LMFDB - Embedded newform 4004.2.m.b.2157.7

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:	$30$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2157.7
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.b.2157.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41023 q^{3} -3.42759i q^{5} +1.00000i q^{7} +2.80920 q^{9} +O(q^{10})$	$q-2.41023 q^{3} -3.42759i q^{5} +1.00000i q^{7} +2.80920 q^{9} +1.00000i q^{11} +(3.35262 - 1.32663i) q^{13} +8.26127i q^{15} -0.705996 q^{17} -0.998774i q^{19} -2.41023i q^{21} +2.09474 q^{23} -6.74835 q^{25} +0.459875 q^{27} +1.73454 q^{29} +8.34796i q^{31} -2.41023i q^{33} +3.42759 q^{35} -6.44693i q^{37} +(-8.08058 + 3.19747i) q^{39} +8.10009i q^{41} -3.25458 q^{43} -9.62877i q^{45} +10.5464i q^{47} -1.00000 q^{49} +1.70161 q^{51} -5.72840 q^{53} +3.42759 q^{55} +2.40727i q^{57} +3.36759i q^{59} -9.47398 q^{61} +2.80920i q^{63} +(-4.54712 - 11.4914i) q^{65} -1.90892i q^{67} -5.04880 q^{69} -0.177480i q^{71} +13.0757i q^{73} +16.2651 q^{75} -1.00000 q^{77} -1.06149 q^{79} -9.53600 q^{81} +4.13544i q^{83} +2.41986i q^{85} -4.18063 q^{87} +1.28092i q^{89} +(1.32663 + 3.35262i) q^{91} -20.1205i q^{93} -3.42339 q^{95} -1.32942i q^{97} +2.80920i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 18 q^{9}+O(q^{10}) $ 30 * q + 18 * q^9	$ 30 q + 18 q^{9} + 4 q^{13} + 12 q^{17} + 6 q^{23} - 6 q^{29} - 2 q^{35} + 8 q^{39} - 22 q^{43} - 30 q^{49} + 60 q^{51} - 38 q^{53} - 2 q^{55} - 36 q^{61} + 10 q^{65} + 36 q^{69} - 20 q^{75} - 30 q^{77} - 10 q^{79} - 42 q^{81} + 36 q^{87} + 6 q^{91} - 2 q^{95}+O(q^{100}) $ 30 * q + 18 * q^9 + 4 * q^13 + 12 * q^17 + 6 * q^23 - 6 * q^29 - 2 * q^35 + 8 * q^39 - 22 * q^43 - 30 * q^49 + 60 * q^51 - 38 * q^53 - 2 * q^55 - 36 * q^61 + 10 * q^65 + 36 * q^69 - 20 * q^75 - 30 * q^77 - 10 * q^79 - 42 * q^81 + 36 * q^87 + 6 * q^91 - 2 * 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
$3$	−2.41023			−1.39155			−0.695773	−	0.718262i	$-0.744938\pi$
$3$	−2.41023			−1.39155			−0.695773	+	0.718262i	$0.744938\pi$
$4$		0			0
$5$		−	3.42759i		−	1.53286i	−0.642326	−	0.766432i	$-0.722030\pi$
$5$		−	3.42759i		−	1.53286i	0.642326	−	0.766432i	$-0.277970\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	2.80920			0.936399
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$	3.35262	−	1.32663i	0.929850	−	0.367940i
$13$	3.35262	−	1.32663i	0.929850	−	0.367940i
$14$		0			0
$15$			8.26127i			2.13305i
$16$		0			0
$17$	−0.705996			−0.171229			−0.0856146	−	0.996328i	$-0.527285\pi$
$17$	−0.705996			−0.171229			−0.0856146	+	0.996328i	$0.527285\pi$
$18$		0			0
$19$		−	0.998774i		−	0.229135i	−0.993415	−	0.114567i	$-0.963452\pi$
$19$		−	0.998774i		−	0.229135i	0.993415	−	0.114567i	$-0.0365481\pi$
$20$		0			0
$21$		−	2.41023i		−	0.525955i
$22$		0			0
$23$	2.09474			0.436783			0.218392	−	0.975861i	$-0.429919\pi$
$23$	2.09474			0.436783			0.218392	+	0.975861i	$0.429919\pi$
$24$		0			0
$25$	−6.74835			−1.34967
$26$		0			0
$27$	0.459875			0.0885031
$28$		0			0
$29$	1.73454			0.322095			0.161048	−	0.986947i	$-0.448513\pi$
$29$	1.73454			0.322095			0.161048	+	0.986947i	$0.448513\pi$
$30$		0			0
$31$			8.34796i			1.49934i	0.661813	+	0.749669i	$0.269787\pi$
$31$			8.34796i			1.49934i	−0.661813	+	0.749669i	$0.730213\pi$
$32$		0			0
$33$		−	2.41023i		−	0.419567i
$34$		0			0
$35$	3.42759			0.579368
$36$		0			0
$37$		−	6.44693i		−	1.05987i	−0.848039	−	0.529934i	$-0.822217\pi$
$37$		−	6.44693i		−	1.05987i	0.848039	−	0.529934i	$-0.177783\pi$
$38$		0			0
$39$	−8.08058	+	3.19747i	−1.29393	+	0.512005i
$40$		0			0
$41$			8.10009i			1.26502i	0.774551	+	0.632511i	$0.217976\pi$
$41$			8.10009i			1.26502i	−0.774551	+	0.632511i	$0.782024\pi$
$42$		0			0
$43$	−3.25458			−0.496319			−0.248160	−	0.968719i	$-0.579826\pi$
$43$	−3.25458			−0.496319			−0.248160	+	0.968719i	$0.579826\pi$
$44$		0			0
$45$		−	9.62877i		−	1.43537i
$46$		0			0
$47$			10.5464i			1.53836i	0.639034	+	0.769179i	$0.279334\pi$
$47$			10.5464i			1.53836i	−0.639034	+	0.769179i	$0.720666\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	1.70161			0.238273
$52$		0			0
$53$	−5.72840			−0.786857			−0.393428	−	0.919355i	$-0.628711\pi$
$53$	−5.72840			−0.786857			−0.393428	+	0.919355i	$0.628711\pi$
$54$		0			0
$55$	3.42759			0.462176
$56$		0			0
$57$			2.40727i			0.318851i
$58$		0			0
$59$			3.36759i			0.438423i	0.975677	+	0.219212i	$0.0703485\pi$
$59$			3.36759i			0.438423i	−0.975677	+	0.219212i	$0.929651\pi$
$60$		0			0
$61$	−9.47398			−1.21302			−0.606509	−	0.795076i	$-0.707431\pi$
$61$	−9.47398			−1.21302			−0.606509	+	0.795076i	$0.707431\pi$
$62$		0			0
$63$			2.80920i			0.353926i
$64$		0			0
$65$	−4.54712	−	11.4914i	−0.564001	−	1.42533i
$66$		0			0
$67$		−	1.90892i		−	0.233211i	−0.993178	−	0.116606i	$-0.962799\pi$
$67$		−	1.90892i		−	0.233211i	0.993178	−	0.116606i	$-0.0372013\pi$
$68$		0			0
$69$	−5.04880			−0.607804
$70$		0			0
$71$		−	0.177480i		−	0.0210630i	−0.999945	−	0.0105315i	$-0.996648\pi$
$71$		−	0.177480i		−	0.0210630i	0.999945	−	0.0105315i	$-0.00335235\pi$
$72$		0			0
$73$			13.0757i			1.53040i	0.643795	+	0.765198i	$0.277359\pi$
$73$			13.0757i			1.53040i	−0.643795	+	0.765198i	$0.722641\pi$
$74$		0			0
$75$	16.2651			1.87813
$76$		0			0
$77$	−1.00000			−0.113961
$78$		0			0
