LMFDB - Embedded newform 312.4.a.f.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [312,4,Mod(1,312)] 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("312.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(312, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 312 = 2^{3} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	312.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,6,0,12,0,44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.4085959218$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{43}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 43 $ x^2 - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.55744$ of defining polynomial
Character	$\chi$	$=$	312.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +19.1149 q^{5} +35.1149 q^{7} +9.00000 q^{9} +26.0000 q^{11} -13.0000 q^{13} +57.3446 q^{15} -36.2298 q^{17} -95.5744 q^{19} +105.345 q^{21} -161.379 q^{23} +240.379 q^{25} +27.0000 q^{27} -91.3785 q^{29} -266.723 q^{31} +78.0000 q^{33} +671.217 q^{35} -149.608 q^{37} -39.0000 q^{39} -77.8041 q^{41} +183.608 q^{43} +172.034 q^{45} -60.6893 q^{47} +890.055 q^{49} -108.689 q^{51} +281.540 q^{53} +496.987 q^{55} -286.723 q^{57} -542.527 q^{59} +65.0810 q^{61} +316.034 q^{63} -248.493 q^{65} -1033.94 q^{67} -484.136 q^{69} +1041.81 q^{71} +483.311 q^{73} +721.136 q^{75} +912.987 q^{77} -1337.05 q^{79} +81.0000 q^{81} +812.825 q^{83} -692.527 q^{85} -274.136 q^{87} +936.885 q^{89} -456.493 q^{91} -800.169 q^{93} -1826.89 q^{95} -954.068 q^{97} +234.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 12 q^{5} + 44 q^{7} + 18 q^{9} + 52 q^{11} - 26 q^{13} + 36 q^{15} - 20 q^{17} - 60 q^{19} + 132 q^{21} - 8 q^{23} + 166 q^{25} + 54 q^{27} + 132 q^{29} - 140 q^{31} + 156 q^{33} + 608 q^{35}+ \cdots + 468 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 12 * q^5 + 44 * q^7 + 18 * q^9 + 52 * q^11 - 26 * q^13 + 36 * q^15 - 20 * q^17 - 60 * q^19 + 132 * q^21 - 8 * q^23 + 166 * q^25 + 54 * q^27 + 132 * q^29 - 140 * q^31 + 156 * q^33 + 608 * q^35 + 68 * q^37 - 78 * q^39 + 28 * q^41 + 108 * q^45 + 36 * q^47 + 626 * q^49 - 60 * q^51 + 668 * q^53 + 312 * q^55 - 180 * q^57 - 508 * q^59 + 340 * q^61 + 396 * q^63 - 156 * q^65 - 940 * q^67 - 24 * q^69 + 300 * q^71 + 1124 * q^73 + 498 * q^75 + 1144 * q^77 - 1520 * q^79 + 162 * q^81 + 524 * q^83 - 808 * q^85 + 396 * q^87 + 1900 * q^89 - 572 * q^91 - 420 * q^93 - 2080 * q^95 - 1436 * q^97 + 468 * 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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	19.1149	1.70969	0.854843	−	0.518886i	$-0.173653\pi$
$5$	19.1149	1.70969	0.854843	+	0.518886i	$0.173653\pi$
$6$	0	0
$7$	35.1149	1.89603	0.948013	−	0.318233i	$-0.103089\pi$
$7$	35.1149	1.89603	0.948013	+	0.318233i	$0.103089\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	26.0000	0.712663	0.356332	−	0.934360i	$-0.384027\pi$
$11$	26.0000	0.712663	0.356332	+	0.934360i	$0.384027\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	57.3446	0.987088
$16$	0	0
$17$	−36.2298	−0.516883	−0.258441	−	0.966027i	$-0.583209\pi$
$17$	−36.2298	−0.516883	−0.258441	+	0.966027i	$0.583209\pi$
$18$	0	0
$19$	−95.5744	−1.15401	−0.577007	−	0.816739i	$-0.695779\pi$
$19$	−95.5744	−1.15401	−0.577007	+	0.816739i	$0.695779\pi$
$20$	0	0
$21$	105.345	1.09467
$22$	0	0
$23$	−161.379	−1.46303	−0.731516	−	0.681824i	$-0.761187\pi$
$23$	−161.379	−1.46303	−0.731516	+	0.681824i	$0.761187\pi$
$24$	0	0
$25$	240.379	1.92303
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−91.3785	−0.585123	−0.292561	−	0.956247i	$-0.594508\pi$
$29$	−91.3785	−0.585123	−0.292561	+	0.956247i	$0.594508\pi$
$30$	0	0
$31$	−266.723	−1.54532	−0.772660	−	0.634821i	$-0.781074\pi$
$31$	−266.723	−1.54532	−0.772660	+	0.634821i	$0.781074\pi$
$32$	0	0
$33$	78.0000	0.411456
$34$	0	0
$35$	671.217	3.24161
$36$	0	0
$37$	−149.608	−0.664742	−0.332371	−	0.943149i	$-0.607849\pi$
$37$	−149.608	−0.664742	−0.332371	+	0.943149i	$0.607849\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	−77.8041	−0.296365	−0.148183	−	0.988960i	$-0.547342\pi$
$41$	−77.8041	−0.296365	−0.148183	+	0.988960i	$0.547342\pi$
$42$	0	0
$43$	183.608	0.651163	0.325581	−	0.945514i	$-0.394440\pi$
$43$	183.608	0.651163	0.325581	+	0.945514i	$0.394440\pi$
$44$	0	0
$45$	172.034	0.569896
$46$	0	0
$47$	−60.6893	−0.188350	−0.0941749	−	0.995556i	$-0.530021\pi$
$47$	−60.6893	−0.188350	−0.0941749	+	0.995556i	$0.530021\pi$
$48$	0	0
$49$	890.055	2.59491
$50$	0	0
$51$	−108.689	−0.298422
$52$	0	0
$53$	281.540	0.729671	0.364835	−	0.931072i	$-0.381125\pi$
$53$	281.540	0.729671	0.364835	+	0.931072i	$0.381125\pi$
$54$	0	0
$55$	496.987	1.21843
$56$	0	0
$57$	−286.723	−0.666270
$58$	0	0
$59$	−542.527	−1.19714	−0.598568	−	0.801072i	$-0.704263\pi$
