LMFDB - Embedded newform 960.3.l.j.641.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,3,Mod(641,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("960.641");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	960.l (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:	$26.1581053786$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 56 x^{14} - 252 x^{13} + 1094 x^{12} - 3652 x^{11} + 5452 x^{10} + 1164 x^{9} + \cdots + 20736 $ x^16 - 8x^15 + 56x^14 - 252x^13 + 1094x^12 - 3652x^11 + 5452x^10 + 1164x^9 - 5879x^8 - 15060x^7 + 42772x^6 - 39536x^5 + 64024x^4 - 97600x^3 + 22080x^2 + 25344*x + 20736
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{28}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			641.2
Root			$2.45733 + 4.45426i$ of defining polynomial
Character	$\chi$	$=$	960.641
Dual form			960.3.l.j.641.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.99655 + 0.143843i) q^{3} -2.23607i q^{5} +2.24307 q^{7} +(8.95862 - 0.862068i) q^{9} +O(q^{10})$	\(q+(-2.99655 + 0.143843i) q^{3} -2.23607i q^{5} +2.24307 q^{7} +(8.95862 - 0.862068i) q^{9} +9.03137i q^{11} -8.03098 q^{13} +(0.321644 + 6.70049i) q^{15} -7.34437i q^{17} -14.0089 q^{19} +(-6.72147 + 0.322651i) q^{21} +12.6041i q^{23} -5.00000 q^{25} +(-26.7209 + 3.87187i) q^{27} -38.9853i q^{29} +44.0396 q^{31} +(-1.29910 - 27.0629i) q^{33} -5.01566i q^{35} +25.3292 q^{37} +(24.0652 - 1.15520i) q^{39} -8.51074i q^{41} -45.9971 q^{43} +(-1.92764 - 20.0321i) q^{45} +51.7389i q^{47} -43.9686 q^{49} +(1.05644 + 22.0078i) q^{51} +60.6457i q^{53} +20.1947 q^{55} +(41.9783 - 2.01508i) q^{57} +86.0177i q^{59} +67.0499 q^{61} +(20.0948 - 1.93368i) q^{63} +17.9578i q^{65} -15.5703 q^{67} +(-1.81302 - 37.7689i) q^{69} +68.6484i q^{71} -90.6777 q^{73} +(14.9827 - 0.719217i) q^{75} +20.2580i q^{77} +31.3181 q^{79} +(79.5137 - 15.4459i) q^{81} +143.046i q^{83} -16.4225 q^{85} +(5.60778 + 116.821i) q^{87} +140.581i q^{89} -18.0140 q^{91} +(-131.967 + 6.33480i) q^{93} +31.3248i q^{95} +0.825684 q^{97} +(7.78565 + 80.9086i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{9}+O(q^{10}) $ 16 * q + 8 * q^9	$ 16 q + 8 q^{9} + 16 q^{13} + 104 q^{21} - 80 q^{25} + 192 q^{33} - 144 q^{37} - 40 q^{45} - 128 q^{49} - 80 q^{57} - 144 q^{61} + 280 q^{69} + 192 q^{81} + 96 q^{93} - 160 q^{97}+O(q^{100}) $ 16 * q + 8 * q^9 + 16 * q^13 + 104 * q^21 - 80 * q^25 + 192 * q^33 - 144 * q^37 - 40 * q^45 - 128 * q^49 - 80 * q^57 - 144 * q^61 + 280 * q^69 + 192 * q^81 + 96 * q^93 - 160 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	−2.99655	+	0.143843i	−0.998850	+	0.0479478i
$3$	−2.99655	+	0.143843i	−0.998850	+	0.0479478i
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$	2.24307			0.320439			0.160219	−	0.987081i	$-0.448780\pi$
$7$	2.24307			0.320439			0.160219	+	0.987081i	$0.448780\pi$
$8$		0			0
$9$	8.95862	−	0.862068i	0.995402	−	0.0957853i
$10$		0			0
$11$			9.03137i			0.821033i	0.911853	+	0.410517i	$0.134652\pi$
$11$			9.03137i			0.821033i	−0.911853	+	0.410517i	$0.865348\pi$
$12$		0			0
$13$	−8.03098			−0.617767			−0.308884	−	0.951100i	$-0.599955\pi$
$13$	−8.03098			−0.617767			−0.308884	+	0.951100i	$0.599955\pi$
$14$		0			0
$15$	0.321644	+	6.70049i	0.0214429	+	0.446699i
$16$		0			0
$17$		−	7.34437i		−	0.432022i	−0.976391	−	0.216011i	$-0.930695\pi$
$17$		−	7.34437i		−	0.432022i	0.976391	−	0.216011i	$-0.0693047\pi$
$18$		0			0
$19$	−14.0089			−0.737310			−0.368655	−	0.929566i	$-0.620182\pi$
$19$	−14.0089			−0.737310			−0.368655	+	0.929566i	$0.620182\pi$
$20$		0			0
$21$	−6.72147	+	0.322651i	−0.320070	+	0.0153643i
$22$		0			0
$23$			12.6041i			0.548006i	0.961729	+	0.274003i	$0.0883478\pi$
$23$			12.6041i			0.548006i	−0.961729	+	0.274003i	$0.911652\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	−26.7209	+	3.87187i	−0.989664	+	0.143402i
$28$		0			0
$29$		−	38.9853i		−	1.34432i	−0.740405	−	0.672161i	$-0.765366\pi$
$29$		−	38.9853i		−	1.34432i	0.740405	−	0.672161i	$-0.234634\pi$
$30$		0			0
$31$	44.0396			1.42063			0.710316	−	0.703883i	$-0.248552\pi$
$31$	44.0396			1.42063			0.710316	+	0.703883i	$0.248552\pi$
$32$		0			0
$33$	−1.29910	−	27.0629i	−0.0393667	−	0.820089i
$34$		0			0
$35$		−	5.01566i		−	0.143304i
$36$		0			0
$37$	25.3292			0.684573			0.342286	−	0.939596i	$-0.388799\pi$
$37$	25.3292			0.684573			0.342286	+	0.939596i	$0.388799\pi$
$38$		0			0
$39$	24.0652	−	1.15520i	0.617057	−	0.0296206i
$40$		0			0
$41$		−	8.51074i		−	0.207579i	−0.994599	−	0.103790i	$-0.966903\pi$
$41$		−	8.51074i		−	0.207579i	0.994599	−	0.103790i	$-0.0330968\pi$
$42$		0			0
$43$	−45.9971			−1.06970			−0.534850	−	0.844947i	$-0.679632\pi$
$43$	−45.9971			−1.06970			−0.534850	+	0.844947i	$0.679632\pi$
$44$		0			0
$45$	−1.92764	−	20.0321i	−0.0428365	−	0.445157i
$46$		0			0
$47$			51.7389i			1.10083i	0.834892	+	0.550414i	$0.185530\pi$
$47$			51.7389i			1.10083i	−0.834892	+	0.550414i	$0.814470\pi$
$48$		0			0
$49$	−43.9686			−0.897319
$50$		0			0
$51$	1.05644	+	22.0078i	0.0207145	+	0.431525i
$52$		0			0
$53$			60.6457i			1.14426i	0.820163	+	0.572130i	$0.193883\pi$
$53$			60.6457i			1.14426i	−0.820163	+	0.572130i	$0.806117\pi$
$54$		0			0
$55$	20.1947			0.367177
$56$		0			0
$57$	41.9783	−	2.01508i	0.736462	−	0.0353524i
$58$		0			0
$59$			86.0177i			1.45793i	0.684552	+	0.728964i	$0.259998\pi$
$59$			86.0177i			1.45793i	−0.684552	+	0.728964i	$0.740002\pi$
$60$		0			0
$61$	67.0499			1.09918			0.549590	−	0.835435i	$-0.314784\pi$
$61$	67.0499			1.09918			0.549590	+	0.835435i	$0.314784\pi$
$62$		0			0
$63$	20.0948	−	1.93368i	0.318965	−	0.0306933i
$64$		0			0
$65$			17.9578i			0.276274i
$66$		0			0
$67$	−15.5703			−0.232392			−0.116196	−	0.993226i	$-0.537070\pi$
$67$	−15.5703			−0.232392			−0.116196	+	0.993226i	$0.537070\pi$
$68$		0			0
