LMFDB - Embedded newform 4004.2.e.b.3849.15

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			3849.15
Character	$\chi$	$=$	4004.3849
Dual form			4004.2.e.b.3849.34

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.12498i q^{3} -0.542613i q^{5} +(-1.59076 + 2.11411i) q^{7} +1.73443 q^{9} +O(q^{10})$	$q-1.12498i q^{3} -0.542613i q^{5} +(-1.59076 + 2.11411i) q^{7} +1.73443 q^{9} +(-3.31290 - 0.157052i) q^{11} -1.00000 q^{13} -0.610427 q^{15} -4.02265 q^{17} +5.33379 q^{19} +(2.37833 + 1.78957i) q^{21} -0.782280 q^{23} +4.70557 q^{25} -5.32612i q^{27} +1.41746i q^{29} +1.44439i q^{31} +(-0.176679 + 3.72694i) q^{33} +(1.14715 + 0.863168i) q^{35} +7.61353 q^{37} +1.12498i q^{39} -2.96215 q^{41} +1.89524i q^{43} -0.941124i q^{45} -12.1838i q^{47} +(-1.93896 - 6.72610i) q^{49} +4.52539i q^{51} -8.00267 q^{53} +(-0.0852184 + 1.79763i) q^{55} -6.00039i q^{57} -11.0063i q^{59} -2.17603 q^{61} +(-2.75906 + 3.66678i) q^{63} +0.542613i q^{65} -4.82061 q^{67} +0.880046i q^{69} -7.39107 q^{71} +2.62679 q^{73} -5.29365i q^{75} +(5.60207 - 6.75403i) q^{77} -11.5302i q^{79} -0.788469 q^{81} -8.51333 q^{83} +2.18274i q^{85} +1.59461 q^{87} -4.14290i q^{89} +(1.59076 - 2.11411i) q^{91} +1.62491 q^{93} -2.89419i q^{95} -2.96598i q^{97} +(-5.74600 - 0.272395i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	1.12498i		−	0.649505i	−0.945799	−	0.324753i	$-0.894719\pi$
$3$		−	1.12498i		−	0.649505i	0.945799	−	0.324753i	$-0.105281\pi$
$4$		0			0
$5$		−	0.542613i		−	0.242664i	−0.992612	−	0.121332i	$-0.961283\pi$
$5$		−	0.542613i		−	0.242664i	0.992612	−	0.121332i	$-0.0387166\pi$
$6$		0			0
$7$	−1.59076	+	2.11411i	−0.601251	+	0.799060i
$7$	−1.59076	+	2.11411i	−0.601251	+	0.799060i
$8$		0			0
$9$	1.73443			0.578143
$10$		0			0
$11$	−3.31290	−	0.157052i	−0.998878	−	0.0473529i
$11$	−3.31290	−	0.157052i	−0.998878	−	0.0473529i
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$	−0.610427			−0.157612
$16$		0			0
$17$	−4.02265			−0.975636			−0.487818	−	0.872945i	$-0.662207\pi$
$17$	−4.02265			−0.975636			−0.487818	+	0.872945i	$0.662207\pi$
$18$		0			0
$19$	5.33379			1.22366			0.611828	−	0.790991i	$-0.290434\pi$
$19$	5.33379			1.22366			0.611828	+	0.790991i	$0.290434\pi$
$20$		0			0
$21$	2.37833	+	1.78957i	0.518994	+	0.390516i
$22$		0			0
$23$	−0.782280			−0.163117			−0.0815583	−	0.996669i	$-0.525990\pi$
$23$	−0.782280			−0.163117			−0.0815583	+	0.996669i	$0.525990\pi$
$24$		0			0
$25$	4.70557			0.941114
$26$		0			0
$27$		−	5.32612i		−	1.02501i
$28$		0			0
$29$			1.41746i			0.263216i	0.991302	+	0.131608i	$0.0420140\pi$
$29$			1.41746i			0.263216i	−0.991302	+	0.131608i	$0.957986\pi$
$30$		0			0
$31$			1.44439i			0.259420i	0.991552	+	0.129710i	$0.0414047\pi$
$31$			1.44439i			0.259420i	−0.991552	+	0.129710i	$0.958595\pi$
$32$		0			0
$33$	−0.176679	+	3.72694i	−0.0307559	+	0.648777i
$34$		0			0
$35$	1.14715	+	0.863168i	0.193903	+	0.145902i
$36$		0			0
$37$	7.61353			1.25166			0.625828	−	0.779961i	$-0.284761\pi$
$37$	7.61353			1.25166			0.625828	+	0.779961i	$0.284761\pi$
$38$		0			0
$39$			1.12498i			0.180140i
$40$		0			0
$41$	−2.96215			−0.462610			−0.231305	−	0.972881i	$-0.574300\pi$
$41$	−2.96215			−0.462610			−0.231305	+	0.972881i	$0.574300\pi$
$42$		0			0
$43$			1.89524i			0.289022i	0.989503	+	0.144511i	$0.0461609\pi$
$43$			1.89524i			0.289022i	−0.989503	+	0.144511i	$0.953839\pi$
$44$		0			0
$45$		−	0.941124i		−	0.140295i
$46$		0			0
$47$		−	12.1838i		−	1.77719i	−0.458695	−	0.888594i	$-0.651683\pi$
$47$		−	12.1838i		−	1.77719i	0.458695	−	0.888594i	$-0.348317\pi$
$48$		0			0
$49$	−1.93896	−	6.72610i	−0.276994	−	0.960872i
$50$		0			0
$51$			4.52539i			0.633681i
$52$		0			0
$53$	−8.00267			−1.09925			−0.549625	−	0.835411i	$-0.685230\pi$
$53$	−8.00267			−1.09925			−0.549625	+	0.835411i	$0.685230\pi$
$54$		0			0
$55$	−0.0852184	+	1.79763i	−0.0114908	+	0.242392i
$56$		0			0
$57$		−	6.00039i		−	0.794771i
$58$		0			0
$59$		−	11.0063i		−	1.43289i	−0.697642	−	0.716447i	$-0.745767\pi$
$59$		−	11.0063i		−	1.43289i	0.697642	−	0.716447i	$-0.254233\pi$
$60$		0			0
$61$	−2.17603			−0.278612			−0.139306	−	0.990249i	$-0.544487\pi$
$61$	−2.17603			−0.278612			−0.139306	+	0.990249i	$0.544487\pi$
$62$		0			0
$63$	−2.75906	+	3.66678i	−0.347609	+	0.461971i
$64$		0			0
$65$			0.542613i			0.0673029i
$66$		0			0
$67$	−4.82061			−0.588931			−0.294466	−	0.955662i	$-0.595142\pi$
$67$	−4.82061			−0.588931			−0.294466	+	0.955662i	$0.595142\pi$
$68$		0			0
$69$			0.880046i			0.105945i
$70$		0			0
$71$	−7.39107			−0.877159			−0.438579	−	0.898692i	$-0.644518\pi$
$71$	−7.39107			−0.877159			−0.438579	+	0.898692i	$0.644518\pi$
$72$		0			0
$73$	2.62679			0.307442			0.153721	−	0.988114i	$-0.450874\pi$
$73$	2.62679			0.307442			0.153721	+	0.988114i	$0.450874\pi$
$74$		0			0
$75$		−	5.29365i		−	0.611259i
$76$		0			0
$77$	5.60207	−	6.75403i	0.638415	−	0.769693i
$78$		0			0
$79$		−	11.5302i		−	1.29725i	−0.761107	−	0.648626i	$-0.775344\pi$
$79$		−	11.5302i		−	1.29725i	0.761107	−	0.648626i	$-0.224656\pi$
$80$		0			0
$81$	−0.788469			−0.0876076
$82$		0			0
$83$	−8.51333			−0.934460			−0.467230	−	0.884136i	$-0.654748\pi$
$83$	−8.51333			−0.934460			−0.467230	+	0.884136i	$0.654748\pi$
$84$		0			0
$85$			2.18274i			0.236752i
$86$		0			0
$87$	1.59461			0.170960
$88$		0			0
$89$		−	4.14290i		−	0.439147i	−0.975596	−	0.219573i	$-0.929533\pi$
