LMFDB - Embedded newform 119.5.d.b.118.3

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			118.3
Root			$-2.63923$ 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)

$f(q)$	$=$	$q+0.794362 q^{2} -12.4018 q^{3} -15.3690 q^{4} -25.8998 q^{5} -9.85151 q^{6} -49.0000 q^{7} -24.9183 q^{8} +72.8042 q^{9} -20.5738 q^{10} +190.603 q^{12} -38.9238 q^{14} +321.204 q^{15} +226.110 q^{16} -289.000 q^{17} +57.8329 q^{18} +398.054 q^{20} +607.687 q^{21} +309.032 q^{24} +45.7995 q^{25} +101.643 q^{27} +753.080 q^{28} +255.152 q^{30} -1062.38 q^{31} +578.306 q^{32} -229.571 q^{34} +1269.09 q^{35} -1118.93 q^{36} +645.380 q^{40} -116.933 q^{41} +482.724 q^{42} -2454.86 q^{43} -1885.61 q^{45} -2804.16 q^{48} +2401.00 q^{49} +36.3814 q^{50} +3584.11 q^{51} -1301.69 q^{53} +80.7410 q^{54} +1221.00 q^{56} -4936.58 q^{60} +7293.40 q^{61} -843.912 q^{62} -3567.41 q^{63} -3158.37 q^{64} -8519.10 q^{67} +4441.64 q^{68} +1008.12 q^{70} -1814.16 q^{72} +5897.40 q^{73} -567.996 q^{75} -5856.19 q^{80} -7157.69 q^{81} -92.8875 q^{82} -9339.54 q^{84} +7485.04 q^{85} -1950.05 q^{86} -1497.86 q^{90} +13175.4 q^{93} -7172.03 q^{96} +39.2350 q^{97} +1907.26 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.794362	0.198591	0.0992953	−	0.995058i	$-0.468341\pi$
$2$	0.794362	0.198591	0.0992953	+	0.995058i	$0.468341\pi$
$3$	−12.4018	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$3$	−12.4018	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$4$	−15.3690	−0.960562
$5$	−25.8998	−1.03599	−0.517996	−	0.855383i	$-0.673322\pi$
$5$	−25.8998	−1.03599	−0.517996	+	0.855383i	$0.673322\pi$
$6$	−9.85151	−0.273653
$7$	−49.0000	−1.00000
$7$	−49.0000	−1.00000
$8$	−24.9183	−0.389349
$9$	72.8042	0.898817
$10$	−20.5738	−0.205738
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	190.603	1.32363
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	−38.9238	−0.198591
$15$	321.204	1.42757
$16$	226.110	0.883241
$17$	−289.000	−1.00000
$17$	−289.000	−1.00000
$18$	57.8329	0.178497
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	398.054	0.995134
$21$	607.687	1.37798
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	309.032	0.536514
$25$	45.7995	0.0732792
$26$	0	0
$27$	101.643	0.139427
$28$	753.080	0.960562
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	255.152	0.283502
$31$	−1062.38	−1.10549	−0.552745	−	0.833350i	$-0.686420\pi$
$31$	−1062.38	−1.10549	−0.552745	+	0.833350i	$0.686420\pi$
$32$	578.306	0.564752
$33$	0	0
$34$	−229.571	−0.198591
$35$	1269.09	1.03599
$36$	−1118.93	−0.863369
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	645.380	0.403363
$41$	−116.933	−0.0695618	−0.0347809	−	0.999395i	$-0.511073\pi$
$41$	−116.933	−0.0695618	−0.0347809	+	0.999395i	$0.511073\pi$
$42$	482.724	0.273653
$43$	−2454.86	−1.32767	−0.663833	−	0.747881i	$-0.731072\pi$
$43$	−2454.86	−1.32767	−0.663833	+	0.747881i	$0.731072\pi$
$44$	0	0
$45$	−1885.61	−0.931167
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−2804.16	−1.21708
$49$	2401.00	1.00000
$50$	36.3814	0.0145526
$51$	3584.11	1.37798
$52$	0	0
$53$	−1301.69	−0.463399	−0.231700	−	0.972787i	$-0.574429\pi$
$53$	−1301.69	−0.463399	−0.231700	+	0.972787i	$0.574429\pi$
$54$	80.7410	0.0276890
$55$	0	0
$56$	1221.00	0.389349
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	−4936.58	−1.37127
$61$	7293.40	1.96006	0.980032	−	0.198839i	$-0.0637172\pi$
$61$	7293.40	1.96006	0.980032	+	0.198839i	$0.0637172\pi$
$62$	−843.912	−0.219540
$63$	−3567.41	−0.898817
$64$	−3158.37	−0.771086
$65$	0	0
$66$	0	0
$67$	−8519.10	−1.89777	−0.948887	−	0.315617i	$-0.897789\pi$
$67$	−8519.10	−1.89777	−0.948887	+	0.315617i	$0.897789\pi$
$68$	4441.64	0.960562
$69$	0	0
