LMFDB - Embedded newform 84.6.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,6,Mod(1,84)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("84.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	84.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$13.4722408643$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{505}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 126 $ x^2 - x - 126
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-10.7361$ of defining polynomial
Character	$\chi$	$=$	84.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} +106.417 q^{5} +49.0000 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} +106.417 q^{5} +49.0000 q^{7} +81.0000 q^{9} -250.083 q^{11} -300.500 q^{13} +957.750 q^{15} +2021.92 q^{17} -2251.00 q^{19} +441.000 q^{21} +3092.75 q^{23} +8199.50 q^{25} +729.000 q^{27} -6603.66 q^{29} +833.002 q^{31} -2250.75 q^{33} +5214.41 q^{35} +8954.50 q^{37} -2704.50 q^{39} -7206.58 q^{41} +14446.0 q^{43} +8619.75 q^{45} -17968.2 q^{47} +2401.00 q^{49} +18197.2 q^{51} -15810.5 q^{53} -26613.0 q^{55} -20259.0 q^{57} -26710.5 q^{59} -26799.0 q^{61} +3969.00 q^{63} -31978.2 q^{65} -44494.5 q^{67} +27834.7 q^{69} -20414.1 q^{71} +38742.5 q^{73} +73795.5 q^{75} -12254.1 q^{77} -67211.5 q^{79} +6561.00 q^{81} -35850.7 q^{83} +215165. q^{85} -59433.0 q^{87} +106267. q^{89} -14724.5 q^{91} +7497.02 q^{93} -239544. q^{95} +98133.5 q^{97} -20256.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18 q^{3} + 78 q^{5} + 98 q^{7} + 162 q^{9}+O(q^{10}) $ 2 * q + 18 * q^3 + 78 * q^5 + 98 * q^7 + 162 * q^9	$ 2 q + 18 q^{3} + 78 q^{5} + 98 q^{7} + 162 q^{9} + 174 q^{11} + 208 q^{13} + 702 q^{15} + 1482 q^{17} + 352 q^{19} + 882 q^{21} + 3354 q^{23} + 5882 q^{25} + 1458 q^{27} + 276 q^{29} + 6520 q^{31} + 1566 q^{33} + 3822 q^{35} + 13864 q^{37} + 1872 q^{39} - 12930 q^{41} + 12712 q^{43} + 6318 q^{45} - 28116 q^{47} + 4802 q^{49} + 13338 q^{51} - 46992 q^{53} - 38664 q^{55} + 3168 q^{57} - 65556 q^{59} - 13148 q^{61} + 7938 q^{63} - 46428 q^{65} - 75236 q^{67} + 30186 q^{69} - 66042 q^{71} + 60496 q^{73} + 52938 q^{75} + 8526 q^{77} - 34916 q^{79} + 13122 q^{81} - 82488 q^{83} + 230508 q^{85} + 2484 q^{87} + 42510 q^{89} + 10192 q^{91} + 58680 q^{93} - 313512 q^{95} + 213256 q^{97} + 14094 q^{99}+O(q^{100}) $ 2 * q + 18 * q^3 + 78 * q^5 + 98 * q^7 + 162 * q^9 + 174 * q^11 + 208 * q^13 + 702 * q^15 + 1482 * q^17 + 352 * q^19 + 882 * q^21 + 3354 * q^23 + 5882 * q^25 + 1458 * q^27 + 276 * q^29 + 6520 * q^31 + 1566 * q^33 + 3822 * q^35 + 13864 * q^37 + 1872 * q^39 - 12930 * q^41 + 12712 * q^43 + 6318 * q^45 - 28116 * q^47 + 4802 * q^49 + 13338 * q^51 - 46992 * q^53 - 38664 * q^55 + 3168 * q^57 - 65556 * q^59 - 13148 * q^61 + 7938 * q^63 - 46428 * q^65 - 75236 * q^67 + 30186 * q^69 - 66042 * q^71 + 60496 * q^73 + 52938 * q^75 + 8526 * q^77 - 34916 * q^79 + 13122 * q^81 - 82488 * q^83 + 230508 * q^85 + 2484 * q^87 + 42510 * q^89 + 10192 * q^91 + 58680 * q^93 - 313512 * q^95 + 213256 * q^97 + 14094 * q^99

Display 100 coefficients 10 coefficients

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
$5$	106.417	1.90364	0.951819	−	0.306660i	$-0.0992114\pi$
$5$	106.417	1.90364	0.951819	+	0.306660i	$0.0992114\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−250.083	−0.623164	−0.311582	−	0.950219i	$-0.600859\pi$
$11$	−250.083	−0.623164	−0.311582	+	0.950219i	$0.600859\pi$
$12$	0	0
$13$	−300.500	−0.493158	−0.246579	−	0.969123i	$-0.579306\pi$
$13$	−300.500	−0.493158	−0.246579	+	0.969123i	$0.579306\pi$
$14$	0	0
$15$	957.750	1.09907
$16$	0	0
$17$	2021.92	1.69684	0.848420	−	0.529324i	$-0.177554\pi$
$17$	2021.92	1.69684	0.848420	+	0.529324i	$0.177554\pi$
$18$	0	0
$19$	−2251.00	−1.43051	−0.715255	−	0.698863i	$-0.753690\pi$
$19$	−2251.00	−1.43051	−0.715255	+	0.698863i	$0.753690\pi$
$20$	0	0
$21$	441.000	0.218218
$22$	0	0
$23$	3092.75	1.21906	0.609530	−	0.792763i	$-0.291358\pi$
$23$	3092.75	1.21906	0.609530	+	0.792763i	$0.291358\pi$
$24$	0	0
$25$	8199.50	2.62384
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−6603.66	−1.45811	−0.729054	−	0.684456i	$-0.760040\pi$
$29$	−6603.66	−1.45811	−0.729054	+	0.684456i	$0.760040\pi$
$30$	0	0
$31$	833.002	0.155683	0.0778416	−	0.996966i	$-0.475197\pi$
$31$	833.002	0.155683	0.0778416	+	0.996966i	$0.475197\pi$
$32$	0	0
$33$	−2250.75	−0.359784
$34$	0	0
$35$	5214.41	0.719508
$36$	0	0
$37$	8954.50	1.07532	0.537659	−	0.843162i	$-0.319309\pi$
$37$	8954.50	1.07532	0.537659	+	0.843162i	$0.319309\pi$
$38$	0	0
$39$	−2704.50	−0.284725
$40$	0	0
$41$	−7206.58	−0.669530	−0.334765	−	0.942302i	$-0.608657\pi$
$41$	−7206.58	−0.669530	−0.334765	+	0.942302i	$0.608657\pi$
$42$	0	0
$43$	14446.0	1.19145	0.595726	−	0.803188i	$-0.296865\pi$
$43$	14446.0	1.19145	0.595726	+	0.803188i	$0.296865\pi$
$44$	0	0
$45$	8619.75	0.634546
$46$	0	0
$47$	−17968.2	−1.18648	−0.593238	−	0.805027i	$-0.702151\pi$
$47$	−17968.2	−1.18648	−0.593238	+	0.805027i	$0.702151\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	18197.2	0.979671
$52$	0	0
