LMFDB - Embedded newform 119.5.d.a.118.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [119,5,Mod(118,119)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("119.118"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(119, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 119 = 7 \cdot 17 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	119.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [5,0,0,80,0,-235] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$12.3010256070$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.44253125.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 10x^{3} + 20x - 3 $ x^5 - 10x^3 + 20x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			118.1
Root			$-1.78288$ of defining polynomial
Character	$\chi$	$=$	119.118

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-7.32538 q^{2} +16.2398 q^{3} +37.6612 q^{4} -46.0925 q^{5} -118.962 q^{6} +49.0000 q^{7} -158.676 q^{8} +182.730 q^{9} +337.645 q^{10} +611.609 q^{12} -358.944 q^{14} -748.530 q^{15} +559.786 q^{16} +289.000 q^{17} -1338.57 q^{18} -1735.90 q^{20} +795.748 q^{21} -2576.87 q^{24} +1499.51 q^{25} +1652.07 q^{27} +1845.40 q^{28} +5483.27 q^{30} +81.9749 q^{31} -1561.82 q^{32} -2117.03 q^{34} -2258.53 q^{35} +6881.82 q^{36} +7313.79 q^{40} +3231.65 q^{41} -5829.16 q^{42} +1871.71 q^{43} -8422.47 q^{45} +9090.79 q^{48} +2401.00 q^{49} -10984.5 q^{50} +4693.29 q^{51} +4265.40 q^{53} -12102.0 q^{54} -7775.15 q^{56} -28190.5 q^{60} -3661.12 q^{61} -600.498 q^{62} +8953.76 q^{63} +2484.37 q^{64} +5226.55 q^{67} +10884.1 q^{68} +16544.6 q^{70} -28994.9 q^{72} -10265.6 q^{73} +24351.8 q^{75} -25801.9 q^{80} +12028.1 q^{81} -23673.1 q^{82} +29968.8 q^{84} -13320.7 q^{85} -13711.0 q^{86} +61697.8 q^{90} +1331.25 q^{93} -25363.6 q^{96} -11029.2 q^{97} -17588.2 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 80 q^{4} - 235 q^{6} + 245 q^{7} + 405 q^{9} + 1280 q^{16} + 1445 q^{17} - 1355 q^{20} - 3760 q^{24} + 3125 q^{25} + 3920 q^{28} + 8405 q^{30} + 4885 q^{32} + 4565 q^{36} + 10645 q^{40} - 11515 q^{42}+ \cdots - 60160 q^{96}+O(q^{100}) $ 5 * q + 80 * q^4 - 235 * q^6 + 245 * q^7 + 405 * q^9 + 1280 * q^16 + 1445 * q^17 - 1355 * q^20 - 3760 * q^24 + 3125 * q^25 + 3920 * q^28 + 8405 * q^30 + 4885 * q^32 + 4565 * q^36 + 10645 * q^40 - 11515 * q^42 - 17830 * q^45 + 8165 * q^48 + 12005 * q^49 - 23195 * q^50 - 19035 * q^54 - 35995 * q^60 + 19845 * q^63 + 20480 * q^64 + 23120 * q^68 - 48715 * q^72 + 18170 * q^75 - 21680 * q^80 + 32805 * q^81 - 66635 * q^82 - 59915 * q^86 + 51845 * q^90 - 60160 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/119\mathbb{Z}\right)^\times$.

$n$	$52$	$71$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−7.32538	−1.83134	−0.915672	−	0.401925i	$-0.868341\pi$
$2$	−7.32538	−1.83134	−0.915672	+	0.401925i	$0.868341\pi$
$3$	16.2398	1.80442	0.902209	−	0.431299i	$-0.141945\pi$
$3$	16.2398	1.80442	0.902209	+	0.431299i	$0.141945\pi$
$4$	37.6612	2.35382
$5$	−46.0925	−1.84370	−0.921849	−	0.387549i	$-0.873322\pi$
$5$	−46.0925	−1.84370	−0.921849	+	0.387549i	$0.873322\pi$
$6$	−118.962	−3.30451
$7$	49.0000	1.00000
$7$	49.0000	1.00000
$8$	−158.676	−2.47932
$9$	182.730	2.25592
$10$	337.645	3.37645
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	611.609	4.24728
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−358.944	−1.83134
$15$	−748.530	−3.32680
$16$	559.786	2.18666
$17$	289.000	1.00000
$17$	289.000	1.00000
$18$	−1338.57	−4.13137
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−1735.90	−4.33974
$21$	795.748	1.80442
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−2576.87	−4.47373
$25$	1499.51	2.39922
$26$	0	0
$27$	1652.07	2.26621
$28$	1845.40	2.35382
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	5483.27	6.09252
$31$	81.9749	0.0853017	0.0426508	−	0.999090i	$-0.486420\pi$
$31$	81.9749	0.0853017	0.0426508	+	0.999090i	$0.486420\pi$
$32$	−1561.82	−1.52522
$33$	0	0
$34$	−2117.03	−1.83134
