LMFDB - Embedded newform 7644.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7644,2,Mod(1,7644)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7644, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7644.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7644 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7644.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:	$61.0376473051$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1092)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	7644.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.37228 q^{5} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.37228 q^{5} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{13} +3.37228 q^{15} -0.627719 q^{19} -1.37228 q^{23} +6.37228 q^{25} -1.00000 q^{27} +1.37228 q^{29} +3.37228 q^{31} -2.00000 q^{33} -4.74456 q^{37} -1.00000 q^{39} +2.74456 q^{41} -6.11684 q^{43} -3.37228 q^{45} +0.627719 q^{47} +5.37228 q^{53} -6.74456 q^{55} +0.627719 q^{57} +8.00000 q^{59} -6.00000 q^{61} -3.37228 q^{65} +1.25544 q^{67} +1.37228 q^{69} -8.74456 q^{71} -14.8614 q^{73} -6.37228 q^{75} -6.11684 q^{79} +1.00000 q^{81} +6.11684 q^{83} -1.37228 q^{87} +7.37228 q^{89} -3.37228 q^{93} +2.11684 q^{95} -17.3723 q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - q^5 + 2 * q^9	$ 2 q - 2 q^{3} - q^{5} + 2 q^{9} + 4 q^{11} + 2 q^{13} + q^{15} - 7 q^{19} + 3 q^{23} + 7 q^{25} - 2 q^{27} - 3 q^{29} + q^{31} - 4 q^{33} + 2 q^{37} - 2 q^{39} - 6 q^{41} + 5 q^{43} - q^{45} + 7 q^{47} + 5 q^{53} - 2 q^{55} + 7 q^{57} + 16 q^{59} - 12 q^{61} - q^{65} + 14 q^{67} - 3 q^{69} - 6 q^{71} - q^{73} - 7 q^{75} + 5 q^{79} + 2 q^{81} - 5 q^{83} + 3 q^{87} + 9 q^{89} - q^{93} - 13 q^{95} - 29 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - q^5 + 2 * q^9 + 4 * q^11 + 2 * q^13 + q^15 - 7 * q^19 + 3 * q^23 + 7 * q^25 - 2 * q^27 - 3 * q^29 + q^31 - 4 * q^33 + 2 * q^37 - 2 * q^39 - 6 * q^41 + 5 * q^43 - q^45 + 7 * q^47 + 5 * q^53 - 2 * q^55 + 7 * q^57 + 16 * q^59 - 12 * q^61 - q^65 + 14 * q^67 - 3 * q^69 - 6 * q^71 - q^73 - 7 * q^75 + 5 * q^79 + 2 * q^81 - 5 * q^83 + 3 * q^87 + 9 * q^89 - q^93 - 13 * q^95 - 29 * q^97 + 4 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−3.37228	−1.50813	−0.754065	−	0.656800i	$-0.771910\pi$
$5$	−3.37228	−1.50813	−0.754065	+	0.656800i	$0.771910\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	3.37228	0.870719
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−0.627719	−0.144009	−0.0720043	−	0.997404i	$-0.522940\pi$
$19$	−0.627719	−0.144009	−0.0720043	+	0.997404i	$0.522940\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.37228	−0.286140	−0.143070	−	0.989713i	$-0.545697\pi$
$23$	−1.37228	−0.286140	−0.143070	+	0.989713i	$0.545697\pi$
$24$	0	0
$25$	6.37228	1.27446
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.37228	0.254826	0.127413	−	0.991850i	$-0.459333\pi$
$29$	1.37228	0.254826	0.127413	+	0.991850i	$0.459333\pi$
$30$	0	0
$31$	3.37228	0.605680	0.302840	−	0.953041i	$-0.402065\pi$
$31$	3.37228	0.605680	0.302840	+	0.953041i	$0.402065\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.74456	−0.780001	−0.390001	−	0.920815i	$-0.627525\pi$
$37$	−4.74456	−0.780001	−0.390001	+	0.920815i	$0.627525\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	2.74456	0.428629	0.214314	−	0.976765i	$-0.431248\pi$
$41$	2.74456	0.428629	0.214314	+	0.976765i	$0.431248\pi$
$42$	0	0
$43$	−6.11684	−0.932810	−0.466405	−	0.884571i	$-0.654451\pi$
$43$	−6.11684	−0.932810	−0.466405	+	0.884571i	$0.654451\pi$
$44$	0	0
$45$	−3.37228	−0.502710
$46$	0	0
$47$	0.627719	0.0915622	0.0457811	−	0.998951i	$-0.485422\pi$
$47$	0.627719	0.0915622	0.0457811	+	0.998951i	$0.485422\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.37228	0.737940	0.368970	−	0.929441i	$-0.379711\pi$
$53$	5.37228	0.737940	0.368970	+	0.929441i	$0.379711\pi$
$54$	0	0
$55$	−6.74456	−0.909437
$56$	0	0
$57$	0.627719	0.0831434
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.37228	−0.418280
$66$	0	0
$67$	1.25544	0.153376	0.0766880	−	0.997055i	$-0.475565\pi$
$67$	1.25544	0.153376	0.0766880	+	0.997055i	$0.475565\pi$
$68$	0	0
$69$	1.37228	0.165203
$70$	0	0
$71$	−8.74456	−1.03779	−0.518894	−	0.854838i	$-0.673656\pi$
$71$	−8.74456	−1.03779	−0.518894	+	0.854838i	$0.673656\pi$
$72$	0	0