$70$	1008.12	0.205738
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	−1814.16	−0.349954
$73$	5897.40	1.10666	0.553330	−	0.832962i	$-0.313357\pi$
$73$	5897.40	1.10666	0.553330	+	0.832962i	$0.313357\pi$
$74$	0	0
$75$	−567.996	−0.100977
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−5856.19	−0.915030
$81$	−7157.69	−1.09094
$82$	−92.8875	−0.0138143
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−9339.54	−1.32363
$85$	7485.04	1.03599
$86$	−1950.05	−0.263662
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	−1497.86	−0.184921
$91$	0	0
$92$	0	0
$93$	13175.4	1.52334
$94$	0	0
$95$	0	0
$96$	−7172.03	−0.778215
$97$	39.2350	0.00416995	0.00208497	−	0.999998i	$-0.499336\pi$
$97$	39.2350	0.00416995	0.00208497	+	0.999998i	$0.499336\pi$
$98$	1907.26	0.198591
$99$	0	0
$100$	−703.892	−0.0703892
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	2847.09	0.273653
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	−15739.0	−1.42757
$106$	−1034.01	−0.0920267
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−1562.14	−0.133929
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	−11079.4	−0.883241
$113$	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
$120$	−8003.86	−0.555824
$121$	14641.0	1.00000
$122$	5793.60	0.389250
$123$	1450.18	0.0958545
$124$	16327.6	1.06189
$125$	15001.2	0.960075
$126$	−2833.81	−0.178497
$127$	30747.4	1.90634	0.953172	−	0.302428i	$-0.0977972\pi$
$127$	30747.4	1.90634	0.953172	+	0.302428i	$0.0977972\pi$
$128$	−11761.8	−0.717883
$129$	30444.6	1.82949
$130$	0	0
$131$	34222.0	1.99417	0.997086	−	0.0762803i	$-0.0243044\pi$
$131$	34222.0	1.99417	0.997086	+	0.0762803i	$0.0243044\pi$
$132$	0	0
$133$	0	0
$134$	−6767.26	−0.376880
$135$	−2632.52	−0.144446
$136$	7201.40	0.389349
$137$	24351.7	1.29744	0.648720	−	0.761027i	$-0.275304\pi$
$137$	24351.7	1.29744	0.648720	+	0.761027i	$0.275304\pi$
$138$	0	0
$139$	−30042.9	−1.55493	−0.777467	−	0.628923i	$-0.783496\pi$
$139$	−30042.9	−1.55493	−0.777467	+	0.628923i	$0.783496\pi$
$140$	−19504.6	−0.995134
$141$	0	0
$142$	0	0
$143$	0	0
$144$	16461.7	0.793872
$145$	0	0
$146$	4684.67	0.219772
$147$	−29776.7	−1.37798
$148$	0	0
$149$	−41345.3	−1.86232	−0.931158	−	0.364615i	$-0.881200\pi$
$149$	−41345.3	−1.86232	−0.931158	+	0.364615i	$0.881200\pi$
$150$	−451.194	−0.0200531
$151$	44743.9	1.96237	0.981183	−	0.193082i	$-0.0618482\pi$
$151$	44743.9	1.96237	0.981183	+	0.193082i	$0.0618482\pi$
$152$	0	0
$153$	−21040.4	−0.898817
$154$	0	0
$155$	27515.3	1.14528
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	16143.3	0.638553
$160$	−14978.0	−0.585079
$161$	0	0
$162$	−5685.80	−0.216651
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	1797.15	0.0668184
$165$	0	0
$166$	0	0
$167$	−55550.1	−1.99183	−0.995914	−	0.0903095i	$-0.971214\pi$
$167$	−55550.1	−1.99183	−0.995914	+	0.0903095i	$0.971214\pi$
$168$	−15142.6	−0.536514
$169$	28561.0	1.00000
$170$	5945.84	0.205738
$171$	0	0
$172$	37728.6	1.27531
$173$	−22434.3	−0.749584	−0.374792	−	0.927109i	$-0.622286\pi$
$173$	−22434.3	−0.749584	−0.374792	+	0.927109i	$0.622286\pi$
$174$	0	0
$175$	−2244.18	−0.0732792
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−58146.3	−1.81475	−0.907374	−	0.420324i	$-0.861916\pi$
$179$	−58146.3	−1.81475	−0.907374	+	0.420324i	$0.861916\pi$
$180$	28980.0	0.894444
$181$	3022.00	0.0922438	0.0461219	−	0.998936i	$-0.485314\pi$
$181$	3022.00	0.0922438	0.0461219	+	0.998936i	$0.485314\pi$
$182$	0	0
$183$	−90451.1	−2.70092
$184$	0	0
$185$	0	0
$186$	10466.0	0.302521
$187$	0	0
$188$	0	0
$189$	−4980.49	−0.139427
$190$	0	0
$191$	−65532.2	−1.79634	−0.898169	−	0.439651i	$-0.855102\pi$
$191$	−65532.2	−1.79634	−0.898169	+	0.439651i	$0.855102\pi$
