LMFDB - Embedded newform 5776.2.a.br.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5776,2,Mod(1,5776)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 5776 = 2^{4} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5776.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:	$46.1215922075$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.87939$ 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)

$f(q)$	$=$	$q+2.87939 q^{3} +0.879385 q^{5} -0.347296 q^{7} +5.29086 q^{9} +O(q^{10})$	$q+2.87939 q^{3} +0.879385 q^{5} -0.347296 q^{7} +5.29086 q^{9} +2.22668 q^{11} +2.57398 q^{13} +2.53209 q^{15} +0.467911 q^{17} -1.00000 q^{21} +2.69459 q^{23} -4.22668 q^{25} +6.59627 q^{27} -6.87939 q^{29} +7.10607 q^{31} +6.41147 q^{33} -0.305407 q^{35} +4.94356 q^{37} +7.41147 q^{39} +2.47565 q^{41} -3.90167 q^{43} +4.65270 q^{45} +7.29086 q^{47} -6.87939 q^{49} +1.34730 q^{51} -2.83750 q^{53} +1.95811 q^{55} +6.30541 q^{59} +9.12836 q^{61} -1.83750 q^{63} +2.26352 q^{65} -7.67499 q^{67} +7.75877 q^{69} +9.30541 q^{71} +1.38919 q^{73} -12.1702 q^{75} -0.773318 q^{77} +11.8452 q^{79} +3.12061 q^{81} +14.8307 q^{83} +0.411474 q^{85} -19.8084 q^{87} -10.2909 q^{89} -0.893933 q^{91} +20.4611 q^{93} +9.45336 q^{97} +11.7811 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^5	$ 3 q + 3 q^{3} - 3 q^{5} + 3 q^{15} + 6 q^{17} - 3 q^{21} + 6 q^{23} - 6 q^{25} + 6 q^{27} - 15 q^{29} + 9 q^{31} + 9 q^{33} - 3 q^{35} + 12 q^{39} - 12 q^{41} + 15 q^{45} + 6 q^{47} - 15 q^{49} + 3 q^{51} - 6 q^{53} + 9 q^{55} + 21 q^{59} + 9 q^{61} - 3 q^{63} + 12 q^{65} - 18 q^{67} + 12 q^{69} + 30 q^{71} - 15 q^{75} - 9 q^{77} + 9 q^{79} + 15 q^{81} - 9 q^{85} - 21 q^{87} - 15 q^{89} - 15 q^{91} + 24 q^{93} + 15 q^{97} + 18 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^15 + 6 * q^17 - 3 * q^21 + 6 * q^23 - 6 * q^25 + 6 * q^27 - 15 * q^29 + 9 * q^31 + 9 * q^33 - 3 * q^35 + 12 * q^39 - 12 * q^41 + 15 * q^45 + 6 * q^47 - 15 * q^49 + 3 * q^51 - 6 * q^53 + 9 * q^55 + 21 * q^59 + 9 * q^61 - 3 * q^63 + 12 * q^65 - 18 * q^67 + 12 * q^69 + 30 * q^71 - 15 * q^75 - 9 * q^77 + 9 * q^79 + 15 * q^81 - 9 * q^85 - 21 * q^87 - 15 * q^89 - 15 * q^91 + 24 * q^93 + 15 * q^97 + 18 * q^99

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
$3$	2.87939	1.66241	0.831207	−	0.555963i	$-0.187650\pi$
$3$	2.87939	1.66241	0.831207	+	0.555963i	$0.187650\pi$
$4$	0	0
$5$	0.879385	0.393273	0.196637	−	0.980476i	$-0.436998\pi$
$5$	0.879385	0.393273	0.196637	+	0.980476i	$0.436998\pi$
$6$	0	0
$7$	−0.347296	−0.131266	−0.0656328	−	0.997844i	$-0.520907\pi$
$7$	−0.347296	−0.131266	−0.0656328	+	0.997844i	$0.520907\pi$
$8$	0	0
$9$	5.29086	1.76362
$10$	0	0
$11$	2.22668	0.671370	0.335685	−	0.941974i	$-0.391032\pi$
$11$	2.22668	0.671370	0.335685	+	0.941974i	$0.391032\pi$
$12$	0	0
$13$	2.57398	0.713893	0.356947	−	0.934125i	$-0.383818\pi$
$13$	2.57398	0.713893	0.356947	+	0.934125i	$0.383818\pi$
$14$	0	0
$15$	2.53209	0.653783
$16$	0	0
$17$	0.467911	0.113485	0.0567426	−	0.998389i	$-0.481929\pi$
$17$	0.467911	0.113485	0.0567426	+	0.998389i	$0.481929\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	2.69459	0.561861	0.280931	−	0.959728i	$-0.409357\pi$
$23$	2.69459	0.561861	0.280931	+	0.959728i	$0.409357\pi$
$24$	0	0
$25$	−4.22668	−0.845336
$26$	0	0
$27$	6.59627	1.26945
$28$	0	0
$29$	−6.87939	−1.27747	−0.638735	−	0.769427i	$-0.720542\pi$
$29$	−6.87939	−1.27747	−0.638735	+	0.769427i	$0.720542\pi$
$30$	0	0
$31$	7.10607	1.27629	0.638144	−	0.769917i	$-0.279703\pi$
$31$	7.10607	1.27629	0.638144	+	0.769917i	$0.279703\pi$
$32$	0	0
$33$	6.41147	1.11609
$34$	0	0
$35$	−0.305407	−0.0516233
$36$	0	0
$37$	4.94356	0.812717	0.406358	−	0.913714i	$-0.366798\pi$
$37$	4.94356	0.812717	0.406358	+	0.913714i	$0.366798\pi$
$38$	0	0
$39$	7.41147	1.18679
$40$	0	0
$41$	2.47565	0.386632	0.193316	−	0.981137i	$-0.438076\pi$
$41$	2.47565	0.386632	0.193316	+	0.981137i	$0.438076\pi$
$42$	0	0
$43$	−3.90167	−0.595000	−0.297500	−	0.954722i	$-0.596153\pi$
$43$	−3.90167	−0.595000	−0.297500	+	0.954722i	$0.596153\pi$
$44$	0	0
$45$	4.65270	0.693584
$46$	0	0
$47$	7.29086	1.06348	0.531741	−	0.846907i	$-0.321538\pi$
$47$	7.29086	1.06348	0.531741	+	0.846907i	$0.321538\pi$
$48$	0	0
$49$	−6.87939	−0.982769
$50$	0	0
$51$	1.34730	0.188659
$52$	0	0
$53$	−2.83750	−0.389760	−0.194880	−	0.980827i	$-0.562432\pi$
$53$	−2.83750	−0.389760	−0.194880	+	0.980827i	$0.562432\pi$
$54$	0	0
$55$	1.95811	0.264032
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.30541	0.820894	0.410447	−	0.911884i	$-0.365373\pi$
$59$	6.30541	0.820894	0.410447	+	0.911884i	$0.365373\pi$
$60$	0	0
$61$	9.12836	1.16877	0.584383	−	0.811478i	$-0.301337\pi$
$61$	9.12836	1.16877	0.584383	+	0.811478i	$0.301337\pi$
$62$	0	0
$63$	−1.83750	−0.231503
$64$	0	0
$65$	2.26352	0.280755
$66$	0	0
$67$	−7.67499	−0.937650	−0.468825	−	0.883291i	$-0.655322\pi$
$67$	−7.67499	−0.937650	−0.468825	+	0.883291i	$0.655322\pi$
$68$	0	0
$69$	7.75877	0.934046
$70$	0	0
$71$	9.30541	1.10435	0.552174	−	0.833729i	$-0.313798\pi$
$71$	9.30541	1.10435	0.552174	+	0.833729i	$0.313798\pi$
