LMFDB - Embedded newform 320.10.a.z.1.4

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			1.4
Root			$-18.1438$ 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)

$f(q)$	$=$	$q+223.725 q^{3} +625.000 q^{5} +7001.03 q^{7} +30370.1 q^{9} -88321.8 q^{11} -167508. q^{13} +139828. q^{15} -129588. q^{17} +411904. q^{19} +1.56631e6 q^{21} -461548. q^{23} +390625. q^{25} +2.39097e6 q^{27} -4.18180e6 q^{29} -3.97105e6 q^{31} -1.97598e7 q^{33} +4.37565e6 q^{35} -1.78098e7 q^{37} -3.74759e7 q^{39} +2.55026e7 q^{41} -4.98702e6 q^{43} +1.89813e7 q^{45} -6.37511e7 q^{47} +8.66085e6 q^{49} -2.89922e7 q^{51} -3.97959e7 q^{53} -5.52011e7 q^{55} +9.21535e7 q^{57} +3.20652e7 q^{59} +1.49458e8 q^{61} +2.12622e8 q^{63} -1.04693e8 q^{65} -1.68066e8 q^{67} -1.03260e8 q^{69} -9.59405e7 q^{71} +2.30366e7 q^{73} +8.73928e7 q^{75} -6.18343e8 q^{77} +1.24934e8 q^{79} -6.28530e7 q^{81} -5.23811e8 q^{83} -8.09926e7 q^{85} -9.35576e8 q^{87} +1.00213e9 q^{89} -1.17273e9 q^{91} -8.88426e8 q^{93} +2.57440e8 q^{95} -1.10468e9 q^{97} -2.68234e9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	223.725	1.59467	0.797333	−	0.603540i	$-0.206243\pi$
$3$	223.725	1.59467	0.797333	+	0.603540i	$0.206243\pi$
$4$	0	0
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	7001.03	1.10210	0.551050	−	0.834472i	$-0.314227\pi$
$7$	7001.03	1.10210	0.551050	+	0.834472i	$0.314227\pi$
$8$	0	0
$9$	30370.1	1.54296
$10$	0	0
$11$	−88321.8	−1.81887	−0.909433	−	0.415851i	$-0.863484\pi$
$11$	−88321.8	−1.81887	−0.909433	+	0.415851i	$0.863484\pi$
$12$	0	0
$13$	−167508.	−1.62664	−0.813321	−	0.581816i	$-0.802342\pi$
$13$	−167508.	−1.62664	−0.813321	+	0.581816i	$0.802342\pi$
$14$	0	0
$15$	139828.	0.713156
$16$	0	0
$17$	−129588.	−0.376310	−0.188155	−	0.982139i	$-0.560251\pi$
$17$	−129588.	−0.376310	−0.188155	+	0.982139i	$0.560251\pi$
$18$	0	0
$19$	411904.	0.725112	0.362556	−	0.931962i	$-0.381904\pi$
$19$	411904.	0.725112	0.362556	+	0.931962i	$0.381904\pi$
$20$	0	0
$21$	1.56631e6	1.75748
$22$	0	0
$23$	−461548.	−0.343908	−0.171954	−	0.985105i	$-0.555008\pi$
$23$	−461548.	−0.343908	−0.171954	+	0.985105i	$0.555008\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	0	0
$27$	2.39097e6	0.865840
$28$	0	0
$29$	−4.18180e6	−1.09793	−0.548963	−	0.835847i	$-0.684977\pi$
$29$	−4.18180e6	−1.09793	−0.548963	+	0.835847i	$0.684977\pi$
$30$	0	0
$31$	−3.97105e6	−0.772286	−0.386143	−	0.922439i	$-0.626193\pi$
$31$	−3.97105e6	−0.772286	−0.386143	+	0.922439i	$0.626193\pi$
$32$	0	0
$33$	−1.97598e7	−2.90048
$34$	0	0
$35$	4.37565e6	0.492874
$36$	0	0
$37$	−1.78098e7	−1.56225	−0.781124	−	0.624375i	$-0.785354\pi$
$37$	−1.78098e7	−1.56225	−0.781124	+	0.624375i	$0.785354\pi$
$38$	0	0
$39$	−3.74759e7	−2.59395
$40$	0	0
$41$	2.55026e7	1.40947	0.704736	−	0.709470i	$-0.251065\pi$
$41$	2.55026e7	1.40947	0.704736	+	0.709470i	$0.251065\pi$
$42$	0	0
$43$	−4.98702e6	−0.222450	−0.111225	−	0.993795i	$-0.535477\pi$
$43$	−4.98702e6	−0.222450	−0.111225	+	0.993795i	$0.535477\pi$
$44$	0	0
$45$	1.89813e7	0.690033
$46$	0	0
$47$	−6.37511e7	−1.90567	−0.952834	−	0.303492i	$-0.901847\pi$
$47$	−6.37511e7	−1.90567	−0.952834	+	0.303492i	$0.901847\pi$
$48$	0	0
$49$	8.66085e6	0.214624
$50$	0	0
$51$	−2.89922e7	−0.600088
$52$	0	0
$53$	−3.97959e7	−0.692783	−0.346391	−	0.938090i	$-0.612593\pi$
$53$	−3.97959e7	−0.692783	−0.346391	+	0.938090i	$0.612593\pi$
$54$	0	0
$55$	−5.52011e7	−0.813421
$56$	0	0
$57$	9.21535e7	1.15631
$58$	0	0
$59$	3.20652e7	0.344508	0.172254	−	0.985053i	$-0.444895\pi$
$59$	3.20652e7	0.344508	0.172254	+	0.985053i	$0.444895\pi$
$60$	0	0
$61$	1.49458e8	1.38209	0.691043	−	0.722814i	$-0.257152\pi$
$61$	1.49458e8	1.38209	0.691043	+	0.722814i	$0.257152\pi$
$62$	0	0
$63$	2.12622e8	1.70050
$64$	0	0
$65$	−1.04693e8	−0.727456
$66$	0	0
$67$	−1.68066e8	−1.01893	−0.509463	−	0.860492i	$-0.670156\pi$
$67$	−1.68066e8	−1.01893	−0.509463	+	0.860492i	$0.670156\pi$
$68$	0	0
$69$	−1.03260e8	−0.548418
$70$	0	0
$71$	−9.59405e7	−0.448063	−0.224032	−	0.974582i	$-0.571922\pi$
$71$	−9.59405e7	−0.448063	−0.224032	+	0.974582i	$0.571922\pi$
$72$	0	0
$73$	2.30366e7	0.0949434	0.0474717	−	0.998873i	$-0.484884\pi$
$73$	2.30366e7	0.0949434	0.0474717	+	0.998873i	$0.484884\pi$
$74$	0	0
$75$	8.73928e7	0.318933
$76$	0	0
$77$	−6.18343e8	−2.00457
$78$	0	0
$79$	1.24934e8	0.360876	0.180438	−	0.983586i	$-0.442249\pi$
$79$	1.24934e8	0.360876	0.180438	+	0.983586i	$0.442249\pi$
$80$	0	0
$81$	−6.28530e7	−0.162235
$82$	0	0
$83$	−5.23811e8	−1.21150	−0.605750	−	0.795655i	$-0.707127\pi$
$83$	−5.23811e8	−1.21150	−0.605750	+	0.795655i	$0.707127\pi$
$84$	0	0
$85$	−8.09926e7	−0.168291
$86$	0	0
$87$	−9.35576e8	−1.75082
$88$	0	0
$89$	1.00213e9	1.69305	0.846526	−	0.532347i	$-0.178690\pi$
$89$	1.00213e9	1.69305	0.846526	+	0.532347i	$0.178690\pi$
