LMFDB - Embedded newform 79.5.b.a.78.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [79,5,Mod(78,79)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(79, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("79.78");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 79 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	79.b (of order $2$, degree $1$, minimal)

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:	$8.16622708362$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.19503125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 10x^{3} + 20x - 7 $ x^5 - 10x^3 + 20x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			78.4
Root			$2.78236$ of defining polynomial
Character	$\chi$	$=$	79.78

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.99886 q^{2} +19.9864 q^{4} +27.1098 q^{5} +23.9137 q^{8} +81.0000 q^{9} +O(q^{10})$	$q+5.99886 q^{2} +19.9864 q^{4} +27.1098 q^{5} +23.9137 q^{8} +81.0000 q^{9} +162.628 q^{10} -106.183 q^{11} +81.6737 q^{13} -176.327 q^{16} +485.908 q^{18} +555.096 q^{19} +541.827 q^{20} -636.975 q^{22} -1029.83 q^{23} +109.943 q^{25} +489.950 q^{26} -1921.62 q^{31} -1440.38 q^{32} +1618.90 q^{36} +3329.94 q^{38} +648.298 q^{40} -2122.21 q^{44} +2195.90 q^{45} -6177.78 q^{46} +2401.00 q^{49} +659.535 q^{50} +1632.36 q^{52} -2878.59 q^{55} -11527.6 q^{62} -5819.42 q^{64} +2214.16 q^{65} +4618.08 q^{67} +1937.01 q^{72} -8527.61 q^{73} +11094.4 q^{76} +6241.00 q^{79} -4780.19 q^{80} +6561.00 q^{81} +8722.00 q^{83} -2539.22 q^{88} +10180.3 q^{89} +13172.9 q^{90} -20582.5 q^{92} +15048.6 q^{95} +805.607 q^{97} +14403.3 q^{98} -8600.79 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 80 q^{4} + 405 q^{9}+O(q^{10}) $ 5 * q + 80 * q^4 + 405 * q^9	$ 5 q + 80 q^{4} + 405 q^{9} + 1280 q^{16} + 3605 q^{20} + 1285 q^{22} + 3125 q^{25} - 3115 q^{26} - 9115 q^{32} + 6480 q^{36} - 15995 q^{40} + 12005 q^{49} - 22795 q^{50} - 28315 q^{62} + 20480 q^{64} - 31115 q^{76} + 31205 q^{79} + 57680 q^{80} + 32805 q^{81} + 43610 q^{83} + 20560 q^{88} - 29515 q^{92} - 10870 q^{95}+O(q^{100}) $ 5 * q + 80 * q^4 + 405 * q^9 + 1280 * q^16 + 3605 * q^20 + 1285 * q^22 + 3125 * q^25 - 3115 * q^26 - 9115 * q^32 + 6480 * q^36 - 15995 * q^40 + 12005 * q^49 - 22795 * q^50 - 28315 * q^62 + 20480 * q^64 - 31115 * q^76 + 31205 * q^79 + 57680 * q^80 + 32805 * q^81 + 43610 * q^83 + 20560 * q^88 - 29515 * q^92 - 10870 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/79\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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 $

$2$	5.99886	1.49972	0.749858	−	0.661599i	$-0.230122\pi$
$2$	5.99886	1.49972	0.749858	+	0.661599i	$0.230122\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	19.9864	1.24915
$5$	27.1098	1.08439	0.542197	−	0.840252i	$-0.317593\pi$
$5$	27.1098	1.08439	0.542197	+	0.840252i	$0.317593\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	23.9137	0.373652
$9$	81.0000	1.00000
$10$	162.628	1.62628
$11$	−106.183	−0.877542	−0.438771	−	0.898599i	$-0.644586\pi$
$11$	−106.183	−0.877542	−0.438771	+	0.898599i	$0.644586\pi$
$12$	0	0
$13$	81.6737	0.483276	0.241638	−	0.970366i	$-0.422315\pi$
$13$	81.6737	0.483276	0.241638	+	0.970366i	$0.422315\pi$
$14$	0	0
$15$	0	0
$16$	−176.327	−0.688776
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	485.908	1.49972
$19$	555.096	1.53766	0.768831	−	0.639452i	$-0.220839\pi$
$19$	555.096	1.53766	0.768831	+	0.639452i	$0.220839\pi$
$20$	541.827	1.35457
$21$	0	0
$22$	−636.975	−1.31606
$23$	−1029.83	−1.94674	−0.973370	−	0.229241i	$-0.926376\pi$
