LMFDB - Embedded newform 7744.2.a.di.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7744,2,Mod(1,7744)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$61.8361513253$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.5225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + x + 11 $ x^4 - x^3 - 8*x^2 + x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.10522$ 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)

$f(q)$	$=$	$q-1.91913 q^{3} -3.40632 q^{5} +0.869151 q^{7} +0.683063 q^{9} +O(q^{10})$	$q-1.91913 q^{3} -3.40632 q^{5} +0.869151 q^{7} +0.683063 q^{9} -2.80412 q^{13} +6.53716 q^{15} -7.29131 q^{17} -2.38197 q^{19} -1.66801 q^{21} +7.44651 q^{23} +6.60299 q^{25} +4.44651 q^{27} +7.34129 q^{29} -2.64238 q^{31} -2.96060 q^{35} -1.03089 q^{37} +5.38148 q^{39} +2.89350 q^{41} -2.18609 q^{43} -2.32673 q^{45} -3.94348 q^{47} -6.24458 q^{49} +13.9930 q^{51} +7.11452 q^{53} +4.57130 q^{57} -8.09017 q^{59} +2.69364 q^{61} +0.593685 q^{63} +9.55172 q^{65} +13.7720 q^{67} -14.2908 q^{69} -1.92971 q^{71} +4.99346 q^{73} -12.6720 q^{75} +13.4550 q^{79} -10.5826 q^{81} +0.194597 q^{83} +24.8365 q^{85} -14.0889 q^{87} +4.18609 q^{89} -2.43720 q^{91} +5.07108 q^{93} +8.11373 q^{95} +4.99021 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - q^{5} + q^{7} + 6 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - q^5 + q^7 + 6 * q^9	$ 4 q - 2 q^{3} - q^{5} + q^{7} + 6 q^{9} - q^{13} + 16 q^{15} - 12 q^{17} - 14 q^{19} + q^{21} - 2 q^{23} + 11 q^{25} - 14 q^{27} + 9 q^{29} + 11 q^{31} - 18 q^{35} - 13 q^{37} - 18 q^{39} - 8 q^{41} - 3 q^{43} - 22 q^{45} + 7 q^{47} - q^{49} - 14 q^{51} - 11 q^{53} + 7 q^{57} - 10 q^{59} + 17 q^{61} + 15 q^{63} - 5 q^{65} + 5 q^{67} + 4 q^{69} - 5 q^{71} - 6 q^{73} - 7 q^{75} + 7 q^{79} - 8 q^{81} - 20 q^{83} + 13 q^{85} - 3 q^{87} + 11 q^{89} - 6 q^{91} + 10 q^{93} + 6 q^{95} + 4 q^{97}+O(q^{100}) $ 4 * q - 2 * q^3 - q^5 + q^7 + 6 * q^9 - q^13 + 16 * q^15 - 12 * q^17 - 14 * q^19 + q^21 - 2 * q^23 + 11 * q^25 - 14 * q^27 + 9 * q^29 + 11 * q^31 - 18 * q^35 - 13 * q^37 - 18 * q^39 - 8 * q^41 - 3 * q^43 - 22 * q^45 + 7 * q^47 - q^49 - 14 * q^51 - 11 * q^53 + 7 * q^57 - 10 * q^59 + 17 * q^61 + 15 * q^63 - 5 * q^65 + 5 * q^67 + 4 * q^69 - 5 * q^71 - 6 * q^73 - 7 * q^75 + 7 * q^79 - 8 * q^81 - 20 * q^83 + 13 * q^85 - 3 * q^87 + 11 * q^89 - 6 * q^91 + 10 * q^93 + 6 * q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.91913	−1.10801	−0.554005	−	0.832513i	$-0.686901\pi$
$3$	−1.91913	−1.10801	−0.554005	+	0.832513i	$0.686901\pi$
$4$	0	0
$5$	−3.40632	−1.52335	−0.761675	−	0.647959i	$-0.775623\pi$
$5$	−3.40632	−1.52335	−0.761675	+	0.647959i	$0.775623\pi$
$6$	0	0
$7$	0.869151	0.328508	0.164254	−	0.986418i	$-0.447478\pi$
$7$	0.869151	0.328508	0.164254	+	0.986418i	$0.447478\pi$
$8$	0	0
$9$	0.683063	0.227688
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.80412	−0.777724	−0.388862	−	0.921296i	$-0.627132\pi$
$13$	−2.80412	−0.777724	−0.388862	+	0.921296i	$0.627132\pi$
$14$	0	0
$15$	6.53716	1.68789
$16$	0	0
$17$	−7.29131	−1.76840	−0.884201	−	0.467107i	$-0.845296\pi$
$17$	−7.29131	−1.76840	−0.884201	+	0.467107i	$0.845296\pi$
$18$	0	0
$19$	−2.38197	−0.546460	−0.273230	−	0.961949i	$-0.588092\pi$
$19$	−2.38197	−0.546460	−0.273230	+	0.961949i	$0.588092\pi$
$20$	0	0
$21$	−1.66801	−0.363990
$22$	0	0
$23$	7.44651	1.55270	0.776352	−	0.630300i	$-0.217068\pi$
$23$	7.44651	1.55270	0.776352	+	0.630300i	$0.217068\pi$
$24$	0	0
$25$	6.60299	1.32060
$26$	0	0
$27$	4.44651	0.855730
$28$	0	0
$29$	7.34129	1.36324	0.681621	−	0.731705i	$-0.261275\pi$
$29$	7.34129	1.36324	0.681621	+	0.731705i	$0.261275\pi$
$30$	0	0
$31$	−2.64238	−0.474586	−0.237293	−	0.971438i	$-0.576260\pi$
$31$	−2.64238	−0.474586	−0.237293	+	0.971438i	$0.576260\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.96060	−0.500433
$36$	0	0
$37$	−1.03089	−0.169477	−0.0847386	−	0.996403i	$-0.527006\pi$
$37$	−1.03089	−0.169477	−0.0847386	+	0.996403i	$0.527006\pi$
$38$	0	0
$39$	5.38148	0.861726
$40$	0	0
$41$	2.89350	0.451889	0.225944	−	0.974140i	$-0.427453\pi$
$41$	2.89350	0.451889	0.225944	+	0.974140i	$0.427453\pi$
$42$	0	0
$43$	−2.18609	−0.333375	−0.166688	−	0.986010i	$-0.553307\pi$
$43$	−2.18609	−0.333375	−0.166688	+	0.986010i	$0.553307\pi$
$44$	0	0
$45$	−2.32673	−0.346848
$46$	0	0
$47$	−3.94348	−0.575216	−0.287608	−	0.957748i	$-0.592860\pi$
$47$	−3.94348	−0.575216	−0.287608	+	0.957748i	$0.592860\pi$
$48$	0	0
$49$	−6.24458	−0.892082
$50$	0	0
$51$	13.9930	1.95941
$52$	0	0
$53$	7.11452	0.977254	0.488627	−	0.872493i	$-0.337498\pi$
$53$	7.11452	0.977254	0.488627	+	0.872493i	$0.337498\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.57130	0.605484
$58$	0	0
$59$	−8.09017	−1.05325	−0.526625	−	0.850098i	$-0.676543\pi$
$59$	−8.09017	−1.05325	−0.526625	+	0.850098i	$0.676543\pi$
$60$	0	0
$61$	2.69364	0.344886	0.172443	−	0.985020i	$-0.444834\pi$
$61$	2.69364	0.344886	0.172443	+	0.985020i	$0.444834\pi$