$53$	−15810.5	−0.773136	−0.386568	−	0.922261i	$-0.626340\pi$
$53$	−15810.5	−0.773136	−0.386568	+	0.922261i	$0.626340\pi$
$54$	0	0
$55$	−26613.0	−1.18628
$56$	0	0
$57$	−20259.0	−0.825906
$58$	0	0
$59$	−26710.5	−0.998969	−0.499485	−	0.866323i	$-0.666477\pi$
$59$	−26710.5	−0.998969	−0.499485	+	0.866323i	$0.666477\pi$
$60$	0	0
$61$	−26799.0	−0.922133	−0.461067	−	0.887365i	$-0.652533\pi$
$61$	−26799.0	−0.922133	−0.461067	+	0.887365i	$0.652533\pi$
$62$	0	0
$63$	3969.00	0.125988
$64$	0	0
$65$	−31978.2	−0.938794
$66$	0	0
$67$	−44494.5	−1.21093	−0.605465	−	0.795872i	$-0.707013\pi$
$67$	−44494.5	−1.21093	−0.605465	+	0.795872i	$0.707013\pi$
$68$	0	0
$69$	27834.7	0.703825
$70$	0	0
$71$	−20414.1	−0.480600	−0.240300	−	0.970699i	$-0.577246\pi$
$71$	−20414.1	−0.480600	−0.240300	+	0.970699i	$0.577246\pi$
$72$	0	0
$73$	38742.5	0.850904	0.425452	−	0.904981i	$-0.360115\pi$
$73$	38742.5	0.850904	0.425452	+	0.904981i	$0.360115\pi$
$74$	0	0
$75$	73795.5	1.51487
$76$	0	0
$77$	−12254.1	−0.235534
$78$	0	0
$79$	−67211.5	−1.21165	−0.605823	−	0.795600i	$-0.707156\pi$
$79$	−67211.5	−1.21165	−0.605823	+	0.795600i	$0.707156\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−35850.7	−0.571218	−0.285609	−	0.958346i	$-0.592196\pi$
$83$	−35850.7	−0.571218	−0.285609	+	0.958346i	$0.592196\pi$
$84$	0	0
$85$	215165.	3.23017
$86$	0	0
$87$	−59433.0	−0.841839
$88$	0	0
$89$	106267.	1.42208	0.711041	−	0.703150i	$-0.248224\pi$
$89$	106267.	1.42208	0.711041	+	0.703150i	$0.248224\pi$
$90$	0	0
$91$	−14724.5	−0.186396
$92$	0	0
$93$	7497.02	0.0898837
$94$	0	0
$95$	−239544.	−2.72318
$96$	0	0
$97$	98133.5	1.05898	0.529490	−	0.848316i	$-0.322383\pi$
$97$	98133.5	1.05898	0.529490	+	0.848316i	$0.322383\pi$
$98$	0	0
$99$	−20256.7	−0.207721
$100$	0	0
$101$	−50480.7	−0.492404	−0.246202	−	0.969219i	$-0.579183\pi$
$101$	−50480.7	−0.492404	−0.246202	+	0.969219i	$0.579183\pi$
$102$	0	0
$103$	−14246.0	−0.132312	−0.0661561	−	0.997809i	$-0.521074\pi$
$103$	−14246.0	−0.132312	−0.0661561	+	0.997809i	$0.521074\pi$
$104$	0	0
$105$	46929.7	0.415408
$106$	0	0
$107$	39457.9	0.333176	0.166588	−	0.986027i	$-0.446725\pi$
$107$	39457.9	0.333176	0.166588	+	0.986027i	$0.446725\pi$
$108$	0	0
$109$	14194.0	0.114430	0.0572148	−	0.998362i	$-0.481778\pi$
$109$	14194.0	0.114430	0.0572148	+	0.998362i	$0.481778\pi$
$110$	0	0
$111$	80590.5	0.620835
$112$	0	0
$113$	−114936.	−0.846761	−0.423380	−	0.905952i	$-0.639157\pi$
$113$	−114936.	−0.846761	−0.423380	+	0.905952i	$0.639157\pi$
$114$	0	0
$115$	329120.	2.32065
$116$	0	0
$117$	−24340.5	−0.164386
$118$	0	0
$119$	99073.9	0.641345
$120$	0	0
$121$	−98509.5	−0.611666
$122$	0	0
$123$	−64859.2	−0.386553
$124$	0	0
$125$	540011.	3.09120
$126$	0	0
$127$	194998.	1.07281	0.536404	−	0.843961i	$-0.319782\pi$
$127$	194998.	1.07281	0.536404	+	0.843961i	$0.319782\pi$
$128$	0	0
$129$	130014.	0.687885
$130$	0	0
$131$	−200981.	−1.02324	−0.511619	−	0.859212i	$-0.670954\pi$
$131$	−200981.	−1.02324	−0.511619	+	0.859212i	$0.670954\pi$
$132$	0	0
$133$	−110299.	−0.540682
$134$	0	0
$135$	77577.7	0.366355
$136$	0	0
$137$	60872.4	0.277089	0.138544	−	0.990356i	$-0.455758\pi$
$137$	60872.4	0.277089	0.138544	+	0.990356i	$0.455758\pi$
$138$	0	0
$139$	−386638.	−1.69733	−0.848667	−	0.528928i	$-0.822594\pi$
$139$	−386638.	−1.69733	−0.848667	+	0.528928i	$0.822594\pi$
$140$	0	0
$141$	−161713.	−0.685012
$142$	0	0
$143$	75149.9	0.307318
$144$	0	0
$145$	−702739.	−2.77571
$146$	0	0
$147$	21609.0	0.0824786
$148$	0	0
$149$	−152878.	−0.564132	−0.282066	−	0.959395i	$-0.591020\pi$
$149$	−152878.	−0.564132	−0.282066	+	0.959395i	$0.591020\pi$
$150$	0	0
$151$	−134322.	−0.479408	−0.239704	−	0.970846i	$-0.577050\pi$
$151$	−134322.	−0.479408	−0.239704	+	0.970846i	$0.577050\pi$
$152$	0	0
$153$	163775.	0.565613
$154$	0	0
$155$	88645.2	0.296364
$156$	0	0
$157$	80730.0	0.261388	0.130694	−	0.991423i	$-0.458279\pi$
$157$	80730.0	0.261388	0.130694	+	0.991423i	$0.458279\pi$
$158$	0	0
$159$	−142295.	−0.446370
$160$	0	0
$161$	151545.	0.460761
$162$	0	0
$163$	90972.2	0.268188	0.134094	−	0.990969i	$-0.457188\pi$
$163$	90972.2	0.268188	0.134094	+	0.990969i	$0.457188\pi$
$164$	0	0
$165$	−239517.	−0.684899
$166$	0	0
$167$	−70407.3	−0.195356	−0.0976779	−	0.995218i	$-0.531142\pi$
$167$	−70407.3	−0.195356	−0.0976779	+	0.995218i	$0.531142\pi$
$168$	0	0
$169$	−280993.	−0.756796
$170$	0	0
$171$	−182331.	−0.476837
$172$	0	0
$173$	−304829.	−0.774356	−0.387178	−	0.922005i	$-0.626550\pi$
$173$	−304829.	−0.774356	−0.387178	+	0.922005i	$0.626550\pi$
$174$	0	0
$175$	401775.	0.991718
$176$	0	0
$177$	−240395.	−0.576755
$178$	0	0
$179$	−586010.	−1.36701	−0.683506	−	0.729945i	$-0.739546\pi$
$179$	−586010.	−1.36701	−0.683506	+	0.729945i	$0.739546\pi$
$180$	0	0
$181$	797296.	1.80894	0.904468	−	0.426542i	$-0.140268\pi$
$181$	797296.	1.80894	0.904468	+	0.426542i	$0.140268\pi$
$182$	0	0
$183$	−241191.	−0.532394
$184$	0	0
$185$	952907.	2.04702
$186$	0	0
$187$	−505647.	−1.05741
$188$	0	0
$189$	35721.0	0.0727393
$190$	0	0
$191$	−150935.	−0.299369	−0.149684	−	0.988734i	$-0.547826\pi$
