LMFDB - Embedded newform 2352.2.k.i.881.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(881,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2352.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.k (of order $2$, degree $1$, not 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:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 65 x^{12} - 48 x^{11} - 94 x^{10} + 444 x^{9} - 962 x^{8} + \cdots + 6561 $ x^16 - 6x^15 + 19x^14 - 42x^13 + 65x^12 - 48x^11 - 94x^10 + 444x^9 - 962x^8 + 1332x^7 - 846x^6 - 1296x^5 + 5265x^4 - 10206x^3 + 13851x^2 - 13122*x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.13
Root			$-0.601642 + 1.62420i$ of defining polynomial
Character	$\chi$	$=$	2352.881
Dual form			2352.2.k.i.881.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.33314 - 1.10578i) q^{3} -0.145339 q^{5} +(0.554510 - 2.94831i) q^{9} +O(q^{10})$	$q+(1.33314 - 1.10578i) q^{3} -0.145339 q^{5} +(0.554510 - 2.94831i) q^{9} +2.46977i q^{11} +2.04143i q^{13} +(-0.193756 + 0.160712i) q^{15} -1.75684 q^{17} -4.25299i q^{19} -8.61812i q^{23} -4.97888 q^{25} +(-2.52094 - 4.54366i) q^{27} -7.08790i q^{29} -3.60050i q^{31} +(2.73102 + 3.29255i) q^{33} +5.86986 q^{37} +(2.25737 + 2.72151i) q^{39} -5.33255 q^{41} +9.19692 q^{43} +(-0.0805917 + 0.428503i) q^{45} +9.30380 q^{47} +(-2.34211 + 1.94267i) q^{51} -5.19128i q^{53} -0.358953i q^{55} +(-4.70286 - 5.66982i) q^{57} +11.2060 q^{59} -5.38498i q^{61} -0.296699i q^{65} +5.14835 q^{67} +(-9.52973 - 11.4891i) q^{69} -7.79323i q^{71} +13.0496i q^{73} +(-6.63753 + 5.50553i) q^{75} +5.72151 q^{79} +(-8.38504 - 3.26973i) q^{81} -15.9818 q^{83} +0.255336 q^{85} +(-7.83764 - 9.44914i) q^{87} -8.68504 q^{89} +(-3.98135 - 4.79996i) q^{93} +0.618123i q^{95} -6.65337i q^{97} +(7.28165 + 1.36951i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{9}+O(q^{10}) $ 16 * q - 4 * q^9	$ 16 q - 4 q^{9} - 8 q^{15} + 36 q^{25} + 4 q^{37} - 44 q^{39} - 20 q^{43} + 12 q^{51} - 8 q^{57} + 28 q^{67} + 56 q^{79} - 60 q^{81} + 16 q^{85} - 32 q^{93} - 20 q^{99}+O(q^{100}) $ 16 * q - 4 * q^9 - 8 * q^15 + 36 * q^25 + 4 * q^37 - 44 * q^39 - 20 * q^43 + 12 * q^51 - 8 * q^57 + 28 * q^67 + 56 * q^79 - 60 * q^81 + 16 * q^85 - 32 * q^93 - 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$-1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.33314	−	1.10578i	0.769687	−	0.638421i
$3$	1.33314	−	1.10578i	0.769687	−	0.638421i
$4$		0			0
$5$	−0.145339			−0.0649974			−0.0324987	−	0.999472i	$-0.510346\pi$
$5$	−0.145339			−0.0649974			−0.0324987	+	0.999472i	$0.510346\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	0.554510	−	2.94831i	0.184837	−	0.982769i
$10$		0			0
$11$			2.46977i			0.744665i	0.928099	+	0.372332i	$0.121442\pi$
$11$			2.46977i			0.744665i	−0.928099	+	0.372332i	$0.878558\pi$
$12$		0			0
$13$			2.04143i			0.566191i	0.959092	+	0.283096i	$0.0913613\pi$
$13$			2.04143i			0.566191i	−0.959092	+	0.283096i	$0.908639\pi$
$14$		0			0
$15$	−0.193756	+	0.160712i	−0.0500276	+	0.0414957i
$16$		0			0
$17$	−1.75684			−0.426096			−0.213048	−	0.977042i	$-0.568339\pi$
$17$	−1.75684			−0.426096			−0.213048	+	0.977042i	$0.568339\pi$
$18$		0			0
$19$		−	4.25299i		−	0.975702i	−0.872927	−	0.487851i	$-0.837781\pi$
$19$		−	4.25299i		−	0.975702i	0.872927	−	0.487851i	$-0.162219\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	8.61812i		−	1.79700i	−0.438971	−	0.898501i	$-0.644657\pi$
$23$		−	8.61812i		−	1.79700i	0.438971	−	0.898501i	$-0.355343\pi$
$24$		0			0
$25$	−4.97888			−0.995775
$26$		0			0
$27$	−2.52094	−	4.54366i	−0.485154	−	0.874429i
$28$		0			0
$29$		−	7.08790i		−	1.31619i	−0.752935	−	0.658095i	$-0.771363\pi$
$29$		−	7.08790i		−	1.31619i	0.752935	−	0.658095i	$-0.228637\pi$
$30$		0			0
$31$		−	3.60050i		−	0.646669i	−0.946285	−	0.323334i	$-0.895196\pi$
$31$		−	3.60050i		−	0.646669i	0.946285	−	0.323334i	$-0.104804\pi$
$32$		0			0
$33$	2.73102	+	3.29255i	0.475410	+	0.573159i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.86986			0.964999			0.482499	−	0.875896i	$-0.339729\pi$
$37$	5.86986			0.964999			0.482499	+	0.875896i	$0.339729\pi$
$38$		0			0
$39$	2.25737	+	2.72151i	0.361468	+	0.435790i
$40$		0			0
$41$	−5.33255			−0.832804			−0.416402	−	0.909181i	$-0.636709\pi$
$41$	−5.33255			−0.832804			−0.416402	+	0.909181i	$0.636709\pi$
$42$		0			0
$43$	9.19692			1.40252			0.701258	−	0.712907i	$-0.252622\pi$
$43$	9.19692			1.40252			0.701258	+	0.712907i	$0.252622\pi$
$44$		0			0
$45$	−0.0805917	+	0.428503i	−0.0120139	+	0.0638774i
$46$		0			0
$47$	9.30380			1.35710			0.678549	−	0.734555i	$-0.262609\pi$
$47$	9.30380			1.35710			0.678549	+	0.734555i	$0.262609\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−2.34211	+	1.94267i	−0.327960	+	0.272029i
$52$		0			0
$53$		−	5.19128i		−	0.713077i	−0.934281	−	0.356539i	$-0.883957\pi$
$53$		−	5.19128i		−	0.713077i	0.934281	−	0.356539i	$-0.116043\pi$
$54$		0			0
$55$		−	0.358953i		−	0.0484013i
$56$		0			0
$57$	−4.70286	−	5.66982i	−0.622909	−	0.750985i
$58$		0			0
$59$	11.2060			1.45889			0.729447	−	0.684037i	$-0.239777\pi$
$59$	11.2060			1.45889			0.729447	+	0.684037i	$0.239777\pi$
$60$		0			0
$61$		−	5.38498i		−	0.689476i	−0.938699	−	0.344738i	$-0.887968\pi$
$61$		−	5.38498i		−	0.689476i	0.938699	−	0.344738i	$-0.112032\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	0.296699i		−	0.0368009i
$66$		0			0
$67$	5.14835			0.628971			0.314485	−	0.949262i	$-0.398168\pi$
$67$	5.14835			0.628971			0.314485	+	0.949262i	$0.398168\pi$
$68$		0			0
$69$	−9.52973	−	11.4891i	−1.14724	−	1.38313i
$70$		0			0
$71$		−	7.79323i		−	0.924886i	−0.886649	−	0.462443i	$-0.846973\pi$
$71$		−	7.79323i		−	0.924886i	0.886649	−	0.462443i	$-0.153027\pi$
