LMFDB - Embedded newform 960.4.f.a.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 120)
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.a.769.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +(-5.00000 - 10.0000i) q^{5} -4.00000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} +(-5.00000 - 10.0000i) q^{5} -4.00000i q^{7} -9.00000 q^{9} -28.0000 q^{11} -16.0000i q^{13} +(30.0000 - 15.0000i) q^{15} +108.000i q^{17} -32.0000 q^{19} +12.0000 q^{21} -28.0000i q^{23} +(-75.0000 + 100.000i) q^{25} -27.0000i q^{27} -238.000 q^{29} +180.000 q^{31} -84.0000i q^{33} +(-40.0000 + 20.0000i) q^{35} +40.0000i q^{37} +48.0000 q^{39} +422.000 q^{41} -276.000i q^{43} +(45.0000 + 90.0000i) q^{45} -60.0000i q^{47} +327.000 q^{49} -324.000 q^{51} +220.000i q^{53} +(140.000 + 280.000i) q^{55} -96.0000i q^{57} +804.000 q^{59} +358.000 q^{61} +36.0000i q^{63} +(-160.000 + 80.0000i) q^{65} -884.000i q^{67} +84.0000 q^{69} +64.0000 q^{71} +152.000i q^{73} +(-300.000 - 225.000i) q^{75} +112.000i q^{77} -932.000 q^{79} +81.0000 q^{81} +1292.00i q^{83} +(1080.00 - 540.000i) q^{85} -714.000i q^{87} +1146.00 q^{89} -64.0000 q^{91} +540.000i q^{93} +(160.000 + 320.000i) q^{95} +824.000i q^{97} +252.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{5} - 18 q^{9}+O(q^{10}) $ 2 * q - 10 * q^5 - 18 * q^9	$ 2 q - 10 q^{5} - 18 q^{9} - 56 q^{11} + 60 q^{15} - 64 q^{19} + 24 q^{21} - 150 q^{25} - 476 q^{29} + 360 q^{31} - 80 q^{35} + 96 q^{39} + 844 q^{41} + 90 q^{45} + 654 q^{49} - 648 q^{51} + 280 q^{55} + 1608 q^{59} + 716 q^{61} - 320 q^{65} + 168 q^{69} + 128 q^{71} - 600 q^{75} - 1864 q^{79} + 162 q^{81} + 2160 q^{85} + 2292 q^{89} - 128 q^{91} + 320 q^{95} + 504 q^{99}+O(q^{100}) $ 2 * q - 10 * q^5 - 18 * q^9 - 56 * q^11 + 60 * q^15 - 64 * q^19 + 24 * q^21 - 150 * q^25 - 476 * q^29 + 360 * q^31 - 80 * q^35 + 96 * q^39 + 844 * q^41 + 90 * q^45 + 654 * q^49 - 648 * q^51 + 280 * q^55 + 1608 * q^59 + 716 * q^61 - 320 * q^65 + 168 * q^69 + 128 * q^71 - 600 * q^75 - 1864 * q^79 + 162 * q^81 + 2160 * q^85 + 2292 * q^89 - 128 * q^91 + 320 * q^95 + 504 * 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
$5$	−5.00000	−	10.0000i	−0.447214	−	0.894427i
$5$	−5.00000	−	10.0000i	−0.447214	−	0.894427i
$6$		0			0
$7$		−	4.00000i		−	0.215980i	−0.994152	−	0.107990i	$-0.965559\pi$
$7$		−	4.00000i		−	0.215980i	0.994152	−	0.107990i	$-0.0344414\pi$
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	−28.0000			−0.767483			−0.383742	−	0.923440i	$-0.625365\pi$
$11$	−28.0000			−0.767483			−0.383742	+	0.923440i	$0.625365\pi$
$12$		0			0
$13$		−	16.0000i		−	0.341354i	−0.985327	−	0.170677i	$-0.945405\pi$
$13$		−	16.0000i		−	0.341354i	0.985327	−	0.170677i	$-0.0545955\pi$
$14$		0			0
$15$	30.0000	−	15.0000i	0.516398	−	0.258199i
$16$		0			0
$17$			108.000i			1.54081i	0.637552	+	0.770407i	$0.279947\pi$
$17$			108.000i			1.54081i	−0.637552	+	0.770407i	$0.720053\pi$
$18$		0			0
$19$	−32.0000			−0.386384			−0.193192	−	0.981161i	$-0.561884\pi$
$19$	−32.0000			−0.386384			−0.193192	+	0.981161i	$0.561884\pi$
$20$		0			0
$21$	12.0000			0.124696
$22$		0			0
$23$		−	28.0000i		−	0.253844i	−0.991913	−	0.126922i	$-0.959490\pi$
$23$		−	28.0000i		−	0.253844i	0.991913	−	0.126922i	$-0.0405097\pi$
$24$		0			0
$25$	−75.0000	+	100.000i	−0.600000	+	0.800000i
$26$		0			0
$27$		−	27.0000i		−	0.192450i
$28$		0			0
$29$	−238.000			−1.52398			−0.761991	−	0.647587i	$-0.775778\pi$
$29$	−238.000			−1.52398			−0.761991	+	0.647587i	$0.775778\pi$
$30$		0			0
$31$	180.000			1.04287			0.521435	−	0.853291i	$-0.325397\pi$
$31$	180.000			1.04287			0.521435	+	0.853291i	$0.325397\pi$
$32$		0			0
$33$		−	84.0000i		−	0.443107i
$34$		0			0
$35$	−40.0000	+	20.0000i	−0.193178	+	0.0965891i
$36$		0			0
$37$			40.0000i			0.177729i	0.996044	+	0.0888643i	$0.0283238\pi$
$37$			40.0000i			0.177729i	−0.996044	+	0.0888643i	$0.971676\pi$
$38$		0			0
$39$	48.0000			0.197081
$40$		0			0
$41$	422.000			1.60745			0.803724	−	0.595003i	$-0.202849\pi$
$41$	422.000			1.60745			0.803724	+	0.595003i	$0.202849\pi$
$42$		0			0
$43$		−	276.000i		−	0.978828i	−0.872052	−	0.489414i	$-0.837211\pi$
$43$		−	276.000i		−	0.978828i	0.872052	−	0.489414i	$-0.162789\pi$
$44$		0			0
$45$	45.0000	+	90.0000i	0.149071	+	0.298142i
$46$		0			0
$47$		−	60.0000i		−	0.186211i	−0.995656	−	0.0931053i	$-0.970321\pi$
$47$		−	60.0000i		−	0.186211i	0.995656	−	0.0931053i	$-0.0296793\pi$
$48$		0			0
$49$	327.000			0.953353
$50$		0			0
$51$	−324.000			−0.889590
$52$		0			0
$53$			220.000i			0.570176i	0.958501	+	0.285088i	$0.0920228\pi$
$53$			220.000i			0.570176i	−0.958501	+	0.285088i	$0.907977\pi$
$54$		0			0
$55$	140.000	+	280.000i	0.343229	+	0.686458i
$56$		0			0
$57$		−	96.0000i		−	0.223079i
$58$		0			0
$59$	804.000			1.77410			0.887050	−	0.461674i	$-0.152751\pi$
$59$	804.000			1.77410			0.887050	+	0.461674i	$0.152751\pi$
$60$		0			0
$61$	358.000			0.751430			0.375715	−	0.926735i	$-0.377397\pi$
$61$	358.000			0.751430			0.375715	+	0.926735i	$0.377397\pi$
$62$		0			0
$63$			36.0000i			0.0719932i
$64$		0			0
$65$	−160.000	+	80.0000i	−0.305316	+	0.152658i
$66$		0			0
$67$		−	884.000i		−	1.61191i	−0.591979	−	0.805954i	$-0.701653\pi$
$67$		−	884.000i		−	1.61191i	0.591979	−	0.805954i	$-0.298347\pi$
$68$		0			0
$69$	84.0000			0.146557
$70$		0			0
$71$	64.0000			0.106978			0.0534888	−	0.998568i	$-0.482966\pi$
$71$	64.0000			0.106978			0.0534888	+	0.998568i	$0.482966\pi$
$72$		0			0
$73$			152.000i			0.243702i	0.992548	+	0.121851i	$0.0388830\pi$
$73$			152.000i			0.243702i	−0.992548	+	0.121851i	$0.961117\pi$
$74$		0			0
$75$	−300.000	−	225.000i	−0.461880	−	0.346410i
$76$		0			0