$62$	0	0
$63$	0.593685	0.0747972
$64$	0	0
$65$	9.55172	1.18475
$66$	0	0
$67$	13.7720	1.68251	0.841256	−	0.540637i	$-0.181817\pi$
$67$	13.7720	1.68251	0.841256	+	0.540637i	$0.181817\pi$
$68$	0	0
$69$	−14.2908	−1.72041
$70$	0	0
$71$	−1.92971	−0.229015	−0.114507	−	0.993422i	$-0.536529\pi$
$71$	−1.92971	−0.229015	−0.114507	+	0.993422i	$0.536529\pi$
$72$	0	0
$73$	4.99346	0.584440	0.292220	−	0.956351i	$-0.405606\pi$
$73$	4.99346	0.584440	0.292220	+	0.956351i	$0.405606\pi$
$74$	0	0
$75$	−12.6720	−1.46324
$76$	0	0
$77$	0	0
$78$	0	0
$79$	13.4550	1.51381	0.756904	−	0.653526i	$-0.226711\pi$
$79$	13.4550	1.51381	0.756904	+	0.653526i	$0.226711\pi$
$80$	0	0
$81$	−10.5826	−1.17585
$82$	0	0
$83$	0.194597	0.0213598	0.0106799	−	0.999943i	$-0.496600\pi$
$83$	0.194597	0.0213598	0.0106799	+	0.999943i	$0.496600\pi$
$84$	0	0
$85$	24.8365	2.69390
$86$	0	0
$87$	−14.0889	−1.51049
$88$	0	0
$89$	4.18609	0.443724	0.221862	−	0.975078i	$-0.428786\pi$
$89$	4.18609	0.443724	0.221862	+	0.975078i	$0.428786\pi$
$90$	0	0
$91$	−2.43720	−0.255488
$92$	0	0
$93$	5.07108	0.525846
$94$	0	0
$95$	8.11373	0.832451
$96$	0	0
$97$	4.99021	0.506679	0.253340	−	0.967377i	$-0.418471\pi$
$97$	4.99021	0.506679	0.253340	+	0.967377i	$0.418471\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.27021	0.822916	0.411458	−	0.911429i	$-0.365020\pi$
$101$	8.27021	0.822916	0.411458	+	0.911429i	$0.365020\pi$
$102$	0	0
$103$	0.820452	0.0808415	0.0404208	−	0.999183i	$-0.487130\pi$
$103$	0.820452	0.0808415	0.0404208	+	0.999183i	$0.487130\pi$
$104$	0	0
$105$	5.68178	0.554485
$106$	0	0
$107$	−12.3913	−1.19791	−0.598954	−	0.800783i	$-0.704417\pi$
$107$	−12.3913	−1.19791	−0.598954	+	0.800783i	$0.704417\pi$
$108$	0	0
$109$	7.20439	0.690055	0.345028	−	0.938593i	$-0.387870\pi$
$109$	7.20439	0.690055	0.345028	+	0.938593i	$0.387870\pi$
$110$	0	0
$111$	1.97841	0.187782
$112$	0	0
$113$	2.05849	0.193646	0.0968232	−	0.995302i	$-0.469132\pi$
$113$	2.05849	0.193646	0.0968232	+	0.995302i	$0.469132\pi$
$114$	0	0
$115$	−25.3651	−2.36531
$116$	0	0
$117$	−1.91539	−0.177078
$118$	0	0
$119$	−6.33724	−0.580934
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−5.55301	−0.500698
$124$	0	0
$125$	−5.46027	−0.488382
$126$	0	0
$127$	8.37017	0.742732	0.371366	−	0.928487i	$-0.378889\pi$
$127$	8.37017	0.742732	0.371366	+	0.928487i	$0.378889\pi$
$128$	0	0
$129$	4.19539	0.369383
$130$	0	0
$131$	−13.3511	−1.16649	−0.583244	−	0.812297i	$-0.698217\pi$
$131$	−13.3511	−1.16649	−0.583244	+	0.812297i	$0.698217\pi$
$132$	0	0
$133$	−2.07029	−0.179517
$134$	0	0
$135$	−15.1462	−1.30358
$136$	0	0
$137$	0.990210	0.0845994	0.0422997	−	0.999105i	$-0.486532\pi$
$137$	0.990210	0.0845994	0.0422997	+	0.999105i	$0.486532\pi$
$138$	0	0
$139$	−5.99474	−0.508467	−0.254234	−	0.967143i	$-0.581823\pi$
$139$	−5.99474	−0.508467	−0.254234	+	0.967143i	$0.581823\pi$
$140$	0	0
$141$	7.56805	0.637345
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−25.0067	−2.07670
$146$	0	0
$147$	11.9842	0.988437
$148$	0	0
$149$	17.2933	1.41672	0.708360	−	0.705851i	$-0.249435\pi$
$149$	17.2933	1.41672	0.708360	+	0.705851i	$0.249435\pi$
$150$	0	0
$151$	−0.790354	−0.0643181	−0.0321591	−	0.999483i	$-0.510238\pi$
$151$	−0.790354	−0.0643181	−0.0321591	+	0.999483i	$0.510238\pi$
$152$	0	0
$153$	−4.98042	−0.402643
$154$	0	0
$155$	9.00079	0.722961
$156$	0	0
$157$	12.7365	1.01649	0.508243	−	0.861214i	$-0.330295\pi$
$157$	12.7365	1.01649	0.508243	+	0.861214i	$0.330295\pi$
$158$	0	0
$159$	−13.6537	−1.08281
$160$	0	0
$161$	6.47214	0.510076
$162$	0	0
$163$	0.165716	0.0129799	0.00648995	−	0.999979i	$-0.497934\pi$
$163$	0.165716	0.0129799	0.00648995	+	0.999979i	$0.497934\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.6866	−1.67816	−0.839080	−	0.544008i	$-0.816906\pi$
$167$	−21.6866	−1.67816	−0.839080	+	0.544008i	$0.816906\pi$
$168$	0	0
$169$	−5.13690	−0.395146
$170$	0	0
$171$	−1.62703	−0.124422
$172$	0	0
$173$	−13.0982	−0.995837	−0.497918	−	0.867224i	$-0.665902\pi$
$173$	−13.0982	−0.995837	−0.497918	+	0.867224i	$0.665902\pi$
$174$	0	0
$175$	5.73899	0.433827
$176$	0	0
$177$	15.5261	1.16701
$178$	0	0
$179$	10.5924	0.791713	0.395857	−	0.918312i	$-0.370448\pi$
$179$	10.5924	0.791713	0.395857	+	0.918312i	$0.370448\pi$
$180$	0	0
$181$	−6.40185	−0.475846	−0.237923	−	0.971284i	$-0.576467\pi$
$181$	−6.40185	−0.475846	−0.237923	+	0.971284i	$0.576467\pi$
$182$	0	0
$183$	−5.16946	−0.382137
$184$	0	0
$185$	3.51153	0.258173
$186$	0	0
$187$	0	0
$188$	0	0
$189$	3.86468	0.281114
$190$	0	0
$191$	6.78779	0.491147	0.245574	−	0.969378i	$-0.421024\pi$
$191$	6.78779	0.491147	0.245574	+	0.969378i	$0.421024\pi$
$192$	0	0
$193$	−8.13759	−0.585756	−0.292878	−	0.956150i	$-0.594613\pi$
$193$	−8.13759	−0.585756	−0.292878	+	0.956150i	$0.594613\pi$
$194$	0	0
$195$	−18.3310	−1.31271
$196$	0	0
$197$	16.4907	1.17492	0.587458	−	0.809255i	$-0.300129\pi$
$197$	16.4907	1.17492	0.587458	+	0.809255i	$0.300129\pi$
$198$	0	0
