LMFDB - Embedded newform 84.6.a.b.1.1

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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q+9.00000 q^{3} -34.0000 q^{5} -49.0000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q+9.00000 q^{3} -34.0000 q^{5} -49.0000 q^{7} +81.0000 q^{9} -332.000 q^{11} -1026.00 q^{13} -306.000 q^{15} +922.000 q^{17} +452.000 q^{19} -441.000 q^{21} -3776.00 q^{23} -1969.00 q^{25} +729.000 q^{27} +1166.00 q^{29} -9792.00 q^{31} -2988.00 q^{33} +1666.00 q^{35} +2390.00 q^{37} -9234.00 q^{39} -7230.00 q^{41} +4652.00 q^{43} -2754.00 q^{45} +24672.0 q^{47} +2401.00 q^{49} +8298.00 q^{51} +1110.00 q^{53} +11288.0 q^{55} +4068.00 q^{57} +46892.0 q^{59} -9762.00 q^{61} -3969.00 q^{63} +34884.0 q^{65} -26252.0 q^{67} -33984.0 q^{69} +65440.0 q^{71} -5606.00 q^{73} -17721.0 q^{75} +16268.0 q^{77} -9840.00 q^{79} +6561.00 q^{81} +61108.0 q^{83} -31348.0 q^{85} +10494.0 q^{87} -62958.0 q^{89} +50274.0 q^{91} -88128.0 q^{93} -15368.0 q^{95} -37838.0 q^{97} -26892.0 q^{99} +O(q^{100})$

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
$5$	−34.0000	−0.608210	−0.304105	−	0.952638i	$-0.598357\pi$
$5$	−34.0000	−0.608210	−0.304105	+	0.952638i	$0.598357\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−332.000	−0.827287	−0.413644	−	0.910439i	$-0.635744\pi$
$11$	−332.000	−0.827287	−0.413644	+	0.910439i	$0.635744\pi$
$12$	0	0
$13$	−1026.00	−1.68379	−0.841897	−	0.539638i	$-0.818561\pi$
$13$	−1026.00	−1.68379	−0.841897	+	0.539638i	$0.818561\pi$
$14$	0	0
$15$	−306.000	−0.351150
$16$	0	0
$17$	922.000	0.773764	0.386882	−	0.922129i	$-0.373552\pi$
$17$	922.000	0.773764	0.386882	+	0.922129i	$0.373552\pi$
$18$	0	0
$19$	452.000	0.287246	0.143623	−	0.989632i	$-0.454125\pi$
$19$	452.000	0.287246	0.143623	+	0.989632i	$0.454125\pi$
$20$	0	0
$21$	−441.000	−0.218218
$22$	0	0
$23$	−3776.00	−1.48838	−0.744188	−	0.667971i	$-0.767163\pi$
$23$	−3776.00	−1.48838	−0.744188	+	0.667971i	$0.767163\pi$
$24$	0	0
$25$	−1969.00	−0.630080
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	1166.00	0.257456	0.128728	−	0.991680i	$-0.458911\pi$
$29$	1166.00	0.257456	0.128728	+	0.991680i	$0.458911\pi$
$30$	0	0
$31$	−9792.00	−1.83007	−0.915034	−	0.403377i	$-0.867836\pi$
$31$	−9792.00	−1.83007	−0.915034	+	0.403377i	$0.867836\pi$
$32$	0	0
$33$	−2988.00	−0.477635
$34$	0	0
$35$	1666.00	0.229882
$36$	0	0
$37$	2390.00	0.287008	0.143504	−	0.989650i	$-0.454163\pi$
$37$	2390.00	0.287008	0.143504	+	0.989650i	$0.454163\pi$
$38$	0	0
$39$	−9234.00	−0.972139
$40$	0	0
$41$	−7230.00	−0.671705	−0.335853	−	0.941915i	$-0.609024\pi$
$41$	−7230.00	−0.671705	−0.335853	+	0.941915i	$0.609024\pi$
$42$	0	0
$43$	4652.00	0.383679	0.191840	−	0.981426i	$-0.438555\pi$
$43$	4652.00	0.383679	0.191840	+	0.981426i	$0.438555\pi$
$44$	0	0
$45$	−2754.00	−0.202737
$46$	0	0
$47$	24672.0	1.62914	0.814572	−	0.580062i	$-0.196972\pi$
$47$	24672.0	1.62914	0.814572	+	0.580062i	$0.196972\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	8298.00	0.446733
$52$	0	0
$53$	1110.00	0.0542792	0.0271396	−	0.999632i	$-0.491360\pi$
$53$	1110.00	0.0542792	0.0271396	+	0.999632i	$0.491360\pi$
$54$	0	0
$55$	11288.0	0.503165
$56$	0	0
$57$	4068.00	0.165842
$58$	0	0
$59$	46892.0	1.75375	0.876877	−	0.480715i	$-0.159623\pi$
$59$	46892.0	1.75375	0.876877	+	0.480715i	$0.159623\pi$
$60$	0	0
$61$	−9762.00	−0.335903	−0.167952	−	0.985795i	$-0.553715\pi$
$61$	−9762.00	−0.335903	−0.167952	+	0.985795i	$0.553715\pi$
$62$	0	0
$63$	−3969.00	−0.125988
$64$	0	0
$65$	34884.0	1.02410
$66$	0	0
$67$	−26252.0	−0.714456	−0.357228	−	0.934017i	$-0.616278\pi$
$67$	−26252.0	−0.714456	−0.357228	+	0.934017i	$0.616278\pi$
$68$	0	0
$69$	−33984.0	−0.859314
$70$	0	0
$71$	65440.0	1.54063	0.770313	−	0.637666i	$-0.220100\pi$
$71$	65440.0	1.54063	0.770313	+	0.637666i	$0.220100\pi$
$72$	0	0
$73$	−5606.00	−0.123125	−0.0615625	−	0.998103i	$-0.519608\pi$
$73$	−5606.00	−0.123125	−0.0615625	+	0.998103i	$0.519608\pi$
$74$	0	0
$75$	−17721.0	−0.363777
$76$	0	0
$77$	16268.0	0.312685
$78$	0	0
$79$	−9840.00	−0.177389	−0.0886946	−	0.996059i	$-0.528270\pi$
$79$	−9840.00	−0.177389	−0.0886946	+	0.996059i	$0.528270\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	61108.0	0.973650	0.486825	−	0.873500i	$-0.338155\pi$
$83$	61108.0	0.973650	0.486825	+	0.873500i	$0.338155\pi$
$84$	0	0
$85$	−31348.0	−0.470611
$86$	0	0
$87$	10494.0	0.148642
$88$	0	0
$89$	−62958.0	−0.842512	−0.421256	−	0.906942i	$-0.638411\pi$
$89$	−62958.0	−0.842512	−0.421256	+	0.906942i	$0.638411\pi$
