LMFDB - Embedded newform 261.6.a.e.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [261,6,Mod(1,261)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(261, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("261.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 261 = 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	261.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:	$41.8601769712$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 184x^{5} + 584x^{4} + 10145x^{3} - 34491x^{2} - 149754x + 524902 $ x^7 - 3x^6 - 184x^5 + 584x^4 + 10145x^3 - 34491x^2 - 149754x + 524902
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$7.83842$ of defining polynomial
Character	$\chi$	$=$	261.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.83842 q^{2} +14.7640 q^{4} +54.3066 q^{5} -37.6697 q^{7} -117.867 q^{8} +O(q^{10})$	\(q+6.83842 q^{2} +14.7640 q^{4} +54.3066 q^{5} -37.6697 q^{7} -117.867 q^{8} +371.372 q^{10} +146.903 q^{11} +162.581 q^{13} -257.601 q^{14} -1278.47 q^{16} +2162.70 q^{17} +2494.06 q^{19} +801.781 q^{20} +1004.59 q^{22} +122.601 q^{23} -175.788 q^{25} +1111.79 q^{26} -556.154 q^{28} -841.000 q^{29} +8674.26 q^{31} -4970.98 q^{32} +14789.4 q^{34} -2045.71 q^{35} +11752.8 q^{37} +17055.4 q^{38} -6400.97 q^{40} -9519.28 q^{41} -14834.7 q^{43} +2168.88 q^{44} +838.394 q^{46} +3827.35 q^{47} -15388.0 q^{49} -1202.11 q^{50} +2400.34 q^{52} +33207.9 q^{53} +7977.83 q^{55} +4440.02 q^{56} -5751.11 q^{58} -19608.0 q^{59} +24024.9 q^{61} +59318.2 q^{62} +6917.49 q^{64} +8829.21 q^{65} +55181.2 q^{67} +31930.0 q^{68} -13989.4 q^{70} +41626.8 q^{71} -15707.6 q^{73} +80370.5 q^{74} +36822.3 q^{76} -5533.80 q^{77} +56772.8 q^{79} -69429.5 q^{80} -65096.8 q^{82} -107406. q^{83} +117449. q^{85} -101446. q^{86} -17315.1 q^{88} -37615.4 q^{89} -6124.36 q^{91} +1810.07 q^{92} +26173.0 q^{94} +135444. q^{95} -43944.4 q^{97} -105230. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 4 q^{2} + 154 q^{4} - 32 q^{5} + 184 q^{7} - 942 q^{8}+O(q^{10}) $ 7 * q - 4 * q^2 + 154 * q^4 - 32 * q^5 + 184 * q^7 - 942 * q^8	$ 7 q - 4 q^{2} + 154 q^{4} - 32 q^{5} + 184 q^{7} - 942 q^{8} + 922 q^{10} - 1106 q^{11} + 408 q^{13} + 2008 q^{14} + 242 q^{16} + 874 q^{17} + 4288 q^{19} + 6350 q^{20} - 6114 q^{22} + 4532 q^{23} + 5527 q^{25} + 19806 q^{26} - 496 q^{28} - 5887 q^{29} + 7794 q^{31} - 7898 q^{32} + 20840 q^{34} - 7088 q^{35} + 5086 q^{37} - 23732 q^{38} + 22906 q^{40} - 19826 q^{41} + 19498 q^{43} + 6074 q^{44} - 12404 q^{46} - 14278 q^{47} + 38431 q^{49} + 41066 q^{50} - 34302 q^{52} + 58644 q^{53} - 25574 q^{55} + 79560 q^{56} + 3364 q^{58} - 12888 q^{59} + 102866 q^{61} + 42654 q^{62} - 10170 q^{64} + 149206 q^{65} + 102996 q^{67} - 85100 q^{68} + 349480 q^{70} + 51596 q^{71} - 17566 q^{73} - 12132 q^{74} + 360740 q^{76} + 94104 q^{77} + 212058 q^{79} - 142510 q^{80} + 201924 q^{82} + 122928 q^{83} - 109336 q^{85} + 63290 q^{86} + 136666 q^{88} + 66510 q^{89} + 194368 q^{91} + 110108 q^{92} + 438926 q^{94} + 131676 q^{95} - 118182 q^{97} + 29132 q^{98}+O(q^{100}) $ 7 * q - 4 * q^2 + 154 * q^4 - 32 * q^5 + 184 * q^7 - 942 * q^8 + 922 * q^10 - 1106 * q^11 + 408 * q^13 + 2008 * q^14 + 242 * q^16 + 874 * q^17 + 4288 * q^19 + 6350 * q^20 - 6114 * q^22 + 4532 * q^23 + 5527 * q^25 + 19806 * q^26 - 496 * q^28 - 5887 * q^29 + 7794 * q^31 - 7898 * q^32 + 20840 * q^34 - 7088 * q^35 + 5086 * q^37 - 23732 * q^38 + 22906 * q^40 - 19826 * q^41 + 19498 * q^43 + 6074 * q^44 - 12404 * q^46 - 14278 * q^47 + 38431 * q^49 + 41066 * q^50 - 34302 * q^52 + 58644 * q^53 - 25574 * q^55 + 79560 * q^56 + 3364 * q^58 - 12888 * q^59 + 102866 * q^61 + 42654 * q^62 - 10170 * q^64 + 149206 * q^65 + 102996 * q^67 - 85100 * q^68 + 349480 * q^70 + 51596 * q^71 - 17566 * q^73 - 12132 * q^74 + 360740 * q^76 + 94104 * q^77 + 212058 * q^79 - 142510 * q^80 + 201924 * q^82 + 122928 * q^83 - 109336 * q^85 + 63290 * q^86 + 136666 * q^88 + 66510 * q^89 + 194368 * q^91 + 110108 * q^92 + 438926 * q^94 + 131676 * q^95 - 118182 * q^97 + 29132 * 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 $

$2$	6.83842	1.20887	0.604436	−	0.796653i	$-0.293398\pi$
$2$	6.83842	1.20887	0.604436	+	0.796653i	$0.293398\pi$
$3$	0	0
$3$	0	0
$4$	14.7640	0.461374
$5$	54.3066	0.971467	0.485733	−	0.874107i	$-0.338553\pi$
$5$	54.3066	0.971467	0.485733	+	0.874107i	$0.338553\pi$
$6$	0	0
$7$	−37.6697	−0.290567	−0.145284	−	0.989390i	$-0.546409\pi$
$7$	−37.6697	−0.290567	−0.145284	+	0.989390i	$0.546409\pi$
$8$	−117.867	−0.651131
$9$	0	0
$10$	371.372	1.17438
$11$	146.903	0.366058	0.183029	−	0.983107i	$-0.441410\pi$
$11$	146.903	0.366058	0.183029	+	0.983107i	$0.441410\pi$
$12$	0	0
$13$	162.581	0.266815	0.133408	−	0.991061i	$-0.457408\pi$
$13$	162.581	0.266815	0.133408	+	0.991061i	$0.457408\pi$
$14$	−257.601	−0.351259
$15$	0	0
$16$	−1278.47	−1.24851
$17$	2162.70	1.81499	0.907493	−	0.420068i	$-0.137994\pi$
$17$	2162.70	1.81499	0.907493	+	0.420068i	$0.137994\pi$
$18$	0	0
$19$	2494.06	1.58498	0.792489	−	0.609886i	$-0.208785\pi$
$19$	2494.06	1.58498	0.792489	+	0.609886i	$0.208785\pi$
$20$	801.781	0.448209
$21$	0	0
$22$	1004.59	0.442518
$23$	122.601	0.0483251	0.0241626	−	0.999708i	$-0.492308\pi$
$23$	122.601	0.0483251	0.0241626	+	0.999708i	$0.492308\pi$
$24$	0	0
$25$	−175.788	−0.0562522
$26$	1111.79	0.322546
$27$	0	0
$28$	−556.154	−0.134060
$29$	−841.000	−0.185695
$29$	−841.000	−0.185695
$30$	0	0
$31$	8674.26	1.62117	0.810584	−	0.585622i	$-0.199150\pi$
$31$	8674.26	1.62117	0.810584	+	0.585622i	$0.199150\pi$
$32$	−4970.98	−0.858157
$33$	0	0
$34$	14789.4	2.19409
