LMFDB - Embedded newform 624.6.a.j.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [624,6,Mod(1,624)] 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("624.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	624.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-18,0,10,0,-138] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$100.079503563$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3241}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 810 $ x^2 - x - 810
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-27.9649$ of defining polynomial
Character	$\chi$	$=$	624.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} +61.9298 q^{5} -12.0702 q^{7} +81.0000 q^{9} +183.438 q^{11} +169.000 q^{13} -557.368 q^{15} -524.596 q^{17} -1389.96 q^{19} +108.632 q^{21} -1916.63 q^{23} +710.298 q^{25} -729.000 q^{27} -2653.37 q^{29} +1441.68 q^{31} -1650.94 q^{33} -747.506 q^{35} -3015.61 q^{37} -1521.00 q^{39} +3302.13 q^{41} -456.484 q^{43} +5016.31 q^{45} +21191.2 q^{47} -16661.3 q^{49} +4721.36 q^{51} -5099.12 q^{53} +11360.3 q^{55} +12509.7 q^{57} -694.164 q^{59} -55101.5 q^{61} -977.688 q^{63} +10466.1 q^{65} +29603.6 q^{67} +17249.7 q^{69} -30371.8 q^{71} -43898.6 q^{73} -6392.68 q^{75} -2214.14 q^{77} +63192.4 q^{79} +6561.00 q^{81} -20100.2 q^{83} -32488.1 q^{85} +23880.3 q^{87} +118120. q^{89} -2039.87 q^{91} -12975.1 q^{93} -86080.1 q^{95} -27497.1 q^{97} +14858.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{3} + 10 q^{5} - 138 q^{7} + 162 q^{9} - 544 q^{11} + 338 q^{13} - 90 q^{15} + 1228 q^{17} + 522 q^{19} + 1242 q^{21} + 1632 q^{23} + 282 q^{25} - 1458 q^{27} + 2208 q^{29} - 874 q^{31} + 4896 q^{33}+ \cdots - 44064 q^{99}+O(q^{100}) $ 2 * q - 18 * q^3 + 10 * q^5 - 138 * q^7 + 162 * q^9 - 544 * q^11 + 338 * q^13 - 90 * q^15 + 1228 * q^17 + 522 * q^19 + 1242 * q^21 + 1632 * q^23 + 282 * q^25 - 1458 * q^27 + 2208 * q^29 - 874 * q^31 + 4896 * q^33 + 5792 * q^35 - 2160 * q^37 - 3042 * q^39 - 12638 * q^41 + 17760 * q^43 + 810 * q^45 - 4300 * q^47 - 17610 * q^49 - 11052 * q^51 - 12020 * q^53 + 49136 * q^55 - 4698 * q^57 + 27532 * q^59 - 38016 * q^61 - 11178 * q^63 + 1690 * q^65 + 90974 * q^67 - 14688 * q^69 + 4384 * q^71 - 95312 * q^73 - 2538 * q^75 + 89392 * q^77 + 62168 * q^79 + 13122 * q^81 - 137892 * q^83 - 123500 * q^85 - 19872 * q^87 + 142534 * q^89 - 23322 * q^91 + 7866 * q^93 - 185368 * q^95 + 4896 * q^97 - 44064 * 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
$5$	61.9298	1.10783	0.553917	−	0.832572i	$-0.313133\pi$
$5$	61.9298	1.10783	0.553917	+	0.832572i	$0.313133\pi$
$6$	0	0
$7$	−12.0702	−0.0931044	−0.0465522	−	0.998916i	$-0.514823\pi$
$7$	−12.0702	−0.0931044	−0.0465522	+	0.998916i	$0.514823\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	183.438	0.457097	0.228548	−	0.973533i	$-0.426602\pi$
$11$	183.438	0.457097	0.228548	+	0.973533i	$0.426602\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	0	0
$15$	−557.368	−0.639608
$16$	0	0
$17$	−524.596	−0.440253	−0.220127	−	0.975471i	$-0.570647\pi$
$17$	−524.596	−0.440253	−0.220127	+	0.975471i	$0.570647\pi$
$18$	0	0
$19$	−1389.96	−0.883323	−0.441661	−	0.897182i	$-0.645611\pi$
$19$	−1389.96	−0.883323	−0.441661	+	0.897182i	$0.645611\pi$
$20$	0	0
$21$	108.632	0.0537538
$22$	0	0
$23$	−1916.63	−0.755472	−0.377736	−	0.925913i	$-0.623297\pi$
$23$	−1916.63	−0.755472	−0.377736	+	0.925913i	$0.623297\pi$
$24$	0	0
$25$	710.298	0.227295
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−2653.37	−0.585871	−0.292936	−	0.956132i	$-0.594632\pi$
$29$	−2653.37	−0.585871	−0.292936	+	0.956132i	$0.594632\pi$
$30$	0	0
$31$	1441.68	0.269442	0.134721	−	0.990884i	$-0.456986\pi$
$31$	1441.68	0.269442	0.134721	+	0.990884i	$0.456986\pi$
$32$	0	0
$33$	−1650.94	−0.263905
$34$	0	0
$35$	−747.506	−0.103144
$36$	0	0
$37$	−3015.61	−0.362136	−0.181068	−	0.983471i	$-0.557955\pi$
$37$	−3015.61	−0.362136	−0.181068	+	0.983471i	$0.557955\pi$
$38$	0	0
$39$	−1521.00	−0.160128
$40$	0	0
$41$	3302.13	0.306786	0.153393	−	0.988165i	$-0.450980\pi$
$41$	3302.13	0.306786	0.153393	+	0.988165i	$0.450980\pi$
$42$	0	0
$43$	−456.484	−0.0376491	−0.0188245	−	0.999823i	$-0.505992\pi$
$43$	−456.484	−0.0376491	−0.0188245	+	0.999823i	$0.505992\pi$
$44$	0	0
$45$	5016.31	0.369278
$46$	0	0
$47$	21191.2	1.39930	0.699650	−	0.714485i	$-0.253339\pi$
$47$	21191.2	1.39930	0.699650	+	0.714485i	$0.253339\pi$
$48$	0	0
$49$	−16661.3	−0.991332
$50$	0	0
$51$	4721.36	0.254180
$52$	0	0
$53$	−5099.12	−0.249348	−0.124674	−	0.992198i	$-0.539788\pi$
$53$	−5099.12	−0.249348	−0.124674	+	0.992198i	$0.539788\pi$
$54$	0	0
$55$	11360.3	0.506387
$56$	0	0
$57$	12509.7	0.509987
$58$	0	0
$59$	−694.164	−0.0259617	−0.0129808	−	0.999916i	$-0.504132\pi$
$59$	−694.164	−0.0259617	−0.0129808	+	0.999916i	$0.504132\pi$
$60$	0	0
$61$	−55101.5	−1.89600	−0.948001	−	0.318268i	$-0.896899\pi$
