LMFDB - Embedded newform 784.6.a.t.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,6,Mod(1,784)] 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("784.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-8,0,38,0,0,0,-380] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$125.740914733$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{37}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9 $ x^2 - x - 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.54138$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-10.0828 q^{3} +79.8276 q^{5} -141.338 q^{9} -351.904 q^{11} +291.683 q^{13} -804.883 q^{15} +370.075 q^{17} +1504.93 q^{19} +425.711 q^{23} +3247.45 q^{25} +3875.19 q^{27} -7783.93 q^{29} -2575.18 q^{31} +3548.16 q^{33} +739.618 q^{37} -2940.97 q^{39} -7029.84 q^{41} -1835.23 q^{43} -11282.7 q^{45} -1532.68 q^{47} -3731.38 q^{51} -9537.46 q^{53} -28091.6 q^{55} -15173.8 q^{57} -29674.1 q^{59} +46510.8 q^{61} +23284.3 q^{65} -26746.1 q^{67} -4292.34 q^{69} +14388.8 q^{71} +70095.1 q^{73} -32743.3 q^{75} +27085.8 q^{79} -4727.49 q^{81} -79755.4 q^{83} +29542.2 q^{85} +78483.6 q^{87} -43577.3 q^{89} +25964.9 q^{93} +120135. q^{95} -103374. q^{97} +49737.3 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{3} + 38 q^{5} - 380 q^{9} - 424 q^{11} + 924 q^{13} - 892 q^{15} + 2346 q^{17} - 360 q^{19} + 12 q^{23} + 1872 q^{25} + 2872 q^{27} - 7052 q^{29} + 3548 q^{31} + 3398 q^{33} + 11090 q^{37} - 1624 q^{39}+ \cdots + 66944 q^{99}+O(q^{100}) $ 2 * q - 8 * q^3 + 38 * q^5 - 380 * q^9 - 424 * q^11 + 924 * q^13 - 892 * q^15 + 2346 * q^17 - 360 * q^19 + 12 * q^23 + 1872 * q^25 + 2872 * q^27 - 7052 * q^29 + 3548 * q^31 + 3398 * q^33 + 11090 * q^37 - 1624 * q^39 - 3500 * q^41 + 12680 * q^43 - 1300 * q^45 - 22956 * q^47 + 384 * q^51 + 3042 * q^53 - 25076 * q^55 - 19058 * q^57 - 65808 * q^59 + 42486 * q^61 - 3164 * q^65 - 42312 * q^67 - 5154 * q^69 + 2208 * q^71 + 50506 * q^73 - 35608 * q^75 - 9004 * q^79 + 51178 * q^81 - 104328 * q^83 - 53106 * q^85 + 80008 * q^87 + 26666 * q^89 + 38718 * q^93 + 198140 * q^95 - 209132 * q^97 + 66944 * 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
$3$	−10.0828	−0.646810	−0.323405	−	0.946261i	$-0.604828\pi$
$3$	−10.0828	−0.646810	−0.323405	+	0.946261i	$0.604828\pi$
$4$	0	0
$5$	79.8276	1.42800	0.714000	−	0.700146i	$-0.246882\pi$
$5$	79.8276	1.42800	0.714000	+	0.700146i	$0.246882\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−141.338	−0.581637
$10$	0	0
$11$	−351.904	−0.876884	−0.438442	−	0.898760i	$-0.644469\pi$
$11$	−351.904	−0.876884	−0.438442	+	0.898760i	$0.644469\pi$
$12$	0	0
$13$	291.683	0.478688	0.239344	−	0.970935i	$-0.423068\pi$
$13$	291.683	0.478688	0.239344	+	0.970935i	$0.423068\pi$
$14$	0	0
$15$	−804.883	−0.923644
$16$	0	0
$17$	370.075	0.310576	0.155288	−	0.987869i	$-0.450369\pi$
$17$	370.075	0.310576	0.155288	+	0.987869i	$0.450369\pi$
$18$	0	0
$19$	1504.93	0.956381	0.478190	−	0.878256i	$-0.341293\pi$
$19$	1504.93	0.956381	0.478190	+	0.878256i	$0.341293\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	425.711	0.167801	0.0839006	−	0.996474i	$-0.473262\pi$
$23$	425.711	0.167801	0.0839006	+	0.996474i	$0.473262\pi$
$24$	0	0
$25$	3247.45	1.03918
$26$	0	0
$27$	3875.19	1.02302
$28$	0	0
$29$	−7783.93	−1.71872	−0.859358	−	0.511374i	$-0.829137\pi$
$29$	−7783.93	−1.71872	−0.859358	+	0.511374i	$0.829137\pi$
$30$	0	0
$31$	−2575.18	−0.481285	−0.240643	−	0.970614i	$-0.577358\pi$
$31$	−2575.18	−0.481285	−0.240643	+	0.970614i	$0.577358\pi$
$32$	0	0
$33$	3548.16	0.567177
$34$	0	0
$35$	0	0
$36$	0	0
$37$	739.618	0.0888184	0.0444092	−	0.999013i	$-0.485859\pi$
$37$	739.618	0.0888184	0.0444092	+	0.999013i	$0.485859\pi$
$38$	0	0
$39$	−2940.97	−0.309620
$40$	0	0
$41$	−7029.84	−0.653109	−0.326554	−	0.945178i	$-0.605888\pi$
$41$	−7029.84	−0.653109	−0.326554	+	0.945178i	$0.605888\pi$
$42$	0	0
$43$	−1835.23	−0.151363	−0.0756816	−	0.997132i	$-0.524113\pi$
$43$	−1835.23	−0.151363	−0.0756816	+	0.997132i	$0.524113\pi$
$44$	0	0
$45$	−11282.7	−0.830578
$46$	0	0
$47$	−1532.68	−0.101206	−0.0506032	−	0.998719i	$-0.516114\pi$
$47$	−1532.68	−0.101206	−0.0506032	+	0.998719i	$0.516114\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−3731.38	−0.200883
$52$	0	0
$53$	−9537.46	−0.466383	−0.233192	−	0.972431i	$-0.574917\pi$
$53$	−9537.46	−0.466383	−0.233192	+	0.972431i	$0.574917\pi$
$54$	0	0
$55$	−28091.6	−1.25219
$56$	0	0
$57$	−15173.8	−0.618596
$58$	0	0
$59$	−29674.1	−1.10981	−0.554903	−	0.831915i	$-0.687245\pi$
$59$	−29674.1	−1.10981	−0.554903	+	0.831915i	$0.687245\pi$
$60$	0	0
$61$	46510.8	1.60040	0.800201	−	0.599732i	$-0.204726\pi$
$61$	46510.8	1.60040	0.800201	+	0.599732i	$0.204726\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	23284.3	0.683566
$66$	0	0
$67$	−26746.1	−0.727902	−0.363951	−	0.931418i	$-0.618572\pi$