$35$	−2258.53	−1.84370
$36$	6881.82	5.31005
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	7313.79	4.57112
$41$	3231.65	1.92246	0.961229	−	0.275751i	$-0.0889267\pi$
$41$	3231.65	1.92246	0.961229	+	0.275751i	$0.0889267\pi$
$42$	−5829.16	−3.30451
$43$	1871.71	1.01228	0.506141	−	0.862451i	$-0.331072\pi$
$43$	1871.71	1.01228	0.506141	+	0.862451i	$0.331072\pi$
$44$	0	0
$45$	−8422.47	−4.15924
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	9090.79	3.94566
$49$	2401.00	1.00000
$50$	−10984.5	−4.39380
$51$	4693.29	1.80442
$52$	0	0
$53$	4265.40	1.51848	0.759239	−	0.650812i	$-0.225571\pi$
$53$	4265.40	1.51848	0.759239	+	0.650812i	$0.225571\pi$
$54$	−12102.0	−4.15021
$55$	0	0
$56$	−7775.15	−2.47932
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−28190.5	−7.83071
$61$	−3661.12	−0.983908	−0.491954	−	0.870621i	$-0.663717\pi$
$61$	−3661.12	−0.983908	−0.491954	+	0.870621i	$0.663717\pi$
$62$	−600.498	−0.156217
$63$	8953.76	2.25592
$64$	2484.37	0.606535
$65$	0	0
$66$	0	0
$67$	5226.55	1.16430	0.582150	−	0.813081i	$-0.302211\pi$
$67$	5226.55	1.16430	0.582150	+	0.813081i	$0.302211\pi$
$68$	10884.1	2.35382
$69$	0	0
$70$	16544.6	3.37645
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−28994.9	−5.59316
$73$	−10265.6	−1.92636	−0.963182	−	0.268849i	$-0.913357\pi$
$73$	−10265.6	−1.92636	−0.963182	+	0.268849i	$0.913357\pi$
$74$	0	0
$75$	24351.8	4.32920
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−25801.9	−4.03155
$81$	12028.1	1.83327
$82$	−23673.1	−3.52068
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	29968.8	4.24728
$85$	−13320.7	−1.84370
$86$	−13711.0	−1.85384
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	61697.8	7.61701
$91$	0	0
$92$	0	0
$93$	1331.25	0.153920
$94$	0	0
$95$	0	0
$96$	−25363.6	−2.75213
$97$	−11029.2	−1.17219	−0.586097	−	0.810241i	$-0.699336\pi$
$97$	−11029.2	−1.17219	−0.586097	+	0.810241i	$0.699336\pi$
$98$	−17588.2	−1.83134
$99$	0	0
$100$	56473.5	5.64735
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	−34380.1	−3.30451
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	−36678.0	−3.32680
$106$	−31245.7	−2.78086
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	62218.8	5.33426
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	27429.5	2.18666
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	14161.0	1.00000
$120$	118774.	8.24820
$121$	14641.0	1.00000
$122$	26819.1	1.80188
$123$	52481.3	3.46892
$124$	3087.27	0.200785
$125$	−40308.5	−2.57974
$126$	−65589.7	−4.13137
$127$	−19140.9	−1.18674	−0.593370	−	0.804930i	$-0.702203\pi$
$127$	−19140.9	−1.18674	−0.593370	+	0.804930i	$0.702203\pi$
$128$	6790.23	0.414442
$129$	30396.1	1.82658
$130$	0	0
$131$	−34222.0	−1.99417	−0.997086	−	0.0762803i	$-0.975696\pi$
$131$	−34222.0	−1.99417	−0.997086	+	0.0762803i	$0.975696\pi$
$132$	0	0
$133$	0	0
$134$	−38286.4	−2.13224
$135$	−76147.9	−4.17821
$136$	−45857.5	−2.47932
$137$	−19644.2	−1.04663	−0.523314	−	0.852140i	$-0.675304\pi$
$137$	−19644.2	−1.04663	−0.523314	+	0.852140i	$0.675304\pi$
$138$	0	0
$139$	−10020.3	−0.518625	−0.259312	−	0.965794i	$-0.583496\pi$
$139$	−10020.3	−0.518625	−0.259312	+	0.965794i	$0.583496\pi$
$140$	−85058.9	−4.33974
$141$	0	0
$142$	0	0
$143$	0	0
$144$	102290.	4.93295
$145$	0	0
$146$	75199.4	3.52784
$147$	38991.7	1.80442
$148$	0	0
$149$	−28173.7	−1.26903	−0.634514	−	0.772912i	$-0.718800\pi$
$149$	−28173.7	−1.26903	−0.634514	+	0.772912i	$0.718800\pi$
$150$	−178386.	−7.92826
$151$	5452.66	0.239141	0.119571	−	0.992826i	$-0.461848\pi$
$151$	5452.66	0.239141	0.119571	+	0.992826i	$0.461848\pi$
$152$	0	0
$153$	52808.9	2.25592
$154$	0	0
$155$	−3778.43	−0.157271
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	69269.1	2.73997
$160$	71988.3	2.81204
$161$	0	0
$162$	−88110.2	−3.35735
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	121708.	4.52513
$165$	0	0
$166$	0	0
$167$	12375.2	0.443730	0.221865	−	0.975077i	$-0.428786\pi$
$167$	12375.2	0.443730	0.221865	+	0.975077i	$0.428786\pi$
