LMFDB - Embedded newform 2268.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,2,Mod(1,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2268.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$18.1100711784$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	2268.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-1.73205 q^{5} +1.00000 q^{7} -3.46410 q^{11} +5.00000 q^{13} -1.73205 q^{17} +2.00000 q^{19} +3.46410 q^{23} -2.00000 q^{25} -1.73205 q^{29} +2.00000 q^{31} -1.73205 q^{35} -7.00000 q^{37} +6.92820 q^{41} +8.00000 q^{43} -10.3923 q^{47} +1.00000 q^{49} +13.8564 q^{53} +6.00000 q^{55} +10.3923 q^{59} -1.00000 q^{61} -8.66025 q^{65} +2.00000 q^{67} +11.0000 q^{73} -3.46410 q^{77} +14.0000 q^{79} +3.46410 q^{83} +3.00000 q^{85} +1.73205 q^{89} +5.00000 q^{91} -3.46410 q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^7	$ 2 q + 2 q^{7} + 10 q^{13} + 4 q^{19} - 4 q^{25} + 4 q^{31} - 14 q^{37} + 16 q^{43} + 2 q^{49} + 12 q^{55} - 2 q^{61} + 4 q^{67} + 22 q^{73} + 28 q^{79} + 6 q^{85} + 10 q^{91} + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 + 10 * q^13 + 4 * q^19 - 4 * q^25 + 4 * q^31 - 14 * q^37 + 16 * q^43 + 2 * q^49 + 12 * q^55 - 2 * q^61 + 4 * q^67 + 22 * q^73 + 28 * q^79 + 6 * q^85 + 10 * q^91 + 4 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−1.73205	−0.774597	−0.387298	−	0.921954i	$-0.626592\pi$
$5$	−1.73205	−0.774597	−0.387298	+	0.921954i	$0.626592\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	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
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.73205	−0.321634	−0.160817	−	0.986984i	$-0.551413\pi$
$29$	−1.73205	−0.321634	−0.160817	+	0.986984i	$0.551413\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.73205	−0.292770
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.92820	1.08200	0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820	1.08200	0.541002	+	0.841021i	$0.318045\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.3923	−1.51587	−0.757937	−	0.652328i	$-0.773792\pi$
$47$	−10.3923	−1.51587	−0.757937	+	0.652328i	$0.773792\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	13.8564	1.90332	0.951662	−	0.307148i	$-0.0993745\pi$
$53$	13.8564	1.90332	0.951662	+	0.307148i	$0.0993745\pi$
$54$	0	0
$55$	6.00000	0.809040
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.3923	1.35296	0.676481	−	0.736460i	$-0.263504\pi$
$59$	10.3923	1.35296	0.676481	+	0.736460i	$0.263504\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−8.66025	−1.07417
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.46410	−0.394771
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.46410	0.380235	0.190117	−	0.981761i	$-0.439113\pi$
$83$	3.46410	0.380235	0.190117	+	0.981761i	$0.439113\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.73205	0.183597	0.0917985	−	0.995778i	$-0.470738\pi$
$89$	1.73205	0.183597	0.0917985	+	0.995778i	$0.470738\pi$
$90$	0	0
$91$	5.00000	0.524142
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.46410	−0.355409
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.8564	−1.37876	−0.689382	−	0.724398i	$-0.742118\pi$
$101$	−13.8564	−1.37876	−0.689382	+	0.724398i	$0.742118\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.92820	0.669775	0.334887	−	0.942258i	$-0.391302\pi$
$107$	6.92820	0.669775	0.334887	+	0.942258i	$0.391302\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.73205	0.162938	0.0814688	−	0.996676i	$-0.474039\pi$
$113$	1.73205	0.162938	0.0814688	+	0.996676i	$0.474039\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.73205	−0.158777
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.1244	1.08444
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.8564	1.21064	0.605320	−	0.795982i	$-0.293045\pi$
$131$	13.8564	1.21064	0.605320	+	0.795982i	$0.293045\pi$
$132$	0	0
