LMFDB - Embedded newform 2268.2.x.j.377.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2268,2,Mod(377,2268)] 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("2268.377"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5, 3])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.x (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$18.1100711784$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 756)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			377.4
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2268.377
Dual form			2268.2.x.j.1889.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 - 1.50000i) q^{5} +(2.62132 - 0.358719i) q^{7} +(-3.67423 + 2.12132i) q^{11} +(2.12132 + 1.22474i) q^{13} -1.73205 q^{17} +2.44949i q^{19} +(7.34847 + 4.24264i) q^{23} +(1.00000 + 1.73205i) q^{25} +(-3.67423 + 2.12132i) q^{29} +(6.36396 + 3.67423i) q^{31} +(1.73205 - 4.24264i) q^{35} +1.00000 q^{37} +(-0.866025 + 1.50000i) q^{41} +(-3.50000 - 6.06218i) q^{43} +(6.06218 + 10.5000i) q^{47} +(6.74264 - 1.88064i) q^{49} +7.34847i q^{55} +(4.33013 - 7.50000i) q^{59} +(2.12132 - 1.22474i) q^{61} +(3.67423 - 2.12132i) q^{65} +(5.00000 - 8.66025i) q^{67} -8.48528i q^{71} +9.79796i q^{73} +(-8.87039 + 6.87868i) q^{77} +(-2.50000 - 4.33013i) q^{79} +(-6.06218 - 10.5000i) q^{83} +(-1.50000 + 2.59808i) q^{85} -10.3923 q^{89} +(6.00000 + 2.44949i) q^{91} +(3.67423 + 2.12132i) q^{95} +(-2.12132 + 1.22474i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{7} + 8 q^{25} + 8 q^{37} - 28 q^{43} + 20 q^{49} + 40 q^{67} - 20 q^{79} - 12 q^{85} + 48 q^{91}+O(q^{100}) $ 8 * q + 4 * q^7 + 8 * q^25 + 8 * q^37 - 28 * q^43 + 20 * q^49 + 40 * q^67 - 20 * q^79 - 12 * q^85 + 48 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$-1$	$1$	$e\left(\frac{5}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.866025	−	1.50000i	0.387298	−	0.670820i	−0.604787	−	0.796387i	$-0.706742\pi$
$5$	0.866025	−	1.50000i	0.387298	−	0.670820i	0.992085	+	0.125567i	$0.0400750\pi$
$6$		0			0
$7$	2.62132	−	0.358719i	0.990766	−	0.135583i
$7$	2.62132	−	0.358719i	0.990766	−	0.135583i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.67423	+	2.12132i	−1.10782	+	0.639602i	−0.938265	−	0.345918i	$-0.887568\pi$
$11$	−3.67423	+	2.12132i	−1.10782	+	0.639602i	−0.169559	+	0.985520i	$0.554234\pi$
$12$		0			0
$13$	2.12132	+	1.22474i	0.588348	+	0.339683i	0.764444	−	0.644690i	$-0.223014\pi$
$13$	2.12132	+	1.22474i	0.588348	+	0.339683i	−0.176096	+	0.984373i	$0.556347\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.73205			−0.420084			−0.210042	−	0.977692i	$-0.567360\pi$
$17$	−1.73205			−0.420084			−0.210042	+	0.977692i	$0.567360\pi$
$18$		0			0
$19$			2.44949i			0.561951i	0.959715	+	0.280976i	$0.0906580\pi$
$19$			2.44949i			0.561951i	−0.959715	+	0.280976i	$0.909342\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	7.34847	+	4.24264i	1.53226	+	0.884652i	0.999257	+	0.0385394i	$0.0122705\pi$
$23$	7.34847	+	4.24264i	1.53226	+	0.884652i	0.533005	+	0.846112i	$0.321063\pi$
$24$		0			0
$25$	1.00000	+	1.73205i	0.200000	+	0.346410i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.67423	+	2.12132i	−0.682288	+	0.393919i	−0.800717	−	0.599043i	$-0.795548\pi$
$29$	−3.67423	+	2.12132i	−0.682288	+	0.393919i	0.118428	+	0.992963i	$0.462214\pi$
$30$		0			0
$31$	6.36396	+	3.67423i	1.14300	+	0.659912i	0.947172	−	0.320726i	$-0.103927\pi$
$31$	6.36396	+	3.67423i	1.14300	+	0.659912i	0.195829	+	0.980638i	$0.437260\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.73205	−	4.24264i	0.292770	−	0.717137i
$36$		0			0
$37$	1.00000			0.164399			0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000			0.164399			0.0821995	+	0.996616i	$0.473806\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.866025	+	1.50000i	−0.135250	+	0.234261i	−0.925693	−	0.378275i	$-0.876517\pi$
$41$	−0.866025	+	1.50000i	−0.135250	+	0.234261i	0.790443	+	0.612536i	$0.209851\pi$
$42$		0			0
$43$	−3.50000	−	6.06218i	−0.533745	−	0.924473i	−0.999223	−	0.0394140i	$-0.987451\pi$
$43$	−3.50000	−	6.06218i	−0.533745	−	0.924473i	0.465478	−	0.885059i	$-0.345882\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.06218	+	10.5000i	0.884260	+	1.53158i	0.846560	+	0.532293i	$0.178670\pi$
$47$	6.06218	+	10.5000i	0.884260	+	1.53158i	0.0376995	+	0.999289i	$0.487997\pi$
$48$		0			0
$49$	6.74264	−	1.88064i	0.963234	−	0.268662i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$			7.34847i			0.990867i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.33013	−	7.50000i	0.563735	−	0.976417i	−0.433432	−	0.901186i	$-0.642697\pi$
$59$	4.33013	−	7.50000i	0.563735	−	0.976417i	0.997166	−	0.0752304i	$-0.0239692\pi$
$60$		0			0
$61$	2.12132	−	1.22474i	0.271607	−	0.156813i	−0.358011	−	0.933718i	$-0.616545\pi$
$61$	2.12132	−	1.22474i	0.271607	−	0.156813i	0.629618	+	0.776905i	$0.283211\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.67423	−	2.12132i	0.455733	−	0.263117i
$66$		0			0
$67$	5.00000	−	8.66025i	0.610847	−	1.05802i	−0.380251	−	0.924883i	$-0.624162\pi$
$67$	5.00000	−	8.66025i	0.610847	−	1.05802i	0.991098	−	0.133135i	$-0.0425044\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	8.48528i		−	1.00702i	−0.863990	−	0.503509i	$-0.832042\pi$
$71$		−	8.48528i		−	1.00702i	0.863990	−	0.503509i	$-0.167958\pi$
$72$		0			0
$73$			9.79796i			1.14676i	0.819288	+	0.573382i	$0.194369\pi$
$73$			9.79796i			1.14676i	−0.819288	+	0.573382i	$0.805631\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−8.87039	+	6.87868i	−1.01087	+	0.783898i
$78$		0			0
$79$	−2.50000	−	4.33013i	−0.281272	−	0.487177i	0.690426	−	0.723403i	$-0.257423\pi$
$79$	−2.50000	−	4.33013i	−0.281272	−	0.487177i	−0.971698	+	0.236225i	$0.924090\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.06218	−	10.5000i	−0.665410	−	1.15252i	−0.979174	−	0.203024i	$-0.934923\pi$