$90$	0	0
$91$	−1.17273e9	−1.79272
$92$	0	0
$93$	−8.88426e8	−1.23154
$94$	0	0
$95$	2.57440e8	0.324280
$96$	0	0
$97$	−1.10468e9	−1.26696	−0.633480	−	0.773759i	$-0.718374\pi$
$97$	−1.10468e9	−1.26696	−0.633480	+	0.773759i	$0.718374\pi$
$98$	0	0
$99$	−2.68234e9	−2.80644
$100$	0	0
$101$	1.06499e9	1.01836	0.509178	−	0.860662i	$-0.329950\pi$
$101$	1.06499e9	1.01836	0.509178	+	0.860662i	$0.329950\pi$
$102$	0	0
$103$	6.97414e8	0.610553	0.305276	−	0.952264i	$-0.401251\pi$
$103$	6.97414e8	0.610553	0.305276	+	0.952264i	$0.401251\pi$
$104$	0	0
$105$	9.78943e8	0.785969
$106$	0	0
$107$	1.06119e9	0.782646	0.391323	−	0.920253i	$-0.372017\pi$
$107$	1.06119e9	0.782646	0.391323	+	0.920253i	$0.372017\pi$
$108$	0	0
$109$	−1.49799e8	−0.101646	−0.0508228	−	0.998708i	$-0.516184\pi$
$109$	−1.49799e8	−0.101646	−0.0508228	+	0.998708i	$0.516184\pi$
$110$	0	0
$111$	−3.98450e9	−2.49127
$112$	0	0
$113$	3.37225e9	1.94566	0.972830	−	0.231523i	$-0.0743707\pi$
$113$	3.37225e9	1.94566	0.972830	+	0.231523i	$0.0743707\pi$
$114$	0	0
$115$	−2.88468e8	−0.153800
$116$	0	0
$117$	−5.08725e9	−2.50984
$118$	0	0
$119$	−9.07251e8	−0.414731
$120$	0	0
$121$	5.44279e9	2.30827
$122$	0	0
$123$	5.70557e9	2.24764
$124$	0	0
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	−2.92465e9	−0.997602	−0.498801	−	0.866717i	$-0.666226\pi$
$127$	−2.92465e9	−0.997602	−0.498801	+	0.866717i	$0.666226\pi$
$128$	0	0
$129$	−1.11572e9	−0.354734
$130$	0	0
$131$	4.10577e9	1.21807	0.609037	−	0.793142i	$-0.291556\pi$
$131$	4.10577e9	1.21807	0.609037	+	0.793142i	$0.291556\pi$
$132$	0	0
$133$	2.88375e9	0.799146
$134$	0	0
$135$	1.49436e9	0.387215
$136$	0	0
$137$	−5.30030e9	−1.28546	−0.642728	−	0.766094i	$-0.722198\pi$
$137$	−5.30030e9	−1.28546	−0.642728	+	0.766094i	$0.722198\pi$
$138$	0	0
$139$	2.26913e9	0.515577	0.257789	−	0.966201i	$-0.417006\pi$
$139$	2.26913e9	0.515577	0.257789	+	0.966201i	$0.417006\pi$
$140$	0	0
$141$	−1.42627e10	−3.03890
$142$	0	0
$143$	1.47946e10	2.95864
$144$	0	0
$145$	−2.61363e9	−0.491007
$146$	0	0
$147$	1.93765e9	0.342253
$148$	0	0
$149$	−3.02575e9	−0.502916	−0.251458	−	0.967868i	$-0.580910\pi$
$149$	−3.02575e9	−0.502916	−0.251458	+	0.967868i	$0.580910\pi$
$150$	0	0
$151$	−1.41916e9	−0.222145	−0.111072	−	0.993812i	$-0.535429\pi$
$151$	−1.41916e9	−0.222145	−0.111072	+	0.993812i	$0.535429\pi$
$152$	0	0
$153$	−3.93560e9	−0.580631
$154$	0	0
$155$	−2.48191e9	−0.345377
$156$	0	0
$157$	−8.44736e8	−0.110962	−0.0554808	−	0.998460i	$-0.517669\pi$
$157$	−8.44736e8	−0.110962	−0.0554808	+	0.998460i	$0.517669\pi$
$158$	0	0
$159$	−8.90336e9	−1.10476
$160$	0	0
$161$	−3.23131e9	−0.379020
$162$	0	0
$163$	−1.31072e10	−1.45434	−0.727171	−	0.686457i	$-0.759165\pi$
$163$	−1.31072e10	−1.45434	−0.727171	+	0.686457i	$0.759165\pi$
$164$	0	0
$165$	−1.23499e10	−1.29714
$166$	0	0
$167$	1.89972e10	1.89001	0.945007	−	0.327051i	$-0.106055\pi$
$167$	1.89972e10	1.89001	0.945007	+	0.327051i	$0.106055\pi$
$168$	0	0
$169$	1.74546e10	1.64596
$170$	0	0
$171$	1.25096e10	1.11882
$172$	0	0
$173$	−1.43419e10	−1.21731	−0.608653	−	0.793436i	$-0.708290\pi$
$173$	−1.43419e10	−1.21731	−0.608653	+	0.793436i	$0.708290\pi$
$174$	0	0
$175$	2.73478e9	0.220420
$176$	0	0
$177$	7.17379e9	0.549375
$178$	0	0
$179$	8.34282e9	0.607399	0.303700	−	0.952768i	$-0.401778\pi$
$179$	8.34282e9	0.607399	0.303700	+	0.952768i	$0.401778\pi$
$180$	0	0
$181$	6.84351e9	0.473942	0.236971	−	0.971517i	$-0.423845\pi$
$181$	6.84351e9	0.473942	0.236971	+	0.971517i	$0.423845\pi$
$182$	0	0
$183$	3.34376e10	2.20396
$184$	0	0
$185$	−1.11311e10	−0.698659
$186$	0	0
$187$	1.14455e10	0.684456
$188$	0	0
$189$	1.67393e10	0.954242
$190$	0	0
$191$	6.44325e9	0.350312	0.175156	−	0.984541i	$-0.443957\pi$
$191$	6.44325e9	0.350312	0.175156	+	0.984541i	$0.443957\pi$
$192$	0	0
$193$	−4.18680e9	−0.217207	−0.108604	−	0.994085i	$-0.534638\pi$
$193$	−4.18680e9	−0.217207	−0.108604	+	0.994085i	$0.534638\pi$
$194$	0	0
$195$	−2.34224e10	−1.16005
$196$	0	0
$197$	−1.94307e10	−0.919161	−0.459580	−	0.888136i	$-0.652000\pi$
$197$	−1.94307e10	−0.919161	−0.459580	+	0.888136i	$0.652000\pi$
$198$	0	0
$199$	−2.45813e10	−1.11113	−0.555566	−	0.831473i	$-0.687498\pi$
$199$	−2.45813e10	−1.11113	−0.555566	+	0.831473i	$0.687498\pi$
$200$	0	0
$201$	−3.76006e10	−1.62485
$202$	0	0
$203$	−2.92769e10	−1.21002
$204$	0	0
$205$	1.59391e10	0.630335
$206$	0	0
$207$	−1.40173e10	−0.530636
$208$	0	0
$209$	−3.63801e10	−1.31888
$210$	0	0
$211$	4.32159e10	1.50097	0.750486	−	0.660886i	$-0.229819\pi$
$211$	4.32159e10	1.50097	0.750486	+	0.660886i	$0.229819\pi$
$212$	0	0
$213$	−2.14643e10	−0.714511
$214$	0	0
$215$	−3.11688e9	−0.0994827
$216$	0	0
$217$	−2.78015e10	−0.851136
$218$	0	0
$219$	5.15386e9	0.151403
$220$	0	0
$221$	2.17071e10	0.612121