$67$	−26746.1	−0.727902	−0.363951	+	0.931418i	$0.618572\pi$
$68$	0	0
$69$	−4292.34	−0.108535
$70$	0	0
$71$	14388.8	0.338748	0.169374	−	0.985552i	$-0.445825\pi$
$71$	14388.8	0.338748	0.169374	+	0.985552i	$0.445825\pi$
$72$	0	0
$73$	70095.1	1.53950	0.769752	−	0.638343i	$-0.220380\pi$
$73$	70095.1	1.53950	0.769752	+	0.638343i	$0.220380\pi$
$74$	0	0
$75$	−32743.3	−0.672154
$76$	0	0
$77$	0	0
$78$	0	0
$79$	27085.8	0.488285	0.244143	−	0.969739i	$-0.421493\pi$
$79$	27085.8	0.488285	0.244143	+	0.969739i	$0.421493\pi$
$80$	0	0
$81$	−4727.49	−0.0800604
$82$	0	0
$83$	−79755.4	−1.27076	−0.635382	−	0.772198i	$-0.719157\pi$
$83$	−79755.4	−1.27076	−0.635382	+	0.772198i	$0.719157\pi$
$84$	0	0
$85$	29542.2	0.443502
$86$	0	0
$87$	78483.6	1.11168
$88$	0	0
$89$	−43577.3	−0.583157	−0.291579	−	0.956547i	$-0.594180\pi$
$89$	−43577.3	−0.583157	−0.291579	+	0.956547i	$0.594180\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	25964.9	0.311300
$94$	0	0
$95$	120135.	1.36571
$96$	0	0
$97$	−103374.	−1.11553	−0.557765	−	0.829999i	$-0.688341\pi$
$97$	−103374.	−1.11553	−0.557765	+	0.829999i	$0.688341\pi$
$98$	0	0
$99$	49737.3	0.510028
$100$	0	0
$101$	−28700.2	−0.279951	−0.139975	−	0.990155i	$-0.544702\pi$
$101$	−28700.2	−0.279951	−0.139975	+	0.990155i	$0.544702\pi$
$102$	0	0
$103$	29227.9	0.271459	0.135730	−	0.990746i	$-0.456662\pi$
$103$	29227.9	0.271459	0.135730	+	0.990746i	$0.456662\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−87858.4	−0.741863	−0.370932	−	0.928660i	$-0.620962\pi$
$107$	−87858.4	−0.741863	−0.370932	+	0.928660i	$0.620962\pi$
$108$	0	0
$109$	220628.	1.77867	0.889333	−	0.457260i	$-0.151169\pi$
$109$	220628.	1.77867	0.889333	+	0.457260i	$0.151169\pi$
$110$	0	0
$111$	−7457.39	−0.0574486
$112$	0	0
$113$	39665.6	0.292225	0.146113	−	0.989268i	$-0.453324\pi$
$113$	39665.6	0.292225	0.146113	+	0.989268i	$0.453324\pi$
$114$	0	0
$115$	33983.5	0.239620
$116$	0	0
$117$	−41225.8	−0.278423
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−37214.9	−0.231075
$122$	0	0
$123$	70880.2	0.422437
$124$	0	0
$125$	9774.87	0.0559546
$126$	0	0
$127$	−51740.3	−0.284655	−0.142328	−	0.989820i	$-0.545459\pi$
$127$	−51740.3	−0.284655	−0.142328	+	0.989820i	$0.545459\pi$
$128$	0	0
$129$	18504.2	0.0979031
$130$	0	0
$131$	−166674.	−0.848572	−0.424286	−	0.905528i	$-0.639475\pi$
$131$	−166674.	−0.848572	−0.424286	+	0.905528i	$0.639475\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	309347.	1.46087
$136$	0	0
$137$	28259.4	0.128636	0.0643178	−	0.997929i	$-0.479513\pi$
$137$	28259.4	0.128636	0.0643178	+	0.997929i	$0.479513\pi$
$138$	0	0
$139$	336393.	1.47676	0.738380	−	0.674384i	$-0.235591\pi$
$139$	336393.	1.47676	0.738380	+	0.674384i	$0.235591\pi$
$140$	0	0
$141$	15453.7	0.0654612
$142$	0	0
$143$	−102644.	−0.419753
$144$	0	0
$145$	−621373.	−2.45433
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−355381.	−1.31138	−0.655691	−	0.755030i	$-0.727622\pi$
$149$	−355381.	−1.31138	−0.655691	+	0.755030i	$0.727622\pi$
$150$	0	0
$151$	358797.	1.28058	0.640290	−	0.768133i	$-0.278814\pi$
$151$	358797.	1.28058	0.640290	+	0.768133i	$0.278814\pi$
$152$	0	0
$153$	−52305.7	−0.180643
$154$	0	0
$155$	−205570.	−0.687275
$156$	0	0
$157$	−458911.	−1.48586	−0.742932	−	0.669367i	$-0.766565\pi$
$157$	−458911.	−1.48586	−0.742932	+	0.669367i	$0.766565\pi$
$158$	0	0
$159$	96164.0	0.301661
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−502441.	−1.48121	−0.740603	−	0.671943i	$-0.765460\pi$
$163$	−502441.	−1.48121	−0.740603	+	0.671943i	$0.765460\pi$
$164$	0	0
$165$	283241.	0.809928
$166$	0	0
$167$	−676652.	−1.87748	−0.938738	−	0.344632i	$-0.888004\pi$
$167$	−676652.	−1.87748	−0.938738	+	0.344632i	$0.888004\pi$
$168$	0	0
$169$	−286214.	−0.770858
$170$	0	0
$171$	−212703.	−0.556267
$172$	0	0
$173$	249160.	0.632941	0.316470	−	0.948602i	$-0.397502\pi$
$173$	249160.	0.632941	0.316470	+	0.948602i	$0.397502\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	299196.	0.717833
$178$	0	0
$179$	−139258.	−0.324853	−0.162427	−	0.986721i	$-0.551932\pi$
$179$	−139258.	−0.324853	−0.162427	+	0.986721i	$0.551932\pi$
$180$	0	0
$181$	−306246.	−0.694823	−0.347412	−	0.937713i	$-0.612939\pi$
$181$	−306246.	−0.694823	−0.347412	+	0.937713i	$0.612939\pi$
$182$	0	0
$183$	−468957.	−1.03516
$184$	0	0
$185$	59041.9	0.126833
$186$	0	0
$187$	−130231.	−0.272339
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−227494.	−0.451218	−0.225609	−	0.974218i	$-0.572437\pi$
$191$	−227494.	−0.451218	−0.225609	+	0.974218i	$0.572437\pi$
$192$	0	0
$193$	−672374.	−1.29933	−0.649663	−	0.760223i	$-0.725090\pi$
$193$	−672374.	−1.29933	−0.649663	+	0.760223i	$0.725090\pi$
$194$	0	0
$195$	−234770.	−0.442137
$196$	0	0
$197$	−1282.76	−0.00235493	−0.00117747	−	0.999999i	$-0.500375\pi$
$197$	−1282.76	−0.00235493	−0.00117747	+	0.999999i	$0.500375\pi$
$198$	0	0
$199$	368898.	0.660349	0.330175	−	0.943920i	$-0.392892\pi$