$133$	2.00000	0.173422
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.66025	−0.739895	−0.369948	−	0.929053i	$-0.620624\pi$
$137$	−8.66025	−0.739895	−0.369948	+	0.929053i	$0.620624\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−17.3205	−1.44841
$144$	0	0
$145$	3.00000	0.249136
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19.0526	1.56085	0.780423	−	0.625252i	$-0.215004\pi$
$149$	19.0526	1.56085	0.780423	+	0.625252i	$0.215004\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.46410	−0.278243
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.46410	0.273009
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.46410	−0.268060	−0.134030	−	0.990977i	$-0.542792\pi$
$167$	−3.46410	−0.268060	−0.134030	+	0.990977i	$0.542792\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−5.19615	−0.395056	−0.197528	−	0.980297i	$-0.563291\pi$
$173$	−5.19615	−0.395056	−0.197528	+	0.980297i	$0.563291\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−17.3205	−1.29460	−0.647298	−	0.762237i	$-0.724101\pi$
$179$	−17.3205	−1.29460	−0.647298	+	0.762237i	$0.724101\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	12.1244	0.891400
$186$	0	0
$187$	6.00000	0.438763
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.3923	0.751961	0.375980	−	0.926628i	$-0.377306\pi$
$191$	10.3923	0.751961	0.375980	+	0.926628i	$0.377306\pi$
$192$	0	0
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	25.9808	1.85105	0.925526	−	0.378683i	$-0.123623\pi$
$197$	25.9808	1.85105	0.925526	+	0.378683i	$0.123623\pi$
$198$	0	0
$199$	−22.0000	−1.55954	−0.779769	−	0.626067i	$-0.784664\pi$
$199$	−22.0000	−1.55954	−0.779769	+	0.626067i	$0.784664\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.73205	−0.121566
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6.92820	−0.479234
$210$	0	0
$211$	26.0000	1.78991	0.894957	−	0.446153i	$-0.147206\pi$
$211$	26.0000	1.78991	0.894957	+	0.446153i	$0.147206\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−13.8564	−0.944999
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−8.66025	−0.582552
$222$	0	0
$223$	20.0000	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.8564	−0.919682	−0.459841	−	0.888001i	$-0.652094\pi$
$227$	−13.8564	−0.919682	−0.459841	+	0.888001i	$0.652094\pi$
$228$	0	0
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.66025	−0.567352	−0.283676	−	0.958920i	$-0.591554\pi$
$233$	−8.66025	−0.567352	−0.283676	+	0.958920i	$0.591554\pi$
$234$	0	0
$235$	18.0000	1.17419
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−24.2487	−1.56852	−0.784259	−	0.620433i	$-0.786957\pi$
$239$	−24.2487	−1.56852	−0.784259	+	0.620433i	$0.786957\pi$
$240$	0	0
$241$	−25.0000	−1.61039	−0.805196	−	0.593009i	$-0.797940\pi$
$241$	−25.0000	−1.61039	−0.805196	+	0.593009i	$0.797940\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.73205	−0.110657
$246$	0	0
$247$	10.0000	0.636285
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.3923	0.655956	0.327978	−	0.944685i	$-0.393633\pi$
$251$	10.3923	0.655956	0.327978	+	0.944685i	$0.393633\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.1244	−0.756297	−0.378148	−	0.925745i	$-0.623439\pi$
$257$	−12.1244	−0.756297	−0.378148	+	0.925745i	$0.623439\pi$
$258$	0	0
$259$	−7.00000	−0.434959
$260$	0	0
$261$	0	0
$262$	0	0
$263$	17.3205	1.06803	0.534014	−	0.845476i	$-0.320683\pi$
$263$	17.3205	1.06803	0.534014	+	0.845476i	$0.320683\pi$
$264$	0	0
$265$	−24.0000	−1.47431
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.73205	−0.105605	−0.0528025	−	0.998605i	$-0.516815\pi$
$269$	−1.73205	−0.105605	−0.0528025	+	0.998605i	$0.516815\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.92820	0.417786
$276$	0	0
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.66025	0.516627	0.258314	−	0.966061i	$-0.416833\pi$
