LMFDB - Embedded newform 507.4.a.q.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.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:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 48x^{7} + 29x^{6} + 772x^{5} - 150x^{4} - 4745x^{3} - 966x^{2} + 9428x + 5144 $ x^9 - x^8 - 48x^7 + 29x^6 + 772x^5 - 150x^4 - 4745x^3 - 966x^2 + 9428*x + 5144
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-3.76649$ of defining polynomial
Character	$\chi$	$=$	507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.76649 q^{2} -3.00000 q^{3} +14.7194 q^{4} +18.8390 q^{5} -14.2995 q^{6} -23.8593 q^{7} +32.0282 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+4.76649 q^{2} -3.00000 q^{3} +14.7194 q^{4} +18.8390 q^{5} -14.2995 q^{6} -23.8593 q^{7} +32.0282 q^{8} +9.00000 q^{9} +89.7961 q^{10} +60.2327 q^{11} -44.1583 q^{12} -113.725 q^{14} -56.5171 q^{15} +34.9065 q^{16} -1.17630 q^{17} +42.8984 q^{18} +29.9930 q^{19} +277.300 q^{20} +71.5780 q^{21} +287.099 q^{22} +159.182 q^{23} -96.0845 q^{24} +229.909 q^{25} -27.0000 q^{27} -351.196 q^{28} +20.8211 q^{29} -269.388 q^{30} -67.2170 q^{31} -89.8439 q^{32} -180.698 q^{33} -5.60681 q^{34} -449.487 q^{35} +132.475 q^{36} +138.799 q^{37} +142.961 q^{38} +603.380 q^{40} -113.297 q^{41} +341.176 q^{42} +32.9644 q^{43} +886.592 q^{44} +169.551 q^{45} +758.740 q^{46} -520.256 q^{47} -104.720 q^{48} +226.268 q^{49} +1095.86 q^{50} +3.52889 q^{51} +467.189 q^{53} -128.695 q^{54} +1134.73 q^{55} -764.171 q^{56} -89.9789 q^{57} +99.2437 q^{58} -409.028 q^{59} -831.900 q^{60} +74.9067 q^{61} -320.389 q^{62} -214.734 q^{63} -707.492 q^{64} -861.296 q^{66} +305.693 q^{67} -17.3144 q^{68} -477.546 q^{69} -2142.47 q^{70} -318.757 q^{71} +288.254 q^{72} +867.378 q^{73} +661.585 q^{74} -689.727 q^{75} +441.480 q^{76} -1437.11 q^{77} -626.973 q^{79} +657.605 q^{80} +81.0000 q^{81} -540.031 q^{82} +1212.06 q^{83} +1053.59 q^{84} -22.1603 q^{85} +157.125 q^{86} -62.4634 q^{87} +1929.14 q^{88} -679.991 q^{89} +808.165 q^{90} +2343.07 q^{92} +201.651 q^{93} -2479.80 q^{94} +565.038 q^{95} +269.532 q^{96} -491.109 q^{97} +1078.51 q^{98} +542.094 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 8 q^{2} - 27 q^{3} + 32 q^{4} + 41 q^{5} - 24 q^{6} + q^{7} + 111 q^{8} + 81 q^{9}+O(q^{10}) $ 9 * q + 8 * q^2 - 27 * q^3 + 32 * q^4 + 41 * q^5 - 24 * q^6 + q^7 + 111 * q^8 + 81 * q^9	\( 9 q + 8 q^{2} - 27 q^{3} + 32 q^{4} + 41 q^{5} - 24 q^{6} + q^{7} + 111 q^{8} + 81 q^{9} + 198 q^{10} + 37 q^{11} - 96 q^{12} + 98 q^{14} - 123 q^{15} + 32 q^{16} - 134 q^{17} + 72 q^{18} - 72 q^{19} + 356 q^{20} - 3 q^{21} + 274 q^{22} + 226 q^{23} - 333 q^{24} + 612 q^{25} - 243 q^{27} + 132 q^{28} - 547 q^{29} - 594 q^{30} - 521 q^{31} + 721 q^{32} - 111 q^{33} - 100 q^{34} + 138 q^{35} + 288 q^{36} + 584 q^{37} - 416 q^{38} + 1342 q^{40} + 482 q^{41} - 294 q^{42} + 158 q^{43} + 1453 q^{44} + 369 q^{45} + 1537 q^{46} + 1500 q^{47} - 96 q^{48} + 642 q^{49} + 2777 q^{50} + 402 q^{51} + 1399 q^{53} - 216 q^{54} - 1408 q^{55} - 616 q^{56} + 216 q^{57} + 1455 q^{58} + 1541 q^{59} - 1068 q^{60} + 2092 q^{61} - 293 q^{62} + 9 q^{63} + 2481 q^{64} - 822 q^{66} + 252 q^{67} - 1579 q^{68} - 678 q^{69} + 2492 q^{70} + 2352 q^{71} + 999 q^{72} + 903 q^{73} + 1037 q^{74} - 1836 q^{75} - 485 q^{76} - 1686 q^{77} - 115 q^{79} + 5701 q^{80} + 729 q^{81} - 5147 q^{82} + 1207 q^{83} - 396 q^{84} + 4308 q^{85} + 5691 q^{86} + 1641 q^{87} - 484 q^{88} + 2336 q^{89} + 1782 q^{90} + 2087 q^{92} + 1563 q^{93} - 468 q^{94} - 222 q^{95} - 2163 q^{96} + 2155 q^{97} + 5593 q^{98} + 333 q^{99}+O(q^{100}) \) 9 * q + 8 * q^2 - 27 * q^3 + 32 * q^4 + 41 * q^5 - 24 * q^6 + q^7 + 111 * q^8 + 81 * q^9 + 198 * q^10 + 37 * q^11 - 96 * q^12 + 98 * q^14 - 123 * q^15 + 32 * q^16 - 134 * q^17 + 72 * q^18 - 72 * q^19 + 356 * q^20 - 3 * q^21 + 274 * q^22 + 226 * q^23 - 333 * q^24 + 612 * q^25 - 243 * q^27 + 132 * q^28 - 547 * q^29 - 594 * q^30 - 521 * q^31 + 721 * q^32 - 111 * q^33 - 100 * q^34 + 138 * q^35 + 288 * q^36 + 584 * q^37 - 416 * q^38 + 1342 * q^40 + 482 * q^41 - 294 * q^42 + 158 * q^43 + 1453 * q^44 + 369 * q^45 + 1537 * q^46 + 1500 * q^47 - 96 * q^48 + 642 * q^49 + 2777 * q^50 + 402 * q^51 + 1399 * q^53 - 216 * q^54 - 1408 * q^55 - 616 * q^56 + 216 * q^57 + 1455 * q^58 + 1541 * q^59 - 1068 * q^60 + 2092 * q^61 - 293 * q^62 + 9 * q^63 + 2481 * q^64 - 822 * q^66 + 252 * q^67 - 1579 * q^68 - 678 * q^69 + 2492 * q^70 + 2352 * q^71 + 999 * q^72 + 903 * q^73 + 1037 * q^74 - 1836 * q^75 - 485 * q^76 - 1686 * q^77 - 115 * q^79 + 5701 * q^80 + 729 * q^81 - 5147 * q^82 + 1207 * q^83 - 396 * q^84 + 4308 * q^85 + 5691 * q^86 + 1641 * q^87 - 484 * q^88 + 2336 * q^89 + 1782 * q^90 + 2087 * q^92 + 1563 * q^93 - 468 * q^94 - 222 * q^95 - 2163 * q^96 + 2155 * q^97 + 5593 * q^98 + 333 * q^99

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 $

$2$	4.76649	1.68521	0.842605	−	0.538533i	$-0.181021\pi$
$2$	4.76649	1.68521	0.842605	+	0.538533i	$0.181021\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	14.7194	1.83993
$5$	18.8390	1.68501	0.842507	−	0.538686i	$-0.181079\pi$
$5$	18.8390	1.68501	0.842507	+	0.538686i	$0.181079\pi$
$6$	−14.2995	−0.972956
$7$	−23.8593	−1.28828	−0.644142	−	0.764906i	$-0.722785\pi$
$7$	−23.8593	−1.28828	−0.644142	+	0.764906i	$0.722785\pi$
$8$	32.0282	1.41546
$9$	9.00000	0.333333
$10$	89.7961	2.83960
$11$	60.2327	1.65099	0.825493	−	0.564412i	$-0.190897\pi$
$11$	60.2327	1.65099	0.825493	+	0.564412i	$0.190897\pi$
$12$	−44.1583	−1.06228
$13$	0	0
$13$	0	0
$14$	−113.725	−2.17103
$15$	−56.5171	−0.972843
$16$	34.9065	0.545414
$17$	−1.17630	−0.0167820	−0.00839099	−	0.999965i	$-0.502671\pi$
$17$	−1.17630	−0.0167820	−0.00839099	+	0.999965i	$0.502671\pi$
$18$	42.8984	0.561736
$19$	29.9930	0.362150	0.181075	−	0.983469i	$-0.442042\pi$
$19$	29.9930	0.362150	0.181075	+	0.983469i	$0.442042\pi$
$20$	277.300	3.10031
$21$	71.5780	0.743791
$22$	287.099	2.78226
$23$	159.182	1.44312	0.721560	−	0.692352i	$-0.243425\pi$
$23$	159.182	1.44312	0.721560	+	0.692352i	$0.243425\pi$
$24$	−96.0845	−0.817216
$25$	229.909	1.83927
$26$	0	0
