LMFDB - Embedded newform 4004.2.m.b.2157.5

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			2157.5
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.b.2157.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.42937 q^{3} +1.20006i q^{5} +1.00000i q^{7} +2.90185 q^{9} +O(q^{10})$	$q-2.42937 q^{3} +1.20006i q^{5} +1.00000i q^{7} +2.90185 q^{9} +1.00000i q^{11} +(-2.92856 - 2.10322i) q^{13} -2.91540i q^{15} -0.0204508 q^{17} -1.17541i q^{19} -2.42937i q^{21} -3.64958 q^{23} +3.55985 q^{25} +0.238439 q^{27} +9.92286 q^{29} +1.43379i q^{31} -2.42937i q^{33} -1.20006 q^{35} +8.76575i q^{37} +(7.11456 + 5.10951i) q^{39} -4.94879i q^{41} +0.228499 q^{43} +3.48240i q^{45} +9.65438i q^{47} -1.00000 q^{49} +0.0496825 q^{51} -4.95235 q^{53} -1.20006 q^{55} +2.85550i q^{57} -7.31985i q^{59} +1.17956 q^{61} +2.90185i q^{63} +(2.52400 - 3.51445i) q^{65} +7.58208i q^{67} +8.86619 q^{69} +11.7879i q^{71} -11.4308i q^{73} -8.64820 q^{75} -1.00000 q^{77} +10.1861 q^{79} -9.28481 q^{81} -3.97300i q^{83} -0.0245422i q^{85} -24.1063 q^{87} -0.313403i q^{89} +(2.10322 - 2.92856i) q^{91} -3.48321i q^{93} +1.41056 q^{95} -6.85758i q^{97} +2.90185i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.42937			−1.40260			−0.701299	−	0.712867i	$-0.747396\pi$
$3$	−2.42937			−1.40260			−0.701299	+	0.712867i	$0.747396\pi$
$4$		0			0
$5$			1.20006i			0.536684i	0.963324	+	0.268342i	$0.0864758\pi$
$5$			1.20006i			0.536684i	−0.963324	+	0.268342i	$0.913524\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	2.90185			0.967284
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$	−2.92856	−	2.10322i	−0.812236	−	0.583329i
$13$	−2.92856	−	2.10322i	−0.812236	−	0.583329i
$14$		0			0
$15$		−	2.91540i		−	0.752753i
$16$		0			0
$17$	−0.0204508			−0.00496004			−0.00248002	−	0.999997i	$-0.500789\pi$
$17$	−0.0204508			−0.00496004			−0.00248002	+	0.999997i	$0.500789\pi$
$18$		0			0
$19$		−	1.17541i		−	0.269657i	−0.990869	−	0.134829i	$-0.956952\pi$
$19$		−	1.17541i		−	0.269657i	0.990869	−	0.134829i	$-0.0430484\pi$
$20$		0			0
$21$		−	2.42937i		−	0.530133i
$22$		0			0
$23$	−3.64958			−0.760990			−0.380495	−	0.924783i	$-0.624246\pi$
$23$	−3.64958			−0.760990			−0.380495	+	0.924783i	$0.624246\pi$
$24$		0			0
$25$	3.55985			0.711970
$26$		0			0
$27$	0.238439			0.0458875
$28$		0			0
$29$	9.92286			1.84263			0.921314	−	0.388819i	$-0.127117\pi$
$29$	9.92286			1.84263			0.921314	+	0.388819i	$0.127117\pi$
$30$		0			0
$31$			1.43379i			0.257516i	0.991676	+	0.128758i	$0.0410991\pi$
$31$			1.43379i			0.257516i	−0.991676	+	0.128758i	$0.958901\pi$
$32$		0			0
$33$		−	2.42937i		−	0.422900i
$34$		0			0
$35$	−1.20006			−0.202848
$36$		0			0
$37$			8.76575i			1.44108i	0.693413	+	0.720541i	$0.256106\pi$
$37$			8.76575i			1.44108i	−0.693413	+	0.720541i	$0.743894\pi$
$38$		0			0
$39$	7.11456	+	5.10951i	1.13924	+	0.818177i
$40$		0			0
$41$		−	4.94879i		−	0.772872i	−0.922316	−	0.386436i	$-0.873706\pi$
$41$		−	4.94879i		−	0.772872i	0.922316	−	0.386436i	$-0.126294\pi$
$42$		0			0
$43$	0.228499			0.0348457			0.0174229	−	0.999848i	$-0.494454\pi$
$43$	0.228499			0.0348457			0.0174229	+	0.999848i	$0.494454\pi$
$44$		0			0
$45$			3.48240i			0.519126i
$46$		0			0
$47$			9.65438i			1.40824i	0.710083	+	0.704118i	$0.248657\pi$
$47$			9.65438i			1.40824i	−0.710083	+	0.704118i	$0.751343\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	0.0496825			0.00695695
$52$		0			0
$53$	−4.95235			−0.680258			−0.340129	−	0.940379i	$-0.610471\pi$
$53$	−4.95235			−0.680258			−0.340129	+	0.940379i	$0.610471\pi$
$54$		0			0
$55$	−1.20006			−0.161816
$56$		0			0
$57$			2.85550i			0.378221i
$58$		0			0
$59$		−	7.31985i		−	0.952963i	−0.879184	−	0.476482i	$-0.841912\pi$
$59$		−	7.31985i		−	0.952963i	0.879184	−	0.476482i	$-0.158088\pi$
$60$		0			0
$61$	1.17956			0.151027			0.0755137	−	0.997145i	$-0.475940\pi$
$61$	1.17956			0.151027			0.0755137	+	0.997145i	$0.475940\pi$
$62$		0			0
$63$			2.90185i			0.365599i
$64$		0			0
$65$	2.52400	−	3.51445i	0.313064	−	0.435914i
$66$		0			0
$67$			7.58208i			0.926299i	0.886280	+	0.463149i	$0.153281\pi$
$67$			7.58208i			0.926299i	−0.886280	+	0.463149i	$0.846719\pi$
$68$		0			0
$69$	8.86619			1.06736
$70$		0			0
$71$			11.7879i			1.39896i	0.714650	+	0.699482i	$0.246586\pi$
$71$			11.7879i			1.39896i	−0.714650	+	0.699482i	$0.753414\pi$
$72$		0			0
$73$		−	11.4308i		−	1.33787i	−0.743322	−	0.668934i	$-0.766751\pi$
$73$		−	11.4308i		−	1.33787i	0.743322	−	0.668934i	$-0.233249\pi$
$74$		0			0
$75$	−8.64820			−0.998608
$76$		0			0
$77$	−1.00000			−0.113961
$78$		0			0
$79$	10.1861			1.14603			0.573014	−	0.819546i	$-0.305774\pi$
$79$	10.1861			1.14603			0.573014	+	0.819546i	$0.305774\pi$
$80$		0			0
$81$	−9.28481			−1.03165
$82$		0			0
$83$		−	3.97300i		−	0.436093i	−0.975938	−	0.218046i	$-0.930032\pi$
$83$		−	3.97300i		−	0.436093i	0.975938	−	0.218046i	$-0.0699684\pi$
$84$		0			0
$85$		−	0.0245422i		−	0.00266198i
$86$		0			0
$87$	−24.1063			−2.58447
$88$		0			0
$89$		−	0.313403i		−	0.0332207i	−0.999862	−	0.0166103i	$-0.994713\pi$
$89$		−	0.313403i		−	0.0332207i	0.999862	−	0.0166103i	$-0.00528748\pi$
$90$		0			0
