LMFDB - Embedded newform 4004.2.m.b.2157.3

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			2157.3
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.b.2157.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.50189 q^{3} -2.36927i q^{5} -1.00000i q^{7} +3.25946 q^{9} +O(q^{10})$	$q-2.50189 q^{3} -2.36927i q^{5} -1.00000i q^{7} +3.25946 q^{9} -1.00000i q^{11} +(0.178681 - 3.60112i) q^{13} +5.92765i q^{15} -2.97226 q^{17} +0.164490i q^{19} +2.50189i q^{21} -8.05078 q^{23} -0.613427 q^{25} -0.649138 q^{27} -4.68898 q^{29} +1.30481i q^{31} +2.50189i q^{33} -2.36927 q^{35} +1.83347i q^{37} +(-0.447042 + 9.00961i) q^{39} -4.62950i q^{41} -7.77895 q^{43} -7.72253i q^{45} +7.81855i q^{47} -1.00000 q^{49} +7.43627 q^{51} +12.6608 q^{53} -2.36927 q^{55} -0.411536i q^{57} -7.01797i q^{59} -8.99004 q^{61} -3.25946i q^{63} +(-8.53202 - 0.423344i) q^{65} +3.03458i q^{67} +20.1422 q^{69} -13.5706i q^{71} -12.8343i q^{73} +1.53473 q^{75} -1.00000 q^{77} -10.0912 q^{79} -8.15430 q^{81} +11.4998i q^{83} +7.04208i q^{85} +11.7313 q^{87} -18.0329i q^{89} +(-3.60112 - 0.178681i) q^{91} -3.26450i q^{93} +0.389721 q^{95} +4.72035i q^{97} -3.25946i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.50189			−1.44447			−0.722234	−	0.691649i	$-0.756885\pi$
$3$	−2.50189			−1.44447			−0.722234	+	0.691649i	$0.756885\pi$
$4$		0			0
$5$		−	2.36927i		−	1.05957i	−0.848132	−	0.529784i	$-0.822273\pi$
$5$		−	2.36927i		−	1.05957i	0.848132	−	0.529784i	$-0.177727\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	3.25946			1.08649
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	0.178681	−	3.60112i	0.0495573	−	0.998771i
$13$	0.178681	−	3.60112i	0.0495573	−	0.998771i
$14$		0			0
$15$			5.92765i			1.53051i
$16$		0			0
$17$	−2.97226			−0.720879			−0.360439	−	0.932783i	$-0.617373\pi$
$17$	−2.97226			−0.720879			−0.360439	+	0.932783i	$0.617373\pi$
$18$		0			0
$19$			0.164490i			0.0377366i	0.999822	+	0.0188683i	$0.00600632\pi$
$19$			0.164490i			0.0377366i	−0.999822	+	0.0188683i	$0.993994\pi$
$20$		0			0
$21$			2.50189i			0.545957i
$22$		0			0
$23$	−8.05078			−1.67870			−0.839351	−	0.543589i	$-0.817065\pi$
$23$	−8.05078			−1.67870			−0.839351	+	0.543589i	$0.817065\pi$
$24$		0			0
$25$	−0.613427			−0.122685
$26$		0			0
$27$	−0.649138			−0.124927
$28$		0			0
$29$	−4.68898			−0.870722			−0.435361	−	0.900256i	$-0.643379\pi$
$29$	−4.68898			−0.870722			−0.435361	+	0.900256i	$0.643379\pi$
$30$		0			0
$31$			1.30481i			0.234352i	0.993111	+	0.117176i	$0.0373841\pi$
$31$			1.30481i			0.234352i	−0.993111	+	0.117176i	$0.962616\pi$
$32$		0			0
$33$			2.50189i			0.435523i
$34$		0			0
$35$	−2.36927			−0.400479
$36$		0			0
$37$			1.83347i			0.301421i	0.988578	+	0.150710i	$0.0481561\pi$
$37$			1.83347i			0.301421i	−0.988578	+	0.150710i	$0.951844\pi$
$38$		0			0
$39$	−0.447042	+	9.00961i	−0.0715839	+	1.44269i
$40$		0			0
$41$		−	4.62950i		−	0.723007i	−0.932371	−	0.361503i	$-0.882264\pi$
$41$		−	4.62950i		−	0.723007i	0.932371	−	0.361503i	$-0.117736\pi$
$42$		0			0
$43$	−7.77895			−1.18628			−0.593139	−	0.805100i	$-0.702112\pi$
$43$	−7.77895			−1.18628			−0.593139	+	0.805100i	$0.702112\pi$
$44$		0			0
$45$		−	7.72253i		−	1.15121i
$46$		0			0
$47$			7.81855i			1.14045i	0.821488	+	0.570226i	$0.193144\pi$
$47$			7.81855i			1.14045i	−0.821488	+	0.570226i	$0.806856\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	7.43627			1.04129
$52$		0			0
$53$	12.6608			1.73909			0.869547	−	0.493849i	$-0.164411\pi$
$53$	12.6608			1.73909			0.869547	+	0.493849i	$0.164411\pi$
$54$		0			0
$55$	−2.36927			−0.319472
$56$		0			0
$57$		−	0.411536i		−	0.0545093i
$58$		0			0
$59$		−	7.01797i		−	0.913662i	−0.889554	−	0.456831i	$-0.848984\pi$
$59$		−	7.01797i		−	0.913662i	0.889554	−	0.456831i	$-0.151016\pi$
$60$		0			0
$61$	−8.99004			−1.15106			−0.575529	−	0.817782i	$-0.695204\pi$
$61$	−8.99004			−1.15106			−0.575529	+	0.817782i	$0.695204\pi$
$62$		0			0
$63$		−	3.25946i		−	0.410653i
$64$		0			0
$65$	−8.53202	−	0.423344i	−1.05827	−	0.0525094i
$66$		0			0
$67$			3.03458i			0.370732i	0.982670	+	0.185366i	$0.0593471\pi$
$67$			3.03458i			0.370732i	−0.982670	+	0.185366i	$0.940653\pi$
$68$		0			0
$69$	20.1422			2.42483
$70$		0			0
$71$		−	13.5706i		−	1.61054i	−0.592909	−	0.805269i	$-0.702021\pi$
$71$		−	13.5706i		−	1.61054i	0.592909	−	0.805269i	$-0.297979\pi$
$72$		0			0
$73$		−	12.8343i		−	1.50214i	−0.660223	−	0.751070i	$-0.729538\pi$
$73$		−	12.8343i		−	1.50214i	0.660223	−	0.751070i	$-0.270462\pi$
$74$		0			0
$75$	1.53473			0.177215
$76$		0			0
$77$	−1.00000			−0.113961
$78$		0			0
$79$	−10.0912			−1.13534			−0.567672	−	0.823255i	$-0.692156\pi$
$79$	−10.0912			−1.13534			−0.567672	+	0.823255i	$0.692156\pi$
$80$		0			0
$81$	−8.15430			−0.906034
$82$		0			0
$83$			11.4998i			1.26226i	0.775675	+	0.631132i	$0.217409\pi$
$83$			11.4998i			1.26226i	−0.775675	+	0.631132i	$0.782591\pi$
$84$		0			0
$85$			7.04208i			0.763820i
$86$		0			0
$87$	11.7313			1.25773
$88$		0			0
$89$		−	18.0329i		−	1.91149i	−0.294202	−	0.955743i	$-0.595054\pi$
$89$		−	18.0329i		−	1.91149i	0.294202	−	0.955743i	$-0.404946\pi$
$90$		0			0