$61$	−55101.5	−1.89600	−0.948001	+	0.318268i	$0.896899\pi$
$62$	0	0
$63$	−977.688	−0.0310348
$64$	0	0
$65$	10466.1	0.307258
$66$	0	0
$67$	29603.6	0.805670	0.402835	−	0.915273i	$-0.368025\pi$
$67$	29603.6	0.805670	0.402835	+	0.915273i	$0.368025\pi$
$68$	0	0
$69$	17249.7	0.436172
$70$	0	0
$71$	−30371.8	−0.715031	−0.357516	−	0.933907i	$-0.616376\pi$
$71$	−30371.8	−0.715031	−0.357516	+	0.933907i	$0.616376\pi$
$72$	0	0
$73$	−43898.6	−0.964148	−0.482074	−	0.876130i	$-0.660116\pi$
$73$	−43898.6	−0.964148	−0.482074	+	0.876130i	$0.660116\pi$
$74$	0	0
$75$	−6392.68	−0.131229
$76$	0	0
$77$	−2214.14	−0.0425577
$78$	0	0
$79$	63192.4	1.13919	0.569596	−	0.821925i	$-0.307100\pi$
$79$	63192.4	1.13919	0.569596	+	0.821925i	$0.307100\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−20100.2	−0.320263	−0.160131	−	0.987096i	$-0.551192\pi$
$83$	−20100.2	−0.320263	−0.160131	+	0.987096i	$0.551192\pi$
$84$	0	0
$85$	−32488.1	−0.487727
$86$	0	0
$87$	23880.3	0.338253
$88$	0	0
$89$	118120.	1.58070	0.790350	−	0.612656i	$-0.209899\pi$
$89$	118120.	1.58070	0.790350	+	0.612656i	$0.209899\pi$
$90$	0	0
$91$	−2039.87	−0.0258225
$92$	0	0
$93$	−12975.1	−0.155562
$94$	0	0
$95$	−86080.1	−0.978575
$96$	0	0
$97$	−27497.1	−0.296727	−0.148363	−	0.988933i	$-0.547401\pi$
$97$	−27497.1	−0.296727	−0.148363	+	0.988933i	$0.547401\pi$
$98$	0	0
$99$	14858.5	0.152366
$100$	0	0
$101$	116434.	1.13574	0.567869	−	0.823119i	$-0.307768\pi$
$101$	116434.	1.13574	0.567869	+	0.823119i	$0.307768\pi$
$102$	0	0
$103$	−79882.4	−0.741921	−0.370961	−	0.928649i	$-0.620972\pi$
$103$	−79882.4	−0.741921	−0.370961	+	0.928649i	$0.620972\pi$
$104$	0	0
$105$	6727.55	0.0595503
$106$	0	0
$107$	−144176.	−1.21740	−0.608699	−	0.793401i	$-0.708308\pi$
$107$	−144176.	−1.21740	−0.608699	+	0.793401i	$0.708308\pi$
$108$	0	0
$109$	141022.	1.13690	0.568449	−	0.822719i	$-0.307544\pi$
$109$	141022.	1.13690	0.568449	+	0.822719i	$0.307544\pi$
$110$	0	0
$111$	27140.5	0.209079
$112$	0	0
$113$	182492.	1.34446	0.672231	−	0.740341i	$-0.265336\pi$
$113$	182492.	1.34446	0.672231	+	0.740341i	$0.265336\pi$
$114$	0	0
$115$	−118696.	−0.836938
$116$	0	0
$117$	13689.0	0.0924500
$118$	0	0
$119$	6331.98	0.0409895
$120$	0	0
$121$	−127401.	−0.791063
$122$	0	0
$123$	−29719.2	−0.177123
$124$	0	0
$125$	−149542.	−0.856028
$126$	0	0
$127$	−245274.	−1.34940	−0.674702	−	0.738091i	$-0.735728\pi$
$127$	−245274.	−1.34940	−0.674702	+	0.738091i	$0.735728\pi$
$128$	0	0
$129$	4108.36	0.0217367
$130$	0	0
$131$	70592.4	0.359401	0.179701	−	0.983721i	$-0.442487\pi$
$131$	70592.4	0.359401	0.179701	+	0.983721i	$0.442487\pi$
$132$	0	0
$133$	16777.2	0.0822412
$134$	0	0
$135$	−45146.8	−0.213203
$136$	0	0
$137$	−193511.	−0.880856	−0.440428	−	0.897788i	$-0.645173\pi$
$137$	−193511.	−0.880856	−0.440428	+	0.897788i	$0.645173\pi$
$138$	0	0
$139$	16550.3	0.0726556	0.0363278	−	0.999340i	$-0.488434\pi$
$139$	16550.3	0.0726556	0.0363278	+	0.999340i	$0.488434\pi$
$140$	0	0
$141$	−190721.	−0.807887
$142$	0	0
$143$	31001.1	0.126776
$144$	0	0
$145$	−164322.	−0.649048
$146$	0	0
$147$	149952.	0.572346
$148$	0	0
$149$	−479101.	−1.76792	−0.883958	−	0.467567i	$-0.845131\pi$
$149$	−479101.	−1.76792	−0.883958	+	0.467567i	$0.845131\pi$
$150$	0	0
$151$	−315548.	−1.12622	−0.563111	−	0.826382i	$-0.690396\pi$
$151$	−315548.	−1.12622	−0.563111	+	0.826382i	$0.690396\pi$
$152$	0	0
$153$	−42492.2	−0.146751
$154$	0	0
$155$	89283.1	0.298497
$156$	0	0
$157$	−540764.	−1.75089	−0.875444	−	0.483319i	$-0.839431\pi$
$157$	−540764.	−1.75089	−0.875444	+	0.483319i	$0.839431\pi$
$158$	0	0
$159$	45892.1	0.143961
$160$	0	0
$161$	23134.1	0.0703378
$162$	0	0
$163$	81219.0	0.239436	0.119718	−	0.992808i	$-0.461801\pi$
$163$	81219.0	0.239436	0.119718	+	0.992808i	$0.461801\pi$
$164$	0	0
$165$	−102243.	−0.292363
$166$	0	0
$167$	−401419.	−1.11380	−0.556899	−	0.830580i	$-0.688009\pi$
$167$	−401419.	−1.11380	−0.556899	+	0.830580i	$0.688009\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	−112587.	−0.294441
$172$	0	0
$173$	−256472.	−0.651515	−0.325757	−	0.945453i	$-0.605619\pi$
$173$	−256472.	−0.651515	−0.325757	+	0.945453i	$0.605619\pi$
$174$	0	0
$175$	−8573.45	−0.0211622
$176$	0	0
$177$	6247.48	0.0149890
$178$	0	0
$179$	−584078.	−1.36251	−0.681253	−	0.732048i	$-0.738565\pi$
$179$	−584078.	−1.36251	−0.681253	+	0.732048i	$0.738565\pi$
$180$	0	0
$181$	−170556.	−0.386965	−0.193482	−	0.981104i	$-0.561978\pi$
$181$	−170556.	−0.386965	−0.193482	+	0.981104i	$0.561978\pi$
$182$	0	0
$183$	495913.	1.09466
$184$	0	0
$185$	−186756.	−0.401186
$186$	0	0
$187$	−96230.9	−0.201238
$188$	0	0
$189$	8799.19	0.0179179
$190$	0	0
$191$	−512527.	−1.01656	−0.508280	−	0.861192i	$-0.669719\pi$
$191$	−512527.	−1.01656	−0.508280	+	0.861192i	$0.669719\pi$
$192$	0	0
$193$	220134.	0.425397	0.212699	−	0.977118i	$-0.431775\pi$
$193$	220134.	0.425397	0.212699	+	0.977118i	$0.431775\pi$
