LMFDB - Embedded newform 4560.2.d.e.2431.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(2431,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4560.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.d (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:	$36.4117833217$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.9821011968.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 20x^{4} + 118x^{2} + 192 $ x^6 + 20x^4 + 118x^2 + 192
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.5
Root			$2.60808i$ of defining polynomial
Character	$\chi$	$=$	4560.2431
Dual form			4560.2.d.e.2431.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +2.60808i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +2.60808i q^{7} +1.00000 q^{9} -1.74632i q^{11} +2.60808i q^{13} +1.00000 q^{15} -0.802094 q^{17} +(0.197906 - 4.35440i) q^{19} -2.60808i q^{21} +1.23017i q^{23} +1.00000 q^{25} -1.00000 q^{27} -1.74632i q^{29} +8.15874 q^{31} +1.74632i q^{33} -2.60808i q^{35} +2.97649i q^{37} -2.60808i q^{39} +6.96249i q^{41} +2.97649i q^{43} -1.00000 q^{45} -9.93897i q^{47} +0.197906 q^{49} +0.802094 q^{51} -1.23017i q^{53} +1.74632i q^{55} +(-0.197906 + 4.35440i) q^{57} +1.60419 q^{59} +10.1587 q^{61} +2.60808i q^{63} -2.60808i q^{65} -9.35664 q^{67} -1.23017i q^{69} +1.60419 q^{71} -10.5650 q^{73} -1.00000 q^{75} +4.55455 q^{77} +16.9608 q^{79} +1.00000 q^{81} -4.72281i q^{83} +0.802094 q^{85} +1.74632i q^{87} +12.5471i q^{89} -6.80209 q^{91} -8.15874 q^{93} +(-0.197906 + 4.35440i) q^{95} +8.85656i q^{97} -1.74632i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} - 6 q^{5} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 - 6 * q^5 + 6 * q^9	$ 6 q - 6 q^{3} - 6 q^{5} + 6 q^{9} + 6 q^{15} - 4 q^{17} + 2 q^{19} + 6 q^{25} - 6 q^{27} - 12 q^{31} - 6 q^{45} + 2 q^{49} + 4 q^{51} - 2 q^{57} + 8 q^{59} + 4 q^{67} + 8 q^{71} - 6 q^{75} - 32 q^{77} + 40 q^{79} + 6 q^{81} + 4 q^{85} - 40 q^{91} + 12 q^{93} - 2 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 - 6 * q^5 + 6 * q^9 + 6 * q^15 - 4 * q^17 + 2 * q^19 + 6 * q^25 - 6 * q^27 - 12 * q^31 - 6 * q^45 + 2 * q^49 + 4 * q^51 - 2 * q^57 + 8 * q^59 + 4 * q^67 + 8 * q^71 - 6 * q^75 - 32 * q^77 + 40 * q^79 + 6 * q^81 + 4 * q^85 - 40 * q^91 + 12 * q^93 - 2 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times$.

$n$	$1141$	$1711$	$1921$	$2737$	$3041$
$\chi(n)$	$1$	$-1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			2.60808i			0.985763i	0.870097	+	0.492881i	$0.164056\pi$
$7$			2.60808i			0.985763i	−0.870097	+	0.492881i	$0.835944\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	1.74632i		−	0.526536i	−0.964723	−	0.263268i	$-0.915200\pi$
$11$		−	1.74632i		−	0.526536i	0.964723	−	0.263268i	$-0.0848003\pi$
$12$		0			0
$13$			2.60808i			0.723352i	0.932304	+	0.361676i	$0.117795\pi$
$13$			2.60808i			0.723352i	−0.932304	+	0.361676i	$0.882205\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$	−0.802094			−0.194536			−0.0972682	−	0.995258i	$-0.531010\pi$
$17$	−0.802094			−0.194536			−0.0972682	+	0.995258i	$0.531010\pi$
$18$		0			0
$19$	0.197906	−	4.35440i	0.0454027	−	0.998969i
$19$	0.197906	−	4.35440i	0.0454027	−	0.998969i
$20$		0			0
$21$		−	2.60808i		−	0.569130i
$22$		0			0
$23$			1.23017i			0.256508i	0.991741	+	0.128254i	$0.0409372\pi$
$23$			1.23017i			0.256508i	−0.991741	+	0.128254i	$0.959063\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$		−	1.74632i		−	0.324284i	−0.986767	−	0.162142i	$-0.948160\pi$
$29$		−	1.74632i		−	0.324284i	0.986767	−	0.162142i	$-0.0518402\pi$
$30$		0			0
$31$	8.15874			1.46535			0.732676	−	0.680577i	$-0.238271\pi$
$31$	8.15874			1.46535			0.732676	+	0.680577i	$0.238271\pi$
$32$		0			0
$33$			1.74632i			0.303996i
$34$		0			0
$35$		−	2.60808i		−	0.440846i
$36$		0			0
$37$			2.97649i			0.489332i	0.969607	+	0.244666i	$0.0786783\pi$
$37$			2.97649i			0.489332i	−0.969607	+	0.244666i	$0.921322\pi$
$38$		0			0
$39$		−	2.60808i		−	0.417627i
$40$		0			0
$41$			6.96249i			1.08736i	0.839293	+	0.543679i	$0.182969\pi$
$41$			6.96249i			1.08736i	−0.839293	+	0.543679i	$0.817031\pi$
$42$		0			0
$43$			2.97649i			0.453910i	0.973905	+	0.226955i	$0.0728771\pi$
$43$			2.97649i			0.453910i	−0.973905	+	0.226955i	$0.927123\pi$
$44$		0			0
$45$	−1.00000			−0.149071
$46$		0			0
$47$		−	9.93897i		−	1.44975i	−0.688881	−	0.724874i	$-0.741898\pi$
$47$		−	9.93897i		−	1.44975i	0.688881	−	0.724874i	$-0.258102\pi$
$48$		0			0
$49$	0.197906			0.0282722
$50$		0			0
$51$	0.802094			0.112316
$52$		0			0
$53$		−	1.23017i		−	0.168976i	−0.996424	−	0.0844882i	$-0.973074\pi$
$53$		−	1.23017i		−	0.168976i	0.996424	−	0.0844882i	$-0.0269255\pi$
$54$		0			0
$55$			1.74632i			0.235474i
$56$		0			0
$57$	−0.197906	+	4.35440i	−0.0262132	+	0.576755i
$58$		0			0
$59$	1.60419			0.208848			0.104424	−	0.994533i	$-0.466700\pi$
$59$	1.60419			0.208848			0.104424	+	0.994533i	$0.466700\pi$
$60$		0			0
$61$	10.1587			1.30069			0.650347	−	0.759638i	$-0.274624\pi$
$61$	10.1587			1.30069			0.650347	+	0.759638i	$0.274624\pi$
$62$		0			0
$63$			2.60808i			0.328588i
$64$		0			0
$65$		−	2.60808i		−	0.323493i
$66$		0			0
$67$	−9.35664			−1.14310			−0.571548	−	0.820569i	$-0.693657\pi$
$67$	−9.35664			−1.14310			−0.571548	+	0.820569i	$0.693657\pi$
$68$		0			0
$69$		−	1.23017i		−	0.148095i
$70$		0			0
$71$	1.60419			0.190382			0.0951911	−	0.995459i	$-0.469654\pi$
