LMFDB - Embedded newform 5776.2.a.br.1.1

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			1.1
Root			$-1.53209$ 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)

$f(q)$	$=$	$q-0.532089 q^{3} -2.53209 q^{5} +1.87939 q^{7} -2.71688 q^{9} +O(q^{10})$	$q-0.532089 q^{3} -2.53209 q^{5} +1.87939 q^{7} -2.71688 q^{9} -3.41147 q^{11} -5.29086 q^{13} +1.34730 q^{15} +1.65270 q^{17} -1.00000 q^{21} -1.75877 q^{23} +1.41147 q^{25} +3.04189 q^{27} -3.46791 q^{29} -1.94356 q^{31} +1.81521 q^{33} -4.75877 q^{35} -0.837496 q^{37} +2.81521 q^{39} -4.49020 q^{41} -4.80066 q^{43} +6.87939 q^{45} -0.716881 q^{47} -3.46791 q^{49} -0.879385 q^{51} -6.10607 q^{53} +8.63816 q^{55} +10.7588 q^{59} +4.38919 q^{61} -5.10607 q^{63} +13.3969 q^{65} -14.2121 q^{67} +0.935822 q^{69} +13.7588 q^{71} -7.51754 q^{73} -0.751030 q^{75} -6.41147 q^{77} +6.96316 q^{79} +6.53209 q^{81} -2.51249 q^{83} -4.18479 q^{85} +1.84524 q^{87} -2.28312 q^{89} -9.94356 q^{91} +1.03415 q^{93} -1.82295 q^{97} +9.26857 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.532089	−0.307202	−0.153601	−	0.988133i	$-0.549087\pi$
$3$	−0.532089	−0.307202	−0.153601	+	0.988133i	$0.549087\pi$
$4$	0	0
$5$	−2.53209	−1.13238	−0.566192	−	0.824273i	$-0.691584\pi$
$5$	−2.53209	−1.13238	−0.566192	+	0.824273i	$0.691584\pi$
$6$	0	0
$7$	1.87939	0.710341	0.355170	−	0.934802i	$-0.384423\pi$
$7$	1.87939	0.710341	0.355170	+	0.934802i	$0.384423\pi$
$8$	0	0
$9$	−2.71688	−0.905627
$10$	0	0
$11$	−3.41147	−1.02860	−0.514299	−	0.857611i	$-0.671948\pi$
$11$	−3.41147	−1.02860	−0.514299	+	0.857611i	$0.671948\pi$
$12$	0	0
$13$	−5.29086	−1.46742	−0.733710	−	0.679463i	$-0.762213\pi$
$13$	−5.29086	−1.46742	−0.733710	+	0.679463i	$0.762213\pi$
$14$	0	0
$15$	1.34730	0.347870
$16$	0	0
$17$	1.65270	0.400840	0.200420	−	0.979710i	$-0.435769\pi$
$17$	1.65270	0.400840	0.200420	+	0.979710i	$0.435769\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.75877	−0.366729	−0.183364	−	0.983045i	$-0.558699\pi$
$23$	−1.75877	−0.366729	−0.183364	+	0.983045i	$0.558699\pi$
$24$	0	0
$25$	1.41147	0.282295
$26$	0	0
$27$	3.04189	0.585412
$28$	0	0
$29$	−3.46791	−0.643975	−0.321987	−	0.946744i	$-0.604351\pi$
$29$	−3.46791	−0.643975	−0.321987	+	0.946744i	$0.604351\pi$
$30$	0	0
$31$	−1.94356	−0.349074	−0.174537	−	0.984651i	$-0.555843\pi$
$31$	−1.94356	−0.349074	−0.174537	+	0.984651i	$0.555843\pi$
$32$	0	0
$33$	1.81521	0.315987
$34$	0	0
$35$	−4.75877	−0.804379
$36$	0	0
$37$	−0.837496	−0.137684	−0.0688418	−	0.997628i	$-0.521930\pi$
$37$	−0.837496	−0.137684	−0.0688418	+	0.997628i	$0.521930\pi$
$38$	0	0
$39$	2.81521	0.450794
$40$	0	0
$41$	−4.49020	−0.701251	−0.350626	−	0.936516i	$-0.614031\pi$
$41$	−4.49020	−0.701251	−0.350626	+	0.936516i	$0.614031\pi$
$42$	0	0
$43$	−4.80066	−0.732094	−0.366047	−	0.930596i	$-0.619289\pi$
$43$	−4.80066	−0.732094	−0.366047	+	0.930596i	$0.619289\pi$
$44$	0	0
$45$	6.87939	1.02552
$46$	0	0
$47$	−0.716881	−0.104568	−0.0522840	−	0.998632i	$-0.516650\pi$
$47$	−0.716881	−0.104568	−0.0522840	+	0.998632i	$0.516650\pi$
$48$	0	0
$49$	−3.46791	−0.495416
$50$	0	0
$51$	−0.879385	−0.123139
$52$	0	0
$53$	−6.10607	−0.838733	−0.419366	−	0.907817i	$-0.637748\pi$
$53$	−6.10607	−0.838733	−0.419366	+	0.907817i	$0.637748\pi$
$54$	0	0
$55$	8.63816	1.16477
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.7588	1.40067	0.700336	−	0.713813i	$-0.253033\pi$
$59$	10.7588	1.40067	0.700336	+	0.713813i	$0.253033\pi$
$60$	0	0
$61$	4.38919	0.561978	0.280989	−	0.959711i	$-0.409338\pi$
$61$	4.38919	0.561978	0.280989	+	0.959711i	$0.409338\pi$
$62$	0	0
$63$	−5.10607	−0.643304
$64$	0	0
$65$	13.3969	1.66168
$66$	0	0
$67$	−14.2121	−1.73629	−0.868144	−	0.496312i	$-0.834687\pi$
$67$	−14.2121	−1.73629	−0.868144	+	0.496312i	$0.834687\pi$
$68$	0	0
$69$	0.935822	0.112660
$70$	0	0
$71$	13.7588	1.63287	0.816433	−	0.577440i	$-0.195948\pi$
$71$	13.7588	1.63287	0.816433	+	0.577440i	$0.195948\pi$
$72$	0	0
$73$	−7.51754	−0.879862	−0.439931	−	0.898032i	$-0.644997\pi$
$73$	−7.51754	−0.879862	−0.439931	+	0.898032i	$0.644997\pi$
$74$	0	0
$75$	−0.751030	−0.0867214
$76$	0	0
$77$	−6.41147	−0.730655
$78$	0	0
$79$	6.96316	0.783417	0.391709	−	0.920089i	$-0.371884\pi$
$79$	6.96316	0.783417	0.391709	+	0.920089i	$0.371884\pi$
$80$	0	0
$81$	6.53209	0.725788
$82$	0	0
$83$	−2.51249	−0.275781	−0.137891	−	0.990447i	$-0.544032\pi$
$83$	−2.51249	−0.275781	−0.137891	+	0.990447i	$0.544032\pi$
$84$	0	0
$85$	−4.18479	−0.453904
$86$	0	0
$87$	1.84524	0.197830
$88$	0	0
$89$	−2.28312	−0.242010	−0.121005	−	0.992652i	$-0.538612\pi$
$89$	−2.28312	−0.242010	−0.121005	+	0.992652i	$0.538612\pi$
$90$	0	0
$91$	−9.94356	−1.04237
$92$	0	0
$93$	1.03415	0.107236
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.82295	−0.185092	−0.0925462	−	0.995708i	$-0.529501\pi$