$91$	−3.60112	−	0.178681i	−0.377500	−	0.0187309i
$92$		0			0
$93$		−	3.26450i		−	0.338513i
$94$		0			0
$95$	0.389721			0.0399845
$96$		0			0
$97$			4.72035i			0.479279i	0.970862	+	0.239640i	$0.0770293\pi$
$97$			4.72035i			0.479279i	−0.970862	+	0.239640i	$0.922971\pi$
$98$		0			0
$99$		−	3.25946i		−	0.327588i
$100$		0			0
$101$	6.61709			0.658426			0.329213	−	0.944256i	$-0.393217\pi$
$101$	6.61709			0.658426			0.329213	+	0.944256i	$0.393217\pi$
$102$		0			0
$103$	4.86143			0.479010			0.239505	−	0.970895i	$-0.423015\pi$
$103$	4.86143			0.479010			0.239505	+	0.970895i	$0.423015\pi$
$104$		0			0
$105$	5.92765			0.578479
$106$		0			0
$107$	−1.59173			−0.153879			−0.0769394	−	0.997036i	$-0.524515\pi$
$107$	−1.59173			−0.153879			−0.0769394	+	0.997036i	$0.524515\pi$
$108$		0			0
$109$			11.9286i			1.14256i	0.820757	+	0.571278i	$0.193552\pi$
$109$			11.9286i			1.14256i	−0.820757	+	0.571278i	$0.806448\pi$
$110$		0			0
$111$		−	4.58715i		−	0.435393i
$112$		0			0
$113$	−8.92513			−0.839605			−0.419803	−	0.907615i	$-0.637901\pi$
$113$	−8.92513			−0.839605			−0.419803	+	0.907615i	$0.637901\pi$
$114$		0			0
$115$			19.0744i			1.77870i
$116$		0			0
$117$	0.582405	−	11.7377i	0.0538434	−	1.08515i
$118$		0			0
$119$			2.97226i			0.272467i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			11.5825i			1.04436i
$124$		0			0
$125$		−	10.3930i		−	0.929575i
$126$		0			0
$127$	17.8269			1.58189			0.790943	−	0.611890i	$-0.209591\pi$
$127$	17.8269			1.58189			0.790943	+	0.611890i	$0.209591\pi$
$128$		0			0
$129$	19.4621			1.71354
$130$		0			0
$131$	−0.00719904			−0.000628983			−0.000314492	−	1.00000i	$-0.500100\pi$
$131$	−0.00719904			−0.000628983			−0.000314492		1.00000i	$0.500100\pi$
$132$		0			0
$133$	0.164490			0.0142631
$134$		0			0
$135$			1.53798i			0.132368i
$136$		0			0
$137$			12.2663i			1.04798i	0.851725	+	0.523989i	$0.175557\pi$
$137$			12.2663i			1.04798i	−0.851725	+	0.523989i	$0.824443\pi$
$138$		0			0
$139$	10.9138			0.925698			0.462849	−	0.886437i	$-0.346827\pi$
$139$	10.9138			0.925698			0.462849	+	0.886437i	$0.346827\pi$
$140$		0			0
$141$		−	19.5612i		−	1.64735i
$142$		0			0
$143$	−3.60112	−	0.178681i	−0.301141	−	0.0149421i
$144$		0			0
$145$			11.1094i			0.922589i
$146$		0			0
$147$	2.50189			0.206352
$148$		0			0
$149$		−	4.61561i		−	0.378125i	−0.981965	−	0.189063i	$-0.939455\pi$
$149$		−	4.61561i		−	0.378125i	0.981965	−	0.189063i	$-0.0605449\pi$
$150$		0			0
$151$			15.3225i			1.24693i	0.781852	+	0.623464i	$0.214275\pi$
$151$			15.3225i			1.24693i	−0.781852	+	0.623464i	$0.785725\pi$
$152$		0			0
$153$	−9.68796			−0.783225
$154$		0			0
$155$	3.09145			0.248312
$156$		0			0
$157$	10.5866			0.844900			0.422450	−	0.906386i	$-0.361170\pi$
$157$	10.5866			0.844900			0.422450	+	0.906386i	$0.361170\pi$
$158$		0			0
$159$	−31.6759			−2.51207
$160$		0			0
$161$			8.05078i			0.634490i
$162$		0			0
$163$			24.4971i			1.91876i	0.282118	+	0.959380i	$0.408963\pi$
$163$			24.4971i			1.91876i	−0.282118	+	0.959380i	$0.591037\pi$
$164$		0			0
$165$	5.92765			0.461467
$166$		0			0
$167$			11.4141i			0.883248i	0.897200	+	0.441624i	$0.145598\pi$
$167$			11.4141i			0.883248i	−0.897200	+	0.441624i	$0.854402\pi$
$168$		0			0
$169$	−12.9361	−	1.28691i	−0.995088	−	0.0989929i
$170$		0			0
$171$			0.536148i			0.0410003i
$172$		0			0
$173$	2.88943			0.219679			0.109840	−	0.993949i	$-0.464966\pi$
$173$	2.88943			0.219679			0.109840	+	0.993949i	$0.464966\pi$
$174$		0			0
$175$			0.613427i			0.0463707i
$176$		0			0
$177$			17.5582i			1.31975i
$178$		0			0
$179$	16.6233			1.24249			0.621243	−	0.783618i	$-0.286628\pi$
$179$	16.6233			1.24249			0.621243	+	0.783618i	$0.286628\pi$
$180$		0			0
$181$	19.9012			1.47924			0.739622	−	0.673022i	$-0.235004\pi$
$181$	19.9012			1.47924			0.739622	+	0.673022i	$0.235004\pi$
$182$		0			0
$183$	22.4921			1.66266
$184$		0			0
$185$	4.34398			0.319376
$186$		0			0
$187$			2.97226i			0.217353i
$188$		0			0
$189$			0.649138i			0.0472179i
$190$		0			0
$191$	−21.4732			−1.55374			−0.776871	−	0.629659i	$-0.783194\pi$
$191$	−21.4732			−1.55374			−0.776871	+	0.629659i	$0.783194\pi$
$192$		0			0
$193$		−	8.57888i		−	0.617521i	−0.951140	−	0.308761i	$-0.900086\pi$
$193$		−	8.57888i		−	0.617521i	0.951140	−	0.308761i	$-0.0999142\pi$
$194$		0			0
$195$	21.3462	+	1.05916i	1.52863	+	0.0758481i
$196$		0			0
$197$		−	7.16450i		−	0.510449i	−0.966882	−	0.255225i	$-0.917851\pi$
$197$		−	7.16450i		−	0.510449i	0.966882	−	0.255225i	$-0.0821494\pi$
$198$		0			0
$199$	−10.9899			−0.779054			−0.389527	−	0.921015i	$-0.627362\pi$
$199$	−10.9899			−0.779054			−0.389527	+	0.921015i	$0.627362\pi$
$200$		0			0
$201$		−	7.59218i		−	0.535511i
$202$		0			0
$203$			4.68898i			0.329102i
$204$		0			0
$205$	−10.9685			−0.766075
$206$		0			0
$207$	−26.2412			−1.82389
$208$		0			0
$209$	0.164490			0.0113780
$210$		0			0
$211$	−21.7948			−1.50042			−0.750209	−	0.661200i	$-0.770047\pi$
$211$	−21.7948			−1.50042			−0.750209	+	0.661200i	$0.770047\pi$
$212$		0			0
$213$			33.9523i			2.32637i
$214$		0			0
$215$			18.4304i			1.25694i
$216$		0			0
$217$	1.30481			0.0885766
$218$		0			0
$219$			32.1100i			2.16979i
$220$		0			0
