LMFDB - Embedded newform 304.4.a.i.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [304,4,Mod(1,304)] 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("304.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	304.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-1,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$17.9365806417$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3144.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 8 $ x^3 - x^2 - 16*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.526440$ of defining polynomial
Character	$\chi$	$=$	304.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+8.66998 q^{3} -2.61710 q^{5} +15.1058 q^{7} +48.1686 q^{9} -12.6171 q^{11} +46.9571 q^{13} -22.6902 q^{15} -28.9745 q^{17} +19.0000 q^{19} +130.967 q^{21} +112.166 q^{23} -118.151 q^{25} +183.531 q^{27} +295.107 q^{29} +57.6979 q^{31} -109.390 q^{33} -39.5333 q^{35} -341.167 q^{37} +407.117 q^{39} +274.056 q^{41} -327.536 q^{43} -126.062 q^{45} -139.140 q^{47} -114.816 q^{49} -251.209 q^{51} +296.715 q^{53} +33.0202 q^{55} +164.730 q^{57} -459.383 q^{59} -232.911 q^{61} +727.623 q^{63} -122.891 q^{65} +320.784 q^{67} +972.475 q^{69} +9.54518 q^{71} +320.868 q^{73} -1024.37 q^{75} -190.591 q^{77} +89.2323 q^{79} +290.661 q^{81} +439.455 q^{83} +75.8293 q^{85} +2558.58 q^{87} +883.164 q^{89} +709.322 q^{91} +500.240 q^{93} -49.7249 q^{95} -1705.87 q^{97} -607.748 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + 14 q^{5} + 35 q^{7} + 48 q^{9} - 16 q^{11} + 65 q^{13} - 140 q^{15} + 29 q^{17} + 57 q^{19} - 25 q^{21} + 101 q^{23} - 37 q^{25} + 377 q^{27} + 377 q^{29} + 140 q^{31} - 130 q^{33} + 438 q^{35}+ \cdots - 250 q^{99}+O(q^{100}) $ 3 * q - q^3 + 14 * q^5 + 35 * q^7 + 48 * q^9 - 16 * q^11 + 65 * q^13 - 140 * q^15 + 29 * q^17 + 57 * q^19 - 25 * q^21 + 101 * q^23 - 37 * q^25 + 377 * q^27 + 377 * q^29 + 140 * q^31 - 130 * q^33 + 438 * q^35 - 290 * q^37 + 371 * q^39 + 956 * q^41 + 570 * q^43 + 230 * q^45 - 66 * q^47 - 98 * q^49 - 187 * q^51 + 817 * q^53 + 198 * q^55 - 19 * q^57 - 265 * q^59 + 988 * q^61 + 1304 * q^63 - 240 * q^65 + 207 * q^67 + 807 * q^69 - 846 * q^71 + 627 * q^73 - 2031 * q^75 + 88 * q^77 - 382 * q^79 - 501 * q^81 + 766 * q^83 - 1242 * q^85 + 1671 * q^87 - 172 * q^89 + 457 * q^91 + 892 * q^93 + 266 * q^95 - 2450 * q^97 - 250 * 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
$3$	8.66998	1.66854	0.834269	−	0.551357i	$-0.185890\pi$
$3$	8.66998	1.66854	0.834269	+	0.551357i	$0.185890\pi$
$4$	0	0
$5$	−2.61710	−0.234081	−0.117040	−	0.993127i	$-0.537341\pi$
$5$	−2.61710	−0.234081	−0.117040	+	0.993127i	$0.537341\pi$
$6$	0	0
$7$	15.1058	0.815634	0.407817	−	0.913064i	$-0.366290\pi$
$7$	15.1058	0.815634	0.407817	+	0.913064i	$0.366290\pi$
$8$	0	0
$9$	48.1686	1.78402
$10$	0	0
$11$	−12.6171	−0.345836	−0.172918	−	0.984936i	$-0.555320\pi$
$11$	−12.6171	−0.345836	−0.172918	+	0.984936i	$0.555320\pi$
$12$	0	0
$13$	46.9571	1.00181	0.500906	−	0.865502i	$-0.333000\pi$
$13$	46.9571	1.00181	0.500906	+	0.865502i	$0.333000\pi$
$14$	0	0
$15$	−22.6902	−0.390573
$16$	0	0
$17$	−28.9745	−0.413374	−0.206687	−	0.978407i	$-0.566268\pi$
$17$	−28.9745	−0.413374	−0.206687	+	0.978407i	$0.566268\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	130.967	1.36092
$22$	0	0
$23$	112.166	1.01688	0.508439	−	0.861098i	$-0.330223\pi$
$23$	112.166	1.01688	0.508439	+	0.861098i	$0.330223\pi$
$24$	0	0
$25$	−118.151	−0.945206
$26$	0	0
$27$	183.531	1.30817
$28$	0	0
$29$	295.107	1.88966	0.944828	−	0.327565i	$-0.106228\pi$
$29$	295.107	1.88966	0.944828	+	0.327565i	$0.106228\pi$
$30$	0	0
$31$	57.6979	0.334285	0.167143	−	0.985933i	$-0.446546\pi$
$31$	57.6979	0.334285	0.167143	+	0.985933i	$0.446546\pi$
$32$	0	0
$33$	−109.390	−0.577041
$34$	0	0
$35$	−39.5333	−0.190924
$36$	0	0
$37$	−341.167	−1.51588	−0.757940	−	0.652324i	$-0.773794\pi$
$37$	−341.167	−1.51588	−0.757940	+	0.652324i	$0.773794\pi$
$38$	0	0
$39$	407.117	1.67156
$40$	0	0
$41$	274.056	1.04391	0.521955	−	0.852973i	$-0.325203\pi$
$41$	274.056	1.04391	0.521955	+	0.852973i	$0.325203\pi$
$42$	0	0
$43$	−327.536	−1.16160	−0.580800	−	0.814046i	$-0.697260\pi$
$43$	−327.536	−1.16160	−0.580800	+	0.814046i	$0.697260\pi$
$44$	0	0
$45$	−126.062	−0.417605
$46$	0	0
$47$	−139.140	−0.431823	−0.215912	−	0.976413i	$-0.569272\pi$
$47$	−139.140	−0.431823	−0.215912	+	0.976413i	$0.569272\pi$
$48$	0	0
$49$	−114.816	−0.334741
$50$	0	0
$51$	−251.209	−0.689730
$52$	0	0
$53$	296.715	0.769000	0.384500	−	0.923125i	$-0.374374\pi$
$53$	296.715	0.769000	0.384500	+	0.923125i	$0.374374\pi$
$54$	0	0
$55$	33.0202	0.0809536
$56$	0	0
$57$	164.730	0.382789
$58$	0	0
$59$	−459.383	−1.01367	−0.506836	−	0.862043i	$-0.669185\pi$
$59$	−459.383	−1.01367	−0.506836	+	0.862043i	$0.669185\pi$
$60$	0	0
