LMFDB - Embedded newform 960.3.l.j.641.10

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			641.10
Root			$2.45733 + 0.803980i$ of defining polynomial
Character	$\chi$	$=$	960.641
Dual form			960.3.l.j.641.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.721589 + 2.91193i) q^{3} +2.23607i q^{5} +1.03633 q^{7} +(-7.95862 + 4.20243i) q^{9} +O(q^{10})$	\(q+(0.721589 + 2.91193i) q^{3} +2.23607i q^{5} +1.03633 q^{7} +(-7.95862 + 4.20243i) q^{9} -19.5478i q^{11} +16.3555 q^{13} +(-6.51126 + 1.61352i) q^{15} -27.1434i q^{17} +24.3792 q^{19} +(0.747803 + 3.01771i) q^{21} +19.9371i q^{23} -5.00000 q^{25} +(-17.9800 - 20.1425i) q^{27} -50.0323i q^{29} +32.9831 q^{31} +(56.9219 - 14.1055i) q^{33} +2.31730i q^{35} +0.942690 q^{37} +(11.8020 + 47.6261i) q^{39} -11.5314i q^{41} +53.0880 q^{43} +(-9.39691 - 17.7960i) q^{45} +68.3474i q^{47} -47.9260 q^{49} +(79.0394 - 19.5864i) q^{51} +48.4140i q^{53} +43.7103 q^{55} +(17.5918 + 70.9905i) q^{57} -14.3858i q^{59} -34.4535 q^{61} +(-8.24774 + 4.35509i) q^{63} +36.5721i q^{65} +9.54367 q^{67} +(-58.0553 + 14.3864i) q^{69} -10.9462i q^{71} +14.7831 q^{73} +(-3.60795 - 14.5596i) q^{75} -20.2580i q^{77} -81.0412 q^{79} +(45.6792 - 66.8910i) q^{81} -58.1989i q^{83} +60.6944 q^{85} +(145.690 - 36.1028i) q^{87} +37.4547i q^{89} +16.9497 q^{91} +(23.8002 + 96.0443i) q^{93} +54.5137i q^{95} -96.7203 q^{97} +(82.1484 + 155.574i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.721589	+	2.91193i	0.240530	+	0.970642i
$3$	0.721589	+	2.91193i	0.240530	+	0.970642i
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$	1.03633			0.148047			0.0740234	−	0.997257i	$-0.476416\pi$
$7$	1.03633			0.148047			0.0740234	+	0.997257i	$0.476416\pi$
$8$		0			0
$9$	−7.95862	+	4.20243i	−0.884291	+	0.466936i
$10$		0			0
$11$		−	19.5478i		−	1.77708i	−0.458802	−	0.888539i	$-0.651721\pi$
$11$		−	19.5478i		−	1.77708i	0.458802	−	0.888539i	$-0.348279\pi$
$12$		0			0
$13$	16.3555			1.25812			0.629059	−	0.777358i	$-0.283440\pi$
$13$	16.3555			1.25812			0.629059	+	0.777358i	$0.283440\pi$
$14$		0			0
$15$	−6.51126	+	1.61352i	−0.434084	+	0.107568i
$16$		0			0
$17$		−	27.1434i		−	1.59667i	−0.602215	−	0.798334i	$-0.705715\pi$
$17$		−	27.1434i		−	1.59667i	0.602215	−	0.798334i	$-0.294285\pi$
$18$		0			0
$19$	24.3792			1.28312			0.641559	−	0.767074i	$-0.278288\pi$
$19$	24.3792			1.28312			0.641559	+	0.767074i	$0.278288\pi$
$20$		0			0
$21$	0.747803	+	3.01771i	0.0356097	+	0.143700i
$22$		0			0
$23$			19.9371i			0.866830i	0.901194	+	0.433415i	$0.142692\pi$
$23$			19.9371i			0.866830i	−0.901194	+	0.433415i	$0.857308\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	−17.9800	−	20.1425i	−0.665926	−	0.746018i
$28$		0			0
$29$		−	50.0323i		−	1.72525i	−0.505843	−	0.862626i	$-0.668818\pi$
$29$		−	50.0323i		−	1.72525i	0.505843	−	0.862626i	$-0.331182\pi$
$30$		0			0
$31$	32.9831			1.06397			0.531985	−	0.846754i	$-0.321446\pi$
$31$	32.9831			1.06397			0.531985	+	0.846754i	$0.321446\pi$
$32$		0			0
$33$	56.9219	−	14.1055i	1.72491	−	0.427440i
$34$		0			0
$35$			2.31730i			0.0662086i
$36$		0			0
$37$	0.942690			0.0254781			0.0127390	−	0.999919i	$-0.495945\pi$
$37$	0.942690			0.0254781			0.0127390	+	0.999919i	$0.495945\pi$
$38$		0			0
$39$	11.8020	+	47.6261i	0.302615	+	1.22118i
$40$		0			0
$41$		−	11.5314i		−	0.281254i	−0.990063	−	0.140627i	$-0.955088\pi$
$41$		−	11.5314i		−	0.281254i	0.990063	−	0.140627i	$-0.0449119\pi$
$42$		0			0
$43$	53.0880			1.23461			0.617303	−	0.786726i	$-0.288225\pi$
$43$	53.0880			1.23461			0.617303	+	0.786726i	$0.288225\pi$
$44$		0			0
$45$	−9.39691	−	17.7960i	−0.208820	−	0.395467i
$46$		0			0
$47$			68.3474i			1.45420i	0.686532	+	0.727100i	$0.259132\pi$
$47$			68.3474i			1.45420i	−0.686532	+	0.727100i	$0.740868\pi$
$48$		0			0
$49$	−47.9260			−0.978082
$50$		0			0
$51$	79.0394	−	19.5864i	1.54979	−	0.384046i
$52$		0			0
$53$			48.4140i			0.913473i	0.889602	+	0.456736i	$0.150982\pi$
$53$			48.4140i			0.913473i	−0.889602	+	0.456736i	$0.849018\pi$
$54$		0			0
$55$	43.7103			0.794733
$56$		0			0
$57$	17.5918	+	70.9905i	0.308628	+	1.24545i
$58$		0			0
$59$		−	14.3858i		−	0.243828i	−0.992541	−	0.121914i	$-0.961097\pi$
$59$		−	14.3858i		−	0.243828i	0.992541	−	0.121914i	$-0.0389032\pi$
$60$		0			0
$61$	−34.4535			−0.564811			−0.282406	−	0.959295i	$-0.591132\pi$
$61$	−34.4535			−0.564811			−0.282406	+	0.959295i	$0.591132\pi$
$62$		0			0
$63$	−8.24774	+	4.35509i	−0.130916	+	0.0691285i
$64$		0			0
$65$			36.5721i			0.562647i
$66$		0			0
$67$	9.54367			0.142443			0.0712214	−	0.997461i	$-0.477310\pi$
$67$	9.54367			0.142443			0.0712214	+	0.997461i	$0.477310\pi$
$68$		0			0
$69$	−58.0553	+	14.3864i	−0.841381	+	0.208498i
$70$		0			0
$71$		−	10.9462i		−	0.154172i	−0.997024	−	0.0770862i	$-0.975438\pi$
$71$		−	10.9462i		−	0.154172i	0.997024	−	0.0770862i	$-0.0245617\pi$
$72$		0			0
$73$	14.7831			0.202508			0.101254	−	0.994861i	$-0.467715\pi$
$73$	14.7831			0.202508			0.101254	+	0.994861i	$0.467715\pi$
$74$		0			0
$75$	−3.60795	−	14.5596i	−0.0481059	−	0.194128i
$76$		0			0
$77$		−	20.2580i		−	0.263091i
$78$		0			0
$79$	−81.0412			−1.02584			−0.512919	−	0.858437i	$-0.671436\pi$
$79$	−81.0412			−1.02584			−0.512919	+	0.858437i	$0.671436\pi$
$80$		0			0
$81$	45.6792	−	66.8910i	0.563941	−	0.825815i
$82$		0			0
$83$		−	58.1989i		−	0.701191i	−0.936527	−	0.350596i	$-0.885979\pi$
$83$		−	58.1989i		−	0.701191i	0.936527	−	0.350596i	$-0.114021\pi$
$84$		0			0
$85$	60.6944			0.714052
$86$		0			0
$87$	145.690	−	36.1028i	1.67460	−	0.414974i
$88$		0			0