$83$	−6.06218	−	10.5000i	−0.665410	−	1.15252i	0.313763	−	0.949501i	$-0.398410\pi$
$84$		0			0
$85$	−1.50000	+	2.59808i	−0.162698	+	0.281801i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.3923			−1.10158			−0.550791	−	0.834643i	$-0.685674\pi$
$89$	−10.3923			−1.10158			−0.550791	+	0.834643i	$0.685674\pi$
$90$		0			0
$91$	6.00000	+	2.44949i	0.628971	+	0.256776i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.67423	+	2.12132i	0.376969	+	0.217643i
$96$		0			0
$97$	−2.12132	+	1.22474i	−0.215387	+	0.124354i	−0.603813	−	0.797126i	$-0.706353\pi$
$97$	−2.12132	+	1.22474i	−0.215387	+	0.124354i	0.388425	+	0.921480i	$0.373019\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.19615	−	9.00000i	−0.517036	−	0.895533i	−0.999804	−	0.0197851i	$-0.993702\pi$
$101$	−5.19615	−	9.00000i	−0.517036	−	0.895533i	0.482768	−	0.875748i	$-0.339632\pi$
$102$		0			0
$103$	16.9706	+	9.79796i	1.67216	+	0.965422i	0.966426	+	0.256943i	$0.0827154\pi$
$103$	16.9706	+	9.79796i	1.67216	+	0.965422i	0.705733	+	0.708478i	$0.250618\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			8.48528i			0.820303i	0.912017	+	0.410152i	$0.134524\pi$
$107$			8.48528i			0.820303i	−0.912017	+	0.410152i	$0.865476\pi$
$108$		0			0
$109$	13.0000			1.24517			0.622587	−	0.782551i	$-0.286082\pi$
$109$	13.0000			1.24517			0.622587	+	0.782551i	$0.286082\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−11.0227	−	6.36396i	−1.03693	−	0.598671i	−0.117967	−	0.993018i	$-0.537638\pi$
$113$	−11.0227	−	6.36396i	−1.03693	−	0.598671i	−0.918962	+	0.394346i	$0.870971\pi$
$114$		0			0
$115$	12.7279	−	7.34847i	1.18688	−	0.685248i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.54026	+	0.621320i	−0.416205	+	0.0569563i
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.1244			1.08444
$126$		0			0
$127$	−13.0000			−1.15356			−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000			−1.15356			−0.576782	+	0.816898i	$0.695692\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.19615	−	9.00000i	0.453990	−	0.786334i	−0.544640	−	0.838670i	$-0.683334\pi$
$131$	5.19615	−	9.00000i	0.453990	−	0.786334i	0.998630	+	0.0523366i	$0.0166669\pi$
$132$		0			0
$133$	0.878680	+	6.42090i	0.0761912	+	0.556762i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	18.3712	−	10.6066i	1.56956	−	0.906183i	0.573335	−	0.819321i	$-0.305649\pi$
$137$	18.3712	−	10.6066i	1.56956	−	0.906183i	0.996220	−	0.0868620i	$-0.0276839\pi$
$138$		0			0
$139$	10.6066	+	6.12372i	0.899640	+	0.519408i	0.877083	−	0.480338i	$-0.159486\pi$
$139$	10.6066	+	6.12372i	0.899640	+	0.519408i	0.0225568	+	0.999746i	$0.492819\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−10.3923			−0.869048
$144$		0			0
$145$			7.34847i			0.610257i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−11.0227	−	6.36396i	−0.903015	−	0.521356i	−0.0248379	−	0.999691i	$-0.507907\pi$
$149$	−11.0227	−	6.36396i	−0.903015	−	0.521356i	−0.878177	+	0.478335i	$0.841240\pi$
$150$		0			0
$151$	2.50000	+	4.33013i	0.203447	+	0.352381i	0.949637	−	0.313353i	$-0.101452\pi$
$151$	2.50000	+	4.33013i	0.203447	+	0.352381i	−0.746190	+	0.665733i	$0.768119\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	11.0227	−	6.36396i	0.885365	−	0.511166i
$156$		0			0
$157$	6.36396	+	3.67423i	0.507899	+	0.293236i	0.731970	−	0.681337i	$-0.238601\pi$
$157$	6.36396	+	3.67423i	0.507899	+	0.293236i	−0.224070	+	0.974573i	$0.571935\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	20.7846	+	8.48528i	1.63806	+	0.668734i
$162$		0			0
$163$	−1.00000			−0.0783260			−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000			−0.0783260			−0.0391630	+	0.999233i	$0.512469\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−9.52628	+	16.5000i	−0.737166	+	1.27681i	0.216601	+	0.976260i	$0.430503\pi$
$167$	−9.52628	+	16.5000i	−0.737166	+	1.27681i	−0.953767	+	0.300548i	$0.902830\pi$
$168$		0			0
$169$	−3.50000	−	6.06218i	−0.269231	−	0.466321i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	10.3923	+	18.0000i	0.790112	+	1.36851i	0.925897	+	0.377776i	$0.123311\pi$
$173$	10.3923	+	18.0000i	0.790112	+	1.36851i	−0.135785	+	0.990738i	$0.543356\pi$
$174$		0			0
$175$	3.24264	+	4.18154i	0.245121	+	0.316095i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			4.24264i			0.317110i	0.987350	+	0.158555i	$0.0506835\pi$
$179$			4.24264i			0.317110i	−0.987350	+	0.158555i	$0.949317\pi$
$180$		0			0
$181$			4.89898i			0.364138i	0.983286	+	0.182069i	$0.0582795\pi$
$181$			4.89898i			0.364138i	−0.983286	+	0.182069i	$0.941721\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.866025	−	1.50000i	0.0636715	−	0.110282i
$186$		0			0
$187$	6.36396	−	3.67423i	0.465379	−	0.268687i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−7.34847	+	4.24264i	−0.531717	+	0.306987i	−0.741715	−	0.670715i	$-0.765987\pi$
$191$	−7.34847	+	4.24264i	−0.531717	+	0.306987i	0.209999	+	0.977702i	$0.432654\pi$
$192$		0			0
$193$	−3.50000	+	6.06218i	−0.251936	+	0.436365i	−0.964059	−	0.265689i	$-0.914400\pi$
$193$	−3.50000	+	6.06218i	−0.251936	+	0.436365i	0.712123	+	0.702055i	$0.247734\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			21.2132i			1.51138i	0.654931	+	0.755689i	$0.272698\pi$
$197$			21.2132i			1.51138i	−0.654931	+	0.755689i	$0.727302\pi$
$198$		0			0
$199$		−	9.79796i		−	0.694559i	−0.937762	−	0.347279i	$-0.887106\pi$
$199$		−	9.79796i		−	0.694559i	0.937762	−	0.347279i	$-0.112894\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−8.87039	+	6.87868i	−0.622579	+	0.482789i
$204$		0			0
$205$	1.50000	+	2.59808i	0.104765	+	0.181458i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.19615	−	9.00000i	−0.359425	−	0.622543i
$210$		0			0
$211$	4.00000	−	6.92820i	0.275371	−	0.476957i	−0.694857	−	0.719148i	$-0.744533\pi$
