LMFDB - Embedded newform 819.6.a.f.1.1

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:	$1$
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} - 3x^{5} - 111x^{4} + 75x^{3} + 2750x^{2} + 1800x - 1600 $ x^6 - 3x^5 - 111x^4 + 75x^3 + 2750x^2 + 1800*x - 1600
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			1.1
Root			$10.3949$ 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)

$f(q)$	$=$	$q-7.39489 q^{2} +22.6844 q^{4} +8.21614 q^{5} -49.0000 q^{7} +68.8876 q^{8} +O(q^{10})$	\(q-7.39489 q^{2} +22.6844 q^{4} +8.21614 q^{5} -49.0000 q^{7} +68.8876 q^{8} -60.7574 q^{10} +573.264 q^{11} +169.000 q^{13} +362.350 q^{14} -1235.32 q^{16} +352.618 q^{17} -464.700 q^{19} +186.378 q^{20} -4239.22 q^{22} +631.590 q^{23} -3057.50 q^{25} -1249.74 q^{26} -1111.54 q^{28} -489.148 q^{29} -1748.90 q^{31} +6930.64 q^{32} -2607.58 q^{34} -402.591 q^{35} -1720.54 q^{37} +3436.41 q^{38} +565.990 q^{40} -11070.5 q^{41} -6800.23 q^{43} +13004.2 q^{44} -4670.54 q^{46} +8351.44 q^{47} +2401.00 q^{49} +22609.8 q^{50} +3833.67 q^{52} +7308.55 q^{53} +4710.01 q^{55} -3375.49 q^{56} +3617.20 q^{58} -645.594 q^{59} +21907.1 q^{61} +12933.0 q^{62} -11721.2 q^{64} +1388.53 q^{65} -21873.0 q^{67} +7998.95 q^{68} +2977.11 q^{70} -39595.3 q^{71} -80110.8 q^{73} +12723.2 q^{74} -10541.5 q^{76} -28089.9 q^{77} +18126.6 q^{79} -10149.5 q^{80} +81865.4 q^{82} -15653.7 q^{83} +2897.16 q^{85} +50286.9 q^{86} +39490.7 q^{88} -10576.5 q^{89} -8281.00 q^{91} +14327.3 q^{92} -61758.0 q^{94} -3818.04 q^{95} +105737. q^{97} -17755.1 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 15 q^{2} + 75 q^{4} - 15 q^{5} - 294 q^{7} + 399 q^{8}+O(q^{10}) $ 6 * q + 15 * q^2 + 75 * q^4 - 15 * q^5 - 294 * q^7 + 399 * q^8	$ 6 q + 15 q^{2} + 75 q^{4} - 15 q^{5} - 294 q^{7} + 399 q^{8} - 265 q^{10} + 90 q^{11} + 1014 q^{13} - 735 q^{14} + 355 q^{16} + 1314 q^{17} - 341 q^{19} - 2085 q^{20} - 6749 q^{22} + 4905 q^{23} - 10231 q^{25} + 2535 q^{26} - 3675 q^{28} + 6771 q^{29} - 14711 q^{31} + 10143 q^{32} + 9595 q^{34} + 735 q^{35} - 13314 q^{37} - 1017 q^{38} - 13195 q^{40} + 5040 q^{41} - 9127 q^{43} - 10041 q^{44} + 23739 q^{46} + 4713 q^{47} + 14406 q^{49} - 18018 q^{50} + 12675 q^{52} + 159 q^{53} - 13590 q^{55} - 19551 q^{56} - 38427 q^{58} - 30288 q^{59} - 78126 q^{61} - 40632 q^{62} - 8461 q^{64} - 2535 q^{65} - 33894 q^{67} + 61863 q^{68} + 12985 q^{70} - 8316 q^{71} - 89861 q^{73} + 16635 q^{74} - 169069 q^{76} - 4410 q^{77} - 239803 q^{79} - 19305 q^{80} - 106624 q^{82} + 42753 q^{83} - 184316 q^{85} - 51609 q^{86} - 205923 q^{88} + 11421 q^{89} - 49686 q^{91} + 252519 q^{92} - 4976 q^{94} + 180807 q^{95} - 199445 q^{97} + 36015 q^{98}+O(q^{100}) $ 6 * q + 15 * q^2 + 75 * q^4 - 15 * q^5 - 294 * q^7 + 399 * q^8 - 265 * q^10 + 90 * q^11 + 1014 * q^13 - 735 * q^14 + 355 * q^16 + 1314 * q^17 - 341 * q^19 - 2085 * q^20 - 6749 * q^22 + 4905 * q^23 - 10231 * q^25 + 2535 * q^26 - 3675 * q^28 + 6771 * q^29 - 14711 * q^31 + 10143 * q^32 + 9595 * q^34 + 735 * q^35 - 13314 * q^37 - 1017 * q^38 - 13195 * q^40 + 5040 * q^41 - 9127 * q^43 - 10041 * q^44 + 23739 * q^46 + 4713 * q^47 + 14406 * q^49 - 18018 * q^50 + 12675 * q^52 + 159 * q^53 - 13590 * q^55 - 19551 * q^56 - 38427 * q^58 - 30288 * q^59 - 78126 * q^61 - 40632 * q^62 - 8461 * q^64 - 2535 * q^65 - 33894 * q^67 + 61863 * q^68 + 12985 * q^70 - 8316 * q^71 - 89861 * q^73 + 16635 * q^74 - 169069 * q^76 - 4410 * q^77 - 239803 * q^79 - 19305 * q^80 - 106624 * q^82 + 42753 * q^83 - 184316 * q^85 - 51609 * q^86 - 205923 * q^88 + 11421 * q^89 - 49686 * q^91 + 252519 * q^92 - 4976 * q^94 + 180807 * q^95 - 199445 * q^97 + 36015 * 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 $

$2$	−7.39489	−1.30724	−0.653622	−	0.756821i	$-0.726752\pi$
$2$	−7.39489	−1.30724	−0.653622	+	0.756821i	$0.726752\pi$
$3$	0	0
$3$	0	0
$4$	22.6844	0.708889
$5$	8.21614	0.146975	0.0734874	−	0.997296i	$-0.476587\pi$
$5$	8.21614	0.146975	0.0734874	+	0.997296i	$0.476587\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	68.8876	0.380554
$9$	0	0
$10$	−60.7574	−0.192132
$11$	573.264	1.42847	0.714237	−	0.699903i	$-0.246774\pi$
$11$	573.264	1.42847	0.714237	+	0.699903i	$0.246774\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	362.350	0.494092
$15$	0	0
$16$	−1235.32	−1.20637
$17$	352.618	0.295926	0.147963	−	0.988993i	$-0.452728\pi$
$17$	352.618	0.295926	0.147963	+	0.988993i	$0.452728\pi$
$18$	0	0
$19$	−464.700	−0.295317	−0.147659	−	0.989038i	$-0.547174\pi$
$19$	−464.700	−0.295317	−0.147659	+	0.989038i	$0.547174\pi$
$20$	186.378	0.104189
$21$	0	0
$22$	−4239.22	−1.86737
$23$	631.590	0.248952	0.124476	−	0.992223i	$-0.460275\pi$
$23$	631.590	0.248952	0.124476	+	0.992223i	$0.460275\pi$
$24$	0	0
$25$	−3057.50	−0.978398
$26$	−1249.74	−0.362564
$27$	0	0
$28$	−1111.54	−0.267935
$29$	−489.148	−0.108005	−0.0540027	−	0.998541i	$-0.517198\pi$
$29$	−489.148	−0.108005	−0.0540027	+	0.998541i	$0.517198\pi$
$30$	0	0
$31$	−1748.90	−0.326860	−0.163430	−	0.986555i	$-0.552256\pi$
$31$	−1748.90	−0.326860	−0.163430	+	0.986555i	$0.552256\pi$
$32$	6930.64	1.19646
$33$	0	0
$34$	−2607.58	−0.386847
$35$	−402.591	−0.0555512
$36$	0	0
$37$	−1720.54	−0.206614	−0.103307	−	0.994650i	$-0.532942\pi$
$37$	−1720.54	−0.206614	−0.103307	+	0.994650i	$0.532942\pi$
$38$	3436.41	0.386052
$39$	0	0
$40$	565.990	0.0559318
$41$	−11070.5	−1.02851	−0.514255	−	0.857637i	$-0.671932\pi$
$41$	−11070.5	−1.02851	−0.514255	+	0.857637i	$0.671932\pi$
$42$	0	0
$43$	−6800.23	−0.560857	−0.280429	−	0.959875i	$-0.590477\pi$
