LMFDB - Embedded newform 3312.2.i.b.2575.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3312,2,Mod(2575,3312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3312, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("3312.2575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3312 = 2^{4} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3312.i (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:	$26.4464531494$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.56070144.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 $ x^8 - 4x^7 + 16x^6 - 34x^5 + 63x^4 - 74x^3 + 70x^2 - 38*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2575.4
Root			$0.500000 + 1.19293i$ of defining polynomial
Character	$\chi$	$=$	3312.2575
Dual form			3312.2.i.b.2575.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{5} +3.38587 q^{7} +O(q^{10})$	$q-2.00000i q^{5} +3.38587 q^{7} -4.73205 q^{11} -3.46410 q^{13} +0.732051i q^{17} +2.47863 q^{19} +(-1.26795 + 4.62518i) q^{23} +1.00000 q^{25} -4.29311 q^{29} -6.00000i q^{31} -6.77174i q^{35} -10.1576i q^{37} -11.7290 q^{41} -2.47863 q^{43} -4.29311i q^{47} +4.46410 q^{49} -0.535898i q^{53} +9.46410i q^{55} -11.7290i q^{59} -1.57139i q^{61} +6.92820i q^{65} -9.25036 q^{67} +1.81448i q^{71} -12.3923 q^{73} -16.0221 q^{77} +8.34312 q^{79} +2.19615 q^{83} +1.46410 q^{85} +13.1244i q^{89} -11.7290 q^{91} -4.95725i q^{95} +11.7290i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 24 q^{11} - 24 q^{23} + 8 q^{25} + 8 q^{49} - 16 q^{73} - 24 q^{83} - 16 q^{85}+O(q^{100}) $ 8 * q - 24 * q^11 - 24 * q^23 + 8 * q^25 + 8 * q^49 - 16 * q^73 - 24 * q^83 - 16 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3312\mathbb{Z}\right)^\times$.

$n$	$415$	$2305$	$2485$	$2945$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	2.00000i		−	0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$		−	2.00000i		−	0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$		0			0
$7$	3.38587			1.27974			0.639869	−	0.768484i	$-0.278989\pi$
$7$	3.38587			1.27974			0.639869	+	0.768484i	$0.278989\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.73205			−1.42677			−0.713384	−	0.700774i	$-0.752838\pi$
$11$	−4.73205			−1.42677			−0.713384	+	0.700774i	$0.752838\pi$
$12$		0			0
$13$	−3.46410			−0.960769			−0.480384	−	0.877058i	$-0.659503\pi$
$13$	−3.46410			−0.960769			−0.480384	+	0.877058i	$0.659503\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			0.732051i			0.177548i	0.996052	+	0.0887742i	$0.0282950\pi$
$17$			0.732051i			0.177548i	−0.996052	+	0.0887742i	$0.971705\pi$
$18$		0			0
$19$	2.47863			0.568636			0.284318	−	0.958730i	$-0.408233\pi$
$19$	2.47863			0.568636			0.284318	+	0.958730i	$0.408233\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.26795	+	4.62518i	−0.264386	+	0.964417i
$23$	−1.26795	+	4.62518i	−0.264386	+	0.964417i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.29311			−0.797210			−0.398605	−	0.917123i	$-0.630506\pi$
$29$	−4.29311			−0.797210			−0.398605	+	0.917123i	$0.630506\pi$
$30$		0			0
$31$		−	6.00000i		−	1.07763i	−0.842424	−	0.538816i	$-0.818872\pi$
$31$		−	6.00000i		−	1.07763i	0.842424	−	0.538816i	$-0.181128\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	6.77174i		−	1.14463i
$36$		0			0
$37$		−	10.1576i		−	1.66990i	−0.550326	−	0.834950i	$-0.685497\pi$
$37$		−	10.1576i		−	1.66990i	0.550326	−	0.834950i	$-0.314503\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−11.7290			−1.83176			−0.915880	−	0.401451i	$-0.868506\pi$
$41$	−11.7290			−1.83176			−0.915880	+	0.401451i	$0.868506\pi$
$42$		0			0
$43$	−2.47863			−0.377987			−0.188994	−	0.981978i	$-0.560523\pi$
$43$	−2.47863			−0.377987			−0.188994	+	0.981978i	$0.560523\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	4.29311i		−	0.626214i	−0.949718	−	0.313107i	$-0.898630\pi$
$47$		−	4.29311i		−	0.626214i	0.949718	−	0.313107i	$-0.101370\pi$
$48$		0			0
$49$	4.46410			0.637729
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	0.535898i		−	0.0736113i	−0.999322	−	0.0368057i	$-0.988282\pi$
$53$		−	0.535898i		−	0.0736113i	0.999322	−	0.0368057i	$-0.0117182\pi$
$54$		0			0
$55$			9.46410i			1.27614i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	11.7290i		−	1.52698i	−0.645817	−	0.763492i	$-0.723483\pi$
$59$		−	11.7290i		−	1.52698i	0.645817	−	0.763492i	$-0.276517\pi$
$60$		0			0
$61$		−	1.57139i		−	0.201195i	−0.994927	−	0.100598i	$-0.967924\pi$
$61$		−	1.57139i		−	0.201195i	0.994927	−	0.100598i	$-0.0320755\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			6.92820i			0.859338i
$66$		0			0
$67$	−9.25036			−1.13011			−0.565056	−	0.825053i	$-0.691145\pi$
$67$	−9.25036			−1.13011			−0.565056	+	0.825053i	$0.691145\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			1.81448i			0.215339i	0.994187	+	0.107670i	$0.0343389\pi$
$71$			1.81448i			0.215339i	−0.994187	+	0.107670i	$0.965661\pi$
$72$		0			0
$73$	−12.3923			−1.45041			−0.725205	−	0.688533i	$-0.758255\pi$
