LMFDB - Embedded newform 5824.2.a.by.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5824,2,Mod(1,5824)] 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("5824.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5824, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5824 = 2^{6} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5824.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,2,0,-2,0,-3,0,7,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$46.5048741372$
Analytic rank:	$1$
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 91)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.81361$ of defining polynomial
Character	$\chi$	$=$	5824.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.10278 q^{3} -2.81361 q^{5} -1.00000 q^{7} +6.62721 q^{9} -3.10278 q^{11} -1.00000 q^{13} -8.72999 q^{15} -0.524438 q^{17} -0.813607 q^{19} -3.10278 q^{21} +7.33804 q^{23} +2.91638 q^{25} +11.2544 q^{27} -8.28917 q^{29} +1.39194 q^{31} -9.62721 q^{33} +2.81361 q^{35} +6.15165 q^{37} -3.10278 q^{39} -4.20555 q^{41} -6.75971 q^{43} -18.6464 q^{45} -5.97028 q^{47} +1.00000 q^{49} -1.62721 q^{51} +2.49472 q^{53} +8.72999 q^{55} -2.52444 q^{57} +4.47054 q^{59} +2.00000 q^{61} -6.62721 q^{63} +2.81361 q^{65} -10.0383 q^{67} +22.7683 q^{69} -8.72999 q^{71} -2.34307 q^{73} +9.04888 q^{75} +3.10278 q^{77} -13.5436 q^{79} +15.0383 q^{81} -16.4791 q^{83} +1.47556 q^{85} -25.7194 q^{87} -10.6464 q^{89} +1.00000 q^{91} +4.31889 q^{93} +2.28917 q^{95} -1.18639 q^{97} -20.5628 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} - 2 q^{5} - 3 q^{7} + 7 q^{9} - 2 q^{11} - 3 q^{13} - 6 q^{15} + 4 q^{17} + 4 q^{19} - 2 q^{21} + 10 q^{23} - 5 q^{25} + 8 q^{27} - 24 q^{29} - 4 q^{31} - 16 q^{33} + 2 q^{35} - 2 q^{39} + 2 q^{41}+ \cdots - 14 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 - 2 * q^5 - 3 * q^7 + 7 * q^9 - 2 * q^11 - 3 * q^13 - 6 * q^15 + 4 * q^17 + 4 * q^19 - 2 * q^21 + 10 * q^23 - 5 * q^25 + 8 * q^27 - 24 * q^29 - 4 * q^31 - 16 * q^33 + 2 * q^35 - 2 * q^39 + 2 * q^41 - 10 * q^43 - 22 * q^45 - 8 * q^47 + 3 * q^49 + 8 * q^51 - 8 * q^53 + 6 * q^55 - 2 * q^57 + 4 * q^59 + 6 * q^61 - 7 * q^63 + 2 * q^65 + 12 * q^67 + 6 * q^69 - 6 * q^71 - 10 * q^73 + 16 * q^75 + 2 * q^77 - 14 * q^79 + 3 * q^81 + 12 * q^83 + 10 * q^85 - 26 * q^87 + 2 * q^89 + 3 * q^91 + 22 * q^93 + 6 * q^95 - 10 * q^97 - 14 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.10278	1.79139	0.895694	−	0.444671i	$-0.146679\pi$
$3$	3.10278	1.79139	0.895694	+	0.444671i	$0.146679\pi$
$4$	0	0
$5$	−2.81361	−1.25828	−0.629142	−	0.777291i	$-0.716593\pi$
$5$	−2.81361	−1.25828	−0.629142	+	0.777291i	$0.716593\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	6.62721	2.20907
$10$	0	0
$11$	−3.10278	−0.935522	−0.467761	−	0.883855i	$-0.654939\pi$
$11$	−3.10278	−0.935522	−0.467761	+	0.883855i	$0.654939\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−8.72999	−2.25407
$16$	0	0
$17$	−0.524438	−0.127195	−0.0635974	−	0.997976i	$-0.520257\pi$
$17$	−0.524438	−0.127195	−0.0635974	+	0.997976i	$0.520257\pi$
$18$	0	0
$19$	−0.813607	−0.186654	−0.0933271	−	0.995636i	$-0.529750\pi$
$19$	−0.813607	−0.186654	−0.0933271	+	0.995636i	$0.529750\pi$
$20$	0	0
$21$	−3.10278	−0.677081
$22$	0	0
$23$	7.33804	1.53009	0.765044	−	0.643978i	$-0.222717\pi$
$23$	7.33804	1.53009	0.765044	+	0.643978i	$0.222717\pi$
$24$	0	0
$25$	2.91638	0.583276
$26$	0	0
$27$	11.2544	2.16592
$28$	0	0
$29$	−8.28917	−1.53926	−0.769630	−	0.638490i	$-0.779559\pi$
$29$	−8.28917	−1.53926	−0.769630	+	0.638490i	$0.779559\pi$
$30$	0	0
$31$	1.39194	0.250000	0.125000	−	0.992157i	$-0.460107\pi$
$31$	1.39194	0.250000	0.125000	+	0.992157i	$0.460107\pi$
$32$	0	0
$33$	−9.62721	−1.67588
$34$	0	0
$35$	2.81361	0.475586
$36$	0	0
$37$	6.15165	1.01133	0.505663	−	0.862731i	$-0.331248\pi$
$37$	6.15165	1.01133	0.505663	+	0.862731i	$0.331248\pi$
$38$	0	0
$39$	−3.10278	−0.496842
$40$	0	0
$41$	−4.20555	−0.656797	−0.328398	−	0.944539i	$-0.606509\pi$
$41$	−4.20555	−0.656797	−0.328398	+	0.944539i	$0.606509\pi$
$42$	0	0
$43$	−6.75971	−1.03085	−0.515423	−	0.856936i	$-0.672365\pi$
$43$	−6.75971	−1.03085	−0.515423	+	0.856936i	$0.672365\pi$
$44$	0	0
$45$	−18.6464	−2.77964
$46$	0	0
$47$	−5.97028	−0.870855	−0.435427	−	0.900224i	$-0.643403\pi$
$47$	−5.97028	−0.870855	−0.435427	+	0.900224i	$0.643403\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.62721	−0.227855
$52$	0	0
$53$	2.49472	0.342676	0.171338	−	0.985212i	$-0.445191\pi$
$53$	2.49472	0.342676	0.171338	+	0.985212i	$0.445191\pi$
$54$	0	0
$55$	8.72999	1.17715
$56$	0	0
$57$	−2.52444	−0.334370
$58$	0	0
$59$	4.47054	0.582015	0.291007	−	0.956721i	$-0.406010\pi$
$59$	4.47054	0.582015	0.291007	+	0.956721i	$0.406010\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	−6.62721	−0.834950
$64$	0	0
