LMFDB - Embedded newform 768.6.a.ba.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [768,6,Mod(1,768)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("768.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	768.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-45,0,-50,0,98,0,405,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$123.174773616$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 65x^{3} + 85x^{2} + 856x - 1692 $ x^5 - x^4 - 65x^3 + 85x^2 + 856*x - 1692
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.28141$ of defining polynomial
Character	$\chi$	$=$	768.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} +20.4846 q^{5} -35.4345 q^{7} +81.0000 q^{9} -486.951 q^{11} +200.684 q^{13} -184.361 q^{15} -692.475 q^{17} +2640.10 q^{19} +318.910 q^{21} +3629.10 q^{23} -2705.38 q^{25} -729.000 q^{27} -8495.84 q^{29} +6268.83 q^{31} +4382.56 q^{33} -725.859 q^{35} +9599.13 q^{37} -1806.15 q^{39} +3773.49 q^{41} -5060.77 q^{43} +1659.25 q^{45} -12234.9 q^{47} -15551.4 q^{49} +6232.27 q^{51} -16864.5 q^{53} -9974.98 q^{55} -23760.9 q^{57} +13173.3 q^{59} +24525.7 q^{61} -2870.19 q^{63} +4110.92 q^{65} -18703.9 q^{67} -32661.9 q^{69} +35604.9 q^{71} +58600.2 q^{73} +24348.4 q^{75} +17254.9 q^{77} +98134.9 q^{79} +6561.00 q^{81} -37722.5 q^{83} -14185.0 q^{85} +76462.5 q^{87} -60955.8 q^{89} -7111.12 q^{91} -56419.4 q^{93} +54081.3 q^{95} -68778.9 q^{97} -39443.1 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 45 q^{3} - 50 q^{5} + 98 q^{7} + 405 q^{9} - 676 q^{13} + 450 q^{15} - 202 q^{17} - 716 q^{19} - 882 q^{21} - 836 q^{23} + 4683 q^{25} - 3645 q^{27} - 5046 q^{29} + 9346 q^{31} - 436 q^{35} - 10952 q^{37}+ \cdots + 29686 q^{97}+O(q^{100}) $ 5 * q - 45 * q^3 - 50 * q^5 + 98 * q^7 + 405 * q^9 - 676 * q^13 + 450 * q^15 - 202 * q^17 - 716 * q^19 - 882 * q^21 - 836 * q^23 + 4683 * q^25 - 3645 * q^27 - 5046 * q^29 + 9346 * q^31 - 436 * q^35 - 10952 * q^37 + 6084 * q^39 + 1238 * q^41 - 1676 * q^43 - 4050 * q^45 + 32628 * q^47 + 4689 * q^49 + 1818 * q^51 + 8490 * q^53 + 51712 * q^55 + 6444 * q^57 + 15884 * q^59 + 13824 * q^61 + 7938 * q^63 + 25768 * q^65 - 26652 * q^67 + 7524 * q^69 + 136116 * q^71 + 13694 * q^73 - 42147 * q^75 - 89856 * q^77 + 126154 * q^79 + 32805 * q^81 + 64720 * q^83 - 164636 * q^85 + 45414 * q^87 - 87790 * q^89 + 179992 * q^91 - 84114 * q^93 + 121592 * q^95 + 29686 * q^97

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
$5$	20.4846	0.366439	0.183219	−	0.983072i	$-0.441348\pi$
$5$	20.4846	0.366439	0.183219	+	0.983072i	$0.441348\pi$
$6$	0	0
$7$	−35.4345	−0.273326	−0.136663	−	0.990618i	$-0.543638\pi$
$7$	−35.4345	−0.273326	−0.136663	+	0.990618i	$0.543638\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−486.951	−1.21340	−0.606700	−	0.794931i	$-0.707507\pi$
$11$	−486.951	−1.21340	−0.606700	+	0.794931i	$0.707507\pi$
$12$	0	0
$13$	200.684	0.329347	0.164674	−	0.986348i	$-0.447343\pi$
$13$	200.684	0.329347	0.164674	+	0.986348i	$0.447343\pi$
$14$	0	0
$15$	−184.361	−0.211564
$16$	0	0
$17$	−692.475	−0.581141	−0.290571	−	0.956854i	$-0.593845\pi$
$17$	−692.475	−0.581141	−0.290571	+	0.956854i	$0.593845\pi$
$18$	0	0
$19$	2640.10	1.67779	0.838894	−	0.544296i	$-0.183203\pi$
$19$	2640.10	1.67779	0.838894	+	0.544296i	$0.183203\pi$
$20$	0	0
$21$	318.910	0.157805
$22$	0	0
$23$	3629.10	1.43047	0.715237	−	0.698882i	$-0.246319\pi$
$23$	3629.10	1.43047	0.715237	+	0.698882i	$0.246319\pi$
$24$	0	0
$25$	−2705.38	−0.865723
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−8495.84	−1.87591	−0.937953	−	0.346762i	$-0.887281\pi$
$29$	−8495.84	−1.87591	−0.937953	+	0.346762i	$0.887281\pi$
$30$	0	0
$31$	6268.83	1.17161	0.585804	−	0.810453i	$-0.300779\pi$
$31$	6268.83	1.17161	0.585804	+	0.810453i	$0.300779\pi$
$32$	0	0
$33$	4382.56	0.700557
$34$	0	0
$35$	−725.859	−0.100157
$36$	0	0
$37$	9599.13	1.15273	0.576365	−	0.817193i	$-0.304471\pi$
$37$	9599.13	1.15273	0.576365	+	0.817193i	$0.304471\pi$
$38$	0	0
$39$	−1806.15	−0.190149
$40$	0	0
$41$	3773.49	0.350577	0.175289	−	0.984517i	$-0.443914\pi$
$41$	3773.49	0.350577	0.175289	+	0.984517i	$0.443914\pi$
$42$	0	0
$43$	−5060.77	−0.417393	−0.208697	−	0.977980i	$-0.566922\pi$
$43$	−5060.77	−0.417393	−0.208697	+	0.977980i	$0.566922\pi$
$44$	0	0
$45$	1659.25	0.122146
$46$	0	0
$47$	−12234.9	−0.807898	−0.403949	−	0.914782i	$-0.632363\pi$
$47$	−12234.9	−0.807898	−0.403949	+	0.914782i	$0.632363\pi$
$48$	0	0
$49$	−15551.4	−0.925293
$50$	0	0
$51$	6232.27	0.335522
$52$	0	0
$53$	−16864.5	−0.824678	−0.412339	−	0.911031i	$-0.635288\pi$
$53$	−16864.5	−0.824678	−0.412339	+	0.911031i	$0.635288\pi$
$54$	0	0
$55$	−9974.98	−0.444637
$56$	0	0
$57$	−23760.9	−0.968671
$58$	0	0
$59$	13173.3	0.492680	0.246340	−	0.969183i	$-0.420772\pi$
$59$	13173.3	0.492680	0.246340	+	0.969183i	$0.420772\pi$
$60$	0	0
$61$	24525.7	0.843912	0.421956	−	0.906616i	$-0.361344\pi$
$61$	24525.7	0.843912	0.421956	+	0.906616i	$0.361344\pi$