$73$	−12.3923			−1.45041			−0.725205	+	0.688533i	$0.758255\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−16.0221			−1.82589
$78$		0			0
$79$	8.34312			0.938675			0.469337	−	0.883019i	$-0.344493\pi$
$79$	8.34312			0.938675			0.469337	+	0.883019i	$0.344493\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.19615			0.241059			0.120530	−	0.992710i	$-0.461541\pi$
$83$	2.19615			0.241059			0.120530	+	0.992710i	$0.461541\pi$
$84$		0			0
$85$	1.46410			0.158804
$86$		0			0
$87$		0			0
$88$		0			0
$89$			13.1244i			1.39118i	0.718439	+	0.695589i	$0.244857\pi$
$89$			13.1244i			1.39118i	−0.718439	+	0.695589i	$0.755143\pi$
$90$		0			0
$91$	−11.7290			−1.22953
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	4.95725i		−	0.508604i
$96$		0			0
$97$			11.7290i			1.19090i	0.803393	+	0.595449i	$0.203026\pi$
$97$			11.7290i			1.19090i	−0.803393	+	0.595449i	$0.796974\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−11.7290			−1.16708			−0.583539	−	0.812085i	$-0.698333\pi$
$101$	−11.7290			−1.16708			−0.583539	+	0.812085i	$0.698333\pi$
$102$		0			0
$103$	−3.38587			−0.333619			−0.166810	−	0.985989i	$-0.553347\pi$
$103$	−3.38587			−0.333619			−0.166810	+	0.985989i	$0.553347\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.12436			0.882085			0.441042	−	0.897486i	$-0.354609\pi$
$107$	9.12436			0.882085			0.441042	+	0.897486i	$0.354609\pi$
$108$		0			0
$109$			1.57139i			0.150512i	0.997164	+	0.0752558i	$0.0239773\pi$
$109$			1.57139i			0.150512i	−0.997164	+	0.0752558i	$0.976023\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			13.1244i			1.23464i	0.786714	+	0.617318i	$0.211781\pi$
$113$			13.1244i			1.23464i	−0.786714	+	0.617318i	$0.788219\pi$
$114$		0			0
$115$	9.25036	+	2.53590i	0.862601	+	0.236474i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			2.47863i			0.227215i
$120$		0			0
$121$	11.3923			1.03566
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	12.0000i		−	1.07331i
$126$		0			0
$127$		−	8.53590i		−	0.757438i	−0.925512	−	0.378719i	$-0.876365\pi$
$127$		−	8.53590i		−	0.757438i	0.925512	−	0.378719i	$-0.123635\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	18.5007i		−	1.61642i	−0.588897	−	0.808208i	$-0.700438\pi$
$131$		−	18.5007i		−	1.61642i	0.588897	−	0.808208i	$-0.299562\pi$
$132$		0			0
$133$	8.39230			0.727705
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	18.1962i		−	1.55460i	−0.629129	−	0.777301i	$-0.716588\pi$
$137$		−	18.1962i		−	1.55460i	0.629129	−	0.777301i	$-0.283412\pi$
$138$		0			0
$139$			8.53590i			0.724005i	0.932177	+	0.362003i	$0.117907\pi$
$139$			8.53590i			0.724005i	−0.932177	+	0.362003i	$0.882093\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	16.3923			1.37079
$144$		0			0
$145$			8.58622i			0.713047i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			13.3205i			1.09126i	0.838027	+	0.545629i	$0.183709\pi$
$149$			13.3205i			1.09126i	−0.838027	+	0.545629i	$0.816291\pi$
$150$		0			0
$151$		−	12.9282i		−	1.05208i	−0.850459	−	0.526041i	$-0.823676\pi$
$151$		−	12.9282i		−	1.05208i	0.850459	−	0.526041i	$-0.176324\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−12.0000			−0.963863
$156$		0			0
$157$			21.8866i			1.74674i	0.487057	+	0.873370i	$0.338070\pi$
$157$			21.8866i			1.74674i	−0.487057	+	0.873370i	$0.661930\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.29311	+	15.6603i	−0.338344	+	1.23420i
$162$		0			0
$163$		−	3.46410i		−	0.271329i	−0.990755	−	0.135665i	$-0.956683\pi$
$163$		−	3.46410i		−	0.271329i	0.990755	−	0.135665i	$-0.0433170\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	1.81448i		−	0.140409i	−0.997533	−	0.0702044i	$-0.977635\pi$
$167$		−	1.81448i		−	0.140409i	0.997533	−	0.0702044i	$-0.0223651\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.29311			−0.326399			−0.163199	−	0.986593i	$-0.552181\pi$
$173$	−4.29311			−0.326399			−0.163199	+	0.986593i	$0.552181\pi$
$174$		0			0
$175$	3.38587			0.255948
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	11.7290i		−	0.876666i	−0.898813	−	0.438333i	$-0.855569\pi$
$179$		−	11.7290i		−	0.876666i	0.898813	−	0.438333i	$-0.144431\pi$
$180$		0			0
$181$		−	13.3004i		−	0.988609i	−0.869289	−	0.494305i	$-0.835423\pi$
$181$		−	13.3004i		−	0.988609i	0.869289	−	0.494305i	$-0.164577\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−20.3152			−1.49360
$186$		0			0
$187$		−	3.46410i		−	0.253320i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−13.8564			−1.00261			−0.501307	−	0.865269i	$-0.667147\pi$
$191$	−13.8564			−1.00261			−0.501307	+	0.865269i	$0.667147\pi$
$192$		0			0
$193$	−9.46410			−0.681241			−0.340620	−	0.940201i	$-0.610637\pi$
$193$	−9.46410			−0.681241			−0.340620	+	0.940201i	$0.610637\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−11.7290			−0.835656			−0.417828	−	0.908526i	$-0.637209\pi$
$197$	−11.7290			−0.835656			−0.417828	+	0.908526i	$0.637209\pi$
$198$		0			0
$199$	11.9721			0.848679			0.424339	−	0.905503i	$-0.360506\pi$