$69$	−1.81302	−	37.7689i	−0.0262757	−	0.547375i
$70$		0			0
$71$			68.6484i			0.966879i	0.875378	+	0.483440i	$0.160613\pi$
$71$			68.6484i			0.966879i	−0.875378	+	0.483440i	$0.839387\pi$
$72$		0			0
$73$	−90.6777			−1.24216			−0.621080	−	0.783747i	$-0.713306\pi$
$73$	−90.6777			−1.24216			−0.621080	+	0.783747i	$0.713306\pi$
$74$		0			0
$75$	14.9827	−	0.719217i	0.199770	−	0.00958956i
$76$		0			0
$77$			20.2580i			0.263091i
$78$		0			0
$79$	31.3181			0.396431			0.198216	−	0.980158i	$-0.436485\pi$
$79$	31.3181			0.396431			0.198216	+	0.980158i	$0.436485\pi$
$80$		0			0
$81$	79.5137	−	15.4459i	0.981650	−	0.190690i
$82$		0			0
$83$			143.046i			1.72344i	0.507383	+	0.861720i	$0.330613\pi$
$83$			143.046i			1.72344i	−0.507383	+	0.861720i	$0.669387\pi$
$84$		0			0
$85$	−16.4225			−0.193206
$86$		0			0
$87$	5.60778	+	116.821i	0.0644572	+	1.34278i
$88$		0			0
$89$			140.581i			1.57956i	0.613392	+	0.789778i	$0.289804\pi$
$89$			140.581i			1.57956i	−0.613392	+	0.789778i	$0.710196\pi$
$90$		0			0
$91$	−18.0140			−0.197956
$92$		0			0
$93$	−131.967	+	6.33480i	−1.41900	+	0.0681161i
$94$		0			0
$95$			31.3248i			0.329735i
$96$		0			0
$97$	0.825684			0.00851221			0.00425610	−	0.999991i	$-0.498645\pi$
$97$	0.825684			0.00851221			0.00425610	+	0.999991i	$0.498645\pi$
$98$		0			0
$99$	7.78565	+	80.9086i	0.0786429	+	0.817258i
$100$		0			0
$101$			149.256i			1.47778i	0.673824	+	0.738892i	$0.264651\pi$
$101$			149.256i			1.47778i	−0.673824	+	0.738892i	$0.735349\pi$
$102$		0			0
$103$	−86.3106			−0.837967			−0.418984	−	0.907994i	$-0.637613\pi$
$103$	−86.3106			−0.837967			−0.418984	+	0.907994i	$0.637613\pi$
$104$		0			0
$105$	0.721469	+	15.0297i	0.00687113	+	0.143140i
$106$		0			0
$107$		−	156.269i		−	1.46046i	−0.683202	−	0.730230i	$-0.739413\pi$
$107$		−	156.269i		−	1.46046i	0.683202	−	0.730230i	$-0.260587\pi$
$108$		0			0
$109$	26.0810			0.239275			0.119638	−	0.992818i	$-0.461827\pi$
$109$	26.0810			0.239275			0.119638	+	0.992818i	$0.461827\pi$
$110$		0			0
$111$	−75.9002	+	3.64344i	−0.683786	+	0.0328238i
$112$		0			0
$113$		−	42.8246i		−	0.378978i	−0.981883	−	0.189489i	$-0.939317\pi$
$113$		−	42.8246i		−	0.378978i	0.981883	−	0.189489i	$-0.0606832\pi$
$114$		0			0
$115$	28.1837			0.245076
$116$		0			0
$117$	−71.9465	+	6.92324i	−0.614927	+	0.0591730i
$118$		0			0
$119$		−	16.4739i		−	0.138436i
$120$		0			0
$121$	39.4344			0.325904
$122$		0			0
$123$	1.22421	+	25.5029i	0.00995296	+	0.207340i
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	−98.4145			−0.774918			−0.387459	−	0.921887i	$-0.626647\pi$
$127$	−98.4145			−0.774918			−0.387459	+	0.921887i	$0.626647\pi$
$128$		0			0
$129$	137.832	−	6.61637i	1.06847	−	0.0512897i
$130$		0			0
$131$			98.5827i			0.752539i	0.926510	+	0.376270i	$0.122793\pi$
$131$			98.5827i			0.752539i	−0.926510	+	0.376270i	$0.877207\pi$
$132$		0			0
$133$	−31.4229			−0.236262
$134$		0			0
$135$	8.65776	+	59.7498i	0.0641315	+	0.442591i
$136$		0			0
$137$			154.756i			1.12961i	0.825225	+	0.564804i	$0.191048\pi$
$137$			154.756i			1.12961i	−0.825225	+	0.564804i	$0.808952\pi$
$138$		0			0
$139$	227.186			1.63443			0.817215	−	0.576333i	$-0.195517\pi$
$139$	227.186			1.63443			0.817215	+	0.576333i	$0.195517\pi$
$140$		0			0
$141$	−7.44230	−	155.038i	−0.0527822	−	1.09956i
$142$		0			0
$143$		−	72.5307i		−	0.507208i
$144$		0			0
$145$	−87.1738			−0.601199
$146$		0			0
$147$	131.754	−	6.32460i	0.896287	−	0.0430245i
$148$		0			0
$149$			104.398i			0.700659i	0.936627	+	0.350330i	$0.113930\pi$
$149$			104.398i			0.700659i	−0.936627	+	0.350330i	$0.886070\pi$
$150$		0			0
$151$	−216.256			−1.43216			−0.716079	−	0.698020i	$-0.754065\pi$
$151$	−216.256			−1.43216			−0.716079	+	0.698020i	$0.754065\pi$
$152$		0			0
$153$	−6.33134	−	65.7954i	−0.0413813	−	0.430035i
$154$		0			0
$155$		−	98.4755i		−	0.635326i
$156$		0			0
$157$	239.367			1.52463			0.762316	−	0.647205i	$-0.224062\pi$
$157$	239.367			1.52463			0.762316	+	0.647205i	$0.224062\pi$
$158$		0			0
$159$	−8.72349	−	181.728i	−0.0548647	−	1.14294i
$160$		0			0
$161$			28.2719i			0.175602i
$162$		0			0
$163$	19.8765			0.121942			0.0609708	−	0.998140i	$-0.480580\pi$
$163$	19.8765			0.121942			0.0609708	+	0.998140i	$0.480580\pi$
$164$		0			0
$165$	−60.5146	+	2.90488i	−0.366755	+	0.0176053i
$166$		0			0
$167$			293.172i			1.75552i	0.479099	+	0.877761i	$0.340964\pi$
$167$			293.172i			1.75552i	−0.479099	+	0.877761i	$0.659036\pi$
$168$		0			0
$169$	−104.503			−0.618363
$170$		0			0
$171$	−125.500	+	12.0766i	−0.733919	+	0.0706234i
$172$		0			0
$173$		−	90.1213i		−	0.520932i	−0.965483	−	0.260466i	$-0.916124\pi$
$173$		−	90.1213i		−	0.520932i	0.965483	−	0.260466i	$-0.0838762\pi$
$174$		0			0
$175$	−11.2153			−0.0640877
$176$		0			0
$177$	−12.3731	−	257.756i	−0.0699044	−	1.45625i
$178$		0			0
$179$			328.689i			1.83625i	0.396292	+	0.918124i	$0.370297\pi$
$179$			328.689i			1.83625i	−0.396292	+	0.918124i	$0.629703\pi$
$180$		0			0
$181$	200.100			1.10552			0.552762	−	0.833339i	$-0.313574\pi$
$181$	200.100			1.10552			0.552762	+	0.833339i	$0.313574\pi$
$182$		0			0
$183$	−200.918	+	9.64469i	−1.09791	+	0.0527032i
$184$		0			0
$185$		−	56.6378i		−	0.306150i
$186$		0			0
$187$	66.3297			0.354704
$188$		0			0
$189$	−59.9369	+	8.68486i	−0.317127	+	0.0459517i
$190$		0			0
$191$		−	197.777i		−	1.03548i	−0.855537	−	0.517742i	$-0.826773\pi$
$191$		−	197.777i		−	1.03548i	0.855537	−	0.517742i	$-0.173227\pi$
$192$		0			0
$193$	−212.677			−1.10195			−0.550977	−	0.834521i	$-0.685745\pi$
$193$	−212.677			−1.10195			−0.550977	+	0.834521i	$0.685745\pi$
$194$		0			0
$195$	−2.58311	−	53.8115i	−0.0132467	−	0.275956i
$196$		0			0
$197$		−	68.5241i		−	0.347838i	−0.984760	−	0.173919i	$-0.944357\pi$
$197$		−	68.5241i		−	0.347838i	0.984760	−	0.173919i	$-0.0556431\pi$