$90$	0	0
$91$	50274.0	0.636414
$92$	0	0
$93$	−88128.0	−1.05659
$94$	0	0
$95$	−15368.0	−0.174706
$96$	0	0
$97$	−37838.0	−0.408318	−0.204159	−	0.978938i	$-0.565446\pi$
$97$	−37838.0	−0.408318	−0.204159	+	0.978938i	$0.565446\pi$
$98$	0	0
$99$	−26892.0	−0.275762
$100$	0	0
$101$	−56146.0	−0.547666	−0.273833	−	0.961777i	$-0.588291\pi$
$101$	−56146.0	−0.547666	−0.273833	+	0.961777i	$0.588291\pi$
$102$	0	0
$103$	−26392.0	−0.245120	−0.122560	−	0.992461i	$-0.539110\pi$
$103$	−26392.0	−0.245120	−0.122560	+	0.992461i	$0.539110\pi$
$104$	0	0
$105$	14994.0	0.132722
$106$	0	0
$107$	47124.0	0.397908	0.198954	−	0.980009i	$-0.436246\pi$
$107$	47124.0	0.397908	0.198954	+	0.980009i	$0.436246\pi$
$108$	0	0
$109$	−221474.	−1.78549	−0.892743	−	0.450566i	$-0.851222\pi$
$109$	−221474.	−1.78549	−0.892743	+	0.450566i	$0.851222\pi$
$110$	0	0
$111$	21510.0	0.165704
$112$	0	0
$113$	54194.0	0.399259	0.199630	−	0.979871i	$-0.436026\pi$
$113$	54194.0	0.399259	0.199630	+	0.979871i	$0.436026\pi$
$114$	0	0
$115$	128384.	0.905245
$116$	0	0
$117$	−83106.0	−0.561265
$118$	0	0
$119$	−45178.0	−0.292455
$120$	0	0
$121$	−50827.0	−0.315596
$122$	0	0
$123$	−65070.0	−0.387809
$124$	0	0
$125$	173196.	0.991432
$126$	0	0
$127$	−245760.	−1.35208	−0.676039	−	0.736866i	$-0.736305\pi$
$127$	−245760.	−1.35208	−0.676039	+	0.736866i	$0.736305\pi$
$128$	0	0
$129$	41868.0	0.221517
$130$	0	0
$131$	−150268.	−0.765047	−0.382524	−	0.923946i	$-0.624945\pi$
$131$	−150268.	−0.765047	−0.382524	+	0.923946i	$0.624945\pi$
$132$	0	0
$133$	−22148.0	−0.108569
$134$	0	0
$135$	−24786.0	−0.117050
$136$	0	0
$137$	−401638.	−1.82824	−0.914120	−	0.405443i	$-0.867117\pi$
$137$	−401638.	−1.82824	−0.914120	+	0.405443i	$0.867117\pi$
$138$	0	0
$139$	374092.	1.64226	0.821129	−	0.570743i	$-0.193345\pi$
$139$	374092.	1.64226	0.821129	+	0.570743i	$0.193345\pi$
$140$	0	0
$141$	222048.	0.940587
$142$	0	0
$143$	340632.	1.39298
$144$	0	0
$145$	−39644.0	−0.156588
$146$	0	0
$147$	21609.0	0.0824786
$148$	0	0
$149$	−456042.	−1.68283	−0.841413	−	0.540393i	$-0.818276\pi$
$149$	−456042.	−1.68283	−0.841413	+	0.540393i	$0.818276\pi$
$150$	0	0
$151$	−8024.00	−0.0286384	−0.0143192	−	0.999897i	$-0.504558\pi$
$151$	−8024.00	−0.0286384	−0.0143192	+	0.999897i	$0.504558\pi$
$152$	0	0
$153$	74682.0	0.257921
$154$	0	0
$155$	332928.	1.11307
$156$	0	0
$157$	110078.	0.356411	0.178206	−	0.983993i	$-0.442971\pi$
$157$	110078.	0.356411	0.178206	+	0.983993i	$0.442971\pi$
$158$	0	0
$159$	9990.00	0.0313381
$160$	0	0
$161$	185024.	0.562553
$162$	0	0
$163$	−3628.00	−0.0106954	−0.00534772	−	0.999986i	$-0.501702\pi$
$163$	−3628.00	−0.0106954	−0.00534772	+	0.999986i	$0.501702\pi$
$164$	0	0
$165$	101592.	0.290502
$166$	0	0
$167$	192824.	0.535020	0.267510	−	0.963555i	$-0.413799\pi$
$167$	192824.	0.535020	0.267510	+	0.963555i	$0.413799\pi$
$168$	0	0
$169$	681383.	1.83516
$170$	0	0
$171$	36612.0	0.0957488
$172$	0	0
$173$	157142.	0.399188	0.199594	−	0.979879i	$-0.436038\pi$
$173$	157142.	0.399188	0.199594	+	0.979879i	$0.436038\pi$
$174$	0	0
$175$	96481.0	0.238148
$176$	0	0
$177$	422028.	1.01253
$178$	0	0
$179$	−446868.	−1.04243	−0.521215	−	0.853426i	$-0.674521\pi$
$179$	−446868.	−1.04243	−0.521215	+	0.853426i	$0.674521\pi$
$180$	0	0
$181$	805638.	1.82786	0.913931	−	0.405869i	$-0.133031\pi$
$181$	805638.	1.82786	0.913931	+	0.405869i	$0.133031\pi$
$182$	0	0
$183$	−87858.0	−0.193934
$184$	0	0
$185$	−81260.0	−0.174561
$186$	0	0
$187$	−306104.	−0.640125
$188$	0	0
$189$	−35721.0	−0.0727393
$190$	0	0
$191$	−747912.	−1.48343	−0.741715	−	0.670715i	$-0.765987\pi$
$191$	−747912.	−1.48343	−0.741715	+	0.670715i	$0.765987\pi$
$192$	0	0
$193$	−577534.	−1.11605	−0.558026	−	0.829824i	$-0.688441\pi$
$193$	−577534.	−1.11605	−0.558026	+	0.829824i	$0.688441\pi$
$194$	0	0
$195$	313956.	0.591265
$196$	0	0
$197$	−771098.	−1.41561	−0.707806	−	0.706407i	$-0.750315\pi$
$197$	−771098.	−1.41561	−0.707806	+	0.706407i	$0.750315\pi$
$198$	0	0
$199$	557240.	0.997492	0.498746	−	0.866748i	$-0.333794\pi$
$199$	557240.	0.997492	0.498746	+	0.866748i	$0.333794\pi$
$200$	0	0
$201$	−236268.	−0.412491
$202$	0	0
$203$	−57134.0	−0.0973093
$204$	0	0
$205$	245820.	0.408538
$206$	0	0
$207$	−305856.	−0.496125
$208$	0	0
$209$	−150064.	−0.237635
$210$	0	0
$211$	−19660.0	−0.0304003	−0.0152001	−	0.999884i	$-0.504839\pi$
$211$	−19660.0	−0.0304003	−0.0152001	+	0.999884i	$0.504839\pi$
$212$	0	0
$213$	588960.	0.889481
$214$	0	0
$215$	−158168.	−0.233358
$216$	0	0
$217$	479808.	0.691701
