LMFDB - Embedded newform 608.4.a.k.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [608,4,Mod(1,608)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(608, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("608.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 608 = 2^{5} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	608.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$35.8731612835$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 121x^{5} + 402x^{4} + 4234x^{3} - 14542x^{2} - 40996x + 141664 $ x^7 - 3x^6 - 121x^5 + 402x^4 + 4234x^3 - 14542x^2 - 40996x + 141664
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.10834$ of defining polynomial
Character	$\chi$	$=$	608.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.20712 q^{3} +7.79991 q^{5} +26.8592 q^{7} +0.114065 q^{9} +40.7763 q^{11} +24.2099 q^{13} -40.6150 q^{15} +70.2535 q^{17} -19.0000 q^{19} -139.859 q^{21} +37.2905 q^{23} -64.1614 q^{25} +139.998 q^{27} -171.838 q^{29} +64.4716 q^{31} -212.327 q^{33} +209.499 q^{35} -432.867 q^{37} -126.064 q^{39} +270.141 q^{41} +49.8344 q^{43} +0.889699 q^{45} +362.230 q^{47} +378.416 q^{49} -365.818 q^{51} -515.513 q^{53} +318.052 q^{55} +98.9352 q^{57} +71.0451 q^{59} +735.093 q^{61} +3.06370 q^{63} +188.835 q^{65} -224.369 q^{67} -194.176 q^{69} +67.6494 q^{71} -89.3246 q^{73} +334.096 q^{75} +1095.22 q^{77} +1219.09 q^{79} -732.067 q^{81} +948.434 q^{83} +547.970 q^{85} +894.780 q^{87} +877.711 q^{89} +650.258 q^{91} -335.711 q^{93} -148.198 q^{95} +812.241 q^{97} +4.65117 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{3} - 17 q^{5} + 42 q^{7} + 86 q^{9} - 33 q^{11} - 35 q^{13} + 120 q^{15} + 66 q^{17} - 133 q^{19} + 33 q^{21} + 389 q^{23} + 44 q^{25} - 39 q^{27} + 233 q^{29} + 158 q^{31} - 206 q^{33} + 123 q^{35}+ \cdots - 1591 q^{99}+O(q^{100}) $ 7 * q + 3 * q^3 - 17 * q^5 + 42 * q^7 + 86 * q^9 - 33 * q^11 - 35 * q^13 + 120 * q^15 + 66 * q^17 - 133 * q^19 + 33 * q^21 + 389 * q^23 + 44 * q^25 - 39 * q^27 + 233 * q^29 + 158 * q^31 - 206 * q^33 + 123 * q^35 - 436 * q^37 + 807 * q^39 - 94 * q^41 + 645 * q^43 + 103 * q^45 + 1451 * q^47 + 93 * q^49 + 1741 * q^51 + 3 * q^53 + 1971 * q^55 - 57 * q^57 + 297 * q^59 + 93 * q^61 + 2999 * q^63 - 788 * q^65 + 1641 * q^67 + 945 * q^69 + 2392 * q^71 + 324 * q^73 + 1909 * q^75 - 711 * q^77 + 2492 * q^79 + 143 * q^81 + 310 * q^83 + 2353 * q^85 + 4795 * q^87 - 440 * q^89 - 107 * q^91 + 900 * q^93 + 323 * q^95 - 532 * q^97 - 1591 * 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
$3$	−5.20712	−1.00211	−0.501055	−	0.865415i	$-0.667055\pi$
$3$	−5.20712	−1.00211	−0.501055	+	0.865415i	$0.667055\pi$
$4$	0	0
$5$	7.79991	0.697645	0.348822	−	0.937189i	$-0.386582\pi$
$5$	7.79991	0.697645	0.348822	+	0.937189i	$0.386582\pi$
$6$	0	0
$7$	26.8592	1.45026	0.725130	−	0.688612i	$-0.241780\pi$
$7$	26.8592	1.45026	0.725130	+	0.688612i	$0.241780\pi$
$8$	0	0
$9$	0.114065	0.00422464
$10$	0	0
$11$	40.7763	1.11768	0.558842	−	0.829274i	$-0.311246\pi$
$11$	40.7763	1.11768	0.558842	+	0.829274i	$0.311246\pi$
$12$	0	0
$13$	24.2099	0.516508	0.258254	−	0.966077i	$-0.416853\pi$
$13$	24.2099	0.516508	0.258254	+	0.966077i	$0.416853\pi$
$14$	0	0
$15$	−40.6150	−0.699117
$16$	0	0
$17$	70.2535	1.00229	0.501146	−	0.865363i	$-0.332912\pi$
$17$	70.2535	1.00229	0.501146	+	0.865363i	$0.332912\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	−139.859	−1.45332
$22$	0	0
$23$	37.2905	0.338069	0.169035	−	0.985610i	$-0.445935\pi$
$23$	37.2905	0.338069	0.169035	+	0.985610i	$0.445935\pi$
$24$	0	0
$25$	−64.1614	−0.513292
$26$	0	0
$27$	139.998	0.997877
$28$	0	0
$29$	−171.838	−1.10033	−0.550164	−	0.835057i	$-0.685435\pi$
$29$	−171.838	−1.10033	−0.550164	+	0.835057i	$0.685435\pi$
$30$	0	0
$31$	64.4716	0.373530	0.186765	−	0.982405i	$-0.440200\pi$
$31$	64.4716	0.373530	0.186765	+	0.982405i	$0.440200\pi$
$32$	0	0
$33$	−212.327	−1.12004
$34$	0	0
$35$	209.499	1.01177
$36$	0	0
$37$	−432.867	−1.92332	−0.961660	−	0.274245i	$-0.911572\pi$
$37$	−432.867	−1.92332	−0.961660	+	0.274245i	$0.911572\pi$
$38$	0	0
$39$	−126.064	−0.517598
$40$	0	0
$41$	270.141	1.02900	0.514498	−	0.857491i	$-0.327978\pi$
$41$	270.141	1.02900	0.514498	+	0.857491i	$0.327978\pi$
$42$	0	0
$43$	49.8344	0.176736	0.0883682	−	0.996088i	$-0.471835\pi$
$43$	49.8344	0.176736	0.0883682	+	0.996088i	$0.471835\pi$
$44$	0	0
$45$	0.889699	0.00294730
$46$	0	0
$47$	362.230	1.12418	0.562092	−	0.827074i	$-0.309997\pi$
$47$	362.230	1.12418	0.562092	+	0.827074i	$0.309997\pi$
$48$	0	0
$49$	378.416	1.10325
$50$	0	0
$51$	−365.818	−1.00441
$52$	0	0
$53$	−515.513	−1.33606	−0.668029	−	0.744135i	$-0.732862\pi$
$53$	−515.513	−1.33606	−0.668029	+	0.744135i	$0.732862\pi$
$54$	0	0
$55$	318.052	0.779747
$56$	0	0
$57$	98.9352	0.229900
