LMFDB - Embedded newform 2912.2.i.a.337.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2912,2,Mod(337,2912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 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("2912.337"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.i (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$84$
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.18
Character	$\chi$	$=$	2912.337
Dual form			2912.2.i.a.337.17

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.944401i q^{3} +3.13735 q^{5} +1.00000i q^{7} +2.10811 q^{9} -0.668990 q^{11} +(-3.33818 - 1.36256i) q^{13} +2.96292i q^{15} -2.89622 q^{17} +5.15356 q^{19} -0.944401 q^{21} +4.18873 q^{23} +4.84297 q^{25} +4.82410i q^{27} -2.22655i q^{29} -7.61759i q^{31} -0.631795i q^{33} +3.13735i q^{35} +10.8069 q^{37} +(1.28681 - 3.15258i) q^{39} +11.2974i q^{41} +6.14812i q^{43} +6.61387 q^{45} +11.4840i q^{47} -1.00000 q^{49} -2.73519i q^{51} +3.09234i q^{53} -2.09886 q^{55} +4.86703i q^{57} +12.7056 q^{59} -11.0620i q^{61} +2.10811i q^{63} +(-10.4730 - 4.27484i) q^{65} -6.94022 q^{67} +3.95584i q^{69} -7.81277i q^{71} +10.3939i q^{73} +4.57371i q^{75} -0.668990i q^{77} +13.1329 q^{79} +1.76843 q^{81} -5.56860 q^{83} -9.08645 q^{85} +2.10276 q^{87} +0.496967i q^{89} +(1.36256 - 3.33818i) q^{91} +7.19406 q^{93} +16.1685 q^{95} +1.71943i q^{97} -1.41030 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 84 q^{9} + 8 q^{17} + 24 q^{23} + 92 q^{25} + 24 q^{39} - 84 q^{49} - 32 q^{55} - 24 q^{65} + 40 q^{79} + 84 q^{81} + 48 q^{87} + 16 q^{95}+O(q^{100}) $ 84 * q - 84 * q^9 + 8 * q^17 + 24 * q^23 + 92 * q^25 + 24 * q^39 - 84 * q^49 - 32 * q^55 - 24 * q^65 + 40 * q^79 + 84 * q^81 + 48 * q^87 + 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2912\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$2017$	$2367$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$			0.944401i			0.545250i	0.962120	+	0.272625i	$0.0878919\pi$
$3$			0.944401i			0.545250i	−0.962120	+	0.272625i	$0.912108\pi$
$4$		0			0
$5$	3.13735			1.40307			0.701533	−	0.712637i	$-0.252499\pi$
$5$	3.13735			1.40307			0.701533	+	0.712637i	$0.252499\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	2.10811			0.702702
$10$		0			0
$11$	−0.668990			−0.201708			−0.100854	−	0.994901i	$-0.532157\pi$
$11$	−0.668990			−0.201708			−0.100854	+	0.994901i	$0.532157\pi$
$12$		0			0
$13$	−3.33818	−	1.36256i	−0.925844	−	0.377907i
$13$	−3.33818	−	1.36256i	−0.925844	−	0.377907i
$14$		0			0
$15$			2.96292i			0.765022i
$16$		0			0
$17$	−2.89622			−0.702436			−0.351218	−	0.936294i	$-0.614232\pi$
$17$	−2.89622			−0.702436			−0.351218	+	0.936294i	$0.614232\pi$
$18$		0			0
$19$	5.15356			1.18231			0.591154	−	0.806559i	$-0.298673\pi$
$19$	5.15356			1.18231			0.591154	+	0.806559i	$0.298673\pi$
$20$		0			0
$21$	−0.944401			−0.206085
$22$		0			0
$23$	4.18873			0.873411			0.436705	−	0.899605i	$-0.356145\pi$
$23$	4.18873			0.873411			0.436705	+	0.899605i	$0.356145\pi$
$24$		0			0
$25$	4.84297			0.968594
$26$		0			0
$27$			4.82410i			0.928399i
$28$		0			0
$29$		−	2.22655i		−	0.413460i	−0.978398	−	0.206730i	$-0.933718\pi$
$29$		−	2.22655i		−	0.413460i	0.978398	−	0.206730i	$-0.0662821\pi$
$30$		0			0
$31$		−	7.61759i		−	1.36816i	−0.729408	−	0.684079i	$-0.760204\pi$
$31$		−	7.61759i		−	1.36816i	0.729408	−	0.684079i	$-0.239796\pi$
$32$		0			0
$33$		−	0.631795i		−	0.109981i
$34$		0			0
$35$			3.13735i			0.530309i
$36$		0			0
$37$	10.8069			1.77665			0.888326	−	0.459214i	$-0.151869\pi$
$37$	10.8069			1.77665			0.888326	+	0.459214i	$0.151869\pi$
$38$		0			0
$39$	1.28681	−	3.15258i	0.206054	−	0.504817i
$40$		0			0
$41$			11.2974i			1.76436i	0.470909	+	0.882182i	$0.343926\pi$
$41$			11.2974i			1.76436i	−0.470909	+	0.882182i	$0.656074\pi$
$42$		0			0
$43$			6.14812i			0.937580i	0.883310	+	0.468790i	$0.155310\pi$
$43$			6.14812i			0.937580i	−0.883310	+	0.468790i	$0.844690\pi$
$44$		0			0
$45$	6.61387			0.985938
$46$		0			0
$47$			11.4840i			1.67511i	0.546349	+	0.837557i	$0.316017\pi$
$47$			11.4840i			1.67511i	−0.546349	+	0.837557i	$0.683983\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		−	2.73519i		−	0.383003i
$52$		0			0
$53$			3.09234i			0.424766i	0.977187	+	0.212383i	$0.0681224\pi$
$53$			3.09234i			0.424766i	−0.977187	+	0.212383i	$0.931878\pi$
$54$		0			0
$55$	−2.09886			−0.283010
$56$		0			0
$57$			4.86703i			0.644654i
$58$		0			0
$59$	12.7056			1.65413			0.827067	−	0.562103i	$-0.190008\pi$
$59$	12.7056			1.65413			0.827067	+	0.562103i	$0.190008\pi$
$60$		0			0
$61$		−	11.0620i		−	1.41634i	−0.706041	−	0.708171i	$-0.749520\pi$
$61$		−	11.0620i		−	1.41634i	0.706041	−	0.708171i	$-0.250480\pi$
$62$		0			0
$63$			2.10811i			0.265596i
$64$		0			0
$65$	−10.4730	−	4.27484i	−1.29902	−	0.530228i
$66$		0			0
$67$	−6.94022			−0.847883			−0.423942	−	0.905690i	$-0.639354\pi$
$67$	−6.94022			−0.847883			−0.423942	+	0.905690i	$0.639354\pi$
$68$		0			0
$69$			3.95584i			0.476227i
$70$		0			0
$71$		−	7.81277i		−	0.927205i	−0.886043	−	0.463602i	$-0.846557\pi$
$71$		−	7.81277i		−	0.927205i	0.886043	−	0.463602i	$-0.153443\pi$
$72$		0			0
$73$			10.3939i			1.21652i	0.793739	+	0.608259i	$0.208132\pi$
$73$			10.3939i			1.21652i	−0.793739	+	0.608259i	$0.791868\pi$
$74$		0			0
$75$			4.57371i			0.528126i
$76$		0			0
$77$		−	0.668990i		−	0.0762385i
