LMFDB - Embedded newform 4004.2.e.b.3849.7

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			3849.7
Character	$\chi$	$=$	4004.3849
Dual form			4004.2.e.b.3849.42

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.42199i q^{3} -0.343428i q^{5} +(-2.23070 + 1.42266i) q^{7} -2.86603 q^{9} +O(q^{10})$	$q-2.42199i q^{3} -0.343428i q^{5} +(-2.23070 + 1.42266i) q^{7} -2.86603 q^{9} +(-2.51132 + 2.16640i) q^{11} -1.00000 q^{13} -0.831780 q^{15} +7.55799 q^{17} -4.71439 q^{19} +(3.44568 + 5.40273i) q^{21} +7.48858 q^{23} +4.88206 q^{25} -0.324476i q^{27} +4.61743i q^{29} -6.22496i q^{31} +(5.24699 + 6.08239i) q^{33} +(0.488584 + 0.766086i) q^{35} -4.35429 q^{37} +2.42199i q^{39} +7.26385 q^{41} -1.07514i q^{43} +0.984276i q^{45} +1.00695i q^{47} +(2.95205 - 6.34708i) q^{49} -18.3054i q^{51} -1.66130 q^{53} +(0.744002 + 0.862459i) q^{55} +11.4182i q^{57} +0.229128i q^{59} +4.47489 q^{61} +(6.39325 - 4.07740i) q^{63} +0.343428i q^{65} -0.898957 q^{67} -18.1373i q^{69} -7.01794 q^{71} -5.83754 q^{73} -11.8243i q^{75} +(2.51995 - 8.40535i) q^{77} -5.70523i q^{79} -9.38396 q^{81} +5.78767 q^{83} -2.59563i q^{85} +11.1834 q^{87} +0.531306i q^{89} +(2.23070 - 1.42266i) q^{91} -15.0768 q^{93} +1.61905i q^{95} -9.91329i q^{97} +(7.19752 - 6.20896i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	2.42199i		−	1.39834i	−0.714957	−	0.699168i	$-0.753554\pi$
$3$		−	2.42199i		−	1.39834i	0.714957	−	0.699168i	$-0.246446\pi$
$4$		0			0
$5$		−	0.343428i		−	0.153586i	−0.997047	−	0.0767929i	$-0.975532\pi$
$5$		−	0.343428i		−	0.153586i	0.997047	−	0.0767929i	$-0.0244680\pi$
$6$		0			0
$7$	−2.23070	+	1.42266i	−0.843126	+	0.537717i
$7$	−2.23070	+	1.42266i	−0.843126	+	0.537717i
$8$		0			0
$9$	−2.86603			−0.955343
$10$		0			0
$11$	−2.51132	+	2.16640i	−0.757191	+	0.653193i
$11$	−2.51132	+	2.16640i	−0.757191	+	0.653193i
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$	−0.831780			−0.214765
$16$		0			0
$17$	7.55799			1.83308			0.916541	−	0.399941i	$-0.130969\pi$
$17$	7.55799			1.83308			0.916541	+	0.399941i	$0.130969\pi$
$18$		0			0
$19$	−4.71439			−1.08155			−0.540777	−	0.841166i	$-0.681870\pi$
$19$	−4.71439			−1.08155			−0.540777	+	0.841166i	$0.681870\pi$
$20$		0			0
$21$	3.44568	+	5.40273i	0.751909	+	1.17897i
$22$		0			0
$23$	7.48858			1.56148			0.780738	−	0.624858i	$-0.214843\pi$
$23$	7.48858			1.56148			0.780738	+	0.624858i	$0.214843\pi$
$24$		0			0
$25$	4.88206			0.976411
$26$		0			0
$27$		−	0.324476i		−	0.0624453i
$28$		0			0
$29$			4.61743i			0.857434i	0.903439	+	0.428717i	$0.141034\pi$
$29$			4.61743i			0.857434i	−0.903439	+	0.428717i	$0.858966\pi$
$30$		0			0
$31$		−	6.22496i		−	1.11804i	−0.829156	−	0.559018i	$-0.811178\pi$
$31$		−	6.22496i		−	1.11804i	0.829156	−	0.559018i	$-0.188822\pi$
$32$		0			0
$33$	5.24699	+	6.08239i	0.913383	+	1.05881i
$34$		0			0
$35$	0.488584	+	0.766086i	0.0825857	+	0.129492i
$36$		0			0
$37$	−4.35429			−0.715842			−0.357921	−	0.933752i	$-0.616514\pi$
$37$	−4.35429			−0.715842			−0.357921	+	0.933752i	$0.616514\pi$
$38$		0			0
$39$			2.42199i			0.387829i
$40$		0			0
$41$	7.26385			1.13442			0.567212	−	0.823572i	$-0.308022\pi$
$41$	7.26385			1.13442			0.567212	+	0.823572i	$0.308022\pi$
$42$		0			0
$43$		−	1.07514i		−	0.163958i	−0.996634	−	0.0819788i	$-0.973876\pi$
$43$		−	1.07514i		−	0.163958i	0.996634	−	0.0819788i	$-0.0261240\pi$
$44$		0			0
$45$			0.984276i			0.146727i
$46$		0			0
$47$			1.00695i			0.146878i	0.997300	+	0.0734392i	$0.0233975\pi$
$47$			1.00695i			0.146878i	−0.997300	+	0.0734392i	$0.976603\pi$
$48$		0			0
$49$	2.95205	−	6.34708i	0.421721	−	0.906725i
$50$		0			0
$51$		−	18.3054i		−	2.56326i
$52$		0			0
$53$	−1.66130			−0.228197			−0.114099	−	0.993469i	$-0.536398\pi$
$53$	−1.66130			−0.228197			−0.114099	+	0.993469i	$0.536398\pi$
$54$		0			0
$55$	0.744002	+	0.862459i	0.100321	+	0.116294i
$56$		0			0
$57$			11.4182i			1.51238i
$58$		0			0
$59$			0.229128i			0.0298300i	0.999889	+	0.0149150i	$0.00474776\pi$
$59$			0.229128i			0.0298300i	−0.999889	+	0.0149150i	$0.995252\pi$
$60$		0			0
$61$	4.47489			0.572950			0.286475	−	0.958088i	$-0.407516\pi$
$61$	4.47489			0.572950			0.286475	+	0.958088i	$0.407516\pi$
$62$		0			0
$63$	6.39325	−	4.07740i	0.805474	−	0.513704i
$64$		0			0
$65$			0.343428i			0.0425971i
$66$		0			0
$67$	−0.898957			−0.109825			−0.0549125	−	0.998491i	$-0.517488\pi$
$67$	−0.898957			−0.109825			−0.0549125	+	0.998491i	$0.517488\pi$
$68$		0			0
$69$		−	18.1373i		−	2.18347i
$70$		0			0
$71$	−7.01794			−0.832876			−0.416438	−	0.909164i	$-0.636722\pi$
$71$	−7.01794			−0.832876			−0.416438	+	0.909164i	$0.636722\pi$
$72$		0			0
$73$	−5.83754			−0.683232			−0.341616	−	0.939840i	$-0.610974\pi$
$73$	−5.83754			−0.683232			−0.341616	+	0.939840i	$0.610974\pi$
$74$		0			0
$75$		−	11.8243i		−	1.36535i
$76$		0			0
$77$	2.51995	−	8.40535i	0.287174	−	0.957878i
$78$		0			0
$79$		−	5.70523i		−	0.641888i	−0.947098	−	0.320944i	$-0.896000\pi$
$79$		−	5.70523i		−	0.641888i	0.947098	−	0.320944i	$-0.104000\pi$
$80$		0			0
$81$	−9.38396			−1.04266
$82$		0			0
$83$	5.78767			0.635279			0.317640	−	0.948212i	$-0.397110\pi$
$83$	5.78767			0.635279			0.317640	+	0.948212i	$0.397110\pi$
$84$		0			0
$85$		−	2.59563i		−	0.281535i
$86$		0			0
$87$	11.1834			1.19898
$88$		0			0
$89$			0.531306i			0.0563183i	0.999603	+	0.0281592i	$0.00896453\pi$