$97$	−1.82295	−0.185092	−0.0925462	+	0.995708i	$0.529501\pi$
$98$	0	0
$99$	9.26857	0.931526
$100$	0	0
$101$	−7.92127	−0.788196	−0.394098	−	0.919068i	$-0.628943\pi$
$101$	−7.92127	−0.788196	−0.394098	+	0.919068i	$0.628943\pi$
$102$	0	0
$103$	0.0145479	0.00143345	0.000716725	−	1.00000i	$-0.499772\pi$
$103$	0.0145479	0.00143345	0.000716725		1.00000i	$0.499772\pi$
$104$	0	0
$105$	2.53209	0.247107
$106$	0	0
$107$	3.55438	0.343615	0.171807	−	0.985131i	$-0.445039\pi$
$107$	3.55438	0.343615	0.171807	+	0.985131i	$0.445039\pi$
$108$	0	0
$109$	7.36959	0.705878	0.352939	−	0.935646i	$-0.385182\pi$
$109$	7.36959	0.705878	0.352939	+	0.935646i	$0.385182\pi$
$110$	0	0
$111$	0.445622	0.0422966
$112$	0	0
$113$	7.37733	0.694000	0.347000	−	0.937865i	$-0.387200\pi$
$113$	7.37733	0.694000	0.347000	+	0.937865i	$0.387200\pi$
$114$	0	0
$115$	4.45336	0.415278
$116$	0	0
$117$	14.3746	1.32894
$118$	0	0
$119$	3.10607	0.284733
$120$	0	0
$121$	0.638156	0.0580142
$122$	0	0
$123$	2.38919	0.215426
$124$	0	0
$125$	9.08647	0.812718
$126$	0	0
$127$	0.101014	0.00896358	0.00448179	−	0.999990i	$-0.498573\pi$
$127$	0.101014	0.00896358	0.00448179	+	0.999990i	$0.498573\pi$
$128$	0	0
$129$	2.55438	0.224900
$130$	0	0
$131$	3.03684	0.265330	0.132665	−	0.991161i	$-0.457647\pi$
$131$	3.03684	0.265330	0.132665	+	0.991161i	$0.457647\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−7.70233	−0.662911
$136$	0	0
$137$	19.5398	1.66940	0.834700	−	0.550705i	$-0.185641\pi$
$137$	19.5398	1.66940	0.834700	+	0.550705i	$0.185641\pi$
$138$	0	0
$139$	−15.3969	−1.30595	−0.652975	−	0.757379i	$-0.726479\pi$
$139$	−15.3969	−1.30595	−0.652975	+	0.757379i	$0.726479\pi$
$140$	0	0
$141$	0.381445	0.0321234
$142$	0	0
$143$	18.0496	1.50939
$144$	0	0
$145$	8.78106	0.729227
$146$	0	0
$147$	1.84524	0.152193
$148$	0	0
$149$	−3.76651	−0.308565	−0.154282	−	0.988027i	$-0.549307\pi$
$149$	−3.76651	−0.308565	−0.154282	+	0.988027i	$0.549307\pi$
$150$	0	0
$151$	14.5963	1.18783	0.593914	−	0.804529i	$-0.297582\pi$
$151$	14.5963	1.18783	0.593914	+	0.804529i	$0.297582\pi$
$152$	0	0
$153$	−4.49020	−0.363011
$154$	0	0
$155$	4.92127	0.395286
$156$	0	0
$157$	−10.3746	−0.827986	−0.413993	−	0.910280i	$-0.635866\pi$
$157$	−10.3746	−0.827986	−0.413993	+	0.910280i	$0.635866\pi$
$158$	0	0
$159$	3.24897	0.257660
$160$	0	0
$161$	−3.30541	−0.260503
$162$	0	0
$163$	2.02229	0.158398	0.0791989	−	0.996859i	$-0.474764\pi$
$163$	2.02229	0.158398	0.0791989	+	0.996859i	$0.474764\pi$
$164$	0	0
$165$	−4.59627	−0.357819
$166$	0	0
$167$	23.2567	1.79966	0.899829	−	0.436242i	$-0.143691\pi$
$167$	23.2567	1.79966	0.899829	+	0.436242i	$0.143691\pi$
$168$	0	0
$169$	14.9932	1.15332
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0.896622	0.0681689	0.0340844	−	0.999419i	$-0.489148\pi$
$173$	0.896622	0.0681689	0.0340844	+	0.999419i	$0.489148\pi$
$174$	0	0
$175$	2.65270	0.200526
$176$	0	0
$177$	−5.72462	−0.430289
$178$	0	0
$179$	21.3182	1.59340	0.796699	−	0.604377i	$-0.206578\pi$
$179$	21.3182	1.59340	0.796699	+	0.604377i	$0.206578\pi$
$180$	0	0
$181$	−16.0993	−1.19665	−0.598324	−	0.801254i	$-0.704167\pi$
$181$	−16.0993	−1.19665	−0.598324	+	0.801254i	$0.704167\pi$
$182$	0	0
$183$	−2.33544	−0.172640
$184$	0	0
$185$	2.12061	0.155911
$186$	0	0
$187$	−5.63816	−0.412303
$188$	0	0
$189$	5.71688	0.415842
$190$	0	0
$191$	−18.9486	−1.37107	−0.685537	−	0.728038i	$-0.740432\pi$
$191$	−18.9486	−1.37107	−0.685537	+	0.728038i	$0.740432\pi$
$192$	0	0
$193$	12.9017	0.928683	0.464341	−	0.885656i	$-0.346291\pi$
$193$	12.9017	0.928683	0.464341	+	0.885656i	$0.346291\pi$
$194$	0	0
$195$	−7.12836	−0.510472
$196$	0	0
$197$	−23.2003	−1.65295	−0.826476	−	0.562973i	$-0.809658\pi$
$197$	−23.2003	−1.65295	−0.826476	+	0.562973i	$0.809658\pi$
$198$	0	0
$199$	−9.22163	−0.653704	−0.326852	−	0.945076i	$-0.605988\pi$
$199$	−9.22163	−0.653704	−0.326852	+	0.945076i	$0.605988\pi$
$200$	0	0
$201$	7.56212	0.533391
$202$	0	0
$203$	−6.51754	−0.457442
$204$	0	0
$205$	11.3696	0.794086
$206$	0	0
$207$	4.77837	0.332120
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−14.6236	−1.00673	−0.503365	−	0.864074i	$-0.667905\pi$
$211$	−14.6236	−1.00673	−0.503365	+	0.864074i	$0.667905\pi$
$212$	0	0
$213$	−7.32089	−0.501619
$214$	0	0
$215$	12.1557	0.829012
$216$	0	0
$217$	−3.65270	−0.247962
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	−8.74422	−0.588200
$222$	0	0
$223$	3.01455	0.201869	0.100935	−	0.994893i	$-0.467817\pi$
$223$	3.01455	0.201869	0.100935	+	0.994893i	$0.467817\pi$
$224$	0	0
$225$	−3.83481	−0.255654
$226$	0	0
$227$	13.7219	0.910757	0.455378	−	0.890298i	$-0.349504\pi$
$227$	13.7219	0.910757	0.455378	+	0.890298i	$0.349504\pi$
$228$	0	0
$229$	9.41416	0.622105	0.311053	−	0.950393i	$-0.399318\pi$
$229$	9.41416	0.622105	0.311053	+	0.950393i	$0.399318\pi$
$230$	0	0
$231$	3.41147	0.224459
$232$	0	0
