LMFDB - Embedded newform 3969.2.a.bh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.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:	$31.6926245622$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 22 x^{10} + 92 x^{9} + 125 x^{8} - 620 x^{7} - 94 x^{6} + 1280 x^{5} - 234 x^{4} + \cdots - 7 $ x^12 - 4x^11 - 22x^10 + 92x^9 + 125x^8 - 620x^7 - 94x^6 + 1280x^5 - 234x^4 - 736x^3 + 96x^2 + 60*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 441)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.30092$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.71513 q^{2} +5.37195 q^{4} +1.58639 q^{5} -9.15528 q^{8} +O(q^{10})$	$q-2.71513 q^{2} +5.37195 q^{4} +1.58639 q^{5} -9.15528 q^{8} -4.30727 q^{10} -1.34875 q^{11} +3.17832 q^{13} +14.1139 q^{16} -2.80054 q^{17} -0.625693 q^{19} +8.52202 q^{20} +3.66204 q^{22} -0.284867 q^{23} -2.48336 q^{25} -8.62957 q^{26} -4.54792 q^{29} -7.43005 q^{31} -20.0106 q^{32} +7.60383 q^{34} +8.02252 q^{37} +1.69884 q^{38} -14.5239 q^{40} +10.0266 q^{41} +6.25873 q^{43} -7.24542 q^{44} +0.773452 q^{46} -11.1477 q^{47} +6.74264 q^{50} +17.0738 q^{52} +2.78698 q^{53} -2.13965 q^{55} +12.3482 q^{58} -4.57469 q^{59} +0.385014 q^{61} +20.1736 q^{62} +26.1036 q^{64} +5.04207 q^{65} -2.53916 q^{67} -15.0443 q^{68} -1.45208 q^{71} +0.468134 q^{73} -21.7822 q^{74} -3.36119 q^{76} -15.7124 q^{79} +22.3902 q^{80} -27.2235 q^{82} -13.9868 q^{83} -4.44275 q^{85} -16.9933 q^{86} +12.3482 q^{88} +2.58706 q^{89} -1.53029 q^{92} +30.2674 q^{94} -0.992595 q^{95} +14.4592 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{2} + 12 q^{4} - 12 q^{8}+O(q^{10}) $ 12 * q - 4 * q^2 + 12 * q^4 - 12 * q^8	$ 12 q - 4 q^{2} + 12 q^{4} - 12 q^{8} - 20 q^{11} + 12 q^{16} - 32 q^{23} + 12 q^{25} - 16 q^{29} - 48 q^{32} + 12 q^{37} - 56 q^{44} - 24 q^{46} + 4 q^{50} - 32 q^{53} + 48 q^{64} - 60 q^{65} + 12 q^{67} - 56 q^{71} - 68 q^{74} - 12 q^{79} - 12 q^{85} - 76 q^{86} - 16 q^{92} - 64 q^{95}+O(q^{100}) $ 12 * q - 4 * q^2 + 12 * q^4 - 12 * q^8 - 20 * q^11 + 12 * q^16 - 32 * q^23 + 12 * q^25 - 16 * q^29 - 48 * q^32 + 12 * q^37 - 56 * q^44 - 24 * q^46 + 4 * q^50 - 32 * q^53 + 48 * q^64 - 60 * q^65 + 12 * q^67 - 56 * q^71 - 68 * q^74 - 12 * q^79 - 12 * q^85 - 76 * q^86 - 16 * q^92 - 64 * q^95

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 $

$2$	−2.71513	−1.91989	−0.959944	−	0.280191i	$-0.909602\pi$
$2$	−2.71513	−1.91989	−0.959944	+	0.280191i	$0.909602\pi$
$3$	0	0
$3$	0	0
$4$	5.37195	2.68597
$5$	1.58639	0.709457	0.354728	−	0.934969i	$-0.384573\pi$
$5$	1.58639	0.709457	0.354728	+	0.934969i	$0.384573\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−9.15528	−3.23688
$9$	0	0
$10$	−4.30727	−1.36208
$11$	−1.34875	−0.406664	−0.203332	−	0.979110i	$-0.565177\pi$
$11$	−1.34875	−0.406664	−0.203332	+	0.979110i	$0.565177\pi$
$12$	0	0
$13$	3.17832	0.881508	0.440754	−	0.897628i	$-0.354711\pi$
$13$	3.17832	0.881508	0.440754	+	0.897628i	$0.354711\pi$
$14$	0	0
$15$	0	0
$16$	14.1139	3.52848
$17$	−2.80054	−0.679230	−0.339615	−	0.940565i	$-0.610297\pi$
$17$	−2.80054	−0.679230	−0.339615	+	0.940565i	$0.610297\pi$
$18$	0	0
$19$	−0.625693	−0.143544	−0.0717719	−	0.997421i	$-0.522865\pi$
$19$	−0.625693	−0.143544	−0.0717719	+	0.997421i	$0.522865\pi$
$20$	8.52202	1.90558
$21$	0	0
$22$	3.66204	0.780750
$23$	−0.284867	−0.0593989	−0.0296995	−	0.999559i	$-0.509455\pi$
$23$	−0.284867	−0.0593989	−0.0296995	+	0.999559i	$0.509455\pi$
$24$	0	0
$25$	−2.48336	−0.496671
$26$	−8.62957	−1.69240
$27$	0	0
$28$	0	0
$29$	−4.54792	−0.844527	−0.422264	−	0.906473i	$-0.638764\pi$
$29$	−4.54792	−0.844527	−0.422264	+	0.906473i	$0.638764\pi$
$30$	0	0
$31$	−7.43005	−1.33448	−0.667238	−	0.744845i	$-0.732524\pi$
$31$	−7.43005	−1.33448	−0.667238	+	0.744845i	$0.732524\pi$
$32$	−20.0106	−3.53741
$33$	0	0
$34$	7.60383	1.30405
$35$	0	0
$36$	0	0
$37$	8.02252	1.31889	0.659447	−	0.751751i	$-0.270791\pi$
$37$	8.02252	1.31889	0.659447	+	0.751751i	$0.270791\pi$
$38$	1.69884	0.275588
$39$	0	0
$40$	−14.5239	−2.29643
$41$	10.0266	1.56589	0.782944	−	0.622092i	$-0.213717\pi$
$41$	10.0266	1.56589	0.782944	+	0.622092i	$0.213717\pi$
$42$	0	0
$43$	6.25873	0.954448	0.477224	−	0.878782i	$-0.341643\pi$
$43$	6.25873	0.954448	0.477224	+	0.878782i	$0.341643\pi$
$44$	−7.24542	−1.09229
$45$	0	0
$46$	0.773452	0.114039
$47$	−11.1477	−1.62605	−0.813026	−	0.582227i	$-0.802181\pi$
$47$	−11.1477	−1.62605	−0.813026	+	0.582227i	$0.802181\pi$
$48$	0	0
$49$	0	0
$50$	6.74264	0.953554
$51$	0	0
$52$	17.0738	2.36771
$53$	2.78698	0.382821	0.191410	−	0.981510i	$-0.438694\pi$
$53$	2.78698	0.382821	0.191410	+	0.981510i	$0.438694\pi$
$54$	0	0
$55$	−2.13965	−0.288510
$56$	0	0
$57$	0	0
$58$	12.3482	1.62140
$59$	−4.57469	−0.595574	−0.297787	−	0.954632i	$-0.596248\pi$
$59$	−4.57469	−0.595574	−0.297787	+	0.954632i	$0.596248\pi$
$60$	0	0
$61$	0.385014	0.0492960	0.0246480	−	0.999696i	$-0.492154\pi$
$61$	0.385014	0.0492960	0.0246480	+	0.999696i	$0.492154\pi$
$62$	20.1736	2.56205
$63$	0	0
$64$	26.1036	3.26295
$65$	5.04207	0.625392
$66$	0	0