$59$	−542.527	−1.19714	−0.598568	+	0.801072i	$0.704263\pi$
$60$	0	0
$61$	65.0810	0.136603	0.0683014	−	0.997665i	$-0.478242\pi$
$61$	65.0810	0.136603	0.0683014	+	0.997665i	$0.478242\pi$
$62$	0	0
$63$	316.034	0.632008
$64$	0	0
$65$	−248.493	−0.474182
$66$	0	0
$67$	−1033.94	−1.88531	−0.942656	−	0.333767i	$-0.891680\pi$
$67$	−1033.94	−1.88531	−0.942656	+	0.333767i	$0.891680\pi$
$68$	0	0
$69$	−484.136	−0.844682
$70$	0	0
$71$	1041.81	1.74141	0.870706	−	0.491803i	$-0.163662\pi$
$71$	1041.81	1.74141	0.870706	+	0.491803i	$0.163662\pi$
$72$	0	0
$73$	483.311	0.774894	0.387447	−	0.921892i	$-0.373357\pi$
$73$	483.311	0.774894	0.387447	+	0.921892i	$0.373357\pi$
$74$	0	0
$75$	721.136	1.11026
$76$	0	0
$77$	912.987	1.35123
$78$	0	0
$79$	−1337.05	−1.90418	−0.952091	−	0.305815i	$-0.901071\pi$
$79$	−1337.05	−1.90418	−0.952091	+	0.305815i	$0.901071\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	812.825	1.07493	0.537465	−	0.843286i	$-0.319382\pi$
$83$	812.825	1.07493	0.537465	+	0.843286i	$0.319382\pi$
$84$	0	0
$85$	−692.527	−0.883707
$86$	0	0
$87$	−274.136	−0.337821
$88$	0	0
$89$	936.885	1.11584	0.557919	−	0.829895i	$-0.311600\pi$
$89$	936.885	1.11584	0.557919	+	0.829895i	$0.311600\pi$
$90$	0	0
$91$	−456.493	−0.525863
$92$	0	0
$93$	−800.169	−0.892190
$94$	0	0
$95$	−1826.89	−1.97300
$96$	0	0
$97$	−954.068	−0.998669	−0.499335	−	0.866409i	$-0.666422\pi$
$97$	−954.068	−0.998669	−0.499335	+	0.866409i	$0.666422\pi$
$98$	0	0
$99$	234.000	0.237554
$100$	0	0
$101$	1401.15	1.38039	0.690196	−	0.723623i	$-0.257524\pi$
$101$	1401.15	1.38039	0.690196	+	0.723623i	$0.257524\pi$
$102$	0	0
$103$	954.960	0.913544	0.456772	−	0.889584i	$-0.349005\pi$
$103$	954.960	0.913544	0.456772	+	0.889584i	$0.349005\pi$
$104$	0	0
$105$	2013.65	1.87154
$106$	0	0
$107$	−924.595	−0.835364	−0.417682	−	0.908593i	$-0.637157\pi$
$107$	−924.595	−0.835364	−0.417682	+	0.908593i	$0.637157\pi$
$108$	0	0
$109$	172.893	0.151928	0.0759638	−	0.997111i	$-0.475797\pi$
$109$	172.893	0.151928	0.0759638	+	0.997111i	$0.475797\pi$
$110$	0	0
$111$	−448.825	−0.383789
$112$	0	0
$113$	1241.81	1.03380	0.516902	−	0.856045i	$-0.327085\pi$
$113$	1241.81	1.03380	0.516902	+	0.856045i	$0.327085\pi$
$114$	0	0
$115$	−3084.73	−2.50133
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	−1272.20	−0.980023
$120$	0	0
$121$	−655.000	−0.492111
$122$	0	0
$123$	−233.412	−0.171106
$124$	0	0
$125$	2205.45	1.57809
$126$	0	0
$127$	−1079.93	−0.754555	−0.377278	−	0.926100i	$-0.623140\pi$
$127$	−1079.93	−0.754555	−0.377278	+	0.926100i	$0.623140\pi$
$128$	0	0
$129$	550.825	0.375949
$130$	0	0
$131$	−918.026	−0.612277	−0.306139	−	0.951987i	$-0.599037\pi$
$131$	−918.026	−0.612277	−0.306139	+	0.951987i	$0.599037\pi$
$132$	0	0
$133$	−3356.08	−2.18804
$134$	0	0
$135$	516.102	0.329029
$136$	0	0
$137$	1311.06	0.817603	0.408801	−	0.912623i	$-0.365947\pi$
$137$	1311.06	0.817603	0.408801	+	0.912623i	$0.365947\pi$
$138$	0	0
$139$	−227.593	−0.138879	−0.0694396	−	0.997586i	$-0.522121\pi$
$139$	−227.593	−0.138879	−0.0694396	+	0.997586i	$0.522121\pi$
$140$	0	0
$141$	−182.068	−0.108744
$142$	0	0
$143$	−338.000	−0.197657
$144$	0	0
$145$	−1746.69	−1.00038
$146$	0	0
$147$	2670.16	1.49817
$148$	0	0
$149$	1325.48	0.728776	0.364388	−	0.931247i	$-0.381278\pi$
$149$	1325.48	0.728776	0.364388	+	0.931247i	$0.381278\pi$
$150$	0	0
$151$	252.426	0.136040	0.0680202	−	0.997684i	$-0.478332\pi$
$151$	252.426	0.136040	0.0680202	+	0.997684i	$0.478332\pi$
$152$	0	0
$153$	−326.068	−0.172294
$154$	0	0
$155$	−5098.38	−2.64201
$156$	0	0
$157$	−2268.62	−1.15322	−0.576611	−	0.817019i	$-0.695625\pi$
$157$	−2268.62	−1.15322	−0.576611	+	0.817019i	$0.695625\pi$
$158$	0	0
$159$	844.621	0.421276
$160$	0	0
$161$	−5666.79	−2.77395
$162$	0	0
$163$	−1083.45	−0.520630	−0.260315	−	0.965524i	$-0.583826\pi$
$163$	−1083.45	−0.520630	−0.260315	+	0.965524i	$0.583826\pi$
$164$	0	0
$165$	1490.96	0.703461
$166$	0	0
$167$	−796.041	−0.368859	−0.184430	−	0.982846i	$-0.559044\pi$
$167$	−796.041	−0.368859	−0.184430	+	0.982846i	$0.559044\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−860.169	−0.384671
$172$	0	0
$173$	−1.09598	−0.000481654 0	−0.000240827	−	1.00000i	$-0.500077\pi$
$173$	−1.09598	−0.000481654 0	−0.000240827		1.00000i	$0.500077\pi$
$174$	0	0
$175$	8440.86	3.64611
$176$	0	0
$177$	−1627.58	−0.691167
$178$	0	0
$179$	3250.06	1.35710	0.678549	−	0.734555i	$-0.262609\pi$
$179$	3250.06	1.35710	0.678549	+	0.734555i	$0.262609\pi$
$180$	0	0
$181$	2980.84	1.22411	0.612055	−	0.790815i	$-0.290343\pi$
$181$	2980.84	1.22411	0.612055	+	0.790815i	$0.290343\pi$
$182$	0	0
$183$	195.243	0.0788676
$184$	0	0
$185$	−2859.74	−1.13650
$186$	0	0
$187$	−941.974	−0.368363
$188$	0	0
$189$	948.102	0.364890
$190$	0	0
$191$	612.595	0.232072	0.116036	−	0.993245i	$-0.462981\pi$