$78$		0			0
$79$	13.1329			1.47757			0.738783	−	0.673944i	$-0.235401\pi$
$79$	13.1329			1.47757			0.738783	+	0.673944i	$0.235401\pi$
$80$		0			0
$81$	1.76843			0.196492
$82$		0			0
$83$	−5.56860			−0.611233			−0.305617	−	0.952155i	$-0.598863\pi$
$83$	−5.56860			−0.611233			−0.305617	+	0.952155i	$0.598863\pi$
$84$		0			0
$85$	−9.08645			−0.985564
$86$		0			0
$87$	2.10276			0.225439
$88$		0			0
$89$			0.496967i			0.0526784i	0.999653	+	0.0263392i	$0.00838500\pi$
$89$			0.496967i			0.0526784i	−0.999653	+	0.0263392i	$0.991615\pi$
$90$		0			0
$91$	1.36256	−	3.33818i	0.142835	−	0.349936i
$92$		0			0
$93$	7.19406			0.745989
$94$		0			0
$95$	16.1685			1.65886
$96$		0			0
$97$			1.71943i			0.174582i	0.996183	+	0.0872911i	$0.0278210\pi$
$97$			1.71943i			0.174582i	−0.996183	+	0.0872911i	$0.972179\pi$
$98$		0			0
$99$	−1.41030			−0.141741
$100$		0			0
$101$		−	19.5040i		−	1.94072i	−0.241661	−	0.970361i	$-0.577692\pi$
$101$		−	19.5040i		−	1.94072i	0.241661	−	0.970361i	$-0.422308\pi$
$102$		0			0
$103$	−8.67472			−0.854745			−0.427373	−	0.904076i	$-0.640561\pi$
$103$	−8.67472			−0.854745			−0.427373	+	0.904076i	$0.640561\pi$
$104$		0			0
$105$	−2.96292			−0.289151
$106$		0			0
$107$		−	11.9973i		−	1.15982i	−0.814681	−	0.579910i	$-0.803088\pi$
$107$		−	11.9973i		−	1.15982i	0.814681	−	0.579910i	$-0.196912\pi$
$108$		0			0
$109$	−0.388646			−0.0372255			−0.0186128	−	0.999827i	$-0.505925\pi$
$109$	−0.388646			−0.0372255			−0.0186128	+	0.999827i	$0.505925\pi$
$110$		0			0
$111$			10.2061i			0.968720i
$112$		0			0
$113$	4.53657			0.426764			0.213382	−	0.976969i	$-0.431552\pi$
$113$	4.53657			0.426764			0.213382	+	0.976969i	$0.431552\pi$
$114$		0			0
$115$	13.1415			1.22545
$116$		0			0
$117$	−7.03723	−	2.87243i	−0.650592	−	0.265556i
$118$		0			0
$119$		−	2.89622i		−	0.265496i
$120$		0			0
$121$	−10.5525			−0.959314
$122$		0			0
$123$	−10.6693			−0.962020
$124$		0			0
$125$	−0.492652			−0.0440641
$126$		0			0
$127$	−6.72796			−0.597010			−0.298505	−	0.954408i	$-0.596488\pi$
$127$	−6.72796			−0.597010			−0.298505	+	0.954408i	$0.596488\pi$
$128$		0			0
$129$	−5.80629			−0.511216
$130$		0			0
$131$		−	2.37392i		−	0.207410i	−0.994608	−	0.103705i	$-0.966930\pi$
$131$		−	2.37392i		−	0.207410i	0.994608	−	0.103705i	$-0.0330698\pi$
$132$		0			0
$133$			5.15356i			0.446870i
$134$		0			0
$135$			15.1349i			1.30260i
$136$		0			0
$137$			16.0179i			1.36850i	0.729246	+	0.684251i	$0.239871\pi$
$137$			16.0179i			1.36850i	−0.729246	+	0.684251i	$0.760129\pi$
$138$		0			0
$139$			0.193116i			0.0163798i	0.999966	+	0.00818992i	$0.00260696\pi$
$139$			0.193116i			0.0163798i	−0.999966	+	0.00818992i	$0.997393\pi$
$140$		0			0
$141$	−10.8455			−0.913357
$142$		0			0
$143$	2.23321	+	0.911540i	0.186750	+	0.0762268i
$144$		0			0
$145$		−	6.98547i		−	0.580111i
$146$		0			0
$147$		−	0.944401i		−	0.0778929i
$148$		0			0
$149$	−13.1609			−1.07819			−0.539093	−	0.842246i	$-0.681233\pi$
$149$	−13.1609			−1.07819			−0.539093	+	0.842246i	$0.681233\pi$
$150$		0			0
$151$			22.8188i			1.85697i	0.371374	+	0.928483i	$0.378887\pi$
$151$			22.8188i			1.85697i	−0.371374	+	0.928483i	$0.621113\pi$
$152$		0			0
$153$	−6.10553			−0.493603
$154$		0			0
$155$		−	23.8990i		−	1.91962i
$156$		0			0
$157$			6.81439i			0.543847i	0.962319	+	0.271924i	$0.0876598\pi$
$157$			6.81439i			0.543847i	−0.962319	+	0.271924i	$0.912340\pi$
$158$		0			0
$159$	−2.92041			−0.231604
$160$		0			0
$161$			4.18873i			0.330118i
$162$		0			0
$163$	19.5692			1.53278			0.766390	−	0.642376i	$-0.222051\pi$
$163$	19.5692			1.53278			0.766390	+	0.642376i	$0.222051\pi$
$164$		0			0
$165$		−	1.98216i		−	0.154311i
$166$		0			0
$167$			12.6160i			0.976257i	0.872772	+	0.488129i	$0.162320\pi$
$167$			12.6160i			0.976257i	−0.872772	+	0.488129i	$0.837680\pi$
$168$		0			0
$169$	9.28685	+	9.09695i	0.714373	+	0.699765i
$170$		0			0
$171$	10.8643			0.830810
$172$		0			0
$173$		−	20.5720i		−	1.56406i	−0.623239	−	0.782031i	$-0.714184\pi$
$173$		−	20.5720i		−	1.56406i	0.623239	−	0.782031i	$-0.285816\pi$
$174$		0			0
$175$			4.84297i			0.366094i
$176$		0			0
$177$			11.9992i			0.901917i
$178$		0			0
$179$			0.669511i			0.0500416i	0.999687	+	0.0250208i	$0.00796519\pi$
$179$			0.669511i			0.0500416i	−0.999687	+	0.0250208i	$0.992035\pi$
$180$		0			0
$181$			6.65770i			0.494863i	0.968905	+	0.247432i	$0.0795865\pi$
$181$			6.65770i			0.494863i	−0.968905	+	0.247432i	$0.920413\pi$
$182$		0			0
$183$	10.4470			0.772261
$184$		0			0
$185$	33.9052			2.49276
$186$		0			0
$187$	1.93754			0.141687
$188$		0			0
$189$	−4.82410			−0.350902
$190$		0			0
$191$	−8.68661			−0.628541			−0.314271	−	0.949333i	$-0.601760\pi$
$191$	−8.68661			−0.628541			−0.314271	+	0.949333i	$0.601760\pi$
$192$		0			0
$193$		−	8.69805i		−	0.626099i	−0.949737	−	0.313050i	$-0.898649\pi$
$193$		−	8.69805i		−	0.626099i	0.949737	−	0.313050i	$-0.101351\pi$
$194$		0			0
$195$	4.03716	−	9.89074i	0.289107	−	0.708291i
$196$		0			0
$197$	0.999335			0.0711997			0.0355999	−	0.999366i	$-0.488666\pi$
$197$	0.999335			0.0711997			0.0355999	+	0.999366i	$0.488666\pi$
$198$		0			0
$199$	−2.44311			−0.173187			−0.0865936	−	0.996244i	$-0.527598\pi$
$199$	−2.44311			−0.173187			−0.0865936	+	0.996244i	$0.527598\pi$
$200$		0			0
$201$		−	6.55435i		−	0.462309i
$202$		0			0
$203$	2.22655			0.156273
$204$		0			0