$43$	−6800.23	−0.560857	−0.280429	+	0.959875i	$0.590477\pi$
$44$	13004.2	1.01263
$45$	0	0
$46$	−4670.54	−0.325441
$47$	8351.44	0.551463	0.275732	−	0.961235i	$-0.411080\pi$
$47$	8351.44	0.551463	0.275732	+	0.961235i	$0.411080\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	22609.8	1.27901
$51$	0	0
$52$	3833.67	0.196610
$53$	7308.55	0.357389	0.178695	−	0.983905i	$-0.442813\pi$
$53$	7308.55	0.357389	0.178695	+	0.983905i	$0.442813\pi$
$54$	0	0
$55$	4710.01	0.209950
$56$	−3375.49	−0.143836
$57$	0	0
$58$	3617.20	0.141190
$59$	−645.594	−0.0241451	−0.0120726	−	0.999927i	$-0.503843\pi$
$59$	−645.594	−0.0241451	−0.0120726	+	0.999927i	$0.503843\pi$
$60$	0	0
$61$	21907.1	0.753807	0.376904	−	0.926253i	$-0.376989\pi$
$61$	21907.1	0.753807	0.376904	+	0.926253i	$0.376989\pi$
$62$	12933.0	0.427286
$63$	0	0
$64$	−11721.2	−0.357702
$65$	1388.53	0.0407635
$66$	0	0
$67$	−21873.0	−0.595279	−0.297640	−	0.954678i	$-0.596199\pi$
$67$	−21873.0	−0.595279	−0.297640	+	0.954678i	$0.596199\pi$
$68$	7998.95	0.209778
$69$	0	0
$70$	2977.11	0.0726190
$71$	−39595.3	−0.932176	−0.466088	−	0.884738i	$-0.654337\pi$
$71$	−39595.3	−0.932176	−0.466088	+	0.884738i	$0.654337\pi$
$72$	0	0
$73$	−80110.8	−1.75948	−0.879739	−	0.475456i	$-0.842283\pi$
$73$	−80110.8	−1.75948	−0.879739	+	0.475456i	$0.842283\pi$
$74$	12723.2	0.270095
$75$	0	0
$76$	−10541.5	−0.209347
$77$	−28089.9	−0.539913
$78$	0	0
$79$	18126.6	0.326775	0.163387	−	0.986562i	$-0.447758\pi$
$79$	18126.6	0.326775	0.163387	+	0.986562i	$0.447758\pi$
$80$	−10149.5	−0.177305
$81$	0	0
$82$	81865.4	1.34452
$83$	−15653.7	−0.249414	−0.124707	−	0.992194i	$-0.539799\pi$
$83$	−15653.7	−0.249414	−0.124707	+	0.992194i	$0.539799\pi$
$84$	0	0
$85$	2897.16	0.0434936
$86$	50286.9	0.733178
$87$	0	0
$88$	39490.7	0.543611
$89$	−10576.5	−0.141535	−0.0707677	−	0.997493i	$-0.522545\pi$
$89$	−10576.5	−0.141535	−0.0707677	+	0.997493i	$0.522545\pi$
$90$	0	0
$91$	−8281.00	−0.104828
$92$	14327.3	0.176479
$93$	0	0
$94$	−61758.0	−0.720897
$95$	−3818.04	−0.0434041
$96$	0	0
$97$	105737.	1.14103	0.570514	−	0.821288i	$-0.306744\pi$
$97$	105737.	1.14103	0.570514	+	0.821288i	$0.306744\pi$
$98$	−17755.1	−0.186749
$99$	0	0
$100$	−69357.6	−0.693576
$101$	−8998.92	−0.0877782	−0.0438891	−	0.999036i	$-0.513975\pi$
$101$	−8998.92	−0.0877782	−0.0438891	+	0.999036i	$0.513975\pi$
$102$	0	0
$103$	−125965.	−1.16992	−0.584959	−	0.811063i	$-0.698889\pi$
$103$	−125965.	−1.16992	−0.584959	+	0.811063i	$0.698889\pi$
$104$	11642.0	0.105547
$105$	0	0
$106$	−54046.0	−0.467195
$107$	89601.5	0.756581	0.378291	−	0.925687i	$-0.376512\pi$
$107$	89601.5	0.756581	0.378291	+	0.925687i	$0.376512\pi$
$108$	0	0
$109$	223246.	1.79977	0.899884	−	0.436129i	$-0.143651\pi$
$109$	223246.	1.79977	0.899884	+	0.436129i	$0.143651\pi$
$110$	−34830.0	−0.274456
$111$	0	0
$112$	60530.6	0.455963
$113$	109418.	0.806107	0.403053	−	0.915176i	$-0.367949\pi$
$113$	109418.	0.806107	0.403053	+	0.915176i	$0.367949\pi$
$114$	0	0
$115$	5189.23	0.0365897
$116$	−11096.1	−0.0765638
$117$	0	0
$118$	4774.10	0.0315636
$119$	−17278.3	−0.111849
$120$	0	0
$121$	167580.	1.04054
$122$	−162001.	−0.985410
$123$	0	0
$124$	−39672.9	−0.231707
$125$	−50796.2	−0.290775
$126$	0	0
$127$	−70008.0	−0.385157	−0.192579	−	0.981282i	$-0.561685\pi$
$127$	−70008.0	−0.385157	−0.192579	+	0.981282i	$0.561685\pi$
$128$	−135104.	−0.728857
$129$	0	0
$130$	−10268.0	−0.0532878
$131$	296866.	1.51141	0.755705	−	0.654912i	$-0.227294\pi$
$131$	296866.	1.51141	0.755705	+	0.654912i	$0.227294\pi$
$132$	0	0
$133$	22770.3	0.111619
$134$	161748.	0.778175
$135$	0	0
$136$	24291.0	0.112616
$137$	−319029.	−1.45221	−0.726104	−	0.687585i	$-0.758671\pi$
$137$	−319029.	−1.45221	−0.726104	+	0.687585i	$0.758671\pi$
$138$	0	0
$139$	94833.7	0.416319	0.208159	−	0.978095i	$-0.433253\pi$
$139$	94833.7	0.416319	0.208159	+	0.978095i	$0.433253\pi$
$140$	−9132.54	−0.0393796
$141$	0	0
$142$	292803.	1.21858
$143$	96881.5	0.396188
$144$	0	0
$145$	−4018.91	−0.0158741
$146$	592411.	2.30007
$147$	0	0
$148$	−39029.5	−0.146466
$149$	232017.	0.856160	0.428080	−	0.903741i	$-0.359190\pi$
$149$	232017.	0.856160	0.428080	+	0.903741i	$0.359190\pi$
$150$	0	0
$151$	1030.72	0.00367874	0.00183937	−	0.999998i	$-0.499415\pi$
$151$	1030.72	0.00367874	0.00183937	+	0.999998i	$0.499415\pi$
$152$	−32012.1	−0.112384
$153$	0	0
$154$	207722.	0.705798
$155$	−14369.2	−0.0480402
$156$	0	0
$157$	159473.	0.516343	0.258171	−	0.966099i	$-0.416880\pi$
$157$	159473.	0.516343	0.258171	+	0.966099i	$0.416880\pi$
$158$	−134044.	−0.427175
$159$	0	0
$160$	56943.1	0.175850
$161$	−30947.9	−0.0940951
$162$	0	0
$163$	−97146.4	−0.286390	−0.143195	−	0.989695i	$-0.545738\pi$
$163$	−97146.4	−0.286390	−0.143195	+	0.989695i	$0.545738\pi$
$164$	−251129.	−0.729100
$165$	0	0
$166$	115757.	0.326045
$167$	118994.	0.330168	0.165084	−	0.986279i	$-0.447210\pi$
$167$	118994.	0.330168	0.165084	+	0.986279i	$0.447210\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	−21424.2	−0.0568568
$171$	0	0
$172$	−154259.	−0.397585
$173$	106218.	0.269825	0.134912	−	0.990858i	$-0.456925\pi$
$173$	106218.	0.269825	0.134912	+	0.990858i	$0.456925\pi$
$174$	0	0
$175$	149817.	0.369800
$176$	−708163.	−1.72326
$177$	0	0
$178$	78211.7	0.185021
$179$	−367158.	−0.856487	−0.428243	−	0.903663i	$-0.640867\pi$
$179$	−367158.	−0.856487	−0.428243	+	0.903663i	$0.640867\pi$
$180$	0	0
$181$	−227080.	−0.515207	−0.257604	−	0.966251i	$-0.582933\pi$
$181$	−227080.	−0.515207	−0.257604	+	0.966251i	$0.582933\pi$
