838258i	$-0.316426\pi$
$569$	8.42220e6	1.09055	0.545274	+	0.838258i	$0.316426\pi$
$570$	0	0
$571$	− 5.62974e6i	− 0.722600i	−0.932450	−	0.361300i	$-0.882333\pi$
$571$	− 5.62974e6i	− 0.722600i	0.932450	−	0.361300i	$-0.117667\pi$
$572$	0	0
$573$	− 1.81363e6i	− 0.230760i
$574$	0	0
$575$	−1.24444e6	−0.156965
$576$	0	0
$577$	6.49266e6	0.811864	0.405932	−	0.913903i	$-0.366947\pi$
$577$	6.49266e6	0.811864	0.405932	+	0.913903i	$0.366947\pi$
$578$	0	0
$579$	− 5.81650e6i	− 0.721050i
$580$	0	0
$581$	907590.i	0.111545i
$582$	0	0
$583$	−1.55448e7	−1.89415
$584$	0	0
$585$	4.74446e6	0.573189
$586$	0	0
$587$	3.45770e6i	0.414183i	0.978322	+	0.207091i	$0.0663997\pi$
$587$	3.45770e6i	0.414183i	−0.978322	+	0.207091i	$0.933600\pi$
$588$	0	0
$589$	1.43593e7i	1.70548i
$590$	0	0
$591$	1.38826e6	0.163494
$592$	0	0
$593$	−3.86898e6	−0.451815	−0.225907	−	0.974149i	$-0.572535\pi$
$593$	−3.86898e6	−0.451815	−0.225907	+	0.974149i	$0.572535\pi$
$594$	0	0
$595$	− 325127.i	− 0.0376496i
$596$	0	0
$597$	4.46873e6i	0.513155i
$598$	0	0
$599$	6.06771e6	0.690967	0.345484	−	0.938425i	$-0.387715\pi$
$599$	6.06771e6	0.690967	0.345484	+	0.938425i	$0.387715\pi$
$600$	0	0
$601$	−6.05369e6	−0.683651	−0.341825	−	0.939764i	$-0.611045\pi$
$601$	−6.05369e6	−0.683651	−0.341825	+	0.939764i	$0.611045\pi$
$602$	0	0
$603$	− 1.35310e6i	− 0.151544i
$604$	0	0
$605$	2.01268e6i	0.223557i
$606$	0	0
$607$	−3.04959e6	−0.335946	−0.167973	−	0.985792i	$-0.553722\pi$
$607$	−3.04959e6	−0.335946	−0.167973	+	0.985792i	$0.553722\pi$
$608$	0	0
$609$	−913098.	−0.0997641
$610$	0	0
$611$	723902.i	0.0784471i
$612$	0	0
$613$	7.56693e6i	0.813333i	0.913577	+	0.406667i	$0.133309\pi$
$613$	7.56693e6i	0.813333i	−0.913577	+	0.406667i	$0.866691\pi$
$614$	0	0
$615$	−1.78235e6	−0.190023
$616$	0	0
$617$	−6.31124e6	−0.667424	−0.333712	−	0.942675i	$-0.608301\pi$
$617$	−6.31124e6	−0.667424	−0.333712	+	0.942675i	$0.608301\pi$
$618$	0	0
$619$	1.15260e7i	1.20907i	0.796578	+	0.604536i	$0.206641\pi$
$619$	1.15260e7i	1.20907i	−0.796578	+	0.604536i	$0.793359\pi$
$620$	0	0
$621$	− 5.86670e6i	− 0.610470i
$622$	0	0
$623$	−4.81011e6	−0.496518
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	6.10942e6i	0.620627i
$628$	0	0
$629$	− 963416.i	− 0.0970928i
$630$	0	0
$631$	−3.69301e6	−0.369238	−0.184619	−	0.982810i	$-0.559105\pi$
$631$	−3.69301e6	−0.369238	−0.184619	+	0.982810i	$0.559105\pi$
$632$	0	0
$633$	−4.12131e6	−0.408814
$634$	0	0
$635$	6.84675e6i	0.673829i
$636$	0	0
$637$	− 1.46711e7i	− 1.43257i
$638$	0	0
$639$	2.20972e6	0.214085
$640$	0	0
$641$	−6.48129e6	−0.623040	−0.311520	−	0.950240i	$-0.600838\pi$