$205$			35.4440i			2.47552i
$206$		0			0
$207$	8.83029			0.613747
$208$		0			0
$209$	−3.44768			−0.238481
$210$		0			0
$211$		−	4.07413i		−	0.280474i	−0.990118	−	0.140237i	$-0.955213\pi$
$211$		−	4.07413i		−	0.280474i	0.990118	−	0.140237i	$-0.0447865\pi$
$212$		0			0
$213$	7.37839			0.505559
$214$		0			0
$215$			19.2888i			1.31549i
$216$		0			0
$217$	7.61759			0.517115
$218$		0			0
$219$	−9.81605			−0.663307
$220$		0			0
$221$	9.66808	+	3.94627i	0.650345	+	0.265455i
$222$		0			0
$223$		−	15.4424i		−	1.03410i	−0.855956	−	0.517049i	$-0.827030\pi$
$223$		−	15.4424i		−	1.03410i	0.855956	−	0.517049i	$-0.172970\pi$
$224$		0			0
$225$	10.2095			0.680633
$226$		0			0
$227$	−18.8950			−1.25411			−0.627054	−	0.778976i	$-0.715739\pi$
$227$	−18.8950			−1.25411			−0.627054	+	0.778976i	$0.715739\pi$
$228$		0			0
$229$	15.5007			1.02432			0.512158	−	0.858891i	$-0.328846\pi$
$229$	15.5007			1.02432			0.512158	+	0.858891i	$0.328846\pi$
$230$		0			0
$231$	0.631795			0.0415690
$232$		0			0
$233$	11.3719			0.744995			0.372497	−	0.928033i	$-0.378502\pi$
$233$	11.3719			0.744995			0.372497	+	0.928033i	$0.378502\pi$
$234$		0			0
$235$			36.0294i			2.35030i
$236$		0			0
$237$			12.4027i			0.805643i
$238$		0			0
$239$			6.26097i			0.404989i	0.979283	+	0.202494i	$0.0649048\pi$
$239$			6.26097i			0.404989i	−0.979283	+	0.202494i	$0.935095\pi$
$240$		0			0
$241$		−	17.2558i		−	1.11155i	−0.831334	−	0.555773i	$-0.812422\pi$
$241$		−	17.2558i		−	1.11155i	0.831334	−	0.555773i	$-0.187578\pi$
$242$		0			0
$243$			16.1424i			1.03554i
$244$		0			0
$245$	−3.13735			−0.200438
$246$		0			0
$247$	−17.2035	−	7.02205i	−1.09463	−	0.446802i
$248$		0			0
$249$		−	5.25899i		−	0.333275i
$250$		0			0
$251$		−	0.276960i		−	0.0174816i	−0.999962	−	0.00874078i	$-0.997218\pi$
$251$		−	0.276960i		−	0.0174816i	0.999962	−	0.00874078i	$-0.00278231\pi$
$252$		0			0
$253$	−2.80222			−0.176174
$254$		0			0
$255$		−	8.58125i		−	0.537379i
$256$		0			0
$257$	2.29506			0.143162			0.0715810	−	0.997435i	$-0.477196\pi$
$257$	2.29506			0.143162			0.0715810	+	0.997435i	$0.477196\pi$
$258$		0			0
$259$			10.8069i			0.671511i
$260$		0			0
$261$		−	4.69380i		−	0.290539i
$262$		0			0
$263$	−17.5631			−1.08299			−0.541495	−	0.840704i	$-0.682142\pi$
$263$	−17.5631			−1.08299			−0.541495	+	0.840704i	$0.682142\pi$
$264$		0			0
$265$			9.70176i			0.595974i
$266$		0			0
$267$	−0.469337			−0.0287229
$268$		0			0
$269$			14.8418i			0.904920i	0.891785	+	0.452460i	$0.149454\pi$
$269$			14.8418i			0.904920i	−0.891785	+	0.452460i	$0.850546\pi$
$270$		0			0
$271$		−	1.11711i		−	0.0678595i	−0.999424	−	0.0339298i	$-0.989198\pi$
$271$		−	1.11711i		−	0.0678595i	0.999424	−	0.0339298i	$-0.0108022\pi$
$272$		0			0
$273$	3.15258	+	1.28681i	0.190803	+	0.0778810i
$274$		0			0
$275$	−3.23990			−0.195373
$276$		0			0
$277$		−	23.9638i		−	1.43985i	−0.694053	−	0.719924i	$-0.744177\pi$
$277$		−	23.9638i		−	1.43985i	0.694053	−	0.719924i	$-0.255823\pi$
$278$		0			0
$279$		−	16.0587i		−	0.961408i
$280$		0			0
$281$			17.3694i			1.03617i	0.855329	+	0.518085i	$0.173355\pi$
$281$			17.3694i			1.03617i	−0.855329	+	0.518085i	$0.826645\pi$
$282$		0			0
$283$		−	11.8138i		−	0.702260i	−0.936327	−	0.351130i	$-0.885798\pi$
$283$		−	11.8138i		−	0.702260i	0.936327	−	0.351130i	$-0.114202\pi$
$284$		0			0
$285$			15.2696i			0.904492i
$286$		0			0
$287$	−11.2974			−0.666867
$288$		0			0
$289$	−8.61193			−0.506584
$290$		0			0
$291$	−1.62384			−0.0951910
$292$		0			0
$293$	2.89026			0.168851			0.0844253	−	0.996430i	$-0.473095\pi$
$293$	2.89026			0.168851			0.0844253	+	0.996430i	$0.473095\pi$
$294$		0			0
$295$	39.8621			2.32086
$296$		0			0
$297$		−	3.22727i		−	0.187265i
$298$		0			0
$299$	−13.9827	−	5.70741i	−0.808642	−	0.330068i
$300$		0			0
$301$	−6.14812			−0.354372
$302$		0			0
$303$	18.4196			1.05818
$304$		0			0
$305$		−	34.7053i		−	1.98722i
$306$		0			0
$307$	−5.74500			−0.327885			−0.163942	−	0.986470i	$-0.552421\pi$
$307$	−5.74500			−0.327885			−0.163942	+	0.986470i	$0.552421\pi$
$308$		0			0
$309$		−	8.19241i		−	0.466050i
$310$		0			0
$311$	−25.1524			−1.42626			−0.713130	−	0.701032i	$-0.752723\pi$
$311$	−25.1524			−1.42626			−0.713130	+	0.701032i	$0.752723\pi$
$312$		0			0
$313$	−16.8673			−0.953394			−0.476697	−	0.879068i	$-0.658166\pi$
$313$	−16.8673			−0.953394			−0.476697	+	0.879068i	$0.658166\pi$
$314$		0			0
$315$			6.61387i			0.372649i
$316$		0			0
$317$	33.6163			1.88808			0.944038	−	0.329835i	$-0.106993\pi$
$317$	33.6163			1.88808			0.944038	+	0.329835i	$0.106993\pi$
$318$		0			0
$319$			1.48954i			0.0833981i
$320$		0			0
$321$	11.3302			0.632392
$322$		0			0
$323$	−14.9258			−0.830495
$324$		0			0
$325$	−16.1667	−	6.59885i	−0.896767	−	0.366038i
$326$		0			0
$327$		−	0.367038i		−	0.0202972i
$328$		0			0
$329$	−11.4840			−0.633134
$330$		0			0
$331$	−3.85492			−0.211885			−0.105943	−	0.994372i	$-0.533786\pi$
$331$	−3.85492			−0.211885			−0.105943	+	0.994372i	$0.533786\pi$
$332$		0			0
$333$	22.7822			1.24846
$334$		0			0
$335$	−21.7739			−1.18964
$336$		0			0
$337$	21.3854			1.16494			0.582469	−	0.812853i	$-0.302087\pi$
$337$	21.3854			1.16494			0.582469	+	0.812853i	$0.302087\pi$
$338$		0			0
$339$			4.28434i			0.232693i
$340$		0			0
$341$			5.09609i			0.275969i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			12.4109i			0.668178i
$346$		0			0
