LMFDB - Embedded newform 320.10.a.z.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [320,10,Mod(1,320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("320.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	320.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,2500,0,0,0,23364] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$164.811467572$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 353x^{2} + 354x + 2313 $ x^4 - 2x^3 - 353x^2 + 354*x + 2313
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{2}\cdot 5 $
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.12853$ of defining polynomial
Character	$\chi$	$=$	320.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+31.5423 q^{3} +625.000 q^{5} +5763.01 q^{7} -18688.1 q^{9} +40083.3 q^{11} +66880.5 q^{13} +19714.0 q^{15} -91431.8 q^{17} -901894. q^{19} +181779. q^{21} -260886. q^{23} +390625. q^{25} -1.21031e6 q^{27} +822128. q^{29} -8.24915e6 q^{31} +1.26432e6 q^{33} +3.60188e6 q^{35} +3.26345e6 q^{37} +2.10957e6 q^{39} -721750. q^{41} +4.71283e6 q^{43} -1.16801e7 q^{45} -1.54235e6 q^{47} -7.14133e6 q^{49} -2.88397e6 q^{51} -4.95803e7 q^{53} +2.50521e7 q^{55} -2.84479e7 q^{57} -1.47617e8 q^{59} -1.31460e8 q^{61} -1.07700e8 q^{63} +4.18003e7 q^{65} +3.13535e8 q^{67} -8.22897e6 q^{69} +1.43087e8 q^{71} -7.49397e6 q^{73} +1.23212e7 q^{75} +2.31000e8 q^{77} +4.72822e7 q^{79} +3.29661e8 q^{81} -5.06399e8 q^{83} -5.71449e7 q^{85} +2.59319e7 q^{87} +2.62827e8 q^{89} +3.85433e8 q^{91} -2.60198e8 q^{93} -5.63684e8 q^{95} +1.33714e9 q^{97} -7.49080e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2500 q^{5} + 23364 q^{9} - 201256 q^{13} - 442040 q^{17} + 3496176 q^{21} + 1562500 q^{25} - 6719352 q^{29} - 36991008 q^{33} - 29092616 q^{37} + 49561624 q^{41} + 14602500 q^{45} + 3039028 q^{49}+ \cdots + 464919272 q^{97}+O(q^{100}) $ 4 * q + 2500 * q^5 + 23364 * q^9 - 201256 * q^13 - 442040 * q^17 + 3496176 * q^21 + 1562500 * q^25 - 6719352 * q^29 - 36991008 * q^33 - 29092616 * q^37 + 49561624 * q^41 + 14602500 * q^45 + 3039028 * q^49 - 178752456 * q^53 + 127411200 * q^57 + 35996024 * q^61 - 125785000 * q^65 - 222978096 * q^69 + 31085160 * q^73 - 774686048 * q^77 + 533616660 * q^81 - 276275000 * q^85 + 2529920360 * q^89 - 2297246688 * q^93 + 464919272 * q^97

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
$3$	31.5423	0.224827	0.112413	−	0.993662i	$-0.464142\pi$
$3$	31.5423	0.224827	0.112413	+	0.993662i	$0.464142\pi$
$4$	0	0
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	5763.01	0.907211	0.453605	−	0.891203i	$-0.350138\pi$
$7$	5763.01	0.907211	0.453605	+	0.891203i	$0.350138\pi$
$8$	0	0
$9$	−18688.1	−0.949453
$10$	0	0
$11$	40083.3	0.825461	0.412730	−	0.910853i	$-0.364575\pi$
$11$	40083.3	0.825461	0.412730	+	0.910853i	$0.364575\pi$
$12$	0	0
$13$	66880.5	0.649463	0.324731	−	0.945806i	$-0.394726\pi$
$13$	66880.5	0.649463	0.324731	+	0.945806i	$0.394726\pi$
$14$	0	0
$15$	19714.0	0.100546
$16$	0	0
$17$	−91431.8	−0.265508	−0.132754	−	0.991149i	$-0.542382\pi$
$17$	−91431.8	−0.265508	−0.132754	+	0.991149i	$0.542382\pi$
$18$	0	0
$19$	−901894.	−1.58768	−0.793842	−	0.608124i	$-0.791923\pi$
$19$	−901894.	−1.58768	−0.793842	+	0.608124i	$0.791923\pi$
$20$	0	0
$21$	181779.	0.203965
$22$	0	0
$23$	−260886.	−0.194391	−0.0971955	−	0.995265i	$-0.530987\pi$
$23$	−260886.	−0.194391	−0.0971955	+	0.995265i	$0.530987\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	0	0
$27$	−1.21031e6	−0.438290
$28$	0	0
$29$	822128.	0.215848	0.107924	−	0.994159i	$-0.465580\pi$
$29$	822128.	0.215848	0.107924	+	0.994159i	$0.465580\pi$
$30$	0	0
$31$	−8.24915e6	−1.60429	−0.802143	−	0.597132i	$-0.796307\pi$
$31$	−8.24915e6	−1.60429	−0.802143	+	0.597132i	$0.796307\pi$
$32$	0	0
$33$	1.26432e6	0.185586
$34$	0	0
$35$	3.60188e6	0.405717
$36$	0	0
$37$	3.26345e6	0.286266	0.143133	−	0.989703i	$-0.454282\pi$
$37$	3.26345e6	0.286266	0.143133	+	0.989703i	$0.454282\pi$
$38$	0	0
$39$	2.10957e6	0.146017
$40$	0	0
$41$	−721750.	−0.0398896	−0.0199448	−	0.999801i	$-0.506349\pi$
$41$	−721750.	−0.0398896	−0.0199448	+	0.999801i	$0.506349\pi$
$42$	0	0
$43$	4.71283e6	0.210220	0.105110	−	0.994461i	$-0.466481\pi$
$43$	4.71283e6	0.210220	0.105110	+	0.994461i	$0.466481\pi$
$44$	0	0
$45$	−1.16801e7	−0.424608
$46$	0	0
$47$	−1.54235e6	−0.0461044	−0.0230522	−	0.999734i	$-0.507338\pi$
$47$	−1.54235e6	−0.0461044	−0.0230522	+	0.999734i	$0.507338\pi$
$48$	0	0
$49$	−7.14133e6	−0.176969
$50$	0	0
$51$	−2.88397e6	−0.0596933
$52$	0	0
$53$	−4.95803e7	−0.863113	−0.431557	−	0.902086i	$-0.642035\pi$
$53$	−4.95803e7	−0.863113	−0.431557	+	0.902086i	$0.642035\pi$
$54$	0	0
$55$	2.50521e7	0.369157
$56$	0	0
$57$	−2.84479e7	−0.356954
$58$	0	0
$59$	−1.47617e8	−1.58600	−0.793000	−	0.609222i	$-0.791482\pi$
$59$	−1.47617e8	−1.58600	−0.793000	+	0.609222i	$0.791482\pi$
$60$	0	0
$61$	−1.31460e8	−1.21565	−0.607826	−	0.794070i	$-0.707958\pi$
$61$	−1.31460e8	−1.21565	−0.607826	+	0.794070i	$0.707958\pi$
$62$	0	0
