LMFDB - Embedded newform 4004.2.e.b.3849.18

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			3849.18
Character	$\chi$	$=$	4004.3849
Dual form			4004.2.e.b.3849.31

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.968020i q^{3} -0.815183i q^{5} +(2.53164 - 0.768635i) q^{7} +2.06294 q^{9} +O(q^{10})$	$q-0.968020i q^{3} -0.815183i q^{5} +(2.53164 - 0.768635i) q^{7} +2.06294 q^{9} +(0.542916 - 3.27189i) q^{11} -1.00000 q^{13} -0.789113 q^{15} +3.02632 q^{17} +2.23283 q^{19} +(-0.744054 - 2.45068i) q^{21} -0.0738458 q^{23} +4.33548 q^{25} -4.90102i q^{27} +2.16020i q^{29} -0.394147i q^{31} +(-3.16725 - 0.525553i) q^{33} +(-0.626578 - 2.06375i) q^{35} -5.14676 q^{37} +0.968020i q^{39} +10.5894 q^{41} -5.58348i q^{43} -1.68167i q^{45} +6.69248i q^{47} +(5.81840 - 3.89181i) q^{49} -2.92954i q^{51} +0.252878 q^{53} +(-2.66719 - 0.442576i) q^{55} -2.16143i q^{57} +5.71703i q^{59} +3.56672 q^{61} +(5.22262 - 1.58565i) q^{63} +0.815183i q^{65} -10.3286 q^{67} +0.0714842i q^{69} -9.86879 q^{71} +3.58494 q^{73} -4.19683i q^{75} +(-1.14042 - 8.70054i) q^{77} +9.98689i q^{79} +1.44453 q^{81} +10.1386 q^{83} -2.46700i q^{85} +2.09112 q^{87} -2.33224i q^{89} +(-2.53164 + 0.768635i) q^{91} -0.381542 q^{93} -1.82017i q^{95} +8.67427i q^{97} +(1.12000 - 6.74970i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 22 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 + 22 * 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
$3$		−	0.968020i		−	0.558886i	−0.960162	−	0.279443i	$-0.909850\pi$
$3$		−	0.968020i		−	0.558886i	0.960162	−	0.279443i	$-0.0901499\pi$
$4$		0			0
$5$		−	0.815183i		−	0.364561i	−0.983247	−	0.182280i	$-0.941652\pi$
$5$		−	0.815183i		−	0.364561i	0.983247	−	0.182280i	$-0.0583479\pi$
$6$		0			0
$7$	2.53164	−	0.768635i	0.956870	−	0.290517i
$7$	2.53164	−	0.768635i	0.956870	−	0.290517i
$8$		0			0
$9$	2.06294			0.687646
$10$		0			0
$11$	0.542916	−	3.27189i	0.163695	−	0.986511i
$11$	0.542916	−	3.27189i	0.163695	−	0.986511i
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$	−0.789113			−0.203748
$16$		0			0
$17$	3.02632			0.733990			0.366995	−	0.930223i	$-0.380387\pi$
$17$	3.02632			0.733990			0.366995	+	0.930223i	$0.380387\pi$
$18$		0			0
$19$	2.23283			0.512247			0.256123	−	0.966644i	$-0.417555\pi$
$19$	2.23283			0.512247			0.256123	+	0.966644i	$0.417555\pi$
$20$		0			0
$21$	−0.744054	−	2.45068i	−0.162366	−	0.534782i
$22$		0			0
$23$	−0.0738458			−0.0153979			−0.00769896	−	0.999970i	$-0.502451\pi$
$23$	−0.0738458			−0.0153979			−0.00769896	+	0.999970i	$0.502451\pi$
$24$		0			0
$25$	4.33548			0.867095
$26$		0			0
$27$		−	4.90102i		−	0.943202i
$28$		0			0
$29$			2.16020i			0.401139i	0.979679	+	0.200570i	$0.0642793\pi$
$29$			2.16020i			0.401139i	−0.979679	+	0.200570i	$0.935721\pi$
$30$		0			0
$31$		−	0.394147i		−	0.0707909i	−0.999373	−	0.0353954i	$-0.988731\pi$
$31$		−	0.394147i		−	0.0707909i	0.999373	−	0.0353954i	$-0.0112691\pi$
$32$		0			0
$33$	−3.16725	−	0.525553i	−0.551348	−	0.0914871i
$34$		0			0
$35$	−0.626578	−	2.06375i	−0.105911	−	0.348837i
$36$		0			0
$37$	−5.14676			−0.846122			−0.423061	−	0.906101i	$-0.639044\pi$
$37$	−5.14676			−0.846122			−0.423061	+	0.906101i	$0.639044\pi$
$38$		0			0
$39$			0.968020i			0.155007i
$40$		0			0
$41$	10.5894			1.65379			0.826894	−	0.562357i	$-0.190105\pi$
$41$	10.5894			1.65379			0.826894	+	0.562357i	$0.190105\pi$
$42$		0			0
$43$		−	5.58348i		−	0.851473i	−0.904847	−	0.425737i	$-0.860015\pi$
$43$		−	5.58348i		−	0.851473i	0.904847	−	0.425737i	$-0.139985\pi$
$44$		0			0
$45$		−	1.68167i		−	0.250689i
$46$		0			0
$47$			6.69248i			0.976198i	0.872788	+	0.488099i	$0.162309\pi$
$47$			6.69248i			0.976198i	−0.872788	+	0.488099i	$0.837691\pi$
$48$		0			0
$49$	5.81840	−	3.89181i	0.831200	−	0.555974i
$50$		0			0
$51$		−	2.92954i		−	0.410217i
$52$		0			0
$53$	0.252878			0.0347355			0.0173678	−	0.999849i	$-0.494471\pi$
$53$	0.252878			0.0347355			0.0173678	+	0.999849i	$0.494471\pi$
$54$		0			0
$55$	−2.66719	−	0.442576i	−0.359643	−	0.0596769i
$56$		0			0
$57$		−	2.16143i		−	0.286288i
$58$		0			0
$59$			5.71703i			0.744294i	0.928174	+	0.372147i	$0.121378\pi$
$59$			5.71703i			0.744294i	−0.928174	+	0.372147i	$0.878622\pi$
$60$		0			0
$61$	3.56672			0.456671			0.228336	−	0.973582i	$-0.426672\pi$
$61$	3.56672			0.456671			0.228336	+	0.973582i	$0.426672\pi$
$62$		0			0
$63$	5.22262	−	1.58565i	0.657988	−	0.199773i
$64$		0			0
$65$			0.815183i			0.101111i
$66$		0			0
$67$	−10.3286			−1.26184			−0.630920	−	0.775848i	$-0.717323\pi$
$67$	−10.3286			−1.26184			−0.630920	+	0.775848i	$0.717323\pi$
$68$		0			0
$69$			0.0714842i			0.00860569i
$70$		0			0
$71$	−9.86879			−1.17121			−0.585605	−	0.810597i	$-0.699143\pi$
$71$	−9.86879			−1.17121			−0.585605	+	0.810597i	$0.699143\pi$
$72$		0			0
$73$	3.58494			0.419586			0.209793	−	0.977746i	$-0.432721\pi$
$73$	3.58494			0.419586			0.209793	+	0.977746i	$0.432721\pi$
$74$		0			0
$75$		−	4.19683i		−	0.484608i
$76$		0			0
$77$	−1.14042	−	8.70054i	−0.129963	−	0.991519i
$78$		0			0
$79$			9.98689i			1.12361i	0.827269	+	0.561806i	$0.189893\pi$
$79$			9.98689i			1.12361i	−0.827269	+	0.561806i	$0.810107\pi$
$80$		0			0
$81$	1.44453			0.160503
$82$		0			0
$83$	10.1386			1.11286			0.556428	−	0.830896i	$-0.312171\pi$
$83$	10.1386			1.11286			0.556428	+	0.830896i	$0.312171\pi$
$84$		0			0
$85$		−	2.46700i		−	0.267584i
$86$		0			0
$87$	2.09112			0.224191
$88$		0			0
$89$		−	2.33224i		−	0.247217i	−0.992331	−	0.123608i	$-0.960553\pi$
