LMFDB - Embedded newform 4004.2.e.a.3849.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(3849,4004)]

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, 1, 1, 0]))

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

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

chi := DirichletCharacter("4004.3849");

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.e (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:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3849.20
Character	$\chi$	$=$	4004.3849
Dual form			4004.2.e.a.3849.29

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.867297i q^{3} -0.687824i q^{5} +(2.28648 + 1.33116i) q^{7} +2.24780 q^{9} +O(q^{10})$	$q-0.867297i q^{3} -0.687824i q^{5} +(2.28648 + 1.33116i) q^{7} +2.24780 q^{9} +(3.31660 - 0.0124497i) q^{11} +1.00000 q^{13} -0.596548 q^{15} -3.72481 q^{17} -1.66492 q^{19} +(1.15451 - 1.98306i) q^{21} +4.80954 q^{23} +4.52690 q^{25} -4.55140i q^{27} -10.1321i q^{29} +10.4660i q^{31} +(-0.0107976 - 2.87648i) q^{33} +(0.915607 - 1.57270i) q^{35} -2.15048 q^{37} -0.867297i q^{39} +5.48239 q^{41} -0.528364i q^{43} -1.54609i q^{45} +1.89404i q^{47} +(3.45601 + 6.08736i) q^{49} +3.23052i q^{51} -4.92027 q^{53} +(-0.00856318 - 2.28124i) q^{55} +1.44398i q^{57} -3.88893i q^{59} -2.19664 q^{61} +(5.13955 + 2.99218i) q^{63} -0.687824i q^{65} +12.5466 q^{67} -4.17130i q^{69} -2.15709 q^{71} +1.25014 q^{73} -3.92617i q^{75} +(7.59993 + 4.38647i) q^{77} +6.76812i q^{79} +2.79597 q^{81} -5.89934 q^{83} +2.56202i q^{85} -8.78755 q^{87} +12.7231i q^{89} +(2.28648 + 1.33116i) q^{91} +9.07712 q^{93} +1.14517i q^{95} +7.61633i q^{97} +(7.45504 - 0.0279843i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 4 q^{7} - 48 q^{9}+O(q^{10}) $ 48 * q - 4 * q^7 - 48 * q^9	$ 48 q - 4 q^{7} - 48 q^{9} + 2 q^{11} + 48 q^{13} + 8 q^{15} + 4 q^{17} + 10 q^{21} + 4 q^{23} - 44 q^{25} + 10 q^{33} - 14 q^{35} - 16 q^{37} + 10 q^{49} - 8 q^{53} - 12 q^{55} + 16 q^{61} + 16 q^{63} + 4 q^{67} - 16 q^{73} + 2 q^{77} + 64 q^{81} - 4 q^{83} - 12 q^{87} - 4 q^{91} + 36 q^{93} - 40 q^{99}+O(q^{100}) $ 48 * q - 4 * q^7 - 48 * q^9 + 2 * q^11 + 48 * q^13 + 8 * q^15 + 4 * q^17 + 10 * q^21 + 4 * q^23 - 44 * q^25 + 10 * q^33 - 14 * q^35 - 16 * q^37 + 10 * q^49 - 8 * q^53 - 12 * q^55 + 16 * q^61 + 16 * q^63 + 4 * q^67 - 16 * q^73 + 2 * q^77 + 64 * q^81 - 4 * q^83 - 12 * q^87 - 4 * q^91 + 36 * q^93 - 40 * q^99

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
$3$		−	0.867297i		−	0.500734i	−0.968151	−	0.250367i	$-0.919449\pi$
$3$		−	0.867297i		−	0.500734i	0.968151	−	0.250367i	$-0.0805513\pi$
$4$		0			0
$5$		−	0.687824i		−	0.307604i	−0.988102	−	0.153802i	$-0.950848\pi$
$5$		−	0.687824i		−	0.307604i	0.988102	−	0.153802i	$-0.0491519\pi$
$6$		0			0
$7$	2.28648	+	1.33116i	0.864209	+	0.503132i
$7$	2.28648	+	1.33116i	0.864209	+	0.503132i
$8$		0			0
$9$	2.24780			0.749265
$10$		0			0
$11$	3.31660	−	0.0124497i	0.999993	−	0.00375371i
$11$	3.31660	−	0.0124497i	0.999993	−	0.00375371i
$12$		0			0
$13$	1.00000			0.277350
$13$	1.00000			0.277350
$14$		0			0
$15$	−0.596548			−0.154028
$16$		0			0
$17$	−3.72481			−0.903400			−0.451700	−	0.892170i	$-0.649182\pi$
$17$	−3.72481			−0.903400			−0.451700	+	0.892170i	$0.649182\pi$
$18$		0			0
$19$	−1.66492			−0.381959			−0.190979	−	0.981594i	$-0.561166\pi$
$19$	−1.66492			−0.381959			−0.190979	+	0.981594i	$0.561166\pi$
$20$		0			0
$21$	1.15451	−	1.98306i	0.251936	−	0.432739i
$22$		0			0
$23$	4.80954			1.00286			0.501429	−	0.865199i	$-0.332808\pi$
$23$	4.80954			1.00286			0.501429	+	0.865199i	$0.332808\pi$
$24$		0			0
$25$	4.52690			0.905379
$26$		0			0
$27$		−	4.55140i		−	0.875917i
$28$		0			0
$29$		−	10.1321i		−	1.88148i	−0.339122	−	0.940742i	$-0.610130\pi$
$29$		−	10.1321i		−	1.88148i	0.339122	−	0.940742i	$-0.389870\pi$
$30$		0			0
$31$			10.4660i			1.87975i	0.341524	+	0.939873i	$0.389057\pi$
$31$			10.4660i			1.87975i	−0.341524	+	0.939873i	$0.610943\pi$
$32$		0			0
$33$	−0.0107976	−	2.87648i	−0.00187961	−	0.500731i
$34$		0			0
$35$	0.915607	−	1.57270i	0.154766	−	0.265835i
$36$		0			0
$37$	−2.15048			−0.353536			−0.176768	−	0.984253i	$-0.556564\pi$
$37$	−2.15048			−0.353536			−0.176768	+	0.984253i	$0.556564\pi$
$38$		0			0
$39$		−	0.867297i		−	0.138879i
$40$		0			0
$41$	5.48239			0.856206			0.428103	−	0.903730i	$-0.359182\pi$
$41$	5.48239			0.856206			0.428103	+	0.903730i	$0.359182\pi$
$42$		0			0
$43$		−	0.528364i		−	0.0805747i	−0.999188	−	0.0402873i	$-0.987173\pi$
$43$		−	0.528364i		−	0.0805747i	0.999188	−	0.0402873i	$-0.0128273\pi$
$44$		0			0
$45$		−	1.54609i		−	0.230477i
$46$		0			0
$47$			1.89404i			0.276275i	0.990413	+	0.138137i	$0.0441116\pi$
$47$			1.89404i			0.276275i	−0.990413	+	0.138137i	$0.955888\pi$
$48$		0			0
$49$	3.45601	+	6.08736i	0.493716	+	0.869623i
$50$		0			0
$51$			3.23052i			0.452363i
$52$		0			0
$53$	−4.92027			−0.675851			−0.337926	−	0.941173i	$-0.609725\pi$
$53$	−4.92027			−0.675851			−0.337926	+	0.941173i	$0.609725\pi$
$54$		0			0
$55$	−0.00856318	−	2.28124i	−0.00115466	−	0.307602i
$56$		0			0
$57$			1.44398i			0.191260i
$58$		0			0
$59$		−	3.88893i		−	0.506295i	−0.967428	−	0.253148i	$-0.918534\pi$
$59$		−	3.88893i		−	0.506295i	0.967428	−	0.253148i	$-0.0814658\pi$
$60$		0			0
$61$	−2.19664			−0.281250			−0.140625	−	0.990063i	$-0.544911\pi$
$61$	−2.19664			−0.281250			−0.140625	+	0.990063i	$0.544911\pi$
$62$		0			0
$63$	5.13955	+	2.99218i	0.647522	+	0.376980i
$64$		0			0
$65$		−	0.687824i		−	0.0853141i
$66$		0			0
$67$	12.5466			1.53281			0.766404	−	0.642358i	$-0.222044\pi$
$67$	12.5466			1.53281			0.766404	+	0.642358i	$0.222044\pi$