$62$	0	0
$63$	−2870.19	−0.0911086
$64$	0	0
$65$	4110.92	0.120686
$66$	0	0
$67$	−18703.9	−0.509032	−0.254516	−	0.967069i	$-0.581916\pi$
$67$	−18703.9	−0.509032	−0.254516	+	0.967069i	$0.581916\pi$
$68$	0	0
$69$	−32661.9	−0.825884
$70$	0	0
$71$	35604.9	0.838231	0.419116	−	0.907933i	$-0.362340\pi$
$71$	35604.9	0.838231	0.419116	+	0.907933i	$0.362340\pi$
$72$	0	0
$73$	58600.2	1.28704	0.643520	−	0.765429i	$-0.277473\pi$
$73$	58600.2	1.28704	0.643520	+	0.765429i	$0.277473\pi$
$74$	0	0
$75$	24348.4	0.499825
$76$	0	0
$77$	17254.9	0.331654
$78$	0	0
$79$	98134.9	1.76911	0.884557	−	0.466432i	$-0.154461\pi$
$79$	98134.9	1.76911	0.884557	+	0.466432i	$0.154461\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−37722.5	−0.601043	−0.300522	−	0.953775i	$-0.597161\pi$
$83$	−37722.5	−0.601043	−0.300522	+	0.953775i	$0.597161\pi$
$84$	0	0
$85$	−14185.0	−0.212953
$86$	0	0
$87$	76462.5	1.08306
$88$	0	0
$89$	−60955.8	−0.815718	−0.407859	−	0.913045i	$-0.633725\pi$
$89$	−60955.8	−0.815718	−0.407859	+	0.913045i	$0.633725\pi$
$90$	0	0
$91$	−7111.12	−0.0900191
$92$	0	0
$93$	−56419.4	−0.676428
$94$	0	0
$95$	54081.3	0.614806
$96$	0	0
$97$	−68778.9	−0.742209	−0.371104	−	0.928591i	$-0.621021\pi$
$97$	−68778.9	−0.742209	−0.371104	+	0.928591i	$0.621021\pi$
$98$	0	0
$99$	−39443.1	−0.404467
$100$	0	0
$101$	−27369.0	−0.266966	−0.133483	−	0.991051i	$-0.542616\pi$
$101$	−27369.0	−0.266966	−0.133483	+	0.991051i	$0.542616\pi$
$102$	0	0
$103$	−58941.9	−0.547433	−0.273717	−	0.961810i	$-0.588253\pi$
$103$	−58941.9	−0.547433	−0.273717	+	0.961810i	$0.588253\pi$
$104$	0	0
$105$	6532.73	0.0578258
$106$	0	0
$107$	−32173.9	−0.271672	−0.135836	−	0.990731i	$-0.543372\pi$
$107$	−32173.9	−0.271672	−0.135836	+	0.990731i	$0.543372\pi$
$108$	0	0
$109$	−185150.	−1.49265	−0.746324	−	0.665583i	$-0.768183\pi$
$109$	−185150.	−1.49265	−0.746324	+	0.665583i	$0.768183\pi$
$110$	0	0
$111$	−86392.1	−0.665529
$112$	0	0
$113$	194419.	1.43233	0.716165	−	0.697931i	$-0.245896\pi$
$113$	194419.	1.43233	0.716165	+	0.697931i	$0.245896\pi$
$114$	0	0
$115$	74340.6	0.524181
$116$	0	0
$117$	16255.4	0.109782
$118$	0	0
$119$	24537.5	0.158841
$120$	0	0
$121$	76070.7	0.472339
$122$	0	0
$123$	−33961.4	−0.202406
$124$	0	0
$125$	−119433.	−0.683673
$126$	0	0
$127$	−106993.	−0.588633	−0.294317	−	0.955708i	$-0.595092\pi$
$127$	−106993.	−0.588633	−0.294317	+	0.955708i	$0.595092\pi$
$128$	0	0
$129$	45546.9	0.240982
$130$	0	0
$131$	288876.	1.47073	0.735366	−	0.677670i	$-0.237010\pi$
$131$	288876.	1.47073	0.735366	+	0.677670i	$0.237010\pi$
$132$	0	0
$133$	−93550.6	−0.458583
$134$	0	0
$135$	−14933.2	−0.0705212
$136$	0	0
$137$	70427.1	0.320582	0.160291	−	0.987070i	$-0.448757\pi$
$137$	70427.1	0.320582	0.160291	+	0.987070i	$0.448757\pi$
$138$	0	0
$139$	−210073.	−0.922217	−0.461108	−	0.887344i	$-0.652548\pi$
$139$	−210073.	−0.922217	−0.461108	+	0.887344i	$0.652548\pi$
$140$	0	0
$141$	110114.	0.466440
$142$	0	0
$143$	−97723.3	−0.399630
$144$	0	0
$145$	−174033.	−0.687405
$146$	0	0
$147$	139963.	0.534218
$148$	0	0
$149$	−233930.	−0.863217	−0.431608	−	0.902061i	$-0.642054\pi$
$149$	−233930.	−0.863217	−0.431608	+	0.902061i	$0.642054\pi$
$150$	0	0
$151$	47860.3	0.170818	0.0854089	−	0.996346i	$-0.472780\pi$
$151$	47860.3	0.170818	0.0854089	+	0.996346i	$0.472780\pi$
$152$	0	0
$153$	−56090.5	−0.193714
$154$	0	0
$155$	128414.	0.429323
$156$	0	0
$157$	−325886.	−1.05516	−0.527578	−	0.849506i	$-0.676900\pi$
$157$	−325886.	−1.05516	−0.527578	+	0.849506i	$0.676900\pi$
$158$	0	0
$159$	151781.	0.476128
$160$	0	0
$161$	−128595.	−0.390985
$162$	0	0
$163$	−423064.	−1.24720	−0.623602	−	0.781742i	$-0.714331\pi$
$163$	−423064.	−1.24720	−0.623602	+	0.781742i	$0.714331\pi$
$164$	0	0
$165$	89774.9	0.256711
$166$	0	0
$167$	−300663.	−0.834236	−0.417118	−	0.908852i	$-0.636960\pi$
$167$	−300663.	−0.834236	−0.417118	+	0.908852i	$0.636960\pi$
$168$	0	0
$169$	−331019.	−0.891530
$170$	0	0
$171$	213848.	0.559262
$172$	0	0
$173$	−486907.	−1.23689	−0.618445	−	0.785828i	$-0.712237\pi$
$173$	−486907.	−1.23689	−0.618445	+	0.785828i	$0.712237\pi$
$174$	0	0
$175$	95863.8	0.236624
$176$	0	0
$177$	−118560.	−0.284449
$178$	0	0
$179$	−499594.	−1.16543	−0.582713	−	0.812678i	$-0.698009\pi$
$179$	−499594.	−1.16543	−0.582713	+	0.812678i	$0.698009\pi$
$180$	0	0
$181$	−840307.	−1.90652	−0.953261	−	0.302149i	$-0.902296\pi$
$181$	−840307.	−1.90652	−0.953261	+	0.302149i	$0.902296\pi$
$182$	0	0
$183$	−220731.	−0.487233
$184$	0	0
$185$	196634.	0.422405
$186$	0	0
$187$	337202.	0.705157
$188$	0	0
$189$	25831.7	0.0526016
$190$	0	0
$191$	−479331.	−0.950718	−0.475359	−	0.879792i	$-0.657682\pi$
$191$	−479331.	−0.950718	−0.475359	+	0.879792i	$0.657682\pi$
$192$	0	0
$193$	1.00793e6	1.94776	0.973880	−	0.227063i	$-0.0729123\pi$
$193$	1.00793e6	1.94776	0.973880	+	0.227063i	$0.0729123\pi$
$194$	0	0
$195$	−36998.3	−0.0696779
$196$	0	0
$197$	−458514.	−0.841757	−0.420879	−	0.907117i	$-0.638278\pi$