$61$	−232.911	−0.488873	−0.244436	−	0.969665i	$-0.578603\pi$
$61$	−232.911	−0.488873	−0.244436	+	0.969665i	$0.578603\pi$
$62$	0	0
$63$	727.623	1.45511
$64$	0	0
$65$	−122.891	−0.234505
$66$	0	0
$67$	320.784	0.584926	0.292463	−	0.956277i	$-0.405525\pi$
$67$	320.784	0.584926	0.292463	+	0.956277i	$0.405525\pi$
$68$	0	0
$69$	972.475	1.69670
$70$	0	0
$71$	9.54518	0.0159550	0.00797749	−	0.999968i	$-0.497461\pi$
$71$	9.54518	0.0159550	0.00797749	+	0.999968i	$0.497461\pi$
$72$	0	0
$73$	320.868	0.514448	0.257224	−	0.966352i	$-0.417192\pi$
$73$	320.868	0.514448	0.257224	+	0.966352i	$0.417192\pi$
$74$	0	0
$75$	−1024.37	−1.57711
$76$	0	0
$77$	−190.591	−0.282076
$78$	0	0
$79$	89.2323	0.127081	0.0635406	−	0.997979i	$-0.479761\pi$
$79$	89.2323	0.127081	0.0635406	+	0.997979i	$0.479761\pi$
$80$	0	0
$81$	290.661	0.398712
$82$	0	0
$83$	439.455	0.581163	0.290581	−	0.956850i	$-0.406151\pi$
$83$	439.455	0.581163	0.290581	+	0.956850i	$0.406151\pi$
$84$	0	0
$85$	75.8293	0.0967628
$86$	0	0
$87$	2558.58	3.15297
$88$	0	0
$89$	883.164	1.05186	0.525928	−	0.850529i	$-0.323718\pi$
$89$	883.164	1.05186	0.525928	+	0.850529i	$0.323718\pi$
$90$	0	0
$91$	709.322	0.817112
$92$	0	0
$93$	500.240	0.557768
$94$	0	0
$95$	−49.7249	−0.0537018
$96$	0	0
$97$	−1705.87	−1.78562	−0.892808	−	0.450437i	$-0.851268\pi$
$97$	−1705.87	−1.78562	−0.892808	+	0.450437i	$0.851268\pi$
$98$	0	0
$99$	−607.748	−0.616979
$100$	0	0
$101$	−961.422	−0.947178	−0.473589	−	0.880746i	$-0.657042\pi$
$101$	−961.422	−0.947178	−0.473589	+	0.880746i	$0.657042\pi$
$102$	0	0
$103$	−173.142	−0.165633	−0.0828166	−	0.996565i	$-0.526392\pi$
$103$	−173.142	−0.165633	−0.0828166	+	0.996565i	$0.526392\pi$
$104$	0	0
$105$	−342.753	−0.318565
$106$	0	0
$107$	−921.087	−0.832194	−0.416097	−	0.909320i	$-0.636602\pi$
$107$	−921.087	−0.832194	−0.416097	+	0.909320i	$0.636602\pi$
$108$	0	0
$109$	552.454	0.485463	0.242731	−	0.970094i	$-0.421957\pi$
$109$	552.454	0.485463	0.242731	+	0.970094i	$0.421957\pi$
$110$	0	0
$111$	−2957.91	−2.52930
$112$	0	0
$113$	−1395.26	−1.16155	−0.580774	−	0.814064i	$-0.697250\pi$
$113$	−1395.26	−1.16155	−0.580774	+	0.814064i	$0.697250\pi$
$114$	0	0
$115$	−293.549	−0.238031
$116$	0	0
$117$	2261.86	1.78725
$118$	0	0
$119$	−437.682	−0.337162
$120$	0	0
$121$	−1171.81	−0.880397
$122$	0	0
$123$	2376.06	1.74180
$124$	0	0
$125$	636.350	0.455335
$126$	0	0
$127$	−793.013	−0.554083	−0.277041	−	0.960858i	$-0.589354\pi$
$127$	−793.013	−0.554083	−0.277041	+	0.960858i	$0.589354\pi$
$128$	0	0
$129$	−2839.73	−1.93818
$130$	0	0
$131$	−25.0544	−0.0167100	−0.00835501	−	0.999965i	$-0.502660\pi$
$131$	−25.0544	−0.0167100	−0.00835501	+	0.999965i	$0.502660\pi$
$132$	0	0
$133$	287.009	0.187119
$134$	0	0
$135$	−480.320	−0.306218
$136$	0	0
$137$	−716.056	−0.446546	−0.223273	−	0.974756i	$-0.571674\pi$
$137$	−716.056	−0.446546	−0.223273	+	0.974756i	$0.571674\pi$
$138$	0	0
$139$	−666.566	−0.406744	−0.203372	−	0.979102i	$-0.565190\pi$
$139$	−666.566	−0.406744	−0.203372	+	0.979102i	$0.565190\pi$
$140$	0	0
$141$	−1206.34	−0.720514
$142$	0	0
$143$	−592.462	−0.346463
$144$	0	0
$145$	−772.326	−0.442332
$146$	0	0
$147$	−995.453	−0.558528
$148$	0	0
$149$	−268.063	−0.147386	−0.0736931	−	0.997281i	$-0.523479\pi$
$149$	−268.063	−0.147386	−0.0736931	+	0.997281i	$0.523479\pi$
$150$	0	0
$151$	−2809.46	−1.51411	−0.757055	−	0.653352i	$-0.773362\pi$
$151$	−2809.46	−1.51411	−0.757055	+	0.653352i	$0.773362\pi$
$152$	0	0
$153$	−1395.66	−0.737468
$154$	0	0
$155$	−151.001	−0.0782498
$156$	0	0
$157$	−1999.77	−1.01656	−0.508278	−	0.861193i	$-0.669718\pi$
$157$	−1999.77	−1.01656	−0.508278	+	0.861193i	$0.669718\pi$
$158$	0	0
$159$	2572.52	1.28311
$160$	0	0
$161$	1694.35	0.829400
$162$	0	0
$163$	−1642.08	−0.789064	−0.394532	−	0.918882i	$-0.629093\pi$
$163$	−1642.08	−0.789064	−0.394532	+	0.918882i	$0.629093\pi$
$164$	0	0
$165$	286.285	0.135074
$166$	0	0
$167$	−2965.92	−1.37431	−0.687155	−	0.726511i	$-0.741141\pi$
$167$	−2965.92	−1.37431	−0.687155	+	0.726511i	$0.741141\pi$
$168$	0	0
$169$	7.96603	0.00362587
$170$	0	0
$171$	915.203	0.409283
$172$	0	0
$173$	92.7872	0.0407773	0.0203887	−	0.999792i	$-0.493510\pi$
$173$	92.7872	0.0407773	0.0203887	+	0.999792i	$0.493510\pi$
$174$	0	0
$175$	−1784.76	−0.770943
$176$	0	0
$177$	−3982.84	−1.69135
$178$	0	0
$179$	3294.07	1.37548	0.687738	−	0.725959i	$-0.258604\pi$
$179$	3294.07	1.37548	0.687738	+	0.725959i	$0.258604\pi$
$180$	0	0
$181$	3590.17	1.47434	0.737168	−	0.675709i	$-0.236162\pi$
$181$	3590.17	1.47434	0.737168	+	0.675709i	$0.236162\pi$
$182$	0	0
$183$	−2019.34	−0.815703
$184$	0	0
$185$	892.870	0.354838
$186$	0	0
$187$	365.574	0.142960
$188$	0	0
$189$	2772.38	1.06699
$190$	0	0
$191$	−480.480	−0.182023	−0.0910114	−	0.995850i	$-0.529010\pi$
$191$	−480.480	−0.182023	−0.0910114	+	0.995850i	$0.529010\pi$
$192$	0	0
$193$	3504.07	1.30688	0.653442	−	0.756977i	$-0.273325\pi$