$194$	0	0
$195$	−94195.2	−0.177395
$196$	0	0
$197$	88389.6	0.162269	0.0811345	−	0.996703i	$-0.474146\pi$
$197$	88389.6	0.162269	0.0811345	+	0.996703i	$0.474146\pi$
$198$	0	0
$199$	−914132.	−1.63635	−0.818175	−	0.574970i	$-0.805014\pi$
$199$	−914132.	−1.63635	−0.818175	+	0.574970i	$0.805014\pi$
$200$	0	0
$201$	−266432.	−0.465154
$202$	0	0
$203$	32026.7	0.0545472
$204$	0	0
$205$	204500.	0.339867
$206$	0	0
$207$	−155247.	−0.251824
$208$	0	0
$209$	−254973.	−0.403764
$210$	0	0
$211$	−293030.	−0.453112	−0.226556	−	0.973998i	$-0.572747\pi$
$211$	−293030.	−0.453112	−0.226556	+	0.973998i	$0.572747\pi$
$212$	0	0
$213$	273347.	0.412824
$214$	0	0
$215$	−28270.0	−0.0417089
$216$	0	0
$217$	−17401.4	−0.0250862
$218$	0	0
$219$	395088.	0.556651
$220$	0	0
$221$	−88656.7	−0.122104
$222$	0	0
$223$	414666.	0.558389	0.279194	−	0.960235i	$-0.409933\pi$
$223$	414666.	0.558389	0.279194	+	0.960235i	$0.409933\pi$
$224$	0	0
$225$	57534.1	0.0757651
$226$	0	0
$227$	709235.	0.913536	0.456768	−	0.889586i	$-0.349007\pi$
$227$	709235.	0.913536	0.456768	+	0.889586i	$0.349007\pi$
$228$	0	0
$229$	−454331.	−0.572511	−0.286256	−	0.958153i	$-0.592411\pi$
$229$	−454331.	−0.572511	−0.286256	+	0.958153i	$0.592411\pi$
$230$	0	0
$231$	19927.3	0.0245707
$232$	0	0
$233$	−347380.	−0.419194	−0.209597	−	0.977788i	$-0.567215\pi$
$233$	−347380.	−0.419194	−0.209597	+	0.977788i	$0.567215\pi$
$234$	0	0
$235$	1.31237e6	1.55019
$236$	0	0
$237$	−568732.	−0.657713
$238$	0	0
$239$	−502585.	−0.569134	−0.284567	−	0.958656i	$-0.591850\pi$
$239$	−502585.	−0.569134	−0.284567	+	0.958656i	$0.591850\pi$
$240$	0	0
$241$	885838.	0.982453	0.491226	−	0.871032i	$-0.336549\pi$
$241$	885838.	0.982453	0.491226	+	0.871032i	$0.336549\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−1.03183e6	−1.09823
$246$	0	0
$247$	−234904.	−0.244990
$248$	0	0
$249$	180902.	0.184904
$250$	0	0
$251$	−1.11705e6	−1.11915	−0.559574	−	0.828781i	$-0.689035\pi$
$251$	−1.11705e6	−1.11915	−0.559574	+	0.828781i	$0.689035\pi$
$252$	0	0
$253$	−351583.	−0.345324
$254$	0	0
$255$	292393.	0.281589
$256$	0	0
$257$	924298.	0.872929	0.436465	−	0.899721i	$-0.356230\pi$
$257$	924298.	0.872929	0.436465	+	0.899721i	$0.356230\pi$
$258$	0	0
$259$	36399.1	0.0337164
$260$	0	0
$261$	−214923.	−0.195290
$262$	0	0
$263$	133242.	0.118782	0.0593911	−	0.998235i	$-0.481084\pi$
$263$	133242.	0.118782	0.0593911	+	0.998235i	$0.481084\pi$
$264$	0	0
$265$	−315788.	−0.276236
$266$	0	0
$267$	−1.06308e6	−0.912617
$268$	0	0
$269$	1.56607e6	1.31956	0.659781	−	0.751458i	$-0.270649\pi$
$269$	1.56607e6	1.31956	0.659781	+	0.751458i	$0.270649\pi$
$270$	0	0
$271$	1.22010e6	1.00919	0.504593	−	0.863357i	$-0.331643\pi$
$271$	1.22010e6	1.00919	0.504593	+	0.863357i	$0.331643\pi$
$272$	0	0
$273$	18358.8	0.0149086
$274$	0	0
$275$	130296.	0.103896
$276$	0	0
$277$	1.86383e6	1.45951	0.729753	−	0.683711i	$-0.239635\pi$
$277$	1.86383e6	1.45951	0.729753	+	0.683711i	$0.239635\pi$
$278$	0	0
$279$	116776.	0.0898140
$280$	0	0
$281$	−848857.	−0.641311	−0.320656	−	0.947196i	$-0.603903\pi$
$281$	−848857.	−0.641311	−0.320656	+	0.947196i	$0.603903\pi$
$282$	0	0
$283$	−478280.	−0.354990	−0.177495	−	0.984122i	$-0.556799\pi$
$283$	−478280.	−0.354990	−0.177495	+	0.984122i	$0.556799\pi$
$284$	0	0
$285$	774721.	0.564980
$286$	0	0
$287$	−39857.5	−0.0285631
$288$	0	0
$289$	−1.14466e6	−0.806177
$290$	0	0
$291$	247474.	0.171315
$292$	0	0
$293$	1.13443e6	0.771984	0.385992	−	0.922502i	$-0.373859\pi$
$293$	1.13443e6	0.771984	0.385992	+	0.922502i	$0.373859\pi$
$294$	0	0
$295$	−42989.5	−0.0287612
$296$	0	0
$297$	−133726.	−0.0879683
$298$	0	0
$299$	−323910.	−0.209530
$300$	0	0
$301$	5509.86	0.00350530
$302$	0	0
$303$	−1.04791e6	−0.655719
$304$	0	0
$305$	−3.41242e6	−2.10045
$306$	0	0
$307$	2.54492e6	1.54109	0.770546	−	0.637384i	$-0.219984\pi$
$307$	2.54492e6	1.54109	0.770546	+	0.637384i	$0.219984\pi$
$308$	0	0
$309$	718941.	0.428348
$310$	0	0
$311$	−508599.	−0.298177	−0.149089	−	0.988824i	$-0.547634\pi$
$311$	−508599.	−0.298177	−0.149089	+	0.988824i	$0.547634\pi$
$312$	0	0
$313$	−1.70498e6	−0.983688	−0.491844	−	0.870683i	$-0.663677\pi$
$313$	−1.70498e6	−0.983688	−0.491844	+	0.870683i	$0.663677\pi$
$314$	0	0
$315$	−60548.0	−0.0343814
$316$	0	0
$317$	−1.13413e6	−0.633892	−0.316946	−	0.948444i	$-0.602657\pi$
$317$	−1.13413e6	−0.633892	−0.316946	+	0.948444i	$0.602657\pi$
$318$	0	0
$319$	−486729.	−0.267800
$320$	0	0
$321$	1.29758e6	0.702865
$322$	0	0
$323$	729169.	0.388886
$324$	0	0
$325$	120040.	0.0630404
$326$	0	0
$327$	−1.26920e6	−0.656388
$328$	0	0
$329$	−255783.	−0.130281
$330$	0	0
$331$	1.47356e6	0.739263	0.369632	−	0.929178i	$-0.379484\pi$
$331$	1.47356e6	0.739263	0.369632	+	0.929178i	$0.379484\pi$
$332$	0	0
$333$	−244265.	−0.120712
$334$	0	0
$335$	1.83334e6	0.892549
$336$	0	0
$337$	2.33105e6	1.11809	0.559045	−	0.829137i	$-0.311168\pi$
