LMFDB - Embedded newform 4140.2.s.a.737.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(737,4140)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 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("4140.737");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.s (of order $4$, degree $2$, 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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$44$
Relative dimension:	$22$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			737.5
Character	$\chi$	$=$	4140.737
Dual form			4140.2.s.a.2393.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.99005 - 1.01966i) q^{5} +(-1.87601 - 1.87601i) q^{7} +O(q^{10})$	$q+(-1.99005 - 1.01966i) q^{5} +(-1.87601 - 1.87601i) q^{7} -1.74047i q^{11} +(3.99776 - 3.99776i) q^{13} +(-0.582171 + 0.582171i) q^{17} -7.02619i q^{19} +(0.707107 + 0.707107i) q^{23} +(2.92060 + 4.05834i) q^{25} -4.66204 q^{29} +2.39492 q^{31} +(1.82046 + 5.64623i) q^{35} +(-7.42183 - 7.42183i) q^{37} -5.25361i q^{41} +(-4.98971 + 4.98971i) q^{43} +(7.04459 - 7.04459i) q^{47} +0.0387972i q^{49} +(7.72818 + 7.72818i) q^{53} +(-1.77468 + 3.46363i) q^{55} +7.38558 q^{59} +0.462830 q^{61} +(-12.0321 + 3.87940i) q^{65} +(5.30026 + 5.30026i) q^{67} +10.2010i q^{71} +(10.1201 - 10.1201i) q^{73} +(-3.26514 + 3.26514i) q^{77} -4.66304i q^{79} +(1.85708 + 1.85708i) q^{83} +(1.75216 - 0.564935i) q^{85} -11.5361 q^{89} -14.9996 q^{91} +(-7.16430 + 13.9825i) q^{95} +(-10.6939 - 10.6939i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 12 q^{7}+O(q^{10}) $ 44 * q - 12 * q^7	$ 44 q - 12 q^{7} - 4 q^{13} + 24 q^{25} - 48 q^{37} + 8 q^{43} + 40 q^{55} - 96 q^{61} - 44 q^{67} + 76 q^{73} + 72 q^{85} - 48 q^{91} - 20 q^{97}+O(q^{100}) $ 44 * q - 12 * q^7 - 4 * q^13 + 24 * q^25 - 48 * q^37 + 8 * q^43 + 40 * q^55 - 96 * q^61 - 44 * q^67 + 76 * q^73 + 72 * q^85 - 48 * q^91 - 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.99005	−	1.01966i	−0.889978	−	0.456004i
$5$	−1.99005	−	1.01966i	−0.889978	−	0.456004i
$6$		0			0
$7$	−1.87601	−	1.87601i	−0.709064	−	0.709064i	0.257275	−	0.966338i	$-0.417175\pi$
$7$	−1.87601	−	1.87601i	−0.709064	−	0.709064i	−0.966338	+	0.257275i	$0.917175\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.74047i		−	0.524772i	−0.964963	−	0.262386i	$-0.915491\pi$
$11$		−	1.74047i		−	0.524772i	0.964963	−	0.262386i	$-0.0845094\pi$
$12$		0			0
$13$	3.99776	−	3.99776i	1.10878	−	1.10878i	0.115467	−	0.993311i	$-0.463163\pi$
$13$	3.99776	−	3.99776i	1.10878	−	1.10878i	0.993311	−	0.115467i	$-0.0368365\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.582171	+	0.582171i	−0.141197	+	0.141197i	−0.774172	−	0.632975i	$-0.781834\pi$
$17$	−0.582171	+	0.582171i	−0.141197	+	0.141197i	0.632975	+	0.774172i	$0.281834\pi$
$18$		0			0
$19$		−	7.02619i		−	1.61192i	−0.591972	−	0.805959i	$-0.701650\pi$
$19$		−	7.02619i		−	1.61192i	0.591972	−	0.805959i	$-0.298350\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.707107	+	0.707107i	0.147442	+	0.147442i
$23$	0.707107	+	0.707107i	0.147442	+	0.147442i
$24$		0			0
$25$	2.92060	+	4.05834i	0.584120	+	0.811667i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.66204			−0.865719			−0.432859	−	0.901461i	$-0.642495\pi$
$29$	−4.66204			−0.865719			−0.432859	+	0.901461i	$0.642495\pi$
$30$		0			0
$31$	2.39492			0.430140			0.215070	−	0.976599i	$-0.431002\pi$
$31$	2.39492			0.430140			0.215070	+	0.976599i	$0.431002\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.82046	+	5.64623i	0.307715	+	0.954387i
$36$		0			0
$37$	−7.42183	−	7.42183i	−1.22014	−	1.22014i	−0.967581	−	0.252559i	$-0.918728\pi$
$37$	−7.42183	−	7.42183i	−1.22014	−	1.22014i	−0.252559	−	0.967581i	$-0.581272\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	5.25361i		−	0.820477i	−0.911978	−	0.410238i	$-0.865446\pi$
$41$		−	5.25361i		−	0.820477i	0.911978	−	0.410238i	$-0.134554\pi$
$42$		0			0
$43$	−4.98971	+	4.98971i	−0.760924	+	0.760924i	−0.976489	−	0.215566i	$-0.930841\pi$
$43$	−4.98971	+	4.98971i	−0.760924	+	0.760924i	0.215566	+	0.976489i	$0.430841\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.04459	−	7.04459i	1.02756	−	1.02756i	0.0279502	−	0.999609i	$-0.491102\pi$
$47$	7.04459	−	7.04459i	1.02756	−	1.02756i	0.999609	−	0.0279502i	$-0.00889800\pi$
$48$		0			0
$49$			0.0387972i			0.00554245i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.72818	+	7.72818i	1.06155	+	1.06155i	0.997977	+	0.0635692i	$0.0202483\pi$
$53$	7.72818	+	7.72818i	1.06155	+	1.06155i	0.0635692	+	0.997977i	$0.479752\pi$
$54$		0			0
$55$	−1.77468	+	3.46363i	−0.239298	+	0.467035i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.38558			0.961521			0.480760	−	0.876852i	$-0.340361\pi$
$59$	7.38558			0.961521			0.480760	+	0.876852i	$0.340361\pi$
$60$		0			0
$61$	0.462830			0.0592593			0.0296296	−	0.999561i	$-0.490567\pi$
$61$	0.462830			0.0592593			0.0296296	+	0.999561i	$0.490567\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−12.0321	+	3.87940i	−1.49240	+	0.481180i
$66$		0			0
$67$	5.30026	+	5.30026i	0.647530	+	0.647530i	0.952395	−	0.304865i	$-0.0986115\pi$
$67$	5.30026	+	5.30026i	0.647530	+	0.647530i	−0.304865	+	0.952395i	$0.598611\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			10.2010i			1.21063i	0.795986	+	0.605315i	$0.206953\pi$
$71$			10.2010i			1.21063i	−0.795986	+	0.605315i	$0.793047\pi$
$72$		0			0
$73$	10.1201	−	10.1201i	1.18447	−	1.18447i	0.205896	−	0.978574i	$-0.433989\pi$
$73$	10.1201	−	10.1201i	1.18447	−	1.18447i	0.978574	−	0.205896i	$-0.0660107\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.26514	+	3.26514i	−0.372097	+	0.372097i
$78$		0			0
$79$		−	4.66304i		−	0.524633i	−0.964982	−	0.262316i	$-0.915514\pi$
