LMFDB - Embedded newform 8100.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8100,2,Mod(1,8100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8100, 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("8100.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8100.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:	$64.6788256372$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 180)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.08613$ of defining polynomial
Character	$\chi$	$=$	8100.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73419 q^{7} +O(q^{10})$	$q-2.73419 q^{7} +5.52420 q^{11} +3.52420 q^{13} +5.52420 q^{17} +7.52420 q^{19} -0.734191 q^{23} -4.46838 q^{29} +7.52420 q^{31} -6.05582 q^{37} -1.05582 q^{41} +3.52420 q^{43} -1.20999 q^{47} +0.475800 q^{49} -1.46838 q^{53} -1.46838 q^{59} +9.05582 q^{61} +4.25839 q^{67} +10.0558 q^{71} -8.00000 q^{73} -15.1042 q^{77} +2.00000 q^{79} +5.26581 q^{83} -3.00000 q^{89} -9.63583 q^{91} -9.46838 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{7}+O(q^{10}) $ 3 * q - 3 * q^7	$ 3 q - 3 q^{7} - 6 q^{13} + 6 q^{19} + 3 q^{23} - 3 q^{29} + 6 q^{31} - 12 q^{37} + 3 q^{41} - 6 q^{43} - 15 q^{47} + 18 q^{49} + 6 q^{53} + 6 q^{59} + 21 q^{61} - 9 q^{67} + 24 q^{71} - 24 q^{73} - 6 q^{77} + 6 q^{79} + 21 q^{83} - 9 q^{89} - 18 q^{97}+O(q^{100}) $ 3 * q - 3 * q^7 - 6 * q^13 + 6 * q^19 + 3 * q^23 - 3 * q^29 + 6 * q^31 - 12 * q^37 + 3 * q^41 - 6 * q^43 - 15 * q^47 + 18 * q^49 + 6 * q^53 + 6 * q^59 + 21 * q^61 - 9 * q^67 + 24 * q^71 - 24 * q^73 - 6 * q^77 + 6 * q^79 + 21 * q^83 - 9 * q^89 - 18 * 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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.73419	−1.03343	−0.516714	−	0.856158i	$-0.672845\pi$
$7$	−2.73419	−1.03343	−0.516714	+	0.856158i	$0.672845\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.52420	1.66561	0.832804	−	0.553567i	$-0.186734\pi$
$11$	5.52420	1.66561	0.832804	+	0.553567i	$0.186734\pi$
$12$	0	0
$13$	3.52420	0.977437	0.488719	−	0.872442i	$-0.337464\pi$
$13$	3.52420	0.977437	0.488719	+	0.872442i	$0.337464\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.52420	1.33982	0.669908	−	0.742444i	$-0.266334\pi$
$17$	5.52420	1.33982	0.669908	+	0.742444i	$0.266334\pi$
$18$	0	0
$19$	7.52420	1.72617	0.863085	−	0.505059i	$-0.168529\pi$
$19$	7.52420	1.72617	0.863085	+	0.505059i	$0.168529\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.734191	−0.153089	−0.0765447	−	0.997066i	$-0.524389\pi$
$23$	−0.734191	−0.153089	−0.0765447	+	0.997066i	$0.524389\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.46838	−0.829758	−0.414879	−	0.909877i	$-0.636176\pi$
$29$	−4.46838	−0.829758	−0.414879	+	0.909877i	$0.636176\pi$
$30$	0	0
$31$	7.52420	1.35139	0.675693	−	0.737183i	$-0.263844\pi$
$31$	7.52420	1.35139	0.675693	+	0.737183i	$0.263844\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.05582	−0.995570	−0.497785	−	0.867300i	$-0.665853\pi$
$37$	−6.05582	−0.995570	−0.497785	+	0.867300i	$0.665853\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.05582	−0.164891	−0.0824455	−	0.996596i	$-0.526273\pi$
$41$	−1.05582	−0.164891	−0.0824455	+	0.996596i	$0.526273\pi$
$42$	0	0
$43$	3.52420	0.537435	0.268718	−	0.963219i	$-0.413400\pi$
$43$	3.52420	0.537435	0.268718	+	0.963219i	$0.413400\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.20999	−0.176495	−0.0882477	−	0.996099i	$-0.528127\pi$
$47$	−1.20999	−0.176495	−0.0882477	+	0.996099i	$0.528127\pi$
$48$	0	0
$49$	0.475800	0.0679715
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.46838	−0.201698	−0.100849	−	0.994902i	$-0.532156\pi$
$53$	−1.46838	−0.201698	−0.100849	+	0.994902i	$0.532156\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.46838	−0.191167	−0.0955835	−	0.995421i	$-0.530472\pi$
$59$	−1.46838	−0.191167	−0.0955835	+	0.995421i	$0.530472\pi$
$60$	0	0
$61$	9.05582	1.15948	0.579739	−	0.814802i	$-0.303154\pi$
$61$	9.05582	1.15948	0.579739	+	0.814802i	$0.303154\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.25839	0.520245	0.260123	−	0.965576i	$-0.416237\pi$
$67$	4.25839	0.520245	0.260123	+	0.965576i	$0.416237\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.0558	1.19341	0.596703	−	0.802462i	$-0.296477\pi$
$71$	10.0558	1.19341	0.596703	+	0.802462i	$0.296477\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−15.1042	−1.72129
