LMFDB - Embedded newform 960.4.h.d.191.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,4,Mod(191,960)]

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([1, 0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("960.191");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	960.h (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:	$56.6418336055$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.5
Character	$\chi$	$=$	960.191
Dual form			960.4.h.d.191.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.56698 - 2.47844i) q^{3} +5.00000i q^{5} +20.9745i q^{7} +(14.7146 + 22.6380i) q^{9} +O(q^{10})$	\(q+(-4.56698 - 2.47844i) q^{3} +5.00000i q^{5} +20.9745i q^{7} +(14.7146 + 22.6380i) q^{9} -30.8097 q^{11} -56.7161 q^{13} +(12.3922 - 22.8349i) q^{15} +88.9672i q^{17} +88.2358i q^{19} +(51.9840 - 95.7899i) q^{21} -138.240 q^{23} -25.0000 q^{25} +(-11.0943 - 139.857i) q^{27} +161.030i q^{29} -197.266i q^{31} +(140.707 + 76.3602i) q^{33} -104.872 q^{35} +179.128 q^{37} +(259.021 + 140.568i) q^{39} -67.7548i q^{41} +41.7076i q^{43} +(-113.190 + 73.5731i) q^{45} -214.016 q^{47} -96.9279 q^{49} +(220.500 - 406.311i) q^{51} +263.225i q^{53} -154.049i q^{55} +(218.688 - 402.971i) q^{57} -103.646 q^{59} +698.376 q^{61} +(-474.820 + 308.631i) q^{63} -283.580i q^{65} +129.743i q^{67} +(631.338 + 342.620i) q^{69} +301.620 q^{71} +1115.84 q^{73} +(114.175 + 61.9611i) q^{75} -646.217i q^{77} -712.880i q^{79} +(-295.960 + 666.220i) q^{81} -336.355 q^{83} -444.836 q^{85} +(399.103 - 735.419i) q^{87} +970.209i q^{89} -1189.59i q^{91} +(-488.913 + 900.911i) q^{93} -441.179 q^{95} -1600.81 q^{97} +(-453.353 - 697.471i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 20 q^{9}+O(q^{10}) $ 24 * q + 20 * q^9	$ 24 q + 20 q^{9} - 72 q^{13} + 68 q^{21} - 600 q^{25} + 848 q^{33} - 504 q^{37} + 220 q^{45} - 2256 q^{49} + 1416 q^{57} - 1992 q^{61} + 1548 q^{69} - 2304 q^{73} + 3840 q^{81} - 240 q^{85} + 4384 q^{93} - 2448 q^{97}+O(q^{100}) $ 24 * q + 20 * q^9 - 72 * q^13 + 68 * q^21 - 600 * q^25 + 848 * q^33 - 504 * q^37 + 220 * q^45 - 2256 * q^49 + 1416 * q^57 - 1992 * q^61 + 1548 * q^69 - 2304 * q^73 + 3840 * q^81 - 240 * q^85 + 4384 * q^93 - 2448 * 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
$3$	−4.56698	−	2.47844i	−0.878916	−	0.476977i
$3$	−4.56698	−	2.47844i	−0.878916	−	0.476977i
$4$		0			0
$5$			5.00000i			0.447214i
$5$			5.00000i			0.447214i
$6$		0			0
$7$			20.9745i			1.13251i	0.824229	+	0.566257i	$0.191609\pi$
$7$			20.9745i			1.13251i	−0.824229	+	0.566257i	$0.808391\pi$
$8$		0			0
$9$	14.7146	+	22.6380i	0.544986	+	0.838445i
$10$		0			0
$11$	−30.8097			−0.844498			−0.422249	−	0.906480i	$-0.638759\pi$
$11$	−30.8097			−0.844498			−0.422249	+	0.906480i	$0.638759\pi$
$12$		0			0
$13$	−56.7161			−1.21002			−0.605008	−	0.796219i	$-0.706830\pi$
$13$	−56.7161			−1.21002			−0.605008	+	0.796219i	$0.706830\pi$
$14$		0			0
$15$	12.3922	−	22.8349i	0.213311	−	0.393063i
$16$		0			0
$17$			88.9672i			1.26928i	0.772809	+	0.634639i	$0.218851\pi$
$17$			88.9672i			1.26928i	−0.772809	+	0.634639i	$0.781149\pi$
$18$		0			0
$19$			88.2358i			1.06540i	0.846303	+	0.532702i	$0.178823\pi$
$19$			88.2358i			1.06540i	−0.846303	+	0.532702i	$0.821177\pi$
$20$		0			0
$21$	51.9840	−	95.7899i	0.540183	−	0.995385i
$22$		0			0
$23$	−138.240			−1.25326			−0.626630	−	0.779317i	$-0.715566\pi$
$23$	−138.240			−1.25326			−0.626630	+	0.779317i	$0.715566\pi$
$24$		0			0
$25$	−25.0000			−0.200000
$26$		0			0
$27$	−11.0943	−	139.857i	−0.0790781	−	0.996868i
$28$		0			0
$29$			161.030i			1.03112i	0.856854	+	0.515560i	$0.172416\pi$
$29$			161.030i			1.03112i	−0.856854	+	0.515560i	$0.827584\pi$
$30$		0			0
$31$		−	197.266i		−	1.14290i	−0.820635	−	0.571452i	$-0.806380\pi$
$31$		−	197.266i		−	1.14290i	0.820635	−	0.571452i	$-0.193620\pi$
$32$		0			0
$33$	140.707	+	76.3602i	0.742243	+	0.402806i
$34$		0			0
$35$	−104.872			−0.506476
$36$		0			0
$37$	179.128			0.795905			0.397953	−	0.917406i	$-0.369721\pi$
$37$	179.128			0.795905			0.397953	+	0.917406i	$0.369721\pi$
$38$		0			0
$39$	259.021	+	140.568i	1.06350	+	0.577150i
$40$		0			0
$41$		−	67.7548i		−	0.258086i	−0.991639	−	0.129043i	$-0.958809\pi$
$41$		−	67.7548i		−	0.258086i	0.991639	−	0.129043i	$-0.0411905\pi$
$42$		0			0
$43$			41.7076i			0.147915i	0.997261	+	0.0739575i	$0.0235629\pi$
$43$			41.7076i			0.147915i	−0.997261	+	0.0739575i	$0.976437\pi$
$44$		0			0
$45$	−113.190	+	73.5731i	−0.374964	+	0.243725i
$46$		0			0
$47$	−214.016			−0.664200			−0.332100	−	0.943244i	$-0.607757\pi$
$47$	−214.016			−0.664200			−0.332100	+	0.943244i	$0.607757\pi$
$48$		0			0
$49$	−96.9279			−0.282589
$50$		0			0
$51$	220.500	−	406.311i	0.605416	−	1.11559i
$52$		0			0
$53$			263.225i			0.682202i	0.940027	+	0.341101i	$0.110800\pi$
$53$			263.225i			0.682202i	−0.940027	+	0.341101i	$0.889200\pi$
$54$		0			0
$55$		−	154.049i		−	0.377671i
$56$		0			0
$57$	218.688	−	402.971i	0.508173	−	0.936401i
$58$		0			0
$59$	−103.646			−0.228704			−0.114352	−	0.993440i	$-0.536479\pi$
$59$	−103.646			−0.228704			−0.114352	+	0.993440i	$0.536479\pi$
$60$		0			0
$61$	698.376			1.46587			0.732934	−	0.680300i	$-0.238151\pi$
$61$	698.376			1.46587			0.732934	+	0.680300i	$0.238151\pi$
$62$		0			0
$63$	−474.820	+	308.631i	−0.949551	+	0.617205i
$64$		0			0
$65$		−	283.580i		−	0.541136i
$66$		0			0
$67$			129.743i			0.236577i	0.992979	+	0.118289i	$0.0377408\pi$
$67$			129.743i			0.236577i	−0.992979	+	0.118289i	$0.962259\pi$
$68$		0			0
$69$	631.338	+	342.620i	1.10151	+	0.597776i
$70$		0			0
$71$	301.620			0.504165			0.252083	−	0.967706i	$-0.418885\pi$
$71$	301.620			0.504165			0.252083	+	0.967706i	$0.418885\pi$
$72$		0			0
$73$	1115.84			1.78903			0.894513	−	0.447041i	$-0.147522\pi$