$79$	−1.06149			−0.119427			−0.0597135	−	0.998216i	$-0.519019\pi$
$79$	−1.06149			−0.119427			−0.0597135	+	0.998216i	$0.519019\pi$
$80$		0			0
$81$	−9.53600			−1.05956
$82$		0			0
$83$			4.13544i			0.453924i	0.973904	+	0.226962i	$0.0728793\pi$
$83$			4.13544i			0.453924i	−0.973904	+	0.226962i	$0.927121\pi$
$84$		0			0
$85$			2.41986i			0.262471i
$86$		0			0
$87$	−4.18063			−0.448211
$88$		0			0
$89$			1.28092i			0.135777i	0.997693	+	0.0678885i	$0.0216262\pi$
$89$			1.28092i			0.135777i	−0.997693	+	0.0678885i	$0.978374\pi$
$90$		0			0
$91$	1.32663	+	3.35262i	0.139068	+	0.351450i
$92$		0			0
$93$		−	20.1205i		−	2.08640i
$94$		0			0
$95$	−3.42339			−0.351232
$96$		0			0
$97$		−	1.32942i		−	0.134982i	−0.997720	−	0.0674910i	$-0.978501\pi$
$97$		−	1.32942i		−	0.134982i	0.997720	−	0.0674910i	$-0.0214994\pi$
$98$		0			0
$99$			2.80920i			0.282335i
$100$		0			0
$101$	6.35320			0.632167			0.316083	−	0.948731i	$-0.397632\pi$
$101$	6.35320			0.632167			0.316083	+	0.948731i	$0.397632\pi$
$102$		0			0
$103$	11.2339			1.10691			0.553456	−	0.832879i	$-0.313309\pi$
$103$	11.2339			1.10691			0.553456	+	0.832879i	$0.313309\pi$
$104$		0			0
$105$	−8.26127			−0.806217
$106$		0			0
$107$	−9.81146			−0.948510			−0.474255	−	0.880388i	$-0.657282\pi$
$107$	−9.81146			−0.948510			−0.474255	+	0.880388i	$0.657282\pi$
$108$		0			0
$109$		−	0.185079i		−	0.0177274i	−0.999961	−	0.00886368i	$-0.997179\pi$
$109$		−	0.185079i		−	0.0177274i	0.999961	−	0.00886368i	$-0.00282143\pi$
$110$		0			0
$111$			15.5386i			1.47486i
$112$		0			0
$113$	1.31495			0.123700			0.0618502	−	0.998085i	$-0.480300\pi$
$113$	1.31495			0.123700			0.0618502	+	0.998085i	$0.480300\pi$
$114$		0			0
$115$		−	7.17990i		−	0.669529i
$116$		0			0
$117$	9.41818	−	3.72675i	0.870711	−	0.344539i
$118$		0			0
$119$		−	0.705996i		−	0.0647185i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	19.5231i		−	1.76034i
$124$		0			0
$125$			5.99263i			0.535997i
$126$		0			0
$127$	3.60631			0.320008			0.160004	−	0.987116i	$-0.448849\pi$
$127$	3.60631			0.320008			0.160004	+	0.987116i	$0.448849\pi$
$128$		0			0
$129$	7.84429			0.690651
$130$		0			0
$131$	4.82053			0.421171			0.210586	−	0.977575i	$-0.432463\pi$
$131$	4.82053			0.421171			0.210586	+	0.977575i	$0.432463\pi$
$132$		0			0
$133$	0.998774			0.0866047
$134$		0			0
$135$		−	1.57626i		−	0.135663i
$136$		0			0
$137$			7.03075i			0.600678i	0.953833	+	0.300339i	$0.0970998\pi$
$137$			7.03075i			0.600678i	−0.953833	+	0.300339i	$0.902900\pi$
$138$		0			0
$139$	10.4700			0.888054			0.444027	−	0.896013i	$-0.353549\pi$
$139$	10.4700			0.888054			0.444027	+	0.896013i	$0.353549\pi$
$140$		0			0
$141$		−	25.4193i		−	2.14069i
$142$		0			0
$143$	1.32663	+	3.35262i	0.110938	+	0.280360i
$144$		0			0
$145$		−	5.94528i		−	0.493728i
$146$		0			0
$147$	2.41023			0.198792
$148$		0			0
$149$			4.59341i			0.376307i	0.982140	+	0.188153i	$0.0602502\pi$
$149$			4.59341i			0.376307i	−0.982140	+	0.188153i	$0.939750\pi$
$150$		0			0
$151$			6.31728i			0.514093i	0.966399	+	0.257046i	$0.0827493\pi$
$151$			6.31728i			0.514093i	−0.966399	+	0.257046i	$0.917251\pi$
$152$		0			0
$153$	−1.98328			−0.160339
$154$		0			0
$155$	28.6134			2.29828
$156$		0			0
$157$	13.7175			1.09478			0.547389	−	0.836878i	$-0.315622\pi$
$157$	13.7175			1.09478			0.547389	+	0.836878i	$0.315622\pi$
$158$		0			0
$159$	13.8068			1.09495
$160$		0			0
$161$			2.09474i			0.165088i
$162$		0			0
$163$			7.01909i			0.549778i	0.961476	+	0.274889i	$0.0886411\pi$
$163$			7.01909i			0.549778i	−0.961476	+	0.274889i	$0.911359\pi$
$164$		0			0
$165$	−8.26127			−0.643139
$166$		0			0
$167$		−	3.52406i		−	0.272700i	−0.990661	−	0.136350i	$-0.956463\pi$
$167$		−	3.52406i		−	0.272700i	0.990661	−	0.136350i	$-0.0435372\pi$
$168$		0			0
$169$	9.48013	−	8.89534i	0.729241	−	0.684257i
$170$		0			0
$171$		−	2.80576i		−	0.214561i
$172$		0			0
$173$	3.63607			0.276445			0.138223	−	0.990401i	$-0.455861\pi$
$173$	3.63607			0.276445			0.138223	+	0.990401i	$0.455861\pi$
$174$		0			0
$175$		−	6.74835i		−	0.510127i
$176$		0			0
$177$		−	8.11667i		−	0.610086i
$178$		0			0
$179$	−14.7103			−1.09950			−0.549748	−	0.835331i	$-0.685276\pi$
$179$	−14.7103			−1.09950			−0.549748	+	0.835331i	$0.685276\pi$
$180$		0			0
$181$	9.94667			0.739330			0.369665	−	0.929165i	$-0.379472\pi$
$181$	9.94667			0.739330			0.369665	+	0.929165i	$0.379472\pi$
$182$		0			0
$183$	22.8344			1.68797
$184$		0			0
$185$	−22.0974			−1.62463
$186$		0			0
$187$		−	0.705996i		−	0.0516275i
$188$		0			0
$189$			0.459875i			0.0334510i
$190$		0			0
$191$	18.9823			1.37351			0.686754	−	0.726890i	$-0.259035\pi$
$191$	18.9823			1.37351			0.686754	+	0.726890i	$0.259035\pi$
$192$		0			0
$193$		−	6.25334i		−	0.450125i	−0.974344	−	0.225063i	$-0.927741\pi$
$193$		−	6.25334i		−	0.450125i	0.974344	−	0.225063i	$-0.0722586\pi$
$194$		0			0
$195$	10.9596	+	27.6969i	0.784834	+	1.98342i
$196$		0			0
$197$		−	3.22463i		−	0.229745i	−0.993380	−	0.114873i	$-0.963354\pi$
$197$		−	3.22463i		−	0.229745i	0.993380	−	0.114873i	$-0.0366460\pi$
$198$		0			0
$199$	21.9129			1.55336			0.776681	−	0.629894i	$-0.216902\pi$
$199$	21.9129			1.55336			0.776681	+	0.629894i	$0.216902\pi$
$200$		0			0
$201$			4.60092i			0.324524i
$202$		0			0
$203$			1.73454i			0.121741i
$204$		0			0
$205$	27.7638			1.93911
$206$		0			0