$641$	−6.48129e6	−0.623040	−0.311520	+	0.950240i	$0.600838\pi$
$642$	0	0
$643$	1.12433e7i	1.07243i	0.844082	+	0.536214i	$0.180146\pi$
$643$	1.12433e7i	1.07243i	−0.844082	+	0.536214i	$0.819854\pi$
$644$	0	0
$645$	3.43419e6i	0.325032i
$646$	0	0
$647$	8.65936e6	0.813252	0.406626	−	0.913595i	$-0.366705\pi$
$647$	8.65936e6	0.813252	0.406626	+	0.913595i	$0.366705\pi$
$648$	0	0
$649$	2.41534e6	0.225095
$650$	0	0
$651$	1.97001e6i	0.182186i
$652$	0	0
$653$	− 1.20375e6i	− 0.110473i	−0.998473	−	0.0552363i	$-0.982409\pi$
$653$	− 1.20375e6i	− 0.110473i	0.998473	−	0.0552363i	$-0.0175912\pi$
$654$	0	0
$655$	3.29575e6	0.300159
$656$	0	0
$657$	1.44988e6	0.131044
$658$	0	0
$659$	3.18525e6i	0.285713i	0.989743	+	0.142856i	$0.0456287\pi$
$659$	3.18525e6i	0.285713i	−0.989743	+	0.142856i	$0.954371\pi$
$660$	0	0
$661$	3.59534e6i	0.320064i	0.987112	+	0.160032i	$0.0511597\pi$
$661$	3.59534e6i	0.320064i	−0.987112	+	0.160032i	$0.948840\pi$
$662$	0	0
$663$	2.16840e6	0.191583
$664$	0	0
$665$	−1.78236e6	−0.156293
$666$	0	0
$667$	− 7.11551e6i	− 0.619286i
$668$	0	0
$669$	− 5.34404e6i	− 0.461641i
$670$	0	0
$671$	−1.05291e7	−0.902783
$672$	0	0
$673$	−5.21823e6	−0.444105	−0.222053	−	0.975035i	$-0.571276\pi$
$673$	−5.21823e6	−0.444105	−0.222053	+	0.975035i	$0.571276\pi$
$674$	0	0
$675$	1.84154e6i	0.155569i
$676$	0	0
$677$	− 1.86028e6i	− 0.155994i	−0.996954	−	0.0779969i	$-0.975148\pi$
$677$	− 1.86028e6i	− 0.155994i	0.996954	−	0.0779969i	$-0.0248524\pi$
$678$	0	0
$679$	4.86149e6	0.404664
$680$	0	0
$681$	−6.24906e6	−0.516353
$682$	0	0
$683$	− 1.83940e7i	− 1.50877i	−0.656431	−	0.754386i	$-0.727935\pi$
$683$	− 1.83940e7i	− 1.50877i	0.656431	−	0.754386i	$-0.272065\pi$
$684$	0	0
$685$	− 9.64334e6i	− 0.785237i
$686$	0	0
$687$	−3.61096e6	−0.291898
$688$	0	0
$689$	3.02460e7	2.42728
$690$	0	0
$691$	9.52732e6i	0.759059i	0.925180	+	0.379530i	$0.123914\pi$
$691$	9.52732e6i	0.759059i	−0.925180	+	0.379530i	$0.876086\pi$
$692$	0	0
$693$	− 3.73379e6i	− 0.295336i
$694$	0	0
$695$	1.11382e7	0.874692
$696$	0	0
$697$	3.62879e6	0.282931
$698$	0	0
$699$	− 3.67383e6i	− 0.284398i
$700$	0	0
$701$	9.81247e6i	0.754194i	0.926174	+	0.377097i	$0.123078\pi$
$701$	9.81247e6i	0.754194i	−0.926174	+	0.377097i	$0.876922\pi$
$702$	0	0
$703$	−5.28149e6	−0.403058
$704$	0	0
$705$	−126312.	−0.00957132
$706$	0	0
$707$	4.93061e6i	0.370981i
$708$	0	0
$709$	− 1.65549e7i	− 1.23683i	−0.785851	−	0.618415i	$-0.787775\pi$
$709$	− 1.65549e7i	− 1.23683i	0.785851	−	0.618415i	$-0.212225\pi$
$710$	0	0
$711$	4.67558e6	0.346866
$712$	0	0