$73$	1115.84			1.78903			0.894513	+	0.447041i	$0.147522\pi$
$74$		0			0
$75$	114.175	+	61.9611i	0.175783	+	0.0953954i
$76$		0			0
$77$		−	646.217i		−	0.956406i
$78$		0			0
$79$		−	712.880i		−	1.01526i	−0.861576	−	0.507628i	$-0.830522\pi$
$79$		−	712.880i		−	1.01526i	0.861576	−	0.507628i	$-0.169478\pi$
$80$		0			0
$81$	−295.960	+	666.220i	−0.405980	+	0.913882i
$82$		0			0
$83$	−336.355			−0.444817			−0.222408	−	0.974954i	$-0.571392\pi$
$83$	−336.355			−0.444817			−0.222408	+	0.974954i	$0.571392\pi$
$84$		0			0
$85$	−444.836			−0.567638
$86$		0			0
$87$	399.103	−	735.419i	0.491820	−	0.906267i
$88$		0			0
$89$			970.209i			1.15553i	0.816204	+	0.577764i	$0.196074\pi$
$89$			970.209i			1.15553i	−0.816204	+	0.577764i	$0.803926\pi$
$90$		0			0
$91$		−	1189.59i		−	1.37036i
$92$		0			0
$93$	−488.913	+	900.911i	−0.545139	+	1.00452i
$94$		0			0
$95$	−441.179			−0.476463
$96$		0			0
$97$	−1600.81			−1.67564			−0.837821	−	0.545945i	$-0.816171\pi$
$97$	−1600.81			−1.67564			−0.837821	+	0.545945i	$0.816171\pi$
$98$		0			0
$99$	−453.353	−	697.471i	−0.460240	−	0.708065i
$100$		0			0
$101$		−	1073.62i		−	1.05772i	−0.848710	−	0.528859i	$-0.822620\pi$
$101$		−	1073.62i		−	1.05772i	0.848710	−	0.528859i	$-0.177380\pi$
$102$		0			0
$103$		−	1708.82i		−	1.63471i	−0.576134	−	0.817355i	$-0.695439\pi$
$103$		−	1708.82i		−	1.63471i	0.576134	−	0.817355i	$-0.304561\pi$
$104$		0			0
$105$	478.950	+	259.920i	0.445150	+	0.241577i
$106$		0			0
$107$	−787.936			−0.711894			−0.355947	−	0.934506i	$-0.615842\pi$
$107$	−787.936			−0.711894			−0.355947	+	0.934506i	$0.615842\pi$
$108$		0			0
$109$	−1200.41			−1.05485			−0.527424	−	0.849602i	$-0.676842\pi$
$109$	−1200.41			−1.05485			−0.527424	+	0.849602i	$0.676842\pi$
$110$		0			0
$111$	−818.075	−	443.959i	−0.699534	−	0.379628i
$112$		0			0
$113$		−	487.048i		−	0.405466i	−0.979234	−	0.202733i	$-0.935018\pi$
$113$		−	487.048i		−	0.405466i	0.979234	−	0.202733i	$-0.0649823\pi$
$114$		0			0
$115$		−	691.199i		−	0.560475i
$116$		0			0
$117$	−834.556	−	1283.94i	−0.659442	−	1.01453i
$118$		0			0
$119$	−1866.04			−1.43747
$120$		0			0
$121$	−381.762			−0.286823
$122$		0			0
$123$	−167.927	+	309.435i	−0.123101	+	0.226836i
$124$		0			0
$125$		−	125.000i		−	0.0894427i
$126$		0			0
$127$			1608.28i			1.12372i	0.827234	+	0.561858i	$0.189913\pi$
$127$			1608.28i			1.12372i	−0.827234	+	0.561858i	$0.810087\pi$
$128$		0			0
$129$	103.370	−	190.478i	0.0705520	−	0.130005i
$130$		0			0
$131$	−1357.63			−0.905468			−0.452734	−	0.891646i	$-0.649551\pi$
$131$	−1357.63			−0.905468			−0.452734	+	0.891646i	$0.649551\pi$
$132$		0			0
$133$	−1850.70			−1.20659
$134$		0			0
$135$	699.284	−	55.4717i	0.445813	−	0.0353648i
$136$		0			0
$137$		−	1684.06i		−	1.05021i	−0.851038	−	0.525105i	$-0.824026\pi$
$137$		−	1684.06i		−	1.05021i	0.851038	−	0.525105i	$-0.175974\pi$
$138$		0			0
$139$		−	1425.55i		−	0.869885i	−0.900458	−	0.434942i	$-0.856769\pi$
$139$		−	1425.55i		−	0.869885i	0.900458	−	0.434942i	$-0.143231\pi$
$140$		0			0
$141$	977.406	+	530.426i	0.583776	+	0.316808i
$142$		0			0
$143$	1747.41			1.02186
$144$		0			0
$145$	−805.148			−0.461130
$146$		0			0
$147$	442.668	+	240.230i	0.248372	+	0.134788i
$148$		0			0
$149$		−	874.024i		−	0.480556i	−0.970704	−	0.240278i	$-0.922761\pi$
$149$		−	874.024i		−	0.480556i	0.970704	−	0.240278i	$-0.0772386\pi$
$150$		0			0
$151$		−	1343.03i		−	0.723800i	−0.932217	−	0.361900i	$-0.882128\pi$
$151$		−	1343.03i		−	0.723800i	0.932217	−	0.361900i	$-0.117872\pi$
$152$		0			0
$153$	−2014.04	+	1309.12i	−1.06422	+	0.691738i
$154$		0			0
$155$	986.331			0.511123
$156$		0			0
$157$	1313.01			0.667450			0.333725	−	0.942670i	$-0.391694\pi$
$157$	1313.01			0.667450			0.333725	+	0.942670i	$0.391694\pi$
$158$		0			0
$159$	652.388	−	1202.14i	0.325395	−	0.599598i
$160$		0			0
$161$		−	2899.50i		−	1.41933i
$162$		0			0
$163$		−	1970.74i		−	0.946995i	−0.880795	−	0.473497i	$-0.842991\pi$
$163$		−	1970.74i		−	0.946995i	0.880795	−	0.473497i	$-0.157009\pi$
$164$		0			0
$165$	−381.801	+	703.537i	−0.180140	+	0.331941i
$166$		0			0
$167$	−1196.07			−0.554221			−0.277110	−	0.960838i	$-0.589377\pi$
$167$	−1196.07			−0.554221			−0.277110	+	0.960838i	$0.589377\pi$
$168$		0			0
$169$	1019.71			0.464139
$170$		0			0
$171$	−1997.48	+	1298.36i	−0.893283	+	0.580631i
$172$		0			0
$173$			3252.29i			1.42929i	0.699487	+	0.714646i	$0.253412\pi$
$173$			3252.29i			1.42929i	−0.699487	+	0.714646i	$0.746588\pi$
$174$		0			0
$175$		−	524.361i		−	0.226503i
$176$		0			0
$177$	473.348	+	256.880i	0.201011	+	0.109086i
$178$		0			0
$179$	4438.90			1.85352			0.926758	−	0.375659i	$-0.122584\pi$
$179$	4438.90			1.85352			0.926758	+	0.375659i	$0.122584\pi$
$180$		0			0
$181$	59.4849			0.0244281			0.0122140	−	0.999925i	$-0.496112\pi$
$181$	59.4849			0.0244281			0.0122140	+	0.999925i	$0.496112\pi$
$182$		0			0
$183$	−3189.47	−	1730.89i	−1.28837	−	0.699185i
$184$		0			0
$185$			895.641i			0.355940i
$186$		0			0
$187$		−	2741.05i		−	1.07190i
$188$		0			0
$189$	2933.42	−	232.698i	1.12897	−	0.0895570i
$190$		0			0
$191$	−1195.27			−0.452810			−0.226405	−	0.974033i	$-0.572697\pi$
$191$	−1195.27			−0.452810			−0.226405	+	0.974033i	$0.572697\pi$
$192$		0			0
$193$	−2886.55			−1.07657			−0.538286	−	0.842762i	$-0.680928\pi$
$193$	−2886.55			−1.07657			−0.538286	+	0.842762i	$0.680928\pi$
$194$		0			0
$195$	−702.838	+	1295.11i	−0.258109	+	0.475613i
$196$		0			0
$197$		−	520.753i		−	0.188336i	−0.995556	−	0.0941679i	$-0.969981\pi$
$197$		−	520.753i		−	0.188336i	0.995556	−	0.0941679i	$-0.0300190\pi$
$198$		0			0
$199$			115.686i			0.0412098i	0.999788	+	0.0206049i	$0.00655920\pi$
$199$			115.686i			0.0412098i	−0.999788	+	0.0206049i	$0.993441\pi$
$200$		0			0
