LMFDB - Embedded newform 4004.2.m.a.2157.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(2157,4004)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4004.2157");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.m (of order $2$, degree $1$, 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:	$31.9721009693$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2157.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.a.2157.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{3} +1.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+2.00000 q^{3} +1.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} -1.00000i q^{11} +(2.00000 + 3.00000i) q^{13} +2.00000i q^{15} -4.00000 q^{17} +5.00000i q^{19} +2.00000i q^{21} -9.00000 q^{23} +4.00000 q^{25} -4.00000 q^{27} +5.00000 q^{29} +7.00000i q^{31} -2.00000i q^{33} -1.00000 q^{35} +2.00000i q^{37} +(4.00000 + 6.00000i) q^{39} -6.00000i q^{41} -11.0000 q^{43} +1.00000i q^{45} -1.00000i q^{47} -1.00000 q^{49} -8.00000 q^{51} -1.00000 q^{53} +1.00000 q^{55} +10.0000i q^{57} +4.00000i q^{59} -6.00000 q^{61} +1.00000i q^{63} +(-3.00000 + 2.00000i) q^{65} +10.0000i q^{67} -18.0000 q^{69} +4.00000i q^{71} +3.00000i q^{73} +8.00000 q^{75} +1.00000 q^{77} +11.0000 q^{79} -11.0000 q^{81} -9.00000i q^{83} -4.00000i q^{85} +10.0000 q^{87} -15.0000i q^{89} +(-3.00000 + 2.00000i) q^{91} +14.0000i q^{93} -5.00000 q^{95} +7.00000i q^{97} -1.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^3 + 2 * q^9	$ 2 q + 4 q^{3} + 2 q^{9} + 4 q^{13} - 8 q^{17} - 18 q^{23} + 8 q^{25} - 8 q^{27} + 10 q^{29} - 2 q^{35} + 8 q^{39} - 22 q^{43} - 2 q^{49} - 16 q^{51} - 2 q^{53} + 2 q^{55} - 12 q^{61} - 6 q^{65} - 36 q^{69} + 16 q^{75} + 2 q^{77} + 22 q^{79} - 22 q^{81} + 20 q^{87} - 6 q^{91} - 10 q^{95}+O(q^{100}) $ 2 * q + 4 * q^3 + 2 * q^9 + 4 * q^13 - 8 * q^17 - 18 * q^23 + 8 * q^25 - 8 * q^27 + 10 * q^29 - 2 * q^35 + 8 * q^39 - 22 * q^43 - 2 * q^49 - 16 * q^51 - 2 * q^53 + 2 * q^55 - 12 * q^61 - 6 * q^65 - 36 * q^69 + 16 * q^75 + 2 * q^77 + 22 * q^79 - 22 * q^81 + 20 * q^87 - 6 * q^91 - 10 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times$.

$n$	$365$	$925$	$2003$	$3433$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	2.00000			1.15470			0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000			1.15470			0.577350	+	0.816497i	$0.304087\pi$
$4$		0			0
$5$			1.00000i			0.447214i	0.974679	+	0.223607i	$0.0717831\pi$
$5$			1.00000i			0.447214i	−0.974679	+	0.223607i	$0.928217\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$14$		0			0
$15$			2.00000i			0.516398i
$16$		0			0
$17$	−4.00000			−0.970143			−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000			−0.970143			−0.485071	+	0.874475i	$0.661206\pi$
$18$		0			0
$19$			5.00000i			1.14708i	0.819178	+	0.573539i	$0.194430\pi$
$19$			5.00000i			1.14708i	−0.819178	+	0.573539i	$0.805570\pi$
$20$		0			0
$21$			2.00000i			0.436436i
$22$		0			0
$23$	−9.00000			−1.87663			−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000			−1.87663			−0.938315	+	0.345782i	$0.887614\pi$
$24$		0			0
$25$	4.00000			0.800000
$26$		0			0
$27$	−4.00000			−0.769800
$28$		0			0
$29$	5.00000			0.928477			0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000			0.928477			0.464238	+	0.885710i	$0.346328\pi$
$30$		0			0
$31$			7.00000i			1.25724i	0.777714	+	0.628619i	$0.216379\pi$
$31$			7.00000i			1.25724i	−0.777714	+	0.628619i	$0.783621\pi$
$32$		0			0
$33$		−	2.00000i		−	0.348155i
$34$		0			0
$35$	−1.00000			−0.169031
$36$		0			0
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$	4.00000	+	6.00000i	0.640513	+	0.960769i
$40$		0			0
$41$		−	6.00000i		−	0.937043i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$		−	6.00000i		−	0.937043i	0.883452	−	0.468521i	$-0.155213\pi$
$42$		0			0
$43$	−11.0000			−1.67748			−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000			−1.67748			−0.838742	+	0.544529i	$0.816708\pi$
$44$		0			0
$45$			1.00000i			0.149071i
$46$		0			0
$47$		−	1.00000i		−	0.145865i	−0.997337	−	0.0729325i	$-0.976764\pi$
$47$		−	1.00000i		−	0.145865i	0.997337	−	0.0729325i	$-0.0232358\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−8.00000			−1.12022
$52$		0			0
$53$	−1.00000			−0.137361			−0.0686803	−	0.997639i	$-0.521879\pi$
$53$	−1.00000			−0.137361			−0.0686803	+	0.997639i	$0.521879\pi$
$54$		0			0
$55$	1.00000			0.134840
$56$		0			0
$57$			10.0000i			1.32453i
$58$		0			0
$59$			4.00000i			0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$			4.00000i			0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$		0			0
$61$	−6.00000			−0.768221			−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000			−0.768221			−0.384111	+	0.923287i	$0.625492\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	−3.00000	+	2.00000i	−0.372104	+	0.248069i
$66$		0			0
$67$			10.0000i			1.22169i	0.791748	+	0.610847i	$0.209171\pi$
$67$			10.0000i			1.22169i	−0.791748	+	0.610847i	$0.790829\pi$
$68$		0			0
$69$	−18.0000			−2.16695
$70$		0			0
$71$			4.00000i			0.474713i	0.971423	+	0.237356i	$0.0762809\pi$
$71$			4.00000i			0.474713i	−0.971423	+	0.237356i	$0.923719\pi$
$72$		0			0