$27$	−27.0000	−0.192450
$28$	−351.196	−2.37035
$29$	20.8211	0.133324	0.0666618	−	0.997776i	$-0.478765\pi$
$29$	20.8211	0.133324	0.0666618	+	0.997776i	$0.478765\pi$
$30$	−269.388	−1.63944
$31$	−67.2170	−0.389436	−0.194718	−	0.980859i	$-0.562379\pi$
$31$	−67.2170	−0.389436	−0.194718	+	0.980859i	$0.562379\pi$
$32$	−89.8439	−0.496322
$33$	−180.698	−0.953197
$34$	−5.60681	−0.0282812
$35$	−449.487	−2.17077
$36$	132.475	0.613310
$37$	138.799	0.616714	0.308357	−	0.951271i	$-0.400221\pi$
$37$	138.799	0.616714	0.308357	+	0.951271i	$0.400221\pi$
$38$	142.961	0.610299
$39$	0	0
$40$	603.380	2.38507
$41$	−113.297	−0.431563	−0.215781	−	0.976442i	$-0.569230\pi$
$41$	−113.297	−0.431563	−0.215781	+	0.976442i	$0.569230\pi$
$42$	341.176	1.25344
$43$	32.9644	0.116908	0.0584538	−	0.998290i	$-0.481383\pi$
$43$	32.9644	0.116908	0.0584538	+	0.998290i	$0.481383\pi$
$44$	886.592	3.03770
$45$	169.551	0.561671
$46$	758.740	2.43196
$47$	−520.256	−1.61462	−0.807310	−	0.590127i	$-0.799078\pi$
$47$	−520.256	−1.61462	−0.807310	+	0.590127i	$0.799078\pi$
$48$	−104.720	−0.314895
$49$	226.268	0.659674
$50$	1095.86	3.09956
$51$	3.52889	0.00968909
$52$	0	0
$53$	467.189	1.21082	0.605409	−	0.795915i	$-0.293010\pi$
$53$	467.189	1.21082	0.605409	+	0.795915i	$0.293010\pi$
$54$	−128.695	−0.324319
$55$	1134.73	2.78193
$56$	−764.171	−1.82351
$57$	−89.9789	−0.209088
$58$	99.2437	0.224678
$59$	−409.028	−0.902558	−0.451279	−	0.892383i	$-0.649032\pi$
$59$	−409.028	−0.902558	−0.451279	+	0.892383i	$0.649032\pi$
$60$	−831.900	−1.78996
$61$	74.9067	0.157226	0.0786132	−	0.996905i	$-0.474951\pi$
$61$	74.9067	0.157226	0.0786132	+	0.996905i	$0.474951\pi$
$62$	−320.389	−0.656282
$63$	−214.734	−0.429428
$64$	−707.492	−1.38182
$65$	0	0
$66$	−861.296	−1.60634
$67$	305.693	0.557408	0.278704	−	0.960377i	$-0.410095\pi$
$67$	305.693	0.557408	0.278704	+	0.960377i	$0.410095\pi$
$68$	−17.3144	−0.0308777
$69$	−477.546	−0.833186
$70$	−2142.47	−3.65821
$71$	−318.757	−0.532810	−0.266405	−	0.963861i	$-0.585836\pi$
$71$	−318.757	−0.532810	−0.266405	+	0.963861i	$0.585836\pi$
$72$	288.254	0.471820
$73$	867.378	1.39067	0.695335	−	0.718686i	$-0.255256\pi$
$73$	867.378	1.39067	0.695335	+	0.718686i	$0.255256\pi$
$74$	661.585	1.03929
$75$	−689.727	−1.06190
$76$	441.480	0.666332
$77$	−1437.11	−2.12694
$78$	0	0
$79$	−626.973	−0.892911	−0.446456	−	0.894806i	$-0.647314\pi$
$79$	−626.973	−0.892911	−0.446456	+	0.894806i	$0.647314\pi$
$80$	657.605	0.919030
$81$	81.0000	0.111111
$82$	−540.031	−0.727274
$83$	1212.06	1.60291	0.801454	−	0.598056i	$-0.204060\pi$
$83$	1212.06	1.60291	0.801454	+	0.598056i	$0.204060\pi$
$84$	1053.59	1.36852
$85$	−22.1603	−0.0282779
$86$	157.125	0.197014
$87$	−62.4634	−0.0769744
$88$	1929.14	2.33690
$89$	−679.991	−0.809875	−0.404938	−	0.914344i	$-0.632707\pi$
$89$	−679.991	−0.809875	−0.404938	+	0.914344i	$0.632707\pi$
$90$	808.165	0.946534
$91$	0	0
$92$	2343.07	2.65524
$93$	201.651	0.224841
$94$	−2479.80	−2.72097
$95$	565.038	0.610228
$96$	269.532	0.286552
$97$	−491.109	−0.514068	−0.257034	−	0.966402i	$-0.582745\pi$
$97$	−491.109	−0.514068	−0.257034	+	0.966402i	$0.582745\pi$
$98$	1078.51	1.11169
$99$	542.094	0.550329
$100$	3384.13	3.38413
$101$	707.245	0.696767	0.348384	−	0.937352i	$-0.386731\pi$
$101$	707.245	0.696767	0.348384	+	0.937352i	$0.386731\pi$
$102$	16.8204	0.0163281
$103$	−1656.38	−1.58454	−0.792269	−	0.610171i	$-0.791101\pi$
$103$	−1656.38	−1.58454	−0.792269	+	0.610171i	$0.791101\pi$
$104$	0	0
$105$	1348.46	1.25330
$106$	2226.85	2.04048
$107$	−1416.79	−1.28006	−0.640029	−	0.768351i	$-0.721078\pi$
$107$	−1416.79	−1.28006	−0.640029	+	0.768351i	$0.721078\pi$
$108$	−397.425	−0.354095
$109$	−855.136	−0.751442	−0.375721	−	0.926733i	$-0.622605\pi$
$109$	−855.136	−0.751442	−0.375721	+	0.926733i	$0.622605\pi$
$110$	5408.66	4.68814
$111$	−416.397	−0.356060
$112$	−832.846	−0.702648
$113$	−2258.04	−1.87981	−0.939905	−	0.341437i	$-0.889086\pi$
$113$	−2258.04	−1.87981	−0.939905	+	0.341437i	$0.889086\pi$
$114$	−428.884	−0.352356
$115$	2998.84	2.43168
$116$	306.475	0.245306
$117$	0	0
$118$	−1949.63	−1.52100
$119$	28.0657	0.0216200
$120$	−1810.14	−1.37702
$121$	2296.98	1.72576
$122$	357.042	0.264960
$123$	339.892	0.249163
$124$	−989.397	−0.716536
$125$	1976.38	1.41418
$126$	−1023.53	−0.723676
$127$	−111.921	−0.0781998	−0.0390999	−	0.999235i	$-0.512449\pi$
$127$	−111.921	−0.0781998	−0.0390999	+	0.999235i	$0.512449\pi$
$128$	−2653.50	−1.83234
$129$	−98.8932	−0.0674966
$130$	0	0
$131$	−2994.35	−1.99708	−0.998540	−	0.0540104i	$-0.982800\pi$
$131$	−2994.35	−1.99708	−0.998540	+	0.0540104i	$0.982800\pi$
$132$	−2659.78	−1.75382
$133$	−715.612	−0.466552
$134$	1457.08	0.939350
$135$	−508.654	−0.324281
$136$	−37.6746	−0.0237542
$137$	559.705	0.349043	0.174521	−	0.984653i	$-0.444162\pi$
$137$	559.705	0.349043	0.174521	+	0.984653i	$0.444162\pi$
$138$	−2276.22	−1.40409
$139$	436.351	0.266265	0.133132	−	0.991098i	$-0.457496\pi$
$139$	436.351	0.266265	0.133132	+	0.991098i	$0.457496\pi$
$140$	−6616.20	−3.99408
$141$	1560.77	0.932201
$142$	−1519.35	−0.897897
$143$	0	0
$144$	314.159	0.181805
$145$	392.250	0.224652
$146$	4134.35	2.34357
$147$	−678.804	−0.380863
$148$	2043.05	1.13471
$149$	489.586	0.269184	0.134592	−	0.990901i	$-0.457028\pi$
$149$	489.586	0.269184	0.134592	+	0.990901i	$0.457028\pi$
$150$	−3287.58	−1.78953
$151$	−1817.82	−0.979682	−0.489841	−	0.871812i	$-0.662945\pi$
$151$	−1817.82	−0.979682	−0.489841	+	0.871812i	$0.662945\pi$
$152$	960.620	0.512609
$153$	−10.5867	−0.00559400
$154$	−6849.99	−3.58434
$155$	−1266.30	−0.656206
$156$	0	0
$157$	1866.37	0.948740	0.474370	−	0.880325i	$-0.342676\pi$
$157$	1866.37	0.948740	0.474370	+	0.880325i	$0.342676\pi$
$158$	−2988.46	−1.50474
$159$	−1401.57	−0.699066
$160$	−1692.57	−0.836309
$161$	−3797.98	−1.85915
$162$	386.086	0.187245
$163$	−1044.58	−0.501948	−0.250974	−	0.967994i	$-0.580751\pi$
$163$	−1044.58	−0.501948	−0.250974	+	0.967994i	$0.580751\pi$
$164$	−1667.67	−0.794046
$165$	−3404.18	−1.60615
$166$	5777.30	2.70124
$167$	936.877	0.434118	0.217059	−	0.976158i	$-0.430354\pi$
$167$	936.877	0.434118	0.217059	+	0.976158i	$0.430354\pi$