$71$	1.60419			0.190382			0.0951911	+	0.995459i	$0.469654\pi$
$72$		0			0
$73$	−10.5650			−1.23654			−0.618271	−	0.785965i	$-0.712167\pi$
$73$	−10.5650			−1.23654			−0.618271	+	0.785965i	$0.712167\pi$
$74$		0			0
$75$	−1.00000			−0.115470
$76$		0			0
$77$	4.55455			0.519039
$78$		0			0
$79$	16.9608			1.90824			0.954121	−	0.299420i	$-0.0967932\pi$
$79$	16.9608			1.90824			0.954121	+	0.299420i	$0.0967932\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	4.72281i		−	0.518396i	−0.965824	−	0.259198i	$-0.916542\pi$
$83$		−	4.72281i		−	0.518396i	0.965824	−	0.259198i	$-0.0834582\pi$
$84$		0			0
$85$	0.802094			0.0869993
$86$		0			0
$87$			1.74632i			0.187225i
$88$		0			0
$89$			12.5471i			1.32999i	0.746850	+	0.664993i	$0.231565\pi$
$89$			12.5471i			1.32999i	−0.746850	+	0.664993i	$0.768435\pi$
$90$		0			0
$91$	−6.80209			−0.713053
$92$		0			0
$93$	−8.15874			−0.846022
$94$		0			0
$95$	−0.197906	+	4.35440i	−0.0203047	+	0.446752i
$96$		0			0
$97$			8.85656i			0.899247i	0.893218	+	0.449624i	$0.148442\pi$
$97$			8.85656i			0.899247i	−0.893218	+	0.449624i	$0.851558\pi$
$98$		0			0
$99$		−	1.74632i		−	0.175512i
$100$		0			0
$101$	−18.3175			−1.82266			−0.911329	−	0.411680i	$-0.864942\pi$
$101$	−18.3175			−1.82266			−0.911329	+	0.411680i	$0.864942\pi$
$102$		0			0
$103$	2.39581			0.236066			0.118033	−	0.993010i	$-0.462341\pi$
$103$	2.39581			0.236066			0.118033	+	0.993010i	$0.462341\pi$
$104$		0			0
$105$			2.60808i			0.254523i
$106$		0			0
$107$	−9.75246			−0.942806			−0.471403	−	0.881918i	$-0.656252\pi$
$107$	−9.75246			−0.942806			−0.471403	+	0.881918i	$0.656252\pi$
$108$		0			0
$109$		−	2.75583i		−	0.263961i	−0.991252	−	0.131980i	$-0.957866\pi$
$109$		−	2.75583i		−	0.263961i	0.991252	−	0.131980i	$-0.0421336\pi$
$110$		0			0
$111$		−	2.97649i		−	0.282516i
$112$		0			0
$113$			13.4316i			1.26354i	0.775156	+	0.631770i	$0.217671\pi$
$113$			13.4316i			1.26354i	−0.775156	+	0.631770i	$0.782329\pi$
$114$		0			0
$115$		−	1.23017i		−	0.114714i
$116$		0			0
$117$			2.60808i			0.241117i
$118$		0			0
$119$		−	2.09193i		−	0.191767i
$120$		0			0
$121$	7.95036			0.722760
$122$		0			0
$123$		−	6.96249i		−	0.627786i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−12.5650			−1.11497			−0.557483	−	0.830189i	$-0.688233\pi$
$127$	−12.5650			−1.11497			−0.557483	+	0.830189i	$0.688233\pi$
$128$		0			0
$129$		−	2.97649i		−	0.262065i
$130$		0			0
$131$		−	15.3029i		−	1.33702i	−0.743703	−	0.668510i	$-0.766932\pi$
$131$		−	15.3029i		−	1.33702i	0.743703	−	0.668510i	$-0.233068\pi$
$132$		0			0
$133$	11.3566	+	0.516154i	0.984746	+	0.0447562i
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$	−8.56502			−0.731759			−0.365880	−	0.930662i	$-0.619232\pi$
$137$	−8.56502			−0.731759			−0.365880	+	0.930662i	$0.619232\pi$
$138$		0			0
$139$			13.9250i			1.18110i	0.807001	+	0.590550i	$0.201089\pi$
$139$			13.9250i			1.18110i	−0.807001	+	0.590550i	$0.798911\pi$
$140$		0			0
$141$			9.93897i			0.837013i
$142$		0			0
$143$	4.55455			0.380871
$144$		0			0
$145$			1.74632i			0.145024i
$146$		0			0
$147$	−0.197906			−0.0163230
$148$		0			0
$149$	7.60419			0.622959			0.311480	−	0.950253i	$-0.399175\pi$
$149$	7.60419			0.622959			0.311480	+	0.950253i	$0.399175\pi$
$150$		0			0
$151$	−13.5154			−1.09987			−0.549933	−	0.835209i	$-0.685347\pi$
$151$	−13.5154			−1.09987			−0.549933	+	0.835209i	$0.685347\pi$
$152$		0			0
$153$	−0.802094			−0.0648455
$154$		0			0
$155$	−8.15874			−0.655326
$156$		0			0
$157$	15.3566			1.22559			0.612797	−	0.790240i	$-0.290044\pi$
$157$	15.3566			1.22559			0.612797	+	0.790240i	$0.290044\pi$
$158$		0			0
$159$			1.23017i			0.0975586i
$160$		0			0
$161$	−3.20838			−0.252856
$162$		0			0
$163$			14.0727i			1.10226i	0.834419	+	0.551130i	$0.185803\pi$
$163$			14.0727i			1.10226i	−0.834419	+	0.551130i	$0.814197\pi$
$164$		0			0
$165$		−	1.74632i		−	0.135951i
$166$		0			0
$167$	−5.19791			−0.402226			−0.201113	−	0.979568i	$-0.564456\pi$
$167$	−5.19791			−0.402226			−0.201113	+	0.979568i	$0.564456\pi$
$168$		0			0
$169$	6.19791			0.476762
$170$		0			0
$171$	0.197906	−	4.35440i	0.0151342	−	0.332990i
$172$		0			0
$173$			22.1404i			1.68331i	0.540019	+	0.841653i	$0.318417\pi$
$173$			22.1404i			1.68331i	−0.540019	+	0.841653i	$0.681583\pi$
$174$		0			0
$175$			2.60808i			0.197153i
$176$		0			0
$177$	−1.60419			−0.120578
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$			6.93969i			0.515823i	0.966169	+	0.257911i	$0.0830343\pi$
$181$			6.93969i			0.515823i	−0.966169	+	0.257911i	$0.916966\pi$
$182$		0			0
$183$	−10.1587			−0.750956
$184$		0			0
$185$		−	2.97649i		−	0.218836i
$186$		0			0
$187$			1.40071i			0.102430i
$188$		0			0
$189$		−	2.60808i		−	0.189710i
$190$		0			0
$191$		−	1.00951i		−	0.0730455i	−0.999333	−	0.0365228i	$-0.988372\pi$
$191$		−	1.00951i		−	0.0730455i	0.999333	−	0.0365228i	$-0.0116281\pi$
$192$		0			0
$193$			26.9654i			1.94101i	0.241079	+	0.970505i	$0.422499\pi$
$193$			26.9654i			1.94101i	−0.241079	+	0.970505i	$0.577501\pi$
$194$		0			0
$195$			2.60808i			0.186769i
$196$		0			0
$197$	11.1979			0.797818			0.398909	−	0.916991i	$-0.369389\pi$
$197$	11.1979			0.797818			0.398909	+	0.916991i	$0.369389\pi$
$198$		0			0
$199$			5.95298i			0.421995i	0.977487	+	0.210998i	$0.0676713\pi$
$199$			5.95298i			0.421995i	−0.977487	+	0.210998i	$0.932329\pi$
