LMFDB - Embedded newform 165.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [165,4,Mod(1,165)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(165, 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("165.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 165 = 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	165.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:	$9.73531515095$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.23612.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 20x + 26 $ x^3 - x^2 - 20*x + 26
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.32906$ of defining polynomial
Character	$\chi$	$=$	165.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.32906 q^{2} -3.00000 q^{3} -2.57547 q^{4} -5.00000 q^{5} +6.98719 q^{6} +22.4672 q^{7} +24.6309 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.32906 q^{2} -3.00000 q^{3} -2.57547 q^{4} -5.00000 q^{5} +6.98719 q^{6} +22.4672 q^{7} +24.6309 q^{8} +9.00000 q^{9} +11.6453 q^{10} +11.0000 q^{11} +7.72640 q^{12} -9.86030 q^{13} -52.3275 q^{14} +15.0000 q^{15} -36.7633 q^{16} -128.137 q^{17} -20.9616 q^{18} +7.04001 q^{19} +12.8773 q^{20} -67.4015 q^{21} -25.6197 q^{22} +0.654969 q^{23} -73.8928 q^{24} +25.0000 q^{25} +22.9653 q^{26} -27.0000 q^{27} -57.8635 q^{28} -229.279 q^{29} -34.9359 q^{30} +155.789 q^{31} -111.423 q^{32} -33.0000 q^{33} +298.438 q^{34} -112.336 q^{35} -23.1792 q^{36} -110.279 q^{37} -16.3966 q^{38} +29.5809 q^{39} -123.155 q^{40} +154.749 q^{41} +156.982 q^{42} -401.014 q^{43} -28.3301 q^{44} -45.0000 q^{45} -1.52546 q^{46} -277.532 q^{47} +110.290 q^{48} +161.774 q^{49} -58.2266 q^{50} +384.410 q^{51} +25.3949 q^{52} -651.566 q^{53} +62.8847 q^{54} -55.0000 q^{55} +553.388 q^{56} -21.1200 q^{57} +534.005 q^{58} -423.869 q^{59} -38.6320 q^{60} +681.851 q^{61} -362.842 q^{62} +202.205 q^{63} +553.618 q^{64} +49.3015 q^{65} +76.8591 q^{66} +374.028 q^{67} +330.011 q^{68} -1.96491 q^{69} +261.637 q^{70} +96.6950 q^{71} +221.678 q^{72} -19.9460 q^{73} +256.848 q^{74} -75.0000 q^{75} -18.1313 q^{76} +247.139 q^{77} -68.8958 q^{78} +24.4286 q^{79} +183.816 q^{80} +81.0000 q^{81} -360.419 q^{82} -1127.35 q^{83} +173.590 q^{84} +640.683 q^{85} +933.987 q^{86} +687.836 q^{87} +270.940 q^{88} -639.624 q^{89} +104.808 q^{90} -221.533 q^{91} -1.68685 q^{92} -467.366 q^{93} +646.389 q^{94} -35.2001 q^{95} +334.270 q^{96} -730.865 q^{97} -376.783 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{2} - 9 q^{3} + 22 q^{4} - 15 q^{5} + 12 q^{6} - 4 q^{7} - 48 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q - 4 * q^2 - 9 * q^3 + 22 * q^4 - 15 * q^5 + 12 * q^6 - 4 * q^7 - 48 * q^8 + 27 * q^9	\( 3 q - 4 q^{2} - 9 q^{3} + 22 q^{4} - 15 q^{5} + 12 q^{6} - 4 q^{7} - 48 q^{8} + 27 q^{9} + 20 q^{10} + 33 q^{11} - 66 q^{12} - 56 q^{14} + 45 q^{15} + 50 q^{16} - 218 q^{17} - 36 q^{18} + 146 q^{19} - 110 q^{20} + 12 q^{21} - 44 q^{22} - 200 q^{23} + 144 q^{24} + 75 q^{25} - 508 q^{26} - 81 q^{27} - 340 q^{28} + 68 q^{29} - 60 q^{30} - 68 q^{31} - 688 q^{32} - 99 q^{33} - 176 q^{34} + 20 q^{35} + 198 q^{36} - 390 q^{37} - 316 q^{38} + 240 q^{40} - 196 q^{41} + 168 q^{42} - 524 q^{43} + 242 q^{44} - 135 q^{45} + 1160 q^{46} - 60 q^{47} - 150 q^{48} - 157 q^{49} - 100 q^{50} + 654 q^{51} + 1020 q^{52} - 158 q^{53} + 108 q^{54} - 165 q^{55} + 1368 q^{56} - 438 q^{57} + 1092 q^{58} - 1044 q^{59} + 330 q^{60} + 642 q^{61} + 88 q^{62} - 36 q^{63} + 1166 q^{64} + 132 q^{66} - 236 q^{67} + 144 q^{68} + 600 q^{69} + 280 q^{70} - 544 q^{71} - 432 q^{72} + 900 q^{73} + 1536 q^{74} - 225 q^{75} + 1996 q^{76} - 44 q^{77} + 1524 q^{78} - 1586 q^{79} - 250 q^{80} + 243 q^{81} + 380 q^{82} - 1582 q^{83} + 1020 q^{84} + 1090 q^{85} + 3568 q^{86} - 204 q^{87} - 528 q^{88} - 2122 q^{89} + 180 q^{90} - 8 q^{91} - 4128 q^{92} + 204 q^{93} - 2152 q^{94} - 730 q^{95} + 2064 q^{96} + 618 q^{97} + 572 q^{98} + 297 q^{99}+O(q^{100}) \) 3 * q - 4 * q^2 - 9 * q^3 + 22 * q^4 - 15 * q^5 + 12 * q^6 - 4 * q^7 - 48 * q^8 + 27 * q^9 + 20 * q^10 + 33 * q^11 - 66 * q^12 - 56 * q^14 + 45 * q^15 + 50 * q^16 - 218 * q^17 - 36 * q^18 + 146 * q^19 - 110 * q^20 + 12 * q^21 - 44 * q^22 - 200 * q^23 + 144 * q^24 + 75 * q^25 - 508 * q^26 - 81 * q^27 - 340 * q^28 + 68 * q^29 - 60 * q^30 - 68 * q^31 - 688 * q^32 - 99 * q^33 - 176 * q^34 + 20 * q^35 + 198 * q^36 - 390 * q^37 - 316 * q^38 + 240 * q^40 - 196 * q^41 + 168 * q^42 - 524 * q^43 + 242 * q^44 - 135 * q^45 + 1160 * q^46 - 60 * q^47 - 150 * q^48 - 157 * q^49 - 100 * q^50 + 654 * q^51 + 1020 * q^52 - 158 * q^53 + 108 * q^54 - 165 * q^55 + 1368 * q^56 - 438 * q^57 + 1092 * q^58 - 1044 * q^59 + 330 * q^60 + 642 * q^61 + 88 * q^62 - 36 * q^63 + 1166 * q^64 + 132 * q^66 - 236 * q^67 + 144 * q^68 + 600 * q^69 + 280 * q^70 - 544 * q^71 - 432 * q^72 + 900 * q^73 + 1536 * q^74 - 225 * q^75 + 1996 * q^76 - 44 * q^77 + 1524 * q^78 - 1586 * q^79 - 250 * q^80 + 243 * q^81 + 380 * q^82 - 1582 * q^83 + 1020 * q^84 + 1090 * q^85 + 3568 * q^86 - 204 * q^87 - 528 * q^88 - 2122 * q^89 + 180 * q^90 - 8 * q^91 - 4128 * q^92 + 204 * q^93 - 2152 * q^94 - 730 * q^95 + 2064 * q^96 + 618 * q^97 + 572 * q^98 + 297 * 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 $

$2$	−2.32906	−0.823448	−0.411724	−	0.911309i	$-0.635073\pi$
$2$	−2.32906	−0.823448	−0.411724	+	0.911309i	$0.635073\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−2.57547	−0.321933
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	6.98719	0.475418
$7$	22.4672	1.21311	0.606557	−	0.795040i	$-0.292550\pi$
$7$	22.4672	1.21311	0.606557	+	0.795040i	$0.292550\pi$
$8$	24.6309	1.08854
$9$	9.00000	0.333333
$10$	11.6453	0.368257
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	7.72640	0.185868
$13$	−9.86030	−0.210366	−0.105183	−	0.994453i	$-0.533543\pi$
$13$	−9.86030	−0.210366	−0.105183	+	0.994453i	$0.533543\pi$
$14$	−52.3275	−0.998936
$15$	15.0000	0.258199
$16$	−36.7633	−0.574426
$17$	−128.137	−1.82810	−0.914049	−	0.405603i	$-0.867062\pi$
$17$	−128.137	−1.82810	−0.914049	+	0.405603i	$0.867062\pi$
$18$	−20.9616	−0.274483
$19$	7.04001	0.0850047	0.0425024	−	0.999096i	$-0.486467\pi$
$19$	7.04001	0.0850047	0.0425024	+	0.999096i	$0.486467\pi$
$20$	12.8773	0.143973
$21$	−67.4015	−0.700392
$22$	−25.6197	−0.248279
$23$	0.654969	0.00593785	0.00296892	−	0.999996i	$-0.499055\pi$