$211$	4.00000	−	6.92820i	0.275371	−	0.476957i	0.970229	+	0.242190i	$0.0778659\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−12.1244			−0.826874
$216$		0			0
$217$	18.0000	+	7.34847i	1.22192	+	0.498847i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.67423	−	2.12132i	−0.247156	−	0.142695i
$222$		0			0
$223$	19.0919	−	11.0227i	1.27849	−	0.738135i	0.301917	−	0.953334i	$-0.402373\pi$
$223$	19.0919	−	11.0227i	1.27849	−	0.738135i	0.976570	+	0.215199i	$0.0690401\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$	−12.7279	−	7.34847i	−0.841085	−	0.485601i	0.0165480	−	0.999863i	$-0.494732\pi$
$229$	−12.7279	−	7.34847i	−0.841085	−	0.485601i	−0.857633	+	0.514263i	$0.828066\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	4.24264i		−	0.277945i	−0.990296	−	0.138972i	$-0.955620\pi$
$233$		−	4.24264i		−	0.277945i	0.990296	−	0.138972i	$-0.0443799\pi$
$234$		0			0
$235$	21.0000			1.36989
$236$		0			0
$237$		0			0
$238$		0			0
$239$	18.3712	+	10.6066i	1.18833	+	0.686084i	0.957927	−	0.287011i	$-0.0926616\pi$
$239$	18.3712	+	10.6066i	1.18833	+	0.686084i	0.230405	+	0.973095i	$0.425995\pi$
$240$		0			0
$241$	−23.3345	+	13.4722i	−1.50311	+	0.867820i	−0.503115	+	0.864219i	$0.667813\pi$
$241$	−23.3345	+	13.4722i	−1.50311	+	0.867820i	−0.999994	+	0.00360098i	$0.998854\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.01834	−	11.7426i	0.192835	−	0.750210i
$246$		0			0
$247$	−3.00000	+	5.19615i	−0.190885	+	0.330623i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	19.0526			1.20259			0.601293	−	0.799028i	$-0.294652\pi$
$251$	19.0526			1.20259			0.601293	+	0.799028i	$0.294652\pi$
$252$		0			0
$253$	−36.0000			−2.26330
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.19615	−	9.00000i	0.324127	−	0.561405i	−0.657208	−	0.753709i	$-0.728263\pi$
$257$	5.19615	−	9.00000i	0.324127	−	0.561405i	0.981335	+	0.192304i	$0.0615961\pi$
$258$		0			0
$259$	2.62132	−	0.358719i	0.162881	−	0.0222897i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−18.3712	+	10.6066i	−1.13282	+	0.654031i	−0.944641	−	0.328105i	$-0.893590\pi$
$263$	−18.3712	+	10.6066i	−1.13282	+	0.654031i	−0.188174	+	0.982136i	$0.560257\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.73205			0.105605			0.0528025	−	0.998605i	$-0.483185\pi$
$269$	1.73205			0.105605			0.0528025	+	0.998605i	$0.483185\pi$
$270$		0			0
$271$			4.89898i			0.297592i	0.988868	+	0.148796i	$0.0475397\pi$
$271$			4.89898i			0.297592i	−0.988868	+	0.148796i	$0.952460\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−7.34847	−	4.24264i	−0.443129	−	0.255841i
$276$		0			0
$277$	3.50000	+	6.06218i	0.210295	+	0.364241i	0.951807	−	0.306699i	$-0.0992243\pi$
$277$	3.50000	+	6.06218i	0.210295	+	0.364241i	−0.741512	+	0.670940i	$0.765891\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−7.34847	+	4.24264i	−0.438373	+	0.253095i	−0.702907	−	0.711282i	$-0.748115\pi$
$281$	−7.34847	+	4.24264i	−0.438373	+	0.253095i	0.264534	+	0.964376i	$0.414782\pi$
$282$		0			0
$283$	−6.36396	−	3.67423i	−0.378298	−	0.218411i	0.298779	−	0.954322i	$-0.403421\pi$
$283$	−6.36396	−	3.67423i	−0.378298	−	0.218411i	−0.677078	+	0.735912i	$0.736754\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.73205	+	4.24264i	−0.102240	+	0.250435i
$288$		0			0
$289$	−14.0000			−0.823529
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4.33013	−	7.50000i	0.252969	−	0.438155i	−0.711373	−	0.702815i	$-0.751926\pi$
$293$	4.33013	−	7.50000i	0.252969	−	0.438155i	0.964342	+	0.264660i	$0.0852597\pi$
$294$		0			0
$295$	−7.50000	−	12.9904i	−0.436667	−	0.756329i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	10.3923	+	18.0000i	0.601003	+	1.04097i
$300$		0			0
$301$	−11.3492	−	14.6354i	−0.654159	−	0.843570i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	4.24264i		−	0.242933i
$306$		0			0
$307$		−	29.3939i		−	1.67760i	−0.544442	−	0.838799i	$-0.683259\pi$
$307$		−	29.3939i		−	1.67760i	0.544442	−	0.838799i	$-0.316741\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	11.2583	−	19.5000i	0.638401	−	1.10574i	−0.347382	−	0.937724i	$-0.612929\pi$
$311$	11.2583	−	19.5000i	0.638401	−	1.10574i	0.985784	−	0.168020i	$-0.0537373\pi$
$312$		0			0
$313$	−21.2132	+	12.2474i	−1.19904	+	0.692267i	−0.960341	−	0.278827i	$-0.910054\pi$
$313$	−21.2132	+	12.2474i	−1.19904	+	0.692267i	−0.238700	+	0.971093i	$0.576721\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$	9.00000	−	15.5885i	0.503903	−	0.872786i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	4.24264i		−	0.236067i
$324$		0			0
$325$			4.89898i			0.271746i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	19.6575	+	25.3492i	1.08375	+	1.39755i
$330$		0			0
$331$	−6.50000	−	11.2583i	−0.357272	−	0.618814i	0.630232	−	0.776407i	$-0.282960\pi$
$331$	−6.50000	−	11.2583i	−0.357272	−	0.618814i	−0.987504	+	0.157593i	$0.949627\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8.66025	−	15.0000i	−0.473160	−	0.819538i
$336$		0			0
$337$	−12.5000	+	21.6506i	−0.680918	+	1.17939i	0.293783	+	0.955872i	$0.405086\pi$
$337$	−12.5000	+	21.6506i	−0.680918	+	1.17939i	−0.974701	+	0.223513i	$0.928247\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−31.1769			−1.68832
$342$		0			0
$343$	17.0000	−	7.34847i	0.917914	−	0.396780i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−18.3712	−	10.6066i	−0.986216	−	0.569392i	−0.0820751	−	0.996626i	$-0.526155\pi$
$347$	−18.3712	−	10.6066i	−0.986216	−	0.569392i	−0.904141	+	0.427234i	$0.859488\pi$
$348$		0			0
$349$	−16.9706	+	9.79796i	−0.908413	+	0.524473i	−0.879920	−	0.475121i	$-0.842404\pi$
$349$	−16.9706	+	9.79796i	−0.908413	+	0.524473i	−0.0284931	+	0.999594i	$0.509071\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.866025	+	1.50000i	0.0460939	+	0.0798369i	0.888152	−	0.459550i	$-0.151989\pi$