$168$	−126266.	−4.47373
$169$	28561.0	1.00000
$170$	97579.3	3.37645
$171$	0	0
$172$	70490.8	2.38273
$173$	−45846.2	−1.53183	−0.765916	−	0.642941i	$-0.777714\pi$
$173$	−45846.2	−1.53183	−0.765916	+	0.642941i	$0.777714\pi$
$174$	0	0
$175$	73476.2	2.39922
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7648.69	0.238716	0.119358	−	0.992851i	$-0.461916\pi$
$179$	7648.69	0.238716	0.119358	+	0.992851i	$0.461916\pi$
$180$	−317200.	−9.79013
$181$	−3022.00	−0.0922438	−0.0461219	−	0.998936i	$-0.514686\pi$
$181$	−3022.00	−0.0922438	−0.0461219	+	0.998936i	$0.514686\pi$
$182$	0	0
$183$	−59455.8	−1.77538
$184$	0	0
$185$	0	0
$186$	−9751.94	−0.281880
$187$	0	0
$188$	0	0
$189$	80951.3	2.26621
$190$	0	0
$191$	71871.5	1.97011	0.985054	−	0.172245i	$-0.0551020\pi$
$191$	71871.5	1.97011	0.985054	+	0.172245i	$0.0551020\pi$
$192$	40345.5	1.09444
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	80792.9	2.14669
$195$	0	0
$196$	90424.5	2.35382
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	−53585.6	−1.35314	−0.676569	−	0.736379i	$-0.736534\pi$
$199$	−53585.6	−1.35314	−0.676569	+	0.736379i	$0.736534\pi$
$200$	−237938.	−5.94844
$201$	84877.9	2.10089
$202$	0	0
$203$	0	0
$204$	176755.	4.24728
$205$	−148955.	−3.54443
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	268680.	6.09252
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	160640.	3.57423
$213$	0	0
$214$	0	0
$215$	−86271.7	−1.86634
$216$	−262144.	−5.61866
$217$	4016.77	0.0853017
$218$	0	0
$219$	−166711.	−3.47597
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−76529.3	−1.52522
$225$	274006.	5.41246
$226$	0	0
$227$	−87259.8	−1.69341	−0.846706	−	0.532061i	$-0.821418\pi$
$227$	−87259.8	−1.69341	−0.846706	+	0.532061i	$0.821418\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	−103735.	−1.83134
$239$	112548.	1.97034	0.985168	−	0.171595i	$-0.0548921\pi$
$239$	112548.	1.97034	0.985168	+	0.171595i	$0.0548921\pi$
$240$	−419017.	−7.27460
$241$	116125.	1.99937	0.999685	−	0.0250910i	$-0.00798756\pi$
$241$	116125.	1.99937	0.999685	+	0.0250910i	$0.00798756\pi$
$242$	−107251.	−1.83134
$243$	61515.5	1.04177
$244$	−137882.	−2.31595
$245$	−110668.	−1.84370
$246$	−384445.	−6.35278
$247$	0	0
$248$	−13007.5	−0.211490
$249$	0	0
$250$	295275.	4.72440
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	337209.	5.31005
$253$	0	0
$254$	140215.	2.17333
$255$	−216325.	−3.32680
$256$	−89490.9	−1.36552
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	−222663.	−3.34510
$259$	0	0
$260$	0	0
$261$	0	0
$262$	250689.	3.65202
$263$	−135838.	−1.96386	−0.981928	−	0.189253i	$-0.939393\pi$
$263$	−135838.	−1.96386	−0.981928	+	0.189253i	$0.939393\pi$
$264$	0	0
$265$	−196603.	−2.79962
$266$	0	0
$267$	0	0
$268$	196838.	2.74056
$269$	76178.0	1.05275	0.526375	−	0.850253i	$-0.323551\pi$
$269$	76178.0	1.05275	0.526375	+	0.850253i	$0.323551\pi$
$270$	557812.	7.65174
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	161778.	2.18666
$273$	0	0
$274$	143901.	1.91674
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	73402.8	0.949780
$279$	14979.3	0.192434
$280$	358376.	4.57112
$281$	−155683.	−1.97164	−0.985821	−	0.167802i	$-0.946333\pi$
$281$	−155683.	−1.97164	−0.985821	+	0.167802i	$0.946333\pi$
$282$	0	0
$283$	652.685	0.00814950	0.00407475	−	0.999992i	$-0.498703\pi$
$283$	652.685	0.00814950	0.00407475	+	0.999992i	$0.498703\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	158351.	1.92246
$288$	−285392.	−3.44077
$289$	83521.0	1.00000
$290$	0	0
$291$	−179111.	−2.11513
$292$	−386615.	−4.53432
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−285629.	−3.30451
$295$	0	0
$296$	0	0
$297$	0	0
$298$	206383.	2.32403
$299$	0	0
$300$	917116.	10.1902
$301$	91713.7	1.01228
$302$	−39942.8	−0.437950
$303$	0	0
$304$	0	0
$305$	168750.	1.81403
$306$	−386845.	−4.13137
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	27678.4	0.288017
$311$	179914.	1.86013	0.930067	−	0.367390i	$-0.119749\pi$