$89$			37.4547i			0.420839i	0.977611	+	0.210420i	$0.0674830\pi$
$89$			37.4547i			0.420839i	−0.977611	+	0.210420i	$0.932517\pi$
$90$		0			0
$91$	16.9497			0.186260
$92$		0			0
$93$	23.8002	+	96.0443i	0.255916	+	1.03273i
$94$		0			0
$95$			54.5137i			0.573828i
$96$		0			0
$97$	−96.7203			−0.997117			−0.498558	−	0.866856i	$-0.666137\pi$
$97$	−96.7203			−0.997117			−0.498558	+	0.866856i	$0.666137\pi$
$98$		0			0
$99$	82.1484	+	155.574i	0.829782	+	1.57145i
$100$		0			0
$101$		−	31.3474i		−	0.310371i	−0.987885	−	0.155185i	$-0.950403\pi$
$101$		−	31.3474i		−	0.310371i	0.987885	−	0.155185i	$-0.0495975\pi$
$102$		0			0
$103$	76.4724			0.742451			0.371225	−	0.928543i	$-0.378938\pi$
$103$	76.4724			0.742451			0.371225	+	0.928543i	$0.378938\pi$
$104$		0			0
$105$	−6.74780	+	1.67214i	−0.0642648	+	0.0159251i
$106$		0			0
$107$			95.8686i			0.895968i	0.894041	+	0.447984i	$0.147858\pi$
$107$			95.8686i			0.895968i	−0.894041	+	0.447984i	$0.852142\pi$
$108$		0			0
$109$	176.357			1.61796			0.808979	−	0.587837i	$-0.200021\pi$
$109$	176.357			1.61796			0.808979	+	0.587837i	$0.200021\pi$
$110$		0			0
$111$	0.680235	+	2.74504i	0.00612824	+	0.0247301i
$112$		0			0
$113$		−	60.6386i		−	0.536625i	−0.963332	−	0.268313i	$-0.913534\pi$
$113$		−	60.6386i		−	0.536625i	0.963332	−	0.268313i	$-0.0864660\pi$
$114$		0			0
$115$	−44.5807			−0.387658
$116$		0			0
$117$	−130.167	+	68.7329i	−1.11254	+	0.587461i
$118$		0			0
$119$		−	28.1294i		−	0.236382i
$120$		0			0
$121$	−261.118			−2.15800
$122$		0			0
$123$	33.5786	−	8.32095i	0.272997	−	0.0676500i
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	243.686			1.91879			0.959394	−	0.282070i	$-0.0910212\pi$
$127$	243.686			1.91879			0.959394	+	0.282070i	$0.0910212\pi$
$128$		0			0
$129$	38.3077	+	154.588i	0.296959	+	1.19836i
$130$		0			0
$131$			108.611i			0.829090i	0.910029	+	0.414545i	$0.136059\pi$
$131$			108.611i			0.829090i	−0.910029	+	0.414545i	$0.863941\pi$
$132$		0			0
$133$	25.2649			0.189962
$134$		0			0
$135$	45.0399	−	40.2045i	0.333629	−	0.297811i
$136$		0			0
$137$		−	160.353i		−	1.17046i	−0.810868	−	0.585229i	$-0.801005\pi$
$137$		−	160.353i		−	1.17046i	0.810868	−	0.585229i	$-0.198995\pi$
$138$		0			0
$139$	200.473			1.44225			0.721127	−	0.692803i	$-0.243624\pi$
$139$	200.473			1.44225			0.721127	+	0.692803i	$0.243624\pi$
$140$		0			0
$141$	−199.022	+	49.3187i	−1.41151	+	0.349778i
$142$		0			0
$143$		−	319.715i		−	2.23577i
$144$		0			0
$145$	111.876			0.771556
$146$		0			0
$147$	−34.5829	−	139.557i	−0.235258	−	0.949367i
$148$		0			0
$149$		−	35.4227i		−	0.237736i	−0.992910	−	0.118868i	$-0.962073\pi$
$149$		−	35.4227i		−	0.237736i	0.992910	−	0.118868i	$-0.0379266\pi$
$150$		0			0
$151$	3.79982			0.0251644			0.0125822	−	0.999921i	$-0.495995\pi$
$151$	3.79982			0.0251644			0.0125822	+	0.999921i	$0.495995\pi$
$152$		0			0
$153$	114.068	+	216.024i	0.745543	+	1.41192i
$154$		0			0
$155$			73.7524i			0.475822i
$156$		0			0
$157$	60.7468			0.386922			0.193461	−	0.981108i	$-0.438029\pi$
$157$	60.7468			0.386922			0.193461	+	0.981108i	$0.438029\pi$
$158$		0			0
$159$	−140.978	+	34.9351i	−0.886655	+	0.219717i
$160$		0			0
$161$			20.6614i			0.128331i
$162$		0			0
$163$	205.783			1.26247			0.631237	−	0.775590i	$-0.282547\pi$
$163$	205.783			1.26247			0.631237	+	0.775590i	$0.282547\pi$
$164$		0			0
$165$	31.5409	+	127.281i	0.191157	+	0.771401i
$166$		0			0
$167$			22.4833i			0.134631i	0.997732	+	0.0673153i	$0.0214433\pi$
$167$			22.4833i			0.134631i	−0.997732	+	0.0673153i	$0.978557\pi$
$168$		0			0
$169$	98.5034			0.582860
$170$		0			0
$171$	−194.025	+	102.452i	−1.13465	+	0.599135i
$172$		0			0
$173$		−	116.805i		−	0.675174i	−0.941294	−	0.337587i	$-0.890389\pi$
$173$		−	116.805i		−	0.675174i	0.941294	−	0.337587i	$-0.109611\pi$
$174$		0			0
$175$	−5.18164			−0.0296094
$176$		0			0
$177$	41.8905	−	10.3807i	0.236670	−	0.0586479i
$178$		0			0
$179$			293.214i			1.63807i	0.573747	+	0.819033i	$0.305489\pi$
$179$			293.214i			1.63807i	−0.573747	+	0.819033i	$0.694511\pi$
$180$		0			0
$181$	−2.90698			−0.0160606			−0.00803032	−	0.999968i	$-0.502556\pi$
$181$	−2.90698			−0.0160606			−0.00803032	+	0.999968i	$0.502556\pi$
$182$		0			0
$183$	−24.8613	−	100.326i	−0.135854	−	0.548229i
$184$		0			0
$185$			2.10792i			0.0113942i
$186$		0			0
$187$	−530.594			−2.83740
$188$		0			0
$189$	−18.6332	−	20.8742i	−0.0985883	−	0.110446i
$190$		0			0
$191$		−	108.907i		−	0.570195i	−0.958499	−	0.285097i	$-0.907974\pi$
$191$		−	108.907i		−	0.570195i	0.958499	−	0.285097i	$-0.0920260\pi$
$192$		0			0
$193$	−66.3579			−0.343823			−0.171912	−	0.985112i	$-0.554994\pi$
$193$	−66.3579			−0.343823			−0.171912	+	0.985112i	$0.554994\pi$
$194$		0			0
$195$	−106.495	+	26.3900i	−0.546129	+	0.135333i
$196$		0			0
$197$		−	109.511i		−	0.555894i	−0.960596	−	0.277947i	$-0.910346\pi$
$197$		−	109.511i		−	0.555894i	0.960596	−	0.277947i	$-0.0896540\pi$
$198$		0			0
$199$	−75.9555			−0.381686			−0.190843	−	0.981621i	$-0.561122\pi$
$199$	−75.9555			−0.381686			−0.190843	+	0.981621i	$0.561122\pi$
$200$		0			0
$201$	6.88661	+	27.7905i	0.0342618	+	0.138261i
$202$		0			0
$203$		−	51.8499i		−	0.255418i
$204$		0			0
$205$	25.7850			0.125781
$206$		0			0
$207$	−83.7842	−	158.672i	−0.404754	−	0.766530i
$208$		0			0
$209$		−	476.562i		−	2.28020i
$210$		0			0
$211$	213.103			1.00997			0.504983	−	0.863129i	$-0.331499\pi$
$211$	213.103			1.00997			0.504983	+	0.863129i	$0.331499\pi$
$212$		0			0
$213$	31.8746	−	7.89869i	0.149646	−	0.0370830i
$214$		0			0
$215$			118.708i			0.552132i
$216$		0			0
$217$	34.1813			0.157517
$218$		0			0