$63$	−1.07700e8	−0.861354
$64$	0	0
$65$	4.18003e7	0.290449
$66$	0	0
$67$	3.13535e8	1.90086	0.950429	−	0.310942i	$-0.100645\pi$
$67$	3.13535e8	1.90086	0.950429	+	0.310942i	$0.100645\pi$
$68$	0	0
$69$	−8.22897e6	−0.0437043
$70$	0	0
$71$	1.43087e8	0.668247	0.334123	−	0.942529i	$-0.391560\pi$
$71$	1.43087e8	0.668247	0.334123	+	0.942529i	$0.391560\pi$
$72$	0	0
$73$	−7.49397e6	−0.0308858	−0.0154429	−	0.999881i	$-0.504916\pi$
$73$	−7.49397e6	−0.0308858	−0.0154429	+	0.999881i	$0.504916\pi$
$74$	0	0
$75$	1.23212e7	0.0449654
$76$	0	0
$77$	2.31000e8	0.748867
$78$	0	0
$79$	4.72822e7	0.136577	0.0682883	−	0.997666i	$-0.478246\pi$
$79$	4.72822e7	0.136577	0.0682883	+	0.997666i	$0.478246\pi$
$80$	0	0
$81$	3.29661e8	0.850914
$82$	0	0
$83$	−5.06399e8	−1.17123	−0.585614	−	0.810590i	$-0.699147\pi$
$83$	−5.06399e8	−1.17123	−0.585614	+	0.810590i	$0.699147\pi$
$84$	0	0
$85$	−5.71449e7	−0.118739
$86$	0	0
$87$	2.59319e7	0.0485285
$88$	0	0
$89$	2.62827e8	0.444032	0.222016	−	0.975043i	$-0.428736\pi$
$89$	2.62827e8	0.444032	0.222016	+	0.975043i	$0.428736\pi$
$90$	0	0
$91$	3.85433e8	0.589200
$92$	0	0
$93$	−2.60198e8	−0.360687
$94$	0	0
$95$	−5.63684e8	−0.710034
$96$	0	0
$97$	1.33714e9	1.53357	0.766784	−	0.641905i	$-0.221856\pi$
$97$	1.33714e9	1.53357	0.766784	+	0.641905i	$0.221856\pi$
$98$	0	0
$99$	−7.49080e8	−0.783736
$100$	0	0
$101$	−1.59609e9	−1.52620	−0.763099	−	0.646282i	$-0.776323\pi$
$101$	−1.59609e9	−1.52620	−0.763099	+	0.646282i	$0.776323\pi$
$102$	0	0
$103$	4.80627e8	0.420766	0.210383	−	0.977619i	$-0.432529\pi$
$103$	4.80627e8	0.420766	0.210383	+	0.977619i	$0.432529\pi$
$104$	0	0
$105$	1.13612e8	0.0912161
$106$	0	0
$107$	3.78945e7	0.0279479	0.0139739	−	0.999902i	$-0.495552\pi$
$107$	3.78945e7	0.0279479	0.0139739	+	0.999902i	$0.495552\pi$
$108$	0	0
$109$	−1.75532e9	−1.19107	−0.595535	−	0.803330i	$-0.703060\pi$
$109$	−1.75532e9	−1.19107	−0.595535	+	0.803330i	$0.703060\pi$
$110$	0	0
$111$	1.02937e8	0.0643602
$112$	0	0
$113$	8.94453e8	0.516065	0.258033	−	0.966136i	$-0.416926\pi$
$113$	8.94453e8	0.516065	0.258033	+	0.966136i	$0.416926\pi$
$114$	0	0
$115$	−1.63054e8	−0.0869343
$116$	0	0
$117$	−1.24987e9	−0.616634
$118$	0	0
$119$	−5.26922e8	−0.240871
$120$	0	0
$121$	−7.51276e8	−0.318615
$122$	0	0
$123$	−2.27657e7	−0.00896825
$124$	0	0
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	−3.04135e9	−1.03741	−0.518705	−	0.854954i	$-0.673586\pi$
$127$	−3.04135e9	−1.03741	−0.518705	+	0.854954i	$0.673586\pi$
$128$	0	0
$129$	1.48654e8	0.0472631
$130$	0	0
$131$	−8.80696e7	−0.0261280	−0.0130640	−	0.999915i	$-0.504159\pi$
$131$	−8.80696e7	−0.0261280	−0.0130640	+	0.999915i	$0.504159\pi$
$132$	0	0
$133$	−5.19762e9	−1.44036
$134$	0	0
$135$	−7.56446e8	−0.196009
$136$	0	0
$137$	5.03962e9	1.22224	0.611118	−	0.791540i	$-0.290720\pi$
$137$	5.03962e9	1.22224	0.611118	+	0.791540i	$0.290720\pi$
$138$	0	0
$139$	3.67230e9	0.834395	0.417197	−	0.908816i	$-0.363012\pi$
$139$	3.67230e9	0.834395	0.417197	+	0.908816i	$0.363012\pi$
$140$	0	0
$141$	−4.86493e7	−0.0103655
$142$	0	0
$143$	2.68079e9	0.536106
$144$	0	0
$145$	5.13830e8	0.0965303
$146$	0	0
$147$	−2.25254e8	−0.0397874
$148$	0	0
$149$	−9.50765e9	−1.58028	−0.790142	−	0.612924i	$-0.789993\pi$
$149$	−9.50765e9	−1.58028	−0.790142	+	0.612924i	$0.789993\pi$
$150$	0	0
$151$	5.53508e9	0.866418	0.433209	−	0.901294i	$-0.357381\pi$
$151$	5.53508e9	0.866418	0.433209	+	0.901294i	$0.357381\pi$
$152$	0	0
$153$	1.70869e9	0.252087
$154$	0	0
$155$	−5.15572e9	−0.717459
$156$	0	0
$157$	−3.26124e9	−0.428385	−0.214193	−	0.976791i	$-0.568712\pi$
$157$	−3.26124e9	−0.428385	−0.214193	+	0.976791i	$0.568712\pi$
$158$	0	0
$159$	−1.56388e9	−0.194051
$160$	0	0
$161$	−1.50349e9	−0.176354
$162$	0	0
$163$	5.90330e9	0.655014	0.327507	−	0.944849i	$-0.393792\pi$
$163$	5.90330e9	0.655014	0.327507	+	0.944849i	$0.393792\pi$
$164$	0	0
$165$	7.90201e8	0.0829965
$166$	0	0
$167$	1.41098e10	1.40377	0.701884	−	0.712291i	$-0.252342\pi$
$167$	1.41098e10	1.40377	0.701884	+	0.712291i	$0.252342\pi$
$168$	0	0
$169$	−6.13150e9	−0.578198
$170$	0	0
$171$	1.68547e10	1.50743
$172$	0	0
$173$	−8.67737e9	−0.736514	−0.368257	−	0.929724i	$-0.620045\pi$
$173$	−8.67737e9	−0.736514	−0.368257	+	0.929724i	$0.620045\pi$
$174$	0	0
$175$	2.25118e9	0.181442
$176$	0	0
$177$	−4.65620e9	−0.356576
$178$	0	0
$179$	−1.99826e10	−1.45483	−0.727417	−	0.686196i	$-0.759279\pi$
$179$	−1.99826e10	−1.45483	−0.727417	+	0.686196i	$0.759279\pi$
$180$	0	0
$181$	−1.09751e10	−0.760072	−0.380036	−	0.924972i	$-0.624088\pi$
$181$	−1.09751e10	−0.760072	−0.380036	+	0.924972i	$0.624088\pi$
$182$	0	0
$183$	−4.14656e9	−0.273311
$184$	0	0
$185$	2.03966e9	0.128022
$186$	0	0
$187$	−3.66489e9	−0.219166
$188$	0	0
$189$	−6.97505e9	−0.397621
$190$	0	0
$191$	−1.88576e10	−1.02527	−0.512633	−	0.858608i	$-0.671330\pi$
$191$	−1.88576e10	−1.02527	−0.512633	+	0.858608i	$0.671330\pi$
$192$	0	0
$193$	−2.12010e10	−1.09989	−0.549944	−	0.835201i	$-0.685351\pi$
$193$	−2.12010e10	−1.09989	−0.549944	+	0.835201i	$0.685351\pi$