$353$	0.866025	+	1.50000i	0.0460939	+	0.0798369i	−0.842058	+	0.539387i	$0.818656\pi$
$354$		0			0
$355$	−12.7279	−	7.34847i	−0.675528	−	0.390016i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	33.9411i		−	1.79134i	−0.444715	−	0.895672i	$-0.646695\pi$
$359$		−	33.9411i		−	1.79134i	0.444715	−	0.895672i	$-0.353305\pi$
$360$		0			0
$361$	13.0000			0.684211
$362$		0			0
$363$		0			0
$364$		0			0
$365$	14.6969	+	8.48528i	0.769273	+	0.444140i
$366$		0			0
$367$	4.24264	−	2.44949i	0.221464	−	0.127862i	−0.385164	−	0.922848i	$-0.625855\pi$
$367$	4.24264	−	2.44949i	0.221464	−	0.127862i	0.606628	+	0.794986i	$0.292522\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−5.50000	+	9.52628i	−0.284779	+	0.493252i	−0.972556	−	0.232671i	$-0.925254\pi$
$373$	−5.50000	+	9.52628i	−0.284779	+	0.493252i	0.687776	+	0.725923i	$0.258587\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−10.3923			−0.535231
$378$		0			0
$379$	−35.0000			−1.79783			−0.898915	−	0.438124i	$-0.855643\pi$
$379$	−35.0000			−1.79783			−0.898915	+	0.438124i	$0.855643\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9.52628	−	16.5000i	0.486770	−	0.843111i	−0.513114	−	0.858320i	$-0.671508\pi$
$383$	9.52628	−	16.5000i	0.486770	−	0.843111i	0.999884	+	0.0152097i	$0.00484160\pi$
$384$		0			0
$385$	2.63604	+	19.2627i	0.134345	+	0.981718i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−25.7196	+	14.8492i	−1.30404	+	0.752886i	−0.981094	−	0.193532i	$-0.938006\pi$
$389$	−25.7196	+	14.8492i	−1.30404	+	0.752886i	−0.322944	+	0.946418i	$0.604672\pi$
$390$		0			0
$391$	−12.7279	−	7.34847i	−0.643679	−	0.371628i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.66025			−0.435745
$396$		0			0
$397$		−	29.3939i		−	1.47524i	−0.675218	−	0.737618i	$-0.735950\pi$
$397$		−	29.3939i		−	1.47524i	0.675218	−	0.737618i	$-0.264050\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.34847	+	4.24264i	0.366965	+	0.211867i	0.672132	−	0.740432i	$-0.265379\pi$
$401$	7.34847	+	4.24264i	0.366965	+	0.211867i	−0.305167	+	0.952299i	$0.598712\pi$
$402$		0			0
$403$	9.00000	+	15.5885i	0.448322	+	0.776516i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.67423	+	2.12132i	−0.182125	+	0.105150i
$408$		0			0
$409$	−33.9411	−	19.5959i	−1.67828	−	0.968956i	−0.962757	−	0.270367i	$-0.912855\pi$
$409$	−33.9411	−	19.5959i	−1.67828	−	0.968956i	−0.715523	−	0.698589i	$-0.753812\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.66025	−	21.2132i	0.426143	−	1.04383i
$414$		0			0
$415$	−21.0000			−1.03085
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−14.7224	+	25.5000i	−0.719238	+	1.24576i	0.242064	+	0.970260i	$0.422176\pi$
$419$	−14.7224	+	25.5000i	−0.719238	+	1.24576i	−0.961302	+	0.275496i	$0.911158\pi$
$420$		0			0
$421$	−10.0000	−	17.3205i	−0.487370	−	0.844150i	0.512524	−	0.858673i	$-0.328710\pi$
$421$	−10.0000	−	17.3205i	−0.487370	−	0.844150i	−0.999895	+	0.0145228i	$0.995377\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.73205	−	3.00000i	−0.0840168	−	0.145521i
$426$		0			0
$427$	5.12132	−	3.97141i	0.247838	−	0.192190i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	16.9706i		−	0.817443i	−0.912659	−	0.408722i	$-0.865975\pi$
$431$		−	16.9706i		−	0.817443i	0.912659	−	0.408722i	$-0.134025\pi$
$432$		0			0
$433$			22.0454i			1.05943i	0.848174	+	0.529717i	$0.177702\pi$
$433$			22.0454i			1.05943i	−0.848174	+	0.529717i	$0.822298\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−10.3923	+	18.0000i	−0.497131	+	0.861057i
$438$		0			0
$439$	25.4558	−	14.6969i	1.21494	−	0.701447i	0.251110	−	0.967959i	$-0.419205\pi$
$439$	25.4558	−	14.6969i	1.21494	−	0.701447i	0.963832	+	0.266512i	$0.0858712\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	11.0227	−	6.36396i	0.523704	−	0.302361i	−0.214745	−	0.976670i	$-0.568892\pi$
$443$	11.0227	−	6.36396i	0.523704	−	0.302361i	0.738449	+	0.674309i	$0.235559\pi$
$444$		0			0
$445$	−9.00000	+	15.5885i	−0.426641	+	0.738964i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	21.2132i		−	1.00111i	−0.865704	−	0.500556i	$-0.833129\pi$
$449$		−	21.2132i		−	1.00111i	0.865704	−	0.500556i	$-0.166871\pi$
$450$		0			0
$451$		−	7.34847i		−	0.346026i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.87039	−	6.87868i	0.415850	−	0.322477i
$456$		0			0
$457$	4.00000	+	6.92820i	0.187112	+	0.324088i	0.944286	−	0.329125i	$-0.106754\pi$
$457$	4.00000	+	6.92820i	0.187112	+	0.324088i	−0.757174	+	0.653213i	$0.773421\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	14.7224	+	25.5000i	0.685692	+	1.18765i	0.973219	+	0.229881i	$0.0738336\pi$
$461$	14.7224	+	25.5000i	0.685692	+	1.18765i	−0.287527	+	0.957773i	$0.592833\pi$
$462$		0			0
$463$	−14.5000	+	25.1147i	−0.673872	+	1.16718i	0.302925	+	0.953014i	$0.402037\pi$
$463$	−14.5000	+	25.1147i	−0.673872	+	1.16718i	−0.976797	+	0.214166i	$0.931297\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.7846			0.961797			0.480899	−	0.876776i	$-0.340311\pi$
$467$	20.7846			0.961797			0.480899	+	0.876776i	$0.340311\pi$
$468$		0			0
$469$	10.0000	−	24.4949i	0.461757	−	1.13107i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	25.7196	+	14.8492i	1.18259	+	0.682769i
$474$		0			0
$475$	−4.24264	+	2.44949i	−0.194666	+	0.112390i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	19.9186	+	34.5000i	0.910103	+	1.57635i	0.813916	+	0.580982i	$0.197331\pi$
$479$	19.9186	+	34.5000i	0.910103	+	1.57635i	0.0961869	+	0.995363i	$0.469335\pi$
$480$		0			0
$481$	2.12132	+	1.22474i	0.0967239	+	0.0558436i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			4.24264i			0.192648i
$486$		0			0
$487$	22.0000			0.996915			0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000			0.996915			0.498458	+	0.866914i	$0.333900\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−11.0227	−	6.36396i	−0.497448	−	0.287202i	0.230211	−	0.973141i	$-0.426058\pi$