$201$	321.562	−	592.535i	0.112842	−	0.207931i
$202$		0			0
$203$	−3377.51			−1.16776
$204$		0			0
$205$	338.774			0.115420
$206$		0			0
$207$	−2034.15	−	3129.47i	−0.683009	−	1.05079i
$208$		0			0
$209$		−	2718.52i		−	0.899732i
$210$		0			0
$211$			671.242i			0.219006i	0.993986	+	0.109503i	$0.0349259\pi$
$211$			671.242i			0.219006i	−0.993986	+	0.109503i	$0.965074\pi$
$212$		0			0
$213$	−1377.49	−	747.549i	−0.443119	−	0.240475i
$214$		0			0
$215$	−208.538			−0.0661496
$216$		0			0
$217$	4137.55			1.29436
$218$		0			0
$219$	−5096.01	−	2765.54i	−1.57240	−	0.853324i
$220$		0			0
$221$		−	5045.87i		−	1.53585i
$222$		0			0
$223$			3012.48i			0.904622i	0.891860	+	0.452311i	$0.149400\pi$
$223$			3012.48i			0.904622i	−0.891860	+	0.452311i	$0.850600\pi$
$224$		0			0
$225$	−367.866	−	565.950i	−0.108997	−	0.167689i
$226$		0			0
$227$	−3106.72			−0.908371			−0.454185	−	0.890907i	$-0.650070\pi$
$227$	−3106.72			−0.908371			−0.454185	+	0.890907i	$0.650070\pi$
$228$		0			0
$229$	−485.058			−0.139972			−0.0699858	−	0.997548i	$-0.522295\pi$
$229$	−485.058			−0.139972			−0.0699858	+	0.997548i	$0.522295\pi$
$230$		0			0
$231$	−1601.61	+	2951.26i	−0.456184	+	0.840600i
$232$		0			0
$233$			6052.81i			1.70186i	0.525281	+	0.850929i	$0.323960\pi$
$233$			6052.81i			1.70186i	−0.525281	+	0.850929i	$0.676040\pi$
$234$		0			0
$235$		−	1070.08i		−	0.297039i
$236$		0			0
$237$	−1766.83	+	3255.71i	−0.484254	+	0.892325i
$238$		0			0
$239$	4846.17			1.31160			0.655800	−	0.754934i	$-0.272331\pi$
$239$	4846.17			1.31160			0.655800	+	0.754934i	$0.272331\pi$
$240$		0			0
$241$	6399.86			1.71058			0.855292	−	0.518146i	$-0.173378\pi$
$241$	6399.86			1.71058			0.855292	+	0.518146i	$0.173378\pi$
$242$		0			0
$243$	3002.83	−	2309.09i	0.792723	−	0.609582i
$244$		0			0
$245$		−	484.639i		−	0.126377i
$246$		0			0
$247$		−	5004.39i		−	1.28916i
$248$		0			0
$249$	1536.13	+	833.638i	0.390957	+	0.212167i
$250$		0			0
$251$	−39.6720			−0.00997640			−0.00498820	−	0.999988i	$-0.501588\pi$
$251$	−39.6720			−0.00997640			−0.00498820	+	0.999988i	$0.501588\pi$
$252$		0			0
$253$	4259.13			1.05838
$254$		0			0
$255$	2031.56	+	1102.50i	0.498906	+	0.270750i
$256$		0			0
$257$			185.740i			0.0450823i	0.999746	+	0.0225412i	$0.00717568\pi$
$257$			185.740i			0.0450823i	−0.999746	+	0.0225412i	$0.992824\pi$
$258$		0			0
$259$			3757.12i			0.901374i
$260$		0			0
$261$	−3645.39	+	2369.49i	−0.864537	+	0.561946i
$262$		0			0
$263$	−1512.06			−0.354516			−0.177258	−	0.984164i	$-0.556723\pi$
$263$	−1512.06			−0.354516			−0.177258	+	0.984164i	$0.556723\pi$
$264$		0			0
$265$	−1316.12			−0.305090
$266$		0			0
$267$	2404.61	−	4430.93i	0.551160	−	1.01561i
$268$		0			0
$269$		−	2951.18i		−	0.668910i	−0.942412	−	0.334455i	$-0.891448\pi$
$269$		−	2951.18i		−	0.668910i	0.942412	−	0.334455i	$-0.108552\pi$
$270$		0			0
$271$			903.495i			0.202522i	0.994860	+	0.101261i	$0.0322877\pi$
$271$			903.495i			0.202522i	−0.994860	+	0.101261i	$0.967712\pi$
$272$		0			0
$273$	−2948.33	+	5432.83i	−0.653630	+	1.20443i
$274$		0			0
$275$	770.243			0.168900
$276$		0			0
$277$	2119.18			0.459671			0.229836	−	0.973229i	$-0.426181\pi$
$277$	2119.18			0.459671			0.229836	+	0.973229i	$0.426181\pi$
$278$		0			0
$279$	4465.71	−	2902.70i	0.958263	−	0.622867i
$280$		0			0
$281$		−	4896.84i		−	1.03958i	−0.854295	−	0.519788i	$-0.826011\pi$
$281$		−	4896.84i		−	1.03958i	0.854295	−	0.519788i	$-0.173989\pi$
$282$		0			0
$283$			4875.98i			1.02419i	0.858928	+	0.512097i	$0.171131\pi$
$283$			4875.98i			1.02419i	−0.858928	+	0.512097i	$0.828869\pi$
$284$		0			0
$285$	2014.86	+	1093.44i	0.418771	+	0.227262i
$286$		0			0
$287$	1421.12			0.292286
$288$		0			0
$289$	−3002.16			−0.611064
$290$		0			0
$291$	7310.85	+	3967.51i	1.47275	+	0.799242i
$292$		0			0
$293$		−	4524.68i		−	0.902166i	−0.892482	−	0.451083i	$-0.851038\pi$
$293$		−	4524.68i		−	0.902166i	0.892482	−	0.451083i	$-0.148962\pi$
$294$		0			0
$295$		−	518.229i		−	0.102279i
$296$		0			0
$297$	341.814	+	4308.95i	0.0667813	+	0.841853i
$298$		0			0
$299$	7840.42			1.51646
$300$		0			0
$301$	−874.793			−0.167516
$302$		0			0
$303$	−2660.92	+	4903.22i	−0.504507	+	0.929645i
$304$		0			0
$305$			3491.88i			0.655556i
$306$		0			0
$307$			6726.35i			1.25047i	0.780438	+	0.625233i	$0.214996\pi$
$307$			6726.35i			1.25047i	−0.780438	+	0.625233i	$0.785004\pi$
$308$		0			0
$309$	−4235.22	+	7804.16i	−0.779719	+	1.43677i
$310$		0			0
$311$	−165.537			−0.0301824			−0.0150912	−	0.999886i	$-0.504804\pi$
$311$	−165.537			−0.0301824			−0.0150912	+	0.999886i	$0.504804\pi$
$312$		0			0
$313$	−3295.86			−0.595185			−0.297593	−	0.954693i	$-0.596184\pi$
$313$	−3295.86			−0.595185			−0.297593	+	0.954693i	$0.596184\pi$
$314$		0			0
$315$	−1543.16	−	2374.10i	−0.276022	−	0.424652i
$316$		0			0
$317$			164.477i			0.0291417i	0.999894	+	0.0145709i	$0.00463822\pi$
$317$			164.477i			0.0291417i	−0.999894	+	0.0145709i	$0.995362\pi$
$318$		0			0
$319$		−	4961.28i		−	0.870778i
$320$		0			0
$321$	3598.49	+	1952.86i	0.625695	+	0.339557i
$322$		0			0
$323$	−7850.09			−1.35229
$324$		0			0
$325$	1417.90			0.242003
$326$		0			0
$327$	5482.25	+	2975.15i	0.927123	+	0.503138i
$328$		0			0
$329$		−	4488.86i		−	0.752216i
$330$		0			0
$331$		−	3358.03i		−	0.557626i	−0.960345	−	0.278813i	$-0.910059\pi$
$331$		−	3358.03i		−	0.557626i	0.960345	−	0.278813i	$-0.0899410\pi$
$332$		0			0
$333$	2635.80	+	4055.11i	0.433757	+	0.667323i
$334$		0			0
$335$	−648.716			−0.105800
$336$		0			0
$337$	3077.19			0.497404			0.248702	−	0.968580i	$-0.419996\pi$
$337$	3077.19			0.497404			0.248702	+	0.968580i	$0.419996\pi$
$338$		0			0
$339$	−1207.12	+	2224.34i	−0.193398	+	0.356370i
$340$		0			0
$341$			6077.71i			0.965181i
$342$		0			0
$343$			5161.23i			0.812479i
$344$		0			0