$219$	10.6673	+	43.0472i	0.0487091	+	0.196563i
$220$		0			0
$221$		−	443.944i		−	2.00880i
$222$		0			0
$223$	−334.912			−1.50185			−0.750924	−	0.660389i	$-0.770391\pi$
$223$	−334.912			−1.50185			−0.750924	+	0.660389i	$0.770391\pi$
$224$		0			0
$225$	39.7931	−	21.0121i	0.176858	−	0.0933873i
$226$		0			0
$227$		−	206.569i		−	0.909995i	−0.890493	−	0.454998i	$-0.849640\pi$
$227$		−	206.569i		−	0.909995i	0.890493	−	0.454998i	$-0.150360\pi$
$228$		0			0
$229$	−249.699			−1.09039			−0.545195	−	0.838309i	$-0.683544\pi$
$229$	−249.699			−1.09039			−0.545195	+	0.838309i	$0.683544\pi$
$230$		0			0
$231$	58.9897	−	14.6179i	0.255367	−	0.0632811i
$232$		0			0
$233$		−	27.2979i		−	0.117159i	−0.998283	−	0.0585793i	$-0.981343\pi$
$233$		−	27.2979i		−	0.117159i	0.998283	−	0.0585793i	$-0.0186570\pi$
$234$		0			0
$235$	−152.829			−0.650338
$236$		0			0
$237$	−58.4784	−	235.986i	−0.246744	−	0.995721i
$238$		0			0
$239$			193.283i			0.808717i	0.914601	+	0.404358i	$0.132505\pi$
$239$			193.283i			0.808717i	−0.914601	+	0.404358i	$0.867495\pi$
$240$		0			0
$241$	156.285			0.648487			0.324244	−	0.945974i	$-0.394890\pi$
$241$	156.285			0.648487			0.324244	+	0.945974i	$0.394890\pi$
$242$		0			0
$243$	227.743	+	84.7466i	0.937215	+	0.348751i
$244$		0			0
$245$		−	107.166i		−	0.437412i
$246$		0			0
$247$	398.736			1.61431
$248$		0			0
$249$	169.471	−	41.9957i	0.680606	−	0.168657i
$250$		0			0
$251$		−	436.697i		−	1.73983i	−0.493203	−	0.869914i	$-0.664174\pi$
$251$		−	436.697i		−	1.73983i	0.493203	−	0.869914i	$-0.335826\pi$
$252$		0			0
$253$	389.727			1.54042
$254$		0			0
$255$	43.7964	+	176.738i	0.171751	+	0.693088i
$256$		0			0
$257$			62.7516i			0.244170i	0.992520	+	0.122085i	$0.0389580\pi$
$257$			62.7516i			0.244170i	−0.992520	+	0.122085i	$0.961042\pi$
$258$		0			0
$259$	0.976935			0.00377195
$260$		0			0
$261$	210.257	+	398.188i	0.805583	+	1.52562i
$262$		0			0
$263$			80.9654i			0.307853i	0.988082	+	0.153927i	$0.0491920\pi$
$263$			80.9654i			0.307853i	−0.988082	+	0.153927i	$0.950808\pi$
$264$		0			0
$265$	−108.257			−0.408517
$266$		0			0
$267$	−109.065	+	27.0269i	−0.408484	+	0.101224i
$268$		0			0
$269$			442.554i			1.64518i	0.568634	+	0.822591i	$0.307472\pi$
$269$			442.554i			1.64518i	−0.568634	+	0.822591i	$0.692528\pi$
$270$		0			0
$271$	−354.542			−1.30827			−0.654137	−	0.756376i	$-0.726968\pi$
$271$	−354.542			−1.30827			−0.654137	+	0.756376i	$0.726968\pi$
$272$		0			0
$273$	12.2307	+	49.3562i	0.0448012	+	0.180792i
$274$		0			0
$275$			97.7392i			0.355415i
$276$		0			0
$277$	214.327			0.773743			0.386871	−	0.922134i	$-0.373556\pi$
$277$	214.327			0.773743			0.386871	+	0.922134i	$0.373556\pi$
$278$		0			0
$279$	−262.500	+	138.609i	−0.940859	+	0.496806i
$280$		0			0
$281$		−	143.922i		−	0.512180i	−0.966653	−	0.256090i	$-0.917566\pi$
$281$		−	143.922i		−	0.512180i	0.966653	−	0.256090i	$-0.0824343\pi$
$282$		0			0
$283$	−431.697			−1.52543			−0.762715	−	0.646734i	$-0.776134\pi$
$283$	−431.697			−1.52543			−0.762715	+	0.646734i	$0.776134\pi$
$284$		0			0
$285$	−158.740	+	39.3365i	−0.556981	+	0.138023i
$286$		0			0
$287$		−	11.9503i		−	0.0416388i
$288$		0			0
$289$	−447.762			−1.54935
$290$		0			0
$291$	−69.7924	−	281.642i	−0.239836	−	0.967843i
$292$		0			0
$293$		−	99.9704i		−	0.341196i	−0.985341	−	0.170598i	$-0.945430\pi$
$293$		−	99.9704i		−	0.341196i	0.985341	−	0.170598i	$-0.0545699\pi$
$294$		0			0
$295$	32.1677			0.109043
$296$		0			0
$297$	−393.742	+	351.470i	−1.32573	+	1.18340i
$298$		0			0
$299$			326.082i			1.09057i
$300$		0			0
$301$	55.0166			0.182779
$302$		0			0
$303$	91.2814	−	22.6200i	0.301259	−	0.0746534i
$304$		0			0
$305$		−	77.0403i		−	0.252591i
$306$		0			0
$307$	−22.3476			−0.0727935			−0.0363968	−	0.999337i	$-0.511588\pi$
$307$	−22.3476			−0.0727935			−0.0363968	+	0.999337i	$0.511588\pi$
$308$		0			0
$309$	55.1817	+	222.682i	0.178581	+	0.720654i
$310$		0			0
$311$		−	253.760i		−	0.815949i	−0.912993	−	0.407974i	$-0.866235\pi$
$311$		−	253.760i		−	0.815949i	0.912993	−	0.407974i	$-0.133765\pi$
$312$		0			0
$313$	212.404			0.678608			0.339304	−	0.940677i	$-0.389808\pi$
$313$	212.404			0.678608			0.339304	+	0.940677i	$0.389808\pi$
$314$		0			0
$315$	−9.73828	−	18.4425i	−0.0309152	−	0.0585476i
$316$		0			0
$317$			363.698i			1.14731i	0.819096	+	0.573656i	$0.194476\pi$
$317$			363.698i			1.14731i	−0.819096	+	0.573656i	$0.805524\pi$
$318$		0			0
$319$	−978.024			−3.06591
$320$		0			0
$321$	−279.162	+	69.1778i	−0.869664	+	0.215507i
$322$		0			0
$323$		−	661.735i		−	2.04871i
$324$		0			0
$325$	−81.7777			−0.251624
$326$		0			0
$327$	127.258	+	513.540i	0.389167	+	1.57046i
$328$		0			0
$329$			70.8303i			0.215290i
$330$		0			0
$331$	−117.936			−0.356302			−0.178151	−	0.984003i	$-0.557012\pi$
$331$	−117.936			−0.356302			−0.178151	+	0.984003i	$0.557012\pi$
$332$		0			0
$333$	−7.50251	+	3.96158i	−0.0225300	+	0.0118967i
$334$		0			0
$335$			21.3403i			0.0637024i
$336$		0			0
$337$	359.570			1.06697			0.533487	−	0.845809i	$-0.320881\pi$
$337$	359.570			1.06697			0.533487	+	0.845809i	$0.320881\pi$
$338$		0			0
$339$	176.575	−	43.7562i	0.520871	−	0.129074i
$340$		0			0
$341$		−	644.748i		−	1.89076i
$342$		0			0
$343$	−100.447			−0.292849
$344$		0			0
$345$	−32.1689	−	129.816i	−0.0932433	−	0.376277i
$346$		0			0
$347$		−	438.402i		−	1.26341i	−0.775210	−	0.631703i	$-0.782356\pi$
$347$		−	438.402i		−	1.26341i	0.775210	−	0.631703i	$-0.217644\pi$
$348$		0			0
$349$	−631.053			−1.80817			−0.904087	−	0.427349i	$-0.859448\pi$
$349$	−631.053			−1.80817			−0.904087	+	0.427349i	$0.859448\pi$