$491$	−11.0227	−	6.36396i	−0.497448	−	0.287202i	−0.727659	+	0.685939i	$0.759392\pi$
$492$		0			0
$493$	6.36396	−	3.67423i	0.286618	−	0.165479i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3.04384	−	22.2426i	−0.136535	−	0.997719i
$498$		0			0
$499$	−9.50000	+	16.4545i	−0.425278	+	0.736604i	−0.996446	−	0.0842294i	$-0.973157\pi$
$499$	−9.50000	+	16.4545i	−0.425278	+	0.736604i	0.571168	+	0.820833i	$0.306490\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.73205			0.0772283			0.0386142	−	0.999254i	$-0.487706\pi$
$503$	1.73205			0.0772283			0.0386142	+	0.999254i	$0.487706\pi$
$504$		0			0
$505$	−18.0000			−0.800989
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.52628	−	16.5000i	0.422245	−	0.731350i	−0.573914	−	0.818916i	$-0.694576\pi$
$509$	9.52628	−	16.5000i	0.422245	−	0.731350i	0.996159	+	0.0875661i	$0.0279089\pi$
$510$		0			0
$511$	3.51472	+	25.6836i	0.155482	+	1.13618i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	29.3939	−	16.9706i	1.29525	−	0.747812i
$516$		0			0
$517$	−44.5477	−	25.7196i	−1.95921	−	1.13115i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.73205			0.0758825			0.0379413	−	0.999280i	$-0.487920\pi$
$521$	1.73205			0.0758825			0.0379413	+	0.999280i	$0.487920\pi$
$522$		0			0
$523$		−	31.8434i		−	1.39241i	−0.717841	−	0.696207i	$-0.754870\pi$
$523$		−	31.8434i		−	1.39241i	0.717841	−	0.696207i	$-0.245130\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−11.0227	−	6.36396i	−0.480157	−	0.277218i
$528$		0			0
$529$	24.5000	+	42.4352i	1.06522	+	1.84501i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.67423	+	2.12132i	−0.159149	+	0.0918846i
$534$		0			0
$535$	12.7279	+	7.34847i	0.550276	+	0.317702i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−20.7846	+	21.2132i	−0.895257	+	0.913717i
$540$		0			0
$541$	17.0000			0.730887			0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000			0.730887			0.365444	+	0.930834i	$0.380917\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	11.2583	−	19.5000i	0.482254	−	0.835288i
$546$		0			0
$547$	−20.5000	−	35.5070i	−0.876517	−	1.51817i	−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−20.5000	−	35.5070i	−0.876517	−	1.51817i	−0.0213785	−	0.999771i	$-0.506805\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−5.19615	−	9.00000i	−0.221364	−	0.383413i
$552$		0			0
$553$	−8.10660	−	10.4539i	−0.344728	−	0.444543i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			8.48528i			0.359533i	0.983709	+	0.179766i	$0.0575342\pi$
$557$			8.48528i			0.359533i	−0.983709	+	0.179766i	$0.942466\pi$
$558$		0			0
$559$		−	17.1464i		−	0.725217i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10.3923	+	18.0000i	−0.437983	+	0.758610i	−0.997534	−	0.0701867i	$-0.977640\pi$
$563$	−10.3923	+	18.0000i	−0.437983	+	0.758610i	0.559550	+	0.828796i	$0.310974\pi$
$564$		0			0
$565$	−19.0919	+	11.0227i	−0.803202	+	0.463729i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−29.3939	+	16.9706i	−1.23226	+	0.711443i	−0.967500	−	0.252872i	$-0.918625\pi$
$569$	−29.3939	+	16.9706i	−1.23226	+	0.711443i	−0.264756	+	0.964315i	$0.585291\pi$
$570$		0			0
$571$	12.5000	−	21.6506i	0.523109	−	0.906051i	−0.476530	−	0.879158i	$-0.658105\pi$
$571$	12.5000	−	21.6506i	0.523109	−	0.906051i	0.999638	−	0.0268925i	$-0.00856117\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			16.9706i			0.707721i
$576$		0			0
$577$		−	19.5959i		−	0.815789i	−0.913029	−	0.407894i	$-0.866263\pi$
$577$		−	19.5959i		−	0.815789i	0.913029	−	0.407894i	$-0.133737\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−19.6575	−	25.3492i	−0.815529	−	1.05166i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−20.7846	−	36.0000i	−0.857873	−	1.48588i	−0.873954	−	0.486008i	$-0.838452\pi$
$587$	−20.7846	−	36.0000i	−0.857873	−	1.48588i	0.0160815	−	0.999871i	$-0.494881\pi$
$588$		0			0
$589$	−9.00000	+	15.5885i	−0.370839	+	0.642311i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−8.66025			−0.355634			−0.177817	−	0.984064i	$-0.556904\pi$
$593$	−8.66025			−0.355634			−0.177817	+	0.984064i	$0.556904\pi$
$594$		0			0
$595$	−3.00000	+	7.34847i	−0.122988	+	0.301258i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	3.67423	+	2.12132i	0.150125	+	0.0866748i	0.573181	−	0.819429i	$-0.305709\pi$
$599$	3.67423	+	2.12132i	0.150125	+	0.0866748i	−0.423056	+	0.906104i	$0.639043\pi$
$600$		0			0
$601$	−19.0919	+	11.0227i	−0.778774	+	0.449625i	−0.835996	−	0.548736i	$-0.815109\pi$
$601$	−19.0919	+	11.0227i	−0.778774	+	0.449625i	0.0572215	+	0.998362i	$0.481776\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6.06218	−	10.5000i	−0.246463	−	0.426886i
$606$		0			0
$607$	−23.3345	−	13.4722i	−0.947119	−	0.546819i	−0.0549343	−	0.998490i	$-0.517495\pi$
$607$	−23.3345	−	13.4722i	−0.947119	−	0.546819i	−0.892185	+	0.451671i	$0.850828\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			29.6985i			1.20147i
$612$		0			0
$613$	−4.00000			−0.161558			−0.0807792	−	0.996732i	$-0.525741\pi$
$613$	−4.00000			−0.161558			−0.0807792	+	0.996732i	$0.525741\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−22.0454	−	12.7279i	−0.887515	−	0.512407i	−0.0143859	−	0.999897i	$-0.504579\pi$
$617$	−22.0454	−	12.7279i	−0.887515	−	0.512407i	−0.873129	+	0.487490i	$0.837913\pi$
$618$		0			0
$619$	−2.12132	+	1.22474i	−0.0852631	+	0.0492267i	−0.542025	−	0.840362i	$-0.682342\pi$
$619$	−2.12132	+	1.22474i	−0.0852631	+	0.0492267i	0.456762	+	0.889589i	$0.349009\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−27.2416	+	3.72792i	−1.09141	+	0.149356i
$624$		0			0
$625$	5.50000	−	9.52628i	0.220000	−	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.73205			−0.0690614
$630$		0			0
$631$	−37.0000			−1.47295			−0.736473	−	0.676467i	$-0.763510\pi$
