LMFDB - Embedded newform 169.8.a.i.1.17

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [169,8,Mod(1,169)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(169, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("169.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	169.a (trivial)

Newform invariants

comment:select newform

sage:traces = [21,31] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$52.7930693068$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	169.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+15.3948 q^{2} -46.3939 q^{3} +108.998 q^{4} +391.457 q^{5} -714.223 q^{6} +1742.39 q^{7} -292.525 q^{8} -34.6060 q^{9} +6026.39 q^{10} +4062.66 q^{11} -5056.86 q^{12} +26823.7 q^{14} -18161.2 q^{15} -18455.1 q^{16} +5498.86 q^{17} -532.750 q^{18} +17102.6 q^{19} +42668.2 q^{20} -80836.3 q^{21} +62543.6 q^{22} -76982.9 q^{23} +13571.4 q^{24} +75113.7 q^{25} +103069. q^{27} +189918. q^{28} +148628. q^{29} -279588. q^{30} -99859.9 q^{31} -246669. q^{32} -188483. q^{33} +84653.6 q^{34} +682071. q^{35} -3771.99 q^{36} +165712. q^{37} +263290. q^{38} -114511. q^{40} +275758. q^{41} -1.24446e6 q^{42} +399389. q^{43} +442823. q^{44} -13546.7 q^{45} -1.18513e6 q^{46} -1.12589e6 q^{47} +856206. q^{48} +2.21238e6 q^{49} +1.15636e6 q^{50} -255114. q^{51} +343549. q^{53} +1.58672e6 q^{54} +1.59036e6 q^{55} -509693. q^{56} -793454. q^{57} +2.28809e6 q^{58} +2.45965e6 q^{59} -1.97954e6 q^{60} +414676. q^{61} -1.53732e6 q^{62} -60297.1 q^{63} -1.43515e6 q^{64} -2.90164e6 q^{66} +3.22512e6 q^{67} +599367. q^{68} +3.57154e6 q^{69} +1.05003e7 q^{70} +3.55984e6 q^{71} +10123.1 q^{72} -368557. q^{73} +2.55109e6 q^{74} -3.48482e6 q^{75} +1.86415e6 q^{76} +7.07874e6 q^{77} -3.22544e6 q^{79} -7.22440e6 q^{80} -4.70609e6 q^{81} +4.24522e6 q^{82} +211812. q^{83} -8.81103e6 q^{84} +2.15257e6 q^{85} +6.14850e6 q^{86} -6.89542e6 q^{87} -1.18843e6 q^{88} -723703. q^{89} -208549. q^{90} -8.39102e6 q^{92} +4.63289e6 q^{93} -1.73328e7 q^{94} +6.69492e6 q^{95} +1.14439e7 q^{96} -1.31082e7 q^{97} +3.40591e7 q^{98} -140592. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 31 q^{2} - 26 q^{3} + 1409 q^{4} + 680 q^{5} + 1470 q^{6} + 2929 q^{7} + 4716 q^{8} + 15465 q^{9} - 5167 q^{10} + 14824 q^{11} + 21795 q^{12} - 179 q^{14} + 36398 q^{15} + 113205 q^{16} + 45016 q^{17}+ \cdots + 37605493 q^{99}+O(q^{100}) $ 21 * q + 31 * q^2 - 26 * q^3 + 1409 * q^4 + 680 * q^5 + 1470 * q^6 + 2929 * q^7 + 4716 * q^8 + 15465 * q^9 - 5167 * q^10 + 14824 * q^11 + 21795 * q^12 - 179 * q^14 + 36398 * q^15 + 113205 * q^16 + 45016 * q^17 + 72967 * q^18 + 119316 * q^19 + 131130 * q^20 + 94178 * q^21 - 274174 * q^22 - 128105 * q^23 + 316902 * q^24 + 366537 * q^25 - 106643 * q^27 + 170352 * q^28 - 62329 * q^29 + 9761 * q^30 + 467367 * q^31 + 677464 * q^32 + 587690 * q^33 + 578614 * q^34 + 848860 * q^35 - 274245 * q^36 + 697064 * q^37 + 225831 * q^38 - 2288827 * q^40 + 2351315 * q^41 + 2736836 * q^42 + 894489 * q^43 - 263025 * q^44 - 1938345 * q^45 - 3170392 * q^46 + 1070934 * q^47 + 6302388 * q^48 + 131530 * q^49 + 3970474 * q^50 - 1184341 * q^51 - 4550027 * q^53 + 7916233 * q^54 + 5524030 * q^55 + 2334386 * q^56 + 5590305 * q^57 + 7510959 * q^58 + 10746233 * q^59 + 15186203 * q^60 + 5717012 * q^61 - 13593653 * q^62 + 22347508 * q^63 + 9767924 * q^64 - 17806977 * q^66 + 11305923 * q^67 + 12252492 * q^68 + 2772206 * q^69 + 28670666 * q^70 + 23267025 * q^71 + 49774322 * q^72 + 4763727 * q^73 + 9625078 * q^74 - 43212397 * q^75 + 63359948 * q^76 + 9446036 * q^77 - 18613281 * q^79 + 57904093 * q^80 + 41055449 * q^81 + 24190327 * q^82 + 21455789 * q^83 + 60768429 * q^84 + 26216249 * q^85 + 35632347 * q^86 + 9027894 * q^87 - 75678180 * q^88 + 54037507 * q^89 + 3335560 * q^90 - 69339390 * q^92 + 20085986 * q^93 + 28955713 * q^94 + 17660031 * q^95 + 66667956 * q^96 + 10916609 * q^97 + 41738840 * q^98 + 37605493 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	15.3948	1.36072	0.680358	−	0.732880i	$-0.261824\pi$
$2$	15.3948	1.36072	0.680358	+	0.732880i	$0.261824\pi$
$3$	−46.3939	−0.992057	−0.496028	−	0.868306i	$-0.665209\pi$
$3$	−46.3939	−0.992057	−0.496028	+	0.868306i	$0.665209\pi$
$4$	108.998	0.851550
$5$	391.457	1.40052	0.700260	−	0.713888i	$-0.253067\pi$
$5$	391.457	1.40052	0.700260	+	0.713888i	$0.253067\pi$
$6$	−714.223	−1.34991
$7$	1742.39	1.92001	0.960003	−	0.279990i	$-0.0903312\pi$
$7$	1742.39	1.92001	0.960003	+	0.279990i	$0.0903312\pi$
$8$	−292.525	−0.201999
$9$	−34.6060	−0.0158235
$10$	6026.39	1.90571
$11$	4062.66	0.920314	0.460157	−	0.887838i	$-0.347793\pi$
$11$	4062.66	0.920314	0.460157	+	0.887838i	$0.347793\pi$
$12$	−5056.86	−0.844786
$13$	0	0
$13$	0	0
$14$	26823.7	2.61258
$15$	−18161.2	−1.38939
$16$	−18455.1	−1.12641
$17$	5498.86	0.271457	0.135729	−	0.990746i	$-0.456662\pi$
$17$	5498.86	0.271457	0.135729	+	0.990746i	$0.456662\pi$
$18$	−532.750	−0.0215313