$168$	2292.51	1.05281
$169$	0	0
$170$	−105.627	−0.0476542
$171$	269.937	0.120717
$172$	485.218	0.215102
$173$	−2989.51	−1.31380	−0.656902	−	0.753976i	$-0.728134\pi$
$173$	−2989.51	−1.31380	−0.656902	+	0.753976i	$0.728134\pi$
$174$	−297.731	−0.129718
$175$	−5485.47	−2.36950
$176$	2102.51	0.900471
$177$	1227.08	0.521092
$178$	−3241.17	−1.36481
$179$	−438.872	−0.183256	−0.0916281	−	0.995793i	$-0.529207\pi$
$179$	−438.872	−0.183256	−0.0916281	+	0.995793i	$0.529207\pi$
$180$	2495.70	1.03344
$181$	991.289	0.407082	0.203541	−	0.979066i	$-0.434755\pi$
$181$	991.289	0.407082	0.203541	+	0.979066i	$0.434755\pi$
$182$	0	0
$183$	−224.720	−0.0907747
$184$	5098.31	2.04268
$185$	2614.84	1.03917
$186$	961.168	0.378905
$187$	−70.8515	−0.0277068
$188$	−7657.88	−2.97079
$189$	644.202	0.247930
$190$	2693.25	1.02836
$191$	−1626.28	−0.616093	−0.308046	−	0.951371i	$-0.599675\pi$
$191$	−1626.28	−0.616093	−0.308046	+	0.951371i	$0.599675\pi$
$192$	2122.48	0.797794
$193$	5282.91	1.97032	0.985160	−	0.171638i	$-0.0549058\pi$
$193$	5282.91	1.97032	0.985160	+	0.171638i	$0.0549058\pi$
$194$	−2340.87	−0.866312
$195$	0	0
$196$	3330.54	1.21375
$197$	−2285.76	−0.826670	−0.413335	−	0.910579i	$-0.635636\pi$
$197$	−2285.76	−0.826670	−0.413335	+	0.910579i	$0.635636\pi$
$198$	2583.89	0.927419
$199$	567.192	0.202046	0.101023	−	0.994884i	$-0.467788\pi$
$199$	567.192	0.202046	0.101023	+	0.994884i	$0.467788\pi$
$200$	7363.56	2.60341
$201$	−917.079	−0.321820
$202$	3371.08	1.17420
$203$	−496.778	−0.171759
$204$	51.9433	0.0178272
$205$	−2134.41	−0.727189
$206$	−7895.10	−2.67028
$207$	1432.64	0.481040
$208$	0	0
$209$	1806.56	0.597905
$210$	6427.42	2.11207
$211$	−3285.95	−1.07210	−0.536052	−	0.844185i	$-0.680085\pi$
$211$	−3285.95	−1.07210	−0.536052	+	0.844185i	$0.680085\pi$
$212$	6876.76	2.22782
$213$	956.272	0.307618
$214$	−6753.11	−2.15716
$215$	621.017	0.196991
$216$	−864.761	−0.272405
$217$	1603.75	0.501704
$218$	−4076.00	−1.26634
$219$	−2602.13	−0.802903
$220$	16702.5	5.11857
$221$	0	0
$222$	−1984.75	−0.600036
$223$	−3801.11	−1.14144	−0.570720	−	0.821145i	$-0.693336\pi$
$223$	−3801.11	−1.14144	−0.570720	+	0.821145i	$0.693336\pi$
$224$	2143.62	0.639403
$225$	2069.18	0.613090
$226$	−10762.9	−3.16787
$227$	−3053.98	−0.892951	−0.446476	−	0.894796i	$-0.647321\pi$
$227$	−3053.98	−0.892951	−0.446476	+	0.894796i	$0.647321\pi$
$228$	−1324.44	−0.384707
$229$	−1338.32	−0.386195	−0.193098	−	0.981180i	$-0.561853\pi$
$229$	−1338.32	−0.386195	−0.193098	+	0.981180i	$0.561853\pi$
$230$	14293.9	4.09789
$231$	4311.34	1.22799
$232$	666.863	0.188714
$233$	1979.59	0.556599	0.278299	−	0.960494i	$-0.410229\pi$
$233$	1979.59	0.556599	0.278299	+	0.960494i	$0.410229\pi$
$234$	0	0
$235$	−9801.12	−2.72066
$236$	−6020.67	−1.66064
$237$	1880.92	0.515523
$238$	133.775	0.0364342
$239$	5326.50	1.44160	0.720800	−	0.693143i	$-0.243774\pi$
$239$	5326.50	1.44160	0.720800	+	0.693143i	$0.243774\pi$
$240$	−1972.81	−0.530602
$241$	−2663.04	−0.711791	−0.355895	−	0.934526i	$-0.615824\pi$
$241$	−2663.04	−0.711791	−0.355895	+	0.934526i	$0.615824\pi$
$242$	10948.5	2.90826
$243$	−243.000	−0.0641500
$244$	1102.58	0.289286
$245$	4262.67	1.11156
$246$	1620.09	0.419892
$247$	0	0
$248$	−2152.84	−0.551231
$249$	−3636.19	−0.925440
$250$	9420.40	2.38319
$251$	−2127.31	−0.534959	−0.267479	−	0.963564i	$-0.586191\pi$
$251$	−2127.31	−0.534959	−0.267479	+	0.963564i	$0.586191\pi$
$252$	−3160.77	−0.790117
$253$	9587.97	2.38257
$254$	−533.470	−0.131783
$255$	66.4808	0.0163262
$256$	−6987.97	−1.70605
$257$	399.185	0.0968891	0.0484446	−	0.998826i	$-0.484574\pi$
$257$	399.185	0.0968891	0.0484446	+	0.998826i	$0.484574\pi$
$258$	−471.374	−0.113746
$259$	−3311.66	−0.794503
$260$	0	0
$261$	187.390	0.0444412
$262$	−14272.5	−3.36550
$263$	1901.20	0.445752	0.222876	−	0.974847i	$-0.428455\pi$
$263$	1901.20	0.445752	0.222876	+	0.974847i	$0.428455\pi$
$264$	−5787.43	−1.34921
$265$	8801.38	2.04024
$266$	−3410.96	−0.786238
$267$	2039.97	0.467582
$268$	4499.63	1.02559
$269$	−3541.04	−0.802606	−0.401303	−	0.915945i	$-0.631443\pi$
$269$	−3541.04	−0.802606	−0.401303	+	0.915945i	$0.631443\pi$
$270$	−2424.49	−0.546481
$271$	3421.98	0.767051	0.383525	−	0.923530i	$-0.374710\pi$
$271$	3421.98	0.767051	0.383525	+	0.923530i	$0.374710\pi$
$272$	−41.0604	−0.00915314
$273$	0	0
$274$	2667.83	0.588210
$275$	13848.0	3.03661
$276$	−7029.22	−1.53300
$277$	5899.10	1.27958	0.639788	−	0.768552i	$-0.279022\pi$
$277$	5899.10	1.27958	0.639788	+	0.768552i	$0.279022\pi$
$278$	2079.86	0.448712
$279$	−604.953	−0.129812
$280$	−14396.2	−3.07264
$281$	7550.89	1.60302	0.801509	−	0.597982i	$-0.204031\pi$
$281$	7550.89	1.60302	0.801509	+	0.597982i	$0.204031\pi$
$282$	7439.39	1.57095
$283$	−4533.54	−0.952266	−0.476133	−	0.879373i	$-0.657962\pi$
$283$	−4533.54	−0.952266	−0.476133	+	0.879373i	$0.657962\pi$
$284$	−4691.93	−0.980334
$285$	−1695.11	−0.352315
$286$	0	0
$287$	2703.20	0.555975
$288$	−808.595	−0.165441
$289$	−4911.62	−0.999718
$290$	1869.65	0.378586
$291$	1473.33	0.296797
$292$	12767.3	2.55874
$293$	1847.62	0.368393	0.184197	−	0.982889i	$-0.441032\pi$
$293$	1847.62	0.368393	0.184197	+	0.982889i	$0.441032\pi$
$294$	−3235.52	−0.641834
$295$	−7705.69	−1.52082
$296$	4445.48	0.872934
$297$	−1626.28	−0.317732
$298$	2333.61	0.453631
$299$	0	0
$300$	−10152.4	−1.95383
$301$	−786.509	−0.150610
$302$	−8664.62	−1.65097
$303$	−2121.74	−0.402279
$304$	1046.95	0.197522
$305$	1411.17	0.264929
$306$	−50.4613	−0.00942706
$307$	370.739	0.0689225	0.0344612	−	0.999406i	$-0.489028\pi$
$307$	370.739	0.0689225	0.0344612	+	0.999406i	$0.489028\pi$
$308$	−21153.5	−3.91342
$309$	4969.13	0.914834
$310$	−6035.82	−1.10584
$311$	5288.17	0.964195	0.482097	−	0.876118i	$-0.339875\pi$
$311$	5288.17	0.964195	0.482097	+	0.876118i	$0.339875\pi$
$312$	0	0
$313$	5220.25	0.942703	0.471351	−	0.881945i	$-0.343766\pi$
$313$	5220.25	0.942703	0.471351	+	0.881945i	$0.343766\pi$
$314$	8896.02	1.59883
$315$	−4045.38	−0.723592
$316$	−9228.70	−1.64290
$317$	8106.97	1.43638	0.718191	−	0.695846i	$-0.244971\pi$
$317$	8106.97	1.43638	0.718191	+	0.695846i	$0.244971\pi$
$318$	−6680.55	−1.17807
$319$	1254.11	0.220115