$58$	0	0
$59$	71.0451	0.156768	0.0783838	−	0.996923i	$-0.475024\pi$
$59$	71.0451	0.156768	0.0783838	+	0.996923i	$0.475024\pi$
$60$	0	0
$61$	735.093	1.54293	0.771467	−	0.636269i	$-0.219523\pi$
$61$	735.093	1.54293	0.771467	+	0.636269i	$0.219523\pi$
$62$	0	0
$63$	3.06370	0.00612683
$64$	0	0
$65$	188.835	0.360340
$66$	0	0
$67$	−224.369	−0.409121	−0.204560	−	0.978854i	$-0.565576\pi$
$67$	−224.369	−0.409121	−0.204560	+	0.978854i	$0.565576\pi$
$68$	0	0
$69$	−194.176	−0.338783
$70$	0	0
$71$	67.6494	0.113078	0.0565388	−	0.998400i	$-0.481994\pi$
$71$	67.6494	0.113078	0.0565388	+	0.998400i	$0.481994\pi$
$72$	0	0
$73$	−89.3246	−0.143214	−0.0716072	−	0.997433i	$-0.522813\pi$
$73$	−89.3246	−0.143214	−0.0716072	+	0.997433i	$0.522813\pi$
$74$	0	0
$75$	334.096	0.514375
$76$	0	0
$77$	1095.22	1.62093
$78$	0	0
$79$	1219.09	1.73619	0.868093	−	0.496401i	$-0.165346\pi$
$79$	1219.09	1.73619	0.868093	+	0.496401i	$0.165346\pi$
$80$	0	0
$81$	−732.067	−1.00421
$82$	0	0
$83$	948.434	1.25427	0.627133	−	0.778912i	$-0.284228\pi$
$83$	948.434	1.25427	0.627133	+	0.778912i	$0.284228\pi$
$84$	0	0
$85$	547.970	0.699244
$86$	0	0
$87$	894.780	1.10265
$88$	0	0
$89$	877.711	1.04536	0.522681	−	0.852528i	$-0.324932\pi$
$89$	877.711	1.04536	0.522681	+	0.852528i	$0.324932\pi$
$90$	0	0
$91$	650.258	0.749072
$92$	0	0
$93$	−335.711	−0.374318
$94$	0	0
$95$	−148.198	−0.160051
$96$	0	0
$97$	812.241	0.850212	0.425106	−	0.905143i	$-0.360237\pi$
$97$	812.241	0.850212	0.425106	+	0.905143i	$0.360237\pi$
$98$	0	0
$99$	4.65117	0.00472182
$100$	0	0
$101$	−348.563	−0.343400	−0.171700	−	0.985149i	$-0.554926\pi$
$101$	−348.563	−0.343400	−0.171700	+	0.985149i	$0.554926\pi$
$102$	0	0
$103$	−44.7869	−0.0428445	−0.0214223	−	0.999771i	$-0.506819\pi$
$103$	−44.7869	−0.0428445	−0.0214223	+	0.999771i	$0.506819\pi$
$104$	0	0
$105$	−1090.89	−1.01390
$106$	0	0
$107$	−1433.73	−1.29537	−0.647683	−	0.761910i	$-0.724262\pi$
$107$	−1433.73	−1.29537	−0.647683	+	0.761910i	$0.724262\pi$
$108$	0	0
$109$	1772.92	1.55794	0.778968	−	0.627063i	$-0.215743\pi$
$109$	1772.92	1.55794	0.778968	+	0.627063i	$0.215743\pi$
$110$	0	0
$111$	2253.99	1.92738
$112$	0	0
$113$	305.042	0.253947	0.126973	−	0.991906i	$-0.459474\pi$
$113$	305.042	0.253947	0.126973	+	0.991906i	$0.459474\pi$
$114$	0	0
$115$	290.862	0.235852
$116$	0	0
$117$	2.76151	0.00218206
$118$	0	0
$119$	1886.95	1.45358
$120$	0	0
$121$	331.709	0.249218
$122$	0	0
$123$	−1406.65	−1.03117
$124$	0	0
$125$	−1475.44	−1.05574
$126$	0	0
$127$	345.706	0.241547	0.120774	−	0.992680i	$-0.461463\pi$
$127$	345.706	0.241547	0.120774	+	0.992680i	$0.461463\pi$
$128$	0	0
$129$	−259.493	−0.177109
$130$	0	0
$131$	2607.14	1.73883	0.869415	−	0.494083i	$-0.164496\pi$
$131$	2607.14	1.73883	0.869415	+	0.494083i	$0.164496\pi$
$132$	0	0
$133$	−510.325	−0.332713
$134$	0	0
$135$	1091.97	0.696164
$136$	0	0
$137$	702.056	0.437815	0.218908	−	0.975746i	$-0.429751\pi$
$137$	702.056	0.437815	0.218908	+	0.975746i	$0.429751\pi$
$138$	0	0
$139$	−2535.33	−1.54708	−0.773538	−	0.633750i	$-0.781515\pi$
$139$	−2535.33	−1.54708	−0.773538	+	0.633750i	$0.781515\pi$
$140$	0	0
$141$	−1886.17	−1.12656
$142$	0	0
$143$	987.190	0.577293
$144$	0	0
$145$	−1340.32	−0.767638
$146$	0	0
$147$	−1970.46	−1.10558
$148$	0	0
$149$	−2056.85	−1.13090	−0.565448	−	0.824784i	$-0.691297\pi$
$149$	−2056.85	−1.13090	−0.565448	+	0.824784i	$0.691297\pi$
$150$	0	0
$151$	532.903	0.287199	0.143599	−	0.989636i	$-0.454132\pi$
$151$	532.903	0.287199	0.143599	+	0.989636i	$0.454132\pi$
$152$	0	0
$153$	8.01348	0.00423433
$154$	0	0
$155$	502.872	0.260591
$156$	0	0
$157$	2554.89	1.29874	0.649370	−	0.760472i	$-0.275032\pi$
$157$	2554.89	1.29874	0.649370	+	0.760472i	$0.275032\pi$
$158$	0	0
$159$	2684.33	1.33888
$160$	0	0
$161$	1001.59	0.490289
$162$	0	0
$163$	−894.659	−0.429909	−0.214954	−	0.976624i	$-0.568960\pi$
$163$	−894.659	−0.429909	−0.214954	+	0.976624i	$0.568960\pi$
$164$	0	0
$165$	−1656.13	−0.781392
$166$	0	0
$167$	−2895.98	−1.34190	−0.670951	−	0.741502i	$-0.734114\pi$
$167$	−2895.98	−1.34190	−0.670951	+	0.741502i	$0.734114\pi$
$168$	0	0
$169$	−1610.88	−0.733219
$170$	0	0
$171$	−2.16724	−0.000969199 0
$172$	0	0
$173$	−927.867	−0.407771	−0.203886	−	0.978995i	$-0.565357\pi$
$173$	−927.867	−0.407771	−0.203886	+	0.978995i	$0.565357\pi$
$174$	0	0
$175$	−1723.32	−0.744406
$176$	0	0
$177$	−369.940	−0.157098
$178$	0	0
$179$	−3219.75	−1.34444	−0.672222	−	0.740350i	$-0.734660\pi$
$179$	−3219.75	−1.34444	−0.672222	+	0.740350i	$0.734660\pi$
$180$	0	0
$181$	823.223	0.338064	0.169032	−	0.985611i	$-0.445936\pi$
$181$	823.223	0.338064	0.169032	+	0.985611i	$0.445936\pi$
$182$	0	0
$183$	−3827.71	−1.54619
$184$	0	0
$185$	−3376.32	−1.34179
$186$	0	0
$187$	2864.68	1.12025
$188$	0	0
$189$	3760.24	1.44718
$190$	0	0