$199$	368898.	0.660349	0.330175	+	0.943920i	$0.392892\pi$
$200$	0	0
$201$	269674.	0.470814
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−561175.	−0.932640
$206$	0	0
$207$	−60169.0	−0.0975994
$208$	0	0
$209$	−529589.	−0.838635
$210$	0	0
$211$	−502168.	−0.776503	−0.388251	−	0.921553i	$-0.626921\pi$
$211$	−502168.	−0.776503	−0.388251	+	0.921553i	$0.626921\pi$
$212$	0	0
$213$	−145078.	−0.219106
$214$	0	0
$215$	−146502.	−0.216147
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−706753.	−0.995765
$220$	0	0
$221$	107945.	0.148669
$222$	0	0
$223$	−1.17328e6	−1.57993	−0.789967	−	0.613149i	$-0.789902\pi$
$223$	−1.17328e6	−1.57993	−0.789967	+	0.613149i	$0.789902\pi$
$224$	0	0
$225$	−458988.	−0.604428
$226$	0	0
$227$	−910159.	−1.17234	−0.586168	−	0.810189i	$-0.699364\pi$
$227$	−910159.	−1.17234	−0.586168	+	0.810189i	$0.699364\pi$
$228$	0	0
$229$	−521924.	−0.657686	−0.328843	−	0.944385i	$-0.606659\pi$
$229$	−521924.	−0.657686	−0.328843	+	0.944385i	$0.606659\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.04279e6	−1.25836	−0.629182	−	0.777258i	$-0.716610\pi$
$233$	−1.04279e6	−1.25836	−0.629182	+	0.777258i	$0.716610\pi$
$234$	0	0
$235$	−122350.	−0.144523
$236$	0	0
$237$	−273100.	−0.315828
$238$	0	0
$239$	1.53447e6	1.73766	0.868830	−	0.495110i	$-0.164872\pi$
$239$	1.53447e6	1.73766	0.868830	+	0.495110i	$0.164872\pi$
$240$	0	0
$241$	1.00758e6	1.11747	0.558735	−	0.829346i	$-0.311287\pi$
$241$	1.00758e6	1.11747	0.558735	+	0.829346i	$0.311287\pi$
$242$	0	0
$243$	−894004.	−0.971234
$244$	0	0
$245$	0	0
$246$	0	0
$247$	438961.	0.457808
$248$	0	0
$249$	804155.	0.821942
$250$	0	0
$251$	8511.89	0.00852789	0.00426394	−	0.999991i	$-0.498643\pi$
$251$	8511.89	0.00852789	0.00426394	+	0.999991i	$0.498643\pi$
$252$	0	0
$253$	−149809.	−0.147142
$254$	0	0
$255$	−297867.	−0.286862
$256$	0	0
$257$	527532.	0.498214	0.249107	−	0.968476i	$-0.419863\pi$
$257$	527532.	0.498214	0.249107	+	0.968476i	$0.419863\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.10016e6	0.999670
$262$	0	0
$263$	352085.	0.313876	0.156938	−	0.987608i	$-0.449838\pi$
$263$	352085.	0.313876	0.156938	+	0.987608i	$0.449838\pi$
$264$	0	0
$265$	−761353.	−0.665996
$266$	0	0
$267$	439380.	0.377192
$268$	0	0
$269$	−479540.	−0.404058	−0.202029	−	0.979380i	$-0.564754\pi$
$269$	−479540.	−0.404058	−0.202029	+	0.979380i	$0.564754\pi$
$270$	0	0
$271$	−977611.	−0.808617	−0.404308	−	0.914623i	$-0.632488\pi$
$271$	−977611.	−0.808617	−0.404308	+	0.914623i	$0.632488\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.14279e6	−0.911243
$276$	0	0
$277$	968723.	0.758578	0.379289	−	0.925278i	$-0.376169\pi$
$277$	968723.	0.758578	0.379289	+	0.925278i	$0.376169\pi$
$278$	0	0
$279$	363970.	0.279934
$280$	0	0
$281$	−318333.	−0.240501	−0.120250	−	0.992744i	$-0.538370\pi$
$281$	−318333.	−0.240501	−0.120250	+	0.992744i	$0.538370\pi$
$282$	0	0
$283$	1.77210e6	1.31529	0.657646	−	0.753327i	$-0.271552\pi$
$283$	1.77210e6	1.31529	0.657646	+	0.753327i	$0.271552\pi$
$284$	0	0
$285$	−1.21129e6	−0.883356
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.28290e6	−0.903543
$290$	0	0
$291$	1.04229e6	0.721535
$292$	0	0
$293$	−1.64148e6	−1.11703	−0.558516	−	0.829494i	$-0.688629\pi$
$293$	−1.64148e6	−1.11703	−0.558516	+	0.829494i	$0.688629\pi$
$294$	0	0
$295$	−2.36881e6	−1.58480
$296$	0	0
$297$	−1.36369e6	−0.897068
$298$	0	0
$299$	124172.	0.0803243
$300$	0	0
$301$	0	0
$302$	0	0
$303$	289377.	0.181075
$304$	0	0
$305$	3.71285e6	2.28537
$306$	0	0
$307$	−466930.	−0.282752	−0.141376	−	0.989956i	$-0.545153\pi$
$307$	−466930.	−0.282752	−0.141376	+	0.989956i	$0.545153\pi$
$308$	0	0
$309$	−294698.	−0.175582
$310$	0	0
$311$	−2.43796e6	−1.42931	−0.714654	−	0.699478i	$-0.753416\pi$
$311$	−2.43796e6	−1.42931	−0.714654	+	0.699478i	$0.753416\pi$
$312$	0	0
$313$	−2.42094e6	−1.39676	−0.698381	−	0.715726i	$-0.746096\pi$
$313$	−2.42094e6	−1.39676	−0.698381	+	0.715726i	$0.746096\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.87611e6	1.04860	0.524301	−	0.851533i	$-0.324327\pi$
$317$	1.87611e6	1.04860	0.524301	+	0.851533i	$0.324327\pi$
$318$	0	0
$319$	2.73919e6	1.50711
$320$	0	0
$321$	885855.	0.479844
$322$	0	0
$323$	556936.	0.297029
$324$	0	0
$325$	947225.	0.497445
$326$	0	0
$327$	−2.22454e6	−1.15046
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.08310e6	0.543373	0.271686	−	0.962386i	$-0.412419\pi$
$331$	1.08310e6	0.543373	0.271686	+	0.962386i	$0.412419\pi$
$332$	0	0
$333$	−104536.	−0.0516601
$334$	0	0
$335$	−2.13507e6	−1.03944
$336$	0	0
$337$	−2.59465e6	−1.24453	−0.622263	−	0.782809i	$-0.713786\pi$
$337$	−2.59465e6	−1.24453	−0.622263	+	0.782809i	$0.713786\pi$
$338$	0	0
$339$	−399939.	−0.189014
$340$	0	0
$341$	906213.	0.422031
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−342647.	−0.154989
$346$	0	0
$347$	1.87051e6	0.833943	0.416972	−	0.908920i	$-0.363091\pi$
