LMFDB - Embedded newform 960.4.f.h.769.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,4,Mod(769,960)]

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, 0, 1]))

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

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

chi := DirichletCharacter("960.769");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	960.f (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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	960.769
Dual form			960.4.f.h.769.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +(10.0000 + 5.00000i) q^{5} -18.0000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} +(10.0000 + 5.00000i) q^{5} -18.0000i q^{7} -9.00000 q^{9} -30.0000 q^{11} +38.0000i q^{13} +(-15.0000 + 30.0000i) q^{15} -70.0000i q^{17} +12.0000 q^{19} +54.0000 q^{21} -72.0000i q^{23} +(75.0000 + 100.000i) q^{25} -27.0000i q^{27} -64.0000 q^{29} -312.000 q^{31} -90.0000i q^{33} +(90.0000 - 180.000i) q^{35} -138.000i q^{37} -114.000 q^{39} -374.000 q^{41} -468.000i q^{43} +(-90.0000 - 45.0000i) q^{45} +132.000i q^{47} +19.0000 q^{49} +210.000 q^{51} +446.000i q^{53} +(-300.000 - 150.000i) q^{55} +36.0000i q^{57} +510.000 q^{59} -754.000 q^{61} +162.000i q^{63} +(-190.000 + 380.000i) q^{65} -384.000i q^{67} +216.000 q^{69} -924.000 q^{71} +340.000i q^{73} +(-300.000 + 225.000i) q^{75} +540.000i q^{77} -72.0000 q^{79} +81.0000 q^{81} -156.000i q^{83} +(350.000 - 700.000i) q^{85} -192.000i q^{87} +290.000 q^{89} +684.000 q^{91} -936.000i q^{93} +(120.000 + 60.0000i) q^{95} -376.000i q^{97} +270.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 20 q^{5} - 18 q^{9}+O(q^{10}) $ 2 * q + 20 * q^5 - 18 * q^9	$ 2 q + 20 q^{5} - 18 q^{9} - 60 q^{11} - 30 q^{15} + 24 q^{19} + 108 q^{21} + 150 q^{25} - 128 q^{29} - 624 q^{31} + 180 q^{35} - 228 q^{39} - 748 q^{41} - 180 q^{45} + 38 q^{49} + 420 q^{51} - 600 q^{55} + 1020 q^{59} - 1508 q^{61} - 380 q^{65} + 432 q^{69} - 1848 q^{71} - 600 q^{75} - 144 q^{79} + 162 q^{81} + 700 q^{85} + 580 q^{89} + 1368 q^{91} + 240 q^{95} + 540 q^{99}+O(q^{100}) $ 2 * q + 20 * q^5 - 18 * q^9 - 60 * q^11 - 30 * q^15 + 24 * q^19 + 108 * q^21 + 150 * q^25 - 128 * q^29 - 624 * q^31 + 180 * q^35 - 228 * q^39 - 748 * q^41 - 180 * q^45 + 38 * q^49 + 420 * q^51 - 600 * q^55 + 1020 * q^59 - 1508 * q^61 - 380 * q^65 + 432 * q^69 - 1848 * q^71 - 600 * q^75 - 144 * q^79 + 162 * q^81 + 700 * q^85 + 580 * q^89 + 1368 * q^91 + 240 * q^95 + 540 * q^99

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
$3$			3.00000i			0.577350i
$3$			3.00000i			0.577350i
$4$		0			0
$5$	10.0000	+	5.00000i	0.894427	+	0.447214i
$5$	10.0000	+	5.00000i	0.894427	+	0.447214i
$6$		0			0
$7$		−	18.0000i		−	0.971909i	−0.873984	−	0.485954i	$-0.838472\pi$
$7$		−	18.0000i		−	0.971909i	0.873984	−	0.485954i	$-0.161528\pi$
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	−30.0000			−0.822304			−0.411152	−	0.911567i	$-0.634873\pi$
$11$	−30.0000			−0.822304			−0.411152	+	0.911567i	$0.634873\pi$
$12$		0			0
$13$			38.0000i			0.810716i	0.914158	+	0.405358i	$0.132853\pi$
$13$			38.0000i			0.810716i	−0.914158	+	0.405358i	$0.867147\pi$
$14$		0			0
$15$	−15.0000	+	30.0000i	−0.258199	+	0.516398i
$16$		0			0
$17$		−	70.0000i		−	0.998676i	−0.866407	−	0.499338i	$-0.833577\pi$
$17$		−	70.0000i		−	0.998676i	0.866407	−	0.499338i	$-0.166423\pi$
$18$		0			0
$19$	12.0000			0.144894			0.0724471	−	0.997372i	$-0.476919\pi$
$19$	12.0000			0.144894			0.0724471	+	0.997372i	$0.476919\pi$
$20$		0			0
$21$	54.0000			0.561132
$22$		0			0
$23$		−	72.0000i		−	0.652741i	−0.945242	−	0.326370i	$-0.894174\pi$
$23$		−	72.0000i		−	0.652741i	0.945242	−	0.326370i	$-0.105826\pi$
$24$		0			0
$25$	75.0000	+	100.000i	0.600000	+	0.800000i
$26$		0			0
$27$		−	27.0000i		−	0.192450i
$28$		0			0
$29$	−64.0000			−0.409810			−0.204905	−	0.978782i	$-0.565689\pi$
$29$	−64.0000			−0.409810			−0.204905	+	0.978782i	$0.565689\pi$
$30$		0			0
$31$	−312.000			−1.80764			−0.903820	−	0.427912i	$-0.859249\pi$
$31$	−312.000			−1.80764			−0.903820	+	0.427912i	$0.859249\pi$
$32$		0			0
$33$		−	90.0000i		−	0.474757i
$34$		0			0
$35$	90.0000	−	180.000i	0.434651	−	0.869302i
$36$		0			0
$37$		−	138.000i		−	0.613164i	−0.951844	−	0.306582i	$-0.900815\pi$
$37$		−	138.000i		−	0.613164i	0.951844	−	0.306582i	$-0.0991853\pi$
$38$		0			0
$39$	−114.000			−0.468067
$40$		0			0
$41$	−374.000			−1.42461			−0.712305	−	0.701870i	$-0.752349\pi$
$41$	−374.000			−1.42461			−0.712305	+	0.701870i	$0.752349\pi$
$42$		0			0
$43$		−	468.000i		−	1.65975i	−0.557948	−	0.829876i	$-0.688411\pi$
$43$		−	468.000i		−	1.65975i	0.557948	−	0.829876i	$-0.311589\pi$
$44$		0			0
$45$	−90.0000	−	45.0000i	−0.298142	−	0.149071i
$46$		0			0
$47$			132.000i			0.409663i	0.978797	+	0.204832i	$0.0656647\pi$
$47$			132.000i			0.409663i	−0.978797	+	0.204832i	$0.934335\pi$
$48$		0			0
$49$	19.0000			0.0553936
$50$		0			0
$51$	210.000			0.576586
$52$		0			0
$53$			446.000i			1.15590i	0.816071	+	0.577951i	$0.196148\pi$
$53$			446.000i			1.15590i	−0.816071	+	0.577951i	$0.803852\pi$
$54$		0			0
$55$	−300.000	−	150.000i	−0.735491	−	0.367745i
$56$		0			0
$57$			36.0000i			0.0836547i
$58$		0			0
$59$	510.000			1.12536			0.562681	−	0.826674i	$-0.309770\pi$
$59$	510.000			1.12536			0.562681	+	0.826674i	$0.309770\pi$
$60$		0			0
$61$	−754.000			−1.58262			−0.791310	−	0.611415i	$-0.790601\pi$
$61$	−754.000			−1.58262			−0.791310	+	0.611415i	$0.790601\pi$
$62$		0			0
$63$			162.000i			0.323970i
$64$		0			0
$65$	−190.000	+	380.000i	−0.362563	+	0.725126i
$66$		0			0
$67$		−	384.000i		−	0.700195i	−0.936713	−	0.350098i	$-0.886148\pi$
$67$		−	384.000i		−	0.700195i	0.936713	−	0.350098i	$-0.113852\pi$
