LMFDB - Embedded newform 784.6.a.bi.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,6,Mod(1,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.1"); S:= CuspForms(chi, 6); N := Newforms(S);

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

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

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,1480] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$125.740914733$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{1177})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 591x^{2} + 592x + 85262 $ x^4 - 2x^3 - 591x^2 + 592*x + 85262
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 196)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$19.0679$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+19.3093 q^{3} -49.9872 q^{5} +129.848 q^{9} +351.152 q^{11} -853.910 q^{13} -965.216 q^{15} -2267.40 q^{17} +201.628 q^{19} +1069.22 q^{23} -626.281 q^{25} -2184.88 q^{27} +6862.06 q^{29} +5402.82 q^{31} +6780.49 q^{33} +9917.10 q^{37} -16488.4 q^{39} +9383.43 q^{41} +22378.6 q^{43} -6490.74 q^{45} -6792.94 q^{47} -43781.9 q^{51} +3494.13 q^{53} -17553.1 q^{55} +3893.29 q^{57} +27543.5 q^{59} +28901.8 q^{61} +42684.6 q^{65} -70459.7 q^{67} +20645.8 q^{69} -24149.9 q^{71} -62230.6 q^{73} -12093.0 q^{75} +79757.9 q^{79} -73741.6 q^{81} +6356.40 q^{83} +113341. q^{85} +132501. q^{87} +60189.5 q^{89} +104324. q^{93} -10078.8 q^{95} +3180.12 q^{97} +45596.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 1480 q^{9} + 444 q^{11} + 3824 q^{15} - 3408 q^{23} + 11904 q^{25} + 20724 q^{29} + 13732 q^{37} - 25608 q^{39} + 28996 q^{43} - 181852 q^{51} + 528 q^{53} + 89540 q^{57} + 110220 q^{65} - 195384 q^{67}+ \cdots - 66412 q^{99}+O(q^{100}) $ 4 * q + 1480 * q^9 + 444 * q^11 + 3824 * q^15 - 3408 * q^23 + 11904 * q^25 + 20724 * q^29 + 13732 * q^37 - 25608 * q^39 + 28996 * q^43 - 181852 * q^51 + 528 * q^53 + 89540 * q^57 + 110220 * q^65 - 195384 * q^67 - 21672 * q^71 + 117304 * q^79 + 182456 * q^81 - 91300 * q^85 + 250152 * q^93 + 255552 * q^95 - 66412 * 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
$3$	19.3093	1.23869	0.619345	−	0.785119i	$-0.287398\pi$
$3$	19.3093	1.23869	0.619345	+	0.785119i	$0.287398\pi$
$4$	0	0
$5$	−49.9872	−0.894198	−0.447099	−	0.894484i	$-0.647543\pi$
$5$	−49.9872	−0.894198	−0.447099	+	0.894484i	$0.647543\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	129.848	0.534354
$10$	0	0
$11$	351.152	0.875011	0.437505	−	0.899216i	$-0.355862\pi$
$11$	351.152	0.875011	0.437505	+	0.899216i	$0.355862\pi$
$12$	0	0
$13$	−853.910	−1.40137	−0.700687	−	0.713469i	$-0.747123\pi$
$13$	−853.910	−1.40137	−0.700687	+	0.713469i	$0.747123\pi$
$14$	0	0
$15$	−965.216	−1.10763
$16$	0	0
$17$	−2267.40	−1.90286	−0.951430	−	0.307866i	$-0.900385\pi$
$17$	−2267.40	−1.90286	−0.951430	+	0.307866i	$0.900385\pi$
$18$	0	0
$19$	201.628	0.128135	0.0640674	−	0.997946i	$-0.479593\pi$
$19$	201.628	0.128135	0.0640674	+	0.997946i	$0.479593\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1069.22	0.421450	0.210725	−	0.977545i	$-0.432418\pi$
$23$	1069.22	0.421450	0.210725	+	0.977545i	$0.432418\pi$
$24$	0	0
$25$	−626.281	−0.200410
$26$	0	0
$27$	−2184.88	−0.576791
$28$	0	0
$29$	6862.06	1.51516	0.757582	−	0.652740i	$-0.226380\pi$
$29$	6862.06	1.51516	0.757582	+	0.652740i	$0.226380\pi$
$30$	0	0
$31$	5402.82	1.00975	0.504877	−	0.863191i	$-0.331538\pi$
$31$	5402.82	1.00975	0.504877	+	0.863191i	$0.331538\pi$
$32$	0	0
$33$	6780.49	1.08387
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9917.10	1.19091	0.595457	−	0.803387i	$-0.296971\pi$
$37$	9917.10	1.19091	0.595457	+	0.803387i	$0.296971\pi$
$38$	0	0
$39$	−16488.4	−1.73587
$40$	0	0
$41$	9383.43	0.871770	0.435885	−	0.900002i	$-0.356435\pi$
$41$	9383.43	0.871770	0.435885	+	0.900002i	$0.356435\pi$
$42$	0	0
$43$	22378.6	1.84570	0.922851	−	0.385158i	$-0.125853\pi$
$43$	22378.6	1.84570	0.922851	+	0.385158i	$0.125853\pi$
$44$	0	0
$45$	−6490.74	−0.477818
$46$	0	0
$47$	−6792.94	−0.448552	−0.224276	−	0.974526i	$-0.572002\pi$
$47$	−6792.94	−0.448552	−0.224276	+	0.974526i	$0.572002\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−43781.9	−2.35705
$52$	0	0
$53$	3494.13	0.170863	0.0854317	−	0.996344i	$-0.472773\pi$
$53$	3494.13	0.170863	0.0854317	+	0.996344i	$0.472773\pi$
$54$	0	0
$55$	−17553.1	−0.782433
$56$	0	0
$57$	3893.29	0.158719
$58$	0	0
$59$	27543.5	1.03012	0.515062	−	0.857153i	$-0.327769\pi$
$59$	27543.5	1.03012	0.515062	+	0.857153i	$0.327769\pi$
$60$	0	0
$61$	28901.8	0.994490	0.497245	−	0.867610i	$-0.334345\pi$
$61$	28901.8	0.994490	0.497245	+	0.867610i	$0.334345\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	42684.6	1.25311
$66$	0	0
$67$	−70459.7	−1.91758	−0.958790	−	0.284114i	$-0.908300\pi$