$89$		−	2.33224i		−	0.247217i	0.992331	−	0.123608i	$-0.0394467\pi$
$90$		0			0
$91$	−2.53164	+	0.768635i	−0.265388	+	0.0805749i
$92$		0			0
$93$	−0.381542			−0.0395640
$94$		0			0
$95$		−	1.82017i		−	0.186745i
$96$		0			0
$97$			8.67427i			0.880739i	0.897817	+	0.440369i	$0.145153\pi$
$97$			8.67427i			0.880739i	−0.897817	+	0.440369i	$0.854847\pi$
$98$		0			0
$99$	1.12000	−	6.74970i	0.112564	−	0.678370i
$100$		0			0
$101$	−5.53060			−0.550316			−0.275158	−	0.961399i	$-0.588730\pi$
$101$	−5.53060			−0.550316			−0.275158	+	0.961399i	$0.588730\pi$
$102$		0			0
$103$			7.91934i			0.780316i	0.920748	+	0.390158i	$0.127580\pi$
$103$			7.91934i			0.780316i	−0.920748	+	0.390158i	$0.872420\pi$
$104$		0			0
$105$	−1.99775	+	0.606540i	−0.194960	+	0.0591922i
$106$		0			0
$107$		−	18.0157i		−	1.74165i	−0.491597	−	0.870823i	$-0.663587\pi$
$107$		−	18.0157i		−	1.74165i	0.491597	−	0.870823i	$-0.336413\pi$
$108$		0			0
$109$			5.18683i			0.496808i	0.968657	+	0.248404i	$0.0799061\pi$
$109$			5.18683i			0.496808i	−0.968657	+	0.248404i	$0.920094\pi$
$110$		0			0
$111$			4.98217i			0.472886i
$112$		0			0
$113$	3.06455			0.288288			0.144144	−	0.989557i	$-0.453957\pi$
$113$	3.06455			0.288288			0.144144	+	0.989557i	$0.453957\pi$
$114$		0			0
$115$			0.0601978i			0.00561348i
$116$		0			0
$117$	−2.06294			−0.190719
$118$		0			0
$119$	7.66155	−	2.32613i	0.702333	−	0.213236i
$120$		0			0
$121$	−10.4105	−	3.55272i	−0.946408	−	0.322975i
$122$		0			0
$123$		−	10.2508i		−	0.924280i
$124$		0			0
$125$		−	7.61012i		−	0.680670i
$126$		0			0
$127$		−	10.3406i		−	0.917579i	−0.888545	−	0.458789i	$-0.848283\pi$
$127$		−	10.3406i		−	0.917579i	0.888545	−	0.458789i	$-0.151717\pi$
$128$		0			0
$129$	−5.40492			−0.475877
$130$		0			0
$131$	−6.09722			−0.532717			−0.266358	−	0.963874i	$-0.585820\pi$
$131$	−6.09722			−0.532717			−0.266358	+	0.963874i	$0.585820\pi$
$132$		0			0
$133$	5.65273	−	1.71623i	0.490154	−	0.148816i
$134$		0			0
$135$	−3.99523			−0.343855
$136$		0			0
$137$	−6.53281			−0.558135			−0.279068	−	0.960271i	$-0.590025\pi$
$137$	−6.53281			−0.558135			−0.279068	+	0.960271i	$0.590025\pi$
$138$		0			0
$139$	9.08852			0.770878			0.385439	−	0.922733i	$-0.374050\pi$
$139$	9.08852			0.770878			0.385439	+	0.922733i	$0.374050\pi$
$140$		0			0
$141$	6.47845			0.545584
$142$		0			0
$143$	−0.542916	+	3.27189i	−0.0454009	+	0.273609i
$144$		0			0
$145$	1.76096			0.146240
$146$		0			0
$147$	−3.76735	−	5.63233i	−0.310726	−	0.464546i
$148$		0			0
$149$		−	1.35600i		−	0.111088i	−0.998456	−	0.0555438i	$-0.982311\pi$
$149$		−	1.35600i		−	0.111088i	0.998456	−	0.0555438i	$-0.0176892\pi$
$150$		0			0
$151$			8.53740i			0.694764i	0.937724	+	0.347382i	$0.112929\pi$
$151$			8.53740i			0.694764i	−0.937724	+	0.347382i	$0.887071\pi$
$152$		0			0
$153$	6.24311			0.504725
$154$		0			0
$155$	−0.321302			−0.0258076
$156$		0			0
$157$			4.64501i			0.370712i	0.982671	+	0.185356i	$0.0593439\pi$
$157$			4.64501i			0.370712i	−0.982671	+	0.185356i	$0.940656\pi$
$158$		0			0
$159$		−	0.244791i		−	0.0194132i
$160$		0			0
$161$	−0.186951	+	0.0567605i	−0.0147338	+	0.00447335i
$162$		0			0
$163$	12.8316			1.00505			0.502525	−	0.864563i	$-0.332404\pi$
$163$	12.8316			1.00505			0.502525	+	0.864563i	$0.332404\pi$
$164$		0			0
$165$	−0.428422	+	2.58189i	−0.0333526	+	0.201000i
$166$		0			0
$167$	−11.9347			−0.923531			−0.461766	−	0.887002i	$-0.652784\pi$
$167$	−11.9347			−0.923531			−0.461766	+	0.887002i	$0.652784\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	4.60619			0.352244
$172$		0			0
$173$	−13.0097			−0.989109			−0.494554	−	0.869147i	$-0.664669\pi$
$173$	−13.0097			−0.989109			−0.494554	+	0.869147i	$0.664669\pi$
$174$		0			0
$175$	10.9759	−	3.33240i	0.829697	−	0.251906i
$176$		0			0
$177$	5.53420			0.415976
$178$		0			0
$179$	−19.9984			−1.49475			−0.747374	−	0.664404i	$-0.768686\pi$
$179$	−19.9984			−1.49475			−0.747374	+	0.664404i	$0.768686\pi$
$180$		0			0
$181$			24.7341i			1.83847i	0.393704	+	0.919237i	$0.371193\pi$
$181$			24.7341i			1.83847i	−0.393704	+	0.919237i	$0.628807\pi$
$182$		0			0
$183$		−	3.45265i		−	0.255227i
$184$		0			0
$185$			4.19555i			0.308463i
$186$		0			0
$187$	1.64304	−	9.90177i	0.120151	−	0.724089i
$188$		0			0
$189$	−3.76710	−	12.4076i	−0.274016	−	0.902522i
$190$		0			0
$191$	−6.14297			−0.444490			−0.222245	−	0.974991i	$-0.571338\pi$
$191$	−6.14297			−0.444490			−0.222245	+	0.974991i	$0.571338\pi$
$192$		0			0
$193$		−	10.5162i		−	0.756975i	−0.925606	−	0.378487i	$-0.876444\pi$
$193$		−	10.5162i		−	0.756975i	0.925606	−	0.378487i	$-0.123556\pi$
$194$		0			0
$195$	0.789113			0.0565096
$196$		0			0
$197$		−	6.57294i		−	0.468302i	−0.972200	−	0.234151i	$-0.924769\pi$
$197$		−	6.57294i		−	0.468302i	0.972200	−	0.234151i	$-0.0752311\pi$
$198$		0			0
$199$		−	0.0862356i		−	0.00611308i	−0.999995	−	0.00305654i	$-0.999027\pi$
$199$		−	0.0862356i		−	0.00611308i	0.999995	−	0.00305654i	$-0.000972929\pi$
$200$		0			0
$201$			9.99830i			0.705226i
$202$		0			0
$203$	1.66041	+	5.46885i	0.116538	+	0.383838i
$204$		0			0
$205$		−	8.63231i		−	0.602907i
$206$		0			0
$207$	−0.152339			−0.0105883
$208$		0			0
$209$	1.21224	−	7.30557i	0.0838524	−	0.505337i
$210$		0			0
$211$		−	17.8946i		−	1.23192i	−0.787779	−	0.615958i	$-0.788769\pi$
$211$		−	17.8946i		−	1.23192i	0.787779	−	0.615958i	$-0.211231\pi$
$212$		0			0
$213$			9.55318i			0.654573i
$214$		0			0
$215$	−4.55156			−0.310414
$216$		0			0
$217$	−0.302955	−	0.997838i	−0.0205659	−	0.0677376i