$192$	39169.4	1.06254
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	31.1668	0.000828112 0
$195$	0	0
$196$	−36900.9	−0.960562
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	−38909.3	−0.982534	−0.491267	−	0.871009i	$-0.663466\pi$
$199$	−38909.3	−0.982534	−0.491267	+	0.871009i	$0.663466\pi$
$200$	−1141.25	−0.0285312
$201$	105652.	2.61509
$202$	0	0
$203$	0	0
$204$	−55084.2	−1.32363
$205$	3028.55	0.0720655
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	−12502.5	−0.283502
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	20005.6	0.445124
$213$	0	0
$214$	0	0
$215$	63580.3	1.37545
$216$	−2532.76	−0.0542859
$217$	52056.4	1.10549
$218$	0	0
$219$	−73138.2	−1.52495
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−28337.0	−0.564752
$225$	3334.40	0.0658646
$226$	0	0
$227$	−102825.	−1.99547	−0.997737	−	0.0672347i	$-0.978582\pi$
$227$	−102825.	−1.99547	−0.997737	+	0.0672347i	$0.978582\pi$
$228$	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
$238$	11249.0	0.198591
$239$	16135.2	0.282474	0.141237	−	0.989976i	$-0.454892\pi$
$239$	16135.2	0.282474	0.141237	+	0.989976i	$0.454892\pi$
$240$	72627.2	1.26089
$241$	95660.6	1.64702	0.823510	−	0.567301i	$-0.192012\pi$
$241$	95660.6	1.64702	0.823510	+	0.567301i	$0.192012\pi$
$242$	11630.3	0.198591
$243$	80535.0	1.36387
$244$	−112092.	−1.88276
$245$	−62185.4	−1.03599
$246$	1151.97	0.0190358
$247$	0	0
$248$	26472.7	0.430422
$249$	0	0
$250$	11916.4	0.190662
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	54827.4	0.863369
$253$	0	0
$254$	24424.6	0.378582
$255$	−92827.9	−1.42757
$256$	41190.8	0.628521
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	24184.0	0.363320
$259$	0	0
$260$	0	0
$261$	0	0
$262$	27184.7	0.396024
$263$	−135838.	−1.96386	−0.981928	−	0.189253i	$-0.939393\pi$
$263$	−135838.	−1.96386	−0.981928	+	0.189253i	$0.939393\pi$
$264$	0	0
$265$	33713.5	0.480078
$266$	0	0
$267$	0	0
$268$	130930.	1.82293
$269$	−76178.0	−1.05275	−0.526375	−	0.850253i	$-0.676449\pi$
$269$	−76178.0	−1.05275	−0.526375	+	0.850253i	$0.676449\pi$
$270$	−2091.18	−0.0286856
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	−65345.7	−0.883241
$273$	0	0
$274$	19344.0	0.257660
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	−23864.9	−0.308795
$279$	−77345.4	−0.993634
$280$	−31623.6	−0.403363
$281$	141526.	1.79235	0.896177	−	0.443697i	$-0.146333\pi$
$281$	141526.	1.79235	0.896177	+	0.443697i	$0.146333\pi$
$282$	0	0
$283$	94677.5	1.18215	0.591077	−	0.806615i	$-0.298703\pi$
$283$	94677.5	1.18215	0.591077	+	0.806615i	$0.298703\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5729.74	0.0695618
$288$	42103.1	0.507609
$289$	83521.0	1.00000
$290$	0	0
$291$	−486.584	−0.00574608
$292$	−90637.0	−1.06302
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	−23653.5	−0.273653
$295$	0	0
$296$	0	0
$297$	0	0
$298$	−32843.1	−0.369839
$299$	0	0
$300$	8729.52	0.0969947
$301$	120288.	1.32767
$302$	35542.9	0.389707
$303$	0	0
$304$	0	0
$305$	−188898.	−2.03061
$306$	−16713.7	−0.178497
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	21857.1	0.227442
$311$	187327.	1.93677	0.968386	−	0.249455i	$-0.0802514\pi$
$311$	187327.	1.93677	0.968386	+	0.249455i	$0.0802514\pi$
$312$	0	0
$313$	−50454.4	−0.515004	−0.257502	−	0.966278i	$-0.582899\pi$
$313$	−50454.4	−0.515004	−0.257502	+	0.966278i	$0.582899\pi$
$314$	0	0
$315$	92395.1	0.931167
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	12823.6	0.126811
$319$	0	0
$320$	81801.1	0.798839
$321$	0	0
$322$	0	0
$323$	0	0
$324$	110006.	1.04792
$325$	0	0
$326$	0	0
$327$	0	0
$328$	2913.79	0.0270838
$329$	0	0
$330$	0	0
$331$	21001.9	0.191691	0.0958457	−	0.995396i	$-0.469444\pi$