$345$	−1713.10	+	3156.69i	−0.267334	+	0.492610i
$346$		0			0
$347$	3274.08			0.506518			0.253259	−	0.967399i	$-0.418498\pi$
$347$	3274.08			0.506518			0.253259	+	0.967399i	$0.418498\pi$
$348$		0			0
$349$	2000.93			0.306898			0.153449	−	0.988157i	$-0.450962\pi$
$349$	2000.93			0.306898			0.153449	+	0.988157i	$0.450962\pi$
$350$		0			0
$351$	629.228	+	7932.13i	0.0956857	+	1.20623i
$352$		0			0
$353$		−	11333.7i		−	1.70888i	−0.519552	−	0.854439i	$-0.673901\pi$
$353$		−	11333.7i		−	1.70888i	0.519552	−	0.854439i	$-0.326099\pi$
$354$		0			0
$355$			1508.10i			0.225470i
$356$		0			0
$357$	8522.16	+	4624.87i	1.26342	+	0.685642i
$358$		0			0
$359$	−4485.62			−0.659449			−0.329724	−	0.944077i	$-0.606956\pi$
$359$	−4485.62			−0.659449			−0.329724	+	0.944077i	$0.606956\pi$
$360$		0			0
$361$	−926.562			−0.135087
$362$		0			0
$363$	1743.50	+	946.175i	0.252094	+	0.136808i
$364$		0			0
$365$			5579.19i			0.800077i
$366$		0			0
$367$			11115.5i			1.58099i	0.612469	+	0.790494i	$0.290176\pi$
$367$			11115.5i			1.58099i	−0.612469	+	0.790494i	$0.709824\pi$
$368$		0			0
$369$	1533.83	−	996.986i	0.216391	−	0.140653i
$370$		0			0
$371$	−5521.00			−0.772603
$372$		0			0
$373$	−2122.71			−0.294664			−0.147332	−	0.989087i	$-0.547069\pi$
$373$	−2122.71			−0.294664			−0.147332	+	0.989087i	$0.547069\pi$
$374$		0			0
$375$	−309.806	+	570.873i	−0.0426621	+	0.0786126i
$376$		0			0
$377$		−	9132.97i		−	1.24767i
$378$		0			0
$379$		−	10724.8i		−	1.45355i	−0.686874	−	0.726776i	$-0.741018\pi$
$379$		−	10724.8i		−	1.45355i	0.686874	−	0.726776i	$-0.258982\pi$
$380$		0			0
$381$	3986.03	−	7344.99i	0.535986	−	0.987651i
$382$		0			0
$383$	−1921.43			−0.256346			−0.128173	−	0.991752i	$-0.540911\pi$
$383$	−1921.43			−0.256346			−0.128173	+	0.991752i	$0.540911\pi$
$384$		0			0
$385$	3231.08			0.427718
$386$		0			0
$387$	−944.176	+	613.711i	−0.124019	+	0.0806116i
$388$		0			0
$389$		−	13145.4i		−	1.71336i	−0.515850	−	0.856679i	$-0.672524\pi$
$389$		−	13145.4i		−	1.71336i	0.515850	−	0.856679i	$-0.327476\pi$
$390$		0			0
$391$		−	12298.8i		−	1.59073i
$392$		0			0
$393$	6200.25	+	3364.80i	0.795831	+	0.431887i
$394$		0			0
$395$	3564.40			0.454037
$396$		0			0
$397$	−7760.16			−0.981036			−0.490518	−	0.871431i	$-0.663192\pi$
$397$	−7760.16			−0.981036			−0.490518	+	0.871431i	$0.663192\pi$
$398$		0			0
$399$	8452.11	+	4586.85i	1.06049	+	0.575514i
$400$		0			0
$401$		−	2039.82i		−	0.254024i	−0.991901	−	0.127012i	$-0.959461\pi$
$401$		−	2039.82i		−	0.254024i	0.991901	−	0.127012i	$-0.0405387\pi$
$402$		0			0
$403$			11188.2i			1.38293i
$404$		0			0
$405$	−3331.10	−	1479.80i	−0.408700	−	0.181560i
$406$		0			0
$407$	−5518.89			−0.672140
$408$		0			0
$409$	−5909.90			−0.714489			−0.357244	−	0.934011i	$-0.616284\pi$
$409$	−5909.90			−0.714489			−0.357244	+	0.934011i	$0.616284\pi$
$410$		0			0
$411$	−4173.84	+	7691.06i	−0.500926	+	0.923046i
$412$		0			0
$413$		−	2173.91i		−	0.259010i
$414$		0			0
$415$		−	1681.78i		−	0.198928i
$416$		0			0
$417$	−3533.16	+	6510.48i	−0.414915	+	0.764556i
$418$		0			0
$419$	12975.3			1.51285			0.756423	−	0.654083i	$-0.226945\pi$
$419$	12975.3			1.51285			0.756423	+	0.654083i	$0.226945\pi$
$420$		0			0
$421$	7858.36			0.909723			0.454861	−	0.890562i	$-0.349689\pi$
$421$	7858.36			0.909723			0.454861	+	0.890562i	$0.349689\pi$
$422$		0			0
$423$	−3149.16	−	4844.89i	−0.361980	−	0.556895i
$424$		0			0
$425$		−	2224.18i		−	0.253855i
$426$		0			0
$427$			14648.1i			1.66012i
$428$		0			0
$429$	−7980.37	−	4330.85i	−0.898125	−	0.487402i
$430$		0			0
$431$	8538.36			0.954242			0.477121	−	0.878838i	$-0.341680\pi$
$431$	8538.36			0.954242			0.477121	+	0.878838i	$0.341680\pi$
$432$		0			0
$433$	−6538.78			−0.725713			−0.362857	−	0.931845i	$-0.618199\pi$
$433$	−6538.78			−0.725713			−0.362857	+	0.931845i	$0.618199\pi$
$434$		0			0
$435$	3677.10	+	1995.52i	0.405295	+	0.219949i
$436$		0			0
$437$		−	12197.7i		−	1.33523i
$438$		0			0
$439$			12295.6i			1.33676i	0.743822	+	0.668378i	$0.233011\pi$
$439$			12295.6i			1.33676i	−0.743822	+	0.668378i	$0.766989\pi$
$440$		0			0
$441$	−1426.26	−	2194.25i	−0.154007	−	0.236935i
$442$		0			0
$443$	−12326.2			−1.32198			−0.660990	−	0.750395i	$-0.729864\pi$
$443$	−12326.2			−1.32198			−0.660990	+	0.750395i	$0.729864\pi$
$444$		0			0
$445$	−4851.04			−0.516768
$446$		0			0
$447$	−2166.22	+	3991.65i	−0.229214	+	0.422368i
$448$		0			0
$449$			4686.65i			0.492598i	0.969194	+	0.246299i	$0.0792146\pi$
$449$			4686.65i			0.492598i	−0.969194	+	0.246299i	$0.920785\pi$
$450$		0			0
$451$			2087.51i			0.217953i
$452$		0			0
$453$	−3328.61	+	6133.57i	−0.345236	+	0.636160i
$454$		0			0
$455$	5947.94			0.612844
$456$		0			0
$457$	5715.08			0.584989			0.292495	−	0.956267i	$-0.405515\pi$
$457$	5715.08			0.584989			0.292495	+	0.956267i	$0.405515\pi$
$458$		0			0
$459$	12442.7	−	987.032i	1.26530	−	0.100372i
$460$		0			0
$461$		−	6752.04i		−	0.682156i	−0.940035	−	0.341078i	$-0.889208\pi$
$461$		−	6752.04i		−	0.682156i	0.940035	−	0.341078i	$-0.110792\pi$
$462$		0			0
$463$			10853.9i			1.08946i	0.838610	+	0.544732i	$0.183369\pi$
$463$			10853.9i			1.08946i	−0.838610	+	0.544732i	$0.816631\pi$
$464$		0			0
$465$	−4504.55	−	2444.57i	−0.449234	−	0.243794i
$466$		0			0
$467$	5170.48			0.512337			0.256169	−	0.966632i	$-0.417540\pi$
$467$	5170.48			0.512337			0.256169	+	0.966632i	$0.417540\pi$
$468$		0			0
$469$	−2721.30			−0.267927
$470$		0			0
$471$	−5996.49	−	3254.22i	−0.586632	−	0.318358i
$472$		0			0
$473$		−	1285.00i		−	0.124914i
$474$		0			0
$475$		−	2205.90i		−	0.213081i
$476$		0			0
$477$	−5958.89	+	3873.25i	−0.571989	+	0.371791i
$478$		0			0
$479$	7707.91			0.735247			0.367623	−	0.929975i	$-0.380172\pi$
$479$	7707.91			0.735247			0.367623	+	0.929975i	$0.380172\pi$