$350$		0			0
$351$	−294.073	−	329.441i	−0.837814	−	0.938578i
$352$		0			0
$353$			377.417i			1.06917i	0.845114	+	0.534585i	$0.179532\pi$
$353$			377.417i			1.06917i	−0.845114	+	0.534585i	$0.820468\pi$
$354$		0			0
$355$	24.4765			0.0689480
$356$		0			0
$357$	81.9108	−	20.2979i	0.229442	−	0.0568568i
$358$		0			0
$359$			671.948i			1.87172i	0.352371	+	0.935860i	$0.385376\pi$
$359$			671.948i			1.87172i	−0.352371	+	0.935860i	$0.614624\pi$
$360$		0			0
$361$	233.348			0.646392
$362$		0			0
$363$	−188.420	−	760.357i	−0.519064	−	2.09465i
$364$		0			0
$365$			33.0559i			0.0905642i
$366$		0			0
$367$	−409.971			−1.11709			−0.558543	−	0.829475i	$-0.688639\pi$
$367$	−409.971			−1.11709			−0.558543	+	0.829475i	$0.688639\pi$
$368$		0			0
$369$	48.4600	+	91.7742i	0.131328	+	0.248710i
$370$		0			0
$371$			50.1728i			0.135237i
$372$		0			0
$373$	−266.527			−0.714551			−0.357275	−	0.933999i	$-0.616294\pi$
$373$	−266.527			−0.714551			−0.357275	+	0.933999i	$0.616294\pi$
$374$		0			0
$375$	32.5563	−	8.06761i	0.0868168	−	0.0215136i
$376$		0			0
$377$		−	818.305i		−	2.17057i
$378$		0			0
$379$	−455.595			−1.20210			−0.601049	−	0.799212i	$-0.705250\pi$
$379$	−455.595			−1.20210			−0.601049	+	0.799212i	$0.705250\pi$
$380$		0			0
$381$	175.841	+	709.595i	0.461525	+	1.86246i
$382$		0			0
$383$		−	457.467i		−	1.19443i	−0.802081	−	0.597215i	$-0.796274\pi$
$383$		−	457.467i		−	1.19443i	0.802081	−	0.597215i	$-0.203726\pi$
$384$		0			0
$385$	45.2982			0.117658
$386$		0			0
$387$	−422.507	+	223.099i	−1.09175	+	0.576482i
$388$		0			0
$389$		−	709.643i		−	1.82427i	−0.409886	−	0.912137i	$-0.634431\pi$
$389$		−	709.643i		−	1.82427i	0.409886	−	0.912137i	$-0.365569\pi$
$390$		0			0
$391$	541.160			1.38404
$392$		0			0
$393$	−316.267	+	78.3724i	−0.804749	+	0.199421i
$394$		0			0
$395$		−	181.214i		−	0.458769i
$396$		0			0
$397$	−489.814			−1.23379			−0.616894	−	0.787046i	$-0.711609\pi$
$397$	−489.814			−1.23379			−0.616894	+	0.787046i	$0.711609\pi$
$398$		0			0
$399$	18.2309	+	73.5695i	0.0456914	+	0.184385i
$400$		0			0
$401$			614.504i			1.53243i	0.642585	+	0.766214i	$0.277862\pi$
$401$			614.504i			1.53243i	−0.642585	+	0.766214i	$0.722138\pi$
$402$		0			0
$403$	539.456			1.33860
$404$		0			0
$405$	149.573	+	102.142i	0.369316	+	0.252202i
$406$		0			0
$407$		−	18.4276i		−	0.0452765i
$408$		0			0
$409$	155.522			0.380248			0.190124	−	0.981760i	$-0.439111\pi$
$409$	155.522			0.380248			0.190124	+	0.981760i	$0.439111\pi$
$410$		0			0
$411$	466.935	−	115.709i	1.13610	−	0.281530i
$412$		0			0
$413$		−	14.9085i		−	0.0360980i
$414$		0			0
$415$	130.137			0.313582
$416$		0			0
$417$	144.659	+	583.763i	0.346905	+	1.39991i
$418$		0			0
$419$			346.568i			0.827132i	0.910474	+	0.413566i	$0.135717\pi$
$419$			346.568i			0.827132i	−0.910474	+	0.413566i	$0.864283\pi$
$420$		0			0
$421$	−257.393			−0.611384			−0.305692	−	0.952131i	$-0.598888\pi$
$421$	−257.393			−0.611384			−0.305692	+	0.952131i	$0.598888\pi$
$422$		0			0
$423$	−287.225	−	543.951i	−0.679019	−	1.28594i
$424$		0			0
$425$			135.717i			0.319334i
$426$		0			0
$427$	−35.7051			−0.0836185
$428$		0			0
$429$	930.988	−	230.703i	2.17013	−	0.537770i
$430$		0			0
$431$			490.699i			1.13851i	0.822160	+	0.569256i	$0.192769\pi$
$431$			490.699i			1.13851i	−0.822160	+	0.569256i	$0.807231\pi$
$432$		0			0
$433$	−321.293			−0.742017			−0.371008	−	0.928629i	$-0.620988\pi$
$433$	−321.293			−0.742017			−0.371008	+	0.928629i	$0.620988\pi$
$434$		0			0
$435$	80.7282	+	325.773i	0.185582	+	0.748904i
$436$		0			0
$437$			486.051i			1.11225i
$438$		0			0
$439$	365.718			0.833070			0.416535	−	0.909120i	$-0.363244\pi$
$439$	365.718			0.833070			0.416535	+	0.909120i	$0.363244\pi$
$440$		0			0
$441$	381.425	−	201.406i	0.864909	−	0.456702i
$442$		0			0
$443$			336.709i			0.760066i	0.924973	+	0.380033i	$0.124087\pi$
$443$			336.709i			0.760066i	−0.924973	+	0.380033i	$0.875913\pi$
$444$		0			0
$445$	−83.7512			−0.188205
$446$		0			0
$447$	103.148	−	25.5607i	0.230757	−	0.0571827i
$448$		0			0
$449$			99.0492i			0.220600i	0.993898	+	0.110300i	$0.0351811\pi$
$449$			99.0492i			0.220600i	−0.993898	+	0.110300i	$0.964819\pi$
$450$		0			0
$451$	−225.414			−0.499810
$452$		0			0
$453$	2.74191	+	11.0648i	0.00605278	+	0.0244256i
$454$		0			0
$455$			37.9007i			0.0832982i
$456$		0			0
$457$	243.254			0.532285			0.266143	−	0.963934i	$-0.414251\pi$
$457$	243.254			0.532285			0.266143	+	0.963934i	$0.414251\pi$
$458$		0			0
$459$	−546.734	+	488.038i	−1.19114	+	1.06326i
$460$		0			0
$461$			422.566i			0.916629i	0.888790	+	0.458315i	$0.151547\pi$
$461$			422.566i			0.916629i	−0.888790	+	0.458315i	$0.848453\pi$
$462$		0			0
$463$	342.059			0.738789			0.369395	−	0.929273i	$-0.379565\pi$
$463$	342.059			0.738789			0.369395	+	0.929273i	$0.379565\pi$
$464$		0			0
$465$	−214.761	+	53.2189i	−0.461853	+	0.114449i
$466$		0			0
$467$			141.394i			0.302770i	0.988475	+	0.151385i	$0.0483733\pi$
$467$			141.394i			0.302770i	−0.988475	+	0.151385i	$0.951627\pi$
$468$		0			0
$469$	9.89037			0.0210882
$470$		0			0
$471$	43.8342	+	176.890i	0.0930663	+	0.375563i
$472$		0			0
$473$		−	1037.76i		−	2.19399i
$474$		0			0
$475$	−121.896			−0.256624
$476$		0			0
$477$	−203.457	−	385.309i	−0.426534	−	0.807775i
$478$		0			0
$479$			571.733i			1.19360i	0.802391	+	0.596799i	$0.203561\pi$
$479$			571.733i			1.19360i	−0.802391	+	0.596799i	$0.796439\pi$
$480$		0			0
$481$	15.4182			0.0320544
$482$		0			0
$483$	−60.1643	+	14.9090i	−0.124564	+	0.0308675i
$484$		0			0
$485$		−	216.273i		−	0.445924i
$486$		0			0
