LMFDB - Embedded newform 7104.2.a.bx.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7104,2,Mod(1,7104)] 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("7104.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7104, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7104 = 2^{6} \cdot 3 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7104.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,3,0,0,0,1,0,3,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$56.7257255959$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 888)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.363328$ of defining polynomial
Character	$\chi$	$=$	7104.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.50466 q^{5} +3.14134 q^{7} +1.00000 q^{9} +5.14134 q^{11} -3.14134 q^{13} +3.50466 q^{15} +3.08998 q^{17} -1.86799 q^{19} +3.14134 q^{21} -3.63667 q^{23} +7.28267 q^{25} +1.00000 q^{27} -3.50466 q^{29} -0.726656 q^{31} +5.14134 q^{33} +11.0093 q^{35} +1.00000 q^{37} -3.14134 q^{39} +2.00000 q^{41} +5.55602 q^{43} +3.50466 q^{45} -11.7360 q^{47} +2.86799 q^{49} +3.08998 q^{51} -1.68802 q^{53} +18.0187 q^{55} -1.86799 q^{57} +14.2313 q^{59} +6.00000 q^{61} +3.14134 q^{63} -11.0093 q^{65} -3.55602 q^{67} -3.63667 q^{69} +14.0187 q^{71} -10.6974 q^{73} +7.28267 q^{75} +16.1507 q^{77} -7.73599 q^{79} +1.00000 q^{81} -8.87732 q^{83} +10.8294 q^{85} -3.50466 q^{87} +11.6553 q^{89} -9.86799 q^{91} -0.726656 q^{93} -6.54669 q^{95} -10.4626 q^{97} +5.14134 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + q^{7} + 3 q^{9} + 7 q^{11} - q^{13} + 3 q^{17} + 7 q^{19} + q^{21} - 13 q^{23} + 5 q^{25} + 3 q^{27} + 2 q^{31} + 7 q^{33} + 12 q^{35} + 3 q^{37} - q^{39} + 6 q^{41} + 4 q^{43} - 10 q^{47}+ \cdots + 7 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + q^7 + 3 * q^9 + 7 * q^11 - q^13 + 3 * q^17 + 7 * q^19 + q^21 - 13 * q^23 + 5 * q^25 + 3 * q^27 + 2 * q^31 + 7 * q^33 + 12 * q^35 + 3 * q^37 - q^39 + 6 * q^41 + 4 * q^43 - 10 * q^47 - 4 * q^49 + 3 * q^51 - 5 * q^53 + 12 * q^55 + 7 * q^57 + 28 * q^59 + 18 * q^61 + q^63 - 12 * q^65 + 2 * q^67 - 13 * q^69 - 11 * q^73 + 5 * q^75 + 19 * q^77 + 2 * q^79 + 3 * q^81 + 7 * q^83 + 24 * q^85 - 5 * q^89 - 17 * q^91 + 2 * q^93 - 28 * q^95 - 2 * q^97 + 7 * q^99

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
$5$	3.50466	1.56733	0.783667	−	0.621181i	$-0.213347\pi$
$5$	3.50466	1.56733	0.783667	+	0.621181i	$0.213347\pi$
$6$	0	0
$7$	3.14134	1.18731	0.593657	−	0.804718i	$-0.297684\pi$
$7$	3.14134	1.18731	0.593657	+	0.804718i	$0.297684\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.14134	1.55017	0.775086	−	0.631856i	$-0.217707\pi$
$11$	5.14134	1.55017	0.775086	+	0.631856i	$0.217707\pi$
$12$	0	0
$13$	−3.14134	−0.871250	−0.435625	−	0.900128i	$-0.643473\pi$
$13$	−3.14134	−0.871250	−0.435625	+	0.900128i	$0.643473\pi$
$14$	0	0
$15$	3.50466	0.904900
$16$	0	0
$17$	3.08998	0.749431	0.374716	−	0.927140i	$-0.377740\pi$
$17$	3.08998	0.749431	0.374716	+	0.927140i	$0.377740\pi$
$18$	0	0
$19$	−1.86799	−0.428547	−0.214273	−	0.976774i	$-0.568738\pi$
$19$	−1.86799	−0.428547	−0.214273	+	0.976774i	$0.568738\pi$
$20$	0	0
$21$	3.14134	0.685496
$22$	0	0
$23$	−3.63667	−0.758298	−0.379149	−	0.925336i	$-0.623783\pi$
$23$	−3.63667	−0.758298	−0.379149	+	0.925336i	$0.623783\pi$
$24$	0	0
$25$	7.28267	1.45653
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.50466	−0.650800	−0.325400	−	0.945576i	$-0.605499\pi$
$29$	−3.50466	−0.650800	−0.325400	+	0.945576i	$0.605499\pi$
$30$	0	0
$31$	−0.726656	−0.130511	−0.0652557	−	0.997869i	$-0.520786\pi$
$31$	−0.726656	−0.130511	−0.0652557	+	0.997869i	$0.520786\pi$
$32$	0	0
$33$	5.14134	0.894992
$34$	0	0
$35$	11.0093	1.86092
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−3.14134	−0.503016
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	5.55602	0.847284	0.423642	−	0.905830i	$-0.360751\pi$
$43$	5.55602	0.847284	0.423642	+	0.905830i	$0.360751\pi$
$44$	0	0
$45$	3.50466	0.522445
$46$	0	0
$47$	−11.7360	−1.71187	−0.855935	−	0.517084i	$-0.827017\pi$
$47$	−11.7360	−1.71187	−0.855935	+	0.517084i	$0.827017\pi$
$48$	0	0
$49$	2.86799	0.409713
$50$	0	0
$51$	3.08998	0.432684
$52$	0	0
$53$	−1.68802	−0.231868	−0.115934	−	0.993257i	$-0.536986\pi$
$53$	−1.68802	−0.231868	−0.115934	+	0.993257i	$0.536986\pi$
$54$	0	0
$55$	18.0187	2.42964
$56$	0	0
$57$	−1.86799	−0.247422
$58$	0	0
$59$	14.2313	1.85276	0.926380	−	0.376590i	$-0.122903\pi$
$59$	14.2313	1.85276	0.926380	+	0.376590i	$0.122903\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	3.14134	0.395771
$64$	0	0
$65$	−11.0093	−1.36554
$66$	0	0
$67$	−3.55602	−0.434436	−0.217218	−	0.976123i	$-0.569698\pi$