$191$	612.595	0.232072	0.116036	+	0.993245i	$0.462981\pi$
$192$	0	0
$193$	−4185.81	−1.56115	−0.780573	−	0.625064i	$-0.785073\pi$
$193$	−4185.81	−1.56115	−0.780573	+	0.625064i	$0.785073\pi$
$194$	0	0
$195$	−745.480	−0.273769
$196$	0	0
$197$	2.06029	0.000745124 0	0.000372562	−	1.00000i	$-0.499881\pi$
$197$	2.06029	0.000745124 0	0.000372562		1.00000i	$0.499881\pi$
$198$	0	0
$199$	1492.39	0.531622	0.265811	−	0.964025i	$-0.414360\pi$
$199$	1492.39	0.531622	0.265811	+	0.964025i	$0.414360\pi$
$200$	0	0
$201$	−3101.82	−1.08848
$202$	0	0
$203$	−3208.75	−1.10941
$204$	0	0
$205$	−1487.22	−0.506691
$206$	0	0
$207$	−1452.41	−0.487678
$208$	0	0
$209$	−2484.93	−0.822423
$210$	0	0
$211$	2748.27	0.896677	0.448338	−	0.893864i	$-0.352016\pi$
$211$	2748.27	0.896677	0.448338	+	0.893864i	$0.352016\pi$
$212$	0	0
$213$	3125.43	1.00541
$214$	0	0
$215$	3509.65	1.11328
$216$	0	0
$217$	−9365.95	−2.92996
$218$	0	0
$219$	1449.93	0.447385
$220$	0	0
$221$	470.987	0.143357
$222$	0	0
$223$	186.655	0.0560510	0.0280255	−	0.999607i	$-0.491078\pi$
$223$	186.655	0.0560510	0.0280255	+	0.999607i	$0.491078\pi$
$224$	0	0
$225$	2163.41	0.641009
$226$	0	0
$227$	3932.57	1.14984	0.574920	−	0.818210i	$-0.305033\pi$
$227$	3932.57	1.14984	0.574920	+	0.818210i	$0.305033\pi$
$228$	0	0
$229$	5951.00	1.71726	0.858632	−	0.512593i	$-0.171315\pi$
$229$	5951.00	1.71726	0.858632	+	0.512593i	$0.171315\pi$
$230$	0	0
$231$	2738.96	0.780131
$232$	0	0
$233$	6548.94	1.84135	0.920676	−	0.390327i	$-0.127638\pi$
$233$	6548.94	1.84135	0.920676	+	0.390327i	$0.127638\pi$
$234$	0	0
$235$	−1160.07	−0.322019
$236$	0	0
$237$	−4011.16	−1.09938
$238$	0	0
$239$	−3462.35	−0.937076	−0.468538	−	0.883443i	$-0.655219\pi$
$239$	−3462.35	−0.937076	−0.468538	+	0.883443i	$0.655219\pi$
$240$	0	0
$241$	4581.96	1.22469	0.612345	−	0.790591i	$-0.290226\pi$
$241$	4581.96	1.22469	0.612345	+	0.790591i	$0.290226\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	17013.3	4.43649
$246$	0	0
$247$	1242.47	0.320066
$248$	0	0
$249$	2438.47	0.620611
$250$	0	0
$251$	4746.76	1.19368	0.596838	−	0.802362i	$-0.296424\pi$
$251$	4746.76	1.19368	0.596838	+	0.802362i	$0.296424\pi$
$252$	0	0
$253$	−4195.84	−1.04265
$254$	0	0
$255$	−2077.58	−0.510209
$256$	0	0
$257$	−3373.64	−0.818840	−0.409420	−	0.912346i	$-0.634269\pi$
$257$	−3373.64	−0.818840	−0.409420	+	0.912346i	$0.634269\pi$
$258$	0	0
$259$	−5253.48	−1.26037
$260$	0	0
$261$	−822.407	−0.195041
$262$	0	0
$263$	3527.19	0.826981	0.413491	−	0.910508i	$-0.364309\pi$
$263$	3527.19	0.826981	0.413491	+	0.910508i	$0.364309\pi$
$264$	0	0
$265$	5381.61	1.24751
$266$	0	0
$267$	2810.66	0.644230
$268$	0	0
$269$	1144.30	0.259365	0.129682	−	0.991556i	$-0.458604\pi$
$269$	1144.30	0.259365	0.129682	+	0.991556i	$0.458604\pi$
$270$	0	0
$271$	2254.25	0.505298	0.252649	−	0.967558i	$-0.418698\pi$
$271$	2254.25	0.505298	0.252649	+	0.967558i	$0.418698\pi$
$272$	0	0
$273$	−1369.48	−0.303607
$274$	0	0
$275$	6249.84	1.37047
$276$	0	0
$277$	7101.62	1.54042	0.770208	−	0.637793i	$-0.220152\pi$
$277$	7101.62	1.54042	0.770208	+	0.637793i	$0.220152\pi$
$278$	0	0
$279$	−2400.51	−0.515106
$280$	0	0
$281$	−3831.81	−0.813475	−0.406738	−	0.913545i	$-0.633334\pi$
$281$	−3831.81	−0.813475	−0.406738	+	0.913545i	$0.633334\pi$
$282$	0	0
$283$	−3032.56	−0.636985	−0.318493	−	0.947925i	$-0.603177\pi$
$283$	−3032.56	−0.636985	−0.318493	+	0.947925i	$0.603177\pi$
$284$	0	0
$285$	−5480.68	−1.13911
$286$	0	0
$287$	−2732.08	−0.561915
$288$	0	0
$289$	−3600.40	−0.732832
$290$	0	0
$291$	−2862.20	−0.576582
$292$	0	0
$293$	−2296.44	−0.457882	−0.228941	−	0.973440i	$-0.573526\pi$
$293$	−2296.44	−0.457882	−0.228941	+	0.973440i	$0.573526\pi$
$294$	0	0
$295$	−10370.3	−2.04673
$296$	0	0
$297$	702.000	0.137152
$298$	0	0
$299$	2097.92	0.405772
$300$	0	0
$301$	6447.38	1.23462
$302$	0	0
$303$	4203.45	0.796969
$304$	0	0
$305$	1244.01	0.233548
$306$	0	0
$307$	5174.26	0.961925	0.480962	−	0.876741i	$-0.340287\pi$
$307$	5174.26	0.961925	0.480962	+	0.876741i	$0.340287\pi$
$308$	0	0
$309$	2864.88	0.527435
$310$	0	0
$311$	−7481.54	−1.36411	−0.682057	−	0.731299i	$-0.738914\pi$
$311$	−7481.54	−1.36411	−0.682057	+	0.731299i	$0.738914\pi$
$312$	0	0
$313$	−4158.97	−0.751051	−0.375526	−	0.926812i	$-0.622538\pi$
$313$	−4158.97	−0.751051	−0.375526	+	0.926812i	$0.622538\pi$
$314$	0	0
$315$	6040.95	1.08054
$316$	0	0
$317$	−370.535	−0.0656508	−0.0328254	−	0.999461i	$-0.510451\pi$
$317$	−370.535	−0.0656508	−0.0328254	+	0.999461i	$0.510451\pi$
$318$	0	0
$319$	−2375.84	−0.416996
$320$	0	0
$321$	−2773.79	−0.482298
$322$	0	0
$323$	3462.64	0.596490
$324$	0	0
$325$	−3124.92	−0.533352
$326$	0	0
$327$	518.678	0.0877154
$328$	0	0
$329$	−2131.10	−0.357116
$330$	0	0
$331$	1820.84	0.302364	0.151182	−	0.988506i	$-0.451692\pi$