$89$		−	4.14290i		−	0.439147i	0.975596	−	0.219573i	$-0.0704665\pi$
$90$		0			0
$91$	1.59076	−	2.11411i	0.166757	−	0.221619i
$92$		0			0
$93$	1.62491			0.168495
$94$		0			0
$95$		−	2.89419i		−	0.296937i
$96$		0			0
$97$		−	2.96598i		−	0.301149i	−0.988599	−	0.150575i	$-0.951888\pi$
$97$		−	2.96598i		−	0.301149i	0.988599	−	0.150575i	$-0.0481124\pi$
$98$		0			0
$99$	−5.74600	−	0.272395i	−0.577494	−	0.0273767i
$100$		0			0
$101$	−5.31205			−0.528568			−0.264284	−	0.964445i	$-0.585136\pi$
$101$	−5.31205			−0.528568			−0.264284	+	0.964445i	$0.585136\pi$
$102$		0			0
$103$			1.15611i			0.113915i	0.998377	+	0.0569577i	$0.0181400\pi$
$103$			1.15611i			0.113915i	−0.998377	+	0.0569577i	$0.981860\pi$
$104$		0			0
$105$	0.971044	−	1.29051i	0.0947641	−	0.125941i
$106$		0			0
$107$		−	12.6495i		−	1.22288i	−0.791292	−	0.611439i	$-0.790591\pi$
$107$		−	12.6495i		−	1.22288i	0.791292	−	0.611439i	$-0.209409\pi$
$108$		0			0
$109$			11.8751i			1.13743i	0.822534	+	0.568716i	$0.192559\pi$
$109$			11.8751i			1.13743i	−0.822534	+	0.568716i	$0.807441\pi$
$110$		0			0
$111$		−	8.56503i		−	0.812957i
$112$		0			0
$113$	4.52126			0.425325			0.212662	−	0.977126i	$-0.431787\pi$
$113$	4.52126			0.425325			0.212662	+	0.977126i	$0.431787\pi$
$114$		0			0
$115$			0.424476i			0.0395825i
$116$		0			0
$117$	−1.73443			−0.160348
$118$		0			0
$119$	6.39908	−	8.50435i	0.586603	−	0.779592i
$120$		0			0
$121$	10.9507	+	1.04059i	0.995515	+	0.0945995i
$122$		0			0
$123$			3.33235i			0.300468i
$124$		0			0
$125$		−	5.26637i		−	0.471039i
$126$		0			0
$127$		−	7.87244i		−	0.698566i	−0.937017	−	0.349283i	$-0.886425\pi$
$127$		−	7.87244i		−	0.698566i	0.937017	−	0.349283i	$-0.113575\pi$
$128$		0			0
$129$	2.13210			0.187721
$130$		0			0
$131$	5.78032			0.505029			0.252514	−	0.967593i	$-0.418743\pi$
$131$	5.78032			0.505029			0.252514	+	0.967593i	$0.418743\pi$
$132$		0			0
$133$	−8.48479	+	11.2762i	−0.735725	+	0.977775i
$134$		0			0
$135$	−2.89002			−0.248734
$136$		0			0
$137$	−1.88191			−0.160783			−0.0803913	−	0.996763i	$-0.525617\pi$
$137$	−1.88191			−0.160783			−0.0803913	+	0.996763i	$0.525617\pi$
$138$		0			0
$139$	9.06829			0.769162			0.384581	−	0.923091i	$-0.374346\pi$
$139$	9.06829			0.769162			0.384581	+	0.923091i	$0.374346\pi$
$140$		0			0
$141$	−13.7065			−1.15429
$142$		0			0
$143$	3.31290	+	0.157052i	0.277039	+	0.0131333i
$144$		0			0
$145$	0.769133			0.0638730
$146$		0			0
$147$	−7.56670	+	2.18128i	−0.624091	+	0.179909i
$148$		0			0
$149$			11.7787i			0.964948i	0.875910	+	0.482474i	$0.160262\pi$
$149$			11.7787i			0.964948i	−0.875910	+	0.482474i	$0.839738\pi$
$150$		0			0
$151$		−	16.6944i		−	1.35857i	−0.733873	−	0.679287i	$-0.762289\pi$
$151$		−	16.6944i		−	1.35857i	0.733873	−	0.679287i	$-0.237711\pi$
$152$		0			0
$153$	−6.97700			−0.564057
$154$		0			0
$155$	0.783746			0.0629520
$156$		0			0
$157$		−	17.0692i		−	1.36227i	−0.732158	−	0.681135i	$-0.761487\pi$
$157$		−	17.0692i		−	1.36227i	0.732158	−	0.681135i	$-0.238513\pi$
$158$		0			0
$159$			9.00281i			0.713969i
$160$		0			0
$161$	1.24442	−	1.65383i	0.0980741	−	0.130340i
$162$		0			0
$163$	−11.7898			−0.923446			−0.461723	−	0.887024i	$-0.652769\pi$
$163$	−11.7898			−0.923446			−0.461723	+	0.887024i	$0.652769\pi$
$164$		0			0
$165$	2.02229	+	0.0958686i	0.157435	+	0.00746336i
$166$		0			0
$167$	−10.9134			−0.844501			−0.422250	−	0.906479i	$-0.638760\pi$
$167$	−10.9134			−0.844501			−0.422250	+	0.906479i	$0.638760\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	9.25109			0.707448
$172$		0			0
$173$	7.00506			0.532585			0.266293	−	0.963892i	$-0.414201\pi$
$173$	7.00506			0.532585			0.266293	+	0.963892i	$0.414201\pi$
$174$		0			0
$175$	−7.48544	+	9.94811i	−0.565846	+	0.752007i
$176$		0			0
$177$	−12.3818			−0.930672
$178$		0			0
$179$	−12.5303			−0.936560			−0.468280	−	0.883580i	$-0.655126\pi$
$179$	−12.5303			−0.936560			−0.468280	+	0.883580i	$0.655126\pi$
$180$		0			0
$181$			0.912460i			0.0678226i	0.999425	+	0.0339113i	$0.0107964\pi$
$181$			0.912460i			0.0678226i	−0.999425	+	0.0339113i	$0.989204\pi$
$182$		0			0
$183$			2.44798i			0.180960i
$184$		0			0
$185$		−	4.13120i		−	0.303732i
$186$		0			0
$187$	13.3267	+	0.631764i	0.974542	+	0.0461992i
$188$		0			0
$189$	11.2600	+	8.47258i	0.819046	+	0.616290i
$190$		0			0
$191$	−6.07399			−0.439498			−0.219749	−	0.975556i	$-0.570524\pi$
$191$	−6.07399			−0.439498			−0.219749	+	0.975556i	$0.570524\pi$
$192$		0			0
$193$		−	19.1088i		−	1.37548i	−0.725955	−	0.687742i	$-0.758602\pi$
$193$		−	19.1088i		−	1.37548i	0.725955	−	0.687742i	$-0.241398\pi$
$194$		0			0
$195$	0.610427			0.0437136
$196$		0			0
$197$			26.7361i			1.90487i	0.304741	+	0.952435i	$0.401430\pi$
$197$			26.7361i			1.90487i	−0.304741	+	0.952435i	$0.598570\pi$
$198$		0			0
$199$		−	4.99775i		−	0.354281i	−0.984186	−	0.177141i	$-0.943315\pi$
$199$		−	4.99775i		−	0.354281i	0.984186	−	0.177141i	$-0.0566847\pi$
$200$		0			0
$201$			5.42307i			0.382514i
$202$		0			0
$203$	−2.99667	−	2.25484i	−0.210325	−	0.158259i
$204$		0			0
$205$			1.60730i			0.112259i
$206$		0			0
$207$	−1.35681			−0.0943048
$208$		0			0
$209$	−17.6703	−	0.837682i	−1.22228	−	0.0579436i
$210$		0			0
$211$		−	3.87824i		−	0.266989i	−0.991050	−	0.133495i	$-0.957380\pi$
$211$		−	3.87824i		−	0.266989i	0.991050	−	0.133495i	$-0.0426199\pi$
$212$		0			0
$213$			8.31478i			0.569719i
$214$		0			0
$215$	1.02838			0.0701352
$216$		0			0
