LMFDB - Embedded newform 784.6.a.be.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:	$4$
Coefficient field:	$\Q(\sqrt{86}, \sqrt{134})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 110x^{2} + 144 $ x^4 - 110*x^2 + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 392)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-10.4247$ 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)

$f(q)$	$=$	$q-20.8495 q^{3} +11.6406 q^{5} +191.700 q^{9} +O(q^{10})$	$q-20.8495 q^{3} +11.6406 q^{5} +191.700 q^{9} -556.099 q^{11} -122.147 q^{13} -242.700 q^{15} +756.480 q^{17} -1237.77 q^{19} +1440.30 q^{23} -2989.50 q^{25} +1069.58 q^{27} +1010.00 q^{29} -10156.4 q^{31} +11594.4 q^{33} +6490.60 q^{37} +2546.70 q^{39} -12532.3 q^{41} -1387.11 q^{43} +2231.50 q^{45} -11639.6 q^{47} -15772.2 q^{51} +17401.8 q^{53} -6473.32 q^{55} +25806.9 q^{57} -34029.9 q^{59} +27767.0 q^{61} -1421.86 q^{65} +44552.1 q^{67} -30029.4 q^{69} -71236.2 q^{71} -84065.0 q^{73} +62329.4 q^{75} -28839.2 q^{79} -68883.2 q^{81} -82782.9 q^{83} +8805.86 q^{85} -21057.9 q^{87} -71184.1 q^{89} +211756. q^{93} -14408.4 q^{95} -7891.05 q^{97} -106604. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 92 q^{9}+O(q^{10}) $ 4 * q - 92 * q^9	$ 4 q - 92 q^{9} + 352 q^{11} - 112 q^{15} - 1968 q^{23} + 924 q^{25} + 4040 q^{29} + 10504 q^{37} + 9328 q^{39} - 28736 q^{43} - 23584 q^{51} - 28296 q^{53} + 62864 q^{57} + 146320 q^{65} + 72576 q^{67} - 176736 q^{71} - 135968 q^{79} - 27340 q^{81} + 298016 q^{85} + 395296 q^{93} + 367472 q^{95} - 561248 q^{99}+O(q^{100}) $ 4 * q - 92 * q^9 + 352 * q^11 - 112 * q^15 - 1968 * q^23 + 924 * q^25 + 4040 * q^29 + 10504 * q^37 + 9328 * q^39 - 28736 * q^43 - 23584 * q^51 - 28296 * q^53 + 62864 * q^57 + 146320 * q^65 + 72576 * q^67 - 176736 * q^71 - 135968 * q^79 - 27340 * q^81 + 298016 * q^85 + 395296 * q^93 + 367472 * q^95 - 561248 * 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
$3$	−20.8495	−1.33749	−0.668747	−	0.743490i	$-0.733169\pi$
$3$	−20.8495	−1.33749	−0.668747	+	0.743490i	$0.733169\pi$
$4$	0	0
$5$	11.6406	0.208233	0.104117	−	0.994565i	$-0.466799\pi$
$5$	11.6406	0.208233	0.104117	+	0.994565i	$0.466799\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	191.700	0.788888
$10$	0	0
$11$	−556.099	−1.38570	−0.692852	−	0.721079i	$-0.743646\pi$
$11$	−556.099	−1.38570	−0.692852	+	0.721079i	$0.743646\pi$
$12$	0	0
$13$	−122.147	−0.200459	−0.100229	−	0.994964i	$-0.531958\pi$
$13$	−122.147	−0.200459	−0.100229	+	0.994964i	$0.531958\pi$
$14$	0	0
$15$	−242.700	−0.278510
$16$	0	0
$17$	756.480	0.634856	0.317428	−	0.948282i	$-0.397181\pi$
$17$	756.480	0.634856	0.317428	+	0.948282i	$0.397181\pi$
$18$	0	0
$19$	−1237.77	−0.786605	−0.393303	−	0.919409i	$-0.628668\pi$
$19$	−1237.77	−0.786605	−0.393303	+	0.919409i	$0.628668\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1440.30	0.567718	0.283859	−	0.958866i	$-0.408385\pi$
$23$	1440.30	0.567718	0.283859	+	0.958866i	$0.408385\pi$
$24$	0	0
$25$	−2989.50	−0.956639
$26$	0	0
$27$	1069.58	0.282361
$28$	0	0
$29$	1010.00	0.223011	0.111506	−	0.993764i	$-0.464433\pi$
$29$	1010.00	0.223011	0.111506	+	0.993764i	$0.464433\pi$
$30$	0	0
$31$	−10156.4	−1.89818	−0.949089	−	0.315008i	$-0.897993\pi$
$31$	−10156.4	−1.89818	−0.949089	+	0.315008i	$0.897993\pi$
$32$	0	0
$33$	11594.4	1.85337
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6490.60	0.779436	0.389718	−	0.920934i	$-0.372573\pi$
$37$	6490.60	0.779436	0.389718	+	0.920934i	$0.372573\pi$
$38$	0	0
$39$	2546.70	0.268112
$40$	0	0
$41$	−12532.3	−1.16432	−0.582159	−	0.813075i	$-0.697792\pi$
$41$	−12532.3	−1.16432	−0.582159	+	0.813075i	$0.697792\pi$
$42$	0	0
$43$	−1387.11	−0.114403	−0.0572016	−	0.998363i	$-0.518218\pi$
$43$	−1387.11	−0.114403	−0.0572016	+	0.998363i	$0.518218\pi$
$44$	0	0
$45$	2231.50	0.164273
$46$	0	0
$47$	−11639.6	−0.768586	−0.384293	−	0.923211i	$-0.625555\pi$
$47$	−11639.6	−0.768586	−0.384293	+	0.923211i	$0.625555\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−15772.2	−0.849115
$52$	0	0
$53$	17401.8	0.850950	0.425475	−	0.904970i	$-0.360107\pi$
$53$	17401.8	0.850950	0.425475	+	0.904970i	$0.360107\pi$
$54$	0	0
$55$	−6473.32	−0.288550
$56$	0	0
$57$	25806.9	1.05208
$58$	0	0
$59$	−34029.9	−1.27271	−0.636356	−	0.771395i	$-0.719559\pi$
$59$	−34029.9	−1.27271	−0.636356	+	0.771395i	$0.719559\pi$
$60$	0	0
$61$	27767.0	0.955442	0.477721	−	0.878512i	$-0.341463\pi$
$61$	27767.0	0.955442	0.477721	+	0.878512i	$0.341463\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1421.86	−0.0417421
$66$	0	0
$67$	44552.1	1.21250	0.606249	−	0.795275i	$-0.292674\pi$
$67$	44552.1	1.21250	0.606249	+	0.795275i	$0.292674\pi$
$68$	0	0
$69$	−30029.4	−0.759319
$70$	0	0
$71$	−71236.2	−1.67708	−0.838542	−	0.544837i	$-0.816591\pi$