$347$		−	12.0550i		−	0.647147i	−0.946203	−	0.323574i	$-0.895116\pi$
$347$		−	12.0550i		−	0.647147i	0.946203	−	0.323574i	$-0.104884\pi$
$348$		0			0
$349$	−6.05799			−0.324277			−0.162138	−	0.986768i	$-0.551839\pi$
$349$	−6.05799			−0.324277			−0.162138	+	0.986768i	$0.551839\pi$
$350$		0			0
$351$	6.57314	−	16.1037i	0.350848	−	0.859552i
$352$		0			0
$353$			24.1344i			1.28454i	0.766477	+	0.642272i	$0.222008\pi$
$353$			24.1344i			1.28454i	−0.766477	+	0.642272i	$0.777992\pi$
$354$		0			0
$355$		−	24.5114i		−	1.30093i
$356$		0			0
$357$	2.73519			0.144762
$358$		0			0
$359$		−	11.6331i		−	0.613972i	−0.951714	−	0.306986i	$-0.900680\pi$
$359$		−	11.6331i		−	0.613972i	0.951714	−	0.306986i	$-0.0993204\pi$
$360$		0			0
$361$	7.55918			0.397852
$362$		0			0
$363$		−	9.96575i		−	0.523066i
$364$		0			0
$365$			32.6094i			1.70686i
$366$		0			0
$367$	25.1559			1.31313			0.656563	−	0.754271i	$-0.272010\pi$
$367$	25.1559			1.31313			0.656563	+	0.754271i	$0.272010\pi$
$368$		0			0
$369$			23.8162i			1.23982i
$370$		0			0
$371$	−3.09234			−0.160546
$372$		0			0
$373$			2.65813i			0.137633i	0.997629	+	0.0688165i	$0.0219223\pi$
$373$			2.65813i			0.137633i	−0.997629	+	0.0688165i	$0.978078\pi$
$374$		0			0
$375$		−	0.465261i		−	0.0240260i
$376$		0			0
$377$	−3.03381	+	7.43261i	−0.156249	+	0.382799i
$378$		0			0
$379$	−1.58415			−0.0813722			−0.0406861	−	0.999172i	$-0.512954\pi$
$379$	−1.58415			−0.0813722			−0.0406861	+	0.999172i	$0.512954\pi$
$380$		0			0
$381$		−	6.35389i		−	0.325520i
$382$		0			0
$383$		−	1.72059i		−	0.0879183i	−0.999033	−	0.0439591i	$-0.986003\pi$
$383$		−	1.72059i		−	0.0879183i	0.999033	−	0.0439591i	$-0.0139971\pi$
$384$		0			0
$385$		−	2.09886i		−	0.106968i
$386$		0			0
$387$			12.9609i			0.658839i
$388$		0			0
$389$		−	1.96218i		−	0.0994865i	−0.998762	−	0.0497433i	$-0.984160\pi$
$389$		−	1.96218i		−	0.0994865i	0.998762	−	0.0497433i	$-0.0158403\pi$
$390$		0			0
$391$	−12.1315			−0.613515
$392$		0			0
$393$	2.24193			0.113091
$394$		0			0
$395$	41.2025			2.07312
$396$		0			0
$397$	0.871142			0.0437214			0.0218607	−	0.999761i	$-0.493041\pi$
$397$	0.871142			0.0437214			0.0218607	+	0.999761i	$0.493041\pi$
$398$		0			0
$399$	−4.86703			−0.243656
$400$		0			0
$401$		−	26.9824i		−	1.34744i	−0.738988	−	0.673718i	$-0.764696\pi$
$401$		−	26.9824i		−	1.34744i	0.738988	−	0.673718i	$-0.235304\pi$
$402$		0			0
$403$	−10.3794	+	25.4288i	−0.517036	+	1.26670i
$404$		0			0
$405$	5.54819			0.275692
$406$		0			0
$407$	−7.22974			−0.358365
$408$		0			0
$409$		−	14.8879i		−	0.736158i	−0.929794	−	0.368079i	$-0.880016\pi$
$409$		−	14.8879i		−	0.736158i	0.929794	−	0.368079i	$-0.119984\pi$
$410$		0			0
$411$	−15.1273			−0.746177
$412$		0			0
$413$			12.7056i			0.625204i
$414$		0			0
$415$	−17.4707			−0.857601
$416$		0			0
$417$	−0.182379			−0.00893112
$418$		0			0
$419$		−	24.4546i		−	1.19468i	−0.801987	−	0.597342i	$-0.796224\pi$
$419$		−	24.4546i		−	1.19468i	0.801987	−	0.597342i	$-0.203776\pi$
$420$		0			0
$421$	−17.3706			−0.846591			−0.423296	−	0.905992i	$-0.639127\pi$
$421$	−17.3706			−0.846591			−0.423296	+	0.905992i	$0.639127\pi$
$422$		0			0
$423$			24.2095i			1.17711i
$424$		0			0
$425$	−14.0263			−0.680375
$426$		0			0
$427$	11.0620			0.535327
$428$		0			0
$429$	−0.860860	+	2.10904i	−0.0415627	+	0.101826i
$430$		0			0
$431$			17.7177i			0.853430i	0.904386	+	0.426715i	$0.140329\pi$
$431$			17.7177i			0.853430i	−0.904386	+	0.426715i	$0.859671\pi$
$432$		0			0
$433$	−29.4265			−1.41415			−0.707074	−	0.707140i	$-0.749985\pi$
$433$	−29.4265			−1.41415			−0.707074	+	0.707140i	$0.749985\pi$
$434$		0			0
$435$	6.59708			0.316306
$436$		0			0
$437$	21.5869			1.03264
$438$		0			0
$439$	−24.2695			−1.15832			−0.579161	−	0.815213i	$-0.696620\pi$
$439$	−24.2695			−1.15832			−0.579161	+	0.815213i	$0.696620\pi$
$440$		0			0
$441$	−2.10811			−0.100386
$442$		0			0
$443$			1.33868i			0.0636027i	0.999494	+	0.0318013i	$0.0101244\pi$
$443$			1.33868i			0.0636027i	−0.999494	+	0.0318013i	$0.989876\pi$
$444$		0			0
$445$			1.55916i			0.0739113i
$446$		0			0
$447$		−	12.4292i		−	0.587881i
$448$		0			0
$449$		−	15.3626i		−	0.725007i	−0.931982	−	0.362503i	$-0.881922\pi$
$449$		−	15.3626i		−	0.725007i	0.931982	−	0.362503i	$-0.118078\pi$
$450$		0			0
$451$		−	7.55787i		−	0.355886i
$452$		0			0
$453$	−21.5501			−1.01251
$454$		0			0
$455$	4.27484	−	10.4730i	0.200407	−	0.490983i
$456$		0			0
$457$		−	27.7847i		−	1.29971i	−0.760057	−	0.649856i	$-0.774829\pi$
$457$		−	27.7847i		−	1.29971i	0.760057	−	0.649856i	$-0.225171\pi$
$458$		0			0
$459$		−	13.9716i		−	0.652140i
$460$		0			0
$461$	−12.3083			−0.573254			−0.286627	−	0.958042i	$-0.592534\pi$
$461$	−12.3083			−0.573254			−0.286627	+	0.958042i	$0.592534\pi$
$462$		0			0
$463$		−	6.36404i		−	0.295762i	−0.989005	−	0.147881i	$-0.952755\pi$
$463$		−	6.36404i		−	0.295762i	0.989005	−	0.147881i	$-0.0472453\pi$
$464$		0			0
$465$	22.5703			1.04667
$466$		0			0
$467$			12.0882i			0.559377i	0.960091	+	0.279688i	$0.0902311\pi$
$467$			12.0882i			0.559377i	−0.960091	+	0.279688i	$0.909769\pi$
$468$		0			0
$469$		−	6.94022i		−	0.320470i
$470$		0			0
$471$	−6.43552			−0.296533
$472$		0			0
$473$		−	4.11303i		−	0.189117i
$474$		0			0
$475$	24.9585			1.14518
$476$		0			0
$477$			6.51898i			0.298484i
$478$		0			0
$479$		−	27.3750i		−	1.25080i	−0.780305	−	0.625399i	$-0.784936\pi$