$73$			3.00000i			0.351123i	0.984468	+	0.175562i	$0.0561742\pi$
$73$			3.00000i			0.351123i	−0.984468	+	0.175562i	$0.943826\pi$
$74$		0			0
$75$	8.00000			0.923760
$76$		0			0
$77$	1.00000			0.113961
$78$		0			0
$79$	11.0000			1.23760			0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000			1.23760			0.618798	+	0.785550i	$0.287620\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$		−	9.00000i		−	0.987878i	−0.869496	−	0.493939i	$-0.835557\pi$
$83$		−	9.00000i		−	0.987878i	0.869496	−	0.493939i	$-0.164443\pi$
$84$		0			0
$85$		−	4.00000i		−	0.433861i
$86$		0			0
$87$	10.0000			1.07211
$88$		0			0
$89$		−	15.0000i		−	1.59000i	−0.606612	−	0.794998i	$-0.707472\pi$
$89$		−	15.0000i		−	1.59000i	0.606612	−	0.794998i	$-0.292528\pi$
$90$		0			0
$91$	−3.00000	+	2.00000i	−0.314485	+	0.209657i
$92$		0			0
$93$			14.0000i			1.45173i
$94$		0			0
$95$	−5.00000			−0.512989
$96$		0			0
$97$			7.00000i			0.710742i	0.934725	+	0.355371i	$0.115646\pi$
$97$			7.00000i			0.710742i	−0.934725	+	0.355371i	$0.884354\pi$
$98$		0			0
$99$		−	1.00000i		−	0.100504i
$100$		0			0
$101$	8.00000			0.796030			0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000			0.796030			0.398015	+	0.917379i	$0.369699\pi$
$102$		0			0
$103$	14.0000			1.37946			0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000			1.37946			0.689730	+	0.724066i	$0.257729\pi$
$104$		0			0
$105$	−2.00000			−0.195180
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$			16.0000i			1.53252i	0.642529	+	0.766261i	$0.277885\pi$
$109$			16.0000i			1.53252i	−0.642529	+	0.766261i	$0.722115\pi$
$110$		0			0
$111$			4.00000i			0.379663i
$112$		0			0
$113$	5.00000			0.470360			0.235180	−	0.971952i	$-0.424432\pi$
$113$	5.00000			0.470360			0.235180	+	0.971952i	$0.424432\pi$
$114$		0			0
$115$		−	9.00000i		−	0.839254i
$116$		0			0
$117$	2.00000	+	3.00000i	0.184900	+	0.277350i
$118$		0			0
$119$		−	4.00000i		−	0.366679i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	12.0000i		−	1.08200i
$124$		0			0
$125$			9.00000i			0.804984i
$126$		0			0
$127$	−20.0000			−1.77471			−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000			−1.77471			−0.887357	+	0.461084i	$0.847461\pi$
$128$		0			0
$129$	−22.0000			−1.93699
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	−5.00000			−0.433555
$134$		0			0
$135$		−	4.00000i		−	0.344265i
$136$		0			0
$137$			18.0000i			1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$			18.0000i			1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$		0			0
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$		0			0
$141$		−	2.00000i		−	0.168430i
$142$		0			0
$143$	3.00000	−	2.00000i	0.250873	−	0.167248i
$144$		0			0
$145$			5.00000i			0.415227i
$146$		0			0
$147$	−2.00000			−0.164957
$148$		0			0
$149$		−	18.0000i		−	1.47462i	−0.675556	−	0.737309i	$-0.736096\pi$
$149$		−	18.0000i		−	1.47462i	0.675556	−	0.737309i	$-0.263904\pi$
$150$		0			0
$151$			4.00000i			0.325515i	0.986666	+	0.162758i	$0.0520389\pi$
$151$			4.00000i			0.325515i	−0.986666	+	0.162758i	$0.947961\pi$
$152$		0			0
$153$	−4.00000			−0.323381
$154$		0			0
$155$	−7.00000			−0.562254
$156$		0			0
$157$	−10.0000			−0.798087			−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000			−0.798087			−0.399043	+	0.916932i	$0.630658\pi$
$158$		0			0
$159$	−2.00000			−0.158610
$160$		0			0
$161$		−	9.00000i		−	0.709299i
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$		0			0
$165$	2.00000			0.155700
$166$		0			0
$167$			23.0000i			1.77979i	0.456162	+	0.889897i	$0.349224\pi$
$167$			23.0000i			1.77979i	−0.456162	+	0.889897i	$0.650776\pi$
$168$		0			0
$169$	−5.00000	+	12.0000i	−0.384615	+	0.923077i
$170$		0			0
$171$			5.00000i			0.382360i
$172$		0			0
$173$	−10.0000			−0.760286			−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000			−0.760286			−0.380143	+	0.924928i	$0.624125\pi$
$174$		0			0
$175$			4.00000i			0.302372i
$176$		0			0
$177$			8.00000i			0.601317i
$178$		0			0
$179$	11.0000			0.822179			0.411089	−	0.911595i	$-0.365148\pi$
$179$	11.0000			0.822179			0.411089	+	0.911595i	$0.365148\pi$
$180$		0			0
$181$	16.0000			1.18927			0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000			1.18927			0.594635	+	0.803996i	$0.297296\pi$
$182$		0			0
$183$	−12.0000			−0.887066
$184$		0			0
$185$	−2.00000			−0.147043
$186$		0			0
$187$			4.00000i			0.292509i
$188$		0			0
$189$		−	4.00000i		−	0.290957i
$190$		0			0
$191$	20.0000			1.44715			0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000			1.44715			0.723575	+	0.690246i	$0.242498\pi$
$192$		0			0
$193$			16.0000i			1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$			16.0000i			1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$		0			0
$195$	−6.00000	+	4.00000i	−0.429669	+	0.286446i
$196$		0			0
$197$		−	26.0000i		−	1.85242i	−0.377004	−	0.926212i	$-0.623046\pi$
$197$		−	26.0000i		−	1.85242i	0.377004	−	0.926212i	$-0.376954\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$			20.0000i			1.41069i
$202$		0			0
$203$			5.00000i			0.350931i
$204$		0			0
$205$	6.00000			0.419058
$206$		0			0
$207$	−9.00000			−0.625543
$208$		0			0
$209$	5.00000			0.345857
$210$		0			0