$218$	0	0
$219$	−50454.0	−0.0710862
$220$	0	0
$221$	−945972.	−1.30286
$222$	0	0
$223$	−896848.	−1.20769	−0.603847	−	0.797100i	$-0.706366\pi$
$223$	−896848.	−1.20769	−0.603847	+	0.797100i	$0.706366\pi$
$224$	0	0
$225$	−159489.	−0.210027
$226$	0	0
$227$	234228.	0.301699	0.150850	−	0.988557i	$-0.451799\pi$
$227$	234228.	0.301699	0.150850	+	0.988557i	$0.451799\pi$
$228$	0	0
$229$	−1.03563e6	−1.30501	−0.652506	−	0.757784i	$-0.726282\pi$
$229$	−1.03563e6	−1.30501	−0.652506	+	0.757784i	$0.726282\pi$
$230$	0	0
$231$	146412.	0.180529
$232$	0	0
$233$	457114.	0.551613	0.275807	−	0.961213i	$-0.411055\pi$
$233$	457114.	0.551613	0.275807	+	0.961213i	$0.411055\pi$
$234$	0	0
$235$	−838848.	−0.990863
$236$	0	0
$237$	−88560.0	−0.102416
$238$	0	0
$239$	676344.	0.765901	0.382951	−	0.923769i	$-0.374908\pi$
$239$	676344.	0.765901	0.382951	+	0.923769i	$0.374908\pi$
$240$	0	0
$241$	−96670.0	−0.107213	−0.0536067	−	0.998562i	$-0.517072\pi$
$241$	−96670.0	−0.107213	−0.0536067	+	0.998562i	$0.517072\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	−81634.0	−0.0868872
$246$	0	0
$247$	−463752.	−0.483664
$248$	0	0
$249$	549972.	0.562137
$250$	0	0
$251$	288876.	0.289419	0.144710	−	0.989474i	$-0.453775\pi$
$251$	288876.	0.289419	0.144710	+	0.989474i	$0.453775\pi$
$252$	0	0
$253$	1.25363e6	1.23131
$254$	0	0
$255$	−282132.	−0.271708
$256$	0	0
$257$	−711846.	−0.672285	−0.336142	−	0.941811i	$-0.609122\pi$
$257$	−711846.	−0.672285	−0.336142	+	0.941811i	$0.609122\pi$
$258$	0	0
$259$	−117110.	−0.108479
$260$	0	0
$261$	94446.0	0.0858188
$262$	0	0
$263$	−1.87368e6	−1.67034	−0.835172	−	0.549988i	$-0.814632\pi$
$263$	−1.87368e6	−1.67034	−0.835172	+	0.549988i	$0.814632\pi$
$264$	0	0
$265$	−37740.0	−0.0330132
$266$	0	0
$267$	−566622.	−0.486424
$268$	0	0
$269$	−1.37660e6	−1.15992	−0.579960	−	0.814645i	$-0.696932\pi$
$269$	−1.37660e6	−1.15992	−0.579960	+	0.814645i	$0.696932\pi$
$270$	0	0
$271$	−781776.	−0.646635	−0.323317	−	0.946291i	$-0.604798\pi$
$271$	−781776.	−0.646635	−0.323317	+	0.946291i	$0.604798\pi$
$272$	0	0
$273$	452466.	0.367434
$274$	0	0
$275$	653708.	0.521257
$276$	0	0
$277$	2.06932e6	1.62042	0.810210	−	0.586139i	$-0.199353\pi$
$277$	2.06932e6	1.62042	0.810210	+	0.586139i	$0.199353\pi$
$278$	0	0
$279$	−793152.	−0.610023
$280$	0	0
$281$	1.87911e6	1.41967	0.709835	−	0.704368i	$-0.248770\pi$
$281$	1.87911e6	1.41967	0.709835	+	0.704368i	$0.248770\pi$
$282$	0	0
$283$	670156.	0.497405	0.248702	−	0.968580i	$-0.419996\pi$
$283$	670156.	0.497405	0.248702	+	0.968580i	$0.419996\pi$
$284$	0	0
$285$	−138312.	−0.100867
$286$	0	0
$287$	354270.	0.253881
$288$	0	0
$289$	−569773.	−0.401289
$290$	0	0
$291$	−340542.	−0.235743
$292$	0	0
$293$	1.69611e6	1.15421	0.577105	−	0.816670i	$-0.304182\pi$
$293$	1.69611e6	1.15421	0.577105	+	0.816670i	$0.304182\pi$
$294$	0	0
$295$	−1.59433e6	−1.06665
$296$	0	0
$297$	−242028.	−0.159212
$298$	0	0
$299$	3.87418e6	2.50612
$300$	0	0
$301$	−227948.	−0.145017
$302$	0	0
$303$	−505314.	−0.316195
$304$	0	0
$305$	331908.	0.204300
$306$	0	0
$307$	−1.09459e6	−0.662834	−0.331417	−	0.943484i	$-0.607527\pi$
$307$	−1.09459e6	−0.662834	−0.331417	+	0.943484i	$0.607527\pi$
$308$	0	0
$309$	−237528.	−0.141520
$310$	0	0
$311$	1.21249e6	0.710848	0.355424	−	0.934705i	$-0.384337\pi$
$311$	1.21249e6	0.710848	0.355424	+	0.934705i	$0.384337\pi$
$312$	0	0
$313$	1.69436e6	0.977564	0.488782	−	0.872406i	$-0.337441\pi$
$313$	1.69436e6	0.977564	0.488782	+	0.872406i	$0.337441\pi$
$314$	0	0
$315$	134946.	0.0766273
$316$	0	0
$317$	333342.	0.186312	0.0931562	−	0.995652i	$-0.470304\pi$
$317$	333342.	0.186312	0.0931562	+	0.995652i	$0.470304\pi$
$318$	0	0
$319$	−387112.	−0.212990
$320$	0	0
$321$	424116.	0.229732
$322$	0	0
$323$	416744.	0.222261
$324$	0	0
$325$	2.02019e6	1.06092
$326$	0	0
$327$	−1.99327e6	−1.03085
$328$	0	0
$329$	−1.20893e6	−0.615759
$330$	0	0
$331$	1.83614e6	0.921162	0.460581	−	0.887618i	$-0.347641\pi$
$331$	1.83614e6	0.921162	0.460581	+	0.887618i	$0.347641\pi$
$332$	0	0
$333$	193590.	0.0956692
$334$	0	0
$335$	892568.	0.434540
$336$	0	0
$337$	−973518.	−0.466949	−0.233474	−	0.972363i	$-0.575009\pi$
$337$	−973518.	−0.466949	−0.233474	+	0.972363i	$0.575009\pi$
$338$	0	0
$339$	487746.	0.230512
$340$	0	0
$341$	3.25094e6	1.51399
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	1.15546e6	0.522644
$346$	0	0
$347$	3.39810e6	1.51500	0.757500	−	0.652835i	$-0.226421\pi$
$347$	3.39810e6	1.51500	0.757500	+	0.652835i	$0.226421\pi$
$348$	0	0
