LMFDB - Embedded newform 6012.2.h.a.3005.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(3005,6012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6012.3005");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.h (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:	$48.0060616952$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3005.4
Character	$\chi$	$=$	6012.3005
Dual form			6012.2.h.a.3005.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.84494 q^{5} -2.14597 q^{7} +O(q^{10})$	$q-3.84494 q^{5} -2.14597 q^{7} -0.389573i q^{11} -5.51059i q^{13} -6.90150 q^{17} -6.97103 q^{19} -4.97768 q^{23} +9.78356 q^{25} -7.02613i q^{29} +0.566791 q^{31} +8.25113 q^{35} +7.77849i q^{37} -8.87875 q^{41} -3.56851i q^{43} -12.8085i q^{47} -2.39480 q^{49} -1.41777 q^{53} +1.49789i q^{55} -7.04798 q^{59} +2.28271 q^{61} +21.1879i q^{65} -12.4511i q^{67} -2.35431 q^{71} -13.7908i q^{73} +0.836013i q^{77} +6.60434i q^{79} +10.0867 q^{83} +26.5359 q^{85} +13.1160i q^{89} +11.8256i q^{91} +26.8032 q^{95} -13.0804 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q+O(q^{10}) $ 56 * q	$ 56 q + 8 q^{19} + 64 q^{25} - 8 q^{31} + 56 q^{49} - 8 q^{61} + 32 q^{85} - 48 q^{97}+O(q^{100}) $ 56 * q + 8 * q^19 + 64 * q^25 - 8 * q^31 + 56 * q^49 - 8 * q^61 + 32 * q^85 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6012\mathbb{Z}\right)^\times$.

$n$	$3007$	$3341$	$4681$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−3.84494			−1.71951			−0.859755	−	0.510707i	$-0.829384\pi$
$5$	−3.84494			−1.71951			−0.859755	+	0.510707i	$0.829384\pi$
$6$		0			0
$7$	−2.14597			−0.811101			−0.405551	−	0.914073i	$-0.632920\pi$
$7$	−2.14597			−0.811101			−0.405551	+	0.914073i	$0.632920\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	0.389573i		−	0.117461i	−0.998274	−	0.0587304i	$-0.981295\pi$
$11$		−	0.389573i		−	0.117461i	0.998274	−	0.0587304i	$-0.0187052\pi$
$12$		0			0
$13$		−	5.51059i		−	1.52836i	−0.645002	−	0.764181i	$-0.723143\pi$
$13$		−	5.51059i		−	1.52836i	0.645002	−	0.764181i	$-0.276857\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.90150			−1.67386			−0.836930	−	0.547310i	$-0.815652\pi$
$17$	−6.90150			−1.67386			−0.836930	+	0.547310i	$0.815652\pi$
$18$		0			0
$19$	−6.97103			−1.59926			−0.799632	−	0.600490i	$-0.794972\pi$
$19$	−6.97103			−1.59926			−0.799632	+	0.600490i	$0.794972\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.97768			−1.03792			−0.518959	−	0.854799i	$-0.673680\pi$
$23$	−4.97768			−1.03792			−0.518959	+	0.854799i	$0.673680\pi$
$24$		0			0
$25$	9.78356			1.95671
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	7.02613i		−	1.30472i	−0.757910	−	0.652359i	$-0.773779\pi$
$29$		−	7.02613i		−	1.30472i	0.757910	−	0.652359i	$-0.226221\pi$
$30$		0			0
$31$	0.566791			0.101799			0.0508993	−	0.998704i	$-0.483791\pi$
$31$	0.566791			0.101799			0.0508993	+	0.998704i	$0.483791\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	8.25113			1.39470
$36$		0			0
$37$			7.77849i			1.27878i	0.768884	+	0.639388i	$0.220812\pi$
$37$			7.77849i			1.27878i	−0.768884	+	0.639388i	$0.779188\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.87875			−1.38663			−0.693314	−	0.720636i	$-0.743850\pi$
$41$	−8.87875			−1.38663			−0.693314	+	0.720636i	$0.743850\pi$
$42$		0			0
$43$		−	3.56851i		−	0.544193i	−0.962270	−	0.272096i	$-0.912283\pi$
$43$		−	3.56851i		−	0.544193i	0.962270	−	0.272096i	$-0.0877170\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	12.8085i		−	1.86831i	−0.356865	−	0.934156i	$-0.616155\pi$
$47$		−	12.8085i		−	1.86831i	0.356865	−	0.934156i	$-0.383845\pi$
$48$		0			0
$49$	−2.39480			−0.342115
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.41777			−0.194746			−0.0973728	−	0.995248i	$-0.531044\pi$
$53$	−1.41777			−0.194746			−0.0973728	+	0.995248i	$0.531044\pi$
$54$		0			0
$55$			1.49789i			0.201975i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.04798			−0.917569			−0.458785	−	0.888548i	$-0.651715\pi$
$59$	−7.04798			−0.917569			−0.458785	+	0.888548i	$0.651715\pi$
$60$		0			0
$61$	2.28271			0.292271			0.146136	−	0.989265i	$-0.453316\pi$
$61$	2.28271			0.292271			0.146136	+	0.989265i	$0.453316\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			21.1879i			2.62803i
$66$		0			0
$67$		−	12.4511i		−	1.52114i	−0.649257	−	0.760569i	$-0.724920\pi$
$67$		−	12.4511i		−	1.52114i	0.649257	−	0.760569i	$-0.275080\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.35431			−0.279405			−0.139703	−	0.990193i	$-0.544615\pi$
$71$	−2.35431			−0.279405			−0.139703	+	0.990193i	$0.544615\pi$
$72$		0			0
$73$		−	13.7908i		−	1.61409i	−0.590489	−	0.807045i	$-0.701065\pi$
$73$		−	13.7908i		−	1.61409i	0.590489	−	0.807045i	$-0.298935\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			0.836013i			0.0952725i
$78$		0			0
$79$			6.60434i			0.743046i	0.928424	+	0.371523i	$0.121164\pi$