$68$		0			0
$69$		−	4.17130i		−	0.502165i
$70$		0			0
$71$	−2.15709			−0.256000			−0.128000	−	0.991774i	$-0.540856\pi$
$71$	−2.15709			−0.256000			−0.128000	+	0.991774i	$0.540856\pi$
$72$		0			0
$73$	1.25014			0.146318			0.0731590	−	0.997320i	$-0.476692\pi$
$73$	1.25014			0.146318			0.0731590	+	0.997320i	$0.476692\pi$
$74$		0			0
$75$		−	3.92617i		−	0.453355i
$76$		0			0
$77$	7.59993	+	4.38647i	0.866092	+	0.499885i
$78$		0			0
$79$			6.76812i			0.761473i	0.924684	+	0.380736i	$0.124329\pi$
$79$			6.76812i			0.761473i	−0.924684	+	0.380736i	$0.875671\pi$
$80$		0			0
$81$	2.79597			0.310664
$82$		0			0
$83$	−5.89934			−0.647536			−0.323768	−	0.946136i	$-0.604950\pi$
$83$	−5.89934			−0.647536			−0.323768	+	0.946136i	$0.604950\pi$
$84$		0			0
$85$			2.56202i			0.277890i
$86$		0			0
$87$	−8.78755			−0.942124
$88$		0			0
$89$			12.7231i			1.34864i	0.738438	+	0.674322i	$0.235564\pi$
$89$			12.7231i			1.34864i	−0.738438	+	0.674322i	$0.764436\pi$
$90$		0			0
$91$	2.28648	+	1.33116i	0.239689	+	0.139544i
$92$		0			0
$93$	9.07712			0.941253
$94$		0			0
$95$			1.14517i			0.117492i
$96$		0			0
$97$			7.61633i			0.773321i	0.922222	+	0.386660i	$0.126371\pi$
$97$			7.61633i			0.773321i	−0.922222	+	0.386660i	$0.873629\pi$
$98$		0			0
$99$	7.45504	−	0.0279843i	0.749260	−	0.00281253i
$100$		0			0
$101$	−8.72311			−0.867981			−0.433991	−	0.900917i	$-0.642895\pi$
$101$	−8.72311			−0.867981			−0.433991	+	0.900917i	$0.642895\pi$
$102$		0			0
$103$		−	12.3421i		−	1.21610i	−0.793898	−	0.608051i	$-0.791952\pi$
$103$		−	12.3421i		−	1.21610i	0.793898	−	0.608051i	$-0.208048\pi$
$104$		0			0
$105$	−1.36400	−	0.794103i	−0.133113	−	0.0774965i
$106$		0			0
$107$		−	4.86121i		−	0.469951i	−0.972001	−	0.234976i	$-0.924499\pi$
$107$		−	4.86121i		−	0.469951i	0.972001	−	0.234976i	$-0.0755010\pi$
$108$		0			0
$109$			5.89584i			0.564719i	0.959309	+	0.282360i	$0.0911171\pi$
$109$			5.89584i			0.564719i	−0.959309	+	0.282360i	$0.908883\pi$
$110$		0			0
$111$			1.86510i			0.177028i
$112$		0			0
$113$	−1.13448			−0.106723			−0.0533616	−	0.998575i	$-0.516994\pi$
$113$	−1.13448			−0.106723			−0.0533616	+	0.998575i	$0.516994\pi$
$114$		0			0
$115$		−	3.30812i		−	0.308484i
$116$		0			0
$117$	2.24780			0.207809
$118$		0			0
$119$	−8.51673	−	4.95834i	−0.780727	−	0.454530i
$120$		0			0
$121$	10.9997	−	0.0825811i	0.999972	−	0.00750737i
$122$		0			0
$123$		−	4.75486i		−	0.428732i
$124$		0			0
$125$		−	6.55283i		−	0.586103i
$126$		0			0
$127$			16.3534i			1.45113i	0.688155	+	0.725564i	$0.258421\pi$
$127$			16.3534i			1.45113i	−0.688155	+	0.725564i	$0.741579\pi$
$128$		0			0
$129$	−0.458248			−0.0403465
$130$		0			0
$131$	−8.96193			−0.783007			−0.391504	−	0.920177i	$-0.628045\pi$
$131$	−8.96193			−0.783007			−0.391504	+	0.920177i	$0.628045\pi$
$132$		0			0
$133$	−3.80681	−	2.21628i	−0.330092	−	0.192176i
$134$		0			0
$135$	−3.13056			−0.269436
$136$		0			0
$137$	−1.50038			−0.128186			−0.0640932	−	0.997944i	$-0.520415\pi$
$137$	−1.50038			−0.128186			−0.0640932	+	0.997944i	$0.520415\pi$
$138$		0			0
$139$	−12.1566			−1.03111			−0.515553	−	0.856858i	$-0.672413\pi$
$139$	−12.1566			−1.03111			−0.515553	+	0.856858i	$0.672413\pi$
$140$		0			0
$141$	1.64270			0.138340
$142$		0			0
$143$	3.31660	−	0.0124497i	0.277348	−	0.00104109i
$144$		0			0
$145$	−6.96911			−0.578753
$146$		0			0
$147$	5.27955	−	2.99739i	0.435450	−	0.247220i
$148$		0			0
$149$		−	6.37383i		−	0.522164i	−0.965317	−	0.261082i	$-0.915921\pi$
$149$		−	6.37383i		−	0.522164i	0.965317	−	0.261082i	$-0.0840793\pi$
$150$		0			0
$151$		−	12.4148i		−	1.01030i	−0.863030	−	0.505152i	$-0.831436\pi$
$151$		−	12.4148i		−	1.01030i	0.863030	−	0.505152i	$-0.168564\pi$
$152$		0			0
$153$	−8.37262			−0.676886
$154$		0			0
$155$	7.19876			0.578218
$156$		0			0
$157$		−	17.1615i		−	1.36963i	−0.728716	−	0.684817i	$-0.759882\pi$
$157$		−	17.1615i		−	1.36963i	0.728716	−	0.684817i	$-0.240118\pi$
$158$		0			0
$159$			4.26734i			0.338422i
$160$		0			0
$161$	10.9969	+	6.40228i	0.866680	+	0.504570i
$162$		0			0
$163$	2.17421			0.170297			0.0851485	−	0.996368i	$-0.472864\pi$
$163$	2.17421			0.170297			0.0851485	+	0.996368i	$0.472864\pi$
$164$		0			0
$165$	−1.97851	+	0.00742682i	−0.154027	+	0.000578177i
$166$		0			0
$167$	14.0925			1.09051			0.545255	−	0.838270i	$-0.316433\pi$
$167$	14.0925			1.09051			0.545255	+	0.838270i	$0.316433\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	−3.74240			−0.286189
$172$		0			0
$173$	17.7795			1.35175			0.675873	−	0.737018i	$-0.263767\pi$
$173$	17.7795			1.35175			0.675873	+	0.737018i	$0.263767\pi$
$174$		0			0
$175$	10.3507	+	6.02604i	0.782437	+	0.455526i
$176$		0			0
$177$	−3.37286			−0.253519
$178$		0			0
$179$	11.2411			0.840198			0.420099	−	0.907478i	$-0.361995\pi$
$179$	11.2411			0.840198			0.420099	+	0.907478i	$0.361995\pi$
$180$		0			0
$181$			2.90128i			0.215651i	0.994170	+	0.107825i	$0.0343887\pi$
$181$			2.90128i			0.215651i	−0.994170	+	0.107825i	$0.965611\pi$
$182$		0			0
$183$			1.90514i			0.140832i
$184$		0			0
$185$			1.47915i			0.108749i
$186$		0			0
$187$	−12.3537	+	0.0463727i	−0.903394	+	0.00339110i
$188$		0			0
$189$	6.05865	−	10.4067i	0.440702	−	0.756976i
$190$		0			0
$191$	18.9588			1.37181			0.685904	−	0.727692i	$-0.259407\pi$
$191$	18.9588			1.37181			0.685904	+	0.727692i	$0.259407\pi$
$192$		0			0
$193$		−	3.69611i		−	0.266052i	−0.991113	−	0.133026i	$-0.957531\pi$
$193$		−	3.69611i		−	0.266052i	0.991113	−	0.133026i	$-0.0424694\pi$
$194$		0			0
$195$	−0.596548			−0.0427197
$196$		0			0
$197$			12.2263i			0.871089i	0.900167	+	0.435544i	$0.143444\pi$
$197$			12.2263i			0.871089i	−0.900167	+	0.435544i	$0.856556\pi$