$221$	−0.531088	+	10.7035i	−0.0357248	+	0.719993i
$222$		0			0
$223$			13.4453i			0.900367i	0.892936	+	0.450183i	$0.148641\pi$
$223$			13.4453i			0.900367i	−0.892936	+	0.450183i	$0.851359\pi$
$224$		0			0
$225$	−1.99944			−0.133296
$226$		0			0
$227$			3.60462i			0.239247i	0.992819	+	0.119624i	$0.0381687\pi$
$227$			3.60462i			0.239247i	−0.992819	+	0.119624i	$0.961831\pi$
$228$		0			0
$229$			4.28374i			0.283078i	0.989933	+	0.141539i	$0.0452050\pi$
$229$			4.28374i			0.283078i	−0.989933	+	0.141539i	$0.954795\pi$
$230$		0			0
$231$	2.50189			0.164612
$232$		0			0
$233$	−20.0642			−1.31445			−0.657224	−	0.753696i	$-0.728269\pi$
$233$	−20.0642			−1.31445			−0.657224	+	0.753696i	$0.728269\pi$
$234$		0			0
$235$	18.5242			1.20839
$236$		0			0
$237$	25.2470			1.63997
$238$		0			0
$239$			15.3069i			0.990121i	0.868859	+	0.495060i	$0.164854\pi$
$239$			15.3069i			0.990121i	−0.868859	+	0.495060i	$0.835146\pi$
$240$		0			0
$241$			13.5221i			0.871032i	0.900181	+	0.435516i	$0.143434\pi$
$241$			13.5221i			0.871032i	−0.900181	+	0.435516i	$0.856566\pi$
$242$		0			0
$243$	22.3486			1.43366
$244$		0			0
$245$			2.36927i			0.151367i
$246$		0			0
$247$	0.592348	+	0.0293913i	0.0376902	+	0.00187012i
$248$		0			0
$249$		−	28.7712i		−	1.82330i
$250$		0			0
$251$	−4.63496			−0.292556			−0.146278	−	0.989244i	$-0.546729\pi$
$251$	−4.63496			−0.292556			−0.146278	+	0.989244i	$0.546729\pi$
$252$		0			0
$253$			8.05078i			0.506148i
$254$		0			0
$255$		−	17.6185i		−	1.10331i
$256$		0			0
$257$	11.2322			0.700643			0.350322	−	0.936630i	$-0.386072\pi$
$257$	11.2322			0.700643			0.350322	+	0.936630i	$0.386072\pi$
$258$		0			0
$259$	1.83347			0.113926
$260$		0			0
$261$	−15.2835			−0.946027
$262$		0			0
$263$	−9.74760			−0.601063			−0.300531	−	0.953772i	$-0.597164\pi$
$263$	−9.74760			−0.601063			−0.300531	+	0.953772i	$0.597164\pi$
$264$		0			0
$265$		−	29.9968i		−	1.84269i
$266$		0			0
$267$			45.1164i			2.76108i
$268$		0			0
$269$	−14.5674			−0.888191			−0.444095	−	0.895980i	$-0.646475\pi$
$269$	−14.5674			−0.888191			−0.444095	+	0.895980i	$0.646475\pi$
$270$		0			0
$271$		−	15.4040i		−	0.935727i	−0.883801	−	0.467864i	$-0.845024\pi$
$271$		−	15.4040i		−	0.935727i	0.883801	−	0.467864i	$-0.154976\pi$
$272$		0			0
$273$	9.00961	+	0.447042i	0.545287	+	0.0270562i
$274$		0			0
$275$			0.613427i			0.0369910i
$276$		0			0
$277$	−22.8944			−1.37559			−0.687794	−	0.725906i	$-0.741421\pi$
$277$	−22.8944			−1.37559			−0.687794	+	0.725906i	$0.741421\pi$
$278$		0			0
$279$			4.25299i			0.254620i
$280$		0			0
$281$			4.87350i			0.290729i	0.989378	+	0.145364i	$0.0464354\pi$
$281$			4.87350i			0.290729i	−0.989378	+	0.145364i	$0.953565\pi$
$282$		0			0
$283$	23.1706			1.37735			0.688674	−	0.725071i	$-0.258193\pi$
$283$	23.1706			1.37735			0.688674	+	0.725071i	$0.258193\pi$
$284$		0			0
$285$	−0.975039			−0.0577563
$286$		0			0
$287$	−4.62950			−0.273271
$288$		0			0
$289$	−8.16567			−0.480334
$290$		0			0
$291$		−	11.8098i		−	0.692303i
$292$		0			0
$293$			7.26487i			0.424419i	0.977224	+	0.212209i	$0.0680658\pi$
$293$			7.26487i			0.424419i	−0.977224	+	0.212209i	$0.931934\pi$
$294$		0			0
$295$	−16.6274			−0.968087
$296$		0			0
$297$			0.649138i			0.0376668i
$298$		0			0
$299$	−1.43852	+	28.9918i	−0.0831920	+	1.67664i
$300$		0			0
$301$			7.77895i			0.448371i
$302$		0			0
$303$	−16.5553			−0.951074
$304$		0			0
$305$			21.2998i			1.21962i
$306$		0			0
$307$			27.6745i			1.57947i	0.613450	+	0.789733i	$0.289781\pi$
$307$			27.6745i			1.57947i	−0.613450	+	0.789733i	$0.710219\pi$
$308$		0			0
$309$	−12.1628			−0.691915
$310$		0			0
$311$	33.0008			1.87130			0.935651	−	0.352926i	$-0.114813\pi$
$311$	33.0008			1.87130			0.935651	+	0.352926i	$0.114813\pi$
$312$		0			0
$313$	10.4905			0.592959			0.296480	−	0.955039i	$-0.404187\pi$
$313$	10.4905			0.592959			0.296480	+	0.955039i	$0.404187\pi$
$314$		0			0
$315$	−7.72253			−0.435115
$316$		0			0
$317$			12.4106i			0.697047i	0.937300	+	0.348523i	$0.113317\pi$
$317$			12.4106i			0.697047i	−0.937300	+	0.348523i	$0.886683\pi$
$318$		0			0
$319$			4.68898i			0.262533i
$320$		0			0
$321$	3.98234			0.222273
$322$		0			0
$323$		−	0.488907i		−	0.0272035i
$324$		0			0
$325$	−0.109608	+	2.20902i	−0.00607996	+	0.122535i
$326$		0			0
$327$		−	29.8441i		−	1.65038i
$328$		0			0
$329$	7.81855			0.431050
$330$		0			0
$331$		−	7.65088i		−	0.420530i	−0.977644	−	0.210265i	$-0.932567\pi$
$331$		−	7.65088i		−	0.420530i	0.977644	−	0.210265i	$-0.0674328\pi$
$332$		0			0
$333$			5.97612i			0.327490i
$334$		0			0
$335$	7.18972			0.392816
$336$		0			0
$337$	3.60897			0.196593			0.0982965	−	0.995157i	$-0.468661\pi$
$337$	3.60897			0.196593			0.0982965	+	0.995157i	$0.468661\pi$
$338$		0			0
$339$	22.3297			1.21278
$340$		0			0
$341$	1.30481			0.0706597
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$		−	47.7222i		−	2.56928i
$346$		0			0
$347$	14.2096			0.762812			0.381406	−	0.924408i	$-0.375440\pi$
$347$	14.2096			0.762812			0.381406	+	0.924408i	$0.375440\pi$
$348$		0			0
$349$		−	16.0230i		−	0.857692i	−0.903378	−	0.428846i	$-0.858920\pi$
$349$		−	16.0230i		−	0.857692i	0.903378	−	0.428846i	$-0.141080\pi$
