LMFDB - Embedded newform 784.6.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,6,Mod(1,784)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.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:	$125.740914733$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{37}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9 $ x^2 - x - 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.54138$ 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)

$f(q)$	$=$	$q-2.08276 q^{3} +41.8276 q^{5} -238.662 q^{9} +O(q^{10})$	$q-2.08276 q^{3} +41.8276 q^{5} -238.662 q^{9} -72.0965 q^{11} -632.317 q^{13} -87.1170 q^{15} -1975.92 q^{17} +1864.93 q^{19} -413.711 q^{23} -1375.45 q^{25} +1003.19 q^{27} +731.934 q^{29} -6123.18 q^{31} +150.160 q^{33} +10350.4 q^{37} +1316.97 q^{39} -3529.84 q^{41} +14515.2 q^{43} -9982.67 q^{45} +21423.3 q^{47} +4115.38 q^{51} +12579.5 q^{53} -3015.62 q^{55} -3884.20 q^{57} +36133.9 q^{59} +4024.80 q^{61} -26448.3 q^{65} -15565.9 q^{67} +861.661 q^{69} -12180.8 q^{71} +19589.1 q^{73} +2864.74 q^{75} -36089.8 q^{79} +55905.5 q^{81} +24572.6 q^{83} -82648.2 q^{85} -1524.44 q^{87} -70243.3 q^{89} +12753.1 q^{93} +78005.4 q^{95} +105758. q^{97} +17206.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{3} - 38 q^{5} - 380 q^{9}+O(q^{10}) $ 2 * q + 8 * q^3 - 38 * q^5 - 380 * q^9	$ 2 q + 8 q^{3} - 38 q^{5} - 380 q^{9} - 424 q^{11} - 924 q^{13} - 892 q^{15} - 2346 q^{17} + 360 q^{19} + 12 q^{23} + 1872 q^{25} - 2872 q^{27} - 7052 q^{29} - 3548 q^{31} - 3398 q^{33} + 11090 q^{37} - 1624 q^{39} + 3500 q^{41} + 12680 q^{43} + 1300 q^{45} + 22956 q^{47} + 384 q^{51} + 3042 q^{53} + 25076 q^{55} - 19058 q^{57} + 65808 q^{59} - 42486 q^{61} - 3164 q^{65} - 42312 q^{67} + 5154 q^{69} + 2208 q^{71} - 50506 q^{73} + 35608 q^{75} - 9004 q^{79} + 51178 q^{81} + 104328 q^{83} - 53106 q^{85} - 80008 q^{87} - 26666 q^{89} + 38718 q^{93} + 198140 q^{95} + 209132 q^{97} + 66944 q^{99}+O(q^{100}) $ 2 * q + 8 * q^3 - 38 * q^5 - 380 * q^9 - 424 * q^11 - 924 * q^13 - 892 * q^15 - 2346 * q^17 + 360 * q^19 + 12 * q^23 + 1872 * q^25 - 2872 * q^27 - 7052 * q^29 - 3548 * q^31 - 3398 * q^33 + 11090 * q^37 - 1624 * q^39 + 3500 * q^41 + 12680 * q^43 + 1300 * q^45 + 22956 * q^47 + 384 * q^51 + 3042 * q^53 + 25076 * q^55 - 19058 * q^57 + 65808 * q^59 - 42486 * q^61 - 3164 * q^65 - 42312 * q^67 + 5154 * q^69 + 2208 * q^71 - 50506 * q^73 + 35608 * q^75 - 9004 * q^79 + 51178 * q^81 + 104328 * q^83 - 53106 * q^85 - 80008 * q^87 - 26666 * q^89 + 38718 * q^93 + 198140 * q^95 + 209132 * q^97 + 66944 * 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
$3$	−2.08276	−0.133609	−0.0668046	−	0.997766i	$-0.521280\pi$
$3$	−2.08276	−0.133609	−0.0668046	+	0.997766i	$0.521280\pi$
$4$	0	0
$5$	41.8276	0.748235	0.374118	−	0.927381i	$-0.377946\pi$
$5$	41.8276	0.748235	0.374118	+	0.927381i	$0.377946\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−238.662	−0.982149
$10$	0	0
$11$	−72.0965	−0.179652	−0.0898260	−	0.995957i	$-0.528631\pi$
$11$	−72.0965	−0.179652	−0.0898260	+	0.995957i	$0.528631\pi$
$12$	0	0
$13$	−632.317	−1.03771	−0.518856	−	0.854862i	$-0.673642\pi$
$13$	−632.317	−1.03771	−0.518856	+	0.854862i	$0.673642\pi$
$14$	0	0
$15$	−87.1170	−0.0999712
$16$	0	0
$17$	−1975.92	−1.65824	−0.829121	−	0.559069i	$-0.811159\pi$
$17$	−1975.92	−1.65824	−0.829121	+	0.559069i	$0.811159\pi$
$18$	0	0
$19$	1864.93	1.18516	0.592581	−	0.805511i	$-0.298109\pi$
$19$	1864.93	1.18516	0.592581	+	0.805511i	$0.298109\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−413.711	−0.163071	−0.0815356	−	0.996670i	$-0.525982\pi$
$23$	−413.711	−0.163071	−0.0815356	+	0.996670i	$0.525982\pi$
$24$	0	0
$25$	−1375.45	−0.440144
$26$	0	0
$27$	1003.19	0.264833
$28$	0	0
$29$	731.934	0.161613	0.0808066	−	0.996730i	$-0.474250\pi$
$29$	731.934	0.161613	0.0808066	+	0.996730i	$0.474250\pi$
$30$	0	0
$31$	−6123.18	−1.14439	−0.572193	−	0.820119i	$-0.693907\pi$
$31$	−6123.18	−1.14439	−0.572193	+	0.820119i	$0.693907\pi$
$32$	0	0
$33$	150.160	0.0240032
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10350.4	1.24295	0.621473	−	0.783436i	$-0.286535\pi$
$37$	10350.4	1.24295	0.621473	+	0.783436i	$0.286535\pi$
$38$	0	0
$39$	1316.97	0.138648
$40$	0	0
$41$	−3529.84	−0.327941	−0.163970	−	0.986465i	$-0.552430\pi$
$41$	−3529.84	−0.327941	−0.163970	+	0.986465i	$0.552430\pi$
$42$	0	0
$43$	14515.2	1.19716	0.598581	−	0.801062i	$-0.295731\pi$
$43$	14515.2	1.19716	0.598581	+	0.801062i	$0.295731\pi$
$44$	0	0
$45$	−9982.67	−0.734878
$46$	0	0
$47$	21423.3	1.41463	0.707314	−	0.706900i	$-0.249907\pi$
$47$	21423.3	1.41463	0.707314	+	0.706900i	$0.249907\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4115.38	0.221557
$52$	0	0
$53$	12579.5	0.615138	0.307569	−	0.951526i	$-0.400485\pi$
$53$	12579.5	0.615138	0.307569	+	0.951526i	$0.400485\pi$
$54$	0	0
$55$	−3015.62	−0.134422
$56$	0	0
$57$	−3884.20	−0.158349
$58$	0	0
$59$	36133.9	1.35140	0.675702	−	0.737175i	$-0.263841\pi$