$23$	−1029.83	−1.94674	−0.973370	+	0.229241i	$0.926376\pi$
$24$	0	0
$25$	109.943	0.175909
$26$	489.950	0.724777
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−1921.62	−1.99961	−0.999804	−	0.0198132i	$-0.993693\pi$
$31$	−1921.62	−1.99961	−0.999804	+	0.0198132i	$0.993693\pi$
$32$	−1440.38	−1.40662
$33$	0	0
$34$	0	0
$35$	0	0
$36$	1618.90	1.24915
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	3329.94	2.30606
$39$	0	0
$40$	648.298	0.405186
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−2122.21	−1.09618
$45$	2195.90	1.08439
$46$	−6177.78	−2.91956
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	2401.00	1.00000
$50$	659.535	0.263814
$51$	0	0
$52$	1632.36	0.603684
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	−2878.59	−0.951601
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	−11527.6	−2.99884
$63$	0	0
$64$	−5819.42	−1.42076
$65$	2214.16	0.524062
$66$	0	0
$67$	4618.08	1.02876	0.514378	−	0.857564i	$-0.328023\pi$
$67$	4618.08	1.02876	0.514378	+	0.857564i	$0.328023\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	1937.01	0.373652
$73$	−8527.61	−1.60023	−0.800114	−	0.599848i	$-0.795228\pi$
$73$	−8527.61	−1.60023	−0.800114	+	0.599848i	$0.795228\pi$
$74$	0	0
$75$	0	0
$76$	11094.4	1.92077
$77$	0	0
$78$	0	0
$79$	6241.00	1.00000
$79$	6241.00	1.00000
$80$	−4780.19	−0.746905
$81$	6561.00	1.00000
$82$	0	0
$83$	8722.00	1.26608	0.633038	−	0.774121i	$-0.281808\pi$
$83$	8722.00	1.26608	0.633038	+	0.774121i	$0.281808\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	−2539.22	−0.327896
$89$	10180.3	1.28523	0.642615	−	0.766189i	$-0.277850\pi$
$89$	10180.3	1.28523	0.642615	+	0.766189i	$0.277850\pi$
$90$	13172.9	1.62628
$91$	0	0
$92$	−20582.5	−2.43177
$93$	0	0
$94$	0	0
$95$	15048.6	1.66743
$96$	0	0
$97$	805.607	0.0856209	0.0428104	−	0.999083i	$-0.486369\pi$
$97$	805.607	0.0856209	0.0428104	+	0.999083i	$0.486369\pi$
$98$	14403.3	1.49972
$99$	−8600.79	−0.877542
$100$	2197.37	0.219737
$101$	17402.5	1.70596	0.852980	−	0.521943i	$-0.174793\pi$
$101$	17402.5	1.70596	0.852980	+	0.521943i	$0.174793\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	1953.12	0.180577
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	−17268.3	−1.42713
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	−27918.4	−2.11103
$116$	0	0
$117$	6615.57	0.483276
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3366.26	−0.229920
$122$	0	0
$123$	0	0
$124$	−38406.3	−2.49781
$125$	−13963.1	−0.893639
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	−11863.8	−0.724110
$129$	0	0
$130$	13282.5	0.785944
$131$	−25687.2	−1.49683	−0.748417	−	0.663229i	$-0.769186\pi$
$131$	−25687.2	−1.49683	−0.748417	+	0.663229i	$0.769186\pi$
$132$	0	0
$133$	0	0
$134$	27703.3	1.54284
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8672.33	−0.424095
$144$	−14282.5	−0.688776
$145$	0	0
$146$	−51156.0	−2.39989
$147$	0	0
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	43289.2	1.89857	0.949283	−	0.314424i	$-0.101811\pi$
$151$	43289.2	1.89857	0.949283	+	0.314424i	$0.101811\pi$
$152$	13274.4	0.574551
$153$	0	0
$154$	0	0
$155$	−52094.9	−2.16836
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	37438.9	1.49972
$159$	0	0
$160$	−39048.5	−1.52533
$161$	0	0
$162$	39358.6	1.49972
$163$	51187.0	1.92657	0.963285	−	0.268482i	$-0.0865219\pi$
$163$	51187.0	1.92657	0.963285	+	0.268482i	$0.0865219\pi$
$164$	0	0
$165$	0	0
$166$	52322.1	1.89876
$167$	−46220.0	−1.65729	−0.828643	−	0.559777i	$-0.810887\pi$
$167$	−46220.0	−1.65729	−0.828643	+	0.559777i	$0.810887\pi$
$168$	0	0
$169$	−21890.4	−0.766444