$350$		0			0
$351$	−0.115989	+	2.33763i	−0.00619103	+	0.124773i
$352$		0			0
$353$			12.6080i			0.671058i	0.942030	+	0.335529i	$0.108915\pi$
$353$			12.6080i			0.671058i	−0.942030	+	0.335529i	$0.891085\pi$
$354$		0			0
$355$	−32.1525			−1.70648
$356$		0			0
$357$		−	7.43627i		−	0.393569i
$358$		0			0
$359$			25.9744i			1.37088i	0.728131	+	0.685438i	$0.240389\pi$
$359$			25.9744i			1.37088i	−0.728131	+	0.685438i	$0.759611\pi$
$360$		0			0
$361$	18.9729			0.998576
$362$		0			0
$363$	2.50189			0.131315
$364$		0			0
$365$	−30.4079			−1.59162
$366$		0			0
$367$	35.6489			1.86086			0.930429	−	0.366471i	$-0.119434\pi$
$367$	35.6489			1.86086			0.930429	+	0.366471i	$0.119434\pi$
$368$		0			0
$369$		−	15.0897i		−	0.785537i
$370$		0			0
$371$		−	12.6608i		−	0.657316i
$372$		0			0
$373$	7.26360			0.376095			0.188048	−	0.982160i	$-0.439784\pi$
$373$	7.26360			0.376095			0.188048	+	0.982160i	$0.439784\pi$
$374$		0			0
$375$			26.0021i			1.34274i
$376$		0			0
$377$	−0.837834	+	16.8856i	−0.0431506	+	0.869652i
$378$		0			0
$379$			17.9583i			0.922457i	0.887281	+	0.461228i	$0.152591\pi$
$379$			17.9583i			0.922457i	−0.887281	+	0.461228i	$0.847409\pi$
$380$		0			0
$381$	−44.6011			−2.28498
$382$		0			0
$383$		−	0.401341i		−	0.0205076i	−0.999947	−	0.0102538i	$-0.996736\pi$
$383$		−	0.401341i		−	0.0205076i	0.999947	−	0.0102538i	$-0.00326394\pi$
$384$		0			0
$385$			2.36927i			0.120749i
$386$		0			0
$387$	−25.3552			−1.28888
$388$		0			0
$389$	14.8693			0.753903			0.376951	−	0.926233i	$-0.376972\pi$
$389$	14.8693			0.753903			0.376951	+	0.926233i	$0.376972\pi$
$390$		0			0
$391$	23.9290			1.21014
$392$		0			0
$393$	0.0180112			0.000908546
$394$		0			0
$395$			23.9086i			1.20297i
$396$		0			0
$397$			21.7373i			1.09096i	0.838123	+	0.545482i	$0.183653\pi$
$397$			21.7373i			1.09096i	−0.838123	+	0.545482i	$0.816347\pi$
$398$		0			0
$399$	−0.411536			−0.0206026
$400$		0			0
$401$		−	11.3620i		−	0.567389i	−0.958915	−	0.283695i	$-0.908440\pi$
$401$		−	11.3620i		−	0.567389i	0.958915	−	0.283695i	$-0.0915602\pi$
$402$		0			0
$403$	4.69879	+	0.233146i	0.234064	+	0.0116138i
$404$		0			0
$405$			19.3197i			0.960005i
$406$		0			0
$407$	1.83347			0.0908818
$408$		0			0
$409$		−	4.69321i		−	0.232064i	−0.993245	−	0.116032i	$-0.962982\pi$
$409$		−	4.69321i		−	0.232064i	0.993245	−	0.116032i	$-0.0370176\pi$
$410$		0			0
$411$		−	30.6889i		−	1.51377i
$412$		0			0
$413$	−7.01797			−0.345332
$414$		0			0
$415$	27.2460			1.33746
$416$		0			0
$417$	−27.3052			−1.33714
$418$		0			0
$419$	−32.8908			−1.60682			−0.803410	−	0.595426i	$-0.796983\pi$
$419$	−32.8908			−1.60682			−0.803410	+	0.595426i	$0.796983\pi$
$420$		0			0
$421$			6.41929i			0.312857i	0.987689	+	0.156428i	$0.0499981\pi$
$421$			6.41929i			0.312857i	−0.987689	+	0.156428i	$0.950002\pi$
$422$		0			0
$423$			25.4842i			1.23909i
$424$		0			0
$425$	1.82326			0.0884413
$426$		0			0
$427$			8.99004i			0.435059i
$428$		0			0
$429$	9.00961	+	0.447042i	0.434988	+	0.0215834i
$430$		0			0
$431$		−	34.1259i		−	1.64378i	−0.569643	−	0.821892i	$-0.692918\pi$
$431$		−	34.1259i		−	1.64378i	0.569643	−	0.821892i	$-0.307082\pi$
$432$		0			0
$433$	11.4968			0.552502			0.276251	−	0.961086i	$-0.410908\pi$
$433$	11.4968			0.552502			0.276251	+	0.961086i	$0.410908\pi$
$434$		0			0
$435$		−	27.7946i		−	1.33265i
$436$		0			0
$437$		−	1.32427i		−	0.0633485i
$438$		0			0
$439$	−38.6451			−1.84443			−0.922216	−	0.386676i	$-0.873623\pi$
$439$	−38.6451			−1.84443			−0.922216	+	0.386676i	$0.873623\pi$
$440$		0			0
$441$	−3.25946			−0.155212
$442$		0			0
$443$	5.60456			0.266280			0.133140	−	0.991097i	$-0.457494\pi$
$443$	5.60456			0.266280			0.133140	+	0.991097i	$0.457494\pi$
$444$		0			0
$445$	−42.7248			−2.02535
$446$		0			0
$447$			11.5477i			0.546189i
$448$		0			0
$449$		−	34.4157i		−	1.62418i	−0.583535	−	0.812088i	$-0.698331\pi$
$449$		−	34.4157i		−	1.62418i	0.583535	−	0.812088i	$-0.301669\pi$
$450$		0			0
$451$	−4.62950			−0.217995
$452$		0			0
$453$		−	38.3352i		−	1.80115i
$454$		0			0
$455$	−0.423344	+	8.53202i	−0.0198467	+	0.399987i
$456$		0			0
$457$			22.7678i			1.06503i	0.846419	+	0.532517i	$0.178754\pi$
$457$			22.7678i			1.06503i	−0.846419	+	0.532517i	$0.821246\pi$
$458$		0			0
$459$	1.92941			0.0900570
$460$		0			0
$461$		−	10.5943i		−	0.493427i	−0.969088	−	0.246713i	$-0.920649\pi$
$461$		−	10.5943i		−	0.493427i	0.969088	−	0.246713i	$-0.0793506\pi$
$462$		0			0
$463$		−	10.6020i		−	0.492718i	−0.969179	−	0.246359i	$-0.920766\pi$
$463$		−	10.6020i		−	0.492718i	0.969179	−	0.246359i	$-0.0792343\pi$
$464$		0			0
$465$	−7.73448			−0.358678
$466$		0			0
$467$	−38.8100			−1.79591			−0.897957	−	0.440084i	$-0.854949\pi$
$467$	−38.8100			−1.79591			−0.897957	+	0.440084i	$0.854949\pi$
$468$		0			0
$469$	3.03458			0.140124
$470$		0			0
$471$	−26.4864			−1.22043
$472$		0			0
$473$			7.77895i			0.357676i
$474$		0			0
$475$		−	0.100903i		−	0.00462973i
$476$		0			0
$477$	41.2674			1.88950
$478$		0			0
$479$			2.10578i			0.0962153i	0.998842	+	0.0481077i	$0.0153191\pi$
$479$			2.10578i			0.0962153i	−0.998842	+	0.0481077i	$0.984681\pi$
$480$		0			0
