LMFDB - Embedded newform 435.4.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [435,4,Mod(1,435)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(435, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("435.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 435 = 3 \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	435.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:	$25.6658308525$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 31x^{4} + 9x^{3} + 230x^{2} + 32x - 128 $ x^6 - x^5 - 31x^4 + 9x^3 + 230x^2 + 32x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.691666$ of defining polynomial
Character	$\chi$	$=$	435.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.691666 q^{2} -3.00000 q^{3} -7.52160 q^{4} +5.00000 q^{5} +2.07500 q^{6} -14.5720 q^{7} +10.7358 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-0.691666 q^{2} -3.00000 q^{3} -7.52160 q^{4} +5.00000 q^{5} +2.07500 q^{6} -14.5720 q^{7} +10.7358 q^{8} +9.00000 q^{9} -3.45833 q^{10} +36.8326 q^{11} +22.5648 q^{12} -23.9199 q^{13} +10.0790 q^{14} -15.0000 q^{15} +52.7472 q^{16} +62.8914 q^{17} -6.22499 q^{18} -94.1599 q^{19} -37.6080 q^{20} +43.7161 q^{21} -25.4759 q^{22} +185.410 q^{23} -32.2073 q^{24} +25.0000 q^{25} +16.5446 q^{26} -27.0000 q^{27} +109.605 q^{28} -29.0000 q^{29} +10.3750 q^{30} -104.971 q^{31} -122.370 q^{32} -110.498 q^{33} -43.4999 q^{34} -72.8602 q^{35} -67.6944 q^{36} -259.799 q^{37} +65.1272 q^{38} +71.7597 q^{39} +53.6788 q^{40} +11.8069 q^{41} -30.2369 q^{42} -61.0501 q^{43} -277.040 q^{44} +45.0000 q^{45} -128.242 q^{46} +281.426 q^{47} -158.242 q^{48} -130.656 q^{49} -17.2916 q^{50} -188.674 q^{51} +179.916 q^{52} -170.846 q^{53} +18.6750 q^{54} +184.163 q^{55} -156.442 q^{56} +282.480 q^{57} +20.0583 q^{58} -636.987 q^{59} +112.824 q^{60} -379.603 q^{61} +72.6051 q^{62} -131.148 q^{63} -337.339 q^{64} -119.600 q^{65} +76.4276 q^{66} -623.333 q^{67} -473.044 q^{68} -556.231 q^{69} +50.3949 q^{70} +298.898 q^{71} +96.6218 q^{72} -524.124 q^{73} +179.694 q^{74} -75.0000 q^{75} +708.233 q^{76} -536.726 q^{77} -49.6337 q^{78} +563.264 q^{79} +263.736 q^{80} +81.0000 q^{81} -8.16645 q^{82} -885.342 q^{83} -328.815 q^{84} +314.457 q^{85} +42.2263 q^{86} +87.0000 q^{87} +395.426 q^{88} -447.652 q^{89} -31.1250 q^{90} +348.562 q^{91} -1394.58 q^{92} +314.914 q^{93} -194.653 q^{94} -470.799 q^{95} +367.109 q^{96} -558.895 q^{97} +90.3702 q^{98} +331.493 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} - 18 q^{3} + 15 q^{4} + 30 q^{5} + 3 q^{6} + 23 q^{7} - 51 q^{8} + 54 q^{9}+O(q^{10}) $ 6 * q - q^2 - 18 * q^3 + 15 * q^4 + 30 * q^5 + 3 * q^6 + 23 * q^7 - 51 * q^8 + 54 * q^9	\( 6 q - q^{2} - 18 q^{3} + 15 q^{4} + 30 q^{5} + 3 q^{6} + 23 q^{7} - 51 q^{8} + 54 q^{9} - 5 q^{10} - 111 q^{11} - 45 q^{12} - 83 q^{13} - 102 q^{14} - 90 q^{15} - 37 q^{16} - 35 q^{17} - 9 q^{18} - 76 q^{19} + 75 q^{20} - 69 q^{21} + 66 q^{22} + 166 q^{23} + 153 q^{24} + 150 q^{25} - 282 q^{26} - 162 q^{27} + 368 q^{28} - 174 q^{29} + 15 q^{30} - 164 q^{31} - 259 q^{32} + 333 q^{33} + 512 q^{34} + 115 q^{35} + 135 q^{36} - 538 q^{37} + 6 q^{38} + 249 q^{39} - 255 q^{40} - 1250 q^{41} + 306 q^{42} - 80 q^{43} - 1004 q^{44} + 270 q^{45} - 1190 q^{46} - 491 q^{47} + 111 q^{48} - 969 q^{49} - 25 q^{50} + 105 q^{51} - 336 q^{52} - 356 q^{53} + 27 q^{54} - 555 q^{55} - 1260 q^{56} + 228 q^{57} + 29 q^{58} - 646 q^{59} - 225 q^{60} - 1476 q^{61} - 4 q^{62} + 207 q^{63} - 845 q^{64} - 415 q^{65} - 198 q^{66} + 87 q^{67} - 558 q^{68} - 498 q^{69} - 510 q^{70} - 1312 q^{71} - 459 q^{72} - 1142 q^{73} - 2270 q^{74} - 450 q^{75} - 1594 q^{76} - 2561 q^{77} + 846 q^{78} - 142 q^{79} - 185 q^{80} + 486 q^{81} - 1544 q^{82} + 346 q^{83} - 1104 q^{84} - 175 q^{85} - 2056 q^{86} + 522 q^{87} + 1400 q^{88} - 2827 q^{89} - 45 q^{90} + 1379 q^{91} - 1434 q^{92} + 492 q^{93} + 978 q^{94} - 380 q^{95} + 777 q^{96} - 4 q^{97} - 1903 q^{98} - 999 q^{99}+O(q^{100}) \) 6 * q - q^2 - 18 * q^3 + 15 * q^4 + 30 * q^5 + 3 * q^6 + 23 * q^7 - 51 * q^8 + 54 * q^9 - 5 * q^10 - 111 * q^11 - 45 * q^12 - 83 * q^13 - 102 * q^14 - 90 * q^15 - 37 * q^16 - 35 * q^17 - 9 * q^18 - 76 * q^19 + 75 * q^20 - 69 * q^21 + 66 * q^22 + 166 * q^23 + 153 * q^24 + 150 * q^25 - 282 * q^26 - 162 * q^27 + 368 * q^28 - 174 * q^29 + 15 * q^30 - 164 * q^31 - 259 * q^32 + 333 * q^33 + 512 * q^34 + 115 * q^35 + 135 * q^36 - 538 * q^37 + 6 * q^38 + 249 * q^39 - 255 * q^40 - 1250 * q^41 + 306 * q^42 - 80 * q^43 - 1004 * q^44 + 270 * q^45 - 1190 * q^46 - 491 * q^47 + 111 * q^48 - 969 * q^49 - 25 * q^50 + 105 * q^51 - 336 * q^52 - 356 * q^53 + 27 * q^54 - 555 * q^55 - 1260 * q^56 + 228 * q^57 + 29 * q^58 - 646 * q^59 - 225 * q^60 - 1476 * q^61 - 4 * q^62 + 207 * q^63 - 845 * q^64 - 415 * q^65 - 198 * q^66 + 87 * q^67 - 558 * q^68 - 498 * q^69 - 510 * q^70 - 1312 * q^71 - 459 * q^72 - 1142 * q^73 - 2270 * q^74 - 450 * q^75 - 1594 * q^76 - 2561 * q^77 + 846 * q^78 - 142 * q^79 - 185 * q^80 + 486 * q^81 - 1544 * q^82 + 346 * q^83 - 1104 * q^84 - 175 * q^85 - 2056 * q^86 + 522 * q^87 + 1400 * q^88 - 2827 * q^89 - 45 * q^90 + 1379 * q^91 - 1434 * q^92 + 492 * q^93 + 978 * q^94 - 380 * q^95 + 777 * q^96 - 4 * q^97 - 1903 * q^98 - 999 * 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 $

$2$	−0.691666	−0.244541	−0.122270	−	0.992497i	$-0.539018\pi$
$2$	−0.691666	−0.244541	−0.122270	+	0.992497i	$0.539018\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.52160	−0.940200
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	2.07500	0.141186
$7$	−14.5720	−0.786816	−0.393408	−	0.919364i	$-0.628704\pi$
$7$	−14.5720	−0.786816	−0.393408	+	0.919364i	$0.628704\pi$
$8$	10.7358	0.474458
$9$	9.00000	0.333333
$10$	−3.45833	−0.109362
$11$	36.8326	1.00959	0.504793	−	0.863240i	$-0.331569\pi$
$11$	36.8326	1.00959	0.504793	+	0.863240i	$0.331569\pi$
$12$	22.5648	0.542825
$13$	−23.9199	−0.510322	−0.255161	−	0.966899i	$-0.582128\pi$
$13$	−23.9199	−0.510322	−0.255161	+	0.966899i	$0.582128\pi$
$14$	10.0790	0.192409
$15$	−15.0000	−0.258199
$16$	52.7472	0.824175
$17$	62.8914	0.897260	0.448630	−	0.893718i	$-0.351912\pi$
$17$	62.8914	0.897260	0.448630	+	0.893718i	$0.351912\pi$
$18$	−6.22499	−0.0815136
$19$	−94.1599	−1.13693	−0.568467	−	0.822706i	$-0.692463\pi$
$19$	−94.1599	−1.13693	−0.568467	+	0.822706i	$0.692463\pi$
