LMFDB - Embedded newform 312.6.a.h.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [312,6,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, 6); 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, 6, names="a")

Level:	$ N $	$=$	$ 312 = 2^{3} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	312.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,36,0,-56] 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:	$50.0397517816$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 147x^{2} - 398x + 828 $ x^4 - 147x^2 - 398x + 828
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-9.92415$ 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)

$f(q)$	$=$	$q+9.00000 q^{3} +12.4897 q^{5} +98.5962 q^{7} +81.0000 q^{9} -616.610 q^{11} -169.000 q^{13} +112.408 q^{15} -1784.10 q^{17} +1126.41 q^{19} +887.366 q^{21} -4720.49 q^{23} -2969.01 q^{25} +729.000 q^{27} +3491.00 q^{29} +3908.28 q^{31} -5549.49 q^{33} +1231.44 q^{35} +4647.10 q^{37} -1521.00 q^{39} +5387.66 q^{41} -11949.7 q^{43} +1011.67 q^{45} -15079.7 q^{47} -7085.79 q^{49} -16056.9 q^{51} -8699.89 q^{53} -7701.29 q^{55} +10137.7 q^{57} -21525.5 q^{59} -2865.97 q^{61} +7986.29 q^{63} -2110.76 q^{65} +15998.4 q^{67} -42484.4 q^{69} +58912.0 q^{71} -59570.0 q^{73} -26721.1 q^{75} -60795.4 q^{77} +67080.5 q^{79} +6561.00 q^{81} -72416.0 q^{83} -22282.9 q^{85} +31419.0 q^{87} +121030. q^{89} -16662.8 q^{91} +35174.5 q^{93} +14068.6 q^{95} -95672.6 q^{97} -49945.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 36 q^{3} - 56 q^{5} + 48 q^{7} + 324 q^{9} - 368 q^{11} - 676 q^{13} - 504 q^{15} - 1976 q^{17} - 1808 q^{19} + 432 q^{21} - 240 q^{23} + 2652 q^{25} + 2916 q^{27} - 4792 q^{29} - 9296 q^{31} - 3312 q^{33}+ \cdots - 29808 q^{99}+O(q^{100}) $ 4 * q + 36 * q^3 - 56 * q^5 + 48 * q^7 + 324 * q^9 - 368 * q^11 - 676 * q^13 - 504 * q^15 - 1976 * q^17 - 1808 * q^19 + 432 * q^21 - 240 * q^23 + 2652 * q^25 + 2916 * q^27 - 4792 * q^29 - 9296 * q^31 - 3312 * q^33 - 21424 * q^35 - 7800 * q^37 - 6084 * q^39 - 8792 * q^41 - 19136 * q^43 - 4536 * q^45 - 37968 * q^47 + 2132 * q^49 - 17784 * q^51 - 15192 * q^53 - 28976 * q^55 - 16272 * q^57 - 69424 * q^59 - 22568 * q^61 + 3888 * q^63 + 9464 * q^65 + 1136 * q^67 - 2160 * q^69 - 26672 * q^71 - 19736 * q^73 + 23868 * q^75 - 76480 * q^77 + 51328 * q^79 + 26244 * q^81 - 22544 * q^83 - 183888 * q^85 - 43128 * q^87 - 216600 * q^89 - 8112 * q^91 - 83664 * q^93 + 85936 * q^95 + 53512 * q^97 - 29808 * 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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	12.4897	0.223423	0.111712	−	0.993741i	$-0.464367\pi$
$5$	12.4897	0.223423	0.111712	+	0.993741i	$0.464367\pi$
$6$	0	0
$7$	98.5962	0.760528	0.380264	−	0.924878i	$-0.375833\pi$
$7$	98.5962	0.760528	0.380264	+	0.924878i	$0.375833\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−616.610	−1.53649	−0.768243	−	0.640158i	$-0.778869\pi$
$11$	−616.610	−1.53649	−0.768243	+	0.640158i	$0.778869\pi$
$12$	0	0
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	0	0
$15$	112.408	0.128993
$16$	0	0
$17$	−1784.10	−1.49726	−0.748630	−	0.662988i	$-0.769288\pi$
$17$	−1784.10	−1.49726	−0.748630	+	0.662988i	$0.769288\pi$
$18$	0	0
$19$	1126.41	0.715837	0.357919	−	0.933753i	$-0.383486\pi$
$19$	1126.41	0.715837	0.357919	+	0.933753i	$0.383486\pi$
$20$	0	0
$21$	887.366	0.439091
$22$	0	0
$23$	−4720.49	−1.86066	−0.930330	−	0.366722i	$-0.880480\pi$
$23$	−4720.49	−1.86066	−0.930330	+	0.366722i	$0.880480\pi$
$24$	0	0
$25$	−2969.01	−0.950082
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	3491.00	0.770822	0.385411	−	0.922745i	$-0.374060\pi$
$29$	3491.00	0.770822	0.385411	+	0.922745i	$0.374060\pi$
$30$	0	0
$31$	3908.28	0.730435	0.365218	−	0.930922i	$-0.380995\pi$
$31$	3908.28	0.730435	0.365218	+	0.930922i	$0.380995\pi$
$32$	0	0
$33$	−5549.49	−0.887091
$34$	0	0
$35$	1231.44	0.169919
$36$	0	0
$37$	4647.10	0.558056	0.279028	−	0.960283i	$-0.409988\pi$
$37$	4647.10	0.558056	0.279028	+	0.960283i	$0.409988\pi$
$38$	0	0
$39$	−1521.00	−0.160128
$40$	0	0
$41$	5387.66	0.500542	0.250271	−	0.968176i	$-0.419480\pi$
$41$	5387.66	0.500542	0.250271	+	0.968176i	$0.419480\pi$
$42$	0	0
$43$	−11949.7	−0.985564	−0.492782	−	0.870153i	$-0.664020\pi$
$43$	−11949.7	−0.985564	−0.492782	+	0.870153i	$0.664020\pi$
$44$	0	0
$45$	1011.67	0.0744744
$46$	0	0
$47$	−15079.7	−0.995745	−0.497873	−	0.867250i	$-0.665885\pi$
$47$	−15079.7	−0.995745	−0.497873	+	0.867250i	$0.665885\pi$
$48$	0	0
$49$	−7085.79	−0.421598
$50$	0	0
$51$	−16056.9	−0.864444
$52$	0	0
$53$	−8699.89	−0.425426	−0.212713	−	0.977115i	$-0.568230\pi$
$53$	−8699.89	−0.425426	−0.212713	+	0.977115i	$0.568230\pi$
$54$	0	0
$55$	−7701.29	−0.343287
$56$	0	0
$57$	10137.7	0.413289
$58$	0	0
$59$	−21525.5	−0.805050	−0.402525	−	0.915409i	$-0.631867\pi$
$59$	−21525.5	−0.805050	−0.402525	+	0.915409i	$0.631867\pi$