$218$		0			0
$219$		−	3.47029i		−	0.234501i
$220$		0			0
$221$	−3.02632			−0.203572
$222$		0			0
$223$			10.3485i			0.692986i	0.938053	+	0.346493i	$0.112627\pi$
$223$			10.3485i			0.692986i	−0.938053	+	0.346493i	$0.887373\pi$
$224$		0			0
$225$	8.94382			0.596255
$226$		0			0
$227$	7.22829			0.479759			0.239879	−	0.970803i	$-0.422892\pi$
$227$	7.22829			0.479759			0.239879	+	0.970803i	$0.422892\pi$
$228$		0			0
$229$		−	14.5729i		−	0.963007i	−0.876444	−	0.481504i	$-0.840091\pi$
$229$		−	14.5729i		−	0.963007i	0.876444	−	0.481504i	$-0.159909\pi$
$230$		0			0
$231$	−8.42230	+	1.10395i	−0.554146	+	0.0726344i
$232$		0			0
$233$		−	19.9523i		−	1.30712i	−0.756875	−	0.653559i	$-0.773275\pi$
$233$		−	19.9523i		−	1.30712i	0.756875	−	0.653559i	$-0.226725\pi$
$234$		0			0
$235$	5.45559			0.355884
$236$		0			0
$237$	9.66751			0.627972
$238$		0			0
$239$		−	18.4514i		−	1.19352i	−0.802419	−	0.596762i	$-0.796454\pi$
$239$		−	18.4514i		−	1.19352i	0.802419	−	0.596762i	$-0.203546\pi$
$240$		0			0
$241$	23.9319			1.54159			0.770795	−	0.637084i	$-0.219859\pi$
$241$	23.9319			1.54159			0.770795	+	0.637084i	$0.219859\pi$
$242$		0			0
$243$		−	16.1014i		−	1.03291i
$244$		0			0
$245$	−3.17254	−	4.74306i	−0.202686	−	0.303023i
$246$		0			0
$247$	−2.23283			−0.142072
$248$		0			0
$249$		−	9.81437i		−	0.621960i
$250$		0			0
$251$			13.8860i			0.876478i	0.898858	+	0.438239i	$0.144398\pi$
$251$			13.8860i			0.876478i	−0.898858	+	0.438239i	$0.855602\pi$
$252$		0			0
$253$	−0.0400921	+	0.241615i	−0.00252057	+	0.0151902i
$254$		0			0
$255$	−2.38811			−0.149549
$256$		0			0
$257$		−	6.43850i		−	0.401622i	−0.979630	−	0.200811i	$-0.935642\pi$
$257$		−	6.43850i		−	0.401622i	0.979630	−	0.200811i	$-0.0643578\pi$
$258$		0			0
$259$	−13.0297	+	3.95598i	−0.809629	+	0.245813i
$260$		0			0
$261$			4.45636i			0.275842i
$262$		0			0
$263$		−	22.9406i		−	1.41458i	−0.706924	−	0.707290i	$-0.749918\pi$
$263$		−	22.9406i		−	1.41458i	0.706924	−	0.707290i	$-0.250082\pi$
$264$		0			0
$265$		−	0.206142i		−	0.0126632i
$266$		0			0
$267$	−2.25765			−0.138166
$268$		0			0
$269$		−	28.4948i		−	1.73736i	−0.495373	−	0.868680i	$-0.664969\pi$
$269$		−	28.4948i		−	1.73736i	0.495373	−	0.868680i	$-0.335031\pi$
$270$		0			0
$271$	1.30425			0.0792274			0.0396137	−	0.999215i	$-0.487387\pi$
$271$	1.30425			0.0792274			0.0396137	+	0.999215i	$0.487387\pi$
$272$		0			0
$273$	0.744054	+	2.45068i	0.0450322	+	0.148322i
$274$		0			0
$275$	2.35380	−	14.1852i	0.141940	−	0.855399i
$276$		0			0
$277$			6.78458i			0.407646i	0.979008	+	0.203823i	$0.0653367\pi$
$277$			6.78458i			0.407646i	−0.979008	+	0.203823i	$0.934663\pi$
$278$		0			0
$279$		−	0.813100i		−	0.0486790i
$280$		0			0
$281$			21.6270i			1.29016i	0.764114	+	0.645081i	$0.223176\pi$
$281$			21.6270i			1.29016i	−0.764114	+	0.645081i	$0.776824\pi$
$282$		0			0
$283$	−22.0090			−1.30830			−0.654149	−	0.756366i	$-0.726973\pi$
$283$	−22.0090			−1.30830			−0.654149	+	0.756366i	$0.726973\pi$
$284$		0			0
$285$	−1.76196			−0.104369
$286$		0			0
$287$	26.8086	−	8.13940i	1.58246	−	0.480453i
$288$		0			0
$289$	−7.84140			−0.461259
$290$		0			0
$291$	8.39686			0.492233
$292$		0			0
$293$	6.88301			0.402110			0.201055	−	0.979580i	$-0.435563\pi$
$293$	6.88301			0.402110			0.201055	+	0.979580i	$0.435563\pi$
$294$		0			0
$295$	4.66043			0.271341
$296$		0			0
$297$	−16.0356	−	2.66084i	−0.930479	−	0.154398i
$298$		0			0
$299$	0.0738458			0.00427061
$300$		0			0
$301$	−4.29166	−	14.1354i	−0.247367	−	0.814749i
$302$		0			0
$303$			5.35373i			0.307564i
$304$		0			0
$305$		−	2.90753i		−	0.166485i
$306$		0			0
$307$	−18.8509			−1.07588			−0.537938	−	0.842985i	$-0.680796\pi$
$307$	−18.8509			−1.07588			−0.537938	+	0.842985i	$0.680796\pi$
$308$		0			0
$309$	7.66607			0.436108
$310$		0			0
$311$		−	18.8992i		−	1.07168i	−0.844321	−	0.535838i	$-0.819996\pi$
$311$		−	18.8992i		−	1.07168i	0.844321	−	0.535838i	$-0.180004\pi$
$312$		0			0
$313$			9.08921i			0.513753i	0.966444	+	0.256876i	$0.0826933\pi$
$313$			9.08921i			0.513753i	−0.966444	+	0.256876i	$0.917307\pi$
$314$		0			0
$315$	−1.29259	−	4.25739i	−0.0728293	−	0.239877i
$316$		0			0
$317$	−6.61847			−0.371730			−0.185865	−	0.982575i	$-0.559509\pi$
$317$	−6.61847			−0.371730			−0.185865	+	0.982575i	$0.559509\pi$
$318$		0			0
$319$	7.06793	+	1.17281i	0.395728	+	0.0656646i
$320$		0			0
$321$	−17.4396			−0.973382
$322$		0			0
$323$	6.75726			0.375984
$324$		0			0
$325$	−4.33548			−0.240489
$326$		0			0
$327$	5.02095			0.277659
$328$		0			0
$329$	5.14407	+	16.9429i	0.283602	+	0.934094i
$330$		0			0
$331$	−8.24674			−0.453282			−0.226641	−	0.973978i	$-0.572774\pi$
$331$	−8.24674			−0.453282			−0.226641	+	0.973978i	$0.572774\pi$
$332$		0			0
$333$	−10.6174			−0.581833
$334$		0			0
$335$			8.41971i			0.460018i
$336$		0			0
$337$			17.3062i			0.942728i	0.881939	+	0.471364i	$0.156238\pi$
$337$			17.3062i			0.942728i	−0.881939	+	0.471364i	$0.843762\pi$
$338$		0			0
$339$		−	2.96654i		−	0.161120i
$340$		0			0
$341$	−1.28960	−	0.213989i	−0.0698360	−	0.0115881i
$342$		0			0
$343$	11.7387	−	14.3249i	0.633831	−	0.773472i
$344$		0			0
$345$	0.0582727			0.00313730
$346$		0			0
$347$		−	23.7044i		−	1.27252i	−0.771476	−	0.636258i	$-0.780481\pi$
$347$		−	23.7044i		−	1.27252i	0.771476	−	0.636258i	$-0.219519\pi$
$348$		0			0
$349$	−2.36711			−0.126708			−0.0633541	−	0.997991i	$-0.520180\pi$
$349$	−2.36711			−0.126708			−0.0633541	+	0.997991i	$0.520180\pi$
$350$		0			0