$67$	−70459.7	−1.91758	−0.958790	+	0.284114i	$0.908300\pi$
$68$	0	0
$69$	20645.8	0.522046
$70$	0	0
$71$	−24149.9	−0.568550	−0.284275	−	0.958743i	$-0.591753\pi$
$71$	−24149.9	−0.568550	−0.284275	+	0.958743i	$0.591753\pi$
$72$	0	0
$73$	−62230.6	−1.36677	−0.683387	−	0.730056i	$-0.739494\pi$
$73$	−62230.6	−1.36677	−0.683387	+	0.730056i	$0.739494\pi$
$74$	0	0
$75$	−12093.0	−0.248246
$76$	0	0
$77$	0	0
$78$	0	0
$79$	79757.9	1.43783	0.718913	−	0.695100i	$-0.244640\pi$
$79$	79757.9	1.43783	0.718913	+	0.695100i	$0.244640\pi$
$80$	0	0
$81$	−73741.6	−1.24882
$82$	0	0
$83$	6356.40	0.101278	0.0506391	−	0.998717i	$-0.483874\pi$
$83$	6356.40	0.101278	0.0506391	+	0.998717i	$0.483874\pi$
$84$	0	0
$85$	113341.	1.70153
$86$	0	0
$87$	132501.	1.87682
$88$	0	0
$89$	60189.5	0.805464	0.402732	−	0.915318i	$-0.368061\pi$
$89$	60189.5	0.805464	0.402732	+	0.915318i	$0.368061\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	104324.	1.25077
$94$	0	0
$95$	−10078.8	−0.114578
$96$	0	0
$97$	3180.12	0.0343174	0.0171587	−	0.999853i	$-0.494538\pi$
$97$	3180.12	0.0343174	0.0171587	+	0.999853i	$0.494538\pi$
$98$	0	0
$99$	45596.4	0.467565
$100$	0	0
$101$	115219.	1.12388	0.561941	−	0.827177i	$-0.310055\pi$
$101$	115219.	1.12388	0.561941	+	0.827177i	$0.310055\pi$
$102$	0	0
$103$	122065.	1.13370	0.566852	−	0.823820i	$-0.308161\pi$
$103$	122065.	1.13370	0.566852	+	0.823820i	$0.308161\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	54130.9	0.457074	0.228537	−	0.973535i	$-0.426606\pi$
$107$	54130.9	0.457074	0.228537	+	0.973535i	$0.426606\pi$
$108$	0	0
$109$	39838.5	0.321171	0.160586	−	0.987022i	$-0.448662\pi$
$109$	39838.5	0.321171	0.160586	+	0.987022i	$0.448662\pi$
$110$	0	0
$111$	191492.	1.47517
$112$	0	0
$113$	−68432.6	−0.504159	−0.252079	−	0.967707i	$-0.581114\pi$
$113$	−68432.6	−0.504159	−0.252079	+	0.967707i	$0.581114\pi$
$114$	0	0
$115$	−53447.1	−0.376860
$116$	0	0
$117$	−110879.	−0.748829
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−37743.2	−0.234356
$122$	0	0
$123$	181187.	1.07985
$124$	0	0
$125$	187516.	1.07340
$126$	0	0
$127$	256765.	1.41262	0.706312	−	0.707901i	$-0.250358\pi$
$127$	256765.	1.41262	0.706312	+	0.707901i	$0.250358\pi$
$128$	0	0
$129$	432114.	2.28625
$130$	0	0
$131$	−222523.	−1.13291	−0.566457	−	0.824091i	$-0.691686\pi$
$131$	−222523.	−1.13291	−0.566457	+	0.824091i	$0.691686\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	109216.	0.515766
$136$	0	0
$137$	88409.0	0.402434	0.201217	−	0.979547i	$-0.435510\pi$
$137$	88409.0	0.402434	0.201217	+	0.979547i	$0.435510\pi$
$138$	0	0
$139$	−105621.	−0.463674	−0.231837	−	0.972755i	$-0.574474\pi$
$139$	−105621.	−0.463674	−0.231837	+	0.972755i	$0.574474\pi$
$140$	0	0
$141$	−131167.	−0.555618
$142$	0	0
$143$	−299852.	−1.22622
$144$	0	0
$145$	−343015.	−1.35486
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−449212.	−1.65762	−0.828811	−	0.559528i	$-0.810982\pi$
$149$	−449212.	−1.65762	−0.828811	+	0.559528i	$0.810982\pi$
$150$	0	0
$151$	−12824.7	−0.0457727	−0.0228863	−	0.999738i	$-0.507286\pi$
$151$	−12824.7	−0.0457727	−0.0228863	+	0.999738i	$0.507286\pi$
$152$	0	0
$153$	−294418.	−1.01680
$154$	0	0
$155$	−270072.	−0.902921
$156$	0	0
$157$	155429.	0.503249	0.251624	−	0.967825i	$-0.419035\pi$
$157$	155429.	0.503249	0.251624	+	0.967825i	$0.419035\pi$
$158$	0	0
$159$	67469.1	0.211647
$160$	0	0
$161$	0	0
$162$	0	0
$163$	474063.	1.39755	0.698774	−	0.715342i	$-0.253729\pi$
$163$	474063.	1.39755	0.698774	+	0.715342i	$0.253729\pi$
$164$	0	0
$165$	−338938.	−0.969192
$166$	0	0
$167$	308200.	0.855147	0.427574	−	0.903981i	$-0.359368\pi$
$167$	308200.	0.855147	0.427574	+	0.903981i	$0.359368\pi$
$168$	0	0
$169$	357870.	0.963847
$170$	0	0
$171$	26181.0	0.0684693
$172$	0	0
$173$	−474941.	−1.20649	−0.603245	−	0.797556i	$-0.706126\pi$
$173$	−474941.	−1.20649	−0.603245	+	0.797556i	$0.706126\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	531846.	1.27601
$178$	0	0
$179$	619706.	1.44562	0.722808	−	0.691049i	$-0.242851\pi$
$179$	619706.	1.44562	0.722808	+	0.691049i	$0.242851\pi$
$180$	0	0
$181$	108568.	0.246324	0.123162	−	0.992387i	$-0.460697\pi$
$181$	108568.	0.246324	0.123162	+	0.992387i	$0.460697\pi$
$182$	0	0
$183$	558073.	1.23187
$184$	0	0
$185$	−495728.	−1.06491
$186$	0	0
$187$	−796204.	−1.66502
$188$	0	0
$189$	0	0
$190$	0	0
$191$	244317.	0.484585	0.242293	−	0.970203i	$-0.422101\pi$
$191$	244317.	0.484585	0.242293	+	0.970203i	$0.422101\pi$
$192$	0	0
$193$	64918.4	0.125451	0.0627255	−	0.998031i	$-0.480021\pi$
$193$	64918.4	0.125451	0.0627255	+	0.998031i	$0.480021\pi$
$194$	0	0
$195$	824208.	1.55221
$196$	0	0
$197$	876652.	1.60939	0.804696	−	0.593687i	$-0.202328\pi$
$197$	876652.	1.60939	0.804696	+	0.593687i	$0.202328\pi$
$198$	0	0