$89$			0.531306i			0.0563183i	−0.999603	+	0.0281592i	$0.991035\pi$
$90$		0			0
$91$	2.23070	−	1.42266i	0.233841	−	0.149136i
$92$		0			0
$93$	−15.0768			−1.56339
$94$		0			0
$95$			1.61905i			0.166111i
$96$		0			0
$97$		−	9.91329i		−	1.00654i	−0.864129	−	0.503271i	$-0.832130\pi$
$97$		−	9.91329i		−	1.00654i	0.864129	−	0.503271i	$-0.167870\pi$
$98$		0			0
$99$	7.19752	−	6.20896i	0.723377	−	0.624024i
$100$		0			0
$101$	0.218890			0.0217803			0.0108902	−	0.999941i	$-0.496533\pi$
$101$	0.218890			0.0217803			0.0108902	+	0.999941i	$0.496533\pi$
$102$		0			0
$103$		−	18.6234i		−	1.83502i	−0.397716	−	0.917509i	$-0.630197\pi$
$103$		−	18.6234i		−	1.83502i	0.397716	−	0.917509i	$-0.369803\pi$
$104$		0			0
$105$	1.85545	−	1.18334i	0.181074	−	0.115483i
$106$		0			0
$107$		−	10.5871i		−	1.02349i	−0.859136	−	0.511747i	$-0.828999\pi$
$107$		−	10.5871i		−	1.02349i	0.859136	−	0.511747i	$-0.171001\pi$
$108$		0			0
$109$		−	16.4489i		−	1.57552i	−0.615983	−	0.787759i	$-0.711241\pi$
$109$		−	16.4489i		−	1.57552i	0.615983	−	0.787759i	$-0.288759\pi$
$110$		0			0
$111$			10.5461i			1.00099i
$112$		0			0
$113$	−1.70419			−0.160317			−0.0801583	−	0.996782i	$-0.525543\pi$
$113$	−1.70419			−0.160317			−0.0801583	+	0.996782i	$0.525543\pi$
$114$		0			0
$115$		−	2.57179i		−	0.239821i
$116$		0			0
$117$	2.86603			0.264965
$118$		0			0
$119$	−16.8596	+	10.7525i	−1.54552	+	0.985679i
$120$		0			0
$121$	1.61345	−	10.8810i	0.146677	−	0.989184i
$122$		0			0
$123$		−	17.5930i		−	1.58630i
$124$		0			0
$125$		−	3.39378i		−	0.303549i
$126$		0			0
$127$			21.0310i			1.86620i	0.359619	+	0.933099i	$0.382907\pi$
$127$			21.0310i			1.86620i	−0.359619	+	0.933099i	$0.617093\pi$
$128$		0			0
$129$	−2.60398			−0.229268
$130$		0			0
$131$	4.02470			0.351640			0.175820	−	0.984422i	$-0.443742\pi$
$131$	4.02470			0.351640			0.175820	+	0.984422i	$0.443742\pi$
$132$		0			0
$133$	10.5164	−	6.70699i	0.911886	−	0.581570i
$134$		0			0
$135$	−0.111434			−0.00959072
$136$		0			0
$137$	20.0884			1.71626			0.858132	−	0.513429i	$-0.171625\pi$
$137$	20.0884			1.71626			0.858132	+	0.513429i	$0.171625\pi$
$138$		0			0
$139$	−9.76759			−0.828476			−0.414238	−	0.910168i	$-0.635952\pi$
$139$	−9.76759			−0.828476			−0.414238	+	0.910168i	$0.635952\pi$
$140$		0			0
$141$	2.43881			0.205385
$142$		0			0
$143$	2.51132	−	2.16640i	0.210007	−	0.181163i
$144$		0			0
$145$	1.58576			0.131690
$146$		0			0
$147$	−15.3726	−	7.14983i	−1.26791	−	0.589708i
$148$		0			0
$149$			2.10134i			0.172148i	0.996289	+	0.0860742i	$0.0274322\pi$
$149$			2.10134i			0.172148i	−0.996289	+	0.0860742i	$0.972568\pi$
$150$		0			0
$151$		−	8.86590i		−	0.721497i	−0.932663	−	0.360748i	$-0.882521\pi$
$151$		−	8.86590i		−	0.721497i	0.932663	−	0.360748i	$-0.117479\pi$
$152$		0			0
$153$	−21.6614			−1.75122
$154$		0			0
$155$	−2.13783			−0.171715
$156$		0			0
$157$			20.4441i			1.63162i	0.578320	+	0.815810i	$0.303708\pi$
$157$			20.4441i			1.63162i	−0.578320	+	0.815810i	$0.696292\pi$
$158$		0			0
$159$			4.02365i			0.319097i
$160$		0			0
$161$	−16.7048	+	10.6537i	−1.31652	+	0.839632i
$162$		0			0
$163$	11.6279			0.910764			0.455382	−	0.890296i	$-0.349503\pi$
$163$	11.6279			0.910764			0.455382	+	0.890296i	$0.349503\pi$
$164$		0			0
$165$	2.08887	−	1.80197i	0.162618	−	0.140283i
$166$		0			0
$167$	14.7411			1.14070			0.570349	−	0.821402i	$-0.306808\pi$
$167$	14.7411			1.14070			0.570349	+	0.821402i	$0.306808\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$	13.5116			1.03326
$172$		0			0
$173$	5.84359			0.444280			0.222140	−	0.975015i	$-0.428696\pi$
$173$	5.84359			0.444280			0.222140	+	0.975015i	$0.428696\pi$
$174$		0			0
$175$	−10.8904	+	6.94553i	−0.823237	+	0.525033i
$176$		0			0
$177$	0.554946			0.0417123
$178$		0			0
$179$	4.41731			0.330165			0.165083	−	0.986280i	$-0.447211\pi$
$179$	4.41731			0.330165			0.165083	+	0.986280i	$0.447211\pi$
$180$		0			0
$181$		−	16.6575i		−	1.23814i	−0.785335	−	0.619071i	$-0.787509\pi$
$181$		−	16.6575i		−	1.23814i	0.785335	−	0.619071i	$-0.212491\pi$
$182$		0			0
$183$		−	10.8381i		−	0.801177i
$184$		0			0
$185$			1.49539i			0.109943i
$186$		0			0
$187$	−18.9805	+	16.3736i	−1.38799	+	1.19736i
$188$		0			0
$189$	0.461620	+	0.723808i	0.0335779	+	0.0526493i
$190$		0			0
$191$	−18.2220			−1.31849			−0.659247	−	0.751926i	$-0.729125\pi$
$191$	−18.2220			−1.31849			−0.659247	+	0.751926i	$0.729125\pi$
$192$		0			0
$193$			21.5717i			1.55276i	0.630263	+	0.776382i	$0.282947\pi$
$193$			21.5717i			1.55276i	−0.630263	+	0.776382i	$0.717053\pi$
$194$		0			0
$195$	0.831780			0.0595650
$196$		0			0
$197$		−	6.05784i		−	0.431603i	−0.976437	−	0.215801i	$-0.930764\pi$
$197$		−	6.05784i		−	0.431603i	0.976437	−	0.215801i	$-0.0692364\pi$
$198$		0			0
$199$			20.5264i			1.45508i	0.686068	+	0.727538i	$0.259335\pi$
$199$			20.5264i			1.45508i	−0.686068	+	0.727538i	$0.740665\pi$
$200$		0			0
$201$			2.17726i			0.153572i
$202$		0			0
$203$	−6.56905	−	10.3001i	−0.461057	−	0.722925i
$204$		0			0
$205$		−	2.49461i		−	0.174231i
$206$		0			0
$207$	−21.4625			−1.49175
$208$		0			0
$209$	11.8393	−	10.2132i	0.818944	−	0.706464i
$210$		0			0
$211$		−	22.5208i		−	1.55039i	−0.631720	−	0.775196i	$-0.717651\pi$
$211$		−	22.5208i		−	1.55039i	0.631720	−	0.775196i	$-0.282349\pi$
$212$		0			0
$213$			16.9974i			1.16464i
$214$		0			0
$215$	−0.369234			−0.0251816
$216$		0			0
$217$	8.85603	+	13.8860i	0.601187	+	0.942645i