$207$	5.88453			0.409003
$208$		0			0
$209$	0.998774			0.0690867
$210$		0			0
$211$	−0.576573			−0.0396929			−0.0198464	−	0.999803i	$-0.506318\pi$
$211$	−0.576573			−0.0396929			−0.0198464	+	0.999803i	$0.506318\pi$
$212$		0			0
$213$			0.427768i			0.0293102i
$214$		0			0
$215$			11.1554i			0.760790i
$216$		0			0
$217$	−8.34796			−0.566697
$218$		0			0
$219$		−	31.5155i		−	2.12962i
$220$		0			0
$221$	−2.36694	+	0.936592i	−0.159217	+	0.0630020i
$222$		0			0
$223$			8.12461i			0.544064i	0.962288	+	0.272032i	$0.0876957\pi$
$223$			8.12461i			0.544064i	−0.962288	+	0.272032i	$0.912304\pi$
$224$		0			0
$225$	−18.9575			−1.26383
$226$		0			0
$227$			25.3671i			1.68368i	0.539731	+	0.841838i	$0.318526\pi$
$227$			25.3671i			1.68368i	−0.539731	+	0.841838i	$0.681474\pi$
$228$		0			0
$229$		−	2.06788i		−	0.136649i	−0.997663	−	0.0683245i	$-0.978235\pi$
$229$		−	2.06788i		−	0.136649i	0.997663	−	0.0683245i	$-0.0217653\pi$
$230$		0			0
$231$	2.41023			0.158581
$232$		0			0
$233$	−3.63376			−0.238056			−0.119028	−	0.992891i	$-0.537978\pi$
$233$	−3.63376			−0.238056			−0.119028	+	0.992891i	$0.537978\pi$
$234$		0			0
$235$	36.1489			2.35809
$236$		0			0
$237$	2.55844			0.166188
$238$		0			0
$239$			13.6623i			0.883740i	0.897079	+	0.441870i	$0.145685\pi$
$239$			13.6623i			0.883740i	−0.897079	+	0.441870i	$0.854315\pi$
$240$		0			0
$241$			2.50903i			0.161621i	0.996729	+	0.0808105i	$0.0257509\pi$
$241$			2.50903i			0.161621i	−0.996729	+	0.0808105i	$0.974249\pi$
$242$		0			0
$243$	21.6043			1.38592
$244$		0			0
$245$			3.42759i			0.218980i
$246$		0			0
$247$	−1.32500	−	3.34851i	−0.0843077	−	0.213061i
$248$		0			0
$249$		−	9.96736i		−	0.631656i
$250$		0			0
$251$	−27.6382			−1.74451			−0.872253	−	0.489054i	$-0.837342\pi$
$251$	−27.6382			−1.74451			−0.872253	+	0.489054i	$0.837342\pi$
$252$		0			0
$253$			2.09474i			0.131695i
$254$		0			0
$255$		−	5.83242i		−	0.365240i
$256$		0			0
$257$	20.2110			1.26073			0.630365	−	0.776299i	$-0.282905\pi$
$257$	20.2110			1.26073			0.630365	+	0.776299i	$0.282905\pi$
$258$		0			0
$259$	6.44693			0.400593
$260$		0			0
$261$	4.87266			0.301610
$262$		0			0
$263$	0.811179			0.0500194			0.0250097	−	0.999687i	$-0.492038\pi$
$263$	0.811179			0.0500194			0.0250097	+	0.999687i	$0.492038\pi$
$264$		0			0
$265$			19.6346i			1.20614i
$266$		0			0
$267$		−	3.08731i		−	0.188940i
$268$		0			0
$269$	−1.26390			−0.0770612			−0.0385306	−	0.999257i	$-0.512268\pi$
$269$	−1.26390			−0.0770612			−0.0385306	+	0.999257i	$0.512268\pi$
$270$		0			0
$271$			7.89302i			0.479467i	0.970839	+	0.239733i	$0.0770600\pi$
$271$			7.89302i			0.479467i	−0.970839	+	0.239733i	$0.922940\pi$
$272$		0			0
$273$	−3.19747	−	8.08058i	−0.193520	−	0.489059i
$274$		0			0
$275$		−	6.74835i		−	0.406941i
$276$		0			0
$277$	23.4405			1.40840			0.704202	−	0.710000i	$-0.251305\pi$
$277$	23.4405			1.40840			0.704202	+	0.710000i	$0.251305\pi$
$278$		0			0
$279$			23.4511i			1.40398i
$280$		0			0
$281$		−	5.39390i		−	0.321773i	−0.986973	−	0.160887i	$-0.948565\pi$
$281$		−	5.39390i		−	0.321773i	0.986973	−	0.160887i	$-0.0514353\pi$
$282$		0			0
$283$	7.73643			0.459883			0.229941	−	0.973204i	$-0.426147\pi$
$283$	7.73643			0.459883			0.229941	+	0.973204i	$0.426147\pi$
$284$		0			0
$285$	8.25114			0.488755
$286$		0			0
$287$	−8.10009			−0.478133
$288$		0			0
$289$	−16.5016			−0.970681
$290$		0			0
$291$			3.20420i			0.187834i
$292$		0			0
$293$			10.8656i			0.634777i	0.948296	+	0.317389i	$0.102806\pi$
$293$			10.8656i			0.634777i	−0.948296	+	0.317389i	$0.897194\pi$
$294$		0			0
$295$	11.5427			0.672043
$296$		0			0
$297$			0.459875i			0.0266847i
$298$		0			0
$299$	7.02286	−	2.77893i	0.406143	−	0.160710i
$300$		0			0
$301$		−	3.25458i		−	0.187591i
$302$		0			0
$303$	−15.3127			−0.879689
$304$		0			0
$305$			32.4729i			1.85939i
$306$		0			0
$307$		−	3.18252i		−	0.181636i	−0.995868	−	0.0908179i	$-0.971052\pi$
$307$		−	3.18252i		−	0.181636i	0.995868	−	0.0908179i	$-0.0289481\pi$
$308$		0			0
$309$	−27.0763			−1.54032
$310$		0			0
$311$	−17.0617			−0.967479			−0.483740	−	0.875212i	$-0.660722\pi$
$311$	−17.0617			−0.967479			−0.483740	+	0.875212i	$0.660722\pi$
$312$		0			0
$313$	1.93270			0.109243			0.0546213	−	0.998507i	$-0.482605\pi$
$313$	1.93270			0.109243			0.0546213	+	0.998507i	$0.482605\pi$
$314$		0			0
$315$	9.62877			0.542520
$316$		0			0
$317$			25.8763i			1.45336i	0.686978	+	0.726678i	$0.258937\pi$
$317$			25.8763i			1.45336i	−0.686978	+	0.726678i	$0.741063\pi$
$318$		0			0
$319$			1.73454i			0.0971154i
$320$		0			0
$321$	23.6479			1.31989
$322$		0			0
$323$			0.705131i			0.0392345i
$324$		0			0
$325$	−22.6247	+	8.95253i	−1.25499	+	0.496597i
$326$		0			0
$327$			0.446083i			0.0246684i
$328$		0			0
$329$	−10.5464			−0.581444
$330$		0			0
$331$			3.21317i			0.176612i	0.996093	+	0.0883058i	$0.0281453\pi$
$331$			3.21317i			0.176612i	−0.996093	+	0.0883058i	$0.971855\pi$
$332$		0			0
$333$		−	18.1107i		−	0.992460i
$334$		0			0
$335$	−6.54297			−0.357481
$336$		0			0
$337$	12.5601			0.684193			0.342096	−	0.939665i	$-0.388863\pi$
$337$	12.5601			0.684193			0.342096	+	0.939665i	$0.388863\pi$
$338$		0			0
$339$	−3.16934			−0.172135
$340$		0			0
$341$	−8.34796			−0.452067
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			17.3052i			0.931680i
$346$		0			0
