LMFDB - Embedded newform 912.2.f.d.113.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [912,2,Mod(113,912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("912.113"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$7.28235666434$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.2
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	912.113
Dual form			912.2.f.d.113.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 + 1.41421i) q^{3} -1.41421i q^{5} +4.00000 q^{7} +(-1.00000 + 2.82843i) q^{9} +5.65685i q^{11} -4.24264i q^{13} +(2.00000 - 1.41421i) q^{15} +2.82843i q^{17} +(-1.00000 - 4.24264i) q^{19} +(4.00000 + 5.65685i) q^{21} +1.41421i q^{23} +3.00000 q^{25} +(-5.00000 + 1.41421i) q^{27} +6.00000 q^{29} +4.24264i q^{31} +(-8.00000 + 5.65685i) q^{33} -5.65685i q^{35} -4.24264i q^{37} +(6.00000 - 4.24264i) q^{39} -2.00000 q^{43} +(4.00000 + 1.41421i) q^{45} +1.41421i q^{47} +9.00000 q^{49} +(-4.00000 + 2.82843i) q^{51} -6.00000 q^{53} +8.00000 q^{55} +(5.00000 - 5.65685i) q^{57} +12.0000 q^{59} +2.00000 q^{61} +(-4.00000 + 11.3137i) q^{63} -6.00000 q^{65} +(-2.00000 + 1.41421i) q^{69} -12.0000 q^{71} -4.00000 q^{73} +(3.00000 + 4.24264i) q^{75} +22.6274i q^{77} +4.24264i q^{79} +(-7.00000 - 5.65685i) q^{81} -2.82843i q^{83} +4.00000 q^{85} +(6.00000 + 8.48528i) q^{87} -6.00000 q^{89} -16.9706i q^{91} +(-6.00000 + 4.24264i) q^{93} +(-6.00000 + 1.41421i) q^{95} +8.48528i q^{97} +(-16.0000 - 5.65685i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 8 q^{7} - 2 q^{9} + 4 q^{15} - 2 q^{19} + 8 q^{21} + 6 q^{25} - 10 q^{27} + 12 q^{29} - 16 q^{33} + 12 q^{39} - 4 q^{43} + 8 q^{45} + 18 q^{49} - 8 q^{51} - 12 q^{53} + 16 q^{55} + 10 q^{57}+ \cdots - 32 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 8 * q^7 - 2 * q^9 + 4 * q^15 - 2 * q^19 + 8 * q^21 + 6 * q^25 - 10 * q^27 + 12 * q^29 - 16 * q^33 + 12 * q^39 - 4 * q^43 + 8 * q^45 + 18 * q^49 - 8 * q^51 - 12 * q^53 + 16 * q^55 + 10 * q^57 + 24 * q^59 + 4 * q^61 - 8 * q^63 - 12 * q^65 - 4 * q^69 - 24 * q^71 - 8 * q^73 + 6 * q^75 - 14 * q^81 + 8 * q^85 + 12 * q^87 - 12 * q^89 - 12 * q^93 - 12 * q^95 - 32 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	1.00000	+	1.41421i	0.577350	+	0.816497i
$3$	1.00000	+	1.41421i	0.577350	+	0.816497i
$4$		0			0
$5$		−	1.41421i		−	0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$		−	1.41421i		−	0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$		0			0
$7$	4.00000			1.51186			0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000			1.51186			0.755929	+	0.654654i	$0.227186\pi$
$8$		0			0
$9$	−1.00000	+	2.82843i	−0.333333	+	0.942809i
$10$		0			0
$11$			5.65685i			1.70561i	0.522233	+	0.852803i	$0.325099\pi$
$11$			5.65685i			1.70561i	−0.522233	+	0.852803i	$0.674901\pi$
$12$		0			0
$13$		−	4.24264i		−	1.17670i	−0.808608	−	0.588348i	$-0.799778\pi$
$13$		−	4.24264i		−	1.17670i	0.808608	−	0.588348i	$-0.200222\pi$
$14$		0			0
$15$	2.00000	−	1.41421i	0.516398	−	0.365148i
$16$		0			0
$17$			2.82843i			0.685994i	0.939336	+	0.342997i	$0.111442\pi$
$17$			2.82843i			0.685994i	−0.939336	+	0.342997i	$0.888558\pi$
$18$		0			0
$19$	−1.00000	−	4.24264i	−0.229416	−	0.973329i
$19$	−1.00000	−	4.24264i	−0.229416	−	0.973329i
$20$		0			0
$21$	4.00000	+	5.65685i	0.872872	+	1.23443i
$22$		0			0
$23$			1.41421i			0.294884i	0.989071	+	0.147442i	$0.0471040\pi$
$23$			1.41421i			0.294884i	−0.989071	+	0.147442i	$0.952896\pi$
$24$		0			0
$25$	3.00000			0.600000
$26$		0			0
$27$	−5.00000	+	1.41421i	−0.962250	+	0.272166i
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$			4.24264i			0.762001i	0.924575	+	0.381000i	$0.124420\pi$
$31$			4.24264i			0.762001i	−0.924575	+	0.381000i	$0.875580\pi$
$32$		0			0
$33$	−8.00000	+	5.65685i	−1.39262	+	0.984732i
$34$		0			0
$35$		−	5.65685i		−	0.956183i
$36$		0			0
$37$		−	4.24264i		−	0.697486i	−0.937218	−	0.348743i	$-0.886609\pi$
$37$		−	4.24264i		−	0.697486i	0.937218	−	0.348743i	$-0.113391\pi$
$38$		0			0
$39$	6.00000	−	4.24264i	0.960769	−	0.679366i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−2.00000			−0.304997			−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000			−0.304997			−0.152499	+	0.988304i	$0.548732\pi$
$44$		0			0
$45$	4.00000	+	1.41421i	0.596285	+	0.210819i
$46$		0			0
$47$			1.41421i			0.206284i	0.994667	+	0.103142i	$0.0328896\pi$
$47$			1.41421i			0.206284i	−0.994667	+	0.103142i	$0.967110\pi$
$48$		0			0
$49$	9.00000			1.28571
$50$		0			0
$51$	−4.00000	+	2.82843i	−0.560112	+	0.396059i
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	8.00000			1.07872
$56$		0			0
$57$	5.00000	−	5.65685i	0.662266	−	0.749269i
$58$		0			0
$59$	12.0000			1.56227			0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000			1.56227			0.781133	+	0.624364i	$0.214642\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$		0			0
$63$	−4.00000	+	11.3137i	−0.503953	+	1.42539i
$64$		0			0
$65$	−6.00000			−0.744208
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	−2.00000	+	1.41421i	−0.240772	+	0.170251i
$70$		0			0
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		0			0
$73$	−4.00000			−0.468165			−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000			−0.468165			−0.234082	+	0.972217i	$0.575209\pi$
$74$		0			0
$75$	3.00000	+	4.24264i	0.346410	+	0.489898i
$76$		0			0
$77$			22.6274i			2.57863i
$78$		0			0
$79$			4.24264i			0.477334i	0.971101	+	0.238667i	$0.0767105\pi$
$79$			4.24264i			0.477334i	−0.971101	+	0.238667i	$0.923290\pi$
$80$		0			0
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$		0			0