$59$	36133.9	1.35140	0.675702	+	0.737175i	$0.263841\pi$
$60$	0	0
$61$	4024.80	0.138490	0.0692451	−	0.997600i	$-0.477941\pi$
$61$	4024.80	0.138490	0.0692451	+	0.997600i	$0.477941\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−26448.3	−0.776453
$66$	0	0
$67$	−15565.9	−0.423632	−0.211816	−	0.977310i	$-0.567938\pi$
$67$	−15565.9	−0.423632	−0.211816	+	0.977310i	$0.567938\pi$
$68$	0	0
$69$	861.661	0.0217878
$70$	0	0
$71$	−12180.8	−0.286766	−0.143383	−	0.989667i	$-0.545798\pi$
$71$	−12180.8	−0.286766	−0.143383	+	0.989667i	$0.545798\pi$
$72$	0	0
$73$	19589.1	0.430237	0.215119	−	0.976588i	$-0.430986\pi$
$73$	19589.1	0.430237	0.215119	+	0.976588i	$0.430986\pi$
$74$	0	0
$75$	2864.74	0.0588073
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−36089.8	−0.650604	−0.325302	−	0.945610i	$-0.605466\pi$
$79$	−36089.8	−0.650604	−0.325302	+	0.945610i	$0.605466\pi$
$80$	0	0
$81$	55905.5	0.946764
$82$	0	0
$83$	24572.6	0.391522	0.195761	−	0.980652i	$-0.437282\pi$
$83$	24572.6	0.391522	0.195761	+	0.980652i	$0.437282\pi$
$84$	0	0
$85$	−82648.2	−1.24076
$86$	0	0
$87$	−1524.44	−0.0215930
$88$	0	0
$89$	−70243.3	−0.940005	−0.470002	−	0.882665i	$-0.655747\pi$
$89$	−70243.3	−0.940005	−0.470002	+	0.882665i	$0.655747\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	12753.1	0.152901
$94$	0	0
$95$	78005.4	0.886779
$96$	0	0
$97$	105758.	1.14126	0.570630	−	0.821207i	$-0.306699\pi$
$97$	105758.	1.14126	0.570630	+	0.821207i	$0.306699\pi$
$98$	0	0
$99$	17206.7	0.176445
$100$	0	0
$101$	36461.8	0.355660	0.177830	−	0.984061i	$-0.443092\pi$
$101$	36461.8	0.355660	0.177830	+	0.984061i	$0.443092\pi$
$102$	0	0
$103$	−64520.1	−0.599242	−0.299621	−	0.954058i	$-0.596860\pi$
$103$	−64520.1	−0.599242	−0.299621	+	0.954058i	$0.596860\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−66045.6	−0.557679	−0.278840	−	0.960338i	$-0.589950\pi$
$107$	−66045.6	−0.557679	−0.278840	+	0.960338i	$0.589950\pi$
$108$	0	0
$109$	−37938.0	−0.305850	−0.152925	−	0.988238i	$-0.548869\pi$
$109$	−37938.0	−0.305850	−0.152925	+	0.988238i	$0.548869\pi$
$110$	0	0
$111$	−21557.4	−0.166069
$112$	0	0
$113$	123802.	0.912080	0.456040	−	0.889959i	$-0.349267\pi$
$113$	123802.	0.912080	0.456040	+	0.889959i	$0.349267\pi$
$114$	0	0
$115$	−17304.5	−0.122016
$116$	0	0
$117$	150910.	1.01919
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−155853.	−0.967725
$122$	0	0
$123$	7351.81	0.0438159
$124$	0	0
$125$	−188243.	−1.07757
$126$	0	0
$127$	−128724.	−0.708189	−0.354095	−	0.935210i	$-0.615211\pi$
$127$	−128724.	−0.708189	−0.354095	+	0.935210i	$0.615211\pi$
$128$	0	0
$129$	−30231.8	−0.159952
$130$	0	0
$131$	147902.	0.753003	0.376501	−	0.926416i	$-0.377127\pi$
$131$	147902.	0.753003	0.376501	+	0.926416i	$0.377127\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	41961.0	0.198158
$136$	0	0
$137$	−91157.4	−0.414945	−0.207472	−	0.978241i	$-0.566524\pi$
$137$	−91157.4	−0.414945	−0.207472	+	0.978241i	$0.566524\pi$
$138$	0	0
$139$	334657.	1.46914	0.734570	−	0.678533i	$-0.237384\pi$
$139$	334657.	1.46914	0.734570	+	0.678533i	$0.237384\pi$
$140$	0	0
$141$	−44619.7	−0.189007
$142$	0	0
$143$	45587.8	0.186427
$144$	0	0
$145$	30615.1	0.120925
$146$	0	0
$147$	0	0
$148$	0	0
$149$	138271.	0.510231	0.255115	−	0.966911i	$-0.417887\pi$
$149$	138271.	0.510231	0.255115	+	0.966911i	$0.417887\pi$
$150$	0	0
$151$	−111169.	−0.396773	−0.198386	−	0.980124i	$-0.563570\pi$
$151$	−111169.	−0.396773	−0.198386	+	0.980124i	$0.563570\pi$
$152$	0	0
$153$	471578.	1.62864
$154$	0	0
$155$	−256118.	−0.856270
$156$	0	0
$157$	−38148.5	−0.123517	−0.0617587	−	0.998091i	$-0.519671\pi$
$157$	−38148.5	−0.123517	−0.0617587	+	0.998091i	$0.519671\pi$
$158$	0	0
$159$	−26200.0	−0.0821881
$160$	0	0
$161$	0	0
$162$	0	0
$163$	212905.	0.627648	0.313824	−	0.949481i	$-0.398390\pi$
$163$	212905.	0.627648	0.313824	+	0.949481i	$0.398390\pi$
$164$	0	0
$165$	6280.83	0.0179600
$166$	0	0
$167$	120396.	0.334057	0.167028	−	0.985952i	$-0.446583\pi$
$167$	120396.	0.334057	0.167028	+	0.985952i	$0.446583\pi$
$168$	0	0
$169$	28532.2	0.0768456
$170$	0	0
$171$	−445087.	−1.16400
$172$	0	0
$173$	712914.	1.81101	0.905507	−	0.424331i	$-0.139491\pi$
$173$	712914.	1.81101	0.905507	+	0.424331i	$0.139491\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−75258.4	−0.180560
$178$	0	0
$179$	749738.	1.74895	0.874474	−	0.485072i	$-0.161207\pi$
$179$	749738.	1.74895	0.874474	+	0.485072i	$0.161207\pi$
$180$	0	0
$181$	623718.	1.41511	0.707557	−	0.706656i	$-0.249797\pi$
$181$	623718.	1.41511	0.707557	+	0.706656i	$0.249797\pi$
$182$	0	0
$183$	−8382.69	−0.0185036
$184$	0	0
$185$	432932.	0.930016
$186$	0	0
$187$	142457.	0.297907
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−417726.	−0.828530	−0.414265	−	0.910156i	$-0.635961\pi$
$191$	−417726.	−0.828530	−0.414265	+	0.910156i	$0.635961\pi$
$192$	0	0
$193$	770700.	1.48933	0.744667	−	0.667436i	$-0.232608\pi$
$193$	770700.	1.48933	0.744667	+	0.667436i	$0.232608\pi$
$194$	0	0