$20$	−37.6080	−0.420470
$21$	43.7161	0.454268
$22$	−25.4759	−0.246885
$23$	185.410	1.68090	0.840450	−	0.541889i	$-0.182291\pi$
$23$	185.410	1.68090	0.840450	+	0.541889i	$0.182291\pi$
$24$	−32.2073	−0.273928
$25$	25.0000	0.200000
$26$	16.5446	0.124795
$27$	−27.0000	−0.192450
$28$	109.605	0.739764
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	10.3750	0.0631402
$31$	−104.971	−0.608174	−0.304087	−	0.952644i	$-0.598351\pi$
$31$	−104.971	−0.608174	−0.304087	+	0.952644i	$0.598351\pi$
$32$	−122.370	−0.676003
$33$	−110.498	−0.582885
$34$	−43.4999	−0.219417
$35$	−72.8602	−0.351875
$36$	−67.6944	−0.313400
$37$	−259.799	−1.15435	−0.577173	−	0.816622i	$-0.695844\pi$
$37$	−259.799	−1.15435	−0.577173	+	0.816622i	$0.695844\pi$
$38$	65.1272	0.278027
$39$	71.7597	0.294635
$40$	53.6788	0.212184
$41$	11.8069	0.0449740	0.0224870	−	0.999747i	$-0.492842\pi$
$41$	11.8069	0.0449740	0.0224870	+	0.999747i	$0.492842\pi$
$42$	−30.2369	−0.111087
$43$	−61.0501	−0.216513	−0.108256	−	0.994123i	$-0.534527\pi$
$43$	−61.0501	−0.216513	−0.108256	+	0.994123i	$0.534527\pi$
$44$	−277.040	−0.949213
$45$	45.0000	0.149071
$46$	−128.242	−0.411049
$47$	281.426	0.873409	0.436705	−	0.899605i	$-0.356145\pi$
$47$	281.426	0.873409	0.436705	+	0.899605i	$0.356145\pi$
$48$	−158.242	−0.475838
$49$	−130.656	−0.380921
$50$	−17.2916	−0.0489082
$51$	−188.674	−0.518033
$52$	179.916	0.479805
$53$	−170.846	−0.442782	−0.221391	−	0.975185i	$-0.571060\pi$
$53$	−170.846	−0.442782	−0.221391	+	0.975185i	$0.571060\pi$
$54$	18.6750	0.0470619
$55$	184.163	0.451501
$56$	−156.442	−0.373311
$57$	282.480	0.656409
$58$	20.0583	0.0454101
$59$	−636.987	−1.40557	−0.702785	−	0.711402i	$-0.748060\pi$
$59$	−636.987	−1.40557	−0.702785	+	0.711402i	$0.748060\pi$
$60$	112.824	0.242759
$61$	−379.603	−0.796774	−0.398387	−	0.917218i	$-0.630430\pi$
$61$	−379.603	−0.796774	−0.398387	+	0.917218i	$0.630430\pi$
$62$	72.6051	0.148723
$63$	−131.148	−0.262272
$64$	−337.339	−0.658865
$65$	−119.600	−0.228223
$66$	76.4276	0.142539
$67$	−623.333	−1.13660	−0.568300	−	0.822821i	$-0.692399\pi$
$67$	−623.333	−1.13660	−0.568300	+	0.822821i	$0.692399\pi$
$68$	−473.044	−0.843603
$69$	−556.231	−0.970468
$70$	50.3949	0.0860477
$71$	298.898	0.499615	0.249808	−	0.968295i	$-0.419633\pi$
$71$	298.898	0.499615	0.249808	+	0.968295i	$0.419633\pi$
$72$	96.6218	0.158153
$73$	−524.124	−0.840329	−0.420165	−	0.907448i	$-0.638028\pi$
$73$	−524.124	−0.840329	−0.420165	+	0.907448i	$0.638028\pi$
$74$	179.694	0.282284
$75$	−75.0000	−0.115470
$76$	708.233	1.06895
$77$	−536.726	−0.794358
$78$	−49.6337	−0.0720502
$79$	563.264	0.802179	0.401090	−	0.916039i	$-0.368632\pi$
$79$	563.264	0.802179	0.401090	+	0.916039i	$0.368632\pi$
$80$	263.736	0.368582
$81$	81.0000	0.111111
$82$	−8.16645	−0.0109980
$83$	−885.342	−1.17083	−0.585415	−	0.810734i	$-0.699068\pi$
$83$	−885.342	−1.17083	−0.585415	+	0.810734i	$0.699068\pi$
$84$	−328.815	−0.427103
$85$	314.457	0.401267
$86$	42.2263	0.0529462
$87$	87.0000	0.107211
$88$	395.426	0.479006
$89$	−447.652	−0.533157	−0.266579	−	0.963813i	$-0.585893\pi$
$89$	−447.652	−0.533157	−0.266579	+	0.963813i	$0.585893\pi$
$90$	−31.1250	−0.0364540
$91$	348.562	0.401529
$92$	−1394.58	−1.58038
$93$	314.914	0.351130
$94$	−194.653	−0.213584
$95$	−470.799	−0.508453
$96$	367.109	0.390290
$97$	−558.895	−0.585022	−0.292511	−	0.956262i	$-0.594491\pi$
$97$	−558.895	−0.585022	−0.292511	+	0.956262i	$0.594491\pi$
$98$	90.3702	0.0931507
$99$	331.493	0.336529
$100$	−188.040	−0.188040
$101$	−1138.78	−1.12191	−0.560956	−	0.827846i	$-0.689566\pi$
$101$	−1138.78	−1.12191	−0.560956	+	0.827846i	$0.689566\pi$
$102$	130.500	0.126680
$103$	1008.31	0.964583	0.482292	−	0.876011i	$-0.339805\pi$
$103$	1008.31	0.964583	0.482292	+	0.876011i	$0.339805\pi$
$104$	−256.798	−0.242126
$105$	218.580	0.203155
$106$	118.168	0.108278
$107$	140.506	0.126946	0.0634730	−	0.997984i	$-0.479782\pi$
$107$	140.506	0.126946	0.0634730	+	0.997984i	$0.479782\pi$
$108$	203.083	0.180942
$109$	−850.530	−0.747394	−0.373697	−	0.927551i	$-0.621910\pi$
$109$	−850.530	−0.747394	−0.373697	+	0.927551i	$0.621910\pi$
$110$	−127.379	−0.110410
$111$	779.398	0.666461
$112$	−768.634	−0.648474
$113$	1851.23	1.54115	0.770573	−	0.637352i	$-0.219970\pi$
$113$	1851.23	1.54115	0.770573	+	0.637352i	$0.219970\pi$
$114$	−195.381	−0.160519
$115$	927.051	0.751721
$116$	218.126	0.174591
$117$	−215.279	−0.170107
$118$	440.582	0.343719
$119$	−916.456	−0.705978
$120$	−161.036	−0.122505
$121$	25.6407	0.0192643
$122$	262.558	0.194844
$123$	−35.4208	−0.0259657
$124$	789.552	0.571805
$125$	125.000	0.0894427
$126$	90.7108	0.0641362
$127$	903.124	0.631018	0.315509	−	0.948923i	$-0.397825\pi$
$127$	903.124	0.631018	0.315509	+	0.948923i	$0.397825\pi$
$128$	1212.28	0.837122
$129$	183.150	0.125004
$130$	82.7229	0.0558098
$131$	−685.958	−0.457499	−0.228750	−	0.973485i	$-0.573464\pi$
$131$	−685.958	−0.457499	−0.228750	+	0.973485i	$0.573464\pi$
$132$	831.120	0.548028
$133$	1372.10	0.894558
$134$	431.138	0.277945
$135$	−135.000	−0.0860663
$136$	675.187	0.425712
$137$	−1152.95	−0.718999	−0.359500	−	0.933145i	$-0.617053\pi$
$137$	−1152.95	−0.718999	−0.359500	+	0.933145i	$0.617053\pi$
$138$	384.726	0.237319
$139$	−438.465	−0.267555	−0.133777	−	0.991011i	$-0.542711\pi$
$139$	−438.465	−0.267555	−0.133777	+	0.991011i	$0.542711\pi$
$140$	548.025	0.330833
$141$	−844.279	−0.504263
$142$	−206.738	−0.122176
$143$	−881.032	−0.515214
$144$	474.725	0.274725
$145$	−145.000	−0.0830455
$146$	362.518	0.205495
$147$	391.968	0.219925
$148$	1954.11	1.08532
$149$	−405.570	−0.222991	−0.111495	−	0.993765i	$-0.535564\pi$
$149$	−405.570	−0.222991	−0.111495	+	0.993765i	$0.535564\pi$
$150$	51.8749	0.0282371
$151$	−78.9559	−0.0425519	−0.0212760	−	0.999774i	$-0.506773\pi$
$151$	−78.9559	−0.0425519	−0.0212760	+	0.999774i	$0.506773\pi$
$152$	−1010.88	−0.539428
$153$	566.023	0.299087
$154$	371.235	0.194253
$155$	−524.856	−0.271984
$156$	−539.748	−0.277015
$157$	314.869	0.160059	0.0800296	−	0.996792i	$-0.474499\pi$
$157$	314.869	0.160059	0.0800296	+	0.996792i	$0.474499\pi$
$158$	−389.590	−0.196166
$159$	512.537	0.255640
$160$	−611.848	−0.302318
$161$	−2701.80	−1.32256
$162$	−56.0249	−0.0271712
$163$	816.812	0.392501	0.196251	−	0.980554i	$-0.437123\pi$
$163$	816.812	0.392501	0.196251	+	0.980554i	$0.437123\pi$
$164$	−88.8070	−0.0422845
$165$	−552.489	−0.260674