$198$		0			0
$199$		−	17.8064i		−	1.26226i	−0.775677	−	0.631130i	$-0.782591\pi$
$199$		−	17.8064i		−	1.26226i	0.775677	−	0.631130i	$-0.217409\pi$
$200$		0			0
$201$		−	10.8816i		−	0.767530i
$202$		0			0
$203$	13.4875	−	23.1669i	0.946636	−	1.62600i
$204$		0			0
$205$		−	3.77092i		−	0.263373i
$206$		0			0
$207$	10.8109			0.751407
$208$		0			0
$209$	−5.52188	+	0.0207277i	−0.381956	+	0.00143376i
$210$		0			0
$211$			12.4849i			0.859493i	0.902950	+	0.429746i	$0.141397\pi$
$211$			12.4849i			0.859493i	−0.902950	+	0.429746i	$0.858603\pi$
$212$		0			0
$213$			1.87084i			0.128188i
$214$		0			0
$215$	−0.363421			−0.0247851
$216$		0			0
$217$	−13.9319	+	23.9303i	−0.945761	+	1.62449i
$218$		0			0
$219$		−	1.08424i		−	0.0732665i
$220$		0			0
$221$	−3.72481			−0.250558
$222$		0			0
$223$		−	13.5734i		−	0.908944i	−0.890761	−	0.454472i	$-0.849828\pi$
$223$		−	13.5734i		−	0.908944i	0.890761	−	0.454472i	$-0.150172\pi$
$224$		0			0
$225$	10.1755			0.678369
$226$		0			0
$227$	13.5091			0.896630			0.448315	−	0.893876i	$-0.352024\pi$
$227$	13.5091			0.896630			0.448315	+	0.893876i	$0.352024\pi$
$228$		0			0
$229$			0.245258i			0.0162071i	0.999967	+	0.00810355i	$0.00257947\pi$
$229$			0.245258i			0.0162071i	−0.999967	+	0.00810355i	$0.997421\pi$
$230$		0			0
$231$	3.80437	−	6.59139i	0.250309	−	0.433682i
$232$		0			0
$233$		−	25.4970i		−	1.67036i	−0.549974	−	0.835182i	$-0.685362\pi$
$233$		−	25.4970i		−	1.67036i	0.549974	−	0.835182i	$-0.314638\pi$
$234$		0			0
$235$	1.30277			0.0849834
$236$		0			0
$237$	5.86997			0.381295
$238$		0			0
$239$		−	20.7513i		−	1.34229i	−0.741326	−	0.671145i	$-0.765803\pi$
$239$		−	20.7513i		−	1.34229i	0.741326	−	0.671145i	$-0.234197\pi$
$240$		0			0
$241$	8.16618			0.526030			0.263015	−	0.964792i	$-0.415283\pi$
$241$	8.16618			0.526030			0.263015	+	0.964792i	$0.415283\pi$
$242$		0			0
$243$		−	16.0791i		−	1.03148i
$244$		0			0
$245$	4.18704	−	2.37713i	0.267500	−	0.151869i
$246$		0			0
$247$	−1.66492			−0.105936
$248$		0			0
$249$			5.11648i			0.324244i
$250$		0			0
$251$		−	2.59574i		−	0.163842i	−0.996639	−	0.0819210i	$-0.973894\pi$
$251$		−	2.59574i		−	0.163842i	0.996639	−	0.0819210i	$-0.0261055\pi$
$252$		0			0
$253$	15.9513	−	0.0598771i	1.00285	−	0.00376444i
$254$		0			0
$255$	2.22203			0.139149
$256$		0			0
$257$		−	10.3457i		−	0.645346i	−0.946510	−	0.322673i	$-0.895419\pi$
$257$		−	10.3457i		−	0.645346i	0.946510	−	0.322673i	$-0.104581\pi$
$258$		0			0
$259$	−4.91703	−	2.86263i	−0.305529	−	0.177875i
$260$		0			0
$261$		−	22.7749i		−	1.40973i
$262$		0			0
$263$			5.34126i			0.329356i	0.986347	+	0.164678i	$0.0526585\pi$
$263$			5.34126i			0.329356i	−0.986347	+	0.164678i	$0.947342\pi$
$264$		0			0
$265$			3.38428i			0.207895i
$266$		0			0
$267$	11.0347			0.675312
$268$		0			0
$269$			4.40061i			0.268310i	0.990960	+	0.134155i	$0.0428320\pi$
$269$			4.40061i			0.268310i	−0.990960	+	0.134155i	$0.957168\pi$
$270$		0			0
$271$	−4.00768			−0.243449			−0.121725	−	0.992564i	$-0.538842\pi$
$271$	−4.00768			−0.243449			−0.121725	+	0.992564i	$0.538842\pi$
$272$		0			0
$273$	1.15451	−	1.98306i	0.0698744	−	0.120020i
$274$		0			0
$275$	15.0139	−	0.0563583i	0.905373	−	0.00339853i
$276$		0			0
$277$		−	21.5491i		−	1.29476i	−0.762167	−	0.647381i	$-0.775864\pi$
$277$		−	21.5491i		−	1.29476i	0.762167	−	0.647381i	$-0.224136\pi$
$278$		0			0
$279$			23.5254i			1.40843i
$280$		0			0
$281$			32.1095i			1.91549i	0.287615	+	0.957746i	$0.407138\pi$
$281$			32.1095i			1.91549i	−0.287615	+	0.957746i	$0.592862\pi$
$282$		0			0
$283$	27.9144			1.65934			0.829669	−	0.558256i	$-0.188529\pi$
$283$	27.9144			1.65934			0.829669	+	0.558256i	$0.188529\pi$
$284$		0			0
$285$	0.993205			0.0588324
$286$		0			0
$287$	12.5354	+	7.29796i	0.739941	+	0.430785i
$288$		0			0
$289$	−3.12576			−0.183868
$290$		0			0
$291$	6.60562			0.387228
$292$		0			0
$293$	−18.6567			−1.08994			−0.544968	−	0.838457i	$-0.683458\pi$
$293$	−18.6567			−1.08994			−0.544968	+	0.838457i	$0.683458\pi$
$294$		0			0
$295$	−2.67490			−0.155739
$296$		0			0
$297$	−0.0566633	−	15.0952i	−0.00328794	−	0.875911i
$298$		0			0
$299$	4.80954			0.278143
$300$		0			0
$301$	0.703338	−	1.20809i	0.0405397	−	0.0696334i
$302$		0			0
$303$			7.56552i			0.434628i
$304$		0			0
$305$			1.51090i			0.0865139i
$306$		0			0
$307$	−14.6280			−0.834861			−0.417431	−	0.908709i	$-0.637069\pi$
$307$	−14.6280			−0.834861			−0.417431	+	0.908709i	$0.637069\pi$
$308$		0			0
$309$	−10.7043			−0.608944
$310$		0			0
$311$		−	18.8288i		−	1.06768i	−0.845585	−	0.533841i	$-0.820748\pi$
$311$		−	18.8288i		−	1.06768i	0.845585	−	0.533841i	$-0.179252\pi$
$312$		0			0
$313$		−	3.87296i		−	0.218913i	−0.993992	−	0.109456i	$-0.965089\pi$
$313$		−	3.87296i		−	0.218913i	0.993992	−	0.109456i	$-0.0349110\pi$
$314$		0			0
$315$	2.05810	−	3.53511i	0.115961	−	0.199181i
$316$		0			0
$317$	−29.0796			−1.63327			−0.816636	−	0.577153i	$-0.804163\pi$
$317$	−29.0796			−1.63327			−0.816636	+	0.577153i	$0.804163\pi$
$318$		0			0
$319$	−0.126141	−	33.6042i	−0.00706255	−	1.88147i
$320$		0			0
$321$	−4.21611			−0.235321
$322$		0			0
$323$	6.20152			0.345062
$324$		0			0
$325$	4.52690			0.251107
$326$		0			0
$327$	5.11345			0.282774
$328$		0			0
$329$	−2.52128	+	4.33070i	−0.139003	+	0.238759i
$330$		0			0
$331$	−23.3178			−1.28166			−0.640831	−	0.767682i	$-0.721410\pi$
$331$	−23.3178			−1.28166			−0.640831	+	0.767682i	$0.721410\pi$
$332$		0			0
$333$	−4.83383			−0.264892
$334$		0			0
$335$		−	8.62984i		−	0.471499i
$336$		0			0
$337$		−	32.0560i		−	1.74620i	−0.487539	−	0.873101i	$-0.662105\pi$
$337$		−	32.0560i		−	1.74620i	0.487539	−	0.873101i	$-0.337895\pi$