$35$	−2045.71	−0.282276
$36$	0	0
$37$	11752.8	1.41136	0.705679	−	0.708532i	$-0.250642\pi$
$37$	11752.8	1.41136	0.705679	+	0.708532i	$0.250642\pi$
$38$	17055.4	1.91604
$39$	0	0
$40$	−6400.97	−0.632552
$41$	−9519.28	−0.884391	−0.442195	−	0.896919i	$-0.645800\pi$
$41$	−9519.28	−0.884391	−0.442195	+	0.896919i	$0.645800\pi$
$42$	0	0
$43$	−14834.7	−1.22351	−0.611756	−	0.791046i	$-0.709537\pi$
$43$	−14834.7	−1.22351	−0.611756	+	0.791046i	$0.709537\pi$
$44$	2168.88	0.168890
$45$	0	0
$46$	838.394	0.0584189
$47$	3827.35	0.252728	0.126364	−	0.991984i	$-0.459669\pi$
$47$	3827.35	0.252728	0.126364	+	0.991984i	$0.459669\pi$
$48$	0	0
$49$	−15388.0	−0.915571
$50$	−1202.11	−0.0680017
$51$	0	0
$52$	2400.34	0.123102
$53$	33207.9	1.62387	0.811936	−	0.583746i	$-0.198414\pi$
$53$	33207.9	1.62387	0.811936	+	0.583746i	$0.198414\pi$
$54$	0	0
$55$	7977.83	0.355613
$56$	4440.02	0.189197
$57$	0	0
$58$	−5751.11	−0.224482
$59$	−19608.0	−0.733336	−0.366668	−	0.930352i	$-0.619502\pi$
$59$	−19608.0	−0.733336	−0.366668	+	0.930352i	$0.619502\pi$
$60$	0	0
$61$	24024.9	0.826679	0.413340	−	0.910577i	$-0.364362\pi$
$61$	24024.9	0.826679	0.413340	+	0.910577i	$0.364362\pi$
$62$	59318.2	1.95979
$63$	0	0
$64$	6917.49	0.211105
$65$	8829.21	0.259202
$66$	0	0
$67$	55181.2	1.50177	0.750886	−	0.660432i	$-0.229627\pi$
$67$	55181.2	1.50177	0.750886	+	0.660432i	$0.229627\pi$
$68$	31930.0	0.837387
$69$	0	0
$70$	−13989.4	−0.341236
$71$	41626.8	0.980002	0.490001	−	0.871722i	$-0.336996\pi$
$71$	41626.8	0.980002	0.490001	+	0.871722i	$0.336996\pi$
$72$	0	0
$73$	−15707.6	−0.344987	−0.172493	−	0.985011i	$-0.555182\pi$
$73$	−15707.6	−0.344987	−0.172493	+	0.985011i	$0.555182\pi$
$74$	80370.5	1.70615
$75$	0	0
$76$	36822.3	0.731268
$77$	−5533.80	−0.106365
$78$	0	0
$79$	56772.8	1.02346	0.511732	−	0.859145i	$-0.329004\pi$
$79$	56772.8	1.02346	0.511732	+	0.859145i	$0.329004\pi$
$80$	−69429.5	−1.21288
$81$	0	0
$82$	−65096.8	−1.06912
$83$	−107406.	−1.71133	−0.855664	−	0.517532i	$-0.826851\pi$
$83$	−107406.	−1.71133	−0.855664	+	0.517532i	$0.826851\pi$
$84$	0	0
$85$	117449.	1.76320
$86$	−101446.	−1.47907
$87$	0	0
$88$	−17315.1	−0.238352
$89$	−37615.4	−0.503374	−0.251687	−	0.967809i	$-0.580985\pi$
$89$	−37615.4	−0.503374	−0.251687	+	0.967809i	$0.580985\pi$
$90$	0	0
$91$	−6124.36	−0.0775278
$92$	1810.07	0.0222960
$93$	0	0
$94$	26173.0	0.305516
$95$	135444.	1.53975
$96$	0	0
$97$	−43944.4	−0.474214	−0.237107	−	0.971484i	$-0.576199\pi$
$97$	−43944.4	−0.474214	−0.237107	+	0.971484i	$0.576199\pi$
$98$	−105230.	−1.10681
$99$	0	0
$100$	−2595.33	−0.0259533
$101$	−71165.6	−0.694171	−0.347086	−	0.937833i	$-0.612829\pi$
$101$	−71165.6	−0.694171	−0.347086	+	0.937833i	$0.612829\pi$
$102$	0	0
$103$	−36173.8	−0.335970	−0.167985	−	0.985790i	$-0.553726\pi$
$103$	−36173.8	−0.335970	−0.167985	+	0.985790i	$0.553726\pi$
$104$	−19162.9	−0.173732
$105$	0	0
$106$	227090.	1.96306
$107$	−44407.3	−0.374969	−0.187484	−	0.982268i	$-0.560033\pi$
$107$	−44407.3	−0.374969	−0.187484	+	0.982268i	$0.560033\pi$
$108$	0	0
$109$	−142245.	−1.14676	−0.573379	−	0.819290i	$-0.694368\pi$
$109$	−142245.	−1.14676	−0.573379	+	0.819290i	$0.694368\pi$
$110$	54555.8	0.429891
$111$	0	0
$112$	48159.6	0.362776
$113$	−29934.7	−0.220536	−0.110268	−	0.993902i	$-0.535171\pi$
$113$	−29934.7	−0.220536	−0.110268	+	0.993902i	$0.535171\pi$
$114$	0	0
$115$	6658.03	0.0469463
$116$	−12416.5	−0.0856750
$117$	0	0
$118$	−134088.	−0.886510
$119$	−81468.0	−0.527375
$120$	0	0
$121$	−139470.	−0.866001
$122$	164292.	0.999350
$123$	0	0
$124$	128066.	0.747965
$125$	−179255.	−1.02611
$126$	0	0
$127$	219282.	1.20641	0.603204	−	0.797587i	$-0.293890\pi$
$127$	219282.	1.20641	0.603204	+	0.797587i	$0.293890\pi$
$128$	206376.	1.11336
$129$	0	0
$130$	60377.8	0.313342
$131$	−207170.	−1.05475	−0.527374	−	0.849633i	$-0.676823\pi$
$131$	−207170.	−1.05475	−0.527374	+	0.849633i	$0.676823\pi$
$132$	0	0
$133$	−93950.5	−0.460543
$134$	377352.	1.81545
$135$	0	0
$136$	−254911.	−1.18179
$137$	−11815.8	−0.0537852	−0.0268926	−	0.999638i	$-0.508561\pi$
$137$	−11815.8	−0.0537852	−0.0268926	+	0.999638i	$0.508561\pi$
$138$	0	0
$139$	−51342.5	−0.225393	−0.112696	−	0.993629i	$-0.535949\pi$
$139$	−51342.5	−0.225393	−0.112696	+	0.993629i	$0.535949\pi$
$140$	−30202.8	−0.130235
$141$	0	0
$142$	284661.	1.18470
$143$	23883.7	0.0976699
$144$	0	0
$145$	−45671.9	−0.180397
$146$	−107415.	−0.417045
$147$	0	0
$148$	173518.	0.651163
$149$	−159538.	−0.588707	−0.294353	−	0.955697i	$-0.595104\pi$
$149$	−159538.	−0.588707	−0.294353	+	0.955697i	$0.595104\pi$
$150$	0	0
$151$	64145.0	0.228939	0.114470	−	0.993427i	$-0.463483\pi$
$151$	64145.0	0.228939	0.114470	+	0.993427i	$0.463483\pi$
$152$	−293968.	−1.03203
$153$	0	0
$154$	−37842.5	−0.128581
$155$	471070.	1.57491
$156$	0	0
$157$	556965.	1.80335	0.901673	−	0.432419i	$-0.142340\pi$
$157$	556965.	1.80335	0.901673	+	0.432419i	$0.142340\pi$
$158$	388236.	1.23724
$159$	0	0
$160$	−269957.	−0.833671
$161$	−4618.32	−0.0140417
$162$	0	0
$163$	−175236.	−0.516599	−0.258299	−	0.966065i	$-0.583162\pi$
$163$	−175236.	−0.516599	−0.258299	+	0.966065i	$0.583162\pi$
$164$	−140542.	−0.408035
$165$	0	0
$166$	−734487.	−2.06878
$167$	−464707.	−1.28940	−0.644700	−	0.764436i	$-0.723018\pi$
$167$	−464707.	−1.28940	−0.644700	+	0.764436i	$0.723018\pi$
$168$	0	0
$169$	−344861.	−0.928810
$170$	803164.	2.13148
$171$	0	0
$172$	−219019.	−0.564497
$173$	92110.9	0.233989	0.116995	−	0.993133i	$-0.462674\pi$
$173$	92110.9	0.233989	0.116995	+	0.993133i	$0.462674\pi$
$174$	0	0
$175$	6621.88	0.0163450
$176$	−187812.	−0.457027
$177$	0	0
$178$	−257230.	−0.608516