$79$		−	4.66304i		−	0.524633i	0.964982	−	0.262316i	$-0.0844864\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.85708	+	1.85708i	0.203841	+	0.203841i	0.801643	−	0.597803i	$-0.203959\pi$
$83$	1.85708	+	1.85708i	0.203841	+	0.203841i	−0.597803	+	0.801643i	$0.703959\pi$
$84$		0			0
$85$	1.75216	−	0.564935i	0.190049	−	0.0612758i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−11.5361			−1.22282			−0.611411	−	0.791313i	$-0.709398\pi$
$89$	−11.5361			−1.22282			−0.611411	+	0.791313i	$0.709398\pi$
$90$		0			0
$91$	−14.9996			−1.57239
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−7.16430	+	13.9825i	−0.735042	+	1.43457i
$96$		0			0
$97$	−10.6939	−	10.6939i	−1.08580	−	1.08580i	−0.995956	−	0.0898446i	$-0.971363\pi$
$97$	−10.6939	−	10.6939i	−1.08580	−	1.08580i	−0.0898446	−	0.995956i	$-0.528637\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			14.4392i			1.43675i	0.695656	+	0.718375i	$0.255114\pi$
$101$			14.4392i			1.43675i	−0.695656	+	0.718375i	$0.744886\pi$
$102$		0			0
$103$	−5.87951	+	5.87951i	−0.579325	+	0.579325i	−0.934717	−	0.355392i	$-0.884347\pi$
$103$	−5.87951	+	5.87951i	−0.579325	+	0.579325i	0.355392	+	0.934717i	$0.384347\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−5.26673	+	5.26673i	−0.509154	+	0.509154i	−0.914267	−	0.405112i	$-0.867232\pi$
$107$	−5.26673	+	5.26673i	−0.509154	+	0.509154i	0.405112	+	0.914267i	$0.367232\pi$
$108$		0			0
$109$			10.6590i			1.02095i	0.859893	+	0.510475i	$0.170530\pi$
$109$			10.6590i			1.02095i	−0.859893	+	0.510475i	$0.829470\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−12.7851	−	12.7851i	−1.20272	−	1.20272i	−0.973335	−	0.229390i	$-0.926327\pi$
$113$	−12.7851	−	12.7851i	−1.20272	−	1.20272i	−0.229390	−	0.973335i	$-0.573673\pi$
$114$		0			0
$115$	−0.686172	−	2.12818i	−0.0639859	−	0.198454i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.18431			0.200236
$120$		0			0
$121$	7.97076			0.724614
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.67403	−	11.0543i	−0.149730	−	0.988727i
$126$		0			0
$127$	−8.74267	−	8.74267i	−0.775786	−	0.775786i	0.203325	−	0.979111i	$-0.434825\pi$
$127$	−8.74267	−	8.74267i	−0.775786	−	0.775786i	−0.979111	+	0.203325i	$0.934825\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			9.83322i			0.859132i	0.903035	+	0.429566i	$0.141334\pi$
$131$			9.83322i			0.859132i	−0.903035	+	0.429566i	$0.858666\pi$
$132$		0			0
$133$	−13.1812	+	13.1812i	−1.14295	+	1.14295i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.24347	−	4.24347i	0.362544	−	0.362544i	−0.502204	−	0.864749i	$-0.667478\pi$
$137$	4.24347	−	4.24347i	0.362544	−	0.362544i	0.864749	+	0.502204i	$0.167478\pi$
$138$		0			0
$139$		−	16.9495i		−	1.43764i	−0.695196	−	0.718820i	$-0.744682\pi$
$139$		−	16.9495i		−	1.43764i	0.695196	−	0.718820i	$-0.255318\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.95798	−	6.95798i	−0.581856	−	0.581856i
$144$		0			0
$145$	9.27769	+	4.75368i	0.770470	+	0.394772i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−20.1720			−1.65256			−0.826279	−	0.563262i	$-0.809546\pi$
$149$	−20.1720			−1.65256			−0.826279	+	0.563262i	$0.809546\pi$
$150$		0			0
$151$	−5.65166			−0.459926			−0.229963	−	0.973199i	$-0.573860\pi$
$151$	−5.65166			−0.459926			−0.229963	+	0.973199i	$0.573860\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.76600	−	2.44199i	−0.382815	−	0.196146i
$156$		0			0
$157$	−6.18004	−	6.18004i	−0.493221	−	0.493221i	0.416099	−	0.909320i	$-0.363397\pi$
$157$	−6.18004	−	6.18004i	−0.493221	−	0.493221i	−0.909320	+	0.416099i	$0.863397\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	2.65307i		−	0.209091i
$162$		0			0
$163$	−6.23866	+	6.23866i	−0.488650	+	0.488650i	−0.907880	−	0.419230i	$-0.862300\pi$
$163$	−6.23866	+	6.23866i	−0.488650	+	0.488650i	0.419230	+	0.907880i	$0.362300\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.8832	+	11.8832i	−0.919548	+	0.919548i	−0.996996	−	0.0774487i	$-0.975323\pi$
$167$	−11.8832	+	11.8832i	−0.919548	+	0.919548i	0.0774487	+	0.996996i	$0.475323\pi$
$168$		0			0
$169$		−	18.9641i		−	1.45878i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.52080	−	1.52080i	−0.115624	−	0.115624i	0.646927	−	0.762552i	$-0.276054\pi$
$173$	−1.52080	−	1.52080i	−0.115624	−	0.115624i	−0.762552	+	0.646927i	$0.776054\pi$
$174$		0			0
$175$	2.13440	−	13.0925i	0.161345	−	0.989702i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−17.6747			−1.32107			−0.660535	−	0.750795i	$-0.729671\pi$
$179$	−17.6747			−1.32107			−0.660535	+	0.750795i	$0.729671\pi$
$180$		0			0
$181$	−6.10044			−0.453442			−0.226721	−	0.973960i	$-0.572801\pi$
$181$	−6.10044			−0.453442			−0.226721	+	0.973960i	$0.572801\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	7.20209	+	22.3375i	0.529508	+	1.64229i
$186$		0			0
$187$	1.01325	+	1.01325i	0.0740963	+	0.0740963i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			0.488584i			0.0353527i	0.999844	+	0.0176764i	$0.00562685\pi$
$191$			0.488584i			0.0353527i	−0.999844	+	0.0176764i	$0.994373\pi$
$192$		0			0
$193$	12.1500	−	12.1500i	0.874573	−	0.874573i	−0.118394	−	0.992967i	$-0.537774\pi$
$193$	12.1500	−	12.1500i	0.874573	−	0.874573i	0.992967	+	0.118394i	$0.0377744\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.8710	+	12.8710i	−0.917017	+	0.917017i	−0.996811	−	0.0797941i	$-0.974574\pi$
$197$	−12.8710	+	12.8710i	−0.917017	+	0.917017i	0.0797941	+	0.996811i	$0.474574\pi$
$198$		0			0
$199$			27.4767i			1.94777i	0.227043	+	0.973885i	$0.427094\pi$
$199$			27.4767i			1.94777i	−0.227043	+	0.973885i	$0.572906\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.74601	+	8.74601i	0.613850	+	0.613850i
$204$		0			0