$713$	−1.53517e7	−1.13092
$714$	0	0
$715$	− 1.17502e7i	− 0.859567i
$716$	0	0
$717$	7.86134e6i	0.571083i
$718$	0	0
$719$	−1.40579e7	−1.01414	−0.507070	−	0.861905i	$-0.669271\pi$
$719$	−1.40579e7	−1.01414	−0.507070	+	0.861905i	$0.669271\pi$
$720$	0	0
$721$	2.69283e6	0.192917
$722$	0	0
$723$	− 6.66689e6i	− 0.474327i
$724$	0	0
$725$	2.23354e6i	0.157815i
$726$	0	0
$727$	1.78648e6	0.125361	0.0626806	−	0.998034i	$-0.480035\pi$
$727$	1.78648e6	0.125361	0.0626806	+	0.998034i	$0.480035\pi$
$728$	0	0
$729$	888368.	0.0619119
$730$	0	0
$731$	− 6.99187e6i	− 0.483950i
$732$	0	0
$733$	− 1.59012e7i	− 1.09313i	−0.837418	−	0.546563i	$-0.815936\pi$
$733$	− 1.59012e7i	− 1.09313i	0.837418	−	0.546563i	$-0.184064\pi$
$734$	0	0
$735$	2.55993e6	0.174787
$736$	0	0
$737$	−3.35111e6	−0.227258
$738$	0	0
$739$	− 5.89659e6i	− 0.397182i	−0.980082	−	0.198591i	$-0.936363\pi$
$739$	− 5.89659e6i	− 0.397182i	0.980082	−	0.198591i	$-0.0636366\pi$
$740$	0	0
$741$	− 1.18873e7i	− 0.795311i
$742$	0	0
$743$	2.21545e6	0.147228	0.0736139	−	0.997287i	$-0.476547\pi$
$743$	2.21545e6	0.147228	0.0736139	+	0.997287i	$0.476547\pi$
$744$	0	0
$745$	−3.14661e6	−0.207708
$746$	0	0
$747$	4.70498e6i	0.308501i
$748$	0	0
$749$	3.44812e6i	0.224583i
$750$	0	0
$751$	1.69953e7	1.09959	0.549793	−	0.835301i	$-0.314706\pi$
$751$	1.69953e7	1.09959	0.549793	+	0.835301i	$0.314706\pi$
$752$	0	0
$753$	7.02370e6	0.451418
$754$	0	0
$755$	− 1.12333e7i	− 0.717199i
$756$	0	0
$757$	5.05028e6i	0.320314i	0.987092	+	0.160157i	$0.0512001\pi$
$757$	5.05028e6i	0.320314i	−0.987092	+	0.160157i	$0.948800\pi$
$758$	0	0
$759$	−6.53164e6	−0.411545
$760$	0	0
$761$	1.21656e7	0.761506	0.380753	−	0.924677i	$-0.375665\pi$
$761$	1.21656e7	0.761506	0.380753	+	0.924677i	$0.375665\pi$
$762$	0	0
$763$	6.74519e6i	0.419453i
$764$	0	0
$765$	− 1.68547e6i	− 0.104128i
$766$	0	0
$767$	−4.69960e6	−0.288451
$768$	0	0
$769$	−2.61059e6	−0.159193	−0.0795964	−	0.996827i	$-0.525363\pi$
$769$	−2.61059e6	−0.159193	−0.0795964	+	0.996827i	$0.525363\pi$
$770$	0	0
$771$	2.77950e6i	0.168396i
$772$	0	0
$773$	− 1.34970e7i	− 0.812434i	−0.913777	−	0.406217i	$-0.866848\pi$
$773$	− 1.34970e7i	− 0.812434i	0.913777	−	0.406217i	$-0.133152\pi$
$774$	0	0
$775$	4.81886e6	0.288197
$776$	0	0
$777$	−724586.	−0.0430564
$778$	0	0
$779$	− 1.98932e7i	− 1.17452i
$780$	0	0
$781$	− 5.47263e6i	− 0.321047i
$782$	0	0
$783$	−1.05297e7	−0.613777
$784$	0	0
$785$	−1.18857e7	−0.688418
$786$	0	0
$787$	− 2.13703e6i	− 0.122991i	−0.998107	−	0.0614956i	$-0.980413\pi$
$787$	− 2.13703e6i	− 0.122991i	0.998107	−	0.0614956i	$-0.0195870\pi$
$788$	0	0