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.26581	0.577998	0.288999	−	0.957329i	$-0.406678\pi$
$83$	5.26581	0.577998	0.288999	+	0.957329i	$0.406678\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	−9.63583	−1.01011
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−9.46838	−0.961369	−0.480684	−	0.876894i	$-0.659612\pi$
$97$	−9.46838	−0.961369	−0.480684	+	0.876894i	$0.659612\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	13.4684	1.34015	0.670077	−	0.742292i	$-0.266261\pi$
$101$	13.4684	1.34015	0.670077	+	0.742292i	$0.266261\pi$
$102$	0	0
$103$	−18.5726	−1.83001	−0.915006	−	0.403440i	$-0.867815\pi$
$103$	−18.5726	−1.83001	−0.915006	+	0.403440i	$0.867815\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−19.2510	−1.86106	−0.930531	−	0.366213i	$-0.880654\pi$
$107$	−19.2510	−1.86106	−0.930531	+	0.366213i	$0.880654\pi$
$108$	0	0
$109$	−0.524200	−0.0502092	−0.0251046	−	0.999685i	$-0.507992\pi$
$109$	−0.524200	−0.0502092	−0.0251046	+	0.999685i	$0.507992\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.46838	0.702566	0.351283	−	0.936269i	$-0.385745\pi$
$113$	7.46838	0.702566	0.351283	+	0.936269i	$0.385745\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−15.1042	−1.38460
$120$	0	0
$121$	19.5168	1.77425
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−8.25839	−0.732814	−0.366407	−	0.930455i	$-0.619412\pi$
$127$	−8.25839	−0.732814	−0.366407	+	0.930455i	$0.619412\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−16.0968	−1.40638	−0.703192	−	0.711000i	$-0.748243\pi$
$131$	−16.0968	−1.40638	−0.703192	+	0.711000i	$0.748243\pi$
$132$	0	0
$133$	−20.5726	−1.78387
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.5168	1.58200	0.790998	−	0.611819i	$-0.209562\pi$
$137$	18.5168	1.58200	0.790998	+	0.611819i	$0.209562\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	19.4684	1.62803
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.5726	1.11191	0.555955	−	0.831212i	$-0.312353\pi$
$149$	13.5726	1.11191	0.555955	+	0.831212i	$0.312353\pi$
$150$	0	0
$151$	−12.4610	−1.01406	−0.507029	−	0.861929i	$-0.669256\pi$
$151$	−12.4610	−1.01406	−0.507029	+	0.861929i	$0.669256\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.58002	0.126099	0.0630496	−	0.998010i	$-0.479917\pi$
$157$	1.58002	0.126099	0.0630496	+	0.998010i	$0.479917\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.00742	0.158207
$162$	0	0
$163$	4.47580	0.350572	0.175286	−	0.984518i	$-0.443915\pi$
$163$	4.47580	0.350572	0.175286	+	0.984518i	$0.443915\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.7900	−1.29925	−0.649625	−	0.760255i	$-0.725074\pi$
$167$	−16.7900	−1.29925	−0.649625	+	0.760255i	$0.725074\pi$
$168$	0	0
$169$	−0.580017	−0.0446167
$170$	0	0
$171$	0	0
$172$	0	0
$173$	23.0484	1.75234	0.876169	−	0.482005i	$-0.160091\pi$
$173$	23.0484	1.75234	0.876169	+	0.482005i	$0.160091\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.9926	0.971111	0.485556	−	0.874206i	$-0.338617\pi$
$179$	12.9926	0.971111	0.485556	+	0.874206i	$0.338617\pi$
$180$	0	0
$181$	−24.5652	−1.82592	−0.912958	−	0.408054i	$-0.866207\pi$
$181$	−24.5652	−1.82592	−0.912958	+	0.408054i	$0.866207\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	30.5168	2.23161
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.1042	1.09290	0.546451	−	0.837491i	$-0.315978\pi$
$191$	15.1042	1.09290	0.546451	+	0.837491i	$0.315978\pi$
$192$	0	0
$193$	14.5726	1.04896	0.524479	−	0.851423i	$-0.324260\pi$
$193$	14.5726	1.04896	0.524479	+	0.851423i	$0.324260\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.53162	0.322864	0.161432	−	0.986884i	$-0.448389\pi$
$197$	4.53162	0.322864	0.161432	+	0.986884i	$0.448389\pi$
$198$	0	0
$199$	−7.41256	−0.525463	−0.262731	−	0.964869i	$-0.584623\pi$
$199$	−7.41256	−0.525463	−0.262731	+	0.964869i	$0.584623\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	12.2174	0.857494
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	41.5652	2.87512
$210$	0	0
$211$	−13.4126	−0.923359	−0.461680	−	0.887047i	$-0.652753\pi$
$211$	−13.4126	−0.923359	−0.461680	+	0.887047i	$0.652753\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−20.5726	−1.39656
$218$	0	0
$219$	0	0
$220$	0	0
$221$	19.4684	1.30959
$222$	0	0
$223$	8.79001	0.588623	0.294311	−	0.955710i	$-0.404910\pi$
