LMFDB - Embedded newform 5776.2.a.bq.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5776,2,Mod(1,5776)] 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("5776.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5776, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5776 = 2^{4} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5776.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,1,0,1,0,2,0,6,0,-4,0,1,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	yes
Analytic conductor:	$46.1215922075$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	5776.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.24914 q^{3} +0.470683 q^{5} +3.30777 q^{7} +7.55691 q^{9} -1.47068 q^{11} +2.83709 q^{13} +1.52932 q^{15} +6.71982 q^{17} +10.7474 q^{21} -0.470683 q^{23} -4.77846 q^{25} +14.8061 q^{27} -4.96896 q^{29} +6.74742 q^{31} -4.77846 q^{33} +1.55691 q^{35} -4.74742 q^{37} +9.21811 q^{39} +0.501719 q^{41} -3.89572 q^{43} +3.55691 q^{45} -11.9103 q^{47} +3.94137 q^{49} +21.8337 q^{51} +4.71982 q^{53} -0.692226 q^{55} +7.24914 q^{59} -12.5259 q^{61} +24.9966 q^{63} +1.33537 q^{65} -0.633593 q^{67} -1.52932 q^{69} -3.77846 q^{71} -3.94137 q^{73} -15.5259 q^{75} -4.86469 q^{77} -8.83709 q^{79} +25.4362 q^{81} -9.14486 q^{83} +3.16291 q^{85} -16.1449 q^{87} -2.95436 q^{89} +9.38445 q^{91} +21.9233 q^{93} -0.0586332 q^{97} -11.1138 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + q^{5} + 2 q^{7} + 6 q^{9} - 4 q^{11} + q^{13} + 5 q^{15} + 11 q^{17} + 6 q^{21} - q^{23} - 6 q^{25} + 19 q^{27} + 3 q^{29} - 6 q^{31} - 6 q^{33} - 12 q^{35} + 12 q^{37} + q^{39} + 19 q^{41}+ \cdots - q^{97}+O(q^{100}) $ 3 * q + q^3 + q^5 + 2 * q^7 + 6 * q^9 - 4 * q^11 + q^13 + 5 * q^15 + 11 * q^17 + 6 * q^21 - q^23 - 6 * q^25 + 19 * q^27 + 3 * q^29 - 6 * q^31 - 6 * q^33 - 12 * q^35 + 12 * q^37 + q^39 + 19 * q^41 - 5 * q^43 - 6 * q^45 - 17 * q^47 + 11 * q^49 + 23 * q^51 + 5 * q^53 - 10 * q^55 + 13 * q^59 - 3 * q^61 + 40 * q^63 - 21 * q^65 - 9 * q^67 - 5 * q^69 - 3 * q^71 - 11 * q^73 - 12 * q^75 + 10 * q^77 - 19 * q^79 + 23 * q^81 - 12 * q^83 + 17 * q^85 - 33 * q^87 - 3 * q^89 + 44 * q^91 + 42 * q^93 - 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
$3$	3.24914	1.87589	0.937946	−	0.346781i	$-0.112725\pi$
$3$	3.24914	1.87589	0.937946	+	0.346781i	$0.112725\pi$
$4$	0	0
$5$	0.470683	0.210496	0.105248	−	0.994446i	$-0.466436\pi$
$5$	0.470683	0.210496	0.105248	+	0.994446i	$0.466436\pi$
$6$	0	0
$7$	3.30777	1.25022	0.625110	−	0.780536i	$-0.285054\pi$
$7$	3.30777	1.25022	0.625110	+	0.780536i	$0.285054\pi$
$8$	0	0
$9$	7.55691	2.51897
$10$	0	0
$11$	−1.47068	−0.443428	−0.221714	−	0.975112i	$-0.571165\pi$
$11$	−1.47068	−0.443428	−0.221714	+	0.975112i	$0.571165\pi$
$12$	0	0
$13$	2.83709	0.786867	0.393434	−	0.919353i	$-0.371287\pi$
$13$	2.83709	0.786867	0.393434	+	0.919353i	$0.371287\pi$
$14$	0	0
$15$	1.52932	0.394868
$16$	0	0
$17$	6.71982	1.62980	0.814898	−	0.579604i	$-0.196793\pi$
$17$	6.71982	1.62980	0.814898	+	0.579604i	$0.196793\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	10.7474	2.34528
$22$	0	0
$23$	−0.470683	−0.0981443	−0.0490721	−	0.998795i	$-0.515626\pi$
$23$	−0.470683	−0.0981443	−0.0490721	+	0.998795i	$0.515626\pi$
$24$	0	0
$25$	−4.77846	−0.955691
$26$	0	0
$27$	14.8061	2.84943
$28$	0	0
$29$	−4.96896	−0.922714	−0.461357	−	0.887215i	$-0.652637\pi$
$29$	−4.96896	−0.922714	−0.461357	+	0.887215i	$0.652637\pi$
$30$	0	0
$31$	6.74742	1.21187	0.605936	−	0.795513i	$-0.292799\pi$
$31$	6.74742	1.21187	0.605936	+	0.795513i	$0.292799\pi$
$32$	0	0
$33$	−4.77846	−0.831823
$34$	0	0
$35$	1.55691	0.263167
$36$	0	0
$37$	−4.74742	−0.780471	−0.390236	−	0.920715i	$-0.627606\pi$
$37$	−4.74742	−0.780471	−0.390236	+	0.920715i	$0.627606\pi$
$38$	0	0
$39$	9.21811	1.47608
$40$	0	0
$41$	0.501719	0.0783553	0.0391777	−	0.999232i	$-0.487526\pi$
$41$	0.501719	0.0783553	0.0391777	+	0.999232i	$0.487526\pi$
$42$	0	0
$43$	−3.89572	−0.594092	−0.297046	−	0.954863i	$-0.596002\pi$
$43$	−3.89572	−0.594092	−0.297046	+	0.954863i	$0.596002\pi$
$44$	0	0
$45$	3.55691	0.530233
$46$	0	0
$47$	−11.9103	−1.73730	−0.868650	−	0.495426i	$-0.835012\pi$
$47$	−11.9103	−1.73730	−0.868650	+	0.495426i	$0.835012\pi$
$48$	0	0
$49$	3.94137	0.563052
$50$	0	0
$51$	21.8337	3.05732
$52$	0	0
$53$	4.71982	0.648318	0.324159	−	0.946003i	$-0.394919\pi$
$53$	4.71982	0.648318	0.324159	+	0.946003i	$0.394919\pi$
$54$	0	0
$55$	−0.692226	−0.0933398
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.24914	0.943758	0.471879	−	0.881663i	$-0.343576\pi$
$59$	7.24914	0.943758	0.471879	+	0.881663i	$0.343576\pi$
$60$	0	0
$61$	−12.5259	−1.60377	−0.801887	−	0.597475i	$-0.796171\pi$
$61$	−12.5259	−1.60377	−0.801887	+	0.597475i	$0.796171\pi$
$62$	0	0
$63$	24.9966	3.14927
$64$	0	0
$65$	1.33537	0.165632
$66$	0	0