$191$	2292.84	0.868607	0.434304	−	0.900767i	$-0.356995\pi$
$191$	2292.84	0.868607	0.434304	+	0.900767i	$0.356995\pi$
$192$	0	0
$193$	910.072	0.339422	0.169711	−	0.985494i	$-0.445717\pi$
$193$	910.072	0.339422	0.169711	+	0.985494i	$0.445717\pi$
$194$	0	0
$195$	−983.285	−0.361100
$196$	0	0
$197$	784.134	0.283590	0.141795	−	0.989896i	$-0.454713\pi$
$197$	784.134	0.283590	0.141795	+	0.989896i	$0.454713\pi$
$198$	0	0
$199$	2725.37	0.970837	0.485419	−	0.874282i	$-0.338667\pi$
$199$	2725.37	0.970837	0.485419	+	0.874282i	$0.338667\pi$
$200$	0	0
$201$	1168.32	0.409984
$202$	0	0
$203$	−4615.43	−1.59576
$204$	0	0
$205$	2107.07	0.717875
$206$	0	0
$207$	4.25355	0.00142822
$208$	0	0
$209$	−774.750	−0.256414
$210$	0	0
$211$	193.856	0.0632493	0.0316247	−	0.999500i	$-0.489932\pi$
$211$	193.856	0.0632493	0.0316247	+	0.999500i	$0.489932\pi$
$212$	0	0
$213$	−352.258	−0.113316
$214$	0	0
$215$	388.703	0.123299
$216$	0	0
$217$	1731.65	0.541716
$218$	0	0
$219$	465.123	0.143517
$220$	0	0
$221$	1700.83	0.517692
$222$	0	0
$223$	−271.131	−0.0814184	−0.0407092	−	0.999171i	$-0.512962\pi$
$223$	−271.131	−0.0814184	−0.0407092	+	0.999171i	$0.512962\pi$
$224$	0	0
$225$	−7.31860	−0.00216847
$226$	0	0
$227$	−1282.75	−0.375061	−0.187531	−	0.982259i	$-0.560048\pi$
$227$	−1282.75	−0.375061	−0.187531	+	0.982259i	$0.560048\pi$
$228$	0	0
$229$	1038.29	0.299618	0.149809	−	0.988715i	$-0.452134\pi$
$229$	1038.29	0.299618	0.149809	+	0.988715i	$0.452134\pi$
$230$	0	0
$231$	−5702.94	−1.62435
$232$	0	0
$233$	−4460.70	−1.25421	−0.627103	−	0.778936i	$-0.715760\pi$
$233$	−4460.70	−1.25421	−0.627103	+	0.778936i	$0.715760\pi$
$234$	0	0
$235$	2825.36	0.784282
$236$	0	0
$237$	−6347.96	−1.73985
$238$	0	0
$239$	2616.79	0.708227	0.354114	−	0.935202i	$-0.384783\pi$
$239$	2616.79	0.708227	0.354114	+	0.935202i	$0.384783\pi$
$240$	0	0
$241$	−4926.25	−1.31671	−0.658356	−	0.752707i	$-0.728748\pi$
$241$	−4926.25	−1.31671	−0.658356	+	0.752707i	$0.728748\pi$
$242$	0	0
$243$	32.0056	0.00844923
$244$	0	0
$245$	2951.61	0.769680
$246$	0	0
$247$	−459.988	−0.118495
$248$	0	0
$249$	−4938.60	−1.25691
$250$	0	0
$251$	−2239.77	−0.563239	−0.281620	−	0.959526i	$-0.590872\pi$
$251$	−2239.77	−0.563239	−0.281620	+	0.959526i	$0.590872\pi$
$252$	0	0
$253$	1520.57	0.377855
$254$	0	0
$255$	−2853.35	−0.700719
$256$	0	0
$257$	8191.14	1.98813	0.994065	−	0.108790i	$-0.0346975\pi$
$257$	8191.14	1.98813	0.994065	+	0.108790i	$0.0346975\pi$
$258$	0	0
$259$	−11626.4	−2.78931
$260$	0	0
$261$	−19.6007	−0.00464849
$262$	0	0
$263$	6874.34	1.61175	0.805875	−	0.592086i	$-0.201696\pi$
$263$	6874.34	1.61175	0.805875	+	0.592086i	$0.201696\pi$
$264$	0	0
$265$	−4020.95	−0.932095
$266$	0	0
$267$	−4570.34	−1.04757
$268$	0	0
$269$	−578.895	−0.131211	−0.0656057	−	0.997846i	$-0.520898\pi$
$269$	−578.895	−0.131211	−0.0656057	+	0.997846i	$0.520898\pi$
$270$	0	0
$271$	−5685.01	−1.27432	−0.637158	−	0.770733i	$-0.719890\pi$
$271$	−5685.01	−1.27432	−0.637158	+	0.770733i	$0.719890\pi$
$272$	0	0
$273$	−3385.97	−0.750652
$274$	0	0
$275$	−2616.27	−0.573698
$276$	0	0
$277$	6612.18	1.43425	0.717125	−	0.696944i	$-0.245457\pi$
$277$	6612.18	1.43425	0.717125	+	0.696944i	$0.245457\pi$
$278$	0	0
$279$	7.35397	0.00157803
$280$	0	0
$281$	2678.07	0.568542	0.284271	−	0.958744i	$-0.408248\pi$
$281$	2678.07	0.568542	0.284271	+	0.958744i	$0.408248\pi$
$282$	0	0
$283$	5219.03	1.09625	0.548126	−	0.836396i	$-0.315341\pi$
$283$	5219.03	1.09625	0.548126	+	0.836396i	$0.315341\pi$
$284$	0	0
$285$	771.686	0.160388
$286$	0	0
$287$	7255.76	1.49231
$288$	0	0
$289$	22.5476	0.00458938
$290$	0	0
$291$	−4229.44	−0.852006
$292$	0	0
$293$	−6337.23	−1.26357	−0.631784	−	0.775145i	$-0.717677\pi$
$293$	−6337.23	−1.26357	−0.631784	+	0.775145i	$0.717677\pi$
$294$	0	0
$295$	554.145	0.109368
$296$	0	0
$297$	5708.61	1.11531
$298$	0	0
$299$	902.797	0.174616
$300$	0	0
$301$	1338.51	0.256314
$302$	0	0
$303$	1815.01	0.344124
$304$	0	0
$305$	5733.66	1.07642
$306$	0	0
$307$	7659.20	1.42389	0.711944	−	0.702236i	$-0.247815\pi$
$307$	7659.20	1.42389	0.711944	+	0.702236i	$0.247815\pi$
$308$	0	0
$309$	233.211	0.0429349
$310$	0	0
$311$	−5374.55	−0.979945	−0.489973	−	0.871738i	$-0.662993\pi$
$311$	−5374.55	−0.979945	−0.489973	+	0.871738i	$0.662993\pi$
$312$	0	0
$313$	−2164.94	−0.390957	−0.195479	−	0.980708i	$-0.562626\pi$
$313$	−2164.94	−0.390957	−0.195479	+	0.980708i	$0.562626\pi$
$314$	0	0
$315$	23.8966	0.00427435
$316$	0	0
$317$	4072.50	0.721559	0.360779	−	0.932651i	$-0.382511\pi$
$317$	4072.50	0.721559	0.360779	+	0.932651i	$0.382511\pi$
$318$	0	0
$319$	−7006.92	−1.22982
$320$	0	0
$321$	7465.61	1.29810
$322$	0	0
$323$	−1334.82	−0.229942
$324$	0	0
$325$	−1553.34	−0.265119
$326$	0	0
$327$	−9231.81	−1.56122
$328$	0	0
$329$	9729.21	1.63036
$330$	0	0
