LMFDB - Embedded newform 7744.2.a.bo.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7744,2,Mod(1,7744)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7744.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7744, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7744 = 2^{6} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7744.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-3,0,1,0,-1,0,1,0,0,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$61.8361513253$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 44)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	7744.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.381966 q^{3} -0.618034 q^{5} +2.85410 q^{7} -2.85410 q^{9} +4.61803 q^{13} +0.236068 q^{15} -4.61803 q^{17} +2.85410 q^{19} -1.09017 q^{21} -6.47214 q^{23} -4.61803 q^{25} +2.23607 q^{27} +6.38197 q^{29} +5.85410 q^{31} -1.76393 q^{35} -4.85410 q^{37} -1.76393 q^{39} -6.38197 q^{41} +1.76393 q^{45} -3.61803 q^{47} +1.14590 q^{49} +1.76393 q^{51} -7.09017 q^{53} -1.09017 q^{57} -10.0902 q^{59} +4.61803 q^{61} -8.14590 q^{63} -2.85410 q^{65} -4.94427 q^{67} +2.47214 q^{69} +8.61803 q^{71} +12.0902 q^{73} +1.76393 q^{75} -13.8541 q^{79} +7.70820 q^{81} +16.0344 q^{83} +2.85410 q^{85} -2.43769 q^{87} -8.47214 q^{89} +13.1803 q^{91} -2.23607 q^{93} -1.76393 q^{95} +5.56231 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} + q^{5} - q^{7} + q^{9} + 7 q^{13} - 4 q^{15} - 7 q^{17} - q^{19} + 9 q^{21} - 4 q^{23} - 7 q^{25} + 15 q^{29} + 5 q^{31} - 8 q^{35} - 3 q^{37} - 8 q^{39} - 15 q^{41} + 8 q^{45} - 5 q^{47}+ \cdots - 9 q^{97}+O(q^{100}) $ 2 * q - 3 * q^3 + q^5 - q^7 + q^9 + 7 * q^13 - 4 * q^15 - 7 * q^17 - q^19 + 9 * q^21 - 4 * q^23 - 7 * q^25 + 15 * q^29 + 5 * q^31 - 8 * q^35 - 3 * q^37 - 8 * q^39 - 15 * q^41 + 8 * q^45 - 5 * q^47 + 9 * q^49 + 8 * q^51 - 3 * q^53 + 9 * q^57 - 9 * q^59 + 7 * q^61 - 23 * q^63 + q^65 + 8 * q^67 - 4 * q^69 + 15 * q^71 + 13 * q^73 + 8 * q^75 - 21 * q^79 + 2 * q^81 + 3 * q^83 - q^85 - 25 * q^87 - 8 * q^89 + 4 * q^91 - 8 * q^95 - 9 * q^97

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
$3$	−0.381966	−0.220528	−0.110264	−	0.993902i	$-0.535170\pi$
$3$	−0.381966	−0.220528	−0.110264	+	0.993902i	$0.535170\pi$
$4$	0	0
$5$	−0.618034	−0.276393	−0.138197	−	0.990405i	$-0.544131\pi$
$5$	−0.618034	−0.276393	−0.138197	+	0.990405i	$0.544131\pi$
$6$	0	0
$7$	2.85410	1.07875	0.539375	−	0.842066i	$-0.318661\pi$
$7$	2.85410	1.07875	0.539375	+	0.842066i	$0.318661\pi$
$8$	0	0
$9$	−2.85410	−0.951367
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.61803	1.28081	0.640406	−	0.768036i	$-0.278766\pi$
$13$	4.61803	1.28081	0.640406	+	0.768036i	$0.278766\pi$
$14$	0	0
$15$	0.236068	0.0609525
$16$	0	0
$17$	−4.61803	−1.12004	−0.560019	−	0.828480i	$-0.689206\pi$
$17$	−4.61803	−1.12004	−0.560019	+	0.828480i	$0.689206\pi$
$18$	0	0
$19$	2.85410	0.654776	0.327388	−	0.944890i	$-0.393832\pi$
$19$	2.85410	0.654776	0.327388	+	0.944890i	$0.393832\pi$
$20$	0	0
$21$	−1.09017	−0.237895
$22$	0	0
$23$	−6.47214	−1.34953	−0.674767	−	0.738031i	$-0.735756\pi$
$23$	−6.47214	−1.34953	−0.674767	+	0.738031i	$0.735756\pi$
$24$	0	0
$25$	−4.61803	−0.923607
$26$	0	0
$27$	2.23607	0.430331
$28$	0	0
$29$	6.38197	1.18510	0.592551	−	0.805533i	$-0.298121\pi$
$29$	6.38197	1.18510	0.592551	+	0.805533i	$0.298121\pi$
$30$	0	0
$31$	5.85410	1.05143	0.525714	−	0.850661i	$-0.323798\pi$
$31$	5.85410	1.05143	0.525714	+	0.850661i	$0.323798\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.76393	−0.298159
$36$	0	0
$37$	−4.85410	−0.798009	−0.399005	−	0.916949i	$-0.630644\pi$
$37$	−4.85410	−0.798009	−0.399005	+	0.916949i	$0.630644\pi$
$38$	0	0
$39$	−1.76393	−0.282455
$40$	0	0
$41$	−6.38197	−0.996696	−0.498348	−	0.866977i	$-0.666060\pi$
$41$	−6.38197	−0.996696	−0.498348	+	0.866977i	$0.666060\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	1.76393	0.262951
$46$	0	0
$47$	−3.61803	−0.527744	−0.263872	−	0.964558i	$-0.585000\pi$
$47$	−3.61803	−0.527744	−0.263872	+	0.964558i	$0.585000\pi$
$48$	0	0
$49$	1.14590	0.163700
$50$	0	0
$51$	1.76393	0.247000
$52$	0	0
$53$	−7.09017	−0.973910	−0.486955	−	0.873427i	$-0.661892\pi$
$53$	−7.09017	−0.973910	−0.486955	+	0.873427i	$0.661892\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.09017	−0.144397
$58$	0	0
$59$	−10.0902	−1.31363	−0.656814	−	0.754053i	$-0.728096\pi$
$59$	−10.0902	−1.31363	−0.656814	+	0.754053i	$0.728096\pi$
$60$	0	0
$61$	4.61803	0.591279	0.295639	−	0.955300i	$-0.404467\pi$
$61$	4.61803	0.591279	0.295639	+	0.955300i	$0.404467\pi$
$62$	0	0
$63$	−8.14590	−1.02629
$64$	0	0
$65$	−2.85410	−0.354008
$66$	0	0
$67$	−4.94427	−0.604039	−0.302019	−	0.953302i	$-0.597661\pi$
$67$	−4.94427	−0.604039	−0.302019	+	0.953302i	$0.597661\pi$
$68$	0	0
$69$	2.47214	0.297610
$70$	0	0
$71$	8.61803	1.02277	0.511386	−	0.859351i	$-0.329132\pi$
