LMFDB - Embedded newform 4004.2.e.b.3849.8

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			3849.8
Character	$\chi$	$=$	4004.3849
Dual form			4004.2.e.b.3849.41

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41288i q^{3} +2.25637i q^{5} +(-2.16066 + 1.52694i) q^{7} -2.82198 q^{9} +O(q^{10})$	$q-2.41288i q^{3} +2.25637i q^{5} +(-2.16066 + 1.52694i) q^{7} -2.82198 q^{9} +(0.870650 - 3.20031i) q^{11} -1.00000 q^{13} +5.44434 q^{15} -1.61172 q^{17} +2.82162 q^{19} +(3.68432 + 5.21341i) q^{21} -1.64741 q^{23} -0.0911906 q^{25} -0.429550i q^{27} +0.879297i q^{29} -0.464165i q^{31} +(-7.72195 - 2.10077i) q^{33} +(-3.44534 - 4.87524i) q^{35} -4.27025 q^{37} +2.41288i q^{39} -6.38848 q^{41} -0.476880i q^{43} -6.36741i q^{45} +11.5986i q^{47} +(2.33690 - 6.59840i) q^{49} +3.88888i q^{51} +10.3139 q^{53} +(7.22107 + 1.96451i) q^{55} -6.80823i q^{57} -6.50814i q^{59} -8.25328 q^{61} +(6.09733 - 4.30899i) q^{63} -2.25637i q^{65} +3.29816 q^{67} +3.97499i q^{69} -4.29070 q^{71} -16.0456 q^{73} +0.220032i q^{75} +(3.00550 + 8.24421i) q^{77} +6.07083i q^{79} -9.50238 q^{81} +3.12544 q^{83} -3.63663i q^{85} +2.12164 q^{87} +11.6698i q^{89} +(2.16066 - 1.52694i) q^{91} -1.11997 q^{93} +6.36662i q^{95} +10.9467i q^{97} +(-2.45695 + 9.03119i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	2.41288i		−	1.39308i	−0.717520	−	0.696538i	$-0.754723\pi$
$3$		−	2.41288i		−	1.39308i	0.717520	−	0.696538i	$-0.245277\pi$
$4$		0			0
$5$			2.25637i			1.00908i	0.863389	+	0.504539i	$0.168338\pi$
$5$			2.25637i			1.00908i	−0.863389	+	0.504539i	$0.831662\pi$
$6$		0			0
$7$	−2.16066	+	1.52694i	−0.816652	+	0.577130i
$7$	−2.16066	+	1.52694i	−0.816652	+	0.577130i
$8$		0			0
$9$	−2.82198			−0.940659
$10$		0			0
$11$	0.870650	−	3.20031i	0.262511	−	0.964929i
$11$	0.870650	−	3.20031i	0.262511	−	0.964929i
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$	5.44434			1.40572
$16$		0			0
$17$	−1.61172			−0.390899			−0.195450	−	0.980714i	$-0.562617\pi$
$17$	−1.61172			−0.390899			−0.195450	+	0.980714i	$0.562617\pi$
$18$		0			0
$19$	2.82162			0.647325			0.323662	−	0.946173i	$-0.395086\pi$
$19$	2.82162			0.647325			0.323662	+	0.946173i	$0.395086\pi$
$20$		0			0
$21$	3.68432	+	5.21341i	0.803985	+	1.13766i
$22$		0			0
$23$	−1.64741			−0.343508			−0.171754	−	0.985140i	$-0.554943\pi$
$23$	−1.64741			−0.343508			−0.171754	+	0.985140i	$0.554943\pi$
$24$		0			0
$25$	−0.0911906			−0.0182381
$26$		0			0
$27$		−	0.429550i		−	0.0826669i
$28$		0			0
$29$			0.879297i			0.163281i	0.996662	+	0.0816407i	$0.0260160\pi$
$29$			0.879297i			0.163281i	−0.996662	+	0.0816407i	$0.973984\pi$
$30$		0			0
$31$		−	0.464165i		−	0.0833666i	−0.999131	−	0.0416833i	$-0.986728\pi$
$31$		−	0.464165i		−	0.0833666i	0.999131	−	0.0416833i	$-0.0132720\pi$
$32$		0			0
$33$	−7.72195	−	2.10077i	−1.34422	−	0.365697i
$34$		0			0
$35$	−3.44534	−	4.87524i	−0.582369	−	0.824066i
$36$		0			0
$37$	−4.27025			−0.702025			−0.351013	−	0.936371i	$-0.614163\pi$
$37$	−4.27025			−0.702025			−0.351013	+	0.936371i	$0.614163\pi$
$38$		0			0
$39$			2.41288i			0.386370i
$40$		0			0
$41$	−6.38848			−0.997713			−0.498857	−	0.866685i	$-0.666247\pi$
$41$	−6.38848			−0.997713			−0.498857	+	0.866685i	$0.666247\pi$
$42$		0			0
$43$		−	0.476880i		−	0.0727235i	−0.999339	−	0.0363618i	$-0.988423\pi$
$43$		−	0.476880i		−	0.0727235i	0.999339	−	0.0363618i	$-0.0115769\pi$
$44$		0			0
$45$		−	6.36741i		−	0.949198i
$46$		0			0
$47$			11.5986i			1.69183i	0.533316	+	0.845916i	$0.320946\pi$
$47$			11.5986i			1.69183i	−0.533316	+	0.845916i	$0.679054\pi$
$48$		0			0
$49$	2.33690	−	6.59840i	0.333843	−	0.942629i
$50$		0			0
$51$			3.88888i			0.544552i
$52$		0			0
$53$	10.3139			1.41673			0.708363	−	0.705848i	$-0.249434\pi$
$53$	10.3139			1.41673			0.708363	+	0.705848i	$0.249434\pi$
$54$		0			0
$55$	7.22107	+	1.96451i	0.973689	+	0.264894i
$56$		0			0
$57$		−	6.80823i		−	0.901772i
$58$		0			0
$59$		−	6.50814i		−	0.847287i	−0.905829	−	0.423644i	$-0.860751\pi$
$59$		−	6.50814i		−	0.847287i	0.905829	−	0.423644i	$-0.139249\pi$
$60$		0			0
$61$	−8.25328			−1.05672			−0.528362	−	0.849019i	$-0.677194\pi$
$61$	−8.25328			−1.05672			−0.528362	+	0.849019i	$0.677194\pi$
$62$		0			0
$63$	6.09733	−	4.30899i	0.768191	−	0.542882i
$64$		0			0
$65$		−	2.25637i		−	0.279868i
$66$		0			0
$67$	3.29816			0.402934			0.201467	−	0.979495i	$-0.435429\pi$
$67$	3.29816			0.402934			0.201467	+	0.979495i	$0.435429\pi$
$68$		0			0
$69$			3.97499i			0.478532i
$70$		0			0
$71$	−4.29070			−0.509213			−0.254606	−	0.967045i	$-0.581946\pi$
$71$	−4.29070			−0.509213			−0.254606	+	0.967045i	$0.581946\pi$
$72$		0			0
$73$	−16.0456			−1.87800			−0.938998	−	0.343921i	$-0.888245\pi$
$73$	−16.0456			−1.87800			−0.938998	+	0.343921i	$0.888245\pi$
$74$		0			0
$75$			0.220032i			0.0254071i
$76$		0			0
$77$	3.00550	+	8.24421i	0.342509	+	0.939515i
$78$		0			0
$79$			6.07083i			0.683022i	0.939878	+	0.341511i	$0.110939\pi$
$79$			6.07083i			0.683022i	−0.939878	+	0.341511i	$0.889061\pi$
$80$		0			0
$81$	−9.50238			−1.05582
$82$		0			0
$83$	3.12544			0.343062			0.171531	−	0.985179i	$-0.445129\pi$
$83$	3.12544			0.343062			0.171531	+	0.985179i	$0.445129\pi$
$84$		0			0
$85$		−	3.63663i		−	0.394448i
$86$		0			0
$87$	2.12164			0.227463
$88$		0			0
$89$			11.6698i			1.23699i	0.785787	+	0.618497i	$0.212258\pi$
$89$			11.6698i			1.23699i	−0.785787	+	0.618497i	$0.787742\pi$
