LMFDB - Embedded newform 320.3.b.a.191.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,3,Mod(191,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("320.191");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	320.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$8.71936845953$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.2
Root			$-0.309017 - 0.535233i$ of defining polynomial
Character	$\chi$	$=$	320.191
Dual form			320.3.b.a.191.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14093i q^{3} -2.23607 q^{5} +9.06914i q^{7} +4.41641 q^{9} +O(q^{10})$	$q-2.14093i q^{3} -2.23607 q^{5} +9.06914i q^{7} +4.41641 q^{9} -4.28187i q^{11} +9.41641 q^{13} +4.78727i q^{15} +18.0000 q^{17} +36.2765i q^{19} +19.4164 q^{21} -22.9255i q^{23} +5.00000 q^{25} -28.7236i q^{27} +44.8328 q^{29} -35.2657i q^{31} -9.16718 q^{33} -20.2792i q^{35} -6.58359 q^{37} -20.1599i q^{39} +52.2492 q^{41} +28.8429i q^{43} -9.87539 q^{45} +90.1860i q^{47} -33.2492 q^{49} -38.5368i q^{51} -52.2492 q^{53} +9.57454i q^{55} +77.6656 q^{57} -17.1275i q^{59} +50.5836 q^{61} +40.0530i q^{63} -21.0557 q^{65} +33.1248i q^{67} -49.0820 q^{69} +20.1599i q^{71} -91.6656 q^{73} -10.7047i q^{75} +38.8328 q^{77} -42.8187i q^{79} -21.7477 q^{81} +22.3008i q^{83} -40.2492 q^{85} -95.9840i q^{87} +47.6656 q^{89} +85.3987i q^{91} -75.5016 q^{93} -81.1168i q^{95} -160.164 q^{97} -18.9105i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{9}+O(q^{10}) $ 4 * q - 36 * q^9	$ 4 q - 36 q^{9} - 16 q^{13} + 72 q^{17} + 24 q^{21} + 20 q^{25} + 72 q^{29} - 144 q^{33} - 80 q^{37} + 48 q^{41} - 120 q^{45} + 28 q^{49} - 48 q^{53} + 96 q^{57} + 256 q^{61} - 120 q^{65} + 72 q^{69} - 152 q^{73} + 48 q^{77} + 396 q^{81} - 24 q^{89} - 624 q^{93} - 104 q^{97}+O(q^{100}) $ 4 * q - 36 * q^9 - 16 * q^13 + 72 * q^17 + 24 * q^21 + 20 * q^25 + 72 * q^29 - 144 * q^33 - 80 * q^37 + 48 * q^41 - 120 * q^45 + 28 * q^49 - 48 * q^53 + 96 * q^57 + 256 * q^61 - 120 * q^65 + 72 * q^69 - 152 * q^73 + 48 * q^77 + 396 * q^81 - 24 * q^89 - 624 * q^93 - 104 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 2.14093i	− 0.713644i	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	− 2.14093i	− 0.713644i	0.934172	−	0.356822i	$-0.116140\pi$
$4$	0	0
$5$	−2.23607	−0.447214
$5$	−2.23607	−0.447214
$6$	0	0
$7$	9.06914i	1.29559i	0.761814	+	0.647795i	$0.224309\pi$
$7$	9.06914i	1.29559i	−0.761814	+	0.647795i	$0.775691\pi$
$8$	0	0
$9$	4.41641	0.490712
$10$	0	0
$11$	− 4.28187i	− 0.389260i	−0.980877	−	0.194630i	$-0.937649\pi$
$11$	− 4.28187i	− 0.389260i	0.980877	−	0.194630i	$-0.0623507\pi$
$12$	0	0
$13$	9.41641	0.724339	0.362170	−	0.932112i	$-0.382036\pi$
$13$	9.41641	0.724339	0.362170	+	0.932112i	$0.382036\pi$
$14$	0	0
$15$	4.78727i	0.319151i
$16$	0	0
$17$	18.0000	1.05882	0.529412	−	0.848365i	$-0.322413\pi$
$17$	18.0000	1.05882	0.529412	+	0.848365i	$0.322413\pi$
$18$	0	0
$19$	36.2765i	1.90929i	0.297744	+	0.954646i	$0.403766\pi$
$19$	36.2765i	1.90929i	−0.297744	+	0.954646i	$0.596234\pi$
$20$	0	0
$21$	19.4164	0.924591
$22$	0	0
$23$	− 22.9255i	− 0.996763i	−0.866958	−	0.498381i	$-0.833928\pi$
$23$	− 22.9255i	− 0.996763i	0.866958	−	0.498381i	$-0.166072\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	− 28.7236i	− 1.06384i
$28$	0	0
$29$	44.8328	1.54596	0.772980	−	0.634431i	$-0.218765\pi$
$29$	44.8328	1.54596	0.772980	+	0.634431i	$0.218765\pi$
$30$	0	0
$31$	− 35.2657i	− 1.13760i	−0.822474	−	0.568802i	$-0.807407\pi$
$31$	− 35.2657i	− 1.13760i	0.822474	−	0.568802i	$-0.192593\pi$
$32$	0	0
$33$	−9.16718	−0.277793
$34$	0	0
$35$	− 20.2792i	− 0.579406i
$36$	0	0
$37$	−6.58359	−0.177935	−0.0889675	−	0.996035i	$-0.528357\pi$
$37$	−6.58359	−0.177935	−0.0889675	+	0.996035i	$0.528357\pi$
$38$	0	0
$39$	− 20.1599i	− 0.516920i
$40$	0	0
$41$	52.2492	1.27437	0.637186	−	0.770710i	$-0.280098\pi$
$41$	52.2492	1.27437	0.637186	+	0.770710i	$0.280098\pi$
$42$	0	0
$43$	28.8429i	0.670766i	0.942082	+	0.335383i	$0.108866\pi$
$43$	28.8429i	0.670766i	−0.942082	+	0.335383i	$0.891134\pi$
$44$	0	0
$45$	−9.87539	−0.219453
$46$	0	0
$47$	90.1860i	1.91885i	0.281964	+	0.959425i	$0.409014\pi$
$47$	90.1860i	1.91885i	−0.281964	+	0.959425i	$0.590986\pi$
$48$	0	0
$49$	−33.2492	−0.678556
$50$	0	0
$51$	− 38.5368i	− 0.755623i
$52$	0	0
$53$	−52.2492	−0.985834	−0.492917	−	0.870076i	$-0.664069\pi$
$53$	−52.2492	−0.985834	−0.492917	+	0.870076i	$0.664069\pi$
$54$	0	0
$55$	9.57454i	0.174083i
$56$	0	0
$57$	77.6656	1.36255
$58$	0	0
$59$	− 17.1275i	− 0.290296i	−0.989410	−	0.145148i	$-0.953634\pi$
$59$	− 17.1275i	− 0.290296i	0.989410	−	0.145148i	$-0.0463658\pi$
$60$	0	0
$61$	50.5836	0.829239	0.414620	−	0.909995i	$-0.363915\pi$
$61$	50.5836	0.829239	0.414620	+	0.909995i	$0.363915\pi$
$62$	0	0
$63$	40.0530i	0.635762i
$64$	0	0
$65$	−21.0557	−0.323934
$66$	0	0
$67$	33.1248i	0.494400i	0.968964	+	0.247200i	$0.0795105\pi$
$67$	33.1248i	0.494400i	−0.968964	+	0.247200i	$0.920490\pi$
$68$	0	0
$69$	−49.0820	−0.711334
$70$	0	0
$71$	20.1599i	0.283942i	0.989871	+	0.141971i	$0.0453440\pi$