$331$	−6388.27	−1.06082	−0.530410	−	0.847742i	$-0.677962\pi$
$331$	−6388.27	−1.06082	−0.530410	+	0.847742i	$0.677962\pi$
$332$	0	0
$333$	−49.3751	−0.00812534
$334$	0	0
$335$	−1750.06	−0.285421
$336$	0	0
$337$	9593.76	1.55076	0.775379	−	0.631496i	$-0.217559\pi$
$337$	9593.76	1.55076	0.775379	+	0.631496i	$0.217559\pi$
$338$	0	0
$339$	−1588.39	−0.254482
$340$	0	0
$341$	2628.91	0.417489
$342$	0	0
$343$	951.257	0.149746
$344$	0	0
$345$	−1514.55	−0.236350
$346$	0	0
$347$	−8456.74	−1.30830	−0.654152	−	0.756363i	$-0.726974\pi$
$347$	−8456.74	−1.30830	−0.654152	+	0.756363i	$0.726974\pi$
$348$	0	0
$349$	−5235.83	−0.803059	−0.401530	−	0.915846i	$-0.631521\pi$
$349$	−5235.83	−0.803059	−0.401530	+	0.915846i	$0.631521\pi$
$350$	0	0
$351$	3389.34	0.515412
$352$	0	0
$353$	4943.57	0.745381	0.372690	−	0.927956i	$-0.378435\pi$
$353$	4943.57	0.745381	0.372690	+	0.927956i	$0.378435\pi$
$354$	0	0
$355$	527.659	0.0788880
$356$	0	0
$357$	−9825.58	−1.45665
$358$	0	0
$359$	−6634.24	−0.975325	−0.487663	−	0.873032i	$-0.662150\pi$
$359$	−6634.24	−0.975325	−0.487663	+	0.873032i	$0.662150\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−1727.25	−0.249744
$364$	0	0
$365$	−696.723	−0.0999128
$366$	0	0
$367$	8379.54	1.19185	0.595924	−	0.803041i	$-0.296786\pi$
$367$	8379.54	1.19185	0.595924	+	0.803041i	$0.296786\pi$
$368$	0	0
$369$	30.8137	0.00434714
$370$	0	0
$371$	−13846.3	−1.93763
$372$	0	0
$373$	−7995.03	−1.10983	−0.554915	−	0.831907i	$-0.687249\pi$
$373$	−7995.03	−1.10983	−0.554915	+	0.831907i	$0.687249\pi$
$374$	0	0
$375$	7682.80	1.05797
$376$	0	0
$377$	−4160.17	−0.568328
$378$	0	0
$379$	10776.2	1.46052	0.730258	−	0.683172i	$-0.239400\pi$
$379$	10776.2	1.46052	0.730258	+	0.683172i	$0.239400\pi$
$380$	0	0
$381$	−1800.13	−0.242057
$382$	0	0
$383$	−5932.82	−0.791522	−0.395761	−	0.918354i	$-0.629519\pi$
$383$	−5932.82	−0.791522	−0.395761	+	0.918354i	$0.629519\pi$
$384$	0	0
$385$	8542.61	1.13084
$386$	0	0
$387$	5.68437	0.000746648 0
$388$	0	0
$389$	2543.56	0.331527	0.165763	−	0.986166i	$-0.446991\pi$
$389$	2543.56	0.331527	0.165763	+	0.986166i	$0.446991\pi$
$390$	0	0
$391$	2619.78	0.338844
$392$	0	0
$393$	−13575.7	−1.74250
$394$	0	0
$395$	9508.82	1.21124
$396$	0	0
$397$	−10082.6	−1.27464	−0.637322	−	0.770598i	$-0.719958\pi$
$397$	−10082.6	−1.27464	−0.637322	+	0.770598i	$0.719958\pi$
$398$	0	0
$399$	2657.32	0.333415
$400$	0	0
$401$	−5602.92	−0.697746	−0.348873	−	0.937170i	$-0.613436\pi$
$401$	−5602.92	−0.697746	−0.348873	+	0.937170i	$0.613436\pi$
$402$	0	0
$403$	1560.85	0.192931
$404$	0	0
$405$	−5710.05	−0.700580
$406$	0	0
$407$	−17650.7	−2.14966
$408$	0	0
$409$	−14919.9	−1.80376	−0.901882	−	0.431983i	$-0.857814\pi$
$409$	−14919.9	−1.80376	−0.901882	+	0.431983i	$0.857814\pi$
$410$	0	0
$411$	−3655.69	−0.438739
$412$	0	0
$413$	1908.22	0.227354
$414$	0	0
$415$	7397.69	0.875033
$416$	0	0
$417$	13201.7	1.55034
$418$	0	0
$419$	3609.85	0.420890	0.210445	−	0.977606i	$-0.432509\pi$
$419$	3609.85	0.420890	0.210445	+	0.977606i	$0.432509\pi$
$420$	0	0
$421$	−1514.54	−0.175331	−0.0876654	−	0.996150i	$-0.527941\pi$
$421$	−1514.54	−0.175331	−0.0876654	+	0.996150i	$0.527941\pi$
$422$	0	0
$423$	41.3179	0.00474928
$424$	0	0
$425$	−4507.56	−0.514468
$426$	0	0
$427$	19744.0	2.23766
$428$	0	0
$429$	−5140.41	−0.578512
$430$	0	0
$431$	−4215.77	−0.471152	−0.235576	−	0.971856i	$-0.575698\pi$
$431$	−4215.77	−0.471152	−0.235576	+	0.971856i	$0.575698\pi$
$432$	0	0
$433$	−10879.5	−1.20747	−0.603737	−	0.797183i	$-0.706322\pi$
$433$	−10879.5	−1.20747	−0.603737	+	0.797183i	$0.706322\pi$
$434$	0	0
$435$	6979.20	0.769257
$436$	0	0
$437$	−708.519	−0.0775584
$438$	0	0
$439$	4973.00	0.540657	0.270328	−	0.962768i	$-0.412868\pi$
$439$	4973.00	0.540657	0.270328	+	0.962768i	$0.412868\pi$
$440$	0	0
$441$	43.1642	0.00466086
$442$	0	0
$443$	6349.20	0.680947	0.340474	−	0.940254i	$-0.389413\pi$
$443$	6349.20	0.680947	0.340474	+	0.940254i	$0.389413\pi$
$444$	0	0
$445$	6846.07	0.729291
$446$	0	0
$447$	10710.2	1.13328
$448$	0	0
$449$	−12179.4	−1.28014	−0.640070	−	0.768316i	$-0.721095\pi$
$449$	−12179.4	−1.28014	−0.640070	+	0.768316i	$0.721095\pi$
$450$	0	0
$451$	11015.3	1.15009
$452$	0	0
$453$	−2774.89	−0.287805
$454$	0	0
$455$	5071.95	0.522586
$456$	0	0
$457$	12539.5	1.28353	0.641767	−	0.766900i	$-0.278202\pi$
$457$	12539.5	1.28353	0.641767	+	0.766900i	$0.278202\pi$
$458$	0	0
$459$	9835.36	1.00016
$460$	0	0
$461$	9809.95	0.991095	0.495548	−	0.868581i	$-0.334967\pi$
$461$	9809.95	0.991095	0.495548	+	0.868581i	$0.334967\pi$
$462$	0	0
$463$	−10361.5	−1.04004	−0.520022	−	0.854153i	$-0.674076\pi$
$463$	−10361.5	−1.04004	−0.520022	+	0.854153i	$0.674076\pi$
$464$	0	0
$465$	−2618.51	−0.261141
$466$	0	0
$467$	−14250.6	−1.41208	−0.706038	−	0.708174i	$-0.749519\pi$
$467$	−14250.6	−1.41208	−0.706038	+	0.708174i	$0.749519\pi$