$191$	−150935.	−0.299369	−0.149684	+	0.988734i	$0.547826\pi$
$192$	0	0
$193$	−701556.	−1.35572	−0.677859	−	0.735192i	$-0.737092\pi$
$193$	−701556.	−1.35572	−0.677859	+	0.735192i	$0.737092\pi$
$194$	0	0
$195$	−287803.	−0.542013
$196$	0	0
$197$	605633.	1.11184	0.555922	−	0.831234i	$-0.312365\pi$
$197$	605633.	1.11184	0.555922	+	0.831234i	$0.312365\pi$
$198$	0	0
$199$	155407.	0.278188	0.139094	−	0.990279i	$-0.455581\pi$
$199$	155407.	0.278188	0.139094	+	0.990279i	$0.455581\pi$
$200$	0	0
$201$	−400450.	−0.699131
$202$	0	0
$203$	−323579.	−0.551113
$204$	0	0
$205$	−766900.	−1.27454
$206$	0	0
$207$	250513.	0.406353
$208$	0	0
$209$	562937.	0.891443
$210$	0	0
$211$	305091.	0.471762	0.235881	−	0.971782i	$-0.424202\pi$
$211$	305091.	0.471762	0.235881	+	0.971782i	$0.424202\pi$
$212$	0	0
$213$	−183727.	−0.277475
$214$	0	0
$215$	1.53729e6	2.26809
$216$	0	0
$217$	40817.1	0.0588427
$218$	0	0
$219$	348682.	0.491269
$220$	0	0
$221$	−607585.	−0.836809
$222$	0	0
$223$	1.35555e6	1.82538	0.912688	−	0.408658i	$-0.134003\pi$
$223$	1.35555e6	1.82538	0.912688	+	0.408658i	$0.134003\pi$
$224$	0	0
$225$	664159.	0.874613
$226$	0	0
$227$	234584.	0.302157	0.151079	−	0.988522i	$-0.451725\pi$
$227$	234584.	0.302157	0.151079	+	0.988522i	$0.451725\pi$
$228$	0	0
$229$	1.47823e6	1.86274	0.931369	−	0.364076i	$-0.118615\pi$
$229$	1.47823e6	1.86274	0.931369	+	0.364076i	$0.118615\pi$
$230$	0	0
$231$	−110287.	−0.135986
$232$	0	0
$233$	262304.	0.316531	0.158265	−	0.987397i	$-0.449410\pi$
$233$	262304.	0.316531	0.158265	+	0.987397i	$0.449410\pi$
$234$	0	0
$235$	−1.91211e6	−2.25862
$236$	0	0
$237$	−604903.	−0.699544
$238$	0	0
$239$	1.62761e6	1.84312	0.921562	−	0.388232i	$-0.126914\pi$
$239$	1.62761e6	1.84312	0.921562	+	0.388232i	$0.126914\pi$
$240$	0	0
$241$	−1.25026e6	−1.38662	−0.693310	−	0.720639i	$-0.743849\pi$
$241$	−1.25026e6	−1.38662	−0.693310	+	0.720639i	$0.743849\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	255506.	0.271948
$246$	0	0
$247$	676424.	0.705467
$248$	0	0
$249$	−322656.	−0.329793
$250$	0	0
$251$	193945.	0.194310	0.0971549	−	0.995269i	$-0.469026\pi$
$251$	193945.	0.194310	0.0971549	+	0.995269i	$0.469026\pi$
$252$	0	0
$253$	−773444.	−0.759675
$254$	0	0
$255$	1.93649e6	1.86494
$256$	0	0
$257$	−254093.	−0.239972	−0.119986	−	0.992776i	$-0.538285\pi$
$257$	−254093.	−0.239972	−0.119986	+	0.992776i	$0.538285\pi$
$258$	0	0
$259$	438770.	0.406432
$260$	0	0
$261$	−534897.	−0.486036
$262$	0	0
$263$	−746211.	−0.665230	−0.332615	−	0.943063i	$-0.607931\pi$
$263$	−746211.	−0.665230	−0.332615	+	0.943063i	$0.607931\pi$
$264$	0	0
$265$	−1.68250e6	−1.47177
$266$	0	0
$267$	956406.	0.821040
$268$	0	0
$269$	−1.87146e6	−1.57689	−0.788444	−	0.615107i	$-0.789113\pi$
$269$	−1.87146e6	−1.57689	−0.788444	+	0.615107i	$0.789113\pi$
$270$	0	0
$271$	98046.4	0.0810977	0.0405488	−	0.999178i	$-0.487089\pi$
$271$	98046.4	0.0810977	0.0405488	+	0.999178i	$0.487089\pi$
$272$	0	0
$273$	−132520.	−0.107616
$274$	0	0
$275$	−2.05056e6	−1.63508
$276$	0	0
$277$	1.57273e6	1.23156	0.615781	−	0.787918i	$-0.288841\pi$
$277$	1.57273e6	1.23156	0.615781	+	0.787918i	$0.288841\pi$
$278$	0	0
$279$	67473.2	0.0518944
$280$	0	0
$281$	1.88960e6	1.42759	0.713797	−	0.700352i	$-0.246974\pi$
$281$	1.88960e6	1.42759	0.713797	+	0.700352i	$0.246974\pi$
$282$	0	0
$283$	1.94676e6	1.44493	0.722464	−	0.691409i	$-0.243010\pi$
$283$	1.94676e6	1.44493	0.722464	+	0.691409i	$0.243010\pi$
$284$	0	0
$285$	−2.15589e6	−1.57223
$286$	0	0
$287$	−353123.	−0.253058
$288$	0	0
$289$	2.66829e6	1.87926
$290$	0	0
$291$	883202.	0.611403
$292$	0	0
$293$	238275.	0.162147	0.0810735	−	0.996708i	$-0.474165\pi$
$293$	238275.	0.162147	0.0810735	+	0.996708i	$0.474165\pi$
$294$	0	0
$295$	−2.84244e6	−1.90168
$296$	0	0
$297$	−182311.	−0.119928
$298$	0	0
$299$	−929370.	−0.601189
$300$	0	0
$301$	707854.	0.450326
$302$	0	0
$303$	−454326.	−0.284290
$304$	0	0
$305$	−2.85186e6	−1.75541
$306$	0	0
$307$	1.70996e6	1.03547	0.517737	−	0.855540i	$-0.326775\pi$
$307$	1.70996e6	1.03547	0.517737	+	0.855540i	$0.326775\pi$
$308$	0	0
$309$	−128214.	−0.0763905
$310$	0	0
$311$	301648.	0.176848	0.0884240	−	0.996083i	$-0.471817\pi$
$311$	301648.	0.176848	0.0884240	+	0.996083i	$0.471817\pi$
$312$	0	0
$313$	−399739.	−0.230630	−0.115315	−	0.993329i	$-0.536788\pi$
$313$	−399739.	−0.230630	−0.115315	+	0.993329i	$0.536788\pi$
$314$	0	0
$315$	422368.	0.239836
$316$	0	0
$317$	−479312.	−0.267899	−0.133949	−	0.990988i	$-0.542766\pi$
$317$	−479312.	−0.267899	−0.133949	+	0.990988i	$0.542766\pi$
$318$	0	0
$319$	1.65146e6	0.908641
$320$	0	0
$321$	355121.	0.192359
$322$	0	0
$323$	−4.55133e6	−2.42735
$324$	0	0
$325$	−2.46395e6	−1.29397
$326$	0	0
$327$	127746.	0.0660660
$328$	0	0
$329$	−880440.	−0.448446
$330$	0	0
$331$	52366.6	0.0262715	0.0131357	−	0.999914i	$-0.495819\pi$
$331$	52366.6	0.0262715	0.0131357	+	0.999914i	$0.495819\pi$
$332$	0	0
$333$	725314.	0.358439
$334$	0	0
$335$	−4.73495e6	−2.30517
$336$	0	0
$337$	−2.31382e6	−1.10983	−0.554913	−	0.831909i	$-0.687248\pi$
$337$	−2.31382e6	−1.10983	−0.554913	+	0.831909i	$0.687248\pi$