$60$	0	0
$61$	−2865.97	−0.0986161	−0.0493080	−	0.998784i	$-0.515702\pi$
$61$	−2865.97	−0.0986161	−0.0493080	+	0.998784i	$0.515702\pi$
$62$	0	0
$63$	7986.29	0.253509
$64$	0	0
$65$	−2110.76	−0.0619664
$66$	0	0
$67$	15998.4	0.435400	0.217700	−	0.976016i	$-0.430144\pi$
$67$	15998.4	0.435400	0.217700	+	0.976016i	$0.430144\pi$
$68$	0	0
$69$	−42484.4	−1.07425
$70$	0	0
$71$	58912.0	1.38694	0.693471	−	0.720485i	$-0.256081\pi$
$71$	58912.0	1.38694	0.693471	+	0.720485i	$0.256081\pi$
$72$	0	0
$73$	−59570.0	−1.30834	−0.654170	−	0.756348i	$-0.726982\pi$
$73$	−59570.0	−1.30834	−0.654170	+	0.756348i	$0.726982\pi$
$74$	0	0
$75$	−26721.1	−0.548530
$76$	0	0
$77$	−60795.4	−1.16854
$78$	0	0
$79$	67080.5	1.20928	0.604642	−	0.796497i	$-0.293316\pi$
$79$	67080.5	1.20928	0.604642	+	0.796497i	$0.293316\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−72416.0	−1.15382	−0.576912	−	0.816807i	$-0.695742\pi$
$83$	−72416.0	−1.15382	−0.576912	+	0.816807i	$0.695742\pi$
$84$	0	0
$85$	−22282.9	−0.334523
$86$	0	0
$87$	31419.0	0.445034
$88$	0	0
$89$	121030.	1.61964	0.809822	−	0.586675i	$-0.199564\pi$
$89$	121030.	1.61964	0.809822	+	0.586675i	$0.199564\pi$
$90$	0	0
$91$	−16662.8	−0.210932
$92$	0	0
$93$	35174.5	0.421717
$94$	0	0
$95$	14068.6	0.159935
$96$	0	0
$97$	−95672.6	−1.03242	−0.516212	−	0.856461i	$-0.672658\pi$
$97$	−95672.6	−1.03242	−0.516212	+	0.856461i	$0.672658\pi$
$98$	0	0
$99$	−49945.4	−0.512162
$100$	0	0
$101$	20048.9	0.195564	0.0977818	−	0.995208i	$-0.468825\pi$
$101$	20048.9	0.195564	0.0977818	+	0.995208i	$0.468825\pi$
$102$	0	0
$103$	−107610.	−0.999445	−0.499722	−	0.866186i	$-0.666565\pi$
$103$	−107610.	−0.999445	−0.499722	+	0.866186i	$0.666565\pi$
$104$	0	0
$105$	11083.0	0.0981030
$106$	0	0
$107$	−214337.	−1.80983	−0.904915	−	0.425593i	$-0.860066\pi$
$107$	−214337.	−1.80983	−0.904915	+	0.425593i	$0.860066\pi$
$108$	0	0
$109$	−155883.	−1.25670	−0.628352	−	0.777929i	$-0.716270\pi$
$109$	−155883.	−1.25670	−0.628352	+	0.777929i	$0.716270\pi$
$110$	0	0
$111$	41823.9	0.322194
$112$	0	0
$113$	−260648.	−1.92025	−0.960127	−	0.279565i	$-0.909810\pi$
$113$	−260648.	−1.92025	−0.960127	+	0.279565i	$0.909810\pi$
$114$	0	0
$115$	−58957.6	−0.415715
$116$	0	0
$117$	−13689.0	−0.0924500
$118$	0	0
$119$	−175906.	−1.13871
$120$	0	0
$121$	219157.	1.36079
$122$	0	0
$123$	48489.0	0.288988
$124$	0	0
$125$	−76112.5	−0.435693
$126$	0	0
$127$	38277.7	0.210590	0.105295	−	0.994441i	$-0.466421\pi$
$127$	38277.7	0.210590	0.105295	+	0.994441i	$0.466421\pi$
$128$	0	0
$129$	−107547.	−0.569016
$130$	0	0
$131$	372611.	1.89704	0.948522	−	0.316712i	$-0.102579\pi$
$131$	372611.	1.89704	0.948522	+	0.316712i	$0.102579\pi$
$132$	0	0
$133$	111060.	0.544414
$134$	0	0
$135$	9105.01	0.0429978
$136$	0	0
$137$	−251087.	−1.14294	−0.571470	−	0.820623i	$-0.693627\pi$
$137$	−251087.	−1.14294	−0.571470	+	0.820623i	$0.693627\pi$
$138$	0	0
$139$	286317.	1.25693	0.628465	−	0.777838i	$-0.283684\pi$
$139$	286317.	1.25693	0.628465	+	0.777838i	$0.283684\pi$
$140$	0	0
$141$	−135717.	−0.574894
$142$	0	0
$143$	104207.	0.426145
$144$	0	0
$145$	43601.6	0.172220
$146$	0	0
$147$	−63772.1	−0.243410
$148$	0	0
$149$	−368403.	−1.35943	−0.679716	−	0.733476i	$-0.737897\pi$
$149$	−368403.	−1.35943	−0.679716	+	0.733476i	$0.737897\pi$
$150$	0	0
$151$	−249669.	−0.891091	−0.445546	−	0.895259i	$-0.646990\pi$
$151$	−249669.	−0.891091	−0.445546	+	0.895259i	$0.646990\pi$
$152$	0	0
$153$	−144512.	−0.499087
$154$	0	0
$155$	48813.4	0.163196
$156$	0	0
$157$	−172234.	−0.557661	−0.278830	−	0.960340i	$-0.589947\pi$
$157$	−172234.	−0.557661	−0.278830	+	0.960340i	$0.589947\pi$
$158$	0	0
$159$	−78299.0	−0.245620
$160$	0	0
$161$	−465422.	−1.41508
$162$	0	0
$163$	395259.	1.16523	0.582617	−	0.812747i	$-0.302029\pi$
$163$	395259.	1.16523	0.582617	+	0.812747i	$0.302029\pi$
$164$	0	0
$165$	−69311.6	−0.198197
$166$	0	0
$167$	333684.	0.925858	0.462929	−	0.886395i	$-0.346798\pi$
$167$	333684.	0.925858	0.462929	+	0.886395i	$0.346798\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	91239.6	0.238612
$172$	0	0
$173$	370115.	0.940202	0.470101	−	0.882613i	$-0.344217\pi$
$173$	370115.	0.940202	0.470101	+	0.882613i	$0.344217\pi$
$174$	0	0
$175$	−292733.	−0.722564
$176$	0	0
$177$	−193729.	−0.464796
$178$	0	0
$179$	205085.	0.478412	0.239206	−	0.970969i	$-0.423113\pi$
$179$	205085.	0.478412	0.239206	+	0.970969i	$0.423113\pi$
$180$	0	0
$181$	55771.6	0.126537	0.0632684	−	0.997997i	$-0.479848\pi$
$181$	55771.6	0.126537	0.0632684	+	0.997997i	$0.479848\pi$
$182$	0	0
$183$	−25793.8	−0.0569360
$184$	0	0
$185$	58041.0	0.124683
$186$	0	0
$187$	1.10009e6	2.30052
$188$	0	0
$189$	71876.6	0.146364
$190$	0	0
$191$	273280.	0.542032	0.271016	−	0.962575i	$-0.412640\pi$
$191$	273280.	0.542032	0.271016	+	0.962575i	$0.412640\pi$
$192$	0	0
$193$	123446.	0.238553	0.119276	−	0.992861i	$-0.461943\pi$