$338$		0			0
$339$			0.983934i			0.0534400i
$340$		0			0
$341$	0.130298	+	34.7115i	0.00705603	+	1.87973i
$342$		0			0
$343$	−0.201164	+	18.5192i	−0.0108619	+	0.999941i
$344$		0			0
$345$	−2.86912			−0.154468
$346$		0			0
$347$			22.1062i			1.18672i	0.804936	+	0.593362i	$0.202200\pi$
$347$			22.1062i			1.18672i	−0.804936	+	0.593362i	$0.797800\pi$
$348$		0			0
$349$	−29.6862			−1.58907			−0.794533	−	0.607221i	$-0.792284\pi$
$349$	−29.6862			−1.58907			−0.794533	+	0.607221i	$0.792284\pi$
$350$		0			0
$351$		−	4.55140i		−	0.242936i
$352$		0			0
$353$			5.11068i			0.272014i	0.990708	+	0.136007i	$0.0434270\pi$
$353$			5.11068i			0.272014i	−0.990708	+	0.136007i	$0.956573\pi$
$354$		0			0
$355$			1.48370i			0.0787466i
$356$		0			0
$357$	−4.30035	+	7.38653i	−0.227599	+	0.390937i
$358$		0			0
$359$			8.66148i			0.457136i	0.973528	+	0.228568i	$0.0734043\pi$
$359$			8.66148i			0.457136i	−0.973528	+	0.228568i	$0.926596\pi$
$360$		0			0
$361$	−16.2280			−0.854107
$362$		0			0
$363$	−0.0716223	−	9.54000i	−0.00375920	−	0.500720i
$364$		0			0
$365$		−	0.859878i		−	0.0450081i
$366$		0			0
$367$			30.1280i			1.57267i	0.617800	+	0.786336i	$0.288024\pi$
$367$			30.1280i			1.57267i	−0.617800	+	0.786336i	$0.711976\pi$
$368$		0			0
$369$	12.3233			0.641525
$370$		0			0
$371$	−11.2501	−	6.54968i	−0.584077	−	0.340043i
$372$		0			0
$373$			20.8059i			1.07729i	0.842533	+	0.538645i	$0.181064\pi$
$373$			20.8059i			1.07729i	−0.842533	+	0.538645i	$0.818936\pi$
$374$		0			0
$375$	−5.68325			−0.293482
$376$		0			0
$377$		−	10.1321i		−	0.521830i
$378$		0			0
$379$	−31.5757			−1.62194			−0.810968	−	0.585091i	$-0.801059\pi$
$379$	−31.5757			−1.62194			−0.810968	+	0.585091i	$0.801059\pi$
$380$		0			0
$381$	14.1832			0.726630
$382$		0			0
$383$		−	24.2352i		−	1.23836i	−0.785249	−	0.619180i	$-0.787465\pi$
$383$		−	24.2352i		−	1.23836i	0.785249	−	0.619180i	$-0.212535\pi$
$384$		0			0
$385$	3.01712	−	5.22742i	0.153767	−	0.266414i
$386$		0			0
$387$		−	1.18765i		−	0.0603718i
$388$		0			0
$389$	−10.0870			−0.511431			−0.255715	−	0.966752i	$-0.582311\pi$
$389$	−10.0870			−0.511431			−0.255715	+	0.966752i	$0.582311\pi$
$390$		0			0
$391$	−17.9146			−0.905982
$392$		0			0
$393$			7.77265i			0.392078i
$394$		0			0
$395$	4.65528			0.234232
$396$		0			0
$397$			16.9418i			0.850283i	0.905127	+	0.425141i	$0.139776\pi$
$397$			16.9418i			0.850283i	−0.905127	+	0.425141i	$0.860224\pi$
$398$		0			0
$399$	−1.92217	+	3.30164i	−0.0962290	+	0.165289i
$400$		0			0
$401$	2.19426			0.109576			0.0547881	−	0.998498i	$-0.482552\pi$
$401$	2.19426			0.109576			0.0547881	+	0.998498i	$0.482552\pi$
$402$		0			0
$403$			10.4660i			0.521348i
$404$		0			0
$405$		−	1.92314i		−	0.0955615i
$406$		0			0
$407$	−7.13227	+	0.0267727i	−0.353534	+	0.00132707i
$408$		0			0
$409$	11.1558			0.551621			0.275811	−	0.961212i	$-0.411054\pi$
$409$	11.1558			0.551621			0.275811	+	0.961212i	$0.411054\pi$
$410$		0			0
$411$			1.30128i			0.0641873i
$412$		0			0
$413$	5.17680	−	8.89197i	0.254733	−	0.437545i
$414$		0			0
$415$			4.05771i			0.199185i
$416$		0			0
$417$			10.5433i			0.516310i
$418$		0			0
$419$		−	16.3555i		−	0.799018i	−0.916729	−	0.399509i	$-0.869181\pi$
$419$		−	16.3555i		−	0.799018i	0.916729	−	0.399509i	$-0.130819\pi$
$420$		0			0
$421$	−0.952221			−0.0464084			−0.0232042	−	0.999731i	$-0.507387\pi$
$421$	−0.952221			−0.0464084			−0.0232042	+	0.999731i	$0.507387\pi$
$422$		0			0
$423$			4.25743i			0.207003i
$424$		0			0
$425$	−16.8619			−0.817920
$426$		0			0
$427$	−5.02257	−	2.92408i	−0.243059	−	0.141506i
$428$		0			0
$429$	−0.0107976	−	2.87648i	−0.000521311	−	0.138878i
$430$		0			0
$431$		−	33.3355i		−	1.60572i	−0.596170	−	0.802858i	$-0.703312\pi$
$431$		−	33.3355i		−	1.60572i	0.596170	−	0.802858i	$-0.296688\pi$
$432$		0			0
$433$			14.2655i			0.685559i	0.939416	+	0.342779i	$0.111368\pi$
$433$			14.2655i			0.685559i	−0.939416	+	0.342779i	$0.888632\pi$
$434$		0			0
$435$			6.04429i			0.289802i
$436$		0			0
$437$	−8.00750			−0.383051
$438$		0			0
$439$	16.6455			0.794449			0.397224	−	0.917722i	$-0.369973\pi$
$439$	16.6455			0.794449			0.397224	+	0.917722i	$0.369973\pi$
$440$		0			0
$441$	7.76840	+	13.6831i	0.369924	+	0.651579i
$442$		0			0
$443$	35.2343			1.67403			0.837015	−	0.547180i	$-0.184299\pi$
$443$	35.2343			1.67403			0.837015	+	0.547180i	$0.184299\pi$
$444$		0			0
$445$	8.75124			0.414849
$446$		0			0
$447$	−5.52800			−0.261466
$448$		0			0
$449$	13.2644			0.625988			0.312994	−	0.949755i	$-0.398668\pi$
$449$	13.2644			0.625988			0.312994	+	0.949755i	$0.398668\pi$
$450$		0			0
$451$	18.1829	−	0.0682539i	0.856200	−	0.00321395i
$452$		0			0
$453$	−10.7673			−0.505894
$454$		0			0
$455$	0.915607	−	1.57270i	0.0429243	−	0.0737293i
$456$		0			0
$457$			13.8767i			0.649123i	0.945865	+	0.324561i	$0.105217\pi$
$457$			13.8767i			0.649123i	−0.945865	+	0.324561i	$0.894783\pi$
$458$		0			0
$459$			16.9531i			0.791304i
$460$		0			0
$461$	12.6982			0.591413			0.295707	−	0.955279i	$-0.404445\pi$
$461$	12.6982			0.591413			0.295707	+	0.955279i	$0.404445\pi$
$462$		0			0
$463$	19.7784			0.919180			0.459590	−	0.888131i	$-0.347996\pi$
$463$	19.7784			0.919180			0.459590	+	0.888131i	$0.347996\pi$
$464$		0			0
$465$		−	6.24346i		−	0.289534i
$466$		0			0
$467$		−	21.8795i		−	1.01246i	−0.862398	−	0.506230i	$-0.831039\pi$
$467$		−	21.8795i		−	1.01246i	0.862398	−	0.506230i	$-0.168961\pi$
$468$		0			0
$469$	28.6875	+	16.7015i	1.32467	+	0.771206i
$470$		0			0
$471$	−14.8841			−0.685822
$472$		0			0
$473$	−0.00657795	−	1.75237i	−0.000302454	−	0.0805741i
$474$		0			0
$475$	−7.53692			−0.345818
$476$		0			0
$477$	−11.0598			−0.506392