$199$	82565.9	0.147798	0.0738989	−	0.997266i	$-0.476456\pi$
$199$	82565.9	0.147798	0.0738989	+	0.997266i	$0.476456\pi$
$200$	0	0
$201$	−1.36053e6	−2.37529
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−469051.	−0.779535
$206$	0	0
$207$	138836.	0.225203
$208$	0	0
$209$	70802.1	0.112119
$210$	0	0
$211$	271448.	0.419740	0.209870	−	0.977729i	$-0.432696\pi$
$211$	271448.	0.419740	0.209870	+	0.977729i	$0.432696\pi$
$212$	0	0
$213$	−466316.	−0.704257
$214$	0	0
$215$	−1.11864e6	−1.65042
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.20163e6	−1.69301
$220$	0	0
$221$	1.93616e6	2.66662
$222$	0	0
$223$	−1.33163e6	−1.79316	−0.896582	−	0.442878i	$-0.853957\pi$
$223$	−1.33163e6	−1.79316	−0.896582	+	0.442878i	$0.853957\pi$
$224$	0	0
$225$	−81321.3	−0.107090
$226$	0	0
$227$	−551492.	−0.710354	−0.355177	−	0.934799i	$-0.615579\pi$
$227$	−551492.	−0.710354	−0.355177	+	0.934799i	$0.615579\pi$
$228$	0	0
$229$	759511.	0.957074	0.478537	−	0.878067i	$-0.341167\pi$
$229$	759511.	0.957074	0.478537	+	0.878067i	$0.341167\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	806195.	0.972860	0.486430	−	0.873720i	$-0.338299\pi$
$233$	806195.	0.972860	0.486430	+	0.873720i	$0.338299\pi$
$234$	0	0
$235$	339560.	0.401095
$236$	0	0
$237$	1.54007e6	1.78102
$238$	0	0
$239$	798850.	0.904629	0.452315	−	0.891858i	$-0.350598\pi$
$239$	798850.	0.904629	0.452315	+	0.891858i	$0.350598\pi$
$240$	0	0
$241$	73687.9	0.0817247	0.0408624	−	0.999165i	$-0.486989\pi$
$241$	73687.9	0.0817247	0.0408624	+	0.999165i	$0.486989\pi$
$242$	0	0
$243$	−892969.	−0.970110
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−172172.	−0.179565
$248$	0	0
$249$	122737.	0.125452
$250$	0	0
$251$	25091.9	0.0251391	0.0125696	−	0.999921i	$-0.495999\pi$
$251$	25091.9	0.0251391	0.0125696	+	0.999921i	$0.495999\pi$
$252$	0	0
$253$	375457.	0.368773
$254$	0	0
$255$	2.18854e6	2.10767
$256$	0	0
$257$	−363530.	−0.343326	−0.171663	−	0.985156i	$-0.554914\pi$
$257$	−363530.	−0.343326	−0.171663	+	0.985156i	$0.554914\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	891025.	0.809634
$262$	0	0
$263$	−2.10649e6	−1.87789	−0.938946	−	0.344064i	$-0.888196\pi$
$263$	−2.10649e6	−1.87789	−0.938946	+	0.344064i	$0.888196\pi$
$264$	0	0
$265$	−174662.	−0.152786
$266$	0	0
$267$	1.16222e6	0.997720
$268$	0	0
$269$	736198.	0.620318	0.310159	−	0.950685i	$-0.399618\pi$
$269$	736198.	0.620318	0.310159	+	0.950685i	$0.399618\pi$
$270$	0	0
$271$	−565623.	−0.467847	−0.233924	−	0.972255i	$-0.575157\pi$
$271$	−565623.	−0.467847	−0.233924	+	0.972255i	$0.575157\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−219920.	−0.175361
$276$	0	0
$277$	−512229.	−0.401111	−0.200555	−	0.979682i	$-0.564275\pi$
$277$	−512229.	−0.401111	−0.200555	+	0.979682i	$0.564275\pi$
$278$	0	0
$279$	701545.	0.539566
$280$	0	0
$281$	541721.	0.409270	0.204635	−	0.978838i	$-0.434399\pi$
$281$	541721.	0.409270	0.204635	+	0.978838i	$0.434399\pi$
$282$	0	0
$283$	1.65919e6	1.23149	0.615744	−	0.787946i	$-0.288856\pi$
$283$	1.65919e6	1.23149	0.615744	+	0.787946i	$0.288856\pi$
$284$	0	0
$285$	−194615.	−0.141927
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3.72127e6	2.62088
$290$	0	0
$291$	61405.8	0.0425086
$292$	0	0
$293$	130633.	0.0888962	0.0444481	−	0.999012i	$-0.485847\pi$
$293$	130633.	0.0888962	0.0444481	+	0.999012i	$0.485847\pi$
$294$	0	0
$295$	−1.37682e6	−0.921136
$296$	0	0
$297$	−767226.	−0.504699
$298$	0	0
$299$	−913015.	−0.590609
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2.22480e6	1.39214
$304$	0	0
$305$	−1.44472e6	−0.889271
$306$	0	0
$307$	789634.	0.478167	0.239084	−	0.970999i	$-0.423153\pi$
$307$	789634.	0.478167	0.239084	+	0.970999i	$0.423153\pi$
$308$	0	0
$309$	2.35699e6	1.40431
$310$	0	0
$311$	1.93597e6	1.13500	0.567502	−	0.823372i	$-0.307910\pi$
$311$	1.93597e6	1.13500	0.567502	+	0.823372i	$0.307910\pi$
$312$	0	0
$313$	−1.82834e6	−1.05486	−0.527430	−	0.849598i	$-0.676844\pi$
$313$	−1.82834e6	−1.05486	−0.527430	+	0.849598i	$0.676844\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.68315e6	−0.940751	−0.470376	−	0.882466i	$-0.655882\pi$
$317$	−1.68315e6	−0.940751	−0.470376	+	0.882466i	$0.655882\pi$
$318$	0	0
$319$	2.40963e6	1.32579
$320$	0	0
$321$	1.04523e6	0.566173
$322$	0	0
$323$	−457173.	−0.243823
$324$	0	0
$325$	534787.	0.280849
$326$	0	0
$327$	769252.	0.397832
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.98161e6	1.49582	0.747912	−	0.663798i	$-0.231057\pi$
$331$	2.98161e6	1.49582	0.747912	+	0.663798i	$0.231057\pi$
$332$	0	0
$333$	1.28772e6	0.636370
$334$	0	0
$335$	3.52208e6	1.71470
$336$	0	0
$337$	−1.45801e6	−0.699335	−0.349668	−	0.936874i	$-0.613705\pi$
$337$	−1.45801e6	−0.699335	−0.349668	+	0.936874i	$0.613705\pi$
$338$	0	0
$339$	−1.32138e6	−0.624496
$340$	0	0