$179$	−46322.2	−0.108058	−0.0540290	−	0.998539i	$-0.517206\pi$
$179$	−46322.2	−0.108058	−0.0540290	+	0.998539i	$0.517206\pi$
$180$	0	0
$181$	118183.	0.268139	0.134069	−	0.990972i	$-0.457196\pi$
$181$	118183.	0.268139	0.134069	+	0.990972i	$0.457196\pi$
$182$	−41880.9	−0.0937212
$183$	0	0
$184$	−14450.6	−0.0314660
$185$	638255.	1.37109
$186$	0	0
$187$	317707.	0.664390
$188$	56506.9	0.116602
$189$	0	0
$190$	926224.	1.86137
$191$	349094.	0.692404	0.346202	−	0.938160i	$-0.387471\pi$
$191$	349094.	0.692404	0.346202	+	0.938160i	$0.387471\pi$
$192$	0	0
$193$	−128726.	−0.248756	−0.124378	−	0.992235i	$-0.539694\pi$
$193$	−128726.	−0.248756	−0.124378	+	0.992235i	$0.539694\pi$
$194$	−300510.	−0.573264
$195$	0	0
$196$	−227188.	−0.422420
$197$	−615954.	−1.13079	−0.565396	−	0.824820i	$-0.691276\pi$
$197$	−615954.	−1.13079	−0.565396	+	0.824820i	$0.691276\pi$
$198$	0	0
$199$	−333500.	−0.596985	−0.298492	−	0.954412i	$-0.596484\pi$
$199$	−333500.	−0.596985	−0.298492	+	0.954412i	$0.596484\pi$
$200$	20719.7	0.0366275
$201$	0	0
$202$	−486660.	−0.839165
$203$	31680.2	0.0539570
$204$	0	0
$205$	−516960.	−0.859156
$206$	−247371.	−0.406145
$207$	0	0
$208$	−207855.	−0.333121
$209$	366386.	0.580195
$210$	0	0
$211$	246416.	0.381033	0.190516	−	0.981684i	$-0.438984\pi$
$211$	246416.	0.381033	0.190516	+	0.981684i	$0.438984\pi$
$212$	490280.	0.749212
$213$	0	0
$214$	−303676.	−0.453290
$215$	−805624.	−1.18860
$216$	0	0
$217$	−326756.	−0.471058
$218$	−972733.	−1.38628
$219$	0	0
$220$	117784.	0.164071
$221$	351613.	0.484266
$222$	0	0
$223$	−275147.	−0.370512	−0.185256	−	0.982690i	$-0.559311\pi$
$223$	−275147.	−0.370512	−0.185256	+	0.982690i	$0.559311\pi$
$224$	187255.	0.249352
$225$	0	0
$226$	−204706.	−0.266600
$227$	461856.	0.594897	0.297448	−	0.954738i	$-0.403864\pi$
$227$	461856.	0.594897	0.297448	+	0.954738i	$0.403864\pi$
$228$	0	0
$229$	−311105.	−0.392029	−0.196015	−	0.980601i	$-0.562800\pi$
$229$	−311105.	−0.392029	−0.196015	+	0.980601i	$0.562800\pi$
$230$	45530.4	0.0567521
$231$	0	0
$232$	99126.3	0.120912
$233$	−1.52969e6	−1.84592	−0.922962	−	0.384891i	$-0.874239\pi$
$233$	−1.52969e6	−1.84592	−0.922962	+	0.384891i	$0.874239\pi$
$234$	0	0
$235$	207851.	0.245517
$236$	−289492.	−0.338342
$237$	0	0
$238$	−557113.	−0.637530
$239$	−1.21977e6	−1.38129	−0.690644	−	0.723195i	$-0.742673\pi$
$239$	−1.21977e6	−1.38129	−0.690644	+	0.723195i	$0.742673\pi$
$240$	0	0
$241$	563284.	0.624719	0.312360	−	0.949964i	$-0.398881\pi$
$241$	563284.	0.624719	0.312360	+	0.949964i	$0.398881\pi$
$242$	−953757.	−1.04689
$243$	0	0
$244$	354703.	0.381408
$245$	−835670.	−0.889447
$246$	0	0
$247$	405486.	0.422896
$248$	−1.02241e6	−1.05559
$249$	0	0
$250$	−1.22582e6	−1.24044
$251$	1.52639e6	1.52926	0.764629	−	0.644470i	$-0.222922\pi$
$251$	1.52639e6	1.52926	0.764629	+	0.644470i	$0.222922\pi$
$252$	0	0
$253$	18010.4	0.0176898
$254$	1.49954e6	1.45839
$255$	0	0
$256$	1.18993e6	1.13480
$257$	−197606.	−0.186624	−0.0933121	−	0.995637i	$-0.529745\pi$
$257$	−197606.	−0.186624	−0.0933121	+	0.995637i	$0.529745\pi$
$258$	0	0
$259$	−442724.	−0.410094
$260$	130354.	0.119589
$261$	0	0
$262$	−1.41671e6	−1.27506
$263$	2.03558e6	1.81468	0.907339	−	0.420400i	$-0.138110\pi$
$263$	2.03558e6	1.81468	0.907339	+	0.420400i	$0.138110\pi$
$264$	0	0
$265$	1.80341e6	1.57754
$266$	−642473.	−0.556738
$267$	0	0
$268$	814693.	0.692878
$269$	728283.	0.613648	0.306824	−	0.951766i	$-0.400734\pi$
$269$	728283.	0.613648	0.306824	+	0.951766i	$0.400734\pi$
$270$	0	0
$271$	−878997.	−0.727049	−0.363525	−	0.931585i	$-0.618427\pi$
$271$	−878997.	−0.727049	−0.363525	+	0.931585i	$0.618427\pi$
$272$	−2.76495e6	−2.26602
$273$	0	0
$274$	−80801.6	−0.0650195
$275$	−25823.9	−0.0205916
$276$	0	0
$277$	−981143.	−0.768304	−0.384152	−	0.923270i	$-0.625506\pi$
$277$	−981143.	−0.768304	−0.384152	+	0.923270i	$0.625506\pi$
$278$	−351102.	−0.272471
$279$	0	0
$280$	241123.	0.183799
$281$	−1.45423e6	−1.09867	−0.549333	−	0.835603i	$-0.685118\pi$
$281$	−1.45423e6	−1.09867	−0.549333	+	0.835603i	$0.685118\pi$
$282$	0	0
$283$	1.81952e6	1.35049	0.675243	−	0.737595i	$-0.264039\pi$
$283$	1.81952e6	1.35049	0.675243	+	0.737595i	$0.264039\pi$
$284$	614576.	0.452147
$285$	0	0
$286$	163326.	0.118070
$287$	358588.	0.256975
$288$	0	0
$289$	3.25740e6	2.29417
$290$	−312323.	−0.218077
$291$	0	0
$292$	−231906.	−0.159168
$293$	−279641.	−0.190297	−0.0951484	−	0.995463i	$-0.530333\pi$
$293$	−279641.	−0.190297	−0.0951484	+	0.995463i	$0.530333\pi$
$294$	0	0
$295$	−1.06484e6	−0.712412
$296$	−1.38527e6	−0.918978
$297$	0	0
$298$	−1.09099e6	−0.711672
$299$	19932.5	0.0128939
$300$	0	0
$301$	558819.	0.355513
$302$	438650.	0.276759
$303$	0	0
$304$	−3.18859e6	−1.97886
$305$	1.30471e6	0.803091
$306$	0	0
$307$	−1.65629e6	−1.00297	−0.501487	−	0.865165i	$-0.667214\pi$
$307$	−1.65629e6	−1.00297	−0.501487	+	0.865165i	$0.667214\pi$
$308$	−81700.9	−0.0490738
$309$	0	0
$310$	3.22137e6	1.90387
$311$	1.33517e6	0.782771	0.391386	−	0.920227i	$-0.371996\pi$
$311$	1.33517e6	0.782771	0.391386	+	0.920227i	$0.371996\pi$
$312$	0	0
$313$	−2.25773e6	−1.30260	−0.651300	−	0.758821i	$-0.725776\pi$
$313$	−2.25773e6	−1.30260	−0.651300	+	0.758821i	$0.725776\pi$
$314$	3.80876e6	2.18002
$315$	0	0
$316$	838191.	0.472199
$317$	−2.76702e6	−1.54655	−0.773274	−	0.634072i	$-0.781382\pi$
$317$	−2.76702e6	−1.54655	−0.773274	+	0.634072i	$0.781382\pi$
$318$	0	0
$319$	−123546.	−0.0679753
$320$	375666.	0.205082
$321$	0	0
$322$	−31582.0	−0.0169746
$323$	5.39390e6	2.87671
$324$	0	0
$325$	−28579.7	−0.0150089
$326$	−1.19833e6	−0.624503
$327$	0	0
$328$	1.12201e6	0.575854
$329$	−144175.	−0.0734346
$330$	0	0