$71$	−71236.2	−1.67708	−0.838542	+	0.544837i	$0.816591\pi$
$72$	0	0
$73$	−84065.0	−1.84632	−0.923162	−	0.384410i	$-0.874405\pi$
$73$	−84065.0	−1.84632	−0.923162	+	0.384410i	$0.874405\pi$
$74$	0	0
$75$	62329.4	1.27950
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−28839.2	−0.519895	−0.259947	−	0.965623i	$-0.583705\pi$
$79$	−28839.2	−0.519895	−0.259947	+	0.965623i	$0.583705\pi$
$80$	0	0
$81$	−68883.2	−1.16654
$82$	0	0
$83$	−82782.9	−1.31900	−0.659501	−	0.751703i	$-0.729232\pi$
$83$	−82782.9	−1.31900	−0.659501	+	0.751703i	$0.729232\pi$
$84$	0	0
$85$	8805.86	0.132198
$86$	0	0
$87$	−21057.9	−0.298276
$88$	0	0
$89$	−71184.1	−0.952594	−0.476297	−	0.879284i	$-0.658021\pi$
$89$	−71184.1	−0.952594	−0.476297	+	0.879284i	$0.658021\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	211756.	2.53880
$94$	0	0
$95$	−14408.4	−0.163797
$96$	0	0
$97$	−7891.05	−0.0851541	−0.0425770	−	0.999093i	$-0.513557\pi$
$97$	−7891.05	−0.0851541	−0.0425770	+	0.999093i	$0.513557\pi$
$98$	0	0
$99$	−106604.	−1.09317
$100$	0	0
$101$	−129613.	−1.26428	−0.632141	−	0.774853i	$-0.717824\pi$
$101$	−129613.	−1.26428	−0.632141	+	0.774853i	$0.717824\pi$
$102$	0	0
$103$	130318.	1.21035	0.605173	−	0.796094i	$-0.293104\pi$
$103$	130318.	1.21035	0.605173	+	0.796094i	$0.293104\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−25343.1	−0.213994	−0.106997	−	0.994259i	$-0.534123\pi$
$107$	−25343.1	−0.213994	−0.106997	+	0.994259i	$0.534123\pi$
$108$	0	0
$109$	18231.8	0.146981	0.0734907	−	0.997296i	$-0.476586\pi$
$109$	18231.8	0.146981	0.0734907	+	0.997296i	$0.476586\pi$
$110$	0	0
$111$	−135325.	−1.04249
$112$	0	0
$113$	227931.	1.67922	0.839609	−	0.543191i	$-0.182784\pi$
$113$	227931.	1.67922	0.839609	+	0.543191i	$0.182784\pi$
$114$	0	0
$115$	16765.9	0.118218
$116$	0	0
$117$	−23415.6	−0.158139
$118$	0	0
$119$	0	0
$120$	0	0
$121$	148196.	0.920178
$122$	0	0
$123$	261292.	1.55727
$124$	0	0
$125$	−71176.3	−0.407437
$126$	0	0
$127$	−266122.	−1.46410	−0.732051	−	0.681249i	$-0.761437\pi$
$127$	−266122.	−1.46410	−0.732051	+	0.681249i	$0.761437\pi$
$128$	0	0
$129$	28920.4	0.153014
$130$	0	0
$131$	−83066.1	−0.422908	−0.211454	−	0.977388i	$-0.567820\pi$
$131$	−83066.1	−0.422908	−0.211454	+	0.977388i	$0.567820\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	12450.6	0.0587969
$136$	0	0
$137$	67549.9	0.307485	0.153742	−	0.988111i	$-0.450867\pi$
$137$	67549.9	0.307485	0.153742	+	0.988111i	$0.450867\pi$
$138$	0	0
$139$	143233.	0.628789	0.314395	−	0.949292i	$-0.398198\pi$
$139$	143233.	0.628789	0.314395	+	0.949292i	$0.398198\pi$
$140$	0	0
$141$	242679.	1.02798
$142$	0	0
$143$	67925.9	0.277776
$144$	0	0
$145$	11757.0	0.0464383
$146$	0	0
$147$	0	0
$148$	0	0
$149$	302043.	1.11456	0.557280	−	0.830325i	$-0.311845\pi$
$149$	302043.	1.11456	0.557280	+	0.830325i	$0.311845\pi$
$150$	0	0
$151$	347296.	1.23953	0.619766	−	0.784787i	$-0.287228\pi$
$151$	347296.	1.23953	0.619766	+	0.784787i	$0.287228\pi$
$152$	0	0
$153$	145017.	0.500830
$154$	0	0
$155$	−118227.	−0.395263
$156$	0	0
$157$	479415.	1.55225	0.776127	−	0.630577i	$-0.217181\pi$
$157$	479415.	1.55225	0.776127	+	0.630577i	$0.217181\pi$
$158$	0	0
$159$	−362818.	−1.13814
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−166808.	−0.491754	−0.245877	−	0.969301i	$-0.579076\pi$
$163$	−166808.	−0.491754	−0.245877	+	0.969301i	$0.579076\pi$
$164$	0	0
$165$	134965.	0.385933
$166$	0	0
$167$	500867.	1.38973	0.694866	−	0.719139i	$-0.255464\pi$
$167$	500867.	1.38973	0.694866	+	0.719139i	$0.255464\pi$
$168$	0	0
$169$	−356373.	−0.959816
$170$	0	0
$171$	−237281.	−0.620544
$172$	0	0
$173$	632727.	1.60732	0.803658	−	0.595091i	$-0.202884\pi$
$173$	632727.	1.60732	0.803658	+	0.595091i	$0.202884\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	709504.	1.70224
$178$	0	0
$179$	200845.	0.468520	0.234260	−	0.972174i	$-0.424733\pi$
$179$	200845.	0.468520	0.234260	+	0.972174i	$0.424733\pi$
$180$	0	0
$181$	−84680.7	−0.192127	−0.0960634	−	0.995375i	$-0.530625\pi$
$181$	−84680.7	−0.192127	−0.0960634	+	0.995375i	$0.530625\pi$
$182$	0	0
$183$	−578927.	−1.27790
$184$	0	0
$185$	75554.3	0.162304
$186$	0	0
$187$	−420678.	−0.879722
$188$	0	0
$189$	0	0
$190$	0	0
$191$	532830.	1.05683	0.528416	−	0.848986i	$-0.322786\pi$
$191$	532830.	1.05683	0.528416	+	0.848986i	$0.322786\pi$
$192$	0	0
$193$	478231.	0.924155	0.462077	−	0.886840i	$-0.347104\pi$
$193$	478231.	0.924155	0.462077	+	0.886840i	$0.347104\pi$
$194$	0	0
$195$	29645.1	0.0558298
$196$	0	0
$197$	−188847.	−0.346692	−0.173346	−	0.984861i	$-0.555458\pi$
$197$	−188847.	−0.346692	−0.173346	+	0.984861i	$0.555458\pi$
$198$	0	0
$199$	530722.	0.950024	0.475012	−	0.879979i	$-0.342444\pi$
$199$	530722.	0.950024	0.475012	+	0.879979i	$0.342444\pi$
$200$	0	0