$222$	0	0
$223$	−7.20262e9	−0.195038	−0.0975189	−	0.995234i	$-0.531091\pi$
$223$	−7.20262e9	−0.195038	−0.0975189	+	0.995234i	$0.531091\pi$
$224$	0	0
$225$	1.18633e10	0.308592
$226$	0	0
$227$	1.80040e10	0.450040	0.225020	−	0.974354i	$-0.427755\pi$
$227$	1.80040e10	0.450040	0.225020	+	0.974354i	$0.427755\pi$
$228$	0	0
$229$	−3.33485e10	−0.801339	−0.400669	−	0.916223i	$-0.631222\pi$
$229$	−3.33485e10	−0.801339	−0.400669	+	0.916223i	$0.631222\pi$
$230$	0	0
$231$	−1.38339e11	−3.19662
$232$	0	0
$233$	−4.41281e9	−0.0980876	−0.0490438	−	0.998797i	$-0.515617\pi$
$233$	−4.41281e9	−0.0980876	−0.0490438	+	0.998797i	$0.515617\pi$
$234$	0	0
$235$	−3.98444e10	−0.852240
$236$	0	0
$237$	2.79508e10	0.575476
$238$	0	0
$239$	6.84172e10	1.35636	0.678179	−	0.734897i	$-0.262769\pi$
$239$	6.84172e10	1.35636	0.678179	+	0.734897i	$0.262769\pi$
$240$	0	0
$241$	−9.25286e10	−1.76685	−0.883425	−	0.468573i	$-0.844768\pi$
$241$	−9.25286e10	−1.76685	−0.883425	+	0.468573i	$0.844768\pi$
$242$	0	0
$243$	−6.11233e10	−1.12455
$244$	0	0
$245$	5.41303e9	0.0959827
$246$	0	0
$247$	−6.89975e10	−1.17950
$248$	0	0
$249$	−1.17190e11	−1.93194
$250$	0	0
$251$	5.74375e10	0.913406	0.456703	−	0.889619i	$-0.349030\pi$
$251$	5.74375e10	0.913406	0.456703	+	0.889619i	$0.349030\pi$
$252$	0	0
$253$	4.07647e10	0.625522
$254$	0	0
$255$	−1.81201e10	−0.268368
$256$	0	0
$257$	1.69810e10	0.242808	0.121404	−	0.992603i	$-0.461260\pi$
$257$	1.69810e10	0.242808	0.121404	+	0.992603i	$0.461260\pi$
$258$	0	0
$259$	−1.24687e11	−1.72175
$260$	0	0
$261$	−1.27002e11	−1.69405
$262$	0	0
$263$	4.13075e9	0.0532388	0.0266194	−	0.999646i	$-0.491526\pi$
$263$	4.13075e9	0.0532388	0.0266194	+	0.999646i	$0.491526\pi$
$264$	0	0
$265$	−2.48725e10	−0.309822
$266$	0	0
$267$	2.24203e11	2.69985
$268$	0	0
$269$	−9.35217e10	−1.08900	−0.544499	−	0.838761i	$-0.683280\pi$
$269$	−9.35217e10	−1.08900	−0.544499	+	0.838761i	$0.683280\pi$
$270$	0	0
$271$	2.54785e10	0.286954	0.143477	−	0.989654i	$-0.454172\pi$
$271$	2.54785e10	0.286954	0.143477	+	0.989654i	$0.454172\pi$
$272$	0	0
$273$	−2.62370e11	−2.85879
$274$	0	0
$275$	−3.45007e10	−0.363773
$276$	0	0
$277$	5.24035e10	0.534813	0.267406	−	0.963584i	$-0.413833\pi$
$277$	5.24035e10	0.534813	0.267406	+	0.963584i	$0.413833\pi$
$278$	0	0
$279$	−1.20601e11	−1.19161
$280$	0	0
$281$	9.33973e10	0.893626	0.446813	−	0.894627i	$-0.352559\pi$
$281$	9.33973e10	0.893626	0.446813	+	0.894627i	$0.352559\pi$
$282$	0	0
$283$	−1.32091e11	−1.22415	−0.612073	−	0.790801i	$-0.709664\pi$
$283$	−1.32091e11	−1.22415	−0.612073	+	0.790801i	$0.709664\pi$
$284$	0	0
$285$	5.75959e10	0.517118
$286$	0	0
$287$	1.78544e11	1.55338
$288$	0	0
$289$	−1.01795e11	−0.858391
$290$	0	0
$291$	−2.47145e11	−2.02038
$292$	0	0
$293$	−6.94338e10	−0.550385	−0.275193	−	0.961389i	$-0.588742\pi$
$293$	−6.94338e10	−0.550385	−0.275193	+	0.961389i	$0.588742\pi$
$294$	0	0
$295$	2.00407e10	0.154069
$296$	0	0
$297$	−2.11175e11	−1.57485
$298$	0	0
$299$	7.73132e10	0.559414
$300$	0	0
$301$	−3.49143e10	−0.245162
$302$	0	0
$303$	2.38265e11	1.62394
$304$	0	0
$305$	9.34112e10	0.618087
$306$	0	0
$307$	−1.68034e11	−1.07963	−0.539814	−	0.841784i	$-0.681505\pi$
$307$	−1.68034e11	−1.07963	−0.539814	+	0.841784i	$0.681505\pi$
$308$	0	0
$309$	1.56029e11	0.973628
$310$	0	0
$311$	−2.14855e11	−1.30234	−0.651169	−	0.758932i	$-0.725721\pi$
$311$	−2.14855e11	−1.30234	−0.651169	+	0.758932i	$0.725721\pi$
$312$	0	0
$313$	3.03173e11	1.78542	0.892712	−	0.450627i	$-0.148799\pi$
$313$	3.03173e11	1.78542	0.892712	+	0.450627i	$0.148799\pi$
$314$	0	0
$315$	1.32889e11	0.760485
$316$	0	0
$317$	−1.06468e11	−0.592177	−0.296089	−	0.955160i	$-0.595682\pi$
$317$	−1.06468e11	−0.592177	−0.296089	+	0.955160i	$0.595682\pi$
$318$	0	0
$319$	3.69344e11	1.99698
$320$	0	0
$321$	2.37415e11	1.24806
$322$	0	0
$323$	−5.33779e10	−0.272867
$324$	0	0
$325$	−6.54330e10	−0.325328
$326$	0	0
$327$	−3.35138e10	−0.162091
$328$	0	0
$329$	−4.46323e11	−2.10024
$330$	0	0
$331$	−4.30772e11	−1.97252	−0.986260	−	0.165200i	$-0.947173\pi$
$331$	−4.30772e11	−1.97252	−0.986260	+	0.165200i	$0.947173\pi$
$332$	0	0
$333$	−5.40884e11	−2.41049
$334$	0	0
$335$	−1.05041e11	−0.455678
$336$	0	0
$337$	2.14791e10	0.0907154	0.0453577	−	0.998971i	$-0.485557\pi$
$337$	2.14791e10	0.0907154	0.0453577	+	0.998971i	$0.485557\pi$
$338$	0	0
$339$	7.54458e11	3.10268
$340$	0	0
$341$	3.50730e11	1.40468
$342$	0	0
$343$	−2.21882e11	−0.865563
$344$	0	0
$345$	−6.45375e10	−0.245260
$346$	0	0
$347$	2.69409e11	0.997537	0.498769	−	0.866735i	$-0.333786\pi$
$347$	2.69409e11	0.997537	0.498769	+	0.866735i	$0.333786\pi$
$348$	0	0
$349$	−2.16349e11	−0.780623	−0.390311	−	0.920683i	$-0.627633\pi$
$349$	−2.16349e11	−0.780623	−0.390311	+	0.920683i	$0.627633\pi$
$350$	0	0