$67$	−0.633593	−0.0774057	−0.0387029	−	0.999251i	$-0.512323\pi$
$67$	−0.633593	−0.0774057	−0.0387029	+	0.999251i	$0.512323\pi$
$68$	0	0
$69$	−1.52932	−0.184108
$70$	0	0
$71$	−3.77846	−0.448420	−0.224210	−	0.974541i	$-0.571980\pi$
$71$	−3.77846	−0.448420	−0.224210	+	0.974541i	$0.571980\pi$
$72$	0	0
$73$	−3.94137	−0.461302	−0.230651	−	0.973037i	$-0.574086\pi$
$73$	−3.94137	−0.461302	−0.230651	+	0.973037i	$0.574086\pi$
$74$	0	0
$75$	−15.5259	−1.79277
$76$	0	0
$77$	−4.86469	−0.554383
$78$	0	0
$79$	−8.83709	−0.994250	−0.497125	−	0.867679i	$-0.665611\pi$
$79$	−8.83709	−0.994250	−0.497125	+	0.867679i	$0.665611\pi$
$80$	0	0
$81$	25.4362	2.82625
$82$	0	0
$83$	−9.14486	−1.00378	−0.501890	−	0.864932i	$-0.667362\pi$
$83$	−9.14486	−1.00378	−0.501890	+	0.864932i	$0.667362\pi$
$84$	0	0
$85$	3.16291	0.343066
$86$	0	0
$87$	−16.1449	−1.73091
$88$	0	0
$89$	−2.95436	−0.313161	−0.156581	−	0.987665i	$-0.550047\pi$
$89$	−2.95436	−0.313161	−0.156581	+	0.987665i	$0.550047\pi$
$90$	0	0
$91$	9.38445	0.983758
$92$	0	0
$93$	21.9233	2.27334
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.0586332	−0.00595330	−0.00297665	−	0.999996i	$-0.500947\pi$
$97$	−0.0586332	−0.00595330	−0.00297665	+	0.999996i	$0.500947\pi$
$98$	0	0
$99$	−11.1138	−1.11698
$100$	0	0
$101$	11.5845	1.15270	0.576351	−	0.817202i	$-0.304476\pi$
$101$	11.5845	1.15270	0.576351	+	0.817202i	$0.304476\pi$
$102$	0	0
$103$	−8.17246	−0.805257	−0.402628	−	0.915364i	$-0.631903\pi$
$103$	−8.17246	−0.805257	−0.402628	+	0.915364i	$0.631903\pi$
$104$	0	0
$105$	5.05863	0.493672
$106$	0	0
$107$	0.498281	0.0481706	0.0240853	−	0.999710i	$-0.492333\pi$
$107$	0.498281	0.0481706	0.0240853	+	0.999710i	$0.492333\pi$
$108$	0	0
$109$	8.39400	0.804000	0.402000	−	0.915640i	$-0.368315\pi$
$109$	8.39400	0.804000	0.402000	+	0.915640i	$0.368315\pi$
$110$	0	0
$111$	−15.4250	−1.46408
$112$	0	0
$113$	−4.83365	−0.454712	−0.227356	−	0.973812i	$-0.573008\pi$
$113$	−4.83365	−0.454712	−0.227356	+	0.973812i	$0.573008\pi$
$114$	0	0
$115$	−0.221543	−0.0206590
$116$	0	0
$117$	21.4396	1.98210
$118$	0	0
$119$	22.2277	2.03761
$120$	0	0
$121$	−8.83709	−0.803372
$122$	0	0
$123$	1.63016	0.146986
$124$	0	0
$125$	−4.60256	−0.411665
$126$	0	0
$127$	−18.4492	−1.63710	−0.818551	−	0.574434i	$-0.805222\pi$
$127$	−18.4492	−1.63710	−0.818551	+	0.574434i	$0.805222\pi$
$128$	0	0
$129$	−12.6578	−1.11445
$130$	0	0
$131$	−0.868126	−0.0758485	−0.0379243	−	0.999281i	$-0.512075\pi$
$131$	−0.868126	−0.0758485	−0.0379243	+	0.999281i	$0.512075\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	6.96896	0.599793
$136$	0	0
$137$	4.55691	0.389323	0.194662	−	0.980870i	$-0.437639\pi$
$137$	4.55691	0.389323	0.194662	+	0.980870i	$0.437639\pi$
$138$	0	0
$139$	16.3078	1.38321	0.691604	−	0.722277i	$-0.256905\pi$
$139$	16.3078	1.38321	0.691604	+	0.722277i	$0.256905\pi$
$140$	0	0
$141$	−38.6983	−3.25899
$142$	0	0
$143$	−4.17246	−0.348919
$144$	0	0
$145$	−2.33881	−0.194228
$146$	0	0
$147$	12.8061	1.05623
$148$	0	0
$149$	20.9069	1.71276	0.856380	−	0.516347i	$-0.172708\pi$
$149$	20.9069	1.71276	0.856380	+	0.516347i	$0.172708\pi$
$150$	0	0
$151$	1.75086	0.142483	0.0712415	−	0.997459i	$-0.477304\pi$
$151$	1.75086	0.142483	0.0712415	+	0.997459i	$0.477304\pi$
$152$	0	0
$153$	50.7811	4.10541
$154$	0	0
$155$	3.17590	0.255094
$156$	0	0
$157$	−0.159472	−0.0127272	−0.00636362	−	0.999980i	$-0.502026\pi$
$157$	−0.159472	−0.0127272	−0.00636362	+	0.999980i	$0.502026\pi$
$158$	0	0
$159$	15.3354	1.21617
$160$	0	0
$161$	−1.55691	−0.122702
$162$	0	0
$163$	0.0862308	0.00675412	0.00337706	−	0.999994i	$-0.498925\pi$
$163$	0.0862308	0.00675412	0.00337706	+	0.999994i	$0.498925\pi$
$164$	0	0
$165$	−2.24914	−0.175095
$166$	0	0
$167$	−6.27674	−0.485709	−0.242854	−	0.970063i	$-0.578084\pi$
$167$	−6.27674	−0.485709	−0.242854	+	0.970063i	$0.578084\pi$
$168$	0	0
$169$	−4.95092	−0.380840
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.8371	0.823929	0.411964	−	0.911200i	$-0.364843\pi$
$173$	10.8371	0.823929	0.411964	+	0.911200i	$0.364843\pi$
$174$	0	0
$175$	−15.8061	−1.19483
$176$	0	0
$177$	23.5535	1.77039
$178$	0	0
$179$	7.08967	0.529907	0.264953	−	0.964261i	$-0.414643\pi$
$179$	7.08967	0.529907	0.264953	+	0.964261i	$0.414643\pi$
$180$	0	0
$181$	12.3534	0.918222	0.459111	−	0.888379i	$-0.348168\pi$
$181$	12.3534	0.918222	0.459111	+	0.888379i	$0.348168\pi$
$182$	0	0
$183$	−40.6983	−3.00851
$184$	0	0
$185$	−2.23453	−0.164286
$186$	0	0
$187$	−9.88273	−0.722697
$188$	0	0
$189$	48.9751	3.56241
$190$	0	0
$191$	−8.99656	−0.650968	−0.325484	−	0.945547i	$-0.605527\pi$
$191$	−8.99656	−0.650968	−0.325484	+	0.945547i	$0.605527\pi$
$192$	0	0
$193$	3.16291	0.227671	0.113836	−	0.993500i	$-0.463686\pi$
$193$	3.16291	0.227671	0.113836	+	0.993500i	$0.463686\pi$
$194$	0	0
$195$	4.33881	0.310709
$196$	0	0