$223$	8.79001	0.588623	0.294311	+	0.955710i	$0.404910\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.05582	−0.269194	−0.134597	−	0.990900i	$-0.542974\pi$
$227$	−4.05582	−0.269194	−0.134597	+	0.990900i	$0.542974\pi$
$228$	0	0
$229$	2.58002	0.170492	0.0852462	−	0.996360i	$-0.472832\pi$
$229$	2.58002	0.170492	0.0852462	+	0.996360i	$0.472832\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.94418	0.127368	0.0636838	−	0.997970i	$-0.479715\pi$
$233$	1.94418	0.127368	0.0636838	+	0.997970i	$0.479715\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	7.46838	0.483089	0.241545	−	0.970390i	$-0.422346\pi$
$239$	7.46838	0.483089	0.241545	+	0.970390i	$0.422346\pi$
$240$	0	0
$241$	2.41256	0.155407	0.0777035	−	0.996977i	$-0.475241\pi$
$241$	2.41256	0.155407	0.0777035	+	0.996977i	$0.475241\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	26.5168	1.68722
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.99258	−0.441368	−0.220684	−	0.975345i	$-0.570829\pi$
$251$	−6.99258	−0.441368	−0.220684	+	0.975345i	$0.570829\pi$
$252$	0	0
$253$	−4.05582	−0.254987
$254$	0	0
$255$	0	0
$256$	0	0
$257$	28.0968	1.75263	0.876315	−	0.481738i	$-0.159994\pi$
$257$	28.0968	1.75263	0.876315	+	0.481738i	$0.159994\pi$
$258$	0	0
$259$	16.5578	1.02885
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.63583	−0.470846	−0.235423	−	0.971893i	$-0.575648\pi$
$263$	−7.63583	−0.470846	−0.235423	+	0.971893i	$0.575648\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−18.6210	−1.13534	−0.567671	−	0.823255i	$-0.692155\pi$
$269$	−18.6210	−1.13534	−0.567671	+	0.823255i	$0.692155\pi$
$270$	0	0
$271$	6.57260	0.399257	0.199628	−	0.979872i	$-0.436026\pi$
$271$	6.57260	0.399257	0.199628	+	0.979872i	$0.436026\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.52420	−0.0915803	−0.0457901	−	0.998951i	$-0.514581\pi$
$277$	−1.52420	−0.0915803	−0.0457901	+	0.998951i	$0.514581\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	14.5242	0.866441	0.433221	−	0.901288i	$-0.357377\pi$
$281$	14.5242	0.866441	0.433221	+	0.901288i	$0.357377\pi$
$282$	0	0
$283$	16.7342	0.994744	0.497372	−	0.867537i	$-0.334298\pi$
$283$	16.7342	0.994744	0.497372	+	0.867537i	$0.334298\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.88681	0.170403
$288$	0	0
$289$	13.5168	0.795105
$290$	0	0
$291$	0	0
$292$	0	0
$293$	20.4610	1.19534	0.597671	−	0.801741i	$-0.296093\pi$
$293$	20.4610	1.19534	0.597671	+	0.801741i	$0.296093\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.58744	−0.149635
$300$	0	0
$301$	−9.63583	−0.555400
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	18.2026	1.03888	0.519438	−	0.854508i	$-0.326141\pi$
$307$	18.2026	1.03888	0.519438	+	0.854508i	$0.326141\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.10422	0.176024	0.0880120	−	0.996119i	$-0.471949\pi$
$311$	3.10422	0.176024	0.0880120	+	0.996119i	$0.471949\pi$
$312$	0	0
$313$	19.1042	1.07983	0.539917	−	0.841718i	$-0.318456\pi$
$313$	19.1042	1.07983	0.539917	+	0.841718i	$0.318456\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.00742	−0.281245	−0.140622	−	0.990063i	$-0.544910\pi$
$317$	−5.00742	−0.281245	−0.140622	+	0.990063i	$0.544910\pi$
$318$	0	0
$319$	−24.6842	−1.38205
$320$	0	0
$321$	0	0
$322$	0	0
$323$	41.5652	2.31275
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.30835	0.182395
$330$	0	0
$331$	−3.48322	−0.191455	−0.0957275	−	0.995408i	$-0.530518\pi$
$331$	−3.48322	−0.191455	−0.0957275	+	0.995408i	$0.530518\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−11.5800	−0.630804	−0.315402	−	0.948958i	$-0.602139\pi$
$337$	−11.5800	−0.630804	−0.315402	+	0.948958i	$0.602139\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	41.5652	2.25088
$342$	0	0
$343$	17.8384	0.963183
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−26.4610	−1.42050	−0.710249	−	0.703950i	$-0.751418\pi$
$347$	−26.4610	−1.42050	−0.710249	+	0.703950i	$0.751418\pi$
$348$	0	0
$349$	−26.0336	−1.39354	−0.696772	−	0.717292i	$-0.745381\pi$
$349$	−26.0336	−1.39354	−0.696772	+	0.717292i	$0.745381\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.04098	0.318831	0.159415	−	0.987212i	$-0.449039\pi$
$359$	6.04098	0.318831	0.159415	+	0.987212i	$0.449039\pi$
$360$	0	0
$361$	37.6136	1.97966
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−22.4610	−1.17245	−0.586226	−	0.810147i	$-0.699387\pi$