$182$	61237.1	0.137036
$183$	0	0
$184$	43508.7	0.0947397
$185$	−14136.2	−0.0303671
$186$	0	0
$187$	202143.	0.422723
$188$	189448.	0.390926
$189$	0	0
$190$	28234.0	0.0567398
$191$	−58951.7	−0.116927	−0.0584633	−	0.998290i	$-0.518620\pi$
$191$	−58951.7	−0.116927	−0.0584633	+	0.998290i	$0.518620\pi$
$192$	0	0
$193$	−43458.4	−0.0839808	−0.0419904	−	0.999118i	$-0.513370\pi$
$193$	−43458.4	−0.0839808	−0.0419904	+	0.999118i	$0.513370\pi$
$194$	−781911.	−1.49160
$195$	0	0
$196$	54465.3	0.101270
$197$	−52089.4	−0.0956278	−0.0478139	−	0.998856i	$-0.515225\pi$
$197$	−52089.4	−0.0956278	−0.0478139	+	0.998856i	$0.515225\pi$
$198$	0	0
$199$	−602708.	−1.07888	−0.539441	−	0.842023i	$-0.681365\pi$
$199$	−602708.	−1.07888	−0.539441	+	0.842023i	$0.681365\pi$
$200$	−210623.	−0.372333
$201$	0	0
$202$	66546.0	0.114748
$203$	23968.3	0.0408222
$204$	0	0
$205$	−90957.0	−0.151165
$206$	931494.	1.52937
$207$	0	0
$208$	−208769.	−0.334586
$209$	−266396.	−0.421853
$210$	0	0
$211$	−448430.	−0.693407	−0.346704	−	0.937975i	$-0.612699\pi$
$211$	−448430.	−0.693407	−0.346704	+	0.937975i	$0.612699\pi$
$212$	165790.	0.253349
$213$	0	0
$214$	−662593.	−0.989037
$215$	−55871.6	−0.0824318
$216$	0	0
$217$	85696.3	0.123542
$218$	−1.65088e6	−2.35274
$219$	0	0
$220$	106844.	0.148831
$221$	59592.5	0.0820750
$222$	0	0
$223$	759118.	1.02223	0.511114	−	0.859513i	$-0.329233\pi$
$223$	759118.	1.02223	0.511114	+	0.859513i	$0.329233\pi$
$224$	−339602.	−0.452220
$225$	0	0
$226$	−809134.	−1.05378
$227$	−132014.	−0.170042	−0.0850209	−	0.996379i	$-0.527096\pi$
$227$	−132014.	−0.170042	−0.0850209	+	0.996379i	$0.527096\pi$
$228$	0	0
$229$	−210032.	−0.264665	−0.132333	−	0.991205i	$-0.542247\pi$
$229$	−210032.	−0.264665	−0.132333	+	0.991205i	$0.542247\pi$
$230$	−38373.8	−0.0478316
$231$	0	0
$232$	−33696.3	−0.0411019
$233$	108465.	0.130888	0.0654438	−	0.997856i	$-0.479154\pi$
$233$	108465.	0.130888	0.0654438	+	0.997856i	$0.479154\pi$
$234$	0	0
$235$	68616.5	0.0810512
$236$	−14644.9	−0.0171162
$237$	0	0
$238$	127771.	0.146215
$239$	−1.48449e6	−1.68106	−0.840531	−	0.541764i	$-0.817757\pi$
$239$	−1.48449e6	−1.68106	−0.840531	+	0.541764i	$0.817757\pi$
$240$	0	0
$241$	−493799.	−0.547655	−0.273828	−	0.961779i	$-0.588290\pi$
$241$	−493799.	−0.547655	−0.273828	+	0.961779i	$0.588290\pi$
$242$	−1.23924e6	−1.36024
$243$	0	0
$244$	496950.	0.534365
$245$	19726.9	0.0209964
$246$	0	0
$247$	−78534.3	−0.0819062
$248$	−120478.	−0.124388
$249$	0	0
$250$	375633.	0.380113
$251$	−619125.	−0.620289	−0.310145	−	0.950689i	$-0.600378\pi$
$251$	−619125.	−0.620289	−0.310145	+	0.950689i	$0.600378\pi$
$252$	0	0
$253$	362068.	0.355622
$254$	517701.	0.503495
$255$	0	0
$256$	1.37416e6	1.31050
$257$	−187402.	−0.176987	−0.0884934	−	0.996077i	$-0.528205\pi$
$257$	−187402.	−0.176987	−0.0884934	+	0.996077i	$0.528205\pi$
$258$	0	0
$259$	84306.4	0.0780928
$260$	31498.0	0.0288967
$261$	0	0
$262$	−2.19529e6	−1.97578
$263$	−1.17917e6	−1.05120	−0.525601	−	0.850731i	$-0.676160\pi$
$263$	−1.17917e6	−1.05120	−0.525601	+	0.850731i	$0.676160\pi$
$264$	0	0
$265$	60048.1	0.0525272
$266$	−168384.	−0.145914
$267$	0	0
$268$	−496176.	−0.421987
$269$	−396969.	−0.334485	−0.167242	−	0.985916i	$-0.553486\pi$
$269$	−396969.	−0.334485	−0.167242	+	0.985916i	$0.553486\pi$
$270$	0	0
$271$	610599.	0.505048	0.252524	−	0.967591i	$-0.418739\pi$
$271$	610599.	0.505048	0.252524	+	0.967591i	$0.418739\pi$
$272$	−435596.	−0.356995
$273$	0	0
$274$	2.35918e6	1.89839
$275$	−1.75275e6	−1.39762
$276$	0	0
$277$	221125.	0.173156	0.0865780	−	0.996245i	$-0.472407\pi$
$277$	221125.	0.173156	0.0865780	+	0.996245i	$0.472407\pi$
$278$	−701285.	−0.544230
$279$	0	0
$280$	−27733.5	−0.0211402
$281$	−2.24701e6	−1.69762	−0.848809	−	0.528700i	$-0.822680\pi$
$281$	−2.24701e6	−1.69762	−0.848809	+	0.528700i	$0.822680\pi$
$282$	0	0
$283$	−1.20289e6	−0.892809	−0.446405	−	0.894831i	$-0.647296\pi$
$283$	−1.20289e6	−0.892809	−0.446405	+	0.894831i	$0.647296\pi$
$284$	−898198.	−0.660809
$285$	0	0
$286$	−716429.	−0.517914
$287$	542456.	0.388741
$288$	0	0
$289$	−1.29552e6	−0.912428
$290$	29719.4	0.0207513
$291$	0	0
$292$	−1.81727e6	−1.24727
$293$	795089.	0.541062	0.270531	−	0.962711i	$-0.412801\pi$
$293$	795089.	0.541062	0.270531	+	0.962711i	$0.412801\pi$
$294$	0	0
$295$	−5304.29	−0.00354872
$296$	−118524.	−0.0786278
$297$	0	0
$298$	−1.71574e6	−1.11921
$299$	106739.	0.0690469
$300$	0	0
$301$	333211.	0.211984
$302$	−7622.07	−0.00480901
$303$	0	0
$304$	574052.	0.356260
$305$	179992.	0.110791
$306$	0	0
$307$	−1.68403e6	−1.01977	−0.509887	−	0.860242i	$-0.670313\pi$
$307$	−1.68403e6	−1.01977	−0.509887	+	0.860242i	$0.670313\pi$
$308$	−637204.	−0.382738
$309$	0	0
$310$	106259.	0.0628003
$311$	1.24972e6	0.732676	0.366338	−	0.930482i	$-0.380611\pi$
$311$	1.24972e6	0.732676	0.366338	+	0.930482i	$0.380611\pi$
$312$	0	0
$313$	−2.18821e6	−1.26249	−0.631247	−	0.775582i	$-0.717456\pi$
$313$	−2.18821e6	−1.26249	−0.631247	+	0.775582i	$0.717456\pi$
$314$	−1.17929e6	−0.674987
$315$	0	0
$316$	411192.	0.231647
$317$	−480686.	−0.268667	−0.134333	−	0.990936i	$-0.542889\pi$
$317$	−480686.	−0.268667	−0.134333	+	0.990936i	$0.542889\pi$
$318$	0	0
$319$	−280411.	−0.154283
$320$	−96302.8	−0.0525732
$321$	0	0
$322$	228857.	0.123005
$323$	−163862.	−0.0873919
$324$	0	0
$325$	−516717.	−0.271359
$326$	718387.	0.374382
$327$	0	0
$328$	−762622.	−0.391404
$329$	−409220.	−0.208434
$330$	0	0
$331$	−2.01991e6	−1.01336	−0.506679	−	0.862135i	$-0.669127\pi$
$331$	−2.01991e6	−1.01336	−0.506679	+	0.862135i	$0.669127\pi$