$170$	0	0
$171$	44962.8	1.53766
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	18722.8	0.604430
$177$	0	0
$178$	61070.3	1.92748
$179$	−62318.0	−1.94495	−0.972473	−	0.233017i	$-0.925140\pi$
$179$	−62318.0	−1.94495	−0.972473	+	0.233017i	$0.925140\pi$
$180$	43888.0	1.35457
$181$	−6226.32	−0.190053	−0.0950264	−	0.995475i	$-0.530294\pi$
$181$	−6226.32	−0.190053	−0.0950264	+	0.995475i	$0.530294\pi$
$182$	0	0
$183$	0	0
$184$	−24627.0	−0.727403
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	90274.3	2.50067
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	4832.73	0.128407
$195$	0	0
$196$	47987.3	1.24915
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−51595.0	−1.31606
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	2629.16	0.0657289
$201$	0	0
$202$	104395.	2.55846
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−83415.8	−1.94674
$208$	−14401.3	−0.332869
$209$	−58941.5	−1.34936
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	−57532.6	−1.18869
$221$	0	0
$222$	0	0
$223$	−82558.0	−1.66016	−0.830079	−	0.557646i	$-0.811705\pi$
$223$	−82558.0	−1.66016	−0.830079	+	0.557646i	$0.811705\pi$
$224$	0	0
$225$	8905.41	0.175909
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	−167479.	−3.16595
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	39685.9	0.724777
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−21666.5	−0.379309	−0.189655	−	0.981851i	$-0.560737\pi$
$239$	−21666.5	−0.379309	−0.189655	+	0.981851i	$0.560737\pi$
$240$	0	0
$241$	115667.	1.99147	0.995736	−	0.0922463i	$-0.0294047\pi$
$241$	115667.	1.99147	0.995736	+	0.0922463i	$0.0294047\pi$
$242$	−20193.7	−0.344815
$243$	0	0
$244$	0	0
$245$	65090.7	1.08439
$246$	0	0
$247$	45336.7	0.743116
$248$	−45953.2	−0.747158
$249$	0	0
$250$	−83762.8	−1.34020
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	109349.	1.70835
$254$	0	0
$255$	0	0
$256$	21941.3	0.334797
$257$	131714.	1.99418	0.997090	−	0.0762310i	$-0.0242886\pi$
$257$	131714.	1.99418	0.997090	+	0.0762310i	$0.0242886\pi$
$258$	0	0
$259$	0	0
$260$	44253.1	0.654631
$261$	0	0
$262$	−154094.	−2.24483
$263$	9375.71	0.135548	0.0677739	−	0.997701i	$-0.478410\pi$
$263$	9375.71	0.135548	0.0677739	+	0.997701i	$0.478410\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	92298.8	1.28507
$269$	−58839.5	−0.813138	−0.406569	−	0.913620i	$-0.633275\pi$
$269$	−58839.5	−0.813138	−0.406569	+	0.913620i	$0.633275\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−11674.1	−0.154368
$276$	0	0
$277$	150323.	1.95914	0.979570	−	0.201102i	$-0.0644523\pi$
$277$	150323.	1.95914	0.979570	+	0.201102i	$0.0644523\pi$
$278$	0	0
$279$	−155651.	−1.99961
$280$	0	0
$281$	−155714.	−1.97204	−0.986020	−	0.166627i	$-0.946712\pi$
$281$	−155714.	−1.97204	−0.986020	+	0.166627i	$0.946712\pi$
$282$	0	0
$283$	118451.	1.47899	0.739496	−	0.673161i	$-0.235064\pi$
$283$	118451.	1.47899	0.739496	+	0.673161i	$0.235064\pi$
$284$	0	0
$285$	0	0
$286$	−52024.1	−0.636023
$287$	0	0
$288$	−116671.	−1.40662
$289$	83521.0	1.00000
$290$	0	0
$291$	0	0
$292$	−170436.	−1.99892
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−84109.6	−0.940813
$300$	0	0
$301$	0	0
$302$	259686.	2.84731
$303$	0	0
$304$	−97878.2	−1.05910
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	−312510.	−3.25193
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−171164.	−1.74713	−0.873564	−	0.486709i	$-0.838197\pi$
$313$	−171164.	−1.74713	−0.873564	+	0.486709i	$0.838197\pi$
$314$	0	0
$315$	0	0
$316$	124735.	1.24915
$317$	195922.	1.94969	0.974843	−	0.222893i	$-0.0715499\pi$