$367$	−22.4610	−1.17245	−0.586226	+	0.810147i	$0.699387\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4.01484	0.208440
$372$	0	0
$373$	2.53162	0.131082	0.0655411	−	0.997850i	$-0.479123\pi$
$373$	2.53162	0.131082	0.0655411	+	0.997850i	$0.479123\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−15.7475	−0.811036
$378$	0	0
$379$	−21.0484	−1.08118	−0.540592	−	0.841285i	$-0.681800\pi$
$379$	−21.0484	−1.08118	−0.540592	+	0.841285i	$0.681800\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7.63583	0.390173	0.195086	−	0.980786i	$-0.437501\pi$
$383$	7.63583	0.390173	0.195086	+	0.980786i	$0.437501\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−18.9293	−0.959756	−0.479878	−	0.877335i	$-0.659319\pi$
$389$	−18.9293	−0.959756	−0.479878	+	0.877335i	$0.659319\pi$
$390$	0	0
$391$	−4.05582	−0.205112
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	29.1600	1.46350	0.731750	−	0.681573i	$-0.238704\pi$
$397$	29.1600	1.46350	0.731750	+	0.681573i	$0.238704\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	32.6284	1.62939	0.814693	−	0.579893i	$-0.196906\pi$
$401$	32.6284	1.62939	0.814693	+	0.579893i	$0.196906\pi$
$402$	0	0
$403$	26.5168	1.32089
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−33.4535	−1.65823
$408$	0	0
$409$	32.2084	1.59260	0.796302	−	0.604899i	$-0.206786\pi$
$409$	32.2084	1.59260	0.796302	+	0.604899i	$0.206786\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.01484	0.197557
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−21.4126	−1.04607	−0.523036	−	0.852311i	$-0.675201\pi$
$419$	−21.4126	−1.04607	−0.523036	+	0.852311i	$0.675201\pi$
$420$	0	0
$421$	−1.88836	−0.0920333	−0.0460166	−	0.998941i	$-0.514653\pi$
$421$	−1.88836	−0.0920333	−0.0460166	+	0.998941i	$0.514653\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−24.7603	−1.19824
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.11164	−0.101714	−0.0508569	−	0.998706i	$-0.516195\pi$
$431$	−2.11164	−0.101714	−0.0508569	+	0.998706i	$0.516195\pi$
$432$	0	0
$433$	9.52420	0.457704	0.228852	−	0.973461i	$-0.426503\pi$
$433$	9.52420	0.457704	0.228852	+	0.973461i	$0.426503\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.52420	−0.264258
$438$	0	0
$439$	−14.4051	−0.687520	−0.343760	−	0.939058i	$-0.611701\pi$
$439$	−14.4051	−0.687520	−0.343760	+	0.939058i	$0.611701\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	19.2100	0.912694	0.456347	−	0.889802i	$-0.349158\pi$
$443$	19.2100	0.912694	0.456347	+	0.889802i	$0.349158\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	33.5800	1.58474	0.792369	−	0.610041i	$-0.208847\pi$
$449$	33.5800	1.58474	0.792369	+	0.610041i	$0.208847\pi$
$450$	0	0
$451$	−5.83255	−0.274644
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2.16745	−0.101389	−0.0506946	−	0.998714i	$-0.516144\pi$
$457$	−2.16745	−0.101389	−0.0506946	+	0.998714i	$0.516144\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−25.7401	−1.19883	−0.599417	−	0.800437i	$-0.704601\pi$
$461$	−25.7401	−1.19883	−0.599417	+	0.800437i	$0.704601\pi$
$462$	0	0
$463$	−12.0558	−0.560281	−0.280141	−	0.959959i	$-0.590381\pi$
$463$	−12.0558	−0.560281	−0.280141	+	0.959959i	$0.590381\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.4610	0.669174	0.334587	−	0.942365i	$-0.391403\pi$
$467$	14.4610	0.669174	0.334587	+	0.942365i	$0.391403\pi$
$468$	0	0
$469$	−11.6433	−0.537635
$470$	0	0
$471$	0	0
$472$	0	0
$473$	19.4684	0.895157
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−19.9852	−0.913145	−0.456573	−	0.889686i	$-0.650923\pi$
$479$	−19.9852	−0.913145	−0.456573	+	0.889686i	$0.650923\pi$
$480$	0	0
$481$	−21.3419	−0.973107
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	18.1526	0.822574	0.411287	−	0.911506i	$-0.365079\pi$
$487$	18.1526	0.822574	0.411287	+	0.911506i	$0.365079\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	39.5800	1.78622	0.893111	−	0.449837i	$-0.148518\pi$
$491$	39.5800	1.78622	0.893111	+	0.449837i	$0.148518\pi$
$492$	0	0
$493$	−24.6842	−1.11172
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−27.4945	−1.23330
$498$	0	0
$499$	−42.6284	−1.90831	−0.954155	−	0.299313i	$-0.903243\pi$
$499$	−42.6284	−1.90831	−0.954155	+	0.299313i	$0.903243\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−21.1952	−0.945045	−0.472523	−	0.881319i	$-0.656657\pi$
$503$	−21.1952	−0.945045	−0.472523	+	0.881319i	$0.656657\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−22.0336	−0.976620	−0.488310	−	0.872670i	$-0.662387\pi$