$72$	0	0
$73$	1.38919	0.162592	0.0812959	−	0.996690i	$-0.474094\pi$
$73$	1.38919	0.162592	0.0812959	+	0.996690i	$0.474094\pi$
$74$	0	0
$75$	−12.1702	−1.40530
$76$	0	0
$77$	−0.773318	−0.0881278
$78$	0	0
$79$	11.8452	1.33269	0.666347	−	0.745642i	$-0.267857\pi$
$79$	11.8452	1.33269	0.666347	+	0.745642i	$0.267857\pi$
$80$	0	0
$81$	3.12061	0.346735
$82$	0	0
$83$	14.8307	1.62788	0.813940	−	0.580949i	$-0.197319\pi$
$83$	14.8307	1.62788	0.813940	+	0.580949i	$0.197319\pi$
$84$	0	0
$85$	0.411474	0.0446306
$86$	0	0
$87$	−19.8084	−2.12368
$88$	0	0
$89$	−10.2909	−1.09083	−0.545414	−	0.838166i	$-0.683628\pi$
$89$	−10.2909	−1.09083	−0.545414	+	0.838166i	$0.683628\pi$
$90$	0	0
$91$	−0.893933	−0.0937097
$92$	0	0
$93$	20.4611	2.12172
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.45336	0.959844	0.479922	−	0.877311i	$-0.340665\pi$
$97$	9.45336	0.959844	0.479922	+	0.877311i	$0.340665\pi$
$98$	0	0
$99$	11.7811	1.18404
$100$	0	0
$101$	−9.24897	−0.920307	−0.460153	−	0.887839i	$-0.652206\pi$
$101$	−9.24897	−0.920307	−0.460153	+	0.887839i	$0.652206\pi$
$102$	0	0
$103$	5.50980	0.542897	0.271448	−	0.962453i	$-0.412497\pi$
$103$	5.50980	0.542897	0.271448	+	0.962453i	$0.412497\pi$
$104$	0	0
$105$	−0.879385	−0.0858192
$106$	0	0
$107$	−10.2344	−0.989399	−0.494699	−	0.869064i	$-0.664722\pi$
$107$	−10.2344	−0.989399	−0.494699	+	0.869064i	$0.664722\pi$
$108$	0	0
$109$	−1.82295	−0.174607	−0.0873034	−	0.996182i	$-0.527825\pi$
$109$	−1.82295	−0.174607	−0.0873034	+	0.996182i	$0.527825\pi$
$110$	0	0
$111$	14.2344	1.35107
$112$	0	0
$113$	−17.6878	−1.66393	−0.831963	−	0.554830i	$-0.812783\pi$
$113$	−17.6878	−1.66393	−0.831963	+	0.554830i	$0.812783\pi$
$114$	0	0
$115$	2.36959	0.220965
$116$	0	0
$117$	13.6186	1.25904
$118$	0	0
$119$	−0.162504	−0.0148967
$120$	0	0
$121$	−6.04189	−0.549263
$122$	0	0
$123$	7.12836	0.642742
$124$	0	0
$125$	−8.11381	−0.725721
$126$	0	0
$127$	−11.6040	−1.02969	−0.514845	−	0.857284i	$-0.672150\pi$
$127$	−11.6040	−1.02969	−0.514845	+	0.857284i	$0.672150\pi$
$128$	0	0
$129$	−11.2344	−0.989136
$130$	0	0
$131$	−1.84524	−0.161219	−0.0806096	−	0.996746i	$-0.525687\pi$
$131$	−1.84524	−0.161219	−0.0806096	+	0.996746i	$0.525687\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	5.80066	0.499241
$136$	0	0
$137$	0.255777	0.0218525	0.0109263	−	0.999940i	$-0.496522\pi$
$137$	0.255777	0.0218525	0.0109263	+	0.999940i	$0.496522\pi$
$138$	0	0
$139$	−4.26352	−0.361627	−0.180813	−	0.983517i	$-0.557873\pi$
$139$	−4.26352	−0.361627	−0.180813	+	0.983517i	$0.557873\pi$
$140$	0	0
$141$	20.9932	1.76795
$142$	0	0
$143$	5.73143	0.479286
$144$	0	0
$145$	−6.04963	−0.502394
$146$	0	0
$147$	−19.8084	−1.63377
$148$	0	0
$149$	16.5594	1.35660	0.678301	−	0.734784i	$-0.262717\pi$
$149$	16.5594	1.35660	0.678301	+	0.734784i	$0.262717\pi$
$150$	0	0
$151$	4.36184	0.354962	0.177481	−	0.984124i	$-0.443205\pi$
$151$	4.36184	0.354962	0.177481	+	0.984124i	$0.443205\pi$
$152$	0	0
$153$	2.47565	0.200145
$154$	0	0
$155$	6.24897	0.501929
$156$	0	0
$157$	−9.61856	−0.767644	−0.383822	−	0.923407i	$-0.625392\pi$
$157$	−9.61856	−0.767644	−0.383822	+	0.923407i	$0.625392\pi$
$158$	0	0
$159$	−8.17024	−0.647943
$160$	0	0
$161$	−0.935822	−0.0737531
$162$	0	0
$163$	−8.35504	−0.654417	−0.327209	−	0.944952i	$-0.606108\pi$
$163$	−8.35504	−0.654417	−0.327209	+	0.944952i	$0.606108\pi$
$164$	0	0
$165$	5.63816	0.438930
$166$	0	0
$167$	−4.03508	−0.312244	−0.156122	−	0.987738i	$-0.549899\pi$
$167$	−4.03508	−0.312244	−0.156122	+	0.987738i	$0.549899\pi$
$168$	0	0
$169$	−6.37464	−0.490357
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−20.1438	−1.53151	−0.765754	−	0.643134i	$-0.777634\pi$
$173$	−20.1438	−1.53151	−0.765754	+	0.643134i	$0.777634\pi$
$174$	0	0
$175$	1.46791	0.110964
$176$	0	0
$177$	18.1557	1.36467
$178$	0	0
$179$	11.5125	0.860484	0.430242	−	0.902714i	$-0.358428\pi$
$179$	11.5125	0.860484	0.430242	+	0.902714i	$0.358428\pi$
$180$	0	0
$181$	8.53714	0.634561	0.317280	−	0.948332i	$-0.397230\pi$
$181$	8.53714	0.634561	0.317280	+	0.948332i	$0.397230\pi$
$182$	0	0
$183$	26.2841	1.94297
$184$	0	0
$185$	4.34730	0.319620
$186$	0	0
$187$	1.04189	0.0761905
$188$	0	0
$189$	−2.29086	−0.166635
$190$	0	0
$191$	−18.3354	−1.32671	−0.663353	−	0.748307i	$-0.730867\pi$
$191$	−18.3354	−1.32671	−0.663353	+	0.748307i	$0.730867\pi$
$192$	0	0
$193$	0.297667	0.0214265	0.0107133	−	0.999943i	$-0.496590\pi$
$193$	0.297667	0.0214265	0.0107133	+	0.999943i	$0.496590\pi$
$194$	0	0
$195$	6.51754	0.466731
$196$	0	0
$197$	13.1411	0.936268	0.468134	−	0.883657i	$-0.344926\pi$
$197$	13.1411	0.936268	0.468134	+	0.883657i	$0.344926\pi$
$198$	0	0
$199$	0.256711	0.0181978	0.00909888	−	0.999959i	$-0.497104\pi$
$199$	0.256711	0.0181978	0.00909888	+	0.999959i	$0.497104\pi$
$200$	0	0
$201$	−22.0993	−1.55876
$202$	0	0
$203$	2.38919	0.167688
$204$	0	0
$205$	2.17705	0.152052
$206$	0	0
$207$	14.2567	0.990910
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−2.44831	−0.168549	−0.0842743	−	0.996443i	$-0.526857\pi$
$211$	−2.44831	−0.168549	−0.0842743	+	0.996443i	$0.526857\pi$
$212$	0	0