$332$	−355095.	−0.176807
$333$	0	0
$334$	−879951.	−0.431611
$335$	−179711.	−0.0874910
$336$	0	0
$337$	−2.90909e6	−1.39535	−0.697673	−	0.716417i	$-0.745781\pi$
$337$	−2.90909e6	−1.39535	−0.697673	+	0.716417i	$0.745781\pi$
$338$	−211206.	−0.100557
$339$	0	0
$340$	65720.5	0.0308321
$341$	−1.00258e6	−0.466911
$342$	0	0
$343$	−117649.	−0.0539949
$344$	−468451.	−0.213436
$345$	0	0
$346$	−785469.	−0.352727
$347$	2.87998e6	1.28400	0.642002	−	0.766703i	$-0.278104\pi$
$347$	2.87998e6	1.28400	0.642002	+	0.766703i	$0.278104\pi$
$348$	0	0
$349$	−3.02889e6	−1.33113	−0.665563	−	0.746341i	$-0.731809\pi$
$349$	−3.02889e6	−1.33113	−0.665563	+	0.746341i	$0.731809\pi$
$350$	−1.10788e6	−0.483419
$351$	0	0
$352$	3.97309e6	1.70911
$353$	4.67664e6	1.99755	0.998774	−	0.0495035i	$-0.0157639\pi$
$353$	4.67664e6	1.99755	0.998774	+	0.0495035i	$0.0157639\pi$
$354$	0	0
$355$	−325321.	−0.137006
$356$	−239921.	−0.100333
$357$	0	0
$358$	2.71510e6	1.11964
$359$	1.07135e6	0.438729	0.219365	−	0.975643i	$-0.429602\pi$
$359$	1.07135e6	0.438729	0.219365	+	0.975643i	$0.429602\pi$
$360$	0	0
$361$	−2.26015e6	−0.912788
$362$	1.67923e6	0.673502
$363$	0	0
$364$	−187850.	−0.0743117
$365$	−658201.	−0.258599
$366$	0	0
$367$	−3.60025e6	−1.39530	−0.697650	−	0.716438i	$-0.745771\pi$
$367$	−3.60025e6	−1.39530	−0.697650	+	0.716438i	$0.745771\pi$
$368$	−780215.	−0.300327
$369$	0	0
$370$	104536.	0.0396972
$371$	−358119.	−0.135080
$372$	0	0
$373$	−2.80634e6	−1.04440	−0.522202	−	0.852822i	$-0.674889\pi$
$373$	−2.80634e6	−1.04440	−0.522202	+	0.852822i	$0.674889\pi$
$374$	−1.49483e6	−0.552602
$375$	0	0
$376$	575310.	0.209861
$377$	−82666.1	−0.0299553
$378$	0	0
$379$	−2.38891e6	−0.854283	−0.427142	−	0.904185i	$-0.640479\pi$
$379$	−2.38891e6	−0.854283	−0.427142	+	0.904185i	$0.640479\pi$
$380$	−86610.0	−0.0307687
$381$	0	0
$382$	435942.	0.152852
$383$	2.36763e6	0.824741	0.412370	−	0.911016i	$-0.364701\pi$
$383$	2.36763e6	0.824741	0.412370	+	0.911016i	$0.364701\pi$
$384$	0	0
$385$	−230791.	−0.0793535
$386$	321370.	0.109784
$387$	0	0
$388$	2.39858e6	0.808861
$389$	1.84463e6	0.618068	0.309034	−	0.951051i	$-0.399994\pi$
$389$	1.84463e6	0.618068	0.309034	+	0.951051i	$0.399994\pi$
$390$	0	0
$391$	222710.	0.0736713
$392$	165399.	0.0543648
$393$	0	0
$394$	385196.	0.125009
$395$	148931.	0.0480276
$396$	0	0
$397$	−569524.	−0.181357	−0.0906787	−	0.995880i	$-0.528904\pi$
$397$	−569524.	−0.181357	−0.0906787	+	0.995880i	$0.528904\pi$
$398$	4.45696e6	1.41036
$399$	0	0
$400$	3.77698e6	1.18031
$401$	3.14837e6	0.977744	0.488872	−	0.872356i	$-0.337409\pi$
$401$	3.14837e6	0.977744	0.488872	+	0.872356i	$0.337409\pi$
$402$	0	0
$403$	−295565.	−0.0906547
$404$	−204135.	−0.0622250
$405$	0	0
$406$	−177243.	−0.0533646
$407$	−986322.	−0.295143
$408$	0	0
$409$	−6.50044e6	−1.92147	−0.960737	−	0.277461i	$-0.910507\pi$
$409$	−6.50044e6	−1.92147	−0.960737	+	0.277461i	$0.910507\pi$
$410$	672617.	0.197610
$411$	0	0
$412$	−2.85743e6	−0.829341
$413$	31634.1	0.00912600
$414$	0	0
$415$	−128613.	−0.0366576
$416$	1.17128e6	0.331839
$417$	0	0
$418$	1.96997e6	0.551465
$419$	−2.29479e6	−0.638570	−0.319285	−	0.947659i	$-0.603443\pi$
$419$	−2.29479e6	−0.638570	−0.319285	+	0.947659i	$0.603443\pi$
$420$	0	0
$421$	5.40450e6	1.48611	0.743054	−	0.669231i	$-0.233376\pi$
$421$	5.40450e6	1.48611	0.743054	+	0.669231i	$0.233376\pi$
$422$	3.31609e6	0.906453
$423$	0	0
$424$	503468.	0.136006
$425$	−1.07813e6	−0.289533
$426$	0	0
$427$	−1.07345e6	−0.284912
$428$	2.03256e6	0.536332
$429$	0	0
$430$	413164.	0.107759
$431$	−2.85092e6	−0.739252	−0.369626	−	0.929181i	$-0.620514\pi$
$431$	−2.85092e6	−0.739252	−0.369626	+	0.929181i	$0.620514\pi$
$432$	0	0
$433$	1.31780e6	0.337776	0.168888	−	0.985635i	$-0.445982\pi$
$433$	1.31780e6	0.337776	0.168888	+	0.985635i	$0.445982\pi$
$434$	−633715.	−0.161499
$435$	0	0
$436$	5.06420e6	1.27584
$437$	−293500.	−0.0735198
$438$	0	0
$439$	−935630.	−0.231709	−0.115854	−	0.993266i	$-0.536961\pi$
$439$	−935630.	−0.231709	−0.115854	+	0.993266i	$0.536961\pi$
$440$	324461.	0.0798971
$441$	0	0
$442$	−440680.	−0.107292
$443$	40668.6	0.00984578	0.00492289	−	0.999988i	$-0.498433\pi$
$443$	40668.6	0.00984578	0.00492289	+	0.999988i	$0.498433\pi$
$444$	0	0
$445$	−86897.6	−0.0208021
$446$	−5.61360e6	−1.33630
$447$	0	0
$448$	574338.	0.135199
$449$	−3.27855e6	−0.767477	−0.383739	−	0.923442i	$-0.625364\pi$
$449$	−3.27855e6	−0.767477	−0.383739	+	0.923442i	$0.625364\pi$
$450$	0	0
$451$	−6.34633e6	−1.46920
$452$	2.48209e6	0.571440
$453$	0	0
$454$	976230.	0.222286
$455$	−68037.8	−0.0154071
$456$	0	0
$457$	6.44769e6	1.44415	0.722077	−	0.691813i	$-0.243188\pi$
$457$	6.44769e6	1.44415	0.722077	+	0.691813i	$0.243188\pi$
$458$	1.55317e6	0.345982
$459$	0	0
$460$	117715.	0.0259380
$461$	5.05662e6	1.10817	0.554087	−	0.832459i	$-0.313068\pi$
$461$	5.05662e6	1.10817	0.554087	+	0.832459i	$0.313068\pi$
$462$	0	0
$463$	−1.45318e6	−0.315041	−0.157520	−	0.987516i	$-0.550350\pi$
$463$	−1.45318e6	−0.315041	−0.157520	+	0.987516i	$0.550350\pi$
$464$	604254.	0.130294
$465$	0	0
$466$	−802085.	−0.171102
$467$	−2.69028e6	−0.570829	−0.285414	−	0.958404i	$-0.592131\pi$
$467$	−2.69028e6	−0.570829	−0.285414	+	0.958404i	$0.592131\pi$
$468$	0	0
$469$	1.07178e6	0.224994
$470$	−507412.	−0.105954
$471$	0	0
$472$	−44473.4	−0.00918852
$473$	−3.89832e6	−0.801170
$474$	0	0
$475$	1.42082e6	0.288938
$476$	−391949.	−0.0792888
$477$	0	0
$478$	1.09777e7	2.19756
$479$	2.08328e6	0.414867	0.207434	−	0.978249i	$-0.433489\pi$
$479$	2.08328e6	0.414867	0.207434	+	0.978249i	$0.433489\pi$