$509$	−22.0336	−0.976620	−0.488310	+	0.872670i	$0.662387\pi$
$510$	0	0
$511$	21.8735	0.967628
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−6.68423	−0.293972
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.40515	0.0615606	0.0307803	−	0.999526i	$-0.490201\pi$
$521$	1.40515	0.0615606	0.0307803	+	0.999526i	$0.490201\pi$
$522$	0	0
$523$	11.8532	0.518306	0.259153	−	0.965836i	$-0.416557\pi$
$523$	11.8532	0.518306	0.259153	+	0.965836i	$0.416557\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	41.5652	1.81061
$528$	0	0
$529$	−22.4610	−0.976564
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3.72091	−0.161171
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.62842	0.113214
$540$	0	0
$541$	−8.98516	−0.386302	−0.193151	−	0.981169i	$-0.561871\pi$
$541$	−8.98516	−0.386302	−0.193151	+	0.981169i	$0.561871\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	6.20257	0.265203	0.132601	−	0.991169i	$-0.457667\pi$
$547$	6.20257	0.265203	0.132601	+	0.991169i	$0.457667\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−33.6210	−1.43230
$552$	0	0
$553$	−5.46838	−0.232539
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	0	0
$559$	12.4200	0.525309
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.27323	0.180095	0.0900475	−	0.995937i	$-0.471298\pi$
$563$	4.27323	0.180095	0.0900475	+	0.995937i	$0.471298\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	16.0968	0.674813	0.337406	−	0.941359i	$-0.390450\pi$
$569$	16.0968	0.674813	0.337406	+	0.941359i	$0.390450\pi$
$570$	0	0
$571$	10.9368	0.457689	0.228845	−	0.973463i	$-0.426505\pi$
$571$	10.9368	0.457689	0.228845	+	0.973463i	$0.426505\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	37.6620	1.56789	0.783944	−	0.620831i	$-0.213205\pi$
$577$	37.6620	1.56789	0.783944	+	0.620831i	$0.213205\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−14.3977	−0.597318
$582$	0	0
$583$	−8.11164	−0.335950
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−21.7974	−0.899676	−0.449838	−	0.893110i	$-0.648518\pi$
$587$	−21.7974	−0.899676	−0.449838	+	0.893110i	$0.648518\pi$
$588$	0	0
$589$	56.6136	2.33272
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−28.2643	−1.16067	−0.580337	−	0.814377i	$-0.697079\pi$
$593$	−28.2643	−1.16067	−0.580337	+	0.814377i	$0.697079\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	14.1116	0.576586	0.288293	−	0.957542i	$-0.406912\pi$
$599$	14.1116	0.576586	0.288293	+	0.957542i	$0.406912\pi$
$600$	0	0
$601$	−28.2084	−1.15065	−0.575323	−	0.817926i	$-0.695124\pi$
$601$	−28.2084	−1.15065	−0.575323	+	0.817926i	$0.695124\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−28.3700	−1.15150	−0.575752	−	0.817624i	$-0.695291\pi$
$607$	−28.3700	−1.15150	−0.575752	+	0.817624i	$0.695291\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.26425	−0.172513
$612$	0	0
$613$	−24.5726	−0.992478	−0.496239	−	0.868186i	$-0.665286\pi$
$613$	−24.5726	−0.992478	−0.496239	+	0.868186i	$0.665286\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26.6284	1.07202	0.536010	−	0.844212i	$-0.319931\pi$
$617$	26.6284	1.07202	0.536010	+	0.844212i	$0.319931\pi$
$618$	0	0
$619$	30.5726	1.22882	0.614408	−	0.788988i	$-0.289395\pi$
$619$	30.5726	1.22882	0.614408	+	0.788988i	$0.289395\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.20257	0.328629
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−33.4535	−1.33388
$630$	0	0
$631$	27.4684	1.09350	0.546750	−	0.837296i	$-0.315865\pi$
$631$	27.4684	1.09350	0.546750	+	0.837296i	$0.315865\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.67682	0.0664379
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.5390	0.495262	0.247631	−	0.968854i	$-0.420348\pi$
$641$	12.5390	0.495262	0.247631	+	0.968854i	$0.420348\pi$
$642$	0	0
$643$	10.2584	0.404551	0.202276	−	0.979329i	$-0.435166\pi$
$643$	10.2584	0.404551	0.202276	+	0.979329i	$0.435166\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16.7900	−0.660083	−0.330042	−	0.943966i	$-0.607063\pi$
$647$	−16.7900	−0.660083	−0.330042	+	0.943966i	$0.607063\pi$
$648$	0	0
$649$	−8.11164	−0.318410
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−50.0261	−1.95767	−0.978837	−	0.204641i	$-0.934397\pi$
$653$	−50.0261	−1.95767	−0.978837	+	0.204641i	$0.934397\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5.56518	0.216789	0.108394	−	0.994108i	$-0.465429\pi$