$193$	3504.07	1.30688	0.653442	+	0.756977i	$0.273325\pi$
$194$	0	0
$195$	−1065.47	−0.391280
$196$	0	0
$197$	603.790	0.218367	0.109183	−	0.994022i	$-0.465176\pi$
$197$	603.790	0.218367	0.109183	+	0.994022i	$0.465176\pi$
$198$	0	0
$199$	−3063.80	−1.09139	−0.545695	−	0.837984i	$-0.683734\pi$
$199$	−3063.80	−1.09139	−0.545695	+	0.837984i	$0.683734\pi$
$200$	0	0
$201$	2781.20	0.975972
$202$	0	0
$203$	4457.82	1.54127
$204$	0	0
$205$	−717.232	−0.244359
$206$	0	0
$207$	5402.87	1.81413
$208$	0	0
$209$	−239.725	−0.0793403
$210$	0	0
$211$	2772.16	0.904470	0.452235	−	0.891899i	$-0.350627\pi$
$211$	2772.16	0.904470	0.452235	+	0.891899i	$0.350627\pi$
$212$	0	0
$213$	82.7565	0.0266215
$214$	0	0
$215$	857.196	0.271908
$216$	0	0
$217$	871.571	0.272655
$218$	0	0
$219$	2781.92	0.858377
$220$	0	0
$221$	−1360.56	−0.414122
$222$	0	0
$223$	5653.47	1.69769	0.848844	−	0.528644i	$-0.177299\pi$
$223$	5653.47	1.69769	0.848844	+	0.528644i	$0.177299\pi$
$224$	0	0
$225$	−5691.16	−1.68627
$226$	0	0
$227$	5083.00	1.48621	0.743106	−	0.669173i	$-0.233352\pi$
$227$	5083.00	1.48621	0.743106	+	0.669173i	$0.233352\pi$
$228$	0	0
$229$	501.966	0.144851	0.0724254	−	0.997374i	$-0.476926\pi$
$229$	501.966	0.144851	0.0724254	+	0.997374i	$0.476926\pi$
$230$	0	0
$231$	−1652.42	−0.470655
$232$	0	0
$233$	2809.38	0.789909	0.394955	−	0.918701i	$-0.370760\pi$
$233$	2809.38	0.789909	0.394955	+	0.918701i	$0.370760\pi$
$234$	0	0
$235$	364.144	0.101082
$236$	0	0
$237$	773.642	0.212040
$238$	0	0
$239$	−2239.56	−0.606130	−0.303065	−	0.952970i	$-0.598010\pi$
$239$	−2239.56	−0.606130	−0.303065	+	0.952970i	$0.598010\pi$
$240$	0	0
$241$	−7214.25	−1.92826	−0.964130	−	0.265431i	$-0.914486\pi$
$241$	−7214.25	−1.92826	−0.964130	+	0.265431i	$0.914486\pi$
$242$	0	0
$243$	−2435.32	−0.642905
$244$	0	0
$245$	300.485	0.0783563
$246$	0	0
$247$	892.184	0.229831
$248$	0	0
$249$	3810.07	0.969693
$250$	0	0
$251$	−6212.82	−1.56235	−0.781174	−	0.624313i	$-0.785379\pi$
$251$	−6212.82	−1.56235	−0.781174	+	0.624313i	$0.785379\pi$
$252$	0	0
$253$	−1415.21	−0.351673
$254$	0	0
$255$	657.438	0.161453
$256$	0	0
$257$	−2796.37	−0.678727	−0.339364	−	0.940655i	$-0.610212\pi$
$257$	−2796.37	−0.678727	−0.339364	+	0.940655i	$0.610212\pi$
$258$	0	0
$259$	−5153.59	−1.23640
$260$	0	0
$261$	14214.9	3.37119
$262$	0	0
$263$	−2976.00	−0.697748	−0.348874	−	0.937170i	$-0.613436\pi$
$263$	−2976.00	−0.697748	−0.348874	+	0.937170i	$0.613436\pi$
$264$	0	0
$265$	−776.534	−0.180008
$266$	0	0
$267$	7657.02	1.75506
$268$	0	0
$269$	−25.4734	−0.00577376	−0.00288688	−	0.999996i	$-0.500919\pi$
$269$	−25.4734	−0.00577376	−0.00288688	+	0.999996i	$0.500919\pi$
$270$	0	0
$271$	4338.19	0.972421	0.486211	−	0.873842i	$-0.338379\pi$
$271$	4338.19	0.972421	0.486211	+	0.873842i	$0.338379\pi$
$272$	0	0
$273$	6149.81	1.36338
$274$	0	0
$275$	1490.72	0.326887
$276$	0	0
$277$	4276.02	0.927514	0.463757	−	0.885962i	$-0.346501\pi$
$277$	4276.02	0.927514	0.463757	+	0.885962i	$0.346501\pi$
$278$	0	0
$279$	2779.23	0.596373
$280$	0	0
$281$	−3716.43	−0.788980	−0.394490	−	0.918900i	$-0.629079\pi$
$281$	−3716.43	−0.788980	−0.394490	+	0.918900i	$0.629079\pi$
$282$	0	0
$283$	−3748.01	−0.787265	−0.393633	−	0.919268i	$-0.628782\pi$
$283$	−3748.01	−0.787265	−0.393633	+	0.919268i	$0.628782\pi$
$284$	0	0
$285$	−431.114	−0.0896035
$286$	0	0
$287$	4139.82	0.851449
$288$	0	0
$289$	−4073.48	−0.829122
$290$	0	0
$291$	−14789.9	−2.97937
$292$	0	0
$293$	8210.10	1.63699	0.818497	−	0.574510i	$-0.194808\pi$
$293$	8210.10	1.63699	0.818497	+	0.574510i	$0.194808\pi$
$294$	0	0
$295$	1202.25	0.237281
$296$	0	0
$297$	−2315.63	−0.452413
$298$	0	0
$299$	5266.98	1.01872
$300$	0	0
$301$	−4947.69	−0.947442
$302$	0	0
$303$	−8335.51	−1.58040
$304$	0	0
$305$	609.553	0.114436
$306$	0	0
$307$	2814.79	0.523285	0.261642	−	0.965165i	$-0.415736\pi$
$307$	2814.79	0.523285	0.261642	+	0.965165i	$0.415736\pi$
$308$	0	0
$309$	−1501.14	−0.276366
$310$	0	0
$311$	8650.75	1.57730	0.788648	−	0.614845i	$-0.210782\pi$
$311$	8650.75	1.57730	0.788648	+	0.614845i	$0.210782\pi$
$312$	0	0
$313$	1623.48	0.293178	0.146589	−	0.989197i	$-0.453171\pi$
$313$	1623.48	0.293178	0.146589	+	0.989197i	$0.453171\pi$
$314$	0	0
$315$	−1904.26	−0.340613
$316$	0	0
$317$	9372.31	1.66057	0.830286	−	0.557337i	$-0.188177\pi$
$317$	9372.31	1.66057	0.830286	+	0.557337i	$0.188177\pi$
$318$	0	0
$319$	−3723.40	−0.653512
$320$	0	0
$321$	−7985.80	−1.38855
$322$	0	0
$323$	−550.516	−0.0948344
$324$	0	0
$325$	−5548.01	−0.946918
$326$	0	0
$327$	4789.76	0.810014
$328$	0	0
$329$	−2101.82	−0.352210
$330$	0	0
$331$	1765.55	0.293182	0.146591	−	0.989197i	$-0.453170\pi$
$331$	1765.55	0.293182	0.146591	+	0.989197i	$0.453170\pi$
$332$	0	0
$333$	−16433.5	−2.70436
$334$	0	0
$335$	−839.526	−0.136920
$336$	0	0
$337$	10189.8	1.64711	0.823555	−	0.567237i	$-0.191988\pi$