$480$	0	0
$481$	−290771.	−0.0573045
$482$	3.65159e6	0.715919
$483$	0	0
$484$	3.80146e6	0.737627
$485$	868746.	0.167702
$486$	0	0
$487$	−8.67085e6	−1.65668	−0.828341	−	0.560224i	$-0.810715\pi$
$487$	−8.67085e6	−1.65668	−0.828341	+	0.560224i	$0.810715\pi$
$488$	1.50913e6	0.286864
$489$	0	0
$490$	−145879.	−0.0274474
$491$	−2.38925e6	−0.447258	−0.223629	−	0.974674i	$-0.571790\pi$
$491$	−2.38925e6	−0.447258	−0.223629	+	0.974674i	$0.571790\pi$
$492$	0	0
$493$	−172483.	−0.0319616
$494$	580753.	0.107071
$495$	0	0
$496$	2.16045e6	0.394313
$497$	1.94017e6	0.352330
$498$	0	0
$499$	4.22705e6	0.759951	0.379975	−	0.924997i	$-0.375932\pi$
$499$	4.22705e6	0.759951	0.379975	+	0.924997i	$0.375932\pi$
$500$	−1.15228e6	−0.206127
$501$	0	0
$502$	4.57837e6	0.810870
$503$	7.93214e6	1.39788	0.698941	−	0.715179i	$-0.253655\pi$
$503$	7.93214e6	1.39788	0.698941	+	0.715179i	$0.253655\pi$
$504$	0	0
$505$	−73936.3	−0.0129012
$506$	−2.67745e6	−0.464885
$507$	0	0
$508$	−1.58809e6	−0.273034
$509$	−2.07944e6	−0.355755	−0.177878	−	0.984053i	$-0.556923\pi$
$509$	−2.07944e6	−0.355755	−0.177878	+	0.984053i	$0.556923\pi$
$510$	0	0
$511$	3.92543e6	0.665020
$512$	−5.83841e6	−0.984283
$513$	0	0
$514$	1.38582e6	0.231365
$515$	−1.03494e6	−0.171948
$516$	0	0
$517$	4.78757e6	0.787751
$518$	−623437.	−0.102086
$519$	0	0
$520$	95652.3	0.0155127
$521$	3.60753e6	0.582258	0.291129	−	0.956684i	$-0.405969\pi$
$521$	3.60753e6	0.582258	0.291129	+	0.956684i	$0.405969\pi$
$522$	0	0
$523$	3.79245e6	0.606269	0.303134	−	0.952948i	$-0.401967\pi$
$523$	3.79245e6	0.606269	0.303134	+	0.952948i	$0.401967\pi$
$524$	6.73424e6	1.07142
$525$	0	0
$526$	8.71982e6	1.37418
$527$	−616696.	−0.0967263
$528$	0	0
$529$	−6.03744e6	−0.938023
$530$	−444049.	−0.0686659
$531$	0	0
$532$	516531.	0.0791257
$533$	−1.87092e6	−0.285258
$534$	0	0
$535$	736178.	0.111198
$536$	−1.50678e6	−0.226536
$537$	0	0
$538$	2.93554e6	0.437253
$539$	1.37641e6	0.204068
$540$	0	0
$541$	8.72291e6	1.28135	0.640676	−	0.767811i	$-0.278654\pi$
$541$	8.72291e6	1.28135	0.640676	+	0.767811i	$0.278654\pi$
$542$	−4.51532e6	−0.660222
$543$	0	0
$544$	2.44387e6	0.354064
$545$	1.83422e6	0.264520
$546$	0	0
$547$	7.94734e6	1.13567	0.567837	−	0.823141i	$-0.307780\pi$
$547$	7.94734e6	1.13567	0.567837	+	0.823141i	$0.307780\pi$
$548$	−7.23699e6	−1.02945
$549$	0	0
$550$	1.29614e7	1.82703
$551$	227307.	0.0318959
$552$	0	0
$553$	−888203.	−0.123509
$554$	−1.63519e6	−0.226357
$555$	0	0
$556$	2.15125e6	0.295124
$557$	−9.86113e6	−1.34676	−0.673378	−	0.739299i	$-0.735157\pi$
$557$	−9.86113e6	−1.34676	−0.673378	+	0.739299i	$0.735157\pi$
$558$	0	0
$559$	−1.14924e6	−0.155554
$560$	497328.	0.0670151
$561$	0	0
$562$	1.66164e7	2.21920
$563$	−9.25492e6	−1.23056	−0.615278	−	0.788310i	$-0.710956\pi$
$563$	−9.25492e6	−1.23056	−0.615278	+	0.788310i	$0.710956\pi$
$564$	0	0
$565$	898993.	0.118477
$566$	8.89522e6	1.16712
$567$	0	0
$568$	−2.72763e6	−0.354743
$569$	−4.62840e6	−0.599308	−0.299654	−	0.954048i	$-0.596871\pi$
$569$	−4.62840e6	−0.599308	−0.299654	+	0.954048i	$0.596871\pi$
$570$	0	0
$571$	−3.54045e6	−0.454431	−0.227215	−	0.973845i	$-0.572962\pi$
$571$	−3.54045e6	−0.454431	−0.227215	+	0.973845i	$0.572962\pi$
$572$	2.19770e6	0.280853
$573$	0	0
$574$	−4.01140e6	−0.508179
$575$	−1.93108e6	−0.243574
$576$	0	0
$577$	−70992.0	−0.00887707	−0.00443854	−	0.999990i	$-0.501413\pi$
$577$	−70992.0	−0.00887707	−0.00443854	+	0.999990i	$0.501413\pi$
$578$	9.58021e6	1.19277
$579$	0	0
$580$	−91166.7	−0.0112529
$581$	767030.	0.0942697
$582$	0	0
$583$	4.18973e6	0.510522
$584$	−5.51864e6	−0.669576
$585$	0	0
$586$	−5.87960e6	−0.707300
$587$	1.56067e7	1.86946	0.934732	−	0.355355i	$-0.115640\pi$
$587$	1.56067e7	1.86946	0.934732	+	0.355355i	$0.115640\pi$
$588$	0	0
$589$	812716.	0.0965274
$590$	39224.7	0.00463905
$591$	0	0
$592$	2.12541e6	0.249252
$593$	−1.34657e7	−1.57251	−0.786253	−	0.617904i	$-0.787982\pi$
$593$	−1.34657e7	−1.57251	−0.786253	+	0.617904i	$0.787982\pi$
$594$	0	0
$595$	−141961.	−0.0164390
$596$	5.26318e6	0.606922
$597$	0	0
$598$	−789322.	−0.0902612
$599$	5.94072e6	0.676507	0.338253	−	0.941055i	$-0.390164\pi$
$599$	5.94072e6	0.676507	0.338253	+	0.941055i	$0.390164\pi$
$600$	0	0
$601$	−4.76850e6	−0.538513	−0.269256	−	0.963069i	$-0.586778\pi$
$601$	−4.76850e6	−0.538513	−0.269256	+	0.963069i	$0.586778\pi$
$602$	−2.46406e6	−0.277115
$603$	0	0
$604$	23381.3	0.00260782
$605$	1.37686e6	0.152933
$606$	0	0
$607$	−4.87682e6	−0.537235	−0.268618	−	0.963247i	$-0.586567\pi$
$607$	−4.87682e6	−0.537235	−0.268618	+	0.963247i	$0.586567\pi$
$608$	−3.22067e6	−0.353335
$609$	0	0
$610$	−1.33102e6	−0.144830
$611$	1.41139e6	0.152948
$612$	0	0
$613$	−6.66508e6	−0.716398	−0.358199	−	0.933645i	$-0.616609\pi$
$613$	−6.66508e6	−0.716398	−0.358199	+	0.933645i	$0.616609\pi$
$614$	1.24532e7	1.33309
$615$	0	0
$616$	−1.93505e6	−0.205466
$617$	1.42572e7	1.50773	0.753863	−	0.657031i	$-0.228188\pi$
$617$	1.42572e7	1.50773	0.753863	+	0.657031i	$0.228188\pi$
$618$	0	0
$619$	1.68231e7	1.76474	0.882369	−	0.470559i	$-0.155948\pi$
$619$	1.68231e7	1.76474	0.882369	+	0.470559i	$0.155948\pi$
$620$	−325958.	−0.0340551
$621$	0	0
$622$	−9.24155e6	−0.957787
$623$	518246.	0.0534953
$624$	0	0
$625$	9.13732e6	0.935662
$626$	1.61816e7	1.65039
$627$	0	0
$628$	3.61756e6	0.366030
$629$	−606694.	−0.0611425
$630$	0	0
$631$	−1.38201e7	−1.38178	−0.690891	−	0.722959i	$-0.742781\pi$
$631$	−1.38201e7	−1.38178	−0.690891	+	0.722959i	$0.742781\pi$
$632$	1.24870e6	0.124355
$633$	0	0
$634$	3.55462e6	0.351213
$635$	−575195.	−0.0566084