$659$	5.56518	0.216789	0.108394	+	0.994108i	$0.465429\pi$
$660$	0	0
$661$	−39.2568	−1.52691	−0.763457	−	0.645859i	$-0.776499\pi$
$661$	−39.2568	−1.52691	−0.763457	+	0.645859i	$0.776499\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.28065	0.127027
$668$	0	0
$669$	0	0
$670$	0	0
$671$	50.0261	1.93124
$672$	0	0
$673$	−15.9442	−0.614603	−0.307302	−	0.951612i	$-0.599426\pi$
$673$	−15.9442	−0.614603	−0.307302	+	0.951612i	$0.599426\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	20.8958	0.803090	0.401545	−	0.915839i	$-0.368473\pi$
$677$	20.8958	0.803090	0.401545	+	0.915839i	$0.368473\pi$
$678$	0	0
$679$	25.8884	0.993504
$680$	0	0
$681$	0	0
$682$	0	0
$683$	19.6358	0.751344	0.375672	−	0.926753i	$-0.377412\pi$
$683$	19.6358	0.751344	0.375672	+	0.926753i	$0.377412\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5.17487	−0.197147
$690$	0	0
$691$	−43.1452	−1.64132	−0.820660	−	0.571416i	$-0.806394\pi$
$691$	−43.1452	−1.64132	−0.820660	+	0.571416i	$0.806394\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−5.83255	−0.220923
$698$	0	0
$699$	0	0
$700$	0	0
$701$	27.0410	1.02132	0.510662	−	0.859782i	$-0.329400\pi$
$701$	27.0410	1.02132	0.510662	+	0.859782i	$0.329400\pi$
$702$	0	0
$703$	−45.5652	−1.71852
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−36.8251	−1.38495
$708$	0	0
$709$	43.5019	1.63375	0.816875	−	0.576815i	$-0.195705\pi$
$709$	43.5019	1.63375	0.816875	+	0.576815i	$0.195705\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5.52420	−0.206883
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−4.40515	−0.164284	−0.0821421	−	0.996621i	$-0.526176\pi$
$719$	−4.40515	−0.164284	−0.0821421	+	0.996621i	$0.526176\pi$
$720$	0	0
$721$	50.7810	1.89118
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−12.4817	−0.462919	−0.231460	−	0.972845i	$-0.574350\pi$
$727$	−12.4817	−0.462919	−0.231460	+	0.972845i	$0.574350\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	19.4684	0.720064
$732$	0	0
$733$	−39.0336	−1.44174	−0.720869	−	0.693072i	$-0.756257\pi$
$733$	−39.0336	−1.44174	−0.720869	+	0.693072i	$0.756257\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	23.5242	0.866525
$738$	0	0
$739$	−12.2935	−0.452224	−0.226112	−	0.974101i	$-0.572602\pi$
$739$	−12.2935	−0.452224	−0.226112	+	0.974101i	$0.572602\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	40.8310	1.49794	0.748972	−	0.662602i	$-0.230548\pi$
$743$	40.8310	1.49794	0.748972	+	0.662602i	$0.230548\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	52.6358	1.92327
$750$	0	0
$751$	−15.0894	−0.550619	−0.275310	−	0.961356i	$-0.588780\pi$
$751$	−15.0894	−0.550619	−0.275310	+	0.961356i	$0.588780\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−34.4610	−1.25251	−0.626253	−	0.779620i	$-0.715412\pi$
$757$	−34.4610	−1.25251	−0.626253	+	0.779620i	$0.715412\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−8.35675	−0.302932	−0.151466	−	0.988462i	$-0.548399\pi$
$761$	−8.35675	−0.302932	−0.151466	+	0.988462i	$0.548399\pi$
$762$	0	0
$763$	1.43326	0.0518876
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.17487	−0.186854
$768$	0	0
$769$	−18.3567	−0.661961	−0.330981	−	0.943638i	$-0.607379\pi$
$769$	−18.3567	−0.661961	−0.330981	+	0.943638i	$0.607379\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	10.0968	0.363157	0.181578	−	0.983376i	$-0.441879\pi$
$773$	10.0968	0.363157	0.181578	+	0.983376i	$0.441879\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.94418	−0.284630
$780$	0	0
$781$	55.5503	1.98775
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	14.5726	0.519457	0.259729	−	0.965682i	$-0.416367\pi$
$787$	14.5726	0.519457	0.259729	+	0.965682i	$0.416367\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−20.4200	−0.726051
$792$	0	0
$793$	31.9145	1.13332
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.07065	0.0733463	0.0366732	−	0.999327i	$-0.488324\pi$
$797$	2.07065	0.0733463	0.0366732	+	0.999327i	$0.488324\pi$
$798$	0	0
$799$	−6.68423	−0.236471
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−44.1936	−1.55956
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.37158	0.118539	0.0592693	−	0.998242i	$-0.481123\pi$
$809$	3.37158	0.118539	0.0592693	+	0.998242i	$0.481123\pi$
$810$	0	0
$811$	−12.9368	−0.454271	−0.227136	−	0.973863i	$-0.572936\pi$