$71$	20.1599i	0.283942i	−0.989871	+	0.141971i	$0.954656\pi$
$72$	0	0
$73$	−91.6656	−1.25569	−0.627847	−	0.778337i	$-0.716064\pi$
$73$	−91.6656	−1.25569	−0.627847	+	0.778337i	$0.716064\pi$
$74$	0	0
$75$	− 10.7047i	− 0.142729i
$76$	0	0
$77$	38.8328	0.504322
$78$	0	0
$79$	− 42.8187i	− 0.542008i	−0.962578	−	0.271004i	$-0.912644\pi$
$79$	− 42.8187i	− 0.542008i	0.962578	−	0.271004i	$-0.0873557\pi$
$80$	0	0
$81$	−21.7477	−0.268490
$82$	0	0
$83$	22.3008i	0.268685i	0.990935	+	0.134342i	$0.0428922\pi$
$83$	22.3008i	0.268685i	−0.990935	+	0.134342i	$0.957108\pi$
$84$	0	0
$85$	−40.2492	−0.473520
$86$	0	0
$87$	− 95.9840i	− 1.10326i
$88$	0	0
$89$	47.6656	0.535569	0.267784	−	0.963479i	$-0.413708\pi$
$89$	47.6656	0.535569	0.267784	+	0.963479i	$0.413708\pi$
$90$	0	0
$91$	85.3987i	0.938447i
$92$	0	0
$93$	−75.5016	−0.811845
$94$	0	0
$95$	− 81.1168i	− 0.853861i
$96$	0	0
$97$	−160.164	−1.65118	−0.825588	−	0.564273i	$-0.809156\pi$
$97$	−160.164	−1.65118	−0.825588	+	0.564273i	$0.809156\pi$
$98$	0	0
$99$	− 18.9105i	− 0.191015i
$100$	0	0
$101$	−23.6656	−0.234313	−0.117157	−	0.993113i	$-0.537378\pi$
$101$	−23.6656	−0.234313	−0.117157	+	0.993113i	$0.537378\pi$
$102$	0	0
$103$	34.5217i	0.335162i	0.985858	+	0.167581i	$0.0535956\pi$
$103$	34.5217i	0.335162i	−0.985858	+	0.167581i	$0.946404\pi$
$104$	0	0
$105$	−43.4164	−0.413490
$106$	0	0
$107$	68.1519i	0.636934i	0.947934	+	0.318467i	$0.103168\pi$
$107$	68.1519i	0.636934i	−0.947934	+	0.318467i	$0.896832\pi$
$108$	0	0
$109$	−29.4164	−0.269875	−0.134938	−	0.990854i	$-0.543083\pi$
$109$	−29.4164	−0.269875	−0.134938	+	0.990854i	$0.543083\pi$
$110$	0	0
$111$	14.0950i	0.126982i
$112$	0	0
$113$	−194.498	−1.72123	−0.860613	−	0.509260i	$-0.829919\pi$
$113$	−194.498	−1.72123	−0.860613	+	0.509260i	$0.829919\pi$
$114$	0	0
$115$	51.2631i	0.445766i
$116$	0	0
$117$	41.5867	0.355442
$118$	0	0
$119$	163.244i	1.37180i
$120$	0	0
$121$	102.666	0.848476
$122$	0	0
$123$	− 111.862i	− 0.909448i
$124$	0	0
$125$	−11.1803	−0.0894427
$126$	0	0
$127$	− 151.915i	− 1.19618i	−0.801428	−	0.598091i	$-0.795926\pi$
$127$	− 151.915i	− 1.19618i	0.801428	−	0.598091i	$-0.204074\pi$
$128$	0	0
$129$	61.7508	0.478688
$130$	0	0
$131$	− 144.868i	− 1.10586i	−0.833228	−	0.552930i	$-0.813510\pi$
$131$	− 144.868i	− 1.10586i	0.833228	−	0.552930i	$-0.186490\pi$
$132$	0	0
$133$	−328.997	−2.47366
$134$	0	0
$135$	64.2280i	0.475763i
$136$	0	0
$137$	75.1672	0.548666	0.274333	−	0.961635i	$-0.411543\pi$
$137$	75.1672	0.548666	0.274333	+	0.961635i	$0.411543\pi$
$138$	0	0
$139$	− 203.031i	− 1.46065i	−0.683099	−	0.730326i	$-0.739368\pi$
$139$	− 203.031i	− 1.46065i	0.683099	−	0.730326i	$-0.260632\pi$
$140$	0	0
$141$	193.082	1.36938
$142$	0	0
$143$	− 40.3198i	− 0.281957i
$144$	0	0
$145$	−100.249	−0.691374
$146$	0	0
$147$	71.1843i	0.484247i
$148$	0	0
$149$	−98.0851	−0.658290	−0.329145	−	0.944279i	$-0.606760\pi$
$149$	−98.0851	−0.658290	−0.329145	+	0.944279i	$0.606760\pi$
$150$	0	0
$151$	43.8295i	0.290261i	0.989412	+	0.145131i	$0.0463603\pi$
$151$	43.8295i	0.290261i	−0.989412	+	0.145131i	$0.953640\pi$
$152$	0	0
$153$	79.4953	0.519577
$154$	0	0
$155$	78.8566i	0.508752i
$156$	0	0
$157$	41.4164	0.263799	0.131899	−	0.991263i	$-0.457892\pi$
$157$	41.4164	0.263799	0.131899	+	0.991263i	$0.457892\pi$
$158$	0	0
$159$	111.862i	0.703535i
$160$	0	0
$161$	207.915	1.29140
$162$	0	0
$163$	− 183.524i	− 1.12591i	−0.826487	−	0.562956i	$-0.809664\pi$
$163$	− 183.524i	− 1.12591i	0.826487	−	0.562956i	$-0.190336\pi$
$164$	0	0
$165$	20.4984	0.124233
$166$	0	0
$167$	− 54.1480i	− 0.324240i	−0.986771	−	0.162120i	$-0.948167\pi$
$167$	− 54.1480i	− 0.324240i	0.986771	−	0.162120i	$-0.0518331\pi$
$168$	0	0
$169$	−80.3313	−0.475333
$170$	0	0
$171$	160.212i	0.936912i
$172$	0	0
$173$	73.4164	0.424372	0.212186	−	0.977229i	$-0.431942\pi$
$173$	73.4164	0.424372	0.212186	+	0.977229i	$0.431942\pi$
$174$	0	0
$175$	45.3457i	0.259118i
$176$	0	0
$177$	−36.6687	−0.207168
$178$	0	0
$179$	117.393i	0.655829i	0.944707	+	0.327914i	$0.106346\pi$
$179$	117.393i	0.655829i	−0.944707	+	0.327914i	$0.893654\pi$
$180$	0	0
$181$	26.6687	0.147341	0.0736705	−	0.997283i	$-0.476529\pi$
$181$	26.6687	0.147341	0.0736705	+	0.997283i	$0.476529\pi$
$182$	0	0
$183$	− 108.296i	− 0.591782i
$184$	0	0
$185$	14.7214	0.0795749
$186$	0	0
$187$	− 77.0736i	− 0.412158i
$188$	0	0
$189$	260.498	1.37830
$190$	0	0
$191$	80.1060i	0.419403i	0.977765	+	0.209702i	$0.0672492\pi$
$191$	80.1060i	0.419403i	−0.977765	+	0.209702i	$0.932751\pi$
$192$	0	0
$193$	−86.9969	−0.450761	−0.225381	−	0.974271i	$-0.572363\pi$
$193$	−86.9969	−0.450761	−0.225381	+	0.974271i	$0.572363\pi$
$194$	0	0
$195$	45.0789i	0.231174i
$196$	0	0
$197$	278.912	1.41580	0.707898	−	0.706315i	$-0.249644\pi$
$197$	278.912	1.41580	0.707898	+	0.706315i	$0.249644\pi$
$198$	0	0
$199$	− 27.7128i	− 0.139260i	−0.997573	−	0.0696302i	$-0.977818\pi$
$199$	− 27.7128i	− 0.139260i	0.997573	−	0.0696302i	$-0.0221819\pi$
$200$	0	0
$201$	70.9180	0.352826
$202$	0	0
$203$	406.595i	2.00293i
$204$	0	0
$205$	−116.833	−0.569916
$206$	0	0
$207$	− 101.249i	− 0.489123i
$208$	0	0