$636$	0	0
$637$	405769.	0.0396214
$638$	2.07361e6	0.201686
$639$	0	0
$640$	−1.11003e6	−0.107124
$641$	1.26367e6	0.121476	0.0607378	−	0.998154i	$-0.480655\pi$
$641$	1.26367e6	0.121476	0.0607378	+	0.998154i	$0.480655\pi$
$642$	0	0
$643$	1.28863e7	1.22914	0.614571	−	0.788862i	$-0.289329\pi$
$643$	1.28863e7	1.22914	0.614571	+	0.788862i	$0.289329\pi$
$644$	−702036.	−0.0667029
$645$	0	0
$646$	1.21174e6	0.114243
$647$	−2.04796e7	−1.92336	−0.961681	−	0.274172i	$-0.911596\pi$
$647$	−2.04796e7	−1.92336	−0.961681	+	0.274172i	$0.911596\pi$
$648$	0	0
$649$	−370096.	−0.0344907
$650$	3.82106e6	0.354732
$651$	0	0
$652$	−2.20371e6	−0.203018
$653$	−9.31094e6	−0.854497	−0.427249	−	0.904134i	$-0.640517\pi$
$653$	−9.31094e6	−0.854497	−0.427249	+	0.904134i	$0.640517\pi$
$654$	0	0
$655$	2.43909e6	0.222139
$656$	1.36756e7	1.24076
$657$	0	0
$658$	3.02614e6	0.272474
$659$	8.57311e6	0.768997	0.384499	−	0.923126i	$-0.374374\pi$
$659$	8.57311e6	0.768997	0.384499	+	0.923126i	$0.374374\pi$
$660$	0	0
$661$	−1.28955e7	−1.14798	−0.573992	−	0.818861i	$-0.694606\pi$
$661$	−1.28955e7	−1.14798	−0.573992	+	0.818861i	$0.694606\pi$
$662$	1.49370e7	1.32471
$663$	0	0
$664$	−1.07834e6	−0.0949155
$665$	187084.	0.0164052
$666$	0	0
$667$	−308941.	−0.0268882
$668$	2.69932e6	0.234053
$669$	0	0
$670$	1.32895e6	0.114372
$671$	1.25585e7	1.07679
$672$	0	0
$673$	1.49491e7	1.27226	0.636130	−	0.771582i	$-0.280534\pi$
$673$	1.49491e7	1.27226	0.636130	+	0.771582i	$0.280534\pi$
$674$	2.15124e7	1.82406
$675$	0	0
$676$	647890.	0.0545299
$677$	6.29805e6	0.528122	0.264061	−	0.964506i	$-0.414938\pi$
$677$	6.29805e6	0.528122	0.264061	+	0.964506i	$0.414938\pi$
$678$	0	0
$679$	−5.18109e6	−0.431268
$680$	199578.	0.0165517
$681$	0	0
$682$	7.41400e6	0.610368
$683$	2.63459e6	0.216104	0.108052	−	0.994145i	$-0.465539\pi$
$683$	2.63459e6	0.216104	0.108052	+	0.994145i	$0.465539\pi$
$684$	0	0
$685$	−2.62118e6	−0.213438
$686$	870002.	0.0705846
$687$	0	0
$688$	8.40044e6	0.676599
$689$	1.23515e6	0.0991220
$690$	0	0
$691$	−2.16023e7	−1.72110	−0.860549	−	0.509368i	$-0.829879\pi$
$691$	−2.16023e7	−1.72110	−0.860549	+	0.509368i	$0.829879\pi$
$692$	2.40949e6	0.191276
$693$	0	0
$694$	−2.12972e7	−1.67851
$695$	779167.	0.0611883
$696$	0	0
$697$	−3.90367e6	−0.304363
$698$	2.23983e7	1.74011
$699$	0	0
$700$	3.39852e6	0.262147
$701$	−1.58443e7	−1.21780	−0.608902	−	0.793245i	$-0.708390\pi$
$701$	−1.58443e7	−1.21780	−0.608902	+	0.793245i	$0.708390\pi$
$702$	0	0
$703$	799534.	0.0610167
$704$	−6.71933e6	−0.510968
$705$	0	0
$706$	−3.45833e7	−2.61128
$707$	440947.	0.0331771
$708$	0	0
$709$	−3.50413e6	−0.261797	−0.130898	−	0.991396i	$-0.541786\pi$
$709$	−3.50413e6	−0.261797	−0.130898	+	0.991396i	$0.541786\pi$
$710$	2.40571e6	0.179101
$711$	0	0
$712$	−728586.	−0.0538618
$713$	−1.10459e6	−0.0813725
$714$	0	0
$715$	795992.	0.0582296
$716$	−8.32878e6	−0.607154
$717$	0	0
$718$	−7.92254e6	−0.573527
$719$	−1.75161e7	−1.26362	−0.631808	−	0.775125i	$-0.717687\pi$
$719$	−1.75161e7	−1.26362	−0.631808	+	0.775125i	$0.717687\pi$
$720$	0	0
$721$	6.17226e6	0.442187
$722$	1.67136e7	1.19324
$723$	0	0
$724$	−5.15118e6	−0.365225
$725$	1.49557e6	0.105672
$726$	0	0
$727$	−9.77302e6	−0.685792	−0.342896	−	0.939373i	$-0.611408\pi$
$727$	−9.77302e6	−0.685792	−0.342896	+	0.939373i	$0.611408\pi$
$728$	−570458.	−0.0398929
$729$	0	0
$730$	4.86733e6	0.338052
$731$	−2.39789e6	−0.165972
$732$	0	0
$733$	1.98845e7	1.36696	0.683479	−	0.729970i	$-0.260466\pi$
$733$	1.98845e7	1.36696	0.683479	+	0.729970i	$0.260466\pi$
$734$	2.66235e7	1.82400
$735$	0	0
$736$	4.37733e6	0.297862
$737$	−1.25390e7	−0.850341
$738$	0	0
$739$	−2.55095e7	−1.71827	−0.859135	−	0.511749i	$-0.828998\pi$
$739$	−2.55095e7	−1.71827	−0.859135	+	0.511749i	$0.828998\pi$
$740$	−320671.	−0.0215269
$741$	0	0
$742$	2.64825e6	0.176583
$743$	−1.90630e7	−1.26683	−0.633415	−	0.773812i	$-0.718347\pi$
$743$	−1.90630e7	−1.26683	−0.633415	+	0.773812i	$0.718347\pi$
$744$	0	0
$745$	1.90629e6	0.125834
$746$	2.07526e7	1.36529
$747$	0	0
$748$	4.58551e6	0.299663
$749$	−4.39047e6	−0.285961
$750$	0	0
$751$	2.86832e7	1.85578	0.927891	−	0.372850i	$-0.121620\pi$
$751$	2.86832e7	1.85578	0.927891	+	0.372850i	$0.121620\pi$
$752$	−1.03167e7	−0.665266
$753$	0	0
$754$	611307.	0.0391589
$755$	8468.55	0.000540682 0
$756$	0	0
$757$	352437.	0.0223533	0.0111766	−	0.999938i	$-0.496442\pi$
$757$	352437.	0.0223533	0.0111766	+	0.999938i	$0.496442\pi$
$758$	1.76657e7	1.11676
$759$	0	0
$760$	−263015.	−0.0165176
$761$	−1.10662e7	−0.692685	−0.346342	−	0.938108i	$-0.612576\pi$
$761$	−1.10662e7	−0.692685	−0.346342	+	0.938108i	$0.612576\pi$
$762$	0	0
$763$	−1.09390e7	−0.680249
$764$	−1.33729e6	−0.0828879
$765$	0	0
$766$	−1.75084e7	−1.07814
$767$	−109105.	−0.00669665
$768$	0	0
$769$	6.34666e6	0.387017	0.193508	−	0.981099i	$-0.438013\pi$
$769$	6.34666e6	0.387017	0.193508	+	0.981099i	$0.438013\pi$
$770$	1.70667e6	0.103734
$771$	0	0
$772$	−985829.	−0.0595331
$773$	−2.15620e7	−1.29790	−0.648950	−	0.760831i	$-0.724792\pi$
$773$	−2.15620e7	−1.29790	−0.648950	+	0.760831i	$0.724792\pi$
$774$	0	0
$775$	5.34727e6	0.319799
$776$	7.28394e6	0.434222
$777$	0	0
$778$	−1.36409e7	−0.807966
$779$	5.14448e6	0.303737
$780$	0	0
$781$	−2.26986e7	−1.33159
$782$	−1.64692e6	−0.0963065
$783$	0	0
$784$	−2.96600e6	−0.172338
$785$	1.31025e6	0.0758894
$786$	0	0
$787$	2.95531e6	0.170085	0.0850426	−	0.996377i	$-0.472897\pi$
$787$	2.95531e6	0.170085	0.0850426	+	0.996377i	$0.472897\pi$
$788$	−1.18162e6	−0.0677895
$789$	0	0
$790$	−1.10133e6	−0.0627839