$337$	2.33105e6	1.11809	0.559045	+	0.829137i	$0.311168\pi$
$338$	0	0
$339$	−1.64243e6	−0.776226
$340$	0	0
$341$	264460.	0.123161
$342$	0	0
$343$	403970.	0.185402
$344$	0	0
$345$	1.06827e6	0.483206
$346$	0	0
$347$	−4.30515e6	−1.91940	−0.959698	−	0.281033i	$-0.909323\pi$
$347$	−4.30515e6	−1.91940	−0.959698	+	0.281033i	$0.909323\pi$
$348$	0	0
$349$	3.63737e6	1.59854	0.799270	−	0.600972i	$-0.205220\pi$
$349$	3.63737e6	1.59854	0.799270	+	0.600972i	$0.205220\pi$
$350$	0	0
$351$	−123201.	−0.0533761
$352$	0	0
$353$	−2.06196e6	−0.880733	−0.440366	−	0.897818i	$-0.645151\pi$
$353$	−2.06196e6	−0.880733	−0.440366	+	0.897818i	$0.645151\pi$
$354$	0	0
$355$	−1.88092e6	−0.792136
$356$	0	0
$357$	−56987.9	−0.0236653
$358$	0	0
$359$	−2.89637e6	−1.18609	−0.593045	−	0.805170i	$-0.702074\pi$
$359$	−2.89637e6	−1.18609	−0.593045	+	0.805170i	$0.702074\pi$
$360$	0	0
$361$	−544100.	−0.219741
$362$	0	0
$363$	1.14661e6	0.456720
$364$	0	0
$365$	−2.71863e6	−1.06812
$366$	0	0
$367$	−4.37778e6	−1.69664	−0.848318	−	0.529487i	$-0.822384\pi$
$367$	−4.37778e6	−1.69664	−0.848318	+	0.529487i	$0.822384\pi$
$368$	0	0
$369$	267473.	0.102262
$370$	0	0
$371$	61547.5	0.0232154
$372$	0	0
$373$	−53191.7	−0.0197957	−0.00989787	−	0.999951i	$-0.503151\pi$
$373$	−53191.7	−0.0197957	−0.00989787	+	0.999951i	$0.503151\pi$
$374$	0	0
$375$	1.34588e6	0.494228
$376$	0	0
$377$	−448419.	−0.162491
$378$	0	0
$379$	4.56757e6	1.63338	0.816690	−	0.577076i	$-0.195807\pi$
$379$	4.56757e6	1.63338	0.816690	+	0.577076i	$0.195807\pi$
$380$	0	0
$381$	2.20746e6	0.779078
$382$	0	0
$383$	−64196.5	−0.0223622	−0.0111811	−	0.999937i	$-0.503559\pi$
$383$	−64196.5	−0.0223622	−0.0111811	+	0.999937i	$0.503559\pi$
$384$	0	0
$385$	−137121.	−0.0471469
$386$	0	0
$387$	−36975.2	−0.0125497
$388$	0	0
$389$	4.56310e6	1.52892	0.764462	−	0.644668i	$-0.223005\pi$
$389$	4.56310e6	1.52892	0.764462	+	0.644668i	$0.223005\pi$
$390$	0	0
$391$	1.00546e6	0.332599
$392$	0	0
$393$	−635331.	−0.207500
$394$	0	0
$395$	3.91349e6	1.26204
$396$	0	0
$397$	3.16929e6	1.00922	0.504610	−	0.863348i	$-0.331636\pi$
$397$	3.16929e6	1.00922	0.504610	+	0.863348i	$0.331636\pi$
$398$	0	0
$399$	−150994.	−0.0474820
$400$	0	0
$401$	−2.57811e6	−0.800646	−0.400323	−	0.916374i	$-0.631102\pi$
$401$	−2.57811e6	−0.800646	−0.400323	+	0.916374i	$0.631102\pi$
$402$	0	0
$403$	243644.	0.0747298
$404$	0	0
$405$	406321.	0.123093
$406$	0	0
$407$	−553179.	−0.165531
$408$	0	0
$409$	−1.93651e6	−0.572417	−0.286208	−	0.958167i	$-0.592395\pi$
$409$	−1.93651e6	−0.572417	−0.286208	+	0.958167i	$0.592395\pi$
$410$	0	0
$411$	1.74160e6	0.508563
$412$	0	0
$413$	8378.72	0.00241714
$414$	0	0
$415$	−1.24480e6	−0.354798
$416$	0	0
$417$	−148953.	−0.0419477
$418$	0	0
$419$	−5.10997e6	−1.42195	−0.710973	−	0.703219i	$-0.751745\pi$
$419$	−5.10997e6	−1.42195	−0.710973	+	0.703219i	$0.751745\pi$
$420$	0	0
$421$	−1.66272e6	−0.457209	−0.228604	−	0.973519i	$-0.573416\pi$
$421$	−1.66272e6	−0.457209	−0.228604	+	0.973519i	$0.573416\pi$
$422$	0	0
$423$	1.71649e6	0.466434
$424$	0	0
$425$	−372619.	−0.100067
$426$	0	0
$427$	665087.	0.176526
$428$	0	0
$429$	−279010.	−0.0731941
$430$	0	0
$431$	3.28622e6	0.852124	0.426062	−	0.904694i	$-0.359900\pi$
$431$	3.28622e6	0.852124	0.426062	+	0.904694i	$0.359900\pi$
$432$	0	0
$433$	−2.15750e6	−0.553007	−0.276503	−	0.961013i	$-0.589176\pi$
$433$	−2.15750e6	−0.553007	−0.276503	+	0.961013i	$0.589176\pi$
$434$	0	0
$435$	1.47890e6	0.374728
$436$	0	0
$437$	2.66405e6	0.667326
$438$	0	0
$439$	−1.64334e6	−0.406974	−0.203487	−	0.979078i	$-0.565227\pi$
$439$	−1.64334e6	−0.406974	−0.203487	+	0.979078i	$0.565227\pi$
$440$	0	0
$441$	−1.34957e6	−0.330444
$442$	0	0
$443$	7.43121e6	1.79908	0.899539	−	0.436840i	$-0.143902\pi$
$443$	7.43121e6	1.79908	0.899539	+	0.436840i	$0.143902\pi$
$444$	0	0
$445$	7.31516e6	1.75115
$446$	0	0
$447$	4.31191e6	1.02071
$448$	0	0
$449$	−2.76955e6	−0.648326	−0.324163	−	0.946001i	$-0.605083\pi$
$449$	−2.76955e6	−0.648326	−0.324163	+	0.946001i	$0.605083\pi$
$450$	0	0
$451$	605738.	0.140231
$452$	0	0
$453$	2.83994e6	0.650224
$454$	0	0
$455$	−126329.	−0.0286070
$456$	0	0
$457$	1.54160e6	0.345287	0.172644	−	0.984984i	$-0.444769\pi$
$457$	1.54160e6	0.345287	0.172644	+	0.984984i	$0.444769\pi$
$458$	0	0
$459$	382430.	0.0847267
$460$	0	0
$461$	−4.78552e6	−1.04876	−0.524381	−	0.851484i	$-0.675703\pi$
$461$	−4.78552e6	−1.04876	−0.524381	+	0.851484i	$0.675703\pi$
$462$	0	0
$463$	8.73387e6	1.89345	0.946726	−	0.322041i	$-0.104369\pi$
$463$	8.73387e6	1.89345	0.946726	+	0.322041i	$0.104369\pi$
$464$	0	0
$465$	−803548.	−0.172337
$466$	0	0
$467$	−448252.	−0.0951109	−0.0475554	−	0.998869i	$-0.515143\pi$
$467$	−448252.	−0.0951109	−0.0475554	+	0.998869i	$0.515143\pi$
$468$	0	0
$469$	−357322.	−0.0750114
$470$	0	0
$471$	4.86688e6	1.01088
$472$	0	0
$473$	−83736.7	−0.0172093
$474$	0	0
$475$	−987288.	−0.200775
$476$	0	0
$477$	−413029.	−0.0831160
$478$	0	0