$201$	−928886.	−1.62171
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−145884.	−0.242450
$206$	0	0
$207$	276105.	0.447866
$208$	0	0
$209$	688325.	1.09000
$210$	0	0
$211$	−527385.	−0.815495	−0.407748	−	0.913095i	$-0.633686\pi$
$211$	−527385.	−0.815495	−0.407748	+	0.913095i	$0.633686\pi$
$212$	0	0
$213$	1.48524e6	2.24309
$214$	0	0
$215$	−16146.7	−0.0238225
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.75271e6	2.46945
$220$	0	0
$221$	−92401.8	−0.127262
$222$	0	0
$223$	595715.	0.802188	0.401094	−	0.916037i	$-0.368630\pi$
$223$	595715.	0.802188	0.401094	+	0.916037i	$0.368630\pi$
$224$	0	0
$225$	−573086.	−0.754681
$226$	0	0
$227$	902497.	1.16247	0.581234	−	0.813736i	$-0.302570\pi$
$227$	902497.	1.16247	0.581234	+	0.813736i	$0.302570\pi$
$228$	0	0
$229$	396205.	0.499265	0.249633	−	0.968341i	$-0.419690\pi$
$229$	396205.	0.499265	0.249633	+	0.968341i	$0.419690\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	965340.	1.16490	0.582452	−	0.812865i	$-0.302093\pi$
$233$	965340.	1.16490	0.582452	+	0.812865i	$0.302093\pi$
$234$	0	0
$235$	−135491.	−0.160045
$236$	0	0
$237$	601282.	0.695356
$238$	0	0
$239$	1.39108e6	1.57528	0.787639	−	0.616138i	$-0.211303\pi$
$239$	1.39108e6	1.57528	0.787639	+	0.616138i	$0.211303\pi$
$240$	0	0
$241$	−1.67302e6	−1.85549	−0.927744	−	0.373218i	$-0.878254\pi$
$241$	−1.67302e6	−1.85549	−0.927744	+	0.373218i	$0.878254\pi$
$242$	0	0
$243$	1.17627e6	1.27788
$244$	0	0
$245$	0	0
$246$	0	0
$247$	151190.	0.157682
$248$	0	0
$249$	1.72598e6	1.76416
$250$	0	0
$251$	−202598.	−0.202979	−0.101489	−	0.994837i	$-0.532361\pi$
$251$	−202598.	−0.202979	−0.101489	+	0.994837i	$0.532361\pi$
$252$	0	0
$253$	−800949.	−0.786690
$254$	0	0
$255$	−183597.	−0.176814
$256$	0	0
$257$	−1.57591e6	−1.48832	−0.744162	−	0.667999i	$-0.767151\pi$
$257$	−1.57591e6	−1.48832	−0.744162	+	0.667999i	$0.767151\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	193617.	0.175931
$262$	0	0
$263$	1.42214e6	1.26781	0.633904	−	0.773412i	$-0.281451\pi$
$263$	1.42214e6	1.26781	0.633904	+	0.773412i	$0.281451\pi$
$264$	0	0
$265$	202567.	0.177196
$266$	0	0
$267$	1.48415e6	1.27409
$268$	0	0
$269$	−732696.	−0.617367	−0.308683	−	0.951165i	$-0.599888\pi$
$269$	−732696.	−0.617367	−0.308683	+	0.951165i	$0.599888\pi$
$270$	0	0
$271$	527076.	0.435963	0.217982	−	0.975953i	$-0.430053\pi$
$271$	527076.	0.435963	0.217982	+	0.975953i	$0.430053\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.66246e6	1.32562
$276$	0	0
$277$	1.73803e6	1.36100	0.680500	−	0.732748i	$-0.261763\pi$
$277$	1.73803e6	1.36100	0.680500	+	0.732748i	$0.261763\pi$
$278$	0	0
$279$	−1.94699e6	−1.49745
$280$	0	0
$281$	−1.01391e6	−0.766006	−0.383003	−	0.923747i	$-0.625110\pi$
$281$	−1.01391e6	−0.766006	−0.383003	+	0.923747i	$0.625110\pi$
$282$	0	0
$283$	461592.	0.342604	0.171302	−	0.985219i	$-0.445203\pi$
$283$	461592.	0.342604	0.171302	+	0.985219i	$0.445203\pi$
$284$	0	0
$285$	300407.	0.219078
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−847595.	−0.596958
$290$	0	0
$291$	164524.	0.113893
$292$	0	0
$293$	−2.10601e6	−1.43315	−0.716573	−	0.697512i	$-0.754290\pi$
$293$	−2.10601e6	−1.43315	−0.716573	+	0.697512i	$0.754290\pi$
$294$	0	0
$295$	−396127.	−0.265021
$296$	0	0
$297$	−594794.	−0.391269
$298$	0	0
$299$	−175928.	−0.113804
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2.70235e6	1.69097
$304$	0	0
$305$	323224.	0.198955
$306$	0	0
$307$	269986.	0.163492	0.0817458	−	0.996653i	$-0.473950\pi$
$307$	269986.	0.163492	0.0817458	+	0.996653i	$0.473950\pi$
$308$	0	0
$309$	−2.71705e6	−1.61883
$310$	0	0
$311$	−425297.	−0.249340	−0.124670	−	0.992198i	$-0.539787\pi$
$311$	−425297.	−0.249340	−0.124670	+	0.992198i	$0.539787\pi$
$312$	0	0
$313$	−568777.	−0.328156	−0.164078	−	0.986447i	$-0.552465\pi$
$313$	−568777.	−0.328156	−0.164078	+	0.986447i	$0.552465\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−584734.	−0.326821	−0.163411	−	0.986558i	$-0.552250\pi$
$317$	−584734.	−0.326821	−0.163411	+	0.986558i	$0.552250\pi$
$318$	0	0
$319$	−561660.	−0.309027
$320$	0	0
$321$	528390.	0.286215
$322$	0	0
$323$	−936350.	−0.499381
$324$	0	0
$325$	365158.	0.191766
$326$	0	0
$327$	−380122.	−0.196587
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.86613e6	0.936209	0.468104	−	0.883673i	$-0.344937\pi$
$331$	1.86613e6	0.936209	0.468104	+	0.883673i	$0.344937\pi$
$332$	0	0
$333$	1.24425e6	0.614887
$334$	0	0
$335$	518612.	0.252482
$336$	0	0
$337$	−1.97456e6	−0.947097	−0.473549	−	0.880768i	$-0.657027\pi$
$337$	−1.97456e6	−0.947097	−0.473549	+	0.880768i	$0.657027\pi$
$338$	0	0
$339$	−4.75224e6	−2.24594
$340$	0	0
$341$	5.64799e6	2.63031
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−349560.	−0.158115
$346$	0	0
$347$	−1.59686e6	−0.711941	−0.355971	−	0.934497i	$-0.615850\pi$