$197$	−21.2457	−1.51369	−0.756847	−	0.653592i	$-0.773261\pi$
$197$	−21.2457	−1.51369	−0.756847	+	0.653592i	$0.773261\pi$
$198$	0	0
$199$	−9.89572	−0.701489	−0.350745	−	0.936471i	$-0.614071\pi$
$199$	−9.89572	−0.701489	−0.350745	+	0.936471i	$0.614071\pi$
$200$	0	0
$201$	−2.05863	−0.145205
$202$	0	0
$203$	−16.4362	−1.15360
$204$	0	0
$205$	0.236151	0.0164935
$206$	0	0
$207$	−3.55691	−0.247223
$208$	0	0
$209$	0	0
$210$	0	0
$211$	19.1008	1.31496	0.657478	−	0.753474i	$-0.271623\pi$
$211$	19.1008	1.31496	0.657478	+	0.753474i	$0.271623\pi$
$212$	0	0
$213$	−12.2767	−0.841188
$214$	0	0
$215$	−1.83365	−0.125054
$216$	0	0
$217$	22.3189	1.51511
$218$	0	0
$219$	−12.8061	−0.865353
$220$	0	0
$221$	19.0647	1.28243
$222$	0	0
$223$	−16.9690	−1.13633	−0.568163	−	0.822916i	$-0.692346\pi$
$223$	−16.9690	−1.13633	−0.568163	+	0.822916i	$0.692346\pi$
$224$	0	0
$225$	−36.1104	−2.40736
$226$	0	0
$227$	0.646583	0.0429152	0.0214576	−	0.999770i	$-0.493169\pi$
$227$	0.646583	0.0429152	0.0214576	+	0.999770i	$0.493169\pi$
$228$	0	0
$229$	−8.55348	−0.565230	−0.282615	−	0.959233i	$-0.591202\pi$
$229$	−8.55348	−0.565230	−0.282615	+	0.959233i	$0.591202\pi$
$230$	0	0
$231$	−15.8061	−1.03996
$232$	0	0
$233$	−3.70683	−0.242843	−0.121421	−	0.992601i	$-0.538745\pi$
$233$	−3.70683	−0.242843	−0.121421	+	0.992601i	$0.538745\pi$
$234$	0	0
$235$	−5.60600	−0.365695
$236$	0	0
$237$	−28.7129	−1.86511
$238$	0	0
$239$	−15.7655	−1.01978	−0.509892	−	0.860239i	$-0.670315\pi$
$239$	−15.7655	−1.01978	−0.509892	+	0.860239i	$0.670315\pi$
$240$	0	0
$241$	16.1138	1.03798	0.518991	−	0.854779i	$-0.326308\pi$
$241$	16.1138	1.03798	0.518991	+	0.854779i	$0.326308\pi$
$242$	0	0
$243$	38.2277	2.45231
$244$	0	0
$245$	1.85514	0.118520
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−29.7129	−1.88298
$250$	0	0
$251$	15.6561	0.988206	0.494103	−	0.869403i	$-0.335496\pi$
$251$	15.6561	0.988206	0.494103	+	0.869403i	$0.335496\pi$
$252$	0	0
$253$	0.692226	0.0435199
$254$	0	0
$255$	10.2767	0.643554
$256$	0	0
$257$	27.8172	1.73519	0.867595	−	0.497271i	$-0.165664\pi$
$257$	27.8172	1.73519	0.867595	+	0.497271i	$0.165664\pi$
$258$	0	0
$259$	−15.7034	−0.975762
$260$	0	0
$261$	−37.5500	−2.32429
$262$	0	0
$263$	−16.8517	−1.03912	−0.519560	−	0.854434i	$-0.673904\pi$
$263$	−16.8517	−1.03912	−0.519560	+	0.854434i	$0.673904\pi$
$264$	0	0
$265$	2.22154	0.136468
$266$	0	0
$267$	−9.59912	−0.587457
$268$	0	0
$269$	29.6251	1.80627	0.903137	−	0.429352i	$-0.141258\pi$
$269$	29.6251	1.80627	0.903137	+	0.429352i	$0.141258\pi$
$270$	0	0
$271$	−14.4707	−0.879031	−0.439516	−	0.898235i	$-0.644850\pi$
$271$	−14.4707	−0.879031	−0.439516	+	0.898235i	$0.644850\pi$
$272$	0	0
$273$	30.4914	1.84542
$274$	0	0
$275$	7.02760	0.423780
$276$	0	0
$277$	−12.4216	−0.746342	−0.373171	−	0.927763i	$-0.621729\pi$
$277$	−12.4216	−0.746342	−0.373171	+	0.927763i	$0.621729\pi$
$278$	0	0
$279$	50.9897	3.05267
$280$	0	0
$281$	10.2932	0.614039	0.307019	−	0.951703i	$-0.400668\pi$
$281$	10.2932	0.614039	0.307019	+	0.951703i	$0.400668\pi$
$282$	0	0
$283$	15.8647	0.943058	0.471529	−	0.881851i	$-0.343702\pi$
$283$	15.8647	0.943058	0.471529	+	0.881851i	$0.343702\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.65957	0.0979615
$288$	0	0
$289$	28.1560	1.65624
$290$	0	0
$291$	−0.190507	−0.0111677
$292$	0	0
$293$	12.1579	0.710269	0.355135	−	0.934815i	$-0.384435\pi$
$293$	12.1579	0.710269	0.355135	+	0.934815i	$0.384435\pi$
$294$	0	0
$295$	3.41205	0.198657
$296$	0	0
$297$	−21.7750	−1.26351
$298$	0	0
$299$	−1.33537	−0.0772265
$300$	0	0
$301$	−12.8862	−0.742747
$302$	0	0
$303$	37.6397	2.16234
$304$	0	0
$305$	−5.89572	−0.337588
$306$	0	0
$307$	−12.3009	−0.702049	−0.351025	−	0.936366i	$-0.614167\pi$
$307$	−12.3009	−0.702049	−0.351025	+	0.936366i	$0.614167\pi$
$308$	0	0
$309$	−26.5535	−1.51057
$310$	0	0
$311$	0.519765	0.0294732	0.0147366	−	0.999891i	$-0.495309\pi$
$311$	0.519765	0.0294732	0.0147366	+	0.999891i	$0.495309\pi$
$312$	0	0
$313$	18.6121	1.05202	0.526009	−	0.850479i	$-0.323688\pi$
$313$	18.6121	1.05202	0.526009	+	0.850479i	$0.323688\pi$
$314$	0	0
$315$	11.7655	0.662909
$316$	0	0
$317$	2.83709	0.159347	0.0796734	−	0.996821i	$-0.474612\pi$
$317$	2.83709	0.159347	0.0796734	+	0.996821i	$0.474612\pi$
$318$	0	0
$319$	7.30777	0.409157
$320$	0	0
$321$	1.61899	0.0903629
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−13.5569	−0.752002
$326$	0	0
$327$	27.2733	1.50822
$328$	0	0
$329$	−39.3967	−2.17201
$330$	0	0
$331$	−19.0276	−1.04585	−0.522926	−	0.852378i	$-0.675159\pi$
$331$	−19.0276	−1.04585	−0.522926	+	0.852378i	$0.675159\pi$
$332$	0	0
$333$	−35.8759	−1.96598
$334$	0	0
$335$	−0.298222	−0.0162936
$336$	0	0
$337$	14.7294	0.802360	0.401180	−	0.915999i	$-0.368600\pi$
$337$	14.7294	0.802360	0.401180	+	0.915999i	$0.368600\pi$
$338$	0	0
$339$	−15.7052	−0.852990