$478$		0			0
$479$	7.35839			0.336214			0.168107	−	0.985769i	$-0.446235\pi$
$479$	7.35839			0.336214			0.168107	+	0.985769i	$0.446235\pi$
$480$		0			0
$481$	−2.15048			−0.0980533
$482$		0			0
$483$	5.55268	−	9.53761i	0.252656	−	0.433976i
$484$		0			0
$485$	5.23870			0.237877
$486$		0			0
$487$	3.42469			0.155188			0.0775939	−	0.996985i	$-0.475276\pi$
$487$	3.42469			0.155188			0.0775939	+	0.996985i	$0.475276\pi$
$488$		0			0
$489$		−	1.88568i		−	0.0852735i
$490$		0			0
$491$			11.5623i			0.521798i	0.965366	+	0.260899i	$0.0840190\pi$
$491$			11.5623i			0.521798i	−0.965366	+	0.260899i	$0.915981\pi$
$492$		0			0
$493$			37.7402i			1.69973i
$494$		0			0
$495$	−0.0192483	−	5.12776i	−0.000865146	−	0.230476i
$496$		0			0
$497$	−4.93215	−	2.87144i	−0.221237	−	0.128802i
$498$		0			0
$499$	6.37229			0.285263			0.142632	−	0.989776i	$-0.454444\pi$
$499$	6.37229			0.285263			0.142632	+	0.989776i	$0.454444\pi$
$500$		0			0
$501$		−	12.2224i		−	0.546056i
$502$		0			0
$503$	−36.4777			−1.62646			−0.813230	−	0.581942i	$-0.802293\pi$
$503$	−36.4777			−1.62646			−0.813230	+	0.581942i	$0.802293\pi$
$504$		0			0
$505$			5.99997i			0.266995i
$506$		0			0
$507$		−	0.867297i		−	0.0385180i
$508$		0			0
$509$			21.7871i			0.965698i	0.875704	+	0.482849i	$0.160398\pi$
$509$			21.7871i			0.965698i	−0.875704	+	0.482849i	$0.839602\pi$
$510$		0			0
$511$	2.85843	+	1.66414i	0.126449	+	0.0736173i
$512$		0			0
$513$			7.57772i			0.334564i
$514$		0			0
$515$	−8.48918			−0.374078
$516$		0			0
$517$	0.0235802	+	6.28179i	0.00103706	+	0.276273i
$518$		0			0
$519$		−	15.4201i		−	0.676866i
$520$		0			0
$521$			14.7033i			0.644162i	0.946712	+	0.322081i	$0.104382\pi$
$521$			14.7033i			0.644162i	−0.946712	+	0.322081i	$0.895618\pi$
$522$		0			0
$523$	−6.36849			−0.278475			−0.139237	−	0.990259i	$-0.544465\pi$
$523$	−6.36849			−0.278475			−0.139237	+	0.990259i	$0.544465\pi$
$524$		0			0
$525$	5.22637	−	8.97711i	0.228097	−	0.391793i
$526$		0			0
$527$		−	38.9838i		−	1.69816i
$528$		0			0
$529$	0.131668			0.00572469
$530$		0			0
$531$		−	8.74151i		−	0.379349i
$532$		0			0
$533$	5.48239			0.237469
$534$		0			0
$535$	−3.34366			−0.144559
$536$		0			0
$537$		−	9.74936i		−	0.420716i
$538$		0			0
$539$	11.5380	+	20.1463i	0.496977	+	0.867764i
$540$		0			0
$541$			18.5052i			0.795600i	0.917472	+	0.397800i	$0.130226\pi$
$541$			18.5052i			0.795600i	−0.917472	+	0.397800i	$0.869774\pi$
$542$		0			0
$543$	2.51627			0.107984
$544$		0			0
$545$	4.05530			0.173710
$546$		0			0
$547$		−	38.0692i		−	1.62772i	−0.581061	−	0.813860i	$-0.697362\pi$
$547$		−	38.0692i		−	1.62772i	0.581061	−	0.813860i	$-0.302638\pi$
$548$		0			0
$549$	−4.93759			−0.210731
$550$		0			0
$551$			16.8691i			0.718650i
$552$		0			0
$553$	−9.00947	+	15.4752i	−0.383121	+	0.658072i
$554$		0			0
$555$	1.28286			0.0544545
$556$		0			0
$557$			15.7873i			0.668930i	0.942408	+	0.334465i	$0.108556\pi$
$557$			15.7873i			0.668930i	−0.942408	+	0.334465i	$0.891444\pi$
$558$		0			0
$559$		−	0.528364i		−	0.0223474i
$560$		0			0
$561$	0.0402189	+	10.7144i	0.00169804	+	0.452360i
$562$		0			0
$563$	25.5413			1.07644			0.538219	−	0.842805i	$-0.319097\pi$
$563$	25.5413			1.07644			0.538219	+	0.842805i	$0.319097\pi$
$564$		0			0
$565$			0.780325i			0.0328285i
$566$		0			0
$567$	6.39294	+	3.72189i	0.268478	+	0.156305i
$568$		0			0
$569$			30.4182i			1.27520i	0.770369	+	0.637598i	$0.220072\pi$
$569$			30.4182i			1.27520i	−0.770369	+	0.637598i	$0.779928\pi$
$570$		0			0
$571$			25.0130i			1.04676i	0.852099	+	0.523381i	$0.175329\pi$
$571$			25.0130i			1.04676i	−0.852099	+	0.523381i	$0.824671\pi$
$572$		0			0
$573$		−	16.4429i		−	0.686912i
$574$		0			0
$575$	21.7723			0.907967
$576$		0			0
$577$		−	1.29518i		−	0.0539192i	−0.999637	−	0.0269596i	$-0.991417\pi$
$577$		−	1.29518i		−	0.0539192i	0.999637	−	0.0269596i	$-0.00858255\pi$
$578$		0			0
$579$	−3.20563			−0.133221
$580$		0			0
$581$	−13.4887	−	7.85298i	−0.559607	−	0.325796i
$582$		0			0
$583$	−16.3186	+	0.0612557i	−0.675846	+	0.00253695i
$584$		0			0
$585$		−	1.54609i		−	0.0639229i
$586$		0			0
$587$		−	28.4291i		−	1.17339i	−0.809807	−	0.586697i	$-0.800428\pi$
$587$		−	28.4291i		−	1.17339i	0.809807	−	0.586697i	$-0.199572\pi$
$588$		0			0
$589$		−	17.4250i		−	0.717986i
$590$		0			0
$591$	10.6038			0.436184
$592$		0			0
$593$	−38.0468			−1.56239			−0.781197	−	0.624285i	$-0.785390\pi$
$593$	−38.0468			−1.56239			−0.781197	+	0.624285i	$0.785390\pi$
$594$		0			0
$595$	−3.41046	+	5.85801i	−0.139815	+	0.240155i
$596$		0			0
$597$	−15.4434			−0.632057
$598$		0			0
$599$	−6.28452			−0.256779			−0.128389	−	0.991724i	$-0.540981\pi$
$599$	−6.28452			−0.256779			−0.128389	+	0.991724i	$0.540981\pi$
$600$		0			0
$601$	−38.1133			−1.55467			−0.777337	−	0.629085i	$-0.783430\pi$
$601$	−38.1133			−1.55467			−0.777337	+	0.629085i	$0.783430\pi$
$602$		0			0
$603$	28.2021			1.14848
$604$		0			0
$605$	−0.0568013	−	7.56586i	−0.00230930	−	0.307596i
$606$		0			0
$607$	4.75573			0.193029			0.0965145	−	0.995332i	$-0.469231\pi$
$607$	4.75573			0.193029			0.0965145	+	0.995332i	$0.469231\pi$
$608$		0			0
$609$	−20.0926	−	11.6977i	−0.814192	−	0.474013i
$610$		0			0
$611$			1.89404i			0.0766248i
$612$		0			0
$613$			11.2122i			0.452856i	0.974028	+	0.226428i	$0.0727049\pi$
$613$			11.2122i			0.452856i	−0.974028	+	0.226428i	$0.927295\pi$
$614$		0			0
$615$	−3.27051			−0.131880
$616$		0			0
$617$	−0.195012			−0.00785090			−0.00392545	−	0.999992i	$-0.501250\pi$
$617$	−0.195012			−0.00785090			−0.00392545	+	0.999992i	$0.501250\pi$
$618$		0			0
$619$		−	27.6605i		−	1.11177i	−0.831259	−	0.555886i	$-0.812379\pi$