$480$		0			0
$481$	−10159.4			−0.963058
$482$		0			0
$483$	−7186.26	+	13242.0i	−0.676990	+	1.24748i
$484$		0			0
$485$		−	8004.03i		−	0.749370i
$486$		0			0
$487$			6961.77i			0.647778i	0.946095	+	0.323889i	$0.104990\pi$
$487$			6961.77i			0.647778i	−0.946095	+	0.323889i	$0.895010\pi$
$488$		0			0
$489$	−4884.37	+	9000.33i	−0.451695	+	0.832329i
$490$		0			0
$491$	−5867.71			−0.539320			−0.269660	−	0.962956i	$-0.586911\pi$
$491$	−5867.71			−0.539320			−0.269660	+	0.962956i	$0.586911\pi$
$492$		0			0
$493$	−14326.4			−1.30878
$494$		0			0
$495$	3487.35	−	2266.77i	0.316656	−	0.205825i
$496$		0			0
$497$			6326.32i			0.570974i
$498$		0			0
$499$			5291.99i			0.474754i	0.971418	+	0.237377i	$0.0762876\pi$
$499$			5291.99i			0.474754i	−0.971418	+	0.237377i	$0.923712\pi$
$500$		0			0
$501$	5462.44	+	2964.40i	0.487113	+	0.264350i
$502$		0			0
$503$	−778.873			−0.0690422			−0.0345211	−	0.999404i	$-0.510991\pi$
$503$	−778.873			−0.0690422			−0.0345211	+	0.999404i	$0.510991\pi$
$504$		0			0
$505$	5368.12			0.473026
$506$		0			0
$507$	−4657.01	−	2527.30i	−0.407939	−	0.221383i
$508$		0			0
$509$			7612.60i			0.662913i	0.943471	+	0.331456i	$0.107540\pi$
$509$			7612.60i			0.662913i	−0.943471	+	0.331456i	$0.892460\pi$
$510$		0			0
$511$			23404.1i			2.02610i
$512$		0			0
$513$	12340.4	−	978.919i	1.06207	−	0.0842501i
$514$		0			0
$515$	8544.11			0.731065
$516$		0			0
$517$	6593.76			0.560916
$518$		0			0
$519$	8060.63	−	14853.2i	0.681739	−	1.25623i
$520$		0			0
$521$			4248.22i			0.357232i	0.983919	+	0.178616i	$0.0571620\pi$
$521$			4248.22i			0.357232i	−0.983919	+	0.178616i	$0.942838\pi$
$522$		0			0
$523$		−	11183.4i		−	0.935022i	−0.883987	−	0.467511i	$-0.845151\pi$
$523$		−	11183.4i		−	0.935022i	0.883987	−	0.467511i	$-0.154849\pi$
$524$		0			0
$525$	−1299.60	+	2394.75i	−0.108037	+	0.199077i
$526$		0			0
$527$	17550.2			1.45066
$528$		0			0
$529$	6943.23			0.570661
$530$		0			0
$531$	−1525.11	−	2346.33i	−0.124640	−	0.191756i
$532$		0			0
$533$			3842.79i			0.312288i
$534$		0			0
$535$		−	3939.68i		−	0.318369i
$536$		0			0
$537$	−20272.4	−	11001.6i	−1.62908	−	0.884084i
$538$		0			0
$539$	2986.32			0.238645
$540$		0			0
$541$	−15537.3			−1.23475			−0.617377	−	0.786667i	$-0.711805\pi$
$541$	−15537.3			−1.23475			−0.617377	+	0.786667i	$0.711805\pi$
$542$		0			0
$543$	−271.666	−	147.430i	−0.0214702	−	0.0116516i
$544$		0			0
$545$		−	6002.05i		−	0.471742i
$546$		0			0
$547$		−	2188.27i		−	0.171049i	−0.996336	−	0.0855244i	$-0.972743\pi$
$547$		−	2188.27i		−	0.171049i	0.996336	−	0.0855244i	$-0.0272565\pi$
$548$		0			0
$549$	10276.3	+	15809.9i	0.798877	+	1.22905i
$550$		0			0
$551$	−14208.6			−1.09856
$552$		0			0
$553$	14952.3			1.14979
$554$		0			0
$555$	2219.80	−	4090.37i	0.169775	−	0.312841i
$556$		0			0
$557$		−	4424.18i		−	0.336550i	−0.985740	−	0.168275i	$-0.946180\pi$
$557$		−	4424.18i		−	0.336550i	0.985740	−	0.168275i	$-0.0538197\pi$
$558$		0			0
$559$		−	2365.49i		−	0.178979i
$560$		0			0
$561$	−6793.55	+	12518.3i	−0.511272	+	0.942111i
$562$		0			0
$563$	24100.3			1.80409			0.902047	−	0.431637i	$-0.142064\pi$
$563$	24100.3			1.80409			0.902047	+	0.431637i	$0.142064\pi$
$564$		0			0
$565$	2435.24			0.181330
$566$		0			0
$567$	−13973.6	−	6207.59i	−1.03498	−	0.459778i
$568$		0			0
$569$		−	5921.64i		−	0.436289i	−0.975917	−	0.218144i	$-0.930000\pi$
$569$		−	5921.64i		−	0.436289i	0.975917	−	0.218144i	$-0.0700004\pi$
$570$		0			0
$571$		−	4120.98i		−	0.302027i	−0.988532	−	0.151014i	$-0.951746\pi$
$571$		−	4120.98i		−	0.302027i	0.988532	−	0.151014i	$-0.0482538\pi$
$572$		0			0
$573$	5458.77	+	2962.41i	0.397982	+	0.215980i
$574$		0			0
$575$	3455.99			0.250652
$576$		0			0
$577$	−14605.1			−1.05376			−0.526878	−	0.849941i	$-0.676637\pi$
$577$	−14605.1			−1.05376			−0.526878	+	0.849941i	$0.676637\pi$
$578$		0			0
$579$	13182.8	+	7154.16i	0.946217	+	0.513500i
$580$		0			0
$581$		−	7054.87i		−	0.503761i
$582$		0			0
$583$		−	8109.88i		−	0.576118i
$584$		0			0
$585$	6419.70	−	4172.78i	0.453712	−	0.294911i
$586$		0			0
$587$	−13164.3			−0.925637			−0.462818	−	0.886453i	$-0.653162\pi$
$587$	−13164.3			−0.925637			−0.462818	+	0.886453i	$0.653162\pi$
$588$		0			0
$589$	17405.9			1.21766
$590$		0			0
$591$	−1290.66	+	2378.27i	−0.0898318	+	0.165531i
$592$		0			0
$593$		−	21694.5i		−	1.50233i	−0.660112	−	0.751167i	$-0.729491\pi$
$593$		−	21694.5i		−	1.50233i	0.660112	−	0.751167i	$-0.270509\pi$
$594$		0			0
$595$		−	9330.19i		−	0.642858i
$596$		0			0
$597$	286.721	−	528.335i	0.0196561	−	0.0362199i
$598$		0			0
$599$	28836.2			1.96697			0.983486	−	0.180986i	$-0.0579289\pi$
$599$	28836.2			1.96697			0.983486	+	0.180986i	$0.0579289\pi$
$600$		0			0
$601$	−25101.6			−1.70369			−0.851844	−	0.523795i	$-0.824516\pi$
$601$	−25101.6			−1.70369			−0.851844	+	0.523795i	$0.824516\pi$
$602$		0			0
$603$	−2937.13	+	1909.12i	−0.198357	+	0.128931i
$604$		0			0
$605$		−	1908.81i		−	0.128271i
$606$		0			0
$607$			1138.66i			0.0761396i	0.999275	+	0.0380698i	$0.0121209\pi$
$607$			1138.66i			0.0761396i	−0.999275	+	0.0380698i	$0.987879\pi$
$608$		0			0
$609$	15425.0	+	8370.97i	1.02636	+	0.556993i
$610$		0			0
$611$	12138.1			0.803693
$612$		0			0
$613$	−3569.26			−0.235173			−0.117586	−	0.993063i	$-0.537516\pi$
$613$	−3569.26			−0.235173			−0.117586	+	0.993063i	$0.537516\pi$
$614$		0			0
$615$	−1547.17	−	839.633i	−0.101444	−	0.0550524i
$616$		0			0
$617$		−	11344.1i		−	0.740190i	−0.928994	−	0.370095i	$-0.879325\pi$
$617$		−	11344.1i		−	0.740190i	0.928994	−	0.370095i	$-0.120675\pi$
$618$		0			0
$619$		−	2452.32i		−	0.159236i	−0.996825	−	0.0796180i	$-0.974630\pi$
$619$		−	2452.32i		−	0.159236i	0.996825	−	0.0796180i	$-0.0253701\pi$
$620$		0			0
$621$	1533.68	+	19333.8i	0.0991054	+	1.24934i