$311$	179914.	1.86013	0.930067	+	0.367390i	$0.119749\pi$
$312$	0	0
$313$	70467.2	0.719281	0.359640	−	0.933091i	$-0.382899\pi$
$313$	70467.2	0.719281	0.359640	+	0.933091i	$0.382899\pi$
$314$	0	0
$315$	−412701.	−4.15924
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	−507423.	−5.01783
$319$	0	0
$320$	−114511.	−1.11827
$321$	0	0
$322$	0	0
$323$	0	0
$324$	452991.	4.31519
$325$	0	0
$326$	0	0
$327$	0	0
$328$	−512787.	−4.76639
$329$	0	0
$330$	0	0
$331$	111213.	1.01508	0.507538	−	0.861629i	$-0.330556\pi$
$331$	111213.	1.01508	0.507538	+	0.861629i	$0.330556\pi$
$332$	0	0
$333$	0	0
$334$	−90652.9	−0.812622
$335$	−240904.	−2.14662
$336$	445449.	3.94566
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	−209220.	−1.83134
$339$	0	0
$340$	−501674.	−4.33974
$341$	0	0
$342$	0	0
$343$	117649.	1.00000
$344$	−296996.	−2.50977
$345$	0	0
$346$	335841.	2.80531
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	−538241.	−4.39380
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	229971.	1.80442
$358$	−56029.6	−0.437171
$359$	249781.	1.93808	0.969038	−	0.246910i	$-0.0794153\pi$
$359$	249781.	1.93808	0.969038	+	0.246910i	$0.0794153\pi$
$360$	1.33645e6	10.3121
$361$	130321.	1.00000
$362$	22137.3	0.168930
$363$	237766.	1.80442
$364$	0	0
$365$	473167.	3.55163
$366$	435536.	3.25134
$367$	−265290.	−1.96965	−0.984825	−	0.173553i	$-0.944475\pi$
$367$	−265290.	−1.96965	−0.984825	+	0.173553i	$0.944475\pi$
$368$	0	0
$369$	590519.	4.33692
$370$	0	0
$371$	209005.	1.51848
$372$	50136.6	0.362300
$373$	−224281.	−1.61203	−0.806017	−	0.591892i	$-0.798381\pi$
$373$	−224281.	−1.61203	−0.806017	+	0.591892i	$0.798381\pi$
$374$	0	0
$375$	−654601.	−4.65494
$376$	0	0
$377$	0	0
$378$	−592999.	−4.15021
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−310844.	−2.14137
$382$	−526486.	−3.60795
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	110272.	0.747827
$385$	0	0
$386$	0	0
$387$	342017.	2.28363
$388$	−415372.	−2.75914
$389$	−223662.	−1.47807	−0.739033	−	0.673669i	$-0.764717\pi$
$389$	−223662.	−1.47807	−0.739033	+	0.673669i	$0.764717\pi$
$390$	0	0
$391$	0	0
$392$	−380982.	−2.47932
$393$	−555757.	−3.59832
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−288379.	−1.82971	−0.914855	−	0.403783i	$-0.867695\pi$
$397$	−288379.	−1.82971	−0.914855	+	0.403783i	$0.867695\pi$
$398$	392535.	2.47806
$399$	0	0
$400$	839407.	5.24630
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	−621763.	−3.84745
$403$	0	0
$404$	0	0
$405$	−554403.	−3.37999
$406$	0	0
$407$	0	0
$408$	−744715.	−4.47373
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	1.09115e6	6.49108
$411$	−319016.	−1.88855
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−162728.	−0.935815
$418$	0	0
$419$	329460.	1.87661	0.938306	−	0.345807i	$-0.112395\pi$
$419$	329460.	1.87661	0.938306	+	0.345807i	$0.112395\pi$
$420$	−1.38134e6	−7.83071
$421$	−138303.	−0.780311	−0.390155	−	0.920749i	$-0.627579\pi$
$421$	−138303.	−0.780311	−0.390155	+	0.920749i	$0.627579\pi$
$422$	0	0
$423$	0	0
$424$	−676819.	−3.76479
$425$	433360.	2.39922
$426$	0	0
$427$	−179395.	−0.983908
$428$	0	0
$429$	0	0
$430$	631973.	3.41792
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	924805.	4.95544
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	−29424.4	−0.156217
$435$	0	0
$436$	0	0
$437$	0	0
$438$	1.22122e6	6.36569
$439$	−205792.	−1.06782	−0.533912	−	0.845540i	$-0.679279\pi$
$439$	−205792.	−1.06782	−0.533912	+	0.845540i	$0.679279\pi$
$440$	0	0
$441$	438734.	2.25592
$442$	0	0
$443$	−224398.	−1.14344	−0.571718	−	0.820451i	$-0.693723\pi$
$443$	−224398.	−1.14344	−0.571718	+	0.820451i	$0.693723\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−457534.	−2.28986
$448$	121734.	0.606535
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−2.00720e6	−9.91209
$451$	0	0
$452$	0	0
$453$	88549.9	0.431511
$454$	639211.	3.10122
$455$	0	0
$456$	0	0
$457$	−307492.	−1.47232	−0.736159	−	0.676808i	$-0.763363\pi$
$457$	−307492.	−1.47232	−0.736159	+	0.676808i	$0.763363\pi$