$340$	0	0
$341$	−9.92332	−0.537378
$342$	0	0
$343$	−10.1173	−0.546281
$344$	0	0
$345$	−0.719824	−0.0387540
$346$	0	0
$347$	10.1353	0.544092	0.272046	−	0.962284i	$-0.412300\pi$
$347$	10.1353	0.544092	0.272046	+	0.962284i	$0.412300\pi$
$348$	0	0
$349$	2.56035	0.137053	0.0685263	−	0.997649i	$-0.478170\pi$
$349$	2.56035	0.137053	0.0685263	+	0.997649i	$0.478170\pi$
$350$	0	0
$351$	42.0061	2.24212
$352$	0	0
$353$	20.4458	1.08822	0.544109	−	0.839015i	$-0.316868\pi$
$353$	20.4458	1.08822	0.544109	+	0.839015i	$0.316868\pi$
$354$	0	0
$355$	−1.77846	−0.0943907
$356$	0	0
$357$	72.2208	3.82233
$358$	0	0
$359$	14.5879	0.769923	0.384961	−	0.922933i	$-0.374215\pi$
$359$	14.5879	0.769923	0.384961	+	0.922933i	$0.374215\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−28.7129	−1.50704
$364$	0	0
$365$	−1.85514	−0.0971023
$366$	0	0
$367$	−29.0794	−1.51793	−0.758965	−	0.651131i	$-0.774295\pi$
$367$	−29.0794	−1.51793	−0.758965	+	0.651131i	$0.774295\pi$
$368$	0	0
$369$	3.79145	0.197375
$370$	0	0
$371$	15.6121	0.810540
$372$	0	0
$373$	−20.7880	−1.07636	−0.538181	−	0.842829i	$-0.680888\pi$
$373$	−20.7880	−1.07636	−0.538181	+	0.842829i	$0.680888\pi$
$374$	0	0
$375$	−14.9544	−0.772240
$376$	0	0
$377$	−14.0974	−0.726053
$378$	0	0
$379$	19.7034	1.01210	0.506048	−	0.862505i	$-0.331106\pi$
$379$	19.7034	1.01210	0.506048	+	0.862505i	$0.331106\pi$
$380$	0	0
$381$	−59.9440	−3.07103
$382$	0	0
$383$	28.4638	1.45443	0.727216	−	0.686408i	$-0.240814\pi$
$383$	28.4638	1.45443	0.727216	+	0.686408i	$0.240814\pi$
$384$	0	0
$385$	−2.28973	−0.116695
$386$	0	0
$387$	−29.4396	−1.49650
$388$	0	0
$389$	3.71639	0.188428	0.0942141	−	0.995552i	$-0.469966\pi$
$389$	3.71639	0.188428	0.0942141	+	0.995552i	$0.469966\pi$
$390$	0	0
$391$	−3.16291	−0.159955
$392$	0	0
$393$	−2.82066	−0.142284
$394$	0	0
$395$	−4.15947	−0.209286
$396$	0	0
$397$	17.6397	0.885312	0.442656	−	0.896692i	$-0.354036\pi$
$397$	17.6397	0.885312	0.442656	+	0.896692i	$0.354036\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.5569	1.52594	0.762970	−	0.646434i	$-0.223741\pi$
$401$	30.5569	1.52594	0.762970	+	0.646434i	$0.223741\pi$
$402$	0	0
$403$	19.1430	0.953583
$404$	0	0
$405$	11.9724	0.594913
$406$	0	0
$407$	6.98195	0.346083
$408$	0	0
$409$	31.2277	1.54411	0.772054	−	0.635557i	$-0.219229\pi$
$409$	31.2277	1.54411	0.772054	+	0.635557i	$0.219229\pi$
$410$	0	0
$411$	14.8061	0.730329
$412$	0	0
$413$	23.9785	1.17991
$414$	0	0
$415$	−4.30434	−0.211292
$416$	0	0
$417$	52.9862	2.59475
$418$	0	0
$419$	28.9966	1.41657	0.708287	−	0.705924i	$-0.249468\pi$
$419$	28.9966	1.41657	0.708287	+	0.705924i	$0.249468\pi$
$420$	0	0
$421$	−31.2587	−1.52346	−0.761728	−	0.647897i	$-0.775649\pi$
$421$	−31.2587	−1.52346	−0.761728	+	0.647897i	$0.775649\pi$
$422$	0	0
$423$	−90.0054	−4.37621
$424$	0	0
$425$	−32.1104	−1.55758
$426$	0	0
$427$	−41.4328	−2.00507
$428$	0	0
$429$	−13.5569	−0.654534
$430$	0	0
$431$	−14.8663	−0.716085	−0.358042	−	0.933705i	$-0.616556\pi$
$431$	−14.8663	−0.716085	−0.358042	+	0.933705i	$0.616556\pi$
$432$	0	0
$433$	−39.5630	−1.90128	−0.950639	−	0.310299i	$-0.899571\pi$
$433$	−39.5630	−1.90128	−0.950639	+	0.310299i	$0.899571\pi$
$434$	0	0
$435$	−7.59912	−0.364350
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−11.5845	−0.552899	−0.276449	−	0.961029i	$-0.589158\pi$
$439$	−11.5845	−0.552899	−0.276449	+	0.961029i	$0.589158\pi$
$440$	0	0
$441$	29.7846	1.41831
$442$	0	0
$443$	−28.9233	−1.37419	−0.687094	−	0.726568i	$-0.741114\pi$
$443$	−28.9233	−1.37419	−0.687094	+	0.726568i	$0.741114\pi$
$444$	0	0
$445$	−1.39057	−0.0659192
$446$	0	0
$447$	67.9294	3.21295
$448$	0	0
$449$	−31.2147	−1.47311	−0.736556	−	0.676377i	$-0.763549\pi$
$449$	−31.2147	−1.47311	−0.736556	+	0.676377i	$0.763549\pi$
$450$	0	0
$451$	−0.737870	−0.0347449
$452$	0	0
$453$	5.68879	0.267283
$454$	0	0
$455$	4.41711	0.207077
$456$	0	0
$457$	17.2215	0.805590	0.402795	−	0.915290i	$-0.368039\pi$
$457$	17.2215	0.805590	0.402795	+	0.915290i	$0.368039\pi$
$458$	0	0
$459$	99.4941	4.64399
$460$	0	0
$461$	−32.8302	−1.52906	−0.764528	−	0.644591i	$-0.777028\pi$
$461$	−32.8302	−1.52906	−0.764528	+	0.644591i	$0.777028\pi$
$462$	0	0
$463$	−6.28973	−0.292308	−0.146154	−	0.989262i	$-0.546690\pi$
$463$	−6.28973	−0.292308	−0.146154	+	0.989262i	$0.546690\pi$
$464$	0	0
$465$	10.3189	0.478530
$466$	0	0
$467$	9.47068	0.438251	0.219125	−	0.975697i	$-0.429680\pi$
$467$	9.47068	0.438251	0.219125	+	0.975697i	$0.429680\pi$
$468$	0	0
$469$	−2.09578	−0.0967743
$470$	0	0
$471$	−0.518147	−0.0238749
$472$	0	0
$473$	5.72938	0.263437
$474$	0	0
$475$	0	0
$476$	0	0
$477$	35.6673	1.63309
$478$	0	0
$479$	−26.0682	−1.19109	−0.595543	−	0.803324i	$-0.703063\pi$
$479$	−26.0682	−1.19109	−0.595543	+	0.803324i	$0.703063\pi$
$480$	0	0
$481$	−13.4689	−0.614127
$482$	0	0
$483$	−5.05863	−0.230176