$218$		0			0
$219$			14.1385i			0.955388i
$220$		0			0
$221$	−7.55799			−0.508405
$222$		0			0
$223$		−	18.4923i		−	1.23833i	−0.785260	−	0.619167i	$-0.787471\pi$
$223$		−	18.4923i		−	1.23833i	0.785260	−	0.619167i	$-0.212529\pi$
$224$		0			0
$225$	−13.9921			−0.932808
$226$		0			0
$227$	−5.63410			−0.373949			−0.186974	−	0.982365i	$-0.559868\pi$
$227$	−5.63410			−0.373949			−0.186974	+	0.982365i	$0.559868\pi$
$228$		0			0
$229$		−	3.66078i		−	0.241911i	−0.992658	−	0.120955i	$-0.961404\pi$
$229$		−	3.66078i		−	0.241911i	0.992658	−	0.120955i	$-0.0385958\pi$
$230$		0			0
$231$	−20.3577	−	6.10328i	−1.33944	−	0.401566i
$232$		0			0
$233$		−	12.9808i		−	0.850401i	−0.905099	−	0.425200i	$-0.860204\pi$
$233$		−	12.9808i		−	0.850401i	0.905099	−	0.425200i	$-0.139796\pi$
$234$		0			0
$235$	0.345814			0.0225584
$236$		0			0
$237$	−13.8180			−0.897575
$238$		0			0
$239$		−	5.08061i		−	0.328637i	−0.986407	−	0.164319i	$-0.947458\pi$
$239$		−	5.08061i		−	0.328637i	0.986407	−	0.164319i	$-0.0525425\pi$
$240$		0			0
$241$	30.3016			1.95189			0.975947	−	0.218006i	$-0.0699552\pi$
$241$	30.3016			1.95189			0.975947	+	0.218006i	$0.0699552\pi$
$242$		0			0
$243$			21.7544i			1.39555i
$244$		0			0
$245$	−2.17977	−	1.01382i	−0.139260	−	0.0647704i
$246$		0			0
$247$	4.71439			0.299969
$248$		0			0
$249$		−	14.0177i		−	0.888334i
$250$		0			0
$251$			20.6457i			1.30314i	0.758588	+	0.651571i	$0.225890\pi$
$251$			20.6457i			1.30314i	−0.758588	+	0.651571i	$0.774110\pi$
$252$		0			0
$253$	−18.8062	+	16.2232i	−1.18234	+	1.01995i
$254$		0			0
$255$	−6.28658			−0.393681
$256$		0			0
$257$		−	3.68775i		−	0.230036i	−0.993363	−	0.115018i	$-0.963307\pi$
$257$		−	3.68775i		−	0.230036i	0.993363	−	0.115018i	$-0.0366925\pi$
$258$		0			0
$259$	9.71313	−	6.19470i	0.603544	−	0.384920i
$260$		0			0
$261$		−	13.2337i		−	0.819144i
$262$		0			0
$263$		−	2.75236i		−	0.169718i	−0.996393	−	0.0848588i	$-0.972956\pi$
$263$		−	2.75236i		−	0.169718i	0.996393	−	0.0848588i	$-0.0270439\pi$
$264$		0			0
$265$			0.570538i			0.0350479i
$266$		0			0
$267$	1.28682			0.0787519
$268$		0			0
$269$		−	16.2365i		−	0.989955i	−0.868906	−	0.494978i	$-0.835176\pi$
$269$		−	16.2365i		−	0.989955i	0.868906	−	0.494978i	$-0.164824\pi$
$270$		0			0
$271$	−2.54220			−0.154428			−0.0772140	−	0.997015i	$-0.524602\pi$
$271$	−2.54220			−0.154428			−0.0772140	+	0.997015i	$0.524602\pi$
$272$		0			0
$273$	−3.44568	−	5.40273i	−0.208542	−	0.326988i
$274$		0			0
$275$	−12.2604	+	10.5765i	−0.739330	+	0.637785i
$276$		0			0
$277$		−	3.06209i		−	0.183983i	−0.995760	−	0.0919915i	$-0.970677\pi$
$277$		−	3.06209i		−	0.183983i	0.995760	−	0.0919915i	$-0.0293233\pi$
$278$		0			0
$279$			17.8409i			1.06811i
$280$		0			0
$281$			5.01730i			0.299307i	0.988739	+	0.149653i	$0.0478158\pi$
$281$			5.01730i			0.299307i	−0.988739	+	0.149653i	$0.952184\pi$
$282$		0			0
$283$	19.4260			1.15476			0.577379	−	0.816477i	$-0.304076\pi$
$283$	19.4260			1.15476			0.577379	+	0.816477i	$0.304076\pi$
$284$		0			0
$285$	3.92133			0.232280
$286$		0			0
$287$	−16.2035	+	10.3340i	−0.956461	+	0.609998i
$288$		0			0
$289$	40.1232			2.36019
$290$		0			0
$291$	−24.0099			−1.40748
$292$		0			0
$293$	1.09041			0.0637022			0.0318511	−	0.999493i	$-0.489860\pi$
$293$	1.09041			0.0637022			0.0318511	+	0.999493i	$0.489860\pi$
$294$		0			0
$295$	0.0786892			0.00458146
$296$		0			0
$297$	0.702943	+	0.814862i	0.0407889	+	0.0472831i
$298$		0			0
$299$	−7.48858			−0.433076
$300$		0			0
$301$	1.52957	+	2.39832i	0.0881628	+	0.138237i
$302$		0			0
$303$		−	0.530148i		−	0.0304562i
$304$		0			0
$305$		−	1.53680i		−	0.0879971i
$306$		0			0
$307$	−8.07190			−0.460687			−0.230344	−	0.973109i	$-0.573985\pi$
$307$	−8.07190			−0.460687			−0.230344	+	0.973109i	$0.573985\pi$
$308$		0			0
$309$	−45.1056			−2.56597
$310$		0			0
$311$		−	23.0434i		−	1.30667i	−0.757070	−	0.653334i	$-0.773370\pi$
$311$		−	23.0434i		−	1.30667i	0.757070	−	0.653334i	$-0.226630\pi$
$312$		0			0
$313$		−	27.9614i		−	1.58047i	−0.612801	−	0.790237i	$-0.709957\pi$
$313$		−	27.9614i		−	1.58047i	0.612801	−	0.790237i	$-0.290043\pi$
$314$		0			0
$315$	−1.40030	−	2.19563i	−0.0788977	−	0.123709i
$316$		0			0
$317$	10.8346			0.608531			0.304266	−	0.952587i	$-0.401589\pi$
$317$	10.8346			0.608531			0.304266	+	0.952587i	$0.401589\pi$
$318$		0			0
$319$	−10.0032	−	11.5958i	−0.560070	−	0.649242i
$320$		0			0
$321$	−25.6418			−1.43119
$322$		0			0
$323$	−35.6313			−1.98258
$324$		0			0
$325$	−4.88206			−0.270808
$326$		0			0
$327$	−39.8390			−2.20310
$328$		0			0
$329$	−1.43255	−	2.24620i	−0.0789789	−	0.123837i
$330$		0			0
$331$	26.1485			1.43725			0.718624	−	0.695398i	$-0.244772\pi$
$331$	26.1485			1.43725			0.718624	+	0.695398i	$0.244772\pi$
$332$		0			0
$333$	12.4795			0.683874
$334$		0			0
$335$			0.308727i			0.0168676i
$336$		0			0
$337$			9.01311i			0.490976i	0.969400	+	0.245488i	$0.0789481\pi$
$337$			9.01311i			0.490976i	−0.969400	+	0.245488i	$0.921052\pi$
$338$		0			0
$339$			4.12752i			0.224176i
$340$		0			0
$341$	13.4857	+	15.6329i	0.730293	+	0.846567i
$342$		0			0
$343$	2.44463	+	18.3582i	0.131997	+	0.991250i
$344$		0			0
$345$	−6.22885			−0.335350
$346$		0			0
$347$		−	27.8362i		−	1.49433i	−0.664641	−	0.747163i	$-0.731416\pi$
$347$		−	27.8362i		−	1.49433i	0.664641	−	0.747163i	$-0.268584\pi$
$348$		0			0
$349$	−24.7273			−1.32362			−0.661811	−	0.749670i	$-0.730212\pi$