$351$			4.90102i			0.261597i
$352$		0			0
$353$		−	8.56589i		−	0.455916i	−0.973671	−	0.227958i	$-0.926795\pi$
$353$		−	8.56589i		−	0.455916i	0.973671	−	0.227958i	$-0.0732050\pi$
$354$		0			0
$355$			8.04487i			0.426977i
$356$		0			0
$357$	−2.25174	−	7.41653i	−0.119175	−	0.392524i
$358$		0			0
$359$			4.79059i			0.252838i	0.991977	+	0.126419i	$0.0403484\pi$
$359$			4.79059i			0.252838i	−0.991977	+	0.126419i	$0.959652\pi$
$360$		0			0
$361$	−14.0145			−0.737603
$362$		0			0
$363$	−3.43910	+	10.0776i	−0.180506	+	0.528934i
$364$		0			0
$365$		−	2.92238i		−	0.152964i
$366$		0			0
$367$		−	29.9682i		−	1.56433i	−0.623072	−	0.782164i	$-0.714116\pi$
$367$		−	29.9682i		−	1.56433i	0.623072	−	0.782164i	$-0.285884\pi$
$368$		0			0
$369$	21.8453			1.13722
$370$		0			0
$371$	0.640197	−	0.194371i	0.0332374	−	0.0100912i
$372$		0			0
$373$			28.8167i			1.49207i	0.665905	+	0.746037i	$0.268046\pi$
$373$			28.8167i			1.49207i	−0.665905	+	0.746037i	$0.731954\pi$
$374$		0			0
$375$	−7.36675			−0.380417
$376$		0			0
$377$		−	2.16020i		−	0.111256i
$378$		0			0
$379$	−2.18904			−0.112444			−0.0562218	−	0.998418i	$-0.517905\pi$
$379$	−2.18904			−0.112444			−0.0562218	+	0.998418i	$0.517905\pi$
$380$		0			0
$381$	−10.0099			−0.512822
$382$		0			0
$383$			13.4028i			0.684850i	0.939545	+	0.342425i	$0.111248\pi$
$383$			13.4028i			0.684850i	−0.939545	+	0.342425i	$0.888752\pi$
$384$		0			0
$385$	−7.09253	+	0.929650i	−0.361469	+	0.0473793i
$386$		0			0
$387$		−	11.5184i		−	0.585512i
$388$		0			0
$389$	11.5705			0.586648			0.293324	−	0.956013i	$-0.405239\pi$
$389$	11.5705			0.586648			0.293324	+	0.956013i	$0.405239\pi$
$390$		0			0
$391$	−0.223481			−0.0113019
$392$		0			0
$393$			5.90223i			0.297728i
$394$		0			0
$395$	8.14114			0.409625
$396$		0			0
$397$			13.7464i			0.689910i	0.938619	+	0.344955i	$0.112106\pi$
$397$			13.7464i			0.689910i	−0.938619	+	0.344955i	$0.887894\pi$
$398$		0			0
$399$	−1.66135	−	5.47195i	−0.0831714	−	0.273940i
$400$		0			0
$401$	−25.4271			−1.26977			−0.634884	−	0.772607i	$-0.718952\pi$
$401$	−25.4271			−1.26977			−0.634884	+	0.772607i	$0.718952\pi$
$402$		0			0
$403$			0.394147i			0.0196339i
$404$		0			0
$405$		−	1.17755i		−	0.0585131i
$406$		0			0
$407$	−2.79426	+	16.8396i	−0.138506	+	0.834709i
$408$		0			0
$409$	−3.65230			−0.180595			−0.0902973	−	0.995915i	$-0.528782\pi$
$409$	−3.65230			−0.180595			−0.0902973	+	0.995915i	$0.528782\pi$
$410$		0			0
$411$			6.32389i			0.311934i
$412$		0			0
$413$	4.39431	+	14.4735i	0.216230	+	0.712193i
$414$		0			0
$415$		−	8.26482i		−	0.405704i
$416$		0			0
$417$		−	8.79786i		−	0.430833i
$418$		0			0
$419$			8.39672i			0.410207i	0.978740	+	0.205103i	$0.0657531\pi$
$419$			8.39672i			0.410207i	−0.978740	+	0.205103i	$0.934247\pi$
$420$		0			0
$421$	−11.4298			−0.557054			−0.278527	−	0.960428i	$-0.589846\pi$
$421$	−11.4298			−0.557054			−0.278527	+	0.960428i	$0.589846\pi$
$422$		0			0
$423$			13.8062i			0.671279i
$424$		0			0
$425$	13.1205			0.636439
$426$		0			0
$427$	9.02965	−	2.74150i	0.436975	−	0.132671i
$428$		0			0
$429$	3.16725	+	0.525553i	0.152916	+	0.0253740i
$430$		0			0
$431$			15.8879i			0.765291i	0.923895	+	0.382646i	$0.124987\pi$
$431$			15.8879i			0.765291i	−0.923895	+	0.382646i	$0.875013\pi$
$432$		0			0
$433$			39.1365i			1.88078i	0.340098	+	0.940390i	$0.389540\pi$
$433$			39.1365i			1.88078i	−0.340098	+	0.940390i	$0.610460\pi$
$434$		0			0
$435$		−	1.70464i		−	0.0817313i
$436$		0			0
$437$	−0.164885			−0.00788753
$438$		0			0
$439$	−0.625727			−0.0298643			−0.0149322	−	0.999889i	$-0.504753\pi$
$439$	−0.625727			−0.0298643			−0.0149322	+	0.999889i	$0.504753\pi$
$440$		0			0
$441$	12.0030	−	8.02857i	0.571571	−	0.382313i
$442$		0			0
$443$	−18.2643			−0.867764			−0.433882	−	0.900970i	$-0.642857\pi$
$443$	−18.2643			−0.867764			−0.433882	+	0.900970i	$0.642857\pi$
$444$		0			0
$445$	−1.90120			−0.0901256
$446$		0			0
$447$	−1.31263			−0.0620853
$448$		0			0
$449$	41.0440			1.93698			0.968492	−	0.249043i	$-0.0801160\pi$
$449$	41.0440			1.93698			0.968492	+	0.249043i	$0.0801160\pi$
$450$		0			0
$451$	5.74916	−	34.6474i	0.270718	−	1.63148i
$452$		0			0
$453$	8.26437			0.388294
$454$		0			0
$455$	0.626578	+	2.06375i	0.0293744	+	0.0967501i
$456$		0			0
$457$			17.3256i			0.810456i	0.914216	+	0.405228i	$0.132808\pi$
$457$			17.3256i			0.810456i	−0.914216	+	0.405228i	$0.867192\pi$
$458$		0			0
$459$		−	14.8321i		−	0.692301i
$460$		0			0
$461$	3.32479			0.154851			0.0774255	−	0.996998i	$-0.475330\pi$
$461$	3.32479			0.154851			0.0774255	+	0.996998i	$0.475330\pi$
$462$		0			0
$463$	−2.65476			−0.123377			−0.0616887	−	0.998095i	$-0.519649\pi$
$463$	−2.65476			−0.123377			−0.0616887	+	0.998095i	$0.519649\pi$
$464$		0			0
$465$			0.311026i			0.0144235i
$466$		0			0
$467$		−	2.78334i		−	0.128798i	−0.997924	−	0.0643988i	$-0.979487\pi$
$467$		−	2.78334i		−	0.128798i	0.997924	−	0.0643988i	$-0.0205130\pi$
$468$		0			0
$469$	−26.1483	+	7.93893i	−1.20742	+	0.366586i
$470$		0			0
$471$	4.49646			0.207186
$472$		0			0
$473$	−18.2685	−	3.03136i	−0.839987	−	0.139382i
$474$		0			0
$475$	9.68039			0.444167
$476$		0			0
$477$	0.521672			0.0238857
$478$		0			0
$479$	14.7670			0.674722			0.337361	−	0.941375i	$-0.390466\pi$
$479$	14.7670			0.674722			0.337361	+	0.941375i	$0.390466\pi$
$480$		0			0
$481$	5.14676			0.234672
$482$		0			0
$483$	0.0549453	+	0.180972i	0.00250010	+	0.00823452i
$484$		0			0
$485$	7.07112			0.321083
$486$		0			0