$619$		−	27.6605i		−	1.11177i	0.831259	−	0.555886i	$-0.187621\pi$
$620$		0			0
$621$		−	21.8901i		−	0.878421i
$622$		0			0
$623$	−16.9365	+	29.0911i	−0.678546	+	1.16551i
$624$		0			0
$625$	18.1273			0.725092
$626$		0			0
$627$	0.0179771	+	4.78911i	0.000717935	+	0.191259i
$628$		0			0
$629$	8.01013			0.319385
$630$		0			0
$631$	−31.1938			−1.24181			−0.620903	−	0.783888i	$-0.713234\pi$
$631$	−31.1938			−1.24181			−0.620903	+	0.783888i	$0.713234\pi$
$632$		0			0
$633$	10.8281			0.430377
$634$		0			0
$635$	11.2483			0.446373
$636$		0			0
$637$	3.45601	+	6.08736i	0.136932	+	0.241190i
$638$		0			0
$639$	−4.84870			−0.191812
$640$		0			0
$641$	−25.1815			−0.994608			−0.497304	−	0.867576i	$-0.665677\pi$
$641$	−25.1815			−0.994608			−0.497304	+	0.867576i	$0.665677\pi$
$642$		0			0
$643$			31.7988i			1.25402i	0.779010	+	0.627011i	$0.215722\pi$
$643$			31.7988i			1.25402i	−0.779010	+	0.627011i	$0.784278\pi$
$644$		0			0
$645$			0.315194i			0.0124108i
$646$		0			0
$647$			20.6424i			0.811538i	0.913976	+	0.405769i	$0.132996\pi$
$647$			20.6424i			0.811538i	−0.913976	+	0.405769i	$0.867004\pi$
$648$		0			0
$649$	−0.0484158	−	12.8980i	−0.00190049	−	0.506292i
$650$		0			0
$651$	20.7547	+	12.0831i	0.813440	+	0.473575i
$652$		0			0
$653$	−29.8589			−1.16847			−0.584234	−	0.811585i	$-0.698605\pi$
$653$	−29.8589			−1.16847			−0.584234	+	0.811585i	$0.698605\pi$
$654$		0			0
$655$			6.16423i			0.240856i
$656$		0			0
$657$	2.81006			0.109631
$658$		0			0
$659$		−	6.54673i		−	0.255024i	−0.991837	−	0.127512i	$-0.959301\pi$
$659$		−	6.54673i		−	0.255024i	0.991837	−	0.127512i	$-0.0406992\pi$
$660$		0			0
$661$			33.0418i			1.28518i	0.766212	+	0.642588i	$0.222139\pi$
$661$			33.0418i			1.28518i	−0.766212	+	0.642588i	$0.777861\pi$
$662$		0			0
$663$			3.23052i			0.125463i
$664$		0			0
$665$	−1.52441	+	2.61842i	−0.0591142	+	0.101538i
$666$		0			0
$667$		−	48.7308i		−	1.88686i
$668$		0			0
$669$	−11.7722			−0.455139
$670$		0			0
$671$	−7.28537	+	0.0273474i	−0.281248	+	0.00105573i
$672$		0			0
$673$			39.2900i			1.51452i	0.653115	+	0.757259i	$0.273462\pi$
$673$			39.2900i			1.51452i	−0.653115	+	0.757259i	$0.726538\pi$
$674$		0			0
$675$		−	20.6037i		−	0.793037i
$676$		0			0
$677$	−40.0376			−1.53877			−0.769385	−	0.638785i	$-0.779437\pi$
$677$	−40.0376			−1.53877			−0.769385	+	0.638785i	$0.779437\pi$
$678$		0			0
$679$	−10.1386	+	17.4146i	−0.389083	+	0.668311i
$680$		0			0
$681$		−	11.7164i		−	0.448973i
$682$		0			0
$683$	43.4869			1.66398			0.831990	−	0.554790i	$-0.187201\pi$
$683$	43.4869			1.66398			0.831990	+	0.554790i	$0.187201\pi$
$684$		0			0
$685$			1.03200i			0.0394307i
$686$		0			0
$687$	0.212712			0.00811545
$688$		0			0
$689$	−4.92027			−0.187447
$690$		0			0
$691$			35.2610i			1.34139i	0.741732	+	0.670697i	$0.234005\pi$
$691$			35.2610i			1.34139i	−0.741732	+	0.670697i	$0.765995\pi$
$692$		0			0
$693$	17.0831	+	9.85989i	0.648933	+	0.374546i
$694$		0			0
$695$			8.36157i			0.317173i
$696$		0			0
$697$	−20.4209			−0.773496
$698$		0			0
$699$	−22.1135			−0.836408
$700$		0			0
$701$			21.7410i			0.821146i	0.911828	+	0.410573i	$0.134671\pi$
$701$			21.7410i			0.821146i	−0.911828	+	0.410573i	$0.865329\pi$
$702$		0			0
$703$	3.58037			0.135036
$704$		0			0
$705$		−	1.12989i		−	0.0425541i
$706$		0			0
$707$	−19.9452	−	11.6119i	−0.750118	−	0.436710i
$708$		0			0
$709$	−35.4964			−1.33310			−0.666548	−	0.745462i	$-0.732229\pi$
$709$	−35.4964			−1.33310			−0.666548	+	0.745462i	$0.732229\pi$
$710$		0			0
$711$			15.2133i			0.570545i
$712$		0			0
$713$			50.3366i			1.88512i
$714$		0			0
$715$	−0.00856318	−	2.28124i	−0.000320245	−	0.0853135i
$716$		0			0
$717$	−17.9976			−0.672131
$718$		0			0
$719$			22.2605i			0.830175i	0.909781	+	0.415088i	$0.136249\pi$
$719$			22.2605i			0.830175i	−0.909781	+	0.415088i	$0.863751\pi$
$720$		0			0
$721$	16.4293	−	28.2200i	0.611860	−	1.05097i
$722$		0			0
$723$		−	7.08250i		−	0.263401i
$724$		0			0
$725$		−	45.8670i		−	1.70346i
$726$		0			0
$727$			43.9408i			1.62968i	0.579689	+	0.814838i	$0.303174\pi$
$727$			43.9408i			1.62968i	−0.579689	+	0.814838i	$0.696826\pi$
$728$		0			0
$729$	−5.55747			−0.205832
$730$		0			0
$731$			1.96806i			0.0727912i
$732$		0			0
$733$	39.8448			1.47170			0.735850	−	0.677144i	$-0.236783\pi$
$733$	39.8448			1.47170			0.735850	+	0.677144i	$0.236783\pi$
$734$		0			0
$735$	−2.06168	−	3.63141i	−0.0760461	−	0.133946i
$736$		0			0
$737$	41.6120	−	0.156201i	1.53280	−	0.00575372i
$738$		0			0
$739$		−	2.67934i		−	0.0985610i	−0.998785	−	0.0492805i	$-0.984307\pi$
$739$		−	2.67934i		−	0.0985610i	0.998785	−	0.0492805i	$-0.0156928\pi$
$740$		0			0
$741$			1.44398i			0.0530460i
$742$		0			0
$743$		−	3.38804i		−	0.124295i	−0.998067	−	0.0621475i	$-0.980205\pi$
$743$		−	3.38804i		−	0.124295i	0.998067	−	0.0621475i	$-0.0197949\pi$
$744$		0			0
$745$	−4.38407			−0.160620
$746$		0			0
$747$	−13.2605			−0.485176
$748$		0			0
$749$	6.47106	−	11.1151i	0.236448	−	0.406136i
$750$		0			0
$751$	2.56340			0.0935397			0.0467698	−	0.998906i	$-0.485107\pi$
$751$	2.56340			0.0935397			0.0467698	+	0.998906i	$0.485107\pi$
$752$		0			0
$753$	−2.25128			−0.0820413
$754$		0			0
$755$	−8.53922			−0.310774
$756$		0			0
$757$	−38.8208			−1.41097			−0.705484	−	0.708726i	$-0.749270\pi$
$757$	−38.8208			−1.41097			−0.705484	+	0.708726i	$0.749270\pi$
$758$		0			0
$759$	−0.0519312	−	13.8345i	−0.00188498	−	0.502162i
$760$		0			0
$761$	36.4683			1.32198			0.660988	−	0.750397i	$-0.270138\pi$
$761$	36.4683			1.32198			0.660988	+	0.750397i	$0.270138\pi$
$762$		0			0
$763$	−7.84833	+	13.4807i	−0.284128	+	0.488036i
$764$		0			0
$765$			5.75889i			0.208213i
$766$		0			0