$337$	10189.8	1.64711	0.823555	+	0.567237i	$0.191988\pi$
$338$	0	0
$339$	−12096.9	−1.93809
$340$	0	0
$341$	−727.980	−0.115608
$342$	0	0
$343$	−6915.66	−1.08866
$344$	0	0
$345$	−2545.07	−0.397165
$346$	0	0
$347$	11350.2	1.75594	0.877971	−	0.478714i	$-0.158897\pi$
$347$	11350.2	1.75594	0.877971	+	0.478714i	$0.158897\pi$
$348$	0	0
$349$	511.619	0.0784709	0.0392354	−	0.999230i	$-0.487508\pi$
$349$	511.619	0.0784709	0.0392354	+	0.999230i	$0.487508\pi$
$350$	0	0
$351$	8618.09	1.31054
$352$	0	0
$353$	816.097	0.123049	0.0615247	−	0.998106i	$-0.480404\pi$
$353$	816.097	0.123049	0.0615247	+	0.998106i	$0.480404\pi$
$354$	0	0
$355$	−24.9807	−0.00373475
$356$	0	0
$357$	−3794.70	−0.562567
$358$	0	0
$359$	−3998.35	−0.587813	−0.293906	−	0.955834i	$-0.594955\pi$
$359$	−3998.35	−0.587813	−0.293906	+	0.955834i	$0.594955\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−10159.6	−1.46898
$364$	0	0
$365$	−839.744	−0.120422
$366$	0	0
$367$	3123.19	0.444221	0.222111	−	0.975021i	$-0.428705\pi$
$367$	3123.19	0.444221	0.222111	+	0.975021i	$0.428705\pi$
$368$	0	0
$369$	13200.9	1.86236
$370$	0	0
$371$	4482.11	0.627223
$372$	0	0
$373$	−7026.28	−0.975354	−0.487677	−	0.873024i	$-0.662156\pi$
$373$	−7026.28	−0.975354	−0.487677	+	0.873024i	$0.662156\pi$
$374$	0	0
$375$	5517.15	0.759745
$376$	0	0
$377$	13857.4	1.89308
$378$	0	0
$379$	11595.9	1.57162	0.785808	−	0.618470i	$-0.212247\pi$
$379$	11595.9	1.57162	0.785808	+	0.618470i	$0.212247\pi$
$380$	0	0
$381$	−6875.41	−0.924508
$382$	0	0
$383$	10962.4	1.46254	0.731268	−	0.682091i	$-0.238929\pi$
$383$	10962.4	1.46254	0.731268	+	0.682091i	$0.238929\pi$
$384$	0	0
$385$	498.796	0.0660286
$386$	0	0
$387$	−15777.0	−2.07232
$388$	0	0
$389$	7126.21	0.928825	0.464413	−	0.885619i	$-0.346265\pi$
$389$	7126.21	0.928825	0.464413	+	0.885619i	$0.346265\pi$
$390$	0	0
$391$	−3249.95	−0.420350
$392$	0	0
$393$	−217.221	−0.0278813
$394$	0	0
$395$	−233.530	−0.0297473
$396$	0	0
$397$	4895.59	0.618899	0.309449	−	0.950916i	$-0.399855\pi$
$397$	4895.59	0.618899	0.309449	+	0.950916i	$0.399855\pi$
$398$	0	0
$399$	2488.37	0.312216
$400$	0	0
$401$	−6148.05	−0.765634	−0.382817	−	0.923824i	$-0.625046\pi$
$401$	−6148.05	−0.765634	−0.382817	+	0.923824i	$0.625046\pi$
$402$	0	0
$403$	2709.32	0.334891
$404$	0	0
$405$	−760.689	−0.0933307
$406$	0	0
$407$	4304.54	0.524246
$408$	0	0
$409$	2968.47	0.358878	0.179439	−	0.983769i	$-0.442572\pi$
$409$	2968.47	0.358878	0.179439	+	0.983769i	$0.442572\pi$
$410$	0	0
$411$	−6208.19	−0.745079
$412$	0	0
$413$	−6939.33	−0.826785
$414$	0	0
$415$	−1150.10	−0.136039
$416$	0	0
$417$	−5779.12	−0.678668
$418$	0	0
$419$	11101.1	1.29432	0.647162	−	0.762352i	$-0.275956\pi$
$419$	11101.1	1.29432	0.647162	+	0.762352i	$0.275956\pi$
$420$	0	0
$421$	3241.73	0.375278	0.187639	−	0.982238i	$-0.439916\pi$
$421$	3241.73	0.375278	0.187639	+	0.982238i	$0.439916\pi$
$422$	0	0
$423$	−6702.19	−0.770382
$424$	0	0
$425$	3423.36	0.390723
$426$	0	0
$427$	−3518.30	−0.398741
$428$	0	0
$429$	−5136.64	−0.578086
$430$	0	0
$431$	−290.271	−0.0324405	−0.0162202	−	0.999868i	$-0.505163\pi$
$431$	−290.271	−0.0324405	−0.0162202	+	0.999868i	$0.505163\pi$
$432$	0	0
$433$	2104.16	0.233533	0.116766	−	0.993159i	$-0.462747\pi$
$433$	2104.16	0.233533	0.116766	+	0.993159i	$0.462747\pi$
$434$	0	0
$435$	−6696.05	−0.738049
$436$	0	0
$437$	2131.15	0.233288
$438$	0	0
$439$	−4013.82	−0.436376	−0.218188	−	0.975907i	$-0.570015\pi$
$439$	−4013.82	−0.436376	−0.218188	+	0.975907i	$0.570015\pi$
$440$	0	0
$441$	−5530.53	−0.597185
$442$	0	0
$443$	−12013.8	−1.28847	−0.644236	−	0.764827i	$-0.722824\pi$
$443$	−12013.8	−1.28847	−0.644236	+	0.764827i	$0.722824\pi$
$444$	0	0
$445$	−2311.33	−0.246219
$446$	0	0
$447$	−2324.10	−0.245920
$448$	0	0
$449$	4281.55	0.450020	0.225010	−	0.974356i	$-0.427759\pi$
$449$	4281.55	0.450020	0.225010	+	0.974356i	$0.427759\pi$
$450$	0	0
$451$	−3457.79	−0.361022
$452$	0	0
$453$	−24358.0	−2.52635
$454$	0	0
$455$	−1856.37	−0.191270
$456$	0	0
$457$	293.330	0.0300249	0.0150125	−	0.999887i	$-0.495221\pi$
$457$	293.330	0.0300249	0.0150125	+	0.999887i	$0.495221\pi$
$458$	0	0
$459$	−5317.73	−0.540763
$460$	0	0
$461$	−5267.87	−0.532210	−0.266105	−	0.963944i	$-0.585737\pi$
$461$	−5267.87	−0.532210	−0.266105	+	0.963944i	$0.585737\pi$
$462$	0	0
$463$	−8076.98	−0.810733	−0.405366	−	0.914154i	$-0.632856\pi$
$463$	−8076.98	−0.810733	−0.405366	+	0.914154i	$0.632856\pi$
$464$	0	0
$465$	−1309.18	−0.130563
$466$	0	0
$467$	3160.43	0.313163	0.156582	−	0.987665i	$-0.449953\pi$
$467$	3160.43	0.313163	0.156582	+	0.987665i	$0.449953\pi$
$468$	0	0
$469$	4845.69	0.477086
$470$	0	0
$471$	−17338.0	−1.69616
$472$	0	0
$473$	4132.56	0.401724
$474$	0	0
$475$	−2244.86	−0.216845
$476$	0	0
$477$	14292.4	1.37191
$478$	0	0
$479$	249.277	0.0237782	0.0118891	−	0.999929i	$-0.496215\pi$