$622$		0			0
$623$	−20349.6			−1.30865
$624$		0			0
$625$	625.000			0.0400000
$626$		0			0
$627$	−6737.70	+	12415.4i	−0.429151	+	0.790789i
$628$		0			0
$629$			15936.5i			1.01022i
$630$		0			0
$631$			6181.89i			0.390011i	0.980802	+	0.195006i	$0.0624725\pi$
$631$			6181.89i			0.390011i	−0.980802	+	0.195006i	$0.937527\pi$
$632$		0			0
$633$	1663.64	−	3065.55i	0.104461	−	0.192488i
$634$		0			0
$635$	−8041.40			−0.502541
$636$		0			0
$637$	5497.37			0.341937
$638$		0			0
$639$	4438.23	+	6828.08i	0.274763	+	0.422715i
$640$		0			0
$641$			15093.8i			0.930064i	0.885294	+	0.465032i	$0.153957\pi$
$641$			15093.8i			0.930064i	−0.885294	+	0.465032i	$0.846043\pi$
$642$		0			0
$643$			30026.8i			1.84159i	0.390049	+	0.920794i	$0.372458\pi$
$643$			30026.8i			1.84159i	−0.390049	+	0.920794i	$0.627542\pi$
$644$		0			0
$645$	952.388	+	516.849i	0.0581399	+	0.0315518i
$646$		0			0
$647$	−11520.7			−0.700038			−0.350019	−	0.936743i	$-0.613825\pi$
$647$	−11520.7			−0.700038			−0.350019	+	0.936743i	$0.613825\pi$
$648$		0			0
$649$	3193.29			0.193140
$650$		0			0
$651$	−18896.1	−	10254.7i	−1.13763	−	0.617378i
$652$		0			0
$653$		−	18334.9i		−	1.09877i	−0.835568	−	0.549386i	$-0.814862\pi$
$653$		−	18334.9i		−	1.09877i	0.835568	−	0.549386i	$-0.185138\pi$
$654$		0			0
$655$		−	6788.13i		−	0.404938i
$656$		0			0
$657$	16419.1	+	25260.4i	0.974995	+	1.50000i
$658$		0			0
$659$	−14497.0			−0.856940			−0.428470	−	0.903556i	$-0.640947\pi$
$659$	−14497.0			−0.856940			−0.428470	+	0.903556i	$0.640947\pi$
$660$		0			0
$661$	9312.87			0.548001			0.274000	−	0.961730i	$-0.411653\pi$
$661$	9312.87			0.548001			0.274000	+	0.961730i	$0.411653\pi$
$662$		0			0
$663$	−12505.9	+	23044.4i	−0.732563	+	1.34988i
$664$		0			0
$665$		−	9253.49i		−	0.539602i
$666$		0			0
$667$		−	22260.7i		−	1.29226i
$668$		0			0
$669$	7466.27	−	13757.9i	0.431484	−	0.795086i
$670$		0			0
$671$	−21516.8			−1.23792
$672$		0			0
$673$	17538.1			1.00453			0.502263	−	0.864715i	$-0.332501\pi$
$673$	17538.1			1.00453			0.502263	+	0.864715i	$0.332501\pi$
$674$		0			0
$675$	277.359	+	3496.42i	0.0158156	+	0.199374i
$676$		0			0
$677$			4502.20i			0.255589i	0.991801	+	0.127794i	$0.0407898\pi$
$677$			4502.20i			0.255589i	−0.991801	+	0.127794i	$0.959210\pi$
$678$		0			0
$679$		−	33576.0i		−	1.89769i
$680$		0			0
$681$	14188.3	+	7699.83i	0.798381	+	0.433272i
$682$		0			0
$683$	−20582.8			−1.15312			−0.576558	−	0.817056i	$-0.695604\pi$
$683$	−20582.8			−1.15312			−0.576558	+	0.817056i	$0.695604\pi$
$684$		0			0
$685$	8420.29			0.469668
$686$		0			0
$687$	2215.25	+	1202.19i	0.123023	+	0.0667632i
$688$		0			0
$689$		−	14929.1i		−	0.825475i
$690$		0			0
$691$			20043.8i			1.10348i	0.834018	+	0.551738i	$0.186035\pi$
$691$			20043.8i			1.10348i	−0.834018	+	0.551738i	$0.813965\pi$
$692$		0			0
$693$	14629.1	−	9508.84i	0.801894	−	0.521228i
$694$		0			0
$695$	7127.77			0.389024
$696$		0			0
$697$	6027.95			0.327582
$698$		0			0
$699$	15001.6	−	27643.1i	0.811747	−	1.49579i
$700$		0			0
$701$			3118.89i			0.168044i	0.996464	+	0.0840221i	$0.0267766\pi$
$701$			3118.89i			0.168044i	−0.996464	+	0.0840221i	$0.973223\pi$
$702$		0			0
$703$			15805.5i			0.847961i
$704$		0			0
$705$	−2652.13	+	4887.03i	−0.141681	+	0.261073i
$706$		0			0
$707$	22518.7			1.19788
$708$		0			0
$709$	−9275.46			−0.491322			−0.245661	−	0.969356i	$-0.579005\pi$
$709$	−9275.46			−0.491322			−0.245661	+	0.969356i	$0.579005\pi$
$710$		0			0
$711$	16138.2	−	10489.8i	0.851237	−	0.553301i
$712$		0			0
$713$			27270.0i			1.43236i
$714$		0			0
$715$			8737.03i			0.456988i
$716$		0			0
$717$	−22132.4	−	12011.0i	−1.15279	−	0.625603i
$718$		0			0
$719$	18100.1			0.938829			0.469415	−	0.882978i	$-0.344465\pi$
$719$	18100.1			0.938829			0.469415	+	0.882978i	$0.344465\pi$
$720$		0			0
$721$	35841.6			1.85133
$722$		0			0
$723$	−29228.0	−	15861.7i	−1.50346	−	0.815909i
$724$		0			0
$725$		−	4025.74i		−	0.206224i
$726$		0			0
$727$		−	30725.5i		−	1.56746i	−0.621101	−	0.783731i	$-0.713314\pi$
$727$		−	30725.5i		−	1.56746i	0.621101	−	0.783731i	$-0.286686\pi$
$728$		0			0
$729$	−19436.8	+	3103.24i	−0.987493	+	0.157661i
$730$		0			0
$731$	−3710.60			−0.187745
$732$		0			0
$733$	20937.2			1.05502			0.527512	−	0.849548i	$-0.323125\pi$
$733$	20937.2			1.05502			0.527512	+	0.849548i	$0.323125\pi$
$734$		0			0
$735$	−1201.15	+	2213.34i	−0.0602791	+	0.111075i
$736$		0			0
$737$		−	3997.35i		−	0.199789i
$738$		0			0
$739$		−	22413.0i		−	1.11566i	−0.829954	−	0.557831i	$-0.811634\pi$
$739$		−	22413.0i		−	1.11566i	0.829954	−	0.557831i	$-0.188366\pi$
$740$		0			0
$741$	−12403.1	+	22855.0i	−0.614898	+	1.13306i
$742$		0			0
$743$	−34862.0			−1.72135			−0.860674	−	0.509156i	$-0.829958\pi$
$743$	−34862.0			−1.72135			−0.860674	+	0.509156i	$0.829958\pi$
$744$		0			0
$745$	4370.12			0.214911
$746$		0			0
$747$	−4949.34	−	7614.42i	−0.242419	−	0.372955i
$748$		0			0
$749$		−	16526.5i		−	0.806230i
$750$		0			0
$751$		−	3994.41i		−	0.194085i	−0.995280	−	0.0970427i	$-0.969062\pi$
$751$		−	3994.41i		−	0.194085i	0.995280	−	0.0970427i	$-0.0309383\pi$
$752$		0			0
$753$	181.181	+	98.3249i	0.00876842	+	0.00475851i
$754$		0			0
$755$	6715.13			0.323693
$756$		0			0
$757$	−2502.29			−0.120142			−0.0600709	−	0.998194i	$-0.519133\pi$
$757$	−2502.29			−0.120142			−0.0600709	+	0.998194i	$0.519133\pi$
$758$		0			0
$759$	−19451.4	−	10556.0i	−0.930223	−	0.504821i
$760$		0			0
$761$		−	5708.61i		−	0.271928i	−0.990714	−	0.135964i	$-0.956587\pi$
$761$		−	5708.61i		−	0.271928i	0.990714	−	0.135964i	$-0.0434131\pi$
$762$		0			0
$763$		−	25178.0i		−	1.19463i
$764$		0			0
$765$	−6545.59	−	10070.2i	−0.309355	−	0.475933i
$766$		0			0
$767$	5878.38			0.276735
$768$		0			0
$769$	8892.22			0.416985			0.208493	−	0.978024i	$-0.433144\pi$