$193$	123446.	0.238553	0.119276	+	0.992861i	$0.461943\pi$
$194$	0	0
$195$	−18996.9	−0.0357763
$196$	0	0
$197$	−202130.	−0.371077	−0.185539	−	0.982637i	$-0.559403\pi$
$197$	−202130.	−0.371077	−0.185539	+	0.982637i	$0.559403\pi$
$198$	0	0
$199$	755484.	1.35236	0.676180	−	0.736736i	$-0.263634\pi$
$199$	755484.	1.35236	0.676180	+	0.736736i	$0.263634\pi$
$200$	0	0
$201$	143985.	0.251379
$202$	0	0
$203$	344199.	0.586232
$204$	0	0
$205$	67290.5	0.111833
$206$	0	0
$207$	−382359.	−0.620220
$208$	0	0
$209$	−694559.	−1.09987
$210$	0	0
$211$	572071.	0.884593	0.442297	−	0.896869i	$-0.354164\pi$
$211$	572071.	0.884593	0.442297	+	0.896869i	$0.354164\pi$
$212$	0	0
$213$	530208.	0.800751
$214$	0	0
$215$	−149248.	−0.220198
$216$	0	0
$217$	385342.	0.555516
$218$	0	0
$219$	−536130.	−0.755370
$220$	0	0
$221$	301513.	0.415265
$222$	0	0
$223$	1.40791e6	1.89588	0.947942	−	0.318444i	$-0.103160\pi$
$223$	1.40791e6	1.89588	0.947942	+	0.318444i	$0.103160\pi$
$224$	0	0
$225$	−240490.	−0.316694
$226$	0	0
$227$	2603.04	0.00335286	0.00167643	−	0.999999i	$-0.499466\pi$
$227$	2603.04	0.00335286	0.00167643	+	0.999999i	$0.499466\pi$
$228$	0	0
$229$	−1.37017e6	−1.72658	−0.863290	−	0.504709i	$-0.831600\pi$
$229$	−1.37017e6	−1.72658	−0.863290	+	0.504709i	$0.831600\pi$
$230$	0	0
$231$	−547158.	−0.674657
$232$	0	0
$233$	787976.	0.950874	0.475437	−	0.879750i	$-0.342290\pi$
$233$	787976.	0.950874	0.475437	+	0.879750i	$0.342290\pi$
$234$	0	0
$235$	−188341.	−0.222472
$236$	0	0
$237$	603724.	0.698181
$238$	0	0
$239$	818370.	0.926734	0.463367	−	0.886166i	$-0.346641\pi$
$239$	818370.	0.926734	0.463367	+	0.886166i	$0.346641\pi$
$240$	0	0
$241$	1.30702e6	1.44958	0.724788	−	0.688972i	$-0.241938\pi$
$241$	1.30702e6	1.44958	0.724788	+	0.688972i	$0.241938\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	−88499.6	−0.0941947
$246$	0	0
$247$	−190364.	−0.198538
$248$	0	0
$249$	−651744.	−0.666160
$250$	0	0
$251$	−366310.	−0.366999	−0.183500	−	0.983020i	$-0.558743\pi$
$251$	−366310.	−0.366999	−0.183500	+	0.983020i	$0.558743\pi$
$252$	0	0
$253$	2.91070e6	2.85888
$254$	0	0
$255$	−200547.	−0.193137
$256$	0	0
$257$	−49390.2	−0.0466453	−0.0233227	−	0.999728i	$-0.507425\pi$
$257$	−49390.2	−0.0466453	−0.0233227	+	0.999728i	$0.507425\pi$
$258$	0	0
$259$	458186.	0.424417
$260$	0	0
$261$	282771.	0.256941
$262$	0	0
$263$	−1.76703e6	−1.57526	−0.787632	−	0.616146i	$-0.788693\pi$
$263$	−1.76703e6	−1.57526	−0.787632	+	0.616146i	$0.788693\pi$
$264$	0	0
$265$	−108659.	−0.0950500
$266$	0	0
$267$	1.08927e6	0.935102
$268$	0	0
$269$	−1.96524e6	−1.65590	−0.827952	−	0.560799i	$-0.810494\pi$
$269$	−1.96524e6	−1.65590	−0.827952	+	0.560799i	$0.810494\pi$
$270$	0	0
$271$	702919.	0.581409	0.290705	−	0.956813i	$-0.406110\pi$
$271$	702919.	0.581409	0.290705	+	0.956813i	$0.406110\pi$
$272$	0	0
$273$	−149965.	−0.121782
$274$	0	0
$275$	1.83072e6	1.45979
$276$	0	0
$277$	−1.26132e6	−0.987700	−0.493850	−	0.869547i	$-0.664411\pi$
$277$	−1.26132e6	−0.987700	−0.493850	+	0.869547i	$0.664411\pi$
$278$	0	0
$279$	316571.	0.243478
$280$	0	0
$281$	1.04007e6	0.785772	0.392886	−	0.919587i	$-0.371477\pi$
$281$	1.04007e6	0.785772	0.392886	+	0.919587i	$0.371477\pi$
$282$	0	0
$283$	−1.12947e6	−0.838321	−0.419160	−	0.907912i	$-0.637676\pi$
$283$	−1.12947e6	−0.838321	−0.419160	+	0.907912i	$0.637676\pi$
$284$	0	0
$285$	126618.	0.0923383
$286$	0	0
$287$	531203.	0.380676
$288$	0	0
$289$	1.76316e6	1.24179
$290$	0	0
$291$	−861053.	−0.596071
$292$	0	0
$293$	−2.56374e6	−1.74464	−0.872319	−	0.488936i	$-0.837385\pi$
$293$	−2.56374e6	−1.74464	−0.872319	+	0.488936i	$0.837385\pi$
$294$	0	0
$295$	−268848.	−0.179867
$296$	0	0
$297$	−449509.	−0.295697
$298$	0	0
$299$	797762.	0.516055
$300$	0	0
$301$	−1.17819e6	−0.749549
$302$	0	0
$303$	180440.	0.112909
$304$	0	0
$305$	−35795.3	−0.0220331
$306$	0	0
$307$	2.95980e6	1.79233	0.896163	−	0.443724i	$-0.146343\pi$
$307$	2.95980e6	1.79233	0.896163	+	0.443724i	$0.146343\pi$
$308$	0	0
$309$	−968488.	−0.577030
$310$	0	0
$311$	1.24424e6	0.729464	0.364732	−	0.931113i	$-0.381161\pi$
$311$	1.24424e6	0.729464	0.364732	+	0.931113i	$0.381161\pi$
$312$	0	0
$313$	1.60429e6	0.925597	0.462798	−	0.886464i	$-0.346845\pi$
$313$	1.60429e6	0.925597	0.462798	+	0.886464i	$0.346845\pi$
$314$	0	0
$315$	99746.6	0.0566398
$316$	0	0
$317$	1.98163e6	1.10758	0.553790	−	0.832656i	$-0.313181\pi$
$317$	1.98163e6	1.10758	0.553790	+	0.832656i	$0.313181\pi$
$318$	0	0
$319$	−2.15258e6	−1.18436
$320$	0	0
$321$	−1.92903e6	−1.04491
$322$	0	0
$323$	−2.00964e6	−1.07179
$324$	0	0
$325$	501762.	0.263505
$326$	0	0
$327$	−1.40295e6	−0.725559
$328$	0	0
$329$	−1.48680e6	−0.757292
$330$	0	0
$331$	−2.45028e6	−1.22927	−0.614633	−	0.788814i	$-0.710696\pi$
$331$	−2.45028e6	−1.22927	−0.614633	+	0.788814i	$0.710696\pi$
$332$	0	0
$333$	376415.	0.186019
$334$	0	0
$335$	199815.	0.0972785
$336$	0	0
$337$	3.06994e6	1.47250	0.736249	−	0.676711i	$-0.236595\pi$