$211$	−13.0000			−0.894957			−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000			−0.894957			−0.447478	+	0.894295i	$0.647678\pi$
$212$		0			0
$213$			8.00000i			0.548151i
$214$		0			0
$215$		−	11.0000i		−	0.750194i
$216$		0			0
$217$	−7.00000			−0.475191
$218$		0			0
$219$			6.00000i			0.405442i
$220$		0			0
$221$	−8.00000	−	12.0000i	−0.538138	−	0.807207i
$222$		0			0
$223$			1.00000i			0.0669650i	0.999439	+	0.0334825i	$0.0106598\pi$
$223$			1.00000i			0.0669650i	−0.999439	+	0.0334825i	$0.989340\pi$
$224$		0			0
$225$	4.00000			0.266667
$226$		0			0
$227$			12.0000i			0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$			12.0000i			0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$		0			0
$229$		−	26.0000i		−	1.71813i	−0.511868	−	0.859064i	$-0.671046\pi$
$229$		−	26.0000i		−	1.71813i	0.511868	−	0.859064i	$-0.328954\pi$
$230$		0			0
$231$	2.00000			0.131590
$232$		0			0
$233$	−21.0000			−1.37576			−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000			−1.37576			−0.687878	+	0.725826i	$0.741458\pi$
$234$		0			0
$235$	1.00000			0.0652328
$236$		0			0
$237$	22.0000			1.42905
$238$		0			0
$239$			2.00000i			0.129369i	0.997906	+	0.0646846i	$0.0206041\pi$
$239$			2.00000i			0.129369i	−0.997906	+	0.0646846i	$0.979396\pi$
$240$		0			0
$241$		−	25.0000i		−	1.61039i	−0.593009	−	0.805196i	$-0.702060\pi$
$241$		−	25.0000i		−	1.61039i	0.593009	−	0.805196i	$-0.297940\pi$
$242$		0			0
$243$	−10.0000			−0.641500
$244$		0			0
$245$		−	1.00000i		−	0.0638877i
$246$		0			0
$247$	−15.0000	+	10.0000i	−0.954427	+	0.636285i
$248$		0			0
$249$		−	18.0000i		−	1.14070i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$			9.00000i			0.565825i
$254$		0			0
$255$		−	8.00000i		−	0.500979i
$256$		0			0
$257$	−6.00000			−0.374270			−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000			−0.374270			−0.187135	+	0.982334i	$0.559920\pi$
$258$		0			0
$259$	−2.00000			−0.124274
$260$		0			0
$261$	5.00000			0.309492
$262$		0			0
$263$	13.0000			0.801614			0.400807	−	0.916162i	$-0.368730\pi$
$263$	13.0000			0.801614			0.400807	+	0.916162i	$0.368730\pi$
$264$		0			0
$265$		−	1.00000i		−	0.0614295i
$266$		0			0
$267$		−	30.0000i		−	1.83597i
$268$		0			0
$269$	6.00000			0.365826			0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000			0.365826			0.182913	+	0.983129i	$0.441447\pi$
$270$		0			0
$271$		−	8.00000i		−	0.485965i	−0.970031	−	0.242983i	$-0.921874\pi$
$271$		−	8.00000i		−	0.485965i	0.970031	−	0.242983i	$-0.0781258\pi$
$272$		0			0
$273$	−6.00000	+	4.00000i	−0.363137	+	0.242091i
$274$		0			0
$275$		−	4.00000i		−	0.241209i
$276$		0			0
$277$	33.0000			1.98278			0.991389	−	0.130950i	$-0.0418029\pi$
$277$	33.0000			1.98278			0.991389	+	0.130950i	$0.0418029\pi$
$278$		0			0
$279$			7.00000i			0.419079i
$280$		0			0
$281$			20.0000i			1.19310i	0.802576	+	0.596550i	$0.203462\pi$
$281$			20.0000i			1.19310i	−0.802576	+	0.596550i	$0.796538\pi$
$282$		0			0
$283$	−6.00000			−0.356663			−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000			−0.356663			−0.178331	+	0.983970i	$0.557070\pi$
$284$		0			0
$285$	−10.0000			−0.592349
$286$		0			0
$287$	6.00000			0.354169
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$			14.0000i			0.820695i
$292$		0			0
$293$			21.0000i			1.22683i	0.789760	+	0.613417i	$0.210205\pi$
$293$			21.0000i			1.22683i	−0.789760	+	0.613417i	$0.789795\pi$
$294$		0			0
$295$	−4.00000			−0.232889
$296$		0			0
$297$			4.00000i			0.232104i
$298$		0			0
$299$	−18.0000	−	27.0000i	−1.04097	−	1.56145i
$300$		0			0
$301$		−	11.0000i		−	0.634029i
$302$		0			0
$303$	16.0000			0.919176
$304$		0			0
$305$		−	6.00000i		−	0.343559i
$306$		0			0
$307$		−	11.0000i		−	0.627803i	−0.949456	−	0.313902i	$-0.898364\pi$
$307$		−	11.0000i		−	0.627803i	0.949456	−	0.313902i	$-0.101636\pi$
$308$		0			0
$309$	28.0000			1.59286
$310$		0			0
$311$	32.0000			1.81455			0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000			1.81455			0.907277	+	0.420534i	$0.138157\pi$
$312$		0			0
$313$	28.0000			1.58265			0.791327	−	0.611393i	$-0.209391\pi$
$313$	28.0000			1.58265			0.791327	+	0.611393i	$0.209391\pi$
$314$		0			0
$315$	−1.00000			−0.0563436
$316$		0			0
$317$			6.00000i			0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$			6.00000i			0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$		0			0
$319$		−	5.00000i		−	0.279946i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	20.0000i		−	1.11283i
$324$		0			0
$325$	8.00000	+	12.0000i	0.443760	+	0.665640i
$326$		0			0
$327$			32.0000i			1.76960i
$328$		0			0
$329$	1.00000			0.0551318
$330$		0			0
$331$		−	6.00000i		−	0.329790i	−0.986311	−	0.164895i	$-0.947272\pi$
$331$		−	6.00000i		−	0.329790i	0.986311	−	0.164895i	$-0.0527285\pi$
$332$		0			0
$333$			2.00000i			0.109599i
$334$		0			0
$335$	−10.0000			−0.546358
$336$		0			0
$337$	−11.0000			−0.599208			−0.299604	−	0.954064i	$-0.596855\pi$
$337$	−11.0000			−0.599208			−0.299604	+	0.954064i	$0.596855\pi$
$338$		0			0
$339$	10.0000			0.543125
$340$		0			0
$341$	7.00000			0.379071
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$		−	18.0000i		−	0.969087i
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$		−	9.00000i		−	0.481759i	−0.970555	−	0.240879i	$-0.922564\pi$