$197$	−458514.	−0.841757	−0.420879	+	0.907117i	$0.638278\pi$
$198$	0	0
$199$	212134.	0.379731	0.189866	−	0.981810i	$-0.439195\pi$
$199$	212134.	0.379731	0.189866	+	0.981810i	$0.439195\pi$
$200$	0	0
$201$	168335.	0.293890
$202$	0	0
$203$	301045.	0.512734
$204$	0	0
$205$	77298.3	0.128465
$206$	0	0
$207$	293957.	0.476825
$208$	0	0
$209$	−1.28560e6	−2.03583
$210$	0	0
$211$	441484.	0.682667	0.341333	−	0.939942i	$-0.389121\pi$
$211$	441484.	0.682667	0.341333	+	0.939942i	$0.389121\pi$
$212$	0	0
$213$	−320444.	−0.483953
$214$	0	0
$215$	−103668.	−0.152949
$216$	0	0
$217$	−222132.	−0.320231
$218$	0	0
$219$	−527402.	−0.743073
$220$	0	0
$221$	−138968.	−0.191397
$222$	0	0
$223$	−477411.	−0.642881	−0.321440	−	0.946930i	$-0.604167\pi$
$223$	−477411.	−0.642881	−0.321440	+	0.946930i	$0.604167\pi$
$224$	0	0
$225$	−219136.	−0.288574
$226$	0	0
$227$	72520.0	0.0934100	0.0467050	−	0.998909i	$-0.485128\pi$
$227$	72520.0	0.0934100	0.0467050	+	0.998909i	$0.485128\pi$
$228$	0	0
$229$	−819922.	−1.03320	−0.516599	−	0.856227i	$-0.672802\pi$
$229$	−819922.	−1.03320	−0.516599	+	0.856227i	$0.672802\pi$
$230$	0	0
$231$	−155294.	−0.191480
$232$	0	0
$233$	1893.51	0.00228495	0.00114248	−	0.999999i	$-0.499636\pi$
$233$	1893.51	0.00228495	0.00114248	+	0.999999i	$0.499636\pi$
$234$	0	0
$235$	−250627.	−0.296045
$236$	0	0
$237$	−883214.	−1.02140
$238$	0	0
$239$	507498.	0.574698	0.287349	−	0.957826i	$-0.407226\pi$
$239$	507498.	0.574698	0.287349	+	0.957826i	$0.407226\pi$
$240$	0	0
$241$	−94613.8	−0.104933	−0.0524665	−	0.998623i	$-0.516708\pi$
$241$	−94613.8	−0.104933	−0.0524665	+	0.998623i	$0.516708\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−318564.	−0.339063
$246$	0	0
$247$	529826.	0.552575
$248$	0	0
$249$	339503.	0.347012
$250$	0	0
$251$	−1.34294e6	−1.34547	−0.672734	−	0.739884i	$-0.734880\pi$
$251$	−1.34294e6	−1.34547	−0.672734	+	0.739884i	$0.734880\pi$
$252$	0	0
$253$	−1.76720e6	−1.73574
$254$	0	0
$255$	127665.	0.122948
$256$	0	0
$257$	1.58681e6	1.49862	0.749311	−	0.662219i	$-0.230385\pi$
$257$	1.58681e6	1.49862	0.749311	+	0.662219i	$0.230385\pi$
$258$	0	0
$259$	−340140.	−0.315071
$260$	0	0
$261$	−688163.	−0.625302
$262$	0	0
$263$	−1.10061e6	−0.981169	−0.490585	−	0.871394i	$-0.663217\pi$
$263$	−1.10061e6	−0.981169	−0.490585	+	0.871394i	$0.663217\pi$
$264$	0	0
$265$	−345462.	−0.302194
$266$	0	0
$267$	548603.	0.470955
$268$	0	0
$269$	−1.70145e6	−1.43363	−0.716817	−	0.697261i	$-0.754402\pi$
$269$	−1.70145e6	−1.43363	−0.716817	+	0.697261i	$0.754402\pi$
$270$	0	0
$271$	−1.67829e6	−1.38817	−0.694085	−	0.719893i	$-0.744191\pi$
$271$	−1.67829e6	−1.38817	−0.694085	+	0.719893i	$0.744191\pi$
$272$	0	0
$273$	64000.1	0.0519726
$274$	0	0
$275$	1.31739e6	1.05047
$276$	0	0
$277$	961399.	0.752842	0.376421	−	0.926449i	$-0.377155\pi$
$277$	961399.	0.752842	0.376421	+	0.926449i	$0.377155\pi$
$278$	0	0
$279$	507775.	0.390536
$280$	0	0
$281$	−576110.	−0.435251	−0.217625	−	0.976032i	$-0.569831\pi$
$281$	−576110.	−0.435251	−0.217625	+	0.976032i	$0.569831\pi$
$282$	0	0
$283$	580242.	0.430668	0.215334	−	0.976540i	$-0.430916\pi$
$283$	580242.	0.430668	0.215334	+	0.976540i	$0.430916\pi$
$284$	0	0
$285$	−486732.	−0.354959
$286$	0	0
$287$	−133712.	−0.0958218
$288$	0	0
$289$	−940336.	−0.662275
$290$	0	0
$291$	619010.	0.428514
$292$	0	0
$293$	−1.17710e6	−0.801021	−0.400510	−	0.916292i	$-0.631167\pi$
$293$	−1.17710e6	−0.801021	−0.400510	+	0.916292i	$0.631167\pi$
$294$	0	0
$295$	269850.	0.180537
$296$	0	0
$297$	354988.	0.233519
$298$	0	0
$299$	728303.	0.471123
$300$	0	0
$301$	179326.	0.114084
$302$	0	0
$303$	246321.	0.154133
$304$	0	0
$305$	502399.	0.309242
$306$	0	0
$307$	316229.	0.191494	0.0957470	−	0.995406i	$-0.469476\pi$
$307$	316229.	0.191494	0.0957470	+	0.995406i	$0.469476\pi$
$308$	0	0
$309$	530477.	0.316061
$310$	0	0
$311$	−435658.	−0.255414	−0.127707	−	0.991812i	$-0.540762\pi$
$311$	−435658.	−0.255414	−0.127707	+	0.991812i	$0.540762\pi$
$312$	0	0
$313$	329209.	0.189938	0.0949688	−	0.995480i	$-0.469725\pi$
$313$	329209.	0.189938	0.0949688	+	0.995480i	$0.469725\pi$
$314$	0	0
$315$	−58794.6	−0.0333857
$316$	0	0
$317$	−874355.	−0.488697	−0.244348	−	0.969687i	$-0.578574\pi$
$317$	−874355.	−0.488697	−0.244348	+	0.969687i	$0.578574\pi$
$318$	0	0
$319$	4.13706e6	2.27622
$320$	0	0
$321$	289565.	0.156850
$322$	0	0
$323$	−1.82820e6	−0.975031
$324$	0	0
$325$	−542927.	−0.285123
$326$	0	0
$327$	1.66635e6	0.861781
$328$	0	0
$329$	433538.	0.220819
$330$	0	0
$331$	−899001.	−0.451014	−0.225507	−	0.974242i	$-0.572404\pi$
$331$	−899001.	−0.451014	−0.225507	+	0.974242i	$0.572404\pi$
$332$	0	0
$333$	777529.	0.384243
$334$	0	0
$335$	−383141.	−0.186529
$336$	0	0
$337$	161680.	0.0775500	0.0387750	−	0.999248i	$-0.487654\pi$
$337$	161680.	0.0775500	0.0387750	+	0.999248i	$0.487654\pi$
$338$	0	0
$339$	−1.74977e6	−0.826956
$340$	0	0
$341$	−3.05261e6	−1.42163
$342$	0	0
$343$	1.14660e6	0.526232
$344$	0	0
$345$	−669065.	−0.302636
