LMFDB - Embedded newform 825.6.a.w.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [825,6,Mod(1,825)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("825.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(825, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	825.a (trivial)

Newform invariants

comment:select newform

sage:traces = [13,-3,117,229] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 3 x^{12} - 318 x^{11} + 776 x^{10} + 37929 x^{9} - 75673 x^{8} - 2114192 x^{7} + \cdots + 1037920000 $ x^13 - 3x^12 - 318x^11 + 776x^10 + 37929x^9 - 75673x^8 - 2114192x^7 + 3527024x^6 + 55736416x^5 - 75128864x^4 - 615221248x^3 + 450708480x^2 + 2256236800x + 1037920000
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 5^{6} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-3.62063$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.62063 q^{2} +9.00000 q^{3} -18.8910 q^{4} +32.5857 q^{6} +187.526 q^{7} -184.258 q^{8} +81.0000 q^{9} -121.000 q^{11} -170.019 q^{12} -145.247 q^{13} +678.961 q^{14} -62.6149 q^{16} -1828.13 q^{17} +293.271 q^{18} +1453.94 q^{19} +1687.73 q^{21} -438.096 q^{22} -66.0886 q^{23} -1658.32 q^{24} -525.886 q^{26} +729.000 q^{27} -3542.55 q^{28} -2596.02 q^{29} -7120.85 q^{31} +5669.54 q^{32} -1089.00 q^{33} -6618.98 q^{34} -1530.17 q^{36} +13727.9 q^{37} +5264.18 q^{38} -1307.22 q^{39} +1924.14 q^{41} +6110.65 q^{42} -247.123 q^{43} +2285.82 q^{44} -239.282 q^{46} -23893.7 q^{47} -563.534 q^{48} +18358.8 q^{49} -16453.2 q^{51} +2743.87 q^{52} -33243.2 q^{53} +2639.44 q^{54} -34553.0 q^{56} +13085.5 q^{57} -9399.23 q^{58} -30883.3 q^{59} -44616.4 q^{61} -25782.0 q^{62} +15189.6 q^{63} +22531.0 q^{64} -3942.86 q^{66} +36268.1 q^{67} +34535.3 q^{68} -594.797 q^{69} +21658.8 q^{71} -14924.9 q^{72} +74800.2 q^{73} +49703.6 q^{74} -27466.5 q^{76} -22690.6 q^{77} -4732.98 q^{78} +3276.39 q^{79} +6561.00 q^{81} +6966.61 q^{82} -64618.5 q^{83} -31883.0 q^{84} -894.741 q^{86} -23364.2 q^{87} +22295.2 q^{88} -10036.5 q^{89} -27237.6 q^{91} +1248.48 q^{92} -64087.7 q^{93} -86510.3 q^{94} +51025.8 q^{96} +176913. q^{97} +66470.6 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 3 q^{2} + 117 q^{3} + 229 q^{4} - 27 q^{6} - 284 q^{7} - 369 q^{8} + 1053 q^{9} - 1573 q^{11} + 2061 q^{12} - 366 q^{13} - 2758 q^{14} + 4141 q^{16} - 2056 q^{17} - 243 q^{18} - 310 q^{19} - 2556 q^{21}+ \cdots - 127413 q^{99}+O(q^{100}) $ 13 * q - 3 * q^2 + 117 * q^3 + 229 * q^4 - 27 * q^6 - 284 * q^7 - 369 * q^8 + 1053 * q^9 - 1573 * q^11 + 2061 * q^12 - 366 * q^13 - 2758 * q^14 + 4141 * q^16 - 2056 * q^17 - 243 * q^18 - 310 * q^19 - 2556 * q^21 + 363 * q^22 - 3612 * q^23 - 3321 * q^24 + 2280 * q^26 + 9477 * q^27 - 7896 * q^28 - 4848 * q^29 - 24 * q^31 - 38111 * q^32 - 14157 * q^33 + 5518 * q^34 + 18549 * q^36 + 8420 * q^37 - 474 * q^38 - 3294 * q^39 - 15120 * q^41 - 24822 * q^42 - 35492 * q^43 - 27709 * q^44 - 20280 * q^46 - 46544 * q^47 + 37269 * q^48 + 81837 * q^49 - 18504 * q^51 - 107194 * q^52 - 42256 * q^53 - 2187 * q^54 - 196602 * q^56 - 2790 * q^57 - 114160 * q^58 - 65592 * q^59 - 52042 * q^61 - 94972 * q^62 - 23004 * q^63 + 185977 * q^64 + 3267 * q^66 - 80580 * q^67 - 61108 * q^68 - 32508 * q^69 - 77820 * q^71 - 29889 * q^72 - 103050 * q^73 - 240028 * q^74 - 271174 * q^76 + 34364 * q^77 + 20520 * q^78 - 112258 * q^79 + 85293 * q^81 - 64060 * q^82 - 292150 * q^83 - 71064 * q^84 - 319250 * q^86 - 43632 * q^87 + 44649 * q^88 - 295810 * q^89 - 24200 * q^91 - 121328 * q^92 - 216 * q^93 - 358144 * q^94 - 342999 * q^96 - 49072 * q^97 - 101815 * q^98 - 127413 * q^99

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 $

$2$	3.62063	0.640043	0.320021	−	0.947410i	$-0.396310\pi$
$2$	3.62063	0.640043	0.320021	+	0.947410i	$0.396310\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−18.8910	−0.590345
$5$	0	0
$5$	0	0
$6$	32.5857	0.369529
$7$	187.526	1.44649	0.723245	−	0.690592i	$-0.242650\pi$
$7$	187.526	1.44649	0.723245	+	0.690592i	$0.242650\pi$
$8$	−184.258	−1.01789
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−170.019	−0.340836
$13$	−145.247	−0.238369	−0.119184	−	0.992872i	$-0.538028\pi$
$13$	−145.247	−0.238369	−0.119184	+	0.992872i	$0.538028\pi$
$14$	678.961	0.925816
$15$	0	0
$16$	−62.6149	−0.0611474
$17$	−1828.13	−1.53421	−0.767105	−	0.641522i	$-0.778303\pi$
$17$	−1828.13	−1.53421	−0.767105	+	0.641522i	$0.778303\pi$
$18$	293.271	0.213348
$19$	1453.94	0.923981	0.461990	−	0.886885i	$-0.347135\pi$
$19$	1453.94	0.923981	0.461990	+	0.886885i	$0.347135\pi$
$20$	0	0
$21$	1687.73	0.835131
$22$	−438.096	−0.192980
$23$	−66.0886	−0.0260499	−0.0130250	−	0.999915i	$-0.504146\pi$
$23$	−66.0886	−0.0260499	−0.0130250	+	0.999915i	$0.504146\pi$
$24$	−1658.32	−0.587679
$25$	0	0
$26$	−525.886	−0.152566
$27$	729.000	0.192450
$28$	−3542.55	−0.853928
$29$	−2596.02	−0.573210	−0.286605	−	0.958049i	$-0.592527\pi$
$29$	−2596.02	−0.573210	−0.286605	+	0.958049i	$0.592527\pi$
$30$	0	0
$31$	−7120.85	−1.33085	−0.665423	−	0.746467i	$-0.731749\pi$
$31$	−7120.85	−1.33085	−0.665423	+	0.746467i	$0.731749\pi$
$32$	5669.54	0.978752
$33$	−1089.00	−0.174078
$34$	−6618.98	−0.981960
$35$	0	0
$36$	−1530.17	−0.196782
$37$	13727.9	1.64854	0.824270	−	0.566197i	$-0.191586\pi$
$37$	13727.9	1.64854	0.824270	+	0.566197i	$0.191586\pi$
$38$	5264.18	0.591387
$39$	−1307.22	−0.137622
$40$	0	0
$41$	1924.14	0.178763	0.0893815	−	0.995997i	$-0.471511\pi$
$41$	1924.14	0.178763	0.0893815	+	0.995997i	$0.471511\pi$
$42$	6110.65	0.534520
$43$	−247.123	−0.0203818	−0.0101909	−	0.999948i	$-0.503244\pi$
$43$	−247.123	−0.0203818	−0.0101909	+	0.999948i	$0.503244\pi$
$44$	2285.82	0.177996
$45$	0	0
$46$	−239.282	−0.0166731