$217$	−3.05361	−	2.29768i	−0.207292	−	0.155977i
$218$		0			0
$219$		−	2.95508i		−	0.199685i
$220$		0			0
$221$	4.02265			0.270593
$222$		0			0
$223$			9.18192i			0.614867i	0.951570	+	0.307434i	$0.0994702\pi$
$223$			9.18192i			0.614867i	−0.951570	+	0.307434i	$0.900530\pi$
$224$		0			0
$225$	8.16148			0.544099
$226$		0			0
$227$	−6.46700			−0.429230			−0.214615	−	0.976699i	$-0.568850\pi$
$227$	−6.46700			−0.429230			−0.214615	+	0.976699i	$0.568850\pi$
$228$		0			0
$229$		−	24.5452i		−	1.62199i	−0.585052	−	0.810996i	$-0.698926\pi$
$229$		−	24.5452i		−	1.62199i	0.585052	−	0.810996i	$-0.301074\pi$
$230$		0			0
$231$	−7.59812	−	6.30219i	−0.499919	−	0.414654i
$232$		0			0
$233$		−	20.2225i		−	1.32482i	−0.749141	−	0.662410i	$-0.769534\pi$
$233$		−	20.2225i		−	1.32482i	0.749141	−	0.662410i	$-0.230466\pi$
$234$		0			0
$235$	−6.61108			−0.431260
$236$		0			0
$237$	−12.9712			−0.842572
$238$		0			0
$239$			22.1695i			1.43403i	0.697060	+	0.717013i	$0.254491\pi$
$239$			22.1695i			1.43403i	−0.697060	+	0.717013i	$0.745509\pi$
$240$		0			0
$241$	28.2847			1.82198			0.910990	−	0.412428i	$-0.135319\pi$
$241$	28.2847			1.82198			0.910990	+	0.412428i	$0.135319\pi$
$242$		0			0
$243$		−	15.0913i		−	0.968110i
$244$		0			0
$245$	−3.64967	+	1.05210i	−0.233169	+	0.0672165i
$246$		0			0
$247$	−5.33379			−0.339381
$248$		0			0
$249$			9.57729i			0.606936i
$250$		0			0
$251$		−	25.7507i		−	1.62537i	−0.582703	−	0.812685i	$-0.698005\pi$
$251$		−	25.7507i		−	1.62537i	0.582703	−	0.812685i	$-0.301995\pi$
$252$		0			0
$253$	2.59162	+	0.122858i	0.162934	+	0.00772404i
$254$		0			0
$255$	2.45554			0.153772
$256$		0			0
$257$		−	5.71650i		−	0.356586i	−0.983977	−	0.178293i	$-0.942943\pi$
$257$		−	5.71650i		−	0.356586i	0.983977	−	0.178293i	$-0.0570574\pi$
$258$		0			0
$259$	−12.1113	+	16.0959i	−0.752560	+	1.00015i
$260$		0			0
$261$			2.45848i			0.152176i
$262$		0			0
$263$		−	26.0902i		−	1.60879i	−0.594096	−	0.804394i	$-0.702490\pi$
$263$		−	26.0902i		−	1.60879i	0.594096	−	0.804394i	$-0.297510\pi$
$264$		0			0
$265$			4.34235i			0.266749i
$266$		0			0
$267$	−4.66066			−0.285228
$268$		0			0
$269$			3.47981i			0.212168i	0.994357	+	0.106084i	$0.0338312\pi$
$269$			3.47981i			0.212168i	−0.994357	+	0.106084i	$0.966169\pi$
$270$		0			0
$271$	−7.53591			−0.457774			−0.228887	−	0.973453i	$-0.573509\pi$
$271$	−7.53591			−0.457774			−0.228887	+	0.973453i	$0.573509\pi$
$272$		0			0
$273$	−2.37833	−	1.78957i	−0.143943	−	0.108310i
$274$		0			0
$275$	−15.5891	−	0.739018i	−0.940058	−	0.0445645i
$276$		0			0
$277$		−	7.04518i		−	0.423304i	−0.977345	−	0.211652i	$-0.932116\pi$
$277$		−	7.04518i		−	0.423304i	0.977345	−	0.211652i	$-0.0678843\pi$
$278$		0			0
$279$			2.50519i			0.149982i
$280$		0			0
$281$			14.8849i			0.887958i	0.896037	+	0.443979i	$0.146434\pi$
$281$			14.8849i			0.887958i	−0.896037	+	0.443979i	$0.853566\pi$
$282$		0			0
$283$	−14.0845			−0.837240			−0.418620	−	0.908162i	$-0.637486\pi$
$283$	−14.0845			−0.837240			−0.418620	+	0.908162i	$0.637486\pi$
$284$		0			0
$285$	−3.25589			−0.192862
$286$		0			0
$287$	4.71207	−	6.26232i	0.278145	−	0.369653i
$288$		0			0
$289$	−0.818271			−0.0481336
$290$		0			0
$291$	−3.33665			−0.195598
$292$		0			0
$293$	7.36150			0.430064			0.215032	−	0.976607i	$-0.431014\pi$
$293$	7.36150			0.430064			0.215032	+	0.976607i	$0.431014\pi$
$294$		0			0
$295$	−5.97215			−0.347712
$296$		0			0
$297$	−0.836476	+	17.6449i	−0.0485373	+	1.02386i
$298$		0			0
$299$	0.782280			0.0452404
$300$		0			0
$301$	−4.00676	−	3.01488i	−0.230946	−	0.173775i
$302$		0			0
$303$			5.97592i			0.343308i
$304$		0			0
$305$			1.18074i			0.0676091i
$306$		0			0
$307$	−30.1790			−1.72241			−0.861204	−	0.508260i	$-0.830289\pi$
$307$	−30.1790			−1.72241			−0.861204	+	0.508260i	$0.830289\pi$
$308$		0			0
$309$	1.30060			0.0739886
$310$		0			0
$311$			5.09899i			0.289137i	0.989495	+	0.144569i	$0.0461794\pi$
$311$			5.09899i			0.289137i	−0.989495	+	0.144569i	$0.953821\pi$
$312$		0			0
$313$		−	2.00608i		−	0.113390i	−0.998392	−	0.0566952i	$-0.981944\pi$
$313$		−	2.00608i		−	0.113390i	0.998392	−	0.0566952i	$-0.0180563\pi$
$314$		0			0
$315$	1.98964	+	1.49710i	0.112104	+	0.0843523i
$316$		0			0
$317$	−29.3106			−1.64625			−0.823123	−	0.567864i	$-0.807770\pi$
$317$	−29.3106			−1.64625			−0.823123	+	0.567864i	$0.807770\pi$
$318$		0			0
$319$	0.222615	−	4.69591i	0.0124640	−	0.262921i
$320$		0			0
$321$	−14.2304			−0.794266
$322$		0			0
$323$	−21.4560			−1.19384
$324$		0			0
$325$	−4.70557			−0.261018
$326$		0			0
$327$	13.3592			0.738767
$328$		0			0
$329$	25.7579	+	19.3815i	1.42008	+	1.06854i
$330$		0			0
$331$	−26.3434			−1.44796			−0.723981	−	0.689820i	$-0.757690\pi$
$331$	−26.3434			−1.44796			−0.723981	+	0.689820i	$0.757690\pi$
$332$		0			0
$333$	13.2051			0.723636
$334$		0			0
$335$			2.61573i			0.142912i
$336$		0			0
$337$		−	15.6781i		−	0.854041i	−0.904242	−	0.427021i	$-0.859563\pi$
$337$		−	15.6781i		−	0.854041i	0.904242	−	0.427021i	$-0.140437\pi$
$338$		0			0
$339$		−	5.08631i		−	0.276251i
$340$		0			0
$341$	0.226844	−	4.78513i	0.0122843	−	0.259129i
$342$		0			0
$343$	17.3042	+	6.60045i	0.934337	+	0.356391i
$344$		0			0
$345$	0.477525			0.0257091
$346$		0			0
$347$			18.5310i			0.994797i	0.867522	+	0.497399i	$0.165711\pi$
$347$			18.5310i			0.994797i	−0.867522	+	0.497399i	$0.834289\pi$
$348$		0			0
$349$	−11.8196			−0.632687			−0.316343	−	0.948645i	$-0.602455\pi$