$79$			6.60434i			0.743046i	−0.928424	+	0.371523i	$0.878836\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	10.0867			1.10716			0.553578	−	0.832797i	$-0.313262\pi$
$83$	10.0867			1.10716			0.553578	+	0.832797i	$0.313262\pi$
$84$		0			0
$85$	26.5359			2.87822
$86$		0			0
$87$		0			0
$88$		0			0
$89$			13.1160i			1.39030i	0.718867	+	0.695148i	$0.244661\pi$
$89$			13.1160i			1.39030i	−0.718867	+	0.695148i	$0.755339\pi$
$90$		0			0
$91$			11.8256i			1.23966i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	26.8032			2.74995
$96$		0			0
$97$	−13.0804			−1.32812			−0.664059	−	0.747680i	$-0.731168\pi$
$97$	−13.0804			−1.32812			−0.664059	+	0.747680i	$0.731168\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.2424			−1.01916			−0.509581	−	0.860423i	$-0.670199\pi$
$101$	−10.2424			−1.01916			−0.509581	+	0.860423i	$0.670199\pi$
$102$		0			0
$103$		−	4.81035i		−	0.473978i	−0.971512	−	0.236989i	$-0.923840\pi$
$103$		−	4.81035i		−	0.473978i	0.971512	−	0.236989i	$-0.0761605\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			17.4053i			1.68264i	0.540540	+	0.841318i	$0.318220\pi$
$107$			17.4053i			1.68264i	−0.540540	+	0.841318i	$0.681780\pi$
$108$		0			0
$109$		−	0.344407i		−	0.0329882i	−0.999864	−	0.0164941i	$-0.994750\pi$
$109$		−	0.344407i		−	0.0329882i	0.999864	−	0.0164941i	$-0.00525048\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.14643			−0.295992			−0.147996	−	0.988988i	$-0.547282\pi$
$113$	−3.14643			−0.295992			−0.147996	+	0.988988i	$0.547282\pi$
$114$		0			0
$115$	19.1389			1.78471
$116$		0			0
$117$		0			0
$118$		0			0
$119$	14.8104			1.35767
$120$		0			0
$121$	10.8482			0.986203
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−18.3925			−1.64508
$126$		0			0
$127$	6.31278			0.560168			0.280084	−	0.959975i	$-0.409638\pi$
$127$	6.31278			0.560168			0.280084	+	0.959975i	$0.409638\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−17.3867			−1.51909			−0.759544	−	0.650456i	$-0.774578\pi$
$131$	−17.3867			−1.51909			−0.759544	+	0.650456i	$0.774578\pi$
$132$		0			0
$133$	14.9596			1.29717
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	8.61256i		−	0.735821i	−0.929861	−	0.367911i	$-0.880073\pi$
$137$		−	8.61256i		−	0.735821i	0.929861	−	0.367911i	$-0.119927\pi$
$138$		0			0
$139$		−	13.3981i		−	1.13641i	−0.822887	−	0.568205i	$-0.807638\pi$
$139$		−	13.3981i		−	1.13641i	0.822887	−	0.568205i	$-0.192362\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.14678			−0.179523
$144$		0			0
$145$			27.0150i			2.24348i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−22.5320			−1.84590			−0.922948	−	0.384925i	$-0.874227\pi$
$149$	−22.5320			−1.84590			−0.922948	+	0.384925i	$0.874227\pi$
$150$		0			0
$151$			5.63828i			0.458837i	0.973328	+	0.229418i	$0.0736824\pi$
$151$			5.63828i			0.458837i	−0.973328	+	0.229418i	$0.926318\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.17928			−0.175044
$156$		0			0
$157$	21.8304			1.74225			0.871126	−	0.491059i	$-0.163390\pi$
$157$	21.8304			1.74225			0.871126	+	0.491059i	$0.163390\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	10.6820			0.841857
$162$		0			0
$163$		−	14.5361i		−	1.13856i	−0.822145	−	0.569278i	$-0.807223\pi$
$163$		−	14.5361i		−	1.13856i	0.822145	−	0.569278i	$-0.192777\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.86706	+	12.3307i	0.299242	+	0.954177i
$167$	3.86706	+	12.3307i	0.299242	+	0.954177i
$168$		0			0
$169$	−17.3666			−1.33589
$170$		0			0
$171$		0			0
$172$		0			0
$173$			4.41761i			0.335865i	0.985798	+	0.167932i	$0.0537091\pi$
$173$			4.41761i			0.335865i	−0.985798	+	0.167932i	$0.946291\pi$
$174$		0			0
$175$	−20.9953			−1.58709
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	5.15992i		−	0.385670i	−0.981231	−	0.192835i	$-0.938232\pi$
$179$		−	5.15992i		−	0.385670i	0.981231	−	0.192835i	$-0.0617683\pi$
$180$		0			0
$181$	−7.15190			−0.531597			−0.265798	−	0.964029i	$-0.585636\pi$
$181$	−7.15190			−0.531597			−0.265798	+	0.964029i	$0.585636\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	29.9078i		−	2.19887i
$186$		0			0
$187$			2.68864i			0.196613i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			10.4303i			0.754709i	0.926069	+	0.377354i	$0.123166\pi$
$191$			10.4303i			0.754709i	−0.926069	+	0.377354i	$0.876834\pi$
$192$		0			0
$193$			5.17362i			0.372405i	0.982511	+	0.186203i	$0.0596181\pi$
$193$			5.17362i			0.372405i	−0.982511	+	0.186203i	$0.940382\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	16.6148			1.18375			0.591877	−	0.806028i	$-0.298387\pi$
$197$	16.6148			1.18375			0.591877	+	0.806028i	$0.298387\pi$
$198$		0			0
$199$	−0.121544			−0.00861603			−0.00430802	−	0.999991i	$-0.501371\pi$
$199$	−0.121544			−0.00861603			−0.00430802	+	0.999991i	$0.501371\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			15.0779i			1.05826i
$204$		0			0
$205$	34.1383			2.38432