$195$	55085.6	0.103741
$196$	0	0
$197$	−479193.	−0.879721	−0.439861	−	0.898066i	$-0.644972\pi$
$197$	−479193.	−0.879721	−0.439861	+	0.898066i	$0.644972\pi$
$198$	0	0
$199$	428686.	0.767373	0.383687	−	0.923463i	$-0.374654\pi$
$199$	428686.	0.767373	0.383687	+	0.923463i	$0.374654\pi$
$200$	0	0
$201$	32420.2	0.0566011
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−147645.	−0.245377
$206$	0	0
$207$	98737.0	0.160160
$208$	0	0
$209$	−134455.	−0.212917
$210$	0	0
$211$	588544.	0.910066	0.455033	−	0.890475i	$-0.349627\pi$
$211$	588544.	0.910066	0.455033	+	0.890475i	$0.349627\pi$
$212$	0	0
$213$	25369.6	0.0383147
$214$	0	0
$215$	607138.	0.895759
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−40799.5	−0.0574837
$220$	0	0
$221$	1.24941e6	1.72078
$222$	0	0
$223$	363249.	0.489151	0.244575	−	0.969630i	$-0.421351\pi$
$223$	363249.	0.489151	0.244575	+	0.969630i	$0.421351\pi$
$224$	0	0
$225$	328268.	0.432287
$226$	0	0
$227$	843041.	1.08589	0.542943	−	0.839770i	$-0.317310\pi$
$227$	843041.	1.08589	0.542943	+	0.839770i	$0.317310\pi$
$228$	0	0
$229$	−568666.	−0.716587	−0.358293	−	0.933609i	$-0.616641\pi$
$229$	−568666.	−0.716587	−0.358293	+	0.933609i	$0.616641\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.05651e6	−1.27492	−0.637461	−	0.770482i	$-0.720015\pi$
$233$	−1.05651e6	−1.27492	−0.637461	+	0.770482i	$0.720015\pi$
$234$	0	0
$235$	896086.	1.05847
$236$	0	0
$237$	75166.5	0.0869267
$238$	0	0
$239$	−853715.	−0.966759	−0.483379	−	0.875411i	$-0.660591\pi$
$239$	−853715.	−0.966759	−0.483379	+	0.875411i	$0.660591\pi$
$240$	0	0
$241$	−388888.	−0.431302	−0.215651	−	0.976470i	$-0.569187\pi$
$241$	−388888.	−0.431302	−0.215651	+	0.976470i	$0.569187\pi$
$242$	0	0
$243$	−360212.	−0.391330
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.17922e6	−1.22986
$248$	0	0
$249$	−51178.9	−0.0523109
$250$	0	0
$251$	839328.	0.840906	0.420453	−	0.907314i	$-0.361871\pi$
$251$	839328.	0.840906	0.420453	+	0.907314i	$0.361871\pi$
$252$	0	0
$253$	29827.1	0.0292961
$254$	0	0
$255$	172137.	0.165776
$256$	0	0
$257$	291986.	0.275759	0.137879	−	0.990449i	$-0.455971\pi$
$257$	291986.	0.275759	0.137879	+	0.990449i	$0.455971\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−174685.	−0.158728
$262$	0	0
$263$	288495.	0.257187	0.128594	−	0.991697i	$-0.458954\pi$
$263$	288495.	0.257187	0.128594	+	0.991697i	$0.458954\pi$
$264$	0	0
$265$	526169.	0.460268
$266$	0	0
$267$	146300.	0.125593
$268$	0	0
$269$	−259370.	−0.218544	−0.109272	−	0.994012i	$-0.534852\pi$
$269$	−259370.	−0.218544	−0.109272	+	0.994012i	$0.534852\pi$
$270$	0	0
$271$	2.19551e6	1.81599	0.907994	−	0.418984i	$-0.137614\pi$
$271$	2.19551e6	1.81599	0.907994	+	0.418984i	$0.137614\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	99165.1	0.0790728
$276$	0	0
$277$	126991.	0.0994426	0.0497213	−	0.998763i	$-0.484167\pi$
$277$	126991.	0.0994426	0.0497213	+	0.998763i	$0.484167\pi$
$278$	0	0
$279$	1.46137e6	1.12396
$280$	0	0
$281$	−2.22759e6	−1.68294	−0.841472	−	0.540301i	$-0.818310\pi$
$281$	−2.22759e6	−1.68294	−0.841472	+	0.540301i	$0.818310\pi$
$282$	0	0
$283$	−1.18895e6	−0.882463	−0.441231	−	0.897393i	$-0.645458\pi$
$283$	−1.18895e6	−0.882463	−0.441231	+	0.897393i	$0.645458\pi$
$284$	0	0
$285$	−162467.	−0.118482
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2.48442e6	1.74977
$290$	0	0
$291$	−220269.	−0.152483
$292$	0	0
$293$	1.83223e6	1.24684	0.623421	−	0.781886i	$-0.285742\pi$
$293$	1.83223e6	1.24684	0.623421	+	0.781886i	$0.285742\pi$
$294$	0	0
$295$	1.51140e6	1.01117
$296$	0	0
$297$	−72326.3	−0.0475779
$298$	0	0
$299$	261596.	0.169221
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−75941.3	−0.0475194
$304$	0	0
$305$	168348.	0.103623
$306$	0	0
$307$	717638.	0.434569	0.217285	−	0.976108i	$-0.430280\pi$
$307$	717638.	0.434569	0.217285	+	0.976108i	$0.430280\pi$
$308$	0	0
$309$	134380.	0.0800642
$310$	0	0
$311$	856892.	0.502372	0.251186	−	0.967939i	$-0.419179\pi$
$311$	856892.	0.502372	0.251186	+	0.967939i	$0.419179\pi$
$312$	0	0
$313$	1.61699e6	0.932924	0.466462	−	0.884541i	$-0.345528\pi$
$313$	1.61699e6	0.932924	0.466462	+	0.884541i	$0.345528\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.26559e6	−1.26629	−0.633145	−	0.774033i	$-0.718236\pi$
$317$	−2.26559e6	−1.26629	−0.633145	+	0.774033i	$0.718236\pi$
$318$	0	0
$319$	−52769.8	−0.0290341
$320$	0	0
$321$	137557.	0.0745111
$322$	0	0
$323$	−3.68495e6	−1.96528
$324$	0	0
$325$	869721.	0.456743
$326$	0	0
$327$	79015.9	0.0408644
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−709650.	−0.356020	−0.178010	−	0.984029i	$-0.556966\pi$
$331$	−709650.	−0.356020	−0.178010	+	0.984029i	$0.556966\pi$
$332$	0	0
$333$	−2.47024e6	−1.22076
$334$	0	0
$335$	−651086.	−0.316976
$336$	0	0
$337$	603572.	0.289504	0.144752	−	0.989468i	$-0.453762\pi$
$337$	603572.	0.289504	0.144752	+	0.989468i	$0.453762\pi$
$338$	0	0
$339$	−257851.	−0.121862
$340$	0	0
$341$	441459.	0.205591
$342$	0	0
$343$	0	0
$344$	0	0