$73$	−14.8614	−1.73940	−0.869698	−	0.493584i	$-0.835686\pi$
$73$	−14.8614	−1.73940	−0.869698	+	0.493584i	$0.835686\pi$
$74$	0	0
$75$	−6.37228	−0.735808
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−6.11684	−0.688199	−0.344099	−	0.938933i	$-0.611816\pi$
$79$	−6.11684	−0.688199	−0.344099	+	0.938933i	$0.611816\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.11684	0.671411	0.335705	−	0.941967i	$-0.391025\pi$
$83$	6.11684	0.671411	0.335705	+	0.941967i	$0.391025\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.37228	−0.147124
$88$	0	0
$89$	7.37228	0.781460	0.390730	−	0.920505i	$-0.372223\pi$
$89$	7.37228	0.781460	0.390730	+	0.920505i	$0.372223\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.37228	−0.349689
$94$	0	0
$95$	2.11684	0.217184
$96$	0	0
$97$	−17.3723	−1.76389	−0.881944	−	0.471354i	$-0.843765\pi$
$97$	−17.3723	−1.76389	−0.881944	+	0.471354i	$0.843765\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	5.25544	0.522936	0.261468	−	0.965212i	$-0.415793\pi$
$101$	5.25544	0.522936	0.261468	+	0.965212i	$0.415793\pi$
$102$	0	0
$103$	−5.48913	−0.540860	−0.270430	−	0.962740i	$-0.587166\pi$
$103$	−5.48913	−0.540860	−0.270430	+	0.962740i	$0.587166\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.744563	0.0719796	0.0359898	−	0.999352i	$-0.488542\pi$
$107$	0.744563	0.0719796	0.0359898	+	0.999352i	$0.488542\pi$
$108$	0	0
$109$	11.4891	1.10046	0.550229	−	0.835014i	$-0.314540\pi$
$109$	11.4891	1.10046	0.550229	+	0.835014i	$0.314540\pi$
$110$	0	0
$111$	4.74456	0.450334
$112$	0	0
$113$	12.1168	1.13986	0.569928	−	0.821694i	$-0.306971\pi$
$113$	12.1168	1.13986	0.569928	+	0.821694i	$0.306971\pi$
$114$	0	0
$115$	4.62772	0.431537
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−2.74456	−0.247469
$124$	0	0
$125$	−4.62772	−0.413916
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	6.11684	0.538558
$130$	0	0
$131$	−9.48913	−0.829069	−0.414534	−	0.910034i	$-0.636056\pi$
$131$	−9.48913	−0.829069	−0.414534	+	0.910034i	$0.636056\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.37228	0.290240
$136$	0	0
$137$	−22.2337	−1.89955	−0.949776	−	0.312930i	$-0.898689\pi$
$137$	−22.2337	−1.89955	−0.949776	+	0.312930i	$0.898689\pi$
$138$	0	0
$139$	−2.74456	−0.232791	−0.116395	−	0.993203i	$-0.537134\pi$
$139$	−2.74456	−0.232791	−0.116395	+	0.993203i	$0.537134\pi$
$140$	0	0
$141$	−0.627719	−0.0528634
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	−4.62772	−0.384311
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	21.4891	1.74876	0.874380	−	0.485242i	$-0.161268\pi$
$151$	21.4891	1.74876	0.874380	+	0.485242i	$0.161268\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−11.3723	−0.913444
$156$	0	0
$157$	8.74456	0.697892	0.348946	−	0.937143i	$-0.386540\pi$
$157$	8.74456	0.697892	0.348946	+	0.937143i	$0.386540\pi$
$158$	0	0
$159$	−5.37228	−0.426050
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−13.4891	−1.05655	−0.528275	−	0.849073i	$-0.677161\pi$
$163$	−13.4891	−1.05655	−0.528275	+	0.849073i	$0.677161\pi$
$164$	0	0
$165$	6.74456	0.525063
$166$	0	0
$167$	1.88316	0.145723	0.0728615	−	0.997342i	$-0.476787\pi$
$167$	1.88316	0.145723	0.0728615	+	0.997342i	$0.476787\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.627719	−0.0480028
$172$	0	0
$173$	22.9783	1.74700	0.873502	−	0.486821i	$-0.161843\pi$
$173$	22.9783	1.74700	0.873502	+	0.486821i	$0.161843\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	−18.8614	−1.40977	−0.704884	−	0.709323i	$-0.749001\pi$
$179$	−18.8614	−1.40977	−0.704884	+	0.709323i	$0.749001\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	16.0000	1.17634
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.25544	−0.235555	−0.117778	−	0.993040i	$-0.537577\pi$
$191$	−3.25544	−0.235555	−0.117778	+	0.993040i	$0.537577\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	3.37228	0.241494
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−1.25544	−0.0885517
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−9.25544	−0.646428
$206$	0	0
$207$	−1.37228	−0.0953801
$208$	0	0
$209$	−1.25544	−0.0868404
$210$	0	0
$211$	7.37228	0.507529	0.253764	−	0.967266i	$-0.418331\pi$
$211$	7.37228	0.507529	0.253764	+	0.967266i	$0.418331\pi$
$212$	0	0
$213$	8.74456	0.599168
$214$	0	0
$215$	20.6277	1.40680
$216$	0	0
$217$	0	0
$218$	0	0
$219$	14.8614	1.00424