$198$		0			0
$199$	252.252			1.26760			0.633798	−	0.773498i	$-0.281495\pi$
$199$	252.252			1.26760			0.633798	+	0.773498i	$0.281495\pi$
$200$		0			0
$201$	46.6572	−	2.23968i	0.232125	−	0.0111427i
$202$		0			0
$203$		−	87.4468i		−	0.430772i
$204$		0			0
$205$	−19.0306			−0.0928322
$206$		0			0
$207$	10.8656	+	112.916i	0.0524909	+	0.545486i
$208$		0			0
$209$		−	126.519i		−	0.605356i
$210$		0			0
$211$	17.9653			0.0851435			0.0425717	−	0.999093i	$-0.486445\pi$
$211$	17.9653			0.0851435			0.0425717	+	0.999093i	$0.486445\pi$
$212$		0			0
$213$	−9.87462	−	205.708i	−0.0463597	−	0.965767i
$214$		0			0
$215$			102.853i			0.478384i
$216$		0			0
$217$	98.7838			0.455225
$218$		0			0
$219$	271.720	−	13.0434i	1.24073	−	0.0595589i
$220$		0			0
$221$			58.9825i			0.266889i
$222$		0			0
$223$	−142.384			−0.638493			−0.319246	−	0.947672i	$-0.603430\pi$
$223$	−142.384			−0.638493			−0.319246	+	0.947672i	$0.603430\pi$
$224$		0			0
$225$	−44.7931	+	4.31034i	−0.199080	+	0.0191571i
$226$		0			0
$227$		−	321.954i		−	1.41830i	−0.705059	−	0.709149i	$-0.749079\pi$
$227$		−	321.954i		−	1.41830i	0.705059	−	0.709149i	$-0.250921\pi$
$228$		0			0
$229$	107.541			0.469613			0.234806	−	0.972042i	$-0.424554\pi$
$229$	107.541			0.469613			0.234806	+	0.972042i	$0.424554\pi$
$230$		0			0
$231$	−2.91398	−	60.7040i	−0.0126146	−	0.262788i
$232$		0			0
$233$			291.098i			1.24935i	0.780885	+	0.624675i	$0.214768\pi$
$233$			291.098i			1.24935i	−0.780885	+	0.624675i	$0.785232\pi$
$234$		0			0
$235$	115.692			0.492305
$236$		0			0
$237$	−93.8461	+	4.50490i	−0.395975	+	0.0190080i
$238$		0			0
$239$		−	452.505i		−	1.89333i	−0.322223	−	0.946664i	$-0.604430\pi$
$239$		−	452.505i		−	1.89333i	0.322223	−	0.946664i	$-0.395570\pi$
$240$		0			0
$241$	62.6968			0.260153			0.130076	−	0.991504i	$-0.458478\pi$
$241$	62.6968			0.260153			0.130076	+	0.991504i	$0.458478\pi$
$242$		0			0
$243$	−236.045	+	57.7218i	−0.971378	+	0.237538i
$244$		0			0
$245$			98.3169i			0.401293i
$246$		0			0
$247$	112.505			0.455486
$248$		0			0
$249$	−20.5762	−	428.643i	−0.0826352	−	1.72146i
$250$		0			0
$251$		−	242.908i		−	0.967759i	−0.875135	−	0.483880i	$-0.839227\pi$
$251$		−	242.908i		−	0.967759i	0.875135	−	0.483880i	$-0.160773\pi$
$252$		0			0
$253$	−113.833			−0.449931
$254$		0			0
$255$	49.2109	−	2.36227i	0.192984	−	0.00926380i
$256$		0			0
$257$			258.831i			1.00712i	0.863959	+	0.503562i	$0.167978\pi$
$257$			258.831i			1.00712i	−0.863959	+	0.503562i	$0.832022\pi$
$258$		0			0
$259$	56.8152			0.219364
$260$		0			0
$261$	−33.6080	−	349.255i	−0.128766	−	1.33814i
$262$		0			0
$263$		−	205.895i		−	0.782869i	−0.920206	−	0.391435i	$-0.871979\pi$
$263$		−	205.895i		−	0.782869i	0.920206	−	0.391435i	$-0.128021\pi$
$264$		0			0
$265$	135.608			0.511728
$266$		0			0
$267$	−20.2216	−	421.257i	−0.0757363	−	1.57774i
$268$		0			0
$269$			200.612i			0.745768i	0.927878	+	0.372884i	$0.121631\pi$
$269$			200.612i			0.745768i	−0.927878	+	0.372884i	$0.878369\pi$
$270$		0			0
$271$	46.0058			0.169763			0.0848816	−	0.996391i	$-0.472949\pi$
$271$	46.0058			0.169763			0.0848816	+	0.996391i	$0.472949\pi$
$272$		0			0
$273$	53.9800	−	2.59120i	0.197729	−	0.00949158i
$274$		0			0
$275$		−	45.1568i		−	0.164207i
$276$		0			0
$277$	−256.932			−0.927551			−0.463776	−	0.885953i	$-0.653506\pi$
$277$	−256.932			−0.927551			−0.463776	+	0.885953i	$0.653506\pi$
$278$		0			0
$279$	394.534	−	37.9651i	1.41410	−	0.136076i
$280$		0			0
$281$		−	42.9617i		−	0.152889i	−0.997074	−	0.0764444i	$-0.975643\pi$
$281$		−	42.9617i		−	0.152889i	0.997074	−	0.0764444i	$-0.0243568\pi$
$282$		0			0
$283$	−519.170			−1.83452			−0.917261	−	0.398287i	$-0.869605\pi$
$283$	−519.170			−1.83452			−0.917261	+	0.398287i	$0.869605\pi$
$284$		0			0
$285$	−4.50587	−	93.8664i	−0.0158101	−	0.329356i
$286$		0			0
$287$		−	19.0902i		−	0.0665163i
$288$		0			0
$289$	235.060			0.813357
$290$		0			0
$291$	−2.47420	+	0.118769i	−0.00850242	+	0.000408142i
$292$		0			0
$293$			88.7773i			0.302994i	0.988458	+	0.151497i	$0.0484094\pi$
$293$			88.7773i			0.302994i	−0.988458	+	0.151497i	$0.951591\pi$
$294$		0			0
$295$	192.341			0.652005
$296$		0			0
$297$	−34.9682	−	241.327i	−0.117738	−	0.812547i
$298$		0			0
$299$		−	101.223i		−	0.338540i
$300$		0			0
$301$	−103.175			−0.342773
$302$		0			0
$303$	−21.4695	−	447.254i	−0.0708565	−	1.47608i
$304$		0			0
$305$		−	149.928i		−	0.491568i
$306$		0			0
$307$	−345.304			−1.12477			−0.562385	−	0.826876i	$-0.690116\pi$
$307$	−345.304			−1.12477			−0.562385	+	0.826876i	$0.690116\pi$
$308$		0			0
$309$	258.634	−	12.4152i	0.837003	−	0.0401787i
$310$		0			0
$311$			65.2492i			0.209804i	0.994483	+	0.104902i	$0.0334529\pi$
$311$			65.2492i			0.209804i	−0.994483	+	0.104902i	$0.966547\pi$
$312$		0			0
$313$	114.858			0.366959			0.183480	−	0.983023i	$-0.441264\pi$
$313$	114.858			0.366959			0.183480	+	0.983023i	$0.441264\pi$
$314$		0			0
$315$	−4.32383	−	44.9333i	−0.0137265	−	0.142646i
$316$		0			0
$317$			10.0703i			0.0317675i	0.999874	+	0.0158837i	$0.00505616\pi$
$317$			10.0703i			0.0317675i	−0.999874	+	0.0158837i	$0.994944\pi$
$318$		0			0
$319$	352.091			1.10373
$320$		0			0
$321$	22.4783	+	468.268i	0.0700258	+	1.45878i
$322$		0			0
$323$			102.886i			0.318534i
$324$		0			0
$325$	40.1549			0.123553
$326$		0			0
$327$	−78.1530	+	3.75158i	−0.239000	+	0.0114727i
$328$		0			0
$329$			116.054i			0.352748i
$330$		0			0
$331$	−365.387			−1.10389			−0.551944	−	0.833881i	$-0.686114\pi$
$331$	−365.387			−1.10389			−0.551944	+	0.833881i	$0.686114\pi$
$332$		0			0
$333$	226.915	−	21.8355i	0.681425	−	0.0655720i
$334$		0			0
$335$			34.8162i			0.103929i
$336$		0			0
$337$	−103.132			−0.306028			−0.153014	−	0.988224i	$-0.548898\pi$
$337$	−103.132			−0.306028			−0.153014	+	0.988224i	$0.548898\pi$
$338$		0			0