$317$	195922.	1.94969	0.974843	+	0.222893i	$0.0715499\pi$
$318$	0	0
$319$	0	0
$320$	−157763.	−1.54066
$321$	0	0
$322$	0	0
$323$	0	0
$324$	131131.	1.24915
$325$	8979.48	0.0850129
$326$	307064.	2.88931
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	174321.	1.58152
$333$	0	0
$334$	−277268.	−2.48546
$335$	125196.	1.11558
$336$	0	0
$337$	159260.	1.40232	0.701160	−	0.713004i	$-0.252666\pi$
$337$	159260.	1.40232	0.701160	+	0.713004i	$0.252666\pi$
$338$	−131318.	−1.14945
$339$	0	0
$340$	0	0
$341$	204043.	1.75474
$342$	269726.	2.30606
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−193882.	−1.61020	−0.805099	−	0.593140i	$-0.797888\pi$
$347$	−193882.	−1.61020	−0.805099	+	0.593140i	$0.797888\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	152943.	1.23437
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	203467.	1.60544
$357$	0	0
$358$	−373837.	−2.91687
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	52512.1	0.405186
$361$	177810.	1.36440
$362$	−37350.8	−0.285025
$363$	0	0
$364$	0	0
$365$	−231182.	−1.73528
$366$	0	0
$367$	−269336.	−1.99969	−0.999844	−	0.0176864i	$-0.994370\pi$
$367$	−269336.	−1.99969	−0.999844	+	0.0176864i	$0.994370\pi$
$368$	181586.	1.34087
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	300766.	2.08287
$381$	0	0
$382$	0	0
$383$	−75744.9	−0.516364	−0.258182	−	0.966096i	$-0.583123\pi$
$383$	−75744.9	−0.516364	−0.258182	+	0.966096i	$0.583123\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	16101.2	0.106953
$389$	89524.8	0.591622	0.295811	−	0.955246i	$-0.404410\pi$
$389$	89524.8	0.591622	0.295811	+	0.955246i	$0.404410\pi$
$390$	0	0
$391$	0	0
$392$	57416.9	0.373652
$393$	0	0
$394$	0	0
$395$	169193.	1.08439
$396$	−171899.	−1.09618
$397$	−94318.0	−0.598430	−0.299215	−	0.954186i	$-0.596725\pi$
$397$	−94318.0	−0.598430	−0.299215	+	0.954186i	$0.596725\pi$
$398$	0	0
$399$	0	0
$400$	−19386.0	−0.121162
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−156946.	−0.966363
$404$	347813.	2.13100
$405$	177868.	1.08439
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	−500400.	−2.91956
$415$	236452.	1.37293
$416$	−117641.	−0.679787
$417$	0	0
$418$	−353582.	−2.02366
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−293942.	−1.65843	−0.829217	−	0.558927i	$-0.811213\pi$
$421$	−293942.	−1.65843	−0.829217	+	0.558927i	$0.811213\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	44874.2	0.241570	0.120785	−	0.992679i	$-0.461459\pi$
$431$	44874.2	0.241570	0.120785	+	0.992679i	$0.461459\pi$
$432$	0	0
$433$	−314513.	−1.67750	−0.838750	−	0.544517i	$-0.816713\pi$
$433$	−314513.	−1.67750	−0.838750	+	0.544517i	$0.816713\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−571652.	−2.99343
$438$	0	0
$439$	374913.	1.94537	0.972684	−	0.232132i	$-0.0745702\pi$
$439$	374913.	1.94537	0.972684	+	0.232132i	$0.0745702\pi$
$440$	−68837.9	−0.355568
$441$	194481.	1.00000
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	275986.	1.39369
$446$	−495254.	−2.48977
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	53422.4	0.263814
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−264862.	−1.26820	−0.634099	−	0.773252i	$-0.718629\pi$
$457$	−264862.	−1.26820	−0.634099	+	0.773252i	$0.718629\pi$
$458$	0	0
$459$	0	0
$460$	−557988.	−2.63699
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−356773.	−1.63590	−0.817952	−	0.575286i	$-0.804891\pi$
$467$	−356773.	−1.63590	−0.817952	+	0.575286i	$0.804891\pi$
$468$	132221.	0.603684
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	61029.1	0.270489
$476$	0	0
$477$	0	0
$478$	−129974.	−0.568856