$77$			112.000i			0.165761i
$78$		0			0
$79$	−932.000			−1.32732			−0.663659	−	0.748035i	$-0.730998\pi$
$79$	−932.000			−1.32732			−0.663659	+	0.748035i	$0.730998\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$			1292.00i			1.70862i	0.519764	+	0.854310i	$0.326020\pi$
$83$			1292.00i			1.70862i	−0.519764	+	0.854310i	$0.673980\pi$
$84$		0			0
$85$	1080.00	−	540.000i	1.37815	−	0.689073i
$86$		0			0
$87$		−	714.000i		−	0.879872i
$88$		0			0
$89$	1146.00			1.36490			0.682448	−	0.730934i	$-0.260915\pi$
$89$	1146.00			1.36490			0.682448	+	0.730934i	$0.260915\pi$
$90$		0			0
$91$	−64.0000			−0.0737255
$92$		0			0
$93$			540.000i			0.602101i
$94$		0			0
$95$	160.000	+	320.000i	0.172796	+	0.345593i
$96$		0			0
$97$			824.000i			0.862521i	0.902227	+	0.431260i	$0.141931\pi$
$97$			824.000i			0.862521i	−0.902227	+	0.431260i	$0.858069\pi$
$98$		0			0
$99$	252.000			0.255828
$100$		0			0
$101$	1290.00			1.27089			0.635445	−	0.772147i	$-0.280817\pi$
$101$	1290.00			1.27089			0.635445	+	0.772147i	$0.280817\pi$
$102$		0			0
$103$		−	1604.00i		−	1.53444i	−0.641387	−	0.767218i	$-0.721641\pi$
$103$		−	1604.00i		−	1.53444i	0.641387	−	0.767218i	$-0.278359\pi$
$104$		0			0
$105$	−60.0000	−	120.000i	−0.0557657	−	0.111531i
$106$		0			0
$107$			892.000i			0.805915i	0.915219	+	0.402957i	$0.132018\pi$
$107$			892.000i			0.805915i	−0.915219	+	0.402957i	$0.867982\pi$
$108$		0			0
$109$	966.000			0.848863			0.424431	−	0.905460i	$-0.360474\pi$
$109$	966.000			0.848863			0.424431	+	0.905460i	$0.360474\pi$
$110$		0			0
$111$	−120.000			−0.102612
$112$		0			0
$113$		−	1124.00i		−	0.935726i	−0.883801	−	0.467863i	$-0.845024\pi$
$113$		−	1124.00i		−	0.935726i	0.883801	−	0.467863i	$-0.154976\pi$
$114$		0			0
$115$	−280.000	+	140.000i	−0.227045	+	0.113522i
$116$		0			0
$117$			144.000i			0.113785i
$118$		0			0
$119$	432.000			0.332785
$120$		0			0
$121$	−547.000			−0.410969
$122$		0			0
$123$			1266.00i			0.928060i
$124$		0			0
$125$	1375.00	+	250.000i	0.983870	+	0.178885i
$126$		0			0
$127$			1884.00i			1.31636i	0.752860	+	0.658181i	$0.228674\pi$
$127$			1884.00i			1.31636i	−0.752860	+	0.658181i	$0.771326\pi$
$128$		0			0
$129$	828.000			0.565127
$130$		0			0
$131$	−588.000			−0.392166			−0.196083	−	0.980587i	$-0.562822\pi$
$131$	−588.000			−0.392166			−0.196083	+	0.980587i	$0.562822\pi$
$132$		0			0
$133$			128.000i			0.0834512i
$134$		0			0
$135$	−270.000	+	135.000i	−0.172133	+	0.0860663i
$136$		0			0
$137$			1060.00i			0.661036i	0.943800	+	0.330518i	$0.107223\pi$
$137$			1060.00i			0.661036i	−0.943800	+	0.330518i	$0.892777\pi$
$138$		0			0
$139$	2864.00			1.74764			0.873818	−	0.486254i	$-0.161637\pi$
$139$	2864.00			1.74764			0.873818	+	0.486254i	$0.161637\pi$
$140$		0			0
$141$	180.000			0.107509
$142$		0			0
$143$			448.000i			0.261984i
$144$		0			0
$145$	1190.00	+	2380.00i	0.681546	+	1.36309i
$146$		0			0
$147$			981.000i			0.550418i
$148$		0			0
$149$	342.000			0.188038			0.0940192	−	0.995570i	$-0.470028\pi$
$149$	342.000			0.188038			0.0940192	+	0.995570i	$0.470028\pi$
$150$		0			0
$151$	−1636.00			−0.881694			−0.440847	−	0.897582i	$-0.645322\pi$
$151$	−1636.00			−0.881694			−0.440847	+	0.897582i	$0.645322\pi$
$152$		0			0
$153$		−	972.000i		−	0.513605i
$154$		0			0
$155$	−900.000	−	1800.00i	−0.466385	−	0.932771i
$156$		0			0
$157$			2072.00i			1.05327i	0.850091	+	0.526636i	$0.176547\pi$
$157$			2072.00i			1.05327i	−0.850091	+	0.526636i	$0.823453\pi$
$158$		0			0
$159$	−660.000			−0.329191
$160$		0			0
$161$	−112.000			−0.0548251
$162$		0			0
$163$		−	772.000i		−	0.370968i	−0.982647	−	0.185484i	$-0.940615\pi$
$163$		−	772.000i		−	0.370968i	0.982647	−	0.185484i	$-0.0593852\pi$
$164$		0			0
$165$	−840.000	+	420.000i	−0.396327	+	0.198163i
$166$		0			0
$167$			1044.00i			0.483755i	0.970307	+	0.241878i	$0.0777633\pi$
$167$			1044.00i			0.483755i	−0.970307	+	0.241878i	$0.922237\pi$
$168$		0			0
$169$	1941.00			0.883477
$170$		0			0
$171$	288.000			0.128795
$172$		0			0
$173$			4404.00i			1.93543i	0.252041	+	0.967717i	$0.418898\pi$
$173$			4404.00i			1.93543i	−0.252041	+	0.967717i	$0.581102\pi$
$174$		0			0
$175$	400.000	+	300.000i	0.172784	+	0.129588i
$176$		0			0
$177$			2412.00i			1.02428i
$178$		0			0
$179$	−3452.00			−1.44142			−0.720711	−	0.693235i	$-0.756185\pi$
$179$	−3452.00			−1.44142			−0.720711	+	0.693235i	$0.756185\pi$
$180$		0			0
$181$	−526.000			−0.216007			−0.108004	−	0.994151i	$-0.534446\pi$
$181$	−526.000			−0.216007			−0.108004	+	0.994151i	$0.534446\pi$
$182$		0			0
$183$			1074.00i			0.433838i
$184$		0			0
$185$	400.000	−	200.000i	0.158965	−	0.0794827i
$186$		0			0
$187$		−	3024.00i		−	1.18255i
$188$		0			0
$189$	−108.000			−0.0415653
$190$		0			0
$191$	−72.0000			−0.0272761			−0.0136381	−	0.999907i	$-0.504341\pi$
$191$	−72.0000			−0.0272761			−0.0136381	+	0.999907i	$0.504341\pi$
$192$		0			0
$193$		−	208.000i		−	0.0775760i	−0.999247	−	0.0387880i	$-0.987650\pi$
$193$		−	208.000i		−	0.0775760i	0.999247	−	0.0387880i	$-0.0123497\pi$
$194$		0			0
$195$	−240.000	−	480.000i	−0.0881372	−	0.176274i
$196$		0			0
$197$		−	372.000i		−	0.134538i	−0.997735	−	0.0672688i	$-0.978571\pi$
$197$		−	372.000i		−	0.134538i	0.997735	−	0.0672688i	$-0.0214285\pi$
$198$		0			0
$199$	4348.00			1.54885			0.774426	−	0.632665i	$-0.218039\pi$
$199$	4348.00			1.54885			0.774426	+	0.632665i	$0.218039\pi$
$200$		0			0
$201$	2652.00			0.930635
$202$		0			0
$203$			952.000i			0.329149i
$204$		0			0
$205$	−2110.00	−	4220.00i	−0.718872	−	1.43774i
$206$		0			0
$207$			252.000i			0.0846145i
$208$		0			0
$209$	896.000			0.296544
$210$		0			0
$211$	−416.000			−0.135728			−0.0678640	−	0.997695i	$-0.521618\pi$