$346$	0	0
$347$	473459.	0.211086	0.105543	−	0.994415i	$-0.466342\pi$
$347$	473459.	0.211086	0.105543	+	0.994415i	$0.466342\pi$
$348$	0	0
$349$	1.98873e6	0.874000	0.437000	−	0.899461i	$-0.356041\pi$
$349$	1.98873e6	0.874000	0.437000	+	0.899461i	$0.356041\pi$
$350$	0	0
$351$	−146299.	−0.0633829
$352$	0	0
$353$	−1.17160e6	−0.500428	−0.250214	−	0.968191i	$-0.580501\pi$
$353$	−1.17160e6	−0.500428	−0.250214	+	0.968191i	$0.580501\pi$
$354$	0	0
$355$	729351.	0.307161
$356$	0	0
$357$	−220837.	−0.0917068
$358$	0	0
$359$	2.22214e6	0.909988	0.454994	−	0.890495i	$-0.349641\pi$
$359$	2.22214e6	0.909988	0.454994	+	0.890495i	$0.349641\pi$
$360$	0	0
$361$	4.49404e6	1.81497
$362$	0	0
$363$	−684636.	−0.272705
$364$	0	0
$365$	1.20040e6	0.471622
$366$	0	0
$367$	3.84620e6	1.49062	0.745310	−	0.666718i	$-0.232301\pi$
$367$	3.84620e6	1.49062	0.745310	+	0.666718i	$0.232301\pi$
$368$	0	0
$369$	305653.	0.116859
$370$	0	0
$371$	597585.	0.225406
$372$	0	0
$373$	−3.45371e6	−1.28533	−0.642664	−	0.766148i	$-0.722171\pi$
$373$	−3.45371e6	−1.28533	−0.642664	+	0.766148i	$0.722171\pi$
$374$	0	0
$375$	1.07490e6	0.394719
$376$	0	0
$377$	−1.70498e6	−0.617825
$378$	0	0
$379$	2.58847e6	0.925646	0.462823	−	0.886451i	$-0.346836\pi$
$379$	2.58847e6	0.925646	0.462823	+	0.886451i	$0.346836\pi$
$380$	0	0
$381$	962934.	0.339848
$382$	0	0
$383$	−4.78283e6	−1.66605	−0.833026	−	0.553234i	$-0.813393\pi$
$383$	−4.78283e6	−1.66605	−0.833026	+	0.553234i	$0.813393\pi$
$384$	0	0
$385$	353458.	0.121531
$386$	0	0
$387$	−409922.	−0.139131
$388$	0	0
$389$	1.09816e6	0.367953	0.183976	−	0.982931i	$-0.441103\pi$
$389$	1.09816e6	0.367953	0.183976	+	0.982931i	$0.441103\pi$
$390$	0	0
$391$	−2.51306e6	−0.831307
$392$	0	0
$393$	−2.59989e6	−0.849128
$394$	0	0
$395$	2.01025e6	0.648272
$396$	0	0
$397$	−4.85409e6	−1.54572	−0.772861	−	0.634575i	$-0.781175\pi$
$397$	−4.85409e6	−1.54572	−0.772861	+	0.634575i	$0.781175\pi$
$398$	0	0
$399$	841956.	0.264763
$400$	0	0
$401$	−752843.	−0.233799	−0.116900	−	0.993144i	$-0.537296\pi$
$401$	−752843.	−0.233799	−0.116900	+	0.993144i	$0.537296\pi$
$402$	0	0
$403$	1.25805e6	0.385866
$404$	0	0
$405$	134399.	0.0407154
$406$	0	0
$407$	−4.67431e6	−1.39872
$408$	0	0
$409$	−3.01319e6	−0.890672	−0.445336	−	0.895364i	$-0.646916\pi$
$409$	−3.01319e6	−0.890672	−0.445336	+	0.895364i	$0.646916\pi$
$410$	0	0
$411$	−633844.	−0.185088
$412$	0	0
$413$	−466789.	−0.134662
$414$	0	0
$415$	−772730.	−0.220246
$416$	0	0
$417$	1.89066e6	0.532442
$418$	0	0
$419$	−4.14924e6	−1.15461	−0.577303	−	0.816530i	$-0.695895\pi$
$419$	−4.14924e6	−1.15461	−0.577303	+	0.816530i	$0.695895\pi$
$420$	0	0
$421$	2.87508e6	0.790578	0.395289	−	0.918557i	$-0.370644\pi$
$421$	2.87508e6	0.790578	0.395289	+	0.918557i	$0.370644\pi$
$422$	0	0
$423$	−991028.	−0.269299
$424$	0	0
$425$	1.87341e6	0.503107
$426$	0	0
$427$	−869056.	−0.230663
$428$	0	0
$429$	879510.	0.230726
$430$	0	0
$431$	−5.02183e6	−1.30217	−0.651086	−	0.759004i	$-0.725686\pi$
$431$	−5.02183e6	−1.30217	−0.651086	+	0.759004i	$0.725686\pi$
$432$	0	0
$433$	−5.92395e6	−1.51842	−0.759210	−	0.650846i	$-0.774414\pi$
$433$	−5.92395e6	−1.51842	−0.759210	+	0.650846i	$0.774414\pi$
$434$	0	0
$435$	1.56630e6	0.396874
$436$	0	0
$437$	9.58121e6	2.40003
$438$	0	0
$439$	−1.14648e6	−0.283926	−0.141963	−	0.989872i	$-0.545341\pi$
$439$	−1.14648e6	−0.283926	−0.141963	+	0.989872i	$0.545341\pi$
$440$	0	0
$441$	−1.25966e6	−0.308431
$442$	0	0
$443$	−4.86693e6	−1.17827	−0.589137	−	0.808033i	$-0.700532\pi$
$443$	−4.86693e6	−1.17827	−0.589137	+	0.808033i	$0.700532\pi$
$444$	0	0
$445$	−1.24865e6	−0.298911
$446$	0	0
$447$	2.10537e6	0.498378
$448$	0	0
$449$	4.33906e6	1.01573	0.507867	−	0.861436i	$-0.330434\pi$
$449$	4.33906e6	1.01573	0.507867	+	0.861436i	$0.330434\pi$
$450$	0	0
$451$	−1.83751e6	−0.425390
$452$	0	0
$453$	−430743.	−0.0986217
$454$	0	0
$455$	−145668.	−0.0329865
$456$	0	0
$457$	−1.40019e6	−0.313615	−0.156808	−	0.987629i	$-0.550120\pi$
$457$	−1.40019e6	−0.313615	−0.156808	+	0.987629i	$0.550120\pi$
$458$	0	0
$459$	504814.	0.111841
$460$	0	0
$461$	4.62462e6	1.01350	0.506750	−	0.862093i	$-0.330847\pi$
$461$	4.62462e6	1.01350	0.506750	+	0.862093i	$0.330847\pi$
$462$	0	0
$463$	592103.	0.128364	0.0641822	−	0.997938i	$-0.479556\pi$
$463$	592103.	0.128364	0.0641822	+	0.997938i	$0.479556\pi$
$464$	0	0
$465$	−1.15573e6	−0.247869
$466$	0	0
$467$	7.12843e6	1.51252	0.756261	−	0.654269i	$-0.227024\pi$
$467$	7.12843e6	1.51252	0.756261	+	0.654269i	$0.227024\pi$
$468$	0	0
$469$	662762.	0.139132
$470$	0	0
$471$	2.93298e6	0.609195
$472$	0	0
$473$	2.46435e6	0.506465
$474$	0	0
$475$	−7.14249e6	−1.45250
$476$	0	0
$477$	−1.36603e6	−0.274893
$478$	0	0
$479$	1.72230e6	0.342980	0.171490	−	0.985186i	$-0.445142\pi$
$479$	1.72230e6	0.342980	0.171490	+	0.985186i	$0.445142\pi$
$480$	0	0
$481$	1.92639e6	0.379648
$482$	0	0
$483$	1.15736e6	0.225736
$484$	0	0
$485$	−1.40891e6	−0.271974
$486$	0	0