$199$	11.9721			0.848679			0.424339	+	0.905503i	$0.360506\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−14.5359			−1.02022
$204$		0			0
$205$			23.4580i			1.63838i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−11.7290			−0.811311
$210$		0			0
$211$			18.9282i			1.30307i	0.758618	+	0.651536i	$0.225875\pi$
$211$			18.9282i			1.30307i	−0.758618	+	0.651536i	$0.774125\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			4.95725i			0.338082i
$216$		0			0
$217$		−	20.3152i		−	1.37909i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	2.53590i		−	0.170583i
$222$		0			0
$223$			24.9282i			1.66932i	0.550769	+	0.834658i	$0.314335\pi$
$223$			24.9282i			1.66932i	−0.550769	+	0.834658i	$0.685665\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−23.6603			−1.57039			−0.785193	−	0.619251i	$-0.787436\pi$
$227$	−23.6603			−1.57039			−0.785193	+	0.619251i	$0.787436\pi$
$228$		0			0
$229$			10.1576i			0.671233i	0.941999	+	0.335617i	$0.108945\pi$
$229$			10.1576i			0.671233i	−0.941999	+	0.335617i	$0.891055\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	−8.58622			−0.560103
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	9.25036i		−	0.598356i	−0.954197	−	0.299178i	$-0.903288\pi$
$239$		−	9.25036i		−	0.598356i	0.954197	−	0.299178i	$-0.0967125\pi$
$240$		0			0
$241$		−	8.58622i		−	0.553087i	−0.961001	−	0.276543i	$-0.910811\pi$
$241$		−	8.58622i		−	0.553087i	0.961001	−	0.276543i	$-0.0891890\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	8.92820i		−	0.570402i
$246$		0			0
$247$	−8.58622			−0.546328
$248$		0			0
$249$		0			0
$250$		0			0
$251$	16.0526			1.01323			0.506614	−	0.862173i	$-0.330897\pi$
$251$	16.0526			1.01323			0.506614	+	0.862173i	$0.330897\pi$
$252$		0			0
$253$	6.00000	−	21.8866i	0.377217	−	1.37600i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−20.3152			−1.26723			−0.633614	−	0.773649i	$-0.718429\pi$
$257$	−20.3152			−1.26723			−0.633614	+	0.773649i	$0.718429\pi$
$258$		0			0
$259$		−	34.3923i		−	2.13703i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	7.60770			0.469111			0.234555	−	0.972103i	$-0.424637\pi$
$263$	7.60770			0.469111			0.234555	+	0.972103i	$0.424637\pi$
$264$		0			0
$265$	−1.07180			−0.0658400
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−11.7290			−0.715129			−0.357565	−	0.933888i	$-0.616393\pi$
$269$	−11.7290			−0.715129			−0.357565	+	0.933888i	$0.616393\pi$
$270$		0			0
$271$		−	1.60770i		−	0.0976605i	−0.998807	−	0.0488303i	$-0.984451\pi$
$271$		−	1.60770i		−	0.0976605i	0.998807	−	0.0488303i	$-0.0155493\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4.73205			−0.285353
$276$		0			0
$277$	−18.7846			−1.12866			−0.564329	−	0.825550i	$-0.690865\pi$
$277$	−18.7846			−1.12866			−0.564329	+	0.825550i	$0.690865\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	22.5885i		−	1.34751i	−0.738953	−	0.673757i	$-0.764679\pi$
$281$		−	22.5885i		−	1.34751i	0.738953	−	0.673757i	$-0.235321\pi$
$282$		0			0
$283$	−9.25036			−0.549877			−0.274939	−	0.961462i	$-0.588657\pi$
$283$	−9.25036			−0.549877			−0.274939	+	0.961462i	$0.588657\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−39.7128			−2.34417
$288$		0			0
$289$	16.4641			0.968477
$290$		0			0
$291$		0			0
$292$		0			0
$293$			4.92820i			0.287909i	0.989584	+	0.143954i	$0.0459818\pi$
$293$			4.92820i			0.287909i	−0.989584	+	0.143954i	$0.954018\pi$
$294$		0			0
$295$	−23.4580			−1.36578
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.39230	−	16.0221i	0.254014	−	0.926582i
$300$		0			0
$301$	−8.39230			−0.483724
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.14277			−0.179955
$306$		0			0
$307$		−	8.53590i		−	0.487169i	−0.969880	−	0.243585i	$-0.921677\pi$
$307$		−	8.53590i		−	0.487169i	0.969880	−	0.243585i	$-0.0783234\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			20.3152i			1.15197i	0.817460	+	0.575985i	$0.195381\pi$
$311$			20.3152i			1.15197i	−0.817460	+	0.575985i	$0.804619\pi$
$312$		0			0
$313$		−	23.4580i		−	1.32592i	−0.748653	−	0.662962i	$-0.769299\pi$
$313$		−	23.4580i		−	1.32592i	0.748653	−	0.662962i	$-0.230701\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	19.1649			1.07641			0.538203	−	0.842815i	$-0.319103\pi$
$317$	19.1649			1.07641			0.538203	+	0.842815i	$0.319103\pi$
$318$		0			0
$319$	20.3152			1.13743
$320$		0			0
$321$		0			0
$322$		0			0
$323$			1.81448i			0.100960i
$324$		0			0
$325$	−3.46410			−0.192154
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	14.5359i		−	0.801390i
$330$		0			0
$331$			13.8564i			0.761617i	0.924654	+	0.380808i	$0.124354\pi$
$331$			13.8564i			0.761617i	−0.924654	+	0.380808i	$0.875646\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			18.5007i			1.01080i
$336$		0			0
$337$		−	35.1870i		−	1.91676i	−0.285502	−	0.958378i	$-0.592160\pi$
$337$		−	35.1870i		−	1.91676i	0.285502	−	0.958378i	$-0.407840\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			28.3923i			1.53753i