$349$	−24.7273			−1.32362			−0.661811	+	0.749670i	$0.730212\pi$
$350$		0			0
$351$			0.324476i			0.0173192i
$352$		0			0
$353$			13.9539i			0.742692i	0.928494	+	0.371346i	$0.121104\pi$
$353$			13.9539i			0.742692i	−0.928494	+	0.371346i	$0.878896\pi$
$354$		0			0
$355$			2.41016i			0.127918i
$356$		0			0
$357$	26.0424	+	40.8338i	1.37831	+	2.16115i
$358$		0			0
$359$		−	20.9755i		−	1.10705i	−0.832834	−	0.553523i	$-0.813283\pi$
$359$		−	20.9755i		−	1.10705i	0.832834	−	0.553523i	$-0.186717\pi$
$360$		0			0
$361$	3.22544			0.169760
$362$		0			0
$363$	−26.3537	−	3.90776i	−1.38321	−	0.205104i
$364$		0			0
$365$			2.00478i			0.104935i
$366$		0			0
$367$			29.4178i			1.53560i	0.640691	+	0.767799i	$0.278648\pi$
$367$			29.4178i			1.53560i	−0.640691	+	0.767799i	$0.721352\pi$
$368$		0			0
$369$	−20.8184			−1.08376
$370$		0			0
$371$	3.70587	−	2.36348i	0.192399	−	0.122706i
$372$		0			0
$373$		−	9.51070i		−	0.492446i	−0.969213	−	0.246223i	$-0.920811\pi$
$373$		−	9.51070i		−	0.492446i	0.969213	−	0.246223i	$-0.0791895\pi$
$374$		0			0
$375$	−8.21970			−0.424463
$376$		0			0
$377$		−	4.61743i		−	0.237809i
$378$		0			0
$379$	−12.5280			−0.643520			−0.321760	−	0.946821i	$-0.604274\pi$
$379$	−12.5280			−0.643520			−0.321760	+	0.946821i	$0.604274\pi$
$380$		0			0
$381$	50.9368			2.60957
$382$		0			0
$383$		−	32.2036i		−	1.64553i	−0.568384	−	0.822764i	$-0.692431\pi$
$383$		−	32.2036i		−	1.64553i	0.568384	−	0.822764i	$-0.307569\pi$
$384$		0			0
$385$	−2.88664	−	0.865421i	−0.147117	−	0.0441059i
$386$		0			0
$387$			3.08139i			0.156636i
$388$		0			0
$389$	26.4995			1.34358			0.671788	−	0.740744i	$-0.265527\pi$
$389$	26.4995			1.34358			0.671788	+	0.740744i	$0.265527\pi$
$390$		0			0
$391$	56.5986			2.86231
$392$		0			0
$393$		−	9.74779i		−	0.491711i
$394$		0			0
$395$	−1.95934			−0.0985850
$396$		0			0
$397$			35.3183i			1.77258i	0.463135	+	0.886288i	$0.346725\pi$
$397$			35.3183i			1.77258i	−0.463135	+	0.886288i	$0.653275\pi$
$398$		0			0
$399$	−16.2443	−	25.4706i	−0.813230	−	1.27512i
$400$		0			0
$401$	31.5605			1.57606			0.788028	−	0.615640i	$-0.211102\pi$
$401$	31.5605			1.57606			0.788028	+	0.615640i	$0.211102\pi$
$402$		0			0
$403$			6.22496i			0.310087i
$404$		0			0
$405$			3.22272i			0.160138i
$406$		0			0
$407$	10.9350	−	9.43313i	0.542029	−	0.467583i
$408$		0			0
$409$	24.0644			1.18991			0.594953	−	0.803760i	$-0.297171\pi$
$409$	24.0644			1.18991			0.594953	+	0.803760i	$0.297171\pi$
$410$		0			0
$411$		−	48.6538i		−	2.39991i
$412$		0			0
$413$	−0.325973	−	0.511116i	−0.0160401	−	0.0251504i
$414$		0			0
$415$		−	1.98765i		−	0.0975699i
$416$		0			0
$417$			23.6570i			1.15849i
$418$		0			0
$419$		−	25.5852i		−	1.24992i	−0.780656	−	0.624960i	$-0.785115\pi$
$419$		−	25.5852i		−	1.24992i	0.780656	−	0.624960i	$-0.214885\pi$
$420$		0			0
$421$	6.89759			0.336168			0.168084	−	0.985773i	$-0.446242\pi$
$421$	6.89759			0.336168			0.168084	+	0.985773i	$0.446242\pi$
$422$		0			0
$423$		−	2.88594i		−	0.140319i
$424$		0			0
$425$	36.8985			1.78984
$426$		0			0
$427$	−9.98213	+	6.36626i	−0.483069	+	0.308085i
$428$		0			0
$429$	−5.24699	−	6.08239i	−0.253327	−	0.293660i
$430$		0			0
$431$		−	1.10707i		−	0.0533256i	−0.999644	−	0.0266628i	$-0.991512\pi$
$431$		−	1.10707i		−	0.0533256i	0.999644	−	0.0266628i	$-0.00848804\pi$
$432$		0			0
$433$		−	11.1319i		−	0.534965i	−0.963563	−	0.267483i	$-0.913808\pi$
$433$		−	11.1319i		−	0.534965i	0.963563	−	0.267483i	$-0.0861918\pi$
$434$		0			0
$435$		−	3.84068i		−	0.184147i
$436$		0			0
$437$	−35.3040			−1.68882
$438$		0			0
$439$	18.6763			0.891372			0.445686	−	0.895189i	$-0.352960\pi$
$439$	18.6763			0.891372			0.445686	+	0.895189i	$0.352960\pi$
$440$		0			0
$441$	−8.46066	+	18.1909i	−0.402889	+	0.866234i
$442$		0			0
$443$	2.03481			0.0966770			0.0483385	−	0.998831i	$-0.484607\pi$
$443$	2.03481			0.0966770			0.0483385	+	0.998831i	$0.484607\pi$
$444$		0			0
$445$	0.182466			0.00864970
$446$		0			0
$447$	5.08942			0.240721
$448$		0			0
$449$	−24.9793			−1.17885			−0.589424	−	0.807824i	$-0.700645\pi$
$449$	−24.9793			−1.17885			−0.589424	+	0.807824i	$0.700645\pi$
$450$		0			0
$451$	−18.2419	+	15.7364i	−0.858976	+	0.740998i
$452$		0			0
$453$	−21.4731			−1.00890
$454$		0			0
$455$	−0.488584	−	0.766086i	−0.0229052	−	0.0359147i
$456$		0			0
$457$		−	21.2878i		−	0.995801i	−0.867234	−	0.497901i	$-0.834104\pi$
$457$		−	21.2878i		−	0.995801i	0.867234	−	0.497901i	$-0.165896\pi$
$458$		0			0
$459$		−	2.45238i		−	0.114467i
$460$		0			0
$461$	−6.39311			−0.297757			−0.148878	−	0.988856i	$-0.547566\pi$
$461$	−6.39311			−0.297757			−0.148878	+	0.988856i	$0.547566\pi$
$462$		0			0
$463$	−36.3232			−1.68808			−0.844041	−	0.536279i	$-0.819830\pi$
$463$	−36.3232			−1.68808			−0.844041	+	0.536279i	$0.819830\pi$
$464$		0			0
$465$			5.17780i			0.240115i
$466$		0			0
$467$			13.1273i			0.607458i	0.952758	+	0.303729i	$0.0982317\pi$
$467$			13.1273i			0.607458i	−0.952758	+	0.303729i	$0.901768\pi$
$468$		0			0
$469$	2.00530	−	1.27891i	0.0925963	−	0.0590548i
$470$		0			0
$471$	49.5155			2.28155
$472$		0			0
$473$	2.32918	+	2.70003i	0.107096	+	0.124147i
$474$		0			0
$475$	−23.0159			−1.05604
$476$		0			0
$477$	4.76134			0.218007
$478$		0			0
$479$	−23.7899			−1.08699			−0.543494	−	0.839413i	$-0.682899\pi$
$479$	−23.7899			−1.08699			−0.543494	+	0.839413i	$0.682899\pi$
$480$		0			0