$347$	29.2500			1.57022			0.785112	−	0.619354i	$-0.212606\pi$
$347$	29.2500			1.57022			0.785112	+	0.619354i	$0.212606\pi$
$348$		0			0
$349$			3.83516i			0.205291i	0.994718	+	0.102646i	$0.0327308\pi$
$349$			3.83516i			0.205291i	−0.994718	+	0.102646i	$0.967269\pi$
$350$		0			0
$351$	1.54179	−	0.610082i	0.0822946	−	0.0325638i
$352$		0			0
$353$			0.571999i			0.0304444i	0.999884	+	0.0152222i	$0.00484557\pi$
$353$			0.571999i			0.0304444i	−0.999884	+	0.0152222i	$0.995154\pi$
$354$		0			0
$355$	−0.608329			−0.0322867
$356$		0			0
$357$			1.70161i			0.0900588i
$358$		0			0
$359$		−	28.7331i		−	1.51648i	−0.651977	−	0.758238i	$-0.726060\pi$
$359$		−	28.7331i		−	1.51648i	0.651977	−	0.758238i	$-0.273940\pi$
$360$		0			0
$361$	18.0024			0.947497
$362$		0			0
$363$	2.41023			0.126504
$364$		0			0
$365$	44.8182			2.34589
$366$		0			0
$367$	−8.20164			−0.428122			−0.214061	−	0.976820i	$-0.568669\pi$
$367$	−8.20164			−0.428122			−0.214061	+	0.976820i	$0.568669\pi$
$368$		0			0
$369$			22.7548i			1.18457i
$370$		0			0
$371$		−	5.72840i		−	0.297404i
$372$		0			0
$373$	26.5105			1.37266			0.686330	−	0.727290i	$-0.259221\pi$
$373$	26.5105			1.37266			0.686330	+	0.727290i	$0.259221\pi$
$374$		0			0
$375$		−	14.4436i		−	0.745864i
$376$		0			0
$377$	5.81525	−	2.30108i	0.299500	−	0.118512i
$378$		0			0
$379$		−	21.8847i		−	1.12414i	−0.827088	−	0.562072i	$-0.810004\pi$
$379$		−	21.8847i		−	1.12414i	0.827088	−	0.562072i	$-0.189996\pi$
$380$		0			0
$381$	−8.69203			−0.445306
$382$		0			0
$383$		−	10.3448i		−	0.528597i	−0.964441	−	0.264298i	$-0.914860\pi$
$383$		−	10.3448i		−	0.528597i	0.964441	−	0.264298i	$-0.0851404\pi$
$384$		0			0
$385$			3.42759i			0.174686i
$386$		0			0
$387$	−9.14277			−0.464753
$388$		0			0
$389$	−4.06985			−0.206350			−0.103175	−	0.994663i	$-0.532900\pi$
$389$	−4.06985			−0.206350			−0.103175	+	0.994663i	$0.532900\pi$
$390$		0			0
$391$	−1.47888			−0.0747900
$392$		0			0
$393$	−11.6186			−0.586079
$394$		0			0
$395$			3.63835i			0.183065i
$396$		0			0
$397$		−	13.5433i		−	0.679719i	−0.940476	−	0.339860i	$-0.889620\pi$
$397$		−	13.5433i		−	0.679719i	0.940476	−	0.339860i	$-0.110380\pi$
$398$		0			0
$399$	−2.40727			−0.120514
$400$		0			0
$401$		−	26.9587i		−	1.34625i	−0.739527	−	0.673127i	$-0.764951\pi$
$401$		−	26.9587i		−	1.34625i	0.739527	−	0.673127i	$-0.235049\pi$
$402$		0			0
$403$	11.0746	+	27.9875i	0.551666	+	1.39416i
$404$		0			0
$405$			32.6855i			1.62415i
$406$		0			0
$407$	6.44693			0.319562
$408$		0			0
$409$		−	21.0401i		−	1.04037i	−0.854054	−	0.520184i	$-0.825863\pi$
$409$		−	21.0401i		−	1.04037i	0.854054	−	0.520184i	$-0.174137\pi$
$410$		0			0
$411$		−	16.9457i		−	0.835870i
$412$		0			0
$413$	−3.36759			−0.165708
$414$		0			0
$415$	14.1746			0.695803
$416$		0			0
$417$	−25.2351			−1.23577
$418$		0			0
$419$	−16.0405			−0.783628			−0.391814	−	0.920044i	$-0.628152\pi$
$419$	−16.0405			−0.783628			−0.391814	+	0.920044i	$0.628152\pi$
$420$		0			0
$421$			27.7048i			1.35025i	0.737703	+	0.675125i	$0.235910\pi$
$421$			27.7048i			1.35025i	−0.737703	+	0.675125i	$0.764090\pi$
$422$		0			0
$423$			29.6271i			1.44052i
$424$		0			0
$425$	4.76431			0.231103
$426$		0			0
$427$		−	9.47398i		−	0.458478i
$428$		0			0
$429$	−3.19747	−	8.08058i	−0.154375	−	0.390134i
$430$		0			0
$431$			2.11299i			0.101779i	0.998704	+	0.0508895i	$0.0162056\pi$
$431$			2.11299i			0.101779i	−0.998704	+	0.0508895i	$0.983794\pi$
$432$		0			0
$433$	−20.3824			−0.979514			−0.489757	−	0.871859i	$-0.662914\pi$
$433$	−20.3824			−0.979514			−0.489757	+	0.871859i	$0.662914\pi$
$434$		0			0
$435$			14.3295i			0.687046i
$436$		0			0
$437$		−	2.09217i		−	0.100082i
$438$		0			0
$439$	26.0325			1.24246			0.621231	−	0.783627i	$-0.286633\pi$
$439$	26.0325			1.24246			0.621231	+	0.783627i	$0.286633\pi$
$440$		0			0
$441$	−2.80920			−0.133771
$442$		0			0
$443$	13.5263			0.642654			0.321327	−	0.946968i	$-0.395871\pi$
$443$	13.5263			0.642654			0.321327	+	0.946968i	$0.395871\pi$
$444$		0			0
$445$	4.39046			0.208128
$446$		0			0
$447$		−	11.0712i		−	0.523648i
$448$		0			0
$449$		−	13.5955i		−	0.641610i	−0.947145	−	0.320805i	$-0.896047\pi$
$449$		−	13.5955i		−	0.641610i	0.947145	−	0.320805i	$-0.103953\pi$
$450$		0			0
$451$	−8.10009			−0.381419
$452$		0			0
$453$		−	15.2261i		−	0.715383i
$454$		0			0
$455$	11.4914	−	4.54712i	0.538725	−	0.213172i
$456$		0			0
$457$		−	2.56117i		−	0.119807i	−0.998204	−	0.0599033i	$-0.980921\pi$
$457$		−	2.56117i		−	0.119807i	0.998204	−	0.0599033i	$-0.0190792\pi$
$458$		0			0
$459$	−0.324670			−0.0151543
$460$		0			0
$461$			39.9660i			1.86140i	0.365779	+	0.930702i	$0.380803\pi$
$461$			39.9660i			1.86140i	−0.365779	+	0.930702i	$0.619197\pi$
$462$		0			0
$463$			19.4354i			0.903239i	0.892211	+	0.451620i	$0.149154\pi$
$463$			19.4354i			0.903239i	−0.892211	+	0.451620i	$0.850846\pi$
$464$		0			0
$465$	−68.9647			−3.19816
$466$		0			0
$467$	24.3478			1.12668			0.563341	−	0.826224i	$-0.309516\pi$
$467$	24.3478			1.12668			0.563341	+	0.826224i	$0.309516\pi$
$468$		0			0
$469$	1.90892			0.0881455
$470$		0			0
$471$	−33.0624			−1.52343
$472$		0			0
$473$		−	3.25458i		−	0.149646i
$474$		0			0
$475$			6.74008i			0.309256i
$476$		0			0
$477$	−16.0922			−0.736812
$478$		0			0
$479$			30.2926i			1.38410i	0.721848	+	0.692052i	$0.243293\pi$