$194$	0	0
$195$	1.31848e9	0.0653007
$196$	0	0
$197$	−4.20226e9	−0.198785	−0.0993927	−	0.995048i	$-0.531690\pi$
$197$	−4.20226e9	−0.198785	−0.0993927	+	0.995048i	$0.531690\pi$
$198$	0	0
$199$	−3.42438e10	−1.54790	−0.773950	−	0.633247i	$-0.781722\pi$
$199$	−3.42438e10	−1.54790	−0.773950	+	0.633247i	$0.781722\pi$
$200$	0	0
$201$	9.88963e9	0.427364
$202$	0	0
$203$	4.73793e9	0.195820
$204$	0	0
$205$	−4.51094e8	−0.0178392
$206$	0	0
$207$	4.87547e9	0.184565
$208$	0	0
$209$	−3.61509e10	−1.31057
$210$	0	0
$211$	−3.15040e10	−1.09419	−0.547097	−	0.837069i	$-0.684267\pi$
$211$	−3.15040e10	−1.09419	−0.547097	+	0.837069i	$0.684267\pi$
$212$	0	0
$213$	4.51329e9	0.150240
$214$	0	0
$215$	2.94552e9	0.0940132
$216$	0	0
$217$	−4.75400e10	−1.45543
$218$	0	0
$219$	−2.36378e8	−0.00694397
$220$	0	0
$221$	−6.11501e9	−0.172437
$222$	0	0
$223$	6.76841e10	1.83280	0.916399	−	0.400267i	$-0.131082\pi$
$223$	6.76841e10	1.83280	0.916399	+	0.400267i	$0.131082\pi$
$224$	0	0
$225$	−7.30003e9	−0.189891
$226$	0	0
$227$	2.32392e10	0.580904	0.290452	−	0.956890i	$-0.406194\pi$
$227$	2.32392e10	0.580904	0.290452	+	0.956890i	$0.406194\pi$
$228$	0	0
$229$	3.87727e10	0.931678	0.465839	−	0.884869i	$-0.345753\pi$
$229$	3.87727e10	0.931678	0.465839	+	0.884869i	$0.345753\pi$
$230$	0	0
$231$	7.28630e9	0.168365
$232$	0	0
$233$	−3.74668e10	−0.832809	−0.416404	−	0.909180i	$-0.636710\pi$
$233$	−3.74668e10	−0.832809	−0.416404	+	0.909180i	$0.636710\pi$
$234$	0	0
$235$	−9.63968e8	−0.0206185
$236$	0	0
$237$	1.49139e9	0.0307061
$238$	0	0
$239$	−1.01374e10	−0.200973	−0.100487	−	0.994938i	$-0.532040\pi$
$239$	−1.01374e10	−0.200973	−0.100487	+	0.994938i	$0.532040\pi$
$240$	0	0
$241$	−6.77474e10	−1.29365	−0.646824	−	0.762639i	$-0.723903\pi$
$241$	−6.77474e10	−1.29365	−0.646824	+	0.762639i	$0.723903\pi$
$242$	0	0
$243$	3.42209e10	0.629598
$244$	0	0
$245$	−4.46333e9	−0.0791429
$246$	0	0
$247$	−6.03191e10	−1.03114
$248$	0	0
$249$	−1.59730e10	−0.263324
$250$	0	0
$251$	1.49908e9	0.0238393	0.0119197	−	0.999929i	$-0.496206\pi$
$251$	1.49908e9	0.0238393	0.0119197	+	0.999929i	$0.496206\pi$
$252$	0	0
$253$	−1.04572e10	−0.160462
$254$	0	0
$255$	−1.80248e9	−0.0266957
$256$	0	0
$257$	−8.73728e10	−1.24933	−0.624665	−	0.780893i	$-0.714764\pi$
$257$	−8.73728e10	−1.24933	−0.624665	+	0.780893i	$0.714764\pi$
$258$	0	0
$259$	1.88073e10	0.259703
$260$	0	0
$261$	−1.53640e10	−0.204938
$262$	0	0
$263$	−8.15504e9	−0.105105	−0.0525527	−	0.998618i	$-0.516736\pi$
$263$	−8.15504e9	−0.105105	−0.0525527	+	0.998618i	$0.516736\pi$
$264$	0	0
$265$	−3.09877e10	−0.385996
$266$	0	0
$267$	8.29018e9	0.0998305
$268$	0	0
$269$	−4.12445e9	−0.0480265	−0.0240133	−	0.999712i	$-0.507644\pi$
$269$	−4.12445e9	−0.0480265	−0.0240133	+	0.999712i	$0.507644\pi$
$270$	0	0
$271$	−4.44948e10	−0.501127	−0.250563	−	0.968100i	$-0.580616\pi$
$271$	−4.44948e10	−0.501127	−0.250563	+	0.968100i	$0.580616\pi$
$272$	0	0
$273$	1.21575e10	0.132468
$274$	0	0
$275$	1.56575e10	0.165092
$276$	0	0
$277$	−9.73881e10	−0.993909	−0.496955	−	0.867777i	$-0.665548\pi$
$277$	−9.73881e10	−0.993909	−0.496955	+	0.867777i	$0.665548\pi$
$278$	0	0
$279$	1.54161e11	1.52319
$280$	0	0
$281$	−1.38133e11	−1.32165	−0.660827	−	0.750539i	$-0.729794\pi$
$281$	−1.38133e11	−1.32165	−0.660827	+	0.750539i	$0.729794\pi$
$282$	0	0
$283$	−1.44541e11	−1.33952	−0.669762	−	0.742575i	$-0.733604\pi$
$283$	−1.44541e11	−1.33952	−0.669762	+	0.742575i	$0.733604\pi$
$284$	0	0
$285$	−1.77799e10	−0.159635
$286$	0	0
$287$	−4.15945e9	−0.0361883
$288$	0	0
$289$	−1.10228e11	−0.929506
$290$	0	0
$291$	4.21765e10	0.344788
$292$	0	0
$293$	−9.30787e10	−0.737813	−0.368906	−	0.929467i	$-0.620268\pi$
$293$	−9.30787e10	−0.737813	−0.368906	+	0.929467i	$0.620268\pi$
$294$	0	0
$295$	−9.22608e10	−0.709281
$296$	0	0
$297$	−4.85134e10	−0.361791
$298$	0	0
$299$	−1.74482e10	−0.126250
$300$	0	0
$301$	2.71601e10	0.190714
$302$	0	0
$303$	−5.03444e10	−0.343130
$304$	0	0
$305$	−8.21625e10	−0.543656
$306$	0	0
$307$	−8.83221e10	−0.567475	−0.283738	−	0.958902i	$-0.591574\pi$
$307$	−8.83221e10	−0.567475	−0.283738	+	0.958902i	$0.591574\pi$
$308$	0	0
$309$	1.51601e10	0.0945996
$310$	0	0
$311$	4.39960e10	0.266681	0.133340	−	0.991070i	$-0.457430\pi$
$311$	4.39960e10	0.266681	0.133340	+	0.991070i	$0.457430\pi$
$312$	0	0
$313$	−9.54876e10	−0.562338	−0.281169	−	0.959658i	$-0.590722\pi$
$313$	−9.54876e10	−0.562338	−0.281169	+	0.959658i	$0.590722\pi$
$314$	0	0
$315$	−6.73122e10	−0.385209
$316$	0	0
$317$	−1.05730e11	−0.588073	−0.294037	−	0.955794i	$-0.594999\pi$
$317$	−1.05730e11	−0.588073	−0.294037	+	0.955794i	$0.594999\pi$
$318$	0	0
$319$	3.29536e10	0.178174
$320$	0	0
$321$	1.19528e9	0.00628344
$322$	0	0
$323$	8.24618e10	0.421543
$324$	0	0
$325$	2.61252e10	0.129893
$326$	0	0
$327$	−5.53669e10	−0.267784
$328$	0	0
$329$	−8.88857e9	−0.0418264
$330$	0	0
$331$	−3.74481e10	−0.171476	−0.0857380	−	0.996318i	$-0.527325\pi$
$331$	−3.74481e10	−0.171476	−0.0857380	+	0.996318i	$0.527325\pi$
$332$	0	0