$233$	−24.1857	−1.58446	−0.792230	−	0.610223i	$-0.791080\pi$
$233$	−24.1857	−1.58446	−0.792230	+	0.610223i	$0.791080\pi$
$234$	0	0
$235$	1.81521	0.118411
$236$	0	0
$237$	−3.70502	−0.240667
$238$	0	0
$239$	23.3259	1.50883	0.754415	−	0.656398i	$-0.227921\pi$
$239$	23.3259	1.50883	0.754415	+	0.656398i	$0.227921\pi$
$240$	0	0
$241$	0.297667	0.0191744	0.00958719	−	0.999954i	$-0.496948\pi$
$241$	0.297667	0.0191744	0.00958719	+	0.999954i	$0.496948\pi$
$242$	0	0
$243$	−12.6013	−0.808375
$244$	0	0
$245$	8.78106	0.561001
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1.33687	0.0847205
$250$	0	0
$251$	16.1976	1.02238	0.511191	−	0.859467i	$-0.329204\pi$
$251$	16.1976	1.02238	0.511191	+	0.859467i	$0.329204\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	0	0
$255$	2.22668	0.139440
$256$	0	0
$257$	15.3550	0.957821	0.478910	−	0.877864i	$-0.341032\pi$
$257$	15.3550	0.957821	0.478910	+	0.877864i	$0.341032\pi$
$258$	0	0
$259$	−1.57398	−0.0978022
$260$	0	0
$261$	9.42190	0.583201
$262$	0	0
$263$	−9.64321	−0.594626	−0.297313	−	0.954780i	$-0.596090\pi$
$263$	−9.64321	−0.594626	−0.297313	+	0.954780i	$0.596090\pi$
$264$	0	0
$265$	15.4611	0.949768
$266$	0	0
$267$	1.21482	0.0743459
$268$	0	0
$269$	−18.2790	−1.11449	−0.557245	−	0.830348i	$-0.688142\pi$
$269$	−18.2790	−1.11449	−0.557245	+	0.830348i	$0.688142\pi$
$270$	0	0
$271$	−18.9641	−1.15199	−0.575993	−	0.817454i	$-0.695385\pi$
$271$	−18.9641	−1.15199	−0.575993	+	0.817454i	$0.695385\pi$
$272$	0	0
$273$	5.29086	0.320217
$274$	0	0
$275$	−4.81521	−0.290368
$276$	0	0
$277$	13.7638	0.826988	0.413494	−	0.910507i	$-0.364308\pi$
$277$	13.7638	0.826988	0.413494	+	0.910507i	$0.364308\pi$
$278$	0	0
$279$	5.28043	0.316131
$280$	0	0
$281$	13.1111	0.782144	0.391072	−	0.920360i	$-0.372104\pi$
$281$	13.1111	0.782144	0.391072	+	0.920360i	$0.372104\pi$
$282$	0	0
$283$	17.3773	1.03297	0.516487	−	0.856295i	$-0.327239\pi$
$283$	17.3773	1.03297	0.516487	+	0.856295i	$0.327239\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.43882	−0.498128
$288$	0	0
$289$	−14.2686	−0.839328
$290$	0	0
$291$	0.969971	0.0568607
$292$	0	0
$293$	15.6040	0.911596	0.455798	−	0.890083i	$-0.349354\pi$
$293$	15.6040	0.911596	0.455798	+	0.890083i	$0.349354\pi$
$294$	0	0
$295$	−27.2422	−1.58610
$296$	0	0
$297$	−10.3773	−0.602154
$298$	0	0
$299$	9.30541	0.538146
$300$	0	0
$301$	−9.02229	−0.520036
$302$	0	0
$303$	4.21482	0.242135
$304$	0	0
$305$	−11.1138	−0.636375
$306$	0	0
$307$	−21.5202	−1.22822	−0.614112	−	0.789219i	$-0.710486\pi$
$307$	−21.5202	−1.22822	−0.614112	+	0.789219i	$0.710486\pi$
$308$	0	0
$309$	−0.00774079	−0.000440358 0
$310$	0	0
$311$	−14.4953	−0.821950	−0.410975	−	0.911647i	$-0.634812\pi$
$311$	−14.4953	−0.821950	−0.410975	+	0.911647i	$0.634812\pi$
$312$	0	0
$313$	19.5185	1.10325	0.551625	−	0.834092i	$-0.314008\pi$
$313$	19.5185	1.10325	0.551625	+	0.834092i	$0.314008\pi$
$314$	0	0
$315$	12.9290	0.728467
$316$	0	0
$317$	28.3473	1.59214	0.796071	−	0.605203i	$-0.206908\pi$
$317$	28.3473	1.59214	0.796071	+	0.605203i	$0.206908\pi$
$318$	0	0
$319$	11.8307	0.662391
$320$	0	0
$321$	−1.89124	−0.105559
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−7.46791	−0.414245
$326$	0	0
$327$	−3.92127	−0.216847
$328$	0	0
$329$	−1.34730	−0.0742789
$330$	0	0
$331$	1.71007	0.0939942	0.0469971	−	0.998895i	$-0.485035\pi$
$331$	1.71007	0.0939942	0.0469971	+	0.998895i	$0.485035\pi$
$332$	0	0
$333$	2.27538	0.124690
$334$	0	0
$335$	35.9864	1.96615
$336$	0	0
$337$	25.4388	1.38574	0.692870	−	0.721062i	$-0.256346\pi$
$337$	25.4388	1.38574	0.692870	+	0.721062i	$0.256346\pi$
$338$	0	0
$339$	−3.92539	−0.213198
$340$	0	0
$341$	6.63041	0.359057
$342$	0	0
$343$	−19.6732	−1.06225
$344$	0	0
$345$	−2.36959	−0.127574
$346$	0	0
$347$	−7.70233	−0.413483	−0.206741	−	0.978396i	$-0.566286\pi$
$347$	−7.70233	−0.413483	−0.206741	+	0.978396i	$0.566286\pi$
$348$	0	0
$349$	22.7570	1.21816	0.609078	−	0.793111i	$-0.291540\pi$
$349$	22.7570	1.21816	0.609078	+	0.793111i	$0.291540\pi$
$350$	0	0
$351$	−16.0942	−0.859045
$352$	0	0
$353$	−11.4456	−0.609189	−0.304595	−	0.952482i	$-0.598521\pi$
$353$	−11.4456	−0.609189	−0.304595	+	0.952482i	$0.598521\pi$
$354$	0	0
$355$	−34.8384	−1.84903
$356$	0	0
$357$	−1.65270	−0.0874704
$358$	0	0
$359$	−10.3841	−0.548054	−0.274027	−	0.961722i	$-0.588356\pi$
$359$	−10.3841	−0.548054	−0.274027	+	0.961722i	$0.588356\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−0.339556	−0.0178220
$364$	0	0
$365$	19.0351	0.996342
$366$	0	0
$367$	32.5330	1.69821	0.849105	−	0.528224i	$-0.177142\pi$
$367$	32.5330	1.69821	0.849105	+	0.528224i	$0.177142\pi$
$368$	0	0
$369$	12.1993	0.635072
$370$	0	0
$371$	−11.4757	−0.595786
$372$	0	0
$373$	30.4858	1.57849	0.789246	−	0.614077i	$-0.210471\pi$
$373$	30.4858	1.57849	0.789246	+	0.614077i	$0.210471\pi$
$374$	0	0