$23$	0.654969	0.00593785	0.00296892	+	0.999996i	$0.499055\pi$
$24$	−73.8928	−0.628471
$25$	25.0000	0.200000
$26$	22.9653	0.173225
$27$	−27.0000	−0.192450
$28$	−57.8635	−0.390542
$29$	−229.279	−1.46814	−0.734069	−	0.679075i	$-0.762381\pi$
$29$	−229.279	−1.46814	−0.734069	+	0.679075i	$0.762381\pi$
$30$	−34.9359	−0.212613
$31$	155.789	0.902596	0.451298	−	0.892373i	$-0.350961\pi$
$31$	155.789	0.902596	0.451298	+	0.892373i	$0.350961\pi$
$32$	−111.423	−0.615534
$33$	−33.0000	−0.174078
$34$	298.438	1.50534
$35$	−112.336	−0.542521
$36$	−23.1792	−0.107311
$37$	−110.279	−0.489995	−0.244998	−	0.969524i	$-0.578787\pi$
$37$	−110.279	−0.489995	−0.244998	+	0.969524i	$0.578787\pi$
$38$	−16.3966	−0.0699970
$39$	29.5809	0.121455
$40$	−123.155	−0.486811
$41$	154.749	0.589456	0.294728	−	0.955581i	$-0.404771\pi$
$41$	154.749	0.589456	0.294728	+	0.955581i	$0.404771\pi$
$42$	156.982	0.576736
$43$	−401.014	−1.42219	−0.711094	−	0.703097i	$-0.751800\pi$
$43$	−401.014	−1.42219	−0.711094	+	0.703097i	$0.751800\pi$
$44$	−28.3301	−0.0970665
$45$	−45.0000	−0.149071
$46$	−1.52546	−0.00488951
$47$	−277.532	−0.861323	−0.430661	−	0.902514i	$-0.641720\pi$
$47$	−277.532	−0.861323	−0.430661	+	0.902514i	$0.641720\pi$
$48$	110.290	0.331645
$49$	161.774	0.471645
$50$	−58.2266	−0.164690
$51$	384.410	1.05545
$52$	25.3949	0.0677237
$53$	−651.566	−1.68867	−0.844334	−	0.535817i	$-0.820004\pi$
$53$	−651.566	−1.68867	−0.844334	+	0.535817i	$0.820004\pi$
$54$	62.8847	0.158473
$55$	−55.0000	−0.134840
$56$	553.388	1.32053
$57$	−21.1200	−0.0490775
$58$	534.005	1.20894
$59$	−423.869	−0.935307	−0.467653	−	0.883912i	$-0.654900\pi$
$59$	−423.869	−0.935307	−0.467653	+	0.883912i	$0.654900\pi$
$60$	−38.6320	−0.0831228
$61$	681.851	1.43118	0.715590	−	0.698520i	$-0.246158\pi$
$61$	681.851	1.43118	0.715590	+	0.698520i	$0.246158\pi$
$62$	−362.842	−0.743241
$63$	202.205	0.404371
$64$	553.618	1.08129
$65$	49.3015	0.0940785
$66$	76.8591	0.143344
$67$	374.028	0.682012	0.341006	−	0.940061i	$-0.389232\pi$
$67$	374.028	0.682012	0.341006	+	0.940061i	$0.389232\pi$
$68$	330.011	0.588526
$69$	−1.96491	−0.00342822
$70$	261.637	0.446738
$71$	96.6950	0.161628	0.0808140	−	0.996729i	$-0.474248\pi$
$71$	96.6950	0.161628	0.0808140	+	0.996729i	$0.474248\pi$
$72$	221.678	0.362848
$73$	−19.9460	−0.0319795	−0.0159897	−	0.999872i	$-0.505090\pi$
$73$	−19.9460	−0.0319795	−0.0159897	+	0.999872i	$0.505090\pi$
$74$	256.848	0.403486
$75$	−75.0000	−0.115470
$76$	−18.1313	−0.0273658
$77$	247.139	0.365768
$78$	−68.8958	−0.100012
$79$	24.4286	0.0347903	0.0173951	−	0.999849i	$-0.494463\pi$
$79$	24.4286	0.0347903	0.0173951	+	0.999849i	$0.494463\pi$
$80$	183.816	0.256891
$81$	81.0000	0.111111
$82$	−360.419	−0.485386
$83$	−1127.35	−1.49088	−0.745439	−	0.666574i	$-0.767760\pi$
$83$	−1127.35	−1.49088	−0.745439	+	0.666574i	$0.767760\pi$
$84$	173.590	0.225479
$85$	640.683	0.817551
$86$	933.987	1.17110
$87$	687.836	0.847630
$88$	270.940	0.328208
$89$	−639.624	−0.761798	−0.380899	−	0.924617i	$-0.624385\pi$
$89$	−639.624	−0.761798	−0.380899	+	0.924617i	$0.624385\pi$
$90$	104.808	0.122752
$91$	−221.533	−0.255198
$92$	−1.68685	−0.00191159
$93$	−467.366	−0.521114
$94$	646.389	0.709255
$95$	−35.2001	−0.0380153
$96$	334.270	0.355378
$97$	−730.865	−0.765032	−0.382516	−	0.923949i	$-0.624942\pi$
$97$	−730.865	−0.765032	−0.382516	+	0.923949i	$0.624942\pi$
$98$	−376.783	−0.388375
$99$	99.0000	0.100504
$100$	−64.3866	−0.0643866
$101$	−810.342	−0.798337	−0.399168	−	0.916878i	$-0.630701\pi$
$101$	−810.342	−0.798337	−0.399168	+	0.916878i	$0.630701\pi$
$102$	−895.314	−0.869111
$103$	−1461.89	−1.39849	−0.699245	−	0.714882i	$-0.746480\pi$
$103$	−1461.89	−1.39849	−0.699245	+	0.714882i	$0.746480\pi$
$104$	−242.868	−0.228992
$105$	337.008	0.313225
$106$	1517.54	1.39053
$107$	1690.40	1.52726	0.763630	−	0.645654i	$-0.223415\pi$
$107$	1690.40	1.52726	0.763630	+	0.645654i	$0.223415\pi$
$108$	69.5376	0.0619561
$109$	−1409.41	−1.23851	−0.619254	−	0.785190i	$-0.712565\pi$
$109$	−1409.41	−1.23851	−0.619254	+	0.785190i	$0.712565\pi$
$110$	128.098	0.111034
$111$	330.838	0.282899
$112$	−825.967	−0.696844
$113$	2185.67	1.81956	0.909780	−	0.415090i	$-0.136250\pi$
$113$	2185.67	1.81956	0.909780	+	0.415090i	$0.136250\pi$
$114$	49.1899	0.0404128
$115$	−3.27485	−0.00265549
$116$	590.499	0.472642
$117$	−88.7427	−0.0701219
$118$	987.219	0.770177
$119$	−2878.87	−2.21769
$120$	369.464	0.281061
$121$	121.000	0.0909091
$122$	−1588.07	−1.17850
$123$	−464.246	−0.340322
$124$	−401.228	−0.290576
$125$	−125.000	−0.0894427
$126$	−470.947	−0.332979
$127$	−1918.85	−1.34071	−0.670357	−	0.742038i	$-0.733859\pi$
$127$	−1918.85	−1.34071	−0.670357	+	0.742038i	$0.733859\pi$
$128$	−398.024	−0.274849
$129$	1203.04	0.821100
$130$	−114.826	−0.0774687
$131$	1339.41	0.893320	0.446660	−	0.894704i	$-0.352613\pi$
$131$	1339.41	0.893320	0.446660	+	0.894704i	$0.352613\pi$
$132$	84.9904	0.0560414
$133$	158.169	0.103120
$134$	−871.135	−0.561602
$135$	135.000	0.0860663
$136$	−3156.12	−1.98996
$137$	−1100.56	−0.686330	−0.343165	−	0.939275i	$-0.611499\pi$
$137$	−1100.56	−0.686330	−0.343165	+	0.939275i	$0.611499\pi$
$138$	4.57639	0.00282296
$139$	−1284.51	−0.783819	−0.391910	−	0.920004i	$-0.628185\pi$
$139$	−1284.51	−0.783819	−0.391910	+	0.920004i	$0.628185\pi$
$140$	289.317	0.174656
$141$	832.595	0.497285
$142$	−225.209	−0.133092
$143$	−108.463	−0.0634277
$144$	−330.869	−0.191475
$145$	1146.39	0.656571
$146$	46.4554	0.0263334
$147$	−485.323	−0.272305
$148$	284.021	0.157746
$149$	1277.21	0.702236	0.351118	−	0.936331i	$-0.385802\pi$
$149$	1277.21	0.702236	0.351118	+	0.936331i	$0.385802\pi$
$150$	174.680	0.0950836
$151$	886.317	0.477665	0.238833	−	0.971061i	$-0.423235\pi$
$151$	886.317	0.477665	0.238833	+	0.971061i	$0.423235\pi$
$152$	173.402	0.0925313
$153$	−1153.23	−0.609366
$154$	−575.602	−0.301191
$155$	−778.944	−0.403653
$156$	−76.1846	−0.0391003
$157$	−1681.12	−0.854575	−0.427288	−	0.904116i	$-0.640531\pi$
$157$	−1681.12	−0.854575	−0.427288	+	0.904116i	$0.640531\pi$
$158$	−56.8958	−0.0286480
$159$	1954.70	0.974953
$160$	557.117	0.275275
$161$	14.7153	0.00720329
$162$	−188.654	−0.0914942
$163$	−622.100	−0.298937	−0.149468	−	0.988767i	$-0.547756\pi$
$163$	−622.100	−0.298937	−0.149468	+	0.988767i	$0.547756\pi$
$164$	−398.550	−0.189765
$165$	165.000	0.0778499
$166$	2625.67	1.22766
$167$	−2611.82	−1.21023	−0.605115	−	0.796138i	$-0.706873\pi$