$333$	−6.09876e10	−0.271796
$334$	0	0
$335$	1.95959e11	0.850089
$336$	0	0
$337$	2.00187e11	0.845474	0.422737	−	0.906252i	$-0.361069\pi$
$337$	2.00187e11	0.845474	0.422737	+	0.906252i	$0.361069\pi$
$338$	0	0
$339$	2.82131e10	0.116025
$340$	0	0
$341$	−3.30653e11	−1.32428
$342$	0	0
$343$	−2.73714e11	−1.06776
$344$	0	0
$345$	−5.14311e9	−0.0195452
$346$	0	0
$347$	3.12228e11	1.15609	0.578043	−	0.816007i	$-0.303817\pi$
$347$	3.12228e11	1.15609	0.578043	+	0.816007i	$0.303817\pi$
$348$	0	0
$349$	1.24537e10	0.0449349	0.0224675	−	0.999748i	$-0.492848\pi$
$349$	1.24537e10	0.0449349	0.0224675	+	0.999748i	$0.492848\pi$
$350$	0	0
$351$	−8.09464e10	−0.284653
$352$	0	0
$353$	4.00381e11	1.37242	0.686210	−	0.727404i	$-0.259273\pi$
$353$	4.00381e11	1.37242	0.686210	+	0.727404i	$0.259273\pi$
$354$	0	0
$355$	8.94292e10	0.298849
$356$	0	0
$357$	−1.66204e10	−0.0541544
$358$	0	0
$359$	−5.64723e10	−0.179437	−0.0897183	−	0.995967i	$-0.528597\pi$
$359$	−5.64723e10	−0.179437	−0.0897183	+	0.995967i	$0.528597\pi$
$360$	0	0
$361$	4.90725e11	1.52074
$362$	0	0
$363$	−2.36970e10	−0.0716331
$364$	0	0
$365$	−4.68373e9	−0.0138126
$366$	0	0
$367$	6.09656e11	1.75424	0.877118	−	0.480276i	$-0.159463\pi$
$367$	6.09656e11	1.75424	0.877118	+	0.480276i	$0.159463\pi$
$368$	0	0
$369$	1.34881e10	0.0378733
$370$	0	0
$371$	−2.85732e11	−0.783025
$372$	0	0
$373$	−3.30775e11	−0.884796	−0.442398	−	0.896819i	$-0.645872\pi$
$373$	−3.30775e11	−0.884796	−0.442398	+	0.896819i	$0.645872\pi$
$374$	0	0
$375$	7.70077e9	0.0201091
$376$	0	0
$377$	5.49843e10	0.140185
$378$	0	0
$379$	5.91235e11	1.47192	0.735959	−	0.677026i	$-0.236732\pi$
$379$	5.91235e11	1.47192	0.735959	+	0.677026i	$0.236732\pi$
$380$	0	0
$381$	−9.59314e10	−0.233238
$382$	0	0
$383$	−1.13650e11	−0.269882	−0.134941	−	0.990854i	$-0.543085\pi$
$383$	−1.13650e11	−0.269882	−0.134941	+	0.990854i	$0.543085\pi$
$384$	0	0
$385$	1.44375e11	0.334903
$386$	0	0
$387$	−8.80737e10	−0.199594
$388$	0	0
$389$	1.04287e11	0.230917	0.115458	−	0.993312i	$-0.463166\pi$
$389$	1.04287e11	0.230917	0.115458	+	0.993312i	$0.463166\pi$
$390$	0	0
$391$	2.38533e10	0.0516123
$392$	0	0
$393$	−2.77792e9	−0.00587427
$394$	0	0
$395$	2.95514e10	0.0610789
$396$	0	0
$397$	6.43948e10	0.130105	0.0650524	−	0.997882i	$-0.479279\pi$
$397$	6.43948e10	0.130105	0.0650524	+	0.997882i	$0.479279\pi$
$398$	0	0
$399$	−1.63945e11	−0.323833
$400$	0	0
$401$	5.25190e11	1.01430	0.507151	−	0.861857i	$-0.330699\pi$
$401$	5.25190e11	1.01430	0.507151	+	0.861857i	$0.330699\pi$
$402$	0	0
$403$	−5.51708e11	−1.04192
$404$	0	0
$405$	2.06038e11	0.380540
$406$	0	0
$407$	1.30810e11	0.236301
$408$	0	0
$409$	−4.93589e11	−0.872189	−0.436094	−	0.899901i	$-0.643639\pi$
$409$	−4.93589e11	−0.872189	−0.436094	+	0.899901i	$0.643639\pi$
$410$	0	0
$411$	1.58961e11	0.274792
$412$	0	0
$413$	−8.50720e11	−1.43884
$414$	0	0
$415$	−3.16500e11	−0.523789
$416$	0	0
$417$	1.15833e11	0.187594
$418$	0	0
$419$	−1.44705e11	−0.229362	−0.114681	−	0.993402i	$-0.536585\pi$
$419$	−1.44705e11	−0.229362	−0.114681	+	0.993402i	$0.536585\pi$
$420$	0	0
$421$	−2.14833e11	−0.333297	−0.166648	−	0.986016i	$-0.553294\pi$
$421$	−2.14833e11	−0.333297	−0.166648	+	0.986016i	$0.553294\pi$
$422$	0	0
$423$	2.88235e10	0.0437740
$424$	0	0
$425$	−3.57156e10	−0.0531016
$426$	0	0
$427$	−7.57605e11	−1.10285
$428$	0	0
$429$	8.45585e10	0.120531
$430$	0	0
$431$	−1.60116e11	−0.223504	−0.111752	−	0.993736i	$-0.535646\pi$
$431$	−1.60116e11	−0.223504	−0.111752	+	0.993736i	$0.535646\pi$
$432$	0	0
$433$	1.08023e12	1.47679	0.738397	−	0.674366i	$-0.235583\pi$
$433$	1.08023e12	1.47679	0.738397	+	0.674366i	$0.235583\pi$
$434$	0	0
$435$	1.62074e10	0.0217026
$436$	0	0
$437$	2.35292e11	0.308632
$438$	0	0
$439$	5.70666e11	0.733317	0.366659	−	0.930356i	$-0.380502\pi$
$439$	5.70666e11	0.733317	0.366659	+	0.930356i	$0.380502\pi$
$440$	0	0
$441$	1.33458e11	0.168024
$442$	0	0
$443$	8.64148e11	1.06604	0.533018	−	0.846104i	$-0.321058\pi$
$443$	8.64148e11	1.06604	0.533018	+	0.846104i	$0.321058\pi$
$444$	0	0
$445$	1.64267e11	0.198577
$446$	0	0
$447$	−2.99894e11	−0.355290
$448$	0	0
$449$	−8.23785e11	−0.956545	−0.478273	−	0.878211i	$-0.658737\pi$
$449$	−8.23785e11	−0.956545	−0.478273	+	0.878211i	$0.658737\pi$
$450$	0	0
$451$	−2.89301e10	−0.0329273
$452$	0	0
$453$	1.74589e11	0.194794
$454$	0	0
$455$	2.40896e11	0.263498
$456$	0	0
$457$	1.18240e12	1.26807	0.634033	−	0.773306i	$-0.281398\pi$
$457$	1.18240e12	1.26807	0.634033	+	0.773306i	$0.281398\pi$
$458$	0	0
$459$	1.10661e11	0.116369
$460$	0	0
$461$	8.13722e11	0.839116	0.419558	−	0.907728i	$-0.362185\pi$
$461$	8.13722e11	0.839116	0.419558	+	0.907728i	$0.362185\pi$
$462$	0	0
$463$	−3.50433e11	−0.354398	−0.177199	−	0.984175i	$-0.556704\pi$
$463$	−3.50433e11	−0.354398	−0.177199	+	0.984175i	$0.556704\pi$
$464$	0	0
$465$	−1.62624e11	−0.161304
$466$	0	0
$467$	8.86558e11	0.862543	0.431272	−	0.902222i	$-0.358065\pi$
$467$	8.86558e11	0.862543	0.431272	+	0.902222i	$0.358065\pi$
$468$	0	0
$469$	1.80691e12	1.72448