$166$	612.361	0.286316
$167$	−1364.36	−0.632200	−0.316100	−	0.948726i	$-0.602374\pi$
$167$	−1364.36	−0.632200	−0.316100	+	0.948726i	$0.602374\pi$
$168$	469.325	0.215531
$169$	−1624.84	−0.739571
$170$	−217.499	−0.0981261
$171$	−847.439	−0.378978
$172$	459.194	0.203565
$173$	430.857	0.189350	0.0946748	−	0.995508i	$-0.469819\pi$
$173$	430.857	0.189350	0.0946748	+	0.995508i	$0.469819\pi$
$174$	−60.1749	−0.0262175
$175$	−364.301	−0.157363
$176$	1942.82	0.832076
$177$	1910.96	0.811506
$178$	309.625	0.130379
$179$	−3192.64	−1.33312	−0.666561	−	0.745451i	$-0.732234\pi$
$179$	−3192.64	−1.33312	−0.666561	+	0.745451i	$0.732234\pi$
$180$	−338.472	−0.140157
$181$	−621.160	−0.255085	−0.127543	−	0.991833i	$-0.540709\pi$
$181$	−621.160	−0.255085	−0.127543	+	0.991833i	$0.540709\pi$
$182$	−241.088	−0.0981903
$183$	1138.81	0.460017
$184$	1990.52	0.797517
$185$	−1299.00	−0.516239
$186$	−217.815	−0.0858655
$187$	2316.46	0.905861
$188$	−2116.78	−0.821179
$189$	393.445	0.151423
$190$	325.636	0.124337
$191$	2943.37	1.11505	0.557526	−	0.830160i	$-0.311751\pi$
$191$	2943.37	1.11505	0.557526	+	0.830160i	$0.311751\pi$
$192$	1012.02	0.380396
$193$	3259.79	1.21578	0.607889	−	0.794022i	$-0.292017\pi$
$193$	3259.79	1.21578	0.607889	+	0.794022i	$0.292017\pi$
$194$	386.568	0.143062
$195$	358.799	0.131765
$196$	982.741	0.358142
$197$	2372.35	0.857983	0.428992	−	0.903308i	$-0.358869\pi$
$197$	2372.35	0.857983	0.428992	+	0.903308i	$0.358869\pi$
$198$	−229.283	−0.0822950
$199$	−1828.75	−0.651441	−0.325721	−	0.945466i	$-0.605607\pi$
$199$	−1828.75	−0.651441	−0.325721	+	0.945466i	$0.605607\pi$
$200$	268.394	0.0948916
$201$	1870.00	0.656216
$202$	787.657	0.274353
$203$	422.589	0.146108
$204$	1419.13	0.487055
$205$	59.0347	0.0201130
$206$	−697.416	−0.235880
$207$	1668.69	0.560300
$208$	−1261.71	−0.420595
$209$	−3468.15	−1.14783
$210$	−151.185	−0.0496797
$211$	4223.45	1.37798	0.688991	−	0.724770i	$-0.258054\pi$
$211$	4223.45	1.37798	0.688991	+	0.724770i	$0.258054\pi$
$212$	1285.03	0.416304
$213$	−896.694	−0.288453
$214$	−97.1832	−0.0310435
$215$	−305.251	−0.0968275
$216$	−289.865	−0.0913095
$217$	1529.65	0.478521
$218$	588.282	0.182768
$219$	1572.37	0.485164
$220$	−1385.20	−0.424501
$221$	−1504.36	−0.457891
$222$	−539.083	−0.162977
$223$	−327.670	−0.0983964	−0.0491982	−	0.998789i	$-0.515667\pi$
$223$	−327.670	−0.0983964	−0.0491982	+	0.998789i	$0.515667\pi$
$224$	1783.17	0.531889
$225$	225.000	0.0666667
$226$	−1280.44	−0.376873
$227$	3662.12	1.07076	0.535382	−	0.844610i	$-0.320168\pi$
$227$	3662.12	1.07076	0.535382	+	0.844610i	$0.320168\pi$
$228$	−2124.70	−0.617156
$229$	−6562.57	−1.89374	−0.946871	−	0.321613i	$-0.895775\pi$
$229$	−6562.57	−1.89374	−0.946871	+	0.321613i	$0.895775\pi$
$230$	−641.210	−0.183827
$231$	1610.18	0.458623
$232$	−311.337	−0.0881046
$233$	2490.28	0.700189	0.350094	−	0.936714i	$-0.386150\pi$
$233$	2490.28	0.700189	0.350094	+	0.936714i	$0.386150\pi$
$234$	148.901	0.0415982
$235$	1407.13	0.390601
$236$	4791.16	1.32152
$237$	−1689.79	−0.463138
$238$	633.881	0.172640
$239$	−3584.40	−0.970106	−0.485053	−	0.874485i	$-0.661200\pi$
$239$	−3584.40	−0.970106	−0.485053	+	0.874485i	$0.661200\pi$
$240$	−791.208	−0.212801
$241$	−6543.94	−1.74910	−0.874548	−	0.484938i	$-0.838842\pi$
$241$	−6543.94	−1.74910	−0.874548	+	0.484938i	$0.838842\pi$
$242$	−17.7348	−0.00471090
$243$	−243.000	−0.0641500
$244$	2855.22	0.749126
$245$	−653.279	−0.170353
$246$	24.4994	0.00634968
$247$	2252.29	0.580203
$248$	−1126.95	−0.288553
$249$	2656.03	0.675979
$250$	−86.4582	−0.0218724
$251$	528.100	0.132802	0.0664011	−	0.997793i	$-0.478848\pi$
$251$	528.100	0.132802	0.0664011	+	0.997793i	$0.478848\pi$
$252$	986.445	0.246588
$253$	6829.14	1.69701
$254$	−624.660	−0.154310
$255$	−943.371	−0.231671
$256$	1860.22	0.454155
$257$	5020.78	1.21863	0.609314	−	0.792929i	$-0.291445\pi$
$257$	5020.78	1.21863	0.609314	+	0.792929i	$0.291445\pi$
$258$	−126.679	−0.0305685
$259$	3785.81	0.908257
$260$	899.579	0.214575
$261$	−261.000	−0.0618984
$262$	474.453	0.111877
$263$	−422.212	−0.0989913	−0.0494957	−	0.998774i	$-0.515761\pi$
$263$	−422.212	−0.0989913	−0.0494957	+	0.998774i	$0.515761\pi$
$264$	−1186.28	−0.276554
$265$	−854.228	−0.198018
$266$	−949.035	−0.218756
$267$	1342.96	0.307818
$268$	4688.46	1.06863
$269$	−1777.19	−0.402815	−0.201407	−	0.979508i	$-0.564551\pi$
$269$	−1777.19	−0.402815	−0.201407	+	0.979508i	$0.564551\pi$
$270$	93.3749	0.0210467
$271$	−1465.51	−0.328499	−0.164250	−	0.986419i	$-0.552520\pi$
$271$	−1465.51	−0.328499	−0.164250	+	0.986419i	$0.552520\pi$
$272$	3317.35	0.739499
$273$	−1045.68	−0.231823
$274$	797.454	0.175825
$275$	920.815	0.201917
$276$	4183.74	0.912434
$277$	−6081.50	−1.31914	−0.659570	−	0.751643i	$-0.729262\pi$
$277$	−6081.50	−1.31914	−0.659570	+	0.751643i	$0.729262\pi$
$278$	303.271	0.0654281
$279$	−944.742	−0.202725
$280$	−782.209	−0.166950
$281$	−1194.50	−0.253587	−0.126794	−	0.991929i	$-0.540469\pi$
$281$	−1194.50	−0.253587	−0.126794	+	0.991929i	$0.540469\pi$
$282$	583.959	0.123313
$283$	3827.53	0.803968	0.401984	−	0.915647i	$-0.368321\pi$
$283$	3827.53	0.803968	0.401984	+	0.915647i	$0.368321\pi$
$284$	−2248.19	−0.469738
$285$	1412.40	0.293555
$286$	609.380	0.125991
$287$	−172.051	−0.0353862
$288$	−1101.33	−0.225334
$289$	−957.668	−0.194925
$290$	100.292	0.0203080
$291$	1676.68	0.337763
$292$	3942.25	0.790077
$293$	−6821.00	−1.36002	−0.680012	−	0.733201i	$-0.738025\pi$
$293$	−6821.00	−1.36002	−0.680012	+	0.733201i	$0.738025\pi$
$294$	−271.111	−0.0537806
$295$	−3184.93	−0.628590
$296$	−2789.14	−0.547688
$297$	−994.480	−0.194295
$298$	280.519	0.0545303
$299$	−4434.99	−0.857801
$300$	564.120	0.108565
$301$	889.624	0.170356
$302$	54.6111	0.0104057
$303$	3416.35	0.647736
$304$	−4966.67	−0.937034
$305$	−1898.02	−0.356328
$306$	−391.499	−0.0731389
$307$	−7350.74	−1.36654	−0.683272	−	0.730164i	$-0.739444\pi$
$307$	−7350.74	−1.36654	−0.683272	+	0.730164i	$0.739444\pi$
$308$	4037.04	0.746856
$309$	−3024.94	−0.556902
$310$	363.025	0.0665111
$311$	3393.14	0.618673	0.309337	−	0.950953i	$-0.399893\pi$
$311$	3393.14	0.618673	0.309337	+	0.950953i	$0.399893\pi$
$312$	770.395	0.139792
$313$	−10732.3	−1.93810	−0.969049	−	0.246867i	$-0.920599\pi$
$313$	−10732.3	−1.93810	−0.969049	+	0.246867i	$0.920599\pi$
$314$	−217.784	−0.0391410
$315$	−655.741	−0.117292
$316$	−4236.65	−0.754209