$281$	8.66025	0.516627	0.258314	+	0.966061i	$0.416833\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.92820	0.408959
$288$	0	0
$289$	−14.0000	−0.823529
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−29.4449	−1.72019	−0.860094	−	0.510136i	$-0.829595\pi$
$293$	−29.4449	−1.72019	−0.860094	+	0.510136i	$0.829595\pi$
$294$	0	0
$295$	−18.0000	−1.04800
$296$	0	0
$297$	0	0
$298$	0	0
$299$	17.3205	1.00167
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.73205	0.0991769
$306$	0	0
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	31.1769	1.76788	0.883940	−	0.467600i	$-0.154881\pi$
$311$	31.1769	1.76788	0.883940	+	0.467600i	$0.154881\pi$
$312$	0	0
$313$	−31.0000	−1.75222	−0.876112	−	0.482108i	$-0.839871\pi$
$313$	−31.0000	−1.75222	−0.876112	+	0.482108i	$0.839871\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.19615	−0.291845	−0.145922	−	0.989296i	$-0.546615\pi$
$317$	−5.19615	−0.291845	−0.145922	+	0.989296i	$0.546615\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.46410	−0.192748
$324$	0	0
$325$	−10.0000	−0.554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−10.3923	−0.572946
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.46410	−0.189264
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−6.92820	−0.375183
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.7846	−1.11578	−0.557888	−	0.829916i	$-0.688388\pi$
$347$	−20.7846	−1.11578	−0.557888	+	0.829916i	$0.688388\pi$
$348$	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
$351$	0	0
$352$	0	0
$353$	−13.8564	−0.737502	−0.368751	−	0.929528i	$-0.620215\pi$
$353$	−13.8564	−0.737502	−0.368751	+	0.929528i	$0.620215\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13.8564	−0.731313	−0.365657	−	0.930750i	$-0.619156\pi$
$359$	−13.8564	−0.731313	−0.365657	+	0.930750i	$0.619156\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−19.0526	−0.997257
$366$	0	0
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	13.8564	0.719389
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.66025	−0.446026
$378$	0	0
$379$	26.0000	1.33553	0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000	1.33553	0.667765	+	0.744372i	$0.267251\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	34.6410	1.77007	0.885037	−	0.465521i	$-0.154133\pi$
$383$	34.6410	1.77007	0.885037	+	0.465521i	$0.154133\pi$
$384$	0	0
$385$	6.00000	0.305788
$386$	0	0
$387$	0	0
$388$	0	0
$389$	27.7128	1.40510	0.702548	−	0.711637i	$-0.252046\pi$
$389$	27.7128	1.40510	0.702548	+	0.711637i	$0.252046\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−24.2487	−1.22009
$396$	0	0
$397$	−25.0000	−1.25471	−0.627357	−	0.778732i	$-0.715863\pi$
$397$	−25.0000	−1.25471	−0.627357	+	0.778732i	$0.715863\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	19.0526	0.951439	0.475720	−	0.879597i	$-0.342188\pi$
$401$	19.0526	0.951439	0.475720	+	0.879597i	$0.342188\pi$
$402$	0	0
$403$	10.0000	0.498135
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.2487	1.20196
$408$	0	0
$409$	−25.0000	−1.23617	−0.618085	−	0.786111i	$-0.712091\pi$
$409$	−25.0000	−1.23617	−0.618085	+	0.786111i	$0.712091\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	10.3923	0.511372
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−13.8564	−0.676930	−0.338465	−	0.940979i	$-0.609908\pi$
$419$	−13.8564	−0.676930	−0.338465	+	0.940979i	$0.609908\pi$
$420$	0	0
$421$	−1.00000	−0.0487370	−0.0243685	−	0.999703i	$-0.507758\pi$
$421$	−1.00000	−0.0487370	−0.0243685	+	0.999703i	$0.507758\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.46410	0.168034
$426$	0	0
$427$	−1.00000	−0.0483934
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13.8564	0.667440	0.333720	−	0.942672i	$-0.391696\pi$
$431$	13.8564	0.667440	0.333720	+	0.942672i	$0.391696\pi$
$432$	0	0
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.92820	0.331421
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	31.1769	1.48126	0.740630	−	0.671913i	$-0.234527\pi$