$481$	6.60255	+	0.327607i	0.301050	+	0.0149376i
$482$		0			0
$483$		−	20.1422i		−	0.916500i
$484$		0			0
$485$	11.1838			0.507829
$486$		0			0
$487$		−	22.8701i		−	1.03634i	−0.855277	−	0.518172i	$-0.826613\pi$
$487$		−	22.8701i		−	1.03634i	0.855277	−	0.518172i	$-0.173387\pi$
$488$		0			0
$489$		−	61.2890i		−	2.77159i
$490$		0			0
$491$	−4.15448			−0.187489			−0.0937444	−	0.995596i	$-0.529884\pi$
$491$	−4.15448			−0.187489			−0.0937444	+	0.995596i	$0.529884\pi$
$492$		0			0
$493$	13.9369			0.627685
$494$		0			0
$495$	−7.72253			−0.347102
$496$		0			0
$497$	−13.5706			−0.608726
$498$		0			0
$499$			15.4382i			0.691111i	0.938398	+	0.345555i	$0.112309\pi$
$499$			15.4382i			0.691111i	−0.938398	+	0.345555i	$0.887691\pi$
$500$		0			0
$501$		−	28.5568i		−	1.27582i
$502$		0			0
$503$	29.4508			1.31315			0.656574	−	0.754262i	$-0.272005\pi$
$503$	29.4508			1.31315			0.656574	+	0.754262i	$0.272005\pi$
$504$		0			0
$505$		−	15.6777i		−	0.697647i
$506$		0			0
$507$	32.3648	+	3.21970i	1.43737	+	0.142992i
$508$		0			0
$509$			6.53473i			0.289647i	0.989458	+	0.144823i	$0.0462614\pi$
$509$			6.53473i			0.289647i	−0.989458	+	0.144823i	$0.953739\pi$
$510$		0			0
$511$	−12.8343			−0.567755
$512$		0			0
$513$		−	0.106777i		−	0.00471431i
$514$		0			0
$515$		−	11.5180i		−	0.507544i
$516$		0			0
$517$	7.81855			0.343859
$518$		0			0
$519$	−7.22904			−0.317320
$520$		0			0
$521$	−9.60615			−0.420853			−0.210427	−	0.977610i	$-0.567485\pi$
$521$	−9.60615			−0.420853			−0.210427	+	0.977610i	$0.567485\pi$
$522$		0			0
$523$	3.36160			0.146992			0.0734962	−	0.997295i	$-0.476584\pi$
$523$	3.36160			0.146992			0.0734962	+	0.997295i	$0.476584\pi$
$524$		0			0
$525$		−	1.53473i		−	0.0669810i
$526$		0			0
$527$		−	3.87825i		−	0.168939i
$528$		0			0
$529$	41.8150			1.81804
$530$		0			0
$531$		−	22.8748i		−	0.992681i
$532$		0			0
$533$	−16.6714	−	0.827206i	−0.722118	−	0.0358303i
$534$		0			0
$535$			3.77124i			0.163045i
$536$		0			0
$537$	−41.5897			−1.79473
$538$		0			0
$539$			1.00000i			0.0430730i
$540$		0			0
$541$		−	0.326562i		−	0.0140400i	−0.999975	−	0.00702001i	$-0.997765\pi$
$541$		−	0.326562i		−	0.0140400i	0.999975	−	0.00702001i	$-0.00223456\pi$
$542$		0			0
$543$	−49.7906			−2.13672
$544$		0			0
$545$	28.2621			1.21062
$546$		0			0
$547$	36.4784			1.55970			0.779851	−	0.625965i	$-0.215295\pi$
$547$	36.4784			1.55970			0.779851	+	0.625965i	$0.215295\pi$
$548$		0			0
$549$	−29.3027			−1.25061
$550$		0			0
$551$		−	0.771290i		−	0.0328581i
$552$		0			0
$553$			10.0912i			0.429120i
$554$		0			0
$555$	−10.8682			−0.461328
$556$		0			0
$557$			15.5953i			0.660794i	0.943842	+	0.330397i	$0.107183\pi$
$557$			15.5953i			0.660794i	−0.943842	+	0.330397i	$0.892817\pi$
$558$		0			0
$559$	−1.38995	+	28.0129i	−0.0587888	+	1.18482i
$560$		0			0
$561$		−	7.43627i		−	0.313960i
$562$		0			0
$563$	−39.3860			−1.65992			−0.829962	−	0.557820i	$-0.811638\pi$
$563$	−39.3860			−1.65992			−0.829962	+	0.557820i	$0.811638\pi$
$564$		0			0
$565$			21.1460i			0.889619i
$566$		0			0
$567$			8.15430i			0.342449i
$568$		0			0
$569$	23.9509			1.00407			0.502037	−	0.864846i	$-0.332584\pi$
$569$	23.9509			1.00407			0.502037	+	0.864846i	$0.332584\pi$
$570$		0			0
$571$	−14.4020			−0.602704			−0.301352	−	0.953513i	$-0.597438\pi$
$571$	−14.4020			−0.602704			−0.301352	+	0.953513i	$0.597438\pi$
$572$		0			0
$573$	53.7235			2.24433
$574$		0			0
$575$	4.93856			0.205952
$576$		0			0
$577$		−	13.6179i		−	0.566922i	−0.958984	−	0.283461i	$-0.908517\pi$
$577$		−	13.6179i		−	0.566922i	0.958984	−	0.283461i	$-0.0914826\pi$
$578$		0			0
$579$			21.4634i			0.891989i
$580$		0			0
$581$	11.4998			0.477091
$582$		0			0
$583$		−	12.6608i		−	0.524357i
$584$		0			0
$585$	−27.8098	−	1.37987i	−1.14979	−	0.0570507i
$586$		0			0
$587$			3.43229i			0.141666i	0.997488	+	0.0708329i	$0.0225657\pi$
$587$			3.43229i			0.141666i	−0.997488	+	0.0708329i	$0.977434\pi$
$588$		0			0
$589$	−0.214629			−0.00884363
$590$		0			0
$591$			17.9248i			0.737327i
$592$		0			0
$593$		−	24.1367i		−	0.991175i	−0.868558	−	0.495587i	$-0.834953\pi$
$593$		−	24.1367i		−	0.991175i	0.868558	−	0.495587i	$-0.165047\pi$
$594$		0			0
$595$	7.04208			0.288697
$596$		0			0
$597$	27.4956			1.12532
$598$		0			0
$599$	−32.7967			−1.34004			−0.670019	−	0.742344i	$-0.733714\pi$
$599$	−32.7967			−1.34004			−0.670019	+	0.742344i	$0.733714\pi$
$600$		0			0
$601$	12.7441			0.519842			0.259921	−	0.965630i	$-0.416304\pi$
$601$	12.7441			0.519842			0.259921	+	0.965630i	$0.416304\pi$
$602$		0			0
$603$			9.89107i			0.402796i
$604$		0			0
$605$			2.36927i			0.0963244i
$606$		0			0
$607$	−38.3348			−1.55596			−0.777982	−	0.628287i	$-0.783756\pi$
$607$	−38.3348			−1.55596			−0.777982	+	0.628287i	$0.783756\pi$
$608$		0			0
$609$		−	11.7313i		−	0.475377i
$610$		0			0
$611$	28.1555	+	1.39703i	1.13905	+	0.0565178i
$612$		0			0
$613$		−	24.2905i		−	0.981083i	−0.871418	−	0.490542i	$-0.836799\pi$
$613$		−	24.2905i		−	0.981083i	0.871418	−	0.490542i	$-0.163201\pi$
$614$		0			0
$615$	27.4421			1.10657
$616$		0			0
$617$			45.4918i			1.83143i	0.401827	+	0.915716i	$0.368375\pi$
$617$			45.4918i			1.83143i	−0.401827	+	0.915716i	$0.631625\pi$