$206$		0			0
$207$		0			0
$208$		0			0
$209$			2.71573i			0.187851i
$210$		0			0
$211$	17.7050			1.21886			0.609431	−	0.792839i	$-0.291398\pi$
$211$	17.7050			1.21886			0.609431	+	0.792839i	$0.291398\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			13.7207i			0.935745i
$216$		0			0
$217$	−1.21632			−0.0825690
$218$		0			0
$219$		0			0
$220$		0			0
$221$			38.0314i			2.55827i
$222$		0			0
$223$	−5.69361			−0.381272			−0.190636	−	0.981661i	$-0.561055\pi$
$223$	−5.69361			−0.381272			−0.190636	+	0.981661i	$0.561055\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.7274			−0.712004			−0.356002	−	0.934485i	$-0.615860\pi$
$227$	−10.7274			−0.712004			−0.356002	+	0.934485i	$0.615860\pi$
$228$		0			0
$229$	−10.8467			−0.716773			−0.358386	−	0.933573i	$-0.616673\pi$
$229$	−10.8467			−0.716773			−0.358386	+	0.933573i	$0.616673\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			0.490036i			0.0321033i	0.999871	+	0.0160517i	$0.00510962\pi$
$233$			0.490036i			0.0321033i	−0.999871	+	0.0160517i	$0.994890\pi$
$234$		0			0
$235$			49.2479i			3.21258i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	15.7501i		−	1.01879i	−0.860532	−	0.509396i	$-0.829869\pi$
$239$		−	15.7501i		−	1.01879i	0.860532	−	0.509396i	$-0.170131\pi$
$240$		0			0
$241$		−	5.55929i		−	0.358105i	−0.983839	−	0.179053i	$-0.942697\pi$
$241$		−	5.55929i		−	0.358105i	0.983839	−	0.179053i	$-0.0573032\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	9.20788			0.588270
$246$		0			0
$247$			38.4145i			2.44426i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	15.0853i		−	0.952176i	−0.879398	−	0.476088i	$-0.842054\pi$
$251$		−	15.0853i		−	0.952176i	0.879398	−	0.476088i	$-0.157946\pi$
$252$		0			0
$253$			1.93917i			0.121915i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	8.73533			0.544895			0.272447	−	0.962171i	$-0.412167\pi$
$257$	8.73533			0.544895			0.272447	+	0.962171i	$0.412167\pi$
$258$		0			0
$259$		−	16.6924i		−	1.03722i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			28.7921i			1.77540i	0.460423	+	0.887700i	$0.347698\pi$
$263$			28.7921i			1.77540i	−0.460423	+	0.887700i	$0.652302\pi$
$264$		0			0
$265$	5.45124			0.334867
$266$		0			0
$267$		0			0
$268$		0			0
$269$	21.4007			1.30483			0.652413	−	0.757864i	$-0.273757\pi$
$269$	21.4007			1.30483			0.652413	+	0.757864i	$0.273757\pi$
$270$		0			0
$271$			18.7899i			1.14141i	0.821156	+	0.570704i	$0.193330\pi$
$271$			18.7899i			1.14141i	−0.821156	+	0.570704i	$0.806670\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	3.81141i		−	0.229837i
$276$		0			0
$277$			16.7280i			1.00509i	0.864551	+	0.502545i	$0.167603\pi$
$277$			16.7280i			1.00509i	−0.864551	+	0.502545i	$0.832397\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	28.6978i		−	1.71197i	−0.517005	−	0.855983i	$-0.672953\pi$
$281$		−	28.6978i		−	1.71197i	0.517005	−	0.855983i	$-0.327047\pi$
$282$		0			0
$283$	−4.08638			−0.242910			−0.121455	−	0.992597i	$-0.538756\pi$
$283$	−4.08638			−0.242910			−0.121455	+	0.992597i	$0.538756\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	19.0535			1.12470
$288$		0			0
$289$	30.6308			1.80181
$290$		0			0
$291$		0			0
$292$		0			0
$293$			11.1321i			0.650341i	0.945655	+	0.325171i	$0.105422\pi$
$293$			11.1321i			0.650341i	−0.945655	+	0.325171i	$0.894578\pi$
$294$		0			0
$295$	27.0991			1.57777
$296$		0			0
$297$		0			0
$298$		0			0
$299$			27.4300i			1.58632i
$300$		0			0
$301$			7.65793i			0.441396i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.77689			−0.502563
$306$		0			0
$307$			32.8026i			1.87214i	0.351811	+	0.936071i	$0.385566\pi$
$307$			32.8026i			1.87214i	−0.351811	+	0.936071i	$0.614434\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	3.24173i		−	0.183822i	−0.995767	−	0.0919109i	$-0.970702\pi$
$311$		−	3.24173i		−	0.183822i	0.995767	−	0.0919109i	$-0.0292975\pi$
$312$		0			0
$313$			21.5298i			1.21693i	0.793579	+	0.608467i	$0.208215\pi$
$313$			21.5298i			1.21693i	−0.793579	+	0.608467i	$0.791785\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			12.4153i			0.697312i	0.937251	+	0.348656i	$0.113362\pi$
$317$			12.4153i			0.697312i	−0.937251	+	0.348656i	$0.886638\pi$
$318$		0			0
$319$	−2.73719			−0.153253
$320$		0			0
$321$		0			0
$322$		0			0
$323$	48.1106			2.67695
$324$		0			0
$325$		−	53.9132i		−	2.99057i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			27.4867i			1.51539i
$330$		0			0
$331$			9.46960i			0.520496i	0.965542	+	0.260248i	$0.0838043\pi$
$331$			9.46960i			0.520496i	−0.965542	+	0.260248i	$0.916196\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			47.8735i			2.61561i
$336$		0			0
$337$	−31.2526			−1.70244			−0.851219	−	0.524811i	$-0.824136\pi$
$337$	−31.2526			−1.70244			−0.851219	+	0.524811i	$0.824136\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	0.220806i		−	0.0119573i
$342$		0			0
$343$	20.1610			1.08859