$351$	−4.00508e11	−1.40841
$352$	0	0
$353$	−1.39649e11	−0.478688	−0.239344	−	0.970935i	$-0.576932\pi$
$353$	−1.39649e11	−0.478688	−0.239344	+	0.970935i	$0.576932\pi$
$354$	0	0
$355$	−5.99628e10	−0.200380
$356$	0	0
$357$	−2.02975e11	−0.661357
$358$	0	0
$359$	7.67571e10	0.243890	0.121945	−	0.992537i	$-0.461087\pi$
$359$	7.67571e10	0.243890	0.121945	+	0.992537i	$0.461087\pi$
$360$	0	0
$361$	−1.53023e11	−0.474213
$362$	0	0
$363$	1.21769e12	3.68092
$364$	0	0
$365$	1.43978e10	0.0424600
$366$	0	0
$367$	5.15249e11	1.48259	0.741293	−	0.671182i	$-0.234213\pi$
$367$	5.15249e11	1.48259	0.741293	+	0.671182i	$0.234213\pi$
$368$	0	0
$369$	7.74515e11	2.17476
$370$	0	0
$371$	−2.78613e11	−0.763516
$372$	0	0
$373$	8.21346e9	0.0219703	0.0109852	−	0.999940i	$-0.496503\pi$
$373$	8.21346e9	0.0219703	0.0109852	+	0.999940i	$0.496503\pi$
$374$	0	0
$375$	5.46205e10	0.142631
$376$	0	0
$377$	7.00488e11	1.78593
$378$	0	0
$379$	2.22340e11	0.553530	0.276765	−	0.960938i	$-0.410738\pi$
$379$	2.22340e11	0.553530	0.276765	+	0.960938i	$0.410738\pi$
$380$	0	0
$381$	−6.54319e11	−1.59084
$382$	0	0
$383$	−7.07610e10	−0.168035	−0.0840175	−	0.996464i	$-0.526775\pi$
$383$	−7.07610e10	−0.168035	−0.0840175	+	0.996464i	$0.526775\pi$
$384$	0	0
$385$	−3.86465e11	−0.896472
$386$	0	0
$387$	−1.51456e11	−0.343232
$388$	0	0
$389$	−2.07926e11	−0.460399	−0.230200	−	0.973143i	$-0.573938\pi$
$389$	−2.07926e11	−0.460399	−0.230200	+	0.973143i	$0.573938\pi$
$390$	0	0
$391$	5.98112e10	0.129416
$392$	0	0
$393$	9.18565e11	1.94242
$394$	0	0
$395$	7.80835e10	0.161388
$396$	0	0
$397$	−1.44824e11	−0.292606	−0.146303	−	0.989240i	$-0.546737\pi$
$397$	−1.44824e11	−0.292606	−0.146303	+	0.989240i	$0.546737\pi$
$398$	0	0
$399$	6.45169e11	1.27437
$400$	0	0
$401$	−2.66461e11	−0.514617	−0.257309	−	0.966329i	$-0.582836\pi$
$401$	−2.66461e11	−0.514617	−0.257309	+	0.966329i	$0.582836\pi$
$402$	0	0
$403$	6.65185e11	1.25623
$404$	0	0
$405$	−3.92831e10	−0.0725535
$406$	0	0
$407$	1.57299e12	2.84152
$408$	0	0
$409$	1.35020e11	0.238585	0.119293	−	0.992859i	$-0.461937\pi$
$409$	1.35020e11	0.238585	0.119293	+	0.992859i	$0.461937\pi$
$410$	0	0
$411$	−1.18581e12	−2.04987
$412$	0	0
$413$	2.24489e11	0.379682
$414$	0	0
$415$	−3.27382e11	−0.541799
$416$	0	0
$417$	5.07663e11	0.822173
$418$	0	0
$419$	−6.88110e11	−1.09067	−0.545337	−	0.838217i	$-0.683598\pi$
$419$	−6.88110e11	−1.09067	−0.545337	+	0.838217i	$0.683598\pi$
$420$	0	0
$421$	−5.43848e11	−0.843739	−0.421869	−	0.906657i	$-0.638626\pi$
$421$	−5.43848e11	−0.843739	−0.421869	+	0.906657i	$0.638626\pi$
$422$	0	0
$423$	−1.93613e12	−2.94037
$424$	0	0
$425$	−5.06204e10	−0.0752619
$426$	0	0
$427$	1.04636e12	1.52320
$428$	0	0
$429$	3.30994e12	4.71805
$430$	0	0
$431$	−2.40997e11	−0.336406	−0.168203	−	0.985752i	$-0.553796\pi$
$431$	−2.40997e11	−0.336406	−0.168203	+	0.985752i	$0.553796\pi$
$432$	0	0
$433$	7.20520e11	0.985033	0.492516	−	0.870303i	$-0.336077\pi$
$433$	7.20520e11	0.985033	0.492516	+	0.870303i	$0.336077\pi$
$434$	0	0
$435$	−5.84735e11	−0.782992
$436$	0	0
$437$	−1.90114e11	−0.249371
$438$	0	0
$439$	−1.24147e11	−0.159531	−0.0797653	−	0.996814i	$-0.525417\pi$
$439$	−1.24147e11	−0.159531	−0.0797653	+	0.996814i	$0.525417\pi$
$440$	0	0
$441$	2.63031e11	0.331156
$442$	0	0
$443$	−1.10043e12	−1.35752	−0.678761	−	0.734360i	$-0.737483\pi$
$443$	−1.10043e12	−1.35752	−0.678761	+	0.734360i	$0.737483\pi$
$444$	0	0
$445$	6.26333e11	0.757156
$446$	0	0
$447$	−6.76938e11	−0.801982
$448$	0	0
$449$	2.84066e11	0.329846	0.164923	−	0.986306i	$-0.447262\pi$
$449$	2.84066e11	0.329846	0.164923	+	0.986306i	$0.447262\pi$
$450$	0	0
$451$	−2.25243e12	−2.56364
$452$	0	0
$453$	−3.17503e11	−0.354247
$454$	0	0
$455$	−7.32958e11	−0.801729
$456$	0	0
$457$	−1.41452e12	−1.51700	−0.758499	−	0.651674i	$-0.774067\pi$
$457$	−1.41452e12	−1.51700	−0.758499	+	0.651674i	$0.774067\pi$
$458$	0	0
$459$	−3.09842e11	−0.325824
$460$	0	0
$461$	−8.15481e11	−0.840930	−0.420465	−	0.907309i	$-0.638133\pi$
$461$	−8.15481e11	−0.840930	−0.420465	+	0.907309i	$0.638133\pi$
$462$	0	0
$463$	−1.82985e11	−0.185055	−0.0925276	−	0.995710i	$-0.529495\pi$
$463$	−1.82985e11	−0.185055	−0.0925276	+	0.995710i	$0.529495\pi$
$464$	0	0
$465$	−5.55266e11	−0.550761
$466$	0	0
$467$	−1.50939e12	−1.46850	−0.734252	−	0.678877i	$-0.762467\pi$
$467$	−1.50939e12	−1.46850	−0.734252	+	0.678877i	$0.762467\pi$
$468$	0	0
$469$	−1.17663e12	−1.12296
$470$	0	0
$471$	−1.88989e11	−0.176947
$472$	0	0
$473$	4.40462e11	0.404607
$474$	0	0
$475$	1.60900e11	0.145022
$476$	0	0
$477$	−1.20861e12	−1.06894
$478$	0	0
$479$	1.02887e12	0.893002	0.446501	−	0.894783i	$-0.352670\pi$
$479$	1.02887e12	0.893002	0.446501	+	0.894783i	$0.352670\pi$
$480$	0	0
$481$	2.98329e12	2.54122
$482$	0	0
$483$	−7.22927e11	−0.604411