$487$	7.89178e6	1.50783	0.753915	−	0.656972i	$-0.228163\pi$
$487$	7.89178e6	1.50783	0.753915	+	0.656972i	$0.228163\pi$
$488$	0	0
$489$	3.80758e6	0.720073
$490$	0	0
$491$	−653436.	−0.122320	−0.0611602	−	0.998128i	$-0.519480\pi$
$491$	−653436.	−0.122320	−0.0611602	+	0.998128i	$0.519480\pi$
$492$	0	0
$493$	5.88315e6	1.09017
$494$	0	0
$495$	−807974.	−0.148212
$496$	0	0
$497$	−1.26164e6	−0.229110
$498$	0	0
$499$	4.37171e6	0.785960	0.392980	−	0.919547i	$-0.371444\pi$
$499$	4.37171e6	0.785960	0.392980	+	0.919547i	$0.371444\pi$
$500$	0	0
$501$	2.70597e6	0.481646
$502$	0	0
$503$	−3.95604e6	−0.697173	−0.348586	−	0.937277i	$-0.613338\pi$
$503$	−3.95604e6	−0.697173	−0.348586	+	0.937277i	$0.613338\pi$
$504$	0	0
$505$	−560642.	−0.0978266
$506$	0	0
$507$	2.97917e6	0.514725
$508$	0	0
$509$	6.01043e6	1.02828	0.514140	−	0.857706i	$-0.328111\pi$
$509$	6.01043e6	1.02828	0.514140	+	0.857706i	$0.328111\pi$
$510$	0	0
$511$	−2.07647e6	−0.351781
$512$	0	0
$513$	−1.92463e6	−0.322890
$514$	0	0
$515$	−1.20740e6	−0.200601
$516$	0	0
$517$	5.95781e6	0.980303
$518$	0	0
$519$	4.38216e6	0.714118
$520$	0	0
$521$	−6.95647e6	−1.12278	−0.561390	−	0.827551i	$-0.689733\pi$
$521$	−6.95647e6	−1.12278	−0.561390	+	0.827551i	$0.689733\pi$
$522$	0	0
$523$	7.73556e6	1.23662	0.618312	−	0.785933i	$-0.287817\pi$
$523$	7.73556e6	1.23662	0.618312	+	0.785933i	$0.287817\pi$
$524$	0	0
$525$	−862774.	−0.136615
$526$	0	0
$527$	−4.34100e6	−0.680869
$528$	0	0
$529$	6.73406e6	1.04625
$530$	0	0
$531$	1.06704e6	0.164227
$532$	0	0
$533$	757278.	0.115462
$534$	0	0
$535$	−659069.	−0.0995511
$536$	0	0
$537$	4.49634e6	0.672858
$538$	0	0
$539$	7.57278e6	1.12275
$540$	0	0
$541$	8.51945e6	1.25146	0.625732	−	0.780038i	$-0.284800\pi$
$541$	8.51945e6	1.25146	0.625732	+	0.780038i	$0.284800\pi$
$542$	0	0
$543$	7.56277e6	1.10073
$544$	0	0
$545$	−3.79271e6	−0.546964
$546$	0	0
$547$	−8.97125e6	−1.28199	−0.640995	−	0.767545i	$-0.721478\pi$
$547$	−8.97125e6	−1.28199	−0.640995	+	0.767545i	$0.721478\pi$
$548$	0	0
$549$	1.98658e6	0.281304
$550$	0	0
$551$	−2.24299e7	−3.14737
$552$	0	0
$553$	−3.47736e6	−0.483545
$554$	0	0
$555$	−1.76970e6	−0.243876
$556$	0	0
$557$	−2.53509e6	−0.346223	−0.173112	−	0.984902i	$-0.555382\pi$
$557$	−2.53509e6	−0.346223	−0.173112	+	0.984902i	$0.555382\pi$
$558$	0	0
$559$	−1.01561e6	−0.137467
$560$	0	0
$561$	−3.03481e6	−0.407122
$562$	0	0
$563$	−2.58664e6	−0.343926	−0.171963	−	0.985103i	$-0.555011\pi$
$563$	−2.58664e6	−0.343926	−0.171963	+	0.985103i	$0.555011\pi$
$564$	0	0
$565$	3.98259e6	0.524861
$566$	0	0
$567$	−232485.	−0.0303695
$568$	0	0
$569$	8.86488e6	1.14787	0.573934	−	0.818901i	$-0.305416\pi$
$569$	8.86488e6	1.14787	0.573934	+	0.818901i	$0.305416\pi$
$570$	0	0
$571$	1.07428e6	0.137888	0.0689439	−	0.997621i	$-0.478037\pi$
$571$	1.07428e6	0.137888	0.0689439	+	0.997621i	$0.478037\pi$
$572$	0	0
$573$	4.31398e6	0.548898
$574$	0	0
$575$	−9.81812e6	−1.23839
$576$	0	0
$577$	2.69414e6	0.336884	0.168442	−	0.985712i	$-0.446126\pi$
$577$	2.69414e6	0.336884	0.168442	+	0.985712i	$0.446126\pi$
$578$	0	0
$579$	−9.07134e6	−1.12454
$580$	0	0
$581$	1.33668e6	0.164281
$582$	0	0
$583$	8.21220e6	1.00066
$584$	0	0
$585$	332985.	0.0402285
$586$	0	0
$587$	−4.97659e6	−0.596124	−0.298062	−	0.954547i	$-0.596340\pi$
$587$	−4.97659e6	−0.596124	−0.298062	+	0.954547i	$0.596340\pi$
$588$	0	0
$589$	1.65503e7	1.96571
$590$	0	0
$591$	4.12662e6	0.485989
$592$	0	0
$593$	4.50238e6	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$593$	4.50238e6	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$594$	0	0
$595$	502639.	0.0582055
$596$	0	0
$597$	−1.90920e6	−0.219238
$598$	0	0
$599$	−6.54197e6	−0.744975	−0.372488	−	0.928037i	$-0.621495\pi$
$599$	−6.54197e6	−0.744975	−0.372488	+	0.928037i	$0.621495\pi$
$600$	0	0
$601$	6.08530e6	0.687221	0.343610	−	0.939112i	$-0.388350\pi$
$601$	6.08530e6	0.687221	0.343610	+	0.939112i	$0.388350\pi$
$602$	0	0
$603$	−1.51502e6	−0.169677
$604$	0	0
$605$	1.55827e6	0.173083
$606$	0	0
$607$	−2.60935e6	−0.287449	−0.143725	−	0.989618i	$-0.545908\pi$
$607$	−2.60935e6	−0.287449	−0.143725	+	0.989618i	$0.545908\pi$
$608$	0	0
$609$	−2.70941e6	−0.296027
$610$	0	0
$611$	−2.45535e6	−0.266079
$612$	0	0
$613$	1.05839e7	1.13762	0.568809	−	0.822470i	$-0.307404\pi$
$613$	1.05839e7	1.13762	0.568809	+	0.822470i	$0.307404\pi$
$614$	0	0
$615$	−695684.	−0.0741693
$616$	0	0
$617$	−4.53956e6	−0.480065	−0.240033	−	0.970765i	$-0.577158\pi$
$617$	−4.53956e6	−0.480065	−0.240033	+	0.970765i	$0.577158\pi$
$618$	0	0
$619$	1.13246e7	1.18794	0.593971	−	0.804487i	$-0.297559\pi$
$619$	1.13246e7	1.18794	0.593971	+	0.804487i	$0.297559\pi$
$620$	0	0
$621$	−2.64562e6	−0.275295
$622$	0	0
$623$	2.15994e6	0.222957
$624$	0	0
$625$	6.00779e6	0.615198
$626$	0	0
$627$	1.15704e7	1.17538
$628$	0	0
$629$	−6.64715e6	−0.669899
$630$	0	0
$631$	1.19337e7	1.19317	0.596584	−	0.802551i	$-0.296524\pi$
$631$	1.19337e7	1.19317	0.596584	+	0.802551i	$0.296524\pi$
$632$	0	0