$72$		0			0
$73$			13.0496i			1.52734i	0.645604	+	0.763672i	$0.276606\pi$
$73$			13.0496i			1.52734i	−0.645604	+	0.763672i	$0.723394\pi$
$74$		0			0
$75$	−6.63753	+	5.50553i	−0.766435	+	0.635724i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	5.72151			0.643720			0.321860	−	0.946787i	$-0.395692\pi$
$79$	5.72151			0.643720			0.321860	+	0.946787i	$0.395692\pi$
$80$		0			0
$81$	−8.38504	−	3.26973i	−0.931671	−	0.363304i
$82$		0			0
$83$	−15.9818			−1.75423			−0.877115	−	0.480280i	$-0.840535\pi$
$83$	−15.9818			−1.75423			−0.877115	+	0.480280i	$0.840535\pi$
$84$		0			0
$85$	0.255336			0.0276951
$86$		0			0
$87$	−7.83764	−	9.44914i	−0.840283	−	1.01305i
$88$		0			0
$89$	−8.68504			−0.920612			−0.460306	−	0.887760i	$-0.652260\pi$
$89$	−8.68504			−0.920612			−0.460306	+	0.887760i	$0.652260\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−3.98135	−	4.79996i	−0.412847	−	0.497733i
$94$		0			0
$95$			0.618123i			0.0634181i
$96$		0			0
$97$		−	6.65337i		−	0.675547i	−0.941227	−	0.337774i	$-0.890326\pi$
$97$		−	6.65337i		−	0.675547i	0.941227	−	0.337774i	$-0.109674\pi$
$98$		0			0
$99$	7.28165	+	1.36951i	0.731834	+	0.137641i
$100$		0			0
$101$	−16.1271			−1.60471			−0.802355	−	0.596847i	$-0.796420\pi$
$101$	−16.1271			−1.60471			−0.802355	+	0.596847i	$0.796420\pi$
$102$		0			0
$103$		−	0.170125i		−	0.0167629i	−0.999965	−	0.00838147i	$-0.997332\pi$
$103$		−	0.170125i		−	0.0167629i	0.999965	−	0.00838147i	$-0.00266794\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	6.97324i		−	0.674129i	−0.941482	−	0.337064i	$-0.890566\pi$
$107$		−	6.97324i		−	0.674129i	0.941482	−	0.337064i	$-0.109434\pi$
$108$		0			0
$109$	1.35512			0.129797			0.0648984	−	0.997892i	$-0.479328\pi$
$109$	1.35512			0.129797			0.0648984	+	0.997892i	$0.479328\pi$
$110$		0			0
$111$	7.82532	−	6.49076i	0.742747	−	0.616076i
$112$		0			0
$113$			4.00000i			0.376288i	0.982141	+	0.188144i	$0.0602472\pi$
$113$			4.00000i			0.376288i	−0.982141	+	0.188144i	$0.939753\pi$
$114$		0			0
$115$			1.25255i			0.116800i
$116$		0			0
$117$	6.01877	+	1.13199i	0.556435	+	0.104653i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	4.90022			0.445474
$122$		0			0
$123$	−7.10902	+	5.89662i	−0.640999	+	0.531680i
$124$		0			0
$125$	1.45032			0.129720
$126$		0			0
$127$	7.33399			0.650787			0.325393	−	0.945579i	$-0.394503\pi$
$127$	7.33399			0.650787			0.325393	+	0.945579i	$0.394503\pi$
$128$		0			0
$129$	12.2608	−	10.1697i	1.07950	−	0.895396i
$130$		0			0
$131$	−6.09665			−0.532667			−0.266333	−	0.963881i	$-0.585812\pi$
$131$	−6.09665			−0.532667			−0.266333	+	0.963881i	$0.585812\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	0.366389	+	0.660370i	0.0315338	+	0.0568356i
$136$		0			0
$137$			20.0724i			1.71490i	0.514566	+	0.857451i	$0.327953\pi$
$137$			20.0724i			1.71490i	−0.514566	+	0.857451i	$0.672047\pi$
$138$		0			0
$139$		−	0.117694i		−	0.00998266i	−0.999988	−	0.00499133i	$-0.998411\pi$
$139$		−	0.117694i		−	0.00998266i	0.999988	−	0.00499133i	$-0.00158880\pi$
$140$		0			0
$141$	12.4032	−	10.2879i	1.04454	−	0.866401i
$142$		0			0
$143$	−5.04187			−0.421623
$144$		0			0
$145$			1.03014i			0.0855489i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			8.95707i			0.733792i	0.930262	+	0.366896i	$0.119579\pi$
$149$			8.95707i			0.733792i	−0.930262	+	0.366896i	$0.880421\pi$
$150$		0			0
$151$	2.74263			0.223192			0.111596	−	0.993754i	$-0.464404\pi$
$151$	2.74263			0.223192			0.111596	+	0.993754i	$0.464404\pi$
$152$		0			0
$153$	−0.974184	+	5.17970i	−0.0787581	+	0.418754i
$154$		0			0
$155$			0.523291i			0.0420318i
$156$		0			0
$157$		−	13.5450i		−	1.08101i	−0.841342	−	0.540504i	$-0.818234\pi$
$157$		−	13.5450i		−	1.08101i	0.841342	−	0.540504i	$-0.181766\pi$
$158$		0			0
$159$	−5.74040	−	6.92069i	−0.455244	−	0.548846i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	4.04857			0.317108			0.158554	−	0.987350i	$-0.449317\pi$
$163$	4.04857			0.317108			0.158554	+	0.987350i	$0.449317\pi$
$164$		0			0
$165$	−0.396923	−	0.478534i	−0.0309004	−	0.0372538i
$166$		0			0
$167$	3.70521			0.286717			0.143359	−	0.989671i	$-0.454210\pi$
$167$	3.70521			0.286717			0.143359	+	0.989671i	$0.454210\pi$
$168$		0			0
$169$	8.83256			0.679428
$170$		0			0
$171$	−12.5391	−	2.35832i	−0.958890	−	0.180346i
$172$		0			0
$173$	−22.4740			−1.70867			−0.854333	−	0.519726i	$-0.826034\pi$
$173$	−22.4740			−1.70867			−0.854333	+	0.519726i	$0.826034\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	14.9391	−	12.3913i	1.12289	−	0.931389i
$178$		0			0
$179$			3.67858i			0.274950i	0.990505	+	0.137475i	$0.0438986\pi$
$179$			3.67858i			0.274950i	−0.990505	+	0.137475i	$0.956101\pi$
$180$		0			0
$181$			8.01062i			0.595425i	0.954656	+	0.297712i	$0.0962237\pi$
$181$			8.01062i			0.595425i	−0.954656	+	0.297712i	$0.903776\pi$
$182$		0			0
$183$	−5.95459	−	7.17892i	−0.440176	−	0.530681i
$184$		0			0
$185$	−0.853117			−0.0627224
$186$		0			0
$187$		−	4.33899i		−	0.317298i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			1.12159i			0.0811554i	0.999176	+	0.0405777i	$0.0129198\pi$
$191$			1.12159i			0.0811554i	−0.999176	+	0.0405777i	$0.987080\pi$
$192$		0			0
$193$	18.3684			1.32219			0.661094	−	0.750303i	$-0.270092\pi$
$193$	18.3684			1.32219			0.661094	+	0.750303i	$0.270092\pi$
$194$		0			0
$195$	−0.328083	−	0.395540i	−0.0234945	−	0.0283252i
$196$		0			0
$197$		−	0.296699i		−	0.0211389i	−0.999944	−	0.0105695i	$-0.996636\pi$
$197$		−	0.296699i		−	0.0211389i	0.999944	−	0.0105695i	$-0.00336442\pi$
$198$		0			0
$199$			27.3518i			1.93892i	0.245251	+	0.969460i	$0.421130\pi$
$199$			27.3518i			1.93892i	−0.245251	+	0.969460i	$0.578870\pi$
$200$		0			0