$91$	2.10322	−	2.92856i	0.220478	−	0.306996i
$92$		0			0
$93$		−	3.48321i		−	0.361192i
$94$		0			0
$95$	1.41056			0.144721
$96$		0			0
$97$		−	6.85758i		−	0.696281i	−0.937442	−	0.348141i	$-0.886813\pi$
$97$		−	6.85758i		−	0.696281i	0.937442	−	0.348141i	$-0.113187\pi$
$98$		0			0
$99$			2.90185i			0.291647i
$100$		0			0
$101$	19.8294			1.97310			0.986550	−	0.163462i	$-0.0522662\pi$
$101$	19.8294			1.97310			0.986550	+	0.163462i	$0.0522662\pi$
$102$		0			0
$103$	4.28758			0.422467			0.211234	−	0.977436i	$-0.432252\pi$
$103$	4.28758			0.422467			0.211234	+	0.977436i	$0.432252\pi$
$104$		0			0
$105$	2.91540			0.284514
$106$		0			0
$107$	−18.8408			−1.82141			−0.910705	−	0.413057i	$-0.864461\pi$
$107$	−18.8408			−1.82141			−0.910705	+	0.413057i	$0.864461\pi$
$108$		0			0
$109$			1.70146i			0.162970i	0.996675	+	0.0814852i	$0.0259663\pi$
$109$			1.70146i			0.162970i	−0.996675	+	0.0814852i	$0.974034\pi$
$110$		0			0
$111$		−	21.2953i		−	2.02126i
$112$		0			0
$113$	4.04916			0.380913			0.190456	−	0.981696i	$-0.439003\pi$
$113$	4.04916			0.380913			0.190456	+	0.981696i	$0.439003\pi$
$114$		0			0
$115$		−	4.37972i		−	0.408411i
$116$		0			0
$117$	−8.49824	−	6.10324i	−0.785663	−	0.564245i
$118$		0			0
$119$		−	0.0204508i		−	0.00187472i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			12.0225i			1.08403i
$124$		0			0
$125$			10.2724i			0.918787i
$126$		0			0
$127$	0.439529			0.0390019			0.0195010	−	0.999810i	$-0.493792\pi$
$127$	0.439529			0.0390019			0.0195010	+	0.999810i	$0.493792\pi$
$128$		0			0
$129$	−0.555109			−0.0488746
$130$		0			0
$131$	7.55087			0.659722			0.329861	−	0.944029i	$-0.392998\pi$
$131$	7.55087			0.659722			0.329861	+	0.944029i	$0.392998\pi$
$132$		0			0
$133$	1.17541			0.101921
$134$		0			0
$135$			0.286141i			0.0246271i
$136$		0			0
$137$		−	6.36284i		−	0.543614i	−0.962352	−	0.271807i	$-0.912379\pi$
$137$		−	6.36284i		−	0.543614i	0.962352	−	0.271807i	$-0.0876213\pi$
$138$		0			0
$139$	−14.7103			−1.24771			−0.623854	−	0.781541i	$-0.714434\pi$
$139$	−14.7103			−1.24771			−0.623854	+	0.781541i	$0.714434\pi$
$140$		0			0
$141$		−	23.4541i		−	1.97519i
$142$		0			0
$143$	2.10322	−	2.92856i	0.175880	−	0.244898i
$144$		0			0
$145$			11.9080i			0.988910i
$146$		0			0
$147$	2.42937			0.200371
$148$		0			0
$149$			10.5595i			0.865070i	0.901617	+	0.432535i	$0.142381\pi$
$149$			10.5595i			0.865070i	−0.901617	+	0.432535i	$0.857619\pi$
$150$		0			0
$151$		−	21.8771i		−	1.78034i	−0.455633	−	0.890168i	$-0.650587\pi$
$151$		−	21.8771i		−	1.78034i	0.455633	−	0.890168i	$-0.349413\pi$
$152$		0			0
$153$	−0.0593451			−0.00479777
$154$		0			0
$155$	−1.72064			−0.138205
$156$		0			0
$157$	−18.3275			−1.46270			−0.731348	−	0.682004i	$-0.761109\pi$
$157$	−18.3275			−1.46270			−0.731348	+	0.682004i	$0.761109\pi$
$158$		0			0
$159$	12.0311			0.954129
$160$		0			0
$161$		−	3.64958i		−	0.287627i
$162$		0			0
$163$			23.8972i			1.87177i	0.352306	+	0.935885i	$0.385398\pi$
$163$			23.8972i			1.87177i	−0.352306	+	0.935885i	$0.614602\pi$
$164$		0			0
$165$	2.91540			0.226964
$166$		0			0
$167$			17.5616i			1.35896i	0.733695	+	0.679479i	$0.237794\pi$
$167$			17.5616i			1.35896i	−0.733695	+	0.679479i	$0.762206\pi$
$168$		0			0
$169$	4.15291	+	12.3188i	0.319455	+	0.947602i
$170$		0			0
$171$		−	3.41086i		−	0.260835i
$172$		0			0
$173$	−14.7846			−1.12405			−0.562027	−	0.827119i	$-0.689978\pi$
$173$	−14.7846			−1.12405			−0.562027	+	0.827119i	$0.689978\pi$
$174$		0			0
$175$			3.55985i			0.269099i
$176$		0			0
$177$			17.7826i			1.33663i
$178$		0			0
$179$	−17.8297			−1.33266			−0.666329	−	0.745658i	$-0.732135\pi$
$179$	−17.8297			−1.33266			−0.666329	+	0.745658i	$0.732135\pi$
$180$		0			0
$181$	−15.9427			−1.18502			−0.592508	−	0.805565i	$-0.701862\pi$
$181$	−15.9427			−1.18502			−0.592508	+	0.805565i	$0.701862\pi$
$182$		0			0
$183$	−2.86559			−0.211831
$184$		0			0
$185$	−10.5195			−0.773406
$186$		0			0
$187$		−	0.0204508i		−	0.00149551i
$188$		0			0
$189$			0.238439i			0.0173439i
$190$		0			0
$191$	−4.70833			−0.340683			−0.170341	−	0.985385i	$-0.554487\pi$
$191$	−4.70833			−0.340683			−0.170341	+	0.985385i	$0.554487\pi$
$192$		0			0
$193$		−	11.1935i		−	0.805724i	−0.915261	−	0.402862i	$-0.868015\pi$
$193$		−	11.1935i		−	0.805724i	0.915261	−	0.402862i	$-0.131985\pi$
$194$		0			0
$195$	−6.13173	+	8.53792i	−0.439103	+	0.611413i
$196$		0			0
$197$			18.4886i			1.31726i	0.752468	+	0.658628i	$0.228863\pi$
$197$			18.4886i			1.31726i	−0.752468	+	0.658628i	$0.771137\pi$
$198$		0			0
$199$	−17.8774			−1.26729			−0.633647	−	0.773622i	$-0.718443\pi$
$199$	−17.8774			−1.26729			−0.633647	+	0.773622i	$0.718443\pi$
$200$		0			0
$201$		−	18.4197i		−	1.29923i
$202$		0			0
$203$			9.92286i			0.696448i
$204$		0			0
$205$	5.93886			0.414788
$206$		0			0
$207$	−10.5905			−0.736093
$208$		0			0
$209$	1.17541			0.0813047
$210$		0			0
$211$	−27.0494			−1.86216			−0.931079	−	0.364817i	$-0.881132\pi$
$211$	−27.0494			−1.86216			−0.931079	+	0.364817i	$0.881132\pi$
$212$		0			0
$213$		−	28.6372i		−	1.96219i
$214$		0			0
$215$			0.274213i			0.0187012i
$216$		0			0
$217$	−1.43379			−0.0973321
$218$		0			0
$219$			27.7696i			1.87649i