$341$	1.89721e6	0.883547
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1.03202e6	−0.466812
$346$	0	0
$347$	−3.34680e6	−1.49213	−0.746065	−	0.665873i	$-0.768059\pi$
$347$	−3.34680e6	−1.49213	−0.746065	+	0.665873i	$0.768059\pi$
$348$	0	0
$349$	3.49755e6	1.53709	0.768546	−	0.639794i	$-0.220980\pi$
$349$	3.49755e6	1.53709	0.768546	+	0.639794i	$0.220980\pi$
$350$	0	0
$351$	1.86569e6	0.808300
$352$	0	0
$353$	3.00600e6	1.28396	0.641981	−	0.766721i	$-0.278113\pi$
$353$	3.00600e6	1.28396	0.641981	+	0.766721i	$0.278113\pi$
$354$	0	0
$355$	1.20718e6	0.508396
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.23247e6	−0.504708	−0.252354	−	0.967635i	$-0.581205\pi$
$359$	−1.23247e6	−0.504708	−0.252354	+	0.967635i	$0.581205\pi$
$360$	0	0
$361$	−2.43545e6	−0.983581
$362$	0	0
$363$	−728795.	−0.290294
$364$	0	0
$365$	3.11073e6	1.22217
$366$	0	0
$367$	−2.16047e6	−0.837305	−0.418653	−	0.908146i	$-0.637498\pi$
$367$	−2.16047e6	−0.837305	−0.418653	+	0.908146i	$0.637498\pi$
$368$	0	0
$369$	1.21842e6	0.465834
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1.33520e6	−0.496908	−0.248454	−	0.968644i	$-0.579922\pi$
$373$	−1.33520e6	−0.496908	−0.248454	+	0.968644i	$0.579922\pi$
$374$	0	0
$375$	3.62080e6	1.32962
$376$	0	0
$377$	−5.85959e6	−2.12331
$378$	0	0
$379$	3.68471e6	1.31767	0.658834	−	0.752289i	$-0.271050\pi$
$379$	3.68471e6	1.31767	0.658834	+	0.752289i	$0.271050\pi$
$380$	0	0
$381$	4.95795e6	1.74980
$382$	0	0
$383$	4.01114e6	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$383$	4.01114e6	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.90581e6	0.986257
$388$	0	0
$389$	−1.80717e6	−0.605515	−0.302757	−	0.953068i	$-0.597907\pi$
$389$	−1.80717e6	−0.605515	−0.302757	+	0.953068i	$0.597907\pi$
$390$	0	0
$391$	−2.42435e6	−0.801960
$392$	0	0
$393$	−4.29676e6	−1.40333
$394$	0	0
$395$	−3.98687e6	−1.28570
$396$	0	0
$397$	−3.28328e6	−1.04552	−0.522759	−	0.852480i	$-0.675097\pi$
$397$	−3.28328e6	−1.04552	−0.522759	+	0.852480i	$0.675097\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.09097e6	0.959918	0.479959	−	0.877291i	$-0.340652\pi$
$401$	3.09097e6	0.959918	0.479959	+	0.877291i	$0.340652\pi$
$402$	0	0
$403$	−4.61352e6	−1.41504
$404$	0	0
$405$	3.68613e6	1.11669
$406$	0	0
$407$	3.48241e6	1.04206
$408$	0	0
$409$	416847.	0.123216	0.0616082	−	0.998100i	$-0.480377\pi$
$409$	416847.	0.123216	0.0616082	+	0.998100i	$0.480377\pi$
$410$	0	0
$411$	1.70711e6	0.498492
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−317739.	−0.0905628
$416$	0	0
$417$	−2.03946e6	−0.574348
$418$	0	0
$419$	4.70440e6	1.30909	0.654544	−	0.756024i	$-0.272861\pi$
$419$	4.70440e6	1.30909	0.654544	+	0.756024i	$0.272861\pi$
$420$	0	0
$421$	−2.75525e6	−0.757626	−0.378813	−	0.925473i	$-0.623668\pi$
$421$	−2.75525e6	−0.757626	−0.378813	+	0.925473i	$0.623668\pi$
$422$	0	0
$423$	−882050.	−0.239686
$424$	0	0
$425$	1.42003e6	0.381352
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−5.78993e6	−1.51890
$430$	0	0
$431$	249909.	0.0648022	0.0324011	−	0.999475i	$-0.489685\pi$
$431$	249909.	0.0648022	0.0324011	+	0.999475i	$0.489685\pi$
$432$	0	0
$433$	−7.61708e6	−1.95240	−0.976199	−	0.216875i	$-0.930414\pi$
$433$	−7.61708e6	−1.95240	−0.976199	+	0.216875i	$0.930414\pi$
$434$	0	0
$435$	−6.62338e6	−1.67825
$436$	0	0
$437$	215584.	0.0540024
$438$	0	0
$439$	−3.30848e6	−0.819345	−0.409672	−	0.912233i	$-0.634357\pi$
$439$	−3.30848e6	−0.819345	−0.409672	+	0.912233i	$0.634357\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.39077e6	−0.336703	−0.168351	−	0.985727i	$-0.553844\pi$
$443$	−1.39077e6	−0.336703	−0.168351	+	0.985727i	$0.553844\pi$
$444$	0	0
$445$	−3.00871e6	−0.720244
$446$	0	0
$447$	−8.67396e6	−2.05328
$448$	0	0
$449$	7.39368e6	1.73079	0.865396	−	0.501089i	$-0.167067\pi$
$449$	7.39368e6	1.73079	0.865396	+	0.501089i	$0.167067\pi$
$450$	0	0
$451$	3.29501e6	0.762809
$452$	0	0
$453$	−247636.	−0.0566982
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.54533e6	−0.794085	−0.397042	−	0.917800i	$-0.629963\pi$
$457$	−3.54533e6	−0.794085	−0.397042	+	0.917800i	$0.629963\pi$
$458$	0	0
$459$	4.95402e6	1.09755
$460$	0	0
$461$	−4.59122e6	−1.00618	−0.503090	−	0.864234i	$-0.667804\pi$
$461$	−4.59122e6	−1.00618	−0.503090	+	0.864234i	$0.667804\pi$
$462$	0	0
$463$	183800.	0.0398468	0.0199234	−	0.999802i	$-0.493658\pi$
$463$	183800.	0.0398468	0.0199234	+	0.999802i	$0.493658\pi$
$464$	0	0
$465$	−5.21489e6	−1.11844
$466$	0	0
$467$	−2.58361e6	−0.548194	−0.274097	−	0.961702i	$-0.588379\pi$
$467$	−2.58361e6	−0.548194	−0.274097	+	0.961702i	$0.588379\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	3.00122e6	0.623370
$472$	0	0
$473$	7.85828e6	1.61501
$474$	0	0
$475$	−126276.	−0.0256795
$476$	0	0
$477$	453705.	0.0913015
$478$	0	0