$338$	0	0
$339$	−1.03443e6	−0.488878
$340$	0	0
$341$	−208320.	−0.0970162
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	2.96208e6	1.33983
$346$	0	0
$347$	3.82303e6	1.70445	0.852225	−	0.523175i	$-0.175253\pi$
$347$	3.82303e6	1.70445	0.852225	+	0.523175i	$0.175253\pi$
$348$	0	0
$349$	−3.19867e6	−1.40574	−0.702872	−	0.711316i	$-0.748099\pi$
$349$	−3.19867e6	−1.40574	−0.702872	+	0.711316i	$0.748099\pi$
$350$	0	0
$351$	−219064.	−0.0949082
$352$	0	0
$353$	1.47712e6	0.630927	0.315464	−	0.948938i	$-0.397840\pi$
$353$	1.47712e6	0.630927	0.315464	+	0.948938i	$0.397840\pi$
$354$	0	0
$355$	−2.17240e6	−0.914890
$356$	0	0
$357$	891665.	0.370281
$358$	0	0
$359$	−1.12162e6	−0.459315	−0.229657	−	0.973272i	$-0.573761\pi$
$359$	−1.12162e6	−0.459315	−0.229657	+	0.973272i	$0.573761\pi$
$360$	0	0
$361$	2.59089e6	1.04636
$362$	0	0
$363$	−886585.	−0.353146
$364$	0	0
$365$	4.12285e6	1.61981
$366$	0	0
$367$	2.40681e6	0.932774	0.466387	−	0.884581i	$-0.345555\pi$
$367$	2.40681e6	0.932774	0.466387	+	0.884581i	$0.345555\pi$
$368$	0	0
$369$	−583733.	−0.223177
$370$	0	0
$371$	−774715.	−0.292218
$372$	0	0
$373$	1.38495e6	0.515421	0.257711	−	0.966222i	$-0.417032\pi$
$373$	1.38495e6	0.515421	0.257711	+	0.966222i	$0.417032\pi$
$374$	0	0
$375$	4.86010e6	1.78471
$376$	0	0
$377$	1.98440e6	0.719077
$378$	0	0
$379$	2.35789e6	0.843190	0.421595	−	0.906784i	$-0.361470\pi$
$379$	2.35789e6	0.843190	0.421595	+	0.906784i	$0.361470\pi$
$380$	0	0
$381$	1.75499e6	0.619386
$382$	0	0
$383$	3.67412e6	1.27984	0.639922	−	0.768440i	$-0.278967\pi$
$383$	3.67412e6	1.27984	0.639922	+	0.768440i	$0.278967\pi$
$384$	0	0
$385$	−1.30404e6	−0.448371
$386$	0	0
$387$	1.17013e6	0.397150
$388$	0	0
$389$	123837.	0.0414932	0.0207466	−	0.999785i	$-0.493396\pi$
$389$	123837.	0.0414932	0.0207466	+	0.999785i	$0.493396\pi$
$390$	0	0
$391$	6.25328e6	2.06855
$392$	0	0
$393$	−1.80883e6	−0.590767
$394$	0	0
$395$	−7.15242e6	−2.30653
$396$	0	0
$397$	−1.66109e6	−0.528954	−0.264477	−	0.964392i	$-0.585199\pi$
$397$	−1.66109e6	−0.528954	−0.264477	+	0.964392i	$0.585199\pi$
$398$	0	0
$399$	−992690.	−0.312163
$400$	0	0
$401$	−291011.	−0.0903750	−0.0451875	−	0.998979i	$-0.514389\pi$
$401$	−291011.	−0.0903750	−0.0451875	+	0.998979i	$0.514389\pi$
$402$	0	0
$403$	−250317.	−0.0767763
$404$	0	0
$405$	698199.	0.211515
$406$	0	0
$407$	−2.23937e6	−0.670100
$408$	0	0
$409$	1.64280e6	0.485597	0.242798	−	0.970077i	$-0.421935\pi$
$409$	1.64280e6	0.485597	0.242798	+	0.970077i	$0.421935\pi$
$410$	0	0
$411$	547851.	0.159977
$412$	0	0
$413$	−1.30881e6	−0.377575
$414$	0	0
$415$	−3.81511e6	−1.08739
$416$	0	0
$417$	−3.47974e6	−0.979956
$418$	0	0
$419$	4.89016e6	1.36078	0.680391	−	0.732850i	$-0.261810\pi$
$419$	4.89016e6	1.36078	0.680391	+	0.732850i	$0.261810\pi$
$420$	0	0
$421$	3.91443e6	1.07637	0.538187	−	0.842825i	$-0.319109\pi$
$421$	3.91443e6	1.07637	0.538187	+	0.842825i	$0.319109\pi$
$422$	0	0
$423$	−1.45542e6	−0.395492
$424$	0	0
$425$	1.65787e7	4.45223
$426$	0	0
$427$	−1.31315e6	−0.348534
$428$	0	0
$429$	676349.	0.177430
$430$	0	0
$431$	−461156.	−0.119579	−0.0597895	−	0.998211i	$-0.519043\pi$
$431$	−461156.	−0.119579	−0.0597895	+	0.998211i	$0.519043\pi$
$432$	0	0
$433$	−7.23092e6	−1.85342	−0.926711	−	0.375776i	$-0.877376\pi$
$433$	−7.23092e6	−1.85342	−0.926711	+	0.375776i	$0.877376\pi$
$434$	0	0
$435$	−6.32465e6	−1.60256
$436$	0	0
$437$	−6.96177e6	−1.74388
$438$	0	0
$439$	243865.	0.0603932	0.0301966	−	0.999544i	$-0.490387\pi$
$439$	243865.	0.0603932	0.0301966	+	0.999544i	$0.490387\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	0	0
$443$	−7.58521e6	−1.83636	−0.918181	−	0.396162i	$-0.870342\pi$
$443$	−7.58521e6	−1.83636	−0.918181	+	0.396162i	$0.870342\pi$
$444$	0	0
$445$	1.13086e7	2.70713
$446$	0	0
$447$	−1.37591e6	−0.325702
$448$	0	0
$449$	−117953.	−0.0276118	−0.0138059	−	0.999905i	$-0.504395\pi$
$449$	−117953.	−0.0276118	−0.0138059	+	0.999905i	$0.504395\pi$
$450$	0	0
$451$	1.80224e6	0.417227
$452$	0	0
$453$	−1.20890e6	−0.276786
$454$	0	0
$455$	−1.56693e6	−0.354831
$456$	0	0
$457$	4.31220e6	0.965848	0.482924	−	0.875662i	$-0.339575\pi$
$457$	4.31220e6	0.965848	0.482924	+	0.875662i	$0.339575\pi$
$458$	0	0
$459$	1.47398e6	0.326557
$460$	0	0
$461$	1.82503e6	0.399960	0.199980	−	0.979800i	$-0.435912\pi$
$461$	1.82503e6	0.399960	0.199980	+	0.979800i	$0.435912\pi$
$462$	0	0
$463$	4.94213e6	1.07143	0.535713	−	0.844400i	$-0.320043\pi$
$463$	4.94213e6	1.07143	0.535713	+	0.844400i	$0.320043\pi$
$464$	0	0
$465$	797807.	0.171106
$466$	0	0
$467$	8.24041e6	1.74846	0.874232	−	0.485508i	$-0.161365\pi$
$467$	8.24041e6	1.74846	0.874232	+	0.485508i	$0.161365\pi$
$468$	0	0
$469$	−2.18023e6	−0.457689
$470$	0	0
$471$	726570.	0.150912
$472$	0	0
$473$	−3.61270e6	−0.742470
$474$	0	0
$475$	−1.84571e7	−3.75343
$476$	0	0
$477$	−1.28065e6	−0.257712
$478$	0	0
$479$	1.06615e6	0.212315	0.106158	−	0.994349i	$-0.466145\pi$
$479$	1.06615e6	0.212315	0.106158	+	0.994349i	$0.466145\pi$
$480$	0	0
$481$	−2.69082e6	−0.530301
$482$	0	0
$483$	1.36390e6	0.266021
$484$	0	0
$485$	1.04430e7	2.01592