$470$	0	0
$471$	−1.02867e11	−0.0963126
$472$	0	0
$473$	1.88906e11	0.173528
$474$	0	0
$475$	−3.52302e11	−0.317537
$476$	0	0
$477$	9.26561e11	0.819485
$478$	0	0
$479$	1.88439e10	0.0163554	0.00817770	−	0.999967i	$-0.497397\pi$
$479$	1.88439e10	0.0163554	0.00817770	+	0.999967i	$0.497397\pi$
$480$	0	0
$481$	2.18261e11	0.185919
$482$	0	0
$483$	−4.74236e10	−0.0396490
$484$	0	0
$485$	8.35711e11	0.685833
$486$	0	0
$487$	−1.88234e12	−1.51642	−0.758208	−	0.652013i	$-0.773925\pi$
$487$	−1.88234e12	−1.51642	−0.758208	+	0.652013i	$0.773925\pi$
$488$	0	0
$489$	1.86204e11	0.147265
$490$	0	0
$491$	−2.33032e12	−1.80946	−0.904729	−	0.425988i	$-0.859927\pi$
$491$	−2.33032e12	−1.80946	−0.904729	+	0.425988i	$0.859927\pi$
$492$	0	0
$493$	−7.51687e10	−0.0573094
$494$	0	0
$495$	−4.68175e11	−0.350497
$496$	0	0
$497$	8.24611e11	0.606241
$498$	0	0
$499$	−1.76574e12	−1.27489	−0.637446	−	0.770495i	$-0.720009\pi$
$499$	−1.76574e12	−1.27489	−0.637446	+	0.770495i	$0.720009\pi$
$500$	0	0
$501$	4.45055e11	0.315605
$502$	0	0
$503$	2.75964e12	1.92219	0.961095	−	0.276219i	$-0.0890817\pi$
$503$	2.75964e12	1.92219	0.961095	+	0.276219i	$0.0890817\pi$
$504$	0	0
$505$	−9.97555e11	−0.682536
$506$	0	0
$507$	−1.93402e11	−0.129994
$508$	0	0
$509$	2.34774e12	1.55031	0.775156	−	0.631770i	$-0.217671\pi$
$509$	2.34774e12	1.55031	0.775156	+	0.631770i	$0.217671\pi$
$510$	0	0
$511$	−4.31878e10	−0.0280200
$512$	0	0
$513$	1.09157e12	0.695866
$514$	0	0
$515$	3.00392e11	0.188172
$516$	0	0
$517$	−6.18225e10	−0.0380574
$518$	0	0
$519$	−2.73705e11	−0.165588
$520$	0	0
$521$	1.00091e12	0.595151	0.297576	−	0.954698i	$-0.403822\pi$
$521$	1.00091e12	0.595151	0.297576	+	0.954698i	$0.403822\pi$
$522$	0	0
$523$	3.01151e12	1.76006	0.880030	−	0.474919i	$-0.157523\pi$
$523$	3.01151e12	1.76006	0.880030	+	0.474919i	$0.157523\pi$
$524$	0	0
$525$	7.10074e10	0.0407931
$526$	0	0
$527$	7.54235e11	0.425950
$528$	0	0
$529$	−1.73309e12	−0.962212
$530$	0	0
$531$	2.75868e12	1.50583
$532$	0	0
$533$	−4.82710e10	−0.0259068
$534$	0	0
$535$	2.36840e10	0.0124987
$536$	0	0
$537$	−6.30298e11	−0.327086
$538$	0	0
$539$	−2.86248e11	−0.146081
$540$	0	0
$541$	1.79083e12	0.898807	0.449404	−	0.893329i	$-0.351637\pi$
$541$	1.79083e12	0.898807	0.449404	+	0.893329i	$0.351637\pi$
$542$	0	0
$543$	−3.46180e11	−0.170885
$544$	0	0
$545$	−1.09707e12	−0.532662
$546$	0	0
$547$	1.42084e12	0.678580	0.339290	−	0.940682i	$-0.389813\pi$
$547$	1.42084e12	0.678580	0.339290	+	0.940682i	$0.389813\pi$
$548$	0	0
$549$	2.45673e12	1.15420
$550$	0	0
$551$	−7.41472e11	−0.342699
$552$	0	0
$553$	2.72488e11	0.123904
$554$	0	0
$555$	6.43355e10	0.0287828
$556$	0	0
$557$	2.06883e12	0.910704	0.455352	−	0.890311i	$-0.349513\pi$
$557$	2.06883e12	0.910704	0.455352	+	0.890311i	$0.349513\pi$
$558$	0	0
$559$	3.15196e11	0.136530
$560$	0	0
$561$	−1.15599e11	−0.0492745
$562$	0	0
$563$	−1.67871e12	−0.704188	−0.352094	−	0.935965i	$-0.614530\pi$
$563$	−1.67871e12	−0.704188	−0.352094	+	0.935965i	$0.614530\pi$
$564$	0	0
$565$	5.59033e11	0.230791
$566$	0	0
$567$	1.89984e12	0.771958
$568$	0	0
$569$	−2.89714e12	−1.15868	−0.579342	−	0.815085i	$-0.696690\pi$
$569$	−2.89714e12	−1.15868	−0.579342	+	0.815085i	$0.696690\pi$
$570$	0	0
$571$	5.19889e11	0.204667	0.102334	−	0.994750i	$-0.467369\pi$
$571$	5.19889e11	0.204667	0.102334	+	0.994750i	$0.467369\pi$
$572$	0	0
$573$	−5.94813e11	−0.230507
$574$	0	0
$575$	−1.01909e11	−0.0388782
$576$	0	0
$577$	−1.35855e12	−0.510252	−0.255126	−	0.966908i	$-0.582117\pi$
$577$	−1.35855e12	−0.510252	−0.255126	+	0.966908i	$0.582117\pi$
$578$	0	0
$579$	−6.68730e11	−0.247285
$580$	0	0
$581$	−2.91838e12	−1.06255
$582$	0	0
$583$	−1.98734e12	−0.712466
$584$	0	0
$585$	−7.81168e11	−0.275767
$586$	0	0
$587$	5.99152e11	0.208289	0.104144	−	0.994562i	$-0.466790\pi$
$587$	5.99152e11	0.208289	0.104144	+	0.994562i	$0.466790\pi$
$588$	0	0
$589$	7.43986e12	2.54710
$590$	0	0
$591$	−1.32549e11	−0.0446923
$592$	0	0
$593$	5.46699e12	1.81553	0.907763	−	0.419483i	$-0.137789\pi$
$593$	5.46699e12	1.81553	0.907763	+	0.419483i	$0.137789\pi$
$594$	0	0
$595$	−3.29327e11	−0.107721
$596$	0	0
$597$	−1.08013e12	−0.348010
$598$	0	0
$599$	−4.52056e12	−1.43473	−0.717367	−	0.696695i	$-0.754653\pi$
$599$	−4.52056e12	−1.43473	−0.717367	+	0.696695i	$0.754653\pi$
$600$	0	0
$601$	3.06316e12	0.957710	0.478855	−	0.877894i	$-0.341052\pi$
$601$	3.06316e12	0.957710	0.478855	+	0.877894i	$0.341052\pi$
$602$	0	0
$603$	−5.85937e12	−1.80477
$604$	0	0
$605$	−4.69548e11	−0.142489
$606$	0	0
$607$	5.44730e12	1.62867	0.814333	−	0.580398i	$-0.197103\pi$
$607$	5.44730e12	1.62867	0.814333	+	0.580398i	$0.197103\pi$
$608$	0	0
$609$	1.49446e11	0.0440256
$610$	0	0
$611$	−1.03153e11	−0.0299431
$612$	0	0
$613$	−4.89823e12	−1.40109	−0.700547	−	0.713607i	$-0.747060\pi$
$613$	−4.89823e12	−1.40109	−0.700547	+	0.713607i	$0.747060\pi$
$614$	0	0
$615$	−1.42286e10	−0.00401072
$616$	0	0
$617$	2.36830e11	0.0657890	0.0328945	−	0.999459i	$-0.489527\pi$
$617$	2.36830e11	0.0657890	0.0328945	+	0.999459i	$0.489527\pi$