$211$	−416.000			−0.135728			−0.0678640	+	0.997695i	$0.521618\pi$
$212$		0			0
$213$			192.000i			0.0617635i
$214$		0			0
$215$	−2760.00	+	1380.00i	−0.875490	+	0.437745i
$216$		0			0
$217$		−	720.000i		−	0.225239i
$218$		0			0
$219$	−456.000			−0.140701
$220$		0			0
$221$	1728.00			0.525963
$222$		0			0
$223$		−	5748.00i		−	1.72607i	−0.505141	−	0.863037i	$-0.668559\pi$
$223$		−	5748.00i		−	1.72607i	0.505141	−	0.863037i	$-0.331441\pi$
$224$		0			0
$225$	675.000	−	900.000i	0.200000	−	0.266667i
$226$		0			0
$227$			1148.00i			0.335663i	0.985816	+	0.167831i	$0.0536764\pi$
$227$			1148.00i			0.335663i	−0.985816	+	0.167831i	$0.946324\pi$
$228$		0			0
$229$	3234.00			0.933226			0.466613	−	0.884462i	$-0.345474\pi$
$229$	3234.00			0.933226			0.466613	+	0.884462i	$0.345474\pi$
$230$		0			0
$231$	−336.000			−0.0957021
$232$		0			0
$233$			228.000i			0.0641063i	0.999486	+	0.0320532i	$0.0102046\pi$
$233$			228.000i			0.0641063i	−0.999486	+	0.0320532i	$0.989795\pi$
$234$		0			0
$235$	−600.000	+	300.000i	−0.166552	+	0.0832759i
$236$		0			0
$237$		−	2796.00i		−	0.766328i
$238$		0			0
$239$	4760.00			1.28828			0.644140	−	0.764908i	$-0.277216\pi$
$239$	4760.00			1.28828			0.644140	+	0.764908i	$0.277216\pi$
$240$		0			0
$241$	3230.00			0.863330			0.431665	−	0.902034i	$-0.357926\pi$
$241$	3230.00			0.863330			0.431665	+	0.902034i	$0.357926\pi$
$242$		0			0
$243$			243.000i			0.0641500i
$244$		0			0
$245$	−1635.00	−	3270.00i	−0.426352	−	0.852705i
$246$		0			0
$247$			512.000i			0.131894i
$248$		0			0
$249$	−3876.00			−0.986472
$250$		0			0
$251$	1708.00			0.429514			0.214757	−	0.976668i	$-0.431104\pi$
$251$	1708.00			0.429514			0.214757	+	0.976668i	$0.431104\pi$
$252$		0			0
$253$			784.000i			0.194821i
$254$		0			0
$255$	1620.00	+	3240.00i	0.397837	+	0.795673i
$256$		0			0
$257$			6372.00i			1.54659i	0.634044	+	0.773297i	$0.281394\pi$
$257$			6372.00i			1.54659i	−0.634044	+	0.773297i	$0.718606\pi$
$258$		0			0
$259$	160.000			0.0383858
$260$		0			0
$261$	2142.00			0.507994
$262$		0			0
$263$		−	3036.00i		−	0.711817i	−0.934521	−	0.355908i	$-0.884172\pi$
$263$		−	3036.00i		−	0.711817i	0.934521	−	0.355908i	$-0.115828\pi$
$264$		0			0
$265$	2200.00	−	1100.00i	0.509981	−	0.254990i
$266$		0			0
$267$			3438.00i			0.788023i
$268$		0			0
$269$	−114.000			−0.0258390			−0.0129195	−	0.999917i	$-0.504113\pi$
$269$	−114.000			−0.0258390			−0.0129195	+	0.999917i	$0.504113\pi$
$270$		0			0
$271$	5236.00			1.17367			0.586835	−	0.809707i	$-0.300374\pi$
$271$	5236.00			1.17367			0.586835	+	0.809707i	$0.300374\pi$
$272$		0			0
$273$		−	192.000i		−	0.0425655i
$274$		0			0
$275$	2100.00	−	2800.00i	0.460490	−	0.613987i
$276$		0			0
$277$			5712.00i			1.23899i	0.785000	+	0.619496i	$0.212663\pi$
$277$			5712.00i			1.23899i	−0.785000	+	0.619496i	$0.787337\pi$
$278$		0			0
$279$	−1620.00			−0.347623
$280$		0			0
$281$	3222.00			0.684016			0.342008	−	0.939697i	$-0.388893\pi$
$281$	3222.00			0.684016			0.342008	+	0.939697i	$0.388893\pi$
$282$		0			0
$283$			4620.00i			0.970426i	0.874396	+	0.485213i	$0.161258\pi$
$283$			4620.00i			0.970426i	−0.874396	+	0.485213i	$0.838742\pi$
$284$		0			0
$285$	−960.000	+	480.000i	−0.199528	+	0.0997640i
$286$		0			0
$287$		−	1688.00i		−	0.347176i
$288$		0			0
$289$	−6751.00			−1.37411
$290$		0			0
$291$	−2472.00			−0.497977
$292$		0			0
$293$		−	5404.00i		−	1.07749i	−0.842468	−	0.538746i	$-0.818898\pi$
$293$		−	5404.00i		−	1.07749i	0.842468	−	0.538746i	$-0.181102\pi$
$294$		0			0
$295$	−4020.00	−	8040.00i	−0.793402	−	1.58680i
$296$		0			0
$297$			756.000i			0.147702i
$298$		0			0
$299$	−448.000			−0.0866505
$300$		0			0
$301$	−1104.00			−0.211407
$302$		0			0
$303$			3870.00i			0.733748i
$304$		0			0
$305$	−1790.00	−	3580.00i	−0.336050	−	0.672099i
$306$		0			0
$307$		−	9700.00i		−	1.80328i	−0.432483	−	0.901642i	$-0.642362\pi$
$307$		−	9700.00i		−	1.80328i	0.432483	−	0.901642i	$-0.357638\pi$
$308$		0			0
$309$	4812.00			0.885907
$310$		0			0
$311$	−9672.00			−1.76350			−0.881750	−	0.471716i	$-0.843635\pi$
$311$	−9672.00			−1.76350			−0.881750	+	0.471716i	$0.843635\pi$
$312$		0			0
$313$		−	4048.00i		−	0.731011i	−0.930809	−	0.365506i	$-0.880896\pi$
$313$		−	4048.00i		−	0.731011i	0.930809	−	0.365506i	$-0.119104\pi$
$314$		0			0
$315$	360.000	−	180.000i	0.0643927	−	0.0321964i
$316$		0			0
$317$			84.0000i			0.0148830i	0.999972	+	0.00744150i	$0.00236872\pi$
$317$			84.0000i			0.0148830i	−0.999972	+	0.00744150i	$0.997631\pi$
$318$		0			0
$319$	6664.00			1.16963
$320$		0			0
$321$	−2676.00			−0.465295
$322$		0			0
$323$		−	3456.00i		−	0.595347i
$324$		0			0
$325$	1600.00	+	1200.00i	0.273083	+	0.204812i
$326$		0			0
$327$			2898.00i			0.490091i
$328$		0			0
$329$	−240.000			−0.0402177
$330$		0			0
$331$	5416.00			0.899366			0.449683	−	0.893188i	$-0.351537\pi$
$331$	5416.00			0.899366			0.449683	+	0.893188i	$0.351537\pi$
$332$		0			0
$333$		−	360.000i		−	0.0592429i
$334$		0			0
$335$	−8840.00	+	4420.00i	−1.44173	+	0.720867i
$336$		0			0
$337$		−	8216.00i		−	1.32805i	−0.747709	−	0.664027i	$-0.768846\pi$
$337$		−	8216.00i		−	1.32805i	0.747709	−	0.664027i	$-0.231154\pi$
$338$		0			0
$339$	3372.00			0.540242
$340$		0			0
$341$	−5040.00			−0.800385
$342$		0			0
$343$		−	2680.00i		−	0.421885i
$344$		0			0
$345$	−420.000	−	840.000i	−0.0655421	−	0.131084i
$346$		0			0
$347$			3836.00i			0.593450i	0.954963	+	0.296725i	$0.0958945\pi$
$347$			3836.00i			0.593450i	−0.954963	+	0.296725i	$0.904105\pi$
$348$		0			0
$349$	−2038.00			−0.312583			−0.156292	−	0.987711i	$-0.549954\pi$
$349$	−2038.00			−0.312583			−0.156292	+	0.987711i	$0.549954\pi$
$350$		0			0
$351$	−432.000			−0.0656936