$90$		0			0
$91$	2.16066	−	1.52694i	0.226499	−	0.160067i
$92$		0			0
$93$	−1.11997			−0.116136
$94$		0			0
$95$			6.36662i			0.653201i
$96$		0			0
$97$			10.9467i			1.11146i	0.831361	+	0.555732i	$0.187562\pi$
$97$			10.9467i			1.11146i	−0.831361	+	0.555732i	$0.812438\pi$
$98$		0			0
$99$	−2.45695	+	9.03119i	−0.246933	+	0.907669i
$100$		0			0
$101$	−19.5867			−1.94895			−0.974474	−	0.224502i	$-0.927925\pi$
$101$	−19.5867			−1.94895			−0.974474	+	0.224502i	$0.927925\pi$
$102$		0			0
$103$			11.0699i			1.09075i	0.838191	+	0.545377i	$0.183613\pi$
$103$			11.0699i			1.09075i	−0.838191	+	0.545377i	$0.816387\pi$
$104$		0			0
$105$	−11.7634	+	8.31318i	−1.14799	+	0.811284i
$106$		0			0
$107$			2.37580i			0.229677i	0.993384	+	0.114839i	$0.0366351\pi$
$107$			2.37580i			0.229677i	−0.993384	+	0.114839i	$0.963365\pi$
$108$		0			0
$109$			3.60112i			0.344925i	0.985016	+	0.172462i	$0.0551723\pi$
$109$			3.60112i			0.344925i	−0.985016	+	0.172462i	$0.944828\pi$
$110$		0			0
$111$			10.3036i			0.977974i
$112$		0			0
$113$	1.18172			0.111167			0.0555833	−	0.998454i	$-0.482298\pi$
$113$	1.18172			0.111167			0.0555833	+	0.998454i	$0.482298\pi$
$114$		0			0
$115$		−	3.71715i		−	0.346626i
$116$		0			0
$117$	2.82198			0.260892
$118$		0			0
$119$	3.48238	−	2.46100i	0.319229	−	0.225600i
$120$		0			0
$121$	−9.48394	−	5.57270i	−0.862176	−	0.506609i
$122$		0			0
$123$			15.4146i			1.38989i
$124$		0			0
$125$			11.0761i			0.990674i
$126$		0			0
$127$			8.97186i			0.796124i	0.917359	+	0.398062i	$0.130317\pi$
$127$			8.97186i			0.796124i	−0.917359	+	0.398062i	$0.869683\pi$
$128$		0			0
$129$	−1.15065			−0.101309
$130$		0			0
$131$	−13.6768			−1.19495			−0.597473	−	0.801889i	$-0.703828\pi$
$131$	−13.6768			−1.19495			−0.597473	+	0.801889i	$0.703828\pi$
$132$		0			0
$133$	−6.09657	+	4.30845i	−0.528639	+	0.373590i
$134$		0			0
$135$	0.969222			0.0834173
$136$		0			0
$137$	−2.43600			−0.208121			−0.104061	−	0.994571i	$-0.533184\pi$
$137$	−2.43600			−0.208121			−0.104061	+	0.994571i	$0.533184\pi$
$138$		0			0
$139$	−11.8008			−1.00093			−0.500467	−	0.865755i	$-0.666838\pi$
$139$	−11.8008			−1.00093			−0.500467	+	0.865755i	$0.666838\pi$
$140$		0			0
$141$	27.9860			2.35685
$142$		0			0
$143$	−0.870650	+	3.20031i	−0.0728074	+	0.267623i
$144$		0			0
$145$	−1.98402			−0.164764
$146$		0			0
$147$	−15.9211	−	5.63865i	−1.31315	−	0.465068i
$148$		0			0
$149$			5.57927i			0.457072i	0.973536	+	0.228536i	$0.0733938\pi$
$149$			5.57927i			0.457072i	−0.973536	+	0.228536i	$0.926606\pi$
$150$		0			0
$151$			15.9345i			1.29673i	0.761328	+	0.648367i	$0.224548\pi$
$151$			15.9345i			1.29673i	−0.761328	+	0.648367i	$0.775452\pi$
$152$		0			0
$153$	4.54823			0.367703
$154$		0			0
$155$	1.04733			0.0841233
$156$		0			0
$157$		−	16.5328i		−	1.31946i	−0.751502	−	0.659730i	$-0.770670\pi$
$157$		−	16.5328i		−	1.31946i	0.751502	−	0.659730i	$-0.229330\pi$
$158$		0			0
$159$		−	24.8862i		−	1.97361i
$160$		0			0
$161$	3.55948	−	2.51549i	0.280527	−	0.198249i
$162$		0			0
$163$	−11.5734			−0.906500			−0.453250	−	0.891383i	$-0.649736\pi$
$163$	−11.5734			−0.906500			−0.453250	+	0.891383i	$0.649736\pi$
$164$		0			0
$165$	4.74011	−	17.4235i	0.369017	−	1.35642i
$166$		0			0
$167$	−1.72674			−0.133619			−0.0668095	−	0.997766i	$-0.521282\pi$
$167$	−1.72674			−0.133619			−0.0668095	+	0.997766i	$0.521282\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	−7.96255			−0.608912
$172$		0			0
$173$	−19.3548			−1.47152			−0.735760	−	0.677242i	$-0.763175\pi$
$173$	−19.3548			−1.47152			−0.735760	+	0.677242i	$0.763175\pi$
$174$		0			0
$175$	0.197032	−	0.139243i	0.0148942	−	0.0105258i
$176$		0			0
$177$	−15.7033			−1.18033
$178$		0			0
$179$	25.9846			1.94218			0.971089	−	0.238716i	$-0.0767266\pi$
$179$	25.9846			1.94218			0.971089	+	0.238716i	$0.0767266\pi$
$180$		0			0
$181$		−	13.7252i		−	1.02019i	−0.860119	−	0.510094i	$-0.829611\pi$
$181$		−	13.7252i		−	1.02019i	0.860119	−	0.510094i	$-0.170389\pi$
$182$		0			0
$183$			19.9141i			1.47210i
$184$		0			0
$185$		−	9.63526i		−	0.708398i
$186$		0			0
$187$	−1.40324	+	5.15800i	−0.102615	+	0.377190i
$188$		0			0
$189$	0.655898	+	0.928111i	0.0477095	+	0.0675101i
$190$		0			0
$191$	−20.8558			−1.50907			−0.754537	−	0.656258i	$-0.772138\pi$
$191$	−20.8558			−1.50907			−0.754537	+	0.656258i	$0.772138\pi$
$192$		0			0
$193$		−	4.02499i		−	0.289725i	−0.989452	−	0.144862i	$-0.953726\pi$
$193$		−	4.02499i		−	0.289725i	0.989452	−	0.144862i	$-0.0462740\pi$
$194$		0			0
$195$	−5.44434			−0.389877
$196$		0			0
$197$			10.7395i			0.765156i	0.923923	+	0.382578i	$0.124964\pi$
$197$			10.7395i			0.765156i	−0.923923	+	0.382578i	$0.875036\pi$
$198$		0			0
$199$			9.36418i			0.663809i	0.943313	+	0.331905i	$0.107691\pi$
$199$			9.36418i			0.663809i	−0.943313	+	0.331905i	$0.892309\pi$
$200$		0			0
$201$		−	7.95805i		−	0.561318i
$202$		0			0
$203$	−1.34264	−	1.89986i	−0.0942345	−	0.133344i
$204$		0			0
$205$		−	14.4148i		−	1.00677i
$206$		0			0
$207$	4.64894			0.323124
$208$		0			0
$209$	2.45665	−	9.03006i	0.169930	−	0.624622i
$210$		0			0
$211$		−	8.48158i		−	0.583896i	−0.956434	−	0.291948i	$-0.905697\pi$
$211$		−	8.48158i		−	0.583896i	0.956434	−	0.291948i	$-0.0943034\pi$
$212$		0			0
$213$			10.3529i			0.709372i
$214$		0			0
$215$	1.07602			0.0733837
$216$		0			0
$217$	0.708753	+	1.00290i	0.0481133	+	0.0680815i
$218$		0			0