$220$		0			0
$221$	0.0598913	+	0.0430125i	0.00402872	+	0.00289333i
$222$		0			0
$223$		−	0.655938i		−	0.0439248i	−0.999759	−	0.0219624i	$-0.993009\pi$
$223$		−	0.655938i		−	0.0439248i	0.999759	−	0.0219624i	$-0.00699142\pi$
$224$		0			0
$225$	10.3302			0.688677
$226$		0			0
$227$			18.8927i			1.25395i	0.779040	+	0.626975i	$0.215707\pi$
$227$			18.8927i			1.25395i	−0.779040	+	0.626975i	$0.784293\pi$
$228$		0			0
$229$			0.353234i			0.0233423i	0.999932	+	0.0116712i	$0.00371513\pi$
$229$			0.353234i			0.0233423i	−0.999932	+	0.0116712i	$0.996285\pi$
$230$		0			0
$231$	2.42937			0.159841
$232$		0			0
$233$	26.9896			1.76815			0.884075	−	0.467345i	$-0.154789\pi$
$233$	26.9896			1.76815			0.884075	+	0.467345i	$0.154789\pi$
$234$		0			0
$235$	−11.5859			−0.755778
$236$		0			0
$237$	−24.7459			−1.60742
$238$		0			0
$239$			2.13323i			0.137987i	0.997617	+	0.0689936i	$0.0219788\pi$
$239$			2.13323i			0.137987i	−0.997617	+	0.0689936i	$0.978021\pi$
$240$		0			0
$241$			21.8407i			1.40688i	0.710753	+	0.703442i	$0.248354\pi$
$241$			21.8407i			1.40688i	−0.710753	+	0.703442i	$0.751646\pi$
$242$		0			0
$243$	21.8410			1.40110
$244$		0			0
$245$		−	1.20006i		−	0.0766692i
$246$		0			0
$247$	−2.47214	+	3.44225i	−0.157299	+	0.219025i
$248$		0			0
$249$			9.65189i			0.611663i
$250$		0			0
$251$	−23.0474			−1.45474			−0.727369	−	0.686247i	$-0.759257\pi$
$251$	−23.0474			−1.45474			−0.727369	+	0.686247i	$0.759257\pi$
$252$		0			0
$253$		−	3.64958i		−	0.229447i
$254$		0			0
$255$			0.0596221i			0.00373368i
$256$		0			0
$257$	−0.141938			−0.00885386			−0.00442693	−	0.999990i	$-0.501409\pi$
$257$	−0.141938			−0.00885386			−0.00442693	+	0.999990i	$0.501409\pi$
$258$		0			0
$259$	−8.76575			−0.544677
$260$		0			0
$261$	28.7947			1.78234
$262$		0			0
$263$	−19.8961			−1.22685			−0.613424	−	0.789754i	$-0.710208\pi$
$263$	−19.8961			−1.22685			−0.613424	+	0.789754i	$0.710208\pi$
$264$		0			0
$265$		−	5.94314i		−	0.365084i
$266$		0			0
$267$			0.761373i			0.0465953i
$268$		0			0
$269$	−2.33419			−0.142318			−0.0711590	−	0.997465i	$-0.522670\pi$
$269$	−2.33419			−0.142318			−0.0711590	+	0.997465i	$0.522670\pi$
$270$		0			0
$271$			25.7776i			1.56587i	0.622101	+	0.782937i	$0.286280\pi$
$271$			25.7776i			1.56587i	−0.622101	+	0.782937i	$0.713720\pi$
$272$		0			0
$273$	−5.10951	+	7.11456i	−0.309242	+	0.430593i
$274$		0			0
$275$			3.55985i			0.214667i
$276$		0			0
$277$	−11.0837			−0.665956			−0.332978	−	0.942935i	$-0.608054\pi$
$277$	−11.0837			−0.665956			−0.332978	+	0.942935i	$0.608054\pi$
$278$		0			0
$279$			4.16065i			0.249092i
$280$		0			0
$281$			11.6884i			0.697274i	0.937258	+	0.348637i	$0.113355\pi$
$281$			11.6884i			0.697274i	−0.937258	+	0.348637i	$0.886645\pi$
$282$		0			0
$283$	−26.0969			−1.55130			−0.775651	−	0.631162i	$-0.782578\pi$
$283$	−26.0969			−1.55130			−0.775651	+	0.631162i	$0.782578\pi$
$284$		0			0
$285$	−3.42678			−0.202985
$286$		0			0
$287$	4.94879			0.292118
$288$		0			0
$289$	−16.9996			−0.999975
$290$		0			0
$291$			16.6596i			0.976604i
$292$		0			0
$293$		−	25.1488i		−	1.46921i	−0.678495	−	0.734605i	$-0.737368\pi$
$293$		−	25.1488i		−	1.46921i	0.678495	−	0.734605i	$-0.262632\pi$
$294$		0			0
$295$	8.78428			0.511440
$296$		0			0
$297$			0.238439i			0.0138356i
$298$		0			0
$299$	10.6880	+	7.67588i	0.618103	+	0.443908i
$300$		0			0
$301$			0.228499i			0.0131704i
$302$		0			0
$303$	−48.1730			−2.76747
$304$		0			0
$305$			1.41555i			0.0810540i
$306$		0			0
$307$			10.2992i			0.587805i	0.955835	+	0.293903i	$0.0949542\pi$
$307$			10.2992i			0.587805i	−0.955835	+	0.293903i	$0.905046\pi$
$308$		0			0
$309$	−10.4161			−0.592552
$310$		0			0
$311$	18.9028			1.07188			0.535940	−	0.844256i	$-0.319957\pi$
$311$	18.9028			1.07188			0.535940	+	0.844256i	$0.319957\pi$
$312$		0			0
$313$	5.45742			0.308472			0.154236	−	0.988034i	$-0.450708\pi$
$313$	5.45742			0.308472			0.154236	+	0.988034i	$0.450708\pi$
$314$		0			0
$315$	−3.48240			−0.196211
$316$		0			0
$317$			20.5743i			1.15557i	0.816189	+	0.577785i	$0.196083\pi$
$317$			20.5743i			1.15557i	−0.816189	+	0.577785i	$0.803917\pi$
$318$		0			0
$319$			9.92286i			0.555573i
$320$		0			0
$321$	45.7714			2.55471
$322$		0			0
$323$			0.0240380i			0.00133751i
$324$		0			0
$325$	−10.4252	−	7.48716i	−0.578288	−	0.415313i
$326$		0			0
$327$		−	4.13348i		−	0.228582i
$328$		0			0
$329$	−9.65438			−0.532263
$330$		0			0
$331$			8.63204i			0.474460i	0.971454	+	0.237230i	$0.0762395\pi$
$331$			8.63204i			0.474460i	−0.971454	+	0.237230i	$0.923761\pi$
$332$		0			0
$333$			25.4369i			1.39393i
$334$		0			0
$335$	−9.09897			−0.497130
$336$		0			0
$337$	21.1565			1.15247			0.576234	−	0.817285i	$-0.304522\pi$
$337$	21.1565			1.15247			0.576234	+	0.817285i	$0.304522\pi$
$338$		0			0
$339$	−9.83691			−0.534268
$340$		0			0
$341$	−1.43379			−0.0776441
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			10.6400i			0.572837i
$346$		0			0
$347$	−4.02685			−0.216173			−0.108086	−	0.994142i	$-0.534472\pi$
$347$	−4.02685			−0.216173			−0.108086	+	0.994142i	$0.534472\pi$
$348$		0			0
$349$		−	29.2902i		−	1.56787i	−0.620844	−	0.783934i	$-0.713210\pi$