$631$	−37.0000			−1.47295			−0.736473	+	0.676467i	$0.763510\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−11.2583	+	19.5000i	−0.446773	+	0.773834i
$636$		0			0
$637$	16.6066	+	4.26858i	0.657978	+	0.169127i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	18.3712	−	10.6066i	0.725618	−	0.418936i	−0.0911991	−	0.995833i	$-0.529070\pi$
$641$	18.3712	−	10.6066i	0.725618	−	0.418936i	0.816817	+	0.576897i	$0.195737\pi$
$642$		0			0
$643$	12.7279	+	7.34847i	0.501940	+	0.289795i	0.729514	−	0.683965i	$-0.239746\pi$
$643$	12.7279	+	7.34847i	0.501940	+	0.289795i	−0.227574	+	0.973761i	$0.573079\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	20.7846			0.817127			0.408564	−	0.912730i	$-0.366030\pi$
$647$	20.7846			0.817127			0.408564	+	0.912730i	$0.366030\pi$
$648$		0			0
$649$			36.7423i			1.44226i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	25.7196	+	14.8492i	1.00649	+	0.581096i	0.910161	−	0.414254i	$-0.135958\pi$
$653$	25.7196	+	14.8492i	1.00649	+	0.581096i	0.0963261	+	0.995350i	$0.469291\pi$
$654$		0			0
$655$	−9.00000	−	15.5885i	−0.351659	−	0.609091i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	14.6969	−	8.48528i	0.572511	−	0.330540i	−0.185640	−	0.982618i	$-0.559436\pi$
$659$	14.6969	−	8.48528i	0.572511	−	0.330540i	0.758152	+	0.652078i	$0.226103\pi$
$660$		0			0
$661$	−10.6066	−	6.12372i	−0.412549	−	0.238185i	0.279335	−	0.960194i	$-0.409886\pi$
$661$	−10.6066	−	6.12372i	−0.412549	−	0.238185i	−0.691884	+	0.722008i	$0.743219\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	10.3923	+	4.24264i	0.402996	+	0.164523i
$666$		0			0
$667$	−36.0000			−1.39393
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−5.19615	+	9.00000i	−0.200595	+	0.347441i
$672$		0			0
$673$	−8.00000	−	13.8564i	−0.308377	−	0.534125i	0.669630	−	0.742695i	$-0.266453\pi$
$673$	−8.00000	−	13.8564i	−0.308377	−	0.534125i	−0.978008	+	0.208569i	$0.933119\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−15.5885	−	27.0000i	−0.599113	−	1.03769i	−0.992952	−	0.118515i	$-0.962187\pi$
$677$	−15.5885	−	27.0000i	−0.599113	−	1.03769i	0.393839	−	0.919179i	$-0.371147\pi$
$678$		0			0
$679$	−5.12132	+	3.97141i	−0.196538	+	0.152409i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			38.1838i			1.46106i	0.682880	+	0.730531i	$0.260727\pi$
$683$			38.1838i			1.46106i	−0.682880	+	0.730531i	$0.739273\pi$
$684$		0			0
$685$		−	36.7423i		−	1.40385i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−19.0919	+	11.0227i	−0.726289	+	0.419323i	−0.817063	−	0.576548i	$-0.804399\pi$
$691$	−19.0919	+	11.0227i	−0.726289	+	0.419323i	0.0907737	+	0.995872i	$0.471066\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	18.3712	−	10.6066i	0.696858	−	0.402331i
$696$		0			0
$697$	1.50000	−	2.59808i	0.0568166	−	0.0984092i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	12.7279i		−	0.480727i	−0.970683	−	0.240363i	$-0.922733\pi$
$701$		−	12.7279i		−	0.480727i	0.970683	−	0.240363i	$-0.0772666\pi$
$702$		0			0
$703$			2.44949i			0.0923843i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−16.8493	−	21.7279i	−0.633681	−	0.817163i
$708$		0			0
$709$	−24.5000	−	42.4352i	−0.920117	−	1.59369i	−0.799232	−	0.601023i	$-0.794760\pi$
$709$	−24.5000	−	42.4352i	−0.920117	−	1.59369i	−0.120885	−	0.992667i	$-0.538573\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	31.1769	+	54.0000i	1.16758	+	2.02232i
$714$		0			0
$715$	−9.00000	+	15.5885i	−0.336581	+	0.582975i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	32.9090			1.22730			0.613649	−	0.789579i	$-0.289701\pi$
$719$	32.9090			1.22730			0.613649	+	0.789579i	$0.289701\pi$
$720$		0			0
$721$	48.0000	+	19.5959i	1.78761	+	0.729790i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−7.34847	−	4.24264i	−0.272915	−	0.157568i
$726$		0			0
$727$	−6.36396	+	3.67423i	−0.236026	+	0.136270i	−0.613349	−	0.789812i	$-0.710178\pi$
$727$	−6.36396	+	3.67423i	−0.236026	+	0.136270i	0.377323	+	0.926082i	$0.376845\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.06218	+	10.5000i	0.224218	+	0.388357i
$732$		0			0
$733$	−25.4558	−	14.6969i	−0.940233	−	0.542844i	−0.0501997	−	0.998739i	$-0.515986\pi$
$733$	−25.4558	−	14.6969i	−0.940233	−	0.542844i	−0.890033	+	0.455895i	$0.849319\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			42.4264i			1.56280i
$738$		0			0
$739$	38.0000			1.39785			0.698926	−	0.715194i	$-0.253662\pi$
$739$	38.0000			1.39785			0.698926	+	0.715194i	$0.253662\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−18.3712	−	10.6066i	−0.673973	−	0.389118i	0.123607	−	0.992331i	$-0.460554\pi$
$743$	−18.3712	−	10.6066i	−0.673973	−	0.389118i	−0.797580	+	0.603213i	$0.793887\pi$
$744$		0			0
$745$	−19.0919	+	11.0227i	−0.699472	+	0.403841i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3.04384	+	22.2426i	0.111219	+	0.812728i
$750$		0			0
$751$	−5.00000	+	8.66025i	−0.182453	+	0.316017i	−0.942715	−	0.333599i	$-0.891737\pi$
$751$	−5.00000	+	8.66025i	−0.182453	+	0.316017i	0.760263	+	0.649616i	$0.225070\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.66025			0.315179
$756$		0			0
$757$	−53.0000			−1.92632			−0.963159	−	0.268933i	$-0.913329\pi$
$757$	−53.0000			−1.92632			−0.963159	+	0.268933i	$0.913329\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11.2583	+	19.5000i	−0.408114	+	0.706874i	−0.994678	−	0.103028i	$-0.967147\pi$
$761$	−11.2583	+	19.5000i	−0.408114	+	0.706874i	0.586564	+	0.809903i	$0.300480\pi$
$762$		0			0
$763$	34.0772	−	4.66335i	1.23368	−	0.168825i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	18.3712	−	10.6066i	0.663345	−	0.382982i
$768$		0			0
$769$	4.24264	+	2.44949i	0.152994	+	0.0883309i	0.574542	−	0.818475i	$-0.305180\pi$
$769$	4.24264	+	2.44949i	0.152994	+	0.0883309i	−0.421549	+	0.906806i	$0.638513\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	39.8372			1.43284			0.716422	−	0.697668i	$-0.245779\pi$