$71$	8.61803	1.02277	0.511386	+	0.859351i	$0.329132\pi$
$72$	0	0
$73$	12.0902	1.41505	0.707524	−	0.706690i	$-0.249812\pi$
$73$	12.0902	1.41505	0.707524	+	0.706690i	$0.249812\pi$
$74$	0	0
$75$	1.76393	0.203681
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−13.8541	−1.55871	−0.779354	−	0.626584i	$-0.784453\pi$
$79$	−13.8541	−1.55871	−0.779354	+	0.626584i	$0.784453\pi$
$80$	0	0
$81$	7.70820	0.856467
$82$	0	0
$83$	16.0344	1.76001	0.880004	−	0.474966i	$-0.157540\pi$
$83$	16.0344	1.76001	0.880004	+	0.474966i	$0.157540\pi$
$84$	0	0
$85$	2.85410	0.309571
$86$	0	0
$87$	−2.43769	−0.261348
$88$	0	0
$89$	−8.47214	−0.898045	−0.449022	−	0.893521i	$-0.648228\pi$
$89$	−8.47214	−0.898045	−0.449022	+	0.893521i	$0.648228\pi$
$90$	0	0
$91$	13.1803	1.38168
$92$	0	0
$93$	−2.23607	−0.231869
$94$	0	0
$95$	−1.76393	−0.180976
$96$	0	0
$97$	5.56231	0.564767	0.282383	−	0.959302i	$-0.408875\pi$
$97$	5.56231	0.564767	0.282383	+	0.959302i	$0.408875\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.79837	−0.676463	−0.338232	−	0.941063i	$-0.609829\pi$
$101$	−6.79837	−0.676463	−0.338232	+	0.941063i	$0.609829\pi$
$102$	0	0
$103$	1.14590	0.112909	0.0564543	−	0.998405i	$-0.482020\pi$
$103$	1.14590	0.112909	0.0564543	+	0.998405i	$0.482020\pi$
$104$	0	0
$105$	0.673762	0.0657524
$106$	0	0
$107$	−8.56231	−0.827749	−0.413875	−	0.910334i	$-0.635825\pi$
$107$	−8.56231	−0.827749	−0.413875	+	0.910334i	$0.635825\pi$
$108$	0	0
$109$	−3.52786	−0.337908	−0.168954	−	0.985624i	$-0.554039\pi$
$109$	−3.52786	−0.337908	−0.168954	+	0.985624i	$0.554039\pi$
$110$	0	0
$111$	1.85410	0.175984
$112$	0	0
$113$	−2.56231	−0.241041	−0.120521	−	0.992711i	$-0.538456\pi$
$113$	−2.56231	−0.241041	−0.120521	+	0.992711i	$0.538456\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	−13.1803	−1.21852
$118$	0	0
$119$	−13.1803	−1.20824
$120$	0	0
$121$	0	0
$122$	0	0
$123$	2.43769	0.219799
$124$	0	0
$125$	5.94427	0.531672
$126$	0	0
$127$	−4.61803	−0.409784	−0.204892	−	0.978785i	$-0.565684\pi$
$127$	−4.61803	−0.409784	−0.204892	+	0.978785i	$0.565684\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−18.4721	−1.61392	−0.806959	−	0.590607i	$-0.798888\pi$
$131$	−18.4721	−1.61392	−0.806959	+	0.590607i	$0.798888\pi$
$132$	0	0
$133$	8.14590	0.706339
$134$	0	0
$135$	−1.38197	−0.118941
$136$	0	0
$137$	−10.7984	−0.922567	−0.461284	−	0.887253i	$-0.652611\pi$
$137$	−10.7984	−0.922567	−0.461284	+	0.887253i	$0.652611\pi$
$138$	0	0
$139$	8.56231	0.726245	0.363123	−	0.931741i	$-0.381711\pi$
$139$	8.56231	0.726245	0.363123	+	0.931741i	$0.381711\pi$
$140$	0	0
$141$	1.38197	0.116383
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−3.94427	−0.327554
$146$	0	0
$147$	−0.437694	−0.0361004
$148$	0	0
$149$	4.61803	0.378324	0.189162	−	0.981946i	$-0.439423\pi$
$149$	4.61803	0.378324	0.189162	+	0.981946i	$0.439423\pi$
$150$	0	0
$151$	−14.2705	−1.16132	−0.580659	−	0.814147i	$-0.697205\pi$
$151$	−14.2705	−1.16132	−0.580659	+	0.814147i	$0.697205\pi$
$152$	0	0
$153$	13.1803	1.06557
$154$	0	0
$155$	−3.61803	−0.290607
$156$	0	0
$157$	2.56231	0.204494	0.102247	−	0.994759i	$-0.467397\pi$
$157$	2.56231	0.204494	0.102247	+	0.994759i	$0.467397\pi$
$158$	0	0
$159$	2.70820	0.214775
$160$	0	0
$161$	−18.4721	−1.45581
$162$	0	0
$163$	−4.32624	−0.338857	−0.169429	−	0.985542i	$-0.554192\pi$
$163$	−4.32624	−0.338857	−0.169429	+	0.985542i	$0.554192\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.79837	−0.526074	−0.263037	−	0.964786i	$-0.584724\pi$
$167$	−6.79837	−0.526074	−0.263037	+	0.964786i	$0.584724\pi$
$168$	0	0
$169$	8.32624	0.640480
$170$	0	0
$171$	−8.14590	−0.622932
$172$	0	0
$173$	17.7984	1.35319	0.676593	−	0.736358i	$-0.263456\pi$
$173$	17.7984	1.35319	0.676593	+	0.736358i	$0.263456\pi$
$174$	0	0
$175$	−13.1803	−0.996340
$176$	0	0
$177$	3.85410	0.289692
$178$	0	0
$179$	−0.381966	−0.0285495	−0.0142747	−	0.999898i	$-0.504544\pi$
$179$	−0.381966	−0.0285495	−0.0142747	+	0.999898i	$0.504544\pi$
$180$	0	0
$181$	−0.618034	−0.0459381	−0.0229691	−	0.999736i	$-0.507312\pi$
$181$	−0.618034	−0.0459381	−0.0229691	+	0.999736i	$0.507312\pi$
$182$	0	0
$183$	−1.76393	−0.130394
$184$	0	0
$185$	3.00000	0.220564
$186$	0	0
$187$	0	0
$188$	0	0
$189$	6.38197	0.464220
$190$	0	0
$191$	3.03444	0.219565	0.109782	−	0.993956i	$-0.464985\pi$
$191$	3.03444	0.219565	0.109782	+	0.993956i	$0.464985\pi$
$192$	0	0
$193$	2.43769	0.175469	0.0877345	−	0.996144i	$-0.472037\pi$
$193$	2.43769	0.175469	0.0877345	+	0.996144i	$0.472037\pi$
$194$	0	0
$195$	1.09017	0.0780687
$196$	0	0
$197$	14.9443	1.06474	0.532368	−	0.846513i	$-0.321302\pi$
$197$	14.9443	1.06474	0.532368	+	0.846513i	$0.321302\pi$
$198$	0	0
$199$	−16.9443	−1.20115	−0.600574	−	0.799569i	$-0.705061\pi$
$199$	−16.9443	−1.20115	−0.600574	+	0.799569i	$0.705061\pi$
$200$	0	0
$201$	1.88854	0.133208
$202$	0	0