$479$			30.2926i			1.38410i	−0.721848	+	0.692052i	$0.756707\pi$
$480$		0			0
$481$	−8.55266	−	21.6141i	−0.389968	−	0.985518i
$482$		0			0
$483$		−	5.04880i		−	0.229728i
$484$		0			0
$485$	−4.55670			−0.206909
$486$		0			0
$487$		−	24.1936i		−	1.09632i	−0.836375	−	0.548158i	$-0.815329\pi$
$487$		−	24.1936i		−	1.09632i	0.836375	−	0.548158i	$-0.184671\pi$
$488$		0			0
$489$		−	16.9176i		−	0.765041i
$490$		0			0
$491$	−32.6262			−1.47240			−0.736200	−	0.676764i	$-0.763382\pi$
$491$	−32.6262			−1.47240			−0.736200	+	0.676764i	$0.763382\pi$
$492$		0			0
$493$	−1.22458			−0.0551521
$494$		0			0
$495$	9.62877			0.432781
$496$		0			0
$497$	0.177480			0.00796107
$498$		0			0
$499$			34.9136i			1.56295i	0.623939	+	0.781473i	$0.285531\pi$
$499$			34.9136i			1.56295i	−0.623939	+	0.781473i	$0.714469\pi$
$500$		0			0
$501$			8.49379i			0.379475i
$502$		0			0
$503$	−40.5634			−1.80863			−0.904317	−	0.426862i	$-0.859619\pi$
$503$	−40.5634			−1.80863			−0.904317	+	0.426862i	$0.859619\pi$
$504$		0			0
$505$		−	21.7761i		−	0.969025i
$506$		0			0
$507$	−22.8493	+	21.4398i	−1.01477	+	0.952175i
$508$		0			0
$509$		−	16.7142i		−	0.740843i	−0.928864	−	0.370422i	$-0.879213\pi$
$509$		−	16.7142i		−	0.740843i	0.928864	−	0.370422i	$-0.120787\pi$
$510$		0			0
$511$	−13.0757			−0.578436
$512$		0			0
$513$		−	0.459312i		−	0.0202791i
$514$		0			0
$515$		−	38.5052i		−	1.69674i
$516$		0			0
$517$	−10.5464			−0.463832
$518$		0			0
$519$	−8.76375			−0.384686
$520$		0			0
$521$	−5.07569			−0.222370			−0.111185	−	0.993800i	$-0.535465\pi$
$521$	−5.07569			−0.222370			−0.111185	+	0.993800i	$0.535465\pi$
$522$		0			0
$523$	4.43662			0.194000			0.0970000	−	0.995284i	$-0.469075\pi$
$523$	4.43662			0.194000			0.0970000	+	0.995284i	$0.469075\pi$
$524$		0			0
$525$			16.2651i			0.709866i
$526$		0			0
$527$		−	5.89363i		−	0.256730i
$528$		0			0
$529$	−18.6121			−0.809221
$530$		0			0
$531$			9.46024i			0.410539i
$532$		0			0
$533$	10.7458	+	27.1565i	0.465452	+	1.17628i
$534$		0			0
$535$			33.6296i			1.45394i
$536$		0			0
$537$	35.4551			1.53000
$538$		0			0
$539$		−	1.00000i		−	0.0430730i
$540$		0			0
$541$			3.91739i			0.168422i	0.996448	+	0.0842109i	$0.0268370\pi$
$541$			3.91739i			0.168422i	−0.996448	+	0.0842109i	$0.973163\pi$
$542$		0			0
$543$	−23.9738			−1.02881
$544$		0			0
$545$	−0.634374			−0.0271736
$546$		0			0
$547$	44.0229			1.88228			0.941142	−	0.338012i	$-0.109754\pi$
$547$	44.0229			1.88228			0.941142	+	0.338012i	$0.109754\pi$
$548$		0			0
$549$	−26.6143			−1.13587
$550$		0			0
$551$		−	1.73241i		−	0.0738032i
$552$		0			0
$553$		−	1.06149i		−	0.0451392i
$554$		0			0
$555$	53.2598			2.26075
$556$		0			0
$557$		−	5.95424i		−	0.252289i	−0.992012	−	0.126145i	$-0.959740\pi$
$557$		−	5.95424i		−	0.252289i	0.992012	−	0.126145i	$-0.0402604\pi$
$558$		0			0
$559$	−10.9114	+	4.31761i	−0.461502	+	0.182616i
$560$		0			0
$561$			1.70161i			0.0718421i
$562$		0			0
$563$	33.4631			1.41030			0.705151	−	0.709057i	$-0.250879\pi$
$563$	33.4631			1.41030			0.705151	+	0.709057i	$0.250879\pi$
$564$		0			0
$565$		−	4.50712i		−	0.189616i
$566$		0			0
$567$		−	9.53600i		−	0.400474i
$568$		0			0
$569$	9.22228			0.386618			0.193309	−	0.981138i	$-0.438078\pi$
$569$	9.22228			0.386618			0.193309	+	0.981138i	$0.438078\pi$
$570$		0			0
$571$	37.8778			1.58514			0.792569	−	0.609783i	$-0.208743\pi$
$571$	37.8778			1.58514			0.792569	+	0.609783i	$0.208743\pi$
$572$		0			0
$573$	−45.7516			−1.91130
$574$		0			0
$575$	−14.1360			−0.589513
$576$		0			0
$577$			1.90958i			0.0794968i	0.999210	+	0.0397484i	$0.0126556\pi$
$577$			1.90958i			0.0794968i	−0.999210	+	0.0397484i	$0.987344\pi$
$578$		0			0
$579$			15.0720i			0.626370i
$580$		0			0
$581$	−4.13544			−0.171567
$582$		0			0
$583$		−	5.72840i		−	0.237246i
$584$		0			0
$585$	−12.7738	−	32.2816i	−0.528130	−	1.33468i
$586$		0			0
$587$			35.9427i			1.48351i	0.670670	+	0.741756i	$0.266007\pi$
$587$			35.9427i			1.48351i	−0.670670	+	0.741756i	$0.733993\pi$
$588$		0			0
$589$	8.33773			0.343550
$590$		0			0
$591$			7.77209i			0.319701i
$592$		0			0
$593$			11.7408i			0.482139i	0.970508	+	0.241069i	$0.0774981\pi$
$593$			11.7408i			0.482139i	−0.970508	+	0.241069i	$0.922502\pi$
$594$		0			0
$595$	−2.41986			−0.0992047
$596$		0			0
$597$	−52.8150			−2.16157
$598$		0			0
$599$	3.75425			0.153395			0.0766973	−	0.997054i	$-0.475563\pi$
$599$	3.75425			0.153395			0.0766973	+	0.997054i	$0.475563\pi$
$600$		0			0
$601$	−17.8456			−0.727936			−0.363968	−	0.931411i	$-0.618578\pi$
$601$	−17.8456			−0.727936			−0.363968	+	0.931411i	$0.618578\pi$
$602$		0			0
$603$		−	5.36252i		−	0.218379i
$604$		0			0
$605$			3.42759i			0.139351i
$606$		0			0
$607$	−7.34451			−0.298104			−0.149052	−	0.988829i	$-0.547622\pi$
$607$	−7.34451			−0.298104			−0.149052	+	0.988829i	$0.547622\pi$
$608$		0			0
$609$		−	4.18063i		−	0.169408i
$610$		0			0
$611$	13.9912	+	35.3582i	0.566023	+	1.43044i
$612$		0			0
$613$			23.0181i			0.929693i	0.885391	+	0.464847i	$0.153891\pi$
$613$			23.0181i			0.929693i	−0.885391	+	0.464847i	$0.846109\pi$
$614$		0			0
$615$	−66.9170			−2.69835
$616$		0			0
$617$			12.5642i			0.505814i	0.967491	+	0.252907i	$0.0813867\pi$
$617$			12.5642i			0.505814i	−0.967491	+	0.252907i	$0.918613\pi$
$618$		0			0
$619$		−	13.1137i		−	0.527083i	−0.964648	−	0.263541i	$-0.915109\pi$