$67$	−2.53916	−0.310208	−0.155104	−	0.987898i	$-0.549571\pi$
$67$	−2.53916	−0.310208	−0.155104	+	0.987898i	$0.549571\pi$
$68$	−15.0443	−1.82439
$69$	0	0
$70$	0	0
$71$	−1.45208	−0.172330	−0.0861651	−	0.996281i	$-0.527461\pi$
$71$	−1.45208	−0.172330	−0.0861651	+	0.996281i	$0.527461\pi$
$72$	0	0
$73$	0.468134	0.0547909	0.0273955	−	0.999625i	$-0.491279\pi$
$73$	0.468134	0.0547909	0.0273955	+	0.999625i	$0.491279\pi$
$74$	−21.7822	−2.53213
$75$	0	0
$76$	−3.36119	−0.385555
$77$	0	0
$78$	0	0
$79$	−15.7124	−1.76778	−0.883892	−	0.467691i	$-0.845086\pi$
$79$	−15.7124	−1.76778	−0.883892	+	0.467691i	$0.845086\pi$
$80$	22.3902	2.50330
$81$	0	0
$82$	−27.2235	−3.00633
$83$	−13.9868	−1.53525	−0.767623	−	0.640901i	$-0.778561\pi$
$83$	−13.9868	−1.53525	−0.767623	+	0.640901i	$0.778561\pi$
$84$	0	0
$85$	−4.44275	−0.481884
$86$	−16.9933	−1.83243
$87$	0	0
$88$	12.3482	1.31632
$89$	2.58706	0.274228	0.137114	−	0.990555i	$-0.456217\pi$
$89$	2.58706	0.274228	0.137114	+	0.990555i	$0.456217\pi$
$90$	0	0
$91$	0	0
$92$	−1.53029	−0.159544
$93$	0	0
$94$	30.2674	3.12184
$95$	−0.992595	−0.101838
$96$	0	0
$97$	14.4592	1.46811	0.734057	−	0.679088i	$-0.237624\pi$
$97$	14.4592	1.46811	0.734057	+	0.679088i	$0.237624\pi$
$98$	0	0
$99$	0	0
$100$	−13.3405	−1.33405
$101$	−9.83776	−0.978894	−0.489447	−	0.872033i	$-0.662801\pi$
$101$	−9.83776	−0.978894	−0.489447	+	0.872033i	$0.662801\pi$
$102$	0	0
$103$	−11.0579	−1.08957	−0.544786	−	0.838575i	$-0.683389\pi$
$103$	−11.0579	−1.08957	−0.544786	+	0.838575i	$0.683389\pi$
$104$	−29.0984	−2.85334
$105$	0	0
$106$	−7.56701	−0.734973
$107$	−1.92431	−0.186030	−0.0930149	−	0.995665i	$-0.529650\pi$
$107$	−1.92431	−0.186030	−0.0930149	+	0.995665i	$0.529650\pi$
$108$	0	0
$109$	−18.6068	−1.78221	−0.891105	−	0.453797i	$-0.850069\pi$
$109$	−18.6068	−1.78221	−0.891105	+	0.453797i	$0.850069\pi$
$110$	5.80944	0.553908
$111$	0	0
$112$	0	0
$113$	−3.18677	−0.299786	−0.149893	−	0.988702i	$-0.547893\pi$
$113$	−3.18677	−0.299786	−0.149893	+	0.988702i	$0.547893\pi$
$114$	0	0
$115$	−0.451911	−0.0421410
$116$	−24.4312	−2.26838
$117$	0	0
$118$	12.4209	1.14344
$119$	0	0
$120$	0	0
$121$	−9.18087	−0.834624
$122$	−1.04536	−0.0946428
$123$	0	0
$124$	−39.9138	−3.58437
$125$	−11.8715	−1.06182
$126$	0	0
$127$	−8.37387	−0.743061	−0.371530	−	0.928421i	$-0.621167\pi$
$127$	−8.37387	−0.743061	−0.371530	+	0.928421i	$0.621167\pi$
$128$	−30.8535	−2.72709
$129$	0	0
$130$	−13.6899	−1.20068
$131$	11.9726	1.04605	0.523024	−	0.852318i	$-0.324804\pi$
$131$	11.9726	1.04605	0.523024	+	0.852318i	$0.324804\pi$
$132$	0	0
$133$	0	0
$134$	6.89415	0.595564
$135$	0	0
$136$	25.6397	2.19859
$137$	16.5505	1.41401	0.707003	−	0.707211i	$-0.250047\pi$
$137$	16.5505	1.41401	0.707003	+	0.707211i	$0.250047\pi$
$138$	0	0
$139$	7.90239	0.670272	0.335136	−	0.942170i	$-0.391218\pi$
$139$	7.90239	0.670272	0.335136	+	0.942170i	$0.391218\pi$
$140$	0	0
$141$	0	0
$142$	3.94259	0.330855
$143$	−4.28677	−0.358478
$144$	0	0
$145$	−7.21479	−0.599156
$146$	−1.27105	−0.105192
$147$	0	0
$148$	43.0965	3.54251
$149$	−13.6685	−1.11977	−0.559885	−	0.828570i	$-0.689155\pi$
$149$	−13.6685	−1.11977	−0.559885	+	0.828570i	$0.689155\pi$
$150$	0	0
$151$	3.89963	0.317348	0.158674	−	0.987331i	$-0.449278\pi$
$151$	3.89963	0.317348	0.158674	+	0.987331i	$0.449278\pi$
$152$	5.72839	0.464634
$153$	0	0
$154$	0	0
$155$	−11.7870	−0.946753
$156$	0	0
$157$	0.294352	0.0234919	0.0117459	−	0.999931i	$-0.496261\pi$
$157$	0.294352	0.0234919	0.0117459	+	0.999931i	$0.496261\pi$
$158$	42.6613	3.39395
$159$	0	0
$160$	−31.7447	−2.50964
$161$	0	0
$162$	0	0
$163$	10.7091	0.838802	0.419401	−	0.907801i	$-0.362240\pi$
$163$	10.7091	0.838802	0.419401	+	0.907801i	$0.362240\pi$
$164$	53.8622	4.20593
$165$	0	0
$166$	37.9759	2.94750
$167$	3.19745	0.247426	0.123713	−	0.992318i	$-0.460520\pi$
$167$	3.19745	0.247426	0.123713	+	0.992318i	$0.460520\pi$
$168$	0	0
$169$	−2.89826	−0.222943
$170$	12.0627	0.925164
$171$	0	0
$172$	33.6216	2.56362
$173$	11.4375	0.869577	0.434789	−	0.900533i	$-0.356823\pi$
$173$	11.4375	0.869577	0.434789	+	0.900533i	$0.356823\pi$
$174$	0	0
$175$	0	0
$176$	−19.0362	−1.43491
$177$	0	0
$178$	−7.02421	−0.526487
$179$	1.09855	0.0821095	0.0410547	−	0.999157i	$-0.486928\pi$
$179$	1.09855	0.0821095	0.0410547	+	0.999157i	$0.486928\pi$
$180$	0	0
$181$	−3.19013	−0.237120	−0.118560	−	0.992947i	$-0.537828\pi$
$181$	−3.19013	−0.237120	−0.118560	+	0.992947i	$0.537828\pi$
$182$	0	0
$183$	0	0
$184$	2.60804	0.192267
$185$	12.7269	0.935698
$186$	0	0
$187$	3.77723	0.276218
$188$	−59.8846	−4.36753
$189$	0	0
$190$	2.69503	0.195518
$191$	3.86815	0.279889	0.139945	−	0.990159i	$-0.455308\pi$
$191$	3.86815	0.279889	0.139945	+	0.990159i	$0.455308\pi$
$192$	0	0
$193$	−4.13585	−0.297705	−0.148853	−	0.988859i	$-0.547558\pi$
$193$	−4.13585	−0.297705	−0.148853	+	0.988859i	$0.547558\pi$
$194$	−39.2588	−2.81862
$195$	0	0
$196$	0	0
$197$	−0.889267	−0.0633576	−0.0316788	−	0.999498i	$-0.510085\pi$
$197$	−0.889267	−0.0633576	−0.0316788	+	0.999498i	$0.510085\pi$
$198$	0	0
$199$	−6.32386	−0.448287	−0.224143	−	0.974556i	$-0.571958\pi$