$618$	0	0
$619$	1.58066e12	0.432743	0.216372	−	0.976311i	$-0.430578\pi$
$619$	1.58066e12	0.432743	0.216372	+	0.976311i	$0.430578\pi$
$620$	0	0
$621$	3.15754e11	0.0851996
$622$	0	0
$623$	1.51467e12	0.402831
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	0	0
$627$	−1.14028e12	−0.294652
$628$	0	0
$629$	−2.98383e11	−0.0760057
$630$	0	0
$631$	5.75117e12	1.44419	0.722094	−	0.691795i	$-0.243180\pi$
$631$	5.75117e12	1.44419	0.722094	+	0.691795i	$0.243180\pi$
$632$	0	0
$633$	−9.93709e11	−0.246004
$634$	0	0
$635$	−1.90085e12	−0.463943
$636$	0	0
$637$	−4.77616e11	−0.114935
$638$	0	0
$639$	−2.67402e12	−0.634469
$640$	0	0
$641$	5.42860e12	1.27007	0.635033	−	0.772485i	$-0.280986\pi$
$641$	5.42860e12	1.27007	0.635033	+	0.772485i	$0.280986\pi$
$642$	0	0
$643$	3.08107e12	0.710808	0.355404	−	0.934713i	$-0.384343\pi$
$643$	3.08107e12	0.710808	0.355404	+	0.934713i	$0.384343\pi$
$644$	0	0
$645$	9.29086e10	0.0211367
$646$	0	0
$647$	−7.88099e12	−1.76812	−0.884059	−	0.467375i	$-0.845200\pi$
$647$	−7.88099e12	−1.76812	−0.884059	+	0.467375i	$0.845200\pi$
$648$	0	0
$649$	−5.91699e12	−1.30918
$650$	0	0
$651$	−1.49952e12	−0.327219
$652$	0	0
$653$	−9.00773e12	−1.93868	−0.969339	−	0.245725i	$-0.920974\pi$
$653$	−9.00773e12	−1.93868	−0.969339	+	0.245725i	$0.920974\pi$
$654$	0	0
$655$	−5.50435e10	−0.0116848
$656$	0	0
$657$	1.40048e11	0.0293246
$658$	0	0
$659$	4.22272e12	0.872183	0.436091	−	0.899902i	$-0.356362\pi$
$659$	4.22272e12	0.872183	0.436091	+	0.899902i	$0.356362\pi$
$660$	0	0
$661$	−4.48023e12	−0.912837	−0.456419	−	0.889765i	$-0.650868\pi$
$661$	−4.48023e12	−0.912837	−0.456419	+	0.889765i	$0.650868\pi$
$662$	0	0
$663$	−1.92882e11	−0.0387686
$664$	0	0
$665$	−3.24851e12	−0.644151
$666$	0	0
$667$	−2.14482e11	−0.0419590
$668$	0	0
$669$	2.13491e12	0.412062
$670$	0	0
$671$	−5.26935e12	−1.00347
$672$	0	0
$673$	1.92638e12	0.361971	0.180986	−	0.983486i	$-0.442071\pi$
$673$	1.92638e12	0.361971	0.180986	+	0.983486i	$0.442071\pi$
$674$	0	0
$675$	−4.72779e11	−0.0876579
$676$	0	0
$677$	−9.32257e12	−1.70564	−0.852819	−	0.522207i	$-0.825109\pi$
$677$	−9.32257e12	−1.70564	−0.852819	+	0.522207i	$0.825109\pi$
$678$	0	0
$679$	7.70594e12	1.39127
$680$	0	0
$681$	7.33018e11	0.130603
$682$	0	0
$683$	4.38750e12	0.771478	0.385739	−	0.922608i	$-0.373946\pi$
$683$	4.38750e12	0.771478	0.385739	+	0.922608i	$0.373946\pi$
$684$	0	0
$685$	3.14976e12	0.546600
$686$	0	0
$687$	1.22298e12	0.209466
$688$	0	0
$689$	−3.31596e12	−0.560560
$690$	0	0
$691$	6.05838e12	1.01089	0.505447	−	0.862858i	$-0.331328\pi$
$691$	6.05838e12	1.01089	0.505447	+	0.862858i	$0.331328\pi$
$692$	0	0
$693$	−4.31696e12	−0.711014
$694$	0	0
$695$	2.29519e12	0.373153
$696$	0	0
$697$	6.59909e10	0.0105910
$698$	0	0
$699$	−1.18179e12	−0.187238
$700$	0	0
$701$	−2.15758e12	−0.337471	−0.168736	−	0.985661i	$-0.553968\pi$
$701$	−2.15758e12	−0.337471	−0.168736	+	0.985661i	$0.553968\pi$
$702$	0	0
$703$	−2.94328e12	−0.454499
$704$	0	0
$705$	−3.04058e10	−0.00463560
$706$	0	0
$707$	−9.19827e12	−1.38458
$708$	0	0
$709$	−2.60621e12	−0.387348	−0.193674	−	0.981066i	$-0.562040\pi$
$709$	−2.60621e12	−0.387348	−0.193674	+	0.981066i	$0.562040\pi$
$710$	0	0
$711$	−8.83614e11	−0.129673
$712$	0	0
$713$	2.15209e12	0.311859
$714$	0	0
$715$	1.67549e12	0.239754
$716$	0	0
$717$	−3.19759e11	−0.0451842
$718$	0	0
$719$	6.17622e12	0.861872	0.430936	−	0.902382i	$-0.358183\pi$
$719$	6.17622e12	0.861872	0.430936	+	0.902382i	$0.358183\pi$
$720$	0	0
$721$	2.76986e12	0.381723
$722$	0	0
$723$	−2.13691e12	−0.290847
$724$	0	0
$725$	3.21144e11	0.0431697
$726$	0	0
$727$	5.75173e11	0.0763649	0.0381825	−	0.999271i	$-0.487843\pi$
$727$	5.75173e11	0.0763649	0.0381825	+	0.999271i	$0.487843\pi$
$728$	0	0
$729$	−5.40932e12	−0.709363
$730$	0	0
$731$	−4.30903e11	−0.0558150
$732$	0	0
$733$	−1.41134e13	−1.80577	−0.902886	−	0.429880i	$-0.858556\pi$
$733$	−1.41134e13	−1.80577	−0.902886	+	0.429880i	$0.858556\pi$
$734$	0	0
$735$	−1.40784e11	−0.0177935
$736$	0	0
$737$	1.25675e13	1.56908
$738$	0	0
$739$	−7.30570e11	−0.0901077	−0.0450538	−	0.998985i	$-0.514346\pi$
$739$	−7.30570e11	−0.0901077	−0.0450538	+	0.998985i	$0.514346\pi$
$740$	0	0
$741$	−1.90261e12	−0.231829
$742$	0	0
$743$	1.38765e12	0.167044	0.0835222	−	0.996506i	$-0.473383\pi$
$743$	1.38765e12	0.167044	0.0835222	+	0.996506i	$0.473383\pi$
$744$	0	0
$745$	−5.94228e12	−0.706724
$746$	0	0
$747$	9.46363e12	1.11203
$748$	0	0
$749$	2.18386e11	0.0253546
$750$	0	0
$751$	−3.54749e12	−0.406951	−0.203475	−	0.979080i	$-0.565224\pi$
$751$	−3.54749e12	−0.406951	−0.203475	+	0.979080i	$0.565224\pi$
$752$	0	0
$753$	4.72846e10	0.00535972
$754$	0	0
$755$	3.45942e12	0.387474
$756$	0	0
$757$	−3.77008e11	−0.0417272	−0.0208636	−	0.999782i	$-0.506642\pi$
$757$	−3.77008e11	−0.0417272	−0.0208636	+	0.999782i	$0.506642\pi$
$758$	0	0
$759$	−3.29844e11	−0.0360762
$760$	0	0
$761$	7.57515e12	0.818767	0.409383	−	0.912362i	$-0.365744\pi$
$761$	7.57515e12	0.818767	0.409383	+	0.912362i	$0.365744\pi$