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−13.8933			−0.745833			−0.372916	−	0.927865i	$-0.621642\pi$
$347$	−13.8933			−0.745833			−0.372916	+	0.927865i	$0.621642\pi$
$348$		0			0
$349$		−	20.4127i		−	1.09267i	−0.837568	−	0.546333i	$-0.816023\pi$
$349$		−	20.4127i		−	1.09267i	0.837568	−	0.546333i	$-0.183977\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			1.97509i			0.105123i	0.998618	+	0.0525617i	$0.0167386\pi$
$353$			1.97509i			0.105123i	−0.998618	+	0.0525617i	$0.983261\pi$
$354$		0			0
$355$	9.05219			0.480440
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	6.84437i		−	0.361232i	−0.983554	−	0.180616i	$-0.942191\pi$
$359$		−	6.84437i		−	0.361232i	0.983554	−	0.180616i	$-0.0578092\pi$
$360$		0			0
$361$	29.5953			1.55765
$362$		0			0
$363$		0			0
$364$		0			0
$365$			53.0248i			2.77544i
$366$		0			0
$367$	−28.5647			−1.49107			−0.745533	−	0.666469i	$-0.767805\pi$
$367$	−28.5647			−1.49107			−0.745533	+	0.666469i	$0.767805\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.04249			0.157958
$372$		0			0
$373$			16.3680i			0.847501i	0.905779	+	0.423750i	$0.139287\pi$
$373$			16.3680i			0.847501i	−0.905779	+	0.423750i	$0.860713\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−38.7181			−1.99408
$378$		0			0
$379$			12.4090i			0.637408i	0.947854	+	0.318704i	$0.103248\pi$
$379$			12.4090i			0.637408i	−0.947854	+	0.318704i	$0.896752\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	26.7233i		−	1.36550i	−0.730653	−	0.682749i	$-0.760784\pi$
$383$		−	26.7233i		−	1.36550i	0.730653	−	0.682749i	$-0.239216\pi$
$384$		0			0
$385$		−	3.21442i		−	0.163822i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.794694			0.0402926			0.0201463	−	0.999797i	$-0.493587\pi$
$389$	0.794694			0.0402926			0.0201463	+	0.999797i	$0.493587\pi$
$390$		0			0
$391$	34.3535			1.73733
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	25.3933i		−	1.27767i
$396$		0			0
$397$	28.4180			1.42626			0.713130	−	0.701032i	$-0.247277\pi$
$397$	28.4180			1.42626			0.713130	+	0.701032i	$0.247277\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.657033			0.0328107			0.0164053	−	0.999865i	$-0.494778\pi$
$401$	0.657033			0.0328107			0.0164053	+	0.999865i	$0.494778\pi$
$402$		0			0
$403$		−	3.12335i		−	0.155585i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.03029			0.150206
$408$		0			0
$409$	−11.1429			−0.550981			−0.275491	−	0.961304i	$-0.588840\pi$
$409$	−11.1429			−0.550981			−0.275491	+	0.961304i	$0.588840\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	15.1248			0.744241
$414$		0			0
$415$	−38.7827			−1.90377
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	4.61669i		−	0.225540i	−0.993621	−	0.112770i	$-0.964028\pi$
$419$		−	4.61669i		−	0.225540i	0.993621	−	0.112770i	$-0.0359724\pi$
$420$		0			0
$421$	11.3568			0.553497			0.276749	−	0.960942i	$-0.410743\pi$
$421$	11.3568			0.553497			0.276749	+	0.960942i	$0.410743\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−67.5213			−3.27526
$426$		0			0
$427$	−4.89864			−0.237062
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	17.6836i		−	0.851789i	−0.904773	−	0.425895i	$-0.859959\pi$
$431$		−	17.6836i		−	0.851789i	0.904773	−	0.425895i	$-0.140041\pi$
$432$		0			0
$433$	−14.9960			−0.720660			−0.360330	−	0.932825i	$-0.617336\pi$
$433$	−14.9960			−0.720660			−0.360330	+	0.932825i	$0.617336\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	34.6996			1.65991
$438$		0			0
$439$		−	24.5369i		−	1.17108i	−0.810642	−	0.585542i	$-0.800882\pi$
$439$		−	24.5369i		−	1.17108i	0.810642	−	0.585542i	$-0.199118\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.39883			0.113972			0.0569858	−	0.998375i	$-0.481851\pi$
$443$	2.39883			0.113972			0.0569858	+	0.998375i	$0.481851\pi$
$444$		0			0
$445$		−	50.4303i		−	2.39063i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	11.2497i		−	0.530906i	−0.964124	−	0.265453i	$-0.914478\pi$
$449$		−	11.2497i		−	0.530906i	0.964124	−	0.265453i	$-0.0855215\pi$
$450$		0			0
$451$			3.45892i			0.162874i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	45.4686i		−	2.13160i
$456$		0			0
$457$		−	30.2665i		−	1.41581i	−0.706309	−	0.707903i	$-0.749641\pi$
$457$		−	30.2665i		−	1.41581i	0.706309	−	0.707903i	$-0.250359\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	3.77808i		−	0.175963i	−0.996122	−	0.0879814i	$-0.971958\pi$
$461$		−	3.77808i		−	0.175963i	0.996122	−	0.0879814i	$-0.0280416\pi$
$462$		0			0
$463$		−	40.9705i		−	1.90406i	−0.306005	−	0.952030i	$-0.598992\pi$
$463$		−	40.9705i		−	1.90406i	0.306005	−	0.952030i	$-0.401008\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	36.7738i		−	1.70169i	−0.525417	−	0.850845i	$-0.676091\pi$
$467$		−	36.7738i		−	1.70169i	0.525417	−	0.850845i	$-0.323909\pi$
$468$		0			0
$469$			26.7196i			1.23380i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.39020			−0.0639213