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−16.8614	−1.12912	−0.564562	−	0.825391i	$-0.690955\pi$
$223$	−16.8614	−1.12912	−0.564562	+	0.825391i	$0.690955\pi$
$224$	0	0
$225$	6.37228	0.424819
$226$	0	0
$227$	29.4891	1.95726	0.978631	−	0.205624i	$-0.0659225\pi$
$227$	29.4891	1.95726	0.978631	+	0.205624i	$0.0659225\pi$
$228$	0	0
$229$	11.4891	0.759223	0.379611	−	0.925146i	$-0.376058\pi$
$229$	11.4891	0.759223	0.379611	+	0.925146i	$0.376058\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.37228	0.0899011	0.0449506	−	0.998989i	$-0.485687\pi$
$233$	1.37228	0.0899011	0.0449506	+	0.998989i	$0.485687\pi$
$234$	0	0
$235$	−2.11684	−0.138088
$236$	0	0
$237$	6.11684	0.397332
$238$	0	0
$239$	−15.2554	−0.986792	−0.493396	−	0.869805i	$-0.664245\pi$
$239$	−15.2554	−0.986792	−0.493396	+	0.869805i	$0.664245\pi$
$240$	0	0
$241$	−1.37228	−0.0883964	−0.0441982	−	0.999023i	$-0.514073\pi$
$241$	−1.37228	−0.0883964	−0.0441982	+	0.999023i	$0.514073\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.627719	−0.0399408
$248$	0	0
$249$	−6.11684	−0.387639
$250$	0	0
$251$	6.74456	0.425713	0.212857	−	0.977083i	$-0.431723\pi$
$251$	6.74456	0.425713	0.212857	+	0.977083i	$0.431723\pi$
$252$	0	0
$253$	−2.74456	−0.172549
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.23369	−0.513603	−0.256802	−	0.966464i	$-0.582669\pi$
$257$	−8.23369	−0.513603	−0.256802	+	0.966464i	$0.582669\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.37228	0.0849421
$262$	0	0
$263$	−10.6277	−0.655333	−0.327667	−	0.944793i	$-0.606262\pi$
$263$	−10.6277	−0.655333	−0.327667	+	0.944793i	$0.606262\pi$
$264$	0	0
$265$	−18.1168	−1.11291
$266$	0	0
$267$	−7.37228	−0.451176
$268$	0	0
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	26.9783	1.63881	0.819406	−	0.573214i	$-0.194303\pi$
$271$	26.9783	1.63881	0.819406	+	0.573214i	$0.194303\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.7446	0.768526
$276$	0	0
$277$	12.1168	0.728031	0.364015	−	0.931393i	$-0.381406\pi$
$277$	12.1168	0.728031	0.364015	+	0.931393i	$0.381406\pi$
$278$	0	0
$279$	3.37228	0.201893
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	0	0
$283$	25.4891	1.51517	0.757586	−	0.652736i	$-0.226379\pi$
$283$	25.4891	1.51517	0.757586	+	0.652736i	$0.226379\pi$
$284$	0	0
$285$	−2.11684	−0.125391
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	17.3723	1.01838
$292$	0	0
$293$	−20.8614	−1.21874	−0.609368	−	0.792887i	$-0.708577\pi$
$293$	−20.8614	−1.21874	−0.609368	+	0.792887i	$0.708577\pi$
$294$	0	0
$295$	−26.9783	−1.57073
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	−1.37228	−0.0793611
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−5.25544	−0.301917
$304$	0	0
$305$	20.2337	1.15858
$306$	0	0
$307$	−12.8614	−0.734039	−0.367020	−	0.930213i	$-0.619622\pi$
$307$	−12.8614	−0.734039	−0.367020	+	0.930213i	$0.619622\pi$
$308$	0	0
$309$	5.48913	0.312265
$310$	0	0
$311$	18.7446	1.06291	0.531453	−	0.847088i	$-0.321646\pi$
$311$	18.7446	1.06291	0.531453	+	0.847088i	$0.321646\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.25544	0.182844	0.0914218	−	0.995812i	$-0.470859\pi$
$317$	3.25544	0.182844	0.0914218	+	0.995812i	$0.470859\pi$
$318$	0	0
$319$	2.74456	0.153666
$320$	0	0
$321$	−0.744563	−0.0415574
$322$	0	0
$323$	0	0
$324$	0	0
$325$	6.37228	0.353471
$326$	0	0
$327$	−11.4891	−0.635350
$328$	0	0
$329$	0	0
$330$	0	0
$331$	24.2337	1.33200	0.666002	−	0.745950i	$-0.268004\pi$
$331$	24.2337	1.33200	0.666002	+	0.745950i	$0.268004\pi$
$332$	0	0
$333$	−4.74456	−0.260000
$334$	0	0
$335$	−4.23369	−0.231311
$336$	0	0
$337$	25.6060	1.39485	0.697423	−	0.716660i	$-0.254330\pi$
$337$	25.6060	1.39485	0.697423	+	0.716660i	$0.254330\pi$
$338$	0	0
$339$	−12.1168	−0.658097
$340$	0	0
$341$	6.74456	0.365239
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−4.62772	−0.249148
$346$	0	0
$347$	4.74456	0.254701	0.127351	−	0.991858i	$-0.459353\pi$
$347$	4.74456	0.254701	0.127351	+	0.991858i	$0.459353\pi$
$348$	0	0
$349$	25.6060	1.37066	0.685328	−	0.728234i	$-0.259659\pi$
$349$	25.6060	1.37066	0.685328	+	0.728234i	$0.259659\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	14.7446	0.784774	0.392387	−	0.919800i	$-0.371649\pi$
$353$	14.7446	0.784774	0.392387	+	0.919800i	$0.371649\pi$
$354$	0	0
$355$	29.4891	1.56512
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.74456	0.250408	0.125204	−	0.992131i	$-0.460041\pi$