$767$		−	3.88893i		−	0.140421i
$768$		0			0
$769$	32.1947			1.16097			0.580485	−	0.814271i	$-0.302863\pi$
$769$	32.1947			1.16097			0.580485	+	0.814271i	$0.302863\pi$
$770$		0			0
$771$	−8.97278			−0.323147
$772$		0			0
$773$		−	7.73132i		−	0.278076i	−0.990287	−	0.139038i	$-0.955599\pi$
$773$		−	7.73132i		−	0.278076i	0.990287	−	0.139038i	$-0.0444011\pi$
$774$		0			0
$775$			47.3784i			1.70188i
$776$		0			0
$777$	−2.48275	+	4.26452i	−0.0890683	+	0.152989i
$778$		0			0
$779$	−9.12774			−0.327035
$780$		0			0
$781$	−7.15421	+	0.0268550i	−0.255998	+	0.000960949i
$782$		0			0
$783$	−46.1152			−1.64802
$784$		0			0
$785$	−11.8041			−0.421305
$786$		0			0
$787$	47.9554			1.70943			0.854713	−	0.519101i	$-0.173733\pi$
$787$	47.9554			1.70943			0.854713	+	0.519101i	$0.173733\pi$
$788$		0			0
$789$	4.63246			0.164920
$790$		0			0
$791$	−2.59398	−	1.51018i	−0.0922312	−	0.0536959i
$792$		0			0
$793$	−2.19664			−0.0780048
$794$		0			0
$795$	2.93518			0.104100
$796$		0			0
$797$			0.725471i			0.0256975i	0.999917	+	0.0128487i	$0.00409000\pi$
$797$			0.725471i			0.0256975i	−0.999917	+	0.0128487i	$0.995910\pi$
$798$		0			0
$799$		−	7.05496i		−	0.249587i
$800$		0			0
$801$			28.5989i			1.01049i
$802$		0			0
$803$	4.14622	−	0.0155638i	0.146317	−	0.000549236i
$804$		0			0
$805$	4.40365	−	7.56396i	0.155208	−	0.266594i
$806$		0			0
$807$	3.81664			0.134352
$808$		0			0
$809$		−	28.8894i		−	1.01570i	−0.861446	−	0.507849i	$-0.830441\pi$
$809$		−	28.8894i		−	1.01570i	0.861446	−	0.507849i	$-0.169559\pi$
$810$		0			0
$811$	−46.2673			−1.62466			−0.812332	−	0.583195i	$-0.801802\pi$
$811$	−46.2673			−1.62466			−0.812332	+	0.583195i	$0.801802\pi$
$812$		0			0
$813$			3.47585i			0.121903i
$814$		0			0
$815$		−	1.49547i		−	0.0523841i
$816$		0			0
$817$			0.879683i			0.0307762i
$818$		0			0
$819$	5.13955	+	2.99218i	0.179590	+	0.104555i
$820$		0			0
$821$		−	37.8342i		−	1.32042i	−0.751080	−	0.660211i	$-0.770467\pi$
$821$		−	37.8342i		−	1.32042i	0.751080	−	0.660211i	$-0.229533\pi$
$822$		0			0
$823$	−43.3185			−1.50999			−0.754993	−	0.655733i	$-0.772360\pi$
$823$	−43.3185			−1.50999			−0.754993	+	0.655733i	$0.772360\pi$
$824$		0			0
$825$	−0.0488794	−	13.0215i	−0.00170176	−	0.453351i
$826$		0			0
$827$			21.9507i			0.763301i	0.924307	+	0.381651i	$0.124644\pi$
$827$			21.9507i			0.763301i	−0.924307	+	0.381651i	$0.875356\pi$
$828$		0			0
$829$			21.8619i			0.759296i	0.925131	+	0.379648i	$0.123955\pi$
$829$			21.8619i			0.759296i	−0.925131	+	0.379648i	$0.876045\pi$
$830$		0			0
$831$	−18.6895			−0.648332
$832$		0			0
$833$	−12.8730	−	22.6743i	−0.446023	−	0.785618i
$834$		0			0
$835$		−	9.69316i		−	0.335446i
$836$		0			0
$837$	47.6349			1.64650
$838$		0			0
$839$		−	42.2878i		−	1.45994i	−0.683481	−	0.729968i	$-0.739535\pi$
$839$		−	42.2878i		−	1.45994i	0.683481	−	0.729968i	$-0.260465\pi$
$840$		0			0
$841$	−73.6596			−2.53998
$842$		0			0
$843$	27.8485			0.959153
$844$		0			0
$845$		−	0.687824i		−	0.0236619i
$846$		0			0
$847$	25.2605	+	14.4536i	0.867962	+	0.496630i
$848$		0			0
$849$		−	24.2101i		−	0.830887i
$850$		0			0
$851$	−10.3428			−0.354547
$852$		0			0
$853$	−50.2567			−1.72076			−0.860379	−	0.509656i	$-0.829773\pi$
$853$	−50.2567			−1.72076			−0.860379	+	0.509656i	$0.829773\pi$
$854$		0			0
$855$			2.57411i			0.0880329i
$856$		0			0
$857$	−7.81069			−0.266808			−0.133404	−	0.991062i	$-0.542591\pi$
$857$	−7.81069			−0.266808			−0.133404	+	0.991062i	$0.542591\pi$
$858$		0			0
$859$		−	26.3351i		−	0.898543i	−0.893395	−	0.449272i	$-0.851684\pi$
$859$		−	26.3351i		−	0.898543i	0.893395	−	0.449272i	$-0.148316\pi$
$860$		0			0
$861$	6.32950	−	10.8719i	0.215709	−	0.370514i
$862$		0			0
$863$	7.14910			0.243358			0.121679	−	0.992569i	$-0.461172\pi$
$863$	7.14910			0.243358			0.121679	+	0.992569i	$0.461172\pi$
$864$		0			0
$865$		−	12.2291i		−	0.415803i
$866$		0			0
$867$			2.71096i			0.0920690i
$868$		0			0
$869$	0.0842607	+	22.4471i	0.00285835	+	0.761467i
$870$		0			0
$871$	12.5466			0.425125
$872$		0			0
$873$			17.1199i			0.579422i
$874$		0			0
$875$	8.72289	−	14.9829i	0.294887	−	0.506516i
$876$		0			0
$877$		−	23.9833i		−	0.809858i	−0.914348	−	0.404929i	$-0.867296\pi$
$877$		−	23.9833i		−	0.809858i	0.914348	−	0.404929i	$-0.132704\pi$
$878$		0			0
$879$			16.1809i			0.545768i
$880$		0			0
$881$			27.9873i			0.942916i	0.881889	+	0.471458i	$0.156272\pi$
$881$			27.9873i			0.942916i	−0.881889	+	0.471458i	$0.843728\pi$
$882$		0			0
$883$	28.2244			0.949826			0.474913	−	0.880033i	$-0.342480\pi$
$883$	28.2244			0.949826			0.474913	+	0.880033i	$0.342480\pi$
$884$		0			0
$885$			2.31993i			0.0779837i
$886$		0			0
$887$	−13.3357			−0.447769			−0.223885	−	0.974616i	$-0.571874\pi$
$887$	−13.3357			−0.447769			−0.223885	+	0.974616i	$0.571874\pi$
$888$		0			0
$889$	−21.7690	+	37.3917i	−0.730109	+	1.25408i
$890$		0			0
$891$	9.27312	−	0.0348089i	0.310661	−	0.00116614i
$892$		0			0
$893$		−	3.15343i		−	0.105526i
$894$		0			0
$895$		−	7.73189i		−	0.258449i
$896$		0			0
$897$		−	4.17130i		−	0.139276i
$898$		0			0
$899$	106.042			3.53671
$900$		0			0
$901$	18.3271			0.610564
$902$		0			0
$903$	−1.04778	−	0.610003i	−0.0348678	−	0.0202996i
$904$		0			0
$905$	1.99557			0.0663351
$906$		0			0
$907$	−7.97494			−0.264804			−0.132402	−	0.991196i	$-0.542269\pi$
$907$	−7.97494			−0.264804			−0.132402	+	0.991196i	$0.542269\pi$
$908$		0			0
$909$	−19.6078			−0.650348
$910$		0			0
$911$	−39.3371			−1.30330			−0.651648	−	0.758522i	$-0.725922\pi$
$911$	−39.3371			−1.30330			−0.651648	+	0.758522i	$0.725922\pi$
$912$		0			0
$913$	−19.5657	+	0.0734447i	−0.647532	+	0.00243067i
$914$		0			0
$915$	1.31040			0.0433205