$352$		0			0
$353$		−	5292.00i		−	0.797917i	−0.916969	−	0.398959i	$-0.869372\pi$
$353$		−	5292.00i		−	0.797917i	0.916969	−	0.398959i	$-0.130628\pi$
$354$		0			0
$355$	−320.000	−	640.000i	−0.0478418	−	0.0956836i
$356$		0			0
$357$			1296.00i			0.192133i
$358$		0			0
$359$	−3896.00			−0.572766			−0.286383	−	0.958115i	$-0.592453\pi$
$359$	−3896.00			−0.572766			−0.286383	+	0.958115i	$0.592453\pi$
$360$		0			0
$361$	−5835.00			−0.850707
$362$		0			0
$363$		−	1641.00i		−	0.237273i
$364$		0			0
$365$	1520.00	−	760.000i	0.217974	−	0.108987i
$366$		0			0
$367$		−	7652.00i		−	1.08837i	−0.838966	−	0.544184i	$-0.816839\pi$
$367$		−	7652.00i		−	1.08837i	0.838966	−	0.544184i	$-0.183161\pi$
$368$		0			0
$369$	−3798.00			−0.535816
$370$		0			0
$371$	880.000			0.123146
$372$		0			0
$373$		−	1576.00i		−	0.218773i	−0.993999	−	0.109386i	$-0.965111\pi$
$373$		−	1576.00i		−	0.218773i	0.993999	−	0.109386i	$-0.0348886\pi$
$374$		0			0
$375$	−750.000	+	4125.00i	−0.103280	+	0.568038i
$376$		0			0
$377$			3808.00i			0.520217i
$378$		0			0
$379$	5416.00			0.734040			0.367020	−	0.930213i	$-0.380378\pi$
$379$	5416.00			0.734040			0.367020	+	0.930213i	$0.380378\pi$
$380$		0			0
$381$	−5652.00			−0.760002
$382$		0			0
$383$			8292.00i			1.10627i	0.833092	+	0.553135i	$0.186569\pi$
$383$			8292.00i			1.10627i	−0.833092	+	0.553135i	$0.813431\pi$
$384$		0			0
$385$	1120.00	−	560.000i	0.148261	−	0.0741305i
$386$		0			0
$387$			2484.00i			0.326276i
$388$		0			0
$389$	−9642.00			−1.25673			−0.628366	−	0.777918i	$-0.716276\pi$
$389$	−9642.00			−1.25673			−0.628366	+	0.777918i	$0.716276\pi$
$390$		0			0
$391$	3024.00			0.391126
$392$		0			0
$393$		−	1764.00i		−	0.226417i
$394$		0			0
$395$	4660.00	+	9320.00i	0.593595	+	1.18719i
$396$		0			0
$397$			13032.0i			1.64750i	0.566954	+	0.823750i	$0.308122\pi$
$397$			13032.0i			1.64750i	−0.566954	+	0.823750i	$0.691878\pi$
$398$		0			0
$399$	−384.000			−0.0481806
$400$		0			0
$401$	−13358.0			−1.66351			−0.831754	−	0.555144i	$-0.812663\pi$
$401$	−13358.0			−1.66351			−0.831754	+	0.555144i	$0.812663\pi$
$402$		0			0
$403$		−	2880.00i		−	0.355988i
$404$		0			0
$405$	−405.000	−	810.000i	−0.0496904	−	0.0993808i
$406$		0			0
$407$		−	1120.00i		−	0.136404i
$408$		0			0
$409$	−6410.00			−0.774949			−0.387474	−	0.921880i	$-0.626652\pi$
$409$	−6410.00			−0.774949			−0.387474	+	0.921880i	$0.626652\pi$
$410$		0			0
$411$	−3180.00			−0.381649
$412$		0			0
$413$		−	3216.00i		−	0.383170i
$414$		0			0
$415$	12920.0	−	6460.00i	1.52824	−	0.764118i
$416$		0			0
$417$			8592.00i			1.00900i
$418$		0			0
$419$	−7644.00			−0.891250			−0.445625	−	0.895220i	$-0.647019\pi$
$419$	−7644.00			−0.891250			−0.445625	+	0.895220i	$0.647019\pi$
$420$		0			0
$421$	−14674.0			−1.69873			−0.849367	−	0.527803i	$-0.823016\pi$
$421$	−14674.0			−1.69873			−0.849367	+	0.527803i	$0.823016\pi$
$422$		0			0
$423$			540.000i			0.0620702i
$424$		0			0
$425$	−10800.0	−	8100.00i	−1.23265	−	0.924489i
$426$		0			0
$427$		−	1432.00i		−	0.162294i
$428$		0			0
$429$	−1344.00			−0.151256
$430$		0			0
$431$	9704.00			1.08451			0.542257	−	0.840213i	$-0.317570\pi$
$431$	9704.00			1.08451			0.542257	+	0.840213i	$0.317570\pi$
$432$		0			0
$433$		−	1296.00i		−	0.143838i	−0.997410	−	0.0719189i	$-0.977088\pi$
$433$		−	1296.00i		−	0.143838i	0.997410	−	0.0719189i	$-0.0229123\pi$
$434$		0			0
$435$	−7140.00	+	3570.00i	−0.786981	+	0.393491i
$436$		0			0
$437$			896.000i			0.0980812i
$438$		0			0
$439$	−15684.0			−1.70514			−0.852570	−	0.522613i	$-0.824957\pi$
$439$	−15684.0			−1.70514			−0.852570	+	0.522613i	$0.824957\pi$
$440$		0			0
$441$	−2943.00			−0.317784
$442$		0			0
$443$			5772.00i			0.619043i	0.950892	+	0.309521i	$0.100169\pi$
$443$			5772.00i			0.619043i	−0.950892	+	0.309521i	$0.899831\pi$
$444$		0			0
$445$	−5730.00	−	11460.0i	−0.610400	−	1.22080i
$446$		0			0
$447$			1026.00i			0.108564i
$448$		0			0
$449$	4782.00			0.502620			0.251310	−	0.967907i	$-0.419139\pi$
$449$	4782.00			0.502620			0.251310	+	0.967907i	$0.419139\pi$
$450$		0			0
$451$	−11816.0			−1.23369
$452$		0			0
$453$		−	4908.00i		−	0.509046i
$454$		0			0
$455$	320.000	+	640.000i	0.0329711	+	0.0659421i
$456$		0			0
$457$			15000.0i			1.53538i	0.640819	+	0.767692i	$0.278595\pi$
$457$			15000.0i			1.53538i	−0.640819	+	0.767692i	$0.721405\pi$
$458$		0			0
$459$	2916.00			0.296530
$460$		0			0
$461$	3762.00			0.380073			0.190037	−	0.981777i	$-0.439139\pi$
$461$	3762.00			0.380073			0.190037	+	0.981777i	$0.439139\pi$
$462$		0			0
$463$			5036.00i			0.505492i	0.967533	+	0.252746i	$0.0813337\pi$
$463$			5036.00i			0.505492i	−0.967533	+	0.252746i	$0.918666\pi$
$464$		0			0
$465$	5400.00	−	2700.00i	0.538535	−	0.269268i
$466$		0			0
$467$			2268.00i			0.224733i	0.993667	+	0.112367i	$0.0358431\pi$
$467$			2268.00i			0.224733i	−0.993667	+	0.112367i	$0.964157\pi$
$468$		0			0
$469$	−3536.00			−0.348139
$470$		0			0
$471$	−6216.00			−0.608106
$472$		0			0
$473$			7728.00i			0.751234i
$474$		0			0
$475$	2400.00	−	3200.00i	0.231831	−	0.309108i
$476$		0			0
$477$		−	1980.00i		−	0.190059i
$478$		0			0
$479$	16208.0			1.54606			0.773030	−	0.634370i	$-0.218740\pi$
$479$	16208.0			1.54606			0.773030	+	0.634370i	$0.218740\pi$
$480$		0			0
$481$	640.000			0.0606684
$482$		0			0
$483$		−	336.000i		−	0.0316533i
$484$		0			0
$485$	8240.00	−	4120.00i	0.771462	−	0.385731i
$486$		0			0
$487$		−	11572.0i		−	1.07675i	−0.842705	−	0.538375i	$-0.819038\pi$
$487$		−	11572.0i		−	1.07675i	0.842705	−	0.538375i	$-0.180962\pi$
$488$		0			0
$489$	2316.00			0.214178
$490$		0			0
$491$	5636.00			0.518023			0.259011	−	0.965874i	$-0.416603\pi$
$491$	5636.00			0.518023			0.259011	+	0.965874i	$0.416603\pi$