$317$	−8721.17	−1.54520	−0.772602	−	0.634890i	$-0.781045\pi$
$317$	−8721.17	−1.54520	−0.772602	+	0.634890i	$0.781045\pi$
$318$	−354.504	−0.0625145
$319$	−1068.15	−0.187475
$320$	−1686.70	−0.294654
$321$	−421.518	−0.0732923
$322$	1868.75	0.323420
$323$	−5921.85	−1.02013
$324$	−609.249	−0.104467
$325$	−597.998	−0.102064
$326$	−564.961	−0.0959825
$327$	2551.59	0.431508
$328$	126.756	0.0213383
$329$	−4100.95	−0.687212
$330$	382.138	0.0637454
$331$	−6093.02	−1.01179	−0.505895	−	0.862595i	$-0.668838\pi$
$331$	−6093.02	−1.01179	−0.505895	+	0.862595i	$0.668838\pi$
$332$	6659.18	1.10081
$333$	−2338.20	−0.384782
$334$	943.682	0.154599
$335$	−3116.66	−0.508303
$336$	2305.90	0.374397
$337$	−6306.92	−1.01947	−0.509733	−	0.860333i	$-0.670256\pi$
$337$	−6306.92	−1.01947	−0.509733	+	0.860333i	$0.670256\pi$
$338$	1123.85	0.180855
$339$	−5553.70	−0.889781
$340$	−2365.22	−0.377271
$341$	−3866.37	−0.614004
$342$	586.144	0.0926756
$343$	6902.13	1.08653
$344$	−655.419	−0.102726
$345$	−2781.15	−0.434007
$346$	−298.009	−0.0463037
$347$	11395.7	1.76298	0.881491	−	0.472201i	$-0.156540\pi$
$347$	11395.7	1.76298	0.881491	+	0.472201i	$0.156540\pi$
$348$	−654.379	−0.100800
$349$	8835.08	1.35510	0.677552	−	0.735475i	$-0.263041\pi$
$349$	8835.08	1.35510	0.677552	+	0.735475i	$0.263041\pi$
$350$	251.974	0.0384817
$351$	645.837	0.0982115
$352$	−4507.19	−0.682483
$353$	−3390.75	−0.511251	−0.255626	−	0.966776i	$-0.582281\pi$
$353$	−3390.75	−0.511251	−0.255626	+	0.966776i	$0.582281\pi$
$354$	−1321.75	−0.198446
$355$	1494.49	0.223435
$356$	3367.06	0.501274
$357$	2749.37	0.407597
$358$	2208.24	0.326003
$359$	−5160.27	−0.758632	−0.379316	−	0.925267i	$-0.623841\pi$
$359$	−5160.27	−0.758632	−0.379316	+	0.925267i	$0.623841\pi$
$360$	483.109	0.0707280
$361$	2007.08	0.292620
$362$	429.635	0.0623788
$363$	−76.9222	−0.0111222
$364$	−2621.74	−0.377518
$365$	−2620.62	−0.375807
$366$	−787.675	−0.112493
$367$	1110.17	0.157902	0.0789512	−	0.996878i	$-0.474843\pi$
$367$	1110.17	0.157902	0.0789512	+	0.996878i	$0.474843\pi$
$368$	9779.88	1.38536
$369$	106.262	0.0149913
$370$	898.472	0.126241
$371$	2489.57	0.348388
$372$	−2368.66	−0.330132
$373$	5801.22	0.805297	0.402648	−	0.915355i	$-0.368090\pi$
$373$	5801.22	0.805297	0.402648	+	0.915355i	$0.368090\pi$
$374$	−1602.21	−0.221520
$375$	−375.000	−0.0516398
$376$	3021.32	0.414396
$377$	693.677	0.0947644
$378$	−272.132	−0.0370290
$379$	−12664.5	−1.71644	−0.858219	−	0.513284i	$-0.828429\pi$
$379$	−12664.5	−1.71644	−0.858219	+	0.513284i	$0.828429\pi$
$380$	3541.16	0.478047
$381$	−2709.37	−0.364318
$382$	−2035.83	−0.272675
$383$	−13408.3	−1.78885	−0.894427	−	0.447215i	$-0.852416\pi$
$383$	−13408.3	−1.78885	−0.894427	+	0.447215i	$0.852416\pi$
$384$	−3636.85	−0.483313
$385$	−2683.63	−0.355248
$386$	−2254.69	−0.297307
$387$	−549.451	−0.0721710
$388$	4203.78	0.550038
$389$	−4676.56	−0.609540	−0.304770	−	0.952426i	$-0.598580\pi$
$389$	−4676.56	−0.609540	−0.304770	+	0.952426i	$0.598580\pi$
$390$	−248.169	−0.0322218
$391$	11660.7	1.50820
$392$	−1402.69	−0.180731
$393$	2057.87	0.264137
$394$	−1640.87	−0.209812
$395$	2816.32	0.358745
$396$	−2493.36	−0.316404
$397$	−5913.71	−0.747608	−0.373804	−	0.927508i	$-0.621947\pi$
$397$	−5913.71	−0.747608	−0.373804	+	0.927508i	$0.621947\pi$
$398$	1264.89	0.159304
$399$	−4116.30	−0.516473
$400$	1318.68	0.164835
$401$	7756.80	0.965975	0.482987	−	0.875627i	$-0.339552\pi$
$401$	7756.80	0.965975	0.482987	+	0.875627i	$0.339552\pi$
$402$	−1293.41	−0.160472
$403$	2510.90	0.310365
$404$	8565.46	1.05482
$405$	405.000	0.0496904
$406$	−292.290	−0.0357294
$407$	−9569.09	−1.16541
$408$	−2025.56	−0.245785
$409$	7674.37	0.927808	0.463904	−	0.885886i	$-0.346448\pi$
$409$	7674.37	0.927808	0.463904	+	0.885886i	$0.346448\pi$
$410$	−40.8323	−0.00491844
$411$	3458.84	0.415114
$412$	−7584.13	−0.906901
$413$	9282.19	1.10592
$414$	−1154.18	−0.137016
$415$	−4426.71	−0.523611
$416$	2927.07	0.344979
$417$	1315.39	0.154473
$418$	2398.80	0.280692
$419$	11316.2	1.31940	0.659702	−	0.751527i	$-0.270682\pi$
$419$	11316.2	1.31940	0.659702	+	0.751527i	$0.270682\pi$
$420$	−1644.07	−0.191006
$421$	−15960.8	−1.84770	−0.923851	−	0.382753i	$-0.874976\pi$
$421$	−15960.8	−1.84770	−0.923851	+	0.382753i	$0.874976\pi$
$422$	−2921.21	−0.336973
$423$	2532.84	0.291136
$424$	−1834.16	−0.210081
$425$	1572.29	0.179452
$426$	620.213	0.0705385
$427$	5531.59	0.626914
$428$	−1056.83	−0.119355
$429$	2643.10	0.297459
$430$	211.131	0.0236783
$431$	9181.83	1.02616	0.513078	−	0.858342i	$-0.328505\pi$
$431$	9181.83	1.02616	0.513078	+	0.858342i	$0.328505\pi$
$432$	−1424.18	−0.158613
$433$	312.822	0.0347188	0.0173594	−	0.999849i	$-0.494474\pi$
$433$	312.822	0.0347188	0.0173594	+	0.999849i	$0.494474\pi$
$434$	−1058.00	−0.117018
$435$	435.000	0.0479463
$436$	6397.34	0.702700
$437$	−17458.2	−1.91107
$438$	−1087.56	−0.118642
$439$	−8018.14	−0.871720	−0.435860	−	0.900014i	$-0.643556\pi$
$439$	−8018.14	−0.871720	−0.435860	+	0.900014i	$0.643556\pi$
$440$	1977.13	0.214218
$441$	−1175.90	−0.126974
$442$	1040.51	0.111973
$443$	553.480	0.0593603	0.0296802	−	0.999559i	$-0.490551\pi$
$443$	553.480	0.0593603	0.0296802	+	0.999559i	$0.490551\pi$
$444$	−5862.32	−0.626607
$445$	−2238.26	−0.238435
$446$	226.638	0.0240619
$447$	1216.71	0.128744
$448$	4915.71	0.518406
$449$	−12055.2	−1.26708	−0.633540	−	0.773710i	$-0.718399\pi$
$449$	−12055.2	−1.26708	−0.633540	+	0.773710i	$0.718399\pi$
$450$	−155.625	−0.0163027
$451$	434.880	0.0454051
$452$	−13924.2	−1.44899
$453$	236.868	0.0245674
$454$	−2532.96	−0.261845
$455$	1742.81	0.179569
$456$	3032.63	0.311439
$457$	−268.261	−0.0274589	−0.0137295	−	0.999906i	$-0.504370\pi$
$457$	−268.261	−0.0274589	−0.0137295	+	0.999906i	$0.504370\pi$
$458$	4539.11	0.463097
$459$	−1698.07	−0.172678
$460$	−6972.91	−0.706768
$461$	3621.49	0.365877	0.182939	−	0.983124i	$-0.441439\pi$
$461$	3621.49	0.365877	0.182939	+	0.983124i	$0.441439\pi$
$462$	−1113.70	−0.112152
$463$	−7751.51	−0.778063	−0.389031	−	0.921225i	$-0.627190\pi$
$463$	−7751.51	−0.778063	−0.389031	+	0.921225i	$0.627190\pi$
$464$	−1529.67	−0.153046
$465$	1574.57	0.157030
$466$	−1722.44	−0.171225
$467$	8838.90	0.875836	0.437918	−	0.899015i	$-0.355716\pi$
$467$	8838.90	0.875836	0.437918	+	0.899015i	$0.355716\pi$
$468$	1619.24	0.159935
$469$	9083.22	0.894295
$470$	−973.265	−0.0955178
$471$	−944.607	−0.0924102
$472$	−6838.54	−0.666884