$205$	−5.35688	+	10.4550i	−0.374141	+	0.730206i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−12.2289			−0.845889
$210$		0			0
$211$	−18.0545			−1.24292			−0.621460	−	0.783446i	$-0.713460\pi$
$211$	−18.0545			−1.24292			−0.621460	+	0.783446i	$0.713460\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	15.0176	−	4.84198i	1.02419	−	0.330221i
$216$		0			0
$217$	−4.49288	−	4.49288i	−0.304996	−	0.304996i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			4.65476i			0.313113i
$222$		0			0
$223$	13.6796	−	13.6796i	0.916055	−	0.916055i	−0.0806849	−	0.996740i	$-0.525711\pi$
$223$	13.6796	−	13.6796i	0.916055	−	0.916055i	0.996740	+	0.0806849i	$0.0257108\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.42166	−	1.42166i	0.0943590	−	0.0943590i	−0.658352	−	0.752711i	$-0.728746\pi$
$227$	1.42166	−	1.42166i	0.0943590	−	0.0943590i	0.752711	+	0.658352i	$0.228746\pi$
$228$		0			0
$229$			24.5323i			1.62114i	0.585642	+	0.810570i	$0.300842\pi$
$229$			24.5323i			1.62114i	−0.585642	+	0.810570i	$0.699158\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	5.62549	+	5.62549i	0.368538	+	0.368538i	0.866944	−	0.498406i	$-0.166081\pi$
$233$	5.62549	+	5.62549i	0.368538	+	0.368538i	−0.498406	+	0.866944i	$0.666081\pi$
$234$		0			0
$235$	−21.2022	+	6.83603i	−1.38308	+	0.445933i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−26.5622			−1.71817			−0.859085	−	0.511834i	$-0.828966\pi$
$239$	−26.5622			−1.71817			−0.859085	+	0.511834i	$0.828966\pi$
$240$		0			0
$241$	6.38296			0.411163			0.205581	−	0.978640i	$-0.434091\pi$
$241$	6.38296			0.411163			0.205581	+	0.978640i	$0.434091\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.0395598	−	0.0772083i	0.00252738	−	0.00493266i
$246$		0			0
$247$	−28.0890	−	28.0890i	−1.78726	−	1.78726i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	2.81409i		−	0.177624i	−0.996048	−	0.0888118i	$-0.971693\pi$
$251$		−	2.81409i		−	0.177624i	0.996048	−	0.0888118i	$-0.0283070\pi$
$252$		0			0
$253$	1.23070	−	1.23070i	0.0773734	−	0.0773734i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	16.8399	−	16.8399i	1.05044	−	1.05044i	0.0517864	−	0.998658i	$-0.483509\pi$
$257$	16.8399	−	16.8399i	1.05044	−	1.05044i	0.998658	−	0.0517864i	$-0.0164915\pi$
$258$		0			0
$259$			27.8468i			1.73031i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.13573	+	1.13573i	0.0700321	+	0.0700321i	0.741255	−	0.671223i	$-0.234231\pi$
$263$	1.13573	+	1.13573i	0.0700321	+	0.0700321i	−0.671223	+	0.741255i	$0.734231\pi$
$264$		0			0
$265$	−7.49937	−	23.2595i	−0.460683	−	1.42882i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.553553			−0.0337507			−0.0168754	−	0.999858i	$-0.505372\pi$
$269$	−0.553553			−0.0337507			−0.0168754	+	0.999858i	$0.505372\pi$
$270$		0			0
$271$	4.72218			0.286852			0.143426	−	0.989661i	$-0.454188\pi$
$271$	4.72218			0.286852			0.143426	+	0.989661i	$0.454188\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	7.06342	−	5.08322i	0.425940	−	0.306530i
$276$		0			0
$277$	−9.42720	−	9.42720i	−0.566425	−	0.566425i	0.364700	−	0.931125i	$-0.381172\pi$
$277$	−9.42720	−	9.42720i	−0.566425	−	0.566425i	−0.931125	+	0.364700i	$0.881172\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			12.6372i			0.753873i	0.926239	+	0.376936i	$0.123022\pi$
$281$			12.6372i			0.753873i	−0.926239	+	0.376936i	$0.876978\pi$
$282$		0			0
$283$	9.50350	−	9.50350i	0.564925	−	0.564925i	−0.365778	−	0.930702i	$-0.619197\pi$
$283$	9.50350	−	9.50350i	0.564925	−	0.564925i	0.930702	+	0.365778i	$0.119197\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−9.85581	+	9.85581i	−0.581770	+	0.581770i
$288$		0			0
$289$			16.3222i			0.960127i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.52474	+	3.52474i	0.205918	+	0.205918i	0.802530	−	0.596612i	$-0.203487\pi$
$293$	3.52474	+	3.52474i	0.205918	+	0.205918i	−0.596612	+	0.802530i	$0.703487\pi$
$294$		0			0
$295$	−14.6977	−	7.53076i	−0.855732	−	0.438458i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.65368			0.326961
$300$		0			0
$301$	18.7215			1.07909
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.921055	−	0.471928i	−0.0527394	−	0.0270225i
$306$		0			0
$307$	21.1113	+	21.1113i	1.20489	+	1.20489i	0.972663	+	0.232223i	$0.0745999\pi$
$307$	21.1113	+	21.1113i	1.20489	+	1.20489i	0.232223	+	0.972663i	$0.425400\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	23.2636i		−	1.31916i	−0.751635	−	0.659579i	$-0.770734\pi$
$311$		−	23.2636i		−	1.31916i	0.751635	−	0.659579i	$-0.229266\pi$
$312$		0			0
$313$	15.6143	−	15.6143i	0.882573	−	0.882573i	−0.111223	−	0.993796i	$-0.535477\pi$
$313$	15.6143	−	15.6143i	0.882573	−	0.882573i	0.993796	+	0.111223i	$0.0354767\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	17.2389	−	17.2389i	0.968230	−	0.968230i	−0.0312803	−	0.999511i	$-0.509958\pi$
$317$	17.2389	−	17.2389i	0.968230	−	0.968230i	0.999511	+	0.0312803i	$0.00995846\pi$
$318$		0			0
$319$			8.11415i			0.454305i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.09044	+	4.09044i	0.227598	+	0.227598i
$324$		0			0
$325$	27.9001	+	4.54839i	1.54762	+	0.252299i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−26.4314			−1.45721
$330$		0			0
$331$	25.3536			1.39356			0.696780	−	0.717285i	$-0.254615\pi$
$331$	25.3536			1.39356			0.696780	+	0.717285i	$0.254615\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.14334	−	15.9522i	−0.281011	−	0.871564i
$336$		0			0
$337$	5.24737	+	5.24737i	0.285843	+	0.285843i	0.835434	−	0.549591i	$-0.185216\pi$
$337$	5.24737	+	5.24737i	0.285843	+	0.285843i	−0.549591	+	0.835434i	$0.685216\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	4.16828i		−	0.225725i
$342$		0			0