$492$		0			0
$493$		−	25704.0i		−	2.34817i
$494$		0			0
$495$	−1260.00	−	2520.00i	−0.114410	−	0.228819i
$496$		0			0
$497$		−	256.000i		−	0.0231050i
$498$		0			0
$499$	5560.00			0.498797			0.249399	−	0.968401i	$-0.419767\pi$
$499$	5560.00			0.498797			0.249399	+	0.968401i	$0.419767\pi$
$500$		0			0
$501$	−3132.00			−0.279296
$502$		0			0
$503$			15172.0i			1.34490i	0.740141	+	0.672451i	$0.234759\pi$
$503$			15172.0i			1.34490i	−0.740141	+	0.672451i	$0.765241\pi$
$504$		0			0
$505$	−6450.00	−	12900.0i	−0.568359	−	1.13672i
$506$		0			0
$507$			5823.00i			0.510076i
$508$		0			0
$509$	−17342.0			−1.51016			−0.755079	−	0.655634i	$-0.772402\pi$
$509$	−17342.0			−1.51016			−0.755079	+	0.655634i	$0.772402\pi$
$510$		0			0
$511$	608.000			0.0526347
$512$		0			0
$513$			864.000i			0.0743597i
$514$		0			0
$515$	−16040.0	+	8020.00i	−1.37244	+	0.686220i
$516$		0			0
$517$			1680.00i			0.142914i
$518$		0			0
$519$	−13212.0			−1.11742
$520$		0			0
$521$	−4886.00			−0.410863			−0.205431	−	0.978672i	$-0.565860\pi$
$521$	−4886.00			−0.410863			−0.205431	+	0.978672i	$0.565860\pi$
$522$		0			0
$523$		−	18548.0i		−	1.55076i	−0.631495	−	0.775380i	$-0.717558\pi$
$523$		−	18548.0i		−	1.55076i	0.631495	−	0.775380i	$-0.282442\pi$
$524$		0			0
$525$	−900.000	+	1200.00i	−0.0748176	+	0.0997567i
$526$		0			0
$527$			19440.0i			1.60687i
$528$		0			0
$529$	11383.0			0.935563
$530$		0			0
$531$	−7236.00			−0.591367
$532$		0			0
$533$		−	6752.00i		−	0.548708i
$534$		0			0
$535$	8920.00	−	4460.00i	0.720832	−	0.360416i
$536$		0			0
$537$		−	10356.0i		−	0.832206i
$538$		0			0
$539$	−9156.00			−0.731682
$540$		0			0
$541$	15770.0			1.25324			0.626622	−	0.779323i	$-0.284437\pi$
$541$	15770.0			1.25324			0.626622	+	0.779323i	$0.284437\pi$
$542$		0			0
$543$		−	1578.00i		−	0.124712i
$544$		0			0
$545$	−4830.00	−	9660.00i	−0.379623	−	0.759246i
$546$		0			0
$547$		−	7700.00i		−	0.601880i	−0.953643	−	0.300940i	$-0.902700\pi$
$547$		−	7700.00i		−	0.601880i	0.953643	−	0.300940i	$-0.0973004\pi$
$548$		0			0
$549$	−3222.00			−0.250477
$550$		0			0
$551$	7616.00			0.588843
$552$		0			0
$553$			3728.00i			0.286674i
$554$		0			0
$555$	600.000	+	1200.00i	0.0458893	+	0.0917787i
$556$		0			0
$557$			19236.0i			1.46330i	0.681683	+	0.731648i	$0.261248\pi$
$557$			19236.0i			1.46330i	−0.681683	+	0.731648i	$0.738752\pi$
$558$		0			0
$559$	−4416.00			−0.334127
$560$		0			0
$561$	9072.00			0.682745
$562$		0			0
$563$		−	8388.00i		−	0.627908i	−0.949438	−	0.313954i	$-0.898346\pi$
$563$		−	8388.00i		−	0.627908i	0.949438	−	0.313954i	$-0.101654\pi$
$564$		0			0
$565$	−11240.0	+	5620.00i	−0.836939	+	0.418469i
$566$		0			0
$567$		−	324.000i		−	0.0239977i
$568$		0			0
$569$	16758.0			1.23468			0.617339	−	0.786697i	$-0.288211\pi$
$569$	16758.0			1.23468			0.617339	+	0.786697i	$0.288211\pi$
$570$		0			0
$571$	8056.00			0.590426			0.295213	−	0.955432i	$-0.404609\pi$
$571$	8056.00			0.590426			0.295213	+	0.955432i	$0.404609\pi$
$572$		0			0
$573$		−	216.000i		−	0.0157479i
$574$		0			0
$575$	2800.00	+	2100.00i	0.203075	+	0.152306i
$576$		0			0
$577$			5728.00i			0.413275i	0.978418	+	0.206638i	$0.0662521\pi$
$577$			5728.00i			0.413275i	−0.978418	+	0.206638i	$0.933748\pi$
$578$		0			0
$579$	624.000			0.0447885
$580$		0			0
$581$	5168.00			0.369027
$582$		0			0
$583$		−	6160.00i		−	0.437601i
$584$		0			0
$585$	1440.00	−	720.000i	0.101772	−	0.0508860i
$586$		0			0
$587$			12172.0i			0.855864i	0.903811	+	0.427932i	$0.140758\pi$
$587$			12172.0i			0.855864i	−0.903811	+	0.427932i	$0.859242\pi$
$588$		0			0
$589$	−5760.00			−0.402948
$590$		0			0
$591$	1116.00			0.0776753
$592$		0			0
$593$		−	10708.0i		−	0.741526i	−0.928728	−	0.370763i	$-0.879096\pi$
$593$		−	10708.0i		−	0.741526i	0.928728	−	0.370763i	$-0.120904\pi$
$594$		0			0
$595$	−2160.00	−	4320.00i	−0.148826	−	0.297652i
$596$		0			0
$597$			13044.0i			0.894230i
$598$		0			0
$599$	−9416.00			−0.642283			−0.321141	−	0.947031i	$-0.604066\pi$
$599$	−9416.00			−0.642283			−0.321141	+	0.947031i	$0.604066\pi$
$600$		0			0
$601$	9270.00			0.629170			0.314585	−	0.949229i	$-0.398135\pi$
$601$	9270.00			0.629170			0.314585	+	0.949229i	$0.398135\pi$
$602$		0			0
$603$			7956.00i			0.537302i
$604$		0			0
$605$	2735.00	+	5470.00i	0.183791	+	0.367582i
$606$		0			0
$607$			7996.00i			0.534675i	0.963603	+	0.267337i	$0.0861438\pi$
$607$			7996.00i			0.534675i	−0.963603	+	0.267337i	$0.913856\pi$
$608$		0			0
$609$	−2856.00			−0.190034
$610$		0			0
$611$	−960.000			−0.0635637
$612$		0			0
$613$		−	232.000i		−	0.0152861i	−0.999971	−	0.00764306i	$-0.997567\pi$
$613$		−	232.000i		−	0.0152861i	0.999971	−	0.00764306i	$-0.00243289\pi$
$614$		0			0
$615$	12660.0	−	6330.00i	0.830082	−	0.415041i
$616$		0			0
$617$			3740.00i			0.244030i	0.992528	+	0.122015i	$0.0389357\pi$
$617$			3740.00i			0.244030i	−0.992528	+	0.122015i	$0.961064\pi$
$618$		0			0
$619$	26000.0			1.68825			0.844126	−	0.536145i	$-0.180120\pi$
$619$	26000.0			1.68825			0.844126	+	0.536145i	$0.180120\pi$
$620$		0			0
$621$	−756.000			−0.0488522
$622$		0			0
$623$		−	4584.00i		−	0.294790i
$624$		0			0
$625$	−4375.00	−	15000.0i	−0.280000	−	0.960000i
$626$		0			0
$627$			2688.00i			0.171210i
$628$		0			0
$629$	−4320.00			−0.273847
$630$		0			0
$631$	−11660.0			−0.735622			−0.367811	−	0.929901i	$-0.619893\pi$
$631$	−11660.0			−0.735622			−0.367811	+	0.929901i	$0.619893\pi$
$632$		0			0
$633$		−	1248.00i		−	0.0783626i
$634$		0			0
$635$	18840.0	−	9420.00i	1.17739	−	0.588695i
$636$		0			0
$637$		−	5232.00i		−	0.325431i
$638$		0			0
$639$	−576.000			−0.0356592
$640$		0			0
$641$	−7602.00			−0.468426			−0.234213	−	0.972185i	$-0.575251\pi$