$67$	−3.55602	−0.434436	−0.217218	+	0.976123i	$0.569698\pi$
$68$	0	0
$69$	−3.63667	−0.437804
$70$	0	0
$71$	14.0187	1.66371	0.831854	−	0.554994i	$-0.187279\pi$
$71$	14.0187	1.66371	0.831854	+	0.554994i	$0.187279\pi$
$72$	0	0
$73$	−10.6974	−1.25203	−0.626015	−	0.779811i	$-0.715315\pi$
$73$	−10.6974	−1.25203	−0.626015	+	0.779811i	$0.715315\pi$
$74$	0	0
$75$	7.28267	0.840931
$76$	0	0
$77$	16.1507	1.84054
$78$	0	0
$79$	−7.73599	−0.870366	−0.435183	−	0.900342i	$-0.643316\pi$
$79$	−7.73599	−0.870366	−0.435183	+	0.900342i	$0.643316\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.87732	−0.974413	−0.487206	−	0.873287i	$-0.661984\pi$
$83$	−8.87732	−0.974413	−0.487206	+	0.873287i	$0.661984\pi$
$84$	0	0
$85$	10.8294	1.17461
$86$	0	0
$87$	−3.50466	−0.375739
$88$	0	0
$89$	11.6553	1.23546	0.617731	−	0.786389i	$-0.288052\pi$
$89$	11.6553	1.23546	0.617731	+	0.786389i	$0.288052\pi$
$90$	0	0
$91$	−9.86799	−1.03445
$92$	0	0
$93$	−0.726656	−0.0753508
$94$	0	0
$95$	−6.54669	−0.671676
$96$	0	0
$97$	−10.4626	−1.06232	−0.531160	−	0.847271i	$-0.678244\pi$
$97$	−10.4626	−1.06232	−0.531160	+	0.847271i	$0.678244\pi$
$98$	0	0
$99$	5.14134	0.516724
$100$	0	0
$101$	18.5653	1.84732	0.923660	−	0.383212i	$-0.125182\pi$
$101$	18.5653	1.84732	0.923660	+	0.383212i	$0.125182\pi$
$102$	0	0
$103$	−16.3013	−1.60622	−0.803109	−	0.595832i	$-0.796822\pi$
$103$	−16.3013	−1.60622	−0.803109	+	0.595832i	$0.796822\pi$
$104$	0	0
$105$	11.0093	1.07440
$106$	0	0
$107$	16.9800	1.64152	0.820760	−	0.571273i	$-0.193550\pi$
$107$	16.9800	1.64152	0.820760	+	0.571273i	$0.193550\pi$
$108$	0	0
$109$	12.5946	1.20635	0.603174	−	0.797609i	$-0.293902\pi$
$109$	12.5946	1.20635	0.603174	+	0.797609i	$0.293902\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	0	0
$113$	12.3340	1.16029	0.580144	−	0.814514i	$-0.302996\pi$
$113$	12.3340	1.16029	0.580144	+	0.814514i	$0.302996\pi$
$114$	0	0
$115$	−12.7453	−1.18851
$116$	0	0
$117$	−3.14134	−0.290417
$118$	0	0
$119$	9.70668	0.889810
$120$	0	0
$121$	15.4333	1.40303
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	8.00000	0.715542
$126$	0	0
$127$	−18.4333	−1.63569	−0.817847	−	0.575436i	$-0.804833\pi$
$127$	−18.4333	−1.63569	−0.817847	+	0.575436i	$0.804833\pi$
$128$	0	0
$129$	5.55602	0.489180
$130$	0	0
$131$	−5.60737	−0.489918	−0.244959	−	0.969533i	$-0.578774\pi$
$131$	−5.60737	−0.489918	−0.244959	+	0.969533i	$0.578774\pi$
$132$	0	0
$133$	−5.86799	−0.508819
$134$	0	0
$135$	3.50466	0.301633
$136$	0	0
$137$	−15.2920	−1.30648	−0.653242	−	0.757149i	$-0.726592\pi$
$137$	−15.2920	−1.30648	−0.653242	+	0.757149i	$0.726592\pi$
$138$	0	0
$139$	−7.55602	−0.640893	−0.320446	−	0.947267i	$-0.603833\pi$
$139$	−7.55602	−0.640893	−0.320446	+	0.947267i	$0.603833\pi$
$140$	0	0
$141$	−11.7360	−0.988348
$142$	0	0
$143$	−16.1507	−1.35059
$144$	0	0
$145$	−12.2827	−1.02002
$146$	0	0
$147$	2.86799	0.236548
$148$	0	0
$149$	4.54669	0.372479	0.186240	−	0.982504i	$-0.440370\pi$
$149$	4.54669	0.372479	0.186240	+	0.982504i	$0.440370\pi$
$150$	0	0
$151$	−10.4333	−0.849053	−0.424526	−	0.905416i	$-0.639559\pi$
$151$	−10.4333	−0.849053	−0.424526	+	0.905416i	$0.639559\pi$
$152$	0	0
$153$	3.08998	0.249810
$154$	0	0
$155$	−2.54669	−0.204555
$156$	0	0
$157$	5.45331	0.435222	0.217611	−	0.976036i	$-0.430174\pi$
$157$	5.45331	0.435222	0.217611	+	0.976036i	$0.430174\pi$
$158$	0	0
$159$	−1.68802	−0.133869
$160$	0	0
$161$	−11.4240	−0.900338
$162$	0	0
$163$	−1.14134	−0.0893963	−0.0446982	−	0.999001i	$-0.514233\pi$
$163$	−1.14134	−0.0893963	−0.0446982	+	0.999001i	$0.514233\pi$
$164$	0	0
$165$	18.0187	1.40275
$166$	0	0
$167$	−1.35400	−0.104776	−0.0523878	−	0.998627i	$-0.516683\pi$
$167$	−1.35400	−0.104776	−0.0523878	+	0.998627i	$0.516683\pi$
$168$	0	0
$169$	−3.13201	−0.240924
$170$	0	0
$171$	−1.86799	−0.142849
$172$	0	0
$173$	−15.4427	−1.17408	−0.587042	−	0.809556i	$-0.699708\pi$
$173$	−15.4427	−1.17408	−0.587042	+	0.809556i	$0.699708\pi$
$174$	0	0
$175$	22.8773	1.72936
$176$	0	0
$177$	14.2313	1.06969
$178$	0	0
$179$	1.76868	0.132197	0.0660986	−	0.997813i	$-0.478945\pi$
$179$	1.76868	0.132197	0.0660986	+	0.997813i	$0.478945\pi$
$180$	0	0
$181$	24.5653	1.82593	0.912964	−	0.408040i	$-0.133788\pi$
$181$	24.5653	1.82593	0.912964	+	0.408040i	$0.133788\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	0	0
$185$	3.50466	0.257668
$186$	0	0
$187$	15.8867	1.16175
$188$	0	0
$189$	3.14134	0.228499
$190$	0	0
$191$	2.91002	0.210561	0.105281	−	0.994443i	$-0.466426\pi$
$191$	2.91002	0.210561	0.105281	+	0.994443i	$0.466426\pi$
$192$	0	0
$193$	16.0187	1.15305	0.576524	−	0.817080i	$-0.304409\pi$
$193$	16.0187	1.15305	0.576524	+	0.817080i	$0.304409\pi$
$194$	0	0
$195$	−11.0093	−0.788394
