LMFDB - Embedded newform 960.3.l.j.641.3

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			641.3
Root			$-0.317839 + 0.355703i$ of defining polynomial
Character	$\chi$	$=$	960.641
Dual form			960.3.l.j.641.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.65305 - 1.40048i) q^{3} +2.23607i q^{5} -12.5386 q^{7} +(5.07731 + 7.43108i) q^{9} +O(q^{10})$	\(q+(-2.65305 - 1.40048i) q^{3} +2.23607i q^{5} -12.5386 q^{7} +(5.07731 + 7.43108i) q^{9} -0.188971i q^{11} +10.5391 q^{13} +(3.13157 - 5.93239i) q^{15} +27.8619i q^{17} +19.7835 q^{19} +(33.2655 + 17.5601i) q^{21} +15.8894i q^{23} -5.00000 q^{25} +(-3.06327 - 26.8257i) q^{27} -27.2759i q^{29} -45.3952 q^{31} +(-0.264651 + 0.501350i) q^{33} -28.0372i q^{35} -43.8373 q^{37} +(-27.9607 - 14.7598i) q^{39} -76.1582i q^{41} +23.6486 q^{43} +(-16.6164 + 11.3532i) q^{45} +25.8265i q^{47} +108.217 q^{49} +(39.0201 - 73.9190i) q^{51} +45.0295i q^{53} +0.422553 q^{55} +(-52.4864 - 27.7063i) q^{57} -7.81352i q^{59} -6.83435 q^{61} +(-63.6624 - 93.1754i) q^{63} +23.5661i q^{65} -24.4197 q^{67} +(22.2528 - 42.1553i) q^{69} -110.457i q^{71} -67.7858 q^{73} +(13.2652 + 7.00240i) q^{75} +2.36944i q^{77} -20.9911 q^{79} +(-29.4418 + 75.4598i) q^{81} -92.8732i q^{83} -62.3012 q^{85} +(-38.1993 + 72.3641i) q^{87} -8.05818i q^{89} -132.145 q^{91} +(120.435 + 63.5750i) q^{93} +44.2371i q^{95} -22.8581 q^{97} +(1.40426 - 0.959466i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.65305	−	1.40048i	−0.884349	−	0.466827i
$3$	−2.65305	−	1.40048i	−0.884349	−	0.466827i
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$	−12.5386			−1.79123			−0.895615	−	0.444831i	$-0.853264\pi$
$7$	−12.5386			−1.79123			−0.895615	+	0.444831i	$0.853264\pi$
$8$		0			0
$9$	5.07731	+	7.43108i	0.564146	+	0.825675i
$10$		0			0
$11$		−	0.188971i		−	0.0171792i	−0.999963	−	0.00858960i	$-0.997266\pi$
$11$		−	0.188971i		−	0.0171792i	0.999963	−	0.00858960i	$-0.00273419\pi$
$12$		0			0
$13$	10.5391			0.810699			0.405349	−	0.914162i	$-0.367150\pi$
$13$	10.5391			0.810699			0.405349	+	0.914162i	$0.367150\pi$
$14$		0			0
$15$	3.13157	−	5.93239i	0.208771	−	0.395493i
$16$		0			0
$17$			27.8619i			1.63894i	0.573124	+	0.819469i	$0.305731\pi$
$17$			27.8619i			1.63894i	−0.573124	+	0.819469i	$0.694269\pi$
$18$		0			0
$19$	19.7835			1.04123			0.520617	−	0.853790i	$-0.325702\pi$
$19$	19.7835			1.04123			0.520617	+	0.853790i	$0.325702\pi$
$20$		0			0
$21$	33.2655	+	17.5601i	1.58407	+	0.836194i
$22$		0			0
$23$			15.8894i			0.690843i	0.938448	+	0.345422i	$0.112264\pi$
$23$			15.8894i			0.690843i	−0.938448	+	0.345422i	$0.887736\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	−3.06327	−	26.8257i	−0.113454	−	0.993543i
$28$		0			0
$29$		−	27.2759i		−	0.940547i	−0.882521	−	0.470274i	$-0.844155\pi$
$29$		−	27.2759i		−	0.940547i	0.882521	−	0.470274i	$-0.155845\pi$
$30$		0			0
$31$	−45.3952			−1.46436			−0.732180	−	0.681111i	$-0.761497\pi$
$31$	−45.3952			−1.46436			−0.732180	+	0.681111i	$0.761497\pi$
$32$		0			0
$33$	−0.264651	+	0.501350i	−0.00801971	+	0.0151924i
$34$		0			0
$35$		−	28.0372i		−	0.801062i
$36$		0			0
$37$	−43.8373			−1.18479			−0.592396	−	0.805647i	$-0.701818\pi$
$37$	−43.8373			−1.18479			−0.592396	+	0.805647i	$0.701818\pi$
$38$		0			0
$39$	−27.9607	−	14.7598i	−0.716940	−	0.378456i
$40$		0			0
$41$		−	76.1582i		−	1.85752i	−0.370687	−	0.928758i	$-0.620878\pi$
$41$		−	76.1582i		−	1.85752i	0.370687	−	0.928758i	$-0.379122\pi$
$42$		0			0
$43$	23.6486			0.549968			0.274984	−	0.961449i	$-0.411327\pi$
$43$	23.6486			0.549968			0.274984	+	0.961449i	$0.411327\pi$
$44$		0			0
$45$	−16.6164	+	11.3532i	−0.369253	+	0.252294i
$46$		0			0
$47$			25.8265i			0.549500i	0.961516	+	0.274750i	$0.0885951\pi$
$47$			25.8265i			0.549500i	−0.961516	+	0.274750i	$0.911405\pi$
$48$		0			0
$49$	108.217			2.20850
$50$		0			0
$51$	39.0201	−	73.9190i	0.765100	−	1.44939i
$52$		0			0
$53$			45.0295i			0.849613i	0.905284	+	0.424807i	$0.139658\pi$
$53$			45.0295i			0.849613i	−0.905284	+	0.424807i	$0.860342\pi$
$54$		0			0
$55$	0.422553			0.00768278
$56$		0			0
$57$	−52.4864	−	27.7063i	−0.920814	−	0.486076i
$58$		0			0
$59$		−	7.81352i		−	0.132432i	−0.997805	−	0.0662162i	$-0.978907\pi$
$59$		−	7.81352i		−	0.132432i	0.997805	−	0.0662162i	$-0.0210927\pi$
$60$		0			0
$61$	−6.83435			−0.112039			−0.0560193	−	0.998430i	$-0.517841\pi$
$61$	−6.83435			−0.112039			−0.0560193	+	0.998430i	$0.517841\pi$
$62$		0			0
$63$	−63.6624	−	93.1754i	−1.01051	−	1.47897i
$64$		0			0
$65$			23.5661i			0.362555i
$66$		0			0
$67$	−24.4197			−0.364474			−0.182237	−	0.983255i	$-0.558334\pi$
$67$	−24.4197			−0.364474			−0.182237	+	0.983255i	$0.558334\pi$
$68$		0			0
$69$	22.2528	−	42.1553i	0.322504	−	0.610946i
$70$		0			0
$71$		−	110.457i		−	1.55574i	−0.628427	−	0.777868i	$-0.716301\pi$
$71$		−	110.457i		−	1.55574i	0.628427	−	0.777868i	$-0.283699\pi$
$72$		0			0
$73$	−67.7858			−0.928573			−0.464287	−	0.885685i	$-0.653689\pi$
$73$	−67.7858			−0.928573			−0.464287	+	0.885685i	$0.653689\pi$
$74$		0			0
$75$	13.2652	+	7.00240i	0.176870	+	0.0933653i
$76$		0			0
$77$			2.36944i			0.0307719i
$78$		0			0
$79$	−20.9911			−0.265710			−0.132855	−	0.991135i	$-0.542414\pi$
$79$	−20.9911			−0.265710			−0.132855	+	0.991135i	$0.542414\pi$
$80$		0			0
$81$	−29.4418	+	75.4598i	−0.363479	+	0.931602i
$82$		0			0
$83$		−	92.8732i		−	1.11895i	−0.828846	−	0.559477i	$-0.811002\pi$
$83$		−	92.8732i		−	1.11895i	0.828846	−	0.559477i	$-0.188998\pi$
$84$		0			0
$85$	−62.3012			−0.732955
$86$		0			0
$87$	−38.1993	+	72.3641i	−0.439072	+	0.831772i
$88$		0			0