$359$	4.74456	0.250408	0.125204	+	0.992131i	$0.460041\pi$
$360$	0	0
$361$	−18.6060	−0.979262
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	50.1168	2.62324
$366$	0	0
$367$	6.74456	0.352063	0.176032	−	0.984385i	$-0.443674\pi$
$367$	6.74456	0.352063	0.176032	+	0.984385i	$0.443674\pi$
$368$	0	0
$369$	2.74456	0.142876
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	4.62772	0.238974
$376$	0	0
$377$	1.37228	0.0706761
$378$	0	0
$379$	24.2337	1.24480	0.622400	−	0.782699i	$-0.286158\pi$
$379$	24.2337	1.24480	0.622400	+	0.782699i	$0.286158\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	0	0
$383$	13.4891	0.689262	0.344631	−	0.938738i	$-0.388004\pi$
$383$	13.4891	0.689262	0.344631	+	0.938738i	$0.388004\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6.11684	−0.310937
$388$	0	0
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	9.48913	0.478663
$394$	0	0
$395$	20.6277	1.03789
$396$	0	0
$397$	22.6277	1.13565	0.567826	−	0.823148i	$-0.307785\pi$
$397$	22.6277	1.13565	0.567826	+	0.823148i	$0.307785\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.7446	1.03593	0.517967	−	0.855401i	$-0.326689\pi$
$401$	20.7446	1.03593	0.517967	+	0.855401i	$0.326689\pi$
$402$	0	0
$403$	3.37228	0.167985
$404$	0	0
$405$	−3.37228	−0.167570
$406$	0	0
$407$	−9.48913	−0.470358
$408$	0	0
$409$	−22.8614	−1.13042	−0.565212	−	0.824946i	$-0.691206\pi$
$409$	−22.8614	−1.13042	−0.565212	+	0.824946i	$0.691206\pi$
$410$	0	0
$411$	22.2337	1.09671
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−20.6277	−1.01257
$416$	0	0
$417$	2.74456	0.134402
$418$	0	0
$419$	−1.25544	−0.0613321	−0.0306661	−	0.999530i	$-0.509763\pi$
$419$	−1.25544	−0.0613321	−0.0306661	+	0.999530i	$0.509763\pi$
$420$	0	0
$421$	3.25544	0.158660	0.0793302	−	0.996848i	$-0.474722\pi$
$421$	3.25544	0.158660	0.0793302	+	0.996848i	$0.474722\pi$
$422$	0	0
$423$	0.627719	0.0305207
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	16.5109	0.795301	0.397650	−	0.917537i	$-0.369826\pi$
$431$	16.5109	0.795301	0.397650	+	0.917537i	$0.369826\pi$
$432$	0	0
$433$	−0.744563	−0.0357814	−0.0178907	−	0.999840i	$-0.505695\pi$
$433$	−0.744563	−0.0357814	−0.0178907	+	0.999840i	$0.505695\pi$
$434$	0	0
$435$	4.62772	0.221882
$436$	0	0
$437$	0.861407	0.0412067
$438$	0	0
$439$	−1.25544	−0.0599188	−0.0299594	−	0.999551i	$-0.509538\pi$
$439$	−1.25544	−0.0599188	−0.0299594	+	0.999551i	$0.509538\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	34.6277	1.64521	0.822606	−	0.568611i	$-0.192519\pi$
$443$	34.6277	1.64521	0.822606	+	0.568611i	$0.192519\pi$
$444$	0	0
$445$	−24.8614	−1.17854
$446$	0	0
$447$	−2.00000	−0.0945968
$448$	0	0
$449$	16.7446	0.790225	0.395112	−	0.918633i	$-0.370706\pi$
$449$	16.7446	0.790225	0.395112	+	0.918633i	$0.370706\pi$
$450$	0	0
$451$	5.48913	0.258473
$452$	0	0
$453$	−21.4891	−1.00965
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.2337	−0.478712	−0.239356	−	0.970932i	$-0.576936\pi$
$457$	−10.2337	−0.478712	−0.239356	+	0.970932i	$0.576936\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13.2554	−0.617367	−0.308684	−	0.951165i	$-0.599888\pi$
$461$	−13.2554	−0.617367	−0.308684	+	0.951165i	$0.599888\pi$
$462$	0	0
$463$	−2.74456	−0.127551	−0.0637753	−	0.997964i	$-0.520314\pi$
$463$	−2.74456	−0.127551	−0.0637753	+	0.997964i	$0.520314\pi$
$464$	0	0
$465$	11.3723	0.527377
$466$	0	0
$467$	20.2337	0.936303	0.468152	−	0.883648i	$-0.344920\pi$
$467$	20.2337	0.936303	0.468152	+	0.883648i	$0.344920\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−8.74456	−0.402928
$472$	0	0
$473$	−12.2337	−0.562506
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	5.37228	0.245980
$478$	0	0
$479$	17.8832	0.817102	0.408551	−	0.912735i	$-0.366034\pi$
$479$	17.8832	0.817102	0.408551	+	0.912735i	$0.366034\pi$
$480$	0	0
$481$	−4.74456	−0.216333
$482$	0	0
$483$	0	0
$484$	0	0
$485$	58.5842	2.66017
$486$	0	0
$487$	22.7446	1.03065	0.515327	−	0.856993i	$-0.327670\pi$
$487$	22.7446	1.03065	0.515327	+	0.856993i	$0.327670\pi$
$488$	0	0
$489$	13.4891	0.609999
$490$	0	0
$491$	7.25544	0.327433	0.163717	−	0.986507i	$-0.447652\pi$
$491$	7.25544	0.327433	0.163717	+	0.986507i	$0.447652\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−6.74456	−0.303146
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	−1.88316	−0.0841332