$331$	1820.84	0.302364	0.151182	+	0.988506i	$0.451692\pi$
$332$	0	0
$333$	−1346.47	−0.221581
$334$	0	0
$335$	−19763.6	−3.22329
$336$	0	0
$337$	−11838.2	−1.91356	−0.956781	−	0.290809i	$-0.906076\pi$
$337$	−11838.2	−1.91356	−0.956781	+	0.290809i	$0.906076\pi$
$338$	0	0
$339$	3725.43	0.596867
$340$	0	0
$341$	−6934.80	−1.10129
$342$	0	0
$343$	19209.8	3.02399
$344$	0	0
$345$	−9254.19	−1.44414
$346$	0	0
$347$	10603.5	1.64043	0.820214	−	0.572057i	$-0.193855\pi$
$347$	10603.5	1.64043	0.820214	+	0.572057i	$0.193855\pi$
$348$	0	0
$349$	−6398.72	−0.981421	−0.490710	−	0.871323i	$-0.663263\pi$
$349$	−6398.72	−0.981421	−0.490710	+	0.871323i	$0.663263\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	−7031.27	−1.06016	−0.530080	−	0.847948i	$-0.677838\pi$
$353$	−7031.27	−1.06016	−0.530080	+	0.847948i	$0.677838\pi$
$354$	0	0
$355$	19914.1	2.97727
$356$	0	0
$357$	−3816.61	−0.565816
$358$	0	0
$359$	5053.75	0.742972	0.371486	−	0.928439i	$-0.378848\pi$
$359$	5053.75	0.742972	0.371486	+	0.928439i	$0.378848\pi$
$360$	0	0
$361$	2275.46	0.331749
$362$	0	0
$363$	−1965.00	−0.284121
$364$	0	0
$365$	9238.43	1.32483
$366$	0	0
$367$	−81.1902	−0.0115479	−0.00577397	−	0.999983i	$-0.501838\pi$
$367$	−81.1902	−0.0115479	−0.00577397	+	0.999983i	$0.501838\pi$
$368$	0	0
$369$	−700.237	−0.0987883
$370$	0	0
$371$	9886.26	1.38347
$372$	0	0
$373$	8295.79	1.15158	0.575790	−	0.817597i	$-0.304695\pi$
$373$	8295.79	1.15158	0.575790	+	0.817597i	$0.304695\pi$
$374$	0	0
$375$	6616.34	0.911110
$376$	0	0
$377$	1187.92	0.162284
$378$	0	0
$379$	−6592.90	−0.893548	−0.446774	−	0.894647i	$-0.647427\pi$
$379$	−6592.90	−0.893548	−0.446774	+	0.894647i	$0.647427\pi$
$380$	0	0
$381$	−3239.80	−0.435643
$382$	0	0
$383$	−7472.96	−0.996999	−0.498499	−	0.866890i	$-0.666115\pi$
$383$	−7472.96	−0.996999	−0.498499	+	0.866890i	$0.666115\pi$
$384$	0	0
$385$	17451.6	2.31018
$386$	0	0
$387$	1652.47	0.217054
$388$	0	0
$389$	−3175.88	−0.413943	−0.206971	−	0.978347i	$-0.566361\pi$
$389$	−3175.88	−0.413943	−0.206971	+	0.978347i	$0.566361\pi$
$390$	0	0
$391$	5846.70	0.756216
$392$	0	0
$393$	−2754.08	−0.353499
$394$	0	0
$395$	−25557.6	−3.25555
$396$	0	0
$397$	11394.6	1.44050	0.720251	−	0.693713i	$-0.244027\pi$
$397$	11394.6	1.44050	0.720251	+	0.693713i	$0.244027\pi$
$398$	0	0
$399$	−10068.2	−1.26327
$400$	0	0
$401$	13046.7	1.62474	0.812370	−	0.583142i	$-0.198177\pi$
$401$	13046.7	1.62474	0.812370	+	0.583142i	$0.198177\pi$
$402$	0	0
$403$	3467.40	0.428594
$404$	0	0
$405$	1548.31	0.189965
$406$	0	0
$407$	−3889.82	−0.473737
$408$	0	0
$409$	−2888.77	−0.349243	−0.174622	−	0.984636i	$-0.555870\pi$
$409$	−2888.77	−0.349243	−0.174622	+	0.984636i	$0.555870\pi$
$410$	0	0
$411$	3933.19	0.472043
$412$	0	0
$413$	−19050.8	−2.26980
$414$	0	0
$415$	15537.0	1.83779
$416$	0	0
$417$	−682.780	−0.0801819
$418$	0	0
$419$	−14260.8	−1.66273	−0.831367	−	0.555724i	$-0.812441\pi$
$419$	−14260.8	−1.66273	−0.831367	+	0.555724i	$0.812441\pi$
$420$	0	0
$421$	890.723	0.103114	0.0515572	−	0.998670i	$-0.483582\pi$
$421$	890.723	0.103114	0.0515572	+	0.998670i	$0.483582\pi$
$422$	0	0
$423$	−546.203	−0.0627833
$424$	0	0
$425$	−8708.85	−0.993980
$426$	0	0
$427$	2285.31	0.259002
$428$	0	0
$429$	−1014.00	−0.114117
$430$	0	0
$431$	−1323.12	−0.147871	−0.0739353	−	0.997263i	$-0.523556\pi$
$431$	−1323.12	−0.147871	−0.0739353	+	0.997263i	$0.523556\pi$
$432$	0	0
$433$	−9399.47	−1.04321	−0.521605	−	0.853187i	$-0.674666\pi$
$433$	−9399.47	−1.04321	−0.521605	+	0.853187i	$0.674666\pi$
$434$	0	0
$435$	−5240.07	−0.577568
$436$	0	0
$437$	15423.7	1.68836
$438$	0	0
$439$	−6115.04	−0.664817	−0.332409	−	0.943135i	$-0.607861\pi$
$439$	−6115.04	−0.664817	−0.332409	+	0.943135i	$0.607861\pi$
$440$	0	0
$441$	8010.49	0.864970
$442$	0	0
$443$	−12931.1	−1.38685	−0.693425	−	0.720529i	$-0.743899\pi$
$443$	−12931.1	−1.38685	−0.693425	+	0.720529i	$0.743899\pi$
$444$	0	0
$445$	17908.4	1.90773
$446$	0	0
$447$	3976.44	0.420759
$448$	0	0
$449$	−2050.73	−0.215546	−0.107773	−	0.994176i	$-0.534372\pi$
$449$	−2050.73	−0.215546	−0.107773	+	0.994176i	$0.534372\pi$
$450$	0	0
$451$	−2022.91	−0.211208
$452$	0	0
$453$	757.277	0.0785430
$454$	0	0
$455$	−8725.82	−0.899061
$456$	0	0
$457$	−11923.7	−1.22049	−0.610247	−	0.792211i	$-0.708930\pi$
$457$	−11923.7	−1.22049	−0.610247	+	0.792211i	$0.708930\pi$
$458$	0	0
$459$	−978.203	−0.0994741
$460$	0	0
$461$	1324.14	0.133777	0.0668886	−	0.997760i	$-0.478693\pi$
$461$	1324.14	0.133777	0.0668886	+	0.997760i	$0.478693\pi$
$462$	0	0
$463$	−6840.65	−0.686635	−0.343318	−	0.939219i	$-0.611551\pi$
$463$	−6840.65	−0.686635	−0.343318	+	0.939219i	$0.611551\pi$
$464$	0	0
$465$	−15295.1	−1.52537
$466$	0	0
$467$	−9041.73	−0.895935	−0.447967	−	0.894050i	$-0.647852\pi$
$467$	−9041.73	−0.895935	−0.447967	+	0.894050i	$0.647852\pi$
$468$	0	0