$349$		−	29.2902i		−	1.56787i	0.620844	−	0.783934i	$-0.286790\pi$
$350$		0			0
$351$	−0.698282	−	0.501490i	−0.0372715	−	0.0267675i
$352$		0			0
$353$		−	20.4545i		−	1.08868i	−0.838863	−	0.544342i	$-0.816779\pi$
$353$		−	20.4545i		−	1.08868i	0.838863	−	0.544342i	$-0.183221\pi$
$354$		0			0
$355$	−14.1462			−0.750802
$356$		0			0
$357$			0.0496825i			0.00262948i
$358$		0			0
$359$		−	27.0083i		−	1.42544i	−0.701447	−	0.712721i	$-0.747463\pi$
$359$		−	27.0083i		−	1.42544i	0.701447	−	0.712721i	$-0.252537\pi$
$360$		0			0
$361$	17.6184			0.927285
$362$		0			0
$363$	2.42937			0.127509
$364$		0			0
$365$	13.7176			0.718013
$366$		0			0
$367$	−3.62088			−0.189009			−0.0945043	−	0.995524i	$-0.530127\pi$
$367$	−3.62088			−0.189009			−0.0945043	+	0.995524i	$0.530127\pi$
$368$		0			0
$369$		−	14.3607i		−	0.747586i
$370$		0			0
$371$		−	4.95235i		−	0.257113i
$372$		0			0
$373$	10.9390			0.566400			0.283200	−	0.959061i	$-0.408604\pi$
$373$	10.9390			0.566400			0.283200	+	0.959061i	$0.408604\pi$
$374$		0			0
$375$		−	24.9554i		−	1.28869i
$376$		0			0
$377$	−29.0597	−	20.8700i	−1.49665	−	1.07486i
$378$		0			0
$379$			25.1470i			1.29172i	0.763458	+	0.645858i	$0.223500\pi$
$379$			25.1470i			1.29172i	−0.763458	+	0.645858i	$0.776500\pi$
$380$		0			0
$381$	−1.06778			−0.0547041
$382$		0			0
$383$			14.8713i			0.759890i	0.925009	+	0.379945i	$0.124057\pi$
$383$			14.8713i			0.759890i	−0.925009	+	0.379945i	$0.875943\pi$
$384$		0			0
$385$		−	1.20006i		−	0.0611609i
$386$		0			0
$387$	0.663069			0.0337057
$388$		0			0
$389$	17.0427			0.864099			0.432050	−	0.901850i	$-0.357791\pi$
$389$	17.0427			0.864099			0.432050	+	0.901850i	$0.357791\pi$
$390$		0			0
$391$	0.0746367			0.00377454
$392$		0			0
$393$	−18.3439			−0.925326
$394$		0			0
$395$			12.2240i			0.615055i
$396$		0			0
$397$			17.0954i			0.857992i	0.903306	+	0.428996i	$0.141133\pi$
$397$			17.0954i			0.857992i	−0.903306	+	0.428996i	$0.858867\pi$
$398$		0			0
$399$	−2.85550			−0.142954
$400$		0			0
$401$		−	6.97528i		−	0.348329i	−0.984717	−	0.174164i	$-0.944278\pi$
$401$		−	6.97528i		−	0.348329i	0.984717	−	0.174164i	$-0.0557224\pi$
$402$		0			0
$403$	3.01558	−	4.19894i	0.150217	−	0.209164i
$404$		0			0
$405$		−	11.1424i		−	0.553668i
$406$		0			0
$407$	−8.76575			−0.434502
$408$		0			0
$409$			14.0481i			0.694634i	0.937748	+	0.347317i	$0.112907\pi$
$409$			14.0481i			0.694634i	−0.937748	+	0.347317i	$0.887093\pi$
$410$		0			0
$411$			15.4577i			0.762473i
$412$		0			0
$413$	7.31985			0.360186
$414$		0			0
$415$	4.76784			0.234044
$416$		0			0
$417$	35.7367			1.75004
$418$		0			0
$419$	−28.2227			−1.37877			−0.689384	−	0.724396i	$-0.742119\pi$
$419$	−28.2227			−1.37877			−0.689384	+	0.724396i	$0.742119\pi$
$420$		0			0
$421$		−	12.6830i		−	0.618130i	−0.951041	−	0.309065i	$-0.899984\pi$
$421$		−	12.6830i		−	0.618130i	0.951041	−	0.309065i	$-0.100016\pi$
$422$		0			0
$423$			28.0156i			1.36216i
$424$		0			0
$425$	−0.0728017			−0.00353140
$426$		0			0
$427$			1.17956i			0.0570830i
$428$		0			0
$429$	−5.10951	+	7.11456i	−0.246690	+	0.343494i
$430$		0			0
$431$			30.5289i			1.47053i	0.677781	+	0.735264i	$0.262942\pi$
$431$			30.5289i			1.47053i	−0.677781	+	0.735264i	$0.737058\pi$
$432$		0			0
$433$	−24.7242			−1.18817			−0.594086	−	0.804402i	$-0.702486\pi$
$433$	−24.7242			−1.18817			−0.594086	+	0.804402i	$0.702486\pi$
$434$		0			0
$435$		−	28.9291i		−	1.38704i
$436$		0			0
$437$			4.28974i			0.205206i
$438$		0			0
$439$	8.63745			0.412243			0.206121	−	0.978526i	$-0.433916\pi$
$439$	8.63745			0.412243			0.206121	+	0.978526i	$0.433916\pi$
$440$		0			0
$441$	−2.90185			−0.138183
$442$		0			0
$443$	8.26077			0.392481			0.196240	−	0.980556i	$-0.437127\pi$
$443$	8.26077			0.392481			0.196240	+	0.980556i	$0.437127\pi$
$444$		0			0
$445$	0.376103			0.0178290
$446$		0			0
$447$		−	25.6530i		−	1.21335i
$448$		0			0
$449$		−	3.76176i		−	0.177528i	−0.996053	−	0.0887642i	$-0.971708\pi$
$449$		−	3.76176i		−	0.177528i	0.996053	−	0.0887642i	$-0.0282918\pi$
$450$		0			0
$451$	4.94879			0.233030
$452$		0			0
$453$			53.1477i			2.49710i
$454$		0			0
$455$	3.51445	+	2.52400i	0.164760	+	0.118327i
$456$		0			0
$457$		−	4.34915i		−	0.203444i	−0.994813	−	0.101722i	$-0.967565\pi$
$457$		−	4.34915i		−	0.203444i	0.994813	−	0.101722i	$-0.0324353\pi$
$458$		0			0
$459$	−0.00487625			−0.000227604
$460$		0			0
$461$		−	23.8585i		−	1.11120i	−0.831449	−	0.555600i	$-0.812488\pi$
$461$		−	23.8585i		−	1.11120i	0.831449	−	0.555600i	$-0.187512\pi$
$462$		0			0
$463$			7.97817i			0.370777i	0.982665	+	0.185389i	$0.0593544\pi$
$463$			7.97817i			0.370777i	−0.982665	+	0.185389i	$0.940646\pi$
$464$		0			0
$465$	4.18007			0.193846
$466$		0			0
$467$	16.0020			0.740484			0.370242	−	0.928935i	$-0.379275\pi$
$467$	16.0020			0.740484			0.370242	+	0.928935i	$0.379275\pi$
$468$		0			0
$469$	−7.58208			−0.350108
$470$		0			0
$471$	44.5244			2.05158
$472$		0			0
$473$			0.228499i			0.0105064i
$474$		0			0
$475$		−	4.18427i		−	0.191988i
$476$		0			0
$477$	−14.3710			−0.658003
$478$		0			0
$479$		−	20.9784i		−	0.958526i	−0.877671	−	0.479263i	$-0.840904\pi$
$479$		−	20.9784i		−	0.958526i	0.877671	−	0.479263i	$-0.159096\pi$
