LMFDB - Embedded newform 819.6.a.c.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,6,Mod(1,819)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("819.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 819 = 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	819.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$131.354348427$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 148x^{4} - 156x^{3} + 4763x^{2} + 5973x - 23760 $ x^6 - x^5 - 148x^4 - 156x^3 + 4763x^2 + 5973x - 23760
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-7.06881$ of defining polynomial
Character	$\chi$	$=$	819.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.06881 q^{2} +33.1056 q^{4} +61.6957 q^{5} -49.0000 q^{7} +8.92103 q^{8} +O(q^{10})$	\(q+8.06881 q^{2} +33.1056 q^{4} +61.6957 q^{5} -49.0000 q^{7} +8.92103 q^{8} +497.810 q^{10} +182.336 q^{11} -169.000 q^{13} -395.371 q^{14} -987.398 q^{16} +1314.48 q^{17} +1167.60 q^{19} +2042.47 q^{20} +1471.23 q^{22} +347.744 q^{23} +681.356 q^{25} -1363.63 q^{26} -1622.18 q^{28} +4364.85 q^{29} -1176.69 q^{31} -8252.59 q^{32} +10606.3 q^{34} -3023.09 q^{35} +11147.9 q^{37} +9421.16 q^{38} +550.389 q^{40} +5057.52 q^{41} -11116.9 q^{43} +6036.34 q^{44} +2805.88 q^{46} +13734.6 q^{47} +2401.00 q^{49} +5497.73 q^{50} -5594.85 q^{52} +21300.9 q^{53} +11249.3 q^{55} -437.130 q^{56} +35219.1 q^{58} +39880.3 q^{59} +20892.4 q^{61} -9494.46 q^{62} -34991.8 q^{64} -10426.6 q^{65} +5151.33 q^{67} +43516.8 q^{68} -24392.7 q^{70} -78765.6 q^{71} +44868.4 q^{73} +89950.1 q^{74} +38654.2 q^{76} -8934.45 q^{77} +26088.6 q^{79} -60918.2 q^{80} +40808.1 q^{82} -65717.2 q^{83} +81098.0 q^{85} -89700.1 q^{86} +1626.62 q^{88} -62986.5 q^{89} +8281.00 q^{91} +11512.3 q^{92} +110822. q^{94} +72036.0 q^{95} +136524. q^{97} +19373.2 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{2} + 109 q^{4} - 112 q^{5} - 294 q^{7} - 339 q^{8}+O(q^{10}) $ 6 * q + 5 * q^2 + 109 * q^4 - 112 * q^5 - 294 * q^7 - 339 * q^8	$ 6 q + 5 q^{2} + 109 q^{4} - 112 q^{5} - 294 q^{7} - 339 q^{8} - 417 q^{10} - 360 q^{11} - 1014 q^{13} - 245 q^{14} + 1353 q^{16} - 2708 q^{17} + 6892 q^{19} - 3261 q^{20} - 33 q^{22} - 9744 q^{23} + 2526 q^{25} - 845 q^{26} - 5341 q^{28} + 70 q^{29} - 130 q^{31} - 31303 q^{32} + 3917 q^{34} + 5488 q^{35} + 11512 q^{37} - 17799 q^{38} + 19339 q^{40} + 912 q^{41} - 34918 q^{43} - 6503 q^{44} + 19333 q^{46} - 8594 q^{47} + 14406 q^{49} + 83560 q^{50} - 18421 q^{52} + 10102 q^{53} - 14552 q^{55} + 16611 q^{56} - 8989 q^{58} + 43260 q^{59} + 137538 q^{61} + 19690 q^{62} + 16129 q^{64} + 18928 q^{65} + 8280 q^{67} + 42295 q^{68} + 20433 q^{70} - 62380 q^{71} + 76670 q^{73} - 10387 q^{74} + 287197 q^{76} + 17640 q^{77} + 21154 q^{79} - 182689 q^{80} + 223876 q^{82} - 223186 q^{83} + 277084 q^{85} + 64747 q^{86} + 126093 q^{88} - 235190 q^{89} + 49686 q^{91} + 121233 q^{92} + 8238 q^{94} - 113200 q^{95} + 108602 q^{97} + 12005 q^{98}+O(q^{100}) $ 6 * q + 5 * q^2 + 109 * q^4 - 112 * q^5 - 294 * q^7 - 339 * q^8 - 417 * q^10 - 360 * q^11 - 1014 * q^13 - 245 * q^14 + 1353 * q^16 - 2708 * q^17 + 6892 * q^19 - 3261 * q^20 - 33 * q^22 - 9744 * q^23 + 2526 * q^25 - 845 * q^26 - 5341 * q^28 + 70 * q^29 - 130 * q^31 - 31303 * q^32 + 3917 * q^34 + 5488 * q^35 + 11512 * q^37 - 17799 * q^38 + 19339 * q^40 + 912 * q^41 - 34918 * q^43 - 6503 * q^44 + 19333 * q^46 - 8594 * q^47 + 14406 * q^49 + 83560 * q^50 - 18421 * q^52 + 10102 * q^53 - 14552 * q^55 + 16611 * q^56 - 8989 * q^58 + 43260 * q^59 + 137538 * q^61 + 19690 * q^62 + 16129 * q^64 + 18928 * q^65 + 8280 * q^67 + 42295 * q^68 + 20433 * q^70 - 62380 * q^71 + 76670 * q^73 - 10387 * q^74 + 287197 * q^76 + 17640 * q^77 + 21154 * q^79 - 182689 * q^80 + 223876 * q^82 - 223186 * q^83 + 277084 * q^85 + 64747 * q^86 + 126093 * q^88 - 235190 * q^89 + 49686 * q^91 + 121233 * q^92 + 8238 * q^94 - 113200 * q^95 + 108602 * q^97 + 12005 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	8.06881	1.42638	0.713188	−	0.700972i	$-0.247250\pi$
$2$	8.06881	1.42638	0.713188	+	0.700972i	$0.247250\pi$
$3$	0	0
$3$	0	0
$4$	33.1056	1.03455
$5$	61.6957	1.10365	0.551823	−	0.833961i	$-0.313932\pi$
$5$	61.6957	1.10365	0.551823	+	0.833961i	$0.313932\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	8.92103	0.0492822
$9$	0	0
$10$	497.810	1.57421
$11$	182.336	0.454350	0.227175	−	0.973854i	$-0.427051\pi$
$11$	182.336	0.454350	0.227175	+	0.973854i	$0.427051\pi$
$12$	0	0
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	−395.371	−0.539120
$15$	0	0
$16$	−987.398	−0.964256
$17$	1314.48	1.10315	0.551573	−	0.834126i	$-0.314028\pi$
$17$	1314.48	1.10315	0.551573	+	0.834126i	$0.314028\pi$
$18$	0	0
$19$	1167.60	0.742012	0.371006	−	0.928630i	$-0.379013\pi$
$19$	1167.60	0.742012	0.371006	+	0.928630i	$0.379013\pi$
$20$	2042.47	1.14178
$21$	0	0
$22$	1471.23	0.648074
$23$	347.744	0.137069	0.0685347	−	0.997649i	$-0.478168\pi$
$23$	347.744	0.137069	0.0685347	+	0.997649i	$0.478168\pi$
$24$	0	0
$25$	681.356	0.218034
$26$	−1363.63	−0.395606
$27$	0	0
$28$	−1622.18	−0.391023
$29$	4364.85	0.963772	0.481886	−	0.876234i	$-0.339952\pi$
$29$	4364.85	0.963772	0.481886	+	0.876234i	$0.339952\pi$
$30$	0	0
$31$	−1176.69	−0.219916	−0.109958	−	0.993936i	$-0.535072\pi$
$31$	−1176.69	−0.219916	−0.109958	+	0.993936i	$0.535072\pi$
$32$	−8252.59	−1.42467
$33$	0	0
$34$	10606.3	1.57350
$35$	−3023.09	−0.417139
$36$	0	0
$37$	11147.9	1.33871	0.669357	−	0.742941i	$-0.266570\pi$
$37$	11147.9	1.33871	0.669357	+	0.742941i	$0.266570\pi$
