LMFDB - Embedded newform 2352.2.k.j.881.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2352,2,Mod(881,2352)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [24,0,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,8,0,0,0,0, 0,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 1176)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.16
Character	$\chi$	$=$	2352.881
Dual form			2352.2.k.j.881.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.615077 + 1.61916i) q^{3} +1.31193 q^{5} +(-2.24336 + 1.99182i) q^{9} -4.97953i q^{11} -0.733252i q^{13} +(0.806939 + 2.12423i) q^{15} +0.728350 q^{17} -6.94457i q^{19} -7.92926i q^{23} -3.27884 q^{25} +(-4.60491 - 2.40724i) q^{27} -4.62960i q^{29} -1.80300i q^{31} +(8.06265 - 3.06280i) q^{33} -5.22423 q^{37} +(1.18725 - 0.451007i) q^{39} +7.83857 q^{41} +1.27217 q^{43} +(-2.94313 + 2.61313i) q^{45} +4.24163 q^{47} +(0.447992 + 1.17932i) q^{51} +14.4166i q^{53} -6.53279i q^{55} +(11.2444 - 4.27145i) q^{57} -9.35318 q^{59} -13.5881i q^{61} -0.961975i q^{65} +7.54811 q^{67} +(12.8387 - 4.87711i) q^{69} +6.92721i q^{71} -3.79913i q^{73} +(-2.01674 - 5.30897i) q^{75} -13.5928 q^{79} +(1.06532 - 8.93673i) q^{81} -6.58972 q^{83} +0.955545 q^{85} +(7.49607 - 2.84757i) q^{87} -4.37845 q^{89} +(2.91935 - 1.10898i) q^{93} -9.11079i q^{95} +15.1510i q^{97} +(9.91831 + 11.1709i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 16 q^{15} + 8 q^{25} + 16 q^{37} + 64 q^{39} - 16 q^{43} - 48 q^{51} + 48 q^{57} - 16 q^{67} + 80 q^{81} - 64 q^{85} - 32 q^{93} + 56 q^{99}+O(q^{100}) $ 24 * q + 16 * q^15 + 8 * q^25 + 16 * q^37 + 64 * q^39 - 16 * q^43 - 48 * q^51 + 48 * q^57 - 16 * q^67 + 80 * q^81 - 64 * q^85 - 32 * q^93 + 56 * q^99

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
$3$	0.615077	+	1.61916i	0.355115	+	0.934823i
$3$	0.615077	+	1.61916i	0.355115	+	0.934823i
$4$		0			0
$5$	1.31193			0.586713			0.293356	−	0.956003i	$-0.405228\pi$
$5$	1.31193			0.586713			0.293356	+	0.956003i	$0.405228\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−2.24336	+	1.99182i	−0.747786	+	0.663939i
$10$		0			0
$11$		−	4.97953i		−	1.50138i	−0.660652	−	0.750692i	$-0.729720\pi$
$11$		−	4.97953i		−	1.50138i	0.660652	−	0.750692i	$-0.270280\pi$
$12$		0			0
$13$		−	0.733252i		−	0.203367i	−0.994817	−	0.101684i	$-0.967577\pi$
$13$		−	0.733252i		−	0.203367i	0.994817	−	0.101684i	$-0.0324230\pi$
$14$		0			0
$15$	0.806939	+	2.12423i	0.208351	+	0.548473i
$16$		0			0
$17$	0.728350			0.176651			0.0883255	−	0.996092i	$-0.471848\pi$
$17$	0.728350			0.176651			0.0883255	+	0.996092i	$0.471848\pi$
$18$		0			0
$19$		−	6.94457i		−	1.59319i	−0.604511	−	0.796597i	$-0.706631\pi$
$19$		−	6.94457i		−	1.59319i	0.604511	−	0.796597i	$-0.293369\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	7.92926i		−	1.65337i	−0.562668	−	0.826683i	$-0.690225\pi$
$23$		−	7.92926i		−	1.65337i	0.562668	−	0.826683i	$-0.309775\pi$
$24$		0			0
$25$	−3.27884			−0.655768
$26$		0			0
$27$	−4.60491	−	2.40724i	−0.886216	−	0.463273i
$28$		0			0
$29$		−	4.62960i		−	0.859696i	−0.902901	−	0.429848i	$-0.858567\pi$
$29$		−	4.62960i		−	0.859696i	0.902901	−	0.429848i	$-0.141433\pi$
$30$		0			0
$31$		−	1.80300i		−	0.323828i	−0.986805	−	0.161914i	$-0.948233\pi$
$31$		−	1.80300i		−	0.323828i	0.986805	−	0.161914i	$-0.0517668\pi$
$32$		0			0
$33$	8.06265	−	3.06280i	1.40353	−	0.533164i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.22423			−0.858858			−0.429429	−	0.903101i	$-0.641285\pi$
$37$	−5.22423			−0.858858			−0.429429	+	0.903101i	$0.641285\pi$
$38$		0			0
$39$	1.18725	−	0.451007i	0.190112	−	0.0722189i
$40$		0			0
$41$	7.83857			1.22418			0.612089	−	0.790789i	$-0.290329\pi$
$41$	7.83857			1.22418			0.612089	+	0.790789i	$0.290329\pi$
$42$		0			0
$43$	1.27217			0.194004			0.0970020	−	0.995284i	$-0.469075\pi$
$43$	1.27217			0.194004			0.0970020	+	0.995284i	$0.469075\pi$
$44$		0			0
$45$	−2.94313	+	2.61313i	−0.438736	+	0.389542i
$46$		0			0
$47$	4.24163			0.618705			0.309352	−	0.950947i	$-0.399888\pi$
$47$	4.24163			0.618705			0.309352	+	0.950947i	$0.399888\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	0.447992	+	1.17932i	0.0627314	+	0.165137i
$52$		0			0
$53$			14.4166i			1.98027i	0.140107	+	0.990136i	$0.455255\pi$
$53$			14.4166i			1.98027i	−0.140107	+	0.990136i	$0.544745\pi$
$54$		0			0
$55$		−	6.53279i		−	0.880882i
$56$		0			0
$57$	11.2444	−	4.27145i	1.48935	−	0.565767i
$58$		0			0
$59$	−9.35318			−1.21768			−0.608840	−	0.793293i	$-0.708365\pi$
$59$	−9.35318			−1.21768			−0.608840	+	0.793293i	$0.708365\pi$
$60$		0			0
$61$		−	13.5881i		−	1.73978i	−0.493245	−	0.869890i	$-0.664189\pi$
$61$		−	13.5881i		−	1.73978i	0.493245	−	0.869890i	$-0.335811\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	0.961975i		−	0.119318i
$66$		0			0
$67$	7.54811			0.922148			0.461074	−	0.887362i	$-0.347464\pi$
$67$	7.54811			0.922148			0.461074	+	0.887362i	$0.347464\pi$
$68$		0			0
$69$	12.8387	−	4.87711i	1.54560	−	0.587135i
$70$		0			0
$71$			6.92721i			0.822109i	0.911611	+	0.411055i	$0.134839\pi$
$71$			6.92721i			0.822109i	−0.911611	+	0.411055i	$0.865161\pi$
$72$		0			0
$73$		−	3.79913i		−	0.444654i	−0.974972	−	0.222327i	$-0.928635\pi$
$73$		−	3.79913i		−	0.444654i	0.974972	−	0.222327i	$-0.0713653\pi$
$74$		0			0
$75$	−2.01674	−	5.30897i	−0.232873	−	0.613027i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−13.5928			−1.52931			−0.764653	−	0.644442i	$-0.777090\pi$
$79$	−13.5928			−1.52931			−0.764653	+	0.644442i	$0.777090\pi$
$80$		0			0
$81$	1.06532	−	8.93673i	0.118369	−	0.992970i