$167$	−2611.82	−1.21023	−0.605115	+	0.796138i	$0.706873\pi$
$168$	−1660.16	−0.762407
$169$	−2099.77	−0.955746
$170$	−1492.19	−0.673210
$171$	63.3601	0.0283349
$172$	1032.80	0.457849
$173$	2342.97	1.02967	0.514835	−	0.857290i	$-0.327853\pi$
$173$	2342.97	1.02967	0.514835	+	0.857290i	$0.327853\pi$
$174$	−1602.01	−0.697979
$175$	561.680	0.242623
$176$	−404.396	−0.173196
$177$	1271.61	0.540000
$178$	1489.72	0.627301
$179$	1314.75	0.548991	0.274495	−	0.961588i	$-0.411489\pi$
$179$	1314.75	0.548991	0.274495	+	0.961588i	$0.411489\pi$
$180$	115.896	0.0479910
$181$	8.69006	0.00356866	0.00178433	−	0.999998i	$-0.499432\pi$
$181$	8.69006	0.00356866	0.00178433	+	0.999998i	$0.499432\pi$
$182$	515.965	0.210142
$183$	−2045.55	−0.826292
$184$	16.1325	0.00646361
$185$	551.397	0.219133
$186$	1088.53	0.429110
$187$	−1409.50	−0.551192
$188$	714.774	0.277288
$189$	−606.614	−0.233464
$190$	81.9832	0.0313036
$191$	644.102	0.244008	0.122004	−	0.992530i	$-0.461068\pi$
$191$	644.102	0.244008	0.122004	+	0.992530i	$0.461068\pi$
$192$	−1660.85	−0.624281
$193$	3970.76	1.48094	0.740470	−	0.672089i	$-0.234603\pi$
$193$	3970.76	1.48094	0.740470	+	0.672089i	$0.234603\pi$
$194$	1702.23	0.629964
$195$	−147.905	−0.0543162
$196$	−416.644	−0.151838
$197$	3756.34	1.35852	0.679260	−	0.733898i	$-0.262301\pi$
$197$	3756.34	1.35852	0.679260	+	0.733898i	$0.262301\pi$
$198$	−230.577	−0.0827596
$199$	4825.48	1.71894	0.859470	−	0.511186i	$-0.170794\pi$
$199$	4825.48	1.71894	0.859470	+	0.511186i	$0.170794\pi$
$200$	615.773	0.217709
$201$	−1122.08	−0.393760
$202$	1887.34	0.657389
$203$	−5151.25	−1.78102
$204$	−990.034	−0.339785
$205$	−773.743	−0.263613
$206$	3404.84	1.15158
$207$	5.89472	0.00197928
$208$	362.497	0.120840
$209$	77.4401	0.0256299
$210$	−784.912	−0.257924
$211$	−4394.02	−1.43363	−0.716817	−	0.697261i	$-0.754402\pi$
$211$	−4394.02	−1.43363	−0.716817	+	0.697261i	$0.754402\pi$
$212$	1678.08	0.543638
$213$	−290.085	−0.0933160
$214$	−3937.04	−1.25762
$215$	2005.07	0.636022
$216$	−665.035	−0.209490
$217$	3500.13	1.09495
$218$	3282.62	1.01985
$219$	59.8379	0.0184633
$220$	141.651	0.0434095
$221$	1263.46	0.384569
$222$	−770.543	−0.232953
$223$	2189.67	0.657538	0.328769	−	0.944410i	$-0.393366\pi$
$223$	2189.67	0.657538	0.328769	+	0.944410i	$0.393366\pi$
$224$	−2503.37	−0.746712
$225$	225.000	0.0666667
$226$	−5090.56	−1.49831
$227$	−1139.27	−0.333110	−0.166555	−	0.986032i	$-0.553264\pi$
$227$	−1139.27	−0.333110	−0.166555	+	0.986032i	$0.553264\pi$
$228$	54.3939	0.0157997
$229$	3416.10	0.985773	0.492886	−	0.870094i	$-0.335942\pi$
$229$	3416.10	0.985773	0.492886	+	0.870094i	$0.335942\pi$
$230$	7.62732	0.00218666
$231$	−741.417	−0.211176
$232$	−5647.35	−1.59813
$233$	6147.08	1.72836	0.864181	−	0.503181i	$-0.167837\pi$
$233$	6147.08	1.72836	0.864181	+	0.503181i	$0.167837\pi$
$234$	206.687	0.0577418
$235$	1387.66	0.385195
$236$	1091.66	0.301106
$237$	−73.2858	−0.0200862
$238$	6705.06	1.82615
$239$	−2080.03	−0.562954	−0.281477	−	0.959568i	$-0.590824\pi$
$239$	−2080.03	−0.562954	−0.281477	+	0.959568i	$0.590824\pi$
$240$	−551.449	−0.148316
$241$	1846.28	0.493484	0.246742	−	0.969081i	$-0.420640\pi$
$241$	1846.28	0.493484	0.246742	+	0.969081i	$0.420640\pi$
$242$	−281.817	−0.0748589
$243$	−243.000	−0.0641500
$244$	−1756.08	−0.460745
$245$	−808.872	−0.210926
$246$	1081.26	0.280238
$247$	−69.4166	−0.0178821
$248$	3837.22	0.982515
$249$	3382.05	0.860758
$250$	291.133	0.0736514
$251$	−2555.32	−0.642592	−0.321296	−	0.946979i	$-0.604118\pi$
$251$	−2555.32	−0.642592	−0.321296	+	0.946979i	$0.604118\pi$
$252$	−520.771	−0.130181
$253$	7.20466	0.00179033
$254$	4469.13	1.10401
$255$	−1922.05	−0.472013
$256$	−3501.92	−0.854962
$257$	−1819.39	−0.441598	−0.220799	−	0.975319i	$-0.570866\pi$
$257$	−1819.39	−0.441598	−0.220799	+	0.975319i	$0.570866\pi$
$258$	−2801.96	−0.676134
$259$	−2477.67	−0.594420
$260$	−126.974	−0.0302870
$261$	−2063.51	−0.489379
$262$	−3119.57	−0.735603
$263$	−6023.03	−1.41215	−0.706076	−	0.708136i	$-0.749536\pi$
$263$	−6023.03	−1.41215	−0.706076	+	0.708136i	$0.749536\pi$
$264$	−812.821	−0.189491
$265$	3257.83	0.755195
$266$	−368.386	−0.0849143
$267$	1918.87	0.439824
$268$	−963.297	−0.219562
$269$	−2978.38	−0.675075	−0.337537	−	0.941312i	$-0.609594\pi$
$269$	−2978.38	−0.675075	−0.337537	+	0.941312i	$0.609594\pi$
$270$	−314.424	−0.0708711
$271$	−524.969	−0.117674	−0.0588369	−	0.998268i	$-0.518739\pi$
$271$	−524.969	−0.117674	−0.0588369	+	0.998268i	$0.518739\pi$
$272$	4710.72	1.05011
$273$	664.600	0.147338
$274$	2563.28	0.565157
$275$	275.000	0.0603023
$276$	5.06055	0.00110366
$277$	−1693.07	−0.367245	−0.183623	−	0.982997i	$-0.558782\pi$
$277$	−1693.07	−0.367245	−0.183623	+	0.982997i	$0.558782\pi$
$278$	2991.71	0.645435
$279$	1402.10	0.300865
$280$	−2766.94	−0.590558
$281$	7346.60	1.55965	0.779824	−	0.625998i	$-0.215308\pi$
$281$	7346.60	1.55965	0.779824	+	0.625998i	$0.215308\pi$
$282$	−1939.17	−0.409488
$283$	−1501.69	−0.315429	−0.157714	−	0.987485i	$-0.550413\pi$
$283$	−1501.69	−0.315429	−0.157714	+	0.987485i	$0.550413\pi$
$284$	−249.035	−0.0520334
$285$	105.600	0.0219481
$286$	252.618	0.0522294
$287$	3476.77	0.715077
$288$	−1002.81	−0.205178
$289$	11506.0	2.34194
$290$	−2670.02	−0.540652
$291$	2192.59	0.441691
$292$	51.3702	0.0102952
$293$	−4481.03	−0.893462	−0.446731	−	0.894668i	$-0.647412\pi$
$293$	−4481.03	−0.893462	−0.446731	+	0.894668i	$0.647412\pi$
$294$	1130.35	0.224229
$295$	2119.35	0.418282
$296$	−2716.28	−0.533381
$297$	−297.000	−0.0580259
$298$	−2974.71	−0.578255
$299$	−6.45819	−0.00124912
$300$	193.160	0.0371736
$301$	−9009.66	−1.72528
$302$	−2064.29	−0.393333
$303$	2431.02	0.460920
$304$	−258.814	−0.0488289
$305$	−3409.25	−0.640043
$306$	2685.94	0.501781
$307$	3052.17	0.567416	0.283708	−	0.958911i	$-0.408435\pi$
$307$	3052.17	0.567416	0.283708	+	0.958911i	$0.408435\pi$
$308$	−636.498	−0.117753
$309$	4385.67	0.807418
$310$	1814.21	0.332387
$311$	10255.1	1.86983	0.934913	−	0.354878i	$-0.115477\pi$
$311$	10255.1	1.86983	0.934913	+	0.354878i	$0.115477\pi$
$312$	728.605	0.132209
$313$	−6190.18	−1.11786	−0.558929	−	0.829215i	$-0.688788\pi$
$313$	−6190.18	−1.11786	−0.558929	+	0.829215i	$0.688788\pi$
$314$	3915.44	0.703698
$315$	−1011.02	−0.180840
$316$	−62.9150	−0.0112001
$317$	−6735.38	−1.19337	−0.596683	−	0.802477i	$-0.703515\pi$
$317$	−6735.38	−1.19337	−0.596683	+	0.802477i	$0.703515\pi$
$318$	−4552.61	−0.802823