$484$	0	0
$485$	−0.0275977	−0.00125315
$486$	0	0
$487$	16.8939	0.765536	0.382768	−	0.923845i	$-0.374971\pi$
$487$	16.8939	0.765536	0.382768	+	0.923845i	$0.374971\pi$
$488$	0	0
$489$	0.280176	0.0126700
$490$	0	0
$491$	−17.8888	−0.807312	−0.403656	−	0.914911i	$-0.632261\pi$
$491$	−17.8888	−0.807312	−0.403656	+	0.914911i	$0.632261\pi$
$492$	0	0
$493$	−33.3906	−1.50384
$494$	0	0
$495$	−5.23109	−0.235120
$496$	0	0
$497$	−12.4983	−0.560625
$498$	0	0
$499$	29.4216	1.31709	0.658546	−	0.752541i	$-0.271172\pi$
$499$	29.4216	1.31709	0.658546	+	0.752541i	$0.271172\pi$
$500$	0	0
$501$	−20.3940	−0.911137
$502$	0	0
$503$	−9.07936	−0.404828	−0.202414	−	0.979300i	$-0.564879\pi$
$503$	−9.07936	−0.404828	−0.202414	+	0.979300i	$0.564879\pi$
$504$	0	0
$505$	5.45264	0.242639
$506$	0	0
$507$	−16.0862	−0.714415
$508$	0	0
$509$	20.3940	0.903948	0.451974	−	0.892031i	$-0.350720\pi$
$509$	20.3940	0.903948	0.451974	+	0.892031i	$0.350720\pi$
$510$	0	0
$511$	−13.0371	−0.576730
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.84664	−0.169503
$516$	0	0
$517$	17.5163	0.770367
$518$	0	0
$519$	35.2112	1.54560
$520$	0	0
$521$	14.8697	0.651455	0.325728	−	0.945464i	$-0.394391\pi$
$521$	14.8697	0.651455	0.325728	+	0.945464i	$0.394391\pi$
$522$	0	0
$523$	−10.8992	−0.476587	−0.238294	−	0.971193i	$-0.576588\pi$
$523$	−10.8992	−0.476587	−0.238294	+	0.971193i	$0.576588\pi$
$524$	0	0
$525$	−51.3561	−2.24136
$526$	0	0
$527$	45.3415	1.97511
$528$	0	0
$529$	−22.7785	−0.990368
$530$	0	0
$531$	54.7811	2.37730
$532$	0	0
$533$	1.42342	0.0616552
$534$	0	0
$535$	0.234533	0.0101397
$536$	0	0
$537$	23.0353	0.994048
$538$	0	0
$539$	−5.79650	−0.249673
$540$	0	0
$541$	11.9509	0.513810	0.256905	−	0.966437i	$-0.417297\pi$
$541$	11.9509	0.513810	0.256905	+	0.966437i	$0.417297\pi$
$542$	0	0
$543$	40.1380	1.72249
$544$	0	0
$545$	3.95092	0.169239
$546$	0	0
$547$	−4.00611	−0.171289	−0.0856445	−	0.996326i	$-0.527295\pi$
$547$	−4.00611	−0.171289	−0.0856445	+	0.996326i	$0.527295\pi$
$548$	0	0
$549$	−94.6570	−4.03986
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−29.2311	−1.24303
$554$	0	0
$555$	−7.26031	−0.308183
$556$	0	0
$557$	10.7198	0.454214	0.227107	−	0.973870i	$-0.427073\pi$
$557$	10.7198	0.454214	0.227107	+	0.973870i	$0.427073\pi$
$558$	0	0
$559$	−11.0525	−0.467472
$560$	0	0
$561$	−32.1104	−1.35570
$562$	0	0
$563$	7.18096	0.302641	0.151321	−	0.988485i	$-0.451647\pi$
$563$	7.18096	0.302641	0.151321	+	0.988485i	$0.451647\pi$
$564$	0	0
$565$	−2.27512	−0.0957150
$566$	0	0
$567$	84.1372	3.53343
$568$	0	0
$569$	−31.4328	−1.31773	−0.658865	−	0.752261i	$-0.728963\pi$
$569$	−31.4328	−1.31773	−0.658865	+	0.752261i	$0.728963\pi$
$570$	0	0
$571$	−8.08623	−0.338398	−0.169199	−	0.985582i	$-0.554118\pi$
$571$	−8.08623	−0.338398	−0.169199	+	0.985582i	$0.554118\pi$
$572$	0	0
$573$	−29.2311	−1.22115
$574$	0	0
$575$	2.24914	0.0937956
$576$	0	0
$577$	4.95779	0.206396	0.103198	−	0.994661i	$-0.467093\pi$
$577$	4.95779	0.206396	0.103198	+	0.994661i	$0.467093\pi$
$578$	0	0
$579$	10.2767	0.427087
$580$	0	0
$581$	−30.2491	−1.25495
$582$	0	0
$583$	−6.94137	−0.287482
$584$	0	0
$585$	10.0913	0.417223
$586$	0	0
$587$	−23.7716	−0.981158	−0.490579	−	0.871397i	$-0.663215\pi$
$587$	−23.7716	−0.981158	−0.490579	+	0.871397i	$0.663215\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−69.0303	−2.83953
$592$	0	0
$593$	−42.4036	−1.74131	−0.870653	−	0.491898i	$-0.836303\pi$
$593$	−42.4036	−1.74131	−0.870653	+	0.491898i	$0.836303\pi$
$594$	0	0
$595$	10.4622	0.428908
$596$	0	0
$597$	−32.1526	−1.31592
$598$	0	0
$599$	42.8517	1.75087	0.875436	−	0.483333i	$-0.160574\pi$
$599$	42.8517	1.75087	0.875436	+	0.483333i	$0.160574\pi$
$600$	0	0
$601$	15.1855	0.619427	0.309714	−	0.950830i	$-0.399767\pi$
$601$	15.1855	0.619427	0.309714	+	0.950830i	$0.399767\pi$
$602$	0	0
$603$	−4.78801	−0.194983
$604$	0	0
$605$	−4.15947	−0.169107
$606$	0	0
$607$	−32.3336	−1.31238	−0.656189	−	0.754596i	$-0.727833\pi$
$607$	−32.3336	−1.31238	−0.656189	+	0.754596i	$0.727833\pi$
$608$	0	0
$609$	−53.4036	−2.16402
$610$	0	0
$611$	−33.7907	−1.36702
$612$	0	0
$613$	−7.83365	−0.316398	−0.158199	−	0.987407i	$-0.550569\pi$
$613$	−7.83365	−0.316398	−0.158199	+	0.987407i	$0.550569\pi$
$614$	0	0
$615$	0.767287	0.0309400
$616$	0	0
$617$	−4.26719	−0.171790	−0.0858952	−	0.996304i	$-0.527375\pi$
$617$	−4.26719	−0.171790	−0.0858952	+	0.996304i	$0.527375\pi$
$618$	0	0
$619$	−42.8363	−1.72174	−0.860869	−	0.508827i	$-0.830079\pi$
$619$	−42.8363	−1.72174	−0.860869	+	0.508827i	$0.830079\pi$
$620$	0	0
$621$	−6.96896	−0.279655
$622$	0	0
$623$	−9.77234	−0.391521
$624$	0	0
$625$	21.7259	0.869038
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−31.9018	−1.27201
$630$	0	0
$631$	36.3104	1.44550	0.722748	−	0.691112i	$-0.242879\pi$
$631$	36.3104	1.44550	0.722748	+	0.691112i	$0.242879\pi$