$641$	−7602.00			−0.468426			−0.234213	+	0.972185i	$0.575251\pi$
$642$		0			0
$643$		−	29268.0i		−	1.79505i	−0.440963	−	0.897525i	$-0.645363\pi$
$643$		−	29268.0i		−	1.79505i	0.440963	−	0.897525i	$-0.354637\pi$
$644$		0			0
$645$	−4140.00	−	8280.00i	−0.252732	−	0.505465i
$646$		0			0
$647$		−	17836.0i		−	1.08378i	−0.840449	−	0.541890i	$-0.817709\pi$
$647$		−	17836.0i		−	1.08378i	0.840449	−	0.541890i	$-0.182291\pi$
$648$		0			0
$649$	−22512.0			−1.36159
$650$		0			0
$651$	2160.00			0.130042
$652$		0			0
$653$		−	19188.0i		−	1.14990i	−0.818189	−	0.574950i	$-0.805022\pi$
$653$		−	19188.0i		−	1.14990i	0.818189	−	0.574950i	$-0.194978\pi$
$654$		0			0
$655$	2940.00	+	5880.00i	0.175382	+	0.350764i
$656$		0			0
$657$		−	1368.00i		−	0.0812340i
$658$		0			0
$659$	−13860.0			−0.819285			−0.409643	−	0.912246i	$-0.634347\pi$
$659$	−13860.0			−0.819285			−0.409643	+	0.912246i	$0.634347\pi$
$660$		0			0
$661$	16558.0			0.974329			0.487165	−	0.873310i	$-0.338031\pi$
$661$	16558.0			0.974329			0.487165	+	0.873310i	$0.338031\pi$
$662$		0			0
$663$			5184.00i			0.303665i
$664$		0			0
$665$	1280.00	−	640.000i	0.0746410	−	0.0373205i
$666$		0			0
$667$			6664.00i			0.386853i
$668$		0			0
$669$	17244.0			0.996549
$670$		0			0
$671$	−10024.0			−0.576710
$672$		0			0
$673$			4640.00i			0.265764i	0.991132	+	0.132882i	$0.0424231\pi$
$673$			4640.00i			0.265764i	−0.991132	+	0.132882i	$0.957577\pi$
$674$		0			0
$675$	2700.00	+	2025.00i	0.153960	+	0.115470i
$676$		0			0
$677$			34260.0i			1.94493i	0.233045	+	0.972466i	$0.425131\pi$
$677$			34260.0i			1.94493i	−0.233045	+	0.972466i	$0.574869\pi$
$678$		0			0
$679$	3296.00			0.186287
$680$		0			0
$681$	−3444.00			−0.193795
$682$		0			0
$683$			19420.0i			1.08797i	0.839094	+	0.543987i	$0.183086\pi$
$683$			19420.0i			1.08797i	−0.839094	+	0.543987i	$0.816914\pi$
$684$		0			0
$685$	10600.0	−	5300.00i	0.591248	−	0.295624i
$686$		0			0
$687$			9702.00i			0.538798i
$688$		0			0
$689$	3520.00			0.194632
$690$		0			0
$691$	4608.00			0.253685			0.126843	−	0.991923i	$-0.459516\pi$
$691$	4608.00			0.253685			0.126843	+	0.991923i	$0.459516\pi$
$692$		0			0
$693$		−	1008.00i		−	0.0552536i
$694$		0			0
$695$	−14320.0	−	28640.0i	−0.781566	−	1.56313i
$696$		0			0
$697$			45576.0i			2.47678i
$698$		0			0
$699$	−684.000			−0.0370118
$700$		0			0
$701$	2318.00			0.124893			0.0624463	−	0.998048i	$-0.480110\pi$
$701$	2318.00			0.124893			0.0624463	+	0.998048i	$0.480110\pi$
$702$		0			0
$703$		−	1280.00i		−	0.0686716i
$704$		0			0
$705$	−900.000	−	1800.00i	−0.0480794	−	0.0961588i
$706$		0			0
$707$		−	5160.00i		−	0.274486i
$708$		0			0
$709$	16834.0			0.891698			0.445849	−	0.895108i	$-0.352902\pi$
$709$	16834.0			0.891698			0.445849	+	0.895108i	$0.352902\pi$
$710$		0			0
$711$	8388.00			0.442440
$712$		0			0
$713$		−	5040.00i		−	0.264726i
$714$		0			0
$715$	4480.00	−	2240.00i	0.234325	−	0.117163i
$716$		0			0
$717$			14280.0i			0.743789i
$718$		0			0
$719$	−7400.00			−0.383830			−0.191915	−	0.981412i	$-0.561470\pi$
$719$	−7400.00			−0.383830			−0.191915	+	0.981412i	$0.561470\pi$
$720$		0			0
$721$	−6416.00			−0.331407
$722$		0			0
$723$			9690.00i			0.498444i
$724$		0			0
$725$	17850.0	−	23800.0i	0.914389	−	1.21919i
$726$		0			0
$727$		−	20340.0i		−	1.03765i	−0.854882	−	0.518823i	$-0.826370\pi$
$727$		−	20340.0i		−	1.03765i	0.854882	−	0.518823i	$-0.173630\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$	29808.0			1.50819
$732$		0			0
$733$			4896.00i			0.246709i	0.992363	+	0.123355i	$0.0393653\pi$
$733$			4896.00i			0.246709i	−0.992363	+	0.123355i	$0.960635\pi$
$734$		0			0
$735$	9810.00	−	4905.00i	0.492309	−	0.246155i
$736$		0			0
$737$			24752.0i			1.23711i
$738$		0			0
$739$	−26040.0			−1.29621			−0.648103	−	0.761552i	$-0.724438\pi$
$739$	−26040.0			−1.29621			−0.648103	+	0.761552i	$0.724438\pi$
$740$		0			0
$741$	−1536.00			−0.0761489
$742$		0			0
$743$		−	6780.00i		−	0.334770i	−0.985892	−	0.167385i	$-0.946468\pi$
$743$		−	6780.00i		−	0.334770i	0.985892	−	0.167385i	$-0.0535323\pi$
$744$		0			0
$745$	−1710.00	−	3420.00i	−0.0840934	−	0.168187i
$746$		0			0
$747$		−	11628.0i		−	0.569540i
$748$		0			0
$749$	3568.00			0.174061
$750$		0			0
$751$	20692.0			1.00541			0.502704	−	0.864458i	$-0.332338\pi$
$751$	20692.0			1.00541			0.502704	+	0.864458i	$0.332338\pi$
$752$		0			0
$753$			5124.00i			0.247980i
$754$		0			0
$755$	8180.00	+	16360.0i	0.394306	+	0.788611i
$756$		0			0
$757$			10816.0i			0.519305i	0.965702	+	0.259653i	$0.0836081\pi$
$757$			10816.0i			0.519305i	−0.965702	+	0.259653i	$0.916392\pi$
$758$		0			0
$759$	−2352.00			−0.112480
$760$		0			0
$761$	13978.0			0.665837			0.332919	−	0.942956i	$-0.391967\pi$
$761$	13978.0			0.665837			0.332919	+	0.942956i	$0.391967\pi$
$762$		0			0
$763$		−	3864.00i		−	0.183337i
$764$		0			0
$765$	−9720.00	+	4860.00i	−0.459382	+	0.229691i
$766$		0			0
$767$		−	12864.0i		−	0.605596i
$768$		0			0
$769$	−2926.00			−0.137210			−0.0686048	−	0.997644i	$-0.521855\pi$
$769$	−2926.00			−0.137210			−0.0686048	+	0.997644i	$0.521855\pi$
$770$		0			0
$771$	−19116.0			−0.892926
$772$		0			0
$773$			13916.0i			0.647508i	0.946141	+	0.323754i	$0.104945\pi$
$773$			13916.0i			0.647508i	−0.946141	+	0.323754i	$0.895055\pi$
$774$		0			0
$775$	−13500.0	+	18000.0i	−0.625722	+	0.834296i
$776$		0			0
$777$			480.000i			0.0221620i
$778$		0			0
$779$	−13504.0			−0.621092
$780$		0			0
$781$	−1792.00			−0.0821035
$782$		0			0
$783$			6426.00i			0.293291i
$784$		0			0
$785$	20720.0	−	10360.0i	0.942075	−	0.471037i
$786$		0			0
$787$			29996.0i			1.35863i	0.733847	+	0.679315i	$0.237723\pi$
$787$			29996.0i			1.35863i	−0.733847	+	0.679315i	$0.762277\pi$