$201$	6.86346	−	5.69293i	0.484111	−	0.401548i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	0.775025			0.0541301
$206$		0			0
$207$	−25.4089	−	4.77884i	−1.76604	−	0.332152i
$208$		0			0
$209$	10.5039			0.726571
$210$		0			0
$211$	−21.0295			−1.44773			−0.723864	−	0.689942i	$-0.757636\pi$
$211$	−21.0295			−1.44773			−0.723864	+	0.689942i	$0.757636\pi$
$212$		0			0
$213$	−8.61758	−	10.3894i	−0.590467	−	0.711873i
$214$		0			0
$215$	−1.33667			−0.0911599
$216$		0			0
$217$		0			0
$218$		0			0
$219$	14.4300	+	17.3970i	0.975089	+	1.17558i
$220$		0			0
$221$		−	3.58646i		−	0.241252i
$222$		0			0
$223$		−	6.89447i		−	0.461688i	−0.972991	−	0.230844i	$-0.925851\pi$
$223$		−	6.89447i		−	0.461688i	0.972991	−	0.230844i	$-0.0741487\pi$
$224$		0			0
$225$	−2.76084	+	14.6793i	−0.184056	+	0.978617i
$226$		0			0
$227$	−13.4147			−0.890363			−0.445182	−	0.895440i	$-0.646861\pi$
$227$	−13.4147			−0.890363			−0.445182	+	0.895440i	$0.646861\pi$
$228$		0			0
$229$			6.36254i			0.420448i	0.977653	+	0.210224i	$0.0674194\pi$
$229$			6.36254i			0.420448i	−0.977653	+	0.210224i	$0.932581\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			4.95707i			0.324748i	0.986729	+	0.162374i	$0.0519152\pi$
$233$			4.95707i			0.324748i	−0.986729	+	0.162374i	$0.948085\pi$
$234$		0			0
$235$	−1.35220			−0.0882079
$236$		0			0
$237$	7.62755	−	6.32672i	0.495463	−	0.410964i
$238$		0			0
$239$		−	17.3756i		−	1.12394i	−0.827159	−	0.561968i	$-0.810044\pi$
$239$		−	17.3756i		−	1.12394i	0.827159	−	0.561968i	$-0.189956\pi$
$240$		0			0
$241$			14.5060i			0.934416i	0.884147	+	0.467208i	$0.154740\pi$
$241$			14.5060i			0.934416i	−0.884147	+	0.467208i	$0.845260\pi$
$242$		0			0
$243$	−14.7940	+	4.91299i	−0.949036	+	0.315168i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	8.68218			0.552434
$248$		0			0
$249$	−21.3059	+	17.6723i	−1.35021	+	1.11994i
$250$		0			0
$251$	−3.49783			−0.220781			−0.110391	−	0.993888i	$-0.535210\pi$
$251$	−3.49783			−0.220781			−0.110391	+	0.993888i	$0.535210\pi$
$252$		0			0
$253$	21.2848			1.33816
$254$		0			0
$255$	0.340398	−	0.282345i	0.0213166	−	0.0176811i
$256$		0			0
$257$	15.9356			0.994036			0.497018	−	0.867740i	$-0.334428\pi$
$257$	15.9356			0.994036			0.497018	+	0.867740i	$0.334428\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−20.8973	−	3.93031i	−1.29351	−	0.243280i
$262$		0			0
$263$		−	14.3578i		−	0.885342i	−0.896684	−	0.442671i	$-0.854031\pi$
$263$		−	14.3578i		−	0.885342i	0.896684	−	0.442671i	$-0.145969\pi$
$264$		0			0
$265$			0.754493i			0.0463482i
$266$		0			0
$267$	−11.5784	+	9.60373i	−0.708584	+	0.587739i
$268$		0			0
$269$	7.36422			0.449004			0.224502	−	0.974474i	$-0.427924\pi$
$269$	7.36422			0.449004			0.224502	+	0.974474i	$0.427924\pi$
$270$		0			0
$271$			12.5327i			0.761309i	0.924717	+	0.380654i	$0.124301\pi$
$271$			12.5327i			0.761309i	−0.924717	+	0.380654i	$0.875699\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	12.2967i		−	0.741519i
$276$		0			0
$277$	32.4817			1.95164			0.975819	−	0.218580i	$-0.0701425\pi$
$277$	32.4817			1.95164			0.975819	+	0.218580i	$0.0701425\pi$
$278$		0			0
$279$	−10.6154	−	1.99651i	−0.635526	−	0.119528i
$280$		0			0
$281$			10.1758i			0.607037i	0.952826	+	0.303518i	$0.0981614\pi$
$281$			10.1758i			0.607037i	−0.952826	+	0.303518i	$0.901839\pi$
$282$		0			0
$283$			1.37024i			0.0814523i	0.999170	+	0.0407261i	$0.0129671\pi$
$283$			1.37024i			0.0814523i	−0.999170	+	0.0407261i	$0.987033\pi$
$284$		0			0
$285$	0.683507	+	0.824043i	0.0404874	+	0.0488121i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−13.9135			−0.818442
$290$		0			0
$291$	−7.35715	−	8.86986i	−0.431284	−	0.519960i
$292$		0			0
$293$	−16.9961			−0.992923			−0.496461	−	0.868059i	$-0.665368\pi$
$293$	−16.9961			−0.992923			−0.496461	+	0.868059i	$0.665368\pi$
$294$		0			0
$295$	−1.62866			−0.0948243
$296$		0			0
$297$	11.2218	−	6.22614i	0.651156	−	0.361277i
$298$		0			0
$299$	17.5933			1.01745
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−21.4997	+	17.8330i	−1.23513	+	1.02448i
$304$		0			0
$305$			0.782645i			0.0448141i
$306$		0			0
$307$			20.9023i			1.19296i	0.802629	+	0.596479i	$0.203434\pi$
$307$			20.9023i			1.19296i	−0.802629	+	0.596479i	$0.796566\pi$
$308$		0			0
$309$	−0.188121	−	0.226800i	−0.0107018	−	0.0129022i
$310$		0			0
$311$	11.4808			0.651017			0.325508	−	0.945539i	$-0.394465\pi$
$311$	11.4808			0.651017			0.325508	+	0.945539i	$0.394465\pi$
$312$		0			0
$313$		−	9.89777i		−	0.559455i	−0.960079	−	0.279728i	$-0.909756\pi$
$313$		−	9.89777i		−	0.559455i	0.960079	−	0.279728i	$-0.0902442\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			6.63430i			0.372619i	0.982491	+	0.186310i	$0.0596527\pi$
$317$			6.63430i			0.372619i	−0.982491	+	0.186310i	$0.940347\pi$
$318$		0			0
$319$	17.5055			0.980120
$320$		0			0
$321$	−7.71086	−	9.29629i	−0.430378	−	0.518868i
$322$		0			0
$323$			7.47181i			0.415742i
$324$		0			0
$325$		−	10.1640i		−	0.563799i
$326$		0			0
$327$	1.80656	−	1.49846i	0.0999029	−	0.0828650i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−14.7307			−0.809675			−0.404837	−	0.914389i	$-0.632672\pi$
$331$	−14.7307			−0.809675			−0.404837	+	0.914389i	$0.632672\pi$
$332$		0			0
$333$	3.25489	−	17.3061i	0.178367	−	0.948371i
$334$		0			0
$335$	−0.748254			−0.0408815
$336$		0			0
$337$	−30.7209			−1.67347			−0.836737	−	0.547605i	$-0.815540\pi$
$337$	−30.7209			−1.67347			−0.836737	+	0.547605i	$0.815540\pi$
$338$		0			0
$339$	4.42311	+	5.33255i	0.240230	+	0.289624i
$340$		0			0
$341$	8.89242			0.481551
$342$		0			0
$343$		0			0
$344$		0			0
$345$	1.38504	+	1.66982i	0.0745679	+	0.0898998i
$346$		0			0
$347$		−	16.7575i		−	0.899590i	−0.893132	−	0.449795i	$-0.851497\pi$