$473$	−2248.63	−0.218588
$474$	1168.77	0.113256
$475$	−2354.00	−0.227387
$476$	6893.21	0.663760
$477$	−1537.61	−0.147594
$478$	2479.20	0.237230
$479$	−9904.02	−0.944731	−0.472366	−	0.881403i	$-0.656600\pi$
$479$	−9904.02	−0.944731	−0.472366	+	0.881403i	$0.656600\pi$
$480$	1835.54	0.174543
$481$	6214.38	0.589088
$482$	4526.22	0.427726
$483$	8105.41	0.763580
$484$	−192.859	−0.0181123
$485$	−2794.47	−0.261630
$486$	168.075	0.0156873
$487$	799.412	0.0743836	0.0371918	−	0.999308i	$-0.488159\pi$
$487$	799.412	0.0743836	0.0371918	+	0.999308i	$0.488159\pi$
$488$	−4075.33	−0.378036
$489$	−2450.44	−0.226611
$490$	451.851	0.0416583
$491$	−10013.3	−0.920351	−0.460176	−	0.887828i	$-0.652214\pi$
$491$	−10013.3	−0.920351	−0.460176	+	0.887828i	$0.652214\pi$
$492$	266.421	0.0244130
$493$	−1823.85	−0.166617
$494$	−1557.84	−0.141883
$495$	1657.47	0.150500
$496$	−5536.94	−0.501242
$497$	−4355.55	−0.393105
$498$	−1837.08	−0.165304
$499$	2556.35	0.229335	0.114667	−	0.993404i	$-0.463420\pi$
$499$	2556.35	0.229335	0.114667	+	0.993404i	$0.463420\pi$
$500$	−940.200	−0.0840940
$501$	4093.08	0.365001
$502$	−365.269	−0.0324756
$503$	11520.2	1.02120	0.510598	−	0.859820i	$-0.329424\pi$
$503$	11520.2	1.02120	0.510598	+	0.859820i	$0.329424\pi$
$504$	−1407.98	−0.124437
$505$	−5693.91	−0.501734
$506$	−4723.48	−0.414989
$507$	4874.51	0.426992
$508$	−6792.94	−0.593283
$509$	8055.10	0.701446	0.350723	−	0.936479i	$-0.385936\pi$
$509$	8055.10	0.701446	0.350723	+	0.936479i	$0.385936\pi$
$510$	652.498	0.0566531
$511$	7637.55	0.661184
$512$	−10984.9	−0.948181
$513$	2542.32	0.218803
$514$	−3472.70	−0.298004
$515$	5041.57	0.431375
$516$	−1377.58	−0.117529
$517$	10365.7	0.881782
$518$	−2618.51	−0.222106
$519$	−1292.57	−0.109321
$520$	−1283.99	−0.108282
$521$	22383.5	1.88222	0.941111	−	0.338097i	$-0.109783\pi$
$521$	22383.5	1.88222	0.941111	+	0.338097i	$0.109783\pi$
$522$	180.525	0.0151367
$523$	5260.45	0.439816	0.219908	−	0.975521i	$-0.429424\pi$
$523$	5260.45	0.439816	0.219908	+	0.975521i	$0.429424\pi$
$524$	5159.50	0.430141
$525$	1092.90	0.0908537
$526$	292.030	0.0242074
$527$	−6601.79	−0.545690
$528$	−5828.45	−0.480399
$529$	22210.0	1.82543
$530$	590.840	0.0484235
$531$	−5732.88	−0.468523
$532$	−10320.4	−0.841063
$533$	−282.421	−0.0229512
$534$	−928.876	−0.0752742
$535$	702.530	0.0567720
$536$	−6691.95	−0.539269
$537$	9577.91	0.769678
$538$	1229.22	0.0985046
$539$	−4812.40	−0.384573
$540$	1015.42	0.0809195
$541$	−15052.8	−1.19625	−0.598123	−	0.801404i	$-0.704087\pi$
$541$	−15052.8	−1.19625	−0.598123	+	0.801404i	$0.704087\pi$
$542$	1013.64	0.0803315
$543$	1863.48	0.147274
$544$	−7695.99	−0.606550
$545$	−4252.65	−0.334245
$546$	723.264	0.0566902
$547$	13743.0	1.07424	0.537121	−	0.843505i	$-0.319512\pi$
$547$	13743.0	1.07424	0.537121	+	0.843505i	$0.319512\pi$
$548$	8672.00	0.676003
$549$	−3416.43	−0.265591
$550$	−636.896	−0.0493770
$551$	2730.64	0.211123
$552$	−5971.56	−0.460446
$553$	−8207.90	−0.631167
$554$	4206.36	0.322584
$555$	3896.99	0.298051
$556$	3297.96	0.251555
$557$	23296.7	1.77219	0.886097	−	0.463500i	$-0.153407\pi$
$557$	23296.7	1.77219	0.886097	+	0.463500i	$0.153407\pi$
$558$	653.446	0.0495745
$559$	1460.31	0.110491
$560$	−3843.17	−0.290006
$561$	−6949.37	−0.522999
$562$	826.196	0.0620124
$563$	6823.79	0.510814	0.255407	−	0.966834i	$-0.417791\pi$
$563$	6823.79	0.510814	0.255407	+	0.966834i	$0.417791\pi$
$564$	6350.33	0.474108
$565$	9256.17	0.689221
$566$	−2647.37	−0.196603
$567$	−1180.33	−0.0874240
$568$	3208.90	0.237046
$569$	17974.8	1.32433	0.662166	−	0.749358i	$-0.269638\pi$
$569$	17974.8	1.32433	0.662166	+	0.749358i	$0.269638\pi$
$570$	−976.907	−0.0717862
$571$	22031.6	1.61470	0.807350	−	0.590072i	$-0.200901\pi$
$571$	22031.6	1.61470	0.807350	+	0.590072i	$0.200901\pi$
$572$	6626.77	0.484404
$573$	−8830.11	−0.643775
$574$	119.002	0.00865338
$575$	4635.26	0.336180
$576$	−3036.05	−0.219622
$577$	14520.1	1.04762	0.523811	−	0.851834i	$-0.324510\pi$
$577$	14520.1	1.04762	0.523811	+	0.851834i	$0.324510\pi$
$578$	662.386	0.0476672
$579$	−9779.38	−0.701929
$580$	1090.63	0.0780793
$581$	12901.2	0.921227
$582$	−1159.71	−0.0825968
$583$	−6292.69	−0.447027
$584$	−5626.87	−0.398701
$585$	−1076.40	−0.0760743
$586$	4717.85	0.332581
$587$	−998.184	−0.0701865	−0.0350932	−	0.999384i	$-0.511173\pi$
$587$	−998.184	−0.0701865	−0.0350932	+	0.999384i	$0.511173\pi$
$588$	−2948.22	−0.206773
$589$	9884.08	0.691454
$590$	2202.91	0.153716
$591$	−7117.04	−0.495357
$592$	−13703.7	−0.951383
$593$	3431.22	0.237611	0.118806	−	0.992918i	$-0.462093\pi$
$593$	3431.22	0.237611	0.118806	+	0.992918i	$0.462093\pi$
$594$	687.848	0.0475130
$595$	−4582.28	−0.315723
$596$	3050.54	0.209656
$597$	5486.26	0.376110
$598$	3067.53	0.209767
$599$	−21059.5	−1.43651	−0.718255	−	0.695780i	$-0.755059\pi$
$599$	−21059.5	−1.43651	−0.718255	+	0.695780i	$0.755059\pi$
$600$	−805.182	−0.0547857
$601$	8534.75	0.579267	0.289634	−	0.957138i	$-0.406467\pi$
$601$	8534.75	0.579267	0.289634	+	0.957138i	$0.406467\pi$
$602$	−615.323	−0.0416589
$603$	−5609.99	−0.378867
$604$	593.874	0.0400073
$605$	128.204	0.00861524
$606$	−2362.97	−0.158398
$607$	−17101.7	−1.14355	−0.571776	−	0.820409i	$-0.693746\pi$
$607$	−17101.7	−1.14355	−0.571776	+	0.820409i	$0.693746\pi$
$608$	11522.3	0.768571
$609$	−1267.77	−0.0843555
$610$	1312.79	0.0871367
$611$	−6731.69	−0.445720
$612$	−4257.40	−0.281201
$613$	−1048.59	−0.0690897	−0.0345449	−	0.999403i	$-0.510998\pi$
$613$	−1048.59	−0.0690897	−0.0345449	+	0.999403i	$0.510998\pi$
$614$	5084.26	0.334176
$615$	−177.104	−0.0116122
$616$	−5762.16	−0.376890
$617$	2212.12	0.144338	0.0721689	−	0.997392i	$-0.477008\pi$
$617$	2212.12	0.144338	0.0721689	+	0.997392i	$0.477008\pi$
$618$	2092.25	0.136185
$619$	25602.7	1.66246	0.831228	−	0.555932i	$-0.187638\pi$
$619$	25602.7	1.66246	0.831228	+	0.555932i	$0.187638\pi$
$620$	3947.76	0.255719
$621$	−5006.08	−0.323489
$622$	−2346.92	−0.151291
$623$	6523.19	0.419496
$624$	3785.13	0.242831
$625$	625.000	0.0400000
$626$	7423.16	0.473944
$627$	10404.5	0.662702
$628$	−2368.32	−0.150488
$629$	−16339.2	−1.03575
$630$	453.554	0.0286826
$631$	−25127.0	−1.58524	−0.792622	−	0.609713i	$-0.791285\pi$
$631$	−25127.0	−1.58524	−0.792622	+	0.609713i	$0.791285\pi$
$632$	6047.07	0.380600
$633$	−12670.3	−0.795578
$634$	6032.14	0.377866
$635$	4515.62	0.282200
$636$	−3855.10	−0.240353