$479$	−46718.0	−0.203617	−0.101808	−	0.994804i	$-0.532463\pi$
$479$	−46718.0	−0.203617	−0.101808	+	0.994804i	$0.532463\pi$
$480$	0	0
$481$	0	0
$482$	693869.	2.98664
$483$	0	0
$484$	−67279.3	−0.287204
$485$	21839.9	0.0928467
$486$	0	0
$487$	238599.	1.00603	0.503015	−	0.864278i	$-0.332224\pi$
$487$	238599.	1.00603	0.503015	+	0.864278i	$0.332224\pi$
$488$	0	0
$489$	0	0
$490$	390470.	1.62628
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	271969.	1.11446
$495$	−233166.	−0.951601
$496$	338833.	1.37728
$497$	0	0
$498$	0	0
$499$	497870.	1.99947	0.999736	−	0.0229820i	$-0.00731604\pi$
$499$	497870.	1.99947	0.999736	+	0.0229820i	$0.00731604\pi$
$500$	−279072.	−1.11629
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	471779.	1.84993
$506$	655973.	2.56203
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	321444.	1.22621
$513$	0	0
$514$	790132.	2.99070
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	52948.9	0.195817
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	−367229.	−1.34256	−0.671280	−	0.741204i	$-0.734255\pi$
$523$	−367229.	−1.34256	−0.671280	+	0.741204i	$0.734255\pi$
$524$	−513393.	−1.86977
$525$	0	0
$526$	56243.6	0.203283
$527$	0	0
$528$	0	0
$529$	780699.	2.78979
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	110436.	0.384397
$537$	0	0
$538$	−352970.	−1.21948
$539$	−254944.	−0.877542
$540$	0	0
$541$	−552238.	−1.88683	−0.943413	−	0.331621i	$-0.892405\pi$
$541$	−552238.	−1.88683	−0.943413	+	0.331621i	$0.892405\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−516941.	−1.72769	−0.863846	−	0.503756i	$-0.831951\pi$
$547$	−516941.	−1.72769	−0.863846	+	0.503756i	$0.831951\pi$
$548$	0	0
$549$	0	0
$550$	−70031.2	−0.231508
$551$	0	0
$552$	0	0
$553$	0	0
$554$	901767.	2.93815
$555$	0	0
$556$	0	0
$557$	−234888.	−0.757095	−0.378547	−	0.925582i	$-0.623576\pi$
$557$	−234888.	−0.757095	−0.378547	+	0.925582i	$0.623576\pi$
$558$	−933732.	−2.99884
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−934109.	−2.95750
$563$	22162.0	0.0699185	0.0349593	−	0.999389i	$-0.488870\pi$
$563$	22162.0	0.0699185	0.0349593	+	0.999389i	$0.488870\pi$
$564$	0	0
$565$	0	0
$566$	710572.	2.21807
$567$	0	0
$568$	0	0
$569$	309331.	0.955431	0.477715	−	0.878515i	$-0.341465\pi$
$569$	309331.	0.955431	0.477715	+	0.878515i	$0.341465\pi$
$570$	0	0
$571$	639674.	1.96194	0.980972	−	0.194150i	$-0.0621948\pi$
$571$	639674.	1.96194	0.980972	+	0.194150i	$0.0621948\pi$
$572$	−173328.	−0.529758
$573$	0	0
$574$	0	0
$575$	−113222.	−0.342450
$576$	−471373.	−1.42076
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	501031.	1.49972
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−203927.	−0.597928
$585$	179347.	0.524062
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	−1.06668e6	−3.07472
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−167588.	−0.476579	−0.238289	−	0.971194i	$-0.576587\pi$
$593$	−167588.	−0.476579	−0.238289	+	0.971194i	$0.576587\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	−504562.	−1.41095
$599$	−56978.3	−0.158802	−0.0794009	−	0.996843i	$-0.525301\pi$
$599$	−56978.3	−0.158802	−0.0794009	+	0.996843i	$0.525301\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	374065.	1.02876
$604$	865194.	2.37159
$605$	−91258.7	−0.249324
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−799549.	−2.16291
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	511467.	1.34353	0.671764	−	0.740765i	$-0.265537\pi$
$617$	511467.	1.34353	0.671764	+	0.740765i	$0.265537\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	−1.04119e6	−2.70861