$65$	2.81361	0.348985
$66$	0	0
$67$	−10.0383	−1.22638	−0.613188	−	0.789937i	$-0.710113\pi$
$67$	−10.0383	−1.22638	−0.613188	+	0.789937i	$0.710113\pi$
$68$	0	0
$69$	22.7683	2.74098
$70$	0	0
$71$	−8.72999	−1.03606	−0.518029	−	0.855363i	$-0.673334\pi$
$71$	−8.72999	−1.03606	−0.518029	+	0.855363i	$0.673334\pi$
$72$	0	0
$73$	−2.34307	−0.274235	−0.137118	−	0.990555i	$-0.543784\pi$
$73$	−2.34307	−0.274235	−0.137118	+	0.990555i	$0.543784\pi$
$74$	0	0
$75$	9.04888	1.04487
$76$	0	0
$77$	3.10278	0.353594
$78$	0	0
$79$	−13.5436	−1.52377	−0.761887	−	0.647710i	$-0.775727\pi$
$79$	−13.5436	−1.52377	−0.761887	+	0.647710i	$0.775727\pi$
$80$	0	0
$81$	15.0383	1.67092
$82$	0	0
$83$	−16.4791	−1.80882	−0.904410	−	0.426665i	$-0.859688\pi$
$83$	−16.4791	−1.80882	−0.904410	+	0.426665i	$0.859688\pi$
$84$	0	0
$85$	1.47556	0.160047
$86$	0	0
$87$	−25.7194	−2.75741
$88$	0	0
$89$	−10.6464	−1.12851	−0.564256	−	0.825600i	$-0.690837\pi$
$89$	−10.6464	−1.12851	−0.564256	+	0.825600i	$0.690837\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	4.31889	0.447848
$94$	0	0
$95$	2.28917	0.234864
$96$	0	0
$97$	−1.18639	−0.120460	−0.0602300	−	0.998185i	$-0.519183\pi$
$97$	−1.18639	−0.120460	−0.0602300	+	0.998185i	$0.519183\pi$
$98$	0	0
$99$	−20.5628	−2.06663
$100$	0	0
$101$	−13.1028	−1.30377	−0.651887	−	0.758316i	$-0.726023\pi$
$101$	−13.1028	−1.30377	−0.651887	+	0.758316i	$0.726023\pi$
$102$	0	0
$103$	4.41110	0.434639	0.217319	−	0.976101i	$-0.430269\pi$
$103$	4.41110	0.434639	0.217319	+	0.976101i	$0.430269\pi$
$104$	0	0
$105$	8.72999	0.851960
$106$	0	0
$107$	−0.578337	−0.0559100	−0.0279550	−	0.999609i	$-0.508900\pi$
$107$	−0.578337	−0.0559100	−0.0279550	+	0.999609i	$0.508900\pi$
$108$	0	0
$109$	−5.57331	−0.533827	−0.266913	−	0.963721i	$-0.586004\pi$
$109$	−5.57331	−0.533827	−0.266913	+	0.963721i	$0.586004\pi$
$110$	0	0
$111$	19.0872	1.81168
$112$	0	0
$113$	5.44584	0.512302	0.256151	−	0.966637i	$-0.417546\pi$
$113$	5.44584	0.512302	0.256151	+	0.966637i	$0.417546\pi$
$114$	0	0
$115$	−20.6464	−1.92528
$116$	0	0
$117$	−6.62721	−0.612686
$118$	0	0
$119$	0.524438	0.0480751
$120$	0	0
$121$	−1.37279	−0.124799
$122$	0	0
$123$	−13.0489	−1.17658
$124$	0	0
$125$	5.86248	0.524356
$126$	0	0
$127$	−12.8816	−1.14306	−0.571530	−	0.820581i	$-0.693650\pi$
$127$	−12.8816	−1.14306	−0.571530	+	0.820581i	$0.693650\pi$
$128$	0	0
$129$	−20.9739	−1.84664
$130$	0	0
$131$	−9.04888	−0.790604	−0.395302	−	0.918551i	$-0.629360\pi$
$131$	−9.04888	−0.790604	−0.395302	+	0.918551i	$0.629360\pi$
$132$	0	0
$133$	0.813607	0.0705486
$134$	0	0
$135$	−31.6655	−2.72533
$136$	0	0
$137$	−6.25945	−0.534781	−0.267390	−	0.963588i	$-0.586161\pi$
$137$	−6.25945	−0.534781	−0.267390	+	0.963588i	$0.586161\pi$
$138$	0	0
$139$	11.5733	0.981636	0.490818	−	0.871262i	$-0.336698\pi$
$139$	11.5733	0.981636	0.490818	+	0.871262i	$0.336698\pi$
$140$	0	0
$141$	−18.5244	−1.56004
$142$	0	0
$143$	3.10278	0.259467
$144$	0	0
$145$	23.3225	1.93682
$146$	0	0
$147$	3.10278	0.255913
$148$	0	0
$149$	−8.52444	−0.698349	−0.349175	−	0.937058i	$-0.613538\pi$
$149$	−8.52444	−0.698349	−0.349175	+	0.937058i	$0.613538\pi$
$150$	0	0
$151$	−11.9844	−0.975278	−0.487639	−	0.873045i	$-0.662142\pi$
$151$	−11.9844	−0.975278	−0.487639	+	0.873045i	$0.662142\pi$
$152$	0	0
$153$	−3.47556	−0.280983
$154$	0	0
$155$	−3.91638	−0.314571
$156$	0	0
$157$	12.8277	1.02377	0.511883	−	0.859055i	$-0.328948\pi$
$157$	12.8277	1.02377	0.511883	+	0.859055i	$0.328948\pi$
$158$	0	0
$159$	7.74055	0.613866
$160$	0	0
$161$	−7.33804	−0.578319
$162$	0	0
$163$	13.4600	1.05427	0.527133	−	0.849783i	$-0.323267\pi$
$163$	13.4600	1.05427	0.527133	+	0.849783i	$0.323267\pi$
$164$	0	0
$165$	27.0872	2.10873
$166$	0	0
$167$	−2.02972	−0.157064	−0.0785322	−	0.996912i	$-0.525023\pi$
$167$	−2.02972	−0.157064	−0.0785322	+	0.996912i	$0.525023\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−5.39194	−0.412332
$172$	0	0
$173$	−20.2978	−1.54321	−0.771605	−	0.636102i	$-0.780546\pi$
$173$	−20.2978	−1.54321	−0.771605	+	0.636102i	$0.780546\pi$
$174$	0	0
$175$	−2.91638	−0.220458
$176$	0	0
$177$	13.8711	1.04261
$178$	0	0
$179$	11.0036	0.822445	0.411223	−	0.911535i	$-0.365102\pi$
$179$	11.0036	0.822445	0.411223	+	0.911535i	$0.365102\pi$
$180$	0	0
$181$	−0.691675	−0.0514118	−0.0257059	−	0.999670i	$-0.508183\pi$
$181$	−0.691675	−0.0514118	−0.0257059	+	0.999670i	$0.508183\pi$
$182$	0	0
$183$	6.20555	0.458727
$184$	0	0
$185$	−17.3083	−1.27253
$186$	0	0
$187$	1.62721	0.118994
$188$	0	0
$189$	−11.2544	−0.818639
$190$	0	0
$191$	7.83276	0.566759	0.283379	−	0.959008i	$-0.408544\pi$
$191$	7.83276	0.566759	0.283379	+	0.959008i	$0.408544\pi$
$192$	0	0
$193$	−12.2056	−0.878575	−0.439287	−	0.898347i	$-0.644769\pi$
$193$	−12.2056	−0.878575	−0.439287	+	0.898347i	$0.644769\pi$
$194$	0	0
$195$	8.72999	0.625167
$196$	0	0