$83$		−	2.82843i		−	0.310460i	−0.987878	−	0.155230i	$-0.950388\pi$
$83$		−	2.82843i		−	0.310460i	0.987878	−	0.155230i	$-0.0496119\pi$
$84$		0			0
$85$	4.00000			0.433861
$86$		0			0
$87$	6.00000	+	8.48528i	0.643268	+	0.909718i
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$		−	16.9706i		−	1.77900i
$92$		0			0
$93$	−6.00000	+	4.24264i	−0.622171	+	0.439941i
$94$		0			0
$95$	−6.00000	+	1.41421i	−0.615587	+	0.145095i
$96$		0			0
$97$			8.48528i			0.861550i	0.902459	+	0.430775i	$0.141760\pi$
$97$			8.48528i			0.861550i	−0.902459	+	0.430775i	$0.858240\pi$
$98$		0			0
$99$	−16.0000	−	5.65685i	−1.60806	−	0.568535i
$100$		0			0
$101$		−	9.89949i		−	0.985037i	−0.870302	−	0.492518i	$-0.836076\pi$
$101$		−	9.89949i		−	0.985037i	0.870302	−	0.492518i	$-0.163924\pi$
$102$		0			0
$103$			12.7279i			1.25412i	0.778971	+	0.627060i	$0.215742\pi$
$103$			12.7279i			1.25412i	−0.778971	+	0.627060i	$0.784258\pi$
$104$		0			0
$105$	8.00000	−	5.65685i	0.780720	−	0.552052i
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$			4.24264i			0.406371i	0.979140	+	0.203186i	$0.0651295\pi$
$109$			4.24264i			0.406371i	−0.979140	+	0.203186i	$0.934871\pi$
$110$		0			0
$111$	6.00000	−	4.24264i	0.569495	−	0.402694i
$112$		0			0
$113$	12.0000			1.12887			0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000			1.12887			0.564433	+	0.825479i	$0.309095\pi$
$114$		0			0
$115$	2.00000			0.186501
$116$		0			0
$117$	12.0000	+	4.24264i	1.10940	+	0.392232i
$118$		0			0
$119$			11.3137i			1.03713i
$120$		0			0
$121$	−21.0000			−1.90909
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.3137i		−	1.01193i
$126$		0			0
$127$		−	21.2132i		−	1.88237i	−0.337895	−	0.941184i	$-0.609715\pi$
$127$		−	21.2132i		−	1.88237i	0.337895	−	0.941184i	$-0.390285\pi$
$128$		0			0
$129$	−2.00000	−	2.82843i	−0.176090	−	0.249029i
$130$		0			0
$131$		−	19.7990i		−	1.72985i	−0.501905	−	0.864923i	$-0.667367\pi$
$131$		−	19.7990i		−	1.72985i	0.501905	−	0.864923i	$-0.332633\pi$
$132$		0			0
$133$	−4.00000	−	16.9706i	−0.346844	−	1.47153i
$134$		0			0
$135$	2.00000	+	7.07107i	0.172133	+	0.608581i
$136$		0			0
$137$		−	14.1421i		−	1.20824i	−0.796892	−	0.604122i	$-0.793524\pi$
$137$		−	14.1421i		−	1.20824i	0.796892	−	0.604122i	$-0.206476\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$	−2.00000	+	1.41421i	−0.168430	+	0.119098i
$142$		0			0
$143$	24.0000			2.00698
$144$		0			0
$145$		−	8.48528i		−	0.704664i
$146$		0			0
$147$	9.00000	+	12.7279i	0.742307	+	1.04978i
$148$		0			0
$149$		−	9.89949i		−	0.810998i	−0.914095	−	0.405499i	$-0.867098\pi$
$149$		−	9.89949i		−	0.810998i	0.914095	−	0.405499i	$-0.132902\pi$
$150$		0			0
$151$		−	12.7279i		−	1.03578i	−0.855446	−	0.517892i	$-0.826717\pi$
$151$		−	12.7279i		−	1.03578i	0.855446	−	0.517892i	$-0.173283\pi$
$152$		0			0
$153$	−8.00000	−	2.82843i	−0.646762	−	0.228665i
$154$		0			0
$155$	6.00000			0.481932
$156$		0			0
$157$	14.0000			1.11732			0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000			1.11732			0.558661	+	0.829396i	$0.311315\pi$
$158$		0			0
$159$	−6.00000	−	8.48528i	−0.475831	−	0.672927i
$160$		0			0
$161$			5.65685i			0.445823i
$162$		0			0
$163$	−20.0000			−1.56652			−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000			−1.56652			−0.783260	+	0.621694i	$0.786445\pi$
$164$		0			0
$165$	8.00000	+	11.3137i	0.622799	+	0.880771i
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$		0			0
$171$	13.0000	+	1.41421i	0.994135	+	0.108148i
$172$		0			0
$173$	−6.00000			−0.456172			−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000			−0.456172			−0.228086	+	0.973641i	$0.573247\pi$
$174$		0			0
$175$	12.0000			0.907115
$176$		0			0
$177$	12.0000	+	16.9706i	0.901975	+	1.27559i
$178$		0			0
$179$	−18.0000			−1.34538			−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000			−1.34538			−0.672692	+	0.739923i	$0.734862\pi$
$180$		0			0
$181$		−	12.7279i		−	0.946059i	−0.881047	−	0.473029i	$-0.843160\pi$
$181$		−	12.7279i		−	0.946059i	0.881047	−	0.473029i	$-0.156840\pi$
$182$		0			0
$183$	2.00000	+	2.82843i	0.147844	+	0.209083i
$184$		0			0
$185$	−6.00000			−0.441129
$186$		0			0
$187$	−16.0000			−1.17004
$188$		0			0
$189$	−20.0000	+	5.65685i	−1.45479	+	0.411476i
$190$		0			0
$191$			18.3848i			1.33028i	0.746721	+	0.665138i	$0.231627\pi$
$191$			18.3848i			1.33028i	−0.746721	+	0.665138i	$0.768373\pi$
$192$		0			0
$193$		−	8.48528i		−	0.610784i	−0.952227	−	0.305392i	$-0.901213\pi$
$193$		−	8.48528i		−	0.610784i	0.952227	−	0.305392i	$-0.0987875\pi$
$194$		0			0
$195$	−6.00000	−	8.48528i	−0.429669	−	0.607644i
$196$		0			0
$197$			7.07107i			0.503793i	0.967754	+	0.251896i	$0.0810542\pi$
$197$			7.07107i			0.503793i	−0.967754	+	0.251896i	$0.918946\pi$
$198$		0			0
$199$	4.00000			0.283552			0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000			0.283552			0.141776	+	0.989899i	$0.454719\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	24.0000			1.68447
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−4.00000	−	1.41421i	−0.278019	−	0.0982946i
$208$		0			0
$209$	24.0000	−	5.65685i	1.66011	−	0.391293i
$210$		0			0
$211$		−	8.48528i		−	0.584151i	−0.956395	−	0.292075i	$-0.905654\pi$
$211$		−	8.48528i		−	0.584151i	0.956395	−	0.292075i	$-0.0943458\pi$
$212$		0			0
$213$	−12.0000	−	16.9706i	−0.822226	−	1.16280i
$214$		0			0
$215$			2.82843i			0.192897i
$216$		0			0
$217$			16.9706i			1.15204i
$218$		0			0
$219$	−4.00000	−	5.65685i	−0.270295	−	0.382255i
$220$		0			0
$221$	12.0000			0.807207
$222$		0			0