$196$	0	0
$197$	−12.4333	−0.885839	−0.442919	−	0.896561i	$-0.646057\pi$
$197$	−12.4333	−0.885839	−0.442919	+	0.896561i	$0.646057\pi$
$198$	0	0
$199$	0.990671	0.0702268	0.0351134	−	0.999383i	$-0.488821\pi$
$199$	0.990671	0.0702268	0.0351134	+	0.999383i	$0.488821\pi$
$200$	0	0
$201$	−3.55602	−0.250822
$202$	0	0
$203$	−11.0093	−0.772703
$204$	0	0
$205$	7.00933	0.489553
$206$	0	0
$207$	−3.63667	−0.252766
$208$	0	0
$209$	−9.60398	−0.664321
$210$	0	0
$211$	−5.45331	−0.375422	−0.187711	−	0.982224i	$-0.560107\pi$
$211$	−5.45331	−0.375422	−0.187711	+	0.982224i	$0.560107\pi$
$212$	0	0
$213$	14.0187	0.960543
$214$	0	0
$215$	19.4720	1.32798
$216$	0	0
$217$	−2.28267	−0.154958
$218$	0	0
$219$	−10.6974	−0.722860
$220$	0	0
$221$	−9.70668	−0.652942
$222$	0	0
$223$	−6.01866	−0.403039	−0.201520	−	0.979484i	$-0.564588\pi$
$223$	−6.01866	−0.403039	−0.201520	+	0.979484i	$0.564588\pi$
$224$	0	0
$225$	7.28267	0.485511
$226$	0	0
$227$	−25.2406	−1.67528	−0.837640	−	0.546222i	$-0.816065\pi$
$227$	−25.2406	−1.67528	−0.837640	+	0.546222i	$0.816065\pi$
$228$	0	0
$229$	−15.3947	−1.01731	−0.508655	−	0.860970i	$-0.669857\pi$
$229$	−15.3947	−1.01731	−0.508655	+	0.860970i	$0.669857\pi$
$230$	0	0
$231$	16.1507	1.06264
$232$	0	0
$233$	7.45331	0.488283	0.244141	−	0.969740i	$-0.421494\pi$
$233$	7.45331	0.488283	0.244141	+	0.969740i	$0.421494\pi$
$234$	0	0
$235$	−41.1307	−2.68307
$236$	0	0
$237$	−7.73599	−0.502506
$238$	0	0
$239$	−19.7873	−1.27994	−0.639968	−	0.768401i	$-0.721052\pi$
$239$	−19.7873	−1.27994	−0.639968	+	0.768401i	$0.721052\pi$
$240$	0	0
$241$	20.7453	1.33632	0.668162	−	0.744016i	$-0.267081\pi$
$241$	20.7453	1.33632	0.668162	+	0.744016i	$0.267081\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	10.0514	0.642157
$246$	0	0
$247$	5.86799	0.373371
$248$	0	0
$249$	−8.87732	−0.562577
$250$	0	0
$251$	−17.9673	−1.13409	−0.567043	−	0.823688i	$-0.691913\pi$
$251$	−17.9673	−1.13409	−0.567043	+	0.823688i	$0.691913\pi$
$252$	0	0
$253$	−18.6974	−1.17549
$254$	0	0
$255$	10.8294	0.678161
$256$	0	0
$257$	8.54330	0.532916	0.266458	−	0.963846i	$-0.414147\pi$
$257$	8.54330	0.532916	0.266458	+	0.963846i	$0.414147\pi$
$258$	0	0
$259$	3.14134	0.195193
$260$	0	0
$261$	−3.50466	−0.216933
$262$	0	0
$263$	−18.7453	−1.15589	−0.577943	−	0.816077i	$-0.696144\pi$
$263$	−18.7453	−1.15589	−0.577943	+	0.816077i	$0.696144\pi$
$264$	0	0
$265$	−5.91595	−0.363414
$266$	0	0
$267$	11.6553	0.713295
$268$	0	0
$269$	−30.4520	−1.85669	−0.928345	−	0.371719i	$-0.878769\pi$
$269$	−30.4520	−1.85669	−0.928345	+	0.371719i	$0.878769\pi$
$270$	0	0
$271$	18.7267	1.13756	0.568782	−	0.822489i	$-0.307415\pi$
$271$	18.7267	1.13756	0.568782	+	0.822489i	$0.307415\pi$
$272$	0	0
$273$	−9.86799	−0.597238
$274$	0	0
$275$	37.4427	2.25788
$276$	0	0
$277$	14.2534	0.856402	0.428201	−	0.903684i	$-0.359148\pi$
$277$	14.2534	0.856402	0.428201	+	0.903684i	$0.359148\pi$
$278$	0	0
$279$	−0.726656	−0.0435038
$280$	0	0
$281$	−17.8353	−1.06396	−0.531982	−	0.846755i	$-0.678553\pi$
$281$	−17.8353	−1.06396	−0.531982	+	0.846755i	$0.678553\pi$
$282$	0	0
$283$	−14.6974	−0.873667	−0.436833	−	0.899542i	$-0.643900\pi$
$283$	−14.6974	−0.873667	−0.436833	+	0.899542i	$0.643900\pi$
$284$	0	0
$285$	−6.54669	−0.387792
$286$	0	0
$287$	6.28267	0.370854
$288$	0	0
$289$	−7.45199	−0.438353
$290$	0	0
$291$	−10.4626	−0.613331
$292$	0	0
$293$	20.3306	1.18773	0.593864	−	0.804565i	$-0.297602\pi$
$293$	20.3306	1.18773	0.593864	+	0.804565i	$0.297602\pi$
$294$	0	0
$295$	49.8760	2.90389
$296$	0	0
$297$	5.14134	0.298331
$298$	0	0
$299$	11.4240	0.660667
$300$	0	0
$301$	17.4533	1.00599
$302$	0	0
$303$	18.5653	1.06655
$304$	0	0
$305$	21.0280	1.20406
$306$	0	0
$307$	−5.45331	−0.311237	−0.155619	−	0.987817i	$-0.549737\pi$
$307$	−5.45331	−0.311237	−0.155619	+	0.987817i	$0.549737\pi$
$308$	0	0
$309$	−16.3013	−0.927350
$310$	0	0
$311$	−3.78734	−0.214760	−0.107380	−	0.994218i	$-0.534246\pi$
$311$	−3.78734	−0.214760	−0.107380	+	0.994218i	$0.534246\pi$
$312$	0	0
$313$	−15.1307	−0.855237	−0.427619	−	0.903959i	$-0.640647\pi$
$313$	−15.1307	−0.855237	−0.427619	+	0.903959i	$0.640647\pi$
$314$	0	0
$315$	11.0093	0.620305
$316$	0	0
$317$	22.4626	1.26163	0.630814	−	0.775934i	$-0.282721\pi$
$317$	22.4626	1.26163	0.630814	+	0.775934i	$0.282721\pi$
$318$	0	0
$319$	−18.0187	−1.00885
$320$	0	0
$321$	16.9800	0.947733
$322$	0	0
$323$	−5.77207	−0.321166
$324$	0	0
$325$	−22.8773	−1.26901
$326$	0	0
$327$	12.5946	0.696486
$328$	0	0
$329$	−36.8667	−2.03253
$330$	0	0
$331$	23.4720	1.29014	0.645068	−	0.764125i	$-0.276829\pi$
$331$	23.4720	1.29014	0.645068	+	0.764125i	$0.276829\pi$
$332$	0	0
$333$	1.00000	0.0547997
$334$	0	0
$335$	−12.4626	−0.680907