$197$	18.8222	1.34103	0.670513	−	0.741898i	$-0.266074\pi$
$197$	18.8222	1.34103	0.670513	+	0.741898i	$0.266074\pi$
$198$	0	0
$199$	21.6116	1.53201	0.766004	−	0.642836i	$-0.222242\pi$
$199$	21.6116	1.53201	0.766004	+	0.642836i	$0.222242\pi$
$200$	0	0
$201$	−31.1466	−2.19691
$202$	0	0
$203$	8.28917	0.581786
$204$	0	0
$205$	11.8328	0.826436
$206$	0	0
$207$	48.6308	3.38007
$208$	0	0
$209$	2.52444	0.174619
$210$	0	0
$211$	17.3764	1.19624	0.598119	−	0.801407i	$-0.295915\pi$
$211$	17.3764	1.19624	0.598119	+	0.801407i	$0.295915\pi$
$212$	0	0
$213$	−27.0872	−1.85598
$214$	0	0
$215$	19.0192	1.29710
$216$	0	0
$217$	−1.39194	−0.0944913
$218$	0	0
$219$	−7.27001	−0.491262
$220$	0	0
$221$	0.524438	0.0352775
$222$	0	0
$223$	−10.5486	−0.706388	−0.353194	−	0.935550i	$-0.614904\pi$
$223$	−10.5486	−0.706388	−0.353194	+	0.935550i	$0.614904\pi$
$224$	0	0
$225$	19.3275	1.28850
$226$	0	0
$227$	6.95112	0.461362	0.230681	−	0.973029i	$-0.425905\pi$
$227$	6.95112	0.461362	0.230681	+	0.973029i	$0.425905\pi$
$228$	0	0
$229$	21.0872	1.39348	0.696740	−	0.717323i	$-0.254633\pi$
$229$	21.0872	1.39348	0.696740	+	0.717323i	$0.254633\pi$
$230$	0	0
$231$	9.62721	0.633424
$232$	0	0
$233$	6.08362	0.398551	0.199276	−	0.979943i	$-0.436141\pi$
$233$	6.08362	0.398551	0.199276	+	0.979943i	$0.436141\pi$
$234$	0	0
$235$	16.7980	1.09578
$236$	0	0
$237$	−42.0227	−2.72967
$238$	0	0
$239$	−14.2056	−0.918881	−0.459440	−	0.888209i	$-0.651950\pi$
$239$	−14.2056	−0.918881	−0.459440	+	0.888209i	$0.651950\pi$
$240$	0	0
$241$	8.44082	0.543721	0.271860	−	0.962337i	$-0.412361\pi$
$241$	8.44082	0.543721	0.271860	+	0.962337i	$0.412361\pi$
$242$	0	0
$243$	12.8972	0.827357
$244$	0	0
$245$	−2.81361	−0.179755
$246$	0	0
$247$	0.813607	0.0517685
$248$	0	0
$249$	−51.1310	−3.24030
$250$	0	0
$251$	23.9844	1.51388	0.756941	−	0.653483i	$-0.226693\pi$
$251$	23.9844	1.51388	0.756941	+	0.653483i	$0.226693\pi$
$252$	0	0
$253$	−22.7683	−1.43143
$254$	0	0
$255$	4.57834	0.286707
$256$	0	0
$257$	15.6116	0.973827	0.486913	−	0.873450i	$-0.338123\pi$
$257$	15.6116	0.973827	0.486913	+	0.873450i	$0.338123\pi$
$258$	0	0
$259$	−6.15165	−0.382245
$260$	0	0
$261$	−54.9341	−3.40033
$262$	0	0
$263$	−15.1708	−0.935472	−0.467736	−	0.883868i	$-0.654930\pi$
$263$	−15.1708	−0.935472	−0.467736	+	0.883868i	$0.654930\pi$
$264$	0	0
$265$	−7.01916	−0.431183
$266$	0	0
$267$	−33.0333	−2.02160
$268$	0	0
$269$	−23.2927	−1.42018	−0.710092	−	0.704109i	$-0.751347\pi$
$269$	−23.2927	−1.42018	−0.710092	+	0.704109i	$0.751347\pi$
$270$	0	0
$271$	12.7456	0.774238	0.387119	−	0.922030i	$-0.373470\pi$
$271$	12.7456	0.774238	0.387119	+	0.922030i	$0.373470\pi$
$272$	0	0
$273$	3.10278	0.187788
$274$	0	0
$275$	−9.04888	−0.545668
$276$	0	0
$277$	−8.12193	−0.488000	−0.244000	−	0.969775i	$-0.578460\pi$
$277$	−8.12193	−0.488000	−0.244000	+	0.969775i	$0.578460\pi$
$278$	0	0
$279$	9.22471	0.552269
$280$	0	0
$281$	19.0333	1.13543	0.567715	−	0.823225i	$-0.307827\pi$
$281$	19.0333	1.13543	0.567715	+	0.823225i	$0.307827\pi$
$282$	0	0
$283$	−11.1466	−0.662598	−0.331299	−	0.943526i	$-0.607487\pi$
$283$	−11.1466	−0.662598	−0.331299	+	0.943526i	$0.607487\pi$
$284$	0	0
$285$	7.10278	0.420732
$286$	0	0
$287$	4.20555	0.248246
$288$	0	0
$289$	−16.7250	−0.983821
$290$	0	0
$291$	−3.68111	−0.215791
$292$	0	0
$293$	14.1758	0.828161	0.414080	−	0.910240i	$-0.364103\pi$
$293$	14.1758	0.828161	0.414080	+	0.910240i	$0.364103\pi$
$294$	0	0
$295$	−12.5783	−0.732339
$296$	0	0
$297$	−34.9200	−2.02626
$298$	0	0
$299$	−7.33804	−0.424370
$300$	0	0
$301$	6.75971	0.389623
$302$	0	0
$303$	−40.6550	−2.33557
$304$	0	0
$305$	−5.62721	−0.322213
$306$	0	0
$307$	−13.5592	−0.773863	−0.386932	−	0.922108i	$-0.626465\pi$
$307$	−13.5592	−0.773863	−0.386932	+	0.922108i	$0.626465\pi$
$308$	0	0
$309$	13.6867	0.778606
$310$	0	0
$311$	0.426686	0.0241952	0.0120976	−	0.999927i	$-0.496149\pi$
$311$	0.426686	0.0241952	0.0120976	+	0.999927i	$0.496149\pi$
$312$	0	0
$313$	−18.1517	−1.02599	−0.512996	−	0.858391i	$-0.671464\pi$
$313$	−18.1517	−1.02599	−0.512996	+	0.858391i	$0.671464\pi$
$314$	0	0
$315$	18.6464	1.05060
$316$	0	0
$317$	−9.42166	−0.529173	−0.264587	−	0.964362i	$-0.585236\pi$
$317$	−9.42166	−0.529173	−0.264587	+	0.964362i	$0.585236\pi$
$318$	0	0
$319$	25.7194	1.44001
$320$	0	0
$321$	−1.79445	−0.100156
$322$	0	0
$323$	0.426686	0.0237415
$324$	0	0
$325$	−2.91638	−0.161772
$326$	0	0
$327$	−17.2927	−0.956291
$328$	0	0
$329$	5.97028	0.329152
$330$	0	0
$331$	17.4005	0.956420	0.478210	−	0.878246i	$-0.341286\pi$
$331$	17.4005	0.956420	0.478210	+	0.878246i	$0.341286\pi$
$332$	0	0
$333$	40.7683	2.23409
$334$	0	0
$335$	28.2439	1.54313
$336$	0	0
$337$	22.0524	1.20127	0.600637	−	0.799522i	$-0.294914\pi$
$337$	22.0524	1.20127	0.600637	+	0.799522i	$0.294914\pi$
$338$	0	0
$339$	16.8972	0.917731
$340$	0	0
$341$	−4.31889	−0.233881