$487$	−370.957			−0.761718			−0.380859	−	0.924633i	$-0.624372\pi$
$487$	−370.957			−0.761718			−0.380859	+	0.924633i	$0.624372\pi$
$488$		0			0
$489$	148.491	+	599.226i	0.303663	+	1.22541i
$490$		0			0
$491$		−	365.317i		−	0.744027i	−0.928227	−	0.372014i	$-0.878667\pi$
$491$		−	365.317i		−	0.744027i	0.928227	−	0.372014i	$-0.121333\pi$
$492$		0			0
$493$	−1358.04			−2.75465
$494$		0			0
$495$	−347.874	+	183.689i	−0.702775	+	0.371090i
$496$		0			0
$497$		−	11.3439i		−	0.0228247i
$498$		0			0
$499$	491.257			0.984482			0.492241	−	0.870459i	$-0.336178\pi$
$499$	491.257			0.984482			0.492241	+	0.870459i	$0.336178\pi$
$500$		0			0
$501$	−65.4697	+	16.2237i	−0.130678	+	0.0323826i
$502$		0			0
$503$		−	492.283i		−	0.978694i	−0.872089	−	0.489347i	$-0.837235\pi$
$503$		−	492.283i		−	0.978694i	0.872089	−	0.489347i	$-0.162765\pi$
$504$		0			0
$505$	70.0950			0.138802
$506$		0			0
$507$	71.0790	+	286.835i	0.140195	+	0.565749i
$508$		0			0
$509$			510.427i			1.00280i	0.865215	+	0.501402i	$0.167182\pi$
$509$			510.427i			1.00280i	−0.865215	+	0.501402i	$0.832818\pi$
$510$		0			0
$511$	15.3201			0.0299806
$512$		0			0
$513$	−438.339	−	491.058i	−0.854462	−	0.957229i
$514$		0			0
$515$			170.998i			0.332034i
$516$		0			0
$517$	1336.04			2.58423
$518$		0			0
$519$	340.128	−	84.2853i	0.655352	−	0.162399i
$520$		0			0
$521$			509.492i			0.977912i	0.872309	+	0.488956i	$0.162622\pi$
$521$			509.492i			0.977912i	−0.872309	+	0.488956i	$0.837378\pi$
$522$		0			0
$523$	546.521			1.04497			0.522487	−	0.852647i	$-0.325004\pi$
$523$	546.521			1.04497			0.522487	+	0.852647i	$0.325004\pi$
$524$		0			0
$525$	−3.73901	−	15.0885i	−0.00712193	−	0.0287401i
$526$		0			0
$527$		−	895.272i		−	1.69881i
$528$		0			0
$529$	131.513			0.248606
$530$		0			0
$531$	60.4555	+	114.491i	0.113852	+	0.215615i
$532$		0			0
$533$		−	188.603i		−	0.353851i
$534$		0			0
$535$	−214.369			−0.400689
$536$		0			0
$537$	−853.816	+	211.580i	−1.58997	+	0.394003i
$538$		0			0
$539$			936.851i			1.73813i
$540$		0			0
$541$	707.439			1.30765			0.653825	−	0.756646i	$-0.273163\pi$
$541$	707.439			1.30765			0.653825	+	0.756646i	$0.273163\pi$
$542$		0			0
$543$	−2.09764	−	8.46490i	−0.00386306	−	0.0155891i
$544$		0			0
$545$			394.347i			0.723573i
$546$		0			0
$547$	502.450			0.918556			0.459278	−	0.888293i	$-0.348108\pi$
$547$	502.450			0.918556			0.459278	+	0.888293i	$0.348108\pi$
$548$		0			0
$549$	274.202	−	144.788i	0.499457	−	0.263731i
$550$		0			0
$551$		−	1219.75i		−	2.21370i
$552$		0			0
$553$	−83.9852			−0.151872
$554$		0			0
$555$	−6.13810	+	1.52105i	−0.0110596	+	0.00274063i
$556$		0			0
$557$		−	348.442i		−	0.625569i	−0.949824	−	0.312784i	$-0.898738\pi$
$557$		−	348.442i		−	0.625569i	0.949824	−	0.312784i	$-0.101262\pi$
$558$		0			0
$559$	868.283			1.55328
$560$		0			0
$561$	−382.871	−	1545.05i	−0.682480	−	2.75410i
$562$		0			0
$563$		−	183.509i		−	0.325949i	−0.986630	−	0.162974i	$-0.947891\pi$
$563$		−	183.509i		−	0.325949i	0.986630	−	0.162974i	$-0.0521088\pi$
$564$		0			0
$565$	135.592			0.239986
$566$		0			0
$567$	47.3386	−	69.3210i	0.0834897	−	0.122259i
$568$		0			0
$569$		−	495.857i		−	0.871454i	−0.900079	−	0.435727i	$-0.856491\pi$
$569$		−	495.857i		−	0.871454i	0.900079	−	0.435727i	$-0.143509\pi$
$570$		0			0
$571$	−431.674			−0.755996			−0.377998	−	0.925806i	$-0.623387\pi$
$571$	−431.674			−0.755996			−0.377998	+	0.925806i	$0.623387\pi$
$572$		0			0
$573$	317.130	−	78.5863i	0.553455	−	0.137149i
$574$		0			0
$575$		−	99.6854i		−	0.173366i
$576$		0			0
$577$	955.465			1.65592			0.827959	−	0.560788i	$-0.189502\pi$
$577$	955.465			1.65592			0.827959	+	0.560788i	$0.189502\pi$
$578$		0			0
$579$	−47.8832	−	193.229i	−0.0826997	−	0.333729i
$580$		0			0
$581$		−	60.3131i		−	0.103809i
$582$		0			0
$583$	946.390			1.62331
$584$		0			0
$585$	−153.692	−	291.063i	−0.262721	−	0.497544i
$586$		0			0
$587$			141.610i			0.241243i	0.992699	+	0.120622i	$0.0384887\pi$
$587$			141.610i			0.241243i	−0.992699	+	0.120622i	$0.961511\pi$
$588$		0			0
$589$	804.103			1.36520
$590$		0			0
$591$	318.888	−	79.0221i	0.539574	−	0.133709i
$592$		0			0
$593$		−	745.182i		−	1.25663i	−0.777959	−	0.628315i	$-0.783745\pi$
$593$		−	745.182i		−	1.25663i	0.777959	−	0.628315i	$-0.216255\pi$
$594$		0			0
$595$	62.8993			0.105713
$596$		0			0
$597$	−54.8086	−	221.177i	−0.0918068	−	0.370480i
$598$		0			0
$599$			220.391i			0.367932i	0.982933	+	0.183966i	$0.0588936\pi$
$599$			220.391i			0.367932i	−0.982933	+	0.183966i	$0.941106\pi$
$600$		0			0
$601$	741.831			1.23433			0.617164	−	0.786835i	$-0.288282\pi$
$601$	741.831			1.23433			0.617164	+	0.786835i	$0.288282\pi$
$602$		0			0
$603$	−75.9545	+	40.1066i	−0.125961	+	0.0665118i
$604$		0			0
$605$		−	583.879i		−	0.965089i
$606$		0			0
$607$	140.449			0.231383			0.115691	−	0.993285i	$-0.463092\pi$
$607$	140.449			0.231383			0.115691	+	0.993285i	$0.463092\pi$
$608$		0			0
$609$	150.983	−	37.4143i	0.247919	−	0.0614356i
$610$		0			0
$611$			1117.86i			1.82955i
$612$		0			0
$613$	−758.713			−1.23770			−0.618852	−	0.785507i	$-0.712402\pi$
$613$	−758.713			−1.23770			−0.618852	+	0.785507i	$0.712402\pi$
$614$		0			0
$615$	18.6062	+	75.0841i	0.0302540	+	0.122088i
$616$		0			0
$617$		−	149.024i		−	0.241530i	−0.992681	−	0.120765i	$-0.961465\pi$
$617$		−	149.024i		−	0.241530i	0.992681	−	0.120765i	$-0.0385348\pi$
$618$		0			0
$619$	−272.205			−0.439749			−0.219875	−	0.975528i	$-0.570565\pi$
$619$	−272.205			−0.439749			−0.219875	+	0.975528i	$0.570565\pi$
$620$		0			0
$621$	401.582	−	358.469i	0.646670	−	0.577245i
$622$		0			0