$487$	−8.17883			−0.370618			−0.185309	−	0.982680i	$-0.559329\pi$
$487$	−8.17883			−0.370618			−0.185309	+	0.982680i	$0.559329\pi$
$488$		0			0
$489$		−	12.4213i		−	0.561709i
$490$		0			0
$491$		−	2.57117i		−	0.116035i	−0.998316	−	0.0580176i	$-0.981522\pi$
$491$		−	2.57117i		−	0.116035i	0.998316	−	0.0580176i	$-0.0184779\pi$
$492$		0			0
$493$			6.53745i			0.294432i
$494$		0			0
$495$	−5.50224	−	0.913007i	−0.247307	−	0.0410366i
$496$		0			0
$497$	−24.9842	+	7.58550i	−1.12070	+	0.340256i
$498$		0			0
$499$	9.88844			0.442667			0.221334	−	0.975198i	$-0.428959\pi$
$499$	9.88844			0.442667			0.221334	+	0.975198i	$0.428959\pi$
$500$		0			0
$501$			11.5530i			0.516149i
$502$		0			0
$503$	−2.73975			−0.122159			−0.0610797	−	0.998133i	$-0.519454\pi$
$503$	−2.73975			−0.122159			−0.0610797	+	0.998133i	$0.519454\pi$
$504$		0			0
$505$			4.50845i			0.200623i
$506$		0			0
$507$		−	0.968020i		−	0.0429913i
$508$		0			0
$509$			20.8021i			0.922037i	0.887391	+	0.461019i	$0.152516\pi$
$509$			20.8021i			0.922037i	−0.887391	+	0.461019i	$0.847484\pi$
$510$		0			0
$511$	9.07578	−	2.75551i	0.401489	−	0.121897i
$512$		0			0
$513$		−	10.9432i		−	0.483152i
$514$		0			0
$515$	6.45571			0.284472
$516$		0			0
$517$	21.8970	+	3.63345i	0.963030	+	0.159799i
$518$		0			0
$519$			12.5936i			0.552799i
$520$		0			0
$521$			12.0959i			0.529930i	0.964258	+	0.264965i	$0.0853605\pi$
$521$			12.0959i			0.529930i	−0.964258	+	0.264965i	$0.914640\pi$
$522$		0			0
$523$	20.8803			0.913030			0.456515	−	0.889716i	$-0.349097\pi$
$523$	20.8803			0.913030			0.456515	+	0.889716i	$0.349097\pi$
$524$		0			0
$525$	−3.22583	−	10.6249i	−0.140787	−	0.463707i
$526$		0			0
$527$		−	1.19281i		−	0.0519598i
$528$		0			0
$529$	−22.9945			−0.999763
$530$		0			0
$531$			11.7939i			0.511811i
$532$		0			0
$533$	−10.5894			−0.458678
$534$		0			0
$535$	−14.6861			−0.634936
$536$		0			0
$537$			19.3588i			0.835394i
$538$		0			0
$539$	−9.57467	−	21.1501i	−0.412410	−	0.910998i
$540$		0			0
$541$			12.9856i			0.558294i	0.960248	+	0.279147i	$0.0900516\pi$
$541$			12.9856i			0.558294i	−0.960248	+	0.279147i	$0.909948\pi$
$542$		0			0
$543$	23.9431			1.02750
$544$		0			0
$545$	4.22821			0.181117
$546$		0			0
$547$			4.67935i			0.200075i	0.994984	+	0.100037i	$0.0318962\pi$
$547$			4.67935i			0.200075i	−0.994984	+	0.100037i	$0.968104\pi$
$548$		0			0
$549$	7.35792			0.314028
$550$		0			0
$551$			4.82336i			0.205482i
$552$		0			0
$553$	7.67627	+	25.2832i	0.326428	+	1.07515i
$554$		0			0
$555$	4.06138			0.172396
$556$		0			0
$557$		−	21.2105i		−	0.898717i	−0.893351	−	0.449359i	$-0.851653\pi$
$557$		−	21.2105i		−	0.898717i	0.893351	−	0.449359i	$-0.148347\pi$
$558$		0			0
$559$			5.58348i			0.236156i
$560$		0			0
$561$	−9.58511	−	1.59049i	−0.404684	−	0.0671506i
$562$		0			0
$563$	35.1839			1.48282			0.741412	−	0.671050i	$-0.234157\pi$
$563$	35.1839			1.48282			0.741412	+	0.671050i	$0.234157\pi$
$564$		0			0
$565$		−	2.49817i		−	0.105099i
$566$		0			0
$567$	3.65702	−	1.11031i	0.153580	−	0.0466288i
$568$		0			0
$569$			36.6831i			1.53784i	0.639347	+	0.768918i	$0.279205\pi$
$569$			36.6831i			1.53784i	−0.639347	+	0.768918i	$0.720795\pi$
$570$		0			0
$571$			42.0114i			1.75812i	0.476708	+	0.879062i	$0.341830\pi$
$571$			42.0114i			1.75812i	−0.476708	+	0.879062i	$0.658170\pi$
$572$		0			0
$573$			5.94651i			0.248419i
$574$		0			0
$575$	−0.320157			−0.0133515
$576$		0			0
$577$			16.7578i			0.697635i	0.937191	+	0.348817i	$0.113417\pi$
$577$			16.7578i			0.697635i	−0.937191	+	0.348817i	$0.886583\pi$
$578$		0			0
$579$	−10.1799			−0.423063
$580$		0			0
$581$	25.6673	−	7.79289i	1.06486	−	0.323303i
$582$		0			0
$583$	0.137292	−	0.827389i	0.00568604	−	0.0342670i
$584$		0			0
$585$			1.68167i			0.0695286i
$586$		0			0
$587$			43.5502i			1.79751i	0.438452	+	0.898755i	$0.355527\pi$
$587$			43.5502i			1.79751i	−0.438452	+	0.898755i	$0.644473\pi$
$588$		0			0
$589$		−	0.880064i		−	0.0362624i
$590$		0			0
$591$	−6.36273			−0.261728
$592$		0			0
$593$	−15.6803			−0.643913			−0.321957	−	0.946754i	$-0.604340\pi$
$593$	−15.6803			−0.643913			−0.321957	+	0.946754i	$0.604340\pi$
$594$		0			0
$595$	−1.89623	−	6.24556i	−0.0777376	−	0.256043i
$596$		0			0
$597$	−0.0834778			−0.00341652
$598$		0			0
$599$	14.8080			0.605038			0.302519	−	0.953143i	$-0.402172\pi$
$599$	14.8080			0.605038			0.302519	+	0.953143i	$0.402172\pi$
$600$		0			0
$601$	12.6609			0.516450			0.258225	−	0.966085i	$-0.416862\pi$
$601$	12.6609			0.516450			0.258225	+	0.966085i	$0.416862\pi$
$602$		0			0
$603$	−21.3073			−0.867700
$604$		0			0
$605$	−2.89612	+	8.48645i	−0.117744	+	0.345023i
$606$		0			0
$607$	12.1019			0.491199			0.245600	−	0.969371i	$-0.421015\pi$
$607$	12.1019			0.491199			0.245600	+	0.969371i	$0.421015\pi$
$608$		0			0
$609$	5.29395	−	1.60731i	0.214522	−	0.0651313i
$610$		0			0
$611$		−	6.69248i		−	0.270749i
$612$		0			0
$613$			30.8781i			1.24716i	0.781761	+	0.623578i	$0.214322\pi$
$613$			30.8781i			1.24716i	−0.781761	+	0.623578i	$0.785678\pi$
$614$		0			0
$615$	−8.35624			−0.336956
$616$		0			0
$617$	−15.3540			−0.618129			−0.309065	−	0.951041i	$-0.600016\pi$
$617$	−15.3540			−0.618129			−0.309065	+	0.951041i	$0.600016\pi$
$618$		0			0
$619$		−	37.4935i		−	1.50699i	−0.657452	−	0.753496i	$-0.728366\pi$
$619$		−	37.4935i		−	1.50699i	0.657452	−	0.753496i	$-0.271634\pi$
$620$		0			0
$621$			0.361920i			0.0145234i
$622$		0			0
$623$	−1.79264	−	5.90439i	−0.0718206	−	0.236554i
$624$		0			0
$625$	15.4737			0.618950
$626$		0			0