$619$		−	13.1137i		−	0.527083i	0.964648	−	0.263541i	$-0.0848906\pi$
$620$		0			0
$621$	0.963319			0.0386566
$622$		0			0
$623$	−1.28092			−0.0513189
$624$		0			0
$625$	−13.2015			−0.528061
$626$		0			0
$627$	−2.40727			−0.0961372
$628$		0			0
$629$			4.55151i			0.181480i
$630$		0			0
$631$		−	22.5268i		−	0.896780i	−0.893838	−	0.448390i	$-0.851998\pi$
$631$		−	22.5268i		−	0.896780i	0.893838	−	0.448390i	$-0.148002\pi$
$632$		0			0
$633$	1.38967			0.0552345
$634$		0			0
$635$		−	12.3609i		−	0.490529i
$636$		0			0
$637$	−3.35262	+	1.32663i	−0.132836	+	0.0525628i
$638$		0			0
$639$		−	0.498577i		−	0.0197234i
$640$		0			0
$641$	−34.2583			−1.35312			−0.676561	−	0.736386i	$-0.736531\pi$
$641$	−34.2583			−1.35312			−0.676561	+	0.736386i	$0.736531\pi$
$642$		0			0
$643$		−	50.4972i		−	1.99142i	−0.0925488	−	0.995708i	$-0.529501\pi$
$643$		−	50.4972i		−	1.99142i	0.0925488	−	0.995708i	$-0.470499\pi$
$644$		0			0
$645$		−	26.8870i		−	1.05867i
$646$		0			0
$647$	25.8700			1.01705			0.508527	−	0.861046i	$-0.330190\pi$
$647$	25.8700			1.01705			0.508527	+	0.861046i	$0.330190\pi$
$648$		0			0
$649$	−3.36759			−0.132190
$650$		0			0
$651$	20.1205			0.788584
$652$		0			0
$653$	−17.7651			−0.695201			−0.347600	−	0.937643i	$-0.613003\pi$
$653$	−17.7651			−0.695201			−0.347600	+	0.937643i	$0.613003\pi$
$654$		0			0
$655$		−	16.5228i		−	0.645598i
$656$		0			0
$657$			36.7323i			1.43306i
$658$		0			0
$659$	−14.8178			−0.577220			−0.288610	−	0.957447i	$-0.593193\pi$
$659$	−14.8178			−0.577220			−0.288610	+	0.957447i	$0.593193\pi$
$660$		0			0
$661$		−	34.5795i		−	1.34499i	−0.740103	−	0.672494i	$-0.765223\pi$
$661$		−	34.5795i		−	1.34499i	0.740103	−	0.672494i	$-0.234777\pi$
$662$		0			0
$663$	5.70486	−	2.25740i	0.221558	−	0.0876702i
$664$		0			0
$665$		−	3.42339i		−	0.132753i
$666$		0			0
$667$	3.63340			0.140686
$668$		0			0
$669$		−	19.5822i		−	0.757090i
$670$		0			0
$671$		−	9.47398i		−	0.365739i
$672$		0			0
$673$	−6.98517			−0.269258			−0.134629	−	0.990896i	$-0.542984\pi$
$673$	−6.98517			−0.269258			−0.134629	+	0.990896i	$0.542984\pi$
$674$		0			0
$675$	−3.10340			−0.119450
$676$		0			0
$677$	5.92313			0.227644			0.113822	−	0.993501i	$-0.463691\pi$
$677$	5.92313			0.227644			0.113822	+	0.993501i	$0.463691\pi$
$678$		0			0
$679$	1.32942			0.0510184
$680$		0			0
$681$		−	61.1406i		−	2.34291i
$682$		0			0
$683$		−	24.2007i		−	0.926015i	−0.886354	−	0.463007i	$-0.846770\pi$
$683$		−	24.2007i		−	0.926015i	0.886354	−	0.463007i	$-0.153230\pi$
$684$		0			0
$685$	24.0985			0.920757
$686$		0			0
$687$			4.98405i			0.190153i
$688$		0			0
$689$	−19.2052	+	7.59945i	−0.731658	+	0.289516i
$690$		0			0
$691$		−	36.0179i		−	1.37018i	−0.728457	−	0.685092i	$-0.759762\pi$
$691$		−	36.0179i		−	1.37018i	0.728457	−	0.685092i	$-0.240238\pi$
$692$		0			0
$693$	−2.80920			−0.106713
$694$		0			0
$695$		−	35.8868i		−	1.36126i
$696$		0			0
$697$		−	5.71863i		−	0.216609i
$698$		0			0
$699$	8.75820			0.331266
$700$		0			0
$701$	21.2812			0.803780			0.401890	−	0.915688i	$-0.368353\pi$
$701$	21.2812			0.803780			0.401890	+	0.915688i	$0.368353\pi$
$702$		0			0
$703$	−6.43903			−0.242852
$704$		0			0
$705$	−87.1270			−3.28139
$706$		0			0
$707$			6.35320i			0.238937i
$708$		0			0
$709$		−	30.9195i		−	1.16121i	−0.814186	−	0.580604i	$-0.802817\pi$
$709$		−	30.9195i		−	1.16121i	0.814186	−	0.580604i	$-0.197183\pi$
$710$		0			0
$711$	−2.98194			−0.111831
$712$		0			0
$713$			17.4868i			0.654885i
$714$		0			0
$715$	11.4914	−	4.54712i	0.429754	−	0.170053i
$716$		0			0
$717$		−	32.9292i		−	1.22977i
$718$		0			0
$719$	−21.4354			−0.799407			−0.399703	−	0.916645i	$-0.630887\pi$
$719$	−21.4354			−0.799407			−0.399703	+	0.916645i	$0.630887\pi$
$720$		0			0
$721$			11.2339i			0.418373i
$722$		0			0
$723$		−	6.04734i		−	0.224903i
$724$		0			0
$725$	−11.7053			−0.434723
$726$		0			0
$727$	−2.38447			−0.0884351			−0.0442175	−	0.999022i	$-0.514079\pi$
$727$	−2.38447			−0.0884351			−0.0442175	+	0.999022i	$0.514079\pi$
$728$		0			0
$729$	−23.4633			−0.869011
$730$		0			0
$731$	2.29772			0.0849843
$732$		0			0
$733$		−	22.9318i		−	0.847005i	−0.905895	−	0.423503i	$-0.860800\pi$
$733$		−	22.9318i		−	0.847005i	0.905895	−	0.423503i	$-0.139200\pi$
$734$		0			0
$735$		−	8.26127i		−	0.304721i
$736$		0			0
$737$	1.90892			0.0703158
$738$		0			0
$739$		−	45.3557i		−	1.66844i	−0.551434	−	0.834219i	$-0.685919\pi$
$739$		−	45.3557i		−	1.66844i	0.551434	−	0.834219i	$-0.314081\pi$
$740$		0			0
$741$	3.19355	+	8.07068i	0.117318	+	0.296484i
$742$		0			0
$743$			50.2538i			1.84363i	0.387627	+	0.921816i	$0.373295\pi$
$743$			50.2538i			1.84363i	−0.387627	+	0.921816i	$0.626705\pi$
$744$		0			0
$745$	15.7443			0.576827
$746$		0			0
$747$			11.6173i			0.425054i
$748$		0			0
$749$		−	9.81146i		−	0.358503i
$750$		0			0
$751$	−39.1042			−1.42693			−0.713466	−	0.700690i	$-0.752876\pi$
$751$	−39.1042			−1.42693			−0.713466	+	0.700690i	$0.752876\pi$
$752$		0			0
$753$	66.6143			2.42756
$754$		0			0
$755$	21.6530			0.788034
$756$		0			0
$757$	−6.94476			−0.252412			−0.126206	−	0.992004i	$-0.540280\pi$
$757$	−6.94476			−0.252412			−0.126206	+	0.992004i	$0.540280\pi$
$758$		0			0
$759$		−	5.04880i		−	0.183260i
$760$		0			0
$761$		−	43.2277i		−	1.56700i	−0.621389	−	0.783502i	$-0.713431\pi$