$47$	−23893.7	−1.57775	−0.788877	−	0.614551i	$-0.789337\pi$
$47$	−23893.7	−1.57775	−0.788877	+	0.614551i	$0.789337\pi$
$48$	−563.534	−0.0353035
$49$	18358.8	1.09233
$50$	0	0
$51$	−16453.2	−0.885776
$52$	2743.87	0.140720
$53$	−33243.2	−1.62560	−0.812798	−	0.582545i	$-0.802057\pi$
$53$	−33243.2	−1.62560	−0.812798	+	0.582545i	$0.802057\pi$
$54$	2639.44	0.123176
$55$	0	0
$56$	−34553.0	−1.47237
$57$	13085.5	0.533461
$58$	−9399.23	−0.366879
$59$	−30883.3	−1.15503	−0.577515	−	0.816380i	$-0.695978\pi$
$59$	−30883.3	−1.15503	−0.577515	+	0.816380i	$0.695978\pi$
$60$	0	0
$61$	−44616.4	−1.53522	−0.767609	−	0.640918i	$-0.778554\pi$
$61$	−44616.4	−1.53522	−0.767609	+	0.640918i	$0.778554\pi$
$62$	−25782.0	−0.851798
$63$	15189.6	0.482163
$64$	22531.0	0.687591
$65$	0	0
$66$	−3942.86	−0.111417
$67$	36268.1	0.987046	0.493523	−	0.869733i	$-0.335709\pi$
$67$	36268.1	0.987046	0.493523	+	0.869733i	$0.335709\pi$
$68$	34535.3	0.905713
$69$	−594.797	−0.0150399
$70$	0	0
$71$	21658.8	0.509905	0.254953	−	0.966954i	$-0.417940\pi$
$71$	21658.8	0.509905	0.254953	+	0.966954i	$0.417940\pi$
$72$	−14924.9	−0.339296
$73$	74800.2	1.64284	0.821420	−	0.570323i	$-0.193182\pi$
$73$	74800.2	1.64284	0.821420	+	0.570323i	$0.193182\pi$
$74$	49703.6	1.05514
$75$	0	0
$76$	−27466.5	−0.545468
$77$	−22690.6	−0.436133
$78$	−4732.98	−0.0880842
$79$	3276.39	0.0590648	0.0295324	−	0.999564i	$-0.490598\pi$
$79$	3276.39	0.0590648	0.0295324	+	0.999564i	$0.490598\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	6966.61	0.114416
$83$	−64618.5	−1.02958	−0.514792	−	0.857315i	$-0.672131\pi$
$83$	−64618.5	−1.02958	−0.514792	+	0.857315i	$0.672131\pi$
$84$	−31883.0	−0.493016
$85$	0	0
$86$	−894.741	−0.0130452
$87$	−23364.2	−0.330943
$88$	22295.2	0.306905
$89$	−10036.5	−0.134309	−0.0671546	−	0.997743i	$-0.521392\pi$
$89$	−10036.5	−0.134309	−0.0671546	+	0.997743i	$0.521392\pi$
$90$	0	0
$91$	−27237.6	−0.344798
$92$	1248.48	0.0153785
$93$	−64087.7	−0.768364
$94$	−86510.3	−1.00983
$95$	0	0
$96$	51025.8	0.565083
$97$	176913.	1.90910	0.954552	−	0.298046i	$-0.0963349\pi$
$97$	176913.	1.90910	0.954552	+	0.298046i	$0.0963349\pi$
$98$	66470.6	0.699140
$99$	−9801.00	−0.100504
$100$	0	0
$101$	−33318.6	−0.325000	−0.162500	−	0.986709i	$-0.551956\pi$
$101$	−33318.6	−0.325000	−0.162500	+	0.986709i	$0.551956\pi$
$102$	−59570.8	−0.566935
$103$	−168417.	−1.56420	−0.782101	−	0.623152i	$-0.785852\pi$
$103$	−168417.	−1.56420	−0.782101	+	0.623152i	$0.785852\pi$
$104$	26762.9	0.242633
$105$	0	0
$106$	−120361.	−1.04045
$107$	103640.	0.875122	0.437561	−	0.899189i	$-0.355842\pi$
$107$	103640.	0.875122	0.437561	+	0.899189i	$0.355842\pi$
$108$	−13771.6	−0.113612
$109$	−163450.	−1.31770	−0.658852	−	0.752273i	$-0.728958\pi$
$109$	−163450.	−1.31770	−0.658852	+	0.752273i	$0.728958\pi$
$110$	0	0
$111$	123551.	0.951785
$112$	−11741.9	−0.0884491
$113$	−9079.82	−0.0668931	−0.0334465	−	0.999441i	$-0.510648\pi$
$113$	−9079.82	−0.0668931	−0.0334465	+	0.999441i	$0.510648\pi$
$114$	47377.7	0.341438
$115$	0	0
$116$	49041.6	0.338391
$117$	−11765.0	−0.0794563
$118$	−111817.	−0.739269
$119$	−342821.	−2.21922
$120$	0	0
$121$	14641.0	0.0909091
$122$	−161539.	−0.982605
$123$	17317.3	0.103209
$124$	134520.	0.785658
$125$	0	0
$126$	54995.8	0.308605
$127$	−117210.	−0.644845	−0.322422	−	0.946596i	$-0.604497\pi$
$127$	−117210.	−0.644845	−0.322422	+	0.946596i	$0.604497\pi$
$128$	−99848.9	−0.538665
$129$	−2224.11	−0.0117674
$130$	0	0
$131$	−120279.	−0.612366	−0.306183	−	0.951973i	$-0.599052\pi$
$131$	−120279.	−0.612366	−0.306183	+	0.951973i	$0.599052\pi$
$132$	20572.3	0.102766
$133$	272651.	1.33653
$134$	131313.	0.631752
$135$	0	0
$136$	336847.	1.56166
$137$	−369850.	−1.68354	−0.841772	−	0.539834i	$-0.818487\pi$
$137$	−369850.	−1.68354	−0.841772	+	0.539834i	$0.818487\pi$
$138$	−2153.54	−0.00962620
$139$	−120159.	−0.527498	−0.263749	−	0.964591i	$-0.584959\pi$
$139$	−120159.	−0.527498	−0.263749	+	0.964591i	$0.584959\pi$
$140$	0	0
$141$	−215044.	−0.910917
$142$	78418.6	0.326361
$143$	17574.9	0.0718709
$144$	−5071.81	−0.0203825
$145$	0	0
$146$	270824.	1.05149
$147$	165230.	0.630659
$148$	−259334.	−0.973207
$149$	−526307.	−1.94211	−0.971055	−	0.238856i	$-0.923227\pi$
$149$	−526307.	−1.94211	−0.971055	+	0.238856i	$0.923227\pi$
$150$	0	0
$151$	−6526.41	−0.0232934	−0.0116467	−	0.999932i	$-0.503707\pi$
$151$	−6526.41	−0.0232934	−0.0116467	+	0.999932i	$0.503707\pi$
$152$	−267900.	−0.940510
$153$	−148079.	−0.511403
$154$	−82154.2	−0.279144
$155$	0	0
$156$	24694.8	0.0812447
$157$	78920.0	0.255528	0.127764	−	0.991805i	$-0.459220\pi$
$157$	78920.0	0.255528	0.127764	+	0.991805i	$0.459220\pi$
$158$	11862.6	0.0378040
$159$	−299189.	−0.938539
$160$	0	0
$161$	−12393.3	−0.0376810
$162$	23754.9	0.0711159
$163$	−334420.	−0.985879	−0.492939	−	0.870064i	$-0.664078\pi$
$163$	−334420.	−0.985879	−0.492939	+	0.870064i	$0.664078\pi$
$164$	−36349.1	−0.105532
$165$	0	0
$166$	−233960.	−0.658978
$167$	−460611.	−1.27804	−0.639018	−	0.769191i	$-0.720659\pi$
$167$	−460611.	−1.27804	−0.639018	+	0.769191i	$0.720659\pi$
$168$	−310977.	−0.850071
$169$	−350196.	−0.943180
$170$	0	0
$171$	117769.	0.307994
$172$	4668.41	0.0120323
$173$	121903.	0.309671	0.154836	−	0.987940i	$-0.450515\pi$
$173$	121903.	0.309671	0.154836	+	0.987940i	$0.450515\pi$
$174$	−84593.1	−0.211817
$175$	0	0
$176$	7576.41	0.0184366
$177$	−277950.	−0.666857
$178$	−36338.3	−0.0859637
$179$	572399.	1.33526	0.667631	−	0.744493i	$-0.267309\pi$
$179$	572399.	1.33526	0.667631	+	0.744493i	$0.267309\pi$
$180$	0	0
$181$	46785.9	0.106150	0.0530749	−	0.998591i	$-0.483098\pi$
$181$	46785.9	0.106150	0.0530749	+	0.998591i	$0.483098\pi$
$182$	−98617.1	−0.220686
$183$	−401548.	−0.886359
$184$	12177.3	0.0265159
$185$	0	0
$186$	−232038.	−0.491786