$339$	6.16003	+	128.326i	0.0181712	+	0.378543i
$340$		0			0
$341$			397.738i			1.16639i
$342$		0			0
$343$	−208.535			−0.607974
$344$		0			0
$345$	−84.4538	+	4.05404i	−0.244794	+	0.0117508i
$346$		0			0
$347$		−	411.888i		−	1.18700i	−0.804836	−	0.593498i	$-0.797747\pi$
$347$		−	411.888i		−	1.18700i	0.804836	−	0.593498i	$-0.202253\pi$
$348$		0			0
$349$	−492.648			−1.41160			−0.705800	−	0.708411i	$-0.749412\pi$
$349$	−492.648			−1.41160			−0.705800	+	0.708411i	$0.749412\pi$
$350$		0			0
$351$	214.595	−	31.0949i	0.611382	−	0.0885894i
$352$		0			0
$353$			444.692i			1.25975i	0.776697	+	0.629875i	$0.216894\pi$
$353$			444.692i			1.25975i	−0.776697	+	0.629875i	$0.783106\pi$
$354$		0			0
$355$	153.503			0.432402
$356$		0			0
$357$	2.36967	+	49.3650i	0.00663772	+	0.138277i
$358$		0			0
$359$		−	137.546i		−	0.383136i	−0.981479	−	0.191568i	$-0.938643\pi$
$359$		−	137.546i		−	0.383136i	0.981479	−	0.191568i	$-0.0613572\pi$
$360$		0			0
$361$	−164.751			−0.456375
$362$		0			0
$363$	−118.167	+	5.67238i	−0.325530	+	0.0156264i
$364$		0			0
$365$			202.762i			0.555511i
$366$		0			0
$367$	−122.989			−0.335119			−0.167559	−	0.985862i	$-0.553589\pi$
$367$	−122.989			−0.335119			−0.167559	+	0.985862i	$0.553589\pi$
$368$		0			0
$369$	−7.33683	−	76.2445i	−0.0198830	−	0.206625i
$370$		0			0
$371$			136.033i			0.366665i
$372$		0			0
$373$	17.5537			0.0470609			0.0235305	−	0.999723i	$-0.492509\pi$
$373$	17.5537			0.0470609			0.0235305	+	0.999723i	$0.492509\pi$
$374$		0			0
$375$	−1.60822	−	33.5024i	−0.00428858	−	0.0893398i
$376$		0			0
$377$			313.090i			0.830478i
$378$		0			0
$379$	295.437			0.779517			0.389759	−	0.920917i	$-0.372558\pi$
$379$	295.437			0.779517			0.389759	+	0.920917i	$0.372558\pi$
$380$		0			0
$381$	294.904	−	14.1563i	0.774026	−	0.0371556i
$382$		0			0
$383$		−	249.327i		−	0.650984i	−0.945545	−	0.325492i	$-0.894470\pi$
$383$		−	249.327i		−	0.650984i	0.945545	−	0.325492i	$-0.105530\pi$
$384$		0			0
$385$	45.2982			0.117658
$386$		0			0
$387$	−412.070	+	39.6526i	−1.06478	+	0.102461i
$388$		0			0
$389$		−	100.365i		−	0.258008i	−0.991644	−	0.129004i	$-0.958822\pi$
$389$		−	100.365i		−	0.258008i	0.991644	−	0.129004i	$-0.0411780\pi$
$390$		0			0
$391$	92.5694			0.236750
$392$		0			0
$393$	−14.1805	−	295.408i	−0.0360826	−	0.751674i
$394$		0			0
$395$		−	70.0293i		−	0.177289i
$396$		0			0
$397$	420.402			1.05895			0.529473	−	0.848327i	$-0.322390\pi$
$397$	420.402			1.05895			0.529473	+	0.848327i	$0.322390\pi$
$398$		0			0
$399$	94.1603	−	4.51998i	0.235991	−	0.0113283i
$400$		0			0
$401$		−	465.360i		−	1.16050i	−0.814439	−	0.580249i	$-0.802955\pi$
$401$		−	465.360i		−	1.16050i	0.814439	−	0.580249i	$-0.197045\pi$
$402$		0			0
$403$	−353.681			−0.877620
$404$		0			0
$405$	−34.5380	−	177.798i	−0.0852790	−	0.439007i
$406$		0			0
$407$			228.757i			0.562057i
$408$		0			0
$409$	−441.627			−1.07977			−0.539886	−	0.841738i	$-0.681533\pi$
$409$	−441.627			−1.07977			−0.539886	+	0.841738i	$0.681533\pi$
$410$		0			0
$411$	−22.2607	−	463.735i	−0.0541622	−	1.12831i
$412$		0			0
$413$			192.944i			0.467176i
$414$		0			0
$415$	319.860			0.770746
$416$		0			0
$417$	−680.773	+	32.6792i	−1.63255	+	0.0783673i
$418$		0			0
$419$		−	133.888i		−	0.319542i	−0.987154	−	0.159771i	$-0.948924\pi$
$419$		−	133.888i		−	0.319542i	0.987154	−	0.159771i	$-0.0510756\pi$
$420$		0			0
$421$	−659.449			−1.56639			−0.783193	−	0.621778i	$-0.786411\pi$
$421$	−659.449			−1.56639			−0.783193	+	0.621778i	$0.786411\pi$
$422$		0			0
$423$	44.6024	+	463.509i	0.105443	+	1.09577i
$424$		0			0
$425$			36.7219i			0.0864044i
$426$		0			0
$427$	150.398			0.352219
$428$		0			0
$429$	10.4331	+	217.342i	0.0243195	+	0.506624i
$430$		0			0
$431$		−	464.823i		−	1.07848i	−0.842153	−	0.539238i	$-0.818712\pi$
$431$		−	464.823i		−	1.07848i	0.842153	−	0.539238i	$-0.181288\pi$
$432$		0			0
$433$	393.188			0.908055			0.454028	−	0.890988i	$-0.349987\pi$
$433$	393.188			0.908055			0.454028	+	0.890988i	$0.349987\pi$
$434$		0			0
$435$	261.221	−	12.5394i	0.600507	−	0.0288262i
$436$		0			0
$437$		−	176.570i		−	0.404050i
$438$		0			0
$439$	406.911			0.926904			0.463452	−	0.886122i	$-0.346611\pi$
$439$	406.911			0.926904			0.463452	+	0.886122i	$0.346611\pi$
$440$		0			0
$441$	−393.898	+	37.9039i	−0.893193	+	0.0859500i
$442$		0			0
$443$		−	531.885i		−	1.20064i	−0.799758	−	0.600322i	$-0.795039\pi$
$443$		−	531.885i		−	1.20064i	0.799758	−	0.600322i	$-0.204961\pi$
$444$		0			0
$445$	314.348			0.706399
$446$		0			0
$447$	−15.0170	−	312.834i	−0.0335951	−	0.699853i
$448$		0			0
$449$			690.916i			1.53879i	0.638774	+	0.769394i	$0.279442\pi$
$449$			690.916i			1.53879i	−0.638774	+	0.769394i	$0.720558\pi$
$450$		0			0
$451$	76.8636			0.170429
$452$		0			0
$453$	648.021	−	31.1070i	1.43051	−	0.0686688i
$454$		0			0
$455$			40.2806i			0.0885288i
$456$		0			0
$457$	885.218			1.93702			0.968510	−	0.248974i	$-0.0800934\pi$
$457$	885.218			1.93702			0.968510	+	0.248974i	$0.0800934\pi$
$458$		0			0
$459$	28.4364	+	196.248i	0.0619530	+	0.427557i
$460$		0			0
$461$			516.543i			1.12048i	0.828329	+	0.560242i	$0.189292\pi$
$461$			516.543i			1.12048i	−0.828329	+	0.560242i	$0.810708\pi$
$462$		0			0
$463$	−465.526			−1.00546			−0.502728	−	0.864445i	$-0.667670\pi$
$463$	−465.526			−1.00546			−0.502728	+	0.864445i	$0.667670\pi$
$464$		0			0
$465$	14.1650	+	295.087i	0.0304625	+	0.634595i
$466$		0			0
$467$			646.351i			1.38405i	0.721874	+	0.692025i	$0.243281\pi$
$467$			646.351i			1.38405i	−0.721874	+	0.692025i	$0.756719\pi$
$468$		0			0
$469$	−34.9253			−0.0744675
$470$		0			0
$471$	−717.275	+	34.4314i	−1.52288	+	0.0731027i
$472$		0			0
$473$		−	415.416i		−	0.878258i
$474$		0			0
$475$	70.0444			0.147462
$476$		0			0
$477$	52.2807	+	543.302i	0.109603	+	1.13900i
$478$		0			0