$479$	249.277	0.0237782	0.0118891	+	0.999929i	$0.496215\pi$
$480$	0	0
$481$	−16020.2	−1.51863
$482$	0	0
$483$	14690.0	1.38389
$484$	0	0
$485$	4464.44	0.417979
$486$	0	0
$487$	−7265.71	−0.676059	−0.338029	−	0.941136i	$-0.609760\pi$
$487$	−7265.71	−0.676059	−0.338029	+	0.941136i	$0.609760\pi$
$488$	0	0
$489$	−14236.8	−1.31658
$490$	0	0
$491$	8456.47	0.777261	0.388630	−	0.921394i	$-0.372948\pi$
$491$	8456.47	0.777261	0.388630	+	0.921394i	$0.372948\pi$
$492$	0	0
$493$	−8550.59	−0.781134
$494$	0	0
$495$	1590.54	0.144423
$496$	0	0
$497$	144.187	0.0130134
$498$	0	0
$499$	−1905.15	−0.170914	−0.0854571	−	0.996342i	$-0.527235\pi$
$499$	−1905.15	−0.170914	−0.0854571	+	0.996342i	$0.527235\pi$
$500$	0	0
$501$	−25714.5	−2.29309
$502$	0	0
$503$	−6082.63	−0.539187	−0.269593	−	0.962974i	$-0.586889\pi$
$503$	−6082.63	−0.539187	−0.269593	+	0.962974i	$0.586889\pi$
$504$	0	0
$505$	2516.14	0.221716
$506$	0	0
$507$	69.0653	0.00604990
$508$	0	0
$509$	13858.9	1.20684	0.603422	−	0.797422i	$-0.293803\pi$
$509$	13858.9	1.20684	0.603422	+	0.797422i	$0.293803\pi$
$510$	0	0
$511$	4846.95	0.419602
$512$	0	0
$513$	3487.09	0.300115
$514$	0	0
$515$	453.131	0.0387716
$516$	0	0
$517$	1755.55	0.149340
$518$	0	0
$519$	804.463	0.0680385
$520$	0	0
$521$	4086.72	0.343651	0.171826	−	0.985127i	$-0.445033\pi$
$521$	4086.72	0.343651	0.171826	+	0.985127i	$0.445033\pi$
$522$	0	0
$523$	−20188.4	−1.68791	−0.843957	−	0.536411i	$-0.819780\pi$
$523$	−20188.4	−1.68791	−0.843957	+	0.536411i	$0.819780\pi$
$524$	0	0
$525$	−15473.8	−1.28635
$526$	0	0
$527$	−1671.77	−0.138185
$528$	0	0
$529$	414.164	0.0340399
$530$	0	0
$531$	−22127.8	−1.80841
$532$	0	0
$533$	12868.9	1.04580
$534$	0	0
$535$	2410.58	0.194801
$536$	0	0
$537$	28559.5	2.29503
$538$	0	0
$539$	1448.65	0.115765
$540$	0	0
$541$	16356.6	1.29986	0.649932	−	0.759992i	$-0.274797\pi$
$541$	16356.6	1.29986	0.649932	+	0.759992i	$0.274797\pi$
$542$	0	0
$543$	31126.7	2.45999
$544$	0	0
$545$	−1445.83	−0.113638
$546$	0	0
$547$	12751.0	0.996697	0.498349	−	0.866977i	$-0.333940\pi$
$547$	12751.0	0.996697	0.498349	+	0.866977i	$0.333940\pi$
$548$	0	0
$549$	−11219.0	−0.872159
$550$	0	0
$551$	5607.04	0.433517
$552$	0	0
$553$	1347.92	0.103652
$554$	0	0
$555$	7741.17	0.592062
$556$	0	0
$557$	−7353.77	−0.559406	−0.279703	−	0.960087i	$-0.590236\pi$
$557$	−7353.77	−0.559406	−0.279703	+	0.960087i	$0.590236\pi$
$558$	0	0
$559$	−15380.1	−1.16370
$560$	0	0
$561$	3169.52	0.238534
$562$	0	0
$563$	5597.74	0.419035	0.209517	−	0.977805i	$-0.432811\pi$
$563$	5597.74	0.419035	0.209517	+	0.977805i	$0.432811\pi$
$564$	0	0
$565$	3651.54	0.271896
$566$	0	0
$567$	4390.65	0.325203
$568$	0	0
$569$	−19640.2	−1.44703	−0.723515	−	0.690309i	$-0.757475\pi$
$569$	−19640.2	−1.44703	−0.723515	+	0.690309i	$0.757475\pi$
$570$	0	0
$571$	9749.71	0.714558	0.357279	−	0.933998i	$-0.383705\pi$
$571$	9749.71	0.714558	0.357279	+	0.933998i	$0.383705\pi$
$572$	0	0
$573$	−4165.76	−0.303712
$574$	0	0
$575$	−13252.5	−0.961159
$576$	0	0
$577$	11197.9	0.807927	0.403964	−	0.914775i	$-0.367632\pi$
$577$	11197.9	0.807927	0.403964	+	0.914775i	$0.367632\pi$
$578$	0	0
$579$	30380.2	2.18059
$580$	0	0
$581$	6638.31	0.474016
$582$	0	0
$583$	−3743.69	−0.265948
$584$	0	0
$585$	−5919.51	−0.418362
$586$	0	0
$587$	18654.4	1.31167	0.655833	−	0.754906i	$-0.272318\pi$
$587$	18654.4	1.31167	0.655833	+	0.754906i	$0.272318\pi$
$588$	0	0
$589$	1096.26	0.0766903
$590$	0	0
$591$	5234.85	0.364354
$592$	0	0
$593$	21513.4	1.48980	0.744900	−	0.667176i	$-0.232497\pi$
$593$	21513.4	1.48980	0.744900	+	0.667176i	$0.232497\pi$
$594$	0	0
$595$	1145.46	0.0789231
$596$	0	0
$597$	−26563.1	−1.82103
$598$	0	0
$599$	−22762.8	−1.55269	−0.776346	−	0.630308i	$-0.782929\pi$
$599$	−22762.8	−1.55269	−0.776346	+	0.630308i	$0.782929\pi$
$600$	0	0
$601$	−13941.0	−0.946195	−0.473098	−	0.881010i	$-0.656864\pi$
$601$	−13941.0	−0.946195	−0.473098	+	0.881010i	$0.656864\pi$
$602$	0	0
$603$	15451.7	1.04352
$604$	0	0
$605$	3066.74	0.206084
$606$	0	0
$607$	−8031.64	−0.537058	−0.268529	−	0.963272i	$-0.586538\pi$
$607$	−8031.64	−0.537058	−0.268529	+	0.963272i	$0.586538\pi$
$608$	0	0
$609$	38649.2	2.57167
$610$	0	0
$611$	−6533.62	−0.432606
$612$	0	0
$613$	−19429.7	−1.28019	−0.640095	−	0.768296i	$-0.721105\pi$
$613$	−19429.7	−1.28019	−0.640095	+	0.768296i	$0.721105\pi$
$614$	0	0
$615$	−6218.39	−0.407723
$616$	0	0
$617$	−11264.0	−0.734959	−0.367480	−	0.930032i	$-0.619779\pi$
$617$	−11264.0	−0.734959	−0.367480	+	0.930032i	$0.619779\pi$
$618$	0	0
$619$	22183.9	1.44046	0.720231	−	0.693734i	$-0.244036\pi$
$619$	22183.9	1.44046	0.720231	+	0.693734i	$0.244036\pi$
$620$	0	0
$621$	20585.9	1.33025
$622$	0	0
$623$	13340.9	0.857930
$624$	0	0
$625$	13103.5	0.838621
$626$	0	0
$627$	−2078.41	−0.132382
$628$	0	0
$629$	9885.16	0.626625
$630$	0	0
$631$	23139.3	1.45984	0.729920	−	0.683532i	$-0.239557\pi$