$443$	31.1769	1.48126	0.740630	+	0.671913i	$0.234527\pi$
$444$	0	0
$445$	−3.00000	−0.142214
$446$	0	0
$447$	0	0
$448$	0	0
$449$	20.7846	0.980886	0.490443	−	0.871473i	$-0.336835\pi$
$449$	20.7846	0.980886	0.490443	+	0.871473i	$0.336835\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−8.66025	−0.405999
$456$	0	0
$457$	−31.0000	−1.45012	−0.725059	−	0.688686i	$-0.758188\pi$
$457$	−31.0000	−1.45012	−0.725059	+	0.688686i	$0.758188\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	13.8564	0.645357	0.322679	−	0.946509i	$-0.395417\pi$
$461$	13.8564	0.645357	0.322679	+	0.946509i	$0.395417\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.46410	−0.160300	−0.0801498	−	0.996783i	$-0.525540\pi$
$467$	−3.46410	−0.160300	−0.0801498	+	0.996783i	$0.525540\pi$
$468$	0	0
$469$	2.00000	0.0923514
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−27.7128	−1.27424
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	0	0
$478$	0	0
$479$	38.1051	1.74107	0.870534	−	0.492109i	$-0.163774\pi$
$479$	38.1051	1.74107	0.870534	+	0.492109i	$0.163774\pi$
$480$	0	0
$481$	−35.0000	−1.59586
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.46410	−0.157297
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−13.8564	−0.625331	−0.312665	−	0.949863i	$-0.601222\pi$
$491$	−13.8564	−0.625331	−0.312665	+	0.949863i	$0.601222\pi$
$492$	0	0
$493$	3.00000	0.135113
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10.3923	−0.463370	−0.231685	−	0.972791i	$-0.574424\pi$
$503$	−10.3923	−0.463370	−0.231685	+	0.972791i	$0.574424\pi$
$504$	0	0
$505$	24.0000	1.06799
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−27.7128	−1.22835	−0.614174	−	0.789170i	$-0.710511\pi$
$509$	−27.7128	−1.22835	−0.614174	+	0.789170i	$0.710511\pi$
$510$	0	0
$511$	11.0000	0.486611
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.92820	0.305293
$516$	0	0
$517$	36.0000	1.58328
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−27.7128	−1.21412	−0.607060	−	0.794656i	$-0.707651\pi$
$521$	−27.7128	−1.21412	−0.607060	+	0.794656i	$0.707651\pi$
$522$	0	0
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.46410	−0.150899
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	34.6410	1.50047
$534$	0	0
$535$	−12.0000	−0.518805
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.46410	−0.149209
$540$	0	0
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−19.0526	−0.816122
$546$	0	0
$547$	38.0000	1.62476	0.812381	−	0.583127i	$-0.198171\pi$
$547$	38.0000	1.62476	0.812381	+	0.583127i	$0.198171\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.46410	−0.147576
$552$	0	0
$553$	14.0000	0.595341
$554$	0	0
$555$	0	0
$556$	0	0
$557$	39.8372	1.68796	0.843978	−	0.536379i	$-0.180208\pi$
$557$	39.8372	1.68796	0.843978	+	0.536379i	$0.180208\pi$
$558$	0	0
$559$	40.0000	1.69182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	41.5692	1.75193	0.875967	−	0.482371i	$-0.160224\pi$
$563$	41.5692	1.75193	0.875967	+	0.482371i	$0.160224\pi$
$564$	0	0
$565$	−3.00000	−0.126211
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−46.7654	−1.96051	−0.980253	−	0.197747i	$-0.936638\pi$
$569$	−46.7654	−1.96051	−0.980253	+	0.197747i	$0.936638\pi$
$570$	0	0
$571$	2.00000	0.0836974	0.0418487	−	0.999124i	$-0.486675\pi$
$571$	2.00000	0.0836974	0.0418487	+	0.999124i	$0.486675\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.92820	−0.288926
$576$	0	0
$577$	−43.0000	−1.79011	−0.895057	−	0.445952i	$-0.852865\pi$
$577$	−43.0000	−1.79011	−0.895057	+	0.445952i	$0.852865\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.46410	0.143715
$582$	0	0
$583$	−48.0000	−1.98796
$584$	0	0
$585$	0	0
$586$	0	0
$587$	45.0333	1.85872	0.929362	−	0.369170i	$-0.120358\pi$
$587$	45.0333	1.85872	0.929362	+	0.369170i	$0.120358\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.73205	0.0711268	0.0355634	−	0.999367i	$-0.488677\pi$