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−64.0610	−3.44893
$346$	0	0
$347$	−25.3522	−1.36098	−0.680488	−	0.732759i	$-0.738232\pi$
$347$	−25.3522	−1.36098	−0.680488	+	0.732759i	$0.738232\pi$
$348$	0	0
$349$	5.70529	0.305397	0.152699	−	0.988273i	$-0.451204\pi$
$349$	5.70529	0.305397	0.152699	+	0.988273i	$0.451204\pi$
$350$	0	0
$351$	−11.2544	−0.600717
$352$	0	0
$353$	−28.6761	−1.52627	−0.763137	−	0.646237i	$-0.776342\pi$
$353$	−28.6761	−1.52627	−0.763137	+	0.646237i	$0.776342\pi$
$354$	0	0
$355$	24.5628	1.30366
$356$	0	0
$357$	1.62721	0.0861212
$358$	0	0
$359$	11.0433	0.582845	0.291423	−	0.956594i	$-0.405871\pi$
$359$	11.0433	0.582845	0.291423	+	0.956594i	$0.405871\pi$
$360$	0	0
$361$	−18.3380	−0.965160
$362$	0	0
$363$	−4.25945	−0.223563
$364$	0	0
$365$	6.59247	0.345066
$366$	0	0
$367$	27.3466	1.42748	0.713741	−	0.700409i	$-0.246999\pi$
$367$	27.3466	1.42748	0.713741	+	0.700409i	$0.246999\pi$
$368$	0	0
$369$	−27.8711	−1.45091
$370$	0	0
$371$	−2.49472	−0.129519
$372$	0	0
$373$	−16.1461	−0.836014	−0.418007	−	0.908444i	$-0.637271\pi$
$373$	−16.1461	−0.836014	−0.418007	+	0.908444i	$0.637271\pi$
$374$	0	0
$375$	18.1900	0.939326
$376$	0	0
$377$	8.28917	0.426914
$378$	0	0
$379$	−26.1305	−1.34223	−0.671117	−	0.741351i	$-0.734185\pi$
$379$	−26.1305	−1.34223	−0.671117	+	0.741351i	$0.734185\pi$
$380$	0	0
$381$	−39.9688	−2.04767
$382$	0	0
$383$	−21.0489	−1.07555	−0.537774	−	0.843089i	$-0.680735\pi$
$383$	−21.0489	−1.07555	−0.537774	+	0.843089i	$0.680735\pi$
$384$	0	0
$385$	−8.72999	−0.444921
$386$	0	0
$387$	−44.7980	−2.27721
$388$	0	0
$389$	21.6061	1.09547	0.547736	−	0.836651i	$-0.315490\pi$
$389$	21.6061	1.09547	0.547736	+	0.836651i	$0.315490\pi$
$390$	0	0
$391$	−3.84835	−0.194619
$392$	0	0
$393$	−28.0766	−1.41628
$394$	0	0
$395$	38.1063	1.91734
$396$	0	0
$397$	27.6952	1.38998	0.694992	−	0.719017i	$-0.255408\pi$
$397$	27.6952	1.38998	0.694992	+	0.719017i	$0.255408\pi$
$398$	0	0
$399$	2.52444	0.126380
$400$	0	0
$401$	−2.57834	−0.128756	−0.0643780	−	0.997926i	$-0.520506\pi$
$401$	−2.57834	−0.128756	−0.0643780	+	0.997926i	$0.520506\pi$
$402$	0	0
$403$	−1.39194	−0.0693376
$404$	0	0
$405$	−42.3119	−2.10250
$406$	0	0
$407$	−19.0872	−0.946117
$408$	0	0
$409$	15.1169	0.747483	0.373742	−	0.927533i	$-0.378075\pi$
$409$	15.1169	0.747483	0.373742	+	0.927533i	$0.378075\pi$
$410$	0	0
$411$	−19.4217	−0.958000
$412$	0	0
$413$	−4.47054	−0.219981
$414$	0	0
$415$	46.3658	2.27601
$416$	0	0
$417$	35.9094	1.75849
$418$	0	0
$419$	9.99446	0.488261	0.244131	−	0.969742i	$-0.421497\pi$
$419$	9.99446	0.488261	0.244131	+	0.969742i	$0.421497\pi$
$420$	0	0
$421$	25.9250	1.26351	0.631753	−	0.775170i	$-0.282336\pi$
$421$	25.9250	1.26351	0.631753	+	0.775170i	$0.282336\pi$
$422$	0	0
$423$	−39.5663	−1.92378
$424$	0	0
$425$	−1.52946	−0.0741898
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	9.62721	0.464806
$430$	0	0
$431$	30.6761	1.47762	0.738808	−	0.673916i	$-0.235389\pi$
$431$	30.6761	1.47762	0.738808	+	0.673916i	$0.235389\pi$
$432$	0	0
$433$	−3.51941	−0.169132	−0.0845661	−	0.996418i	$-0.526950\pi$
$433$	−3.51941	−0.169132	−0.0845661	+	0.996418i	$0.526950\pi$
$434$	0	0
$435$	72.3643	3.46960
$436$	0	0
$437$	−5.97028	−0.285597
$438$	0	0
$439$	32.3517	1.54406	0.772030	−	0.635586i	$-0.219241\pi$
$439$	32.3517	1.54406	0.772030	+	0.635586i	$0.219241\pi$
$440$	0	0
$441$	6.62721	0.315582
$442$	0	0
$443$	−15.4458	−0.733854	−0.366927	−	0.930250i	$-0.619590\pi$
$443$	−15.4458	−0.733854	−0.366927	+	0.930250i	$0.619590\pi$
$444$	0	0
$445$	29.9547	1.41999
$446$	0	0
$447$	−26.4494	−1.25101
$448$	0	0
$449$	−14.4705	−0.682907	−0.341453	−	0.939899i	$-0.610919\pi$
$449$	−14.4705	−0.682907	−0.341453	+	0.939899i	$0.610919\pi$
$450$	0	0
$451$	13.0489	0.614448
$452$	0	0
$453$	−37.1849	−1.74710
$454$	0	0
$455$	−2.81361	−0.131904
$456$	0	0
$457$	−34.6705	−1.62182	−0.810910	−	0.585171i	$-0.801027\pi$
$457$	−34.6705	−1.62182	−0.810910	+	0.585171i	$0.801027\pi$
$458$	0	0
$459$	−5.90225	−0.275493
$460$	0	0
$461$	−12.5400	−0.584047	−0.292024	−	0.956411i	$-0.594329\pi$
$461$	−12.5400	−0.584047	−0.292024	+	0.956411i	$0.594329\pi$
$462$	0	0
$463$	12.1517	0.564735	0.282368	−	0.959306i	$-0.408880\pi$
$463$	12.1517	0.564735	0.282368	+	0.959306i	$0.408880\pi$
$464$	0	0
$465$	−12.1517	−0.563519
$466$	0	0
$467$	37.0333	1.71370	0.856848	−	0.515569i	$-0.172419\pi$
$467$	37.0333	1.71370	0.856848	+	0.515569i	$0.172419\pi$
$468$	0	0
$469$	10.0383	0.463526
$470$	0	0
$471$	39.8016	1.83396
$472$	0	0
$473$	20.9739	0.964379
$474$	0	0
$475$	−2.37279	−0.108871
$476$	0	0
$477$	16.5330	0.756996
$478$	0	0
$479$	12.0086	0.548687	0.274343	−	0.961632i	$-0.411540\pi$
$479$	12.0086	0.548687	0.274343	+	0.961632i	$0.411540\pi$
$480$	0	0
$481$	−6.15165	−0.280491
$482$	0	0
$483$	−22.7683	−1.03599
$484$	0	0
$485$	3.33804	0.151573