$773$	39.8372			1.43284			0.716422	+	0.697668i	$0.245779\pi$
$774$		0			0
$775$			14.6969i			0.527930i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3.67423	−	2.12132i	−0.131643	−	0.0760042i
$780$		0			0
$781$	18.0000	+	31.1769i	0.644091	+	1.11560i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	11.0227	−	6.36396i	0.393417	−	0.227140i
$786$		0			0
$787$	−4.24264	−	2.44949i	−0.151234	−	0.0873149i	0.422473	−	0.906375i	$-0.361162\pi$
$787$	−4.24264	−	2.44949i	−0.151234	−	0.0873149i	−0.573707	+	0.819060i	$0.694495\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−31.1769	−	12.7279i	−1.10852	−	0.452553i
$792$		0			0
$793$	6.00000			0.213066
$794$		0			0
$795$		0			0
$796$		0			0
$797$	5.19615	−	9.00000i	0.184057	−	0.318796i	−0.759201	−	0.650856i	$-0.774410\pi$
$797$	5.19615	−	9.00000i	0.184057	−	0.318796i	0.943258	+	0.332060i	$0.107744\pi$
$798$		0			0
$799$	−10.5000	−	18.1865i	−0.371463	−	0.643393i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−20.7846	−	36.0000i	−0.733473	−	1.27041i
$804$		0			0
$805$	30.7279	−	23.8284i	1.08302	−	0.839842i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	25.4558i		−	0.894980i	−0.894289	−	0.447490i	$-0.852318\pi$
$809$		−	25.4558i		−	0.894980i	0.894289	−	0.447490i	$-0.147682\pi$
$810$		0			0
$811$			51.4393i			1.80628i	0.429349	+	0.903139i	$0.358743\pi$
$811$			51.4393i			1.80628i	−0.429349	+	0.903139i	$0.641257\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.866025	+	1.50000i	−0.0303355	+	0.0525427i
$816$		0			0
$817$	14.8492	−	8.57321i	0.519509	−	0.299939i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	22.0454	−	12.7279i	0.769390	−	0.444208i	−0.0632669	−	0.997997i	$-0.520152\pi$
$821$	22.0454	−	12.7279i	0.769390	−	0.444208i	0.832657	+	0.553789i	$0.186819\pi$
$822$		0			0
$823$	12.5000	−	21.6506i	0.435723	−	0.754694i	−0.561632	−	0.827387i	$-0.689826\pi$
$823$	12.5000	−	21.6506i	0.435723	−	0.754694i	0.997354	+	0.0726937i	$0.0231595\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			38.1838i			1.32778i	0.747830	+	0.663890i	$0.231096\pi$
$827$			38.1838i			1.32778i	−0.747830	+	0.663890i	$0.768904\pi$
$828$		0			0
$829$		−	31.8434i		−	1.10597i	−0.833193	−	0.552983i	$-0.813489\pi$
$829$		−	31.8434i		−	1.10597i	0.833193	−	0.552983i	$-0.186511\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−11.6786	+	3.25736i	−0.404639	+	0.112861i
$834$		0			0
$835$	16.5000	+	28.5788i	0.571006	+	0.989011i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	0.866025	+	1.50000i	0.0298985	+	0.0517858i	0.880587	−	0.473884i	$-0.157148\pi$
$839$	0.866025	+	1.50000i	0.0298985	+	0.0517858i	−0.850689	+	0.525669i	$0.823815\pi$
$840$		0			0
$841$	−5.50000	+	9.52628i	−0.189655	+	0.328492i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12.1244			−0.417091
$846$		0			0
$847$	7.00000	−	17.1464i	0.240523	−	0.589158i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	7.34847	+	4.24264i	0.251902	+	0.145436i
$852$		0			0
$853$	16.9706	−	9.79796i	0.581061	−	0.335476i	−0.180494	−	0.983576i	$-0.557770\pi$
$853$	16.9706	−	9.79796i	0.581061	−	0.335476i	0.761555	+	0.648100i	$0.224436\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6.06218	+	10.5000i	0.207080	+	0.358673i	0.950793	−	0.309825i	$-0.100271\pi$
$857$	6.06218	+	10.5000i	0.207080	+	0.358673i	−0.743713	+	0.668499i	$0.766937\pi$
$858$		0			0
$859$	33.9411	+	19.5959i	1.15806	+	0.668604i	0.950837	−	0.309691i	$-0.100225\pi$
$859$	33.9411	+	19.5959i	1.15806	+	0.668604i	0.207219	+	0.978295i	$0.433559\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			16.9706i			0.577685i	0.957377	+	0.288842i	$0.0932703\pi$
$863$			16.9706i			0.577685i	−0.957377	+	0.288842i	$0.906730\pi$
$864$		0			0
$865$	36.0000			1.22404
$866$		0			0
$867$		0			0
$868$		0			0
$869$	18.3712	+	10.6066i	0.623199	+	0.359804i
$870$		0			0
$871$	21.2132	−	12.2474i	0.718782	−	0.414989i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	31.7818	−	4.34924i	1.07442	−	0.147031i
$876$		0			0
$877$	3.50000	−	6.06218i	0.118187	−	0.204705i	−0.800862	−	0.598848i	$-0.795625\pi$
$877$	3.50000	−	6.06218i	0.118187	−	0.204705i	0.919049	+	0.394143i	$0.128959\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−31.1769			−1.05038			−0.525188	−	0.850986i	$-0.676005\pi$
$881$	−31.1769			−1.05038			−0.525188	+	0.850986i	$0.676005\pi$
$882$		0			0
$883$	−17.0000			−0.572096			−0.286048	−	0.958215i	$-0.592342\pi$
$883$	−17.0000			−0.572096			−0.286048	+	0.958215i	$0.592342\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−9.52628	+	16.5000i	−0.319861	+	0.554016i	−0.980459	−	0.196724i	$-0.936970\pi$
$887$	−9.52628	+	16.5000i	−0.319861	+	0.554016i	0.660598	+	0.750740i	$0.270303\pi$
$888$		0			0
$889$	−34.0772	+	4.66335i	−1.14291	+	0.156404i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−25.7196	+	14.8492i	−0.860675	+	0.496911i
$894$		0			0
$895$	6.36396	+	3.67423i	0.212724	+	0.122816i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−31.1769			−1.03981
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	7.34847	+	4.24264i	0.244271	+	0.141030i
$906$		0			0
$907$	−6.50000	−	11.2583i	−0.215829	−	0.373827i	0.737700	−	0.675129i	$-0.235912\pi$
$907$	−6.50000	−	11.2583i	−0.215829	−	0.373827i	−0.953529	+	0.301302i	$0.902579\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−25.7196	+	14.8492i	−0.852130	+	0.491977i	−0.861369	−	0.507980i	$-0.830392\pi$
$911$	−25.7196	+	14.8492i	−0.852130	+	0.491977i	0.00923912	+	0.999957i	$0.497059\pi$
$912$		0			0
$913$	44.5477	+	25.7196i	1.47431	+	0.851196i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	10.3923	−	25.4558i	0.343184	−	0.840626i
$918$		0			0
$919$	−5.00000			−0.164935			−0.0824674	−	0.996594i	$-0.526280\pi$