$347$		−	16.7575i		−	0.899590i	0.893132	−	0.449795i	$-0.148503\pi$
$348$		0			0
$349$		−	3.12385i		−	0.167216i	−0.996499	−	0.0836080i	$-0.973356\pi$
$349$		−	3.12385i		−	0.167216i	0.996499	−	0.0836080i	$-0.0266443\pi$
$350$		0			0
$351$	9.27558	−	5.14632i	0.495094	−	0.274690i
$352$		0			0
$353$	35.4901			1.88895			0.944473	−	0.328590i	$-0.106574\pi$
$353$	35.4901			1.88895			0.944473	+	0.328590i	$0.106574\pi$
$354$		0			0
$355$			1.13266i			0.0601152i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			6.26791i			0.330807i	0.986226	+	0.165404i	$0.0528927\pi$
$359$			6.26791i			0.330807i	−0.986226	+	0.165404i	$0.947107\pi$
$360$		0			0
$361$	0.912103			0.0480054
$362$		0			0
$363$	6.53266	−	5.41855i	0.342876	−	0.284400i
$364$		0			0
$365$		−	1.89662i		−	0.0992734i
$366$		0			0
$367$			16.8381i			0.878944i	0.898256	+	0.439472i	$0.144834\pi$
$367$			16.8381i			0.878944i	−0.898256	+	0.439472i	$0.855166\pi$
$368$		0			0
$369$	−2.95695	+	15.7220i	−0.153933	+	0.818454i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1.23421			−0.0639051			−0.0319526	−	0.999489i	$-0.510173\pi$
$373$	−1.23421			−0.0639051			−0.0319526	+	0.999489i	$0.510173\pi$
$374$		0			0
$375$	1.93347	−	1.60373i	0.0998439	−	0.0828161i
$376$		0			0
$377$	14.4695			0.745215
$378$		0			0
$379$	14.3895			0.739141			0.369571	−	0.929203i	$-0.379505\pi$
$379$	14.3895			0.739141			0.369571	+	0.929203i	$0.379505\pi$
$380$		0			0
$381$	9.77722	−	8.10977i	0.500902	−	0.415476i
$382$		0			0
$383$	9.91684			0.506727			0.253364	−	0.967371i	$-0.418463\pi$
$383$	9.91684			0.506727			0.253364	+	0.967371i	$0.418463\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	5.09978	−	27.1153i	0.259236	−	1.37835i
$388$		0			0
$389$		−	13.2861i		−	0.673633i	−0.941570	−	0.336816i	$-0.890650\pi$
$389$		−	13.2861i		−	0.673633i	0.941570	−	0.336816i	$-0.109350\pi$
$390$		0			0
$391$			15.1406i			0.765695i
$392$		0			0
$393$	−8.12767	+	6.74154i	−0.409987	+	0.340066i
$394$		0			0
$395$	−0.831556			−0.0418401
$396$		0			0
$397$		−	24.2961i		−	1.21939i	−0.792638	−	0.609693i	$-0.791293\pi$
$397$		−	24.2961i		−	1.21939i	0.792638	−	0.609693i	$-0.208707\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	14.3327i		−	0.715741i	−0.933771	−	0.357870i	$-0.883503\pi$
$401$		−	14.3327i		−	0.715741i	0.933771	−	0.357870i	$-0.116497\pi$
$402$		0			0
$403$	7.35017			0.366138
$404$		0			0
$405$	1.21867	+	0.475218i	0.0605562	+	0.0236138i
$406$		0			0
$407$			14.4972i			0.718600i
$408$		0			0
$409$			20.0059i			0.989226i	0.869113	+	0.494613i	$0.164690\pi$
$409$			20.0059i			0.989226i	−0.869113	+	0.494613i	$0.835310\pi$
$410$		0			0
$411$	22.1956	+	26.7593i	1.09483	+	1.31994i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	2.32277			0.114020
$416$		0			0
$417$	−0.130143	−	0.156902i	−0.00637314	−	0.00768353i
$418$		0			0
$419$	27.7445			1.35541			0.677704	−	0.735335i	$-0.262975\pi$
$419$	27.7445			1.35541			0.677704	+	0.735335i	$0.262975\pi$
$420$		0			0
$421$	−1.53586			−0.0748533			−0.0374267	−	0.999299i	$-0.511916\pi$
$421$	−1.53586			−0.0748533			−0.0374267	+	0.999299i	$0.511916\pi$
$422$		0			0
$423$	5.15905	−	27.4305i	0.250842	−	1.33371i
$424$		0			0
$425$	8.74708			0.424296
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−6.72151	+	5.57519i	−0.324517	+	0.269173i
$430$		0			0
$431$			17.1216i			0.824718i	0.911021	+	0.412359i	$0.135295\pi$
$431$			17.1216i			0.824718i	−0.911021	+	0.412359i	$0.864705\pi$
$432$		0			0
$433$			27.5219i			1.32262i	0.750113	+	0.661310i	$0.229999\pi$
$433$			27.5219i			1.32262i	−0.750113	+	0.661310i	$0.770001\pi$
$434$		0			0
$435$	1.13911	+	1.37332i	0.0546162	+	0.0658459i
$436$		0			0
$437$	−36.6528			−1.75334
$438$		0			0
$439$		−	21.9303i		−	1.04667i	−0.852126	−	0.523337i	$-0.824687\pi$
$439$		−	21.9303i		−	1.04667i	0.852126	−	0.523337i	$-0.175313\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			20.5062i			0.974278i	0.873324	+	0.487139i	$0.161960\pi$
$443$			20.5062i			0.974278i	−0.873324	+	0.487139i	$0.838040\pi$
$444$		0			0
$445$	1.26227			0.0598374
$446$		0			0
$447$	9.90453	+	11.9410i	0.468468	+	0.564790i
$448$		0			0
$449$			18.7692i			0.885773i	0.896578	+	0.442886i	$0.146046\pi$
$449$			18.7692i			0.885773i	−0.896578	+	0.442886i	$0.853954\pi$
$450$		0			0
$451$		−	13.1702i		−	0.620160i
$452$		0			0
$453$	3.65630	−	3.03274i	0.171788	−	0.142491i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	7.59340			0.355204			0.177602	−	0.984102i	$-0.443166\pi$
$457$	7.59340			0.355204			0.177602	+	0.984102i	$0.443166\pi$
$458$		0			0
$459$	4.42887	+	7.98248i	0.206722	+	0.372590i
$460$		0			0
$461$	29.2727			1.36337			0.681683	−	0.731648i	$-0.261248\pi$
$461$	29.2727			1.36337			0.681683	+	0.731648i	$0.261248\pi$
$462$		0			0
$463$	−11.8326			−0.549906			−0.274953	−	0.961458i	$-0.588662\pi$
$463$	−11.8326			−0.549906			−0.274953	+	0.961458i	$0.588662\pi$
$464$		0			0
$465$	0.578644	+	0.697619i	0.0268340	+	0.0323513i
$466$		0			0
$467$	−5.16565			−0.239038			−0.119519	−	0.992832i	$-0.538135\pi$
$467$	−5.16565			−0.239038			−0.119519	+	0.992832i	$0.538135\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−14.9777	−	18.0573i	−0.690138	−	0.832038i
$472$		0			0
$473$			22.7143i			1.04440i
$474$		0			0
$475$			21.1751i			0.971580i
$476$		0			0
$477$	−15.3055	−	2.87862i	−0.700790	−	0.131803i
$478$		0			0
$479$	19.7099			0.900569			0.450284	−	0.892885i	$-0.351323\pi$
$479$	19.7099			0.900569			0.450284	+	0.892885i	$0.351323\pi$
$480$		0			0
$481$			11.9829i			0.546373i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			0.966991i			0.0439088i
$486$		0			0
$487$	5.00720			0.226898			0.113449	−	0.993544i	$-0.463810\pi$