$479$		−	55.5045i		−	0.115876i	−0.998320	−	0.0579379i	$-0.981547\pi$
$479$		−	55.5045i		−	0.115876i	0.998320	−	0.0579379i	$-0.0184526\pi$
$480$		0			0
$481$	−203.418			−0.422907
$482$		0			0
$483$	−4.06673	−	84.7183i	−0.00841973	−	0.175400i
$484$		0			0
$485$		−	1.84629i		−	0.00380678i
$486$		0			0
$487$	−188.411			−0.386880			−0.193440	−	0.981112i	$-0.561964\pi$
$487$	−188.411			−0.386880			−0.193440	+	0.981112i	$0.561964\pi$
$488$		0			0
$489$	−59.5608	+	2.85910i	−0.121801	+	0.00584683i
$490$		0			0
$491$			154.621i			0.314910i	0.987526	+	0.157455i	$0.0503290\pi$
$491$			154.621i			0.314910i	−0.987526	+	0.157455i	$0.949671\pi$
$492$		0			0
$493$	−286.323			−0.580776
$494$		0			0
$495$	180.917	−	17.4092i	0.365489	−	0.0351702i
$496$		0			0
$497$			153.983i			0.309825i
$498$		0			0
$499$	961.012			1.92588			0.962938	−	0.269722i	$-0.0869318\pi$
$499$	961.012			1.92588			0.962938	+	0.269722i	$0.0869318\pi$
$500$		0			0
$501$	−42.1709	−	878.505i	−0.0841734	−	1.75350i
$502$		0			0
$503$			435.850i			0.866500i	0.901274	+	0.433250i	$0.142633\pi$
$503$			435.850i			0.866500i	−0.901274	+	0.433250i	$0.857367\pi$
$504$		0			0
$505$	333.747			0.660885
$506$		0			0
$507$	313.150	−	15.0321i	0.617652	−	0.0296492i
$508$		0			0
$509$		−	748.589i		−	1.47070i	−0.677685	−	0.735352i	$-0.737017\pi$
$509$		−	748.589i		−	1.47070i	0.677685	−	0.735352i	$-0.262983\pi$
$510$		0			0
$511$	−203.396			−0.398036
$512$		0			0
$513$	374.330	−	54.2405i	0.729689	−	0.105732i
$514$		0			0
$515$			192.996i			0.374750i
$516$		0			0
$517$	−467.273			−0.903816
$518$		0			0
$519$	12.9633	+	270.053i	0.0249775	+	0.520333i
$520$		0			0
$521$		−	331.457i		−	0.636193i	−0.948058	−	0.318097i	$-0.896956\pi$
$521$		−	331.457i		−	0.636193i	0.948058	−	0.318097i	$-0.103044\pi$
$522$		0			0
$523$	223.173			0.426717			0.213359	−	0.976974i	$-0.431560\pi$
$523$	223.173			0.426717			0.213359	+	0.976974i	$0.431560\pi$
$524$		0			0
$525$	33.6073	−	1.61325i	0.0640140	−	0.00307286i
$526$		0			0
$527$		−	323.443i		−	0.613744i
$528$		0			0
$529$	370.136			0.699690
$530$		0			0
$531$	74.1531	+	770.600i	0.139648	+	1.45122i
$532$		0			0
$533$			68.3495i			0.128236i
$534$		0			0
$535$	−349.428			−0.653137
$536$		0			0
$537$	−47.2797	−	984.931i	−0.0880441	−	1.83414i
$538$		0			0
$539$		−	397.097i		−	0.736729i
$540$		0			0
$541$	−674.035			−1.24591			−0.622953	−	0.782259i	$-0.714067\pi$
$541$	−674.035			−1.24591			−0.622953	+	0.782259i	$0.714067\pi$
$542$		0			0
$543$	−599.609	+	28.7830i	−1.10425	+	0.0530074i
$544$		0			0
$545$		−	58.3189i		−	0.107007i
$546$		0			0
$547$	205.195			0.375128			0.187564	−	0.982252i	$-0.439941\pi$
$547$	205.195			0.375128			0.187564	+	0.982252i	$0.439941\pi$
$548$		0			0
$549$	600.675	−	57.8016i	1.09413	−	0.105285i
$550$		0			0
$551$			546.141i			0.991181i
$552$		0			0
$553$	70.2486			0.127032
$554$		0			0
$555$	8.14697	+	169.718i	0.0146792	+	0.305798i
$556$		0			0
$557$		−	886.612i		−	1.59176i	−0.605453	−	0.795881i	$-0.707008\pi$
$557$		−	886.612i		−	1.59176i	0.605453	−	0.795881i	$-0.292992\pi$
$558$		0			0
$559$	369.401			0.660825
$560$		0			0
$561$	−198.760	+	9.54109i	−0.354296	+	0.0170073i
$562$		0			0
$563$			453.686i			0.805836i	0.915236	+	0.402918i	$0.132004\pi$
$563$			453.686i			0.805836i	−0.915236	+	0.402918i	$0.867996\pi$
$564$		0			0
$565$	−95.7586			−0.169484
$566$		0			0
$567$	178.355	−	34.6462i	0.314559	−	0.0611043i
$568$		0			0
$569$			313.662i			0.551251i	0.961265	+	0.275625i	$0.0888849\pi$
$569$			313.662i			0.551251i	−0.961265	+	0.275625i	$0.911115\pi$
$570$		0			0
$571$	−418.596			−0.733092			−0.366546	−	0.930400i	$-0.619460\pi$
$571$	−418.596			−0.733092			−0.366546	+	0.930400i	$0.619460\pi$
$572$		0			0
$573$	28.4490	+	592.650i	0.0496492	+	1.03429i
$574$		0			0
$575$		−	63.0207i		−	0.109601i
$576$		0			0
$577$	−18.7107			−0.0324276			−0.0162138	−	0.999869i	$-0.505161\pi$
$577$	−18.7107			−0.0324276			−0.0162138	+	0.999869i	$0.505161\pi$
$578$		0			0
$579$	637.297	−	30.5922i	1.10069	−	0.0528362i
$580$		0			0
$581$			320.861i			0.552257i
$582$		0			0
$583$	−547.714			−0.939475
$584$		0			0
$585$	15.4808	+	160.877i	0.0264630	+	0.275004i
$586$		0			0
$587$			937.505i			1.59711i	0.601920	+	0.798556i	$0.294403\pi$
$587$			937.505i			1.59711i	−0.601920	+	0.798556i	$0.705597\pi$
$588$		0			0
$589$	−616.945			−1.04745
$590$		0			0
$591$	9.85674	+	205.336i	0.0166781	+	0.347438i
$592$		0			0
$593$		−	902.817i		−	1.52246i	−0.648484	−	0.761228i	$-0.724597\pi$
$593$		−	902.817i		−	1.52246i	0.648484	−	0.761228i	$-0.275403\pi$
$594$		0			0
$595$	−36.8368			−0.0619106
$596$		0			0
$597$	−755.885	+	36.2847i	−1.26614	+	0.0607784i
$598$		0			0
$599$			991.033i			1.65448i	0.561849	+	0.827240i	$0.310090\pi$
$599$			991.033i			1.65448i	−0.561849	+	0.827240i	$0.689910\pi$
$600$		0			0
$601$	218.484			0.363534			0.181767	−	0.983342i	$-0.441818\pi$
$601$	218.484			0.363534			0.181767	+	0.983342i	$0.441818\pi$
$602$		0			0
$603$	−139.488	+	13.4226i	−0.231324	+	0.0222598i
$604$		0			0
$605$		−	88.1781i		−	0.145749i
$606$		0			0
$607$	−438.702			−0.722738			−0.361369	−	0.932423i	$-0.617690\pi$
$607$	−438.702			−0.722738			−0.361369	+	0.932423i	$0.617690\pi$
$608$		0			0
$609$	12.5786	+	262.039i	0.0206546	+	0.430277i
$610$		0			0
$611$		−	415.514i		−	0.680055i
$612$		0			0
$613$	597.090			0.974046			0.487023	−	0.873389i	$-0.338083\pi$
$613$	597.090			0.974046			0.487023	+	0.873389i	$0.338083\pi$
$614$		0			0
$615$	57.0261	−	2.73742i	0.0927254	−	0.00445110i
$616$		0			0
$617$			321.463i			0.521010i	0.965473	+	0.260505i	$0.0838890\pi$
$617$			321.463i			0.521010i	−0.965473	+	0.260505i	$0.916111\pi$
$618$		0			0
$619$	511.960			0.827076			0.413538	−	0.910487i	$-0.364293\pi$
$619$	511.960			0.827076			0.413538	+	0.910487i	$0.364293\pi$