$502$	0	0
$503$	25.7228	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$503$	25.7228	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$504$	0	0
$505$	−17.7228	−0.788655
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−22.3505	−0.990670	−0.495335	−	0.868702i	$-0.664955\pi$
$509$	−22.3505	−0.990670	−0.495335	+	0.868702i	$0.664955\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0.627719	0.0277145
$514$	0	0
$515$	18.5109	0.815687
$516$	0	0
$517$	1.25544	0.0552141
$518$	0	0
$519$	−22.9783	−1.00863
$520$	0	0
$521$	−16.0000	−0.700973	−0.350486	−	0.936568i	$-0.613984\pi$
$521$	−16.0000	−0.700973	−0.350486	+	0.936568i	$0.613984\pi$
$522$	0	0
$523$	−29.7228	−1.29969	−0.649844	−	0.760068i	$-0.725166\pi$
$523$	−29.7228	−1.29969	−0.649844	+	0.760068i	$0.725166\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−21.1168	−0.918124
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	2.74456	0.118880
$534$	0	0
$535$	−2.51087	−0.108555
$536$	0	0
$537$	18.8614	0.813930
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−7.25544	−0.311936	−0.155968	−	0.987762i	$-0.549850\pi$
$541$	−7.25544	−0.311936	−0.155968	+	0.987762i	$0.549850\pi$
$542$	0	0
$543$	2.00000	0.0858282
$544$	0	0
$545$	−38.7446	−1.65963
$546$	0	0
$547$	28.6277	1.22403	0.612016	−	0.790845i	$-0.290359\pi$
$547$	28.6277	1.22403	0.612016	+	0.790845i	$0.290359\pi$
$548$	0	0
$549$	−6.00000	−0.256074
$550$	0	0
$551$	−0.861407	−0.0366972
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−16.0000	−0.679162
$556$	0	0
$557$	35.7228	1.51362	0.756812	−	0.653633i	$-0.226756\pi$
$557$	35.7228	1.51362	0.756812	+	0.653633i	$0.226756\pi$
$558$	0	0
$559$	−6.11684	−0.258715
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−40.4674	−1.70550	−0.852748	−	0.522322i	$-0.825066\pi$
$563$	−40.4674	−1.70550	−0.852748	+	0.522322i	$0.825066\pi$
$564$	0	0
$565$	−40.8614	−1.71905
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−25.6060	−1.07346	−0.536729	−	0.843755i	$-0.680340\pi$
$569$	−25.6060	−1.07346	−0.536729	+	0.843755i	$0.680340\pi$
$570$	0	0
$571$	35.3723	1.48028	0.740142	−	0.672451i	$-0.234758\pi$
$571$	35.3723	1.48028	0.740142	+	0.672451i	$0.234758\pi$
$572$	0	0
$573$	3.25544	0.135998
$574$	0	0
$575$	−8.74456	−0.364673
$576$	0	0
$577$	−20.9783	−0.873336	−0.436668	−	0.899623i	$-0.643842\pi$
$577$	−20.9783	−0.873336	−0.436668	+	0.899623i	$0.643842\pi$
$578$	0	0
$579$	−6.00000	−0.249351
$580$	0	0
$581$	0	0
$582$	0	0
$583$	10.7446	0.444994
$584$	0	0
$585$	−3.37228	−0.139427
$586$	0	0
$587$	−27.3723	−1.12977	−0.564887	−	0.825168i	$-0.691080\pi$
$587$	−27.3723	−1.12977	−0.564887	+	0.825168i	$0.691080\pi$
$588$	0	0
$589$	−2.11684	−0.0872230
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	21.0951	0.866272	0.433136	−	0.901329i	$-0.357407\pi$
$593$	21.0951	0.866272	0.433136	+	0.901329i	$0.357407\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−16.0000	−0.654836
$598$	0	0
$599$	−10.8614	−0.443785	−0.221893	−	0.975071i	$-0.571223\pi$
$599$	−10.8614	−0.443785	−0.221893	+	0.975071i	$0.571223\pi$
$600$	0	0
$601$	−31.4891	−1.28447	−0.642234	−	0.766509i	$-0.721992\pi$
$601$	−31.4891	−1.28447	−0.642234	+	0.766509i	$0.721992\pi$
$602$	0	0
$603$	1.25544	0.0511254
$604$	0	0
$605$	23.6060	0.959719
$606$	0	0
$607$	25.2554	1.02509	0.512543	−	0.858661i	$-0.328703\pi$
$607$	25.2554	1.02509	0.512543	+	0.858661i	$0.328703\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.627719	0.0253948
$612$	0	0
$613$	−3.25544	−0.131486	−0.0657429	−	0.997837i	$-0.520942\pi$
$613$	−3.25544	−0.131486	−0.0657429	+	0.997837i	$0.520942\pi$
$614$	0	0
$615$	9.25544	0.373215
$616$	0	0
$617$	35.4891	1.42874	0.714369	−	0.699769i	$-0.246714\pi$
$617$	35.4891	1.42874	0.714369	+	0.699769i	$0.246714\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	1.37228	0.0550678
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−16.2554	−0.650217
$626$	0	0
$627$	1.25544	0.0501373
$628$	0	0
$629$	0	0
$630$	0	0
$631$	35.2119	1.40177	0.700883	−	0.713277i	$-0.252790\pi$
$631$	35.2119	1.40177	0.700883	+	0.713277i	$0.252790\pi$
$632$	0	0
$633$	−7.37228	−0.293022
$634$	0	0
$635$	13.4891	0.535300
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−8.74456	−0.345930
$640$	0	0
$641$	16.1168	0.636577	0.318289	−	0.947994i	$-0.396892\pi$
$641$	16.1168	0.636577	0.318289	+	0.947994i	$0.396892\pi$