$223$		−	4.24264i		−	0.284108i	−0.989859	−	0.142054i	$-0.954629\pi$
$223$		−	4.24264i		−	0.284108i	0.989859	−	0.142054i	$-0.0453707\pi$
$224$		0			0
$225$	−3.00000	+	8.48528i	−0.200000	+	0.565685i
$226$		0			0
$227$	−18.0000			−1.19470			−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000			−1.19470			−0.597351	+	0.801980i	$0.703780\pi$
$228$		0			0
$229$	2.00000			0.132164			0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000			0.132164			0.0660819	+	0.997814i	$0.478950\pi$
$230$		0			0
$231$	−32.0000	+	22.6274i	−2.10545	+	1.48877i
$232$		0			0
$233$			11.3137i			0.741186i	0.928795	+	0.370593i	$0.120845\pi$
$233$			11.3137i			0.741186i	−0.928795	+	0.370593i	$0.879155\pi$
$234$		0			0
$235$	2.00000			0.130466
$236$		0			0
$237$	−6.00000	+	4.24264i	−0.389742	+	0.275589i
$238$		0			0
$239$		−	15.5563i		−	1.00626i	−0.864212	−	0.503128i	$-0.832182\pi$
$239$		−	15.5563i		−	1.00626i	0.864212	−	0.503128i	$-0.167818\pi$
$240$		0			0
$241$		−	8.48528i		−	0.546585i	−0.961931	−	0.273293i	$-0.911887\pi$
$241$		−	8.48528i		−	0.546585i	0.961931	−	0.273293i	$-0.0881127\pi$
$242$		0			0
$243$	1.00000	−	15.5563i	0.0641500	−	0.997940i
$244$		0			0
$245$		−	12.7279i		−	0.813157i
$246$		0			0
$247$	−18.0000	+	4.24264i	−1.14531	+	0.269953i
$248$		0			0
$249$	4.00000	−	2.82843i	0.253490	−	0.179244i
$250$		0			0
$251$		−	19.7990i		−	1.24970i	−0.780744	−	0.624851i	$-0.785160\pi$
$251$		−	19.7990i		−	1.24970i	0.780744	−	0.624851i	$-0.214840\pi$
$252$		0			0
$253$	−8.00000			−0.502956
$254$		0			0
$255$	4.00000	+	5.65685i	0.250490	+	0.354246i
$256$		0			0
$257$	−6.00000			−0.374270			−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000			−0.374270			−0.187135	+	0.982334i	$0.559920\pi$
$258$		0			0
$259$		−	16.9706i		−	1.05450i
$260$		0			0
$261$	−6.00000	+	16.9706i	−0.371391	+	1.05045i
$262$		0			0
$263$		−	15.5563i		−	0.959246i	−0.877475	−	0.479623i	$-0.840774\pi$
$263$		−	15.5563i		−	0.959246i	0.877475	−	0.479623i	$-0.159226\pi$
$264$		0			0
$265$			8.48528i			0.521247i
$266$		0			0
$267$	−6.00000	−	8.48528i	−0.367194	−	0.519291i
$268$		0			0
$269$	−18.0000			−1.09748			−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000			−1.09748			−0.548740	+	0.835993i	$0.684892\pi$
$270$		0			0
$271$	16.0000			0.971931			0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000			0.971931			0.485965	+	0.873978i	$0.338468\pi$
$272$		0			0
$273$	24.0000	−	16.9706i	1.45255	−	1.02711i
$274$		0			0
$275$			16.9706i			1.02336i
$276$		0			0
$277$	−22.0000			−1.32185			−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000			−1.32185			−0.660926	+	0.750451i	$0.729836\pi$
$278$		0			0
$279$	−12.0000	−	4.24264i	−0.718421	−	0.254000i
$280$		0			0
$281$	12.0000			0.715860			0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000			0.715860			0.357930	+	0.933748i	$0.383483\pi$
$282$		0			0
$283$	−20.0000			−1.18888			−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000			−1.18888			−0.594438	+	0.804141i	$0.702626\pi$
$284$		0			0
$285$	−8.00000	−	7.07107i	−0.473879	−	0.418854i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	9.00000			0.529412
$290$		0			0
$291$	−12.0000	+	8.48528i	−0.703452	+	0.497416i
$292$		0			0
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$		0			0
$295$		−	16.9706i		−	0.988064i
$296$		0			0
$297$	−8.00000	−	28.2843i	−0.464207	−	1.64122i
$298$		0			0
$299$	6.00000			0.346989
$300$		0			0
$301$	−8.00000			−0.461112
$302$		0			0
$303$	14.0000	−	9.89949i	0.804279	−	0.568711i
$304$		0			0
$305$		−	2.82843i		−	0.161955i
$306$		0			0
$307$			16.9706i			0.968561i	0.874913	+	0.484281i	$0.160919\pi$
$307$			16.9706i			0.968561i	−0.874913	+	0.484281i	$0.839081\pi$
$308$		0			0
$309$	−18.0000	+	12.7279i	−1.02398	+	0.724066i
$310$		0			0
$311$			1.41421i			0.0801927i	0.999196	+	0.0400963i	$0.0127665\pi$
$311$			1.41421i			0.0801927i	−0.999196	+	0.0400963i	$0.987234\pi$
$312$		0			0
$313$	8.00000			0.452187			0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000			0.452187			0.226093	+	0.974106i	$0.427405\pi$
$314$		0			0
$315$	16.0000	+	5.65685i	0.901498	+	0.318728i
$316$		0			0
$317$	−18.0000			−1.01098			−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000			−1.01098			−0.505490	+	0.862832i	$0.668688\pi$
$318$		0			0
$319$			33.9411i			1.90034i
$320$		0			0
$321$	−12.0000	−	16.9706i	−0.669775	−	0.947204i
$322$		0			0
$323$	12.0000	−	2.82843i	0.667698	−	0.157378i
$324$		0			0
$325$		−	12.7279i		−	0.706018i
$326$		0			0
$327$	−6.00000	+	4.24264i	−0.331801	+	0.234619i
$328$		0			0
$329$			5.65685i			0.311872i
$330$		0			0
$331$			16.9706i			0.932786i	0.884577	+	0.466393i	$0.154447\pi$
$331$			16.9706i			0.932786i	−0.884577	+	0.466393i	$0.845553\pi$
$332$		0			0
$333$	12.0000	+	4.24264i	0.657596	+	0.232495i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			16.9706i			0.924445i	0.886764	+	0.462223i	$0.152948\pi$
$337$			16.9706i			0.924445i	−0.886764	+	0.462223i	$0.847052\pi$
$338$		0			0
$339$	12.0000	+	16.9706i	0.651751	+	0.921714i
$340$		0			0
$341$	−24.0000			−1.29967
$342$		0			0
$343$	8.00000			0.431959
$344$		0			0
$345$	2.00000	+	2.82843i	0.107676	+	0.152277i
$346$		0			0
$347$		−	11.3137i		−	0.607352i	−0.952775	−	0.303676i	$-0.901786\pi$
$347$		−	11.3137i		−	0.607352i	0.952775	−	0.303676i	$-0.0982140\pi$
$348$		0			0
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$		0			0
$351$	6.00000	+	21.2132i	0.320256	+	1.13228i
$352$		0			0
$353$			28.2843i			1.50542i	0.658352	+	0.752710i	$0.271254\pi$