$337$	3.06994e6	1.47250	0.736249	+	0.676711i	$0.236595\pi$
$338$	0	0
$339$	−2.34583e6	−1.10866
$340$	0	0
$341$	−2.40989e6	−1.12230
$342$	0	0
$343$	−2.35574e6	−1.08116
$344$	0	0
$345$	−530619.	−0.240013
$346$	0	0
$347$	1.69414e6	0.755309	0.377654	−	0.925947i	$-0.376731\pi$
$347$	1.69414e6	0.755309	0.377654	+	0.925947i	$0.376731\pi$
$348$	0	0
$349$	1.01115e6	0.444377	0.222188	−	0.975004i	$-0.428680\pi$
$349$	1.01115e6	0.444377	0.222188	+	0.975004i	$0.428680\pi$
$350$	0	0
$351$	−123201.	−0.0533761
$352$	0	0
$353$	742397.	0.317102	0.158551	−	0.987351i	$-0.449318\pi$
$353$	742397.	0.317102	0.158551	+	0.987351i	$0.449318\pi$
$354$	0	0
$355$	735796.	0.309875
$356$	0	0
$357$	−1.58315e6	−0.657433
$358$	0	0
$359$	−1.43654e6	−0.588278	−0.294139	−	0.955763i	$-0.595033\pi$
$359$	−1.43654e6	−0.588278	−0.294139	+	0.955763i	$0.595033\pi$
$360$	0	0
$361$	−1.20729e6	−0.487577
$362$	0	0
$363$	1.97241e6	0.785653
$364$	0	0
$365$	−744014.	−0.292313
$366$	0	0
$367$	805385.	0.312132	0.156066	−	0.987747i	$-0.450119\pi$
$367$	805385.	0.312132	0.156066	+	0.987747i	$0.450119\pi$
$368$	0	0
$369$	436401.	0.166847
$370$	0	0
$371$	−857776.	−0.323548
$372$	0	0
$373$	2.10938e6	0.785025	0.392512	−	0.919747i	$-0.371606\pi$
$373$	2.10938e6	0.785025	0.392512	+	0.919747i	$0.371606\pi$
$374$	0	0
$375$	−685013.	−0.251548
$376$	0	0
$377$	−589978.	−0.213788
$378$	0	0
$379$	285684.	0.102162	0.0510809	−	0.998695i	$-0.483733\pi$
$379$	285684.	0.102162	0.0510809	+	0.998695i	$0.483733\pi$
$380$	0	0
$381$	344500.	0.121584
$382$	0	0
$383$	−2.31466e6	−0.806287	−0.403144	−	0.915137i	$-0.632082\pi$
$383$	−2.31466e6	−0.806287	−0.403144	+	0.915137i	$0.632082\pi$
$384$	0	0
$385$	−759318.	−0.261079
$386$	0	0
$387$	−967923.	−0.328521
$388$	0	0
$389$	−3.35677e6	−1.12473	−0.562363	−	0.826890i	$-0.690108\pi$
$389$	−3.35677e6	−1.12473	−0.562363	+	0.826890i	$0.690108\pi$
$390$	0	0
$391$	8.42183e6	2.78589
$392$	0	0
$393$	3.35350e6	1.09526
$394$	0	0
$395$	837817.	0.270182
$396$	0	0
$397$	2.09116e6	0.665904	0.332952	−	0.942944i	$-0.391955\pi$
$397$	2.09116e6	0.665904	0.332952	+	0.942944i	$0.391955\pi$
$398$	0	0
$399$	999542.	0.314318
$400$	0	0
$401$	−1.54044e6	−0.478393	−0.239197	−	0.970971i	$-0.576884\pi$
$401$	−1.54044e6	−0.478393	−0.239197	+	0.970971i	$0.576884\pi$
$402$	0	0
$403$	−660500.	−0.202586
$404$	0	0
$405$	81945.1	0.0248248
$406$	0	0
$407$	−2.86545e6	−0.857446
$408$	0	0
$409$	−2.35747e6	−0.696848	−0.348424	−	0.937337i	$-0.613283\pi$
$409$	−2.35747e6	−0.696848	−0.348424	+	0.937337i	$0.613283\pi$
$410$	0	0
$411$	−2.25979e6	−0.659876
$412$	0	0
$413$	−2.12233e6	−0.612263
$414$	0	0
$415$	−904456.	−0.257791
$416$	0	0
$417$	2.57686e6	0.725688
$418$	0	0
$419$	−2.30460e6	−0.641300	−0.320650	−	0.947198i	$-0.603901\pi$
$419$	−2.30460e6	−0.641300	−0.320650	+	0.947198i	$0.603901\pi$
$420$	0	0
$421$	1.60143e6	0.440354	0.220177	−	0.975460i	$-0.429337\pi$
$421$	1.60143e6	0.440354	0.220177	+	0.975460i	$0.429337\pi$
$422$	0	0
$423$	−1.22146e6	−0.331915
$424$	0	0
$425$	5.29701e6	1.42252
$426$	0	0
$427$	−282574.	−0.0750003
$428$	0	0
$429$	937864.	0.246035
$430$	0	0
$431$	−1.32525e6	−0.343640	−0.171820	−	0.985128i	$-0.554965\pi$
$431$	−1.32525e6	−0.343640	−0.171820	+	0.985128i	$0.554965\pi$
$432$	0	0
$433$	−69801.4	−0.0178914	−0.00894570	−	0.999960i	$-0.502848\pi$
$433$	−69801.4	−0.0178914	−0.00894570	+	0.999960i	$0.502848\pi$
$434$	0	0
$435$	392414.	0.0994310
$436$	0	0
$437$	−5.31723e6	−1.33193
$438$	0	0
$439$	−1.42955e6	−0.354028	−0.177014	−	0.984208i	$-0.556644\pi$
$439$	−1.42955e6	−0.354028	−0.177014	+	0.984208i	$0.556644\pi$
$440$	0	0
$441$	−573949.	−0.140533
$442$	0	0
$443$	−6.90924e6	−1.67271	−0.836356	−	0.548187i	$-0.815318\pi$
$443$	−6.90924e6	−1.67271	−0.836356	+	0.548187i	$0.815318\pi$
$444$	0	0
$445$	1.51164e6	0.361866
$446$	0	0
$447$	−3.31563e6	−0.784868
$448$	0	0
$449$	−3.00300e6	−0.702974	−0.351487	−	0.936193i	$-0.614324\pi$
$449$	−3.00300e6	−0.702974	−0.351487	+	0.936193i	$0.614324\pi$
$450$	0	0
$451$	−3.32209e6	−0.769077
$452$	0	0
$453$	−2.24702e6	−0.514472
$454$	0	0
$455$	−208113.	−0.0471272
$456$	0	0
$457$	−4.93899e6	−1.10624	−0.553118	−	0.833103i	$-0.686562\pi$
$457$	−4.93899e6	−1.10624	−0.553118	+	0.833103i	$0.686562\pi$
$458$	0	0
$459$	−1.30061e6	−0.288148
$460$	0	0
$461$	−3.34186e6	−0.732378	−0.366189	−	0.930540i	$-0.619338\pi$
$461$	−3.34186e6	−0.732378	−0.366189	+	0.930540i	$0.619338\pi$
$462$	0	0
$463$	−8.18534e6	−1.77453	−0.887267	−	0.461256i	$-0.847399\pi$
$463$	−8.18534e6	−1.77453	−0.887267	+	0.461256i	$0.847399\pi$
$464$	0	0
$465$	439321.	0.0942213
$466$	0	0
$467$	2.73162e6	0.579599	0.289800	−	0.957087i	$-0.406411\pi$
$467$	2.73162e6	0.579599	0.289800	+	0.957087i	$0.406411\pi$
$468$	0	0
$469$	1.57738e6	0.331134
$470$	0	0
$471$	−1.55011e6	−0.321966
$472$	0	0
$473$	7.36829e6	1.51431
$474$	0	0
$475$	−3.34433e6	−0.680104