$19$	17102.6	0.572036	0.286018	−	0.958224i	$-0.407668\pi$
$19$	17102.6	0.572036	0.286018	+	0.958224i	$0.407668\pi$
$20$	42668.2	1.19261
$21$	−80836.3	−1.90475
$22$	62543.6	1.25229
$23$	−76982.9	−1.31931	−0.659655	−	0.751568i	$-0.729298\pi$
$23$	−76982.9	−1.31931	−0.659655	+	0.751568i	$0.729298\pi$
$24$	13571.4	0.200394
$25$	75113.7	0.961455
$26$	0	0
$27$	103069.	1.00775
$28$	189918.	1.63498
$29$	148628.	1.13164	0.565818	−	0.824530i	$-0.308561\pi$
$29$	148628.	1.13164	0.565818	+	0.824530i	$0.308561\pi$
$30$	−279588.	−1.89057
$31$	−99859.9	−0.602040	−0.301020	−	0.953618i	$-0.597327\pi$
$31$	−99859.9	−0.602040	−0.301020	+	0.953618i	$0.597327\pi$
$32$	−246669.	−1.33073
$33$	−188483.	−0.913004
$34$	84653.6	0.369376
$35$	682071.	2.68901
$36$	−3771.99	−0.0134745
$37$	165712.	0.537832	0.268916	−	0.963164i	$-0.413335\pi$
$37$	165712.	0.537832	0.268916	+	0.963164i	$0.413335\pi$
$38$	263290.	0.778379
$39$	0	0
$40$	−114511.	−0.282903
$41$	275758.	0.624862	0.312431	−	0.949941i	$-0.398857\pi$
$41$	275758.	0.624862	0.312431	+	0.949941i	$0.398857\pi$
$42$	−1.24446e6	−2.59183
$43$	399389.	0.766050	0.383025	−	0.923738i	$-0.374882\pi$
$43$	399389.	0.766050	0.383025	+	0.923738i	$0.374882\pi$
$44$	442823.	0.783693
$45$	−13546.7	−0.0221611
$46$	−1.18513e6	−1.79521
$47$	−1.12589e6	−1.58180	−0.790902	−	0.611943i	$-0.790388\pi$
$47$	−1.12589e6	−1.58180	−0.790902	+	0.611943i	$0.790388\pi$
$48$	856206.	1.11747
$49$	2.21238e6	2.68642
$50$	1.15636e6	1.30827
$51$	−255114.	−0.269301
$52$	0	0
$53$	343549.	0.316974	0.158487	−	0.987361i	$-0.449338\pi$
$53$	343549.	0.316974	0.158487	+	0.987361i	$0.449338\pi$
$54$	1.58672e6	1.37127
$55$	1.59036e6	1.28892
$56$	−509693.	−0.387838
$57$	−793454.	−0.567492
$58$	2.28809e6	1.53984
$59$	2.45965e6	1.55916	0.779580	−	0.626302i	$-0.215432\pi$
$59$	2.45965e6	1.55916	0.779580	+	0.626302i	$0.215432\pi$
$60$	−1.97954e6	−1.18314
$61$	414676.	0.233913	0.116957	−	0.993137i	$-0.462686\pi$
$61$	414676.	0.233913	0.116957	+	0.993137i	$0.462686\pi$
$62$	−1.53732e6	−0.819205
$63$	−60297.1	−0.0303812
$64$	−1.43515e6	−0.684334
$65$	0	0
$66$	−2.90164e6	−1.24234
$67$	3.22512e6	1.31004	0.655020	−	0.755612i	$-0.272660\pi$
$67$	3.22512e6	1.31004	0.655020	+	0.755612i	$0.272660\pi$
$68$	599367.	0.231159
$69$	3.57154e6	1.30883
$70$	1.05003e7	3.65897
$71$	3.55984e6	1.18039	0.590196	−	0.807260i	$-0.299050\pi$
$71$	3.55984e6	1.18039	0.590196	+	0.807260i	$0.299050\pi$
$72$	10123.1	0.00319632
$73$	−368557.	−0.110885	−0.0554427	−	0.998462i	$-0.517657\pi$