$353$			28.2843i			1.50542i	−0.658352	+	0.752710i	$0.728746\pi$
$354$		0			0
$355$			16.9706i			0.900704i
$356$		0			0
$357$	−16.0000	+	11.3137i	−0.846810	+	0.598785i
$358$		0			0
$359$			18.3848i			0.970311i	0.874428	+	0.485156i	$0.161237\pi$
$359$			18.3848i			0.970311i	−0.874428	+	0.485156i	$0.838763\pi$
$360$		0			0
$361$	−17.0000	+	8.48528i	−0.894737	+	0.446594i
$362$		0			0
$363$	−21.0000	−	29.6985i	−1.10221	−	1.55877i
$364$		0			0
$365$			5.65685i			0.296093i
$366$		0			0
$367$	28.0000			1.46159			0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000			1.46159			0.730794	+	0.682598i	$0.239150\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−24.0000			−1.24602
$372$		0			0
$373$		−	29.6985i		−	1.53773i	−0.639412	−	0.768865i	$-0.720822\pi$
$373$		−	29.6985i		−	1.53773i	0.639412	−	0.768865i	$-0.279178\pi$
$374$		0			0
$375$	16.0000	−	11.3137i	0.826236	−	0.584237i
$376$		0			0
$377$		−	25.4558i		−	1.31104i
$378$		0			0
$379$			33.9411i			1.74344i	0.490006	+	0.871719i	$0.336995\pi$
$379$			33.9411i			1.74344i	−0.490006	+	0.871719i	$0.663005\pi$
$380$		0			0
$381$	30.0000	−	21.2132i	1.53695	−	1.08679i
$382$		0			0
$383$	−24.0000			−1.22634			−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000			−1.22634			−0.613171	+	0.789950i	$0.710106\pi$
$384$		0			0
$385$	32.0000			1.63087
$386$		0			0
$387$	2.00000	−	5.65685i	0.101666	−	0.287554i
$388$		0			0
$389$			15.5563i			0.788738i	0.918952	+	0.394369i	$0.129037\pi$
$389$			15.5563i			0.788738i	−0.918952	+	0.394369i	$0.870963\pi$
$390$		0			0
$391$	−4.00000			−0.202289
$392$		0			0
$393$	28.0000	−	19.7990i	1.41241	−	0.998727i
$394$		0			0
$395$	6.00000			0.301893
$396$		0			0
$397$	38.0000			1.90717			0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.0000			1.90717			0.953583	+	0.301131i	$0.0973643\pi$
$398$		0			0
$399$	20.0000	−	22.6274i	1.00125	−	1.13279i
$400$		0			0
$401$	−24.0000			−1.19850			−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000			−1.19850			−0.599251	+	0.800561i	$0.704535\pi$
$402$		0			0
$403$	18.0000			0.896644
$404$		0			0
$405$	−8.00000	+	9.89949i	−0.397523	+	0.491910i
$406$		0			0
$407$	24.0000			1.18964
$408$		0			0
$409$			8.48528i			0.419570i	0.977748	+	0.209785i	$0.0672764\pi$
$409$			8.48528i			0.419570i	−0.977748	+	0.209785i	$0.932724\pi$
$410$		0			0
$411$	20.0000	−	14.1421i	0.986527	−	0.697580i
$412$		0			0
$413$	48.0000			2.36193
$414$		0			0
$415$	−4.00000			−0.196352
$416$		0			0
$417$	4.00000	+	5.65685i	0.195881	+	0.277017i
$418$		0			0
$419$			39.5980i			1.93449i	0.253849	+	0.967244i	$0.418303\pi$
$419$			39.5980i			1.93449i	−0.253849	+	0.967244i	$0.581697\pi$
$420$		0			0
$421$			12.7279i			0.620321i	0.950684	+	0.310160i	$0.100383\pi$
$421$			12.7279i			0.620321i	−0.950684	+	0.310160i	$0.899617\pi$
$422$		0			0
$423$	−4.00000	−	1.41421i	−0.194487	−	0.0687614i
$424$		0			0
$425$			8.48528i			0.411597i
$426$		0			0
$427$	8.00000			0.387147
$428$		0			0
$429$	24.0000	+	33.9411i	1.15873	+	1.63869i
$430$		0			0
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$		0			0
$433$			25.4558i			1.22333i	0.791117	+	0.611665i	$0.209500\pi$
$433$			25.4558i			1.22333i	−0.791117	+	0.611665i	$0.790500\pi$
$434$		0			0
$435$	12.0000	−	8.48528i	0.575356	−	0.406838i
$436$		0			0
$437$	6.00000	−	1.41421i	0.287019	−	0.0676510i
$438$		0			0
$439$		−	12.7279i		−	0.607471i	−0.952756	−	0.303735i	$-0.901766\pi$
$439$		−	12.7279i		−	0.607471i	0.952756	−	0.303735i	$-0.0982338\pi$
$440$		0			0
$441$	−9.00000	+	25.4558i	−0.428571	+	1.21218i
$442$		0			0
$443$		−	28.2843i		−	1.34383i	−0.740630	−	0.671913i	$-0.765473\pi$
$443$		−	28.2843i		−	1.34383i	0.740630	−	0.671913i	$-0.234527\pi$
$444$		0			0
$445$			8.48528i			0.402241i
$446$		0			0
$447$	14.0000	−	9.89949i	0.662177	−	0.468230i
$448$		0			0
$449$	30.0000			1.41579			0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000			1.41579			0.707894	+	0.706319i	$0.249646\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	18.0000	−	12.7279i	0.845714	−	0.598010i
$454$		0			0
$455$	−24.0000			−1.12514
$456$		0			0
$457$	−10.0000			−0.467780			−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000			−0.467780			−0.233890	+	0.972263i	$0.575146\pi$
$458$		0			0
$459$	−4.00000	−	14.1421i	−0.186704	−	0.660098i
$460$		0			0
$461$		−	1.41421i		−	0.0658665i	−0.999458	−	0.0329332i	$-0.989515\pi$
$461$		−	1.41421i		−	0.0658665i	0.999458	−	0.0329332i	$-0.0104849\pi$
$462$		0			0
$463$	−20.0000			−0.929479			−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000			−0.929479			−0.464739	+	0.885448i	$0.653852\pi$
$464$		0			0
$465$	6.00000	+	8.48528i	0.278243	+	0.393496i
$466$		0			0
$467$		−	2.82843i		−	0.130884i	−0.997856	−	0.0654420i	$-0.979154\pi$
$467$		−	2.82843i		−	0.130884i	0.997856	−	0.0654420i	$-0.0208457\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	14.0000	+	19.7990i	0.645086	+	0.912289i
$472$		0			0
$473$		−	11.3137i		−	0.520205i
$474$		0			0
$475$	−3.00000	−	12.7279i	−0.137649	−	0.583997i
$476$		0			0
$477$	6.00000	−	16.9706i	0.274721	−	0.777029i
$478$		0			0
$479$			9.89949i			0.452319i	0.974090	+	0.226160i	$0.0726171\pi$
$479$			9.89949i			0.452319i	−0.974090	+	0.226160i	$0.927383\pi$
$480$		0			0
$481$	−18.0000			−0.820729
$482$		0			0
$483$	−8.00000	+	5.65685i	−0.364013	+	0.257396i
$484$		0			0
$485$	12.0000			0.544892
$486$		0			0
$487$		−	12.7279i		−	0.576757i	−0.957516	−	0.288379i	$-0.906884\pi$
$487$		−	12.7279i		−	0.576757i	0.957516	−	0.288379i	$-0.0931162\pi$
$488$		0			0