$187$	221204.	0.462582
$188$	451378.	0.931420
$189$	136706.	0.278377
$190$	0	0
$191$	−581929.	−1.15421	−0.577107	−	0.816668i	$-0.695819\pi$
$191$	−581929.	−1.15421	−0.577107	+	0.816668i	$0.695819\pi$
$192$	202779.	0.396981
$193$	−586600.	−1.13357	−0.566786	−	0.823865i	$-0.691813\pi$
$193$	−586600.	−1.13357	−0.566786	+	0.823865i	$0.691813\pi$
$194$	640535.	1.22191
$195$	0	0
$196$	−346818.	−0.644854
$197$	−446858.	−0.820359	−0.410179	−	0.912005i	$-0.634534\pi$
$197$	−446858.	−0.820359	−0.410179	+	0.912005i	$0.634534\pi$
$198$	−35485.8	−0.0643267
$199$	−259397.	−0.464336	−0.232168	−	0.972676i	$-0.574582\pi$
$199$	−259397.	−0.464336	−0.232168	+	0.972676i	$0.574582\pi$
$200$	0	0
$201$	326413.	0.569871
$202$	−120634.	−0.208014
$203$	−486821.	−0.829142
$204$	310818.	0.522914
$205$	0	0
$206$	−609776.	−1.00116
$207$	−5353.17	−0.00868331
$208$	9094.65	0.0145756
$209$	−175927.	−0.278591
$210$	0	0
$211$	494961.	0.765358	0.382679	−	0.923881i	$-0.375002\pi$
$211$	494961.	0.765358	0.382679	+	0.923881i	$0.375002\pi$
$212$	627998.	0.959663
$213$	194930.	0.294394
$214$	375243.	0.560116
$215$	0	0
$216$	−134324.	−0.195893
$217$	−1.33534e6	−1.92505
$218$	−591791.	−0.843387
$219$	673202.	0.948495
$220$	0	0
$221$	265531.	0.365708
$222$	447332.	0.609183
$223$	−812706.	−1.09439	−0.547194	−	0.837006i	$-0.684304\pi$
$223$	−812706.	−1.09439	−0.547194	+	0.837006i	$0.684304\pi$
$224$	1.06318e6	1.41576
$225$	0	0
$226$	−32874.7	−0.0428144
$227$	1.25202e6	1.61267	0.806337	−	0.591456i	$-0.201447\pi$
$227$	1.25202e6	1.61267	0.806337	+	0.591456i	$0.201447\pi$
$228$	−247198.	−0.314926
$229$	−434753.	−0.547841	−0.273920	−	0.961752i	$-0.588320\pi$
$229$	−434753.	−0.547841	−0.273920	+	0.961752i	$0.588320\pi$
$230$	0	0
$231$	−204215.	−0.251802
$232$	478337.	0.583464
$233$	−453587.	−0.547358	−0.273679	−	0.961821i	$-0.588241\pi$
$233$	−453587.	−0.547358	−0.273679	+	0.961821i	$0.588241\pi$
$234$	−42596.8	−0.0508554
$235$	0	0
$236$	583418.	0.681867
$237$	29487.6	0.0341011
$238$	−1.24123e6	−1.42040
$239$	860298.	0.974214	0.487107	−	0.873342i	$-0.338052\pi$
$239$	860298.	0.974214	0.487107	+	0.873342i	$0.338052\pi$
$240$	0	0
$241$	−374215.	−0.415029	−0.207515	−	0.978232i	$-0.566537\pi$
$241$	−374215.	−0.415029	−0.207515	+	0.978232i	$0.566537\pi$
$242$	53009.6	0.0581857
$243$	59049.0	0.0641500
$244$	842851.	0.906309
$245$	0	0
$246$	62699.4	0.0660581
$247$	−211181.	−0.220248
$248$	1.31207e6	1.35465
$249$	−581567.	−0.594431
$250$	0	0
$251$	1.45908e6	1.46182	0.730912	−	0.682472i	$-0.239095\pi$
$251$	1.45908e6	1.46182	0.730912	+	0.682472i	$0.239095\pi$
$252$	−286947.	−0.284643
$253$	7996.72	0.00785435
$254$	−424374.	−0.412728
$255$	0	0
$256$	−1.08251e6	−1.03236
$257$	1.04725e6	0.989053	0.494526	−	0.869163i	$-0.335342\pi$
$257$	1.04725e6	0.989053	0.494526	+	0.869163i	$0.335342\pi$
$258$	−8052.67	−0.00753166
$259$	2.57433e6	2.38460
$260$	0	0
$261$	−210278.	−0.191070
$262$	−435485.	−0.391940
$263$	−150062.	−0.133777	−0.0668885	−	0.997760i	$-0.521307\pi$
$263$	−150062.	−0.133777	−0.0668885	+	0.997760i	$0.521307\pi$
$264$	200657.	0.177192
$265$	0	0
$266$	987169.	0.855436
$267$	−90328.2	−0.0775435
$268$	−685142.	−0.582698
$269$	−1.45785e6	−1.22838	−0.614191	−	0.789158i	$-0.710518\pi$
$269$	−1.45785e6	−1.22838	−0.614191	+	0.789158i	$0.710518\pi$
$270$	0	0
$271$	583177.	0.482367	0.241183	−	0.970480i	$-0.422465\pi$
$271$	583177.	0.482367	0.241183	+	0.970480i	$0.422465\pi$
$272$	114468.	0.0938129
$273$	−245138.	−0.199069
$274$	−1.33909e6	−1.07754
$275$	0	0
$276$	11236.3	0.00887876
$277$	−2.35418e6	−1.84349	−0.921743	−	0.387802i	$-0.873234\pi$
$277$	−2.35418e6	−1.84349	−0.921743	+	0.387802i	$0.873234\pi$
$278$	−435053.	−0.337621
$279$	−576789.	−0.443615
$280$	0	0
$281$	177355.	0.133991	0.0669957	−	0.997753i	$-0.478659\pi$
$281$	177355.	0.133991	0.0669957	+	0.997753i	$0.478659\pi$
$282$	−778593.	−0.583026
$283$	−114604.	−0.0850616	−0.0425308	−	0.999095i	$-0.513542\pi$
$283$	−114604.	−0.0850616	−0.0425308	+	0.999095i	$0.513542\pi$
$284$	−409158.	−0.301020
$285$	0	0
$286$	63632.2	0.0460005
$287$	360826.	0.258579
$288$	459233.	0.326251
$289$	1.92220e6	1.35380
$290$	0	0
$291$	1.59221e6	1.10222
$292$	−1.41305e6	−0.969843
$293$	634916.	0.432063	0.216032	−	0.976386i	$-0.430689\pi$
$293$	634916.	0.432063	0.216032	+	0.976386i	$0.430689\pi$
$294$	598235.	0.403649
$295$	0	0
$296$	−2.52947e6	−1.67803
$297$	−88209.0	−0.0580259
$298$	−1.90556e6	−1.24303
$299$	9599.18	0.00620949
$300$	0	0
$301$	−46341.9	−0.0294820
$302$	−23629.7	−0.0149087
$303$	−299868.	−0.187639
$304$	−91038.5	−0.0564990
$305$	0	0
$306$	−536137.	−0.327320
$307$	−1.16348e6	−0.704555	−0.352277	−	0.935896i	$-0.614593\pi$
$307$	−1.16348e6	−0.704555	−0.352277	+	0.935896i	$0.614593\pi$
$308$	428649.	0.257469
$309$	−1.51575e6	−0.903093
$310$	0	0
$311$	2.80571e6	1.64491	0.822455	−	0.568830i	$-0.192604\pi$
$311$	2.80571e6	1.64491	0.822455	+	0.568830i	$0.192604\pi$
$312$	240866.	0.140084
$313$	799439.	0.461238	0.230619	−	0.973044i	$-0.425925\pi$
$313$	799439.	0.461238	0.230619	+	0.973044i	$0.425925\pi$
$314$	285740.	0.163549
$315$	0	0
$316$	−61894.5	−0.0348686
$317$	1.23775e6	0.691808	0.345904	−	0.938270i	$-0.387572\pi$
$317$	1.23775e6	0.691808	0.345904	+	0.938270i	$0.387572\pi$
$318$	−1.08325e6	−0.600705
$319$	314119.	0.172829
$320$	0	0
$321$	932762.	0.505252
$322$	−44871.5	−0.0241174
$323$	−2.65799e6	−1.41758
$324$	−123944.	−0.0655939
$325$	0	0
$326$	−1.21081e6	−0.631005
$327$	−1.47105e6	−0.760777
$328$	−354538.	−0.181961
$329$	−4.48069e6	−2.28221
$330$	0	0
$331$	1.27850e6	0.641402	0.320701	−	0.947180i	$-0.396082\pi$
$331$	1.27850e6	0.641402	0.320701	+	0.947180i	$0.396082\pi$
$332$	1.22071e6	0.607810