$38$	9421.16	1.05839
$39$	0	0
$40$	550.389	0.0543901
$41$	5057.52	0.469870	0.234935	−	0.972011i	$-0.424512\pi$
$41$	5057.52	0.469870	0.234935	+	0.972011i	$0.424512\pi$
$42$	0	0
$43$	−11116.9	−0.916880	−0.458440	−	0.888725i	$-0.651592\pi$
$43$	−11116.9	−0.916880	−0.458440	+	0.888725i	$0.651592\pi$
$44$	6036.34	0.470048
$45$	0	0
$46$	2805.88	0.195513
$47$	13734.6	0.906925	0.453463	−	0.891275i	$-0.350189\pi$
$47$	13734.6	0.906925	0.453463	+	0.891275i	$0.350189\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	5497.73	0.310999
$51$	0	0
$52$	−5594.85	−0.286933
$53$	21300.9	1.04162	0.520809	−	0.853673i	$-0.325630\pi$
$53$	21300.9	1.04162	0.520809	+	0.853673i	$0.325630\pi$
$54$	0	0
$55$	11249.3	0.501441
$56$	−437.130	−0.0186269
$57$	0	0
$58$	35219.1	1.37470
$59$	39880.3	1.49152	0.745759	−	0.666216i	$-0.232087\pi$
$59$	39880.3	1.49152	0.745759	+	0.666216i	$0.232087\pi$
$60$	0	0
$61$	20892.4	0.718892	0.359446	−	0.933166i	$-0.382966\pi$
$61$	20892.4	0.718892	0.359446	+	0.933166i	$0.382966\pi$
$62$	−9494.46	−0.313683
$63$	0	0
$64$	−34991.8	−1.06787
$65$	−10426.6	−0.306096
$66$	0	0
$67$	5151.33	0.140195	0.0700975	−	0.997540i	$-0.477669\pi$
$67$	5151.33	0.140195	0.0700975	+	0.997540i	$0.477669\pi$
$68$	43516.8	1.14126
$69$	0	0
$70$	−24392.7	−0.594997
$71$	−78765.6	−1.85435	−0.927173	−	0.374634i	$-0.877768\pi$
$71$	−78765.6	−1.85435	−0.927173	+	0.374634i	$0.877768\pi$
$72$	0	0
$73$	44868.4	0.985448	0.492724	−	0.870186i	$-0.336001\pi$
$73$	44868.4	0.985448	0.492724	+	0.870186i	$0.336001\pi$
$74$	89950.1	1.90951
$75$	0	0
$76$	38654.2	0.767649
$77$	−8934.45	−0.171728
$78$	0	0
$79$	26088.6	0.470309	0.235154	−	0.971958i	$-0.424440\pi$
$79$	26088.6	0.470309	0.235154	+	0.971958i	$0.424440\pi$
$80$	−60918.2	−1.06420
$81$	0	0
$82$	40808.1	0.670212
$83$	−65717.2	−1.04709	−0.523545	−	0.851998i	$-0.675391\pi$
$83$	−65717.2	−1.04709	−0.523545	+	0.851998i	$0.675391\pi$
$84$	0	0
$85$	81098.0	1.21748
$86$	−89700.1	−1.30782
$87$	0	0
$88$	1626.62	0.0223913
$89$	−62986.5	−0.842892	−0.421446	−	0.906853i	$-0.638477\pi$
$89$	−62986.5	−0.842892	−0.421446	+	0.906853i	$0.638477\pi$
$90$	0	0
$91$	8281.00	0.104828
$92$	11512.3	0.141805
$93$	0	0
$94$	110822.	1.29362
$95$	72036.0	0.818918
$96$	0	0
$97$	136524.	1.47326	0.736630	−	0.676296i	$-0.236416\pi$
$97$	136524.	1.47326	0.736630	+	0.676296i	$0.236416\pi$
$98$	19373.2	0.203768
$99$	0	0
$100$	22556.7	0.225567
$101$	85261.2	0.831664	0.415832	−	0.909441i	$-0.363490\pi$
$101$	85261.2	0.831664	0.415832	+	0.909441i	$0.363490\pi$
$102$	0	0
$103$	127614.	1.18524	0.592621	−	0.805482i	$-0.298093\pi$
$103$	127614.	1.18524	0.592621	+	0.805482i	$0.298093\pi$
$104$	−1507.65	−0.0136684
$105$	0	0
$106$	171873.	1.48574
$107$	−88129.7	−0.744154	−0.372077	−	0.928202i	$-0.621354\pi$
$107$	−88129.7	−0.744154	−0.372077	+	0.928202i	$0.621354\pi$
$108$	0	0
$109$	75654.4	0.609913	0.304956	−	0.952366i	$-0.401358\pi$
$109$	75654.4	0.609913	0.304956	+	0.952366i	$0.401358\pi$
$110$	90768.6	0.715244
$111$	0	0
$112$	48382.5	0.364454
$113$	55094.4	0.405893	0.202947	−	0.979190i	$-0.434948\pi$
$113$	55094.4	0.405893	0.202947	+	0.979190i	$0.434948\pi$
$114$	0	0
$115$	21454.3	0.151276
$116$	144501.	0.997071
$117$	0	0
$118$	321786.	2.12747
$119$	−64409.8	−0.416950
$120$	0	0
$121$	−127805.	−0.793566
$122$	168577.	1.02541
$123$	0	0
$124$	−38955.0	−0.227514
$125$	−150762.	−0.863014
$126$	0	0
$127$	−56574.4	−0.311251	−0.155625	−	0.987816i	$-0.549739\pi$
$127$	−56574.4	−0.311251	−0.155625	+	0.987816i	$0.549739\pi$
$128$	−18259.4	−0.0985055
$129$	0	0
$130$	−84130.0	−0.436609
$131$	−225053.	−1.14579	−0.572896	−	0.819628i	$-0.694180\pi$
$131$	−225053.	−1.14579	−0.572896	+	0.819628i	$0.694180\pi$
$132$	0	0
$133$	−57212.5	−0.280454
$134$	41565.1	0.199971
$135$	0	0
$136$	11726.6	0.0543655
$137$	−89095.9	−0.405561	−0.202781	−	0.979224i	$-0.564998\pi$
$137$	−89095.9	−0.405561	−0.202781	+	0.979224i	$0.564998\pi$
$138$	0	0
$139$	244435.	1.07307	0.536534	−	0.843879i	$-0.319733\pi$
$139$	244435.	1.07307	0.536534	+	0.843879i	$0.319733\pi$
$140$	−100081.	−0.431551
$141$	0	0
$142$	−635544.	−2.64500
$143$	−30814.7	−0.126014
$144$	0	0
$145$	269292.	1.06366
$146$	362035.	1.40562
$147$	0	0
$148$	369058.	1.38497
$149$	11739.4	0.0433193	0.0216596	−	0.999765i	$-0.493105\pi$
$149$	11739.4	0.0433193	0.0216596	+	0.999765i	$0.493105\pi$
$150$	0	0
$151$	63446.7	0.226447	0.113223	−	0.993570i	$-0.463882\pi$
$151$	63446.7	0.226447	0.113223	+	0.993570i	$0.463882\pi$
$152$	10416.2	0.0365680
$153$	0	0
$154$	−72090.4	−0.244949
$155$	−72596.5	−0.242709
$156$	0	0
$157$	−81145.3	−0.262733	−0.131366	−	0.991334i	$-0.541936\pi$
$157$	−81145.3	−0.262733	−0.131366	+	0.991334i	$0.541936\pi$
$158$	210504.	0.670837
$159$	0	0
$160$	−509149.	−1.57234
$161$	−17039.5	−0.0518074
$162$	0	0
$163$	−206023.	−0.607361	−0.303680	−	0.952774i	$-0.598216\pi$
$163$	−206023.	−0.607361	−0.303680	+	0.952774i	$0.598216\pi$
$164$	167432.	0.486104
$165$	0	0
$166$	−530259.	−1.49354
$167$	−2086.02	−0.00578797	−0.00289399	−	0.999996i	$-0.500921\pi$
$167$	−2086.02	−0.00578797	−0.00289399	+	0.999996i	$0.500921\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	654364.	1.73659
$171$	0	0
$172$	−368032.	−0.948559
$173$	−333147.	−0.846292	−0.423146	−	0.906062i	$-0.639074\pi$
$173$	−333147.	−0.846292	−0.423146	+	0.906062i	$0.639074\pi$
$174$	0	0
$175$	−33386.4	−0.0824091
$176$	−180038.	−0.438109
$177$	0	0
$178$	−508225.	−1.20228
$179$	565383.	1.31889	0.659447	−	0.751751i	$-0.270790\pi$