$479$	9.23494e6	1.83906	0.919529	−	0.393022i	$-0.128570\pi$
$479$	9.23494e6	1.83906	0.919529	+	0.393022i	$0.128570\pi$
$480$	0	0
$481$	−509639.	−0.100438
$482$	0	0
$483$	−208207.	−0.0406095
$484$	0	0
$485$	−1.70289e6	−0.328724
$486$	0	0
$487$	−4.87963e6	−0.932318	−0.466159	−	0.884701i	$-0.654363\pi$
$487$	−4.87963e6	−0.932318	−0.466159	+	0.884701i	$0.654363\pi$
$488$	0	0
$489$	−730971.	−0.138238
$490$	0	0
$491$	−576710.	−0.107958	−0.0539789	−	0.998542i	$-0.517190\pi$
$491$	−576710.	−0.107958	−0.0539789	+	0.998542i	$0.517190\pi$
$492$	0	0
$493$	1.39194e6	0.257932
$494$	0	0
$495$	920184.	0.168796
$496$	0	0
$497$	366595.	0.0665726
$498$	0	0
$499$	1.34239e6	0.241339	0.120669	−	0.992693i	$-0.461496\pi$
$499$	1.34239e6	0.241339	0.120669	+	0.992693i	$0.461496\pi$
$500$	0	0
$501$	3.61277e6	0.643052
$502$	0	0
$503$	−17469.0	−0.00307857	−0.00153928	−	0.999999i	$-0.500490\pi$
$503$	−17469.0	−0.00307857	−0.00153928	+	0.999999i	$0.500490\pi$
$504$	0	0
$505$	7.21076e6	1.25821
$506$	0	0
$507$	−257049.	−0.0444116
$508$	0	0
$509$	6.23272e6	1.06631	0.533155	−	0.846018i	$-0.321006\pi$
$509$	6.23272e6	1.06631	0.533155	+	0.846018i	$0.321006\pi$
$510$	0	0
$511$	529866.	0.0897664
$512$	0	0
$513$	1.01328e6	0.169996
$514$	0	0
$515$	−4.94710e6	−0.821925
$516$	0	0
$517$	3.88728e6	0.639616
$518$	0	0
$519$	2.30824e6	0.376152
$520$	0	0
$521$	−6.45233e6	−1.04141	−0.520705	−	0.853736i	$-0.674331\pi$
$521$	−6.45233e6	−1.04141	−0.520705	+	0.853736i	$0.674331\pi$
$522$	0	0
$523$	−3.39647e6	−0.542967	−0.271483	−	0.962443i	$-0.587514\pi$
$523$	−3.39647e6	−0.542967	−0.271483	+	0.962443i	$0.587514\pi$
$524$	0	0
$525$	77161.0	0.0122180
$526$	0	0
$527$	−756300.	−0.118623
$528$	0	0
$529$	−2.76287e6	−0.429262
$530$	0	0
$531$	−56227.3	−0.00865388
$532$	0	0
$533$	558060.	0.0850870
$534$	0	0
$535$	−8.92877e6	−1.34868
$536$	0	0
$537$	5.25670e6	0.786643
$538$	0	0
$539$	−3.05632e6	−0.453134
$540$	0	0
$541$	−7.37387e6	−1.08318	−0.541592	−	0.840641i	$-0.682178\pi$
$541$	−7.37387e6	−1.08318	−0.541592	+	0.840641i	$0.682178\pi$
$542$	0	0
$543$	1.53501e6	0.223414
$544$	0	0
$545$	8.73347e6	1.25949
$546$	0	0
$547$	6.57677e6	0.939820	0.469910	−	0.882714i	$-0.344286\pi$
$547$	6.57677e6	0.939820	0.469910	+	0.882714i	$0.344286\pi$
$548$	0	0
$549$	−4.46322e6	−0.632001
$550$	0	0
$551$	3.68808e6	0.517513
$552$	0	0
$553$	−762746.	−0.106064
$554$	0	0
$555$	1.68081e6	0.231625
$556$	0	0
$557$	7.51699e6	1.02661	0.513305	−	0.858206i	$-0.328421\pi$
$557$	7.51699e6	1.02661	0.513305	+	0.858206i	$0.328421\pi$
$558$	0	0
$559$	−77145.8	−0.0104420
$560$	0	0
$561$	866078.	0.116185
$562$	0	0
$563$	−1.09315e7	−1.45347	−0.726737	−	0.686916i	$-0.758964\pi$
$563$	−1.09315e7	−1.45347	−0.726737	+	0.686916i	$0.758964\pi$
$564$	0	0
$565$	1.13017e7	1.48944
$566$	0	0
$567$	−79192.7	−0.0103449
$568$	0	0
$569$	−3.95020e6	−0.511491	−0.255746	−	0.966744i	$-0.582321\pi$
$569$	−3.95020e6	−0.511491	−0.255746	+	0.966744i	$0.582321\pi$
$570$	0	0
$571$	1.15680e7	1.48480	0.742399	−	0.669958i	$-0.233688\pi$
$571$	1.15680e7	1.48480	0.742399	+	0.669958i	$0.233688\pi$
$572$	0	0
$573$	4.61274e6	0.586911
$574$	0	0
$575$	−1.36138e6	−0.171715
$576$	0	0
$577$	3.91551e6	0.489609	0.244804	−	0.969572i	$-0.421276\pi$
$577$	3.91551e6	0.489609	0.244804	+	0.969572i	$0.421276\pi$
$578$	0	0
$579$	−1.98121e6	−0.245603
$580$	0	0
$581$	242614.	0.0298178
$582$	0	0
$583$	−935374.	−0.113976
$584$	0	0
$585$	847757.	0.102419
$586$	0	0
$587$	−1.47390e7	−1.76552	−0.882762	−	0.469820i	$-0.844319\pi$
$587$	−1.47390e7	−1.76552	−0.882762	+	0.469820i	$0.844319\pi$
$588$	0	0
$589$	−2.00389e6	−0.238004
$590$	0	0
$591$	−795506.	−0.0936861
$592$	0	0
$593$	3.49709e6	0.408386	0.204193	−	0.978931i	$-0.434543\pi$
$593$	3.49709e6	0.408386	0.204193	+	0.978931i	$0.434543\pi$
$594$	0	0
$595$	392138.	0.0454095
$596$	0	0
$597$	8.22719e6	0.944747
$598$	0	0
$599$	−1.14482e7	−1.30368	−0.651842	−	0.758355i	$-0.726003\pi$
$599$	−1.14482e7	−1.30368	−0.651842	+	0.758355i	$0.726003\pi$
$600$	0	0
$601$	755669.	0.0853386	0.0426693	−	0.999089i	$-0.486414\pi$
$601$	755669.	0.0853386	0.0426693	+	0.999089i	$0.486414\pi$
$602$	0	0
$603$	2.39789e6	0.268557
$604$	0	0
$605$	−7.88994e6	−0.876366
$606$	0	0
$607$	−165519.	−0.0182337	−0.00911687	−	0.999958i	$-0.502902\pi$
$607$	−165519.	−0.0182337	−0.00911687	+	0.999958i	$0.502902\pi$
$608$	0	0
$609$	−288240.	−0.0314928
$610$	0	0
$611$	3.58131e6	0.388096
$612$	0	0
$613$	8.97746e6	0.964945	0.482472	−	0.875911i	$-0.339739\pi$
$613$	8.97746e6	0.964945	0.482472	+	0.875911i	$0.339739\pi$
$614$	0	0
$615$	−1.84050e6	−0.196223
$616$	0	0
$617$	2.13304e6	0.225573	0.112786	−	0.993619i	$-0.464022\pi$
$617$	2.13304e6	0.225573	0.112786	+	0.993619i	$0.464022\pi$
$618$	0	0
$619$	1.55236e7	1.62841	0.814206	−	0.580576i	$-0.197172\pi$
$619$	1.55236e7	1.62841	0.814206	+	0.580576i	$0.197172\pi$
$620$	0	0
$621$	1.39722e6	0.145391