$331$	1.50762e6	0.756350	0.378175	−	0.925734i	$-0.376552\pi$
$331$	1.50762e6	0.756350	0.378175	+	0.925734i	$0.376552\pi$
$332$	−1.58574e6	−0.789562
$333$	0	0
$334$	−3.17786e6	−1.55872
$335$	2.99670e6	1.45892
$336$	0	0
$337$	2.65301e6	1.27252	0.636259	−	0.771476i	$-0.280481\pi$
$337$	2.65301e6	1.27252	0.636259	+	0.771476i	$0.280481\pi$
$338$	−2.35830e6	−1.12281
$339$	0	0
$340$	1.73401e6	0.813493
$341$	1.27428e6	0.593442
$342$	0	0
$343$	1.21277e6	0.556602
$344$	1.74853e6	0.796666
$345$	0	0
$346$	629893.	0.282863
$347$	1.30221e6	0.580574	0.290287	−	0.956940i	$-0.406249\pi$
$347$	1.30221e6	0.580574	0.290287	+	0.956940i	$0.406249\pi$
$348$	0	0
$349$	−1.20351e6	−0.528914	−0.264457	−	0.964397i	$-0.585193\pi$
$349$	−1.20351e6	−0.528914	−0.264457	+	0.964397i	$0.585193\pi$
$350$	45283.2	0.0197591
$351$	0	0
$352$	−730253.	−0.314135
$353$	1.97691e6	0.844404	0.422202	−	0.906502i	$-0.361257\pi$
$353$	1.97691e6	0.844404	0.422202	+	0.906502i	$0.361257\pi$
$354$	0	0
$355$	2.26061e6	0.952039
$356$	−555353.	−0.232244
$357$	0	0
$358$	−316771.	−0.130628
$359$	2.29106e6	0.938209	0.469105	−	0.883143i	$-0.344577\pi$
$359$	2.29106e6	0.938209	0.469105	+	0.883143i	$0.344577\pi$
$360$	0	0
$361$	3.74425e6	1.51216
$362$	808187.	0.324146
$363$	0	0
$364$	−90419.8	−0.0357693
$365$	−853027.	−0.335143
$366$	0	0
$367$	755682.	0.292869	0.146435	−	0.989220i	$-0.453220\pi$
$367$	755682.	0.292869	0.146435	+	0.989220i	$0.453220\pi$
$368$	−156741.	−0.0603343
$369$	0	0
$370$	4.36466e6	1.65747
$371$	−1.25093e6	−0.471844
$372$	0	0
$373$	−4.45947e6	−1.65963	−0.829815	−	0.558038i	$-0.811554\pi$
$373$	−4.45947e6	−1.65963	−0.829815	+	0.558038i	$0.811554\pi$
$374$	2.17262e6	0.803164
$375$	0	0
$376$	−451119.	−0.164559
$377$	−136730.	−0.0495463
$378$	0	0
$379$	4.02987e6	1.44109	0.720547	−	0.693406i	$-0.243891\pi$
$379$	4.02987e6	1.44109	0.720547	+	0.693406i	$0.243891\pi$
$380$	1.99969e6	0.710402
$381$	0	0
$382$	2.38725e6	0.837028
$383$	−4.12660e6	−1.43746	−0.718729	−	0.695291i	$-0.755276\pi$
$383$	−4.12660e6	−1.43746	−0.718729	+	0.695291i	$0.755276\pi$
$384$	0	0
$385$	−300522.	−0.103330
$386$	−880284.	−0.300715
$387$	0	0
$388$	−648793.	−0.218790
$389$	1.37522e6	0.460784	0.230392	−	0.973098i	$-0.425999\pi$
$389$	1.37522e6	0.460784	0.230392	+	0.973098i	$0.425999\pi$
$390$	0	0
$391$	265148.	0.0877094
$392$	1.81374e6	0.596156
$393$	0	0
$394$	−4.21215e6	−1.36698
$395$	3.08314e6	0.994261
$396$	0	0
$397$	3.95058e6	1.25801	0.629006	−	0.777400i	$-0.283462\pi$
$397$	3.95058e6	1.25801	0.629006	+	0.777400i	$0.283462\pi$
$398$	−2.28061e6	−0.721679
$399$	0	0
$400$	224740.	0.0702313
$401$	−601236.	−0.186717	−0.0933585	−	0.995633i	$-0.529760\pi$
$401$	−601236.	−0.186717	−0.0933585	+	0.995633i	$0.529760\pi$
$402$	0	0
$403$	1.41027e6	0.432552
$404$	−1.05069e6	−0.320272
$405$	0	0
$406$	216642.	0.0652272
$407$	1.72653e6	0.516639
$408$	0	0
$409$	6.38666e6	1.88784	0.943921	−	0.330171i	$-0.107106\pi$
$409$	6.38666e6	1.88784	0.943921	+	0.330171i	$0.107106\pi$
$410$	−3.53519e6	−1.03861
$411$	0	0
$412$	−534068.	−0.155008
$413$	738627.	0.213084
$414$	0	0
$415$	−5.83286e6	−1.66250
$416$	−808185.	−0.228969
$417$	0	0
$418$	2.50550e6	0.701381
$419$	−4.44705e6	−1.23748	−0.618738	−	0.785597i	$-0.712356\pi$
$419$	−4.44705e6	−1.23748	−0.618738	+	0.785597i	$0.712356\pi$
$420$	0	0
$421$	413525.	0.113710	0.0568548	−	0.998382i	$-0.481893\pi$
$421$	413525.	0.113710	0.0568548	+	0.998382i	$0.481893\pi$
$422$	1.68509e6	0.460620
$423$	0	0
$424$	−3.91412e6	−1.05735
$425$	−380176.	−0.102097
$426$	0	0
$427$	−905010.	−0.240206
$428$	−655628.	−0.173001
$429$	0	0
$430$	−5.50920e6	−1.43687
$431$	3.32051e6	0.861016	0.430508	−	0.902587i	$-0.358334\pi$
$431$	3.32051e6	0.861016	0.430508	+	0.902587i	$0.358334\pi$
$432$	0	0
$433$	1.14444e6	0.293341	0.146670	−	0.989185i	$-0.453144\pi$
$433$	1.14444e6	0.293341	0.146670	+	0.989185i	$0.453144\pi$
$434$	−2.23450e6	−0.569450
$435$	0	0
$436$	−2.10011e6	−0.529084
$437$	305774.	0.0765943
$438$	0	0
$439$	900181.	0.222930	0.111465	−	0.993768i	$-0.464446\pi$
$439$	900181.	0.222930	0.111465	+	0.993768i	$0.464446\pi$
$440$	−940325.	−0.231551
$441$	0	0
$442$	2.40447e6	0.585416
$443$	−3.62823e6	−0.878386	−0.439193	−	0.898393i	$-0.644736\pi$
$443$	−3.62823e6	−0.878386	−0.439193	+	0.898393i	$0.644736\pi$
$444$	0	0
$445$	−2.04277e6	−0.489012
$446$	−1.88157e6	−0.447902
$447$	0	0
$448$	−260580.	−0.0613403
$449$	2.10851e6	0.493583	0.246791	−	0.969069i	$-0.420624\pi$
$449$	2.10851e6	0.493583	0.246791	+	0.969069i	$0.420624\pi$
$450$	0	0
$451$	−1.39841e6	−0.323739
$452$	−441955.	−0.101749
$453$	0	0
$454$	3.15836e6	0.719155
$455$	−332593.	−0.0753157
$456$	0	0
$457$	157338.	0.0352407	0.0176203	−	0.999845i	$-0.494391\pi$
$457$	157338.	0.0352407	0.0176203	+	0.999845i	$0.494391\pi$
$458$	−2.12747e6	−0.473914
$459$	0	0
$460$	98298.9	0.0216598
$461$	79527.0	0.0174286	0.00871429	−	0.999962i	$-0.497226\pi$
$461$	79527.0	0.0174286	0.00871429	+	0.999962i	$0.497226\pi$
$462$	0	0
$463$	421922.	0.0914703	0.0457352	−	0.998954i	$-0.485437\pi$
$463$	421922.	0.0914703	0.0457352	+	0.998954i	$0.485437\pi$
$464$	1.07520e6	0.231842
$465$	0	0
$466$	−1.04607e7	−2.23149
$467$	−3.56062e6	−0.755498	−0.377749	−	0.925908i	$-0.623302\pi$
$467$	−3.56062e6	−0.755498	−0.377749	+	0.925908i	$0.623302\pi$
$468$	0	0
$469$	−2.07866e6	−0.436366
$470$	1.42137e6	0.296799
$471$	0	0
$472$	2.31114e6	0.477498
$473$	−2.17927e6	−0.447877
$474$	0	0
$475$	−438427.	−0.0891585
$476$	−1.20279e6	−0.243317
$477$	0	0
$478$	−8.34132e6	−1.66980
$479$	−3.58247e6	−0.713417	−0.356708	−	0.934216i	$-0.616101\pi$
$479$	−3.58247e6	−0.713417	−0.356708	+	0.934216i	$0.616101\pi$