$633$	−3.97336e6	−0.394138
$634$	0	0
$635$	−2.19170e6	−0.215698
$636$	0	0
$637$	−3.12091e6	−0.304743
$638$	0	0
$639$	2.88400e6	0.279410
$640$	0	0
$641$	−9.98183e6	−0.959545	−0.479772	−	0.877393i	$-0.659281\pi$
$641$	−9.98183e6	−0.959545	−0.479772	+	0.877393i	$0.659281\pi$
$642$	0	0
$643$	−6.13762e6	−0.585427	−0.292713	−	0.956200i	$-0.594558\pi$
$643$	−6.13762e6	−0.585427	−0.292713	+	0.956200i	$0.594558\pi$
$644$	0	0
$645$	933008.	0.0883052
$646$	0	0
$647$	6.41984e6	0.602925	0.301463	−	0.953478i	$-0.402525\pi$
$647$	6.41984e6	0.602925	0.301463	+	0.953478i	$0.402525\pi$
$648$	0	0
$649$	−6.41477e6	−0.597818
$650$	0	0
$651$	1.99919e6	0.184885
$652$	0	0
$653$	8.30695e6	0.762357	0.381179	−	0.924501i	$-0.375518\pi$
$653$	8.30695e6	0.762357	0.381179	+	0.924501i	$0.375518\pi$
$654$	0	0
$655$	5.91750e6	0.538934
$656$	0	0
$657$	4.74662e6	0.429014
$658$	0	0
$659$	9.79907e6	0.878965	0.439482	−	0.898251i	$-0.355162\pi$
$659$	9.79907e6	0.878965	0.439482	+	0.898251i	$0.355162\pi$
$660$	0	0
$661$	5.74303e6	0.511255	0.255628	−	0.966775i	$-0.417718\pi$
$661$	5.74303e6	0.511255	0.255628	+	0.966775i	$0.417718\pi$
$662$	0	0
$663$	1.25072e6	0.110503
$664$	0	0
$665$	−1.91634e6	−0.168042
$666$	0	0
$667$	−3.08323e7	−2.68343
$668$	0	0
$669$	4.29670e6	0.371167
$670$	0	0
$671$	−1.19428e7	−1.02400
$672$	0	0
$673$	−5.79225e6	−0.492957	−0.246479	−	0.969148i	$-0.579274\pi$
$673$	−5.79225e6	−0.492957	−0.246479	+	0.969148i	$0.579274\pi$
$674$	0	0
$675$	1.97222e6	0.166608
$676$	0	0
$677$	2.17998e7	1.82802	0.914010	−	0.405692i	$-0.132970\pi$
$677$	2.17998e7	1.82802	0.914010	+	0.405692i	$0.132970\pi$
$678$	0	0
$679$	2.43714e6	0.202865
$680$	0	0
$681$	−652680.	−0.0539303
$682$	0	0
$683$	5.02655e6	0.412305	0.206152	−	0.978520i	$-0.433906\pi$
$683$	5.02655e6	0.412305	0.206152	+	0.978520i	$0.433906\pi$
$684$	0	0
$685$	1.44267e6	0.117474
$686$	0	0
$687$	7.37929e6	0.596517
$688$	0	0
$689$	−3.38444e6	−0.271605
$690$	0	0
$691$	1.17895e7	0.939291	0.469645	−	0.882855i	$-0.344382\pi$
$691$	1.17895e7	0.939291	0.469645	+	0.882855i	$0.344382\pi$
$692$	0	0
$693$	1.39764e6	0.110551
$694$	0	0
$695$	−4.30325e6	−0.337936
$696$	0	0
$697$	−2.61305e6	−0.203735
$698$	0	0
$699$	−17041.6	−0.00131922
$700$	0	0
$701$	−1.26771e6	−0.0974370	−0.0487185	−	0.998813i	$-0.515514\pi$
$701$	−1.26771e6	−0.0974370	−0.0487185	+	0.998813i	$0.515514\pi$
$702$	0	0
$703$	2.53427e7	1.93403
$704$	0	0
$705$	2.25564e6	0.170922
$706$	0	0
$707$	969805.	0.0729686
$708$	0	0
$709$	−9.52142e6	−0.711354	−0.355677	−	0.934609i	$-0.615750\pi$
$709$	−9.52142e6	−0.711354	−0.355677	+	0.934609i	$0.615750\pi$
$710$	0	0
$711$	7.94893e6	0.589705
$712$	0	0
$713$	2.27502e7	1.67595
$714$	0	0
$715$	−2.00182e6	−0.146440
$716$	0	0
$717$	−4.56748e6	−0.331802
$718$	0	0
$719$	−2.41159e7	−1.73973	−0.869864	−	0.493292i	$-0.835793\pi$
$719$	−2.41159e7	−1.73973	−0.869864	+	0.493292i	$0.835793\pi$
$720$	0	0
$721$	2.08858e6	0.149628
$722$	0	0
$723$	851524.	0.0605831
$724$	0	0
$725$	2.29845e7	1.62401
$726$	0	0
$727$	4.23660e6	0.297291	0.148646	−	0.988891i	$-0.452509\pi$
$727$	4.23660e6	0.297291	0.148646	+	0.988891i	$0.452509\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	3.50445e6	0.242564
$732$	0	0
$733$	−1.18809e7	−0.816750	−0.408375	−	0.912814i	$-0.633904\pi$
$733$	−1.18809e7	−0.816750	−0.408375	+	0.912814i	$0.633904\pi$
$734$	0	0
$735$	2.86707e6	0.195758
$736$	0	0
$737$	9.10789e6	0.617659
$738$	0	0
$739$	−1.04722e7	−0.705384	−0.352692	−	0.935739i	$-0.614734\pi$
$739$	−1.04722e7	−0.705384	−0.352692	+	0.935739i	$0.614734\pi$
$740$	0	0
$741$	−4.76843e6	−0.319029
$742$	0	0
$743$	−2.31594e6	−0.153906	−0.0769530	−	0.997035i	$-0.524519\pi$
$743$	−2.31594e6	−0.153906	−0.0769530	+	0.997035i	$0.524519\pi$
$744$	0	0
$745$	−4.79195e6	−0.316316
$746$	0	0
$747$	−3.05553e6	−0.200348
$748$	0	0
$749$	1.14007e6	0.0742549
$750$	0	0
$751$	1.24358e7	0.804592	0.402296	−	0.915510i	$-0.368212\pi$
$751$	1.24358e7	0.804592	0.402296	+	0.915510i	$0.368212\pi$
$752$	0	0
$753$	1.20865e7	0.776806
$754$	0	0
$755$	980397.	0.0625943
$756$	0	0
$757$	−2.94671e6	−0.186895	−0.0934475	−	0.995624i	$-0.529789\pi$
$757$	−2.94671e6	−0.186895	−0.0934475	+	0.995624i	$0.529789\pi$
$758$	0	0
$759$	1.59048e7	1.00213
$760$	0	0
$761$	−2.71350e7	−1.69851	−0.849257	−	0.527980i	$-0.822949\pi$
$761$	−2.71350e7	−1.69851	−0.849257	+	0.527980i	$0.822949\pi$
$762$	0	0
$763$	6.56069e6	0.407979
$764$	0	0
$765$	−1.14899e6	−0.0709842
$766$	0	0
$767$	2.64367e6	0.162263
$768$	0	0
$769$	−1.55459e7	−0.947982	−0.473991	−	0.880530i	$-0.657187\pi$
$769$	−1.55459e7	−0.947982	−0.473991	+	0.880530i	$0.657187\pi$
$770$	0	0
$771$	−1.42813e7	−0.865230
$772$	0	0
$773$	−1.13507e7	−0.683240	−0.341620	−	0.939838i	$-0.610976\pi$
$773$	−1.13507e7	−0.683240	−0.341620	+	0.939838i	$0.610976\pi$
$774$	0	0
$775$	−1.69596e7	−1.01429
$776$	0	0
$777$	3.06126e6	0.181906
$778$	0	0
$779$	9.96240e6	0.588194
$780$	0	0
$781$	−1.73379e7	−1.01711