$68$		0			0
$69$	216.000			0.376860
$70$		0			0
$71$	−924.000			−1.54449			−0.772244	−	0.635326i	$-0.780866\pi$
$71$	−924.000			−1.54449			−0.772244	+	0.635326i	$0.780866\pi$
$72$		0			0
$73$			340.000i			0.545123i	0.962138	+	0.272562i	$0.0878708\pi$
$73$			340.000i			0.545123i	−0.962138	+	0.272562i	$0.912129\pi$
$74$		0			0
$75$	−300.000	+	225.000i	−0.461880	+	0.346410i
$76$		0			0
$77$			540.000i			0.799204i
$78$		0			0
$79$	−72.0000			−0.102540			−0.0512698	−	0.998685i	$-0.516327\pi$
$79$	−72.0000			−0.102540			−0.0512698	+	0.998685i	$0.516327\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$		−	156.000i		−	0.206304i	−0.994666	−	0.103152i	$-0.967107\pi$
$83$		−	156.000i		−	0.206304i	0.994666	−	0.103152i	$-0.0328928\pi$
$84$		0			0
$85$	350.000	−	700.000i	0.446622	−	0.893243i
$86$		0			0
$87$		−	192.000i		−	0.236604i
$88$		0			0
$89$	290.000			0.345393			0.172696	−	0.984975i	$-0.444752\pi$
$89$	290.000			0.345393			0.172696	+	0.984975i	$0.444752\pi$
$90$		0			0
$91$	684.000			0.787942
$92$		0			0
$93$		−	936.000i		−	1.04364i
$94$		0			0
$95$	120.000	+	60.0000i	0.129597	+	0.0647986i
$96$		0			0
$97$		−	376.000i		−	0.393577i	−0.980446	−	0.196789i	$-0.936949\pi$
$97$		−	376.000i		−	0.393577i	0.980446	−	0.196789i	$-0.0630513\pi$
$98$		0			0
$99$	270.000			0.274101
$100$		0			0
$101$	−104.000			−0.102459			−0.0512296	−	0.998687i	$-0.516314\pi$
$101$	−104.000			−0.102459			−0.0512296	+	0.998687i	$0.516314\pi$
$102$		0			0
$103$		−	1014.00i		−	0.970023i	−0.874508	−	0.485012i	$-0.838815\pi$
$103$		−	1014.00i		−	0.970023i	0.874508	−	0.485012i	$-0.161185\pi$
$104$		0			0
$105$	540.000	+	270.000i	0.501891	+	0.250946i
$106$		0			0
$107$		−	1284.00i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	1284.00i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$		0			0
$109$	570.000			0.500882			0.250441	−	0.968132i	$-0.419424\pi$
$109$	570.000			0.500882			0.250441	+	0.968132i	$0.419424\pi$
$110$		0			0
$111$	414.000			0.354010
$112$		0			0
$113$		−	254.000i		−	0.211454i	−0.994395	−	0.105727i	$-0.966283\pi$
$113$		−	254.000i		−	0.211454i	0.994395	−	0.105727i	$-0.0337170\pi$
$114$		0			0
$115$	360.000	−	720.000i	0.291915	−	0.583829i
$116$		0			0
$117$		−	342.000i		−	0.270239i
$118$		0			0
$119$	−1260.00			−0.970622
$120$		0			0
$121$	−431.000			−0.323817
$122$		0			0
$123$		−	1122.00i		−	0.822499i
$124$		0			0
$125$	250.000	+	1375.00i	0.178885	+	0.983870i
$126$		0			0
$127$		−	138.000i		−	0.0964214i	−0.998837	−	0.0482107i	$-0.984648\pi$
$127$		−	138.000i		−	0.0964214i	0.998837	−	0.0482107i	$-0.0153519\pi$
$128$		0			0
$129$	1404.00			0.958258
$130$		0			0
$131$	894.000			0.596253			0.298126	−	0.954526i	$-0.403638\pi$
$131$	894.000			0.596253			0.298126	+	0.954526i	$0.403638\pi$
$132$		0			0
$133$		−	216.000i		−	0.140824i
$134$		0			0
$135$	135.000	−	270.000i	0.0860663	−	0.172133i
$136$		0			0
$137$		−	3050.00i		−	1.90204i	−0.309134	−	0.951019i	$-0.600039\pi$
$137$		−	3050.00i		−	1.90204i	0.309134	−	0.951019i	$-0.399961\pi$
$138$		0			0
$139$	−2352.00			−1.43521			−0.717604	−	0.696451i	$-0.754761\pi$
$139$	−2352.00			−1.43521			−0.717604	+	0.696451i	$0.754761\pi$
$140$		0			0
$141$	−396.000			−0.236519
$142$		0			0
$143$		−	1140.00i		−	0.666654i
$144$		0			0
$145$	−640.000	−	320.000i	−0.366546	−	0.183273i
$146$		0			0
$147$			57.0000i			0.0319815i
$148$		0			0
$149$	2476.00			1.36135			0.680677	−	0.732583i	$-0.261686\pi$
$149$	2476.00			1.36135			0.680677	+	0.732583i	$0.261686\pi$
$150$		0			0
$151$	−3192.00			−1.72027			−0.860137	−	0.510064i	$-0.829622\pi$
$151$	−3192.00			−1.72027			−0.860137	+	0.510064i	$0.829622\pi$
$152$		0			0
$153$			630.000i			0.332892i
$154$		0			0
$155$	−3120.00	−	1560.00i	−1.61680	−	0.808401i
$156$		0			0
$157$			1218.00i			0.619153i	0.950875	+	0.309576i	$0.100187\pi$
$157$			1218.00i			0.619153i	−0.950875	+	0.309576i	$0.899813\pi$
$158$		0			0
$159$	−1338.00			−0.667360
$160$		0			0
$161$	−1296.00			−0.634404
$162$		0			0
$163$		−	2328.00i		−	1.11867i	−0.828942	−	0.559334i	$-0.811057\pi$
$163$		−	2328.00i		−	1.11867i	0.828942	−	0.559334i	$-0.188943\pi$
$164$		0			0
$165$	450.000	−	900.000i	0.212318	−	0.424636i
$166$		0			0
$167$		−	2928.00i		−	1.35674i	−0.734721	−	0.678370i	$-0.762687\pi$
$167$		−	2928.00i		−	1.35674i	0.734721	−	0.678370i	$-0.237313\pi$
$168$		0			0
$169$	753.000			0.342740
$170$		0			0
$171$	−108.000			−0.0482980
$172$		0			0
$173$		−	502.000i		−	0.220615i	−0.993898	−	0.110307i	$-0.964816\pi$
$173$		−	502.000i		−	0.220615i	0.993898	−	0.110307i	$-0.0351835\pi$
$174$		0			0
$175$	1800.00	−	1350.00i	0.777527	−	0.583145i
$176$		0			0
$177$			1530.00i			0.649728i
$178$		0			0
$179$	−4026.00			−1.68110			−0.840551	−	0.541732i	$-0.817769\pi$
$179$	−4026.00			−1.68110			−0.840551	+	0.541732i	$0.817769\pi$
$180$		0			0
$181$	−594.000			−0.243932			−0.121966	−	0.992534i	$-0.538920\pi$
$181$	−594.000			−0.243932			−0.121966	+	0.992534i	$0.538920\pi$
$182$		0			0
$183$		−	2262.00i		−	0.913726i
$184$		0			0
$185$	690.000	−	1380.00i	0.274215	−	0.548430i
$186$		0			0
$187$			2100.00i			0.821215i
$188$		0			0
$189$	−486.000			−0.187044
$190$		0			0
$191$	1044.00			0.395504			0.197752	−	0.980252i	$-0.436636\pi$
$191$	1044.00			0.395504			0.197752	+	0.980252i	$0.436636\pi$
$192$		0			0
$193$			3252.00i			1.21287i	0.795133	+	0.606435i	$0.207401\pi$
$193$			3252.00i			1.21287i	−0.795133	+	0.606435i	$0.792599\pi$
$194$		0			0
$195$	−1140.00	−	570.000i	−0.418652	−	0.209326i
$196$		0			0
$197$			2914.00i			1.05388i	0.849903	+	0.526939i	$0.176660\pi$
$197$			2914.00i			1.05388i	−0.849903	+	0.526939i	$0.823340\pi$
$198$		0			0
$199$	4872.00			1.73551			0.867756	−	0.496990i	$-0.165562\pi$