$484$	0	0
$485$	−6.90424e11	−0.566602
$486$	0	0
$487$	1.17861e12	0.949489	0.474745	−	0.880124i	$-0.342540\pi$
$487$	1.17861e12	0.949489	0.474745	+	0.880124i	$0.342540\pi$
$488$	0	0
$489$	−2.93242e12	−2.31919
$490$	0	0
$491$	6.83487e11	0.530718	0.265359	−	0.964150i	$-0.414510\pi$
$491$	6.83487e11	0.530718	0.265359	+	0.964150i	$0.414510\pi$
$492$	0	0
$493$	5.41912e11	0.413160
$494$	0	0
$495$	−1.67646e12	−1.25508
$496$	0	0
$497$	−6.71682e11	−0.493810
$498$	0	0
$499$	1.30507e12	0.942280	0.471140	−	0.882059i	$-0.343843\pi$
$499$	1.30507e12	0.942280	0.471140	+	0.882059i	$0.343843\pi$
$500$	0	0
$501$	4.25015e12	3.01394
$502$	0	0
$503$	7.41937e11	0.516786	0.258393	−	0.966040i	$-0.416807\pi$
$503$	7.41937e11	0.516786	0.258393	+	0.966040i	$0.416807\pi$
$504$	0	0
$505$	6.65619e11	0.455422
$506$	0	0
$507$	3.90504e12	2.62476
$508$	0	0
$509$	2.68908e12	1.77572	0.887858	−	0.460119i	$-0.152193\pi$
$509$	2.68908e12	1.77572	0.887858	+	0.460119i	$0.152193\pi$
$510$	0	0
$511$	1.61280e11	0.104637
$512$	0	0
$513$	9.84851e11	0.627831
$514$	0	0
$515$	4.35884e11	0.273047
$516$	0	0
$517$	5.63061e12	3.46615
$518$	0	0
$519$	−3.20865e12	−1.94120
$520$	0	0
$521$	4.09113e11	0.243262	0.121631	−	0.992575i	$-0.461188\pi$
$521$	4.09113e11	0.243262	0.121631	+	0.992575i	$0.461188\pi$
$522$	0	0
$523$	3.30598e11	0.193216	0.0966078	−	0.995323i	$-0.469201\pi$
$523$	3.30598e11	0.193216	0.0966078	+	0.995323i	$0.469201\pi$
$524$	0	0
$525$	6.11840e11	0.351496
$526$	0	0
$527$	5.14601e11	0.290619
$528$	0	0
$529$	−1.58813e12	−0.881728
$530$	0	0
$531$	9.73821e11	0.531562
$532$	0	0
$533$	−4.27190e12	−2.29271
$534$	0	0
$535$	6.63243e11	0.350010
$536$	0	0
$537$	1.86650e12	0.968599
$538$	0	0
$539$	−7.64941e11	−0.390372
$540$	0	0
$541$	1.33596e12	0.670513	0.335256	−	0.942127i	$-0.391177\pi$
$541$	1.33596e12	0.670513	0.335256	+	0.942127i	$0.391177\pi$
$542$	0	0
$543$	1.53107e12	0.755780
$544$	0	0
$545$	−9.36242e10	−0.0454573
$546$	0	0
$547$	2.77719e12	1.32636	0.663181	−	0.748459i	$-0.269206\pi$
$547$	2.77719e12	1.32636	0.663181	+	0.748459i	$0.269206\pi$
$548$	0	0
$549$	4.53905e12	2.13250
$550$	0	0
$551$	−1.72250e12	−0.796119
$552$	0	0
$553$	8.74664e11	0.397721
$554$	0	0
$555$	−2.49031e12	−1.11413
$556$	0	0
$557$	1.79639e12	0.790775	0.395387	−	0.918514i	$-0.370610\pi$
$557$	1.79639e12	0.790775	0.395387	+	0.918514i	$0.370610\pi$
$558$	0	0
$559$	8.35367e11	0.361847
$560$	0	0
$561$	2.56064e12	1.09148
$562$	0	0
$563$	3.40712e12	1.42922	0.714611	−	0.699522i	$-0.246604\pi$
$563$	3.40712e12	1.42922	0.714611	+	0.699522i	$0.246604\pi$
$564$	0	0
$565$	2.10766e12	0.870125
$566$	0	0
$567$	−4.40036e11	−0.178799
$568$	0	0
$569$	−1.63595e12	−0.654281	−0.327140	−	0.944976i	$-0.606085\pi$
$569$	−1.63595e12	−0.654281	−0.327140	+	0.944976i	$0.606085\pi$
$570$	0	0
$571$	−2.78503e12	−1.09639	−0.548197	−	0.836349i	$-0.684686\pi$
$571$	−2.78503e12	−1.09639	−0.548197	+	0.836349i	$0.684686\pi$
$572$	0	0
$573$	1.44152e12	0.558631
$574$	0	0
$575$	−1.80292e11	−0.0687815
$576$	0	0
$577$	3.12732e12	1.17458	0.587288	−	0.809378i	$-0.300196\pi$
$577$	3.12732e12	1.17458	0.587288	+	0.809378i	$0.300196\pi$
$578$	0	0
$579$	−9.36695e11	−0.346373
$580$	0	0
$581$	−3.66722e12	−1.33519
$582$	0	0
$583$	3.51485e12	1.26008
$584$	0	0
$585$	−3.17953e12	−1.12244
$586$	0	0
$587$	2.24503e10	0.00780460	0.00390230	−	0.999992i	$-0.498758\pi$
$587$	2.24503e10	0.00780460	0.00390230	+	0.999992i	$0.498758\pi$
$588$	0	0
$589$	−1.63569e12	−0.559994
$590$	0	0
$591$	−4.34715e12	−1.46575
$592$	0	0
$593$	−2.98315e12	−0.990671	−0.495336	−	0.868702i	$-0.664955\pi$
$593$	−2.98315e12	−0.990671	−0.495336	+	0.868702i	$0.664955\pi$
$594$	0	0
$595$	−5.67032e11	−0.185473
$596$	0	0
$597$	−5.49945e12	−1.77188
$598$	0	0
$599$	5.37802e12	1.70687	0.853437	−	0.521196i	$-0.174514\pi$
$599$	5.37802e12	1.70687	0.853437	+	0.521196i	$0.174514\pi$
$600$	0	0
$601$	−3.95138e12	−1.23542	−0.617709	−	0.786407i	$-0.711939\pi$
$601$	−3.95138e12	−1.23542	−0.617709	+	0.786407i	$0.711939\pi$
$602$	0	0
$603$	−5.10417e12	−1.57216
$604$	0	0
$605$	3.40174e12	1.03229
$606$	0	0
$607$	1.13850e12	0.340396	0.170198	−	0.985410i	$-0.445559\pi$
$607$	1.13850e12	0.340396	0.170198	+	0.985410i	$0.445559\pi$
$608$	0	0
$609$	−6.55000e12	−1.92958
$610$	0	0
$611$	1.06788e13	3.09984
$612$	0	0
$613$	−4.84605e12	−1.38617	−0.693083	−	0.720857i	$-0.743748\pi$
$613$	−4.84605e12	−1.38617	−0.693083	+	0.720857i	$0.743748\pi$
$614$	0	0
$615$	3.56598e12	1.00517
$616$	0	0
$617$	2.57919e12	0.716474	0.358237	−	0.933631i	$-0.383378\pi$
$617$	2.57919e12	0.716474	0.358237	+	0.933631i	$0.383378\pi$
$618$	0	0
$619$	−6.27422e11	−0.171772	−0.0858859	−	0.996305i	$-0.527372\pi$
$619$	−6.27422e11	−0.171772	−0.0858859	+	0.996305i	$0.527372\pi$
$620$	0	0