$637$	3125.28	0.194392
$638$	738.800	0.0458454
$639$	2690.08	0.166538
$640$	6061.41	0.374372
$641$	2231.80	0.137521	0.0687605	−	0.997633i	$-0.478096\pi$
$641$	2231.80	0.137521	0.0687605	+	0.997633i	$0.478096\pi$
$642$	291.550	0.0179230
$643$	−15105.0	−0.926415	−0.463207	−	0.886250i	$-0.653301\pi$
$643$	−15105.0	−0.926415	−0.463207	+	0.886250i	$0.653301\pi$
$644$	20321.9	1.24347
$645$	915.752	0.0559034
$646$	4095.94	0.249462
$647$	21013.9	1.27688	0.638441	−	0.769671i	$-0.279580\pi$
$647$	21013.9	1.27688	0.638441	+	0.769671i	$0.279580\pi$
$648$	869.596	0.0527176
$649$	−23461.9	−1.41904
$650$	413.614	0.0249589
$651$	−4588.94	−0.276274
$652$	−6143.73	−0.369029
$653$	25780.5	1.54498	0.772489	−	0.635028i	$-0.219011\pi$
$653$	25780.5	1.54498	0.772489	+	0.635028i	$0.219011\pi$
$654$	−1764.85	−0.105521
$655$	−3429.79	−0.204600
$656$	622.783	0.0370665
$657$	−4717.11	−0.280110
$658$	2836.49	0.168051
$659$	14065.8	0.831448	0.415724	−	0.909491i	$-0.363528\pi$
$659$	14065.8	0.831448	0.415724	+	0.909491i	$0.363528\pi$
$660$	4155.60	0.245086
$661$	−13363.3	−0.786342	−0.393171	−	0.919465i	$-0.628622\pi$
$661$	−13363.3	−0.786342	−0.393171	+	0.919465i	$0.628622\pi$
$662$	4214.33	0.247424
$663$	4513.07	0.264364
$664$	−9504.81	−0.555510
$665$	6860.50	0.400058
$666$	1617.25	0.0940948
$667$	−5376.90	−0.312135
$668$	10262.2	0.594395
$669$	983.010	0.0568092
$670$	2155.69	0.124301
$671$	−13981.8	−0.804412
$672$	−5349.52	−0.307087
$673$	−17727.5	−1.01537	−0.507686	−	0.861542i	$-0.669499\pi$
$673$	−17727.5	−1.01537	−0.507686	+	0.861542i	$0.669499\pi$
$674$	4362.28	0.249301
$675$	−675.000	−0.0384900
$676$	12221.4	0.695345
$677$	15776.1	0.895605	0.447803	−	0.894132i	$-0.352207\pi$
$677$	15776.1	0.895605	0.447803	+	0.894132i	$0.352207\pi$
$678$	3841.31	0.217588
$679$	8144.23	0.460305
$680$	3375.94	0.190384
$681$	−10986.4	−0.618206
$682$	2674.23	0.150149
$683$	14482.3	0.811350	0.405675	−	0.914018i	$-0.367037\pi$
$683$	14482.3	0.811350	0.405675	+	0.914018i	$0.367037\pi$
$684$	6374.09	0.356315
$685$	−5764.73	−0.321546
$686$	−4773.97	−0.265701
$687$	19687.7	1.09335
$688$	−3220.22	−0.178445
$689$	4086.61	0.225961
$690$	1923.63	0.106132
$691$	19721.5	1.08573	0.542867	−	0.839819i	$-0.317339\pi$
$691$	19721.5	1.08573	0.542867	+	0.839819i	$0.317339\pi$
$692$	−3240.74	−0.178026
$693$	−4830.53	−0.264786
$694$	−7882.04	−0.431121
$695$	−2192.32	−0.119654
$696$	934.011	0.0508672
$697$	742.555	0.0403533
$698$	−6110.92	−0.331378
$699$	−7470.85	−0.404254
$700$	2740.12	0.147953
$701$	2004.45	0.107998	0.0539992	−	0.998541i	$-0.482803\pi$
$701$	2004.45	0.107998	0.0539992	+	0.998541i	$0.482803\pi$
$702$	−446.704	−0.0240167
$703$	24462.7	1.31241
$704$	−12425.1	−0.665181
$705$	−4221.39	−0.225513
$706$	2345.27	0.125022
$707$	16594.4	0.882738
$708$	−14373.5	−0.762978
$709$	10037.8	0.531705	0.265853	−	0.964014i	$-0.414347\pi$
$709$	10037.8	0.531705	0.265853	+	0.964014i	$0.414347\pi$
$710$	−1033.69	−0.0546389
$711$	5069.38	0.267393
$712$	−4805.88	−0.252961
$713$	−19462.8	−1.02228
$714$	−1901.64	−0.0996740
$715$	−4405.16	−0.230411
$716$	24013.7	1.25340
$717$	10753.2	0.560091
$718$	3569.18	0.185516
$719$	10367.8	0.537768	0.268884	−	0.963173i	$-0.413345\pi$
$719$	10367.8	0.537768	0.268884	+	0.963173i	$0.413345\pi$
$720$	2373.63	0.122861
$721$	−14693.2	−0.758949
$722$	−1388.23	−0.0715576
$723$	19631.8	1.00984
$724$	4672.11	0.239831
$725$	−725.000	−0.0371391
$726$	53.2044	0.00271984
$727$	−32648.0	−1.66554	−0.832770	−	0.553619i	$-0.813246\pi$
$727$	−32648.0	−1.66554	−0.832770	+	0.553619i	$0.813246\pi$
$728$	3742.07	0.190509
$729$	729.000	0.0370370
$730$	1812.59	0.0919001
$731$	−3839.53	−0.194268
$732$	−8565.66	−0.432508
$733$	21036.1	1.06001	0.530004	−	0.847995i	$-0.322191\pi$
$733$	21036.1	1.06001	0.530004	+	0.847995i	$0.322191\pi$
$734$	−767.863	−0.0386136
$735$	1959.84	0.0983534
$736$	−22688.6	−1.13629
$737$	−22959.0	−1.14750
$738$	−73.4981	−0.00366599
$739$	24591.3	1.22410	0.612048	−	0.790820i	$-0.290346\pi$
$739$	24591.3	1.22410	0.612048	+	0.790820i	$0.290346\pi$
$740$	9770.54	0.485368
$741$	−6756.88	−0.334980
$742$	−1721.95	−0.0851950
$743$	3352.77	0.165547	0.0827734	−	0.996568i	$-0.473622\pi$
$743$	3352.77	0.165547	0.0827734	+	0.996568i	$0.473622\pi$
$744$	3380.84	0.166596
$745$	−2027.85	−0.0997244
$746$	−4012.50	−0.196928
$747$	−7968.08	−0.390277
$748$	−17423.4	−0.851690
$749$	−2047.46	−0.0998831
$750$	259.375	0.0126280
$751$	10898.0	0.529525	0.264762	−	0.964314i	$-0.414706\pi$
$751$	10898.0	0.529525	0.264762	+	0.964314i	$0.414706\pi$
$752$	14844.5	0.719843
$753$	−1584.30	−0.0766734
$754$	−479.793	−0.0231738
$755$	−394.779	−0.0190298
$756$	−2959.33	−0.142368
$757$	−34910.6	−1.67615	−0.838075	−	0.545555i	$-0.816319\pi$
$757$	−34910.6	−1.67615	−0.838075	+	0.545555i	$0.816319\pi$
$758$	8759.58	0.419739
$759$	−20487.4	−0.979771
$760$	−5054.39	−0.241239
$761$	−17093.2	−0.814229	−0.407115	−	0.913377i	$-0.633465\pi$
$761$	−17093.2	−0.814229	−0.407115	+	0.913377i	$0.633465\pi$
$762$	1873.98	0.0890907
$763$	12393.9	0.588062
$764$	−22138.8	−1.04837
$765$	2830.11	0.133756
$766$	9274.05	0.437448
$767$	15236.7	0.717293
$768$	−5580.65	−0.262206
$769$	29371.4	1.37732	0.688660	−	0.725084i	$-0.258199\pi$
$769$	29371.4	1.37732	0.688660	+	0.725084i	$0.258199\pi$
$770$	1856.17	0.0868726
$771$	−15062.3	−0.703575
$772$	−24518.9	−1.14307
$773$	−25646.0	−1.19330	−0.596651	−	0.802501i	$-0.703502\pi$
$773$	−25646.0	−1.19330	−0.596651	+	0.802501i	$0.703502\pi$
$774$	380.036	0.0176487
$775$	−2624.28	−0.121635
$776$	−6000.16	−0.277568
$777$	−11357.4	−0.524382
$778$	3234.62	0.149057
$779$	−1111.74	−0.0511325
$780$	−2698.74	−0.123885
$781$	11009.2	0.504405
$782$	−8065.32	−0.368817
$783$	783.000	0.0357371
$784$	−6891.74	−0.313946
$785$	1574.35	0.0715806
$786$	−1423.36	−0.0645923
$787$	18188.7	0.823834	0.411917	−	0.911221i	$-0.364859\pi$
$787$	18188.7	0.823834	0.411917	+	0.911221i	$0.364859\pi$
$788$	−17843.8	−0.806676
$789$	1266.64	0.0571527
$790$	−1947.95	−0.0877279
$791$	−26976.2	−1.21260
$792$	3558.83	0.159669
$793$	9080.07	0.406611
$794$	4090.31	0.182821
$795$	2562.68	0.114326
$796$	13755.1	0.612485
$797$	12068.7	0.536379	0.268190	−	0.963366i	$-0.413575\pi$
$797$	12068.7	0.536379	0.268190	+	0.963366i	$0.413575\pi$
$798$	2847.11	0.126299
$799$	17699.3	0.783675
$800$	−3059.24	−0.135201
$801$	−4028.87	−0.177719