$199$	4872.00			1.73551			0.867756	+	0.496990i	$0.165562\pi$
$200$		0			0
$201$	1152.00			0.404258
$202$		0			0
$203$			1152.00i			0.398298i
$204$		0			0
$205$	−3740.00	−	1870.00i	−1.27421	−	0.637105i
$206$		0			0
$207$			648.000i			0.217580i
$208$		0			0
$209$	−360.000			−0.119147
$210$		0			0
$211$	−1872.00			−0.610776			−0.305388	−	0.952228i	$-0.598786\pi$
$211$	−1872.00			−0.610776			−0.305388	+	0.952228i	$0.598786\pi$
$212$		0			0
$213$		−	2772.00i		−	0.891710i
$214$		0			0
$215$	2340.00	−	4680.00i	0.742264	−	1.48453i
$216$		0			0
$217$			5616.00i			1.75686i
$218$		0			0
$219$	−1020.00			−0.314727
$220$		0			0
$221$	2660.00			0.809642
$222$		0			0
$223$		−	1302.00i		−	0.390979i	−0.980706	−	0.195490i	$-0.937370\pi$
$223$		−	1302.00i		−	0.390979i	0.980706	−	0.195490i	$-0.0626296\pi$
$224$		0			0
$225$	−675.000	−	900.000i	−0.200000	−	0.266667i
$226$		0			0
$227$			708.000i			0.207011i	0.994629	+	0.103506i	$0.0330060\pi$
$227$			708.000i			0.207011i	−0.994629	+	0.103506i	$0.966994\pi$
$228$		0			0
$229$	−5082.00			−1.46650			−0.733249	−	0.679960i	$-0.761997\pi$
$229$	−5082.00			−1.46650			−0.733249	+	0.679960i	$0.761997\pi$
$230$		0			0
$231$	−1620.00			−0.461421
$232$		0			0
$233$		−	5786.00i		−	1.62684i	−0.581678	−	0.813419i	$-0.697603\pi$
$233$		−	5786.00i		−	1.62684i	0.581678	−	0.813419i	$-0.302397\pi$
$234$		0			0
$235$	−660.000	+	1320.00i	−0.183207	+	0.366414i
$236$		0			0
$237$		−	216.000i		−	0.0592013i
$238$		0			0
$239$	3156.00			0.854162			0.427081	−	0.904213i	$-0.359542\pi$
$239$	3156.00			0.854162			0.427081	+	0.904213i	$0.359542\pi$
$240$		0			0
$241$	−3306.00			−0.883644			−0.441822	−	0.897103i	$-0.645668\pi$
$241$	−3306.00			−0.883644			−0.441822	+	0.897103i	$0.645668\pi$
$242$		0			0
$243$			243.000i			0.0641500i
$244$		0			0
$245$	190.000	+	95.0000i	0.0495455	+	0.0247728i
$246$		0			0
$247$			456.000i			0.117468i
$248$		0			0
$249$	468.000			0.119110
$250$		0			0
$251$	4134.00			1.03958			0.519792	−	0.854293i	$-0.326009\pi$
$251$	4134.00			1.03958			0.519792	+	0.854293i	$0.326009\pi$
$252$		0			0
$253$			2160.00i			0.536751i
$254$		0			0
$255$	2100.00	+	1050.00i	0.515714	+	0.257857i
$256$		0			0
$257$		−	3482.00i		−	0.845141i	−0.906330	−	0.422570i	$-0.861128\pi$
$257$		−	3482.00i		−	0.845141i	0.906330	−	0.422570i	$-0.138872\pi$
$258$		0			0
$259$	−2484.00			−0.595939
$260$		0			0
$261$	576.000			0.136603
$262$		0			0
$263$			2856.00i			0.669614i	0.942287	+	0.334807i	$0.108671\pi$
$263$			2856.00i			0.669614i	−0.942287	+	0.334807i	$0.891329\pi$
$264$		0			0
$265$	−2230.00	+	4460.00i	−0.516935	+	1.03387i
$266$		0			0
$267$			870.000i			0.199412i
$268$		0			0
$269$	6464.00			1.46512			0.732560	−	0.680703i	$-0.238326\pi$
$269$	6464.00			1.46512			0.732560	+	0.680703i	$0.238326\pi$
$270$		0			0
$271$	−5184.00			−1.16201			−0.581007	−	0.813899i	$-0.697341\pi$
$271$	−5184.00			−1.16201			−0.581007	+	0.813899i	$0.697341\pi$
$272$		0			0
$273$			2052.00i			0.454918i
$274$		0			0
$275$	−2250.00	−	3000.00i	−0.493382	−	0.657843i
$276$		0			0
$277$		−	3818.00i		−	0.828164i	−0.910240	−	0.414082i	$-0.864103\pi$
$277$		−	3818.00i		−	0.828164i	0.910240	−	0.414082i	$-0.135897\pi$
$278$		0			0
$279$	2808.00			0.602547
$280$		0			0
$281$	6094.00			1.29373			0.646864	−	0.762605i	$-0.276080\pi$
$281$	6094.00			1.29373			0.646864	+	0.762605i	$0.276080\pi$
$282$		0			0
$283$			4440.00i			0.932617i	0.884622	+	0.466308i	$0.154416\pi$
$283$			4440.00i			0.932617i	−0.884622	+	0.466308i	$0.845584\pi$
$284$		0			0
$285$	−180.000	+	360.000i	−0.0374115	+	0.0748230i
$286$		0			0
$287$			6732.00i			1.38459i
$288$		0			0
$289$	13.0000			0.00264604
$290$		0			0
$291$	1128.00			0.227232
$292$		0			0
$293$			5950.00i			1.18636i	0.805071	+	0.593179i	$0.202127\pi$
$293$			5950.00i			1.18636i	−0.805071	+	0.593179i	$0.797873\pi$
$294$		0			0
$295$	5100.00	+	2550.00i	1.00655	+	0.503277i
$296$		0			0
$297$			810.000i			0.158252i
$298$		0			0
$299$	2736.00			0.529187
$300$		0			0
$301$	−8424.00			−1.61313
$302$		0			0
$303$		−	312.000i		−	0.0591549i
$304$		0			0
$305$	−7540.00	−	3770.00i	−1.41554	−	0.707769i
$306$		0			0
$307$		−	924.000i		−	0.171777i	−0.996305	−	0.0858884i	$-0.972627\pi$
$307$		−	924.000i		−	0.171777i	0.996305	−	0.0858884i	$-0.0273728\pi$
$308$		0			0
$309$	3042.00			0.560043
$310$		0			0
$311$	−7020.00			−1.27996			−0.639980	−	0.768391i	$-0.721057\pi$
$311$	−7020.00			−1.27996			−0.639980	+	0.768391i	$0.721057\pi$
$312$		0			0
$313$		−	5244.00i		−	0.946992i	−0.880796	−	0.473496i	$-0.842992\pi$
$313$		−	5244.00i		−	0.946992i	0.880796	−	0.473496i	$-0.157008\pi$
$314$		0			0
$315$	−810.000	+	1620.00i	−0.144884	+	0.289767i
$316$		0			0
$317$		−	5194.00i		−	0.920265i	−0.887850	−	0.460133i	$-0.847802\pi$
$317$		−	5194.00i		−	0.920265i	0.887850	−	0.460133i	$-0.152198\pi$
$318$		0			0
$319$	1920.00			0.336989
$320$		0			0
$321$	3852.00			0.669775
$322$		0			0
$323$		−	840.000i		−	0.144702i
$324$		0			0
$325$	−3800.00	+	2850.00i	−0.648573	+	0.486429i
$326$		0			0
$327$			1710.00i			0.289184i
$328$		0			0
$329$	2376.00			0.398155
$330$		0			0
$331$	11628.0			1.93091			0.965457	−	0.260562i	$-0.0839077\pi$
$331$	11628.0			1.93091			0.965457	+	0.260562i	$0.0839077\pi$
$332$		0			0
$333$			1242.00i			0.204388i
$334$		0			0
$335$	1920.00	−	3840.00i	0.313137	−	0.626273i
$336$		0			0
$337$			4372.00i			0.706700i	0.935491	+	0.353350i	$0.114958\pi$
$337$			4372.00i			0.706700i	−0.935491	+	0.353350i	$0.885042\pi$
$338$		0			0
$339$	762.000			0.122083
$340$		0			0
$341$	9360.00			1.48643
$342$		0			0
$343$		−	6516.00i		−	1.02575i
$344$		0			0
$345$	2160.00	+	1080.00i	0.337074	+	0.168537i
$346$		0			0