$769$	8892.22			0.416985			0.208493	+	0.978024i	$0.433144\pi$
$770$		0			0
$771$	460.347	−	848.272i	0.0215032	−	0.0396236i
$772$		0			0
$773$		−	6494.04i		−	0.302166i	−0.988521	−	0.151083i	$-0.951724\pi$
$773$		−	6494.04i		−	0.302166i	0.988521	−	0.151083i	$-0.0482761\pi$
$774$		0			0
$775$			4931.65i			0.228581i
$776$		0			0
$777$	9311.80	−	17158.7i	0.429935	−	0.792232i
$778$		0			0
$779$	5978.40			0.274966
$780$		0			0
$781$	−9292.83			−0.425766
$782$		0			0
$783$	22521.1	−	1786.52i	1.02789	−	0.0815389i
$784$		0			0
$785$			6565.05i			0.298493i
$786$		0			0
$787$		−	8799.08i		−	0.398543i	−0.979944	−	0.199272i	$-0.936142\pi$
$787$		−	8799.08i		−	0.398543i	0.979944	−	0.199272i	$-0.0638576\pi$
$788$		0			0
$789$	6905.56	+	3747.56i	0.311590	+	0.169096i
$790$		0			0
$791$	10215.6			0.459195
$792$		0			0
$793$	−39609.2			−1.77372
$794$		0			0
$795$	6010.71	+	3261.94i	0.268148	+	0.145521i
$796$		0			0
$797$			32892.0i			1.46185i	0.682459	+	0.730924i	$0.260911\pi$
$797$			32892.0i			1.46185i	−0.682459	+	0.730924i	$0.739089\pi$
$798$		0			0
$799$		−	19040.4i		−	0.843054i
$800$		0			0
$801$	−21963.6	+	14276.3i	−0.968846	+	0.629746i
$802$		0			0
$803$	−34378.6			−1.51083
$804$		0			0
$805$	14497.5			0.634746
$806$		0			0
$807$	−7314.35	+	13478.0i	−0.319055	+	0.587916i
$808$		0			0
$809$			40905.7i			1.77771i	0.458187	+	0.888856i	$0.348499\pi$
$809$			40905.7i			1.77771i	−0.458187	+	0.888856i	$0.651501\pi$
$810$		0			0
$811$			30156.7i			1.30573i	0.757475	+	0.652865i	$0.226433\pi$
$811$			30156.7i			1.30573i	−0.757475	+	0.652865i	$0.773567\pi$
$812$		0			0
$813$	2239.26	−	4126.24i	0.0965982	−	0.178000i
$814$		0			0
$815$	9853.69			0.423509
$816$		0			0
$817$	−3680.10			−0.157589
$818$		0			0
$819$	26929.9	−	17504.4i	1.14897	−	0.746827i
$820$		0			0
$821$			10559.2i			0.448867i	0.974489	+	0.224433i	$0.0720531\pi$
$821$			10559.2i			0.448867i	−0.974489	+	0.224433i	$0.927947\pi$
$822$		0			0
$823$		−	31067.1i		−	1.31583i	−0.753091	−	0.657917i	$-0.771438\pi$
$823$		−	31067.1i		−	1.31583i	0.753091	−	0.657917i	$-0.228562\pi$
$824$		0			0
$825$	−3517.68	−	1909.00i	−0.148449	−	0.0805612i
$826$		0			0
$827$	−5421.39			−0.227957			−0.113978	−	0.993483i	$-0.536359\pi$
$827$	−5421.39			−0.227957			−0.113978	+	0.993483i	$0.536359\pi$
$828$		0			0
$829$	−38426.3			−1.60989			−0.804946	−	0.593348i	$-0.797806\pi$
$829$	−38426.3			−1.60989			−0.804946	+	0.593348i	$0.797806\pi$
$830$		0			0
$831$	−9678.23	−	5252.26i	−0.404012	−	0.219253i
$832$		0			0
$833$		−	8623.40i		−	0.358683i
$834$		0			0
$835$		−	5980.36i		−	0.247855i
$836$		0			0
$837$	−27589.0	+	2188.54i	−1.13933	+	0.0903787i
$838$		0			0
$839$	14422.0			0.593448			0.296724	−	0.954963i	$-0.404106\pi$
$839$	14422.0			0.593448			0.296724	+	0.954963i	$0.404106\pi$
$840$		0			0
$841$	−1541.54			−0.0632066
$842$		0			0
$843$	−12136.5	+	22363.8i	−0.495854	+	0.913700i
$844$		0			0
$845$			5098.56i			0.207569i
$846$		0			0
$847$		−	8007.25i		−	0.324831i
$848$		0			0
$849$	12084.8	−	22268.5i	0.488516	−	0.900179i
$850$		0			0
$851$	−24762.6			−0.997476
$852$		0			0
$853$	17955.9			0.720750			0.360375	−	0.932807i	$-0.382649\pi$
$853$	17955.9			0.720750			0.360375	+	0.932807i	$0.382649\pi$
$854$		0			0
$855$	−6491.79	−	9987.42i	−0.259666	−	0.399488i
$856$		0			0
$857$			26611.0i			1.06070i	0.847780	+	0.530348i	$0.177939\pi$
$857$			26611.0i			1.06070i	−0.847780	+	0.530348i	$0.822061\pi$
$858$		0			0
$859$		−	19264.8i		−	0.765201i	−0.923914	−	0.382601i	$-0.875028\pi$
$859$		−	19264.8i		−	0.765201i	0.923914	−	0.382601i	$-0.124972\pi$
$860$		0			0
$861$	−6490.23	−	3522.17i	−0.256895	−	0.139414i
$862$		0			0
$863$	−47910.6			−1.88980			−0.944899	−	0.327363i	$-0.893840\pi$
$863$	−47910.6			−1.88980			−0.944899	+	0.327363i	$0.893840\pi$
$864$		0			0
$865$	−16261.5			−0.639198
$866$		0			0
$867$	13710.8	+	7440.68i	0.537074	+	0.291463i
$868$		0			0
$869$			21963.6i			0.857382i
$870$		0			0
$871$		−	7358.53i		−	0.286262i
$872$		0			0
$873$	−23555.3	−	36239.1i	−0.913202	−	1.40493i
$874$		0			0
$875$	2621.81			0.101295
$876$		0			0
$877$	38173.3			1.46981			0.734904	−	0.678171i	$-0.237227\pi$
$877$	38173.3			1.46981			0.734904	+	0.678171i	$0.237227\pi$
$878$		0			0
$879$	−11214.2	+	20664.1i	−0.430312	+	0.792928i
$880$		0			0
$881$		−	31583.6i		−	1.20781i	−0.797057	−	0.603904i	$-0.793611\pi$
$881$		−	31583.6i		−	1.20781i	0.797057	−	0.603904i	$-0.206389\pi$
$882$		0			0
$883$			3771.86i			0.143752i	0.997414	+	0.0718762i	$0.0228986\pi$
$883$			3771.86i			0.143752i	−0.997414	+	0.0718762i	$0.977101\pi$
$884$		0			0
$885$	−1284.40	+	2366.74i	−0.0487849	+	0.0898950i
$886$		0			0
$887$	−27483.2			−1.04036			−0.520178	−	0.854058i	$-0.674134\pi$
$887$	−27483.2			−1.04036			−0.520178	+	0.854058i	$0.674134\pi$
$888$		0			0
$889$	−33732.8			−1.27262
$890$		0			0
$891$	9118.43	−	20526.0i	0.342849	−	0.771771i
$892$		0			0
$893$		−	18883.9i		−	0.707642i
$894$		0			0
$895$			22194.5i			0.828917i
$896$		0			0
$897$	−35807.0	−	19432.0i	−1.33284	−	0.723319i
$898$		0			0
$899$	31765.7			1.17847
$900$		0			0
$901$	−23418.4			−0.865903
$902$		0			0
$903$	3995.16	+	2168.13i	0.147232	+	0.0799011i
$904$		0			0
$905$			297.425i			0.0109246i
$906$		0			0
$907$		−	28400.8i		−	1.03973i	−0.854249	−	0.519864i	$-0.825983\pi$
$907$		−	28400.8i		−	1.03973i	0.854249	−	0.519864i	$-0.174017\pi$
$908$		0			0
$909$	24304.7	−	15798.0i	0.886839	−	0.576442i
$910$		0			0
$911$	−35770.7			−1.30092			−0.650458	−	0.759542i	$-0.725423\pi$
$911$	−35770.7			−1.30092			−0.650458	+	0.759542i	$0.725423\pi$
$912$		0			0
$913$	10363.0			0.375647
$914$		0			0
$915$	8654.43	−	15947.4i	0.312685	−	0.576178i
$916$		0			0
$917$		−	28475.5i		−	1.02546i
$918$		0			0