$179$	565383.	1.31889	0.659447	+	0.751751i	$0.270790\pi$
$180$	0	0
$181$	532184.	1.20744	0.603720	−	0.797197i	$-0.293685\pi$
$181$	532184.	1.20744	0.603720	+	0.797197i	$0.293685\pi$
$182$	66817.8	0.149525
$183$	0	0
$184$	3102.24	0.00675508
$185$	687776.	1.47747
$186$	0	0
$187$	239678.	0.501214
$188$	454693.	0.938260
$189$	0	0
$190$	581245.	1.16809
$191$	−693816.	−1.37613	−0.688067	−	0.725647i	$-0.741541\pi$
$191$	−693816.	−1.37613	−0.688067	+	0.725647i	$0.741541\pi$
$192$	0	0
$193$	655564.	1.26684	0.633420	−	0.773808i	$-0.281651\pi$
$193$	655564.	1.26684	0.633420	+	0.773808i	$0.281651\pi$
$194$	1.10159e6	2.10142
$195$	0	0
$196$	79486.6	0.147793
$197$	−444450.	−0.815939	−0.407970	−	0.912996i	$-0.633763\pi$
$197$	−444450.	−0.815939	−0.407970	+	0.912996i	$0.633763\pi$
$198$	0	0
$199$	−768892.	−1.37636	−0.688181	−	0.725539i	$-0.741590\pi$
$199$	−768892.	−1.37636	−0.688181	+	0.725539i	$0.741590\pi$
$200$	6078.39	0.0107452
$201$	0	0
$202$	687956.	1.18627
$203$	−213878.	−0.364272
$204$	0	0
$205$	312027.	0.518570
$206$	1.02970e6	1.69060
$207$	0	0
$208$	166870.	0.267436
$209$	212896.	0.337133
$210$	0	0
$211$	741744.	1.14696	0.573480	−	0.819220i	$-0.305593\pi$
$211$	741744.	1.14696	0.573480	+	0.819220i	$0.305593\pi$
$212$	705180.	1.07761
$213$	0	0
$214$	−711101.	−1.06144
$215$	−685865.	−1.01191
$216$	0	0
$217$	57657.7	0.0831204
$218$	610441.	0.869966
$219$	0	0
$220$	372416.	0.518766
$221$	−222148.	−0.305958
$222$	0	0
$223$	90230.1	0.121504	0.0607518	−	0.998153i	$-0.480650\pi$
$223$	90230.1	0.121504	0.0607518	+	0.998153i	$0.480650\pi$
$224$	404377.	0.538476
$225$	0	0
$226$	444546.	0.578957
$227$	1.40947e6	1.81548	0.907738	−	0.419539i	$-0.137808\pi$
$227$	1.40947e6	1.81548	0.907738	+	0.419539i	$0.137808\pi$
$228$	0	0
$229$	−145158.	−0.182916	−0.0914580	−	0.995809i	$-0.529153\pi$
$229$	−145158.	−0.182916	−0.0914580	+	0.995809i	$0.529153\pi$
$230$	173111.	0.215777
$231$	0	0
$232$	38939.0	0.0474968
$233$	899946.	1.08599	0.542996	−	0.839735i	$-0.317290\pi$
$233$	899946.	1.08599	0.542996	+	0.839735i	$0.317290\pi$
$234$	0	0
$235$	847366.	1.00092
$236$	1.32026e6	1.54305
$237$	0	0
$238$	−519710.	−0.594728
$239$	1.29041e6	1.46128	0.730641	−	0.682761i	$-0.239221\pi$
$239$	1.29041e6	1.46128	0.730641	+	0.682761i	$0.239221\pi$
$240$	0	0
$241$	−810584.	−0.898991	−0.449496	−	0.893283i	$-0.648396\pi$
$241$	−810584.	−0.898991	−0.449496	+	0.893283i	$0.648396\pi$
$242$	−1.03123e6	−1.13192
$243$	0	0
$244$	691656.	0.743730
$245$	148131.	0.157664
$246$	0	0
$247$	−197325.	−0.205797
$248$	−10497.3	−0.0108379
$249$	0	0
$250$	−1.21647e6	−1.23098
$251$	472623.	0.473511	0.236756	−	0.971569i	$-0.423916\pi$
$251$	472623.	0.473511	0.236756	+	0.971569i	$0.423916\pi$
$252$	0	0
$253$	63406.2	0.0622774
$254$	−456488.	−0.443961
$255$	0	0
$256$	972408.	0.927360
$257$	−483260.	−0.456402	−0.228201	−	0.973614i	$-0.573284\pi$
$257$	−483260.	−0.456402	−0.228201	+	0.973614i	$0.573284\pi$
$258$	0	0
$259$	−546246.	−0.505987
$260$	−345178.	−0.316672
$261$	0	0
$262$	−1.81591e6	−1.63433
$263$	1.04768e6	0.933983	0.466992	−	0.884262i	$-0.345338\pi$
$263$	1.04768e6	0.933983	0.466992	+	0.884262i	$0.345338\pi$
$264$	0	0
$265$	1.31418e6	1.14958
$266$	−461637.	−0.400033
$267$	0	0
$268$	170538.	0.145039
$269$	−1.53525e6	−1.29360	−0.646799	−	0.762661i	$-0.723893\pi$
$269$	−1.53525e6	−1.29360	−0.646799	+	0.762661i	$0.723893\pi$
$270$	0	0
$271$	−331442.	−0.274148	−0.137074	−	0.990561i	$-0.543770\pi$
$271$	−331442.	−0.274148	−0.137074	+	0.990561i	$0.543770\pi$
$272$	−1.29792e6	−1.06372
$273$	0	0
$274$	−718897.	−0.578483
$275$	124236.	0.0990636
$276$	0	0
$277$	648449.	0.507781	0.253891	−	0.967233i	$-0.418290\pi$
$277$	648449.	0.507781	0.253891	+	0.967233i	$0.418290\pi$
$278$	1.97230e6	1.53060
$279$	0	0
$280$	−26969.0	−0.0205575
$281$	−1.11297e6	−0.840848	−0.420424	−	0.907328i	$-0.638119\pi$
$281$	−1.11297e6	−0.840848	−0.420424	+	0.907328i	$0.638119\pi$
$282$	0	0
$283$	−548640.	−0.407213	−0.203606	−	0.979053i	$-0.565266\pi$
$283$	−548640.	−0.407213	−0.203606	+	0.979053i	$0.565266\pi$
$284$	−2.60758e6	−1.91841
$285$	0	0
$286$	−248638.	−0.179743
$287$	−247818.	−0.177594
$288$	0	0
$289$	308013.	0.216933
$290$	2.17287e6	1.51718
$291$	0	0
$292$	1.48540e6	1.01950
$293$	−2.21097e6	−1.50457	−0.752287	−	0.658836i	$-0.771049\pi$
$293$	−2.21097e6	−1.50457	−0.752287	+	0.658836i	$0.771049\pi$
$294$	0	0
$295$	2.46044e6	1.64611
$296$	99450.5	0.0659748
$297$	0	0
$298$	94723.2	0.0617896
$299$	−58768.8	−0.0380162
$300$	0	0
$301$	544728.	0.346548
$302$	511939.	0.322999
$303$	0	0
$304$	−1.15289e6	−0.715489
$305$	1.28897e6	0.793402
$306$	0	0
$307$	−3.16454e6	−1.91630	−0.958152	−	0.286260i	$-0.907588\pi$
$307$	−3.16454e6	−1.91630	−0.958152	+	0.286260i	$0.907588\pi$
$308$	−295781.	−0.177661
$309$	0	0
$310$	−585767.	−0.346195
$311$	960074.	0.562864	0.281432	−	0.959581i	$-0.409191\pi$
$311$	960074.	0.562864	0.281432	+	0.959581i	$0.409191\pi$
$312$	0	0
$313$	321392.	0.185428	0.0927138	−	0.995693i	$-0.470446\pi$
$313$	321392.	0.185428	0.0927138	+	0.995693i	$0.470446\pi$
$314$	−654746.	−0.374756
$315$	0	0
$316$	863679.	0.486558
$317$	360153.	0.201298	0.100649	−	0.994922i	$-0.467908\pi$
$317$	360153.	0.201298	0.100649	+	0.994922i	$0.467908\pi$
$318$	0	0
$319$	795869.	0.437890
$320$	−2.15885e6	−1.17855
$321$	0	0
$322$	−137488.	−0.0738968
$323$	1.53480e6	0.818548
$324$	0	0
$325$	−115149.	−0.0604717
$326$	−1.66236e6	−0.866325
$327$	0	0
$328$	45118.3	0.0231562
$329$	−672996.	−0.342786
$330$	0	0
$331$	−272147.	−0.136532	−0.0682659	−	0.997667i	$-0.521747\pi$