$82$		0			0
$83$	−6.58972			−0.723316			−0.361658	−	0.932311i	$-0.617789\pi$
$83$	−6.58972			−0.723316			−0.361658	+	0.932311i	$0.617789\pi$
$84$		0			0
$85$	0.955545			0.103643
$86$		0			0
$87$	7.49607	−	2.84757i	0.803663	−	0.305291i
$88$		0			0
$89$	−4.37845			−0.464115			−0.232057	−	0.972702i	$-0.574546\pi$
$89$	−4.37845			−0.464115			−0.232057	+	0.972702i	$0.574546\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	2.91935	−	1.10898i	0.302722	−	0.114996i
$94$		0			0
$95$		−	9.11079i		−	0.934748i
$96$		0			0
$97$			15.1510i			1.53835i	0.639039	+	0.769174i	$0.279332\pi$
$97$			15.1510i			1.53835i	−0.639039	+	0.769174i	$0.720668\pi$
$98$		0			0
$99$	9.91831	+	11.1709i	0.996828	+	1.12271i
$100$		0			0
$101$	10.9959			1.09414			0.547068	−	0.837088i	$-0.315744\pi$
$101$	10.9959			1.09414			0.547068	+	0.837088i	$0.315744\pi$
$102$		0			0
$103$		−	10.4292i		−	1.02762i	−0.857903	−	0.513812i	$-0.828233\pi$
$103$		−	10.4292i		−	1.02762i	0.857903	−	0.513812i	$-0.171767\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			3.08012i			0.297767i	0.988855	+	0.148883i	$0.0475679\pi$
$107$			3.08012i			0.297767i	−0.988855	+	0.148883i	$0.952432\pi$
$108$		0			0
$109$	13.7512			1.31713			0.658565	−	0.752524i	$-0.271164\pi$
$109$	13.7512			1.31713			0.658565	+	0.752524i	$0.271164\pi$
$110$		0			0
$111$	−3.21330	−	8.45886i	−0.304993	−	0.802879i
$112$		0			0
$113$			12.6414i			1.18920i	0.804022	+	0.594599i	$0.202689\pi$
$113$			12.6414i			1.18920i	−0.804022	+	0.594599i	$0.797311\pi$
$114$		0			0
$115$		−	10.4026i		−	0.970051i
$116$		0			0
$117$	1.46050	+	1.64495i	0.135024	+	0.152075i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−13.7957			−1.25415
$122$		0			0
$123$	4.82133	+	12.6919i	0.434724	+	1.14439i
$124$		0			0
$125$	−10.8613			−0.971461
$126$		0			0
$127$	−3.31989			−0.294593			−0.147296	−	0.989092i	$-0.547057\pi$
$127$	−3.31989			−0.294593			−0.147296	+	0.989092i	$0.547057\pi$
$128$		0			0
$129$	0.782483	+	2.05985i	0.0688937	+	0.181359i
$130$		0			0
$131$	16.0122			1.39899			0.699497	−	0.714635i	$-0.253407\pi$
$131$	16.0122			1.39899			0.699497	+	0.714635i	$0.253407\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−6.04132	−	3.15813i	−0.519954	−	0.271808i
$136$		0			0
$137$			10.7533i			0.918720i	0.888250	+	0.459360i	$0.151921\pi$
$137$			10.7533i			0.918720i	−0.888250	+	0.459360i	$0.848079\pi$
$138$		0			0
$139$			18.5535i			1.57369i	0.617152	+	0.786844i	$0.288286\pi$
$139$			18.5535i			1.57369i	−0.617152	+	0.786844i	$0.711714\pi$
$140$		0			0
$141$	2.60893	+	6.86787i	0.219711	+	0.578379i
$142$		0			0
$143$	−3.65125			−0.305333
$144$		0			0
$145$		−	6.07372i		−	0.504395i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			4.86849i			0.398843i	0.979914	+	0.199421i	$0.0639062\pi$
$149$			4.86849i			0.398843i	−0.979914	+	0.199421i	$0.936094\pi$
$150$		0			0
$151$	−1.70120			−0.138442			−0.0692209	−	0.997601i	$-0.522051\pi$
$151$	−1.70120			−0.138442			−0.0692209	+	0.997601i	$0.522051\pi$
$152$		0			0
$153$	−1.63395	+	1.45074i	−0.132097	+	0.117285i
$154$		0			0
$155$		−	2.36541i		−	0.189994i
$156$		0			0
$157$		−	18.1776i		−	1.45073i	−0.688363	−	0.725366i	$-0.741670\pi$
$157$		−	18.1776i		−	1.45073i	0.688363	−	0.725366i	$-0.258330\pi$
$158$		0			0
$159$	−23.3428	+	8.86733i	−1.85120	+	0.703225i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	5.02332			0.393457			0.196728	−	0.980458i	$-0.436968\pi$
$163$	5.02332			0.393457			0.196728	+	0.980458i	$0.436968\pi$
$164$		0			0
$165$	10.5776	−	4.01817i	0.823468	−	0.312814i
$166$		0			0
$167$	16.4374			1.27196			0.635982	−	0.771704i	$-0.280595\pi$
$167$	16.4374			1.27196			0.635982	+	0.771704i	$0.280595\pi$
$168$		0			0
$169$	12.4623			0.958642
$170$		0			0
$171$	13.8323	+	15.5792i	1.05778	+	1.19137i
$172$		0			0
$173$	−4.20447			−0.319660			−0.159830	−	0.987145i	$-0.551095\pi$
$173$	−4.20447			−0.319660			−0.159830	+	0.987145i	$0.551095\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−5.75293	−	15.1443i	−0.432417	−	1.13831i
$178$		0			0
$179$			8.14004i			0.608415i	0.952606	+	0.304208i	$0.0983917\pi$
$179$			8.14004i			0.608415i	−0.952606	+	0.304208i	$0.901608\pi$
$180$		0			0
$181$		−	17.8884i		−	1.32963i	−0.747007	−	0.664817i	$-0.768510\pi$
$181$		−	17.8884i		−	1.32963i	0.747007	−	0.664817i	$-0.231490\pi$
$182$		0			0
$183$	22.0013	−	8.35775i	1.62639	−	0.617823i
$184$		0			0
$185$	−6.85382			−0.503903
$186$		0			0
$187$		−	3.62684i		−	0.265221i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	4.11041i		−	0.297418i	−0.988881	−	0.148709i	$-0.952488\pi$
$191$		−	4.11041i		−	0.297418i	0.988881	−	0.148709i	$-0.0475119\pi$
$192$		0			0
$193$	−3.22919			−0.232442			−0.116221	−	0.993223i	$-0.537078\pi$
$193$	−3.22919			−0.232442			−0.116221	+	0.993223i	$0.537078\pi$
$194$		0			0
$195$	1.55759	−	0.591689i	0.111541	−	0.0423717i
$196$		0			0
$197$			4.70768i			0.335408i	0.985837	+	0.167704i	$0.0536353\pi$
$197$			4.70768i			0.335408i	−0.985837	+	0.167704i	$0.946365\pi$
$198$		0			0
$199$		−	16.9835i		−	1.20393i	−0.798523	−	0.601964i	$-0.794385\pi$
$199$		−	16.9835i		−	1.20393i	0.798523	−	0.601964i	$-0.205615\pi$
$200$		0			0
$201$	4.64267	+	12.2216i	0.327469	+	0.862045i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	10.2837			0.718241
$206$		0			0
$207$	15.7936	+	17.7882i	1.09773	+	1.23636i
$208$		0			0
$209$	−34.5807			−2.39200
$210$		0			0
$211$	1.89126			0.130199			0.0650997	−	0.997879i	$-0.479263\pi$
$211$	1.89126			0.130199			0.0650997	+	0.997879i	$0.479263\pi$
$212$		0			0