$481$	4.35429			0.198539
$482$		0			0
$483$	25.8032	+	40.4588i	1.17409	+	1.84094i
$484$		0			0
$485$	−3.40451			−0.154591
$486$		0			0
$487$	2.84765			0.129039			0.0645196	−	0.997916i	$-0.479449\pi$
$487$	2.84765			0.129039			0.0645196	+	0.997916i	$0.479449\pi$
$488$		0			0
$489$		−	28.1625i		−	1.27355i
$490$		0			0
$491$			32.7878i			1.47969i	0.672775	+	0.739847i	$0.265102\pi$
$491$			32.7878i			1.47969i	−0.672775	+	0.739847i	$0.734898\pi$
$492$		0			0
$493$			34.8984i			1.57175i
$494$		0			0
$495$	−2.13233	−	2.47183i	−0.0958412	−	0.111101i
$496$		0			0
$497$	15.6549	−	9.98418i	0.702219	−	0.447852i
$498$		0			0
$499$	−19.3327			−0.865449			−0.432725	−	0.901526i	$-0.642448\pi$
$499$	−19.3327			−0.865449			−0.432725	+	0.901526i	$0.642448\pi$
$500$		0			0
$501$		−	35.7027i		−	1.59508i
$502$		0			0
$503$	−11.9588			−0.533215			−0.266607	−	0.963805i	$-0.585903\pi$
$503$	−11.9588			−0.533215			−0.266607	+	0.963805i	$0.585903\pi$
$504$		0			0
$505$		−	0.0751730i		−	0.00334515i
$506$		0			0
$507$		−	2.42199i		−	0.107564i
$508$		0			0
$509$		−	15.3956i		−	0.682400i	−0.939991	−	0.341200i	$-0.889167\pi$
$509$		−	15.3956i		−	0.682400i	0.939991	−	0.341200i	$-0.110833\pi$
$510$		0			0
$511$	13.0218	−	8.30486i	0.576051	−	0.367385i
$512$		0			0
$513$			1.52970i			0.0675380i
$514$		0			0
$515$	−6.39580			−0.281833
$516$		0			0
$517$	−2.18145	−	2.52877i	−0.0959399	−	0.111215i
$518$		0			0
$519$		−	14.1531i		−	0.621252i
$520$		0			0
$521$		−	32.9052i		−	1.44160i	−0.693141	−	0.720802i	$-0.743774\pi$
$521$		−	32.9052i		−	1.44160i	0.693141	−	0.720802i	$-0.256226\pi$
$522$		0			0
$523$	−35.5858			−1.55606			−0.778028	−	0.628229i	$-0.783780\pi$
$523$	−35.5858			−1.55606			−0.778028	+	0.628229i	$0.783780\pi$
$524$		0			0
$525$	16.8220	+	26.3764i	0.734172	+	1.15116i
$526$		0			0
$527$		−	47.0482i		−	2.04945i
$528$		0			0
$529$	33.0788			1.43821
$530$		0			0
$531$		−	0.656688i		−	0.0284978i
$532$		0			0
$533$	−7.26385			−0.314632
$534$		0			0
$535$	−3.63591			−0.157194
$536$		0			0
$537$		−	10.6987i		−	0.461682i
$538$		0			0
$539$	6.33675	+	22.3349i	0.272943	+	0.962030i
$540$		0			0
$541$			5.47142i			0.235235i	0.993059	+	0.117617i	$0.0375256\pi$
$541$			5.47142i			0.235235i	−0.993059	+	0.117617i	$0.962474\pi$
$542$		0			0
$543$	−40.3442			−1.73134
$544$		0			0
$545$	−5.64902			−0.241977
$546$		0			0
$547$		−	3.61236i		−	0.154453i	−0.997014	−	0.0772267i	$-0.975393\pi$
$547$		−	3.61236i		−	0.154453i	0.997014	−	0.0772267i	$-0.0246065\pi$
$548$		0			0
$549$	−12.8252			−0.547364
$550$		0			0
$551$		−	21.7683i		−	0.927362i
$552$		0			0
$553$	8.11662	+	12.7267i	0.345154	+	0.541192i
$554$		0			0
$555$	3.62181			0.153737
$556$		0			0
$557$			24.4094i			1.03426i	0.855907	+	0.517131i	$0.173000\pi$
$557$			24.4094i			1.03426i	−0.855907	+	0.517131i	$0.827000\pi$
$558$		0			0
$559$			1.07514i			0.0454737i
$560$		0			0
$561$	39.6567	+	45.9706i	1.67431	+	1.94088i
$562$		0			0
$563$	22.5282			0.949449			0.474724	−	0.880135i	$-0.342548\pi$
$563$	22.5282			0.949449			0.474724	+	0.880135i	$0.342548\pi$
$564$		0			0
$565$			0.585267i			0.0246224i
$566$		0			0
$567$	20.9328	−	13.3502i	0.879096	−	0.560657i
$568$		0			0
$569$			34.4160i			1.44280i	0.692521	+	0.721398i	$0.256500\pi$
$569$			34.4160i			1.44280i	−0.692521	+	0.721398i	$0.743500\pi$
$570$		0			0
$571$			31.7939i			1.33053i	0.746605	+	0.665267i	$0.231682\pi$
$571$			31.7939i			1.33053i	−0.746605	+	0.665267i	$0.768318\pi$
$572$		0			0
$573$			44.1334i			1.84370i
$574$		0			0
$575$	36.5597			1.52464
$576$		0			0
$577$		−	16.8941i		−	0.703313i	−0.936129	−	0.351656i	$-0.885619\pi$
$577$		−	16.8941i		−	0.703313i	0.936129	−	0.351656i	$-0.114381\pi$
$578$		0			0
$579$	52.2464			2.17129
$580$		0			0
$581$	−12.9106	+	8.23391i	−0.535620	+	0.341600i
$582$		0			0
$583$	4.17206	−	3.59904i	0.172789	−	0.149057i
$584$		0			0
$585$		−	0.984276i		−	0.0406948i
$586$		0			0
$587$			11.2083i			0.462616i	0.972881	+	0.231308i	$0.0743004\pi$
$587$			11.2083i			0.462616i	−0.972881	+	0.231308i	$0.925700\pi$
$588$		0			0
$589$			29.3469i			1.20922i
$590$		0			0
$591$	−14.6720			−0.603526
$592$		0			0
$593$	8.65684			0.355494			0.177747	−	0.984076i	$-0.443119\pi$
$593$	8.65684			0.355494			0.177747	+	0.984076i	$0.443119\pi$
$594$		0			0
$595$	3.69271	+	5.79007i	0.151386	+	0.237370i
$596$		0			0
$597$	49.7146			2.03468
$598$		0			0
$599$	48.1211			1.96617			0.983087	−	0.183138i	$-0.0586256\pi$
$599$	48.1211			1.96617			0.983087	+	0.183138i	$0.0586256\pi$
$600$		0			0
$601$	−19.7690			−0.806395			−0.403198	−	0.915113i	$-0.632101\pi$
$601$	−19.7690			−0.806395			−0.403198	+	0.915113i	$0.632101\pi$
$602$		0			0
$603$	2.57644			0.104921
$604$		0			0
$605$	−3.73686	−	0.554105i	−0.151925	−	0.0225276i
$606$		0			0
$607$	−3.90372			−0.158447			−0.0792236	−	0.996857i	$-0.525244\pi$
$607$	−3.90372			−0.158447			−0.0792236	+	0.996857i	$0.525244\pi$
$608$		0			0
$609$	−24.9467	+	15.9102i	−1.01089	+	0.644712i
$610$		0			0
$611$		−	1.00695i		−	0.0407367i
$612$		0			0
$613$			24.0286i			0.970508i	0.874373	+	0.485254i	$0.161273\pi$
$613$			24.0286i			0.970508i	−0.874373	+	0.485254i	$0.838727\pi$
$614$		0			0
$615$	−6.04193			−0.243634
$616$		0			0
$617$	7.22805			0.290990			0.145495	−	0.989359i	$-0.453523\pi$
$617$	7.22805			0.290990			0.145495	+	0.989359i	$0.453523\pi$
$618$		0			0