$476$	0	0
$477$	−704691.	−0.141809
$478$	0	0
$479$	5.87735e6	1.17042	0.585211	−	0.810881i	$-0.301012\pi$
$479$	5.87735e6	1.17042	0.585211	+	0.810881i	$0.301012\pi$
$480$	0	0
$481$	−785360.	−0.154777
$482$	0	0
$483$	−4.18880e6	−0.816999
$484$	0	0
$485$	−1.19493e6	−0.230667
$486$	0	0
$487$	−3.31383e6	−0.633151	−0.316575	−	0.948567i	$-0.602533\pi$
$487$	−3.31383e6	−0.633151	−0.316575	+	0.948567i	$0.602533\pi$
$488$	0	0
$489$	3.55733e6	0.672748
$490$	0	0
$491$	9.43187e6	1.76561	0.882803	−	0.469743i	$-0.155653\pi$
$491$	9.43187e6	1.76561	0.882803	+	0.469743i	$0.155653\pi$
$492$	0	0
$493$	−6.22829e6	−1.15412
$494$	0	0
$495$	−623805.	−0.114429
$496$	0	0
$497$	5.80850e6	1.05481
$498$	0	0
$499$	−303155.	−0.0545021	−0.0272511	−	0.999629i	$-0.508675\pi$
$499$	−303155.	−0.0545021	−0.0272511	+	0.999629i	$0.508675\pi$
$500$	0	0
$501$	3.00316e6	0.534545
$502$	0	0
$503$	5.76211e6	1.01546	0.507728	−	0.861517i	$-0.330485\pi$
$503$	5.76211e6	1.01546	0.507728	+	0.861517i	$0.330485\pi$
$504$	0	0
$505$	250406.	0.0436934
$506$	0	0
$507$	257049.	0.0444116
$508$	0	0
$509$	6.52865e6	1.11694	0.558469	−	0.829526i	$-0.311389\pi$
$509$	6.52865e6	1.11694	0.558469	+	0.829526i	$0.311389\pi$
$510$	0	0
$511$	−5.87338e6	−0.995029
$512$	0	0
$513$	821156.	0.137763
$514$	0	0
$515$	−1.34402e6	−0.223299
$516$	0	0
$517$	9.29829e6	1.52995
$518$	0	0
$519$	3.33103e6	0.542826
$520$	0	0
$521$	−3.68227e6	−0.594321	−0.297160	−	0.954828i	$-0.596040\pi$
$521$	−3.68227e6	−0.594321	−0.297160	+	0.954828i	$0.596040\pi$
$522$	0	0
$523$	−2.51775e6	−0.402494	−0.201247	−	0.979541i	$-0.564499\pi$
$523$	−2.51775e6	−0.402494	−0.201247	+	0.979541i	$0.564499\pi$
$524$	0	0
$525$	−2.63459e6	−0.417172
$526$	0	0
$527$	−6.97277e6	−1.09365
$528$	0	0
$529$	1.58467e7	2.46206
$530$	0	0
$531$	−1.74356e6	−0.268350
$532$	0	0
$533$	−910515.	−0.138826
$534$	0	0
$535$	−2.67701e6	−0.404358
$536$	0	0
$537$	1.84577e6	0.276211
$538$	0	0
$539$	4.36917e6	0.647779
$540$	0	0
$541$	−8.92318e6	−1.31077	−0.655385	−	0.755295i	$-0.727494\pi$
$541$	−8.92318e6	−1.31077	−0.655385	+	0.755295i	$0.727494\pi$
$542$	0	0
$543$	501944.	0.0730560
$544$	0	0
$545$	−1.94694e6	−0.280777
$546$	0	0
$547$	919340.	0.131374	0.0656868	−	0.997840i	$-0.479076\pi$
$547$	919340.	0.131374	0.0656868	+	0.997840i	$0.479076\pi$
$548$	0	0
$549$	−232144.	−0.0328720
$550$	0	0
$551$	3.93231e6	0.551783
$552$	0	0
$553$	6.61388e6	0.919694
$554$	0	0
$555$	522369.	0.0719855
$556$	0	0
$557$	−4.02040e6	−0.549075	−0.274538	−	0.961576i	$-0.588525\pi$
$557$	−4.02040e6	−0.549075	−0.274538	+	0.961576i	$0.588525\pi$
$558$	0	0
$559$	2.01949e6	0.273346
$560$	0	0
$561$	9.90085e6	1.32821
$562$	0	0
$563$	−3.06726e6	−0.407830	−0.203915	−	0.978989i	$-0.565367\pi$
$563$	−3.06726e6	−0.407830	−0.203915	+	0.978989i	$0.565367\pi$
$564$	0	0
$565$	−3.25543e6	−0.429029
$566$	0	0
$567$	646890.	0.0845031
$568$	0	0
$569$	−7.84400e6	−1.01568	−0.507840	−	0.861451i	$-0.669556\pi$
$569$	−7.84400e6	−1.01568	−0.507840	+	0.861451i	$0.669556\pi$
$570$	0	0
$571$	−3.33691e6	−0.428306	−0.214153	−	0.976800i	$-0.568699\pi$
$571$	−3.33691e6	−0.428306	−0.214153	+	0.976800i	$0.568699\pi$
$572$	0	0
$573$	2.45952e6	0.312942
$574$	0	0
$575$	1.40152e7	1.76778
$576$	0	0
$577$	−7.49913e6	−0.937717	−0.468858	−	0.883273i	$-0.655335\pi$
$577$	−7.49913e6	−0.937717	−0.468858	+	0.883273i	$0.655335\pi$
$578$	0	0
$579$	1.11101e6	0.137728
$580$	0	0
$581$	−7.13994e6	−0.877515
$582$	0	0
$583$	5.36444e6	0.653662
$584$	0	0
$585$	−170972.	−0.0206555
$586$	0	0
$587$	−5.45467e6	−0.653391	−0.326696	−	0.945130i	$-0.605935\pi$
$587$	−5.45467e6	−0.653391	−0.326696	+	0.945130i	$0.605935\pi$
$588$	0	0
$589$	4.40235e6	0.522873
$590$	0	0
$591$	−1.81917e6	−0.214241
$592$	0	0
$593$	3.88062e6	0.453174	0.226587	−	0.973991i	$-0.427243\pi$
$593$	3.88062e6	0.453174	0.226587	+	0.973991i	$0.427243\pi$
$594$	0	0
$595$	−2.19701e6	−0.254414
$596$	0	0
$597$	6.79935e6	0.780785
$598$	0	0
$599$	1.12527e7	1.28141	0.640705	−	0.767787i	$-0.278642\pi$
$599$	1.12527e7	1.28141	0.640705	+	0.767787i	$0.278642\pi$
$600$	0	0
$601$	3.37601e6	0.381257	0.190628	−	0.981662i	$-0.438947\pi$
$601$	3.37601e6	0.381257	0.190628	+	0.981662i	$0.438947\pi$
$602$	0	0
$603$	1.29587e6	0.145133
$604$	0	0
$605$	2.73721e6	0.304032
$606$	0	0
$607$	836171.	0.0921135	0.0460568	−	0.998939i	$-0.485334\pi$
$607$	836171.	0.0921135	0.0460568	+	0.998939i	$0.485334\pi$
$608$	0	0
$609$	3.09779e6	0.338461
$610$	0	0
$611$	2.54847e6	0.276170
$612$	0	0
$613$	5.01065e6	0.538571	0.269286	−	0.963060i	$-0.413212\pi$
$613$	5.01065e6	0.538571	0.269286	+	0.963060i	$0.413212\pi$
$614$	0	0
$615$	605614.	0.0645667
$616$	0	0
$617$	1.47301e7	1.55773	0.778864	−	0.627193i	$-0.215796\pi$
$617$	1.47301e7	1.55773	0.778864	+	0.627193i	$0.215796\pi$
$618$	0	0
$619$	6.91542e6	0.725424	0.362712	−	0.931901i	$-0.381851\pi$
$619$	6.91542e6	0.725424	0.362712	+	0.931901i	$0.381851\pi$