$349$	−11.8196			−0.632687			−0.316343	+	0.948645i	$0.602455\pi$
$350$		0			0
$351$			5.32612i			0.284287i
$352$		0			0
$353$		−	6.15971i		−	0.327848i	−0.986473	−	0.163924i	$-0.947585\pi$
$353$		−	6.15971i		−	0.327848i	0.986473	−	0.163924i	$-0.0524153\pi$
$354$		0			0
$355$			4.01049i			0.212855i
$356$		0			0
$357$	−9.56718	−	7.19881i	−0.506349	−	0.381001i
$358$		0			0
$359$			18.9343i			0.999313i	0.866224	+	0.499656i	$0.166540\pi$
$359$			18.9343i			0.999313i	−0.866224	+	0.499656i	$0.833460\pi$
$360$		0			0
$361$	9.44936			0.497335
$362$		0			0
$363$	1.17064	−	12.3192i	0.0614429	−	0.646592i
$364$		0			0
$365$		−	1.42533i		−	0.0746052i
$366$		0			0
$367$		−	1.56290i		−	0.0815825i	−0.999168	−	0.0407912i	$-0.987012\pi$
$367$		−	1.56290i		−	0.0815825i	0.999168	−	0.0407912i	$-0.0129879\pi$
$368$		0			0
$369$	−5.13764			−0.267455
$370$		0			0
$371$	12.7303	−	16.9186i	0.660926	−	0.878367i
$372$		0			0
$373$		−	21.2673i		−	1.10118i	−0.834776	−	0.550590i	$-0.814403\pi$
$373$		−	21.2673i		−	1.10118i	0.834776	−	0.550590i	$-0.185597\pi$
$374$		0			0
$375$	−5.92454			−0.305942
$376$		0			0
$377$		−	1.41746i		−	0.0730029i
$378$		0			0
$379$	24.1466			1.24033			0.620163	−	0.784473i	$-0.287066\pi$
$379$	24.1466			1.24033			0.620163	+	0.784473i	$0.287066\pi$
$380$		0			0
$381$	−8.85631			−0.453722
$382$		0			0
$383$			24.3252i			1.24296i	0.783431	+	0.621479i	$0.213468\pi$
$383$			24.3252i			1.24296i	−0.783431	+	0.621479i	$0.786532\pi$
$384$		0			0
$385$	−3.66482	−	3.03976i	−0.186777	−	0.154920i
$386$		0			0
$387$			3.28717i			0.167096i
$388$		0			0
$389$	−1.68893			−0.0856322			−0.0428161	−	0.999083i	$-0.513633\pi$
$389$	−1.68893			−0.0856322			−0.0428161	+	0.999083i	$0.513633\pi$
$390$		0			0
$391$	3.14684			0.159143
$392$		0			0
$393$		−	6.50272i		−	0.328019i
$394$		0			0
$395$	−6.25645			−0.314796
$396$		0			0
$397$			28.7448i			1.44266i	0.692591	+	0.721331i	$0.256469\pi$
$397$			28.7448i			1.44266i	−0.692591	+	0.721331i	$0.743531\pi$
$398$		0			0
$399$	12.6855	+	9.54519i	0.635070	+	0.477857i
$400$		0			0
$401$	32.5750			1.62672			0.813360	−	0.581761i	$-0.197636\pi$
$401$	32.5750			1.62672			0.813360	+	0.581761i	$0.197636\pi$
$402$		0			0
$403$		−	1.44439i		−	0.0719502i
$404$		0			0
$405$			0.427834i			0.0212592i
$406$		0			0
$407$	−25.2229	−	1.19572i	−1.25025	−	0.0592695i
$408$		0			0
$409$	36.2013			1.79004			0.895021	−	0.446025i	$-0.147161\pi$
$409$	36.2013			1.79004			0.895021	+	0.446025i	$0.147161\pi$
$410$		0			0
$411$			2.11711i			0.104429i
$412$		0			0
$413$	23.2685	+	17.5083i	1.14497	+	0.861529i
$414$		0			0
$415$			4.61945i			0.226760i
$416$		0			0
$417$		−	10.2016i		−	0.499575i
$418$		0			0
$419$			31.8643i			1.55667i	0.627848	+	0.778336i	$0.283936\pi$
$419$			31.8643i			1.55667i	−0.627848	+	0.778336i	$0.716064\pi$
$420$		0			0
$421$	30.1231			1.46811			0.734054	−	0.679091i	$-0.237626\pi$
$421$	30.1231			1.46811			0.734054	+	0.679091i	$0.237626\pi$
$422$		0			0
$423$		−	21.1319i		−	1.02747i
$424$		0			0
$425$	−18.9289			−0.918185
$426$		0			0
$427$	3.46154	−	4.60037i	0.167516	−	0.222628i
$428$		0			0
$429$	0.176679	−	3.72694i	0.00853016	−	0.179938i
$430$		0			0
$431$			19.7786i			0.952701i	0.879255	+	0.476351i	$0.158041\pi$
$431$			19.7786i			0.952701i	−0.879255	+	0.476351i	$0.841959\pi$
$432$		0			0
$433$		−	15.8404i		−	0.761242i	−0.924731	−	0.380621i	$-0.875710\pi$
$433$		−	15.8404i		−	0.761242i	0.924731	−	0.380621i	$-0.124290\pi$
$434$		0			0
$435$		−	0.865256i		−	0.0414859i
$436$		0			0
$437$	−4.17252			−0.199599
$438$		0			0
$439$	−19.3796			−0.924937			−0.462469	−	0.886636i	$-0.653036\pi$
$439$	−19.3796			−0.924937			−0.462469	+	0.886636i	$0.653036\pi$
$440$		0			0
$441$	−3.36298	−	11.6659i	−0.160142	−	0.555521i
$442$		0			0
$443$	21.9332			1.04208			0.521039	−	0.853533i	$-0.325545\pi$
$443$	21.9332			1.04208			0.521039	+	0.853533i	$0.325545\pi$
$444$		0			0
$445$	−2.24799			−0.106565
$446$		0			0
$447$	13.2507			0.626739
$448$		0			0
$449$	6.75505			0.318790			0.159395	−	0.987215i	$-0.449046\pi$
$449$	6.75505			0.318790			0.159395	+	0.987215i	$0.449046\pi$
$450$		0			0
$451$	9.81332	+	0.465211i	0.462091	+	0.0219059i
$452$		0			0
$453$	−18.7808			−0.882400
$454$		0			0
$455$	−1.14715	−	0.863168i	−0.0537791	−	0.0404660i
$456$		0			0
$457$			1.24446i			0.0582135i	0.999576	+	0.0291068i	$0.00926628\pi$
$457$			1.24446i			0.0582135i	−0.999576	+	0.0291068i	$0.990734\pi$
$458$		0			0
$459$			21.4251i			1.00004i
$460$		0			0
$461$	−16.1607			−0.752680			−0.376340	−	0.926482i	$-0.622818\pi$
$461$	−16.1607			−0.752680			−0.376340	+	0.926482i	$0.622818\pi$
$462$		0			0
$463$	23.2427			1.08018			0.540090	−	0.841607i	$-0.318390\pi$
$463$	23.2427			1.08018			0.540090	+	0.841607i	$0.318390\pi$
$464$		0			0
$465$		−	0.881695i		−	0.0408876i
$466$		0			0
$467$		−	37.1943i		−	1.72115i	−0.509324	−	0.860575i	$-0.670105\pi$
$467$		−	37.1943i		−	1.72115i	0.509324	−	0.860575i	$-0.329895\pi$
$468$		0			0
$469$	7.66844	−	10.1913i	0.354096	−	0.470591i
$470$		0			0
$471$	−19.2024			−0.884801
$472$		0			0
$473$	0.297651	−	6.27876i	0.0136860	−	0.288698i
$474$		0			0
$475$	25.0985			1.15160
$476$		0			0
$477$	−13.8801			−0.635524
$478$		0			0
$479$	24.2609			1.10851			0.554254	−	0.832347i	$-0.313004\pi$
$479$	24.2609			1.10851			0.554254	+	0.832347i	$0.313004\pi$
$480$		0			0
$481$	−7.61353			−0.347147