$319$	−2522.07	−0.442660
$320$	−2768.09	−0.483566
$321$	−5071.19	−0.881764
$322$	−34.2729	−0.00593153
$323$	−902.083	−0.155397
$324$	−208.613	−0.0357704
$325$	−246.508	−0.0420732
$326$	1448.91	0.246159
$327$	4228.24	0.715053
$328$	3811.60	0.641648
$329$	−6235.36	−1.04488
$330$	−384.295	−0.0641053
$331$	−4780.83	−0.793891	−0.396946	−	0.917842i	$-0.629930\pi$
$331$	−4780.83	−0.793891	−0.396946	+	0.917842i	$0.629930\pi$
$332$	2903.45	0.479963
$333$	−992.515	−0.163332
$334$	6083.09	0.996562
$335$	−1870.14	−0.305005
$336$	2477.90	0.402323
$337$	11890.3	1.92197	0.960984	−	0.276604i	$-0.0892090\pi$
$337$	11890.3	1.92197	0.960984	+	0.276604i	$0.0892090\pi$
$338$	4890.51	0.787007
$339$	−6557.00	−1.05052
$340$	−1650.06	−0.263197
$341$	1713.68	0.272143
$342$	−147.570	−0.0233323
$343$	−4071.63	−0.640954
$344$	−9877.35	−1.54811
$345$	9.82454	0.00153315
$346$	−5456.93	−0.847879
$347$	8462.47	1.30919	0.654595	−	0.755979i	$-0.272839\pi$
$347$	8462.47	1.30919	0.654595	+	0.755979i	$0.272839\pi$
$348$	−1771.50	−0.272880
$349$	−3291.90	−0.504903	−0.252452	−	0.967610i	$-0.581237\pi$
$349$	−3291.90	−0.504903	−0.252452	+	0.967610i	$0.581237\pi$
$350$	−1308.19	−0.199787
$351$	266.228	0.0404849
$352$	−1225.66	−0.185590
$353$	−8193.52	−1.23540	−0.617701	−	0.786413i	$-0.711936\pi$
$353$	−8193.52	−1.23540	−0.617701	+	0.786413i	$0.711936\pi$
$354$	−2961.66	−0.444662
$355$	−483.475	−0.0722822
$356$	1647.33	0.245248
$357$	8636.60	1.28038
$358$	−3062.15	−0.452066
$359$	12817.6	1.88437	0.942185	−	0.335093i	$-0.108768\pi$
$359$	12817.6	1.88437	0.942185	+	0.335093i	$0.108768\pi$
$360$	−1108.39	−0.162270
$361$	−6809.44	−0.992774
$362$	−20.2397	−0.00293861
$363$	−363.000	−0.0524864
$364$	570.551	0.0821566
$365$	99.7299	0.0143016
$366$	4764.22	0.680409
$367$	−2801.22	−0.398427	−0.199213	−	0.979956i	$-0.563839\pi$
$367$	−2801.22	−0.398427	−0.199213	+	0.979956i	$0.563839\pi$
$368$	−24.0788	−0.00341085
$369$	1392.74	0.196485
$370$	−1284.24	−0.180444
$371$	−14638.8	−2.04855
$372$	1203.69	0.167764
$373$	6838.03	0.949222	0.474611	−	0.880196i	$-0.342589\pi$
$373$	6838.03	0.949222	0.474611	+	0.880196i	$0.342589\pi$
$374$	3282.82	0.453878
$375$	375.000	0.0516398
$376$	−6835.86	−0.937587
$377$	2260.76	0.308846
$378$	1412.84	0.192245
$379$	−7465.79	−1.01185	−0.505926	−	0.862577i	$-0.668849\pi$
$379$	−7465.79	−1.01185	−0.505926	+	0.862577i	$0.668849\pi$
$380$	90.6565	0.0122384
$381$	5756.56	0.774062
$382$	−1500.16	−0.200928
$383$	−8646.55	−1.15357	−0.576786	−	0.816895i	$-0.695693\pi$
$383$	−8646.55	−1.15357	−0.576786	+	0.816895i	$0.695693\pi$
$384$	1194.07	0.158684
$385$	−1235.70	−0.163576
$386$	−9248.15	−1.21948
$387$	−3609.13	−0.474063
$388$	1882.32	0.246289
$389$	4382.78	0.571248	0.285624	−	0.958342i	$-0.407799\pi$
$389$	4382.78	0.571248	0.285624	+	0.958342i	$0.407799\pi$
$390$	344.479	0.0447266
$391$	−83.9255	−0.0108550
$392$	3984.65	0.513406
$393$	−4018.24	−0.515759
$394$	−8748.76	−1.11867
$395$	−122.143	−0.0155587
$396$	−254.971	−0.0323555
$397$	−4432.58	−0.560365	−0.280182	−	0.959947i	$-0.590395\pi$
$397$	−4432.58	−0.560365	−0.280182	+	0.959947i	$0.590395\pi$
$398$	−11238.8	−1.41546
$399$	−474.508	−0.0595366
$400$	−919.081	−0.114885
$401$	−5034.93	−0.627013	−0.313507	−	0.949586i	$-0.601504\pi$
$401$	−5034.93	−0.627013	−0.313507	+	0.949586i	$0.601504\pi$
$402$	2613.41	0.324241
$403$	−1536.12	−0.189875
$404$	2087.01	0.257011
$405$	−405.000	−0.0496904
$406$	11997.6	1.46658
$407$	−1213.07	−0.147739
$408$	9468.36	1.14891
$409$	6474.64	0.782764	0.391382	−	0.920228i	$-0.371997\pi$
$409$	6474.64	0.782764	0.391382	+	0.920228i	$0.371997\pi$
$410$	1802.10	0.217071
$411$	3301.68	0.396253
$412$	3765.05	0.450220
$413$	−9523.15	−1.13463
$414$	−13.7292	−0.00162984
$415$	5636.75	0.666741
$416$	1098.67	0.129487
$417$	3853.54	0.452538
$418$	−180.363	−0.0211049
$419$	−8257.80	−0.962816	−0.481408	−	0.876497i	$-0.659874\pi$
$419$	−8257.80	−0.962816	−0.481408	+	0.876497i	$0.659874\pi$
$420$	−867.952	−0.100837
$421$	−3429.36	−0.397000	−0.198500	−	0.980101i	$-0.563607\pi$
$421$	−3429.36	−0.397000	−0.198500	+	0.980101i	$0.563607\pi$
$422$	10233.9	1.18052
$423$	−2497.79	−0.287108
$424$	−16048.7	−1.83819
$425$	−3203.41	−0.365620
$426$	675.626	0.0768408
$427$	15319.3	1.73618
$428$	−4353.56	−0.491676
$429$	325.390	0.0366200
$430$	−4669.94	−0.523731
$431$	−11260.4	−1.25846	−0.629230	−	0.777219i	$-0.716630\pi$
$431$	−11260.4	−1.25846	−0.629230	+	0.777219i	$0.716630\pi$
$432$	992.608	0.110548
$433$	12598.5	1.39826	0.699128	−	0.714996i	$-0.253572\pi$
$433$	12598.5	1.39826	0.699128	+	0.714996i	$0.253572\pi$
$434$	−8152.03	−0.901636
$435$	−3439.18	−0.379071
$436$	3629.90	0.398717
$437$	4.61099	0.000504745 0
$438$	−139.366	−0.0152036
$439$	4176.90	0.454106	0.227053	−	0.973882i	$-0.427091\pi$
$439$	4176.90	0.454106	0.227053	+	0.973882i	$0.427091\pi$
$440$	−1354.70	−0.146779
$441$	1455.97	0.157215
$442$	−2942.69	−0.316673
$443$	2354.16	0.252482	0.126241	−	0.992000i	$-0.459709\pi$
$443$	2354.16	0.252482	0.126241	+	0.992000i	$0.459709\pi$
$444$	−852.062	−0.0910745
$445$	3198.12	0.340686
$446$	−5099.88	−0.541449
$447$	−3831.63	−0.405436
$448$	12438.2	1.31172
$449$	9286.18	0.976040	0.488020	−	0.872832i	$-0.337719\pi$
$449$	9286.18	0.976040	0.488020	+	0.872832i	$0.337719\pi$
$450$	−524.039	−0.0548965
$451$	1702.24	0.177728
$452$	−5629.11	−0.585777
$453$	−2658.95	−0.275780
$454$	2653.43	0.274299
$455$	1107.67	0.114128
$456$	−520.206	−0.0534230
$457$	14378.4	1.47176	0.735878	−	0.677115i	$-0.236770\pi$
$457$	14378.4	1.47176	0.735878	+	0.677115i	$0.236770\pi$
$458$	−7956.30	−0.811733
$459$	3459.69	0.351818
$460$	8.43425	0.000854889 0
$461$	−5383.19	−0.543861	−0.271931	−	0.962317i	$-0.587662\pi$
$461$	−5383.19	−0.543861	−0.271931	+	0.962317i	$0.587662\pi$
$462$	1726.81	0.173892
$463$	18360.5	1.84294	0.921471	−	0.388446i	$-0.126988\pi$
$463$	18360.5	1.84294	0.921471	+	0.388446i	$0.126988\pi$
$464$	8429.03	0.843336
$465$	2336.83	0.233049
$466$	−14316.9	−1.42322
$467$	9063.65	0.898106	0.449053	−	0.893505i	$-0.351761\pi$
$467$	9063.65	0.898106	0.449053	+	0.893505i	$0.351761\pi$
$468$	228.554	0.0225746
$469$	8403.36	0.827358
$470$	−3231.94	−0.317188
$471$	5043.37	0.493389
$472$	−10440.3	−1.01812
$473$	−4411.15	−0.428806
$474$	170.687	0.0165399
$475$	176.000	0.0170009
$476$	7414.42	0.713949
$477$	−5864.09	−0.562889
$478$	4844.53	0.463564