$627$	−7.07194	−	1.17347i	−0.282426	−	0.0468640i
$628$		0			0
$629$	−15.5757			−0.621045
$630$		0			0
$631$	11.5041			0.457972			0.228986	−	0.973430i	$-0.426459\pi$
$631$	11.5041			0.457972			0.228986	+	0.973430i	$0.426459\pi$
$632$		0			0
$633$	−17.3223			−0.688501
$634$		0			0
$635$	−8.42947			−0.334513
$636$		0			0
$637$	−5.81840	+	3.89181i	−0.230533	+	0.154199i
$638$		0			0
$639$	−20.3587			−0.805378
$640$		0			0
$641$	−10.1773			−0.401980			−0.200990	−	0.979593i	$-0.564416\pi$
$641$	−10.1773			−0.401980			−0.200990	+	0.979593i	$0.564416\pi$
$642$		0			0
$643$			1.93571i			0.0763369i	0.999271	+	0.0381685i	$0.0121523\pi$
$643$			1.93571i			0.0763369i	−0.999271	+	0.0381685i	$0.987848\pi$
$644$		0			0
$645$			4.40600i			0.173486i
$646$		0			0
$647$		−	23.8842i		−	0.938986i	−0.882936	−	0.469493i	$-0.844437\pi$
$647$		−	23.8842i		−	0.938986i	0.882936	−	0.469493i	$-0.155563\pi$
$648$		0			0
$649$	18.7055	+	3.10387i	0.734255	+	0.121838i
$650$		0			0
$651$	−0.965927	+	0.293266i	−0.0378576	+	0.0114940i
$652$		0			0
$653$	16.7057			0.653746			0.326873	−	0.945068i	$-0.394005\pi$
$653$	16.7057			0.653746			0.326873	+	0.945068i	$0.394005\pi$
$654$		0			0
$655$			4.97035i			0.194208i
$656$		0			0
$657$	7.39551			0.288526
$658$		0			0
$659$		−	35.2080i		−	1.37151i	−0.727833	−	0.685754i	$-0.759473\pi$
$659$		−	35.2080i		−	1.37151i	0.727833	−	0.685754i	$-0.240527\pi$
$660$		0			0
$661$		−	29.5699i		−	1.15013i	−0.818106	−	0.575067i	$-0.804976\pi$
$661$		−	29.5699i		−	1.15013i	0.818106	−	0.575067i	$-0.195024\pi$
$662$		0			0
$663$			2.92954i			0.113774i
$664$		0			0
$665$	−1.39904	−	4.60801i	−0.0542526	−	0.178691i
$666$		0			0
$667$		−	0.159522i		−	0.00617671i
$668$		0			0
$669$	10.0175			0.387300
$670$		0			0
$671$	1.93643	−	11.6699i	0.0747550	−	0.450511i
$672$		0			0
$673$			20.1724i			0.777590i	0.921324	+	0.388795i	$0.127109\pi$
$673$			20.1724i			0.777590i	−0.921324	+	0.388795i	$0.872891\pi$
$674$		0			0
$675$		−	21.2483i		−	0.817846i
$676$		0			0
$677$	22.4574			0.863109			0.431554	−	0.902087i	$-0.357965\pi$
$677$	22.4574			0.863109			0.431554	+	0.902087i	$0.357965\pi$
$678$		0			0
$679$	6.66735	+	21.9601i	0.255869	+	0.842752i
$680$		0			0
$681$		−	6.99713i		−	0.268131i
$682$		0			0
$683$	8.93918			0.342048			0.171024	−	0.985267i	$-0.445292\pi$
$683$	8.93918			0.342048			0.171024	+	0.985267i	$0.445292\pi$
$684$		0			0
$685$			5.32543i			0.203474i
$686$		0			0
$687$	−14.1069			−0.538212
$688$		0			0
$689$	−0.252878			−0.00963389
$690$		0			0
$691$			19.0480i			0.724619i	0.932058	+	0.362309i	$0.118012\pi$
$691$			19.0480i			0.724619i	−0.932058	+	0.362309i	$0.881988\pi$
$692$		0			0
$693$	−2.35261	−	17.9487i	−0.0893684	−	0.681814i
$694$		0			0
$695$		−	7.40880i		−	0.281032i
$696$		0			0
$697$	32.0469			1.21386
$698$		0			0
$699$	−19.3142			−0.730531
$700$		0			0
$701$		−	11.8009i		−	0.445716i	−0.974851	−	0.222858i	$-0.928461\pi$
$701$		−	11.8009i		−	0.445716i	0.974851	−	0.222858i	$-0.0715386\pi$
$702$		0			0
$703$	−11.4919			−0.433423
$704$		0			0
$705$		−	5.28112i		−	0.198898i
$706$		0			0
$707$	−14.0015	+	4.25102i	−0.526580	+	0.159876i
$708$		0			0
$709$	−5.61119			−0.210733			−0.105366	−	0.994433i	$-0.533602\pi$
$709$	−5.61119			−0.210733			−0.105366	+	0.994433i	$0.533602\pi$
$710$		0			0
$711$			20.6023i			0.772648i
$712$		0			0
$713$			0.0291061i			0.00109003i
$714$		0			0
$715$	2.66719	+	0.442576i	0.0997471	+	0.0165514i
$716$		0			0
$717$	−17.8613			−0.667044
$718$		0			0
$719$		−	32.7831i		−	1.22260i	−0.791398	−	0.611301i	$-0.790646\pi$
$719$		−	32.7831i		−	1.22260i	0.791398	−	0.611301i	$-0.209354\pi$
$720$		0			0
$721$	6.08708	+	20.0489i	0.226695	+	0.746660i
$722$		0			0
$723$		−	23.1666i		−	0.861573i
$724$		0			0
$725$			9.36550i			0.347826i
$726$		0			0
$727$		−	11.1584i		−	0.413843i	−0.978358	−	0.206921i	$-0.933656\pi$
$727$		−	11.1584i		−	0.413843i	0.978358	−	0.206921i	$-0.0663444\pi$
$728$		0			0
$729$	−11.2529			−0.416774
$730$		0			0
$731$		−	16.8974i		−	0.624973i
$732$		0			0
$733$	−14.9440			−0.551968			−0.275984	−	0.961162i	$-0.589004\pi$
$733$	−14.9440			−0.551968			−0.275984	+	0.961162i	$0.589004\pi$
$734$		0			0
$735$	−4.59138	+	3.07108i	−0.169355	+	0.113279i
$736$		0			0
$737$	−5.60757	+	33.7940i	−0.206557	+	1.24482i
$738$		0			0
$739$			30.7199i			1.13005i	0.825074	+	0.565025i	$0.191133\pi$
$739$			30.7199i			1.13005i	−0.825074	+	0.565025i	$0.808867\pi$
$740$		0			0
$741$			2.16143i			0.0794019i
$742$		0			0
$743$			11.5012i			0.421937i	0.977493	+	0.210968i	$0.0676617\pi$
$743$			11.5012i			0.421937i	−0.977493	+	0.210968i	$0.932338\pi$
$744$		0			0
$745$	−1.10539			−0.0404982
$746$		0			0
$747$	20.9153			0.765251
$748$		0			0
$749$	−13.8475	−	45.6093i	−0.505977	−	1.66653i
$750$		0			0
$751$	−49.4469			−1.80434			−0.902172	−	0.431377i	$-0.858028\pi$
$751$	−49.4469			−1.80434			−0.902172	+	0.431377i	$0.858028\pi$
$752$		0			0
$753$	13.4419			0.489852
$754$		0			0
$755$	6.95954			0.253284
$756$		0			0
$757$	53.0309			1.92744			0.963720	−	0.266915i	$-0.0860043\pi$
$757$	53.0309			1.92744			0.963720	+	0.266915i	$0.0860043\pi$
$758$		0			0
$759$	0.233888	+	0.0388099i	0.00848960	+	0.00140871i
$760$		0			0
$761$	11.3438			0.411211			0.205606	−	0.978635i	$-0.434084\pi$
$761$	11.3438			0.411211			0.205606	+	0.978635i	$0.434084\pi$
$762$		0			0
$763$	3.98678	+	13.1312i	0.144331	+	0.475381i
$764$		0			0
$765$		−	5.08927i		−	0.184003i
$766$		0			0
$767$		−	5.71703i		−	0.206430i