$482$		0			0
$483$	−1.86052	−	1.39994i	−0.0846565	−	0.0636996i
$484$		0			0
$485$	−1.60938			−0.0730781
$486$		0			0
$487$	32.7210			1.48273			0.741365	−	0.671102i	$-0.234179\pi$
$487$	32.7210			1.48273			0.741365	+	0.671102i	$0.234179\pi$
$488$		0			0
$489$			13.2632i			0.599783i
$490$		0			0
$491$		−	2.33499i		−	0.105377i	−0.998611	−	0.0526884i	$-0.983221\pi$
$491$		−	2.33499i		−	0.105377i	0.998611	−	0.0526884i	$-0.0167790\pi$
$492$		0			0
$493$		−	5.70195i		−	0.256803i
$494$		0			0
$495$	−0.147805	+	3.11785i	−0.00664335	+	0.140137i
$496$		0			0
$497$	11.7574	−	15.6256i	0.527393	−	0.700903i
$498$		0			0
$499$	−7.74831			−0.346862			−0.173431	−	0.984846i	$-0.555485\pi$
$499$	−7.74831			−0.346862			−0.173431	+	0.984846i	$0.555485\pi$
$500$		0			0
$501$			12.2773i			0.548508i
$502$		0			0
$503$	−13.0916			−0.583728			−0.291864	−	0.956460i	$-0.594275\pi$
$503$	−13.0916			−0.583728			−0.291864	+	0.956460i	$0.594275\pi$
$504$		0			0
$505$			2.88239i			0.128265i
$506$		0			0
$507$		−	1.12498i		−	0.0499619i
$508$		0			0
$509$			8.92323i			0.395515i	0.980251	+	0.197758i	$0.0633659\pi$
$509$			8.92323i			0.395515i	−0.980251	+	0.197758i	$0.936634\pi$
$510$		0			0
$511$	−4.17860	+	5.55333i	−0.184850	+	0.245665i
$512$		0			0
$513$		−	28.4084i		−	1.25426i
$514$		0			0
$515$	0.627323			0.0276431
$516$		0			0
$517$	−1.91348	+	40.3637i	−0.0841550	+	1.77519i
$518$		0			0
$519$		−	7.88053i		−	0.345917i
$520$		0			0
$521$		−	9.24530i		−	0.405044i	−0.979278	−	0.202522i	$-0.935086\pi$
$521$		−	9.24530i		−	0.405044i	0.979278	−	0.202522i	$-0.0649138\pi$
$522$		0			0
$523$	−7.70462			−0.336899			−0.168450	−	0.985710i	$-0.553876\pi$
$523$	−7.70462			−0.336899			−0.168450	+	0.985710i	$0.553876\pi$
$524$		0			0
$525$	11.1914	+	8.42094i	0.488432	+	0.367520i
$526$		0			0
$527$		−	5.81028i		−	0.253100i
$528$		0			0
$529$	−22.3880			−0.973393
$530$		0			0
$531$		−	19.0896i		−	0.828417i
$532$		0			0
$533$	2.96215			0.128305
$534$		0			0
$535$	−6.86381			−0.296749
$536$		0			0
$537$			14.0963i			0.608301i
$538$		0			0
$539$	5.36723	+	22.5874i	0.231183	+	0.972910i
$540$		0			0
$541$		−	34.1488i		−	1.46817i	−0.679057	−	0.734086i	$-0.737611\pi$
$541$		−	34.1488i		−	1.46817i	0.679057	−	0.734086i	$-0.262389\pi$
$542$		0			0
$543$	1.02650			0.0440511
$544$		0			0
$545$	6.44360			0.276014
$546$		0			0
$547$		−	8.98549i		−	0.384192i	−0.981376	−	0.192096i	$-0.938472\pi$
$547$		−	8.98549i		−	0.384192i	0.981376	−	0.192096i	$-0.0615285\pi$
$548$		0			0
$549$	−3.77417			−0.161077
$550$		0			0
$551$			7.56044i			0.322086i
$552$		0			0
$553$	24.3762	+	18.3418i	1.03658	+	0.779974i
$554$		0			0
$555$	−4.64750			−0.197275
$556$		0			0
$557$		−	16.5296i		−	0.700380i	−0.936679	−	0.350190i	$-0.886117\pi$
$557$		−	16.5296i		−	0.700380i	0.936679	−	0.350190i	$-0.113883\pi$
$558$		0			0
$559$		−	1.89524i		−	0.0801603i
$560$		0			0
$561$	0.710720	−	14.9922i	0.0300066	−	0.632970i
$562$		0			0
$563$	7.31919			0.308467			0.154233	−	0.988034i	$-0.450709\pi$
$563$	7.31919			0.308467			0.154233	+	0.988034i	$0.450709\pi$
$564$		0			0
$565$		−	2.45330i		−	0.103211i
$566$		0			0
$567$	1.25427	−	1.66691i	0.0526742	−	0.0700037i
$568$		0			0
$569$		−	20.1180i		−	0.843390i	−0.906738	−	0.421695i	$-0.861435\pi$
$569$		−	20.1180i		−	0.843390i	0.906738	−	0.421695i	$-0.138565\pi$
$570$		0			0
$571$		−	9.80273i		−	0.410232i	−0.978738	−	0.205116i	$-0.934243\pi$
$571$		−	9.80273i		−	0.410232i	0.978738	−	0.205116i	$-0.0657571\pi$
$572$		0			0
$573$			6.83309i			0.285456i
$574$		0			0
$575$	−3.68107			−0.153511
$576$		0			0
$577$			27.6553i			1.15131i	0.817694	+	0.575653i	$0.195252\pi$
$577$			27.6553i			1.15131i	−0.817694	+	0.575653i	$0.804748\pi$
$578$		0			0
$579$	−21.4970			−0.893384
$580$		0			0
$581$	13.5427	−	17.9982i	0.561845	−	0.746689i
$582$		0			0
$583$	26.5121	+	1.25683i	1.09802	+	0.0520527i
$584$		0			0
$585$			0.941124i			0.0389107i
$586$		0			0
$587$			2.43464i			0.100488i	0.998737	+	0.0502442i	$0.0160000\pi$
$587$			2.43464i			0.100488i	−0.998737	+	0.0502442i	$0.984000\pi$
$588$		0			0
$589$			7.70408i			0.317441i
$590$		0			0
$591$	30.0775			1.23722
$592$		0			0
$593$	−30.1067			−1.23634			−0.618168	−	0.786046i	$-0.712125\pi$
$593$	−30.1067			−1.23634			−0.618168	+	0.786046i	$0.712125\pi$
$594$		0			0
$595$	−4.61457	−	3.47223i	−0.189179	−	0.142347i
$596$		0			0
$597$	−5.62235			−0.230107
$598$		0			0
$599$	−1.06062			−0.0433358			−0.0216679	−	0.999765i	$-0.506898\pi$
$599$	−1.06062			−0.0433358			−0.0216679	+	0.999765i	$0.506898\pi$
$600$		0			0
$601$	4.46521			0.182140			0.0910698	−	0.995845i	$-0.470971\pi$
$601$	4.46521			0.182140			0.0910698	+	0.995845i	$0.470971\pi$
$602$		0			0
$603$	−8.36100			−0.340486
$604$		0			0
$605$	0.564641	−	5.94198i	0.0229559	−	0.241576i
$606$		0			0
$607$	−46.2106			−1.87563			−0.937816	−	0.347134i	$-0.887155\pi$
$607$	−46.2106			−1.87563			−0.937816	+	0.347134i	$0.887155\pi$
$608$		0			0
$609$	−2.53664	+	3.37119i	−0.102790	+	0.136607i
$610$		0			0
$611$			12.1838i			0.492903i
$612$		0			0
$613$		−	19.8369i		−	0.801204i	−0.916252	−	0.400602i	$-0.868801\pi$
$613$		−	19.8369i		−	0.801204i	0.916252	−	0.400602i	$-0.131199\pi$
$614$		0			0
$615$	1.80818			0.0729127
$616$		0			0
$617$	23.7739			0.957102			0.478551	−	0.878060i	$-0.341162\pi$
$617$	23.7739			0.957102			0.478551	+	0.878060i	$0.341162\pi$
$618$		0			0