$203$	18.2148	1.27843
$204$	0	0
$205$	3.94427	0.275480
$206$	0	0
$207$	18.4721	1.28390
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−23.0902	−1.58959	−0.794796	−	0.606876i	$-0.792422\pi$
$211$	−23.0902	−1.58959	−0.794796	+	0.606876i	$0.792422\pi$
$212$	0	0
$213$	−3.29180	−0.225550
$214$	0	0
$215$	0	0
$216$	0	0
$217$	16.7082	1.13423
$218$	0	0
$219$	−4.61803	−0.312058
$220$	0	0
$221$	−21.3262	−1.43456
$222$	0	0
$223$	18.8541	1.26256	0.631282	−	0.775553i	$-0.282529\pi$
$223$	18.8541	1.26256	0.631282	+	0.775553i	$0.282529\pi$
$224$	0	0
$225$	13.1803	0.878689
$226$	0	0
$227$	−21.3262	−1.41547	−0.707736	−	0.706477i	$-0.750283\pi$
$227$	−21.3262	−1.41547	−0.707736	+	0.706477i	$0.750283\pi$
$228$	0	0
$229$	17.2705	1.14127	0.570634	−	0.821205i	$-0.306698\pi$
$229$	17.2705	1.14127	0.570634	+	0.821205i	$0.306698\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.61803	−0.302537	−0.151269	−	0.988493i	$-0.548336\pi$
$233$	−4.61803	−0.302537	−0.151269	+	0.988493i	$0.548336\pi$
$234$	0	0
$235$	2.23607	0.145865
$236$	0	0
$237$	5.29180	0.343739
$238$	0	0
$239$	8.56231	0.553850	0.276925	−	0.960892i	$-0.410685\pi$
$239$	8.56231	0.553850	0.276925	+	0.960892i	$0.410685\pi$
$240$	0	0
$241$	−3.52786	−0.227250	−0.113625	−	0.993524i	$-0.536246\pi$
$241$	−3.52786	−0.227250	−0.113625	+	0.993524i	$0.536246\pi$
$242$	0	0
$243$	−9.65248	−0.619207
$244$	0	0
$245$	−0.708204	−0.0452455
$246$	0	0
$247$	13.1803	0.838645
$248$	0	0
$249$	−6.12461	−0.388132
$250$	0	0
$251$	1.20163	0.0758460	0.0379230	−	0.999281i	$-0.487926\pi$
$251$	1.20163	0.0758460	0.0379230	+	0.999281i	$0.487926\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−1.09017	−0.0682691
$256$	0	0
$257$	−16.0902	−1.00368	−0.501839	−	0.864961i	$-0.667343\pi$
$257$	−16.0902	−1.00368	−0.501839	+	0.864961i	$0.667343\pi$
$258$	0	0
$259$	−13.8541	−0.860852
$260$	0	0
$261$	−18.2148	−1.12747
$262$	0	0
$263$	−29.8885	−1.84301	−0.921503	−	0.388371i	$-0.873038\pi$
$263$	−29.8885	−1.84301	−0.921503	+	0.388371i	$0.873038\pi$
$264$	0	0
$265$	4.38197	0.269182
$266$	0	0
$267$	3.23607	0.198044
$268$	0	0
$269$	12.9098	0.787126	0.393563	−	0.919298i	$-0.371242\pi$
$269$	12.9098	0.787126	0.393563	+	0.919298i	$0.371242\pi$
$270$	0	0
$271$	14.2705	0.866872	0.433436	−	0.901184i	$-0.357301\pi$
$271$	14.2705	0.866872	0.433436	+	0.901184i	$0.357301\pi$
$272$	0	0
$273$	−5.03444	−0.304698
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2.43769	−0.146467	−0.0732334	−	0.997315i	$-0.523332\pi$
$277$	−2.43769	−0.146467	−0.0732334	+	0.997315i	$0.523332\pi$
$278$	0	0
$279$	−16.7082	−1.00029
$280$	0	0
$281$	−11.6738	−0.696398	−0.348199	−	0.937421i	$-0.613207\pi$
$281$	−11.6738	−0.696398	−0.348199	+	0.937421i	$0.613207\pi$
$282$	0	0
$283$	−8.56231	−0.508976	−0.254488	−	0.967076i	$-0.581907\pi$
$283$	−8.56231	−0.508976	−0.254488	+	0.967076i	$0.581907\pi$
$284$	0	0
$285$	0.673762	0.0399102
$286$	0	0
$287$	−18.2148	−1.07518
$288$	0	0
$289$	4.32624	0.254485
$290$	0	0
$291$	−2.12461	−0.124547
$292$	0	0
$293$	6.38197	0.372838	0.186419	−	0.982470i	$-0.440312\pi$
$293$	6.38197	0.372838	0.186419	+	0.982470i	$0.440312\pi$
$294$	0	0
$295$	6.23607	0.363078
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−29.8885	−1.72850
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2.59675	0.149179
$304$	0	0
$305$	−2.85410	−0.163425
$306$	0	0
$307$	−18.4721	−1.05426	−0.527130	−	0.849785i	$-0.676732\pi$
$307$	−18.4721	−1.05426	−0.527130	+	0.849785i	$0.676732\pi$
$308$	0	0
$309$	−0.437694	−0.0248995
$310$	0	0
$311$	7.79837	0.442205	0.221103	−	0.975251i	$-0.429034\pi$
$311$	7.79837	0.442205	0.221103	+	0.975251i	$0.429034\pi$
$312$	0	0
$313$	30.5066	1.72433	0.862167	−	0.506624i	$-0.169107\pi$
$313$	30.5066	1.72433	0.862167	+	0.506624i	$0.169107\pi$
$314$	0	0
$315$	5.03444	0.283659
$316$	0	0
$317$	−5.56231	−0.312410	−0.156205	−	0.987725i	$-0.549926\pi$
$317$	−5.56231	−0.312410	−0.156205	+	0.987725i	$0.549926\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	3.27051	0.182542
$322$	0	0
$323$	−13.1803	−0.733374
$324$	0	0
$325$	−21.3262	−1.18297
$326$	0	0
$327$	1.34752	0.0745183
$328$	0	0
$329$	−10.3262	−0.569304
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	0	0
$333$	13.8541	0.759200
$334$	0	0
$335$	3.05573	0.166952
$336$	0	0
$337$	5.03444	0.274244	0.137122	−	0.990554i	$-0.456215\pi$
$337$	5.03444	0.274244	0.137122	+	0.990554i	$0.456215\pi$
$338$	0	0
$339$	0.978714	0.0531564
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−16.7082	−0.902158
$344$	0	0
$345$	−1.52786	−0.0822574
$346$	0	0
$347$	−6.79837	−0.364956	−0.182478	−	0.983210i	$-0.558412\pi$
$347$	−6.79837	−0.364956	−0.182478	+	0.983210i	$0.558412\pi$
$348$	0	0
$349$	−0.673762	−0.0360657	−0.0180328	−	0.999837i	$-0.505740\pi$
$349$	−0.673762	−0.0360657	−0.0180328	+	0.999837i	$0.505740\pi$
$350$	0	0
$351$	10.3262	0.551174