$199$	−6.32386	−0.448287	−0.224143	+	0.974556i	$0.571958\pi$
$200$	22.7358	1.60767
$201$	0	0
$202$	26.7108	1.87937
$203$	0	0
$204$	0	0
$205$	15.9061	1.11093
$206$	30.0238	2.09186
$207$	0	0
$208$	44.8586	3.11038
$209$	0.843905	0.0583741
$210$	0	0
$211$	−11.4258	−0.786586	−0.393293	−	0.919413i	$-0.628664\pi$
$211$	−11.4258	−0.786586	−0.393293	+	0.919413i	$0.628664\pi$
$212$	14.9715	1.02825
$213$	0	0
$214$	5.22475	0.357156
$215$	9.92881	0.677139
$216$	0	0
$217$	0	0
$218$	50.5200	3.42164
$219$	0	0
$220$	−11.4941	−0.774931
$221$	−8.90101	−0.598747
$222$	0	0
$223$	16.7191	1.11959	0.559796	−	0.828631i	$-0.310880\pi$
$223$	16.7191	1.11959	0.559796	+	0.828631i	$0.310880\pi$
$224$	0	0
$225$	0	0
$226$	8.65250	0.575555
$227$	−17.0700	−1.13298	−0.566489	−	0.824070i	$-0.691698\pi$
$227$	−17.0700	−1.13298	−0.566489	+	0.824070i	$0.691698\pi$
$228$	0	0
$229$	−19.7894	−1.30772	−0.653861	−	0.756615i	$-0.726852\pi$
$229$	−19.7894	−1.30772	−0.653861	+	0.756615i	$0.726852\pi$
$230$	1.22700	0.0809059
$231$	0	0
$232$	41.6375	2.73363
$233$	5.93159	0.388591	0.194296	−	0.980943i	$-0.437758\pi$
$233$	5.93159	0.388591	0.194296	+	0.980943i	$0.437758\pi$
$234$	0	0
$235$	−17.6846	−1.15361
$236$	−24.5750	−1.59969
$237$	0	0
$238$	0	0
$239$	20.0554	1.29727	0.648637	−	0.761098i	$-0.275339\pi$
$239$	20.0554	1.29727	0.648637	+	0.761098i	$0.275339\pi$
$240$	0	0
$241$	−29.2887	−1.88665	−0.943326	−	0.331869i	$-0.892321\pi$
$241$	−29.2887	−1.88665	−0.943326	+	0.331869i	$0.892321\pi$
$242$	24.9273	1.60239
$243$	0	0
$244$	2.06827	0.132408
$245$	0	0
$246$	0	0
$247$	−1.98865	−0.126535
$248$	68.0242	4.31954
$249$	0	0
$250$	32.2328	2.03858
$251$	22.7856	1.43821	0.719106	−	0.694901i	$-0.244552\pi$
$251$	22.7856	1.43821	0.719106	+	0.694901i	$0.244552\pi$
$252$	0	0
$253$	0.384215	0.0241554
$254$	22.7362	1.42659
$255$	0	0
$256$	31.5642	1.97276
$257$	24.2889	1.51510	0.757550	−	0.652778i	$-0.226396\pi$
$257$	24.2889	1.51510	0.757550	+	0.652778i	$0.226396\pi$
$258$	0	0
$259$	0	0
$260$	27.0857	1.67979
$261$	0	0
$262$	−32.5071	−2.00830
$263$	−8.61155	−0.531011	−0.265506	−	0.964109i	$-0.585539\pi$
$263$	−8.61155	−0.531011	−0.265506	+	0.964109i	$0.585539\pi$
$264$	0	0
$265$	4.42124	0.271595
$266$	0	0
$267$	0	0
$268$	−13.6402	−0.833209
$269$	15.2312	0.928664	0.464332	−	0.885661i	$-0.346294\pi$
$269$	15.2312	0.928664	0.464332	+	0.885661i	$0.346294\pi$
$270$	0	0
$271$	4.67820	0.284181	0.142090	−	0.989854i	$-0.454618\pi$
$271$	4.67820	0.284181	0.142090	+	0.989854i	$0.454618\pi$
$272$	−39.5265	−2.39665
$273$	0	0
$274$	−44.9368	−2.71473
$275$	3.34943	0.201978
$276$	0	0
$277$	−16.3907	−0.984824	−0.492412	−	0.870362i	$-0.663885\pi$
$277$	−16.3907	−0.984824	−0.492412	+	0.870362i	$0.663885\pi$
$278$	−21.4560	−1.28685
$279$	0	0
$280$	0	0
$281$	−3.51404	−0.209630	−0.104815	−	0.994492i	$-0.533425\pi$
$281$	−3.51404	−0.209630	−0.104815	+	0.994492i	$0.533425\pi$
$282$	0	0
$283$	−26.0708	−1.54975	−0.774874	−	0.632116i	$-0.782187\pi$
$283$	−26.0708	−1.54975	−0.774874	+	0.632116i	$0.782187\pi$
$284$	−7.80050	−0.462874
$285$	0	0
$286$	11.6392	0.688237
$287$	0	0
$288$	0	0
$289$	−9.15699	−0.538647
$290$	19.5891	1.15031
$291$	0	0
$292$	2.51479	0.147167
$293$	−18.8838	−1.10321	−0.551603	−	0.834107i	$-0.685984\pi$
$293$	−18.8838	−1.10321	−0.551603	+	0.834107i	$0.685984\pi$
$294$	0	0
$295$	−7.25725	−0.422534
$296$	−73.4484	−4.26910
$297$	0	0
$298$	37.1119	2.14983
$299$	−0.905400	−0.0523606
$300$	0	0
$301$	0	0
$302$	−10.5880	−0.609272
$303$	0	0
$304$	−8.83097	−0.506491
$305$	0.610783	0.0349734
$306$	0	0
$307$	21.6407	1.23510	0.617551	−	0.786531i	$-0.288125\pi$
$307$	21.6407	1.23510	0.617551	+	0.786531i	$0.288125\pi$
$308$	0	0
$309$	0	0
$310$	32.0032	1.81766
$311$	−4.49448	−0.254859	−0.127429	−	0.991848i	$-0.540673\pi$
$311$	−4.49448	−0.254859	−0.127429	+	0.991848i	$0.540673\pi$
$312$	0	0
$313$	−8.60204	−0.486216	−0.243108	−	0.969999i	$-0.578167\pi$
$313$	−8.60204	−0.486216	−0.243108	+	0.969999i	$0.578167\pi$
$314$	−0.799206	−0.0451018
$315$	0	0
$316$	−84.4062	−4.74822
$317$	−8.06255	−0.452838	−0.226419	−	0.974030i	$-0.572702\pi$
$317$	−8.06255	−0.452838	−0.226419	+	0.974030i	$0.572702\pi$
$318$	0	0
$319$	6.13402	0.343439
$320$	41.4105	2.31492
$321$	0	0
$322$	0	0
$323$	1.75228	0.0974992
$324$	0	0
$325$	−7.89291	−0.437820
$326$	−29.0766	−1.61041
$327$	0	0
$328$	−91.7961	−5.06859
$329$	0	0
$330$	0	0
$331$	−22.9026	−1.25884	−0.629419	−	0.777066i	$-0.716707\pi$
$331$	−22.9026	−1.25884	−0.629419	+	0.777066i	$0.716707\pi$
$332$	−75.1361	−4.12363
$333$	0	0
$334$	−8.68150	−0.475030
$335$	−4.02810	−0.220079
$336$	0	0
$337$	13.6378	0.742899	0.371450	−	0.928453i	$-0.378861\pi$
$337$	13.6378	0.742899	0.371450	+	0.928453i	$0.378861\pi$
$338$	7.86916	0.428026
$339$	0	0
$340$	−23.8662	−1.29433
$341$	10.0213	0.542683
$342$	0	0
$343$	0	0
$344$	−57.3005	−3.08943
$345$	0	0
$346$	−31.0543	−1.66949
$347$	−2.82563	−0.151688	−0.0758440	−	0.997120i	$-0.524165\pi$
$347$	−2.82563	−0.151688	−0.0758440	+	0.997120i	$0.524165\pi$
$348$	0	0
$349$	−3.62404	−0.193990	−0.0969951	−	0.995285i	$-0.530923\pi$