$349$		−	9.00000i		−	0.481759i	0.970555	−	0.240879i	$-0.0774359\pi$
$350$		0			0
$351$	−8.00000	−	12.0000i	−0.427008	−	0.640513i
$352$		0			0
$353$			30.0000i			1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$			30.0000i			1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$		0			0
$355$	−4.00000			−0.212298
$356$		0			0
$357$		−	8.00000i		−	0.423405i
$358$		0			0
$359$		−	4.00000i		−	0.211112i	−0.994413	−	0.105556i	$-0.966338\pi$
$359$		−	4.00000i		−	0.211112i	0.994413	−	0.105556i	$-0.0336622\pi$
$360$		0			0
$361$	−6.00000			−0.315789
$362$		0			0
$363$	−2.00000			−0.104973
$364$		0			0
$365$	−3.00000			−0.157027
$366$		0			0
$367$	14.0000			0.730794			0.365397	−	0.930852i	$-0.380933\pi$
$367$	14.0000			0.730794			0.365397	+	0.930852i	$0.380933\pi$
$368$		0			0
$369$		−	6.00000i		−	0.312348i
$370$		0			0
$371$		−	1.00000i		−	0.0519174i
$372$		0			0
$373$	−30.0000			−1.55334			−0.776671	−	0.629907i	$-0.783093\pi$
$373$	−30.0000			−1.55334			−0.776671	+	0.629907i	$0.783093\pi$
$374$		0			0
$375$			18.0000i			0.929516i
$376$		0			0
$377$	10.0000	+	15.0000i	0.515026	+	0.772539i
$378$		0			0
$379$			6.00000i			0.308199i	0.988055	+	0.154100i	$0.0492477\pi$
$379$			6.00000i			0.308199i	−0.988055	+	0.154100i	$0.950752\pi$
$380$		0			0
$381$	−40.0000			−2.04926
$382$		0			0
$383$			24.0000i			1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$			24.0000i			1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$		0			0
$385$			1.00000i			0.0509647i
$386$		0			0
$387$	−11.0000			−0.559161
$388$		0			0
$389$	34.0000			1.72387			0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000			1.72387			0.861934	+	0.507020i	$0.169253\pi$
$390$		0			0
$391$	36.0000			1.82060
$392$		0			0
$393$		0			0
$394$		0			0
$395$			11.0000i			0.553470i
$396$		0			0
$397$		−	3.00000i		−	0.150566i	−0.997162	−	0.0752828i	$-0.976014\pi$
$397$		−	3.00000i		−	0.150566i	0.997162	−	0.0752828i	$-0.0239860\pi$
$398$		0			0
$399$	−10.0000			−0.500626
$400$		0			0
$401$			22.0000i			1.09863i	0.835616	+	0.549314i	$0.185111\pi$
$401$			22.0000i			1.09863i	−0.835616	+	0.549314i	$0.814889\pi$
$402$		0			0
$403$	−21.0000	+	14.0000i	−1.04608	+	0.697390i
$404$		0			0
$405$		−	11.0000i		−	0.546594i
$406$		0			0
$407$	2.00000			0.0991363
$408$		0			0
$409$		−	31.0000i		−	1.53285i	−0.642333	−	0.766426i	$-0.722033\pi$
$409$		−	31.0000i		−	1.53285i	0.642333	−	0.766426i	$-0.277967\pi$
$410$		0			0
$411$			36.0000i			1.77575i
$412$		0			0
$413$	−4.00000			−0.196827
$414$		0			0
$415$	9.00000			0.441793
$416$		0			0
$417$	24.0000			1.17529
$418$		0			0
$419$	34.0000			1.66101			0.830504	−	0.557012i	$-0.188052\pi$
$419$	34.0000			1.66101			0.830504	+	0.557012i	$0.188052\pi$
$420$		0			0
$421$		−	28.0000i		−	1.36464i	−0.731055	−	0.682318i	$-0.760972\pi$
$421$		−	28.0000i		−	1.36464i	0.731055	−	0.682318i	$-0.239028\pi$
$422$		0			0
$423$		−	1.00000i		−	0.0486217i
$424$		0			0
$425$	−16.0000			−0.776114
$426$		0			0
$427$		−	6.00000i		−	0.290360i
$428$		0			0
$429$	6.00000	−	4.00000i	0.289683	−	0.193122i
$430$		0			0
$431$			10.0000i			0.481683i	0.970564	+	0.240842i	$0.0774234\pi$
$431$			10.0000i			0.481683i	−0.970564	+	0.240842i	$0.922577\pi$
$432$		0			0
$433$	20.0000			0.961139			0.480569	−	0.876957i	$-0.340430\pi$
$433$	20.0000			0.961139			0.480569	+	0.876957i	$0.340430\pi$
$434$		0			0
$435$			10.0000i			0.479463i
$436$		0			0
$437$		−	45.0000i		−	2.15264i
$438$		0			0
$439$	−14.0000			−0.668184			−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000			−0.668184			−0.334092	+	0.942541i	$0.608430\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	5.00000			0.237557			0.118779	−	0.992921i	$-0.462102\pi$
$443$	5.00000			0.237557			0.118779	+	0.992921i	$0.462102\pi$
$444$		0			0
$445$	15.0000			0.711068
$446$		0			0
$447$		−	36.0000i		−	1.70274i
$448$		0			0
$449$			16.0000i			0.755087i	0.925992	+	0.377543i	$0.123231\pi$
$449$			16.0000i			0.755087i	−0.925992	+	0.377543i	$0.876769\pi$
$450$		0			0
$451$	−6.00000			−0.282529
$452$		0			0
$453$			8.00000i			0.375873i
$454$		0			0
$455$	−2.00000	−	3.00000i	−0.0937614	−	0.140642i
$456$		0			0
$457$		−	32.0000i		−	1.49690i	−0.663193	−	0.748448i	$-0.730799\pi$
$457$		−	32.0000i		−	1.49690i	0.663193	−	0.748448i	$-0.269201\pi$
$458$		0			0
$459$	16.0000			0.746816
$460$		0			0
$461$			6.00000i			0.279448i	0.990190	+	0.139724i	$0.0446215\pi$
$461$			6.00000i			0.279448i	−0.990190	+	0.139724i	$0.955378\pi$
$462$		0			0
$463$		−	38.0000i		−	1.76601i	−0.469364	−	0.883005i	$-0.655517\pi$
$463$		−	38.0000i		−	1.76601i	0.469364	−	0.883005i	$-0.344483\pi$
$464$		0			0
$465$	−14.0000			−0.649234
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	−10.0000			−0.461757
$470$		0			0
$471$	−20.0000			−0.921551
$472$		0			0
$473$			11.0000i			0.505781i
$474$		0			0
$475$			20.0000i			0.917663i
$476$		0			0
$477$	−1.00000			−0.0457869
$478$		0			0
$479$			11.0000i			0.502603i	0.967909	+	0.251301i	$0.0808585\pi$
$479$			11.0000i			0.502603i	−0.967909	+	0.251301i	$0.919141\pi$
$480$		0			0
$481$	−6.00000	+	4.00000i	−0.273576	+	0.182384i
$482$		0			0
$483$		−	18.0000i		−	0.819028i
$484$		0			0