$622$	0	0
$623$	−1.42574e6	−0.147170
$624$	0	0
$625$	−1.14808e7	−1.17563
$626$	0	0
$627$	2.29475e6	0.233113
$628$	0	0
$629$	1.58198e6	0.159431
$630$	0	0
$631$	1.53539e7	1.53513	0.767567	−	0.640969i	$-0.221467\pi$
$631$	1.53539e7	1.53513	0.767567	+	0.640969i	$0.221467\pi$
$632$	0	0
$633$	2.63727e6	0.261604
$634$	0	0
$635$	−1.51897e7	−1.49491
$636$	0	0
$637$	−2.81576e6	−0.274946
$638$	0	0
$639$	−2.46012e6	−0.238344
$640$	0	0
$641$	−1.61787e7	−1.55524	−0.777621	−	0.628734i	$-0.783574\pi$
$641$	−1.61787e7	−1.55524	−0.777621	+	0.628734i	$0.783574\pi$
$642$	0	0
$643$	−1.13012e7	−1.07795	−0.538973	−	0.842323i	$-0.681188\pi$
$643$	−1.13012e7	−1.07795	−0.538973	+	0.842323i	$0.681188\pi$
$644$	0	0
$645$	254430.	0.0240807
$646$	0	0
$647$	4.91218e6	0.461332	0.230666	−	0.973033i	$-0.425910\pi$
$647$	4.91218e6	0.461332	0.230666	+	0.973033i	$0.425910\pi$
$648$	0	0
$649$	−127336.	−0.0118670
$650$	0	0
$651$	156613.	0.0144835
$652$	0	0
$653$	−1.27515e7	−1.17025	−0.585123	−	0.810944i	$-0.698954\pi$
$653$	−1.27515e7	−1.17025	−0.585123	+	0.810944i	$0.698954\pi$
$654$	0	0
$655$	4.37177e6	0.398157
$656$	0	0
$657$	−3.55579e6	−0.321383
$658$	0	0
$659$	−4.57515e6	−0.410385	−0.205193	−	0.978722i	$-0.565782\pi$
$659$	−4.57515e6	−0.410385	−0.205193	+	0.978722i	$0.565782\pi$
$660$	0	0
$661$	−1.94596e6	−0.173233	−0.0866164	−	0.996242i	$-0.527605\pi$
$661$	−1.94596e6	−0.173233	−0.0866164	+	0.996242i	$0.527605\pi$
$662$	0	0
$663$	797910.	0.0704969
$664$	0	0
$665$	1.03901e6	0.0911096
$666$	0	0
$667$	5.08552e6	0.442609
$668$	0	0
$669$	−3.73200e6	−0.322386
$670$	0	0
$671$	−1.01077e7	−0.866656
$672$	0	0
$673$	824071.	0.0701337	0.0350669	−	0.999385i	$-0.488836\pi$
$673$	824071.	0.0701337	0.0350669	+	0.999385i	$0.488836\pi$
$674$	0	0
$675$	−517807.	−0.0437430
$676$	0	0
$677$	1.06076e7	0.889496	0.444748	−	0.895656i	$-0.353293\pi$
$677$	1.06076e7	0.889496	0.444748	+	0.895656i	$0.353293\pi$
$678$	0	0
$679$	331896.	0.0276266
$680$	0	0
$681$	−6.38312e6	−0.527430
$682$	0	0
$683$	1.54948e7	1.27097	0.635483	−	0.772115i	$-0.280801\pi$
$683$	1.54948e7	1.27097	0.635483	+	0.772115i	$0.280801\pi$
$684$	0	0
$685$	−1.19841e7	−0.975842
$686$	0	0
$687$	4.08898e6	0.330539
$688$	0	0
$689$	−861752.	−0.0691567
$690$	0	0
$691$	−2.90507e6	−0.231452	−0.115726	−	0.993281i	$-0.536919\pi$
$691$	−2.90507e6	−0.231452	−0.115726	+	0.993281i	$0.536919\pi$
$692$	0	0
$693$	−179345.	−0.0141859
$694$	0	0
$695$	1.02496e6	0.0804903
$696$	0	0
$697$	−1.73228e6	−0.135063
$698$	0	0
$699$	3.12642e6	0.242022
$700$	0	0
$701$	8.10828e6	0.623209	0.311605	−	0.950212i	$-0.399134\pi$
$701$	8.10828e6	0.623209	0.311605	+	0.950212i	$0.399134\pi$
$702$	0	0
$703$	4.19159e6	0.319883
$704$	0	0
$705$	−1.18113e7	−0.895004
$706$	0	0
$707$	−1.40539e6	−0.105742
$708$	0	0
$709$	−1.64144e7	−1.22634	−0.613168	−	0.789953i	$-0.710105\pi$
$709$	−1.64144e7	−1.22634	−0.613168	+	0.789953i	$0.710105\pi$
$710$	0	0
$711$	5.11858e6	0.379731
$712$	0	0
$713$	−2.76317e6	−0.203556
$714$	0	0
$715$	1.91989e6	0.140447
$716$	0	0
$717$	4.52327e6	0.328590
$718$	0	0
$719$	−6.27334e6	−0.452561	−0.226280	−	0.974062i	$-0.572657\pi$
$719$	−6.27334e6	−0.452561	−0.226280	+	0.974062i	$0.572657\pi$
$720$	0	0
$721$	964198.	0.0690761
$722$	0	0
$723$	−7.97254e6	−0.567219
$724$	0	0
$725$	−1.88468e6	−0.133166
$726$	0	0
$727$	−6.27335e6	−0.440214	−0.220107	−	0.975476i	$-0.570641\pi$
$727$	−6.27335e6	−0.440214	−0.220107	+	0.975476i	$0.570641\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	239470.	0.0165751
$732$	0	0
$733$	2.38673e7	1.64075	0.820376	−	0.571825i	$-0.193764\pi$
$733$	2.38673e7	1.64075	0.820376	+	0.571825i	$0.193764\pi$
$734$	0	0
$735$	9.28648e6	0.634064
$736$	0	0
$737$	5.43043e6	0.368269
$738$	0	0
$739$	1.74901e7	1.17810	0.589050	−	0.808097i	$-0.299502\pi$
$739$	1.74901e7	1.17810	0.589050	+	0.808097i	$0.299502\pi$
$740$	0	0
$741$	2.11413e6	0.141445
$742$	0	0
$743$	−1.76450e7	−1.17260	−0.586299	−	0.810095i	$-0.699416\pi$
$743$	−1.76450e7	−1.17260	−0.586299	+	0.810095i	$0.699416\pi$
$744$	0	0
$745$	−2.96706e7	−1.95856
$746$	0	0
$747$	−1.62812e6	−0.106754
$748$	0	0
$749$	1.74023e6	0.113345
$750$	0	0
$751$	2.66005e7	1.72104	0.860519	−	0.509418i	$-0.170139\pi$
$751$	2.66005e7	1.72104	0.860519	+	0.509418i	$0.170139\pi$
$752$	0	0
$753$	1.00534e7	0.646140
$754$	0	0
$755$	−1.95418e7	−1.24767
$756$	0	0
$757$	8.15204e6	0.517043	0.258521	−	0.966006i	$-0.416765\pi$
$757$	8.15204e6	0.517043	0.258521	+	0.966006i	$0.416765\pi$
$758$	0	0
$759$	3.16425e6	0.199373
$760$	0	0
$761$	−2.70623e7	−1.69396	−0.846978	−	0.531627i	$-0.821581\pi$
$761$	−2.70623e7	−1.69396	−0.846978	+	0.531627i	$0.821581\pi$
$762$	0	0
$763$	−1.70217e6	−0.105850
$764$	0	0
$765$	−2.63154e6	−0.162576
$766$	0	0
$767$	−117314.	−0.00720047
$768$	0	0
$769$	3.05350e7	1.86201	0.931004	−	0.365009i	$-0.118934\pi$
$769$	3.05350e7	1.86201	0.931004	+	0.365009i	$0.118934\pi$