$347$	−1.59686e6	−0.711941	−0.355971	+	0.934497i	$0.615850\pi$
$348$	0	0
$349$	−3.32227e6	−1.46006	−0.730030	−	0.683415i	$-0.760494\pi$
$349$	−3.32227e6	−1.46006	−0.730030	+	0.683415i	$0.760494\pi$
$350$	0	0
$351$	−130646.	−0.0566017
$352$	0	0
$353$	−2.79852e6	−1.19534	−0.597670	−	0.801742i	$-0.703907\pi$
$353$	−2.79852e6	−1.19534	−0.597670	+	0.801742i	$0.703907\pi$
$354$	0	0
$355$	−829231.	−0.349224
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−918843.	−0.376275	−0.188137	−	0.982143i	$-0.560245\pi$
$359$	−918843.	−0.376275	−0.188137	+	0.982143i	$0.560245\pi$
$360$	0	0
$361$	−944017.	−0.381252
$362$	0	0
$363$	−3.08980e6	−1.23073
$364$	0	0
$365$	−978566.	−0.384466
$366$	0	0
$367$	−1.40210e6	−0.543393	−0.271696	−	0.962383i	$-0.587585\pi$
$367$	−1.40210e6	−0.543393	−0.271696	+	0.962383i	$0.587585\pi$
$368$	0	0
$369$	−2.40244e6	−0.918517
$370$	0	0
$371$	0	0
$372$	0	0
$373$	4.39061e6	1.63400	0.817002	−	0.576635i	$-0.195635\pi$
$373$	4.39061e6	1.63400	0.817002	+	0.576635i	$0.195635\pi$
$374$	0	0
$375$	1.48399e6	0.544944
$376$	0	0
$377$	−123369.	−0.0447045
$378$	0	0
$379$	4.94450e6	1.76817	0.884086	−	0.467325i	$-0.154782\pi$
$379$	4.94450e6	1.76817	0.884086	+	0.467325i	$0.154782\pi$
$380$	0	0
$381$	5.54850e6	1.95823
$382$	0	0
$383$	−348199.	−0.121291	−0.0606457	−	0.998159i	$-0.519316\pi$
$383$	−348199.	−0.121291	−0.0606457	+	0.998159i	$0.519316\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−265908.	−0.0902514
$388$	0	0
$389$	1.78801e6	0.599097	0.299548	−	0.954081i	$-0.403164\pi$
$389$	1.78801e6	0.599097	0.299548	+	0.954081i	$0.403164\pi$
$390$	0	0
$391$	1.08956e6	0.360419
$392$	0	0
$393$	1.73188e6	0.565636
$394$	0	0
$395$	−335705.	−0.108259
$396$	0	0
$397$	5.24647e6	1.67067	0.835336	−	0.549740i	$-0.185273\pi$
$397$	5.24647e6	1.67067	0.835336	+	0.549740i	$0.185273\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.26723e6	−1.32521	−0.662606	−	0.748968i	$-0.730550\pi$
$401$	−4.26723e6	−1.32521	−0.662606	+	0.748968i	$0.730550\pi$
$402$	0	0
$403$	1.24058e6	0.380506
$404$	0	0
$405$	−801841.	−0.242913
$406$	0	0
$407$	−3.60942e6	−1.08007
$408$	0	0
$409$	−3.94192e6	−1.16520	−0.582598	−	0.812760i	$-0.697964\pi$
$409$	−3.94192e6	−1.16520	−0.582598	+	0.812760i	$0.697964\pi$
$410$	0	0
$411$	−1.40838e6	−0.411259
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−963642.	−0.274660
$416$	0	0
$417$	−2.98632e6	−0.841002
$418$	0	0
$419$	3.95829e6	1.10147	0.550735	−	0.834680i	$-0.314348\pi$
$419$	3.95829e6	1.10147	0.550735	+	0.834680i	$0.314348\pi$
$420$	0	0
$421$	4.36155e6	1.19932	0.599661	−	0.800254i	$-0.295302\pi$
$421$	4.36155e6	1.19932	0.599661	+	0.800254i	$0.295302\pi$
$422$	0	0
$423$	−2.23130e6	−0.606329
$424$	0	0
$425$	−2.26149e6	−0.607328
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.41622e6	−0.371524
$430$	0	0
$431$	15369.3	0.00398530	0.00199265	−	0.999998i	$-0.499366\pi$
$431$	15369.3	0.00398530	0.00199265	+	0.999998i	$0.499366\pi$
$432$	0	0
$433$	967585.	0.248010	0.124005	−	0.992282i	$-0.460426\pi$
$433$	967585.	0.248010	0.124005	+	0.992282i	$0.460426\pi$
$434$	0	0
$435$	−245127.	−0.0621109
$436$	0	0
$437$	−1.78276e6	−0.446570
$438$	0	0
$439$	2.95188e6	0.731032	0.365516	−	0.930805i	$-0.380892\pi$
$439$	2.95188e6	0.731032	0.365516	+	0.930805i	$0.380892\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	721038.	0.174562	0.0872809	−	0.996184i	$-0.472182\pi$
$443$	721038.	0.174562	0.0872809	+	0.996184i	$0.472182\pi$
$444$	0	0
$445$	−828624.	−0.198362
$446$	0	0
$447$	−6.29744e6	−1.49072
$448$	0	0
$449$	5.48059e6	1.28295	0.641477	−	0.767142i	$-0.278322\pi$
$449$	5.48059e6	1.28295	0.641477	+	0.767142i	$0.278322\pi$
$450$	0	0
$451$	6.96922e6	1.61340
$452$	0	0
$453$	−7.24093e6	−1.65786
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.75663e6	1.28937	0.644686	−	0.764448i	$-0.276988\pi$
$457$	5.75663e6	1.28937	0.644686	+	0.764448i	$0.276988\pi$
$458$	0	0
$459$	809117.	0.179258
$460$	0	0
$461$	4.64082e6	1.01705	0.508525	−	0.861047i	$-0.330191\pi$
$461$	4.64082e6	1.01705	0.508525	+	0.861047i	$0.330191\pi$
$462$	0	0
$463$	−2.72920e6	−0.591674	−0.295837	−	0.955238i	$-0.595599\pi$
$463$	−2.72920e6	−0.591674	−0.295837	+	0.955238i	$0.595599\pi$
$464$	0	0
$465$	2.46496e6	0.528662
$466$	0	0
$467$	2.44068e6	0.517867	0.258934	−	0.965895i	$-0.416629\pi$
$467$	2.44068e6	0.517867	0.258934	+	0.965895i	$0.416629\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−9.99555e6	−2.07613
$472$	0	0
$473$	771369.	0.158529
$474$	0	0
$475$	3.70032e6	0.752497
$476$	0	0
$477$	3.33592e6	0.671304
$478$	0	0
$479$	6.26551e6	1.24772	0.623861	−	0.781536i	$-0.285563\pi$
$479$	6.26551e6	1.24772	0.623861	+	0.781536i	$0.285563\pi$
$480$	0	0
$481$	−792807.	−0.156245