$347$			12564.0i			1.94372i	0.235559	+	0.971860i	$0.424308\pi$
$347$			12564.0i			1.94372i	−0.235559	+	0.971860i	$0.575692\pi$
$348$		0			0
$349$	970.000			0.148776			0.0743881	−	0.997229i	$-0.476300\pi$
$349$	970.000			0.148776			0.0743881	+	0.997229i	$0.476300\pi$
$350$		0			0
$351$	1026.00			0.156022
$352$		0			0
$353$			1958.00i			0.295223i	0.989045	+	0.147612i	$0.0471586\pi$
$353$			1958.00i			0.295223i	−0.989045	+	0.147612i	$0.952841\pi$
$354$		0			0
$355$	−9240.00	−	4620.00i	−1.38143	−	0.690716i
$356$		0			0
$357$		−	3780.00i		−	0.560389i
$358$		0			0
$359$	1920.00			0.282267			0.141133	−	0.989991i	$-0.454925\pi$
$359$	1920.00			0.282267			0.141133	+	0.989991i	$0.454925\pi$
$360$		0			0
$361$	−6715.00			−0.979006
$362$		0			0
$363$		−	1293.00i		−	0.186956i
$364$		0			0
$365$	−1700.00	+	3400.00i	−0.243786	+	0.487573i
$366$		0			0
$367$			8886.00i			1.26388i	0.775016	+	0.631942i	$0.217742\pi$
$367$			8886.00i			1.26388i	−0.775016	+	0.631942i	$0.782258\pi$
$368$		0			0
$369$	3366.00			0.474870
$370$		0			0
$371$	8028.00			1.12343
$372$		0			0
$373$			7470.00i			1.03695i	0.855093	+	0.518474i	$0.173500\pi$
$373$			7470.00i			1.03695i	−0.855093	+	0.518474i	$0.826500\pi$
$374$		0			0
$375$	−4125.00	+	750.000i	−0.568038	+	0.103280i
$376$		0			0
$377$		−	2432.00i		−	0.332240i
$378$		0			0
$379$	6312.00			0.855477			0.427738	−	0.903903i	$-0.359310\pi$
$379$	6312.00			0.855477			0.427738	+	0.903903i	$0.359310\pi$
$380$		0			0
$381$	414.000			0.0556689
$382$		0			0
$383$			12876.0i			1.71784i	0.512109	+	0.858920i	$0.328864\pi$
$383$			12876.0i			1.71784i	−0.512109	+	0.858920i	$0.671136\pi$
$384$		0			0
$385$	−2700.00	+	5400.00i	−0.357415	+	0.714830i
$386$		0			0
$387$			4212.00i			0.553251i
$388$		0			0
$389$	−8780.00			−1.14438			−0.572190	−	0.820121i	$-0.693906\pi$
$389$	−8780.00			−1.14438			−0.572190	+	0.820121i	$0.693906\pi$
$390$		0			0
$391$	−5040.00			−0.651877
$392$		0			0
$393$			2682.00i			0.344247i
$394$		0			0
$395$	−720.000	−	360.000i	−0.0917143	−	0.0458571i
$396$		0			0
$397$			3246.00i			0.410358i	0.978724	+	0.205179i	$0.0657776\pi$
$397$			3246.00i			0.410358i	−0.978724	+	0.205179i	$0.934222\pi$
$398$		0			0
$399$	648.000			0.0813047
$400$		0			0
$401$	1870.00			0.232876			0.116438	−	0.993198i	$-0.462852\pi$
$401$	1870.00			0.232876			0.116438	+	0.993198i	$0.462852\pi$
$402$		0			0
$403$		−	11856.0i		−	1.46548i
$404$		0			0
$405$	810.000	+	405.000i	0.0993808	+	0.0496904i
$406$		0			0
$407$			4140.00i			0.504207i
$408$		0			0
$409$	−15990.0			−1.93314			−0.966570	−	0.256401i	$-0.917463\pi$
$409$	−15990.0			−1.93314			−0.966570	+	0.256401i	$0.917463\pi$
$410$		0			0
$411$	9150.00			1.09814
$412$		0			0
$413$		−	9180.00i		−	1.09375i
$414$		0			0
$415$	780.000	−	1560.00i	0.0922619	−	0.184524i
$416$		0			0
$417$		−	7056.00i		−	0.828618i
$418$		0			0
$419$	−3054.00			−0.356080			−0.178040	−	0.984023i	$-0.556976\pi$
$419$	−3054.00			−0.356080			−0.178040	+	0.984023i	$0.556976\pi$
$420$		0			0
$421$	1194.00			0.138223			0.0691116	−	0.997609i	$-0.477984\pi$
$421$	1194.00			0.138223			0.0691116	+	0.997609i	$0.477984\pi$
$422$		0			0
$423$		−	1188.00i		−	0.136554i
$424$		0			0
$425$	7000.00	−	5250.00i	0.798941	−	0.599206i
$426$		0			0
$427$			13572.0i			1.53816i
$428$		0			0
$429$	3420.00			0.384893
$430$		0			0
$431$	−15264.0			−1.70590			−0.852948	−	0.521996i	$-0.825188\pi$
$431$	−15264.0			−1.70590			−0.852948	+	0.521996i	$0.825188\pi$
$432$		0			0
$433$		−	8220.00i		−	0.912305i	−0.889902	−	0.456152i	$-0.849227\pi$
$433$		−	8220.00i		−	0.912305i	0.889902	−	0.456152i	$-0.150773\pi$
$434$		0			0
$435$	960.000	−	1920.00i	0.105813	−	0.211625i
$436$		0			0
$437$		−	864.000i		−	0.0945783i
$438$		0			0
$439$	−16248.0			−1.76646			−0.883229	−	0.468943i	$-0.844635\pi$
$439$	−16248.0			−1.76646			−0.883229	+	0.468943i	$0.844635\pi$
$440$		0			0
$441$	−171.000			−0.0184645
$442$		0			0
$443$			8220.00i			0.881589i	0.897608	+	0.440795i	$0.145303\pi$
$443$			8220.00i			0.881589i	−0.897608	+	0.440795i	$0.854697\pi$
$444$		0			0
$445$	2900.00	+	1450.00i	0.308929	+	0.154464i
$446$		0			0
$447$			7428.00i			0.785978i
$448$		0			0
$449$	9490.00			0.997463			0.498731	−	0.866757i	$-0.333799\pi$
$449$	9490.00			0.997463			0.498731	+	0.866757i	$0.333799\pi$
$450$		0			0
$451$	11220.0			1.17146
$452$		0			0
$453$		−	9576.00i		−	0.993200i
$454$		0			0
$455$	6840.00	+	3420.00i	0.704756	+	0.352378i
$456$		0			0
$457$			5776.00i			0.591225i	0.955308	+	0.295613i	$0.0955237\pi$
$457$			5776.00i			0.591225i	−0.955308	+	0.295613i	$0.904476\pi$
$458$		0			0
$459$	−1890.00			−0.192195
$460$		0			0
$461$	−2812.00			−0.284095			−0.142048	−	0.989860i	$-0.545369\pi$
$461$	−2812.00			−0.284095			−0.142048	+	0.989860i	$0.545369\pi$
$462$		0			0
$463$			13710.0i			1.37615i	0.725639	+	0.688075i	$0.241544\pi$
$463$			13710.0i			1.37615i	−0.725639	+	0.688075i	$0.758456\pi$
$464$		0			0
$465$	4680.00	−	9360.00i	0.466731	−	0.933462i
$466$		0			0
$467$		−	6252.00i		−	0.619503i	−0.950817	−	0.309752i	$-0.899754\pi$
$467$		−	6252.00i		−	0.619503i	0.950817	−	0.309752i	$-0.100246\pi$
$468$		0			0
$469$	−6912.00			−0.680526
$470$		0			0
$471$	−3654.00			−0.357468
$472$		0			0
$473$			14040.0i			1.36482i
$474$		0			0
$475$	900.000	+	1200.00i	0.0869365	+	0.115915i
$476$		0			0
$477$		−	4014.00i		−	0.385301i
$478$		0			0
$479$	7908.00			0.754333			0.377167	−	0.926145i	$-0.376898\pi$
$479$	7908.00			0.754333			0.377167	+	0.926145i	$0.376898\pi$
$480$		0			0
$481$	5244.00			0.497101
$482$		0			0
$483$		−	3888.00i		−	0.366274i
$484$		0			0
$485$	1880.00	−	3760.00i	0.176013	−	0.352026i
$486$		0			0
$487$			17562.0i			1.63411i	0.576562	+	0.817054i	$0.304394\pi$