$209$	155.331	0.743212
$210$	0	0
$211$	232.526i	1.10202i	0.834498	+	0.551011i	$0.185758\pi$
$211$	232.526i	1.10202i	−0.834498	+	0.551011i	$0.814242\pi$
$212$	0	0
$213$	43.1610	0.202634
$214$	0	0
$215$	− 64.4948i	− 0.299976i
$216$	0	0
$217$	319.830	1.47387
$218$	0	0
$219$	196.250i	0.896118i
$220$	0	0
$221$	169.495	0.766947
$222$	0	0
$223$	242.073i	1.08553i	0.839885	+	0.542764i	$0.182622\pi$
$223$	242.073i	1.08553i	−0.839885	+	0.542764i	$0.817378\pi$
$224$	0	0
$225$	22.0820	0.0981424
$226$	0	0
$227$	− 247.274i	− 1.08931i	−0.838659	−	0.544657i	$-0.816660\pi$
$227$	− 247.274i	− 1.08931i	0.838659	−	0.544657i	$-0.183340\pi$
$228$	0	0
$229$	−71.4953	−0.312207	−0.156103	−	0.987741i	$-0.549893\pi$
$229$	−71.4953	−0.312207	−0.156103	+	0.987741i	$0.549893\pi$
$230$	0	0
$231$	− 83.1384i	− 0.359907i
$232$	0	0
$233$	−279.167	−1.19814	−0.599071	−	0.800696i	$-0.704463\pi$
$233$	−279.167	−1.19814	−0.599071	+	0.800696i	$0.704463\pi$
$234$	0	0
$235$	− 201.662i	− 0.858136i
$236$	0	0
$237$	−91.6718	−0.386801
$238$	0	0
$239$	− 446.381i	− 1.86770i	−0.357660	−	0.933852i	$-0.616425\pi$
$239$	− 446.381i	− 1.86770i	0.357660	−	0.933852i	$-0.383575\pi$
$240$	0	0
$241$	−105.416	−0.437412	−0.218706	−	0.975791i	$-0.570184\pi$
$241$	−105.416	−0.437412	−0.218706	+	0.975791i	$0.570184\pi$
$242$	0	0
$243$	− 211.952i	− 0.872232i
$244$	0	0
$245$	74.3475	0.303459
$246$	0	0
$247$	341.595i	1.38297i
$248$	0	0
$249$	47.7446	0.191745
$250$	0	0
$251$	− 419.974i	− 1.67320i	−0.547812	−	0.836602i	$-0.684539\pi$
$251$	− 419.974i	− 1.67320i	0.547812	−	0.836602i	$-0.315461\pi$
$252$	0	0
$253$	−98.1641	−0.388000
$254$	0	0
$255$	86.1709i	0.337925i
$256$	0	0
$257$	289.830	1.12774	0.563871	−	0.825863i	$-0.309311\pi$
$257$	289.830	1.12774	0.563871	+	0.825863i	$0.309311\pi$
$258$	0	0
$259$	− 59.7075i	− 0.230531i
$260$	0	0
$261$	198.000	0.758621
$262$	0	0
$263$	− 262.528i	− 0.998204i	−0.866543	−	0.499102i	$-0.833663\pi$
$263$	− 262.528i	− 0.998204i	0.866543	−	0.499102i	$-0.166337\pi$
$264$	0	0
$265$	116.833	0.440879
$266$	0	0
$267$	− 102.049i	− 0.382206i
$268$	0	0
$269$	178.584	0.663880	0.331940	−	0.943301i	$-0.392297\pi$
$269$	178.584	0.663880	0.331940	+	0.943301i	$0.392297\pi$
$270$	0	0
$271$	217.126i	0.801202i	0.916253	+	0.400601i	$0.131199\pi$
$271$	217.126i	0.801202i	−0.916253	+	0.400601i	$0.868801\pi$
$272$	0	0
$273$	182.833	0.669717
$274$	0	0
$275$	− 21.4093i	− 0.0778521i
$276$	0	0
$277$	−159.410	−0.575488	−0.287744	−	0.957707i	$-0.592905\pi$
$277$	−159.410	−0.575488	−0.287744	+	0.957707i	$0.592905\pi$
$278$	0	0
$279$	− 155.748i	− 0.558236i
$280$	0	0
$281$	196.249	0.698396	0.349198	−	0.937049i	$-0.386454\pi$
$281$	196.249	0.698396	0.349198	+	0.937049i	$0.386454\pi$
$282$	0	0
$283$	− 37.9402i	− 0.134064i	−0.997751	−	0.0670322i	$-0.978647\pi$
$283$	− 37.9402i	− 0.134064i	0.997751	−	0.0670322i	$-0.0213530\pi$
$284$	0	0
$285$	−173.666	−0.609353
$286$	0	0
$287$	473.855i	1.65106i
$288$	0	0
$289$	35.0000	0.121107
$290$	0	0
$291$	342.900i	1.17835i
$292$	0	0
$293$	258.413	0.881957	0.440978	−	0.897518i	$-0.354632\pi$
$293$	258.413	0.881957	0.440978	+	0.897518i	$0.354632\pi$
$294$	0	0
$295$	38.2982i	0.129824i
$296$	0	0
$297$	−122.991	−0.414110
$298$	0	0
$299$	− 215.876i	− 0.721994i
$300$	0	0
$301$	−261.580	−0.869038
$302$	0	0
$303$	50.6665i	0.167216i
$304$	0	0
$305$	−113.108	−0.370847
$306$	0	0
$307$	− 143.737i	− 0.468200i	−0.972213	−	0.234100i	$-0.924786\pi$
$307$	− 143.737i	− 0.468200i	0.972213	−	0.234100i	$-0.0752143\pi$
$308$	0	0
$309$	73.9086	0.239187
$310$	0	0
$311$	− 242.817i	− 0.780762i	−0.920653	−	0.390381i	$-0.872343\pi$
$311$	− 242.817i	− 0.780762i	0.920653	−	0.390381i	$-0.127657\pi$
$312$	0	0
$313$	2.00000	0.00638978	0.00319489	−	0.999995i	$-0.498983\pi$
$313$	2.00000	0.00638978	0.00319489	+	0.999995i	$0.498983\pi$
$314$	0	0
$315$	− 89.5612i	− 0.284321i
$316$	0	0
$317$	−490.741	−1.54808	−0.774040	−	0.633137i	$-0.781767\pi$
$317$	−490.741	−1.54808	−0.774040	+	0.633137i	$0.781767\pi$
$318$	0	0
$319$	− 191.968i	− 0.601781i
$320$	0	0
$321$	145.909	0.454544
$322$	0	0
$323$	652.978i	2.02160i
$324$	0	0
$325$	47.0820	0.144868
$326$	0	0
$327$	62.9785i	0.192595i
$328$	0	0
$329$	−817.909	−2.48604
$330$	0	0
$331$	1.78300i	0.00538671i	0.999996	+	0.00269336i	$0.000857323\pi$
$331$	1.78300i	0.00538671i	−0.999996	+	0.00269336i	$0.999143\pi$
$332$	0	0
$333$	−29.0758	−0.0873148
$334$	0	0
$335$	− 74.0693i	− 0.221102i
$336$	0	0
$337$	−623.994	−1.85161	−0.925807	−	0.377997i	$-0.876613\pi$
$337$	−623.994	−1.85161	−0.925807	+	0.377997i	$0.876613\pi$
$338$	0	0
$339$	416.408i	1.22834i
$340$	0	0
$341$	−151.003	−0.442824
$342$	0	0
$343$	142.846i	0.416460i
$344$	0	0
$345$	109.751	0.318118
$346$	0	0
$347$	266.367i	0.767629i	0.923410	+	0.383814i	$0.125390\pi$
$347$	266.367i	0.767629i	−0.923410	+	0.383814i	$0.874610\pi$
$348$	0	0
$349$	12.8328	0.0367702	0.0183851	−	0.999831i	$-0.494148\pi$
$349$	12.8328	0.0367702	0.0183851	+	0.999831i	$0.494148\pi$
$350$	0	0
$351$	− 270.473i	− 0.770579i
$352$	0	0
$353$	303.325	0.859278	0.429639	−	0.903001i	$-0.358641\pi$