$336$	0	0
$337$	−4.43334	−0.241499	−0.120750	−	0.992683i	$-0.538530\pi$
$337$	−4.43334	−0.241499	−0.120750	+	0.992683i	$0.538530\pi$
$338$	0	0
$339$	12.3340	0.669892
$340$	0	0
$341$	−3.73599	−0.202315
$342$	0	0
$343$	−12.9800	−0.700855
$344$	0	0
$345$	−12.7453	−0.686185
$346$	0	0
$347$	22.5327	1.20962	0.604808	−	0.796371i	$-0.293250\pi$
$347$	22.5327	1.20962	0.604808	+	0.796371i	$0.293250\pi$
$348$	0	0
$349$	27.4134	1.46740	0.733702	−	0.679472i	$-0.237791\pi$
$349$	27.4134	1.46740	0.733702	+	0.679472i	$0.237791\pi$
$350$	0	0
$351$	−3.14134	−0.167672
$352$	0	0
$353$	−11.8060	−0.628370	−0.314185	−	0.949362i	$-0.601731\pi$
$353$	−11.8060	−0.628370	−0.314185	+	0.949362i	$0.601731\pi$
$354$	0	0
$355$	49.1307	2.60759
$356$	0	0
$357$	9.70668	0.513732
$358$	0	0
$359$	−5.82003	−0.307169	−0.153585	−	0.988135i	$-0.549082\pi$
$359$	−5.82003	−0.307169	−0.153585	+	0.988135i	$0.549082\pi$
$360$	0	0
$361$	−15.5106	−0.816348
$362$	0	0
$363$	15.4333	0.810040
$364$	0	0
$365$	−37.4906	−1.96235
$366$	0	0
$367$	8.77462	0.458031	0.229016	−	0.973423i	$-0.426449\pi$
$367$	8.77462	0.458031	0.229016	+	0.973423i	$0.426449\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	−5.30265	−0.275300
$372$	0	0
$373$	−37.4906	−1.94119	−0.970596	−	0.240716i	$-0.922618\pi$
$373$	−37.4906	−1.94119	−0.970596	+	0.240716i	$0.922618\pi$
$374$	0	0
$375$	8.00000	0.413118
$376$	0	0
$377$	11.0093	0.567009
$378$	0	0
$379$	−26.4040	−1.35628	−0.678142	−	0.734931i	$-0.737215\pi$
$379$	−26.4040	−1.35628	−0.678142	+	0.734931i	$0.737215\pi$
$380$	0	0
$381$	−18.4333	−0.944368
$382$	0	0
$383$	−23.4754	−1.19953	−0.599767	−	0.800174i	$-0.704740\pi$
$383$	−23.4754	−1.19953	−0.599767	+	0.800174i	$0.704740\pi$
$384$	0	0
$385$	56.6027	2.88474
$386$	0	0
$387$	5.55602	0.282428
$388$	0	0
$389$	5.89004	0.298637	0.149318	−	0.988789i	$-0.452292\pi$
$389$	5.89004	0.298637	0.149318	+	0.988789i	$0.452292\pi$
$390$	0	0
$391$	−11.2373	−0.568293
$392$	0	0
$393$	−5.60737	−0.282854
$394$	0	0
$395$	−27.1120	−1.36415
$396$	0	0
$397$	−13.4533	−0.675202	−0.337601	−	0.941289i	$-0.609616\pi$
$397$	−13.4533	−0.675202	−0.337601	+	0.941289i	$0.609616\pi$
$398$	0	0
$399$	−5.86799	−0.293767
$400$	0	0
$401$	27.0900	1.35281	0.676405	−	0.736530i	$-0.263537\pi$
$401$	27.0900	1.35281	0.676405	+	0.736530i	$0.263537\pi$
$402$	0	0
$403$	2.28267	0.113708
$404$	0	0
$405$	3.50466	0.174148
$406$	0	0
$407$	5.14134	0.254847
$408$	0	0
$409$	33.0093	1.63221	0.816103	−	0.577906i	$-0.196130\pi$
$409$	33.0093	1.63221	0.816103	+	0.577906i	$0.196130\pi$
$410$	0	0
$411$	−15.2920	−0.754299
$412$	0	0
$413$	44.7054	2.19981
$414$	0	0
$415$	−31.1120	−1.52723
$416$	0	0
$417$	−7.55602	−0.370020
$418$	0	0
$419$	−21.1787	−1.03464	−0.517322	−	0.855791i	$-0.673071\pi$
$419$	−21.1787	−1.03464	−0.517322	+	0.855791i	$0.673071\pi$
$420$	0	0
$421$	−17.5747	−0.856537	−0.428269	−	0.903651i	$-0.640876\pi$
$421$	−17.5747	−0.856537	−0.428269	+	0.903651i	$0.640876\pi$
$422$	0	0
$423$	−11.7360	−0.570623
$424$	0	0
$425$	22.5033	1.09157
$426$	0	0
$427$	18.8480	0.912119
$428$	0	0
$429$	−16.1507	−0.779761
$430$	0	0
$431$	23.7767	1.14528	0.572641	−	0.819806i	$-0.305919\pi$
$431$	23.7767	1.14528	0.572641	+	0.819806i	$0.305919\pi$
$432$	0	0
$433$	22.1320	1.06360	0.531798	−	0.846871i	$-0.321516\pi$
$433$	22.1320	1.06360	0.531798	+	0.846871i	$0.321516\pi$
$434$	0	0
$435$	−12.2827	−0.588909
$436$	0	0
$437$	6.79328	0.324966
$438$	0	0
$439$	20.4040	0.973831	0.486916	−	0.873449i	$-0.338122\pi$
$439$	20.4040	0.973831	0.486916	+	0.873449i	$0.338122\pi$
$440$	0	0
$441$	2.86799	0.136571
$442$	0	0
$443$	15.6774	0.744855	0.372427	−	0.928061i	$-0.378526\pi$
$443$	15.6774	0.744855	0.372427	+	0.928061i	$0.378526\pi$
$444$	0	0
$445$	40.8480	1.93638
$446$	0	0
$447$	4.54669	0.215051
$448$	0	0
$449$	−18.1473	−0.856423	−0.428211	−	0.903679i	$-0.640856\pi$
$449$	−18.1473	−0.856423	−0.428211	+	0.903679i	$0.640856\pi$
$450$	0	0
$451$	10.2827	0.484192
$452$	0	0
$453$	−10.4333	−0.490201
$454$	0	0
$455$	−34.5840	−1.62132
$456$	0	0
$457$	9.83869	0.460234	0.230117	−	0.973163i	$-0.426089\pi$
$457$	9.83869	0.460234	0.230117	+	0.973163i	$0.426089\pi$
$458$	0	0
$459$	3.08998	0.144228
$460$	0	0
$461$	27.7033	1.29027	0.645135	−	0.764068i	$-0.276801\pi$
$461$	27.7033	1.29027	0.645135	+	0.764068i	$0.276801\pi$
$462$	0	0
$463$	−0.689342	−0.0320364	−0.0160182	−	0.999872i	$-0.505099\pi$
$463$	−0.689342	−0.0320364	−0.0160182	+	0.999872i	$0.505099\pi$
$464$	0	0
$465$	−2.54669	−0.118100
$466$	0	0
$467$	−34.9953	−1.61939	−0.809694	−	0.586852i	$-0.800367\pi$
$467$	−34.9953	−1.61939	−0.809694	+	0.586852i	$0.800367\pi$
$468$	0	0
$469$	−11.1706	−0.515812
$470$	0	0
$471$	5.45331	0.251275