$479$	12608.2	1.20268	0.601341	−	0.798992i	$-0.294633\pi$
$479$	12608.2	1.20268	0.601341	+	0.798992i	$0.294633\pi$
$480$	−1671.35	−0.158930
$481$	1087.39	0.103078
$482$	−4300.11	−0.406358
$483$	−44.1459	−0.00415882
$484$	−311.631	−0.0292667
$485$	3654.32	0.342133
$486$	565.962	0.0528242
$487$	−13214.2	−1.22955	−0.614775	−	0.788703i	$-0.710753\pi$
$487$	−13214.2	−1.22955	−0.614775	+	0.788703i	$0.710753\pi$
$488$	16794.6	1.55790
$489$	1866.30	0.172591
$490$	1883.91	0.173687
$491$	6553.27	0.602332	0.301166	−	0.953572i	$-0.402624\pi$
$491$	6553.27	0.602332	0.301166	+	0.953572i	$0.402624\pi$
$492$	1195.65	0.109561
$493$	29379.0	2.68390
$494$	161.676	0.0147250
$495$	−495.000	−0.0449467
$496$	−5727.30	−0.518474
$497$	2172.46	0.196073
$498$	−7877.01	−0.708790
$499$	−2596.63	−0.232948	−0.116474	−	0.993194i	$-0.537159\pi$
$499$	−2596.63	−0.232948	−0.116474	+	0.993194i	$0.537159\pi$
$500$	321.933	0.0287946
$501$	7835.45	0.698727
$502$	5951.51	0.529141
$503$	−659.714	−0.0584795	−0.0292398	−	0.999572i	$-0.509309\pi$
$503$	−659.714	−0.0584795	−0.0292398	+	0.999572i	$0.509309\pi$
$504$	4980.49	0.440176
$505$	4051.71	0.357027
$506$	−16.7801	−0.00147424
$507$	6299.32	0.551800
$508$	4941.94	0.431621
$509$	4825.41	0.420201	0.210101	−	0.977680i	$-0.432621\pi$
$509$	4825.41	0.420201	0.210101	+	0.977680i	$0.432621\pi$
$510$	4476.57	0.388678
$511$	−448.130	−0.0387947
$512$	11340.4	0.978866
$513$	−190.080	−0.0163592
$514$	4237.48	0.363633
$515$	7309.46	0.625424
$516$	−3098.39	−0.264339
$517$	−3052.85	−0.259699
$518$	5770.64	0.489474
$519$	−7028.91	−0.594480
$520$	1214.34	0.102408
$521$	−2329.24	−0.195866	−0.0979328	−	0.995193i	$-0.531223\pi$
$521$	−2329.24	−0.195866	−0.0979328	+	0.995193i	$0.531223\pi$
$522$	4806.04	0.402978
$523$	−15104.6	−1.26287	−0.631434	−	0.775429i	$-0.717533\pi$
$523$	−15104.6	−1.26287	−0.631434	+	0.775429i	$0.717533\pi$
$524$	−3449.61	−0.287589
$525$	−1685.04	−0.140078
$526$	14028.0	1.16283
$527$	−19962.2	−1.65003
$528$	1213.19	0.0999947
$529$	−12166.6	−0.999965
$530$	−7587.69	−0.621864
$531$	−3814.82	−0.311769
$532$	−407.359	−0.0331979
$533$	−1525.87	−0.124001
$534$	−4469.17	−0.362172
$535$	−8451.99	−0.683012
$536$	9212.66	0.742400
$537$	−3944.26	−0.316960
$538$	6936.84	0.555889
$539$	1779.52	0.142206
$540$	−347.688	−0.0277076
$541$	10712.1	0.851293	0.425647	−	0.904889i	$-0.360047\pi$
$541$	10712.1	0.851293	0.425647	+	0.904889i	$0.360047\pi$
$542$	1222.69	0.0968983
$543$	−26.0702	−0.00206037
$544$	14277.4	1.12526
$545$	7047.07	0.553878
$546$	−1547.89	−0.121326
$547$	−17251.6	−1.34849	−0.674245	−	0.738508i	$-0.735531\pi$
$547$	−17251.6	−1.34849	−0.674245	+	0.738508i	$0.735531\pi$
$548$	2834.46	0.220953
$549$	6136.65	0.477060
$550$	−640.492	−0.0496558
$551$	−1614.12	−0.124799
$552$	−48.3975	−0.00373177
$553$	548.842	0.0422046
$554$	3943.27	0.302407
$555$	−1654.19	−0.126516
$556$	3308.22	0.252337
$557$	−8179.34	−0.622208	−0.311104	−	0.950376i	$-0.600699\pi$
$557$	−8179.34	−0.622208	−0.311104	+	0.950376i	$0.600699\pi$
$558$	−3265.58	−0.247747
$559$	3954.12	0.299180
$560$	4129.83	0.311638
$561$	4228.51	0.318231
$562$	−17110.7	−1.28429
$563$	4939.38	0.369752	0.184876	−	0.982762i	$-0.440812\pi$
$563$	4939.38	0.369752	0.184876	+	0.982762i	$0.440812\pi$
$564$	−2144.32	−0.160093
$565$	−10928.3	−0.813732
$566$	3497.54	0.259739
$567$	1819.84	0.134790
$568$	2381.69	0.175939
$569$	−7658.76	−0.564274	−0.282137	−	0.959374i	$-0.591043\pi$
$569$	−7658.76	−0.564274	−0.282137	+	0.959374i	$0.591043\pi$
$570$	−245.949	−0.0180731
$571$	−1744.16	−0.127830	−0.0639149	−	0.997955i	$-0.520359\pi$
$571$	−1744.16	−0.127830	−0.0639149	+	0.997955i	$0.520359\pi$
$572$	279.344	0.0204195
$573$	−1932.31	−0.140878
$574$	−8097.61	−0.588829
$575$	16.3742	0.00118757
$576$	4982.56	0.360429
$577$	1264.61	0.0912414	0.0456207	−	0.998959i	$-0.485473\pi$
$577$	1264.61	0.0912414	0.0456207	+	0.998959i	$0.485473\pi$
$578$	−26798.1	−1.92847
$579$	−11912.3	−0.855022
$580$	−2952.50	−0.211372
$581$	−25328.4	−1.80860
$582$	−5106.69	−0.363710
$583$	−7167.22	−0.509153
$584$	−491.288	−0.0348110
$585$	443.714	0.0313595
$586$	10436.6	0.735720
$587$	−17167.4	−1.20711	−0.603557	−	0.797320i	$-0.706250\pi$
$587$	−17167.4	−1.20711	−0.603557	+	0.797320i	$0.706250\pi$
$588$	1249.93	0.0876639
$589$	1096.75	0.0767249
$590$	−4936.09	−0.344433
$591$	−11269.0	−0.784342
$592$	4054.23	0.281466
$593$	−21429.5	−1.48399	−0.741995	−	0.670406i	$-0.766120\pi$
$593$	−21429.5	−1.48399	−0.741995	+	0.670406i	$0.766120\pi$
$594$	691.732	0.0477813
$595$	14394.3	0.991782
$596$	−3289.41	−0.226073
$597$	−14476.4	−0.992431
$598$	15.0415	0.00102859
$599$	−7994.61	−0.545327	−0.272664	−	0.962109i	$-0.587905\pi$
$599$	−7994.61	−0.545327	−0.272664	+	0.962109i	$0.587905\pi$
$600$	−1847.32	−0.125694
$601$	−24313.4	−1.65019	−0.825094	−	0.564996i	$-0.808878\pi$
$601$	−24313.4	−1.65019	−0.825094	+	0.564996i	$0.808878\pi$
$602$	20984.1	1.42067
$603$	3366.25	0.227337
$604$	−2282.68	−0.153776
$605$	−605.000	−0.0406558
$606$	−5662.01	−0.379544
$607$	24569.7	1.64292	0.821460	−	0.570266i	$-0.193160\pi$
$607$	24569.7	1.64292	0.821460	+	0.570266i	$0.193160\pi$
$608$	−784.423	−0.0523232
$609$	15453.7	1.02827
$610$	7940.36	0.527043
$611$	2736.55	0.181193
$612$	2970.10	0.196175
$613$	−12746.7	−0.839859	−0.419929	−	0.907557i	$-0.637945\pi$
$613$	−12746.7	−0.839859	−0.419929	+	0.907557i	$0.637945\pi$
$614$	−7108.70	−0.467237
$615$	2321.23	0.152197
$616$	6087.26	0.398154
$617$	−15607.4	−1.01837	−0.509183	−	0.860658i	$-0.670052\pi$
$617$	−15607.4	−1.01837	−0.509183	+	0.860658i	$0.670052\pi$
$618$	−10214.5	−0.664867
$619$	−11909.7	−0.773329	−0.386665	−	0.922220i	$-0.626373\pi$
$619$	−11909.7	−0.773329	−0.386665	+	0.922220i	$0.626373\pi$
$620$	2006.14	0.129949
$621$	−17.6842	−0.00114274
$622$	−23884.9	−1.53970
$623$	−14370.5	−0.924147
$624$	−1087.49	−0.0697667
$625$	625.000	0.0400000
$626$	14417.3	0.920499
$627$	−232.320	−0.0147974
$628$	4329.68	0.275116
$629$	14130.8	0.895759
$630$	2354.74	0.148913
$631$	−19304.2	−1.21789	−0.608946	−	0.793212i	$-0.708407\pi$
$631$	−19304.2	−1.21789	−0.608946	+	0.793212i	$0.708407\pi$
$632$	601.699	0.0378707
$633$	13182.1	0.827709
$634$	15687.1	0.982674
$635$	9594.27	0.599586
$636$	−5034.25	−0.313870
$637$	−1595.14	−0.0992180
$638$	5874.05	0.364508
$639$	870.255	0.0538760
$640$	1990.12	0.122916
$641$	26678.6	1.64390	0.821950	−	0.569560i	$-0.192886\pi$