$353$	303.325	0.859278	0.429639	+	0.903001i	$0.358641\pi$
$354$	0	0
$355$	− 45.0789i	− 0.126983i
$356$	0	0
$357$	349.495	0.978979
$358$	0	0
$359$	449.947i	1.25333i	0.779287	+	0.626667i	$0.215581\pi$
$359$	449.947i	1.25333i	−0.779287	+	0.626667i	$0.784419\pi$
$360$	0	0
$361$	−954.988	−2.64539
$362$	0	0
$363$	− 219.800i	− 0.605510i
$364$	0	0
$365$	204.971	0.561563
$366$	0	0
$367$	− 474.361i	− 1.29254i	−0.763111	−	0.646268i	$-0.776329\pi$
$367$	− 474.361i	− 1.29254i	0.763111	−	0.646268i	$-0.223671\pi$
$368$	0	0
$369$	230.754	0.625349
$370$	0	0
$371$	− 473.855i	− 1.27724i
$372$	0	0
$373$	−104.748	−0.280825	−0.140412	−	0.990093i	$-0.544843\pi$
$373$	−104.748	−0.280825	−0.140412	+	0.990093i	$0.544843\pi$
$374$	0	0
$375$	23.9364i	0.0638303i
$376$	0	0
$377$	422.164	1.11980
$378$	0	0
$379$	465.053i	1.22705i	0.789674	+	0.613526i	$0.210250\pi$
$379$	465.053i	1.22705i	−0.789674	+	0.613526i	$0.789750\pi$
$380$	0	0
$381$	−325.240	−0.853648
$382$	0	0
$383$	− 421.490i	− 1.10050i	−0.835001	−	0.550248i	$-0.814533\pi$
$383$	− 421.490i	− 1.10050i	0.835001	−	0.550248i	$-0.185467\pi$
$384$	0	0
$385$	−86.8328	−0.225540
$386$	0	0
$387$	127.382i	0.329153i
$388$	0	0
$389$	−468.079	−1.20329	−0.601644	−	0.798764i	$-0.705487\pi$
$389$	−468.079	−1.20329	−0.601644	+	0.798764i	$0.705487\pi$
$390$	0	0
$391$	− 412.660i	− 1.05540i
$392$	0	0
$393$	−310.152	−0.789190
$394$	0	0
$395$	95.7454i	0.242393i
$396$	0	0
$397$	−653.246	−1.64546	−0.822728	−	0.568435i	$-0.807549\pi$
$397$	−653.246	−1.64546	−0.822728	+	0.568435i	$0.807549\pi$
$398$	0	0
$399$	704.360i	1.76531i
$400$	0	0
$401$	13.6718	0.0340944	0.0170472	−	0.999855i	$-0.494573\pi$
$401$	13.6718	0.0340944	0.0170472	+	0.999855i	$0.494573\pi$
$402$	0	0
$403$	− 332.077i	− 0.824011i
$404$	0	0
$405$	48.6293	0.120072
$406$	0	0
$407$	28.1901i	0.0692630i
$408$	0	0
$409$	107.082	0.261814	0.130907	−	0.991395i	$-0.458211\pi$
$409$	107.082	0.261814	0.130907	+	0.991395i	$0.458211\pi$
$410$	0	0
$411$	− 160.928i	− 0.391552i
$412$	0	0
$413$	155.331	0.376105
$414$	0	0
$415$	− 49.8662i	− 0.120159i
$416$	0	0
$417$	−434.675	−1.04239
$418$	0	0
$419$	− 383.936i	− 0.916315i	−0.888871	−	0.458158i	$-0.848510\pi$
$419$	− 383.936i	− 0.916315i	0.888871	−	0.458158i	$-0.151490\pi$
$420$	0	0
$421$	−180.420	−0.428550	−0.214275	−	0.976773i	$-0.568739\pi$
$421$	−180.420	−0.428550	−0.214275	+	0.976773i	$0.568739\pi$
$422$	0	0
$423$	398.298i	0.941603i
$424$	0	0
$425$	90.0000	0.211765
$426$	0	0
$427$	458.749i	1.07435i
$428$	0	0
$429$	−86.3219	−0.201217
$430$	0	0
$431$	703.827i	1.63301i	0.577340	+	0.816504i	$0.304091\pi$
$431$	703.827i	1.63301i	−0.577340	+	0.816504i	$0.695909\pi$
$432$	0	0
$433$	461.502	1.06582	0.532912	−	0.846171i	$-0.321098\pi$
$433$	461.502	1.06582	0.532912	+	0.846171i	$0.321098\pi$
$434$	0	0
$435$	214.627i	0.493395i
$436$	0	0
$437$	831.659	1.90311
$438$	0	0
$439$	56.9700i	0.129772i	0.997893	+	0.0648861i	$0.0206684\pi$
$439$	56.9700i	0.129772i	−0.997893	+	0.0648861i	$0.979332\pi$
$440$	0	0
$441$	−146.842	−0.332975
$442$	0	0
$443$	− 485.276i	− 1.09543i	−0.836665	−	0.547715i	$-0.815498\pi$
$443$	− 485.276i	− 1.09543i	0.836665	−	0.547715i	$-0.184502\pi$
$444$	0	0
$445$	−106.584	−0.239514
$446$	0	0
$447$	209.994i	0.469785i
$448$	0	0
$449$	712.407	1.58665	0.793326	−	0.608797i	$-0.208347\pi$
$449$	712.407	1.58665	0.793326	+	0.608797i	$0.208347\pi$
$450$	0	0
$451$	− 223.724i	− 0.496062i
$452$	0	0
$453$	93.8359	0.207143
$454$	0	0
$455$	− 190.957i	− 0.419686i
$456$	0	0
$457$	−427.666	−0.935811	−0.467906	−	0.883779i	$-0.654991\pi$
$457$	−427.666	−0.935811	−0.467906	+	0.883779i	$0.654991\pi$
$458$	0	0
$459$	− 517.025i	− 1.12642i
$460$	0	0
$461$	−592.663	−1.28560	−0.642801	−	0.766033i	$-0.722228\pi$
$461$	−592.663	−1.28560	−0.642801	+	0.766033i	$0.722228\pi$
$462$	0	0
$463$	70.0261i	0.151244i	0.997137	+	0.0756221i	$0.0240943\pi$
$463$	70.0261i	0.151244i	−0.997137	+	0.0756221i	$0.975906\pi$
$464$	0	0
$465$	168.827	0.363068
$466$	0	0
$467$	79.2145i	0.169624i	0.996397	+	0.0848121i	$0.0270290\pi$
$467$	79.2145i	0.169624i	−0.996397	+	0.0848121i	$0.972971\pi$
$468$	0	0
$469$	−300.413	−0.640540
$470$	0	0
$471$	− 88.6697i	− 0.188258i
$472$	0	0
$473$	123.502	0.261103
$474$	0	0
$475$	181.383i	0.381858i
$476$	0	0
$477$	−230.754	−0.483761
$478$	0	0
$479$	− 140.586i	− 0.293498i	−0.989174	−	0.146749i	$-0.953119\pi$
$479$	− 140.586i	− 0.293498i	0.989174	−	0.146749i	$-0.0468810\pi$
$480$	0	0
$481$	−61.9938	−0.128885
$482$	0	0
$483$	− 445.132i	− 0.921598i
$484$	0	0
$485$	358.138	0.738428
$486$	0	0
$487$	− 644.386i	− 1.32317i	−0.749868	−	0.661587i	$-0.769883\pi$
$487$	− 644.386i	− 1.32317i	0.749868	−	0.661587i	$-0.230117\pi$
$488$	0	0
$489$	−392.912	−0.803501
$490$	0	0
$491$	416.408i	0.848082i	0.905643	+	0.424041i	$0.139389\pi$
$491$	416.408i	0.848082i	−0.905643	+	0.424041i	$0.860611\pi$
$492$	0	0
$493$	806.991	1.63690
$494$	0	0
$495$	42.2851i	0.0854244i
$496$	0	0
$497$	−182.833	−0.367873
$498$	0	0
$499$	− 447.925i	− 0.897646i	−0.893621	−	0.448823i	$-0.851843\pi$
$499$	− 447.925i	− 0.897646i	0.893621	−	0.448823i	$-0.148157\pi$