$489$	−20.0000	−	28.2843i	−0.904431	−	1.27906i
$490$		0			0
$491$			14.1421i			0.638226i	0.947717	+	0.319113i	$0.103385\pi$
$491$			14.1421i			0.638226i	−0.947717	+	0.319113i	$0.896615\pi$
$492$		0			0
$493$			16.9706i			0.764316i
$494$		0			0
$495$	−8.00000	+	22.6274i	−0.359573	+	1.01703i
$496$		0			0
$497$	−48.0000			−2.15309
$498$		0			0
$499$	10.0000			0.447661			0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000			0.447661			0.223831	+	0.974628i	$0.428144\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	7.07107i		−	0.315283i	−0.987496	−	0.157642i	$-0.949611\pi$
$503$		−	7.07107i		−	0.315283i	0.987496	−	0.157642i	$-0.0503891\pi$
$504$		0			0
$505$	−14.0000			−0.622992
$506$		0			0
$507$	−5.00000	−	7.07107i	−0.222058	−	0.314037i
$508$		0			0
$509$	−6.00000			−0.265945			−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000			−0.265945			−0.132973	+	0.991120i	$0.542452\pi$
$510$		0			0
$511$	−16.0000			−0.707798
$512$		0			0
$513$	11.0000	+	19.7990i	0.485662	+	0.874147i
$514$		0			0
$515$	18.0000			0.793175
$516$		0			0
$517$	−8.00000			−0.351840
$518$		0			0
$519$	−6.00000	−	8.48528i	−0.263371	−	0.372463i
$520$		0			0
$521$	−6.00000			−0.262865			−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000			−0.262865			−0.131432	+	0.991325i	$0.541958\pi$
$522$		0			0
$523$			16.9706i			0.742071i	0.928619	+	0.371035i	$0.120997\pi$
$523$			16.9706i			0.742071i	−0.928619	+	0.371035i	$0.879003\pi$
$524$		0			0
$525$	12.0000	+	16.9706i	0.523723	+	0.740656i
$526$		0			0
$527$	−12.0000			−0.522728
$528$		0			0
$529$	21.0000			0.913043
$530$		0			0
$531$	−12.0000	+	33.9411i	−0.520756	+	1.47292i
$532$		0			0
$533$		0			0
$534$		0			0
$535$			16.9706i			0.733701i
$536$		0			0
$537$	−18.0000	−	25.4558i	−0.776757	−	1.09850i
$538$		0			0
$539$			50.9117i			2.19292i
$540$		0			0
$541$	38.0000			1.63375			0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000			1.63375			0.816874	+	0.576816i	$0.195705\pi$
$542$		0			0
$543$	18.0000	−	12.7279i	0.772454	−	0.546207i
$544$		0			0
$545$	6.00000			0.257012
$546$		0			0
$547$		−	16.9706i		−	0.725609i	−0.931865	−	0.362804i	$-0.881819\pi$
$547$		−	16.9706i		−	0.725609i	0.931865	−	0.362804i	$-0.118181\pi$
$548$		0			0
$549$	−2.00000	+	5.65685i	−0.0853579	+	0.241429i
$550$		0			0
$551$	−6.00000	−	25.4558i	−0.255609	−	1.08446i
$552$		0			0
$553$			16.9706i			0.721662i
$554$		0			0
$555$	−6.00000	−	8.48528i	−0.254686	−	0.360180i
$556$		0			0
$557$			24.0416i			1.01868i	0.860566	+	0.509338i	$0.170110\pi$
$557$			24.0416i			1.01868i	−0.860566	+	0.509338i	$0.829890\pi$
$558$		0			0
$559$			8.48528i			0.358889i
$560$		0			0
$561$	−16.0000	−	22.6274i	−0.675521	−	0.955330i
$562$		0			0
$563$	36.0000			1.51722			0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000			1.51722			0.758610	+	0.651546i	$0.225879\pi$
$564$		0			0
$565$		−	16.9706i		−	0.713957i
$566$		0			0
$567$	−28.0000	−	22.6274i	−1.17589	−	0.950262i
$568$		0			0
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$	−14.0000			−0.585882			−0.292941	−	0.956131i	$-0.594634\pi$
$571$	−14.0000			−0.585882			−0.292941	+	0.956131i	$0.594634\pi$
$572$		0			0
$573$	−26.0000	+	18.3848i	−1.08617	+	0.768035i
$574$		0			0
$575$			4.24264i			0.176930i
$576$		0			0
$577$	−10.0000			−0.416305			−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000			−0.416305			−0.208153	+	0.978096i	$0.566745\pi$
$578$		0			0
$579$	12.0000	−	8.48528i	0.498703	−	0.352636i
$580$		0			0
$581$		−	11.3137i		−	0.469372i
$582$		0			0
$583$		−	33.9411i		−	1.40570i
$584$		0			0
$585$	6.00000	−	16.9706i	0.248069	−	0.701646i
$586$		0			0
$587$			22.6274i			0.933933i	0.884275	+	0.466967i	$0.154653\pi$
$587$			22.6274i			0.933933i	−0.884275	+	0.466967i	$0.845347\pi$
$588$		0			0
$589$	18.0000	−	4.24264i	0.741677	−	0.174815i
$590$		0			0
$591$	−10.0000	+	7.07107i	−0.411345	+	0.290865i
$592$		0			0
$593$		−	5.65685i		−	0.232299i	−0.993232	−	0.116150i	$-0.962945\pi$
$593$		−	5.65685i		−	0.232299i	0.993232	−	0.116150i	$-0.0370552\pi$
$594$		0			0
$595$	16.0000			0.655936
$596$		0			0
$597$	4.00000	+	5.65685i	0.163709	+	0.231520i
$598$		0			0
$599$	12.0000			0.490307			0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000			0.490307			0.245153	+	0.969484i	$0.421162\pi$
$600$		0			0
$601$		−	16.9706i		−	0.692244i	−0.938190	−	0.346122i	$-0.887498\pi$
$601$		−	16.9706i		−	0.692244i	0.938190	−	0.346122i	$-0.112502\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			29.6985i			1.20742i
$606$		0			0
$607$			38.1838i			1.54983i	0.632065	+	0.774916i	$0.282208\pi$
$607$			38.1838i			1.54983i	−0.632065	+	0.774916i	$0.717792\pi$
$608$		0			0
$609$	24.0000	+	33.9411i	0.972529	+	1.37536i
$610$		0			0
$611$	6.00000			0.242734
$612$		0			0
$613$	−22.0000			−0.888572			−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000			−0.888572			−0.444286	+	0.895885i	$0.646543\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	22.6274i		−	0.910946i	−0.890250	−	0.455473i	$-0.849470\pi$
$617$		−	22.6274i		−	0.910946i	0.890250	−	0.455473i	$-0.150530\pi$
$618$		0			0
$619$	46.0000			1.84890			0.924448	−	0.381308i	$-0.124526\pi$
$619$	46.0000			1.84890			0.924448	+	0.381308i	$0.124526\pi$
$620$		0			0
$621$	−2.00000	−	7.07107i	−0.0802572	−	0.283752i
$622$		0			0
$623$	−24.0000			−0.961540
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$	32.0000	+	28.2843i	1.27796	+	1.12956i
$628$		0			0
$629$	12.0000			0.478471
$630$		0			0
$631$	4.00000			0.159237			0.0796187	−	0.996825i	$-0.474630\pi$