$618$		0			0
$619$		−	8.95028i		−	0.359742i	−0.983690	−	0.179871i	$-0.942432\pi$
$619$		−	8.95028i		−	0.359742i	0.983690	−	0.179871i	$-0.0575681\pi$
$620$		0			0
$621$	5.22607			0.209715
$622$		0			0
$623$	−18.0329			−0.722474
$624$		0			0
$625$	−27.6908			−1.10763
$626$		0			0
$627$	−0.411536			−0.0164352
$628$		0			0
$629$		−	5.44955i		−	0.217288i
$630$		0			0
$631$		−	13.8628i		−	0.551868i	−0.961177	−	0.275934i	$-0.911013\pi$
$631$		−	13.8628i		−	0.551868i	0.961177	−	0.275934i	$-0.0889871\pi$
$632$		0			0
$633$	54.5283			2.16731
$634$		0			0
$635$		−	42.2368i		−	1.67612i
$636$		0			0
$637$	−0.178681	+	3.60112i	−0.00707962	+	0.142682i
$638$		0			0
$639$		−	44.2329i		−	1.74983i
$640$		0			0
$641$	25.7931			1.01877			0.509383	−	0.860540i	$-0.329874\pi$
$641$	25.7931			1.01877			0.509383	+	0.860540i	$0.329874\pi$
$642$		0			0
$643$			26.6880i			1.05247i	0.850339	+	0.526236i	$0.176397\pi$
$643$			26.6880i			1.05247i	−0.850339	+	0.526236i	$0.823603\pi$
$644$		0			0
$645$		−	46.1109i		−	1.81561i
$646$		0			0
$647$	−36.0549			−1.41746			−0.708732	−	0.705477i	$-0.750733\pi$
$647$	−36.0549			−1.41746			−0.708732	+	0.705477i	$0.750733\pi$
$648$		0			0
$649$	−7.01797			−0.275479
$650$		0			0
$651$	−3.26450			−0.127946
$652$		0			0
$653$	−44.6230			−1.74623			−0.873117	−	0.487510i	$-0.837905\pi$
$653$	−44.6230			−1.74623			−0.873117	+	0.487510i	$0.837905\pi$
$654$		0			0
$655$			0.0170565i			0.000666451i
$656$		0			0
$657$		−	41.8328i		−	1.63205i
$658$		0			0
$659$	46.5571			1.81361			0.906803	−	0.421554i	$-0.138515\pi$
$659$	46.5571			1.81361			0.906803	+	0.421554i	$0.138515\pi$
$660$		0			0
$661$			42.7449i			1.66258i	0.555836	+	0.831292i	$0.312398\pi$
$661$			42.7449i			1.66258i	−0.555836	+	0.831292i	$0.687602\pi$
$662$		0			0
$663$	1.32872	−	26.7789i	0.0516033	−	1.04001i
$664$		0			0
$665$		−	0.389721i		−	0.0151127i
$666$		0			0
$667$	37.7499			1.46168
$668$		0			0
$669$		−	33.6388i		−	1.30055i
$670$		0			0
$671$			8.99004i			0.347057i
$672$		0			0
$673$	31.3637			1.20898			0.604491	−	0.796612i	$-0.293377\pi$
$673$	31.3637			1.20898			0.604491	+	0.796612i	$0.293377\pi$
$674$		0			0
$675$	0.398199			0.0153267
$676$		0			0
$677$	−13.6234			−0.523589			−0.261794	−	0.965124i	$-0.584314\pi$
$677$	−13.6234			−0.523589			−0.261794	+	0.965124i	$0.584314\pi$
$678$		0			0
$679$	4.72035			0.181151
$680$		0			0
$681$		−	9.01837i		−	0.345585i
$682$		0			0
$683$		−	49.7405i		−	1.90327i	−0.307235	−	0.951634i	$-0.599404\pi$
$683$		−	49.7405i		−	1.90327i	0.307235	−	0.951634i	$-0.400596\pi$
$684$		0			0
$685$	29.0621			1.11040
$686$		0			0
$687$		−	10.7175i		−	0.408896i
$688$		0			0
$689$	2.26225	−	45.5931i	0.0861849	−	1.73696i
$690$		0			0
$691$			23.5987i			0.897736i	0.893598	+	0.448868i	$0.148173\pi$
$691$			23.5987i			0.897736i	−0.893598	+	0.448868i	$0.851827\pi$
$692$		0			0
$693$	−3.25946			−0.123817
$694$		0			0
$695$		−	25.8578i		−	0.980840i
$696$		0			0
$697$			13.7601i			0.521200i
$698$		0			0
$699$	50.1984			1.89868
$700$		0			0
$701$	−4.47902			−0.169170			−0.0845852	−	0.996416i	$-0.526957\pi$
$701$	−4.47902			−0.169170			−0.0845852	+	0.996416i	$0.526957\pi$
$702$		0			0
$703$	−0.301588			−0.0113746
$704$		0			0
$705$	−46.3456			−1.74548
$706$		0			0
$707$		−	6.61709i		−	0.248861i
$708$		0			0
$709$			4.96454i			0.186447i	0.995645	+	0.0932237i	$0.0297172\pi$
$709$			4.96454i			0.186447i	−0.995645	+	0.0932237i	$0.970283\pi$
$710$		0			0
$711$	−32.8917			−1.23354
$712$		0			0
$713$		−	10.5048i		−	0.393407i
$714$		0			0
$715$	−0.423344	+	8.53202i	−0.0158322	+	0.319079i
$716$		0			0
$717$		−	38.2962i		−	1.43020i
$718$		0			0
$719$	15.2599			0.569100			0.284550	−	0.958661i	$-0.408156\pi$
$719$	15.2599			0.569100			0.284550	+	0.958661i	$0.408156\pi$
$720$		0			0
$721$		−	4.86143i		−	0.181049i
$722$		0			0
$723$		−	33.8307i		−	1.25818i
$724$		0			0
$725$	2.87635			0.106825
$726$		0			0
$727$	32.2747			1.19700			0.598500	−	0.801122i	$-0.295763\pi$
$727$	32.2747			1.19700			0.598500	+	0.801122i	$0.295763\pi$
$728$		0			0
$729$	−31.4508			−1.16485
$730$		0			0
$731$	23.1211			0.855163
$732$		0			0
$733$			36.5281i			1.34919i	0.738186	+	0.674597i	$0.235683\pi$
$733$			36.5281i			1.34919i	−0.738186	+	0.674597i	$0.764317\pi$
$734$		0			0
$735$		−	5.92765i		−	0.218645i
$736$		0			0
$737$	3.03458			0.111780
$738$		0			0
$739$			39.6571i			1.45881i	0.684082	+	0.729405i	$0.260203\pi$
$739$			39.6571i			1.45881i	−0.684082	+	0.729405i	$0.739797\pi$
$740$		0			0
$741$	−1.48199	−	0.0735338i	−0.0544423	−	0.00270133i
$742$		0			0
$743$		−	36.3842i		−	1.33481i	−0.744696	−	0.667404i	$-0.767406\pi$
$743$		−	36.3842i		−	1.33481i	0.744696	−	0.667404i	$-0.232594\pi$
$744$		0			0
$745$	−10.9356			−0.400649
$746$		0			0
$747$			37.4830i			1.37143i
$748$		0			0
$749$			1.59173i			0.0581607i
$750$		0			0
$751$	−2.89530			−0.105651			−0.0528255	−	0.998604i	$-0.516823\pi$
$751$	−2.89530			−0.105651			−0.0528255	+	0.998604i	$0.516823\pi$
$752$		0			0
$753$	11.5962			0.422587
$754$		0			0
$755$	36.3031			1.32121
$756$		0			0
$757$	−27.3046			−0.992402			−0.496201	−	0.868208i	$-0.665272\pi$