$468$	0	0
$469$	−6026.38	−0.593331
$470$	0	0
$471$	−13303.6	−1.30148
$472$	0	0
$473$	2032.06	0.197536
$474$	0	0
$475$	1219.07	0.117757
$476$	0	0
$477$	−58.8021	−0.00564437
$478$	0	0
$479$	−9231.75	−0.880604	−0.440302	−	0.897850i	$-0.645129\pi$
$479$	−9231.75	−0.880604	−0.440302	+	0.897850i	$0.645129\pi$
$480$	0	0
$481$	−10479.6	−0.993411
$482$	0	0
$483$	−5215.40	−0.491323
$484$	0	0
$485$	6335.41	0.593146
$486$	0	0
$487$	12264.5	1.14119	0.570595	−	0.821231i	$-0.306713\pi$
$487$	12264.5	1.14119	0.570595	+	0.821231i	$0.306713\pi$
$488$	0	0
$489$	4658.59	0.430816
$490$	0	0
$491$	7462.60	0.685911	0.342956	−	0.939352i	$-0.388572\pi$
$491$	7462.60	0.685911	0.342956	+	0.939352i	$0.388572\pi$
$492$	0	0
$493$	−12072.2	−1.10285
$494$	0	0
$495$	36.2787	0.00329415
$496$	0	0
$497$	1817.01	0.163992
$498$	0	0
$499$	6515.51	0.584517	0.292259	−	0.956339i	$-0.405593\pi$
$499$	6515.51	0.584517	0.292259	+	0.956339i	$0.405593\pi$
$500$	0	0
$501$	15079.7	1.34473
$502$	0	0
$503$	19970.3	1.77024	0.885119	−	0.465365i	$-0.154077\pi$
$503$	19970.3	1.77024	0.885119	+	0.465365i	$0.154077\pi$
$504$	0	0
$505$	−2718.76	−0.239571
$506$	0	0
$507$	8388.05	0.734766
$508$	0	0
$509$	12882.7	1.12184	0.560921	−	0.827869i	$-0.310447\pi$
$509$	12882.7	1.12184	0.560921	+	0.827869i	$0.310447\pi$
$510$	0	0
$511$	−2399.19	−0.207698
$512$	0	0
$513$	−2659.97	−0.228929
$514$	0	0
$515$	−349.334	−0.0298903
$516$	0	0
$517$	14770.4	1.25648
$518$	0	0
$519$	4831.51	0.408632
$520$	0	0
$521$	−9248.25	−0.777684	−0.388842	−	0.921304i	$-0.627125\pi$
$521$	−9248.25	−0.777684	−0.388842	+	0.921304i	$0.627125\pi$
$522$	0	0
$523$	12189.8	1.01916	0.509582	−	0.860422i	$-0.329800\pi$
$523$	12189.8	1.01916	0.509582	+	0.860422i	$0.329800\pi$
$524$	0	0
$525$	8973.55	0.745977
$526$	0	0
$527$	4529.35	0.374386
$528$	0	0
$529$	−10776.4	−0.885709
$530$	0	0
$531$	8.10379	0.000662287 0
$532$	0	0
$533$	6540.07	0.531486
$534$	0	0
$535$	−11183.0	−0.903705
$536$	0	0
$537$	16765.6	1.34728
$538$	0	0
$539$	15430.4	1.23309
$540$	0	0
$541$	11873.6	0.943597	0.471798	−	0.881706i	$-0.343605\pi$
$541$	11873.6	0.943597	0.471798	+	0.881706i	$0.343605\pi$
$542$	0	0
$543$	−4286.62	−0.338778
$544$	0	0
$545$	13828.6	1.08689
$546$	0	0
$547$	14476.6	1.13158	0.565791	−	0.824549i	$-0.308571\pi$
$547$	14476.6	1.13158	0.565791	+	0.824549i	$0.308571\pi$
$548$	0	0
$549$	83.8486	0.00651834
$550$	0	0
$551$	3264.92	0.252432
$552$	0	0
$553$	32743.9	2.51792
$554$	0	0
$555$	17580.9	1.34463
$556$	0	0
$557$	−8641.98	−0.657401	−0.328701	−	0.944434i	$-0.606611\pi$
$557$	−8641.98	−0.657401	−0.328701	+	0.944434i	$0.606611\pi$
$558$	0	0
$559$	1206.48	0.0912859
$560$	0	0
$561$	−14916.7	−1.12261
$562$	0	0
$563$	−10421.6	−0.780136	−0.390068	−	0.920786i	$-0.627549\pi$
$563$	−10421.6	−0.780136	−0.390068	+	0.920786i	$0.627549\pi$
$564$	0	0
$565$	2379.30	0.177164
$566$	0	0
$567$	−19662.7	−1.45636
$568$	0	0
$569$	−19122.4	−1.40888	−0.704440	−	0.709764i	$-0.748802\pi$
$569$	−19122.4	−1.40888	−0.704440	+	0.709764i	$0.748802\pi$
$570$	0	0
$571$	22719.0	1.66508	0.832538	−	0.553968i	$-0.186887\pi$
$571$	22719.0	1.66508	0.832538	+	0.553968i	$0.186887\pi$
$572$	0	0
$573$	−11939.1	−0.870440
$574$	0	0
$575$	−2392.61	−0.173528
$576$	0	0
$577$	−14566.0	−1.05094	−0.525468	−	0.850813i	$-0.676110\pi$
$577$	−14566.0	−1.05094	−0.525468	+	0.850813i	$0.676110\pi$
$578$	0	0
$579$	−4738.85	−0.340138
$580$	0	0
$581$	25474.2	1.81901
$582$	0	0
$583$	−21020.7	−1.49329
$584$	0	0
$585$	21.5395	0.00152231
$586$	0	0
$587$	1884.52	0.132509	0.0662544	−	0.997803i	$-0.478895\pi$
$587$	1884.52	0.132509	0.0662544	+	0.997803i	$0.478895\pi$
$588$	0	0
$589$	−1224.96	−0.0856937
$590$	0	0
$591$	−4083.08	−0.284189
$592$	0	0
$593$	−16442.1	−1.13861	−0.569305	−	0.822126i	$-0.692788\pi$
$593$	−16442.1	−1.13861	−0.569305	+	0.822126i	$0.692788\pi$
$594$	0	0
$595$	14718.0	1.01409
$596$	0	0
$597$	−14191.3	−0.972886
$598$	0	0
$599$	−22081.2	−1.50620	−0.753101	−	0.657905i	$-0.771443\pi$
$599$	−22081.2	−1.50620	−0.753101	+	0.657905i	$0.771443\pi$
$600$	0	0
$601$	22054.4	1.49687	0.748434	−	0.663209i	$-0.230806\pi$
$601$	22054.4	1.49687	0.748434	+	0.663209i	$0.230806\pi$
$602$	0	0
$603$	−25.5928	−0.00172839
$604$	0	0
$605$	2587.30	0.173866
$606$	0	0
$607$	26705.7	1.78575	0.892876	−	0.450302i	$-0.148684\pi$
$607$	26705.7	1.78575	0.892876	+	0.450302i	$0.148684\pi$
$608$	0	0
$609$	24033.1	1.59913
$610$	0	0
$611$	8769.54	0.580651
$612$	0	0
$613$	−4759.75	−0.313613	−0.156806	−	0.987629i	$-0.550120\pi$
$613$	−4759.75	−0.313613	−0.156806	+	0.987629i	$0.550120\pi$
$614$	0	0
$615$	−10971.8	−0.719389
$616$	0	0
$617$	−360.734	−0.0235375	−0.0117687	−	0.999931i	$-0.503746\pi$
$617$	−360.734	−0.0235375	−0.0117687	+	0.999931i	$0.503746\pi$
$618$	0	0