$762$	0	0
$763$	−1.01159e13	−1.08055
$764$	0	0
$765$	1.06793e12	0.112737
$766$	0	0
$767$	−9.87272e12	−1.03005
$768$	0	0
$769$	−6.31842e12	−0.651539	−0.325769	−	0.945449i	$-0.605623\pi$
$769$	−6.31842e12	−0.651539	−0.325769	+	0.945449i	$0.605623\pi$
$770$	0	0
$771$	−2.75594e12	−0.280883
$772$	0	0
$773$	−1.62451e13	−1.63650	−0.818249	−	0.574864i	$-0.805055\pi$
$773$	−1.62451e13	−1.63650	−0.818249	+	0.574864i	$0.805055\pi$
$774$	0	0
$775$	−3.22233e12	−0.320857
$776$	0	0
$777$	5.93226e11	0.0583883
$778$	0	0
$779$	6.50942e11	0.0633321
$780$	0	0
$781$	5.73539e12	0.551612
$782$	0	0
$783$	−9.95033e11	−0.0946040
$784$	0	0
$785$	−2.03828e12	−0.191580
$786$	0	0
$787$	−6.25892e12	−0.581585	−0.290793	−	0.956786i	$-0.593919\pi$
$787$	−6.25892e12	−0.581585	−0.290793	+	0.956786i	$0.593919\pi$
$788$	0	0
$789$	−2.57229e11	−0.0236305
$790$	0	0
$791$	5.15474e12	0.468180
$792$	0	0
$793$	−8.79211e12	−0.789521
$794$	0	0
$795$	−9.77425e11	−0.0867823
$796$	0	0
$797$	1.35039e12	0.118549	0.0592744	−	0.998242i	$-0.481121\pi$
$797$	1.35039e12	0.118549	0.0592744	+	0.998242i	$0.481121\pi$
$798$	0	0
$799$	1.41020e11	0.0122411
$800$	0	0
$801$	−4.91173e12	−0.421588
$802$	0	0
$803$	−3.00383e11	−0.0254950
$804$	0	0
$805$	−9.39682e11	−0.0788677
$806$	0	0
$807$	−1.30095e11	−0.0107977
$808$	0	0
$809$	−1.96020e13	−1.60891	−0.804455	−	0.594014i	$-0.797543\pi$
$809$	−1.96020e13	−1.60891	−0.804455	+	0.594014i	$0.797543\pi$
$810$	0	0
$811$	−1.40454e13	−1.14009	−0.570046	−	0.821612i	$-0.693075\pi$
$811$	−1.40454e13	−1.14009	−0.570046	+	0.821612i	$0.693075\pi$
$812$	0	0
$813$	−1.40347e12	−0.112667
$814$	0	0
$815$	3.68956e12	0.292931
$816$	0	0
$817$	−4.25047e12	−0.333763
$818$	0	0
$819$	−7.20300e12	−0.559417
$820$	0	0
$821$	−6.06800e12	−0.466124	−0.233062	−	0.972462i	$-0.574875\pi$
$821$	−6.06800e12	−0.466124	−0.233062	+	0.972462i	$0.574875\pi$
$822$	0	0
$823$	−1.46415e13	−1.11246	−0.556232	−	0.831027i	$-0.687753\pi$
$823$	−1.46415e13	−1.11246	−0.556232	+	0.831027i	$0.687753\pi$
$824$	0	0
$825$	4.93876e11	0.0371172
$826$	0	0
$827$	1.85325e13	1.37772	0.688859	−	0.724895i	$-0.258112\pi$
$827$	1.85325e13	1.37772	0.688859	+	0.724895i	$0.258112\pi$
$828$	0	0
$829$	1.09144e13	0.802612	0.401306	−	0.915944i	$-0.368556\pi$
$829$	1.09144e13	0.802612	0.401306	+	0.915944i	$0.368556\pi$
$830$	0	0
$831$	−3.07185e12	−0.223458
$832$	0	0
$833$	6.52945e11	0.0469866
$834$	0	0
$835$	8.81860e12	0.627784
$836$	0	0
$837$	9.98407e12	0.703142
$838$	0	0
$839$	−2.71319e12	−0.189039	−0.0945196	−	0.995523i	$-0.530132\pi$
$839$	−2.71319e12	−0.189039	−0.0945196	+	0.995523i	$0.530132\pi$
$840$	0	0
$841$	−1.38313e13	−0.953410
$842$	0	0
$843$	−4.35702e12	−0.297143
$844$	0	0
$845$	−3.83219e12	−0.258578
$846$	0	0
$847$	−4.32961e12	−0.289050
$848$	0	0
$849$	−4.55915e12	−0.301161
$850$	0	0
$851$	−8.51389e11	−0.0556474
$852$	0	0
$853$	−1.54689e13	−1.00043	−0.500217	−	0.865900i	$-0.666747\pi$
$853$	−1.54689e13	−1.00043	−0.500217	+	0.865900i	$0.666747\pi$
$854$	0	0
$855$	1.05342e13	0.674144
$856$	0	0
$857$	−6.70023e12	−0.424303	−0.212151	−	0.977237i	$-0.568047\pi$
$857$	−6.70023e12	−0.424303	−0.212151	+	0.977237i	$0.568047\pi$
$858$	0	0
$859$	1.12634e13	0.705832	0.352916	−	0.935655i	$-0.385190\pi$
$859$	1.12634e13	0.705832	0.352916	+	0.935655i	$0.385190\pi$
$860$	0	0
$861$	−1.31199e11	−0.00813610
$862$	0	0
$863$	2.44794e13	1.50229	0.751144	−	0.660139i	$-0.229503\pi$
$863$	2.44794e13	1.50229	0.751144	+	0.660139i	$0.229503\pi$
$864$	0	0
$865$	−5.42336e12	−0.329379
$866$	0	0
$867$	−3.47685e12	−0.208978
$868$	0	0
$869$	1.89523e12	0.112739
$870$	0	0
$871$	2.09694e13	1.23454
$872$	0	0
$873$	−2.49885e13	−1.45605
$874$	0	0
$875$	1.40698e12	0.0811434
$876$	0	0
$877$	2.25259e13	1.28583	0.642914	−	0.765938i	$-0.277725\pi$
$877$	2.25259e13	1.28583	0.642914	+	0.765938i	$0.277725\pi$
$878$	0	0
$879$	−2.93592e12	−0.165880
$880$	0	0
$881$	2.29048e13	1.28096	0.640480	−	0.767975i	$-0.278736\pi$
$881$	2.29048e13	1.28096	0.640480	+	0.767975i	$0.278736\pi$
$882$	0	0
$883$	2.33032e13	1.29001	0.645005	−	0.764178i	$-0.276855\pi$
$883$	2.33032e13	1.29001	0.645005	+	0.764178i	$0.276855\pi$
$884$	0	0
$885$	−2.91012e12	−0.159465
$886$	0	0
$887$	3.30975e13	1.79531	0.897655	−	0.440699i	$-0.145269\pi$
$887$	3.30975e13	1.79531	0.897655	+	0.440699i	$0.145269\pi$
$888$	0	0
$889$	−1.75273e13	−0.941149
$890$	0	0
$891$	1.32139e13	0.702396
$892$	0	0
$893$	1.39104e12	0.0731993
$894$	0	0
$895$	−1.24891e13	−0.650621
$896$	0	0
$897$	−5.50358e11	−0.0283844
$898$	0	0
$899$	−6.78186e12	−0.346282
$900$	0	0
$901$	4.53322e12	0.229163
$902$	0	0
$903$	8.56693e11	0.0428776
$904$	0	0
$905$	−6.85943e12	−0.339914
$906$	0	0
$907$	−2.75737e13	−1.35289	−0.676445	−	0.736493i	$-0.736480\pi$
$907$	−2.75737e13	−1.35289	−0.676445	+	0.736493i	$0.736480\pi$
$908$	0	0
$909$	2.98278e13	1.44905
$910$	0	0
$911$	−2.32541e13	−1.11858	−0.559291	−	0.828972i	$-0.688926\pi$
$911$	−2.32541e13	−1.11858	−0.559291	+	0.828972i	$0.688926\pi$
$912$	0	0