$472$	0	0
$473$	28.5653	1.31344
$474$	0	0
$475$	−13.6040	−0.624193
$476$	0	0
$477$	−1.68802	−0.0772893
$478$	0	0
$479$	−40.5033	−1.85065	−0.925323	−	0.379181i	$-0.876206\pi$
$479$	−40.5033	−1.85065	−0.925323	+	0.379181i	$0.876206\pi$
$480$	0	0
$481$	−3.14134	−0.143233
$482$	0	0
$483$	−11.4240	−0.519810
$484$	0	0
$485$	−36.6680	−1.66501
$486$	0	0
$487$	2.81070	0.127365	0.0636825	−	0.997970i	$-0.479715\pi$
$487$	2.81070	0.127365	0.0636825	+	0.997970i	$0.479715\pi$
$488$	0	0
$489$	−1.14134	−0.0516130
$490$	0	0
$491$	−12.3120	−0.555632	−0.277816	−	0.960634i	$-0.589611\pi$
$491$	−12.3120	−0.555632	−0.277816	+	0.960634i	$0.589611\pi$
$492$	0	0
$493$	−10.8294	−0.487730
$494$	0	0
$495$	18.0187	0.809878
$496$	0	0
$497$	44.0373	1.97534
$498$	0	0
$499$	25.4054	1.13730	0.568650	−	0.822580i	$-0.307466\pi$
$499$	25.4054	1.13730	0.568650	+	0.822580i	$0.307466\pi$
$500$	0	0
$501$	−1.35400	−0.0604922
$502$	0	0
$503$	20.9180	0.932689	0.466344	−	0.884603i	$-0.345571\pi$
$503$	20.9180	0.932689	0.466344	+	0.884603i	$0.345571\pi$
$504$	0	0
$505$	65.0653	2.89537
$506$	0	0
$507$	−3.13201	−0.139097
$508$	0	0
$509$	35.0827	1.55501	0.777507	−	0.628874i	$-0.216484\pi$
$509$	35.0827	1.55501	0.777507	+	0.628874i	$0.216484\pi$
$510$	0	0
$511$	−33.6040	−1.48655
$512$	0	0
$513$	−1.86799	−0.0824739
$514$	0	0
$515$	−57.1307	−2.51748
$516$	0	0
$517$	−60.3386	−2.65369
$518$	0	0
$519$	−15.4427	−0.677858
$520$	0	0
$521$	−44.5840	−1.95326	−0.976630	−	0.214926i	$-0.931049\pi$
$521$	−44.5840	−1.95326	−0.976630	+	0.214926i	$0.931049\pi$
$522$	0	0
$523$	−7.00933	−0.306497	−0.153248	−	0.988188i	$-0.548973\pi$
$523$	−7.00933	−0.306497	−0.153248	+	0.988188i	$0.548973\pi$
$524$	0	0
$525$	22.8773	0.998448
$526$	0	0
$527$	−2.24536	−0.0978093
$528$	0	0
$529$	−9.77462	−0.424983
$530$	0	0
$531$	14.2313	0.617587
$532$	0	0
$533$	−6.28267	−0.272133
$534$	0	0
$535$	59.5093	2.57281
$536$	0	0
$537$	1.76868	0.0763241
$538$	0	0
$539$	14.7453	0.635126
$540$	0	0
$541$	−10.2534	−0.440827	−0.220413	−	0.975407i	$-0.570741\pi$
$541$	−10.2534	−0.440827	−0.220413	+	0.975407i	$0.570741\pi$
$542$	0	0
$543$	24.5653	1.05420
$544$	0	0
$545$	44.1400	1.89075
$546$	0	0
$547$	1.76529	0.0754783	0.0377392	−	0.999288i	$-0.487984\pi$
$547$	1.76529	0.0754783	0.0377392	+	0.999288i	$0.487984\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	6.54669	0.278898
$552$	0	0
$553$	−24.3013	−1.03340
$554$	0	0
$555$	3.50466	0.148765
$556$	0	0
$557$	6.88071	0.291545	0.145773	−	0.989318i	$-0.453433\pi$
$557$	6.88071	0.291545	0.145773	+	0.989318i	$0.453433\pi$
$558$	0	0
$559$	−17.4533	−0.738196
$560$	0	0
$561$	15.8867	0.670735
$562$	0	0
$563$	−27.9274	−1.17700	−0.588499	−	0.808498i	$-0.700281\pi$
$563$	−27.9274	−1.17700	−0.588499	+	0.808498i	$0.700281\pi$
$564$	0	0
$565$	43.2266	1.81856
$566$	0	0
$567$	3.14134	0.131924
$568$	0	0
$569$	−6.62734	−0.277833	−0.138916	−	0.990304i	$-0.544362\pi$
$569$	−6.62734	−0.277833	−0.138916	+	0.990304i	$0.544362\pi$
$570$	0	0
$571$	11.8200	0.494653	0.247326	−	0.968932i	$-0.420448\pi$
$571$	11.8200	0.494653	0.247326	+	0.968932i	$0.420448\pi$
$572$	0	0
$573$	2.91002	0.121568
$574$	0	0
$575$	−26.4847	−1.10449
$576$	0	0
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	0	0
$579$	16.0187	0.665713
$580$	0	0
$581$	−27.8867	−1.15693
$582$	0	0
$583$	−8.67869	−0.359435
$584$	0	0
$585$	−11.0093	−0.455180
$586$	0	0
$587$	−11.3247	−0.467420	−0.233710	−	0.972306i	$-0.575087\pi$
$587$	−11.3247	−0.467420	−0.233710	+	0.972306i	$0.575087\pi$
$588$	0	0
$589$	1.35739	0.0559302
$590$	0	0
$591$	−12.4333	−0.511439
$592$	0	0
$593$	−40.2827	−1.65421	−0.827106	−	0.562047i	$-0.810014\pi$
$593$	−40.2827	−1.65421	−0.827106	+	0.562047i	$0.810014\pi$
$594$	0	0
$595$	34.0187	1.39463
$596$	0	0
$597$	0.990671	0.0405455
$598$	0	0
$599$	−25.3947	−1.03760	−0.518800	−	0.854896i	$-0.673621\pi$
$599$	−25.3947	−1.03760	−0.518800	+	0.854896i	$0.673621\pi$
$600$	0	0
$601$	20.5106	0.836645	0.418322	−	0.908299i	$-0.362618\pi$
$601$	20.5106	0.836645	0.418322	+	0.908299i	$0.362618\pi$
$602$	0	0
$603$	−3.55602	−0.144812
$604$	0	0
$605$	54.0887	2.19902
$606$	0	0
$607$	39.4134	1.59974	0.799869	−	0.600174i	$-0.204902\pi$
$607$	39.4134	1.59974	0.799869	+	0.600174i	$0.204902\pi$
$608$	0	0
$609$	−11.0093	−0.446121
$610$	0	0
$611$	36.8667	1.49147
$612$	0	0
$613$	40.9439	1.65371	0.826855	−	0.562415i	$-0.190128\pi$
$613$	40.9439	1.65371	0.826855	+	0.562415i	$0.190128\pi$
$614$	0	0
$615$	7.00933	0.282643
$616$	0	0
$617$	−29.8760	−1.20276	−0.601381	−	0.798962i	$-0.705383\pi$
$617$	−29.8760	−1.20276	−0.601381	+	0.798962i	$0.705383\pi$
$618$	0	0
$619$	−26.6426	−1.07086	−0.535428	−	0.844581i	$-0.679850\pi$