$791$	−5.36148e6	−0.304680
$792$	0	0
$793$	3.70230e6	0.209068
$794$	4.21157e6	0.237079
$795$	0	0
$796$	−1.36721e7	−0.764808
$797$	2.42235e6	0.135080	0.0675401	−	0.997717i	$-0.478485\pi$
$797$	2.42235e6	0.135080	0.0675401	+	0.997717i	$0.478485\pi$
$798$	0	0
$799$	2.94487e6	0.163192
$800$	−2.11904e7	−1.17062
$801$	0	0
$802$	−2.32819e7	−1.27815
$803$	−4.59246e7	−2.51337
$804$	0	0
$805$	−254272.	−0.0138296
$806$	2.18567e6	0.118508
$807$	0	0
$808$	−619914.	−0.0334043
$809$	−1.82361e7	−0.979625	−0.489813	−	0.871828i	$-0.662935\pi$
$809$	−1.82361e7	−0.979625	−0.489813	+	0.871828i	$0.662935\pi$
$810$	0	0
$811$	3.44594e7	1.83974	0.919868	−	0.392227i	$-0.128295\pi$
$811$	3.44594e7	1.83974	0.919868	+	0.392227i	$0.128295\pi$
$812$	543707.	0.0289384
$813$	0	0
$814$	7.29375e6	0.385824
$815$	−798168.	−0.0420921
$816$	0	0
$817$	3.16006e6	0.165631
$818$	4.80701e7	2.51184
$819$	0	0
$820$	−2.06331e6	−0.107159
$821$	2.13235e7	1.10408	0.552039	−	0.833818i	$-0.313850\pi$
$821$	2.13235e7	1.10408	0.552039	+	0.833818i	$0.313850\pi$
$822$	0	0
$823$	−1.70435e7	−0.877120	−0.438560	−	0.898702i	$-0.644511\pi$
$823$	−1.70435e7	−0.877120	−0.438560	+	0.898702i	$0.644511\pi$
$824$	−8.67739e6	−0.445216
$825$	0	0
$826$	−233931.	−0.0119299
$827$	1.60526e7	0.816170	0.408085	−	0.912944i	$-0.366197\pi$
$827$	1.60526e7	0.816170	0.408085	+	0.912944i	$0.366197\pi$
$828$	0	0
$829$	−1.69996e7	−0.859115	−0.429558	−	0.903039i	$-0.641330\pi$
$829$	−1.69996e7	−0.859115	−0.429558	+	0.903039i	$0.641330\pi$
$830$	951077.	0.0479204
$831$	0	0
$832$	−1.98088e6	−0.0992087
$833$	846637.	0.0422751
$834$	0	0
$835$	977674.	0.0485264
$836$	−6.04303e6	−0.299047
$837$	0	0
$838$	1.69698e7	0.834768
$839$	−1.70509e7	−0.836260	−0.418130	−	0.908387i	$-0.637314\pi$
$839$	−1.70509e7	−0.836260	−0.418130	+	0.908387i	$0.637314\pi$
$840$	0	0
$841$	−2.02719e7	−0.988335
$842$	−3.99657e7	−1.94271
$843$	0	0
$844$	−1.01724e7	−0.491548
$845$	234661.	0.0113057
$846$	0	0
$847$	−8.21143e6	−0.393287
$848$	−9.02839e6	−0.431142
$849$	0	0
$850$	7.97265e6	0.378491
$851$	−1.08668e6	−0.0514371
$852$	0	0
$853$	6.73420e6	0.316894	0.158447	−	0.987368i	$-0.449351\pi$
$853$	6.73420e6	0.316894	0.158447	+	0.987368i	$0.449351\pi$
$854$	7.93803e6	0.372450
$855$	0	0
$856$	6.17243e6	0.287920
$857$	−1.04845e7	−0.487635	−0.243818	−	0.969821i	$-0.578400\pi$
$857$	−1.04845e7	−0.487635	−0.243818	+	0.969821i	$0.578400\pi$
$858$	0	0
$859$	3.90857e6	0.180732	0.0903661	−	0.995909i	$-0.471196\pi$
$859$	3.90857e6	0.180732	0.0903661	+	0.995909i	$0.471196\pi$
$860$	−1.26742e6	−0.0584350
$861$	0	0
$862$	2.10823e7	0.966383
$863$	2.43163e7	1.11140	0.555700	−	0.831383i	$-0.312450\pi$
$863$	2.43163e7	1.11140	0.555700	+	0.831383i	$0.312450\pi$
$864$	0	0
$865$	872700.	0.0396574
$866$	−9.74496e6	−0.441556
$867$	0	0
$868$	1.94397e6	0.0875772
$869$	1.03913e7	0.466790
$870$	0	0
$871$	−3.69653e6	−0.165101
$872$	1.53789e7	0.684909
$873$	0	0
$874$	2.17040e6	0.0961084
$875$	2.48901e6	0.109902
$876$	0	0
$877$	−1.17476e7	−0.515762	−0.257881	−	0.966177i	$-0.583024\pi$
$877$	−1.17476e7	−0.515762	−0.257881	+	0.966177i	$0.583024\pi$
$878$	6.91888e6	0.302900
$879$	0	0
$880$	−5.81836e6	−0.253276
$881$	−3.23460e7	−1.40404	−0.702022	−	0.712156i	$-0.747719\pi$
$881$	−3.23460e7	−1.40404	−0.702022	+	0.712156i	$0.747719\pi$
$882$	0	0
$883$	5.94162e6	0.256450	0.128225	−	0.991745i	$-0.459072\pi$
$883$	5.94162e6	0.256450	0.128225	+	0.991745i	$0.459072\pi$
$884$	1.35182e6	0.0581821
$885$	0	0
$886$	−300740.	−0.0128708
$887$	−4.90615e6	−0.209378	−0.104689	−	0.994505i	$-0.533385\pi$
$887$	−4.90615e6	−0.209378	−0.104689	+	0.994505i	$0.533385\pi$
$888$	0	0
$889$	3.43039e6	0.145576
$890$	642598.	0.0271935
$891$	0	0
$892$	1.72202e7	0.724645
$893$	−3.88091e6	−0.162857
$894$	0	0
$895$	−3.01662e6	−0.125882
$896$	6.62008e6	0.275482
$897$	0	0
$898$	2.42445e7	1.00328
$899$	855474.	0.0353027
$900$	0	0
$901$	2.57713e6	0.105761
$902$	4.69304e7	1.92061
$903$	0	0
$904$	7.53754e6	0.306767
$905$	−1.86572e6	−0.0757224
$906$	0	0
$907$	−1.96483e7	−0.793060	−0.396530	−	0.918022i	$-0.629786\pi$
$907$	−1.96483e7	−0.793060	−0.396530	+	0.918022i	$0.629786\pi$
$908$	−2.99467e6	−0.120541
$909$	0	0
$910$	503132.	0.0201409
$911$	3.36988e7	1.34530	0.672650	−	0.739961i	$-0.265156\pi$
$911$	3.36988e7	1.34530	0.672650	+	0.739961i	$0.265156\pi$
$912$	0	0
$913$	−8.97368e6	−0.356282
$914$	−4.76800e7	−1.88786
$915$	0	0
$916$	−4.76446e6	−0.187618
$917$	−1.45464e7	−0.571259
$918$	0	0
$919$	1.01210e7	0.395308	0.197654	−	0.980272i	$-0.436668\pi$
$919$	1.01210e7	0.395308	0.197654	+	0.980272i	$0.436668\pi$
$920$	357474.	0.0139243
$921$	0	0
$922$	−3.73931e7	−1.44865
$923$	−6.69161e6	−0.258539
$924$	0	0
$925$	5.26054e6	0.202151
$926$	1.07461e7	0.411835
$927$	0	0
$928$	−3.39011e6	−0.129224
$929$	−4.96443e7	−1.88725	−0.943627	−	0.331011i	$-0.892610\pi$
$929$	−4.96443e7	−1.88725	−0.943627	+	0.331011i	$0.892610\pi$
$930$	0	0
$931$	−1.11574e6	−0.0421882
$932$	2.46046e6	0.0927847
$933$	0	0
$934$	1.98944e7	0.746213
$935$	1.66084e6	0.0621295
$936$	0	0
$937$	5.35292e6	0.199178	0.0995891	−	0.995029i	$-0.468247\pi$
$937$	5.35292e6	0.199178	0.0995891	+	0.995029i	$0.468247\pi$
$938$	−7.92566e6	−0.294123
$939$	0	0
$940$	1.55653e6	0.0574562
$941$	1.25573e7	0.462299	0.231149	−	0.972918i	$-0.425751\pi$
$941$	1.25573e7	0.462299	0.231149	+	0.972918i	$0.425751\pi$
$942$	0	0
$943$	−6.99204e6	−0.256050
$944$	797514.	0.0291279
$945$	0	0
$946$	2.88277e7	1.04733
$947$	8.90888e6	0.322811	0.161405	−	0.986888i	$-0.448397\pi$