$345$	36041.2	0.0163024
$346$	0	0
$347$	−1.75731e6	−0.783474	−0.391737	−	0.920077i	$-0.628126\pi$
$347$	−1.75731e6	−0.783474	−0.391737	+	0.920077i	$0.628126\pi$
$348$	0	0
$349$	391875.	0.172220	0.0861102	−	0.996286i	$-0.472556\pi$
$349$	391875.	0.172220	0.0861102	+	0.996286i	$0.472556\pi$
$350$	0	0
$351$	−634333.	−0.274821
$352$	0	0
$353$	492407.	0.210323	0.105162	−	0.994455i	$-0.466464\pi$
$353$	492407.	0.210323	0.105162	+	0.994455i	$0.466464\pi$
$354$	0	0
$355$	−509492.	−0.214569
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.77032e6	1.54398	0.771991	−	0.635634i	$-0.219261\pi$
$359$	3.77032e6	1.54398	0.771991	+	0.635634i	$0.219261\pi$
$360$	0	0
$361$	1.00185e6	0.404607
$362$	0	0
$363$	324605.	0.129297
$364$	0	0
$365$	819367.	0.321919
$366$	0	0
$367$	2.19768e6	0.851726	0.425863	−	0.904788i	$-0.359970\pi$
$367$	2.19768e6	0.851726	0.425863	+	0.904788i	$0.359970\pi$
$368$	0	0
$369$	842439.	0.322086
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1.65636e6	−0.616427	−0.308213	−	0.951317i	$-0.599731\pi$
$373$	−1.65636e6	−0.616427	−0.308213	+	0.951317i	$0.599731\pi$
$374$	0	0
$375$	392066.	0.143973
$376$	0	0
$377$	−462814.	−0.167708
$378$	0	0
$379$	2.82050e6	1.00862	0.504310	−	0.863523i	$-0.331747\pi$
$379$	2.82050e6	1.00862	0.504310	+	0.863523i	$0.331747\pi$
$380$	0	0
$381$	268101.	0.0946206
$382$	0	0
$383$	−3.24845e6	−1.13156	−0.565781	−	0.824555i	$-0.691425\pi$
$383$	−3.24845e6	−1.13156	−0.565781	+	0.824555i	$0.691425\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.46424e6	−1.17579
$388$	0	0
$389$	4.65348e6	1.55921	0.779604	−	0.626273i	$-0.215420\pi$
$389$	4.65348e6	1.55921	0.779604	+	0.626273i	$0.215420\pi$
$390$	0	0
$391$	817461.	0.270411
$392$	0	0
$393$	−308045.	−0.100608
$394$	0	0
$395$	−1.50955e6	−0.486805
$396$	0	0
$397$	−1.16361e6	−0.370536	−0.185268	−	0.982688i	$-0.559315\pi$
$397$	−1.16361e6	−0.370536	−0.185268	+	0.982688i	$0.559315\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	322380.	0.100117	0.0500584	−	0.998746i	$-0.484059\pi$
$401$	322380.	0.100117	0.0500584	+	0.998746i	$0.484059\pi$
$402$	0	0
$403$	3.87179e6	1.18754
$404$	0	0
$405$	2.33839e6	0.708403
$406$	0	0
$407$	−746226.	−0.223298
$408$	0	0
$409$	1.38690e6	0.409956	0.204978	−	0.978767i	$-0.434288\pi$
$409$	1.38690e6	0.409956	0.204978	+	0.978767i	$0.434288\pi$
$410$	0	0
$411$	189859.	0.0554405
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1.02781e6	0.292950
$416$	0	0
$417$	−697011.	−0.196291
$418$	0	0
$419$	−4.90871e6	−1.36594	−0.682971	−	0.730446i	$-0.739312\pi$
$419$	−4.90871e6	−1.36594	−0.682971	+	0.730446i	$0.739312\pi$
$420$	0	0
$421$	2.43924e6	0.670733	0.335367	−	0.942088i	$-0.391140\pi$
$421$	2.43924e6	0.670733	0.335367	+	0.942088i	$0.391140\pi$
$422$	0	0
$423$	−5.11293e6	−1.38937
$424$	0	0
$425$	2.71779e6	0.729865
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−94948.7	−0.0249084
$430$	0	0
$431$	5.22752e6	1.35551	0.677755	−	0.735288i	$-0.262953\pi$
$431$	5.22752e6	1.35551	0.677755	+	0.735288i	$0.262953\pi$
$432$	0	0
$433$	−2.63022e6	−0.674174	−0.337087	−	0.941473i	$-0.609442\pi$
$433$	−2.63022e6	−0.674174	−0.337087	+	0.941473i	$0.609442\pi$
$434$	0	0
$435$	−63763.9	−0.0161567
$436$	0	0
$437$	−771539.	−0.193266
$438$	0	0
$439$	−2.55412e6	−0.632527	−0.316264	−	0.948671i	$-0.602428\pi$
$439$	−2.55412e6	−0.632527	−0.316264	+	0.948671i	$0.602428\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.83900e6	−0.929414	−0.464707	−	0.885465i	$-0.653840\pi$
$443$	−3.83900e6	−0.929414	−0.464707	+	0.885465i	$0.653840\pi$
$444$	0	0
$445$	−2.93811e6	−0.703345
$446$	0	0
$447$	−287986.	−0.0681715
$448$	0	0
$449$	1.49369e6	0.349658	0.174829	−	0.984599i	$-0.444063\pi$
$449$	1.49369e6	0.349658	0.174829	+	0.984599i	$0.444063\pi$
$450$	0	0
$451$	254489.	0.0589152
$452$	0	0
$453$	231539.	0.0530126
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2.16221e6	−0.484293	−0.242146	−	0.970240i	$-0.577851\pi$
$457$	−2.16221e6	−0.484293	−0.242146	+	0.970240i	$0.577851\pi$
$458$	0	0
$459$	−1.98222e6	−0.439158
$460$	0	0
$461$	−6.11949e6	−1.34111	−0.670553	−	0.741862i	$-0.733943\pi$
$461$	−6.11949e6	−1.34111	−0.670553	+	0.741862i	$0.733943\pi$
$462$	0	0
$463$	−3.93615e6	−0.853335	−0.426667	−	0.904409i	$-0.640313\pi$
$463$	−3.93615e6	−0.853335	−0.426667	+	0.904409i	$0.640313\pi$
$464$	0	0
$465$	533433.	0.114406
$466$	0	0
$467$	5.47044e6	1.16073	0.580363	−	0.814358i	$-0.302911\pi$
$467$	5.47044e6	1.16073	0.580363	+	0.814358i	$0.302911\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	79454.3	0.0165031
$472$	0	0
$473$	−1.04650e6	−0.215073
$474$	0	0
$475$	−2.56511e6	−0.521641
$476$	0	0
$477$	−3.00224e6	−0.604157
$478$	0	0
$479$	6.66289e6	1.32686	0.663428	−	0.748240i	$-0.269101\pi$
$479$	6.66289e6	1.32686	0.663428	+	0.748240i	$0.269101\pi$
$480$	0	0
$481$	−6.54473e6	−1.28982
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.42362e6	0.853931
$486$	0	0
$487$	−9.53693e6	−1.82216	−0.911079	−	0.412232i	$-0.864749\pi$