$352$	0	0
$353$	2.94427	0.156708	0.0783539	−	0.996926i	$-0.475034\pi$
$353$	2.94427	0.156708	0.0783539	+	0.996926i	$0.475034\pi$
$354$	0	0
$355$	−5.32624	−0.282687
$356$	0	0
$357$	5.03444	0.266451
$358$	0	0
$359$	−8.56231	−0.451901	−0.225951	−	0.974139i	$-0.572549\pi$
$359$	−8.56231	−0.451901	−0.225951	+	0.974139i	$0.572549\pi$
$360$	0	0
$361$	−10.8541	−0.571269
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.47214	−0.391109
$366$	0	0
$367$	−26.8541	−1.40177	−0.700886	−	0.713273i	$-0.747212\pi$
$367$	−26.8541	−1.40177	−0.700886	+	0.713273i	$0.747212\pi$
$368$	0	0
$369$	18.2148	0.948224
$370$	0	0
$371$	−20.2361	−1.05060
$372$	0	0
$373$	−33.4164	−1.73024	−0.865118	−	0.501568i	$-0.832757\pi$
$373$	−33.4164	−1.73024	−0.865118	+	0.501568i	$0.832757\pi$
$374$	0	0
$375$	−2.27051	−0.117249
$376$	0	0
$377$	29.4721	1.51789
$378$	0	0
$379$	9.85410	0.506171	0.253086	−	0.967444i	$-0.418555\pi$
$379$	9.85410	0.506171	0.253086	+	0.967444i	$0.418555\pi$
$380$	0	0
$381$	1.76393	0.0903690
$382$	0	0
$383$	−31.0902	−1.58863	−0.794317	−	0.607504i	$-0.792171\pi$
$383$	−31.0902	−1.58863	−0.794317	+	0.607504i	$0.792171\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.20163	0.111627	0.0558134	−	0.998441i	$-0.482225\pi$
$389$	2.20163	0.111627	0.0558134	+	0.998441i	$0.482225\pi$
$390$	0	0
$391$	29.8885	1.51153
$392$	0	0
$393$	7.05573	0.355914
$394$	0	0
$395$	8.56231	0.430816
$396$	0	0
$397$	20.8328	1.04557	0.522785	−	0.852465i	$-0.324893\pi$
$397$	20.8328	1.04557	0.522785	+	0.852465i	$0.324893\pi$
$398$	0	0
$399$	−3.11146	−0.155768
$400$	0	0
$401$	−35.7426	−1.78490	−0.892451	−	0.451144i	$-0.851016\pi$
$401$	−35.7426	−1.78490	−0.892451	+	0.451144i	$0.851016\pi$
$402$	0	0
$403$	27.0344	1.34668
$404$	0	0
$405$	−4.76393	−0.236722
$406$	0	0
$407$	0	0
$408$	0	0
$409$	13.8541	0.685041	0.342521	−	0.939510i	$-0.388719\pi$
$409$	13.8541	0.685041	0.342521	+	0.939510i	$0.388719\pi$
$410$	0	0
$411$	4.12461	0.203452
$412$	0	0
$413$	−28.7984	−1.41708
$414$	0	0
$415$	−9.90983	−0.486454
$416$	0	0
$417$	−3.27051	−0.160158
$418$	0	0
$419$	−31.4164	−1.53479	−0.767396	−	0.641173i	$-0.778448\pi$
$419$	−31.4164	−1.53479	−0.767396	+	0.641173i	$0.778448\pi$
$420$	0	0
$421$	−24.2705	−1.18287	−0.591436	−	0.806352i	$-0.701439\pi$
$421$	−24.2705	−1.18287	−0.591436	+	0.806352i	$0.701439\pi$
$422$	0	0
$423$	10.3262	0.502079
$424$	0	0
$425$	21.3262	1.03447
$426$	0	0
$427$	13.1803	0.637841
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−32.3262	−1.55710	−0.778550	−	0.627583i	$-0.784044\pi$
$431$	−32.3262	−1.55710	−0.778550	+	0.627583i	$0.784044\pi$
$432$	0	0
$433$	−36.4508	−1.75172	−0.875858	−	0.482569i	$-0.839704\pi$
$433$	−36.4508	−1.75172	−0.875858	+	0.482569i	$0.839704\pi$
$434$	0	0
$435$	1.50658	0.0722349
$436$	0	0
$437$	−18.4721	−0.883642
$438$	0	0
$439$	11.4164	0.544875	0.272438	−	0.962173i	$-0.412170\pi$
$439$	11.4164	0.544875	0.272438	+	0.962173i	$0.412170\pi$
$440$	0	0
$441$	−3.27051	−0.155739
$442$	0	0
$443$	−30.2705	−1.43820	−0.719098	−	0.694909i	$-0.755445\pi$
$443$	−30.2705	−1.43820	−0.719098	+	0.694909i	$0.755445\pi$
$444$	0	0
$445$	5.23607	0.248213
$446$	0	0
$447$	−1.76393	−0.0834311
$448$	0	0
$449$	5.56231	0.262501	0.131251	−	0.991349i	$-0.458101\pi$
$449$	5.56231	0.262501	0.131251	+	0.991349i	$0.458101\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	5.45085	0.256103
$454$	0	0
$455$	−8.14590	−0.381886
$456$	0	0
$457$	13.8541	0.648068	0.324034	−	0.946046i	$-0.394961\pi$
$457$	13.8541	0.648068	0.324034	+	0.946046i	$0.394961\pi$
$458$	0	0
$459$	−10.3262	−0.481988
$460$	0	0
$461$	37.7771	1.75945	0.879727	−	0.475479i	$-0.157725\pi$
$461$	37.7771	1.75945	0.879727	+	0.475479i	$0.157725\pi$
$462$	0	0
$463$	12.0000	0.557687	0.278844	−	0.960337i	$-0.410049\pi$
$463$	12.0000	0.557687	0.278844	+	0.960337i	$0.410049\pi$
$464$	0	0
$465$	1.38197	0.0640871
$466$	0	0
$467$	28.3262	1.31078	0.655391	−	0.755290i	$-0.272504\pi$
$467$	28.3262	1.31078	0.655391	+	0.755290i	$0.272504\pi$
$468$	0	0
$469$	−14.1115	−0.651607
$470$	0	0
$471$	−0.978714	−0.0450967
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−13.1803	−0.604755
$476$	0	0
$477$	20.2361	0.926546
$478$	0	0
$479$	−11.6738	−0.533388	−0.266694	−	0.963781i	$-0.585931\pi$
$479$	−11.6738	−0.533388	−0.266694	+	0.963781i	$0.585931\pi$
$480$	0	0
$481$	−22.4164	−1.02210
$482$	0	0
$483$	7.05573	0.321047
$484$	0	0
$485$	−3.43769	−0.156098
$486$	0	0
$487$	−11.0344	−0.500018	−0.250009	−	0.968243i	$-0.580434\pi$
$487$	−11.0344	−0.500018	−0.250009	+	0.968243i	$0.580434\pi$
$488$	0	0
$489$	1.65248	0.0747275
$490$	0	0
$491$	8.56231	0.386411	0.193206	−	0.981158i	$-0.438112\pi$
$491$	8.56231	0.386411	0.193206	+	0.981158i	$0.438112\pi$
$492$	0	0
$493$	−29.4721	−1.32736
$494$	0	0
$495$	0	0
$496$	0	0
$497$	24.5967	1.10331
$498$	0	0
$499$	10.2705	0.459771	0.229886	−	0.973218i	$-0.426165\pi$