$768$		0			0
$769$	17.3785			0.626684			0.313342	−	0.949640i	$-0.398551\pi$
$769$	17.3785			0.626684			0.313342	+	0.949640i	$0.398551\pi$
$770$		0			0
$771$	−6.23259			−0.224461
$772$		0			0
$773$			1.77065i			0.0636858i	0.999493	+	0.0318429i	$0.0101376\pi$
$773$			1.77065i			0.0636858i	−0.999493	+	0.0318429i	$0.989862\pi$
$774$		0			0
$775$		−	1.70881i		−	0.0613824i
$776$		0			0
$777$	3.82947	+	12.6130i	0.137381	+	0.452491i
$778$		0			0
$779$	23.6444			0.847148
$780$		0			0
$781$	−5.35792	+	32.2896i	−0.191722	+	1.15541i
$782$		0			0
$783$	10.5872			0.378355
$784$		0			0
$785$	3.78654			0.135147
$786$		0			0
$787$	−29.5600			−1.05370			−0.526850	−	0.849958i	$-0.676627\pi$
$787$	−29.5600			−1.05370			−0.526850	+	0.849958i	$0.676627\pi$
$788$		0			0
$789$	−22.2070			−0.790589
$790$		0			0
$791$	7.75833	−	2.35552i	0.275854	−	0.0837526i
$792$		0			0
$793$	−3.56672			−0.126658
$794$		0			0
$795$	−0.199550			−0.00707729
$796$		0			0
$797$			41.4483i			1.46817i	0.679056	+	0.734087i	$0.262389\pi$
$797$			41.4483i			1.46817i	−0.679056	+	0.734087i	$0.737611\pi$
$798$		0			0
$799$			20.2536i			0.716520i
$800$		0			0
$801$		−	4.81126i		−	0.169998i
$802$		0			0
$803$	1.94632	−	11.7295i	0.0686842	−	0.413926i
$804$		0			0
$805$	0.0462702	+	0.152399i	0.00163081	+	0.00537137i
$806$		0			0
$807$	−27.5836			−0.970987
$808$		0			0
$809$			19.3655i			0.680855i	0.940271	+	0.340428i	$0.110572\pi$
$809$			19.3655i			0.680855i	−0.940271	+	0.340428i	$0.889428\pi$
$810$		0			0
$811$	−15.5530			−0.546139			−0.273070	−	0.961994i	$-0.588039\pi$
$811$	−15.5530			−0.546139			−0.273070	+	0.961994i	$0.588039\pi$
$812$		0			0
$813$		−	1.26254i		−	0.0442791i
$814$		0			0
$815$		−	10.4601i		−	0.366402i
$816$		0			0
$817$		−	12.4670i		−	0.436164i
$818$		0			0
$819$	−5.22262	+	1.58565i	−0.182493	+	0.0554070i
$820$		0			0
$821$			26.3660i			0.920180i	0.887872	+	0.460090i	$0.152183\pi$
$821$			26.3660i			0.920180i	−0.887872	+	0.460090i	$0.847817\pi$
$822$		0			0
$823$	−23.8141			−0.830107			−0.415053	−	0.909797i	$-0.636237\pi$
$823$	−23.8141			−0.830107			−0.415053	+	0.909797i	$0.636237\pi$
$824$		0			0
$825$	−13.7315	−	2.27853i	−0.478071	−	0.0793281i
$826$		0			0
$827$		−	17.5655i		−	0.610814i	−0.952222	−	0.305407i	$-0.901208\pi$
$827$		−	17.5655i		−	0.610814i	0.952222	−	0.305407i	$-0.0987925\pi$
$828$		0			0
$829$			7.99991i			0.277848i	0.990303	+	0.138924i	$0.0443644\pi$
$829$			7.99991i			0.277848i	−0.990303	+	0.138924i	$0.955636\pi$
$830$		0			0
$831$	6.56760			0.227828
$832$		0			0
$833$	17.6083	−	11.7779i	0.610093	−	0.408079i
$834$		0			0
$835$			9.72892i			0.336683i
$836$		0			0
$837$	−1.93172			−0.0667701
$838$		0			0
$839$			51.7122i			1.78530i	0.450746	+	0.892652i	$0.351158\pi$
$839$			51.7122i			1.78530i	−0.450746	+	0.892652i	$0.648842\pi$
$840$		0			0
$841$	24.3335			0.839087
$842$		0			0
$843$	20.9354			0.721054
$844$		0			0
$845$		−	0.815183i		−	0.0280431i
$846$		0			0
$847$	−29.0863	−	0.992344i	−0.999419	−	0.0340973i
$848$		0			0
$849$			21.3051i			0.731190i
$850$		0			0
$851$	0.380067			0.0130285
$852$		0			0
$853$	−28.9818			−0.992317			−0.496159	−	0.868232i	$-0.665257\pi$
$853$	−28.9818			−0.992317			−0.496159	+	0.868232i	$0.665257\pi$
$854$		0			0
$855$		−	3.75489i		−	0.128415i
$856$		0			0
$857$	−25.0555			−0.855880			−0.427940	−	0.903807i	$-0.640761\pi$
$857$	−25.0555			−0.855880			−0.427940	+	0.903807i	$0.640761\pi$
$858$		0			0
$859$			30.6875i			1.04704i	0.852012	+	0.523522i	$0.175382\pi$
$859$			30.6875i			1.04704i	−0.852012	+	0.523522i	$0.824618\pi$
$860$		0			0
$861$	−7.87909	−	25.9512i	−0.268519	−	0.884416i
$862$		0			0
$863$	43.9984			1.49772			0.748861	−	0.662727i	$-0.230601\pi$
$863$	43.9984			1.49772			0.748861	+	0.662727i	$0.230601\pi$
$864$		0			0
$865$			10.6053i			0.360590i
$866$		0			0
$867$			7.59063i			0.257791i
$868$		0			0
$869$	32.6760	+	5.42204i	1.10846	+	0.183930i
$870$		0			0
$871$	10.3286			0.349972
$872$		0			0
$873$			17.8945i			0.605636i
$874$		0			0
$875$	−5.84941	−	19.2661i	−0.197746	−	0.651313i
$876$		0			0
$877$			32.1354i			1.08513i	0.840013	+	0.542567i	$0.182547\pi$
$877$			32.1354i			1.08513i	−0.840013	+	0.542567i	$0.817453\pi$
$878$		0			0
$879$		−	6.66289i		−	0.224734i
$880$		0			0
$881$			4.33094i			0.145913i	0.997335	+	0.0729565i	$0.0232434\pi$
$881$			4.33094i			0.145913i	−0.997335	+	0.0729565i	$0.976757\pi$
$882$		0			0
$883$	−19.3947			−0.652682			−0.326341	−	0.945252i	$-0.605816\pi$
$883$	−19.3947			−0.652682			−0.326341	+	0.945252i	$0.605816\pi$
$884$		0			0
$885$		−	4.51139i		−	0.151649i
$886$		0			0
$887$	−11.6256			−0.390349			−0.195175	−	0.980769i	$-0.562527\pi$
$887$	−11.6256			−0.390349			−0.195175	+	0.980769i	$0.562527\pi$
$888$		0			0
$889$	−7.94814	−	26.1786i	−0.266572	−	0.878004i
$890$		0			0
$891$	0.784257	−	4.72633i	0.0262736	−	0.158338i
$892$		0			0
$893$			14.9432i			0.500054i
$894$		0			0
$895$			16.3023i			0.544927i
$896$		0			0
$897$		−	0.0714842i		−	0.00238679i
$898$		0			0
$899$	0.851436			0.0283970
$900$		0			0
$901$	0.765290			0.0254955
$902$		0			0
$903$	−13.6833	+	4.15441i	−0.455352	+	0.138250i
$904$		0			0
$905$	20.1628			0.670236
$906$		0			0
$907$	24.9601			0.828785			0.414393	−	0.910098i	$-0.363994\pi$
$907$	24.9601			0.828785			0.414393	+	0.910098i	$0.363994\pi$
$908$		0			0
$909$	−11.4093			−0.378422
$910$		0			0
$911$	32.5570			1.07866			0.539331	−	0.842094i	$-0.318677\pi$
$911$	32.5570			1.07866			0.539331	+	0.842094i	$0.318677\pi$