$486$	0	0
$487$	2.25993e6	0.431790	0.215895	−	0.976417i	$-0.430733\pi$
$487$	2.25993e6	0.431790	0.215895	+	0.976417i	$0.430733\pi$
$488$	0	0
$489$	818750.	0.154839
$490$	0	0
$491$	−3.15916e6	−0.591382	−0.295691	−	0.955284i	$-0.595550\pi$
$491$	−3.15916e6	−0.591382	−0.295691	+	0.955284i	$0.595550\pi$
$492$	0	0
$493$	−1.33520e7	−2.47418
$494$	0	0
$495$	−2.15565e6	−0.395426
$496$	0	0
$497$	−1.00029e6	−0.181650
$498$	0	0
$499$	−7.40623e6	−1.33151	−0.665757	−	0.746168i	$-0.731891\pi$
$499$	−7.40623e6	−1.33151	−0.665757	+	0.746168i	$0.731891\pi$
$500$	0	0
$501$	−633666.	−0.112789
$502$	0	0
$503$	−1.82580e6	−0.321761	−0.160881	−	0.986974i	$-0.551433\pi$
$503$	−1.82580e6	−0.321761	−0.160881	+	0.986974i	$0.551433\pi$
$504$	0	0
$505$	−5.37198e6	−0.937359
$506$	0	0
$507$	−2.52894e6	−0.436936
$508$	0	0
$509$	−2.68759e6	−0.459800	−0.229900	−	0.973214i	$-0.573840\pi$
$509$	−2.68759e6	−0.459800	−0.229900	+	0.973214i	$0.573840\pi$
$510$	0	0
$511$	1.89838e6	0.321611
$512$	0	0
$513$	−1.64098e6	−0.275302
$514$	0	0
$515$	−1.51601e6	−0.251875
$516$	0	0
$517$	4.49353e6	0.739370
$518$	0	0
$519$	−2.74346e6	−0.447075
$520$	0	0
$521$	5.27987e6	0.852174	0.426087	−	0.904682i	$-0.359892\pi$
$521$	5.27987e6	0.852174	0.426087	+	0.904682i	$0.359892\pi$
$522$	0	0
$523$	−8.39485e6	−1.34202	−0.671010	−	0.741449i	$-0.734139\pi$
$523$	−8.39485e6	−1.34202	−0.671010	+	0.741449i	$0.734139\pi$
$524$	0	0
$525$	3.61598e6	0.572569
$526$	0	0
$527$	1.68426e6	0.264169
$528$	0	0
$529$	3.12875e6	0.486107
$530$	0	0
$531$	−2.16355e6	−0.332990
$532$	0	0
$533$	2.16558e6	0.330184
$534$	0	0
$535$	4.19897e6	0.634247
$536$	0	0
$537$	−5.27409e6	−0.789245
$538$	0	0
$539$	−600449.	−0.0890235
$540$	0	0
$541$	6.11826e6	0.898741	0.449370	−	0.893346i	$-0.351648\pi$
$541$	6.11826e6	0.898741	0.449370	+	0.893346i	$0.351648\pi$
$542$	0	0
$543$	7.17566e6	1.04439
$544$	0	0
$545$	1.51048e6	0.217833
$546$	0	0
$547$	−5.59301e6	−0.799241	−0.399620	−	0.916681i	$-0.630858\pi$
$547$	−5.59301e6	−0.799241	−0.399620	+	0.916681i	$0.630858\pi$
$548$	0	0
$549$	−2.17072e6	−0.307378
$550$	0	0
$551$	1.48648e7	2.08584
$552$	0	0
$553$	−3.29336e6	−0.457959
$554$	0	0
$555$	8.57617e6	1.18185
$556$	0	0
$557$	2.52668e6	0.345074	0.172537	−	0.985003i	$-0.444804\pi$
$557$	2.52668e6	0.345074	0.172537	+	0.985003i	$0.444804\pi$
$558$	0	0
$559$	−4.34102e6	−0.587573
$560$	0	0
$561$	−4.55082e6	−0.610496
$562$	0	0
$563$	−5.32508e6	−0.708036	−0.354018	−	0.935239i	$-0.615185\pi$
$563$	−5.32508e6	−0.708036	−0.354018	+	0.935239i	$0.615185\pi$
$564$	0	0
$565$	−1.22311e7	−1.61193
$566$	0	0
$567$	321489.	0.0419961
$568$	0	0
$569$	−9.05147e6	−1.17203	−0.586015	−	0.810300i	$-0.699304\pi$
$569$	−9.05147e6	−1.17203	−0.586015	+	0.810300i	$0.699304\pi$
$570$	0	0
$571$	6.41867e6	0.823863	0.411932	−	0.911215i	$-0.364854\pi$
$571$	6.41867e6	0.823863	0.411932	+	0.911215i	$0.364854\pi$
$572$	0	0
$573$	−1.35842e6	−0.172841
$574$	0	0
$575$	2.53590e7	3.19862
$576$	0	0
$577$	1.40363e7	1.75515	0.877574	−	0.479442i	$-0.159161\pi$
$577$	1.40363e7	1.75515	0.877574	+	0.479442i	$0.159161\pi$
$578$	0	0
$579$	−6.31401e6	−0.782724
$580$	0	0
$581$	−1.75668e6	−0.215900
$582$	0	0
$583$	3.95394e6	0.481791
$584$	0	0
$585$	−2.59023e6	−0.312931
$586$	0	0
$587$	−1.39012e7	−1.66516	−0.832579	−	0.553906i	$-0.813137\pi$
$587$	−1.39012e7	−1.66516	−0.832579	+	0.553906i	$0.813137\pi$
$588$	0	0
$589$	−1.87509e6	−0.222706
$590$	0	0
$591$	5.45070e6	0.641924
$592$	0	0
$593$	−1.17354e7	−1.37045	−0.685224	−	0.728332i	$-0.740296\pi$
$593$	−1.17354e7	−1.37045	−0.685224	+	0.728332i	$0.740296\pi$
$594$	0	0
$595$	1.05431e7	1.22089
$596$	0	0
$597$	1.39867e6	0.160612
$598$	0	0
$599$	2.01682e6	0.229668	0.114834	−	0.993385i	$-0.463366\pi$
$599$	2.01682e6	0.229668	0.114834	+	0.993385i	$0.463366\pi$
$600$	0	0
$601$	3.72242e6	0.420378	0.210189	−	0.977661i	$-0.432592\pi$
$601$	3.72242e6	0.420378	0.210189	+	0.977661i	$0.432592\pi$
$602$	0	0
$603$	−3.60405e6	−0.403644
$604$	0	0
$605$	−1.04830e7	−1.16439
$606$	0	0
$607$	6.18076e6	0.680879	0.340440	−	0.940266i	$-0.389424\pi$
$607$	6.18076e6	0.680879	0.340440	+	0.940266i	$0.389424\pi$
$608$	0	0
$609$	−2.91221e6	−0.318185
$610$	0	0
$611$	5.39943e6	0.585120
$612$	0	0
$613$	−2.43775e6	−0.262022	−0.131011	−	0.991381i	$-0.541822\pi$
$613$	−2.43775e6	−0.262022	−0.131011	+	0.991381i	$0.541822\pi$
$614$	0	0
$615$	−6.90210e6	−0.735857
$616$	0	0
$617$	−1.19168e7	−1.26022	−0.630109	−	0.776506i	$-0.716990\pi$
$617$	−1.19168e7	−1.26022	−0.630109	+	0.776506i	$0.716990\pi$
$618$	0	0
$619$	9.61238e6	1.00833	0.504167	−	0.863606i	$-0.331800\pi$
$619$	9.61238e6	1.00833	0.504167	+	0.863606i	$0.331800\pi$
$620$	0	0
$621$	2.25461e6	0.234608
$622$	0	0
$623$	5.20710e6	0.537497
$624$	0	0
$625$	3.18427e7	3.26069
$626$	0	0
$627$	5.06643e6	0.514675
$628$	0	0
$629$	1.81052e7	1.82464
$630$	0	0
$631$	−4.44826e6	−0.444750	−0.222375	−	0.974961i	$-0.571381\pi$
$631$	−4.44826e6	−0.444750	−0.222375	+	0.974961i	$0.571381\pi$
$632$	0	0
$633$	2.74582e6	0.272372
$634$	0	0
$635$	2.07511e7	2.04224
$636$	0	0