$623$			38.8153i			0.0623039i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	1387.71	−	343.882i	2.21326	−	0.548456i
$628$		0			0
$629$		−	25.5878i		−	0.0406801i
$630$		0			0
$631$	−38.8333			−0.0615425			−0.0307712	−	0.999526i	$-0.509796\pi$
$631$	−38.8333			−0.0615425			−0.0307712	+	0.999526i	$0.509796\pi$
$632$		0			0
$633$	153.773	+	620.539i	0.242927	+	0.980315i
$634$		0			0
$635$			544.898i			0.858108i
$636$		0			0
$637$	−783.856			−1.23054
$638$		0			0
$639$	46.0008	+	87.1169i	0.0719887	+	0.136333i
$640$		0			0
$641$			736.722i			1.14933i	0.818388	+	0.574666i	$0.194868\pi$
$641$			736.722i			1.14933i	−0.818388	+	0.574666i	$0.805132\pi$
$642$		0			0
$643$	−940.145			−1.46212			−0.731062	−	0.682311i	$-0.760975\pi$
$643$	−940.145			−1.46212			−0.731062	+	0.682311i	$0.760975\pi$
$644$		0			0
$645$	−345.670	+	85.6587i	−0.535923	+	0.132804i
$646$		0			0
$647$			246.934i			0.381660i	0.981623	+	0.190830i	$0.0611179\pi$
$647$			246.934i			0.381660i	−0.981623	+	0.190830i	$0.938882\pi$
$648$		0			0
$649$	−281.212			−0.433301
$650$		0			0
$651$	24.6648	+	99.5333i	0.0378876	+	0.152893i
$652$		0			0
$653$			686.694i			1.05160i	0.850609	+	0.525799i	$0.176234\pi$
$653$			686.694i			1.05160i	−0.850609	+	0.525799i	$0.823766\pi$
$654$		0			0
$655$	−242.861			−0.370780
$656$		0			0
$657$	−117.653	+	62.1248i	−0.179076	+	0.0945583i
$658$		0			0
$659$			19.6644i			0.0298397i	0.999889	+	0.0149199i	$0.00474932\pi$
$659$			19.6644i			0.0298397i	−0.999889	+	0.0149199i	$0.995251\pi$
$660$		0			0
$661$	−788.369			−1.19269			−0.596346	−	0.802728i	$-0.703381\pi$
$661$	−788.369			−1.19269			−0.596346	+	0.802728i	$0.703381\pi$
$662$		0			0
$663$	1292.73	−	320.345i	1.94982	−	0.483175i
$664$		0			0
$665$			56.4940i			0.0849534i
$666$		0			0
$667$	997.498			1.49550
$668$		0			0
$669$	−241.669	−	975.239i	−0.361239	−	1.45776i
$670$		0			0
$671$			673.492i			1.00371i
$672$		0			0
$673$	−192.921			−0.286659			−0.143329	−	0.989675i	$-0.545781\pi$
$673$	−192.921			−0.286659			−0.143329	+	0.989675i	$0.545781\pi$
$674$		0			0
$675$	89.9000	+	100.712i	0.133185	+	0.149204i
$676$		0			0
$677$			852.933i			1.25987i	0.776647	+	0.629936i	$0.216919\pi$
$677$			852.933i			1.25987i	−0.776647	+	0.629936i	$0.783081\pi$
$678$		0			0
$679$	−100.234			−0.147620
$680$		0			0
$681$	601.513	−	149.058i	0.883279	−	0.218881i
$682$		0			0
$683$		−	778.286i		−	1.13951i	−0.821814	−	0.569755i	$-0.807038\pi$
$683$		−	778.286i		−	1.13951i	0.821814	−	0.569755i	$-0.192962\pi$
$684$		0			0
$685$	358.560			0.523445
$686$		0			0
$687$	−180.180	−	727.106i	−0.262271	−	1.05838i
$688$		0			0
$689$			791.837i			1.14926i
$690$		0			0
$691$	790.376			1.14382			0.571908	−	0.820318i	$-0.306204\pi$
$691$	790.376			1.14382			0.571908	+	0.820318i	$0.306204\pi$
$692$		0			0
$693$	85.1327	+	161.226i	0.122847	+	0.232649i
$694$		0			0
$695$			448.272i			0.644996i
$696$		0			0
$697$	−313.001			−0.449070
$698$		0			0
$699$	79.4896	−	19.6979i	0.113719	−	0.0281801i
$700$		0			0
$701$		−	254.995i		−	0.363759i	−0.983321	−	0.181879i	$-0.941782\pi$
$701$		−	254.995i		−	0.363759i	0.983321	−	0.181879i	$-0.0582181\pi$
$702$		0			0
$703$	22.9821			0.0326914
$704$		0			0
$705$	−110.280	−	445.028i	−0.156426	−	0.631245i
$706$		0			0
$707$		−	32.4862i		−	0.0459494i
$708$		0			0
$709$	−1099.40			−1.55063			−0.775315	−	0.631575i	$-0.782409\pi$
$709$	−1099.40			−1.55063			−0.775315	+	0.631575i	$0.782409\pi$
$710$		0			0
$711$	644.976	−	340.570i	0.907139	−	0.479001i
$712$		0			0
$713$			657.586i			0.922281i
$714$		0			0
$715$	714.906			0.999868
$716$		0			0
$717$	−562.827	+	139.471i	−0.784974	+	0.194520i
$718$		0			0
$719$		−	413.345i		−	0.574888i	−0.957797	−	0.287444i	$-0.907194\pi$
$719$		−	413.345i		−	0.574888i	0.957797	−	0.287444i	$-0.0928056\pi$
$720$		0			0
$721$	79.2505			0.109917
$722$		0			0
$723$	112.774	+	455.092i	0.155980	+	0.629449i
$724$		0			0
$725$			250.161i			0.345050i
$726$		0			0
$727$	−103.627			−0.142541			−0.0712706	−	0.997457i	$-0.522705\pi$
$727$	−103.627			−0.142541			−0.0712706	+	0.997457i	$0.522705\pi$
$728$		0			0
$729$	−82.4386	+	724.324i	−0.113085	+	0.993585i
$730$		0			0
$731$		−	1440.99i		−	1.97126i
$732$		0			0
$733$	1407.40			1.92006			0.960029	−	0.279901i	$-0.0903015\pi$
$733$	1407.40			1.92006			0.960029	+	0.279901i	$0.0903015\pi$
$734$		0			0
$735$	312.059	−	77.3297i	0.424570	−	0.105210i
$736$		0			0
$737$		−	186.558i		−	0.253132i
$738$		0			0
$739$	−1328.22			−1.79732			−0.898659	−	0.438649i	$-0.855457\pi$
$739$	−1328.22			−1.79732			−0.898659	+	0.438649i	$0.855457\pi$
$740$		0			0
$741$	287.723	+	1161.09i	0.388290	+	1.56692i
$742$		0			0
$743$			770.118i			1.03650i	0.855230	+	0.518249i	$0.173416\pi$
$743$			770.118i			1.03650i	−0.855230	+	0.518249i	$0.826584\pi$
$744$		0			0
$745$	79.2076			0.106319
$746$		0			0
$747$	244.577	+	463.183i	0.327412	+	0.620057i
$748$		0			0
$749$			99.3513i			0.132645i
$750$		0			0
$751$	−252.564			−0.336304			−0.168152	−	0.985761i	$-0.553780\pi$
$751$	−252.564			−0.336304			−0.168152	+	0.985761i	$0.553780\pi$
$752$		0			0
$753$	1271.63	−	315.116i	1.68875	−	0.418480i
$754$		0			0
$755$			8.49666i			0.0112539i
$756$		0			0
$757$	977.325			1.29105			0.645525	−	0.763739i	$-0.276639\pi$
$757$	977.325			1.29105			0.645525	+	0.763739i	$0.276639\pi$
$758$		0			0
$759$	281.223	+	1134.86i	0.370518	+	1.49520i
$760$		0			0
$761$			849.704i			1.11656i	0.829652	+	0.558281i	$0.188539\pi$
$761$			849.704i			1.11656i	−0.829652	+	0.558281i	$0.811461\pi$
$762$		0			0
$763$	182.764			0.239534
$764$		0			0