$782$	0	0
$783$	6.19347e6	0.361018
$784$	0	0
$785$	−6.67564e6	−0.386650
$786$	0	0
$787$	1.74714e7	1.00552	0.502759	−	0.864426i	$-0.332318\pi$
$787$	1.74714e7	1.00552	0.502759	+	0.864426i	$0.332318\pi$
$788$	0	0
$789$	9.90549e6	0.566478
$790$	0	0
$791$	−6.88914e6	−0.391493
$792$	0	0
$793$	4.92192e6	0.277940
$794$	0	0
$795$	3.10916e6	0.174472
$796$	0	0
$797$	1.94474e7	1.08446	0.542231	−	0.840229i	$-0.317580\pi$
$797$	1.94474e7	1.08446	0.542231	+	0.840229i	$0.317580\pi$
$798$	0	0
$799$	8.47237e6	0.469503
$800$	0	0
$801$	−4.93742e6	−0.271906
$802$	0	0
$803$	−2.85355e7	−1.56169
$804$	0	0
$805$	−2.63422e6	−0.143272
$806$	0	0
$807$	1.53131e7	0.827709
$808$	0	0
$809$	−2.51681e7	−1.35201	−0.676005	−	0.736897i	$-0.736290\pi$
$809$	−2.51681e7	−1.35201	−0.676005	+	0.736897i	$0.736290\pi$
$810$	0	0
$811$	2.78670e7	1.48778	0.743890	−	0.668302i	$-0.232979\pi$
$811$	2.78670e7	1.48778	0.743890	+	0.668302i	$0.232979\pi$
$812$	0	0
$813$	1.51046e7	0.801460
$814$	0	0
$815$	−8.66628e6	−0.457024
$816$	0	0
$817$	−1.33609e7	−0.700297
$818$	0	0
$819$	−576001.	−0.0300064
$820$	0	0
$821$	−1.86040e7	−0.963270	−0.481635	−	0.876372i	$-0.659957\pi$
$821$	−1.86040e7	−0.963270	−0.481635	+	0.876372i	$0.659957\pi$
$822$	0	0
$823$	1.39034e7	0.715521	0.357761	−	0.933813i	$-0.383540\pi$
$823$	1.39034e7	0.715521	0.357761	+	0.933813i	$0.383540\pi$
$824$	0	0
$825$	−1.18565e7	−0.606488
$826$	0	0
$827$	−2.85989e7	−1.45407	−0.727034	−	0.686601i	$-0.759102\pi$
$827$	−2.85989e7	−1.45407	−0.727034	+	0.686601i	$0.759102\pi$
$828$	0	0
$829$	2.09644e7	1.05949	0.529743	−	0.848158i	$-0.322288\pi$
$829$	2.09644e7	1.05949	0.529743	+	0.848158i	$0.322288\pi$
$830$	0	0
$831$	−8.65259e6	−0.434654
$832$	0	0
$833$	1.07690e7	0.537726
$834$	0	0
$835$	−6.15895e6	−0.305697
$836$	0	0
$837$	−4.56998e6	−0.225476
$838$	0	0
$839$	−3.24925e7	−1.59359	−0.796797	−	0.604247i	$-0.793474\pi$
$839$	−3.24925e7	−1.59359	−0.796797	+	0.604247i	$0.793474\pi$
$840$	0	0
$841$	5.16681e7	2.51903
$842$	0	0
$843$	5.18499e6	0.251292
$844$	0	0
$845$	−6.78078e6	−0.326691
$846$	0	0
$847$	−2.69552e6	−0.129102
$848$	0	0
$849$	−5.22218e6	−0.248646
$850$	0	0
$851$	3.48362e7	1.64895
$852$	0	0
$853$	−2.77561e7	−1.30613	−0.653065	−	0.757302i	$-0.726517\pi$
$853$	−2.77561e7	−1.30613	−0.653065	+	0.757302i	$0.726517\pi$
$854$	0	0
$855$	4.38059e6	0.204935
$856$	0	0
$857$	3.26911e7	1.52047	0.760234	−	0.649649i	$-0.225084\pi$
$857$	3.26911e7	1.52047	0.760234	+	0.649649i	$0.225084\pi$
$858$	0	0
$859$	8.75427e6	0.404797	0.202398	−	0.979303i	$-0.435126\pi$
$859$	8.75427e6	0.404797	0.202398	+	0.979303i	$0.435126\pi$
$860$	0	0
$861$	1.20340e6	0.0553227
$862$	0	0
$863$	−3.58575e7	−1.63890	−0.819452	−	0.573148i	$-0.805722\pi$
$863$	−3.58575e7	−1.63890	−0.819452	+	0.573148i	$0.805722\pi$
$864$	0	0
$865$	−9.97408e6	−0.453244
$866$	0	0
$867$	8.46302e6	0.382365
$868$	0	0
$869$	−4.77869e7	−2.14664
$870$	0	0
$871$	−3.75357e6	−0.167648
$872$	0	0
$873$	−5.57109e6	−0.247403
$874$	0	0
$875$	4.23204e6	0.186866
$876$	0	0
$877$	−2.12181e7	−0.931552	−0.465776	−	0.884903i	$-0.654225\pi$
$877$	−2.12181e7	−0.931552	−0.465776	+	0.884903i	$0.654225\pi$
$878$	0	0
$879$	1.05939e7	0.462470
$880$	0	0
$881$	3.68571e7	1.59986	0.799928	−	0.600095i	$-0.204871\pi$
$881$	3.68571e7	1.59986	0.799928	+	0.600095i	$0.204871\pi$
$882$	0	0
$883$	1.72459e7	0.744360	0.372180	−	0.928161i	$-0.378610\pi$
$883$	1.72459e7	0.744360	0.372180	+	0.928161i	$0.378610\pi$
$884$	0	0
$885$	−2.42865e6	−0.104233
$886$	0	0
$887$	−2.90215e7	−1.23854	−0.619270	−	0.785178i	$-0.712571\pi$
$887$	−2.90215e7	−1.23854	−0.619270	+	0.785178i	$0.712571\pi$
$888$	0	0
$889$	3.79123e6	0.160889
$890$	0	0
$891$	−3.19489e6	−0.134822
$892$	0	0
$893$	−3.23014e7	−1.35548
$894$	0	0
$895$	−1.02340e7	−0.427057
$896$	0	0
$897$	−6.55472e6	−0.272003
$898$	0	0
$899$	−5.32589e7	−2.19783
$900$	0	0
$901$	1.16783e7	0.479254
$902$	0	0
$903$	−1.61393e6	−0.0658666
$904$	0	0
$905$	−1.72133e7	−0.698624
$906$	0	0
$907$	−2.17240e7	−0.876842	−0.438421	−	0.898770i	$-0.644462\pi$
$907$	−2.17240e7	−0.876842	−0.438421	+	0.898770i	$0.644462\pi$
$908$	0	0
$909$	−2.21689e6	−0.0889885
$910$	0	0
$911$	1.18167e7	0.471736	0.235868	−	0.971785i	$-0.424207\pi$
$911$	1.18167e7	0.471736	0.235868	+	0.971785i	$0.424207\pi$
$912$	0	0
$913$	1.83690e7	0.729306
$914$	0	0
$915$	−4.52159e6	−0.178541
$916$	0	0
$917$	−1.02362e7	−0.401989
$918$	0	0
$919$	−1.08515e7	−0.423838	−0.211919	−	0.977287i	$-0.567971\pi$
$919$	−1.08515e7	−0.423838	−0.211919	+	0.977287i	$0.567971\pi$
$920$	0	0
$921$	−2.84606e6	−0.110559
$922$	0	0
$923$	7.14533e6	0.276069
$924$	0	0
$925$	−2.59693e7	−0.997944
$926$	0	0
$927$	−4.77430e6	−0.182478
$928$	0	0
$929$	−2.49688e7	−0.949203	−0.474601	−	0.880201i	$-0.657408\pi$
$929$	−2.49688e7	−0.949203	−0.474601	+	0.880201i	$0.657408\pi$
$930$	0	0
$931$	−4.10573e7	−1.55244
$932$	0	0
$933$	3.92093e6	0.147464
$934$	0	0
$935$	6.90743e6	0.258397
$936$	0	0