$213$	−11.2163	+	4.26077i	−0.768526	+	0.291943i
$214$		0			0
$215$	1.66900			0.113825
$216$		0			0
$217$		0			0
$218$		0			0
$219$	6.15140	−	2.33676i	0.415673	−	0.157904i
$220$		0			0
$221$		−	0.534064i		−	0.0359250i
$222$		0			0
$223$			9.29671i			0.622554i	0.950319	+	0.311277i	$0.100757\pi$
$223$			9.29671i			0.622554i	−0.950319	+	0.311277i	$0.899243\pi$
$224$		0			0
$225$	7.35562	−	6.53085i	0.490374	−	0.435390i
$226$		0			0
$227$	4.96994			0.329867			0.164933	−	0.986305i	$-0.447259\pi$
$227$	4.96994			0.329867			0.164933	+	0.986305i	$0.447259\pi$
$228$		0			0
$229$		−	5.33157i		−	0.352320i	−0.984362	−	0.176160i	$-0.943632\pi$
$229$		−	5.33157i		−	0.352320i	0.984362	−	0.176160i	$-0.0563676\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	3.50235i		−	0.229447i	−0.993397	−	0.114723i	$-0.963402\pi$
$233$		−	3.50235i		−	0.229447i	0.993397	−	0.114723i	$-0.0365981\pi$
$234$		0			0
$235$	5.56472			0.363002
$236$		0			0
$237$	−8.36061	−	22.0089i	−0.543080	−	1.42963i
$238$		0			0
$239$		−	23.4769i		−	1.51859i	−0.650744	−	0.759297i	$-0.725543\pi$
$239$		−	23.4769i		−	1.51859i	0.650744	−	0.759297i	$-0.274457\pi$
$240$		0			0
$241$		−	10.0674i		−	0.648500i	−0.945971	−	0.324250i	$-0.894888\pi$
$241$		−	10.0674i		−	0.648500i	0.945971	−	0.324250i	$-0.105112\pi$
$242$		0			0
$243$	15.1252	−	3.77185i	0.970285	−	0.241964i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−5.09212			−0.324004
$248$		0			0
$249$	−4.05319	−	10.6698i	−0.256860	−	0.676172i
$250$		0			0
$251$	19.7932			1.24933			0.624666	−	0.780892i	$-0.285235\pi$
$251$	19.7932			1.24933			0.624666	+	0.780892i	$0.285235\pi$
$252$		0			0
$253$	−39.4840			−2.48234
$254$		0			0
$255$	0.587734	+	1.54718i	0.0368053	+	0.0968882i
$256$		0			0
$257$	−12.6163			−0.786984			−0.393492	−	0.919328i	$-0.628733\pi$
$257$	−12.6163			−0.786984			−0.393492	+	0.919328i	$0.628733\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	9.22133	+	10.3859i	0.570786	+	0.642869i
$262$		0			0
$263$			6.33199i			0.390447i	0.980759	+	0.195224i	$0.0625433\pi$
$263$			6.33199i			0.390447i	−0.980759	+	0.195224i	$0.937457\pi$
$264$		0			0
$265$			18.9136i			1.16185i
$266$		0			0
$267$	−2.69309	−	7.08941i	−0.164814	−	0.433865i
$268$		0			0
$269$	24.3839			1.48671			0.743356	−	0.668896i	$-0.233233\pi$
$269$	24.3839			1.48671			0.743356	+	0.668896i	$0.233233\pi$
$270$		0			0
$271$			0.344072i			0.0209009i	0.999945	+	0.0104504i	$0.00332654\pi$
$271$			0.344072i			0.0209009i	−0.999945	+	0.0104504i	$0.996673\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			16.3271i			0.984559i
$276$		0			0
$277$	12.2360			0.735187			0.367594	−	0.929987i	$-0.380182\pi$
$277$	12.2360			0.735187			0.367594	+	0.929987i	$0.380182\pi$
$278$		0			0
$279$	3.59125	+	4.04478i	0.215002	+	0.242155i
$280$		0			0
$281$			16.6761i			0.994810i	0.867519	+	0.497405i	$0.165714\pi$
$281$			16.6761i			0.994810i	−0.867519	+	0.497405i	$0.834286\pi$
$282$		0			0
$283$			3.10159i			0.184371i	0.995742	+	0.0921853i	$0.0293852\pi$
$283$			3.10159i			0.184371i	−0.995742	+	0.0921853i	$0.970615\pi$
$284$		0			0
$285$	14.7518	−	5.60384i	0.873823	−	0.331943i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−16.4695			−0.968794
$290$		0			0
$291$	−24.5319	+	9.31902i	−1.43808	+	0.546291i
$292$		0			0
$293$	16.2490			0.949278			0.474639	−	0.880181i	$-0.342579\pi$
$293$	16.2490			0.949278			0.474639	+	0.880181i	$0.342579\pi$
$294$		0			0
$295$	−12.2707			−0.714429
$296$		0			0
$297$	−11.9869	+	22.9303i	−0.695550	+	1.33055i
$298$		0			0
$299$	−5.81415			−0.336241
$300$		0			0
$301$		0			0
$302$		0			0
$303$	6.76334	+	17.8042i	0.388544	+	1.02282i
$304$		0			0
$305$		−	17.8267i		−	1.02075i
$306$		0			0
$307$			2.82559i			0.161265i	0.996744	+	0.0806326i	$0.0256940\pi$
$307$			2.82559i			0.161265i	−0.996744	+	0.0806326i	$0.974306\pi$
$308$		0			0
$309$	16.8866	−	6.41479i	0.960646	−	0.364925i
$310$		0			0
$311$	16.4036			0.930163			0.465081	−	0.885268i	$-0.346025\pi$
$311$	16.4036			0.930163			0.465081	+	0.885268i	$0.346025\pi$
$312$		0			0
$313$			0.324800i			0.0183588i	0.999958	+	0.00917938i	$0.00292193\pi$
$313$			0.324800i			0.0183588i	−0.999958	+	0.00917938i	$0.997078\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	12.0488i		−	0.676726i	−0.941016	−	0.338363i	$-0.890127\pi$
$317$		−	12.0488i		−	0.676726i	0.941016	−	0.338363i	$-0.109873\pi$
$318$		0			0
$319$	−23.0532			−1.29073
$320$		0			0
$321$	−4.98721	+	1.89451i	−0.278359	+	0.105742i
$322$		0			0
$323$		−	5.05808i		−	0.281439i
$324$		0			0
$325$			2.40421i			0.133362i
$326$		0			0
$327$	8.45808	+	22.2655i	0.467733	+	1.23128i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−7.00005			−0.384758			−0.192379	−	0.981321i	$-0.561620\pi$
$331$	−7.00005			−0.384758			−0.192379	+	0.981321i	$0.561620\pi$
$332$		0			0
$333$	11.7198	−	10.4057i	0.642242	−	0.570229i
$334$		0			0
$335$	9.90259			0.541036
$336$		0			0
$337$	−12.2829			−0.669094			−0.334547	−	0.942379i	$-0.608583\pi$
$337$	−12.2829			−0.669094			−0.334547	+	0.942379i	$0.608583\pi$
$338$		0			0
$339$	−20.4684	+	7.77541i	−1.11169	+	0.422302i
$340$		0			0
$341$	−8.97809			−0.486191
$342$		0			0
$343$		0			0
$344$		0			0
$345$	16.8435	−	6.39843i	0.906826	−	0.344480i
$346$		0			0
$347$		−	20.7785i		−	1.11545i	−0.830027	−	0.557724i	$-0.811675\pi$
$347$		−	20.7785i		−	1.11545i	0.830027	−	0.557724i	$-0.188325\pi$
$348$		0			0
$349$		−	1.72878i		−	0.0925396i	−0.998929	−	0.0462698i	$-0.985267\pi$
$349$		−	1.72878i		−	0.0925396i	0.998929	−	0.0462698i	$-0.0147334\pi$
$350$		0			0