$200$		0			0
$201$	9.35664			0.659967
$202$		0			0
$203$	4.55455			0.319667
$204$		0			0
$205$		−	6.96249i		−	0.486281i
$206$		0			0
$207$			1.23017i			0.0855025i
$208$		0			0
$209$	−7.60419	−	0.345607i	−0.525993	−	0.0239061i
$210$		0			0
$211$	1.35664			0.0933953			0.0466976	−	0.998909i	$-0.485130\pi$
$211$	1.35664			0.0933953			0.0466976	+	0.998909i	$0.485130\pi$
$212$		0			0
$213$	−1.60419			−0.109917
$214$		0			0
$215$		−	2.97649i		−	0.202995i
$216$		0			0
$217$			21.2787i			1.44449i
$218$		0			0
$219$	10.5650			0.713918
$220$		0			0
$221$		−	2.09193i		−	0.140718i
$222$		0			0
$223$	−0.812566			−0.0544135			−0.0272067	−	0.999630i	$-0.508661\pi$
$223$	−0.812566			−0.0544135			−0.0272067	+	0.999630i	$0.508661\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	22.3279			1.48196			0.740979	−	0.671528i	$-0.234362\pi$
$227$	22.3279			1.48196			0.740979	+	0.671528i	$0.234362\pi$
$228$		0			0
$229$	−16.8021			−1.11031			−0.555157	−	0.831746i	$-0.687342\pi$
$229$	−16.8021			−1.11031			−0.555157	+	0.831746i	$0.687342\pi$
$230$		0			0
$231$	−4.55455			−0.299667
$232$		0			0
$233$	25.5049			1.67088			0.835441	−	0.549580i	$-0.185212\pi$
$233$	25.5049			1.67088			0.835441	+	0.549580i	$0.185212\pi$
$234$		0			0
$235$			9.93897i			0.648347i
$236$		0			0
$237$	−16.9608			−1.10172
$238$		0			0
$239$		−	8.36320i		−	0.540970i	−0.962724	−	0.270485i	$-0.912816\pi$
$239$		−	8.36320i		−	0.540970i	0.962724	−	0.270485i	$-0.0871841\pi$
$240$		0			0
$241$			26.8632i			1.73041i	0.501416	+	0.865207i	$0.332813\pi$
$241$			26.8632i			1.73041i	−0.501416	+	0.865207i	$0.667187\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	−0.197906			−0.0126437
$246$		0			0
$247$	11.3566	+	0.516154i	0.722606	+	0.0328421i
$248$		0			0
$249$			4.72281i			0.299296i
$250$		0			0
$251$			3.10144i			0.195761i	0.995198	+	0.0978805i	$0.0312063\pi$
$251$			3.10144i			0.195761i	−0.995198	+	0.0978805i	$0.968794\pi$
$252$		0			0
$253$	2.14827			0.135060
$254$		0			0
$255$	−0.802094			−0.0502291
$256$		0			0
$257$			17.6155i			1.09882i	0.835552	+	0.549412i	$0.185148\pi$
$257$			17.6155i			1.09882i	−0.835552	+	0.549412i	$0.814852\pi$
$258$		0			0
$259$	−7.76293			−0.482365
$260$		0			0
$261$		−	1.74632i		−	0.108095i
$262$		0			0
$263$			17.6155i			1.08622i	0.839663	+	0.543108i	$0.182753\pi$
$263$			17.6155i			1.08622i	−0.839663	+	0.543108i	$0.817247\pi$
$264$		0			0
$265$			1.23017i			0.0755686i
$266$		0			0
$267$		−	12.5471i		−	0.767867i
$268$		0			0
$269$		−	13.2839i		−	0.809932i	−0.914332	−	0.404966i	$-0.867283\pi$
$269$		−	13.2839i		−	0.809932i	0.914332	−	0.404966i	$-0.132717\pi$
$270$		0			0
$271$			22.6338i			1.37490i	0.726230	+	0.687452i	$0.241271\pi$
$271$			22.6338i			1.37490i	−0.726230	+	0.687452i	$0.758729\pi$
$272$		0			0
$273$	6.80209			0.411681
$274$		0			0
$275$		−	1.74632i		−	0.105307i
$276$		0			0
$277$	−10.3175			−0.619917			−0.309959	−	0.950750i	$-0.600315\pi$
$277$	−10.3175			−0.619917			−0.309959	+	0.950750i	$0.600315\pi$
$278$		0			0
$279$	8.15874			0.488451
$280$		0			0
$281$			27.5043i			1.64077i	0.571811	+	0.820386i	$0.306241\pi$
$281$			27.5043i			1.64077i	−0.571811	+	0.820386i	$0.693759\pi$
$282$		0			0
$283$			0.884560i			0.0525817i	0.999654	+	0.0262908i	$0.00836959\pi$
$283$			0.884560i			0.0525817i	−0.999654	+	0.0262908i	$0.991630\pi$
$284$		0			0
$285$	0.197906	−	4.35440i	0.0117229	−	0.257933i
$286$		0			0
$287$	−18.1587			−1.07188
$288$		0			0
$289$	−16.3566			−0.962156
$290$		0			0
$291$		−	8.85656i		−	0.519181i
$292$		0			0
$293$			2.26248i			0.132175i	0.997814	+	0.0660876i	$0.0210517\pi$
$293$			2.26248i			0.132175i	−0.997814	+	0.0660876i	$0.978948\pi$
$294$		0			0
$295$	−1.60419			−0.0933995
$296$		0			0
$297$			1.74632i			0.101332i
$298$		0			0
$299$	−3.20838			−0.185545
$300$		0			0
$301$	−7.76293			−0.447448
$302$		0			0
$303$	18.3175			1.05231
$304$		0			0
$305$	−10.1587			−0.581688
$306$		0			0
$307$	14.3958			0.821612			0.410806	−	0.911723i	$-0.365247\pi$
$307$	14.3958			0.821612			0.410806	+	0.911723i	$0.365247\pi$
$308$		0			0
$309$	−2.39581			−0.136293
$310$		0			0
$311$		−	8.06770i		−	0.457478i	−0.973488	−	0.228739i	$-0.926540\pi$
$311$		−	8.06770i		−	0.457478i	0.973488	−	0.228739i	$-0.0734602\pi$
$312$		0			0
$313$	−12.1692			−0.687844			−0.343922	−	0.938998i	$-0.611756\pi$
$313$	−12.1692			−0.687844			−0.343922	+	0.938998i	$0.611756\pi$
$314$		0			0
$315$		−	2.60808i		−	0.146949i
$316$		0			0
$317$			10.9257i			0.613648i	0.951766	+	0.306824i	$0.0992662\pi$
$317$			10.9257i			0.613648i	−0.951766	+	0.306824i	$0.900734\pi$
$318$		0			0
$319$	−3.04964			−0.170747
$320$		0			0
$321$	9.75246			0.544329
$322$		0			0
$323$	−0.158739	+	3.49264i	−0.00883247	+	0.194336i
$324$		0			0
$325$			2.60808i			0.144670i
$326$		0			0
$327$			2.75583i			0.152398i
$328$		0			0
$329$	25.9217			1.42911
$330$		0			0
$331$	7.59372			0.417388			0.208694	−	0.977981i	$-0.433079\pi$
$331$	7.59372			0.417388			0.208694	+	0.977981i	$0.433079\pi$
$332$		0			0
$333$			2.97649i			0.163111i
$334$		0			0
$335$	9.35664			0.511208
$336$		0			0
$337$		−	27.0383i		−	1.47287i	−0.676508	−	0.736435i	$-0.736508\pi$
$337$		−	27.0383i		−	1.47287i	0.676508	−	0.736435i	$-0.263492\pi$
$338$		0			0
$339$		−	13.4316i		−	0.729505i
$340$		0			0
$341$		−	14.2478i		−	0.771561i
$342$		0			0
$343$			18.7727i			1.01363i