$788$		0			0
$789$	9108.00			0.410968
$790$		0			0
$791$	−4496.00			−0.202098
$792$		0			0
$793$		−	5728.00i		−	0.256503i
$794$		0			0
$795$	3300.00	+	6600.00i	0.147219	+	0.294438i
$796$		0			0
$797$			8940.00i			0.397329i	0.980068	+	0.198664i	$0.0636604\pi$
$797$			8940.00i			0.397329i	−0.980068	+	0.198664i	$0.936340\pi$
$798$		0			0
$799$	6480.00			0.286916
$800$		0			0
$801$	−10314.0			−0.454965
$802$		0			0
$803$		−	4256.00i		−	0.187037i
$804$		0			0
$805$	560.000	+	1120.00i	0.0245185	+	0.0490370i
$806$		0			0
$807$		−	342.000i		−	0.0149182i
$808$		0			0
$809$	10698.0			0.464922			0.232461	−	0.972606i	$-0.425322\pi$
$809$	10698.0			0.464922			0.232461	+	0.972606i	$0.425322\pi$
$810$		0			0
$811$	−6408.00			−0.277454			−0.138727	−	0.990331i	$-0.544301\pi$
$811$	−6408.00			−0.277454			−0.138727	+	0.990331i	$0.544301\pi$
$812$		0			0
$813$			15708.0i			0.677618i
$814$		0			0
$815$	−7720.00	+	3860.00i	−0.331803	+	0.165902i
$816$		0			0
$817$			8832.00i			0.378204i
$818$		0			0
$819$	576.000			0.0245752
$820$		0			0
$821$	−23130.0			−0.983243			−0.491622	−	0.870809i	$-0.663596\pi$
$821$	−23130.0			−0.983243			−0.491622	+	0.870809i	$0.663596\pi$
$822$		0			0
$823$			11852.0i			0.501986i	0.967989	+	0.250993i	$0.0807572\pi$
$823$			11852.0i			0.501986i	−0.967989	+	0.250993i	$0.919243\pi$
$824$		0			0
$825$	8400.00	+	6300.00i	0.354485	+	0.265864i
$826$		0			0
$827$		−	32628.0i		−	1.37193i	−0.727634	−	0.685965i	$-0.759380\pi$
$827$		−	32628.0i		−	1.37193i	0.727634	−	0.685965i	$-0.240620\pi$
$828$		0			0
$829$	36694.0			1.53732			0.768658	−	0.639660i	$-0.220925\pi$
$829$	36694.0			1.53732			0.768658	+	0.639660i	$0.220925\pi$
$830$		0			0
$831$	−17136.0			−0.715332
$832$		0			0
$833$			35316.0i			1.46894i
$834$		0			0
$835$	10440.0	−	5220.00i	0.432684	−	0.216342i
$836$		0			0
$837$		−	4860.00i		−	0.200700i
$838$		0			0
$839$	−1704.00			−0.0701175			−0.0350588	−	0.999385i	$-0.511162\pi$
$839$	−1704.00			−0.0701175			−0.0350588	+	0.999385i	$0.511162\pi$
$840$		0			0
$841$	32255.0			1.32252
$842$		0			0
$843$			9666.00i			0.394917i
$844$		0			0
$845$	−9705.00	−	19410.0i	−0.395103	−	0.790206i
$846$		0			0
$847$			2188.00i			0.0887610i
$848$		0			0
$849$	−13860.0			−0.560276
$850$		0			0
$851$	1120.00			0.0451153
$852$		0			0
$853$			31880.0i			1.27966i	0.768516	+	0.639830i	$0.220995\pi$
$853$			31880.0i			1.27966i	−0.768516	+	0.639830i	$0.779005\pi$
$854$		0			0
$855$	−1440.00	−	2880.00i	−0.0575988	−	0.115198i
$856$		0			0
$857$		−	7972.00i		−	0.317758i	−0.987298	−	0.158879i	$-0.949212\pi$
$857$		−	7972.00i		−	0.317758i	0.987298	−	0.158879i	$-0.0507879\pi$
$858$		0			0
$859$	6008.00			0.238638			0.119319	−	0.992856i	$-0.461929\pi$
$859$	6008.00			0.238638			0.119319	+	0.992856i	$0.461929\pi$
$860$		0			0
$861$	5064.00			0.200442
$862$		0			0
$863$			1716.00i			0.0676863i	0.999427	+	0.0338432i	$0.0107747\pi$
$863$			1716.00i			0.0676863i	−0.999427	+	0.0338432i	$0.989225\pi$
$864$		0			0
$865$	44040.0	−	22020.0i	1.73110	−	0.865552i
$866$		0			0
$867$		−	20253.0i		−	0.793342i
$868$		0			0
$869$	26096.0			1.01870
$870$		0			0
$871$	−14144.0			−0.550231
$872$		0			0
$873$		−	7416.00i		−	0.287507i
$874$		0			0
$875$	1000.00	−	5500.00i	0.0386356	−	0.212496i
$876$		0			0
$877$			9032.00i			0.347764i	0.984767	+	0.173882i	$0.0556311\pi$
$877$			9032.00i			0.347764i	−0.984767	+	0.173882i	$0.944369\pi$
$878$		0			0
$879$	16212.0			0.622090
$880$		0			0
$881$	27838.0			1.06457			0.532285	−	0.846565i	$-0.321334\pi$
$881$	27838.0			1.06457			0.532285	+	0.846565i	$0.321334\pi$
$882$		0			0
$883$			4316.00i			0.164490i	0.996612	+	0.0822452i	$0.0262091\pi$
$883$			4316.00i			0.164490i	−0.996612	+	0.0822452i	$0.973791\pi$
$884$		0			0
$885$	24120.0	−	12060.0i	0.916141	−	0.458071i
$886$		0			0
$887$			43524.0i			1.64757i	0.566904	+	0.823784i	$0.308141\pi$
$887$			43524.0i			1.64757i	−0.566904	+	0.823784i	$0.691859\pi$
$888$		0			0
$889$	7536.00			0.284307
$890$		0			0
$891$	−2268.00			−0.0852759
$892$		0			0
$893$			1920.00i			0.0719489i
$894$		0			0
$895$	17260.0	+	34520.0i	0.644624	+	1.28925i
$896$		0			0
$897$		−	1344.00i		−	0.0500277i
$898$		0			0
$899$	−42840.0			−1.58931
$900$		0			0
$901$	−23760.0			−0.878535
$902$		0			0
$903$		−	3312.00i		−	0.122056i
$904$		0			0
$905$	2630.00	+	5260.00i	0.0966013	+	0.193203i
$906$		0			0
$907$			10556.0i			0.386446i	0.981155	+	0.193223i	$0.0618940\pi$
$907$			10556.0i			0.386446i	−0.981155	+	0.193223i	$0.938106\pi$
$908$		0			0
$909$	−11610.0			−0.423630
$910$		0			0
$911$	47472.0			1.72647			0.863237	−	0.504799i	$-0.168433\pi$
$911$	47472.0			1.72647			0.863237	+	0.504799i	$0.168433\pi$
$912$		0			0
$913$		−	36176.0i		−	1.31134i
$914$		0			0
$915$	10740.0	−	5370.00i	0.388037	−	0.194018i
$916$		0			0
$917$			2352.00i			0.0847000i
$918$		0			0
$919$	11964.0			0.429441			0.214720	−	0.976676i	$-0.431116\pi$
$919$	11964.0			0.429441			0.214720	+	0.976676i	$0.431116\pi$
$920$		0			0
$921$	29100.0			1.04113
$922$		0			0
$923$		−	1024.00i		−	0.0365172i
$924$		0			0
$925$	−4000.00	−	3000.00i	−0.142183	−	0.106637i
$926$		0			0
$927$			14436.0i			0.511478i
$928$		0			0
$929$	15214.0			0.537304			0.268652	−	0.963237i	$-0.413422\pi$
$929$	15214.0			0.537304			0.268652	+	0.963237i	$0.413422\pi$
$930$		0			0
$931$	−10464.0			−0.368361
$932$		0			0
$933$		−	29016.0i		−	1.01816i
$934$		0			0
$935$	−30240.0	+	15120.0i	−1.05770	+	0.528852i
$936$		0			0
$937$			39712.0i			1.38456i	0.721628	+	0.692281i	$0.243394\pi$
$937$			39712.0i			1.38456i	−0.721628	+	0.692281i	$0.756606\pi$
$938$		0			0
$939$	12144.0			0.422049
$940$		0			0