$469$	−36306.7	−3.57460
$470$	0	0
$471$	−6805.86	−0.665812
$472$	0	0
$473$	4773.82	0.464060
$474$	0	0
$475$	−22974.0	−2.21920
$476$	0	0
$477$	2533.86	0.243224
$478$	0	0
$479$	4650.71	0.443625	0.221812	−	0.975089i	$-0.428803\pi$
$479$	4650.71	0.443625	0.221812	+	0.975089i	$0.428803\pi$
$480$	0	0
$481$	1944.91	0.184366
$482$	0	0
$483$	−17000.4	−1.60154
$484$	0	0
$485$	−18236.9	−1.70741
$486$	0	0
$487$	−19632.4	−1.82675	−0.913375	−	0.407120i	$-0.866533\pi$
$487$	−19632.4	−1.82675	−0.913375	+	0.407120i	$0.866533\pi$
$488$	0	0
$489$	−3250.36	−0.300586
$490$	0	0
$491$	−14472.0	−1.33017	−0.665084	−	0.746769i	$-0.731604\pi$
$491$	−14472.0	−1.33017	−0.665084	+	0.746769i	$0.731604\pi$
$492$	0	0
$493$	3310.62	0.302440
$494$	0	0
$495$	4472.88	0.406144
$496$	0	0
$497$	36583.1	3.30176
$498$	0	0
$499$	−7065.91	−0.633895	−0.316947	−	0.948443i	$-0.602658\pi$
$499$	−7065.91	−0.633895	−0.316947	+	0.948443i	$0.602658\pi$
$500$	0	0
$501$	−2388.12	−0.212961
$502$	0	0
$503$	11955.1	1.05974	0.529871	−	0.848078i	$-0.322240\pi$
$503$	11955.1	1.05974	0.529871	+	0.848078i	$0.322240\pi$
$504$	0	0
$505$	26782.8	2.36004
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	−8321.15	−0.724614	−0.362307	−	0.932059i	$-0.618011\pi$
$509$	−8321.15	−0.724614	−0.362307	+	0.932059i	$0.618011\pi$
$510$	0	0
$511$	16971.4	1.46922
$512$	0	0
$513$	−2580.51	−0.222090
$514$	0	0
$515$	18254.0	1.56187
$516$	0	0
$517$	−1577.92	−0.134230
$518$	0	0
$519$	−3.28795	−0.000278083 0
$520$	0	0
$521$	−20130.2	−1.69274	−0.846372	−	0.532592i	$-0.821218\pi$
$521$	−20130.2	−1.69274	−0.846372	+	0.532592i	$0.821218\pi$
$522$	0	0
$523$	8810.52	0.736630	0.368315	−	0.929701i	$-0.379935\pi$
$523$	8810.52	0.736630	0.368315	+	0.929701i	$0.379935\pi$
$524$	0	0
$525$	25322.6	2.10508
$526$	0	0
$527$	9663.31	0.798749
$528$	0	0
$529$	13876.0	1.14046
$530$	0	0
$531$	−4882.75	−0.399045
$532$	0	0
$533$	1011.45	0.0821969
$534$	0	0
$535$	−17673.5	−1.42821
$536$	0	0
$537$	9750.17	0.783521
$538$	0	0
$539$	23141.4	1.84930
$540$	0	0
$541$	7765.62	0.617135	0.308567	−	0.951202i	$-0.400150\pi$
$541$	7765.62	0.617135	0.308567	+	0.951202i	$0.400150\pi$
$542$	0	0
$543$	8942.52	0.706741
$544$	0	0
$545$	3304.82	0.259749
$546$	0	0
$547$	19782.4	1.54631	0.773156	−	0.634216i	$-0.218677\pi$
$547$	19782.4	1.54631	0.773156	+	0.634216i	$0.218677\pi$
$548$	0	0
$549$	585.729	0.0455342
$550$	0	0
$551$	8733.45	0.675240
$552$	0	0
$553$	−46950.5	−3.61038
$554$	0	0
$555$	−8579.23	−0.656159
$556$	0	0
$557$	−10153.9	−0.772413	−0.386206	−	0.922412i	$-0.626215\pi$
$557$	−10153.9	−0.772413	−0.386206	+	0.922412i	$0.626215\pi$
$558$	0	0
$559$	−2386.91	−0.180600
$560$	0	0
$561$	−2825.92	−0.212675
$562$	0	0
$563$	18388.2	1.37650	0.688252	−	0.725472i	$-0.258378\pi$
$563$	18388.2	1.37650	0.688252	+	0.725472i	$0.258378\pi$
$564$	0	0
$565$	23737.1	1.76748
$566$	0	0
$567$	2844.31	0.210669
$568$	0	0
$569$	16365.0	1.20573	0.602863	−	0.797845i	$-0.294027\pi$
$569$	16365.0	1.20573	0.602863	+	0.797845i	$0.294027\pi$
$570$	0	0
$571$	544.241	0.0398875	0.0199437	−	0.999801i	$-0.493651\pi$
$571$	544.241	0.0398875	0.0199437	+	0.999801i	$0.493651\pi$
$572$	0	0
$573$	1837.79	0.133987
$574$	0	0
$575$	−38791.9	−2.81345
$576$	0	0
$577$	−798.738	−0.0576289	−0.0288145	−	0.999585i	$-0.509173\pi$
$577$	−798.738	−0.0576289	−0.0288145	+	0.999585i	$0.509173\pi$
$578$	0	0
$579$	−12557.4	−0.901328
$580$	0	0
$581$	28542.2	2.03809
$582$	0	0
$583$	7320.05	0.520010
$584$	0	0
$585$	−2236.44	−0.158061
$586$	0	0
$587$	9187.53	0.646013	0.323007	−	0.946397i	$-0.395306\pi$
$587$	9187.53	0.646013	0.323007	+	0.946397i	$0.395306\pi$
$588$	0	0
$589$	25491.9	1.78332
$590$	0	0
$591$	6.18086	0.000430197 0
$592$	0	0
$593$	−4371.80	−0.302746	−0.151373	−	0.988477i	$-0.548369\pi$
$593$	−4371.80	−0.302746	−0.151373	+	0.988477i	$0.548369\pi$
$594$	0	0
$595$	−24318.0	−1.67553
$596$	0	0
$597$	4477.18	0.306932
$598$	0	0
$599$	−6780.78	−0.462530	−0.231265	−	0.972891i	$-0.574286\pi$
$599$	−6780.78	−0.462530	−0.231265	+	0.972891i	$0.574286\pi$
$600$	0	0
$601$	−21348.5	−1.44895	−0.724477	−	0.689299i	$-0.757919\pi$
$601$	−21348.5	−1.44895	−0.724477	+	0.689299i	$0.757919\pi$
$602$	0	0
$603$	−9305.46	−0.628437
$604$	0	0
$605$	−12520.2	−0.841356
$606$	0	0
$607$	11161.8	0.746362	0.373181	−	0.927758i	$-0.378267\pi$
$607$	11161.8	0.746362	0.373181	+	0.927758i	$0.378267\pi$
$608$	0	0
$609$	−9626.24	−0.640517
$610$	0	0
$611$	788.960	0.0522388
$612$	0	0
$613$	12407.8	0.817528	0.408764	−	0.912640i	$-0.365960\pi$
$613$	12407.8	0.817528	0.408764	+	0.912640i	$0.365960\pi$
$614$	0	0
$615$	−4461.65	−0.292538
$616$	0	0
$617$	5035.01	0.328528	0.164264	−	0.986416i	$-0.447475\pi$
$617$	5035.01	0.328528	0.164264	+	0.986416i	$0.447475\pi$
$618$	0	0
$619$	−1623.67	−0.105430	−0.0527148	−	0.998610i	$-0.516787\pi$