$631$	4.00000			0.159237			0.0796187	+	0.996825i	$0.474630\pi$
$632$		0			0
$633$	12.0000	−	8.48528i	0.476957	−	0.337260i
$634$		0			0
$635$	−30.0000			−1.19051
$636$		0			0
$637$		−	38.1838i		−	1.51290i
$638$		0			0
$639$	12.0000	−	33.9411i	0.474713	−	1.34269i
$640$		0			0
$641$	−30.0000			−1.18493			−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000			−1.18493			−0.592464	+	0.805597i	$0.701845\pi$
$642$		0			0
$643$	28.0000			1.10421			0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000			1.10421			0.552106	+	0.833774i	$0.313824\pi$
$644$		0			0
$645$	−4.00000	+	2.82843i	−0.157500	+	0.111369i
$646$		0			0
$647$		−	24.0416i		−	0.945174i	−0.881284	−	0.472587i	$-0.843320\pi$
$647$		−	24.0416i		−	0.945174i	0.881284	−	0.472587i	$-0.156680\pi$
$648$		0			0
$649$			67.8823i			2.66461i
$650$		0			0
$651$	−24.0000	+	16.9706i	−0.940634	+	0.665129i
$652$		0			0
$653$		−	35.3553i		−	1.38356i	−0.722108	−	0.691781i	$-0.756827\pi$
$653$		−	35.3553i		−	1.38356i	0.722108	−	0.691781i	$-0.243173\pi$
$654$		0			0
$655$	−28.0000			−1.09405
$656$		0			0
$657$	4.00000	−	11.3137i	0.156055	−	0.441390i
$658$		0			0
$659$	−18.0000			−0.701180			−0.350590	−	0.936529i	$-0.614019\pi$
$659$	−18.0000			−0.701180			−0.350590	+	0.936529i	$0.614019\pi$
$660$		0			0
$661$			29.6985i			1.15514i	0.816342	+	0.577569i	$0.195998\pi$
$661$			29.6985i			1.15514i	−0.816342	+	0.577569i	$0.804002\pi$
$662$		0			0
$663$	12.0000	+	16.9706i	0.466041	+	0.659082i
$664$		0			0
$665$	−24.0000	+	5.65685i	−0.930680	+	0.219363i
$666$		0			0
$667$			8.48528i			0.328551i
$668$		0			0
$669$	6.00000	−	4.24264i	0.231973	−	0.164030i
$670$		0			0
$671$			11.3137i			0.436761i
$672$		0			0
$673$			8.48528i			0.327084i	0.986536	+	0.163542i	$0.0522919\pi$
$673$			8.48528i			0.327084i	−0.986536	+	0.163542i	$0.947708\pi$
$674$		0			0
$675$	−15.0000	+	4.24264i	−0.577350	+	0.163299i
$676$		0			0
$677$	30.0000			1.15299			0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000			1.15299			0.576497	+	0.817099i	$0.304419\pi$
$678$		0			0
$679$			33.9411i			1.30254i
$680$		0			0
$681$	−18.0000	−	25.4558i	−0.689761	−	0.975470i
$682$		0			0
$683$	−18.0000			−0.688751			−0.344375	−	0.938832i	$-0.611909\pi$
$683$	−18.0000			−0.688751			−0.344375	+	0.938832i	$0.611909\pi$
$684$		0			0
$685$	−20.0000			−0.764161
$686$		0			0
$687$	2.00000	+	2.82843i	0.0763048	+	0.107911i
$688$		0			0
$689$			25.4558i			0.969790i
$690$		0			0
$691$	10.0000			0.380418			0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000			0.380418			0.190209	+	0.981744i	$0.439083\pi$
$692$		0			0
$693$	−64.0000	−	22.6274i	−2.43116	−	0.859544i
$694$		0			0
$695$		−	5.65685i		−	0.214577i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−16.0000	+	11.3137i	−0.605176	+	0.427924i
$700$		0			0
$701$			7.07107i			0.267071i	0.991044	+	0.133535i	$0.0426329\pi$
$701$			7.07107i			0.267071i	−0.991044	+	0.133535i	$0.957367\pi$
$702$		0			0
$703$	−18.0000	+	4.24264i	−0.678883	+	0.160014i
$704$		0			0
$705$	2.00000	+	2.82843i	0.0753244	+	0.106525i
$706$		0			0
$707$		−	39.5980i		−	1.48924i
$708$		0			0
$709$	26.0000			0.976450			0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000			0.976450			0.488225	+	0.872718i	$0.337644\pi$
$710$		0			0
$711$	−12.0000	−	4.24264i	−0.450035	−	0.159111i
$712$		0			0
$713$	−6.00000			−0.224702
$714$		0			0
$715$		−	33.9411i		−	1.26933i
$716$		0			0
$717$	22.0000	−	15.5563i	0.821605	−	0.580963i
$718$		0			0
$719$		−	7.07107i		−	0.263706i	−0.991269	−	0.131853i	$-0.957907\pi$
$719$		−	7.07107i		−	0.263706i	0.991269	−	0.131853i	$-0.0420927\pi$
$720$		0			0
$721$			50.9117i			1.89605i
$722$		0			0
$723$	12.0000	−	8.48528i	0.446285	−	0.315571i
$724$		0			0
$725$	18.0000			0.668503
$726$		0			0
$727$	−8.00000			−0.296704			−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000			−0.296704			−0.148352	+	0.988935i	$0.547397\pi$
$728$		0			0
$729$	23.0000	−	14.1421i	0.851852	−	0.523783i
$730$		0			0
$731$		−	5.65685i		−	0.209226i
$732$		0			0
$733$	−22.0000			−0.812589			−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000			−0.812589			−0.406294	+	0.913742i	$0.633179\pi$
$734$		0			0
$735$	18.0000	−	12.7279i	0.663940	−	0.469476i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−20.0000			−0.735712			−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000			−0.735712			−0.367856	+	0.929883i	$0.619908\pi$
$740$		0			0
$741$	−24.0000	−	21.2132i	−0.881662	−	0.779287i
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	−14.0000			−0.512920
$746$		0			0
$747$	8.00000	+	2.82843i	0.292705	+	0.103487i
$748$		0			0
$749$	−48.0000			−1.75388
$750$		0			0
$751$		−	38.1838i		−	1.39335i	−0.717389	−	0.696673i	$-0.754663\pi$
$751$		−	38.1838i		−	1.39335i	0.717389	−	0.696673i	$-0.245337\pi$
$752$		0			0
$753$	28.0000	−	19.7990i	1.02038	−	0.721515i
$754$		0			0
$755$	−18.0000			−0.655087
$756$		0			0
$757$	−34.0000			−1.23575			−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000			−1.23575			−0.617876	+	0.786276i	$0.712006\pi$
$758$		0			0
$759$	−8.00000	−	11.3137i	−0.290382	−	0.410662i
$760$		0			0
$761$		−	14.1421i		−	0.512652i	−0.966590	−	0.256326i	$-0.917488\pi$
$761$		−	14.1421i		−	0.512652i	0.966590	−	0.256326i	$-0.0825121\pi$
$762$		0			0
$763$			16.9706i			0.614376i
$764$		0			0
$765$	−4.00000	+	11.3137i	−0.144620	+	0.409048i
$766$		0			0
$767$		−	50.9117i		−	1.83831i
$768$		0			0
$769$	−16.0000			−0.576975			−0.288487	−	0.957484i	$-0.593152\pi$
$769$	−16.0000			−0.576975			−0.288487	+	0.957484i	$0.593152\pi$