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−447252.	−1.14497
$626$	−1.02679e6	−2.62020
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	149246.	0.373652
$633$	0	0
$634$	1.17531e6	2.92398
$635$	0	0
$636$	0	0
$637$	196099.	0.483276
$638$	0	0
$639$	0	0
$640$	−321626.	−0.785220
$641$	765770.	1.86373	0.931864	−	0.362808i	$-0.118182\pi$
$641$	765770.	1.86373	0.931864	+	0.362808i	$0.118182\pi$
$642$	0	0
$643$	771498.	1.86601	0.933003	−	0.359869i	$-0.117179\pi$
$643$	771498.	1.86601	0.933003	+	0.359869i	$0.117179\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	156898.	0.373652
$649$	0	0
$650$	53866.7	0.127495
$651$	0	0
$652$	1.02304e6	2.40657
$653$	−830315.	−1.94723	−0.973614	−	0.228202i	$-0.926715\pi$
$653$	−830315.	−1.94723	−0.973614	+	0.228202i	$0.926715\pi$
$654$	0	0
$655$	−696375.	−1.62316
$656$	0	0
$657$	−690737.	−1.60023
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	208576.	0.473072
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−923771.	−2.07020
$669$	0	0
$670$	751031.	1.67305
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	955380.	2.10308
$675$	0	0
$676$	−437510.	−0.957402
$677$	−908558.	−1.98233	−0.991164	−	0.132645i	$-0.957653\pi$
$677$	−908558.	−1.98233	−0.991164	+	0.132645i	$0.957653\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	1.22403e6	2.63161
$683$	−904304.	−1.93853	−0.969266	−	0.246015i	$-0.920879\pi$
$683$	−904304.	−1.93853	−0.969266	+	0.246015i	$0.920879\pi$
$684$	898643.	1.92077
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	−1.16307e6	−2.41484
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	617921.	1.24677
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	505521.	1.00000
$712$	243449.	0.480229
$713$	1.97894e6	3.89271
$714$	0	0
$715$	−235105.	−0.459886
$716$	−1.24551e6	−2.42953
$717$	0	0
$718$	0	0
$719$	−574538.	−1.11138	−0.555688	−	0.831391i	$-0.687545\pi$
$719$	−574538.	−1.11138	−0.555688	+	0.831391i	$0.687545\pi$
$720$	−387195.	−0.746905
$721$	0	0
$722$	1.06666e6	2.04622
$723$	0	0
$724$	−124442.	−0.237404
$725$	0	0
$726$	0	0
$727$	875042.	1.65562	0.827809	−	0.561010i	$-0.189587\pi$
$727$	875042.	1.65562	0.827809	+	0.561010i	$0.189587\pi$
$728$	0	0
$729$	531441.	1.00000
$730$	−1.38683e6	−2.60242
$731$	0	0
$732$	0	0
$733$	62102.4	0.115585	0.0577924	−	0.998329i	$-0.481594\pi$
$733$	62102.4	0.115585	0.0577924	+	0.998329i	$0.481594\pi$
$734$	−1.61571e6	−2.99896
$735$	0	0
$736$	1.48334e6	2.73832
$737$	−490360.	−0.902776
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−179882.	−0.325844	−0.162922	−	0.986639i	$-0.552092\pi$
$743$	−179882.	−0.325844	−0.162922	+	0.986639i	$0.552092\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	706482.	1.26608
$748$	0	0
$749$	0	0
$750$	0	0
$751$	426930.	0.756968	0.378484	−	0.925608i	$-0.376446\pi$
$751$	426930.	0.756968	0.378484	+	0.925608i	$0.376446\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.17356e6	2.05879
$756$	0	0
$757$	−1.08360e6	−1.89093	−0.945467	−	0.325717i	$-0.894394\pi$
$757$	−1.08360e6	−1.89093	−0.945467	+	0.325717i	$0.894394\pi$
$758$	0	0
$759$	0	0
$760$	359867.	0.623039
$761$	−1.15742e6	−1.99858	−0.999292	−	0.0376265i	$-0.988020\pi$
$761$	−1.15742e6	−1.99858	−0.999292	+	0.0376265i	$0.988020\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−454383.	−0.774399
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.14471e6	−1.91574	−0.957872	−	0.287196i	$-0.907277\pi$
$773$	−1.14471e6	−1.91574	−0.957872	+	0.287196i	$0.907277\pi$
$774$	0	0
$775$	−211270.	−0.351750
$776$	19265.1	0.0319924
$777$	0	0
$778$	537047.	0.887265