$375$	−4.83481	−0.249668
$376$	0	0
$377$	18.3482	0.944982
$378$	0	0
$379$	−17.8598	−0.917396	−0.458698	−	0.888592i	$-0.651684\pi$
$379$	−17.8598	−0.917396	−0.458698	+	0.888592i	$0.651684\pi$
$380$	0	0
$381$	−0.0537486	−0.00275363
$382$	0	0
$383$	23.4561	1.19855	0.599274	−	0.800544i	$-0.295456\pi$
$383$	23.4561	1.19855	0.599274	+	0.800544i	$0.295456\pi$
$384$	0	0
$385$	16.2344	0.827383
$386$	0	0
$387$	13.0428	0.663004
$388$	0	0
$389$	3.90941	0.198215	0.0991076	−	0.995077i	$-0.468401\pi$
$389$	3.90941	0.198215	0.0991076	+	0.995077i	$0.468401\pi$
$390$	0	0
$391$	−2.90673	−0.146999
$392$	0	0
$393$	−1.61587	−0.0815097
$394$	0	0
$395$	−17.6313	−0.887129
$396$	0	0
$397$	8.95904	0.449642	0.224821	−	0.974400i	$-0.427820\pi$
$397$	8.95904	0.449642	0.224821	+	0.974400i	$0.427820\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.02734	−0.101241	−0.0506203	−	0.998718i	$-0.516120\pi$
$401$	−2.02734	−0.101241	−0.0506203	+	0.998718i	$0.516120\pi$
$402$	0	0
$403$	10.2831	0.512239
$404$	0	0
$405$	−16.5398	−0.821871
$406$	0	0
$407$	2.85710	0.141621
$408$	0	0
$409$	−32.2080	−1.59258	−0.796292	−	0.604913i	$-0.793208\pi$
$409$	−32.2080	−1.59258	−0.796292	+	0.604913i	$0.793208\pi$
$410$	0	0
$411$	−10.3969	−0.512843
$412$	0	0
$413$	20.2199	0.994955
$414$	0	0
$415$	6.36184	0.312291
$416$	0	0
$417$	8.19253	0.401190
$418$	0	0
$419$	23.2499	1.13583	0.567916	−	0.823086i	$-0.307750\pi$
$419$	23.2499	1.13583	0.567916	+	0.823086i	$0.307750\pi$
$420$	0	0
$421$	6.45336	0.314518	0.157259	−	0.987557i	$-0.449734\pi$
$421$	6.45336	0.314518	0.157259	+	0.987557i	$0.449734\pi$
$422$	0	0
$423$	1.94768	0.0946995
$424$	0	0
$425$	2.33275	0.113155
$426$	0	0
$427$	8.24897	0.399196
$428$	0	0
$429$	−9.60401	−0.463686
$430$	0	0
$431$	13.9973	0.674227	0.337113	−	0.941464i	$-0.390549\pi$
$431$	13.9973	0.674227	0.337113	+	0.941464i	$0.390549\pi$
$432$	0	0
$433$	−28.6928	−1.37889	−0.689445	−	0.724338i	$-0.742145\pi$
$433$	−28.6928	−1.37889	−0.689445	+	0.724338i	$0.742145\pi$
$434$	0	0
$435$	−4.67230	−0.224020
$436$	0	0
$437$	0	0
$438$	0	0
$439$	13.3422	0.636791	0.318395	−	0.947958i	$-0.396856\pi$
$439$	13.3422	0.636791	0.318395	+	0.947958i	$0.396856\pi$
$440$	0	0
$441$	9.42190	0.448662
$442$	0	0
$443$	−33.8830	−1.60983	−0.804915	−	0.593390i	$-0.797789\pi$
$443$	−33.8830	−1.60983	−0.804915	+	0.593390i	$0.797789\pi$
$444$	0	0
$445$	5.78106	0.274048
$446$	0	0
$447$	2.00412	0.0947916
$448$	0	0
$449$	18.8402	0.889123	0.444562	−	0.895748i	$-0.353359\pi$
$449$	18.8402	0.889123	0.444562	+	0.895748i	$0.353359\pi$
$450$	0	0
$451$	15.3182	0.721306
$452$	0	0
$453$	−7.76651	−0.364903
$454$	0	0
$455$	25.1780	1.18036
$456$	0	0
$457$	14.2790	0.667943	0.333972	−	0.942583i	$-0.391611\pi$
$457$	14.2790	0.667943	0.333972	+	0.942583i	$0.391611\pi$
$458$	0	0
$459$	5.02734	0.234656
$460$	0	0
$461$	−13.9281	−0.648695	−0.324348	−	0.945938i	$-0.605145\pi$
$461$	−13.9281	−0.648695	−0.324348	+	0.945938i	$0.605145\pi$
$462$	0	0
$463$	1.76289	0.0819284	0.0409642	−	0.999161i	$-0.486957\pi$
$463$	1.76289	0.0819284	0.0409642	+	0.999161i	$0.486957\pi$
$464$	0	0
$465$	−2.61856	−0.121433
$466$	0	0
$467$	−22.0419	−1.01998	−0.509988	−	0.860181i	$-0.670350\pi$
$467$	−22.0419	−1.01998	−0.509988	+	0.860181i	$0.670350\pi$
$468$	0	0
$469$	−26.7101	−1.23336
$470$	0	0
$471$	5.52023	0.254359
$472$	0	0
$473$	16.3773	0.753030
$474$	0	0
$475$	0	0
$476$	0	0
$477$	16.5895	0.759579
$478$	0	0
$479$	25.4570	1.16316	0.581580	−	0.813489i	$-0.302435\pi$
$479$	25.4570	1.16316	0.581580	+	0.813489i	$0.302435\pi$
$480$	0	0
$481$	4.43107	0.202040
$482$	0	0
$483$	1.75877	0.0800268
$484$	0	0
$485$	4.61587	0.209596
$486$	0	0
$487$	22.5107	1.02006	0.510029	−	0.860157i	$-0.329635\pi$
$487$	22.5107	1.02006	0.510029	+	0.860157i	$0.329635\pi$
$488$	0	0
$489$	−1.07604	−0.0486601
$490$	0	0
$491$	−15.6340	−0.705554	−0.352777	−	0.935707i	$-0.614763\pi$
$491$	−15.6340	−0.705554	−0.352777	+	0.935707i	$0.614763\pi$
$492$	0	0
$493$	−5.73143	−0.258131
$494$	0	0
$495$	−23.4688	−1.05485
$496$	0	0
$497$	25.8580	1.15989
$498$	0	0
$499$	28.6168	1.28106	0.640532	−	0.767932i	$-0.278714\pi$
$499$	28.6168	1.28106	0.640532	+	0.767932i	$0.278714\pi$
$500$	0	0
$501$	−12.3746	−0.552858
$502$	0	0
$503$	25.0455	1.11672	0.558362	−	0.829597i	$-0.311430\pi$
$503$	25.0455	1.11672	0.558362	+	0.829597i	$0.311430\pi$
$504$	0	0
$505$	20.0574	0.892541
$506$	0	0
$507$	−7.97771	−0.354303
$508$	0	0
$509$	33.5212	1.48580	0.742900	−	0.669403i	$-0.233450\pi$
$509$	33.5212	1.48580	0.742900	+	0.669403i	$0.233450\pi$
$510$	0	0
$511$	−14.1284	−0.625002
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−0.0368366	−0.00162322
$516$	0	0
$517$	2.44562	0.107558
$518$	0	0
$519$	−0.477082	−0.0209416
$520$	0	0
$521$	−27.4783	−1.20385	−0.601924	−	0.798553i	$-0.705599\pi$