$619$	−14562.1	−0.945560	−0.472780	−	0.881180i	$-0.656750\pi$
$619$	−14562.1	−0.945560	−0.472780	+	0.881180i	$0.656750\pi$
$620$	0	0
$621$	5220.60	0.337352
$622$	0	0
$623$	23574.6	1.51605
$624$	0	0
$625$	−3488.13	−0.223240
$626$	0	0
$627$	4034.22	0.256955
$628$	0	0
$629$	−30410.4	−1.92773
$630$	0	0
$631$	328.881	0.0207489	0.0103744	−	0.999946i	$-0.496698\pi$
$631$	328.881	0.0207489	0.0103744	+	0.999946i	$0.496698\pi$
$632$	0	0
$633$	−1009.43	−0.0633828
$634$	0	0
$635$	2696.48	0.168514
$636$	0	0
$637$	9161.41	0.569840
$638$	0	0
$639$	7.71646	0.000477712 0
$640$	0	0
$641$	−7278.40	−0.448486	−0.224243	−	0.974533i	$-0.571991\pi$
$641$	−7278.40	−0.448486	−0.224243	+	0.974533i	$0.571991\pi$
$642$	0	0
$643$	−6629.96	−0.406625	−0.203313	−	0.979114i	$-0.565171\pi$
$643$	−6629.96	−0.406625	−0.203313	+	0.979114i	$0.565171\pi$
$644$	0	0
$645$	−2024.02	−0.123559
$646$	0	0
$647$	−17987.5	−1.09298	−0.546492	−	0.837464i	$-0.684037\pi$
$647$	−17987.5	−1.09298	−0.546492	+	0.837464i	$0.684037\pi$
$648$	0	0
$649$	2896.96	0.175217
$650$	0	0
$651$	−9016.92	−0.542859
$652$	0	0
$653$	−8638.65	−0.517698	−0.258849	−	0.965918i	$-0.583343\pi$
$653$	−8638.65	−0.517698	−0.258849	+	0.965918i	$0.583343\pi$
$654$	0	0
$655$	20335.4	1.21309
$656$	0	0
$657$	−10.1888	−0.000605029 0
$658$	0	0
$659$	−18722.0	−1.10669	−0.553343	−	0.832953i	$-0.686648\pi$
$659$	−18722.0	−1.10669	−0.553343	+	0.832953i	$0.686648\pi$
$660$	0	0
$661$	6277.59	0.369395	0.184697	−	0.982795i	$-0.440870\pi$
$661$	6277.59	0.369395	0.184697	+	0.982795i	$0.440870\pi$
$662$	0	0
$663$	−8856.41	−0.518785
$664$	0	0
$665$	−3980.49	−0.232115
$666$	0	0
$667$	−6407.91	−0.371987
$668$	0	0
$669$	1411.81	0.0815902
$670$	0	0
$671$	29974.4	1.72451
$672$	0	0
$673$	−14862.3	−0.851265	−0.425632	−	0.904896i	$-0.639948\pi$
$673$	−14862.3	−0.851265	−0.425632	+	0.904896i	$0.639948\pi$
$674$	0	0
$675$	−8982.49	−0.512202
$676$	0	0
$677$	5571.65	0.316301	0.158151	−	0.987415i	$-0.449447\pi$
$677$	5571.65	0.316301	0.158151	+	0.987415i	$0.449447\pi$
$678$	0	0
$679$	21816.1	1.23303
$680$	0	0
$681$	6679.41	0.375853
$682$	0	0
$683$	−29053.9	−1.62770	−0.813849	−	0.581077i	$-0.802632\pi$
$683$	−29053.9	−1.62770	−0.813849	+	0.581077i	$0.802632\pi$
$684$	0	0
$685$	5475.97	0.305440
$686$	0	0
$687$	−5406.52	−0.300250
$688$	0	0
$689$	−12480.5	−0.690086
$690$	0	0
$691$	27785.8	1.52970	0.764849	−	0.644210i	$-0.222814\pi$
$691$	27785.8	1.52970	0.764849	+	0.644210i	$0.222814\pi$
$692$	0	0
$693$	124.927	0.00684786
$694$	0	0
$695$	−19775.3	−1.07931
$696$	0	0
$697$	18978.3	1.03136
$698$	0	0
$699$	23227.4	1.25685
$700$	0	0
$701$	6058.01	0.326402	0.163201	−	0.986593i	$-0.447818\pi$
$701$	6058.01	0.326402	0.163201	+	0.986593i	$0.447818\pi$
$702$	0	0
$703$	8224.46	0.441240
$704$	0	0
$705$	−14712.0	−0.785937
$706$	0	0
$707$	−9362.14	−0.498019
$708$	0	0
$709$	30416.6	1.61117	0.805586	−	0.592479i	$-0.201851\pi$
$709$	30416.6	1.61117	0.805586	+	0.592479i	$0.201851\pi$
$710$	0	0
$711$	139.056	0.00733477
$712$	0	0
$713$	2404.17	0.126279
$714$	0	0
$715$	7699.99	0.402746
$716$	0	0
$717$	−13626.0	−0.709722
$718$	0	0
$719$	−7483.56	−0.388164	−0.194082	−	0.980985i	$-0.562173\pi$
$719$	−7483.56	−0.388164	−0.194082	+	0.980985i	$0.562173\pi$
$720$	0	0
$721$	−1202.94	−0.0621357
$722$	0	0
$723$	25651.5	1.31949
$724$	0	0
$725$	11025.4	0.564789
$726$	0	0
$727$	7721.59	0.393918	0.196959	−	0.980412i	$-0.436893\pi$
$727$	7721.59	0.393918	0.196959	+	0.980412i	$0.436893\pi$
$728$	0	0
$729$	19599.1	0.995740
$730$	0	0
$731$	3501.04	0.177142
$732$	0	0
$733$	−4540.36	−0.228788	−0.114394	−	0.993435i	$-0.536493\pi$
$733$	−4540.36	−0.228788	−0.114394	+	0.993435i	$0.536493\pi$
$734$	0	0
$735$	−15369.4	−0.771304
$736$	0	0
$737$	−9148.96	−0.457268
$738$	0	0
$739$	−36117.7	−1.79785	−0.898925	−	0.438103i	$-0.855651\pi$
$739$	−36117.7	−1.79785	−0.898925	+	0.438103i	$0.855651\pi$
$740$	0	0
$741$	2395.21	0.118745
$742$	0	0
$743$	−14158.2	−0.699079	−0.349539	−	0.936922i	$-0.613662\pi$
$743$	−14158.2	−0.699079	−0.349539	+	0.936922i	$0.613662\pi$
$744$	0	0
$745$	−16043.2	−0.788964
$746$	0	0
$747$	108.183	0.00529883
$748$	0	0
$749$	−38508.9	−1.87862
$750$	0	0
$751$	−8166.40	−0.396799	−0.198400	−	0.980121i	$-0.563574\pi$
$751$	−8166.40	−0.396799	−0.198400	+	0.980121i	$0.563574\pi$
$752$	0	0
$753$	11662.7	0.564428
$754$	0	0
$755$	4156.60	0.200363
$756$	0	0
$757$	−25616.1	−1.22990	−0.614948	−	0.788567i	$-0.710823\pi$
$757$	−25616.1	−1.22990	−0.614948	+	0.788567i	$0.710823\pi$
$758$	0	0
$759$	−7917.77	−0.378652
$760$	0	0
$761$	−30062.7	−1.43203	−0.716013	−	0.698087i	$-0.754035\pi$
$761$	−30062.7	−1.43203	−0.716013	+	0.698087i	$0.754035\pi$
$762$	0	0
$763$	47619.3	2.25941
$764$	0	0
$765$	62.5044	0.00295406
$766$	0	0
$767$	1719.99	0.0809718
$768$	0	0