$479$		−	27.3750i		−	1.25080i	0.780305	−	0.625399i	$-0.215064\pi$
$480$		0			0
$481$	−36.0755	−	14.7251i	−1.64490	−	0.671409i
$482$		0			0
$483$	−3.95584			−0.179997
$484$		0			0
$485$			5.39447i			0.244950i
$486$		0			0
$487$			5.41381i			0.245323i	0.992449	+	0.122662i	$0.0391430\pi$
$487$			5.41381i			0.245323i	−0.992449	+	0.122662i	$0.960857\pi$
$488$		0			0
$489$			18.4812i			0.835749i
$490$		0			0
$491$		−	13.9327i		−	0.628774i	−0.949295	−	0.314387i	$-0.898201\pi$
$491$		−	13.9327i		−	0.628774i	0.949295	−	0.314387i	$-0.101799\pi$
$492$		0			0
$493$			6.44857i			0.290429i
$494$		0			0
$495$	−4.42461			−0.198871
$496$		0			0
$497$	7.81277			0.350451
$498$		0			0
$499$	−31.8347			−1.42512			−0.712558	−	0.701613i	$-0.752464\pi$
$499$	−31.8347			−1.42512			−0.712558	+	0.701613i	$0.752464\pi$
$500$		0			0
$501$	−11.9146			−0.532305
$502$		0			0
$503$	8.66521			0.386362			0.193181	−	0.981163i	$-0.438119\pi$
$503$	8.66521			0.386362			0.193181	+	0.981163i	$0.438119\pi$
$504$		0			0
$505$		−	61.1909i		−	2.72296i
$506$		0			0
$507$	−8.59117	+	8.77051i	−0.381547	+	0.389512i
$508$		0			0
$509$	25.7198			1.14001			0.570004	−	0.821642i	$-0.306942\pi$
$509$	25.7198			1.14001			0.570004	+	0.821642i	$0.306942\pi$
$510$		0			0
$511$	−10.3939			−0.459801
$512$		0			0
$513$			24.8613i			1.09765i
$514$		0			0
$515$	−27.2156			−1.19926
$516$		0			0
$517$		−	7.68268i		−	0.337884i
$518$		0			0
$519$	19.4282			0.852806
$520$		0			0
$521$	16.7303			0.732968			0.366484	−	0.930424i	$-0.380561\pi$
$521$	16.7303			0.732968			0.366484	+	0.930424i	$0.380561\pi$
$522$		0			0
$523$		−	0.415884i		−	0.0181853i	−0.999959	−	0.00909267i	$-0.997106\pi$
$523$		−	0.415884i		−	0.0181853i	0.999959	−	0.00909267i	$-0.00289433\pi$
$524$		0			0
$525$	−4.57371			−0.199613
$526$		0			0
$527$			22.0622i			0.961043i
$528$		0			0
$529$	−5.45454			−0.237154
$530$		0			0
$531$	26.7849			1.16236
$532$		0			0
$533$	15.3935	−	37.7128i	0.666765	−	1.63352i
$534$		0			0
$535$		−	37.6396i		−	1.62730i
$536$		0			0
$537$	−0.632287			−0.0272852
$538$		0			0
$539$	0.668990			0.0288154
$540$		0			0
$541$	0.749319			0.0322158			0.0161079	−	0.999870i	$-0.494872\pi$
$541$	0.749319			0.0322158			0.0161079	+	0.999870i	$0.494872\pi$
$542$		0			0
$543$	−6.28754			−0.269824
$544$		0			0
$545$	−1.21932			−0.0522299
$546$		0			0
$547$		−	17.6399i		−	0.754227i	−0.926167	−	0.377114i	$-0.876917\pi$
$547$		−	17.6399i		−	0.754227i	0.926167	−	0.377114i	$-0.123083\pi$
$548$		0			0
$549$		−	23.3198i		−	0.995267i
$550$		0			0
$551$		−	11.4747i		−	0.488837i
$552$		0			0
$553$			13.1329i			0.558467i
$554$		0			0
$555$			32.0201i			1.35918i
$556$		0			0
$557$	17.7594			0.752491			0.376246	−	0.926520i	$-0.377215\pi$
$557$	17.7594			0.752491			0.376246	+	0.926520i	$0.377215\pi$
$558$		0			0
$559$	8.37720	−	20.5235i	0.354318	−	0.868052i
$560$		0			0
$561$			1.82981i			0.0772548i
$562$		0			0
$563$		−	8.32427i		−	0.350826i	−0.984495	−	0.175413i	$-0.943874\pi$
$563$		−	8.32427i		−	0.350826i	0.984495	−	0.175413i	$-0.0561261\pi$
$564$		0			0
$565$	14.2328			0.598779
$566$		0			0
$567$			1.76843i			0.0742672i
$568$		0			0
$569$	−0.240261			−0.0100723			−0.00503613	−	0.999987i	$-0.501603\pi$
$569$	−0.240261			−0.0100723			−0.00503613	+	0.999987i	$0.501603\pi$
$570$		0			0
$571$			6.53041i			0.273289i	0.990620	+	0.136644i	$0.0436318\pi$
$571$			6.53041i			0.273289i	−0.990620	+	0.136644i	$0.956368\pi$
$572$		0			0
$573$		−	8.20365i		−	0.342712i
$574$		0			0
$575$	20.2859			0.845981
$576$		0			0
$577$		−	21.3879i		−	0.890391i	−0.895433	−	0.445196i	$-0.853134\pi$
$577$		−	21.3879i		−	0.890391i	0.895433	−	0.445196i	$-0.146866\pi$
$578$		0			0
$579$	8.21445			0.341381
$580$		0			0
$581$		−	5.56860i		−	0.231025i
$582$		0			0
$583$		−	2.06874i		−	0.0856786i
$584$		0			0
$585$	−22.0783	−	9.01181i	−0.912824	−	0.372592i
$586$		0			0
$587$	−21.3987			−0.883221			−0.441610	−	0.897207i	$-0.645593\pi$
$587$	−21.3987			−0.883221			−0.441610	+	0.897207i	$0.645593\pi$
$588$		0			0
$589$		−	39.2577i		−	1.61758i
$590$		0			0
$591$			0.943774i			0.0388217i
$592$		0			0
$593$			17.8910i			0.734695i	0.930084	+	0.367348i	$0.119734\pi$
$593$			17.8910i			0.734695i	−0.930084	+	0.367348i	$0.880266\pi$
$594$		0			0
$595$		−	9.08645i		−	0.372508i
$596$		0			0
$597$		−	2.30727i		−	0.0944304i
$598$		0			0
$599$	20.3457			0.831304			0.415652	−	0.909524i	$-0.363553\pi$
$599$	20.3457			0.831304			0.415652	+	0.909524i	$0.363553\pi$
$600$		0			0
$601$	−6.96098			−0.283944			−0.141972	−	0.989871i	$-0.545344\pi$
$601$	−6.96098			−0.283944			−0.141972	+	0.989871i	$0.545344\pi$
$602$		0			0
$603$	−14.6307			−0.595809
$604$		0			0
$605$	−33.1067			−1.34598
$606$		0			0
$607$	−16.4197			−0.666454			−0.333227	−	0.942847i	$-0.608138\pi$
$607$	−16.4197			−0.666454			−0.333227	+	0.942847i	$0.608138\pi$
$608$		0			0
$609$			2.10276i			0.0852079i
$610$		0			0
$611$	15.6477	−	38.3357i	0.633037	−	1.55089i
$612$		0			0
$613$	−18.6762			−0.754324			−0.377162	−	0.926147i	$-0.623100\pi$
$613$	−18.6762			−0.754324			−0.377162	+	0.926147i	$0.623100\pi$
$614$		0			0
$615$	−33.4734			−1.34978
$616$		0			0
$617$		−	23.3338i		−	0.939383i	−0.882830	−	0.469692i	$-0.844365\pi$
$617$		−	23.3338i		−	0.939383i	0.882830	−	0.469692i	$-0.155635\pi$
$618$		0			0