$521$	−27.4783	−1.20385	−0.601924	+	0.798553i	$0.705599\pi$
$522$	0	0
$523$	10.3574	0.452898	0.226449	−	0.974023i	$-0.427288\pi$
$523$	10.3574	0.452898	0.226449	+	0.974023i	$0.427288\pi$
$524$	0	0
$525$	−1.41147	−0.0616018
$526$	0	0
$527$	−3.21213	−0.139923
$528$	0	0
$529$	−19.9067	−0.865510
$530$	0	0
$531$	−29.2303	−1.26849
$532$	0	0
$533$	23.7570	1.02903
$534$	0	0
$535$	−9.00000	−0.389104
$536$	0	0
$537$	−11.3432	−0.489494
$538$	0	0
$539$	11.8307	0.509584
$540$	0	0
$541$	2.52435	0.108530	0.0542651	−	0.998527i	$-0.482718\pi$
$541$	2.52435	0.108530	0.0542651	+	0.998527i	$0.482718\pi$
$542$	0	0
$543$	8.56624	0.367612
$544$	0	0
$545$	−18.6604	−0.799326
$546$	0	0
$547$	−7.67499	−0.328159	−0.164079	−	0.986447i	$-0.552465\pi$
$547$	−7.67499	−0.328159	−0.164079	+	0.986447i	$0.552465\pi$
$548$	0	0
$549$	−11.9249	−0.508942
$550$	0	0
$551$	0	0
$552$	0	0
$553$	13.0865	0.556493
$554$	0	0
$555$	−1.12836	−0.0478960
$556$	0	0
$557$	3.25578	0.137952	0.0689759	−	0.997618i	$-0.478027\pi$
$557$	3.25578	0.137952	0.0689759	+	0.997618i	$0.478027\pi$
$558$	0	0
$559$	25.3996	1.07429
$560$	0	0
$561$	3.00000	0.126660
$562$	0	0
$563$	−5.25908	−0.221644	−0.110822	−	0.993840i	$-0.535348\pi$
$563$	−5.25908	−0.221644	−0.110822	+	0.993840i	$0.535348\pi$
$564$	0	0
$565$	−18.6800	−0.785875
$566$	0	0
$567$	12.2763	0.515557
$568$	0	0
$569$	−29.9564	−1.25584	−0.627918	−	0.778280i	$-0.716093\pi$
$569$	−29.9564	−1.25584	−0.627918	+	0.778280i	$0.716093\pi$
$570$	0	0
$571$	16.7101	0.699295	0.349647	−	0.936881i	$-0.386301\pi$
$571$	16.7101	0.699295	0.349647	+	0.936881i	$0.386301\pi$
$572$	0	0
$573$	10.0823	0.421196
$574$	0	0
$575$	−2.48246	−0.103526
$576$	0	0
$577$	−13.6800	−0.569508	−0.284754	−	0.958601i	$-0.591912\pi$
$577$	−13.6800	−0.569508	−0.284754	+	0.958601i	$0.591912\pi$
$578$	0	0
$579$	−6.86484	−0.285293
$580$	0	0
$581$	−4.72193	−0.195899
$582$	0	0
$583$	20.8307	0.862719
$584$	0	0
$585$	−36.3979	−1.50487
$586$	0	0
$587$	24.0368	0.992106	0.496053	−	0.868292i	$-0.334782\pi$
$587$	24.0368	0.992106	0.496053	+	0.868292i	$0.334782\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	12.3446	0.507789
$592$	0	0
$593$	−4.24123	−0.174166	−0.0870832	−	0.996201i	$-0.527755\pi$
$593$	−4.24123	−0.174166	−0.0870832	+	0.996201i	$0.527755\pi$
$594$	0	0
$595$	−7.86484	−0.322427
$596$	0	0
$597$	4.90673	0.200819
$598$	0	0
$599$	26.2739	1.07352	0.536762	−	0.843734i	$-0.319647\pi$
$599$	26.2739	1.07352	0.536762	+	0.843734i	$0.319647\pi$
$600$	0	0
$601$	−42.2395	−1.72298	−0.861492	−	0.507771i	$-0.830470\pi$
$601$	−42.2395	−1.72298	−0.861492	+	0.507771i	$0.830470\pi$
$602$	0	0
$603$	38.6127	1.57243
$604$	0	0
$605$	−1.61587	−0.0656943
$606$	0	0
$607$	22.0969	0.896885	0.448443	−	0.893812i	$-0.351979\pi$
$607$	22.0969	0.896885	0.448443	+	0.893812i	$0.351979\pi$
$608$	0	0
$609$	3.46791	0.140527
$610$	0	0
$611$	3.79292	0.153445
$612$	0	0
$613$	7.17705	0.289878	0.144939	−	0.989441i	$-0.453701\pi$
$613$	7.17705	0.289878	0.144939	+	0.989441i	$0.453701\pi$
$614$	0	0
$615$	−6.04963	−0.243945
$616$	0	0
$617$	−49.3729	−1.98768	−0.993839	−	0.110836i	$-0.964647\pi$
$617$	−49.3729	−1.98768	−0.993839	+	0.110836i	$0.964647\pi$
$618$	0	0
$619$	−26.4979	−1.06504	−0.532521	−	0.846417i	$-0.678755\pi$
$619$	−26.4979	−1.06504	−0.532521	+	0.846417i	$0.678755\pi$
$620$	0	0
$621$	−5.34998	−0.214687
$622$	0	0
$623$	−4.29086	−0.171910
$624$	0	0
$625$	−30.0651	−1.20260
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.38413	−0.0551890
$630$	0	0
$631$	−33.3209	−1.32648	−0.663242	−	0.748405i	$-0.730820\pi$
$631$	−33.3209	−1.32648	−0.663242	+	0.748405i	$0.730820\pi$
$632$	0	0
$633$	7.78106	0.309269
$634$	0	0
$635$	−0.255777	−0.0101502
$636$	0	0
$637$	18.3482	0.726983
$638$	0	0
$639$	−37.3809	−1.47877
$640$	0	0
$641$	−0.136096	−0.00537548	−0.00268774	−	0.999996i	$-0.500856\pi$
$641$	−0.136096	−0.00537548	−0.00268774	+	0.999996i	$0.500856\pi$
$642$	0	0
$643$	−48.1780	−1.89995	−0.949977	−	0.312319i	$-0.898894\pi$
$643$	−48.1780	−1.89995	−0.949977	+	0.312319i	$0.898894\pi$
$644$	0	0
$645$	−6.46791	−0.254674
$646$	0	0
$647$	36.9718	1.45351	0.726756	−	0.686895i	$-0.241027\pi$
$647$	36.9718	1.45351	0.726756	+	0.686895i	$0.241027\pi$
$648$	0	0
$649$	−36.7033	−1.44073
$650$	0	0
$651$	1.94356	0.0761742
$652$	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
$655$	−7.68954	−0.300455
$656$	0	0
$657$	20.4243	0.796827
$658$	0	0
$659$	−18.8749	−0.735263	−0.367632	−	0.929971i	$-0.619831\pi$
$659$	−18.8749	−0.735263	−0.367632	+	0.929971i	$0.619831\pi$
$660$	0	0
$661$	−30.6117	−1.19066	−0.595330	−	0.803482i	$-0.702978\pi$
$661$	−30.6117	−1.19066	−0.595330	+	0.803482i	$0.702978\pi$
$662$	0	0
$663$	4.65270	0.180696
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.09926	0.236164
$668$	0	0
$669$	−1.60401	−0.0620145
$670$	0	0