$811$	−12.9368	−0.454271	−0.227136	+	0.973863i	$0.572936\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	26.5168	0.927705
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−21.9926	−0.767546	−0.383773	−	0.923427i	$-0.625376\pi$
$821$	−21.9926	−0.767546	−0.383773	+	0.923427i	$0.625376\pi$
$822$	0	0
$823$	−14.9016	−0.519439	−0.259719	−	0.965684i	$-0.583630\pi$
$823$	−14.9016	−0.519439	−0.259719	+	0.965684i	$0.583630\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−56.5785	−1.96743	−0.983713	−	0.179747i	$-0.942472\pi$
$827$	−56.5785	−1.96743	−0.983713	+	0.179747i	$0.942472\pi$
$828$	0	0
$829$	27.0558	0.939687	0.469844	−	0.882750i	$-0.344310\pi$
$829$	27.0558	0.939687	0.469844	+	0.882750i	$0.344310\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.62842	0.0910692
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−34.5726	−1.19358	−0.596789	−	0.802398i	$-0.703557\pi$
$839$	−34.5726	−1.19358	−0.596789	+	0.802398i	$0.703557\pi$
$840$	0	0
$841$	−9.03356	−0.311502
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−53.3626	−1.83356
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.44613	0.152411
$852$	0	0
$853$	−16.9777	−0.581307	−0.290653	−	0.956828i	$-0.593873\pi$
$853$	−16.9777	−0.581307	−0.290653	+	0.956828i	$0.593873\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.41256	−0.116571	−0.0582855	−	0.998300i	$-0.518563\pi$
$857$	−3.41256	−0.116571	−0.0582855	+	0.998300i	$0.518563\pi$
$858$	0	0
$859$	−9.83255	−0.335482	−0.167741	−	0.985831i	$-0.553647\pi$
$859$	−9.83255	−0.335482	−0.167741	+	0.985831i	$0.553647\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	48.9836	1.66742	0.833711	−	0.552202i	$-0.186212\pi$
$863$	48.9836	1.66742	0.833711	+	0.552202i	$0.186212\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	11.0484	0.374791
$870$	0	0
$871$	15.0074	0.508507
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	12.6694	0.427815	0.213908	−	0.976854i	$-0.431381\pi$
$877$	12.6694	0.427815	0.213908	+	0.976854i	$0.431381\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−26.0894	−0.878974	−0.439487	−	0.898249i	$-0.644840\pi$
$881$	−26.0894	−0.878974	−0.439487	+	0.898249i	$0.644840\pi$
$882$	0	0
$883$	−2.69321	−0.0906337	−0.0453169	−	0.998973i	$-0.514430\pi$
$883$	−2.69321	−0.0906337	−0.0453169	+	0.998973i	$0.514430\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.5578	0.824569	0.412284	−	0.911055i	$-0.364731\pi$
$887$	24.5578	0.824569	0.412284	+	0.911055i	$0.364731\pi$
$888$	0	0
$889$	22.5800	0.757309
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9.10422	−0.304661
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−33.6210	−1.12132
$900$	0	0
$901$	−8.11164	−0.270238
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−34.3700	−1.14124	−0.570619	−	0.821215i	$-0.693297\pi$
$907$	−34.3700	−1.14124	−0.570619	+	0.821215i	$0.693297\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	37.5503	1.24410	0.622049	−	0.782978i	$-0.286300\pi$
$911$	37.5503	1.24410	0.622049	+	0.782978i	$0.286300\pi$
$912$	0	0
$913$	29.0894	0.962718
$914$	0	0
$915$	0	0
$916$	0	0
$917$	44.0117	1.45340
$918$	0	0
$919$	10.1116	0.333552	0.166776	−	0.985995i	$-0.446664\pi$
$919$	10.1116	0.333552	0.166776	+	0.985995i	$0.446664\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	35.4387	1.16648
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	24.8251	0.814486	0.407243	−	0.913320i	$-0.366490\pi$
$929$	24.8251	0.814486	0.407243	+	0.913320i	$0.366490\pi$
$930$	0	0
$931$	3.58002	0.117330
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1.56518	−0.0511322	−0.0255661	−	0.999673i	$-0.508139\pi$
$937$	−1.56518	−0.0511322	−0.0255661	+	0.999673i	$0.508139\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−10.4684	−0.341259	−0.170630	−	0.985335i	$-0.554580\pi$
$941$	−10.4684	−0.341259	−0.170630	+	0.985335i	$0.554580\pi$
$942$	0	0
$943$	0.775172	0.0252431
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.19515	0.103829	0.0519143	−	0.998652i	$-0.483468\pi$
$947$	3.19515	0.103829	0.0519143	+	0.998652i	$0.483468\pi$
$948$	0	0
$949$	−28.1936	−0.915203
$950$	0	0
$951$	0	0
$952$	0	0
$953$	33.9293	1.09908	0.549540	−	0.835468i	$-0.314803\pi$
$953$	33.9293	1.09908	0.549540	+	0.835468i	$0.314803\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−50.6284	−1.63488
$960$	0	0
$961$	25.6136	0.826245