$344$		0			0
$345$			1.23017i			0.0662300i
$346$		0			0
$347$		−	31.8359i		−	1.70904i	−0.519416	−	0.854521i	$-0.673850\pi$
$347$		−	31.8359i		−	1.70904i	0.519416	−	0.854521i	$-0.326150\pi$
$348$		0			0
$349$	−6.14827			−0.329109			−0.164555	−	0.986368i	$-0.552619\pi$
$349$	−6.14827			−0.329109			−0.164555	+	0.986368i	$0.552619\pi$
$350$		0			0
$351$		−	2.60808i		−	0.139209i
$352$		0			0
$353$	2.79162			0.148583			0.0742915	−	0.997237i	$-0.476330\pi$
$353$	2.79162			0.148583			0.0742915	+	0.997237i	$0.476330\pi$
$354$		0			0
$355$	−1.60419			−0.0851415
$356$		0			0
$357$			2.09193i			0.110717i
$358$		0			0
$359$		−	2.43754i		−	0.128648i	−0.997929	−	0.0643241i	$-0.979511\pi$
$359$		−	2.43754i		−	0.128648i	0.997929	−	0.0643241i	$-0.0204891\pi$
$360$		0			0
$361$	−18.9217	−	1.72352i	−0.995877	−	0.0907117i
$362$		0			0
$363$	−7.95036			−0.417286
$364$		0			0
$365$	10.5650			0.552999
$366$		0			0
$367$			29.0573i			1.51678i	0.651801	+	0.758390i	$0.274014\pi$
$367$			29.0573i			1.51678i	−0.651801	+	0.758390i	$0.725986\pi$
$368$		0			0
$369$			6.96249i			0.362453i
$370$		0			0
$371$	3.20838			0.166571
$372$		0			0
$373$			17.2699i			0.894200i	0.894484	+	0.447100i	$0.147543\pi$
$373$			17.2699i			0.894200i	−0.894484	+	0.447100i	$0.852457\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	4.55455			0.234571
$378$		0			0
$379$	7.59372			0.390063			0.195032	−	0.980797i	$-0.437519\pi$
$379$	7.59372			0.390063			0.195032	+	0.980797i	$0.437519\pi$
$380$		0			0
$381$	12.5650			0.643726
$382$		0			0
$383$	24.0000			1.22634			0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000			1.22634			0.613171	+	0.789950i	$0.289894\pi$
$384$		0			0
$385$	−4.55455			−0.232121
$386$		0			0
$387$			2.97649i			0.151303i
$388$		0			0
$389$	−1.18743			−0.0602053			−0.0301026	−	0.999547i	$-0.509583\pi$
$389$	−1.18743			−0.0602053			−0.0301026	+	0.999547i	$0.509583\pi$
$390$		0			0
$391$		−	0.986710i		−	0.0499001i
$392$		0			0
$393$			15.3029i			0.771929i
$394$		0			0
$395$	−16.9608			−0.853392
$396$		0			0
$397$	2.00000			0.100377			0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000			0.100377			0.0501886	+	0.998740i	$0.484018\pi$
$398$		0			0
$399$	−11.3566	−	0.516154i	−0.568543	−	0.0258400i
$400$		0			0
$401$			11.4418i			0.571378i	0.958322	+	0.285689i	$0.0922225\pi$
$401$			11.4418i			0.571378i	−0.958322	+	0.285689i	$0.907778\pi$
$402$		0			0
$403$			21.2787i			1.05997i
$404$		0			0
$405$	−1.00000			−0.0496904
$406$		0			0
$407$	5.19791			0.257651
$408$		0			0
$409$		−	4.22945i		−	0.209133i	−0.994518	−	0.104567i	$-0.966654\pi$
$409$		−	4.22945i		−	0.209133i	0.994518	−	0.104567i	$-0.0333455\pi$
$410$		0			0
$411$	8.56502			0.422481
$412$		0			0
$413$			4.18386i			0.205874i
$414$		0			0
$415$			4.72281i			0.231834i
$416$		0			0
$417$		−	13.9250i		−	0.681909i
$418$		0			0
$419$			18.1316i			0.885788i	0.896574	+	0.442894i	$0.146048\pi$
$419$			18.1316i			0.885788i	−0.896574	+	0.442894i	$0.853952\pi$
$420$		0			0
$421$		−	9.44562i		−	0.460351i	−0.973149	−	0.230176i	$-0.926070\pi$
$421$		−	9.44562i		−	0.460351i	0.973149	−	0.230176i	$-0.0739301\pi$
$422$		0			0
$423$		−	9.93897i		−	0.483249i
$424$		0			0
$425$	−0.802094			−0.0389073
$426$		0			0
$427$			26.4948i			1.28217i
$428$		0			0
$429$	−4.55455			−0.219896
$430$		0			0
$431$	1.60419			0.0772711			0.0386355	−	0.999253i	$-0.487699\pi$
$431$	1.60419			0.0772711			0.0386355	+	0.999253i	$0.487699\pi$
$432$		0			0
$433$			0.884560i			0.0425093i	0.999774	+	0.0212546i	$0.00676607\pi$
$433$			0.884560i			0.0425093i	−0.999774	+	0.0212546i	$0.993234\pi$
$434$		0			0
$435$		−	1.74632i		−	0.0837297i
$436$		0			0
$437$	5.35664	+	0.243457i	0.256243	+	0.0116461i
$438$		0			0
$439$	−8.00000			−0.381819			−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000			−0.381819			−0.190910	+	0.981608i	$0.561144\pi$
$440$		0			0
$441$	0.197906			0.00942408
$442$		0			0
$443$			14.4639i			0.687202i	0.939116	+	0.343601i	$0.111647\pi$
$443$			14.4639i			0.687202i	−0.939116	+	0.343601i	$0.888353\pi$
$444$		0			0
$445$		−	12.5471i		−	0.594788i
$446$		0			0
$447$	−7.60419			−0.359666
$448$		0			0
$449$		−	33.3844i		−	1.57551i	−0.615990	−	0.787754i	$-0.711244\pi$
$449$		−	33.3844i		−	1.57551i	0.615990	−	0.787754i	$-0.288756\pi$
$450$		0			0
$451$	12.1587			0.572533
$452$		0			0
$453$	13.5154			0.635008
$454$		0			0
$455$	6.80209			0.318887
$456$		0			0
$457$	−7.35664			−0.344129			−0.172065	−	0.985086i	$-0.555044\pi$
$457$	−7.35664			−0.344129			−0.172065	+	0.985086i	$0.555044\pi$
$458$		0			0
$459$	0.802094			0.0374386
$460$		0			0
$461$	−30.6350			−1.42681			−0.713406	−	0.700751i	$-0.752848\pi$
$461$	−30.6350			−1.42681			−0.713406	+	0.700751i	$0.752848\pi$
$462$		0			0
$463$			15.1779i			0.705379i	0.935740	+	0.352689i	$0.114733\pi$
$463$			15.1779i			0.705379i	−0.935740	+	0.352689i	$0.885267\pi$
$464$		0			0
$465$	8.15874			0.378352
$466$		0			0
$467$		−	13.7271i		−	0.635215i	−0.948222	−	0.317608i	$-0.897121\pi$
$467$		−	13.7271i		−	0.635215i	0.948222	−	0.317608i	$-0.102879\pi$
$468$		0			0
$469$		−	24.4029i		−	1.12682i
$470$		0			0
$471$	−15.3566			−0.707597
$472$		0			0
$473$	5.19791			0.239000
$474$		0			0
$475$	0.197906	−	4.35440i	0.00908053	−	0.199794i
$476$		0			0
$477$		−	1.23017i		−	0.0563255i
$478$		0			0
$479$			14.2706i			0.652039i	0.945363	+	0.326020i	$0.105708\pi$