$770$		0			0
$771$	−6.00000	−	8.48528i	−0.216085	−	0.305590i
$772$		0			0
$773$	18.0000			0.647415			0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000			0.647415			0.323708	+	0.946157i	$0.395071\pi$
$774$		0			0
$775$			12.7279i			0.457200i
$776$		0			0
$777$	24.0000	−	16.9706i	0.860995	−	0.608816i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		−	67.8823i		−	2.42902i
$782$		0			0
$783$	−30.0000	+	8.48528i	−1.07211	+	0.303239i
$784$		0			0
$785$		−	19.7990i		−	0.706656i
$786$		0			0
$787$		−	8.48528i		−	0.302468i	−0.988498	−	0.151234i	$-0.951675\pi$
$787$		−	8.48528i		−	0.302468i	0.988498	−	0.151234i	$-0.0483246\pi$
$788$		0			0
$789$	22.0000	−	15.5563i	0.783221	−	0.553821i
$790$		0			0
$791$	48.0000			1.70668
$792$		0			0
$793$		−	8.48528i		−	0.301321i
$794$		0			0
$795$	−12.0000	+	8.48528i	−0.425596	+	0.300942i
$796$		0			0
$797$	6.00000			0.212531			0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000			0.212531			0.106265	+	0.994338i	$0.466111\pi$
$798$		0			0
$799$	−4.00000			−0.141510
$800$		0			0
$801$	6.00000	−	16.9706i	0.212000	−	0.599625i
$802$		0			0
$803$		−	22.6274i		−	0.798504i
$804$		0			0
$805$	8.00000			0.281963
$806$		0			0
$807$	−18.0000	−	25.4558i	−0.633630	−	0.896088i
$808$		0			0
$809$		−	31.1127i		−	1.09386i	−0.837177	−	0.546932i	$-0.815796\pi$
$809$		−	31.1127i		−	1.09386i	0.837177	−	0.546932i	$-0.184204\pi$
$810$		0			0
$811$			33.9411i			1.19183i	0.803046	+	0.595917i	$0.203211\pi$
$811$			33.9411i			1.19183i	−0.803046	+	0.595917i	$0.796789\pi$
$812$		0			0
$813$	16.0000	+	22.6274i	0.561144	+	0.793578i
$814$		0			0
$815$			28.2843i			0.990755i
$816$		0			0
$817$	2.00000	+	8.48528i	0.0699711	+	0.296862i
$818$		0			0
$819$	48.0000	+	16.9706i	1.67726	+	0.592999i
$820$		0			0
$821$			7.07107i			0.246782i	0.992358	+	0.123391i	$0.0393769\pi$
$821$			7.07107i			0.246782i	−0.992358	+	0.123391i	$0.960623\pi$
$822$		0			0
$823$	−32.0000			−1.11545			−0.557725	−	0.830026i	$-0.688326\pi$
$823$	−32.0000			−1.11545			−0.557725	+	0.830026i	$0.688326\pi$
$824$		0			0
$825$	−24.0000	+	16.9706i	−0.835573	+	0.590839i
$826$		0			0
$827$	42.0000			1.46048			0.730242	−	0.683189i	$-0.239408\pi$
$827$	42.0000			1.46048			0.730242	+	0.683189i	$0.239408\pi$
$828$		0			0
$829$			46.6690i			1.62088i	0.585820	+	0.810442i	$0.300773\pi$
$829$			46.6690i			1.62088i	−0.585820	+	0.810442i	$0.699227\pi$
$830$		0			0
$831$	−22.0000	−	31.1127i	−0.763172	−	1.07929i
$832$		0			0
$833$			25.4558i			0.881993i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−6.00000	−	21.2132i	−0.207390	−	0.733236i
$838$		0			0
$839$	12.0000			0.414286			0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000			0.414286			0.207143	+	0.978311i	$0.433583\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$	12.0000	+	16.9706i	0.413302	+	0.584497i
$844$		0			0
$845$			7.07107i			0.243252i
$846$		0			0
$847$	−84.0000			−2.88627
$848$		0			0
$849$	−20.0000	−	28.2843i	−0.686398	−	0.970714i
$850$		0			0
$851$	6.00000			0.205677
$852$		0			0
$853$	−46.0000			−1.57501			−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000			−1.57501			−0.787505	+	0.616308i	$0.788628\pi$
$854$		0			0
$855$	2.00000	−	18.3848i	0.0683986	−	0.628746i
$856$		0			0
$857$	54.0000			1.84460			0.922302	−	0.386469i	$-0.126305\pi$
$857$	54.0000			1.84460			0.922302	+	0.386469i	$0.126305\pi$
$858$		0			0
$859$	10.0000			0.341196			0.170598	−	0.985341i	$-0.445430\pi$
$859$	10.0000			0.341196			0.170598	+	0.985341i	$0.445430\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−36.0000			−1.22545			−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000			−1.22545			−0.612727	+	0.790295i	$0.709928\pi$
$864$		0			0
$865$			8.48528i			0.288508i
$866$		0			0
$867$	9.00000	+	12.7279i	0.305656	+	0.432263i
$868$		0			0
$869$	−24.0000			−0.814144
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−24.0000	−	8.48528i	−0.812277	−	0.287183i
$874$		0			0
$875$		−	45.2548i		−	1.52989i
$876$		0			0
$877$			38.1838i			1.28937i	0.764447	+	0.644687i	$0.223012\pi$
$877$			38.1838i			1.28937i	−0.764447	+	0.644687i	$0.776988\pi$
$878$		0			0
$879$	6.00000	+	8.48528i	0.202375	+	0.286201i
$880$		0			0
$881$			28.2843i			0.952921i	0.879196	+	0.476461i	$0.158081\pi$
$881$			28.2843i			0.952921i	−0.879196	+	0.476461i	$0.841919\pi$
$882$		0			0
$883$	−26.0000			−0.874970			−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000			−0.874970			−0.437485	+	0.899226i	$0.644131\pi$
$884$		0			0
$885$	24.0000	−	16.9706i	0.806751	−	0.570459i
$886$		0			0
$887$	24.0000			0.805841			0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000			0.805841			0.402921	+	0.915235i	$0.367995\pi$
$888$		0			0
$889$		−	84.8528i		−	2.84587i
$890$		0			0
$891$	32.0000	−	39.5980i	1.07204	−	1.32658i
$892$		0			0
$893$	6.00000	−	1.41421i	0.200782	−	0.0473249i
$894$		0			0
$895$			25.4558i			0.850895i
$896$		0			0
$897$	6.00000	+	8.48528i	0.200334	+	0.283315i
$898$		0			0
$899$			25.4558i			0.849000i
$900$		0			0
$901$		−	16.9706i		−	0.565371i
$902$		0			0
$903$	−8.00000	−	11.3137i	−0.266223	−	0.376497i
$904$		0			0
$905$	−18.0000			−0.598340
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$	28.0000	+	9.89949i	0.928701	+	0.328346i
$910$		0			0
$911$	60.0000			1.98789			0.993944	−	0.109885i	$-0.0350482\pi$
$911$	60.0000			1.98789			0.993944	+	0.109885i	$0.0350482\pi$
$912$		0			0
$913$	16.0000			0.529523
$914$		0			0
$915$	4.00000	−	2.82843i	0.132236	−	0.0935049i
$916$		0			0