$485$	−7.00000			−0.317854
$486$		0			0
$487$		−	16.0000i		−	0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$		−	16.0000i		−	0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$		0			0
$489$			8.00000i			0.361773i
$490$		0			0
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$		0			0
$493$	−20.0000			−0.900755
$494$		0			0
$495$	1.00000			0.0449467
$496$		0			0
$497$	−4.00000			−0.179425
$498$		0			0
$499$			34.0000i			1.52205i	0.648723	+	0.761025i	$0.275303\pi$
$499$			34.0000i			1.52205i	−0.648723	+	0.761025i	$0.724697\pi$
$500$		0			0
$501$			46.0000i			2.05513i
$502$		0			0
$503$	−22.0000			−0.980932			−0.490466	−	0.871460i	$-0.663173\pi$
$503$	−22.0000			−0.980932			−0.490466	+	0.871460i	$0.663173\pi$
$504$		0			0
$505$			8.00000i			0.355995i
$506$		0			0
$507$	−10.0000	+	24.0000i	−0.444116	+	1.06588i
$508$		0			0
$509$		−	35.0000i		−	1.55135i	−0.631134	−	0.775674i	$-0.717410\pi$
$509$		−	35.0000i		−	1.55135i	0.631134	−	0.775674i	$-0.282590\pi$
$510$		0			0
$511$	−3.00000			−0.132712
$512$		0			0
$513$		−	20.0000i		−	0.883022i
$514$		0			0
$515$			14.0000i			0.616914i
$516$		0			0
$517$	−1.00000			−0.0439799
$518$		0			0
$519$	−20.0000			−0.877903
$520$		0			0
$521$	8.00000			0.350486			0.175243	−	0.984525i	$-0.443929\pi$
$521$	8.00000			0.350486			0.175243	+	0.984525i	$0.443929\pi$
$522$		0			0
$523$	−22.0000			−0.961993			−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000			−0.961993			−0.480996	+	0.876723i	$0.659725\pi$
$524$		0			0
$525$			8.00000i			0.349149i
$526$		0			0
$527$		−	28.0000i		−	1.21970i
$528$		0			0
$529$	58.0000			2.52174
$530$		0			0
$531$			4.00000i			0.173585i
$532$		0			0
$533$	18.0000	−	12.0000i	0.779667	−	0.519778i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	22.0000			0.949370
$538$		0			0
$539$			1.00000i			0.0430730i
$540$		0			0
$541$			22.0000i			0.945854i	0.881102	+	0.472927i	$0.156803\pi$
$541$			22.0000i			0.945854i	−0.881102	+	0.472927i	$0.843197\pi$
$542$		0			0
$543$	32.0000			1.37325
$544$		0			0
$545$	−16.0000			−0.685365
$546$		0			0
$547$	37.0000			1.58201			0.791003	−	0.611812i	$-0.209559\pi$
$547$	37.0000			1.58201			0.791003	+	0.611812i	$0.209559\pi$
$548$		0			0
$549$	−6.00000			−0.256074
$550$		0			0
$551$			25.0000i			1.06504i
$552$		0			0
$553$			11.0000i			0.467768i
$554$		0			0
$555$	−4.00000			−0.169791
$556$		0			0
$557$		−	6.00000i		−	0.254228i	−0.991888	−	0.127114i	$-0.959429\pi$
$557$		−	6.00000i		−	0.254228i	0.991888	−	0.127114i	$-0.0405714\pi$
$558$		0			0
$559$	−22.0000	−	33.0000i	−0.930501	−	1.39575i
$560$		0			0
$561$			8.00000i			0.337760i
$562$		0			0
$563$	−4.00000			−0.168580			−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000			−0.168580			−0.0842900	+	0.996441i	$0.526862\pi$
$564$		0			0
$565$			5.00000i			0.210352i
$566$		0			0
$567$		−	11.0000i		−	0.461957i
$568$		0			0
$569$	−37.0000			−1.55112			−0.775560	−	0.631273i	$-0.782533\pi$
$569$	−37.0000			−1.55112			−0.775560	+	0.631273i	$0.782533\pi$
$570$		0			0
$571$	5.00000			0.209243			0.104622	−	0.994512i	$-0.466637\pi$
$571$	5.00000			0.209243			0.104622	+	0.994512i	$0.466637\pi$
$572$		0			0
$573$	40.0000			1.67102
$574$		0			0
$575$	−36.0000			−1.50130
$576$		0			0
$577$		−	10.0000i		−	0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$		−	10.0000i		−	0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$		0			0
$579$			32.0000i			1.32987i
$580$		0			0
$581$	9.00000			0.373383
$582$		0			0
$583$			1.00000i			0.0414158i
$584$		0			0
$585$	−3.00000	+	2.00000i	−0.124035	+	0.0826898i
$586$		0			0
$587$		−	27.0000i		−	1.11441i	−0.830375	−	0.557205i	$-0.811874\pi$
$587$		−	27.0000i		−	1.11441i	0.830375	−	0.557205i	$-0.188126\pi$
$588$		0			0
$589$	−35.0000			−1.44215
$590$		0			0
$591$		−	52.0000i		−	2.13899i
$592$		0			0
$593$		−	19.0000i		−	0.780236i	−0.920765	−	0.390118i	$-0.872434\pi$
$593$		−	19.0000i		−	0.780236i	0.920765	−	0.390118i	$-0.127566\pi$
$594$		0			0
$595$	4.00000			0.163984
$596$		0			0
$597$	−40.0000			−1.63709
$598$		0			0
$599$	33.0000			1.34834			0.674172	−	0.738575i	$-0.264501\pi$
$599$	33.0000			1.34834			0.674172	+	0.738575i	$0.264501\pi$
$600$		0			0
$601$	−42.0000			−1.71322			−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000			−1.71322			−0.856608	+	0.515968i	$0.827432\pi$
$602$		0			0
$603$			10.0000i			0.407231i
$604$		0			0
$605$		−	1.00000i		−	0.0406558i
$606$		0			0
$607$	38.0000			1.54237			0.771186	−	0.636610i	$-0.219664\pi$
$607$	38.0000			1.54237			0.771186	+	0.636610i	$0.219664\pi$
$608$		0			0
$609$			10.0000i			0.405220i
$610$		0			0
$611$	3.00000	−	2.00000i	0.121367	−	0.0809113i
$612$		0			0
$613$			14.0000i			0.565455i	0.959200	+	0.282727i	$0.0912392\pi$
$613$			14.0000i			0.565455i	−0.959200	+	0.282727i	$0.908761\pi$
$614$		0			0
$615$	12.0000			0.483887
$616$		0			0
$617$		−	18.0000i		−	0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$		−	18.0000i		−	0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$		0			0
$619$			12.0000i			0.482321i	0.970485	+	0.241160i	$0.0775280\pi$
$619$			12.0000i			0.482321i	−0.970485	+	0.241160i	$0.922472\pi$
$620$		0			0
$621$	36.0000			1.44463