$637$	−721500.	−0.0704511
$638$	0	0
$639$	−1.65354e6	−0.160200
$640$	0	0
$641$	−8.20161e6	−0.788414	−0.394207	−	0.919022i	$-0.628981\pi$
$641$	−8.20161e6	−0.788414	−0.394207	+	0.919022i	$0.628981\pi$
$642$	0	0
$643$	1.38379e7	1.31990	0.659952	−	0.751308i	$-0.270577\pi$
$643$	1.38379e7	1.31990	0.659952	+	0.751308i	$0.270577\pi$
$644$	0	0
$645$	1.38356e7	1.30948
$646$	0	0
$647$	−3.57496e6	−0.335746	−0.167873	−	0.985809i	$-0.553690\pi$
$647$	−3.57496e6	−0.335746	−0.167873	+	0.985809i	$0.553690\pi$
$648$	0	0
$649$	6.67985e6	0.622522
$650$	0	0
$651$	367354.	0.0339729
$652$	0	0
$653$	−2.08055e7	−1.90939	−0.954695	−	0.297587i	$-0.903818\pi$
$653$	−2.08055e7	−1.90939	−0.954695	+	0.297587i	$0.903818\pi$
$654$	0	0
$655$	−2.13877e7	−1.94787
$656$	0	0
$657$	3.13814e6	0.283635
$658$	0	0
$659$	4.34521e6	0.389760	0.194880	−	0.980827i	$-0.437568\pi$
$659$	4.34521e6	0.389760	0.194880	+	0.980827i	$0.437568\pi$
$660$	0	0
$661$	−3.59984e6	−0.320464	−0.160232	−	0.987079i	$-0.551224\pi$
$661$	−3.59984e6	−0.320464	−0.160232	+	0.987079i	$0.551224\pi$
$662$	0	0
$663$	−5.46827e6	−0.483132
$664$	0	0
$665$	−1.17376e7	−1.02926
$666$	0	0
$667$	−2.04235e7	−1.77752
$668$	0	0
$669$	1.21999e7	1.05388
$670$	0	0
$671$	6.70197e6	0.574641
$672$	0	0
$673$	−1.35130e7	−1.15005	−0.575023	−	0.818137i	$-0.695007\pi$
$673$	−1.35130e7	−1.15005	−0.575023	+	0.818137i	$0.695007\pi$
$674$	0	0
$675$	5.97743e6	0.504958
$676$	0	0
$677$	7.06427e6	0.592373	0.296187	−	0.955130i	$-0.404285\pi$
$677$	7.06427e6	0.592373	0.296187	+	0.955130i	$0.404285\pi$
$678$	0	0
$679$	4.80854e6	0.400257
$680$	0	0
$681$	2.11125e6	0.174451
$682$	0	0
$683$	−1.75676e7	−1.44099	−0.720494	−	0.693461i	$-0.756085\pi$
$683$	−1.75676e7	−1.44099	−0.720494	+	0.693461i	$0.756085\pi$
$684$	0	0
$685$	6.47783e6	0.527477
$686$	0	0
$687$	1.33040e7	1.07545
$688$	0	0
$689$	4.75105e6	0.381278
$690$	0	0
$691$	−617820.	−0.0492229	−0.0246114	−	0.999697i	$-0.507835\pi$
$691$	−617820.	−0.0492229	−0.0246114	+	0.999697i	$0.507835\pi$
$692$	0	0
$693$	−992580.	−0.0785113
$694$	0	0
$695$	−4.11447e7	−3.23111
$696$	0	0
$697$	−1.45711e7	−1.13608
$698$	0	0
$699$	2.36074e6	0.182749
$700$	0	0
$701$	1.54273e7	1.18576	0.592879	−	0.805292i	$-0.297991\pi$
$701$	1.54273e7	1.18576	0.592879	+	0.805292i	$0.297991\pi$
$702$	0	0
$703$	−2.01566e7	−1.53825
$704$	0	0
$705$	−1.72090e7	−1.30402
$706$	0	0
$707$	−2.47355e6	−0.186111
$708$	0	0
$709$	−2.42780e7	−1.81383	−0.906917	−	0.421309i	$-0.861571\pi$
$709$	−2.42780e7	−1.81383	−0.906917	+	0.421309i	$0.861571\pi$
$710$	0	0
$711$	−5.44413e6	−0.403882
$712$	0	0
$713$	2.57627e6	0.189787
$714$	0	0
$715$	7.99720e6	0.585023
$716$	0	0
$717$	1.46485e7	1.06413
$718$	0	0
$719$	−8.00425e6	−0.577429	−0.288714	−	0.957415i	$-0.593228\pi$
$719$	−8.00425e6	−0.577429	−0.288714	+	0.957415i	$0.593228\pi$
$720$	0	0
$721$	−698054.	−0.0500093
$722$	0	0
$723$	−1.12523e7	−0.800566
$724$	0	0
$725$	−5.41467e7	−3.82584
$726$	0	0
$727$	2.03222e7	1.42605	0.713025	−	0.701139i	$-0.247325\pi$
$727$	2.03222e7	1.42605	0.713025	+	0.701139i	$0.247325\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	2.92086e7	2.02170
$732$	0	0
$733$	6.77940e6	0.466049	0.233024	−	0.972471i	$-0.425138\pi$
$733$	6.77940e6	0.466049	0.233024	+	0.972471i	$0.425138\pi$
$734$	0	0
$735$	2.29956e6	0.157009
$736$	0	0
$737$	1.11273e7	0.754609
$738$	0	0
$739$	1.62382e7	1.09377	0.546887	−	0.837207i	$-0.315813\pi$
$739$	1.62382e7	1.09377	0.546887	+	0.837207i	$0.315813\pi$
$740$	0	0
$741$	6.08782e6	0.407302
$742$	0	0
$743$	−2.34767e6	−0.156014	−0.0780072	−	0.996953i	$-0.524856\pi$
$743$	−2.34767e6	−0.156014	−0.0780072	+	0.996953i	$0.524856\pi$
$744$	0	0
$745$	−1.62688e7	−1.07390
$746$	0	0
$747$	−2.90390e6	−0.190406
$748$	0	0
$749$	1.93343e6	0.125929
$750$	0	0
$751$	2.15068e7	1.39147	0.695737	−	0.718296i	$-0.255078\pi$
$751$	2.15068e7	1.39147	0.695737	+	0.718296i	$0.255078\pi$
$752$	0	0
$753$	1.74551e6	0.112185
$754$	0	0
$755$	−1.42941e7	−0.912619
$756$	0	0
$757$	−4.88736e6	−0.309981	−0.154990	−	0.987916i	$-0.549535\pi$
$757$	−4.88736e6	−0.309981	−0.154990	+	0.987916i	$0.549535\pi$
$758$	0	0
$759$	−6.96100e6	−0.438598
$760$	0	0
$761$	−1.09272e7	−0.683988	−0.341994	−	0.939702i	$-0.611102\pi$
$761$	−1.09272e7	−0.683988	−0.341994	+	0.939702i	$0.611102\pi$
$762$	0	0
$763$	695506.	0.0432503
$764$	0	0
$765$	1.74284e7	1.07672
$766$	0	0
$767$	8.02650e6	0.492649
$768$	0	0
$769$	8.31780e6	0.507216	0.253608	−	0.967307i	$-0.418383\pi$
$769$	8.31780e6	0.507216	0.253608	+	0.967307i	$0.418383\pi$
$770$	0	0
$771$	−2.28684e6	−0.138548
$772$	0	0
$773$	−2.78229e7	−1.67476	−0.837381	−	0.546620i	$-0.815914\pi$
$773$	−2.78229e7	−1.67476	−0.837381	+	0.546620i	$0.815914\pi$
$774$	0	0
$775$	6.83020e6	0.408488
$776$	0	0
$777$	3.94893e6	0.234654
$778$	0	0
$779$	1.62220e7	0.957769
$780$	0	0
$781$	5.10522e6	0.299493
$782$	0	0
$783$	−4.81407e6	−0.280613
$784$	0	0
$785$	8.59101e6	0.497588
$786$	0	0
$787$	3.14847e7	1.81202	0.906009	−	0.423258i	$-0.139114\pi$
$787$	3.14847e7	1.81202	0.906009	+	0.423258i	$0.139114\pi$
$788$	0	0
$789$	−6.71590e6	−0.384071