$474$		0			0
$475$	−68.2016			−3.12930
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−34.4612			−1.57457			−0.787285	−	0.616589i	$-0.788514\pi$
$479$	−34.4612			−1.57457			−0.787285	+	0.616589i	$0.788514\pi$
$480$		0			0
$481$	42.8641			1.95443
$482$		0			0
$483$		0			0
$484$		0			0
$485$	50.2935			2.28371
$486$		0			0
$487$		−	17.7395i		−	0.803855i	−0.915672	−	0.401927i	$-0.868340\pi$
$487$		−	17.7395i		−	0.803855i	0.915672	−	0.401927i	$-0.131660\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	7.84701i		−	0.354130i	−0.984199	−	0.177065i	$-0.943340\pi$
$491$		−	7.84701i		−	0.354130i	0.984199	−	0.177065i	$-0.0566604\pi$
$492$		0			0
$493$			48.4908i			2.18392i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5.05229			0.226626
$498$		0			0
$499$		−	31.5691i		−	1.41323i	−0.707600	−	0.706613i	$-0.750222\pi$
$499$		−	31.5691i		−	1.41323i	0.707600	−	0.706613i	$-0.249778\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	12.4593i		−	0.555532i	−0.960649	−	0.277766i	$-0.910406\pi$
$503$		−	12.4593i		−	0.555532i	0.960649	−	0.277766i	$-0.0895939\pi$
$504$		0			0
$505$	39.3816			1.75246
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	39.7294i		−	1.76097i	−0.474071	−	0.880486i	$-0.657216\pi$
$509$		−	39.7294i		−	1.76097i	0.474071	−	0.880486i	$-0.342784\pi$
$510$		0			0
$511$			29.5947i			1.30919i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			18.4955i			0.815009i
$516$		0			0
$517$	−4.98985			−0.219453
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12.8198			0.561648			0.280824	−	0.959759i	$-0.409392\pi$
$521$	12.8198			0.561648			0.280824	+	0.959759i	$0.409392\pi$
$522$		0			0
$523$	43.6321			1.90790			0.953950	−	0.299965i	$-0.0969749\pi$
$523$	43.6321			1.90790			0.953950	+	0.299965i	$0.0969749\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.91171			−0.170397
$528$		0			0
$529$	1.77731			0.0772744
$530$		0			0
$531$		0			0
$532$		0			0
$533$			48.9272i			2.11927i
$534$		0			0
$535$		−	66.9224i		−	2.89331i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			0.932952i			0.0401851i
$540$		0			0
$541$			4.44318i			0.191027i	0.995428	+	0.0955135i	$0.0304493\pi$
$541$			4.44318i			0.191027i	−0.995428	+	0.0955135i	$0.969551\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			1.32423i			0.0567236i
$546$		0			0
$547$			0.608706i			0.0260264i	0.999915	+	0.0130132i	$0.00414234\pi$
$547$			0.608706i			0.0260264i	−0.999915	+	0.0130132i	$0.995858\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			48.9794i			2.08659i
$552$		0			0
$553$		−	14.1727i		−	0.602685i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			24.0939i			1.02089i	0.859910	+	0.510446i	$0.170519\pi$
$557$			24.0939i			1.02089i	−0.859910	+	0.510446i	$0.829481\pi$
$558$		0			0
$559$	−19.6646			−0.831724
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	37.0205i		−	1.56023i	−0.625638	−	0.780114i	$-0.715161\pi$
$563$		−	37.0205i		−	1.56023i	0.625638	−	0.780114i	$-0.284839\pi$
$564$		0			0
$565$	12.0979			0.508960
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−12.6770			−0.531447			−0.265724	−	0.964049i	$-0.585611\pi$
$569$	−12.6770			−0.531447			−0.265724	+	0.964049i	$0.585611\pi$
$570$		0			0
$571$		−	36.6157i		−	1.53232i	−0.642649	−	0.766160i	$-0.722165\pi$
$571$		−	36.6157i		−	1.53232i	0.642649	−	0.766160i	$-0.277835\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−48.6995			−2.03091
$576$		0			0
$577$	−35.0676			−1.45988			−0.729942	−	0.683509i	$-0.760453\pi$
$577$	−35.0676			−1.45988			−0.729942	+	0.683509i	$0.760453\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−21.6457			−0.898016
$582$		0			0
$583$			0.552325i			0.0228750i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	35.5617			1.46779			0.733894	−	0.679264i	$-0.237701\pi$
$587$	35.5617			1.46779			0.733894	+	0.679264i	$0.237701\pi$
$588$		0			0
$589$	−3.95112			−0.162803
$590$		0			0
$591$		0			0
$592$		0			0
$593$	8.86416			0.364007			0.182004	−	0.983298i	$-0.441742\pi$
$593$	8.86416			0.364007			0.182004	+	0.983298i	$0.441742\pi$
$594$		0			0
$595$	−56.9452			−2.33453
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	22.9526i		−	0.937816i	−0.883247	−	0.468908i	$-0.844648\pi$
$599$		−	22.9526i		−	0.937816i	0.883247	−	0.468908i	$-0.155352\pi$
$600$		0			0
$601$	−40.5175			−1.65275			−0.826373	−	0.563124i	$-0.809599\pi$
$601$	−40.5175			−1.65275			−0.826373	+	0.563124i	$0.809599\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−41.7108			−1.69579
$606$		0			0
$607$			39.3296i			1.59634i	0.602434	+	0.798169i	$0.294198\pi$
$607$			39.3296i			1.59634i	−0.602434	+	0.798169i	$0.705802\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−70.5824			−2.85546
$612$		0			0
$613$	−1.06987			−0.0432116			−0.0216058	−	0.999767i	$-0.506878\pi$