$622$		0			0
$623$	15.0000			0.600962
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$	10.0000			0.399362
$628$		0			0
$629$		−	8.00000i		−	0.318981i
$630$		0			0
$631$			26.0000i			1.03504i	0.855670	+	0.517522i	$0.173145\pi$
$631$			26.0000i			1.03504i	−0.855670	+	0.517522i	$0.826855\pi$
$632$		0			0
$633$	−26.0000			−1.03341
$634$		0			0
$635$		−	20.0000i		−	0.793676i
$636$		0			0
$637$	−2.00000	−	3.00000i	−0.0792429	−	0.118864i
$638$		0			0
$639$			4.00000i			0.158238i
$640$		0			0
$641$	−9.00000			−0.355479			−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000			−0.355479			−0.177739	+	0.984078i	$0.556878\pi$
$642$		0			0
$643$			28.0000i			1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$			28.0000i			1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$		0			0
$645$		−	22.0000i		−	0.866249i
$646$		0			0
$647$	−10.0000			−0.393141			−0.196570	−	0.980490i	$-0.562980\pi$
$647$	−10.0000			−0.393141			−0.196570	+	0.980490i	$0.562980\pi$
$648$		0			0
$649$	4.00000			0.157014
$650$		0			0
$651$	−14.0000			−0.548703
$652$		0			0
$653$	26.0000			1.01746			0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000			1.01746			0.508729	+	0.860927i	$0.330115\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			3.00000i			0.117041i
$658$		0			0
$659$	9.00000			0.350590			0.175295	−	0.984516i	$-0.443912\pi$
$659$	9.00000			0.350590			0.175295	+	0.984516i	$0.443912\pi$
$660$		0			0
$661$			13.0000i			0.505641i	0.967513	+	0.252821i	$0.0813583\pi$
$661$			13.0000i			0.505641i	−0.967513	+	0.252821i	$0.918642\pi$
$662$		0			0
$663$	−16.0000	−	24.0000i	−0.621389	−	0.932083i
$664$		0			0
$665$		−	5.00000i		−	0.193892i
$666$		0			0
$667$	−45.0000			−1.74241
$668$		0			0
$669$			2.00000i			0.0773245i
$670$		0			0
$671$			6.00000i			0.231627i
$672$		0			0
$673$	17.0000			0.655302			0.327651	−	0.944799i	$-0.393743\pi$
$673$	17.0000			0.655302			0.327651	+	0.944799i	$0.393743\pi$
$674$		0			0
$675$	−16.0000			−0.615840
$676$		0			0
$677$	30.0000			1.15299			0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000			1.15299			0.576497	+	0.817099i	$0.304419\pi$
$678$		0			0
$679$	−7.00000			−0.268635
$680$		0			0
$681$			24.0000i			0.919682i
$682$		0			0
$683$			20.0000i			0.765279i	0.923898	+	0.382639i	$0.124985\pi$
$683$			20.0000i			0.765279i	−0.923898	+	0.382639i	$0.875015\pi$
$684$		0			0
$685$	−18.0000			−0.687745
$686$		0			0
$687$		−	52.0000i		−	1.98392i
$688$		0			0
$689$	−2.00000	−	3.00000i	−0.0761939	−	0.114291i
$690$		0			0
$691$		−	19.0000i		−	0.722794i	−0.932412	−	0.361397i	$-0.882300\pi$
$691$		−	19.0000i		−	0.722794i	0.932412	−	0.361397i	$-0.117700\pi$
$692$		0			0
$693$	1.00000			0.0379869
$694$		0			0
$695$			12.0000i			0.455186i
$696$		0			0
$697$			24.0000i			0.909065i
$698$		0			0
$699$	−42.0000			−1.58859
$700$		0			0
$701$	−3.00000			−0.113308			−0.0566542	−	0.998394i	$-0.518043\pi$
$701$	−3.00000			−0.113308			−0.0566542	+	0.998394i	$0.518043\pi$
$702$		0			0
$703$	−10.0000			−0.377157
$704$		0			0
$705$	2.00000			0.0753244
$706$		0			0
$707$			8.00000i			0.300871i
$708$		0			0
$709$			4.00000i			0.150223i	0.997175	+	0.0751116i	$0.0239313\pi$
$709$			4.00000i			0.150223i	−0.997175	+	0.0751116i	$0.976069\pi$
$710$		0			0
$711$	11.0000			0.412532
$712$		0			0
$713$		−	63.0000i		−	2.35937i
$714$		0			0
$715$	2.00000	+	3.00000i	0.0747958	+	0.112194i
$716$		0			0
$717$			4.00000i			0.149383i
$718$		0			0
$719$	−2.00000			−0.0745874			−0.0372937	−	0.999304i	$-0.511874\pi$
$719$	−2.00000			−0.0745874			−0.0372937	+	0.999304i	$0.511874\pi$
$720$		0			0
$721$			14.0000i			0.521387i
$722$		0			0
$723$		−	50.0000i		−	1.85952i
$724$		0			0
$725$	20.0000			0.742781
$726$		0			0
$727$	−14.0000			−0.519231			−0.259616	−	0.965712i	$-0.583596\pi$
$727$	−14.0000			−0.519231			−0.259616	+	0.965712i	$0.583596\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	44.0000			1.62740
$732$		0			0
$733$			1.00000i			0.0369358i	0.999829	+	0.0184679i	$0.00587886\pi$
$733$			1.00000i			0.0369358i	−0.999829	+	0.0184679i	$0.994121\pi$
$734$		0			0
$735$		−	2.00000i		−	0.0737711i
$736$		0			0
$737$	10.0000			0.368355
$738$		0			0
$739$			20.0000i			0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$			20.0000i			0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$		0			0
$741$	−30.0000	+	20.0000i	−1.10208	+	0.734718i
$742$		0			0
$743$			48.0000i			1.76095i	0.474093	+	0.880475i	$0.342776\pi$
$743$			48.0000i			1.76095i	−0.474093	+	0.880475i	$0.657224\pi$
$744$		0			0
$745$	18.0000			0.659469
$746$		0			0
$747$		−	9.00000i		−	0.329293i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	3.00000			0.109472			0.0547358	−	0.998501i	$-0.482568\pi$
$751$	3.00000			0.109472			0.0547358	+	0.998501i	$0.482568\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−4.00000			−0.145575
$756$		0			0
$757$	−19.0000			−0.690567			−0.345283	−	0.938498i	$-0.612217\pi$
$757$	−19.0000			−0.690567			−0.345283	+	0.938498i	$0.612217\pi$
$758$		0			0
$759$			18.0000i			0.653359i
$760$		0			0
$761$			5.00000i			0.181250i	0.995885	+	0.0906249i	$0.0288864\pi$
$761$			5.00000i			0.181250i	−0.995885	+	0.0906249i	$0.971114\pi$