$331$	21001.9	0.191691	0.0958457	+	0.995396i	$0.469444\pi$
$332$	0	0
$333$	0	0
$334$	−44126.9	−0.395558
$335$	220643.	1.96608
$336$	137404.	1.21708
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	22687.8	0.198591
$339$	0	0
$340$	−115038.	−0.995134
$341$	0	0
$342$	0	0
$343$	−117649.	−1.00000
$344$	61170.9	0.516926
$345$	0	0
$346$	−17821.0	−0.148860
$347$	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			$\pi$
$350$	−1782.69	−0.0145526
$351$	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
$357$	−175622.	−1.37798
$358$	−46189.3	−0.360392
$359$	−239486.	−1.85820	−0.929099	−	0.369832i	$-0.879415\pi$
$359$	−239486.	−1.85820	−0.929099	+	0.369832i	$0.879415\pi$
$360$	46986.4	0.362549
$361$	130321.	1.00000
$362$	2400.56	0.0183188
$363$	−181574.	−1.37798
$364$	0	0
$365$	−152741.	−1.14649
$366$	−71851.0	−0.536377
$367$	126442.	0.938772	0.469386	−	0.882993i	$-0.344475\pi$
$367$	126442.	0.938772	0.469386	+	0.882993i	$0.344475\pi$
$368$	0	0
$369$	−8513.25	−0.0625234
$370$	0	0
$371$	63782.7	0.463399
$372$	−202492.	−1.46326
$373$	87331.3	0.627700	0.313850	−	0.949473i	$-0.398381\pi$
$373$	87331.3	0.627700	0.313850	+	0.949473i	$0.398381\pi$
$374$	0	0
$375$	−186041.	−1.32296
$376$	0	0
$377$	0	0
$378$	−3956.31	−0.0276890
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−381323.	−2.62690
$382$	−52056.3	−0.356736
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	145867.	0.989225
$385$	0	0
$386$	0	0
$387$	−178724.	−1.19333
$388$	−603.002	−0.00400549
$389$	300785.	1.98773	0.993863	−	0.110617i	$-0.0352826\pi$
$389$	300785.	1.98773	0.993863	+	0.110617i	$0.0352826\pi$
$390$	0	0
$391$	0	0
$392$	−59828.9	−0.389349
$393$	−424414.	−2.74792
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−308116.	−1.95494	−0.977471	−	0.211071i	$-0.932305\pi$
$397$	−308116.	−1.95494	−0.977471	+	0.211071i	$0.932305\pi$
$398$	−30908.1	−0.195122
$399$	0	0
$400$	10355.7	0.0647232
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	83926.0	0.519331
$403$	0	0
$404$	0	0
$405$	185383.	1.13021
$406$	0	0
$407$	0	0
$408$	−89310.2	−0.536514
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	2405.77	0.0143115
$411$	−302004.	−1.78784
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	372585.	2.14266
$418$	0	0
$419$	195169.	1.11169	0.555845	−	0.831286i	$-0.312395\pi$
$419$	195169.	1.11169	0.555845	+	0.831286i	$0.312395\pi$
$420$	241892.	1.37127
$421$	267676.	1.51024	0.755120	−	0.655587i	$-0.227579\pi$
$421$	267676.	1.51024	0.755120	+	0.655587i	$0.227579\pi$
$422$	0	0
$423$	0	0
$424$	32435.9	0.180424
$425$	−13236.1	−0.0732792
$426$	0	0
$427$	−357377.	−1.96006
$428$	0	0
$429$	0	0
$430$	50505.8	0.273152
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	22982.4	0.123148
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	41351.7	0.219540
$435$	0	0
$436$	0	0
$437$	0	0
$438$	−58098.2	−0.302841
$439$	373549.	1.93829	0.969144	−	0.246495i	$-0.0792788\pi$
$439$	373549.	1.93829	0.969144	+	0.246495i	$0.0792788\pi$
$440$	0	0
$441$	174803.	0.898817
$442$	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
$447$	512755.	2.56623
$448$	154760.	0.771086
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	2648.72	0.0130801
$451$	0	0
$452$	0	0
$453$	−554904.	−2.70409
$454$	−81680.2	−0.396282
$455$	0	0
$456$	0	0
$457$	82598.7	0.395495	0.197747	−	0.980253i	$-0.436637\pi$
$457$	82598.7	0.395495	0.197747	+	0.980253i	$0.436637\pi$
$458$	0	0
$459$	−29374.7	−0.139427
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	400145.	1.86662	0.933309	−	0.359075i	$-0.116908\pi$
$463$	400145.	1.86662	0.933309	+	0.359075i	$0.116908\pi$
$464$	0	0
$465$	−341239.	−1.57817
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	417436.	1.89777