$480$	0	0
$481$	1.91078e6	0.376572
$482$	3.85197e6	0.755206
$483$	0	0
$484$	−2.05914e6	−0.399550
$485$	−2.38647e6	−0.460683
$486$	0	0
$487$	8.65020e6	1.65274	0.826368	−	0.563130i	$-0.190403\pi$
$487$	8.65020e6	1.65274	0.826368	+	0.563130i	$0.190403\pi$
$488$	−2.83175e6	−0.538276
$489$	0	0
$490$	−5.71466e6	−1.07523
$491$	−146402.	−0.0274059	−0.0137030	−	0.999906i	$-0.504362\pi$
$491$	−146402.	−0.0274059	−0.0137030	+	0.999906i	$0.504362\pi$
$492$	0	0
$493$	−1.81883e6	−0.337034
$494$	2.77289e6	0.511228
$495$	0	0
$496$	−1.10898e7	−2.02404
$497$	−1.56807e6	−0.284756
$498$	0	0
$499$	−4.24620e6	−0.763394	−0.381697	−	0.924287i	$-0.624660\pi$
$499$	−4.24620e6	−0.763394	−0.381697	+	0.924287i	$0.624660\pi$
$500$	−2.64651e6	−0.473422
$501$	0	0
$502$	1.04381e7	1.84868
$503$	−2.20106e6	−0.387892	−0.193946	−	0.981012i	$-0.562129\pi$
$503$	−2.20106e6	−0.387892	−0.193946	+	0.981012i	$0.562129\pi$
$504$	0	0
$505$	−3.86476e6	−0.674364
$506$	123163.	0.0213847
$507$	0	0
$508$	3.23748e6	0.556605
$509$	1.03089e6	0.176367	0.0881834	−	0.996104i	$-0.471894\pi$
$509$	1.03089e6	0.176367	0.0881834	+	0.996104i	$0.471894\pi$
$510$	0	0
$511$	591700.	0.100242
$512$	1.53318e6	0.258474
$513$	0	0
$514$	−1.35131e6	−0.225605
$515$	−1.96448e6	−0.326384
$516$	0	0
$517$	562251.	0.0925133
$518$	−3.02753e6	−0.495752
$519$	0	0
$520$	−1.04067e6	−0.168774
$521$	−1.09737e7	−1.77117	−0.885583	−	0.464480i	$-0.846241\pi$
$521$	−1.09737e7	−1.77117	−0.885583	+	0.464480i	$0.846241\pi$
$522$	0	0
$523$	−1.91358e6	−0.305909	−0.152954	−	0.988233i	$-0.548879\pi$
$523$	−1.91358e6	−0.305909	−0.152954	+	0.988233i	$0.548879\pi$
$524$	−3.05865e6	−0.486633
$525$	0	0
$526$	1.39202e7	2.19372
$527$	1.87598e7	2.94240
$528$	0	0
$529$	−6.42131e6	−0.997665
$530$	1.23325e7	1.90704
$531$	0	0
$532$	−1.38708e6	−0.212482
$533$	−1.54765e6	−0.235969
$534$	0	0
$535$	−2.41161e6	−0.364270
$536$	−6.50405e6	−0.977849
$537$	0	0
$538$	4.98030e6	0.741822
$539$	−2.26055e6	−0.335152
$540$	0	0
$541$	−3.63173e6	−0.533483	−0.266741	−	0.963768i	$-0.585947\pi$
$541$	−3.63173e6	−0.533483	−0.266741	+	0.963768i	$0.585947\pi$
$542$	−6.01095e6	−0.878910
$543$	0	0
$544$	−1.07507e7	−1.55754
$545$	−7.72487e6	−1.11404
$546$	0	0
$547$	−7.98387e6	−1.14089	−0.570447	−	0.821335i	$-0.693230\pi$
$547$	−7.98387e6	−1.14089	−0.570447	+	0.821335i	$0.693230\pi$
$548$	−174449.	−0.0248151
$549$	0	0
$550$	−176594.	−0.0248926
$551$	−2.09751e6	−0.294323
$552$	0	0
$553$	−2.13861e6	−0.297385
$554$	−6.70947e6	−0.928782
$555$	0	0
$556$	−758019.	−0.103990
$557$	4.40511e6	0.601614	0.300807	−	0.953685i	$-0.402744\pi$
$557$	4.40511e6	0.601614	0.300807	+	0.953685i	$0.402744\pi$
$558$	0	0
$559$	−2.41184e6	−0.326452
$560$	2.61539e6	0.352424
$561$	0	0
$562$	−9.94460e6	−1.32815
$563$	−8.26618e6	−1.09909	−0.549545	−	0.835464i	$-0.685199\pi$
$563$	−8.26618e6	−1.09909	−0.549545	+	0.835464i	$0.685199\pi$
$564$	0	0
$565$	−1.62565e6	−0.214243
$566$	1.24426e7	1.63257
$567$	0	0
$568$	−4.90643e6	−0.638109
$569$	1.12505e7	1.45677	0.728385	−	0.685168i	$-0.240271\pi$
$569$	1.12505e7	1.45677	0.728385	+	0.685168i	$0.240271\pi$
$570$	0	0
$571$	2.47253e6	0.317360	0.158680	−	0.987330i	$-0.449276\pi$
$571$	2.47253e6	0.317360	0.158680	+	0.987330i	$0.449276\pi$
$572$	352617.	0.0450623
$573$	0	0
$574$	2.45217e6	0.310650
$575$	−21551.7	−0.00271839
$576$	0	0
$577$	1.24000e7	1.55054	0.775270	−	0.631630i	$-0.217614\pi$
$577$	1.24000e7	1.55054	0.775270	+	0.631630i	$0.217614\pi$
$578$	2.22754e7	2.77336
$579$	0	0
$580$	−674298.	−0.0832304
$581$	4.04595e6	0.497256
$582$	0	0
$583$	4.87836e6	0.594432
$584$	1.85141e6	0.224631
$585$	0	0
$586$	−1.91230e6	−0.230045
$587$	−5.73588e6	−0.687076	−0.343538	−	0.939139i	$-0.611625\pi$
$587$	−5.73588e6	−0.687076	−0.343538	+	0.939139i	$0.611625\pi$
$588$	0	0
$589$	2.16341e7	2.56952
$590$	−7.28185e6	−0.861215
$591$	0	0
$592$	−1.50256e7	−1.76209
$593$	8.23346e6	0.961492	0.480746	−	0.876860i	$-0.340366\pi$
$593$	8.23346e6	0.961492	0.480746	+	0.876860i	$0.340366\pi$
$594$	0	0
$595$	−4.42426e6	−0.512328
$596$	−2.35542e6	−0.271614
$597$	0	0
$598$	136307.	0.0155871
$599$	2.06935e6	0.235650	0.117825	−	0.993034i	$-0.462408\pi$
$599$	2.06935e6	0.235650	0.117825	+	0.993034i	$0.462408\pi$
$600$	0	0
$601$	2.18287e6	0.246514	0.123257	−	0.992375i	$-0.460666\pi$
$601$	2.18287e6	0.246514	0.123257	+	0.992375i	$0.460666\pi$
$602$	3.82144e6	0.429770
$603$	0	0
$604$	947034.	0.105627
$605$	−7.57417e6	−0.841292
$606$	0	0
$607$	−7.87116e6	−0.867095	−0.433548	−	0.901131i	$-0.642738\pi$
$607$	−7.87116e6	−0.867095	−0.433548	+	0.901131i	$0.642738\pi$
$608$	−1.23979e7	−1.36016
$609$	0	0
$610$	8.92217e6	0.970836
$611$	622253.	0.0674317
$612$	0	0
$613$	1.01857e7	1.09481	0.547406	−	0.836867i	$-0.315615\pi$
$613$	1.01857e7	1.09481	0.547406	+	0.836867i	$0.315615\pi$
$614$	−1.13264e7	−1.21247
$615$	0	0
$616$	652254.	0.0692572
$617$	−1.28805e7	−1.36213	−0.681067	−	0.732221i	$-0.738484\pi$
$617$	−1.28805e7	−1.36213	−0.681067	+	0.732221i	$0.738484\pi$
$618$	0	0
$619$	2.22093e6	0.232974	0.116487	−	0.993192i	$-0.462837\pi$
$619$	2.22093e6	0.232974	0.116487	+	0.993192i	$0.462837\pi$
$620$	6.95486e6	0.726623
$621$	0	0
$622$	9.13043e6	0.946271
$623$	1.41696e6	0.146264
$624$	0	0
$625$	−9.18539e6	−0.940584
$626$	−1.54393e7	−1.57468
$627$	0	0
$628$	8.22301e6	0.832017
$629$	2.54177e7	2.56159
$630$	0	0
$631$	−1.12269e7	−1.12250	−0.561252	−	0.827645i	$-0.689680\pi$
$631$	−1.12269e7	−1.12250	−0.561252	+	0.827645i	$0.689680\pi$
$632$	−6.69165e6	−0.666408
$633$	0	0
$634$	−1.89220e7	−1.86958
$635$	1.19085e7	1.17199
$636$	0	0
$637$	−2.50179e6	−0.244288
$638$	−844858.	−0.0821735