$331$	−272147.	−0.136532	−0.0682659	+	0.997667i	$0.521747\pi$
$332$	−2.17561e6	−1.08327
$333$	0	0
$334$	−16831.7	−0.00825583
$335$	317815.	0.154726
$336$	0	0
$337$	−2.01513e6	−0.966558	−0.483279	−	0.875466i	$-0.660554\pi$
$337$	−2.01513e6	−0.966558	−0.483279	+	0.875466i	$0.660554\pi$
$338$	230453.	0.109721
$339$	0	0
$340$	2.68480e6	1.25955
$341$	−214552.	−0.0999187
$342$	0	0
$343$	−117649.	−0.0539949
$344$	−99174.1	−0.0451858
$345$	0	0
$346$	−2.68810e6	−1.20713
$347$	−3.85206e6	−1.71739	−0.858696	−	0.512485i	$-0.828725\pi$
$347$	−3.85206e6	−1.71739	−0.858696	+	0.512485i	$0.828725\pi$
$348$	0	0
$349$	−2.40588e6	−1.05733	−0.528665	−	0.848830i	$-0.677307\pi$
$349$	−2.40588e6	−1.05733	−0.528665	+	0.848830i	$0.677307\pi$
$350$	−269389.	−0.117546
$351$	0	0
$352$	−1.50474e6	−0.647300
$353$	−798411.	−0.341028	−0.170514	−	0.985355i	$-0.554543\pi$
$353$	−798411.	−0.341028	−0.170514	+	0.985355i	$0.554543\pi$
$354$	0	0
$355$	−4.85950e6	−2.04654
$356$	−2.08521e6	−0.872015
$357$	0	0
$358$	4.56197e6	1.88124
$359$	1.87791e6	0.769020	0.384510	−	0.923121i	$-0.374370\pi$
$359$	1.87791e6	0.769020	0.384510	+	0.923121i	$0.374370\pi$
$360$	0	0
$361$	−1.11280e6	−0.449418
$362$	4.29409e6	1.72226
$363$	0	0
$364$	274148.	0.108450
$365$	2.76819e6	1.08759
$366$	0	0
$367$	−1.01598e6	−0.393751	−0.196876	−	0.980428i	$-0.563080\pi$
$367$	−1.01598e6	−0.393751	−0.196876	+	0.980428i	$0.563080\pi$
$368$	−343362.	−0.132170
$369$	0	0
$370$	5.54953e6	2.10742
$371$	−1.04375e6	−0.393695
$372$	0	0
$373$	−1.46869e6	−0.546584	−0.273292	−	0.961931i	$-0.588113\pi$
$373$	−1.46869e6	−0.546584	−0.273292	+	0.961931i	$0.588113\pi$
$374$	1.93391e6	0.714920
$375$	0	0
$376$	122527.	0.0446952
$377$	−737660.	−0.267302
$378$	0	0
$379$	678350.	0.242581	0.121290	−	0.992617i	$-0.461297\pi$
$379$	678350.	0.242581	0.121290	+	0.992617i	$0.461297\pi$
$380$	2.38480e6	0.847212
$381$	0	0
$382$	−5.59827e6	−1.96289
$383$	−886827.	−0.308917	−0.154459	−	0.987999i	$-0.549363\pi$
$383$	−886827.	−0.308917	−0.154459	+	0.987999i	$0.549363\pi$
$384$	0	0
$385$	−551217.	−0.189527
$386$	5.28962e6	1.80699
$387$	0	0
$388$	4.51971e6	1.52416
$389$	−3.67149e6	−1.23018	−0.615090	−	0.788457i	$-0.710880\pi$
$389$	−3.67149e6	−1.23018	−0.615090	+	0.788457i	$0.710880\pi$
$390$	0	0
$391$	457105.	0.151208
$392$	21419.4	0.00704031
$393$	0	0
$394$	−3.58618e6	−1.16384
$395$	1.60955e6	0.519054
$396$	0	0
$397$	5.68374e6	1.80991	0.904957	−	0.425504i	$-0.139903\pi$
$397$	5.68374e6	1.80991	0.904957	+	0.425504i	$0.139903\pi$
$398$	−6.20404e6	−1.96321
$399$	0	0
$400$	−672769.	−0.210240
$401$	−3.04422e6	−0.945398	−0.472699	−	0.881224i	$-0.656720\pi$
$401$	−3.04422e6	−0.945398	−0.472699	+	0.881224i	$0.656720\pi$
$402$	0	0
$403$	198860.	0.0609937
$404$	2.82263e6	0.860399
$405$	0	0
$406$	−1.72574e6	−0.519589
$407$	2.03266e6	0.608245
$408$	0	0
$409$	2.06260e6	0.609686	0.304843	−	0.952403i	$-0.401396\pi$
$409$	2.06260e6	0.609686	0.304843	+	0.952403i	$0.401396\pi$
$410$	2.51769e6	0.739677
$411$	0	0
$412$	4.22476e6	1.22619
$413$	−1.95414e6	−0.563741
$414$	0	0
$415$	−4.05447e6	−1.15562
$416$	1.39469e6	0.395133
$417$	0	0
$418$	1.71781e6	0.480878
$419$	1.98119e6	0.551303	0.275652	−	0.961258i	$-0.411106\pi$
$419$	1.98119e6	0.551303	0.275652	+	0.961258i	$0.411106\pi$
$420$	0	0
$421$	−4.29886e6	−1.18208	−0.591042	−	0.806641i	$-0.701283\pi$
$421$	−4.29886e6	−1.18208	−0.591042	+	0.806641i	$0.701283\pi$
$422$	5.98499e6	1.63600
$423$	0	0
$424$	190026.	0.0513332
$425$	895632.	0.240523
$426$	0	0
$427$	−1.02373e6	−0.271716
$428$	−2.91759e6	−0.769865
$429$	0	0
$430$	−5.53411e6	−1.44337
$431$	−7.45226e6	−1.93239	−0.966195	−	0.257814i	$-0.916998\pi$
$431$	−7.45226e6	−1.93239	−0.966195	+	0.257814i	$0.916998\pi$
$432$	0	0
$433$	1.96735e6	0.504267	0.252134	−	0.967692i	$-0.418868\pi$
$433$	1.96735e6	0.504267	0.252134	+	0.967692i	$0.418868\pi$
$434$	465229.	0.118561
$435$	0	0
$436$	2.50459e6	0.630986
$437$	406027.	0.101707
$438$	0	0
$439$	−5.94216e6	−1.47158	−0.735788	−	0.677211i	$-0.763188\pi$
$439$	−5.94216e6	−1.47158	−0.735788	+	0.677211i	$0.763188\pi$
$440$	100356.	0.0247121
$441$	0	0
$442$	−1.79247e6	−0.436411
$443$	283455.	0.0686238	0.0343119	−	0.999411i	$-0.489076\pi$
$443$	283455.	0.0686238	0.0343119	+	0.999411i	$0.489076\pi$
$444$	0	0
$445$	−3.88599e6	−0.930255
$446$	728049.	0.173310
$447$	0	0
$448$	1.71460e6	0.403615
$449$	2.46590e6	0.577244	0.288622	−	0.957443i	$-0.406803\pi$
$449$	2.46590e6	0.577244	0.288622	+	0.957443i	$0.406803\pi$
$450$	0	0
$451$	922167.	0.213485
$452$	1.82394e6	0.419917
$453$	0	0
$454$	1.13727e7	2.58955
$455$	510902.	0.115694
$456$	0	0
$457$	2.39982e6	0.537511	0.268756	−	0.963208i	$-0.413388\pi$
$457$	2.39982e6	0.537511	0.268756	+	0.963208i	$0.413388\pi$
$458$	−1.17125e6	−0.260907
$459$	0	0
$460$	710259.	0.156503
$461$	−2.40784e6	−0.527686	−0.263843	−	0.964566i	$-0.584990\pi$
$461$	−2.40784e6	−0.527686	−0.263843	+	0.964566i	$0.584990\pi$
$462$	0	0
$463$	−2.86881e6	−0.621942	−0.310971	−	0.950419i	$-0.600654\pi$
$463$	−2.86881e6	−0.621942	−0.310971	+	0.950419i	$0.600654\pi$
$464$	−4.30984e6	−0.929323
$465$	0	0
$466$	7.26149e6	1.54903
$467$	−6.12958e6	−1.30058	−0.650292	−	0.759684i	$-0.725353\pi$
$467$	−6.12958e6	−1.30058	−0.650292	+	0.759684i	$0.725353\pi$
$468$	0	0
$469$	−252415.	−0.0529887
$470$	6.83723e6	1.42769
$471$	0	0
$472$	355773.	0.0735052
$473$	−2.02701e6	−0.416584
$474$	0	0
$475$	795553.	0.161784
$476$	−2.13232e6	−0.431356
$477$	0	0
$478$	1.04121e7	2.08434
$479$	−6.41737e6	−1.27796	−0.638982	−	0.769222i	$-0.720644\pi$