$343$	−13.0593	+	13.0593i	−0.705134	+	0.705134i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	11.9553	−	11.9553i	0.641796	−	0.641796i	−0.309201	−	0.950997i	$-0.600061\pi$
$347$	11.9553	−	11.9553i	0.641796	−	0.641796i	0.950997	+	0.309201i	$0.100061\pi$
$348$		0			0
$349$		−	8.55447i		−	0.457910i	−0.973437	−	0.228955i	$-0.926469\pi$
$349$		−	8.55447i		−	0.457910i	0.973437	−	0.228955i	$-0.0735309\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−3.58791	−	3.58791i	−0.190965	−	0.190965i	0.605148	−	0.796113i	$-0.293114\pi$
$353$	−3.58791	−	3.58791i	−0.190965	−	0.190965i	−0.796113	+	0.605148i	$0.793114\pi$
$354$		0			0
$355$	10.4015	−	20.3004i	0.552053	−	1.07743i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	31.2766			1.65072			0.825358	−	0.564609i	$-0.190973\pi$
$359$	31.2766			1.65072			0.825358	+	0.564609i	$0.190973\pi$
$360$		0			0
$361$	−30.3673			−1.59828
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−30.4586	+	9.82049i	−1.59427	+	0.514028i
$366$		0			0
$367$	−2.33963	−	2.33963i	−0.122128	−	0.122128i	0.643401	−	0.765529i	$-0.277523\pi$
$367$	−2.33963	−	2.33963i	−0.122128	−	0.122128i	−0.765529	+	0.643401i	$0.777523\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	28.9962i		−	1.50541i
$372$		0			0
$373$	19.2971	−	19.2971i	0.999167	−	0.999167i	−0.000832768	−	1.00000i	$-0.500265\pi$
$373$	19.2971	−	19.2971i	0.999167	−	0.999167i	1.00000		0.000832768i	$0.000265078\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−18.6377	+	18.6377i	−0.959890	+	0.959890i
$378$		0			0
$379$			17.8601i			0.917414i	0.888588	+	0.458707i	$0.151687\pi$
$379$			17.8601i			0.917414i	−0.888588	+	0.458707i	$0.848313\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	20.3614	+	20.3614i	1.04042	+	1.04042i	0.999148	+	0.0412710i	$0.0131407\pi$
$383$	20.3614	+	20.3614i	1.04042	+	1.04042i	0.0412710	+	0.999148i	$0.486859\pi$
$384$		0			0
$385$	9.82710	−	3.16847i	0.500835	−	0.161480i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.56492			−0.0793446			−0.0396723	−	0.999213i	$-0.512631\pi$
$389$	−1.56492			−0.0793446			−0.0396723	+	0.999213i	$0.512631\pi$
$390$		0			0
$391$	−0.823314			−0.0416368
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.75470	+	9.27968i	−0.239235	+	0.466912i
$396$		0			0
$397$	−7.40811	−	7.40811i	−0.371803	−	0.371803i	0.496331	−	0.868133i	$-0.334680\pi$
$397$	−7.40811	−	7.40811i	−0.371803	−	0.371803i	−0.868133	+	0.496331i	$0.834680\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			21.3078i			1.06406i	0.846725	+	0.532031i	$0.178571\pi$
$401$			21.3078i			1.06406i	−0.846725	+	0.532031i	$0.821429\pi$
$402$		0			0
$403$	9.57430	−	9.57430i	0.476930	−	0.476930i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−12.9175	+	12.9175i	−0.640296	+	0.640296i
$408$		0			0
$409$		−	7.45948i		−	0.368847i	−0.982847	−	0.184424i	$-0.940958\pi$
$409$		−	7.45948i		−	0.368847i	0.982847	−	0.184424i	$-0.0590419\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−13.8554	−	13.8554i	−0.681779	−	0.681779i
$414$		0			0
$415$	−1.80210	−	5.58926i	−0.0884615	−	0.274366i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−18.2231			−0.890257			−0.445128	−	0.895467i	$-0.646842\pi$
$419$	−18.2231			−0.890257			−0.445128	+	0.895467i	$0.646842\pi$
$420$		0			0
$421$	−24.6457			−1.20116			−0.600580	−	0.799565i	$-0.705063\pi$
$421$	−24.6457			−1.20116			−0.600580	+	0.799565i	$0.705063\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−4.06293	−	0.662357i	−0.197081	−	0.0321290i
$426$		0			0
$427$	−0.868271	−	0.868271i	−0.0420186	−	0.0420186i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			8.08673i			0.389524i	0.980851	+	0.194762i	$0.0623935\pi$
$431$			8.08673i			0.389524i	−0.980851	+	0.194762i	$0.937607\pi$
$432$		0			0
$433$	−2.18507	+	2.18507i	−0.105008	+	0.105008i	−0.757659	−	0.652651i	$-0.773657\pi$
$433$	−2.18507	+	2.18507i	−0.105008	+	0.105008i	0.652651	+	0.757659i	$0.273657\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.96826	−	4.96826i	0.237664	−	0.237664i
$438$		0			0
$439$		−	10.8326i		−	0.517014i	−0.966009	−	0.258507i	$-0.916769\pi$
$439$		−	10.8326i		−	0.517014i	0.966009	−	0.258507i	$-0.0832305\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	18.7370	+	18.7370i	0.890221	+	0.890221i	0.994543	−	0.104323i	$-0.0332675\pi$
$443$	18.7370	+	18.7370i	0.890221	+	0.890221i	−0.104323	+	0.994543i	$0.533267\pi$
$444$		0			0
$445$	22.9574	+	11.7628i	1.08828	+	0.557612i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	29.9284			1.41241			0.706203	−	0.708009i	$-0.250407\pi$
$449$	29.9284			1.41241			0.706203	+	0.708009i	$0.250407\pi$
$450$		0			0
$451$	−9.14377			−0.430563
$452$		0			0
$453$		0			0
$454$		0			0
$455$	29.8500	+	15.2945i	1.39939	+	0.717016i
$456$		0			0
$457$	−3.90878	−	3.90878i	−0.182845	−	0.182845i	0.609749	−	0.792594i	$-0.291270\pi$
$457$	−3.90878	−	3.90878i	−0.182845	−	0.182845i	−0.792594	+	0.609749i	$0.791270\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	31.9890i		−	1.48988i	−0.667134	−	0.744938i	$-0.732479\pi$
$461$		−	31.9890i		−	1.48988i	0.667134	−	0.744938i	$-0.267521\pi$
$462$		0			0
$463$	1.66707	−	1.66707i	0.0774751	−	0.0774751i	−0.667307	−	0.744782i	$-0.732553\pi$
$463$	1.66707	−	1.66707i	0.0774751	−	0.0774751i	0.744782	+	0.667307i	$0.232553\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−23.4171	+	23.4171i	−1.08361	+	1.08361i	−0.0874428	+	0.996170i	$0.527870\pi$
$467$	−23.4171	+	23.4171i	−1.08361	+	1.08361i	−0.996170	+	0.0874428i	$0.972130\pi$
$468$		0			0
$469$		−	19.8866i		−	0.918280i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	8.68445	+	8.68445i	0.399312	+	0.399312i
$474$		0			0