$199$	3.39781	0.240864	0.120432	−	0.992722i	$-0.461572\pi$
$199$	3.39781	0.240864	0.120432	+	0.992722i	$0.461572\pi$
$200$	0	0
$201$	−26.4302	−1.86424
$202$	0	0
$203$	6.38069	0.447836
$204$	0	0
$205$	−9.85617	−0.688385
$206$	0	0
$207$	5.08643	0.353531
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−15.9832	−1.10033	−0.550164	−	0.835057i	$-0.685435\pi$
$211$	−15.9832	−1.10033	−0.550164	+	0.835057i	$0.685435\pi$
$212$	0	0
$213$	3.70337	0.253751
$214$	0	0
$215$	7.44651	0.507847
$216$	0	0
$217$	−2.29663	−0.155905
$218$	0	0
$219$	−9.58310	−0.647566
$220$	0	0
$221$	20.4457	1.37533
$222$	0	0
$223$	7.58182	0.507716	0.253858	−	0.967241i	$-0.418300\pi$
$223$	7.58182	0.507716	0.253858	+	0.967241i	$0.418300\pi$
$224$	0	0
$225$	4.51025	0.300684
$226$	0	0
$227$	−15.0133	−0.996466	−0.498233	−	0.867043i	$-0.666018\pi$
$227$	−15.0133	−0.996466	−0.498233	+	0.867043i	$0.666018\pi$
$228$	0	0
$229$	−2.40185	−0.158719	−0.0793593	−	0.996846i	$-0.525287\pi$
$229$	−2.40185	−0.158719	−0.0793593	+	0.996846i	$0.525287\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.24330	0.409012	0.204506	−	0.978865i	$-0.434441\pi$
$233$	6.24330	0.409012	0.204506	+	0.978865i	$0.434441\pi$
$234$	0	0
$235$	13.4327	0.876255
$236$	0	0
$237$	−25.8219	−1.67731
$238$	0	0
$239$	−17.0600	−1.10352	−0.551760	−	0.834003i	$-0.686044\pi$
$239$	−17.0600	−1.10352	−0.551760	+	0.834003i	$0.686044\pi$
$240$	0	0
$241$	0.0166322	0.00107138	0.000535688	−	1.00000i	$-0.499829\pi$
$241$	0.0166322	0.00107138	0.000535688		1.00000i	$0.499829\pi$
$242$	0	0
$243$	6.96990	0.447119
$244$	0	0
$245$	21.2710	1.35895
$246$	0	0
$247$	6.67932	0.424995
$248$	0	0
$249$	−0.373457	−0.0236669
$250$	0	0
$251$	24.1888	1.52679	0.763393	−	0.645934i	$-0.223532\pi$
$251$	24.1888	1.52679	0.763393	+	0.645934i	$0.223532\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−47.6645	−2.98487
$256$	0	0
$257$	−28.9291	−1.80455	−0.902274	−	0.431162i	$-0.858104\pi$
$257$	−28.9291	−1.80455	−0.902274	+	0.431162i	$0.858104\pi$
$258$	0	0
$259$	−0.895998	−0.0556746
$260$	0	0
$261$	5.01456	0.310393
$262$	0	0
$263$	−17.2531	−1.06387	−0.531935	−	0.846785i	$-0.678535\pi$
$263$	−17.2531	−1.06387	−0.531935	+	0.846785i	$0.678535\pi$
$264$	0	0
$265$	−24.2343	−1.48870
$266$	0	0
$267$	−8.03365	−0.491651
$268$	0	0
$269$	−2.80412	−0.170970	−0.0854852	−	0.996339i	$-0.527244\pi$
$269$	−2.80412	−0.170970	−0.0854852	+	0.996339i	$0.527244\pi$
$270$	0	0
$271$	22.7577	1.38243	0.691216	−	0.722648i	$-0.257075\pi$
$271$	22.7577	1.38243	0.691216	+	0.722648i	$0.257075\pi$
$272$	0	0
$273$	4.67731	0.283084
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−3.32752	−0.199931	−0.0999656	−	0.994991i	$-0.531873\pi$
$277$	−3.32752	−0.199931	−0.0999656	+	0.994991i	$0.531873\pi$
$278$	0	0
$279$	−1.80491	−0.108057
$280$	0	0
$281$	8.71543	0.519919	0.259960	−	0.965620i	$-0.416291\pi$
$281$	8.71543	0.519919	0.259960	+	0.965620i	$0.416291\pi$
$282$	0	0
$283$	29.1087	1.73033	0.865167	−	0.501485i	$-0.167213\pi$
$283$	29.1087	1.73033	0.865167	+	0.501485i	$0.167213\pi$
$284$	0	0
$285$	−15.5713	−0.922364
$286$	0	0
$287$	2.51489	0.148449
$288$	0	0
$289$	36.1632	2.12724
$290$	0	0
$291$	−9.57687	−0.561406
$292$	0	0
$293$	21.9556	1.28266	0.641329	−	0.767266i	$-0.278383\pi$
$293$	21.9556	1.28266	0.641329	+	0.767266i	$0.278383\pi$
$294$	0	0
$295$	27.5577	1.60447
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−20.8809	−1.20757
$300$	0	0
$301$	−1.90004	−0.109516
$302$	0	0
$303$	−15.8716	−0.911800
$304$	0	0
$305$	−9.17540	−0.525382
$306$	0	0
$307$	24.1835	1.38023	0.690113	−	0.723701i	$-0.257561\pi$
$307$	24.1835	1.38023	0.690113	+	0.723701i	$0.257561\pi$
$308$	0	0
$309$	−1.57455	−0.0895733
$310$	0	0
$311$	−3.57130	−0.202510	−0.101255	−	0.994861i	$-0.532286\pi$
$311$	−3.57130	−0.202510	−0.101255	+	0.994861i	$0.532286\pi$
$312$	0	0
$313$	8.70869	0.492244	0.246122	−	0.969239i	$-0.420844\pi$
$313$	8.70869	0.492244	0.246122	+	0.969239i	$0.420844\pi$
$314$	0	0
$315$	−2.02228	−0.113942
$316$	0	0
$317$	−22.0206	−1.23680	−0.618400	−	0.785863i	$-0.712219\pi$
$317$	−22.0206	−1.23680	−0.618400	+	0.785863i	$0.712219\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	23.7805	1.32730
$322$	0	0
$323$	17.3676	0.966362
$324$	0	0
$325$	−18.5156	−1.02706
$326$	0	0
$327$	−13.8262	−0.764588
$328$	0	0
$329$	−3.42748	−0.188963
$330$	0	0
$331$	2.07719	0.114173	0.0570865	−	0.998369i	$-0.481819\pi$
$331$	2.07719	0.114173	0.0570865	+	0.998369i	$0.481819\pi$
$332$	0	0
$333$	−0.704162	−0.0385878
$334$	0	0
$335$	−46.9116	−2.56306
$336$	0	0
$337$	25.2750	1.37682	0.688408	−	0.725324i	$-0.258310\pi$
$337$	25.2750	1.37682	0.688408	+	0.725324i	$0.258310\pi$
$338$	0	0
$339$	−3.95051	−0.214562
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−11.5115	−0.621564
$344$	0	0
$345$	48.6790	2.62079
$346$	0	0
$347$	−21.0466	−1.12984	−0.564921	−	0.825145i	$-0.691093\pi$
$347$	−21.0466	−1.12984	−0.564921	+	0.825145i	$0.691093\pi$
$348$	0	0