$757$	−27.3046			−0.992402			−0.496201	+	0.868208i	$0.665272\pi$
$758$		0			0
$759$		−	20.1422i		−	0.731114i
$760$		0			0
$761$		−	47.5415i		−	1.72338i	−0.507437	−	0.861689i	$-0.669407\pi$
$761$		−	47.5415i		−	1.72338i	0.507437	−	0.861689i	$-0.330593\pi$
$762$		0			0
$763$	11.9286			0.431845
$764$		0			0
$765$			22.9534i			0.829880i
$766$		0			0
$767$	−25.2726	−	1.25398i	−0.912539	−	0.0452786i
$768$		0			0
$769$		−	36.4418i		−	1.31412i	−0.753837	−	0.657062i	$-0.771799\pi$
$769$		−	36.4418i		−	1.31412i	0.753837	−	0.657062i	$-0.228201\pi$
$770$		0			0
$771$	−28.1016			−1.01206
$772$		0			0
$773$			4.28819i			0.154235i	0.997022	+	0.0771177i	$0.0245717\pi$
$773$			4.28819i			0.154235i	−0.997022	+	0.0771177i	$0.975428\pi$
$774$		0			0
$775$		−	0.800408i		−	0.0287515i
$776$		0			0
$777$	−4.58715			−0.164563
$778$		0			0
$779$	0.761507			0.0272838
$780$		0			0
$781$	−13.5706			−0.485596
$782$		0			0
$783$	3.04380			0.108776
$784$		0			0
$785$		−	25.0824i		−	0.895229i
$786$		0			0
$787$		−	34.9205i		−	1.24478i	−0.782707	−	0.622391i	$-0.786162\pi$
$787$		−	34.9205i		−	1.24478i	0.782707	−	0.622391i	$-0.213838\pi$
$788$		0			0
$789$	24.3874			0.868216
$790$		0			0
$791$			8.92513i			0.317341i
$792$		0			0
$793$	−1.60635	+	32.3742i	−0.0570433	+	1.14964i
$794$		0			0
$795$			75.0488i			2.66171i
$796$		0			0
$797$	12.8511			0.455209			0.227605	−	0.973754i	$-0.426911\pi$
$797$	12.8511			0.455209			0.227605	+	0.973754i	$0.426911\pi$
$798$		0			0
$799$		−	23.2388i		−	0.822128i
$800$		0			0
$801$		−	58.7776i		−	2.07680i
$802$		0			0
$803$	−12.8343			−0.452912
$804$		0			0
$805$	19.0744			0.672286
$806$		0			0
$807$	36.4461			1.28296
$808$		0			0
$809$	−8.47340			−0.297909			−0.148954	−	0.988844i	$-0.547591\pi$
$809$	−8.47340			−0.297909			−0.148954	+	0.988844i	$0.547591\pi$
$810$		0			0
$811$		−	0.902208i		−	0.0316808i	−0.999875	−	0.0158404i	$-0.994958\pi$
$811$		−	0.902208i		−	0.0316808i	0.999875	−	0.0158404i	$-0.00504237\pi$
$812$		0			0
$813$			38.5392i			1.35163i
$814$		0			0
$815$	58.0401			2.03306
$816$		0			0
$817$		−	1.27956i		−	0.0447661i
$818$		0			0
$819$	−11.7377	−	0.582405i	−0.410149	−	0.0203509i
$820$		0			0
$821$		−	8.79482i		−	0.306941i	−0.988153	−	0.153471i	$-0.950955\pi$
$821$		−	8.79482i		−	0.306941i	0.988153	−	0.153471i	$-0.0490451\pi$
$822$		0			0
$823$	−10.3232			−0.359843			−0.179922	−	0.983681i	$-0.557584\pi$
$823$	−10.3232			−0.359843			−0.179922	+	0.983681i	$0.557584\pi$
$824$		0			0
$825$		−	1.53473i		−	0.0534323i
$826$		0			0
$827$			12.3101i			0.428064i	0.976827	+	0.214032i	$0.0686597\pi$
$827$			12.3101i			0.428064i	−0.976827	+	0.214032i	$0.931340\pi$
$828$		0			0
$829$	−12.9505			−0.449788			−0.224894	−	0.974383i	$-0.572204\pi$
$829$	−12.9505			−0.449788			−0.224894	+	0.974383i	$0.572204\pi$
$830$		0			0
$831$	57.2792			1.98699
$832$		0			0
$833$	2.97226			0.102983
$834$		0			0
$835$	27.0430			0.935862
$836$		0			0
$837$		−	0.847005i		−	0.0292768i
$838$		0			0
$839$		−	23.5540i		−	0.813174i	−0.913612	−	0.406587i	$-0.866719\pi$
$839$		−	23.5540i		−	0.813174i	0.913612	−	0.406587i	$-0.133281\pi$
$840$		0			0
$841$	−7.01346			−0.241843
$842$		0			0
$843$		−	12.1930i		−	0.419948i
$844$		0			0
$845$	−3.04903	+	30.6492i	−0.104890	+	1.05436i
$846$		0			0
$847$			1.00000i			0.0343604i
$848$		0			0
$849$	−57.9702			−1.98953
$850$		0			0
$851$		−	14.7609i		−	0.505996i
$852$		0			0
$853$		−	4.51743i		−	0.154674i	−0.997005	−	0.0773370i	$-0.975358\pi$
$853$		−	4.51743i		−	0.154674i	0.997005	−	0.0773370i	$-0.0246417\pi$
$854$		0			0
$855$	1.27028			0.0434426
$856$		0			0
$857$	−3.10214			−0.105967			−0.0529835	−	0.998595i	$-0.516873\pi$
$857$	−3.10214			−0.105967			−0.0529835	+	0.998595i	$0.516873\pi$
$858$		0			0
$859$	53.3660			1.82082			0.910412	−	0.413703i	$-0.135765\pi$
$859$	53.3660			1.82082			0.910412	+	0.413703i	$0.135765\pi$
$860$		0			0
$861$	11.5825			0.394731
$862$		0			0
$863$		−	29.2395i		−	0.995324i	−0.867371	−	0.497662i	$-0.834192\pi$
$863$		−	29.2395i		−	0.995324i	0.867371	−	0.497662i	$-0.165808\pi$
$864$		0			0
$865$		−	6.84583i		−	0.232765i
$866$		0			0
$867$	20.4296			0.693827
$868$		0			0
$869$			10.0912i			0.342319i
$870$		0			0
$871$	10.9279	+	0.542222i	0.370277	+	0.0183725i
$872$		0			0
$873$			15.3858i			0.520730i
$874$		0			0
$875$	−10.3930			−0.351346
$876$		0			0
$877$			18.1440i			0.612681i	0.951922	+	0.306340i	$0.0991045\pi$
$877$			18.1440i			0.612681i	−0.951922	+	0.306340i	$0.900895\pi$
$878$		0			0
$879$		−	18.1759i		−	0.613059i
$880$		0			0
$881$	1.96203			0.0661023			0.0330512	−	0.999454i	$-0.489478\pi$
$881$	1.96203			0.0661023			0.0330512	+	0.999454i	$0.489478\pi$
$882$		0			0
$883$	−2.35033			−0.0790949			−0.0395474	−	0.999218i	$-0.512592\pi$
$883$	−2.35033			−0.0790949			−0.0395474	+	0.999218i	$0.512592\pi$
$884$		0			0
$885$	41.6001			1.39837
$886$		0			0
$887$	−25.2974			−0.849403			−0.424702	−	0.905333i	$-0.639621\pi$
$887$	−25.2974			−0.849403			−0.424702	+	0.905333i	$0.639621\pi$
$888$		0			0
$889$		−	17.8269i		−	0.597897i
$890$		0			0
$891$			8.15430i			0.273179i
$892$		0			0
$893$	−1.28607			−0.0430368