$349$	−34370.0	−0.0151048	−0.00755242	−	0.999971i	$-0.502404\pi$
$349$	−34370.0	−0.0151048	−0.00755242	+	0.999971i	$0.502404\pi$
$350$	0	0
$351$	−747954.	−0.324046
$352$	0	0
$353$	−2.50239e6	−1.06885	−0.534427	−	0.845215i	$-0.679472\pi$
$353$	−2.50239e6	−1.06885	−0.534427	+	0.845215i	$0.679472\pi$
$354$	0	0
$355$	−2.22496e6	−0.937025
$356$	0	0
$357$	−406602.	−0.168849
$358$	0	0
$359$	−3.02800e6	−1.23999	−0.619997	−	0.784604i	$-0.712866\pi$
$359$	−3.02800e6	−1.23999	−0.619997	+	0.784604i	$0.712866\pi$
$360$	0	0
$361$	−2.27180e6	−0.917490
$362$	0	0
$363$	−457443.	−0.182209
$364$	0	0
$365$	190604.	0.0748859
$366$	0	0
$367$	−3.20944e6	−1.24384	−0.621919	−	0.783081i	$-0.713647\pi$
$367$	−3.20944e6	−1.24384	−0.621919	+	0.783081i	$0.713647\pi$
$368$	0	0
$369$	−585630.	−0.223902
$370$	0	0
$371$	−54390.0	−0.0205156
$372$	0	0
$373$	1.51505e6	0.563837	0.281919	−	0.959438i	$-0.409029\pi$
$373$	1.51505e6	0.563837	0.281919	+	0.959438i	$0.409029\pi$
$374$	0	0
$375$	1.55876e6	0.572403
$376$	0	0
$377$	−1.19632e6	−0.433503
$378$	0	0
$379$	643516.	0.230124	0.115062	−	0.993358i	$-0.463293\pi$
$379$	643516.	0.230124	0.115062	+	0.993358i	$0.463293\pi$
$380$	0	0
$381$	−2.21184e6	−0.780623
$382$	0	0
$383$	4.75082e6	1.65490	0.827449	−	0.561541i	$-0.189791\pi$
$383$	4.75082e6	1.65490	0.827449	+	0.561541i	$0.189791\pi$
$384$	0	0
$385$	−553112.	−0.190178
$386$	0	0
$387$	376812.	0.127893
$388$	0	0
$389$	379574.	0.127181	0.0635905	−	0.997976i	$-0.479745\pi$
$389$	379574.	0.127181	0.0635905	+	0.997976i	$0.479745\pi$
$390$	0	0
$391$	−3.48147e6	−1.15165
$392$	0	0
$393$	−1.35241e6	−0.441700
$394$	0	0
$395$	334560.	0.107890
$396$	0	0
$397$	−5.42133e6	−1.72635	−0.863176	−	0.504902i	$-0.831528\pi$
$397$	−5.42133e6	−1.72635	−0.863176	+	0.504902i	$0.831528\pi$
$398$	0	0
$399$	−199332.	−0.0626823
$400$	0	0
$401$	−6.20643e6	−1.92744	−0.963720	−	0.266915i	$-0.913996\pi$
$401$	−6.20643e6	−1.92744	−0.963720	+	0.266915i	$0.913996\pi$
$402$	0	0
$403$	1.00466e7	3.08146
$404$	0	0
$405$	−223074.	−0.0675789
$406$	0	0
$407$	−793480.	−0.237438
$408$	0	0
$409$	−4.25397e6	−1.25744	−0.628719	−	0.777633i	$-0.716420\pi$
$409$	−4.25397e6	−1.25744	−0.628719	+	0.777633i	$0.716420\pi$
$410$	0	0
$411$	−3.61474e6	−1.05554
$412$	0	0
$413$	−2.29771e6	−0.662857
$414$	0	0
$415$	−2.07767e6	−0.592184
$416$	0	0
$417$	3.36683e6	0.948158
$418$	0	0
$419$	−725484.	−0.201880	−0.100940	−	0.994893i	$-0.532185\pi$
$419$	−725484.	−0.201880	−0.100940	+	0.994893i	$0.532185\pi$
$420$	0	0
$421$	−6.49867e6	−1.78698	−0.893489	−	0.449086i	$-0.851750\pi$
$421$	−6.49867e6	−1.78698	−0.893489	+	0.449086i	$0.851750\pi$
$422$	0	0
$423$	1.99843e6	0.543048
$424$	0	0
$425$	−1.81542e6	−0.487533
$426$	0	0
$427$	478338.	0.126959
$428$	0	0
$429$	3.06569e6	0.804238
$430$	0	0
$431$	−1.96524e6	−0.509592	−0.254796	−	0.966995i	$-0.582008\pi$
$431$	−1.96524e6	−0.509592	−0.254796	+	0.966995i	$0.582008\pi$
$432$	0	0
$433$	4.33531e6	1.11122	0.555611	−	0.831442i	$-0.312484\pi$
$433$	4.33531e6	1.11122	0.555611	+	0.831442i	$0.312484\pi$
$434$	0	0
$435$	−356796.	−0.0904059
$436$	0	0
$437$	−1.70675e6	−0.427530
$438$	0	0
$439$	6.47748e6	1.60415	0.802075	−	0.597224i	$-0.203730\pi$
$439$	6.47748e6	1.60415	0.802075	+	0.597224i	$0.203730\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	0	0
$443$	4.32696e6	1.04755	0.523774	−	0.851857i	$-0.324524\pi$
$443$	4.32696e6	1.04755	0.523774	+	0.851857i	$0.324524\pi$
$444$	0	0
$445$	2.14057e6	0.512424
$446$	0	0
$447$	−4.10438e6	−0.971580
$448$	0	0
$449$	482210.	0.112881	0.0564404	−	0.998406i	$-0.482025\pi$
$449$	482210.	0.112881	0.0564404	+	0.998406i	$0.482025\pi$
$450$	0	0
$451$	2.40036e6	0.555693
$452$	0	0
$453$	−72216.0	−0.0165344
$454$	0	0
$455$	−1.70932e6	−0.387074
$456$	0	0
$457$	8.52164e6	1.90868	0.954339	−	0.298725i	$-0.0965613\pi$
$457$	8.52164e6	1.90868	0.954339	+	0.298725i	$0.0965613\pi$
$458$	0	0
$459$	672138.	0.148911
$460$	0	0
$461$	−5.99857e6	−1.31461	−0.657303	−	0.753627i	$-0.728303\pi$
$461$	−5.99857e6	−1.31461	−0.657303	+	0.753627i	$0.728303\pi$
$462$	0	0
$463$	−4.59483e6	−0.996133	−0.498066	−	0.867139i	$-0.665956\pi$
$463$	−4.59483e6	−0.996133	−0.498066	+	0.867139i	$0.665956\pi$
$464$	0	0
$465$	2.99635e6	0.642629
$466$	0	0
$467$	8.84330e6	1.87639	0.938193	−	0.346113i	$-0.112499\pi$
$467$	8.84330e6	1.87639	0.938193	+	0.346113i	$0.112499\pi$
$468$	0	0
$469$	1.28635e6	0.270039
$470$	0	0
$471$	990702.	0.205774
$472$	0	0
$473$	−1.54446e6	−0.317413
$474$	0	0
$475$	−889988.	−0.180988
$476$	0	0
$477$	89910.0	0.0180931