$671$	−14.9736	−0.578049
$672$	0	0
$673$	11.9094	0.459074	0.229537	−	0.973300i	$-0.426279\pi$
$673$	11.9094	0.459074	0.229537	+	0.973300i	$0.426279\pi$
$674$	0	0
$675$	4.29355	0.165259
$676$	0	0
$677$	5.78106	0.222184	0.111092	−	0.993810i	$-0.464565\pi$
$677$	5.78106	0.222184	0.111092	+	0.993810i	$0.464565\pi$
$678$	0	0
$679$	−3.42602	−0.131479
$680$	0	0
$681$	−7.30129	−0.279786
$682$	0	0
$683$	−21.0496	−0.805442	−0.402721	−	0.915323i	$-0.631935\pi$
$683$	−21.0496	−0.805442	−0.402721	+	0.915323i	$0.631935\pi$
$684$	0	0
$685$	−49.4766	−1.89040
$686$	0	0
$687$	−5.00917	−0.191112
$688$	0	0
$689$	32.3063	1.23077
$690$	0	0
$691$	32.9377	1.25301	0.626504	−	0.779418i	$-0.284485\pi$
$691$	32.9377	1.25301	0.626504	+	0.779418i	$0.284485\pi$
$692$	0	0
$693$	17.4192	0.661701
$694$	0	0
$695$	38.9864	1.47884
$696$	0	0
$697$	−7.42097	−0.281089
$698$	0	0
$699$	12.8690	0.486749
$700$	0	0
$701$	21.3574	0.806658	0.403329	−	0.915055i	$-0.367853\pi$
$701$	21.3574	0.806658	0.403329	+	0.915055i	$0.367853\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−0.965852	−0.0363761
$706$	0	0
$707$	−14.8871	−0.559888
$708$	0	0
$709$	−15.7706	−0.592278	−0.296139	−	0.955145i	$-0.595699\pi$
$709$	−15.7706	−0.592278	−0.296139	+	0.955145i	$0.595699\pi$
$710$	0	0
$711$	−18.9181	−0.709484
$712$	0	0
$713$	3.41828	0.128016
$714$	0	0
$715$	−45.7033	−1.70921
$716$	0	0
$717$	−12.4115	−0.463515
$718$	0	0
$719$	35.3283	1.31752	0.658762	−	0.752352i	$-0.271081\pi$
$719$	35.3283	1.31752	0.658762	+	0.752352i	$0.271081\pi$
$720$	0	0
$721$	0.0273411	0.00101824
$722$	0	0
$723$	−0.158385	−0.00589040
$724$	0	0
$725$	−4.89487	−0.181791
$726$	0	0
$727$	−40.4216	−1.49915	−0.749577	−	0.661918i	$-0.769743\pi$
$727$	−40.4216	−1.49915	−0.749577	+	0.661918i	$0.769743\pi$
$728$	0	0
$729$	−12.8912	−0.477454
$730$	0	0
$731$	−7.93407	−0.293452
$732$	0	0
$733$	−36.2763	−1.33990	−0.669948	−	0.742408i	$-0.733684\pi$
$733$	−36.2763	−1.33990	−0.669948	+	0.742408i	$0.733684\pi$
$734$	0	0
$735$	−4.67230	−0.172341
$736$	0	0
$737$	48.4843	1.78594
$738$	0	0
$739$	20.6759	0.760576	0.380288	−	0.924868i	$-0.375825\pi$
$739$	20.6759	0.760576	0.380288	+	0.924868i	$0.375825\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−6.70409	−0.245949	−0.122975	−	0.992410i	$-0.539243\pi$
$743$	−6.70409	−0.245949	−0.122975	+	0.992410i	$0.539243\pi$
$744$	0	0
$745$	9.53714	0.349414
$746$	0	0
$747$	6.82613	0.249755
$748$	0	0
$749$	6.68004	0.244084
$750$	0	0
$751$	−10.7074	−0.390718	−0.195359	−	0.980732i	$-0.562587\pi$
$751$	−10.7074	−0.390718	−0.195359	+	0.980732i	$0.562587\pi$
$752$	0	0
$753$	−8.61856	−0.314078
$754$	0	0
$755$	−36.9590	−1.34508
$756$	0	0
$757$	−4.06242	−0.147651	−0.0738256	−	0.997271i	$-0.523521\pi$
$757$	−4.06242	−0.147651	−0.0738256	+	0.997271i	$0.523521\pi$
$758$	0	0
$759$	−3.19253	−0.115882
$760$	0	0
$761$	−11.0077	−0.399030	−0.199515	−	0.979895i	$-0.563937\pi$
$761$	−11.0077	−0.399030	−0.199515	+	0.979895i	$0.563937\pi$
$762$	0	0
$763$	13.8503	0.501414
$764$	0	0
$765$	11.3696	0.411068
$766$	0	0
$767$	−56.9231	−2.05538
$768$	0	0
$769$	21.3473	0.769803	0.384902	−	0.922958i	$-0.374235\pi$
$769$	21.3473	0.769803	0.384902	+	0.922958i	$0.374235\pi$
$770$	0	0
$771$	−8.17024	−0.294244
$772$	0	0
$773$	17.9172	0.644435	0.322218	−	0.946666i	$-0.395572\pi$
$773$	17.9172	0.644435	0.322218	+	0.946666i	$0.395572\pi$
$774$	0	0
$775$	−2.74329	−0.0985418
$776$	0	0
$777$	0.837496	0.0300450
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−46.9377	−1.67956
$782$	0	0
$783$	−10.5490	−0.376991
$784$	0	0
$785$	26.2695	0.937599
$786$	0	0
$787$	−48.8316	−1.74066	−0.870330	−	0.492470i	$-0.836094\pi$
$787$	−48.8316	−1.74066	−0.870330	+	0.492470i	$0.836094\pi$
$788$	0	0
$789$	5.13104	0.182670
$790$	0	0
$791$	13.8648	0.492977
$792$	0	0
$793$	−23.2226	−0.824657
$794$	0	0
$795$	−8.22668	−0.291770
$796$	0	0
$797$	−28.5262	−1.01045	−0.505225	−	0.862988i	$-0.668591\pi$
$797$	−28.5262	−1.01045	−0.505225	+	0.862988i	$0.668591\pi$
$798$	0	0
$799$	−1.18479	−0.0419149
$800$	0	0
$801$	6.20296	0.219171
$802$	0	0
$803$	25.6459	0.905024
$804$	0	0
$805$	8.36959	0.294989
$806$	0	0
$807$	9.72605	0.342373
$808$	0	0
$809$	−14.8367	−0.521630	−0.260815	−	0.965389i	$-0.583991\pi$
$809$	−14.8367	−0.521630	−0.260815	+	0.965389i	$0.583991\pi$
$810$	0	0
$811$	8.37733	0.294168	0.147084	−	0.989124i	$-0.453011\pi$
$811$	8.37733	0.294168	0.147084	+	0.989124i	$0.453011\pi$
$812$	0	0
$813$	10.0906	0.353892
$814$	0	0
$815$	−5.12061	−0.179367
$816$	0	0
$817$	0	0
$818$	0	0
$819$	27.0155	0.943997
$820$	0	0
$821$	6.27900	0.219139	0.109569	−	0.993979i	$-0.465053\pi$
$821$	6.27900	0.219139	0.109569	+	0.993979i	$0.465053\pi$
$822$	0	0
$823$	−11.0264	−0.384356	−0.192178	−	0.981360i	$-0.561555\pi$
$823$	−11.0264	−0.384356	−0.192178	+	0.981360i	$0.561555\pi$
$824$	0	0