$333$	1.11196e6	0.549513
$334$	−1.66770e6	−0.817998
$335$	0	0
$336$	−105677.	−0.0510661
$337$	2.54954e6	1.22289	0.611446	−	0.791286i	$-0.290588\pi$
$337$	2.54954e6	1.22289	0.611446	+	0.791286i	$0.290588\pi$
$338$	−1.26793e6	−0.603676
$339$	−81718.4	−0.0386207
$340$	0	0
$341$	861623.	0.401265
$342$	426399.	0.197129
$343$	291011.	0.133559
$344$	45534.3	0.0207464
$345$	0	0
$346$	441367.	0.198203
$347$	2.80363e6	1.24996	0.624982	−	0.780639i	$-0.285106\pi$
$347$	2.80363e6	1.24996	0.624982	+	0.780639i	$0.285106\pi$
$348$	441374.	0.195370
$349$	1.94021e6	0.852678	0.426339	−	0.904563i	$-0.359803\pi$
$349$	1.94021e6	0.852678	0.426339	+	0.904563i	$0.359803\pi$
$350$	0	0
$351$	−105885.	−0.0458741
$352$	−686014.	−0.295105
$353$	−2.21823e6	−0.947478	−0.473739	−	0.880665i	$-0.657096\pi$
$353$	−2.21823e6	−0.947478	−0.473739	+	0.880665i	$0.657096\pi$
$354$	−1.00635e6	−0.426817
$355$	0	0
$356$	189599.	0.0792888
$357$	−3.08539e6	−1.28127
$358$	2.07244e6	0.854624
$359$	2.03978e6	0.835309	0.417655	−	0.908606i	$-0.362852\pi$
$359$	2.03978e6	0.835309	0.417655	+	0.908606i	$0.362852\pi$
$360$	0	0
$361$	−362152.	−0.146259
$362$	169395.	0.0679404
$363$	131769.	0.0524864
$364$	514546.	0.203550
$365$	0	0
$366$	−1.45386e6	−0.567307
$367$	−332022.	−0.128677	−0.0643387	−	0.997928i	$-0.520494\pi$
$367$	−332022.	−0.128677	−0.0643387	+	0.997928i	$0.520494\pi$
$368$	4138.13	0.00159289
$369$	155856.	0.0595877
$370$	0	0
$371$	−6.23395e6	−2.35141
$372$	1.21068e6	0.453600
$373$	2.34078e6	0.871142	0.435571	−	0.900154i	$-0.356546\pi$
$373$	2.34078e6	0.871142	0.435571	+	0.900154i	$0.356546\pi$
$374$	800897.	0.296072
$375$	0	0
$376$	4.40260e6	1.60598
$377$	377065.	0.136635
$378$	494962.	0.178173
$379$	−2.36625e6	−0.846179	−0.423089	−	0.906088i	$-0.639054\pi$
$379$	−2.36625e6	−0.846179	−0.423089	+	0.906088i	$0.639054\pi$
$380$	0	0
$381$	−1.05489e6	−0.372301
$382$	−2.10695e6	−0.738747
$383$	−913385.	−0.318168	−0.159084	−	0.987265i	$-0.550854\pi$
$383$	−913385.	−0.318168	−0.159084	+	0.987265i	$0.550854\pi$
$384$	−898640.	−0.310998
$385$	0	0
$386$	−2.12386e6	−0.725535
$387$	−20017.0	−0.00679393
$388$	−3.34206e6	−1.12703
$389$	−2.01815e6	−0.676208	−0.338104	−	0.941109i	$-0.609786\pi$
$389$	−2.01815e6	−0.676208	−0.338104	+	0.941109i	$0.609786\pi$
$390$	0	0
$391$	120818.	0.0399661
$392$	−3.38276e6	−1.11187
$393$	−1.08251e6	−0.353549
$394$	−1.61791e6	−0.525065
$395$	0	0
$396$	185151.	0.0593319
$397$	−919738.	−0.292879	−0.146439	−	0.989220i	$-0.546781\pi$
$397$	−919738.	−0.292879	−0.146439	+	0.989220i	$0.546781\pi$
$398$	−939181.	−0.297195
$399$	2.45386e6	0.771646
$400$	0	0
$401$	3.22661e6	1.00204	0.501020	−	0.865436i	$-0.332958\pi$
$401$	3.22661e6	1.00204	0.501020	+	0.865436i	$0.332958\pi$
$402$	1.18182e6	0.364742
$403$	1.03428e6	0.317232
$404$	629424.	0.191862
$405$	0	0
$406$	−1.76260e6	−0.530686
$407$	−1.66107e6	−0.497053
$408$	3.03162e6	0.901622
$409$	2.73228e6	0.807638	0.403819	−	0.914839i	$-0.367682\pi$
$409$	2.73228e6	0.807638	0.403819	+	0.914839i	$0.367682\pi$
$410$	0	0
$411$	−3.32865e6	−0.971994
$412$	3.18157e6	0.923419
$413$	−5.79141e6	−1.67074
$414$	−19381.9	−0.00555769
$415$	0	0
$416$	−823485.	−0.233304
$417$	−1.08143e6	−0.304551
$418$	−636966.	−0.178310
$419$	1.74485e6	0.485538	0.242769	−	0.970084i	$-0.421944\pi$
$419$	1.74485e6	0.485538	0.242769	+	0.970084i	$0.421944\pi$
$420$	0	0
$421$	−3.54354e6	−0.974389	−0.487194	−	0.873294i	$-0.661980\pi$
$421$	−3.54354e6	−0.974389	−0.487194	+	0.873294i	$0.661980\pi$
$422$	1.79207e6	0.489862
$423$	−1.93539e6	−0.525918
$424$	6.12531e6	1.65468
$425$	0	0
$426$	705768.	0.188425
$427$	−8.36672e6	−2.22068
$428$	−1.95787e6	−0.516624
$429$	158174.	0.0414947
$430$	0	0
$431$	93351.9	0.0242064	0.0121032	−	0.999927i	$-0.496147\pi$
$431$	93351.9	0.0242064	0.0121032	+	0.999927i	$0.496147\pi$
$432$	−45646.3	−0.0117678
$433$	−4.10096e6	−1.05115	−0.525576	−	0.850746i	$-0.676150\pi$
$433$	−4.10096e6	−1.05115	−0.525576	+	0.850746i	$0.676150\pi$
$434$	−4.83478e6	−1.23212
$435$	0	0
$436$	3.08773e6	0.777900
$437$	−96088.9	−0.0240696
$438$	2.43741e6	0.607077
$439$	1.90862e6	0.472671	0.236335	−	0.971672i	$-0.424054\pi$
$439$	1.90862e6	0.472671	0.236335	+	0.971672i	$0.424054\pi$
$440$	0	0
$441$	1.48707e6	0.364111
$442$	961388.	0.234069
$443$	714784.	0.173048	0.0865238	−	0.996250i	$-0.472424\pi$
$443$	714784.	0.173048	0.0865238	+	0.996250i	$0.472424\pi$
$444$	−2.33401e6	−0.561882
$445$	0	0
$446$	−2.94251e6	−0.700455
$447$	−4.73677e6	−1.12128
$448$	4.22513e6	0.994593
$449$	−1.96182e6	−0.459243	−0.229621	−	0.973280i	$-0.573749\pi$
$449$	−1.96182e6	−0.459243	−0.229621	+	0.973280i	$0.573749\pi$
$450$	0	0
$451$	−232821.	−0.0538991
$452$	171527.	0.0394900
$453$	−58737.7	−0.0134484
$454$	4.53310e6	1.03218
$455$	0	0
$456$	−2.41110e6	−0.543004
$457$	8.72831e6	1.95497	0.977484	−	0.211012i	$-0.0676758\pi$
$457$	8.72831e6	1.95497	0.977484	+	0.211012i	$0.0676758\pi$
$458$	−1.57408e6	−0.350642
$459$	−1.33271e6	−0.295259
$460$	0	0
$461$	1.76308e6	0.386384	0.193192	−	0.981161i	$-0.438116\pi$
$461$	1.76308e6	0.386384	0.193192	+	0.981161i	$0.438116\pi$
$462$	−739388.	−0.161164
$463$	2.70535e6	0.586505	0.293252	−	0.956035i	$-0.405262\pi$
$463$	2.70535e6	0.586505	0.293252	+	0.956035i	$0.405262\pi$
$464$	162550.	0.0350503
$465$	0	0
$466$	−1.64227e6	−0.350332
$467$	1.15801e6	0.245708	0.122854	−	0.992425i	$-0.460795\pi$
$467$	1.15801e6	0.245708	0.122854	+	0.992425i	$0.460795\pi$
$468$	222254.	0.0469066
$469$	6.80119e6	1.42775
$470$	0	0
$471$	710280.	0.147529
$472$	5.69048e6	1.17569
$473$	29901.9	0.00614534
$474$	106763.	0.0218261
$475$	0	0
$476$	6.47625e6	1.31011
$477$	−2.69270e6	−0.541866
$478$	3.11482e6	0.623538
$479$	8.60379e6	1.71337	0.856685	−	0.515840i	$-0.172520\pi$