$937$	−4.20064e7	−1.56303	−0.781513	−	0.623889i	$-0.785552\pi$
$937$	−4.20064e7	−1.56303	−0.781513	+	0.623889i	$0.785552\pi$
$938$	0	0
$939$	−2.96288e6	−0.109660
$940$	0	0
$941$	−3.01821e7	−1.11116	−0.555578	−	0.831464i	$-0.687503\pi$
$941$	−3.01821e7	−1.11116	−0.555578	+	0.831464i	$0.687503\pi$
$942$	0	0
$943$	1.36944e7	0.501491
$944$	0	0
$945$	529151.	0.0192753
$946$	0	0
$947$	4.84746e7	1.75646	0.878232	−	0.478235i	$-0.158723\pi$
$947$	4.84746e7	1.75646	0.878232	+	0.478235i	$0.158723\pi$
$948$	0	0
$949$	1.17601e7	0.423883
$950$	0	0
$951$	7.86920e6	0.282149
$952$	0	0
$953$	1.12917e7	0.402741	0.201370	−	0.979515i	$-0.435460\pi$
$953$	1.12917e7	0.402741	0.201370	+	0.979515i	$0.435460\pi$
$954$	0	0
$955$	−9.81888e6	−0.348380
$956$	0	0
$957$	−3.72335e7	−1.31418
$958$	0	0
$959$	−2.49555e6	−0.0876232
$960$	0	0
$961$	1.06690e7	0.372664
$962$	0	0
$963$	−2.60609e6	−0.0905573
$964$	0	0
$965$	2.06469e7	0.713735
$966$	0	0
$967$	4.05647e7	1.39503	0.697513	−	0.716572i	$-0.254290\pi$
$967$	4.05647e7	1.39503	0.697513	+	0.716572i	$0.254290\pi$
$968$	0	0
$969$	1.64538e7	0.562934
$970$	0	0
$971$	1.23608e7	0.420724	0.210362	−	0.977624i	$-0.432536\pi$
$971$	1.23608e7	0.420724	0.210362	+	0.977624i	$0.432536\pi$
$972$	0	0
$973$	7.44382e6	0.252066
$974$	0	0
$975$	4.88634e6	0.164616
$976$	0	0
$977$	7.98701e6	0.267699	0.133850	−	0.991002i	$-0.457266\pi$
$977$	7.98701e6	0.267699	0.133850	+	0.991002i	$0.457266\pi$
$978$	0	0
$979$	2.96825e7	0.989793
$980$	0	0
$981$	−1.49971e7	−0.497549
$982$	0	0
$983$	5.17855e6	0.170932	0.0854661	−	0.996341i	$-0.472762\pi$
$983$	5.17855e6	0.170932	0.0854661	+	0.996341i	$0.472762\pi$
$984$	0	0
$985$	−9.39245e6	−0.308453
$986$	0	0
$987$	−3.90184e6	−0.127490
$988$	0	0
$989$	−1.83661e7	−0.597070
$990$	0	0
$991$	1.43782e7	0.465072	0.232536	−	0.972588i	$-0.425298\pi$
$991$	1.43782e7	0.465072	0.232536	+	0.972588i	$0.425298\pi$
$992$	0	0
$993$	8.09101e6	0.260393
$994$	0	0
$995$	4.34546e6	0.139148
$996$	0	0
$997$	−1.42316e7	−0.453435	−0.226717	−	0.973961i	$-0.572799\pi$
$997$	−1.42316e7	−0.453435	−0.226717	+	0.973961i	$0.572799\pi$
$998$	0	0
$999$	−6.99776e6	−0.221843

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.6.a.ba.1.3		5
4.3	odd	2		768.6.a.bc.1.3		5
8.3	odd	2		768.6.a.bb.1.3		5
8.5	even	2		768.6.a.bd.1.3		5
16.3	odd	4		96.6.d.a.49.8		10
16.5	even	4		24.6.d.a.13.10	yes	10
16.11	odd	4		96.6.d.a.49.3		10
16.13	even	4		24.6.d.a.13.9	✓	10
48.5	odd	4		72.6.d.d.37.1		10
48.11	even	4		288.6.d.d.145.5		10
48.29	odd	4		72.6.d.d.37.2		10
48.35	even	4		288.6.d.d.145.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.6.d.a.13.9	✓	10	16.13	even	4
24.6.d.a.13.10	yes	10	16.5	even	4
72.6.d.d.37.1		10	48.5	odd	4
72.6.d.d.37.2		10	48.29	odd	4
96.6.d.a.49.3		10	16.11	odd	4
96.6.d.a.49.8		10	16.3	odd	4
288.6.d.d.145.5		10	48.11	even	4
288.6.d.d.145.6		10	48.35	even	4
768.6.a.ba.1.3		5	1.1	even	1	trivial
768.6.a.bb.1.3		5	8.3	odd	2
768.6.a.bc.1.3		5	4.3	odd	2
768.6.a.bd.1.3		5	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +20.4846 q^{5} -35.4345 q^{7} +81.0000 q^{9} -486.951 q^{11} +200.684 q^{13} -184.361 q^{15} -692.475 q^{17} +2640.10 q^{19} +318.910 q^{21} +3629.10 q^{23} -2705.38 q^{25} -729.000 q^{27} -8495.84 q^{29} +6268.83 q^{31} +4382.56 q^{33} -725.859 q^{35} +9599.13 q^{37} -1806.15 q^{39} +3773.49 q^{41} -5060.77 q^{43} +1659.25 q^{45} -12234.9 q^{47} -15551.4 q^{49} +6232.27 q^{51} -16864.5 q^{53} -9974.98 q^{55} -23760.9 q^{57} +13173.3 q^{59} +24525.7 q^{61} -2870.19 q^{63} +4110.92 q^{65} -18703.9 q^{67} -32661.9 q^{69} +35604.9 q^{71} +58600.2 q^{73} +24348.4 q^{75} +17254.9 q^{77} +98134.9 q^{79} +6561.00 q^{81} -37722.5 q^{83} -14185.0 q^{85} +76462.5 q^{87} -60955.8 q^{89} -7111.12 q^{91} -56419.4 q^{93} +54081.3 q^{95} -68778.9 q^{97} -39443.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 45 q^{3} - 50 q^{5} + 98 q^{7} + 405 q^{9} - 676 q^{13} + 450 q^{15} - 202 q^{17} - 716 q^{19} - 882 q^{21} - 836 q^{23} + 4683 q^{25} - 3645 q^{27} - 5046 q^{29} + 9346 q^{31} - 436 q^{35} - 10952 q^{37}+ \cdots + 29686 q^{97}+O(q^{100}) \) 5 * q - 45 * q^3 - 50 * q^5 + 98 * q^7 + 405 * q^9 - 676 * q^13 + 450 * q^15 - 202 * q^17 - 716 * q^19 - 882 * q^21 - 836 * q^23 + 4683 * q^25 - 3645 * q^27 - 5046 * q^29 + 9346 * q^31 - 436 * q^35 - 10952 * q^37 + 6084 * q^39 + 1238 * q^41 - 1676 * q^43 - 4050 * q^45 + 32628 * q^47 + 4689 * q^49 + 1818 * q^51 + 8490 * q^53 + 51712 * q^55 + 6444 * q^57 + 15884 * q^59 + 13824 * q^61 + 7938 * q^63 + 25768 * q^65 - 26652 * q^67 + 7524 * q^69 + 136116 * q^71 + 13694 * q^73 - 42147 * q^75 - 89856 * q^77 + 126154 * q^79 + 32805 * q^81 + 64720 * q^83 - 164636 * q^85 + 45414 * q^87 - 87790 * q^89 + 179992 * q^91 - 84114 * q^93 + 121592 * q^95 + 29686 * q^97

Introduction

L-functions

Modular forms

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Database

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