$770$	0	0
$771$	−8.31868e6	−0.503986
$772$	0	0
$773$	−1.86484e7	−1.12252	−0.561259	−	0.827640i	$-0.689683\pi$
$773$	−1.86484e7	−1.12252	−0.561259	+	0.827640i	$0.689683\pi$
$774$	0	0
$775$	1.02402e6	0.0612429
$776$	0	0
$777$	−327592.	−0.0194662
$778$	0	0
$779$	−4.58984e6	−0.270991
$780$	0	0
$781$	−5.57136e6	−0.326839
$782$	0	0
$783$	1.93430e6	0.112751
$784$	0	0
$785$	−3.34894e7	−1.93969
$786$	0	0
$787$	−1.71709e7	−0.988225	−0.494113	−	0.869398i	$-0.664507\pi$
$787$	−1.71709e7	−0.988225	−0.494113	+	0.869398i	$0.664507\pi$
$788$	0	0
$789$	−1.19918e6	−0.0685789
$790$	0	0
$791$	−2.20272e6	−0.125175
$792$	0	0
$793$	−9.31215e6	−0.525856
$794$	0	0
$795$	2.84209e6	0.159485
$796$	0	0
$797$	1.22037e7	0.680526	0.340263	−	0.940330i	$-0.389484\pi$
$797$	1.22037e7	0.680526	0.340263	+	0.940330i	$0.389484\pi$
$798$	0	0
$799$	−1.11168e7	−0.616046
$800$	0	0
$801$	9.56774e6	0.526900
$802$	0	0
$803$	−8.05269e6	−0.440709
$804$	0	0
$805$	1.43269e6	0.0779225
$806$	0	0
$807$	−1.40946e7	−0.761850
$808$	0	0
$809$	−1.20229e7	−0.645862	−0.322931	−	0.946423i	$-0.604668\pi$
$809$	−1.20229e7	−0.645862	−0.322931	+	0.946423i	$0.604668\pi$
$810$	0	0
$811$	−1.73884e7	−0.928339	−0.464169	−	0.885747i	$-0.653647\pi$
$811$	−1.73884e7	−0.928339	−0.464169	+	0.885747i	$0.653647\pi$
$812$	0	0
$813$	−1.09809e7	−0.582654
$814$	0	0
$815$	5.02988e6	0.265255
$816$	0	0
$817$	634496.	0.0332563
$818$	0	0
$819$	−165229.	−0.00860750
$820$	0	0
$821$	865202.	0.0447981	0.0223990	−	0.999749i	$-0.492870\pi$
$821$	865202.	0.0447981	0.0223990	+	0.999749i	$0.492870\pi$
$822$	0	0
$823$	−2.06811e7	−1.06432	−0.532162	−	0.846643i	$-0.678620\pi$
$823$	−2.06811e7	−1.06432	−0.532162	+	0.846643i	$0.678620\pi$
$824$	0	0
$825$	−1.17266e6	−0.0599844
$826$	0	0
$827$	−9.77186e6	−0.496836	−0.248418	−	0.968653i	$-0.579911\pi$
$827$	−9.77186e6	−0.496836	−0.248418	+	0.968653i	$0.579911\pi$
$828$	0	0
$829$	1.48880e7	0.752404	0.376202	−	0.926538i	$-0.377230\pi$
$829$	1.48880e7	0.752404	0.376202	+	0.926538i	$0.377230\pi$
$830$	0	0
$831$	−1.67744e7	−0.842647
$832$	0	0
$833$	8.74045e6	0.436437
$834$	0	0
$835$	−2.48598e7	−1.23390
$836$	0	0
$837$	−1.05099e6	−0.0518542
$838$	0	0
$839$	4.00819e6	0.196582	0.0982908	−	0.995158i	$-0.468662\pi$
$839$	4.00819e6	0.196582	0.0982908	+	0.995158i	$0.468662\pi$
$840$	0	0
$841$	−1.34708e7	−0.656755
$842$	0	0
$843$	7.63972e6	0.370261
$844$	0	0
$845$	1.76878e6	0.0852180
$846$	0	0
$847$	1.53776e6	0.0736514
$848$	0	0
$849$	4.30452e6	0.204953
$850$	0	0
$851$	5.77981e6	0.273583
$852$	0	0
$853$	2.59920e7	1.22312	0.611558	−	0.791200i	$-0.290543\pi$
$853$	2.59920e7	1.22312	0.611558	+	0.791200i	$0.290543\pi$
$854$	0	0
$855$	−6.97249e6	−0.326192
$856$	0	0
$857$	−2.99606e7	−1.39347	−0.696736	−	0.717327i	$-0.745365\pi$
$857$	−2.99606e7	−1.39347	−0.696736	+	0.717327i	$0.745365\pi$
$858$	0	0
$859$	−3.34024e6	−0.154452	−0.0772261	−	0.997014i	$-0.524606\pi$
$859$	−3.34024e6	−0.154452	−0.0772261	+	0.997014i	$0.524606\pi$
$860$	0	0
$861$	358717.	0.0164909
$862$	0	0
$863$	2.95313e7	1.34976	0.674879	−	0.737928i	$-0.264196\pi$
$863$	2.95313e7	1.34976	0.674879	+	0.737928i	$0.264196\pi$
$864$	0	0
$865$	−1.58832e7	−0.721770
$866$	0	0
$867$	1.03019e7	0.465447
$868$	0	0
$869$	1.15919e7	0.520721
$870$	0	0
$871$	5.00301e6	0.223453
$872$	0	0
$873$	−2.22726e6	−0.0989090
$874$	0	0
$875$	1.80500e6	0.0797000
$876$	0	0
$877$	3.82361e7	1.67871	0.839353	−	0.543587i	$-0.182934\pi$
$877$	3.82361e7	1.67871	0.839353	+	0.543587i	$0.182934\pi$
$878$	0	0
$879$	−1.02099e7	−0.445705
$880$	0	0
$881$	−2.68726e7	−1.16646	−0.583229	−	0.812308i	$-0.698211\pi$
$881$	−2.68726e7	−1.16646	−0.583229	+	0.812308i	$0.698211\pi$
$882$	0	0
$883$	−1.63532e7	−0.705832	−0.352916	−	0.935655i	$-0.614810\pi$
$883$	−1.63532e7	−0.705832	−0.352916	+	0.935655i	$0.614810\pi$
$884$	0	0
$885$	386905.	0.0166053
$886$	0	0
$887$	1.48121e6	0.0632133	0.0316066	−	0.999500i	$-0.489938\pi$
$887$	1.48121e6	0.0632133	0.0316066	+	0.999500i	$0.489938\pi$
$888$	0	0
$889$	2.96051e6	0.125635
$890$	0	0
$891$	1.20354e6	0.0507885
$892$	0	0
$893$	−2.94550e7	−1.23603
$894$	0	0
$895$	−3.61718e7	−1.50943
$896$	0	0
$897$	2.91519e6	0.120972
$898$	0	0
$899$	−3.82531e6	−0.157858
$900$	0	0
$901$	2.67498e6	0.109776
$902$	0	0
$903$	−49588.8	−0.00202378
$904$	0	0
$905$	−1.05625e7	−0.428693
$906$	0	0
$907$	1.14259e7	0.461181	0.230590	−	0.973051i	$-0.425934\pi$
$907$	1.14259e7	0.461181	0.230590	+	0.973051i	$0.425934\pi$
$908$	0	0
$909$	9.43119e6	0.378579
$910$	0	0
$911$	−3.27578e7	−1.30773	−0.653866	−	0.756610i	$-0.726854\pi$
$911$	−3.27578e7	−1.30773	−0.653866	+	0.756610i	$0.726854\pi$
$912$	0	0
$913$	−3.68715e6	−0.146391
$914$	0	0
$915$	3.07118e7	1.21270
$916$	0	0
$917$	−852065.	−0.0334618
$918$	0	0
$919$	9.44857e6	0.369043	0.184522	−	0.982828i	$-0.440926\pi$
$919$	9.44857e6	0.369043	0.184522	+	0.982828i	$0.440926\pi$