$351$	−1.76511	+	3.37656i	−0.0942146	+	0.180227i
$352$		0			0
$353$	10.6311			0.565837			0.282919	−	0.959144i	$-0.408697\pi$
$353$	10.6311			0.565837			0.282919	+	0.959144i	$0.408697\pi$
$354$		0			0
$355$			9.08802i			0.482342i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			4.22497i			0.222985i	0.993765	+	0.111493i	$0.0355632\pi$
$359$			4.22497i			0.222985i	−0.993765	+	0.111493i	$0.964437\pi$
$360$		0			0
$361$	−29.2271			−1.53827
$362$		0			0
$363$	−8.48542	−	22.3374i	−0.445369	−	1.17241i
$364$		0			0
$365$		−	4.98419i		−	0.260885i
$366$		0			0
$367$			24.0423i			1.25500i	0.778617	+	0.627499i	$0.215921\pi$
$367$			24.0423i			1.25500i	−0.778617	+	0.627499i	$0.784079\pi$
$368$		0			0
$369$	−17.5847	+	15.6130i	−0.915424	+	0.812780i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	32.5901			1.68745			0.843727	−	0.536773i	$-0.180357\pi$
$373$	32.5901			1.68745			0.843727	+	0.536773i	$0.180357\pi$
$374$		0			0
$375$	−6.68052	−	17.5861i	−0.344980	−	0.908143i
$376$		0			0
$377$	−3.39466			−0.174834
$378$		0			0
$379$	−19.2333			−0.987948			−0.493974	−	0.869477i	$-0.664456\pi$
$379$	−19.2333			−0.987948			−0.493974	+	0.869477i	$0.664456\pi$
$380$		0			0
$381$	−2.04199	−	5.37544i	−0.104614	−	0.275392i
$382$		0			0
$383$	25.9404			1.32549			0.662745	−	0.748845i	$-0.269391\pi$
$383$	25.9404			1.32549			0.662745	+	0.748845i	$0.269391\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−2.85393	+	2.53393i	−0.145074	+	0.128807i
$388$		0			0
$389$		−	22.5765i		−	1.14467i	−0.820019	−	0.572336i	$-0.806037\pi$
$389$		−	22.5765i		−	1.14467i	0.820019	−	0.572336i	$-0.193963\pi$
$390$		0			0
$391$		−	5.77528i		−	0.292069i
$392$		0			0
$393$	9.84876	+	25.9264i	0.496804	+	1.30781i
$394$		0			0
$395$	−17.8328			−0.897264
$396$		0			0
$397$			6.72959i			0.337748i	0.985638	+	0.168874i	$0.0540132\pi$
$397$			6.72959i			0.337748i	−0.985638	+	0.168874i	$0.945987\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	6.54640i		−	0.326911i	−0.986551	−	0.163456i	$-0.947736\pi$
$401$		−	6.54640i		−	0.326911i	0.986551	−	0.163456i	$-0.0522641\pi$
$402$		0			0
$403$	−1.32205			−0.0658562
$404$		0			0
$405$	1.39763	−	11.7244i	0.0694487	−	0.582588i
$406$		0			0
$407$			26.0142i			1.28948i
$408$		0			0
$409$			20.2168i			0.999654i	0.866125	+	0.499827i	$0.166603\pi$
$409$			20.2168i			0.999654i	−0.866125	+	0.499827i	$0.833397\pi$
$410$		0			0
$411$	−17.4114	+	6.61414i	−0.858840	+	0.326251i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−8.64525			−0.424379
$416$		0			0
$417$	−30.0411	+	11.4118i	−1.47112	+	0.558840i
$418$		0			0
$419$	−13.0519			−0.637627			−0.318813	−	0.947817i	$-0.603284\pi$
$419$	−13.0519			−0.637627			−0.318813	+	0.947817i	$0.603284\pi$
$420$		0			0
$421$	−33.1602			−1.61613			−0.808065	−	0.589093i	$-0.799485\pi$
$421$	−33.1602			−1.61613			−0.808065	+	0.589093i	$0.799485\pi$
$422$		0			0
$423$	−9.51549	+	8.44855i	−0.462659	+	0.410782i
$424$		0			0
$425$	−2.38814			−0.115842
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−2.24580	−	5.91195i	−0.108428	−	0.285432i
$430$		0			0
$431$			1.12004i			0.0539506i	0.999636	+	0.0269753i	$0.00858755\pi$
$431$			1.12004i			0.0539506i	−0.999636	+	0.0269753i	$0.991412\pi$
$432$		0			0
$433$			29.9594i			1.43976i	0.694100	+	0.719878i	$0.255802\pi$
$433$			29.9594i			1.43976i	−0.694100	+	0.719878i	$0.744198\pi$
$434$		0			0
$435$	9.83432	−	3.73581i	0.471520	−	0.179118i
$436$		0			0
$437$	−55.0653			−2.63413
$438$		0			0
$439$		−	21.6375i		−	1.03270i	−0.856378	−	0.516350i	$-0.827290\pi$
$439$		−	21.6375i		−	1.03270i	0.856378	−	0.516350i	$-0.172710\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	7.88010i		−	0.374395i	−0.982322	−	0.187198i	$-0.940060\pi$
$443$		−	7.88010i		−	0.374395i	0.982322	−	0.187198i	$-0.0599404\pi$
$444$		0			0
$445$	−5.74422			−0.272302
$446$		0			0
$447$	−7.88287	+	2.99450i	−0.372847	+	0.141635i
$448$		0			0
$449$		−	29.5174i		−	1.39301i	−0.717550	−	0.696507i	$-0.754737\pi$
$449$		−	29.5174i		−	1.39301i	0.717550	−	0.696507i	$-0.245263\pi$
$450$		0			0
$451$		−	39.0324i		−	1.83796i
$452$		0			0
$453$	−1.04637	−	2.75452i	−0.0491628	−	0.129419i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	2.57745			0.120568			0.0602839	−	0.998181i	$-0.480799\pi$
$457$	2.57745			0.120568			0.0602839	+	0.998181i	$0.480799\pi$
$458$		0			0
$459$	−3.35399	−	1.75331i	−0.156551	−	0.0818376i
$460$		0			0
$461$	30.8387			1.43630			0.718151	−	0.695888i	$-0.244989\pi$
$461$	30.8387			1.43630			0.718151	+	0.695888i	$0.244989\pi$
$462$		0			0
$463$	−36.6548			−1.70349			−0.851745	−	0.523956i	$-0.824456\pi$
$463$	−36.6548			−1.70349			−0.851745	+	0.523956i	$0.824456\pi$
$464$		0			0
$465$	3.82998	−	1.45491i	0.177611	−	0.0674699i
$466$		0			0
$467$	−35.0771			−1.62318			−0.811588	−	0.584231i	$-0.801396\pi$
$467$	−35.0771			−1.62318			−0.811588	+	0.584231i	$0.801396\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	29.4325	−	11.1806i	1.35618	−	0.515177i
$472$		0			0
$473$		−	6.33480i		−	0.291274i
$474$		0			0
$475$			22.7701i			1.04477i
$476$		0			0
$477$	−28.7152	−	32.3416i	−1.31478	−	1.48082i
$478$		0			0
$479$	−28.7683			−1.31446			−0.657228	−	0.753691i	$-0.728271\pi$
$479$	−28.7683			−1.31446			−0.657228	+	0.753691i	$0.728271\pi$
$480$		0			0
$481$			3.83067i			0.174664i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			19.8770i			0.902569i
$486$		0			0
$487$	−36.0750			−1.63471			−0.817356	−	0.576133i	$-0.804561\pi$
$487$	−36.0750			−1.63471			−0.817356	+	0.576133i	$0.804561\pi$
$488$		0			0