$479$			14.2706i			0.652039i	−0.945363	+	0.326020i	$0.894292\pi$
$480$		0			0
$481$	−7.76293			−0.353959
$482$		0			0
$483$	3.20838			0.145986
$484$		0			0
$485$		−	8.85656i		−	0.402155i
$486$		0			0
$487$	11.7525			0.532555			0.266277	−	0.963896i	$-0.414206\pi$
$487$	11.7525			0.532555			0.266277	+	0.963896i	$0.414206\pi$
$488$		0			0
$489$		−	14.0727i		−	0.636390i
$490$		0			0
$491$		−	5.23896i		−	0.236431i	−0.992988	−	0.118216i	$-0.962283\pi$
$491$		−	5.23896i		−	0.236431i	0.992988	−	0.118216i	$-0.0377174\pi$
$492$		0			0
$493$			1.40071i			0.0630850i
$494$		0			0
$495$			1.74632i			0.0784913i
$496$		0			0
$497$			4.18386i			0.187672i
$498$		0			0
$499$		−	14.6618i		−	0.656352i	−0.944617	−	0.328176i	$-0.893566\pi$
$499$		−	14.6618i		−	0.656352i	0.944617	−	0.328176i	$-0.106434\pi$
$500$		0			0
$501$	5.19791			0.232225
$502$		0			0
$503$		−	17.6155i		−	0.785435i	−0.919659	−	0.392718i	$-0.871535\pi$
$503$		−	17.6155i		−	0.785435i	0.919659	−	0.392718i	$-0.128465\pi$
$504$		0			0
$505$	18.3175			0.815117
$506$		0			0
$507$	−6.19791			−0.275259
$508$		0			0
$509$		−	27.1359i		−	1.20278i	−0.798956	−	0.601390i	$-0.794614\pi$
$509$		−	27.1359i		−	1.20278i	0.798956	−	0.601390i	$-0.205386\pi$
$510$		0			0
$511$		−	27.5544i		−	1.21894i
$512$		0			0
$513$	−0.197906	+	4.35440i	−0.00873775	+	0.192252i
$514$		0			0
$515$	−2.39581			−0.105572
$516$		0			0
$517$	−17.3566			−0.763344
$518$		0			0
$519$		−	22.1404i		−	0.971857i
$520$		0			0
$521$			40.3970i			1.76982i	0.465757	+	0.884912i	$0.345782\pi$
$521$			40.3970i			1.76982i	−0.465757	+	0.884912i	$0.654218\pi$
$522$		0			0
$523$	−17.3776			−0.759869			−0.379934	−	0.925013i	$-0.624053\pi$
$523$	−17.3776			−0.759869			−0.379934	+	0.925013i	$0.624053\pi$
$524$		0			0
$525$		−	2.60808i		−	0.113826i
$526$		0			0
$527$	−6.54408			−0.285065
$528$		0			0
$529$	21.4867			0.934204
$530$		0			0
$531$	1.60419			0.0696159
$532$		0			0
$533$	−18.1587			−0.786542
$534$		0			0
$535$	9.75246			0.421635
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	0.345607i		−	0.0148863i
$540$		0			0
$541$	28.6350			1.23111			0.615556	−	0.788093i	$-0.288931\pi$
$541$	28.6350			1.23111			0.615556	+	0.788093i	$0.288931\pi$
$542$		0			0
$543$		−	6.93969i		−	0.297810i
$544$		0			0
$545$			2.75583i			0.118047i
$546$		0			0
$547$	16.3175			0.697685			0.348842	−	0.937181i	$-0.386575\pi$
$547$	16.3175			0.697685			0.348842	+	0.937181i	$0.386575\pi$
$548$		0			0
$549$	10.1587			0.433564
$550$		0			0
$551$	−7.60419	−	0.345607i	−0.323949	−	0.0147233i
$552$		0			0
$553$			44.2353i			1.88107i
$554$		0			0
$555$			2.97649i			0.126345i
$556$		0			0
$557$	22.4867			0.952792			0.476396	−	0.879231i	$-0.341943\pi$
$557$	22.4867			0.952792			0.476396	+	0.879231i	$0.341943\pi$
$558$		0			0
$559$	−7.76293			−0.328337
$560$		0			0
$561$		−	1.40071i		−	0.0591382i
$562$		0			0
$563$	19.1196			0.805794			0.402897	−	0.915245i	$-0.368003\pi$
$563$	19.1196			0.805794			0.402897	+	0.915245i	$0.368003\pi$
$564$		0			0
$565$		−	13.4316i		−	0.565072i
$566$		0			0
$567$			2.60808i			0.109529i
$568$		0			0
$569$		−	29.3008i		−	1.22835i	−0.789169	−	0.614176i	$-0.789488\pi$
$569$		−	29.3008i		−	1.22835i	0.789169	−	0.614176i	$-0.210512\pi$
$570$		0			0
$571$			26.0808i			1.09145i	0.837965	+	0.545724i	$0.183745\pi$
$571$			26.0808i			1.09145i	−0.837965	+	0.545724i	$0.816255\pi$
$572$		0			0
$573$			1.00951i			0.0421729i
$574$		0			0
$575$			1.23017i			0.0513015i
$576$		0			0
$577$	−5.75246			−0.239478			−0.119739	−	0.992805i	$-0.538206\pi$
$577$	−5.75246			−0.239478			−0.119739	+	0.992805i	$0.538206\pi$
$578$		0			0
$579$		−	26.9654i		−	1.12064i
$580$		0			0
$581$	12.3175			0.511015
$582$		0			0
$583$	−2.14827			−0.0889721
$584$		0			0
$585$		−	2.60808i		−	0.107831i
$586$		0			0
$587$			1.48007i			0.0610888i	0.999533	+	0.0305444i	$0.00972410\pi$
$587$			1.48007i			0.0610888i	−0.999533	+	0.0305444i	$0.990276\pi$
$588$		0			0
$589$	1.61466	−	35.5264i	0.0665309	−	1.46384i
$590$		0			0
$591$	−11.1979			−0.460620
$592$		0			0
$593$	2.14827			0.0882188			0.0441094	−	0.999027i	$-0.485955\pi$
$593$	2.14827			0.0882188			0.0441094	+	0.999027i	$0.485955\pi$
$594$		0			0
$595$			2.09193i			0.0857607i
$596$		0			0
$597$		−	5.95298i		−	0.243639i
$598$		0			0
$599$	−36.3175			−1.48389			−0.741946	−	0.670460i	$-0.766097\pi$
$599$	−36.3175			−1.48389			−0.741946	+	0.670460i	$0.766097\pi$
$600$		0			0
$601$			19.1867i			0.782643i	0.920254	+	0.391322i	$0.127982\pi$
$601$			19.1867i			0.782643i	−0.920254	+	0.391322i	$0.872018\pi$
$602$		0			0
$603$	−9.35664			−0.381032
$604$		0			0
$605$	−7.95036			−0.323228
$606$		0			0
$607$	−15.7734			−0.640223			−0.320111	−	0.947380i	$-0.603720\pi$
$607$	−15.7734			−0.640223			−0.320111	+	0.947380i	$0.603720\pi$
$608$		0			0
$609$	−4.55455			−0.184560
$610$		0			0
$611$	25.9217			1.04868
$612$		0			0
$613$	−19.6741			−0.794630			−0.397315	−	0.917682i	$-0.630058\pi$
$613$	−19.6741			−0.794630			−0.397315	+	0.917682i	$0.630058\pi$
$614$		0			0
$615$			6.96249i			0.280755i
$616$		0			0
$617$	2.40628			0.0968733			0.0484367	−	0.998826i	$-0.484576\pi$
$617$	2.40628			0.0968733			0.0484367	+	0.998826i	$0.484576\pi$
$618$		0			0
$619$			23.6205i			0.949388i	0.880151	+	0.474694i	$0.157441\pi$
$619$			23.6205i			0.949388i	−0.880151	+	0.474694i	$0.842559\pi$