$487$	−9.53693e6	−1.82216	−0.911079	+	0.412232i	$0.864749\pi$
$488$	0	0
$489$	−443430.	−0.0838595
$490$	0	0
$491$	8.19294e6	1.53369	0.766843	−	0.641835i	$-0.221827\pi$
$491$	8.19294e6	1.53369	0.766843	+	0.641835i	$0.221827\pi$
$492$	0	0
$493$	−1.44625e6	−0.267994
$494$	0	0
$495$	719715.	0.132022
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.31437e6	0.775650	0.387825	−	0.921733i	$-0.373226\pi$
$499$	4.31437e6	0.775650	0.387825	+	0.921733i	$0.373226\pi$
$500$	0	0
$501$	−250756.	−0.0446331
$502$	0	0
$503$	1.04015e7	1.83306	0.916529	−	0.399968i	$-0.130979\pi$
$503$	1.04015e7	1.83306	0.916529	+	0.399968i	$0.130979\pi$
$504$	0	0
$505$	1.52511e6	0.266117
$506$	0	0
$507$	−59425.9	−0.0102673
$508$	0	0
$509$	−3.09396e6	−0.529322	−0.264661	−	0.964342i	$-0.585260\pi$
$509$	−3.09396e6	−0.529322	−0.264661	+	0.964342i	$0.585260\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.87087e6	0.313870
$514$	0	0
$515$	−2.69872e6	−0.448374
$516$	0	0
$517$	−1.54455e6	−0.254141
$518$	0	0
$519$	−1.48483e6	−0.241968
$520$	0	0
$521$	−7.60175e6	−1.22693	−0.613464	−	0.789723i	$-0.710224\pi$
$521$	−7.60175e6	−1.22693	−0.613464	+	0.789723i	$0.710224\pi$
$522$	0	0
$523$	−4.75669e6	−0.760415	−0.380208	−	0.924901i	$-0.624147\pi$
$523$	−4.75669e6	−0.760415	−0.380208	+	0.924901i	$0.624147\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.20989e7	1.89767
$528$	0	0
$529$	−6.26519e6	−0.973408
$530$	0	0
$531$	−8.62380e6	−1.32728
$532$	0	0
$533$	2.23198e6	0.340308
$534$	0	0
$535$	−2.76253e6	−0.417275
$536$	0	0
$537$	−1.56153e6	−0.233676
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1.10052e7	1.61661	0.808305	−	0.588764i	$-0.200385\pi$
$541$	1.10052e7	1.61661	0.808305	+	0.588764i	$0.200385\pi$
$542$	0	0
$543$	−1.29906e6	−0.189072
$544$	0	0
$545$	−1.58686e6	−0.228848
$546$	0	0
$547$	4.46311e6	0.637778	0.318889	−	0.947792i	$-0.396690\pi$
$547$	4.46311e6	0.637778	0.318889	+	0.947792i	$0.396690\pi$
$548$	0	0
$549$	−960566.	−0.136018
$550$	0	0
$551$	1.36500e6	0.191538
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−901694.	−0.124259
$556$	0	0
$557$	−6.45222e6	−0.881194	−0.440597	−	0.897705i	$-0.645233\pi$
$557$	−6.45222e6	−0.881194	−0.440597	+	0.897705i	$0.645233\pi$
$558$	0	0
$559$	−9.17823e6	−1.24231
$560$	0	0
$561$	−296704.	−0.0398031
$562$	0	0
$563$	−1.74748e6	−0.232349	−0.116175	−	0.993229i	$-0.537063\pi$
$563$	−1.74748e6	−0.232349	−0.116175	+	0.993229i	$0.537063\pi$
$564$	0	0
$565$	5.17836e6	0.682450
$566$	0	0
$567$	0	0
$568$	0	0
$569$	512789.	0.0663985	0.0331992	−	0.999449i	$-0.489430\pi$
$569$	512789.	0.0663985	0.0331992	+	0.999449i	$0.489430\pi$
$570$	0	0
$571$	−5.22364e6	−0.670475	−0.335238	−	0.942134i	$-0.608817\pi$
$571$	−5.22364e6	−0.670475	−0.335238	+	0.942134i	$0.608817\pi$
$572$	0	0
$573$	870024.	0.110699
$574$	0	0
$575$	569038.	0.0717748
$576$	0	0
$577$	6.63973e6	0.830254	0.415127	−	0.909763i	$-0.363737\pi$
$577$	6.63973e6	0.830254	0.415127	+	0.909763i	$0.363737\pi$
$578$	0	0
$579$	−1.60519e6	−0.198989
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−906935.	−0.110511
$584$	0	0
$585$	6.31221e6	0.762592
$586$	0	0
$587$	−774096.	−0.0927256	−0.0463628	−	0.998925i	$-0.514763\pi$
$587$	−774096.	−0.0927256	−0.0463628	+	0.998925i	$0.514763\pi$
$588$	0	0
$589$	−1.14193e7	−1.35628
$590$	0	0
$591$	998046.	0.117539
$592$	0	0
$593$	−1.43756e7	−1.67876	−0.839379	−	0.543546i	$-0.817081\pi$
$593$	−1.43756e7	−1.67876	−0.839379	+	0.543546i	$0.817081\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−892851.	−0.102528
$598$	0	0
$599$	−1.20835e7	−1.37602	−0.688010	−	0.725701i	$-0.741516\pi$
$599$	−1.20835e7	−1.37602	−0.688010	+	0.725701i	$0.741516\pi$
$600$	0	0
$601$	5.75607e6	0.650040	0.325020	−	0.945707i	$-0.394629\pi$
$601$	5.75607e6	0.650040	0.325020	+	0.945707i	$0.394629\pi$
$602$	0	0
$603$	3.71500e6	0.416069
$604$	0	0
$605$	−6.51897e6	−0.724086
$606$	0	0
$607$	−4.20121e6	−0.462810	−0.231405	−	0.972858i	$-0.574332\pi$
$607$	−4.20121e6	−0.462810	−0.231405	+	0.972858i	$0.574332\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.35463e7	−1.46798
$612$	0	0
$613$	−2.64543e6	−0.284344	−0.142172	−	0.989842i	$-0.545409\pi$
$613$	−2.64543e6	−0.284344	−0.142172	+	0.989842i	$0.545409\pi$
$614$	0	0
$615$	307509.	0.0327846
$616$	0	0
$617$	6.43533e6	0.680546	0.340273	−	0.940327i	$-0.389480\pi$
$617$	6.43533e6	0.680546	0.340273	+	0.940327i	$0.389480\pi$
$618$	0	0
$619$	−1.41177e7	−1.48094	−0.740469	−	0.672090i	$-0.765397\pi$
$619$	−1.41177e7	−1.48094	−0.740469	+	0.672090i	$0.765397\pi$
$620$	0	0
$621$	−415029.	−0.0431867
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−3.57548e6	−0.366129
$626$	0	0
$627$	280037.	0.0284476
$628$	0	0
$629$	−2.04516e7	−2.06111
$630$	0	0
$631$	4.70856e6	0.470777	0.235388	−	0.971901i	$-0.424364\pi$
$631$	4.70856e6	0.470777	0.235388	+	0.971901i	$0.424364\pi$
$632$	0	0
$633$	−1.22580e6	−0.121593
$634$	0	0
$635$	−5.38421e6	−0.529892