$947$	8.90888e6	0.322811	0.161405	+	0.986888i	$0.448397\pi$
$948$	0	0
$949$	−1.35387e7	−0.487992
$950$	−1.05068e7	−0.377712
$951$	0	0
$952$	−1.19026e6	−0.0425647
$953$	−3.93448e7	−1.40332	−0.701658	−	0.712514i	$-0.747556\pi$
$953$	−3.93448e7	−1.40332	−0.701658	+	0.712514i	$0.747556\pi$
$954$	0	0
$955$	−484355.	−0.0171852
$956$	−3.36749e7	−1.19169
$957$	0	0
$958$	−1.54056e7	−0.542333
$959$	1.56324e7	0.548883
$960$	0	0
$961$	−2.55705e7	−0.893162
$962$	2.15022e6	0.0749110
$963$	0	0
$964$	−1.12015e7	−0.388227
$965$	−357060.	−0.0123431
$966$	0	0
$967$	−2.19680e7	−0.755482	−0.377741	−	0.925911i	$-0.623299\pi$
$967$	−2.19680e7	−0.755482	−0.377741	+	0.925911i	$0.623299\pi$
$968$	1.15442e7	0.395982
$969$	0	0
$970$	−6.42429e6	−0.219228
$971$	−2.57512e7	−0.876493	−0.438247	−	0.898855i	$-0.644400\pi$
$971$	−2.57512e7	−0.876493	−0.438247	+	0.898855i	$0.644400\pi$
$972$	0	0
$973$	−4.64685e6	−0.157354
$974$	6.41200e7	2.16569
$975$	0	0
$976$	−2.70622e7	−0.909367
$977$	−3.60773e7	−1.20920	−0.604599	−	0.796530i	$-0.706667\pi$
$977$	−3.60773e7	−1.20920	−0.604599	+	0.796530i	$0.706667\pi$
$978$	0	0
$979$	−6.06309e6	−0.202180
$980$	447495.	0.0148841
$981$	0	0
$982$	1.76683e7	0.584676
$983$	1.08558e7	0.358324	0.179162	−	0.983820i	$-0.442661\pi$
$983$	1.08558e7	0.358324	0.179162	+	0.983820i	$0.442661\pi$
$984$	0	0
$985$	−427974.	−0.0140549
$986$	1.27549e6	0.0417816
$987$	0	0
$988$	−1.78151e6	−0.0580624
$989$	−4.29496e6	−0.139627
$990$	0	0
$991$	−4.37367e7	−1.41469	−0.707346	−	0.706867i	$-0.750108\pi$
$991$	−4.37367e7	−1.41469	−0.707346	+	0.706867i	$0.750108\pi$
$992$	−1.21210e7	−0.391075
$993$	0	0
$994$	−1.43474e7	−0.460581
$995$	−4.95193e6	−0.158568
$996$	0	0
$997$	−2.11524e7	−0.673941	−0.336970	−	0.941515i	$-0.609402\pi$
$997$	−2.11524e7	−0.673941	−0.336970	+	0.941515i	$0.609402\pi$
$998$	−3.12585e7	−0.993442
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.6.a.f.1.1		6
3.2	odd	2		273.6.a.b.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.6.a.b.1.6	✓	6	3.2	odd	2
819.6.a.f.1.1		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)

\(f(q)\)	\(=\)	\(q-7.39489 q^{2} +22.6844 q^{4} +8.21614 q^{5} -49.0000 q^{7} +68.8876 q^{8} +O(q^{10})\)	\(q-7.39489 q^{2} +22.6844 q^{4} +8.21614 q^{5} -49.0000 q^{7} +68.8876 q^{8} -60.7574 q^{10} +573.264 q^{11} +169.000 q^{13} +362.350 q^{14} -1235.32 q^{16} +352.618 q^{17} -464.700 q^{19} +186.378 q^{20} -4239.22 q^{22} +631.590 q^{23} -3057.50 q^{25} -1249.74 q^{26} -1111.54 q^{28} -489.148 q^{29} -1748.90 q^{31} +6930.64 q^{32} -2607.58 q^{34} -402.591 q^{35} -1720.54 q^{37} +3436.41 q^{38} +565.990 q^{40} -11070.5 q^{41} -6800.23 q^{43} +13004.2 q^{44} -4670.54 q^{46} +8351.44 q^{47} +2401.00 q^{49} +22609.8 q^{50} +3833.67 q^{52} +7308.55 q^{53} +4710.01 q^{55} -3375.49 q^{56} +3617.20 q^{58} -645.594 q^{59} +21907.1 q^{61} +12933.0 q^{62} -11721.2 q^{64} +1388.53 q^{65} -21873.0 q^{67} +7998.95 q^{68} +2977.11 q^{70} -39595.3 q^{71} -80110.8 q^{73} +12723.2 q^{74} -10541.5 q^{76} -28089.9 q^{77} +18126.6 q^{79} -10149.5 q^{80} +81865.4 q^{82} -15653.7 q^{83} +2897.16 q^{85} +50286.9 q^{86} +39490.7 q^{88} -10576.5 q^{89} -8281.00 q^{91} +14327.3 q^{92} -61758.0 q^{94} -3818.04 q^{95} +105737. q^{97} -17755.1 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 15 q^{2} + 75 q^{4} - 15 q^{5} - 294 q^{7} + 399 q^{8}+O(q^{10}) \) 6 * q + 15 * q^2 + 75 * q^4 - 15 * q^5 - 294 * q^7 + 399 * q^8	\( 6 q + 15 q^{2} + 75 q^{4} - 15 q^{5} - 294 q^{7} + 399 q^{8} - 265 q^{10} + 90 q^{11} + 1014 q^{13} - 735 q^{14} + 355 q^{16} + 1314 q^{17} - 341 q^{19} - 2085 q^{20} - 6749 q^{22} + 4905 q^{23} - 10231 q^{25} + 2535 q^{26} - 3675 q^{28} + 6771 q^{29} - 14711 q^{31} + 10143 q^{32} + 9595 q^{34} + 735 q^{35} - 13314 q^{37} - 1017 q^{38} - 13195 q^{40} + 5040 q^{41} - 9127 q^{43} - 10041 q^{44} + 23739 q^{46} + 4713 q^{47} + 14406 q^{49} - 18018 q^{50} + 12675 q^{52} + 159 q^{53} - 13590 q^{55} - 19551 q^{56} - 38427 q^{58} - 30288 q^{59} - 78126 q^{61} - 40632 q^{62} - 8461 q^{64} - 2535 q^{65} - 33894 q^{67} + 61863 q^{68} + 12985 q^{70} - 8316 q^{71} - 89861 q^{73} + 16635 q^{74} - 169069 q^{76} - 4410 q^{77} - 239803 q^{79} - 19305 q^{80} - 106624 q^{82} + 42753 q^{83} - 184316 q^{85} - 51609 q^{86} - 205923 q^{88} + 11421 q^{89} - 49686 q^{91} + 252519 q^{92} - 4976 q^{94} + 180807 q^{95} - 199445 q^{97} + 36015 q^{98}+O(q^{100}) \) 6 * q + 15 * q^2 + 75 * q^4 - 15 * q^5 - 294 * q^7 + 399 * q^8 - 265 * q^10 + 90 * q^11 + 1014 * q^13 - 735 * q^14 + 355 * q^16 + 1314 * q^17 - 341 * q^19 - 2085 * q^20 - 6749 * q^22 + 4905 * q^23 - 10231 * q^25 + 2535 * q^26 - 3675 * q^28 + 6771 * q^29 - 14711 * q^31 + 10143 * q^32 + 9595 * q^34 + 735 * q^35 - 13314 * q^37 - 1017 * q^38 - 13195 * q^40 + 5040 * q^41 - 9127 * q^43 - 10041 * q^44 + 23739 * q^46 + 4713 * q^47 + 14406 * q^49 - 18018 * q^50 + 12675 * q^52 + 159 * q^53 - 13590 * q^55 - 19551 * q^56 - 38427 * q^58 - 30288 * q^59 - 78126 * q^61 - 40632 * q^62 - 8461 * q^64 - 2535 * q^65 - 33894 * q^67 + 61863 * q^68 + 12985 * q^70 - 8316 * q^71 - 89861 * q^73 + 16635 * q^74 - 169069 * q^76 - 4410 * q^77 - 239803 * q^79 - 19305 * q^80 - 106624 * q^82 + 42753 * q^83 - 184316 * q^85 - 51609 * q^86 - 205923 * q^88 + 11421 * q^89 - 49686 * q^91 + 252519 * q^92 - 4976 * q^94 + 180807 * q^95 - 199445 * q^97 + 36015 * q^98

Introduction

L-functions

Modular forms

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Database

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