$620$		0			0
$621$		−	1.23017i		−	0.0493649i
$622$		0			0
$623$	−32.7238			−1.31105
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	7.60419	+	0.345607i	0.303682	+	0.0138022i
$628$		0			0
$629$		−	2.38742i		−	0.0951929i
$630$		0			0
$631$		−	6.64419i		−	0.264501i	−0.991216	−	0.132251i	$-0.957780\pi$
$631$		−	6.64419i		−	0.264501i	0.991216	−	0.132251i	$-0.0422203\pi$
$632$		0			0
$633$	−1.35664			−0.0539218
$634$		0			0
$635$	12.5650			0.498628
$636$		0			0
$637$			0.516154i			0.0204508i
$638$		0			0
$639$	1.60419			0.0634607
$640$		0			0
$641$		−	30.9514i		−	1.22251i	−0.791435	−	0.611253i	$-0.790666\pi$
$641$		−	30.9514i		−	1.22251i	0.791435	−	0.611253i	$-0.209334\pi$
$642$		0			0
$643$		−	13.4088i		−	0.528792i	−0.964414	−	0.264396i	$-0.914827\pi$
$643$		−	13.4088i		−	0.528792i	0.964414	−	0.264396i	$-0.0851726\pi$
$644$		0			0
$645$			2.97649i			0.117199i
$646$		0			0
$647$		−	23.1271i		−	0.909222i	−0.890690	−	0.454611i	$-0.849778\pi$
$647$		−	23.1271i		−	0.909222i	0.890690	−	0.454611i	$-0.150222\pi$
$648$		0			0
$649$		−	2.80143i		−	0.109966i
$650$		0			0
$651$		−	21.2787i		−	0.833977i
$652$		0			0
$653$	−17.9321			−0.701739			−0.350869	−	0.936424i	$-0.614114\pi$
$653$	−17.9321			−0.701739			−0.350869	+	0.936424i	$0.614114\pi$
$654$		0			0
$655$			15.3029i			0.597933i
$656$		0			0
$657$	−10.5650			−0.412181
$658$		0			0
$659$	−37.9217			−1.47722			−0.738609	−	0.674134i	$-0.764517\pi$
$659$	−37.9217			−1.47722			−0.738609	+	0.674134i	$0.764517\pi$
$660$		0			0
$661$		−	0.295496i		−	0.0114935i	−0.999983	−	0.00574674i	$-0.998171\pi$
$661$		−	0.295496i		−	0.0114935i	0.999983	−	0.00574674i	$-0.00182925\pi$
$662$		0			0
$663$			2.09193i			0.0812438i
$664$		0			0
$665$	−11.3566	−	0.516154i	−0.440392	−	0.0200156i
$666$		0			0
$667$	2.14827			0.0831812
$668$		0			0
$669$	0.812566			0.0314156
$670$		0			0
$671$		−	17.7404i		−	0.684861i
$672$		0			0
$673$		−	39.4896i		−	1.52221i	−0.648626	−	0.761107i	$-0.724656\pi$
$673$		−	39.4896i		−	1.52221i	0.648626	−	0.761107i	$-0.275344\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$		−	13.1361i		−	0.504862i	−0.967615	−	0.252431i	$-0.918770\pi$
$677$		−	13.1361i		−	0.504862i	0.967615	−	0.252431i	$-0.0812301\pi$
$678$		0			0
$679$	−23.0986			−0.886444
$680$		0			0
$681$	−22.3279			−0.855609
$682$		0			0
$683$	−49.5958			−1.89773			−0.948865	−	0.315682i	$-0.897767\pi$
$683$	−49.5958			−1.89773			−0.948865	+	0.315682i	$0.897767\pi$
$684$		0			0
$685$	8.56502			0.327253
$686$		0			0
$687$	16.8021			0.641040
$688$		0			0
$689$	3.20838			0.122229
$690$		0			0
$691$			28.2913i			1.07625i	0.842865	+	0.538125i	$0.180867\pi$
$691$			28.2913i			1.07625i	−0.842865	+	0.538125i	$0.819133\pi$
$692$		0			0
$693$	4.55455			0.173013
$694$		0			0
$695$		−	13.9250i		−	0.528204i
$696$		0			0
$697$		−	5.58457i		−	0.211531i
$698$		0			0
$699$	−25.5049			−0.964684
$700$		0			0
$701$	−1.18743			−0.0448488			−0.0224244	−	0.999749i	$-0.507138\pi$
$701$	−1.18743			−0.0448488			−0.0224244	+	0.999749i	$0.507138\pi$
$702$		0			0
$703$	12.9608	+	0.589064i	0.488827	+	0.0222170i
$704$		0			0
$705$		−	9.93897i		−	0.374323i
$706$		0			0
$707$		−	47.7735i		−	1.79671i
$708$		0			0
$709$	−7.74198			−0.290756			−0.145378	−	0.989376i	$-0.546440\pi$
$709$	−7.74198			−0.290756			−0.145378	+	0.989376i	$0.546440\pi$
$710$		0			0
$711$	16.9608			0.636081
$712$		0			0
$713$			10.0366i			0.375874i
$714$		0			0
$715$	−4.55455			−0.170331
$716$		0			0
$717$			8.36320i			0.312329i
$718$		0			0
$719$			36.9044i			1.37630i	0.725568	+	0.688150i	$0.241577\pi$
$719$			36.9044i			1.37630i	−0.725568	+	0.688150i	$0.758423\pi$
$720$		0			0
$721$			6.24847i			0.232705i
$722$		0			0
$723$		−	26.8632i		−	0.999054i
$724$		0			0
$725$		−	1.74632i		−	0.0648567i
$726$		0			0
$727$		−	0.147748i		−	0.00547968i	−0.999996	−	0.00273984i	$-0.999128\pi$
$727$		−	0.147748i		−	0.00547968i	0.999996	−	0.00273984i	$-0.000872119\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		−	2.38742i		−	0.0883021i
$732$		0			0
$733$	30.5650			1.12894			0.564472	−	0.825452i	$-0.309080\pi$
$733$	30.5650			1.12894			0.564472	+	0.825452i	$0.309080\pi$
$734$		0			0
$735$	0.197906			0.00729986
$736$		0			0
$737$			16.3397i			0.601881i
$738$		0			0
$739$			38.0324i			1.39904i	0.714611	+	0.699522i	$0.246604\pi$
$739$			38.0324i			1.39904i	−0.714611	+	0.699522i	$0.753396\pi$
$740$		0			0
$741$	−11.3566	−	0.516154i	−0.417197	−	0.0189614i
$742$		0			0
$743$	−10.0783			−0.369738			−0.184869	−	0.982763i	$-0.559186\pi$
$743$	−10.0783			−0.369738			−0.184869	+	0.982763i	$0.559186\pi$
$744$		0			0
$745$	−7.60419			−0.278596
$746$		0			0
$747$		−	4.72281i		−	0.172799i
$748$		0			0
$749$		−	25.4352i		−	0.929382i
$750$		0			0
$751$	−10.3070			−0.376108			−0.188054	−	0.982159i	$-0.560218\pi$
$751$	−10.3070			−0.376108			−0.188054	+	0.982159i	$0.560218\pi$
$752$		0			0
$753$		−	3.10144i		−	0.113023i
$754$		0			0
$755$	13.5154			0.491875
$756$		0			0
$757$	2.31748			0.0842302			0.0421151	−	0.999113i	$-0.486590\pi$
$757$	2.31748			0.0842302			0.0421151	+	0.999113i	$0.486590\pi$
$758$		0			0
$759$	−2.14827			−0.0779772
$760$		0			0
$761$	−39.9426			−1.44792			−0.723959	−	0.689843i	$-0.757680\pi$
$761$	−39.9426			−1.44792			−0.723959	+	0.689843i	$0.757680\pi$
$762$		0			0
$763$	7.18743			0.260203
$764$		0			0