$619$	−16.9403			−0.680888			−0.340444	−	0.940265i	$-0.610577\pi$
$619$	−16.9403			−0.680888			−0.340444	+	0.940265i	$0.610577\pi$
$620$		0			0
$621$			20.2069i			0.810873i
$622$		0			0
$623$	−0.496967			−0.0199106
$624$		0			0
$625$	−25.7605			−1.03042
$626$		0			0
$627$		−	3.25599i		−	0.130032i
$628$		0			0
$629$	−31.2993			−1.24798
$630$		0			0
$631$		−	18.3836i		−	0.731838i	−0.930647	−	0.365919i	$-0.880755\pi$
$631$		−	18.3836i		−	0.731838i	0.930647	−	0.365919i	$-0.119245\pi$
$632$		0			0
$633$	3.84761			0.152929
$634$		0			0
$635$	−21.1080			−0.837644
$636$		0			0
$637$	3.33818	+	1.36256i	0.132263	+	0.0539867i
$638$		0			0
$639$		−	16.4701i		−	0.651549i
$640$		0			0
$641$	36.6206			1.44643			0.723213	−	0.690625i	$-0.242665\pi$
$641$	36.6206			1.44643			0.723213	+	0.690625i	$0.242665\pi$
$642$		0			0
$643$	29.9927			1.18280			0.591399	−	0.806379i	$-0.298576\pi$
$643$	29.9927			1.18280			0.591399	+	0.806379i	$0.298576\pi$
$644$		0			0
$645$	−18.2164			−0.717269
$646$		0			0
$647$	−34.5613			−1.35875			−0.679373	−	0.733793i	$-0.737748\pi$
$647$	−34.5613			−1.35875			−0.679373	+	0.733793i	$0.737748\pi$
$648$		0			0
$649$	−8.49995			−0.333652
$650$		0			0
$651$			7.19406i			0.281957i
$652$		0			0
$653$		−	33.2212i		−	1.30004i	−0.759915	−	0.650022i	$-0.774760\pi$
$653$		−	33.2212i		−	1.30004i	0.759915	−	0.650022i	$-0.225240\pi$
$654$		0			0
$655$		−	7.44782i		−	0.291011i
$656$		0			0
$657$			21.9115i			0.854850i
$658$		0			0
$659$		−	2.56000i		−	0.0997233i	−0.998756	−	0.0498616i	$-0.984122\pi$
$659$		−	2.56000i		−	0.0997233i	0.998756	−	0.0498616i	$-0.0158780\pi$
$660$		0			0
$661$	−33.5760			−1.30596			−0.652978	−	0.757377i	$-0.726481\pi$
$661$	−33.5760			−1.30596			−0.652978	+	0.757377i	$0.726481\pi$
$662$		0			0
$663$	−3.72687	+	9.13055i	−0.144740	+	0.354601i
$664$		0			0
$665$			16.1685i			0.626989i
$666$		0			0
$667$		−	9.32641i		−	0.361120i
$668$		0			0
$669$	14.5838			0.563843
$670$		0			0
$671$			7.40036i			0.285688i
$672$		0			0
$673$	12.7789			0.492590			0.246295	−	0.969195i	$-0.420787\pi$
$673$	12.7789			0.492590			0.246295	+	0.969195i	$0.420787\pi$
$674$		0			0
$675$			23.3630i			0.899242i
$676$		0			0
$677$		−	2.32427i		−	0.0893289i	−0.999002	−	0.0446645i	$-0.985778\pi$
$677$		−	2.32427i		−	0.0893289i	0.999002	−	0.0446645i	$-0.0142219\pi$
$678$		0			0
$679$	−1.71943			−0.0659858
$680$		0			0
$681$		−	17.8445i		−	0.683802i
$682$		0			0
$683$	−3.42642			−0.131108			−0.0655541	−	0.997849i	$-0.520881\pi$
$683$	−3.42642			−0.131108			−0.0655541	+	0.997849i	$0.520881\pi$
$684$		0			0
$685$			50.2538i			1.92010i
$686$		0			0
$687$			14.6389i			0.558509i
$688$		0			0
$689$	4.21351	−	10.3228i	0.160522	−	0.393267i
$690$		0			0
$691$	9.36666			0.356324			0.178162	−	0.984001i	$-0.442985\pi$
$691$	9.36666			0.356324			0.178162	+	0.984001i	$0.442985\pi$
$692$		0			0
$693$		−	1.41030i		−	0.0535729i
$694$		0			0
$695$			0.605871i			0.0229820i
$696$		0			0
$697$		−	32.7198i		−	1.23935i
$698$		0			0
$699$			10.7396i			0.406209i
$700$		0			0
$701$			28.0448i			1.05924i	0.848235	+	0.529619i	$0.177665\pi$
$701$			28.0448i			1.05924i	−0.848235	+	0.529619i	$0.822335\pi$
$702$		0			0
$703$	55.6943			2.10055
$704$		0			0
$705$	−34.0262			−1.28150
$706$		0			0
$707$	19.5040			0.733524
$708$		0			0
$709$	−18.6985			−0.702238			−0.351119	−	0.936331i	$-0.614199\pi$
$709$	−18.6985			−0.702238			−0.351119	+	0.936331i	$0.614199\pi$
$710$		0			0
$711$	27.6855			1.03829
$712$		0			0
$713$		−	31.9080i		−	1.19496i
$714$		0			0
$715$	7.00635	+	2.85982i	0.262023	+	0.106951i
$716$		0			0
$717$	−5.91287			−0.220820
$718$		0			0
$719$	−13.1469			−0.490298			−0.245149	−	0.969485i	$-0.578837\pi$
$719$	−13.1469			−0.490298			−0.245149	+	0.969485i	$0.578837\pi$
$720$		0			0
$721$		−	8.67472i		−	0.323063i
$722$		0			0
$723$	16.2964			0.606071
$724$		0			0
$725$		−	10.7831i		−	0.400475i
$726$		0			0
$727$	−19.0751			−0.707458			−0.353729	−	0.935348i	$-0.615086\pi$
$727$	−19.0751			−0.707458			−0.353729	+	0.935348i	$0.615086\pi$
$728$		0			0
$729$	−9.93962			−0.368134
$730$		0			0
$731$		−	17.8063i		−	0.658589i
$732$		0			0
$733$	−26.4358			−0.976428			−0.488214	−	0.872724i	$-0.662352\pi$
$733$	−26.4358			−0.976428			−0.488214	+	0.872724i	$0.662352\pi$
$734$		0			0
$735$		−	2.96292i		−	0.109289i
$736$		0			0
$737$	4.64294			0.171025
$738$		0			0
$739$	12.9305			0.475658			0.237829	−	0.971307i	$-0.423564\pi$
$739$	12.9305			0.475658			0.237829	+	0.971307i	$0.423564\pi$
$740$		0			0
$741$	6.63163	−	16.2470i	0.243619	−	0.596848i
$742$		0			0
$743$		−	29.1976i		−	1.07116i	−0.844485	−	0.535578i	$-0.820094\pi$
$743$		−	29.1976i		−	1.07116i	0.844485	−	0.535578i	$-0.179906\pi$
$744$		0			0
$745$	−41.2905			−1.51277
$746$		0			0
$747$	−11.7392			−0.429515
$748$		0			0
$749$	11.9973			0.438371
$750$		0			0
$751$	−22.8429			−0.833548			−0.416774	−	0.909010i	$-0.636839\pi$
$751$	−22.8429			−0.833548			−0.416774	+	0.909010i	$0.636839\pi$
$752$		0			0
$753$	0.261561			0.00953182
$754$		0			0
$755$			71.5906i			2.60545i
$756$		0			0
$757$		−	47.2876i		−	1.71870i	−0.511391	−	0.859348i	$-0.670870\pi$
$757$		−	47.2876i		−	1.71870i	0.511391	−	0.859348i	$-0.329130\pi$
$758$		0			0
$759$		−	2.64642i		−	0.0960589i
$760$		0			0
$761$		−	24.2034i		−	0.877371i	−0.898641	−	0.438686i	$-0.855444\pi$