$475$	28.5146	−	20.5207i	1.30834	−	0.941553i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	18.9040			0.863747			0.431873	−	0.901934i	$-0.357853\pi$
$479$	18.9040			0.863747			0.431873	+	0.901934i	$0.357853\pi$
$480$		0			0
$481$	−59.3413			−2.70573
$482$		0			0
$483$		0			0
$484$		0			0
$485$	10.3773	+	32.1855i	0.471208	+	1.46147i
$486$		0			0
$487$	−5.62271	−	5.62271i	−0.254789	−	0.254789i	0.568142	−	0.822931i	$-0.307663\pi$
$487$	−5.62271	−	5.62271i	−0.254789	−	0.254789i	−0.822931	+	0.568142i	$0.807663\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			36.4273i			1.64394i	0.569532	+	0.821969i	$0.307125\pi$
$491$			36.4273i			1.64394i	−0.569532	+	0.821969i	$0.692875\pi$
$492$		0			0
$493$	2.71410	−	2.71410i	0.122237	−	0.122237i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	19.1371	−	19.1371i	0.858414	−	0.858414i
$498$		0			0
$499$			5.33084i			0.238641i	0.992856	+	0.119321i	$0.0380716\pi$
$499$			5.33084i			0.238641i	−0.992856	+	0.119321i	$0.961928\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.28950	+	1.28950i	0.0574961	+	0.0574961i	0.735270	−	0.677774i	$-0.237055\pi$
$503$	1.28950	+	1.28950i	0.0574961	+	0.0574961i	−0.677774	+	0.735270i	$0.737055\pi$
$504$		0			0
$505$	14.7230	−	28.7347i	0.655164	−	1.27868i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9.98879			−0.442745			−0.221373	−	0.975189i	$-0.571054\pi$
$509$	−9.98879			−0.442745			−0.221373	+	0.975189i	$0.571054\pi$
$510$		0			0
$511$	−37.9708			−1.67973
$512$		0			0
$513$		0			0
$514$		0			0
$515$	17.6956	−	5.70544i	0.779761	−	0.251412i
$516$		0			0
$517$	−12.2609	−	12.2609i	−0.539234	−	0.539234i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	35.4834i		−	1.55456i	−0.629158	−	0.777278i	$-0.716600\pi$
$521$		−	35.4834i		−	1.55456i	0.629158	−	0.777278i	$-0.283400\pi$
$522$		0			0
$523$	−13.8341	+	13.8341i	−0.604924	+	0.604924i	−0.941615	−	0.336691i	$-0.890692\pi$
$523$	−13.8341	+	13.8341i	−0.604924	+	0.604924i	0.336691	+	0.941615i	$0.390692\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.39425	+	1.39425i	−0.0607345	+	0.0607345i
$528$		0			0
$529$			1.00000i			0.0434783i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−21.0027	−	21.0027i	−0.909727	−	0.909727i
$534$		0			0
$535$	15.8513	−	5.11081i	0.685313	−	0.220959i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	0.0675254			0.00290852
$540$		0			0
$541$	−2.65955			−0.114343			−0.0571716	−	0.998364i	$-0.518208\pi$
$541$	−2.65955			−0.114343			−0.0571716	+	0.998364i	$0.518208\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10.8685	−	21.2120i	0.465557	−	0.908622i
$546$		0			0
$547$	25.7480	+	25.7480i	1.10090	+	1.10090i	0.994302	+	0.106601i	$0.0339969\pi$
$547$	25.7480	+	25.7480i	1.10090	+	1.10090i	0.106601	+	0.994302i	$0.466003\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			32.7563i			1.39547i
$552$		0			0
$553$	−8.74789	+	8.74789i	−0.371998	+	0.371998i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5.46821	−	5.46821i	0.231695	−	0.231695i	−0.581705	−	0.813400i	$-0.697614\pi$
$557$	5.46821	−	5.46821i	0.231695	−	0.231695i	0.813400	+	0.581705i	$0.197614\pi$
$558$		0			0
$559$			39.8953i			1.68739i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−16.8121	−	16.8121i	−0.708544	−	0.708544i	0.257685	−	0.966229i	$-0.417040\pi$
$563$	−16.8121	−	16.8121i	−0.708544	−	0.708544i	−0.966229	+	0.257685i	$0.917040\pi$
$564$		0			0
$565$	12.4066	+	38.4795i	0.521950	+	1.61885i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−25.4910			−1.06864			−0.534319	−	0.845283i	$-0.679432\pi$
$569$	−25.4910			−1.06864			−0.534319	+	0.845283i	$0.679432\pi$
$570$		0			0
$571$	−14.5152			−0.607441			−0.303721	−	0.952761i	$-0.598229\pi$
$571$	−14.5152			−0.607441			−0.303721	+	0.952761i	$0.598229\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.804501	+	4.93485i	−0.0335500	+	0.205798i
$576$		0			0
$577$	24.6531	+	24.6531i	1.02632	+	1.02632i	0.999644	+	0.0266781i	$0.00849290\pi$
$577$	24.6531	+	24.6531i	1.02632	+	1.02632i	0.0266781	+	0.999644i	$0.491507\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	6.96778i		−	0.289072i
$582$		0			0
$583$	13.4507	−	13.4507i	0.557070	−	0.557070i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.178333	+	0.178333i	−0.00736059	+	0.00736059i	−0.710777	−	0.703417i	$-0.751657\pi$
$587$	−0.178333	+	0.178333i	−0.00736059	+	0.00736059i	0.703417	+	0.710777i	$0.251657\pi$
$588$		0			0
$589$		−	16.8271i		−	0.693350i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	3.01255	+	3.01255i	0.123711	+	0.123711i	0.766252	−	0.642541i	$-0.222120\pi$
$593$	3.01255	+	3.01255i	0.123711	+	0.123711i	−0.642541	+	0.766252i	$0.722120\pi$
$594$		0			0
$595$	−4.34689	−	2.22725i	−0.178205	−	0.0913083i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−31.1503			−1.27277			−0.636383	−	0.771373i	$-0.719570\pi$
$599$	−31.1503			−1.27277			−0.636383	+	0.771373i	$0.719570\pi$
$600$		0			0
$601$	32.3250			1.31856			0.659281	−	0.751896i	$-0.270861\pi$
$601$	32.3250			1.31856			0.659281	+	0.751896i	$0.270861\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−15.8622	−	8.12744i	−0.644891	−	0.330427i
$606$		0			0
$607$	−12.6548	−	12.6548i	−0.513642	−	0.513642i	0.401998	−	0.915640i	$-0.368316\pi$
$607$	−12.6548	−	12.6548i	−0.513642	−	0.513642i	−0.915640	+	0.401998i	$0.868316\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	56.3252i		−	2.27867i
$612$		0			0
$613$	−1.32124	+	1.32124i	−0.0533643	+	0.0533643i	−0.733285	−	0.679921i	$-0.762014\pi$
$613$	−1.32124	+	1.32124i	−0.0533643	+	0.0533643i	0.679921	+	0.733285i	$0.262014\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	7.12241	−	7.12241i	0.286737	−	0.286737i	−0.549051	−	0.835789i	$-0.685011\pi$