$631$	23139.3	1.45984	0.729920	+	0.683532i	$0.239557\pi$
$632$	0	0
$633$	24034.5	1.50914
$634$	0	0
$635$	2075.40	0.129700
$636$	0	0
$637$	−5391.42	−0.335347
$638$	0	0
$639$	459.778	0.0284640
$640$	0	0
$641$	−4988.45	−0.307382	−0.153691	−	0.988119i	$-0.549116\pi$
$641$	−4988.45	−0.307382	−0.153691	+	0.988119i	$0.549116\pi$
$642$	0	0
$643$	11115.2	0.681712	0.340856	−	0.940115i	$-0.389283\pi$
$643$	11115.2	0.681712	0.340856	+	0.940115i	$0.389283\pi$
$644$	0	0
$645$	7431.88	0.453690
$646$	0	0
$647$	−11916.2	−0.724071	−0.362036	−	0.932164i	$-0.617918\pi$
$647$	−11916.2	−0.724071	−0.362036	+	0.932164i	$0.617918\pi$
$648$	0	0
$649$	5796.09	0.350564
$650$	0	0
$651$	7556.50	0.454935
$652$	0	0
$653$	−18100.5	−1.08473	−0.542363	−	0.840144i	$-0.682470\pi$
$653$	−18100.5	−1.08473	−0.542363	+	0.840144i	$0.682470\pi$
$654$	0	0
$655$	65.5699	0.00391149
$656$	0	0
$657$	15455.7	0.917787
$658$	0	0
$659$	−331.740	−0.0196096	−0.00980481	−	0.999952i	$-0.503121\pi$
$659$	−331.740	−0.0196096	−0.00980481	+	0.999952i	$0.503121\pi$
$660$	0	0
$661$	−30555.8	−1.79801	−0.899004	−	0.437939i	$-0.855708\pi$
$661$	−30555.8	−1.79801	−0.899004	+	0.437939i	$0.855708\pi$
$662$	0	0
$663$	−11796.0	−0.690979
$664$	0	0
$665$	−751.133	−0.0438010
$666$	0	0
$667$	33100.9	1.92155
$668$	0	0
$669$	49015.5	2.83266
$670$	0	0
$671$	2938.67	0.169070
$672$	0	0
$673$	3261.14	0.186787	0.0933936	−	0.995629i	$-0.470228\pi$
$673$	3261.14	0.186787	0.0933936	+	0.995629i	$0.470228\pi$
$674$	0	0
$675$	−21684.4	−1.23649
$676$	0	0
$677$	11556.6	0.656063	0.328031	−	0.944667i	$-0.393615\pi$
$677$	11556.6	0.656063	0.328031	+	0.944667i	$0.393615\pi$
$678$	0	0
$679$	−25768.5	−1.45641
$680$	0	0
$681$	44069.5	2.47980
$682$	0	0
$683$	−19704.1	−1.10389	−0.551944	−	0.833881i	$-0.686114\pi$
$683$	−19704.1	−1.10389	−0.551944	+	0.833881i	$0.686114\pi$
$684$	0	0
$685$	1873.99	0.104528
$686$	0	0
$687$	4352.03	0.241689
$688$	0	0
$689$	13932.9	0.770393
$690$	0	0
$691$	2956.51	0.162765	0.0813827	−	0.996683i	$-0.474066\pi$
$691$	2956.51	0.162765	0.0813827	+	0.996683i	$0.474066\pi$
$692$	0	0
$693$	−9180.49	−0.503230
$694$	0	0
$695$	1744.47	0.0952109
$696$	0	0
$697$	−7940.63	−0.431525
$698$	0	0
$699$	24357.3	1.31799
$700$	0	0
$701$	29022.1	1.56370	0.781848	−	0.623469i	$-0.214277\pi$
$701$	29022.1	1.56370	0.781848	+	0.623469i	$0.214277\pi$
$702$	0	0
$703$	−6482.18	−0.347767
$704$	0	0
$705$	3157.13	0.168658
$706$	0	0
$707$	−14523.0	−0.772551
$708$	0	0
$709$	−5110.64	−0.270711	−0.135356	−	0.990797i	$-0.543218\pi$
$709$	−5110.64	−0.270711	−0.135356	+	0.990797i	$0.543218\pi$
$710$	0	0
$711$	4298.19	0.226716
$712$	0	0
$713$	6471.73	0.339927
$714$	0	0
$715$	1550.53	0.0811003
$716$	0	0
$717$	−19416.9	−1.01135
$718$	0	0
$719$	25225.9	1.30844	0.654219	−	0.756305i	$-0.272998\pi$
$719$	25225.9	1.30844	0.654219	+	0.756305i	$0.272998\pi$
$720$	0	0
$721$	−2615.45	−0.135096
$722$	0	0
$723$	−62547.4	−3.21738
$724$	0	0
$725$	−34867.2	−1.78612
$726$	0	0
$727$	12817.7	0.653893	0.326947	−	0.945043i	$-0.393980\pi$
$727$	12817.7	0.653893	0.326947	+	0.945043i	$0.393980\pi$
$728$	0	0
$729$	−28962.0	−1.47142
$730$	0	0
$731$	9490.21	0.480175
$732$	0	0
$733$	−15307.5	−0.771342	−0.385671	−	0.922636i	$-0.626030\pi$
$733$	−15307.5	−0.771342	−0.385671	+	0.922636i	$0.626030\pi$
$734$	0	0
$735$	2605.20	0.130741
$736$	0	0
$737$	−4047.37	−0.202289
$738$	0	0
$739$	−34340.1	−1.70936	−0.854682	−	0.519152i	$-0.826248\pi$
$739$	−34340.1	−1.70936	−0.854682	+	0.519152i	$0.826248\pi$
$740$	0	0
$741$	7735.22	0.383482
$742$	0	0
$743$	1883.40	0.0929948	0.0464974	−	0.998918i	$-0.485194\pi$
$743$	1883.40	0.0929948	0.0464974	+	0.998918i	$0.485194\pi$
$744$	0	0
$745$	701.547	0.0345003
$746$	0	0
$747$	21167.9	1.03681
$748$	0	0
$749$	−13913.7	−0.678766
$750$	0	0
$751$	9431.98	0.458292	0.229146	−	0.973392i	$-0.426407\pi$
$751$	9431.98	0.458292	0.229146	+	0.973392i	$0.426407\pi$
$752$	0	0
$753$	−53865.0	−2.60684
$754$	0	0
$755$	7352.64	0.354424
$756$	0	0
$757$	−12355.0	−0.593196	−0.296598	−	0.955002i	$-0.595852\pi$
$757$	−12355.0	−0.593196	−0.296598	+	0.955002i	$0.595852\pi$
$758$	0	0
$759$	−12269.8	−0.586780
$760$	0	0
$761$	−27257.6	−1.29841	−0.649204	−	0.760614i	$-0.724898\pi$
$761$	−27257.6	−1.29841	−0.649204	+	0.760614i	$0.724898\pi$
$762$	0	0
$763$	8345.23	0.395960
$764$	0	0
$765$	3652.59	0.172627
$766$	0	0
$767$	−21571.3	−1.01551
$768$	0	0
$769$	−1191.72	−0.0558835	−0.0279418	−	0.999610i	$-0.508895\pi$
$769$	−1191.72	−0.0558835	−0.0279418	+	0.999610i	$0.508895\pi$
$770$	0	0
$771$	−24244.5	−1.13248
$772$	0	0
$773$	−7481.03	−0.348091	−0.174045	−	0.984738i	$-0.555684\pi$
$773$	−7481.03	−0.348091	−0.174045	+	0.984738i	$0.555684\pi$
$774$	0	0
$775$	−6817.05	−0.315969
$776$	0	0
$777$	−44681.5	−2.06299
$778$	0	0
$779$	5207.06	0.239489
$780$	0	0