$89$		−	8.05818i		−	0.0905413i	−0.998975	−	0.0452707i	$-0.985585\pi$
$89$		−	8.05818i		−	0.0905413i	0.998975	−	0.0452707i	$-0.0144150\pi$
$90$		0			0
$91$	−132.145			−1.45215
$92$		0			0
$93$	120.435	+	63.5750i	1.29500	+	0.683602i
$94$		0			0
$95$			44.2371i			0.465654i
$96$		0			0
$97$	−22.8581			−0.235651			−0.117825	−	0.993034i	$-0.537592\pi$
$97$	−22.8581			−0.235651			−0.117825	+	0.993034i	$0.537592\pi$
$98$		0			0
$99$	1.40426	−	0.959466i	0.0141844	−	0.00969158i
$100$		0			0
$101$		−	157.603i		−	1.56043i	−0.625512	−	0.780215i	$-0.715110\pi$
$101$		−	157.603i		−	1.56043i	0.625512	−	0.780215i	$-0.284890\pi$
$102$		0			0
$103$	−95.3007			−0.925250			−0.462625	−	0.886554i	$-0.653092\pi$
$103$	−95.3007			−0.925250			−0.462625	+	0.886554i	$0.653092\pi$
$104$		0			0
$105$	−39.2655	+	74.3839i	−0.373957	+	0.708418i
$106$		0			0
$107$		−	146.404i		−	1.36826i	−0.729360	−	0.684130i	$-0.760182\pi$
$107$		−	146.404i		−	1.36826i	0.729360	−	0.684130i	$-0.239818\pi$
$108$		0			0
$109$	−65.2804			−0.598902			−0.299451	−	0.954112i	$-0.596804\pi$
$109$	−65.2804			−0.598902			−0.299451	+	0.954112i	$0.596804\pi$
$110$		0			0
$111$	116.302	+	61.3933i	1.04777	+	0.553093i
$112$		0			0
$113$		−	8.75984i		−	0.0775207i	−0.999249	−	0.0387603i	$-0.987659\pi$
$113$		−	8.75984i		−	0.0775207i	0.999249	−	0.0387603i	$-0.0123409\pi$
$114$		0			0
$115$	−35.5298			−0.308954
$116$		0			0
$117$	53.5102	+	78.3167i	0.457352	+	0.669374i
$118$		0			0
$119$		−	349.350i		−	2.93571i
$120$		0			0
$121$	120.964			0.999705
$122$		0			0
$123$	−106.658	+	202.051i	−0.867138	+	1.64269i
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	240.723			1.89545			0.947727	−	0.319081i	$-0.103374\pi$
$127$	240.723			1.89545			0.947727	+	0.319081i	$0.103374\pi$
$128$		0			0
$129$	−62.7409	−	33.1194i	−0.486364	−	0.256740i
$130$		0			0
$131$			73.3057i			0.559585i	0.960060	+	0.279793i	$0.0902657\pi$
$131$			73.3057i			0.559585i	−0.960060	+	0.279793i	$0.909734\pi$
$132$		0			0
$133$	−248.057			−1.86509
$134$		0			0
$135$	59.9840	−	6.84967i	0.444326	−	0.0507383i
$136$		0			0
$137$		−	140.965i		−	1.02894i	−0.857508	−	0.514470i	$-0.827989\pi$
$137$		−	140.965i		−	1.02894i	0.857508	−	0.514470i	$-0.172011\pi$
$138$		0			0
$139$	161.995			1.16543			0.582715	−	0.812677i	$-0.301991\pi$
$139$	161.995			1.16543			0.582715	+	0.812677i	$0.301991\pi$
$140$		0			0
$141$	36.1695	−	68.5189i	0.256521	−	0.485949i
$142$		0			0
$143$		−	1.99158i		−	0.0139272i
$144$		0			0
$145$	60.9907			0.420625
$146$		0			0
$147$	−287.104	−	151.555i	−1.95309	−	1.03099i
$148$		0			0
$149$		−	100.178i		−	0.672339i	−0.941802	−	0.336169i	$-0.890869\pi$
$149$		−	100.178i		−	0.672339i	0.941802	−	0.336169i	$-0.109131\pi$
$150$		0			0
$151$	−204.108			−1.35171			−0.675855	−	0.737035i	$-0.736225\pi$
$151$	−204.108			−1.35171			−0.675855	+	0.737035i	$0.736225\pi$
$152$		0			0
$153$	−207.044	+	141.464i	−1.35323	+	0.924599i
$154$		0			0
$155$		−	101.507i		−	0.654882i
$156$		0			0
$157$	59.5722			0.379440			0.189720	−	0.981838i	$-0.439242\pi$
$157$	59.5722			0.379440			0.189720	+	0.981838i	$0.439242\pi$
$158$		0			0
$159$	63.0629	−	119.465i	0.396622	−	0.751355i
$160$		0			0
$161$		−	199.231i		−	1.23746i
$162$		0			0
$163$	−244.940			−1.50270			−0.751350	−	0.659904i	$-0.770597\pi$
$163$	−244.940			−1.50270			−0.751350	+	0.659904i	$0.770597\pi$
$164$		0			0
$165$	−1.12105	−	0.591777i	−0.00679425	−	0.00358652i
$166$		0			0
$167$			14.7227i			0.0881599i	0.999028	+	0.0440799i	$0.0140356\pi$
$167$			14.7227i			0.0881599i	−0.999028	+	0.0440799i	$0.985964\pi$
$168$		0			0
$169$	−57.9277			−0.342768
$170$		0			0
$171$	100.447	+	147.012i	0.587408	+	0.859721i
$172$		0			0
$173$		−	13.0475i		−	0.0754193i	−0.999289	−	0.0377096i	$-0.987994\pi$
$173$		−	13.0475i		−	0.0754193i	0.999289	−	0.0377096i	$-0.0120062\pi$
$174$		0			0
$175$	62.6930			0.358246
$176$		0			0
$177$	−10.9427	+	20.7296i	−0.0618230	+	0.117117i
$178$		0			0
$179$			168.994i			0.944099i	0.881572	+	0.472049i	$0.156486\pi$
$179$			168.994i			0.944099i	−0.881572	+	0.472049i	$0.843514\pi$
$180$		0			0
$181$	52.3313			0.289123			0.144562	−	0.989496i	$-0.453823\pi$
$181$	52.3313			0.289123			0.144562	+	0.989496i	$0.453823\pi$
$182$		0			0
$183$	18.1319	+	9.57138i	0.0990812	+	0.0523026i
$184$		0			0
$185$		−	98.0232i		−	0.529855i
$186$		0			0
$187$	5.26511			0.0281556
$188$		0			0
$189$	38.4091	+	336.356i	0.203223	+	1.77966i
$190$		0			0
$191$			71.3300i			0.373456i	0.982412	+	0.186728i	$0.0597883\pi$
$191$			71.3300i			0.373456i	−0.982412	+	0.186728i	$0.940212\pi$
$192$		0			0
$193$	151.726			0.786143			0.393072	−	0.919508i	$-0.371412\pi$
$193$	151.726			0.786143			0.393072	+	0.919508i	$0.371412\pi$
$194$		0			0
$195$	33.0039	−	62.5220i	0.169251	−	0.320626i
$196$		0			0
$197$		−	358.155i		−	1.81804i	−0.416748	−	0.909022i	$-0.636830\pi$
$197$		−	358.155i		−	1.81804i	0.416748	−	0.909022i	$-0.363170\pi$
$198$		0			0
$199$	314.749			1.58165			0.790827	−	0.612039i	$-0.209651\pi$
$199$	314.749			1.58165			0.790827	+	0.612039i	$0.209651\pi$
$200$		0			0
$201$	64.7867	+	34.1994i	0.322322	+	0.170146i
$202$		0			0
$203$			342.001i			1.68474i
$204$		0			0
$205$	170.295			0.830706
$206$		0			0
$207$	−118.075	+	80.6754i	−0.570412	+	0.389736i
$208$		0			0
$209$		−	3.73850i		−	0.0178876i
$210$		0			0
$211$	−224.327			−1.06316			−0.531581	−	0.847007i	$-0.678402\pi$
$211$	−224.327			−1.06316			−0.531581	+	0.847007i	$0.678402\pi$
$212$		0			0
$213$	−154.693	+	293.048i	−0.726259	+	1.37581i
$214$		0			0
$215$			52.8799i			0.245953i
$216$		0			0
$217$	569.192			2.62300
$218$		0			0