$789$	− 7.04174e6i	− 0.402705i
$790$	0	0
$791$	−209246.	−0.0118909
$792$	0	0
$793$	2.04867e7	1.15688
$794$	0	0
$795$	5.27756e6i	0.296153i
$796$	0	0
$797$	− 2.73479e7i	− 1.52503i	−0.646969	−	0.762516i	$-0.723964\pi$
$797$	− 2.73479e7i	− 1.52503i	0.646969	−	0.762516i	$-0.276036\pi$
$798$	0	0
$799$	257166.	0.0142510
$800$	0	0
$801$	−2.49358e7	−1.37323
$802$	0	0
$803$	− 3.59078e6i	− 0.196517i
$804$	0	0
$805$	− 1.90554e6i	− 0.103640i
$806$	0	0
$807$	1.15688e7	0.625321
$808$	0	0
$809$	9.30041e6	0.499609	0.249805	−	0.968296i	$-0.419634\pi$
$809$	9.30041e6	0.499609	0.249805	+	0.968296i	$0.419634\pi$
$810$	0	0
$811$	− 9.39914e6i	− 0.501806i	−0.968012	−	0.250903i	$-0.919272\pi$
$811$	− 9.39914e6i	− 0.501806i	0.968012	−	0.250903i	$-0.0807275\pi$
$812$	0	0
$813$	8.81242e6i	0.467594i
$814$	0	0
$815$	5.26286e6	0.277542
$816$	0	0
$817$	−3.83298e7	−2.00900
$818$	0	0
$819$	7.26494e6i	0.378462i
$820$	0	0
$821$	1.40799e7i	0.729024i	0.931199	+	0.364512i	$0.118764\pi$
$821$	1.40799e7i	0.729024i	−0.931199	+	0.364512i	$0.881236\pi$
$822$	0	0
$823$	−2.71740e7	−1.39847	−0.699236	−	0.714891i	$-0.746476\pi$
$823$	−2.71740e7	−1.39847	−0.699236	+	0.714891i	$0.746476\pi$
$824$	0	0
$825$	2.05026e6	0.104876
$826$	0	0
$827$	− 3.74963e7i	− 1.90645i	−0.302264	−	0.953224i	$-0.597743\pi$
$827$	− 3.74963e7i	− 1.90645i	0.302264	−	0.953224i	$-0.402257\pi$
$828$	0	0
$829$	1.94820e7i	0.984572i	0.870433	+	0.492286i	$0.163839\pi$
$829$	1.94820e7i	0.984572i	−0.870433	+	0.492286i	$0.836161\pi$
$830$	0	0
$831$	−39063.7	−0.00196232
$832$	0	0
$833$	−5.21191e6	−0.260246
$834$	0	0
$835$	9.16946e6i	0.455122i
$836$	0	0
$837$	2.27178e7i	1.12086i
$838$	0	0
$839$	−2.75705e7	−1.35220	−0.676099	−	0.736811i	$-0.736331\pi$
$839$	−2.75705e7	−1.35220	−0.676099	+	0.736811i	$0.736331\pi$
$840$	0	0
$841$	7.74008e6	0.377359
$842$	0	0
$843$	− 9.77864e6i	− 0.473925i
$844$	0	0
$845$	1.35804e7i	0.654290i
$846$	0	0
$847$	−3.08192e6	−0.147609
$848$	0	0
$849$	1.31849e7	0.627780
$850$	0	0
$851$	− 5.64649e6i	− 0.267273i
$852$	0	0
$853$	1.21852e7i	0.573403i	0.958020	+	0.286701i	$0.0925588\pi$
$853$	1.21852e7i	0.573403i	−0.958020	+	0.286701i	$0.907441\pi$
$854$	0	0
$855$	−9.23982e6	−0.432263
$856$	0	0
$857$	3.11095e6	0.144691	0.0723455	−	0.997380i	$-0.476952\pi$
$857$	3.11095e6	0.144691	0.0723455	+	0.997380i	$0.476952\pi$
$858$	0	0
$859$	3.87955e6i	0.179390i	0.995969	+	0.0896951i	$0.0285893\pi$
$859$	3.87955e6i	0.179390i	−0.995969	+	0.0896951i	$0.971411\pi$
$860$	0	0
$861$	− 2.72922e6i	− 0.125467i
$862$	0	0
$863$	4.14946e7	1.89655	0.948275	−	0.317451i	$-0.102827\pi$