$620$		0			0
$621$	−48.8015	−	336.794i	−0.0785854	−	0.542342i
$622$		0			0
$623$			315.332i			0.506151i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	18.1990	+	379.121i	0.0290255	+	0.604659i
$628$		0			0
$629$		−	186.027i		−	0.295750i
$630$		0			0
$631$	−1070.42			−1.69639			−0.848195	−	0.529684i	$-0.822311\pi$
$631$	−1070.42			−1.69639			−0.848195	+	0.529684i	$0.822311\pi$
$632$		0			0
$633$	−53.8338	+	2.58418i	−0.0850455	+	0.00408244i
$634$		0			0
$635$			220.062i			0.346554i
$636$		0			0
$637$	353.111			0.554335
$638$		0			0
$639$	59.1796	+	614.995i	0.0926128	+	0.962434i
$640$		0			0
$641$		−	865.824i		−	1.35074i	−0.737479	−	0.675370i	$-0.763984\pi$
$641$		−	865.824i		−	1.35074i	0.737479	−	0.675370i	$-0.236016\pi$
$642$		0			0
$643$	−731.670			−1.13790			−0.568950	−	0.822372i	$-0.692650\pi$
$643$	−731.670			−1.13790			−0.568950	+	0.822372i	$0.692650\pi$
$644$		0			0
$645$	−14.7947	−	308.203i	−0.0229375	−	0.477834i
$646$		0			0
$647$		−	230.583i		−	0.356388i	−0.983995	−	0.178194i	$-0.942975\pi$
$647$		−	230.583i		−	0.356388i	0.983995	−	0.178194i	$-0.0570254\pi$
$648$		0			0
$649$	−776.857			−1.19701
$650$		0			0
$651$	−296.011	+	14.2094i	−0.454701	+	0.0218270i
$652$		0			0
$653$		−	617.718i		−	0.945970i	−0.881070	−	0.472985i	$-0.843176\pi$
$653$		−	617.718i		−	0.945970i	0.881070	−	0.472985i	$-0.156824\pi$
$654$		0			0
$655$	220.438			0.336546
$656$		0			0
$657$	−812.347	+	78.1703i	−1.23645	+	0.118981i
$658$		0			0
$659$			265.710i			0.403202i	0.979468	+	0.201601i	$0.0646145\pi$
$659$			265.710i			0.403202i	−0.979468	+	0.201601i	$0.935386\pi$
$660$		0			0
$661$	987.246			1.49356			0.746782	−	0.665069i	$-0.231598\pi$
$661$	987.246			1.49356			0.746782	+	0.665069i	$0.231598\pi$
$662$		0			0
$663$	−8.48424	−	176.744i	−0.0127967	−	0.266582i
$664$		0			0
$665$			70.2637i			0.105660i
$666$		0			0
$667$	491.376			0.736696
$668$		0			0
$669$	426.660	−	20.4810i	0.637758	−	0.0306143i
$670$		0			0
$671$			605.552i			0.902463i
$672$		0			0
$673$	587.447			0.872878			0.436439	−	0.899734i	$-0.356240\pi$
$673$	587.447			0.872878			0.436439	+	0.899734i	$0.356240\pi$
$674$		0			0
$675$	133.605	−	19.3593i	0.197933	−	0.0286805i
$676$		0			0
$677$			168.689i			0.249172i	0.992209	+	0.124586i	$0.0397602\pi$
$677$			168.689i			0.249172i	−0.992209	+	0.124586i	$0.960240\pi$
$678$		0			0
$679$	1.85207			0.00272764
$680$		0			0
$681$	46.3109	+	964.750i	0.0680043	+	1.41667i
$682$		0			0
$683$			363.184i			0.531749i	0.964008	+	0.265874i	$0.0856606\pi$
$683$			363.184i			0.531749i	−0.964008	+	0.265874i	$0.914339\pi$
$684$		0			0
$685$	346.045			0.505176
$686$		0			0
$687$	−322.253	+	15.4691i	−0.469073	+	0.0225169i
$688$		0			0
$689$		−	487.044i		−	0.706886i
$690$		0			0
$691$	395.111			0.571795			0.285898	−	0.958260i	$-0.407708\pi$
$691$	395.111			0.571795			0.285898	+	0.958260i	$0.407708\pi$
$692$		0			0
$693$	17.4637	+	181.484i	0.0252002	+	0.261881i
$694$		0			0
$695$		−	508.003i		−	0.730939i
$696$		0			0
$697$	−62.5060			−0.0896787
$698$		0			0
$699$	−41.8726	−	872.291i	−0.0599035	−	1.24791i
$700$		0			0
$701$			473.115i			0.674914i	0.941341	+	0.337457i	$0.109567\pi$
$701$			473.115i			0.674914i	−0.941341	+	0.337457i	$0.890433\pi$
$702$		0			0
$703$	−354.834			−0.504742
$704$		0			0
$705$	−346.676	+	16.6415i	−0.491739	+	0.0236049i
$706$		0			0
$707$			334.792i			0.473539i
$708$		0			0
$709$	−441.603			−0.622853			−0.311427	−	0.950270i	$-0.600807\pi$
$709$	−441.603			−0.622853			−0.311427	+	0.950270i	$0.600807\pi$
$710$		0			0
$711$	280.567	−	26.9983i	0.394608	−	0.0379723i
$712$		0			0
$713$			555.081i			0.778514i
$714$		0			0
$715$	−162.184			−0.226830
$716$		0			0
$717$	65.0899	+	1355.95i	0.0907809	+	1.89115i
$718$		0			0
$719$			1096.09i			1.52446i	0.647307	+	0.762229i	$0.275895\pi$
$719$			1096.09i			1.52446i	−0.647307	+	0.762229i	$0.724105\pi$
$720$		0			0
$721$	−193.601			−0.268517
$722$		0			0
$723$	−187.874	+	9.01852i	−0.259853	+	0.0124737i
$724$		0			0
$725$			194.927i			0.268864i
$726$		0			0
$727$	1331.47			1.83146			0.915731	−	0.401791i	$-0.131612\pi$
$727$	1331.47			1.83146			0.915731	+	0.401791i	$0.131612\pi$
$728$		0			0
$729$	699.017	−	206.920i	0.958871	−	0.283841i
$730$		0			0
$731$			337.819i			0.462133i
$732$		0			0
$733$	295.464			0.403089			0.201545	−	0.979479i	$-0.435404\pi$
$733$	295.464			0.403089			0.201545	+	0.979479i	$0.435404\pi$
$734$		0			0
$735$	−14.1422	−	294.611i	−0.0192411	−	0.400832i
$736$		0			0
$737$		−	140.621i		−	0.190802i
$738$		0			0
$739$	−512.085			−0.692943			−0.346471	−	0.938061i	$-0.612620\pi$
$739$	−512.085			−0.692943			−0.346471	+	0.938061i	$0.612620\pi$
$740$		0			0
$741$	−337.127	+	16.1831i	−0.454962	+	0.0218395i
$742$		0			0
$743$			117.627i			0.158313i	0.996862	+	0.0791567i	$0.0252227\pi$
$743$			117.627i			0.158313i	−0.996862	+	0.0791567i	$0.974777\pi$
$744$		0			0
$745$	233.441			0.313344
$746$		0			0
$747$	123.315	+	1281.49i	0.165080	+	1.71552i
$748$		0			0
$749$		−	350.523i		−	0.467987i
$750$		0			0
$751$	313.995			0.418103			0.209051	−	0.977905i	$-0.432962\pi$
$751$	313.995			0.418103			0.209051	+	0.977905i	$0.432962\pi$
$752$		0			0
$753$	34.9406	+	727.884i	0.0464019	+	0.966646i
$754$		0			0
$755$			483.563i			0.640480i
$756$		0			0
$757$	880.421			1.16304			0.581520	−	0.813532i	$-0.302458\pi$
$757$	880.421			1.16304			0.581520	+	0.813532i	$0.302458\pi$
$758$		0			0
$759$	341.105	−	16.3741i	0.449413	−	0.0215732i
$760$		0			0
$761$		−	177.647i		−	0.233439i	−0.993165	−	0.116720i	$-0.962762\pi$
$761$		−	177.647i		−	0.233439i	0.993165	−	0.116720i	$-0.0372379\pi$
$762$		0			0
$763$	58.5015			0.0766730
$764$		0			0
$765$	−147.123	+	14.1573i	−0.192318	+	0.0185063i
$766$		0			0
$767$		−	690.806i		−	0.900660i
$768$		0			0