$632$	0	0
$633$	62.0613	2.46672
$634$	0	0
$635$	−8.68373	−0.344603
$636$	0	0
$637$	11.1820	0.443048
$638$	0	0
$639$	−28.5535	−1.12956
$640$	0	0
$641$	49.6087	1.95942	0.979712	−	0.200409i	$-0.0642270\pi$
$641$	49.6087	1.95942	0.979712	+	0.200409i	$0.0642270\pi$
$642$	0	0
$643$	4.98539	0.196605	0.0983023	−	0.995157i	$-0.468659\pi$
$643$	4.98539	0.196605	0.0983023	+	0.995157i	$0.468659\pi$
$644$	0	0
$645$	−5.95779	−0.234588
$646$	0	0
$647$	43.2863	1.70176	0.850880	−	0.525360i	$-0.176069\pi$
$647$	43.2863	1.70176	0.850880	+	0.525360i	$0.176069\pi$
$648$	0	0
$649$	−10.6612	−0.418488
$650$	0	0
$651$	72.5174	2.84218
$652$	0	0
$653$	−5.43965	−0.212870	−0.106435	−	0.994320i	$-0.533944\pi$
$653$	−5.43965	−0.212870	−0.106435	+	0.994320i	$0.533944\pi$
$654$	0	0
$655$	−0.408612	−0.0159658
$656$	0	0
$657$	−29.7846	−1.16201
$658$	0	0
$659$	−5.74924	−0.223959	−0.111979	−	0.993711i	$-0.535719\pi$
$659$	−5.74924	−0.223959	−0.111979	+	0.993711i	$0.535719\pi$
$660$	0	0
$661$	10.2553	0.398883	0.199442	−	0.979910i	$-0.436087\pi$
$661$	10.2553	0.398883	0.199442	+	0.979910i	$0.436087\pi$
$662$	0	0
$663$	61.9440	2.40571
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.33881	0.0905591
$668$	0	0
$669$	−55.1346	−2.13162
$670$	0	0
$671$	18.4216	0.711158
$672$	0	0
$673$	29.9379	1.15402	0.577011	−	0.816736i	$-0.304219\pi$
$673$	29.9379	1.15402	0.577011	+	0.816736i	$0.304219\pi$
$674$	0	0
$675$	−70.7501	−2.72317
$676$	0	0
$677$	11.8827	0.456691	0.228345	−	0.973580i	$-0.426668\pi$
$677$	11.8827	0.456691	0.228345	+	0.973580i	$0.426668\pi$
$678$	0	0
$679$	−0.193945	−0.00744293
$680$	0	0
$681$	2.10084	0.0805043
$682$	0	0
$683$	6.35180	0.243045	0.121522	−	0.992589i	$-0.461222\pi$
$683$	6.35180	0.243045	0.121522	+	0.992589i	$0.461222\pi$
$684$	0	0
$685$	2.14486	0.0819510
$686$	0	0
$687$	−27.7914	−1.06031
$688$	0	0
$689$	13.3906	0.510140
$690$	0	0
$691$	−28.9966	−1.10308	−0.551541	−	0.834148i	$-0.685960\pi$
$691$	−28.9966	−1.10308	−0.551541	+	0.834148i	$0.685960\pi$
$692$	0	0
$693$	−36.7620	−1.39647
$694$	0	0
$695$	7.67580	0.291160
$696$	0	0
$697$	3.37146	0.127703
$698$	0	0
$699$	−12.0440	−0.455547
$700$	0	0
$701$	4.00162	0.151139	0.0755695	−	0.997141i	$-0.475923\pi$
$701$	4.00162	0.151139	0.0755695	+	0.997141i	$0.475923\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−18.2147	−0.686004
$706$	0	0
$707$	38.3189	1.44113
$708$	0	0
$709$	−16.4638	−0.618311	−0.309156	−	0.951011i	$-0.600046\pi$
$709$	−16.4638	−0.618311	−0.309156	+	0.951011i	$0.600046\pi$
$710$	0	0
$711$	−66.7811	−2.50449
$712$	0	0
$713$	−3.17590	−0.118938
$714$	0	0
$715$	−1.96391	−0.0734460
$716$	0	0
$717$	−51.2242	−1.91300
$718$	0	0
$719$	−19.2112	−0.716458	−0.358229	−	0.933634i	$-0.616619\pi$
$719$	−19.2112	−0.716458	−0.358229	+	0.933634i	$0.616619\pi$
$720$	0	0
$721$	−27.0327	−1.00675
$722$	0	0
$723$	52.3561	1.94714
$724$	0	0
$725$	23.7440	0.881829
$726$	0	0
$727$	−18.1663	−0.673753	−0.336876	−	0.941549i	$-0.609370\pi$
$727$	−18.1663	−0.673753	−0.336876	+	0.941549i	$0.609370\pi$
$728$	0	0
$729$	47.8984	1.77401
$730$	0	0
$731$	−26.1786	−0.968250
$732$	0	0
$733$	50.2423	1.85574	0.927870	−	0.372903i	$-0.121638\pi$
$733$	50.2423	1.85574	0.927870	+	0.372903i	$0.121638\pi$
$734$	0	0
$735$	6.02760	0.222331
$736$	0	0
$737$	0.931815	0.0343238
$738$	0	0
$739$	29.7766	1.09535	0.547676	−	0.836691i	$-0.315513\pi$
$739$	29.7766	1.09535	0.547676	+	0.836691i	$0.315513\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	39.5156	1.44969	0.724843	−	0.688914i	$-0.241912\pi$
$743$	39.5156	1.44969	0.724843	+	0.688914i	$0.241912\pi$
$744$	0	0
$745$	9.84053	0.360529
$746$	0	0
$747$	−69.1070	−2.52849
$748$	0	0
$749$	1.64820	0.0602240
$750$	0	0
$751$	−18.1043	−0.660634	−0.330317	−	0.943870i	$-0.607156\pi$
$751$	−18.1043	−0.660634	−0.330317	+	0.943870i	$0.607156\pi$
$752$	0	0
$753$	50.8690	1.85377
$754$	0	0
$755$	0.824101	0.0299921
$756$	0	0
$757$	−23.5991	−0.857725	−0.428862	−	0.903370i	$-0.641085\pi$
$757$	−23.5991	−0.857725	−0.428862	+	0.903370i	$0.641085\pi$
$758$	0	0
$759$	2.24914	0.0816386
$760$	0	0
$761$	−1.69061	−0.0612845	−0.0306422	−	0.999530i	$-0.509755\pi$
$761$	−1.69061	−0.0612845	−0.0306422	+	0.999530i	$0.509755\pi$
$762$	0	0
$763$	27.7655	1.00518
$764$	0	0
$765$	23.9018	0.864173
$766$	0	0
$767$	20.5665	0.742612
$768$	0	0
$769$	−52.1165	−1.87937	−0.939685	−	0.342042i	$-0.888881\pi$
$769$	−52.1165	−1.87937	−0.939685	+	0.342042i	$0.888881\pi$
$770$	0	0
$771$	90.3821	3.25503
$772$	0	0
$773$	−45.0794	−1.62139	−0.810696	−	0.585468i	$-0.800911\pi$
$773$	−45.0794	−1.62139	−0.810696	+	0.585468i	$0.800911\pi$
$774$	0	0
$775$	−32.2423	−1.15818
$776$	0	0
$777$	−51.0225	−1.83042
$778$	0	0
$779$	0	0
$780$	0	0
$781$	5.55691	0.198842
$782$	0	0
$783$	−73.5708	−2.62920
$784$	0	0