$761$		−	43.2277i		−	1.56700i	0.621389	−	0.783502i	$-0.286569\pi$
$762$		0			0
$763$	0.185079			0.00670031
$764$		0			0
$765$			6.79787i			0.245778i
$766$		0			0
$767$	4.46753	+	11.2903i	0.161313	+	0.407668i
$768$		0			0
$769$			4.70669i			0.169727i	0.996393	+	0.0848637i	$0.0270455\pi$
$769$			4.70669i			0.169727i	−0.996393	+	0.0848637i	$0.972955\pi$
$770$		0			0
$771$	−48.7132			−1.75436
$772$		0			0
$773$			22.8333i			0.821257i	0.911803	+	0.410628i	$0.134691\pi$
$773$			22.8333i			0.821257i	−0.911803	+	0.410628i	$0.865309\pi$
$774$		0			0
$775$		−	56.3350i		−	2.02361i
$776$		0			0
$777$	−15.5386			−0.557443
$778$		0			0
$779$	8.09017			0.289860
$780$		0			0
$781$	0.177480			0.00635074
$782$		0			0
$783$	0.797671			0.0285064
$784$		0			0
$785$		−	47.0180i		−	1.67815i
$786$		0			0
$787$			4.12773i			0.147138i	0.997290	+	0.0735688i	$0.0234389\pi$
$787$			4.12773i			0.147138i	−0.997290	+	0.0735688i	$0.976561\pi$
$788$		0			0
$789$	−1.95513			−0.0696043
$790$		0			0
$791$			1.31495i			0.0467544i
$792$		0			0
$793$	−31.7626	+	12.5684i	−1.12792	+	0.446318i
$794$		0			0
$795$		−	47.3239i		−	1.67840i
$796$		0			0
$797$	−30.0500			−1.06443			−0.532214	−	0.846610i	$-0.678640\pi$
$797$	−30.0500			−1.06443			−0.532214	+	0.846610i	$0.678640\pi$
$798$		0			0
$799$		−	7.44575i		−	0.263412i
$800$		0			0
$801$			3.59835i			0.127142i
$802$		0			0
$803$	−13.0757			−0.461432
$804$		0			0
$805$	7.17990			0.253058
$806$		0			0
$807$	3.04628			0.107234
$808$		0			0
$809$	−12.3574			−0.434463			−0.217231	−	0.976120i	$-0.569703\pi$
$809$	−12.3574			−0.434463			−0.217231	+	0.976120i	$0.569703\pi$
$810$		0			0
$811$			10.9273i			0.383708i	0.981423	+	0.191854i	$0.0614500\pi$
$811$			10.9273i			0.383708i	−0.981423	+	0.191854i	$0.938550\pi$
$812$		0			0
$813$		−	19.0240i		−	0.667200i
$814$		0			0
$815$	24.0586			0.842734
$816$		0			0
$817$			3.25059i			0.113724i
$818$		0			0
$819$	3.72675	+	9.41818i	0.130223	+	0.329098i
$820$		0			0
$821$		−	54.6095i		−	1.90588i	−0.303151	−	0.952942i	$-0.598039\pi$
$821$		−	54.6095i		−	1.90588i	0.303151	−	0.952942i	$-0.401961\pi$
$822$		0			0
$823$	−15.6090			−0.544097			−0.272048	−	0.962284i	$-0.587701\pi$
$823$	−15.6090			−0.544097			−0.272048	+	0.962284i	$0.587701\pi$
$824$		0			0
$825$			16.2651i			0.566277i
$826$		0			0
$827$		−	18.1899i		−	0.632524i	−0.948672	−	0.316262i	$-0.897572\pi$
$827$		−	18.1899i		−	0.632524i	0.948672	−	0.316262i	$-0.102428\pi$
$828$		0			0
$829$	28.5464			0.991458			0.495729	−	0.868477i	$-0.334901\pi$
$829$	28.5464			0.991458			0.495729	+	0.868477i	$0.334901\pi$
$830$		0			0
$831$	−56.4970			−1.95986
$832$		0			0
$833$	0.705996			0.0244613
$834$		0			0
$835$	−12.0790			−0.418012
$836$		0			0
$837$			3.83902i			0.132696i
$838$		0			0
$839$			15.8297i			0.546503i	0.961943	+	0.273252i	$0.0880991\pi$
$839$			15.8297i			0.546503i	−0.961943	+	0.273252i	$0.911901\pi$
$840$		0			0
$841$	−25.9914			−0.896254
$842$		0			0
$843$			13.0005i			0.447762i
$844$		0			0
$845$	−30.4896	−	32.4940i	−1.04887	−	1.11783i
$846$		0			0
$847$		−	1.00000i		−	0.0343604i
$848$		0			0
$849$	−18.6466			−0.639948
$850$		0			0
$851$		−	13.5046i		−	0.462933i
$852$		0			0
$853$			24.1139i			0.825643i	0.910812	+	0.412822i	$0.135457\pi$
$853$			24.1139i			0.825643i	−0.910812	+	0.412822i	$0.864543\pi$
$854$		0			0
$855$	−9.61697			−0.328893
$856$		0			0
$857$	45.8840			1.56737			0.783683	−	0.621161i	$-0.213339\pi$
$857$	45.8840			1.56737			0.783683	+	0.621161i	$0.213339\pi$
$858$		0			0
$859$	19.2053			0.655275			0.327637	−	0.944804i	$-0.393748\pi$
$859$	19.2053			0.655275			0.327637	+	0.944804i	$0.393748\pi$
$860$		0			0
$861$	19.5231			0.665345
$862$		0			0
$863$			23.5101i			0.800292i	0.916451	+	0.400146i	$0.131041\pi$
$863$			23.5101i			0.800292i	−0.916451	+	0.400146i	$0.868959\pi$
$864$		0			0
$865$		−	12.4629i		−	0.423753i
$866$		0			0
$867$	39.7725			1.35075
$868$		0			0
$869$		−	1.06149i		−	0.0360086i
$870$		0			0
$871$	−2.53242	−	6.39987i	−0.0858076	−	0.216851i
$872$		0			0
$873$		−	3.73460i		−	0.126397i
$874$		0			0
$875$	−5.99263			−0.202588
$876$		0			0
$877$			28.0542i			0.947321i	0.880707	+	0.473661i	$0.157068\pi$
$877$			28.0542i			0.947321i	−0.880707	+	0.473661i	$0.842932\pi$
$878$		0			0
$879$		−	26.1887i		−	0.883322i
$880$		0			0
$881$	−6.44905			−0.217274			−0.108637	−	0.994081i	$-0.534649\pi$
$881$	−6.44905			−0.217274			−0.108637	+	0.994081i	$0.534649\pi$
$882$		0			0
$883$	18.3967			0.619100			0.309550	−	0.950883i	$-0.399822\pi$
$883$	18.3967			0.619100			0.309550	+	0.950883i	$0.399822\pi$
$884$		0			0
$885$	−27.8206			−0.935178
$886$		0			0
$887$	−44.8273			−1.50515			−0.752576	−	0.658505i	$-0.771189\pi$
$887$	−44.8273			−1.50515			−0.752576	+	0.658505i	$0.771189\pi$
$888$		0			0
$889$			3.60631i			0.120952i
$890$		0			0
$891$		−	9.53600i		−	0.319468i
$892$		0			0
$893$	10.5335			0.352491
$894$		0			0
$895$			50.4207i			1.68538i
$896$		0			0
$897$	−16.9267	+	6.69786i	−0.565166	+	0.223635i
$898$		0			0
$899$			14.4798i			0.482930i
$900$		0			0
$901$	4.04423			0.134733
$902$		0			0
$903$			7.84429i			0.261042i
$904$		0			0
$905$		−	34.0931i		−	1.13329i
$906$		0			0
$907$	−41.5188			−1.37861			−0.689305	−	0.724471i	$-0.742084\pi$
$907$	−41.5188			−1.37861			−0.689305	+	0.724471i	$0.742084\pi$