$478$	0	0
$479$	−6.56062e6	−1.30649	−0.653245	−	0.757146i	$-0.726593\pi$
$479$	−6.56062e6	−1.30649	−0.653245	+	0.757146i	$0.726593\pi$
$480$	0	0
$481$	−2.45214e6	−0.483262
$482$	0	0
$483$	1.66522e6	0.324790
$484$	0	0
$485$	1.28649e6	0.248343
$486$	0	0
$487$	7.87772e6	1.50514	0.752572	−	0.658510i	$-0.228813\pi$
$487$	7.87772e6	1.50514	0.752572	+	0.658510i	$0.228813\pi$
$488$	0	0
$489$	−32652.0	−0.00617501
$490$	0	0
$491$	637860.	0.119405	0.0597024	−	0.998216i	$-0.480985\pi$
$491$	637860.	0.119405	0.0597024	+	0.998216i	$0.480985\pi$
$492$	0	0
$493$	1.07505e6	0.199210
$494$	0	0
$495$	914328.	0.167722
$496$	0	0
$497$	−3.20656e6	−0.582302
$498$	0	0
$499$	−4.93646e6	−0.887492	−0.443746	−	0.896153i	$-0.646351\pi$
$499$	−4.93646e6	−0.887492	−0.443746	+	0.896153i	$0.646351\pi$
$500$	0	0
$501$	1.73542e6	0.308894
$502$	0	0
$503$	226872.	0.0399817	0.0199908	−	0.999800i	$-0.493636\pi$
$503$	226872.	0.0399817	0.0199908	+	0.999800i	$0.493636\pi$
$504$	0	0
$505$	1.90896e6	0.333096
$506$	0	0
$507$	6.13245e6	1.05953
$508$	0	0
$509$	5.37404e6	0.919404	0.459702	−	0.888073i	$-0.347956\pi$
$509$	5.37404e6	0.919404	0.459702	+	0.888073i	$0.347956\pi$
$510$	0	0
$511$	274694.	0.0465368
$512$	0	0
$513$	329508.	0.0552806
$514$	0	0
$515$	897328.	0.149085
$516$	0	0
$517$	−8.19110e6	−1.34777
$518$	0	0
$519$	1.41428e6	0.230471
$520$	0	0
$521$	−9.61419e6	−1.55174	−0.775869	−	0.630894i	$-0.782688\pi$
$521$	−9.61419e6	−1.55174	−0.775869	+	0.630894i	$0.782688\pi$
$522$	0	0
$523$	4.96430e6	0.793604	0.396802	−	0.917904i	$-0.370120\pi$
$523$	4.96430e6	0.793604	0.396802	+	0.917904i	$0.370120\pi$
$524$	0	0
$525$	868329.	0.137495
$526$	0	0
$527$	−9.02822e6	−1.41604
$528$	0	0
$529$	7.82183e6	1.21526
$530$	0	0
$531$	3.79825e6	0.584585
$532$	0	0
$533$	7.41798e6	1.13101
$534$	0	0
$535$	−1.60222e6	−0.242012
$536$	0	0
$537$	−4.02181e6	−0.601847
$538$	0	0
$539$	−797132.	−0.118184
$540$	0	0
$541$	1.20449e7	1.76934	0.884668	−	0.466221i	$-0.154385\pi$
$541$	1.20449e7	1.76934	0.884668	+	0.466221i	$0.154385\pi$
$542$	0	0
$543$	7.25074e6	1.05532
$544$	0	0
$545$	7.53012e6	1.08595
$546$	0	0
$547$	4.23695e6	0.605459	0.302730	−	0.953077i	$-0.402102\pi$
$547$	4.23695e6	0.605459	0.302730	+	0.953077i	$0.402102\pi$
$548$	0	0
$549$	−790722.	−0.111968
$550$	0	0
$551$	527032.	0.0739534
$552$	0	0
$553$	482160.	0.0670468
$554$	0	0
$555$	−731340.	−0.100783
$556$	0	0
$557$	−1.02575e7	−1.40089	−0.700444	−	0.713708i	$-0.747014\pi$
$557$	−1.02575e7	−1.40089	−0.700444	+	0.713708i	$0.747014\pi$
$558$	0	0
$559$	−4.77295e6	−0.646037
$560$	0	0
$561$	−2.75494e6	−0.369577
$562$	0	0
$563$	−5.40777e6	−0.719031	−0.359515	−	0.933139i	$-0.617058\pi$
$563$	−5.40777e6	−0.719031	−0.359515	+	0.933139i	$0.617058\pi$
$564$	0	0
$565$	−1.84260e6	−0.242834
$566$	0	0
$567$	−321489.	−0.0419961
$568$	0	0
$569$	5.38967e6	0.697882	0.348941	−	0.937145i	$-0.386541\pi$
$569$	5.38967e6	0.697882	0.348941	+	0.937145i	$0.386541\pi$
$570$	0	0
$571$	−8.24552e6	−1.05835	−0.529173	−	0.848514i	$-0.677498\pi$
$571$	−8.24552e6	−1.05835	−0.529173	+	0.848514i	$0.677498\pi$
$572$	0	0
$573$	−6.73121e6	−0.856459
$574$	0	0
$575$	7.43494e6	0.937795
$576$	0	0
$577$	−1.15408e6	−0.144310	−0.0721549	−	0.997393i	$-0.522988\pi$
$577$	−1.15408e6	−0.144310	−0.0721549	+	0.997393i	$0.522988\pi$
$578$	0	0
$579$	−5.19781e6	−0.644353
$580$	0	0
$581$	−2.99429e6	−0.368005
$582$	0	0
$583$	−368520.	−0.0449045
$584$	0	0
$585$	2.82560e6	0.341367
$586$	0	0
$587$	−7.16464e6	−0.858221	−0.429111	−	0.903252i	$-0.641173\pi$
$587$	−7.16464e6	−0.858221	−0.429111	+	0.903252i	$0.641173\pi$
$588$	0	0
$589$	−4.42598e6	−0.525680
$590$	0	0
$591$	−6.93988e6	−0.817304
$592$	0	0
$593$	1.45534e7	1.69953	0.849763	−	0.527165i	$-0.176745\pi$
$593$	1.45534e7	1.69953	0.849763	+	0.527165i	$0.176745\pi$
$594$	0	0
$595$	1.53605e6	0.177874
$596$	0	0
$597$	5.01516e6	0.575903
$598$	0	0
$599$	−1.04320e7	−1.18795	−0.593977	−	0.804482i	$-0.702443\pi$
$599$	−1.04320e7	−1.18795	−0.593977	+	0.804482i	$0.702443\pi$
$600$	0	0
$601$	416858.	0.0470763	0.0235381	−	0.999723i	$-0.492507\pi$
$601$	416858.	0.0470763	0.0235381	+	0.999723i	$0.492507\pi$
$602$	0	0
$603$	−2.12641e6	−0.238152
$604$	0	0
$605$	1.72812e6	0.191949
$606$	0	0
$607$	−7.90834e6	−0.871191	−0.435596	−	0.900143i	$-0.643462\pi$
$607$	−7.90834e6	−0.871191	−0.435596	+	0.900143i	$0.643462\pi$
$608$	0	0
$609$	−514206.	−0.0561816
$610$	0	0
$611$	−2.53135e7	−2.74314
$612$	0	0
$613$	−1.13761e7	−1.22277	−0.611383	−	0.791335i	$-0.709387\pi$