$342$		0			0
$343$	−8.58622			−0.463612
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	28.9014i		−	1.55151i	−0.631034	−	0.775755i	$-0.717369\pi$
$347$		−	28.9014i		−	1.55151i	0.631034	−	0.775755i	$-0.282631\pi$
$348$		0			0
$349$	10.3923			0.556287			0.278144	−	0.960539i	$-0.410281\pi$
$349$	10.3923			0.556287			0.278144	+	0.960539i	$0.410281\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	32.0442			1.70554			0.852770	−	0.522286i	$-0.174921\pi$
$353$	32.0442			1.70554			0.852770	+	0.522286i	$0.174921\pi$
$354$		0			0
$355$	3.62896			0.192605
$356$		0			0
$357$		0			0
$358$		0			0
$359$	28.3923			1.49849			0.749244	−	0.662294i	$-0.230417\pi$
$359$	28.3923			1.49849			0.749244	+	0.662294i	$0.230417\pi$
$360$		0			0
$361$	−12.8564			−0.676653
$362$		0			0
$363$		0			0
$364$		0			0
$365$			24.7846i			1.29729i
$366$		0			0
$367$	20.0721			1.04776			0.523878	−	0.851793i	$-0.324485\pi$
$367$	20.0721			1.04776			0.523878	+	0.851793i	$0.324485\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	1.81448i		−	0.0942032i
$372$		0			0
$373$			13.3004i			0.688667i	0.938847	+	0.344334i	$0.111895\pi$
$373$			13.3004i			0.688667i	−0.938847	+	0.344334i	$0.888105\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	14.8718			0.765935
$378$		0			0
$379$	20.9794			1.07764			0.538819	−	0.842422i	$-0.318871\pi$
$379$	20.9794			1.07764			0.538819	+	0.842422i	$0.318871\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	23.3205			1.19162			0.595811	−	0.803125i	$-0.296831\pi$
$383$	23.3205			1.19162			0.595811	+	0.803125i	$0.296831\pi$
$384$		0			0
$385$			32.0442i			1.63312i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			31.4641i			1.59529i	0.603125	+	0.797647i	$0.293922\pi$
$389$			31.4641i			1.59529i	−0.603125	+	0.797647i	$0.706078\pi$
$390$		0			0
$391$	−3.38587	−	0.928203i	−0.171231	−	0.0469413i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	16.6862i		−	0.839576i
$396$		0			0
$397$	14.0000			0.702640			0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000			0.702640			0.351320	+	0.936255i	$0.385733\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	25.1244i		−	1.25465i	−0.778757	−	0.627325i	$-0.784150\pi$
$401$		−	25.1244i		−	1.25465i	0.778757	−	0.627325i	$-0.215850\pi$
$402$		0			0
$403$			20.7846i			1.03536i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			48.0663i			2.38256i
$408$		0			0
$409$	−16.3923			−0.810547			−0.405274	−	0.914195i	$-0.632824\pi$
$409$	−16.3923			−0.810547			−0.405274	+	0.914195i	$0.632824\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	39.7128i		−	1.95414i
$414$		0			0
$415$		−	4.39230i		−	0.215610i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	32.4449			1.58504			0.792518	−	0.609849i	$-0.208770\pi$
$419$	32.4449			1.58504			0.792518	+	0.609849i	$0.208770\pi$
$420$		0			0
$421$		−	30.4728i		−	1.48515i	−0.669761	−	0.742577i	$-0.733603\pi$
$421$		−	30.4728i		−	1.48515i	0.669761	−	0.742577i	$-0.266397\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			0.732051i			0.0355097i
$426$		0			0
$427$		−	5.32051i		−	0.257477i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	5.07180			0.244300			0.122150	−	0.992512i	$-0.461021\pi$
$431$	5.07180			0.244300			0.122150	+	0.992512i	$0.461021\pi$
$432$		0			0
$433$			3.14277i			0.151032i	0.997145	+	0.0755160i	$0.0240604\pi$
$433$			3.14277i			0.151032i	−0.997145	+	0.0755160i	$0.975940\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.14277	+	11.4641i	−0.150339	+	0.548402i
$438$		0			0
$439$		−	5.32051i		−	0.253934i	−0.991907	−	0.126967i	$-0.959476\pi$
$439$		−	5.32051i		−	0.253934i	0.991907	−	0.126967i	$-0.0405242\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			3.62896i			0.172417i	0.996277	+	0.0862086i	$0.0274752\pi$
$443$			3.62896i			0.172417i	−0.996277	+	0.0862086i	$0.972525\pi$
$444$		0			0
$445$	26.2487			1.24431
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−3.14277			−0.148317			−0.0741583	−	0.997246i	$-0.523627\pi$
$449$	−3.14277			−0.148317			−0.0741583	+	0.997246i	$0.523627\pi$
$450$		0			0
$451$	55.5022			2.61350
$452$		0			0
$453$		0			0
$454$		0			0
$455$			23.4580i			1.09973i
$456$		0			0
$457$		−	3.14277i		−	0.147013i	−0.997295	−	0.0735064i	$-0.976581\pi$
$457$		−	3.14277i		−	0.147013i	0.997295	−	0.0735064i	$-0.0234189\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	28.9014			1.34607			0.673037	−	0.739609i	$-0.264990\pi$
$461$	28.9014			1.34607			0.673037	+	0.739609i	$0.264990\pi$
$462$		0			0
$463$		−	27.4641i		−	1.27637i	−0.769885	−	0.638183i	$-0.779687\pi$
$463$		−	27.4641i		−	1.27637i	0.769885	−	0.638183i	$-0.220313\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−16.0526			−0.742824			−0.371412	−	0.928468i	$-0.621126\pi$
$467$	−16.0526			−0.742824			−0.371412	+	0.928468i	$0.621126\pi$
$468$		0			0
$469$	−31.3205			−1.44625
$470$		0			0
$471$		0			0
$472$		0			0
$473$	11.7290			0.539300
$474$		0			0
$475$	2.47863			0.113727