$320$	−13328.5	−2.32839
$321$	4250.37	0.739041
$322$	−18103.0	−3.13305
$323$	−35.2806	−0.00607760
$324$	1192.28	0.204437
$325$	0	0
$326$	−4978.97	−0.845888
$327$	2565.41	0.433845
$328$	−3628.71	−0.610859
$329$	12413.0	2.08009
$330$	−16226.0	−2.70670
$331$	−8131.30	−1.35026	−0.675131	−	0.737698i	$-0.735913\pi$
$331$	−8131.30	−1.35026	−0.675131	+	0.737698i	$0.735913\pi$
$332$	17840.9	2.94924
$333$	1249.19	0.205571
$334$	4465.62	0.731580
$335$	5758.96	0.939240
$336$	2498.54	0.405674
$337$	−3109.19	−0.502577	−0.251288	−	0.967912i	$-0.580854\pi$
$337$	−3109.19	−0.502577	−0.251288	+	0.967912i	$0.580854\pi$
$338$	0	0
$339$	6774.12	1.08531
$340$	−326.187	−0.0520293
$341$	−4048.66	−0.642954
$342$	1286.65	0.203433
$343$	2785.15	0.438436
$344$	1055.79	0.165478
$345$	−8996.51	−1.40393
$346$	−14249.5	−2.21404
$347$	851.124	0.131674	0.0658368	−	0.997830i	$-0.479028\pi$
$347$	851.124	0.131674	0.0658368	+	0.997830i	$0.479028\pi$
$348$	−919.426	−0.141628
$349$	−4114.16	−0.631019	−0.315510	−	0.948922i	$-0.602175\pi$
$349$	−4114.16	−0.631019	−0.315510	+	0.948922i	$0.602175\pi$
$350$	−26146.5	−3.99311
$351$	0	0
$352$	−5411.54	−0.819421
$353$	4659.80	0.702595	0.351297	−	0.936264i	$-0.385741\pi$
$353$	4659.80	0.702595	0.351297	+	0.936264i	$0.385741\pi$
$354$	5848.89	0.878149
$355$	−6005.08	−0.897793
$356$	−10009.1	−1.49011
$357$	−84.1970	−0.0124823
$358$	−2091.88	−0.308825
$359$	2811.33	0.413305	0.206653	−	0.978414i	$-0.433743\pi$
$359$	2811.33	0.413305	0.206653	+	0.978414i	$0.433743\pi$
$360$	5430.42	0.795023
$361$	−5959.42	−0.868847
$362$	4724.97	0.686019
$363$	−6890.94	−0.996365
$364$	0	0
$365$	16340.5	2.34330
$366$	−1071.13	−0.152974
$367$	−1159.77	−0.164958	−0.0824791	−	0.996593i	$-0.526284\pi$
$367$	−1159.77	−0.164958	−0.0824791	+	0.996593i	$0.526284\pi$
$368$	5556.49	0.787098
$369$	−1019.68	−0.143854
$370$	12463.6	1.75122
$371$	−11146.8	−1.55988
$372$	2968.19	0.413692
$373$	8876.80	1.23223	0.616117	−	0.787655i	$-0.288705\pi$
$373$	8876.80	1.23223	0.616117	+	0.787655i	$0.288705\pi$
$374$	−337.713	−0.0466918
$375$	−5929.14	−0.816479
$376$	−16662.9	−2.28543
$377$	0	0
$378$	3070.58	0.417814
$379$	2256.21	0.305788	0.152894	−	0.988243i	$-0.451141\pi$
$379$	2256.21	0.305788	0.152894	+	0.988243i	$0.451141\pi$
$380$	8317.05	1.12278
$381$	335.763	0.0451487
$382$	−7751.67	−1.03825
$383$	10331.9	1.37843	0.689213	−	0.724559i	$-0.257956\pi$
$383$	10331.9	1.37843	0.689213	+	0.724559i	$0.257956\pi$
$384$	7960.51	1.05790
$385$	−27073.8	−3.58392
$386$	25180.9	3.32040
$387$	296.680	0.0389692
$388$	−7228.85	−0.945849
$389$	5548.06	0.723131	0.361565	−	0.932347i	$-0.382242\pi$
$389$	5548.06	0.723131	0.361565	+	0.932347i	$0.382242\pi$
$390$	0	0
$391$	−187.245	−0.0242184
$392$	7246.96	0.933741
$393$	8983.05	1.15302
$394$	−10895.1	−1.39311
$395$	−11811.6	−1.50457
$396$	7979.33	1.01257
$397$	15168.9	1.91764	0.958821	−	0.284010i	$-0.0916650\pi$
$397$	15168.9	1.91764	0.958821	+	0.284010i	$0.0916650\pi$
$398$	2703.52	0.340490
$399$	2146.84	0.269364
$400$	8025.32	1.00316
$401$	4305.72	0.536204	0.268102	−	0.963391i	$-0.413604\pi$
$401$	4305.72	0.536204	0.268102	+	0.963391i	$0.413604\pi$
$402$	−4371.25	−0.542334
$403$	0	0
$404$	10410.3	1.28200
$405$	1525.96	0.187224
$406$	−2367.89	−0.289449
$407$	8360.25	1.01819
$408$	113.024	0.0137145
$409$	5100.24	0.616603	0.308301	−	0.951289i	$-0.400239\pi$
$409$	5100.24	0.616603	0.308301	+	0.951289i	$0.400239\pi$
$410$	−10173.7	−1.22547
$411$	−1679.12	−0.201520
$412$	−24380.9	−2.91544
$413$	9759.14	1.16275
$414$	6828.66	0.810653
$415$	22834.1	2.70092
$416$	0	0
$417$	−1309.05	−0.153728
$418$	8610.94	1.00760
$419$	−10210.2	−1.19045	−0.595226	−	0.803558i	$-0.702938\pi$
$419$	−10210.2	−1.19045	−0.595226	+	0.803558i	$0.702938\pi$
$420$	19848.6	2.30598
$421$	−14590.5	−1.68906	−0.844531	−	0.535506i	$-0.820121\pi$
$421$	−14590.5	−1.68906	−0.844531	+	0.535506i	$0.820121\pi$
$422$	−15662.4	−1.80672
$423$	−4682.31	−0.538207
$424$	14963.2	1.71386
$425$	−270.441	−0.0308666
$426$	4558.06	0.518401
$427$	−1787.22	−0.202552
$428$	−20854.3	−2.35522
$429$	0	0
$430$	2960.07	0.331971
$431$	−2653.77	−0.296584	−0.148292	−	0.988944i	$-0.547378\pi$
$431$	−2653.77	−0.296584	−0.148292	+	0.988944i	$0.547378\pi$
$432$	−942.476	−0.104965
$433$	−4081.18	−0.452953	−0.226477	−	0.974017i	$-0.572721\pi$
$433$	−4081.18	−0.452953	−0.226477	+	0.974017i	$0.572721\pi$
$434$	7644.28	0.845477
$435$	−1176.75	−0.129703
$436$	−12587.1	−1.38260
$437$	4774.34	0.522627
$438$	−12403.0	−1.35306
$439$	13002.4	1.41360	0.706802	−	0.707411i	$-0.250137\pi$
$439$	13002.4	1.41360	0.706802	+	0.707411i	$0.250137\pi$
$440$	36343.2	3.93771
$441$	2036.41	0.219891
$442$	0	0
$443$	−2681.37	−0.287575	−0.143787	−	0.989609i	$-0.545928\pi$
$443$	−2681.37	−0.287575	−0.143787	+	0.989609i	$0.545928\pi$
$444$	−6129.14	−0.655126
$445$	−12810.4	−1.36465
$446$	−18118.0	−1.92357
$447$	−1468.76	−0.155413
$448$	16880.3	1.78018
$449$	14525.2	1.52669	0.763346	−	0.645990i	$-0.223555\pi$
$449$	14525.2	1.52669	0.763346	+	0.645990i	$0.223555\pi$
$450$	9862.73	1.03319
$451$	−6824.21	−0.712504
$452$	−33237.1	−3.45872
$453$	5453.46	0.565620
$454$	−14556.8	−1.50481
$455$	0	0
$456$	−2881.86	−0.295955
$457$	11054.8	1.13156	0.565779	−	0.824557i	$-0.308576\pi$
$457$	11054.8	1.13156	0.565779	+	0.824557i	$0.308576\pi$
$458$	−6379.09	−0.650819
$459$	31.7600	0.00322970
$460$	44141.2	4.47412
$461$	−2168.26	−0.219059	−0.109529	−	0.993984i	$-0.534934\pi$
$461$	−2168.26	−0.219059	−0.109529	+	0.993984i	$0.534934\pi$
$462$	20550.0	2.06942
$463$	16618.0	1.66804	0.834021	−	0.551733i	$-0.186033\pi$
$463$	16618.0	1.66804	0.834021	+	0.551733i	$0.186033\pi$
$464$	726.793	0.0727166
$465$	3798.91	0.378861
$466$	9435.72	0.937985
$467$	6857.86	0.679537	0.339769	−	0.940509i	$-0.389651\pi$
$467$	6857.86	0.679537	0.339769	+	0.940509i	$0.389651\pi$
$468$	0	0
$469$	−7293.63	−0.718100
$470$	−46717.0	−4.58488
$471$	−5599.10	−0.547755
$472$	−13100.4	−1.27753
$473$	1985.53	0.193013
$474$	8965.39	0.868764
$475$	6895.65	0.666093
$476$	413.111	0.0397792
$477$	4204.70	0.403606
$478$	25388.7	2.42940
$479$	−17395.1	−1.65930	−0.829648	−	0.558287i	$-0.811459\pi$