$781$	−120.432	−0.00551781
$782$	0	0
$783$	54161.4	2.47199
$784$	0	0
$785$	5233.61	0.237956
$786$	0	0
$787$	−27403.6	−1.24121	−0.620604	−	0.784124i	$-0.713113\pi$
$787$	−27403.6	−1.24121	−0.620604	+	0.784124i	$0.713113\pi$
$788$	0	0
$789$	−25801.8	−1.16422
$790$	0	0
$791$	−21076.5	−0.947399
$792$	0	0
$793$	−10936.8	−0.489758
$794$	0	0
$795$	−6732.54	−0.300350
$796$	0	0
$797$	30558.5	1.35814	0.679070	−	0.734073i	$-0.262383\pi$
$797$	30558.5	1.35814	0.679070	+	0.734073i	$0.262383\pi$
$798$	0	0
$799$	4031.52	0.178504
$800$	0	0
$801$	42540.8	1.87654
$802$	0	0
$803$	−4048.42	−0.177915
$804$	0	0
$805$	−4434.29	−0.194147
$806$	0	0
$807$	−220.854	−0.00963374
$808$	0	0
$809$	−28018.7	−1.21766	−0.608829	−	0.793302i	$-0.708360\pi$
$809$	−28018.7	−1.21766	−0.608829	+	0.793302i	$0.708360\pi$
$810$	0	0
$811$	−2520.45	−0.109131	−0.0545654	−	0.998510i	$-0.517377\pi$
$811$	−2520.45	−0.109131	−0.0545654	+	0.998510i	$0.517377\pi$
$812$	0	0
$813$	37612.0	1.62252
$814$	0	0
$815$	4297.49	0.184705
$816$	0	0
$817$	−6223.19	−0.266490
$818$	0	0
$819$	34167.0	1.45774
$820$	0	0
$821$	21887.7	0.930434	0.465217	−	0.885197i	$-0.345976\pi$
$821$	21887.7	0.930434	0.465217	+	0.885197i	$0.345976\pi$
$822$	0	0
$823$	−8149.95	−0.345188	−0.172594	−	0.984993i	$-0.555215\pi$
$823$	−8149.95	−0.345188	−0.172594	+	0.984993i	$0.555215\pi$
$824$	0	0
$825$	12924.5	0.545423
$826$	0	0
$827$	−22007.0	−0.925345	−0.462672	−	0.886529i	$-0.653109\pi$
$827$	−22007.0	−0.925345	−0.462672	+	0.886529i	$0.653109\pi$
$828$	0	0
$829$	8082.69	0.338629	0.169314	−	0.985562i	$-0.445845\pi$
$829$	8082.69	0.338629	0.169314	+	0.985562i	$0.445845\pi$
$830$	0	0
$831$	37073.0	1.54759
$832$	0	0
$833$	3326.74	0.138373
$834$	0	0
$835$	7762.12	0.321700
$836$	0	0
$837$	10589.4	0.437302
$838$	0	0
$839$	1144.88	0.0471105	0.0235552	−	0.999723i	$-0.492501\pi$
$839$	1144.88	0.0471105	0.0235552	+	0.999723i	$0.492501\pi$
$840$	0	0
$841$	62699.3	2.57080
$842$	0	0
$843$	−32221.3	−1.31644
$844$	0	0
$845$	−20.8479	−0.000848746 0
$846$	0	0
$847$	−17701.1	−0.718082
$848$	0	0
$849$	−32495.2	−1.31358
$850$	0	0
$851$	−38267.3	−1.54146
$852$	0	0
$853$	−12853.7	−0.515948	−0.257974	−	0.966152i	$-0.583055\pi$
$853$	−12853.7	−0.515948	−0.257974	+	0.966152i	$0.583055\pi$
$854$	0	0
$855$	−2395.18	−0.0958052
$856$	0	0
$857$	24528.1	0.977669	0.488834	−	0.872377i	$-0.337422\pi$
$857$	24528.1	0.977669	0.488834	+	0.872377i	$0.337422\pi$
$858$	0	0
$859$	37536.3	1.49095	0.745473	−	0.666536i	$-0.232224\pi$
$859$	37536.3	1.49095	0.745473	+	0.666536i	$0.232224\pi$
$860$	0	0
$861$	35892.2	1.42068
$862$	0	0
$863$	2098.30	0.0827659	0.0413830	−	0.999143i	$-0.486824\pi$
$863$	2098.30	0.0827659	0.0413830	+	0.999143i	$0.486824\pi$
$864$	0	0
$865$	−242.833	−0.00954519
$866$	0	0
$867$	−35317.0	−1.38342
$868$	0	0
$869$	−1125.85	−0.0439493
$870$	0	0
$871$	15063.1	0.585986
$872$	0	0
$873$	−82169.3	−3.18558
$874$	0	0
$875$	9612.56	0.371387
$876$	0	0
$877$	−9857.12	−0.379534	−0.189767	−	0.981829i	$-0.560773\pi$
$877$	−9857.12	−0.379534	−0.189767	+	0.981829i	$0.560773\pi$
$878$	0	0
$879$	71181.5	2.73139
$880$	0	0
$881$	2301.91	0.0880289	0.0440144	−	0.999031i	$-0.485985\pi$
$881$	2301.91	0.0880289	0.0440144	+	0.999031i	$0.485985\pi$
$882$	0	0
$883$	−25401.4	−0.968093	−0.484047	−	0.875042i	$-0.660834\pi$
$883$	−25401.4	−0.968093	−0.484047	+	0.875042i	$0.660834\pi$
$884$	0	0
$885$	10423.5	0.395912
$886$	0	0
$887$	11451.6	0.433493	0.216746	−	0.976228i	$-0.430456\pi$
$887$	11451.6	0.433493	0.216746	+	0.976228i	$0.430456\pi$
$888$	0	0
$889$	−11979.1	−0.451929
$890$	0	0
$891$	−3667.30	−0.137889
$892$	0	0
$893$	−2643.67	−0.0990671
$894$	0	0
$895$	−8620.91	−0.321972
$896$	0	0
$897$	45664.6	1.69977
$898$	0	0
$899$	17027.1	0.631685
$900$	0	0
$901$	−8597.18	−0.317884
$902$	0	0
$903$	−42896.4	−1.58084
$904$	0	0
$905$	−9395.83	−0.345114
$906$	0	0
$907$	−53074.7	−1.94302	−0.971508	−	0.237005i	$-0.923834\pi$
$907$	−53074.7	−1.94302	−0.971508	+	0.237005i	$0.923834\pi$
$908$	0	0
$909$	−46310.3	−1.68979
$910$	0	0
$911$	20284.7	0.737718	0.368859	−	0.929485i	$-0.379749\pi$
$911$	20284.7	0.737718	0.368859	+	0.929485i	$0.379749\pi$
$912$	0	0
$913$	−5544.65	−0.200987
$914$	0	0
$915$	5284.81	0.190940
$916$	0	0
$917$	−378.466	−0.0136293
$918$	0	0
$919$	−34782.0	−1.24848	−0.624240	−	0.781233i	$-0.714591\pi$
$919$	−34782.0	−1.24848	−0.624240	+	0.781233i	$0.714591\pi$
$920$	0	0
$921$	24404.2	0.873121
$922$	0	0
$923$	448.213	0.0159839
$924$	0	0
$925$	40309.2	1.43282
$926$	0	0
$927$	−8340.02	−0.295493
$928$	0	0
$929$	−15586.2	−0.550448	−0.275224	−	0.961380i	$-0.588752\pi$
$929$	−15586.2	−0.550448	−0.275224	+	0.961380i	$0.588752\pi$
$930$	0	0
$931$	−2181.50	−0.0767948
$932$	0	0
$933$	75001.8	2.63178
$934$	0	0
$935$	−956.746	−0.0334641
$936$	0	0
$937$	−15194.9	−0.529770	−0.264885	−	0.964280i	$-0.585334\pi$