$642$	0	0
$643$	−14.5109	−0.572253	−0.286127	−	0.958192i	$-0.592368\pi$
$643$	−14.5109	−0.572253	−0.286127	+	0.958192i	$0.592368\pi$
$644$	0	0
$645$	−20.6277	−0.812216
$646$	0	0
$647$	−5.25544	−0.206613	−0.103306	−	0.994650i	$-0.532942\pi$
$647$	−5.25544	−0.206613	−0.103306	+	0.994650i	$0.532942\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−42.4674	−1.66188	−0.830938	−	0.556364i	$-0.812196\pi$
$653$	−42.4674	−1.66188	−0.830938	+	0.556364i	$0.812196\pi$
$654$	0	0
$655$	32.0000	1.25034
$656$	0	0
$657$	−14.8614	−0.579799
$658$	0	0
$659$	18.8614	0.734736	0.367368	−	0.930076i	$-0.380259\pi$
$659$	18.8614	0.734736	0.367368	+	0.930076i	$0.380259\pi$
$660$	0	0
$661$	1.13859	0.0442861	0.0221431	−	0.999755i	$-0.492951\pi$
$661$	1.13859	0.0442861	0.0221431	+	0.999755i	$0.492951\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.88316	−0.0729161
$668$	0	0
$669$	16.8614	0.651900
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	27.0951	1.04444	0.522220	−	0.852811i	$-0.325104\pi$
$673$	27.0951	1.04444	0.522220	+	0.852811i	$0.325104\pi$
$674$	0	0
$675$	−6.37228	−0.245269
$676$	0	0
$677$	−48.2337	−1.85377	−0.926886	−	0.375344i	$-0.877525\pi$
$677$	−48.2337	−1.85377	−0.926886	+	0.375344i	$0.877525\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−29.4891	−1.13003
$682$	0	0
$683$	2.23369	0.0854697	0.0427348	−	0.999086i	$-0.486393\pi$
$683$	2.23369	0.0854697	0.0427348	+	0.999086i	$0.486393\pi$
$684$	0	0
$685$	74.9783	2.86477
$686$	0	0
$687$	−11.4891	−0.438337
$688$	0	0
$689$	5.37228	0.204668
$690$	0	0
$691$	−50.3505	−1.91542	−0.957712	−	0.287728i	$-0.907100\pi$
$691$	−50.3505	−1.91542	−0.957712	+	0.287728i	$0.907100\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.25544	0.351079
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−1.37228	−0.0519044
$700$	0	0
$701$	13.1386	0.496238	0.248119	−	0.968730i	$-0.420188\pi$
$701$	13.1386	0.496238	0.248119	+	0.968730i	$0.420188\pi$
$702$	0	0
$703$	2.97825	0.112327
$704$	0	0
$705$	2.11684	0.0797250
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−12.5109	−0.469856	−0.234928	−	0.972013i	$-0.575485\pi$
$709$	−12.5109	−0.469856	−0.234928	+	0.972013i	$0.575485\pi$
$710$	0	0
$711$	−6.11684	−0.229400
$712$	0	0
$713$	−4.62772	−0.173309
$714$	0	0
$715$	−6.74456	−0.252232
$716$	0	0
$717$	15.2554	0.569725
$718$	0	0
$719$	52.2337	1.94799	0.973994	−	0.226574i	$-0.0727525\pi$
$719$	52.2337	1.94799	0.973994	+	0.226574i	$0.0727525\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1.37228	0.0510357
$724$	0	0
$725$	8.74456	0.324765
$726$	0	0
$727$	28.2337	1.04713	0.523565	−	0.851986i	$-0.324602\pi$
$727$	28.2337	1.04713	0.523565	+	0.851986i	$0.324602\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−15.8832	−0.586658	−0.293329	−	0.956012i	$-0.594763\pi$
$733$	−15.8832	−0.586658	−0.293329	+	0.956012i	$0.594763\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.51087	0.0924893
$738$	0	0
$739$	47.2119	1.73672	0.868360	−	0.495935i	$-0.165175\pi$
$739$	47.2119	1.73672	0.868360	+	0.495935i	$0.165175\pi$
$740$	0	0
$741$	0.627719	0.0230598
$742$	0	0
$743$	18.2337	0.668929	0.334465	−	0.942408i	$-0.391445\pi$
$743$	18.2337	0.668929	0.334465	+	0.942408i	$0.391445\pi$
$744$	0	0
$745$	−6.74456	−0.247102
$746$	0	0
$747$	6.11684	0.223804
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−29.0951	−1.06170	−0.530848	−	0.847467i	$-0.678126\pi$
$751$	−29.0951	−1.06170	−0.530848	+	0.847467i	$0.678126\pi$
$752$	0	0
$753$	−6.74456	−0.245786
$754$	0	0
$755$	−72.4674	−2.63736
$756$	0	0
$757$	−32.3505	−1.17580	−0.587900	−	0.808934i	$-0.700045\pi$
$757$	−32.3505	−1.17580	−0.587900	+	0.808934i	$0.700045\pi$
$758$	0	0
$759$	2.74456	0.0996213
$760$	0	0
$761$	3.13859	0.113774	0.0568870	−	0.998381i	$-0.481883\pi$
$761$	3.13859	0.113774	0.0568870	+	0.998381i	$0.481883\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	3.88316	0.140030	0.0700151	−	0.997546i	$-0.477695\pi$
$769$	3.88316	0.140030	0.0700151	+	0.997546i	$0.477695\pi$
$770$	0	0
$771$	8.23369	0.296529
$772$	0	0
$773$	−37.7228	−1.35680	−0.678398	−	0.734695i	$-0.737325\pi$
$773$	−37.7228	−1.35680	−0.678398	+	0.734695i	$0.737325\pi$
$774$	0	0
$775$	21.4891	0.771912
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.72281	−0.0617262
$780$	0	0
$781$	−17.4891	−0.625810
$782$	0	0