$487$			17562.0i			1.63411i	−0.576562	+	0.817054i	$0.695606\pi$
$488$		0			0
$489$	6984.00			0.645864
$490$		0			0
$491$	13062.0			1.20057			0.600285	−	0.799786i	$-0.295054\pi$
$491$	13062.0			1.20057			0.600285	+	0.799786i	$0.295054\pi$
$492$		0			0
$493$			4480.00i			0.409268i
$494$		0			0
$495$	2700.00	+	1350.00i	0.245164	+	0.122582i
$496$		0			0
$497$			16632.0i			1.50110i
$498$		0			0
$499$	8556.00			0.767573			0.383787	−	0.923422i	$-0.374620\pi$
$499$	8556.00			0.767573			0.383787	+	0.923422i	$0.374620\pi$
$500$		0			0
$501$	8784.00			0.783314
$502$		0			0
$503$		−	14436.0i		−	1.27966i	−0.768516	−	0.639830i	$-0.779005\pi$
$503$		−	14436.0i		−	1.27966i	0.768516	−	0.639830i	$-0.220995\pi$
$504$		0			0
$505$	−1040.00	−	520.000i	−0.0916424	−	0.0458212i
$506$		0			0
$507$			2259.00i			0.197881i
$508$		0			0
$509$	9536.00			0.830404			0.415202	−	0.909729i	$-0.363711\pi$
$509$	9536.00			0.830404			0.415202	+	0.909729i	$0.363711\pi$
$510$		0			0
$511$	6120.00			0.529810
$512$		0			0
$513$		−	324.000i		−	0.0278849i
$514$		0			0
$515$	5070.00	−	10140.0i	0.433808	−	0.867615i
$516$		0			0
$517$		−	3960.00i		−	0.336868i
$518$		0			0
$519$	1506.00			0.127372
$520$		0			0
$521$	−1130.00			−0.0950215			−0.0475107	−	0.998871i	$-0.515129\pi$
$521$	−1130.00			−0.0950215			−0.0475107	+	0.998871i	$0.515129\pi$
$522$		0			0
$523$			4104.00i			0.343127i	0.985173	+	0.171563i	$0.0548819\pi$
$523$			4104.00i			0.343127i	−0.985173	+	0.171563i	$0.945118\pi$
$524$		0			0
$525$	4050.00	+	5400.00i	0.336679	+	0.448905i
$526$		0			0
$527$			21840.0i			1.80525i
$528$		0			0
$529$	6983.00			0.573929
$530$		0			0
$531$	−4590.00			−0.375121
$532$		0			0
$533$		−	14212.0i		−	1.15495i
$534$		0			0
$535$	6420.00	−	12840.0i	0.518805	−	1.03761i
$536$		0			0
$537$		−	12078.0i		−	0.970585i
$538$		0			0
$539$	−570.000			−0.0455503
$540$		0			0
$541$	−6810.00			−0.541192			−0.270596	−	0.962693i	$-0.587221\pi$
$541$	−6810.00			−0.541192			−0.270596	+	0.962693i	$0.587221\pi$
$542$		0			0
$543$		−	1782.00i		−	0.140834i
$544$		0			0
$545$	5700.00	+	2850.00i	0.448002	+	0.224001i
$546$		0			0
$547$			23772.0i			1.85817i	0.369870	+	0.929083i	$0.379402\pi$
$547$			23772.0i			1.85817i	−0.369870	+	0.929083i	$0.620598\pi$
$548$		0			0
$549$	6786.00			0.527540
$550$		0			0
$551$	−768.000			−0.0593791
$552$		0			0
$553$			1296.00i			0.0996592i
$554$		0			0
$555$	4140.00	+	2070.00i	0.316636	+	0.158318i
$556$		0			0
$557$			10762.0i			0.818672i	0.912384	+	0.409336i	$0.134240\pi$
$557$			10762.0i			0.818672i	−0.912384	+	0.409336i	$0.865760\pi$
$558$		0			0
$559$	17784.0			1.34559
$560$		0			0
$561$	−6300.00			−0.474129
$562$		0			0
$563$			1716.00i			0.128456i	0.997935	+	0.0642280i	$0.0204585\pi$
$563$			1716.00i			0.128456i	−0.997935	+	0.0642280i	$0.979541\pi$
$564$		0			0
$565$	1270.00	−	2540.00i	0.0945651	−	0.189130i
$566$		0			0
$567$		−	1458.00i		−	0.107990i
$568$		0			0
$569$	−6670.00			−0.491425			−0.245713	−	0.969343i	$-0.579022\pi$
$569$	−6670.00			−0.491425			−0.245713	+	0.969343i	$0.579022\pi$
$570$		0			0
$571$	5580.00			0.408959			0.204480	−	0.978871i	$-0.434450\pi$
$571$	5580.00			0.408959			0.204480	+	0.978871i	$0.434450\pi$
$572$		0			0
$573$			3132.00i			0.228344i
$574$		0			0
$575$	7200.00	−	5400.00i	0.522193	−	0.391644i
$576$		0			0
$577$		−	17808.0i		−	1.28485i	−0.766350	−	0.642424i	$-0.777929\pi$
$577$		−	17808.0i		−	1.28485i	0.766350	−	0.642424i	$-0.222071\pi$
$578$		0			0
$579$	−9756.00			−0.700251
$580$		0			0
$581$	−2808.00			−0.200509
$582$		0			0
$583$		−	13380.0i		−	0.950503i
$584$		0			0
$585$	1710.00	−	3420.00i	0.120854	−	0.241709i
$586$		0			0
$587$			23004.0i			1.61751i	0.588148	+	0.808754i	$0.299857\pi$
$587$			23004.0i			1.61751i	−0.588148	+	0.808754i	$0.700143\pi$
$588$		0			0
$589$	−3744.00			−0.261917
$590$		0			0
$591$	−8742.00			−0.608457
$592$		0			0
$593$			3266.00i			0.226170i	0.993585	+	0.113085i	$0.0360732\pi$
$593$			3266.00i			0.226170i	−0.993585	+	0.113085i	$0.963927\pi$
$594$		0			0
$595$	−12600.0	−	6300.00i	−0.868151	−	0.434075i
$596$		0			0
$597$			14616.0i			1.00200i
$598$		0			0
$599$	2208.00			0.150612			0.0753059	−	0.997160i	$-0.476007\pi$
$599$	2208.00			0.150612			0.0753059	+	0.997160i	$0.476007\pi$
$600$		0			0
$601$	−21166.0			−1.43657			−0.718285	−	0.695748i	$-0.755073\pi$
$601$	−21166.0			−1.43657			−0.718285	+	0.695748i	$0.755073\pi$
$602$		0			0
$603$			3456.00i			0.233398i
$604$		0			0
$605$	−4310.00	−	2155.00i	−0.289630	−	0.144815i
$606$		0			0
$607$			17610.0i			1.17754i	0.808300	+	0.588771i	$0.200388\pi$
$607$			17610.0i			1.17754i	−0.808300	+	0.588771i	$0.799612\pi$
$608$		0			0
$609$	−3456.00			−0.229958
$610$		0			0
$611$	−5016.00			−0.332121
$612$		0			0
$613$			29610.0i			1.95096i	0.220095	+	0.975478i	$0.429363\pi$
$613$			29610.0i			1.95096i	−0.220095	+	0.975478i	$0.570637\pi$
$614$		0			0
$615$	5610.00	−	11220.0i	0.367833	−	0.735665i
$616$		0			0
$617$		−	2294.00i		−	0.149681i	−0.997196	−	0.0748403i	$-0.976155\pi$
$617$		−	2294.00i		−	0.149681i	0.997196	−	0.0748403i	$-0.0238447\pi$
$618$		0			0
$619$	−19704.0			−1.27944			−0.639718	−	0.768610i	$-0.720949\pi$
$619$	−19704.0			−1.27944			−0.639718	+	0.768610i	$0.720949\pi$
$620$		0			0
$621$	−1944.00			−0.125620
$622$		0			0
$623$		−	5220.00i		−	0.335690i
$624$		0			0
$625$	−4375.00	+	15000.0i	−0.280000	+	0.960000i
$626$		0			0
$627$		−	1080.00i		−	0.0687895i
$628$		0			0
$629$	−9660.00			−0.612352
$630$		0			0
$631$	4392.00			0.277088			0.138544	−	0.990356i	$-0.455758\pi$
$631$	4392.00			0.277088			0.138544	+	0.990356i	$0.455758\pi$
$632$		0			0
$633$		−	5616.00i		−	0.352632i
$634$		0			0
$635$	690.000	−	1380.00i	0.0431210	−	0.0862419i