$790$	0	0
$791$	−5.63187e6	−0.320046
$792$	0	0
$793$	8.05309e6	0.454757
$794$	0	0
$795$	−1.51425e7	−0.849728
$796$	0	0
$797$	9.29354e6	0.518245	0.259123	−	0.965844i	$-0.416567\pi$
$797$	9.29354e6	0.518245	0.259123	+	0.965844i	$0.416567\pi$
$798$	0	0
$799$	−3.63301e7	−2.01326
$800$	0	0
$801$	8.60766e6	0.474028
$802$	0	0
$803$	−9.68884e6	−0.530253
$804$	0	0
$805$	1.61269e7	0.877123
$806$	0	0
$807$	−1.68432e7	−0.910417
$808$	0	0
$809$	−1.36908e7	−0.735459	−0.367730	−	0.929933i	$-0.619865\pi$
$809$	−1.36908e7	−0.735459	−0.367730	+	0.929933i	$0.619865\pi$
$810$	0	0
$811$	7.83843e6	0.418482	0.209241	−	0.977864i	$-0.432901\pi$
$811$	7.83843e6	0.418482	0.209241	+	0.977864i	$0.432901\pi$
$812$	0	0
$813$	882418.	0.0468218
$814$	0	0
$815$	9.68096e6	0.510534
$816$	0	0
$817$	−3.25179e7	−1.70438
$818$	0	0
$819$	−1.19268e6	−0.0621320
$820$	0	0
$821$	−2.77295e7	−1.43577	−0.717883	−	0.696164i	$-0.754889\pi$
$821$	−2.77295e7	−1.43577	−0.717883	+	0.696164i	$0.754889\pi$
$822$	0	0
$823$	−2.04827e6	−0.105411	−0.0527057	−	0.998610i	$-0.516785\pi$
$823$	−2.04827e6	−0.105411	−0.0527057	+	0.998610i	$0.516785\pi$
$824$	0	0
$825$	−1.84550e7	−0.944015
$826$	0	0
$827$	2.35186e7	1.19577	0.597884	−	0.801582i	$-0.296008\pi$
$827$	2.35186e7	1.19577	0.597884	+	0.801582i	$0.296008\pi$
$828$	0	0
$829$	1.93665e7	0.978734	0.489367	−	0.872078i	$-0.337228\pi$
$829$	1.93665e7	0.978734	0.489367	+	0.872078i	$0.337228\pi$
$830$	0	0
$831$	1.41546e7	0.711042
$832$	0	0
$833$	4.85462e6	0.242406
$834$	0	0
$835$	−7.49250e6	−0.371887
$836$	0	0
$837$	607258.	0.0299612
$838$	0	0
$839$	1.84417e7	0.904473	0.452236	−	0.891898i	$-0.350626\pi$
$839$	1.84417e7	0.904473	0.452236	+	0.891898i	$0.350626\pi$
$840$	0	0
$841$	2.30972e7	1.12608
$842$	0	0
$843$	1.70064e7	0.824222
$844$	0	0
$845$	−2.99023e7	−1.44067
$846$	0	0
$847$	−4.82696e6	−0.231188
$848$	0	0
$849$	1.75208e7	0.834229
$850$	0	0
$851$	2.76940e7	1.31088
$852$	0	0
$853$	1.91827e7	0.902689	0.451345	−	0.892350i	$-0.350945\pi$
$853$	1.91827e7	0.902689	0.451345	+	0.892350i	$0.350945\pi$
$854$	0	0
$855$	−1.94030e7	−0.907725
$856$	0	0
$857$	2.65637e7	1.23548	0.617741	−	0.786381i	$-0.288048\pi$
$857$	2.65637e7	1.23548	0.617741	+	0.786381i	$0.288048\pi$
$858$	0	0
$859$	−1.68342e7	−0.778410	−0.389205	−	0.921151i	$-0.627250\pi$
$859$	−1.68342e7	−0.778410	−0.389205	+	0.921151i	$0.627250\pi$
$860$	0	0
$861$	−3.17810e6	−0.146103
$862$	0	0
$863$	−4.26732e6	−0.195042	−0.0975210	−	0.995233i	$-0.531091\pi$
$863$	−4.26732e6	−0.195042	−0.0975210	+	0.995233i	$0.531091\pi$
$864$	0	0
$865$	−3.24389e7	−1.47409
$866$	0	0
$867$	2.40146e7	1.08499
$868$	0	0
$869$	1.68084e7	0.755054
$870$	0	0
$871$	1.33706e7	0.597180
$872$	0	0
$873$	7.94881e6	0.352994
$874$	0	0
$875$	2.64605e7	1.16836
$876$	0	0
$877$	2.58174e7	1.13348	0.566739	−	0.823897i	$-0.308205\pi$
$877$	2.58174e7	1.13348	0.566739	+	0.823897i	$0.308205\pi$
$878$	0	0
$879$	2.14447e6	0.0936156
$880$	0	0
$881$	2.99398e7	1.29960	0.649799	−	0.760106i	$-0.274853\pi$
$881$	2.99398e7	1.29960	0.649799	+	0.760106i	$0.274853\pi$
$882$	0	0
$883$	−4.46525e6	−0.192727	−0.0963637	−	0.995346i	$-0.530721\pi$
$883$	−4.46525e6	−0.192727	−0.0963637	+	0.995346i	$0.530721\pi$
$884$	0	0
$885$	−2.55820e7	−1.09793
$886$	0	0
$887$	2.89113e7	1.23384	0.616919	−	0.787027i	$-0.288381\pi$
$887$	2.89113e7	1.23384	0.616919	+	0.787027i	$0.288381\pi$
$888$	0	0
$889$	9.55492e6	0.405483
$890$	0	0
$891$	−1.64080e6	−0.0692405
$892$	0	0
$893$	4.04463e7	1.69727
$894$	0	0
$895$	−6.23612e7	−2.60230
$896$	0	0
$897$	−8.36433e6	−0.347096
$898$	0	0
$899$	−5.50086e6	−0.227003
$900$	0	0
$901$	−3.19675e7	−1.31189
$902$	0	0
$903$	6.37068e6	0.259996
$904$	0	0
$905$	8.48456e7	3.44356
$906$	0	0
$907$	−7.38019e6	−0.297885	−0.148943	−	0.988846i	$-0.547587\pi$
$907$	−7.38019e6	−0.297885	−0.148943	+	0.988846i	$0.547587\pi$
$908$	0	0
$909$	−4.08893e6	−0.164135
$910$	0	0
$911$	−4.25542e7	−1.69882	−0.849409	−	0.527736i	$-0.823041\pi$
$911$	−4.25542e7	−1.69882	−0.849409	+	0.527736i	$0.823041\pi$
$912$	0	0
$913$	8.96565e6	0.355963
$914$	0	0
$915$	−2.56667e7	−1.01349
$916$	0	0
$917$	−9.84807e6	−0.386748
$918$	0	0
$919$	−2.88119e7	−1.12534	−0.562669	−	0.826682i	$-0.690225\pi$
$919$	−2.88119e7	−1.12534	−0.562669	+	0.826682i	$0.690225\pi$
$920$	0	0
$921$	1.53896e7	0.597831
$922$	0	0
$923$	6.13443e6	0.237012
$924$	0	0
$925$	7.34224e7	2.82146
$926$	0	0
$927$	−1.15393e6	−0.0441041
$928$	0	0
$929$	4.82043e7	1.83251	0.916255	−	0.400595i	$-0.131196\pi$
$929$	4.82043e7	1.83251	0.916255	+	0.400595i	$0.131196\pi$
$930$	0	0
$931$	−5.40465e6	−0.204359
$932$	0	0
$933$	2.71484e6	0.102103
$934$	0	0
$935$	−5.38092e7	−2.01293
$936$	0	0
$937$	−2.04317e7	−0.760247	−0.380123	−	0.924936i	$-0.624118\pi$
$937$	−2.04317e7	−0.760247	−0.380123	+	0.924936i	$0.624118\pi$
$938$	0	0
$939$	−3.59765e6	−0.133154
$940$	0	0
$941$	−2.62728e7	−0.967237	−0.483618	−	0.875279i	$-0.660678\pi$
$941$	−2.62728e7	−0.967237	−0.483618	+	0.875279i	$0.660678\pi$
$942$	0	0
$943$	−2.22882e7	−0.816197
$944$	0	0
$945$	3.80131e6	0.138469
$946$	0	0