$619$	−26.6426	−1.07086	−0.535428	+	0.844581i	$0.679850\pi$
$620$	0	0
$621$	−3.63667	−0.145935
$622$	0	0
$623$	36.6133	1.46688
$624$	0	0
$625$	−8.37605	−0.335042
$626$	0	0
$627$	−9.60398	−0.383546
$628$	0	0
$629$	3.08998	0.123206
$630$	0	0
$631$	9.45331	0.376330	0.188165	−	0.982137i	$-0.439746\pi$
$631$	9.45331	0.376330	0.188165	+	0.982137i	$0.439746\pi$
$632$	0	0
$633$	−5.45331	−0.216750
$634$	0	0
$635$	−64.6027	−2.56368
$636$	0	0
$637$	−9.00933	−0.356963
$638$	0	0
$639$	14.0187	0.554570
$640$	0	0
$641$	42.3386	1.67228	0.836138	−	0.548519i	$-0.184808\pi$
$641$	42.3386	1.67228	0.836138	+	0.548519i	$0.184808\pi$
$642$	0	0
$643$	24.3561	0.960510	0.480255	−	0.877129i	$-0.340544\pi$
$643$	24.3561	0.960510	0.480255	+	0.877129i	$0.340544\pi$
$644$	0	0
$645$	19.4720	0.766708
$646$	0	0
$647$	3.01272	0.118442	0.0592211	−	0.998245i	$-0.481138\pi$
$647$	3.01272	0.118442	0.0592211	+	0.998245i	$0.481138\pi$
$648$	0	0
$649$	73.1680	2.87210
$650$	0	0
$651$	−2.28267	−0.0894650
$652$	0	0
$653$	8.95798	0.350553	0.175276	−	0.984519i	$-0.443918\pi$
$653$	8.95798	0.350553	0.175276	+	0.984519i	$0.443918\pi$
$654$	0	0
$655$	−19.6519	−0.767865
$656$	0	0
$657$	−10.6974	−0.417343
$658$	0	0
$659$	23.5747	0.918339	0.459169	−	0.888349i	$-0.348147\pi$
$659$	23.5747	0.918339	0.459169	+	0.888349i	$0.348147\pi$
$660$	0	0
$661$	44.7933	1.74226	0.871128	−	0.491056i	$-0.163389\pi$
$661$	44.7933	1.74226	0.871128	+	0.491056i	$0.163389\pi$
$662$	0	0
$663$	−9.70668	−0.376976
$664$	0	0
$665$	−20.5653	−0.797490
$666$	0	0
$667$	12.7453	0.493501
$668$	0	0
$669$	−6.01866	−0.232695
$670$	0	0
$671$	30.8480	1.19087
$672$	0	0
$673$	8.77462	0.338237	0.169118	−	0.985596i	$-0.445908\pi$
$673$	8.77462	0.338237	0.169118	+	0.985596i	$0.445908\pi$
$674$	0	0
$675$	7.28267	0.280310
$676$	0	0
$677$	14.6133	0.561635	0.280817	−	0.959761i	$-0.409394\pi$
$677$	14.6133	0.561635	0.280817	+	0.959761i	$0.409394\pi$
$678$	0	0
$679$	−32.8667	−1.26131
$680$	0	0
$681$	−25.2406	−0.967224
$682$	0	0
$683$	28.5513	1.09249	0.546243	−	0.837627i	$-0.316058\pi$
$683$	28.5513	1.09249	0.546243	+	0.837627i	$0.316058\pi$
$684$	0	0
$685$	−53.5933	−2.04770
$686$	0	0
$687$	−15.3947	−0.587345
$688$	0	0
$689$	5.30265	0.202015
$690$	0	0
$691$	6.72666	0.255894	0.127947	−	0.991781i	$-0.459161\pi$
$691$	6.72666	0.255894	0.127947	+	0.991781i	$0.459161\pi$
$692$	0	0
$693$	16.1507	0.613513
$694$	0	0
$695$	−26.4813	−1.00449
$696$	0	0
$697$	6.17997	0.234083
$698$	0	0
$699$	7.45331	0.281910
$700$	0	0
$701$	1.48601	0.0561257	0.0280629	−	0.999606i	$-0.491066\pi$
$701$	1.48601	0.0561257	0.0280629	+	0.999606i	$0.491066\pi$
$702$	0	0
$703$	−1.86799	−0.0704527
$704$	0	0
$705$	−41.1307	−1.54907
$706$	0	0
$707$	58.3200	2.19335
$708$	0	0
$709$	32.5360	1.22192	0.610958	−	0.791663i	$-0.290784\pi$
$709$	32.5360	1.22192	0.610958	+	0.791663i	$0.290784\pi$
$710$	0	0
$711$	−7.73599	−0.290122
$712$	0	0
$713$	2.64261	0.0989666
$714$	0	0
$715$	−56.6027	−2.11682
$716$	0	0
$717$	−19.7873	−0.738972
$718$	0	0
$719$	−39.0093	−1.45480	−0.727401	−	0.686212i	$-0.759272\pi$
$719$	−39.0093	−1.45480	−0.727401	+	0.686212i	$0.759272\pi$
$720$	0	0
$721$	−51.2080	−1.90708
$722$	0	0
$723$	20.7453	0.771527
$724$	0	0
$725$	−25.5233	−0.947912
$726$	0	0
$727$	18.8039	0.697399	0.348699	−	0.937235i	$-0.386623\pi$
$727$	18.8039	0.697399	0.348699	+	0.937235i	$0.386623\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	17.1680	0.634982
$732$	0	0
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	0	0
$735$	10.0514	0.370750
$736$	0	0
$737$	−18.2827	−0.673451
$738$	0	0
$739$	−18.8480	−0.693336	−0.346668	−	0.937988i	$-0.612687\pi$
$739$	−18.8480	−0.693336	−0.346668	+	0.937988i	$0.612687\pi$
$740$	0	0
$741$	5.86799	0.215566
$742$	0	0
$743$	37.5933	1.37917	0.689583	−	0.724207i	$-0.257794\pi$
$743$	37.5933	1.37917	0.689583	+	0.724207i	$0.257794\pi$
$744$	0	0
$745$	15.9346	0.583799
$746$	0	0
$747$	−8.87732	−0.324804
$748$	0	0
$749$	53.3400	1.94900
$750$	0	0
$751$	−24.5399	−0.895474	−0.447737	−	0.894165i	$-0.647770\pi$
$751$	−24.5399	−0.895474	−0.447737	+	0.894165i	$0.647770\pi$
$752$	0	0
$753$	−17.9673	−0.654765
$754$	0	0
$755$	−36.5653	−1.33075
$756$	0	0
$757$	−36.3306	−1.32046	−0.660230	−	0.751064i	$-0.729541\pi$
$757$	−36.3306	−1.32046	−0.660230	+	0.751064i	$0.729541\pi$
$758$	0	0
$759$	−18.6974	−0.678671
$760$	0	0
$761$	−2.42533	−0.0879180	−0.0439590	−	0.999033i	$-0.513997\pi$
$761$	−2.42533	−0.0879180	−0.0439590	+	0.999033i	$0.513997\pi$
$762$	0	0
$763$	39.5640	1.43231
$764$	0	0
$765$	10.8294	0.391536
$766$	0	0
$767$	−44.7054	−1.61422
$768$	0	0
$769$	−43.7546	−1.57783	−0.788916	−	0.614501i	$-0.789358\pi$
$769$	−43.7546	−1.57783	−0.788916	+	0.614501i	$0.789358\pi$