$487$	5.00720			0.226898			0.113449	+	0.993544i	$0.463810\pi$
$488$		0			0
$489$	5.39730	−	4.47682i	0.244074	−	0.202449i
$490$		0			0
$491$		−	3.55902i		−	0.160616i	−0.996770	−	0.0803081i	$-0.974410\pi$
$491$		−	3.55902i		−	0.160616i	0.996770	−	0.0803081i	$-0.0255904\pi$
$492$		0			0
$493$			12.4523i			0.560823i
$494$		0			0
$495$	−1.05830	−	0.199043i	−0.0475673	−	0.00894633i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.809404			0.0362339			0.0181170	−	0.999836i	$-0.494233\pi$
$499$	0.809404			0.0362339			0.0181170	+	0.999836i	$0.494233\pi$
$500$		0			0
$501$	4.93955	−	4.09713i	0.220683	−	0.183046i
$502$		0			0
$503$	−9.47070			−0.422278			−0.211139	−	0.977456i	$-0.567717\pi$
$503$	−9.47070			−0.422278			−0.211139	+	0.977456i	$0.567717\pi$
$504$		0			0
$505$	2.34390			0.104302
$506$		0			0
$507$	11.7750	−	9.76685i	0.522947	−	0.433761i
$508$		0			0
$509$	10.4881			0.464876			0.232438	−	0.972611i	$-0.425330\pi$
$509$	10.4881			0.464876			0.232438	+	0.972611i	$0.425330\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−19.3241	+	10.7215i	−0.853182	+	0.473366i
$514$		0			0
$515$			0.0247258i			0.00108955i
$516$		0			0
$517$			22.9783i			1.01058i
$518$		0			0
$519$	−29.9609	+	24.8512i	−1.31514	+	1.09085i
$520$		0			0
$521$	9.55707			0.418703			0.209351	−	0.977840i	$-0.432865\pi$
$521$	9.55707			0.418703			0.209351	+	0.977840i	$0.432865\pi$
$522$		0			0
$523$		−	27.7480i		−	1.21334i	−0.794955	−	0.606668i	$-0.792506\pi$
$523$		−	27.7480i		−	1.21334i	0.794955	−	0.606668i	$-0.207494\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			6.32549i			0.275543i
$528$		0			0
$529$	−51.2720			−2.22922
$530$		0			0
$531$	6.21383	−	33.0387i	0.269657	−	1.43376i
$532$		0			0
$533$		−	10.8860i		−	0.471526i
$534$		0			0
$535$			1.01348i			0.0438166i
$536$		0			0
$537$	4.06769	+	4.90405i	0.175534	+	0.211625i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.15555			−0.0496812			−0.0248406	−	0.999691i	$-0.507908\pi$
$541$	−1.15555			−0.0496812			−0.0248406	+	0.999691i	$0.507908\pi$
$542$		0			0
$543$	8.85797	+	10.6793i	0.380132	+	0.458291i
$544$		0			0
$545$	−0.196951			−0.00843645
$546$		0			0
$547$	−16.1394			−0.690070			−0.345035	−	0.938590i	$-0.612133\pi$
$547$	−16.1394			−0.690070			−0.345035	+	0.938590i	$0.612133\pi$
$548$		0			0
$549$	−15.8766	−	2.98603i	−0.677596	−	0.127440i
$550$		0			0
$551$	−30.1447			−1.28421
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−1.13732	+	0.943358i	−0.0482766	+	0.0400433i
$556$		0			0
$557$		−	37.0744i		−	1.57089i	−0.618929	−	0.785447i	$-0.712433\pi$
$557$		−	37.0744i		−	1.57089i	0.618929	−	0.785447i	$-0.287567\pi$
$558$		0			0
$559$			18.7749i			0.794092i
$560$		0			0
$561$	−4.79796	−	5.78447i	−0.202570	−	0.244221i
$562$		0			0
$563$	15.5917			0.657111			0.328556	−	0.944485i	$-0.393438\pi$
$563$	15.5917			0.657111			0.328556	+	0.944485i	$0.393438\pi$
$564$		0			0
$565$		−	0.581354i		−	0.0244578i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	15.0429i		−	0.630633i	−0.948987	−	0.315316i	$-0.897889\pi$
$569$		−	15.0429i		−	0.630633i	0.948987	−	0.315316i	$-0.102111\pi$
$570$		0			0
$571$	−5.62668			−0.235469			−0.117735	−	0.993045i	$-0.537563\pi$
$571$	−5.62668			−0.235469			−0.117735	+	0.993045i	$0.537563\pi$
$572$		0			0
$573$	1.24023	+	1.49523i	0.0518114	+	0.0624643i
$574$		0			0
$575$			42.9086i			1.78941i
$576$		0			0
$577$			22.2024i			0.924297i	0.886802	+	0.462149i	$0.152921\pi$
$577$			22.2024i			0.924297i	−0.886802	+	0.462149i	$0.847079\pi$
$578$		0			0
$579$	24.4876	−	20.3114i	1.01767	−	0.844113i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	12.8213			0.531004
$584$		0			0
$585$	−0.874759	−	0.164522i	−0.0361668	−	0.00680216i
$586$		0			0
$587$	20.9245			0.863648			0.431824	−	0.901958i	$-0.357870\pi$
$587$	20.9245			0.863648			0.431824	+	0.901958i	$0.357870\pi$
$588$		0			0
$589$	−15.3129			−0.630956
$590$		0			0
$591$	−0.328083	−	0.395540i	−0.0134955	−	0.0162703i
$592$		0			0
$593$	−21.1690			−0.869307			−0.434654	−	0.900598i	$-0.643129\pi$
$593$	−21.1690			−0.869307			−0.434654	+	0.900598i	$0.643129\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	30.2450	+	36.4637i	1.23785	+	1.49236i
$598$		0			0
$599$		−	5.29601i		−	0.216389i	−0.994130	−	0.108195i	$-0.965493\pi$
$599$		−	5.29601i		−	0.216389i	0.994130	−	0.108195i	$-0.0345070\pi$
$600$		0			0
$601$			37.5346i			1.53107i	0.643396	+	0.765533i	$0.277525\pi$
$601$			37.5346i			1.53107i	−0.643396	+	0.765533i	$0.722475\pi$
$602$		0			0
$603$	2.85481	−	15.1789i	0.116257	−	0.618133i
$604$		0			0
$605$	−0.712191			−0.0289547
$606$		0			0
$607$			39.5050i			1.60346i	0.597687	+	0.801729i	$0.296086\pi$
$607$			39.5050i			1.60346i	−0.597687	+	0.801729i	$0.703914\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			18.9931i			0.768377i
$612$		0			0
$613$	14.3947			0.581397			0.290699	−	0.956815i	$-0.406112\pi$
$613$	14.3947			0.581397			0.290699	+	0.956815i	$0.406112\pi$
$614$		0			0
$615$	1.03321	−	0.857006i	0.0416632	−	0.0345578i
$616$		0			0
$617$			12.1573i			0.489435i	0.969594	+	0.244718i	$0.0786952\pi$
$617$			12.1573i			0.489435i	−0.969594	+	0.244718i	$0.921305\pi$
$618$		0			0
$619$			19.4835i			0.783107i	0.920155	+	0.391553i	$0.128062\pi$
$619$			19.4835i			0.783107i	−0.920155	+	0.391553i	$0.871938\pi$
$620$		0			0
$621$	−39.1579	+	21.7257i	−1.57135	+	0.871824i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	24.6836			0.987344
$626$		0			0
$627$	14.0032	−	11.6150i	0.559232	−	0.463858i
$628$		0			0
$629$	−10.3124			−0.411182
$630$		0			0
$631$	31.3846			1.24940			0.624701	−	0.780864i	$-0.285221\pi$
$631$	31.3846			1.24940			0.624701	+	0.780864i	$0.285221\pi$