$486$	0	0
$487$	11.1184	0.503821	0.251911	−	0.967751i	$-0.418941\pi$
$487$	11.1184	0.503821	0.251911	+	0.967751i	$0.418941\pi$
$488$	0	0
$489$	41.7633	1.88860
$490$	0	0
$491$	−0.0594386	−0.00268243	−0.00134121	−	0.999999i	$-0.500427\pi$
$491$	−0.0594386	−0.00268243	−0.00134121	+	0.999999i	$0.500427\pi$
$492$	0	0
$493$	4.34715	0.195786
$494$	0	0
$495$	57.8555	2.60041
$496$	0	0
$497$	8.72999	0.391593
$498$	0	0
$499$	−10.2978	−0.460991	−0.230496	−	0.973073i	$-0.574035\pi$
$499$	−10.2978	−0.460991	−0.230496	+	0.973073i	$0.574035\pi$
$500$	0	0
$501$	−6.29776	−0.281363
$502$	0	0
$503$	−9.32391	−0.415733	−0.207866	−	0.978157i	$-0.566652\pi$
$503$	−9.32391	−0.415733	−0.207866	+	0.978157i	$0.566652\pi$
$504$	0	0
$505$	36.8661	1.64052
$506$	0	0
$507$	3.10278	0.137799
$508$	0	0
$509$	−39.6952	−1.75946	−0.879730	−	0.475473i	$-0.842277\pi$
$509$	−39.6952	−1.75946	−0.879730	+	0.475473i	$0.842277\pi$
$510$	0	0
$511$	2.34307	0.103651
$512$	0	0
$513$	−9.15667	−0.404277
$514$	0	0
$515$	−12.4111	−0.546898
$516$	0	0
$517$	18.5244	0.814704
$518$	0	0
$519$	−62.9794	−2.76449
$520$	0	0
$521$	22.3627	0.979729	0.489865	−	0.871798i	$-0.337046\pi$
$521$	22.3627	0.979729	0.489865	+	0.871798i	$0.337046\pi$
$522$	0	0
$523$	20.6550	0.903178	0.451589	−	0.892226i	$-0.350857\pi$
$523$	20.6550	0.903178	0.451589	+	0.892226i	$0.350857\pi$
$524$	0	0
$525$	−9.04888	−0.394925
$526$	0	0
$527$	−0.729988	−0.0317988
$528$	0	0
$529$	30.8469	1.34117
$530$	0	0
$531$	29.6272	1.28571
$532$	0	0
$533$	4.20555	0.182163
$534$	0	0
$535$	1.62721	0.0703506
$536$	0	0
$537$	34.1416	1.47332
$538$	0	0
$539$	−3.10278	−0.133646
$540$	0	0
$541$	−5.62167	−0.241695	−0.120847	−	0.992671i	$-0.538561\pi$
$541$	−5.62167	−0.241695	−0.120847	+	0.992671i	$0.538561\pi$
$542$	0	0
$543$	−2.14611	−0.0920985
$544$	0	0
$545$	15.6811	0.671705
$546$	0	0
$547$	10.3970	0.444542	0.222271	−	0.974985i	$-0.428653\pi$
$547$	10.3970	0.444542	0.222271	+	0.974985i	$0.428653\pi$
$548$	0	0
$549$	13.2544	0.565685
$550$	0	0
$551$	6.74412	0.287309
$552$	0	0
$553$	13.5436	0.575932
$554$	0	0
$555$	−53.7038	−2.27960
$556$	0	0
$557$	−14.6550	−0.620951	−0.310475	−	0.950581i	$-0.600488\pi$
$557$	−14.6550	−0.620951	−0.310475	+	0.950581i	$0.600488\pi$
$558$	0	0
$559$	6.75971	0.285905
$560$	0	0
$561$	5.04888	0.213164
$562$	0	0
$563$	24.7456	1.04290	0.521451	−	0.853281i	$-0.325391\pi$
$563$	24.7456	1.04290	0.521451	+	0.853281i	$0.325391\pi$
$564$	0	0
$565$	−15.3225	−0.644621
$566$	0	0
$567$	−15.0383	−0.631550
$568$	0	0
$569$	−20.5330	−0.860789	−0.430395	−	0.902641i	$-0.641626\pi$
$569$	−20.5330	−0.860789	−0.430395	+	0.902641i	$0.641626\pi$
$570$	0	0
$571$	41.8953	1.75326	0.876631	−	0.481163i	$-0.159786\pi$
$571$	41.8953	1.75326	0.876631	+	0.481163i	$0.159786\pi$
$572$	0	0
$573$	24.3033	1.01529
$574$	0	0
$575$	21.4005	0.892464
$576$	0	0
$577$	−20.1744	−0.839870	−0.419935	−	0.907554i	$-0.637947\pi$
$577$	−20.1744	−0.839870	−0.419935	+	0.907554i	$0.637947\pi$
$578$	0	0
$579$	−37.8711	−1.57387
$580$	0	0
$581$	16.4791	0.683670
$582$	0	0
$583$	−7.74055	−0.320581
$584$	0	0
$585$	18.6464	0.770933
$586$	0	0
$587$	18.7441	0.773653	0.386826	−	0.922153i	$-0.373571\pi$
$587$	18.7441	0.773653	0.386826	+	0.922153i	$0.373571\pi$
$588$	0	0
$589$	−1.13249	−0.0466636
$590$	0	0
$591$	58.4011	2.40230
$592$	0	0
$593$	−2.98084	−0.122409	−0.0612043	−	0.998125i	$-0.519494\pi$
$593$	−2.98084	−0.122409	−0.0612043	+	0.998125i	$0.519494\pi$
$594$	0	0
$595$	−1.47556	−0.0604921
$596$	0	0
$597$	67.0560	2.74442
$598$	0	0
$599$	7.47411	0.305384	0.152692	−	0.988274i	$-0.451206\pi$
$599$	7.47411	0.305384	0.152692	+	0.988274i	$0.451206\pi$
$600$	0	0
$601$	21.4700	0.875780	0.437890	−	0.899028i	$-0.355726\pi$
$601$	21.4700	0.875780	0.437890	+	0.899028i	$0.355726\pi$
$602$	0	0
$603$	−66.5260	−2.70915
$604$	0	0
$605$	3.86248	0.157032
$606$	0	0
$607$	−22.9044	−0.929660	−0.464830	−	0.885400i	$-0.653884\pi$
$607$	−22.9044	−0.929660	−0.464830	+	0.885400i	$0.653884\pi$
$608$	0	0
$609$	25.7194	1.04220
$610$	0	0
$611$	5.97028	0.241532
$612$	0	0
$613$	20.1461	0.813694	0.406847	−	0.913496i	$-0.366628\pi$
$613$	20.1461	0.813694	0.406847	+	0.913496i	$0.366628\pi$
$614$	0	0
$615$	36.7144	1.48047
$616$	0	0
$617$	−13.7844	−0.554939	−0.277470	−	0.960734i	$-0.589496\pi$
$617$	−13.7844	−0.554939	−0.277470	+	0.960734i	$0.589496\pi$
$618$	0	0
$619$	19.6655	0.790424	0.395212	−	0.918590i	$-0.370671\pi$
$619$	19.6655	0.790424	0.395212	+	0.918590i	$0.370671\pi$
$620$	0	0
$621$	82.5855	3.31404
$622$	0	0
$623$	10.6464	0.426538
$624$	0	0
$625$	−31.0766	−1.24307
$626$	0	0
$627$	7.83276	0.312810
$628$	0	0
$629$	−3.22616	−0.128635
$630$	0	0
$631$	−16.1672	−0.643608	−0.321804	−	0.946806i	$-0.604289\pi$
$631$	−16.1672	−0.643608	−0.321804	+	0.946806i	$0.604289\pi$
$632$	0	0
$633$	53.9149	2.14293
$634$	0	0
$635$	36.2439	1.43829