$894$		0			0
$895$		−	39.3851i		−	1.31650i
$896$		0			0
$897$	3.59903	−	72.5344i	0.120168	−	2.42185i
$898$		0			0
$899$		−	6.11825i		−	0.204055i
$900$		0			0
$901$	−37.6312			−1.25368
$902$		0			0
$903$		−	19.4621i		−	0.647658i
$904$		0			0
$905$		−	47.1512i		−	1.56736i
$906$		0			0
$907$	−15.6495			−0.519632			−0.259816	−	0.965658i	$-0.583662\pi$
$907$	−15.6495			−0.519632			−0.259816	+	0.965658i	$0.583662\pi$
$908$		0			0
$909$	21.5681			0.715370
$910$		0			0
$911$	−18.8609			−0.624891			−0.312445	−	0.949936i	$-0.601148\pi$
$911$	−18.8609			−0.624891			−0.312445	+	0.949936i	$0.601148\pi$
$912$		0			0
$913$	11.4998			0.380587
$914$		0			0
$915$		−	53.2898i		−	1.76171i
$916$		0			0
$917$			0.00719904i			0.000237733i
$918$		0			0
$919$	−7.24619			−0.239030			−0.119515	−	0.992832i	$-0.538134\pi$
$919$	−7.24619			−0.239030			−0.119515	+	0.992832i	$0.538134\pi$
$920$		0			0
$921$		−	69.2386i		−	2.28149i
$922$		0			0
$923$	−48.8695	−	2.42482i	−1.60856	−	0.0798140i
$924$		0			0
$925$		−	1.12470i		−	0.0369799i
$926$		0			0
$927$	15.8456			0.520438
$928$		0			0
$929$			22.4099i			0.735245i	0.929975	+	0.367622i	$0.119828\pi$
$929$			22.4099i			0.735245i	−0.929975	+	0.367622i	$0.880172\pi$
$930$		0			0
$931$		−	0.164490i		−	0.00539094i
$932$		0			0
$933$	−82.5644			−2.70304
$934$		0			0
$935$	7.04208			0.230301
$936$		0			0
$937$	−39.6590			−1.29560			−0.647802	−	0.761809i	$-0.724312\pi$
$937$	−39.6590			−1.29560			−0.647802	+	0.761809i	$0.724312\pi$
$938$		0			0
$939$	−26.2461			−0.856511
$940$		0			0
$941$		−	47.2499i		−	1.54030i	−0.637862	−	0.770151i	$-0.720181\pi$
$941$		−	47.2499i		−	1.54030i	0.637862	−	0.770151i	$-0.279819\pi$
$942$		0			0
$943$			37.2711i			1.21371i
$944$		0			0
$945$	1.53798			0.0500305
$946$		0			0
$947$		−	28.6199i		−	0.930021i	−0.885305	−	0.465010i	$-0.846051\pi$
$947$		−	28.6199i		−	0.930021i	0.885305	−	0.465010i	$-0.153949\pi$
$948$		0			0
$949$	−46.2178	−	2.29325i	−1.50029	−	0.0744420i
$950$		0			0
$951$		−	31.0499i		−	1.00686i
$952$		0			0
$953$	18.2730			0.591921			0.295960	−	0.955200i	$-0.404360\pi$
$953$	18.2730			0.591921			0.295960	+	0.955200i	$0.404360\pi$
$954$		0			0
$955$			50.8756i			1.64630i
$956$		0			0
$957$		−	11.7313i		−	0.379220i
$958$		0			0
$959$	12.2663			0.396098
$960$		0			0
$961$	29.2975			0.945079
$962$		0			0
$963$	−5.18819			−0.167187
$964$		0			0
$965$	−20.3257			−0.654306
$966$		0			0
$967$		−	1.67485i		−	0.0538594i	−0.999637	−	0.0269297i	$-0.991427\pi$
$967$		−	1.67485i		−	0.0538594i	0.999637	−	0.0269297i	$-0.00857303\pi$
$968$		0			0
$969$			1.22319i			0.0392946i
$970$		0			0
$971$	−20.3305			−0.652437			−0.326219	−	0.945294i	$-0.605775\pi$
$971$	−20.3305			−0.652437			−0.326219	+	0.945294i	$0.605775\pi$
$972$		0			0
$973$		−	10.9138i		−	0.349881i
$974$		0			0
$975$	0.274227	−	5.52674i	0.00878230	−	0.176997i
$976$		0			0
$977$			37.7050i			1.20629i	0.797631	+	0.603145i	$0.206086\pi$
$977$			37.7050i			1.20629i	−0.797631	+	0.603145i	$0.793914\pi$
$978$		0			0
$979$	−18.0329			−0.576335
$980$		0			0
$981$			38.8809i			1.24137i
$982$		0			0
$983$		−	51.2569i		−	1.63484i	−0.576041	−	0.817420i	$-0.695403\pi$
$983$		−	51.2569i		−	1.63484i	0.576041	−	0.817420i	$-0.304597\pi$
$984$		0			0
$985$	−16.9746			−0.540856
$986$		0			0
$987$	−19.5612			−0.622638
$988$		0			0
$989$	62.6266			1.99141
$990$		0			0
$991$	−2.46826			−0.0784067			−0.0392034	−	0.999231i	$-0.512482\pi$
$991$	−2.46826			−0.0784067			−0.0392034	+	0.999231i	$0.512482\pi$
$992$		0			0
$993$			19.1417i			0.607443i
$994$		0			0
$995$			26.0380i			0.825461i
$996$		0			0
$997$	1.29190			0.0409147			0.0204574	−	0.999791i	$-0.493488\pi$
$997$	1.29190			0.0409147			0.0204574	+	0.999791i	$0.493488\pi$
$998$		0			0
$999$		−	1.19018i		−	0.0376555i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.b.2157.3	✓	30
13.12	even	2	inner	4004.2.m.b.2157.4	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.b.2157.3	✓	30	1.1	even	1	trivial
4004.2.m.b.2157.4	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)

\(f(q)\)	\(=\)	\(q-2.50189 q^{3} -2.36927i q^{5} -1.00000i q^{7} +3.25946 q^{9} +O(q^{10})\)	\(q-2.50189 q^{3} -2.36927i q^{5} -1.00000i q^{7} +3.25946 q^{9} -1.00000i q^{11} +(0.178681 - 3.60112i) q^{13} +5.92765i q^{15} -2.97226 q^{17} +0.164490i q^{19} +2.50189i q^{21} -8.05078 q^{23} -0.613427 q^{25} -0.649138 q^{27} -4.68898 q^{29} +1.30481i q^{31} +2.50189i q^{33} -2.36927 q^{35} +1.83347i q^{37} +(-0.447042 + 9.00961i) q^{39} -4.62950i q^{41} -7.77895 q^{43} -7.72253i q^{45} +7.81855i q^{47} -1.00000 q^{49} +7.43627 q^{51} +12.6608 q^{53} -2.36927 q^{55} -0.411536i q^{57} -7.01797i q^{59} -8.99004 q^{61} -3.25946i q^{63} +(-8.53202 - 0.423344i) q^{65} +3.03458i q^{67} +20.1422 q^{69} -13.5706i q^{71} -12.8343i q^{73} +1.53473 q^{75} -1.00000 q^{77} -10.0912 q^{79} -8.15430 q^{81} +11.4998i q^{83} +7.04208i q^{85} +11.7313 q^{87} -18.0329i q^{89} +(-3.60112 - 0.178681i) q^{91} -3.26450i q^{93} +0.389721 q^{95} +4.72035i q^{97} -3.25946i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 18 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more