$219$	179.839	+	94.9327i	0.821183	+	0.433483i
$220$		0			0
$221$			293.639i			1.32868i
$222$		0			0
$223$	−67.5604			−0.302961			−0.151481	−	0.988460i	$-0.548404\pi$
$223$	−67.5604			−0.302961			−0.151481	+	0.988460i	$0.548404\pi$
$224$		0			0
$225$	−25.3866	−	37.1554i	−0.112829	−	0.165135i
$226$		0			0
$227$		−	330.836i		−	1.45743i	−0.684819	−	0.728713i	$-0.740119\pi$
$227$		−	330.836i		−	1.45743i	0.684819	−	0.728713i	$-0.259881\pi$
$228$		0			0
$229$	−49.6628			−0.216868			−0.108434	−	0.994104i	$-0.534584\pi$
$229$	−49.6628			−0.216868			−0.108434	+	0.994104i	$0.534584\pi$
$230$		0			0
$231$	3.31835	−	6.28623i	0.0143651	−	0.0272131i
$232$		0			0
$233$			279.445i			1.19934i	0.800249	+	0.599668i	$0.204701\pi$
$233$			279.445i			1.19934i	−0.800249	+	0.599668i	$0.795299\pi$
$234$		0			0
$235$	−57.7498			−0.245744
$236$		0			0
$237$	55.6903	+	29.3976i	0.234980	+	0.124040i
$238$		0			0
$239$			106.652i			0.446241i	0.974791	+	0.223121i	$0.0716244\pi$
$239$			106.652i			0.446241i	−0.974791	+	0.223121i	$0.928376\pi$
$240$		0			0
$241$	−14.4164			−0.0598189			−0.0299095	−	0.999553i	$-0.509522\pi$
$241$	−14.4164			−0.0598189			−0.0299095	+	0.999553i	$0.509522\pi$
$242$		0			0
$243$	183.790	−	158.966i	0.756339	−	0.654180i
$244$		0			0
$245$			241.980i			0.987673i
$246$		0			0
$247$	208.499			0.844127
$248$		0			0
$249$	−130.067	+	246.397i	−0.522358	+	0.989546i
$250$		0			0
$251$			228.057i			0.908592i	0.890851	+	0.454296i	$0.150109\pi$
$251$			228.057i			0.908592i	−0.890851	+	0.454296i	$0.849891\pi$
$252$		0			0
$253$	3.00264			0.0118681
$254$		0			0
$255$	165.288	+	87.2515i	0.648188	+	0.342163i
$256$		0			0
$257$			139.868i			0.544235i	0.962264	+	0.272117i	$0.0877239\pi$
$257$			139.868i			0.544235i	−0.962264	+	0.272117i	$0.912276\pi$
$258$		0			0
$259$	549.659			2.12223
$260$		0			0
$261$	202.689	−	138.488i	0.776586	−	0.530606i
$262$		0			0
$263$		−	183.087i		−	0.696149i	−0.937467	−	0.348074i	$-0.886836\pi$
$263$		−	183.087i		−	0.696149i	0.937467	−	0.348074i	$-0.113164\pi$
$264$		0			0
$265$	−100.689			−0.379959
$266$		0			0
$267$	−11.2853	+	21.3787i	−0.0422671	+	0.0800701i
$268$		0			0
$269$		−	82.0353i		−	0.304964i	−0.988306	−	0.152482i	$-0.951273\pi$
$269$		−	82.0353i		−	0.304964i	0.988306	−	0.152482i	$-0.0487266\pi$
$270$		0			0
$271$	−46.3882			−0.171174			−0.0855872	−	0.996331i	$-0.527277\pi$
$271$	−46.3882			−0.171174			−0.0855872	+	0.996331i	$0.527277\pi$
$272$		0			0
$273$	350.588	+	185.067i	1.28420	+	0.677901i
$274$		0			0
$275$			0.944856i			0.00343584i
$276$		0			0
$277$	348.090			1.25664			0.628321	−	0.777954i	$-0.283743\pi$
$277$	348.090			1.25664			0.628321	+	0.777954i	$0.283743\pi$
$278$		0			0
$279$	−230.485	−	337.335i	−0.826112	−	1.20909i
$280$		0			0
$281$		−	247.067i		−	0.879243i	−0.898183	−	0.439621i	$-0.855113\pi$
$281$		−	247.067i		−	0.879243i	0.898183	−	0.439621i	$-0.144887\pi$
$282$		0			0
$283$	240.654			0.850367			0.425183	−	0.905107i	$-0.360210\pi$
$283$	240.654			0.850367			0.425183	+	0.905107i	$0.360210\pi$
$284$		0			0
$285$	61.9532	−	117.363i	0.217380	−	0.411801i
$286$		0			0
$287$			954.917i			3.32724i
$288$		0			0
$289$	−487.287			−1.68611
$290$		0			0
$291$	60.6436	+	32.0123i	0.208397	+	0.110008i
$292$		0			0
$293$		−	332.723i		−	1.13557i	−0.823175	−	0.567787i	$-0.807800\pi$
$293$		−	332.723i		−	1.13557i	0.823175	−	0.567787i	$-0.192200\pi$
$294$		0			0
$295$	17.4716			0.0592256
$296$		0			0
$297$	−5.06928	+	0.578870i	−0.0170683	+	0.00194906i
$298$		0			0
$299$			167.460i			0.560066i
$300$		0			0
$301$	−296.521			−0.985119
$302$		0			0
$303$	−220.720	+	418.129i	−0.728450	+	1.37996i
$304$		0			0
$305$		−	15.2821i		−	0.0501052i
$306$		0			0
$307$	−22.8530			−0.0744398			−0.0372199	−	0.999307i	$-0.511850\pi$
$307$	−22.8530			−0.0744398			−0.0372199	+	0.999307i	$0.511850\pi$
$308$		0			0
$309$	252.837	+	133.467i	0.818243	+	0.431931i
$310$		0			0
$311$			512.365i			1.64748i	0.566971	+	0.823738i	$0.308115\pi$
$311$			512.365i			1.64748i	−0.566971	+	0.823738i	$0.691885\pi$
$312$		0			0
$313$	−316.826			−1.01222			−0.506112	−	0.862468i	$-0.668918\pi$
$313$	−316.826			−1.01222			−0.506112	+	0.862468i	$0.668918\pi$
$314$		0			0
$315$	208.346	−	142.353i	0.661417	−	0.451916i
$316$		0			0
$317$		−	538.756i		−	1.69955i	−0.527149	−	0.849773i	$-0.676739\pi$
$317$		−	538.756i		−	1.69955i	0.527149	−	0.849773i	$-0.323261\pi$
$318$		0			0
$319$	−5.15436			−0.0161579
$320$		0			0
$321$	−205.036	+	388.416i	−0.638740	+	1.21002i
$322$		0			0
$323$			551.205i			1.70652i
$324$		0			0
$325$	−52.6954			−0.162140
$326$		0			0
$327$	173.192	+	91.4239i	0.529639	+	0.279584i
$328$		0			0
$329$		−	323.828i		−	0.984280i
$330$		0			0
$331$	176.668			0.533740			0.266870	−	0.963733i	$-0.414011\pi$
$331$	176.668			0.533740			0.266870	+	0.963733i	$0.414011\pi$
$332$		0			0
$333$	−222.576	−	325.758i	−0.668395	−	0.978253i
$334$		0			0
$335$		−	54.6042i		−	0.162998i
$336$		0			0
$337$	−277.211			−0.822583			−0.411292	−	0.911504i	$-0.634922\pi$
$337$	−277.211			−0.822583			−0.411292	+	0.911504i	$0.634922\pi$
$338$		0			0
$339$	−12.2680	+	23.2403i	−0.0361887	+	0.0685553i
$340$		0			0
$341$			8.57838i			0.0251565i
$342$		0			0
$343$	−742.494			−2.16471
$344$		0			0
$345$	94.2621	+	49.7587i	0.273223	+	0.144228i
$346$		0			0
$347$			362.220i			1.04386i	0.852988	+	0.521931i	$0.174788\pi$
$347$			362.220i			1.04386i	−0.852988	+	0.521931i	$0.825212\pi$
$348$		0			0
$349$	−98.8962			−0.283370			−0.141685	−	0.989912i	$-0.545252\pi$
$349$	−98.8962			−0.283370			−0.141685	+	0.989912i	$0.545252\pi$
$350$		0			0