$619$		−	38.6447i		−	1.55326i	−0.629955	−	0.776631i	$-0.716927\pi$
$619$		−	38.6447i		−	1.55326i	0.629955	−	0.776631i	$-0.283073\pi$
$620$		0			0
$621$		−	2.42986i		−	0.0975069i
$622$		0			0
$623$	−0.755870	−	1.18518i	−0.0302833	−	0.0474834i
$624$		0			0
$625$	23.2448			0.929791
$626$		0			0
$627$	−24.7363	−	28.6747i	−0.987874	−	1.14516i
$628$		0			0
$629$	−32.9097			−1.31220
$630$		0			0
$631$	37.4737			1.49181			0.745903	−	0.666055i	$-0.232018\pi$
$631$	37.4737			1.49181			0.745903	+	0.666055i	$0.232018\pi$
$632$		0			0
$633$	−54.5450			−2.16797
$634$		0			0
$635$	7.22264			0.286622
$636$		0			0
$637$	−2.95205	+	6.34708i	−0.116964	+	0.251480i
$638$		0			0
$639$	20.1136			0.795683
$640$		0			0
$641$	24.2532			0.957945			0.478972	−	0.877830i	$-0.341009\pi$
$641$	24.2532			0.957945			0.478972	+	0.877830i	$0.341009\pi$
$642$		0			0
$643$			4.98567i			0.196616i	0.995156	+	0.0983078i	$0.0313430\pi$
$643$			4.98567i			0.196616i	−0.995156	+	0.0983078i	$0.968657\pi$
$644$		0			0
$645$			0.894282i			0.0352123i
$646$		0			0
$647$		−	26.4281i		−	1.03900i	−0.854471	−	0.519499i	$-0.826119\pi$
$647$		−	26.4281i		−	1.03900i	0.854471	−	0.519499i	$-0.173881\pi$
$648$		0			0
$649$	−0.496383	−	0.575414i	−0.0194847	−	0.0225870i
$650$		0			0
$651$	33.6318	−	21.4492i	1.31813	−	0.840661i
$652$		0			0
$653$	8.60717			0.336825			0.168412	−	0.985717i	$-0.446136\pi$
$653$	8.60717			0.336825			0.168412	+	0.985717i	$0.446136\pi$
$654$		0			0
$655$		−	1.38220i		−	0.0540070i
$656$		0			0
$657$	16.7306			0.652721
$658$		0			0
$659$		−	4.18494i		−	0.163022i	−0.996672	−	0.0815110i	$-0.974025\pi$
$659$		−	4.18494i		−	0.163022i	0.996672	−	0.0815110i	$-0.0259746\pi$
$660$		0			0
$661$		−	14.3851i		−	0.559514i	−0.960071	−	0.279757i	$-0.909746\pi$
$661$		−	14.3851i		−	0.559514i	0.960071	−	0.279757i	$-0.0902540\pi$
$662$		0			0
$663$			18.3054i			0.710921i
$664$		0			0
$665$	−2.30337	−	3.61163i	−0.0893209	−	0.140053i
$666$		0			0
$667$			34.5779i			1.33886i
$668$		0			0
$669$	−44.7880			−1.73161
$670$		0			0
$671$	−11.2379	+	9.69438i	−0.433833	+	0.374247i
$672$		0			0
$673$		−	19.0433i		−	0.734067i	−0.930208	−	0.367033i	$-0.880374\pi$
$673$		−	19.0433i		−	0.734067i	0.930208	−	0.367033i	$-0.119626\pi$
$674$		0			0
$675$		−	1.58411i		−	0.0609723i
$676$		0			0
$677$	−26.8185			−1.03072			−0.515359	−	0.856975i	$-0.672341\pi$
$677$	−26.8185			−1.03072			−0.515359	+	0.856975i	$0.672341\pi$
$678$		0			0
$679$	14.1033	+	22.1136i	0.541235	+	0.848641i
$680$		0			0
$681$			13.6457i			0.522906i
$682$		0			0
$683$	41.9514			1.60523			0.802613	−	0.596501i	$-0.203443\pi$
$683$	41.9514			1.60523			0.802613	+	0.596501i	$0.203443\pi$
$684$		0			0
$685$		−	6.89891i		−	0.263594i
$686$		0			0
$687$	−8.86636			−0.338273
$688$		0			0
$689$	1.66130			0.0632906
$690$		0			0
$691$		−	22.0885i		−	0.840288i	−0.907458	−	0.420144i	$-0.861980\pi$
$691$		−	22.0885i		−	0.840288i	0.907458	−	0.420144i	$-0.138020\pi$
$692$		0			0
$693$	−7.22224	+	24.0900i	−0.274350	+	0.915102i
$694$		0			0
$695$			3.35447i			0.127242i
$696$		0			0
$697$	54.9001			2.07949
$698$		0			0
$699$	−31.4394			−1.18915
$700$		0			0
$701$			32.4562i			1.22586i	0.790139	+	0.612928i	$0.210008\pi$
$701$			32.4562i			1.22586i	−0.790139	+	0.612928i	$0.789992\pi$
$702$		0			0
$703$	20.5278			0.774222
$704$		0			0
$705$		−	0.837558i		−	0.0315443i
$706$		0			0
$707$	−0.488277	+	0.311407i	−0.0183636	+	0.0117117i
$708$		0			0
$709$	−35.4149			−1.33003			−0.665017	−	0.746828i	$-0.731576\pi$
$709$	−35.4149			−1.33003			−0.665017	+	0.746828i	$0.731576\pi$
$710$		0			0
$711$			16.3513i			0.613223i
$712$		0			0
$713$		−	46.6161i		−	1.74579i
$714$		0			0
$715$	−0.744002	−	0.862459i	−0.0278241	−	0.0322541i
$716$		0			0
$717$	−12.3052			−0.459545
$718$		0			0
$719$			50.4286i			1.88067i	0.340252	+	0.940334i	$0.389488\pi$
$719$			50.4286i			1.88067i	−0.340252	+	0.940334i	$0.610512\pi$
$720$		0			0
$721$	26.4948	+	41.5432i	0.986720	+	1.54715i
$722$		0			0
$723$		−	73.3900i		−	2.72940i
$724$		0			0
$725$			22.5425i			0.837209i
$726$		0			0
$727$		−	18.5001i		−	0.686131i	−0.939311	−	0.343066i	$-0.888535\pi$
$727$		−	18.5001i		−	0.686131i	0.939311	−	0.343066i	$-0.111465\pi$
$728$		0			0
$729$	24.5371			0.908781
$730$		0			0
$731$		−	8.12591i		−	0.300548i
$732$		0			0
$733$	10.0690			0.371905			0.185953	−	0.982559i	$-0.440463\pi$
$733$	10.0690			0.371905			0.185953	+	0.982559i	$0.440463\pi$
$734$		0			0
$735$	−2.45546	+	5.27937i	−0.0905708	+	0.194733i
$736$		0			0
$737$	2.25757	−	1.94750i	0.0831586	−	0.0717370i
$738$		0			0
$739$			24.6489i			0.906724i	0.891326	+	0.453362i	$0.149776\pi$
$739$			24.6489i			0.906724i	−0.891326	+	0.453362i	$0.850224\pi$
$740$		0			0
$741$		−	11.4182i		−	0.419458i
$742$		0			0
$743$			43.2761i			1.58765i	0.608148	+	0.793824i	$0.291913\pi$
$743$			43.2761i			1.58765i	−0.608148	+	0.793824i	$0.708087\pi$
$744$		0			0
$745$	0.721659			0.0264395
$746$		0			0
$747$	−16.5876			−0.606910
$748$		0			0
$749$	15.0619	+	23.6166i	0.550349	+	0.862933i
$750$		0			0
$751$	2.89910			0.105790			0.0528949	−	0.998600i	$-0.483155\pi$
$751$	2.89910			0.105790			0.0528949	+	0.998600i	$0.483155\pi$
$752$		0			0
$753$	50.0035			1.82223
$754$		0			0
$755$	−3.04480			−0.110812
$756$		0			0
$757$	12.6242			0.458834			0.229417	−	0.973328i	$-0.426318\pi$
$757$	12.6242			0.458834			0.229417	+	0.973328i	$0.426318\pi$
$758$		0			0