$219$			38.7161i			2.61619i
$220$		0			0
$221$	1.61172			0.108416
$222$		0			0
$223$			1.58876i			0.106391i	0.998584	+	0.0531956i	$0.0169407\pi$
$223$			1.58876i			0.106391i	−0.998584	+	0.0531956i	$0.983059\pi$
$224$		0			0
$225$	0.257338			0.0171559
$226$		0			0
$227$	11.3353			0.752350			0.376175	−	0.926549i	$-0.377239\pi$
$227$	11.3353			0.752350			0.376175	+	0.926549i	$0.377239\pi$
$228$		0			0
$229$			20.3078i			1.34198i	0.741467	+	0.670989i	$0.234130\pi$
$229$			20.3078i			1.34198i	−0.741467	+	0.670989i	$0.765870\pi$
$230$		0			0
$231$	19.8923	−	7.25191i	1.30881	−	0.477141i
$232$		0			0
$233$			2.33496i			0.152969i	0.997071	+	0.0764843i	$0.0243695\pi$
$233$			2.33496i			0.152969i	−0.997071	+	0.0764843i	$0.975630\pi$
$234$		0			0
$235$	−26.1707			−1.70719
$236$		0			0
$237$	14.6482			0.951501
$238$		0			0
$239$		−	12.4365i		−	0.804448i	−0.915541	−	0.402224i	$-0.868237\pi$
$239$		−	12.4365i		−	0.804448i	0.915541	−	0.402224i	$-0.131763\pi$
$240$		0			0
$241$	11.4528			0.737743			0.368871	−	0.929480i	$-0.379744\pi$
$241$	11.4528			0.737743			0.368871	+	0.929480i	$0.379744\pi$
$242$		0			0
$243$			21.6394i			1.38817i
$244$		0			0
$245$	14.8884	+	5.27290i	0.951186	+	0.336873i
$246$		0			0
$247$	−2.82162			−0.179536
$248$		0			0
$249$		−	7.54131i		−	0.477911i
$250$		0			0
$251$		−	11.5573i		−	0.729493i	−0.931107	−	0.364746i	$-0.881156\pi$
$251$		−	11.5573i		−	0.729493i	0.931107	−	0.364746i	$-0.118844\pi$
$252$		0			0
$253$	−1.43431	+	5.27221i	−0.0901746	+	0.331461i
$254$		0			0
$255$	−8.77474			−0.549495
$256$		0			0
$257$			16.8289i			1.04976i	0.851177	+	0.524879i	$0.175889\pi$
$257$			16.8289i			1.04976i	−0.851177	+	0.524879i	$0.824111\pi$
$258$		0			0
$259$	9.22656	−	6.52043i	0.573311	−	0.405160i
$260$		0			0
$261$		−	2.48136i		−	0.153592i
$262$		0			0
$263$			10.5972i			0.653449i	0.945120	+	0.326725i	$0.105945\pi$
$263$			10.5972i			0.653449i	−0.945120	+	0.326725i	$0.894055\pi$
$264$		0			0
$265$			23.2720i			1.42959i
$266$		0			0
$267$	28.1577			1.72322
$268$		0			0
$269$		−	27.1980i		−	1.65829i	−0.559031	−	0.829146i	$-0.688827\pi$
$269$		−	27.1980i		−	1.65829i	0.559031	−	0.829146i	$-0.311173\pi$
$270$		0			0
$271$	17.9320			1.08929			0.544644	−	0.838667i	$-0.316665\pi$
$271$	17.9320			1.08929			0.544644	+	0.838667i	$0.316665\pi$
$272$		0			0
$273$	−3.68432	−	5.21341i	−0.222985	−	0.315530i
$274$		0			0
$275$	−0.0793951	+	0.291838i	−0.00478771	+	0.0175985i
$276$		0			0
$277$			0.452266i			0.0271740i	0.999908	+	0.0135870i	$0.00432501\pi$
$277$			0.452266i			0.0271740i	−0.999908	+	0.0135870i	$0.995675\pi$
$278$		0			0
$279$			1.30986i			0.0784195i
$280$		0			0
$281$		−	15.2131i		−	0.907536i	−0.891120	−	0.453768i	$-0.850079\pi$
$281$		−	15.2131i		−	0.907536i	0.891120	−	0.453768i	$-0.149921\pi$
$282$		0			0
$283$	20.8203			1.23764			0.618818	−	0.785534i	$-0.287612\pi$
$283$	20.8203			1.23764			0.618818	+	0.785534i	$0.287612\pi$
$284$		0			0
$285$	15.3619			0.909958
$286$		0			0
$287$	13.8033	−	9.75484i	0.814785	−	0.575810i
$288$		0			0
$289$	−14.4024			−0.847198
$290$		0			0
$291$	26.4129			1.54835
$292$		0			0
$293$	−9.80082			−0.572570			−0.286285	−	0.958144i	$-0.592420\pi$
$293$	−9.80082			−0.572570			−0.286285	+	0.958144i	$0.592420\pi$
$294$		0			0
$295$	14.6847			0.854979
$296$		0			0
$297$	−1.37469	−	0.373988i	−0.0797677	−	0.0217010i
$298$		0			0
$299$	1.64741			0.0952720
$300$		0			0
$301$	0.728168	+	1.03038i	0.0419709	+	0.0593898i
$302$		0			0
$303$			47.2602i			2.71503i
$304$		0			0
$305$		−	18.6224i		−	1.06632i
$306$		0			0
$307$	12.2907			0.701469			0.350734	−	0.936475i	$-0.385932\pi$
$307$	12.2907			0.701469			0.350734	+	0.936475i	$0.385932\pi$
$308$		0			0
$309$	26.7104			1.51950
$310$		0			0
$311$			7.45172i			0.422548i	0.977427	+	0.211274i	$0.0677613\pi$
$311$			7.45172i			0.422548i	−0.977427	+	0.211274i	$0.932239\pi$
$312$		0			0
$313$		−	22.9461i		−	1.29699i	−0.761219	−	0.648495i	$-0.775399\pi$
$313$		−	22.9461i		−	1.29699i	0.761219	−	0.648495i	$-0.224601\pi$
$314$		0			0
$315$	9.72267	+	13.7578i	0.547810	+	0.775165i
$316$		0			0
$317$	−10.7856			−0.605778			−0.302889	−	0.953026i	$-0.597951\pi$
$317$	−10.7856			−0.605778			−0.302889	+	0.953026i	$0.597951\pi$
$318$		0			0
$319$	2.81402	+	0.765560i	0.157555	+	0.0428631i
$320$		0			0
$321$	5.73252			0.319958
$322$		0			0
$323$	−4.54766			−0.253039
$324$		0			0
$325$	0.0911906			0.00505835
$326$		0			0
$327$	8.68906			0.480506
$328$		0			0
$329$	−17.7104	−	25.0607i	−0.976407	−	1.38164i
$330$		0			0
$331$	−24.5895			−1.35156			−0.675780	−	0.737103i	$-0.736193\pi$
$331$	−24.5895			−1.35156			−0.675780	+	0.737103i	$0.736193\pi$
$332$		0			0
$333$	12.0506			0.660366
$334$		0			0
$335$			7.44185i			0.406592i
$336$		0			0
$337$			1.84164i			0.100320i	0.998741	+	0.0501602i	$0.0159732\pi$
$337$			1.84164i			0.100320i	−0.998741	+	0.0501602i	$0.984027\pi$
$338$		0			0
$339$		−	2.85134i		−	0.154864i
$340$		0			0
$341$	−1.48547	−	0.404126i	−0.0804428	−	0.0218846i
$342$		0			0
$343$	5.02613	+	17.8252i	0.271386	+	0.962471i
$344$		0			0
$345$	−8.96903			−0.482876
$346$		0			0
$347$			30.0985i			1.61577i	0.589337	+	0.807887i	$0.299389\pi$
$347$			30.0985i			1.61577i	−0.589337	+	0.807887i	$0.700611\pi$
$348$		0			0
$349$	−4.52920			−0.242442			−0.121221	−	0.992626i	$-0.538681\pi$
$349$	−4.52920			−0.242442			−0.121221	+	0.992626i	$0.538681\pi$