$500$	0	0
$501$	−115.927	−0.231392
$502$	0	0
$503$	5.26451i	0.0104662i	0.999986	+	0.00523311i	$0.00166576\pi$
$503$	5.26451i	0.0104662i	−0.999986	+	0.00523311i	$0.998334\pi$
$504$	0	0
$505$	52.9180	0.104788
$506$	0	0
$507$	171.984i	0.339219i
$508$	0	0
$509$	−416.833	−0.818925	−0.409462	−	0.912327i	$-0.634284\pi$
$509$	−416.833	−0.818925	−0.409462	+	0.912327i	$0.634284\pi$
$510$	0	0
$511$	− 831.328i	− 1.62687i
$512$	0	0
$513$	1041.99	2.03118
$514$	0	0
$515$	− 77.1929i	− 0.149889i
$516$	0	0
$517$	386.164	0.746932
$518$	0	0
$519$	− 157.180i	− 0.302851i
$520$	0	0
$521$	−384.334	−0.737686	−0.368843	−	0.929492i	$-0.620246\pi$
$521$	−384.334	−0.737686	−0.368843	+	0.929492i	$0.620246\pi$
$522$	0	0
$523$	663.921i	1.26945i	0.772739	+	0.634724i	$0.218886\pi$
$523$	663.921i	1.26945i	−0.772739	+	0.634724i	$0.781114\pi$
$524$	0	0
$525$	97.0820	0.184918
$526$	0	0
$527$	− 634.783i	− 1.20452i
$528$	0	0
$529$	3.41951	0.00646411
$530$	0	0
$531$	− 75.6419i	− 0.142452i
$532$	0	0
$533$	492.000	0.923077
$534$	0	0
$535$	− 152.392i	− 0.284845i
$536$	0	0
$537$	251.331	0.468028
$538$	0	0
$539$	142.369i	0.264135i
$540$	0	0
$541$	732.663	1.35427	0.677137	−	0.735857i	$-0.263220\pi$
$541$	732.663	1.35427	0.677137	+	0.735857i	$0.263220\pi$
$542$	0	0
$543$	− 57.0960i	− 0.105149i
$544$	0	0
$545$	65.7771	0.120692
$546$	0	0
$547$	655.119i	1.19766i	0.800877	+	0.598829i	$0.204367\pi$
$547$	655.119i	1.19766i	−0.800877	+	0.598829i	$0.795633\pi$
$548$	0	0
$549$	223.398	0.406918
$550$	0	0
$551$	1626.38i	2.95169i
$552$	0	0
$553$	388.328	0.702221
$554$	0	0
$555$	− 31.5174i	− 0.0567882i
$556$	0	0
$557$	−394.741	−0.708692	−0.354346	−	0.935114i	$-0.615296\pi$
$557$	−394.741	−0.708692	−0.354346	+	0.935114i	$0.615296\pi$
$558$	0	0
$559$	271.597i	0.485862i
$560$	0	0
$561$	−165.009	−0.294134
$562$	0	0
$563$	492.057i	0.873990i	0.899464	+	0.436995i	$0.143957\pi$
$563$	492.057i	0.873990i	−0.899464	+	0.436995i	$0.856043\pi$
$564$	0	0
$565$	434.912	0.769755
$566$	0	0
$567$	− 197.233i	− 0.347853i
$568$	0	0
$569$	−340.918	−0.599153	−0.299576	−	0.954072i	$-0.596845\pi$
$569$	−340.918	−0.599153	−0.299576	+	0.954072i	$0.596845\pi$
$570$	0	0
$571$	452.685i	0.792793i	0.918079	+	0.396396i	$0.129739\pi$
$571$	452.685i	0.792793i	−0.918079	+	0.396396i	$0.870261\pi$
$572$	0	0
$573$	171.502	0.299305
$574$	0	0
$575$	− 114.628i	− 0.199353i
$576$	0	0
$577$	−372.833	−0.646157	−0.323079	−	0.946372i	$-0.604718\pi$
$577$	−372.833	−0.646157	−0.323079	+	0.946372i	$0.604718\pi$
$578$	0	0
$579$	186.254i	0.321683i
$580$	0	0
$581$	−202.249	−0.348105
$582$	0	0
$583$	223.724i	0.383746i
$584$	0	0
$585$	−92.9907	−0.158958
$586$	0	0
$587$	− 977.508i	− 1.66526i	−0.553829	−	0.832630i	$-0.686834\pi$
$587$	− 977.508i	− 1.66526i	0.553829	−	0.832630i	$-0.313166\pi$
$588$	0	0
$589$	1279.32	2.17202
$590$	0	0
$591$	− 597.131i	− 1.01037i
$592$	0	0
$593$	111.325	0.187732	0.0938660	−	0.995585i	$-0.470077\pi$
$593$	111.325	0.187732	0.0938660	+	0.995585i	$0.470077\pi$
$594$	0	0
$595$	− 365.026i	− 0.613489i
$596$	0	0
$597$	−59.3313	−0.0993823
$598$	0	0
$599$	− 333.985i	− 0.557572i	−0.960353	−	0.278786i	$-0.910068\pi$
$599$	− 333.985i	− 0.557572i	0.960353	−	0.278786i	$-0.0899320\pi$
$600$	0	0
$601$	913.234	1.51952	0.759762	−	0.650202i	$-0.225316\pi$
$601$	913.234	1.51952	0.759762	+	0.650202i	$0.225316\pi$
$602$	0	0
$603$	146.293i	0.242608i
$604$	0	0
$605$	−229.567	−0.379450
$606$	0	0
$607$	738.166i	1.21609i	0.793903	+	0.608044i	$0.208046\pi$
$607$	738.166i	1.21609i	−0.793903	+	0.608044i	$0.791954\pi$
$608$	0	0
$609$	870.492	1.42938
$610$	0	0
$611$	849.228i	1.38990i
$612$	0	0
$613$	−123.082	−0.200786	−0.100393	−	0.994948i	$-0.532010\pi$
$613$	−123.082	−0.200786	−0.100393	+	0.994948i	$0.532010\pi$
$614$	0	0
$615$	250.131i	0.406717i
$616$	0	0
$617$	598.839	0.970566	0.485283	−	0.874357i	$-0.338717\pi$
$617$	598.839	0.970566	0.485283	+	0.874357i	$0.338717\pi$
$618$	0	0
$619$	− 201.009i	− 0.324732i	−0.986731	−	0.162366i	$-0.948087\pi$
$619$	− 201.009i	− 0.324732i	0.986731	−	0.162366i	$-0.0519125\pi$
$620$	0	0
$621$	−658.505	−1.06039
$622$	0	0
$623$	432.286i	0.693878i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	− 332.554i	− 0.530389i
$628$	0	0
$629$	−118.505	−0.188402
$630$	0	0
$631$	549.202i	0.870368i	0.900342	+	0.435184i	$0.143317\pi$
$631$	549.202i	0.870368i	−0.900342	+	0.435184i	$0.856683\pi$
$632$	0	0
$633$	497.823	0.786451
$634$	0	0
$635$	339.692i	0.534949i
$636$	0	0
$637$	−313.088	−0.491504
$638$	0	0
$639$	89.0343i	0.139334i
$640$	0	0
$641$	−498.413	−0.777556	−0.388778	−	0.921331i	$-0.627103\pi$
$641$	−498.413	−0.777556	−0.388778	+	0.921331i	$0.627103\pi$
$642$	0	0
$643$	246.025i	0.382620i	0.981530	+	0.191310i	$0.0612737\pi$
$643$	246.025i	0.382620i	−0.981530	+	0.191310i	$0.938726\pi$
$644$	0	0
$645$	−138.079	−0.214076
$646$	0	0
$647$	390.984i	0.604302i	0.953260	+	0.302151i	$0.0977047\pi$
$647$	390.984i	0.604302i	−0.953260	+	0.302151i	$0.902295\pi$
$648$	0	0
$649$	−73.3375	−0.113001
$650$	0	0
$651$	− 684.734i	− 1.05182i
$652$	0	0