$73$	−368557.	−0.110885	−0.0554427	+	0.998462i	$0.517657\pi$
$74$	2.55109e6	0.731837
$75$	−3.48482e6	−0.953818
$76$	1.86415e6	0.487117
$77$	7.07874e6	1.76701
$78$	0	0
$79$	−3.22544e6	−0.736027	−0.368014	−	0.929820i	$-0.619962\pi$
$79$	−3.22544e6	−0.736027	−0.368014	+	0.929820i	$0.619962\pi$
$80$	−7.22440e6	−1.57756
$81$	−4.70609e6	−0.983926
$82$	4.24522e6	0.850260
$83$	211812.	0.0406610	0.0203305	−	0.999793i	$-0.493528\pi$
$83$	211812.	0.0406610	0.0203305	+	0.999793i	$0.493528\pi$
$84$	−8.81103e6	−1.62199
$85$	2.15257e6	0.380181
$86$	6.14850e6	1.04238
$87$	−6.89542e6	−1.12265
$88$	−1.18843e6	−0.185902
$89$	−723703.	−0.108817	−0.0544083	−	0.998519i	$-0.517327\pi$
$89$	−723703.	−0.108817	−0.0544083	+	0.998519i	$0.517327\pi$
$90$	−208549.	−0.0301550
$91$	0	0
$92$	−8.39102e6	−1.12346
$93$	4.63289e6	0.597258
$94$	−1.73328e7	−2.15239
$95$	6.69492e6	0.801148
$96$	1.14439e7	1.32016
$97$	−1.31082e7	−1.45828	−0.729140	−	0.684365i	$-0.760079\pi$
$97$	−1.31082e7	−1.45828	−0.729140	+	0.684365i	$0.760079\pi$
$98$	3.40591e7	3.65546
$99$	−140592.	−0.0145626
$100$	8.18727e6	0.818727
$101$	−8.64751e6	−0.835154	−0.417577	−	0.908642i	$-0.637121\pi$
$101$	−8.64751e6	−0.835154	−0.417577	+	0.908642i	$0.637121\pi$
$102$	−3.92741e6	−0.366442
$103$	−1.64656e7	−1.48473	−0.742365	−	0.669996i	$-0.766296\pi$
$103$	−1.64656e7	−1.48473	−0.742365	+	0.669996i	$0.766296\pi$
$104$	0	0
$105$	−3.16440e7	−2.66765
$106$	5.28886e6	0.431312
$107$	−2.75512e6	−0.217419	−0.108709	−	0.994074i	$-0.534672\pi$
$107$	−2.75512e6	−0.217419	−0.108709	+	0.994074i	$0.534672\pi$
$108$	1.12344e7	0.858153
$109$	−1.01827e7	−0.753133	−0.376566	−	0.926390i	$-0.622895\pi$
$109$	−1.01827e7	−0.753133	−0.376566	+	0.926390i	$0.622895\pi$
$110$	2.44832e7	1.75385
$111$	−7.68801e6	−0.533560
$112$	−3.21561e7	−2.16272
$113$	387303.	0.0252509	0.0126255	−	0.999920i	$-0.495981\pi$
$113$	387303.	0.0252509	0.0126255	+	0.999920i	$0.495981\pi$
$114$	−1.22150e7	−0.772196
$115$	−3.01355e7	−1.84772
$116$	1.62002e7	0.963644
$117$	0	0
$118$	3.78657e7	2.12158
$119$	9.58117e6	0.521200
$120$	5.31261e6	0.280656
$121$	−2.98196e6	−0.153022
$122$	6.38384e6	0.318290
$123$	−1.27935e7	−0.619898
$124$	−1.08846e7	−0.512667
$125$	−1.17880e6	−0.0539828
$126$	−928259.	−0.0413402
$127$	293630.	0.0127200	0.00636001	−	0.999980i	$-0.497976\pi$
$127$	293630.	0.0127200	0.00636001	+	0.999980i	$0.497976\pi$
$128$	9.47984e6	0.399546
$129$	−1.85292e7	−0.759965