$499$	10.2705	0.459771	0.229886	+	0.973218i	$0.426165\pi$
$500$	0	0
$501$	2.59675	0.116014
$502$	0	0
$503$	1.50658	0.0671750	0.0335875	−	0.999436i	$-0.489307\pi$
$503$	1.50658	0.0671750	0.0335875	+	0.999436i	$0.489307\pi$
$504$	0	0
$505$	4.20163	0.186970
$506$	0	0
$507$	−3.18034	−0.141244
$508$	0	0
$509$	−31.6869	−1.40450	−0.702249	−	0.711931i	$-0.747821\pi$
$509$	−31.6869	−1.40450	−0.702249	+	0.711931i	$0.747821\pi$
$510$	0	0
$511$	34.5066	1.52648
$512$	0	0
$513$	6.38197	0.281771
$514$	0	0
$515$	−0.708204	−0.0312072
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−6.79837	−0.298415
$520$	0	0
$521$	23.3262	1.02194	0.510971	−	0.859598i	$-0.329286\pi$
$521$	23.3262	1.02194	0.510971	+	0.859598i	$0.329286\pi$
$522$	0	0
$523$	27.4508	1.20034	0.600171	−	0.799872i	$-0.295099\pi$
$523$	27.4508	1.20034	0.600171	+	0.799872i	$0.295099\pi$
$524$	0	0
$525$	5.03444	0.219721
$526$	0	0
$527$	−27.0344	−1.17764
$528$	0	0
$529$	18.8885	0.821241
$530$	0	0
$531$	28.7984	1.24974
$532$	0	0
$533$	−29.4721	−1.27658
$534$	0	0
$535$	5.29180	0.228784
$536$	0	0
$537$	0.145898	0.00629596
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−25.2705	−1.08646	−0.543232	−	0.839583i	$-0.682800\pi$
$541$	−25.2705	−1.08646	−0.543232	+	0.839583i	$0.682800\pi$
$542$	0	0
$543$	0.236068	0.0101306
$544$	0	0
$545$	2.18034	0.0933955
$546$	0	0
$547$	−22.6738	−0.969460	−0.484730	−	0.874664i	$-0.661082\pi$
$547$	−22.6738	−0.969460	−0.484730	+	0.874664i	$0.661082\pi$
$548$	0	0
$549$	−13.1803	−0.562523
$550$	0	0
$551$	18.2148	0.775976
$552$	0	0
$553$	−39.5410	−1.68146
$554$	0	0
$555$	−1.14590	−0.0486407
$556$	0	0
$557$	10.7426	0.455181	0.227590	−	0.973757i	$-0.426915\pi$
$557$	10.7426	0.455181	0.227590	+	0.973757i	$0.426915\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−16.0344	−0.675771	−0.337886	−	0.941187i	$-0.609712\pi$
$563$	−16.0344	−0.675771	−0.337886	+	0.941187i	$0.609712\pi$
$564$	0	0
$565$	1.58359	0.0666222
$566$	0	0
$567$	22.0000	0.923913
$568$	0	0
$569$	34.9230	1.46405	0.732024	−	0.681279i	$-0.238576\pi$
$569$	34.9230	1.46405	0.732024	+	0.681279i	$0.238576\pi$
$570$	0	0
$571$	−18.4721	−0.773035	−0.386517	−	0.922282i	$-0.626322\pi$
$571$	−18.4721	−0.773035	−0.386517	+	0.922282i	$0.626322\pi$
$572$	0	0
$573$	−1.15905	−0.0484202
$574$	0	0
$575$	29.8885	1.24644
$576$	0	0
$577$	−24.9098	−1.03701	−0.518505	−	0.855075i	$-0.673511\pi$
$577$	−24.9098	−1.03701	−0.518505	+	0.855075i	$0.673511\pi$
$578$	0	0
$579$	−0.931116	−0.0386959
$580$	0	0
$581$	45.7639	1.89861
$582$	0	0
$583$	0	0
$584$	0	0
$585$	8.14590	0.336791
$586$	0	0
$587$	−1.14590	−0.0472963	−0.0236481	−	0.999720i	$-0.507528\pi$
$587$	−1.14590	−0.0472963	−0.0236481	+	0.999720i	$0.507528\pi$
$588$	0	0
$589$	16.7082	0.688450
$590$	0	0
$591$	−5.70820	−0.234804
$592$	0	0
$593$	−10.5836	−0.434616	−0.217308	−	0.976103i	$-0.569728\pi$
$593$	−10.5836	−0.434616	−0.217308	+	0.976103i	$0.569728\pi$
$594$	0	0
$595$	8.14590	0.333949
$596$	0	0
$597$	6.47214	0.264887
$598$	0	0
$599$	0.978714	0.0399892	0.0199946	−	0.999800i	$-0.493635\pi$
$599$	0.978714	0.0399892	0.0199946	+	0.999800i	$0.493635\pi$
$600$	0	0
$601$	−13.4377	−0.548135	−0.274067	−	0.961710i	$-0.588369\pi$
$601$	−13.4377	−0.548135	−0.274067	+	0.961710i	$0.588369\pi$
$602$	0	0
$603$	14.1115	0.574663
$604$	0	0
$605$	0	0
$606$	0	0
$607$	41.5623	1.68696	0.843481	−	0.537159i	$-0.180503\pi$
$607$	41.5623	1.68696	0.843481	+	0.537159i	$0.180503\pi$
$608$	0	0
$609$	−6.95743	−0.281929
$610$	0	0
$611$	−16.7082	−0.675942
$612$	0	0
$613$	−30.5623	−1.23440	−0.617200	−	0.786806i	$-0.711733\pi$
$613$	−30.5623	−1.23440	−0.617200	+	0.786806i	$0.711733\pi$
$614$	0	0
$615$	−1.50658	−0.0607511
$616$	0	0
$617$	−16.4721	−0.663143	−0.331572	−	0.943430i	$-0.607579\pi$
$617$	−16.4721	−0.663143	−0.331572	+	0.943430i	$0.607579\pi$
$618$	0	0
$619$	6.67376	0.268241	0.134121	−	0.990965i	$-0.457179\pi$
$619$	6.67376	0.268241	0.134121	+	0.990965i	$0.457179\pi$
$620$	0	0
$621$	−14.4721	−0.580747
$622$	0	0
$623$	−24.1803	−0.968765
$624$	0	0
$625$	19.4164	0.776656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	22.4164	0.893801
$630$	0	0
$631$	17.1459	0.682567	0.341284	−	0.939960i	$-0.389138\pi$
$631$	17.1459	0.682567	0.341284	+	0.939960i	$0.389138\pi$
$632$	0	0
$633$	8.81966	0.350550
$634$	0	0
$635$	2.85410	0.113262
$636$	0	0
$637$	5.29180	0.209669
$638$	0	0
$639$	−24.5967	−0.973032
$640$	0	0
$641$	15.9098	0.628401	0.314200	−	0.949357i	$-0.398264\pi$
$641$	15.9098	0.628401	0.314200	+	0.949357i	$0.398264\pi$
$642$	0	0
$643$	21.2705	0.838827	0.419414	−	0.907795i	$-0.362236\pi$
$643$	21.2705	0.838827	0.419414	+	0.907795i	$0.362236\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	35.7426	1.40519	0.702594	−	0.711591i	$-0.252025\pi$
$647$	35.7426	1.40519	0.702594	+	0.711591i	$0.252025\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−6.38197	−0.250129