$762$		0			0
$763$	−16.0000			−0.579239
$764$		0			0
$765$		−	4.00000i		−	0.144620i
$766$		0			0
$767$	−12.0000	+	8.00000i	−0.433295	+	0.288863i
$768$		0			0
$769$		−	9.00000i		−	0.324548i	−0.986746	−	0.162274i	$-0.948117\pi$
$769$		−	9.00000i		−	0.324548i	0.986746	−	0.162274i	$-0.0518829\pi$
$770$		0			0
$771$	−12.0000			−0.432169
$772$		0			0
$773$		−	18.0000i		−	0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$		−	18.0000i		−	0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$		0			0
$775$			28.0000i			1.00579i
$776$		0			0
$777$	−4.00000			−0.143499
$778$		0			0
$779$	30.0000			1.07486
$780$		0			0
$781$	4.00000			0.143131
$782$		0			0
$783$	−20.0000			−0.714742
$784$		0			0
$785$		−	10.0000i		−	0.356915i
$786$		0			0
$787$		−	5.00000i		−	0.178231i	−0.996021	−	0.0891154i	$-0.971596\pi$
$787$		−	5.00000i		−	0.178231i	0.996021	−	0.0891154i	$-0.0284040\pi$
$788$		0			0
$789$	26.0000			0.925625
$790$		0			0
$791$			5.00000i			0.177780i
$792$		0			0
$793$	−12.0000	−	18.0000i	−0.426132	−	0.639199i
$794$		0			0
$795$		−	2.00000i		−	0.0709327i
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$			4.00000i			0.141510i
$800$		0			0
$801$		−	15.0000i		−	0.529999i
$802$		0			0
$803$	3.00000			0.105868
$804$		0			0
$805$	9.00000			0.317208
$806$		0			0
$807$	12.0000			0.422420
$808$		0			0
$809$	15.0000			0.527372			0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000			0.527372			0.263686	+	0.964609i	$0.415062\pi$
$810$		0			0
$811$			44.0000i			1.54505i	0.634985	+	0.772524i	$0.281006\pi$
$811$			44.0000i			1.54505i	−0.634985	+	0.772524i	$0.718994\pi$
$812$		0			0
$813$		−	16.0000i		−	0.561144i
$814$		0			0
$815$	−4.00000			−0.140114
$816$		0			0
$817$		−	55.0000i		−	1.92421i
$818$		0			0
$819$	−3.00000	+	2.00000i	−0.104828	+	0.0698857i
$820$		0			0
$821$		−	40.0000i		−	1.39601i	−0.716093	−	0.698005i	$-0.754071\pi$
$821$		−	40.0000i		−	1.39601i	0.716093	−	0.698005i	$-0.245929\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		−	8.00000i		−	0.278524i
$826$		0			0
$827$		−	6.00000i		−	0.208640i	−0.994544	−	0.104320i	$-0.966733\pi$
$827$		−	6.00000i		−	0.208640i	0.994544	−	0.104320i	$-0.0332667\pi$
$828$		0			0
$829$	46.0000			1.59765			0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000			1.59765			0.798823	+	0.601566i	$0.205456\pi$
$830$		0			0
$831$	66.0000			2.28951
$832$		0			0
$833$	4.00000			0.138592
$834$		0			0
$835$	−23.0000			−0.795948
$836$		0			0
$837$		−	28.0000i		−	0.967822i
$838$		0			0
$839$		−	16.0000i		−	0.552381i	−0.961103	−	0.276191i	$-0.910928\pi$
$839$		−	16.0000i		−	0.552381i	0.961103	−	0.276191i	$-0.0890721\pi$
$840$		0			0
$841$	−4.00000			−0.137931
$842$		0			0
$843$			40.0000i			1.37767i
$844$		0			0
$845$	−12.0000	−	5.00000i	−0.412813	−	0.172005i
$846$		0			0
$847$		−	1.00000i		−	0.0343604i
$848$		0			0
$849$	−12.0000			−0.411839
$850$		0			0
$851$		−	18.0000i		−	0.617032i
$852$		0			0
$853$		−	19.0000i		−	0.650548i	−0.945620	−	0.325274i	$-0.894544\pi$
$853$		−	19.0000i		−	0.650548i	0.945620	−	0.325274i	$-0.105456\pi$
$854$		0			0
$855$	−5.00000			−0.170996
$856$		0			0
$857$	30.0000			1.02478			0.512390	−	0.858753i	$-0.328760\pi$
$857$	30.0000			1.02478			0.512390	+	0.858753i	$0.328760\pi$
$858$		0			0
$859$	−48.0000			−1.63774			−0.818869	−	0.573980i	$-0.805399\pi$
$859$	−48.0000			−1.63774			−0.818869	+	0.573980i	$0.805399\pi$
$860$		0			0
$861$	12.0000			0.408959
$862$		0			0
$863$		−	38.0000i		−	1.29354i	−0.762687	−	0.646768i	$-0.776120\pi$
$863$		−	38.0000i		−	1.29354i	0.762687	−	0.646768i	$-0.223880\pi$
$864$		0			0
$865$		−	10.0000i		−	0.340010i
$866$		0			0
$867$	−2.00000			−0.0679236
$868$		0			0
$869$		−	11.0000i		−	0.373149i
$870$		0			0
$871$	−30.0000	+	20.0000i	−1.01651	+	0.677674i
$872$		0			0
$873$			7.00000i			0.236914i
$874$		0			0
$875$	−9.00000			−0.304256
$876$		0			0
$877$		−	6.00000i		−	0.202606i	−0.994856	−	0.101303i	$-0.967699\pi$
$877$		−	6.00000i		−	0.202606i	0.994856	−	0.101303i	$-0.0323011\pi$
$878$		0			0
$879$			42.0000i			1.41662i
$880$		0			0
$881$	24.0000			0.808581			0.404290	−	0.914631i	$-0.367519\pi$
$881$	24.0000			0.808581			0.404290	+	0.914631i	$0.367519\pi$
$882$		0			0
$883$	−52.0000			−1.74994			−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000			−1.74994			−0.874970	+	0.484178i	$0.839119\pi$
$884$		0			0
$885$	−8.00000			−0.268917
$886$		0			0
$887$	54.0000			1.81314			0.906571	−	0.422053i	$-0.138690\pi$
$887$	54.0000			1.81314			0.906571	+	0.422053i	$0.138690\pi$
$888$		0			0
$889$		−	20.0000i		−	0.670778i
$890$		0			0
$891$			11.0000i			0.368514i
$892$		0			0
$893$	5.00000			0.167319
$894$		0			0
$895$			11.0000i			0.367689i
$896$		0			0
$897$	−36.0000	−	54.0000i	−1.20201	−	1.80301i
$898$		0			0
$899$			35.0000i			1.16732i
$900$		0			0
$901$	4.00000			0.133259
$902$		0			0
$903$		−	22.0000i		−	0.732114i
$904$		0			0
$905$			16.0000i			0.531858i
$906$		0			0
$907$	23.0000			0.763702			0.381851	−	0.924224i	$-0.375287\pi$
$907$	23.0000			0.763702			0.381851	+	0.924224i	$0.375287\pi$
$908$		0			0
$909$	8.00000			0.265343
$910$		0			0
$911$	−9.00000			−0.298183			−0.149092	−	0.988823i	$-0.547635\pi$