$769$	−21116.2	−0.990209	−0.495105	−	0.868833i	$-0.664870\pi$
$769$	−21116.2	−0.990209	−0.495105	+	0.868833i	$0.664870\pi$
$770$	0	0
$771$	−42652.2	−1.99232
$772$	0	0
$773$	−5086.88	−0.236691	−0.118346	−	0.992972i	$-0.537759\pi$
$773$	−5086.88	−0.236691	−0.118346	+	0.992972i	$0.537759\pi$
$774$	0	0
$775$	−4136.59	−0.191730
$776$	0	0
$777$	60540.3	2.79520
$778$	0	0
$779$	−5132.67	−0.236068
$780$	0	0
$781$	2758.50	0.126385
$782$	0	0
$783$	−24057.0	−1.09799
$784$	0	0
$785$	19927.9	0.906060
$786$	0	0
$787$	−12350.0	−0.559377	−0.279688	−	0.960091i	$-0.590231\pi$
$787$	−12350.0	−0.559377	−0.279688	+	0.960091i	$0.590231\pi$
$788$	0	0
$789$	−35795.5	−1.61515
$790$	0	0
$791$	8193.19	0.368289
$792$	0	0
$793$	17796.5	0.796939
$794$	0	0
$795$	20937.6	0.934061
$796$	0	0
$797$	−10263.6	−0.456155	−0.228078	−	0.973643i	$-0.573244\pi$
$797$	−10263.6	−0.456155	−0.228078	+	0.973643i	$0.573244\pi$
$798$	0	0
$799$	25447.9	1.12676
$800$	0	0
$801$	100.116	0.00441628
$802$	0	0
$803$	−3642.33	−0.160068
$804$	0	0
$805$	7812.32	0.342047
$806$	0	0
$807$	3014.37	0.131488
$808$	0	0
$809$	−42090.0	−1.82918	−0.914590	−	0.404383i	$-0.867486\pi$
$809$	−42090.0	−1.82918	−0.914590	+	0.404383i	$0.867486\pi$
$810$	0	0
$811$	−33315.6	−1.44250	−0.721250	−	0.692674i	$-0.756432\pi$
$811$	−33315.6	−1.44250	−0.721250	+	0.692674i	$0.756432\pi$
$812$	0	0
$813$	29602.5	1.27701
$814$	0	0
$815$	−6978.26	−0.299924
$816$	0	0
$817$	−946.853	−0.0405461
$818$	0	0
$819$	74.1719	0.00316456
$820$	0	0
$821$	−32210.1	−1.36923	−0.684616	−	0.728903i	$-0.740030\pi$
$821$	−32210.1	−1.36923	−0.684616	+	0.728903i	$0.740030\pi$
$822$	0	0
$823$	−15834.7	−0.670672	−0.335336	−	0.942099i	$-0.608850\pi$
$823$	−15834.7	−0.670672	−0.335336	+	0.942099i	$0.608850\pi$
$824$	0	0
$825$	13623.2	0.574908
$826$	0	0
$827$	−9712.04	−0.408369	−0.204184	−	0.978932i	$-0.565454\pi$
$827$	−9712.04	−0.408369	−0.204184	+	0.978932i	$0.565454\pi$
$828$	0	0
$829$	8471.30	0.354910	0.177455	−	0.984129i	$-0.443214\pi$
$829$	8471.30	0.354910	0.177455	+	0.984129i	$0.443214\pi$
$830$	0	0
$831$	−34430.4	−1.43728
$832$	0	0
$833$	26585.1	1.10578
$834$	0	0
$835$	−22588.4	−0.936171
$836$	0	0
$837$	9025.90	0.372737
$838$	0	0
$839$	−332.139	−0.0136671	−0.00683356	−	0.999977i	$-0.502175\pi$
$839$	−332.139	−0.0136671	−0.00683356	+	0.999977i	$0.502175\pi$
$840$	0	0
$841$	5139.25	0.210720
$842$	0	0
$843$	−13945.0	−0.569741
$844$	0	0
$845$	−12564.7	−0.511527
$846$	0	0
$847$	8909.45	0.361431
$848$	0	0
$849$	−27176.1	−1.09857
$850$	0	0
$851$	−16141.8	−0.650216
$852$	0	0
$853$	15669.4	0.628970	0.314485	−	0.949262i	$-0.398168\pi$
$853$	15669.4	0.628970	0.314485	+	0.949262i	$0.398168\pi$
$854$	0	0
$855$	−16.9043	−0.000676157 0
$856$	0	0
$857$	−16464.7	−0.656270	−0.328135	−	0.944631i	$-0.606420\pi$
$857$	−16464.7	−0.656270	−0.328135	+	0.944631i	$0.606420\pi$
$858$	0	0
$859$	7766.89	0.308502	0.154251	−	0.988032i	$-0.450704\pi$
$859$	7766.89	0.308502	0.154251	+	0.988032i	$0.450704\pi$
$860$	0	0
$861$	−37781.6	−1.49546
$862$	0	0
$863$	7812.54	0.308160	0.154080	−	0.988058i	$-0.450759\pi$
$863$	7812.54	0.308160	0.154080	+	0.988058i	$0.450759\pi$
$864$	0	0
$865$	−7237.28	−0.284479
$866$	0	0
$867$	−117.408	−0.00459906
$868$	0	0
$869$	49710.2	1.94051
$870$	0	0
$871$	−5431.95	−0.211314
$872$	0	0
$873$	92.6486	0.00359184
$874$	0	0
$875$	−39629.2	−1.53110
$876$	0	0
$877$	−31616.2	−1.21734	−0.608668	−	0.793425i	$-0.708296\pi$
$877$	−31616.2	−1.21734	−0.608668	+	0.793425i	$0.708296\pi$
$878$	0	0
$879$	32998.7	1.26623
$880$	0	0
$881$	−7258.59	−0.277580	−0.138790	−	0.990322i	$-0.544321\pi$
$881$	−7258.59	−0.277580	−0.138790	+	0.990322i	$0.544321\pi$
$882$	0	0
$883$	−26858.2	−1.02361	−0.511806	−	0.859101i	$-0.671023\pi$
$883$	−26858.2	−1.02361	−0.511806	+	0.859101i	$0.671023\pi$
$884$	0	0
$885$	−2885.50	−0.109599
$886$	0	0
$887$	−26005.8	−0.984431	−0.492215	−	0.870473i	$-0.663813\pi$
$887$	−26005.8	−0.984431	−0.492215	+	0.870473i	$0.663813\pi$
$888$	0	0
$889$	9285.39	0.350306
$890$	0	0
$891$	−29851.0	−1.12239
$892$	0	0
$893$	−6882.37	−0.257906
$894$	0	0
$895$	−25113.7	−0.937944
$896$	0	0
$897$	−4700.97	−0.174984
$898$	0	0
$899$	−11078.7	−0.411005
$900$	0	0
$901$	−36216.5	−1.33912
$902$	0	0
$903$	−6969.78	−0.256855
$904$	0	0
$905$	6421.06	0.235849
$906$	0	0
$907$	41499.6	1.51926	0.759631	−	0.650354i	$-0.225380\pi$
$907$	41499.6	1.51926	0.759631	+	0.650354i	$0.225380\pi$
$908$	0	0
$909$	−39.7590	−0.00145074
$910$	0	0
$911$	−11858.0	−0.431254	−0.215627	−	0.976476i	$-0.569179\pi$
$911$	−11858.0	−0.431254	−0.215627	+	0.976476i	$0.569179\pi$
$912$	0	0
$913$	38673.6	1.40187
$914$	0	0
$915$	−29855.8	−1.07869
$916$	0	0
$917$	70025.6	2.52175
$918$	0	0
$919$	−20359.3	−0.730784	−0.365392	−	0.930854i	$-0.619065\pi$
$919$	−20359.3	−0.730784	−0.365392	+	0.930854i	$0.619065\pi$