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−423361.	−0.688776
$785$	0	0
$786$	0	0
$787$	1.09517e6	1.76820	0.884102	−	0.467294i	$-0.154771\pi$
$787$	1.09517e6	1.76820	0.884102	+	0.467294i	$0.154771\pi$
$788$	0	0
$789$	0	0
$790$	1.01496e6	1.62628
$791$	0	0
$792$	−205677.	−0.327896
$793$	0	0
$794$	−565801.	−0.897476
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−158360.	−0.247438
$801$	824605.	1.28523
$802$	0	0
$803$	905484.	1.40427
$804$	0	0
$805$	0	0
$806$	−941498.	−1.44927
$807$	0	0
$808$	416159.	0.637436
$809$	−572680.	−0.875014	−0.437507	−	0.899215i	$-0.644138\pi$
$809$	−572680.	−0.875014	−0.437507	+	0.899215i	$0.644138\pi$
$810$	1.06700e6	1.62628
$811$	177842.	0.270391	0.135196	−	0.990819i	$-0.456834\pi$
$811$	177842.	0.270391	0.135196	+	0.990819i	$0.456834\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.38767e6	2.08916
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.34393e6	1.99384	0.996918	−	0.0784510i	$-0.0249974\pi$
$821$	1.34393e6	1.99384	0.996918	+	0.0784510i	$0.0249974\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	−1.66718e6	−2.43177
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	1.41844e6	2.05900
$831$	0	0
$832$	−475293.	−0.686618
$833$	0	0
$834$	0	0
$835$	−1.25302e6	−1.79715
$836$	−1.17803e6	−1.68555
$837$	0	0
$838$	0	0
$839$	1.39619e6	1.98345	0.991726	−	0.128375i	$-0.0409761\pi$
$839$	1.39619e6	1.98345	0.991726	+	0.128375i	$0.0409761\pi$
$840$	0	0
$841$	707281.	1.00000
$842$	−1.76332e6	−2.48718
$843$	0	0
$844$	0	0
$845$	−593445.	−0.831127
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	1.21893e6	1.66743
$856$	0	0
$857$	−738785.	−1.00590	−0.502952	−	0.864314i	$-0.667753\pi$
$857$	−738785.	−1.00590	−0.502952	+	0.864314i	$0.667753\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	269195.	0.362286
$863$	498562.	0.669418	0.334709	−	0.942321i	$-0.391362\pi$
$863$	498562.	0.669418	0.334709	+	0.942321i	$0.391362\pi$
$864$	0	0
$865$	0	0
$866$	−1.88672e6	−2.51577
$867$	0	0
$868$	0	0
$869$	−662686.	−0.877542
$870$	0	0
$871$	377176.	0.497173
$872$	0	0
$873$	65254.1	0.0856209
$874$	−3.42926e6	−4.48929
$875$	0	0
$876$	0	0
$877$	87136.0	0.113292	0.0566459	−	0.998394i	$-0.481959\pi$
$877$	87136.0	0.113292	0.0566459	+	0.998394i	$0.481959\pi$
$878$	2.24905e6	2.91750
$879$	0	0
$880$	507573.	0.655440
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	1.16667e6	1.49972
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.40837e6	1.79007	0.895033	−	0.445999i	$-0.147151\pi$
$887$	1.40837e6	1.79007	0.895033	+	0.445999i	$0.147151\pi$
$888$	0	0
$889$	0	0
$890$	1.65560e6	2.09015
$891$	−696664.	−0.877542
$892$	−1.65004e6	−2.07378
$893$	0	0
$894$	0	0
$895$	−1.68943e6	−2.10909
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	177987.	0.219737
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−168794.	−0.206092
$906$	0	0
$907$	−584398.	−0.710386	−0.355193	−	0.934793i	$-0.615585\pi$
$907$	−584398.	−0.710386	−0.355193	+	0.934793i	$0.615585\pi$
$908$	0	0
$909$	1.40960e6	1.70596
$910$	0	0
$911$	−1.44110e6	−1.73643	−0.868214	−	0.496189i	$-0.834732\pi$
$911$	−1.44110e6	−1.73643	−0.868214	+	0.496189i	$0.834732\pi$
$912$	0	0
$913$	−926125.	−1.11104
$914$	−1.58887e6	−1.90194
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1.14108e6	−1.35109	−0.675545	−	0.737318i	$-0.736092\pi$
$919$	−1.14108e6	−1.35109	−0.675545	+	0.737318i	$0.736092\pi$
$920$	−667633.	−0.788792
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	1.33279e6	1.53766
$932$	0	0
$933$	0	0
$934$	−2.14023e6	−2.45339
$935$	0	0
$936$	158203.	0.180577