$458$	0	0
$459$	477448.	2.26621
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	270066.	1.25982	0.629909	−	0.776669i	$-0.283092\pi$
$463$	270066.	1.25982	0.629909	+	0.776669i	$0.283092\pi$
$464$	0	0
$465$	−61360.7	−0.283782
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	256101.	1.16430
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	533320.	2.35382
$477$	779417.	3.42557
$478$	−824453.	−3.60836
$479$	−445444.	−1.94143	−0.970715	−	0.240233i	$-0.922776\pi$
$479$	−445444.	−1.94143	−0.970715	+	0.240233i	$0.922776\pi$
$480$	1.16907e6	5.07410
$481$	0	0
$482$	−850663.	−3.66154
$483$	0	0
$484$	551397.	2.35382
$485$	508362.	2.16117
$486$	−450624.	−1.90784
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	580934.	2.43942
$489$	0	0
$490$	810685.	3.37645
$491$	−473998.	−1.96614	−0.983069	−	0.183237i	$-0.941342\pi$
$491$	−473998.	−1.96614	−0.983069	+	0.183237i	$0.941342\pi$
$492$	1.97651e6	8.16522
$493$	0	0
$494$	0	0
$495$	0	0
$496$	45888.4	0.186526
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−1.51807e6	−6.07227
$501$	200970.	0.800674
$502$	0	0
$503$	503477.	1.98996	0.994979	−	0.100085i	$-0.0319115\pi$
$503$	503477.	1.98996	0.994979	+	0.100085i	$0.0319115\pi$
$504$	−1.42075e6	−5.59316
$505$	0	0
$506$	0	0
$507$	463824.	1.80442
$508$	−720870.	−2.79338
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	1.58466e6	6.09252
$511$	−503014.	−1.92636
$512$	546911.	2.08630
$513$	0	0
$514$	0	0
$515$	0	0
$516$	1.14475e6	4.29945
$517$	0	0
$518$	0	0
$519$	−744531.	−2.76406
$520$	0	0
$521$	3609.29	0.0132968	0.00664838	−	0.999978i	$-0.497884\pi$
$521$	3609.29	0.0132968	0.00664838	+	0.999978i	$0.497884\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−1.28884e6	−4.69393
$525$	1.19324e6	4.32920
$526$	995065.	3.59650
$527$	23690.8	0.0853017
$528$	0	0
$529$	279841.	1.00000
$530$	1.44019e6	5.12706
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−829330.	−2.88667
$537$	124213.	0.430743
$538$	−558033.	−1.92795
$539$	0	0
$540$	−2.86782e6	−9.83477
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	−49076.6	−0.166446
$544$	−451367.	−1.52522
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−739822.	−2.46358
$549$	−668996.	−2.21962
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	−377378.	−1.22075
$557$	3602.00	0.0116100	0.00580501	−	0.999983i	$-0.498152\pi$
$557$	3602.00	0.0116100	0.00580501	+	0.999983i	$0.498152\pi$
$558$	−109729.	−0.352413
$559$	0	0
$560$	−1.26429e6	−4.03155
$561$	0	0
$562$	1.14044e6	3.61076
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	−4781.17	−0.0149245
$567$	589375.	1.83327
$568$	0	0
$569$	494310.	1.52677	0.763387	−	0.645942i	$-0.223535\pi$
$569$	494310.	1.52677	0.763387	+	0.645942i	$0.223535\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	1.16718e6	3.55490
$574$	−1.15998e6	−3.52068
$575$	0	0
$576$	453968.	1.36830
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−611823.	−1.83134
$579$	0	0
$580$	0	0
$581$	0	0
$582$	1.31206e6	3.87353
$583$	0	0
$584$	1.62891e6	4.77607
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	1.46847e6	4.24728
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	−652715.	−1.84370
$596$	−1.06105e6	−2.98707
$597$	−870218.	−2.44163
$598$	0	0
$599$	−10588.1	−0.0295097	−0.0147548	−	0.999891i	$-0.504697\pi$
$599$	−10588.1	−0.0295097	−0.0147548	+	0.999891i	$0.504697\pi$
$600$	−3.86405e6	−10.7335
$601$	−374302.	−1.03627	−0.518135	−	0.855299i	$-0.673374\pi$
$601$	−374302.	−1.03627	−0.518135	+	0.855299i	$0.673374\pi$
$602$	−671838.	−1.85384
$603$	955046.	2.62657
$604$	205354.	0.562897
$605$	−674840.	−1.84370
$606$	0	0
$607$	−678914.	−1.84263	−0.921314	−	0.388820i	$-0.872883\pi$
$607$	−678914.	−1.84263	−0.921314	+	0.388820i	$0.872883\pi$
$608$	0	0
$609$	0	0
$610$	−1.23616e6	−3.32211
$611$	0	0
$612$	1.98885e6	5.31005
$613$	717827.	1.91029	0.955144	−	0.296143i	$-0.0957006\pi$
$613$	717827.	1.91029	0.955144	+	0.296143i	$0.0957006\pi$
$614$	0	0
$615$	−2.41899e6	−6.39564