$863$	4.14946e7	1.89655	0.948275	+	0.317451i	$0.102827\pi$
$864$	0	0
$865$	6.05631e6	0.275212
$866$	0	0
$867$	8.70651e6i	0.393366i
$868$	0	0
$869$	− 1.15796e7i	− 0.520168i
$870$	0	0
$871$	6.52036e6	0.291223
$872$	0	0
$873$	2.52022e7	1.11919
$874$	0	0
$875$	598143.i	0.0264110i
$876$	0	0
$877$	2.44099e6i	0.107168i	0.998563	+	0.0535842i	$0.0170646\pi$
$877$	2.44099e6i	0.107168i	−0.998563	+	0.0535842i	$0.982935\pi$
$878$	0	0
$879$	9.22925e6	0.402897
$880$	0	0
$881$	1.29196e7	0.560801	0.280400	−	0.959883i	$-0.409533\pi$
$881$	1.29196e7	0.560801	0.280400	+	0.959883i	$0.409533\pi$
$882$	0	0
$883$	1.07860e7i	0.465540i	0.972532	+	0.232770i	$0.0747789\pi$
$883$	1.07860e7i	0.465540i	−0.972532	+	0.232770i	$0.925221\pi$
$884$	0	0
$885$	− 820022.i	− 0.0351939i
$886$	0	0
$887$	3.36076e7	1.43426	0.717132	−	0.696938i	$-0.245455\pi$
$887$	3.36076e7	1.43426	0.717132	+	0.696938i	$0.245455\pi$
$888$	0	0
$889$	−1.04841e7	−0.444913
$890$	0	0
$891$	− 1.40356e7i	− 0.592292i
$892$	0	0
$893$	− 1.40979e6i	− 0.0591599i
$894$	0	0
$895$	−1.46991e7	−0.613384
$896$	0	0
$897$	1.27088e7	0.527380
$898$	0	0
$899$	2.75536e7i	1.13705i
$900$	0	0
$901$	− 1.07449e7i	− 0.440951i
$902$	0	0
$903$	−5.25860e6	−0.214610
$904$	0	0
$905$	−1.07311e7	−0.435534
$906$	0	0
$907$	2.51579e7i	1.01545i	0.861520	+	0.507723i	$0.169513\pi$
$907$	2.51579e7i	1.01545i	−0.861520	+	0.507723i	$0.830487\pi$
$908$	0	0
$909$	2.55605e7i	1.02603i
$910$	0	0
$911$	−3.38933e7	−1.35306	−0.676531	−	0.736414i	$-0.736518\pi$
$911$	−3.38933e7	−1.35306	−0.676531	+	0.736414i	$0.736518\pi$
$912$	0	0
$913$	1.16524e7	0.462635
$914$	0	0
$915$	3.57468e6i	0.141151i
$916$	0	0
$917$	5.04660e6i	0.198187i
$918$	0	0
$919$	−3.09549e7	−1.20904	−0.604519	−	0.796590i	$-0.706635\pi$
$919$	−3.09549e7	−1.20904	−0.604519	+	0.796590i	$0.706635\pi$
$920$	0	0
$921$	−363917.	−0.0141369
$922$	0	0
$923$	1.06483e7i	0.411409i
$924$	0	0
$925$	1.77242e6i	0.0681102i
$926$	0	0
$927$	1.39598e7	0.533554
$928$	0	0
$929$	−4.44068e7	−1.68815	−0.844074	−	0.536227i	$-0.819849\pi$
$929$	−4.44068e7	−1.68815	−0.844074	+	0.536227i	$0.819849\pi$
$930$	0	0
$931$	2.85719e7i	1.08035i
$932$	0	0
$933$	− 7.16663e6i	− 0.269532i
$934$	0	0
$935$	−4.17425e6	−0.156153
$936$	0	0
$937$	−2.56682e7	−0.955094	−0.477547	−	0.878606i	$-0.658474\pi$
$937$	−2.56682e7	−0.955094	−0.477547	+	0.878606i	$0.658474\pi$
$938$	0	0
$939$	− 1.80695e7i	− 0.668777i
$940$	0	0
$941$	− 5.12276e6i	− 0.188595i	−0.995544	−	0.0942975i	$-0.969940\pi$
$941$	− 5.12276e6i	− 0.188595i	0.995544	−	0.0942975i	$-0.0300605\pi$
$942$	0	0