$480$		0			0
$481$	18.4363	−	25.6710i	0.840624	−	1.17050i
$482$		0			0
$483$			8.86619i			0.403426i
$484$		0			0
$485$	8.22952			0.373683
$486$		0			0
$487$			22.4028i			1.01517i	0.861602	+	0.507584i	$0.169461\pi$
$487$			22.4028i			1.01517i	−0.861602	+	0.507584i	$0.830539\pi$
$488$		0			0
$489$		−	58.0551i		−	2.62534i
$490$		0			0
$491$	7.27066			0.328120			0.164060	−	0.986450i	$-0.447541\pi$
$491$	7.27066			0.328120			0.164060	+	0.986450i	$0.447541\pi$
$492$		0			0
$493$	−0.202930			−0.00913951
$494$		0			0
$495$	−3.48240			−0.156522
$496$		0			0
$497$	−11.7879			−0.528759
$498$		0			0
$499$		−	2.48384i		−	0.111192i	−0.998453	−	0.0555959i	$-0.982294\pi$
$499$		−	2.48384i		−	0.111192i	0.998453	−	0.0555959i	$-0.0177059\pi$
$500$		0			0
$501$		−	42.6637i		−	1.90607i
$502$		0			0
$503$	−7.93949			−0.354004			−0.177002	−	0.984210i	$-0.556640\pi$
$503$	−7.93949			−0.354004			−0.177002	+	0.984210i	$0.556640\pi$
$504$		0			0
$505$			23.7965i			1.05893i
$506$		0			0
$507$	−10.0890	−	29.9270i	−0.448067	−	1.32911i
$508$		0			0
$509$		−	20.4678i		−	0.907219i	−0.891201	−	0.453610i	$-0.850136\pi$
$509$		−	20.4678i		−	0.907219i	0.891201	−	0.453610i	$-0.149864\pi$
$510$		0			0
$511$	11.4308			0.505667
$512$		0			0
$513$		−	0.280263i		−	0.0123739i
$514$		0			0
$515$			5.14536i			0.226732i
$516$		0			0
$517$	−9.65438			−0.424599
$518$		0			0
$519$	35.9173			1.57660
$520$		0			0
$521$	23.0856			1.01140			0.505700	−	0.862709i	$-0.331234\pi$
$521$	23.0856			1.01140			0.505700	+	0.862709i	$0.331234\pi$
$522$		0			0
$523$	8.89791			0.389078			0.194539	−	0.980895i	$-0.437679\pi$
$523$	8.89791			0.389078			0.194539	+	0.980895i	$0.437679\pi$
$524$		0			0
$525$		−	8.64820i		−	0.377438i
$526$		0			0
$527$		−	0.0293221i		−	0.00127729i
$528$		0			0
$529$	−9.68056			−0.420894
$530$		0			0
$531$		−	21.2411i		−	0.921786i
$532$		0			0
$533$	−10.4084	+	14.4928i	−0.450838	+	0.627754i
$534$		0			0
$535$		−	22.6102i		−	0.977522i
$536$		0			0
$537$	43.3151			1.86918
$538$		0			0
$539$		−	1.00000i		−	0.0430730i
$540$		0			0
$541$			30.7262i			1.32102i	0.750816	+	0.660511i	$0.229661\pi$
$541$			30.7262i			1.32102i	−0.750816	+	0.660511i	$0.770339\pi$
$542$		0			0
$543$	38.7309			1.66210
$544$		0			0
$545$	−2.04186			−0.0874637
$546$		0			0
$547$	−27.4057			−1.17178			−0.585892	−	0.810389i	$-0.699256\pi$
$547$	−27.4057			−1.17178			−0.585892	+	0.810389i	$0.699256\pi$
$548$		0			0
$549$	3.42291			0.146086
$550$		0			0
$551$		−	11.6634i		−	0.496878i
$552$		0			0
$553$			10.1861i			0.433158i
$554$		0			0
$555$	25.5557			1.08478
$556$		0			0
$557$			39.2683i			1.66385i	0.554889	+	0.831925i	$0.312761\pi$
$557$			39.2683i			1.66385i	−0.554889	+	0.831925i	$0.687239\pi$
$558$		0			0
$559$	−0.669172	−	0.480584i	−0.0283030	−	0.0203265i
$560$		0			0
$561$			0.0496825i			0.00209760i
$562$		0			0
$563$	−19.6003			−0.826055			−0.413028	−	0.910719i	$-0.635529\pi$
$563$	−19.6003			−0.826055			−0.413028	+	0.910719i	$0.635529\pi$
$564$		0			0
$565$			4.85924i			0.204430i
$566$		0			0
$567$		−	9.28481i		−	0.389925i
$568$		0			0
$569$	12.0368			0.504607			0.252304	−	0.967648i	$-0.418812\pi$
$569$	12.0368			0.504607			0.252304	+	0.967648i	$0.418812\pi$
$570$		0			0
$571$	−18.4858			−0.773607			−0.386804	−	0.922162i	$-0.626421\pi$
$571$	−18.4858			−0.773607			−0.386804	+	0.922162i	$0.626421\pi$
$572$		0			0
$573$	11.4383			0.477841
$574$		0			0
$575$	−12.9920			−0.541802
$576$		0			0
$577$			25.0364i			1.04228i	0.853472	+	0.521139i	$0.174493\pi$
$577$			25.0364i			1.04228i	−0.853472	+	0.521139i	$0.825507\pi$
$578$		0			0
$579$			27.1931i			1.13011i
$580$		0			0
$581$	3.97300			0.164828
$582$		0			0
$583$		−	4.95235i		−	0.205106i
$584$		0			0
$585$	7.32427	−	10.1984i	0.302821	−	0.421653i
$586$		0			0
$587$		−	13.9386i		−	0.575307i	−0.957735	−	0.287653i	$-0.907125\pi$
$587$		−	13.9386i		−	0.575307i	0.957735	−	0.287653i	$-0.0928751\pi$
$588$		0			0
$589$	1.68529			0.0694411
$590$		0			0
$591$		−	44.9156i		−	1.84758i
$592$		0			0
$593$			11.0669i			0.454464i	0.973841	+	0.227232i	$0.0729675\pi$
$593$			11.0669i			0.454464i	−0.973841	+	0.227232i	$0.927033\pi$
$594$		0			0
$595$	0.0245422			0.00100613
$596$		0			0
$597$	43.4308			1.77750
$598$		0			0
$599$	46.2752			1.89075			0.945376	−	0.325983i	$-0.105695\pi$
$599$	46.2752			1.89075			0.945376	+	0.325983i	$0.105695\pi$
$600$		0			0
$601$	−40.7946			−1.66404			−0.832022	−	0.554742i	$-0.812817\pi$
$601$	−40.7946			−1.66404			−0.832022	+	0.554742i	$0.812817\pi$
$602$		0			0
$603$			22.0021i			0.895994i
$604$		0			0
$605$		−	1.20006i		−	0.0487895i
$606$		0			0
$607$	14.2725			0.579303			0.289651	−	0.957132i	$-0.406461\pi$
$607$	14.2725			0.579303			0.289651	+	0.957132i	$0.406461\pi$
$608$		0			0
$609$		−	24.1063i		−	0.976837i
$610$		0			0
$611$	20.3053	−	28.2734i	0.821465	−	1.14382i
$612$		0			0
$613$			7.76321i			0.313553i	0.987634	+	0.156777i	$0.0501102\pi$
$613$			7.76321i			0.313553i	−0.987634	+	0.156777i	$0.949890\pi$
$614$		0			0
$615$	−14.4277			−0.581781
$616$		0			0
$617$		−	37.3161i		−	1.50229i	−0.660137	−	0.751145i	$-0.729502\pi$
$617$		−	37.3161i		−	1.50229i	0.660137	−	0.751145i	$-0.270498\pi$