$920$	0	0
$921$	−2.29043e7	−0.889750
$922$	0	0
$923$	−5.13284e6	−0.198314
$924$	0	0
$925$	−2.14198e6	−0.0823117
$926$	0	0
$927$	−6.47047e6	−0.247307
$928$	0	0
$929$	−1.64252e7	−0.624413	−0.312207	−	0.950014i	$-0.601068\pi$
$929$	−1.64252e7	−0.624413	−0.312207	+	0.950014i	$0.601068\pi$
$930$	0	0
$931$	2.31586e7	0.875666
$932$	0	0
$933$	4.57739e6	0.172153
$934$	0	0
$935$	−5.95956e6	−0.222939
$936$	0	0
$937$	−2.32993e7	−0.866948	−0.433474	−	0.901166i	$-0.642712\pi$
$937$	−2.32993e7	−0.866948	−0.433474	+	0.901166i	$0.642712\pi$
$938$	0	0
$939$	1.53448e7	0.567933
$940$	0	0
$941$	−9.27042e6	−0.341291	−0.170646	−	0.985332i	$-0.554585\pi$
$941$	−9.27042e6	−0.341291	−0.170646	+	0.985332i	$0.554585\pi$
$942$	0	0
$943$	−6.32897e6	−0.231768
$944$	0	0
$945$	544932.	0.0198501
$946$	0	0
$947$	1.92640e7	0.698025	0.349012	−	0.937118i	$-0.386517\pi$
$947$	1.92640e7	0.698025	0.349012	+	0.937118i	$0.386517\pi$
$948$	0	0
$949$	−7.41887e6	−0.267407
$950$	0	0
$951$	1.02072e7	0.365977
$952$	0	0
$953$	2.81599e7	1.00438	0.502191	−	0.864756i	$-0.332527\pi$
$953$	2.81599e7	1.00438	0.502191	+	0.864756i	$0.332527\pi$
$954$	0	0
$955$	−3.17407e7	−1.12618
$956$	0	0
$957$	4.38056e6	0.154614
$958$	0	0
$959$	2.33572e6	0.0820116
$960$	0	0
$961$	−2.65507e7	−0.927401
$962$	0	0
$963$	−1.16782e7	−0.405800
$964$	0	0
$965$	1.36329e7	0.471269
$966$	0	0
$967$	−3.67528e7	−1.26393	−0.631967	−	0.774996i	$-0.717752\pi$
$967$	−3.67528e7	−1.26393	−0.631967	+	0.774996i	$0.717752\pi$
$968$	0	0
$969$	−6.56252e6	−0.224523
$970$	0	0
$971$	5.55259e7	1.88994	0.944970	−	0.327158i	$-0.106091\pi$
$971$	5.55259e7	1.88994	0.944970	+	0.327158i	$0.106091\pi$
$972$	0	0
$973$	−199766.	−0.00676455
$974$	0	0
$975$	−1.08036e6	−0.0363964
$976$	0	0
$977$	1.59739e7	0.535394	0.267697	−	0.963503i	$-0.413737\pi$
$977$	1.59739e7	0.535394	0.267697	+	0.963503i	$0.413737\pi$
$978$	0	0
$979$	2.16678e7	0.722532
$980$	0	0
$981$	1.14228e7	0.378966
$982$	0	0
$983$	4.69856e7	1.55089	0.775446	−	0.631414i	$-0.217525\pi$
$983$	4.69856e7	1.55089	0.775446	+	0.631414i	$0.217525\pi$
$984$	0	0
$985$	5.47395e6	0.179767
$986$	0	0
$987$	2.30204e6	0.0752178
$988$	0	0
$989$	874911.	0.0284429
$990$	0	0
$991$	−3.05086e7	−0.986821	−0.493411	−	0.869797i	$-0.664250\pi$
$991$	−3.05086e7	−0.986821	−0.493411	+	0.869797i	$0.664250\pi$
$992$	0	0
$993$	−1.32621e7	−0.426814
$994$	0	0
$995$	−5.66120e7	−1.81280
$996$	0	0
$997$	1.52773e7	0.486754	0.243377	−	0.969932i	$-0.421745\pi$
$997$	1.52773e7	0.486754	0.243377	+	0.969932i	$0.421745\pi$
$998$	0	0
$999$	2.19838e6	0.0696930

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.6.a.j.1.2		2
4.3	odd	2		78.6.a.h.1.2	✓	2
12.11	even	2		234.6.a.i.1.1		2
52.51	odd	2		1014.6.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.6.a.h.1.2	✓	2	4.3	odd	2
234.6.a.i.1.1		2	12.11	even	2
624.6.a.j.1.2		2	1.1	even	1	trivial
1014.6.a.i.1.1		2	52.51	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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +61.9298 q^{5} -12.0702 q^{7} +81.0000 q^{9} +183.438 q^{11} +169.000 q^{13} -557.368 q^{15} -524.596 q^{17} -1389.96 q^{19} +108.632 q^{21} -1916.63 q^{23} +710.298 q^{25} -729.000 q^{27} -2653.37 q^{29} +1441.68 q^{31} -1650.94 q^{33} -747.506 q^{35} -3015.61 q^{37} -1521.00 q^{39} +3302.13 q^{41} -456.484 q^{43} +5016.31 q^{45} +21191.2 q^{47} -16661.3 q^{49} +4721.36 q^{51} -5099.12 q^{53} +11360.3 q^{55} +12509.7 q^{57} -694.164 q^{59} -55101.5 q^{61} -977.688 q^{63} +10466.1 q^{65} +29603.6 q^{67} +17249.7 q^{69} -30371.8 q^{71} -43898.6 q^{73} -6392.68 q^{75} -2214.14 q^{77} +63192.4 q^{79} +6561.00 q^{81} -20100.2 q^{83} -32488.1 q^{85} +23880.3 q^{87} +118120. q^{89} -2039.87 q^{91} -12975.1 q^{93} -86080.1 q^{95} -27497.1 q^{97} +14858.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{3} + 10 q^{5} - 138 q^{7} + 162 q^{9} - 544 q^{11} + 338 q^{13} - 90 q^{15} + 1228 q^{17} + 522 q^{19} + 1242 q^{21} + 1632 q^{23} + 282 q^{25} - 1458 q^{27} + 2208 q^{29} - 874 q^{31} + 4896 q^{33}+ \cdots - 44064 q^{99}+O(q^{100}) \) 2 * q - 18 * q^3 + 10 * q^5 - 138 * q^7 + 162 * q^9 - 544 * q^11 + 338 * q^13 - 90 * q^15 + 1228 * q^17 + 522 * q^19 + 1242 * q^21 + 1632 * q^23 + 282 * q^25 - 1458 * q^27 + 2208 * q^29 - 874 * q^31 + 4896 * q^33 + 5792 * q^35 - 2160 * q^37 - 3042 * q^39 - 12638 * q^41 + 17760 * q^43 + 810 * q^45 - 4300 * q^47 - 17610 * q^49 - 11052 * q^51 - 12020 * q^53 + 49136 * q^55 - 4698 * q^57 + 27532 * q^59 - 38016 * q^61 - 11178 * q^63 + 1690 * q^65 + 90974 * q^67 - 14688 * q^69 + 4384 * q^71 - 95312 * q^73 - 2538 * q^75 + 89392 * q^77 + 62168 * q^79 + 13122 * q^81 - 137892 * q^83 - 123500 * q^85 - 19872 * q^87 + 142534 * q^89 - 23322 * q^91 + 7866 * q^93 - 185368 * q^95 + 4896 * q^97 - 44064 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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