$802$	−5365.11	−0.236220
$803$	−19304.8	−0.848385
$804$	−14065.4	−0.616974
$805$	−13509.0	−0.591466
$806$	−1736.71	−0.0758968
$807$	5331.57	0.232565
$808$	−12225.7	−0.532300
$809$	−18274.6	−0.794191	−0.397096	−	0.917777i	$-0.629982\pi$
$809$	−18274.6	−0.794191	−0.397096	+	0.917777i	$0.629982\pi$
$810$	−280.125	−0.0121513
$811$	−30620.5	−1.32581	−0.662903	−	0.748705i	$-0.730676\pi$
$811$	−30620.5	−1.32581	−0.662903	+	0.748705i	$0.730676\pi$
$812$	−3178.54	−0.137371
$813$	4396.53	0.189659
$814$	6618.61	0.284990
$815$	4084.06	0.175532
$816$	−9952.05	−0.426950
$817$	5748.47	0.246161
$818$	−5308.10	−0.226887
$819$	3137.05	0.133843
$820$	−444.035	−0.0189102
$821$	−10076.0	−0.428326	−0.214163	−	0.976798i	$-0.568702\pi$
$821$	−10076.0	−0.428326	−0.214163	+	0.976798i	$0.568702\pi$
$822$	−2392.36	−0.101512
$823$	37593.9	1.59227	0.796137	−	0.605116i	$-0.206873\pi$
$823$	37593.9	1.59227	0.796137	+	0.605116i	$0.206873\pi$
$824$	10825.0	0.457654
$825$	−2762.45	−0.116577
$826$	−6420.18	−0.270444
$827$	34830.2	1.46453	0.732264	−	0.681021i	$-0.238464\pi$
$827$	34830.2	1.46453	0.732264	+	0.681021i	$0.238464\pi$
$828$	−12551.2	−0.526794
$829$	−19925.3	−0.834784	−0.417392	−	0.908727i	$-0.637056\pi$
$829$	−19925.3	−0.834784	−0.417392	+	0.908727i	$0.637056\pi$
$830$	3061.80	0.128044
$831$	18244.5	0.761606
$832$	8069.12	0.336233
$833$	−8217.14	−0.341785
$834$	−909.814	−0.0377749
$835$	−6821.81	−0.282729
$836$	26086.1	1.07919
$837$	2834.22	0.117043
$838$	−7827.00	−0.322648
$839$	26090.0	1.07357	0.536786	−	0.843718i	$-0.319638\pi$
$839$	26090.0	1.07357	0.536786	+	0.843718i	$0.319638\pi$
$840$	2346.63	0.0963885
$841$	841.000	0.0344828
$842$	11039.5	0.451838
$843$	3583.51	0.146409
$844$	−31767.1	−1.29558
$845$	−8124.19	−0.330746
$846$	−1751.88	−0.0711947
$847$	−373.637	−0.0151574
$848$	−9011.63	−0.364930
$849$	−11482.6	−0.464171
$850$	−1087.50	−0.0438833
$851$	−48169.5	−1.94034
$852$	6744.57	0.271203
$853$	−37626.4	−1.51032	−0.755160	−	0.655540i	$-0.772441\pi$
$853$	−37626.4	−1.51032	−0.755160	+	0.655540i	$0.772441\pi$
$854$	−3826.01	−0.153306
$855$	−4237.19	−0.169484
$856$	1508.44	0.0602306
$857$	25833.1	1.02969	0.514844	−	0.857284i	$-0.327850\pi$
$857$	25833.1	1.02969	0.514844	+	0.857284i	$0.327850\pi$
$858$	−1828.14	−0.0727409
$859$	49111.2	1.95070	0.975351	−	0.220661i	$-0.0708215\pi$
$859$	49111.2	1.95070	0.975351	+	0.220661i	$0.0708215\pi$
$860$	2295.97	0.0910372
$861$	516.153	0.0204303
$862$	−6350.76	−0.250937
$863$	14549.3	0.573886	0.286943	−	0.957948i	$-0.407361\pi$
$863$	14549.3	0.573886	0.286943	+	0.957948i	$0.407361\pi$
$864$	3303.98	0.130097
$865$	2154.29	0.0846797
$866$	−216.368	−0.00849017
$867$	2873.00	0.112540
$868$	−11505.4	−0.449905
$869$	20746.5	0.809869
$870$	−300.875	−0.0117248
$871$	14910.1	0.580032
$872$	−9131.08	−0.354607
$873$	−5030.05	−0.195007
$874$	12075.2	0.467335
$875$	−1821.50	−0.0703749
$876$	−11826.7	−0.456151
$877$	501.781	0.0193203	0.00966017	−	0.999953i	$-0.496925\pi$
$877$	501.781	0.0193203	0.00966017	+	0.999953i	$0.496925\pi$
$878$	5545.88	0.213171
$879$	20463.0	0.785210
$880$	9714.09	0.372116
$881$	1569.47	0.0600189	0.0300095	−	0.999550i	$-0.490446\pi$
$881$	1569.47	0.0600189	0.0300095	+	0.999550i	$0.490446\pi$
$882$	813.332	0.0310502
$883$	−26159.0	−0.996967	−0.498483	−	0.866899i	$-0.666110\pi$
$883$	−26159.0	−0.996967	−0.498483	+	0.866899i	$0.666110\pi$
$884$	11315.2	0.430509
$885$	9554.80	0.362917
$886$	−382.823	−0.0145160
$887$	−29723.0	−1.12514	−0.562571	−	0.826749i	$-0.690188\pi$
$887$	−29723.0	−1.12514	−0.562571	+	0.826749i	$0.690188\pi$
$888$	8367.43	0.316208
$889$	−13160.4	−0.496495
$890$	1548.13	0.0583071
$891$	2983.44	0.112176
$892$	2464.60	0.0925123
$893$	−26499.1	−0.993009
$894$	−841.557	−0.0314831
$895$	−15963.2	−0.596190
$896$	−17665.4	−0.658661
$897$	13305.0	0.495251
$898$	8338.16	0.309853
$899$	3044.17	0.112935
$900$	−1692.36	−0.0626800
$901$	−10744.7	−0.397290
$902$	−300.792	−0.0111034
$903$	−2668.87	−0.0983549
$904$	19874.4	0.731209
$905$	−3105.80	−0.114078
$906$	−163.833	−0.00600772
$907$	−9772.21	−0.357752	−0.178876	−	0.983872i	$-0.557246\pi$
$907$	−9772.21	−0.357752	−0.178876	+	0.983872i	$0.557246\pi$
$908$	−27545.0	−1.00673
$909$	−10249.0	−0.373971
$910$	−1205.44	−0.0439120
$911$	−49823.7	−1.81200	−0.906001	−	0.423275i	$-0.860880\pi$
$911$	−49823.7	−1.81200	−0.906001	+	0.423275i	$0.860880\pi$
$912$	14900.0	0.540997
$913$	−32609.4	−1.18205
$914$	185.547	0.00671483
$915$	5694.05	0.205726
$916$	49361.0	1.78050
$917$	9995.80	0.359968
$918$	1174.50	0.0422267
$919$	28235.9	1.01351	0.506756	−	0.862090i	$-0.330845\pi$
$919$	28235.9	1.01351	0.506756	+	0.862090i	$0.330845\pi$
$920$	9952.60	0.356660
$921$	22052.2	0.788974
$922$	−2504.86	−0.0894720
$923$	−7149.61	−0.254965
$924$	−12111.1	−0.431197
$925$	−6494.99	−0.230869
$926$	5361.45	0.190268
$927$	9074.82	0.321528
$928$	3548.72	0.125531
$929$	−22846.1	−0.806842	−0.403421	−	0.915014i	$-0.632179\pi$
$929$	−22846.1	−0.806842	−0.403421	+	0.915014i	$0.632179\pi$
$930$	−1089.08	−0.0384002
$931$	12302.5	0.433082
$932$	−18730.9	−0.658317
$933$	−10179.4	−0.357191
$934$	−6113.56	−0.214178
$935$	11582.3	0.405113
$936$	−2311.18	−0.0807088
$937$	−45416.9	−1.58346	−0.791732	−	0.610868i	$-0.790820\pi$
$937$	−45416.9	−1.58346	−0.791732	+	0.610868i	$0.790820\pi$
$938$	−6282.55	−0.218692
$939$	32196.9	1.11896
$940$	−10583.9	−0.367243
$941$	−39333.3	−1.36262	−0.681311	−	0.731994i	$-0.738590\pi$
$941$	−39333.3	−1.36262	−0.681311	+	0.731994i	$0.738590\pi$
$942$	653.352	0.0225981
$943$	2189.13	0.0755968
$944$	−33599.3	−1.15844
$945$	1967.22	0.0677183
$946$	1555.30	0.0534538
$947$	31274.8	1.07317	0.536587	−	0.843845i	$-0.319713\pi$
$947$	31274.8	1.07317	0.536587	+	0.843845i	$0.319713\pi$
$948$	12709.9	0.435443
$949$	12537.0	0.428839
$950$	1628.18	0.0556054
$951$	26163.5	0.892124
$952$	−9838.85	−0.334957
$953$	−34174.1	−1.16160	−0.580802	−	0.814045i	$-0.697261\pi$
$953$	−34174.1	−1.16160	−0.580802	+	0.814045i	$0.697261\pi$
$954$	1063.51	0.0360928
$955$	14716.8	0.498666
$956$	26960.4	0.912093
$957$	3204.44	0.108239
$958$	6850.27	0.231025
$959$	16800.8	0.565720
$960$	5060.09	0.170118
$961$	−18772.0	−0.630124
$962$	−4298.27	−0.144056
$963$	1264.55	0.0423153
$964$	49220.9	1.64450
$965$	16299.0	0.543712
$966$	−5606.24	−0.186726
$967$	37894.7	1.26020	0.630098	−	0.776515i	$-0.283015\pi$