$479$	−17395.1	−1.65930	−0.829648	+	0.558287i	$0.811459\pi$
$480$	5077.71	0.482843
$481$	0	0
$482$	−12693.4	−1.19952
$483$	11393.9	1.07338
$484$	33810.3	3.17527
$485$	−9252.01	−0.866211
$486$	−1158.26	−0.108106
$487$	−11698.7	−1.08854	−0.544270	−	0.838910i	$-0.683193\pi$
$487$	−11698.7	−1.08854	−0.544270	+	0.838910i	$0.683193\pi$
$488$	2399.12	0.222548
$489$	3133.73	0.289800
$490$	20318.0	1.87321
$491$	6600.47	0.606670	0.303335	−	0.952884i	$-0.401900\pi$
$491$	6600.47	0.606670	0.303335	+	0.952884i	$0.401900\pi$
$492$	5003.02	0.458442
$493$	−24.4918	−0.00223744
$494$	0	0
$495$	10212.5	0.927311
$496$	−2346.31	−0.212404
$497$	7605.34	0.686411
$498$	−17331.9	−1.55956
$499$	21149.0	1.89731	0.948657	−	0.316308i	$-0.102443\pi$
$499$	21149.0	1.89731	0.948657	+	0.316308i	$0.102443\pi$
$500$	29091.2	2.60200
$501$	−2810.63	−0.250638
$502$	−10139.8	−0.901517
$503$	9591.55	0.850231	0.425116	−	0.905139i	$-0.360233\pi$
$503$	9591.55	0.850231	0.425116	+	0.905139i	$0.360233\pi$
$504$	−6877.54	−0.607837
$505$	13323.8	1.17406
$506$	45701.0	4.01513
$507$	0	0
$508$	−1647.41	−0.143882
$509$	−2306.28	−0.200833	−0.100416	−	0.994946i	$-0.532017\pi$
$509$	−2306.28	−0.200833	−0.100416	+	0.994946i	$0.532017\pi$
$510$	316.880	0.0275131
$511$	−20695.1	−1.79158
$512$	−12080.1	−1.04271
$513$	−809.810	−0.0696959
$514$	1902.71	0.163278
$515$	−31204.5	−2.66997
$516$	−1455.65	−0.124189
$517$	−31336.4	−2.66572
$518$	−15785.0	−1.33890
$519$	8968.53	0.758525
$520$	0	0
$521$	−12511.5	−1.05209	−0.526045	−	0.850457i	$-0.676326\pi$
$521$	−12511.5	−1.05209	−0.526045	+	0.850457i	$0.676326\pi$
$522$	893.193	0.0748927
$523$	−12924.3	−1.08058	−0.540289	−	0.841480i	$-0.681685\pi$
$523$	−12924.3	−1.08058	−0.540289	+	0.841480i	$0.681685\pi$
$524$	−44075.2	−3.67449
$525$	16456.4	1.36803
$526$	9062.04	0.751186
$527$	79.0671	0.00653552
$528$	−6307.54	−0.519887
$529$	13172.0	1.08260
$530$	41951.7	3.43824
$531$	−3681.25	−0.300853
$532$	−10533.4	−0.858424
$533$	0	0
$534$	9723.51	0.787973
$535$	−26690.9	−2.15691
$536$	9790.79	0.788988
$537$	1316.62	0.105803
$538$	−16878.3	−1.35256
$539$	13628.7	1.08911
$540$	−7487.10	−0.596655
$541$	15384.3	1.22260	0.611298	−	0.791400i	$-0.290648\pi$
$541$	15384.3	1.22260	0.611298	+	0.791400i	$0.290648\pi$
$542$	16310.9	1.29264
$543$	−2973.87	−0.235029
$544$	105.683	0.00832927
$545$	−16109.9	−1.26619
$546$	0	0
$547$	7658.37	0.598626	0.299313	−	0.954155i	$-0.403243\pi$
$547$	7658.37	0.598626	0.299313	+	0.954155i	$0.403243\pi$
$548$	8238.55	0.642215
$549$	674.160	0.0524088
$550$	66006.5	5.11732
$551$	624.487	0.0482832
$552$	−15294.9	−1.17934
$553$	14959.2	1.15032
$554$	28118.0	2.15635
$555$	−7844.52	−0.599966
$556$	6422.84	0.489909
$557$	−13922.9	−1.05913	−0.529563	−	0.848271i	$-0.677644\pi$
$557$	−13922.9	−1.05913	−0.529563	+	0.848271i	$0.677644\pi$
$558$	−2883.50	−0.218761
$559$	0	0
$560$	−15690.0	−1.18397
$561$	212.555	0.0159965
$562$	35991.2	2.70142
$563$	20611.9	1.54296	0.771481	−	0.636252i	$-0.219516\pi$
$563$	20611.9	1.54296	0.771481	+	0.636252i	$0.219516\pi$
$564$	22973.6	1.71519
$565$	−42539.3	−3.16750
$566$	−21609.1	−1.60477
$567$	−1932.61	−0.143143
$568$	−10209.2	−0.754171
$569$	2204.50	0.162421	0.0812105	−	0.996697i	$-0.474121\pi$
$569$	2204.50	0.162421	0.0812105	+	0.996697i	$0.474121\pi$
$570$	−8079.75	−0.593725
$571$	1018.86	0.0746723	0.0373362	−	0.999303i	$-0.488113\pi$
$571$	1018.86	0.0746723	0.0373362	+	0.999303i	$0.488113\pi$
$572$	0	0
$573$	4878.85	0.355701
$574$	12884.8	0.936935
$575$	36597.4	2.65429
$576$	−6367.43	−0.460607
$577$	20426.8	1.47379	0.736896	−	0.676006i	$-0.236291\pi$
$577$	20426.8	1.47379	0.736896	+	0.676006i	$0.236291\pi$
$578$	−23411.2	−1.68473
$579$	−15848.7	−1.13756
$580$	5773.70	0.413344
$581$	−28919.1	−2.06500
$582$	7022.60	0.500165
$583$	28140.0	1.99904
$584$	27780.5	1.96844
$585$	0	0
$586$	8806.68	0.620820
$587$	11674.0	0.820846	0.410423	−	0.911895i	$-0.365381\pi$
$587$	11674.0	0.820846	0.410423	+	0.911895i	$0.365381\pi$
$588$	−9991.62	−0.700761
$589$	−2016.04	−0.141035
$590$	−36729.1	−2.56290
$591$	6857.29	0.477278
$592$	4844.99	0.336365
$593$	−21447.4	−1.48523	−0.742613	−	0.669721i	$-0.766414\pi$
$593$	−21447.4	−1.48523	−0.742613	+	0.669721i	$0.766414\pi$
$594$	−7751.67	−0.535446
$595$	528.730	0.0364299
$596$	7206.43	0.495280
$597$	−1701.58	−0.116651
$598$	0	0
$599$	11316.8	0.771937	0.385968	−	0.922512i	$-0.373867\pi$
$599$	11316.8	0.771937	0.385968	+	0.922512i	$0.373867\pi$
$600$	−22090.7	−1.50308
$601$	−9753.39	−0.661979	−0.330989	−	0.943635i	$-0.607382\pi$
$601$	−9753.39	−0.661979	−0.330989	+	0.943635i	$0.607382\pi$
$602$	−3748.89	−0.253809
$603$	2751.24	0.185803
$604$	−26757.3	−1.80255
$605$	43272.9	2.90792
$606$	−10113.2	−0.677924
$607$	17947.7	1.20012	0.600061	−	0.799954i	$-0.295143\pi$
$607$	17947.7	1.20012	0.600061	+	0.799954i	$0.295143\pi$
$608$	−2694.68	−0.179743
$609$	1490.33	0.0991649
$610$	6726.32	0.446460
$611$	0	0
$612$	−155.830	−0.0102926
$613$	6494.46	0.427910	0.213955	−	0.976844i	$-0.431365\pi$
$613$	6494.46	0.427910	0.213955	+	0.976844i	$0.431365\pi$
$614$	1767.13	0.116149
$615$	6403.24	0.419843
$616$	−46028.1	−3.01059
$617$	2746.15	0.179183	0.0895914	−	0.995979i	$-0.471444\pi$
$617$	2746.15	0.179183	0.0895914	+	0.995979i	$0.471444\pi$
$618$	23685.3	1.54169
$619$	−16319.9	−1.05969	−0.529847	−	0.848093i	$-0.677751\pi$
$619$	−16319.9	−1.05969	−0.529847	+	0.848093i	$0.677751\pi$
$620$	−18639.3	−1.20737
$621$	−4297.92	−0.277729
$622$	25206.0	1.62487
$623$	16224.1	1.04335
$624$	0	0
$625$	8494.48	0.543647
$626$	24882.3	1.58865
$627$	−5419.67	−0.345201
$628$	27471.9	1.74562
$629$	−163.269	−0.0103497
$630$	−19282.3	−1.21940
$631$	−1381.57	−0.0871621	−0.0435810	−	0.999050i	$-0.513877\pi$
$631$	−1381.57	−0.0871621	−0.0435810	+	0.999050i	$0.513877\pi$
$632$	−20080.8	−1.26388
$633$	9857.84	0.618979
$634$	38641.8	2.42060
$635$	−2108.48	−0.131768
$636$	−20630.3	−1.28623
$637$	0	0
$638$	5977.72	0.370941
$639$	−2868.82	−0.177603
$640$	−49989.4	−3.08751
$641$	−15055.0	−0.927672	−0.463836	−	0.885921i	$-0.653527\pi$
$641$	−15055.0	−0.927672	−0.463836	+	0.885921i	$0.653527\pi$