$613$	−1.06987			−0.0432116			−0.0216058	+	0.999767i	$0.506878\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			44.8837i			1.80695i	0.428640	+	0.903476i	$0.358993\pi$
$617$			44.8837i			1.80695i	−0.428640	+	0.903476i	$0.641007\pi$
$618$		0			0
$619$			27.7345i			1.11475i	0.830262	+	0.557373i	$0.188191\pi$
$619$			27.7345i			1.11475i	−0.830262	+	0.557373i	$0.811809\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	28.1466i		−	1.12767i
$624$		0			0
$625$	21.8003			0.872012
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	53.6833i		−	2.14049i
$630$		0			0
$631$	−24.3534			−0.969495			−0.484747	−	0.874654i	$-0.661088\pi$
$631$	−24.3534			−0.969495			−0.484747	+	0.874654i	$0.661088\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−24.2722			−0.963215
$636$		0			0
$637$			13.1968i			0.522876i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	25.8492			1.02098			0.510492	−	0.859883i	$-0.329463\pi$
$641$	25.8492			1.02098			0.510492	+	0.859883i	$0.329463\pi$
$642$		0			0
$643$		−	7.67452i		−	0.302654i	−0.988484	−	0.151327i	$-0.951645\pi$
$643$		−	7.67452i		−	0.302654i	0.988484	−	0.151327i	$-0.0483546\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−27.2381			−1.07084			−0.535420	−	0.844586i	$-0.679847\pi$
$647$	−27.2381			−1.07084			−0.535420	+	0.844586i	$0.679847\pi$
$648$		0			0
$649$			2.74571i			0.107778i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			11.5414i			0.451650i	0.974168	+	0.225825i	$0.0725077\pi$
$653$			11.5414i			0.451650i	−0.974168	+	0.225825i	$0.927492\pi$
$654$		0			0
$655$	66.8510			2.61208
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.76163			0.380259			0.190130	−	0.981759i	$-0.439109\pi$
$659$	9.76163			0.380259			0.190130	+	0.981759i	$0.439109\pi$
$660$		0			0
$661$			1.42557i			0.0554482i	0.999616	+	0.0277241i	$0.00882598\pi$
$661$			1.42557i			0.0554482i	−0.999616	+	0.0277241i	$0.991174\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−57.5189			−2.23049
$666$		0			0
$667$			34.9738i			1.35419i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	0.889284i		−	0.0343304i
$672$		0			0
$673$			15.6409i			0.602911i	0.953480	+	0.301456i	$0.0974725\pi$
$673$			15.6409i			0.602911i	−0.953480	+	0.301456i	$0.902527\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			32.2251i			1.23851i	0.785190	+	0.619255i	$0.212565\pi$
$677$			32.2251i			1.23851i	−0.785190	+	0.619255i	$0.787435\pi$
$678$		0			0
$679$	28.0703			1.07724
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−19.8122			−0.758093			−0.379047	−	0.925378i	$-0.623748\pi$
$683$	−19.8122			−0.758093			−0.379047	+	0.925378i	$0.623748\pi$
$684$		0			0
$685$			33.1148i			1.26525i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			7.81275i			0.297642i
$690$		0			0
$691$		−	1.57096i		−	0.0597621i	−0.999553	−	0.0298811i	$-0.990487\pi$
$691$		−	1.57096i		−	0.0597621i	0.999553	−	0.0298811i	$-0.00951285\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			51.5148i			1.95407i
$696$		0			0
$697$	61.2767			2.32102
$698$		0			0
$699$		0			0
$700$		0			0
$701$			32.3493i			1.22182i	0.791701	+	0.610909i	$0.209196\pi$
$701$			32.3493i			1.22182i	−0.791701	+	0.610909i	$0.790804\pi$
$702$		0			0
$703$		−	54.2241i		−	2.04510i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	21.9800			0.826643
$708$		0			0
$709$			0.370820i			0.0139264i	0.999976	+	0.00696322i	$0.00221648\pi$
$709$			0.370820i			0.0139264i	−0.999976	+	0.00696322i	$0.997784\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.82130			−0.105659
$714$		0			0
$715$	8.25423			0.308691
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−7.74411			−0.288806			−0.144403	−	0.989519i	$-0.546126\pi$
$719$	−7.74411			−0.288806			−0.144403	+	0.989519i	$0.546126\pi$
$720$		0			0
$721$			10.3229i			0.384444i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	68.7406i		−	2.55296i
$726$		0			0
$727$			10.4401i			0.387203i	0.981080	+	0.193602i	$0.0620169\pi$
$727$			10.4401i			0.387203i	−0.981080	+	0.193602i	$0.937983\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			24.6281i			0.910903i
$732$		0			0
$733$	−12.6827			−0.468445			−0.234222	−	0.972183i	$-0.575254\pi$
$733$	−12.6827			−0.468445			−0.234222	+	0.972183i	$0.575254\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.85060			−0.178674
$738$		0			0
$739$		−	28.8453i		−	1.06109i	−0.847656	−	0.530547i	$-0.821987\pi$
$739$		−	28.8453i		−	1.06109i	0.847656	−	0.530547i	$-0.178013\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	19.3349i		−	0.709329i	−0.934994	−	0.354665i	$-0.884595\pi$
$743$		−	19.3349i		−	0.709329i	0.934994	−	0.354665i	$-0.115405\pi$
$744$		0			0
$745$	86.6343			3.17403
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	37.3513i		−	1.36479i
$750$		0			0
$751$		−	14.2753i		−	0.520913i	−0.965486	−	0.260457i	$-0.916127\pi$
$751$		−	14.2753i		−	0.520913i	0.965486	−	0.260457i	$-0.0838731\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	21.6789i		−	0.788974i