$785$	−0.0750608	−0.00267903
$786$	0	0
$787$	−38.1414	−1.35960	−0.679798	−	0.733400i	$-0.737932\pi$
$787$	−38.1414	−1.35960	−0.679798	+	0.733400i	$0.737932\pi$
$788$	0	0
$789$	−54.7535	−1.94928
$790$	0	0
$791$	−15.9886	−0.568490
$792$	0	0
$793$	−35.5370	−1.26196
$794$	0	0
$795$	7.21811	0.256000
$796$	0	0
$797$	13.7148	0.485802	0.242901	−	0.970051i	$-0.421901\pi$
$797$	13.7148	0.485802	0.242901	+	0.970051i	$0.421901\pi$
$798$	0	0
$799$	−80.0353	−2.83145
$800$	0	0
$801$	−22.3258	−0.788844
$802$	0	0
$803$	5.79650	0.204554
$804$	0	0
$805$	−0.732814	−0.0258283
$806$	0	0
$807$	96.2561	3.38838
$808$	0	0
$809$	−21.6285	−0.760419	−0.380209	−	0.924900i	$-0.624148\pi$
$809$	−21.6285	−0.760419	−0.380209	+	0.924900i	$0.624148\pi$
$810$	0	0
$811$	−6.54049	−0.229668	−0.114834	−	0.993385i	$-0.536634\pi$
$811$	−6.54049	−0.229668	−0.114834	+	0.993385i	$0.536634\pi$
$812$	0	0
$813$	−47.0173	−1.64897
$814$	0	0
$815$	0.0405874	0.00142172
$816$	0	0
$817$	0	0
$818$	0	0
$819$	70.9175	2.47806
$820$	0	0
$821$	41.6182	1.45249	0.726243	−	0.687438i	$-0.241265\pi$
$821$	41.6182	1.45249	0.726243	+	0.687438i	$0.241265\pi$
$822$	0	0
$823$	5.45264	0.190067	0.0950335	−	0.995474i	$-0.469704\pi$
$823$	5.45264	0.190067	0.0950335	+	0.995474i	$0.469704\pi$
$824$	0	0
$825$	22.8337	0.794966
$826$	0	0
$827$	−2.68879	−0.0934983	−0.0467492	−	0.998907i	$-0.514886\pi$
$827$	−2.68879	−0.0934983	−0.0467492	+	0.998907i	$0.514886\pi$
$828$	0	0
$829$	27.4328	0.952780	0.476390	−	0.879234i	$-0.341945\pi$
$829$	27.4328	0.952780	0.476390	+	0.879234i	$0.341945\pi$
$830$	0	0
$831$	−40.3595	−1.40006
$832$	0	0
$833$	26.4853	0.917661
$834$	0	0
$835$	−2.95436	−0.102240
$836$	0	0
$837$	99.9027	3.45314
$838$	0	0
$839$	−27.4413	−0.947378	−0.473689	−	0.880692i	$-0.657078\pi$
$839$	−27.4413	−0.947378	−0.473689	+	0.880692i	$0.657078\pi$
$840$	0	0
$841$	−4.30939	−0.148600
$842$	0	0
$843$	33.4439	1.15187
$844$	0	0
$845$	−2.33032	−0.0801653
$846$	0	0
$847$	−29.2311	−1.00439
$848$	0	0
$849$	51.5466	1.76907
$850$	0	0
$851$	2.23453	0.0765988
$852$	0	0
$853$	36.3871	1.24587	0.622936	−	0.782273i	$-0.285940\pi$
$853$	36.3871	1.24587	0.622936	+	0.782273i	$0.285940\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	49.1295	1.67823	0.839116	−	0.543953i	$-0.183073\pi$
$857$	49.1295	1.67823	0.839116	+	0.543953i	$0.183073\pi$
$858$	0	0
$859$	−51.9130	−1.77125	−0.885624	−	0.464402i	$-0.846269\pi$
$859$	−51.9130	−1.77125	−0.885624	+	0.464402i	$0.846269\pi$
$860$	0	0
$861$	5.39218	0.183765
$862$	0	0
$863$	27.9785	0.952400	0.476200	−	0.879337i	$-0.342014\pi$
$863$	27.9785	0.952400	0.476200	+	0.879337i	$0.342014\pi$
$864$	0	0
$865$	5.10084	0.173434
$866$	0	0
$867$	91.4829	3.10692
$868$	0	0
$869$	12.9966	0.440878
$870$	0	0
$871$	−1.79756	−0.0609080
$872$	0	0
$873$	−0.443086	−0.0149962
$874$	0	0
$875$	−15.2242	−0.514673
$876$	0	0
$877$	12.4017	0.418777	0.209388	−	0.977833i	$-0.432853\pi$
$877$	12.4017	0.418777	0.209388	+	0.977833i	$0.432853\pi$
$878$	0	0
$879$	39.5026	1.33239
$880$	0	0
$881$	17.8923	0.602806	0.301403	−	0.953497i	$-0.402545\pi$
$881$	17.8923	0.602806	0.301403	+	0.953497i	$0.402545\pi$
$882$	0	0
$883$	11.0337	0.371314	0.185657	−	0.982615i	$-0.440559\pi$
$883$	11.0337	0.371314	0.185657	+	0.982615i	$0.440559\pi$
$884$	0	0
$885$	11.0862	0.372660
$886$	0	0
$887$	−40.3940	−1.35630	−0.678149	−	0.734924i	$-0.737218\pi$
$887$	−40.3940	−1.35630	−0.678149	+	0.734924i	$0.737218\pi$
$888$	0	0
$889$	−61.0258	−2.04674
$890$	0	0
$891$	−37.4086	−1.25324
$892$	0	0
$893$	0	0
$894$	0	0
$895$	3.33699	0.111543
$896$	0	0
$897$	−4.33881	−0.144869
$898$	0	0
$899$	−33.5277	−1.11821
$900$	0	0
$901$	31.7164	1.05663
$902$	0	0
$903$	−41.8690	−1.39331
$904$	0	0
$905$	5.81455	0.193282
$906$	0	0
$907$	−39.1510	−1.29999	−0.649993	−	0.759940i	$-0.725228\pi$
$907$	−39.1510	−1.29999	−0.649993	+	0.759940i	$0.725228\pi$
$908$	0	0
$909$	87.5432	2.90362
$910$	0	0
$911$	−27.3561	−0.906348	−0.453174	−	0.891422i	$-0.649708\pi$
$911$	−27.3561	−0.906348	−0.453174	+	0.891422i	$0.649708\pi$
$912$	0	0
$913$	13.4492	0.445104
$914$	0	0
$915$	−19.1560	−0.633279
$916$	0	0
$917$	−2.87156	−0.0948274
$918$	0	0
$919$	0.948243	0.0312796	0.0156398	−	0.999878i	$-0.495021\pi$
$919$	0.948243	0.0312796	0.0156398	+	0.999878i	$0.495021\pi$
$920$	0	0
$921$	−39.9673	−1.31697
$922$	0	0
$923$	−10.7198	−0.352847
$924$	0	0
$925$	22.6854	0.745890
$926$	0	0
$927$	−61.7586	−2.02842
$928$	0	0
$929$	5.26051	0.172592	0.0862959	−	0.996270i	$-0.472497\pi$
$929$	5.26051	0.172592	0.0862959	+	0.996270i	$0.472497\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1.68879	0.0552885
$934$	0	0
$935$	−4.65164	−0.152125
$936$	0	0
$937$	45.9276	1.50039	0.750195	−	0.661217i	$-0.229960\pi$
$937$	45.9276	1.50039	0.750195	+	0.661217i	$0.229960\pi$
$938$	0	0
$939$	60.4734	1.97347
$940$	0	0