$967$	37894.7	1.26020	0.630098	+	0.776515i	$0.283015\pi$
$968$	275.273	0.00914008
$969$	17765.5	0.588970
$970$	1932.84	0.0639792
$971$	−39869.1	−1.31767	−0.658837	−	0.752286i	$-0.728951\pi$
$971$	−39869.1	−1.31767	−0.658837	+	0.752286i	$0.728951\pi$
$972$	1827.75	0.0603138
$973$	6389.33	0.210516
$974$	−552.926	−0.0181898
$975$	1793.99	0.0589269
$976$	−20023.0	−0.656681
$977$	−34257.3	−1.12179	−0.560894	−	0.827887i	$-0.689543\pi$
$977$	−34257.3	−1.12179	−0.560894	+	0.827887i	$0.689543\pi$
$978$	1694.88	0.0554155
$979$	−16488.2	−0.538268
$980$	4913.71	0.160166
$981$	−7654.77	−0.249131
$982$	6925.83	0.225063
$983$	8665.65	0.281171	0.140586	−	0.990069i	$-0.455101\pi$
$983$	8665.65	0.281171	0.140586	+	0.990069i	$0.455101\pi$
$984$	−380.269	−0.0123197
$985$	11861.7	0.383702
$986$	1261.50	0.0407446
$987$	12302.9	0.396762
$988$	−16940.9	−0.545507
$989$	−11319.3	−0.363937
$990$	−1146.41	−0.0368034
$991$	22129.9	0.709362	0.354681	−	0.934987i	$-0.384589\pi$
$991$	22129.9	0.709362	0.354681	+	0.934987i	$0.384589\pi$
$992$	12845.3	0.411127
$993$	18279.0	0.584157
$994$	3012.59	0.0961302
$995$	−9143.76	−0.291333
$996$	−19977.6	−0.635555
$997$	−14159.7	−0.449791	−0.224895	−	0.974383i	$-0.572204\pi$
$997$	−14159.7	−0.449791	−0.224895	+	0.974383i	$0.572204\pi$
$998$	−1768.14	−0.0560817
$999$	7014.59	0.222154

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	435.4.a.g.1.3	✓	6
3.2	odd	2		1305.4.a.i.1.4		6
5.4	even	2		2175.4.a.l.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.4.a.g.1.3	✓	6	1.1	even	1	trivial
1305.4.a.i.1.4		6	3.2	odd	2
2175.4.a.l.1.4		6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 435 = 3 \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	435.a (trivial)

\(f(q)\)	\(=\)	\(q-0.691666 q^{2} -3.00000 q^{3} -7.52160 q^{4} +5.00000 q^{5} +2.07500 q^{6} -14.5720 q^{7} +10.7358 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-0.691666 q^{2} -3.00000 q^{3} -7.52160 q^{4} +5.00000 q^{5} +2.07500 q^{6} -14.5720 q^{7} +10.7358 q^{8} +9.00000 q^{9} -3.45833 q^{10} +36.8326 q^{11} +22.5648 q^{12} -23.9199 q^{13} +10.0790 q^{14} -15.0000 q^{15} +52.7472 q^{16} +62.8914 q^{17} -6.22499 q^{18} -94.1599 q^{19} -37.6080 q^{20} +43.7161 q^{21} -25.4759 q^{22} +185.410 q^{23} -32.2073 q^{24} +25.0000 q^{25} +16.5446 q^{26} -27.0000 q^{27} +109.605 q^{28} -29.0000 q^{29} +10.3750 q^{30} -104.971 q^{31} -122.370 q^{32} -110.498 q^{33} -43.4999 q^{34} -72.8602 q^{35} -67.6944 q^{36} -259.799 q^{37} +65.1272 q^{38} +71.7597 q^{39} +53.6788 q^{40} +11.8069 q^{41} -30.2369 q^{42} -61.0501 q^{43} -277.040 q^{44} +45.0000 q^{45} -128.242 q^{46} +281.426 q^{47} -158.242 q^{48} -130.656 q^{49} -17.2916 q^{50} -188.674 q^{51} +179.916 q^{52} -170.846 q^{53} +18.6750 q^{54} +184.163 q^{55} -156.442 q^{56} +282.480 q^{57} +20.0583 q^{58} -636.987 q^{59} +112.824 q^{60} -379.603 q^{61} +72.6051 q^{62} -131.148 q^{63} -337.339 q^{64} -119.600 q^{65} +76.4276 q^{66} -623.333 q^{67} -473.044 q^{68} -556.231 q^{69} +50.3949 q^{70} +298.898 q^{71} +96.6218 q^{72} -524.124 q^{73} +179.694 q^{74} -75.0000 q^{75} +708.233 q^{76} -536.726 q^{77} -49.6337 q^{78} +563.264 q^{79} +263.736 q^{80} +81.0000 q^{81} -8.16645 q^{82} -885.342 q^{83} -328.815 q^{84} +314.457 q^{85} +42.2263 q^{86} +87.0000 q^{87} +395.426 q^{88} -447.652 q^{89} -31.1250 q^{90} +348.562 q^{91} -1394.58 q^{92} +314.914 q^{93} -194.653 q^{94} -470.799 q^{95} +367.109 q^{96} -558.895 q^{97} +90.3702 q^{98} +331.493 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} - 18 q^{3} + 15 q^{4} + 30 q^{5} + 3 q^{6} + 23 q^{7} - 51 q^{8} + 54 q^{9}+O(q^{10}) \) 6 * q - q^2 - 18 * q^3 + 15 * q^4 + 30 * q^5 + 3 * q^6 + 23 * q^7 - 51 * q^8 + 54 * q^9	\( 6 q - q^{2} - 18 q^{3} + 15 q^{4} + 30 q^{5} + 3 q^{6} + 23 q^{7} - 51 q^{8} + 54 q^{9} - 5 q^{10} - 111 q^{11} - 45 q^{12} - 83 q^{13} - 102 q^{14} - 90 q^{15} - 37 q^{16} - 35 q^{17} - 9 q^{18} - 76 q^{19} + 75 q^{20} - 69 q^{21} + 66 q^{22} + 166 q^{23} + 153 q^{24} + 150 q^{25} - 282 q^{26} - 162 q^{27} + 368 q^{28} - 174 q^{29} + 15 q^{30} - 164 q^{31} - 259 q^{32} + 333 q^{33} + 512 q^{34} + 115 q^{35} + 135 q^{36} - 538 q^{37} + 6 q^{38} + 249 q^{39} - 255 q^{40} - 1250 q^{41} + 306 q^{42} - 80 q^{43} - 1004 q^{44} + 270 q^{45} - 1190 q^{46} - 491 q^{47} + 111 q^{48} - 969 q^{49} - 25 q^{50} + 105 q^{51} - 336 q^{52} - 356 q^{53} + 27 q^{54} - 555 q^{55} - 1260 q^{56} + 228 q^{57} + 29 q^{58} - 646 q^{59} - 225 q^{60} - 1476 q^{61} - 4 q^{62} + 207 q^{63} - 845 q^{64} - 415 q^{65} - 198 q^{66} + 87 q^{67} - 558 q^{68} - 498 q^{69} - 510 q^{70} - 1312 q^{71} - 459 q^{72} - 1142 q^{73} - 2270 q^{74} - 450 q^{75} - 1594 q^{76} - 2561 q^{77} + 846 q^{78} - 142 q^{79} - 185 q^{80} + 486 q^{81} - 1544 q^{82} + 346 q^{83} - 1104 q^{84} - 175 q^{85} - 2056 q^{86} + 522 q^{87} + 1400 q^{88} - 2827 q^{89} - 45 q^{90} + 1379 q^{91} - 1434 q^{92} + 492 q^{93} + 978 q^{94} - 380 q^{95} + 777 q^{96} - 4 q^{97} - 1903 q^{98} - 999 q^{99}+O(q^{100}) \) 6 * q - q^2 - 18 * q^3 + 15 * q^4 + 30 * q^5 + 3 * q^6 + 23 * q^7 - 51 * q^8 + 54 * q^9 - 5 * q^10 - 111 * q^11 - 45 * q^12 - 83 * q^13 - 102 * q^14 - 90 * q^15 - 37 * q^16 - 35 * q^17 - 9 * q^18 - 76 * q^19 + 75 * q^20 - 69 * q^21 + 66 * q^22 + 166 * q^23 + 153 * q^24 + 150 * q^25 - 282 * q^26 - 162 * q^27 + 368 * q^28 - 174 * q^29 + 15 * q^30 - 164 * q^31 - 259 * q^32 + 333 * q^33 + 512 * q^34 + 115 * q^35 + 135 * q^36 - 538 * q^37 + 6 * q^38 + 249 * q^39 - 255 * q^40 - 1250 * q^41 + 306 * q^42 - 80 * q^43 - 1004 * q^44 + 270 * q^45 - 1190 * q^46 - 491 * q^47 + 111 * q^48 - 969 * q^49 - 25 * q^50 + 105 * q^51 - 336 * q^52 - 356 * q^53 + 27 * q^54 - 555 * q^55 - 1260 * q^56 + 228 * q^57 + 29 * q^58 - 646 * q^59 - 225 * q^60 - 1476 * q^61 - 4 * q^62 + 207 * q^63 - 845 * q^64 - 415 * q^65 - 198 * q^66 + 87 * q^67 - 558 * q^68 - 498 * q^69 - 510 * q^70 - 1312 * q^71 - 459 * q^72 - 1142 * q^73 - 2270 * q^74 - 450 * q^75 - 1594 * q^76 - 2561 * q^77 + 846 * q^78 - 142 * q^79 - 185 * q^80 + 486 * q^81 - 1544 * q^82 + 346 * q^83 - 1104 * q^84 - 175 * q^85 - 2056 * q^86 + 522 * q^87 + 1400 * q^88 - 2827 * q^89 - 45 * q^90 + 1379 * q^91 - 1434 * q^92 + 492 * q^93 + 978 * q^94 - 380 * q^95 + 777 * q^96 - 4 * q^97 - 1903 * q^98 - 999 * q^99

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