$479$	−7.30745e6	−1.45522	−0.727608	−	0.685993i	$-0.759368\pi$
$479$	−7.30745e6	−1.45522	−0.727608	+	0.685993i	$0.759368\pi$
$480$	0	0
$481$	−8.46832e6	−1.66892
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−158965.	−0.0306866
$486$	0	0
$487$	5.08456e6	0.971473	0.485737	−	0.874105i	$-0.338551\pi$
$487$	5.08456e6	0.971473	0.485737	+	0.874105i	$0.338551\pi$
$488$	0	0
$489$	9.15381e6	1.73113
$490$	0	0
$491$	−5.83908e6	−1.09305	−0.546526	−	0.837442i	$-0.684050\pi$
$491$	−5.83908e6	−1.09305	−0.546526	+	0.837442i	$0.684050\pi$
$492$	0	0
$493$	−1.55591e7	−2.88315
$494$	0	0
$495$	−2.27923e6	−0.418096
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−8.78201e6	−1.57886	−0.789428	−	0.613843i	$-0.789623\pi$
$499$	−8.78201e6	−1.57886	−0.789428	+	0.613843i	$0.789623\pi$
$500$	0	0
$501$	5.95111e6	1.05926
$502$	0	0
$503$	2.64568e6	0.466248	0.233124	−	0.972447i	$-0.425105\pi$
$503$	2.64568e6	0.466248	0.233124	+	0.972447i	$0.425105\pi$
$504$	0	0
$505$	−5.75948e6	−1.00497
$506$	0	0
$507$	6.91020e6	1.19391
$508$	0	0
$509$	1.11720e7	1.91134	0.955668	−	0.294445i	$-0.0951349\pi$
$509$	1.11720e7	1.91134	0.955668	+	0.294445i	$0.0951349\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−440534.	−0.0739071
$514$	0	0
$515$	−6.10171e6	−1.01376
$516$	0	0
$517$	−2.38535e6	−0.392488
$518$	0	0
$519$	−9.17076e6	−1.49447
$520$	0	0
$521$	−2.17607e6	−0.351219	−0.175610	−	0.984460i	$-0.556190\pi$
$521$	−2.17607e6	−0.351219	−0.175610	+	0.984460i	$0.556190\pi$
$522$	0	0
$523$	7.26978e6	1.16216	0.581082	−	0.813845i	$-0.302630\pi$
$523$	7.26978e6	1.16216	0.581082	+	0.813845i	$0.302630\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.22504e7	−1.92142
$528$	0	0
$529$	−5.29312e6	−0.822380
$530$	0	0
$531$	3.57647e6	0.550451
$532$	0	0
$533$	−8.01261e6	−1.22168
$534$	0	0
$535$	−2.70585e6	−0.408714
$536$	0	0
$537$	1.19661e7	1.79067
$538$	0	0
$539$	0	0
$540$	0	0
$541$	3.66943e6	0.539021	0.269510	−	0.962998i	$-0.413138\pi$
$541$	3.66943e6	0.539021	0.269510	+	0.962998i	$0.413138\pi$
$542$	0	0
$543$	2.09637e6	0.305119
$544$	0	0
$545$	−1.99141e6	−0.287191
$546$	0	0
$547$	2.30515e6	0.329406	0.164703	−	0.986343i	$-0.447333\pi$
$547$	2.30515e6	0.329406	0.164703	+	0.986343i	$0.447333\pi$
$548$	0	0
$549$	3.75284e6	0.531410
$550$	0	0
$551$	1.38359e6	0.194145
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−9.57215e6	−1.31910
$556$	0	0
$557$	1.13750e7	1.55350	0.776752	−	0.629807i	$-0.216866\pi$
$557$	1.13750e7	1.55350	0.776752	+	0.629807i	$0.216866\pi$
$558$	0	0
$559$	−1.91093e7	−2.58652
$560$	0	0
$561$	−1.53741e7	−2.06245
$562$	0	0
$563$	−3.48375e6	−0.463208	−0.231604	−	0.972810i	$-0.574397\pi$
$563$	−3.48375e6	−0.463208	−0.231604	+	0.972810i	$0.574397\pi$
$564$	0	0
$565$	3.42076e6	0.450818
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.96029e6	0.642283	0.321142	−	0.947031i	$-0.395933\pi$
$569$	4.96029e6	0.642283	0.321142	+	0.947031i	$0.395933\pi$
$570$	0	0
$571$	−1.08592e7	−1.39382	−0.696912	−	0.717156i	$-0.745443\pi$
$571$	−1.08592e7	−1.39382	−0.696912	+	0.717156i	$0.745443\pi$
$572$	0	0
$573$	4.71758e6	0.600251
$574$	0	0
$575$	−669629.	−0.0844627
$576$	0	0
$577$	−5.84580e6	−0.730978	−0.365489	−	0.930816i	$-0.619098\pi$
$577$	−5.84580e6	−0.730978	−0.365489	+	0.930816i	$0.619098\pi$
$578$	0	0
$579$	1.25353e6	0.155395
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.22697e6	0.149507
$584$	0	0
$585$	5.54251e6	0.669602
$586$	0	0
$587$	4.30716e6	0.515936	0.257968	−	0.966153i	$-0.416947\pi$
$587$	4.30716e6	0.515936	0.257968	+	0.966153i	$0.416947\pi$
$588$	0	0
$589$	1.08936e6	0.129385
$590$	0	0
$591$	1.69275e7	1.99354
$592$	0	0
$593$	−1.98311e6	−0.231585	−0.115792	−	0.993273i	$-0.536941\pi$
$593$	−1.98311e6	−0.231585	−0.115792	+	0.993273i	$0.536941\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.59429e6	0.183076
$598$	0	0
$599$	5.16092e6	0.587706	0.293853	−	0.955851i	$-0.405062\pi$
$599$	5.16092e6	0.587706	0.293853	+	0.955851i	$0.405062\pi$
$600$	0	0
$601$	9.35142e6	1.05607	0.528033	−	0.849224i	$-0.322930\pi$
$601$	9.35142e6	1.05607	0.528033	+	0.849224i	$0.322930\pi$
$602$	0	0
$603$	−9.14905e6	−1.02467
$604$	0	0
$605$	1.88668e6	0.209561
$606$	0	0
$607$	−4.99700e6	−0.550475	−0.275237	−	0.961376i	$-0.588756\pi$
$607$	−4.99700e6	−0.550475	−0.275237	+	0.961376i	$0.588756\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.80056e6	0.628589
$612$	0	0
$613$	−3.63239e6	−0.390428	−0.195214	−	0.980761i	$-0.562540\pi$
$613$	−3.63239e6	−0.390428	−0.195214	+	0.980761i	$0.562540\pi$
$614$	0	0
$615$	−9.05704e6	−0.965603
$616$	0	0
$617$	8.56763e6	0.906041	0.453021	−	0.891500i	$-0.350346\pi$
$617$	8.56763e6	0.906041	0.453021	+	0.891500i	$0.350346\pi$
$618$	0	0
$619$	−1.70312e7	−1.78656	−0.893282	−	0.449497i	$-0.851603\pi$