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−217640.	−0.960562
$477$	−94768.4	−0.416511
$478$	12817.2	0.0560966
$479$	32806.6	0.142985	0.0714925	−	0.997441i	$-0.477224\pi$
$479$	32806.6	0.142985	0.0714925	+	0.997441i	$0.477224\pi$
$480$	185754.	0.806225
$481$	0	0
$482$	75989.2	0.327083
$483$	0	0
$484$	−225017.	−0.960562
$485$	−1016.18	−0.00432003
$486$	63974.0	0.270851
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	−181739.	−0.763149
$489$	0	0
$490$	−49397.8	−0.205738
$491$	331542.	1.37523	0.687615	−	0.726075i	$-0.258658\pi$
$491$	331542.	1.37523	0.687615	+	0.726075i	$0.258658\pi$
$492$	−22287.8	−0.0920742
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−240213.	−0.976414
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−230553.	−0.922212
$501$	688920.	2.74469
$502$	0	0
$503$	377553.	1.49225	0.746126	−	0.665804i	$-0.231911\pi$
$503$	377553.	1.49225	0.746126	+	0.665804i	$0.231911\pi$
$504$	88893.8	0.349954
$505$	0	0
$506$	0	0
$507$	−354207.	−1.37798
$508$	−472557.	−1.83116
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	−73739.0	−0.283502
$511$	−288972.	−1.10666
$512$	220909.	0.842701
$513$	0	0
$514$	0	0
$515$	0	0
$516$	−467902.	−1.75734
$517$	0	0
$518$	0	0
$519$	278225.	1.03291
$520$	0	0
$521$	−517415.	−1.90618	−0.953090	−	0.302687i	$-0.902116\pi$
$521$	−517415.	−1.90618	−0.953090	+	0.302687i	$0.902116\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	−525958.	−1.91553
$525$	27831.8	0.100977
$526$	−107905.	−0.390003
$527$	307027.	1.10549
$528$	0	0
$529$	279841.	1.00000
$530$	26780.7	0.0953389
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	212282.	0.738896
$537$	721118.	2.50068
$538$	−60512.9	−0.209066
$539$	0	0
$540$	40459.2	0.138749
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	−37478.2	−0.127110
$544$	−167131.	−0.564752
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	−374261.	−1.24627
$549$	530990.	1.76174
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	461729.	1.49361
$557$	3602.00	0.0116100	0.00580501	−	0.999983i	$-0.498152\pi$
$557$	3602.00	0.0116100	0.00580501	+	0.999983i	$0.498152\pi$
$558$	−61440.3	−0.197326
$559$	0	0
$560$	286953.	0.915030
$561$	0	0
$562$	112423.	0.355945
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	75208.3	0.234765
$567$	350727.	1.09094
$568$	0	0
$569$	550541.	1.70045	0.850227	−	0.526417i	$-0.176465\pi$
$569$	550541.	1.70045	0.850227	+	0.526417i	$0.176465\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	812716.	2.47531
$574$	4551.49	0.0138143
$575$	0	0
$576$	−229943.	−0.693066
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	66345.9	0.198591
$579$	0	0
$580$	0	0
$581$	0	0
$582$	−386.524	−0.00114112
$583$	0	0
$584$	−146953.	−0.430877
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	457637.	1.32363
$589$	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
$595$	−366767.	−1.03599
$596$	635435.	1.78887
$597$	482545.	1.35391
$598$	0	0
$599$	−685678.	−1.91102	−0.955512	−	0.294951i	$-0.904697\pi$
$599$	−685678.	−1.91102	−0.955512	+	0.294951i	$0.904697\pi$
$600$	14153.5	0.0393153
$601$	374302.	1.03627	0.518135	−	0.855299i	$-0.326626\pi$
$601$	374302.	1.03627	0.518135	+	0.855299i	$0.326626\pi$
$602$	95552.2	0.263662
$603$	−620227.	−1.70575
$604$	−687668.	−1.88497
$605$	−379199.	−1.03599
$606$	0	0
$607$	−717666.	−1.94780	−0.973901	−	0.226972i	$-0.927117\pi$
$607$	−717666.	−1.94780	−0.973901	+	0.226972i	$0.927117\pi$
$608$	0	0
$609$	0	0
$610$	−150053.	−0.403260
$611$	0	0
$612$	323370.	0.863369
$613$	−711553.	−1.89359	−0.946796	−	0.321834i	$-0.895701\pi$
$613$	−711553.	−1.89359	−0.946796	+	0.321834i	$0.895701\pi$
$614$	0	0
$615$	−37559.4	−0.0993045