$351$	−32.2840	−	282.718i	−0.0919773	−	0.805464i
$352$		0			0
$353$		−	606.538i		−	1.71824i	−0.511776	−	0.859119i	$-0.671012\pi$
$353$		−	606.538i		−	1.71824i	0.511776	−	0.859119i	$-0.328988\pi$
$354$		0			0
$355$	246.990			0.695747
$356$		0			0
$357$	−489.257	+	926.841i	−1.37047	+	2.59619i
$358$		0			0
$359$		−	350.441i		−	0.976158i	−0.872799	−	0.488079i	$-0.837698\pi$
$359$		−	350.441i		−	0.976158i	0.872799	−	0.488079i	$-0.162302\pi$
$360$		0			0
$361$	30.3849			0.0841688
$362$		0			0
$363$	−320.924	−	169.408i	−0.884088	−	0.466689i
$364$		0			0
$365$		−	151.574i		−	0.415271i
$366$		0			0
$367$	−121.789			−0.331850			−0.165925	−	0.986138i	$-0.553061\pi$
$367$	−121.789			−0.331850			−0.165925	+	0.986138i	$0.553061\pi$
$368$		0			0
$369$	565.937	−	386.679i	1.53370	−	1.04791i
$370$		0			0
$371$		−	564.607i		−	1.52185i
$372$		0			0
$373$	67.8491			0.181901			0.0909505	−	0.995855i	$-0.471009\pi$
$373$	67.8491			0.181901			0.0909505	+	0.995855i	$0.471009\pi$
$374$		0			0
$375$	−15.6578	+	29.6620i	−0.0417542	+	0.0790986i
$376$		0			0
$377$		−	287.463i		−	0.762500i
$378$		0			0
$379$	686.323			1.81088			0.905439	−	0.424476i	$-0.139542\pi$
$379$	686.323			1.81088			0.905439	+	0.424476i	$0.139542\pi$
$380$		0			0
$381$	−638.649	−	337.127i	−1.67624	−	0.884849i
$382$		0			0
$383$			431.125i			1.12565i	0.826575	+	0.562826i	$0.190286\pi$
$383$			431.125i			1.12565i	−0.826575	+	0.562826i	$0.809714\pi$
$384$		0			0
$385$	−5.29822			−0.0137616
$386$		0			0
$387$	120.071	+	175.735i	0.310262	+	0.454095i
$388$		0			0
$389$		−	133.062i		−	0.342061i	−0.985266	−	0.171030i	$-0.945290\pi$
$389$		−	133.062i		−	0.342061i	0.985266	−	0.171030i	$-0.0547096\pi$
$390$		0			0
$391$	−442.709			−1.13225
$392$		0			0
$393$	102.663	−	194.483i	0.261229	−	0.494869i
$394$		0			0
$395$		−	46.9375i		−	0.118829i
$396$		0			0
$397$	−164.706			−0.414876			−0.207438	−	0.978248i	$-0.566512\pi$
$397$	−164.706			−0.414876			−0.207438	+	0.978248i	$0.566512\pi$
$398$		0			0
$399$	658.106	+	347.399i	1.64939	+	0.870674i
$400$		0			0
$401$			38.9227i			0.0970641i	0.998822	+	0.0485320i	$0.0154543\pi$
$401$			38.9227i			0.0970641i	−0.998822	+	0.0485320i	$0.984546\pi$
$402$		0			0
$403$	−478.423			−1.18715
$404$		0			0
$405$	−168.733	−	65.8339i	−0.416625	−	0.162553i
$406$		0			0
$407$			8.28399i			0.0203538i
$408$		0			0
$409$	−148.923			−0.364114			−0.182057	−	0.983288i	$-0.558276\pi$
$409$	−148.923			−0.364114			−0.182057	+	0.983288i	$0.558276\pi$
$410$		0			0
$411$	−197.418	+	373.986i	−0.480337	+	0.909942i
$412$		0			0
$413$			97.9706i			0.237217i
$414$		0			0
$415$	207.671			0.500412
$416$		0			0
$417$	−429.779	−	226.870i	−1.03065	−	0.544053i
$418$		0			0
$419$		−	560.005i		−	1.33653i	−0.743925	−	0.668263i	$-0.767038\pi$
$419$		−	560.005i		−	1.33653i	0.743925	−	0.668263i	$-0.232962\pi$
$420$		0			0
$421$	344.834			0.819084			0.409542	−	0.912291i	$-0.365688\pi$
$421$	344.834			0.819084			0.409542	+	0.912291i	$0.365688\pi$
$422$		0			0
$423$	−191.919	+	131.129i	−0.453708	+	0.309998i
$424$		0			0
$425$		−	139.310i		−	0.327787i
$426$		0			0
$427$	85.6933			0.200687
$428$		0			0
$429$	−2.78917	+	5.28377i	−0.00650157	+	0.0123165i
$430$		0			0
$431$		−	0.916843i		−	0.00212725i	−0.999999	−	0.00106362i	$-0.999661\pi$
$431$		−	0.916843i		−	0.00212725i	0.999999	−	0.00106362i	$-0.000338562\pi$
$432$		0			0
$433$	230.569			0.532492			0.266246	−	0.963905i	$-0.414217\pi$
$433$	230.569			0.532492			0.266246	+	0.963905i	$0.414217\pi$
$434$		0			0
$435$	−161.811	−	85.4162i	−0.371980	−	0.196359i
$436$		0			0
$437$			314.347i			0.719329i
$438$		0			0
$439$	651.139			1.48323			0.741616	−	0.670825i	$-0.234060\pi$
$439$	651.139			1.48323			0.741616	+	0.670825i	$0.234060\pi$
$440$		0			0
$441$	549.450	+	804.166i	1.24592	+	1.82351i
$442$		0			0
$443$			399.277i			0.901302i	0.892700	+	0.450651i	$0.148808\pi$
$443$			399.277i			0.901302i	−0.892700	+	0.450651i	$0.851192\pi$
$444$		0			0
$445$	18.0186			0.0404913
$446$		0			0
$447$	−140.298	+	265.778i	−0.313866	+	0.594582i
$448$		0			0
$449$			181.919i			0.405165i	0.979265	+	0.202582i	$0.0649334\pi$
$449$			181.919i			0.405165i	−0.979265	+	0.202582i	$0.935067\pi$
$450$		0			0
$451$	−14.3917			−0.0319107
$452$		0			0
$453$	541.508	+	285.849i	1.19538	+	0.631014i
$454$		0			0
$455$		−	295.486i		−	0.649420i
$456$		0			0
$457$	−709.933			−1.55346			−0.776732	−	0.629831i	$-0.783124\pi$
$457$	−709.933			−1.55346			−0.776732	+	0.629831i	$0.783124\pi$
$458$		0			0
$459$	747.415	−	85.3485i	1.62835	−	0.185945i
$460$		0			0
$461$			359.776i			0.780425i	0.920725	+	0.390212i	$0.127598\pi$
$461$			359.776i			0.780425i	−0.920725	+	0.390212i	$0.872402\pi$
$462$		0			0
$463$	318.022			0.686873			0.343437	−	0.939176i	$-0.388409\pi$
$463$	318.022			0.686873			0.343437	+	0.939176i	$0.388409\pi$
$464$		0			0
$465$	−142.158	+	269.302i	−0.305716	+	0.579144i
$466$		0			0
$467$		−	366.006i		−	0.783739i	−0.920021	−	0.391870i	$-0.871828\pi$
$467$		−	366.006i		−	0.783739i	0.920021	−	0.391870i	$-0.128172\pi$
$468$		0			0
$469$	306.189			0.652856
$470$		0			0
$471$	−158.048	−	83.4296i	−0.335558	−	0.177133i
$472$		0			0
$473$		−	4.46891i		−	0.00944801i
$474$		0			0
$475$	−98.9173			−0.208247
$476$		0			0
$477$	−334.618	+	228.629i	−0.701505	+	0.479306i
$478$		0			0
$479$		−	395.258i		−	0.825173i	−0.910918	−	0.412587i	$-0.864625\pi$
$479$		−	395.258i		−	0.825173i	0.910918	−	0.412587i	$-0.135375\pi$
$480$		0			0
$481$	−462.005			−0.960509
$482$		0			0
$483$	−279.019	+	528.569i	−0.577679	+	1.09434i
$484$		0			0
$485$		−	51.1123i		−	0.105386i