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	697778.	1.82111	0.910555	−	0.413389i	$-0.135655\pi$
$619$	697778.	1.82111	0.910555	+	0.413389i	$0.135655\pi$
$620$	−142300.	−0.370187
$621$	0	0
$622$	−1.31794e6	−3.40655
$623$	0	0
$624$	0	0
$625$	920722.	2.35705
$626$	−516199.	−1.31725
$627$	0	0
$628$	0	0
$629$	0	0
$630$	3.02319e6	7.61701
$631$	−438703.	−1.10182	−0.550911	−	0.834564i	$-0.685720\pi$
$631$	−438703.	−1.10182	−0.550911	+	0.834564i	$0.685720\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	882252.	2.18799
$636$	2.60876e6	6.44940
$637$	0	0
$638$	0	0
$639$	0	0
$640$	−312978.	−0.764107
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	826853.	1.99989	0.999946	−	0.0104262i	$-0.00331883\pi$
$643$	826853.	1.99989	0.999946	+	0.0104262i	$0.00331883\pi$
$644$	0	0
$645$	−1.40103e6	−3.36766
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	−1.90857e6	−4.54526
$649$	0	0
$650$	0	0
$651$	65231.4	0.153920
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	1.57738e6	3.67665
$656$	1.80903e6	4.20377
$657$	−1.87583e6	−4.34573
$658$	0	0
$659$	136721.	0.314821	0.157410	−	0.987533i	$-0.449685\pi$
$659$	136721.	0.314821	0.157410	+	0.987533i	$0.449685\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	−814676.	−1.85896
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	466064.	1.04446
$669$	0	0
$670$	1.76472e6	3.93120
$671$	0	0
$672$	−1.24282e6	−2.75213
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	2.47730e6	5.43715
$676$	1.07564e6	2.35382
$677$	−796942.	−1.73880	−0.869399	−	0.494110i	$-0.835494\pi$
$677$	−796942.	−1.73880	−0.869399	+	0.494110i	$0.835494\pi$
$678$	0	0
$679$	−540430.	−1.17219
$680$	2.11368e6	4.57112
$681$	−1.41708e6	−3.05562
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	905447.	1.92967
$686$	−861824.	−1.83134
$687$	0	0
$688$	1.04776e6	2.21352
$689$	0	0
$690$	0	0
$691$	−31843.6	−0.0666909	−0.0333454	−	0.999444i	$-0.510616\pi$
$691$	−31843.6	−0.0666909	−0.0333454	+	0.999444i	$0.510616\pi$
$692$	−1.72662e6	−3.60566
$693$	0	0
$694$	0	0
$695$	461862.	0.956187
$696$	0	0
$697$	933947.	1.92246
$698$	0	0
$699$	0	0
$700$	2.76720e6	5.64735
$701$	−730798.	−1.48717	−0.743586	−	0.668640i	$-0.766877\pi$
$701$	−730798.	−1.48717	−0.743586	+	0.668640i	$0.766877\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	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
$711$	0	0
$712$	0	0
$713$	0	0
$714$	−1.68463e6	−3.30451
$715$	0	0
$716$	288059.	0.561895
$717$	1.82774e6	3.55531
$718$	−1.82974e6	−3.54929
$719$	62660.2	0.121209	0.0606044	−	0.998162i	$-0.480697\pi$
$719$	62660.2	0.121209	0.0606044	+	0.998162i	$0.480697\pi$
$720$	−4.71478e6	−9.09487
$721$	0	0
$722$	−954651.	−1.83134
$723$	1.88585e6	3.60770
$724$	−113812.	−0.217126
$725$	0	0
$726$	−1.74173e6	−3.30451
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	24723.2	0.0465210
$730$	−3.46612e6	−6.50427
$731$	540924.	1.01228
$732$	−2.23917e6	−4.17894
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	1.94335e6	3.60711
$735$	−1.79722e6	−3.32680
$736$	0	0
$737$	0	0
$738$	−4.32578e6	−7.94239
$739$	−693480.	−1.26983	−0.634915	−	0.772582i	$-0.718965\pi$
$739$	−693480.	−1.26983	−0.634915	+	0.772582i	$0.718965\pi$
$740$	0	0
$741$	0	0
$742$	−1.53104e6	−2.78086
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	−211239.	−0.381617
$745$	1.29859e6	2.33970
$746$	1.64294e6	2.95219
$747$	0	0
$748$	0	0
$749$	0	0
$750$	4.79520e6	8.52480
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−251327.	−0.440904
$756$	3.04872e6	5.33426
$757$	−1.07172e6	−1.87021	−0.935104	−	0.354374i	$-0.884694\pi$
$757$	−1.07172e6	−1.87021	−0.935104	+	0.354374i	$0.884694\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	2.27705e6	3.92159
$763$	0	0
$764$	2.70677e6	4.63729
$765$	−2.43409e6	−4.15924
$766$	0	0
$767$	0	0
$768$	−1.45331e6	−2.46397
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	−2.50540e6	−4.18212
$775$	122923.	0.204658
$776$	1.75007e6	2.90624
$777$	0	0