$636$		0			0
$637$			722.000i			0.0449084i
$638$		0			0
$639$	8316.00			0.514829
$640$		0			0
$641$	19586.0			1.20687			0.603433	−	0.797414i	$-0.293799\pi$
$641$	19586.0			1.20687			0.603433	+	0.797414i	$0.293799\pi$
$642$		0			0
$643$			624.000i			0.0382709i	0.999817	+	0.0191354i	$0.00609137\pi$
$643$			624.000i			0.0382709i	−0.999817	+	0.0191354i	$0.993909\pi$
$644$		0			0
$645$	14040.0	+	7020.00i	0.857092	+	0.428546i
$646$		0			0
$647$		−	8556.00i		−	0.519893i	−0.965623	−	0.259947i	$-0.916295\pi$
$647$		−	8556.00i		−	0.519893i	0.965623	−	0.259947i	$-0.0837050\pi$
$648$		0			0
$649$	−15300.0			−0.925389
$650$		0			0
$651$	−16848.0			−1.01432
$652$		0			0
$653$		−	22910.0i		−	1.37295i	−0.727153	−	0.686476i	$-0.759157\pi$
$653$		−	22910.0i		−	1.37295i	0.727153	−	0.686476i	$-0.240843\pi$
$654$		0			0
$655$	8940.00	+	4470.00i	0.533305	+	0.266652i
$656$		0			0
$657$		−	3060.00i		−	0.181708i
$658$		0			0
$659$	−7590.00			−0.448656			−0.224328	−	0.974514i	$-0.572019\pi$
$659$	−7590.00			−0.448656			−0.224328	+	0.974514i	$0.572019\pi$
$660$		0			0
$661$	−31970.0			−1.88122			−0.940612	−	0.339484i	$-0.889748\pi$
$661$	−31970.0			−1.88122			−0.940612	+	0.339484i	$0.889748\pi$
$662$		0			0
$663$			7980.00i			0.467447i
$664$		0			0
$665$	1080.00	−	2160.00i	0.0629784	−	0.125957i
$666$		0			0
$667$			4608.00i			0.267500i
$668$		0			0
$669$	3906.00			0.225732
$670$		0			0
$671$	22620.0			1.30139
$672$		0			0
$673$		−	24996.0i		−	1.43169i	−0.698261	−	0.715843i	$-0.746043\pi$
$673$		−	24996.0i		−	1.43169i	0.698261	−	0.715843i	$-0.253957\pi$
$674$		0			0
$675$	2700.00	−	2025.00i	0.153960	−	0.115470i
$676$		0			0
$677$		−	3226.00i		−	0.183139i	−0.995799	−	0.0915696i	$-0.970812\pi$
$677$		−	3226.00i		−	0.183139i	0.995799	−	0.0915696i	$-0.0291884\pi$
$678$		0			0
$679$	−6768.00			−0.382521
$680$		0			0
$681$	−2124.00			−0.119518
$682$		0			0
$683$		−	23412.0i		−	1.31162i	−0.754927	−	0.655809i	$-0.772328\pi$
$683$		−	23412.0i		−	1.31162i	0.754927	−	0.655809i	$-0.227672\pi$
$684$		0			0
$685$	15250.0	−	30500.0i	0.850617	−	1.70123i
$686$		0			0
$687$		−	15246.0i		−	0.846683i
$688$		0			0
$689$	−16948.0			−0.937108
$690$		0			0
$691$	−16572.0			−0.912342			−0.456171	−	0.889892i	$-0.650780\pi$
$691$	−16572.0			−0.912342			−0.456171	+	0.889892i	$0.650780\pi$
$692$		0			0
$693$		−	4860.00i		−	0.266401i
$694$		0			0
$695$	−23520.0	−	11760.0i	−1.28369	−	0.641845i
$696$		0			0
$697$			26180.0i			1.42272i
$698$		0			0
$699$	17358.0			0.939256
$700$		0			0
$701$	448.000			0.0241380			0.0120690	−	0.999927i	$-0.496158\pi$
$701$	448.000			0.0241380			0.0120690	+	0.999927i	$0.496158\pi$
$702$		0			0
$703$		−	1656.00i		−	0.0888438i
$704$		0			0
$705$	−3960.00	−	1980.00i	−0.211549	−	0.105775i
$706$		0			0
$707$			1872.00i			0.0995811i
$708$		0			0
$709$	−11922.0			−0.631509			−0.315755	−	0.948841i	$-0.602258\pi$
$709$	−11922.0			−0.631509			−0.315755	+	0.948841i	$0.602258\pi$
$710$		0			0
$711$	648.000			0.0341799
$712$		0			0
$713$			22464.0i			1.17992i
$714$		0			0
$715$	5700.00	−	11400.0i	0.298137	−	0.596274i
$716$		0			0
$717$			9468.00i			0.493151i
$718$		0			0
$719$	−4344.00			−0.225318			−0.112659	−	0.993634i	$-0.535937\pi$
$719$	−4344.00			−0.225318			−0.112659	+	0.993634i	$0.535937\pi$
$720$		0			0
$721$	−18252.0			−0.942774
$722$		0			0
$723$		−	9918.00i		−	0.510172i
$724$		0			0
$725$	−4800.00	−	6400.00i	−0.245886	−	0.327848i
$726$		0			0
$727$		−	26934.0i		−	1.37404i	−0.726639	−	0.687020i	$-0.758919\pi$
$727$		−	26934.0i		−	1.37404i	0.726639	−	0.687020i	$-0.241081\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$	−32760.0			−1.65755
$732$		0			0
$733$			35962.0i			1.81212i	0.423145	+	0.906062i	$0.360926\pi$
$733$			35962.0i			1.81212i	−0.423145	+	0.906062i	$0.639074\pi$
$734$		0			0
$735$	−285.000	+	570.000i	−0.0143026	+	0.0286051i
$736$		0			0
$737$			11520.0i			0.575773i
$738$		0			0
$739$	−4860.00			−0.241919			−0.120959	−	0.992657i	$-0.538597\pi$
$739$	−4860.00			−0.241919			−0.120959	+	0.992657i	$0.538597\pi$
$740$		0			0
$741$	−1368.00			−0.0678202
$742$		0			0
$743$		−	29844.0i		−	1.47358i	−0.676121	−	0.736790i	$-0.736341\pi$
$743$		−	29844.0i		−	1.47358i	0.676121	−	0.736790i	$-0.263659\pi$
$744$		0			0
$745$	24760.0	+	12380.0i	1.21763	+	0.608816i
$746$		0			0
$747$			1404.00i			0.0687680i
$748$		0			0
$749$	−23112.0			−1.12750
$750$		0			0
$751$	1248.00			0.0606394			0.0303197	−	0.999540i	$-0.490347\pi$
$751$	1248.00			0.0606394			0.0303197	+	0.999540i	$0.490347\pi$
$752$		0			0
$753$			12402.0i			0.600205i
$754$		0			0
$755$	−31920.0	−	15960.0i	−1.53866	−	0.769330i
$756$		0			0
$757$			25094.0i			1.20483i	0.798183	+	0.602415i	$0.205795\pi$
$757$			25094.0i			1.20483i	−0.798183	+	0.602415i	$0.794205\pi$
$758$		0			0
$759$	−6480.00			−0.309893
$760$		0			0
$761$	23294.0			1.10960			0.554801	−	0.831983i	$-0.312794\pi$
$761$	23294.0			1.10960			0.554801	+	0.831983i	$0.312794\pi$
$762$		0			0
$763$		−	10260.0i		−	0.486811i
$764$		0			0
$765$	−3150.00	+	6300.00i	−0.148874	+	0.297748i
$766$		0			0
$767$			19380.0i			0.912348i
$768$		0			0
$769$	254.000			0.0119109			0.00595544	−	0.999982i	$-0.498104\pi$
$769$	254.000			0.0119109			0.00595544	+	0.999982i	$0.498104\pi$
$770$		0			0
$771$	10446.0			0.487942
$772$		0			0
$773$		−	21886.0i		−	1.01835i	−0.860663	−	0.509175i	$-0.829951\pi$
$773$		−	21886.0i		−	1.01835i	0.860663	−	0.509175i	$-0.170049\pi$
$774$		0			0
$775$	−23400.0	−	31200.0i	−1.08458	−	1.44611i
$776$		0			0
$777$		−	7452.00i		−	0.344066i
$778$		0			0
$779$	−4488.00			−0.206418
$780$		0			0
$781$	27720.0			1.27004
$782$		0			0
$783$			1728.00i			0.0788680i
$784$		0			0