$613$	−1.13761e7	−1.22277	−0.611383	+	0.791335i	$0.709387\pi$
$614$	0	0
$615$	2.21238e6	0.235870
$616$	0	0
$617$	−8.77271e6	−0.927728	−0.463864	−	0.885906i	$-0.653537\pi$
$617$	−8.77271e6	−0.927728	−0.463864	+	0.885906i	$0.653537\pi$
$618$	0	0
$619$	1.44110e7	1.51171	0.755854	−	0.654740i	$-0.227222\pi$
$619$	1.44110e7	1.51171	0.755854	+	0.654740i	$0.227222\pi$
$620$	0	0
$621$	−2.75270e6	−0.286438
$622$	0	0
$623$	3.08494e6	0.318439
$624$	0	0
$625$	264461.	0.0270808
$626$	0	0
$627$	−1.35058e6	−0.137199
$628$	0	0
$629$	2.20358e6	0.222076
$630$	0	0
$631$	−1.29466e7	−1.29444	−0.647221	−	0.762303i	$-0.724069\pi$
$631$	−1.29466e7	−1.29444	−0.647221	+	0.762303i	$0.724069\pi$
$632$	0	0
$633$	−176940.	−0.0175516
$634$	0	0
$635$	8.35584e6	0.822348
$636$	0	0
$637$	−2.46343e6	−0.240542
$638$	0	0
$639$	5.30064e6	0.513542
$640$	0	0
$641$	−4.89035e6	−0.470105	−0.235052	−	0.971983i	$-0.575526\pi$
$641$	−4.89035e6	−0.470105	−0.235052	+	0.971983i	$0.575526\pi$
$642$	0	0
$643$	1.22604e6	0.116943	0.0584717	−	0.998289i	$-0.481377\pi$
$643$	1.22604e6	0.116943	0.0584717	+	0.998289i	$0.481377\pi$
$644$	0	0
$645$	−1.42351e6	−0.134729
$646$	0	0
$647$	1.21098e7	1.13731	0.568654	−	0.822577i	$-0.307464\pi$
$647$	1.21098e7	1.13731	0.568654	+	0.822577i	$0.307464\pi$
$648$	0	0
$649$	−1.55681e7	−1.45086
$650$	0	0
$651$	4.31827e6	0.399354
$652$	0	0
$653$	9.28697e6	0.852298	0.426149	−	0.904653i	$-0.359870\pi$
$653$	9.28697e6	0.852298	0.426149	+	0.904653i	$0.359870\pi$
$654$	0	0
$655$	5.10911e6	0.465310
$656$	0	0
$657$	−454086.	−0.0410416
$658$	0	0
$659$	451612.	0.0405090	0.0202545	−	0.999795i	$-0.493552\pi$
$659$	451612.	0.0405090	0.0202545	+	0.999795i	$0.493552\pi$
$660$	0	0
$661$	−1.85508e6	−0.165143	−0.0825714	−	0.996585i	$-0.526313\pi$
$661$	−1.85508e6	−0.165143	−0.0825714	+	0.996585i	$0.526313\pi$
$662$	0	0
$663$	−8.51375e6	−0.752206
$664$	0	0
$665$	753032.	0.0660327
$666$	0	0
$667$	−4.40282e6	−0.383192
$668$	0	0
$669$	−8.07163e6	−0.697262
$670$	0	0
$671$	3.24098e6	0.277889
$672$	0	0
$673$	2.14534e7	1.82582	0.912911	−	0.408158i	$-0.133829\pi$
$673$	2.14534e7	1.82582	0.912911	+	0.408158i	$0.133829\pi$
$674$	0	0
$675$	−1.43540e6	−0.121259
$676$	0	0
$677$	1.56987e7	1.31641	0.658205	−	0.752839i	$-0.271316\pi$
$677$	1.56987e7	1.31641	0.658205	+	0.752839i	$0.271316\pi$
$678$	0	0
$679$	1.85406e6	0.154330
$680$	0	0
$681$	2.10805e6	0.174186
$682$	0	0
$683$	1.40250e7	1.15040	0.575201	−	0.818012i	$-0.304924\pi$
$683$	1.40250e7	1.15040	0.575201	+	0.818012i	$0.304924\pi$
$684$	0	0
$685$	1.36557e7	1.11196
$686$	0	0
$687$	−9.32063e6	−0.753449
$688$	0	0
$689$	−1.13886e6	−0.0913950
$690$	0	0
$691$	−1.89819e7	−1.51232	−0.756160	−	0.654387i	$-0.772927\pi$
$691$	−1.89819e7	−1.51232	−0.756160	+	0.654387i	$0.772927\pi$
$692$	0	0
$693$	1.31771e6	0.104228
$694$	0	0
$695$	−1.27191e7	−0.998839
$696$	0	0
$697$	−6.66606e6	−0.519741
$698$	0	0
$699$	4.11403e6	0.318474
$700$	0	0
$701$	2.22806e7	1.71250	0.856251	−	0.516560i	$-0.172788\pi$
$701$	2.22806e7	1.71250	0.856251	+	0.516560i	$0.172788\pi$
$702$	0	0
$703$	1.08028e6	0.0824419
$704$	0	0
$705$	−7.54963e6	−0.572075
$706$	0	0
$707$	2.75115e6	0.206998
$708$	0	0
$709$	−476266.	−0.0355823	−0.0177911	−	0.999842i	$-0.505663\pi$
$709$	−476266.	−0.0355823	−0.0177911	+	0.999842i	$0.505663\pi$
$710$	0	0
$711$	−797040.	−0.0591298
$712$	0	0
$713$	3.69746e7	2.72383
$714$	0	0
$715$	−1.15815e7	−0.847226
$716$	0	0
$717$	6.08710e6	0.442193
$718$	0	0
$719$	−263568.	−0.0190139	−0.00950693	−	0.999955i	$-0.503026\pi$
$719$	−263568.	−0.0190139	−0.00950693	+	0.999955i	$0.503026\pi$
$720$	0	0
$721$	1.29321e6	0.0926468
$722$	0	0
$723$	−870030.	−0.0618997
$724$	0	0
$725$	−2.29585e6	−0.162218
$726$	0	0
$727$	9.28319e6	0.651420	0.325710	−	0.945470i	$-0.394397\pi$
$727$	9.28319e6	0.651420	0.325710	+	0.945470i	$0.394397\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	4.28914e6	0.296877
$732$	0	0
$733$	−1.89547e7	−1.30304	−0.651520	−	0.758631i	$-0.725868\pi$
$733$	−1.89547e7	−1.30304	−0.651520	+	0.758631i	$0.725868\pi$
$734$	0	0
$735$	−734706.	−0.0501644
$736$	0	0
$737$	8.71566e6	0.591060
$738$	0	0
$739$	1.95454e7	1.31654	0.658269	−	0.752783i	$-0.271289\pi$
$739$	1.95454e7	1.31654	0.658269	+	0.752783i	$0.271289\pi$
$740$	0	0
$741$	−4.17377e6	−0.279243
$742$	0	0
$743$	1.54683e7	1.02795	0.513973	−	0.857806i	$-0.328173\pi$
$743$	1.54683e7	1.02795	0.513973	+	0.857806i	$0.328173\pi$
$744$	0	0
$745$	1.55054e7	1.02351
$746$	0	0
$747$	4.94975e6	0.324550
$748$	0	0