$350$		0			0
$351$			0.429550i			0.0229277i
$352$		0			0
$353$			22.3160i			1.18776i	0.804553	+	0.593881i	$0.202405\pi$
$353$			22.3160i			1.18776i	−0.804553	+	0.593881i	$0.797595\pi$
$354$		0			0
$355$		−	9.68140i		−	0.513835i
$356$		0			0
$357$	−5.93809	−	8.40254i	−0.314277	−	0.444710i
$358$		0			0
$359$			16.8000i			0.886670i	0.896356	+	0.443335i	$0.146205\pi$
$359$			16.8000i			0.886670i	−0.896356	+	0.443335i	$0.853795\pi$
$360$		0			0
$361$	−11.0384			−0.580971
$362$		0			0
$363$	−13.4462	+	22.8836i	−0.705744	+	1.20108i
$364$		0			0
$365$		−	36.2048i		−	1.89505i
$366$		0			0
$367$		−	14.6272i		−	0.763535i	−0.924258	−	0.381768i	$-0.875315\pi$
$367$		−	14.6272i		−	0.763535i	0.924258	−	0.381768i	$-0.124685\pi$
$368$		0			0
$369$	18.0281			0.938508
$370$		0			0
$371$	−22.2849	+	15.7488i	−1.15697	+	0.817635i
$372$		0			0
$373$			25.1007i			1.29966i	0.760078	+	0.649832i	$0.225161\pi$
$373$			25.1007i			1.29966i	−0.760078	+	0.649832i	$0.774839\pi$
$374$		0			0
$375$	26.7252			1.38008
$376$		0			0
$377$		−	0.879297i		−	0.0452861i
$378$		0			0
$379$	32.9275			1.69137			0.845687	−	0.533679i	$-0.179191\pi$
$379$	32.9275			1.69137			0.845687	+	0.533679i	$0.179191\pi$
$380$		0			0
$381$	21.6480			1.10906
$382$		0			0
$383$		−	15.7017i		−	0.802317i	−0.916009	−	0.401158i	$-0.868608\pi$
$383$		−	15.7017i		−	0.802317i	0.916009	−	0.401158i	$-0.131392\pi$
$384$		0			0
$385$	−18.6020	+	6.78152i	−0.948043	+	0.345618i
$386$		0			0
$387$			1.34574i			0.0684080i
$388$		0			0
$389$	14.0643			0.713088			0.356544	−	0.934279i	$-0.383955\pi$
$389$	14.0643			0.713088			0.356544	+	0.934279i	$0.383955\pi$
$390$		0			0
$391$	2.65516			0.134277
$392$		0			0
$393$			33.0004i			1.66465i
$394$		0			0
$395$	−13.6980			−0.689222
$396$		0			0
$397$		−	12.8389i		−	0.644368i	−0.946677	−	0.322184i	$-0.895583\pi$
$397$		−	12.8389i		−	0.644368i	0.946677	−	0.322184i	$-0.104417\pi$
$398$		0			0
$399$	10.3958	+	14.7103i	0.520439	+	0.736434i
$400$		0			0
$401$	29.0268			1.44953			0.724765	−	0.688996i	$-0.241948\pi$
$401$	29.0268			1.44953			0.724765	+	0.688996i	$0.241948\pi$
$402$		0			0
$403$			0.464165i			0.0231217i
$404$		0			0
$405$		−	21.4409i		−	1.06540i
$406$		0			0
$407$	−3.71790	+	13.6661i	−0.184289	+	0.677405i
$408$		0			0
$409$	−6.11456			−0.302346			−0.151173	−	0.988507i	$-0.548305\pi$
$409$	−6.11456			−0.302346			−0.151173	+	0.988507i	$0.548305\pi$
$410$		0			0
$411$			5.87777i			0.289929i
$412$		0			0
$413$	9.93755	+	14.0619i	0.488995	+	0.691939i
$414$		0			0
$415$			7.05214i			0.346176i
$416$		0			0
$417$			28.4740i			1.39438i
$418$		0			0
$419$		−	13.6914i		−	0.668871i	−0.942419	−	0.334435i	$-0.891454\pi$
$419$		−	13.6914i		−	0.668871i	0.942419	−	0.334435i	$-0.108546\pi$
$420$		0			0
$421$	12.2602			0.597524			0.298762	−	0.954328i	$-0.403426\pi$
$421$	12.2602			0.597524			0.298762	+	0.954328i	$0.403426\pi$
$422$		0			0
$423$		−	32.7310i		−	1.59144i
$424$		0			0
$425$	0.146974			0.00712927
$426$		0			0
$427$	17.8325	−	12.6023i	0.862976	−	0.609867i
$428$		0			0
$429$	7.72195	+	2.10077i	0.372819	+	0.101426i
$430$		0			0
$431$		−	29.5449i		−	1.42313i	−0.702622	−	0.711564i	$-0.747987\pi$
$431$		−	29.5449i		−	1.42313i	0.702622	−	0.711564i	$-0.252013\pi$
$432$		0			0
$433$		−	8.25303i		−	0.396615i	−0.980140	−	0.198308i	$-0.936455\pi$
$433$		−	8.25303i		−	0.396615i	0.980140	−	0.198308i	$-0.0635445\pi$
$434$		0			0
$435$			4.78719i			0.229528i
$436$		0			0
$437$	−4.64836			−0.222361
$438$		0			0
$439$	−19.9525			−0.952280			−0.476140	−	0.879369i	$-0.657964\pi$
$439$	−19.9525			−0.952280			−0.476140	+	0.879369i	$0.657964\pi$
$440$		0			0
$441$	−6.59467	+	18.6205i	−0.314032	+	0.886692i
$442$		0			0
$443$	−26.4530			−1.25682			−0.628411	−	0.777882i	$-0.716294\pi$
$443$	−26.4530			−1.25682			−0.628411	+	0.777882i	$0.716294\pi$
$444$		0			0
$445$	−26.3313			−1.24822
$446$		0			0
$447$	13.4621			0.636735
$448$		0			0
$449$	−9.39207			−0.443239			−0.221620	−	0.975133i	$-0.571134\pi$
$449$	−9.39207			−0.443239			−0.221620	+	0.975133i	$0.571134\pi$
$450$		0			0
$451$	−5.56213	+	20.4451i	−0.261911	+	0.962723i
$452$		0			0
$453$	38.4481			1.80645
$454$		0			0
$455$	3.44534	+	4.87524i	0.161520	+	0.228555i
$456$		0			0
$457$		−	13.5957i		−	0.635981i	−0.948094	−	0.317990i	$-0.896992\pi$
$457$		−	13.5957i		−	0.635981i	0.948094	−	0.317990i	$-0.103008\pi$
$458$		0			0
$459$			0.692314i			0.0323144i
$460$		0			0
$461$	7.97633			0.371495			0.185747	−	0.982598i	$-0.440529\pi$
$461$	7.97633			0.371495			0.185747	+	0.982598i	$0.440529\pi$
$462$		0			0
$463$	−3.31131			−0.153890			−0.0769449	−	0.997035i	$-0.524517\pi$
$463$	−3.31131			−0.153890			−0.0769449	+	0.997035i	$0.524517\pi$
$464$		0			0
$465$		−	2.52707i		−	0.117190i
$466$		0			0
$467$			25.3661i			1.17380i	0.809658	+	0.586902i	$0.199653\pi$
$467$			25.3661i			1.17380i	−0.809658	+	0.586902i	$0.800347\pi$
$468$		0			0
$469$	−7.12620	+	5.03610i	−0.329057	+	0.232545i
$470$		0			0
$471$	−39.8916			−1.83811
$472$		0			0
$473$	−1.52616	−	0.415196i	−0.0701730	−	0.0190907i
$474$		0			0
$475$	−0.257306			−0.0118060
$476$		0			0
$477$	−29.1056			−1.33266
$478$		0			0
$479$	−5.19660			−0.237439			−0.118719	−	0.992928i	$-0.537879\pi$
$479$	−5.19660			−0.237439			−0.118719	+	0.992928i	$0.537879\pi$
$480$		0			0
$481$	4.27025			0.194707
$482$		0			0
$483$	−6.06958	−	8.58860i	−0.276175	−	0.390795i