$476$		0			0
$477$		0			0
$478$		0			0
$479$	21.4641			0.980720			0.490360	−	0.871520i	$-0.336865\pi$
$479$	21.4641			0.980720			0.490360	+	0.871520i	$0.336865\pi$
$480$		0			0
$481$			35.1870i			1.60439i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	23.4580			1.06517
$486$		0			0
$487$			22.3923i			1.01469i	0.861742	+	0.507346i	$0.169373\pi$
$487$			22.3923i			1.01469i	−0.861742	+	0.507346i	$0.830627\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	35.1870i		−	1.58797i	−0.607940	−	0.793983i	$-0.708004\pi$
$491$		−	35.1870i		−	1.58797i	0.607940	−	0.793983i	$-0.291996\pi$
$492$		0			0
$493$		−	3.14277i		−	0.141543i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			6.14359i			0.275578i
$498$		0			0
$499$		−	20.5359i		−	0.919313i	−0.888097	−	0.459657i	$-0.847973\pi$
$499$		−	20.5359i		−	0.919313i	0.888097	−	0.459657i	$-0.152027\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	17.0718			0.761194			0.380597	−	0.924741i	$-0.375719\pi$
$503$	17.0718			0.761194			0.380597	+	0.924741i	$0.375719\pi$
$504$		0			0
$505$			23.4580i			1.04387i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−27.7511			−1.23005			−0.615023	−	0.788509i	$-0.710853\pi$
$509$	−27.7511			−1.23005			−0.615023	+	0.788509i	$0.710853\pi$
$510$		0			0
$511$	−41.9587			−1.85614
$512$		0			0
$513$		0			0
$514$		0			0
$515$			6.77174i			0.298398i
$516$		0			0
$517$			20.3152i			0.893462i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			9.41154i			0.412327i	0.978518	+	0.206164i	$0.0660979\pi$
$521$			9.41154i			0.412327i	−0.978518	+	0.206164i	$0.933902\pi$
$522$		0			0
$523$	−12.3931			−0.541914			−0.270957	−	0.962591i	$-0.587340\pi$
$523$	−12.3931			−0.541914			−0.270957	+	0.962591i	$0.587340\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.39230			0.191332
$528$		0			0
$529$	−19.7846	−	11.7290i	−0.860200	−	0.509956i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	40.6304			1.75990
$534$		0			0
$535$		−	18.2487i		−	0.788961i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−21.1244			−0.909890
$540$		0			0
$541$	−8.53590			−0.366987			−0.183493	−	0.983021i	$-0.558741\pi$
$541$	−8.53590			−0.366987			−0.183493	+	0.983021i	$0.558741\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.14277			0.134622
$546$		0			0
$547$			20.7846i			0.888686i	0.895857	+	0.444343i	$0.146563\pi$
$547$			20.7846i			0.888686i	−0.895857	+	0.444343i	$0.853437\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−10.6410			−0.453323
$552$		0			0
$553$	28.2487			1.20126
$554$		0			0
$555$		0			0
$556$		0			0
$557$			8.14359i			0.345055i	0.985005	+	0.172528i	$0.0551934\pi$
$557$			8.14359i			0.345055i	−0.985005	+	0.172528i	$0.944807\pi$
$558$		0			0
$559$	8.58622			0.363158
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−16.0526			−0.676535			−0.338267	−	0.941050i	$-0.609841\pi$
$563$	−16.0526			−0.676535			−0.338267	+	0.941050i	$0.609841\pi$
$564$		0			0
$565$	26.2487			1.10429
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	14.5885i		−	0.611580i	−0.952099	−	0.305790i	$-0.901079\pi$
$569$		−	14.5885i		−	0.611580i	0.952099	−	0.305790i	$-0.0989205\pi$
$570$		0			0
$571$	−31.3801			−1.31321			−0.656607	−	0.754233i	$-0.728009\pi$
$571$	−31.3801			−1.31321			−0.656607	+	0.754233i	$0.728009\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.26795	+	4.62518i	−0.0528771	+	0.192883i
$576$		0			0
$577$	38.7846			1.61462			0.807312	−	0.590125i	$-0.200921\pi$
$577$	38.7846			1.61462			0.807312	+	0.590125i	$0.200921\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	7.43588			0.308492
$582$		0			0
$583$			2.53590i			0.105026i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	18.5007i		−	0.763607i	−0.924244	−	0.381803i	$-0.875303\pi$
$587$		−	18.5007i		−	0.763607i	0.924244	−	0.381803i	$-0.124697\pi$
$588$		0			0
$589$		−	14.8718i		−	0.612780i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	20.3152			0.834246			0.417123	−	0.908850i	$-0.363038\pi$
$593$	20.3152			0.834246			0.417123	+	0.908850i	$0.363038\pi$
$594$		0			0
$595$	4.95725			0.203228
$596$		0			0
$597$		0			0
$598$		0			0
$599$			31.3801i			1.28215i	0.767476	+	0.641077i	$0.221512\pi$
$599$			31.3801i			1.28215i	−0.767476	+	0.641077i	$0.778488\pi$
$600$		0			0
$601$	12.9282			0.527352			0.263676	−	0.964611i	$-0.415065\pi$
$601$	12.9282			0.527352			0.263676	+	0.964611i	$0.415065\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	22.7846i		−	0.926326i
$606$		0			0
$607$			45.7128i			1.85543i	0.373294	+	0.927713i	$0.378228\pi$
$607$			45.7128i			1.85543i	−0.373294	+	0.927713i	$0.621772\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			14.8718i			0.601647i
$612$		0			0
$613$		−	21.8866i		−	0.883991i	−0.897017	−	0.441996i	$-0.854271\pi$
$613$		−	21.8866i		−	0.883991i	0.897017	−	0.441996i	$-0.145729\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	12.7321i		−	0.512573i	−0.966601	−	0.256287i	$-0.917501\pi$