$479$	−6.41737e6	−1.27796	−0.638982	+	0.769222i	$0.720644\pi$
$480$	0	0
$481$	−1.88399e6	−0.371293
$482$	−6.54044e6	−1.28230
$483$	0	0
$484$	−4.23105e6	−0.820985
$485$	8.42294e6	1.62596
$486$	0	0
$487$	9.81834e6	1.87593	0.937963	−	0.346735i	$-0.112710\pi$
$487$	9.81834e6	1.87593	0.937963	+	0.346735i	$0.112710\pi$
$488$	186382.	0.0354286
$489$	0	0
$490$	1.19524e6	0.224888
$491$	−4.06986e6	−0.761861	−0.380930	−	0.924604i	$-0.624396\pi$
$491$	−4.06986e6	−0.761861	−0.380930	+	0.924604i	$0.624396\pi$
$492$	0	0
$493$	5.73753e6	1.06318
$494$	−1.59218e6	−0.293544
$495$	0	0
$496$	1.16186e6	0.212055
$497$	3.85951e6	0.700877
$498$	0	0
$499$	−5.27989e6	−0.949235	−0.474617	−	0.880192i	$-0.657414\pi$
$499$	−5.27989e6	−0.949235	−0.474617	+	0.880192i	$0.657414\pi$
$500$	−4.99108e6	−0.892831
$501$	0	0
$502$	3.81350e6	0.675406
$503$	−1.07478e7	−1.89409	−0.947045	−	0.321100i	$-0.895947\pi$
$503$	−1.07478e7	−1.89409	−0.947045	+	0.321100i	$0.895947\pi$
$504$	0	0
$505$	5.26025e6	0.917863
$506$	511613.	0.0888311
$507$	0	0
$508$	−1.87293e6	−0.322005
$509$	969081.	0.165793	0.0828964	−	0.996558i	$-0.473583\pi$
$509$	969081.	0.165793	0.0828964	+	0.996558i	$0.473583\pi$
$510$	0	0
$511$	−2.19855e6	−0.372464
$512$	8.43047e6	1.42127
$513$	0	0
$514$	−3.89933e6	−0.651002
$515$	7.87326e6	1.30809
$516$	0	0
$517$	2.50431e6	0.412061
$518$	−4.40755e6	−0.721728
$519$	0	0
$520$	−93015.7	−0.0150851
$521$	2.21395e6	0.357333	0.178667	−	0.983910i	$-0.442822\pi$
$521$	2.21395e6	0.357333	0.178667	+	0.983910i	$0.442822\pi$
$522$	0	0
$523$	8.63454e6	1.38034	0.690168	−	0.723649i	$-0.257536\pi$
$523$	8.63454e6	1.38034	0.690168	+	0.723649i	$0.257536\pi$
$524$	−7.45051e6	−1.18538
$525$	0	0
$526$	8.45352e6	1.33221
$527$	−1.54674e6	−0.242600
$528$	0	0
$529$	−6.31542e6	−0.981212
$530$	1.06038e7	1.63973
$531$	0	0
$532$	−1.89406e6	−0.290144
$533$	−854721.	−0.130319
$534$	0	0
$535$	−5.43722e6	−0.821282
$536$	45955.2	0.00690911
$537$	0	0
$538$	−1.23877e7	−1.84516
$539$	437788.	0.0649071
$540$	0	0
$541$	563987.	0.0828469	0.0414234	−	0.999142i	$-0.486811\pi$
$541$	563987.	0.0828469	0.0414234	+	0.999142i	$0.486811\pi$
$542$	−2.67434e6	−0.391038
$543$	0	0
$544$	−1.08479e7	−1.57162
$545$	4.66755e6	0.673128
$546$	0	0
$547$	6.31063e6	0.901788	0.450894	−	0.892578i	$-0.351105\pi$
$547$	6.31063e6	0.901788	0.450894	+	0.892578i	$0.351105\pi$
$548$	−2.94957e6	−0.419573
$549$	0	0
$550$	1.00243e6	0.141302
$551$	5.09641e6	0.715131
$552$	0	0
$553$	−1.27834e6	−0.177760
$554$	5.23221e6	0.724287
$555$	0	0
$556$	8.09218e6	1.11014
$557$	3.01937e6	0.412362	0.206181	−	0.978514i	$-0.433896\pi$
$557$	3.01937e6	0.412362	0.206181	+	0.978514i	$0.433896\pi$
$558$	0	0
$559$	1.87876e6	0.254297
$560$	2.98499e6	0.402229
$561$	0	0
$562$	−8.98034e6	−1.19937
$563$	−966973.	−0.128571	−0.0642855	−	0.997932i	$-0.520477\pi$
$563$	−966973.	−0.128571	−0.0642855	+	0.997932i	$0.520477\pi$
$564$	0	0
$565$	3.39909e6	0.447962
$566$	−4.42687e6	−0.580839
$567$	0	0
$568$	−702670.	−0.0913862
$569$	7.83080e6	1.01397	0.506986	−	0.861954i	$-0.330760\pi$
$569$	7.83080e6	1.01397	0.506986	+	0.861954i	$0.330760\pi$
$570$	0	0
$571$	−4.61168e6	−0.591928	−0.295964	−	0.955199i	$-0.595641\pi$
$571$	−4.61168e6	−0.591928	−0.295964	+	0.955199i	$0.595641\pi$
$572$	−1.02014e6	−0.130368
$573$	0	0
$574$	−1.99960e6	−0.253316
$575$	236938.	0.0298858
$576$	0	0
$577$	−4.45239e6	−0.556742	−0.278371	−	0.960474i	$-0.589794\pi$
$577$	−4.45239e6	−0.556742	−0.278371	+	0.960474i	$0.589794\pi$
$578$	2.48530e6	0.309428
$579$	0	0
$580$	8.91509e6	1.10041
$581$	3.22014e6	0.395763
$582$	0	0
$583$	3.88392e6	0.473259
$584$	400272.	0.0485650
$585$	0	0
$586$	−1.78399e7	−2.14609
$587$	3.29893e6	0.395165	0.197582	−	0.980286i	$-0.436691\pi$
$587$	3.29893e6	0.395165	0.197582	+	0.980286i	$0.436691\pi$
$588$	0	0
$589$	−1.37390e6	−0.163180
$590$	1.98528e7	2.34797
$591$	0	0
$592$	−1.10074e7	−1.29086
$593$	−3.19169e6	−0.372721	−0.186360	−	0.982481i	$-0.559669\pi$
$593$	−3.19169e6	−0.372721	−0.186360	+	0.982481i	$0.559669\pi$
$594$	0	0
$595$	−3.97380e6	−0.460165
$596$	388641.	0.0448160
$597$	0	0
$598$	−474194.	−0.0542254
$599$	1.44661e7	1.64735	0.823674	−	0.567064i	$-0.191921\pi$
$599$	1.44661e7	1.64735	0.823674	+	0.567064i	$0.191921\pi$
$600$	0	0
$601$	1.75722e6	0.198445	0.0992223	−	0.995065i	$-0.468365\pi$
$601$	1.75722e6	0.198445	0.0992223	+	0.995065i	$0.468365\pi$
$602$	4.39530e6	0.494308
$603$	0	0
$604$	2.10044e6	0.234271
$605$	−7.88499e6	−0.875816
$606$	0	0
$607$	−9.00987e6	−0.992538	−0.496269	−	0.868169i	$-0.665297\pi$
$607$	−9.00987e6	−0.992538	−0.496269	+	0.868169i	$0.665297\pi$
$608$	−9.63575e6	−1.05713
$609$	0	0
$610$	1.04005e7	1.13169
$611$	−2.32115e6	−0.251536
$612$	0	0
$613$	104926.	0.0112780	0.00563899	−	0.999984i	$-0.498205\pi$
$613$	104926.	0.0112780	0.00563899	+	0.999984i	$0.498205\pi$
$614$	−2.55340e7	−2.73337
$615$	0	0
$616$	−79704.5	−0.00846313
$617$	72477.1	0.00766457	0.00383229	−	0.999993i	$-0.498780\pi$
$617$	72477.1	0.00766457	0.00383229	+	0.999993i	$0.498780\pi$
$618$	0	0
$619$	−1.20843e6	−0.126764	−0.0633821	−	0.997989i	$-0.520189\pi$
$619$	−1.20843e6	−0.126764	−0.0633821	+	0.997989i	$0.520189\pi$
$620$	−2.40335e6	−0.251095
$621$	0	0
$622$	7.74665e6	0.802857
$623$	3.08634e6	0.318583
$624$	0	0
$625$	−1.14306e7	−1.17050
$626$	2.59325e6	0.264489
$627$	0	0
$628$	−2.68636e6	−0.271810
$629$	1.46537e7	1.47680
$630$	0	0
$631$	7.26084e6	0.725962	0.362981	−	0.931797i	$-0.381759\pi$
$631$	7.26084e6	0.725962	0.362981	+	0.931797i	$0.381759\pi$