$653$	459.580	0.703799	0.351899	−	0.936038i	$-0.385536\pi$
$653$	459.580	0.703799	0.351899	+	0.936038i	$0.385536\pi$
$654$	0	0
$655$	323.934i	0.494555i
$656$	0	0
$657$	−404.833	−0.616184
$658$	0	0
$659$	− 743.613i	− 1.12840i	−0.825640	−	0.564198i	$-0.809186\pi$
$659$	− 743.613i	− 1.12840i	0.825640	−	0.564198i	$-0.190814\pi$
$660$	0	0
$661$	−342.073	−0.517508	−0.258754	−	0.965943i	$-0.583312\pi$
$661$	−342.073	−0.517508	−0.258754	+	0.965943i	$0.583312\pi$
$662$	0	0
$663$	− 362.878i	− 0.547327i
$664$	0	0
$665$	735.659	1.10625
$666$	0	0
$667$	− 1027.82i	− 1.54095i
$668$	0	0
$669$	518.262	0.774681
$670$	0	0
$671$	− 216.592i	− 0.322790i
$672$	0	0
$673$	95.8359	0.142401	0.0712005	−	0.997462i	$-0.477317\pi$
$673$	95.8359	0.142401	0.0712005	+	0.997462i	$0.477317\pi$
$674$	0	0
$675$	− 143.618i	− 0.212768i
$676$	0	0
$677$	251.921	0.372114	0.186057	−	0.982539i	$-0.440429\pi$
$677$	251.921	0.372114	0.186057	+	0.982539i	$0.440429\pi$
$678$	0	0
$679$	− 1452.55i	− 2.13925i
$680$	0	0
$681$	−529.398	−0.777383
$682$	0	0
$683$	− 767.697i	− 1.12401i	−0.827135	−	0.562003i	$-0.810031\pi$
$683$	− 767.697i	− 1.12401i	0.827135	−	0.562003i	$-0.189969\pi$
$684$	0	0
$685$	−168.079	−0.245371
$686$	0	0
$687$	153.067i	0.222805i
$688$	0	0
$689$	−492.000	−0.714078
$690$	0	0
$691$	− 70.2928i	− 0.101726i	−0.998706	−	0.0508631i	$-0.983803\pi$
$691$	− 70.2928i	− 0.101726i	0.998706	−	0.0508631i	$-0.0161972\pi$
$692$	0	0
$693$	171.502	0.247477
$694$	0	0
$695$	453.990i	0.653224i
$696$	0	0
$697$	940.486	1.34933
$698$	0	0
$699$	597.678i	0.855047i
$700$	0	0
$701$	−102.413	−0.146096	−0.0730480	−	0.997328i	$-0.523273\pi$
$701$	−102.413	−0.146096	−0.0730480	+	0.997328i	$0.523273\pi$
$702$	0	0
$703$	− 238.830i	− 0.339730i
$704$	0	0
$705$	−431.745	−0.612404
$706$	0	0
$707$	− 214.627i	− 0.303574i
$708$	0	0
$709$	−788.820	−1.11258	−0.556291	−	0.830988i	$-0.687776\pi$
$709$	−788.820	−1.11258	−0.556291	+	0.830988i	$0.687776\pi$
$710$	0	0
$711$	− 189.105i	− 0.265970i
$712$	0	0
$713$	−808.486	−1.13392
$714$	0	0
$715$	90.1578i	0.126095i
$716$	0	0
$717$	−955.672	−1.33288
$718$	0	0
$719$	− 612.658i	− 0.852097i	−0.904700	−	0.426049i	$-0.859905\pi$
$719$	− 612.658i	− 0.852097i	0.904700	−	0.426049i	$-0.140095\pi$
$720$	0	0
$721$	−313.082	−0.434233
$722$	0	0
$723$	225.689i	0.312157i
$724$	0	0
$725$	224.164	0.309192
$726$	0	0
$727$	537.522i	0.739369i	0.929157	+	0.369685i	$0.120534\pi$
$727$	537.522i	0.739369i	−0.929157	+	0.369685i	$0.879466\pi$
$728$	0	0
$729$	−649.505	−0.890953
$730$	0	0
$731$	519.173i	0.710223i
$732$	0	0
$733$	612.407	0.835480	0.417740	−	0.908567i	$-0.362822\pi$
$733$	612.407	0.835480	0.417740	+	0.908567i	$0.362822\pi$
$734$	0	0
$735$	− 159.173i	− 0.216562i
$736$	0	0
$737$	141.836	0.192450
$738$	0	0
$739$	1195.69i	1.61799i	0.587816	+	0.808995i	$0.299988\pi$
$739$	1195.69i	1.61799i	−0.587816	+	0.808995i	$0.700012\pi$
$740$	0	0
$741$	731.331	0.986952
$742$	0	0
$743$	− 762.790i	− 1.02664i	−0.858199	−	0.513318i	$-0.828416\pi$
$743$	− 762.790i	− 1.02664i	0.858199	−	0.513318i	$-0.171584\pi$
$744$	0	0
$745$	219.325	0.294396
$746$	0	0
$747$	98.4895i	0.131847i
$748$	0	0
$749$	−618.079	−0.825206
$750$	0	0
$751$	761.274i	1.01368i	0.862040	+	0.506840i	$0.169187\pi$
$751$	761.274i	1.01368i	−0.862040	+	0.506840i	$0.830813\pi$
$752$	0	0
$753$	−899.136	−1.19407
$754$	0	0
$755$	− 98.0057i	− 0.129809i
$756$	0	0
$757$	−416.091	−0.549658	−0.274829	−	0.961493i	$-0.588621\pi$
$757$	−416.091	−0.549658	−0.274829	+	0.961493i	$0.588621\pi$
$758$	0	0
$759$	210.163i	0.276894i
$760$	0	0
$761$	−765.659	−1.00612	−0.503061	−	0.864251i	$-0.667793\pi$
$761$	−765.659	−1.00612	−0.503061	+	0.864251i	$0.667793\pi$
$762$	0	0
$763$	− 266.781i	− 0.349648i
$764$	0	0
$765$	−177.757	−0.232362
$766$	0	0
$767$	− 161.279i	− 0.210273i
$768$	0	0
$769$	71.0093	0.0923398	0.0461699	−	0.998934i	$-0.485298\pi$
$769$	71.0093	0.0923398	0.0461699	+	0.998934i	$0.485298\pi$
$770$	0	0
$771$	− 620.506i	− 0.804807i
$772$	0	0
$773$	−1010.09	−1.30671	−0.653354	−	0.757053i	$-0.726639\pi$
$773$	−1010.09	−1.30671	−0.653354	+	0.757053i	$0.726639\pi$
$774$	0	0
$775$	− 176.329i	− 0.227521i
$776$	0	0
$777$	−127.830	−0.164517
$778$	0	0
$779$	1895.42i	2.43315i
$780$	0	0
$781$	86.3219	0.110527
$782$	0	0
$783$	− 1287.76i	− 1.64465i
$784$	0	0
$785$	−92.6099	−0.117974
$786$	0	0
$787$	− 171.927i	− 0.218459i	−0.994017	−	0.109230i	$-0.965162\pi$
$787$	− 171.927i	− 0.218459i	0.994017	−	0.109230i	$-0.0348384\pi$
$788$	0	0
$789$	−562.054	−0.712363
$790$	0	0
$791$	− 1763.93i	− 2.23000i
$792$	0	0
$793$	476.316	0.600650
$794$	0	0
$795$	− 250.131i	− 0.314630i
$796$	0	0
$797$	907.240	1.13832	0.569159	−	0.822227i	$-0.307269\pi$
$797$	907.240	1.13832	0.569159	+	0.822227i	$0.307269\pi$
$798$	0	0
$799$	1623.35i	2.03172i
$800$	0	0
$801$	210.511	0.262810
$802$	0	0
$803$	392.500i	0.488792i
$804$	0	0
$805$	−464.912	−0.577530
$806$	0	0
$807$	− 382.335i	− 0.473774i
$808$	0	0
$809$	−991.653	−1.22578	−0.612888	−	0.790170i	$-0.709992\pi$
$809$	−991.653	−1.22578	−0.612888	+	0.790170i	$0.709992\pi$
$810$	0	0