$770$	0	0
$771$	8.54330	0.307679
$772$	0	0
$773$	−6.62009	−0.238108	−0.119054	−	0.992888i	$-0.537986\pi$
$773$	−6.62009	−0.238108	−0.119054	+	0.992888i	$0.537986\pi$
$774$	0	0
$775$	−5.29200	−0.190094
$776$	0	0
$777$	3.14134	0.112695
$778$	0	0
$779$	−3.73599	−0.133856
$780$	0	0
$781$	72.0746	2.57903
$782$	0	0
$783$	−3.50466	−0.125246
$784$	0	0
$785$	19.1120	0.682138
$786$	0	0
$787$	21.5492	0.768147	0.384074	−	0.923302i	$-0.374521\pi$
$787$	21.5492	0.768147	0.384074	+	0.923302i	$0.374521\pi$
$788$	0	0
$789$	−18.7453	−0.667351
$790$	0	0
$791$	38.7453	1.37762
$792$	0	0
$793$	−18.8480	−0.669313
$794$	0	0
$795$	−5.91595	−0.209817
$796$	0	0
$797$	−36.4953	−1.29273	−0.646366	−	0.763028i	$-0.723712\pi$
$797$	−36.4953	−1.29273	−0.646366	+	0.763028i	$0.723712\pi$
$798$	0	0
$799$	−36.2640	−1.28293
$800$	0	0
$801$	11.6553	0.411821
$802$	0	0
$803$	−54.9987	−1.94086
$804$	0	0
$805$	−40.0373	−1.41113
$806$	0	0
$807$	−30.4520	−1.07196
$808$	0	0
$809$	−5.69528	−0.200235	−0.100118	−	0.994976i	$-0.531922\pi$
$809$	−5.69528	−0.200235	−0.100118	+	0.994976i	$0.531922\pi$
$810$	0	0
$811$	−18.1613	−0.637730	−0.318865	−	0.947800i	$-0.603302\pi$
$811$	−18.1613	−0.637730	−0.318865	+	0.947800i	$0.603302\pi$
$812$	0	0
$813$	18.7267	0.656773
$814$	0	0
$815$	−4.00000	−0.140114
$816$	0	0
$817$	−10.3786	−0.363101
$818$	0	0
$819$	−9.86799	−0.344816
$820$	0	0
$821$	46.5547	1.62477	0.812385	−	0.583121i	$-0.198169\pi$
$821$	46.5547	1.62477	0.812385	+	0.583121i	$0.198169\pi$
$822$	0	0
$823$	28.1880	0.982571	0.491286	−	0.870999i	$-0.336527\pi$
$823$	28.1880	0.982571	0.491286	+	0.870999i	$0.336527\pi$
$824$	0	0
$825$	37.4427	1.30359
$826$	0	0
$827$	−45.0793	−1.56756	−0.783781	−	0.621037i	$-0.786712\pi$
$827$	−45.0793	−1.56756	−0.783781	+	0.621037i	$0.786712\pi$
$828$	0	0
$829$	31.1787	1.08288	0.541440	−	0.840740i	$-0.317879\pi$
$829$	31.1787	1.08288	0.541440	+	0.840740i	$0.317879\pi$
$830$	0	0
$831$	14.2534	0.494444
$832$	0	0
$833$	8.86205	0.307052
$834$	0	0
$835$	−4.74531	−0.164218
$836$	0	0
$837$	−0.726656	−0.0251169
$838$	0	0
$839$	−22.6867	−0.783232	−0.391616	−	0.920129i	$-0.628084\pi$
$839$	−22.6867	−0.783232	−0.391616	+	0.920129i	$0.628084\pi$
$840$	0	0
$841$	−16.7173	−0.576460
$842$	0	0
$843$	−17.8353	−0.614280
$844$	0	0
$845$	−10.9766	−0.377608
$846$	0	0
$847$	48.4813	1.66584
$848$	0	0
$849$	−14.6974	−0.504412
$850$	0	0
$851$	−3.63667	−0.124664
$852$	0	0
$853$	−13.2254	−0.452828	−0.226414	−	0.974031i	$-0.572700\pi$
$853$	−13.2254	−0.452828	−0.226414	+	0.974031i	$0.572700\pi$
$854$	0	0
$855$	−6.54669	−0.223892
$856$	0	0
$857$	28.6901	0.980035	0.490017	−	0.871713i	$-0.336990\pi$
$857$	28.6901	0.980035	0.490017	+	0.871713i	$0.336990\pi$
$858$	0	0
$859$	38.8587	1.32584	0.662920	−	0.748690i	$-0.269317\pi$
$859$	38.8587	1.32584	0.662920	+	0.748690i	$0.269317\pi$
$860$	0	0
$861$	6.28267	0.214113
$862$	0	0
$863$	−29.6519	−1.00936	−0.504682	−	0.863305i	$-0.668390\pi$
$863$	−29.6519	−1.00936	−0.504682	+	0.863305i	$0.668390\pi$
$864$	0	0
$865$	−54.1214	−1.84018
$866$	0	0
$867$	−7.45199	−0.253083
$868$	0	0
$869$	−39.7733	−1.34922
$870$	0	0
$871$	11.1706	0.378503
$872$	0	0
$873$	−10.4626	−0.354107
$874$	0	0
$875$	25.1307	0.849572
$876$	0	0
$877$	5.75464	0.194320	0.0971602	−	0.995269i	$-0.469024\pi$
$877$	5.75464	0.194320	0.0971602	+	0.995269i	$0.469024\pi$
$878$	0	0
$879$	20.3306	0.685735
$880$	0	0
$881$	27.6146	0.930360	0.465180	−	0.885216i	$-0.345990\pi$
$881$	27.6146	0.930360	0.465180	+	0.885216i	$0.345990\pi$
$882$	0	0
$883$	23.3213	0.784824	0.392412	−	0.919789i	$-0.371641\pi$
$883$	23.3213	0.784824	0.392412	+	0.919789i	$0.371641\pi$
$884$	0	0
$885$	49.8760	1.67656
$886$	0	0
$887$	26.9507	0.904917	0.452458	−	0.891786i	$-0.350547\pi$
$887$	26.9507	0.904917	0.452458	+	0.891786i	$0.350547\pi$
$888$	0	0
$889$	−57.9053	−1.94208
$890$	0	0
$891$	5.14134	0.172241
$892$	0	0
$893$	21.9227	0.733616
$894$	0	0
$895$	6.19863	0.207197
$896$	0	0
$897$	11.4240	0.381437
$898$	0	0
$899$	2.54669	0.0849368
$900$	0	0
$901$	−5.21597	−0.173769
$902$	0	0
$903$	17.4533	0.580810
$904$	0	0
$905$	86.0933	2.86184
$906$	0	0
$907$	−14.2720	−0.473895	−0.236947	−	0.971522i	$-0.576147\pi$
$907$	−14.2720	−0.473895	−0.236947	+	0.971522i	$0.576147\pi$
$908$	0	0
$909$	18.5653	0.615774
$910$	0	0
$911$	42.9367	1.42256	0.711278	−	0.702911i	$-0.248117\pi$
$911$	42.9367	1.42256	0.711278	+	0.702911i	$0.248117\pi$
$912$	0	0
$913$	−45.6413	−1.51051
$914$	0	0
$915$	21.0280	0.695164
$916$	0	0
$917$	−17.6146	−0.581686
$918$	0	0
$919$	−2.64261	−0.0871717	−0.0435858	−	0.999050i	$-0.513878\pi$
$919$	−2.64261	−0.0871717	−0.0435858	+	0.999050i	$0.513878\pi$
$920$	0	0
$921$	−5.45331	−0.179693