$616$	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	$-0.864345\pi$
$619$	−697778.	−1.82111	−0.910555	+	0.413389i	$0.864345\pi$
$620$	−422883.	−1.10011
$621$	0	0
$622$	148805.	0.384625
$623$	0	0
$624$	0	0
$625$	−417152.	−1.06791
$626$	−40079.1	−0.102275
$627$	0	0
$628$	0	0
$629$	0	0
$630$	73395.2	0.184921
$631$	−35713.1	−0.0896951	−0.0448476	−	0.998994i	$-0.514280\pi$
$631$	−35713.1	−0.0896951	−0.0448476	+	0.998994i	$0.514280\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−796352.	−1.97496
$636$	−248105.	−0.613369
$637$	0	0
$638$	0	0
$639$	0	0
$640$	304628.	0.743721
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	−247312.	−0.598169	−0.299084	−	0.954227i	$-0.596681\pi$
$643$	−247312.	−0.598169	−0.299084	+	0.954227i	$0.596681\pi$
$644$	0	0
$645$	−788509.	−1.89534
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	178358.	0.424758
$649$	0	0
$650$	0	0
$651$	−645593.	−1.52334
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	−886343.	−2.06595
$656$	−26439.8	−0.0614398
$657$	429355.	0.994686
$658$	0	0
$659$	393554.	0.906220	0.453110	−	0.891455i	$-0.350315\pi$
$659$	393554.	0.906220	0.453110	+	0.891455i	$0.350315\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	16683.1	0.0380681
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	853748.	1.91327
$669$	0	0
$670$	175271.	0.390445
$671$	0	0
$672$	351430.	0.778215
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	4655.18	0.0102171
$676$	−438954.	−0.960562
$677$	796942.	1.73880	0.869399	−	0.494110i	$-0.164506\pi$
$677$	796942.	1.73880	0.869399	+	0.494110i	$0.164506\pi$
$678$	0	0
$679$	−1922.52	−0.00416995
$680$	−186515.	−0.403363
$681$	1.27521e6	2.74972
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	−630703.	−1.34414
$686$	−93455.9	−0.198591
$687$	0	0
$688$	−555066.	−1.17265
$689$	0	0
$690$	0	0
$691$	−897878.	−1.88045	−0.940223	−	0.340559i	$-0.889384\pi$
$691$	−897878.	−1.88045	−0.940223	+	0.340559i	$0.889384\pi$
$692$	344793.	0.720022
$693$	0	0
$694$	0	0
$695$	778105.	1.61090
$696$	0	0
$697$	33793.8	0.0695618
$698$	0	0
$699$	0	0
$700$	34490.7	0.0703892
$701$	−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
$714$	−139507.	−0.273653
$715$	0	0
$716$	893651.	1.74318
$717$	−200105.	−0.389242
$718$	−190239.	−0.369021
$719$	962148.	1.86116	0.930581	−	0.366087i	$-0.119303\pi$
$719$	962148.	1.86116	0.930581	+	0.366087i	$0.119303\pi$
$720$	−426355.	−0.822445
$721$	0	0
$722$	103522.	0.198591
$723$	−1.18636e6	−2.26955
$724$	−46445.1	−0.0886059
$725$	0	0
$726$	−144236.	−0.273653
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−419005.	−0.788432
$730$	−121332.	−0.227682
$731$	709453.	1.32767
$732$	1.39014e6	2.59440
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	100441.	0.186431
$735$	771210.	1.42757
$736$	0	0
$737$	0	0
$738$	−6762.60	−0.0124166
$739$	588249.	1.07714	0.538570	−	0.842581i	$-0.318965\pi$
$739$	588249.	1.07714	0.538570	+	0.842581i	$0.318965\pi$
$740$	0	0
$741$	0	0
$742$	50666.6	0.0920267
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	−328308.	−0.593111
$745$	1.07083e6	1.92934
$746$	69372.7	0.124655
$747$	0	0
$748$	0	0
$749$	0	0
$750$	−147784.	−0.262727
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.15886e6	−2.03299
$756$	76545.0	0.133929
$757$	−717449.	−1.25199	−0.625993	−	0.779829i	$-0.715306\pi$
$757$	−717449.	−1.25199	−0.625993	+	0.779829i	$0.715306\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	−302909.	−0.521677
$763$	0	0
$764$	1.00716e6	1.72549
$765$	544942.	0.931167
$766$	0	0
$767$	0	0
$768$	−510839.	−0.866087
$769$	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			$\pi$
$774$	−141971.	−0.236984
$775$	−48656.3	−0.0810095