$765$	0.802094			0.0289998
$766$		0			0
$767$			4.18386i			0.151070i
$768$		0			0
$769$	−15.5049			−0.559121			−0.279561	−	0.960128i	$-0.590189\pi$
$769$	−15.5049			−0.559121			−0.279561	+	0.960128i	$0.590189\pi$
$770$		0			0
$771$		−	17.6155i		−	0.634406i
$772$		0			0
$773$			32.5272i			1.16992i	0.811062	+	0.584960i	$0.198890\pi$
$773$			32.5272i			1.16992i	−0.811062	+	0.584960i	$0.801110\pi$
$774$		0			0
$775$	8.15874			0.293071
$776$		0			0
$777$	7.76293			0.278493
$778$		0			0
$779$	30.3175	+	1.37792i	1.08624	+	0.0493689i
$780$		0			0
$781$		−	2.80143i		−	0.100243i
$782$		0			0
$783$			1.74632i			0.0624084i
$784$		0			0
$785$	−15.3566			−0.548102
$786$		0			0
$787$	1.03917			0.0370423			0.0185211	−	0.999828i	$-0.494104\pi$
$787$	1.03917			0.0370423			0.0185211	+	0.999828i	$0.494104\pi$
$788$		0			0
$789$		−	17.6155i		−	0.627127i
$790$		0			0
$791$	−35.0308			−1.24555
$792$		0			0
$793$			26.4948i			0.940859i
$794$		0			0
$795$		−	1.23017i		−	0.0436295i
$796$		0			0
$797$		−	31.0991i		−	1.10159i	−0.834641	−	0.550794i	$-0.814325\pi$
$797$		−	31.0991i		−	1.10159i	0.834641	−	0.550794i	$-0.185675\pi$
$798$		0			0
$799$			7.97200i			0.282029i
$800$		0			0
$801$			12.5471i			0.443328i
$802$		0			0
$803$			18.4499i			0.651084i
$804$		0			0
$805$	3.20838			0.113080
$806$		0			0
$807$			13.2839i			0.467614i
$808$		0			0
$809$	−11.9007			−0.418407			−0.209204	−	0.977872i	$-0.567087\pi$
$809$	−11.9007			−0.418407			−0.209204	+	0.977872i	$0.567087\pi$
$810$		0			0
$811$	−2.41675			−0.0848637			−0.0424319	−	0.999099i	$-0.513511\pi$
$811$	−2.41675			−0.0848637			−0.0424319	+	0.999099i	$0.513511\pi$
$812$		0			0
$813$		−	22.6338i		−	0.793801i
$814$		0			0
$815$		−	14.0727i		−	0.492946i
$816$		0			0
$817$	12.9608	+	0.589064i	0.453442	+	0.0206087i
$818$		0			0
$819$	−6.80209			−0.237684
$820$		0			0
$821$	7.92167			0.276468			0.138234	−	0.990400i	$-0.455857\pi$
$821$	7.92167			0.276468			0.138234	+	0.990400i	$0.455857\pi$
$822$		0			0
$823$		−	24.6236i		−	0.858323i	−0.903228	−	0.429162i	$-0.858809\pi$
$823$		−	24.6236i		−	0.858323i	0.903228	−	0.429162i	$-0.141191\pi$
$824$		0			0
$825$			1.74632i			0.0607991i
$826$		0			0
$827$	36.2496			1.26052			0.630261	−	0.776383i	$-0.282948\pi$
$827$	36.2496			1.26052			0.630261	+	0.776383i	$0.282948\pi$
$828$		0			0
$829$		−	27.8955i		−	0.968852i	−0.874832	−	0.484426i	$-0.839028\pi$
$829$		−	27.8955i		−	0.968852i	0.874832	−	0.484426i	$-0.160972\pi$
$830$		0			0
$831$	10.3175			0.357909
$832$		0			0
$833$	−0.158739			−0.00549998
$834$		0			0
$835$	5.19791			0.179881
$836$		0			0
$837$	−8.15874			−0.282007
$838$		0			0
$839$	23.6825			0.817611			0.408806	−	0.912621i	$-0.365945\pi$
$839$	23.6825			0.817611			0.408806	+	0.912621i	$0.365945\pi$
$840$		0			0
$841$	25.9504			0.894840
$842$		0			0
$843$		−	27.5043i		−	0.947300i
$844$		0			0
$845$	−6.19791			−0.213214
$846$		0			0
$847$			20.7352i			0.712470i
$848$		0			0
$849$		−	0.884560i		−	0.0303580i
$850$		0			0
$851$	−3.66158			−0.125517
$852$		0			0
$853$	35.1091			1.20211			0.601056	−	0.799207i	$-0.294747\pi$
$853$	35.1091			1.20211			0.601056	+	0.799207i	$0.294747\pi$
$854$		0			0
$855$	−0.197906	+	4.35440i	−0.00676823	+	0.148917i
$856$		0			0
$857$		−	22.0948i		−	0.754745i	−0.926062	−	0.377372i	$-0.876828\pi$
$857$		−	22.0948i		−	0.754745i	0.926062	−	0.377372i	$-0.123172\pi$
$858$		0			0
$859$		−	21.8514i		−	0.745559i	−0.927920	−	0.372780i	$-0.878405\pi$
$859$		−	21.8514i		−	0.745559i	0.927920	−	0.372780i	$-0.121595\pi$
$860$		0			0
$861$	18.1587			0.618848
$862$		0			0
$863$	40.1692			1.36738			0.683688	−	0.729775i	$-0.260375\pi$
$863$	40.1692			1.36738			0.683688	+	0.729775i	$0.260375\pi$
$864$		0			0
$865$		−	22.1404i		−	0.752797i
$866$		0			0
$867$	16.3566			0.555501
$868$		0			0
$869$		−	29.6191i		−	1.00476i
$870$		0			0
$871$		−	24.4029i		−	0.826861i
$872$		0			0
$873$			8.85656i			0.299749i
$874$		0			0
$875$		−	2.60808i		−	0.0881693i
$876$		0			0
$877$			14.8824i			0.502544i	0.967916	+	0.251272i	$0.0808489\pi$
$877$			14.8824i			0.502544i	−0.967916	+	0.251272i	$0.919151\pi$
$878$		0			0
$879$		−	2.26248i		−	0.0763114i
$880$		0			0
$881$	43.9217			1.47976			0.739879	−	0.672740i	$-0.234883\pi$
$881$	43.9217			1.47976			0.739879	+	0.672740i	$0.234883\pi$
$882$		0			0
$883$			24.5051i			0.824661i	0.911034	+	0.412330i	$0.135285\pi$
$883$			24.5051i			0.824661i	−0.911034	+	0.412330i	$0.864715\pi$
$884$		0			0
$885$	1.60419			0.0539242
$886$		0			0
$887$	20.3384			0.682897			0.341449	−	0.939900i	$-0.389082\pi$
$887$	20.3384			0.682897			0.341449	+	0.939900i	$0.389082\pi$
$888$		0			0
$889$		−	32.7706i		−	1.09909i
$890$		0			0
$891$		−	1.74632i		−	0.0585040i
$892$		0			0
$893$	−43.2783	−	1.96698i	−1.44825	−	0.0658225i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	3.20838			0.107125
$898$		0			0
$899$		−	14.2478i		−	0.475190i
$900$		0			0
$901$			0.986710i			0.0328721i
$902$		0			0
$903$	7.76293			0.258334
$904$		0			0
$905$		−	6.93969i		−	0.230683i
$906$		0			0
$907$	−5.06011			−0.168018			−0.0840091	−	0.996465i	$-0.526772\pi$
$907$	−5.06011			−0.168018			−0.0840091	+	0.996465i	$0.526772\pi$
$908$		0			0
$909$	−18.3175			−0.607552
$910$		0			0
$911$	43.8224			1.45190			0.725950	−	0.687747i	$-0.241400\pi$
$911$	43.8224			1.45190			0.725950	+	0.687747i	$0.241400\pi$
$912$		0			0