$632$	232737.	0.0231778
$633$	0	0
$634$	2.90600e6	0.287126
$635$	−3.49039e6	−0.343511
$636$	0	0
$637$	−405769.	−0.0396214
$638$	6.42171e6	0.624596
$639$	0	0
$640$	−1.12652e6	−0.108715
$641$	1.50425e7	1.44602	0.723010	−	0.690837i	$-0.242758\pi$
$641$	1.50425e7	1.44602	0.723010	+	0.690837i	$0.242758\pi$
$642$	0	0
$643$	1.95516e6	0.186490	0.0932450	−	0.995643i	$-0.470276\pi$
$643$	1.95516e6	0.186490	0.0932450	+	0.995643i	$0.470276\pi$
$644$	−564102.	−0.0535973
$645$	0	0
$646$	1.23840e7	1.16756
$647$	−1.73300e7	−1.62756	−0.813782	−	0.581171i	$-0.802595\pi$
$647$	−1.73300e7	−1.62756	−0.813782	+	0.581171i	$0.802595\pi$
$648$	0	0
$649$	7.27161e6	0.677671
$650$	−929116.	−0.0862555
$651$	0	0
$652$	−6.82052e6	−0.628346
$653$	−1.21384e7	−1.11398	−0.556990	−	0.830519i	$-0.688044\pi$
$653$	−1.21384e7	−1.11398	−0.556990	+	0.830519i	$0.688044\pi$
$654$	0	0
$655$	−1.38848e7	−1.26455
$656$	−4.99378e6	−0.453075
$657$	0	0
$658$	−5.43027e6	−0.488941
$659$	6.23051e6	0.558869	0.279435	−	0.960165i	$-0.409853\pi$
$659$	6.23051e6	0.558869	0.279435	+	0.960165i	$0.409853\pi$
$660$	0	0
$661$	−5.44485e6	−0.484710	−0.242355	−	0.970188i	$-0.577920\pi$
$661$	−5.44485e6	−0.484710	−0.242355	+	0.970188i	$0.577920\pi$
$662$	−2.19590e6	−0.194746
$663$	0	0
$664$	−586265.	−0.0516028
$665$	−3.52976e6	−0.309522
$666$	0	0
$667$	1.51785e6	0.132104
$668$	−69058.8	−0.00598795
$669$	0	0
$670$	2.56439e6	0.220697
$671$	3.80943e6	0.326628
$672$	0	0
$673$	−8.47948e6	−0.721659	−0.360829	−	0.932632i	$-0.617506\pi$
$673$	−8.47948e6	−0.721659	−0.360829	+	0.932632i	$0.617506\pi$
$674$	−1.62597e7	−1.37868
$675$	0	0
$676$	945530.	0.0795808
$677$	−7.86437e6	−0.659466	−0.329733	−	0.944074i	$-0.606959\pi$
$677$	−7.86437e6	−0.659466	−0.329733	+	0.944074i	$0.606959\pi$
$678$	0	0
$679$	−6.68967e6	−0.556840
$680$	723478.	0.0600002
$681$	0	0
$682$	−1.73118e6	−0.142522
$683$	−1.36568e7	−1.12020	−0.560102	−	0.828424i	$-0.689238\pi$
$683$	−1.36568e7	−1.12020	−0.560102	+	0.828424i	$0.689238\pi$
$684$	0	0
$685$	−5.49683e6	−0.447596
$686$	−949287.	−0.0770171
$687$	0	0
$688$	1.09768e7	0.884107
$689$	−3.59986e6	−0.288893
$690$	0	0
$691$	−8.81900e6	−0.702626	−0.351313	−	0.936258i	$-0.614265\pi$
$691$	−8.81900e6	−0.702626	−0.351313	+	0.936258i	$0.614265\pi$
$692$	−1.10290e7	−0.875532
$693$	0	0
$694$	−3.10815e7	−2.44965
$695$	1.50806e7	1.18429
$696$	0	0
$697$	6.64803e6	0.518336
$698$	−1.94126e7	−1.50815
$699$	0	0
$700$	−1.10528e6	−0.0852564
$701$	1.69582e7	1.30342	0.651712	−	0.758466i	$-0.274051\pi$
$701$	1.69582e7	1.30342	0.651712	+	0.758466i	$0.274051\pi$
$702$	0	0
$703$	1.30163e7	0.993342
$704$	−6.38026e6	−0.485185
$705$	0	0
$706$	−6.44222e6	−0.486434
$707$	−4.17780e6	−0.314340
$708$	0	0
$709$	564936.	0.0422069	0.0211035	−	0.999777i	$-0.493282\pi$
$709$	564936.	0.0422069	0.0211035	+	0.999777i	$0.493282\pi$
$710$	−3.92103e7	−2.91914
$711$	0	0
$712$	−561904.	−0.0415396
$713$	−409186.	−0.0301438
$714$	0	0
$715$	−1.90114e6	−0.139075
$716$	1.87174e7	1.36446
$717$	0	0
$718$	1.51525e7	1.09691
$719$	−1.95556e7	−1.41075	−0.705373	−	0.708836i	$-0.749221\pi$
$719$	−1.95556e7	−1.41075	−0.705373	+	0.708836i	$0.749221\pi$
$720$	0	0
$721$	−6.25311e6	−0.447979
$722$	−8.97900e6	−0.641040
$723$	0	0
$724$	1.76183e7	1.24916
$725$	2.97402e6	0.210135
$726$	0	0
$727$	−966911.	−0.0678501	−0.0339250	−	0.999424i	$-0.510801\pi$
$727$	−966911.	−0.0678501	−0.0339250	+	0.999424i	$0.510801\pi$
$728$	73875.0	0.00516617
$729$	0	0
$730$	2.23360e7	1.55131
$731$	−1.46130e7	−1.01145
$732$	0	0
$733$	−1.46397e7	−1.00640	−0.503202	−	0.864169i	$-0.667845\pi$
$733$	−1.46397e7	−1.00640	−0.503202	+	0.864169i	$0.667845\pi$
$734$	−8.19778e6	−0.561637
$735$	0	0
$736$	−2.86979e6	−0.195279
$737$	939272.	0.0636976
$738$	0	0
$739$	1.81287e7	1.22112	0.610558	−	0.791972i	$-0.290945\pi$
$739$	1.81287e7	1.22112	0.610558	+	0.791972i	$0.290945\pi$
$740$	2.27693e7	1.52851
$741$	0	0
$742$	−8.42178e6	−0.561557
$743$	−1.24511e6	−0.0827436	−0.0413718	−	0.999144i	$-0.513173\pi$
$743$	−1.24511e6	−0.0827436	−0.0413718	+	0.999144i	$0.513173\pi$
$744$	0	0
$745$	724272.	0.0478092
$746$	−1.18505e7	−0.779635
$747$	0	0
$748$	7.93468e6	0.518532
$749$	4.31835e6	0.281264
$750$	0	0
$751$	8.21193e6	0.531307	0.265653	−	0.964069i	$-0.414412\pi$
$751$	8.21193e6	0.531307	0.265653	+	0.964069i	$0.414412\pi$
$752$	−1.35615e7	−0.874508
$753$	0	0
$754$	−5.95203e6	−0.381274
$755$	3.91438e6	0.249917
$756$	0	0
$757$	5.49249e6	0.348361	0.174181	−	0.984714i	$-0.444272\pi$
$757$	5.49249e6	0.348361	0.174181	+	0.984714i	$0.444272\pi$
$758$	5.47348e6	0.346011
$759$	0	0
$760$	642635.	0.0403581
$761$	−3.16489e7	−1.98106	−0.990529	−	0.137307i	$-0.956155\pi$
$761$	−3.16489e7	−1.98106	−0.990529	+	0.137307i	$0.956155\pi$
$762$	0	0
$763$	−3.70706e6	−0.230525
$764$	−2.29692e7	−1.42368
$765$	0	0
$766$	−7.15563e6	−0.440632
$767$	−6.73977e6	−0.413673
$768$	0	0
$769$	1.73759e7	1.05957	0.529786	−	0.848132i	$-0.322272\pi$
$769$	1.73759e7	1.05957	0.529786	+	0.848132i	$0.322272\pi$
$770$	−4.44766e6	−0.270337
$771$	0	0
$772$	2.17028e7	1.31061
$773$	−8.65259e6	−0.520832	−0.260416	−	0.965497i	$-0.583860\pi$
$773$	−8.65259e6	−0.520832	−0.260416	+	0.965497i	$0.583860\pi$
$774$	0	0
$775$	−801743.	−0.0479491
$776$	1.21793e6	0.0726055
$777$	0	0
$778$	−2.96245e7	−1.75470
$779$	5.90517e6	0.348649
$780$	0	0
$781$	−1.43618e7	−0.842521
$782$	3.68829e6	0.215679
$783$	0	0
$784$	−2.37074e6	−0.137751
$785$	−5.00631e6	−0.289964
$786$	0	0
$787$	−2.02122e7	−1.16326	−0.581630	−	0.813454i	$-0.697585\pi$