$619$	−1623.67	−0.105430	−0.0527148	+	0.998610i	$0.516787\pi$
$620$	0	0
$621$	−4357.22	−0.281561
$622$	0	0
$623$	32898.6	2.11566
$624$	0	0
$625$	12109.5	0.775009
$626$	0	0
$627$	−7454.80	−0.474826
$628$	0	0
$629$	5420.27	0.343594
$630$	0	0
$631$	24809.0	1.56519	0.782593	−	0.622534i	$-0.213897\pi$
$631$	24809.0	1.56519	0.782593	+	0.622534i	$0.213897\pi$
$632$	0	0
$633$	8244.81	0.517697
$634$	0	0
$635$	−20642.8	−1.29005
$636$	0	0
$637$	−11570.7	−0.719699
$638$	0	0
$639$	9376.30	0.580471
$640$	0	0
$641$	18076.6	1.11386	0.556928	−	0.830561i	$-0.311980\pi$
$641$	18076.6	1.11386	0.556928	+	0.830561i	$0.311980\pi$
$642$	0	0
$643$	19954.2	1.22382	0.611910	−	0.790928i	$-0.290402\pi$
$643$	19954.2	1.22382	0.611910	+	0.790928i	$0.290402\pi$
$644$	0	0
$645$	10528.9	0.642755
$646$	0	0
$647$	−11206.1	−0.680921	−0.340461	−	0.940259i	$-0.610583\pi$
$647$	−11206.1	−0.680921	−0.340461	+	0.940259i	$0.610583\pi$
$648$	0	0
$649$	−14105.7	−0.853155
$650$	0	0
$651$	−28097.9	−1.69162
$652$	0	0
$653$	2456.10	0.147189	0.0735947	−	0.997288i	$-0.476553\pi$
$653$	2456.10	0.147189	0.0735947	+	0.997288i	$0.476553\pi$
$654$	0	0
$655$	−17548.0	−1.04680
$656$	0	0
$657$	4349.80	0.258298
$658$	0	0
$659$	−23280.8	−1.37616	−0.688081	−	0.725633i	$-0.741547\pi$
$659$	−23280.8	−1.37616	−0.688081	+	0.725633i	$0.741547\pi$
$660$	0	0
$661$	−6999.49	−0.411874	−0.205937	−	0.978565i	$-0.566024\pi$
$661$	−6999.49	−0.411874	−0.205937	+	0.978565i	$0.566024\pi$
$662$	0	0
$663$	1412.96	0.0827675
$664$	0	0
$665$	−64151.1	−3.74086
$666$	0	0
$667$	14746.5	0.856054
$668$	0	0
$669$	559.966	0.0323610
$670$	0	0
$671$	1692.11	0.0973517
$672$	0	0
$673$	33166.1	1.89964	0.949820	−	0.312798i	$-0.101266\pi$
$673$	33166.1	1.89964	0.949820	+	0.312798i	$0.101266\pi$
$674$	0	0
$675$	6490.22	0.370087
$676$	0	0
$677$	19998.6	1.13532	0.567658	−	0.823265i	$-0.307850\pi$
$677$	19998.6	1.13532	0.567658	+	0.823265i	$0.307850\pi$
$678$	0	0
$679$	−33502.0	−1.89350
$680$	0	0
$681$	11797.7	0.663861
$682$	0	0
$683$	22857.9	1.28058	0.640289	−	0.768134i	$-0.278815\pi$
$683$	22857.9	1.28058	0.640289	+	0.768134i	$0.278815\pi$
$684$	0	0
$685$	25060.8	1.39784
$686$	0	0
$687$	17853.0	0.991462
$688$	0	0
$689$	−3660.03	−0.202374
$690$	0	0
$691$	−2866.56	−0.157813	−0.0789066	−	0.996882i	$-0.525143\pi$
$691$	−2866.56	−0.157813	−0.0789066	+	0.996882i	$0.525143\pi$
$692$	0	0
$693$	8216.88	0.450409
$694$	0	0
$695$	−4350.42	−0.237440
$696$	0	0
$697$	2818.82	0.153186
$698$	0	0
$699$	19646.8	1.06311
$700$	0	0
$701$	30788.1	1.65885	0.829424	−	0.558620i	$-0.188669\pi$
$701$	30788.1	1.65885	0.829424	+	0.558620i	$0.188669\pi$
$702$	0	0
$703$	14298.7	0.767121
$704$	0	0
$705$	−3480.20	−0.185918
$706$	0	0
$707$	49201.2	2.61726
$708$	0	0
$709$	−4389.07	−0.232489	−0.116245	−	0.993221i	$-0.537086\pi$
$709$	−4389.07	−0.232489	−0.116245	+	0.993221i	$0.537086\pi$
$710$	0	0
$711$	−12033.5	−0.634727
$712$	0	0
$713$	43043.4	2.26085
$714$	0	0
$715$	−6460.83	−0.337932
$716$	0	0
$717$	−10387.1	−0.541021
$718$	0	0
$719$	−8729.09	−0.452768	−0.226384	−	0.974038i	$-0.572690\pi$
$719$	−8729.09	−0.452768	−0.226384	+	0.974038i	$0.572690\pi$
$720$	0	0
$721$	33533.3	1.73210
$722$	0	0
$723$	13745.9	0.707075
$724$	0	0
$725$	−21965.4	−1.12521
$726$	0	0
$727$	1471.81	0.0750845	0.0375423	−	0.999295i	$-0.488047\pi$
$727$	1471.81	0.0750845	0.0375423	+	0.999295i	$0.488047\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−6652.08	−0.336575
$732$	0	0
$733$	−14082.8	−0.709632	−0.354816	−	0.934936i	$-0.615457\pi$
$733$	−14082.8	−0.709632	−0.354816	+	0.934936i	$0.615457\pi$
$734$	0	0
$735$	51039.9	2.56141
$736$	0	0
$737$	−26882.4	−1.34359
$738$	0	0
$739$	12529.5	0.623687	0.311843	−	0.950133i	$-0.399054\pi$
$739$	12529.5	0.623687	0.311843	+	0.950133i	$0.399054\pi$
$740$	0	0
$741$	3727.40	0.184790
$742$	0	0
$743$	−19044.9	−0.940363	−0.470182	−	0.882570i	$-0.655812\pi$
$743$	−19044.9	−0.940363	−0.470182	+	0.882570i	$0.655812\pi$
$744$	0	0
$745$	25336.4	1.24598
$746$	0	0
$747$	7315.42	0.358310
$748$	0	0
$749$	−32467.0	−1.58387
$750$	0	0
$751$	−22009.9	−1.06944	−0.534722	−	0.845028i	$-0.679584\pi$
$751$	−22009.9	−1.06944	−0.534722	+	0.845028i	$0.679584\pi$
$752$	0	0
$753$	14240.3	0.689169
$754$	0	0
$755$	4825.08	0.232587
$756$	0	0
$757$	−8198.54	−0.393634	−0.196817	−	0.980440i	$-0.563061\pi$
$757$	−8198.54	−0.393634	−0.196817	+	0.980440i	$0.563061\pi$
$758$	0	0
$759$	−12587.5	−0.601974
$760$	0	0
$761$	−31929.6	−1.52096	−0.760478	−	0.649363i	$-0.775036\pi$
$761$	−31929.6	−1.52096	−0.760478	+	0.649363i	$0.775036\pi$
$762$	0	0
$763$	6071.10	0.288059
$764$	0	0
$765$	−6232.75	−0.294569
$766$	0	0
$767$	7052.85	0.332026
$768$	0	0
$769$	−26725.8	−1.25326	−0.626630	−	0.779317i	$-0.715566\pi$