$632$		0			0
$633$	−28.0352	+	23.2539i	−1.11430	+	0.924261i
$634$		0			0
$635$	−1.06591			−0.0422994
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−22.9768	−	4.32142i	−0.908950	−	0.170953i
$640$		0			0
$641$		−	41.5828i		−	1.64242i	−0.570624	−	0.821211i	$-0.693299\pi$
$641$		−	41.5828i		−	1.64242i	0.570624	−	0.821211i	$-0.306701\pi$
$642$		0			0
$643$		−	13.5290i		−	0.533531i	−0.963761	−	0.266766i	$-0.914045\pi$
$643$		−	13.5290i		−	0.533531i	0.963761	−	0.266766i	$-0.0859549\pi$
$644$		0			0
$645$	−1.78196	+	1.47806i	−0.0701646	+	0.0581984i
$646$		0			0
$647$	−30.0884			−1.18290			−0.591449	−	0.806343i	$-0.701444\pi$
$647$	−30.0884			−1.18290			−0.591449	+	0.806343i	$0.701444\pi$
$648$		0			0
$649$			27.6762i			1.08639i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			21.2138i			0.830161i	0.909785	+	0.415081i	$0.136247\pi$
$653$			21.2138i			0.830161i	−0.909785	+	0.415081i	$0.863753\pi$
$654$		0			0
$655$	0.886078			0.0346219
$656$		0			0
$657$	38.4743	+	7.23615i	1.50103	+	0.282309i
$658$		0			0
$659$			2.67926i			0.104369i	0.998637	+	0.0521846i	$0.0166184\pi$
$659$			2.67926i			0.104369i	−0.998637	+	0.0521846i	$0.983382\pi$
$660$		0			0
$661$			5.54008i			0.215484i	0.994179	+	0.107742i	$0.0343621\pi$
$661$			5.54008i			0.215484i	−0.994179	+	0.107742i	$0.965638\pi$
$662$		0			0
$663$	−3.96583	−	4.78125i	−0.154020	−	0.185688i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−61.0844			−2.36520
$668$		0			0
$669$	−7.62376	−	9.19128i	−0.294752	−	0.355355i
$670$		0			0
$671$	13.2997			0.513429
$672$		0			0
$673$	−17.1946			−0.662804			−0.331402	−	0.943490i	$-0.607522\pi$
$673$	−17.1946			−0.662804			−0.331402	+	0.943490i	$0.607522\pi$
$674$		0			0
$675$	12.5514	+	22.6223i	0.483105	+	0.870734i
$676$		0			0
$677$	14.4664			0.555988			0.277994	−	0.960583i	$-0.410330\pi$
$677$	14.4664			0.555988			0.277994	+	0.960583i	$0.410330\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−17.8836	+	14.8337i	−0.685301	+	0.568427i
$682$		0			0
$683$			32.4882i			1.24313i	0.783363	+	0.621564i	$0.213502\pi$
$683$			32.4882i			1.24313i	−0.783363	+	0.621564i	$0.786498\pi$
$684$		0			0
$685$		−	2.91730i		−	0.111464i
$686$		0			0
$687$	7.03555	+	8.48214i	0.268423	+	0.323614i
$688$		0			0
$689$	10.5976			0.403738
$690$		0			0
$691$		−	32.4426i		−	1.23417i	−0.786895	−	0.617087i	$-0.788312\pi$
$691$		−	32.4426i		−	1.23417i	0.786895	−	0.617087i	$-0.211688\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			0.0171055i			0.000648847i
$696$		0			0
$697$	9.36842			0.354854
$698$		0			0
$699$	5.48142	+	6.60845i	0.207326	+	0.249955i
$700$		0			0
$701$		−	30.3777i		−	1.14735i	−0.819084	−	0.573674i	$-0.805518\pi$
$701$		−	30.3777i		−	1.14735i	0.819084	−	0.573674i	$-0.194482\pi$
$702$		0			0
$703$		−	24.9644i		−	0.941551i
$704$		0			0
$705$	−1.80267	+	1.49523i	−0.0678925	+	0.0563138i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−32.5139			−1.22108			−0.610542	−	0.791984i	$-0.709048\pi$
$709$	−32.5139			−1.22108			−0.610542	+	0.791984i	$0.709048\pi$
$710$		0			0
$711$	3.17263	−	16.8688i	0.118983	−	0.632628i
$712$		0			0
$713$	−31.0295			−1.16207
$714$		0			0
$715$	0.732778			0.0274044
$716$		0			0
$717$	−19.2136	−	23.1641i	−0.717545	−	0.865079i
$718$		0			0
$719$	10.5973			0.395214			0.197607	−	0.980281i	$-0.436683\pi$
$719$	10.5973			0.395214			0.197607	+	0.980281i	$0.436683\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	16.0405	+	19.3385i	0.596551	+	0.719208i
$724$		0			0
$725$			35.2898i			1.31063i
$726$		0			0
$727$		−	31.3600i		−	1.16308i	−0.813518	−	0.581540i	$-0.802451\pi$
$727$		−	31.3600i		−	1.16308i	0.813518	−	0.581540i	$-0.197549\pi$
$728$		0			0
$729$	−14.2898	+	22.9086i	−0.529251	+	0.848466i
$730$		0			0
$731$	−16.1575			−0.597606
$732$		0			0
$733$			3.61634i			0.133573i	0.997767	+	0.0667863i	$0.0212746\pi$
$733$			3.61634i			0.133573i	−0.997767	+	0.0667863i	$0.978725\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			12.7153i			0.468373i
$738$		0			0
$739$	38.6926			1.42333			0.711665	−	0.702519i	$-0.247942\pi$
$739$	38.6926			1.42333			0.711665	+	0.702519i	$0.247942\pi$
$740$		0			0
$741$	11.5745	−	9.60056i	0.425201	−	0.352685i
$742$		0			0
$743$			45.1194i			1.65527i	0.561266	+	0.827635i	$0.310314\pi$
$743$			45.1194i			1.65527i	−0.561266	+	0.827635i	$0.689686\pi$
$744$		0			0
$745$		−	1.30181i		−	0.0476945i
$746$		0			0
$747$	−8.86207	+	47.1193i	−0.324246	+	1.72400i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	23.5683			0.860018			0.430009	−	0.902825i	$-0.358510\pi$
$751$	23.5683			0.860018			0.430009	+	0.902825i	$0.358510\pi$
$752$		0			0
$753$	−4.66309	+	3.86782i	−0.169932	+	0.140951i
$754$		0			0
$755$	−0.398610			−0.0145069
$756$		0			0
$757$	26.2967			0.955770			0.477885	−	0.878422i	$-0.341404\pi$
$757$	26.2967			0.955770			0.477885	+	0.878422i	$0.341404\pi$
$758$		0			0
$759$	28.3756	−	23.5363i	1.02997	−	0.854313i
$760$		0			0
$761$	25.9560			0.940903			0.470452	−	0.882426i	$-0.344091\pi$
$761$	25.9560			0.940903			0.470452	+	0.882426i	$0.344091\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	0.141586	−	0.752810i	0.00511907	−	0.0272179i
$766$		0			0
$767$			22.8762i			0.826013i
$768$		0			0
$769$		−	36.9215i		−	1.33142i	−0.746209	−	0.665712i	$-0.768128\pi$
$769$		−	36.9215i		−	1.33142i	0.746209	−	0.665712i	$-0.231872\pi$
$770$		0			0
$771$	21.2444	−	17.6212i	0.765097	−	0.634614i
$772$		0			0
$773$	11.5857			0.416708			0.208354	−	0.978054i	$-0.433189\pi$
$773$	11.5857			0.416708			0.208354	+	0.978054i	$0.433189\pi$
$774$		0			0
$775$			17.9264i			0.643937i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			22.6793i			0.812569i
$780$		0			0