$825$	2.56212	0.0892015
$826$	0	0
$827$	−33.9358	−1.18006	−0.590032	−	0.807380i	$-0.700885\pi$
$827$	−33.9358	−1.18006	−0.590032	+	0.807380i	$0.700885\pi$
$828$	0	0
$829$	−20.3669	−0.707372	−0.353686	−	0.935364i	$-0.615072\pi$
$829$	−20.3669	−0.707372	−0.353686	+	0.935364i	$0.615072\pi$
$830$	0	0
$831$	−7.32358	−0.254052
$832$	0	0
$833$	−5.73143	−0.198582
$834$	0	0
$835$	−58.8881	−2.03791
$836$	0	0
$837$	−5.91210	−0.204352
$838$	0	0
$839$	−15.8093	−0.545799	−0.272899	−	0.962043i	$-0.587983\pi$
$839$	−15.8093	−0.545799	−0.272899	+	0.962043i	$0.587983\pi$
$840$	0	0
$841$	−16.9736	−0.585296
$842$	0	0
$843$	−6.97628	−0.240276
$844$	0	0
$845$	−37.9641	−1.30600
$846$	0	0
$847$	1.19934	0.0412098
$848$	0	0
$849$	−9.24628	−0.317332
$850$	0	0
$851$	1.47296	0.0504925
$852$	0	0
$853$	−52.6492	−1.80267	−0.901337	−	0.433118i	$-0.857413\pi$
$853$	−52.6492	−1.80267	−0.901337	+	0.433118i	$0.857413\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−23.0273	−0.786599	−0.393299	−	0.919410i	$-0.628666\pi$
$857$	−23.0273	−0.786599	−0.393299	+	0.919410i	$0.628666\pi$
$858$	0	0
$859$	−8.90941	−0.303985	−0.151993	−	0.988382i	$-0.548569\pi$
$859$	−8.90941	−0.303985	−0.151993	+	0.988382i	$0.548569\pi$
$860$	0	0
$861$	4.49020	0.153026
$862$	0	0
$863$	−29.7698	−1.01338	−0.506688	−	0.862129i	$-0.669130\pi$
$863$	−29.7698	−1.01338	−0.506688	+	0.862129i	$0.669130\pi$
$864$	0	0
$865$	−2.27033	−0.0771934
$866$	0	0
$867$	7.59215	0.257843
$868$	0	0
$869$	−23.7547	−0.805821
$870$	0	0
$871$	75.1944	2.54787
$872$	0	0
$873$	4.95273	0.167625
$874$	0	0
$875$	17.0770	0.577307
$876$	0	0
$877$	−25.0300	−0.845204	−0.422602	−	0.906315i	$-0.638883\pi$
$877$	−25.0300	−0.845204	−0.422602	+	0.906315i	$0.638883\pi$
$878$	0	0
$879$	−8.30272	−0.280044
$880$	0	0
$881$	20.3960	0.687158	0.343579	−	0.939124i	$-0.388361\pi$
$881$	20.3960	0.687158	0.343579	+	0.939124i	$0.388361\pi$
$882$	0	0
$883$	10.6281	0.357662	0.178831	−	0.983880i	$-0.442768\pi$
$883$	10.6281	0.357662	0.178831	+	0.983880i	$0.442768\pi$
$884$	0	0
$885$	14.4953	0.487253
$886$	0	0
$887$	56.3387	1.89167	0.945835	−	0.324648i	$-0.105246\pi$
$887$	56.3387	1.89167	0.945835	+	0.324648i	$0.105246\pi$
$888$	0	0
$889$	0.189845	0.00636720
$890$	0	0
$891$	−22.2841	−0.746544
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−53.9796	−1.80434
$896$	0	0
$897$	−4.95130	−0.165319
$898$	0	0
$899$	6.74010	0.224795
$900$	0	0
$901$	−10.0915	−0.336197
$902$	0	0
$903$	4.80066	0.159756
$904$	0	0
$905$	40.7648	1.35507
$906$	0	0
$907$	5.92364	0.196691	0.0983456	−	0.995152i	$-0.468645\pi$
$907$	5.92364	0.196691	0.0983456	+	0.995152i	$0.468645\pi$
$908$	0	0
$909$	21.5212	0.713812
$910$	0	0
$911$	34.0591	1.12843	0.564215	−	0.825628i	$-0.309179\pi$
$911$	34.0591	1.12843	0.564215	+	0.825628i	$0.309179\pi$
$912$	0	0
$913$	8.57129	0.283668
$914$	0	0
$915$	5.91353	0.195495
$916$	0	0
$917$	5.70739	0.188474
$918$	0	0
$919$	6.27395	0.206958	0.103479	−	0.994632i	$-0.467002\pi$
$919$	6.27395	0.206958	0.103479	+	0.994632i	$0.467002\pi$
$920$	0	0
$921$	11.4507	0.377313
$922$	0	0
$923$	−72.7957	−2.39610
$924$	0	0
$925$	−1.18210	−0.0388673
$926$	0	0
$927$	−0.0395250	−0.00129817
$928$	0	0
$929$	28.3233	0.929256	0.464628	−	0.885506i	$-0.346188\pi$
$929$	28.3233	0.929256	0.464628	+	0.885506i	$0.346188\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	7.71276	0.252505
$934$	0	0
$935$	14.2763	0.466885
$936$	0	0
$937$	−20.0719	−0.655721	−0.327860	−	0.944726i	$-0.606328\pi$
$937$	−20.0719	−0.655721	−0.327860	+	0.944726i	$0.606328\pi$
$938$	0	0
$939$	−10.3856	−0.338920
$940$	0	0
$941$	5.39517	0.175878	0.0879388	−	0.996126i	$-0.471972\pi$
$941$	5.39517	0.175878	0.0879388	+	0.996126i	$0.471972\pi$
$942$	0	0
$943$	7.89723	0.257169
$944$	0	0
$945$	−14.4757	−0.470893
$946$	0	0
$947$	−6.64321	−0.215875	−0.107938	−	0.994158i	$-0.534425\pi$
$947$	−6.64321	−0.215875	−0.107938	+	0.994158i	$0.534425\pi$
$948$	0	0
$949$	39.7743	1.29113
$950$	0	0
$951$	−15.0833	−0.489109
$952$	0	0
$953$	−14.8239	−0.480193	−0.240096	−	0.970749i	$-0.577179\pi$
$953$	−14.8239	−0.480193	−0.240096	+	0.970749i	$0.577179\pi$
$954$	0	0
$955$	47.9796	1.55258
$956$	0	0
$957$	−6.29498	−0.203488
$958$	0	0
$959$	36.7229	1.18584
$960$	0	0
$961$	−27.2226	−0.878147
$962$	0	0
$963$	−9.65682	−0.311187
$964$	0	0
$965$	−32.6682	−1.05163
$966$	0	0
$967$	−21.2216	−0.682442	−0.341221	−	0.939983i	$-0.610840\pi$
$967$	−21.2216	−0.682442	−0.341221	+	0.939983i	$0.610840\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	37.6067	1.20686	0.603428	−	0.797417i	$-0.293801\pi$
$971$	37.6067	1.20686	0.603428	+	0.797417i	$0.293801\pi$
$972$	0	0
$973$	−28.9368	−0.927670
$974$	0	0
$975$	3.97359	0.127257
$976$	0	0
$977$	−46.0215	−1.47236	−0.736179	−	0.676787i	$-0.763372\pi$
$977$	−46.0215	−1.47236	−0.736179	+	0.676787i	$0.763372\pi$