$920$	0	0
$921$	−39882.4	−1.42689
$922$	0	0
$923$	1637.78	0.0584055
$924$	0	0
$925$	27773.3	0.987224
$926$	0	0
$927$	−5.10864	−0.000181003 0
$928$	0	0
$929$	11453.8	0.404508	0.202254	−	0.979333i	$-0.435173\pi$
$929$	11453.8	0.404508	0.202254	+	0.979333i	$0.435173\pi$
$930$	0	0
$931$	−7189.91	−0.253104
$932$	0	0
$933$	27985.9	0.982013
$934$	0	0
$935$	22344.2	0.781534
$936$	0	0
$937$	−26514.3	−0.924422	−0.462211	−	0.886770i	$-0.652944\pi$
$937$	−26514.3	−0.924422	−0.462211	+	0.886770i	$0.652944\pi$
$938$	0	0
$939$	11273.1	0.391782
$940$	0	0
$941$	−32948.7	−1.14144	−0.570721	−	0.821144i	$-0.693336\pi$
$941$	−32948.7	−1.14144	−0.570721	+	0.821144i	$0.693336\pi$
$942$	0	0
$943$	10073.7	0.347872
$944$	0	0
$945$	29329.5	1.00962
$946$	0	0
$947$	22527.0	0.772998	0.386499	−	0.922290i	$-0.373684\pi$
$947$	22527.0	0.772998	0.386499	+	0.922290i	$0.373684\pi$
$948$	0	0
$949$	−2162.54	−0.0739714
$950$	0	0
$951$	−21206.0	−0.723081
$952$	0	0
$953$	−20808.5	−0.707297	−0.353648	−	0.935378i	$-0.615059\pi$
$953$	−20808.5	−0.707297	−0.353648	+	0.935378i	$0.615059\pi$
$954$	0	0
$955$	17883.9	0.605979
$956$	0	0
$957$	36485.8	1.23241
$958$	0	0
$959$	18856.7	0.634946
$960$	0	0
$961$	−25634.4	−0.860475
$962$	0	0
$963$	−163.539	−0.00547245
$964$	0	0
$965$	7098.48	0.236796
$966$	0	0
$967$	19794.4	0.658269	0.329135	−	0.944283i	$-0.393243\pi$
$967$	19794.4	0.658269	0.329135	+	0.944283i	$0.393243\pi$
$968$	0	0
$969$	6950.54	0.230427
$970$	0	0
$971$	−50395.8	−1.66558	−0.832789	−	0.553590i	$-0.813257\pi$
$971$	−50395.8	−1.66558	−0.832789	+	0.553590i	$0.813257\pi$
$972$	0	0
$973$	−68096.8	−2.24366
$974$	0	0
$975$	8088.42	0.265679
$976$	0	0
$977$	45621.3	1.49391	0.746957	−	0.664872i	$-0.231514\pi$
$977$	45621.3	1.49391	0.746957	+	0.664872i	$0.231514\pi$
$978$	0	0
$979$	35789.8	1.16838
$980$	0	0
$981$	202.229	0.00658173
$982$	0	0
$983$	29123.9	0.944972	0.472486	−	0.881338i	$-0.343357\pi$
$983$	29123.9	0.944972	0.472486	+	0.881338i	$0.343357\pi$
$984$	0	0
$985$	6116.18	0.197845
$986$	0	0
$987$	−50661.1	−1.63380
$988$	0	0
$989$	1858.35	0.0597492
$990$	0	0
$991$	28418.2	0.910933	0.455466	−	0.890253i	$-0.349473\pi$
$991$	28418.2	0.910933	0.455466	+	0.890253i	$0.349473\pi$
$992$	0	0
$993$	33264.5	1.06306
$994$	0	0
$995$	21257.7	0.677300
$996$	0	0
$997$	−38478.3	−1.22229	−0.611143	−	0.791520i	$-0.709290\pi$
$997$	−38478.3	−1.22229	−0.611143	+	0.791520i	$0.709290\pi$
$998$	0	0
$999$	−60600.5	−1.91924

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.4.a.k.1.2	yes	7
4.3	odd	2		608.4.a.j.1.6	✓	7
8.3	odd	2		1216.4.a.bg.1.2		7
8.5	even	2		1216.4.a.bf.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.4.a.j.1.6	✓	7	4.3	odd	2
608.4.a.k.1.2	yes	7	1.1	even	1	trivial
1216.4.a.bf.1.6		7	8.5	even	2
1216.4.a.bg.1.2		7	8.3	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:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	608.a (trivial)

\(f(q)\)	\(=\)	\(q-5.20712 q^{3} +7.79991 q^{5} +26.8592 q^{7} +0.114065 q^{9} +40.7763 q^{11} +24.2099 q^{13} -40.6150 q^{15} +70.2535 q^{17} -19.0000 q^{19} -139.859 q^{21} +37.2905 q^{23} -64.1614 q^{25} +139.998 q^{27} -171.838 q^{29} +64.4716 q^{31} -212.327 q^{33} +209.499 q^{35} -432.867 q^{37} -126.064 q^{39} +270.141 q^{41} +49.8344 q^{43} +0.889699 q^{45} +362.230 q^{47} +378.416 q^{49} -365.818 q^{51} -515.513 q^{53} +318.052 q^{55} +98.9352 q^{57} +71.0451 q^{59} +735.093 q^{61} +3.06370 q^{63} +188.835 q^{65} -224.369 q^{67} -194.176 q^{69} +67.6494 q^{71} -89.3246 q^{73} +334.096 q^{75} +1095.22 q^{77} +1219.09 q^{79} -732.067 q^{81} +948.434 q^{83} +547.970 q^{85} +894.780 q^{87} +877.711 q^{89} +650.258 q^{91} -335.711 q^{93} -148.198 q^{95} +812.241 q^{97} +4.65117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} - 17 q^{5} + 42 q^{7} + 86 q^{9} - 33 q^{11} - 35 q^{13} + 120 q^{15} + 66 q^{17} - 133 q^{19} + 33 q^{21} + 389 q^{23} + 44 q^{25} - 39 q^{27} + 233 q^{29} + 158 q^{31} - 206 q^{33} + 123 q^{35}+ \cdots - 1591 q^{99}+O(q^{100}) \) 7 * q + 3 * q^3 - 17 * q^5 + 42 * q^7 + 86 * q^9 - 33 * q^11 - 35 * q^13 + 120 * q^15 + 66 * q^17 - 133 * q^19 + 33 * q^21 + 389 * q^23 + 44 * q^25 - 39 * q^27 + 233 * q^29 + 158 * q^31 - 206 * q^33 + 123 * q^35 - 436 * q^37 + 807 * q^39 - 94 * q^41 + 645 * q^43 + 103 * q^45 + 1451 * q^47 + 93 * q^49 + 1741 * q^51 + 3 * q^53 + 1971 * q^55 - 57 * q^57 + 297 * q^59 + 93 * q^61 + 2999 * q^63 - 788 * q^65 + 1641 * q^67 + 945 * q^69 + 2392 * q^71 + 324 * q^73 + 1909 * q^75 - 711 * q^77 + 2492 * q^79 + 143 * q^81 + 310 * q^83 + 2353 * q^85 + 4795 * q^87 - 440 * q^89 - 107 * q^91 + 900 * q^93 + 323 * q^95 - 532 * q^97 - 1591 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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