$489$	3.08973	+	8.13356i	0.139723	+	0.367812i
$490$		0			0
$491$		−	26.0949i		−	1.17764i	−0.808263	−	0.588822i	$-0.799592\pi$
$491$		−	26.0949i		−	1.17764i	0.808263	−	0.588822i	$-0.200408\pi$
$492$		0			0
$493$		−	3.37197i		−	0.151866i
$494$		0			0
$495$	13.0121	+	14.6554i	0.584852	+	0.658711i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	18.5440			0.830143			0.415072	−	0.909789i	$-0.363757\pi$
$499$	18.5440			0.830143			0.415072	+	0.909789i	$0.363757\pi$
$500$		0			0
$501$	10.1103	+	26.6148i	0.451694	+	1.18906i
$502$		0			0
$503$	−19.2682			−0.859126			−0.429563	−	0.903037i	$-0.641332\pi$
$503$	−19.2682			−0.859126			−0.429563	+	0.903037i	$0.641332\pi$
$504$		0			0
$505$	14.4259			0.641943
$506$		0			0
$507$	7.66531	+	20.1785i	0.340428	+	0.896160i
$508$		0			0
$509$	25.9311			1.14938			0.574688	−	0.818373i	$-0.305123\pi$
$509$	25.9311			1.14938			0.574688	+	0.818373i	$0.305123\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−16.7172	+	31.9791i	−0.738084	+	1.41191i
$514$		0			0
$515$		−	13.6824i		−	0.602920i
$516$		0			0
$517$		−	21.1213i		−	0.928914i
$518$		0			0
$519$	−2.58607	−	6.80771i	−0.113516	−	0.298825i
$520$		0			0
$521$	11.0717			0.485059			0.242530	−	0.970144i	$-0.422023\pi$
$521$	11.0717			0.485059			0.242530	+	0.970144i	$0.422023\pi$
$522$		0			0
$523$			35.4586i			1.55050i	0.631656	+	0.775249i	$0.282376\pi$
$523$			35.4586i			1.55050i	−0.631656	+	0.775249i	$0.717624\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	1.31322i		−	0.0572046i
$528$		0			0
$529$	−39.8732			−1.73362
$530$		0			0
$531$	20.9825	−	18.6298i	0.910565	−	0.808466i
$532$		0			0
$533$		−	5.74764i		−	0.248958i
$534$		0			0
$535$			4.04091i			0.174704i
$536$		0			0
$537$	−13.1800	+	5.00676i	−0.568760	+	0.216057i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−8.04913			−0.346059			−0.173029	−	0.984917i	$-0.555356\pi$
$541$	−8.04913			−0.346059			−0.173029	+	0.984917i	$0.555356\pi$
$542$		0			0
$543$	28.9642	−	11.0027i	1.24297	−	0.472173i
$544$		0			0
$545$	18.0407			0.772777
$546$		0			0
$547$	−15.3821			−0.657690			−0.328845	−	0.944384i	$-0.606659\pi$
$547$	−15.3821			−0.657690			−0.328845	+	0.944384i	$0.606659\pi$
$548$		0			0
$549$	27.0651	+	30.4830i	1.15511	+	1.30098i
$550$		0			0
$551$	−32.1506			−1.36966
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−4.21563	−	11.0974i	−0.178944	−	0.471060i
$556$		0			0
$557$			34.1048i			1.44507i	0.691336	+	0.722534i	$0.257023\pi$
$557$			34.1048i			1.44507i	−0.691336	+	0.722534i	$0.742977\pi$
$558$		0			0
$559$		−	0.932820i		−	0.0394541i
$560$		0			0
$561$	5.87244	−	2.23079i	0.247934	−	0.0941840i
$562$		0			0
$563$	32.6655			1.37669			0.688343	−	0.725385i	$-0.258338\pi$
$563$	32.6655			1.37669			0.688343	+	0.725385i	$0.258338\pi$
$564$		0			0
$565$			16.5846i			0.697718i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			23.9380i			1.00353i	0.865004	+	0.501766i	$0.167316\pi$
$569$			23.9380i			1.00353i	−0.865004	+	0.501766i	$0.832684\pi$
$570$		0			0
$571$	24.9501			1.04413			0.522065	−	0.852906i	$-0.325162\pi$
$571$	24.9501			1.04413			0.522065	+	0.852906i	$0.325162\pi$
$572$		0			0
$573$	6.65540	−	2.52822i	0.278034	−	0.105618i
$574$		0			0
$575$			25.9988i			1.08422i
$576$		0			0
$577$		−	20.7925i		−	0.865601i	−0.901490	−	0.432801i	$-0.857525\pi$
$577$		−	20.7925i		−	0.865601i	0.901490	−	0.432801i	$-0.142475\pi$
$578$		0			0
$579$	−1.98620	−	5.22858i	−0.0825438	−	0.217292i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	71.7879			2.97315
$584$		0			0
$585$	1.91608	+	2.15806i	0.0792201	+	0.0892246i
$586$		0			0
$587$	−11.7010			−0.482951			−0.241475	−	0.970407i	$-0.577631\pi$
$587$	−11.7010			−0.482951			−0.241475	+	0.970407i	$0.577631\pi$
$588$		0			0
$589$	−12.5211			−0.515922
$590$		0			0
$591$	−7.62249	+	2.89559i	−0.313547	+	0.119109i
$592$		0			0
$593$	33.6454			1.38165			0.690826	−	0.723021i	$-0.257247\pi$
$593$	33.6454			1.38165			0.690826	+	0.723021i	$0.257247\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	27.4990	−	10.4462i	1.12546	−	0.427533i
$598$		0			0
$599$			11.2044i			0.457800i	0.973450	+	0.228900i	$0.0735129\pi$
$599$			11.2044i			0.457800i	−0.973450	+	0.228900i	$0.926487\pi$
$600$		0			0
$601$			6.08245i			0.248108i	0.992275	+	0.124054i	$0.0395897\pi$
$601$			6.08245i			0.248108i	−0.992275	+	0.124054i	$0.960410\pi$
$602$		0			0
$603$	−16.9331	+	15.0345i	−0.689570	+	0.612251i
$604$		0			0
$605$	−18.0990			−0.735829
$606$		0			0
$607$		−	6.92845i		−	0.281217i	−0.990065	−	0.140608i	$-0.955094\pi$
$607$		−	6.92845i		−	0.281217i	0.990065	−	0.140608i	$-0.0449059\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	3.11018i		−	0.125824i
$612$		0			0
$613$	13.5231			0.546194			0.273097	−	0.961987i	$-0.411952\pi$
$613$	13.5231			0.546194			0.273097	+	0.961987i	$0.411952\pi$
$614$		0			0
$615$	6.32524	+	16.6509i	0.255058	+	0.671428i
$616$		0			0
$617$		−	3.82786i		−	0.154104i	−0.997027	−	0.0770519i	$-0.975449\pi$
$617$		−	3.82786i		−	0.154104i	0.997027	−	0.0770519i	$-0.0245507\pi$
$618$		0			0
$619$			5.89220i			0.236827i	0.992964	+	0.118414i	$0.0377809\pi$
$619$			5.89220i			0.236827i	−0.992964	+	0.118414i	$0.962219\pi$
$620$		0			0
$621$	−19.0876	+	36.5136i	−0.765959	+	1.46524i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	2.14498			0.0857994
$626$		0			0
$627$	−21.2698	−	55.9917i	−0.849434	−	2.23609i
$628$		0			0
$629$	−3.80507			−0.151718
$630$		0			0
$631$	3.83697			0.152747			0.0763737	−	0.997079i	$-0.475666\pi$
$631$	3.83697			0.152747			0.0763737	+	0.997079i	$0.475666\pi$