$652$	0	0
$653$	−23.3262	−0.912826	−0.456413	−	0.889768i	$-0.650866\pi$
$653$	−23.3262	−0.912826	−0.456413	+	0.889768i	$0.650866\pi$
$654$	0	0
$655$	11.4164	0.446076
$656$	0	0
$657$	−34.5066	−1.34623
$658$	0	0
$659$	4.36068	0.169868	0.0849340	−	0.996387i	$-0.472932\pi$
$659$	4.36068	0.169868	0.0849340	+	0.996387i	$0.472932\pi$
$660$	0	0
$661$	−14.3607	−0.558566	−0.279283	−	0.960209i	$-0.590097\pi$
$661$	−14.3607	−0.558566	−0.279283	+	0.960209i	$0.590097\pi$
$662$	0	0
$663$	8.14590	0.316360
$664$	0	0
$665$	−5.03444	−0.195227
$666$	0	0
$667$	−41.3050	−1.59933
$668$	0	0
$669$	−7.20163	−0.278431
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−30.1459	−1.16204	−0.581019	−	0.813890i	$-0.697346\pi$
$673$	−30.1459	−1.16204	−0.581019	+	0.813890i	$0.697346\pi$
$674$	0	0
$675$	−10.3262	−0.397457
$676$	0	0
$677$	34.5066	1.32620	0.663098	−	0.748533i	$-0.269241\pi$
$677$	34.5066	1.32620	0.663098	+	0.748533i	$0.269241\pi$
$678$	0	0
$679$	15.8754	0.609241
$680$	0	0
$681$	8.14590	0.312151
$682$	0	0
$683$	16.9443	0.648355	0.324177	−	0.945996i	$-0.394913\pi$
$683$	16.9443	0.648355	0.324177	+	0.945996i	$0.394913\pi$
$684$	0	0
$685$	6.67376	0.254991
$686$	0	0
$687$	−6.59675	−0.251682
$688$	0	0
$689$	−32.7426	−1.24740
$690$	0	0
$691$	−12.9098	−0.491113	−0.245557	−	0.969382i	$-0.578971\pi$
$691$	−12.9098	−0.491113	−0.245557	+	0.969382i	$0.578971\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.29180	−0.200729
$696$	0	0
$697$	29.4721	1.11634
$698$	0	0
$699$	1.76393	0.0667180
$700$	0	0
$701$	2.02129	0.0763429	0.0381715	−	0.999271i	$-0.487847\pi$
$701$	2.02129	0.0763429	0.0381715	+	0.999271i	$0.487847\pi$
$702$	0	0
$703$	−13.8541	−0.522517
$704$	0	0
$705$	−0.854102	−0.0321673
$706$	0	0
$707$	−19.4033	−0.729734
$708$	0	0
$709$	−7.67376	−0.288194	−0.144097	−	0.989564i	$-0.546028\pi$
$709$	−7.67376	−0.288194	−0.144097	+	0.989564i	$0.546028\pi$
$710$	0	0
$711$	39.5410	1.48290
$712$	0	0
$713$	−37.8885	−1.41894
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3.27051	−0.122139
$718$	0	0
$719$	5.72949	0.213674	0.106837	−	0.994277i	$-0.465928\pi$
$719$	5.72949	0.213674	0.106837	+	0.994277i	$0.465928\pi$
$720$	0	0
$721$	3.27051	0.121800
$722$	0	0
$723$	1.34752	0.0501150
$724$	0	0
$725$	−29.4721	−1.09457
$726$	0	0
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	−19.4377	−0.719915
$730$	0	0
$731$	0	0
$732$	0	0
$733$	13.4377	0.496333	0.248166	−	0.968717i	$-0.420172\pi$
$733$	13.4377	0.496333	0.248166	+	0.968717i	$0.420172\pi$
$734$	0	0
$735$	0.270510	0.00997791
$736$	0	0
$737$	0	0
$738$	0	0
$739$	36.6869	1.34955	0.674775	−	0.738023i	$-0.264241\pi$
$739$	36.6869	1.34955	0.674775	+	0.738023i	$0.264241\pi$
$740$	0	0
$741$	−5.03444	−0.184945
$742$	0	0
$743$	−34.5066	−1.26592	−0.632962	−	0.774183i	$-0.718161\pi$
$743$	−34.5066	−1.26592	−0.632962	+	0.774183i	$0.718161\pi$
$744$	0	0
$745$	−2.85410	−0.104566
$746$	0	0
$747$	−45.7639	−1.67441
$748$	0	0
$749$	−24.4377	−0.892934
$750$	0	0
$751$	−18.0902	−0.660120	−0.330060	−	0.943960i	$-0.607069\pi$
$751$	−18.0902	−0.660120	−0.330060	+	0.943960i	$0.607069\pi$
$752$	0	0
$753$	−0.458980	−0.0167262
$754$	0	0
$755$	8.81966	0.320980
$756$	0	0
$757$	15.7426	0.572176	0.286088	−	0.958203i	$-0.407645\pi$
$757$	15.7426	0.572176	0.286088	+	0.958203i	$0.407645\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−45.9230	−1.66471	−0.832353	−	0.554245i	$-0.813007\pi$
$761$	−45.9230	−1.66471	−0.832353	+	0.554245i	$0.813007\pi$
$762$	0	0
$763$	−10.0689	−0.364518
$764$	0	0
$765$	−8.14590	−0.294516
$766$	0	0
$767$	−46.5967	−1.68251
$768$	0	0
$769$	−10.5836	−0.381654	−0.190827	−	0.981624i	$-0.561117\pi$
$769$	−10.5836	−0.381654	−0.190827	+	0.981624i	$0.561117\pi$
$770$	0	0
$771$	6.14590	0.221339
$772$	0	0
$773$	12.6738	0.455844	0.227922	−	0.973679i	$-0.426807\pi$
$773$	12.6738	0.455844	0.227922	+	0.973679i	$0.426807\pi$
$774$	0	0
$775$	−27.0344	−0.971106
$776$	0	0
$777$	5.29180	0.189842
$778$	0	0
$779$	−18.2148	−0.652612
$780$	0	0
$781$	0	0
$782$	0	0
$783$	14.2705	0.509986
$784$	0	0
$785$	−1.58359	−0.0565208
$786$	0	0
$787$	−32.3262	−1.15231	−0.576153	−	0.817342i	$-0.695447\pi$
$787$	−32.3262	−1.15231	−0.576153	+	0.817342i	$0.695447\pi$
$788$	0	0
$789$	11.4164	0.406435
$790$	0	0
$791$	−7.31308	−0.260023
$792$	0	0
$793$	21.3262	0.757317
$794$	0	0
$795$	−1.67376	−0.0593622
$796$	0	0
$797$	10.7984	0.382498	0.191249	−	0.981542i	$-0.438746\pi$
$797$	10.7984	0.382498	0.191249	+	0.981542i	$0.438746\pi$
$798$	0	0
$799$	16.7082	0.591094
$800$	0	0
$801$	24.1803	0.854370
$802$	0	0
$803$	0	0
$804$	0	0
$805$	11.4164	0.402376
$806$	0	0
$807$	−4.93112	−0.173584
$808$	0	0
$809$	−45.9230	−1.61457	−0.807283	−	0.590164i	$-0.799063\pi$
$809$	−45.9230	−1.61457	−0.807283	+	0.590164i	$0.799063\pi$
$810$	0	0