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−91856.4	−0.0177319
$486$	0	0
$487$	−5.67752e6	−1.08477	−0.542383	−	0.840131i	$-0.682478\pi$
$487$	−5.67752e6	−1.08477	−0.542383	+	0.840131i	$0.682478\pi$
$488$	0	0
$489$	3.47786e6	0.657717
$490$	0	0
$491$	489721.	0.0916738	0.0458369	−	0.998949i	$-0.485405\pi$
$491$	489721.	0.0916738	0.0458369	+	0.998949i	$0.485405\pi$
$492$	0	0
$493$	764045.	0.141580
$494$	0	0
$495$	−1.24093e6	−0.227633
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−7.41525e6	−1.33314	−0.666568	−	0.745444i	$-0.732237\pi$
$499$	−7.41525e6	−1.33314	−0.666568	+	0.745444i	$0.732237\pi$
$500$	0	0
$501$	−1.04428e7	−1.85876
$502$	0	0
$503$	9.82290e6	1.73109	0.865545	−	0.500830i	$-0.166972\pi$
$503$	9.82290e6	1.73109	0.865545	+	0.500830i	$0.166972\pi$
$504$	0	0
$505$	−1.50877e6	−0.263265
$506$	0	0
$507$	7.43018e6	1.28375
$508$	0	0
$509$	−2.75947e6	−0.472096	−0.236048	−	0.971741i	$-0.575852\pi$
$509$	−2.75947e6	−0.472096	−0.236048	+	0.971741i	$0.575852\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1.32390e6	−0.222107
$514$	0	0
$515$	1.51697e6	0.252034
$516$	0	0
$517$	6.47276e6	1.06503
$518$	0	0
$519$	−1.31920e7	−2.14977
$520$	0	0
$521$	406899.	0.0656739	0.0328369	−	0.999461i	$-0.489546\pi$
$521$	406899.	0.0656739	0.0328369	+	0.999461i	$0.489546\pi$
$522$	0	0
$523$	3.40895e6	0.544963	0.272481	−	0.962161i	$-0.412156\pi$
$523$	3.40895e6	0.544963	0.272481	+	0.962161i	$0.412156\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7.68314e6	−1.20507
$528$	0	0
$529$	−4.36188e6	−0.677696
$530$	0	0
$531$	−6.52352e6	−1.00403
$532$	0	0
$533$	1.53079e6	0.233398
$534$	0	0
$535$	−295009.	−0.0445605
$536$	0	0
$537$	−4.18750e6	−0.626642
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4.42159e6	−0.649510	−0.324755	−	0.945798i	$-0.605282\pi$
$541$	−4.42159e6	−0.649510	−0.324755	+	0.945798i	$0.605282\pi$
$542$	0	0
$543$	1.76555e6	0.256968
$544$	0	0
$545$	212228.	0.0306064
$546$	0	0
$547$	3.18558e6	0.455219	0.227610	−	0.973752i	$-0.426909\pi$
$547$	3.18558e6	0.455219	0.227610	+	0.973752i	$0.426909\pi$
$548$	0	0
$549$	5.32293e6	0.753737
$550$	0	0
$551$	−1.25015e6	−0.175422
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.57527e6	−0.217081
$556$	0	0
$557$	−1.20859e7	−1.65060	−0.825301	−	0.564693i	$-0.808995\pi$
$557$	−1.20859e7	−1.65060	−0.825301	+	0.564693i	$0.808995\pi$
$558$	0	0
$559$	169431.	0.0229331
$560$	0	0
$561$	8.77091e6	1.17662
$562$	0	0
$563$	−4.94492e6	−0.657488	−0.328744	−	0.944419i	$-0.606625\pi$
$563$	−4.94492e6	−0.657488	−0.328744	+	0.944419i	$0.606625\pi$
$564$	0	0
$565$	2.65325e6	0.349669
$566$	0	0
$567$	0	0
$568$	0	0
$569$	106511.	0.0137915	0.00689576	−	0.999976i	$-0.497805\pi$
$569$	106511.	0.0137915	0.00689576	+	0.999976i	$0.497805\pi$
$570$	0	0
$571$	−1.20485e7	−1.54647	−0.773236	−	0.634118i	$-0.781363\pi$
$571$	−1.20485e7	−1.54647	−0.773236	+	0.634118i	$0.781363\pi$
$572$	0	0
$573$	−1.11092e7	−1.41350
$574$	0	0
$575$	−4.30577e6	−0.543101
$576$	0	0
$577$	5.89751e6	0.737445	0.368722	−	0.929540i	$-0.379795\pi$
$577$	5.89751e6	0.737445	0.368722	+	0.929540i	$0.379795\pi$
$578$	0	0
$579$	−9.97086e6	−1.23605
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−9.67712e6	−1.17917
$584$	0	0
$585$	−272571.	−0.0329298
$586$	0	0
$587$	238041.	0.0285139	0.0142570	−	0.999898i	$-0.495462\pi$
$587$	238041.	0.0285139	0.0142570	+	0.999898i	$0.495462\pi$
$588$	0	0
$589$	1.25714e7	1.49312
$590$	0	0
$591$	3.93735e6	0.463698
$592$	0	0
$593$	3.60081e6	0.420498	0.210249	−	0.977648i	$-0.432573\pi$
$593$	3.60081e6	0.420498	0.210249	+	0.977648i	$0.432573\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.10653e7	−1.27065
$598$	0	0
$599$	1.03662e7	1.18046	0.590232	−	0.807233i	$-0.299036\pi$
$599$	1.03662e7	1.18046	0.590232	+	0.807233i	$0.299036\pi$
$600$	0	0
$601$	5.04311e6	0.569524	0.284762	−	0.958598i	$-0.408085\pi$
$601$	5.04311e6	0.569524	0.284762	+	0.958598i	$0.408085\pi$
$602$	0	0
$603$	8.54062e6	0.956525
$604$	0	0
$605$	1.72508e6	0.191611
$606$	0	0
$607$	−8.10009e6	−0.892314	−0.446157	−	0.894955i	$-0.647208\pi$
$607$	−8.10009e6	−0.892314	−0.446157	+	0.894955i	$0.647208\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.42174e6	0.154070
$612$	0	0
$613$	1.14179e7	1.22726	0.613628	−	0.789596i	$-0.289710\pi$
$613$	1.14179e7	1.22726	0.613628	+	0.789596i	$0.289710\pi$
$614$	0	0
$615$	3.04159e6	0.324275
$616$	0	0
$617$	4.67937e6	0.494851	0.247426	−	0.968907i	$-0.420415\pi$
$617$	4.67937e6	0.494851	0.247426	+	0.968907i	$0.420415\pi$
$618$	0	0
$619$	−6.42397e6	−0.673871	−0.336936	−	0.941528i	$-0.609390\pi$
$619$	−6.42397e6	−0.673871	−0.336936	+	0.941528i	$0.609390\pi$
$620$	0	0
$621$	1.54052e6	0.160301
$622$	0	0
$623$	0	0
$624$	0	0
$625$	8.51364e6	0.871797
$626$	0	0
$627$	−1.43512e7	−1.45787
$628$	0	0