$941$	44.5957	1.45378	0.726889	−	0.686755i	$-0.240966\pi$
$941$	44.5957	1.45378	0.726889	+	0.686755i	$0.240966\pi$
$942$	0	0
$943$	−0.236151	−0.00769013
$944$	0	0
$945$	23.0518	0.749874
$946$	0	0
$947$	14.0162	0.455466	0.227733	−	0.973724i	$-0.426869\pi$
$947$	14.0162	0.455466	0.227733	+	0.973724i	$0.426869\pi$
$948$	0	0
$949$	−11.1820	−0.362984
$950$	0	0
$951$	9.21811	0.298918
$952$	0	0
$953$	−2.84664	−0.0922118	−0.0461059	−	0.998937i	$-0.514681\pi$
$953$	−2.84664	−0.0922118	−0.0461059	+	0.998937i	$0.514681\pi$
$954$	0	0
$955$	−4.23453	−0.137026
$956$	0	0
$957$	23.7440	0.767534
$958$	0	0
$959$	15.0732	0.486740
$960$	0	0
$961$	14.5277	0.468635
$962$	0	0
$963$	3.76547	0.121340
$964$	0	0
$965$	1.48873	0.0479239
$966$	0	0
$967$	1.66119	0.0534203	0.0267101	−	0.999643i	$-0.491497\pi$
$967$	1.66119	0.0534203	0.0267101	+	0.999643i	$0.491497\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	40.4182	1.29708	0.648540	−	0.761180i	$-0.275380\pi$
$971$	40.4182	1.29708	0.648540	+	0.761180i	$0.275380\pi$
$972$	0	0
$973$	53.9424	1.72931
$974$	0	0
$975$	−44.0483	−1.41068
$976$	0	0
$977$	−10.5439	−0.337330	−0.168665	−	0.985673i	$-0.553946\pi$
$977$	−10.5439	−0.337330	−0.168665	+	0.985673i	$0.553946\pi$
$978$	0	0
$979$	4.34492	0.138864
$980$	0	0
$981$	63.4328	2.02525
$982$	0	0
$983$	−9.83365	−0.313645	−0.156822	−	0.987627i	$-0.550125\pi$
$983$	−9.83365	−0.313645	−0.156822	+	0.987627i	$0.550125\pi$
$984$	0	0
$985$	−10.0000	−0.318626
$986$	0	0
$987$	−128.005	−4.07446
$988$	0	0
$989$	1.83365	0.0583068
$990$	0	0
$991$	20.2699	0.643893	0.321947	−	0.946758i	$-0.395663\pi$
$991$	20.2699	0.643893	0.321947	+	0.946758i	$0.395663\pi$
$992$	0	0
$993$	−61.8233	−1.96190
$994$	0	0
$995$	−4.65775	−0.147661
$996$	0	0
$997$	15.3208	0.485213	0.242607	−	0.970125i	$-0.421998\pi$
$997$	15.3208	0.485213	0.242607	+	0.970125i	$0.421998\pi$
$998$	0	0
$999$	−70.2906	−2.22390

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5776.2.a.bq.1.3		3
4.3	odd	2		2888.2.a.n.1.1		3
19.8	odd	6		304.2.i.f.273.3		6
19.12	odd	6		304.2.i.f.49.3		6
19.18	odd	2		5776.2.a.bk.1.1		3
57.8	even	6		2736.2.s.y.577.2		6
57.50	even	6		2736.2.s.y.1873.2		6
76.27	even	6		152.2.i.c.121.1	yes	6
76.31	even	6		152.2.i.c.49.1	✓	6
76.75	even	2		2888.2.a.r.1.3		3
152.27	even	6		1216.2.i.n.577.3		6
152.69	odd	6		1216.2.i.m.961.1		6
152.107	even	6		1216.2.i.n.961.3		6
152.141	odd	6		1216.2.i.m.577.1		6
228.107	odd	6		1368.2.s.k.505.2		6
228.179	odd	6		1368.2.s.k.577.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.i.c.49.1	✓	6	76.31	even	6
152.2.i.c.121.1	yes	6	76.27	even	6
304.2.i.f.49.3		6	19.12	odd	6
304.2.i.f.273.3		6	19.8	odd	6
1216.2.i.m.577.1		6	152.141	odd	6
1216.2.i.m.961.1		6	152.69	odd	6
1216.2.i.n.577.3		6	152.27	even	6
1216.2.i.n.961.3		6	152.107	even	6
1368.2.s.k.505.2		6	228.107	odd	6
1368.2.s.k.577.2		6	228.179	odd	6
2736.2.s.y.577.2		6	57.8	even	6
2736.2.s.y.1873.2		6	57.50	even	6
2888.2.a.n.1.1		3	4.3	odd	2
2888.2.a.r.1.3		3	76.75	even	2
5776.2.a.bk.1.1		3	19.18	odd	2
5776.2.a.bq.1.3		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5776 = 2^{4} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5776.a (trivial)

\(f(q)\)	\(=\)	\(q+3.24914 q^{3} +0.470683 q^{5} +3.30777 q^{7} +7.55691 q^{9} -1.47068 q^{11} +2.83709 q^{13} +1.52932 q^{15} +6.71982 q^{17} +10.7474 q^{21} -0.470683 q^{23} -4.77846 q^{25} +14.8061 q^{27} -4.96896 q^{29} +6.74742 q^{31} -4.77846 q^{33} +1.55691 q^{35} -4.74742 q^{37} +9.21811 q^{39} +0.501719 q^{41} -3.89572 q^{43} +3.55691 q^{45} -11.9103 q^{47} +3.94137 q^{49} +21.8337 q^{51} +4.71982 q^{53} -0.692226 q^{55} +7.24914 q^{59} -12.5259 q^{61} +24.9966 q^{63} +1.33537 q^{65} -0.633593 q^{67} -1.52932 q^{69} -3.77846 q^{71} -3.94137 q^{73} -15.5259 q^{75} -4.86469 q^{77} -8.83709 q^{79} +25.4362 q^{81} -9.14486 q^{83} +3.16291 q^{85} -16.1449 q^{87} -2.95436 q^{89} +9.38445 q^{91} +21.9233 q^{93} -0.0586332 q^{97} -11.1138 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + q^{5} + 2 q^{7} + 6 q^{9} - 4 q^{11} + q^{13} + 5 q^{15} + 11 q^{17} + 6 q^{21} - q^{23} - 6 q^{25} + 19 q^{27} + 3 q^{29} - 6 q^{31} - 6 q^{33} - 12 q^{35} + 12 q^{37} + q^{39} + 19 q^{41}+ \cdots - q^{97}+O(q^{100}) \) 3 * q + q^3 + q^5 + 2 * q^7 + 6 * q^9 - 4 * q^11 + q^13 + 5 * q^15 + 11 * q^17 + 6 * q^21 - q^23 - 6 * q^25 + 19 * q^27 + 3 * q^29 - 6 * q^31 - 6 * q^33 - 12 * q^35 + 12 * q^37 + q^39 + 19 * q^41 - 5 * q^43 - 6 * q^45 - 17 * q^47 + 11 * q^49 + 23 * q^51 + 5 * q^53 - 10 * q^55 + 13 * q^59 - 3 * q^61 + 40 * q^63 - 21 * q^65 - 9 * q^67 - 5 * q^69 - 3 * q^71 - 11 * q^73 - 12 * q^75 + 10 * q^77 - 19 * q^79 + 23 * q^81 - 12 * q^83 + 17 * q^85 - 33 * q^87 - 3 * q^89 + 44 * q^91 + 42 * q^93 - q^97

Introduction

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