$619$			28.3290i			1.13864i	0.822117	+	0.569319i	$0.192793\pi$
$619$			28.3290i			1.13864i	−0.822117	+	0.569319i	$0.807207\pi$
$620$		0			0
$621$			4.16652i			0.167197i
$622$		0			0
$623$	8.75857	+	6.59037i	0.350905	+	0.264037i
$624$		0			0
$625$	20.6703			0.826810
$626$		0			0
$627$	−0.942372	+	19.8787i	−0.0376347	+	0.793880i
$628$		0			0
$629$	−30.6266			−1.22116
$630$		0			0
$631$	9.08124			0.361519			0.180759	−	0.983527i	$-0.442144\pi$
$631$	9.08124			0.361519			0.180759	+	0.983527i	$0.442144\pi$
$632$		0			0
$633$	−4.36293			−0.173411
$634$		0			0
$635$	−4.27169			−0.169517
$636$		0			0
$637$	1.93896	+	6.72610i	0.0768243	+	0.266498i
$638$		0			0
$639$	−12.8193			−0.507123
$640$		0			0
$641$	34.0837			1.34622			0.673112	−	0.739541i	$-0.264957\pi$
$641$	34.0837			1.34622			0.673112	+	0.739541i	$0.264957\pi$
$642$		0			0
$643$		−	24.8023i		−	0.978108i	−0.872254	−	0.489054i	$-0.837342\pi$
$643$		−	24.8023i		−	0.978108i	0.872254	−	0.489054i	$-0.162658\pi$
$644$		0			0
$645$		−	1.15691i		−	0.0455532i
$646$		0			0
$647$		−	6.05031i		−	0.237862i	−0.992902	−	0.118931i	$-0.962053\pi$
$647$		−	6.05031i		−	0.237862i	0.992902	−	0.118931i	$-0.0379468\pi$
$648$		0			0
$649$	−1.72855	+	36.4627i	−0.0678516	+	1.43129i
$650$		0			0
$651$	−2.58484	+	3.43523i	−0.101308	+	0.134637i
$652$		0			0
$653$	24.8955			0.974235			0.487118	−	0.873336i	$-0.338048\pi$
$653$	24.8955			0.974235			0.487118	+	0.873336i	$0.338048\pi$
$654$		0			0
$655$		−	3.13648i		−	0.122552i
$656$		0			0
$657$	4.55598			0.177746
$658$		0			0
$659$		−	32.3881i		−	1.26166i	−0.775921	−	0.630830i	$-0.782715\pi$
$659$		−	32.3881i		−	1.26166i	0.775921	−	0.630830i	$-0.217285\pi$
$660$		0			0
$661$			31.7723i			1.23580i	0.786257	+	0.617899i	$0.212016\pi$
$661$			31.7723i			1.23580i	−0.786257	+	0.617899i	$0.787984\pi$
$662$		0			0
$663$		−	4.52539i		−	0.175751i
$664$		0			0
$665$	6.11864	+	4.60396i	0.237271	+	0.178534i
$666$		0			0
$667$		−	1.10885i		−	0.0429349i
$668$		0			0
$669$	10.3294			0.399359
$670$		0			0
$671$	7.20897	+	0.341749i	0.278299	+	0.0131931i
$672$		0			0
$673$			49.3140i			1.90091i	0.310857	+	0.950457i	$0.399384\pi$
$673$			49.3140i			1.90091i	−0.310857	+	0.950457i	$0.600616\pi$
$674$		0			0
$675$		−	25.0624i		−	0.964653i
$676$		0			0
$677$	42.4625			1.63196			0.815982	−	0.578077i	$-0.196197\pi$
$677$	42.4625			1.63196			0.815982	+	0.578077i	$0.196197\pi$
$678$		0			0
$679$	6.27041	+	4.71816i	0.240636	+	0.181066i
$680$		0			0
$681$			7.27522i			0.278787i
$682$		0			0
$683$	−18.5517			−0.709862			−0.354931	−	0.934892i	$-0.615496\pi$
$683$	−18.5517			−0.709862			−0.354931	+	0.934892i	$0.615496\pi$
$684$		0			0
$685$			1.02115i			0.0390162i
$686$		0			0
$687$	−27.6128			−1.05349
$688$		0			0
$689$	8.00267			0.304877
$690$		0			0
$691$		−	29.6112i		−	1.12646i	−0.826300	−	0.563231i	$-0.809558\pi$
$691$		−	29.6112i		−	1.12646i	0.826300	−	0.563231i	$-0.190442\pi$
$692$		0			0
$693$	9.71638	−	11.7144i	0.369095	−	0.444992i
$694$		0			0
$695$		−	4.92057i		−	0.186648i
$696$		0			0
$697$	11.9157			0.451339
$698$		0			0
$699$	−22.7498			−0.860478
$700$		0			0
$701$			12.2153i			0.461366i	0.973029	+	0.230683i	$0.0740960\pi$
$701$			12.2153i			0.461366i	−0.973029	+	0.230683i	$0.925904\pi$
$702$		0			0
$703$	40.6090			1.53160
$704$		0			0
$705$			7.43731i			0.280105i
$706$		0			0
$707$	8.45020	−	11.2303i	0.317802	−	0.422358i
$708$		0			0
$709$	21.0916			0.792112			0.396056	−	0.918226i	$-0.370379\pi$
$709$	21.0916			0.792112			0.396056	+	0.918226i	$0.370379\pi$
$710$		0			0
$711$		−	19.9984i		−	0.749997i
$712$		0			0
$713$		−	1.12992i		−	0.0423158i
$714$		0			0
$715$	0.0852184	−	1.79763i	0.00318699	−	0.0672274i
$716$		0			0
$717$	24.9401			0.931407
$718$		0			0
$719$		−	16.1610i		−	0.602702i	−0.953513	−	0.301351i	$-0.902562\pi$
$719$		−	16.1610i		−	0.602702i	0.953513	−	0.301351i	$-0.0974376\pi$
$720$		0			0
$721$	−2.44416	−	1.83910i	−0.0910252	−	0.0684917i
$722$		0			0
$723$		−	31.8197i		−	1.18339i
$724$		0			0
$725$			6.66996i			0.247716i
$726$		0			0
$727$		−	2.50393i		−	0.0928655i	−0.998921	−	0.0464327i	$-0.985215\pi$
$727$		−	2.50393i		−	0.0928655i	0.998921	−	0.0464327i	$-0.0147853\pi$
$728$		0			0
$729$	−19.3428			−0.716400
$730$		0			0
$731$		−	7.62391i		−	0.281980i
$732$		0			0
$733$	−10.0670			−0.371835			−0.185917	−	0.982565i	$-0.559526\pi$
$733$	−10.0670			−0.371835			−0.185917	+	0.982565i	$0.559526\pi$
$734$		0			0
$735$	1.18359	+	4.10579i	0.0436574	+	0.151444i
$736$		0			0
$737$	15.9702	+	0.757085i	0.588270	+	0.0278876i
$738$		0			0
$739$			52.1934i			1.91997i	0.280058	+	0.959983i	$0.409646\pi$
$739$			52.1934i			1.91997i	−0.280058	+	0.959983i	$0.590354\pi$
$740$		0			0
$741$			6.00039i			0.220430i
$742$		0			0
$743$			13.8058i			0.506485i	0.967403	+	0.253242i	$0.0814970\pi$
$743$			13.8058i			0.506485i	−0.967403	+	0.253242i	$0.918503\pi$
$744$		0			0
$745$	6.39127			0.234158
$746$		0			0
$747$	−14.7658			−0.540251
$748$		0			0
$749$	26.7426	+	20.1224i	0.977153	+	0.735257i
$750$		0			0
$751$	45.9560			1.67696			0.838479	−	0.544934i	$-0.183445\pi$
$751$	45.9560			1.67696			0.838479	+	0.544934i	$0.183445\pi$
$752$		0			0
$753$	−28.9689			−1.05569
$754$		0			0
$755$	−9.05862			−0.329677
$756$		0			0
$757$	−1.79515			−0.0652459			−0.0326230	−	0.999468i	$-0.510386\pi$
$757$	−1.79515			−0.0652459			−0.0326230	+	0.999468i	$0.510386\pi$
$758$		0			0