$617$		−	12.7321i		−	0.512573i	0.966601	−	0.256287i	$-0.0824991\pi$
$618$		0			0
$619$	0.664146			0.0266943			0.0133471	−	0.999911i	$-0.495751\pi$
$619$	0.664146			0.0266943			0.0133471	+	0.999911i	$0.495751\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			44.4373i			1.78034i
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	7.43588			0.296488
$630$		0			0
$631$	−25.5156			−1.01576			−0.507879	−	0.861428i	$-0.669570\pi$
$631$	−25.5156			−1.01576			−0.507879	+	0.861428i	$0.669570\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−17.0718			−0.677474
$636$		0			0
$637$	−15.4641			−0.612710
$638$		0			0
$639$		0			0
$640$		0			0
$641$			3.94744i			0.155915i	0.996957	+	0.0779573i	$0.0248398\pi$
$641$			3.94744i			0.155915i	−0.996957	+	0.0779573i	$0.975160\pi$
$642$		0			0
$643$	−11.0648			−0.436355			−0.218177	−	0.975909i	$-0.570011\pi$
$643$	−11.0648			−0.436355			−0.218177	+	0.975909i	$0.570011\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			26.6008i			1.04578i	0.852399	+	0.522892i	$0.175147\pi$
$647$			26.6008i			1.04578i	−0.852399	+	0.522892i	$0.824853\pi$
$648$		0			0
$649$			55.5022i			2.17865i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	28.9014			1.13100			0.565500	−	0.824748i	$-0.308683\pi$
$653$	28.9014			1.13100			0.565500	+	0.824748i	$0.308683\pi$
$654$		0			0
$655$	−37.0015			−1.44577
$656$		0			0
$657$		0			0
$658$		0			0
$659$	44.4449			1.73133			0.865663	−	0.500627i	$-0.166897\pi$
$659$	44.4449			1.73133			0.865663	+	0.500627i	$0.166897\pi$
$660$		0			0
$661$			1.57139i			0.0611199i	0.999533	+	0.0305599i	$0.00972904\pi$
$661$			1.57139i			0.0611199i	−0.999533	+	0.0305599i	$0.990271\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	16.7846i		−	0.650879i
$666$		0			0
$667$	5.44344	−	19.8564i	0.210771	−	0.768843i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			7.43588i			0.287059i
$672$		0			0
$673$	−14.7846			−0.569905			−0.284952	−	0.958542i	$-0.591978\pi$
$673$	−14.7846			−0.569905			−0.284952	+	0.958542i	$0.591978\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	28.2487i		−	1.08569i	−0.839834	−	0.542843i	$-0.817348\pi$
$677$		−	28.2487i		−	1.08569i	0.839834	−	0.542843i	$-0.182652\pi$
$678$		0			0
$679$			39.7128i			1.52404i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	3.62896i		−	0.138858i	−0.997587	−	0.0694292i	$-0.977882\pi$
$683$		−	3.62896i		−	0.138858i	0.997587	−	0.0694292i	$-0.0221178\pi$
$684$		0			0
$685$	−36.3923			−1.39048
$686$		0			0
$687$		0			0
$688$		0			0
$689$			1.85641i			0.0707235i
$690$		0			0
$691$		−	6.92820i		−	0.263561i	−0.991279	−	0.131781i	$-0.957931\pi$
$691$		−	6.92820i		−	0.263561i	0.991279	−	0.131781i	$-0.0420694\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	17.0718			0.647570
$696$		0			0
$697$		−	8.58622i		−	0.325226i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	27.8564i		−	1.05212i	−0.850447	−	0.526061i	$-0.823668\pi$
$701$		−	27.8564i		−	1.05212i	0.850447	−	0.526061i	$-0.176332\pi$
$702$		0			0
$703$		−	25.1769i		−	0.949565i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−39.7128			−1.49355
$708$		0			0
$709$			21.8866i			0.821968i	0.911642	+	0.410984i	$0.134815\pi$
$709$			21.8866i			0.821968i	−0.911642	+	0.410984i	$0.865185\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	27.7511	+	7.60770i	1.03929	+	0.284910i
$714$		0			0
$715$		−	32.7846i		−	1.22607i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	3.14277i		−	0.117206i	−0.998281	−	0.0586028i	$-0.981335\pi$
$719$		−	3.14277i		−	0.117206i	0.998281	−	0.0586028i	$-0.0186646\pi$
$720$		0			0
$721$	−11.4641			−0.426945
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.29311			−0.159442
$726$		0			0
$727$	−41.7156			−1.54715			−0.773573	−	0.633707i	$-0.781533\pi$
$727$	−41.7156			−1.54715			−0.773573	+	0.633707i	$0.781533\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	1.81448i		−	0.0671110i
$732$		0			0
$733$		−	27.3300i		−	1.00946i	−0.863278	−	0.504729i	$-0.831592\pi$
$733$		−	27.3300i		−	1.00946i	0.863278	−	0.504729i	$-0.168408\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	43.7732			1.61241
$738$		0			0
$739$			24.0000i			0.882854i	0.897297	+	0.441427i	$0.145528\pi$
$739$			24.0000i			0.882854i	−0.897297	+	0.441427i	$0.854472\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−32.1051			−1.17782			−0.588911	−	0.808198i	$-0.700443\pi$
$743$	−32.1051			−1.17782			−0.588911	+	0.808198i	$0.700443\pi$
$744$		0			0
$745$	26.6410			0.976051
$746$		0			0
$747$		0			0
$748$		0			0
$749$	30.8939			1.12884
$750$		0			0
$751$	47.1591			1.72086			0.860429	−	0.509570i	$-0.170196\pi$
$751$	47.1591			1.72086			0.860429	+	0.509570i	$0.170196\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−25.8564			−0.941011
$756$		0			0
$757$			10.1576i			0.369184i	0.982815	+	0.184592i	$0.0590964\pi$
$757$			10.1576i			0.369184i	−0.982815	+	0.184592i	$0.940904\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−8.58622			−0.311250			−0.155625	−	0.987816i	$-0.549739\pi$