$213$	26.7939	1.83588
$214$	0	0
$215$	−3.43107	−0.233997
$216$	0	0
$217$	−2.46791	−0.167533
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	1.20439	0.0810162
$222$	0	0
$223$	8.50980	0.569858	0.284929	−	0.958549i	$-0.408030\pi$
$223$	8.50980	0.569858	0.284929	+	0.958549i	$0.408030\pi$
$224$	0	0
$225$	−22.3628	−1.49085
$226$	0	0
$227$	14.1506	0.939211	0.469606	−	0.882876i	$-0.344396\pi$
$227$	14.1506	0.939211	0.469606	+	0.882876i	$0.344396\pi$
$228$	0	0
$229$	−20.5330	−1.35686	−0.678430	−	0.734665i	$-0.737339\pi$
$229$	−20.5330	−1.35686	−0.678430	+	0.734665i	$0.737339\pi$
$230$	0	0
$231$	−2.22668	−0.146505
$232$	0	0
$233$	17.6509	1.15635	0.578176	−	0.815912i	$-0.303765\pi$
$233$	17.6509	1.15635	0.578176	+	0.815912i	$0.303765\pi$
$234$	0	0
$235$	6.41147	0.418238
$236$	0	0
$237$	34.1070	2.21549
$238$	0	0
$239$	−2.35235	−0.152161	−0.0760804	−	0.997102i	$-0.524241\pi$
$239$	−2.35235	−0.152161	−0.0760804	+	0.997102i	$0.524241\pi$
$240$	0	0
$241$	13.8007	0.888979	0.444489	−	0.895784i	$-0.353385\pi$
$241$	13.8007	0.888979	0.444489	+	0.895784i	$0.353385\pi$
$242$	0	0
$243$	−10.8033	−0.693035
$244$	0	0
$245$	−6.04963	−0.386497
$246$	0	0
$247$	0	0
$248$	0	0
$249$	42.7033	2.70621
$250$	0	0
$251$	4.16519	0.262905	0.131452	−	0.991322i	$-0.458036\pi$
$251$	4.16519	0.262905	0.131452	+	0.991322i	$0.458036\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	0	0
$255$	1.18479	0.0741946
$256$	0	0
$257$	0.667252	0.0416220	0.0208110	−	0.999783i	$-0.493375\pi$
$257$	0.667252	0.0416220	0.0208110	+	0.999783i	$0.493375\pi$
$258$	0	0
$259$	−1.71688	−0.106682
$260$	0	0
$261$	−36.3979	−2.25297
$262$	0	0
$263$	−11.3996	−0.702930	−0.351465	−	0.936201i	$-0.614316\pi$
$263$	−11.3996	−0.702930	−0.351465	+	0.936201i	$0.614316\pi$
$264$	0	0
$265$	−2.49525	−0.153282
$266$	0	0
$267$	−29.6313	−1.81341
$268$	0	0
$269$	19.3901	1.18224	0.591118	−	0.806585i	$-0.298687\pi$
$269$	19.3901	1.18224	0.591118	+	0.806585i	$0.298687\pi$
$270$	0	0
$271$	13.3942	0.813642	0.406821	−	0.913508i	$-0.366637\pi$
$271$	13.3942	0.813642	0.406821	+	0.913508i	$0.366637\pi$
$272$	0	0
$273$	−2.57398	−0.155784
$274$	0	0
$275$	−9.41147	−0.567533
$276$	0	0
$277$	17.7469	1.06631	0.533154	−	0.846018i	$-0.321007\pi$
$277$	17.7469	1.06631	0.533154	+	0.846018i	$0.321007\pi$
$278$	0	0
$279$	37.5972	2.25089
$280$	0	0
$281$	18.2790	1.09043	0.545217	−	0.838295i	$-0.316447\pi$
$281$	18.2790	1.09043	0.545217	+	0.838295i	$0.316447\pi$
$282$	0	0
$283$	−7.68779	−0.456991	−0.228496	−	0.973545i	$-0.573381\pi$
$283$	−7.68779	−0.456991	−0.228496	+	0.973545i	$0.573381\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.859785	−0.0507515
$288$	0	0
$289$	−16.7811	−0.987121
$290$	0	0
$291$	27.2199	1.59566
$292$	0	0
$293$	−10.5030	−0.613591	−0.306796	−	0.951775i	$-0.599257\pi$
$293$	−10.5030	−0.613591	−0.306796	+	0.951775i	$0.599257\pi$
$294$	0	0
$295$	5.54488	0.322836
$296$	0	0
$297$	14.6878	0.852272
$298$	0	0
$299$	6.93582	0.401109
$300$	0	0
$301$	1.35504	0.0781030
$302$	0	0
$303$	−26.6313	−1.52993
$304$	0	0
$305$	8.02734	0.459644
$306$	0	0
$307$	11.6955	0.667499	0.333749	−	0.942662i	$-0.391686\pi$
$307$	11.6955	0.667499	0.333749	+	0.942662i	$0.391686\pi$
$308$	0	0
$309$	15.8648	0.902519
$310$	0	0
$311$	−15.9659	−0.905340	−0.452670	−	0.891678i	$-0.649529\pi$
$311$	−15.9659	−0.905340	−0.452670	+	0.891678i	$0.649529\pi$
$312$	0	0
$313$	−26.6287	−1.50514	−0.752570	−	0.658512i	$-0.771186\pi$
$313$	−26.6287	−1.50514	−0.752570	+	0.658512i	$0.771186\pi$
$314$	0	0
$315$	−1.61587	−0.0910438
$316$	0	0
$317$	29.5321	1.65869	0.829344	−	0.558739i	$-0.188715\pi$
$317$	29.5321	1.65869	0.829344	+	0.558739i	$0.188715\pi$
$318$	0	0
$319$	−15.3182	−0.857655
$320$	0	0
$321$	−29.4688	−1.64479
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−10.8794	−0.603480
$326$	0	0
$327$	−5.24897	−0.290269
$328$	0	0
$329$	−2.53209	−0.139599
$330$	0	0
$331$	−27.6655	−1.52063	−0.760317	−	0.649553i	$-0.774956\pi$
$331$	−27.6655	−1.52063	−0.760317	+	0.649553i	$0.774956\pi$
$332$	0	0
$333$	26.1557	1.43332
$334$	0	0
$335$	−6.74928	−0.368752
$336$	0	0
$337$	17.8598	0.972884	0.486442	−	0.873713i	$-0.338294\pi$
$337$	17.8598	0.972884	0.486442	+	0.873713i	$0.338294\pi$
$338$	0	0
$339$	−50.9299	−2.76614
$340$	0	0
$341$	15.8229	0.856861
$342$	0	0
$343$	4.82026	0.260270
$344$	0	0
$345$	6.82295	0.367335
$346$	0	0
$347$	5.80066	0.311396	0.155698	−	0.987805i	$-0.450237\pi$
$347$	5.80066	0.311396	0.155698	+	0.987805i	$0.450237\pi$
$348$	0	0
$349$	5.37227	0.287571	0.143786	−	0.989609i	$-0.454072\pi$
$349$	5.37227	0.287571	0.143786	+	0.989609i	$0.454072\pi$
$350$	0	0
$351$	16.9786	0.906253
$352$	0	0
$353$	−25.2344	−1.34309	−0.671546	−	0.740963i	$-0.734370\pi$
$353$	−25.2344	−1.34309	−0.671546	+	0.740963i	$0.734370\pi$
$354$	0	0
$355$	8.18304	0.434311
$356$	0	0
$357$	−0.467911	−0.0247645
$358$	0	0
$359$	−6.68685	−0.352919	−0.176459	−	0.984308i	$-0.556464\pi$
$359$	−6.68685	−0.352919	−0.176459	+	0.984308i	$0.556464\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−17.3969	−0.913102
$364$	0	0
$365$	1.22163	0.0639430
$366$	0	0