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−57.8555	−2.28873
$640$	0	0
$641$	−29.0036	−1.14557	−0.572786	−	0.819705i	$-0.694137\pi$
$641$	−29.0036	−1.14557	−0.572786	+	0.819705i	$0.694137\pi$
$642$	0	0
$643$	39.2233	1.54681	0.773407	−	0.633910i	$-0.218551\pi$
$643$	39.2233	1.54681	0.773407	+	0.633910i	$0.218551\pi$
$644$	0	0
$645$	59.0122	2.32360
$646$	0	0
$647$	−11.9844	−0.471156	−0.235578	−	0.971855i	$-0.575698\pi$
$647$	−11.9844	−0.471156	−0.235578	+	0.971855i	$0.575698\pi$
$648$	0	0
$649$	−13.8711	−0.544487
$650$	0	0
$651$	−4.31889	−0.169271
$652$	0	0
$653$	−45.3311	−1.77394	−0.886971	−	0.461826i	$-0.847194\pi$
$653$	−45.3311	−1.77394	−0.886971	+	0.461826i	$0.847194\pi$
$654$	0	0
$655$	25.4600	0.994804
$656$	0	0
$657$	−15.5280	−0.605805
$658$	0	0
$659$	−6.12193	−0.238477	−0.119238	−	0.992866i	$-0.538045\pi$
$659$	−6.12193	−0.238477	−0.119238	+	0.992866i	$0.538045\pi$
$660$	0	0
$661$	27.5280	1.07072	0.535358	−	0.844625i	$-0.320177\pi$
$661$	27.5280	1.07072	0.535358	+	0.844625i	$0.320177\pi$
$662$	0	0
$663$	1.62721	0.0631957
$664$	0	0
$665$	−2.28917	−0.0887701
$666$	0	0
$667$	−60.8263	−2.35520
$668$	0	0
$669$	−32.7300	−1.26541
$670$	0	0
$671$	−6.20555	−0.239563
$672$	0	0
$673$	27.9547	1.07757	0.538787	−	0.842442i	$-0.318883\pi$
$673$	27.9547	1.07757	0.538787	+	0.842442i	$0.318883\pi$
$674$	0	0
$675$	32.8222	1.26333
$676$	0	0
$677$	12.6605	0.486583	0.243291	−	0.969953i	$-0.421773\pi$
$677$	12.6605	0.486583	0.243291	+	0.969953i	$0.421773\pi$
$678$	0	0
$679$	1.18639	0.0455296
$680$	0	0
$681$	21.5678	0.826479
$682$	0	0
$683$	28.3033	1.08300	0.541498	−	0.840702i	$-0.317857\pi$
$683$	28.3033	1.08300	0.541498	+	0.840702i	$0.317857\pi$
$684$	0	0
$685$	17.6116	0.672906
$686$	0	0
$687$	65.4288	2.49626
$688$	0	0
$689$	−2.49472	−0.0950412
$690$	0	0
$691$	−12.2353	−0.465452	−0.232726	−	0.972542i	$-0.574764\pi$
$691$	−12.2353	−0.465452	−0.232726	+	0.972542i	$0.574764\pi$
$692$	0	0
$693$	20.5628	0.781114
$694$	0	0
$695$	−32.5628	−1.23518
$696$	0	0
$697$	2.20555	0.0835412
$698$	0	0
$699$	18.8761	0.713960
$700$	0	0
$701$	−51.0419	−1.92783	−0.963913	−	0.266219i	$-0.914226\pi$
$701$	−51.0419	−1.92783	−0.963913	+	0.266219i	$0.914226\pi$
$702$	0	0
$703$	−5.00502	−0.188768
$704$	0	0
$705$	52.1205	1.96297
$706$	0	0
$707$	13.1028	0.492781
$708$	0	0
$709$	−42.5910	−1.59954	−0.799770	−	0.600307i	$-0.795045\pi$
$709$	−42.5910	−1.59954	−0.799770	+	0.600307i	$0.795045\pi$
$710$	0	0
$711$	−89.7563	−3.36612
$712$	0	0
$713$	10.2141	0.382523
$714$	0	0
$715$	−8.72999	−0.326483
$716$	0	0
$717$	−44.0766	−1.64607
$718$	0	0
$719$	−42.4933	−1.58473	−0.792366	−	0.610046i	$-0.791151\pi$
$719$	−42.4933	−1.58473	−0.792366	+	0.610046i	$0.791151\pi$
$720$	0	0
$721$	−4.41110	−0.164278
$722$	0	0
$723$	26.1900	0.974015
$724$	0	0
$725$	−24.1744	−0.897814
$726$	0	0
$727$	−3.75614	−0.139307	−0.0696537	−	0.997571i	$-0.522189\pi$
$727$	−3.75614	−0.139307	−0.0696537	+	0.997571i	$0.522189\pi$
$728$	0	0
$729$	−5.09775	−0.188806
$730$	0	0
$731$	3.54505	0.131118
$732$	0	0
$733$	−45.7819	−1.69099	−0.845497	−	0.533980i	$-0.820696\pi$
$733$	−45.7819	−1.69099	−0.845497	+	0.533980i	$0.820696\pi$
$734$	0	0
$735$	−8.72999	−0.322010
$736$	0	0
$737$	31.1466	1.14730
$738$	0	0
$739$	14.0539	0.516981	0.258491	−	0.966014i	$-0.416775\pi$
$739$	14.0539	0.516981	0.258491	+	0.966014i	$0.416775\pi$
$740$	0	0
$741$	2.52444	0.0927375
$742$	0	0
$743$	4.74557	0.174098	0.0870491	−	0.996204i	$-0.472256\pi$
$743$	4.74557	0.174098	0.0870491	+	0.996204i	$0.472256\pi$
$744$	0	0
$745$	23.9844	0.878721
$746$	0	0
$747$	−109.211	−3.99581
$748$	0	0
$749$	0.578337	0.0211320
$750$	0	0
$751$	36.1008	1.31734	0.658669	−	0.752433i	$-0.271120\pi$
$751$	36.1008	1.31734	0.658669	+	0.752433i	$0.271120\pi$
$752$	0	0
$753$	74.4182	2.71195
$754$	0	0
$755$	33.7194	1.22718
$756$	0	0
$757$	1.03474	0.0376084	0.0188042	−	0.999823i	$-0.494014\pi$
$757$	1.03474	0.0376084	0.0188042	+	0.999823i	$0.494014\pi$
$758$	0	0
$759$	−70.6449	−2.56425
$760$	0	0
$761$	29.8414	1.08175	0.540874	−	0.841104i	$-0.318094\pi$
$761$	29.8414	1.08175	0.540874	+	0.841104i	$0.318094\pi$
$762$	0	0
$763$	5.57331	0.201768
$764$	0	0
$765$	9.77886	0.353556
$766$	0	0
$767$	−4.47054	−0.161422
$768$	0	0
$769$	23.6358	0.852329	0.426164	−	0.904646i	$-0.359864\pi$
$769$	23.6358	0.852329	0.426164	+	0.904646i	$0.359864\pi$
$770$	0	0
$771$	48.4394	1.74450
$772$	0	0
$773$	−13.0278	−0.468576	−0.234288	−	0.972167i	$-0.575276\pi$
$773$	−13.0278	−0.468576	−0.234288	+	0.972167i	$0.575276\pi$
$774$	0	0
$775$	4.05944	0.145819
$776$	0	0
$777$	−19.0872	−0.684749
$778$	0	0
$779$	3.42166	0.122594
$780$	0	0
$781$	27.0872	0.969256
$782$	0	0
$783$	−93.2898	−3.33391
$784$	0	0
$785$	−36.0922	−1.28819
$786$	0	0
$787$	−46.2141	−1.64736	−0.823678	−	0.567058i	$-0.808082\pi$