$761$	−8.58622			−0.311250			−0.155625	+	0.987816i	$0.549739\pi$
$762$		0			0
$763$			5.32051i			0.192615i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			40.6304i			1.46708i
$768$		0			0
$769$		−	14.8718i		−	0.536290i	−0.963379	−	0.268145i	$-0.913589\pi$
$769$		−	14.8718i		−	0.536290i	0.963379	−	0.268145i	$-0.0864106\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	25.7128i		−	0.924826i	−0.886665	−	0.462413i	$-0.846984\pi$
$773$		−	25.7128i		−	0.924826i	0.886665	−	0.462413i	$-0.153016\pi$
$774$		0			0
$775$		−	6.00000i		−	0.215526i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−29.0718			−1.04161
$780$		0			0
$781$		−	8.58622i		−	0.307239i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	43.7732			1.56233
$786$		0			0
$787$	20.9794			0.747833			0.373917	−	0.927462i	$-0.378015\pi$
$787$	20.9794			0.747833			0.373917	+	0.927462i	$0.378015\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			44.4373i			1.58001i
$792$		0			0
$793$			5.44344i			0.193302i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			6.39230i			0.226427i	0.993571	+	0.113214i	$0.0361144\pi$
$797$			6.39230i			0.226427i	−0.993571	+	0.113214i	$0.963886\pi$
$798$		0			0
$799$	3.14277			0.111183
$800$		0			0
$801$		0			0
$802$		0			0
$803$	58.6410			2.06940
$804$		0			0
$805$	31.3205	+	8.58622i	1.10390	+	0.302624i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	8.58622			0.301875			0.150938	−	0.988543i	$-0.451771\pi$
$809$	8.58622			0.301875			0.150938	+	0.988543i	$0.451771\pi$
$810$		0			0
$811$		−	15.7128i		−	0.551751i	−0.961193	−	0.275876i	$-0.911032\pi$
$811$		−	15.7128i		−	0.551751i	0.961193	−	0.275876i	$-0.0889678\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.92820			−0.242684
$816$		0			0
$817$	−6.14359			−0.214937
$818$		0			0
$819$		0			0
$820$		0			0
$821$	52.3594			1.82736			0.913678	−	0.406439i	$-0.133230\pi$
$821$	52.3594			1.82736			0.913678	+	0.406439i	$0.133230\pi$
$822$		0			0
$823$			21.7128i			0.756861i	0.925630	+	0.378431i	$0.123536\pi$
$823$			21.7128i			0.756861i	−0.925630	+	0.378431i	$0.876464\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−6.58846			−0.229103			−0.114552	−	0.993417i	$-0.536543\pi$
$827$	−6.58846			−0.229103			−0.114552	+	0.993417i	$0.536543\pi$
$828$		0			0
$829$	20.5359			0.713241			0.356621	−	0.934249i	$-0.383929\pi$
$829$	20.5359			0.713241			0.356621	+	0.934249i	$0.383929\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			3.26795i			0.113228i
$834$		0			0
$835$	−3.62896			−0.125585
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−29.5692			−1.02084			−0.510421	−	0.859924i	$-0.670511\pi$
$839$	−29.5692			−1.02084			−0.510421	+	0.859924i	$0.670511\pi$
$840$		0			0
$841$	−10.5692			−0.364456
$842$		0			0
$843$		0			0
$844$		0			0
$845$			2.00000i			0.0688021i
$846$		0			0
$847$	38.5728			1.32538
$848$		0			0
$849$		0			0
$850$		0			0
$851$	46.9808	+	12.8793i	1.61048	+	0.441498i
$852$		0			0
$853$	10.0000			0.342393			0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000			0.342393			0.171197	+	0.985237i	$0.445237\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	32.0442			1.09461			0.547304	−	0.836934i	$-0.315654\pi$
$857$	32.0442			1.09461			0.547304	+	0.836934i	$0.315654\pi$
$858$		0			0
$859$			12.0000i			0.409435i	0.978821	+	0.204717i	$0.0656275\pi$
$859$			12.0000i			0.409435i	−0.978821	+	0.204717i	$0.934372\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			3.14277i			0.106981i	0.998568	+	0.0534906i	$0.0170347\pi$
$863$			3.14277i			0.106981i	−0.998568	+	0.0534906i	$0.982965\pi$
$864$		0			0
$865$			8.58622i			0.291940i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−39.4801			−1.33927
$870$		0			0
$871$	32.0442			1.08578
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	40.6304i		−	1.37356i
$876$		0			0
$877$	−22.7846			−0.769382			−0.384691	−	0.923045i	$-0.625692\pi$
$877$	−22.7846			−0.769382			−0.384691	+	0.923045i	$0.625692\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			12.7321i			0.428954i	0.976729	+	0.214477i	$0.0688046\pi$
$881$			12.7321i			0.428954i	−0.976729	+	0.214477i	$0.931195\pi$
$882$		0			0
$883$			10.6410i			0.358099i	0.983840	+	0.179049i	$0.0573022\pi$
$883$			10.6410i			0.358099i	−0.983840	+	0.179049i	$0.942698\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			4.29311i			0.144148i	0.997399	+	0.0720742i	$0.0229619\pi$
$887$			4.29311i			0.144148i	−0.997399	+	0.0720742i	$0.977038\pi$
$888$		0			0
$889$		−	28.9014i		−	0.969323i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	10.6410i		−	0.356088i
$894$		0			0
$895$	−23.4580			−0.784114
$896$		0			0
$897$		0			0
$898$		0			0
$899$			25.7587i			0.859099i
$900$		0			0
$901$	0.392305			0.0130696
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−26.6008			−0.884239
$906$		0			0
$907$	51.6953			1.71651			0.858256	−	0.513221i	$-0.171548\pi$