$778$	1.63841e6	2.70685
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	1.34405e6	2.18666
$785$	0	0
$786$	4.07113e6	6.58977
$787$	−474862.	−0.766687	−0.383343	−	0.923606i	$-0.625227\pi$
$787$	−474862.	−0.766687	−0.383343	+	0.923606i	$0.625227\pi$
$788$	0	0
$789$	−2.20598e6	−3.54362
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	2.11248e6	3.35083
$795$	−3.19279e6	−5.05168
$796$	−2.01810e6	−3.18505
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	−2.34198e6	−3.65934
$801$	0	0
$802$	0	0
$803$	0	0
$804$	3.19660e6	4.94511
$805$	0	0
$806$	0	0
$807$	1.23711e6	1.89960
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	4.06121e6	6.18993
$811$	100345.	0.152564	0.0762821	−	0.997086i	$-0.475695\pi$
$811$	100345.	0.152564	0.0762821	+	0.997086i	$0.475695\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	2.62724e6	3.94566
$817$	0	0
$818$	0	0
$819$	0	0
$820$	−5.60981e6	−8.34297
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	2.33692e6	3.45859
$823$	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			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	693889.	1.00000
$834$	1.19204e6	1.71380
$835$	−570402.	−0.818104
$836$	0	0
$837$	135428.	0.193312
$838$	−2.41342e6	−3.43672
$839$	−1.05974e6	−1.50548	−0.752742	−	0.658315i	$-0.771269\pi$
$839$	−1.05974e6	−1.50548	−0.752742	+	0.658315i	$0.771269\pi$
$840$	5.81993e6	8.24820
$841$	707281.	1.00000
$842$	1.01312e6	1.42902
$843$	−2.52825e6	−3.55767
$844$	0	0
$845$	−1.31645e6	−1.84370
$846$	0	0
$847$	717409.	1.00000
$848$	2.38771e6	3.32040
$849$	10599.5	0.0147051
$850$	−3.17452e6	−4.39380
$851$	0	0
$852$	0	0
$853$	−258382.	−0.355111	−0.177556	−	0.984111i	$-0.556819\pi$
$853$	−258382.	−0.355111	−0.177556	+	0.984111i	$0.556819\pi$
$854$	1.31414e6	1.80188
$855$	0	0
$856$	0	0
$857$	−1.38634e6	−1.88759	−0.943796	−	0.330528i	$-0.892773\pi$
$857$	−1.38634e6	−1.88759	−0.943796	+	0.330528i	$0.892773\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	−3.24909e6	−4.39304
$861$	2.57158e6	3.46892
$862$	0	0
$863$	540982.	0.726376	0.363188	−	0.931716i	$-0.381688\pi$
$863$	540982.	0.726376	0.363188	+	0.931716i	$0.381688\pi$
$864$	−2.58024e6	−3.45647
$865$	2.11316e6	2.82424
$866$	0	0
$867$	1.35636e6	1.80442
$868$	151276.	0.200785
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−2.01536e6	−2.64438
$874$	0	0
$875$	−1.97512e6	−2.57974
$876$	−6.27853e6	−8.18181
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	1.50751e6	1.95556
$879$	0	0
$880$	0	0
$881$	−1.49689e6	−1.92858	−0.964289	−	0.264852i	$-0.914677\pi$
$881$	−1.49689e6	−1.92858	−0.964289	+	0.264852i	$0.914677\pi$
$882$	−3.21390e6	−4.13137
$883$	−1.55427e6	−1.99344	−0.996722	−	0.0809045i	$-0.974219\pi$
$883$	−1.55427e6	−1.99344	−0.996722	+	0.0809045i	$0.974219\pi$
$884$	0	0
$885$	0	0
$886$	1.64380e6	2.09402
$887$	1.21482e6	1.54406	0.772029	−	0.635587i	$-0.219242\pi$
$887$	1.21482e6	1.54406	0.772029	+	0.635587i	$0.219242\pi$
$888$	0	0
$889$	−937905.	−1.18674
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	3.35161e6	4.19351
$895$	−352547.	−0.440120
$896$	332721.	0.414442
$897$	0	0
$898$	0	0
$899$	0	0
$900$	1.03194e7	12.7400
$901$	1.23270e6	1.51848
$902$	0	0
$903$	1.48941e6	1.82658
$904$	0	0
$905$	139291.	0.170070
$906$	−648662.	−0.790245
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	−3.28631e6	−3.98599
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	2.25250e6	2.69632
$915$	2.74046e6	3.27327
$916$	0	0
$917$	−1.67688e6	−1.99417
$918$	−3.49749e6	−4.15021
$919$	−161284.	−0.190968	−0.0954838	−	0.995431i	$-0.530440\pi$
$919$	−161284.	−0.190968	−0.0954838	+	0.995431i	$0.530440\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−1.97833e6	−2.30716
$927$	0	0
$928$	0	0
$929$	−73738.6	−0.0854405	−0.0427202	−	0.999087i	$-0.513602\pi$
$929$	−73738.6	−0.0854405	−0.0427202	+	0.999087i	$0.513602\pi$
$930$	449491.	0.519702
$931$	0	0
$932$	0	0
$933$	2.92176e6	3.35646
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	−1.87604e6	−2.13224