$632$		0			0
$633$	1.16327	+	3.06225i	0.0462358	+	0.121713i
$634$		0			0
$635$	−4.35547			−0.172841
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−13.7977	−	15.5402i	−0.545830	−	0.614762i
$640$		0			0
$641$		−	3.98598i		−	0.157437i	−0.996897	−	0.0787183i	$-0.974917\pi$
$641$		−	3.98598i		−	0.157437i	0.996897	−	0.0787183i	$-0.0250828\pi$
$642$		0			0
$643$		−	15.4887i		−	0.610814i	−0.952222	−	0.305407i	$-0.901207\pi$
$643$		−	15.4887i		−	0.610814i	0.952222	−	0.305407i	$-0.0987925\pi$
$644$		0			0
$645$	1.02656	+	2.70237i	0.0404209	+	0.106406i
$646$		0			0
$647$	20.6256			0.810874			0.405437	−	0.914123i	$-0.367119\pi$
$647$	20.6256			0.810874			0.405437	+	0.914123i	$0.367119\pi$
$648$		0			0
$649$			46.5744i			1.82821i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			50.2427i			1.96615i	0.183203	+	0.983075i	$0.441354\pi$
$653$			50.2427i			1.96615i	−0.183203	+	0.983075i	$0.558646\pi$
$654$		0			0
$655$	21.0069			0.820808
$656$		0			0
$657$	7.56717	+	8.52281i	0.295224	+	0.332507i
$658$		0			0
$659$			34.6160i			1.34845i	0.738528	+	0.674223i	$0.235521\pi$
$659$			34.6160i			1.34845i	−0.738528	+	0.674223i	$0.764479\pi$
$660$		0			0
$661$		−	9.98831i		−	0.388500i	−0.980952	−	0.194250i	$-0.937773\pi$
$661$		−	9.98831i		−	0.388500i	0.980952	−	0.194250i	$-0.0622274\pi$
$662$		0			0
$663$	0.864736	−	0.328491i	0.0335835	−	0.0127575i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−36.7093			−1.42139
$668$		0			0
$669$	−15.0529	+	5.71819i	−0.581977	+	0.221078i
$670$		0			0
$671$	−67.6624			−2.61208
$672$		0			0
$673$	−8.78901			−0.338791			−0.169396	−	0.985548i	$-0.554182\pi$
$673$	−8.78901			−0.338791			−0.169396	+	0.985548i	$0.554182\pi$
$674$		0			0
$675$	15.0988	+	7.89294i	0.581152	+	0.303799i
$676$		0			0
$677$	−4.75875			−0.182894			−0.0914468	−	0.995810i	$-0.529149\pi$
$677$	−4.75875			−0.182894			−0.0914468	+	0.995810i	$0.529149\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	3.05690	+	8.04714i	0.117141	+	0.308367i
$682$		0			0
$683$			26.0958i			0.998529i	0.866450	+	0.499265i	$0.166396\pi$
$683$			26.0958i			0.998529i	−0.866450	+	0.499265i	$0.833604\pi$
$684$		0			0
$685$			14.1076i			0.539025i
$686$		0			0
$687$	8.63267	−	3.27933i	0.329357	−	0.125114i
$688$		0			0
$689$	10.5710			0.402723
$690$		0			0
$691$			30.6050i			1.16427i	0.813093	+	0.582134i	$0.197782\pi$
$691$			30.6050i			1.16427i	−0.813093	+	0.582134i	$0.802218\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			24.3409i			0.923303i
$696$		0			0
$697$	5.70922			0.216252
$698$		0			0
$699$	5.67087	−	2.15422i	0.214492	−	0.0814800i
$700$		0			0
$701$		−	45.8510i		−	1.73177i	−0.500246	−	0.865884i	$-0.666757\pi$
$701$		−	45.8510i		−	1.73177i	0.500246	−	0.865884i	$-0.333243\pi$
$702$		0			0
$703$			36.2800i			1.36833i
$704$		0			0
$705$	3.42273	+	9.01017i	0.128908	+	0.339343i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−7.32458			−0.275080			−0.137540	−	0.990496i	$-0.543920\pi$
$709$	−7.32458			−0.275080			−0.137540	+	0.990496i	$0.543920\pi$
$710$		0			0
$711$	30.4935	−	27.0743i	1.14359	−	1.01537i
$712$		0			0
$713$	−14.2965			−0.535407
$714$		0			0
$715$	−4.79018			−0.179143
$716$		0			0
$717$	38.0128	−	14.4401i	1.41962	−	0.539276i
$718$		0			0
$719$	−18.6425			−0.695246			−0.347623	−	0.937634i	$-0.613011\pi$
$719$	−18.6425			−0.695246			−0.347623	+	0.937634i	$0.613011\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	16.3008	−	6.19225i	0.606233	−	0.230292i
$724$		0			0
$725$			15.1797i			0.563761i
$726$		0			0
$727$			4.44113i			0.164712i	0.996603	+	0.0823562i	$0.0262445\pi$
$727$			4.44113i			0.164712i	−0.996603	+	0.0823562i	$0.973755\pi$
$728$		0			0
$729$	15.4104	+	22.1702i	0.570757	+	0.821119i
$730$		0			0
$731$	0.926585			0.0342710
$732$		0			0
$733$		−	37.3288i		−	1.37877i	−0.724395	−	0.689385i	$-0.757881\pi$
$733$		−	37.3288i		−	1.37877i	0.724395	−	0.689385i	$-0.242119\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	37.5860i		−	1.38450i
$738$		0			0
$739$	−15.7067			−0.577780			−0.288890	−	0.957362i	$-0.593286\pi$
$739$	−15.7067			−0.577780			−0.288890	+	0.957362i	$0.593286\pi$
$740$		0			0
$741$	−3.13205	−	8.24496i	−0.115059	−	0.302886i
$742$		0			0
$743$		−	41.4427i		−	1.52039i	−0.649697	−	0.760193i	$-0.725104\pi$
$743$		−	41.4427i		−	1.52039i	0.649697	−	0.760193i	$-0.274896\pi$
$744$		0			0
$745$			6.38712i			0.234006i
$746$		0			0
$747$	14.7831	−	13.1255i	0.540886	−	0.480238i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−11.9195			−0.434948			−0.217474	−	0.976066i	$-0.569782\pi$
$751$	−11.9195			−0.434948			−0.217474	+	0.976066i	$0.569782\pi$
$752$		0			0
$753$	12.1743	+	32.0483i	0.443657	+	1.16790i
$754$		0			0
$755$	−2.23186			−0.0812256
$756$		0			0
$757$	20.9855			0.762731			0.381365	−	0.924424i	$-0.375454\pi$
$757$	20.9855			0.762731			0.381365	+	0.924424i	$0.375454\pi$
$758$		0			0
$759$	−24.2857	−	63.9309i	−0.881516	−	2.32054i
$760$		0			0
$761$	−26.4963			−0.960489			−0.480245	−	0.877135i	$-0.659452\pi$
$761$	−26.4963			−0.960489			−0.480245	+	0.877135i	$0.659452\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−2.14363	+	1.90327i	−0.0775031	+	0.0688129i
$766$		0			0
$767$			6.85823i			0.247636i
$768$		0			0
$769$			49.5271i			1.78599i	0.450062	+	0.892997i	$0.351402\pi$
$769$			49.5271i			1.78599i	−0.450062	+	0.892997i	$0.648598\pi$
$770$		0			0
$771$	−7.76001	−	20.4278i	−0.279470	−	0.735690i
$772$		0			0
$773$	7.37448			0.265242			0.132621	−	0.991167i	$-0.457661\pi$
$773$	7.37448			0.265242			0.132621	+	0.991167i	$0.457661\pi$