$367$	−8.11886	−0.423801	−0.211901	−	0.977291i	$-0.567965\pi$
$367$	−8.11886	−0.423801	−0.211901	+	0.977291i	$0.567965\pi$
$368$	0	0
$369$	13.0983	0.681872
$370$	0	0
$371$	0.985452	0.0511621
$372$	0	0
$373$	34.8976	1.80693	0.903463	−	0.428665i	$-0.141016\pi$
$373$	34.8976	1.80693	0.903463	+	0.428665i	$0.141016\pi$
$374$	0	0
$375$	−23.3628	−1.20645
$376$	0	0
$377$	−17.7074	−0.911977
$378$	0	0
$379$	−1.70140	−0.0873950	−0.0436975	−	0.999045i	$-0.513914\pi$
$379$	−1.70140	−0.0873950	−0.0436975	+	0.999045i	$0.513914\pi$
$380$	0	0
$381$	−33.4124	−1.71177
$382$	0	0
$383$	−2.93676	−0.150061	−0.0750306	−	0.997181i	$-0.523905\pi$
$383$	−2.93676	−0.150061	−0.0750306	+	0.997181i	$0.523905\pi$
$384$	0	0
$385$	−0.680045	−0.0346583
$386$	0	0
$387$	−20.6432	−1.04935
$388$	0	0
$389$	−24.5672	−1.24561	−0.622803	−	0.782379i	$-0.714006\pi$
$389$	−24.5672	−1.24561	−0.622803	+	0.782379i	$0.714006\pi$
$390$	0	0
$391$	1.26083	0.0637629
$392$	0	0
$393$	−5.31315	−0.268013
$394$	0	0
$395$	10.4165	0.524112
$396$	0	0
$397$	−31.8357	−1.59779	−0.798895	−	0.601470i	$-0.794582\pi$
$397$	−31.8357	−1.59779	−0.798895	+	0.601470i	$0.794582\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−0.0864665	−0.00431793	−0.00215896	−	0.999998i	$-0.500687\pi$
$401$	−0.0864665	−0.00431793	−0.00215896	+	0.999998i	$0.500687\pi$
$402$	0	0
$403$	18.2909	0.911133
$404$	0	0
$405$	2.74422	0.136362
$406$	0	0
$407$	11.0077	0.545633
$408$	0	0
$409$	20.0060	0.989232	0.494616	−	0.869112i	$-0.335309\pi$
$409$	20.0060	0.989232	0.494616	+	0.869112i	$0.335309\pi$
$410$	0	0
$411$	0.736482	0.0363280
$412$	0	0
$413$	−2.18984	−0.107755
$414$	0	0
$415$	13.0419	0.640201
$416$	0	0
$417$	−12.2763	−0.601174
$418$	0	0
$419$	−25.4097	−1.24135	−0.620673	−	0.784070i	$-0.713141\pi$
$419$	−25.4097	−1.24135	−0.620673	+	0.784070i	$0.713141\pi$
$420$	0	0
$421$	4.36959	0.212961	0.106480	−	0.994315i	$-0.466042\pi$
$421$	4.36959	0.212961	0.106480	+	0.994315i	$0.466042\pi$
$422$	0	0
$423$	38.5749	1.87558
$424$	0	0
$425$	−1.97771	−0.0959331
$426$	0	0
$427$	−3.17024	−0.153419
$428$	0	0
$429$	16.5030	0.796772
$430$	0	0
$431$	38.3063	1.84515	0.922576	−	0.385816i	$-0.126080\pi$
$431$	38.3063	1.84515	0.922576	+	0.385816i	$0.126080\pi$
$432$	0	0
$433$	−18.1310	−0.871322	−0.435661	−	0.900111i	$-0.643485\pi$
$433$	−18.1310	−0.871322	−0.435661	+	0.900111i	$0.643485\pi$
$434$	0	0
$435$	−17.4192	−0.835187
$436$	0	0
$437$	0	0
$438$	0	0
$439$	6.09059	0.290688	0.145344	−	0.989381i	$-0.453571\pi$
$439$	6.09059	0.290688	0.145344	+	0.989381i	$0.453571\pi$
$440$	0	0
$441$	−36.3979	−1.73323
$442$	0	0
$443$	29.8931	1.42026	0.710132	−	0.704068i	$-0.248635\pi$
$443$	29.8931	1.42026	0.710132	+	0.704068i	$0.248635\pi$
$444$	0	0
$445$	−9.04963	−0.428994
$446$	0	0
$447$	47.6810	2.25523
$448$	0	0
$449$	−11.2499	−0.530916	−0.265458	−	0.964122i	$-0.585523\pi$
$449$	−11.2499	−0.530916	−0.265458	+	0.964122i	$0.585523\pi$
$450$	0	0
$451$	5.51249	0.259573
$452$	0	0
$453$	12.5594	0.590093
$454$	0	0
$455$	−0.786112	−0.0368535
$456$	0	0
$457$	−23.3901	−1.09414	−0.547072	−	0.837086i	$-0.684258\pi$
$457$	−23.3901	−1.09414	−0.547072	+	0.837086i	$0.684258\pi$
$458$	0	0
$459$	3.08647	0.144064
$460$	0	0
$461$	−36.6236	−1.70573	−0.852866	−	0.522130i	$-0.825137\pi$
$461$	−36.6236	−1.70573	−0.852866	+	0.522130i	$0.825137\pi$
$462$	0	0
$463$	42.9864	1.99775	0.998873	−	0.0474549i	$-0.0151110\pi$
$463$	42.9864	1.99775	0.998873	+	0.0474549i	$0.0151110\pi$
$464$	0	0
$465$	17.9932	0.834414
$466$	0	0
$467$	−25.5963	−1.18445	−0.592227	−	0.805771i	$-0.701751\pi$
$467$	−25.5963	−1.18445	−0.592227	+	0.805771i	$0.701751\pi$
$468$	0	0
$469$	2.66550	0.123081
$470$	0	0
$471$	−27.6955	−1.27614
$472$	0	0
$473$	−8.68779	−0.399465
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−15.0128	−0.687389
$478$	0	0
$479$	−38.1762	−1.74432	−0.872158	−	0.489224i	$-0.837280\pi$
$479$	−38.1762	−1.74432	−0.872158	+	0.489224i	$0.837280\pi$
$480$	0	0
$481$	12.7246	0.580193
$482$	0	0
$483$	−2.69459	−0.122608
$484$	0	0
$485$	8.31315	0.377481
$486$	0	0
$487$	−7.76382	−0.351812	−0.175906	−	0.984407i	$-0.556286\pi$
$487$	−7.76382	−0.351812	−0.175906	+	0.984407i	$0.556286\pi$
$488$	0	0
$489$	−24.0574	−1.08791
$490$	0	0
$491$	36.7229	1.65728	0.828640	−	0.559782i	$-0.189115\pi$
$491$	36.7229	1.65728	0.828640	+	0.559782i	$0.189115\pi$
$492$	0	0
$493$	−3.21894	−0.144974
$494$	0	0
$495$	10.3601	0.465651
$496$	0	0
$497$	−3.23173	−0.144963
$498$	0	0
$499$	−4.92633	−0.220533	−0.110266	−	0.993902i	$-0.535170\pi$
$499$	−4.92633	−0.220533	−0.110266	+	0.993902i	$0.535170\pi$
$500$	0	0
$501$	−11.6186	−0.519079
$502$	0	0
$503$	−32.9495	−1.46915	−0.734574	−	0.678529i	$-0.762618\pi$
$503$	−32.9495	−1.46915	−0.734574	+	0.678529i	$0.762618\pi$
$504$	0	0
$505$	−8.13341	−0.361932
$506$	0	0
$507$	−18.3550	−0.815176
$508$	0	0
$509$	−36.9350	−1.63712	−0.818558	−	0.574424i	$-0.805226\pi$
$509$	−36.9350	−1.63712	−0.818558	+	0.574424i	$0.805226\pi$
$510$	0	0
$511$	−0.482459	−0.0213427
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.84524	0.213507
$516$	0	0
$517$	16.2344	0.713989