$937$	−15194.9	−0.529770	−0.264885	+	0.964280i	$0.585334\pi$
$938$	0	0
$939$	14075.6	0.489179
$940$	0	0
$941$	−48650.1	−1.68538	−0.842692	−	0.538396i	$-0.819031\pi$
$941$	−48650.1	−1.68538	−0.842692	+	0.538396i	$0.819031\pi$
$942$	0	0
$943$	30739.7	1.06153
$944$	0	0
$945$	−7255.60	−0.249762
$946$	0	0
$947$	−3258.29	−0.111806	−0.0559030	−	0.998436i	$-0.517804\pi$
$947$	−3258.29	−0.111806	−0.0559030	+	0.998436i	$0.517804\pi$
$948$	0	0
$949$	15067.0	0.515380
$950$	0	0
$951$	81257.8	2.77073
$952$	0	0
$953$	−44488.8	−1.51221	−0.756104	−	0.654451i	$-0.772900\pi$
$953$	−44488.8	−1.51221	−0.756104	+	0.654451i	$0.772900\pi$
$954$	0	0
$955$	1257.47	0.0426080
$956$	0	0
$957$	−32281.8	−1.09041
$958$	0	0
$959$	−10816.6	−0.364218
$960$	0	0
$961$	−26462.0	−0.888253
$962$	0	0
$963$	−44367.4	−1.48465
$964$	0	0
$965$	−9170.51	−0.305916
$966$	0	0
$967$	5791.53	0.192599	0.0962994	−	0.995352i	$-0.469299\pi$
$967$	5791.53	0.192599	0.0962994	+	0.995352i	$0.469299\pi$
$968$	0	0
$969$	−4772.96	−0.158235
$970$	0	0
$971$	17829.5	0.589265	0.294632	−	0.955611i	$-0.404803\pi$
$971$	17829.5	0.589265	0.294632	+	0.955611i	$0.404803\pi$
$972$	0	0
$973$	−10069.0	−0.331754
$974$	0	0
$975$	−48101.2	−1.57997
$976$	0	0
$977$	−2645.66	−0.0866349	−0.0433174	−	0.999061i	$-0.513793\pi$
$977$	−2645.66	−0.0866349	−0.0433174	+	0.999061i	$0.513793\pi$
$978$	0	0
$979$	−11143.0	−0.363770
$980$	0	0
$981$	26610.9	0.866076
$982$	0	0
$983$	−3880.50	−0.125909	−0.0629545	−	0.998016i	$-0.520052\pi$
$983$	−3880.50	−0.125909	−0.0629545	+	0.998016i	$0.520052\pi$
$984$	0	0
$985$	−1580.18	−0.0511155
$986$	0	0
$987$	−18222.7	−0.587676
$988$	0	0
$989$	−36738.4	−1.18121
$990$	0	0
$991$	−57787.1	−1.85234	−0.926170	−	0.377106i	$-0.876919\pi$
$991$	−57787.1	−1.85234	−0.926170	+	0.377106i	$0.876919\pi$
$992$	0	0
$993$	15307.3	0.489186
$994$	0	0
$995$	8018.27	0.255474
$996$	0	0
$997$	30649.0	0.973585	0.486793	−	0.873518i	$-0.338167\pi$
$997$	30649.0	0.973585	0.486793	+	0.873518i	$0.338167\pi$
$998$	0	0
$999$	−62614.9	−1.98303

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.4.a.i.1.3		3
4.3	odd	2		19.4.a.b.1.3	✓	3
8.3	odd	2		1216.4.a.s.1.3		3
8.5	even	2		1216.4.a.u.1.1		3
12.11	even	2		171.4.a.f.1.1		3
20.3	even	4		475.4.b.f.324.1		6
20.7	even	4		475.4.b.f.324.6		6
20.19	odd	2		475.4.a.f.1.1		3
28.27	even	2		931.4.a.c.1.3		3
44.43	even	2		2299.4.a.h.1.1		3
76.75	even	2		361.4.a.i.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.a.b.1.3	✓	3	4.3	odd	2
171.4.a.f.1.1		3	12.11	even	2
304.4.a.i.1.3		3	1.1	even	1	trivial
361.4.a.i.1.1		3	76.75	even	2
475.4.a.f.1.1		3	20.19	odd	2
475.4.b.f.324.1		6	20.3	even	4
475.4.b.f.324.6		6	20.7	even	4
931.4.a.c.1.3		3	28.27	even	2
1216.4.a.s.1.3		3	8.3	odd	2
1216.4.a.u.1.1		3	8.5	even	2
2299.4.a.h.1.1		3	44.43	even	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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	304.a (trivial)

\(f(q)\)	\(=\)	\(q+8.66998 q^{3} -2.61710 q^{5} +15.1058 q^{7} +48.1686 q^{9} -12.6171 q^{11} +46.9571 q^{13} -22.6902 q^{15} -28.9745 q^{17} +19.0000 q^{19} +130.967 q^{21} +112.166 q^{23} -118.151 q^{25} +183.531 q^{27} +295.107 q^{29} +57.6979 q^{31} -109.390 q^{33} -39.5333 q^{35} -341.167 q^{37} +407.117 q^{39} +274.056 q^{41} -327.536 q^{43} -126.062 q^{45} -139.140 q^{47} -114.816 q^{49} -251.209 q^{51} +296.715 q^{53} +33.0202 q^{55} +164.730 q^{57} -459.383 q^{59} -232.911 q^{61} +727.623 q^{63} -122.891 q^{65} +320.784 q^{67} +972.475 q^{69} +9.54518 q^{71} +320.868 q^{73} -1024.37 q^{75} -190.591 q^{77} +89.2323 q^{79} +290.661 q^{81} +439.455 q^{83} +75.8293 q^{85} +2558.58 q^{87} +883.164 q^{89} +709.322 q^{91} +500.240 q^{93} -49.7249 q^{95} -1705.87 q^{97} -607.748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + 14 q^{5} + 35 q^{7} + 48 q^{9} - 16 q^{11} + 65 q^{13} - 140 q^{15} + 29 q^{17} + 57 q^{19} - 25 q^{21} + 101 q^{23} - 37 q^{25} + 377 q^{27} + 377 q^{29} + 140 q^{31} - 130 q^{33} + 438 q^{35}+ \cdots - 250 q^{99}+O(q^{100}) \) 3 * q - q^3 + 14 * q^5 + 35 * q^7 + 48 * q^9 - 16 * q^11 + 65 * q^13 - 140 * q^15 + 29 * q^17 + 57 * q^19 - 25 * q^21 + 101 * q^23 - 37 * q^25 + 377 * q^27 + 377 * q^29 + 140 * q^31 - 130 * q^33 + 438 * q^35 - 290 * q^37 + 371 * q^39 + 956 * q^41 + 570 * q^43 + 230 * q^45 - 66 * q^47 - 98 * q^49 - 187 * q^51 + 817 * q^53 + 198 * q^55 - 19 * q^57 - 265 * q^59 + 988 * q^61 + 1304 * q^63 - 240 * q^65 + 207 * q^67 + 807 * q^69 - 846 * q^71 + 627 * q^73 - 2031 * q^75 + 88 * q^77 - 382 * q^79 - 501 * q^81 + 766 * q^83 - 1242 * q^85 + 1671 * q^87 - 172 * q^89 + 457 * q^91 + 892 * q^93 + 266 * q^95 - 2450 * q^97 - 250 * q^99

Introduction

L-functions

Modular forms

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Database

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