$941$	36034.0			1.24833			0.624163	−	0.781294i	$-0.285440\pi$
$941$	36034.0			1.24833			0.624163	+	0.781294i	$0.285440\pi$
$942$		0			0
$943$		−	11816.0i		−	0.408040i
$944$		0			0
$945$	540.000	+	1080.00i	0.0185886	+	0.0371771i
$946$		0			0
$947$		−	2532.00i		−	0.0868838i	−0.999056	−	0.0434419i	$-0.986168\pi$
$947$		−	2532.00i		−	0.0868838i	0.999056	−	0.0434419i	$-0.0138323\pi$
$948$		0			0
$949$	2432.00			0.0831887
$950$		0			0
$951$	−252.000			−0.00859270
$952$		0			0
$953$		−	55284.0i		−	1.87914i	−0.342351	−	0.939572i	$-0.611223\pi$
$953$		−	55284.0i		−	1.87914i	0.342351	−	0.939572i	$-0.388777\pi$
$954$		0			0
$955$	360.000	+	720.000i	0.0121982	+	0.0243965i
$956$		0			0
$957$			19992.0i			0.675287i
$958$		0			0
$959$	4240.00			0.142770
$960$		0			0
$961$	2609.00			0.0875768
$962$		0			0
$963$		−	8028.00i		−	0.268638i
$964$		0			0
$965$	−2080.00	+	1040.00i	−0.0693861	+	0.0346930i
$966$		0			0
$967$		−	4372.00i		−	0.145392i	−0.997354	−	0.0726960i	$-0.976840\pi$
$967$		−	4372.00i		−	0.145392i	0.997354	−	0.0726960i	$-0.0231603\pi$
$968$		0			0
$969$	10368.0			0.343724
$970$		0			0
$971$	24300.0			0.803114			0.401557	−	0.915834i	$-0.368469\pi$
$971$	24300.0			0.803114			0.401557	+	0.915834i	$0.368469\pi$
$972$		0			0
$973$		−	11456.0i		−	0.377454i
$974$		0			0
$975$	−3600.00	+	4800.00i	−0.118248	+	0.157665i
$976$		0			0
$977$			46204.0i			1.51300i	0.653996	+	0.756498i	$0.273091\pi$
$977$			46204.0i			1.51300i	−0.653996	+	0.756498i	$0.726909\pi$
$978$		0			0
$979$	−32088.0			−1.04754
$980$		0			0
$981$	−8694.00			−0.282954
$982$		0			0
$983$		−	25468.0i		−	0.826351i	−0.910651	−	0.413176i	$-0.864420\pi$
$983$		−	25468.0i		−	0.826351i	0.910651	−	0.413176i	$-0.135580\pi$
$984$		0			0
$985$	−3720.00	+	1860.00i	−0.120334	+	0.0601670i
$986$		0			0
$987$		−	720.000i		−	0.0232197i
$988$		0			0
$989$	−7728.00			−0.248469
$990$		0			0
$991$	−11668.0			−0.374012			−0.187006	−	0.982359i	$-0.559878\pi$
$991$	−11668.0			−0.374012			−0.187006	+	0.982359i	$0.559878\pi$
$992$		0			0
$993$			16248.0i			0.519249i
$994$		0			0
$995$	−21740.0	−	43480.0i	−0.692668	−	1.38534i
$996$		0			0
$997$			7224.00i			0.229475i	0.993396	+	0.114737i	$0.0366027\pi$
$997$			7224.00i			0.229475i	−0.993396	+	0.114737i	$0.963397\pi$
$998$		0			0
$999$	1080.00			0.0342039

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.4.f.a.769.2		2
4.3	odd	2		960.4.f.b.769.1		2
5.4	even	2	inner	960.4.f.a.769.1		2
8.3	odd	2		120.4.f.c.49.2	yes	2
8.5	even	2		240.4.f.e.49.1		2
20.19	odd	2		960.4.f.b.769.2		2
24.5	odd	2		720.4.f.b.289.1		2
24.11	even	2		360.4.f.a.289.1		2
40.3	even	4		600.4.a.f.1.1		1
40.13	odd	4		1200.4.a.z.1.1		1
40.19	odd	2		120.4.f.c.49.1	✓	2
40.27	even	4		600.4.a.k.1.1		1
40.29	even	2		240.4.f.e.49.2		2
40.37	odd	4		1200.4.a.l.1.1		1
120.29	odd	2		720.4.f.b.289.2		2
120.59	even	2		360.4.f.a.289.2		2
120.83	odd	4		1800.4.a.u.1.1		1
120.107	odd	4		1800.4.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.f.c.49.1	✓	2	40.19	odd	2
120.4.f.c.49.2	yes	2	8.3	odd	2
240.4.f.e.49.1		2	8.5	even	2
240.4.f.e.49.2		2	40.29	even	2
360.4.f.a.289.1		2	24.11	even	2
360.4.f.a.289.2		2	120.59	even	2
600.4.a.f.1.1		1	40.3	even	4
600.4.a.k.1.1		1	40.27	even	4
720.4.f.b.289.1		2	24.5	odd	2
720.4.f.b.289.2		2	120.29	odd	2
960.4.f.a.769.1		2	5.4	even	2	inner
960.4.f.a.769.2		2	1.1	even	1	trivial
960.4.f.b.769.1		2	4.3	odd	2
960.4.f.b.769.2		2	20.19	odd	2
1200.4.a.l.1.1		1	40.37	odd	4
1200.4.a.z.1.1		1	40.13	odd	4
1800.4.a.o.1.1		1	120.107	odd	4
1800.4.a.u.1.1		1	120.83	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)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +(-5.00000 - 10.0000i) q^{5} -4.00000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} +(-5.00000 - 10.0000i) q^{5} -4.00000i q^{7} -9.00000 q^{9} -28.0000 q^{11} -16.0000i q^{13} +(30.0000 - 15.0000i) q^{15} +108.000i q^{17} -32.0000 q^{19} +12.0000 q^{21} -28.0000i q^{23} +(-75.0000 + 100.000i) q^{25} -27.0000i q^{27} -238.000 q^{29} +180.000 q^{31} -84.0000i q^{33} +(-40.0000 + 20.0000i) q^{35} +40.0000i q^{37} +48.0000 q^{39} +422.000 q^{41} -276.000i q^{43} +(45.0000 + 90.0000i) q^{45} -60.0000i q^{47} +327.000 q^{49} -324.000 q^{51} +220.000i q^{53} +(140.000 + 280.000i) q^{55} -96.0000i q^{57} +804.000 q^{59} +358.000 q^{61} +36.0000i q^{63} +(-160.000 + 80.0000i) q^{65} -884.000i q^{67} +84.0000 q^{69} +64.0000 q^{71} +152.000i q^{73} +(-300.000 - 225.000i) q^{75} +112.000i q^{77} -932.000 q^{79} +81.0000 q^{81} +1292.00i q^{83} +(1080.00 - 540.000i) q^{85} -714.000i q^{87} +1146.00 q^{89} -64.0000 q^{91} +540.000i q^{93} +(160.000 + 320.000i) q^{95} +824.000i q^{97} +252.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{5} - 18 q^{9}+O(q^{10}) \) 2 * q - 10 * q^5 - 18 * q^9	\( 2 q - 10 q^{5} - 18 q^{9} - 56 q^{11} + 60 q^{15} - 64 q^{19} + 24 q^{21} - 150 q^{25} - 476 q^{29} + 360 q^{31} - 80 q^{35} + 96 q^{39} + 844 q^{41} + 90 q^{45} + 654 q^{49} - 648 q^{51} + 280 q^{55} + 1608 q^{59} + 716 q^{61} - 320 q^{65} + 168 q^{69} + 128 q^{71} - 600 q^{75} - 1864 q^{79} + 162 q^{81} + 2160 q^{85} + 2292 q^{89} - 128 q^{91} + 320 q^{95} + 504 q^{99}+O(q^{100}) \) 2 * q - 10 * q^5 - 18 * q^9 - 56 * q^11 + 60 * q^15 - 64 * q^19 + 24 * q^21 - 150 * q^25 - 476 * q^29 + 360 * q^31 - 80 * q^35 + 96 * q^39 + 844 * q^41 + 90 * q^45 + 654 * q^49 - 648 * q^51 + 280 * q^55 + 1608 * q^59 + 716 * q^61 - 320 * q^65 + 168 * q^69 + 128 * q^71 - 600 * q^75 - 1864 * q^79 + 162 * q^81 + 2160 * q^85 + 2292 * q^89 - 128 * q^91 + 320 * q^95 + 504 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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