$479$	8.60379e6	1.71337	0.856685	+	0.515840i	$0.172520\pi$
$480$	0	0
$481$	−1.99394e6	−0.392960
$482$	−1.35489e6	−0.265637
$483$	−111540.	−0.0217551
$484$	−276584.	−0.0536677
$485$	0	0
$486$	213795.	0.0410588
$487$	−5.23139e6	−0.999527	−0.499764	−	0.866162i	$-0.666580\pi$
$487$	−5.23139e6	−0.999527	−0.499764	+	0.866162i	$0.666580\pi$
$488$	8.22091e6	1.56268
$489$	−3.00978e6	−0.569197
$490$	0	0
$491$	−4.91924e6	−0.920862	−0.460431	−	0.887695i	$-0.652305\pi$
$491$	−4.91924e6	−0.920862	−0.460431	+	0.887695i	$0.652305\pi$
$492$	−327142.	−0.0609288
$493$	4.74587e6	0.879424
$494$	−764608.	−0.140968
$495$	0	0
$496$	445872.	0.0813777
$497$	4.06159e6	0.737572
$498$	−2.10564e6	−0.380461
$499$	255957.	0.0460167	0.0230083	−	0.999735i	$-0.492676\pi$
$499$	255957.	0.0460167	0.0230083	+	0.999735i	$0.492676\pi$
$500$	0	0
$501$	−4.14550e6	−0.737875
$502$	5.28279e6	0.935630
$503$	−3.39889e6	−0.598987	−0.299493	−	0.954098i	$-0.596818\pi$
$503$	−3.39889e6	−0.598987	−0.299493	+	0.954098i	$0.596818\pi$
$504$	−2.79879e6	−0.490789
$505$	0	0
$506$	28953.1	0.00502712
$507$	−3.15177e6	−0.544545
$508$	2.21422e6	0.380681
$509$	1.51733e6	0.259588	0.129794	−	0.991541i	$-0.458568\pi$
$509$	1.51733e6	0.259588	0.129794	+	0.991541i	$0.458568\pi$
$510$	0	0
$511$	1.40269e7	2.37635
$512$	−724191.	−0.122089
$513$	1.05992e6	0.177820
$514$	3.79172e6	0.633036
$515$	0	0
$516$	42015.7	0.00694684
$517$	2.89114e6	0.475711
$518$	9.32069e6	1.52624
$519$	1.09713e6	0.178789
$520$	0	0
$521$	5.98003e6	0.965181	0.482591	−	0.875846i	$-0.339696\pi$
$521$	5.98003e6	0.965181	0.482591	+	0.875846i	$0.339696\pi$
$522$	−761338.	−0.122293
$523$	6.02099e6	0.962528	0.481264	−	0.876576i	$-0.340178\pi$
$523$	6.02099e6	0.962528	0.481264	+	0.876576i	$0.340178\pi$
$524$	2.27219e6	0.361507
$525$	0	0
$526$	−543319.	−0.0856230
$527$	1.30178e7	2.04180
$528$	68187.7	0.0106444
$529$	−6.43198e6	−0.999321
$530$	0	0
$531$	−2.50155e6	−0.385010
$532$	−5.15067e6	−0.789014
$533$	−279476.	−0.0426115
$534$	−327045.	−0.0496312
$535$	0	0
$536$	−6.68267e6	−1.00470
$537$	5.15159e6	0.770913
$538$	−5.27835e6	−0.786217
$539$	−2.22142e6	−0.329351
$540$	0	0
$541$	8.42013e6	1.23687	0.618437	−	0.785834i	$-0.287766\pi$
$541$	8.42013e6	1.23687	0.618437	+	0.785834i	$0.287766\pi$
$542$	2.11147e6	0.308735
$543$	421074.	0.0612856
$544$	−1.03647e7	−1.50161
$545$	0	0
$546$	−887554.	−0.127413
$547$	3.91280e6	0.559138	0.279569	−	0.960126i	$-0.409808\pi$
$547$	3.91280e6	0.559138	0.279569	+	0.960126i	$0.409808\pi$
$548$	6.98686e6	0.993872
$549$	−3.61393e6	−0.511739
$550$	0	0
$551$	−3.77447e6	−0.529635
$552$	109596.	0.0153090
$553$	614408.	0.0854366
$554$	−8.52360e6	−1.17991
$555$	0	0
$556$	2.26994e6	0.311406
$557$	−6.89573e6	−0.941765	−0.470882	−	0.882196i	$-0.656064\pi$
$557$	−6.89573e6	−0.941765	−0.470882	+	0.882196i	$0.656064\pi$
$558$	−2.08834e6	−0.283933
$559$	35893.9	0.00485838
$560$	0	0
$561$	1.99083e6	0.267072
$562$	642135.	0.0857602
$563$	6.25769e6	0.832038	0.416019	−	0.909356i	$-0.363425\pi$
$563$	6.25769e6	0.832038	0.416019	+	0.909356i	$0.363425\pi$
$564$	4.06240e6	0.537755
$565$	0	0
$566$	−414938.	−0.0544431
$567$	1.23036e6	0.160721
$568$	−3.99081e6	−0.519027
$569$	−4.10333e6	−0.531320	−0.265660	−	0.964067i	$-0.585590\pi$
$569$	−4.10333e6	−0.531320	−0.265660	+	0.964067i	$0.585590\pi$
$570$	0	0
$571$	−1.32616e7	−1.70218	−0.851092	−	0.525017i	$-0.824059\pi$
$571$	−1.32616e7	−1.70218	−0.851092	+	0.525017i	$0.824059\pi$
$572$	−332008.	−0.0424286
$573$	−5.23736e6	−0.666386
$574$	1.30642e6	0.165502
$575$	0	0
$576$	1.82501e6	0.229197
$577$	−4.22991e6	−0.528923	−0.264461	−	0.964396i	$-0.585194\pi$
$577$	−4.22991e6	−0.528923	−0.264461	+	0.964396i	$0.585194\pi$
$578$	6.95958e6	0.866490
$579$	−5.27940e6	−0.654468
$580$	0	0
$581$	−1.21176e7	−1.48928
$582$	5.76481e6	0.705469
$583$	4.02242e6	0.490136
$584$	−1.37825e7	−1.67223
$585$	0	0
$586$	2.29880e6	0.276539
$587$	3.48400e6	0.417333	0.208666	−	0.977987i	$-0.433088\pi$
$587$	3.48400e6	0.417333	0.208666	+	0.977987i	$0.433088\pi$
$588$	−3.12136e6	−0.372306
$589$	−1.03533e7	−1.22968
$590$	0	0
$591$	−4.02172e6	−0.473634
$592$	−859571.	−0.100804
$593$	1.44723e7	1.69006	0.845028	−	0.534722i	$-0.179584\pi$
$593$	1.44723e7	1.69006	0.845028	+	0.534722i	$0.179584\pi$
$594$	−319372.	−0.0371391
$595$	0	0
$596$	9.94250e6	1.14652
$597$	−2.33457e6	−0.268085
$598$	34755.1	0.00397434
$599$	−2.29915e6	−0.261819	−0.130909	−	0.991394i	$-0.541790\pi$
$599$	−2.29915e6	−0.261819	−0.130909	+	0.991394i	$0.541790\pi$
$600$	0	0
$601$	3.18751e6	0.359969	0.179984	−	0.983669i	$-0.442395\pi$
$601$	3.18751e6	0.359969	0.179984	+	0.983669i	$0.442395\pi$
$602$	−167787.	−0.0188698
$603$	2.93771e6	0.329015
$604$	123291.	0.0137511
$605$	0	0
$606$	−1.08571e6	−0.120097
$607$	−3.06556e6	−0.337706	−0.168853	−	0.985641i	$-0.554006\pi$
$607$	−3.06556e6	−0.337706	−0.168853	+	0.985641i	$0.554006\pi$
$608$	8.24318e6	0.904348
$609$	−4.38139e6	−0.478705
$610$	0	0
$611$	3.47050e6	0.376087
$612$	2.79736e6	0.301904
$613$	−3.64199e6	−0.391460	−0.195730	−	0.980658i	$-0.562708\pi$
$613$	−3.64199e6	−0.391460	−0.195730	+	0.980658i	$0.562708\pi$
$614$	−4.21255e6	−0.450945
$615$	0	0
$616$	4.18091e6	0.443935
$617$	6.08176e6	0.643156	0.321578	−	0.946883i	$-0.395787\pi$
$617$	6.08176e6	0.643156	0.321578	+	0.946883i	$0.395787\pi$
$618$	−5.48798e6	−0.578018
$619$	1.33232e7	1.39759	0.698796	−	0.715321i	$-0.253719\pi$
$619$	1.33232e7	1.39759	0.698796	+	0.715321i	$0.253719\pi$
$620$	0	0
$621$	−48178.6	−0.00501331
$622$	1.01584e7	1.05281
$623$	−1.88210e6	−0.194277
$624$	81851.8	0.00841525
$625$	0	0
$626$	2.89447e6	0.295212
$627$	−1.58334e6	−0.160844
$628$	−1.49088e6	−0.150850
$629$	−2.50964e7	−2.52921