$912$		0			0
$913$	5.50441	−	33.1724i	0.182169	−	1.09785i
$914$		0			0
$915$	−2.81454			−0.0930459
$916$		0			0
$917$	−15.4360	+	4.68654i	−0.509741	+	0.154763i
$918$		0			0
$919$		−	1.59576i		−	0.0526394i	−0.999654	−	0.0263197i	$-0.991621\pi$
$919$		−	1.59576i		−	0.0526394i	0.999654	−	0.0263197i	$-0.00837878\pi$
$920$		0			0
$921$			18.2480i			0.601292i
$922$		0			0
$923$	9.86879			0.324835
$924$		0			0
$925$	−22.3137			−0.733669
$926$		0			0
$927$			16.3371i			0.536581i
$928$		0			0
$929$			36.6141i			1.20127i	0.799524	+	0.600635i	$0.205085\pi$
$929$			36.6141i			1.20127i	−0.799524	+	0.600635i	$0.794915\pi$
$930$		0			0
$931$	12.9915	−	8.68977i	0.425780	−	0.284796i
$932$		0			0
$933$	−18.2948			−0.598945
$934$		0			0
$935$	−8.07175	−	1.33938i	−0.263975	−	0.0438023i
$936$		0			0
$937$	−16.6500			−0.543930			−0.271965	−	0.962307i	$-0.587674\pi$
$937$	−16.6500			−0.543930			−0.271965	+	0.962307i	$0.587674\pi$
$938$		0			0
$939$	8.79853			0.287129
$940$		0			0
$941$	−0.852263			−0.0277830			−0.0138915	−	0.999904i	$-0.504422\pi$
$941$	−0.852263			−0.0277830			−0.0138915	+	0.999904i	$0.504422\pi$
$942$		0			0
$943$	−0.781984			−0.0254649
$944$		0			0
$945$	−10.1145	+	3.07087i	−0.329024	+	0.0998955i
$946$		0			0
$947$	−44.3899			−1.44248			−0.721239	−	0.692686i	$-0.756427\pi$
$947$	−44.3899			−1.44248			−0.721239	+	0.692686i	$0.756427\pi$
$948$		0			0
$949$	−3.58494			−0.116372
$950$		0			0
$951$			6.40681i			0.207755i
$952$		0			0
$953$			20.1491i			0.652692i	0.945250	+	0.326346i	$0.105817\pi$
$953$			20.1491i			0.652692i	−0.945250	+	0.326346i	$0.894183\pi$
$954$		0			0
$955$			5.00764i			0.162043i
$956$		0			0
$957$	1.13530	−	6.84189i	0.0366991	−	0.221167i
$958$		0			0
$959$	−16.5387	+	5.02135i	−0.534063	+	0.162148i
$960$		0			0
$961$	30.8446			0.994989
$962$		0			0
$963$		−	37.1653i		−	1.19764i
$964$		0			0
$965$	−8.57265			−0.275963
$966$		0			0
$967$			53.6310i			1.72466i	0.506350	+	0.862328i	$0.330994\pi$
$967$			53.6310i			1.72466i	−0.506350	+	0.862328i	$0.669006\pi$
$968$		0			0
$969$		−	6.54116i		−	0.210132i
$970$		0			0
$971$			11.6508i			0.373893i	0.982370	+	0.186947i	$0.0598592\pi$
$971$			11.6508i			0.373893i	−0.982370	+	0.186947i	$0.940141\pi$
$972$		0			0
$973$	23.0089	−	6.98575i	0.737630	−	0.223953i
$974$		0			0
$975$			4.19683i			0.134406i
$976$		0			0
$977$	17.4141			0.557127			0.278564	−	0.960418i	$-0.410142\pi$
$977$	17.4141			0.557127			0.278564	+	0.960418i	$0.410142\pi$
$978$		0			0
$979$	−7.63082	−	1.26621i	−0.243882	−	0.0404682i
$980$		0			0
$981$			10.7001i			0.341628i
$982$		0			0
$983$		−	47.3090i		−	1.50892i	−0.656344	−	0.754462i	$-0.727898\pi$
$983$		−	47.3090i		−	1.50892i	0.656344	−	0.754462i	$-0.272102\pi$
$984$		0			0
$985$	−5.35815			−0.170725
$986$		0			0
$987$	16.4011	−	4.97956i	0.522053	−	0.158501i
$988$		0			0
$989$			0.412317i			0.0131109i
$990$		0			0
$991$	53.3527			1.69481			0.847403	−	0.530950i	$-0.178165\pi$
$991$	53.3527			1.69481			0.847403	+	0.530950i	$0.178165\pi$
$992$		0			0
$993$			7.98300i			0.253333i
$994$		0			0
$995$	−0.0702978			−0.00222859
$996$		0			0
$997$	−48.0246			−1.52095			−0.760477	−	0.649365i	$-0.775035\pi$
$997$	−48.0246			−1.52095			−0.760477	+	0.649365i	$0.775035\pi$
$998$		0			0
$999$			25.2244i			0.798065i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.e.b.3849.18	yes	48
7.6	odd	2		4004.2.e.a.3849.31	yes	48
11.10	odd	2		4004.2.e.a.3849.18	✓	48
77.76	even	2	inner	4004.2.e.b.3849.31	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.e.a.3849.18	✓	48	11.10	odd	2
4004.2.e.a.3849.31	yes	48	7.6	odd	2
4004.2.e.b.3849.18	yes	48	1.1	even	1	trivial
4004.2.e.b.3849.31	yes	48	77.76	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.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.968020i q^{3} -0.815183i q^{5} +(2.53164 - 0.768635i) q^{7} +2.06294 q^{9} +O(q^{10})\)	\(q-0.968020i q^{3} -0.815183i q^{5} +(2.53164 - 0.768635i) q^{7} +2.06294 q^{9} +(0.542916 - 3.27189i) q^{11} -1.00000 q^{13} -0.789113 q^{15} +3.02632 q^{17} +2.23283 q^{19} +(-0.744054 - 2.45068i) q^{21} -0.0738458 q^{23} +4.33548 q^{25} -4.90102i q^{27} +2.16020i q^{29} -0.394147i q^{31} +(-3.16725 - 0.525553i) q^{33} +(-0.626578 - 2.06375i) q^{35} -5.14676 q^{37} +0.968020i q^{39} +10.5894 q^{41} -5.58348i q^{43} -1.68167i q^{45} +6.69248i q^{47} +(5.81840 - 3.89181i) q^{49} -2.92954i q^{51} +0.252878 q^{53} +(-2.66719 - 0.442576i) q^{55} -2.16143i q^{57} +5.71703i q^{59} +3.56672 q^{61} +(5.22262 - 1.58565i) q^{63} +0.815183i q^{65} -10.3286 q^{67} +0.0714842i q^{69} -9.86879 q^{71} +3.58494 q^{73} -4.19683i q^{75} +(-1.14042 - 8.70054i) q^{77} +9.98689i q^{79} +1.44453 q^{81} +10.1386 q^{83} -2.46700i q^{85} +2.09112 q^{87} -2.33224i q^{89} +(-2.53164 + 0.768635i) q^{91} -0.381542 q^{93} -1.82017i q^{95} +8.67427i q^{97} +(1.12000 - 6.74970i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{7} - 48 q^{9}+O(q^{10}) \) 48 * q + 4 * q^7 - 48 * q^9	\( 48 q + 4 q^{7} - 48 q^{9} + 2 q^{11} - 48 q^{13} + 8 q^{15} - 4 q^{17} - 10 q^{21} + 4 q^{23} - 44 q^{25} - 10 q^{33} + 14 q^{35} - 16 q^{37} + 10 q^{49} - 8 q^{53} + 12 q^{55} - 16 q^{61} - 16 q^{63} + 4 q^{67} + 16 q^{73} + 22 q^{77} + 64 q^{81} + 4 q^{83} + 12 q^{87} - 4 q^{91} + 36 q^{93} - 40 q^{99}+O(q^{100}) \) 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 + 22 * 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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