$785$	−6090.00	+	12180.0i	−0.276894	+	0.553787i
$786$		0			0
$787$		−	14904.0i		−	0.675057i	−0.941315	−	0.337529i	$-0.890409\pi$
$787$		−	14904.0i		−	0.675057i	0.941315	−	0.337529i	$-0.109591\pi$
$788$		0			0
$789$	−8568.00			−0.386602
$790$		0			0
$791$	−4572.00			−0.205514
$792$		0			0
$793$		−	28652.0i		−	1.28305i
$794$		0			0
$795$	−13380.0	−	6690.00i	−0.596905	−	0.298453i
$796$		0			0
$797$		−	23834.0i		−	1.05928i	−0.848224	−	0.529638i	$-0.822328\pi$
$797$		−	23834.0i		−	1.05928i	0.848224	−	0.529638i	$-0.177672\pi$
$798$		0			0
$799$	9240.00			0.409121
$800$		0			0
$801$	−2610.00			−0.115131
$802$		0			0
$803$		−	10200.0i		−	0.448257i
$804$		0			0
$805$	−12960.0	−	6480.00i	−0.567429	−	0.283714i
$806$		0			0
$807$			19392.0i			0.845887i
$808$		0			0
$809$	31502.0			1.36904			0.684519	−	0.728995i	$-0.260012\pi$
$809$	31502.0			1.36904			0.684519	+	0.728995i	$0.260012\pi$
$810$		0			0
$811$	−21072.0			−0.912377			−0.456189	−	0.889883i	$-0.650786\pi$
$811$	−21072.0			−0.912377			−0.456189	+	0.889883i	$0.650786\pi$
$812$		0			0
$813$		−	15552.0i		−	0.670889i
$814$		0			0
$815$	11640.0	−	23280.0i	0.500284	−	1.00057i
$816$		0			0
$817$		−	5616.00i		−	0.240488i
$818$		0			0
$819$	−6156.00			−0.262647
$820$		0			0
$821$	38072.0			1.61842			0.809209	−	0.587520i	$-0.199896\pi$
$821$	38072.0			1.61842			0.809209	+	0.587520i	$0.199896\pi$
$822$		0			0
$823$		−	31578.0i		−	1.33747i	−0.743500	−	0.668736i	$-0.766836\pi$
$823$		−	31578.0i		−	1.33747i	0.743500	−	0.668736i	$-0.233164\pi$
$824$		0			0
$825$	9000.00	−	6750.00i	0.379806	−	0.284854i
$826$		0			0
$827$			41532.0i			1.74632i	0.487431	+	0.873162i	$0.337934\pi$
$827$			41532.0i			1.74632i	−0.487431	+	0.873162i	$0.662066\pi$
$828$		0			0
$829$	24786.0			1.03842			0.519212	−	0.854646i	$-0.326226\pi$
$829$	24786.0			1.03842			0.519212	+	0.854646i	$0.326226\pi$
$830$		0			0
$831$	11454.0			0.478141
$832$		0			0
$833$		−	1330.00i		−	0.0553203i
$834$		0			0
$835$	14640.0	−	29280.0i	0.606752	−	1.21350i
$836$		0			0
$837$			8424.00i			0.347881i
$838$		0			0
$839$	−1608.00			−0.0661673			−0.0330836	−	0.999453i	$-0.510533\pi$
$839$	−1608.00			−0.0661673			−0.0330836	+	0.999453i	$0.510533\pi$
$840$		0			0
$841$	−20293.0			−0.832055
$842$		0			0
$843$			18282.0i			0.746934i
$844$		0			0
$845$	7530.00	+	3765.00i	0.306556	+	0.153278i
$846$		0			0
$847$			7758.00i			0.314720i
$848$		0			0
$849$	−13320.0			−0.538447
$850$		0			0
$851$	−9936.00			−0.400237
$852$		0			0
$853$			8514.00i			0.341751i	0.985293	+	0.170876i	$0.0546596\pi$
$853$			8514.00i			0.341751i	−0.985293	+	0.170876i	$0.945340\pi$
$854$		0			0
$855$	−1080.00	−	540.000i	−0.0431991	−	0.0215995i
$856$		0			0
$857$		−	29726.0i		−	1.18486i	−0.805624	−	0.592428i	$-0.798170\pi$
$857$		−	29726.0i		−	1.18486i	0.805624	−	0.592428i	$-0.201830\pi$
$858$		0			0
$859$	2136.00			0.0848421			0.0424211	−	0.999100i	$-0.486493\pi$
$859$	2136.00			0.0848421			0.0424211	+	0.999100i	$0.486493\pi$
$860$		0			0
$861$	−20196.0			−0.799394
$862$		0			0
$863$		−	11928.0i		−	0.470491i	−0.971936	−	0.235246i	$-0.924411\pi$
$863$		−	11928.0i		−	0.470491i	0.971936	−	0.235246i	$-0.0755894\pi$
$864$		0			0
$865$	2510.00	−	5020.00i	0.0986619	−	0.197324i
$866$		0			0
$867$			39.0000i			0.00152769i
$868$		0			0
$869$	2160.00			0.0843187
$870$		0			0
$871$	14592.0			0.567659
$872$		0			0
$873$			3384.00i			0.131192i
$874$		0			0
$875$	24750.0	−	4500.00i	0.956232	−	0.173860i
$876$		0			0
$877$		−	27006.0i		−	1.03983i	−0.854219	−	0.519913i	$-0.825964\pi$
$877$		−	27006.0i		−	1.03983i	0.854219	−	0.519913i	$-0.174036\pi$
$878$		0			0
$879$	−17850.0			−0.684944
$880$		0			0
$881$	29318.0			1.12117			0.560584	−	0.828098i	$-0.310577\pi$
$881$	29318.0			1.12117			0.560584	+	0.828098i	$0.310577\pi$
$882$		0			0
$883$		−	7008.00i		−	0.267087i	−0.991043	−	0.133544i	$-0.957364\pi$
$883$		−	7008.00i		−	0.267087i	0.991043	−	0.133544i	$-0.0426356\pi$
$884$		0			0
$885$	−7650.00	+	15300.0i	−0.290567	+	0.581134i
$886$		0			0
$887$			15024.0i			0.568722i	0.958717	+	0.284361i	$0.0917814\pi$
$887$			15024.0i			0.568722i	−0.958717	+	0.284361i	$0.908219\pi$
$888$		0			0
$889$	−2484.00			−0.0937128
$890$		0			0
$891$	−2430.00			−0.0913671
$892$		0			0
$893$			1584.00i			0.0593578i
$894$		0			0
$895$	−40260.0	−	20130.0i	−1.50362	−	0.751812i
$896$		0			0
$897$			8208.00i			0.305526i
$898$		0			0
$899$	19968.0			0.740790
$900$		0			0
$901$	31220.0			1.15437
$902$		0			0
$903$		−	25272.0i		−	0.931339i
$904$		0			0
$905$	−5940.00	−	2970.00i	−0.218179	−	0.109090i
$906$		0			0
$907$			1380.00i			0.0505206i	0.999681	+	0.0252603i	$0.00804145\pi$
$907$			1380.00i			0.0505206i	−0.999681	+	0.0252603i	$0.991959\pi$
$908$		0			0
$909$	936.000			0.0341531
$910$		0			0
$911$	17952.0			0.652883			0.326441	−	0.945217i	$-0.394150\pi$
$911$	17952.0			0.652883			0.326441	+	0.945217i	$0.394150\pi$
$912$		0			0
$913$			4680.00i			0.169644i
$914$		0			0
$915$	11310.0	−	22620.0i	0.408631	−	0.817261i
$916$		0			0
$917$		−	16092.0i		−	0.579503i
$918$		0			0
$919$	−19056.0			−0.684004			−0.342002	−	0.939699i	$-0.611105\pi$
$919$	−19056.0			−0.684004			−0.342002	+	0.939699i	$0.611105\pi$
$920$		0			0
$921$	2772.00			0.0991754
$922$		0			0
$923$		−	35112.0i		−	1.25214i
$924$		0			0
$925$	13800.0	−	10350.0i	0.490531	−	0.367898i
$926$		0			0
$927$			9126.00i			0.323341i
$928$		0			0
$929$	−26398.0			−0.932282			−0.466141	−	0.884710i	$-0.654356\pi$
$929$	−26398.0			−0.932282			−0.466141	+	0.884710i	$0.654356\pi$
$930$		0			0
$931$	228.000			0.00802621
$932$		0			0
$933$		−	21060.0i		−	0.738985i
$934$		0			0
$935$	−10500.0	+	21000.0i	−0.367259	+	0.734517i
$936$		0			0
$937$		−	21192.0i		−	0.738861i	−0.929258	−	0.369430i	$-0.879553\pi$