$759$	0.138213	−	2.91551i	0.00501681	−	0.105826i
$760$		0			0
$761$	19.0229			0.689581			0.344790	−	0.938680i	$-0.387950\pi$
$761$	19.0229			0.689581			0.344790	+	0.938680i	$0.387950\pi$
$762$		0			0
$763$	−25.1054	−	18.8905i	−0.908876	−	0.683882i
$764$		0			0
$765$			3.78582i			0.136876i
$766$		0			0
$767$			11.0063i			0.397413i
$768$		0			0
$769$	−28.7707			−1.03750			−0.518750	−	0.854926i	$-0.673602\pi$
$769$	−28.7707			−1.03750			−0.518750	+	0.854926i	$0.673602\pi$
$770$		0			0
$771$	−6.43093			−0.231604
$772$		0			0
$773$			21.4193i			0.770397i	0.922834	+	0.385199i	$0.125867\pi$
$773$			21.4193i			0.770397i	−0.922834	+	0.385199i	$0.874133\pi$
$774$		0			0
$775$			6.79668i			0.244144i
$776$		0			0
$777$	18.1075	+	13.6249i	0.649601	+	0.488791i
$778$		0			0
$779$	−15.7995			−0.566076
$780$		0			0
$781$	24.4859	+	1.16078i	0.876175	+	0.0415360i
$782$		0			0
$783$	7.54956			0.269799
$784$		0			0
$785$	−9.26197			−0.330574
$786$		0			0
$787$	39.0125			1.39065			0.695323	−	0.718697i	$-0.255261\pi$
$787$	39.0125			1.39065			0.695323	+	0.718697i	$0.255261\pi$
$788$		0			0
$789$	−29.3508			−1.04492
$790$		0			0
$791$	−7.19225	+	9.55846i	−0.255727	+	0.339860i
$792$		0			0
$793$	2.17603			0.0772730
$794$		0			0
$795$	4.88504			0.173255
$796$		0			0
$797$		−	9.16026i		−	0.324473i	−0.986752	−	0.162237i	$-0.948129\pi$
$797$		−	9.16026i		−	0.324473i	0.986752	−	0.162237i	$-0.0518708\pi$
$798$		0			0
$799$			49.0111i			1.73389i
$800$		0			0
$801$		−	7.18557i		−	0.253890i
$802$		0			0
$803$	−8.70230	−	0.412542i	−0.307098	−	0.0145583i
$804$		0			0
$805$	−0.897390	−	0.675239i	−0.0316288	−	0.0237991i
$806$		0			0
$807$	3.91470			0.137804
$808$		0			0
$809$		−	35.3689i		−	1.24350i	−0.783215	−	0.621751i	$-0.786421\pi$
$809$		−	35.3689i		−	1.24350i	0.783215	−	0.621751i	$-0.213579\pi$
$810$		0			0
$811$	−36.3648			−1.27694			−0.638470	−	0.769647i	$-0.720432\pi$
$811$	−36.3648			−1.27694			−0.638470	+	0.769647i	$0.720432\pi$
$812$		0			0
$813$			8.47772i			0.297327i
$814$		0			0
$815$			6.39728i			0.224087i
$816$		0			0
$817$			10.1088i			0.353663i
$818$		0			0
$819$	2.75906	−	3.66678i	0.0964095	−	0.128128i
$820$		0			0
$821$			4.66901i			0.162950i	0.996675	+	0.0814748i	$0.0259630\pi$
$821$			4.66901i			0.162950i	−0.996675	+	0.0814748i	$0.974037\pi$
$822$		0			0
$823$	−10.3486			−0.360728			−0.180364	−	0.983600i	$-0.557728\pi$
$823$	−10.3486			−0.360728			−0.180364	+	0.983600i	$0.557728\pi$
$824$		0			0
$825$	−0.831378	+	17.5374i	−0.0289449	+	0.610573i
$826$		0			0
$827$		−	49.1288i		−	1.70837i	−0.519966	−	0.854187i	$-0.674055\pi$
$827$		−	49.1288i		−	1.70837i	0.519966	−	0.854187i	$-0.325945\pi$
$828$		0			0
$829$			22.3148i			0.775026i	0.921864	+	0.387513i	$0.126666\pi$
$829$			22.3148i			0.775026i	−0.921864	+	0.387513i	$0.873334\pi$
$830$		0			0
$831$	−7.92566			−0.274938
$832$		0			0
$833$	7.79975	+	27.0568i	0.270245	+	0.937461i
$834$		0			0
$835$			5.92173i			0.204930i
$836$		0			0
$837$	7.69300			0.265909
$838$		0			0
$839$			3.00657i			0.103798i	0.998652	+	0.0518991i	$0.0165274\pi$
$839$			3.00657i			0.103798i	−0.998652	+	0.0518991i	$0.983473\pi$
$840$		0			0
$841$	26.9908			0.930717
$842$		0			0
$843$	16.7451			0.576734
$844$		0			0
$845$		−	0.542613i		−	0.0186665i
$846$		0			0
$847$	−19.6198	+	21.4956i	−0.674146	+	0.738598i
$848$		0			0
$849$			15.8448i			0.543791i
$850$		0			0
$851$	−5.95591			−0.204166
$852$		0			0
$853$	−20.6544			−0.707193			−0.353596	−	0.935398i	$-0.615041\pi$
$853$	−20.6544			−0.707193			−0.353596	+	0.935398i	$0.615041\pi$
$854$		0			0
$855$		−	5.01976i		−	0.171672i
$856$		0			0
$857$	24.4847			0.836382			0.418191	−	0.908359i	$-0.362664\pi$
$857$	24.4847			0.836382			0.418191	+	0.908359i	$0.362664\pi$
$858$		0			0
$859$			24.5758i			0.838515i	0.907867	+	0.419258i	$0.137710\pi$
$859$			24.5758i			0.838515i	−0.907867	+	0.419258i	$0.862290\pi$
$860$		0			0
$861$	−7.04496	−	5.30097i	−0.240092	−	0.180657i
$862$		0			0
$863$	17.4854			0.595209			0.297605	−	0.954689i	$-0.403812\pi$
$863$	17.4854			0.595209			0.297605	+	0.954689i	$0.403812\pi$
$864$		0			0
$865$		−	3.80104i		−	0.129239i
$866$		0			0
$867$			0.920535i			0.0312630i
$868$		0			0
$869$	−1.81084	+	38.1985i	−0.0614286	+	1.29580i
$870$		0			0
$871$	4.82061			0.163340
$872$		0			0
$873$		−	5.14428i		−	0.174107i
$874$		0			0
$875$	11.1337	+	8.37754i	0.376388	+	0.283213i
$876$		0			0
$877$			54.7318i			1.84816i	0.382198	+	0.924081i	$0.375167\pi$
$877$			54.7318i			1.84816i	−0.382198	+	0.924081i	$0.624833\pi$
$878$		0			0
$879$		−	8.28151i		−	0.279329i
$880$		0			0
$881$			0.616267i			0.0207626i	0.999946	+	0.0103813i	$0.00330453\pi$
$881$			0.616267i			0.0207626i	−0.999946	+	0.0103813i	$0.996695\pi$
$882$		0			0
$883$	30.8811			1.03923			0.519616	−	0.854400i	$-0.326075\pi$
$883$	30.8811			1.03923			0.519616	+	0.854400i	$0.326075\pi$
$884$		0			0
$885$			6.71852i			0.225841i
$886$		0			0
$887$	−20.5326			−0.689419			−0.344709	−	0.938709i	$-0.612023\pi$
$887$	−20.5326			−0.689419			−0.344709	+	0.938709i	$0.612023\pi$
$888$		0			0
$889$	16.6432	+	12.5232i	0.558196	+	0.420014i
$890$		0			0
$891$	2.61212	+	0.123830i	0.0875093	+	0.00414847i
$892$		0			0
$893$		−	64.9858i		−	2.17467i
$894$		0			0
$895$			6.79912i			0.227269i
$896$		0			0
$897$		−	0.880046i		−	0.0293839i
$898$		0			0
$899$	−2.04737			−0.0682835
$900$		0			0
$901$	32.1919			1.07247
$902$		0			0