$630$	0	0
$631$	1.16585e7	1.16565	0.582825	−	0.812598i	$-0.301947\pi$
$631$	1.16585e7	1.16565	0.582825	+	0.812598i	$0.301947\pi$
$632$	−603701.	−0.0601214
$633$	4.45465e6	0.441880
$634$	4.48144e6	0.442787
$635$	0	0
$636$	5.65198e6	0.554062
$637$	−2.66657e6	−0.260378
$638$	1.13731e6	0.110618
$639$	1.75437e6	0.169968
$640$	0	0
$641$	−1.08123e7	−1.03937	−0.519687	−	0.854357i	$-0.673951\pi$
$641$	−1.08123e7	−1.03937	−0.519687	+	0.854357i	$0.673951\pi$
$642$	3.37718e6	0.323383
$643$	−2.10439e6	−0.200723	−0.100362	−	0.994951i	$-0.532000\pi$
$643$	−2.10439e6	−0.200723	−0.100362	+	0.994951i	$0.532000\pi$
$644$	234122.	0.0222448
$645$	0	0
$646$	−9.62361e6	−0.907312
$647$	−8.68332e6	−0.815502	−0.407751	−	0.913093i	$-0.633687\pi$
$647$	−8.68332e6	−0.815502	−0.407751	+	0.913093i	$0.633687\pi$
$648$	−1.20891e6	−0.113099
$649$	3.73688e6	0.348255
$650$	0	0
$651$	−1.20181e7	−1.11143
$652$	6.31755e6	0.582009
$653$	1.33057e7	1.22111	0.610555	−	0.791974i	$-0.290946\pi$
$653$	1.33057e7	1.22111	0.610555	+	0.791974i	$0.290946\pi$
$654$	−5.32611e6	−0.486930
$655$	0	0
$656$	−120480.	−0.0109309
$657$	6.05881e6	0.547614
$658$	−1.62229e7	−1.46071
$659$	9.70036e6	0.870111	0.435055	−	0.900404i	$-0.356729\pi$
$659$	9.70036e6	0.870111	0.435055	+	0.900404i	$0.356729\pi$
$660$	0	0
$661$	2.12502e7	1.89173	0.945864	−	0.324562i	$-0.105217\pi$
$661$	2.12502e7	1.89173	0.945864	+	0.324562i	$0.105217\pi$
$662$	4.62897e6	0.410525
$663$	2.38978e6	0.211141
$664$	1.19065e7	1.04800
$665$	0	0
$666$	4.02599e6	0.351712
$667$	171567.	0.0149321
$668$	8.70143e6	0.754483
$669$	−7.31435e6	−0.631845
$670$	0	0
$671$	5.39859e6	0.462886
$672$	9.56865e6	0.817387
$673$	5.24543e6	0.446420	0.223210	−	0.974770i	$-0.428346\pi$
$673$	5.24543e6	0.446420	0.223210	+	0.974770i	$0.428346\pi$
$674$	9.23095e6	0.782703
$675$	0	0
$676$	6.61557e6	0.556802
$677$	−2.18502e7	−1.83225	−0.916124	−	0.400894i	$-0.868700\pi$
$677$	−2.18502e7	−1.83225	−0.916124	+	0.400894i	$0.868700\pi$
$678$	−295872.	−0.0247189
$679$	3.31756e7	2.76150
$680$	0	0
$681$	1.12682e7	0.931078
$682$	3.11962e6	0.256827
$683$	−1.61102e7	−1.32145	−0.660724	−	0.750629i	$-0.729750\pi$
$683$	−1.61102e7	−1.32145	−0.660724	+	0.750629i	$0.729750\pi$
$684$	−2.22478e6	−0.181823
$685$	0	0
$686$	1.05364e6	0.0854836
$687$	−3.91278e6	−0.316296
$688$	15473.6	0.00124629
$689$	4.82848e6	0.387492
$690$	0	0
$691$	4.06476e6	0.323847	0.161923	−	0.986803i	$-0.448230\pi$
$691$	4.06476e6	0.323847	0.161923	+	0.986803i	$0.448230\pi$
$692$	−2.30288e6	−0.182813
$693$	−1.83794e6	−0.145378
$694$	1.01509e7	0.800030
$695$	0	0
$696$	4.30503e6	0.336863
$697$	−3.51758e6	−0.274260
$698$	7.02478e6	0.545751
$699$	−4.08229e6	−0.316017
$700$	0	0
$701$	−3.76410e6	−0.289312	−0.144656	−	0.989482i	$-0.546207\pi$
$701$	−3.76410e6	−0.289312	−0.144656	+	0.989482i	$0.546207\pi$
$702$	−383371.	−0.0293614
$703$	1.99596e7	1.52322
$704$	−2.72625e6	−0.207316
$705$	0	0
$706$	−8.03138e6	−0.606427
$707$	−6.24810e6	−0.470110
$708$	5.25076e6	0.393676
$709$	8.48557e6	0.633965	0.316982	−	0.948431i	$-0.397330\pi$
$709$	8.48557e6	0.633965	0.316982	+	0.948431i	$0.397330\pi$
$710$	0	0
$711$	265388.	0.0196883
$712$	1.84930e6	0.136712
$713$	470607.	0.0346684
$714$	−1.11711e7	−0.820066
$715$	0	0
$716$	−1.08132e7	−0.788265
$717$	7.74268e6	0.562462
$718$	7.38529e6	0.534634
$719$	−5.44794e6	−0.393016	−0.196508	−	0.980502i	$-0.562960\pi$
$719$	−5.44794e6	−0.393016	−0.196508	+	0.980502i	$0.562960\pi$
$720$	0	0
$721$	−3.15825e7	−2.26260
$722$	−1.31122e6	−0.0936121
$723$	−3.36794e6	−0.239617
$724$	−883835.	−0.0626650
$725$	0	0
$726$	477087.	0.0335935
$727$	1.52365e7	1.06918	0.534589	−	0.845112i	$-0.320467\pi$
$727$	1.52365e7	1.06918	0.534589	+	0.845112i	$0.320467\pi$
$728$	5.01873e6	0.350966
$729$	531441.	0.0370370
$730$	0	0
$731$	451773.	0.0312699
$732$	7.58566e6	0.523258
$733$	2.32268e7	1.59673	0.798363	−	0.602177i	$-0.205700\pi$
$733$	2.32268e7	1.59673	0.798363	+	0.602177i	$0.205700\pi$
$734$	−1.20213e6	−0.0823590
$735$	0	0
$736$	−374692.	−0.0254964
$737$	−4.38844e6	−0.297606
$738$	564295.	0.0381387
$739$	−8.86078e6	−0.596844	−0.298422	−	0.954434i	$-0.596460\pi$
$739$	−8.86078e6	−0.596844	−0.298422	+	0.954434i	$0.596460\pi$
$740$	0	0
$741$	−1.90063e6	−0.127160
$742$	−2.25708e7	−1.50500
$743$	1.16363e7	0.773289	0.386644	−	0.922229i	$-0.373634\pi$
$743$	1.16363e7	0.773289	0.386644	+	0.922229i	$0.373634\pi$
$744$	1.18086e7	0.782109
$745$	0	0
$746$	8.47511e6	0.557568
$747$	−5.23410e6	−0.343195
$748$	−4.17877e6	−0.273083
$749$	1.94352e7	1.26586
$750$	0	0
$751$	−1.39412e7	−0.901989	−0.450995	−	0.892527i	$-0.648931\pi$
$751$	−1.39412e7	−0.901989	−0.450995	+	0.892527i	$0.648931\pi$
$752$	1.49610e6	0.0964756
$753$	1.31317e7	0.843984
$754$	1.36521e6	0.0874524
$755$	0	0
$756$	−2.58252e6	−0.164339
$757$	−1.50634e7	−0.955394	−0.477697	−	0.878525i	$-0.658528\pi$
$757$	−1.50634e7	−0.955394	−0.477697	+	0.878525i	$0.658528\pi$
$758$	−8.56730e6	−0.541591
$759$	71970.4	0.00453471
$760$	0	0
$761$	1.92157e7	1.20280	0.601402	−	0.798947i	$-0.294609\pi$
$761$	1.92157e7	1.20280	0.601402	+	0.798947i	$0.294609\pi$
$762$	−3.81936e6	−0.238289
$763$	−3.06510e7	−1.90605
$764$	1.09932e7	0.681385
$765$	0	0
$766$	−3.30703e6	−0.203641
$767$	4.48571e6	0.275323
$768$	−9.74256e6	−0.596033
$769$	7.66150e6	0.467195	0.233597	−	0.972333i	$-0.424950\pi$
$769$	7.66150e6	0.467195	0.233597	+	0.972333i	$0.424950\pi$
$770$	0	0
$771$	9.42529e6	0.571030
$772$	1.10815e7	0.669199
$773$	−4.99696e6	−0.300786	−0.150393	−	0.988626i	$-0.548054\pi$
$773$	−4.99696e6	−0.300786	−0.150393	+	0.988626i	$0.548054\pi$
$774$	−72474.0	−0.00434840
$775$	0	0
$776$	−3.25975e7	−1.94326
$777$	2.31690e7	1.37675
$778$	−7.30699e6	−0.432802