$130$	0	0
$131$	−4.00277e6	−0.155565	−0.0777823	−	0.996970i	$-0.524784\pi$
$131$	−4.00277e6	−0.155565	−0.0777823	+	0.996970i	$0.524784\pi$
$132$	−2.05443e7	−0.777468
$133$	2.97994e7	1.09831
$134$	4.96500e7	1.78259
$135$	4.03471e7	1.41138
$136$	−1.60855e6	−0.0548340
$137$	−1.93045e6	−0.0641410	−0.0320705	−	0.999486i	$-0.510210\pi$
$137$	−1.93045e6	−0.0641410	−0.0320705	+	0.999486i	$0.510210\pi$
$138$	5.49830e7	1.78095
$139$	4.84164e7	1.52912	0.764558	−	0.644555i	$-0.222957\pi$
$139$	4.84164e7	1.52912	0.764558	+	0.644555i	$0.222957\pi$
$140$	7.43447e7	2.28982
$141$	5.22343e7	1.56924
$142$	5.48029e7	1.60618
$143$	0	0
$144$	638658.	0.0178238
$145$	5.81813e7	1.58488
$146$	−5.67384e6	−0.150884
$147$	−1.02641e8	−2.66508
$148$	1.80623e7	0.457991
$149$	6.37957e7	1.57993	0.789967	−	0.613149i	$-0.210097\pi$
$149$	6.37957e7	1.57993	0.789967	+	0.613149i	$0.210097\pi$
$150$	−5.36479e7	−1.29788
$151$	−4.18711e7	−0.989681	−0.494841	−	0.868984i	$-0.664774\pi$
$151$	−4.18711e7	−0.989681	−0.494841	+	0.868984i	$0.664774\pi$
$152$	−5.00293e6	−0.115550
$153$	−190293.	−0.00429540
$154$	1.08976e8	2.40440
$155$	−3.90909e7	−0.843168
$156$	0	0
$157$	3.62001e7	0.746554	0.373277	−	0.927720i	$-0.378234\pi$
$157$	3.62001e7	0.746554	0.373277	+	0.927720i	$0.378234\pi$
$158$	−4.96548e7	−1.00152
$159$	−1.59386e7	−0.314456
$160$	−9.65604e7	−1.86371
$161$	−1.34134e8	−2.53308
$162$	−7.24491e7	−1.33884
$163$	−6.89262e7	−1.24660	−0.623301	−	0.781982i	$-0.714209\pi$
$163$	−6.89262e7	−1.24660	−0.623301	+	0.781982i	$0.714209\pi$
$164$	3.00571e7	0.532101
$165$	−7.37829e7	−1.27868
$166$	3.26080e6	0.0553281
$167$	−6.74572e7	−1.12078	−0.560390	−	0.828229i	$-0.689349\pi$
$167$	−6.74572e7	−1.12078	−0.560390	+	0.828229i	$0.689349\pi$
$168$	2.36467e7	0.384758
$169$	0	0
$170$	3.31383e7	0.517319
$171$	−591851.	−0.00905161
$172$	4.35328e7	0.652329
$173$	9.65797e6	0.141816	0.0709079	−	0.997483i	$-0.477410\pi$
$173$	9.65797e6	0.141816	0.0709079	+	0.997483i	$0.477410\pi$
$174$	−1.06153e8	−1.52760
$175$	1.30877e8	1.84600
$176$	−7.49770e7	−1.03665
$177$	−1.14113e8	−1.54678
$178$	−1.11412e7	−0.148069
$179$	−7.17176e7	−0.934630	−0.467315	−	0.884091i	$-0.654779\pi$
$179$	−7.17176e7	−0.934630	−0.467315	+	0.884091i	$0.654779\pi$
$180$	−1.47657e6	−0.0188713
$181$	2.62286e7	0.328776	0.164388	−	0.986396i	$-0.447435\pi$
$181$	2.62286e7	0.328776	0.164388	+	0.986396i	$0.447435\pi$
$182$	0	0
$183$	−1.92385e7	−0.232055
$184$	2.25194e7	0.266499