$913$	−2.02982e13	−0.966803
$914$	0	0
$915$	−2.59160e12	−0.122229
$916$	0	0
$917$	−5.07546e11	−0.0237036
$918$	0	0
$919$	−2.51992e13	−1.16538	−0.582689	−	0.812695i	$-0.697999\pi$
$919$	−2.51992e13	−1.16538	−0.582689	+	0.812695i	$0.697999\pi$
$920$	0	0
$921$	−2.78589e12	−0.127584
$922$	0	0
$923$	9.56972e12	0.434002
$924$	0	0
$925$	1.27478e12	0.0572531
$926$	0	0
$927$	−8.98200e12	−0.399498
$928$	0	0
$929$	3.57780e13	1.57596	0.787980	−	0.615701i	$-0.211127\pi$
$929$	3.57780e13	1.57596	0.787980	+	0.615701i	$0.211127\pi$
$930$	0	0
$931$	6.44072e12	0.280971
$932$	0	0
$933$	1.38774e12	0.0599570
$934$	0	0
$935$	−2.29056e12	−0.0980141
$936$	0	0
$937$	−2.23802e13	−0.948498	−0.474249	−	0.880391i	$-0.657280\pi$
$937$	−2.23802e13	−0.948498	−0.474249	+	0.880391i	$0.657280\pi$
$938$	0	0
$939$	−3.01190e12	−0.126429
$940$	0	0
$941$	3.36203e13	1.39781	0.698906	−	0.715214i	$-0.253671\pi$
$941$	3.36203e13	1.39781	0.698906	+	0.715214i	$0.253671\pi$
$942$	0	0
$943$	1.88295e11	0.00775418
$944$	0	0
$945$	−4.35941e12	−0.177821
$946$	0	0
$947$	−1.09703e13	−0.443245	−0.221622	−	0.975133i	$-0.571135\pi$
$947$	−1.09703e13	−0.443245	−0.221622	+	0.975133i	$0.571135\pi$
$948$	0	0
$949$	−5.01201e11	−0.0200592
$950$	0	0
$951$	−3.33497e12	−0.132215
$952$	0	0
$953$	2.40941e12	0.0946220	0.0473110	−	0.998880i	$-0.484935\pi$
$953$	2.40941e12	0.0946220	0.0473110	+	0.998880i	$0.484935\pi$
$954$	0	0
$955$	−1.17860e13	−0.458512
$956$	0	0
$957$	1.03943e12	0.0400584
$958$	0	0
$959$	2.90434e13	1.10883
$960$	0	0
$961$	4.16089e13	1.57373
$962$	0	0
$963$	−7.08175e11	−0.0265352
$964$	0	0
$965$	−1.32506e13	−0.491885
$966$	0	0
$967$	−4.16349e13	−1.53122	−0.765611	−	0.643304i	$-0.777563\pi$
$967$	−4.16349e13	−1.53122	−0.765611	+	0.643304i	$0.777563\pi$
$968$	0	0
$969$	2.60104e12	0.0947742
$970$	0	0
$971$	2.19480e13	0.792334	0.396167	−	0.918178i	$-0.370340\pi$
$971$	2.19480e13	0.792334	0.396167	+	0.918178i	$0.370340\pi$
$972$	0	0
$973$	2.11635e13	0.756972
$974$	0	0
$975$	8.24050e11	0.0292034
$976$	0	0
$977$	5.29108e13	1.85788	0.928942	−	0.370225i	$-0.120719\pi$
$977$	5.29108e13	1.85788	0.928942	+	0.370225i	$0.120719\pi$
$978$	0	0
$979$	1.05350e13	0.366531
$980$	0	0
$981$	3.28036e13	1.13086
$982$	0	0
$983$	−4.37280e13	−1.49372	−0.746860	−	0.664981i	$-0.768440\pi$
$983$	−4.37280e13	−1.49372	−0.746860	+	0.664981i	$0.768440\pi$
$984$	0	0
$985$	−2.62641e12	−0.0888995
$986$	0	0
$987$	−2.80366e11	−0.00940370
$988$	0	0
$989$	−1.22951e12	−0.0408649
$990$	0	0
$991$	5.51408e13	1.81611	0.908055	−	0.418852i	$-0.137567\pi$
$991$	5.51408e13	1.81611	0.908055	+	0.418852i	$0.137567\pi$
$992$	0	0
$993$	−1.18120e12	−0.0385524
$994$	0	0
$995$	−2.14024e13	−0.692242
$996$	0	0
$997$	−1.52889e13	−0.490060	−0.245030	−	0.969516i	$-0.578798\pi$
$997$	−1.52889e13	−0.490060	−0.245030	+	0.969516i	$0.578798\pi$
$998$	0	0
$999$	−3.94980e12	−0.125467

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.10.a.z.1.3		4
4.3	odd	2	inner	320.10.a.z.1.2		4
8.3	odd	2		160.10.a.c.1.3	yes	4
8.5	even	2		160.10.a.c.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.10.a.c.1.2	✓	4	8.5	even	2
160.10.a.c.1.3	yes	4	8.3	odd	2
320.10.a.z.1.2		4	4.3	odd	2	inner
320.10.a.z.1.3		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	320.a (trivial)

\(f(q)\)	\(=\)	\(q+31.5423 q^{3} +625.000 q^{5} +5763.01 q^{7} -18688.1 q^{9} +40083.3 q^{11} +66880.5 q^{13} +19714.0 q^{15} -91431.8 q^{17} -901894. q^{19} +181779. q^{21} -260886. q^{23} +390625. q^{25} -1.21031e6 q^{27} +822128. q^{29} -8.24915e6 q^{31} +1.26432e6 q^{33} +3.60188e6 q^{35} +3.26345e6 q^{37} +2.10957e6 q^{39} -721750. q^{41} +4.71283e6 q^{43} -1.16801e7 q^{45} -1.54235e6 q^{47} -7.14133e6 q^{49} -2.88397e6 q^{51} -4.95803e7 q^{53} +2.50521e7 q^{55} -2.84479e7 q^{57} -1.47617e8 q^{59} -1.31460e8 q^{61} -1.07700e8 q^{63} +4.18003e7 q^{65} +3.13535e8 q^{67} -8.22897e6 q^{69} +1.43087e8 q^{71} -7.49397e6 q^{73} +1.23212e7 q^{75} +2.31000e8 q^{77} +4.72822e7 q^{79} +3.29661e8 q^{81} -5.06399e8 q^{83} -5.71449e7 q^{85} +2.59319e7 q^{87} +2.62827e8 q^{89} +3.85433e8 q^{91} -2.60198e8 q^{93} -5.63684e8 q^{95} +1.33714e9 q^{97} -7.49080e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2500 q^{5} + 23364 q^{9} - 201256 q^{13} - 442040 q^{17} + 3496176 q^{21} + 1562500 q^{25} - 6719352 q^{29} - 36991008 q^{33} - 29092616 q^{37} + 49561624 q^{41} + 14602500 q^{45} + 3039028 q^{49}+ \cdots + 464919272 q^{97}+O(q^{100}) \) 4 * q + 2500 * q^5 + 23364 * q^9 - 201256 * q^13 - 442040 * q^17 + 3496176 * q^21 + 1562500 * q^25 - 6719352 * q^29 - 36991008 * q^33 - 29092616 * q^37 + 49561624 * q^41 + 14602500 * q^45 + 3039028 * q^49 - 178752456 * q^53 + 127411200 * q^57 + 35996024 * q^61 - 125785000 * q^65 - 222978096 * q^69 + 31085160 * q^73 - 774686048 * q^77 + 533616660 * q^81 - 276275000 * q^85 + 2529920360 * q^89 - 2297246688 * q^93 + 464919272 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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