$756$		0			0
$757$	9.35269			0.339929			0.169965	−	0.985450i	$-0.445635\pi$
$757$	9.35269			0.339929			0.169965	+	0.985450i	$0.445635\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	6.70767i		−	0.243153i	−0.992582	−	0.121577i	$-0.961205\pi$
$761$		−	6.70767i		−	0.243153i	0.992582	−	0.121577i	$-0.0387950\pi$
$762$		0			0
$763$			0.739088i			0.0267568i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			38.8385i			1.40238i
$768$		0			0
$769$		−	8.17188i		−	0.294686i	−0.989085	−	0.147343i	$-0.952928\pi$
$769$		−	8.17188i		−	0.294686i	0.989085	−	0.147343i	$-0.0470721\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.64691			0.239073			0.119536	−	0.992830i	$-0.461859\pi$
$773$	6.64691			0.239073			0.119536	+	0.992830i	$0.461859\pi$
$774$		0			0
$775$	5.54523			0.199191
$776$		0			0
$777$		0			0
$778$		0			0
$779$	61.8941			2.21759
$780$		0			0
$781$			0.917177i			0.0328192i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−83.9364			−2.99582
$786$		0			0
$787$		−	14.9964i		−	0.534563i	−0.963618	−	0.267282i	$-0.913875\pi$
$787$		−	14.9964i		−	0.534563i	0.963618	−	0.267282i	$-0.0861254\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6.75216			0.240079
$792$		0			0
$793$		−	12.5791i		−	0.446697i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−13.1470			−0.465690			−0.232845	−	0.972514i	$-0.574803\pi$
$797$	−13.1470			−0.465690			−0.232845	+	0.972514i	$0.574803\pi$
$798$		0			0
$799$			88.3979i			3.12729i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−5.37253			−0.189592
$804$		0			0
$805$	−41.0715			−1.44758
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	16.4313i		−	0.577695i	−0.957375	−	0.288847i	$-0.906728\pi$
$809$		−	16.4313i		−	0.577695i	0.957375	−	0.288847i	$-0.0932720\pi$
$810$		0			0
$811$		−	1.49533i		−	0.0525083i	−0.999655	−	0.0262541i	$-0.991642\pi$
$811$		−	1.49533i		−	0.0525083i	0.999655	−	0.0262541i	$-0.00835791\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			55.8904i			1.95776i
$816$		0			0
$817$			24.8762i			0.870309i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	34.0593			1.18868			0.594339	−	0.804215i	$-0.297414\pi$
$821$	34.0593			1.18868			0.594339	+	0.804215i	$0.297414\pi$
$822$		0			0
$823$		−	16.5580i		−	0.577177i	−0.957453	−	0.288589i	$-0.906814\pi$
$823$		−	16.5580i		−	0.577177i	0.957453	−	0.288589i	$-0.0931860\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	10.2447			0.356244			0.178122	−	0.984008i	$-0.442998\pi$
$827$	10.2447			0.356244			0.178122	+	0.984008i	$0.442998\pi$
$828$		0			0
$829$		−	20.4080i		−	0.708799i	−0.935094	−	0.354400i	$-0.884685\pi$
$829$		−	20.4080i		−	0.708799i	0.935094	−	0.354400i	$-0.115315\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	16.5278			0.572653
$834$		0			0
$835$	−14.8686	−	47.4107i	−0.514550	−	1.64072i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	23.6286i		−	0.815749i	−0.913038	−	0.407875i	$-0.866270\pi$
$839$		−	23.6286i		−	0.815749i	0.913038	−	0.407875i	$-0.133730\pi$
$840$		0			0
$841$	−20.3664			−0.702291
$842$		0			0
$843$		0			0
$844$		0			0
$845$	66.7735			2.29708
$846$		0			0
$847$	−23.2800			−0.799910
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	38.7188i		−	1.32726i
$852$		0			0
$853$	−22.0584			−0.755266			−0.377633	−	0.925955i	$-0.623262\pi$
$853$	−22.0584			−0.755266			−0.377633	+	0.925955i	$0.623262\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	17.4917i		−	0.597507i	−0.954330	−	0.298753i	$-0.903429\pi$
$857$		−	17.4917i		−	0.597507i	0.954330	−	0.298753i	$-0.0965708\pi$
$858$		0			0
$859$	−31.3423			−1.06939			−0.534693	−	0.845047i	$-0.679573\pi$
$859$	−31.3423			−1.06939			−0.534693	+	0.845047i	$0.679573\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	36.9509i		−	1.25782i	−0.777477	−	0.628911i	$-0.783501\pi$
$863$		−	36.9509i		−	1.25782i	0.777477	−	0.628911i	$-0.216499\pi$
$864$		0			0
$865$		−	16.9855i		−	0.577523i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	2.57287			0.0872787
$870$		0			0
$871$	−68.6126			−2.32485
$872$		0			0
$873$		0			0
$874$		0			0
$875$	39.4698			1.33432
$876$		0			0
$877$	46.9347			1.58487			0.792436	−	0.609955i	$-0.208812\pi$
$877$	46.9347			1.58487			0.792436	+	0.609955i	$0.208812\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−36.3396			−1.22431			−0.612157	−	0.790736i	$-0.709698\pi$
$881$	−36.3396			−1.22431			−0.612157	+	0.790736i	$0.709698\pi$
$882$		0			0
$883$	−54.6825			−1.84021			−0.920106	−	0.391669i	$-0.871898\pi$
$883$	−54.6825			−1.84021			−0.920106	+	0.391669i	$0.871898\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	26.2189			0.880345			0.440172	−	0.897913i	$-0.354917\pi$
$887$	26.2189			0.880345			0.440172	+	0.897913i	$0.354917\pi$
$888$		0			0
$889$	−13.5470			−0.454353
$890$		0			0
$891$		0			0
$892$		0			0
$893$			89.2885i			2.98792i
$894$		0			0