$922$	0	0
$923$	−44.0373	−1.44951
$924$	0	0
$925$	7.28267	0.239453
$926$	0	0
$927$	−16.3013	−0.535406
$928$	0	0
$929$	12.3786	0.406129	0.203064	−	0.979165i	$-0.434910\pi$
$929$	12.3786	0.406129	0.203064	+	0.979165i	$0.434910\pi$
$930$	0	0
$931$	−5.35739	−0.175581
$932$	0	0
$933$	−3.78734	−0.123992
$934$	0	0
$935$	55.6774	1.82084
$936$	0	0
$937$	−27.1307	−0.886321	−0.443160	−	0.896442i	$-0.646143\pi$
$937$	−27.1307	−0.886321	−0.443160	+	0.896442i	$0.646143\pi$
$938$	0	0
$939$	−15.1307	−0.493771
$940$	0	0
$941$	5.57467	0.181729	0.0908646	−	0.995863i	$-0.471037\pi$
$941$	5.57467	0.181729	0.0908646	+	0.995863i	$0.471037\pi$
$942$	0	0
$943$	−7.27334	−0.236853
$944$	0	0
$945$	11.0093	0.358133
$946$	0	0
$947$	23.4860	0.763193	0.381596	−	0.924329i	$-0.375374\pi$
$947$	23.4860	0.763193	0.381596	+	0.924329i	$0.375374\pi$
$948$	0	0
$949$	33.6040	1.09083
$950$	0	0
$951$	22.4626	0.728401
$952$	0	0
$953$	−12.7080	−0.411652	−0.205826	−	0.978589i	$-0.565988\pi$
$953$	−12.7080	−0.411652	−0.205826	+	0.978589i	$0.565988\pi$
$954$	0	0
$955$	10.1986	0.330020
$956$	0	0
$957$	−18.0187	−0.582460
$958$	0	0
$959$	−48.0373	−1.55121
$960$	0	0
$961$	−30.4720	−0.982967
$962$	0	0
$963$	16.9800	0.547174
$964$	0	0
$965$	56.1400	1.80721
$966$	0	0
$967$	−11.7801	−0.378822	−0.189411	−	0.981898i	$-0.560658\pi$
$967$	−11.7801	−0.378822	−0.189411	+	0.981898i	$0.560658\pi$
$968$	0	0
$969$	−5.77207	−0.185426
$970$	0	0
$971$	−7.89730	−0.253436	−0.126718	−	0.991939i	$-0.540444\pi$
$971$	−7.89730	−0.253436	−0.126718	+	0.991939i	$0.540444\pi$
$972$	0	0
$973$	−23.7360	−0.760941
$974$	0	0
$975$	−22.8773	−0.732661
$976$	0	0
$977$	44.2207	1.41474	0.707372	−	0.706841i	$-0.249880\pi$
$977$	44.2207	1.41474	0.707372	+	0.706841i	$0.249880\pi$
$978$	0	0
$979$	59.9240	1.91518
$980$	0	0
$981$	12.5946	0.402116
$982$	0	0
$983$	−10.1800	−0.324691	−0.162345	−	0.986734i	$-0.551906\pi$
$983$	−10.1800	−0.324691	−0.162345	+	0.986734i	$0.551906\pi$
$984$	0	0
$985$	−43.5747	−1.38840
$986$	0	0
$987$	−36.8667	−1.17348
$988$	0	0
$989$	−20.2054	−0.642495
$990$	0	0
$991$	49.3879	1.56886	0.784430	−	0.620218i	$-0.212956\pi$
$991$	49.3879	1.56886	0.784430	+	0.620218i	$0.212956\pi$
$992$	0	0
$993$	23.4720	0.744860
$994$	0	0
$995$	3.47197	0.110069
$996$	0	0
$997$	−47.0173	−1.48905	−0.744527	−	0.667592i	$-0.767325\pi$
$997$	−47.0173	−1.48905	−0.744527	+	0.667592i	$0.767325\pi$
$998$	0	0
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7104.2.a.bx.1.3		3
4.3	odd	2		7104.2.a.br.1.3		3
8.3	odd	2		888.2.a.j.1.1	✓	3
8.5	even	2		1776.2.a.r.1.1		3
24.5	odd	2		5328.2.a.bm.1.3		3
24.11	even	2		2664.2.a.o.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
888.2.a.j.1.1	✓	3	8.3	odd	2
1776.2.a.r.1.1		3	8.5	even	2
2664.2.a.o.1.3		3	24.11	even	2
5328.2.a.bm.1.3		3	24.5	odd	2
7104.2.a.br.1.3		3	4.3	odd	2
7104.2.a.bx.1.3		3	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 7104 = 2^{6} \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7104.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.50466 q^{5} +3.14134 q^{7} +1.00000 q^{9} +5.14134 q^{11} -3.14134 q^{13} +3.50466 q^{15} +3.08998 q^{17} -1.86799 q^{19} +3.14134 q^{21} -3.63667 q^{23} +7.28267 q^{25} +1.00000 q^{27} -3.50466 q^{29} -0.726656 q^{31} +5.14134 q^{33} +11.0093 q^{35} +1.00000 q^{37} -3.14134 q^{39} +2.00000 q^{41} +5.55602 q^{43} +3.50466 q^{45} -11.7360 q^{47} +2.86799 q^{49} +3.08998 q^{51} -1.68802 q^{53} +18.0187 q^{55} -1.86799 q^{57} +14.2313 q^{59} +6.00000 q^{61} +3.14134 q^{63} -11.0093 q^{65} -3.55602 q^{67} -3.63667 q^{69} +14.0187 q^{71} -10.6974 q^{73} +7.28267 q^{75} +16.1507 q^{77} -7.73599 q^{79} +1.00000 q^{81} -8.87732 q^{83} +10.8294 q^{85} -3.50466 q^{87} +11.6553 q^{89} -9.86799 q^{91} -0.726656 q^{93} -6.54669 q^{95} -10.4626 q^{97} +5.14134 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + q^{7} + 3 q^{9} + 7 q^{11} - q^{13} + 3 q^{17} + 7 q^{19} + q^{21} - 13 q^{23} + 5 q^{25} + 3 q^{27} + 2 q^{31} + 7 q^{33} + 12 q^{35} + 3 q^{37} - q^{39} + 6 q^{41} + 4 q^{43} - 10 q^{47}+ \cdots + 7 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + q^7 + 3 * q^9 + 7 * q^11 - q^13 + 3 * q^17 + 7 * q^19 + q^21 - 13 * q^23 + 5 * q^25 + 3 * q^27 + 2 * q^31 + 7 * q^33 + 12 * q^35 + 3 * q^37 - q^39 + 6 * q^41 + 4 * q^43 - 10 * q^47 - 4 * q^49 + 3 * q^51 - 5 * q^53 + 12 * q^55 + 7 * q^57 + 28 * q^59 + 18 * q^61 + q^63 - 12 * q^65 + 2 * q^67 - 13 * q^69 - 11 * q^73 + 5 * q^75 + 19 * q^77 + 2 * q^79 + 3 * q^81 + 7 * q^83 + 24 * q^85 - 5 * q^89 - 17 * q^91 + 2 * q^93 - 28 * q^95 - 2 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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