$347$	1.87051e6	0.833943	0.416972	+	0.908920i	$0.363091\pi$
$348$	0	0
$349$	1.61685e6	0.710568	0.355284	−	0.934758i	$-0.384384\pi$
$349$	1.61685e6	0.710568	0.355284	+	0.934758i	$0.384384\pi$
$350$	0	0
$351$	1.13033e6	0.489706
$352$	0	0
$353$	578305.	0.247013	0.123507	−	0.992344i	$-0.460586\pi$
$353$	578305.	0.247013	0.123507	+	0.992344i	$0.460586\pi$
$354$	0	0
$355$	1.14862e6	0.483733
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.96818e6	0.805988	0.402994	−	0.915203i	$-0.367970\pi$
$359$	1.96818e6	0.805988	0.402994	+	0.915203i	$0.367970\pi$
$360$	0	0
$361$	−211299.	−0.0853355
$362$	0	0
$363$	375229.	0.149462
$364$	0	0
$365$	5.59553e6	2.19841
$366$	0	0
$367$	2.17452e6	0.842749	0.421375	−	0.906887i	$-0.361548\pi$
$367$	2.17452e6	0.842749	0.421375	+	0.906887i	$0.361548\pi$
$368$	0	0
$369$	993583.	0.379873
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1.38476e6	0.515349	0.257675	−	0.966232i	$-0.417044\pi$
$373$	1.38476e6	0.515349	0.257675	+	0.966232i	$0.417044\pi$
$374$	0	0
$375$	−98557.7	−0.0361920
$376$	0	0
$377$	−2.27044e6	−0.822728
$378$	0	0
$379$	−3.37190e6	−1.20580	−0.602902	−	0.797815i	$-0.705989\pi$
$379$	−3.37190e6	−1.20580	−0.602902	+	0.797815i	$0.705989\pi$
$380$	0	0
$381$	521685.	0.184118
$382$	0	0
$383$	3.28060e6	1.14276	0.571382	−	0.820685i	$-0.306408\pi$
$383$	3.28060e6	1.14276	0.571382	+	0.820685i	$0.306408\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	259388.	0.0880385
$388$	0	0
$389$	−2.94810e6	−0.987797	−0.493899	−	0.869520i	$-0.664429\pi$
$389$	−2.94810e6	−0.987797	−0.493899	+	0.869520i	$0.664429\pi$
$390$	0	0
$391$	157545.	0.0521150
$392$	0	0
$393$	1.68053e6	0.548865
$394$	0	0
$395$	2.16219e6	0.697271
$396$	0	0
$397$	69270.2	0.0220582	0.0110291	−	0.999939i	$-0.496489\pi$
$397$	69270.2	0.0220582	0.0110291	+	0.999939i	$0.496489\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.34786e6	−1.03970	−0.519848	−	0.854259i	$-0.674012\pi$
$401$	−3.34786e6	−1.03970	−0.519848	+	0.854259i	$0.674012\pi$
$402$	0	0
$403$	−751134.	−0.230385
$404$	0	0
$405$	−377384.	−0.114326
$406$	0	0
$407$	−260274.	−0.0778834
$408$	0	0
$409$	−2.91217e6	−0.860812	−0.430406	−	0.902636i	$-0.641630\pi$
$409$	−2.91217e6	−0.860812	−0.430406	+	0.902636i	$0.641630\pi$
$410$	0	0
$411$	−284933.	−0.0832028
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−6.36669e6	−1.81465
$416$	0	0
$417$	−3.39177e6	−0.955183
$418$	0	0
$419$	−4.62361e6	−1.28661	−0.643304	−	0.765611i	$-0.722437\pi$
$419$	−4.62361e6	−1.28661	−0.643304	+	0.765611i	$0.722437\pi$
$420$	0	0
$421$	−2.63042e6	−0.723303	−0.361652	−	0.932313i	$-0.617787\pi$
$421$	−2.63042e6	−0.723303	−0.361652	+	0.932313i	$0.617787\pi$
$422$	0	0
$423$	216626.	0.0588654
$424$	0	0
$425$	1.20180e6	0.322746
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.03494e6	0.271500
$430$	0	0
$431$	−7.54128e6	−1.95547	−0.977736	−	0.209837i	$-0.932707\pi$
$431$	−7.54128e6	−1.95547	−0.977736	+	0.209837i	$0.932707\pi$
$432$	0	0
$433$	−5.83558e6	−1.49577	−0.747883	−	0.663830i	$-0.768930\pi$
$433$	−5.83558e6	−1.49577	−0.747883	+	0.663830i	$0.768930\pi$
$434$	0	0
$435$	6.26516e6	1.58748
$436$	0	0
$437$	640663.	0.160482
$438$	0	0
$439$	168104.	0.0416310	0.0208155	−	0.999783i	$-0.493374\pi$
$439$	168104.	0.0416310	0.0208155	+	0.999783i	$0.493374\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.84151e6	0.687924	0.343962	−	0.938984i	$-0.388231\pi$
$443$	2.84151e6	0.687924	0.343962	+	0.938984i	$0.388231\pi$
$444$	0	0
$445$	−3.47867e6	−0.832748
$446$	0	0
$447$	3.58323e6	0.848214
$448$	0	0
$449$	−1.41567e6	−0.331396	−0.165698	−	0.986177i	$-0.552988\pi$
$449$	−1.41567e6	−0.331396	−0.165698	+	0.986177i	$0.552988\pi$
$450$	0	0
$451$	2.47382e6	0.572701
$452$	0	0
$453$	−3.61767e6	−0.828291
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.55727e6	0.348799	0.174399	−	0.984675i	$-0.444202\pi$
$457$	1.55727e6	0.348799	0.174399	+	0.984675i	$0.444202\pi$
$458$	0	0
$459$	1.43411e6	0.317725
$460$	0	0
$461$	4.45345e6	0.975987	0.487994	−	0.872847i	$-0.337729\pi$
$461$	4.45345e6	0.975987	0.487994	+	0.872847i	$0.337729\pi$
$462$	0	0
$463$	−4.92263e6	−1.06720	−0.533599	−	0.845738i	$-0.679161\pi$
$463$	−4.92263e6	−1.06720	−0.533599	+	0.845738i	$0.679161\pi$
$464$	0	0
$465$	2.07271e6	0.444536
$466$	0	0
$467$	5.09090e6	1.08020	0.540098	−	0.841602i	$-0.318387\pi$
$467$	5.09090e6	1.08020	0.540098	+	0.841602i	$0.318387\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	4.62709e6	0.961071
$472$	0	0
$473$	645825.	0.132728
$474$	0	0
$475$	4.88717e6	0.993856
$476$	0	0
$477$	1.34800e6	0.271266
$478$	0	0
$479$	−8.30085e6	−1.65304	−0.826521	−	0.562907i	$-0.809683\pi$
$479$	−8.30085e6	−1.65304	−0.826521	+	0.562907i	$0.809683\pi$
$480$	0	0
$481$	215734.	0.0425163
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−8.25208e6	−1.59298
$486$	0	0