$486$		0			0
$487$	−234.984			−0.482513			−0.241257	−	0.970461i	$-0.577560\pi$
$487$	−234.984			−0.482513			−0.241257	+	0.970461i	$0.577560\pi$
$488$		0			0
$489$	649.838	+	343.034i	1.32891	+	0.701501i
$490$		0			0
$491$			381.082i			0.776134i	0.921631	+	0.388067i	$0.126857\pi$
$491$			381.082i			0.776134i	−0.921631	+	0.388067i	$0.873143\pi$
$492$		0			0
$493$	759.958			1.54150
$494$		0			0
$495$	2.14543	+	3.14002i	0.00433420	+	0.00634348i
$496$		0			0
$497$			1384.98i			2.78668i
$498$		0			0
$499$	−335.389			−0.672122			−0.336061	−	0.941840i	$-0.609095\pi$
$499$	−335.389			−0.672122			−0.336061	+	0.941840i	$0.609095\pi$
$500$		0			0
$501$	20.6188	−	39.0600i	0.0411554	−	0.0779641i
$502$		0			0
$503$		−	61.3530i		−	0.121974i	−0.998139	−	0.0609871i	$-0.980575\pi$
$503$		−	61.3530i		−	0.121974i	0.998139	−	0.0609871i	$-0.0194248\pi$
$504$		0			0
$505$	352.412			0.697845
$506$		0			0
$507$	153.685	+	81.1266i	0.303126	+	0.160013i
$508$		0			0
$509$		−	203.626i		−	0.400051i	−0.979791	−	0.200025i	$-0.935898\pi$
$509$		−	203.626i		−	0.400051i	0.979791	−	0.200025i	$-0.0641025\pi$
$510$		0			0
$511$	849.940			1.66329
$512$		0			0
$513$	−60.6020	−	530.704i	−0.118133	−	1.03451i
$514$		0			0
$515$		−	213.099i		−	0.413784i
$516$		0			0
$517$	4.88047			0.00943997
$518$		0			0
$519$	−18.2728	+	34.6157i	−0.0352077	+	0.0666970i
$520$		0			0
$521$		−	712.647i		−	1.36784i	−0.729555	−	0.683922i	$-0.760273\pi$
$521$		−	712.647i		−	1.36784i	0.729555	−	0.683922i	$-0.239727\pi$
$522$		0			0
$523$	−580.738			−1.11040			−0.555199	−	0.831717i	$-0.687358\pi$
$523$	−580.738			−1.11040			−0.555199	+	0.831717i	$0.687358\pi$
$524$		0			0
$525$	−166.328	−	87.8003i	−0.316814	−	0.167239i
$526$		0			0
$527$		−	1264.80i		−	2.39999i
$528$		0			0
$529$	276.527			0.522736
$530$		0			0
$531$	58.0628	−	39.6717i	0.109346	−	0.0747112i
$532$		0			0
$533$		−	802.637i		−	1.50589i
$534$		0			0
$535$	327.369			0.611904
$536$		0			0
$537$	236.672	−	448.348i	0.440731	−	0.834913i
$538$		0			0
$539$		−	20.4498i		−	0.0379403i
$540$		0			0
$541$	465.497			0.860438			0.430219	−	0.902724i	$-0.358436\pi$
$541$	465.497			0.860438			0.430219	+	0.902724i	$0.358436\pi$
$542$		0			0
$543$	−138.837	−	73.2889i	−0.255686	−	0.134970i
$544$		0			0
$545$		−	145.971i		−	0.267837i
$546$		0			0
$547$	−858.664			−1.56977			−0.784885	−	0.619641i	$-0.787278\pi$
$547$	−858.664			−1.56977			−0.784885	+	0.619641i	$0.787278\pi$
$548$		0			0
$549$	−34.7001	−	50.7866i	−0.0632061	−	0.0925075i
$550$		0			0
$551$		−	539.611i		−	0.979330i
$552$		0			0
$553$	263.199			0.475947
$554$		0			0
$555$	−137.280	+	260.060i	−0.247350	+	0.468577i
$556$		0			0
$557$		−	68.6276i		−	0.123209i	−0.998101	−	0.0616047i	$-0.980378\pi$
$557$		−	68.6276i		−	0.123209i	0.998101	−	0.0616047i	$-0.0196218\pi$
$558$		0			0
$559$	249.235			0.445858
$560$		0			0
$561$	−13.9686	−	7.37367i	−0.0248994	−	0.0131438i
$562$		0			0
$563$		−	941.510i		−	1.67231i	−0.548494	−	0.836155i	$-0.684799\pi$
$563$		−	941.510i		−	1.67231i	0.548494	−	0.836155i	$-0.315201\pi$
$564$		0			0
$565$	19.5876			0.0346683
$566$		0			0
$567$	369.159	−	946.161i	0.651075	−	1.66871i
$568$		0			0
$569$			926.834i			1.62888i	0.580246	+	0.814441i	$0.302956\pi$
$569$			926.834i			1.62888i	−0.580246	+	0.814441i	$0.697044\pi$
$570$		0			0
$571$	−485.307			−0.849925			−0.424963	−	0.905211i	$-0.639713\pi$
$571$	−485.307			−0.849925			−0.424963	+	0.905211i	$0.639713\pi$
$572$		0			0
$573$	99.8963	−	189.242i	0.174339	−	0.330265i
$574$		0			0
$575$		−	79.4469i		−	0.138169i
$576$		0			0
$577$	75.0428			0.130057			0.0650284	−	0.997883i	$-0.479286\pi$
$577$	75.0428			0.130057			0.0650284	+	0.997883i	$0.479286\pi$
$578$		0			0
$579$	−402.535	−	212.489i	−0.695225	−	0.366993i
$580$		0			0
$581$			1164.50i			2.00430i
$582$		0			0
$583$	8.50928			0.0145957
$584$		0			0
$585$	−175.122	+	119.652i	−0.299353	+	0.204534i
$586$		0			0
$587$			80.6015i			0.137311i	0.997640	+	0.0686555i	$0.0218709\pi$
$587$			80.6015i			0.137311i	−0.997640	+	0.0686555i	$0.978129\pi$
$588$		0			0
$589$	−898.073			−1.52474
$590$		0			0
$591$	−501.589	+	950.201i	−0.848712	+	1.60779i
$592$		0			0
$593$		−	264.412i		−	0.445889i	−0.974831	−	0.222944i	$-0.928433\pi$
$593$		−	264.412i		−	0.445889i	0.974831	−	0.222944i	$-0.0715668\pi$
$594$		0			0
$595$	781.170			1.31289
$596$		0			0
$597$	−835.044	−	440.800i	−1.39873	−	0.738359i
$598$		0			0
$599$			480.698i			0.802501i	0.915968	+	0.401250i	$0.131424\pi$
$599$			480.698i			0.802501i	−0.915968	+	0.401250i	$0.868576\pi$
$600$		0			0
$601$	−1184.55			−1.97097			−0.985486	−	0.169758i	$-0.945701\pi$
$601$	−1184.55			−1.97097			−0.985486	+	0.169758i	$0.945701\pi$
$602$		0			0
$603$	−123.987	−	181.465i	−0.205616	−	0.300937i
$604$		0			0
$605$			270.484i			0.447082i
$606$		0			0
$607$	−592.820			−0.976640			−0.488320	−	0.872665i	$-0.662390\pi$
$607$	−592.820			−0.976640			−0.488320	+	0.872665i	$0.662390\pi$
$608$		0			0
$609$	478.966	−	907.345i	0.786479	−	1.48989i
$610$		0			0
$611$			272.188i			0.445479i
$612$		0			0
$613$	529.948			0.864516			0.432258	−	0.901750i	$-0.357717\pi$
$613$	529.948			0.864516			0.432258	+	0.901750i	$0.357717\pi$
$614$		0			0
$615$	−451.800	−	238.494i	−0.734634	−	0.387796i
$616$		0			0
$617$			843.085i			1.36643i	0.730219	+	0.683213i	$0.239418\pi$
$617$			843.085i			1.36643i	−0.730219	+	0.683213i	$0.760582\pi$
$618$		0			0
$619$	−949.995			−1.53472			−0.767362	−	0.641214i	$-0.778431\pi$
$619$	−949.995			−1.53472			−0.767362	+	0.641214i	$0.778431\pi$
$620$		0			0
$621$	426.243	−	48.6734i	0.686382	−	0.0783791i