$787$	−2.02122e7	−1.16326	−0.581630	+	0.813454i	$0.697585\pi$
$788$	−1.47138e7	−0.844130
$789$	0	0
$790$	1.29872e7	0.740367
$791$	−2.69963e6	−0.153413
$792$	0	0
$793$	−3.53081e6	−0.199385
$794$	4.58610e7	2.58162
$795$	0	0
$796$	−2.54546e7	−1.42392
$797$	1.83089e7	1.02098	0.510490	−	0.859884i	$-0.329464\pi$
$797$	1.83089e7	1.02098	0.510490	+	0.859884i	$0.329464\pi$
$798$	0	0
$799$	1.80539e7	1.00047
$800$	−5.62295e6	−0.310627
$801$	0	0
$802$	−2.45632e7	−1.34849
$803$	8.18112e6	0.447738
$804$	0	0
$805$	−1.05126e6	−0.0571770
$806$	1.60456e6	0.0870000
$807$	0	0
$808$	760618.	0.0409862
$809$	2.60802e7	1.40100	0.700502	−	0.713651i	$-0.252960\pi$
$809$	2.60802e7	1.40100	0.700502	+	0.713651i	$0.252960\pi$
$810$	0	0
$811$	−2.26608e7	−1.20983	−0.604915	−	0.796290i	$-0.706793\pi$
$811$	−2.26608e7	−1.20983	−0.604915	+	0.796290i	$0.706793\pi$
$812$	−7.08055e6	−0.376858
$813$	0	0
$814$	1.64011e7	0.867586
$815$	−1.27107e7	−0.670311
$816$	0	0
$817$	−1.29801e7	−0.680336
$818$	1.66427e7	0.869641
$819$	0	0
$820$	1.03298e7	0.536487
$821$	2.86215e7	1.48195	0.740976	−	0.671532i	$-0.234363\pi$
$821$	2.86215e7	1.48195	0.740976	+	0.671532i	$0.234363\pi$
$822$	0	0
$823$	2.21929e7	1.14213	0.571063	−	0.820907i	$-0.306531\pi$
$823$	2.21929e7	1.14213	0.571063	+	0.820907i	$0.306531\pi$
$824$	1.13845e6	0.0584113
$825$	0	0
$826$	−1.57675e7	−0.804107
$827$	7.24598e6	0.368412	0.184206	−	0.982888i	$-0.441029\pi$
$827$	7.24598e6	0.368412	0.184206	+	0.982888i	$0.441029\pi$
$828$	0	0
$829$	1.85123e7	0.935563	0.467781	−	0.883844i	$-0.345053\pi$
$829$	1.85123e7	0.935563	0.467781	+	0.883844i	$0.345053\pi$
$830$	−3.27147e7	−1.64834
$831$	0	0
$832$	5.91362e6	0.296173
$833$	3.15608e6	0.157592
$834$	0	0
$835$	−128698.	−0.00638787
$836$	7.04804e6	0.348781
$837$	0	0
$838$	1.59858e7	0.786366
$839$	−2.29993e7	−1.12800	−0.564002	−	0.825774i	$-0.690739\pi$
$839$	−2.29993e7	−1.12800	−0.564002	+	0.825774i	$0.690739\pi$
$840$	0	0
$841$	−1.45922e6	−0.0711427
$842$	−3.46867e7	−1.68610
$843$	0	0
$844$	2.45559e7	1.18659
$845$	1.76209e6	0.0848958
$846$	0	0
$847$	6.26243e6	0.299940
$848$	−2.10325e7	−1.00439
$849$	0	0
$850$	7.22668e6	0.343077
$851$	3.87661e6	0.183497
$852$	0	0
$853$	8.15515e6	0.383760	0.191880	−	0.981418i	$-0.438542\pi$
$853$	8.15515e6	0.383760	0.191880	+	0.981418i	$0.438542\pi$
$854$	−8.26026e6	−0.387569
$855$	0	0
$856$	−786207.	−0.0366735
$857$	1.51571e7	0.704960	0.352480	−	0.935819i	$-0.385339\pi$
$857$	1.51571e7	0.704960	0.352480	+	0.935819i	$0.385339\pi$
$858$	0	0
$859$	−9.65204e6	−0.446309	−0.223155	−	0.974783i	$-0.571636\pi$
$859$	−9.65204e6	−0.446309	−0.223155	+	0.974783i	$0.571636\pi$
$860$	−2.27060e7	−1.04687
$861$	0	0
$862$	−6.01308e7	−2.75631
$863$	−2.69217e7	−1.23048	−0.615242	−	0.788338i	$-0.710942\pi$
$863$	−2.69217e7	−1.23048	−0.615242	+	0.788338i	$0.710942\pi$
$864$	0	0
$865$	−2.05537e7	−0.934006
$866$	1.58741e7	0.719275
$867$	0	0
$868$	1.90879e6	0.0859923
$869$	4.75688e6	0.213685
$870$	0	0
$871$	−870575.	−0.0388831
$872$	674915.	0.0300578
$873$	0	0
$874$	3.27615e6	0.145073
$875$	7.38735e6	0.326188
$876$	0	0
$877$	4.06168e7	1.78323	0.891614	−	0.452797i	$-0.149574\pi$
$877$	4.06168e7	1.78323	0.891614	+	0.452797i	$0.149574\pi$
$878$	−4.79461e7	−2.09902
$879$	0	0
$880$	−1.11076e7	−0.483517
$881$	−1.93921e7	−0.841756	−0.420878	−	0.907117i	$-0.638278\pi$
$881$	−1.93921e7	−0.841756	−0.420878	+	0.907117i	$0.638278\pi$
$882$	0	0
$883$	−3.08097e7	−1.32980	−0.664899	−	0.746933i	$-0.731526\pi$
$883$	−3.08097e7	−1.32980	−0.664899	+	0.746933i	$0.731526\pi$
$884$	−7.35435e6	−0.316529
$885$	0	0
$886$	2.28714e6	0.0978834
$887$	−1.22600e7	−0.523216	−0.261608	−	0.965174i	$-0.584253\pi$
$887$	−1.22600e7	−0.523216	−0.261608	+	0.965174i	$0.584253\pi$
$888$	0	0
$889$	2.77214e6	0.117642
$890$	−3.13553e7	−1.32689
$891$	0	0
$892$	2.98712e6	0.125702
$893$	1.60366e7	0.672949
$894$	0	0
$895$	3.48817e7	1.45559
$896$	894708.	0.0372316
$897$	0	0
$898$	1.98969e7	0.823367
$899$	−5.13607e6	−0.211949
$900$	0	0
$901$	2.79997e7	1.14906
$902$	7.44078e6	0.304511
$903$	0	0
$904$	491499.	0.0200033
$905$	3.28335e7	1.33259
$906$	0	0
$907$	−1.09928e7	−0.443699	−0.221850	−	0.975081i	$-0.571209\pi$
$907$	−1.09928e7	−0.443699	−0.221850	+	0.975081i	$0.571209\pi$
$908$	4.66613e7	1.87820
$909$	0	0
$910$	4.12237e6	0.165023
$911$	3.20528e6	0.127959	0.0639794	−	0.997951i	$-0.479621\pi$
$911$	3.20528e6	0.127959	0.0639794	+	0.997951i	$0.479621\pi$
$912$	0	0
$913$	−1.19826e7	−0.475745
$914$	1.93637e7	0.766693
$915$	0	0
$916$	−4.80554e6	−0.189236
$917$	1.10276e7	0.433069
$918$	0	0
$919$	4.76057e7	1.85939	0.929695	−	0.368330i	$-0.120070\pi$
$919$	4.76057e7	1.85939	0.929695	+	0.368330i	$0.120070\pi$
$920$	191395.	0.00745521
$921$	0	0
$922$	−1.94284e7	−0.752679
$923$	1.33114e7	0.514303
$924$	0	0
$925$	7.59568e6	0.291885
$926$	−2.31479e7	−0.887124
$927$	0	0
$928$	−3.60213e7	−1.37306
$929$	4.60688e7	1.75133	0.875664	−	0.482920i	$-0.160424\pi$
$929$	4.60688e7	1.75133	0.875664	+	0.482920i	$0.160424\pi$
$930$	0	0
$931$	2.80341e6	0.106002
$932$	2.97933e7	1.12351
$933$	0	0
$934$	−4.94584e7	−1.85512
$935$	1.47871e7	0.553163
$936$	0	0
$937$	5.21102e7	1.93898	0.969491	−	0.245126i	$-0.0788294\pi$
$937$	5.21102e7	1.93898	0.969491	+	0.245126i	$0.0788294\pi$
$938$	−2.03669e6	−0.0755819
$939$	0	0
$940$	2.80526e7	1.03551
$941$	3.10171e6	0.114190	0.0570949	−	0.998369i	$-0.481816\pi$
$941$	3.10171e6	0.114190	0.0570949	+	0.998369i	$0.481816\pi$
$942$	0	0
$943$	1.75872e6	0.0644048
$944$	−3.93777e7	−1.43820
$945$	0	0
$946$	−1.63555e7	−0.594206