$621$	−1.10355e12	−0.297769
$622$	0	0
$623$	7.01597e12	1.86591
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	0	0
$627$	−8.13916e12	−2.10317
$628$	0	0
$629$	2.30793e12	0.587889
$630$	0	0
$631$	3.78374e12	0.950144	0.475072	−	0.879947i	$-0.342422\pi$
$631$	3.78374e12	0.950144	0.475072	+	0.879947i	$0.342422\pi$
$632$	0	0
$633$	9.66850e12	2.39355
$634$	0	0
$635$	−1.82791e12	−0.446141
$636$	0	0
$637$	−1.45077e12	−0.349116
$638$	0	0
$639$	−2.91372e12	−0.691343
$640$	0	0
$641$	2.72795e12	0.638226	0.319113	−	0.947717i	$-0.396615\pi$
$641$	2.72795e12	0.638226	0.319113	+	0.947717i	$0.396615\pi$
$642$	0	0
$643$	6.22277e12	1.43560	0.717801	−	0.696248i	$-0.245149\pi$
$643$	6.22277e12	1.43560	0.717801	+	0.696248i	$0.245149\pi$
$644$	0	0
$645$	−6.97326e11	−0.158642
$646$	0	0
$647$	8.57440e10	0.0192369	0.00961844	−	0.999954i	$-0.496938\pi$
$647$	8.57440e10	0.0192369	0.00961844	+	0.999954i	$0.496938\pi$
$648$	0	0
$649$	−2.83205e12	−0.626614
$650$	0	0
$651$	−6.21990e12	−1.35728
$652$	0	0
$653$	−1.14476e12	−0.246381	−0.123190	−	0.992383i	$-0.539313\pi$
$653$	−1.14476e12	−0.246381	−0.123190	+	0.992383i	$0.539313\pi$
$654$	0	0
$655$	2.56610e12	0.544739
$656$	0	0
$657$	6.99622e11	0.146494
$658$	0	0
$659$	5.51626e12	1.13936	0.569679	−	0.821867i	$-0.307068\pi$
$659$	5.51626e12	1.13936	0.569679	+	0.821867i	$0.307068\pi$
$660$	0	0
$661$	1.26891e12	0.258539	0.129269	−	0.991610i	$-0.458737\pi$
$661$	1.26891e12	0.258539	0.129269	+	0.991610i	$0.458737\pi$
$662$	0	0
$663$	4.85644e12	0.976128
$664$	0	0
$665$	1.80235e12	0.357389
$666$	0	0
$667$	1.93010e12	0.377585
$668$	0	0
$669$	−1.61141e12	−0.311020
$670$	0	0
$671$	−1.32004e13	−2.51383
$672$	0	0
$673$	−1.86276e12	−0.350016	−0.175008	−	0.984567i	$-0.555995\pi$
$673$	−1.86276e12	−0.350016	−0.175008	+	0.984567i	$0.555995\pi$
$674$	0	0
$675$	9.33973e11	0.173168
$676$	0	0
$677$	−2.47298e12	−0.452450	−0.226225	−	0.974075i	$-0.572639\pi$
$677$	−2.47298e12	−0.452450	−0.226225	+	0.974075i	$0.572639\pi$
$678$	0	0
$679$	−7.73389e12	−1.39632
$680$	0	0
$681$	4.02794e12	0.717664
$682$	0	0
$683$	−5.42850e12	−0.954524	−0.477262	−	0.878761i	$-0.658371\pi$
$683$	−5.42850e12	−0.954524	−0.477262	+	0.878761i	$0.658371\pi$
$684$	0	0
$685$	−3.31268e12	−0.574874
$686$	0	0
$687$	−7.46090e12	−1.27787
$688$	0	0
$689$	6.66616e12	1.12691
$690$	0	0
$691$	6.80097e12	1.13480	0.567400	−	0.823442i	$-0.307949\pi$
$691$	6.80097e12	1.13480	0.567400	+	0.823442i	$0.307949\pi$
$692$	0	0
$693$	−1.87791e13	−3.09297
$694$	0	0
$695$	1.41821e12	0.230573
$696$	0	0
$697$	−3.30483e12	−0.530398
$698$	0	0
$699$	−9.87259e11	−0.156417
$700$	0	0
$701$	−1.01070e13	−1.58085	−0.790427	−	0.612557i	$-0.790141\pi$
$701$	−1.01070e13	−1.58085	−0.790427	+	0.612557i	$0.790141\pi$
$702$	0	0
$703$	−7.33591e12	−1.13281
$704$	0	0
$705$	−8.91421e12	−1.35904
$706$	0	0
$707$	7.45603e12	1.12233
$708$	0	0
$709$	1.23040e13	1.82868	0.914342	−	0.404943i	$-0.132709\pi$
$709$	1.23040e13	1.82868	0.914342	+	0.404943i	$0.132709\pi$
$710$	0	0
$711$	3.79424e12	0.556817
$712$	0	0
$713$	1.83283e12	0.265595
$714$	0	0
$715$	9.24665e12	1.32314
$716$	0	0
$717$	1.53067e13	2.16294
$718$	0	0
$719$	−8.29578e12	−1.15765	−0.578825	−	0.815452i	$-0.696488\pi$
$719$	−8.29578e12	−1.15765	−0.578825	+	0.815452i	$0.696488\pi$
$720$	0	0
$721$	4.88262e12	0.672890
$722$	0	0
$723$	−2.07010e13	−2.81753
$724$	0	0
$725$	−1.63352e12	−0.219585
$726$	0	0
$727$	−7.42195e12	−0.985401	−0.492701	−	0.870199i	$-0.663990\pi$
$727$	−7.42195e12	−0.985401	−0.492701	+	0.870199i	$0.663990\pi$
$728$	0	0
$729$	−1.24377e13	−1.63105
$730$	0	0
$731$	6.46258e11	0.0837101
$732$	0	0
$733$	−7.91808e12	−1.01310	−0.506550	−	0.862211i	$-0.669079\pi$
$733$	−7.91808e12	−1.01310	−0.506550	+	0.862211i	$0.669079\pi$
$734$	0	0
$735$	1.21103e12	0.153060
$736$	0	0
$737$	1.48439e13	1.85329
$738$	0	0
$739$	−3.03898e12	−0.374825	−0.187412	−	0.982281i	$-0.560010\pi$
$739$	−3.03898e12	−0.374825	−0.187412	+	0.982281i	$0.560010\pi$
$740$	0	0
$741$	−1.54365e13	−1.88090
$742$	0	0
$743$	−4.69140e12	−0.564746	−0.282373	−	0.959305i	$-0.591122\pi$
$743$	−4.69140e12	−0.564746	−0.282373	+	0.959305i	$0.591122\pi$
$744$	0	0
$745$	−1.89109e12	−0.224911
$746$	0	0
$747$	−1.59082e13	−1.86930
$748$	0	0
$749$	7.42941e12	0.862554
$750$	0	0
$751$	1.00434e13	1.15213	0.576063	−	0.817405i	$-0.304588\pi$
$751$	1.00434e13	1.15213	0.576063	+	0.817405i	$0.304588\pi$
$752$	0	0
$753$	1.28502e13	1.45658
$754$	0	0
$755$	−8.86977e11	−0.0993462
$756$	0	0
$757$	−1.57346e13	−1.74151	−0.870753	−	0.491721i	$-0.836368\pi$
$757$	−1.57346e13	−1.74151	−0.870753	+	0.491721i	$0.836368\pi$
$758$	0	0
$759$	9.12011e12	0.997498
$760$	0	0
$761$	−6.23160e12	−0.673548	−0.336774	−	0.941586i	$-0.609336\pi$
$761$	−6.23160e12	−0.673548	−0.336774	+	0.941586i	$0.609336\pi$