$787$	−46.2141	−1.64736	−0.823678	+	0.567058i	$0.808082\pi$
$788$	0	0
$789$	−47.0716	−1.67579
$790$	0	0
$791$	−5.44584	−0.193632
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	0	0
$795$	−21.7789	−0.772417
$796$	0	0
$797$	−53.1155	−1.88145	−0.940723	−	0.339176i	$-0.889852\pi$
$797$	−53.1155	−1.88145	−0.940723	+	0.339176i	$0.889852\pi$
$798$	0	0
$799$	3.13104	0.110768
$800$	0	0
$801$	−70.5558	−2.49297
$802$	0	0
$803$	7.27001	0.256553
$804$	0	0
$805$	20.6464	0.727689
$806$	0	0
$807$	−72.2721	−2.54410
$808$	0	0
$809$	−54.4635	−1.91484	−0.957418	−	0.288705i	$-0.906775\pi$
$809$	−54.4635	−1.91484	−0.957418	+	0.288705i	$0.906775\pi$
$810$	0	0
$811$	−38.0978	−1.33779	−0.668897	−	0.743356i	$-0.733233\pi$
$811$	−38.0978	−1.33779	−0.668897	+	0.743356i	$0.733233\pi$
$812$	0	0
$813$	39.5466	1.38696
$814$	0	0
$815$	−37.8711	−1.32657
$816$	0	0
$817$	5.49974	0.192412
$818$	0	0
$819$	6.62721	0.231574
$820$	0	0
$821$	2.30330	0.0803858	0.0401929	−	0.999192i	$-0.487203\pi$
$821$	2.30330	0.0803858	0.0401929	+	0.999192i	$0.487203\pi$
$822$	0	0
$823$	23.6172	0.823243	0.411621	−	0.911355i	$-0.364963\pi$
$823$	23.6172	0.823243	0.411621	+	0.911355i	$0.364963\pi$
$824$	0	0
$825$	−28.0766	−0.977503
$826$	0	0
$827$	48.1643	1.67484	0.837419	−	0.546562i	$-0.184064\pi$
$827$	48.1643	1.67484	0.837419	+	0.546562i	$0.184064\pi$
$828$	0	0
$829$	−13.0716	−0.453996	−0.226998	−	0.973895i	$-0.572891\pi$
$829$	−13.0716	−0.453996	−0.226998	+	0.973895i	$0.572891\pi$
$830$	0	0
$831$	−25.2005	−0.874197
$832$	0	0
$833$	−0.524438	−0.0181707
$834$	0	0
$835$	5.71083	0.197631
$836$	0	0
$837$	15.6655	0.541480
$838$	0	0
$839$	−17.6756	−0.610229	−0.305114	−	0.952316i	$-0.598695\pi$
$839$	−17.6756	−0.610229	−0.305114	+	0.952316i	$0.598695\pi$
$840$	0	0
$841$	39.7103	1.36932
$842$	0	0
$843$	59.0560	2.03400
$844$	0	0
$845$	−2.81361	−0.0967910
$846$	0	0
$847$	1.37279	0.0471695
$848$	0	0
$849$	−34.5855	−1.18697
$850$	0	0
$851$	45.1411	1.54742
$852$	0	0
$853$	5.48970	0.187964	0.0939818	−	0.995574i	$-0.470040\pi$
$853$	5.48970	0.187964	0.0939818	+	0.995574i	$0.470040\pi$
$854$	0	0
$855$	15.1708	0.518831
$856$	0	0
$857$	−11.0489	−0.377422	−0.188711	−	0.982033i	$-0.560431\pi$
$857$	−11.0489	−0.377422	−0.188711	+	0.982033i	$0.560431\pi$
$858$	0	0
$859$	−45.2616	−1.54430	−0.772152	−	0.635437i	$-0.780820\pi$
$859$	−45.2616	−1.54430	−0.772152	+	0.635437i	$0.780820\pi$
$860$	0	0
$861$	13.0489	0.444705
$862$	0	0
$863$	−5.90225	−0.200915	−0.100457	−	0.994941i	$-0.532031\pi$
$863$	−5.90225	−0.200915	−0.100457	+	0.994941i	$0.532031\pi$
$864$	0	0
$865$	57.1099	1.94180
$866$	0	0
$867$	−51.8938	−1.76241
$868$	0	0
$869$	42.0227	1.42552
$870$	0	0
$871$	10.0383	0.340135
$872$	0	0
$873$	−7.86248	−0.266105
$874$	0	0
$875$	−5.86248	−0.198188
$876$	0	0
$877$	4.90727	0.165707	0.0828534	−	0.996562i	$-0.473597\pi$
$877$	4.90727	0.165707	0.0828534	+	0.996562i	$0.473597\pi$
$878$	0	0
$879$	43.9844	1.48356
$880$	0	0
$881$	44.2822	1.49190	0.745952	−	0.665999i	$-0.231995\pi$
$881$	44.2822	1.49190	0.745952	+	0.665999i	$0.231995\pi$
$882$	0	0
$883$	58.8605	1.98081	0.990407	−	0.138181i	$-0.0441256\pi$
$883$	58.8605	1.98081	0.990407	+	0.138181i	$0.0441256\pi$
$884$	0	0
$885$	−39.0278	−1.31190
$886$	0	0
$887$	−10.1289	−0.340096	−0.170048	−	0.985436i	$-0.554392\pi$
$887$	−10.1289	−0.340096	−0.170048	+	0.985436i	$0.554392\pi$
$888$	0	0
$889$	12.8816	0.432036
$890$	0	0
$891$	−46.6605	−1.56319
$892$	0	0
$893$	4.85746	0.162549
$894$	0	0
$895$	−30.9597	−1.03487
$896$	0	0
$897$	−22.7683	−0.760211
$898$	0	0
$899$	−11.5381	−0.384816
$900$	0	0
$901$	−1.30833	−0.0435866
$902$	0	0
$903$	20.9739	0.697966
$904$	0	0
$905$	1.94610	0.0646906
$906$	0	0
$907$	−37.9547	−1.26026	−0.630132	−	0.776488i	$-0.716999\pi$
$907$	−37.9547	−1.26026	−0.630132	+	0.776488i	$0.716999\pi$
$908$	0	0
$909$	−86.8349	−2.88013
$910$	0	0
$911$	5.57477	0.184700	0.0923501	−	0.995727i	$-0.470562\pi$
$911$	5.57477	0.184700	0.0923501	+	0.995727i	$0.470562\pi$
$912$	0	0
$913$	51.1310	1.69219
$914$	0	0
$915$	−17.4600	−0.577209
$916$	0	0
$917$	9.04888	0.298820
$918$	0	0
$919$	15.7844	0.520679	0.260340	−	0.965517i	$-0.416165\pi$
$919$	15.7844	0.520679	0.260340	+	0.965517i	$0.416165\pi$
$920$	0	0
$921$	−42.0711	−1.38629
$922$	0	0
$923$	8.72999	0.287351
$924$	0	0
$925$	17.9406	0.589882
$926$	0	0
$927$	29.2333	0.960148
$928$	0	0
$929$	−45.2630	−1.48503	−0.742516	−	0.669829i	$-0.766368\pi$
$929$	−45.2630	−1.48503	−0.742516	+	0.669829i	$0.766368\pi$
$930$	0	0
$931$	−0.813607	−0.0266649
$932$	0	0
$933$	1.32391	0.0433429
$934$	0	0
$935$	−4.57834	−0.149728
$936$	0	0
$937$	53.6188	1.75165	0.875824	−	0.482630i	$-0.160318\pi$
$937$	53.6188	1.75165	0.875824	+	0.482630i	$0.160318\pi$
$938$	0	0
$939$	−56.3205	−1.83795
$940$	0	0
$941$	−20.7753	−0.677255	−0.338628	−	0.940920i	$-0.609963\pi$