$484$		0			0
$485$	−24.6997			−1.12155
$486$		0			0
$487$	9.31799			0.422239			0.211119	−	0.977460i	$-0.432289\pi$
$487$	9.31799			0.422239			0.211119	+	0.977460i	$0.432289\pi$
$488$		0			0
$489$			27.9252i			1.26282i
$490$		0			0
$491$		−	2.53794i		−	0.114536i	−0.998359	−	0.0572678i	$-0.981761\pi$
$491$		−	2.53794i		−	0.114536i	0.998359	−	0.0572678i	$-0.0182389\pi$
$492$		0			0
$493$		−	1.41718i		−	0.0638266i
$494$		0			0
$495$	−20.3777	−	5.54379i	−0.915909	−	0.249175i
$496$		0			0
$497$	9.27075	−	6.55165i	0.415850	−	0.293882i
$498$		0			0
$499$	13.6324			0.610270			0.305135	−	0.952309i	$-0.401298\pi$
$499$	13.6324			0.610270			0.305135	+	0.952309i	$0.401298\pi$
$500$		0			0
$501$			4.16640i			0.186141i
$502$		0			0
$503$	−13.9891			−0.623742			−0.311871	−	0.950125i	$-0.600956\pi$
$503$	−13.9891			−0.623742			−0.311871	+	0.950125i	$0.600956\pi$
$504$		0			0
$505$		−	44.1947i		−	1.96664i
$506$		0			0
$507$		−	2.41288i		−	0.107160i
$508$		0			0
$509$		−	5.82129i		−	0.258024i	−0.991643	−	0.129012i	$-0.958819\pi$
$509$		−	5.82129i		−	0.258024i	0.991643	−	0.129012i	$-0.0411806\pi$
$510$		0			0
$511$	34.6691	−	24.5007i	1.53367	−	1.08385i
$512$		0			0
$513$		−	1.21203i		−	0.0535123i
$514$		0			0
$515$	−24.9778			−1.10066
$516$		0			0
$517$	37.1192	+	10.0983i	1.63250	+	0.444124i
$518$		0			0
$519$			46.7008i			2.04994i
$520$		0			0
$521$			1.72986i			0.0757866i	0.999282	+	0.0378933i	$0.0120647\pi$
$521$			1.72986i			0.0757866i	−0.999282	+	0.0378933i	$0.987935\pi$
$522$		0			0
$523$	7.29889			0.319158			0.159579	−	0.987185i	$-0.448986\pi$
$523$	7.29889			0.319158			0.159579	+	0.987185i	$0.448986\pi$
$524$		0			0
$525$	−0.335976	−	0.475414i	−0.0146632	−	0.0207488i
$526$		0			0
$527$			0.748104i			0.0325879i
$528$		0			0
$529$	−20.2861			−0.882002
$530$		0			0
$531$			18.3658i			0.797008i
$532$		0			0
$533$	6.38848			0.276716
$534$		0			0
$535$	−5.36068			−0.231762
$536$		0			0
$537$		−	62.6976i		−	2.70560i
$538$		0			0
$539$	−19.0823	−	13.2237i	−0.821933	−	0.569585i
$540$		0			0
$541$		−	18.2252i		−	0.783562i	−0.920058	−	0.391781i	$-0.871859\pi$
$541$		−	18.2252i		−	0.783562i	0.920058	−	0.391781i	$-0.128141\pi$
$542$		0			0
$543$	−33.1173			−1.42120
$544$		0			0
$545$	−8.12545			−0.348056
$546$		0			0
$547$		−	37.9640i		−	1.62322i	−0.584198	−	0.811611i	$-0.698591\pi$
$547$		−	37.9640i		−	1.62322i	0.584198	−	0.811611i	$-0.301409\pi$
$548$		0			0
$549$	23.2906			0.994016
$550$		0			0
$551$			2.48104i			0.105696i
$552$		0			0
$553$	−9.26981	−	13.1170i	−0.394192	−	0.557792i
$554$		0			0
$555$	−23.2487			−0.986852
$556$		0			0
$557$		−	25.9796i		−	1.10079i	−0.834904	−	0.550395i	$-0.814477\pi$
$557$		−	25.9796i		−	1.10079i	0.834904	−	0.550395i	$-0.185523\pi$
$558$		0			0
$559$			0.476880i			0.0201699i
$560$		0			0
$561$	12.4456	+	3.38585i	0.525454	+	0.142951i
$562$		0			0
$563$	−9.06582			−0.382079			−0.191039	−	0.981582i	$-0.561186\pi$
$563$	−9.06582			−0.382079			−0.191039	+	0.981582i	$0.561186\pi$
$564$		0			0
$565$			2.66639i			0.112176i
$566$		0			0
$567$	20.5314	−	14.5096i	0.862238	−	0.609345i
$568$		0			0
$569$			13.3156i			0.558220i	0.960259	+	0.279110i	$0.0900394\pi$
$569$			13.3156i			0.558220i	−0.960259	+	0.279110i	$0.909961\pi$
$570$		0			0
$571$		−	35.9969i		−	1.50642i	−0.657778	−	0.753212i	$-0.728504\pi$
$571$		−	35.9969i		−	1.50642i	0.657778	−	0.753212i	$-0.271496\pi$
$572$		0			0
$573$			50.3225i			2.10225i
$574$		0			0
$575$	0.150228			0.00626494
$576$		0			0
$577$			17.4591i			0.726833i	0.931627	+	0.363416i	$0.118390\pi$
$577$			17.4591i			0.726833i	−0.931627	+	0.363416i	$0.881610\pi$
$578$		0			0
$579$	−9.71180			−0.403609
$580$		0			0
$581$	−6.75301	+	4.77237i	−0.280162	+	0.197991i
$582$		0			0
$583$	8.97982	−	33.0077i	0.371906	−	1.36704i
$584$		0			0
$585$			6.36741i			0.263260i
$586$		0			0
$587$			36.4888i			1.50606i	0.657989	+	0.753028i	$0.271407\pi$
$587$			36.4888i			1.50606i	−0.657989	+	0.753028i	$0.728593\pi$
$588$		0			0
$589$		−	1.30970i		−	0.0539652i
$590$		0			0
$591$	25.9130			1.06592
$592$		0			0
$593$	−9.36597			−0.384614			−0.192307	−	0.981335i	$-0.561597\pi$
$593$	−9.36597			−0.384614			−0.192307	+	0.981335i	$0.561597\pi$
$594$		0			0
$595$	5.55292	+	7.85752i	0.227648	+	0.322127i
$596$		0			0
$597$	22.5946			0.924737
$598$		0			0
$599$	12.7074			0.519211			0.259605	−	0.965715i	$-0.416408\pi$
$599$	12.7074			0.519211			0.259605	+	0.965715i	$0.416408\pi$
$600$		0			0
$601$	−32.5310			−1.32697			−0.663483	−	0.748192i	$-0.730922\pi$
$601$	−32.5310			−1.32697			−0.663483	+	0.748192i	$0.730922\pi$
$602$		0			0
$603$	−9.30732			−0.379024
$604$		0			0
$605$	12.5740	−	21.3992i	0.511208	−	0.870003i
$606$		0			0
$607$	48.9766			1.98790			0.993950	−	0.109836i	$-0.0350324\pi$
$607$	48.9766			1.98790			0.993950	+	0.109836i	$0.0350324\pi$
$608$		0			0
$609$	−4.58413	+	3.23961i	−0.185758	+	0.131276i
$610$		0			0
$611$		−	11.5986i		−	0.469230i
$612$		0			0
$613$			8.16404i			0.329742i	0.986315	+	0.164871i	$0.0527209\pi$
$613$			8.16404i			0.329742i	−0.986315	+	0.164871i	$0.947279\pi$
$614$		0			0
$615$	−34.7810			−1.40251
$616$		0			0
$617$	−10.9578			−0.441142			−0.220571	−	0.975371i	$-0.570792\pi$
$617$	−10.9578			−0.441142			−0.220571	+	0.975371i	$0.570792\pi$
$618$		0			0
$619$		−	5.95790i		−	0.239468i	−0.992806	−	0.119734i	$-0.961796\pi$