$622$		0			0
$623$			101.038i			0.162180i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−5.23570	+	9.91843i	−0.00835040	+	0.0158189i
$628$		0			0
$629$		−	1221.39i		−	1.94180i
$630$		0			0
$631$	−630.436			−0.999107			−0.499553	−	0.866283i	$-0.666503\pi$
$631$	−630.436			−0.999107			−0.499553	+	0.866283i	$0.666503\pi$
$632$		0			0
$633$	595.151	+	314.166i	0.940207	+	0.496313i
$634$		0			0
$635$			538.272i			0.847673i
$636$		0			0
$637$	1140.50			1.79043
$638$		0			0
$639$	820.817	−	560.826i	1.28453	−	0.877662i
$640$		0			0
$641$		−	569.028i		−	0.887720i	−0.896096	−	0.443860i	$-0.853609\pi$
$641$		−	569.028i		−	0.887720i	0.896096	−	0.443860i	$-0.146391\pi$
$642$		0			0
$643$	−598.681			−0.931075			−0.465538	−	0.885028i	$-0.654139\pi$
$643$	−598.681			−0.931075			−0.465538	+	0.885028i	$0.654139\pi$
$644$		0			0
$645$	74.0573	−	140.293i	0.114817	−	0.217508i
$646$		0			0
$647$		−	648.494i		−	1.00231i	−0.865358	−	0.501155i	$-0.832909\pi$
$647$		−	648.494i		−	1.00231i	0.865358	−	0.501155i	$-0.167091\pi$
$648$		0			0
$649$	−1.47653			−0.00227509
$650$		0			0
$651$	−1510.09	−	797.142i	−2.31965	−	1.22449i
$652$		0			0
$653$			644.566i			0.987083i	0.869722	+	0.493542i	$0.164298\pi$
$653$			644.566i			0.987083i	−0.869722	+	0.493542i	$0.835702\pi$
$654$		0			0
$655$	−163.916			−0.250254
$656$		0			0
$657$	−344.170	−	503.722i	−0.523851	−	0.766700i
$658$		0			0
$659$		−	1094.46i		−	1.66079i	−0.557177	−	0.830394i	$-0.688116\pi$
$659$		−	1094.46i		−	1.66079i	0.557177	−	0.830394i	$-0.311884\pi$
$660$		0			0
$661$	−752.590			−1.13856			−0.569281	−	0.822143i	$-0.692778\pi$
$661$	−752.590			−1.13856			−0.569281	+	0.822143i	$0.692778\pi$
$662$		0			0
$663$	411.236	−	779.038i	0.620265	−	1.17502i
$664$		0			0
$665$		−	554.672i		−	0.834093i
$666$		0			0
$667$	433.397			0.649770
$668$		0			0
$669$	179.241	+	94.6170i	0.267923	+	0.141430i
$670$		0			0
$671$			1.29150i			0.00192473i
$672$		0			0
$673$	−1271.71			−1.88961			−0.944804	−	0.327636i	$-0.893748\pi$
$673$	−1271.71			−1.88961			−0.944804	+	0.327636i	$0.893748\pi$
$674$		0			0
$675$	15.3163	+	134.128i	0.0226909	+	0.198709i
$676$		0			0
$677$			91.8839i			0.135722i	0.997695	+	0.0678611i	$0.0216175\pi$
$677$			91.8839i			0.135722i	−0.997695	+	0.0678611i	$0.978383\pi$
$678$		0			0
$679$	286.609			0.422104
$680$		0			0
$681$	−463.329	+	877.722i	−0.680365	+	1.28887i
$682$		0			0
$683$		−	727.437i		−	1.06506i	−0.846411	−	0.532531i	$-0.821241\pi$
$683$		−	727.437i		−	1.06506i	0.846411	−	0.532531i	$-0.178759\pi$
$684$		0			0
$685$	315.207			0.460156
$686$		0			0
$687$	131.758	+	69.5518i	0.191787	+	0.101240i
$688$		0			0
$689$			474.570i			0.688780i
$690$		0			0
$691$	176.915			0.256028			0.128014	−	0.991772i	$-0.459140\pi$
$691$	176.915			0.256028			0.128014	+	0.991772i	$0.459140\pi$
$692$		0			0
$693$	−17.6075	+	12.0304i	−0.0254076	+	0.0173598i
$694$		0			0
$695$			362.231i			0.521196i
$696$		0			0
$697$	2121.91			3.04435
$698$		0			0
$699$	391.357	−	741.381i	0.559882	−	1.06063i
$700$		0			0
$701$			1005.37i			1.43420i	0.696972	+	0.717098i	$0.254530\pi$
$701$			1005.37i			1.43420i	−0.696972	+	0.717098i	$0.745470\pi$
$702$		0			0
$703$	−867.253			−1.23365
$704$		0			0
$705$	153.213	+	80.8774i	0.217323	+	0.114720i
$706$		0			0
$707$			1976.13i			2.79509i
$708$		0			0
$709$	89.8801			0.126770			0.0633851	−	0.997989i	$-0.479810\pi$
$709$	89.8801			0.126770			0.0633851	+	0.997989i	$0.479810\pi$
$710$		0			0
$711$	−106.578	−	155.986i	−0.149899	−	0.219390i
$712$		0			0
$713$		−	721.301i		−	1.01164i
$714$		0			0
$715$	4.45332			0.00622842
$716$		0			0
$717$	149.364	−	282.952i	0.208317	−	0.394633i
$718$		0			0
$719$		−	361.095i		−	0.502218i	−0.967959	−	0.251109i	$-0.919205\pi$
$719$		−	361.095i		−	0.502218i	0.967959	−	0.251109i	$-0.0807952\pi$
$720$		0			0
$721$	1194.94			1.65733
$722$		0			0
$723$	38.2473	+	20.1898i	0.0529008	+	0.0279251i
$724$		0			0
$725$			136.379i			0.188109i
$726$		0			0
$727$	−637.478			−0.876861			−0.438431	−	0.898765i	$-0.644466\pi$
$727$	−637.478			−0.876861			−0.438431	+	0.898765i	$0.644466\pi$
$728$		0			0
$729$	−710.233	+	164.348i	−0.974256	+	0.225444i
$730$		0			0
$731$			658.896i			0.901363i
$732$		0			0
$733$	−1391.58			−1.89848			−0.949239	−	0.314557i	$-0.898144\pi$
$733$	−1391.58			−1.89848			−0.949239	+	0.314557i	$0.898144\pi$
$734$		0			0
$735$	338.888	−	641.984i	0.461072	−	0.873447i
$736$		0			0
$737$			4.61463i			0.00626137i
$738$		0			0
$739$	1018.45			1.37814			0.689070	−	0.724695i	$-0.258019\pi$
$739$	1018.45			1.37814			0.689070	+	0.724695i	$0.258019\pi$
$740$		0			0
$741$	−553.159	−	291.999i	−0.746503	−	0.394061i
$742$		0			0
$743$			951.250i			1.28028i	0.768257	+	0.640141i	$0.221124\pi$
$743$			951.250i			1.28028i	−0.768257	+	0.640141i	$0.778876\pi$
$744$		0			0
$745$	224.006			0.300679
$746$		0			0
$747$	690.148	−	471.546i	0.923893	−	0.631253i
$748$		0			0
$749$			1835.70i			2.45087i
$750$		0			0
$751$	859.234			1.14412			0.572060	−	0.820212i	$-0.306145\pi$
$751$	859.234			1.14412			0.572060	+	0.820212i	$0.306145\pi$
$752$		0			0
$753$	319.389	−	605.045i	0.424155	−	0.803512i
$754$		0			0
$755$		−	456.400i		−	0.604503i
$756$		0			0
$757$	73.6156			0.0972465			0.0486233	−	0.998817i	$-0.484517\pi$
$757$	73.6156			0.0972465			0.0486233	+	0.998817i	$0.484517\pi$
$758$		0			0
$759$	−7.96614	−	4.20514i	−0.0104956	−	0.00554036i
$760$		0			0
$761$			40.5958i			0.0533453i	0.999644	+	0.0266727i	$0.00849118\pi$
$761$			40.5958i			0.0533453i	−0.999644	+	0.0266727i	$0.991509\pi$
$762$		0			0
$763$	818.525			1.07277
$764$		0			0