$919$		−	36103.6i		−	1.29592i	−0.761675	−	0.647959i	$-0.775623\pi$
$919$		−	36103.6i		−	1.29592i	0.761675	−	0.647959i	$-0.224377\pi$
$920$		0			0
$921$	16670.9	−	30719.1i	0.596444	−	1.09905i
$922$		0			0
$923$	−17106.7			−0.610048
$924$		0			0
$925$	−4478.20			−0.159181
$926$		0			0
$927$	38684.3	−	25144.7i	1.37062	−	0.890895i
$928$		0			0
$929$		−	38275.6i		−	1.35176i	−0.737014	−	0.675878i	$-0.763765\pi$
$929$		−	38275.6i		−	1.35176i	0.737014	−	0.675878i	$-0.236235\pi$
$930$		0			0
$931$		−	8552.51i		−	0.301071i
$932$		0			0
$933$	756.003	+	410.274i	0.0265278	+	0.0143963i
$934$		0			0
$935$	13705.3			0.479369
$936$		0			0
$937$	30657.8			1.06889			0.534443	−	0.845205i	$-0.320521\pi$
$937$	30657.8			1.06889			0.534443	+	0.845205i	$0.320521\pi$
$938$		0			0
$939$	15052.1	+	8168.61i	0.523118	+	0.283890i
$940$		0			0
$941$			24896.8i			0.862498i	0.902233	+	0.431249i	$0.141927\pi$
$941$			24896.8i			0.862498i	−0.902233	+	0.431249i	$0.858073\pi$
$942$		0			0
$943$			9366.41i			0.323449i
$944$		0			0
$945$	1163.49	+	14667.1i	0.0400511	+	0.504890i
$946$		0			0
$947$	−53361.2			−1.83105			−0.915526	−	0.402259i	$-0.868225\pi$
$947$	−53361.2			−1.83105			−0.915526	+	0.402259i	$0.868225\pi$
$948$		0			0
$949$	−63285.9			−2.16475
$950$		0			0
$951$	407.646	−	751.162i	0.0138999	−	0.0256131i
$952$		0			0
$953$		−	25939.5i		−	0.881703i	−0.897580	−	0.440851i	$-0.854677\pi$
$953$		−	25939.5i		−	0.881703i	0.897580	−	0.440851i	$-0.145323\pi$
$954$		0			0
$955$		−	5976.35i		−	0.202503i
$956$		0			0
$957$	−12296.2	+	22658.1i	−0.415341	+	0.765341i
$958$		0			0
$959$	35322.2			1.18938
$960$		0			0
$961$	−9122.93			−0.306231
$962$		0			0
$963$	−11594.2	−	17837.3i	−0.387972	−	0.596884i
$964$		0			0
$965$		−	14432.8i		−	0.481458i
$966$		0			0
$967$			24435.6i			0.812612i	0.913737	+	0.406306i	$0.133183\pi$
$967$			24435.6i			0.812612i	−0.913737	+	0.406306i	$0.866817\pi$
$968$		0			0
$969$	35851.2	+	19456.0i	1.18855	+	0.645013i
$970$		0			0
$971$	−21477.2			−0.709822			−0.354911	−	0.934900i	$-0.615489\pi$
$971$	−21477.2			−0.709822			−0.354911	+	0.934900i	$0.615489\pi$
$972$		0			0
$973$	29900.2			0.985157
$974$		0			0
$975$	−6475.53	−	3514.19i	−0.212700	−	0.115430i
$976$		0			0
$977$			33734.4i			1.10467i	0.833623	+	0.552333i	$0.186262\pi$
$977$			33734.4i			1.10467i	−0.833623	+	0.552333i	$0.813738\pi$
$978$		0			0
$979$		−	29891.9i		−	0.975841i
$980$		0			0
$981$	−17663.6	−	27174.9i	−0.574878	−	0.884432i
$982$		0			0
$983$	9151.98			0.296951			0.148475	−	0.988916i	$-0.452563\pi$
$983$	9151.98			0.296951			0.148475	+	0.988916i	$0.452563\pi$
$984$		0			0
$985$	2603.77			0.0842263
$986$		0			0
$987$	−11125.4	+	20500.6i	−0.358790	+	0.661135i
$988$		0			0
$989$		−	5765.64i		−	0.185376i
$990$		0			0
$991$			11823.3i			0.378990i	0.981882	+	0.189495i	$0.0606850\pi$
$991$			11823.3i			0.378990i	−0.981882	+	0.189495i	$0.939315\pi$
$992$		0			0
$993$	−8322.70	+	15336.1i	−0.265975	+	0.490107i
$994$		0			0
$995$	−578.429			−0.0184296
$996$		0			0
$997$	10234.6			0.325108			0.162554	−	0.986700i	$-0.448027\pi$
$997$	10234.6			0.325108			0.162554	+	0.986700i	$0.448027\pi$
$998$		0			0
$999$	−1987.31	−	25052.3i	−0.0629386	−	0.793413i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.4.h.d.191.5		24
3.2	odd	2	inner	960.4.h.d.191.19		24
4.3	odd	2	inner	960.4.h.d.191.20		24
8.3	odd	2		60.4.e.a.11.21	yes	24
8.5	even	2		60.4.e.a.11.3	✓	24
12.11	even	2	inner	960.4.h.d.191.6		24
24.5	odd	2		60.4.e.a.11.22	yes	24
24.11	even	2		60.4.e.a.11.4	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.e.a.11.3	✓	24	8.5	even	2
60.4.e.a.11.4	yes	24	24.11	even	2
60.4.e.a.11.21	yes	24	8.3	odd	2
60.4.e.a.11.22	yes	24	24.5	odd	2
960.4.h.d.191.5		24	1.1	even	1	trivial
960.4.h.d.191.6		24	12.11	even	2	inner
960.4.h.d.191.19		24	3.2	odd	2	inner
960.4.h.d.191.20		24	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	960.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.56698 - 2.47844i) q^{3} +5.00000i q^{5} +20.9745i q^{7} +(14.7146 + 22.6380i) q^{9} +O(q^{10})\)	\(q+(-4.56698 - 2.47844i) q^{3} +5.00000i q^{5} +20.9745i q^{7} +(14.7146 + 22.6380i) q^{9} -30.8097 q^{11} -56.7161 q^{13} +(12.3922 - 22.8349i) q^{15} +88.9672i q^{17} +88.2358i q^{19} +(51.9840 - 95.7899i) q^{21} -138.240 q^{23} -25.0000 q^{25} +(-11.0943 - 139.857i) q^{27} +161.030i q^{29} -197.266i q^{31} +(140.707 + 76.3602i) q^{33} -104.872 q^{35} +179.128 q^{37} +(259.021 + 140.568i) q^{39} -67.7548i q^{41} +41.7076i q^{43} +(-113.190 + 73.5731i) q^{45} -214.016 q^{47} -96.9279 q^{49} +(220.500 - 406.311i) q^{51} +263.225i q^{53} -154.049i q^{55} +(218.688 - 402.971i) q^{57} -103.646 q^{59} +698.376 q^{61} +(-474.820 + 308.631i) q^{63} -283.580i q^{65} +129.743i q^{67} +(631.338 + 342.620i) q^{69} +301.620 q^{71} +1115.84 q^{73} +(114.175 + 61.9611i) q^{75} -646.217i q^{77} -712.880i q^{79} +(-295.960 + 666.220i) q^{81} -336.355 q^{83} -444.836 q^{85} +(399.103 - 735.419i) q^{87} +970.209i q^{89} -1189.59i q^{91} +(-488.913 + 900.911i) q^{93} -441.179 q^{95} -1600.81 q^{97} +(-453.353 - 697.471i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 20 q^{9}+O(q^{10}) \) 24 * q + 20 * q^9	\( 24 q + 20 q^{9} - 72 q^{13} + 68 q^{21} - 600 q^{25} + 848 q^{33} - 504 q^{37} + 220 q^{45} - 2256 q^{49} + 1416 q^{57} - 1992 q^{61} + 1548 q^{69} - 2304 q^{73} + 3840 q^{81} - 240 q^{85} + 4384 q^{93} - 2448 q^{97}+O(q^{100}) \) 24 * q + 20 * q^9 - 72 * q^13 + 68 * q^21 - 600 * q^25 + 848 * q^33 - 504 * q^37 + 220 * q^45 - 2256 * q^49 + 1416 * q^57 - 1992 * q^61 + 1548 * q^69 - 2304 * q^73 + 3840 * q^81 - 240 * q^85 + 4384 * q^93 - 2448 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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