$619$		−	5.95790i		−	0.239468i	0.992806	−	0.119734i	$-0.0382042\pi$
$620$		0			0
$621$			0.707643i			0.0283967i
$622$		0			0
$623$	−17.8191	−	25.2144i	−0.713906	−	1.01019i
$624$		0			0
$625$	−25.4476			−1.01791
$626$		0			0
$627$	−21.7884	−	5.92758i	−0.870146	−	0.236725i
$628$		0			0
$629$	6.88245			0.274421
$630$		0			0
$631$	−42.0634			−1.67452			−0.837258	−	0.546808i	$-0.815843\pi$
$631$	−42.0634			−1.67452			−0.837258	+	0.546808i	$0.815843\pi$
$632$		0			0
$633$	−20.4650			−0.813411
$634$		0			0
$635$	−20.2438			−0.803351
$636$		0			0
$637$	−2.33690	+	6.59840i	−0.0925913	+	0.261438i
$638$		0			0
$639$	12.1083			0.478995
$640$		0			0
$641$	−26.9874			−1.06594			−0.532969	−	0.846135i	$-0.678924\pi$
$641$	−26.9874			−1.06594			−0.532969	+	0.846135i	$0.678924\pi$
$642$		0			0
$643$		−	19.9346i		−	0.786145i	−0.919507	−	0.393073i	$-0.871412\pi$
$643$		−	19.9346i		−	0.786145i	0.919507	−	0.393073i	$-0.128588\pi$
$644$		0			0
$645$		−	2.59629i		−	0.102229i
$646$		0			0
$647$		−	4.37875i		−	0.172146i	−0.996289	−	0.0860732i	$-0.972568\pi$
$647$		−	4.37875i		−	0.172146i	0.996289	−	0.0860732i	$-0.0274319\pi$
$648$		0			0
$649$	−20.8280	−	5.66631i	−0.817572	−	0.222422i
$650$		0			0
$651$	2.41988	−	1.71013i	0.0948427	−	0.0670255i
$652$		0			0
$653$	−21.3695			−0.836253			−0.418126	−	0.908389i	$-0.637313\pi$
$653$	−21.3695			−0.836253			−0.418126	+	0.908389i	$0.637313\pi$
$654$		0			0
$655$		−	30.8598i		−	1.20579i
$656$		0			0
$657$	45.2803			1.76655
$658$		0			0
$659$		−	3.89775i		−	0.151835i	−0.997114	−	0.0759174i	$-0.975811\pi$
$659$		−	3.89775i		−	0.151835i	0.997114	−	0.0759174i	$-0.0241885\pi$
$660$		0			0
$661$			14.4013i			0.560147i	0.959979	+	0.280074i	$0.0903588\pi$
$661$			14.4013i			0.560147i	−0.959979	+	0.280074i	$0.909641\pi$
$662$		0			0
$663$		−	3.88888i		−	0.151032i
$664$		0			0
$665$	−9.72145	−	13.7561i	−0.376982	−	0.533438i
$666$		0			0
$667$		−	1.44856i		−	0.0560884i
$668$		0			0
$669$	3.83348			0.148211
$670$		0			0
$671$	−7.18572	+	26.4130i	−0.277401	+	1.01966i
$672$		0			0
$673$		−	41.4852i		−	1.59914i	−0.600574	−	0.799569i	$-0.705061\pi$
$673$		−	41.4852i		−	1.59914i	0.600574	−	0.799569i	$-0.294939\pi$
$674$		0			0
$675$			0.0391709i			0.00150769i
$676$		0			0
$677$	2.05887			0.0791288			0.0395644	−	0.999217i	$-0.487403\pi$
$677$	2.05887			0.0791288			0.0395644	+	0.999217i	$0.487403\pi$
$678$		0			0
$679$	−16.7149	−	23.6520i	−0.641459	−	0.907680i
$680$		0			0
$681$		−	27.3507i		−	1.04808i
$682$		0			0
$683$	−15.6964			−0.600606			−0.300303	−	0.953844i	$-0.597088\pi$
$683$	−15.6964			−0.600606			−0.300303	+	0.953844i	$0.597088\pi$
$684$		0			0
$685$		−	5.49651i		−	0.210011i
$686$		0			0
$687$	49.0002			1.86948
$688$		0			0
$689$	−10.3139			−0.392929
$690$		0			0
$691$			3.44317i			0.130984i	0.997853	+	0.0654922i	$0.0208618\pi$
$691$			3.44317i			0.130984i	−0.997853	+	0.0654922i	$0.979138\pi$
$692$		0			0
$693$	−8.48146	−	23.2650i	−0.322184	−	0.883763i
$694$		0			0
$695$		−	26.6270i		−	1.01002i
$696$		0			0
$697$	10.2964			0.390005
$698$		0			0
$699$	5.63398			0.213097
$700$		0			0
$701$			3.00559i			0.113520i	0.998388	+	0.0567598i	$0.0180769\pi$
$701$			3.00559i			0.113520i	−0.998388	+	0.0567598i	$0.981923\pi$
$702$		0			0
$703$	−12.0490			−0.454438
$704$		0			0
$705$			63.1468i			2.37825i
$706$		0			0
$707$	42.3201	−	29.9077i	1.59161	−	1.12480i
$708$		0			0
$709$	18.4722			0.693737			0.346869	−	0.937914i	$-0.387245\pi$
$709$	18.4722			0.693737			0.346869	+	0.937914i	$0.387245\pi$
$710$		0			0
$711$		−	17.1317i		−	0.642491i
$712$		0			0
$713$			0.764669i			0.0286371i
$714$		0			0
$715$	−7.22107	−	1.96451i	−0.270053	−	0.0734683i
$716$		0			0
$717$	−30.0077			−1.12066
$718$		0			0
$719$		−	11.6277i		−	0.433639i	−0.976212	−	0.216820i	$-0.930432\pi$
$719$		−	11.6277i		−	0.433639i	0.976212	−	0.216820i	$-0.0695684\pi$
$720$		0			0
$721$	−16.9032	−	23.9184i	−0.629506	−	0.890767i
$722$		0			0
$723$		−	27.6343i		−	1.02773i
$724$		0			0
$725$		−	0.0801837i		−	0.00297795i
$726$		0			0
$727$			33.5548i			1.24448i	0.782827	+	0.622239i	$0.213777\pi$
$727$			33.5548i			1.24448i	−0.782827	+	0.622239i	$0.786223\pi$
$728$		0			0
$729$	23.7061			0.878005
$730$		0			0
$731$			0.768596i			0.0284276i
$732$		0			0
$733$	46.6585			1.72337			0.861686	−	0.507442i	$-0.169409\pi$
$733$	46.6585			1.72337			0.861686	+	0.507442i	$0.169409\pi$
$734$		0			0
$735$	12.7229	−	35.9239i	0.469290	−	1.32507i
$736$		0			0
$737$	2.87154	−	10.5551i	0.105775	−	0.388803i
$738$		0			0
$739$		−	8.77562i		−	0.322816i	−0.986888	−	0.161408i	$-0.948396\pi$
$739$		−	8.77562i		−	0.322816i	0.986888	−	0.161408i	$-0.0516036\pi$
$740$		0			0
$741$			6.80823i			0.250107i
$742$		0			0
$743$			11.0594i			0.405729i	0.979207	+	0.202864i	$0.0650251\pi$
$743$			11.0594i			0.405729i	−0.979207	+	0.202864i	$0.934975\pi$
$744$		0			0
$745$	−12.5889			−0.461221
$746$		0			0
$747$	−8.81992			−0.322704
$748$		0			0
$749$	−3.62771	−	5.13330i	−0.132554	−	0.187567i
$750$		0			0
$751$	−16.1743			−0.590210			−0.295105	−	0.955465i	$-0.595355\pi$
$751$	−16.1743			−0.590210			−0.295105	+	0.955465i	$0.595355\pi$
$752$		0			0
$753$	−27.8864			−1.01624
$754$		0			0
$755$	−35.9541			−1.30851
$756$		0			0
$757$	−40.6445			−1.47725			−0.738625	−	0.674117i	$-0.764524\pi$
$757$	−40.6445			−1.47725			−0.738625	+	0.674117i	$0.764524\pi$
$758$		0			0
$759$	12.7212	+	3.46082i	0.461750	+	0.125620i