$487$	−8.63401e6	−1.64964	−0.824822	−	0.565392i	$-0.808725\pi$
$487$	−8.63401e6	−1.64964	−0.824822	+	0.565392i	$0.808725\pi$
$488$	0	0
$489$	5.06599e6	0.958059
$490$	0	0
$491$	−95039.5	−0.0177910	−0.00889550	−	0.999960i	$-0.502832\pi$
$491$	−95039.5	−0.0177910	−0.00889550	+	0.999960i	$0.502832\pi$
$492$	0	0
$493$	−2.88064e6	−0.533792
$494$	0	0
$495$	3.97041e6	0.728320
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−2.14203e6	−0.385101	−0.192551	−	0.981287i	$-0.561676\pi$
$499$	−2.14203e6	−0.385101	−0.192551	+	0.981287i	$0.561676\pi$
$500$	0	0
$501$	6.82252e6	1.21437
$502$	0	0
$503$	5.24794e6	0.924844	0.462422	−	0.886660i	$-0.346981\pi$
$503$	5.24794e6	0.924844	0.462422	+	0.886660i	$0.346981\pi$
$504$	0	0
$505$	−2.29107e6	−0.399769
$506$	0	0
$507$	2.88583e6	0.498598
$508$	0	0
$509$	1.05891e7	1.81160	0.905802	−	0.423702i	$-0.139270\pi$
$509$	1.05891e7	1.81160	0.905802	+	0.423702i	$0.139270\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	5.83187e6	0.978395
$514$	0	0
$515$	2.33319e6	0.387644
$516$	0	0
$517$	539357.	0.0887462
$518$	0	0
$519$	−2.51222e6	−0.409392
$520$	0	0
$521$	4.54465e6	0.733510	0.366755	−	0.930318i	$-0.380469\pi$
$521$	4.54465e6	0.733510	0.366755	+	0.930318i	$0.380469\pi$
$522$	0	0
$523$	−5.27197e6	−0.842789	−0.421394	−	0.906877i	$-0.638459\pi$
$523$	−5.27197e6	−0.842789	−0.421394	+	0.906877i	$0.638459\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−953009.	−0.149476
$528$	0	0
$529$	−6.25511e6	−0.971843
$530$	0	0
$531$	4.19407e6	0.645504
$532$	0	0
$533$	−2.05048e6	−0.312635
$534$	0	0
$535$	−7.01353e6	−1.05938
$536$	0	0
$537$	1.40410e6	0.210118
$538$	0	0
$539$	0	0
$540$	0	0
$541$	5.93445e6	0.871741	0.435871	−	0.900009i	$-0.356441\pi$
$541$	5.93445e6	0.871741	0.435871	+	0.900009i	$0.356441\pi$
$542$	0	0
$543$	3.08781e6	0.449418
$544$	0	0
$545$	1.76122e7	2.53994
$546$	0	0
$547$	8.82017e6	1.26040	0.630200	−	0.776433i	$-0.282973\pi$
$547$	8.82017e6	1.26040	0.630200	+	0.776433i	$0.282973\pi$
$548$	0	0
$549$	−6.57374e6	−0.930854
$550$	0	0
$551$	−1.17142e7	−1.64375
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−595306.	−0.0820366
$556$	0	0
$557$	−1.18224e6	−0.161461	−0.0807304	−	0.996736i	$-0.525725\pi$
$557$	−1.18224e6	−0.161461	−0.0807304	+	0.996736i	$0.525725\pi$
$558$	0	0
$559$	−535306.	−0.0724556
$560$	0	0
$561$	1.31309e6	0.176151
$562$	0	0
$563$	4.07741e6	0.542142	0.271071	−	0.962559i	$-0.412622\pi$
$563$	4.07741e6	0.542142	0.271071	+	0.962559i	$0.412622\pi$
$564$	0	0
$565$	3.16641e6	0.417298
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.17615e6	1.05869	0.529344	−	0.848407i	$-0.322438\pi$
$569$	8.17615e6	1.05869	0.529344	+	0.848407i	$0.322438\pi$
$570$	0	0
$571$	−3.30615e6	−0.424357	−0.212179	−	0.977231i	$-0.568056\pi$
$571$	−3.30615e6	−0.424357	−0.212179	+	0.977231i	$0.568056\pi$
$572$	0	0
$573$	2.29377e6	0.291852
$574$	0	0
$575$	1.38247e6	0.174376
$576$	0	0
$577$	7.14994e6	0.894052	0.447026	−	0.894521i	$-0.352483\pi$
$577$	7.14994e6	0.894052	0.447026	+	0.894521i	$0.352483\pi$
$578$	0	0
$579$	6.77939e6	0.840416
$580$	0	0
$581$	0	0
$582$	0	0
$583$	3.35627e6	0.408964
$584$	0	0
$585$	−3.29096e6	−0.397588
$586$	0	0
$587$	9.69191e6	1.16095	0.580476	−	0.814277i	$-0.302867\pi$
$587$	9.69191e6	1.16095	0.580476	+	0.814277i	$0.302867\pi$
$588$	0	0
$589$	−3.87545e6	−0.460292
$590$	0	0
$591$	12933.7	0.00152319
$592$	0	0
$593$	−6.63960e6	−0.775363	−0.387682	−	0.921793i	$-0.626724\pi$
$593$	−6.63960e6	−0.775363	−0.387682	+	0.921793i	$0.626724\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.71951e6	−0.427120
$598$	0	0
$599$	3.24191e6	0.369177	0.184588	−	0.982816i	$-0.440905\pi$
$599$	3.24191e6	0.369177	0.184588	+	0.982816i	$0.440905\pi$
$600$	0	0
$601$	5.65076e6	0.638147	0.319074	−	0.947730i	$-0.396628\pi$
$601$	5.65076e6	0.638147	0.319074	+	0.947730i	$0.396628\pi$
$602$	0	0
$603$	3.78023e6	0.423375
$604$	0	0
$605$	−2.97078e6	−0.329975
$606$	0	0
$607$	235674.	0.0259621	0.0129811	−	0.999916i	$-0.495868\pi$
$607$	235674.	0.0259621	0.0129811	+	0.999916i	$0.495868\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−447057.	−0.0484462
$612$	0	0
$613$	788877.	0.0847926	0.0423963	−	0.999101i	$-0.486501\pi$
$613$	788877.	0.0847926	0.0423963	+	0.999101i	$0.486501\pi$
$614$	0	0
$615$	5.65820e6	0.603240
$616$	0	0
$617$	1.67739e7	1.77387	0.886935	−	0.461894i	$-0.152830\pi$
$617$	1.67739e7	1.77387	0.886935	+	0.461894i	$0.152830\pi$
$618$	0	0
$619$	8.22300e6	0.862588	0.431294	−	0.902211i	$-0.358057\pi$
$619$	8.22300e6	0.862588	0.431294	+	0.902211i	$0.358057\pi$
$620$	0	0
$621$	1.64971e6	0.171664
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−9.36798e6	−0.959281
$626$	0	0
$627$	5.33972e6	0.542437
$628$	0	0
$629$	273714.	0.0275849
$630$	0	0
$631$	5.94507e6	0.594406	0.297203	−	0.954814i	$-0.403946\pi$
$631$	5.94507e6	0.594406	0.297203	+	0.954814i	$0.403946\pi$