$641$	26678.6	1.64390	0.821950	+	0.569560i	$0.192886\pi$
$642$	11811.1	0.726087
$643$	−26456.2	−1.62260	−0.811299	−	0.584631i	$-0.801239\pi$
$643$	−26456.2	−1.62260	−0.811299	+	0.584631i	$0.801239\pi$
$644$	−37.8988	−0.00231898
$645$	−6015.21	−0.367207
$646$	2101.01	0.127961
$647$	−23523.7	−1.42939	−0.714694	−	0.699438i	$-0.753434\pi$
$647$	−23523.7	−1.42939	−0.714694	+	0.699438i	$0.753434\pi$
$648$	1995.10	0.120949
$649$	−4662.56	−0.282006
$650$	574.132	0.0346451
$651$	−10500.4	−0.632171
$652$	1602.20	0.0962376
$653$	18071.1	1.08296	0.541482	−	0.840712i	$-0.317863\pi$
$653$	18071.1	1.08296	0.541482	+	0.840712i	$0.317863\pi$
$654$	−9847.85	−0.588809
$655$	−6697.06	−0.399505
$656$	−5689.07	−0.338599
$657$	−179.514	−0.0106598
$658$	14522.5	0.860407
$659$	17023.1	1.00626	0.503130	−	0.864210i	$-0.332182\pi$
$659$	17023.1	1.00626	0.503130	+	0.864210i	$0.332182\pi$
$660$	−424.952	−0.0250625
$661$	2137.74	0.125792	0.0628959	−	0.998020i	$-0.479966\pi$
$661$	2137.74	0.125792	0.0628959	+	0.998020i	$0.479966\pi$
$662$	11134.8	0.653728
$663$	−3790.39	−0.222031
$664$	−27767.7	−1.62288
$665$	−790.846	−0.0461168
$666$	2311.63	0.134495
$667$	−150.171	−0.00871758
$668$	6726.65	0.389614
$669$	−6569.01	−0.379630
$670$	4355.68	0.251156
$671$	7500.36	0.431517
$672$	7510.11	0.431115
$673$	31790.1	1.82083	0.910414	−	0.413698i	$-0.135763\pi$
$673$	31790.1	1.82083	0.910414	+	0.413698i	$0.135763\pi$
$674$	−27693.1	−1.58264
$675$	−675.000	−0.0384900
$676$	5407.90	0.307686
$677$	−10225.1	−0.580476	−0.290238	−	0.956955i	$-0.593734\pi$
$677$	−10225.1	−0.580476	−0.290238	+	0.956955i	$0.593734\pi$
$678$	15271.7	0.865052
$679$	−16420.5	−0.928071
$680$	15780.6	0.889939
$681$	3417.81	0.192321
$682$	−3991.26	−0.224096
$683$	21274.0	1.19184	0.595919	−	0.803044i	$-0.296788\pi$
$683$	21274.0	1.19184	0.595919	+	0.803044i	$0.296788\pi$
$684$	−163.182	−0.00912195
$685$	5502.80	0.306936
$686$	9483.08	0.527793
$687$	−10248.3	−0.569136
$688$	14742.6	0.816941
$689$	6424.63	0.355238
$690$	−22.8820	−0.00126247
$691$	−22568.1	−1.24245	−0.621224	−	0.783633i	$-0.713364\pi$
$691$	−22568.1	−1.24245	−0.621224	+	0.783633i	$0.713364\pi$
$692$	−6034.24	−0.331485
$693$	2224.25	0.121923
$694$	−19709.6	−1.07805
$695$	6422.56	0.350535
$696$	16942.0	0.922682
$697$	−19829.0	−1.07758
$698$	7667.03	0.415761
$699$	−18441.2	−0.997870
$700$	−1446.59	−0.0781083
$701$	−7735.03	−0.416759	−0.208380	−	0.978048i	$-0.566819\pi$
$701$	−7735.03	−0.416759	−0.208380	+	0.978048i	$0.566819\pi$
$702$	−620.062	−0.0333372
$703$	−776.368	−0.0416519
$704$	6089.80	0.326020
$705$	−4162.98	−0.222393
$706$	19083.2	1.01729
$707$	−18206.1	−0.968473
$708$	−3274.98	−0.173844
$709$	35115.1	1.86005	0.930025	−	0.367497i	$-0.119785\pi$
$709$	35115.1	1.86005	0.930025	+	0.367497i	$0.119785\pi$
$710$	1126.04	0.0595207
$711$	219.857	0.0115968
$712$	−15754.5	−0.829250
$713$	102.037	0.00535948
$714$	−20115.2	−1.05433
$715$	542.317	0.0283657
$716$	−3386.11	−0.176738
$717$	6240.10	0.325022
$718$	−29853.1	−1.55168
$719$	15334.3	0.795370	0.397685	−	0.917522i	$-0.369814\pi$
$719$	15334.3	0.795370	0.397685	+	0.917522i	$0.369814\pi$
$720$	1654.35	0.0856303
$721$	−32844.6	−1.69653
$722$	15859.6	0.817498
$723$	−5538.85	−0.284913
$724$	−22.3810	−0.00114887
$725$	−5731.97	−0.293628
$726$	845.450	0.0432198
$727$	−12360.4	−0.630567	−0.315284	−	0.948997i	$-0.602100\pi$
$727$	−12360.4	−0.630567	−0.315284	+	0.948997i	$0.602100\pi$
$728$	−5456.57	−0.277794
$729$	729.000	0.0370370
$730$	−232.277	−0.0117767
$731$	51384.6	2.59990
$732$	5268.25	0.266011
$733$	−15097.6	−0.760769	−0.380384	−	0.924828i	$-0.624208\pi$
$733$	−15097.6	−0.760769	−0.380384	+	0.924828i	$0.624208\pi$
$734$	6524.23	0.328084
$735$	2426.61	0.121778
$736$	−72.9790	−0.00365495
$737$	4114.31	0.205634
$738$	−3243.78	−0.161795
$739$	−3667.49	−0.182559	−0.0912793	−	0.995825i	$-0.529096\pi$
$739$	−3667.49	−0.182559	−0.0912793	+	0.995825i	$0.529096\pi$
$740$	−1420.10	−0.0705460
$741$	208.250	0.0103242
$742$	34094.8	1.68687
$743$	10172.1	0.502257	0.251128	−	0.967954i	$-0.419198\pi$
$743$	10172.1	0.502257	0.251128	+	0.967954i	$0.419198\pi$
$744$	−11511.7	−0.567255
$745$	−6386.06	−0.314050
$746$	−15926.2	−0.781635
$747$	−10146.2	−0.496959
$748$	3630.12	0.177447
$749$	37978.5	1.85274
$750$	−873.399	−0.0425227
$751$	31430.3	1.52718	0.763588	−	0.645704i	$-0.223436\pi$
$751$	31430.3	1.52718	0.763588	+	0.645704i	$0.223436\pi$
$752$	10203.0	0.494766
$753$	7665.97	0.371000
$754$	−5265.45	−0.254319
$755$	−4431.59	−0.213618
$756$	1562.31	0.0751598
$757$	−28362.0	−1.36174	−0.680868	−	0.732406i	$-0.738397\pi$
$757$	−28362.0	−1.36174	−0.680868	+	0.732406i	$0.738397\pi$
$758$	17388.3	0.833207
$759$	−21.6140	−0.00103365
$760$	−867.010	−0.0413813
$761$	30722.6	1.46346	0.731731	−	0.681594i	$-0.238713\pi$
$761$	30722.6	1.46346	0.731731	+	0.681594i	$0.238713\pi$
$762$	−13407.4	−0.637400
$763$	−31665.6	−1.50245
$764$	−1658.86	−0.0785544
$765$	5766.14	0.272517
$766$	20138.4	0.949906
$767$	4179.48	0.196757
$768$	10505.8	0.493612
$769$	−18443.1	−0.864855	−0.432427	−	0.901669i	$-0.642343\pi$
$769$	−18443.1	−0.864855	−0.432427	+	0.901669i	$0.642343\pi$
$770$	2878.01	0.134697
$771$	5458.18	0.254956
$772$	−10226.6	−0.476764
$773$	−545.742	−0.0253932	−0.0126966	−	0.999919i	$-0.504042\pi$
$773$	−545.742	−0.0253932	−0.0126966	+	0.999919i	$0.504042\pi$
$774$	8405.88	0.390366
$775$	3894.72	0.180519
$776$	−18001.9	−0.832770
$777$	7433.00	0.343189
$778$	−10207.8	−0.470393
$779$	1089.43	0.0501065
$780$	380.923	0.0174862
$781$	1063.65	0.0487327
$782$	195.468	0.00893851
$783$	6190.53	0.282543
$784$	−5947.35	−0.270925
$785$	8405.62	0.382178
$786$	9358.72	0.424701
$787$	17365.2	0.786536	0.393268	−	0.919424i	$-0.371345\pi$
$787$	17365.2	0.786536	0.393268	+	0.919424i	$0.371345\pi$
$788$	−9674.33	−0.437352
$789$	18069.1	0.815306
$790$	284.479	0.0128118
$791$	49105.8	2.20733
$792$	2438.46	0.109403
$793$	−6723.25	−0.301071
$794$	10323.8	0.461431
$795$	−9773.48	−0.436012
$796$	−12427.9	−0.553384
$797$	7055.12	0.313557	0.156779	−	0.987634i	$-0.449889\pi$
$797$	7055.12	0.313557	0.156779	+	0.987634i	$0.449889\pi$
$798$	1105.16	0.0490253
$799$	35562.0	1.57458
$800$	−2785.59	−0.123107
$801$	−5756.61	−0.253933
$802$	11726.7	0.516313
$803$	−219.406	−0.00964217
$804$	2889.89	0.126764
$805$	−73.5766	−0.00322141
$806$	3577.73	0.156353