$349$	−23.2583	−1.24499	−0.622495	−	0.782624i	$-0.713881\pi$
$349$	−23.2583	−1.24499	−0.622495	+	0.782624i	$0.713881\pi$
$350$	0	0
$351$	−12.4685	−0.665522
$352$	0	0
$353$	−18.6464	−0.992446	−0.496223	−	0.868195i	$-0.665280\pi$
$353$	−18.6464	−0.992446	−0.496223	+	0.868195i	$0.665280\pi$
$354$	0	0
$355$	6.57321	0.348870
$356$	0	0
$357$	12.1620	0.643681
$358$	0	0
$359$	−37.6280	−1.98593	−0.992964	−	0.118418i	$-0.962218\pi$
$359$	−37.6280	−1.98593	−0.992964	+	0.118418i	$0.962218\pi$
$360$	0	0
$361$	−13.3262	−0.701381
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−17.0093	−0.890307
$366$	0	0
$367$	−8.54470	−0.446029	−0.223015	−	0.974815i	$-0.571590\pi$
$367$	−8.54470	−0.446029	−0.223015	+	0.974815i	$0.571590\pi$
$368$	0	0
$369$	1.97644	0.102889
$370$	0	0
$371$	6.18359	0.321036
$372$	0	0
$373$	−18.1461	−0.939569	−0.469785	−	0.882781i	$-0.655668\pi$
$373$	−18.1461	−0.939569	−0.469785	+	0.882781i	$0.655668\pi$
$374$	0	0
$375$	10.4790	0.541132
$376$	0	0
$377$	−20.5859	−1.06023
$378$	0	0
$379$	−28.1027	−1.44354	−0.721770	−	0.692133i	$-0.756671\pi$
$379$	−28.1027	−1.44354	−0.721770	+	0.692133i	$0.756671\pi$
$380$	0	0
$381$	−16.0634	−0.822955
$382$	0	0
$383$	−17.7167	−0.905282	−0.452641	−	0.891693i	$-0.649518\pi$
$383$	−17.7167	−0.905282	−0.452641	+	0.891693i	$0.649518\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.49324	−0.0759054
$388$	0	0
$389$	1.32012	0.0669329	0.0334665	−	0.999440i	$-0.489345\pi$
$389$	1.32012	0.0669329	0.0334665	+	0.999440i	$0.489345\pi$
$390$	0	0
$391$	−54.2948	−2.74580
$392$	0	0
$393$	25.6225	1.29248
$394$	0	0
$395$	−45.8320	−2.30606
$396$	0	0
$397$	−13.7700	−0.691096	−0.345548	−	0.938401i	$-0.612307\pi$
$397$	−13.7700	−0.691096	−0.345548	+	0.938401i	$0.612307\pi$
$398$	0	0
$399$	3.97315	0.198906
$400$	0	0
$401$	−17.1795	−0.857902	−0.428951	−	0.903328i	$-0.641117\pi$
$401$	−17.1795	−0.857902	−0.428951	+	0.903328i	$0.641117\pi$
$402$	0	0
$403$	7.40957	0.369097
$404$	0	0
$405$	36.0477	1.79123
$406$	0	0
$407$	0	0
$408$	0	0
$409$	8.44405	0.417531	0.208766	−	0.977966i	$-0.433055\pi$
$409$	8.44405	0.417531	0.208766	+	0.977966i	$0.433055\pi$
$410$	0	0
$411$	−1.90034	−0.0937370
$412$	0	0
$413$	−7.03158	−0.346001
$414$	0	0
$415$	−0.662859	−0.0325385
$416$	0	0
$417$	11.5047	0.563387
$418$	0	0
$419$	−30.9791	−1.51343	−0.756715	−	0.653745i	$-0.773197\pi$
$419$	−30.9791	−1.51343	−0.756715	+	0.653745i	$0.773197\pi$
$420$	0	0
$421$	−25.5582	−1.24563	−0.622816	−	0.782368i	$-0.714011\pi$
$421$	−25.5582	−1.24563	−0.622816	+	0.782368i	$0.714011\pi$
$422$	0	0
$423$	−2.69364	−0.130969
$424$	0	0
$425$	−48.1444	−2.33535
$426$	0	0
$427$	2.34118	0.113298
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−18.0402	−0.868965	−0.434483	−	0.900680i	$-0.643069\pi$
$431$	−18.0402	−0.868965	−0.434483	+	0.900680i	$0.643069\pi$
$432$	0	0
$433$	25.3051	1.21608	0.608042	−	0.793905i	$-0.291955\pi$
$433$	25.3051	1.21608	0.608042	+	0.793905i	$0.291955\pi$
$434$	0	0
$435$	47.9912	2.30100
$436$	0	0
$437$	−17.7373	−0.848491
$438$	0	0
$439$	−28.5630	−1.36324	−0.681620	−	0.731707i	$-0.738724\pi$
$439$	−28.5630	−1.36324	−0.681620	+	0.731707i	$0.738724\pi$
$440$	0	0
$441$	−4.26544	−0.203116
$442$	0	0
$443$	−37.1728	−1.76613	−0.883067	−	0.469247i	$-0.844525\pi$
$443$	−37.1728	−1.76613	−0.883067	+	0.469247i	$0.844525\pi$
$444$	0	0
$445$	−14.2591	−0.675948
$446$	0	0
$447$	−33.1881	−1.56974
$448$	0	0
$449$	10.4365	0.492528	0.246264	−	0.969203i	$-0.420797\pi$
$449$	10.4365	0.492528	0.246264	+	0.969203i	$0.420797\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	1.51679	0.0712652
$454$	0	0
$455$	8.30189	0.389199
$456$	0	0
$457$	26.1344	1.22252	0.611258	−	0.791431i	$-0.290664\pi$
$457$	26.1344	1.22252	0.611258	+	0.791431i	$0.290664\pi$
$458$	0	0
$459$	−32.4208	−1.51328
$460$	0	0
$461$	−11.1743	−0.520439	−0.260219	−	0.965550i	$-0.583795\pi$
$461$	−11.1743	−0.520439	−0.260219	+	0.965550i	$0.583795\pi$
$462$	0	0
$463$	38.7695	1.80177	0.900885	−	0.434059i	$-0.142919\pi$
$463$	38.7695	1.80177	0.900885	+	0.434059i	$0.142919\pi$
$464$	0	0
$465$	−17.2737	−0.801048
$466$	0	0
$467$	2.32049	0.107379	0.0536897	−	0.998558i	$-0.482902\pi$
$467$	2.32049	0.107379	0.0536897	+	0.998558i	$0.482902\pi$
$468$	0	0
$469$	11.9699	0.552719
$470$	0	0
$471$	−24.4431	−1.12628
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−15.7281	−0.721654
$476$	0	0
$477$	4.85966	0.222509
$478$	0	0
$479$	24.8096	1.13358	0.566791	−	0.823862i	$-0.308185\pi$
$479$	24.8096	1.13358	0.566791	+	0.823862i	$0.308185\pi$
$480$	0	0
$481$	2.89074	0.131806
$482$	0	0
$483$	−12.4209	−0.565169
$484$	0	0
$485$	−16.9982	−0.771850
$486$	0	0
$487$	18.3564	0.831808	0.415904	−	0.909409i	$-0.363465\pi$
$487$	18.3564	0.831808	0.415904	+	0.909409i	$0.363465\pi$
$488$	0	0
$489$	−0.318031	−0.0143819
$490$	0	0
$491$	−21.6062	−0.975073	−0.487536	−	0.873103i	$-0.662104\pi$
$491$	−21.6062	−0.975073	−0.487536	+	0.873103i	$0.662104\pi$
$492$	0	0