$642$	20259.3	1.24544
$643$	−5942.18	−0.364443	−0.182221	−	0.983258i	$-0.558329\pi$
$643$	−5942.18	−0.364443	−0.182221	+	0.983258i	$0.558329\pi$
$644$	−55904.2	−3.42070
$645$	−1863.05	−0.113733
$646$	−168.165	−0.0102420
$647$	−26854.0	−1.63175	−0.815874	−	0.578229i	$-0.803744\pi$
$647$	−26854.0	−1.63175	−0.815874	+	0.578229i	$0.803744\pi$
$648$	2594.28	0.157273
$649$	−24636.9	−1.49011
$650$	0	0
$651$	−4811.26	−0.289659
$652$	−15375.6	−0.923550
$653$	−6885.76	−0.412650	−0.206325	−	0.978483i	$-0.566150\pi$
$653$	−6885.76	−0.412650	−0.206325	+	0.978483i	$0.566150\pi$
$654$	12228.0	0.731120
$655$	−56410.6	−3.36511
$656$	−3954.82	−0.235380
$657$	7806.40	0.463556
$658$	59166.3	3.50538
$659$	−14745.5	−0.871627	−0.435813	−	0.900037i	$-0.643539\pi$
$659$	−14745.5	−0.871627	−0.435813	+	0.900037i	$0.643539\pi$
$660$	−50107.6	−2.95521
$661$	−28891.3	−1.70006	−0.850031	−	0.526733i	$-0.823417\pi$
$661$	−28891.3	−1.70006	−0.850031	+	0.526733i	$0.823417\pi$
$662$	−38757.8	−2.27547
$663$	0	0
$664$	38820.2	2.26885
$665$	−13481.4	−0.786147
$666$	5954.26	0.346431
$667$	3314.35	0.192402
$668$	13790.3	0.798747
$669$	11403.3	0.659011
$670$	27450.0	1.58282
$671$	4511.83	0.259579
$672$	−6430.85	−0.369160
$673$	−18362.8	−1.05176	−0.525880	−	0.850559i	$-0.676264\pi$
$673$	−18362.8	−1.05176	−0.525880	+	0.850559i	$0.676264\pi$
$674$	−14819.9	−0.846947
$675$	−6207.54	−0.353968
$676$	0	0
$677$	−25937.6	−1.47247	−0.736237	−	0.676724i	$-0.763399\pi$
$677$	−25937.6	−1.47247	−0.736237	+	0.676724i	$0.763399\pi$
$678$	32288.8	1.82897
$679$	11717.5	0.662265
$680$	−709.754	−0.0400262
$681$	9161.95	0.515546
$682$	−19297.9	−1.08351
$683$	−34538.1	−1.93494	−0.967469	−	0.252989i	$-0.918586\pi$
$683$	−34538.1	−1.93494	−0.967469	+	0.252989i	$0.918586\pi$
$684$	3973.32	0.222111
$685$	10544.3	0.588142
$686$	13275.4	0.738857
$687$	4014.96	0.222970
$688$	1150.67	0.0637630
$689$	0	0
$690$	−42881.8	−2.36592
$691$	−10009.5	−0.551056	−0.275528	−	0.961293i	$-0.588853\pi$
$691$	−10009.5	−0.551056	−0.275528	+	0.961293i	$0.588853\pi$
$692$	−44003.9	−2.41731
$693$	−12934.0	−0.708979
$694$	4056.87	0.221897
$695$	8220.42	0.448660
$696$	−2000.59	−0.108954
$697$	133.271	0.00724248
$698$	−19610.1	−1.06340
$699$	−5938.78	−0.321352
$700$	−80743.1	−4.35972
$701$	7115.37	0.383372	0.191686	−	0.981456i	$-0.438605\pi$
$701$	7115.37	0.383372	0.191686	+	0.981456i	$0.438605\pi$
$702$	0	0
$703$	4163.00	0.223343
$704$	−42614.2	−2.28137
$705$	29403.4	1.57077
$706$	22210.9	1.18402
$707$	−16874.4	−0.897634
$708$	18062.0	0.958773
$709$	−19728.3	−1.04501	−0.522505	−	0.852636i	$-0.675002\pi$
$709$	−19728.3	−1.04501	−0.522505	+	0.852636i	$0.675002\pi$
$710$	−28623.2	−1.51297
$711$	−5642.76	−0.297637
$712$	−21778.9	−1.14635
$713$	−10699.7	−0.562004
$714$	−401.324	−0.0210353
$715$	0	0
$716$	−6459.96	−0.337179
$717$	−15979.5	−0.832309
$718$	13400.2	0.696506
$719$	28248.7	1.46523	0.732614	−	0.680644i	$-0.238300\pi$
$719$	28248.7	1.46523	0.732614	+	0.680644i	$0.238300\pi$
$720$	5918.44	0.306343
$721$	39520.0	2.04134
$722$	−28405.5	−1.46419
$723$	7989.12	0.410952
$724$	14591.2	0.749003
$725$	4786.96	0.245218
$726$	−32845.6	−1.67908
$727$	15082.9	0.769457	0.384728	−	0.923030i	$-0.374295\pi$
$727$	15082.9	0.769457	0.384728	+	0.923030i	$0.374295\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	77887.1	3.94895
$731$	−38.7759	−0.00196194
$732$	−3307.75	−0.167019
$733$	−13245.3	−0.667432	−0.333716	−	0.942674i	$-0.608303\pi$
$733$	−13245.3	−0.667432	−0.333716	+	0.942674i	$0.608303\pi$
$734$	−5528.05	−0.277989
$735$	−12788.0	−0.641759
$736$	−14301.5	−0.716252
$737$	18412.7	0.920273
$738$	−4860.28	−0.242425
$739$	−9334.12	−0.464629	−0.232315	−	0.972641i	$-0.574630\pi$
$739$	−9334.12	−0.464629	−0.232315	+	0.972641i	$0.574630\pi$
$740$	38489.0	1.91200
$741$	0	0
$742$	−53131.2	−2.62872
$743$	29628.9	1.46296	0.731480	−	0.681863i	$-0.238830\pi$
$743$	29628.9	1.46296	0.731480	+	0.681863i	$0.238830\pi$
$744$	6458.51	0.318254
$745$	9223.32	0.453579
$746$	42311.2	2.07657
$747$	10908.6	0.534303
$748$	−1042.90	−0.0509786
$749$	33803.6	1.64908
$750$	−28261.2	−1.37594
$751$	1682.85	0.0817682	0.0408841	−	0.999164i	$-0.486983\pi$
$751$	1682.85	0.0817682	0.0408841	+	0.999164i	$0.486983\pi$
$752$	−18160.3	−0.880637
$753$	6381.93	0.308859
$754$	0	0
$755$	−34245.9	−1.65078
$756$	9482.30	0.456174
$757$	−34195.3	−1.64181	−0.820905	−	0.571064i	$-0.806531\pi$
$757$	−34195.3	−1.64181	−0.820905	+	0.571064i	$0.806531\pi$
$758$	10754.2	0.515316
$759$	−28763.9	−1.37558
$760$	18097.1	0.863753
$761$	−36265.5	−1.72750	−0.863748	−	0.503924i	$-0.831889\pi$
$761$	−36265.5	−1.72750	−0.863748	+	0.503924i	$0.831889\pi$
$762$	1600.41	0.0760850
$763$	20403.0	0.968070
$764$	−23938.0	−1.13357
$765$	−199.443	−0.00942596
$766$	49247.1	2.32294
$767$	0	0
$768$	20963.9	0.984987
$769$	4632.34	0.217226	0.108613	−	0.994084i	$-0.465359\pi$
$769$	4632.34	0.217226	0.108613	+	0.994084i	$0.465359\pi$
$770$	−129047.	−6.03965
$771$	−1197.56	−0.0559390
$772$	77761.4	3.62525
$773$	−243.045	−0.0113088	−0.00565442	−	0.999984i	$-0.501800\pi$
$773$	−243.045	−0.0113088	−0.00565442	+	0.999984i	$0.501800\pi$
$774$	1414.12	0.0656712
$775$	−15453.8	−0.716279
$776$	−15729.3	−0.727642
$777$	9934.97	0.458706
$778$	26444.8	1.21863
$779$	−3398.12	−0.156291
$780$	0	0
$781$	−19199.6	−0.879662
$782$	−892.504	−0.0408131
$783$	−562.170	−0.0256581
$784$	7898.23	0.359796
$785$	35160.5	1.59864
$786$	42817.6	1.94307
$787$	17245.4	0.781108	0.390554	−	0.920580i	$-0.372284\pi$
$787$	17245.4	0.781108	0.390554	+	0.920580i	$0.372284\pi$
$788$	−33645.2	−1.52102
$789$	−5703.59	−0.257355
$790$	−56299.7	−2.53551
$791$	53875.3	2.42173
$792$	17362.3	0.778968
$793$	0	0
$794$	72302.3	3.23163
$795$	−26404.1	−1.17794
$796$	8348.75	0.371751
$797$	−36984.6	−1.64374	−0.821871	−	0.569673i	$-0.807070\pi$
$797$	−36984.6	−1.64374	−0.821871	+	0.569673i	$0.807070\pi$
$798$	10232.9	0.453935
$799$	611.975	0.0270965
$800$	−20655.9	−0.912871
$801$	−6119.92	−0.269958
$802$	20523.2	0.903615
$803$	52244.5	2.29598
$804$	−13498.9	−0.592126
$805$	−71550.3	−3.13269
$806$	0	0