$639$	0	0
$640$	1.12076e7	1.08159
$641$	−1.73945e6	−0.167212	−0.0836060	−	0.996499i	$-0.526644\pi$
$641$	−1.73945e6	−0.167212	−0.0836060	+	0.996499i	$0.526644\pi$
$642$	0	0
$643$	6.08165e6	0.580088	0.290044	−	0.957013i	$-0.406330\pi$
$643$	6.08165e6	0.580088	0.290044	+	0.957013i	$0.406330\pi$
$644$	−68184.8	−0.00647847
$645$	0	0
$646$	3.68857e7	3.47758
$647$	2.01823e6	0.189544	0.0947721	−	0.995499i	$-0.469788\pi$
$647$	2.01823e6	0.189544	0.0947721	+	0.995499i	$0.469788\pi$
$648$	0	0
$649$	−2.88048e6	−0.268444
$650$	−195440.	−0.0181439
$651$	0	0
$652$	−2.58717e6	−0.238345
$653$	−5.21279e6	−0.478396	−0.239198	−	0.970971i	$-0.576884\pi$
$653$	−5.21279e6	−0.478396	−0.239198	+	0.970971i	$0.576884\pi$
$654$	0	0
$655$	−1.12507e7	−1.02465
$656$	1.21701e7	1.10417
$657$	0	0
$658$	−985930.	−0.0887731
$659$	−7.29296e6	−0.654170	−0.327085	−	0.944995i	$-0.606066\pi$
$659$	−7.29296e6	−0.654170	−0.327085	+	0.944995i	$0.606066\pi$
$660$	0	0
$661$	−1.08067e7	−0.962034	−0.481017	−	0.876711i	$-0.659732\pi$
$661$	−1.08067e7	−0.962034	−0.481017	+	0.876711i	$0.659732\pi$
$662$	1.03098e7	0.914331
$663$	0	0
$664$	1.26596e7	1.11430
$665$	−5.10214e6	−0.447402
$666$	0	0
$667$	−103107.	−0.00897375
$668$	−6.86091e6	−0.594895
$669$	0	0
$670$	2.04927e7	1.76365
$671$	3.52934e6	0.302613
$672$	0	0
$673$	3.23697e6	0.275487	0.137743	−	0.990468i	$-0.456015\pi$
$673$	3.23697e6	0.275487	0.137743	+	0.990468i	$0.456015\pi$
$674$	1.81424e7	1.53831
$675$	0	0
$676$	−5.09151e6	−0.428529
$677$	6.00931e6	0.503910	0.251955	−	0.967739i	$-0.418927\pi$
$677$	6.00931e6	0.503910	0.251955	+	0.967739i	$0.418927\pi$
$678$	0	0
$679$	1.65537e6	0.137791
$680$	−1.38434e7	−1.14807
$681$	0	0
$682$	8.71405e6	0.717396
$683$	7.39622e6	0.606678	0.303339	−	0.952883i	$-0.401899\pi$
$683$	7.39622e6	0.606678	0.303339	+	0.952883i	$0.401899\pi$
$684$	0	0
$685$	−641678.	−0.0522506
$686$	8.29346e6	0.672861
$687$	0	0
$688$	1.89658e7	1.52757
$689$	5.39896e6	0.433274
$690$	0	0
$691$	−1.10125e7	−0.877383	−0.438692	−	0.898638i	$-0.644558\pi$
$691$	−1.10125e7	−0.877383	−0.438692	+	0.898638i	$0.644558\pi$
$692$	1.35992e6	0.107956
$693$	0	0
$694$	8.90506e6	0.701840
$695$	−2.78824e6	−0.218962
$696$	0	0
$697$	−2.05873e7	−1.60516
$698$	−8.23008e6	−0.639390
$699$	0	0
$700$	97765.2	0.00754118
$701$	−1.97899e7	−1.52106	−0.760532	−	0.649300i	$-0.775062\pi$
$701$	−1.97899e7	−1.52106	−0.760532	+	0.649300i	$0.775062\pi$
$702$	0	0
$703$	2.93122e7	2.23697
$704$	1.01620e6	0.0772768
$705$	0	0
$706$	1.35189e7	1.02078
$707$	2.68078e6	0.201703
$708$	0	0
$709$	−1.20373e7	−0.899317	−0.449658	−	0.893201i	$-0.648454\pi$
$709$	−1.20373e7	−0.899317	−0.449658	+	0.893201i	$0.648454\pi$
$710$	1.54590e7	1.15089
$711$	0	0
$712$	4.43363e6	0.327762
$713$	1.06347e6	0.0783431
$714$	0	0
$715$	1.29704e6	0.0948831
$716$	−683900.	−0.0498551
$717$	0	0
$718$	1.56672e7	1.13418
$719$	1.01004e7	0.728643	0.364322	−	0.931273i	$-0.381301\pi$
$719$	1.01004e7	0.728643	0.364322	+	0.931273i	$0.381301\pi$
$720$	0	0
$721$	1.36265e6	0.0976219
$722$	2.56048e7	1.82801
$723$	0	0
$724$	1.74485e6	0.123712
$725$	147838.	0.0104458
$726$	0	0
$727$	2.01441e7	1.41356	0.706778	−	0.707436i	$-0.250148\pi$
$727$	2.01441e7	1.41356	0.706778	+	0.707436i	$0.250148\pi$
$728$	721861.	0.0504807
$729$	0	0
$730$	−5.83335e6	−0.405146
$731$	−3.20830e7	−2.22066
$732$	0	0
$733$	−2.42377e7	−1.66622	−0.833110	−	0.553108i	$-0.813442\pi$
$733$	−2.42377e7	−1.66622	−0.833110	+	0.553108i	$0.813442\pi$
$734$	5.16767e6	0.354042
$735$	0	0
$736$	−609445.	−0.0414705
$737$	8.10630e6	0.549736
$738$	0	0
$739$	−6.78569e6	−0.457070	−0.228535	−	0.973536i	$-0.573394\pi$
$739$	−6.78569e6	−0.457070	−0.228535	+	0.973536i	$0.573394\pi$
$740$	9.42317e6	0.632584
$741$	0	0
$742$	−8.55439e6	−0.570400
$743$	−4.71605e6	−0.313406	−0.156703	−	0.987646i	$-0.550086\pi$
$743$	−4.71605e6	−0.313406	−0.156703	+	0.987646i	$0.550086\pi$
$744$	0	0
$745$	−8.66398e6	−0.571909
$746$	−3.04957e7	−2.00628
$747$	0	0
$748$	4.69062e6	0.306532
$749$	1.67281e6	0.108954
$750$	0	0
$751$	−1.69859e7	−1.09898	−0.549490	−	0.835501i	$-0.685178\pi$
$751$	−1.69859e7	−1.09898	−0.549490	+	0.835501i	$0.685178\pi$
$752$	−4.89316e6	−0.315533
$753$	0	0
$754$	−935019.	−0.0598952
$755$	3.48350e6	0.222407
$756$	0	0
$757$	−2.07327e7	−1.31497	−0.657486	−	0.753467i	$-0.728380\pi$
$757$	−2.07327e7	−1.31497	−0.657486	+	0.753467i	$0.728380\pi$
$758$	2.75579e7	1.74210
$759$	0	0
$760$	−1.59644e7	−1.00258
$761$	52400.3	0.00327999	0.00163999	−	0.999999i	$-0.499478\pi$
$761$	52400.3	0.00327999	0.00163999	+	0.999999i	$0.499478\pi$
$762$	0	0
$763$	5.35834e6	0.333210
$764$	5.15402e6	0.319457
$765$	0	0
$766$	−2.82194e7	−1.73770
$767$	−3.18788e6	−0.195665
$768$	0	0
$769$	−2.09357e7	−1.27665	−0.638325	−	0.769767i	$-0.720373\pi$
$769$	−2.09357e7	−1.27665	−0.638325	+	0.769767i	$0.720373\pi$
$770$	−2.05510e6	−0.124912
$771$	0	0
$772$	−1.90051e6	−0.114770
$773$	−2.78692e7	−1.67755	−0.838777	−	0.544476i	$-0.816729\pi$
$773$	−2.78692e7	−1.67755	−0.838777	+	0.544476i	$0.816729\pi$
$774$	0	0
$775$	−1.52483e6	−0.0911942
$776$	5.17960e6	0.308775
$777$	0	0
$778$	9.40431e6	0.557029
$779$	−2.37417e7	−1.40174
$780$	0	0
$781$	6.11512e6	0.358738
$782$	1.81319e6	0.106030
$783$	0	0
$784$	1.96731e7	1.14310
$785$	3.02469e7	1.75189
$786$	0	0
$787$	1.87393e7	1.07849	0.539245	−	0.842149i	$-0.318710\pi$
$787$	1.87393e7	1.07849	0.539245	+	0.842149i	$0.318710\pi$
$788$	−9.09392e6	−0.521717
$789$	0	0
$790$	2.10838e7	1.20194
$791$	1.12763e6	0.0640805
$792$	0	0
$793$	3.90598e6	0.220571
$794$	2.70157e7	1.52078
$795$	0	0
$796$	−4.92378e6	−0.275433