$629$	4.91000e6	0.494829
$630$	0	0
$631$	8.22064e6	0.821925	0.410963	−	0.911652i	$-0.365193\pi$
$631$	8.22064e6	0.821925	0.410963	+	0.911652i	$0.365193\pi$
$632$	0	0
$633$	1.09957e7	1.09072
$634$	0	0
$635$	−3.09782e6	−0.304875
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−1.36560e7	−1.32303
$640$	0	0
$641$	−2.13697e6	−0.205425	−0.102713	−	0.994711i	$-0.532752\pi$
$641$	−2.13697e6	−0.205425	−0.102713	+	0.994711i	$0.532752\pi$
$642$	0	0
$643$	1.79555e7	1.71265	0.856327	−	0.516434i	$-0.172741\pi$
$643$	1.79555e7	1.71265	0.856327	+	0.516434i	$0.172741\pi$
$644$	0	0
$645$	336650.	0.0318625
$646$	0	0
$647$	−1.16274e7	−1.09200	−0.545999	−	0.837786i	$-0.683850\pi$
$647$	−1.16274e7	−1.09200	−0.545999	+	0.837786i	$0.683850\pi$
$648$	0	0
$649$	1.89240e7	1.76360
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8.28773e6	−0.760594	−0.380297	−	0.924865i	$-0.624178\pi$
$653$	−8.28773e6	−0.760594	−0.380297	+	0.924865i	$0.624178\pi$
$654$	0	0
$655$	−966938.	−0.0880634
$656$	0	0
$657$	−1.61152e7	−1.45654
$658$	0	0
$659$	1.81934e7	1.63192	0.815962	−	0.578106i	$-0.196208\pi$
$659$	1.81934e7	1.63192	0.815962	+	0.578106i	$0.196208\pi$
$660$	0	0
$661$	−1.46833e7	−1.30713	−0.653566	−	0.756869i	$-0.726728\pi$
$661$	−1.46833e7	−1.30713	−0.653566	+	0.756869i	$0.726728\pi$
$662$	0	0
$663$	1.92653e6	0.170212
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.45470e6	0.126607
$668$	0	0
$669$	−1.24203e7	−1.07292
$670$	0	0
$671$	−1.54412e7	−1.32396
$672$	0	0
$673$	−1.20331e7	−1.02409	−0.512047	−	0.858958i	$-0.671113\pi$
$673$	−1.20331e7	−1.02409	−0.512047	+	0.858958i	$0.671113\pi$
$674$	0	0
$675$	−3.19751e6	−0.270117
$676$	0	0
$677$	−2.29426e6	−0.192385	−0.0961923	−	0.995363i	$-0.530666\pi$
$677$	−2.29426e6	−0.192385	−0.0961923	+	0.995363i	$0.530666\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1.88166e7	−1.55479
$682$	0	0
$683$	−1.65401e6	−0.135671	−0.0678355	−	0.997697i	$-0.521609\pi$
$683$	−1.65401e6	−0.135671	−0.0678355	+	0.997697i	$0.521609\pi$
$684$	0	0
$685$	786321.	0.0640285
$686$	0	0
$687$	−8.26066e6	−0.667764
$688$	0	0
$689$	−2.12558e6	−0.170580
$690$	0	0
$691$	1.33522e7	1.06380	0.531899	−	0.846808i	$-0.321479\pi$
$691$	1.33522e7	1.06380	0.531899	+	0.846808i	$0.321479\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.66731e6	0.130935
$696$	0	0
$697$	−9.48045e6	−0.739174
$698$	0	0
$699$	−2.01268e7	−1.55805
$700$	0	0
$701$	−6.37044e6	−0.489637	−0.244819	−	0.969569i	$-0.578728\pi$
$701$	−6.37044e6	−0.489637	−0.244819	+	0.969569i	$0.578728\pi$
$702$	0	0
$703$	−8.03388e6	−0.613108
$704$	0	0
$705$	2.82492e6	0.214059
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2.36380e6	0.176601	0.0883007	−	0.996094i	$-0.471856\pi$
$709$	2.36380e6	0.176601	0.0883007	+	0.996094i	$0.471856\pi$
$710$	0	0
$711$	−5.52847e6	−0.410139
$712$	0	0
$713$	−1.46283e7	−1.07763
$714$	0	0
$715$	790697.	0.0578422
$716$	0	0
$717$	−2.90032e7	−2.10692
$718$	0	0
$719$	9.92627e6	0.716084	0.358042	−	0.933706i	$-0.383445\pi$
$719$	9.92627e6	0.716084	0.358042	+	0.933706i	$0.383445\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	3.48815e7	2.48170
$724$	0	0
$725$	−3.01939e6	−0.213341
$726$	0	0
$727$	−5.69945e6	−0.399942	−0.199971	−	0.979802i	$-0.564085\pi$
$727$	−5.69945e6	−0.399942	−0.199971	+	0.979802i	$0.564085\pi$
$728$	0	0
$729$	−7.78596e6	−0.542617
$730$	0	0
$731$	−1.04932e6	−0.0726296
$732$	0	0
$733$	−2.49319e7	−1.71394	−0.856969	−	0.515368i	$-0.827655\pi$
$733$	−2.49319e7	−1.71394	−0.856969	+	0.515368i	$0.827655\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.47754e7	−1.68016
$738$	0	0
$739$	−2.31431e7	−1.55887	−0.779436	−	0.626482i	$-0.784494\pi$
$739$	−2.31431e7	−1.55887	−0.779436	+	0.626482i	$0.784494\pi$
$740$	0	0
$741$	−3.15224e6	−0.210898
$742$	0	0
$743$	−2.21340e6	−0.147091	−0.0735457	−	0.997292i	$-0.523431\pi$
$743$	−2.21340e6	−0.147091	−0.0735457	+	0.997292i	$0.523431\pi$
$744$	0	0
$745$	3.51596e6	0.232088
$746$	0	0
$747$	−1.58695e7	−1.04055
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−3.00456e7	−1.94393	−0.971967	−	0.235117i	$-0.924453\pi$
$751$	−3.00456e7	−1.94393	−0.971967	+	0.235117i	$0.924453\pi$
$752$	0	0
$753$	4.22405e6	0.271483
$754$	0	0
$755$	4.04273e6	0.258111
$756$	0	0
$757$	−2.71166e7	−1.71987	−0.859935	−	0.510403i	$-0.829496\pi$
$757$	−2.71166e7	−1.71987	−0.859935	+	0.510403i	$0.829496\pi$
$758$	0	0
$759$	1.66993e7	1.05219
$760$	0	0
$761$	2.71883e7	1.70185	0.850923	−	0.525290i	$-0.176043\pi$
$761$	2.71883e7	1.70185	0.850923	+	0.525290i	$0.176043\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.68808e6	0.104289
$766$	0	0
$767$	4.15665e6	0.255126
$768$	0	0
$769$	9.92251e6	0.605070	0.302535	−	0.953138i	$-0.402167\pi$
$769$	9.92251e6	0.605070	0.302535	+	0.953138i	$0.402167\pi$