$349$	−3.62404	−0.193990	−0.0969951	+	0.995285i	$0.530923\pi$
$350$	0	0
$351$	0	0
$352$	26.9893	1.43854
$353$	2.75401	0.146581	0.0732907	−	0.997311i	$-0.476650\pi$
$353$	2.75401	0.146581	0.0732907	+	0.997311i	$0.476650\pi$
$354$	0	0
$355$	−2.30357	−0.122261
$356$	13.8975	0.736568
$357$	0	0
$358$	−2.98271	−0.157641
$359$	−16.8015	−0.886750	−0.443375	−	0.896336i	$-0.646219\pi$
$359$	−16.8015	−0.886750	−0.443375	+	0.896336i	$0.646219\pi$
$360$	0	0
$361$	−18.6085	−0.979395
$362$	8.66163	0.455245
$363$	0	0
$364$	0	0
$365$	0.742644	0.0388718
$366$	0	0
$367$	23.9339	1.24934	0.624670	−	0.780889i	$-0.285233\pi$
$367$	23.9339	1.24934	0.624670	+	0.780889i	$0.285233\pi$
$368$	−4.02059	−0.209588
$369$	0	0
$370$	−34.5551	−1.79644
$371$	0	0
$372$	0	0
$373$	−19.1606	−0.992098	−0.496049	−	0.868295i	$-0.665216\pi$
$373$	−19.1606	−0.992098	−0.496049	+	0.868295i	$0.665216\pi$
$374$	−10.2557	−0.530309
$375$	0	0
$376$	102.060	5.26334
$377$	−14.4548	−0.744458
$378$	0	0
$379$	10.0770	0.517622	0.258811	−	0.965928i	$-0.416669\pi$
$379$	10.0770	0.517622	0.258811	+	0.965928i	$0.416669\pi$
$380$	−5.33217	−0.273534
$381$	0	0
$382$	−10.5025	−0.537357
$383$	20.1435	1.02929	0.514643	−	0.857405i	$-0.327925\pi$
$383$	20.1435	1.02929	0.514643	+	0.857405i	$0.327925\pi$
$384$	0	0
$385$	0	0
$386$	11.2294	0.571561
$387$	0	0
$388$	77.6743	3.94332
$389$	13.3947	0.679139	0.339570	−	0.940581i	$-0.389719\pi$
$389$	13.3947	0.679139	0.339570	+	0.940581i	$0.389719\pi$
$390$	0	0
$391$	0.797781	0.0403455
$392$	0	0
$393$	0	0
$394$	2.41448	0.121640
$395$	−24.9261	−1.25417
$396$	0	0
$397$	18.0133	0.904061	0.452031	−	0.892002i	$-0.350700\pi$
$397$	18.0133	0.904061	0.452031	+	0.892002i	$0.350700\pi$
$398$	17.1701	0.860661
$399$	0	0
$400$	−35.0499	−1.75249
$401$	28.8675	1.44157	0.720787	−	0.693157i	$-0.243781\pi$
$401$	28.8675	1.44157	0.720787	+	0.693157i	$0.243781\pi$
$402$	0	0
$403$	−23.6151	−1.17635
$404$	−52.8479	−2.62928
$405$	0	0
$406$	0	0
$407$	−10.8204	−0.536347
$408$	0	0
$409$	10.8587	0.536931	0.268465	−	0.963289i	$-0.413484\pi$
$409$	10.8587	0.536931	0.268465	+	0.963289i	$0.413484\pi$
$410$	−43.1872	−2.13286
$411$	0	0
$412$	−59.4027	−2.92656
$413$	0	0
$414$	0	0
$415$	−22.1885	−1.08919
$416$	−63.6001	−3.11825
$417$	0	0
$418$	−2.29131	−0.112072
$419$	−0.495144	−0.0241893	−0.0120947	−	0.999927i	$-0.503850\pi$
$419$	−0.495144	−0.0241893	−0.0120947	+	0.999927i	$0.503850\pi$
$420$	0	0
$421$	−19.0064	−0.926315	−0.463158	−	0.886276i	$-0.653284\pi$
$421$	−19.0064	−0.926315	−0.463158	+	0.886276i	$0.653284\pi$
$422$	31.0226	1.51016
$423$	0	0
$424$	−25.5155	−1.23914
$425$	6.95473	0.337354
$426$	0	0
$427$	0	0
$428$	−10.3373	−0.499671
$429$	0	0
$430$	−26.9580	−1.30003
$431$	−16.9215	−0.815078	−0.407539	−	0.913188i	$-0.633613\pi$
$431$	−16.9215	−0.815078	−0.407539	+	0.913188i	$0.633613\pi$
$432$	0	0
$433$	33.4740	1.60866	0.804330	−	0.594183i	$-0.202524\pi$
$433$	33.4740	1.60866	0.804330	+	0.594183i	$0.202524\pi$
$434$	0	0
$435$	0	0
$436$	−99.9548	−4.78697
$437$	0.178239	0.00852634
$438$	0	0
$439$	−20.9315	−0.999005	−0.499502	−	0.866313i	$-0.666484\pi$
$439$	−20.9315	−0.999005	−0.499502	+	0.866313i	$0.666484\pi$
$440$	19.5891	0.933874
$441$	0	0
$442$	24.1674	1.14953
$443$	−30.8580	−1.46611	−0.733054	−	0.680170i	$-0.761906\pi$
$443$	−30.8580	−1.46611	−0.733054	+	0.680170i	$0.761906\pi$
$444$	0	0
$445$	4.10409	0.194553
$446$	−45.3945	−2.14949
$447$	0	0
$448$	0	0
$449$	−33.2789	−1.57053	−0.785263	−	0.619162i	$-0.787472\pi$
$449$	−33.2789	−1.57053	−0.785263	+	0.619162i	$0.787472\pi$
$450$	0	0
$451$	−13.5234	−0.636791
$452$	−17.1191	−0.805217
$453$	0	0
$454$	46.3474	2.17519
$455$	0	0
$456$	0	0
$457$	23.7904	1.11287	0.556434	−	0.830892i	$-0.312169\pi$
$457$	23.7904	1.11287	0.556434	+	0.830892i	$0.312169\pi$
$458$	53.7309	2.51068
$459$	0	0
$460$	−2.42764	−0.113189
$461$	17.0624	0.794677	0.397339	−	0.917672i	$-0.369934\pi$
$461$	17.0624	0.794677	0.397339	+	0.917672i	$0.369934\pi$
$462$	0	0
$463$	−36.2485	−1.68461	−0.842306	−	0.538999i	$-0.818803\pi$
$463$	−36.2485	−1.68461	−0.842306	+	0.538999i	$0.818803\pi$
$464$	−64.1889	−2.97990
$465$	0	0
$466$	−16.1051	−0.746052
$467$	8.19160	0.379062	0.189531	−	0.981875i	$-0.439303\pi$
$467$	8.19160	0.379062	0.189531	+	0.981875i	$0.439303\pi$
$468$	0	0
$469$	0	0
$470$	48.0159	2.21481
$471$	0	0
$472$	41.8826	1.92780
$473$	−8.44148	−0.388140
$474$	0	0
$475$	1.55382	0.0712941
$476$	0	0
$477$	0	0
$478$	−54.4530	−2.49062
$479$	25.5549	1.16763	0.583817	−	0.811885i	$-0.301559\pi$
$479$	25.5549	1.16763	0.583817	+	0.811885i	$0.301559\pi$
$480$	0	0
$481$	25.4982	1.16262
$482$	79.5227	3.62216
$483$	0	0
$484$	−49.3191	−2.24178
$485$	22.9381	1.04156
$486$	0	0
$487$	−6.92281	−0.313702	−0.156851	−	0.987622i	$-0.550134\pi$
$487$	−6.92281	−0.313702	−0.156851	+	0.987622i	$0.550134\pi$
$488$	−3.52491	−0.159565
$489$	0	0
$490$	0	0
$491$	−37.4524	−1.69021	−0.845103	−	0.534604i	$-0.820461\pi$
$491$	−37.4524	−1.69021	−0.845103	+	0.534604i	$0.820461\pi$
$492$	0	0
$493$	12.7366	0.573628
$494$	5.39946	0.242933
$495$	0	0