$749$	−2.30908e6	−0.150395
$750$	0	0
$751$	−1.45188e7	−0.939354	−0.469677	−	0.882838i	$-0.655630\pi$
$751$	−1.45188e7	−0.939354	−0.469677	+	0.882838i	$0.655630\pi$
$752$	0	0
$753$	2.59988e6	0.167096
$754$	0	0
$755$	272816.	0.0174182
$756$	0	0
$757$	8.54477e6	0.541952	0.270976	−	0.962586i	$-0.412654\pi$
$757$	8.54477e6	0.541952	0.270976	+	0.962586i	$0.412654\pi$
$758$	0	0
$759$	1.12827e7	0.710899
$760$	0	0
$761$	−8.50398e6	−0.532305	−0.266153	−	0.963931i	$-0.585753\pi$
$761$	−8.50398e6	−0.532305	−0.266153	+	0.963931i	$0.585753\pi$
$762$	0	0
$763$	1.08522e7	0.674850
$764$	0	0
$765$	−2.53919e6	−0.156870
$766$	0	0
$767$	−4.81112e7	−2.95296
$768$	0	0
$769$	−1.66581e7	−1.01580	−0.507901	−	0.861415i	$-0.669578\pi$
$769$	−1.66581e7	−1.01580	−0.507901	+	0.861415i	$0.669578\pi$
$770$	0	0
$771$	−6.40661e6	−0.388144
$772$	0	0
$773$	2.17326e7	1.30817	0.654083	−	0.756423i	$-0.273055\pi$
$773$	2.17326e7	1.30817	0.654083	+	0.756423i	$0.273055\pi$
$774$	0	0
$775$	1.92804e7	1.15309
$776$	0	0
$777$	−1.05399e6	−0.0626302
$778$	0	0
$779$	−3.26796e6	−0.192945
$780$	0	0
$781$	−2.17261e7	−1.27454
$782$	0	0
$783$	850014.	0.0495475
$784$	0	0
$785$	−3.74265e6	−0.216773
$786$	0	0
$787$	2.05602e6	0.118329	0.0591644	−	0.998248i	$-0.481156\pi$
$787$	2.05602e6	0.118329	0.0591644	+	0.998248i	$0.481156\pi$
$788$	0	0
$789$	−1.68631e7	−0.964374
$790$	0	0
$791$	−2.65551e6	−0.150906
$792$	0	0
$793$	1.00158e7	0.565592
$794$	0	0
$795$	−339660.	−0.0190602
$796$	0	0
$797$	3.17641e7	1.77129	0.885647	−	0.464359i	$-0.153715\pi$
$797$	3.17641e7	1.77129	0.885647	+	0.464359i	$0.153715\pi$
$798$	0	0
$799$	2.27476e7	1.26057
$800$	0	0
$801$	−5.09960e6	−0.280837
$802$	0	0
$803$	1.86119e6	0.101860
$804$	0	0
$805$	−6.29082e6	−0.342151
$806$	0	0
$807$	−1.23894e7	−0.669680
$808$	0	0
$809$	−2.14050e7	−1.14986	−0.574930	−	0.818203i	$-0.694971\pi$
$809$	−2.14050e7	−1.14986	−0.574930	+	0.818203i	$0.694971\pi$
$810$	0	0
$811$	−6.61432e6	−0.353129	−0.176564	−	0.984289i	$-0.556498\pi$
$811$	−6.61432e6	−0.353129	−0.176564	+	0.984289i	$0.556498\pi$
$812$	0	0
$813$	−7.03598e6	−0.373335
$814$	0	0
$815$	123352.	0.00650507
$816$	0	0
$817$	2.10270e6	0.110211
$818$	0	0
$819$	4.07219e6	0.212138
$820$	0	0
$821$	1.78006e7	0.921674	0.460837	−	0.887485i	$-0.347549\pi$
$821$	1.78006e7	0.921674	0.460837	+	0.887485i	$0.347549\pi$
$822$	0	0
$823$	−1.23818e7	−0.637212	−0.318606	−	0.947887i	$-0.603215\pi$
$823$	−1.23818e7	−0.637212	−0.318606	+	0.947887i	$0.603215\pi$
$824$	0	0
$825$	5.88337e6	0.300948
$826$	0	0
$827$	−2.17279e7	−1.10473	−0.552363	−	0.833604i	$-0.686274\pi$
$827$	−2.17279e7	−1.10473	−0.552363	+	0.833604i	$0.686274\pi$
$828$	0	0
$829$	−1.35893e7	−0.686771	−0.343385	−	0.939195i	$-0.611574\pi$
$829$	−1.35893e7	−0.686771	−0.343385	+	0.939195i	$0.611574\pi$
$830$	0	0
$831$	1.86239e7	0.935550
$832$	0	0
$833$	2.21372e6	0.110538
$834$	0	0
$835$	−6.55602e6	−0.325405
$836$	0	0
$837$	−7.13837e6	−0.352197
$838$	0	0
$839$	−171272.	−0.00840004	−0.00420002	−	0.999991i	$-0.501337\pi$
$839$	−171272.	−0.00840004	−0.00420002	+	0.999991i	$0.501337\pi$
$840$	0	0
$841$	−1.91516e7	−0.933716
$842$	0	0
$843$	1.69120e7	0.819647
$844$	0	0
$845$	−2.31670e7	−1.11617
$846$	0	0
$847$	2.49052e6	0.119284
$848$	0	0
$849$	6.03140e6	0.287177
$850$	0	0
$851$	−9.02464e6	−0.427175
$852$	0	0
$853$	2.90172e7	1.36547	0.682737	−	0.730664i	$-0.260789\pi$
$853$	2.90172e7	1.36547	0.682737	+	0.730664i	$0.260789\pi$
$854$	0	0
$855$	−1.24481e6	−0.0582354
$856$	0	0
$857$	−3.84967e7	−1.79049	−0.895243	−	0.445578i	$-0.852998\pi$
$857$	−3.84967e7	−1.79049	−0.895243	+	0.445578i	$0.852998\pi$
$858$	0	0
$859$	−1.98458e7	−0.917670	−0.458835	−	0.888521i	$-0.651733\pi$
$859$	−1.98458e7	−0.917670	−0.458835	+	0.888521i	$0.651733\pi$
$860$	0	0
$861$	3.18843e6	0.146578
$862$	0	0
$863$	−1.70833e7	−0.780808	−0.390404	−	0.920644i	$-0.627665\pi$
$863$	−1.70833e7	−0.780808	−0.390404	+	0.920644i	$0.627665\pi$
$864$	0	0
$865$	−5.34283e6	−0.242790
$866$	0	0
$867$	−5.12796e6	−0.231684
$868$	0	0
$869$	3.26688e6	0.146752
$870$	0	0
$871$	2.69346e7	1.20300
$872$	0	0
$873$	−3.06488e6	−0.136106
$874$	0	0
$875$	−8.48660e6	−0.374726
$876$	0	0
$877$	−2.53810e7	−1.11432	−0.557159	−	0.830406i	$-0.688109\pi$
$877$	−2.53810e7	−1.11432	−0.557159	+	0.830406i	$0.688109\pi$
$878$	0	0
$879$	1.52650e7	0.666384
$880$	0	0
$881$	−2.59580e7	−1.12676	−0.563381	−	0.826198i	$-0.690499\pi$
$881$	−2.59580e7	−1.12676	−0.563381	+	0.826198i	$0.690499\pi$
$882$	0	0