$185$	6.48690e7	0.753244
$186$	7.13222e7	0.812698
$187$	2.23400e7	0.249826
$188$	−1.22720e8	−1.34698
$189$	1.79586e8	1.93489
$190$	1.03067e8	1.09014
$191$	6.39932e7	0.664534	0.332267	−	0.943185i	$-0.392187\pi$
$191$	6.39932e7	0.664534	0.332267	+	0.943185i	$0.392187\pi$
$192$	6.65823e7	0.678898
$193$	−1.72506e8	−1.72725	−0.863623	−	0.504138i	$-0.831810\pi$
$193$	−1.72506e8	−1.72725	−0.863623	+	0.504138i	$0.831810\pi$
$194$	−2.01797e8	−1.98430
$195$	0	0
$196$	2.41146e8	2.28762
$197$	2.61866e7	0.244032	0.122016	−	0.992528i	$-0.461064\pi$
$197$	2.61866e7	0.244032	0.122016	+	0.992528i	$0.461064\pi$
$198$	−2.16438e6	−0.0198155
$199$	−5.60235e7	−0.503947	−0.251973	−	0.967734i	$-0.581080\pi$
$199$	−5.60235e7	−0.503947	−0.251973	+	0.967734i	$0.581080\pi$
$200$	−2.19726e7	−0.194213
$201$	−1.49626e8	−1.29963
$202$	−1.33126e8	−1.13641
$203$	2.58967e8	2.17275
$204$	−2.78070e7	−0.229323
$205$	1.07947e8	0.875131
$206$	−2.53484e8	−2.02030
$207$	2.66407e6	0.0208761
$208$	0	0
$209$	6.94819e7	0.526453
$210$	−4.87151e8	−3.62991
$211$	2.11600e8	1.55070	0.775349	−	0.631533i	$-0.217574\pi$
$211$	2.11600e8	1.55070	0.775349	+	0.631533i	$0.217574\pi$
$212$	3.74463e7	0.269919
$213$	−1.65155e8	−1.17102
$214$	−4.24143e7	−0.295845
$215$	1.56344e8	1.07287
$216$	−3.01503e7	−0.203565
$217$	−1.73995e8	−1.15592
$218$	−1.56761e8	−1.02480
$219$	1.70988e7	0.110005
$220$	1.73346e8	1.09758
$221$	0	0
$222$	−1.18355e8	−0.726024
$223$	−2.35833e7	−0.142409	−0.0712046	−	0.997462i	$-0.522684\pi$
$223$	−2.35833e7	−0.142409	−0.0712046	+	0.997462i	$0.522684\pi$
$224$	−4.29794e8	−2.55501
$225$	−2.59938e6	−0.0152136
$226$	5.96244e6	0.0343593
$227$	2.56116e6	0.0145327	0.00726634	−	0.999974i	$-0.497687\pi$
$227$	2.56116e6	0.0145327	0.00726634	+	0.999974i	$0.497687\pi$
$228$	−8.64853e7	−0.483248
$229$	4.19687e7	0.230941	0.115471	−	0.993311i	$-0.463162\pi$
$229$	4.19687e7	0.230941	0.115471	+	0.993311i	$0.463162\pi$
$230$	−4.63929e8	−2.51422
$231$	−3.28411e8	−1.75297
$232$	−4.34773e7	−0.228589
$233$	−3.85893e7	−0.199858	−0.0999289	−	0.994995i	$-0.531862\pi$
$233$	−3.85893e7	−0.199858	−0.0999289	+	0.994995i	$0.531862\pi$
$234$	0	0
$235$	−4.40737e8	−2.21535
$236$	2.68098e8	1.32770
$237$	1.49641e8	0.730181
$238$	1.47500e8	0.709205
$239$	6.11018e7	0.289508	0.144754	−	0.989468i	$-0.453761\pi$
$239$	6.11018e7	0.289508	0.144754	+	0.989468i	$0.453761\pi$
$240$	3.35168e8	1.56503
$241$	1.81320e8	0.834422	0.417211	−	0.908810i	$-0.363008\pi$