$632$	0	0
$633$	5.06324e6	0.502249
$634$	0	0
$635$	−4.13030e6	−0.406488
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−2.03368e6	−0.197029
$640$	0	0
$641$	−1.06761e7	−1.02628	−0.513141	−	0.858304i	$-0.671518\pi$
$641$	−1.06761e7	−1.02628	−0.513141	+	0.858304i	$0.671518\pi$
$642$	0	0
$643$	−3.13159e6	−0.298701	−0.149351	−	0.988784i	$-0.547718\pi$
$643$	−3.13159e6	−0.298701	−0.149351	+	0.988784i	$0.547718\pi$
$644$	0	0
$645$	1.47715e6	0.139806
$646$	0	0
$647$	−4.93457e6	−0.463435	−0.231717	−	0.972783i	$-0.574434\pi$
$647$	−4.93457e6	−0.463435	−0.231717	+	0.972783i	$0.574434\pi$
$648$	0	0
$649$	1.04424e7	0.973170
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.72224e6	0.525150	0.262575	−	0.964912i	$-0.415428\pi$
$653$	5.72224e6	0.525150	0.262575	+	0.964912i	$0.415428\pi$
$654$	0	0
$655$	−1.33052e7	−1.21176
$656$	0	0
$657$	−9.90710e6	−0.895433
$658$	0	0
$659$	−362477.	−0.0325137	−0.0162569	−	0.999868i	$-0.505175\pi$
$659$	−362477.	−0.0325137	−0.0162569	+	0.999868i	$0.505175\pi$
$660$	0	0
$661$	1.91211e7	1.70219	0.851096	−	0.525011i	$-0.175939\pi$
$661$	1.91211e7	1.70219	0.851096	+	0.525011i	$0.175939\pi$
$662$	0	0
$663$	−1.08838e6	−0.0961604
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.31370e6	−0.288403
$668$	0	0
$669$	1.18299e7	1.02192
$670$	0	0
$671$	−1.63673e7	−1.40337
$672$	0	0
$673$	−573374.	−0.0487978	−0.0243989	−	0.999702i	$-0.507767\pi$
$673$	−573374.	−0.0487978	−0.0243989	+	0.999702i	$0.507767\pi$
$674$	0	0
$675$	1.25845e7	1.06310
$676$	0	0
$677$	−1.16903e7	−0.980291	−0.490146	−	0.871641i	$-0.663056\pi$
$677$	−1.16903e7	−0.980291	−0.490146	+	0.871641i	$0.663056\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	9.17691e6	0.758279
$682$	0	0
$683$	−1.83674e7	−1.50659	−0.753297	−	0.657681i	$-0.771538\pi$
$683$	−1.83674e7	−1.50659	−0.753297	+	0.657681i	$0.771538\pi$
$684$	0	0
$685$	2.25588e6	0.183692
$686$	0	0
$687$	5.26244e6	0.425398
$688$	0	0
$689$	−2.78191e6	−0.223252
$690$	0	0
$691$	2.35611e7	1.87716	0.938579	−	0.345066i	$-0.112143\pi$
$691$	2.35611e7	1.87716	0.938579	+	0.345066i	$0.112143\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.68535e7	2.10881
$696$	0	0
$697$	−2.60157e6	−0.202840
$698$	0	0
$699$	1.05142e7	0.813921
$700$	0	0
$701$	1.32980e7	1.02210	0.511048	−	0.859552i	$-0.329257\pi$
$701$	1.32980e7	1.02210	0.511048	+	0.859552i	$0.329257\pi$
$702$	0	0
$703$	1.11307e6	0.0849442
$704$	0	0
$705$	1.23363e6	0.0934786
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−6.85353e6	−0.512034	−0.256017	−	0.966672i	$-0.582410\pi$
$709$	−6.85353e6	−0.512034	−0.256017	+	0.966672i	$0.582410\pi$
$710$	0	0
$711$	−3.82825e6	−0.284005
$712$	0	0
$713$	−1.09628e6	−0.0807602
$714$	0	0
$715$	−8.19384e6	−0.599408
$716$	0	0
$717$	−1.54717e7	−1.12394
$718$	0	0
$719$	2.65729e7	1.91698	0.958490	−	0.285127i	$-0.0920357\pi$
$719$	2.65729e7	1.91698	0.958490	+	0.285127i	$0.0920357\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.01592e7	−0.722791
$724$	0	0
$725$	−2.52779e7	−1.78606
$726$	0	0
$727$	2.16991e6	0.152267	0.0761335	−	0.997098i	$-0.475742\pi$
$727$	2.16991e6	0.152267	0.0761335	+	0.997098i	$0.475742\pi$
$728$	0	0
$729$	1.01628e7	0.708264
$730$	0	0
$731$	−679174.	−0.0470097
$732$	0	0
$733$	1.74653e7	1.20065	0.600324	−	0.799757i	$-0.295038\pi$
$733$	1.74653e7	1.20065	0.600324	+	0.799757i	$0.295038\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.41203e6	0.638285
$738$	0	0
$739$	1.36461e7	0.919172	0.459586	−	0.888133i	$-0.347998\pi$
$739$	1.36461e7	0.919172	0.459586	+	0.888133i	$0.347998\pi$
$740$	0	0
$741$	−4.42594e6	−0.296114
$742$	0	0
$743$	−1.48965e7	−0.989944	−0.494972	−	0.868909i	$-0.664822\pi$
$743$	−1.48965e7	−0.989944	−0.494972	+	0.868909i	$0.664822\pi$
$744$	0	0
$745$	−2.83692e7	−1.87265
$746$	0	0
$747$	1.12725e7	0.739124
$748$	0	0
$749$	0	0
$750$	0	0
$751$	2.53463e7	1.63989	0.819944	−	0.572443i	$-0.194004\pi$
$751$	2.53463e7	1.63989	0.819944	+	0.572443i	$0.194004\pi$
$752$	0	0
$753$	−85823.3	−0.00551592
$754$	0	0
$755$	2.86419e7	1.82867
$756$	0	0
$757$	−2.66725e7	−1.69170	−0.845852	−	0.533417i	$-0.820908\pi$
$757$	−2.66725e7	−1.69170	−0.845852	+	0.533417i	$0.820908\pi$
$758$	0	0
$759$	1.51049e6	0.0951729
$760$	0	0
$761$	−579829.	−0.0362943	−0.0181471	−	0.999835i	$-0.505777\pi$
$761$	−579829.	−0.0362943	−0.0181471	+	0.999835i	$0.505777\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−4.17544e6	−0.257958
$766$	0	0
$767$	−8.65541e6	−0.531250
$768$	0	0
$769$	−1.52438e7	−0.929562	−0.464781	−	0.885426i	$-0.653867\pi$
$769$	−1.52438e7	−0.929562	−0.464781	+	0.885426i	$0.653867\pi$
$770$	0	0
$771$	−5.31898e6	−0.322250
$772$	0	0
$773$	1.94926e7	1.17333	0.586665	−	0.809830i	$-0.300441\pi$
$773$	1.94926e7	1.17333	0.586665	+	0.809830i	$0.300441\pi$
$774$	0	0
$775$	−8.36275e6	−0.500144
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.05794e7	−0.624621