$762$	0	0
$763$	−1.04875e12	−0.112024
$764$	0	0
$765$	−2.45975e12	−0.259666
$766$	0	0
$767$	−5.37119e12	−0.560391
$768$	0	0
$769$	−1.49345e13	−1.54001	−0.770004	−	0.638040i	$-0.779746\pi$
$769$	−1.49345e13	−1.54001	−0.770004	+	0.638040i	$0.779746\pi$
$770$	0	0
$771$	3.79907e12	0.387198
$772$	0	0
$773$	−1.83308e13	−1.84661	−0.923303	−	0.384071i	$-0.874522\pi$
$773$	−1.83308e13	−1.84661	−0.923303	+	0.384071i	$0.874522\pi$
$774$	0	0
$775$	−1.55119e12	−0.154457
$776$	0	0
$777$	−2.78956e13	−2.74562
$778$	0	0
$779$	1.05046e13	1.02202
$780$	0	0
$781$	8.47363e12	0.814967
$782$	0	0
$783$	−9.99858e12	−0.950627
$784$	0	0
$785$	−5.27960e11	−0.0496235
$786$	0	0
$787$	2.36245e11	0.0219521	0.0109760	−	0.999940i	$-0.496506\pi$
$787$	2.36245e11	0.0219521	0.0109760	+	0.999940i	$0.496506\pi$
$788$	0	0
$789$	9.24155e11	0.0848981
$790$	0	0
$791$	2.36092e13	2.14431
$792$	0	0
$793$	−2.50355e13	−2.24816
$794$	0	0
$795$	−5.56460e12	−0.494063
$796$	0	0
$797$	−1.33800e13	−1.17461	−0.587305	−	0.809366i	$-0.699811\pi$
$797$	−1.33800e13	−1.17461	−0.587305	+	0.809366i	$0.699811\pi$
$798$	0	0
$799$	8.26139e12	0.717121
$800$	0	0
$801$	3.04349e13	2.61231
$802$	0	0
$803$	−2.03463e12	−0.172689
$804$	0	0
$805$	−2.01957e12	−0.169503
$806$	0	0
$807$	−2.09232e13	−1.73659
$808$	0	0
$809$	2.10586e13	1.72847	0.864234	−	0.503090i	$-0.167803\pi$
$809$	2.10586e13	1.72847	0.864234	+	0.503090i	$0.167803\pi$
$810$	0	0
$811$	1.05253e13	0.854361	0.427181	−	0.904166i	$-0.359507\pi$
$811$	1.05253e13	0.854361	0.427181	+	0.904166i	$0.359507\pi$
$812$	0	0
$813$	5.70019e12	0.457595
$814$	0	0
$815$	−8.19201e12	−0.650401
$816$	0	0
$817$	−2.05417e12	−0.161301
$818$	0	0
$819$	−3.56160e13	−2.76610
$820$	0	0
$821$	1.02235e13	0.785337	0.392668	−	0.919680i	$-0.371552\pi$
$821$	1.02235e13	0.785337	0.392668	+	0.919680i	$0.371552\pi$
$822$	0	0
$823$	−2.03818e13	−1.54861	−0.774307	−	0.632810i	$-0.781901\pi$
$823$	−2.03818e13	−1.54861	−0.774307	+	0.632810i	$0.781901\pi$
$824$	0	0
$825$	−7.71868e12	−0.580097
$826$	0	0
$827$	−1.26044e13	−0.937020	−0.468510	−	0.883458i	$-0.655209\pi$
$827$	−1.26044e13	−0.937020	−0.468510	+	0.883458i	$0.655209\pi$
$828$	0	0
$829$	1.44545e12	0.106293	0.0531467	−	0.998587i	$-0.483075\pi$
$829$	1.44545e12	0.106293	0.0531467	+	0.998587i	$0.483075\pi$
$830$	0	0
$831$	1.17240e13	0.852848
$832$	0	0
$833$	−1.12234e12	−0.0807650
$834$	0	0
$835$	1.18732e13	0.845240
$836$	0	0
$837$	−9.49468e12	−0.668676
$838$	0	0
$839$	2.78805e13	1.94255	0.971273	−	0.237968i	$-0.0764813\pi$
$839$	2.78805e13	1.94255	0.971273	+	0.237968i	$0.0764813\pi$
$840$	0	0
$841$	2.98034e12	0.205439
$842$	0	0
$843$	2.08953e13	1.42504
$844$	0	0
$845$	1.09091e13	0.736096
$846$	0	0
$847$	3.81051e13	2.54395
$848$	0	0
$849$	−2.95520e13	−1.95210
$850$	0	0
$851$	8.22006e12	0.537269
$852$	0	0
$853$	2.07647e13	1.34294	0.671469	−	0.741033i	$-0.265664\pi$
$853$	2.07647e13	1.34294	0.671469	+	0.741033i	$0.265664\pi$
$854$	0	0
$855$	7.81848e12	0.500351
$856$	0	0
$857$	−1.47635e13	−0.934920	−0.467460	−	0.884014i	$-0.654831\pi$
$857$	−1.47635e13	−0.934920	−0.467460	+	0.884014i	$0.654831\pi$
$858$	0	0
$859$	−3.61756e12	−0.226697	−0.113349	−	0.993555i	$-0.536158\pi$
$859$	−3.61756e12	−0.226697	−0.113349	+	0.993555i	$0.536158\pi$
$860$	0	0
$861$	3.99449e13	2.47712
$862$	0	0
$863$	−1.01093e13	−0.620401	−0.310201	−	0.950671i	$-0.600396\pi$
$863$	−1.01093e13	−0.620401	−0.310201	+	0.950671i	$0.600396\pi$
$864$	0	0
$865$	−8.96370e12	−0.544396
$866$	0	0
$867$	−2.27741e13	−1.36885
$868$	0	0
$869$	−1.10344e13	−0.656384
$870$	0	0
$871$	2.81525e13	1.65743
$872$	0	0
$873$	−3.35492e13	−1.95487
$874$	0	0
$875$	1.70924e12	0.0985748
$876$	0	0
$877$	−1.45609e13	−0.831170	−0.415585	−	0.909554i	$-0.636423\pi$
$877$	−1.45609e13	−0.831170	−0.415585	+	0.909554i	$0.636423\pi$
$878$	0	0
$879$	−1.55341e13	−0.877681
$880$	0	0
$881$	−9.17004e12	−0.512838	−0.256419	−	0.966566i	$-0.582543\pi$
$881$	−9.17004e12	−0.512838	−0.256419	+	0.966566i	$0.582543\pi$
$882$	0	0
$883$	2.49177e13	1.37938	0.689692	−	0.724103i	$-0.257746\pi$
$883$	2.49177e13	1.37938	0.689692	+	0.724103i	$0.257746\pi$
$884$	0	0
$885$	4.48362e12	0.245688
$886$	0	0
$887$	−3.02449e13	−1.64057	−0.820287	−	0.571952i	$-0.806186\pi$
$887$	−3.02449e13	−1.64057	−0.820287	+	0.571952i	$0.806186\pi$
$888$	0	0
$889$	−2.04756e13	−1.09946
$890$	0	0
$891$	5.55129e12	0.295083
$892$	0	0
$893$	−2.62593e13	−1.38182
$894$	0	0
$895$	5.21426e12	0.271637
$896$	0	0
$897$	1.72969e13	0.892079
$898$	0	0
$899$	1.66062e13	0.847912
$900$	0	0
$901$	5.15708e12	0.260701
$902$	0	0
$903$	−7.81121e12	−0.390952
$904$	0	0
$905$	4.27719e12	0.211953
$906$	0	0
$907$	2.65098e13	1.30069	0.650344	−	0.759640i	$-0.274625\pi$