$947$	−2.94399e7	−1.06675	−0.533373	−	0.845880i	$-0.679076\pi$
$947$	−2.94399e7	−1.06675	−0.533373	+	0.845880i	$0.679076\pi$
$948$	0	0
$949$	−1.16421e7	−0.419630
$950$	0	0
$951$	−4.31381e6	−0.154671
$952$	0	0
$953$	−3.07999e7	−1.09854	−0.549272	−	0.835644i	$-0.685095\pi$
$953$	−3.07999e7	−1.09854	−0.549272	+	0.835644i	$0.685095\pi$
$954$	0	0
$955$	−1.60620e7	−0.569890
$956$	0	0
$957$	1.48632e7	0.524604
$958$	0	0
$959$	2.98275e6	0.104730
$960$	0	0
$961$	−2.79353e7	−0.975763
$962$	0	0
$963$	3.19609e6	0.111059
$964$	0	0
$965$	−7.46573e7	−2.58080
$966$	0	0
$967$	3.02028e7	1.03868	0.519338	−	0.854569i	$-0.326178\pi$
$967$	3.02028e7	1.03868	0.519338	+	0.854569i	$0.326178\pi$
$968$	0	0
$969$	−4.09620e7	−1.40143
$970$	0	0
$971$	2.75051e6	0.0936192	0.0468096	−	0.998904i	$-0.485095\pi$
$971$	2.75051e6	0.0936192	0.0468096	+	0.998904i	$0.485095\pi$
$972$	0	0
$973$	−1.89452e7	−0.641532
$974$	0	0
$975$	−2.21755e7	−0.747071
$976$	0	0
$977$	1.71690e7	0.575452	0.287726	−	0.957713i	$-0.407101\pi$
$977$	1.71690e7	0.575452	0.287726	+	0.957713i	$0.407101\pi$
$978$	0	0
$979$	−2.65757e7	−0.886191
$980$	0	0
$981$	1.14971e6	0.0381432
$982$	0	0
$983$	5.13066e7	1.69352	0.846758	−	0.531979i	$-0.178551\pi$
$983$	5.13066e7	1.69352	0.846758	+	0.531979i	$0.178551\pi$
$984$	0	0
$985$	6.44494e7	2.11655
$986$	0	0
$987$	−7.92396e6	−0.258910
$988$	0	0
$989$	4.46778e7	1.45245
$990$	0	0
$991$	2.39045e7	0.773207	0.386603	−	0.922246i	$-0.373648\pi$
$991$	2.39045e7	0.773207	0.386603	+	0.922246i	$0.373648\pi$
$992$	0	0
$993$	471300.	0.0151678
$994$	0	0
$995$	1.65379e7	0.529570
$996$	0	0
$997$	8.08351e6	0.257550	0.128775	−	0.991674i	$-0.458895\pi$
$997$	8.08351e6	0.257550	0.128775	+	0.991674i	$0.458895\pi$
$998$	0	0
$999$	6.52783e6	0.206945

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.6.a.d.1.2	✓	2
3.2	odd	2		252.6.a.e.1.1		2
4.3	odd	2		336.6.a.u.1.2		2
7.2	even	3		588.6.i.h.361.1		4
7.3	odd	6		588.6.i.n.373.2		4
7.4	even	3		588.6.i.h.373.1		4
7.5	odd	6		588.6.i.n.361.2		4
7.6	odd	2		588.6.a.g.1.1		2
12.11	even	2		1008.6.a.bf.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.a.d.1.2	✓	2	1.1	even	1	trivial
252.6.a.e.1.1		2	3.2	odd	2
336.6.a.u.1.2		2	4.3	odd	2
588.6.a.g.1.1		2	7.6	odd	2
588.6.i.h.361.1		4	7.2	even	3
588.6.i.h.373.1		4	7.4	even	3
588.6.i.n.361.2		4	7.5	odd	6
588.6.i.n.373.2		4	7.3	odd	6
1008.6.a.bf.1.1		2	12.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	84.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +106.417 q^{5} +49.0000 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} +106.417 q^{5} +49.0000 q^{7} +81.0000 q^{9} -250.083 q^{11} -300.500 q^{13} +957.750 q^{15} +2021.92 q^{17} -2251.00 q^{19} +441.000 q^{21} +3092.75 q^{23} +8199.50 q^{25} +729.000 q^{27} -6603.66 q^{29} +833.002 q^{31} -2250.75 q^{33} +5214.41 q^{35} +8954.50 q^{37} -2704.50 q^{39} -7206.58 q^{41} +14446.0 q^{43} +8619.75 q^{45} -17968.2 q^{47} +2401.00 q^{49} +18197.2 q^{51} -15810.5 q^{53} -26613.0 q^{55} -20259.0 q^{57} -26710.5 q^{59} -26799.0 q^{61} +3969.00 q^{63} -31978.2 q^{65} -44494.5 q^{67} +27834.7 q^{69} -20414.1 q^{71} +38742.5 q^{73} +73795.5 q^{75} -12254.1 q^{77} -67211.5 q^{79} +6561.00 q^{81} -35850.7 q^{83} +215165. q^{85} -59433.0 q^{87} +106267. q^{89} -14724.5 q^{91} +7497.02 q^{93} -239544. q^{95} +98133.5 q^{97} -20256.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{3} + 78 q^{5} + 98 q^{7} + 162 q^{9}+O(q^{10}) \) 2 * q + 18 * q^3 + 78 * q^5 + 98 * q^7 + 162 * q^9	\( 2 q + 18 q^{3} + 78 q^{5} + 98 q^{7} + 162 q^{9} + 174 q^{11} + 208 q^{13} + 702 q^{15} + 1482 q^{17} + 352 q^{19} + 882 q^{21} + 3354 q^{23} + 5882 q^{25} + 1458 q^{27} + 276 q^{29} + 6520 q^{31} + 1566 q^{33} + 3822 q^{35} + 13864 q^{37} + 1872 q^{39} - 12930 q^{41} + 12712 q^{43} + 6318 q^{45} - 28116 q^{47} + 4802 q^{49} + 13338 q^{51} - 46992 q^{53} - 38664 q^{55} + 3168 q^{57} - 65556 q^{59} - 13148 q^{61} + 7938 q^{63} - 46428 q^{65} - 75236 q^{67} + 30186 q^{69} - 66042 q^{71} + 60496 q^{73} + 52938 q^{75} + 8526 q^{77} - 34916 q^{79} + 13122 q^{81} - 82488 q^{83} + 230508 q^{85} + 2484 q^{87} + 42510 q^{89} + 10192 q^{91} + 58680 q^{93} - 313512 q^{95} + 213256 q^{97} + 14094 q^{99}+O(q^{100}) \) 2 * q + 18 * q^3 + 78 * q^5 + 98 * q^7 + 162 * q^9 + 174 * q^11 + 208 * q^13 + 702 * q^15 + 1482 * q^17 + 352 * q^19 + 882 * q^21 + 3354 * q^23 + 5882 * q^25 + 1458 * q^27 + 276 * q^29 + 6520 * q^31 + 1566 * q^33 + 3822 * q^35 + 13864 * q^37 + 1872 * q^39 - 12930 * q^41 + 12712 * q^43 + 6318 * q^45 - 28116 * q^47 + 4802 * q^49 + 13338 * q^51 - 46992 * q^53 - 38664 * q^55 + 3168 * q^57 - 65556 * q^59 - 13148 * q^61 + 7938 * q^63 - 46428 * q^65 - 75236 * q^67 + 30186 * q^69 - 66042 * q^71 + 60496 * q^73 + 52938 * q^75 + 8526 * q^77 - 34916 * q^79 + 13122 * q^81 - 82488 * q^83 + 230508 * q^85 + 2484 * q^87 + 42510 * q^89 + 10192 * q^91 + 58680 * q^93 - 313512 * q^95 + 213256 * q^97 + 14094 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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