$776$	−977.672	−0.00162356
$777$	0	0
$778$	238932.	0.394744
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	542889.	0.883241
$785$	0	0
$786$	−337138.	−0.545711
$787$	474862.	0.766687	0.383343	−	0.923606i	$-0.374773\pi$
$787$	474862.	0.766687	0.383343	+	0.923606i	$0.374773\pi$
$788$	0	0
$789$	1.68463e6	2.70615
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	−244756.	−0.388233
$795$	−418107.	−0.661536
$796$	597997.	0.943784
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	26486.2	0.0413846
$801$	0	0
$802$	0	0
$803$	0	0
$804$	−1.62377e6	−2.51195
$805$	0	0
$806$	0	0
$807$	944743.	1.45066
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	147261.	0.224449
$811$	1.21641e6	1.84943	0.924713	−	0.380665i	$-0.124305\pi$
$811$	1.21641e6	1.84943	0.924713	+	0.380665i	$0.124305\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	810403.	1.21708
$817$	0	0
$818$	0	0
$819$	0	0
$820$	−46545.8	−0.0692234
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	−239901.	−0.355049
$823$	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
$834$	295968.	0.425513
$835$	1.43874e6	2.06352
$836$	0	0
$837$	−107983.	−0.154136
$838$	155035.	0.220771
$839$	1.05974e6	1.50548	0.752742	−	0.658315i	$-0.228731\pi$
$839$	1.05974e6	1.50548	0.752742	+	0.658315i	$0.228731\pi$
$840$	392189.	0.555824
$841$	707281.	1.00000
$842$	212632.	0.299919
$843$	−1.75518e6	−2.46982
$844$	0	0
$845$	−739724.	−1.03599
$846$	0	0
$847$	−717409.	−1.00000
$848$	−294324.	−0.409293
$849$	−1.17417e6	−1.62898
$850$	−10514.2	−0.0145526
$851$	0	0
$852$	0	0
$853$	258382.	0.355111	0.177556	−	0.984111i	$-0.443181\pi$
$853$	258382.	0.355111	0.177556	+	0.984111i	$0.443181\pi$
$854$	−283886.	−0.389250
$855$	0	0
$856$	0	0
$857$	−836196.	−1.13854	−0.569268	−	0.822152i	$-0.692773\pi$
$857$	−836196.	−1.13854	−0.569268	+	0.822152i	$0.692773\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	−977164.	−1.32121
$861$	−71059.0	−0.0958545
$862$	0	0
$863$	−1.25341e6	−1.68295	−0.841474	−	0.540298i	$-0.818312\pi$
$863$	−1.25341e6	−1.68295	−0.841474	+	0.540298i	$0.818312\pi$
$864$	58780.6	0.0787420
$865$	581044.	0.776563
$866$	0	0
$867$	−1.03581e6	−1.37798
$868$	−800055.	−1.06189
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	2856.47	0.00374802
$874$	0	0
$875$	−735058.	−0.960075
$876$	1.12406e6	1.46481
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	296733.	0.384926
$879$	0	0
$880$	0	0
$881$	−1.45267e6	−1.87160	−0.935802	−	0.352525i	$-0.885323\pi$
$881$	−1.45267e6	−1.87160	−0.935802	+	0.352525i	$0.885323\pi$
$882$	138857.	0.178497
$883$	1.33158e6	1.70784	0.853919	−	0.520405i	$-0.174219\pi$
$883$	1.33158e6	1.70784	0.853919	+	0.520405i	$0.174219\pi$
$884$	0	0
$885$	0	0
$886$	−178253.	−0.227075
$887$	1.57066e6	1.99635	0.998173	−	0.0604136i	$-0.0192420\pi$
$887$	1.57066e6	1.99635	0.998173	+	0.0604136i	$0.0192420\pi$
$888$	0	0
$889$	−1.50662e6	−1.90634
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	407313.	0.509629
$895$	1.50598e6	1.88006
$896$	576328.	0.717883
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−51246.3	−0.0632671
$901$	376188.	0.463399
$902$	0	0
$903$	−1.49178e6	−1.82949
$904$	0	0
$905$	−78269.2	−0.0955639
$906$	−440795.	−0.537007
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	1.58031e6	1.91678
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	65613.3	0.0785416
$915$	2.34267e6	2.79813
$916$	0	0
$917$	−1.67688e6	−1.99417
$918$	−23334.2	−0.0276890
$919$	−1.64895e6	−1.95243	−0.976217	−	0.216795i	$-0.930440\pi$
$919$	−1.64895e6	−1.95243	−0.976217	+	0.216795i	$0.930440\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	317860.	0.370693
$927$	0	0
$928$	0	0
$929$	953984.	1.10537	0.552687	−	0.833389i	$-0.313602\pi$