$493$	−53.5276	−2.41076
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.67721	−0.0752332
$498$	0	0
$499$	−30.8593	−1.38145	−0.690725	−	0.723118i	$-0.742708\pi$
$499$	−30.8593	−1.38145	−0.690725	+	0.723118i	$0.742708\pi$
$500$	0	0
$501$	41.6194	1.85942
$502$	0	0
$503$	−3.24783	−0.144813	−0.0724067	−	0.997375i	$-0.523068\pi$
$503$	−3.24783	−0.144813	−0.0724067	+	0.997375i	$0.523068\pi$
$504$	0	0
$505$	−28.1709	−1.25359
$506$	0	0
$507$	9.85838	0.437826
$508$	0	0
$509$	4.65773	0.206450	0.103225	−	0.994658i	$-0.467084\pi$
$509$	4.65773	0.206450	0.103225	+	0.994658i	$0.467084\pi$
$510$	0	0
$511$	4.34007	0.191993
$512$	0	0
$513$	−10.5914	−0.467623
$514$	0	0
$515$	−2.79472	−0.123150
$516$	0	0
$517$	0	0
$518$	0	0
$519$	25.1371	1.10340
$520$	0	0
$521$	42.3891	1.85710	0.928549	−	0.371209i	$-0.121057\pi$
$521$	42.3891	1.85710	0.928549	+	0.371209i	$0.121057\pi$
$522$	0	0
$523$	−0.302274	−0.0132175	−0.00660876	−	0.999978i	$-0.502104\pi$
$523$	−0.302274	−0.0132175	−0.00660876	+	0.999978i	$0.502104\pi$
$524$	0	0
$525$	−11.0139	−0.480685
$526$	0	0
$527$	19.2664	0.839259
$528$	0	0
$529$	32.4504	1.41089
$530$	0	0
$531$	−5.52609	−0.239812
$532$	0	0
$533$	−8.11373	−0.351445
$534$	0	0
$535$	42.2086	1.82484
$536$	0	0
$537$	−20.3282	−0.877227
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−16.8894	−0.726133	−0.363066	−	0.931763i	$-0.618270\pi$
$541$	−16.8894	−0.726133	−0.363066	+	0.931763i	$0.618270\pi$
$542$	0	0
$543$	12.2860	0.527242
$544$	0	0
$545$	−24.5404	−1.05120
$546$	0	0
$547$	17.4221	0.744913	0.372457	−	0.928050i	$-0.378516\pi$
$547$	17.4221	0.744913	0.372457	+	0.928050i	$0.378516\pi$
$548$	0	0
$549$	1.83993	0.0785262
$550$	0	0
$551$	−17.4867	−0.744958
$552$	0	0
$553$	11.6944	0.497298
$554$	0	0
$555$	−6.73909	−0.286059
$556$	0	0
$557$	−20.6235	−0.873845	−0.436923	−	0.899499i	$-0.643932\pi$
$557$	−20.6235	−0.873845	−0.436923	+	0.899499i	$0.643932\pi$
$558$	0	0
$559$	6.13006	0.259274
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−19.6225	−0.826990	−0.413495	−	0.910506i	$-0.635692\pi$
$563$	−19.6225	−0.826990	−0.413495	+	0.910506i	$0.635692\pi$
$564$	0	0
$565$	−7.01186	−0.294991
$566$	0	0
$567$	−9.19789	−0.386275
$568$	0	0
$569$	−33.7455	−1.41469	−0.707343	−	0.706870i	$-0.750106\pi$
$569$	−33.7455	−1.41469	−0.707343	+	0.706870i	$0.750106\pi$
$570$	0	0
$571$	−19.8373	−0.830164	−0.415082	−	0.909784i	$-0.636247\pi$
$571$	−19.8373	−0.830164	−0.415082	+	0.909784i	$0.636247\pi$
$572$	0	0
$573$	−13.0267	−0.544197
$574$	0	0
$575$	49.1692	2.05050
$576$	0	0
$577$	31.7292	1.32090	0.660452	−	0.750868i	$-0.270365\pi$
$577$	31.7292	1.32090	0.660452	+	0.750868i	$0.270365\pi$
$578$	0	0
$579$	15.6171	0.649024
$580$	0	0
$581$	0.169134	0.00701687
$582$	0	0
$583$	0	0
$584$	0	0
$585$	6.52443	0.269752
$586$	0	0
$587$	19.2328	0.793822	0.396911	−	0.917857i	$-0.370082\pi$
$587$	19.2328	0.793822	0.396911	+	0.917857i	$0.370082\pi$
$588$	0	0
$589$	6.29407	0.259343
$590$	0	0
$591$	−31.6479	−1.30182
$592$	0	0
$593$	−36.8924	−1.51499	−0.757495	−	0.652841i	$-0.773577\pi$
$593$	−36.8924	−1.51499	−0.757495	+	0.652841i	$0.773577\pi$
$594$	0	0
$595$	21.5867	0.884967
$596$	0	0
$597$	−6.52083	−0.266880
$598$	0	0
$599$	−4.09303	−0.167237	−0.0836184	−	0.996498i	$-0.526648\pi$
$599$	−4.09303	−0.167237	−0.0836184	+	0.996498i	$0.526648\pi$
$600$	0	0
$601$	−22.4570	−0.916039	−0.458020	−	0.888942i	$-0.651441\pi$
$601$	−22.4570	−0.916039	−0.458020	+	0.888942i	$0.651441\pi$
$602$	0	0
$603$	9.40711	0.383087
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−14.2310	−0.577620	−0.288810	−	0.957386i	$-0.593260\pi$
$607$	−14.2310	−0.577620	−0.288810	+	0.957386i	$0.593260\pi$
$608$	0	0
$609$	−12.2454	−0.496207
$610$	0	0
$611$	11.0580	0.447359
$612$	0	0
$613$	15.3298	0.619164	0.309582	−	0.950873i	$-0.399811\pi$
$613$	15.3298	0.619164	0.309582	+	0.950873i	$0.399811\pi$
$614$	0	0
$615$	18.9153	0.762738
$616$	0	0
$617$	12.4466	0.501080	0.250540	−	0.968106i	$-0.419392\pi$
$617$	12.4466	0.501080	0.250540	+	0.968106i	$0.419392\pi$
$618$	0	0
$619$	−32.4281	−1.30340	−0.651698	−	0.758479i	$-0.725943\pi$
$619$	−32.4281	−1.30340	−0.651698	+	0.758479i	$0.725943\pi$
$620$	0	0
$621$	33.1109	1.32870
$622$	0	0
$623$	3.63834	0.145767
$624$	0	0
$625$	−14.4155	−0.576621
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7.51653	0.299704
$630$	0	0
$631$	−16.6683	−0.663553	−0.331776	−	0.943358i	$-0.607648\pi$
$631$	−16.6683	−0.663553	−0.331776	+	0.943358i	$0.607648\pi$
$632$	0	0
$633$	30.6738	1.21917
$634$	0	0
$635$	−28.5114	−1.13144
$636$	0	0
$637$	17.5106	0.693793
$638$	0	0
$639$	−1.31811	−0.0521438
$640$	0	0
$641$	−18.3109	−0.723237	−0.361618	−	0.932326i	$-0.617776\pi$
$641$	−18.3109	−0.723237	−0.361618	+	0.932326i	$0.617776\pi$
$642$	0	0
$643$	−5.12185	−0.201986	−0.100993	−	0.994887i	$-0.532202\pi$
$643$	−5.12185	−0.201986	−0.100993	+	0.994887i	$0.532202\pi$
$644$	0	0
$645$	−14.2908	−0.562700
$646$	0	0
$647$	39.4069	1.54925	0.774623	−	0.632423i	$-0.217940\pi$