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.90708e6	0.281647
$640$	0	0
$641$	−1.04174e7	−1.00141	−0.500707	−	0.865617i	$-0.666927\pi$
$641$	−1.04174e7	−1.00141	−0.500707	+	0.865617i	$0.666927\pi$
$642$	0	0
$643$	−1.27284e7	−1.21407	−0.607037	−	0.794674i	$-0.707642\pi$
$643$	−1.27284e7	−1.21407	−0.607037	+	0.794674i	$0.707642\pi$
$644$	0	0
$645$	−1.26452e6	−0.119682
$646$	0	0
$647$	1.61348e7	1.51531	0.757657	−	0.652653i	$-0.226344\pi$
$647$	1.61348e7	1.51531	0.757657	+	0.652653i	$0.226344\pi$
$648$	0	0
$649$	−2.60513e6	−0.242783
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.50295e7	−1.37931	−0.689654	−	0.724139i	$-0.742237\pi$
$653$	−1.50295e7	−1.37931	−0.689654	+	0.724139i	$0.742237\pi$
$654$	0	0
$655$	6.18640e6	0.563423
$656$	0	0
$657$	−4.67518e6	−0.422557
$658$	0	0
$659$	−1.67927e7	−1.50628	−0.753140	−	0.657860i	$-0.771462\pi$
$659$	−1.67927e7	−1.50628	−0.753140	+	0.657860i	$0.771462\pi$
$660$	0	0
$661$	1.08540e7	0.966246	0.483123	−	0.875552i	$-0.339502\pi$
$661$	1.08540e7	0.966246	0.483123	+	0.875552i	$0.339502\pi$
$662$	0	0
$663$	−2.60223e6	−0.229912
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−302809.	−0.0263544
$668$	0	0
$669$	−756562.	−0.0653551
$670$	0	0
$671$	−290174.	−0.0248801
$672$	0	0
$673$	1.23697e7	1.05274	0.526371	−	0.850255i	$-0.323552\pi$
$673$	1.23697e7	1.05274	0.526371	+	0.850255i	$0.323552\pi$
$674$	0	0
$675$	−1.37983e6	−0.116565
$676$	0	0
$677$	−1.00501e6	−0.0842746	−0.0421373	−	0.999112i	$-0.513417\pi$
$677$	−1.00501e6	−0.0842746	−0.0421373	+	0.999112i	$0.513417\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1.75586e6	−0.145084
$682$	0	0
$683$	−1.87019e6	−0.153403	−0.0767014	−	0.997054i	$-0.524439\pi$
$683$	−1.87019e6	−0.153403	−0.0767014	+	0.997054i	$0.524439\pi$
$684$	0	0
$685$	−3.81290e6	−0.310476
$686$	0	0
$687$	1.18440e6	0.0957426
$688$	0	0
$689$	−7.95421e6	−0.638336
$690$	0	0
$691$	−1.93867e7	−1.54457	−0.772286	−	0.635275i	$-0.780887\pi$
$691$	−1.93867e7	−1.54457	−0.772286	+	0.635275i	$0.780887\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.39979e7	1.09926
$696$	0	0
$697$	6.97469e6	0.543805
$698$	0	0
$699$	2.20046e6	0.170342
$700$	0	0
$701$	1.17488e7	0.903024	0.451512	−	0.892265i	$-0.350885\pi$
$701$	1.17488e7	0.903024	0.451512	+	0.892265i	$0.350885\pi$
$702$	0	0
$703$	1.93027e7	1.47309
$704$	0	0
$705$	−1.86634e6	−0.141422
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.67948e7	1.25475	0.627377	−	0.778716i	$-0.284129\pi$
$709$	1.67948e7	1.25475	0.627377	+	0.778716i	$0.284129\pi$
$710$	0	0
$711$	8.61326e6	0.638990
$712$	0	0
$713$	2.53322e6	0.186616
$714$	0	0
$715$	1.90683e6	0.139491
$716$	0	0
$717$	1.77809e6	0.129168
$718$	0	0
$719$	1.65130e7	1.19126	0.595628	−	0.803261i	$-0.296903\pi$
$719$	1.65130e7	1.19126	0.595628	+	0.803261i	$0.296903\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	809961.	0.0576260
$724$	0	0
$725$	−1.00674e6	−0.0711331
$726$	0	0
$727$	1.25756e6	0.0882453	0.0441227	−	0.999026i	$-0.485951\pi$
$727$	1.25756e6	0.0882453	0.0441227	+	0.999026i	$0.485951\pi$
$728$	0	0
$729$	−1.28348e7	−0.894479
$730$	0	0
$731$	−2.86810e7	−1.98518
$732$	0	0
$733$	−1.98332e6	−0.136343	−0.0681716	−	0.997674i	$-0.521717\pi$
$733$	−1.98332e6	−0.136343	−0.0681716	+	0.997674i	$0.521717\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.12225e6	0.0761063
$738$	0	0
$739$	2.38807e7	1.60856	0.804278	−	0.594253i	$-0.202552\pi$
$739$	2.38807e7	1.60856	0.804278	+	0.594253i	$0.202552\pi$
$740$	0	0
$741$	2.45604e6	0.164320
$742$	0	0
$743$	1.90819e7	1.26809	0.634043	−	0.773298i	$-0.281394\pi$
$743$	1.90819e7	1.26809	0.634043	+	0.773298i	$0.281394\pi$
$744$	0	0
$745$	5.78356e6	0.381773
$746$	0	0
$747$	−5.86455e6	−0.384532
$748$	0	0
$749$	0	0
$750$	0	0
$751$	3.75805e6	0.243144	0.121572	−	0.992583i	$-0.461206\pi$
$751$	3.75805e6	0.243144	0.121572	+	0.992583i	$0.461206\pi$
$752$	0	0
$753$	−1.74812e6	−0.112353
$754$	0	0
$755$	−4.64994e6	−0.296880
$756$	0	0
$757$	1.69904e7	1.07761	0.538807	−	0.842429i	$-0.318875\pi$
$757$	1.69904e7	1.07761	0.538807	+	0.842429i	$0.318875\pi$
$758$	0	0
$759$	−62122.7	−0.00391423
$760$	0	0
$761$	2.23998e7	1.40211	0.701056	−	0.713106i	$-0.252712\pi$
$761$	2.23998e7	1.40211	0.701056	+	0.713106i	$0.252712\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.97250e7	1.21861
$766$	0	0
$767$	−2.28481e7	−1.40237
$768$	0	0
$769$	−1.87866e7	−1.14560	−0.572799	−	0.819696i	$-0.694142\pi$
$769$	−1.87866e7	−1.14560	−0.572799	+	0.819696i	$0.694142\pi$
$770$	0	0
$771$	−608138.	−0.0368439
$772$	0	0
$773$	9.30837e6	0.560306	0.280153	−	0.959955i	$-0.409615\pi$
$773$	9.30837e6	0.560306	0.280153	+	0.959955i	$0.409615\pi$
$774$	0	0
$775$	8.42212e6	0.503694
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.58288e6	−0.388662
$780$	0	0
$781$	878189.	0.0515182
$782$	0	0
$783$	734267.	0.0428006
$784$	0	0
$785$	−1.59566e6	−0.0924201
$786$	0	0
$787$	−1.73427e7	−0.998111	−0.499056	−	0.866570i	$-0.666320\pi$
$787$	−1.73427e7	−0.998111	−0.499056	+	0.866570i	$0.666320\pi$