$765$	−483.044	+	255.064i	−0.631429	+	0.333417i
$766$		0			0
$767$		−	235.288i		−	0.306764i
$768$		0			0
$769$	−1027.29			−1.33587			−0.667936	−	0.744219i	$-0.732822\pi$
$769$	−1027.29			−1.33587			−0.667936	+	0.744219i	$0.732822\pi$
$770$		0			0
$771$	−182.728	+	45.2809i	−0.237001	+	0.0587301i
$772$		0			0
$773$		−	630.462i		−	0.815604i	−0.913070	−	0.407802i	$-0.866295\pi$
$773$		−	630.462i		−	0.815604i	0.913070	−	0.407802i	$-0.133705\pi$
$774$		0			0
$775$	−164.915			−0.212794
$776$		0			0
$777$	0.704946	+	2.84476i	0.000907266	+	0.00366121i
$778$		0			0
$779$		−	281.127i		−	0.360882i
$780$		0			0
$781$	−213.975			−0.273976
$782$		0			0
$783$	−1007.77	+	899.581i	−1.28707	+	1.14889i
$784$		0			0
$785$			135.834i			0.173037i
$786$		0			0
$787$	1153.89			1.46618			0.733092	−	0.680130i	$-0.238077\pi$
$787$	1153.89			1.46618			0.733092	+	0.680130i	$0.238077\pi$
$788$		0			0
$789$	−235.765	+	58.4238i	−0.298815	+	0.0740479i
$790$		0			0
$791$		−	62.8415i		−	0.0794456i
$792$		0			0
$793$	−563.505			−0.710599
$794$		0			0
$795$	−78.1171	−	315.237i	−0.0982606	−	0.396524i
$796$		0			0
$797$			719.315i			0.902528i	0.892391	+	0.451264i	$0.149027\pi$
$797$			719.315i			0.902528i	−0.892391	+	0.451264i	$0.850973\pi$
$798$		0			0
$799$	1855.18			2.32187
$800$		0			0
$801$	−157.401	−	298.088i	−0.196505	−	0.372144i
$802$		0			0
$803$		−	288.977i		−	0.359872i
$804$		0			0
$805$	−46.2002			−0.0573916
$806$		0			0
$807$	−1288.68	+	319.342i	−1.59688	+	0.395715i
$808$		0			0
$809$		−	830.702i		−	1.02683i	−0.858142	−	0.513413i	$-0.828381\pi$
$809$		−	830.702i		−	1.02683i	0.858142	−	0.513413i	$-0.171619\pi$
$810$		0			0
$811$	1241.57			1.53091			0.765457	−	0.643487i	$-0.222513\pi$
$811$	1241.57			1.53091			0.765457	+	0.643487i	$0.222513\pi$
$812$		0			0
$813$	−255.834	−	1032.40i	−0.314679	−	1.26987i
$814$		0			0
$815$			460.146i			0.564596i
$816$		0			0
$817$	1294.25			1.58414
$818$		0			0
$819$	−134.896	+	71.2299i	−0.164708	+	0.0869717i
$820$		0			0
$821$		−	703.105i		−	0.856401i	−0.903684	−	0.428200i	$-0.859148\pi$
$821$		−	703.105i		−	0.856401i	0.903684	−	0.428200i	$-0.140852\pi$
$822$		0			0
$823$	67.9779			0.0825977			0.0412988	−	0.999147i	$-0.486850\pi$
$823$	67.9779			0.0825977			0.0412988	+	0.999147i	$0.486850\pi$
$824$		0			0
$825$	−284.609	+	70.5276i	−0.344981	+	0.0854880i
$826$		0			0
$827$			999.731i			1.20886i	0.796657	+	0.604432i	$0.206600\pi$
$827$			999.731i			1.20886i	−0.796657	+	0.604432i	$0.793400\pi$
$828$		0			0
$829$	−778.096			−0.938596			−0.469298	−	0.883040i	$-0.655493\pi$
$829$	−778.096			−0.938596			−0.469298	+	0.883040i	$0.655493\pi$
$830$		0			0
$831$	154.656	+	624.103i	0.186108	+	0.751027i
$832$		0			0
$833$			1300.87i			1.56167i
$834$		0			0
$835$	−50.2742			−0.0602086
$836$		0			0
$837$	−593.036	−	664.361i	−0.708526	−	0.793741i
$838$		0			0
$839$			1005.39i			1.19832i	0.800628	+	0.599162i	$0.204499\pi$
$839$			1005.39i			1.19832i	−0.800628	+	0.599162i	$0.795501\pi$
$840$		0			0
$841$	−1662.23			−1.97649
$842$		0			0
$843$	419.092	−	103.853i	0.497143	−	0.123194i
$844$		0			0
$845$			220.260i			0.260663i
$846$		0			0
$847$	−270.604			−0.319486
$848$		0			0
$849$	−311.508	−	1257.07i	−0.366911	−	1.48065i
$850$		0			0
$851$			18.7945i			0.0220852i
$852$		0			0
$853$	182.815			0.214320			0.107160	−	0.994242i	$-0.465824\pi$
$853$	182.815			0.214320			0.107160	+	0.994242i	$0.465824\pi$
$854$		0			0
$855$	−229.090	−	433.853i	−0.267941	−	0.507431i
$856$		0			0
$857$		−	332.010i		−	0.387409i	−0.981060	−	0.193705i	$-0.937950\pi$
$857$		−	332.010i		−	0.387409i	0.981060	−	0.193705i	$-0.0620503\pi$
$858$		0			0
$859$	−764.137			−0.889566			−0.444783	−	0.895638i	$-0.646719\pi$
$859$	−764.137			−0.889566			−0.444783	+	0.895638i	$0.646719\pi$
$860$		0			0
$861$	34.7985	−	8.62323i	0.0404163	−	0.0100154i
$862$		0			0
$863$		−	1159.73i		−	1.34384i	−0.740625	−	0.671919i	$-0.765470\pi$
$863$		−	1159.73i		−	1.34384i	0.740625	−	0.671919i	$-0.234530\pi$
$864$		0			0
$865$	261.184			0.301947
$866$		0			0
$867$	−323.100	−	1303.85i	−0.372665	−	1.50386i
$868$		0			0
$869$			1584.18i			1.82299i
$870$		0			0
$871$	156.092			0.179210
$872$		0			0
$873$	769.760	−	406.460i	0.881741	−	0.465590i
$874$		0			0
$875$		−	11.5865i		−	0.0132417i
$876$		0			0
$877$	1135.48			1.29473			0.647367	−	0.762179i	$-0.275870\pi$
$877$	1135.48			1.29473			0.647367	+	0.762179i	$0.275870\pi$
$878$		0			0
$879$	291.106	−	72.1376i	0.331179	−	0.0820678i
$880$		0			0
$881$			414.565i			0.470562i	0.971927	+	0.235281i	$0.0756010\pi$
$881$			414.565i			0.470562i	−0.971927	+	0.235281i	$0.924399\pi$
$882$		0			0
$883$	−136.032			−0.154056			−0.0770281	−	0.997029i	$-0.524543\pi$
$883$	−136.032			−0.154056			−0.0770281	+	0.997029i	$0.524543\pi$
$884$		0			0
$885$	23.2119	+	93.6700i	0.0262281	+	0.105842i
$886$		0			0
$887$			1215.49i			1.37033i	0.728386	+	0.685167i	$0.240271\pi$
$887$			1215.49i			1.37033i	−0.728386	+	0.685167i	$0.759729\pi$
$888$		0			0
$889$	252.539			0.284070
$890$		0			0
$891$	−1307.58	−	892.930i	−1.46754	−	1.00217i
$892$		0			0
$893$			1666.26i			1.86591i
$894$		0			0
$895$	−655.646			−0.732565
$896$		0			0
$897$	−949.525	+	235.297i	−1.05856	+	0.262315i
$898$		0			0
$899$		−	1650.22i		−	1.83562i
$900$		0			0
$901$	1314.12			1.45851
$902$		0			0
$903$	39.6994	+	160.204i	0.0439639	+	0.177413i
$904$		0			0
$905$		−	6.50019i		−	0.00718254i
$906$		0			0
$907$	219.419			0.241917			0.120959	−	0.992658i	$-0.461403\pi$
$907$	219.419			0.241917			0.120959	+	0.992658i	$0.461403\pi$
$908$		0			0
$909$	131.735	+	249.482i	0.144923	+	0.274458i