$647$	39.4069	1.54925	0.774623	+	0.632423i	$0.217940\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	4.40753	0.172745
$652$	0	0
$653$	3.03247	0.118670	0.0593349	−	0.998238i	$-0.481102\pi$
$653$	3.03247	0.118670	0.0593349	+	0.998238i	$0.481102\pi$
$654$	0	0
$655$	45.4780	1.77697
$656$	0	0
$657$	3.41085	0.133070
$658$	0	0
$659$	1.09096	0.0424978	0.0212489	−	0.999774i	$-0.493236\pi$
$659$	1.09096	0.0424978	0.0212489	+	0.999774i	$0.493236\pi$
$660$	0	0
$661$	46.1346	1.79443	0.897214	−	0.441596i	$-0.145588\pi$
$661$	46.1346	1.79443	0.897214	+	0.441596i	$0.145588\pi$
$662$	0	0
$663$	−39.2380	−1.52388
$664$	0	0
$665$	7.05205	0.273467
$666$	0	0
$667$	54.6669	2.11671
$668$	0	0
$669$	−14.5505	−0.562555
$670$	0	0
$671$	0	0
$672$	0	0
$673$	19.7593	0.761665	0.380832	−	0.924644i	$-0.375637\pi$
$673$	19.7593	0.761665	0.380832	+	0.924644i	$0.375637\pi$
$674$	0	0
$675$	29.3602	1.13008
$676$	0	0
$677$	−25.3883	−0.975752	−0.487876	−	0.872913i	$-0.662228\pi$
$677$	−25.3883	−0.975752	−0.487876	+	0.872913i	$0.662228\pi$
$678$	0	0
$679$	4.33724	0.166448
$680$	0	0
$681$	28.8124	1.10410
$682$	0	0
$683$	3.69756	0.141483	0.0707416	−	0.997495i	$-0.477463\pi$
$683$	3.69756	0.141483	0.0707416	+	0.997495i	$0.477463\pi$
$684$	0	0
$685$	−3.37297	−0.128874
$686$	0	0
$687$	4.60946	0.175862
$688$	0	0
$689$	−19.9500	−0.760034
$690$	0	0
$691$	−2.55747	−0.0972908	−0.0486454	−	0.998816i	$-0.515490\pi$
$691$	−2.55747	−0.0972908	−0.0486454	+	0.998816i	$0.515490\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	20.4200	0.774574
$696$	0	0
$697$	−21.0974	−0.799121
$698$	0	0
$699$	−11.9817	−0.453190
$700$	0	0
$701$	−36.9746	−1.39651	−0.698256	−	0.715848i	$-0.746040\pi$
$701$	−36.9746	−1.39651	−0.698256	+	0.715848i	$0.746040\pi$
$702$	0	0
$703$	2.45554	0.0926126
$704$	0	0
$705$	−25.7792	−0.970900
$706$	0	0
$707$	7.18806	0.270335
$708$	0	0
$709$	33.5027	1.25822	0.629111	−	0.777316i	$-0.283419\pi$
$709$	33.5027	1.25822	0.629111	+	0.777316i	$0.283419\pi$
$710$	0	0
$711$	9.19062	0.344675
$712$	0	0
$713$	−19.6765	−0.736892
$714$	0	0
$715$	0	0
$716$	0	0
$717$	32.7404	1.22271
$718$	0	0
$719$	30.0530	1.12079	0.560394	−	0.828226i	$-0.310650\pi$
$719$	30.0530	1.12079	0.560394	+	0.828226i	$0.310650\pi$
$720$	0	0
$721$	0.713096	0.0265571
$722$	0	0
$723$	−0.0319194	−0.00118710
$724$	0	0
$725$	48.4744	1.80029
$726$	0	0
$727$	22.4183	0.831449	0.415725	−	0.909491i	$-0.363528\pi$
$727$	22.4183	0.831449	0.415725	+	0.909491i	$0.363528\pi$
$728$	0	0
$729$	18.3717	0.680433
$730$	0	0
$731$	15.9394	0.589541
$732$	0	0
$733$	−24.5352	−0.906227	−0.453113	−	0.891453i	$-0.649687\pi$
$733$	−24.5352	−0.906227	−0.453113	+	0.891453i	$0.649687\pi$
$734$	0	0
$735$	−40.8218	−1.50574
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−3.37425	−0.124124	−0.0620619	−	0.998072i	$-0.519768\pi$
$739$	−3.37425	−0.124124	−0.0620619	+	0.998072i	$0.519768\pi$
$740$	0	0
$741$	−12.8185	−0.470899
$742$	0	0
$743$	−6.10591	−0.224004	−0.112002	−	0.993708i	$-0.535726\pi$
$743$	−6.10591	−0.224004	−0.112002	+	0.993708i	$0.535726\pi$
$744$	0	0
$745$	−58.9064	−2.15816
$746$	0	0
$747$	0.132922	0.00486336
$748$	0	0
$749$	−10.7699	−0.393523
$750$	0	0
$751$	−50.1107	−1.82857	−0.914283	−	0.405075i	$-0.867245\pi$
$751$	−50.1107	−1.82857	−0.914283	+	0.405075i	$0.867245\pi$
$752$	0	0
$753$	−46.4216	−1.69170
$754$	0	0
$755$	2.69220	0.0979790
$756$	0	0
$757$	−6.09941	−0.221687	−0.110843	−	0.993838i	$-0.535355\pi$
$757$	−6.09941	−0.221687	−0.110843	+	0.993838i	$0.535355\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−44.0697	−1.59753	−0.798763	−	0.601646i	$-0.794512\pi$
$761$	−44.0697	−1.59753	−0.798763	+	0.601646i	$0.794512\pi$
$762$	0	0
$763$	6.26170	0.226689
$764$	0	0
$765$	16.9649	0.613367
$766$	0	0
$767$	22.6858	0.819138
$768$	0	0
$769$	53.7688	1.93895	0.969476	−	0.245185i	$-0.0788488\pi$
$769$	53.7688	1.93895	0.969476	+	0.245185i	$0.0788488\pi$
$770$	0	0
$771$	55.5188	1.99946
$772$	0	0
$773$	31.5994	1.13655	0.568277	−	0.822838i	$-0.307610\pi$
$773$	31.5994	1.13655	0.568277	+	0.822838i	$0.307610\pi$
$774$	0	0
$775$	−17.4476	−0.626737
$776$	0	0
$777$	1.71954	0.0616881
$778$	0	0
$779$	−6.89222	−0.246939
$780$	0	0
$781$	0	0
$782$	0	0
$783$	32.6431	1.16657
$784$	0	0
$785$	−43.3846	−1.54846
$786$	0	0
$787$	0.0219743	0.000783301 0	0.000391650	−	1.00000i	$-0.499875\pi$
$787$	0.0219743	0.000783301 0	0.000391650		1.00000i	$0.499875\pi$
$788$	0	0
$789$	33.1109	1.17878
$790$	0	0
$791$	1.78914	0.0636144
$792$	0	0
$793$	−7.55331	−0.268226
$794$	0	0
$795$	46.5088	1.64950
$796$	0	0
$797$	7.54689	0.267325	0.133662	−	0.991027i	$-0.457326\pi$
$797$	7.54689	0.267325	0.133662	+	0.991027i	$0.457326\pi$
$798$	0	0
$799$	28.7531	1.01721
$800$	0	0
$801$	2.85936	0.101031
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−22.0461	−0.777024
$806$	0	0
$807$	5.38148	0.189437
$808$	0	0
$809$	−45.4008	−1.59621	−0.798104	−	0.602520i	$-0.794163\pi$