$593$	1.73205	0.0711268	0.0355634	+	0.999367i	$0.488677\pi$
$594$	0	0
$595$	3.00000	0.122988
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−31.1769	−1.27385	−0.636927	−	0.770924i	$-0.719795\pi$
$599$	−31.1769	−1.27385	−0.636927	+	0.770924i	$0.719795\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.73205	−0.0704179
$606$	0	0
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−51.9615	−2.10214
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−29.4449	−1.18541	−0.592703	−	0.805421i	$-0.701939\pi$
$617$	−29.4449	−1.18541	−0.592703	+	0.805421i	$0.701939\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.73205	0.0693932
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.1244	0.483430
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13.8564	−0.549875
$636$	0	0
$637$	5.00000	0.198107
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−29.4449	−1.16300	−0.581501	−	0.813546i	$-0.697534\pi$
$641$	−29.4449	−1.16300	−0.581501	+	0.813546i	$0.697534\pi$
$642$	0	0
$643$	−22.0000	−0.867595	−0.433798	−	0.901010i	$-0.642827\pi$
$643$	−22.0000	−0.867595	−0.433798	+	0.901010i	$0.642827\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−48.4974	−1.90663	−0.953315	−	0.301977i	$-0.902353\pi$
$647$	−48.4974	−1.90663	−0.953315	+	0.301977i	$0.902353\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−27.7128	−1.08449	−0.542243	−	0.840222i	$-0.682425\pi$
$653$	−27.7128	−1.08449	−0.542243	+	0.840222i	$0.682425\pi$
$654$	0	0
$655$	−24.0000	−0.937758
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−27.7128	−1.07954	−0.539769	−	0.841813i	$-0.681488\pi$
$659$	−27.7128	−1.07954	−0.539769	+	0.841813i	$0.681488\pi$
$660$	0	0
$661$	−25.0000	−0.972387	−0.486194	−	0.873851i	$-0.661615\pi$
$661$	−25.0000	−0.972387	−0.486194	+	0.873851i	$0.661615\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.46410	−0.134332
$666$	0	0
$667$	−6.00000	−0.232321
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.46410	0.133730
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	20.7846	0.798817	0.399409	−	0.916773i	$-0.369215\pi$
$677$	20.7846	0.798817	0.399409	+	0.916773i	$0.369215\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	13.8564	0.530201	0.265100	−	0.964221i	$-0.414595\pi$
$683$	13.8564	0.530201	0.265100	+	0.964221i	$0.414595\pi$
$684$	0	0
$685$	15.0000	0.573121
$686$	0	0
$687$	0	0
$688$	0	0
$689$	69.2820	2.63944
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−24.2487	−0.919806
$696$	0	0
$697$	−12.0000	−0.454532
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−5.19615	−0.196256	−0.0981280	−	0.995174i	$-0.531285\pi$
$701$	−5.19615	−0.196256	−0.0981280	+	0.995174i	$0.531285\pi$
$702$	0	0
$703$	−14.0000	−0.528020
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−13.8564	−0.521124
$708$	0	0
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.92820	0.259463
$714$	0	0
$715$	30.0000	1.12194
$716$	0	0
$717$	0	0
$718$	0	0
$719$	27.7128	1.03351	0.516757	−	0.856132i	$-0.327139\pi$
$719$	27.7128	1.03351	0.516757	+	0.856132i	$0.327139\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.46410	0.128654
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−13.8564	−0.512498
$732$	0	0
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.92820	−0.255204
$738$	0	0
$739$	−34.0000	−1.25071	−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000	−1.25071	−0.625355	+	0.780340i	$0.715046\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−34.6410	−1.27086	−0.635428	−	0.772160i	$-0.719176\pi$
$743$	−34.6410	−1.27086	−0.635428	+	0.772160i	$0.719176\pi$
$744$	0	0
$745$	−33.0000	−1.20903
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6.92820	0.253151
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−3.46410	−0.126072