$769$	−26725.8	−1.25326	−0.626630	+	0.779317i	$0.715566\pi$
$770$	0	0
$771$	−10120.9	−0.472757
$772$	0	0
$773$	23760.8	1.10558	0.552792	−	0.833320i	$-0.313563\pi$
$773$	23760.8	1.10558	0.552792	+	0.833320i	$0.313563\pi$
$774$	0	0
$775$	−64114.5	−2.97169
$776$	0	0
$777$	−15760.4	−0.727673
$778$	0	0
$779$	7436.08	0.342009
$780$	0	0
$781$	27087.1	1.24104
$782$	0	0
$783$	−2467.22	−0.112607
$784$	0	0
$785$	−43364.4	−1.97165
$786$	0	0
$787$	−23647.2	−1.07107	−0.535536	−	0.844513i	$-0.679890\pi$
$787$	−23647.2	−1.07107	−0.535536	+	0.844513i	$0.679890\pi$
$788$	0	0
$789$	10581.6	0.477458
$790$	0	0
$791$	43606.1	1.96012
$792$	0	0
$793$	−846.053	−0.0378868
$794$	0	0
$795$	16144.8	0.720249
$796$	0	0
$797$	21067.5	0.936324	0.468162	−	0.883643i	$-0.344916\pi$
$797$	21067.5	0.936324	0.468162	+	0.883643i	$0.344916\pi$
$798$	0	0
$799$	2198.76	0.0973547
$800$	0	0
$801$	8431.97	0.371946
$802$	0	0
$803$	12566.1	0.552238
$804$	0	0
$805$	−108320.	−4.74258
$806$	0	0
$807$	3432.89	0.149744
$808$	0	0
$809$	1877.67	0.0816011	0.0408005	−	0.999167i	$-0.487009\pi$
$809$	1877.67	0.0816011	0.0408005	+	0.999167i	$0.487009\pi$
$810$	0	0
$811$	2178.85	0.0943401	0.0471700	−	0.998887i	$-0.484980\pi$
$811$	2178.85	0.0943401	0.0471700	+	0.998887i	$0.484980\pi$
$812$	0	0
$813$	6762.75	0.291734
$814$	0	0
$815$	−20710.1	−0.890114
$816$	0	0
$817$	−17548.2	−0.751451
$818$	0	0
$819$	−4108.44	−0.175288
$820$	0	0
$821$	11731.4	0.498696	0.249348	−	0.968414i	$-0.419784\pi$
$821$	11731.4	0.498696	0.249348	+	0.968414i	$0.419784\pi$
$822$	0	0
$823$	−6925.69	−0.293334	−0.146667	−	0.989186i	$-0.546855\pi$
$823$	−6925.69	−0.293334	−0.146667	+	0.989186i	$0.546855\pi$
$824$	0	0
$825$	18749.5	0.791242
$826$	0	0
$827$	−35912.7	−1.51004	−0.755022	−	0.655700i	$-0.772374\pi$
$827$	−35912.7	−1.51004	−0.755022	+	0.655700i	$0.772374\pi$
$828$	0	0
$829$	−7107.67	−0.297780	−0.148890	−	0.988854i	$-0.547570\pi$
$829$	−7107.67	−0.297780	−0.148890	+	0.988854i	$0.547570\pi$
$830$	0	0
$831$	21304.9	0.889360
$832$	0	0
$833$	−32246.5	−1.34126
$834$	0	0
$835$	−15216.2	−0.630634
$836$	0	0
$837$	−7201.53	−0.297397
$838$	0	0
$839$	37844.0	1.55724	0.778618	−	0.627498i	$-0.215921\pi$
$839$	37844.0	1.55724	0.778618	+	0.627498i	$0.215921\pi$
$840$	0	0
$841$	−16039.0	−0.657631
$842$	0	0
$843$	−11495.4	−0.469660
$844$	0	0
$845$	3230.41	0.131514
$846$	0	0
$847$	−23000.2	−0.933055
$848$	0	0
$849$	−9097.67	−0.367764
$850$	0	0
$851$	24143.6	0.972539
$852$	0	0
$853$	23280.9	0.934492	0.467246	−	0.884127i	$-0.345246\pi$
$853$	23280.9	0.934492	0.467246	+	0.884127i	$0.345246\pi$
$854$	0	0
$855$	−16442.0	−0.657667
$856$	0	0
$857$	10395.1	0.414342	0.207171	−	0.978305i	$-0.433574\pi$
$857$	10395.1	0.414342	0.207171	+	0.978305i	$0.433574\pi$
$858$	0	0
$859$	−6698.89	−0.266080	−0.133040	−	0.991111i	$-0.542474\pi$
$859$	−6698.89	−0.266080	−0.133040	+	0.991111i	$0.542474\pi$
$860$	0	0
$861$	−8196.25	−0.324422
$862$	0	0
$863$	31844.8	1.25609	0.628047	−	0.778175i	$-0.283854\pi$
$863$	31844.8	1.25609	0.628047	+	0.778175i	$0.283854\pi$
$864$	0	0
$865$	−20.9496	−0.000823477 0
$866$	0	0
$867$	−10801.2	−0.423101
$868$	0	0
$869$	−34763.4	−1.35704
$870$	0	0
$871$	13441.2	0.522891
$872$	0	0
$873$	−8586.61	−0.332890
$874$	0	0
$875$	77444.0	2.99210
$876$	0	0
$877$	−20114.1	−0.774465	−0.387233	−	0.921982i	$-0.626569\pi$
$877$	−20114.1	−0.774465	−0.387233	+	0.921982i	$0.626569\pi$
$878$	0	0
$879$	−6889.32	−0.264358
$880$	0	0
$881$	−2596.61	−0.0992987	−0.0496493	−	0.998767i	$-0.515810\pi$
$881$	−2596.61	−0.0992987	−0.0496493	+	0.998767i	$0.515810\pi$
$882$	0	0
$883$	−46374.7	−1.76742	−0.883710	−	0.468035i	$-0.844962\pi$
$883$	−46374.7	−1.76742	−0.883710	+	0.468035i	$0.844962\pi$
$884$	0	0
$885$	−31111.0	−1.18168
$886$	0	0
$887$	−31863.1	−1.20615	−0.603077	−	0.797683i	$-0.706059\pi$
$887$	−31863.1	−1.20615	−0.603077	+	0.797683i	$0.706059\pi$
$888$	0	0
$889$	−37921.7	−1.43066
$890$	0	0
$891$	2106.00	0.0791848
$892$	0	0
$893$	5800.34	0.217358
$894$	0	0
$895$	62124.4	2.32021
$896$	0	0
$897$	6293.76	0.234273
$898$	0	0
$899$	24372.8	0.904202
$900$	0	0
$901$	−10200.1	−0.377154
$902$	0	0
$903$	19342.1	0.712809
$904$	0	0
$905$	56978.4	2.09285
$906$	0	0
$907$	3004.29	0.109984	0.0549922	−	0.998487i	$-0.482487\pi$
$907$	3004.29	0.109984	0.0549922	+	0.998487i	$0.482487\pi$
$908$	0	0
$909$	12610.3	0.460130
$910$	0	0
$911$	−9985.05	−0.363139	−0.181569	−	0.983378i	$-0.558118\pi$
$911$	−9985.05	−0.363139	−0.181569	+	0.983378i	$0.558118\pi$
$912$	0	0
$913$	21133.4	0.766062
$914$	0	0
$915$	3732.04	0.134839
$916$	0	0
$917$	−32236.4	−1.16089
$918$	0	0
$919$	−8216.29	−0.294919	−0.147459	−	0.989068i	$-0.547110\pi$
$919$	−8216.29	−0.294919	−0.147459	+	0.989068i	$0.547110\pi$
$920$	0	0
$921$	15522.8	0.555367
$922$	0	0
$923$	−13543.6	−0.482981
$924$	0	0