$910$		0			0
$911$			914.407i			1.00374i	0.864943	+	0.501870i	$0.167354\pi$
$911$			914.407i			1.00374i	−0.864943	+	0.501870i	$0.832646\pi$
$912$		0			0
$913$	−1137.66			−1.24607
$914$		0			0
$915$	224.336	−	55.5915i	0.245176	−	0.0607557i
$916$		0			0
$917$			112.556i			0.122744i
$918$		0			0
$919$	−690.517			−0.751379			−0.375689	−	0.926746i	$-0.622594\pi$
$919$	−690.517			−0.751379			−0.375689	+	0.926746i	$0.622594\pi$
$920$		0			0
$921$	−16.1258	−	65.0746i	−0.0175090	−	0.0706565i
$922$		0			0
$923$		−	179.032i		−	0.193967i
$924$		0			0
$925$	−4.71345			−0.00509562
$926$		0			0
$927$	−608.615	+	321.370i	−0.656542	+	0.346677i
$928$		0			0
$929$		−	346.974i		−	0.373492i	−0.982408	−	0.186746i	$-0.940206\pi$
$929$		−	346.974i		−	0.373492i	0.982408	−	0.186746i	$-0.0597942\pi$
$930$		0			0
$931$	−1168.40			−1.25500
$932$		0			0
$933$	738.930	−	183.110i	0.791994	−	0.196260i
$934$		0			0
$935$		−	1186.44i		−	1.26893i
$936$		0			0
$937$	−1235.71			−1.31879			−0.659396	−	0.751796i	$-0.729188\pi$
$937$	−1235.71			−1.31879			−0.659396	+	0.751796i	$0.729188\pi$
$938$		0			0
$939$	153.269	+	618.506i	0.163225	+	0.658685i
$940$		0			0
$941$		−	61.1959i		−	0.0650329i	−0.999471	−	0.0325164i	$-0.989648\pi$
$941$		−	61.1959i		−	0.0650329i	0.999471	−	0.0325164i	$-0.0103521\pi$
$942$		0			0
$943$	229.903			0.243799
$944$		0			0
$945$	46.6761	−	41.6651i	0.0493927	−	0.0440900i
$946$		0			0
$947$		−	357.044i		−	0.377027i	−0.982071	−	0.188513i	$-0.939633\pi$
$947$		−	357.044i		−	0.377027i	0.982071	−	0.188513i	$-0.0603669\pi$
$948$		0			0
$949$	241.785			0.254779
$950$		0			0
$951$	−1059.06	+	262.441i	−1.11363	+	0.275963i
$952$		0			0
$953$			1105.44i			1.15996i	0.814630	+	0.579980i	$0.196940\pi$
$953$			1105.44i			1.15996i	−0.814630	+	0.579980i	$0.803060\pi$
$954$		0			0
$955$	243.524			0.254999
$956$		0			0
$957$	−705.731	−	2847.93i	−0.737441	−	2.97590i
$958$		0			0
$959$		−	166.178i		−	0.173283i
$960$		0			0
$961$	126.883			0.132033
$962$		0			0
$963$	−402.881	−	762.982i	−0.418360	−	0.792297i
$964$		0			0
$965$		−	148.381i		−	0.153762i
$966$		0			0
$967$	−1375.08			−1.42200			−0.711001	−	0.703191i	$-0.751758\pi$
$967$	−1375.08			−1.42200			−0.711001	+	0.703191i	$0.751758\pi$
$968$		0			0
$969$	1926.92	−	477.501i	1.98857	−	0.492777i
$970$		0			0
$971$			402.979i			0.415014i	0.978234	+	0.207507i	$0.0665350\pi$
$971$			402.979i			0.415014i	−0.978234	+	0.207507i	$0.933465\pi$
$972$		0			0
$973$	207.756			0.213521
$974$		0			0
$975$	−59.0099	−	238.130i	−0.0605229	−	0.244236i
$976$		0			0
$977$			386.921i			0.396030i	0.980199	+	0.198015i	$0.0634494\pi$
$977$			386.921i			0.396030i	−0.980199	+	0.198015i	$0.936551\pi$
$978$		0			0
$979$	732.159			0.747864
$980$		0			0
$981$	−1403.56	+	741.129i	−1.43075	+	0.755484i
$982$		0			0
$983$			140.386i			0.142814i	0.997447	+	0.0714068i	$0.0227489\pi$
$983$			140.386i			0.142814i	−0.997447	+	0.0714068i	$0.977251\pi$
$984$		0			0
$985$	244.874			0.248603
$986$		0			0
$987$	−206.253	+	51.1104i	−0.208969	+	0.0517836i
$988$		0			0
$989$			1058.42i			1.07019i
$990$		0			0
$991$	1048.10			1.05761			0.528807	−	0.848742i	$-0.322639\pi$
$991$	1048.10			1.05761			0.528807	+	0.848742i	$0.322639\pi$
$992$		0			0
$993$	−85.1012	−	343.420i	−0.0857011	−	0.345841i
$994$		0			0
$995$		−	169.842i		−	0.170695i
$996$		0			0
$997$	−314.771			−0.315718			−0.157859	−	0.987462i	$-0.550459\pi$
$997$	−314.771			−0.315718			−0.157859	+	0.987462i	$0.550459\pi$
$998$		0			0
$999$	−16.9496	−	18.9881i	−0.0169665	−	0.0190071i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.3.l.j.641.10		16
3.2	odd	2	inner	960.3.l.j.641.9		16
4.3	odd	2	inner	960.3.l.j.641.7		16
8.3	odd	2		480.3.l.b.161.10	yes	16
8.5	even	2		480.3.l.b.161.7	✓	16
12.11	even	2	inner	960.3.l.j.641.8		16
24.5	odd	2		480.3.l.b.161.8	yes	16
24.11	even	2		480.3.l.b.161.9	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.3.l.b.161.7	✓	16	8.5	even	2
480.3.l.b.161.8	yes	16	24.5	odd	2
480.3.l.b.161.9	yes	16	24.11	even	2
480.3.l.b.161.10	yes	16	8.3	odd	2
960.3.l.j.641.7		16	4.3	odd	2	inner
960.3.l.j.641.8		16	12.11	even	2	inner
960.3.l.j.641.9		16	3.2	odd	2	inner
960.3.l.j.641.10		16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(0.721589 + 2.91193i) q^{3} +2.23607i q^{5} +1.03633 q^{7} +(-7.95862 + 4.20243i) q^{9} +O(q^{10})\)	\(q+(0.721589 + 2.91193i) q^{3} +2.23607i q^{5} +1.03633 q^{7} +(-7.95862 + 4.20243i) q^{9} -19.5478i q^{11} +16.3555 q^{13} +(-6.51126 + 1.61352i) q^{15} -27.1434i q^{17} +24.3792 q^{19} +(0.747803 + 3.01771i) q^{21} +19.9371i q^{23} -5.00000 q^{25} +(-17.9800 - 20.1425i) q^{27} -50.0323i q^{29} +32.9831 q^{31} +(56.9219 - 14.1055i) q^{33} +2.31730i q^{35} +0.942690 q^{37} +(11.8020 + 47.6261i) q^{39} -11.5314i q^{41} +53.0880 q^{43} +(-9.39691 - 17.7960i) q^{45} +68.3474i q^{47} -47.9260 q^{49} +(79.0394 - 19.5864i) q^{51} +48.4140i q^{53} +43.7103 q^{55} +(17.5918 + 70.9905i) q^{57} -14.3858i q^{59} -34.4535 q^{61} +(-8.24774 + 4.35509i) q^{63} +36.5721i q^{65} +9.54367 q^{67} +(-58.0553 + 14.3864i) q^{69} -10.9462i q^{71} +14.7831 q^{73} +(-3.60795 - 14.5596i) q^{75} -20.2580i q^{77} -81.0412 q^{79} +(45.6792 - 66.8910i) q^{81} -58.1989i q^{83} +60.6944 q^{85} +(145.690 - 36.1028i) q^{87} +37.4547i q^{89} +16.9497 q^{91} +(23.8002 + 96.0443i) q^{93} +54.5137i q^{95} -96.7203 q^{97} +(82.1484 + 155.574i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{9}+O(q^{10}) \) 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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