$809$	−45.4008	−1.59621	−0.798104	+	0.602520i	$0.794163\pi$
$810$	0	0
$811$	14.1149	0.495641	0.247821	−	0.968806i	$-0.420286\pi$
$811$	14.1149	0.495641	0.247821	+	0.968806i	$0.420286\pi$
$812$	0	0
$813$	−43.6750	−1.53175
$814$	0	0
$815$	−0.564482	−0.0197729
$816$	0	0
$817$	5.20719	0.182176
$818$	0	0
$819$	−1.66476	−0.0581716
$820$	0	0
$821$	0.360417	0.0125786	0.00628932	−	0.999980i	$-0.497998\pi$
$821$	0.360417	0.0125786	0.00628932	+	0.999980i	$0.497998\pi$
$822$	0	0
$823$	−11.4168	−0.397966	−0.198983	−	0.980003i	$-0.563764\pi$
$823$	−11.4168	−0.397966	−0.198983	+	0.980003i	$0.563764\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	39.1953	1.36295	0.681477	−	0.731840i	$-0.261338\pi$
$827$	39.1953	1.36295	0.681477	+	0.731840i	$0.261338\pi$
$828$	0	0
$829$	29.6321	1.02917	0.514583	−	0.857441i	$-0.327947\pi$
$829$	29.6321	1.02917	0.514583	+	0.857441i	$0.327947\pi$
$830$	0	0
$831$	6.38594	0.221526
$832$	0	0
$833$	45.5311	1.57756
$834$	0	0
$835$	73.8715	2.55643
$836$	0	0
$837$	−11.7494	−0.406118
$838$	0	0
$839$	−27.9831	−0.966084	−0.483042	−	0.875597i	$-0.660468\pi$
$839$	−27.9831	−0.966084	−0.483042	+	0.875597i	$0.660468\pi$
$840$	0	0
$841$	24.8945	0.858431
$842$	0	0
$843$	−16.7261	−0.576076
$844$	0	0
$845$	17.4979	0.601946
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−55.8634	−1.91723
$850$	0	0
$851$	−7.67652	−0.263148
$852$	0	0
$853$	−40.8542	−1.39882	−0.699411	−	0.714720i	$-0.746554\pi$
$853$	−40.8542	−1.39882	−0.699411	+	0.714720i	$0.746554\pi$
$854$	0	0
$855$	5.54219	0.189539
$856$	0	0
$857$	−28.8702	−0.986189	−0.493094	−	0.869976i	$-0.664134\pi$
$857$	−28.8702	−0.986189	−0.493094	+	0.869976i	$0.664134\pi$
$858$	0	0
$859$	8.76432	0.299035	0.149517	−	0.988759i	$-0.452228\pi$
$859$	8.76432	0.299035	0.149517	+	0.988759i	$0.452228\pi$
$860$	0	0
$861$	−4.82640	−0.164483
$862$	0	0
$863$	36.8999	1.25609	0.628044	−	0.778178i	$-0.283856\pi$
$863$	36.8999	1.25609	0.628044	+	0.778178i	$0.283856\pi$
$864$	0	0
$865$	44.6166	1.51701
$866$	0	0
$867$	−69.4018	−2.35701
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−38.6182	−1.30853
$872$	0	0
$873$	3.40863	0.115365
$874$	0	0
$875$	−4.74580	−0.160437
$876$	0	0
$877$	−24.9259	−0.841689	−0.420844	−	0.907133i	$-0.638266\pi$
$877$	−24.9259	−0.841689	−0.420844	+	0.907133i	$0.638266\pi$
$878$	0	0
$879$	−42.1356	−1.42120
$880$	0	0
$881$	25.1252	0.846491	0.423245	−	0.906015i	$-0.360891\pi$
$881$	25.1252	0.846491	0.423245	+	0.906015i	$0.360891\pi$
$882$	0	0
$883$	4.56767	0.153714	0.0768571	−	0.997042i	$-0.475511\pi$
$883$	4.56767	0.153714	0.0768571	+	0.997042i	$0.475511\pi$
$884$	0	0
$885$	−52.8868	−1.77777
$886$	0	0
$887$	3.03884	0.102034	0.0510172	−	0.998698i	$-0.483754\pi$
$887$	3.03884	0.102034	0.0510172	+	0.998698i	$0.483754\pi$
$888$	0	0
$889$	7.27494	0.243994
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9.39324	0.314333
$894$	0	0
$895$	−36.0811	−1.20606
$896$	0	0
$897$	40.0732	1.33801
$898$	0	0
$899$	−19.3985	−0.646976
$900$	0	0
$901$	−51.8741	−1.72818
$902$	0	0
$903$	3.64643	0.121345
$904$	0	0
$905$	21.8067	0.724880
$906$	0	0
$907$	14.0372	0.466099	0.233049	−	0.972465i	$-0.425130\pi$
$907$	14.0372	0.466099	0.233049	+	0.972465i	$0.425130\pi$
$908$	0	0
$909$	5.64907	0.187368
$910$	0	0
$911$	−30.0703	−0.996273	−0.498137	−	0.867099i	$-0.665982\pi$
$911$	−30.0703	−0.996273	−0.498137	+	0.867099i	$0.665982\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	17.6088	0.582129
$916$	0	0
$917$	−11.6041	−0.383201
$918$	0	0
$919$	38.5244	1.27080	0.635402	−	0.772182i	$-0.280834\pi$
$919$	38.5244	1.27080	0.635402	+	0.772182i	$0.280834\pi$
$920$	0	0
$921$	−46.4113	−1.52931
$922$	0	0
$923$	5.41115	0.178110
$924$	0	0
$925$	−6.80695	−0.223811
$926$	0	0
$927$	0.560420	0.0184066
$928$	0	0
$929$	31.0130	1.01750	0.508752	−	0.860913i	$-0.330107\pi$
$929$	31.0130	1.01750	0.508752	+	0.860913i	$0.330107\pi$
$930$	0	0
$931$	14.8744	0.487488
$932$	0	0
$933$	6.85380	0.224383
$934$	0	0
$935$	0	0
$936$	0	0
$937$	34.8221	1.13759	0.568795	−	0.822480i	$-0.307410\pi$
$937$	34.8221	1.13759	0.568795	+	0.822480i	$0.307410\pi$
$938$	0	0
$939$	−16.7131	−0.545412
$940$	0	0
$941$	34.1196	1.11227	0.556133	−	0.831093i	$-0.312284\pi$
$941$	34.1196	1.11227	0.556133	+	0.831093i	$0.312284\pi$
$942$	0	0
$943$	21.5465	0.701649
$944$	0	0
$945$	−13.1643	−0.428236
$946$	0	0
$947$	−14.3465	−0.466198	−0.233099	−	0.972453i	$-0.574887\pi$
$947$	−14.3465	−0.466198	−0.233099	+	0.972453i	$0.574887\pi$
$948$	0	0
$949$	−14.0023	−0.454533
$950$	0	0
$951$	42.2604	1.37039
$952$	0	0
$953$	36.4534	1.18084	0.590421	−	0.807095i	$-0.298962\pi$
$953$	36.4534	1.18084	0.590421	+	0.807095i	$0.298962\pi$
$954$	0	0
$955$	−23.1214	−0.748190
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.860642	0.0277916
$960$	0	0
$961$	−24.0178	−0.774768
$962$	0	0
$963$	−8.46401	−0.272749
$964$	0	0
$965$	27.7192	0.892312
$966$	0	0
$967$	−48.0950	−1.54663	−0.773315	−	0.634022i	$-0.781403\pi$