$783$	−1.37228	−0.0490413
$784$	0	0
$785$	−29.4891	−1.05251
$786$	0	0
$787$	−27.6060	−0.984047	−0.492023	−	0.870582i	$-0.663743\pi$
$787$	−27.6060	−0.984047	−0.492023	+	0.870582i	$0.663743\pi$
$788$	0	0
$789$	10.6277	0.378357
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−6.00000	−0.213066
$794$	0	0
$795$	18.1168	0.642538
$796$	0	0
$797$	13.7228	0.486087	0.243043	−	0.970015i	$-0.421854\pi$
$797$	13.7228	0.486087	0.243043	+	0.970015i	$0.421854\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	7.37228	0.260487
$802$	0	0
$803$	−29.7228	−1.04890
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−24.0000	−0.844840
$808$	0	0
$809$	−49.3723	−1.73584	−0.867918	−	0.496707i	$-0.834542\pi$
$809$	−49.3723	−1.73584	−0.867918	+	0.496707i	$0.834542\pi$
$810$	0	0
$811$	38.9783	1.36871	0.684356	−	0.729148i	$-0.260084\pi$
$811$	38.9783	1.36871	0.684356	+	0.729148i	$0.260084\pi$
$812$	0	0
$813$	−26.9783	−0.946169
$814$	0	0
$815$	45.4891	1.59341
$816$	0	0
$817$	3.83966	0.134333
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−0.978251	−0.0341412	−0.0170706	−	0.999854i	$-0.505434\pi$
$821$	−0.978251	−0.0341412	−0.0170706	+	0.999854i	$0.505434\pi$
$822$	0	0
$823$	−22.5109	−0.784680	−0.392340	−	0.919820i	$-0.628334\pi$
$823$	−22.5109	−0.784680	−0.392340	+	0.919820i	$0.628334\pi$
$824$	0	0
$825$	−12.7446	−0.443709
$826$	0	0
$827$	−29.2119	−1.01580	−0.507899	−	0.861416i	$-0.669578\pi$
$827$	−29.2119	−1.01580	−0.507899	+	0.861416i	$0.669578\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	−12.1168	−0.420329
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−6.35053	−0.219769
$836$	0	0
$837$	−3.37228	−0.116563
$838$	0	0
$839$	9.48913	0.327601	0.163800	−	0.986493i	$-0.447625\pi$
$839$	9.48913	0.327601	0.163800	+	0.986493i	$0.447625\pi$
$840$	0	0
$841$	−27.1168	−0.935064
$842$	0	0
$843$	−22.0000	−0.757720
$844$	0	0
$845$	−3.37228	−0.116010
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−25.4891	−0.874785
$850$	0	0
$851$	6.51087	0.223190
$852$	0	0
$853$	21.8397	0.747776	0.373888	−	0.927474i	$-0.378025\pi$
$853$	21.8397	0.747776	0.373888	+	0.927474i	$0.378025\pi$
$854$	0	0
$855$	2.11684	0.0723945
$856$	0	0
$857$	−32.2337	−1.10108	−0.550541	−	0.834808i	$-0.685578\pi$
$857$	−32.2337	−1.10108	−0.550541	+	0.834808i	$0.685578\pi$
$858$	0	0
$859$	−0.233688	−0.00797333	−0.00398666	−	0.999992i	$-0.501269\pi$
$859$	−0.233688	−0.00797333	−0.00398666	+	0.999992i	$0.501269\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	22.2337	0.756844	0.378422	−	0.925633i	$-0.376467\pi$
$863$	22.2337	0.756844	0.378422	+	0.925633i	$0.376467\pi$
$864$	0	0
$865$	−77.4891	−2.63471
$866$	0	0
$867$	17.0000	0.577350
$868$	0	0
$869$	−12.2337	−0.414999
$870$	0	0
$871$	1.25544	0.0425389
$872$	0	0
$873$	−17.3723	−0.587963
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−8.74456	−0.295283	−0.147641	−	0.989041i	$-0.547168\pi$
$877$	−8.74456	−0.295283	−0.147641	+	0.989041i	$0.547168\pi$
$878$	0	0
$879$	20.8614	0.703638
$880$	0	0
$881$	33.2554	1.12040	0.560202	−	0.828356i	$-0.310723\pi$
$881$	33.2554	1.12040	0.560202	+	0.828356i	$0.310723\pi$
$882$	0	0
$883$	22.9783	0.773280	0.386640	−	0.922231i	$-0.373636\pi$
$883$	22.9783	0.773280	0.386640	+	0.922231i	$0.373636\pi$
$884$	0	0
$885$	26.9783	0.906864
$886$	0	0
$887$	1.02175	0.0343070	0.0171535	−	0.999853i	$-0.494540\pi$
$887$	1.02175	0.0343070	0.0171535	+	0.999853i	$0.494540\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	−0.394031	−0.0131857
$894$	0	0
$895$	63.6060	2.12611
$896$	0	0
$897$	1.37228	0.0458191
$898$	0	0
$899$	4.62772	0.154343
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.74456	0.224197
$906$	0	0
$907$	25.8832	0.859436	0.429718	−	0.902963i	$-0.358613\pi$
$907$	25.8832	0.859436	0.429718	+	0.902963i	$0.358613\pi$
$908$	0	0
$909$	5.25544	0.174312
$910$	0	0
$911$	14.8614	0.492380	0.246190	−	0.969222i	$-0.420821\pi$
$911$	14.8614	0.492380	0.246190	+	0.969222i	$0.420821\pi$
$912$	0	0
$913$	12.2337	0.404876
$914$	0	0
$915$	−20.2337	−0.668905
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	12.8614	0.423798
$922$	0	0
$923$	−8.74456	−0.287831
$924$	0	0
$925$	−30.2337	−0.994078
$926$	0	0
$927$	−5.48913	−0.180287
$928$	0	0
$929$	59.6060	1.95561	0.977804	−	0.209521i	$-0.0671904\pi$