$770$	0	0
$771$	3.28568e7	1.99062
$772$	0	0
$773$	−1.13927e7	−0.685772	−0.342886	−	0.939377i	$-0.611404\pi$
$773$	−1.13927e7	−0.685772	−0.342886	+	0.939377i	$0.611404\pi$
$774$	0	0
$775$	3.03626e7	1.81587
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.55122e7	0.915860
$780$	0	0
$781$	3.96144e7	2.32394
$782$	0	0
$783$	1.08028e6	0.0629696
$784$	0	0
$785$	5.58067e6	0.323231
$786$	0	0
$787$	−1.57570e7	−0.906852	−0.453426	−	0.891294i	$-0.649798\pi$
$787$	−1.57570e7	−0.906852	−0.453426	+	0.891294i	$0.649798\pi$
$788$	0	0
$789$	−2.96509e7	−1.69568
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−3.39166e6	−0.191527
$794$	0	0
$795$	−4.22341e6	−0.236998
$796$	0	0
$797$	3.19736e7	1.78298	0.891490	−	0.453041i	$-0.149661\pi$
$797$	3.19736e7	1.78298	0.891490	+	0.453041i	$0.149661\pi$
$798$	0	0
$799$	−8.80511e6	−0.487941
$800$	0	0
$801$	−1.36460e7	−0.751490
$802$	0	0
$803$	4.67485e7	2.55846
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.52763e7	0.825724
$808$	0	0
$809$	5.88860e6	0.316330	0.158165	−	0.987413i	$-0.449442\pi$
$809$	5.88860e6	0.316330	0.158165	+	0.987413i	$0.449442\pi$
$810$	0	0
$811$	−3.28557e7	−1.75412	−0.877058	−	0.480384i	$-0.840497\pi$
$811$	−3.28557e7	−1.75412	−0.877058	+	0.480384i	$0.840497\pi$
$812$	0	0
$813$	−1.09892e7	−0.583098
$814$	0	0
$815$	−1.94174e6	−0.102399
$816$	0	0
$817$	1.71692e6	0.0899902
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.25643e6	−0.0650551	−0.0325275	−	0.999471i	$-0.510356\pi$
$821$	−1.25643e6	−0.0650551	−0.0325275	+	0.999471i	$0.510356\pi$
$822$	0	0
$823$	3.01328e6	0.155074	0.0775372	−	0.996989i	$-0.475294\pi$
$823$	3.01328e6	0.155074	0.0775372	+	0.996989i	$0.475294\pi$
$824$	0	0
$825$	−3.46613e7	−1.77301
$826$	0	0
$827$	1.78685e7	0.908501	0.454250	−	0.890874i	$-0.349907\pi$
$827$	1.78685e7	0.908501	0.454250	+	0.890874i	$0.349907\pi$
$828$	0	0
$829$	−2.71138e6	−0.137026	−0.0685132	−	0.997650i	$-0.521826\pi$
$829$	−2.71138e6	−0.137026	−0.0685132	+	0.997650i	$0.521826\pi$
$830$	0	0
$831$	−3.62370e7	−1.82033
$832$	0	0
$833$	0	0
$834$	0	0
$835$	5.83038e6	0.289388
$836$	0	0
$837$	−1.08631e7	−0.535971
$838$	0	0
$839$	1.98996e7	0.975976	0.487988	−	0.872850i	$-0.337731\pi$
$839$	1.98996e7	0.975976	0.487988	+	0.872850i	$0.337731\pi$
$840$	0	0
$841$	−1.94910e7	−0.950266
$842$	0	0
$843$	2.11394e7	1.02453
$844$	0	0
$845$	−4.14839e6	−0.199865
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−9.62394e6	−0.458230
$850$	0	0
$851$	9.34839e6	0.442500
$852$	0	0
$853$	343719.	0.0161745	0.00808726	−	0.999967i	$-0.497426\pi$
$853$	343719.	0.0161745	0.00808726	+	0.999967i	$0.497426\pi$
$854$	0	0
$855$	−2.76209e6	−0.129218
$856$	0	0
$857$	1.38911e7	0.646075	0.323038	−	0.946386i	$-0.395296\pi$
$857$	1.38911e7	0.646075	0.323038	+	0.946386i	$0.395296\pi$
$858$	0	0
$859$	−2.24835e7	−1.03963	−0.519817	−	0.854278i	$-0.674000\pi$
$859$	−2.24835e7	−1.03963	−0.519817	+	0.854278i	$0.674000\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.11963e7	0.511737	0.255868	−	0.966712i	$-0.417639\pi$
$863$	1.11963e7	0.511737	0.255868	+	0.966712i	$0.417639\pi$
$864$	0	0
$865$	7.36532e6	0.334696
$866$	0	0
$867$	1.76719e7	0.798428
$868$	0	0
$869$	1.60375e7	0.720421
$870$	0	0
$871$	−5.44191e6	−0.243056
$872$	0	0
$873$	−1.51271e6	−0.0671770
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.32776e7	1.02197	0.510986	−	0.859589i	$-0.329281\pi$
$877$	2.32776e7	1.02197	0.510986	+	0.859589i	$0.329281\pi$
$878$	0	0
$879$	4.39091e7	1.91682
$880$	0	0
$881$	2.83389e7	1.23011	0.615055	−	0.788485i	$-0.289134\pi$
$881$	2.83389e7	1.23011	0.615055	+	0.788485i	$0.289134\pi$
$882$	0	0
$883$	3.60153e7	1.55448	0.777240	−	0.629204i	$-0.216619\pi$
$883$	3.60153e7	1.55448	0.777240	+	0.629204i	$0.216619\pi$
$884$	0	0
$885$	8.25904e6	0.354463
$886$	0	0
$887$	−8.48559e6	−0.362137	−0.181069	−	0.983470i	$-0.557956\pi$
$887$	−8.48559e6	−0.362137	−0.181069	+	0.983470i	$0.557956\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3.83059e7	1.61649
$892$	0	0
$893$	1.44072e7	0.604574
$894$	0	0
$895$	2.33795e6	0.0975613
$896$	0	0
$897$	3.66801e6	0.152212
$898$	0	0
$899$	−1.02580e7	−0.423315
$900$	0	0
$901$	1.31641e7	0.540230
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−985732.	−0.0400072
$906$	0	0
$907$	−1.24864e7	−0.503985	−0.251993	−	0.967729i	$-0.581086\pi$
$907$	−1.24864e7	−0.503985	−0.251993	+	0.967729i	$0.581086\pi$
$908$	0	0
$909$	−2.48467e7	−0.997377
$910$	0	0
$911$	−4.09862e7	−1.63622	−0.818110	−	0.575062i	$-0.804978\pi$
$911$	−4.09862e7	−1.63622	−0.818110	+	0.575062i	$0.804978\pi$
$912$	0	0
$913$	4.60355e7	1.82775
$914$	0	0
$915$	−6.73904e6	−0.266100
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1.99027e7	−0.777361	−0.388680	−	0.921373i	$-0.627069\pi$