$779$	2.79759e6	0.165174
$780$	0	0
$781$	−2.62072e6	−0.153742
$782$	437439.	0.0255800
$783$	−1.89250e6	−0.110314
$784$	−1.14954e6	−0.0667933
$785$	0	0
$786$	−3.91936e6	−0.226287
$787$	−1.52048e7	−0.875073	−0.437537	−	0.899201i	$-0.644149\pi$
$787$	−1.52048e7	−0.875073	−0.437537	+	0.899201i	$0.644149\pi$
$788$	8.44161e6	0.484295
$789$	−1.35056e6	−0.0772361
$790$	0	0
$791$	−1.70270e6	−0.0967601
$792$	1.80591e6	0.102302
$793$	6.48041e6	0.365948
$794$	−3.33003e6	−0.187455
$795$	0	0
$796$	4.90028e6	0.274119
$797$	1.71869e6	0.0958411	0.0479205	−	0.998851i	$-0.484741\pi$
$797$	1.71869e6	0.0958411	0.0479205	+	0.998851i	$0.484741\pi$
$798$	8.88452e6	0.493886
$799$	4.36808e7	2.42061
$800$	0	0
$801$	−812954.	−0.0447698
$802$	1.16823e7	0.641348
$803$	−9.05082e6	−0.495335
$804$	−6.16628e6	−0.336421
$805$	0	0
$806$	3.74476e6	0.203042
$807$	−1.31207e7	−0.709207
$808$	6.13921e6	0.330814
$809$	−2.42828e7	−1.30445	−0.652226	−	0.758025i	$-0.726165\pi$
$809$	−2.42828e7	−1.30445	−0.652226	+	0.758025i	$0.726165\pi$
$810$	0	0
$811$	2.20685e7	1.17820	0.589101	−	0.808059i	$-0.299482\pi$
$811$	2.20685e7	1.17820	0.589101	+	0.808059i	$0.299482\pi$
$812$	9.19655e6	0.489480
$813$	5.24859e6	0.278494
$814$	−6.01413e6	−0.318135
$815$	0	0
$816$	1.03021e6	0.0541629
$817$	−359302.	−0.0188324
$818$	9.89257e6	0.516923
$819$	−2.20624e6	−0.114933
$820$	0	0
$821$	2.07673e7	1.07528	0.537640	−	0.843175i	$-0.319316\pi$
$821$	2.07673e7	1.07528	0.537640	+	0.843175i	$0.319316\pi$
$822$	−1.20518e7	−0.622118
$823$	1.75590e7	0.903650	0.451825	−	0.892107i	$-0.350773\pi$
$823$	1.75590e7	0.903650	0.451825	+	0.892107i	$0.350773\pi$
$824$	3.10321e7	1.59218
$825$	0	0
$826$	−2.09685e7	−1.06935
$827$	807814.	0.0410722	0.0205361	−	0.999789i	$-0.493463\pi$
$827$	807814.	0.0410722	0.0205361	+	0.999789i	$0.493463\pi$
$828$	101127.	0.00512615
$829$	−6.29584e6	−0.318176	−0.159088	−	0.987264i	$-0.550855\pi$
$829$	−6.29584e6	−0.318176	−0.159088	+	0.987264i	$0.550855\pi$
$830$	0	0
$831$	−2.11876e7	−1.06434
$832$	−3.27256e6	−0.163900
$833$	−3.35624e7	−1.67587
$834$	−3.91547e6	−0.194926
$835$	0	0
$836$	3.32344e6	0.164465
$837$	−5.19110e6	−0.256121
$838$	6.31745e6	0.310765
$839$	−1.62539e7	−0.797173	−0.398587	−	0.917131i	$-0.630499\pi$
$839$	−1.62539e7	−0.797173	−0.398587	+	0.917131i	$0.630499\pi$
$840$	0	0
$841$	−1.37718e7	−0.671431
$842$	−1.28298e7	−0.623651
$843$	1.59619e6	0.0773599
$844$	−9.35032e6	−0.451825
$845$	0	0
$846$	−7.00734e6	−0.336610
$847$	2.74556e6	0.131499
$848$	2.08152e6	0.0994010
$849$	−1.03144e6	−0.0491103
$850$	0	0
$851$	−907256.	−0.0429444
$852$	−3.68242e6	−0.173794
$853$	−3.26091e7	−1.53450	−0.767248	−	0.641350i	$-0.778375\pi$
$853$	−3.26091e7	−1.53450	−0.767248	+	0.641350i	$0.778375\pi$
$854$	−3.02928e7	−1.42133
$855$	0	0
$856$	−1.90965e7	−0.890777
$857$	−1.10056e7	−0.511872	−0.255936	−	0.966694i	$-0.582384\pi$
$857$	−1.10056e7	−0.511872	−0.255936	+	0.966694i	$0.582384\pi$
$858$	572690.	0.0265584
$859$	5.98862e6	0.276914	0.138457	−	0.990368i	$-0.455786\pi$
$859$	5.98862e6	0.276914	0.138457	+	0.990368i	$0.455786\pi$
$860$	0	0
$861$	3.24743e6	0.149291
$862$	337993.	0.0154931
$863$	−2.17360e7	−0.993465	−0.496733	−	0.867904i	$-0.665467\pi$
$863$	−2.17360e7	−0.993465	−0.496733	+	0.867904i	$0.665467\pi$
$864$	4.13309e6	0.188361
$865$	0	0
$866$	−1.48481e7	−0.672783
$867$	1.72998e7	0.781616
$868$	2.52260e7	1.13645
$869$	−396444.	−0.0178087
$870$	0	0
$871$	−5.26784e6	−0.235281
$872$	3.01168e7	1.34128
$873$	1.43299e7	0.636368
$874$	−347902.	−0.0154056
$875$	0	0
$876$	−1.27175e7	−0.559939
$877$	−2.33923e7	−1.02701	−0.513504	−	0.858087i	$-0.671653\pi$
$877$	−2.33923e7	−1.02701	−0.513504	+	0.858087i	$0.671653\pi$
$878$	6.91041e6	0.302529
$879$	5.71425e6	0.249452
$880$	0	0
$881$	9.31413e6	0.404299	0.202149	−	0.979355i	$-0.435207\pi$
$881$	9.31413e6	0.404299	0.202149	+	0.979355i	$0.435207\pi$
$882$	5.38412e6	0.233047
$883$	−9.33127e6	−0.402753	−0.201377	−	0.979514i	$-0.564542\pi$
$883$	−9.33127e6	−0.402753	−0.201377	+	0.979514i	$0.564542\pi$
$884$	−5.01615e6	−0.215894
$885$	0	0
$886$	2.58797e6	0.110758
$887$	9.36515e6	0.399674	0.199837	−	0.979829i	$-0.435959\pi$
$887$	9.36515e6	0.399674	0.199837	+	0.979829i	$0.435959\pi$
$888$	−2.27652e7	−0.968811
$889$	−2.19799e7	−0.932762
$890$	0	0
$891$	−793881.	−0.0335013
$892$	1.53529e7	0.646067
$893$	−3.47401e7	−1.45781
$894$	−1.71501e7	−0.717666
$895$	0	0
$896$	−1.87242e7	−0.779173
$897$	86392.6	0.00358505
$898$	−7.10300e6	−0.293935
$899$	1.84859e7	0.762853
$900$	0	0
$901$	6.07728e7	2.49401
$902$	−842959.	−0.0344977
$903$	−417077.	−0.0170215
$904$	1.67303e6	0.0680897
$905$	0	0
$906$	−212667.	−0.00860757
$907$	4.47639e7	1.80680	0.903398	−	0.428802i	$-0.141064\pi$
$907$	4.47639e7	1.80680	0.903398	+	0.428802i	$0.141064\pi$
$908$	−2.36520e7	−0.952034
$909$	−2.69881e6	−0.108333
$910$	0	0
$911$	−7.58451e6	−0.302783	−0.151392	−	0.988474i	$-0.548375\pi$
$911$	−7.58451e6	−0.302783	−0.151392	+	0.988474i	$0.548375\pi$
$912$	−819346.	−0.0326197
$913$	7.81884e6	0.310431
$914$	3.16020e7	1.25126
$915$	0	0
$916$	8.21295e6	0.323415
$917$	−2.25554e7	−0.885781
$918$	−4.82524e6	−0.188978
$919$	4.51373e7	1.76298	0.881488	−	0.472206i	$-0.156542\pi$
$919$	4.51373e7	1.76298	0.881488	+	0.472206i	$0.156542\pi$
$920$	0	0
$921$	−1.04714e7	−0.406775
$922$	6.38345e6	0.247302
$923$	−3.14589e6	−0.121545
$924$	3.85784e6	0.148650
$925$	0	0
$926$	9.79508e6	0.375388
$927$	−1.36418e7	−0.521401
$928$	−1.47182e7	−0.561030
$929$	2.10501e6	0.0800231	0.0400115	−	0.999199i	$-0.487261\pi$
$929$	2.10501e6	0.0800231	0.0400115	+	0.999199i	$0.487261\pi$
$930$	0	0
$931$	2.66927e7	1.00930
$932$	8.56874e6	0.323130
$933$	2.52514e7	0.949690
$934$	4.19271e6	0.157263
$935$	0	0