$913$	−8.24754			−0.272954
$914$		0			0
$915$	10.1587			0.335838
$916$		0			0
$917$	39.9112			1.31798
$918$		0			0
$919$		−	45.6633i		−	1.50629i	−0.657853	−	0.753146i	$-0.728535\pi$
$919$		−	45.6633i		−	1.50629i	0.657853	−	0.753146i	$-0.271465\pi$
$920$		0			0
$921$	−14.3958			−0.474358
$922$		0			0
$923$			4.18386i			0.137713i
$924$		0			0
$925$			2.97649i			0.0978663i
$926$		0			0
$927$	2.39581			0.0786888
$928$		0			0
$929$	−8.37487			−0.274770			−0.137385	−	0.990518i	$-0.543870\pi$
$929$	−8.37487			−0.274770			−0.137385	+	0.990518i	$0.543870\pi$
$930$		0			0
$931$	0.0391666	−	0.861761i	0.00128363	−	0.0282431i
$932$		0			0
$933$			8.06770i			0.264125i
$934$		0			0
$935$		−	1.40071i		−	0.0458083i
$936$		0			0
$937$	−0.890899			−0.0291044			−0.0145522	−	0.999894i	$-0.504632\pi$
$937$	−0.890899			−0.0291044			−0.0145522	+	0.999894i	$0.504632\pi$
$938$		0			0
$939$	12.1692			0.397127
$940$		0			0
$941$		−	29.7148i		−	0.968674i	−0.874882	−	0.484337i	$-0.839061\pi$
$941$		−	29.7148i		−	0.968674i	0.874882	−	0.484337i	$-0.160939\pi$
$942$		0			0
$943$	−8.56502			−0.278915
$944$		0			0
$945$			2.60808i			0.0848409i
$946$		0			0
$947$		−	41.5315i		−	1.34959i	−0.738005	−	0.674796i	$-0.764232\pi$
$947$		−	41.5315i		−	1.34959i	0.738005	−	0.674796i	$-0.235768\pi$
$948$		0			0
$949$		−	27.5544i		−	0.894455i
$950$		0			0
$951$		−	10.9257i		−	0.354290i
$952$		0			0
$953$			47.9258i			1.55247i	0.630445	+	0.776234i	$0.282872\pi$
$953$			47.9258i			1.55247i	−0.630445	+	0.776234i	$0.717128\pi$
$954$		0			0
$955$			1.00951i			0.0326670i
$956$		0			0
$957$	3.04964			0.0985808
$958$		0			0
$959$		−	22.3383i		−	0.721341i
$960$		0			0
$961$	35.5650			1.14726
$962$		0			0
$963$	−9.75246			−0.314269
$964$		0			0
$965$		−	26.9654i		−	0.868046i
$966$		0			0
$967$			23.1772i			0.745330i	0.927966	+	0.372665i	$0.121556\pi$
$967$			23.1772i			0.745330i	−0.927966	+	0.372665i	$0.878444\pi$
$968$		0			0
$969$	0.158739	−	3.49264i	0.00509943	−	0.112200i
$970$		0			0
$971$	−44.3384			−1.42289			−0.711444	−	0.702743i	$-0.751958\pi$
$971$	−44.3384			−1.42289			−0.711444	+	0.702743i	$0.751958\pi$
$972$		0			0
$973$	−36.3175			−1.16428
$974$		0			0
$975$		−	2.60808i		−	0.0835255i
$976$		0			0
$977$		−	3.69050i		−	0.118070i	−0.998256	−	0.0590348i	$-0.981198\pi$
$977$		−	3.69050i		−	0.118070i	0.998256	−	0.0590348i	$-0.0188023\pi$
$978$		0			0
$979$	21.9112			0.700285
$980$		0			0
$981$		−	2.75583i		−	0.0879869i
$982$		0			0
$983$	−8.08881			−0.257993			−0.128996	−	0.991645i	$-0.541176\pi$
$983$	−8.08881			−0.257993			−0.128996	+	0.991645i	$0.541176\pi$
$984$		0			0
$985$	−11.1979			−0.356795
$986$		0			0
$987$	−25.9217			−0.825096
$988$		0			0
$989$	−3.66158			−0.116431
$990$		0			0
$991$	27.6042			0.876876			0.438438	−	0.898761i	$-0.355532\pi$
$991$	27.6042			0.876876			0.438438	+	0.898761i	$0.355532\pi$
$992$		0			0
$993$	−7.59372			−0.240979
$994$		0			0
$995$		−	5.95298i		−	0.188722i
$996$		0			0
$997$	−33.2783			−1.05393			−0.526967	−	0.849886i	$-0.676671\pi$
$997$	−33.2783			−1.05393			−0.526967	+	0.849886i	$0.676671\pi$
$998$		0			0
$999$		−	2.97649i		−	0.0941719i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.d.e.2431.5	yes	6
4.3	odd	2		4560.2.d.g.2431.2	yes	6
19.18	odd	2		4560.2.d.g.2431.5	yes	6
76.75	even	2	inner	4560.2.d.e.2431.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4560.2.d.e.2431.2	✓	6	76.75	even	2	inner
4560.2.d.e.2431.5	yes	6	1.1	even	1	trivial
4560.2.d.g.2431.2	yes	6	4.3	odd	2
4560.2.d.g.2431.5	yes	6	19.18	odd	2

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:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +2.60808i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +2.60808i q^{7} +1.00000 q^{9} -1.74632i q^{11} +2.60808i q^{13} +1.00000 q^{15} -0.802094 q^{17} +(0.197906 - 4.35440i) q^{19} -2.60808i q^{21} +1.23017i q^{23} +1.00000 q^{25} -1.00000 q^{27} -1.74632i q^{29} +8.15874 q^{31} +1.74632i q^{33} -2.60808i q^{35} +2.97649i q^{37} -2.60808i q^{39} +6.96249i q^{41} +2.97649i q^{43} -1.00000 q^{45} -9.93897i q^{47} +0.197906 q^{49} +0.802094 q^{51} -1.23017i q^{53} +1.74632i q^{55} +(-0.197906 + 4.35440i) q^{57} +1.60419 q^{59} +10.1587 q^{61} +2.60808i q^{63} -2.60808i q^{65} -9.35664 q^{67} -1.23017i q^{69} +1.60419 q^{71} -10.5650 q^{73} -1.00000 q^{75} +4.55455 q^{77} +16.9608 q^{79} +1.00000 q^{81} -4.72281i q^{83} +0.802094 q^{85} +1.74632i q^{87} +12.5471i q^{89} -6.80209 q^{91} -8.15874 q^{93} +(-0.197906 + 4.35440i) q^{95} +8.85656i q^{97} -1.74632i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} - 6 q^{5} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 - 6 * q^5 + 6 * q^9	\( 6 q - 6 q^{3} - 6 q^{5} + 6 q^{9} + 6 q^{15} - 4 q^{17} + 2 q^{19} + 6 q^{25} - 6 q^{27} - 12 q^{31} - 6 q^{45} + 2 q^{49} + 4 q^{51} - 2 q^{57} + 8 q^{59} + 4 q^{67} + 8 q^{71} - 6 q^{75} - 32 q^{77} + 40 q^{79} + 6 q^{81} + 4 q^{85} - 40 q^{91} + 12 q^{93} - 2 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 - 6 * q^5 + 6 * q^9 + 6 * q^15 - 4 * q^17 + 2 * q^19 + 6 * q^25 - 6 * q^27 - 12 * q^31 - 6 * q^45 + 2 * q^49 + 4 * q^51 - 2 * q^57 + 8 * q^59 + 4 * q^67 + 8 * q^71 - 6 * q^75 - 32 * q^77 + 40 * q^79 + 6 * q^81 + 4 * q^85 - 40 * q^91 + 12 * q^93 - 2 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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