$807$	10623.1	0.463385
$808$	22651.8	0.986246
$809$	26646.8	1.15804	0.579019	−	0.815314i	$-0.303436\pi$
$809$	26646.8	1.15804	0.579019	+	0.815314i	$0.303436\pi$
$810$	7273.48	0.315511
$811$	−18331.3	−0.793709	−0.396855	−	0.917881i	$-0.629898\pi$
$811$	−18331.3	−0.793709	−0.396855	+	0.917881i	$0.629898\pi$
$812$	−7312.30	−0.316024
$813$	−10266.0	−0.442857
$814$	39849.0	1.71586
$815$	−19678.8	−0.845790
$816$	123.181	0.00528457
$817$	988.700	0.0423381
$818$	24310.3	1.03911
$819$	0	0
$820$	−31417.4	−1.33798
$821$	28227.4	1.19993	0.599966	−	0.800026i	$-0.295181\pi$
$821$	28227.4	1.19993	0.599966	+	0.800026i	$0.295181\pi$
$822$	−8003.49	−0.339603
$823$	27183.2	1.15133	0.575666	−	0.817685i	$-0.304743\pi$
$823$	27183.2	1.15133	0.575666	+	0.817685i	$0.304743\pi$
$824$	−53050.7	−2.24285
$825$	−41544.1	−1.75319
$826$	46516.9	1.95948
$827$	32419.4	1.36316	0.681580	−	0.731744i	$-0.261293\pi$
$827$	32419.4	1.36316	0.681580	+	0.731744i	$0.261293\pi$
$828$	21087.7	0.885081
$829$	22920.2	0.960255	0.480128	−	0.877199i	$-0.340590\pi$
$829$	22920.2	0.960255	0.480128	+	0.877199i	$0.340590\pi$
$830$	108839.	4.55162
$831$	−17697.3	−0.738763
$832$	0	0
$833$	−266.158	−0.0110706
$834$	−6239.59	−0.259064
$835$	17649.9	0.731495
$836$	26591.5	1.10010
$837$	1814.86	0.0749471
$838$	−48666.7	−2.00616
$839$	−958.186	−0.0394282	−0.0197141	−	0.999806i	$-0.506276\pi$
$839$	−958.186	−0.0394282	−0.0197141	+	0.999806i	$0.506276\pi$
$840$	43188.7	1.77399
$841$	−23955.5	−0.982225
$842$	−69545.3	−2.84642
$843$	−22652.7	−0.925503
$844$	−48367.3	−1.97260
$845$	0	0
$846$	−22318.2	−0.906991
$847$	−54804.4	−2.22326
$848$	16307.9	0.660397
$849$	13600.6	0.549791
$850$	−1289.05	−0.0520167
$851$	22094.3	0.889993
$852$	14075.8	0.565996
$853$	40410.6	1.62208	0.811040	−	0.584991i	$-0.198902\pi$
$853$	40410.6	1.62208	0.811040	+	0.584991i	$0.198902\pi$
$854$	−8518.79	−0.341343
$855$	5085.34	0.203409
$856$	−45377.2	−1.81187
$857$	−5170.15	−0.206078	−0.103039	−	0.994677i	$-0.532857\pi$
$857$	−5170.15	−0.206078	−0.103039	+	0.994677i	$0.532857\pi$
$858$	0	0
$859$	−1674.04	−0.0664931	−0.0332465	−	0.999447i	$-0.510585\pi$
$859$	−1674.04	−0.0664931	−0.0332465	+	0.999447i	$0.510585\pi$
$860$	9141.03	0.362449
$861$	−8109.60	−0.320992
$862$	−12649.2	−0.499806
$863$	−43213.2	−1.70451	−0.852257	−	0.523124i	$-0.824766\pi$
$863$	−43213.2	−1.70451	−0.852257	+	0.523124i	$0.824766\pi$
$864$	2425.78	0.0955172
$865$	−56319.4	−2.21378
$866$	−19452.9	−0.763321
$867$	14734.8	0.577188
$868$	23606.4	0.923101
$869$	−37764.3	−1.47418
$870$	−5608.96	−0.218577
$871$	0	0
$872$	−27388.5	−1.06364
$873$	−4419.98	−0.171356
$874$	22756.9	0.880735
$875$	−47155.1	−1.82187
$876$	−38301.9	−1.47729
$877$	3665.73	0.141144	0.0705718	−	0.997507i	$-0.477518\pi$
$877$	3665.73	0.141144	0.0705718	+	0.997507i	$0.477518\pi$
$878$	61976.0	2.38222
$879$	−5542.87	−0.212692
$880$	39609.3	1.51731
$881$	−24997.5	−0.955944	−0.477972	−	0.878375i	$-0.658628\pi$
$881$	−24997.5	−0.955944	−0.477972	+	0.878375i	$0.658628\pi$
$882$	9706.55	0.370563
$883$	28259.1	1.07700	0.538501	−	0.842625i	$-0.318991\pi$
$883$	28259.1	1.07700	0.538501	+	0.842625i	$0.318991\pi$
$884$	0	0
$885$	23117.1	0.878047
$886$	−12780.7	−0.484623
$887$	−32725.2	−1.23879	−0.619394	−	0.785081i	$-0.712622\pi$
$887$	−32725.2	−1.23879	−0.619394	+	0.785081i	$0.712622\pi$
$888$	−13336.4	−0.503989
$889$	2670.36	0.100744
$890$	−61060.5	−2.29972
$891$	4878.85	0.183443
$892$	−55950.2	−2.10017
$893$	−15604.0	−0.584735
$894$	−7000.82	−0.261904
$895$	−8267.92	−0.308789
$896$	63310.9	2.36057
$897$	0	0
$898$	69234.1	2.57280
$899$	−1399.53	−0.0519211
$900$	30457.2	1.12804
$901$	−549.553	−0.0203199
$902$	−32527.5	−1.20072
$903$	2359.53	0.0869547
$904$	−72320.9	−2.66079
$905$	18674.9	0.685939
$906$	25993.9	0.953188
$907$	41341.4	1.51347	0.756736	−	0.653721i	$-0.226793\pi$
$907$	41341.4	1.51347	0.756736	+	0.653721i	$0.226793\pi$
$908$	−44952.9	−1.64297
$909$	6365.21	0.232256
$910$	0	0
$911$	−12496.5	−0.454475	−0.227237	−	0.973839i	$-0.572969\pi$
$911$	−12496.5	−0.454475	−0.227237	+	0.973839i	$0.572969\pi$
$912$	−3140.85	−0.114039
$913$	73006.0	2.64638
$914$	52692.6	1.90691
$915$	−4233.51	−0.152957
$916$	−19699.3	−0.710572
$917$	71443.2	2.57281
$918$	151.384	0.00544271
$919$	−19871.5	−0.713274	−0.356637	−	0.934243i	$-0.616077\pi$
$919$	−19871.5	−0.713274	−0.356637	+	0.934243i	$0.616077\pi$
$920$	96047.3	3.44194
$921$	−1112.22	−0.0397924
$922$	−10335.0	−0.369160
$923$	0	0
$924$	63460.5	2.25941
$925$	31911.1	1.13430
$926$	79209.5	2.81100
$927$	−14907.4	−0.528180
$928$	−1870.65	−0.0661714
$929$	−30850.8	−1.08954	−0.544769	−	0.838586i	$-0.683383\pi$
$929$	−30850.8	−1.08954	−0.544769	+	0.838586i	$0.683383\pi$
$930$	18107.5	0.638459
$931$	6786.45	0.238901
$932$	29138.5	1.02410
$933$	−15864.5	−0.556678
$934$	32687.9	1.14516
$935$	−1334.77	−0.0466864
$936$	0	0
$937$	−31212.1	−1.08821	−0.544106	−	0.839017i	$-0.683131\pi$
$937$	−31212.1	−1.08821	−0.544106	+	0.839017i	$0.683131\pi$
$938$	−34765.1	−1.21015
$939$	−15660.8	−0.544270
$940$	−144267.	−5.00582
$941$	1150.30	0.0398500	0.0199250	−	0.999801i	$-0.493657\pi$
$941$	1150.30	0.0398500	0.0199250	+	0.999801i	$0.493657\pi$
$942$	−26688.1	−0.923083
$943$	−18034.9	−0.622797
$944$	−14277.7	−0.492268
$945$	12136.1	0.417766
$946$	9464.04	0.325267
$947$	39077.9	1.34093	0.670465	−	0.741941i	$-0.266095\pi$
$947$	39077.9	1.34093	0.670465	+	0.741941i	$0.266095\pi$
$948$	27686.1	0.948526
$949$	0	0
$950$	32868.1	1.12251
$951$	−24320.9	−0.829295
$952$	898.892	0.0306022
$953$	51475.3	1.74968	0.874841	−	0.484409i	$-0.160965\pi$
$953$	51475.3	1.74968	0.874841	+	0.484409i	$0.160965\pi$
$954$	20041.7	0.680160
$955$	−30637.6	−1.03812
$956$	78403.1	2.65245
$957$	−3762.34	−0.127084
$958$	−82913.6	−2.79626
$959$	−13354.2	−0.449666
$960$	39985.4	1.34429
$961$	−25272.9	−0.848339
$962$	0	0
$963$	−12751.1	−0.426686
$964$	−39198.5	−1.30965
$965$	99524.8	3.32002
$966$	54309.1	1.80887
$967$	19803.2	0.658562	0.329281	−	0.944232i	$-0.393194\pi$
$967$	19803.2	0.658562	0.329281	+	0.944232i	$0.393194\pi$
$968$	73568.1	2.44274
$969$	105.842	0.00350891
$970$	−44099.6	−1.45975