$769$	−539.556			−0.701633			−0.350816	−	0.936444i	$-0.614096\pi$
$769$	−539.556			−0.701633			−0.350816	+	0.936444i	$0.614096\pi$
$770$		0			0
$771$	−37.2311	−	775.600i	−0.0482894	−	1.00597i
$772$		0			0
$773$			648.160i			0.838499i	0.907871	+	0.419250i	$0.137707\pi$
$773$			648.160i			0.838499i	−0.907871	+	0.419250i	$0.862293\pi$
$774$		0			0
$775$	−220.198			−0.284126
$776$		0			0
$777$	−170.249	+	8.17248i	−0.219111	+	0.0105180i
$778$		0			0
$779$			119.226i			0.153050i
$780$		0			0
$781$	−619.989			−0.793840
$782$		0			0
$783$	150.946	+	1041.72i	0.192779	+	1.33043i
$784$		0			0
$785$		−	535.241i		−	0.681836i
$786$		0			0
$787$	−1149.99			−1.46123			−0.730615	−	0.682789i	$-0.760767\pi$
$787$	−1149.99			−1.46123			−0.730615	+	0.682789i	$0.760767\pi$
$788$		0			0
$789$	29.6166	+	616.973i	0.0375368	+	0.781969i
$790$		0			0
$791$		−	96.0585i		−	0.121439i
$792$		0			0
$793$	−538.476			−0.679037
$794$		0			0
$795$	−406.356	+	19.5063i	−0.511140	+	0.0245362i
$796$		0			0
$797$		−	1436.14i		−	1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$797$		−	1436.14i		−	1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$798$		0			0
$799$	379.990			0.475581
$800$		0			0
$801$	121.190	+	1259.41i	0.151298	+	1.57229i
$802$		0			0
$803$		−	818.944i		−	1.01986i
$804$		0			0
$805$	63.2180			0.0785317
$806$		0			0
$807$	−28.8567	−	601.143i	−0.0357579	−	0.744910i
$808$		0			0
$809$			9.50158i			0.0117449i	0.999983	+	0.00587243i	$0.00186926\pi$
$809$			9.50158i			0.0117449i	−0.999983	+	0.00587243i	$0.998131\pi$
$810$		0			0
$811$	−938.529			−1.15725			−0.578624	−	0.815594i	$-0.696410\pi$
$811$	−938.529			−1.15725			−0.578624	+	0.815594i	$0.696410\pi$
$812$		0			0
$813$	−137.859	+	6.61763i	−0.169568	+	0.00813977i
$814$		0			0
$815$		−	44.4451i		−	0.0545339i
$816$		0			0
$817$	644.367			0.788699
$818$		0			0
$819$	−161.381	+	15.5293i	−0.197046	+	0.0189613i
$820$		0			0
$821$			469.104i			0.571381i	0.958322	+	0.285691i	$0.0922230\pi$
$821$			469.104i			0.571381i	−0.958322	+	0.285691i	$0.907777\pi$
$822$		0			0
$823$	1016.98			1.23570			0.617852	−	0.786294i	$-0.288003\pi$
$823$	1016.98			1.23570			0.617852	+	0.786294i	$0.288003\pi$
$824$		0			0
$825$	6.49551	+	135.315i	0.00787335	+	0.164018i
$826$		0			0
$827$			460.051i			0.556289i	0.960539	+	0.278145i	$0.0897194\pi$
$827$			460.051i			0.556289i	−0.960539	+	0.278145i	$0.910281\pi$
$828$		0			0
$829$	−18.7992			−0.0226770			−0.0113385	−	0.999936i	$-0.503609\pi$
$829$	−18.7992			−0.0226770			−0.0113385	+	0.999936i	$0.503609\pi$
$830$		0			0
$831$	769.909	−	36.9579i	0.926484	−	0.0444740i
$832$		0			0
$833$			322.922i			0.387661i
$834$		0			0
$835$	655.553			0.785093
$836$		0			0
$837$	−1176.78	+	170.515i	−1.40595	+	0.203722i
$838$		0			0
$839$			496.989i			0.592359i	0.955132	+	0.296179i	$0.0957126\pi$
$839$			496.989i			0.592359i	−0.955132	+	0.296179i	$0.904287\pi$
$840$		0			0
$841$	−678.856			−0.807200
$842$		0			0
$843$	6.17976	+	128.737i	0.00733068	+	0.152713i
$844$		0			0
$845$			233.677i			0.276541i
$846$		0			0
$847$	88.4542			0.104432
$848$		0			0
$849$	1555.72	−	74.6791i	1.83241	−	0.0879613i
$850$		0			0
$851$			319.253i			0.375150i
$852$		0			0
$853$	−1432.68			−1.67958			−0.839791	−	0.542911i	$-0.817322\pi$
$853$	−1432.68			−1.67958			−0.839791	+	0.542911i	$0.817322\pi$
$854$		0			0
$855$	27.0041	+	280.627i	0.0315837	+	0.328219i
$856$		0			0
$857$			1211.90i			1.41412i	0.707154	+	0.707060i	$0.249979\pi$
$857$			1211.90i			1.41412i	−0.707154	+	0.707060i	$0.750021\pi$
$858$		0			0
$859$	−18.5883			−0.0216394			−0.0108197	−	0.999941i	$-0.503444\pi$
$859$	−18.5883			−0.0216394			−0.0108197	+	0.999941i	$0.503444\pi$
$860$		0			0
$861$	2.74600	+	57.2047i	0.00318931	+	0.0664398i
$862$		0			0
$863$			936.142i			1.08475i	0.840135	+	0.542377i	$0.182475\pi$
$863$			936.142i			1.08475i	−0.840135	+	0.542377i	$0.817525\pi$
$864$		0			0
$865$	−201.517			−0.232968
$866$		0			0
$867$	−704.370	+	33.8119i	−0.812422	+	0.0389987i
$868$		0			0
$869$			282.845i			0.325483i
$870$		0			0
$871$	125.045			0.143564
$872$		0			0
$873$	7.39699	−	0.711796i	0.00847307	−	0.000815344i
$874$		0			0
$875$			25.0783i			0.0286609i
$876$		0			0
$877$	873.860			0.996419			0.498210	−	0.867057i	$-0.333991\pi$
$877$	873.860			0.996419			0.498210	+	0.867057i	$0.333991\pi$
$878$		0			0
$879$	−12.7700	−	266.025i	−0.0145279	−	0.302646i
$880$		0			0
$881$		−	808.376i		−	0.917566i	−0.888548	−	0.458783i	$-0.848286\pi$
$881$		−	808.376i		−	0.917566i	0.888548	−	0.458783i	$-0.151714\pi$
$882$		0			0
$883$	−177.913			−0.201487			−0.100743	−	0.994912i	$-0.532122\pi$
$883$	−177.913			−0.201487			−0.100743	+	0.994912i	$0.532122\pi$
$884$		0			0
$885$	−576.361	+	27.6670i	−0.651255	+	0.0312622i
$886$		0			0
$887$		−	1157.53i		−	1.30500i	−0.757789	−	0.652499i	$-0.773721\pi$
$887$		−	1157.53i		−	1.30500i	0.757789	−	0.652499i	$-0.226279\pi$
$888$		0			0
$889$	−220.751			−0.248313
$890$		0			0
$891$	139.497	+	718.117i	0.156563	+	0.805968i
$892$		0			0
$893$		−	724.804i		−	0.811651i
$894$		0			0
$895$	734.970			0.821195
$896$		0			0
$897$	14.5603	+	303.321i	0.0162322	+	0.338151i
$898$		0			0
$899$		−	1716.90i		−	1.90979i
$900$		0			0
$901$	445.405			0.494345
$902$		0			0
$903$	309.168	−	14.8410i	0.342379	−	0.0164352i
$904$		0			0
$905$		−	447.437i		−	0.494405i
$906$		0			0
$907$	−89.1205			−0.0982586			−0.0491293	−	0.998792i	$-0.515645\pi$
$907$	−89.1205			−0.0982586			−0.0491293	+	0.998792i	$0.515645\pi$
$908$		0			0
$909$	128.669	+	1337.13i	0.141550	+	1.47099i
$910$		0			0
$911$			280.274i			0.307655i	0.988098	+	0.153827i	$0.0491600\pi$
$911$			280.274i			0.307655i	−0.988098	+	0.153827i	$0.950840\pi$
$912$		0			0
$913$	−1291.90			−1.41500
$914$		0			0
$915$	21.5662	+	449.267i	0.0235696	+	0.491002i