$618$		0			0
$619$		−	32.6159i		−	1.31094i	−0.755219	−	0.655472i	$-0.772470\pi$
$619$		−	32.6159i		−	1.31094i	0.755219	−	0.655472i	$-0.227530\pi$
$620$		0			0
$621$	−0.870201			−0.0349200
$622$		0			0
$623$	0.313403			0.0125562
$624$		0			0
$625$	5.47178			0.218871
$626$		0			0
$627$	−2.85550			−0.114038
$628$		0			0
$629$		−	0.179266i		−	0.00714782i
$630$		0			0
$631$		−	38.7671i		−	1.54329i	−0.636051	−	0.771647i	$-0.719433\pi$
$631$		−	38.7671i		−	1.54329i	0.636051	−	0.771647i	$-0.280567\pi$
$632$		0			0
$633$	65.7131			2.61186
$634$		0			0
$635$			0.527463i			0.0209317i
$636$		0			0
$637$	2.92856	+	2.10322i	0.116034	+	0.0833327i
$638$		0			0
$639$			34.2067i			1.35320i
$640$		0			0
$641$	−40.0663			−1.58252			−0.791262	−	0.611477i	$-0.790576\pi$
$641$	−40.0663			−1.58252			−0.791262	+	0.611477i	$0.790576\pi$
$642$		0			0
$643$			31.6432i			1.24789i	0.781470	+	0.623943i	$0.214470\pi$
$643$			31.6432i			1.24789i	−0.781470	+	0.623943i	$0.785530\pi$
$644$		0			0
$645$		−	0.666165i		−	0.0262302i
$646$		0			0
$647$	6.28286			0.247005			0.123502	−	0.992344i	$-0.460587\pi$
$647$	6.28286			0.247005			0.123502	+	0.992344i	$0.460587\pi$
$648$		0			0
$649$	7.31985			0.287329
$650$		0			0
$651$	3.48321			0.136518
$652$		0			0
$653$	−44.5996			−1.74532			−0.872658	−	0.488332i	$-0.837606\pi$
$653$	−44.5996			−1.74532			−0.872658	+	0.488332i	$0.837606\pi$
$654$		0			0
$655$			9.06151i			0.354063i
$656$		0			0
$657$		−	33.1704i		−	1.29410i
$658$		0			0
$659$	30.0801			1.17175			0.585877	−	0.810400i	$-0.300750\pi$
$659$	30.0801			1.17175			0.585877	+	0.810400i	$0.300750\pi$
$660$		0			0
$661$			40.2771i			1.56660i	0.621645	+	0.783299i	$0.286465\pi$
$661$			40.2771i			1.56660i	−0.621645	+	0.783299i	$0.713535\pi$
$662$		0			0
$663$	−0.145498	−	0.104493i	−0.00565068	−	0.00405819i
$664$		0			0
$665$			1.41056i			0.0546993i
$666$		0			0
$667$	−36.2143			−1.40222
$668$		0			0
$669$			1.59352i			0.0616089i
$670$		0			0
$671$			1.17956i			0.0455364i
$672$		0			0
$673$	0.904823			0.0348784			0.0174392	−	0.999848i	$-0.494449\pi$
$673$	0.904823			0.0348784			0.0174392	+	0.999848i	$0.494449\pi$
$674$		0			0
$675$	0.848806			0.0326706
$676$		0			0
$677$	−5.29861			−0.203642			−0.101821	−	0.994803i	$-0.532467\pi$
$677$	−5.29861			−0.203642			−0.101821	+	0.994803i	$0.532467\pi$
$678$		0			0
$679$	6.85758			0.263170
$680$		0			0
$681$		−	45.8973i		−	1.75879i
$682$		0			0
$683$			2.70844i			0.103636i	0.998657	+	0.0518178i	$0.0165015\pi$
$683$			2.70844i			0.103636i	−0.998657	+	0.0518178i	$0.983498\pi$
$684$		0			0
$685$	7.63581			0.291749
$686$		0			0
$687$		−	0.858136i		−	0.0327399i
$688$		0			0
$689$	14.5033	+	10.4159i	0.552530	+	0.396814i
$690$		0			0
$691$			11.7377i			0.446522i	0.974759	+	0.223261i	$0.0716703\pi$
$691$			11.7377i			0.446522i	−0.974759	+	0.223261i	$0.928330\pi$
$692$		0			0
$693$	−2.90185			−0.110232
$694$		0			0
$695$		−	17.6532i		−	0.669626i
$696$		0			0
$697$			0.101207i			0.00383347i
$698$		0			0
$699$	−65.5679			−2.48001
$700$		0			0
$701$	−36.9015			−1.39375			−0.696875	−	0.717192i	$-0.745427\pi$
$701$	−36.9015			−1.39375			−0.696875	+	0.717192i	$0.745427\pi$
$702$		0			0
$703$	10.3033			0.388598
$704$		0			0
$705$	28.1464			1.06005
$706$		0			0
$707$			19.8294i			0.745761i
$708$		0			0
$709$			28.9545i			1.08741i	0.839276	+	0.543705i	$0.182979\pi$
$709$			28.9545i			1.08741i	−0.839276	+	0.543705i	$0.817021\pi$
$710$		0			0
$711$	29.5586			1.10853
$712$		0			0
$713$		−	5.23274i		−	0.195967i
$714$		0			0
$715$	3.51445	+	2.52400i	0.131433	+	0.0943922i
$716$		0			0
$717$		−	5.18241i		−	0.193541i
$718$		0			0
$719$	1.92935			0.0719525			0.0359763	−	0.999353i	$-0.488546\pi$
$719$	1.92935			0.0719525			0.0359763	+	0.999353i	$0.488546\pi$
$720$		0			0
$721$			4.28758i			0.159678i
$722$		0			0
$723$		−	53.0592i		−	1.97329i
$724$		0			0
$725$	35.3239			1.31190
$726$		0			0
$727$	−27.8010			−1.03108			−0.515542	−	0.856864i	$-0.672409\pi$
$727$	−27.8010			−1.03108			−0.515542	+	0.856864i	$0.672409\pi$
$728$		0			0
$729$	−25.2054			−0.933533
$730$		0			0
$731$	−0.00467297			−0.000172836
$732$		0			0
$733$			23.9835i			0.885850i	0.896559	+	0.442925i	$0.146059\pi$
$733$			23.9835i			0.885850i	−0.896559	+	0.442925i	$0.853941\pi$
$734$		0			0
$735$			2.91540i			0.107536i
$736$		0			0
$737$	−7.58208			−0.279290
$738$		0			0
$739$		−	15.6523i		−	0.575778i	−0.957664	−	0.287889i	$-0.907047\pi$
$739$		−	15.6523i		−	0.575778i	0.957664	−	0.287889i	$-0.0929534\pi$
$740$		0			0
$741$	6.00576	−	8.36251i	0.220627	−	0.307204i
$742$		0			0
$743$		−	14.6276i		−	0.536633i	−0.963331	−	0.268317i	$-0.913533\pi$
$743$		−	14.6276i		−	0.536633i	0.963331	−	0.268317i	$-0.0864673\pi$
$744$		0			0
$745$	−12.6721			−0.464270
$746$		0			0
$747$		−	11.5290i		−	0.421826i
$748$		0			0
$749$		−	18.8408i		−	0.688428i
$750$		0			0
$751$	−15.3746			−0.561025			−0.280513	−	0.959850i	$-0.590504\pi$
$751$	−15.3746			−0.561025			−0.280513	+	0.959850i	$0.590504\pi$
$752$		0			0
$753$	55.9907			2.04041
$754$		0			0
$755$	26.2539			0.955478
$756$		0			0
$757$	−5.10646			−0.185597			−0.0927987	−	0.995685i	$-0.529581\pi$