$496$	−104.867	−4.70867
$497$	0	0
$498$	0	0
$499$	25.6250	1.14713	0.573566	−	0.819159i	$-0.305560\pi$
$499$	25.6250	1.14713	0.573566	+	0.819159i	$0.305560\pi$
$500$	−63.7733	−2.85203
$501$	0	0
$502$	−61.8658	−2.76121
$503$	5.79692	0.258472	0.129236	−	0.991614i	$-0.458748\pi$
$503$	5.79692	0.258472	0.129236	+	0.991614i	$0.458748\pi$
$504$	0	0
$505$	−15.6066	−0.694483
$506$	−1.04320	−0.0463757
$507$	0	0
$508$	−44.9840	−1.99584
$509$	−25.1395	−1.11429	−0.557144	−	0.830416i	$-0.688103\pi$
$509$	−25.1395	−1.11429	−0.557144	+	0.830416i	$0.688103\pi$
$510$	0	0
$511$	0	0
$512$	−23.9940	−1.06039
$513$	0	0
$514$	−65.9475	−2.90882
$515$	−17.5423	−0.773004
$516$	0	0
$517$	15.0354	0.661257
$518$	0	0
$519$	0	0
$520$	−46.1616	−2.02432
$521$	−7.29656	−0.319668	−0.159834	−	0.987144i	$-0.551096\pi$
$521$	−7.29656	−0.319668	−0.159834	+	0.987144i	$0.551096\pi$
$522$	0	0
$523$	−16.7727	−0.733421	−0.366710	−	0.930335i	$-0.619516\pi$
$523$	−16.7727	−0.733421	−0.366710	+	0.930335i	$0.619516\pi$
$524$	64.3160	2.80966
$525$	0	0
$526$	23.3815	1.01948
$527$	20.8081	0.906416
$528$	0	0
$529$	−22.9189	−0.996472
$530$	−12.0042	−0.521431
$531$	0	0
$532$	0	0
$533$	31.8677	1.38034
$534$	0	0
$535$	−3.05271	−0.131980
$536$	23.2467	1.00411
$537$	0	0
$538$	−41.3548	−1.78293
$539$	0	0
$540$	0	0
$541$	−5.29816	−0.227786	−0.113893	−	0.993493i	$-0.536332\pi$
$541$	−5.29816	−0.227786	−0.113893	+	0.993493i	$0.536332\pi$
$542$	−12.7019	−0.545595
$543$	0	0
$544$	56.0404	2.40271
$545$	−29.5177	−1.26440
$546$	0	0
$547$	−32.8650	−1.40521	−0.702603	−	0.711582i	$-0.747979\pi$
$547$	−32.8650	−1.40521	−0.702603	+	0.711582i	$0.747979\pi$
$548$	88.9084	3.79798
$549$	0	0
$550$	−9.09416	−0.387776
$551$	2.84560	0.121227
$552$	0	0
$553$	0	0
$554$	44.5030	1.89075
$555$	0	0
$556$	42.4512	1.80033
$557$	−18.8160	−0.797258	−0.398629	−	0.917112i	$-0.630514\pi$
$557$	−18.8160	−0.797258	−0.398629	+	0.917112i	$0.630514\pi$
$558$	0	0
$559$	19.8923	0.841354
$560$	0	0
$561$	0	0
$562$	9.54108	0.402466
$563$	−27.6650	−1.16594	−0.582970	−	0.812494i	$-0.698109\pi$
$563$	−27.6650	−1.16594	−0.582970	+	0.812494i	$0.698109\pi$
$564$	0	0
$565$	−5.05547	−0.212685
$566$	70.7856	2.97534
$567$	0	0
$568$	13.2942	0.557812
$569$	−40.1831	−1.68456	−0.842282	−	0.539037i	$-0.818788\pi$
$569$	−40.1831	−1.68456	−0.842282	+	0.539037i	$0.818788\pi$
$570$	0	0
$571$	−6.81129	−0.285044	−0.142522	−	0.989792i	$-0.545521\pi$
$571$	−6.81129	−0.285044	−0.142522	+	0.989792i	$0.545521\pi$
$572$	−23.0283	−0.962862
$573$	0	0
$574$	0	0
$575$	0.707427	0.0295017
$576$	0	0
$577$	36.4222	1.51628	0.758138	−	0.652094i	$-0.226109\pi$
$577$	36.4222	1.51628	0.758138	+	0.652094i	$0.226109\pi$
$578$	24.8625	1.03414
$579$	0	0
$580$	−38.7575	−1.60932
$581$	0	0
$582$	0	0
$583$	−3.75894	−0.155679
$584$	−4.28590	−0.177352
$585$	0	0
$586$	51.2721	2.11803
$587$	11.1589	0.460575	0.230288	−	0.973123i	$-0.426033\pi$
$587$	11.1589	0.460575	0.230288	+	0.973123i	$0.426033\pi$
$588$	0	0
$589$	4.64893	0.191556
$590$	19.7044	0.811218
$591$	0	0
$592$	113.229	4.65369
$593$	−19.8085	−0.813439	−0.406720	−	0.913553i	$-0.633327\pi$
$593$	−19.8085	−0.813439	−0.406720	+	0.913553i	$0.633327\pi$
$594$	0	0
$595$	0	0
$596$	−73.4266	−3.00767
$597$	0	0
$598$	2.45828	0.100527
$599$	−18.1320	−0.740853	−0.370427	−	0.928862i	$-0.620789\pi$
$599$	−18.1320	−0.740853	−0.370427	+	0.928862i	$0.620789\pi$
$600$	0	0
$601$	−24.6570	−1.00578	−0.502889	−	0.864351i	$-0.667730\pi$
$601$	−24.6570	−1.00578	−0.502889	+	0.864351i	$0.667730\pi$
$602$	0	0
$603$	0	0
$604$	20.9486	0.852387
$605$	−14.5645	−0.592130
$606$	0	0
$607$	−17.2775	−0.701273	−0.350637	−	0.936512i	$-0.614035\pi$
$607$	−17.2775	−0.701273	−0.350637	+	0.936512i	$0.614035\pi$
$608$	12.5205	0.507772
$609$	0	0
$610$	−1.65836	−0.0671450
$611$	−35.4308	−1.43338
$612$	0	0
$613$	19.5566	0.789882	0.394941	−	0.918707i	$-0.370765\pi$
$613$	19.5566	0.789882	0.394941	+	0.918707i	$0.370765\pi$
$614$	−58.7575	−2.37126
$615$	0	0
$616$	0	0
$617$	−21.7446	−0.875404	−0.437702	−	0.899120i	$-0.644208\pi$
$617$	−21.7446	−0.875404	−0.437702	+	0.899120i	$0.644208\pi$
$618$	0	0
$619$	−33.8048	−1.35873	−0.679366	−	0.733800i	$-0.737745\pi$
$619$	−33.8048	−1.35873	−0.679366	+	0.733800i	$0.737745\pi$
$620$	−63.3190	−2.54295
$621$	0	0
$622$	12.2031	0.489300
$623$	0	0
$624$	0	0
$625$	−6.41615	−0.256646
$626$	23.3557	0.933481
$627$	0	0
$628$	1.58125	0.0630986
$629$	−22.4674	−0.895832
$630$	0	0
$631$	−23.6410	−0.941134	−0.470567	−	0.882364i	$-0.655951\pi$
$631$	−23.6410	−0.941134	−0.470567	+	0.882364i	$0.655951\pi$
$632$	143.852	5.72211
$633$	0	0
$634$	21.8909	0.869399
$635$	−13.2843	−0.527169
$636$	0	0
$637$	0	0
$638$	−16.6547	−0.659365
$639$	0	0
$640$	−48.9458	−1.93475
$641$	15.9180	0.628724	0.314362	−	0.949303i	$-0.398209\pi$
$641$	15.9180	0.628724	0.314362	+	0.949303i	$0.398209\pi$
$642$	0	0
$643$	−26.5054	−1.04527	−0.522636	−	0.852556i	$-0.675051\pi$
$643$	−26.5054	−1.04527	−0.522636	+	0.852556i	$0.675051\pi$
$644$	0	0
$645$	0	0
$646$	−4.75766	−0.187188
$647$	−0.0160392	−0.000630565 0	−0.000315282	−	1.00000i	$-0.500100\pi$