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	50.2295	1.82082	0.910408	−	0.413710i	$-0.135767\pi$
$761$	50.2295	1.82082	0.910408	+	0.413710i	$0.135767\pi$
$762$	0	0
$763$	11.0000	0.398227
$764$	0	0
$765$	0	0
$766$	0	0
$767$	51.9615	1.87622
$768$	0	0
$769$	41.0000	1.47850	0.739249	−	0.673432i	$-0.235181\pi$
$769$	41.0000	1.47850	0.739249	+	0.673432i	$0.235181\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.66025	0.311488	0.155744	−	0.987797i	$-0.450223\pi$
$773$	8.66025	0.311488	0.155744	+	0.987797i	$0.450223\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	0	0
$778$	0	0
$779$	13.8564	0.496457
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	22.5167	0.803654
$786$	0	0
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.73205	0.0615846
$792$	0	0
$793$	−5.00000	−0.177555
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−19.0526	−0.674876	−0.337438	−	0.941348i	$-0.609560\pi$
$797$	−19.0526	−0.674876	−0.337438	+	0.941348i	$0.609560\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−38.1051	−1.34470
$804$	0	0
$805$	−6.00000	−0.211472
$806$	0	0
$807$	0	0
$808$	0	0
$809$	39.8372	1.40060	0.700300	−	0.713849i	$-0.253050\pi$
$809$	39.8372	1.40060	0.700300	+	0.713849i	$0.253050\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6.92820	0.242684
$816$	0	0
$817$	16.0000	0.559769
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−5.19615	−0.181347	−0.0906735	−	0.995881i	$-0.528902\pi$
$821$	−5.19615	−0.181347	−0.0906735	+	0.995881i	$0.528902\pi$
$822$	0	0
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.7846	0.722752	0.361376	−	0.932420i	$-0.382307\pi$
$827$	20.7846	0.722752	0.361376	+	0.932420i	$0.382307\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.73205	−0.0600120
$834$	0	0
$835$	6.00000	0.207639
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−6.92820	−0.239188	−0.119594	−	0.992823i	$-0.538159\pi$
$839$	−6.92820	−0.239188	−0.119594	+	0.992823i	$0.538159\pi$
$840$	0	0
$841$	−26.0000	−0.896552
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−20.7846	−0.715012
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−24.2487	−0.831235
$852$	0	0
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−19.0526	−0.650823	−0.325412	−	0.945572i	$-0.605503\pi$
$857$	−19.0526	−0.650823	−0.325412	+	0.945572i	$0.605503\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.46410	−0.117919	−0.0589597	−	0.998260i	$-0.518778\pi$
$863$	−3.46410	−0.117919	−0.0589597	+	0.998260i	$0.518778\pi$
$864$	0	0
$865$	9.00000	0.306009
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−48.4974	−1.64516
$870$	0	0
$871$	10.0000	0.338837
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12.1244	0.409878
$876$	0	0
$877$	17.0000	0.574049	0.287025	−	0.957923i	$-0.407334\pi$
$877$	17.0000	0.574049	0.287025	+	0.957923i	$0.407334\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20.7846	0.700251	0.350126	−	0.936703i	$-0.386139\pi$
$881$	20.7846	0.700251	0.350126	+	0.936703i	$0.386139\pi$
$882$	0	0
$883$	26.0000	0.874970	0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000	0.874970	0.437485	+	0.899226i	$0.355869\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45.0333	1.51207	0.756035	−	0.654531i	$-0.227134\pi$
$887$	45.0333	1.51207	0.756035	+	0.654531i	$0.227134\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−20.7846	−0.695530
$894$	0	0
$895$	30.0000	1.00279
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.46410	−0.115534
$900$	0	0
$901$	−24.0000	−0.799556
$902$	0	0
$903$	0	0
$904$	0	0
$905$	17.3205	0.575753
$906$	0	0
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−27.7128	−0.918166	−0.459083	−	0.888393i	$-0.651822\pi$
$911$	−27.7128	−0.918166	−0.459083	+	0.888393i	$0.651822\pi$
$912$	0	0
$913$	−12.0000	−0.397142
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13.8564	0.457579
$918$	0	0