$971$	−9889.70	−0.326854	−0.163427	−	0.986555i	$-0.552255\pi$
$971$	−9889.70	−0.326854	−0.163427	+	0.986555i	$0.552255\pi$
$972$	−3576.83	−0.118032
$973$	−10411.0	−0.343024
$974$	−55761.7	−1.83442
$975$	0	0
$976$	2614.73	0.0857535
$977$	28202.9	0.923532	0.461766	−	0.887002i	$-0.347216\pi$
$977$	28202.9	0.923532	0.461766	+	0.887002i	$0.347216\pi$
$978$	14936.9	0.488374
$979$	−40957.7	−1.33709
$980$	62744.2	2.04519
$981$	−7696.22	−0.250481
$982$	31461.1	1.02237
$983$	5757.36	0.186807	0.0934035	−	0.995628i	$-0.470225\pi$
$983$	5757.36	0.186807	0.0934035	+	0.995628i	$0.470225\pi$
$984$	10886.1	0.352680
$985$	−43061.6	−1.39295
$986$	−116.740	−0.00377055
$987$	−37238.9	−1.20094
$988$	0	0
$989$	5247.34	0.168712
$990$	48677.9	1.56271
$991$	24311.1	0.779281	0.389641	−	0.920967i	$-0.372599\pi$
$991$	24311.1	0.779281	0.389641	+	0.920967i	$0.372599\pi$
$992$	6039.04	0.193286
$993$	24393.9	0.779574
$994$	36250.8	1.15675
$995$	10685.3	0.340450
$996$	−53522.8	−1.70274
$997$	−7390.90	−0.234776	−0.117388	−	0.993086i	$-0.537452\pi$
$997$	−7390.90	−0.234776	−0.117388	+	0.993086i	$0.537452\pi$
$998$	100807.	3.19737
$999$	−3747.58	−0.118687

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.q.1.8	yes	9
3.2	odd	2		1521.4.a.be.1.2		9
13.5	odd	4		507.4.b.j.337.2		18
13.8	odd	4		507.4.b.j.337.17		18
13.12	even	2		507.4.a.n.1.2	✓	9
39.38	odd	2		1521.4.a.bj.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.n.1.2	✓	9	13.12	even	2
507.4.a.q.1.8	yes	9	1.1	even	1	trivial
507.4.b.j.337.2		18	13.5	odd	4
507.4.b.j.337.17		18	13.8	odd	4
1521.4.a.be.1.2		9	3.2	odd	2
1521.4.a.bj.1.8		9	39.38	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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

\(f(q)\)	\(=\)	\(q+4.76649 q^{2} -3.00000 q^{3} +14.7194 q^{4} +18.8390 q^{5} -14.2995 q^{6} -23.8593 q^{7} +32.0282 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+4.76649 q^{2} -3.00000 q^{3} +14.7194 q^{4} +18.8390 q^{5} -14.2995 q^{6} -23.8593 q^{7} +32.0282 q^{8} +9.00000 q^{9} +89.7961 q^{10} +60.2327 q^{11} -44.1583 q^{12} -113.725 q^{14} -56.5171 q^{15} +34.9065 q^{16} -1.17630 q^{17} +42.8984 q^{18} +29.9930 q^{19} +277.300 q^{20} +71.5780 q^{21} +287.099 q^{22} +159.182 q^{23} -96.0845 q^{24} +229.909 q^{25} -27.0000 q^{27} -351.196 q^{28} +20.8211 q^{29} -269.388 q^{30} -67.2170 q^{31} -89.8439 q^{32} -180.698 q^{33} -5.60681 q^{34} -449.487 q^{35} +132.475 q^{36} +138.799 q^{37} +142.961 q^{38} +603.380 q^{40} -113.297 q^{41} +341.176 q^{42} +32.9644 q^{43} +886.592 q^{44} +169.551 q^{45} +758.740 q^{46} -520.256 q^{47} -104.720 q^{48} +226.268 q^{49} +1095.86 q^{50} +3.52889 q^{51} +467.189 q^{53} -128.695 q^{54} +1134.73 q^{55} -764.171 q^{56} -89.9789 q^{57} +99.2437 q^{58} -409.028 q^{59} -831.900 q^{60} +74.9067 q^{61} -320.389 q^{62} -214.734 q^{63} -707.492 q^{64} -861.296 q^{66} +305.693 q^{67} -17.3144 q^{68} -477.546 q^{69} -2142.47 q^{70} -318.757 q^{71} +288.254 q^{72} +867.378 q^{73} +661.585 q^{74} -689.727 q^{75} +441.480 q^{76} -1437.11 q^{77} -626.973 q^{79} +657.605 q^{80} +81.0000 q^{81} -540.031 q^{82} +1212.06 q^{83} +1053.59 q^{84} -22.1603 q^{85} +157.125 q^{86} -62.4634 q^{87} +1929.14 q^{88} -679.991 q^{89} +808.165 q^{90} +2343.07 q^{92} +201.651 q^{93} -2479.80 q^{94} +565.038 q^{95} +269.532 q^{96} -491.109 q^{97} +1078.51 q^{98} +542.094 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 8 q^{2} - 27 q^{3} + 32 q^{4} + 41 q^{5} - 24 q^{6} + q^{7} + 111 q^{8} + 81 q^{9}+O(q^{10}) \) 9 * q + 8 * q^2 - 27 * q^3 + 32 * q^4 + 41 * q^5 - 24 * q^6 + q^7 + 111 * q^8 + 81 * q^9	\( 9 q + 8 q^{2} - 27 q^{3} + 32 q^{4} + 41 q^{5} - 24 q^{6} + q^{7} + 111 q^{8} + 81 q^{9} + 198 q^{10} + 37 q^{11} - 96 q^{12} + 98 q^{14} - 123 q^{15} + 32 q^{16} - 134 q^{17} + 72 q^{18} - 72 q^{19} + 356 q^{20} - 3 q^{21} + 274 q^{22} + 226 q^{23} - 333 q^{24} + 612 q^{25} - 243 q^{27} + 132 q^{28} - 547 q^{29} - 594 q^{30} - 521 q^{31} + 721 q^{32} - 111 q^{33} - 100 q^{34} + 138 q^{35} + 288 q^{36} + 584 q^{37} - 416 q^{38} + 1342 q^{40} + 482 q^{41} - 294 q^{42} + 158 q^{43} + 1453 q^{44} + 369 q^{45} + 1537 q^{46} + 1500 q^{47} - 96 q^{48} + 642 q^{49} + 2777 q^{50} + 402 q^{51} + 1399 q^{53} - 216 q^{54} - 1408 q^{55} - 616 q^{56} + 216 q^{57} + 1455 q^{58} + 1541 q^{59} - 1068 q^{60} + 2092 q^{61} - 293 q^{62} + 9 q^{63} + 2481 q^{64} - 822 q^{66} + 252 q^{67} - 1579 q^{68} - 678 q^{69} + 2492 q^{70} + 2352 q^{71} + 999 q^{72} + 903 q^{73} + 1037 q^{74} - 1836 q^{75} - 485 q^{76} - 1686 q^{77} - 115 q^{79} + 5701 q^{80} + 729 q^{81} - 5147 q^{82} + 1207 q^{83} - 396 q^{84} + 4308 q^{85} + 5691 q^{86} + 1641 q^{87} - 484 q^{88} + 2336 q^{89} + 1782 q^{90} + 2087 q^{92} + 1563 q^{93} - 468 q^{94} - 222 q^{95} - 2163 q^{96} + 2155 q^{97} + 5593 q^{98} + 333 q^{99}+O(q^{100}) \) 9 * q + 8 * q^2 - 27 * q^3 + 32 * q^4 + 41 * q^5 - 24 * q^6 + q^7 + 111 * q^8 + 81 * q^9 + 198 * q^10 + 37 * q^11 - 96 * q^12 + 98 * q^14 - 123 * q^15 + 32 * q^16 - 134 * q^17 + 72 * q^18 - 72 * q^19 + 356 * q^20 - 3 * q^21 + 274 * q^22 + 226 * q^23 - 333 * q^24 + 612 * q^25 - 243 * q^27 + 132 * q^28 - 547 * q^29 - 594 * q^30 - 521 * q^31 + 721 * q^32 - 111 * q^33 - 100 * q^34 + 138 * q^35 + 288 * q^36 + 584 * q^37 - 416 * q^38 + 1342 * q^40 + 482 * q^41 - 294 * q^42 + 158 * q^43 + 1453 * q^44 + 369 * q^45 + 1537 * q^46 + 1500 * q^47 - 96 * q^48 + 642 * q^49 + 2777 * q^50 + 402 * q^51 + 1399 * q^53 - 216 * q^54 - 1408 * q^55 - 616 * q^56 + 216 * q^57 + 1455 * q^58 + 1541 * q^59 - 1068 * q^60 + 2092 * q^61 - 293 * q^62 + 9 * q^63 + 2481 * q^64 - 822 * q^66 + 252 * q^67 - 1579 * q^68 - 678 * q^69 + 2492 * q^70 + 2352 * q^71 + 999 * q^72 + 903 * q^73 + 1037 * q^74 - 1836 * q^75 - 485 * q^76 - 1686 * q^77 - 115 * q^79 + 5701 * q^80 + 729 * q^81 - 5147 * q^82 + 1207 * q^83 - 396 * q^84 + 4308 * q^85 + 5691 * q^86 + 1641 * q^87 - 484 * q^88 + 2336 * q^89 + 1782 * q^90 + 2087 * q^92 + 1563 * q^93 - 468 * q^94 - 222 * q^95 - 2163 * q^96 + 2155 * q^97 + 5593 * q^98 + 333 * q^99

Introduction

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