$807$	8935.14	0.389755
$808$	−19959.5	−0.869024
$809$	−6937.17	−0.301481	−0.150740	−	0.988573i	$-0.548166\pi$
$809$	−6937.17	−0.301481	−0.150740	+	0.988573i	$0.548166\pi$
$810$	943.271	0.0409175
$811$	5610.44	0.242921	0.121461	−	0.992596i	$-0.461242\pi$
$811$	5610.44	0.242921	0.121461	+	0.992596i	$0.461242\pi$
$812$	13266.9	0.573369
$813$	1574.91	0.0679390
$814$	2825.32	0.121655
$815$	3110.50	0.133688
$816$	−14132.1	−0.606280
$817$	−2823.14	−0.120893
$818$	−15079.8	−0.644565
$819$	−1993.80	−0.0850659
$820$	1992.75	0.0848657
$821$	−17001.8	−0.722735	−0.361368	−	0.932423i	$-0.617690\pi$
$821$	−17001.8	−0.722735	−0.361368	+	0.932423i	$0.617690\pi$
$822$	−7689.83	−0.326294
$823$	14567.3	0.616991	0.308496	−	0.951226i	$-0.400175\pi$
$823$	14567.3	0.616991	0.308496	+	0.951226i	$0.400175\pi$
$824$	−36007.7	−1.52232
$825$	−825.000	−0.0348155
$826$	22180.0	0.934312
$827$	−7345.87	−0.308877	−0.154438	−	0.988002i	$-0.549357\pi$
$827$	−7345.87	−0.308877	−0.154438	+	0.988002i	$0.549357\pi$
$828$	−15.1817	−0.000637197 0
$829$	−13903.2	−0.582482	−0.291241	−	0.956650i	$-0.594068\pi$
$829$	−13903.2	−0.582482	−0.291241	+	0.956650i	$0.594068\pi$
$830$	−13128.4	−0.549026
$831$	5079.22	0.212029
$832$	−5458.84	−0.227466
$833$	−20729.2	−0.862214
$834$	−8975.13	−0.372642
$835$	13059.1	0.541232
$836$	−199.444	−0.00825111
$837$	−4206.30	−0.173705
$838$	19232.9	0.792829
$839$	−25111.9	−1.03332	−0.516662	−	0.856190i	$-0.672825\pi$
$839$	−25111.9	−1.03332	−0.516662	+	0.856190i	$0.672825\pi$
$840$	8300.81	0.340959
$841$	28179.7	1.15543
$842$	7987.20	0.326909
$843$	−22039.8	−0.900464
$844$	11316.6	0.461534
$845$	10498.9	0.427423
$846$	5817.50	0.236418
$847$	2718.53	0.110283
$848$	23953.7	0.970014
$849$	4505.08	0.182113
$850$	7460.95	0.301069
$851$	−72.2296	−0.00290952
$852$	747.104	0.0300415
$853$	−27545.3	−1.10567	−0.552833	−	0.833292i	$-0.686453\pi$
$853$	−27545.3	−1.10567	−0.552833	+	0.833292i	$0.686453\pi$
$854$	−35679.5	−1.42966
$855$	−316.801	−0.0126718
$856$	41636.1	1.66249
$857$	1808.04	0.0720669	0.0360334	−	0.999351i	$-0.488528\pi$
$857$	1808.04	0.0720669	0.0360334	+	0.999351i	$0.488528\pi$
$858$	−757.854	−0.0301547
$859$	32160.8	1.27743	0.638716	−	0.769443i	$-0.279466\pi$
$859$	32160.8	1.27743	0.638716	+	0.769443i	$0.279466\pi$
$860$	−5163.99	−0.204756
$861$	−10430.3	−0.412850
$862$	26226.3	1.03628
$863$	33734.8	1.33064	0.665321	−	0.746557i	$-0.268294\pi$
$863$	33734.8	1.33064	0.665321	+	0.746557i	$0.268294\pi$
$864$	3008.43	0.118459
$865$	−11714.9	−0.460482
$866$	−29342.7	−1.15139
$867$	−34517.9	−1.35212
$868$	−9014.47	−0.352501
$869$	268.715	0.0104897
$870$	8010.07	0.312146
$871$	−3688.03	−0.143472
$872$	−34715.2	−1.34817
$873$	−6577.78	−0.255011
$874$	−10.7393	−0.000415631 0
$875$	−2808.40	−0.108504
$876$	−154.111	−0.00594397
$877$	46573.5	1.79325	0.896623	−	0.442795i	$-0.146013\pi$
$877$	46573.5	1.79325	0.896623	+	0.442795i	$0.146013\pi$
$878$	−9728.26	−0.373933
$879$	13443.1	0.515841
$880$	2021.98	0.0774556
$881$	−9949.72	−0.380493	−0.190247	−	0.981736i	$-0.560929\pi$
$881$	−9949.72	−0.380493	−0.190247	+	0.981736i	$0.560929\pi$
$882$	−3391.04	−0.129458
$883$	−49269.1	−1.87773	−0.938866	−	0.344282i	$-0.888122\pi$
$883$	−49269.1	−1.87773	−0.938866	+	0.344282i	$0.888122\pi$
$884$	−3254.01	−0.123806
$885$	−6358.04	−0.241495
$886$	−5483.00	−0.207906
$887$	27347.5	1.03522	0.517609	−	0.855617i	$-0.326822\pi$
$887$	27347.5	1.03522	0.517609	+	0.855617i	$0.326822\pi$
$888$	8148.85	0.307948
$889$	−43111.2	−1.62644
$890$	−7448.62	−0.280537
$891$	891.000	0.0335013
$892$	−5639.42	−0.211683
$893$	−1953.83	−0.0732165
$894$	8924.12	0.333856
$895$	−6573.77	−0.245516
$896$	−8942.48	−0.333423
$897$	19.3746	0.000721180 0
$898$	−21628.1	−0.803718
$899$	−35719.0	−1.32514
$900$	−579.480	−0.0214622
$901$	83489.4	3.08705
$902$	−3964.61	−0.146349
$903$	27029.0	0.996088
$904$	53835.0	1.98067
$905$	−43.4503	−0.00159595
$906$	6192.86	0.227091
$907$	−25516.6	−0.934141	−0.467071	−	0.884220i	$-0.654691\pi$
$907$	−25516.6	−0.934141	−0.467071	+	0.884220i	$0.654691\pi$
$908$	2934.15	0.107239
$909$	−7293.07	−0.266112
$910$	−2579.82	−0.0939784
$911$	22379.0	0.813885	0.406942	−	0.913454i	$-0.366595\pi$
$911$	22379.0	0.813885	0.406942	+	0.913454i	$0.366595\pi$
$912$	776.441	0.0281914
$913$	−12400.9	−0.449516
$914$	−33488.1	−1.21191
$915$	10227.8	0.369529
$916$	−8798.04	−0.317353
$917$	30092.8	1.08370
$918$	−8057.83	−0.289704
$919$	−6244.97	−0.224159	−0.112080	−	0.993699i	$-0.535751\pi$
$919$	−6244.97	−0.224159	−0.112080	+	0.993699i	$0.535751\pi$
$920$	−80.6625	−0.00289061
$921$	−9156.52	−0.327598
$922$	12537.8	0.447841
$923$	−953.442	−0.0340010
$924$	1909.49	0.0679846
$925$	−2756.98	−0.0979990
$926$	−42762.6	−1.51757
$927$	−13157.0	−0.466163
$928$	25547.0	0.903688
$929$	16122.2	0.569378	0.284689	−	0.958620i	$-0.408110\pi$
$929$	16122.2	0.569378	0.284689	+	0.958620i	$0.408110\pi$
$930$	−5442.63	−0.191904
$931$	1138.89	0.0400921
$932$	−15831.6	−0.556417
$933$	−30765.4	−1.07954
$934$	−21109.8	−0.739544
$935$	7047.51	0.246501
$936$	−2185.82	−0.0763308
$937$	56379.6	1.96568	0.982839	−	0.184466i	$-0.0590554\pi$
$937$	56379.6	1.96568	0.982839	+	0.184466i	$0.0590554\pi$
$938$	−19572.0	−0.681287
$939$	18570.5	0.645396
$940$	−3573.87	−0.124007
$941$	25527.0	0.884332	0.442166	−	0.896933i	$-0.354210\pi$
$941$	25527.0	0.884332	0.442166	+	0.896933i	$0.354210\pi$
$942$	−11746.3	−0.406280
$943$	101.356	0.00350010
$944$	15582.8	0.537264
$945$	3033.07	0.104408
$946$	10273.9	0.353099
$947$	46411.3	1.59257	0.796285	−	0.604921i	$-0.206795\pi$
$947$	46411.3	1.59257	0.796285	+	0.604921i	$0.206795\pi$
$948$	188.745	0.00646641
$949$	196.673	0.00672738
$950$	−409.916	−0.0139994
$951$	20206.1	0.688990
$952$	−70909.2	−2.41405
$953$	−21266.1	−0.722850	−0.361425	−	0.932401i	$-0.617710\pi$
$953$	−21266.1	−0.722850	−0.361425	+	0.932401i	$0.617710\pi$
$954$	13657.8	0.463510
$955$	−3220.51	−0.109124
$956$	5357.05	0.181234
$957$	7566.20	0.255570
$958$	−29365.4	−0.990347
$959$	−24726.5	−0.832597
$960$	8304.27	0.279187
$961$	−5520.88	−0.185320
$962$	−2532.60	−0.0848796
$963$	15213.6	0.509087
$964$	−4755.04	−0.158869
$965$	−19853.8	−0.662297
$966$	102.819	0.00342457
$967$	20035.9	0.666300	0.333150	−	0.942874i	$-0.391888\pi$
$967$	20035.9	0.666300	0.333150	+	0.942874i	$0.391888\pi$
$968$	2980.34	0.0989585
$969$	2706.25	0.0897185
$970$	−8511.15	−0.281728