$761$		−	24.2034i		−	0.877371i	0.898641	−	0.438686i	$-0.144556\pi$
$762$		0			0
$763$		−	0.388646i		−	0.0140699i
$764$		0			0
$765$	−19.1552			−0.692558
$766$		0			0
$767$	−42.4137	−	17.3122i	−1.53147	−	0.625109i
$768$		0			0
$769$		−	8.16082i		−	0.294287i	−0.989115	−	0.147143i	$-0.952992\pi$
$769$		−	8.16082i		−	0.294287i	0.989115	−	0.147143i	$-0.0470078\pi$
$770$		0			0
$771$			2.16746i			0.0780591i
$772$		0			0
$773$	−50.7014			−1.82360			−0.911801	−	0.410632i	$-0.865308\pi$
$773$	−50.7014			−1.82360			−0.911801	+	0.410632i	$0.865308\pi$
$774$		0			0
$775$		−	36.8918i		−	1.32519i
$776$		0			0
$777$	−10.2061			−0.366142
$778$		0			0
$779$			58.2220i			2.08602i
$780$		0			0
$781$			5.22666i			0.187025i
$782$		0			0
$783$	10.7411			0.383856
$784$		0			0
$785$			21.3791i			0.763054i
$786$		0			0
$787$	−14.8854			−0.530607			−0.265303	−	0.964165i	$-0.585472\pi$
$787$	−14.8854			−0.530607			−0.265303	+	0.964165i	$0.585472\pi$
$788$		0			0
$789$		−	16.5867i		−	0.590501i
$790$		0			0
$791$			4.53657i			0.161302i
$792$		0			0
$793$	−15.0726	+	36.9269i	−0.535245	+	1.31131i
$794$		0			0
$795$	−9.16235			−0.324955
$796$		0			0
$797$			38.4832i			1.36315i	0.731750	+	0.681573i	$0.238704\pi$
$797$			38.4832i			1.36315i	−0.731750	+	0.681573i	$0.761296\pi$
$798$		0			0
$799$		−	33.2602i		−	1.17666i
$800$		0			0
$801$			1.04766i			0.0370173i
$802$		0			0
$803$		−	6.95344i		−	0.245381i
$804$		0			0
$805$			13.1415i			0.463178i
$806$		0			0
$807$	−14.0166			−0.493408
$808$		0			0
$809$	−50.7774			−1.78524			−0.892619	−	0.450812i	$-0.851134\pi$
$809$	−50.7774			−1.78524			−0.892619	+	0.450812i	$0.851134\pi$
$810$		0			0
$811$	−2.26857			−0.0796604			−0.0398302	−	0.999206i	$-0.512682\pi$
$811$	−2.26857			−0.0796604			−0.0398302	+	0.999206i	$0.512682\pi$
$812$		0			0
$813$	1.05500			0.0370004
$814$		0			0
$815$	61.3955			2.15059
$816$		0			0
$817$			31.6847i			1.10851i
$818$		0			0
$819$	2.87243	−	7.03723i	0.100371	−	0.245901i
$820$		0			0
$821$	48.9416			1.70808			0.854038	−	0.520211i	$-0.174147\pi$
$821$	48.9416			1.70808			0.854038	+	0.520211i	$0.174147\pi$
$822$		0			0
$823$	49.5972			1.72885			0.864425	−	0.502762i	$-0.167683\pi$
$823$	49.5972			1.72885			0.864425	+	0.502762i	$0.167683\pi$
$824$		0			0
$825$		−	3.05976i		−	0.106527i
$826$		0			0
$827$	37.1376			1.29140			0.645700	−	0.763591i	$-0.276566\pi$
$827$	37.1376			1.29140			0.645700	+	0.763591i	$0.276566\pi$
$828$		0			0
$829$			23.4998i			0.816181i	0.912942	+	0.408090i	$0.133805\pi$
$829$			23.4998i			0.816181i	−0.912942	+	0.408090i	$0.866195\pi$
$830$		0			0
$831$	22.6315			0.785078
$832$		0			0
$833$	2.89622			0.100348
$834$		0			0
$835$			39.5809i			1.36975i
$836$		0			0
$837$	36.7480			1.27020
$838$		0			0
$839$		−	3.01174i		−	0.103977i	−0.998648	−	0.0519884i	$-0.983444\pi$
$839$		−	3.01174i		−	0.103977i	0.998648	−	0.0519884i	$-0.0165559\pi$
$840$		0			0
$841$	24.0425			0.829051
$842$		0			0
$843$	−16.4036			−0.564971
$844$		0			0
$845$	29.1361	+	28.5403i	1.00231	+	0.981817i
$846$		0			0
$847$		−	10.5525i		−	0.362587i
$848$		0			0
$849$	11.1570			0.382907
$850$		0			0
$851$	45.2674			1.55175
$852$		0			0
$853$	−32.8970			−1.12637			−0.563186	−	0.826330i	$-0.690424\pi$
$853$	−32.8970			−1.12637			−0.563186	+	0.826330i	$0.690424\pi$
$854$		0			0
$855$	34.0850			1.16568
$856$		0			0
$857$	31.4372			1.07388			0.536938	−	0.843622i	$-0.319581\pi$
$857$	31.4372			1.07388			0.536938	+	0.843622i	$0.319581\pi$
$858$		0			0
$859$			49.4426i			1.68696i	0.537162	+	0.843479i	$0.319496\pi$
$859$			49.4426i			1.68696i	−0.537162	+	0.843479i	$0.680504\pi$
$860$		0			0
$861$		−	10.6693i		−	0.363609i
$862$		0			0
$863$			45.5941i			1.55204i	0.630708	+	0.776020i	$0.282764\pi$
$863$			45.5941i			1.55204i	−0.630708	+	0.776020i	$0.717236\pi$
$864$		0			0
$865$		−	64.5417i		−	2.19448i
$866$		0			0
$867$		−	8.13312i		−	0.276215i
$868$		0			0
$869$	−8.78577			−0.298037
$870$		0			0
$871$	23.1677	+	9.45649i	0.785007	+	0.320421i
$872$		0			0
$873$			3.62475i			0.122679i
$874$		0			0
$875$		−	0.492652i		−	0.0166547i
$876$		0			0
$877$	−54.9310			−1.85489			−0.927444	−	0.373963i	$-0.877999\pi$
$877$	−54.9310			−1.85489			−0.927444	+	0.373963i	$0.877999\pi$
$878$		0			0
$879$			2.72956i			0.0920658i
$880$		0			0
$881$	−24.5274			−0.826349			−0.413175	−	0.910652i	$-0.635580\pi$
$881$	−24.5274			−0.826349			−0.413175	+	0.910652i	$0.635580\pi$
$882$		0			0
$883$			1.66721i			0.0561060i	0.999606	+	0.0280530i	$0.00893072\pi$
$883$			1.66721i			0.0561060i	−0.999606	+	0.0280530i	$0.991069\pi$
$884$		0			0
$885$			37.6458i			1.26545i
$886$		0			0
$887$	−29.2746			−0.982946			−0.491473	−	0.870893i	$-0.663541\pi$
$887$	−29.2746			−0.982946			−0.491473	+	0.870893i	$0.663541\pi$
$888$		0			0
$889$		−	6.72796i		−	0.225649i
$890$		0			0
$891$	−1.18306			−0.0396341
$892$		0			0
$893$			59.1835i			1.98050i
$894$		0			0
$895$			2.10049i			0.0702116i
$896$		0			0
$897$	5.39008	−	13.2053i	0.179970	−	0.440912i
$898$		0			0
$899$	−16.9609			−0.565679
$900$		0			0
$901$		−	8.95609i		−	0.298371i
$902$		0			0
$903$		−	5.80629i		−	0.193221i
$904$		0			0
$905$			20.8875i			0.694325i
$906$		0			0
$907$		−	33.3296i		−	1.10669i	−0.832951	−	0.553346i	$-0.813351\pi$
$907$		−	33.3296i		−	1.10669i	0.832951	−	0.553346i	$-0.186649\pi$
$908$		0			0
$909$		−	41.1165i		−	1.36375i
$910$		0			0