$518$	0	0
$519$	−58.0019	−2.54600
$520$	0	0
$521$	9.29179	0.407081	0.203540	−	0.979067i	$-0.434755\pi$
$521$	9.29179	0.407081	0.203540	+	0.979067i	$0.434755\pi$
$522$	0	0
$523$	28.4151	1.24251	0.621253	−	0.783610i	$-0.286624\pi$
$523$	28.4151	1.24251	0.621253	+	0.783610i	$0.286624\pi$
$524$	0	0
$525$	4.22668	0.184468
$526$	0	0
$527$	3.32501	0.144840
$528$	0	0
$529$	−15.7392	−0.684312
$530$	0	0
$531$	33.3610	1.44775
$532$	0	0
$533$	6.37227	0.276014
$534$	0	0
$535$	−9.00000	−0.389104
$536$	0	0
$537$	33.1489	1.43048
$538$	0	0
$539$	−15.3182	−0.659802
$540$	0	0
$541$	14.9855	0.644275	0.322137	−	0.946693i	$-0.395599\pi$
$541$	14.9855	0.644275	0.322137	+	0.946693i	$0.395599\pi$
$542$	0	0
$543$	24.5817	1.05490
$544$	0	0
$545$	−1.60307	−0.0686681
$546$	0	0
$547$	3.88713	0.166202	0.0831008	−	0.996541i	$-0.473518\pi$
$547$	3.88713	0.166202	0.0831008	+	0.996541i	$0.473518\pi$
$548$	0	0
$549$	48.2968	2.06126
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−4.11381	−0.174937
$554$	0	0
$555$	12.5175	0.531340
$556$	0	0
$557$	13.2044	0.559488	0.279744	−	0.960075i	$-0.409750\pi$
$557$	13.2044	0.559488	0.279744	+	0.960075i	$0.409750\pi$
$558$	0	0
$559$	−10.0428	−0.424766
$560$	0	0
$561$	3.00000	0.126660
$562$	0	0
$563$	−10.7128	−0.451489	−0.225745	−	0.974187i	$-0.572481\pi$
$563$	−10.7128	−0.451489	−0.225745	+	0.974187i	$0.572481\pi$
$564$	0	0
$565$	−15.5544	−0.654378
$566$	0	0
$567$	−1.08378	−0.0455144
$568$	0	0
$569$	−13.4706	−0.564717	−0.282358	−	0.959309i	$-0.591117\pi$
$569$	−13.4706	−0.564717	−0.282358	+	0.959309i	$0.591117\pi$
$570$	0	0
$571$	−12.6655	−0.530035	−0.265017	−	0.964244i	$-0.585378\pi$
$571$	−12.6655	−0.530035	−0.265017	+	0.964244i	$0.585378\pi$
$572$	0	0
$573$	−52.7948	−2.20553
$574$	0	0
$575$	−11.3892	−0.474962
$576$	0	0
$577$	−10.5544	−0.439384	−0.219692	−	0.975569i	$-0.570505\pi$
$577$	−10.5544	−0.439384	−0.219692	+	0.975569i	$0.570505\pi$
$578$	0	0
$579$	0.857097	0.0356197
$580$	0	0
$581$	−5.15064	−0.213685
$582$	0	0
$583$	−6.31820	−0.261673
$584$	0	0
$585$	11.9760	0.495145
$586$	0	0
$587$	19.1548	0.790602	0.395301	−	0.918552i	$-0.370640\pi$
$587$	19.1548	0.790602	0.395301	+	0.918552i	$0.370640\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	37.8384	1.55647
$592$	0	0
$593$	−8.69459	−0.357044	−0.178522	−	0.983936i	$-0.557132\pi$
$593$	−8.69459	−0.357044	−0.178522	+	0.983936i	$0.557132\pi$
$594$	0	0
$595$	−0.142903	−0.00585847
$596$	0	0
$597$	0.739170	0.0302522
$598$	0	0
$599$	−19.8316	−0.810298	−0.405149	−	0.914251i	$-0.632780\pi$
$599$	−19.8316	−0.810298	−0.405149	+	0.914251i	$0.632780\pi$
$600$	0	0
$601$	−33.7615	−1.37716	−0.688579	−	0.725161i	$-0.741765\pi$
$601$	−33.7615	−1.37716	−0.688579	+	0.725161i	$0.741765\pi$
$602$	0	0
$603$	−40.6073	−1.65366
$604$	0	0
$605$	−5.31315	−0.216010
$606$	0	0
$607$	−35.2850	−1.43217	−0.716087	−	0.698011i	$-0.754068\pi$
$607$	−35.2850	−1.43217	−0.716087	+	0.698011i	$0.754068\pi$
$608$	0	0
$609$	6.87939	0.278767
$610$	0	0
$611$	18.7665	0.759212
$612$	0	0
$613$	18.4534	0.745324	0.372662	−	0.927967i	$-0.378445\pi$
$613$	18.4534	0.745324	0.372662	+	0.927967i	$0.378445\pi$
$614$	0	0
$615$	6.26857	0.252773
$616$	0	0
$617$	−35.6854	−1.43664	−0.718320	−	0.695712i	$-0.755089\pi$
$617$	−35.6854	−1.43664	−0.718320	+	0.695712i	$0.755089\pi$
$618$	0	0
$619$	−3.65951	−0.147088	−0.0735441	−	0.997292i	$-0.523431\pi$
$619$	−3.65951	−0.147088	−0.0735441	+	0.997292i	$0.523431\pi$
$620$	0	0
$621$	17.7743	0.713256
$622$	0	0
$623$	3.57398	0.143188
$624$	0	0
$625$	13.9982	0.559930
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.31315	0.0922313
$630$	0	0
$631$	0.793852	0.0316028	0.0158014	−	0.999875i	$-0.494970\pi$
$631$	0.793852	0.0316028	0.0158014	+	0.999875i	$0.494970\pi$
$632$	0	0
$633$	−7.04963	−0.280198
$634$	0	0
$635$	−10.2044	−0.404949
$636$	0	0
$637$	−17.7074	−0.701592
$638$	0	0
$639$	49.2336	1.94765
$640$	0	0
$641$	29.3824	1.16053	0.580267	−	0.814426i	$-0.302948\pi$
$641$	29.3824	1.16053	0.580267	+	0.814426i	$0.302948\pi$
$642$	0	0
$643$	−22.2139	−0.876030	−0.438015	−	0.898968i	$-0.644318\pi$
$643$	−22.2139	−0.876030	−0.438015	+	0.898968i	$0.644318\pi$
$644$	0	0
$645$	−9.87939	−0.389000
$646$	0	0
$647$	−11.2591	−0.442640	−0.221320	−	0.975201i	$-0.571037\pi$
$647$	−11.2591	−0.442640	−0.221320	+	0.975201i	$0.571037\pi$
$648$	0	0
$649$	14.0401	0.551123
$650$	0	0
$651$	−7.10607	−0.278509
$652$	0	0
$653$	27.0000	1.05659	0.528296	−	0.849060i	$-0.322831\pi$
$653$	27.0000	1.05659	0.528296	+	0.849060i	$0.322831\pi$
$654$	0	0
$655$	−1.62267	−0.0634031
$656$	0	0
$657$	7.34998	0.286750
$658$	0	0
$659$	−28.0259	−1.09173	−0.545867	−	0.837872i	$-0.683800\pi$
$659$	−28.0259	−1.09173	−0.545867	+	0.837872i	$0.683800\pi$
$660$	0	0
$661$	11.3678	0.442157	0.221079	−	0.975256i	$-0.429042\pi$
$661$	11.3678	0.442157	0.221079	+	0.975256i	$0.429042\pi$
$662$	0	0
$663$	3.46791	0.134683
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−18.5371	−0.717761
$668$	0	0
$669$	24.5030	0.947340
$670$	0	0
$671$	20.3259	0.784674
$672$	0	0
$673$	−16.5672	−0.638618	−0.319309	−	0.947651i	$-0.603451\pi$