$620$	0	0
$621$	−3.44124e6	−0.358084
$622$	0	0
$623$	1.19331e7	1.23178
$624$	0	0
$625$	8.32752e6	0.852738
$626$	0	0
$627$	−6.25103e6	−0.635013
$628$	0	0
$629$	−8.29090e6	−0.835555
$630$	0	0
$631$	−1.33542e7	−1.33519	−0.667596	−	0.744524i	$-0.732677\pi$
$631$	−1.33542e7	−1.33519	−0.667596	+	0.744524i	$0.732677\pi$
$632$	0	0
$633$	5.14864e6	0.510720
$634$	0	0
$635$	478079.	0.0470506
$636$	0	0
$637$	1.19750e6	0.116930
$638$	0	0
$639$	4.77187e6	0.462314
$640$	0	0
$641$	918127.	0.0882587	0.0441293	−	0.999026i	$-0.485949\pi$
$641$	918127.	0.0882587	0.0441293	+	0.999026i	$0.485949\pi$
$642$	0	0
$643$	−2.89311e6	−0.275954	−0.137977	−	0.990435i	$-0.544060\pi$
$643$	−2.89311e6	−0.275954	−0.137977	+	0.990435i	$0.544060\pi$
$644$	0	0
$645$	−1.34323e6	−0.127131
$646$	0	0
$647$	5.69334e6	0.534695	0.267348	−	0.963600i	$-0.413853\pi$
$647$	5.69334e6	0.534695	0.267348	+	0.963600i	$0.413853\pi$
$648$	0	0
$649$	1.32728e7	1.23695
$650$	0	0
$651$	3.46808e6	0.320727
$652$	0	0
$653$	−1.15993e6	−0.106451	−0.0532254	−	0.998583i	$-0.516950\pi$
$653$	−1.15993e6	−0.106451	−0.0532254	+	0.998583i	$0.516950\pi$
$654$	0	0
$655$	4.65381e6	0.423843
$656$	0	0
$657$	−4.82517e6	−0.436113
$658$	0	0
$659$	−2.00691e7	−1.80017	−0.900085	−	0.435714i	$-0.856496\pi$
$659$	−2.00691e7	−1.80017	−0.900085	+	0.435714i	$0.856496\pi$
$660$	0	0
$661$	−1.48412e7	−1.32119	−0.660594	−	0.750743i	$-0.729696\pi$
$661$	−1.48412e7	−1.32119	−0.660594	+	0.750743i	$0.729696\pi$
$662$	0	0
$663$	2.71362e6	0.239754
$664$	0	0
$665$	1.38711e6	0.121635
$666$	0	0
$667$	−1.64792e7	−1.43424
$668$	0	0
$669$	1.26712e7	1.09459
$670$	0	0
$671$	1.76719e6	0.151522
$672$	0	0
$673$	1.13728e7	0.967900	0.483950	−	0.875096i	$-0.339202\pi$
$673$	1.13728e7	0.967900	0.483950	+	0.875096i	$0.339202\pi$
$674$	0	0
$675$	−2.16441e6	−0.182843
$676$	0	0
$677$	1.97484e7	1.65600	0.827998	−	0.560731i	$-0.189480\pi$
$677$	1.97484e7	1.65600	0.827998	+	0.560731i	$0.189480\pi$
$678$	0	0
$679$	−9.43295e6	−0.785187
$680$	0	0
$681$	23427.3	0.00193578
$682$	0	0
$683$	1.61752e7	1.32677	0.663387	−	0.748277i	$-0.269119\pi$
$683$	1.61752e7	1.32677	0.663387	+	0.748277i	$0.269119\pi$
$684$	0	0
$685$	−3.13601e6	−0.255359
$686$	0	0
$687$	−1.23316e7	−0.996841
$688$	0	0
$689$	1.47028e6	0.117992
$690$	0	0
$691$	2.00679e7	1.59885	0.799424	−	0.600767i	$-0.205138\pi$
$691$	2.00679e7	1.59885	0.799424	+	0.600767i	$0.205138\pi$
$692$	0	0
$693$	−4.92443e6	−0.389514
$694$	0	0
$695$	3.57603e6	0.280827
$696$	0	0
$697$	−9.61214e6	−0.749442
$698$	0	0
$699$	7.09178e6	0.548988
$700$	0	0
$701$	3.41979e6	0.262847	0.131424	−	0.991326i	$-0.458045\pi$
$701$	3.41979e6	0.262847	0.131424	+	0.991326i	$0.458045\pi$
$702$	0	0
$703$	5.23456e6	0.399477
$704$	0	0
$705$	−1.69507e6	−0.128445
$706$	0	0
$707$	1.97675e6	0.148732
$708$	0	0
$709$	−3.26171e6	−0.243685	−0.121843	−	0.992549i	$-0.538880\pi$
$709$	−3.26171e6	−0.243685	−0.121843	+	0.992549i	$0.538880\pi$
$710$	0	0
$711$	5.43352e6	0.403095
$712$	0	0
$713$	−1.84490e7	−1.35909
$714$	0	0
$715$	1.30152e6	0.0952106
$716$	0	0
$717$	7.36533e6	0.535050
$718$	0	0
$719$	1.11854e7	0.806917	0.403459	−	0.914998i	$-0.367808\pi$
$719$	1.11854e7	0.806917	0.403459	+	0.914998i	$0.367808\pi$
$720$	0	0
$721$	−1.06099e7	−0.760105
$722$	0	0
$723$	1.17632e7	0.836913
$724$	0	0
$725$	−1.03648e7	−0.732344
$726$	0	0
$727$	−7.39578e6	−0.518977	−0.259488	−	0.965746i	$-0.583554\pi$
$727$	−7.39578e6	−0.518977	−0.259488	+	0.965746i	$0.583554\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	2.13194e7	1.47565
$732$	0	0
$733$	−2.20521e6	−0.151597	−0.0757985	−	0.997123i	$-0.524151\pi$
$733$	−2.20521e6	−0.151597	−0.0757985	+	0.997123i	$0.524151\pi$
$734$	0	0
$735$	−796497.	−0.0543833
$736$	0	0
$737$	−9.86476e6	−0.668987
$738$	0	0
$739$	−5.08425e6	−0.342465	−0.171232	−	0.985231i	$-0.554775\pi$
$739$	−5.08425e6	−0.342465	−0.171232	+	0.985231i	$0.554775\pi$
$740$	0	0
$741$	−1.71328e6	−0.114626
$742$	0	0
$743$	6.19234e6	0.411513	0.205756	−	0.978603i	$-0.434035\pi$
$743$	6.19234e6	0.411513	0.205756	+	0.978603i	$0.434035\pi$
$744$	0	0
$745$	−4.60125e6	−0.303728
$746$	0	0
$747$	−5.86570e6	−0.384608
$748$	0	0
$749$	−2.11328e7	−1.37643
$750$	0	0
$751$	−1.08001e7	−0.698762	−0.349381	−	0.936981i	$-0.613608\pi$
$751$	−1.08001e7	−0.698762	−0.349381	+	0.936981i	$0.613608\pi$
$752$	0	0
$753$	−3.29679e6	−0.211887
$754$	0	0
$755$	−3.11830e6	−0.199090
$756$	0	0
$757$	−2.54604e7	−1.61482	−0.807412	−	0.589988i	$-0.799133\pi$
$757$	−2.54604e7	−1.61482	−0.807412	+	0.589988i	$0.799133\pi$
$758$	0	0
$759$	2.61963e7	1.65058
$760$	0	0
$761$	1.21948e7	0.763332	0.381666	−	0.924300i	$-0.375350\pi$
$761$	1.21948e7	0.763332	0.381666	+	0.924300i	$0.375350\pi$
$762$	0	0
$763$	−1.53695e7	−0.955759
$764$	0	0
$765$	−1.80492e6	−0.111508
$766$	0	0
$767$	3.63781e6	0.223281
$768$	0	0
$769$	−877872.	−0.0535322	−0.0267661	−	0.999642i	$-0.508521\pi$