$757$	−5.10646			−0.185597			−0.0927987	+	0.995685i	$0.529581\pi$
$758$		0			0
$759$			8.86619i			0.321822i
$760$		0			0
$761$		−	14.9905i		−	0.543407i	−0.962381	−	0.271703i	$-0.912413\pi$
$761$		−	14.9905i		−	0.543407i	0.962381	−	0.271703i	$-0.0875870\pi$
$762$		0			0
$763$	−1.70146			−0.0615971
$764$		0			0
$765$		−	0.0712178i		−	0.00257489i
$766$		0			0
$767$	−15.3953	+	21.4366i	−0.555891	+	0.774031i
$768$		0			0
$769$		−	6.75031i		−	0.243422i	−0.992566	−	0.121711i	$-0.961162\pi$
$769$		−	6.75031i		−	0.243422i	0.992566	−	0.121711i	$-0.0388382\pi$
$770$		0			0
$771$	0.344821			0.0124184
$772$		0			0
$773$		−	9.18675i		−	0.330424i	−0.986258	−	0.165212i	$-0.947169\pi$
$773$		−	9.18675i		−	0.330424i	0.986258	−	0.165212i	$-0.0528309\pi$
$774$		0			0
$775$			5.10408i			0.183344i
$776$		0			0
$777$	21.2953			0.763964
$778$		0			0
$779$	−5.81685			−0.208410
$780$		0			0
$781$	−11.7879			−0.421804
$782$		0			0
$783$	2.36599			0.0845537
$784$		0			0
$785$		−	21.9942i		−	0.785006i
$786$		0			0
$787$		−	34.1320i		−	1.21668i	−0.793678	−	0.608338i	$-0.791837\pi$
$787$		−	34.1320i		−	1.21668i	0.793678	−	0.608338i	$-0.208163\pi$
$788$		0			0
$789$	48.3351			1.72078
$790$		0			0
$791$			4.04916i			0.143971i
$792$		0			0
$793$	−3.45441	−	2.48088i	−0.122670	−	0.0880986i
$794$		0			0
$795$			14.4381i			0.512066i
$796$		0			0
$797$	47.4518			1.68083			0.840414	−	0.541945i	$-0.182312\pi$
$797$	47.4518			1.68083			0.840414	+	0.541945i	$0.182312\pi$
$798$		0			0
$799$		−	0.197439i		−	0.00698491i
$800$		0			0
$801$		−	0.909450i		−	0.0321338i
$802$		0			0
$803$	11.4308			0.403383
$804$		0			0
$805$	4.37972			0.154365
$806$		0			0
$807$	5.67062			0.199615
$808$		0			0
$809$	12.2481			0.430621			0.215311	−	0.976546i	$-0.430924\pi$
$809$	12.2481			0.430621			0.215311	+	0.976546i	$0.430924\pi$
$810$		0			0
$811$		−	3.79609i		−	0.133299i	−0.997776	−	0.0666493i	$-0.978769\pi$
$811$		−	3.79609i		−	0.133299i	0.997776	−	0.0666493i	$-0.0212309\pi$
$812$		0			0
$813$		−	62.6233i		−	2.19629i
$814$		0			0
$815$	−28.6781			−1.00455
$816$		0			0
$817$		−	0.268579i		−	0.00939640i
$818$		0			0
$819$	6.10324	−	8.49824i	0.213264	−	0.296953i
$820$		0			0
$821$		−	10.0041i		−	0.349144i	−0.984644	−	0.174572i	$-0.944146\pi$
$821$		−	10.0041i		−	0.349144i	0.984644	−	0.174572i	$-0.0558542\pi$
$822$		0			0
$823$	24.7089			0.861298			0.430649	−	0.902519i	$-0.358285\pi$
$823$	24.7089			0.861298			0.430649	+	0.902519i	$0.358285\pi$
$824$		0			0
$825$		−	8.64820i		−	0.301092i
$826$		0			0
$827$			48.1175i			1.67321i	0.547808	+	0.836604i	$0.315462\pi$
$827$			48.1175i			1.67321i	−0.547808	+	0.836604i	$0.684538\pi$
$828$		0			0
$829$	−16.9133			−0.587422			−0.293711	−	0.955894i	$-0.594890\pi$
$829$	−16.9133			−0.587422			−0.293711	+	0.955894i	$0.594890\pi$
$830$		0			0
$831$	26.9265			0.934070
$832$		0			0
$833$	0.0204508			0.000708577
$834$		0			0
$835$	−21.0750			−0.729331
$836$		0			0
$837$			0.341871i			0.0118168i
$838$		0			0
$839$		−	2.80904i		−	0.0969788i	−0.998824	−	0.0484894i	$-0.984559\pi$
$839$		−	2.80904i		−	0.0969788i	0.998824	−	0.0484894i	$-0.0154407\pi$
$840$		0			0
$841$	69.4631			2.39528
$842$		0			0
$843$		−	28.3956i		−	0.977996i
$844$		0			0
$845$	−14.7834	+	4.98375i	−0.508563	+	0.171446i
$846$		0			0
$847$		−	1.00000i		−	0.0343604i
$848$		0			0
$849$	63.3992			2.17586
$850$		0			0
$851$		−	31.9913i		−	1.09665i
$852$		0			0
$853$		−	35.9929i		−	1.23237i	−0.787600	−	0.616187i	$-0.788677\pi$
$853$		−	35.9929i		−	1.23237i	0.787600	−	0.616187i	$-0.211323\pi$
$854$		0			0
$855$	4.09324			0.139986
$856$		0			0
$857$	24.9474			0.852188			0.426094	−	0.904679i	$-0.359889\pi$
$857$	24.9474			0.852188			0.426094	+	0.904679i	$0.359889\pi$
$858$		0			0
$859$	−3.56992			−0.121804			−0.0609020	−	0.998144i	$-0.519398\pi$
$859$	−3.56992			−0.121804			−0.0609020	+	0.998144i	$0.519398\pi$
$860$		0			0
$861$	−12.0225			−0.409724
$862$		0			0
$863$			29.8374i			1.01568i	0.861452	+	0.507839i	$0.169555\pi$
$863$			29.8374i			1.01568i	−0.861452	+	0.507839i	$0.830445\pi$
$864$		0			0
$865$		−	17.7425i		−	0.603262i
$866$		0			0
$867$	41.2983			1.40256
$868$		0			0
$869$			10.1861i			0.345540i
$870$		0			0
$871$	15.9468	−	22.2046i	0.540337	−	0.752373i
$872$		0			0
$873$		−	19.8997i		−	0.673502i
$874$		0			0
$875$	−10.2724			−0.347269
$876$		0			0
$877$			1.26076i			0.0425729i	0.999773	+	0.0212865i	$0.00677620\pi$
$877$			1.26076i			0.0425729i	−0.999773	+	0.0212865i	$0.993224\pi$
$878$		0			0
$879$			61.0958i			2.06071i
$880$		0			0
$881$	40.2707			1.35676			0.678378	−	0.734713i	$-0.262683\pi$
$881$	40.2707			1.35676			0.678378	+	0.734713i	$0.262683\pi$
$882$		0			0
$883$	13.0093			0.437797			0.218898	−	0.975748i	$-0.429754\pi$
$883$	13.0093			0.437797			0.218898	+	0.975748i	$0.429754\pi$
$884$		0			0
$885$	−21.3403			−0.717346
$886$		0			0
$887$	−23.1362			−0.776836			−0.388418	−	0.921483i	$-0.626978\pi$
$887$	−23.1362			−0.776836			−0.388418	+	0.921483i	$0.626978\pi$
$888$		0			0
$889$			0.439529i			0.0147413i
$890$		0			0
$891$		−	9.28481i		−	0.311053i
$892$		0			0
$893$	11.3478			0.379741
$894$		0			0