$943$	2.12680e7	0.778839
$944$	0	0
$945$	−2.81985e6	−0.102718
$946$	0	0
$947$	1.50587e7i	0.545650i	0.962064	+	0.272825i	$0.0879580\pi$
$947$	1.50587e7i	0.545650i	−0.962064	+	0.272825i	$0.912042\pi$
$948$	0	0
$949$	6.98670e6i	0.251829i
$950$	0	0
$951$	1.01494e7	0.363905
$952$	0	0
$953$	−7.86811e6	−0.280633	−0.140316	−	0.990107i	$-0.544812\pi$
$953$	−7.86811e6	−0.280633	−0.140316	+	0.990107i	$0.544812\pi$
$954$	0	0
$955$	− 6.79312e6i	− 0.241024i
$956$	0	0
$957$	1.17231e7i	0.413774i
$958$	0	0
$959$	1.47663e7	0.518473
$960$	0	0
$961$	3.08176e7	1.07644
$962$	0	0
$963$	1.78752e7i	0.621134i
$964$	0	0
$965$	− 2.17863e7i	− 0.753121i
$966$	0	0
$967$	3.04818e7	1.04827	0.524137	−	0.851634i	$-0.324388\pi$
$967$	3.04818e7	1.04827	0.524137	+	0.851634i	$0.324388\pi$
$968$	0	0
$969$	−4.22295e6	−0.144480
$970$	0	0
$971$	2.94765e7i	1.00329i	0.865073	+	0.501647i	$0.167272\pi$
$971$	2.94765e7i	1.00329i	−0.865073	+	0.501647i	$0.832728\pi$
$972$	0	0
$973$	1.70554e7i	0.577537i
$974$	0	0
$975$	−3.98926e6	−0.134394
$976$	0	0
$977$	3.24171e7	1.08652	0.543260	−	0.839565i	$-0.317190\pi$
$977$	3.24171e7	1.08652	0.543260	+	0.839565i	$0.317190\pi$
$978$	0	0
$979$	6.17563e7i	2.05932i
$980$	0	0
$981$	3.49673e7i	1.16009i
$982$	0	0
$983$	1.21680e7	0.401640	0.200820	−	0.979628i	$-0.435639\pi$
$983$	1.21680e7	0.401640	0.200820	+	0.979628i	$0.435639\pi$
$984$	0	0
$985$	5.19985e6	0.170766
$986$	0	0
$987$	− 193415.i	− 0.00631971i
$988$	0	0
$989$	− 4.09787e7i	− 1.33219i
$990$	0	0
$991$	1.84802e7	0.597754	0.298877	−	0.954292i	$-0.403388\pi$
$991$	1.84802e7	0.597754	0.298877	+	0.954292i	$0.403388\pi$
$992$	0	0
$993$	2.21034e7	0.711356
$994$	0	0
$995$	1.67381e7i	0.535979i
$996$	0	0
$997$	− 1.96954e7i	− 0.627519i	−0.949503	−	0.313759i	$-0.898411\pi$
$997$	− 1.96954e7i	− 0.627519i	0.949503	−	0.313759i	$-0.101589\pi$
$998$	0	0
$999$	−8.35579e6	−0.264895

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.6.d.a.81.10		20
4.3	odd	2		40.6.d.a.21.1	✓	20
5.2	odd	4		800.6.f.b.49.12		20
5.3	odd	4		800.6.f.c.49.9		20
5.4	even	2		800.6.d.c.401.11		20
8.3	odd	2		40.6.d.a.21.2	yes	20
8.5	even	2	inner	160.6.d.a.81.11		20
12.11	even	2		360.6.k.b.181.20		20
20.3	even	4		200.6.f.b.149.10		20
20.7	even	4		200.6.f.c.149.11		20
20.19	odd	2		200.6.d.b.101.20		20
24.11	even	2		360.6.k.b.181.19		20
40.3	even	4		200.6.f.c.149.12		20
40.13	odd	4		800.6.f.b.49.11		20
40.19	odd	2		200.6.d.b.101.19		20
40.27	even	4		200.6.f.b.149.9		20
40.29	even	2		800.6.d.c.401.10		20
40.37	odd	4		800.6.f.c.49.10		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.d.a.21.1	✓	20	4.3	odd	2