$978$	0	0
$979$	7.78880	0.248931
$980$	0	0
$981$	−20.0223	−0.639262
$982$	0	0
$983$	−60.5964	−1.93272	−0.966362	−	0.257185i	$-0.917205\pi$
$983$	−60.5964	−1.93272	−0.966362	+	0.257185i	$0.917205\pi$
$984$	0	0
$985$	58.7452	1.87178
$986$	0	0
$987$	0.716881	0.0228186
$988$	0	0
$989$	8.44326	0.268480
$990$	0	0
$991$	−41.8982	−1.33094	−0.665470	−	0.746425i	$-0.731769\pi$
$991$	−41.8982	−1.33094	−0.665470	+	0.746425i	$0.731769\pi$
$992$	0	0
$993$	−0.909912	−0.0288752
$994$	0	0
$995$	23.3500	0.740244
$996$	0	0
$997$	33.7151	1.06777	0.533884	−	0.845557i	$-0.320732\pi$
$997$	33.7151	1.06777	0.533884	+	0.845557i	$0.320732\pi$
$998$	0	0
$999$	−2.54757	−0.0806016

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5776.2.a.br.1.1		3
4.3	odd	2		361.2.a.g.1.3		3
12.11	even	2		3249.2.a.z.1.1		3
19.9	even	9		304.2.u.b.81.1		6
19.17	even	9		304.2.u.b.289.1		6
19.18	odd	2		5776.2.a.bi.1.3		3
20.19	odd	2		9025.2.a.bd.1.1		3
76.3	even	18		361.2.e.a.28.1		6
76.7	odd	6		361.2.c.i.68.1		6
76.11	odd	6		361.2.c.i.292.1		6
76.15	even	18		361.2.e.b.54.1		6
76.23	odd	18		361.2.e.f.54.1		6
76.27	even	6		361.2.c.h.292.3		6
76.31	even	6		361.2.c.h.68.3		6
76.35	odd	18		361.2.e.g.28.1		6
76.43	odd	18		361.2.e.f.234.1		6
76.47	odd	18		19.2.e.a.5.1	yes	6
76.51	even	18		361.2.e.a.245.1		6
76.55	odd	18		19.2.e.a.4.1	✓	6
76.59	even	18		361.2.e.h.99.1		6
76.63	odd	18		361.2.e.g.245.1		6
76.67	even	18		361.2.e.h.62.1		6
76.71	even	18		361.2.e.b.234.1		6
76.75	even	2		361.2.a.h.1.1		3
228.47	even	18		171.2.u.c.100.1		6
228.131	even	18		171.2.u.c.118.1		6
228.227	odd	2		3249.2.a.s.1.3		3
380.47	even	36		475.2.u.a.24.1		12
380.123	even	36		475.2.u.a.24.2		12
380.199	odd	18		475.2.l.a.176.1		6
380.207	even	36		475.2.u.a.99.2		12
380.283	even	36		475.2.u.a.99.1		12
380.359	odd	18		475.2.l.a.251.1		6
380.379	even	2		9025.2.a.x.1.3		3
532.47	even	18		931.2.v.a.214.1		6
532.55	even	18		931.2.w.a.99.1		6
532.123	odd	18		931.2.x.a.765.1		6
532.131	even	18		931.2.x.b.802.1		6
532.199	even	18		931.2.x.b.765.1		6
532.207	odd	18		931.2.v.b.422.1		6
532.275	odd	18		931.2.v.b.214.1		6
532.283	even	18		931.2.v.a.422.1		6
532.359	odd	18		931.2.x.a.802.1		6
532.503	even	18		931.2.w.a.442.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.2.e.a.4.1	✓	6	76.55	odd	18
19.2.e.a.5.1	yes	6	76.47	odd	18
171.2.u.c.100.1		6	228.47	even	18
171.2.u.c.118.1		6	228.131	even	18
304.2.u.b.81.1		6	19.9	even	9
304.2.u.b.289.1		6	19.17	even	9
361.2.a.g.1.3		3	4.3	odd	2
361.2.a.h.1.1		3	76.75	even	2
361.2.c.h.68.3		6	76.31	even	6
361.2.c.h.292.3		6	76.27	even	6
361.2.c.i.68.1		6	76.7	odd	6
361.2.c.i.292.1		6	76.11	odd	6
361.2.e.a.28.1		6	76.3	even	18
361.2.e.a.245.1		6	76.51	even	18
361.2.e.b.54.1		6	76.15	even	18
361.2.e.b.234.1		6	76.71	even	18
361.2.e.f.54.1		6	76.23	odd	18
361.2.e.f.234.1		6	76.43	odd	18
361.2.e.g.28.1		6	76.35	odd	18
361.2.e.g.245.1		6	76.63	odd	18
361.2.e.h.62.1		6	76.67	even	18
361.2.e.h.99.1		6	76.59	even	18
475.2.l.a.176.1		6	380.199	odd	18
475.2.l.a.251.1		6	380.359	odd	18
475.2.u.a.24.1		12	380.47	even	36
475.2.u.a.24.2		12	380.123	even	36
475.2.u.a.99.1		12	380.283	even	36
475.2.u.a.99.2		12	380.207	even	36
931.2.v.a.214.1		6	532.47	even	18
931.2.v.a.422.1		6	532.283	even	18
931.2.v.b.214.1		6	532.275	odd	18
931.2.v.b.422.1		6	532.207	odd	18
931.2.w.a.99.1		6	532.55	even	18
931.2.w.a.442.1		6	532.503	even	18
931.2.x.a.765.1		6	532.123	odd	18
931.2.x.a.802.1		6	532.359	odd	18
931.2.x.b.765.1		6	532.199	even	18
931.2.x.b.802.1		6	532.131	even	18
3249.2.a.s.1.3		3	228.227	odd	2
3249.2.a.z.1.1		3	12.11	even	2
5776.2.a.bi.1.3		3	19.18	odd	2
5776.2.a.br.1.1		3	1.1	even	1	trivial
9025.2.a.x.1.3		3	380.379	even	2
9025.2.a.bd.1.1		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)

\(f(q)\)	\(=\)	\(q-0.532089 q^{3} -2.53209 q^{5} +1.87939 q^{7} -2.71688 q^{9} +O(q^{10})\)	\(q-0.532089 q^{3} -2.53209 q^{5} +1.87939 q^{7} -2.71688 q^{9} -3.41147 q^{11} -5.29086 q^{13} +1.34730 q^{15} +1.65270 q^{17} -1.00000 q^{21} -1.75877 q^{23} +1.41147 q^{25} +3.04189 q^{27} -3.46791 q^{29} -1.94356 q^{31} +1.81521 q^{33} -4.75877 q^{35} -0.837496 q^{37} +2.81521 q^{39} -4.49020 q^{41} -4.80066 q^{43} +6.87939 q^{45} -0.716881 q^{47} -3.46791 q^{49} -0.879385 q^{51} -6.10607 q^{53} +8.63816 q^{55} +10.7588 q^{59} +4.38919 q^{61} -5.10607 q^{63} +13.3969 q^{65} -14.2121 q^{67} +0.935822 q^{69} +13.7588 q^{71} -7.51754 q^{73} -0.751030 q^{75} -6.41147 q^{77} +6.96316 q^{79} +6.53209 q^{81} -2.51249 q^{83} -4.18479 q^{85} +1.84524 q^{87} -2.28312 q^{89} -9.94356 q^{91} +1.03415 q^{93} -1.82295 q^{97} +9.26857 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5}+O(q^{10}) \) 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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