$971$	21354.1	0.705751	0.352876	−	0.935670i	$-0.385204\pi$
$971$	21354.1	0.705751	0.352876	+	0.935670i	$0.385204\pi$
$972$	625.838	0.0206520
$973$	−28859.4	−0.950862
$974$	30776.6	1.01247
$975$	739.523	0.0242910
$976$	−25067.0	−0.822107
$977$	−34057.7	−1.11525	−0.557626	−	0.830092i	$-0.688288\pi$
$977$	−34057.7	−1.11525	−0.557626	+	0.830092i	$0.688288\pi$
$978$	−4346.73	−0.142120
$979$	−7035.86	−0.229691
$980$	2083.22	0.0679041
$981$	−12684.7	−0.412836
$982$	−15263.0	−0.495989
$983$	31846.5	1.03331	0.516657	−	0.856193i	$-0.327176\pi$
$983$	31846.5	1.03331	0.516657	+	0.856193i	$0.327176\pi$
$984$	−11434.8	−0.370456
$985$	−18781.7	−0.607548
$986$	−68425.5	−2.21005
$987$	18706.1	0.603263
$988$	178.780	0.00575684
$989$	−262.652	−0.00844474
$990$	1152.89	0.0370112
$991$	−20462.5	−0.655915	−0.327957	−	0.944693i	$-0.606360\pi$
$991$	−20462.5	−0.655915	−0.327957	+	0.944693i	$0.606360\pi$
$992$	−17358.5	−0.555578
$993$	14342.5	0.458353
$994$	−5059.81	−0.161456
$995$	−24127.4	−0.768733
$996$	−8710.36	−0.277107
$997$	35227.4	1.11902	0.559510	−	0.828824i	$-0.310989\pi$
$997$	35227.4	1.11902	0.559510	+	0.828824i	$0.310989\pi$
$998$	6047.71	0.191820
$999$	2977.54	0.0942996

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	165.4.a.d.1.2	✓	3
3.2	odd	2		495.4.a.l.1.2		3
5.2	odd	4		825.4.c.l.199.3		6
5.3	odd	4		825.4.c.l.199.4		6
5.4	even	2		825.4.a.s.1.2		3
11.10	odd	2		1815.4.a.s.1.2		3
15.14	odd	2		2475.4.a.s.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.d.1.2	✓	3	1.1	even	1	trivial
495.4.a.l.1.2		3	3.2	odd	2
825.4.a.s.1.2		3	5.4	even	2
825.4.c.l.199.3		6	5.2	odd	4
825.4.c.l.199.4		6	5.3	odd	4
1815.4.a.s.1.2		3	11.10	odd	2
2475.4.a.s.1.2		3	15.14	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:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	165.a (trivial)

\(f(q)\)	\(=\)	\(q-2.32906 q^{2} -3.00000 q^{3} -2.57547 q^{4} -5.00000 q^{5} +6.98719 q^{6} +22.4672 q^{7} +24.6309 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.32906 q^{2} -3.00000 q^{3} -2.57547 q^{4} -5.00000 q^{5} +6.98719 q^{6} +22.4672 q^{7} +24.6309 q^{8} +9.00000 q^{9} +11.6453 q^{10} +11.0000 q^{11} +7.72640 q^{12} -9.86030 q^{13} -52.3275 q^{14} +15.0000 q^{15} -36.7633 q^{16} -128.137 q^{17} -20.9616 q^{18} +7.04001 q^{19} +12.8773 q^{20} -67.4015 q^{21} -25.6197 q^{22} +0.654969 q^{23} -73.8928 q^{24} +25.0000 q^{25} +22.9653 q^{26} -27.0000 q^{27} -57.8635 q^{28} -229.279 q^{29} -34.9359 q^{30} +155.789 q^{31} -111.423 q^{32} -33.0000 q^{33} +298.438 q^{34} -112.336 q^{35} -23.1792 q^{36} -110.279 q^{37} -16.3966 q^{38} +29.5809 q^{39} -123.155 q^{40} +154.749 q^{41} +156.982 q^{42} -401.014 q^{43} -28.3301 q^{44} -45.0000 q^{45} -1.52546 q^{46} -277.532 q^{47} +110.290 q^{48} +161.774 q^{49} -58.2266 q^{50} +384.410 q^{51} +25.3949 q^{52} -651.566 q^{53} +62.8847 q^{54} -55.0000 q^{55} +553.388 q^{56} -21.1200 q^{57} +534.005 q^{58} -423.869 q^{59} -38.6320 q^{60} +681.851 q^{61} -362.842 q^{62} +202.205 q^{63} +553.618 q^{64} +49.3015 q^{65} +76.8591 q^{66} +374.028 q^{67} +330.011 q^{68} -1.96491 q^{69} +261.637 q^{70} +96.6950 q^{71} +221.678 q^{72} -19.9460 q^{73} +256.848 q^{74} -75.0000 q^{75} -18.1313 q^{76} +247.139 q^{77} -68.8958 q^{78} +24.4286 q^{79} +183.816 q^{80} +81.0000 q^{81} -360.419 q^{82} -1127.35 q^{83} +173.590 q^{84} +640.683 q^{85} +933.987 q^{86} +687.836 q^{87} +270.940 q^{88} -639.624 q^{89} +104.808 q^{90} -221.533 q^{91} -1.68685 q^{92} -467.366 q^{93} +646.389 q^{94} -35.2001 q^{95} +334.270 q^{96} -730.865 q^{97} -376.783 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{2} - 9 q^{3} + 22 q^{4} - 15 q^{5} + 12 q^{6} - 4 q^{7} - 48 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q - 4 * q^2 - 9 * q^3 + 22 * q^4 - 15 * q^5 + 12 * q^6 - 4 * q^7 - 48 * q^8 + 27 * q^9	\( 3 q - 4 q^{2} - 9 q^{3} + 22 q^{4} - 15 q^{5} + 12 q^{6} - 4 q^{7} - 48 q^{8} + 27 q^{9} + 20 q^{10} + 33 q^{11} - 66 q^{12} - 56 q^{14} + 45 q^{15} + 50 q^{16} - 218 q^{17} - 36 q^{18} + 146 q^{19} - 110 q^{20} + 12 q^{21} - 44 q^{22} - 200 q^{23} + 144 q^{24} + 75 q^{25} - 508 q^{26} - 81 q^{27} - 340 q^{28} + 68 q^{29} - 60 q^{30} - 68 q^{31} - 688 q^{32} - 99 q^{33} - 176 q^{34} + 20 q^{35} + 198 q^{36} - 390 q^{37} - 316 q^{38} + 240 q^{40} - 196 q^{41} + 168 q^{42} - 524 q^{43} + 242 q^{44} - 135 q^{45} + 1160 q^{46} - 60 q^{47} - 150 q^{48} - 157 q^{49} - 100 q^{50} + 654 q^{51} + 1020 q^{52} - 158 q^{53} + 108 q^{54} - 165 q^{55} + 1368 q^{56} - 438 q^{57} + 1092 q^{58} - 1044 q^{59} + 330 q^{60} + 642 q^{61} + 88 q^{62} - 36 q^{63} + 1166 q^{64} + 132 q^{66} - 236 q^{67} + 144 q^{68} + 600 q^{69} + 280 q^{70} - 544 q^{71} - 432 q^{72} + 900 q^{73} + 1536 q^{74} - 225 q^{75} + 1996 q^{76} - 44 q^{77} + 1524 q^{78} - 1586 q^{79} - 250 q^{80} + 243 q^{81} + 380 q^{82} - 1582 q^{83} + 1020 q^{84} + 1090 q^{85} + 3568 q^{86} - 204 q^{87} - 528 q^{88} - 2122 q^{89} + 180 q^{90} - 8 q^{91} - 4128 q^{92} + 204 q^{93} - 2152 q^{94} - 730 q^{95} + 2064 q^{96} + 618 q^{97} + 572 q^{98} + 297 q^{99}+O(q^{100}) \) 3 * q - 4 * q^2 - 9 * q^3 + 22 * q^4 - 15 * q^5 + 12 * q^6 - 4 * q^7 - 48 * q^8 + 27 * q^9 + 20 * q^10 + 33 * q^11 - 66 * q^12 - 56 * q^14 + 45 * q^15 + 50 * q^16 - 218 * q^17 - 36 * q^18 + 146 * q^19 - 110 * q^20 + 12 * q^21 - 44 * q^22 - 200 * q^23 + 144 * q^24 + 75 * q^25 - 508 * q^26 - 81 * q^27 - 340 * q^28 + 68 * q^29 - 60 * q^30 - 68 * q^31 - 688 * q^32 - 99 * q^33 - 176 * q^34 + 20 * q^35 + 198 * q^36 - 390 * q^37 - 316 * q^38 + 240 * q^40 - 196 * q^41 + 168 * q^42 - 524 * q^43 + 242 * q^44 - 135 * q^45 + 1160 * q^46 - 60 * q^47 - 150 * q^48 - 157 * q^49 - 100 * q^50 + 654 * q^51 + 1020 * q^52 - 158 * q^53 + 108 * q^54 - 165 * q^55 + 1368 * q^56 - 438 * q^57 + 1092 * q^58 - 1044 * q^59 + 330 * q^60 + 642 * q^61 + 88 * q^62 - 36 * q^63 + 1166 * q^64 + 132 * q^66 - 236 * q^67 + 144 * q^68 + 600 * q^69 + 280 * q^70 - 544 * q^71 - 432 * q^72 + 900 * q^73 + 1536 * q^74 - 225 * q^75 + 1996 * q^76 - 44 * q^77 + 1524 * q^78 - 1586 * q^79 - 250 * q^80 + 243 * q^81 + 380 * q^82 - 1582 * q^83 + 1020 * q^84 + 1090 * q^85 + 3568 * q^86 - 204 * q^87 - 528 * q^88 - 2122 * q^89 + 180 * q^90 - 8 * q^91 - 4128 * q^92 + 204 * q^93 - 2152 * q^94 - 730 * q^95 + 2064 * q^96 + 618 * q^97 + 572 * q^98 + 297 * q^99

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