$939$	1.14437e6	1.29788
$940$	0	0
$941$	1.64384e6	1.85643	0.928217	−	0.372039i	$-0.121341\pi$
$941$	1.64384e6	1.85643	0.928217	+	0.372039i	$0.121341\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−3.73125e6	−4.17821
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	−2.24702e6	−2.47932
$953$	1.61665e6	1.78004	0.890020	−	0.455922i	$-0.150690\pi$
$953$	1.61665e6	1.78004	0.890020	+	0.455922i	$0.150690\pi$
$954$	−5.70952e6	−6.27340
$955$	−3.31273e6	−3.63229
$956$	4.23867e6	4.63782
$957$	0	0
$958$	3.26304e6	3.55543
$959$	−962564.	−1.04663
$960$	−1.85963e6	−2.01782
$961$	−916801.	−0.992724
$962$	0	0
$963$	0	0
$964$	4.37342e6	4.70617
$965$	0	0
$966$	0	0
$967$	1.86762e6	1.99726	0.998632	−	0.0522984i	$-0.0166547\pi$
$967$	1.86762e6	1.99726	0.998632	+	0.0522984i	$0.0166547\pi$
$968$	−2.32318e6	−2.47932
$969$	0	0
$970$	−3.72394e6	−3.95785
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	2.31675e6	2.45214
$973$	−490997.	−0.518625
$974$	0	0
$975$	0	0
$976$	−2.04945e6	−2.15148
$977$	−1.81590e6	−1.90241	−0.951205	−	0.308561i	$-0.900153\pi$
$977$	−1.81590e6	−1.90241	−0.951205	+	0.308561i	$0.900153\pi$
$978$	0	0
$979$	0	0
$980$	−4.16789e6	−4.33974
$981$	0	0
$982$	3.47222e6	3.60068
$983$	−50399.5	−0.0521577	−0.0260789	−	0.999660i	$-0.508302\pi$
$983$	−50399.5	−0.0521577	−0.0260789	+	0.999660i	$0.508302\pi$
$984$	−8.32754e6	−8.60055
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	−128030.	−0.130104
$993$	1.80607e6	1.83162
$994$	0	0
$995$	2.46989e6	2.49478
$996$	0	0
$997$	1.96476e6	1.97660	0.988301	−	0.152516i	$-0.0487374\pi$
$997$	1.96476e6	1.97660	0.988301	+	0.152516i	$0.0487374\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	119.5.d.a.118.1	✓	5
7.6	odd	2		119.5.d.b.118.1	yes	5
17.16	even	2		119.5.d.b.118.1	yes	5
119.118	odd	2	CM	119.5.d.a.118.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
119.5.d.a.118.1	✓	5	1.1	even	1	trivial
119.5.d.a.118.1	✓	5	119.118	odd	2	CM
119.5.d.b.118.1	yes	5	7.6	odd	2
119.5.d.b.118.1	yes	5	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 119 = 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	119.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-7.32538 q^{2} +16.2398 q^{3} +37.6612 q^{4} -46.0925 q^{5} -118.962 q^{6} +49.0000 q^{7} -158.676 q^{8} +182.730 q^{9} +337.645 q^{10} +611.609 q^{12} -358.944 q^{14} -748.530 q^{15} +559.786 q^{16} +289.000 q^{17} -1338.57 q^{18} -1735.90 q^{20} +795.748 q^{21} -2576.87 q^{24} +1499.51 q^{25} +1652.07 q^{27} +1845.40 q^{28} +5483.27 q^{30} +81.9749 q^{31} -1561.82 q^{32} -2117.03 q^{34} -2258.53 q^{35} +6881.82 q^{36} +7313.79 q^{40} +3231.65 q^{41} -5829.16 q^{42} +1871.71 q^{43} -8422.47 q^{45} +9090.79 q^{48} +2401.00 q^{49} -10984.5 q^{50} +4693.29 q^{51} +4265.40 q^{53} -12102.0 q^{54} -7775.15 q^{56} -28190.5 q^{60} -3661.12 q^{61} -600.498 q^{62} +8953.76 q^{63} +2484.37 q^{64} +5226.55 q^{67} +10884.1 q^{68} +16544.6 q^{70} -28994.9 q^{72} -10265.6 q^{73} +24351.8 q^{75} -25801.9 q^{80} +12028.1 q^{81} -23673.1 q^{82} +29968.8 q^{84} -13320.7 q^{85} -13711.0 q^{86} +61697.8 q^{90} +1331.25 q^{93} -25363.6 q^{96} -11029.2 q^{97} -17588.2 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 80 q^{4} - 235 q^{6} + 245 q^{7} + 405 q^{9} + 1280 q^{16} + 1445 q^{17} - 1355 q^{20} - 3760 q^{24} + 3125 q^{25} + 3920 q^{28} + 8405 q^{30} + 4885 q^{32} + 4565 q^{36} + 10645 q^{40} - 11515 q^{42}+ \cdots - 60160 q^{96}+O(q^{100}) \) 5 * q + 80 * q^4 - 235 * q^6 + 245 * q^7 + 405 * q^9 + 1280 * q^16 + 1445 * q^17 - 1355 * q^20 - 3760 * q^24 + 3125 * q^25 + 3920 * q^28 + 8405 * q^30 + 4885 * q^32 + 4565 * q^36 + 10645 * q^40 - 11515 * q^42 - 17830 * q^45 + 8165 * q^48 + 12005 * q^49 - 23195 * q^50 - 19035 * q^54 - 35995 * q^60 + 19845 * q^63 + 20480 * q^64 + 23120 * q^68 - 48715 * q^72 + 18170 * q^75 - 21680 * q^80 + 32805 * q^81 - 66635 * q^82 - 59915 * q^86 + 51845 * q^90 - 60160 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more