$929$	953984.	1.10537	0.552687	+	0.833389i	$0.313602\pi$
$930$	−271067.	−0.313409
$931$	0	0
$932$	0	0
$933$	−2.32318e6	−2.66883
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	331596.	0.376880
$939$	625725.	0.709663
$940$	0	0
$941$	118646.	0.133990	0.0669952	−	0.997753i	$-0.478659\pi$
$941$	118646.	0.133990	0.0669952	+	0.997753i	$0.478659\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	128994.	0.144446
$946$	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
$952$	−352869.	−0.389349
$953$	−1.79467e6	−1.97605	−0.988025	−	0.154292i	$-0.950690\pi$
$953$	−1.79467e6	−1.97605	−0.988025	+	0.154292i	$0.950690\pi$
$954$	−75280.4	−0.0827152
$955$	1.69727e6	1.86099
$956$	−247981.	−0.271333
$957$	0	0
$958$	26060.4	0.0283955
$959$	−1.19323e6	−1.29744
$960$	−1.01448e6	−1.10078
$961$	205122.	0.222109
$962$	0	0
$963$	0	0
$964$	−1.47021e6	−1.58207
$965$	0	0
$966$	0	0
$967$	670146.	0.716666	0.358333	−	0.933594i	$-0.383345\pi$
$967$	670146.	0.716666	0.358333	+	0.933594i	$0.383345\pi$
$968$	−364829.	−0.389349
$969$	0	0
$970$	−807.214	−0.000857917 0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−1.23774e6	−1.31008
$973$	1.47210e6	1.55493
$974$	0	0
$975$	0	0
$976$	1.64911e6	1.73121
$977$	1.81534e6	1.90182	0.950908	−	0.309473i	$-0.100153\pi$
$977$	1.81534e6	1.90182	0.950908	+	0.309473i	$0.100153\pi$
$978$	0	0
$979$	0	0
$980$	955727.	0.995134
$981$	0	0
$982$	263364.	0.273108
$983$	−1.17633e6	−1.21737	−0.608684	−	0.793413i	$-0.708302\pi$
$983$	−1.17633e6	−1.21737	−0.608684	+	0.793413i	$0.708302\pi$
$984$	−36136.2	−0.0373209
$985$	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			$\pi$
$992$	−614379.	−0.624328
$993$	−260461.	−0.264146
$994$	0	0
$995$	1.00774e6	1.01790
$996$	0	0
$997$	−895508.	−0.900905	−0.450453	−	0.892800i	$-0.648737\pi$
$997$	−895508.	−0.900905	−0.450453	+	0.892800i	$0.648737\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	119.5.d.b.118.3	yes	5
7.6	odd	2		119.5.d.a.118.3	✓	5
17.16	even	2		119.5.d.a.118.3	✓	5
119.118	odd	2	CM	119.5.d.b.118.3	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
119.5.d.a.118.3	✓	5	7.6	odd	2
119.5.d.a.118.3	✓	5	17.16	even	2
119.5.d.b.118.3	yes	5	1.1	even	1	trivial
119.5.d.b.118.3	yes	5	119.118	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+0.794362 q^{2} -12.4018 q^{3} -15.3690 q^{4} -25.8998 q^{5} -9.85151 q^{6} -49.0000 q^{7} -24.9183 q^{8} +72.8042 q^{9} -20.5738 q^{10} +190.603 q^{12} -38.9238 q^{14} +321.204 q^{15} +226.110 q^{16} -289.000 q^{17} +57.8329 q^{18} +398.054 q^{20} +607.687 q^{21} +309.032 q^{24} +45.7995 q^{25} +101.643 q^{27} +753.080 q^{28} +255.152 q^{30} -1062.38 q^{31} +578.306 q^{32} -229.571 q^{34} +1269.09 q^{35} -1118.93 q^{36} +645.380 q^{40} -116.933 q^{41} +482.724 q^{42} -2454.86 q^{43} -1885.61 q^{45} -2804.16 q^{48} +2401.00 q^{49} +36.3814 q^{50} +3584.11 q^{51} -1301.69 q^{53} +80.7410 q^{54} +1221.00 q^{56} -4936.58 q^{60} +7293.40 q^{61} -843.912 q^{62} -3567.41 q^{63} -3158.37 q^{64} -8519.10 q^{67} +4441.64 q^{68} +1008.12 q^{70} -1814.16 q^{72} +5897.40 q^{73} -567.996 q^{75} -5856.19 q^{80} -7157.69 q^{81} -92.8875 q^{82} -9339.54 q^{84} +7485.04 q^{85} -1950.05 q^{86} -1497.86 q^{90} +13175.4 q^{93} -7172.03 q^{96} +39.2350 q^{97} +1907.26 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 80 q^{4} + 235 q^{6} - 245 q^{7} + 405 q^{9} + 1280 q^{16} - 1445 q^{17} + 1355 q^{20} + 3760 q^{24} + 3125 q^{25} - 3920 q^{28} + 8405 q^{30} + 4885 q^{32} + 4565 q^{36} - 10645 q^{40} - 11515 q^{42}+ \cdots + 60160 q^{96}+O(q^{100}) \) 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

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