$760$		0			0
$761$	−31.6907			−1.14879			−0.574394	−	0.818579i	$-0.694762\pi$
$761$	−31.6907			−1.14879			−0.574394	+	0.818579i	$0.694762\pi$
$762$		0			0
$763$	−5.49870	−	7.78079i	−0.199066	−	0.281684i
$764$		0			0
$765$			10.2625i			0.371041i
$766$		0			0
$767$			6.50814i			0.234995i
$768$		0			0
$769$	20.7918			0.749771			0.374885	−	0.927071i	$-0.377682\pi$
$769$	20.7918			0.749771			0.374885	+	0.927071i	$0.377682\pi$
$770$		0			0
$771$	40.6060			1.46239
$772$		0			0
$773$		−	6.97997i		−	0.251052i	−0.992090	−	0.125526i	$-0.959938\pi$
$773$		−	6.97997i		−	0.251052i	0.992090	−	0.125526i	$-0.0400619\pi$
$774$		0			0
$775$			0.0423275i			0.00152045i
$776$		0			0
$777$	−15.7330	−	22.2626i	−0.564418	−	0.798665i
$778$		0			0
$779$	−18.0259			−0.645844
$780$		0			0
$781$	−3.73570	+	13.7316i	−0.133674	+	0.491354i
$782$		0			0
$783$	0.377702			0.0134980
$784$		0			0
$785$	37.3040			1.33144
$786$		0			0
$787$	−1.53171			−0.0545995			−0.0272997	−	0.999627i	$-0.508691\pi$
$787$	−1.53171			−0.0545995			−0.0272997	+	0.999627i	$0.508691\pi$
$788$		0			0
$789$	25.5697			0.910304
$790$		0			0
$791$	−2.55329	+	1.80441i	−0.0907845	+	0.0641576i
$792$		0			0
$793$	8.25328			0.293082
$794$		0			0
$795$	56.1525			1.99152
$796$		0			0
$797$		−	16.4952i		−	0.584291i	−0.956374	−	0.292145i	$-0.905631\pi$
$797$		−	16.4952i		−	0.584291i	0.956374	−	0.292145i	$-0.0943692\pi$
$798$		0			0
$799$		−	18.6937i		−	0.661336i
$800$		0			0
$801$		−	32.9318i		−	1.16359i
$802$		0			0
$803$	−13.9701	+	51.3509i	−0.492995	+	1.81213i
$804$		0			0
$805$	5.67587	+	8.03150i	0.200048	+	0.283073i
$806$		0			0
$807$	−65.6255			−2.31013
$808$		0			0
$809$		−	0.243697i		−	0.00856795i	−0.999991	−	0.00428397i	$-0.998636\pi$
$809$		−	0.243697i		−	0.00856795i	0.999991	−	0.00428397i	$-0.00136364\pi$
$810$		0			0
$811$	−31.6612			−1.11178			−0.555888	−	0.831257i	$-0.687622\pi$
$811$	−31.6612			−1.11178			−0.555888	+	0.831257i	$0.687622\pi$
$812$		0			0
$813$		−	43.2676i		−	1.51746i
$814$		0			0
$815$		−	26.1139i		−	0.914729i
$816$		0			0
$817$		−	1.34558i		−	0.0470757i
$818$		0			0
$819$	−6.09733	+	4.30899i	−0.213058	+	0.150568i
$820$		0			0
$821$			0.908545i			0.0317084i	0.999874	+	0.0158542i	$0.00504677\pi$
$821$			0.908545i			0.0317084i	−0.999874	+	0.0158542i	$0.994953\pi$
$822$		0			0
$823$	2.24414			0.0782260			0.0391130	−	0.999235i	$-0.487547\pi$
$823$	2.24414			0.0782260			0.0391130	+	0.999235i	$0.487547\pi$
$824$		0			0
$825$	0.704169	+	0.191571i	0.0245160	+	0.00666964i
$826$		0			0
$827$			31.6043i			1.09899i	0.835497	+	0.549495i	$0.185180\pi$
$827$			31.6043i			1.09899i	−0.835497	+	0.549495i	$0.814820\pi$
$828$		0			0
$829$			43.4334i			1.50851i	0.656584	+	0.754253i	$0.272001\pi$
$829$			43.4334i			1.50851i	−0.656584	+	0.754253i	$0.727999\pi$
$830$		0			0
$831$	1.09126			0.0378555
$832$		0			0
$833$	−3.76642	+	10.6348i	−0.130499	+	0.368473i
$834$		0			0
$835$		−	3.89615i		−	0.134832i
$836$		0			0
$837$	−0.199382			−0.00689165
$838$		0			0
$839$		−	19.9842i		−	0.689933i	−0.938615	−	0.344966i	$-0.887890\pi$
$839$		−	19.9842i		−	0.689933i	0.938615	−	0.344966i	$-0.112110\pi$
$840$		0			0
$841$	28.2268			0.973339
$842$		0			0
$843$	−36.7073			−1.26427
$844$		0			0
$845$			2.25637i			0.0776214i
$846$		0			0
$847$	29.0007	−	2.44072i	0.996477	−	0.0838642i
$848$		0			0
$849$		−	50.2367i		−	1.72412i
$850$		0			0
$851$	7.03484			0.241151
$852$		0			0
$853$	16.4287			0.562506			0.281253	−	0.959634i	$-0.409250\pi$
$853$	16.4287			0.562506			0.281253	+	0.959634i	$0.409250\pi$
$854$		0			0
$855$		−	17.9664i		−	0.614439i
$856$		0			0
$857$	42.6144			1.45568			0.727841	−	0.685746i	$-0.240524\pi$
$857$	42.6144			1.45568			0.727841	+	0.685746i	$0.240524\pi$
$858$		0			0
$859$		−	11.6592i		−	0.397807i	−0.980019	−	0.198903i	$-0.936262\pi$
$859$		−	11.6592i		−	0.397807i	0.980019	−	0.198903i	$-0.0637380\pi$
$860$		0			0
$861$	−23.5372	−	33.3057i	−0.802147	−	1.13506i
$862$		0			0
$863$	19.9627			0.679539			0.339769	−	0.940509i	$-0.389651\pi$
$863$	19.9627			0.679539			0.339769	+	0.940509i	$0.389651\pi$
$864$		0			0
$865$		−	43.6716i		−	1.48488i
$866$		0			0
$867$			34.7511i			1.18021i
$868$		0			0
$869$	19.4285	+	5.28557i	0.659068	+	0.179301i
$870$		0			0
$871$	−3.29816			−0.111754
$872$		0			0
$873$		−	30.8912i		−	1.04551i
$874$		0			0
$875$	−16.9125	−	23.9316i	−0.571747	−	0.809037i
$876$		0			0
$877$		−	29.0276i		−	0.980192i	−0.871668	−	0.490096i	$-0.836962\pi$
$877$		−	29.0276i		−	0.980192i	0.871668	−	0.490096i	$-0.163038\pi$
$878$		0			0
$879$			23.6482i			0.797633i
$880$		0			0
$881$			49.9590i			1.68316i	0.540132	+	0.841580i	$0.318374\pi$
$881$			49.9590i			1.68316i	−0.540132	+	0.841580i	$0.681626\pi$
$882$		0			0
$883$	27.3319			0.919792			0.459896	−	0.887973i	$-0.347887\pi$
$883$	27.3319			0.919792			0.459896	+	0.887973i	$0.347887\pi$
$884$		0			0
$885$		−	35.4325i		−	1.19105i
$886$		0			0
$887$	6.03601			0.202670			0.101335	−	0.994852i	$-0.467689\pi$
$887$	6.03601			0.202670			0.101335	+	0.994852i	$0.467689\pi$
$888$		0			0
$889$	−13.6995	−	19.3851i	−0.459467	−	0.650156i
$890$		0			0
$891$	−8.27325	+	30.4105i	−0.277164	+	1.01879i
$892$		0			0
$893$			32.7269i			1.09517i
$894$		0			0
$895$			58.6307i			1.95981i
$896$		0			0
$897$		−	3.97499i		−	0.132721i
$898$		0			0
$899$	0.408139			0.0136122
$900$		0			0
$901$	−16.6231			−0.553797