$673$	−16.5672	−0.638618	−0.319309	+	0.947651i	$0.603451\pi$
$674$	0	0
$675$	−27.8803	−1.07311
$676$	0	0
$677$	−9.04963	−0.347806	−0.173903	−	0.984763i	$-0.555638\pi$
$677$	−9.04963	−0.347806	−0.173903	+	0.984763i	$0.555638\pi$
$678$	0	0
$679$	−3.28312	−0.125995
$680$	0	0
$681$	40.7452	1.56136
$682$	0	0
$683$	−8.73143	−0.334099	−0.167049	−	0.985949i	$-0.553424\pi$
$683$	−8.73143	−0.334099	−0.167049	+	0.985949i	$0.553424\pi$
$684$	0	0
$685$	0.224927	0.00859402
$686$	0	0
$687$	−59.1225	−2.25566
$688$	0	0
$689$	−7.30365	−0.278247
$690$	0	0
$691$	−34.7202	−1.32082	−0.660409	−	0.750906i	$-0.729617\pi$
$691$	−34.7202	−1.32082	−0.660409	+	0.750906i	$0.729617\pi$
$692$	0	0
$693$	−4.09152	−0.155424
$694$	0	0
$695$	−3.74928	−0.142218
$696$	0	0
$697$	1.15839	0.0438770
$698$	0	0
$699$	50.8239	1.92234
$700$	0	0
$701$	39.4151	1.48869	0.744344	−	0.667797i	$-0.232762\pi$
$701$	39.4151	1.48869	0.744344	+	0.667797i	$0.232762\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	18.4611	0.695285
$706$	0	0
$707$	3.21213	0.120805
$708$	0	0
$709$	−41.1215	−1.54435	−0.772176	−	0.635409i	$-0.780832\pi$
$709$	−41.1215	−1.54435	−0.772176	+	0.635409i	$0.780832\pi$
$710$	0	0
$711$	62.6715	2.35036
$712$	0	0
$713$	19.1480	0.717097
$714$	0	0
$715$	5.04013	0.188490
$716$	0	0
$717$	−6.77332	−0.252954
$718$	0	0
$719$	42.3955	1.58109	0.790543	−	0.612407i	$-0.209799\pi$
$719$	42.3955	1.58109	0.790543	+	0.612407i	$0.209799\pi$
$720$	0	0
$721$	−1.91353	−0.0712637
$722$	0	0
$723$	39.7374	1.47785
$724$	0	0
$725$	29.0770	1.07989
$726$	0	0
$727$	−51.6563	−1.91583	−0.957914	−	0.287057i	$-0.907323\pi$
$727$	−51.6563	−1.91583	−0.957914	+	0.287057i	$0.907323\pi$
$728$	0	0
$729$	−40.4688	−1.49885
$730$	0	0
$731$	−1.82564	−0.0675236
$732$	0	0
$733$	−22.9162	−0.846430	−0.423215	−	0.906029i	$-0.639098\pi$
$733$	−22.9162	−0.846430	−0.423215	+	0.906029i	$0.639098\pi$
$734$	0	0
$735$	−17.4192	−0.642517
$736$	0	0
$737$	−17.0898	−0.629510
$738$	0	0
$739$	−28.1266	−1.03465	−0.517327	−	0.855788i	$-0.673073\pi$
$739$	−28.1266	−1.03465	−0.517327	+	0.855788i	$0.673073\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−6.13247	−0.224979	−0.112489	−	0.993653i	$-0.535882\pi$
$743$	−6.13247	−0.224979	−0.112489	+	0.993653i	$0.535882\pi$
$744$	0	0
$745$	14.5621	0.533515
$746$	0	0
$747$	78.4671	2.87096
$748$	0	0
$749$	3.55438	0.129874
$750$	0	0
$751$	−5.64084	−0.205837	−0.102919	−	0.994690i	$-0.532818\pi$
$751$	−5.64084	−0.205837	−0.102919	+	0.994690i	$0.532818\pi$
$752$	0	0
$753$	11.9932	0.437056
$754$	0	0
$755$	3.83574	0.139597
$756$	0	0
$757$	15.6919	0.570332	0.285166	−	0.958478i	$-0.407951\pi$
$757$	15.6919	0.570332	0.285166	+	0.958478i	$0.407951\pi$
$758$	0	0
$759$	17.2763	0.627090
$760$	0	0
$761$	4.86484	0.176350	0.0881751	−	0.996105i	$-0.471896\pi$
$761$	4.86484	0.176350	0.0881751	+	0.996105i	$0.471896\pi$
$762$	0	0
$763$	0.633103	0.0229199
$764$	0	0
$765$	2.17705	0.0787115
$766$	0	0
$767$	16.2300	0.586031
$768$	0	0
$769$	22.5321	0.812528	0.406264	−	0.913756i	$-0.366831\pi$
$769$	22.5321	0.812528	0.406264	+	0.913756i	$0.366831\pi$
$770$	0	0
$771$	1.92127	0.0691930
$772$	0	0
$773$	−26.4320	−0.950693	−0.475347	−	0.879799i	$-0.657677\pi$
$773$	−26.4320	−0.950693	−0.475347	+	0.879799i	$0.657677\pi$
$774$	0	0
$775$	−30.0351	−1.07889
$776$	0	0
$777$	−4.94356	−0.177349
$778$	0	0
$779$	0	0
$780$	0	0
$781$	20.7202	0.741426
$782$	0	0
$783$	−45.3783	−1.62169
$784$	0	0
$785$	−8.45842	−0.301894
$786$	0	0
$787$	15.5577	0.554571	0.277286	−	0.960788i	$-0.410565\pi$
$787$	15.5577	0.554571	0.277286	+	0.960788i	$0.410565\pi$
$788$	0	0
$789$	−32.8239	−1.16856
$790$	0	0
$791$	6.14290	0.218417
$792$	0	0
$793$	23.4962	0.834374
$794$	0	0
$795$	−7.18479	−0.254818
$796$	0	0
$797$	33.4935	1.18640	0.593200	−	0.805055i	$-0.297864\pi$
$797$	33.4935	1.18640	0.593200	+	0.805055i	$0.297864\pi$
$798$	0	0
$799$	3.41147	0.120689
$800$	0	0
$801$	−54.4475	−1.92381
$802$	0	0
$803$	3.09327	0.109159
$804$	0	0
$805$	−0.822948	−0.0290051
$806$	0	0
$807$	55.8316	1.96537
$808$	0	0
$809$	41.1162	1.44557	0.722784	−	0.691074i	$-0.242862\pi$
$809$	41.1162	1.44557	0.722784	+	0.691074i	$0.242862\pi$
$810$	0	0
$811$	−16.6878	−0.585987	−0.292994	−	0.956114i	$-0.594651\pi$
$811$	−16.6878	−0.585987	−0.292994	+	0.956114i	$0.594651\pi$
$812$	0	0
$813$	38.5672	1.35261
$814$	0	0
$815$	−7.34730	−0.257365
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−4.72967	−0.165268
$820$	0	0
$821$	−31.3901	−1.09552	−0.547761	−	0.836635i	$-0.684520\pi$
$821$	−31.3901	−1.09552	−0.547761	+	0.836635i	$0.684520\pi$
$822$	0	0
$823$	−46.3259	−1.61482	−0.807410	−	0.589990i	$-0.799132\pi$
$823$	−46.3259	−1.61482	−0.807410	+	0.589990i	$0.799132\pi$
$824$	0	0
$825$	−27.0993	−0.943475
$826$	0	0
$827$	−40.7588	−1.41732	−0.708661	−	0.705549i	$-0.750700\pi$
$827$	−40.7588	−1.41732	−0.708661	+	0.705549i	$0.750700\pi$
$828$	0	0
$829$	−35.4834	−1.23239	−0.616195	−	0.787594i	$-0.711327\pi$
$829$	−35.4834	−1.23239	−0.616195	+	0.787594i	$0.711327\pi$
$830$	0	0
$831$	51.1002	1.77265
$832$	0	0
$833$	−3.21894	−0.111530
$834$	0	0