$911$	8.37641			0.277523			0.138761	−	0.990326i	$-0.455688\pi$
$911$	8.37641			0.277523			0.138761	+	0.990326i	$0.455688\pi$
$912$		0			0
$913$	3.72534			0.123291
$914$		0			0
$915$	32.7758			1.08353
$916$		0			0
$917$	2.37392			0.0783938
$918$		0			0
$919$	−13.0688			−0.431099			−0.215549	−	0.976493i	$-0.569154\pi$
$919$	−13.0688			−0.431099			−0.215549	+	0.976493i	$0.569154\pi$
$920$		0			0
$921$		−	5.42559i		−	0.178779i
$922$		0			0
$923$	−10.6454	+	26.0804i	−0.350397	+	0.858447i
$924$		0			0
$925$	52.3378			1.72085
$926$		0			0
$927$	−18.2872			−0.600631
$928$		0			0
$929$			6.02154i			0.197560i	0.995109	+	0.0987802i	$0.0314941\pi$
$929$			6.02154i			0.197560i	−0.995109	+	0.0987802i	$0.968506\pi$
$930$		0			0
$931$	−5.15356			−0.168901
$932$		0			0
$933$		−	23.7539i		−	0.777668i
$934$		0			0
$935$	6.07874			0.198796
$936$		0			0
$937$	−14.7717			−0.482571			−0.241285	−	0.970454i	$-0.577569\pi$
$937$	−14.7717			−0.482571			−0.241285	+	0.970454i	$0.577569\pi$
$938$		0			0
$939$		−	15.9295i		−	0.519838i
$940$		0			0
$941$	−14.3166			−0.466706			−0.233353	−	0.972392i	$-0.574970\pi$
$941$	−14.3166			−0.466706			−0.233353	+	0.972392i	$0.574970\pi$
$942$		0			0
$943$			47.3219i			1.54101i
$944$		0			0
$945$	−15.1349			−0.492338
$946$		0			0
$947$	−3.61569			−0.117494			−0.0587470	−	0.998273i	$-0.518711\pi$
$947$	−3.61569			−0.117494			−0.0587470	+	0.998273i	$0.518711\pi$
$948$		0			0
$949$	14.1624	−	34.6968i	0.459731	−	1.12631i
$950$		0			0
$951$			31.7472i			1.02947i
$952$		0			0
$953$	47.7347			1.54628			0.773140	−	0.634236i	$-0.218685\pi$
$953$	47.7347			1.54628			0.773140	+	0.634236i	$0.218685\pi$
$954$		0			0
$955$	−27.2529			−0.881885
$956$		0			0
$957$	−1.40672			−0.0454729
$958$		0			0
$959$	−16.0179			−0.517245
$960$		0			0
$961$	−27.0276			−0.871858
$962$		0			0
$963$		−	25.2915i		−	0.815008i
$964$		0			0
$965$		−	27.2888i		−	0.878458i
$966$		0			0
$967$		−	47.2805i		−	1.52044i	−0.649667	−	0.760219i	$-0.725092\pi$
$967$		−	47.2805i		−	1.52044i	0.649667	−	0.760219i	$-0.274908\pi$
$968$		0			0
$969$		−	14.0960i		−	0.452828i
$970$		0			0
$971$			22.2449i			0.713872i	0.934129	+	0.356936i	$0.116179\pi$
$971$			22.2449i			0.713872i	−0.934129	+	0.356936i	$0.883821\pi$
$972$		0			0
$973$	−0.193116			−0.00619100
$974$		0			0
$975$	6.23196	−	15.2678i	0.199583	−	0.488962i
$976$		0			0
$977$			54.4866i			1.74318i	0.490235	+	0.871590i	$0.336911\pi$
$977$			54.4866i			1.74318i	−0.490235	+	0.871590i	$0.663089\pi$
$978$		0			0
$979$		−	0.332466i		−	0.0106257i
$980$		0			0
$981$	−0.819307			−0.0261585
$982$		0			0
$983$			24.2235i			0.772608i	0.922371	+	0.386304i	$0.126249\pi$
$983$			24.2235i			0.772608i	−0.922371	+	0.386304i	$0.873751\pi$
$984$		0			0
$985$	3.13527			0.0998979
$986$		0			0
$987$		−	10.8455i		−	0.345216i
$988$		0			0
$989$			25.7528i			0.818892i
$990$		0			0
$991$	−16.4289			−0.521882			−0.260941	−	0.965355i	$-0.584033\pi$
$991$	−16.4289			−0.521882			−0.260941	+	0.965355i	$0.584033\pi$
$992$		0			0
$993$		−	3.64059i		−	0.115531i
$994$		0			0
$995$	−7.66488			−0.242993
$996$		0			0
$997$			3.87312i			0.122663i	0.998117	+	0.0613315i	$0.0195347\pi$
$997$			3.87312i			0.122663i	−0.998117	+	0.0613315i	$0.980465\pi$
$998$		0			0
$999$			52.1338i			1.64944i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.i.a.337.18		84
4.3	odd	2		728.2.i.a.701.74	yes	84
8.3	odd	2		728.2.i.a.701.12	yes	84
8.5	even	2	inner	2912.2.i.a.337.67		84
13.12	even	2	inner	2912.2.i.a.337.68		84
52.51	odd	2		728.2.i.a.701.11	✓	84
104.51	odd	2		728.2.i.a.701.73	yes	84
104.77	even	2	inner	2912.2.i.a.337.17		84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.i.a.701.11	✓	84	52.51	odd	2
728.2.i.a.701.12	yes	84	8.3	odd	2
728.2.i.a.701.73	yes	84	104.51	odd	2
728.2.i.a.701.74	yes	84	4.3	odd	2
2912.2.i.a.337.17		84	104.77	even	2	inner
2912.2.i.a.337.18		84	1.1	even	1	trivial
2912.2.i.a.337.67		84	8.5	even	2	inner
2912.2.i.a.337.68		84	13.12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.944401i q^{3} +3.13735 q^{5} +1.00000i q^{7} +2.10811 q^{9} -0.668990 q^{11} +(-3.33818 - 1.36256i) q^{13} +2.96292i q^{15} -2.89622 q^{17} +5.15356 q^{19} -0.944401 q^{21} +4.18873 q^{23} +4.84297 q^{25} +4.82410i q^{27} -2.22655i q^{29} -7.61759i q^{31} -0.631795i q^{33} +3.13735i q^{35} +10.8069 q^{37} +(1.28681 - 3.15258i) q^{39} +11.2974i q^{41} +6.14812i q^{43} +6.61387 q^{45} +11.4840i q^{47} -1.00000 q^{49} -2.73519i q^{51} +3.09234i q^{53} -2.09886 q^{55} +4.86703i q^{57} +12.7056 q^{59} -11.0620i q^{61} +2.10811i q^{63} +(-10.4730 - 4.27484i) q^{65} -6.94022 q^{67} +3.95584i q^{69} -7.81277i q^{71} +10.3939i q^{73} +4.57371i q^{75} -0.668990i q^{77} +13.1329 q^{79} +1.76843 q^{81} -5.56860 q^{83} -9.08645 q^{85} +2.10276 q^{87} +0.496967i q^{89} +(1.36256 - 3.33818i) q^{91} +7.19406 q^{93} +16.1685 q^{95} +1.71943i q^{97} -1.41030 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 84 q^{9} + 8 q^{17} + 24 q^{23} + 92 q^{25} + 24 q^{39} - 84 q^{49} - 32 q^{55} - 24 q^{65} + 40 q^{79} + 84 q^{81} + 48 q^{87} + 16 q^{95}+O(q^{100}) \) 84 * q - 84 * q^9 + 8 * q^17 + 24 * q^23 + 92 * q^25 + 24 * q^39 - 84 * q^49 - 32 * q^55 - 24 * q^65 + 40 * q^79 + 84 * q^81 + 48 * q^87 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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