$781$	19.2475			0.688730
$782$		0			0
$783$	−32.2050	+	17.8681i	−1.15091	+	0.638555i
$784$		0			0
$785$			1.96861i			0.0702627i
$786$		0			0
$787$		−	29.9244i		−	1.06669i	−0.845898	−	0.533344i	$-0.820935\pi$
$787$		−	29.9244i		−	1.06669i	0.845898	−	0.533344i	$-0.179065\pi$
$788$		0			0
$789$	−15.8766	−	19.1410i	−0.565221	−	0.681436i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	10.9931			0.390375
$794$		0			0
$795$	0.834302	+	1.00584i	0.0295896	+	0.0356736i
$796$		0			0
$797$	−20.2866			−0.718587			−0.359293	−	0.933225i	$-0.616982\pi$
$797$	−20.2866			−0.718587			−0.359293	+	0.933225i	$0.616982\pi$
$798$		0			0
$799$	−16.3453			−0.578254
$800$		0			0
$801$	−4.81594	+	25.6062i	−0.170163	+	0.904750i
$802$		0			0
$803$	−32.2297			−1.13736
$804$		0			0
$805$		0			0
$806$		0			0
$807$	9.81751	−	8.14319i	0.345593	−	0.286654i
$808$		0			0
$809$			2.49001i			0.0875442i	0.999042	+	0.0437721i	$0.0139375\pi$
$809$			2.49001i			0.0875442i	−0.999042	+	0.0437721i	$0.986062\pi$
$810$		0			0
$811$			15.9838i			0.561269i	0.959815	+	0.280634i	$0.0905448\pi$
$811$			15.9838i			0.561269i	−0.959815	+	0.280634i	$0.909455\pi$
$812$		0			0
$813$	13.8584	+	16.7078i	0.486036	+	0.585970i
$814$		0			0
$815$	−0.588413			−0.0206112
$816$		0			0
$817$		−	39.1144i		−	1.36844i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			2.75927i			0.0962991i	0.998840	+	0.0481495i	$0.0153324\pi$
$821$			2.75927i			0.0962991i	−0.998840	+	0.0481495i	$0.984668\pi$
$822$		0			0
$823$	28.4424			0.991440			0.495720	−	0.868482i	$-0.334904\pi$
$823$	28.4424			0.991440			0.495720	+	0.868482i	$0.334904\pi$
$824$		0			0
$825$	−13.5974	−	16.3932i	−0.473401	−	0.570738i
$826$		0			0
$827$		−	30.6070i		−	1.06431i	−0.846647	−	0.532154i	$-0.821383\pi$
$827$		−	30.6070i		−	1.06431i	0.846647	−	0.532154i	$-0.178617\pi$
$828$		0			0
$829$			10.0620i			0.349469i	0.984616	+	0.174734i	$0.0559067\pi$
$829$			10.0620i			0.349469i	−0.984616	+	0.174734i	$0.944093\pi$
$830$		0			0
$831$	43.3026	−	35.9176i	1.50215	−	1.24597i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−0.538509			−0.0186359
$836$		0			0
$837$	−16.3595	+	9.07663i	−0.565465	+	0.313734i
$838$		0			0
$839$	13.3067			0.459400			0.229700	−	0.973262i	$-0.426226\pi$
$839$	13.3067			0.459400			0.229700	+	0.973262i	$0.426226\pi$
$840$		0			0
$841$	−21.2383			−0.732354
$842$		0			0
$843$	11.2522	+	13.5657i	0.387545	+	0.467229i
$844$		0			0
$845$	−1.28371			−0.0441610
$846$		0			0
$847$		0			0
$848$		0			0
$849$	1.51518	+	1.82672i	0.0520009	+	0.0626928i
$850$		0			0
$851$		−	50.5871i		−	1.73411i
$852$		0			0
$853$		−	42.4736i		−	1.45427i	−0.686495	−	0.727134i	$-0.740852\pi$
$853$		−	42.4736i		−	1.45427i	0.686495	−	0.727134i	$-0.259148\pi$
$854$		0			0
$855$	1.82242	+	0.342755i	0.0623253	+	0.0117220i
$856$		0			0
$857$	9.55707			0.326463			0.163232	−	0.986588i	$-0.447808\pi$
$857$	9.55707			0.326463			0.163232	+	0.986588i	$0.447808\pi$
$858$		0			0
$859$		−	2.11241i		−	0.0720745i	−0.999350	−	0.0360372i	$-0.988527\pi$
$859$		−	2.11241i		−	0.0720745i	0.999350	−	0.0360372i	$-0.0114735\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			35.2974i			1.20154i	0.799423	+	0.600768i	$0.205139\pi$
$863$			35.2974i			1.20154i	−0.799423	+	0.600768i	$0.794861\pi$
$864$		0			0
$865$	3.26634			0.111059
$866$		0			0
$867$	−18.5486	+	15.3853i	−0.629945	+	0.522511i
$868$		0			0
$869$			14.1308i			0.479356i
$870$		0			0
$871$			10.5100i			0.356118i
$872$		0			0
$873$	−19.6162	−	3.68936i	−0.663907	−	0.124866i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−16.7377			−0.565193			−0.282596	−	0.959239i	$-0.591196\pi$
$877$	−16.7377			−0.565193			−0.282596	+	0.959239i	$0.591196\pi$
$878$		0			0
$879$	−22.6581	+	18.7939i	−0.764240	+	0.633903i
$880$		0			0
$881$	42.6152			1.43574			0.717871	−	0.696176i	$-0.245117\pi$
$881$	42.6152			1.43574			0.717871	+	0.696176i	$0.245117\pi$
$882$		0			0
$883$	15.2392			0.512839			0.256419	−	0.966566i	$-0.417457\pi$
$883$	15.2392			0.512839			0.256419	+	0.966566i	$0.417457\pi$
$884$		0			0
$885$	−2.17123	+	1.80094i	−0.0729850	+	0.0605378i
$886$		0			0
$887$	−56.1267			−1.88455			−0.942275	−	0.334841i	$-0.891317\pi$
$887$	−56.1267			−1.88455			−0.942275	+	0.334841i	$0.891317\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	8.07550	−	20.7091i	0.270539	−	0.693782i
$892$		0			0
$893$		−	39.5689i		−	1.32412i
$894$		0			0
$895$		−	0.534639i		−	0.0178710i
$896$		0			0
$897$	23.4543	−	19.4543i	0.783116	−	0.649560i
$898$		0			0
$899$	−25.5200			−0.851138
$900$		0			0
$901$			9.12024i			0.303839i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	1.16425i		−	0.0387011i
$906$		0			0
$907$	−40.7249			−1.35225			−0.676124	−	0.736788i	$-0.736342\pi$
$907$	−40.7249			−1.35225			−0.676124	+	0.736788i	$0.736342\pi$
$908$		0			0
$909$	−8.94266	+	47.5478i	−0.296609	+	1.57706i
$910$		0			0
$911$		−	31.0308i		−	1.02810i	−0.857761	−	0.514049i	$-0.828145\pi$
$911$		−	31.0308i		−	1.02810i	0.857761	−	0.514049i	$-0.171855\pi$
$912$		0			0
$913$		−	39.4714i		−	1.30631i
$914$		0			0
$915$	0.865432	+	1.04337i	0.0286103	+	0.0344929i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	51.2356			1.69011			0.845053	−	0.534683i	$-0.179569\pi$
$919$	51.2356			1.69011			0.845053	+	0.534683i	$0.179569\pi$
$920$		0			0
$921$	23.1133	+	27.8656i	0.761609	+	0.918204i
$922$		0			0
$923$	15.9093			0.523662
$924$		0			0
$925$	−29.2253			−0.960922
$926$		0			0
$927$	−0.501582	−	0.0943362i	−0.0164741	−	0.00309841i
$928$		0			0
$929$	−3.74232			−0.122781			−0.0613907	−	0.998114i	$-0.519554\pi$
$929$	−3.74232			−0.122781			−0.0613907	+	0.998114i	$0.519554\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	15.3055	−	12.6952i	0.501079	−	0.415623i