$929$	59.6060	1.95561	0.977804	+	0.209521i	$0.0671904\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−18.7446	−0.613669
$934$	0	0
$935$	0	0
$936$	0	0
$937$	18.2337	0.595669	0.297834	−	0.954618i	$-0.403736\pi$
$937$	18.2337	0.595669	0.297834	+	0.954618i	$0.403736\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	0	0
$941$	13.8832	0.452578	0.226289	−	0.974060i	$-0.427341\pi$
$941$	13.8832	0.452578	0.226289	+	0.974060i	$0.427341\pi$
$942$	0	0
$943$	−3.76631	−0.122648
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52.9783	1.72156	0.860781	−	0.508976i	$-0.169976\pi$
$947$	52.9783	1.72156	0.860781	+	0.508976i	$0.169976\pi$
$948$	0	0
$949$	−14.8614	−0.482422
$950$	0	0
$951$	−3.25544	−0.105565
$952$	0	0
$953$	14.6277	0.473838	0.236919	−	0.971529i	$-0.423862\pi$
$953$	14.6277	0.473838	0.236919	+	0.971529i	$0.423862\pi$
$954$	0	0
$955$	10.9783	0.355248
$956$	0	0
$957$	−2.74456	−0.0887191
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−19.6277	−0.633152
$962$	0	0
$963$	0.744563	0.0239932
$964$	0	0
$965$	−20.2337	−0.651345
$966$	0	0
$967$	5.72281	0.184033	0.0920166	−	0.995757i	$-0.470669\pi$
$967$	5.72281	0.184033	0.0920166	+	0.995757i	$0.470669\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	10.7446	0.344809	0.172405	−	0.985026i	$-0.444846\pi$
$971$	10.7446	0.344809	0.172405	+	0.985026i	$0.444846\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−6.37228	−0.204076
$976$	0	0
$977$	−28.9783	−0.927096	−0.463548	−	0.886072i	$-0.653424\pi$
$977$	−28.9783	−0.927096	−0.463548	+	0.886072i	$0.653424\pi$
$978$	0	0
$979$	14.7446	0.471238
$980$	0	0
$981$	11.4891	0.366820
$982$	0	0
$983$	−37.0951	−1.18315	−0.591575	−	0.806250i	$-0.701494\pi$
$983$	−37.0951	−1.18315	−0.591575	+	0.806250i	$0.701494\pi$
$984$	0	0
$985$	−60.7011	−1.93410
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.39403	0.266915
$990$	0	0
$991$	10.5109	0.333889	0.166944	−	0.985966i	$-0.446610\pi$
$991$	10.5109	0.333889	0.166944	+	0.985966i	$0.446610\pi$
$992$	0	0
$993$	−24.2337	−0.769033
$994$	0	0
$995$	−53.9565	−1.71054
$996$	0	0
$997$	0.744563	0.0235805	0.0117903	−	0.999930i	$-0.496247\pi$
$997$	0.744563	0.0235805	0.0117903	+	0.999930i	$0.496247\pi$
$998$	0	0
$999$	4.74456	0.150111

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7644.2.a.n.1.1		2
7.6	odd	2		1092.2.a.g.1.2	✓	2
21.20	even	2		3276.2.a.m.1.1		2
28.27	even	2		4368.2.a.bc.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1092.2.a.g.1.2	✓	2	7.6	odd	2
3276.2.a.m.1.1		2	21.20	even	2
4368.2.a.bc.1.2		2	28.27	even	2
7644.2.a.n.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7644 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7644.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.37228 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.37228 q^{5} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{13} +3.37228 q^{15} -0.627719 q^{19} -1.37228 q^{23} +6.37228 q^{25} -1.00000 q^{27} +1.37228 q^{29} +3.37228 q^{31} -2.00000 q^{33} -4.74456 q^{37} -1.00000 q^{39} +2.74456 q^{41} -6.11684 q^{43} -3.37228 q^{45} +0.627719 q^{47} +5.37228 q^{53} -6.74456 q^{55} +0.627719 q^{57} +8.00000 q^{59} -6.00000 q^{61} -3.37228 q^{65} +1.25544 q^{67} +1.37228 q^{69} -8.74456 q^{71} -14.8614 q^{73} -6.37228 q^{75} -6.11684 q^{79} +1.00000 q^{81} +6.11684 q^{83} -1.37228 q^{87} +7.37228 q^{89} -3.37228 q^{93} +2.11684 q^{95} -17.3723 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - q^5 + 2 * q^9	\( 2 q - 2 q^{3} - q^{5} + 2 q^{9} + 4 q^{11} + 2 q^{13} + q^{15} - 7 q^{19} + 3 q^{23} + 7 q^{25} - 2 q^{27} - 3 q^{29} + q^{31} - 4 q^{33} + 2 q^{37} - 2 q^{39} - 6 q^{41} + 5 q^{43} - q^{45} + 7 q^{47} + 5 q^{53} - 2 q^{55} + 7 q^{57} + 16 q^{59} - 12 q^{61} - q^{65} + 14 q^{67} - 3 q^{69} - 6 q^{71} - q^{73} - 7 q^{75} + 5 q^{79} + 2 q^{81} - 5 q^{83} + 3 q^{87} + 9 q^{89} - q^{93} - 13 q^{95} - 29 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - q^5 + 2 * q^9 + 4 * q^11 + 2 * q^13 + q^15 - 7 * q^19 + 3 * q^23 + 7 * q^25 - 2 * q^27 - 3 * q^29 + q^31 - 4 * q^33 + 2 * q^37 - 2 * q^39 - 6 * q^41 + 5 * q^43 - q^45 + 7 * q^47 + 5 * q^53 - 2 * q^55 + 7 * q^57 + 16 * q^59 - 12 * q^61 - q^65 + 14 * q^67 - 3 * q^69 - 6 * q^71 - q^73 - 7 * q^75 + 5 * q^79 + 2 * q^81 - 5 * q^83 + 3 * q^87 + 9 * q^89 - q^93 - 13 * q^95 - 29 * q^97 + 4 * q^99

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