$797$	2.79661e6	0.155950	0.0779751	−	0.996955i	$-0.475155\pi$
$797$	2.79661e6	0.155950	0.0779751	+	0.996955i	$0.475155\pi$
$798$	0	0
$799$	8.27740e6	0.458698
$800$	873838.	0.0482732
$801$	0	0
$802$	−4.11150e6	−0.225717
$803$	−2.30750e6	−0.126285
$804$	0	0
$805$	−250806.	−0.0136410
$806$	9.64399e6	0.522901
$807$	0	0
$808$	8.38809e6	0.451996
$809$	−2.31444e7	−1.24330	−0.621648	−	0.783296i	$-0.713537\pi$
$809$	−2.31444e7	−1.24330	−0.621648	+	0.783296i	$0.713537\pi$
$810$	0	0
$811$	1.44543e7	0.771694	0.385847	−	0.922563i	$-0.373909\pi$
$811$	1.44543e7	0.771694	0.385847	+	0.922563i	$0.373909\pi$
$812$	467725.	0.0248943
$813$	0	0
$814$	1.18067e7	0.624551
$815$	−9.51646e6	−0.501859
$816$	0	0
$817$	−3.69987e7	−1.93924
$818$	4.36747e7	2.28216
$819$	0	0
$820$	−7.63238e6	−0.396392
$821$	−1.80178e7	−0.932920	−0.466460	−	0.884542i	$-0.654471\pi$
$821$	−1.80178e7	−0.932920	−0.466460	+	0.884542i	$0.654471\pi$
$822$	0	0
$823$	6.65519e6	0.342500	0.171250	−	0.985228i	$-0.445219\pi$
$823$	6.65519e6	0.342500	0.171250	+	0.985228i	$0.445219\pi$
$824$	4.26370e6	0.218760
$825$	0	0
$826$	5.05104e6	0.257591
$827$	−5.27683e6	−0.268293	−0.134146	−	0.990962i	$-0.542829\pi$
$827$	−5.27683e6	−0.268293	−0.134146	+	0.990962i	$0.542829\pi$
$828$	0	0
$829$	−724962.	−0.0366378	−0.0183189	−	0.999832i	$-0.505831\pi$
$829$	−724962.	−0.0366378	−0.0183189	+	0.999832i	$0.505831\pi$
$830$	−3.98875e7	−2.00975
$831$	0	0
$832$	1.12465e6	0.0563261
$833$	−3.32796e7	−1.66175
$834$	0	0
$835$	−2.52367e7	−1.25261
$836$	5.40932e6	0.267687
$837$	0	0
$838$	−3.04108e7	−1.49595
$839$	−1.71120e7	−0.839256	−0.419628	−	0.907696i	$-0.637840\pi$
$839$	−1.71120e7	−0.839256	−0.419628	+	0.907696i	$0.637840\pi$
$840$	0	0
$841$	707281.	0.0344828
$842$	2.82786e6	0.137460
$843$	0	0
$844$	3.63808e6	0.175799
$845$	−1.87282e7	−0.902308
$846$	0	0
$847$	5.25380e6	0.251632
$848$	−4.24554e7	−2.02742
$849$	0	0
$850$	−2.59980e6	−0.123422
$851$	1.44090e6	0.0682040
$852$	0	0
$853$	−3.96032e6	−0.186362	−0.0931810	−	0.995649i	$-0.529704\pi$
$853$	−3.96032e6	−0.186362	−0.0931810	+	0.995649i	$0.529704\pi$
$854$	−6.18884e6	−0.290378
$855$	0	0
$856$	5.23417e6	0.244154
$857$	−1.78958e7	−0.832338	−0.416169	−	0.909287i	$-0.636627\pi$
$857$	−1.78958e7	−0.832338	−0.416169	+	0.909287i	$0.636627\pi$
$858$	0	0
$859$	3.52591e7	1.63038	0.815188	−	0.579197i	$-0.196634\pi$
$859$	3.52591e7	1.63038	0.815188	+	0.579197i	$0.196634\pi$
$860$	−1.18942e7	−0.548390
$861$	0	0
$862$	2.27070e7	1.04086
$863$	4.23888e7	1.93742	0.968711	−	0.248192i	$-0.0798365\pi$
$863$	4.23888e7	1.93742	0.968711	+	0.248192i	$0.0798365\pi$
$864$	0	0
$865$	5.00223e6	0.227313
$866$	7.82614e6	0.354612
$867$	0	0
$868$	−4.82422e6	−0.217334
$869$	8.34011e6	0.374647
$870$	0	0
$871$	8.97139e6	0.400695
$872$	1.67661e7	0.746689
$873$	0	0
$874$	2.09101e6	0.0925928
$875$	6.75247e6	0.298155
$876$	0	0
$877$	3.53937e6	0.155391	0.0776956	−	0.996977i	$-0.475244\pi$
$877$	3.53937e6	0.155391	0.0776956	+	0.996977i	$0.475244\pi$
$878$	6.15582e6	0.269494
$879$	0	0
$880$	−1.01994e7	−0.443986
$881$	1.82655e7	0.792851	0.396426	−	0.918067i	$-0.370250\pi$
$881$	1.82655e7	0.792851	0.396426	+	0.918067i	$0.370250\pi$
$882$	0	0
$883$	−2.56061e7	−1.10520	−0.552602	−	0.833446i	$-0.686365\pi$
$883$	−2.56061e7	−1.10520	−0.552602	+	0.833446i	$0.686365\pi$
$884$	5.19119e6	0.223428
$885$	0	0
$886$	−2.48114e7	−1.06186
$887$	4.46206e6	0.190426	0.0952130	−	0.995457i	$-0.469647\pi$
$887$	4.46206e6	0.190426	0.0952130	+	0.995457i	$0.469647\pi$
$888$	0	0
$889$	−8.26030e6	−0.350543
$890$	−1.39693e7	−0.591153
$891$	0	0
$892$	−4.06226e6	−0.170945
$893$	9.54566e6	0.400569
$894$	0	0
$895$	−2.51561e6	−0.104975
$896$	−7.77412e6	−0.323505
$897$	0	0
$898$	1.44189e7	0.596679
$899$	−7.29505e6	−0.301043
$900$	0	0
$901$	7.18186e7	2.94730
$902$	−9.56294e6	−0.391359
$903$	0	0
$904$	3.52832e6	0.143598
$905$	6.41814e6	0.260488
$906$	0	0
$907$	4.62490e7	1.86674	0.933371	−	0.358914i	$-0.116853\pi$
$907$	4.62490e7	1.86674	0.933371	+	0.358914i	$0.116853\pi$
$908$	6.81882e6	0.274470
$909$	0	0
$910$	−2.27441e6	−0.0910471
$911$	3.42662e7	1.36795	0.683974	−	0.729507i	$-0.260250\pi$
$911$	3.42662e7	1.36795	0.683974	+	0.729507i	$0.260250\pi$
$912$	0	0
$913$	−1.57783e7	−0.626446
$914$	1.07595e6	0.0426015
$915$	0	0
$916$	−4.59315e6	−0.180872
$917$	7.80402e6	0.306475
$918$	0	0
$919$	1.31785e7	0.514728	0.257364	−	0.966315i	$-0.417146\pi$
$919$	1.31785e7	0.514728	0.257364	+	0.966315i	$0.417146\pi$
$920$	−784763.	−0.0305681
$921$	0	0
$922$	543839.	0.0210690
$923$	6.76771e6	0.261479
$924$	0	0
$925$	−2.06600e6	−0.0793919
$926$	2.88528e6	0.110576
$927$	0	0
$928$	4.18059e6	0.159356
$929$	1.62539e7	0.617899	0.308949	−	0.951078i	$-0.400023\pi$
$929$	1.62539e7	0.617899	0.308949	+	0.951078i	$0.400023\pi$
$930$	0	0
$931$	−3.83786e7	−1.45116
$932$	−2.25843e7	−0.851661
$933$	0	0
$934$	−2.43490e7	−0.913301
$935$	1.72536e7	0.645433
$936$	0	0
$937$	−2.53494e7	−0.943233	−0.471617	−	0.881804i	$-0.656329\pi$
$937$	−2.53494e7	−0.943233	−0.471617	+	0.881804i	$0.656329\pi$
$938$	−1.42147e7	−0.527511
$939$	0	0
$940$	3.06870e6	0.113275
$941$	−9.02296e6	−0.332181	−0.166091	−	0.986110i	$-0.553114\pi$
$941$	−9.02296e6	−0.332181	−0.166091	+	0.986110i	$0.553114\pi$
$942$	0	0
$943$	−1.16707e6	−0.0427383
$944$	2.50683e7	0.915576
$945$	0	0
$946$	−1.49028e7	−0.541426
$947$	338514.	0.0122660	0.00613298	−	0.999981i	$-0.498048\pi$
$947$	338514.	0.0122660	0.00613298	+	0.999981i	$0.498048\pi$
$948$	0	0
$949$	−2.55375e6	−0.0920477
$950$	−2.99814e6	−0.107781
$951$	0	0
$952$	9.60241e6	0.343390
$953$	1.25549e7	0.447797	0.223899	−	0.974612i	$-0.428122\pi$