$919$	−5.00000			−0.164935			−0.0824674	+	0.996594i	$0.526280\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	10.3923	−	18.0000i	0.342067	−	0.592477i
$924$		0			0
$925$	1.00000	+	1.73205i	0.0328798	+	0.0569495i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−19.9186	−	34.5000i	−0.653508	−	1.13191i	−0.982266	−	0.187494i	$-0.939963\pi$
$929$	−19.9186	−	34.5000i	−0.653508	−	1.13191i	0.328758	−	0.944414i	$-0.393370\pi$
$930$		0			0
$931$	4.60660	+	16.5160i	0.150975	+	0.541291i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	12.7279i		−	0.416248i
$936$		0			0
$937$			2.44949i			0.0800213i	0.999199	+	0.0400107i	$0.0127392\pi$
$937$			2.44949i			0.0800213i	−0.999199	+	0.0400107i	$0.987261\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−26.8468	+	46.5000i	−0.875180	+	1.51586i	−0.0186097	+	0.999827i	$0.505924\pi$
$941$	−26.8468	+	46.5000i	−0.875180	+	1.51586i	−0.856571	+	0.516030i	$0.827409\pi$
$942$		0			0
$943$	−12.7279	+	7.34847i	−0.414478	+	0.239299i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	14.6969	−	8.48528i	0.477586	−	0.275735i	−0.241824	−	0.970320i	$-0.577746\pi$
$947$	14.6969	−	8.48528i	0.477586	−	0.275735i	0.719410	+	0.694586i	$0.244412\pi$
$948$		0			0
$949$	−12.0000	+	20.7846i	−0.389536	+	0.674697i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	50.9117i		−	1.64919i	−0.565723	−	0.824596i	$-0.691403\pi$
$953$		−	50.9117i		−	1.64919i	0.565723	−	0.824596i	$-0.308597\pi$
$954$		0			0
$955$			14.6969i			0.475582i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	44.3519	−	34.3934i	1.43220	−	1.11062i
$960$		0			0
$961$	11.5000	+	19.9186i	0.370968	+	0.642535i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6.06218	+	10.5000i	0.195148	+	0.338007i
$966$		0			0
$967$	23.0000	−	39.8372i	0.739630	−	1.28108i	−0.213032	−	0.977045i	$-0.568334\pi$
$967$	23.0000	−	39.8372i	0.739630	−	1.28108i	0.952662	−	0.304032i	$-0.0983329\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	32.9090			1.05610			0.528049	−	0.849214i	$-0.322924\pi$
$971$	32.9090			1.05610			0.528049	+	0.849214i	$0.322924\pi$
$972$		0			0
$973$	30.0000	+	12.2474i	0.961756	+	0.392635i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−36.7423	−	21.2132i	−1.17549	−	0.678671i	−0.220524	−	0.975381i	$-0.570777\pi$
$977$	−36.7423	−	21.2132i	−1.17549	−	0.678671i	−0.954967	+	0.296711i	$0.904110\pi$
$978$		0			0
$979$	38.1838	−	22.0454i	1.22036	−	0.704574i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−0.866025	−	1.50000i	−0.0276219	−	0.0478426i	0.851884	−	0.523731i	$-0.175460\pi$
$983$	−0.866025	−	1.50000i	−0.0276219	−	0.0478426i	−0.879506	+	0.475888i	$0.842127\pi$
$984$		0			0
$985$	31.8198	+	18.3712i	1.01386	+	0.585354i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	59.3970i		−	1.88871i
$990$		0			0
$991$	55.0000			1.74713			0.873566	−	0.486705i	$-0.161801\pi$
$991$	55.0000			1.74713			0.873566	+	0.486705i	$0.161801\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−14.6969	−	8.48528i	−0.465924	−	0.269002i
$996$		0			0
$997$	31.8198	−	18.3712i	1.00774	−	0.581821i	0.0972132	−	0.995264i	$-0.469007\pi$
$997$	31.8198	−	18.3712i	1.00774	−	0.581821i	0.910530	+	0.413443i	$0.135674\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.x.j.377.4		8
3.2	odd	2	inner	2268.2.x.j.377.2		8
7.6	odd	2	inner	2268.2.x.j.377.1		8
9.2	odd	6	inner	2268.2.x.j.1889.1		8
9.4	even	3		756.2.f.d.377.2	yes	4
9.5	odd	6		756.2.f.d.377.4	yes	4
9.7	even	3	inner	2268.2.x.j.1889.3		8
21.20	even	2	inner	2268.2.x.j.377.3		8
36.23	even	6		3024.2.k.h.1889.3		4
36.31	odd	6		3024.2.k.h.1889.1		4
63.13	odd	6		756.2.f.d.377.3	yes	4
63.20	even	6	inner	2268.2.x.j.1889.4		8
63.34	odd	6	inner	2268.2.x.j.1889.2		8
63.41	even	6		756.2.f.d.377.1	✓	4
252.139	even	6		3024.2.k.h.1889.4		4
252.167	odd	6		3024.2.k.h.1889.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.f.d.377.1	✓	4	63.41	even	6
756.2.f.d.377.2	yes	4	9.4	even	3
756.2.f.d.377.3	yes	4	63.13	odd	6
756.2.f.d.377.4	yes	4	9.5	odd	6
2268.2.x.j.377.1		8	7.6	odd	2	inner
2268.2.x.j.377.2		8	3.2	odd	2	inner
2268.2.x.j.377.3		8	21.20	even	2	inner
2268.2.x.j.377.4		8	1.1	even	1	trivial
2268.2.x.j.1889.1		8	9.2	odd	6	inner
2268.2.x.j.1889.2		8	63.34	odd	6	inner
2268.2.x.j.1889.3		8	9.7	even	3	inner
2268.2.x.j.1889.4		8	63.20	even	6	inner
3024.2.k.h.1889.1		4	36.31	odd	6
3024.2.k.h.1889.2		4	252.167	odd	6
3024.2.k.h.1889.3		4	36.23	even	6
3024.2.k.h.1889.4		4	252.139	even	6

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:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.x (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 - 1.50000i) q^{5} +(2.62132 - 0.358719i) q^{7} +(-3.67423 + 2.12132i) q^{11} +(2.12132 + 1.22474i) q^{13} -1.73205 q^{17} +2.44949i q^{19} +(7.34847 + 4.24264i) q^{23} +(1.00000 + 1.73205i) q^{25} +(-3.67423 + 2.12132i) q^{29} +(6.36396 + 3.67423i) q^{31} +(1.73205 - 4.24264i) q^{35} +1.00000 q^{37} +(-0.866025 + 1.50000i) q^{41} +(-3.50000 - 6.06218i) q^{43} +(6.06218 + 10.5000i) q^{47} +(6.74264 - 1.88064i) q^{49} +7.34847i q^{55} +(4.33013 - 7.50000i) q^{59} +(2.12132 - 1.22474i) q^{61} +(3.67423 - 2.12132i) q^{65} +(5.00000 - 8.66025i) q^{67} -8.48528i q^{71} +9.79796i q^{73} +(-8.87039 + 6.87868i) q^{77} +(-2.50000 - 4.33013i) q^{79} +(-6.06218 - 10.5000i) q^{83} +(-1.50000 + 2.59808i) q^{85} -10.3923 q^{89} +(6.00000 + 2.44949i) q^{91} +(3.67423 + 2.12132i) q^{95} +(-2.12132 + 1.22474i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{7} + 8 q^{25} + 8 q^{37} - 28 q^{43} + 20 q^{49} + 40 q^{67} - 20 q^{79} - 12 q^{85} + 48 q^{91}+O(q^{100}) \) 8 * q + 4 * q^7 + 8 * q^25 + 8 * q^37 - 28 * q^43 + 20 * q^49 + 40 * q^67 - 20 * q^79 - 12 * q^85 + 48 * q^91

Introduction

L-functions

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