$811$	121.198i	0.149443i	0.997204	+	0.0747213i	$0.0238067\pi$
$811$	121.198i	0.149443i	−0.997204	+	0.0747213i	$0.976193\pi$
$812$	0	0
$813$	464.851	0.571773
$814$	0	0
$815$	410.371i	0.503523i
$816$	0	0
$817$	−1046.32	−1.28069
$818$	0	0
$819$	377.155i	0.460507i
$820$	0	0
$821$	−853.732	−1.03987	−0.519934	−	0.854206i	$-0.674044\pi$
$821$	−853.732	−1.03987	−0.519934	+	0.854206i	$0.674044\pi$
$822$	0	0
$823$	834.037i	1.01341i	0.862119	+	0.506706i	$0.169137\pi$
$823$	834.037i	1.01341i	−0.862119	+	0.506706i	$0.830863\pi$
$824$	0	0
$825$	−45.8359	−0.0555587
$826$	0	0
$827$	− 205.705i	− 0.248737i	−0.992236	−	0.124368i	$-0.960310\pi$
$827$	− 205.705i	− 0.248737i	0.992236	−	0.124368i	$-0.0396904\pi$
$828$	0	0
$829$	1334.57	1.60986	0.804928	−	0.593372i	$-0.202204\pi$
$829$	1334.57	1.60986	0.804928	+	0.593372i	$0.202204\pi$
$830$	0	0
$831$	341.286i	0.410694i
$832$	0	0
$833$	−598.486	−0.718471
$834$	0	0
$835$	121.079i	0.145004i
$836$	0	0
$837$	−1012.96	−1.21023
$838$	0	0
$839$	130.955i	0.156084i	0.996950	+	0.0780422i	$0.0248669\pi$
$839$	130.955i	0.156084i	−0.996950	+	0.0780422i	$0.975133\pi$
$840$	0	0
$841$	1168.98	1.38999
$842$	0	0
$843$	− 420.156i	− 0.498406i
$844$	0	0
$845$	179.626	0.212575
$846$	0	0
$847$	931.089i	1.09928i
$848$	0	0
$849$	−81.2275	−0.0956743
$850$	0	0
$851$	150.932i	0.177359i
$852$	0	0
$853$	−319.410	−0.374455	−0.187228	−	0.982317i	$-0.559950\pi$
$853$	−319.410	−0.374455	−0.187228	+	0.982317i	$0.559950\pi$
$854$	0	0
$855$	− 358.245i	− 0.419000i
$856$	0	0
$857$	413.842	0.482896	0.241448	−	0.970414i	$-0.422378\pi$
$857$	413.842	0.482896	0.241448	+	0.970414i	$0.422378\pi$
$858$	0	0
$859$	− 1707.02i	− 1.98722i	−0.112882	−	0.993608i	$-0.536008\pi$
$859$	− 1707.02i	− 1.98722i	0.112882	−	0.993608i	$-0.463992\pi$
$860$	0	0
$861$	1014.49	1.17827
$862$	0	0
$863$	973.486i	1.12803i	0.825766	+	0.564013i	$0.190743\pi$
$863$	973.486i	1.12803i	−0.825766	+	0.564013i	$0.809257\pi$
$864$	0	0
$865$	−164.164	−0.189785
$866$	0	0
$867$	− 74.9326i	− 0.0864275i
$868$	0	0
$869$	−183.344	−0.210982
$870$	0	0
$871$	311.917i	0.358113i
$872$	0	0
$873$	−707.350	−0.810252
$874$	0	0
$875$	− 101.396i	− 0.115881i
$876$	0	0
$877$	9.07583	0.0103487	0.00517436	−	0.999987i	$-0.498353\pi$
$877$	9.07583	0.0103487	0.00517436	+	0.999987i	$0.498353\pi$
$878$	0	0
$879$	− 553.245i	− 0.629403i
$880$	0	0
$881$	851.568	0.966593	0.483296	−	0.875457i	$-0.339439\pi$
$881$	851.568	0.966593	0.483296	+	0.875457i	$0.339439\pi$
$882$	0	0
$883$	128.927i	0.146010i	0.997332	+	0.0730048i	$0.0232589\pi$
$883$	128.927i	0.146010i	−0.997332	+	0.0730048i	$0.976741\pi$
$884$	0	0
$885$	81.9938	0.0926483
$886$	0	0
$887$	− 451.463i	− 0.508978i	−0.967076	−	0.254489i	$-0.918093\pi$
$887$	− 451.463i	− 0.508978i	0.967076	−	0.254489i	$-0.0819072\pi$
$888$	0	0
$889$	1377.74	1.54976
$890$	0	0
$891$	93.1206i	0.104512i
$892$	0	0
$893$	−3271.63	−3.66364
$894$	0	0
$895$	− 262.500i	− 0.293296i
$896$	0	0
$897$	−462.177	−0.515247
$898$	0	0
$899$	− 1581.06i	− 1.75869i
$900$	0	0
$901$	−940.486	−1.04382
$902$	0	0
$903$	560.026i	0.620184i
$904$	0	0
$905$	−59.6331	−0.0658929
$906$	0	0
$907$	873.845i	0.963446i	0.876324	+	0.481723i	$0.159989\pi$
$907$	873.845i	0.963446i	−0.876324	+	0.481723i	$0.840011\pi$
$908$	0	0
$909$	−104.517	−0.114980
$910$	0	0
$911$	− 204.631i	− 0.224623i	−0.993673	−	0.112311i	$-0.964175\pi$
$911$	− 204.631i	− 0.224623i	0.993673	−	0.112311i	$-0.0358254\pi$
$912$	0	0
$913$	95.4891	0.104588
$914$	0	0
$915$	242.157i	0.264653i
$916$	0	0
$917$	1313.82	1.43274
$918$	0	0
$919$	526.431i	0.572830i	0.958106	+	0.286415i	$0.0924636\pi$
$919$	526.431i	0.572830i	−0.958106	+	0.286415i	$0.907536\pi$
$920$	0	0
$921$	−307.732	−0.334128
$922$	0	0
$923$	189.834i	0.205670i
$924$	0	0
$925$	−32.9180	−0.0355870
$926$	0	0
$927$	152.462i	0.164468i
$928$	0	0
$929$	−1316.58	−1.41720	−0.708599	−	0.705611i	$-0.750673\pi$
$929$	−1316.58	−1.41720	−0.708599	+	0.705611i	$0.750673\pi$
$930$	0	0
$931$	− 1206.17i	− 1.29556i
$932$	0	0
$933$	−519.855	−0.557186
$934$	0	0
$935$	172.342i	0.184323i
$936$	0	0
$937$	1224.15	1.30646	0.653229	−	0.757160i	$-0.273414\pi$
$937$	1224.15	1.30646	0.653229	+	0.757160i	$0.273414\pi$
$938$	0	0
$939$	− 4.28187i	− 0.00456003i
$940$	0	0
$941$	1039.34	1.10450	0.552252	−	0.833678i	$-0.313769\pi$
$941$	1039.34	1.10450	0.552252	+	0.833678i	$0.313769\pi$
$942$	0	0
$943$	− 1197.84i	− 1.27025i
$944$	0	0
$945$	−582.492	−0.616394
$946$	0	0
$947$	1540.57i	1.62679i	0.581714	+	0.813393i	$0.302382\pi$
$947$	1540.57i	1.62679i	−0.581714	+	0.813393i	$0.697618\pi$
$948$	0	0
$949$	−863.161	−0.909548
$950$	0	0
$951$	1050.64i	1.10478i
$952$	0	0
$953$	−148.663	−0.155994	−0.0779971	−	0.996954i	$-0.524852\pi$
$953$	−148.663	−0.155994	−0.0779971	+	0.996954i	$0.524852\pi$
$954$	0	0
$955$	− 179.122i	− 0.187563i
$956$	0	0
$957$	−410.991	−0.429457
$958$	0	0
$959$	681.701i	0.710846i
$960$	0	0
$961$	−282.672	−0.294143
$962$	0	0
$963$	300.987i	0.312551i
$964$	0	0
$965$	194.531	0.201586
$966$	0	0
$967$	402.229i	0.415955i	0.978134	+	0.207978i	$0.0666881\pi$