$617$	7.12241	−	7.12241i	0.286737	−	0.286737i	0.835789	+	0.549051i	$0.185011\pi$
$618$		0			0
$619$			10.6072i			0.426339i	0.977015	+	0.213169i	$0.0683786\pi$
$619$			10.6072i			0.426339i	−0.977015	+	0.213169i	$0.931621\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	21.6417	+	21.6417i	0.867058	+	0.867058i
$624$		0			0
$625$	−7.94019	+	23.7056i	−0.317608	+	0.948222i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	8.64154			0.344561
$630$		0			0
$631$	−17.3998			−0.692675			−0.346338	−	0.938110i	$-0.612575\pi$
$631$	−17.3998			−0.692675			−0.346338	+	0.938110i	$0.612575\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.48383	+	26.3129i	0.336670	+	1.04419i
$636$		0			0
$637$	0.155102	+	0.155102i	0.00614535	+	0.00614535i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	38.0273i		−	1.50199i	−0.660308	−	0.750995i	$-0.729574\pi$
$641$		−	38.0273i		−	1.50199i	0.660308	−	0.750995i	$-0.270426\pi$
$642$		0			0
$643$	20.2777	−	20.2777i	0.799676	−	0.799676i	−0.183368	−	0.983044i	$-0.558700\pi$
$643$	20.2777	−	20.2777i	0.799676	−	0.799676i	0.983044	+	0.183368i	$0.0587001\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−16.4085	+	16.4085i	−0.645084	+	0.645084i	−0.951801	−	0.306717i	$-0.900770\pi$
$647$	−16.4085	+	16.4085i	−0.645084	+	0.645084i	0.306717	+	0.951801i	$0.400770\pi$
$648$		0			0
$649$		−	12.8544i		−	0.504579i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	19.4329	+	19.4329i	0.760469	+	0.760469i	0.976407	−	0.215938i	$-0.0692809\pi$
$653$	19.4329	+	19.4329i	0.760469	+	0.760469i	−0.215938	+	0.976407i	$0.569281\pi$
$654$		0			0
$655$	10.0265	−	19.5686i	0.391768	−	0.764608i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	22.5835			0.879728			0.439864	−	0.898064i	$-0.355027\pi$
$659$	22.5835			0.879728			0.439864	+	0.898064i	$0.355027\pi$
$660$		0			0
$661$	20.6625			0.803677			0.401838	−	0.915711i	$-0.368371\pi$
$661$	20.6625			0.803677			0.401838	+	0.915711i	$0.368371\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	39.6715	−	12.7909i	1.53839	−	0.496011i
$666$		0			0
$667$	−3.29656	−	3.29656i	−0.127643	−	0.127643i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	0.805542i		−	0.0310976i
$672$		0			0
$673$	19.7032	−	19.7032i	0.759503	−	0.759503i	−0.216729	−	0.976232i	$-0.569539\pi$
$673$	19.7032	−	19.7032i	0.759503	−	0.759503i	0.976232	+	0.216729i	$0.0695386\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	5.97159	−	5.97159i	0.229507	−	0.229507i	−0.582980	−	0.812487i	$-0.698113\pi$
$677$	5.97159	−	5.97159i	0.229507	−	0.229507i	0.812487	+	0.582980i	$0.198113\pi$
$678$		0			0
$679$			40.1236i			1.53980i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−5.87047	−	5.87047i	−0.224627	−	0.224627i	0.585816	−	0.810444i	$-0.300774\pi$
$683$	−5.87047	−	5.87047i	−0.224627	−	0.224627i	−0.810444	+	0.585816i	$0.800774\pi$
$684$		0			0
$685$	−12.7716	+	4.11784i	−0.487978	+	0.157335i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	61.7908			2.35404
$690$		0			0
$691$	−35.4616			−1.34902			−0.674512	−	0.738264i	$-0.735646\pi$
$691$	−35.4616			−1.34902			−0.674512	+	0.738264i	$0.735646\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−17.2827	+	33.7304i	−0.655570	+	1.27947i
$696$		0			0
$697$	3.05850	+	3.05850i	0.115849	+	0.115849i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	16.8355i		−	0.635869i	−0.948113	−	0.317935i	$-0.897011\pi$
$701$		−	16.8355i		−	0.635869i	0.948113	−	0.317935i	$-0.102989\pi$
$702$		0			0
$703$	−52.1471	+	52.1471i	−1.96677	+	1.96677i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	27.0880	−	27.0880i	1.01875	−	1.01875i
$708$		0			0
$709$		−	0.549701i		−	0.0206445i	−0.999947	−	0.0103222i	$-0.996714\pi$
$709$		−	0.549701i		−	0.0206445i	0.999947	−	0.0103222i	$-0.00328573\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.69346	+	1.69346i	0.0634206	+	0.0634206i
$714$		0			0
$715$	6.75198	+	20.9415i	0.252510	+	0.783168i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−29.6400			−1.10538			−0.552692	−	0.833385i	$-0.686400\pi$
$719$	−29.6400			−1.10538			−0.552692	+	0.833385i	$0.686400\pi$
$720$		0			0
$721$	22.0600			0.821557
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−13.6159	−	18.9201i	−0.505684	−	0.702676i
$726$		0			0
$727$	16.3336	+	16.3336i	0.605778	+	0.605778i	0.941840	−	0.336062i	$-0.109095\pi$
$727$	16.3336	+	16.3336i	0.605778	+	0.605778i	−0.336062	+	0.941840i	$0.609095\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	5.80973i		−	0.214881i
$732$		0			0
$733$	−26.7493	+	26.7493i	−0.988009	+	0.988009i	−0.999929	−	0.0119203i	$-0.996206\pi$
$733$	−26.7493	+	26.7493i	−0.988009	+	0.988009i	0.0119203	+	0.999929i	$0.496206\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	9.22496	−	9.22496i	0.339806	−	0.339806i
$738$		0			0
$739$		−	25.7796i		−	0.948317i	−0.880439	−	0.474159i	$-0.842752\pi$
$739$		−	25.7796i		−	0.948317i	0.880439	−	0.474159i	$-0.157248\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	16.7266	+	16.7266i	0.613638	+	0.613638i	0.943892	−	0.330254i	$-0.107135\pi$
$743$	16.7266	+	16.7266i	0.613638	+	0.613638i	−0.330254	+	0.943892i	$0.607135\pi$
$744$		0			0
$745$	40.1434	+	20.5685i	1.47074	+	0.753573i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	19.7609			0.722046
$750$		0			0
$751$	31.5495			1.15126			0.575630	−	0.817711i	$-0.304757\pi$
$751$	31.5495			1.15126			0.575630	+	0.817711i	$0.304757\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	11.2471	+	5.76275i	0.409323	+	0.209728i
$756$		0			0
$757$	−17.2962	−	17.2962i	−0.628642	−	0.628642i	0.319084	−	0.947726i	$-0.396625\pi$