$765$	−316.322	−	462.965i	−0.413493	−	0.605183i
$766$		0			0
$767$		−	82.3473i		−	0.107363i
$768$		0			0
$769$	−202.607			−0.263468			−0.131734	−	0.991285i	$-0.542054\pi$
$769$	−202.607			−0.263468			−0.131734	+	0.991285i	$0.542054\pi$
$770$		0			0
$771$	195.883	−	371.077i	0.254063	−	0.481293i
$772$		0			0
$773$		−	265.265i		−	0.343163i	−0.985170	−	0.171581i	$-0.945112\pi$
$773$		−	265.265i		−	0.343163i	0.985170	−	0.171581i	$-0.0548876\pi$
$774$		0			0
$775$	226.976			0.292872
$776$		0			0
$777$	−1458.27	−	769.786i	−1.87680	−	0.990716i
$778$		0			0
$779$		−	1506.67i		−	1.93411i
$780$		0			0
$781$	−20.8733			−0.0267263
$782$		0			0
$783$	−731.693	+	83.5533i	−0.934474	+	0.106709i
$784$		0			0
$785$			133.207i			0.169691i
$786$		0			0
$787$	−240.099			−0.305082			−0.152541	−	0.988297i	$-0.548746\pi$
$787$	−240.099			−0.305082			−0.152541	+	0.988297i	$0.548746\pi$
$788$		0			0
$789$	−256.410	+	485.739i	−0.324981	+	0.615638i
$790$		0			0
$791$			109.836i			0.138857i
$792$		0			0
$793$	−72.0278			−0.0908295
$794$		0			0
$795$	267.133	+	141.013i	0.336016	+	0.177375i
$796$		0			0
$797$		−	737.607i		−	0.925479i	−0.886494	−	0.462740i	$-0.846867\pi$
$797$		−	737.607i		−	0.925479i	0.886494	−	0.462740i	$-0.153133\pi$
$798$		0			0
$799$	−719.576			−0.900596
$800$		0			0
$801$	59.8809	−	40.9139i	0.0747577	−	0.0510785i
$802$		0			0
$803$			12.8096i			0.0159522i
$804$		0			0
$805$	445.494			0.553408
$806$		0			0
$807$	−114.889	+	217.644i	−0.142365	+	0.269695i
$808$		0			0
$809$			56.1836i			0.0694482i	0.999397	+	0.0347241i	$0.0110552\pi$
$809$			56.1836i			0.0694482i	−0.999397	+	0.0347241i	$0.988945\pi$
$810$		0			0
$811$	433.114			0.534050			0.267025	−	0.963690i	$-0.413959\pi$
$811$	433.114			0.534050			0.267025	+	0.963690i	$0.413959\pi$
$812$		0			0
$813$	123.070	+	64.9658i	0.151378	+	0.0799087i
$814$		0			0
$815$		−	547.703i		−	0.672028i
$816$		0			0
$817$	467.851			0.572645
$818$		0			0
$819$	−670.943	−	981.983i	−0.819223	−	1.19900i
$820$		0			0
$821$		−	1006.77i		−	1.22627i	−0.789977	−	0.613137i	$-0.789908\pi$
$821$		−	1006.77i		−	1.22627i	0.789977	−	0.613137i	$-0.210092\pi$
$822$		0			0
$823$	−280.759			−0.341140			−0.170570	−	0.985346i	$-0.554561\pi$
$823$	−280.759			−0.341140			−0.170570	+	0.985346i	$0.554561\pi$
$824$		0			0
$825$	1.32325	−	2.50675i	0.00160394	−	0.00303848i
$826$		0			0
$827$		−	400.280i		−	0.484015i	−0.970274	−	0.242007i	$-0.922194\pi$
$827$		−	400.280i		−	0.484015i	0.970274	−	0.242007i	$-0.0778058\pi$
$828$		0			0
$829$	−876.708			−1.05755			−0.528774	−	0.848762i	$-0.677348\pi$
$829$	−876.708			−1.05755			−0.528774	+	0.848762i	$0.677348\pi$
$830$		0			0
$831$	−923.498	−	487.493i	−1.11131	−	0.586634i
$832$		0			0
$833$			3015.12i			3.61960i
$834$		0			0
$835$	−32.9210			−0.0394263
$836$		0			0
$837$	139.057	+	1217.76i	0.166138	+	1.45490i
$838$		0			0
$839$			1029.90i			1.22753i	0.789487	+	0.613767i	$0.210347\pi$
$839$			1029.90i			1.22753i	−0.789487	+	0.613767i	$0.789653\pi$
$840$		0			0
$841$	97.0273			0.115371
$842$		0			0
$843$	−346.013	+	655.481i	−0.410454	+	0.777557i
$844$		0			0
$845$		−	129.530i		−	0.153290i
$846$		0			0
$847$	−1516.72			−1.79070
$848$		0			0
$849$	−638.466	−	337.031i	−0.752021	−	0.396974i
$850$		0			0
$851$		−	696.548i		−	0.818505i
$852$		0			0
$853$	−574.267			−0.673232			−0.336616	−	0.941642i	$-0.609282\pi$
$853$	−574.267			−0.673232			−0.336616	+	0.941642i	$0.609282\pi$
$854$		0			0
$855$	−328.730	+	224.606i	−0.384479	+	0.262697i
$856$		0			0
$857$		−	1341.94i		−	1.56586i	−0.622108	−	0.782931i	$-0.713724\pi$
$857$		−	1341.94i		−	1.56586i	0.622108	−	0.782931i	$-0.286276\pi$
$858$		0			0
$859$	−688.107			−0.801056			−0.400528	−	0.916284i	$-0.631173\pi$
$859$	−688.107			−0.801056			−0.400528	+	0.916284i	$0.631173\pi$
$860$		0			0
$861$	1337.34	−	2533.44i	1.55324	−	2.94244i
$862$		0			0
$863$			576.026i			0.667469i	0.942667	+	0.333735i	$0.108309\pi$
$863$			576.026i			0.667469i	−0.942667	+	0.333735i	$0.891691\pi$
$864$		0			0
$865$	29.1752			0.0337285
$866$		0			0
$867$	1292.80	+	682.436i	1.49111	+	0.787123i
$868$		0			0
$869$			3.96671i			0.00456468i
$870$		0			0
$871$	−257.362			−0.295478
$872$		0			0
$873$	−116.058	−	169.860i	−0.132941	−	0.194571i
$874$		0			0
$875$			140.186i			0.160212i
$876$		0			0
$877$	1070.22			1.22032			0.610158	−	0.792280i	$-0.291106\pi$
$877$	1070.22			1.22032			0.610158	+	0.792280i	$0.291106\pi$
$878$		0			0
$879$	−465.973	+	882.731i	−0.530117	+	1.00424i
$880$		0			0
$881$			415.178i			0.471257i	0.971843	+	0.235629i	$0.0757149\pi$
$881$			415.178i			0.471257i	−0.971843	+	0.235629i	$0.924285\pi$
$882$		0			0
$883$	−1284.82			−1.45506			−0.727531	−	0.686075i	$-0.759332\pi$
$883$	−1284.82			−1.45506			−0.727531	+	0.686075i	$0.759332\pi$
$884$		0			0
$885$	−46.3528	−	24.4686i	−0.0523761	−	0.0276481i
$886$		0			0
$887$		−	1548.54i		−	1.74582i	−0.487881	−	0.872910i	$-0.662230\pi$
$887$		−	1548.54i		−	1.74582i	0.487881	−	0.872910i	$-0.337770\pi$
$888$		0			0
$889$	−3018.33			−3.39519
$890$		0			0
$891$	14.2597	+	5.56366i	0.0160042	+	0.00624429i
$892$		0			0
$893$			510.937i			0.572158i
$894$		0			0
$895$	−377.881			−0.422214
$896$		0			0
$897$	234.524	−	444.278i	0.261454	−	0.495293i
$898$		0			0
$899$			1238.19i			1.37730i
$900$		0			0
$901$	−1254.61			−1.39246
$902$		0			0
$903$	786.683	+	415.271i	0.871189	+	0.459880i
$904$		0			0
$905$			117.016i			0.129300i
$906$		0			0
$907$	848.631			0.935647			0.467823	−	0.883822i	$-0.345038\pi$
$907$	848.631			0.935647			0.467823	+	0.883822i	$0.345038\pi$
$908$		0			0
$909$	1171.16	−	800.202i	1.28841	−	0.880310i