$788$	0	0
$789$	−600867.	−0.0343626
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.54495e6	−0.143713
$794$	0	0
$795$	−1.09589e6	−0.0614960
$796$	0	0
$797$	3.10445e7	1.73117	0.865584	−	0.500764i	$-0.166948\pi$
$797$	3.10445e7	1.73117	0.865584	+	0.500764i	$0.166948\pi$
$798$	0	0
$799$	−4.23309e7	−2.34580
$800$	0	0
$801$	1.67644e7	0.923224
$802$	0	0
$803$	−1.41231e6	−0.0772930
$804$	0	0
$805$	0	0
$806$	0	0
$807$	540206.	0.0291995
$808$	0	0
$809$	−2.47038e7	−1.32707	−0.663533	−	0.748147i	$-0.730944\pi$
$809$	−2.47038e7	−1.32707	−0.663533	+	0.748147i	$0.730944\pi$
$810$	0	0
$811$	−8.42005e6	−0.449534	−0.224767	−	0.974413i	$-0.572162\pi$
$811$	−8.42005e6	−0.449534	−0.224767	+	0.974413i	$0.572162\pi$
$812$	0	0
$813$	−4.57273e6	−0.242633
$814$	0	0
$815$	8.90529e6	0.469628
$816$	0	0
$817$	2.70698e7	1.41883
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.58827e7	−1.34014	−0.670071	−	0.742297i	$-0.733736\pi$
$821$	−2.58827e7	−1.34014	−0.670071	+	0.742297i	$0.733736\pi$
$822$	0	0
$823$	−1.72004e7	−0.885195	−0.442597	−	0.896720i	$-0.645943\pi$
$823$	−1.72004e7	−0.885195	−0.442597	+	0.896720i	$0.645943\pi$
$824$	0	0
$825$	−206537.	−0.0105649
$826$	0	0
$827$	2.40337e7	1.22196	0.610979	−	0.791647i	$-0.290776\pi$
$827$	2.40337e7	1.22196	0.610979	+	0.791647i	$0.290776\pi$
$828$	0	0
$829$	−3.24736e7	−1.64113	−0.820567	−	0.571550i	$-0.806342\pi$
$829$	−3.24736e7	−1.64113	−0.820567	+	0.571550i	$0.806342\pi$
$830$	0	0
$831$	−264491.	−0.0132865
$832$	0	0
$833$	0	0
$834$	0	0
$835$	5.03587e6	0.249953
$836$	0	0
$837$	−6.14269e6	−0.303072
$838$	0	0
$839$	1.24404e7	0.610139	0.305069	−	0.952330i	$-0.401320\pi$
$839$	1.24404e7	0.610139	0.305069	+	0.952330i	$0.401320\pi$
$840$	0	0
$841$	−1.99754e7	−0.973881
$842$	0	0
$843$	4.63954e6	0.224857
$844$	0	0
$845$	1.19344e6	0.0574986
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2.47629e6	0.117905
$850$	0	0
$851$	−4.28206e6	−0.202689
$852$	0	0
$853$	−999355.	−0.0470270	−0.0235135	−	0.999724i	$-0.507485\pi$
$853$	−999355.	−0.0470270	−0.0235135	+	0.999724i	$0.507485\pi$
$854$	0	0
$855$	−1.86169e7	−0.870949
$856$	0	0
$857$	−2.64465e7	−1.23003	−0.615016	−	0.788514i	$-0.710851\pi$
$857$	−2.64465e7	−1.23003	−0.615016	+	0.788514i	$0.710851\pi$
$858$	0	0
$859$	2.86716e7	1.32577	0.662887	−	0.748719i	$-0.269331\pi$
$859$	2.86716e7	1.32577	0.662887	+	0.748719i	$0.269331\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−4.08173e6	−0.186560	−0.0932798	−	0.995640i	$-0.529735\pi$
$863$	−4.08173e6	−0.186560	−0.0932798	+	0.995640i	$0.529735\pi$
$864$	0	0
$865$	2.98195e7	1.35506
$866$	0	0
$867$	−5.17446e6	−0.233785
$868$	0	0
$869$	2.60195e6	0.116882
$870$	0	0
$871$	9.84261e6	0.439608
$872$	0	0
$873$	−2.52405e7	−1.12089
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.82405e7	−1.23986	−0.619931	−	0.784657i	$-0.712839\pi$
$877$	−2.82405e7	−1.23986	−0.619931	+	0.784657i	$0.712839\pi$
$878$	0	0
$879$	−3.81610e6	−0.166590
$880$	0	0
$881$	1.61480e7	0.700936	0.350468	−	0.936575i	$-0.386022\pi$
$881$	1.61480e7	0.700936	0.350468	+	0.936575i	$0.386022\pi$
$882$	0	0
$883$	3.86021e7	1.66613	0.833065	−	0.553174i	$-0.186584\pi$
$883$	3.86021e7	1.66613	0.833065	+	0.553174i	$0.186584\pi$
$884$	0	0
$885$	−3.14788e6	−0.135102
$886$	0	0
$887$	−7.29088e6	−0.311151	−0.155575	−	0.987824i	$-0.549723\pi$
$887$	−7.29088e6	−0.311151	−0.155575	+	0.987824i	$0.549723\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.03059e6	−0.170088
$892$	0	0
$893$	3.99529e7	1.67656
$894$	0	0
$895$	3.13598e7	1.30862
$896$	0	0
$897$	−544843.	−0.0226095
$898$	0	0
$899$	−4.48176e6	−0.184948
$900$	0	0
$901$	−2.48561e7	−1.02005
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.60886e7	1.05884
$906$	0	0
$907$	3.73335e7	1.50689	0.753443	−	0.657514i	$-0.228392\pi$
$907$	3.73335e7	1.50689	0.753443	+	0.657514i	$0.228392\pi$
$908$	0	0
$909$	−8.70205e6	−0.349311
$910$	0	0
$911$	2475.17	9.88120e−5 0	4.94060e−5	−	1.00000i	$-0.499984\pi$
$911$	2475.17	9.88120e−5 0	4.94060e−5		1.00000i	$0.499984\pi$
$912$	0	0
$913$	−1.77160e6	−0.0703377
$914$	0	0
$915$	−350628.	−0.0138450
$916$	0	0
$917$	0	0
$918$	0	0
$919$	4.48238e6	0.175073	0.0875366	−	0.996161i	$-0.472101\pi$
$919$	4.48238e6	0.175073	0.0875366	+	0.996161i	$0.472101\pi$
$920$	0	0
$921$	−1.49467e6	−0.0580625
$922$	0	0
$923$	7.70210e6	0.297581
$924$	0	0
$925$	−1.42364e7	−0.547075
$926$	0	0
$927$	1.53985e7	0.588544
$928$	0	0
$929$	2.12859e7	0.809193	0.404596	−	0.914495i	$-0.367412\pi$
$929$	2.12859e7	0.809193	0.404596	+	0.914495i	$0.367412\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.78470e6	−0.0671215
$934$	0	0
$935$	5.95865e6	0.222904
$936$	0	0
$937$	6.79757e6	0.252932	0.126466	−	0.991971i	$-0.459636\pi$
$937$	6.79757e6	0.252932	0.126466	+	0.991971i	$0.459636\pi$
$938$	0	0
$939$	−3.36781e6	−0.124647
$940$	0	0
$941$	4.90883e7	1.80719	0.903595	−	0.428388i	$-0.140918\pi$
$941$	4.90883e7	1.80719	0.903595	+	0.428388i	$0.140918\pi$