$919$	38.0000	1.25350	0.626752	−	0.779219i	$-0.284384\pi$
$919$	38.0000	1.25350	0.626752	+	0.779219i	$0.284384\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	14.0000	0.460317
$926$	0	0
$927$	0	0
$928$	0	0
$929$	25.9808	0.852401	0.426201	−	0.904629i	$-0.359852\pi$
$929$	25.9808	0.852401	0.426201	+	0.904629i	$0.359852\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−10.3923	−0.339865
$936$	0	0
$937$	53.0000	1.73143	0.865717	−	0.500533i	$-0.166863\pi$
$937$	53.0000	1.73143	0.865717	+	0.500533i	$0.166863\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	57.1577	1.86329	0.931644	−	0.363374i	$-0.118375\pi$
$941$	57.1577	1.86329	0.931644	+	0.363374i	$0.118375\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	55.0000	1.78538
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5.19615	0.168320	0.0841599	−	0.996452i	$-0.473179\pi$
$953$	5.19615	0.168320	0.0841599	+	0.996452i	$0.473179\pi$
$954$	0	0
$955$	−18.0000	−0.582466
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.66025	−0.279654
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−19.0526	−0.613324
$966$	0	0
$967$	14.0000	0.450210	0.225105	−	0.974335i	$-0.427728\pi$
$967$	14.0000	0.450210	0.225105	+	0.974335i	$0.427728\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−24.2487	−0.778178	−0.389089	−	0.921200i	$-0.627210\pi$
$971$	−24.2487	−0.778178	−0.389089	+	0.921200i	$0.627210\pi$
$972$	0	0
$973$	14.0000	0.448819
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20.7846	0.664959	0.332479	−	0.943111i	$-0.392115\pi$
$977$	20.7846	0.664959	0.332479	+	0.943111i	$0.392115\pi$
$978$	0	0
$979$	−6.00000	−0.191761
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−34.6410	−1.10488	−0.552438	−	0.833554i	$-0.686303\pi$
$983$	−34.6410	−1.10488	−0.552438	+	0.833554i	$0.686303\pi$
$984$	0	0
$985$	−45.0000	−1.43382
$986$	0	0
$987$	0	0
$988$	0	0
$989$	27.7128	0.881216
$990$	0	0
$991$	38.0000	1.20711	0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000	1.20711	0.603555	+	0.797321i	$0.293750\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	38.1051	1.20801
$996$	0	0
$997$	−1.00000	−0.0316703	−0.0158352	−	0.999875i	$-0.505041\pi$
$997$	−1.00000	−0.0316703	−0.0158352	+	0.999875i	$0.505041\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.a.f.1.1	✓	2
3.2	odd	2	inner	2268.2.a.f.1.2	yes	2
4.3	odd	2		9072.2.a.bh.1.1		2
9.2	odd	6		2268.2.j.o.757.1		4
9.4	even	3		2268.2.j.o.1513.2		4
9.5	odd	6		2268.2.j.o.1513.1		4
9.7	even	3		2268.2.j.o.757.2		4
12.11	even	2		9072.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2268.2.a.f.1.1	✓	2	1.1	even	1	trivial
2268.2.a.f.1.2	yes	2	3.2	odd	2	inner
2268.2.j.o.757.1		4	9.2	odd	6
2268.2.j.o.757.2		4	9.7	even	3
2268.2.j.o.1513.1		4	9.5	odd	6
2268.2.j.o.1513.2		4	9.4	even	3
9072.2.a.bh.1.1		2	4.3	odd	2
9072.2.a.bh.1.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-1.73205 q^{5} +1.00000 q^{7} -3.46410 q^{11} +5.00000 q^{13} -1.73205 q^{17} +2.00000 q^{19} +3.46410 q^{23} -2.00000 q^{25} -1.73205 q^{29} +2.00000 q^{31} -1.73205 q^{35} -7.00000 q^{37} +6.92820 q^{41} +8.00000 q^{43} -10.3923 q^{47} +1.00000 q^{49} +13.8564 q^{53} +6.00000 q^{55} +10.3923 q^{59} -1.00000 q^{61} -8.66025 q^{65} +2.00000 q^{67} +11.0000 q^{73} -3.46410 q^{77} +14.0000 q^{79} +3.46410 q^{83} +3.00000 q^{85} +1.73205 q^{89} +5.00000 q^{91} -3.46410 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^7	\( 2 q + 2 q^{7} + 10 q^{13} + 4 q^{19} - 4 q^{25} + 4 q^{31} - 14 q^{37} + 16 q^{43} + 2 q^{49} + 12 q^{55} - 2 q^{61} + 4 q^{67} + 22 q^{73} + 28 q^{79} + 6 q^{85} + 10 q^{91} + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 + 10 * q^13 + 4 * q^19 - 4 * q^25 + 4 * q^31 - 14 * q^37 + 16 * q^43 + 2 * q^49 + 12 * q^55 - 2 * q^61 + 4 * q^67 + 22 * q^73 + 28 * q^79 + 6 * q^85 + 10 * q^91 + 4 * q^97

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