$967$	−48.0950	−1.54663	−0.773315	+	0.634022i	$0.781403\pi$
$968$	0	0
$969$	−33.3308	−1.07074
$970$	0	0
$971$	23.7467	0.762069	0.381034	−	0.924561i	$-0.375568\pi$
$971$	23.7467	0.762069	0.381034	+	0.924561i	$0.375568\pi$
$972$	0	0
$973$	−5.21033	−0.167036
$974$	0	0
$975$	35.5338	1.13799
$976$	0	0
$977$	26.9894	0.863467	0.431733	−	0.902001i	$-0.357902\pi$
$977$	26.9894	0.863467	0.431733	+	0.902001i	$0.357902\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	4.92105	0.157117
$982$	0	0
$983$	23.8385	0.760331	0.380165	−	0.924919i	$-0.375867\pi$
$983$	23.8385	0.760331	0.380165	+	0.924919i	$0.375867\pi$
$984$	0	0
$985$	−56.1727	−1.78981
$986$	0	0
$987$	6.57778	0.209373
$988$	0	0
$989$	−16.2787	−0.517633
$990$	0	0
$991$	−10.6048	−0.336871	−0.168436	−	0.985713i	$-0.553872\pi$
$991$	−10.6048	−0.336871	−0.168436	+	0.985713i	$0.553872\pi$
$992$	0	0
$993$	−3.98641	−0.126505
$994$	0	0
$995$	−11.5740	−0.366920
$996$	0	0
$997$	−2.69848	−0.0854617	−0.0427308	−	0.999087i	$-0.513606\pi$
$997$	−2.69848	−0.0854617	−0.0427308	+	0.999087i	$0.513606\pi$
$998$	0	0
$999$	−4.58385	−0.145027

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7744.2.a.di.1.2		4
4.3	odd	2		7744.2.a.dr.1.3		4
8.3	odd	2		1936.2.a.bb.1.2		4
8.5	even	2		968.2.a.n.1.3		4
11.3	even	5		704.2.m.l.449.2		8
11.4	even	5		704.2.m.l.577.2		8
11.10	odd	2		7744.2.a.dh.1.2		4
24.5	odd	2		8712.2.a.ce.1.1		4
44.3	odd	10		704.2.m.i.449.1		8
44.15	odd	10		704.2.m.i.577.1		8
44.43	even	2		7744.2.a.ds.1.3		4
88.3	odd	10		176.2.m.d.97.2		8
88.5	even	10		968.2.i.s.729.2		8
88.13	odd	10		968.2.i.t.81.2		8
88.21	odd	2		968.2.a.m.1.3		4
88.29	odd	10		968.2.i.p.753.1		8
88.37	even	10		88.2.i.b.49.1	yes	8
88.43	even	2		1936.2.a.bc.1.2		4
88.53	even	10		968.2.i.s.81.2		8
88.59	odd	10		176.2.m.d.49.2		8
88.61	odd	10		968.2.i.t.729.2		8
88.69	even	10		88.2.i.b.9.1	✓	8
88.85	odd	10		968.2.i.p.9.1		8
264.125	odd	10		792.2.r.g.577.1		8
264.197	even	2		8712.2.a.cd.1.1		4
264.245	odd	10		792.2.r.g.361.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.i.b.9.1	✓	8	88.69	even	10
88.2.i.b.49.1	yes	8	88.37	even	10
176.2.m.d.49.2		8	88.59	odd	10
176.2.m.d.97.2		8	88.3	odd	10
704.2.m.i.449.1		8	44.3	odd	10
704.2.m.i.577.1		8	44.15	odd	10
704.2.m.l.449.2		8	11.3	even	5
704.2.m.l.577.2		8	11.4	even	5
792.2.r.g.361.1		8	264.245	odd	10
792.2.r.g.577.1		8	264.125	odd	10
968.2.a.m.1.3		4	88.21	odd	2
968.2.a.n.1.3		4	8.5	even	2
968.2.i.p.9.1		8	88.85	odd	10
968.2.i.p.753.1		8	88.29	odd	10
968.2.i.s.81.2		8	88.53	even	10
968.2.i.s.729.2		8	88.5	even	10
968.2.i.t.81.2		8	88.13	odd	10
968.2.i.t.729.2		8	88.61	odd	10
1936.2.a.bb.1.2		4	8.3	odd	2
1936.2.a.bc.1.2		4	88.43	even	2
7744.2.a.dh.1.2		4	11.10	odd	2
7744.2.a.di.1.2		4	1.1	even	1	trivial
7744.2.a.dr.1.3		4	4.3	odd	2
7744.2.a.ds.1.3		4	44.43	even	2
8712.2.a.cd.1.1		4	264.197	even	2
8712.2.a.ce.1.1		4	24.5	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.91913 q^{3} -3.40632 q^{5} +0.869151 q^{7} +0.683063 q^{9} +O(q^{10})\)	\(q-1.91913 q^{3} -3.40632 q^{5} +0.869151 q^{7} +0.683063 q^{9} -2.80412 q^{13} +6.53716 q^{15} -7.29131 q^{17} -2.38197 q^{19} -1.66801 q^{21} +7.44651 q^{23} +6.60299 q^{25} +4.44651 q^{27} +7.34129 q^{29} -2.64238 q^{31} -2.96060 q^{35} -1.03089 q^{37} +5.38148 q^{39} +2.89350 q^{41} -2.18609 q^{43} -2.32673 q^{45} -3.94348 q^{47} -6.24458 q^{49} +13.9930 q^{51} +7.11452 q^{53} +4.57130 q^{57} -8.09017 q^{59} +2.69364 q^{61} +0.593685 q^{63} +9.55172 q^{65} +13.7720 q^{67} -14.2908 q^{69} -1.92971 q^{71} +4.99346 q^{73} -12.6720 q^{75} +13.4550 q^{79} -10.5826 q^{81} +0.194597 q^{83} +24.8365 q^{85} -14.0889 q^{87} +4.18609 q^{89} -2.43720 q^{91} +5.07108 q^{93} +8.11373 q^{95} +4.99021 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - q^{5} + q^{7} + 6 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - q^5 + q^7 + 6 * q^9	\( 4 q - 2 q^{3} - q^{5} + q^{7} + 6 q^{9} - q^{13} + 16 q^{15} - 12 q^{17} - 14 q^{19} + q^{21} - 2 q^{23} + 11 q^{25} - 14 q^{27} + 9 q^{29} + 11 q^{31} - 18 q^{35} - 13 q^{37} - 18 q^{39} - 8 q^{41} - 3 q^{43} - 22 q^{45} + 7 q^{47} - q^{49} - 14 q^{51} - 11 q^{53} + 7 q^{57} - 10 q^{59} + 17 q^{61} + 15 q^{63} - 5 q^{65} + 5 q^{67} + 4 q^{69} - 5 q^{71} - 6 q^{73} - 7 q^{75} + 7 q^{79} - 8 q^{81} - 20 q^{83} + 13 q^{85} - 3 q^{87} + 11 q^{89} - 6 q^{91} + 10 q^{93} + 6 q^{95} + 4 q^{97}+O(q^{100}) \) 4 * q - 2 * q^3 - q^5 + q^7 + 6 * q^9 - q^13 + 16 * q^15 - 12 * q^17 - 14 * q^19 + q^21 - 2 * q^23 + 11 * q^25 - 14 * q^27 + 9 * q^29 + 11 * q^31 - 18 * q^35 - 13 * q^37 - 18 * q^39 - 8 * q^41 - 3 * q^43 - 22 * q^45 + 7 * q^47 - q^49 - 14 * q^51 - 11 * q^53 + 7 * q^57 - 10 * q^59 + 17 * q^61 + 15 * q^63 - 5 * q^65 + 5 * q^67 + 4 * q^69 - 5 * q^71 - 6 * q^73 - 7 * q^75 + 7 * q^79 - 8 * q^81 - 20 * q^83 + 13 * q^85 - 3 * q^87 + 11 * q^89 - 6 * q^91 + 10 * q^93 + 6 * q^95 + 4 * q^97

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