$757$	−17.2962	−	17.2962i	−0.628642	−	0.628642i	−0.947726	+	0.319084i	$0.896625\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	47.8224i		−	1.73356i	−0.498690	−	0.866781i	$-0.666185\pi$
$761$		−	47.8224i		−	1.73356i	0.498690	−	0.866781i	$-0.333815\pi$
$762$		0			0
$763$	19.9964	−	19.9964i	0.723918	−	0.723918i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	29.5258	−	29.5258i	1.06611	−	1.06611i
$768$		0			0
$769$			44.9897i			1.62237i	0.584790	+	0.811185i	$0.301177\pi$
$769$			44.9897i			1.62237i	−0.584790	+	0.811185i	$0.698823\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−27.2159	−	27.2159i	−0.978887	−	0.978887i	0.0208950	−	0.999782i	$-0.493348\pi$
$773$	−27.2159	−	27.2159i	−0.978887	−	0.978887i	−0.999782	+	0.0208950i	$0.993348\pi$
$774$		0			0
$775$	6.99459	+	9.71938i	0.251253	+	0.349130i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−36.9129			−1.32254
$780$		0			0
$781$	17.7545			0.635305
$782$		0			0
$783$		0			0
$784$		0			0
$785$	5.99707	+	18.6001i	0.214045	+	0.663866i
$786$		0			0
$787$	12.1494	+	12.1494i	0.433080	+	0.433080i	0.889675	−	0.456595i	$-0.150931\pi$
$787$	12.1494	+	12.1494i	0.433080	+	0.433080i	−0.456595	+	0.889675i	$0.650931\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			47.9700i			1.70562i
$792$		0			0
$793$	1.85028	−	1.85028i	0.0657054	−	0.0657054i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8.00646	−	8.00646i	0.283603	−	0.283603i	−0.550941	−	0.834544i	$-0.685731\pi$
$797$	8.00646	−	8.00646i	0.283603	−	0.283603i	0.834544	+	0.550941i	$0.185731\pi$
$798$		0			0
$799$			8.20231i			0.290177i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−17.6138	−	17.6138i	−0.621576	−	0.621576i
$804$		0			0
$805$	−2.70522	+	5.27975i	−0.0953466	+	0.186087i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−6.25084			−0.219768			−0.109884	−	0.993944i	$-0.535048\pi$
$809$	−6.25084			−0.219768			−0.109884	+	0.993944i	$0.535048\pi$
$810$		0			0
$811$	−47.0556			−1.65235			−0.826173	−	0.563416i	$-0.809487\pi$
$811$	−47.0556			−1.65235			−0.826173	+	0.563416i	$0.809487\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	18.7765	−	6.05396i	0.657713	−	0.212061i
$816$		0			0
$817$	35.0586	+	35.0586i	1.22655	+	1.22655i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			3.96291i			0.138306i	0.997606	+	0.0691532i	$0.0220297\pi$
$821$			3.96291i			0.138306i	−0.997606	+	0.0691532i	$0.977970\pi$
$822$		0			0
$823$	−9.51054	+	9.51054i	−0.331517	+	0.331517i	−0.853162	−	0.521646i	$-0.825318\pi$
$823$	−9.51054	+	9.51054i	−0.331517	+	0.331517i	0.521646	+	0.853162i	$0.325318\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	14.2491	−	14.2491i	0.495490	−	0.495490i	−0.414541	−	0.910031i	$-0.636058\pi$
$827$	14.2491	−	14.2491i	0.495490	−	0.495490i	0.910031	+	0.414541i	$0.136058\pi$
$828$		0			0
$829$			40.7088i			1.41388i	0.707276	+	0.706938i	$0.249924\pi$
$829$			40.7088i			1.41388i	−0.707276	+	0.706938i	$0.750076\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.0225866	−	0.0225866i	−0.000782578	−	0.000782578i
$834$		0			0
$835$	35.7649	−	11.5314i	1.23769	−	0.399059i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	19.2158			0.663404			0.331702	−	0.943384i	$-0.392377\pi$
$839$	19.2158			0.663404			0.331702	+	0.943384i	$0.392377\pi$
$840$		0			0
$841$	−7.26540			−0.250531
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−19.3369	+	37.7396i	−0.665210	+	1.29828i
$846$		0			0
$847$	−14.9532	−	14.9532i	−0.513798	−	0.513798i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	10.4960i		−	0.359800i
$852$		0			0
$853$	−11.1423	+	11.1423i	−0.381505	+	0.381505i	−0.871644	−	0.490139i	$-0.836946\pi$
$853$	−11.1423	+	11.1423i	−0.381505	+	0.381505i	0.490139	+	0.871644i	$0.336946\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	15.8744	−	15.8744i	0.542261	−	0.542261i	−0.381930	−	0.924191i	$-0.624741\pi$
$857$	15.8744	−	15.8744i	0.542261	−	0.542261i	0.924191	+	0.381930i	$0.124741\pi$
$858$		0			0
$859$			38.8630i			1.32599i	0.748624	+	0.662995i	$0.230715\pi$
$859$			38.8630i			1.32599i	−0.748624	+	0.662995i	$0.769285\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.588537	−	0.588537i	−0.0200340	−	0.0200340i	0.697019	−	0.717053i	$-0.254509\pi$
$863$	−0.588537	−	0.588537i	−0.0200340	−	0.0200340i	−0.717053	+	0.697019i	$0.754509\pi$
$864$		0			0
$865$	1.47578	+	4.57717i	0.0501779	+	0.155628i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−8.11589			−0.275313
$870$		0			0
$871$	42.3783			1.43594
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−17.5974	+	23.8784i	−0.594902	+	0.807238i
$876$		0			0
$877$	−10.7277	−	10.7277i	−0.362249	−	0.362249i	0.502391	−	0.864640i	$-0.332454\pi$
$877$	−10.7277	−	10.7277i	−0.362249	−	0.362249i	−0.864640	+	0.502391i	$0.832454\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	47.1940i		−	1.59001i	−0.606606	−	0.795003i	$-0.707469\pi$
$881$		−	47.1940i		−	1.59001i	0.606606	−	0.795003i	$-0.292531\pi$
$882$		0			0
$883$	−4.53480	+	4.53480i	−0.152608	+	0.152608i	−0.779282	−	0.626674i	$-0.784416\pi$
$883$	−4.53480	+	4.53480i	−0.152608	+	0.152608i	0.626674	+	0.779282i	$0.284416\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	38.0052	−	38.0052i	1.27609	−	1.27609i	0.333252	−	0.942838i	$-0.391854\pi$
$887$	38.0052	−	38.0052i	1.27609	−	1.27609i	0.942838	−	0.333252i	$-0.108146\pi$
$888$		0			0
$889$			32.8026i			1.10016i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−49.4966	−	49.4966i	−1.65634	−	1.65634i
$894$		0			0
$895$	35.1736	+	18.0221i	1.17572	+	0.602414i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−11.1652			−0.372380
$900$		0			0
$901$	−8.99824			−0.299775
$902$		0			0
$903$		0			0
$904$		0			0
$905$	12.1402	+	6.22036i	0.403553	+	0.206772i
$906$		0			0