$759$	39.2925	+	45.5484i	1.42623	+	1.65330i
$760$		0			0
$761$	21.1550			0.766869			0.383434	−	0.923568i	$-0.374741\pi$
$761$	21.1550			0.766869			0.383434	+	0.923568i	$0.374741\pi$
$762$		0			0
$763$	23.4013	+	36.6926i	0.847183	+	1.32836i
$764$		0			0
$765$			7.43915i			0.268963i
$766$		0			0
$767$		−	0.229128i		−	0.00827334i
$768$		0			0
$769$	−38.2998			−1.38113			−0.690563	−	0.723272i	$-0.742637\pi$
$769$	−38.2998			−1.38113			−0.690563	+	0.723272i	$0.742637\pi$
$770$		0			0
$771$	−8.93170			−0.321667
$772$		0			0
$773$		−	47.0424i		−	1.69200i	−0.533184	−	0.845999i	$-0.679005\pi$
$773$		−	47.0424i		−	1.69200i	0.533184	−	0.845999i	$-0.320995\pi$
$774$		0			0
$775$		−	30.3906i		−	1.09166i
$776$		0			0
$777$	−15.0035	−	23.5251i	−0.538247	−	0.843958i
$778$		0			0
$779$	−34.2446			−1.22694
$780$		0			0
$781$	17.6243	−	15.2036i	0.630647	−	0.544029i
$782$		0			0
$783$	1.49824			0.0535428
$784$		0			0
$785$	7.02110			0.250594
$786$		0			0
$787$	42.3117			1.50825			0.754124	−	0.656732i	$-0.228062\pi$
$787$	42.3117			1.50825			0.754124	+	0.656732i	$0.228062\pi$
$788$		0			0
$789$	−6.66618			−0.237322
$790$		0			0
$791$	3.80153	−	2.42449i	0.135167	−	0.0862049i
$792$		0			0
$793$	−4.47489			−0.158908
$794$		0			0
$795$	1.38184			0.0490087
$796$		0			0
$797$			9.95457i			0.352609i	0.984336	+	0.176304i	$0.0564143\pi$
$797$			9.95457i			0.352609i	−0.984336	+	0.176304i	$0.943586\pi$
$798$		0			0
$799$			7.61050i			0.269240i
$800$		0			0
$801$		−	1.52274i		−	0.0538033i
$802$		0			0
$803$	14.6599	−	12.6464i	0.517338	−	0.446283i
$804$		0			0
$805$	3.65880	+	5.73690i	0.128956	+	0.202199i
$806$		0			0
$807$	−39.3246			−1.38429
$808$		0			0
$809$		−	19.2695i		−	0.677479i	−0.940880	−	0.338739i	$-0.889999\pi$
$809$		−	19.2695i		−	0.677479i	0.940880	−	0.338739i	$-0.110001\pi$
$810$		0			0
$811$	3.88066			0.136268			0.0681342	−	0.997676i	$-0.478295\pi$
$811$	3.88066			0.136268			0.0681342	+	0.997676i	$0.478295\pi$
$812$		0			0
$813$			6.15719i			0.215942i
$814$		0			0
$815$		−	3.99334i		−	0.139880i
$816$		0			0
$817$			5.06864i			0.177329i
$818$		0			0
$819$	−6.39325	+	4.07740i	−0.223398	+	0.142476i
$820$		0			0
$821$			20.5695i			0.717881i	0.933360	+	0.358940i	$0.116862\pi$
$821$			20.5695i			0.717881i	−0.933360	+	0.358940i	$0.883138\pi$
$822$		0			0
$823$	−22.4616			−0.782964			−0.391482	−	0.920186i	$-0.628037\pi$
$823$	−22.4616			−0.782964			−0.391482	+	0.920186i	$0.628037\pi$
$824$		0			0
$825$	25.6161	+	29.6946i	0.891838	+	1.03383i
$826$		0			0
$827$			17.1221i			0.595394i	0.954660	+	0.297697i	$0.0962185\pi$
$827$			17.1221i			0.595394i	−0.954660	+	0.297697i	$0.903782\pi$
$828$		0			0
$829$			8.30177i			0.288332i	0.989553	+	0.144166i	$0.0460500\pi$
$829$			8.30177i			0.288332i	−0.989553	+	0.144166i	$0.953950\pi$
$830$		0			0
$831$	−7.41634			−0.257270
$832$		0			0
$833$	22.3116	−	47.9711i	0.773050	−	1.66210i
$834$		0			0
$835$		−	5.06250i		−	0.175195i
$836$		0			0
$837$	−2.01985			−0.0698161
$838$		0			0
$839$			11.9855i			0.413786i	0.978364	+	0.206893i	$0.0663351\pi$
$839$			11.9855i			0.413786i	−0.978364	+	0.206893i	$0.933665\pi$
$840$		0			0
$841$	7.67939			0.264806
$842$		0			0
$843$	12.1518			0.418532
$844$		0			0
$845$		−	0.343428i		−	0.0118143i
$846$		0			0
$847$	11.8809	+	26.5677i	0.408234	+	0.912877i
$848$		0			0
$849$		−	47.0496i		−	1.61474i
$850$		0			0
$851$	−32.6075			−1.11777
$852$		0			0
$853$	−6.99359			−0.239456			−0.119728	−	0.992807i	$-0.538202\pi$
$853$	−6.99359			−0.239456			−0.119728	+	0.992807i	$0.538202\pi$
$854$		0			0
$855$		−	4.64026i		−	0.158693i
$856$		0			0
$857$	9.19618			0.314135			0.157068	−	0.987588i	$-0.449796\pi$
$857$	9.19618			0.314135			0.157068	+	0.987588i	$0.449796\pi$
$858$		0			0
$859$			27.0828i			0.924053i	0.886866	+	0.462026i	$0.152877\pi$
$859$			27.0828i			0.924053i	−0.886866	+	0.462026i	$0.847123\pi$
$860$		0			0
$861$	25.0289	+	39.2447i	0.852983	+	1.33745i
$862$		0			0
$863$	−13.9378			−0.474448			−0.237224	−	0.971455i	$-0.576237\pi$
$863$	−13.9378			−0.474448			−0.237224	+	0.971455i	$0.576237\pi$
$864$		0			0
$865$		−	2.00685i		−	0.0682351i
$866$		0			0
$867$		−	97.1779i		−	3.30034i
$868$		0			0
$869$	12.3598	+	14.3276i	0.419277	+	0.486032i
$870$		0			0
$871$	0.898957			0.0304600
$872$		0			0
$873$			28.4118i			0.961593i
$874$		0			0
$875$	4.82821	+	7.57051i	0.163223	+	0.255930i
$876$		0			0
$877$			7.96469i			0.268949i	0.990917	+	0.134474i	$0.0429346\pi$
$877$			7.96469i			0.268949i	−0.990917	+	0.134474i	$0.957065\pi$
$878$		0			0
$879$		−	2.64095i		−	0.0890771i
$880$		0			0
$881$		−	6.17575i		−	0.208066i	−0.994574	−	0.104033i	$-0.966825\pi$
$881$		−	6.17575i		−	0.208066i	0.994574	−	0.104033i	$-0.0331748\pi$
$882$		0			0
$883$	48.4286			1.62975			0.814876	−	0.579635i	$-0.196805\pi$
$883$	48.4286			1.62975			0.814876	+	0.579635i	$0.196805\pi$
$884$		0			0
$885$		−	0.190584i		−	0.00640642i
$886$		0			0
$887$	−7.14760			−0.239993			−0.119997	−	0.992774i	$-0.538288\pi$
$887$	−7.14760			−0.239993			−0.119997	+	0.992774i	$0.538288\pi$
$888$		0			0
$889$	−29.9200	−	46.9138i	−1.00349	−	1.57344i
$890$		0			0
$891$	23.5661	−	20.3294i	0.789495	−	0.681060i
$892$		0			0
$893$		−	4.74714i		−	0.158857i
$894$		0			0
$895$		−	1.51703i		−	0.0507087i
$896$		0			0
$897$			18.1373i			0.605585i
$898$		0			0
$899$	28.7433			0.958642
$900$		0			0
$901$	−12.5561			−0.418304