$811$	28.3820	0.996626	0.498313	−	0.866997i	$-0.333953\pi$
$811$	28.3820	0.996626	0.498313	+	0.866997i	$0.333953\pi$
$812$	0	0
$813$	−5.45085	−0.191170
$814$	0	0
$815$	2.67376	0.0936578
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−37.6180	−1.31448
$820$	0	0
$821$	−19.1459	−0.668196	−0.334098	−	0.942538i	$-0.608432\pi$
$821$	−19.1459	−0.668196	−0.334098	+	0.942538i	$0.608432\pi$
$822$	0	0
$823$	1.49342	0.0520574	0.0260287	−	0.999661i	$-0.491714\pi$
$823$	1.49342	0.0520574	0.0260287	+	0.999661i	$0.491714\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−16.0344	−0.557572	−0.278786	−	0.960353i	$-0.589932\pi$
$827$	−16.0344	−0.557572	−0.278786	+	0.960353i	$0.589932\pi$
$828$	0	0
$829$	47.8673	1.66250	0.831249	−	0.555900i	$-0.187626\pi$
$829$	47.8673	1.66250	0.831249	+	0.555900i	$0.187626\pi$
$830$	0	0
$831$	0.931116	0.0323001
$832$	0	0
$833$	−5.29180	−0.183350
$834$	0	0
$835$	4.20163	0.145403
$836$	0	0
$837$	13.0902	0.452462
$838$	0	0
$839$	0.381966	0.0131869	0.00659347	−	0.999978i	$-0.497901\pi$
$839$	0.381966	0.0131869	0.00659347	+	0.999978i	$0.497901\pi$
$840$	0	0
$841$	11.7295	0.404465
$842$	0	0
$843$	4.45898	0.153575
$844$	0	0
$845$	−5.14590	−0.177024
$846$	0	0
$847$	0	0
$848$	0	0
$849$	3.27051	0.112244
$850$	0	0
$851$	31.4164	1.07694
$852$	0	0
$853$	−6.79837	−0.232772	−0.116386	−	0.993204i	$-0.537131\pi$
$853$	−6.79837	−0.232772	−0.116386	+	0.993204i	$0.537131\pi$
$854$	0	0
$855$	5.03444	0.172174
$856$	0	0
$857$	14.9443	0.510487	0.255243	−	0.966877i	$-0.417844\pi$
$857$	14.9443	0.510487	0.255243	+	0.966877i	$0.417844\pi$
$858$	0	0
$859$	−48.9443	−1.66996	−0.834979	−	0.550283i	$-0.814520\pi$
$859$	−48.9443	−1.66996	−0.834979	+	0.550283i	$0.814520\pi$
$860$	0	0
$861$	6.95743	0.237108
$862$	0	0
$863$	−26.2148	−0.892362	−0.446181	−	0.894943i	$-0.647216\pi$
$863$	−26.2148	−0.892362	−0.446181	+	0.894943i	$0.647216\pi$
$864$	0	0
$865$	−11.0000	−0.374011
$866$	0	0
$867$	−1.65248	−0.0561210
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−22.8328	−0.773660
$872$	0	0
$873$	−15.8754	−0.537300
$874$	0	0
$875$	16.9656	0.573541
$876$	0	0
$877$	43.3262	1.46302	0.731512	−	0.681829i	$-0.238815\pi$
$877$	43.3262	1.46302	0.731512	+	0.681829i	$0.238815\pi$
$878$	0	0
$879$	−2.43769	−0.0822214
$880$	0	0
$881$	−19.8885	−0.670062	−0.335031	−	0.942207i	$-0.608747\pi$
$881$	−19.8885	−0.670062	−0.335031	+	0.942207i	$0.608747\pi$
$882$	0	0
$883$	51.9787	1.74922	0.874611	−	0.484824i	$-0.161117\pi$
$883$	51.9787	1.74922	0.874611	+	0.484824i	$0.161117\pi$
$884$	0	0
$885$	−2.38197	−0.0800689
$886$	0	0
$887$	−27.0344	−0.907728	−0.453864	−	0.891071i	$-0.649955\pi$
$887$	−27.0344	−0.907728	−0.453864	+	0.891071i	$0.649955\pi$
$888$	0	0
$889$	−13.1803	−0.442054
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−10.3262	−0.345554
$894$	0	0
$895$	0.236068	0.00789088
$896$	0	0
$897$	11.4164	0.381183
$898$	0	0
$899$	37.3607	1.24605
$900$	0	0
$901$	32.7426	1.09082
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0.381966	0.0126970
$906$	0	0
$907$	−44.9787	−1.49349	−0.746747	−	0.665108i	$-0.768385\pi$
$907$	−44.9787	−1.49349	−0.746747	+	0.665108i	$0.768385\pi$
$908$	0	0
$909$	19.4033	0.643565
$910$	0	0
$911$	−18.5066	−0.613150	−0.306575	−	0.951846i	$-0.599183\pi$
$911$	−18.5066	−0.613150	−0.306575	+	0.951846i	$0.599183\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	1.09017	0.0360399
$916$	0	0
$917$	−52.7214	−1.74101
$918$	0	0
$919$	−41.5623	−1.37101	−0.685507	−	0.728066i	$-0.740419\pi$
$919$	−41.5623	−1.37101	−0.685507	+	0.728066i	$0.740419\pi$
$920$	0	0
$921$	7.05573	0.232494
$922$	0	0
$923$	39.7984	1.30998
$924$	0	0
$925$	22.4164	0.737047
$926$	0	0
$927$	−3.27051	−0.107418
$928$	0	0
$929$	31.0902	1.02004	0.510018	−	0.860164i	$-0.329639\pi$
$929$	31.0902	1.02004	0.510018	+	0.860164i	$0.329639\pi$
$930$	0	0
$931$	3.27051	0.107187
$932$	0	0
$933$	−2.97871	−0.0975187
$934$	0	0
$935$	0	0
$936$	0	0
$937$	13.8541	0.452594	0.226297	−	0.974058i	$-0.427338\pi$
$937$	13.8541	0.452594	0.226297	+	0.974058i	$0.427338\pi$
$938$	0	0
$939$	−11.6525	−0.380264
$940$	0	0
$941$	23.0902	0.752718	0.376359	−	0.926474i	$-0.377176\pi$
$941$	23.0902	0.752718	0.376359	+	0.926474i	$0.377176\pi$
$942$	0	0
$943$	41.3050	1.34507
$944$	0	0
$945$	−3.94427	−0.128307
$946$	0	0
$947$	47.7771	1.55255	0.776273	−	0.630396i	$-0.217108\pi$
$947$	47.7771	1.55255	0.776273	+	0.630396i	$0.217108\pi$
$948$	0	0
$949$	55.8328	1.81241
$950$	0	0
$951$	2.12461	0.0688953
$952$	0	0
$953$	46.3394	1.50108	0.750540	−	0.660825i	$-0.229793\pi$
$953$	46.3394	1.50108	0.750540	+	0.660825i	$0.229793\pi$
$954$	0	0
$955$	−1.87539	−0.0606861
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−30.8197	−0.995219
$960$	0	0
$961$	3.27051	0.105500
$962$	0	0
$963$	24.4377	0.787494
$964$	0	0
$965$	−1.50658	−0.0484985
$966$	0	0