$925$	−35962.6	−1.27832
$926$	0	0
$927$	8594.64	0.304515
$928$	0	0
$929$	−17832.8	−0.629791	−0.314896	−	0.949126i	$-0.601970\pi$
$929$	−17832.8	−0.629791	−0.314896	+	0.949126i	$0.601970\pi$
$930$	0	0
$931$	−85066.4	−2.99456
$932$	0	0
$933$	−22444.6	−0.787572
$934$	0	0
$935$	−18005.7	−0.629786
$936$	0	0
$937$	−444.289	−0.0154902	−0.00774509	−	0.999970i	$-0.502465\pi$
$937$	−444.289	−0.0154902	−0.00774509	+	0.999970i	$0.502465\pi$
$938$	0	0
$939$	−12476.9	−0.433620
$940$	0	0
$941$	30762.5	1.06571	0.532853	−	0.846208i	$-0.321120\pi$
$941$	30762.5	1.06571	0.532853	+	0.846208i	$0.321120\pi$
$942$	0	0
$943$	12555.9	0.433592
$944$	0	0
$945$	18122.8	0.623848
$946$	0	0
$947$	−8791.45	−0.301672	−0.150836	−	0.988559i	$-0.548197\pi$
$947$	−8791.45	−0.301672	−0.150836	+	0.988559i	$0.548197\pi$
$948$	0	0
$949$	−6283.04	−0.214917
$950$	0	0
$951$	−1111.60	−0.0379035
$952$	0	0
$953$	1754.41	0.0596336	0.0298168	−	0.999555i	$-0.490508\pi$
$953$	1754.41	0.0596336	0.0298168	+	0.999555i	$0.490508\pi$
$954$	0	0
$955$	11709.7	0.396771
$956$	0	0
$957$	−7127.52	−0.240753
$958$	0	0
$959$	46037.8	1.55020
$960$	0	0
$961$	41350.2	1.38801
$962$	0	0
$963$	−8321.36	−0.278455
$964$	0	0
$965$	−80011.3	−2.66907
$966$	0	0
$967$	26655.9	0.886447	0.443224	−	0.896411i	$-0.353835\pi$
$967$	26655.9	0.886447	0.443224	+	0.896411i	$0.353835\pi$
$968$	0	0
$969$	10387.9	0.344384
$970$	0	0
$971$	−42151.7	−1.39311	−0.696556	−	0.717502i	$-0.745285\pi$
$971$	−42151.7	−1.39311	−0.696556	+	0.717502i	$0.745285\pi$
$972$	0	0
$973$	−7991.91	−0.263318
$974$	0	0
$975$	−9374.76	−0.307931
$976$	0	0
$977$	14594.7	0.477919	0.238959	−	0.971030i	$-0.423194\pi$
$977$	14594.7	0.477919	0.238959	+	0.971030i	$0.423194\pi$
$978$	0	0
$979$	24359.0	0.795217
$980$	0	0
$981$	1556.03	0.0506425
$982$	0	0
$983$	57077.8	1.85198	0.925991	−	0.377546i	$-0.123232\pi$
$983$	57077.8	1.85198	0.925991	+	0.377546i	$0.123232\pi$
$984$	0	0
$985$	39.3821	0.00127393
$986$	0	0
$987$	−6393.29	−0.206181
$988$	0	0
$989$	−29630.4	−0.952672
$990$	0	0
$991$	11986.6	0.384224	0.192112	−	0.981373i	$-0.438466\pi$
$991$	11986.6	0.384224	0.192112	+	0.981373i	$0.438466\pi$
$992$	0	0
$993$	5462.52	0.174570
$994$	0	0
$995$	28526.9	0.908908
$996$	0	0
$997$	32210.1	1.02317	0.511587	−	0.859231i	$-0.329058\pi$
$997$	32210.1	1.02317	0.511587	+	0.859231i	$0.329058\pi$
$998$	0	0
$999$	−4039.42	−0.127930

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	312.4.a.f.1.2	✓	2
3.2	odd	2		936.4.a.c.1.1		2
4.3	odd	2		624.4.a.l.1.2		2
8.3	odd	2		2496.4.a.bd.1.1		2
8.5	even	2		2496.4.a.u.1.1		2
12.11	even	2		1872.4.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.f.1.2	✓	2	1.1	even	1	trivial
624.4.a.l.1.2		2	4.3	odd	2
936.4.a.c.1.1		2	3.2	odd	2
1872.4.a.v.1.1		2	12.11	even	2
2496.4.a.u.1.1		2	8.5	even	2
2496.4.a.bd.1.1		2	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:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	312.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +19.1149 q^{5} +35.1149 q^{7} +9.00000 q^{9} +26.0000 q^{11} -13.0000 q^{13} +57.3446 q^{15} -36.2298 q^{17} -95.5744 q^{19} +105.345 q^{21} -161.379 q^{23} +240.379 q^{25} +27.0000 q^{27} -91.3785 q^{29} -266.723 q^{31} +78.0000 q^{33} +671.217 q^{35} -149.608 q^{37} -39.0000 q^{39} -77.8041 q^{41} +183.608 q^{43} +172.034 q^{45} -60.6893 q^{47} +890.055 q^{49} -108.689 q^{51} +281.540 q^{53} +496.987 q^{55} -286.723 q^{57} -542.527 q^{59} +65.0810 q^{61} +316.034 q^{63} -248.493 q^{65} -1033.94 q^{67} -484.136 q^{69} +1041.81 q^{71} +483.311 q^{73} +721.136 q^{75} +912.987 q^{77} -1337.05 q^{79} +81.0000 q^{81} +812.825 q^{83} -692.527 q^{85} -274.136 q^{87} +936.885 q^{89} -456.493 q^{91} -800.169 q^{93} -1826.89 q^{95} -954.068 q^{97} +234.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 12 q^{5} + 44 q^{7} + 18 q^{9} + 52 q^{11} - 26 q^{13} + 36 q^{15} - 20 q^{17} - 60 q^{19} + 132 q^{21} - 8 q^{23} + 166 q^{25} + 54 q^{27} + 132 q^{29} - 140 q^{31} + 156 q^{33} + 608 q^{35}+ \cdots + 468 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 12 * q^5 + 44 * q^7 + 18 * q^9 + 52 * q^11 - 26 * q^13 + 36 * q^15 - 20 * q^17 - 60 * q^19 + 132 * q^21 - 8 * q^23 + 166 * q^25 + 54 * q^27 + 132 * q^29 - 140 * q^31 + 156 * q^33 + 608 * q^35 + 68 * q^37 - 78 * q^39 + 28 * q^41 + 108 * q^45 + 36 * q^47 + 626 * q^49 - 60 * q^51 + 668 * q^53 + 312 * q^55 - 180 * q^57 - 508 * q^59 + 340 * q^61 + 396 * q^63 - 156 * q^65 - 940 * q^67 - 24 * q^69 + 300 * q^71 + 1124 * q^73 + 498 * q^75 + 1144 * q^77 - 1520 * q^79 + 162 * q^81 + 524 * q^83 - 808 * q^85 + 396 * q^87 + 1900 * q^89 - 572 * q^91 - 420 * q^93 - 2080 * q^95 - 1436 * q^97 + 468 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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