$910$		0			0
$911$		−	412.110i		−	0.452371i	−0.974084	−	0.226185i	$-0.927374\pi$
$911$		−	412.110i		−	0.452371i	0.974084	−	0.226185i	$-0.0726255\pi$
$912$		0			0
$913$	−17.5504			−0.0192227
$914$		0			0
$915$	−21.4022	+	40.5441i	−0.0233904	+	0.0443105i
$916$		0			0
$917$		−	919.151i		−	1.00235i
$918$		0			0
$919$	1629.86			1.77351			0.886756	−	0.462237i	$-0.152953\pi$
$919$	1629.86			1.77351			0.886756	+	0.462237i	$0.152953\pi$
$920$		0			0
$921$	60.6301	+	32.0052i	0.0658307	+	0.0347505i
$922$		0			0
$923$		−	1164.12i		−	1.26123i
$924$		0			0
$925$	219.187			0.236958
$926$		0			0
$927$	−483.871	−	708.187i	−0.521976	−	0.763956i
$928$		0			0
$929$		−	286.730i		−	0.308644i	−0.988021	−	0.154322i	$-0.950681\pi$
$929$		−	286.730i		−	0.308644i	0.988021	−	0.154322i	$-0.0493193\pi$
$930$		0			0
$931$	2140.90			2.29957
$932$		0			0
$933$	717.557	−	1359.33i	0.769085	−	1.45694i
$934$		0			0
$935$			11.7731i			0.0125916i
$936$		0			0
$937$	−497.070			−0.530491			−0.265245	−	0.964181i	$-0.585453\pi$
$937$	−497.070			−0.530491			−0.265245	+	0.964181i	$0.585453\pi$
$938$		0			0
$939$	840.554	+	443.708i	0.895158	+	0.472533i
$940$		0			0
$941$			448.438i			0.476555i	0.971197	+	0.238277i	$0.0765827\pi$
$941$			448.438i			0.476555i	−0.971197	+	0.238277i	$0.923417\pi$
$942$		0			0
$943$	1210.11			1.28325
$944$		0			0
$945$	−752.116	+	85.8854i	−0.795890	+	0.0908840i
$946$		0			0
$947$			439.789i			0.464402i	0.972668	+	0.232201i	$0.0745927\pi$
$947$			439.789i			0.464402i	−0.972668	+	0.232201i	$0.925407\pi$
$948$		0			0
$949$	−714.401			−0.752793
$950$		0			0
$951$	−754.517	+	1429.34i	−0.793393	+	1.50299i
$952$		0			0
$953$		−	597.982i		−	0.627473i	−0.949510	−	0.313736i	$-0.898419\pi$
$953$		−	597.982i		−	0.627473i	0.949510	−	0.313736i	$-0.101581\pi$
$954$		0			0
$955$	−159.499			−0.167014
$956$		0			0
$957$	13.6747	+	7.21857i	0.0142892	+	0.00754292i
$958$		0			0
$959$			1767.50i			1.84307i
$960$		0			0
$961$	1099.72			1.14435
$962$		0			0
$963$	1087.94	−	743.338i	1.12974	−	0.771898i
$964$		0			0
$965$			339.269i			0.351574i
$966$		0			0
$967$	850.502			0.879527			0.439763	−	0.898114i	$-0.355062\pi$
$967$	850.502			0.879527			0.439763	+	0.898114i	$0.355062\pi$
$968$		0			0
$969$	771.952	−	1462.37i	0.796648	−	1.50916i
$970$		0			0
$971$		−	1419.61i		−	1.46201i	−0.682373	−	0.731004i	$-0.739052\pi$
$971$		−	1419.61i		−	1.46201i	0.682373	−	0.731004i	$-0.260948\pi$
$972$		0			0
$973$	−2031.19			−2.08755
$974$		0			0
$975$	139.803	+	73.7989i	0.143388	+	0.0756912i
$976$		0			0
$977$			1076.78i			1.10213i	0.834461	+	0.551067i	$0.185779\pi$
$977$			1076.78i			1.10213i	−0.834461	+	0.551067i	$0.814221\pi$
$978$		0			0
$979$	−1.52276			−0.00155543
$980$		0			0
$981$	−331.449	−	485.103i	−0.337868	−	0.494499i
$982$		0			0
$983$			312.061i			0.317458i	0.987322	+	0.158729i	$0.0507396\pi$
$983$			312.061i			0.317458i	−0.987322	+	0.158729i	$0.949260\pi$
$984$		0			0
$985$	800.858			0.813054
$986$		0			0
$987$	−453.515	+	859.131i	−0.459488	+	0.870447i
$988$		0			0
$989$			375.762i			0.379942i
$990$		0			0
$991$	755.545			0.762406			0.381203	−	0.924491i	$-0.375510\pi$
$991$	755.545			0.762406			0.381203	+	0.924491i	$0.375510\pi$
$992$		0			0
$993$	−468.708	−	247.420i	−0.472012	−	0.249164i
$994$		0			0
$995$			703.801i			0.707337i
$996$		0			0
$997$	−169.287			−0.169797			−0.0848984	−	0.996390i	$-0.527057\pi$
$997$	−169.287			−0.169797			−0.0848984	+	0.996390i	$0.527057\pi$
$998$		0			0
$999$	134.285	+	1175.96i	0.134420	+	1.17714i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.3.l.j.641.3		16
3.2	odd	2	inner	960.3.l.j.641.4		16
4.3	odd	2	inner	960.3.l.j.641.14		16
8.3	odd	2		480.3.l.b.161.3	✓	16
8.5	even	2		480.3.l.b.161.14	yes	16
12.11	even	2	inner	960.3.l.j.641.13		16
24.5	odd	2		480.3.l.b.161.13	yes	16
24.11	even	2		480.3.l.b.161.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.3.l.b.161.3	✓	16	8.3	odd	2
480.3.l.b.161.4	yes	16	24.11	even	2
480.3.l.b.161.13	yes	16	24.5	odd	2
480.3.l.b.161.14	yes	16	8.5	even	2
960.3.l.j.641.3		16	1.1	even	1	trivial
960.3.l.j.641.4		16	3.2	odd	2	inner
960.3.l.j.641.13		16	12.11	even	2	inner
960.3.l.j.641.14		16	4.3	odd	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)

\(f(q)\)	\(=\)	\(q+(-2.65305 - 1.40048i) q^{3} +2.23607i q^{5} -12.5386 q^{7} +(5.07731 + 7.43108i) q^{9} +O(q^{10})\)	\(q+(-2.65305 - 1.40048i) q^{3} +2.23607i q^{5} -12.5386 q^{7} +(5.07731 + 7.43108i) q^{9} -0.188971i q^{11} +10.5391 q^{13} +(3.13157 - 5.93239i) q^{15} +27.8619i q^{17} +19.7835 q^{19} +(33.2655 + 17.5601i) q^{21} +15.8894i q^{23} -5.00000 q^{25} +(-3.06327 - 26.8257i) q^{27} -27.2759i q^{29} -45.3952 q^{31} +(-0.264651 + 0.501350i) q^{33} -28.0372i q^{35} -43.8373 q^{37} +(-27.9607 - 14.7598i) q^{39} -76.1582i q^{41} +23.6486 q^{43} +(-16.6164 + 11.3532i) q^{45} +25.8265i q^{47} +108.217 q^{49} +(39.0201 - 73.9190i) q^{51} +45.0295i q^{53} +0.422553 q^{55} +(-52.4864 - 27.7063i) q^{57} -7.81352i q^{59} -6.83435 q^{61} +(-63.6624 - 93.1754i) q^{63} +23.5661i q^{65} -24.4197 q^{67} +(22.2528 - 42.1553i) q^{69} -110.457i q^{71} -67.7858 q^{73} +(13.2652 + 7.00240i) q^{75} +2.36944i q^{77} -20.9911 q^{79} +(-29.4418 + 75.4598i) q^{81} -92.8732i q^{83} -62.3012 q^{85} +(-38.1993 + 72.3641i) q^{87} -8.05818i q^{89} -132.145 q^{91} +(120.435 + 63.5750i) q^{93} +44.2371i q^{95} -22.8581 q^{97} +(1.40426 - 0.959466i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{9}+O(q^{10}) \) 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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