$895$		−	21.3968i		−	0.715217i
$896$		0			0
$897$	−25.9652	−	18.6476i	−0.866951	−	0.622624i
$898$		0			0
$899$			14.2273i			0.474507i
$900$		0			0
$901$	0.101279			0.00337411
$902$		0			0
$903$		−	0.555109i		−	0.0184729i
$904$		0			0
$905$		−	19.1323i		−	0.635979i
$906$		0			0
$907$	−34.0951			−1.13211			−0.566054	−	0.824368i	$-0.691531\pi$
$907$	−34.0951			−1.13211			−0.566054	+	0.824368i	$0.691531\pi$
$908$		0			0
$909$	57.5420			1.90855
$910$		0			0
$911$	54.8436			1.81705			0.908524	−	0.417832i	$-0.137210\pi$
$911$	54.8436			1.81705			0.908524	+	0.417832i	$0.137210\pi$
$912$		0			0
$913$	3.97300			0.131487
$914$		0			0
$915$		−	3.43889i		−	0.113686i
$916$		0			0
$917$			7.55087i			0.249352i
$918$		0			0
$919$	−24.4928			−0.807943			−0.403971	−	0.914772i	$-0.632370\pi$
$919$	−24.4928			−0.807943			−0.403971	+	0.914772i	$0.632370\pi$
$920$		0			0
$921$		−	25.0206i		−	0.824455i
$922$		0			0
$923$	24.7925	−	34.5215i	0.816057	−	1.13629i
$924$		0			0
$925$			31.2048i			1.02601i
$926$		0			0
$927$	12.4419			0.408646
$928$		0			0
$929$			47.0325i			1.54309i	0.636177	+	0.771543i	$0.280515\pi$
$929$			47.0325i			1.54309i	−0.636177	+	0.771543i	$0.719485\pi$
$930$		0			0
$931$			1.17541i			0.0385224i
$932$		0			0
$933$	−45.9220			−1.50342
$934$		0			0
$935$	0.0245422			0.000802616
$936$		0			0
$937$	−52.3828			−1.71127			−0.855636	−	0.517579i	$-0.826833\pi$
$937$	−52.3828			−1.71127			−0.855636	+	0.517579i	$0.826833\pi$
$938$		0			0
$939$	−13.2581			−0.432662
$940$		0			0
$941$		−	48.0370i		−	1.56596i	−0.622047	−	0.782980i	$-0.713699\pi$
$941$		−	48.0370i		−	1.56596i	0.622047	−	0.782980i	$-0.286301\pi$
$942$		0			0
$943$			18.0610i			0.588148i
$944$		0			0
$945$	−0.286141			−0.00930818
$946$		0			0
$947$		−	56.2351i		−	1.82740i	−0.406395	−	0.913698i	$-0.633214\pi$
$947$		−	56.2351i		−	1.82740i	0.406395	−	0.913698i	$-0.366786\pi$
$948$		0			0
$949$	−24.0414	+	33.4756i	−0.780418	+	1.08666i
$950$		0			0
$951$		−	49.9827i		−	1.62080i
$952$		0			0
$953$	−6.67473			−0.216216			−0.108108	−	0.994139i	$-0.534479\pi$
$953$	−6.67473			−0.216216			−0.108108	+	0.994139i	$0.534479\pi$
$954$		0			0
$955$		−	5.65029i		−	0.182839i
$956$		0			0
$957$		−	24.1063i		−	0.779247i
$958$		0			0
$959$	6.36284			0.205467
$960$		0			0
$961$	28.9442			0.933685
$962$		0			0
$963$	−54.6732			−1.76182
$964$		0			0
$965$	13.4329			0.432419
$966$		0			0
$967$		−	49.6483i		−	1.59658i	−0.602271	−	0.798291i	$-0.705737\pi$
$967$		−	49.6483i		−	1.59658i	0.602271	−	0.798291i	$-0.294263\pi$
$968$		0			0
$969$		−	0.0583972i		−	0.00187599i
$970$		0			0
$971$	44.1872			1.41803			0.709017	−	0.705191i	$-0.249139\pi$
$971$	44.1872			1.41803			0.709017	+	0.705191i	$0.249139\pi$
$972$		0			0
$973$		−	14.7103i		−	0.471590i
$974$		0			0
$975$	25.3268	+	18.1891i	0.811106	+	0.582517i
$976$		0			0
$977$		−	34.8299i		−	1.11431i	−0.830409	−	0.557154i	$-0.811893\pi$
$977$		−	34.8299i		−	1.11431i	0.830409	−	0.557154i	$-0.188107\pi$
$978$		0			0
$979$	0.313403			0.0100164
$980$		0			0
$981$			4.93739i			0.157639i
$982$		0			0
$983$			25.4023i			0.810208i	0.914271	+	0.405104i	$0.132765\pi$
$983$			25.4023i			0.810208i	−0.914271	+	0.405104i	$0.867235\pi$
$984$		0			0
$985$	−22.1875			−0.706951
$986$		0			0
$987$	23.4541			0.746552
$988$		0			0
$989$	−0.833924			−0.0265173
$990$		0			0
$991$	49.5574			1.57424			0.787122	−	0.616798i	$-0.211570\pi$
$991$	49.5574			1.57424			0.787122	+	0.616798i	$0.211570\pi$
$992$		0			0
$993$		−	20.9704i		−	0.665477i
$994$		0			0
$995$		−	21.4540i		−	0.680137i
$996$		0			0
$997$	11.8761			0.376119			0.188060	−	0.982158i	$-0.439780\pi$
$997$	11.8761			0.376119			0.188060	+	0.982158i	$0.439780\pi$
$998$		0			0
$999$			2.09009i			0.0661277i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.b.2157.5	✓	30
13.12	even	2	inner	4004.2.m.b.2157.6	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.b.2157.5	✓	30	1.1	even	1	trivial
4004.2.m.b.2157.6	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)

\(f(q)\)	\(=\)	\(q-2.42937 q^{3} +1.20006i q^{5} +1.00000i q^{7} +2.90185 q^{9} +O(q^{10})\)	\(q-2.42937 q^{3} +1.20006i q^{5} +1.00000i q^{7} +2.90185 q^{9} +1.00000i q^{11} +(-2.92856 - 2.10322i) q^{13} -2.91540i q^{15} -0.0204508 q^{17} -1.17541i q^{19} -2.42937i q^{21} -3.64958 q^{23} +3.55985 q^{25} +0.238439 q^{27} +9.92286 q^{29} +1.43379i q^{31} -2.42937i q^{33} -1.20006 q^{35} +8.76575i q^{37} +(7.11456 + 5.10951i) q^{39} -4.94879i q^{41} +0.228499 q^{43} +3.48240i q^{45} +9.65438i q^{47} -1.00000 q^{49} +0.0496825 q^{51} -4.95235 q^{53} -1.20006 q^{55} +2.85550i q^{57} -7.31985i q^{59} +1.17956 q^{61} +2.90185i q^{63} +(2.52400 - 3.51445i) q^{65} +7.58208i q^{67} +8.86619 q^{69} +11.7879i q^{71} -11.4308i q^{73} -8.64820 q^{75} -1.00000 q^{77} +10.1861 q^{79} -9.28481 q^{81} -3.97300i q^{83} -0.0245422i q^{85} -24.1063 q^{87} -0.313403i q^{89} +(2.10322 - 2.92856i) q^{91} -3.48321i q^{93} +1.41056 q^{95} -6.85758i q^{97} +2.90185i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 18 q^{9}+O(q^{10}) \) 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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