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.38904e6	−1.56868	−0.784341	−	0.620330i	$-0.786999\pi$
$941$	−1.38904e6	−1.56868	−0.784341	+	0.620330i	$0.786999\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	−696482.	−0.773352
$950$	366105.	0.405657
$951$	0	0
$952$	0	0
$953$	−1.30543e6	−1.43737	−0.718685	−	0.695336i	$-0.755256\pi$
$953$	−1.30543e6	−1.43737	−0.718685	+	0.695336i	$0.755256\pi$
$954$	0	0
$955$	0	0
$956$	−433035.	−0.473813
$957$	0	0
$958$	−280255.	−0.305367
$959$	0	0
$960$	0	0
$961$	2.76911e6	2.99843
$962$	0	0
$963$	0	0
$964$	2.31176e6	2.48765
$965$	0	0
$966$	0	0
$967$	−1.04208e6	−1.11442	−0.557208	−	0.830373i	$-0.688127\pi$
$967$	−1.04208e6	−1.11442	−0.557208	+	0.830373i	$0.688127\pi$
$968$	−80499.8	−0.0859101
$969$	0	0
$970$	131014.	0.139244
$971$	1.00297e6	1.06377	0.531885	−	0.846816i	$-0.321484\pi$
$971$	1.00297e6	1.06377	0.531885	+	0.846816i	$0.321484\pi$
$972$	0	0
$973$	0	0
$974$	1.43132e6	1.50876
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	−1.08097e6	−1.12784
$980$	1.30093e6	1.35457
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	906117.	0.928262
$989$	0	0
$990$	−1.39873e6	−1.42713
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	2.76787e6	2.81269
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.98693e6	−1.99891	−0.999454	−	0.0330371i	$-0.989482\pi$
$997$	−1.98693e6	−1.99891	−0.999454	+	0.0330371i	$0.989482\pi$
$998$	2.98666e6	2.99864
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	79.5.b.a.78.4	✓	5
79.78	odd	2	CM	79.5.b.a.78.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
79.5.b.a.78.4	✓	5	1.1	even	1	trivial
79.5.b.a.78.4	✓	5	79.78	odd	2	CM

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:	\( N \)	\(=\)	\( 79 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	79.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.99886 q^{2} +19.9864 q^{4} +27.1098 q^{5} +23.9137 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+5.99886 q^{2} +19.9864 q^{4} +27.1098 q^{5} +23.9137 q^{8} +81.0000 q^{9} +162.628 q^{10} -106.183 q^{11} +81.6737 q^{13} -176.327 q^{16} +485.908 q^{18} +555.096 q^{19} +541.827 q^{20} -636.975 q^{22} -1029.83 q^{23} +109.943 q^{25} +489.950 q^{26} -1921.62 q^{31} -1440.38 q^{32} +1618.90 q^{36} +3329.94 q^{38} +648.298 q^{40} -2122.21 q^{44} +2195.90 q^{45} -6177.78 q^{46} +2401.00 q^{49} +659.535 q^{50} +1632.36 q^{52} -2878.59 q^{55} -11527.6 q^{62} -5819.42 q^{64} +2214.16 q^{65} +4618.08 q^{67} +1937.01 q^{72} -8527.61 q^{73} +11094.4 q^{76} +6241.00 q^{79} -4780.19 q^{80} +6561.00 q^{81} +8722.00 q^{83} -2539.22 q^{88} +10180.3 q^{89} +13172.9 q^{90} -20582.5 q^{92} +15048.6 q^{95} +805.607 q^{97} +14403.3 q^{98} -8600.79 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 80 q^{4} + 405 q^{9}+O(q^{10}) \) 5 * q + 80 * q^4 + 405 * q^9	\( 5 q + 80 q^{4} + 405 q^{9} + 1280 q^{16} + 3605 q^{20} + 1285 q^{22} + 3125 q^{25} - 3115 q^{26} - 9115 q^{32} + 6480 q^{36} - 15995 q^{40} + 12005 q^{49} - 22795 q^{50} - 28315 q^{62} + 20480 q^{64} - 31115 q^{76} + 31205 q^{79} + 57680 q^{80} + 32805 q^{81} + 43610 q^{83} + 20560 q^{88} - 29515 q^{92} - 10870 q^{95}+O(q^{100}) \) 5 * q + 80 * q^4 + 405 * q^9 + 1280 * q^16 + 3605 * q^20 + 1285 * q^22 + 3125 * q^25 - 3115 * q^26 - 9115 * q^32 + 6480 * q^36 - 15995 * q^40 + 12005 * q^49 - 22795 * q^50 - 28315 * q^62 + 20480 * q^64 - 31115 * q^76 + 31205 * q^79 + 57680 * q^80 + 32805 * q^81 + 43610 * q^83 + 20560 * q^88 - 29515 * q^92 - 10870 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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