$619$	−1.70312e7	−1.78656	−0.893282	+	0.449497i	$0.851603\pi$
$620$	0	0
$621$	−2.33611e6	−0.243089
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−7.41627e6	−0.759426
$626$	0	0
$627$	1.36714e6	0.138881
$628$	0	0
$629$	−2.24861e7	−2.26614
$630$	0	0
$631$	−5.97254e6	−0.597153	−0.298576	−	0.954386i	$-0.596512\pi$
$631$	−5.97254e6	−0.597153	−0.298576	+	0.954386i	$0.596512\pi$
$632$	0	0
$633$	5.24146e6	0.519928
$634$	0	0
$635$	−1.28350e7	−1.26317
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−3.13581e6	−0.303807
$640$	0	0
$641$	6.39027e6	0.614291	0.307146	−	0.951663i	$-0.400626\pi$
$641$	6.39027e6	0.614291	0.307146	+	0.951663i	$0.400626\pi$
$642$	0	0
$643$	−4.19353e6	−0.399993	−0.199996	−	0.979797i	$-0.564093\pi$
$643$	−4.19353e6	−0.399993	−0.199996	+	0.979797i	$0.564093\pi$
$644$	0	0
$645$	−2.16002e7	−2.04436
$646$	0	0
$647$	9.03770e6	0.848784	0.424392	−	0.905478i	$-0.360488\pi$
$647$	9.03770e6	0.848784	0.424392	+	0.905478i	$0.360488\pi$
$648$	0	0
$649$	9.67197e6	0.901370
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8.76742e6	−0.804616	−0.402308	−	0.915504i	$-0.631792\pi$
$653$	−8.76742e6	−0.804616	−0.402308	+	0.915504i	$0.631792\pi$
$654$	0	0
$655$	1.11233e7	1.01305
$656$	0	0
$657$	−8.08051e6	−0.730341
$658$	0	0
$659$	−1.87668e7	−1.68336	−0.841680	−	0.539977i	$-0.818433\pi$
$659$	−1.87668e7	−1.68336	−0.841680	+	0.539977i	$0.818433\pi$
$660$	0	0
$661$	−7.01945e6	−0.624884	−0.312442	−	0.949937i	$-0.601147\pi$
$661$	−7.01945e6	−0.624884	−0.312442	+	0.949937i	$0.601147\pi$
$662$	0	0
$663$	3.73858e7	3.30311
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.33703e6	0.638566
$668$	0	0
$669$	−2.57127e7	−2.22117
$670$	0	0
$671$	1.01489e7	0.870190
$672$	0	0
$673$	1.21574e7	1.03468	0.517338	−	0.855781i	$-0.326923\pi$
$673$	1.21574e7	1.03468	0.517338	+	0.855781i	$0.326923\pi$
$674$	0	0
$675$	1.36835e6	0.115595
$676$	0	0
$677$	890977.	0.0747127	0.0373564	−	0.999302i	$-0.488106\pi$
$677$	890977.	0.0747127	0.0373564	+	0.999302i	$0.488106\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1.06489e7	−0.879909
$682$	0	0
$683$	−1.97098e7	−1.61670	−0.808351	−	0.588700i	$-0.799640\pi$
$683$	−1.97098e7	−1.61670	−0.808351	+	0.588700i	$0.799640\pi$
$684$	0	0
$685$	−4.41932e6	−0.359856
$686$	0	0
$687$	1.46656e7	1.18552
$688$	0	0
$689$	−2.98367e6	−0.239444
$690$	0	0
$691$	1.37488e7	1.09539	0.547697	−	0.836677i	$-0.315505\pi$
$691$	1.37488e7	1.09539	0.547697	+	0.836677i	$0.315505\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.27969e6	0.414616
$696$	0	0
$697$	−2.12760e7	−1.65886
$698$	0	0
$699$	1.55670e7	1.20507
$700$	0	0
$701$	1.40261e7	1.07806	0.539031	−	0.842286i	$-0.318791\pi$
$701$	1.40261e7	1.07806	0.539031	+	0.842286i	$0.318791\pi$
$702$	0	0
$703$	1.99957e6	0.152598
$704$	0	0
$705$	6.55666e6	0.496832
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.19993e7	0.896479	0.448239	−	0.893914i	$-0.352051\pi$
$709$	1.19993e7	0.896479	0.448239	+	0.893914i	$0.352051\pi$
$710$	0	0
$711$	1.03564e7	0.768307
$712$	0	0
$713$	5.77678e6	0.425561
$714$	0	0
$715$	1.49888e7	1.09648
$716$	0	0
$717$	1.54252e7	1.12056
$718$	0	0
$719$	2.31346e7	1.66894	0.834470	−	0.551054i	$-0.185774\pi$
$719$	2.31346e7	1.66894	0.834470	+	0.551054i	$0.185774\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1.42286e6	0.101232
$724$	0	0
$725$	−4.29758e6	−0.303654
$726$	0	0
$727$	117783.	0.00826505	0.00413252	−	0.999991i	$-0.498685\pi$
$727$	117783.	0.00826505	0.00413252	+	0.999991i	$0.498685\pi$
$728$	0	0
$729$	676615.	0.0471545
$730$	0	0
$731$	−5.07413e7	−3.51211
$732$	0	0
$733$	1.09660e7	0.753854	0.376927	−	0.926243i	$-0.376981\pi$
$733$	1.09660e7	0.753854	0.376927	+	0.926243i	$0.376981\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.47421e7	−1.67790
$738$	0	0
$739$	−4.66161e6	−0.313996	−0.156998	−	0.987599i	$-0.550182\pi$
$739$	−4.66161e6	−0.313996	−0.156998	+	0.987599i	$0.550182\pi$
$740$	0	0
$741$	−3.32452e6	−0.222425
$742$	0	0
$743$	1.32766e7	0.882296	0.441148	−	0.897434i	$-0.354571\pi$
$743$	1.32766e7	0.882296	0.441148	+	0.897434i	$0.354571\pi$
$744$	0	0
$745$	2.24549e7	1.48224
$746$	0	0
$747$	825366.	0.0541184
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1.26804e7	0.820412	0.410206	−	0.911993i	$-0.365457\pi$
$751$	1.26804e7	0.820412	0.410206	+	0.911993i	$0.365457\pi$
$752$	0	0
$753$	484507.	0.0311396
$754$	0	0
$755$	641073.	0.0409298
$756$	0	0
$757$	−9.32262e6	−0.591287	−0.295643	−	0.955298i	$-0.595534\pi$
$757$	−9.32262e6	−0.591287	−0.295643	+	0.955298i	$0.595534\pi$
$758$	0	0
$759$	7.24981e6	0.456796
$760$	0	0
$761$	1.89841e7	1.18831	0.594153	−	0.804352i	$-0.297487\pi$
$761$	1.89841e7	1.18831	0.594153	+	0.804352i	$0.297487\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.47171e7	0.909221
$766$	0	0