$919$	−1.99027e7	−0.777361	−0.388680	+	0.921373i	$0.627069\pi$
$920$	0	0
$921$	−5.62907e6	−0.218669
$922$	0	0
$923$	8.70129e6	0.336186
$924$	0	0
$925$	−1.94036e7	−0.745639
$926$	0	0
$927$	2.49818e7	0.954828
$928$	0	0
$929$	1.78375e7	0.678101	0.339050	−	0.940768i	$-0.389894\pi$
$929$	1.78375e7	0.678101	0.339050	+	0.940768i	$0.389894\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	8.86722e6	0.333490
$934$	0	0
$935$	−4.89694e6	−0.183187
$936$	0	0
$937$	1.34407e7	0.500120	0.250060	−	0.968230i	$-0.419550\pi$
$937$	1.34407e7	0.500120	0.250060	+	0.968230i	$0.419550\pi$
$938$	0	0
$939$	1.18587e7	0.438907
$940$	0	0
$941$	1.57357e7	0.579313	0.289657	−	0.957131i	$-0.406459\pi$
$941$	1.57357e7	0.579313	0.289657	+	0.957131i	$0.406459\pi$
$942$	0	0
$943$	−1.80503e7	−0.661005
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.78101e7	1.37004	0.685020	−	0.728525i	$-0.259794\pi$
$947$	3.78101e7	1.37004	0.685020	+	0.728525i	$0.259794\pi$
$948$	0	0
$949$	1.02683e7	0.370112
$950$	0	0
$951$	1.21914e7	0.437121
$952$	0	0
$953$	2.30854e7	0.823390	0.411695	−	0.911322i	$-0.364937\pi$
$953$	2.30854e7	0.823390	0.411695	+	0.911322i	$0.364937\pi$
$954$	0	0
$955$	6.20246e6	0.220067
$956$	0	0
$957$	1.17103e7	0.413322
$958$	0	0
$959$	0	0
$960$	0	0
$961$	7.45240e7	2.60308
$962$	0	0
$963$	−4.85827e6	−0.168817
$964$	0	0
$965$	5.56689e6	0.192440
$966$	0	0
$967$	−5.00012e7	−1.71955	−0.859773	−	0.510676i	$-0.829395\pi$
$967$	−5.00012e7	−1.71955	−0.859773	+	0.510676i	$0.829395\pi$
$968$	0	0
$969$	1.95224e7	0.667919
$970$	0	0
$971$	−4.24614e7	−1.44526	−0.722630	−	0.691235i	$-0.757067\pi$
$971$	−4.24614e7	−1.44526	−0.722630	+	0.691235i	$0.757067\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−7.61335e6	−0.256486
$976$	0	0
$977$	2.68403e7	0.899605	0.449802	−	0.893128i	$-0.351494\pi$
$977$	2.68403e7	0.899605	0.449802	+	0.893128i	$0.351494\pi$
$978$	0	0
$979$	3.95854e7	1.32001
$980$	0	0
$981$	3.49503e6	0.115952
$982$	0	0
$983$	−3.17878e7	−1.04925	−0.524623	−	0.851335i	$-0.675794\pi$
$983$	−3.17878e7	−1.04925	−0.524623	+	0.851335i	$0.675794\pi$
$984$	0	0
$985$	−2.19829e6	−0.0721927
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.99785e6	−0.0649488
$990$	0	0
$991$	−1.93520e7	−0.625952	−0.312976	−	0.949761i	$-0.601326\pi$
$991$	−1.93520e7	−0.625952	−0.312976	+	0.949761i	$0.601326\pi$
$992$	0	0
$993$	−3.89079e7	−1.25217
$994$	0	0
$995$	6.17792e6	0.197826
$996$	0	0
$997$	−9.63171e6	−0.306878	−0.153439	−	0.988158i	$-0.549035\pi$
$997$	−9.63171e6	−0.306878	−0.153439	+	0.988158i	$0.549035\pi$
$998$	0	0
$999$	6.94222e6	0.220082

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.be.1.1		4
4.3	odd	2		392.6.a.g.1.4	yes	4
7.6	odd	2	inner	784.6.a.be.1.4		4
28.3	even	6		392.6.i.n.177.4		8
28.11	odd	6		392.6.i.n.177.1		8
28.19	even	6		392.6.i.n.361.4		8
28.23	odd	6		392.6.i.n.361.1		8
28.27	even	2		392.6.a.g.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.6.a.g.1.1	✓	4	28.27	even	2
392.6.a.g.1.4	yes	4	4.3	odd	2
392.6.i.n.177.1		8	28.11	odd	6
392.6.i.n.177.4		8	28.3	even	6
392.6.i.n.361.1		8	28.23	odd	6
392.6.i.n.361.4		8	28.19	even	6
784.6.a.be.1.1		4	1.1	even	1	trivial
784.6.a.be.1.4		4	7.6	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-20.8495 q^{3} +11.6406 q^{5} +191.700 q^{9} +O(q^{10})\)	\(q-20.8495 q^{3} +11.6406 q^{5} +191.700 q^{9} -556.099 q^{11} -122.147 q^{13} -242.700 q^{15} +756.480 q^{17} -1237.77 q^{19} +1440.30 q^{23} -2989.50 q^{25} +1069.58 q^{27} +1010.00 q^{29} -10156.4 q^{31} +11594.4 q^{33} +6490.60 q^{37} +2546.70 q^{39} -12532.3 q^{41} -1387.11 q^{43} +2231.50 q^{45} -11639.6 q^{47} -15772.2 q^{51} +17401.8 q^{53} -6473.32 q^{55} +25806.9 q^{57} -34029.9 q^{59} +27767.0 q^{61} -1421.86 q^{65} +44552.1 q^{67} -30029.4 q^{69} -71236.2 q^{71} -84065.0 q^{73} +62329.4 q^{75} -28839.2 q^{79} -68883.2 q^{81} -82782.9 q^{83} +8805.86 q^{85} -21057.9 q^{87} -71184.1 q^{89} +211756. q^{93} -14408.4 q^{95} -7891.05 q^{97} -106604. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 92 q^{9}+O(q^{10}) \) 4 * q - 92 * q^9	\( 4 q - 92 q^{9} + 352 q^{11} - 112 q^{15} - 1968 q^{23} + 924 q^{25} + 4040 q^{29} + 10504 q^{37} + 9328 q^{39} - 28736 q^{43} - 23584 q^{51} - 28296 q^{53} + 62864 q^{57} + 146320 q^{65} + 72576 q^{67} - 176736 q^{71} - 135968 q^{79} - 27340 q^{81} + 298016 q^{85} + 395296 q^{93} + 367472 q^{95} - 561248 q^{99}+O(q^{100}) \) 4 * q - 92 * q^9 + 352 * q^11 - 112 * q^15 - 1968 * q^23 + 924 * q^25 + 4040 * q^29 + 10504 * q^37 + 9328 * q^39 - 28736 * q^43 - 23584 * q^51 - 28296 * q^53 + 62864 * q^57 + 146320 * q^65 + 72576 * q^67 - 176736 * q^71 - 135968 * q^79 - 27340 * q^81 + 298016 * q^85 + 395296 * q^93 + 367472 * q^95 - 561248 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more