$937$		−	21192.0i		−	0.738861i	0.929258	−	0.369430i	$-0.120447\pi$
$938$		0			0
$939$	15732.0			0.546746
$940$		0			0
$941$	−7300.00			−0.252894			−0.126447	−	0.991973i	$-0.540357\pi$
$941$	−7300.00			−0.252894			−0.126447	+	0.991973i	$0.540357\pi$
$942$		0			0
$943$			26928.0i			0.929901i
$944$		0			0
$945$	−4860.00	−	2430.00i	−0.167297	−	0.0836486i
$946$		0			0
$947$		−	33540.0i		−	1.15090i	−0.817836	−	0.575451i	$-0.804827\pi$
$947$		−	33540.0i		−	1.15090i	0.817836	−	0.575451i	$-0.195173\pi$
$948$		0			0
$949$	−12920.0			−0.441940
$950$		0			0
$951$	15582.0			0.531315
$952$		0			0
$953$			32714.0i			1.11197i	0.831191	+	0.555987i	$0.187659\pi$
$953$			32714.0i			1.11197i	−0.831191	+	0.555987i	$0.812341\pi$
$954$		0			0
$955$	10440.0	+	5220.00i	0.353749	+	0.176875i
$956$		0			0
$957$			5760.00i			0.194560i
$958$		0			0
$959$	−54900.0			−1.84861
$960$		0			0
$961$	67553.0			2.26756
$962$		0			0
$963$			11556.0i			0.386695i
$964$		0			0
$965$	−16260.0	+	32520.0i	−0.542412	+	1.08482i
$966$		0			0
$967$			35334.0i			1.17504i	0.809209	+	0.587521i	$0.199896\pi$
$967$			35334.0i			1.17504i	−0.809209	+	0.587521i	$0.800104\pi$
$968$		0			0
$969$	2520.00			0.0835439
$970$		0			0
$971$	17094.0			0.564956			0.282478	−	0.959274i	$-0.408844\pi$
$971$	17094.0			0.564956			0.282478	+	0.959274i	$0.408844\pi$
$972$		0			0
$973$			42336.0i			1.39489i
$974$		0			0
$975$	−8550.00	−	11400.0i	−0.280840	−	0.374454i
$976$		0			0
$977$		−	46814.0i		−	1.53297i	−0.642262	−	0.766485i	$-0.722004\pi$
$977$		−	46814.0i		−	1.53297i	0.642262	−	0.766485i	$-0.277996\pi$
$978$		0			0
$979$	−8700.00			−0.284018
$980$		0			0
$981$	−5130.00			−0.166961
$982$		0			0
$983$			5076.00i			0.164699i	0.996604	+	0.0823496i	$0.0262424\pi$
$983$			5076.00i			0.164699i	−0.996604	+	0.0823496i	$0.973758\pi$
$984$		0			0
$985$	−14570.0	+	29140.0i	−0.471308	+	0.942617i
$986$		0			0
$987$			7128.00i			0.229875i
$988$		0			0
$989$	−33696.0			−1.08339
$990$		0			0
$991$	−9456.00			−0.303108			−0.151554	−	0.988449i	$-0.548428\pi$
$991$	−9456.00			−0.303108			−0.151554	+	0.988449i	$0.548428\pi$
$992$		0			0
$993$			34884.0i			1.11481i
$994$		0			0
$995$	48720.0	+	24360.0i	1.55229	+	0.776145i
$996$		0			0
$997$		−	33486.0i		−	1.06370i	−0.846838	−	0.531852i	$-0.821496\pi$
$997$		−	33486.0i		−	1.06370i	0.846838	−	0.531852i	$-0.178504\pi$
$998$		0			0
$999$	−3726.00			−0.118003

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.4.f.h.769.2		2
4.3	odd	2		960.4.f.k.769.1		2
5.4	even	2	inner	960.4.f.h.769.1		2
8.3	odd	2		480.4.f.a.289.2	yes	2
8.5	even	2		480.4.f.b.289.1	yes	2
20.19	odd	2		960.4.f.k.769.2		2
24.5	odd	2		1440.4.f.c.289.2		2
24.11	even	2		1440.4.f.d.289.2		2
40.3	even	4		2400.4.a.i.1.1		1
40.13	odd	4		2400.4.a.n.1.1		1
40.19	odd	2		480.4.f.a.289.1	✓	2
40.27	even	4		2400.4.a.m.1.1		1
40.29	even	2		480.4.f.b.289.2	yes	2
40.37	odd	4		2400.4.a.j.1.1		1
120.29	odd	2		1440.4.f.c.289.1		2
120.59	even	2		1440.4.f.d.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.f.a.289.1	✓	2	40.19	odd	2
480.4.f.a.289.2	yes	2	8.3	odd	2
480.4.f.b.289.1	yes	2	8.5	even	2
480.4.f.b.289.2	yes	2	40.29	even	2
960.4.f.h.769.1		2	5.4	even	2	inner
960.4.f.h.769.2		2	1.1	even	1	trivial
960.4.f.k.769.1		2	4.3	odd	2
960.4.f.k.769.2		2	20.19	odd	2
1440.4.f.c.289.1		2	120.29	odd	2
1440.4.f.c.289.2		2	24.5	odd	2
1440.4.f.d.289.1		2	120.59	even	2
1440.4.f.d.289.2		2	24.11	even	2
2400.4.a.i.1.1		1	40.3	even	4
2400.4.a.j.1.1		1	40.37	odd	4
2400.4.a.m.1.1		1	40.27	even	4
2400.4.a.n.1.1		1	40.13	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	960.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +(10.0000 + 5.00000i) q^{5} -18.0000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} +(10.0000 + 5.00000i) q^{5} -18.0000i q^{7} -9.00000 q^{9} -30.0000 q^{11} +38.0000i q^{13} +(-15.0000 + 30.0000i) q^{15} -70.0000i q^{17} +12.0000 q^{19} +54.0000 q^{21} -72.0000i q^{23} +(75.0000 + 100.000i) q^{25} -27.0000i q^{27} -64.0000 q^{29} -312.000 q^{31} -90.0000i q^{33} +(90.0000 - 180.000i) q^{35} -138.000i q^{37} -114.000 q^{39} -374.000 q^{41} -468.000i q^{43} +(-90.0000 - 45.0000i) q^{45} +132.000i q^{47} +19.0000 q^{49} +210.000 q^{51} +446.000i q^{53} +(-300.000 - 150.000i) q^{55} +36.0000i q^{57} +510.000 q^{59} -754.000 q^{61} +162.000i q^{63} +(-190.000 + 380.000i) q^{65} -384.000i q^{67} +216.000 q^{69} -924.000 q^{71} +340.000i q^{73} +(-300.000 + 225.000i) q^{75} +540.000i q^{77} -72.0000 q^{79} +81.0000 q^{81} -156.000i q^{83} +(350.000 - 700.000i) q^{85} -192.000i q^{87} +290.000 q^{89} +684.000 q^{91} -936.000i q^{93} +(120.000 + 60.0000i) q^{95} -376.000i q^{97} +270.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 20 q^{5} - 18 q^{9}+O(q^{10}) \) 2 * q + 20 * q^5 - 18 * q^9	\( 2 q + 20 q^{5} - 18 q^{9} - 60 q^{11} - 30 q^{15} + 24 q^{19} + 108 q^{21} + 150 q^{25} - 128 q^{29} - 624 q^{31} + 180 q^{35} - 228 q^{39} - 748 q^{41} - 180 q^{45} + 38 q^{49} + 420 q^{51} - 600 q^{55} + 1020 q^{59} - 1508 q^{61} - 380 q^{65} + 432 q^{69} - 1848 q^{71} - 600 q^{75} - 144 q^{79} + 162 q^{81} + 700 q^{85} + 580 q^{89} + 1368 q^{91} + 240 q^{95} + 540 q^{99}+O(q^{100}) \) 2 * q + 20 * q^5 - 18 * q^9 - 60 * q^11 - 30 * q^15 + 24 * q^19 + 108 * q^21 + 150 * q^25 - 128 * q^29 - 624 * q^31 + 180 * q^35 - 228 * q^39 - 748 * q^41 - 180 * q^45 + 38 * q^49 + 420 * q^51 - 600 * q^55 + 1020 * q^59 - 1508 * q^61 - 380 * q^65 + 432 * q^69 - 1848 * q^71 - 600 * q^75 - 144 * q^79 + 162 * q^81 + 700 * q^85 + 580 * q^89 + 1368 * q^91 + 240 * q^95 + 540 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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