$916$		0			0
$917$	−20.4913	−	11.9298i	−0.676682	−	0.393956i
$918$		0			0
$919$		−	51.4839i		−	1.69830i	−0.528153	−	0.849149i	$-0.677115\pi$
$919$		−	51.4839i		−	1.69830i	0.528153	−	0.849149i	$-0.322885\pi$
$920$		0			0
$921$			12.6868i			0.418044i
$922$		0			0
$923$	−2.15709			−0.0710015
$924$		0			0
$925$	−9.73499			−0.320084
$926$		0			0
$927$		−	27.7425i		−	0.911182i
$928$		0			0
$929$			7.90515i			0.259360i	0.991556	+	0.129680i	$0.0413949\pi$
$929$			7.90515i			0.259360i	−0.991556	+	0.129680i	$0.958605\pi$
$930$		0			0
$931$	−5.75398	−	10.1350i	−0.188579	−	0.332160i
$932$		0			0
$933$	−16.3301			−0.534625
$934$		0			0
$935$	0.0318962	+	8.49719i	0.00104312	+	0.277888i
$936$		0			0
$937$	30.2584			0.988498			0.494249	−	0.869320i	$-0.335443\pi$
$937$	30.2584			0.988498			0.494249	+	0.869320i	$0.335443\pi$
$938$		0			0
$939$	−3.35901			−0.109617
$940$		0			0
$941$	39.5049			1.28782			0.643912	−	0.765100i	$-0.277310\pi$
$941$	39.5049			1.28782			0.643912	+	0.765100i	$0.277310\pi$
$942$		0			0
$943$	26.3678			0.858653
$944$		0			0
$945$	−7.15798	−	4.16729i	−0.232849	−	0.135562i
$946$		0			0
$947$	31.5631			1.02566			0.512832	−	0.858489i	$-0.328597\pi$
$947$	31.5631			1.02566			0.512832	+	0.858489i	$0.328597\pi$
$948$		0			0
$949$	1.25014			0.0405813
$950$		0			0
$951$			25.2206i			0.817835i
$952$		0			0
$953$		−	0.802002i		−	0.0259794i	−0.999916	−	0.0129897i	$-0.995865\pi$
$953$		−	0.802002i		−	0.0259794i	0.999916	−	0.0129897i	$-0.00413487\pi$
$954$		0			0
$955$		−	13.0403i		−	0.421974i
$956$		0			0
$957$	−29.1448	+	0.109402i	−0.942117	+	0.00353646i
$958$		0			0
$959$	−3.43060	−	1.99725i	−0.110780	−	0.0644947i
$960$		0			0
$961$	−78.5368			−2.53344
$962$		0			0
$963$		−	10.9270i		−	0.352118i
$964$		0			0
$965$	−2.54228			−0.0818388
$966$		0			0
$967$			9.76435i			0.314000i	0.987599	+	0.157000i	$0.0501823\pi$
$967$			9.76435i			0.314000i	−0.987599	+	0.157000i	$0.949818\pi$
$968$		0			0
$969$		−	5.37856i		−	0.172784i
$970$		0			0
$971$		−	34.5447i		−	1.10859i	−0.832320	−	0.554296i	$-0.812988\pi$
$971$		−	34.5447i		−	1.10859i	0.832320	−	0.554296i	$-0.187012\pi$
$972$		0			0
$973$	−27.7957	−	16.1824i	−0.891091	−	0.518782i
$974$		0			0
$975$		−	3.92617i		−	0.125738i
$976$		0			0
$977$	−37.2926			−1.19310			−0.596548	−	0.802578i	$-0.703461\pi$
$977$	−37.2926			−1.19310			−0.596548	+	0.802578i	$0.703461\pi$
$978$		0			0
$979$	0.158398	+	42.1974i	0.00506242	+	1.34863i
$980$		0			0
$981$			13.2526i			0.423124i
$982$		0			0
$983$			4.08111i			0.130167i	0.997880	+	0.0650836i	$0.0207314\pi$
$983$			4.08111i			0.130167i	−0.997880	+	0.0650836i	$0.979269\pi$
$984$		0			0
$985$	8.40956			0.267951
$986$		0			0
$987$	3.75600	+	2.18670i	0.119555	+	0.0696035i
$988$		0			0
$989$		−	2.54119i		−	0.0808050i
$990$		0			0
$991$	−10.4493			−0.331932			−0.165966	−	0.986132i	$-0.553074\pi$
$991$	−10.4493			−0.331932			−0.165966	+	0.986132i	$0.553074\pi$
$992$		0			0
$993$			20.2234i			0.641772i
$994$		0			0
$995$	−12.2477			−0.388277
$996$		0			0
$997$	−17.9521			−0.568550			−0.284275	−	0.958743i	$-0.591753\pi$
$997$	−17.9521			−0.568550			−0.284275	+	0.958743i	$0.591753\pi$
$998$		0			0
$999$			9.78767i			0.309668i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.e.a.3849.20	✓	48
7.6	odd	2		4004.2.e.b.3849.29	yes	48
11.10	odd	2		4004.2.e.b.3849.20	yes	48
77.76	even	2	inner	4004.2.e.a.3849.29	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.e.a.3849.20	✓	48	1.1	even	1	trivial
4004.2.e.a.3849.29	yes	48	77.76	even	2	inner
4004.2.e.b.3849.20	yes	48	11.10	odd	2
4004.2.e.b.3849.29	yes	48	7.6	odd	2

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.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.867297i q^{3} -0.687824i q^{5} +(2.28648 + 1.33116i) q^{7} +2.24780 q^{9} +O(q^{10})\)	\(q-0.867297i q^{3} -0.687824i q^{5} +(2.28648 + 1.33116i) q^{7} +2.24780 q^{9} +(3.31660 - 0.0124497i) q^{11} +1.00000 q^{13} -0.596548 q^{15} -3.72481 q^{17} -1.66492 q^{19} +(1.15451 - 1.98306i) q^{21} +4.80954 q^{23} +4.52690 q^{25} -4.55140i q^{27} -10.1321i q^{29} +10.4660i q^{31} +(-0.0107976 - 2.87648i) q^{33} +(0.915607 - 1.57270i) q^{35} -2.15048 q^{37} -0.867297i q^{39} +5.48239 q^{41} -0.528364i q^{43} -1.54609i q^{45} +1.89404i q^{47} +(3.45601 + 6.08736i) q^{49} +3.23052i q^{51} -4.92027 q^{53} +(-0.00856318 - 2.28124i) q^{55} +1.44398i q^{57} -3.88893i q^{59} -2.19664 q^{61} +(5.13955 + 2.99218i) q^{63} -0.687824i q^{65} +12.5466 q^{67} -4.17130i q^{69} -2.15709 q^{71} +1.25014 q^{73} -3.92617i q^{75} +(7.59993 + 4.38647i) q^{77} +6.76812i q^{79} +2.79597 q^{81} -5.89934 q^{83} +2.56202i q^{85} -8.78755 q^{87} +12.7231i q^{89} +(2.28648 + 1.33116i) q^{91} +9.07712 q^{93} +1.14517i q^{95} +7.61633i q^{97} +(7.45504 - 0.0279843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 4 q^{7} - 48 q^{9}+O(q^{10}) \) 48 * q - 4 * q^7 - 48 * q^9	\( 48 q - 4 q^{7} - 48 q^{9} + 2 q^{11} + 48 q^{13} + 8 q^{15} + 4 q^{17} + 10 q^{21} + 4 q^{23} - 44 q^{25} + 10 q^{33} - 14 q^{35} - 16 q^{37} + 10 q^{49} - 8 q^{53} - 12 q^{55} + 16 q^{61} + 16 q^{63} + 4 q^{67} - 16 q^{73} + 2 q^{77} + 64 q^{81} - 4 q^{83} - 12 q^{87} - 4 q^{91} + 36 q^{93} - 40 q^{99}+O(q^{100}) \) 48 * q - 4 * q^7 - 48 * q^9 + 2 * q^11 + 48 * q^13 + 8 * q^15 + 4 * q^17 + 10 * q^21 + 4 * q^23 - 44 * q^25 + 10 * q^33 - 14 * q^35 - 16 * q^37 + 10 * q^49 - 8 * q^53 - 12 * q^55 + 16 * q^61 + 16 * q^63 + 4 * q^67 - 16 * q^73 + 2 * q^77 + 64 * q^81 - 4 * q^83 - 12 * q^87 - 4 * q^91 + 36 * q^93 - 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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