$953$	1.25549e7	0.447797	0.223899	+	0.974612i	$0.428122\pi$
$954$	0	0
$955$	1.89581e7	0.672647
$956$	−1.80087e7	−0.637291
$957$	0	0
$958$	−2.44984e7	−0.862430
$959$	445099.	0.0156282
$960$	0	0
$961$	4.66136e7	1.62819
$962$	1.30667e7	0.455227
$963$	0	0
$964$	8.31630e6	0.288229
$965$	−6.99069e6	−0.241658
$966$	0	0
$967$	−1.51562e6	−0.0521224	−0.0260612	−	0.999660i	$-0.508296\pi$
$967$	−1.51562e6	−0.0521224	−0.0260612	+	0.999660i	$0.508296\pi$
$968$	1.64390e7	0.563880
$969$	0	0
$970$	−1.63197e7	−0.556907
$971$	−3.97841e7	−1.35413	−0.677067	−	0.735921i	$-0.736749\pi$
$971$	−3.97841e7	−1.35413	−0.677067	+	0.735921i	$0.736749\pi$
$972$	0	0
$973$	1.93406e6	0.0654918
$974$	5.91537e7	1.99795
$975$	0	0
$976$	−3.07152e7	−1.03212
$977$	1.08895e7	0.364982	0.182491	−	0.983208i	$-0.441584\pi$
$977$	1.08895e7	0.364982	0.182491	+	0.983208i	$0.441584\pi$
$978$	0	0
$979$	−5.52584e6	−0.184264
$980$	−1.23378e7	−0.410367
$981$	0	0
$982$	−1.00116e6	−0.0331303
$983$	−1.51586e7	−0.500353	−0.250177	−	0.968200i	$-0.580489\pi$
$983$	−1.51586e7	−0.500353	−0.250177	+	0.968200i	$0.580489\pi$
$984$	0	0
$985$	−3.34504e7	−1.09853
$986$	−1.24379e7	−0.407432
$987$	0	0
$988$	5.98659e6	0.195113
$989$	−1.81875e6	−0.0591264
$990$	0	0
$991$	−4.43710e7	−1.43521	−0.717605	−	0.696451i	$-0.754762\pi$
$991$	−4.43710e7	−1.43521	−0.717605	+	0.696451i	$0.754762\pi$
$992$	−4.31195e7	−1.39122
$993$	0	0
$994$	−1.07231e7	−0.344234
$995$	−1.81113e7	−0.579951
$996$	0	0
$997$	−4.48040e7	−1.42751	−0.713754	−	0.700397i	$-0.753006\pi$
$997$	−4.48040e7	−1.42751	−0.713754	+	0.700397i	$0.753006\pi$
$998$	−2.90373e7	−0.922847
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	261.6.a.e.1.6		7
3.2	odd	2		29.6.a.b.1.2	✓	7
12.11	even	2		464.6.a.k.1.7		7
15.14	odd	2		725.6.a.b.1.6		7
87.86	odd	2		841.6.a.b.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.a.b.1.2	✓	7	3.2	odd	2
261.6.a.e.1.6		7	1.1	even	1	trivial
464.6.a.k.1.7		7	12.11	even	2
725.6.a.b.1.6		7	15.14	odd	2
841.6.a.b.1.6		7	87.86	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 261 = 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	261.a (trivial)

\(f(q)\)	\(=\)	\(q+6.83842 q^{2} +14.7640 q^{4} +54.3066 q^{5} -37.6697 q^{7} -117.867 q^{8} +O(q^{10})\)	\(q+6.83842 q^{2} +14.7640 q^{4} +54.3066 q^{5} -37.6697 q^{7} -117.867 q^{8} +371.372 q^{10} +146.903 q^{11} +162.581 q^{13} -257.601 q^{14} -1278.47 q^{16} +2162.70 q^{17} +2494.06 q^{19} +801.781 q^{20} +1004.59 q^{22} +122.601 q^{23} -175.788 q^{25} +1111.79 q^{26} -556.154 q^{28} -841.000 q^{29} +8674.26 q^{31} -4970.98 q^{32} +14789.4 q^{34} -2045.71 q^{35} +11752.8 q^{37} +17055.4 q^{38} -6400.97 q^{40} -9519.28 q^{41} -14834.7 q^{43} +2168.88 q^{44} +838.394 q^{46} +3827.35 q^{47} -15388.0 q^{49} -1202.11 q^{50} +2400.34 q^{52} +33207.9 q^{53} +7977.83 q^{55} +4440.02 q^{56} -5751.11 q^{58} -19608.0 q^{59} +24024.9 q^{61} +59318.2 q^{62} +6917.49 q^{64} +8829.21 q^{65} +55181.2 q^{67} +31930.0 q^{68} -13989.4 q^{70} +41626.8 q^{71} -15707.6 q^{73} +80370.5 q^{74} +36822.3 q^{76} -5533.80 q^{77} +56772.8 q^{79} -69429.5 q^{80} -65096.8 q^{82} -107406. q^{83} +117449. q^{85} -101446. q^{86} -17315.1 q^{88} -37615.4 q^{89} -6124.36 q^{91} +1810.07 q^{92} +26173.0 q^{94} +135444. q^{95} -43944.4 q^{97} -105230. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 4 q^{2} + 154 q^{4} - 32 q^{5} + 184 q^{7} - 942 q^{8}+O(q^{10}) \) 7 * q - 4 * q^2 + 154 * q^4 - 32 * q^5 + 184 * q^7 - 942 * q^8	\( 7 q - 4 q^{2} + 154 q^{4} - 32 q^{5} + 184 q^{7} - 942 q^{8} + 922 q^{10} - 1106 q^{11} + 408 q^{13} + 2008 q^{14} + 242 q^{16} + 874 q^{17} + 4288 q^{19} + 6350 q^{20} - 6114 q^{22} + 4532 q^{23} + 5527 q^{25} + 19806 q^{26} - 496 q^{28} - 5887 q^{29} + 7794 q^{31} - 7898 q^{32} + 20840 q^{34} - 7088 q^{35} + 5086 q^{37} - 23732 q^{38} + 22906 q^{40} - 19826 q^{41} + 19498 q^{43} + 6074 q^{44} - 12404 q^{46} - 14278 q^{47} + 38431 q^{49} + 41066 q^{50} - 34302 q^{52} + 58644 q^{53} - 25574 q^{55} + 79560 q^{56} + 3364 q^{58} - 12888 q^{59} + 102866 q^{61} + 42654 q^{62} - 10170 q^{64} + 149206 q^{65} + 102996 q^{67} - 85100 q^{68} + 349480 q^{70} + 51596 q^{71} - 17566 q^{73} - 12132 q^{74} + 360740 q^{76} + 94104 q^{77} + 212058 q^{79} - 142510 q^{80} + 201924 q^{82} + 122928 q^{83} - 109336 q^{85} + 63290 q^{86} + 136666 q^{88} + 66510 q^{89} + 194368 q^{91} + 110108 q^{92} + 438926 q^{94} + 131676 q^{95} - 118182 q^{97} + 29132 q^{98}+O(q^{100}) \) 7 * q - 4 * q^2 + 154 * q^4 - 32 * q^5 + 184 * q^7 - 942 * q^8 + 922 * q^10 - 1106 * q^11 + 408 * q^13 + 2008 * q^14 + 242 * q^16 + 874 * q^17 + 4288 * q^19 + 6350 * q^20 - 6114 * q^22 + 4532 * q^23 + 5527 * q^25 + 19806 * q^26 - 496 * q^28 - 5887 * q^29 + 7794 * q^31 - 7898 * q^32 + 20840 * q^34 - 7088 * q^35 + 5086 * q^37 - 23732 * q^38 + 22906 * q^40 - 19826 * q^41 + 19498 * q^43 + 6074 * q^44 - 12404 * q^46 - 14278 * q^47 + 38431 * q^49 + 41066 * q^50 - 34302 * q^52 + 58644 * q^53 - 25574 * q^55 + 79560 * q^56 + 3364 * q^58 - 12888 * q^59 + 102866 * q^61 + 42654 * q^62 - 10170 * q^64 + 149206 * q^65 + 102996 * q^67 - 85100 * q^68 + 349480 * q^70 + 51596 * q^71 - 17566 * q^73 - 12132 * q^74 + 360740 * q^76 + 94104 * q^77 + 212058 * q^79 - 142510 * q^80 + 201924 * q^82 + 122928 * q^83 - 109336 * q^85 + 63290 * q^86 + 136666 * q^88 + 66510 * q^89 + 194368 * q^91 + 110108 * q^92 + 438926 * q^94 + 131676 * q^95 - 118182 * q^97 + 29132 * q^98

Introduction

L-functions

Modular forms

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Database

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