$780$	0	0
$781$	−5.06345e6	−0.297043
$782$	0	0
$783$	−3.01642e7	−1.75828
$784$	0	0
$785$	−3.66337e7	−2.12181
$786$	0	0
$787$	1.36277e7	0.784305	0.392153	−	0.919900i	$-0.371730\pi$
$787$	1.36277e7	0.784305	0.392153	+	0.919900i	$0.371730\pi$
$788$	0	0
$789$	−3.54999e6	−0.203018
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1.35664e7	0.766093
$794$	0	0
$795$	7.67654e6	0.430772
$796$	0	0
$797$	−3.62853e6	−0.202341	−0.101171	−	0.994869i	$-0.532259\pi$
$797$	−3.62853e6	−0.202341	−0.101171	+	0.994869i	$0.532259\pi$
$798$	0	0
$799$	−567208.	−0.0314323
$800$	0	0
$801$	6.15913e6	0.339186
$802$	0	0
$803$	−2.46667e7	−1.34997
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.83509e6	0.261349
$808$	0	0
$809$	2.98780e7	1.60502	0.802509	−	0.596640i	$-0.203498\pi$
$809$	2.98780e7	1.60502	0.802509	+	0.596640i	$0.203498\pi$
$810$	0	0
$811$	−1.02643e6	−0.0547995	−0.0273998	−	0.999625i	$-0.508723\pi$
$811$	−1.02643e6	−0.0547995	−0.0273998	+	0.999625i	$0.508723\pi$
$812$	0	0
$813$	9.85702e6	0.523021
$814$	0	0
$815$	−4.01086e7	−2.11516
$816$	0	0
$817$	−2.76189e6	−0.144761
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.15062e7	−0.595766	−0.297883	−	0.954602i	$-0.596281\pi$
$821$	−1.15062e7	−0.595766	−0.297883	+	0.954602i	$0.596281\pi$
$822$	0	0
$823$	2.51210e7	1.29282	0.646408	−	0.762992i	$-0.276270\pi$
$823$	2.51210e7	1.29282	0.646408	+	0.762992i	$0.276270\pi$
$824$	0	0
$825$	1.15225e7	0.589401
$826$	0	0
$827$	2.14447e6	0.109032	0.0545162	−	0.998513i	$-0.482638\pi$
$827$	2.14447e6	0.109032	0.0545162	+	0.998513i	$0.482638\pi$
$828$	0	0
$829$	−818065.	−0.0413430	−0.0206715	−	0.999786i	$-0.506580\pi$
$829$	−818065.	−0.0413430	−0.0206715	+	0.999786i	$0.506580\pi$
$830$	0	0
$831$	−9.76741e6	−0.490656
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−5.40155e7	−2.68104
$836$	0	0
$837$	−9.97929e6	−0.492364
$838$	0	0
$839$	−2.11279e7	−1.03622	−0.518110	−	0.855314i	$-0.673364\pi$
$839$	−2.11279e7	−1.03622	−0.518110	+	0.855314i	$0.673364\pi$
$840$	0	0
$841$	4.00785e7	1.95398
$842$	0	0
$843$	3.20968e6	0.155558
$844$	0	0
$845$	−2.28478e7	−1.10079
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.78677e7	−0.850744
$850$	0	0
$851$	314863.	0.0149038
$852$	0	0
$853$	1.89000e7	0.889386	0.444693	−	0.895683i	$-0.353313\pi$
$853$	1.89000e7	0.889386	0.444693	+	0.895683i	$0.353313\pi$
$854$	0	0
$855$	−1.69796e7	−0.794349
$856$	0	0
$857$	−3.72286e7	−1.73151	−0.865753	−	0.500471i	$-0.833160\pi$
$857$	−3.72286e7	−1.73151	−0.865753	+	0.500471i	$0.833160\pi$
$858$	0	0
$859$	3.02064e6	0.139674	0.0698371	−	0.997558i	$-0.477752\pi$
$859$	3.02064e6	0.139674	0.0698371	+	0.997558i	$0.477752\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.71843e7	1.24248	0.621242	−	0.783619i	$-0.286629\pi$
$863$	2.71843e7	1.24248	0.621242	+	0.783619i	$0.286629\pi$
$864$	0	0
$865$	1.98899e7	0.903839
$866$	0	0
$867$	1.29352e7	0.584420
$868$	0	0
$869$	−9.53158e6	−0.428169
$870$	0	0
$871$	−7.80136e6	−0.348438
$872$	0	0
$873$	1.46106e7	0.648834
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.73868e7	0.763344	0.381672	−	0.924298i	$-0.375348\pi$
$877$	1.73868e7	0.763344	0.381672	+	0.924298i	$0.375348\pi$
$878$	0	0
$879$	1.65506e7	0.722507
$880$	0	0
$881$	8.14472e6	0.353538	0.176769	−	0.984252i	$-0.443435\pi$
$881$	8.14472e6	0.353538	0.176769	+	0.984252i	$0.443435\pi$
$882$	0	0
$883$	3.10298e7	1.33930	0.669649	−	0.742678i	$-0.266445\pi$
$883$	3.10298e7	1.33930	0.669649	+	0.742678i	$0.266445\pi$
$884$	0	0
$885$	2.38841e7	1.02507
$886$	0	0
$887$	−1.47028e7	−0.627465	−0.313733	−	0.949511i	$-0.601580\pi$
$887$	−1.47028e7	−0.627465	−0.313733	+	0.949511i	$0.601580\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.66362e6	0.0702037
$892$	0	0
$893$	−2.30657e6	−0.0967918
$894$	0	0
$895$	−1.11166e7	−0.463890
$896$	0	0
$897$	−1.25200e6	−0.0519545
$898$	0	0
$899$	2.00450e7	0.827193
$900$	0	0
$901$	−3.52958e6	−0.144848
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.44469e7	−0.992208
$906$	0	0
$907$	−1.30940e7	−0.528512	−0.264256	−	0.964453i	$-0.585126\pi$
$907$	−1.30940e7	−0.528512	−0.264256	+	0.964453i	$0.585126\pi$
$908$	0	0
$909$	4.05643e6	0.162830
$910$	0	0
$911$	−2.68695e7	−1.07266	−0.536332	−	0.844007i	$-0.680191\pi$
$911$	−2.68695e7	−1.07266	−0.536332	+	0.844007i	$0.680191\pi$
$912$	0	0
$913$	2.80662e7	1.11431
$914$	0	0
$915$	−3.74357e7	−1.47820
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1.27317e6	−0.0497277	−0.0248638	−	0.999691i	$-0.507915\pi$
$919$	−1.27317e6	−0.0497277	−0.0248638	+	0.999691i	$0.507915\pi$
$920$	0	0
$921$	4.70795e6	0.182887
$922$	0	0
$923$	4.19695e6	0.162155
$924$	0	0
$925$	2.40187e6	0.0922986
$926$	0	0
$927$	−4.13101e6	−0.157891
$928$	0	0
$929$	9.31705e6	0.354192	0.177096	−	0.984194i	$-0.443330\pi$
$929$	9.31705e6	0.354192	0.177096	+	0.984194i	$0.443330\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	2.45814e7	0.924490