$941$	−20.7753	−0.677255	−0.338628	+	0.940920i	$0.609963\pi$
$942$	0	0
$943$	−30.8605	−1.00496
$944$	0	0
$945$	31.6655	1.03008
$946$	0	0
$947$	10.8605	0.352919	0.176460	−	0.984308i	$-0.443535\pi$
$947$	10.8605	0.352919	0.176460	+	0.984308i	$0.443535\pi$
$948$	0	0
$949$	2.34307	0.0760592
$950$	0	0
$951$	−29.2333	−0.947955
$952$	0	0
$953$	−25.7180	−0.833087	−0.416543	−	0.909116i	$-0.636759\pi$
$953$	−25.7180	−0.833087	−0.416543	+	0.909116i	$0.636759\pi$
$954$	0	0
$955$	−22.0383	−0.713143
$956$	0	0
$957$	79.8016	2.57962
$958$	0	0
$959$	6.25945	0.202128
$960$	0	0
$961$	−29.0625	−0.937500
$962$	0	0
$963$	−3.83276	−0.123509
$964$	0	0
$965$	34.3416	1.10550
$966$	0	0
$967$	33.5038	1.07741	0.538705	−	0.842494i	$-0.318914\pi$
$967$	33.5038	1.07741	0.538705	+	0.842494i	$0.318914\pi$
$968$	0	0
$969$	1.32391	0.0425302
$970$	0	0
$971$	−2.03831	−0.0654126	−0.0327063	−	0.999465i	$-0.510413\pi$
$971$	−2.03831	−0.0654126	−0.0327063	+	0.999465i	$0.510413\pi$
$972$	0	0
$973$	−11.5733	−0.371023
$974$	0	0
$975$	−9.04888	−0.289796
$976$	0	0
$977$	15.1411	0.484406	0.242203	−	0.970226i	$-0.422130\pi$
$977$	15.1411	0.484406	0.242203	+	0.970226i	$0.422130\pi$
$978$	0	0
$979$	33.0333	1.05575
$980$	0	0
$981$	−36.9355	−1.17926
$982$	0	0
$983$	49.3124	1.57282	0.786411	−	0.617704i	$-0.211937\pi$
$983$	49.3124	1.57282	0.786411	+	0.617704i	$0.211937\pi$
$984$	0	0
$985$	−52.9583	−1.68739
$986$	0	0
$987$	18.5244	0.589639
$988$	0	0
$989$	−49.6030	−1.57728
$990$	0	0
$991$	5.43171	0.172544	0.0862720	−	0.996272i	$-0.472505\pi$
$991$	5.43171	0.172544	0.0862720	+	0.996272i	$0.472505\pi$
$992$	0	0
$993$	53.9900	1.71332
$994$	0	0
$995$	−60.8066	−1.92770
$996$	0	0
$997$	53.6061	1.69772	0.848861	−	0.528616i	$-0.177289\pi$
$997$	53.6061	1.69772	0.848861	+	0.528616i	$0.177289\pi$
$998$	0	0
$999$	69.2333	2.19044

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5824.2.a.by.1.3		3
4.3	odd	2		5824.2.a.bs.1.1		3
8.3	odd	2		1456.2.a.t.1.3		3
8.5	even	2		91.2.a.d.1.1	✓	3
24.5	odd	2		819.2.a.i.1.3		3
40.29	even	2		2275.2.a.m.1.3		3
56.5	odd	6		637.2.e.i.508.3		6
56.13	odd	2		637.2.a.j.1.1		3
56.37	even	6		637.2.e.j.508.3		6
56.45	odd	6		637.2.e.i.79.3		6
56.53	even	6		637.2.e.j.79.3		6
104.5	odd	4		1183.2.c.f.337.5		6
104.21	odd	4		1183.2.c.f.337.2		6
104.77	even	2		1183.2.a.i.1.3		3
168.125	even	2		5733.2.a.x.1.3		3
728.181	odd	2		8281.2.a.bg.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.a.d.1.1	✓	3	8.5	even	2
637.2.a.j.1.1		3	56.13	odd	2
637.2.e.i.79.3		6	56.45	odd	6
637.2.e.i.508.3		6	56.5	odd	6
637.2.e.j.79.3		6	56.53	even	6
637.2.e.j.508.3		6	56.37	even	6
819.2.a.i.1.3		3	24.5	odd	2
1183.2.a.i.1.3		3	104.77	even	2
1183.2.c.f.337.2		6	104.21	odd	4
1183.2.c.f.337.5		6	104.5	odd	4
1456.2.a.t.1.3		3	8.3	odd	2
2275.2.a.m.1.3		3	40.29	even	2
5733.2.a.x.1.3		3	168.125	even	2
5824.2.a.bs.1.1		3	4.3	odd	2
5824.2.a.by.1.3		3	1.1	even	1	trivial
8281.2.a.bg.1.3		3	728.181	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5824 = 2^{6} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5824.a (trivial)

\(f(q)\)	\(=\)	\(q+3.10278 q^{3} -2.81361 q^{5} -1.00000 q^{7} +6.62721 q^{9} -3.10278 q^{11} -1.00000 q^{13} -8.72999 q^{15} -0.524438 q^{17} -0.813607 q^{19} -3.10278 q^{21} +7.33804 q^{23} +2.91638 q^{25} +11.2544 q^{27} -8.28917 q^{29} +1.39194 q^{31} -9.62721 q^{33} +2.81361 q^{35} +6.15165 q^{37} -3.10278 q^{39} -4.20555 q^{41} -6.75971 q^{43} -18.6464 q^{45} -5.97028 q^{47} +1.00000 q^{49} -1.62721 q^{51} +2.49472 q^{53} +8.72999 q^{55} -2.52444 q^{57} +4.47054 q^{59} +2.00000 q^{61} -6.62721 q^{63} +2.81361 q^{65} -10.0383 q^{67} +22.7683 q^{69} -8.72999 q^{71} -2.34307 q^{73} +9.04888 q^{75} +3.10278 q^{77} -13.5436 q^{79} +15.0383 q^{81} -16.4791 q^{83} +1.47556 q^{85} -25.7194 q^{87} -10.6464 q^{89} +1.00000 q^{91} +4.31889 q^{93} +2.28917 q^{95} -1.18639 q^{97} -20.5628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} - 2 q^{5} - 3 q^{7} + 7 q^{9} - 2 q^{11} - 3 q^{13} - 6 q^{15} + 4 q^{17} + 4 q^{19} - 2 q^{21} + 10 q^{23} - 5 q^{25} + 8 q^{27} - 24 q^{29} - 4 q^{31} - 16 q^{33} + 2 q^{35} - 2 q^{39} + 2 q^{41}+ \cdots - 14 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 - 2 * q^5 - 3 * q^7 + 7 * q^9 - 2 * q^11 - 3 * q^13 - 6 * q^15 + 4 * q^17 + 4 * q^19 - 2 * q^21 + 10 * q^23 - 5 * q^25 + 8 * q^27 - 24 * q^29 - 4 * q^31 - 16 * q^33 + 2 * q^35 - 2 * q^39 + 2 * q^41 - 10 * q^43 - 22 * q^45 - 8 * q^47 + 3 * q^49 + 8 * q^51 - 8 * q^53 + 6 * q^55 - 2 * q^57 + 4 * q^59 + 6 * q^61 - 7 * q^63 + 2 * q^65 + 12 * q^67 + 6 * q^69 - 6 * q^71 - 10 * q^73 + 16 * q^75 + 2 * q^77 - 14 * q^79 + 3 * q^81 + 12 * q^83 + 10 * q^85 - 26 * q^87 + 2 * q^89 + 3 * q^91 + 22 * q^93 + 6 * q^95 - 10 * q^97 - 14 * q^99

Introduction

L-functions

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