$947$	−1.39181e7	−0.504319	−0.252159	−	0.967686i	$-0.581141\pi$
$947$	−1.39181e7	−0.504319	−0.252159	+	0.967686i	$0.581141\pi$
$948$	0	0
$949$	−7.58276e6	−0.273314
$950$	6.41916e6	0.230765
$951$	0	0
$952$	−574601.	−0.0205482
$953$	−2.54423e7	−0.907451	−0.453726	−	0.891141i	$-0.649905\pi$
$953$	−2.54423e7	−0.907451	−0.453726	+	0.891141i	$0.649905\pi$
$954$	0	0
$955$	−4.28055e7	−1.51877
$956$	4.27200e7	1.51177
$957$	0	0
$958$	−5.17805e7	−1.82286
$959$	4.36570e6	0.153288
$960$	0	0
$961$	−2.72446e7	−0.951637
$962$	−1.52016e7	−0.529603
$963$	0	0
$964$	−2.68349e7	−0.930052
$965$	4.04454e7	1.39814
$966$	0	0
$967$	1.18091e7	0.406115	0.203057	−	0.979167i	$-0.434912\pi$
$967$	1.18091e7	0.406115	0.203057	+	0.979167i	$0.434912\pi$
$968$	−1.14015e6	−0.0391087
$969$	0	0
$970$	6.79630e7	2.31923
$971$	7.94677e6	0.270485	0.135242	−	0.990813i	$-0.456819\pi$
$971$	7.94677e6	0.270485	0.135242	+	0.990813i	$0.456819\pi$
$972$	0	0
$973$	−1.19773e7	−0.405581
$974$	7.92223e7	2.67578
$975$	0	0
$976$	−2.06291e7	−0.693196
$977$	3.82730e7	1.28279	0.641395	−	0.767211i	$-0.278356\pi$
$977$	3.82730e7	1.28279	0.641395	+	0.767211i	$0.278356\pi$
$978$	0	0
$979$	−1.14847e7	−0.382968
$980$	4.90398e6	0.163111
$981$	0	0
$982$	−3.28389e7	−1.08670
$983$	−2.29567e7	−0.757750	−0.378875	−	0.925448i	$-0.623689\pi$
$983$	−2.29567e7	−0.757750	−0.378875	+	0.925448i	$0.623689\pi$
$984$	0	0
$985$	−2.74207e7	−0.900508
$986$	4.62950e7	1.51650
$987$	0	0
$988$	−6.53256e6	−0.212908
$989$	−3.86584e6	−0.125676
$990$	0	0
$991$	−3.86910e7	−1.25149	−0.625743	−	0.780030i	$-0.715204\pi$
$991$	−3.86910e7	−1.25149	−0.625743	+	0.780030i	$0.715204\pi$
$992$	9.71072e6	0.313309
$993$	0	0
$994$	3.11417e7	0.999714
$995$	−4.74373e7	−1.51902
$996$	0	0
$997$	−5.14906e7	−1.64055	−0.820276	−	0.571968i	$-0.806180\pi$
$997$	−5.14906e7	−1.64055	−0.820276	+	0.571968i	$0.806180\pi$
$998$	−4.26024e7	−1.35397
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.6.a.c.1.5		6
3.2	odd	2		273.6.a.d.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.6.a.d.1.2	✓	6	3.2	odd	2
819.6.a.c.1.5		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	819.a (trivial)

\(f(q)\)	\(=\)	\(q+8.06881 q^{2} +33.1056 q^{4} +61.6957 q^{5} -49.0000 q^{7} +8.92103 q^{8} +O(q^{10})\)	\(q+8.06881 q^{2} +33.1056 q^{4} +61.6957 q^{5} -49.0000 q^{7} +8.92103 q^{8} +497.810 q^{10} +182.336 q^{11} -169.000 q^{13} -395.371 q^{14} -987.398 q^{16} +1314.48 q^{17} +1167.60 q^{19} +2042.47 q^{20} +1471.23 q^{22} +347.744 q^{23} +681.356 q^{25} -1363.63 q^{26} -1622.18 q^{28} +4364.85 q^{29} -1176.69 q^{31} -8252.59 q^{32} +10606.3 q^{34} -3023.09 q^{35} +11147.9 q^{37} +9421.16 q^{38} +550.389 q^{40} +5057.52 q^{41} -11116.9 q^{43} +6036.34 q^{44} +2805.88 q^{46} +13734.6 q^{47} +2401.00 q^{49} +5497.73 q^{50} -5594.85 q^{52} +21300.9 q^{53} +11249.3 q^{55} -437.130 q^{56} +35219.1 q^{58} +39880.3 q^{59} +20892.4 q^{61} -9494.46 q^{62} -34991.8 q^{64} -10426.6 q^{65} +5151.33 q^{67} +43516.8 q^{68} -24392.7 q^{70} -78765.6 q^{71} +44868.4 q^{73} +89950.1 q^{74} +38654.2 q^{76} -8934.45 q^{77} +26088.6 q^{79} -60918.2 q^{80} +40808.1 q^{82} -65717.2 q^{83} +81098.0 q^{85} -89700.1 q^{86} +1626.62 q^{88} -62986.5 q^{89} +8281.00 q^{91} +11512.3 q^{92} +110822. q^{94} +72036.0 q^{95} +136524. q^{97} +19373.2 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{2} + 109 q^{4} - 112 q^{5} - 294 q^{7} - 339 q^{8}+O(q^{10}) \) 6 * q + 5 * q^2 + 109 * q^4 - 112 * q^5 - 294 * q^7 - 339 * q^8	\( 6 q + 5 q^{2} + 109 q^{4} - 112 q^{5} - 294 q^{7} - 339 q^{8} - 417 q^{10} - 360 q^{11} - 1014 q^{13} - 245 q^{14} + 1353 q^{16} - 2708 q^{17} + 6892 q^{19} - 3261 q^{20} - 33 q^{22} - 9744 q^{23} + 2526 q^{25} - 845 q^{26} - 5341 q^{28} + 70 q^{29} - 130 q^{31} - 31303 q^{32} + 3917 q^{34} + 5488 q^{35} + 11512 q^{37} - 17799 q^{38} + 19339 q^{40} + 912 q^{41} - 34918 q^{43} - 6503 q^{44} + 19333 q^{46} - 8594 q^{47} + 14406 q^{49} + 83560 q^{50} - 18421 q^{52} + 10102 q^{53} - 14552 q^{55} + 16611 q^{56} - 8989 q^{58} + 43260 q^{59} + 137538 q^{61} + 19690 q^{62} + 16129 q^{64} + 18928 q^{65} + 8280 q^{67} + 42295 q^{68} + 20433 q^{70} - 62380 q^{71} + 76670 q^{73} - 10387 q^{74} + 287197 q^{76} + 17640 q^{77} + 21154 q^{79} - 182689 q^{80} + 223876 q^{82} - 223186 q^{83} + 277084 q^{85} + 64747 q^{86} + 126093 q^{88} - 235190 q^{89} + 49686 q^{91} + 121233 q^{92} + 8238 q^{94} - 113200 q^{95} + 108602 q^{97} + 12005 q^{98}+O(q^{100}) \) 6 * q + 5 * q^2 + 109 * q^4 - 112 * q^5 - 294 * q^7 - 339 * q^8 - 417 * q^10 - 360 * q^11 - 1014 * q^13 - 245 * q^14 + 1353 * q^16 - 2708 * q^17 + 6892 * q^19 - 3261 * q^20 - 33 * q^22 - 9744 * q^23 + 2526 * q^25 - 845 * q^26 - 5341 * q^28 + 70 * q^29 - 130 * q^31 - 31303 * q^32 + 3917 * q^34 + 5488 * q^35 + 11512 * q^37 - 17799 * q^38 + 19339 * q^40 + 912 * q^41 - 34918 * q^43 - 6503 * q^44 + 19333 * q^46 - 8594 * q^47 + 14406 * q^49 + 83560 * q^50 - 18421 * q^52 + 10102 * q^53 - 14552 * q^55 + 16611 * q^56 - 8989 * q^58 + 43260 * q^59 + 137538 * q^61 + 19690 * q^62 + 16129 * q^64 + 18928 * q^65 + 8280 * q^67 + 42295 * q^68 + 20433 * q^70 - 62380 * q^71 + 76670 * q^73 - 10387 * q^74 + 287197 * q^76 + 17640 * q^77 + 21154 * q^79 - 182689 * q^80 + 223876 * q^82 - 223186 * q^83 + 277084 * q^85 + 64747 * q^86 + 126093 * q^88 - 235190 * q^89 + 49686 * q^91 + 121233 * q^92 + 8238 * q^94 - 113200 * q^95 + 108602 * q^97 + 12005 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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