$967$	402.229i	0.415955i	−0.978134	+	0.207978i	$0.933312\pi$
$968$	0	0
$969$	1397.98	1.44271
$970$	0	0
$971$	− 734.333i	− 0.756265i	−0.925752	−	0.378132i	$-0.876566\pi$
$971$	− 734.333i	− 0.756265i	0.925752	−	0.378132i	$-0.123434\pi$
$972$	0	0
$973$	1841.31	1.89241
$974$	0	0
$975$	− 100.799i	− 0.103384i
$976$	0	0
$977$	−251.155	−0.257067	−0.128534	−	0.991705i	$-0.541027\pi$
$977$	−251.155	−0.257067	−0.128534	+	0.991705i	$0.541027\pi$
$978$	0	0
$979$	− 204.098i	− 0.208476i
$980$	0	0
$981$	−129.915	−0.132431
$982$	0	0
$983$	1405.24i	1.42954i	0.699359	+	0.714771i	$0.253469\pi$
$983$	1405.24i	1.42954i	−0.699359	+	0.714771i	$0.746531\pi$
$984$	0	0
$985$	−623.666	−0.633163
$986$	0	0
$987$	1751.09i	1.77415i
$988$	0	0
$989$	661.240	0.668594
$990$	0	0
$991$	− 1094.30i	− 1.10424i	−0.833764	−	0.552122i	$-0.813818\pi$
$991$	− 1094.30i	− 1.10424i	0.833764	−	0.552122i	$-0.186182\pi$
$992$	0	0
$993$	3.81729	0.00384420
$994$	0	0
$995$	61.9677i	0.0622791i
$996$	0	0
$997$	1045.09	1.04823	0.524116	−	0.851647i	$-0.324396\pi$
$997$	1045.09	1.04823	0.524116	+	0.851647i	$0.324396\pi$
$998$	0	0
$999$	189.105i	0.189294i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.3.b.a.191.2		4
3.2	odd	2		2880.3.e.b.2431.4		4
4.3	odd	2	inner	320.3.b.a.191.3		4
5.2	odd	4		1600.3.h.p.1599.3		8
5.3	odd	4		1600.3.h.p.1599.5		8
5.4	even	2		1600.3.b.k.1151.3		4
8.3	odd	2		80.3.b.a.31.2	✓	4
8.5	even	2		80.3.b.a.31.3	yes	4
12.11	even	2		2880.3.e.b.2431.3		4
16.3	odd	4		1280.3.g.f.1151.4		8
16.5	even	4		1280.3.g.f.1151.3		8
16.11	odd	4		1280.3.g.f.1151.5		8
16.13	even	4		1280.3.g.f.1151.6		8
20.3	even	4		1600.3.h.p.1599.4		8
20.7	even	4		1600.3.h.p.1599.6		8
20.19	odd	2		1600.3.b.k.1151.2		4
24.5	odd	2		720.3.e.c.271.2		4
24.11	even	2		720.3.e.c.271.1		4
40.3	even	4		400.3.h.d.399.5		8
40.13	odd	4		400.3.h.d.399.4		8
40.19	odd	2		400.3.b.g.351.3		4
40.27	even	4		400.3.h.d.399.3		8
40.29	even	2		400.3.b.g.351.2		4
40.37	odd	4		400.3.h.d.399.6		8
120.29	odd	2		3600.3.e.bb.3151.1		4
120.53	even	4		3600.3.j.k.1999.7		8
120.59	even	2		3600.3.e.bb.3151.4		4
120.77	even	4		3600.3.j.k.1999.1		8
120.83	odd	4		3600.3.j.k.1999.2		8
120.107	odd	4		3600.3.j.k.1999.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.3.b.a.31.2	✓	4	8.3	odd	2
80.3.b.a.31.3	yes	4	8.5	even	2
320.3.b.a.191.2		4	1.1	even	1	trivial
320.3.b.a.191.3		4	4.3	odd	2	inner
400.3.b.g.351.2		4	40.29	even	2
400.3.b.g.351.3		4	40.19	odd	2
400.3.h.d.399.3		8	40.27	even	4
400.3.h.d.399.4		8	40.13	odd	4
400.3.h.d.399.5		8	40.3	even	4
400.3.h.d.399.6		8	40.37	odd	4
720.3.e.c.271.1		4	24.11	even	2
720.3.e.c.271.2		4	24.5	odd	2
1280.3.g.f.1151.3		8	16.5	even	4
1280.3.g.f.1151.4		8	16.3	odd	4
1280.3.g.f.1151.5		8	16.11	odd	4
1280.3.g.f.1151.6		8	16.13	even	4
1600.3.b.k.1151.2		4	20.19	odd	2
1600.3.b.k.1151.3		4	5.4	even	2
1600.3.h.p.1599.3		8	5.2	odd	4
1600.3.h.p.1599.4		8	20.3	even	4
1600.3.h.p.1599.5		8	5.3	odd	4
1600.3.h.p.1599.6		8	20.7	even	4
2880.3.e.b.2431.3		4	12.11	even	2
2880.3.e.b.2431.4		4	3.2	odd	2
3600.3.e.bb.3151.1		4	120.29	odd	2
3600.3.e.bb.3151.4		4	120.59	even	2
3600.3.j.k.1999.1		8	120.77	even	4
3600.3.j.k.1999.2		8	120.83	odd	4
3600.3.j.k.1999.7		8	120.53	even	4
3600.3.j.k.1999.8		8	120.107	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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	320.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.14093i q^{3} -2.23607 q^{5} +9.06914i q^{7} +4.41641 q^{9} +O(q^{10})\)	\(q-2.14093i q^{3} -2.23607 q^{5} +9.06914i q^{7} +4.41641 q^{9} -4.28187i q^{11} +9.41641 q^{13} +4.78727i q^{15} +18.0000 q^{17} +36.2765i q^{19} +19.4164 q^{21} -22.9255i q^{23} +5.00000 q^{25} -28.7236i q^{27} +44.8328 q^{29} -35.2657i q^{31} -9.16718 q^{33} -20.2792i q^{35} -6.58359 q^{37} -20.1599i q^{39} +52.2492 q^{41} +28.8429i q^{43} -9.87539 q^{45} +90.1860i q^{47} -33.2492 q^{49} -38.5368i q^{51} -52.2492 q^{53} +9.57454i q^{55} +77.6656 q^{57} -17.1275i q^{59} +50.5836 q^{61} +40.0530i q^{63} -21.0557 q^{65} +33.1248i q^{67} -49.0820 q^{69} +20.1599i q^{71} -91.6656 q^{73} -10.7047i q^{75} +38.8328 q^{77} -42.8187i q^{79} -21.7477 q^{81} +22.3008i q^{83} -40.2492 q^{85} -95.9840i q^{87} +47.6656 q^{89} +85.3987i q^{91} -75.5016 q^{93} -81.1168i q^{95} -160.164 q^{97} -18.9105i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{9}+O(q^{10}) \) 4 * q - 36 * q^9	\( 4 q - 36 q^{9} - 16 q^{13} + 72 q^{17} + 24 q^{21} + 20 q^{25} + 72 q^{29} - 144 q^{33} - 80 q^{37} + 48 q^{41} - 120 q^{45} + 28 q^{49} - 48 q^{53} + 96 q^{57} + 256 q^{61} - 120 q^{65} + 72 q^{69} - 152 q^{73} + 48 q^{77} + 396 q^{81} - 24 q^{89} - 624 q^{93} - 104 q^{97}+O(q^{100}) \) 4 * q - 36 * q^9 - 16 * q^13 + 72 * q^17 + 24 * q^21 + 20 * q^25 + 72 * q^29 - 144 * q^33 - 80 * q^37 + 48 * q^41 - 120 * q^45 + 28 * q^49 - 48 * q^53 + 96 * q^57 + 256 * q^61 - 120 * q^65 + 72 * q^69 - 152 * q^73 + 48 * q^77 + 396 * q^81 - 24 * q^89 - 624 * q^93 - 104 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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