$967$	44.0000	1.41494	0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000	1.41494	0.707472	+	0.706741i	$0.249835\pi$
$968$	0	0
$969$	5.03444	0.161730
$970$	0	0
$971$	18.0902	0.580541	0.290271	−	0.956945i	$-0.406255\pi$
$971$	18.0902	0.580541	0.290271	+	0.956945i	$0.406255\pi$
$972$	0	0
$973$	24.4377	0.783437
$974$	0	0
$975$	8.14590	0.260878
$976$	0	0
$977$	35.4508	1.13417	0.567087	−	0.823658i	$-0.308070\pi$
$977$	35.4508	1.13417	0.567087	+	0.823658i	$0.308070\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	10.0689	0.321475
$982$	0	0
$983$	5.72949	0.182742	0.0913712	−	0.995817i	$-0.470875\pi$
$983$	5.72949	0.182742	0.0913712	+	0.995817i	$0.470875\pi$
$984$	0	0
$985$	−9.23607	−0.294286
$986$	0	0
$987$	3.94427	0.125548
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	0	0
$993$	4.58359	0.145456
$994$	0	0
$995$	10.4721	0.331989
$996$	0	0
$997$	−34.9230	−1.10602	−0.553011	−	0.833174i	$-0.686521\pi$
$997$	−34.9230	−1.10602	−0.553011	+	0.833174i	$0.686521\pi$
$998$	0	0
$999$	−10.8541	−0.343409

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7744.2.a.bo.1.2		2
4.3	odd	2		7744.2.a.db.1.1		2
8.3	odd	2		484.2.a.c.1.2		2
8.5	even	2		1936.2.a.z.1.1		2
11.7	odd	10		704.2.m.d.577.1		4
11.8	odd	10		704.2.m.d.449.1		4
11.10	odd	2		7744.2.a.bp.1.2		2
24.11	even	2		4356.2.a.u.1.1		2
44.7	even	10		704.2.m.e.577.1		4
44.19	even	10		704.2.m.e.449.1		4
44.43	even	2		7744.2.a.da.1.1		2
88.3	odd	10		484.2.e.c.9.1		4
88.19	even	10		44.2.e.a.9.1	yes	4
88.21	odd	2		1936.2.a.ba.1.1		2
88.27	odd	10		484.2.e.d.245.1		4
88.29	odd	10		176.2.m.b.49.1		4
88.35	even	10		484.2.e.e.81.1		4
88.43	even	2		484.2.a.b.1.2		2
88.51	even	10		44.2.e.a.5.1	✓	4
88.59	odd	10		484.2.e.c.269.1		4
88.75	odd	10		484.2.e.d.81.1		4
88.83	even	10		484.2.e.e.245.1		4
88.85	odd	10		176.2.m.b.97.1		4
264.107	odd	10		396.2.j.a.361.1		4
264.131	odd	2		4356.2.a.t.1.1		2
264.227	odd	10		396.2.j.a.181.1		4
440.19	even	10		1100.2.n.a.801.1		4
440.107	odd	20		1100.2.cb.a.449.1		8
440.139	even	10		1100.2.n.a.401.1		4
440.227	odd	20		1100.2.cb.a.49.2		8
440.283	odd	20		1100.2.cb.a.449.2		8
440.403	odd	20		1100.2.cb.a.49.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.2.e.a.5.1	✓	4	88.51	even	10
44.2.e.a.9.1	yes	4	88.19	even	10
176.2.m.b.49.1		4	88.29	odd	10
176.2.m.b.97.1		4	88.85	odd	10
396.2.j.a.181.1		4	264.227	odd	10
396.2.j.a.361.1		4	264.107	odd	10
484.2.a.b.1.2		2	88.43	even	2
484.2.a.c.1.2		2	8.3	odd	2
484.2.e.c.9.1		4	88.3	odd	10
484.2.e.c.269.1		4	88.59	odd	10
484.2.e.d.81.1		4	88.75	odd	10
484.2.e.d.245.1		4	88.27	odd	10
484.2.e.e.81.1		4	88.35	even	10
484.2.e.e.245.1		4	88.83	even	10
704.2.m.d.449.1		4	11.8	odd	10
704.2.m.d.577.1		4	11.7	odd	10
704.2.m.e.449.1		4	44.19	even	10
704.2.m.e.577.1		4	44.7	even	10
1100.2.n.a.401.1		4	440.139	even	10
1100.2.n.a.801.1		4	440.19	even	10
1100.2.cb.a.49.1		8	440.403	odd	20
1100.2.cb.a.49.2		8	440.227	odd	20
1100.2.cb.a.449.1		8	440.107	odd	20
1100.2.cb.a.449.2		8	440.283	odd	20
1936.2.a.z.1.1		2	8.5	even	2
1936.2.a.ba.1.1		2	88.21	odd	2
4356.2.a.t.1.1		2	264.131	odd	2
4356.2.a.u.1.1		2	24.11	even	2
7744.2.a.bo.1.2		2	1.1	even	1	trivial
7744.2.a.bp.1.2		2	11.10	odd	2
7744.2.a.da.1.1		2	44.43	even	2
7744.2.a.db.1.1		2	4.3	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7744 = 2^{6} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7744.a (trivial)

\(f(q)\)	\(=\)	\(q-0.381966 q^{3} -0.618034 q^{5} +2.85410 q^{7} -2.85410 q^{9} +4.61803 q^{13} +0.236068 q^{15} -4.61803 q^{17} +2.85410 q^{19} -1.09017 q^{21} -6.47214 q^{23} -4.61803 q^{25} +2.23607 q^{27} +6.38197 q^{29} +5.85410 q^{31} -1.76393 q^{35} -4.85410 q^{37} -1.76393 q^{39} -6.38197 q^{41} +1.76393 q^{45} -3.61803 q^{47} +1.14590 q^{49} +1.76393 q^{51} -7.09017 q^{53} -1.09017 q^{57} -10.0902 q^{59} +4.61803 q^{61} -8.14590 q^{63} -2.85410 q^{65} -4.94427 q^{67} +2.47214 q^{69} +8.61803 q^{71} +12.0902 q^{73} +1.76393 q^{75} -13.8541 q^{79} +7.70820 q^{81} +16.0344 q^{83} +2.85410 q^{85} -2.43769 q^{87} -8.47214 q^{89} +13.1803 q^{91} -2.23607 q^{93} -1.76393 q^{95} +5.56231 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} + q^{5} - q^{7} + q^{9} + 7 q^{13} - 4 q^{15} - 7 q^{17} - q^{19} + 9 q^{21} - 4 q^{23} - 7 q^{25} + 15 q^{29} + 5 q^{31} - 8 q^{35} - 3 q^{37} - 8 q^{39} - 15 q^{41} + 8 q^{45} - 5 q^{47}+ \cdots - 9 q^{97}+O(q^{100}) \) 2 * q - 3 * q^3 + q^5 - q^7 + q^9 + 7 * q^13 - 4 * q^15 - 7 * q^17 - q^19 + 9 * q^21 - 4 * q^23 - 7 * q^25 + 15 * q^29 + 5 * q^31 - 8 * q^35 - 3 * q^37 - 8 * q^39 - 15 * q^41 + 8 * q^45 - 5 * q^47 + 9 * q^49 + 8 * q^51 - 3 * q^53 + 9 * q^57 - 9 * q^59 + 7 * q^61 - 23 * q^63 + q^65 + 8 * q^67 - 4 * q^69 + 15 * q^71 + 13 * q^73 + 8 * q^75 - 21 * q^79 + 2 * q^81 + 3 * q^83 - q^85 - 25 * q^87 - 8 * q^89 + 4 * q^91 - 8 * q^95 - 9 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more