$936$	2.16780e6	0.0808777
$937$	−1.16498e7	−0.433481	−0.216740	−	0.976229i	$-0.569542\pi$
$937$	−1.16498e7	−0.433481	−0.216740	+	0.976229i	$0.569542\pi$
$938$	2.46246e7	0.913823
$939$	7.19495e6	0.266296
$940$	0	0
$941$	8.26754e6	0.304370	0.152185	−	0.988352i	$-0.451369\pi$
$941$	8.26754e6	0.304370	0.152185	+	0.988352i	$0.451369\pi$
$942$	2.57166e6	0.0944249
$943$	−127164.	−0.00465676
$944$	1.93375e6	0.0706271
$945$	0	0
$946$	108264.	0.00393328
$947$	−1.00789e7	−0.365205	−0.182602	−	0.983187i	$-0.558452\pi$
$947$	−1.00789e7	−0.365205	−0.182602	+	0.983187i	$0.558452\pi$
$948$	−557051.	−0.0201314
$949$	−1.08645e7	−0.391602
$950$	0	0
$951$	1.11398e7	0.399416
$952$	6.31674e7	2.25892
$953$	−4.14245e7	−1.47749	−0.738747	−	0.673983i	$-0.764582\pi$
$953$	−4.14245e7	−1.47749	−0.738747	+	0.673983i	$0.764582\pi$
$954$	−9.74926e6	−0.346817
$955$	0	0
$956$	−1.62519e7	−0.575122
$957$	2.82707e6	0.0997830
$958$	3.11511e7	1.09663
$959$	−6.93564e7	−2.43523
$960$	0	0
$961$	2.20774e7	0.771149
$962$	−7.21931e6	−0.251512
$963$	8.39485e6	0.291707
$964$	7.06932e6	0.245011
$965$	0	0
$966$	−403844.	−0.0139242
$967$	−2.26789e7	−0.779930	−0.389965	−	0.920830i	$-0.627513\pi$
$967$	−2.26789e7	−0.779930	−0.389965	+	0.920830i	$0.627513\pi$
$968$	−2.69772e6	−0.0925354
$969$	−2.39219e7	−0.818441
$970$	0	0
$971$	−2.89929e7	−0.986833	−0.493416	−	0.869793i	$-0.664252\pi$
$971$	−2.89929e7	−0.986833	−0.493416	+	0.869793i	$0.664252\pi$
$972$	−1.11550e6	−0.0378707
$973$	−2.25330e7	−0.763020
$974$	−1.89409e7	−0.639740
$975$	0	0
$976$	2.79365e6	0.0938746
$977$	−5.72118e7	−1.91756	−0.958781	−	0.284146i	$-0.908290\pi$
$977$	−5.72118e7	−1.91756	−0.958781	+	0.284146i	$0.908290\pi$
$978$	−1.08973e7	−0.364311
$979$	1.21441e6	0.0404958
$980$	0	0
$981$	−1.32394e7	−0.439235
$982$	−1.78108e7	−0.589391
$983$	8.82345e6	0.291242	0.145621	−	0.989340i	$-0.453482\pi$
$983$	8.82345e6	0.291242	0.145621	+	0.989340i	$0.453482\pi$
$984$	−3.19084e6	−0.105055
$985$	0	0
$986$	1.71830e7	0.562869
$987$	−4.03262e7	−1.31763
$988$	3.98943e6	0.130022
$989$	16332.0	0.000530944 0
$990$	0	0
$991$	4.55123e7	1.47213	0.736063	−	0.676913i	$-0.236683\pi$
$991$	4.55123e7	1.47213	0.736063	+	0.676913i	$0.236683\pi$
$992$	−4.03719e7	−1.30257
$993$	1.15065e7	0.370314
$994$	1.47055e7	0.472078
$995$	0	0
$996$	1.09864e7	0.350919
$997$	−4.74409e7	−1.51152	−0.755762	−	0.654847i	$-0.772733\pi$
$997$	−4.74409e7	−1.51152	−0.755762	+	0.654847i	$0.772733\pi$
$998$	926725.	0.0294527
$999$	1.00076e7	0.317262

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.w.1.9		13
5.2	odd	4		165.6.c.a.34.16	yes	26
5.3	odd	4		165.6.c.a.34.11	✓	26
5.4	even	2		825.6.a.x.1.5		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.c.a.34.11	✓	26	5.3	odd	4
165.6.c.a.34.16	yes	26	5.2	odd	4
825.6.a.w.1.9		13	1.1	even	1	trivial
825.6.a.x.1.5		13	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q+3.62063 q^{2} +9.00000 q^{3} -18.8910 q^{4} +32.5857 q^{6} +187.526 q^{7} -184.258 q^{8} +81.0000 q^{9} -121.000 q^{11} -170.019 q^{12} -145.247 q^{13} +678.961 q^{14} -62.6149 q^{16} -1828.13 q^{17} +293.271 q^{18} +1453.94 q^{19} +1687.73 q^{21} -438.096 q^{22} -66.0886 q^{23} -1658.32 q^{24} -525.886 q^{26} +729.000 q^{27} -3542.55 q^{28} -2596.02 q^{29} -7120.85 q^{31} +5669.54 q^{32} -1089.00 q^{33} -6618.98 q^{34} -1530.17 q^{36} +13727.9 q^{37} +5264.18 q^{38} -1307.22 q^{39} +1924.14 q^{41} +6110.65 q^{42} -247.123 q^{43} +2285.82 q^{44} -239.282 q^{46} -23893.7 q^{47} -563.534 q^{48} +18358.8 q^{49} -16453.2 q^{51} +2743.87 q^{52} -33243.2 q^{53} +2639.44 q^{54} -34553.0 q^{56} +13085.5 q^{57} -9399.23 q^{58} -30883.3 q^{59} -44616.4 q^{61} -25782.0 q^{62} +15189.6 q^{63} +22531.0 q^{64} -3942.86 q^{66} +36268.1 q^{67} +34535.3 q^{68} -594.797 q^{69} +21658.8 q^{71} -14924.9 q^{72} +74800.2 q^{73} +49703.6 q^{74} -27466.5 q^{76} -22690.6 q^{77} -4732.98 q^{78} +3276.39 q^{79} +6561.00 q^{81} +6966.61 q^{82} -64618.5 q^{83} -31883.0 q^{84} -894.741 q^{86} -23364.2 q^{87} +22295.2 q^{88} -10036.5 q^{89} -27237.6 q^{91} +1248.48 q^{92} -64087.7 q^{93} -86510.3 q^{94} +51025.8 q^{96} +176913. q^{97} +66470.6 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 3 q^{2} + 117 q^{3} + 229 q^{4} - 27 q^{6} - 284 q^{7} - 369 q^{8} + 1053 q^{9} - 1573 q^{11} + 2061 q^{12} - 366 q^{13} - 2758 q^{14} + 4141 q^{16} - 2056 q^{17} - 243 q^{18} - 310 q^{19} - 2556 q^{21}+ \cdots - 127413 q^{99}+O(q^{100}) \) 13 * q - 3 * q^2 + 117 * q^3 + 229 * q^4 - 27 * q^6 - 284 * q^7 - 369 * q^8 + 1053 * q^9 - 1573 * q^11 + 2061 * q^12 - 366 * q^13 - 2758 * q^14 + 4141 * q^16 - 2056 * q^17 - 243 * q^18 - 310 * q^19 - 2556 * q^21 + 363 * q^22 - 3612 * q^23 - 3321 * q^24 + 2280 * q^26 + 9477 * q^27 - 7896 * q^28 - 4848 * q^29 - 24 * q^31 - 38111 * q^32 - 14157 * q^33 + 5518 * q^34 + 18549 * q^36 + 8420 * q^37 - 474 * q^38 - 3294 * q^39 - 15120 * q^41 - 24822 * q^42 - 35492 * q^43 - 27709 * q^44 - 20280 * q^46 - 46544 * q^47 + 37269 * q^48 + 81837 * q^49 - 18504 * q^51 - 107194 * q^52 - 42256 * q^53 - 2187 * q^54 - 196602 * q^56 - 2790 * q^57 - 114160 * q^58 - 65592 * q^59 - 52042 * q^61 - 94972 * q^62 - 23004 * q^63 + 185977 * q^64 + 3267 * q^66 - 80580 * q^67 - 61108 * q^68 - 32508 * q^69 - 77820 * q^71 - 29889 * q^72 - 103050 * q^73 - 240028 * q^74 - 271174 * q^76 + 34364 * q^77 + 20520 * q^78 - 112258 * q^79 + 85293 * q^81 - 64060 * q^82 - 292150 * q^83 - 71064 * q^84 - 319250 * q^86 - 43632 * q^87 + 44649 * q^88 - 295810 * q^89 - 24200 * q^91 - 121328 * q^92 - 216 * q^93 - 358144 * q^94 - 342999 * q^96 - 49072 * q^97 - 101815 * q^98 - 127413 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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