$835$	−3.54839	−0.122797
$836$	0	0
$837$	46.8735	1.62019
$838$	0	0
$839$	38.2026	1.31890	0.659451	−	0.751748i	$-0.270789\pi$
$839$	38.2026	1.31890	0.659451	+	0.751748i	$0.270789\pi$
$840$	0	0
$841$	18.3259	0.631929
$842$	0	0
$843$	52.6323	1.81275
$844$	0	0
$845$	−5.60576	−0.192844
$846$	0	0
$847$	2.09833	0.0720993
$848$	0	0
$849$	−22.1361	−0.759709
$850$	0	0
$851$	13.3209	0.456634
$852$	0	0
$853$	−25.6016	−0.876584	−0.438292	−	0.898833i	$-0.644416\pi$
$853$	−25.6016	−0.876584	−0.438292	+	0.898833i	$0.644416\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−21.0865	−0.720300	−0.360150	−	0.932894i	$-0.617274\pi$
$857$	−21.0865	−0.720300	−0.360150	+	0.932894i	$0.617274\pi$
$858$	0	0
$859$	19.5672	0.667623	0.333812	−	0.942640i	$-0.391665\pi$
$859$	19.5672	0.667623	0.333812	+	0.942640i	$0.391665\pi$
$860$	0	0
$861$	−2.47565	−0.0843700
$862$	0	0
$863$	−4.94894	−0.168464	−0.0842319	−	0.996446i	$-0.526844\pi$
$863$	−4.94894	−0.168464	−0.0842319	+	0.996446i	$0.526844\pi$
$864$	0	0
$865$	−17.7142	−0.602301
$866$	0	0
$867$	−48.3191	−1.64100
$868$	0	0
$869$	26.3756	0.894730
$870$	0	0
$871$	−19.7553	−0.669381
$872$	0	0
$873$	50.0164	1.69280
$874$	0	0
$875$	2.81790	0.0952623
$876$	0	0
$877$	1.21987	0.0411922	0.0205961	−	0.999788i	$-0.493444\pi$
$877$	1.21987	0.0411922	0.0205961	+	0.999788i	$0.493444\pi$
$878$	0	0
$879$	−30.2422	−1.02004
$880$	0	0
$881$	46.5030	1.56673	0.783363	−	0.621565i	$-0.213503\pi$
$881$	46.5030	1.56673	0.783363	+	0.621565i	$0.213503\pi$
$882$	0	0
$883$	−12.9249	−0.434957	−0.217479	−	0.976065i	$-0.569783\pi$
$883$	−12.9249	−0.434957	−0.217479	+	0.976065i	$0.569783\pi$
$884$	0	0
$885$	15.9659	0.536686
$886$	0	0
$887$	23.2243	0.779796	0.389898	−	0.920858i	$-0.372510\pi$
$887$	23.2243	0.779796	0.389898	+	0.920858i	$0.372510\pi$
$888$	0	0
$889$	4.03003	0.135163
$890$	0	0
$891$	6.94862	0.232787
$892$	0	0
$893$	0	0
$894$	0	0
$895$	10.1239	0.338405
$896$	0	0
$897$	19.9709	0.666809
$898$	0	0
$899$	−48.8854	−1.63042
$900$	0	0
$901$	−1.32770	−0.0442320
$902$	0	0
$903$	3.90167	0.129840
$904$	0	0
$905$	7.50744	0.249556
$906$	0	0
$907$	39.9968	1.32807	0.664036	−	0.747700i	$-0.268842\pi$
$907$	39.9968	1.32807	0.664036	+	0.747700i	$0.268842\pi$
$908$	0	0
$909$	−48.9350	−1.62307
$910$	0	0
$911$	18.7997	0.622863	0.311431	−	0.950269i	$-0.399192\pi$
$911$	18.7997	0.622863	0.311431	+	0.950269i	$0.399192\pi$
$912$	0	0
$913$	33.0232	1.09291
$914$	0	0
$915$	23.1138	0.764119
$916$	0	0
$917$	0.640844	0.0211625
$918$	0	0
$919$	−39.8316	−1.31392	−0.656962	−	0.753924i	$-0.728159\pi$
$919$	−39.8316	−1.31392	−0.656962	+	0.753924i	$0.728159\pi$
$920$	0	0
$921$	33.6759	1.10966
$922$	0	0
$923$	23.9519	0.788387
$924$	0	0
$925$	−20.8949	−0.687019
$926$	0	0
$927$	29.1516	0.957463
$928$	0	0
$929$	26.9540	0.884332	0.442166	−	0.896933i	$-0.354210\pi$
$929$	26.9540	0.884332	0.442166	+	0.896933i	$0.354210\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−45.9718	−1.50505
$934$	0	0
$935$	0.916222	0.0299637
$936$	0	0
$937$	2.62361	0.0857095	0.0428548	−	0.999081i	$-0.486355\pi$
$937$	2.62361	0.0857095	0.0428548	+	0.999081i	$0.486355\pi$
$938$	0	0
$939$	−76.6742	−2.50217
$940$	0	0
$941$	−18.6696	−0.608612	−0.304306	−	0.952574i	$-0.598425\pi$
$941$	−18.6696	−0.608612	−0.304306	+	0.952574i	$0.598425\pi$
$942$	0	0
$943$	6.67087	0.217234
$944$	0	0
$945$	−2.01455	−0.0655332
$946$	0	0
$947$	−8.39961	−0.272951	−0.136475	−	0.990643i	$-0.543577\pi$
$947$	−8.39961	−0.272951	−0.136475	+	0.990643i	$0.543577\pi$
$948$	0	0
$949$	3.57573	0.116073
$950$	0	0
$951$	85.0343	2.75742
$952$	0	0
$953$	33.6928	1.09142	0.545709	−	0.837975i	$-0.316260\pi$
$953$	33.6928	1.09142	0.545709	+	0.837975i	$0.316260\pi$
$954$	0	0
$955$	−16.1239	−0.521758
$956$	0	0
$957$	−44.1070	−1.42578
$958$	0	0
$959$	−0.0888306	−0.00286849
$960$	0	0
$961$	19.4962	0.628909
$962$	0	0
$963$	−54.1489	−1.74492
$964$	0	0
$965$	0.261764	0.00842647
$966$	0	0
$967$	−11.7433	−0.377639	−0.188819	−	0.982012i	$-0.560466\pi$
$967$	−11.7433	−0.377639	−0.188819	+	0.982012i	$0.560466\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−12.8093	−0.411071	−0.205536	−	0.978650i	$-0.565894\pi$
$971$	−12.8093	−0.411071	−0.205536	+	0.978650i	$0.565894\pi$
$972$	0	0
$973$	1.48070	0.0474692
$974$	0	0
$975$	−31.3259	−1.00323
$976$	0	0
$977$	14.5276	0.464781	0.232390	−	0.972623i	$-0.425345\pi$
$977$	14.5276	0.464781	0.232390	+	0.972623i	$0.425345\pi$
$978$	0	0
$979$	−22.9145	−0.732350
$980$	0	0
$981$	−9.64496	−0.307940
$982$	0	0
$983$	37.0502	1.18172	0.590860	−	0.806774i	$-0.298789\pi$
$983$	37.0502	1.18172	0.590860	+	0.806774i	$0.298789\pi$
$984$	0	0
$985$	11.5561	0.368209
$986$	0	0
$987$	−7.29086	−0.232071
$988$	0	0
$989$	−10.5134	−0.334307
$990$	0	0
$991$	−3.43140	−0.109002	−0.0545010	−	0.998514i	$-0.517357\pi$
$991$	−3.43140	−0.109002	−0.0545010	+	0.998514i	$0.517357\pi$
$992$	0	0
$993$	−79.6596	−2.52792
$994$	0	0
$995$	0.225748	0.00715669
$996$	0	0
$997$	12.7760	0.404620	0.202310	−	0.979322i	$-0.435155\pi$
$997$	12.7760	0.404620	0.202310	+	0.979322i	$0.435155\pi$
$998$	0	0
$999$	32.6091	1.03170