$647$	−0.0160392	−0.000630565 0	−0.000315282		1.00000i	$0.500100\pi$
$648$	0	0
$649$	6.17012	0.242198
$650$	21.4303	0.840566
$651$	0	0
$652$	57.5287	2.25300
$653$	−33.2879	−1.30266	−0.651328	−	0.758796i	$-0.725788\pi$
$653$	−33.2879	−1.30266	−0.651328	+	0.758796i	$0.725788\pi$
$654$	0	0
$655$	18.9932	0.742126
$656$	141.514	5.52520
$657$	0	0
$658$	0	0
$659$	−38.8312	−1.51265	−0.756324	−	0.654197i	$-0.773007\pi$
$659$	−38.8312	−1.51265	−0.756324	+	0.654197i	$0.773007\pi$
$660$	0	0
$661$	−5.30644	−0.206397	−0.103198	−	0.994661i	$-0.532908\pi$
$661$	−5.30644	−0.206397	−0.103198	+	0.994661i	$0.532908\pi$
$662$	62.1835	2.41683
$663$	0	0
$664$	128.053	4.96941
$665$	0	0
$666$	0	0
$667$	1.29555	0.0501640
$668$	17.1765	0.664579
$669$	0	0
$670$	10.9368	0.422527
$671$	−0.519288	−0.0200469
$672$	0	0
$673$	6.07129	0.234031	0.117016	−	0.993130i	$-0.462667\pi$
$673$	6.07129	0.234031	0.117016	+	0.993130i	$0.462667\pi$
$674$	−37.0285	−1.42628
$675$	0	0
$676$	−15.5693	−0.598819
$677$	34.7850	1.33690	0.668449	−	0.743758i	$-0.266959\pi$
$677$	34.7850	1.33690	0.668449	+	0.743758i	$0.266959\pi$
$678$	0	0
$679$	0	0
$680$	40.6746	1.55980
$681$	0	0
$682$	−27.2091	−1.04189
$683$	19.4241	0.743243	0.371622	−	0.928384i	$-0.378802\pi$
$683$	19.4241	0.743243	0.371622	+	0.928384i	$0.378802\pi$
$684$	0	0
$685$	26.2556	1.00318
$686$	0	0
$687$	0	0
$688$	88.3352	3.36775
$689$	8.85791	0.337459
$690$	0	0
$691$	6.63675	0.252474	0.126237	−	0.992000i	$-0.459710\pi$
$691$	6.63675	0.252474	0.126237	+	0.992000i	$0.459710\pi$
$692$	61.4416	2.33566
$693$	0	0
$694$	7.67197	0.291224
$695$	12.5363	0.475529
$696$	0	0
$697$	−28.0798	−1.06360
$698$	9.83974	0.372440
$699$	0	0
$700$	0	0
$701$	−13.9153	−0.525574	−0.262787	−	0.964854i	$-0.584642\pi$
$701$	−13.9153	−0.525574	−0.262787	+	0.964854i	$0.584642\pi$
$702$	0	0
$703$	−5.01963	−0.189319
$704$	−35.2072	−1.32692
$705$	0	0
$706$	−7.47751	−0.281420
$707$	0	0
$708$	0	0
$709$	34.1556	1.28274	0.641370	−	0.767231i	$-0.278366\pi$
$709$	34.1556	1.28274	0.641370	+	0.767231i	$0.278366\pi$
$710$	6.25450	0.234727
$711$	0	0
$712$	−23.6852	−0.887642
$713$	2.11658	0.0792664
$714$	0	0
$715$	−6.80050	−0.254324
$716$	5.90135	0.220544
$717$	0	0
$718$	45.6183	1.70246
$719$	44.2900	1.65174	0.825870	−	0.563861i	$-0.190684\pi$
$719$	44.2900	1.65174	0.825870	+	0.563861i	$0.190684\pi$
$720$	0	0
$721$	0	0
$722$	50.5246	1.88033
$723$	0	0
$724$	−17.1372	−0.636899
$725$	11.2941	0.419453
$726$	0	0
$727$	28.2494	1.04771	0.523857	−	0.851806i	$-0.324493\pi$
$727$	28.2494	1.04771	0.523857	+	0.851806i	$0.324493\pi$
$728$	0	0
$729$	0	0
$730$	−2.01638	−0.0746295
$731$	−17.5278	−0.648290
$732$	0	0
$733$	25.0169	0.924020	0.462010	−	0.886875i	$-0.347128\pi$
$733$	25.0169	0.924020	0.462010	+	0.886875i	$0.347128\pi$
$734$	−64.9838	−2.39859
$735$	0	0
$736$	5.70036	0.210118
$737$	3.42470	0.126150
$738$	0	0
$739$	32.0230	1.17798	0.588992	−	0.808139i	$-0.299525\pi$
$739$	32.0230	1.17798	0.588992	+	0.808139i	$0.299525\pi$
$740$	68.3680	2.51326
$741$	0	0
$742$	0	0
$743$	−38.8063	−1.42367	−0.711833	−	0.702349i	$-0.752135\pi$
$743$	−38.8063	−1.42367	−0.711833	+	0.702349i	$0.752135\pi$
$744$	0	0
$745$	−21.6837	−0.794428
$746$	52.0236	1.90472
$747$	0	0
$748$	20.2911	0.741915
$749$	0	0
$750$	0	0
$751$	21.6991	0.791811	0.395905	−	0.918291i	$-0.370431\pi$
$751$	21.6991	0.791811	0.395905	+	0.918291i	$0.370431\pi$
$752$	−157.337	−5.73749
$753$	0	0
$754$	39.2466	1.42928
$755$	6.18635	0.225144
$756$	0	0
$757$	33.5242	1.21846	0.609229	−	0.792995i	$-0.291479\pi$
$757$	33.5242	1.21846	0.609229	+	0.792995i	$0.291479\pi$
$758$	−27.3605	−0.993778
$759$	0	0
$760$	9.08748	0.329638
$761$	13.3210	0.482884	0.241442	−	0.970415i	$-0.422380\pi$
$761$	13.3210	0.482884	0.241442	+	0.970415i	$0.422380\pi$
$762$	0	0
$763$	0	0
$764$	20.7795	0.751775
$765$	0	0
$766$	−54.6923	−1.97611
$767$	−14.5398	−0.525003
$768$	0	0
$769$	54.7135	1.97302	0.986510	−	0.163698i	$-0.0523423\pi$
$769$	54.7135	1.97302	0.986510	+	0.163698i	$0.0523423\pi$
$770$	0	0
$771$	0	0
$772$	−22.2176	−0.799628
$773$	−2.36042	−0.0848983	−0.0424491	−	0.999099i	$-0.513516\pi$
$773$	−2.36042	−0.0848983	−0.0424491	+	0.999099i	$0.513516\pi$
$774$	0	0
$775$	18.4515	0.662796
$776$	−132.378	−4.75211
$777$	0	0
$778$	−36.3684	−1.30387
$779$	−6.27356	−0.224774
$780$	0	0
$781$	1.95850	0.0700805
$782$	−2.16608	−0.0774589
$783$	0	0
$784$	0	0
$785$	0.466959	0.0166665
$786$	0	0
$787$	1.66794	0.0594557	0.0297278	−	0.999558i	$-0.490536\pi$
$787$	1.66794	0.0594557	0.0297278	+	0.999558i	$0.490536\pi$
$788$	−4.77709	−0.170177
$789$	0	0
$790$	67.6775	2.40786
$791$	0	0
$792$	0	0
$793$	1.22370	0.0434548
$794$	−48.9085	−1.73570
$795$	0	0
$796$	−33.9715	−1.20409
$797$	−28.6295	−1.01411	−0.507055	−	0.861914i	$-0.669266\pi$
$797$	−28.6295	−1.01411	−0.507055	+	0.861914i	$0.669266\pi$
$798$	0	0
$799$	31.2194	1.10446
$800$	49.6934	1.75693
$801$	0	0
$802$	−78.3791	−2.76766
$803$	−0.631397	−0.0222815
$804$	0	0
$805$	0	0
$806$	64.1181	2.25846
$807$	0	0
$808$	90.0675	3.16856
$809$	−2.85691	−0.100444	−0.0502219	−	0.998738i	$-0.515993\pi$