$903$	−3.39167	+	4.50751i	−0.112868	+	0.150001i
$904$		0			0
$905$	0.495113			0.0164581
$906$		0			0
$907$	−23.2872			−0.773239			−0.386620	−	0.922239i	$-0.626357\pi$
$907$	−23.2872			−0.773239			−0.386620	+	0.922239i	$0.626357\pi$
$908$		0			0
$909$	−9.21337			−0.305588
$910$		0			0
$911$	−11.0093			−0.364753			−0.182377	−	0.983229i	$-0.558379\pi$
$911$	−11.0093			−0.364753			−0.182377	+	0.983229i	$0.558379\pi$
$912$		0			0
$913$	28.2039	+	1.33703i	0.933411	+	0.0442494i
$914$		0			0
$915$	1.32831			0.0439124
$916$		0			0
$917$	−9.19511	+	12.2203i	−0.303649	+	0.403548i
$918$		0			0
$919$		−	39.1893i		−	1.29274i	−0.763025	−	0.646368i	$-0.776287\pi$
$919$		−	39.1893i		−	1.29274i	0.763025	−	0.646368i	$-0.223713\pi$
$920$		0			0
$921$			33.9507i			1.11871i
$922$		0			0
$923$	7.39107			0.243280
$924$		0			0
$925$	35.8260			1.17795
$926$		0			0
$927$			2.00520i			0.0658593i
$928$		0			0
$929$		−	9.17997i		−	0.301185i	−0.988596	−	0.150593i	$-0.951882\pi$
$929$		−	9.17997i		−	0.301185i	0.988596	−	0.150593i	$-0.0481182\pi$
$930$		0			0
$931$	−10.3420	−	35.8756i	−0.338945	−	1.17578i
$932$		0			0
$933$	5.73624			0.187796
$934$		0			0
$935$	0.342804	−	7.23122i	0.0112109	−	0.236486i
$936$		0			0
$937$	−29.3221			−0.957910			−0.478955	−	0.877839i	$-0.658984\pi$
$937$	−29.3221			−0.957910			−0.478955	+	0.877839i	$0.658984\pi$
$938$		0			0
$939$	−2.25679			−0.0736477
$940$		0			0
$941$	−56.0781			−1.82810			−0.914048	−	0.405607i	$-0.867060\pi$
$941$	−56.0781			−1.82810			−0.914048	+	0.405607i	$0.867060\pi$
$942$		0			0
$943$	2.31723			0.0754594
$944$		0			0
$945$	4.59734	−	6.10984i	0.149551	−	0.198753i
$946$		0			0
$947$	41.6490			1.35341			0.676705	−	0.736254i	$-0.263407\pi$
$947$	41.6490			1.35341			0.676705	+	0.736254i	$0.263407\pi$
$948$		0			0
$949$	−2.62679			−0.0852692
$950$		0			0
$951$			32.9737i			1.06924i
$952$		0			0
$953$			30.8813i			1.00034i	0.865927	+	0.500171i	$0.166729\pi$
$953$			30.8813i			1.00034i	−0.865927	+	0.500171i	$0.833271\pi$
$954$		0			0
$955$			3.29583i			0.106650i
$956$		0			0
$957$	−5.28279	−	0.250436i	−0.170768	−	0.00809545i
$958$		0			0
$959$	2.99367	−	3.97858i	0.0966707	−	0.128475i
$960$		0			0
$961$	28.9137			0.932701
$962$		0			0
$963$		−	21.9397i		−	0.706998i
$964$		0			0
$965$	−10.3687			−0.333780
$966$		0			0
$967$		−	35.1191i		−	1.12936i	−0.825311	−	0.564678i	$-0.809000\pi$
$967$		−	35.1191i		−	1.12936i	0.825311	−	0.564678i	$-0.191000\pi$
$968$		0			0
$969$			24.1375i			0.775408i
$970$		0			0
$971$			39.3833i			1.26387i	0.775022	+	0.631934i	$0.217739\pi$
$971$			39.3833i			1.26387i	−0.775022	+	0.631934i	$0.782261\pi$
$972$		0			0
$973$	−14.4255	+	19.1714i	−0.462460	+	0.614607i
$974$		0			0
$975$			5.29365i			0.169533i
$976$		0			0
$977$	−25.3910			−0.812331			−0.406166	−	0.913799i	$-0.633134\pi$
$977$	−25.3910			−0.812331			−0.406166	+	0.913799i	$0.633134\pi$
$978$		0			0
$979$	−0.650650	+	13.7250i	−0.0207949	+	0.438654i
$980$		0			0
$981$			20.5966i			0.657598i
$982$		0			0
$983$			9.84305i			0.313944i	0.987603	+	0.156972i	$0.0501733\pi$
$983$			9.84305i			0.313944i	−0.987603	+	0.156972i	$0.949827\pi$
$984$		0			0
$985$	14.5074			0.462244
$986$		0			0
$987$	21.8037	−	28.9770i	0.694020	−	0.922349i
$988$		0			0
$989$		−	1.48261i		−	0.0471443i
$990$		0			0
$991$	44.9082			1.42656			0.713278	−	0.700881i	$-0.247209\pi$
$991$	44.9082			1.42656			0.713278	+	0.700881i	$0.247209\pi$
$992$		0			0
$993$			29.6357i			0.940459i
$994$		0			0
$995$	−2.71185			−0.0859713
$996$		0			0
$997$	26.6119			0.842806			0.421403	−	0.906873i	$-0.361538\pi$
$997$	26.6119			0.842806			0.421403	+	0.906873i	$0.361538\pi$
$998$		0			0
$999$		−	40.5505i		−	1.28296i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.e.b.3849.15	yes	48
7.6	odd	2		4004.2.e.a.3849.34	yes	48
11.10	odd	2		4004.2.e.a.3849.15	✓	48
77.76	even	2	inner	4004.2.e.b.3849.34	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.e.a.3849.15	✓	48	11.10	odd	2
4004.2.e.a.3849.34	yes	48	7.6	odd	2
4004.2.e.b.3849.15	yes	48	1.1	even	1	trivial
4004.2.e.b.3849.34	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)

\(f(q)\)	\(=\)	\(q-1.12498i q^{3} -0.542613i q^{5} +(-1.59076 + 2.11411i) q^{7} +1.73443 q^{9} +O(q^{10})\)	\(q-1.12498i q^{3} -0.542613i q^{5} +(-1.59076 + 2.11411i) q^{7} +1.73443 q^{9} +(-3.31290 - 0.157052i) q^{11} -1.00000 q^{13} -0.610427 q^{15} -4.02265 q^{17} +5.33379 q^{19} +(2.37833 + 1.78957i) q^{21} -0.782280 q^{23} +4.70557 q^{25} -5.32612i q^{27} +1.41746i q^{29} +1.44439i q^{31} +(-0.176679 + 3.72694i) q^{33} +(1.14715 + 0.863168i) q^{35} +7.61353 q^{37} +1.12498i q^{39} -2.96215 q^{41} +1.89524i q^{43} -0.941124i q^{45} -12.1838i q^{47} +(-1.93896 - 6.72610i) q^{49} +4.52539i q^{51} -8.00267 q^{53} +(-0.0852184 + 1.79763i) q^{55} -6.00039i q^{57} -11.0063i q^{59} -2.17603 q^{61} +(-2.75906 + 3.66678i) q^{63} +0.542613i q^{65} -4.82061 q^{67} +0.880046i q^{69} -7.39107 q^{71} +2.62679 q^{73} -5.29365i q^{75} +(5.60207 - 6.75403i) q^{77} -11.5302i q^{79} -0.788469 q^{81} -8.51333 q^{83} +2.18274i q^{85} +1.59461 q^{87} -4.14290i q^{89} +(1.59076 - 2.11411i) q^{91} +1.62491 q^{93} -2.89419i q^{95} -2.96598i q^{97} +(-5.74600 - 0.272395i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{7} - 48 q^{9}+O(q^{10}) \) 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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