$902$		0			0
$903$	5.80870	−	3.70459i	0.193302	−	0.123281i
$904$		0			0
$905$	−5.72066			−0.190161
$906$		0			0
$907$	22.5984			0.750367			0.375184	−	0.926950i	$-0.377580\pi$
$907$	22.5984			0.750367			0.375184	+	0.926950i	$0.377580\pi$
$908$		0			0
$909$	−0.627344			−0.0208077
$910$		0			0
$911$	24.6423			0.816435			0.408217	−	0.912885i	$-0.366151\pi$
$911$	24.6423			0.816435			0.408217	+	0.912885i	$0.366151\pi$
$912$		0			0
$913$	−14.5347	+	12.5384i	−0.481028	+	0.414960i
$914$		0			0
$915$	−3.72212			−0.123049
$916$		0			0
$917$	−8.97791	+	5.72581i	−0.296477	+	0.189083i
$918$		0			0
$919$			52.9214i			1.74571i	0.487975	+	0.872857i	$0.337736\pi$
$919$			52.9214i			1.74571i	−0.487975	+	0.872857i	$0.662264\pi$
$920$		0			0
$921$			19.5500i			0.644196i
$922$		0			0
$923$	7.01794			0.230998
$924$		0			0
$925$	−21.2579			−0.698956
$926$		0			0
$927$			53.3752i			1.75307i
$928$		0			0
$929$		−	7.08326i		−	0.232394i	−0.993226	−	0.116197i	$-0.962930\pi$
$929$		−	7.08326i		−	0.232394i	0.993226	−	0.116197i	$-0.0370705\pi$
$930$		0			0
$931$	−13.9171	+	29.9226i	−0.456115	+	0.980673i
$932$		0			0
$933$	−55.8107			−1.82716
$934$		0			0
$935$	5.62316	+	6.51845i	0.183897	+	0.213176i
$936$		0			0
$937$	−27.2817			−0.891253			−0.445627	−	0.895219i	$-0.647019\pi$
$937$	−27.2817			−0.891253			−0.445627	+	0.895219i	$0.647019\pi$
$938$		0			0
$939$	−67.7223			−2.21003
$940$		0			0
$941$	−37.4022			−1.21928			−0.609638	−	0.792680i	$-0.708685\pi$
$941$	−37.4022			−1.21928			−0.609638	+	0.792680i	$0.708685\pi$
$942$		0			0
$943$	54.3959			1.77138
$944$		0			0
$945$	0.248576	−	0.158533i	0.00808618	−	0.00515709i
$946$		0			0
$947$	−54.0907			−1.75771			−0.878856	−	0.477087i	$-0.841693\pi$
$947$	−54.0907			−1.75771			−0.878856	+	0.477087i	$0.841693\pi$
$948$		0			0
$949$	5.83754			0.189495
$950$		0			0
$951$		−	26.2413i		−	0.850931i
$952$		0			0
$953$		−	20.1703i		−	0.653381i	−0.945131	−	0.326691i	$-0.894067\pi$
$953$		−	20.1703i		−	0.653381i	0.945131	−	0.326691i	$-0.105933\pi$
$954$		0			0
$955$			6.25794i			0.202502i
$956$		0			0
$957$	−28.0850	+	24.2276i	−0.907858	+	0.783166i
$958$		0			0
$959$	−44.8111	+	28.5790i	−1.44703	+	0.922864i
$960$		0			0
$961$	−7.75014			−0.250004
$962$		0			0
$963$			30.3429i			0.977787i
$964$		0			0
$965$	7.40833			0.238483
$966$		0			0
$967$			0.0812469i			0.00261272i	0.999999	+	0.00130636i	$0.000415828\pi$
$967$			0.0812469i			0.00261272i	−0.999999	+	0.00130636i	$0.999584\pi$
$968$		0			0
$969$			86.2986i			2.77231i
$970$		0			0
$971$		−	40.5094i		−	1.30001i	−0.759930	−	0.650005i	$-0.774767\pi$
$971$		−	40.5094i		−	1.30001i	0.759930	−	0.650005i	$-0.225233\pi$
$972$		0			0
$973$	21.7886	−	13.8960i	0.698510	−	0.445486i
$974$		0			0
$975$			11.8243i			0.378680i
$976$		0			0
$977$	−15.9705			−0.510942			−0.255471	−	0.966817i	$-0.582231\pi$
$977$	−15.9705			−0.510942			−0.255471	+	0.966817i	$0.582231\pi$
$978$		0			0
$979$	−1.15102	−	1.33428i	−0.0367868	−	0.0426438i
$980$		0			0
$981$			47.1430i			1.50516i
$982$		0			0
$983$		−	1.68108i		−	0.0536180i	−0.999641	−	0.0268090i	$-0.991465\pi$
$983$		−	1.68108i		−	0.0536180i	0.999641	−	0.0268090i	$-0.00853460\pi$
$984$		0			0
$985$	−2.08043			−0.0662881
$986$		0			0
$987$	−5.44026	+	3.46962i	−0.173166	+	0.110439i
$988$		0			0
$989$		−	8.05129i		−	0.256016i
$990$		0			0
$991$	2.49875			0.0793753			0.0396877	−	0.999212i	$-0.487364\pi$
$991$	2.49875			0.0793753			0.0396877	+	0.999212i	$0.487364\pi$
$992$		0			0
$993$		−	63.3313i		−	2.00976i
$994$		0			0
$995$	7.04934			0.223479
$996$		0			0
$997$	−33.1781			−1.05076			−0.525380	−	0.850868i	$-0.676077\pi$
$997$	−33.1781			−1.05076			−0.525380	+	0.850868i	$0.676077\pi$
$998$		0			0
$999$			1.41286i			0.0447010i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.e.b.3849.7	yes	48
7.6	odd	2		4004.2.e.a.3849.42	yes	48
11.10	odd	2		4004.2.e.a.3849.7	✓	48
77.76	even	2	inner	4004.2.e.b.3849.42	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.e.a.3849.7	✓	48	11.10	odd	2
4004.2.e.a.3849.42	yes	48	7.6	odd	2
4004.2.e.b.3849.7	yes	48	1.1	even	1	trivial
4004.2.e.b.3849.42	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)

\(f(q)\)	\(=\)	\(q-2.42199i q^{3} -0.343428i q^{5} +(-2.23070 + 1.42266i) q^{7} -2.86603 q^{9} +O(q^{10})\)	\(q-2.42199i q^{3} -0.343428i q^{5} +(-2.23070 + 1.42266i) q^{7} -2.86603 q^{9} +(-2.51132 + 2.16640i) q^{11} -1.00000 q^{13} -0.831780 q^{15} +7.55799 q^{17} -4.71439 q^{19} +(3.44568 + 5.40273i) q^{21} +7.48858 q^{23} +4.88206 q^{25} -0.324476i q^{27} +4.61743i q^{29} -6.22496i q^{31} +(5.24699 + 6.08239i) q^{33} +(0.488584 + 0.766086i) q^{35} -4.35429 q^{37} +2.42199i q^{39} +7.26385 q^{41} -1.07514i q^{43} +0.984276i q^{45} +1.00695i q^{47} +(2.95205 - 6.34708i) q^{49} -18.3054i q^{51} -1.66130 q^{53} +(0.744002 + 0.862459i) q^{55} +11.4182i q^{57} +0.229128i q^{59} +4.47489 q^{61} +(6.39325 - 4.07740i) q^{63} +0.343428i q^{65} -0.898957 q^{67} -18.1373i q^{69} -7.01794 q^{71} -5.83754 q^{73} -11.8243i q^{75} +(2.51995 - 8.40535i) q^{77} -5.70523i q^{79} -9.38396 q^{81} +5.78767 q^{83} -2.59563i q^{85} +11.1834 q^{87} +0.531306i q^{89} +(2.23070 - 1.42266i) q^{91} -15.0768 q^{93} +1.61905i q^{95} -9.91329i q^{97} +(7.19752 - 6.20896i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 4 q^{7} - 48 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more