$809$	−2.85691	−0.100444	−0.0502219	+	0.998738i	$0.515993\pi$
$810$	0	0
$811$	−26.2917	−0.923225	−0.461613	−	0.887082i	$-0.652729\pi$
$811$	−26.2917	−0.923225	−0.461613	+	0.887082i	$0.652729\pi$
$812$	0	0
$813$	0	0
$814$	29.3788	1.02973
$815$	16.9889	0.595094
$816$	0	0
$817$	−3.91605	−0.137005
$818$	−29.4829	−1.03085
$819$	0	0
$820$	85.4467	2.98393
$821$	−2.65849	−0.0927821	−0.0463910	−	0.998923i	$-0.514772\pi$
$821$	−2.65849	−0.0927821	−0.0463910	+	0.998923i	$0.514772\pi$
$822$	0	0
$823$	−12.2154	−0.425801	−0.212901	−	0.977074i	$-0.568291\pi$
$823$	−12.2154	−0.425801	−0.212901	+	0.977074i	$0.568291\pi$
$824$	101.239	3.52681
$825$	0	0
$826$	0	0
$827$	9.15812	0.318459	0.159230	−	0.987242i	$-0.449099\pi$
$827$	9.15812	0.318459	0.159230	+	0.987242i	$0.449099\pi$
$828$	0	0
$829$	−18.3431	−0.637083	−0.318541	−	0.947909i	$-0.603193\pi$
$829$	−18.3431	−0.637083	−0.318541	+	0.947909i	$0.603193\pi$
$830$	60.2447	2.09113
$831$	0	0
$832$	82.9656	2.87631
$833$	0	0
$834$	0	0
$835$	5.07241	0.175538
$836$	4.53341	0.156791
$837$	0	0
$838$	1.34438	0.0464409
$839$	−18.9411	−0.653920	−0.326960	−	0.945038i	$-0.606024\pi$
$839$	−18.9411	−0.653920	−0.326960	+	0.945038i	$0.606024\pi$
$840$	0	0
$841$	−8.31643	−0.286773
$842$	51.6049	1.77842
$843$	0	0
$844$	−61.3789	−2.11275
$845$	−4.59778	−0.158168
$846$	0	0
$847$	0	0
$848$	39.3351	1.35077
$849$	0	0
$850$	−18.8830	−0.647682
$851$	−2.28535	−0.0783408
$852$	0	0
$853$	−19.9584	−0.683364	−0.341682	−	0.939816i	$-0.610997\pi$
$853$	−19.9584	−0.683364	−0.341682	+	0.939816i	$0.610997\pi$
$854$	0	0
$855$	0	0
$856$	17.6176	0.602156
$857$	16.4000	0.560214	0.280107	−	0.959969i	$-0.409630\pi$
$857$	16.4000	0.560214	0.280107	+	0.959969i	$0.409630\pi$
$858$	0	0
$859$	−33.7151	−1.15034	−0.575172	−	0.818033i	$-0.695065\pi$
$859$	−33.7151	−1.15034	−0.575172	+	0.818033i	$0.695065\pi$
$860$	53.3371	1.81878
$861$	0	0
$862$	45.9440	1.56486
$863$	−28.6831	−0.976383	−0.488191	−	0.872737i	$-0.662343\pi$
$863$	−28.6831	−0.976383	−0.488191	+	0.872737i	$0.662343\pi$
$864$	0	0
$865$	18.1444	0.616927
$866$	−90.8865	−3.08845
$867$	0	0
$868$	0	0
$869$	21.1921	0.718894
$870$	0	0
$871$	−8.07027	−0.273451
$872$	170.351	5.76880
$873$	0	0
$874$	−0.483944	−0.0163696
$875$	0	0
$876$	0	0
$877$	−29.5243	−0.996964	−0.498482	−	0.866900i	$-0.666109\pi$
$877$	−29.5243	−0.996964	−0.498482	+	0.866900i	$0.666109\pi$
$878$	56.8317	1.91798
$879$	0	0
$880$	−30.1988	−1.01800
$881$	−57.5032	−1.93733	−0.968666	−	0.248366i	$-0.920107\pi$
$881$	−57.5032	−1.93733	−0.968666	+	0.248366i	$0.920107\pi$
$882$	0	0
$883$	19.8715	0.668730	0.334365	−	0.942444i	$-0.391478\pi$
$883$	19.8715	0.668730	0.334365	+	0.942444i	$0.391478\pi$
$884$	−47.8158	−1.60822
$885$	0	0
$886$	83.7836	2.81477
$887$	−37.0951	−1.24553	−0.622766	−	0.782408i	$-0.713991\pi$
$887$	−37.0951	−1.24553	−0.622766	+	0.782408i	$0.713991\pi$
$888$	0	0
$889$	0	0
$890$	−11.1432	−0.373519
$891$	0	0
$892$	89.8139	3.00719
$893$	6.97501	0.233410
$894$	0	0
$895$	1.74273	0.0582531
$896$	0	0
$897$	0	0
$898$	90.3565	3.01524
$899$	33.7913	1.12700
$900$	0	0
$901$	−7.80503	−0.260023
$902$	36.7177	1.22257
$903$	0	0
$904$	29.1757	0.970371
$905$	−5.06080	−0.168227
$906$	0	0
$907$	24.4088	0.810479	0.405240	−	0.914210i	$-0.367188\pi$
$907$	24.4088	0.810479	0.405240	+	0.914210i	$0.367188\pi$
$908$	−91.6993	−3.04315
$909$	0	0
$910$	0	0
$911$	25.0986	0.831553	0.415776	−	0.909467i	$-0.363510\pi$
$911$	25.0986	0.831553	0.415776	+	0.909467i	$0.363510\pi$
$912$	0	0
$913$	18.8647	0.624330
$914$	−64.5941	−2.13658
$915$	0	0
$916$	−106.308	−3.51250
$917$	0	0
$918$	0	0
$919$	−28.5976	−0.943348	−0.471674	−	0.881773i	$-0.656350\pi$
$919$	−28.5976	−0.943348	−0.471674	+	0.881773i	$0.656350\pi$
$920$	4.13738	0.136405
$921$	0	0
$922$	−46.3268	−1.52569
$923$	−4.61518	−0.151911
$924$	0	0
$925$	−19.9228	−0.655057
$926$	98.4196	3.23427
$927$	0	0
$928$	91.0065	2.98744
$929$	45.4570	1.49140	0.745698	−	0.666284i	$-0.232116\pi$
$929$	45.4570	1.49140	0.745698	+	0.666284i	$0.232116\pi$
$930$	0	0
$931$	0	0
$932$	31.8642	1.04375
$933$	0	0
$934$	−22.2413	−0.727757
$935$	5.99217	0.195965
$936$	0	0
$937$	−27.0083	−0.882322	−0.441161	−	0.897428i	$-0.645433\pi$
$937$	−27.0083	−0.882322	−0.441161	+	0.897428i	$0.645433\pi$
$938$	0	0
$939$	0	0
$940$	−95.0005	−3.09857
$941$	−12.7131	−0.414436	−0.207218	−	0.978295i	$-0.566441\pi$
$941$	−12.7131	−0.414436	−0.207218	+	0.978295i	$0.566441\pi$
$942$	0	0
$943$	−2.85624	−0.0930121
$944$	−64.5667	−2.10147
$945$	0	0
$946$	22.9197	0.745185
$947$	47.5447	1.54500	0.772498	−	0.635017i	$-0.219007\pi$
$947$	47.5447	1.54500	0.772498	+	0.635017i	$0.219007\pi$
$948$	0	0
$949$	1.48788	0.0482987
$950$	−4.21882	−0.136877
$951$	0	0
$952$	0	0
$953$	38.2355	1.23857	0.619285	−	0.785166i	$-0.287423\pi$
$953$	38.2355	1.23857	0.619285	+	0.785166i	$0.287423\pi$
$954$	0	0
$955$	6.13640	0.198569
$956$	107.736	3.48444
$957$	0	0
$958$	−69.3850	−2.24173
$959$	0	0
$960$	0	0
$961$	24.2056	0.780826
$962$	−69.2309	−2.23209
$963$	0	0
$964$	−157.337	−5.06749
$965$	−6.56109	−0.211209