$774$		0			0
$775$			5.91175i			0.212356i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	54.4355i		−	1.95035i
$780$		0			0
$781$	34.4943			1.23430
$782$		0			0
$783$	−11.1445	+	21.3189i	−0.398274	+	0.761876i
$784$		0			0
$785$		−	23.8478i		−	0.851164i
$786$		0			0
$787$		−	36.1045i		−	1.28699i	−0.765452	−	0.643493i	$-0.777484\pi$
$787$		−	36.1045i		−	1.28699i	0.765452	−	0.643493i	$-0.222516\pi$
$788$		0			0
$789$	−10.2525	+	3.89467i	−0.364999	+	0.138654i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−9.96351			−0.353815
$794$		0			0
$795$	−30.6241	+	11.6333i	−1.08613	+	0.412591i
$796$		0			0
$797$	25.6502			0.908576			0.454288	−	0.890855i	$-0.349894\pi$
$797$	25.6502			0.908576			0.454288	+	0.890855i	$0.349894\pi$
$798$		0			0
$799$	3.08939			0.109295
$800$		0			0
$801$	9.82244	−	8.72108i	0.347059	−	0.308144i
$802$		0			0
$803$	−18.9179			−0.667597
$804$		0			0
$805$		0			0
$806$		0			0
$807$	14.9980	+	39.4814i	0.527954	+	1.38981i
$808$		0			0
$809$		−	32.6220i		−	1.14693i	−0.819230	−	0.573464i	$-0.805599\pi$
$809$		−	32.6220i		−	1.14693i	0.819230	−	0.573464i	$-0.194401\pi$
$810$		0			0
$811$		−	12.7234i		−	0.446779i	−0.974729	−	0.223390i	$-0.928288\pi$
$811$		−	12.7234i		−	0.446779i	0.974729	−	0.223390i	$-0.0717122\pi$
$812$		0			0
$813$	−0.557107	+	0.211631i	−0.0195386	+	0.00742221i
$814$		0			0
$815$	6.59025			0.230846
$816$		0			0
$817$		−	8.83467i		−	0.309086i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	52.7965i		−	1.84261i	−0.388842	−	0.921305i	$-0.627125\pi$
$821$		−	52.7965i		−	1.84261i	0.388842	−	0.921305i	$-0.372875\pi$
$822$		0			0
$823$	−12.6738			−0.441783			−0.220891	−	0.975298i	$-0.570897\pi$
$823$	−12.6738			−0.441783			−0.220891	+	0.975298i	$0.570897\pi$
$824$		0			0
$825$	−26.4361	+	10.0424i	−0.920388	+	0.349632i
$826$		0			0
$827$			30.4839i			1.06003i	0.847988	+	0.530015i	$0.177814\pi$
$827$			30.4839i			1.06003i	−0.847988	+	0.530015i	$0.822186\pi$
$828$		0			0
$829$			7.95851i			0.276410i	0.990404	+	0.138205i	$0.0441334\pi$
$829$			7.95851i			0.276410i	−0.990404	+	0.138205i	$0.955867\pi$
$830$		0			0
$831$	7.52606	+	19.8120i	0.261076	+	0.687270i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	21.5647			0.746278
$836$		0			0
$837$	−4.34025	+	8.30266i	−0.150021	+	0.286982i
$838$		0			0
$839$	−15.9255			−0.549808			−0.274904	−	0.961472i	$-0.588646\pi$
$839$	−15.9255			−0.549808			−0.274904	+	0.961472i	$0.588646\pi$
$840$		0			0
$841$	7.56677			0.260923
$842$		0			0
$843$	−27.0012	+	10.2571i	−0.929971	+	0.353272i
$844$		0			0
$845$	16.3497			0.562448
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−5.02198	+	1.90772i	−0.172354	+	0.0654728i
$850$		0			0
$851$			41.4243i			1.42001i
$852$		0			0
$853$		−	24.3170i		−	0.832597i	−0.909228	−	0.416298i	$-0.863327\pi$
$853$		−	24.3170i		−	0.832597i	0.909228	−	0.416298i	$-0.136673\pi$
$854$		0			0
$855$	18.1470	+	20.4388i	0.620616	+	0.698992i
$856$		0			0
$857$	−44.2919			−1.51298			−0.756491	−	0.654005i	$-0.773088\pi$
$857$	−44.2919			−1.51298			−0.756491	+	0.654005i	$0.773088\pi$
$858$		0			0
$859$			22.8833i			0.780769i	0.920652	+	0.390385i	$0.127658\pi$
$859$			22.8833i			0.780769i	−0.920652	+	0.390385i	$0.872342\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			23.6628i			0.805490i	0.915312	+	0.402745i	$0.131944\pi$
$863$			23.6628i			0.805490i	−0.915312	+	0.402745i	$0.868056\pi$
$864$		0			0
$865$	−5.51597			−0.187549
$866$		0			0
$867$	−10.1300	−	26.6668i	−0.344034	−	0.905651i
$868$		0			0
$869$			67.6856i			2.29608i
$870$		0			0
$871$		−	5.53466i		−	0.187535i
$872$		0			0
$873$	−30.1780	−	33.9891i	−1.02137	−	1.15036i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	29.5159			0.996680			0.498340	−	0.866982i	$-0.333943\pi$
$877$	29.5159			0.996680			0.498340	+	0.866982i	$0.333943\pi$
$878$		0			0
$879$	9.99441	+	26.3098i	0.337103	+	0.887406i
$880$		0			0
$881$	12.0113			0.404673			0.202336	−	0.979316i	$-0.435147\pi$
$881$	12.0113			0.404673			0.202336	+	0.979316i	$0.435147\pi$
$882$		0			0
$883$	26.4428			0.889871			0.444935	−	0.895563i	$-0.353227\pi$
$883$	26.4428			0.889871			0.444935	+	0.895563i	$0.353227\pi$
$884$		0			0
$885$	−7.54744	−	19.8683i	−0.253704	−	0.667864i
$886$		0			0
$887$	−3.12476			−0.104919			−0.0524596	−	0.998623i	$-0.516706\pi$
$887$	−3.12476			−0.104919			−0.0524596	+	0.998623i	$0.516706\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−44.5007	−	5.30480i	−1.49083	−	0.177718i
$892$		0			0
$893$		−	29.4563i		−	0.985717i
$894$		0			0
$895$			10.6792i			0.356965i
$896$		0			0
$897$	−3.57615	−	9.41403i	−0.119404	−	0.314325i
$898$		0			0
$899$	−8.34718			−0.278394
$900$		0			0
$901$			10.5003i			0.349817i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	23.4683i		−	0.780113i
$906$		0			0
$907$	57.9808			1.92522			0.962610	−	0.270890i	$-0.0873179\pi$
$907$	57.9808			1.92522			0.962610	+	0.270890i	$0.0873179\pi$
$908$		0			0
$909$	−24.6678	+	21.9019i	−0.818180	+	0.726439i
$910$		0			0
$911$		−	0.434872i		−	0.0144079i	−0.999974	−	0.00720397i	$-0.997707\pi$
$911$		−	0.434872i		−	0.0144079i	0.999974	−	0.00720397i	$-0.00229312\pi$
$912$		0			0
$913$			32.8137i			1.08597i
$914$		0			0
$915$	28.8642	−	10.9648i	0.954222	−	0.362485i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−40.1535			−1.32454			−0.662271	−	0.749264i	$-0.730407\pi$
$919$	−40.1535			−1.32454			−0.662271	+	0.749264i	$0.730407\pi$
$920$		0			0
$921$	−4.57509	+	1.73796i	−0.150754	+	0.0572677i
$922$		0			0
$923$	5.07939			0.167190
$924$		0			0