$883$	4.37666e6	0.188904	0.0944520	−	0.995529i	$-0.469890\pi$
$883$	4.37666e6	0.188904	0.0944520	+	0.995529i	$0.469890\pi$
$884$	0	0
$885$	−1.43490e7	−0.615832
$886$	0	0
$887$	3.23310e7	1.37978	0.689889	−	0.723915i	$-0.257659\pi$
$887$	3.23310e7	1.37978	0.689889	+	0.723915i	$0.257659\pi$
$888$	0	0
$889$	1.20422e7	0.511038
$890$	0	0
$891$	−2.17825e6	−0.0919208
$892$	0	0
$893$	1.11517e7	0.467966
$894$	0	0
$895$	1.51935e7	0.634017
$896$	0	0
$897$	3.48676e7	1.44691
$898$	0	0
$899$	−1.14175e7	−0.471163
$900$	0	0
$901$	1.02342e6	0.0419993
$902$	0	0
$903$	−2.05153e6	−0.0837257
$904$	0	0
$905$	−2.73917e7	−1.11173
$906$	0	0
$907$	−3.67108e7	−1.48175	−0.740876	−	0.671642i	$-0.765589\pi$
$907$	−3.67108e7	−1.48175	−0.740876	+	0.671642i	$0.765589\pi$
$908$	0	0
$909$	−4.54783e6	−0.182555
$910$	0	0
$911$	3.52261e7	1.40627	0.703135	−	0.711056i	$-0.251783\pi$
$911$	3.52261e7	1.40627	0.703135	+	0.711056i	$0.251783\pi$
$912$	0	0
$913$	−2.02879e7	−0.805488
$914$	0	0
$915$	2.98717e6	0.117953
$916$	0	0
$917$	7.36313e6	0.289161
$918$	0	0
$919$	−3.16978e6	−0.123806	−0.0619029	−	0.998082i	$-0.519717\pi$
$919$	−3.16978e6	−0.123806	−0.0619029	+	0.998082i	$0.519717\pi$
$920$	0	0
$921$	−9.85129e6	−0.382687
$922$	0	0
$923$	−6.71414e7	−2.59410
$924$	0	0
$925$	−4.70591e6	−0.180838
$926$	0	0
$927$	−2.13775e6	−0.0817068
$928$	0	0
$929$	−2.98030e7	−1.13298	−0.566488	−	0.824070i	$-0.691698\pi$
$929$	−2.98030e7	−1.13298	−0.566488	+	0.824070i	$0.691698\pi$
$930$	0	0
$931$	1.08525e6	0.0410352
$932$	0	0
$933$	1.09124e7	0.410408
$934$	0	0
$935$	1.04075e7	0.389331
$936$	0	0
$937$	−1.62312e7	−0.603952	−0.301976	−	0.953316i	$-0.597646\pi$
$937$	−1.62312e7	−0.603952	−0.301976	+	0.953316i	$0.597646\pi$
$938$	0	0
$939$	1.52493e7	0.564397
$940$	0	0
$941$	1.09759e7	0.404079	0.202040	−	0.979377i	$-0.435243\pi$
$941$	1.09759e7	0.404079	0.202040	+	0.979377i	$0.435243\pi$
$942$	0	0
$943$	2.73005e7	0.999749
$944$	0	0
$945$	1.21451e6	0.0442408
$946$	0	0
$947$	3.50304e6	0.126932	0.0634658	−	0.997984i	$-0.479785\pi$
$947$	3.50304e6	0.126932	0.0634658	+	0.997984i	$0.479785\pi$
$948$	0	0
$949$	5.75176e6	0.207317
$950$	0	0
$951$	3.00008e6	0.107568
$952$	0	0
$953$	−1.00120e7	−0.357098	−0.178549	−	0.983931i	$-0.557140\pi$
$953$	−1.00120e7	−0.357098	−0.178549	+	0.983931i	$0.557140\pi$
$954$	0	0
$955$	2.54290e7	0.902238
$956$	0	0
$957$	−3.48401e6	−0.122970
$958$	0	0
$959$	1.96803e7	0.691010
$960$	0	0
$961$	6.72541e7	2.34915
$962$	0	0
$963$	3.81704e6	0.132636
$964$	0	0
$965$	1.96362e7	0.678794
$966$	0	0
$967$	−652984.	−0.0224562	−0.0112281	−	0.999937i	$-0.503574\pi$
$967$	−652984.	−0.0224562	−0.0112281	+	0.999937i	$0.503574\pi$
$968$	0	0
$969$	3.75070e6	0.128322
$970$	0	0
$971$	−1.24897e7	−0.425112	−0.212556	−	0.977149i	$-0.568179\pi$
$971$	−1.24897e7	−0.425112	−0.212556	+	0.977149i	$0.568179\pi$
$972$	0	0
$973$	−1.83305e7	−0.620715
$974$	0	0
$975$	1.81817e7	0.612525
$976$	0	0
$977$	−7.43408e6	−0.249167	−0.124584	−	0.992209i	$-0.539759\pi$
$977$	−7.43408e6	−0.249167	−0.124584	+	0.992209i	$0.539759\pi$
$978$	0	0
$979$	2.09021e7	0.696999
$980$	0	0
$981$	−1.79394e7	−0.595162
$982$	0	0
$983$	2.44904e7	0.808373	0.404187	−	0.914677i	$-0.367555\pi$
$983$	2.44904e7	0.808373	0.404187	+	0.914677i	$0.367555\pi$
$984$	0	0
$985$	2.62173e7	0.860990
$986$	0	0
$987$	−1.08804e7	−0.355509
$988$	0	0
$989$	−1.75660e7	−0.571059
$990$	0	0
$991$	4.87464e7	1.57673	0.788367	−	0.615206i	$-0.210927\pi$
$991$	4.87464e7	1.57673	0.788367	+	0.615206i	$0.210927\pi$
$992$	0	0
$993$	1.65253e7	0.531833
$994$	0	0
$995$	−1.89462e7	−0.606685
$996$	0	0
$997$	2.35242e6	0.0749510	0.0374755	−	0.999298i	$-0.488068\pi$
$997$	2.35242e6	0.0749510	0.0374755	+	0.999298i	$0.488068\pi$
$998$	0	0
$999$	1.74231e6	0.0552347

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.6.a.b.1.1	✓	1
3.2	odd	2		252.6.a.c.1.1		1
4.3	odd	2		336.6.a.e.1.1		1
7.2	even	3		588.6.i.c.361.1		2
7.3	odd	6		588.6.i.e.373.1		2
7.4	even	3		588.6.i.c.373.1		2
7.5	odd	6		588.6.i.e.361.1		2
7.6	odd	2		588.6.a.b.1.1		1
12.11	even	2		1008.6.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.a.b.1.1	✓	1	1.1	even	1	trivial
252.6.a.c.1.1		1	3.2	odd	2
336.6.a.e.1.1		1	4.3	odd	2
588.6.a.b.1.1		1	7.6	odd	2
588.6.i.c.361.1		2	7.2	even	3
588.6.i.c.373.1		2	7.4	even	3
588.6.i.e.361.1		2	7.5	odd	6
588.6.i.e.373.1		2	7.3	odd	6
1008.6.a.u.1.1		1	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)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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