$934$		0			0
$935$			0.630623i			0.0206236i
$936$		0			0
$937$		−	3.23951i		−	0.105830i	−0.998599	−	0.0529150i	$-0.983149\pi$
$937$		−	3.23951i		−	0.105830i	0.998599	−	0.0529150i	$-0.0168512\pi$
$938$		0			0
$939$	−10.9447	−	13.1951i	−0.357168	−	0.430606i
$940$		0			0
$941$	−22.0683			−0.719407			−0.359703	−	0.933067i	$-0.617122\pi$
$941$	−22.0683			−0.719407			−0.359703	+	0.933067i	$0.617122\pi$
$942$		0			0
$943$			45.9566i			1.49655i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	36.2595i		−	1.17827i	−0.808033	−	0.589137i	$-0.799468\pi$
$947$		−	36.2595i		−	1.17827i	0.808033	−	0.589137i	$-0.200532\pi$
$948$		0			0
$949$	−26.6399			−0.864769
$950$		0			0
$951$	7.33606	+	8.84443i	0.237888	+	0.286800i
$952$		0			0
$953$		−	13.4656i		−	0.436192i	−0.975927	−	0.218096i	$-0.930015\pi$
$953$		−	13.4656i		−	0.436192i	0.975927	−	0.218096i	$-0.0699846\pi$
$954$		0			0
$955$		−	0.163010i		−	0.00527489i
$956$		0			0
$957$	23.3372	−	19.3572i	0.754386	−	0.625729i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	18.0364			0.581820
$962$		0			0
$963$	−20.5593	−	3.86673i	−0.662513	−	0.124604i
$964$		0			0
$965$	−2.66964			−0.0859388
$966$		0			0
$967$	9.25940			0.297762			0.148881	−	0.988855i	$-0.452433\pi$
$967$	9.25940			0.297762			0.148881	+	0.988855i	$0.452433\pi$
$968$		0			0
$969$	8.26216	+	9.96094i	0.265419	+	0.319992i
$970$		0			0
$971$	39.9290			1.28138			0.640691	−	0.767798i	$-0.278648\pi$
$971$	39.9290			1.28138			0.640691	+	0.767798i	$0.278648\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−11.2392	−	13.5501i	−0.359941	−	0.433949i
$976$		0			0
$977$			60.5408i			1.93687i	0.249261	+	0.968436i	$0.419812\pi$
$977$			60.5408i			1.93687i	−0.249261	+	0.968436i	$0.580188\pi$
$978$		0			0
$979$		−	21.4501i		−	0.685548i
$980$		0			0
$981$	0.751427	−	3.99531i	0.0239912	−	0.127560i
$982$		0			0
$983$	−39.4054			−1.25684			−0.628419	−	0.777875i	$-0.716298\pi$
$983$	−39.4054			−1.25684			−0.628419	+	0.777875i	$0.716298\pi$
$984$		0			0
$985$			0.0431217i			0.00137397i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	79.2602i		−	2.52033i
$990$		0			0
$991$	−30.1015			−0.956205			−0.478102	−	0.878304i	$-0.658675\pi$
$991$	−30.1015			−0.956205			−0.478102	+	0.878304i	$0.658675\pi$
$992$		0			0
$993$	−19.6381	+	16.2889i	−0.623196	+	0.516914i
$994$		0			0
$995$		−	3.97527i		−	0.126025i
$996$		0			0
$997$		−	7.04907i		−	0.223246i	−0.993751	−	0.111623i	$-0.964395\pi$
$997$		−	7.04907i		−	0.223246i	0.993751	−	0.111623i	$-0.0356049\pi$
$998$		0			0
$999$	−14.7975	−	26.6707i	−0.468173	−	0.843822i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.k.i.881.13		16
3.2	odd	2	inner	2352.2.k.i.881.3		16
4.3	odd	2		1176.2.k.a.881.4		16
7.4	even	3		336.2.bc.f.257.2		16
7.5	odd	6		336.2.bc.f.17.4		16
7.6	odd	2	inner	2352.2.k.i.881.4		16
12.11	even	2		1176.2.k.a.881.14		16
21.5	even	6		336.2.bc.f.17.2		16
21.11	odd	6		336.2.bc.f.257.4		16
21.20	even	2	inner	2352.2.k.i.881.14		16
28.3	even	6		1176.2.u.b.1097.2		16
28.11	odd	6		168.2.u.a.89.7	yes	16
28.19	even	6		168.2.u.a.17.5	✓	16
28.23	odd	6		1176.2.u.b.521.4		16
28.27	even	2		1176.2.k.a.881.13		16
84.11	even	6		168.2.u.a.89.5	yes	16
84.23	even	6		1176.2.u.b.521.2		16
84.47	odd	6		168.2.u.a.17.7	yes	16
84.59	odd	6		1176.2.u.b.1097.4		16
84.83	odd	2		1176.2.k.a.881.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.u.a.17.5	✓	16	28.19	even	6
168.2.u.a.17.7	yes	16	84.47	odd	6
168.2.u.a.89.5	yes	16	84.11	even	6
168.2.u.a.89.7	yes	16	28.11	odd	6
336.2.bc.f.17.2		16	21.5	even	6
336.2.bc.f.17.4		16	7.5	odd	6
336.2.bc.f.257.2		16	7.4	even	3
336.2.bc.f.257.4		16	21.11	odd	6
1176.2.k.a.881.3		16	84.83	odd	2
1176.2.k.a.881.4		16	4.3	odd	2
1176.2.k.a.881.13		16	28.27	even	2
1176.2.k.a.881.14		16	12.11	even	2
1176.2.u.b.521.2		16	84.23	even	6
1176.2.u.b.521.4		16	28.23	odd	6
1176.2.u.b.1097.2		16	28.3	even	6
1176.2.u.b.1097.4		16	84.59	odd	6
2352.2.k.i.881.3		16	3.2	odd	2	inner
2352.2.k.i.881.4		16	7.6	odd	2	inner
2352.2.k.i.881.13		16	1.1	even	1	trivial
2352.2.k.i.881.14		16	21.20	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.33314 - 1.10578i) q^{3} -0.145339 q^{5} +(0.554510 - 2.94831i) q^{9} +O(q^{10})\)	\(q+(1.33314 - 1.10578i) q^{3} -0.145339 q^{5} +(0.554510 - 2.94831i) q^{9} +2.46977i q^{11} +2.04143i q^{13} +(-0.193756 + 0.160712i) q^{15} -1.75684 q^{17} -4.25299i q^{19} -8.61812i q^{23} -4.97888 q^{25} +(-2.52094 - 4.54366i) q^{27} -7.08790i q^{29} -3.60050i q^{31} +(2.73102 + 3.29255i) q^{33} +5.86986 q^{37} +(2.25737 + 2.72151i) q^{39} -5.33255 q^{41} +9.19692 q^{43} +(-0.0805917 + 0.428503i) q^{45} +9.30380 q^{47} +(-2.34211 + 1.94267i) q^{51} -5.19128i q^{53} -0.358953i q^{55} +(-4.70286 - 5.66982i) q^{57} +11.2060 q^{59} -5.38498i q^{61} -0.296699i q^{65} +5.14835 q^{67} +(-9.52973 - 11.4891i) q^{69} -7.79323i q^{71} +13.0496i q^{73} +(-6.63753 + 5.50553i) q^{75} +5.72151 q^{79} +(-8.38504 - 3.26973i) q^{81} -15.9818 q^{83} +0.255336 q^{85} +(-7.83764 - 9.44914i) q^{87} -8.68504 q^{89} +(-3.98135 - 4.79996i) q^{93} +0.618123i q^{95} -6.65337i q^{97} +(7.28165 + 1.36951i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{9}+O(q^{10}) \) 16 * q - 4 * q^9	\( 16 q - 4 q^{9} - 8 q^{15} + 36 q^{25} + 4 q^{37} - 44 q^{39} - 20 q^{43} + 12 q^{51} - 8 q^{57} + 28 q^{67} + 56 q^{79} - 60 q^{81} + 16 q^{85} - 32 q^{93} - 20 q^{99}+O(q^{100}) \) 16 * q - 4 * q^9 - 8 * q^15 + 36 * q^25 + 4 * q^37 - 44 * q^39 - 20 * q^43 + 12 * q^51 - 8 * q^57 + 28 * q^67 + 56 * q^79 - 60 * q^81 + 16 * q^85 - 32 * q^93 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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