$966$	0	0
$967$	40.8185	1.31263	0.656317	−	0.754485i	$-0.272113\pi$
$967$	40.8185	1.31263	0.656317	+	0.754485i	$0.272113\pi$
$968$	84.0534	2.70158
$969$	0	0
$970$	−62.2799	−1.99969
$971$	−44.9471	−1.44242	−0.721210	−	0.692717i	$-0.756414\pi$
$971$	−44.9471	−1.44242	−0.721210	+	0.692717i	$0.756414\pi$
$972$	0	0
$973$	0	0
$974$	18.7963	0.602274
$975$	0	0
$976$	5.43405	0.173940
$977$	53.5104	1.71195	0.855974	−	0.517018i	$-0.172958\pi$
$977$	53.5104	1.71195	0.855974	+	0.517018i	$0.172958\pi$
$978$	0	0
$979$	−3.48930	−0.111519
$980$	0	0
$981$	0	0
$982$	101.688	3.24501
$983$	−11.6056	−0.370160	−0.185080	−	0.982723i	$-0.559254\pi$
$983$	−11.6056	−0.370160	−0.185080	+	0.982723i	$0.559254\pi$
$984$	0	0
$985$	−1.41073	−0.0449495
$986$	−34.5816	−1.10130
$987$	0	0
$988$	−10.6829	−0.339870
$989$	−1.78291	−0.0566932
$990$	0	0
$991$	26.0091	0.826208	0.413104	−	0.910684i	$-0.364445\pi$
$991$	26.0091	0.826208	0.413104	+	0.910684i	$0.364445\pi$
$992$	148.680	4.72058
$993$	0	0
$994$	0	0
$995$	−10.0321	−0.318040
$996$	0	0
$997$	46.8998	1.48533	0.742666	−	0.669662i	$-0.233561\pi$
$997$	46.8998	1.48533	0.742666	+	0.669662i	$0.233561\pi$
$998$	−69.5753	−2.20237
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.bh.1.2		12
3.2	odd	2		3969.2.a.bi.1.11		12
7.6	odd	2	inner	3969.2.a.bh.1.1		12
9.2	odd	6		1323.2.f.h.442.1		24
9.4	even	3		441.2.f.h.295.11	yes	24
9.5	odd	6		1323.2.f.h.883.1		24
9.7	even	3		441.2.f.h.148.11	✓	24
21.20	even	2		3969.2.a.bi.1.12		12
63.2	odd	6		1323.2.g.h.361.2		24
63.4	even	3		441.2.g.h.79.12		24
63.5	even	6		1323.2.h.h.802.12		24
63.11	odd	6		1323.2.h.h.226.11		24
63.13	odd	6		441.2.f.h.295.12	yes	24
63.16	even	3		441.2.g.h.67.12		24
63.20	even	6		1323.2.f.h.442.2		24
63.23	odd	6		1323.2.h.h.802.11		24
63.25	even	3		441.2.h.h.373.1		24
63.31	odd	6		441.2.g.h.79.11		24
63.32	odd	6		1323.2.g.h.667.2		24
63.34	odd	6		441.2.f.h.148.12	yes	24
63.38	even	6		1323.2.h.h.226.12		24
63.40	odd	6		441.2.h.h.214.2		24
63.41	even	6		1323.2.f.h.883.2		24
63.47	even	6		1323.2.g.h.361.1		24
63.52	odd	6		441.2.h.h.373.2		24
63.58	even	3		441.2.h.h.214.1		24
63.59	even	6		1323.2.g.h.667.1		24
63.61	odd	6		441.2.g.h.67.11		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.f.h.148.11	✓	24	9.7	even	3
441.2.f.h.148.12	yes	24	63.34	odd	6
441.2.f.h.295.11	yes	24	9.4	even	3
441.2.f.h.295.12	yes	24	63.13	odd	6
441.2.g.h.67.11		24	63.61	odd	6
441.2.g.h.67.12		24	63.16	even	3
441.2.g.h.79.11		24	63.31	odd	6
441.2.g.h.79.12		24	63.4	even	3
441.2.h.h.214.1		24	63.58	even	3
441.2.h.h.214.2		24	63.40	odd	6
441.2.h.h.373.1		24	63.25	even	3
441.2.h.h.373.2		24	63.52	odd	6
1323.2.f.h.442.1		24	9.2	odd	6
1323.2.f.h.442.2		24	63.20	even	6
1323.2.f.h.883.1		24	9.5	odd	6
1323.2.f.h.883.2		24	63.41	even	6
1323.2.g.h.361.1		24	63.47	even	6
1323.2.g.h.361.2		24	63.2	odd	6
1323.2.g.h.667.1		24	63.59	even	6
1323.2.g.h.667.2		24	63.32	odd	6
1323.2.h.h.226.11		24	63.11	odd	6
1323.2.h.h.226.12		24	63.38	even	6
1323.2.h.h.802.11		24	63.23	odd	6
1323.2.h.h.802.12		24	63.5	even	6
3969.2.a.bh.1.1		12	7.6	odd	2	inner
3969.2.a.bh.1.2		12	1.1	even	1	trivial
3969.2.a.bi.1.11		12	3.2	odd	2
3969.2.a.bi.1.12		12	21.20	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:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q-2.71513 q^{2} +5.37195 q^{4} +1.58639 q^{5} -9.15528 q^{8} +O(q^{10})\)	\(q-2.71513 q^{2} +5.37195 q^{4} +1.58639 q^{5} -9.15528 q^{8} -4.30727 q^{10} -1.34875 q^{11} +3.17832 q^{13} +14.1139 q^{16} -2.80054 q^{17} -0.625693 q^{19} +8.52202 q^{20} +3.66204 q^{22} -0.284867 q^{23} -2.48336 q^{25} -8.62957 q^{26} -4.54792 q^{29} -7.43005 q^{31} -20.0106 q^{32} +7.60383 q^{34} +8.02252 q^{37} +1.69884 q^{38} -14.5239 q^{40} +10.0266 q^{41} +6.25873 q^{43} -7.24542 q^{44} +0.773452 q^{46} -11.1477 q^{47} +6.74264 q^{50} +17.0738 q^{52} +2.78698 q^{53} -2.13965 q^{55} +12.3482 q^{58} -4.57469 q^{59} +0.385014 q^{61} +20.1736 q^{62} +26.1036 q^{64} +5.04207 q^{65} -2.53916 q^{67} -15.0443 q^{68} -1.45208 q^{71} +0.468134 q^{73} -21.7822 q^{74} -3.36119 q^{76} -15.7124 q^{79} +22.3902 q^{80} -27.2235 q^{82} -13.9868 q^{83} -4.44275 q^{85} -16.9933 q^{86} +12.3482 q^{88} +2.58706 q^{89} -1.53029 q^{92} +30.2674 q^{94} -0.992595 q^{95} +14.4592 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} + 12 q^{4} - 12 q^{8}+O(q^{10}) \) 12 * q - 4 * q^2 + 12 * q^4 - 12 * q^8	\( 12 q - 4 q^{2} + 12 q^{4} - 12 q^{8} - 20 q^{11} + 12 q^{16} - 32 q^{23} + 12 q^{25} - 16 q^{29} - 48 q^{32} + 12 q^{37} - 56 q^{44} - 24 q^{46} + 4 q^{50} - 32 q^{53} + 48 q^{64} - 60 q^{65} + 12 q^{67} - 56 q^{71} - 68 q^{74} - 12 q^{79} - 12 q^{85} - 76 q^{86} - 16 q^{92} - 64 q^{95}+O(q^{100}) \) 12 * q - 4 * q^2 + 12 * q^4 - 12 * q^8 - 20 * q^11 + 12 * q^16 - 32 * q^23 + 12 * q^25 - 16 * q^29 - 48 * q^32 + 12 * q^37 - 56 * q^44 - 24 * q^46 + 4 * q^50 - 32 * q^53 + 48 * q^64 - 60 * q^65 + 12 * q^67 - 56 * q^71 - 68 * q^74 - 12 * q^79 - 12 * q^85 - 76 * q^86 - 16 * q^92 - 64 * q^95

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