$911$	−9.00000			−0.298183			−0.149092	+	0.988823i	$0.547635\pi$
$912$		0			0
$913$	−9.00000			−0.297857
$914$		0			0
$915$		−	12.0000i		−	0.396708i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	24.0000			0.791687			0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000			0.791687			0.395843	+	0.918318i	$0.370452\pi$
$920$		0			0
$921$		−	22.0000i		−	0.724925i
$922$		0			0
$923$	−12.0000	+	8.00000i	−0.394985	+	0.263323i
$924$		0			0
$925$			8.00000i			0.263038i
$926$		0			0
$927$	14.0000			0.459820
$928$		0			0
$929$			45.0000i			1.47640i	0.674581	+	0.738201i	$0.264324\pi$
$929$			45.0000i			1.47640i	−0.674581	+	0.738201i	$0.735676\pi$
$930$		0			0
$931$		−	5.00000i		−	0.163868i
$932$		0			0
$933$	64.0000			2.09527
$934$		0			0
$935$	−4.00000			−0.130814
$936$		0			0
$937$	38.0000			1.24141			0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000			1.24141			0.620703	+	0.784046i	$0.286847\pi$
$938$		0			0
$939$	56.0000			1.82749
$940$		0			0
$941$		−	39.0000i		−	1.27136i	−0.771951	−	0.635682i	$-0.780719\pi$
$941$		−	39.0000i		−	1.27136i	0.771951	−	0.635682i	$-0.219281\pi$
$942$		0			0
$943$			54.0000i			1.75848i
$944$		0			0
$945$	4.00000			0.130120
$946$		0			0
$947$			36.0000i			1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$			36.0000i			1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$		0			0
$949$	−9.00000	+	6.00000i	−0.292152	+	0.194768i
$950$		0			0
$951$			12.0000i			0.389127i
$952$		0			0
$953$	−15.0000			−0.485898			−0.242949	−	0.970039i	$-0.578115\pi$
$953$	−15.0000			−0.485898			−0.242949	+	0.970039i	$0.578115\pi$
$954$		0			0
$955$			20.0000i			0.647185i
$956$		0			0
$957$		−	10.0000i		−	0.323254i
$958$		0			0
$959$	−18.0000			−0.581250
$960$		0			0
$961$	−18.0000			−0.580645
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−16.0000			−0.515058
$966$		0			0
$967$			16.0000i			0.514525i	0.966342	+	0.257263i	$0.0828206\pi$
$967$			16.0000i			0.514525i	−0.966342	+	0.257263i	$0.917179\pi$
$968$		0			0
$969$		−	40.0000i		−	1.28499i
$970$		0			0
$971$	−20.0000			−0.641831			−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000			−0.641831			−0.320915	+	0.947108i	$0.603990\pi$
$972$		0			0
$973$			12.0000i			0.384702i
$974$		0			0
$975$	16.0000	+	24.0000i	0.512410	+	0.768615i
$976$		0			0
$977$		−	16.0000i		−	0.511885i	−0.966692	−	0.255943i	$-0.917614\pi$
$977$		−	16.0000i		−	0.511885i	0.966692	−	0.255943i	$-0.0823858\pi$
$978$		0			0
$979$	−15.0000			−0.479402
$980$		0			0
$981$			16.0000i			0.510841i
$982$		0			0
$983$		−	11.0000i		−	0.350846i	−0.984493	−	0.175423i	$-0.943871\pi$
$983$		−	11.0000i		−	0.350846i	0.984493	−	0.175423i	$-0.0561292\pi$
$984$		0			0
$985$	26.0000			0.828429
$986$		0			0
$987$	2.00000			0.0636607
$988$		0			0
$989$	99.0000			3.14802
$990$		0			0
$991$	−16.0000			−0.508257			−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000			−0.508257			−0.254128	+	0.967170i	$0.581789\pi$
$992$		0			0
$993$		−	12.0000i		−	0.380808i
$994$		0			0
$995$		−	20.0000i		−	0.634043i
$996$		0			0
$997$	24.0000			0.760088			0.380044	−	0.924968i	$-0.375909\pi$
$997$	24.0000			0.760088			0.380044	+	0.924968i	$0.375909\pi$
$998$		0			0
$999$		−	8.00000i		−	0.253109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.a.2157.2	yes	2
13.12	even	2	inner	4004.2.m.a.2157.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.a.2157.1	✓	2	13.12	even	2	inner
4004.2.m.a.2157.2	yes	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000 q^{3} +1.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{3} +1.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} -1.00000i q^{11} +(2.00000 + 3.00000i) q^{13} +2.00000i q^{15} -4.00000 q^{17} +5.00000i q^{19} +2.00000i q^{21} -9.00000 q^{23} +4.00000 q^{25} -4.00000 q^{27} +5.00000 q^{29} +7.00000i q^{31} -2.00000i q^{33} -1.00000 q^{35} +2.00000i q^{37} +(4.00000 + 6.00000i) q^{39} -6.00000i q^{41} -11.0000 q^{43} +1.00000i q^{45} -1.00000i q^{47} -1.00000 q^{49} -8.00000 q^{51} -1.00000 q^{53} +1.00000 q^{55} +10.0000i q^{57} +4.00000i q^{59} -6.00000 q^{61} +1.00000i q^{63} +(-3.00000 + 2.00000i) q^{65} +10.0000i q^{67} -18.0000 q^{69} +4.00000i q^{71} +3.00000i q^{73} +8.00000 q^{75} +1.00000 q^{77} +11.0000 q^{79} -11.0000 q^{81} -9.00000i q^{83} -4.00000i q^{85} +10.0000 q^{87} -15.0000i q^{89} +(-3.00000 + 2.00000i) q^{91} +14.0000i q^{93} -5.00000 q^{95} +7.00000i q^{97} -1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^3 + 2 * q^9	\( 2 q + 4 q^{3} + 2 q^{9} + 4 q^{13} - 8 q^{17} - 18 q^{23} + 8 q^{25} - 8 q^{27} + 10 q^{29} - 2 q^{35} + 8 q^{39} - 22 q^{43} - 2 q^{49} - 16 q^{51} - 2 q^{53} + 2 q^{55} - 12 q^{61} - 6 q^{65} - 36 q^{69} + 16 q^{75} + 2 q^{77} + 22 q^{79} - 22 q^{81} + 20 q^{87} - 6 q^{91} - 10 q^{95}+O(q^{100}) \) 2 * q + 4 * q^3 + 2 * q^9 + 4 * q^13 - 8 * q^17 - 18 * q^23 + 8 * q^25 - 8 * q^27 + 10 * q^29 - 2 * q^35 + 8 * q^39 - 22 * q^43 - 2 * q^49 - 16 * q^51 - 2 * q^53 + 2 * q^55 - 12 * q^61 - 6 * q^65 - 36 * q^69 + 16 * q^75 + 2 * q^77 + 22 * q^79 - 22 * q^81 + 20 * q^87 - 6 * q^91 - 10 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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