$934$	0	0
$935$	−1.03960e7	−0.388900
$936$	0	0
$937$	−1.18158e7	−0.439657	−0.219829	−	0.975539i	$-0.570550\pi$
$937$	−1.18158e7	−0.439657	−0.219829	+	0.975539i	$0.570550\pi$
$938$	0	0
$939$	2.44097e7	0.903439
$940$	0	0
$941$	2.53529e7	0.933371	0.466685	−	0.884423i	$-0.345448\pi$
$941$	2.53529e7	0.933371	0.466685	+	0.884423i	$0.345448\pi$
$942$	0	0
$943$	−2.99268e6	−0.109592
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.64941e7	−0.960008	−0.480004	−	0.877266i	$-0.659365\pi$
$947$	−2.64941e7	−0.960008	−0.480004	+	0.877266i	$0.659365\pi$
$948$	0	0
$949$	2.04455e7	0.736941
$950$	0	0
$951$	−1.89164e7	−0.678246
$952$	0	0
$953$	−1.88335e7	−0.671735	−0.335868	−	0.941909i	$-0.609030\pi$
$953$	−1.88335e7	−0.671735	−0.335868	+	0.941909i	$0.609030\pi$
$954$	0	0
$955$	−1.81603e7	−0.644339
$956$	0	0
$957$	−2.76186e7	−0.974816
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.19976e7	−0.768365
$962$	0	0
$963$	1.24177e7	0.431495
$964$	0	0
$965$	−5.36740e7	−1.85544
$966$	0	0
$967$	3.14956e7	1.08314	0.541569	−	0.840656i	$-0.317831\pi$
$967$	3.14956e7	1.08314	0.541569	+	0.840656i	$0.317831\pi$
$968$	0	0
$969$	−5.61545e6	−0.192121
$970$	0	0
$971$	−6.85669e6	−0.233381	−0.116691	−	0.993168i	$-0.537229\pi$
$971$	−6.85669e6	−0.233381	−0.116691	+	0.993168i	$0.537229\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−9.55064e6	−0.321752
$976$	0	0
$977$	2.81471e7	0.943402	0.471701	−	0.881759i	$-0.343640\pi$
$977$	2.81471e7	0.943402	0.471701	+	0.881759i	$0.343640\pi$
$978$	0	0
$979$	1.53350e7	0.511361
$980$	0	0
$981$	−3.11831e7	−1.03454
$982$	0	0
$983$	2.34916e7	0.775406	0.387703	−	0.921784i	$-0.373269\pi$
$983$	2.34916e7	0.775406	0.387703	+	0.921784i	$0.373269\pi$
$984$	0	0
$985$	−102399.	−0.00336285
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−781278.	−0.0253989
$990$	0	0
$991$	2.14412e6	0.0693530	0.0346765	−	0.999399i	$-0.488960\pi$
$991$	2.14412e6	0.0693530	0.0346765	+	0.999399i	$0.488960\pi$
$992$	0	0
$993$	−1.09206e7	−0.351459
$994$	0	0
$995$	2.94483e7	0.942979
$996$	0	0
$997$	−2.50872e7	−0.799307	−0.399654	−	0.916666i	$-0.630870\pi$
$997$	−2.50872e7	−0.799307	−0.399654	+	0.916666i	$0.630870\pi$
$998$	0	0
$999$	2.86616e6	0.0908628

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.t.1.1		2
4.3	odd	2		49.6.a.e.1.2		2
7.3	odd	6		112.6.i.c.65.1		4
7.5	odd	6		112.6.i.c.81.1		4
7.6	odd	2		784.6.a.ba.1.2		2
12.11	even	2		441.6.a.m.1.1		2
28.3	even	6		7.6.c.a.2.1	✓	4
28.11	odd	6		49.6.c.f.30.1		4
28.19	even	6		7.6.c.a.4.1	yes	4
28.23	odd	6		49.6.c.f.18.1		4
28.27	even	2		49.6.a.d.1.2		2
84.47	odd	6		63.6.e.d.46.2		4
84.59	odd	6		63.6.e.d.37.2		4
84.83	odd	2		441.6.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.6.c.a.2.1	✓	4	28.3	even	6
7.6.c.a.4.1	yes	4	28.19	even	6
49.6.a.d.1.2		2	28.27	even	2
49.6.a.e.1.2		2	4.3	odd	2
49.6.c.f.18.1		4	28.23	odd	6
49.6.c.f.30.1		4	28.11	odd	6
63.6.e.d.37.2		4	84.59	odd	6
63.6.e.d.46.2		4	84.47	odd	6
112.6.i.c.65.1		4	7.3	odd	6
112.6.i.c.81.1		4	7.5	odd	6
441.6.a.m.1.1		2	12.11	even	2
441.6.a.n.1.1		2	84.83	odd	2
784.6.a.t.1.1		2	1.1	even	1	trivial
784.6.a.ba.1.2		2	7.6	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q-10.0828 q^{3} +79.8276 q^{5} -141.338 q^{9} -351.904 q^{11} +291.683 q^{13} -804.883 q^{15} +370.075 q^{17} +1504.93 q^{19} +425.711 q^{23} +3247.45 q^{25} +3875.19 q^{27} -7783.93 q^{29} -2575.18 q^{31} +3548.16 q^{33} +739.618 q^{37} -2940.97 q^{39} -7029.84 q^{41} -1835.23 q^{43} -11282.7 q^{45} -1532.68 q^{47} -3731.38 q^{51} -9537.46 q^{53} -28091.6 q^{55} -15173.8 q^{57} -29674.1 q^{59} +46510.8 q^{61} +23284.3 q^{65} -26746.1 q^{67} -4292.34 q^{69} +14388.8 q^{71} +70095.1 q^{73} -32743.3 q^{75} +27085.8 q^{79} -4727.49 q^{81} -79755.4 q^{83} +29542.2 q^{85} +78483.6 q^{87} -43577.3 q^{89} +25964.9 q^{93} +120135. q^{95} -103374. q^{97} +49737.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{3} + 38 q^{5} - 380 q^{9} - 424 q^{11} + 924 q^{13} - 892 q^{15} + 2346 q^{17} - 360 q^{19} + 12 q^{23} + 1872 q^{25} + 2872 q^{27} - 7052 q^{29} + 3548 q^{31} + 3398 q^{33} + 11090 q^{37} - 1624 q^{39}+ \cdots + 66944 q^{99}+O(q^{100}) \) 2 * q - 8 * q^3 + 38 * q^5 - 380 * q^9 - 424 * q^11 + 924 * q^13 - 892 * q^15 + 2346 * q^17 - 360 * q^19 + 12 * q^23 + 1872 * q^25 + 2872 * q^27 - 7052 * q^29 + 3548 * q^31 + 3398 * q^33 + 11090 * q^37 - 1624 * q^39 - 3500 * q^41 + 12680 * q^43 - 1300 * q^45 - 22956 * q^47 + 384 * q^51 + 3042 * q^53 - 25076 * q^55 - 19058 * q^57 - 65808 * q^59 + 42486 * q^61 - 3164 * q^65 - 42312 * q^67 - 5154 * q^69 + 2208 * q^71 + 50506 * q^73 - 35608 * q^75 - 9004 * q^79 + 51178 * q^81 - 104328 * q^83 - 53106 * q^85 + 80008 * q^87 + 26666 * q^89 + 38718 * q^93 + 198140 * q^95 - 209132 * q^97 + 66944 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more