40.6.d.a.21.2	yes	20	8.3	odd	2
160.6.d.a.81.10		20	1.1	even	1	trivial
160.6.d.a.81.11		20	8.5	even	2	inner
200.6.d.b.101.19		20	40.19	odd	2
200.6.d.b.101.20		20	20.19	odd	2
200.6.f.b.149.9		20	40.27	even	4
200.6.f.b.149.10		20	20.3	even	4
200.6.f.c.149.11		20	20.7	even	4
200.6.f.c.149.12		20	40.3	even	4
360.6.k.b.181.19		20	24.11	even	2
360.6.k.b.181.20		20	12.11	even	2
800.6.d.c.401.10		20	40.29	even	2
800.6.d.c.401.11		20	5.4	even	2
800.6.f.b.49.11		20	40.13	odd	4
800.6.f.b.49.12		20	5.2	odd	4
800.6.f.c.49.9		20	5.3	odd	4
800.6.f.c.49.10		20	40.37	odd	4

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:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	160.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-6.67450i q^{3} -25.0000i q^{5} +38.2812 q^{7} +198.451 q^{9} +O(q^{10})\)	\(q-6.67450i q^{3} -25.0000i q^{5} +38.2812 q^{7} +198.451 q^{9} -491.486i q^{11} +956.299i q^{13} -166.863 q^{15} +339.725 q^{17} -1862.39i q^{19} -255.508i q^{21} +1991.10 q^{23} -625.000 q^{25} -2946.47i q^{27} -3573.66i q^{29} -7710.17 q^{31} -3280.42 q^{33} -957.029i q^{35} -2835.87i q^{37} +6382.82 q^{39} +10681.6 q^{41} -20581.0i q^{43} -4961.28i q^{45} +756.983 q^{47} -15341.6 q^{49} -2267.49i q^{51} -31628.2i q^{53} -12287.1 q^{55} -12430.5 q^{57} +4914.36i q^{59} -21422.9i q^{61} +7596.94 q^{63} +23907.5 q^{65} -6818.32i q^{67} -13289.6i q^{69} +11134.9 q^{71} +7305.98 q^{73} +4171.56i q^{75} -18814.6i q^{77} +23560.4 q^{79} +28557.4 q^{81} +23708.5i q^{83} -8493.12i q^{85} -23852.4 q^{87} -125652. q^{89} +36608.2i q^{91} +51461.6i q^{93} -46559.7 q^{95} +126994. q^{97} -97535.9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 196 q^{7} - 1620 q^{9}+O(q^{10}) \) 20 * q + 196 * q^7 - 1620 * q^9	\( 20 q + 196 q^{7} - 1620 q^{9} - 900 q^{15} + 4676 q^{23} - 12500 q^{25} - 7160 q^{31} + 5672 q^{33} + 44904 q^{39} + 11608 q^{41} - 44180 q^{47} + 18756 q^{49} + 24200 q^{55} + 5032 q^{57} - 240620 q^{63} + 200312 q^{71} - 105136 q^{73} - 282080 q^{79} + 65172 q^{81} + 332592 q^{87} - 3160 q^{89} - 144400 q^{95} + 147376 q^{97}+O(q^{100}) \) 20 * q + 196 * q^7 - 1620 * q^9 - 900 * q^15 + 4676 * q^23 - 12500 * q^25 - 7160 * q^31 + 5672 * q^33 + 44904 * q^39 + 11608 * q^41 - 44180 * q^47 + 18756 * q^49 + 24200 * q^55 + 5032 * q^57 - 240620 * q^63 + 200312 * q^71 - 105136 * q^73 - 282080 * q^79 + 65172 * q^81 + 332592 * q^87 - 3160 * q^89 - 144400 * q^95 + 147376 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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