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−38.4258	−1.23569	−0.617846	−	0.786299i	$-0.711994\pi$
$967$	−38.4258	−1.23569	−0.617846	+	0.786299i	$0.711994\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	29.7326	0.954166	0.477083	−	0.878858i	$-0.341694\pi$
$971$	29.7326	0.954166	0.477083	+	0.878858i	$0.341694\pi$
$972$	0	0
$973$	10.9368	0.350617
$974$	0	0
$975$	0	0
$976$	0	0
$977$	32.9777	1.05505	0.527526	−	0.849539i	$-0.323120\pi$
$977$	32.9777	1.05505	0.527526	+	0.849539i	$0.323120\pi$
$978$	0	0
$979$	−16.5726	−0.529663
$980$	0	0
$981$	0	0
$982$	0	0
$983$	7.37744	0.235304	0.117652	−	0.993055i	$-0.462463\pi$
$983$	7.37744	0.235304	0.117652	+	0.993055i	$0.462463\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.58744	−0.0822757
$990$	0	0
$991$	−4.51678	−0.143480	−0.0717401	−	0.997423i	$-0.522855\pi$
$991$	−4.51678	−0.143480	−0.0717401	+	0.997423i	$0.522855\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	48.6284	1.54008	0.770039	−	0.637997i	$-0.220237\pi$
$997$	48.6284	1.54008	0.770039	+	0.637997i	$0.220237\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.a.v.1.2		3
3.2	odd	2		8100.2.a.u.1.2		3
5.2	odd	4		8100.2.d.p.649.3		6
5.3	odd	4		8100.2.d.p.649.4		6
5.4	even	2		1620.2.a.i.1.2		3
9.2	odd	6		2700.2.i.c.901.2		6
9.4	even	3		900.2.i.c.601.3		6
9.5	odd	6		2700.2.i.c.1801.2		6
9.7	even	3		900.2.i.c.301.3		6
15.2	even	4		8100.2.d.o.649.3		6
15.8	even	4		8100.2.d.o.649.4		6
15.14	odd	2		1620.2.a.j.1.2		3
20.19	odd	2		6480.2.a.bt.1.2		3
45.2	even	12		2700.2.s.c.1549.3		12
45.4	even	6		180.2.i.b.61.1	✓	6
45.7	odd	12		900.2.s.c.49.4		12
45.13	odd	12		900.2.s.c.349.4		12
45.14	odd	6		540.2.i.b.181.2		6
45.22	odd	12		900.2.s.c.349.3		12
45.23	even	12		2700.2.s.c.2449.3		12
45.29	odd	6		540.2.i.b.361.2		6
45.32	even	12		2700.2.s.c.2449.4		12
45.34	even	6		180.2.i.b.121.1	yes	6
45.38	even	12		2700.2.s.c.1549.4		12
45.43	odd	12		900.2.s.c.49.3		12
60.59	even	2		6480.2.a.bw.1.2		3
180.59	even	6		2160.2.q.i.721.2		6
180.79	odd	6		720.2.q.k.481.3		6
180.119	even	6		2160.2.q.i.1441.2		6
180.139	odd	6		720.2.q.k.241.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.i.b.61.1	✓	6	45.4	even	6
180.2.i.b.121.1	yes	6	45.34	even	6
540.2.i.b.181.2		6	45.14	odd	6
540.2.i.b.361.2		6	45.29	odd	6
720.2.q.k.241.3		6	180.139	odd	6
720.2.q.k.481.3		6	180.79	odd	6
900.2.i.c.301.3		6	9.7	even	3
900.2.i.c.601.3		6	9.4	even	3
900.2.s.c.49.3		12	45.43	odd	12
900.2.s.c.49.4		12	45.7	odd	12
900.2.s.c.349.3		12	45.22	odd	12
900.2.s.c.349.4		12	45.13	odd	12
1620.2.a.i.1.2		3	5.4	even	2
1620.2.a.j.1.2		3	15.14	odd	2
2160.2.q.i.721.2		6	180.59	even	6
2160.2.q.i.1441.2		6	180.119	even	6
2700.2.i.c.901.2		6	9.2	odd	6
2700.2.i.c.1801.2		6	9.5	odd	6
2700.2.s.c.1549.3		12	45.2	even	12
2700.2.s.c.1549.4		12	45.38	even	12
2700.2.s.c.2449.3		12	45.23	even	12
2700.2.s.c.2449.4		12	45.32	even	12
6480.2.a.bt.1.2		3	20.19	odd	2
6480.2.a.bw.1.2		3	60.59	even	2
8100.2.a.u.1.2		3	3.2	odd	2
8100.2.a.v.1.2		3	1.1	even	1	trivial
8100.2.d.o.649.3		6	15.2	even	4
8100.2.d.o.649.4		6	15.8	even	4
8100.2.d.p.649.3		6	5.2	odd	4
8100.2.d.p.649.4		6	5.3	odd	4

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:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73419 q^{7} +O(q^{10})\)	\(q-2.73419 q^{7} +5.52420 q^{11} +3.52420 q^{13} +5.52420 q^{17} +7.52420 q^{19} -0.734191 q^{23} -4.46838 q^{29} +7.52420 q^{31} -6.05582 q^{37} -1.05582 q^{41} +3.52420 q^{43} -1.20999 q^{47} +0.475800 q^{49} -1.46838 q^{53} -1.46838 q^{59} +9.05582 q^{61} +4.25839 q^{67} +10.0558 q^{71} -8.00000 q^{73} -15.1042 q^{77} +2.00000 q^{79} +5.26581 q^{83} -3.00000 q^{89} -9.63583 q^{91} -9.46838 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{7}+O(q^{10}) \) 3 * q - 3 * q^7	\( 3 q - 3 q^{7} - 6 q^{13} + 6 q^{19} + 3 q^{23} - 3 q^{29} + 6 q^{31} - 12 q^{37} + 3 q^{41} - 6 q^{43} - 15 q^{47} + 18 q^{49} + 6 q^{53} + 6 q^{59} + 21 q^{61} - 9 q^{67} + 24 q^{71} - 24 q^{73} - 6 q^{77} + 6 q^{79} + 21 q^{83} - 9 q^{89} - 18 q^{97}+O(q^{100}) \) 3 * q - 3 * q^7 - 6 * q^13 + 6 * q^19 + 3 * q^23 - 3 * q^29 + 6 * q^31 - 12 * q^37 + 3 * q^41 - 6 * q^43 - 15 * q^47 + 18 * q^49 + 6 * q^53 + 6 * q^59 + 21 * q^61 - 9 * q^67 + 24 * q^71 - 24 * q^73 - 6 * q^77 + 6 * q^79 + 21 * q^83 - 9 * q^89 - 18 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more