$942$	0	0
$943$	1.46033e6	0.0534776
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.45484e7	−0.889505	−0.444753	−	0.895653i	$-0.646708\pi$
$947$	−2.45484e7	−0.889505	−0.444753	+	0.895653i	$0.646708\pi$
$948$	0	0
$949$	−1.23865e7	−0.446462
$950$	0	0
$951$	4.71869e6	0.169188
$952$	0	0
$953$	−513120.	−0.0183015	−0.00915075	−	0.999958i	$-0.502913\pi$
$953$	−513120.	−0.0183015	−0.00915075	+	0.999958i	$0.502913\pi$
$954$	0	0
$955$	−1.74725e7	−0.619935
$956$	0	0
$957$	109907.	0.00387923
$958$	0	0
$959$	0	0
$960$	0	0
$961$	8.86412e6	0.309619
$962$	0	0
$963$	1.57626e7	0.547724
$964$	0	0
$965$	3.22366e7	1.11437
$966$	0	0
$967$	−3.34818e7	−1.15144	−0.575722	−	0.817645i	$-0.695279\pi$
$967$	−3.34818e7	−1.15144	−0.575722	+	0.817645i	$0.695279\pi$
$968$	0	0
$969$	7.67488e6	0.262580
$970$	0	0
$971$	−4.76036e6	−0.162029	−0.0810143	−	0.996713i	$-0.525816\pi$
$971$	−4.76036e6	−0.162029	−0.0810143	+	0.996713i	$0.525816\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−1.81142e6	−0.0610250
$976$	0	0
$977$	2.87338e7	0.963067	0.481534	−	0.876428i	$-0.340080\pi$
$977$	2.87338e7	0.963067	0.481534	+	0.876428i	$0.340080\pi$
$978$	0	0
$979$	5.06430e6	0.168874
$980$	0	0
$981$	9.05437e6	0.300390
$982$	0	0
$983$	4.97072e7	1.64072	0.820362	−	0.571845i	$-0.193772\pi$
$983$	4.97072e7	1.64072	0.820362	+	0.571845i	$0.193772\pi$
$984$	0	0
$985$	−2.00435e7	−0.658239
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.00511e6	−0.195223
$990$	0	0
$991$	2.91066e6	0.0941471	0.0470736	−	0.998891i	$-0.485010\pi$
$991$	2.91066e6	0.0941471	0.0470736	+	0.998891i	$0.485010\pi$
$992$	0	0
$993$	1.47803e6	0.0475676
$994$	0	0
$995$	1.79309e7	0.574176
$996$	0	0
$997$	1.43353e7	0.456740	0.228370	−	0.973574i	$-0.426660\pi$
$997$	1.43353e7	0.456740	0.228370	+	0.973574i	$0.426660\pi$
$998$	0	0
$999$	1.03834e7	0.329174

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.ba.1.1		2
4.3	odd	2		49.6.a.d.1.1		2
7.2	even	3		112.6.i.c.81.2		4
7.4	even	3		112.6.i.c.65.2		4
7.6	odd	2		784.6.a.t.1.2		2
12.11	even	2		441.6.a.n.1.2		2
28.3	even	6		49.6.c.f.30.2		4
28.11	odd	6		7.6.c.a.2.2	✓	4
28.19	even	6		49.6.c.f.18.2		4
28.23	odd	6		7.6.c.a.4.2	yes	4
28.27	even	2		49.6.a.e.1.1		2
84.11	even	6		63.6.e.d.37.1		4
84.23	even	6		63.6.e.d.46.1		4
84.83	odd	2		441.6.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.6.c.a.2.2	✓	4	28.11	odd	6
7.6.c.a.4.2	yes	4	28.23	odd	6
49.6.a.d.1.1		2	4.3	odd	2
49.6.a.e.1.1		2	28.27	even	2
49.6.c.f.18.2		4	28.19	even	6
49.6.c.f.30.2		4	28.3	even	6
63.6.e.d.37.1		4	84.11	even	6
63.6.e.d.46.1		4	84.23	even	6
112.6.i.c.65.2		4	7.4	even	3
112.6.i.c.81.2		4	7.2	even	3
441.6.a.m.1.2		2	84.83	odd	2
441.6.a.n.1.2		2	12.11	even	2
784.6.a.t.1.2		2	7.6	odd	2
784.6.a.ba.1.1		2	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.08276 q^{3} +41.8276 q^{5} -238.662 q^{9} +O(q^{10})\)	\(q-2.08276 q^{3} +41.8276 q^{5} -238.662 q^{9} -72.0965 q^{11} -632.317 q^{13} -87.1170 q^{15} -1975.92 q^{17} +1864.93 q^{19} -413.711 q^{23} -1375.45 q^{25} +1003.19 q^{27} +731.934 q^{29} -6123.18 q^{31} +150.160 q^{33} +10350.4 q^{37} +1316.97 q^{39} -3529.84 q^{41} +14515.2 q^{43} -9982.67 q^{45} +21423.3 q^{47} +4115.38 q^{51} +12579.5 q^{53} -3015.62 q^{55} -3884.20 q^{57} +36133.9 q^{59} +4024.80 q^{61} -26448.3 q^{65} -15565.9 q^{67} +861.661 q^{69} -12180.8 q^{71} +19589.1 q^{73} +2864.74 q^{75} -36089.8 q^{79} +55905.5 q^{81} +24572.6 q^{83} -82648.2 q^{85} -1524.44 q^{87} -70243.3 q^{89} +12753.1 q^{93} +78005.4 q^{95} +105758. q^{97} +17206.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{3} - 38 q^{5} - 380 q^{9}+O(q^{10}) \) 2 * q + 8 * q^3 - 38 * q^5 - 380 * q^9	\( 2 q + 8 q^{3} - 38 q^{5} - 380 q^{9} - 424 q^{11} - 924 q^{13} - 892 q^{15} - 2346 q^{17} + 360 q^{19} + 12 q^{23} + 1872 q^{25} - 2872 q^{27} - 7052 q^{29} - 3548 q^{31} - 3398 q^{33} + 11090 q^{37} - 1624 q^{39} + 3500 q^{41} + 12680 q^{43} + 1300 q^{45} + 22956 q^{47} + 384 q^{51} + 3042 q^{53} + 25076 q^{55} - 19058 q^{57} + 65808 q^{59} - 42486 q^{61} - 3164 q^{65} - 42312 q^{67} + 5154 q^{69} + 2208 q^{71} - 50506 q^{73} + 35608 q^{75} - 9004 q^{79} + 51178 q^{81} + 104328 q^{83} - 53106 q^{85} - 80008 q^{87} - 26666 q^{89} + 38718 q^{93} + 198140 q^{95} + 209132 q^{97} + 66944 q^{99}+O(q^{100}) \) 2 * q + 8 * q^3 - 38 * q^5 - 380 * q^9 - 424 * q^11 - 924 * q^13 - 892 * q^15 - 2346 * q^17 + 360 * q^19 + 12 * q^23 + 1872 * q^25 - 2872 * q^27 - 7052 * q^29 - 3548 * q^31 - 3398 * q^33 + 11090 * q^37 - 1624 * q^39 + 3500 * q^41 + 12680 * q^43 + 1300 * q^45 + 22956 * q^47 + 384 * q^51 + 3042 * q^53 + 25076 * q^55 - 19058 * q^57 + 65808 * q^59 - 42486 * q^61 - 3164 * q^65 - 42312 * q^67 + 5154 * q^69 + 2208 * q^71 - 50506 * q^73 + 35608 * q^75 - 9004 * q^79 + 51178 * q^81 + 104328 * q^83 - 53106 * q^85 - 80008 * q^87 - 26666 * q^89 + 38718 * q^93 + 198140 * q^95 + 209132 * q^97 + 66944 * q^99

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