$907$	−21.3663	−	21.3663i	−0.709455	−	0.709455i	0.256966	−	0.966421i	$-0.417277\pi$
$907$	−21.3663	−	21.3663i	−0.709455	−	0.709455i	−0.966421	+	0.256966i	$0.917277\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			43.9218i			1.45519i	0.686005	+	0.727597i	$0.259363\pi$
$911$			43.9218i			1.45519i	−0.686005	+	0.727597i	$0.740637\pi$
$912$		0			0
$913$	3.23219	−	3.23219i	0.106970	−	0.106970i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	18.4472	−	18.4472i	0.609179	−	0.609179i
$918$		0			0
$919$			50.4605i			1.66454i	0.554373	+	0.832269i	$0.312958\pi$
$919$			50.4605i			1.66454i	−0.554373	+	0.832269i	$0.687042\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	40.7810	+	40.7810i	1.34232	+	1.34232i
$924$		0			0
$925$	8.44408	−	51.7965i	0.277640	−	1.70306i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−25.7091			−0.843490			−0.421745	−	0.906715i	$-0.638582\pi$
$929$	−25.7091			−0.843490			−0.421745	+	0.906715i	$0.638582\pi$
$930$		0			0
$931$	0.272596			0.00893398
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−0.983253	−	3.04959i	−0.0321558	−	0.0997323i
$936$		0			0
$937$	−13.5029	−	13.5029i	−0.441120	−	0.441120i	0.451269	−	0.892388i	$-0.350972\pi$
$937$	−13.5029	−	13.5029i	−0.441120	−	0.441120i	−0.892388	+	0.451269i	$0.850972\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	34.0122i		−	1.10877i	−0.832262	−	0.554383i	$-0.812954\pi$
$941$		−	34.0122i		−	1.10877i	0.832262	−	0.554383i	$-0.187046\pi$
$942$		0			0
$943$	3.71487	−	3.71487i	0.120973	−	0.120973i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	9.90948	−	9.90948i	0.322015	−	0.322015i	−0.527525	−	0.849540i	$-0.676880\pi$
$947$	9.90948	−	9.90948i	0.322015	−	0.322015i	0.849540	+	0.527525i	$0.176880\pi$
$948$		0			0
$949$		−	80.9155i		−	2.62663i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−29.2703	−	29.2703i	−0.948159	−	0.948159i	0.0505616	−	0.998721i	$-0.483899\pi$
$953$	−29.2703	−	29.2703i	−0.948159	−	0.948159i	−0.998721	+	0.0505616i	$0.983899\pi$
$954$		0			0
$955$	0.498188	−	0.972307i	0.0161210	−	0.0314631i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−15.9216			−0.514134
$960$		0			0
$961$	−25.2644			−0.814980
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−36.5678	+	11.7902i	−1.17716	+	0.379541i
$966$		0			0
$967$	26.3207	+	26.3207i	0.846418	+	0.846418i	0.989684	−	0.143266i	$-0.0457606\pi$
$967$	26.3207	+	26.3207i	0.846418	+	0.846418i	−0.143266	+	0.989684i	$0.545761\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	3.22570i		−	0.103518i	−0.998660	−	0.0517589i	$-0.983517\pi$
$971$		−	3.22570i		−	0.103518i	0.998660	−	0.0517589i	$-0.0164827\pi$
$972$		0			0
$973$	−31.7974	+	31.7974i	−1.01938	+	1.01938i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.65445	−	1.65445i	0.0529305	−	0.0529305i	−0.680146	−	0.733077i	$-0.738084\pi$
$977$	1.65445	−	1.65445i	0.0529305	−	0.0529305i	0.733077	+	0.680146i	$0.238084\pi$
$978$		0			0
$979$			20.0782i			0.641702i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−29.9198	−	29.9198i	−0.954293	−	0.954293i	0.0447069	−	0.999000i	$-0.485765\pi$
$983$	−29.9198	−	29.9198i	−0.954293	−	0.954293i	−0.999000	+	0.0447069i	$0.985765\pi$
$984$		0			0
$985$	38.7378	−	12.4899i	1.23429	−	0.397961i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−7.05652			−0.224384
$990$		0			0
$991$	41.5604			1.32021			0.660104	−	0.751174i	$-0.270512\pi$
$991$	41.5604			1.32021			0.660104	+	0.751174i	$0.270512\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	28.0168	−	54.6800i	0.888191	−	1.73347i
$996$		0			0
$997$	−7.87557	−	7.87557i	−0.249422	−	0.249422i	0.571311	−	0.820733i	$-0.306435\pi$
$997$	−7.87557	−	7.87557i	−0.249422	−	0.249422i	−0.820733	+	0.571311i	$0.806435\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.s.a.737.5	✓	44
3.2	odd	2	inner	4140.2.s.a.737.18	yes	44
5.3	odd	4	inner	4140.2.s.a.2393.18	yes	44
15.8	even	4	inner	4140.2.s.a.2393.5	yes	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.s.a.737.5	✓	44	1.1	even	1	trivial
4140.2.s.a.737.18	yes	44	3.2	odd	2	inner
4140.2.s.a.2393.5	yes	44	15.8	even	4	inner
4140.2.s.a.2393.18	yes	44	5.3	odd	4	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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.s (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.99005 - 1.01966i) q^{5} +(-1.87601 - 1.87601i) q^{7} +O(q^{10})\)	\(q+(-1.99005 - 1.01966i) q^{5} +(-1.87601 - 1.87601i) q^{7} -1.74047i q^{11} +(3.99776 - 3.99776i) q^{13} +(-0.582171 + 0.582171i) q^{17} -7.02619i q^{19} +(0.707107 + 0.707107i) q^{23} +(2.92060 + 4.05834i) q^{25} -4.66204 q^{29} +2.39492 q^{31} +(1.82046 + 5.64623i) q^{35} +(-7.42183 - 7.42183i) q^{37} -5.25361i q^{41} +(-4.98971 + 4.98971i) q^{43} +(7.04459 - 7.04459i) q^{47} +0.0387972i q^{49} +(7.72818 + 7.72818i) q^{53} +(-1.77468 + 3.46363i) q^{55} +7.38558 q^{59} +0.462830 q^{61} +(-12.0321 + 3.87940i) q^{65} +(5.30026 + 5.30026i) q^{67} +10.2010i q^{71} +(10.1201 - 10.1201i) q^{73} +(-3.26514 + 3.26514i) q^{77} -4.66304i q^{79} +(1.85708 + 1.85708i) q^{83} +(1.75216 - 0.564935i) q^{85} -11.5361 q^{89} -14.9996 q^{91} +(-7.16430 + 13.9825i) q^{95} +(-10.6939 - 10.6939i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 12 q^{7}+O(q^{10}) \) 44 * q - 12 * q^7	\( 44 q - 12 q^{7} - 4 q^{13} + 24 q^{25} - 48 q^{37} + 8 q^{43} + 40 q^{55} - 96 q^{61} - 44 q^{67} + 76 q^{73} + 72 q^{85} - 48 q^{91} - 20 q^{97}+O(q^{100}) \) 44 * q - 12 * q^7 - 4 * q^13 + 24 * q^25 - 48 * q^37 + 8 * q^43 + 40 * q^55 - 96 * q^61 - 44 * q^67 + 76 * q^73 + 72 * q^85 - 48 * q^91 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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