$895$			19.8396i			0.663164i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	3.98234i		−	0.132819i
$900$		0			0
$901$	9.78474			0.325977
$902$		0			0
$903$		0			0
$904$		0			0
$905$	27.4986			0.914085
$906$		0			0
$907$	−13.4105			−0.445289			−0.222644	−	0.974900i	$-0.571469\pi$
$907$	−13.4105			−0.445289			−0.222644	+	0.974900i	$0.571469\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	31.2573i		−	1.03560i	−0.855501	−	0.517800i	$-0.826751\pi$
$911$		−	31.2573i		−	1.03560i	0.855501	−	0.517800i	$-0.173249\pi$
$912$		0			0
$913$		−	3.92950i		−	0.130047i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	37.3115			1.23213
$918$		0			0
$919$	−7.47586			−0.246606			−0.123303	−	0.992369i	$-0.539349\pi$
$919$	−7.47586			−0.246606			−0.123303	+	0.992369i	$0.539349\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			12.9736i			0.427033i
$924$		0			0
$925$			76.1014i			2.50220i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	51.5518i		−	1.69136i	−0.533692	−	0.845679i	$-0.679196\pi$
$929$		−	51.5518i		−	1.69136i	0.533692	−	0.845679i	$-0.320804\pi$
$930$		0			0
$931$	16.6943			0.547132
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	10.3377i		−	0.338078i
$936$		0			0
$937$		−	57.8861i		−	1.89106i	−0.325540	−	0.945528i	$-0.605546\pi$
$937$		−	57.8861i		−	1.89106i	0.325540	−	0.945528i	$-0.394454\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	22.5630			0.735534			0.367767	−	0.929918i	$-0.380122\pi$
$941$	22.5630			0.735534			0.367767	+	0.929918i	$0.380122\pi$
$942$		0			0
$943$	44.1956			1.43921
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	12.3205i		−	0.400364i	−0.979759	−	0.200182i	$-0.935847\pi$
$947$		−	12.3205i		−	0.400364i	0.979759	−	0.200182i	$-0.0641533\pi$
$948$		0			0
$949$	−75.9954			−2.46692
$950$		0			0
$951$		0			0
$952$		0			0
$953$	18.5993			0.602490			0.301245	−	0.953547i	$-0.402598\pi$
$953$	18.5993			0.602490			0.301245	+	0.953547i	$0.402598\pi$
$954$		0			0
$955$		−	40.1038i		−	1.29773i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			18.4823i			0.596825i
$960$		0			0
$961$	−30.6787			−0.989637
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	19.8923i		−	0.640354i
$966$		0			0
$967$	−21.7935			−0.700831			−0.350415	−	0.936594i	$-0.613960\pi$
$967$	−21.7935			−0.700831			−0.350415	+	0.936594i	$0.613960\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.44236			0.0462876			0.0231438	−	0.999732i	$-0.492632\pi$
$971$	1.44236			0.0462876			0.0231438	+	0.999732i	$0.492632\pi$
$972$		0			0
$973$			28.7519i			0.921744i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	11.8828			0.380165			0.190083	−	0.981768i	$-0.439124\pi$
$977$	11.8828			0.380165			0.190083	+	0.981768i	$0.439124\pi$
$978$		0			0
$979$	5.10965			0.163305
$980$		0			0
$981$		0			0
$982$		0			0
$983$	25.8552			0.824651			0.412326	−	0.911037i	$-0.364717\pi$
$983$	25.8552			0.824651			0.412326	+	0.911037i	$0.364717\pi$
$984$		0			0
$985$	−63.8828			−2.03548
$986$		0			0
$987$		0			0
$988$		0			0
$989$			17.7629i			0.564828i
$990$		0			0
$991$			37.1094i			1.17882i	0.807835	+	0.589409i	$0.200639\pi$
$991$			37.1094i			1.17882i	−0.807835	+	0.589409i	$0.799361\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.467330			0.0148153
$996$		0			0
$997$	44.1981			1.39977			0.699884	−	0.714257i	$-0.253235\pi$
$997$	44.1981			1.39977			0.699884	+	0.714257i	$0.253235\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.h.a.3005.4	yes	56
3.2	odd	2	inner	6012.2.h.a.3005.53	yes	56
167.166	odd	2	inner	6012.2.h.a.3005.54	yes	56
501.500	even	2	inner	6012.2.h.a.3005.3	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6012.2.h.a.3005.3	✓	56	501.500	even	2	inner
6012.2.h.a.3005.4	yes	56	1.1	even	1	trivial
6012.2.h.a.3005.53	yes	56	3.2	odd	2	inner
6012.2.h.a.3005.54	yes	56	167.166	odd	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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.84494 q^{5} -2.14597 q^{7} +O(q^{10})\)	\(q-3.84494 q^{5} -2.14597 q^{7} -0.389573i q^{11} -5.51059i q^{13} -6.90150 q^{17} -6.97103 q^{19} -4.97768 q^{23} +9.78356 q^{25} -7.02613i q^{29} +0.566791 q^{31} +8.25113 q^{35} +7.77849i q^{37} -8.87875 q^{41} -3.56851i q^{43} -12.8085i q^{47} -2.39480 q^{49} -1.41777 q^{53} +1.49789i q^{55} -7.04798 q^{59} +2.28271 q^{61} +21.1879i q^{65} -12.4511i q^{67} -2.35431 q^{71} -13.7908i q^{73} +0.836013i q^{77} +6.60434i q^{79} +10.0867 q^{83} +26.5359 q^{85} +13.1160i q^{89} +11.8256i q^{91} +26.8032 q^{95} -13.0804 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q+O(q^{10}) \) 56 * q	\( 56 q + 8 q^{19} + 64 q^{25} - 8 q^{31} + 56 q^{49} - 8 q^{61} + 32 q^{85} - 48 q^{97}+O(q^{100}) \) 56 * q + 8 * q^19 + 64 * q^25 - 8 * q^31 + 56 * q^49 - 8 * q^61 + 32 * q^85 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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