$767$	−2.35197e7	−1.44359
$768$	0	0
$769$	9.22276e6	0.562400	0.281200	−	0.959649i	$-0.409268\pi$
$769$	9.22276e6	0.562400	0.281200	+	0.959649i	$0.409268\pi$
$770$	0	0
$771$	−7.01949e6	−0.425275
$772$	0	0
$773$	−7.55017e6	−0.454473	−0.227236	−	0.973840i	$-0.572969\pi$
$773$	−7.55017e6	−0.454473	−0.227236	+	0.973840i	$0.572969\pi$
$774$	0	0
$775$	−3.38368e6	−0.202365
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.89196e6	0.111704
$780$	0	0
$781$	−8.48027e6	−0.497488
$782$	0	0
$783$	−1.49928e7	−0.873934
$784$	0	0
$785$	−7.76946e6	−0.450004
$786$	0	0
$787$	5.66483e6	0.326024	0.163012	−	0.986624i	$-0.447879\pi$
$787$	5.66483e6	0.326024	0.163012	+	0.986624i	$0.447879\pi$
$788$	0	0
$789$	−4.06748e7	−2.32613
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.46796e7	−1.39365
$794$	0	0
$795$	−3.37259e6	−0.189254
$796$	0	0
$797$	2.25166e6	0.125561	0.0627807	−	0.998027i	$-0.480003\pi$
$797$	2.25166e6	0.125561	0.0627807	+	0.998027i	$0.480003\pi$
$798$	0	0
$799$	1.54023e7	0.853532
$800$	0	0
$801$	7.81549e6	0.430403
$802$	0	0
$803$	−2.18524e7	−1.19594
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.42155e7	0.768381
$808$	0	0
$809$	1.19636e7	0.642675	0.321338	−	0.946965i	$-0.395868\pi$
$809$	1.19636e7	0.642675	0.321338	+	0.946965i	$0.395868\pi$
$810$	0	0
$811$	−958303.	−0.0511624	−0.0255812	−	0.999673i	$-0.508144\pi$
$811$	−958303.	−0.0511624	−0.0255812	+	0.999673i	$0.508144\pi$
$812$	0	0
$813$	−1.09218e7	−0.579518
$814$	0	0
$815$	−2.36971e7	−1.24969
$816$	0	0
$817$	4.51215e6	0.236499
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.56029e7	1.32566	0.662828	−	0.748772i	$-0.269356\pi$
$821$	2.56029e7	1.32566	0.662828	+	0.748772i	$0.269356\pi$
$822$	0	0
$823$	1.15955e7	0.596748	0.298374	−	0.954449i	$-0.403556\pi$
$823$	1.15955e7	0.596748	0.298374	+	0.954449i	$0.403556\pi$
$824$	0	0
$825$	−4.24649e6	−0.217218
$826$	0	0
$827$	4.22668e6	0.214900	0.107450	−	0.994211i	$-0.465731\pi$
$827$	4.22668e6	0.214900	0.107450	+	0.994211i	$0.465731\pi$
$828$	0	0
$829$	1.33000e7	0.672148	0.336074	−	0.941836i	$-0.390901\pi$
$829$	1.33000e7	0.672148	0.336074	+	0.941836i	$0.390901\pi$
$830$	0	0
$831$	−9.89076e6	−0.496852
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.54060e7	−0.764671
$836$	0	0
$837$	−1.18045e7	−0.582418
$838$	0	0
$839$	−3.03054e7	−1.48633	−0.743166	−	0.669107i	$-0.766677\pi$
$839$	−3.03054e7	−1.48633	−0.743166	+	0.669107i	$0.766677\pi$
$840$	0	0
$841$	2.65768e7	1.29572
$842$	0	0
$843$	1.04602e7	0.506959
$844$	0	0
$845$	−1.78889e7	−0.861870
$846$	0	0
$847$	0	0
$848$	0	0
$849$	3.20378e7	1.52543
$850$	0	0
$851$	1.06035e7	0.501911
$852$	0	0
$853$	−1.04047e7	−0.489616	−0.244808	−	0.969572i	$-0.578725\pi$
$853$	−1.04047e7	−0.489616	−0.244808	+	0.969572i	$0.578725\pi$
$854$	0	0
$855$	−1.30872e6	−0.0612251
$856$	0	0
$857$	−2.19500e7	−1.02090	−0.510448	−	0.859909i	$-0.670520\pi$
$857$	−2.19500e7	−1.02090	−0.510448	+	0.859909i	$0.670520\pi$
$858$	0	0
$859$	−9.08936e6	−0.420291	−0.210146	−	0.977670i	$-0.567394\pi$
$859$	−9.08936e6	−0.420291	−0.210146	+	0.977670i	$0.567394\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.82502e6	−0.0834141	−0.0417071	−	0.999130i	$-0.513280\pi$
$863$	−1.82502e6	−0.0834141	−0.0417071	+	0.999130i	$0.513280\pi$
$864$	0	0
$865$	2.37409e7	1.07884
$866$	0	0
$867$	7.18550e7	3.24645
$868$	0	0
$869$	2.80072e7	1.25811
$870$	0	0
$871$	6.01662e7	2.68725
$872$	0	0
$873$	412932.	0.0183376
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.49949e7	−0.658331	−0.329166	−	0.944272i	$-0.606767\pi$
$877$	−1.49949e7	−0.658331	−0.329166	+	0.944272i	$0.606767\pi$
$878$	0	0
$879$	2.52242e6	0.110115
$880$	0	0
$881$	−1.71560e6	−0.0744692	−0.0372346	−	0.999307i	$-0.511855\pi$
$881$	−1.71560e6	−0.0744692	−0.0372346	+	0.999307i	$0.511855\pi$
$882$	0	0
$883$	−3.86989e7	−1.67031	−0.835155	−	0.550015i	$-0.814622\pi$
$883$	−3.86989e7	−1.67031	−0.835155	+	0.550015i	$0.814622\pi$
$884$	0	0
$885$	−2.65855e7	−1.14100
$886$	0	0
$887$	−5.82614e6	−0.248641	−0.124320	−	0.992242i	$-0.539675\pi$
$887$	−5.82614e6	−0.248641	−0.124320	+	0.992242i	$0.539675\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−2.58945e7	−1.09273
$892$	0	0
$893$	−1.36965e6	−0.0574752
$894$	0	0
$895$	−3.09774e7	−1.29267
$896$	0	0
$897$	−1.76297e7	−0.731581
$898$	0	0
$899$	3.70745e7	1.52994
$900$	0	0
$901$	−7.92260e6	−0.325129
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.42702e6	−0.220262
$906$	0	0
$907$	−9.83957e6	−0.397153	−0.198577	−	0.980085i	$-0.563632\pi$
$907$	−9.83957e6	−0.397153	−0.198577	+	0.980085i	$0.563632\pi$
$908$	0	0
$909$	1.49610e7	0.600551
$910$	0	0
$911$	2.53673e7	1.01269	0.506347	−	0.862330i	$-0.330996\pi$
$911$	2.53673e7	1.01269	0.506347	+	0.862330i	$0.330996\pi$
$912$	0	0
$913$	2.23206e6	0.0886196