$769$	−877872.	−0.0535322	−0.0267661	+	0.999642i	$0.508521\pi$
$770$	0	0
$771$	−444512.	−0.0269307
$772$	0	0
$773$	−3.34247e6	−0.201196	−0.100598	−	0.994927i	$-0.532076\pi$
$773$	−3.34247e6	−0.201196	−0.100598	+	0.994927i	$0.532076\pi$
$774$	0	0
$775$	−1.16037e7	−0.693973
$776$	0	0
$777$	4.12368e6	0.245037
$778$	0	0
$779$	6.06875e6	0.358307
$780$	0	0
$781$	−3.63257e7	−2.13102
$782$	0	0
$783$	2.54494e6	0.148345
$784$	0	0
$785$	−2.15116e6	−0.124594
$786$	0	0
$787$	1.91537e7	1.10234	0.551169	−	0.834394i	$-0.314182\pi$
$787$	1.91537e7	1.10234	0.551169	+	0.834394i	$0.314182\pi$
$788$	0	0
$789$	−1.59032e7	−0.909479
$790$	0	0
$791$	−2.56989e7	−1.46041
$792$	0	0
$793$	484350.	0.0273512
$794$	0	0
$795$	−977934.	−0.0548772
$796$	0	0
$797$	−1.40570e7	−0.783877	−0.391939	−	0.919991i	$-0.628195\pi$
$797$	−1.40570e7	−0.783877	−0.391939	+	0.919991i	$0.628195\pi$
$798$	0	0
$799$	2.69037e7	1.49089
$800$	0	0
$801$	9.80347e6	0.539881
$802$	0	0
$803$	3.67315e7	2.01025
$804$	0	0
$805$	−5.81300e6	−0.316163
$806$	0	0
$807$	−1.76872e7	−0.956036
$808$	0	0
$809$	2.93356e6	0.157588	0.0787941	−	0.996891i	$-0.474893\pi$
$809$	2.93356e6	0.157588	0.0787941	+	0.996891i	$0.474893\pi$
$810$	0	0
$811$	−1.39811e7	−0.746430	−0.373215	−	0.927745i	$-0.621745\pi$
$811$	−1.39811e7	−0.746430	−0.373215	+	0.927745i	$0.621745\pi$
$812$	0	0
$813$	6.32627e6	0.335677
$814$	0	0
$815$	4.93668e6	0.260340
$816$	0	0
$817$	−1.34603e7	−0.705504
$818$	0	0
$819$	−1.34968e6	−0.0703108
$820$	0	0
$821$	−4.33466e6	−0.224438	−0.112219	−	0.993683i	$-0.535796\pi$
$821$	−4.33466e6	−0.224438	−0.112219	+	0.993683i	$0.535796\pi$
$822$	0	0
$823$	−2.82999e7	−1.45642	−0.728208	−	0.685356i	$-0.759647\pi$
$823$	−2.82999e7	−1.45642	−0.728208	+	0.685356i	$0.759647\pi$
$824$	0	0
$825$	1.64765e7	0.842809
$826$	0	0
$827$	−8.41388e6	−0.427792	−0.213896	−	0.976856i	$-0.568615\pi$
$827$	−8.41388e6	−0.427792	−0.213896	+	0.976856i	$0.568615\pi$
$828$	0	0
$829$	1.48710e6	0.0751545	0.0375772	−	0.999294i	$-0.488036\pi$
$829$	1.48710e6	0.0751545	0.0375772	+	0.999294i	$0.488036\pi$
$830$	0	0
$831$	−1.13519e7	−0.570249
$832$	0	0
$833$	1.26418e7	0.631241
$834$	0	0
$835$	4.16763e6	0.206858
$836$	0	0
$837$	2.84914e6	0.140572
$838$	0	0
$839$	−3.09088e6	−0.151593	−0.0757963	−	0.997123i	$-0.524150\pi$
$839$	−3.09088e6	−0.151593	−0.0757963	+	0.997123i	$0.524150\pi$
$840$	0	0
$841$	−8.32410e6	−0.405833
$842$	0	0
$843$	9.36063e6	0.453666
$844$	0	0
$845$	356719.	0.0171864
$846$	0	0
$847$	2.16080e7	1.03492
$848$	0	0
$849$	−1.01653e7	−0.484005
$850$	0	0
$851$	−2.19366e7	−1.03835
$852$	0	0
$853$	−2.17219e7	−1.02218	−0.511088	−	0.859529i	$-0.670757\pi$
$853$	−2.17219e7	−1.02218	−0.511088	+	0.859529i	$0.670757\pi$
$854$	0	0
$855$	1.13956e6	0.0533115
$856$	0	0
$857$	−8.65291e6	−0.402448	−0.201224	−	0.979545i	$-0.564492\pi$
$857$	−8.65291e6	−0.402448	−0.201224	+	0.979545i	$0.564492\pi$
$858$	0	0
$859$	−2.49282e7	−1.15268	−0.576340	−	0.817210i	$-0.695520\pi$
$859$	−2.49282e7	−1.15268	−0.576340	+	0.817210i	$0.695520\pi$
$860$	0	0
$861$	4.78083e6	0.219784
$862$	0	0
$863$	7.85566e6	0.359050	0.179525	−	0.983753i	$-0.442544\pi$
$863$	7.85566e6	0.359050	0.179525	+	0.983753i	$0.442544\pi$
$864$	0	0
$865$	4.62264e6	0.210063
$866$	0	0
$867$	1.58685e7	0.716947
$868$	0	0
$869$	−4.13625e7	−1.85805
$870$	0	0
$871$	−2.70373e6	−0.120758
$872$	0	0
$873$	−7.74948e6	−0.344141
$874$	0	0
$875$	−7.50440e6	−0.331357
$876$	0	0
$877$	−4.19542e7	−1.84194	−0.920972	−	0.389630i	$-0.872603\pi$
$877$	−4.19542e7	−1.84194	−0.920972	+	0.389630i	$0.872603\pi$
$878$	0	0
$879$	−2.30737e7	−1.00727
$880$	0	0
$881$	2.75936e7	1.19776	0.598878	−	0.800840i	$-0.295613\pi$
$881$	2.75936e7	1.19776	0.598878	+	0.800840i	$0.295613\pi$
$882$	0	0
$883$	−8.37978e6	−0.361685	−0.180843	−	0.983512i	$-0.557882\pi$
$883$	−8.37978e6	−0.361685	−0.180843	+	0.983512i	$0.557882\pi$
$884$	0	0
$885$	−2.41963e6	−0.103846
$886$	0	0
$887$	8.51154e6	0.363245	0.181622	−	0.983368i	$-0.441865\pi$
$887$	8.51154e6	0.363245	0.181622	+	0.983368i	$0.441865\pi$
$888$	0	0
$889$	3.77404e6	0.160159
$890$	0	0
$891$	−4.04558e6	−0.170721
$892$	0	0
$893$	−1.69860e7	−0.712792
$894$	0	0
$895$	2.56146e6	0.106888
$896$	0	0
$897$	7.17986e6	0.297944
$898$	0	0
$899$	1.36438e7	0.563036
$900$	0	0
$901$	1.55215e7	0.636974
$902$	0	0
$903$	−1.06037e7	−0.432752
$904$	0	0
$905$	696572.	0.0282712
$906$	0	0
$907$	4.32628e7	1.74621	0.873105	−	0.487532i	$-0.162103\pi$
$907$	4.32628e7	1.74621	0.873105	+	0.487532i	$0.162103\pi$
$908$	0	0
$909$	1.62396e6	0.0651879
$910$	0	0
$911$	−8.31414e6	−0.331911	−0.165955	−	0.986133i	$-0.553071\pi$
$911$	−8.31414e6	−0.331911	−0.165955	+	0.986133i	$0.553071\pi$
$912$	0	0
$913$	4.46524e7	1.77283
$914$	0	0
$915$	−322157.	−0.0127208
$916$	0	0
$917$	3.67380e7	1.44275
$918$	0	0
$919$	3.03484e7	1.18535	0.592675	−	0.805442i	$-0.298072\pi$