$907$	2.65098e13	1.30069	0.650344	+	0.759640i	$0.274625\pi$
$908$	0	0
$909$	3.23438e13	1.57128
$910$	0	0
$911$	3.99211e13	1.92030	0.960152	−	0.279480i	$-0.0901619\pi$
$911$	3.99211e13	1.92030	0.960152	+	0.279480i	$0.0901619\pi$
$912$	0	0
$913$	4.62639e13	2.20356
$914$	0	0
$915$	2.08985e13	0.985643
$916$	0	0
$917$	2.87446e13	1.34244
$918$	0	0
$919$	−2.47031e13	−1.14243	−0.571217	−	0.820799i	$-0.693529\pi$
$919$	−2.47031e13	−1.14243	−0.571217	+	0.820799i	$0.693529\pi$
$920$	0	0
$921$	−3.75934e13	−1.72165
$922$	0	0
$923$	1.60708e13	0.728838
$924$	0	0
$925$	−6.95694e12	−0.312450
$926$	0	0
$927$	2.11805e13	0.942058
$928$	0	0
$929$	3.37721e13	1.48760	0.743801	−	0.668401i	$-0.233021\pi$
$929$	3.37721e13	1.48760	0.743801	+	0.668401i	$0.233021\pi$
$930$	0	0
$931$	3.56744e12	0.155626
$932$	0	0
$933$	−4.80686e13	−2.07680
$934$	0	0
$935$	7.15341e12	0.306098
$936$	0	0
$937$	2.88150e13	1.22121	0.610605	−	0.791936i	$-0.290927\pi$
$937$	2.88150e13	1.22121	0.610605	+	0.791936i	$0.290927\pi$
$938$	0	0
$939$	6.78276e13	2.84716
$940$	0	0
$941$	−5.45811e12	−0.226929	−0.113464	−	0.993542i	$-0.536195\pi$
$941$	−5.45811e12	−0.226929	−0.113464	+	0.993542i	$0.536195\pi$
$942$	0	0
$943$	−1.17707e13	−0.484728
$944$	0	0
$945$	1.04620e13	0.426750
$946$	0	0
$947$	−2.35331e11	−0.00950832	−0.00475416	−	0.999989i	$-0.501513\pi$
$947$	−2.35331e11	−0.00950832	−0.00475416	+	0.999989i	$0.501513\pi$
$948$	0	0
$949$	−3.85882e12	−0.154439
$950$	0	0
$951$	−2.38196e13	−0.944325
$952$	0	0
$953$	−1.55591e13	−0.611034	−0.305517	−	0.952187i	$-0.598829\pi$
$953$	−1.55591e13	−0.611034	−0.305517	+	0.952187i	$0.598829\pi$
$954$	0	0
$955$	4.02703e12	0.156664
$956$	0	0
$957$	8.26317e13	3.18451
$958$	0	0
$959$	−3.71075e13	−1.41670
$960$	0	0
$961$	−1.06704e13	−0.403575
$962$	0	0
$963$	3.22284e13	1.20759
$964$	0	0
$965$	−2.61675e12	−0.0971381
$966$	0	0
$967$	1.02802e13	0.378079	0.189040	−	0.981969i	$-0.439463\pi$
$967$	1.02802e13	0.378079	0.189040	+	0.981969i	$0.439463\pi$
$968$	0	0
$969$	−1.19420e13	−0.435131
$970$	0	0
$971$	−2.54088e13	−0.917271	−0.458636	−	0.888624i	$-0.651662\pi$
$971$	−2.54088e13	−0.917271	−0.458636	+	0.888624i	$0.651662\pi$
$972$	0	0
$973$	1.58863e13	0.568217
$974$	0	0
$975$	−1.46390e13	−0.518790
$976$	0	0
$977$	−5.36288e12	−0.188310	−0.0941549	−	0.995558i	$-0.530015\pi$
$977$	−5.36288e12	−0.188310	−0.0941549	+	0.995558i	$0.530015\pi$
$978$	0	0
$979$	−8.85102e13	−3.07944
$980$	0	0
$981$	−4.54940e12	−0.156835
$982$	0	0
$983$	3.51911e13	1.20210	0.601052	−	0.799210i	$-0.294748\pi$
$983$	3.51911e13	1.20210	0.601052	+	0.799210i	$0.294748\pi$
$984$	0	0
$985$	−1.21442e13	−0.411061
$986$	0	0
$987$	−9.98539e13	−3.34917
$988$	0	0
$989$	2.30175e12	0.0765023
$990$	0	0
$991$	3.54083e12	0.116620	0.0583100	−	0.998299i	$-0.481429\pi$
$991$	3.54083e12	0.116620	0.0583100	+	0.998299i	$0.481429\pi$
$992$	0	0
$993$	−9.63746e13	−3.14551
$994$	0	0
$995$	−1.53633e13	−0.496913
$996$	0	0
$997$	4.47899e13	1.43566	0.717830	−	0.696218i	$-0.245136\pi$
$997$	4.47899e13	1.43566	0.717830	+	0.696218i	$0.245136\pi$
$998$	0	0
$999$	−4.25826e13	−1.35266

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.10.a.z.1.4		4
4.3	odd	2	inner	320.10.a.z.1.1		4
8.3	odd	2		160.10.a.c.1.4	yes	4
8.5	even	2		160.10.a.c.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.10.a.c.1.1	✓	4	8.5	even	2
160.10.a.c.1.4	yes	4	8.3	odd	2
320.10.a.z.1.1		4	4.3	odd	2	inner
320.10.a.z.1.4		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)

\(f(q)\)	\(=\)	\(q+223.725 q^{3} +625.000 q^{5} +7001.03 q^{7} +30370.1 q^{9} -88321.8 q^{11} -167508. q^{13} +139828. q^{15} -129588. q^{17} +411904. q^{19} +1.56631e6 q^{21} -461548. q^{23} +390625. q^{25} +2.39097e6 q^{27} -4.18180e6 q^{29} -3.97105e6 q^{31} -1.97598e7 q^{33} +4.37565e6 q^{35} -1.78098e7 q^{37} -3.74759e7 q^{39} +2.55026e7 q^{41} -4.98702e6 q^{43} +1.89813e7 q^{45} -6.37511e7 q^{47} +8.66085e6 q^{49} -2.89922e7 q^{51} -3.97959e7 q^{53} -5.52011e7 q^{55} +9.21535e7 q^{57} +3.20652e7 q^{59} +1.49458e8 q^{61} +2.12622e8 q^{63} -1.04693e8 q^{65} -1.68066e8 q^{67} -1.03260e8 q^{69} -9.59405e7 q^{71} +2.30366e7 q^{73} +8.73928e7 q^{75} -6.18343e8 q^{77} +1.24934e8 q^{79} -6.28530e7 q^{81} -5.23811e8 q^{83} -8.09926e7 q^{85} -9.35576e8 q^{87} +1.00213e9 q^{89} -1.17273e9 q^{91} -8.88426e8 q^{93} +2.57440e8 q^{95} -1.10468e9 q^{97} -2.68234e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2500 q^{5} + 23364 q^{9} - 201256 q^{13} - 442040 q^{17} + 3496176 q^{21} + 1562500 q^{25} - 6719352 q^{29} - 36991008 q^{33} - 29092616 q^{37} + 49561624 q^{41} + 14602500 q^{45} + 3039028 q^{49}+ \cdots + 464919272 q^{97}+O(q^{100}) \) 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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