$925$	17.1294			0.563211
$926$		0			0
$927$	20.7732	+	23.3965i	0.682280	+	0.768443i
$928$		0			0
$929$	28.1366			0.923132			0.461566	−	0.887106i	$-0.347288\pi$
$929$	28.1366			0.923132			0.461566	+	0.887106i	$0.347288\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	10.0895	+	26.5600i	0.330315	+	0.869537i
$934$		0			0
$935$		−	4.75816i		−	0.155609i
$936$		0			0
$937$		−	57.1209i		−	1.86606i	−0.359801	−	0.933029i	$-0.617156\pi$
$937$		−	57.1209i		−	1.86606i	0.359801	−	0.933029i	$-0.382844\pi$
$938$		0			0
$939$	−0.525902	+	0.199777i	−0.0171622	+	0.00651947i
$940$		0			0
$941$	10.0352			0.327138			0.163569	−	0.986532i	$-0.447699\pi$
$941$	10.0352			0.327138			0.163569	+	0.986532i	$0.447699\pi$
$942$		0			0
$943$		−	62.1541i		−	2.02401i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			17.9443i			0.583110i	0.956554	+	0.291555i	$0.0941726\pi$
$947$			17.9443i			0.583110i	−0.956554	+	0.291555i	$0.905827\pi$
$948$		0			0
$949$	−2.78572			−0.0904282
$950$		0			0
$951$	19.5089	−	7.41092i	0.632618	−	0.240316i
$952$		0			0
$953$		−	1.76188i		−	0.0570730i	−0.999593	−	0.0285365i	$-0.990915\pi$
$953$		−	1.76188i		−	0.0570730i	0.999593	−	0.0285365i	$-0.00908469\pi$
$954$		0			0
$955$		−	5.39256i		−	0.174499i
$956$		0			0
$957$	−14.1795	−	37.3269i	−0.458359	−	1.20661i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	27.7492			0.895135
$962$		0			0
$963$	−6.13505	−	6.90983i	−0.197699	−	0.222666i
$964$		0			0
$965$	−4.23647			−0.136377
$966$		0			0
$967$	20.8379			0.670102			0.335051	−	0.942200i	$-0.391246\pi$
$967$	20.8379			0.670102			0.335051	+	0.942200i	$0.391246\pi$
$968$		0			0
$969$	8.18985	−	3.11111i	0.263096	−	0.0999433i
$970$		0			0
$971$	22.7633			0.730509			0.365254	−	0.930908i	$-0.380982\pi$
$971$	22.7633			0.730509			0.365254	+	0.930908i	$0.380982\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−3.89281	+	1.47878i	−0.124670	+	0.0473588i
$976$		0			0
$977$			0.531619i			0.0170080i	0.999964	+	0.00850401i	$0.00270694\pi$
$977$			0.531619i			0.0170080i	−0.999964	+	0.00850401i	$0.997293\pi$
$978$		0			0
$979$			21.8026i			0.696815i
$980$		0			0
$981$	−30.8490	+	27.3900i	−0.984932	+	0.874495i
$982$		0			0
$983$	−9.67319			−0.308527			−0.154263	−	0.988030i	$-0.549300\pi$
$983$	−9.67319			−0.308527			−0.154263	+	0.988030i	$0.549300\pi$
$984$		0			0
$985$			6.17615i			0.196788i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	10.0874i		−	0.320759i
$990$		0			0
$991$	−4.14717			−0.131739			−0.0658695	−	0.997828i	$-0.520982\pi$
$991$	−4.14717			−0.131739			−0.0658695	+	0.997828i	$0.520982\pi$
$992$		0			0
$993$	−4.30557	−	11.3342i	−0.136633	−	0.359680i
$994$		0			0
$995$		−	22.2812i		−	0.706361i
$996$		0			0
$997$			4.38581i			0.138900i	0.997585	+	0.0694500i	$0.0221244\pi$
$997$			4.38581i			0.138900i	−0.997585	+	0.0694500i	$0.977876\pi$
$998$		0			0
$999$	24.0571	+	12.5759i	0.761133	+	0.397885i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.k.j.881.16		24
3.2	odd	2	inner	2352.2.k.j.881.10		24
4.3	odd	2		1176.2.k.b.881.9	✓	24
7.6	odd	2	inner	2352.2.k.j.881.9		24
12.11	even	2		1176.2.k.b.881.15	yes	24
21.20	even	2	inner	2352.2.k.j.881.15		24
28.3	even	6		1176.2.u.c.1097.19		48
28.11	odd	6		1176.2.u.c.1097.6		48
28.19	even	6		1176.2.u.c.521.2		48
28.23	odd	6		1176.2.u.c.521.23		48
28.27	even	2		1176.2.k.b.881.16	yes	24
84.11	even	6		1176.2.u.c.1097.2		48
84.23	even	6		1176.2.u.c.521.19		48
84.47	odd	6		1176.2.u.c.521.6		48
84.59	odd	6		1176.2.u.c.1097.23		48
84.83	odd	2		1176.2.k.b.881.10	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.2.k.b.881.9	✓	24	4.3	odd	2
1176.2.k.b.881.10	yes	24	84.83	odd	2
1176.2.k.b.881.15	yes	24	12.11	even	2
1176.2.k.b.881.16	yes	24	28.27	even	2
1176.2.u.c.521.2		48	28.19	even	6
1176.2.u.c.521.6		48	84.47	odd	6
1176.2.u.c.521.19		48	84.23	even	6
1176.2.u.c.521.23		48	28.23	odd	6
1176.2.u.c.1097.2		48	84.11	even	6
1176.2.u.c.1097.6		48	28.11	odd	6
1176.2.u.c.1097.19		48	28.3	even	6
1176.2.u.c.1097.23		48	84.59	odd	6
2352.2.k.j.881.9		24	7.6	odd	2	inner
2352.2.k.j.881.10		24	3.2	odd	2	inner
2352.2.k.j.881.15		24	21.20	even	2	inner
2352.2.k.j.881.16		24	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q+(0.615077 + 1.61916i) q^{3} +1.31193 q^{5} +(-2.24336 + 1.99182i) q^{9} -4.97953i q^{11} -0.733252i q^{13} +(0.806939 + 2.12423i) q^{15} +0.728350 q^{17} -6.94457i q^{19} -7.92926i q^{23} -3.27884 q^{25} +(-4.60491 - 2.40724i) q^{27} -4.62960i q^{29} -1.80300i q^{31} +(8.06265 - 3.06280i) q^{33} -5.22423 q^{37} +(1.18725 - 0.451007i) q^{39} +7.83857 q^{41} +1.27217 q^{43} +(-2.94313 + 2.61313i) q^{45} +4.24163 q^{47} +(0.447992 + 1.17932i) q^{51} +14.4166i q^{53} -6.53279i q^{55} +(11.2444 - 4.27145i) q^{57} -9.35318 q^{59} -13.5881i q^{61} -0.961975i q^{65} +7.54811 q^{67} +(12.8387 - 4.87711i) q^{69} +6.92721i q^{71} -3.79913i q^{73} +(-2.01674 - 5.30897i) q^{75} -13.5928 q^{79} +(1.06532 - 8.93673i) q^{81} -6.58972 q^{83} +0.955545 q^{85} +(7.49607 - 2.84757i) q^{87} -4.37845 q^{89} +(2.91935 - 1.10898i) q^{93} -9.11079i q^{95} +15.1510i q^{97} +(9.91831 + 11.1709i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 16 q^{15} + 8 q^{25} + 16 q^{37} + 64 q^{39} - 16 q^{43} - 48 q^{51} + 48 q^{57} - 16 q^{67} + 80 q^{81} - 64 q^{85} - 32 q^{93} + 56 q^{99}+O(q^{100}) \) 24 * q + 16 * q^15 + 8 * q^25 + 16 * q^37 + 64 * q^39 - 16 * q^43 - 48 * q^51 + 48 * q^57 - 16 * q^67 + 80 * q^81 - 64 * q^85 - 32 * q^93 + 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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