$919$	3.03484e7	1.18535	0.592675	+	0.805442i	$0.298072\pi$
$920$	0	0
$921$	2.66382e7	1.03480
$922$	0	0
$923$	−9.95613e6	−0.384668
$924$	0	0
$925$	−1.37973e7	−0.530199
$926$	0	0
$927$	−8.71640e6	−0.333148
$928$	0	0
$929$	−4.88666e7	−1.85769	−0.928844	−	0.370472i	$-0.879196\pi$
$929$	−4.88666e7	−1.85769	−0.928844	+	0.370472i	$0.879196\pi$
$930$	0	0
$931$	−7.98154e6	−0.301795
$932$	0	0
$933$	1.11982e7	0.421156
$934$	0	0
$935$	1.37399e7	0.513989
$936$	0	0
$937$	4.72400e7	1.75776	0.878882	−	0.477038i	$-0.158290\pi$
$937$	4.72400e7	1.75776	0.878882	+	0.477038i	$0.158290\pi$
$938$	0	0
$939$	1.44386e7	0.534393
$940$	0	0
$941$	−4.29259e7	−1.58032	−0.790160	−	0.612901i	$-0.790003\pi$
$941$	−4.29259e7	−1.58032	−0.790160	+	0.612901i	$0.790003\pi$
$942$	0	0
$943$	−2.54324e7	−0.931340
$944$	0	0
$945$	897720.	0.0327010
$946$	0	0
$947$	5.26004e6	0.190596	0.0952980	−	0.995449i	$-0.469620\pi$
$947$	5.26004e6	0.190596	0.0952980	+	0.995449i	$0.469620\pi$
$948$	0	0
$949$	1.00673e7	0.362868
$950$	0	0
$951$	1.78347e7	0.639462
$952$	0	0
$953$	−2.86111e7	−1.02047	−0.510237	−	0.860034i	$-0.670442\pi$
$953$	−2.86111e7	−1.02047	−0.510237	+	0.860034i	$0.670442\pi$
$954$	0	0
$955$	3.41320e6	0.121103
$956$	0	0
$957$	−1.93732e7	−0.683789
$958$	0	0
$959$	−2.47562e7	−0.869237
$960$	0	0
$961$	−1.33545e7	−0.466464
$962$	0	0
$963$	−1.73613e7	−0.603277
$964$	0	0
$965$	1.54181e6	0.0532981
$966$	0	0
$967$	−1.15733e7	−0.398007	−0.199003	−	0.979999i	$-0.563770\pi$
$967$	−1.15733e7	−0.398007	−0.199003	+	0.979999i	$0.563770\pi$
$968$	0	0
$969$	−1.80867e7	−0.618801
$970$	0	0
$971$	2.36472e7	0.804880	0.402440	−	0.915446i	$-0.368162\pi$
$971$	2.36472e7	0.804880	0.402440	+	0.915446i	$0.368162\pi$
$972$	0	0
$973$	2.82298e7	0.955930
$974$	0	0
$975$	4.51586e6	0.152135
$976$	0	0
$977$	1.50308e7	0.503786	0.251893	−	0.967755i	$-0.418947\pi$
$977$	1.50308e7	0.503786	0.251893	+	0.967755i	$0.418947\pi$
$978$	0	0
$979$	−7.46286e7	−2.48856
$980$	0	0
$981$	−1.26265e7	−0.418902
$982$	0	0
$983$	1.58366e7	0.522731	0.261366	−	0.965240i	$-0.415827\pi$
$983$	1.58366e7	0.522731	0.261366	+	0.965240i	$0.415827\pi$
$984$	0	0
$985$	−2.52454e6	−0.0829072
$986$	0	0
$987$	−1.33812e7	−0.437223
$988$	0	0
$989$	5.64083e7	1.83380
$990$	0	0
$991$	−1.37246e7	−0.443931	−0.221965	−	0.975055i	$-0.571247\pi$
$991$	−1.37246e7	−0.443931	−0.221965	+	0.975055i	$0.571247\pi$
$992$	0	0
$993$	−2.20525e7	−0.709717
$994$	0	0
$995$	9.43579e6	0.302148
$996$	0	0
$997$	−6.01789e7	−1.91737	−0.958685	−	0.284469i	$-0.908183\pi$
$997$	−6.01789e7	−1.91737	−0.958685	+	0.284469i	$0.908183\pi$
$998$	0	0
$999$	3.38774e6	0.107398

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	312.6.a.h.1.3	✓	4
3.2	odd	2		936.6.a.n.1.2		4
4.3	odd	2		624.6.a.u.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.6.a.h.1.3	✓	4	1.1	even	1	trivial
624.6.a.u.1.3		4	4.3	odd	2
936.6.a.n.1.2		4	3.2	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	312.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +12.4897 q^{5} +98.5962 q^{7} +81.0000 q^{9} -616.610 q^{11} -169.000 q^{13} +112.408 q^{15} -1784.10 q^{17} +1126.41 q^{19} +887.366 q^{21} -4720.49 q^{23} -2969.01 q^{25} +729.000 q^{27} +3491.00 q^{29} +3908.28 q^{31} -5549.49 q^{33} +1231.44 q^{35} +4647.10 q^{37} -1521.00 q^{39} +5387.66 q^{41} -11949.7 q^{43} +1011.67 q^{45} -15079.7 q^{47} -7085.79 q^{49} -16056.9 q^{51} -8699.89 q^{53} -7701.29 q^{55} +10137.7 q^{57} -21525.5 q^{59} -2865.97 q^{61} +7986.29 q^{63} -2110.76 q^{65} +15998.4 q^{67} -42484.4 q^{69} +58912.0 q^{71} -59570.0 q^{73} -26721.1 q^{75} -60795.4 q^{77} +67080.5 q^{79} +6561.00 q^{81} -72416.0 q^{83} -22282.9 q^{85} +31419.0 q^{87} +121030. q^{89} -16662.8 q^{91} +35174.5 q^{93} +14068.6 q^{95} -95672.6 q^{97} -49945.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 36 q^{3} - 56 q^{5} + 48 q^{7} + 324 q^{9} - 368 q^{11} - 676 q^{13} - 504 q^{15} - 1976 q^{17} - 1808 q^{19} + 432 q^{21} - 240 q^{23} + 2652 q^{25} + 2916 q^{27} - 4792 q^{29} - 9296 q^{31} - 3312 q^{33}+ \cdots - 29808 q^{99}+O(q^{100}) \) 4 * q + 36 * q^3 - 56 * q^5 + 48 * q^7 + 324 * q^9 - 368 * q^11 - 676 * q^13 - 504 * q^15 - 1976 * q^17 - 1808 * q^19 + 432 * q^21 - 240 * q^23 + 2652 * q^25 + 2916 * q^27 - 4792 * q^29 - 9296 * q^31 - 3312 * q^33 - 21424 * q^35 - 7800 * q^37 - 6084 * q^39 - 8792 * q^41 - 19136 * q^43 - 4536 * q^45 - 37968 * q^47 + 2132 * q^49 - 17784 * q^51 - 15192 * q^53 - 28976 * q^55 - 16272 * q^57 - 69424 * q^59 - 22568 * q^61 + 3888 * q^63 + 9464 * q^65 + 1136 * q^67 - 2160 * q^69 - 26672 * q^71 - 19736 * q^73 + 23868 * q^75 - 76480 * q^77 + 51328 * q^79 + 26244 * q^81 - 22544 * q^83 - 183888 * q^85 - 43128 * q^87 - 216600 * q^89 - 8112 * q^91 - 83664 * q^93 + 85936 * q^95 + 53512 * q^97 - 29808 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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