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5776.2.a.br.1.3		3
4.3	odd	2		361.2.a.g.1.2		3
12.11	even	2		3249.2.a.z.1.2		3
19.6	even	9		304.2.u.b.17.1		6
19.16	even	9		304.2.u.b.161.1		6
19.18	odd	2		5776.2.a.bi.1.1		3
20.19	odd	2		9025.2.a.bd.1.2		3
76.3	even	18		361.2.e.h.28.1		6
76.7	odd	6		361.2.c.i.68.2		6
76.11	odd	6		361.2.c.i.292.2		6
76.15	even	18		361.2.e.a.54.1		6
76.23	odd	18		361.2.e.g.54.1		6
76.27	even	6		361.2.c.h.292.2		6
76.31	even	6		361.2.c.h.68.2		6
76.35	odd	18		19.2.e.a.9.1	✓	6
76.43	odd	18		361.2.e.g.234.1		6
76.47	odd	18		361.2.e.f.62.1		6
76.51	even	18		361.2.e.h.245.1		6
76.55	odd	18		361.2.e.f.99.1		6
76.59	even	18		361.2.e.b.99.1		6
76.63	odd	18		19.2.e.a.17.1	yes	6
76.67	even	18		361.2.e.b.62.1		6
76.71	even	18		361.2.e.a.234.1		6
76.75	even	2		361.2.a.h.1.2		3
228.35	even	18		171.2.u.c.28.1		6
228.215	even	18		171.2.u.c.55.1		6
228.227	odd	2		3249.2.a.s.1.2		3
380.63	even	36		475.2.u.a.74.2		12
380.139	odd	18		475.2.l.a.226.1		6
380.187	even	36		475.2.u.a.199.2		12
380.263	even	36		475.2.u.a.199.1		12
380.339	odd	18		475.2.l.a.351.1		6
380.367	even	36		475.2.u.a.74.1		12
380.379	even	2		9025.2.a.x.1.2		3
532.111	even	18		931.2.w.a.883.1		6
532.139	even	18		931.2.w.a.834.1		6
532.187	even	18		931.2.x.b.655.1		6
532.215	even	18		931.2.v.a.606.1		6
532.263	odd	18		931.2.v.b.275.1		6
532.291	odd	18		931.2.x.a.226.1		6
532.339	even	18		931.2.v.a.275.1		6
532.367	even	18		931.2.x.b.226.1		6
532.415	odd	18		931.2.x.a.655.1		6
532.443	odd	18		931.2.v.b.606.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.2.e.a.9.1	✓	6	76.35	odd	18
19.2.e.a.17.1	yes	6	76.63	odd	18
171.2.u.c.28.1		6	228.35	even	18
171.2.u.c.55.1		6	228.215	even	18
304.2.u.b.17.1		6	19.6	even	9
304.2.u.b.161.1		6	19.16	even	9
361.2.a.g.1.2		3	4.3	odd	2
361.2.a.h.1.2		3	76.75	even	2
361.2.c.h.68.2		6	76.31	even	6
361.2.c.h.292.2		6	76.27	even	6
361.2.c.i.68.2		6	76.7	odd	6
361.2.c.i.292.2		6	76.11	odd	6
361.2.e.a.54.1		6	76.15	even	18
361.2.e.a.234.1		6	76.71	even	18
361.2.e.b.62.1		6	76.67	even	18
361.2.e.b.99.1		6	76.59	even	18
361.2.e.f.62.1		6	76.47	odd	18
361.2.e.f.99.1		6	76.55	odd	18
361.2.e.g.54.1		6	76.23	odd	18
361.2.e.g.234.1		6	76.43	odd	18
361.2.e.h.28.1		6	76.3	even	18
361.2.e.h.245.1		6	76.51	even	18
475.2.l.a.226.1		6	380.139	odd	18
475.2.l.a.351.1		6	380.339	odd	18
475.2.u.a.74.1		12	380.367	even	36
475.2.u.a.74.2		12	380.63	even	36
475.2.u.a.199.1		12	380.263	even	36
475.2.u.a.199.2		12	380.187	even	36
931.2.v.a.275.1		6	532.339	even	18
931.2.v.a.606.1		6	532.215	even	18
931.2.v.b.275.1		6	532.263	odd	18
931.2.v.b.606.1		6	532.443	odd	18
931.2.w.a.834.1		6	532.139	even	18
931.2.w.a.883.1		6	532.111	even	18
931.2.x.a.226.1		6	532.291	odd	18
931.2.x.a.655.1		6	532.415	odd	18
931.2.x.b.226.1		6	532.367	even	18
931.2.x.b.655.1		6	532.187	even	18
3249.2.a.s.1.2		3	228.227	odd	2
3249.2.a.z.1.2		3	12.11	even	2
5776.2.a.bi.1.1		3	19.18	odd	2
5776.2.a.br.1.3		3	1.1	even	1	trivial
9025.2.a.x.1.2		3	380.379	even	2
9025.2.a.bd.1.2		3	20.19	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:	\( N \)	\(=\)	\( 5776 = 2^{4} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5776.a (trivial)

\(f(q)\)	\(=\)	\(q+2.87939 q^{3} +0.879385 q^{5} -0.347296 q^{7} +5.29086 q^{9} +O(q^{10})\)	\(q+2.87939 q^{3} +0.879385 q^{5} -0.347296 q^{7} +5.29086 q^{9} +2.22668 q^{11} +2.57398 q^{13} +2.53209 q^{15} +0.467911 q^{17} -1.00000 q^{21} +2.69459 q^{23} -4.22668 q^{25} +6.59627 q^{27} -6.87939 q^{29} +7.10607 q^{31} +6.41147 q^{33} -0.305407 q^{35} +4.94356 q^{37} +7.41147 q^{39} +2.47565 q^{41} -3.90167 q^{43} +4.65270 q^{45} +7.29086 q^{47} -6.87939 q^{49} +1.34730 q^{51} -2.83750 q^{53} +1.95811 q^{55} +6.30541 q^{59} +9.12836 q^{61} -1.83750 q^{63} +2.26352 q^{65} -7.67499 q^{67} +7.75877 q^{69} +9.30541 q^{71} +1.38919 q^{73} -12.1702 q^{75} -0.773318 q^{77} +11.8452 q^{79} +3.12061 q^{81} +14.8307 q^{83} +0.411474 q^{85} -19.8084 q^{87} -10.2909 q^{89} -0.893933 q^{91} +20.4611 q^{93} +9.45336 q^{97} +11.7811 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^5	\( 3 q + 3 q^{3} - 3 q^{5} + 3 q^{15} + 6 q^{17} - 3 q^{21} + 6 q^{23} - 6 q^{25} + 6 q^{27} - 15 q^{29} + 9 q^{31} + 9 q^{33} - 3 q^{35} + 12 q^{39} - 12 q^{41} + 15 q^{45} + 6 q^{47} - 15 q^{49} + 3 q^{51} - 6 q^{53} + 9 q^{55} + 21 q^{59} + 9 q^{61} - 3 q^{63} + 12 q^{65} - 18 q^{67} + 12 q^{69} + 30 q^{71} - 15 q^{75} - 9 q^{77} + 9 q^{79} + 15 q^{81} - 9 q^{85} - 21 q^{87} - 15 q^{89} - 15 q^{91} + 24 q^{93} + 15 q^{97} + 18 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^15 + 6 * q^17 - 3 * q^21 + 6 * q^23 - 6 * q^25 + 6 * q^27 - 15 * q^29 + 9 * q^31 + 9 * q^33 - 3 * q^35 + 12 * q^39 - 12 * q^41 + 15 * q^45 + 6 * q^47 - 15 * q^49 + 3 * q^51 - 6 * q^53 + 9 * q^55 + 21 * q^59 + 9 * q^61 - 3 * q^63 + 12 * q^65 - 18 * q^67 + 12 * q^69 + 30 * q^71 - 15 * q^75 - 9 * q^77 + 9 * q^79 + 15 * q^81 - 9 * q^85 - 21 * q^87 - 15 * q^89 - 15 * q^91 + 24 * q^93 + 15 * q^97 + 18 * q^99

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