$914$	0	0
$915$	−2.78965e7	−1.10153
$916$	0	0
$917$	0	0
$918$	0	0
$919$	5.85847e6	0.228821	0.114410	−	0.993434i	$-0.463502\pi$
$919$	5.85847e6	0.228821	0.114410	+	0.993434i	$0.463502\pi$
$920$	0	0
$921$	1.52472e7	0.592301
$922$	0	0
$923$	2.06218e7	0.796751
$924$	0	0
$925$	−6.21089e6	−0.238671
$926$	0	0
$927$	1.58500e7	0.605799
$928$	0	0
$929$	18528.6	0.000704374 0	0.000352187	−	1.00000i	$-0.499888\pi$
$929$	18528.6	0.000704374 0	0.000352187		1.00000i	$0.499888\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.73821e7	1.40592
$934$	0	0
$935$	3.98000e7	1.48886
$936$	0	0
$937$	3.91119e7	1.45533	0.727663	−	0.685935i	$-0.240606\pi$
$937$	3.91119e7	1.45533	0.727663	+	0.685935i	$0.240606\pi$
$938$	0	0
$939$	−3.53038e7	−1.30665
$940$	0	0
$941$	−8.90516e6	−0.327844	−0.163922	−	0.986473i	$-0.552415\pi$
$941$	−8.90516e6	−0.327844	−0.163922	+	0.986473i	$0.552415\pi$
$942$	0	0
$943$	1.00329e7	0.367408
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.43836e6	0.269527	0.134764	−	0.990878i	$-0.456973\pi$
$947$	7.43836e6	0.269527	0.134764	+	0.990878i	$0.456973\pi$
$948$	0	0
$949$	5.31393e7	1.91536
$950$	0	0
$951$	−3.25004e7	−1.16530
$952$	0	0
$953$	3.87886e7	1.38348	0.691739	−	0.722148i	$-0.256845\pi$
$953$	3.87886e7	1.38348	0.691739	+	0.722148i	$0.256845\pi$
$954$	0	0
$955$	−1.22127e7	−0.433315
$956$	0	0
$957$	4.65282e7	1.64224
$958$	0	0
$959$	0	0
$960$	0	0
$961$	561277.	0.0196051
$962$	0	0
$963$	7.02879e6	0.244239
$964$	0	0
$965$	−3.24509e6	−0.112178
$966$	0	0
$967$	2.80184e7	0.963557	0.481779	−	0.876293i	$-0.339991\pi$
$967$	2.80184e7	0.963557	0.481779	+	0.876293i	$0.339991\pi$
$968$	0	0
$969$	−8.82767e6	−0.302021
$970$	0	0
$971$	−4.45242e7	−1.51547	−0.757736	−	0.652561i	$-0.773695\pi$
$971$	−4.45242e7	−1.51547	−0.757736	+	0.652561i	$0.773695\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.03264e7	0.347885
$976$	0	0
$977$	922043.	0.0309040	0.0154520	−	0.999881i	$-0.495081\pi$
$977$	922043.	0.0309040	0.0154520	+	0.999881i	$0.495081\pi$
$978$	0	0
$979$	2.11357e7	0.704790
$980$	0	0
$981$	5.17294e6	0.171619
$982$	0	0
$983$	4.12706e6	0.136225	0.0681125	−	0.997678i	$-0.478302\pi$
$983$	4.12706e6	0.136225	0.0681125	+	0.997678i	$0.478302\pi$
$984$	0	0
$985$	−4.38214e7	−1.43912
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.39275e7	0.777871
$990$	0	0
$991$	−3.49392e6	−0.113013	−0.0565065	−	0.998402i	$-0.517996\pi$
$991$	−3.49392e6	−0.113013	−0.0565065	+	0.998402i	$0.517996\pi$
$992$	0	0
$993$	5.75726e7	1.85286
$994$	0	0
$995$	−4.12724e6	−0.132161
$996$	0	0
$997$	−1.16105e7	−0.369924	−0.184962	−	0.982746i	$-0.559216\pi$
$997$	−1.16105e7	−0.369924	−0.184962	+	0.982746i	$0.559216\pi$
$998$	0	0
$999$	−2.16677e7	−0.686909

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.bi.1.3		4
4.3	odd	2		196.6.a.k.1.2	✓	4
7.6	odd	2	inner	784.6.a.bi.1.2		4
28.3	even	6		196.6.e.l.177.2		8
28.11	odd	6		196.6.e.l.177.3		8
28.19	even	6		196.6.e.l.165.2		8
28.23	odd	6		196.6.e.l.165.3		8
28.27	even	2		196.6.a.k.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
196.6.a.k.1.2	✓	4	4.3	odd	2
196.6.a.k.1.3	yes	4	28.27	even	2
196.6.e.l.165.2		8	28.19	even	6
196.6.e.l.165.3		8	28.23	odd	6
196.6.e.l.177.2		8	28.3	even	6
196.6.e.l.177.3		8	28.11	odd	6
784.6.a.bi.1.2		4	7.6	odd	2	inner
784.6.a.bi.1.3		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+19.3093 q^{3} -49.9872 q^{5} +129.848 q^{9} +351.152 q^{11} -853.910 q^{13} -965.216 q^{15} -2267.40 q^{17} +201.628 q^{19} +1069.22 q^{23} -626.281 q^{25} -2184.88 q^{27} +6862.06 q^{29} +5402.82 q^{31} +6780.49 q^{33} +9917.10 q^{37} -16488.4 q^{39} +9383.43 q^{41} +22378.6 q^{43} -6490.74 q^{45} -6792.94 q^{47} -43781.9 q^{51} +3494.13 q^{53} -17553.1 q^{55} +3893.29 q^{57} +27543.5 q^{59} +28901.8 q^{61} +42684.6 q^{65} -70459.7 q^{67} +20645.8 q^{69} -24149.9 q^{71} -62230.6 q^{73} -12093.0 q^{75} +79757.9 q^{79} -73741.6 q^{81} +6356.40 q^{83} +113341. q^{85} +132501. q^{87} +60189.5 q^{89} +104324. q^{93} -10078.8 q^{95} +3180.12 q^{97} +45596.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 1480 q^{9} + 444 q^{11} + 3824 q^{15} - 3408 q^{23} + 11904 q^{25} + 20724 q^{29} + 13732 q^{37} - 25608 q^{39} + 28996 q^{43} - 181852 q^{51} + 528 q^{53} + 89540 q^{57} + 110220 q^{65} - 195384 q^{67}+ \cdots - 66412 q^{99}+O(q^{100}) \) 4 * q + 1480 * q^9 + 444 * q^11 + 3824 * q^15 - 3408 * q^23 + 11904 * q^25 + 20724 * q^29 + 13732 * q^37 - 25608 * q^39 + 28996 * q^43 - 181852 * q^51 + 528 * q^53 + 89540 * q^57 + 110220 * q^65 - 195384 * q^67 - 21672 * q^71 + 117304 * q^79 + 182456 * q^81 - 91300 * q^85 + 250152 * q^93 + 255552 * q^95 - 66412 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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