$902$		0			0
$903$	2.48617	−	1.75698i	0.0827345	−	0.0584686i
$904$		0			0
$905$	30.9691			1.02945
$906$		0			0
$907$	45.7912			1.52047			0.760235	−	0.649648i	$-0.225084\pi$
$907$	45.7912			1.52047			0.760235	+	0.649648i	$0.225084\pi$
$908$		0			0
$909$	55.2731			1.83329
$910$		0			0
$911$	−48.8404			−1.61816			−0.809078	−	0.587702i	$-0.800033\pi$
$911$	−48.8404			−1.61816			−0.809078	+	0.587702i	$0.800033\pi$
$912$		0			0
$913$	2.72117	−	10.0024i	0.0900574	−	0.331030i
$914$		0			0
$915$	−44.9336			−1.48546
$916$		0			0
$917$	29.5509	−	20.8836i	0.975855	−	0.689639i
$918$		0			0
$919$		−	9.94764i		−	0.328142i	−0.986448	−	0.164071i	$-0.947537\pi$
$919$		−	9.94764i		−	0.328142i	0.986448	−	0.164071i	$-0.0524627\pi$
$920$		0			0
$921$		−	29.6560i		−	0.977199i
$922$		0			0
$923$	4.29070			0.141230
$924$		0			0
$925$	0.389407			0.0128036
$926$		0			0
$927$		−	31.2391i		−	1.02603i
$928$		0			0
$929$		−	32.7813i		−	1.07552i	−0.843098	−	0.537761i	$-0.819270\pi$
$929$		−	32.7813i		−	1.07552i	0.843098	−	0.537761i	$-0.180730\pi$
$930$		0			0
$931$	6.59385	−	18.6182i	0.216105	−	0.610187i
$932$		0			0
$933$	17.9801			0.588641
$934$		0			0
$935$	−11.6383	−	3.16623i	−0.380614	−	0.103547i
$936$		0			0
$937$	−21.6683			−0.707873			−0.353936	−	0.935270i	$-0.615157\pi$
$937$	−21.6683			−0.707873			−0.353936	+	0.935270i	$0.615157\pi$
$938$		0			0
$939$	−55.3661			−1.80680
$940$		0			0
$941$	40.4243			1.31779			0.658897	−	0.752233i	$-0.271023\pi$
$941$	40.4243			1.31779			0.658897	+	0.752233i	$0.271023\pi$
$942$		0			0
$943$	10.5244			0.342722
$944$		0			0
$945$	−2.09416	+	1.47995i	−0.0681230	+	0.0481426i
$946$		0			0
$947$	−41.5518			−1.35025			−0.675125	−	0.737703i	$-0.735910\pi$
$947$	−41.5518			−1.35025			−0.675125	+	0.737703i	$0.735910\pi$
$948$		0			0
$949$	16.0456			0.520863
$950$		0			0
$951$			26.0243i			0.843895i
$952$		0			0
$953$			59.9299i			1.94132i	0.240459	+	0.970659i	$0.422702\pi$
$953$			59.9299i			1.94132i	−0.240459	+	0.970659i	$0.577298\pi$
$954$		0			0
$955$		−	47.0584i		−	1.52277i
$956$		0			0
$957$	1.84720	−	6.78989i	0.0597116	−	0.219486i
$958$		0			0
$959$	5.26337	−	3.71963i	0.169963	−	0.120113i
$960$		0			0
$961$	30.7846			0.993050
$962$		0			0
$963$		−	6.70445i		−	0.216048i
$964$		0			0
$965$	9.08185			0.292355
$966$		0			0
$967$		−	58.5776i		−	1.88373i	−0.335992	−	0.941865i	$-0.609072\pi$
$967$		−	58.5776i		−	1.88373i	0.335992	−	0.941865i	$-0.390928\pi$
$968$		0			0
$969$			10.9730i			0.352502i
$970$		0			0
$971$			53.0782i			1.70336i	0.524063	+	0.851680i	$0.324416\pi$
$971$			53.0782i			1.70336i	−0.524063	+	0.851680i	$0.675584\pi$
$972$		0			0
$973$	25.4976	−	18.0192i	0.817416	−	0.577669i
$974$		0			0
$975$		−	0.220032i		−	0.00704666i
$976$		0			0
$977$	−36.1719			−1.15724			−0.578620	−	0.815597i	$-0.696409\pi$
$977$	−36.1719			−1.15724			−0.578620	+	0.815597i	$0.696409\pi$
$978$		0			0
$979$	37.3469	+	10.1603i	1.19361	+	0.324724i
$980$		0			0
$981$		−	10.1623i		−	0.324456i
$982$		0			0
$983$			20.2286i			0.645193i	0.946537	+	0.322596i	$0.104556\pi$
$983$			20.2286i			0.645193i	−0.946537	+	0.322596i	$0.895444\pi$
$984$		0			0
$985$	−24.2322			−0.772102
$986$		0			0
$987$	−60.4683	+	42.7331i	−1.92473	+	1.36021i
$988$		0			0
$989$			0.785615i			0.0249811i
$990$		0			0
$991$	−22.7984			−0.724214			−0.362107	−	0.932136i	$-0.617943\pi$
$991$	−22.7984			−0.724214			−0.362107	+	0.932136i	$0.617943\pi$
$992$		0			0
$993$			59.3314i			1.88283i
$994$		0			0
$995$	−21.1290			−0.669835
$996$		0			0
$997$	14.6656			0.464465			0.232232	−	0.972660i	$-0.425397\pi$
$997$	14.6656			0.464465			0.232232	+	0.972660i	$0.425397\pi$
$998$		0			0
$999$			1.83429i			0.0580343i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.e.b.3849.8	yes	48
7.6	odd	2		4004.2.e.a.3849.41	yes	48
11.10	odd	2		4004.2.e.a.3849.8	✓	48
77.76	even	2	inner	4004.2.e.b.3849.41	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.e.a.3849.8	✓	48	11.10	odd	2
4004.2.e.a.3849.41	yes	48	7.6	odd	2
4004.2.e.b.3849.8	yes	48	1.1	even	1	trivial
4004.2.e.b.3849.41	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)

\(f(q)\)	\(=\)	\(q-2.41288i q^{3} +2.25637i q^{5} +(-2.16066 + 1.52694i) q^{7} -2.82198 q^{9} +O(q^{10})\)	\(q-2.41288i q^{3} +2.25637i q^{5} +(-2.16066 + 1.52694i) q^{7} -2.82198 q^{9} +(0.870650 - 3.20031i) q^{11} -1.00000 q^{13} +5.44434 q^{15} -1.61172 q^{17} +2.82162 q^{19} +(3.68432 + 5.21341i) q^{21} -1.64741 q^{23} -0.0911906 q^{25} -0.429550i q^{27} +0.879297i q^{29} -0.464165i q^{31} +(-7.72195 - 2.10077i) q^{33} +(-3.44534 - 4.87524i) q^{35} -4.27025 q^{37} +2.41288i q^{39} -6.38848 q^{41} -0.476880i q^{43} -6.36741i q^{45} +11.5986i q^{47} +(2.33690 - 6.59840i) q^{49} +3.88888i q^{51} +10.3139 q^{53} +(7.22107 + 1.96451i) q^{55} -6.80823i q^{57} -6.50814i q^{59} -8.25328 q^{61} +(6.09733 - 4.30899i) q^{63} -2.25637i q^{65} +3.29816 q^{67} +3.97499i q^{69} -4.29070 q^{71} -16.0456 q^{73} +0.220032i q^{75} +(3.00550 + 8.24421i) q^{77} +6.07083i q^{79} -9.50238 q^{81} +3.12544 q^{83} -3.63663i q^{85} +2.12164 q^{87} +11.6698i q^{89} +(2.16066 - 1.52694i) q^{91} -1.11997 q^{93} +6.36662i q^{95} +10.9467i q^{97} +(-2.45695 + 9.03119i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{7} - 48 q^{9}+O(q^{10}) \) 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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