$916$		0			0
$917$			221.128i			0.241143i
$918$		0			0
$919$	240.880			0.262111			0.131056	−	0.991375i	$-0.458163\pi$
$919$	240.880			0.262111			0.131056	+	0.991375i	$0.458163\pi$
$920$		0			0
$921$	1034.72	−	49.6697i	1.12348	−	0.0539302i
$922$		0			0
$923$		−	551.314i		−	0.597307i
$924$		0			0
$925$	−126.646			−0.136915
$926$		0			0
$927$	−773.224	+	74.4056i	−0.834114	+	0.0802649i
$928$		0			0
$929$		−	1119.74i		−	1.20531i	−0.798001	−	0.602657i	$-0.794109\pi$
$929$		−	1119.74i		−	1.20531i	0.798001	−	0.602657i	$-0.205891\pi$
$930$		0			0
$931$	615.951			0.661602
$932$		0			0
$933$	−9.38566	−	195.522i	−0.0100597	−	0.209563i
$934$		0			0
$935$		−	148.318i		−	0.158629i
$936$		0			0
$937$	771.901			0.823800			0.411900	−	0.911229i	$-0.364865\pi$
$937$	771.901			0.823800			0.411900	+	0.911229i	$0.364865\pi$
$938$		0			0
$939$	−344.179	+	16.5216i	−0.366537	+	0.0175949i
$940$		0			0
$941$		−	1423.21i		−	1.51245i	−0.654313	−	0.756223i	$-0.727042\pi$
$941$		−	1423.21i		−	1.51245i	0.654313	−	0.756223i	$-0.272958\pi$
$942$		0			0
$943$	107.270			0.113754
$944$		0			0
$945$	19.4199	+	134.023i	0.0205502	+	0.141823i
$946$		0			0
$947$			492.999i			0.520590i	0.965529	+	0.260295i	$0.0838199\pi$
$947$			492.999i			0.520590i	−0.965529	+	0.260295i	$0.916180\pi$
$948$		0			0
$949$	728.231			0.767366
$950$		0			0
$951$	−1.44854	−	30.1761i	−0.00152318	−	0.0317309i
$952$		0			0
$953$			1076.66i			1.12976i	0.825173	+	0.564880i	$0.191078\pi$
$953$			1076.66i			1.12976i	−0.825173	+	0.564880i	$0.808922\pi$
$954$		0			0
$955$	−442.244			−0.463082
$956$		0			0
$957$	−1055.06	+	50.6459i	−1.10246	+	0.0529215i
$958$		0			0
$959$			347.129i			0.361970i
$960$		0			0
$961$	978.485			1.01819
$962$		0			0
$963$	−134.715	−	1399.96i	−0.139891	−	1.45374i
$964$		0			0
$965$			475.560i			0.492808i
$966$		0			0
$967$	−7.16082			−0.00740519			−0.00370260	−	0.999993i	$-0.501179\pi$
$967$	−7.16082			−0.00740519			−0.00370260	+	0.999993i	$0.501179\pi$
$968$		0			0
$969$	−14.7995	−	308.304i	−0.0152730	−	0.318167i
$970$		0			0
$971$		−	161.099i		−	0.165911i	−0.996553	−	0.0829554i	$-0.973564\pi$
$971$		−	161.099i		−	0.165911i	0.996553	−	0.0829554i	$-0.0264359\pi$
$972$		0			0
$973$	509.593			0.523734
$974$		0			0
$975$	−120.326	+	5.77601i	−0.123411	+	0.00592412i
$976$		0			0
$977$		−	1311.58i		−	1.34246i	−0.741248	−	0.671231i	$-0.765766\pi$
$977$		−	1311.58i		−	1.34246i	0.741248	−	0.671231i	$-0.234234\pi$
$978$		0			0
$979$	−1269.63			−1.29687
$980$		0			0
$981$	233.650	−	22.4836i	0.238175	−	0.0229190i
$982$		0			0
$983$		−	674.305i		−	0.685966i	−0.939342	−	0.342983i	$-0.888563\pi$
$983$		−	674.305i		−	0.685966i	0.939342	−	0.342983i	$-0.111437\pi$
$984$		0			0
$985$	−153.225			−0.155558
$986$		0			0
$987$	−16.6936	−	347.761i	−0.0169135	−	0.352342i
$988$		0			0
$989$		−	579.753i		−	0.586201i
$990$		0			0
$991$	1291.05			1.30277			0.651386	−	0.758746i	$-0.274188\pi$
$991$	1291.05			1.30277			0.651386	+	0.758746i	$0.274188\pi$
$992$		0			0
$993$	1094.90	−	52.5584i	1.10262	−	0.0529289i
$994$		0			0
$995$		−	564.052i		−	0.566886i
$996$		0			0
$997$	409.552			0.410784			0.205392	−	0.978680i	$-0.434153\pi$
$997$	409.552			0.410784			0.205392	+	0.978680i	$0.434153\pi$
$998$		0			0
$999$	−676.820	+	98.0713i	−0.677497	+	0.0981694i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.3.l.j.641.2		16
3.2	odd	2	inner	960.3.l.j.641.1		16
4.3	odd	2	inner	960.3.l.j.641.15		16
8.3	odd	2		480.3.l.b.161.2	yes	16
8.5	even	2		480.3.l.b.161.15	yes	16
12.11	even	2	inner	960.3.l.j.641.16		16
24.5	odd	2		480.3.l.b.161.16	yes	16
24.11	even	2		480.3.l.b.161.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.3.l.b.161.1	✓	16	24.11	even	2
480.3.l.b.161.2	yes	16	8.3	odd	2
480.3.l.b.161.15	yes	16	8.5	even	2
480.3.l.b.161.16	yes	16	24.5	odd	2
960.3.l.j.641.1		16	3.2	odd	2	inner
960.3.l.j.641.2		16	1.1	even	1	trivial
960.3.l.j.641.15		16	4.3	odd	2	inner
960.3.l.j.641.16		16	12.11	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	960.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.99655 + 0.143843i) q^{3} -2.23607i q^{5} +2.24307 q^{7} +(8.95862 - 0.862068i) q^{9} +O(q^{10})\)	\(q+(-2.99655 + 0.143843i) q^{3} -2.23607i q^{5} +2.24307 q^{7} +(8.95862 - 0.862068i) q^{9} +9.03137i q^{11} -8.03098 q^{13} +(0.321644 + 6.70049i) q^{15} -7.34437i q^{17} -14.0089 q^{19} +(-6.72147 + 0.322651i) q^{21} +12.6041i q^{23} -5.00000 q^{25} +(-26.7209 + 3.87187i) q^{27} -38.9853i q^{29} +44.0396 q^{31} +(-1.29910 - 27.0629i) q^{33} -5.01566i q^{35} +25.3292 q^{37} +(24.0652 - 1.15520i) q^{39} -8.51074i q^{41} -45.9971 q^{43} +(-1.92764 - 20.0321i) q^{45} +51.7389i q^{47} -43.9686 q^{49} +(1.05644 + 22.0078i) q^{51} +60.6457i q^{53} +20.1947 q^{55} +(41.9783 - 2.01508i) q^{57} +86.0177i q^{59} +67.0499 q^{61} +(20.0948 - 1.93368i) q^{63} +17.9578i q^{65} -15.5703 q^{67} +(-1.81302 - 37.7689i) q^{69} +68.6484i q^{71} -90.6777 q^{73} +(14.9827 - 0.719217i) q^{75} +20.2580i q^{77} +31.3181 q^{79} +(79.5137 - 15.4459i) q^{81} +143.046i q^{83} -16.4225 q^{85} +(5.60778 + 116.821i) q^{87} +140.581i q^{89} -18.0140 q^{91} +(-131.967 + 6.33480i) q^{93} +31.3248i q^{95} +0.825684 q^{97} +(7.78565 + 80.9086i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{9}+O(q^{10}) \) 16 * q + 8 * q^9	\( 16 q + 8 q^{9} + 16 q^{13} + 104 q^{21} - 80 q^{25} + 192 q^{33} - 144 q^{37} - 40 q^{45} - 128 q^{49} - 80 q^{57} - 144 q^{61} + 280 q^{69} + 192 q^{81} + 96 q^{93} - 160 q^{97}+O(q^{100}) \) 16 * q + 8 * q^9 + 16 * q^13 + 104 * q^21 - 80 * q^25 + 192 * q^33 - 144 * q^37 - 40 * q^45 - 128 * q^49 - 80 * q^57 - 144 * q^61 + 280 * q^69 + 192 * q^81 + 96 * q^93 - 160 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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