$917$		−	79.1960i		−	2.61528i
$918$		0			0
$919$	16.0000			0.527791			0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000			0.527791			0.263896	+	0.964551i	$0.414993\pi$
$920$		0			0
$921$	−24.0000	+	16.9706i	−0.790827	+	0.559199i
$922$		0			0
$923$			50.9117i			1.67578i
$924$		0			0
$925$		−	12.7279i		−	0.418491i
$926$		0			0
$927$	−36.0000	−	12.7279i	−1.18240	−	0.418040i
$928$		0			0
$929$			53.7401i			1.76316i	0.472038	+	0.881578i	$0.343518\pi$
$929$			53.7401i			1.76316i	−0.472038	+	0.881578i	$0.656482\pi$
$930$		0			0
$931$	−9.00000	−	38.1838i	−0.294963	−	1.25142i
$932$		0			0
$933$	−2.00000	+	1.41421i	−0.0654771	+	0.0462993i
$934$		0			0
$935$			22.6274i			0.739996i
$936$		0			0
$937$	−52.0000			−1.69877			−0.849383	−	0.527777i	$-0.823026\pi$
$937$	−52.0000			−1.69877			−0.849383	+	0.527777i	$0.823026\pi$
$938$		0			0
$939$	8.00000	+	11.3137i	0.261070	+	0.369209i
$940$		0			0
$941$	6.00000			0.195594			0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000			0.195594			0.0977972	+	0.995206i	$0.468820\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	8.00000	+	28.2843i	0.260240	+	0.920087i
$946$		0			0
$947$			48.0833i			1.56250i	0.624221	+	0.781248i	$0.285417\pi$
$947$			48.0833i			1.56250i	−0.624221	+	0.781248i	$0.714583\pi$
$948$		0			0
$949$			16.9706i			0.550888i
$950$		0			0
$951$	−18.0000	−	25.4558i	−0.583690	−	0.825462i
$952$		0			0
$953$	24.0000			0.777436			0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000			0.777436			0.388718	+	0.921357i	$0.372918\pi$
$954$		0			0
$955$	26.0000			0.841340
$956$		0			0
$957$	−48.0000	+	33.9411i	−1.55162	+	1.09716i
$958$		0			0
$959$		−	56.5685i		−	1.82669i
$960$		0			0
$961$	13.0000			0.419355
$962$		0			0
$963$	12.0000	−	33.9411i	0.386695	−	1.09374i
$964$		0			0
$965$	−12.0000			−0.386294
$966$		0			0
$967$	4.00000			0.128631			0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000			0.128631			0.0643157	+	0.997930i	$0.479514\pi$
$968$		0			0
$969$	16.0000	+	14.1421i	0.513994	+	0.454311i
$970$		0			0
$971$	−42.0000			−1.34784			−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000			−1.34784			−0.673922	+	0.738802i	$0.735392\pi$
$972$		0			0
$973$	16.0000			0.512936
$974$		0			0
$975$	18.0000	−	12.7279i	0.576461	−	0.407620i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		−	33.9411i		−	1.08476i
$980$		0			0
$981$	−12.0000	−	4.24264i	−0.383131	−	0.135457i
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	10.0000			0.318626
$986$		0			0
$987$	−8.00000	+	5.65685i	−0.254643	+	0.180060i
$988$		0			0
$989$		−	2.82843i		−	0.0899388i
$990$		0			0
$991$			29.6985i			0.943403i	0.881758	+	0.471702i	$0.156360\pi$
$991$			29.6985i			0.943403i	−0.881758	+	0.471702i	$0.843640\pi$
$992$		0			0
$993$	−24.0000	+	16.9706i	−0.761617	+	0.538545i
$994$		0			0
$995$		−	5.65685i		−	0.179334i
$996$		0			0
$997$	62.0000			1.96356			0.981780	−	0.190022i	$-0.0608559\pi$
$997$	62.0000			1.96356			0.981780	+	0.190022i	$0.0608559\pi$
$998$		0			0
$999$	6.00000	+	21.2132i	0.189832	+	0.671156i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.2.f.d.113.2		2
3.2	odd	2		912.2.f.b.113.2		2
4.3	odd	2		114.2.b.a.113.1	✓	2
12.11	even	2		114.2.b.d.113.1	yes	2
19.18	odd	2		912.2.f.b.113.1		2
57.56	even	2	inner	912.2.f.d.113.1		2
76.75	even	2		114.2.b.d.113.2	yes	2
228.227	odd	2		114.2.b.a.113.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.b.a.113.1	✓	2	4.3	odd	2
114.2.b.a.113.2	yes	2	228.227	odd	2
114.2.b.d.113.1	yes	2	12.11	even	2
114.2.b.d.113.2	yes	2	76.75	even	2
912.2.f.b.113.1		2	19.18	odd	2
912.2.f.b.113.2		2	3.2	odd	2
912.2.f.d.113.1		2	57.56	even	2	inner
912.2.f.d.113.2		2	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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.41421i) q^{3} -1.41421i q^{5} +4.00000 q^{7} +(-1.00000 + 2.82843i) q^{9} +5.65685i q^{11} -4.24264i q^{13} +(2.00000 - 1.41421i) q^{15} +2.82843i q^{17} +(-1.00000 - 4.24264i) q^{19} +(4.00000 + 5.65685i) q^{21} +1.41421i q^{23} +3.00000 q^{25} +(-5.00000 + 1.41421i) q^{27} +6.00000 q^{29} +4.24264i q^{31} +(-8.00000 + 5.65685i) q^{33} -5.65685i q^{35} -4.24264i q^{37} +(6.00000 - 4.24264i) q^{39} -2.00000 q^{43} +(4.00000 + 1.41421i) q^{45} +1.41421i q^{47} +9.00000 q^{49} +(-4.00000 + 2.82843i) q^{51} -6.00000 q^{53} +8.00000 q^{55} +(5.00000 - 5.65685i) q^{57} +12.0000 q^{59} +2.00000 q^{61} +(-4.00000 + 11.3137i) q^{63} -6.00000 q^{65} +(-2.00000 + 1.41421i) q^{69} -12.0000 q^{71} -4.00000 q^{73} +(3.00000 + 4.24264i) q^{75} +22.6274i q^{77} +4.24264i q^{79} +(-7.00000 - 5.65685i) q^{81} -2.82843i q^{83} +4.00000 q^{85} +(6.00000 + 8.48528i) q^{87} -6.00000 q^{89} -16.9706i q^{91} +(-6.00000 + 4.24264i) q^{93} +(-6.00000 + 1.41421i) q^{95} +8.48528i q^{97} +(-16.0000 - 5.65685i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 8 q^{7} - 2 q^{9} + 4 q^{15} - 2 q^{19} + 8 q^{21} + 6 q^{25} - 10 q^{27} + 12 q^{29} - 16 q^{33} + 12 q^{39} - 4 q^{43} + 8 q^{45} + 18 q^{49} - 8 q^{51} - 12 q^{53} + 16 q^{55} + 10 q^{57}+ \cdots - 32 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 8 * q^7 - 2 * q^9 + 4 * q^15 - 2 * q^19 + 8 * q^21 + 6 * q^25 - 10 * q^27 + 12 * q^29 - 16 * q^33 + 12 * q^39 - 4 * q^43 + 8 * q^45 + 18 * q^49 - 8 * q^51 - 12 * q^53 + 16 * q^55 + 10 * q^57 + 24 * q^59 + 4 * q^61 - 8 * q^63 - 12 * q^65 - 4 * q^69 - 24 * q^71 - 8 * q^73 + 6 * q^75 - 14 * q^81 + 8 * q^85 + 12 * q^87 - 12 * q^89 - 12 * q^93 - 12 * q^95 - 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more