$908$		0			0
$909$	17.8474			0.591960
$910$		0			0
$911$	−47.1151			−1.56099			−0.780497	−	0.625160i	$-0.785034\pi$
$911$	−47.1151			−1.56099			−0.780497	+	0.625160i	$0.785034\pi$
$912$		0			0
$913$	−4.13544			−0.136863
$914$		0			0
$915$		−	78.2670i		−	2.58743i
$916$		0			0
$917$			4.82053i			0.159188i
$918$		0			0
$919$	43.0246			1.41925			0.709626	−	0.704579i	$-0.248864\pi$
$919$	43.0246			1.41925			0.709626	+	0.704579i	$0.248864\pi$
$920$		0			0
$921$			7.67059i			0.252755i
$922$		0			0
$923$	−0.235450	−	0.595024i	−0.00774992	−	0.0195854i
$924$		0			0
$925$			43.5061i			1.43047i
$926$		0			0
$927$	31.5583			1.03651
$928$		0			0
$929$			46.7085i			1.53245i	0.642570	+	0.766227i	$0.277868\pi$
$929$			46.7085i			1.53245i	−0.642570	+	0.766227i	$0.722132\pi$
$930$		0			0
$931$			0.998774i			0.0327335i
$932$		0			0
$933$	41.1225			1.34629
$934$		0			0
$935$	−2.41986			−0.0791380
$936$		0			0
$937$	46.0219			1.50347			0.751735	−	0.659465i	$-0.229217\pi$
$937$	46.0219			1.50347			0.751735	+	0.659465i	$0.229217\pi$
$938$		0			0
$939$	−4.65824			−0.152016
$940$		0			0
$941$			33.8841i			1.10459i	0.833649	+	0.552295i	$0.186248\pi$
$941$			33.8841i			1.10459i	−0.833649	+	0.552295i	$0.813752\pi$
$942$		0			0
$943$			16.9676i			0.552540i
$944$		0			0
$945$	1.57626			0.0512758
$946$		0			0
$947$			55.9692i			1.81875i	0.415972	+	0.909377i	$0.363441\pi$
$947$			55.9692i			1.81875i	−0.415972	+	0.909377i	$0.636559\pi$
$948$		0			0
$949$	17.3466	+	43.8379i	0.563094	+	1.42304i
$950$		0			0
$951$		−	62.3677i		−	2.02241i
$952$		0			0
$953$	−60.7224			−1.96699			−0.983495	−	0.180934i	$-0.942088\pi$
$953$	−60.7224			−1.96699			−0.983495	+	0.180934i	$0.942088\pi$
$954$		0			0
$955$		−	65.0634i		−	2.10540i
$956$		0			0
$957$		−	4.18063i		−	0.135141i
$958$		0			0
$959$	−7.03075			−0.227035
$960$		0			0
$961$	−38.6885			−1.24801
$962$		0			0
$963$	−27.5623			−0.888184
$964$		0			0
$965$	−21.4339			−0.689980
$966$		0			0
$967$		−	9.83978i		−	0.316426i	−0.987405	−	0.158213i	$-0.949427\pi$
$967$		−	9.83978i		−	0.316426i	0.987405	−	0.158213i	$-0.0505733\pi$
$968$		0			0
$969$		−	1.69953i		−	0.0545966i
$970$		0			0
$971$	6.93232			0.222469			0.111234	−	0.993794i	$-0.464520\pi$
$971$	6.93232			0.222469			0.111234	+	0.993794i	$0.464520\pi$
$972$		0			0
$973$			10.4700i			0.335653i
$974$		0			0
$975$	54.5306	−	21.5776i	1.74638	−	0.691038i
$976$		0			0
$977$		−	16.5868i		−	0.530660i	−0.964158	−	0.265330i	$-0.914519\pi$
$977$		−	16.5868i		−	0.530660i	0.964158	−	0.265330i	$-0.0854809\pi$
$978$		0			0
$979$	−1.28092			−0.0409383
$980$		0			0
$981$		−	0.519924i		−	0.0165999i
$982$		0			0
$983$			33.8131i			1.07847i	0.842155	+	0.539235i	$0.181287\pi$
$983$			33.8131i			1.07847i	−0.842155	+	0.539235i	$0.818713\pi$
$984$		0			0
$985$	−11.0527			−0.352168
$986$		0			0
$987$	25.4193			0.809106
$988$		0			0
$989$	−6.81750			−0.216784
$990$		0			0
$991$	33.8070			1.07391			0.536957	−	0.843610i	$-0.319574\pi$
$991$	33.8070			1.07391			0.536957	+	0.843610i	$0.319574\pi$
$992$		0			0
$993$		−	7.74446i		−	0.245763i
$994$		0			0
$995$		−	75.1082i		−	2.38109i
$996$		0			0
$997$	3.90022			0.123521			0.0617606	−	0.998091i	$-0.480328\pi$
$997$	3.90022			0.123521			0.0617606	+	0.998091i	$0.480328\pi$
$998$		0			0
$999$		−	2.96478i		−	0.0938016i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.b.2157.7	✓	30
13.12	even	2	inner	4004.2.m.b.2157.8	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.b.2157.7	✓	30	1.1	even	1	trivial
4004.2.m.b.2157.8	yes	30	13.12	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.41023 q^{3} -3.42759i q^{5} +1.00000i q^{7} +2.80920 q^{9} +O(q^{10})\)	\(q-2.41023 q^{3} -3.42759i q^{5} +1.00000i q^{7} +2.80920 q^{9} +1.00000i q^{11} +(3.35262 - 1.32663i) q^{13} +8.26127i q^{15} -0.705996 q^{17} -0.998774i q^{19} -2.41023i q^{21} +2.09474 q^{23} -6.74835 q^{25} +0.459875 q^{27} +1.73454 q^{29} +8.34796i q^{31} -2.41023i q^{33} +3.42759 q^{35} -6.44693i q^{37} +(-8.08058 + 3.19747i) q^{39} +8.10009i q^{41} -3.25458 q^{43} -9.62877i q^{45} +10.5464i q^{47} -1.00000 q^{49} +1.70161 q^{51} -5.72840 q^{53} +3.42759 q^{55} +2.40727i q^{57} +3.36759i q^{59} -9.47398 q^{61} +2.80920i q^{63} +(-4.54712 - 11.4914i) q^{65} -1.90892i q^{67} -5.04880 q^{69} -0.177480i q^{71} +13.0757i q^{73} +16.2651 q^{75} -1.00000 q^{77} -1.06149 q^{79} -9.53600 q^{81} +4.13544i q^{83} +2.41986i q^{85} -4.18063 q^{87} +1.28092i q^{89} +(1.32663 + 3.35262i) q^{91} -20.1205i q^{93} -3.42339 q^{95} -1.32942i q^{97} +2.80920i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 18 q^{9}+O(q^{10}) \) 30 * q + 18 * q^9	\( 30 q + 18 q^{9} + 4 q^{13} + 12 q^{17} + 6 q^{23} - 6 q^{29} - 2 q^{35} + 8 q^{39} - 22 q^{43} - 30 q^{49} + 60 q^{51} - 38 q^{53} - 2 q^{55} - 36 q^{61} + 10 q^{65} + 36 q^{69} - 20 q^{75} - 30 q^{77} - 10 q^{79} - 42 q^{81} + 36 q^{87} + 6 q^{91} - 2 q^{95}+O(q^{100}) \) 30 * q + 18 * q^9 + 4 * q^13 + 12 * q^17 + 6 * q^23 - 6 * q^29 - 2 * q^35 + 8 * q^39 - 22 * q^43 - 30 * q^49 + 60 * q^51 - 38 * q^53 - 2 * q^55 - 36 * q^61 + 10 * q^65 + 36 * q^69 - 20 * q^75 - 30 * q^77 - 10 * q^79 - 42 * q^81 + 36 * q^87 + 6 * q^91 - 2 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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