$907$	51.6953			1.71651			0.858256	+	0.513221i	$0.171548\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	35.3205			1.17022			0.585110	−	0.810954i	$-0.301051\pi$
$911$	35.3205			1.17022			0.585110	+	0.810954i	$0.301051\pi$
$912$		0			0
$913$	−10.3923			−0.343935
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	62.6410i		−	2.06859i
$918$		0			0
$919$	−20.0721			−0.662118			−0.331059	−	0.943610i	$-0.607406\pi$
$919$	−20.0721			−0.662118			−0.331059	+	0.943610i	$0.607406\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	6.28555i		−	0.206891i
$924$		0			0
$925$		−	10.1576i		−	0.333980i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−26.6008			−0.872742			−0.436371	−	0.899767i	$-0.643737\pi$
$929$	−26.6008			−0.872742			−0.436371	+	0.899767i	$0.643737\pi$
$930$		0			0
$931$	11.0648			0.362636
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−6.92820			−0.226576
$936$		0			0
$937$		−	8.58622i		−	0.280499i	−0.990116	−	0.140250i	$-0.955209\pi$
$937$		−	8.58622i		−	0.280499i	0.990116	−	0.140250i	$-0.0447905\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			31.4641i			1.02570i	0.858478	+	0.512850i	$0.171410\pi$
$941$			31.4641i			1.02570i	−0.858478	+	0.512850i	$0.828590\pi$
$942$		0			0
$943$	14.8718	−	54.2487i	0.484291	−	1.76658i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			43.7732i			1.42244i	0.702971	+	0.711219i	$0.251856\pi$
$947$			43.7732i			1.42244i	−0.702971	+	0.711219i	$0.748144\pi$
$948$		0			0
$949$	42.9282			1.39351
$950$		0			0
$951$		0			0
$952$		0			0
$953$			39.6603i			1.28472i	0.766402	+	0.642361i	$0.222045\pi$
$953$			39.6603i			1.28472i	−0.766402	+	0.642361i	$0.777955\pi$
$954$		0			0
$955$			27.7128i			0.896766i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	61.6098i		−	1.98948i
$960$		0			0
$961$	−5.00000			−0.161290
$962$		0			0
$963$		0			0
$964$		0			0
$965$			18.9282i			0.609320i
$966$		0			0
$967$		−	43.8564i		−	1.41033i	−0.709045	−	0.705163i	$-0.750874\pi$
$967$		−	43.8564i		−	1.41033i	0.709045	−	0.705163i	$-0.249126\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−24.3397			−0.781100			−0.390550	−	0.920582i	$-0.627715\pi$
$971$	−24.3397			−0.781100			−0.390550	+	0.920582i	$0.627715\pi$
$972$		0			0
$973$			28.9014i			0.926537i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	22.1962i		−	0.710118i	−0.934844	−	0.355059i	$-0.884461\pi$
$977$		−	22.1962i		−	0.710118i	0.934844	−	0.355059i	$-0.115539\pi$
$978$		0			0
$979$		−	62.1051i		−	1.98489i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	50.5359			1.61184			0.805922	−	0.592021i	$-0.201670\pi$
$983$	50.5359			1.61184			0.805922	+	0.592021i	$0.201670\pi$
$984$		0			0
$985$			23.4580i			0.747433i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3.14277	−	11.4641i	0.0999344	−	0.364537i
$990$		0			0
$991$		−	26.7846i		−	0.850841i	−0.904996	−	0.425421i	$-0.860126\pi$
$991$		−	26.7846i		−	0.850841i	0.904996	−	0.425421i	$-0.139874\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	23.9442i		−	0.759081i
$996$		0			0
$997$	−33.0333			−1.04618			−0.523088	−	0.852279i	$-0.675220\pi$
$997$	−33.0333			−1.04618			−0.523088	+	0.852279i	$0.675220\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3312.2.i.b.2575.4	yes	8
3.2	odd	2		3312.2.i.f.2575.8	yes	8
4.3	odd	2		3312.2.i.f.2575.1	yes	8
12.11	even	2	inner	3312.2.i.b.2575.5	yes	8
23.22	odd	2		3312.2.i.f.2575.5	yes	8
69.68	even	2	inner	3312.2.i.b.2575.1	✓	8
92.91	even	2	inner	3312.2.i.b.2575.8	yes	8
276.275	odd	2		3312.2.i.f.2575.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3312.2.i.b.2575.1	✓	8	69.68	even	2	inner
3312.2.i.b.2575.4	yes	8	1.1	even	1	trivial
3312.2.i.b.2575.5	yes	8	12.11	even	2	inner
3312.2.i.b.2575.8	yes	8	92.91	even	2	inner
3312.2.i.f.2575.1	yes	8	4.3	odd	2
3312.2.i.f.2575.4	yes	8	276.275	odd	2
3312.2.i.f.2575.5	yes	8	23.22	odd	2
3312.2.i.f.2575.8	yes	8	3.2	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:	\( N \)	\(=\)	\( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3312.i (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{5} +3.38587 q^{7} +O(q^{10})\)	\(q-2.00000i q^{5} +3.38587 q^{7} -4.73205 q^{11} -3.46410 q^{13} +0.732051i q^{17} +2.47863 q^{19} +(-1.26795 + 4.62518i) q^{23} +1.00000 q^{25} -4.29311 q^{29} -6.00000i q^{31} -6.77174i q^{35} -10.1576i q^{37} -11.7290 q^{41} -2.47863 q^{43} -4.29311i q^{47} +4.46410 q^{49} -0.535898i q^{53} +9.46410i q^{55} -11.7290i q^{59} -1.57139i q^{61} +6.92820i q^{65} -9.25036 q^{67} +1.81448i q^{71} -12.3923 q^{73} -16.0221 q^{77} +8.34312 q^{79} +2.19615 q^{83} +1.46410 q^{85} +13.1244i q^{89} -11.7290 q^{91} -4.95725i q^{95} +11.7290i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 24 q^{11} - 24 q^{23} + 8 q^{25} + 8 q^{49} - 16 q^{73} - 24 q^{83} - 16 q^{85}+O(q^{100}) \) 8 * q - 24 * q^11 - 24 * q^23 + 8 * q^25 + 8 * q^49 - 16 * q^73 - 24 * q^83 - 16 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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