$241$	1.81320e8	0.834422	0.417211	+	0.908810i	$0.363008\pi$
$242$	−4.59066e7	−0.208219
$243$	−7.07804e6	−0.0316440
$244$	4.51991e7	0.199189
$245$	8.66054e8	3.76239
$246$	−1.96952e8	−0.843506
$247$	0	0
$248$	2.92115e7	0.121611
$249$	−9.82680e6	−0.0403380
$250$	−1.81474e7	−0.0734553
$251$	−4.38290e8	−1.74946	−0.874730	−	0.484611i	$-0.838961\pi$
$251$	−4.38290e8	−1.74946	−0.874730	+	0.484611i	$0.838961\pi$
$252$	−6.57229e6	−0.0258711
$253$	−3.12756e8	−1.21418
$254$	4.52036e6	0.0173083
$255$	−9.98660e7	−0.377161
$256$	3.29639e8	1.22800
$257$	−7.24465e7	−0.266227	−0.133113	−	0.991101i	$-0.542497\pi$
$257$	−7.24465e7	−0.266227	−0.133113	+	0.991101i	$0.542497\pi$
$258$	−2.85253e8	−1.03410
$259$	2.88734e8	1.03264
$260$	0	0
$261$	−5.14340e6	−0.0179064
$262$	−6.16216e7	−0.211679
$263$	1.06467e8	0.360886	0.180443	−	0.983585i	$-0.442247\pi$
$263$	1.06467e8	0.360886	0.180443	+	0.983585i	$0.442247\pi$
$264$	5.51359e7	0.184425
$265$	1.34485e8	0.443928
$266$	4.58754e8	1.49449
$267$	3.35754e7	0.107952
$268$	3.51533e8	1.11556
$269$	−3.49491e8	−1.09472	−0.547359	−	0.836898i	$-0.684367\pi$
$269$	−3.49491e8	−1.09472	−0.547359	+	0.836898i	$0.684367\pi$
$270$	6.21133e8	1.92049
$271$	−4.77567e8	−1.45761	−0.728806	−	0.684720i	$-0.759925\pi$
$271$	−4.77567e8	−1.45761	−0.728806	+	0.684720i	$0.759925\pi$
$272$	−1.01482e8	−0.305773
$273$	0	0
$274$	−2.97187e7	−0.0872777
$275$	3.05161e8	0.884841
$276$	3.89292e8	1.11453
$277$	2.30507e8	0.651636	0.325818	−	0.945432i	$-0.394360\pi$
$277$	2.30507e8	0.651636	0.325818	+	0.945432i	$0.394360\pi$
$278$	7.45358e8	2.08069
$279$	3.45575e6	0.00952637
$280$	−1.99523e8	−0.543175
$281$	−1.05592e8	−0.283896	−0.141948	−	0.989874i	$-0.545337\pi$
$281$	−1.05592e8	−0.283896	−0.141948	+	0.989874i	$0.545337\pi$
$282$	8.04134e8	2.13529
$283$	1.68144e8	0.440991	0.220496	−	0.975388i	$-0.429233\pi$
$283$	1.68144e8	0.440991	0.220496	+	0.975388i	$0.429233\pi$
$284$	3.88017e8	1.00516
$285$	−3.10603e8	−0.794784
$286$	0	0
$287$	4.80477e8	1.19974
$288$	8.53622e6	0.0210568
$289$	−3.80101e8	−0.926311
$290$	8.95687e8	2.15657
$291$	6.08139e8	1.44670
$292$	−4.01721e7	−0.0944244
$293$	−3.47837e8	−0.807866	−0.403933	−	0.914789i	$-0.632357\pi$
$293$	−3.47837e8	−0.807866	−0.403933	+	0.914789i	$0.632357\pi$
$294$	−1.58013e9	−3.62642
$295$	9.62846e8	2.18363
$296$	−4.84748e7	−0.108641
$297$	4.18734e8	0.927451
$298$	9.82118e8	2.14984
$299$	0	0
$300$	−3.79839e8	−0.812224
$301$	6.95892e8	1.47082
$302$	−6.44596e8	−1.34668
$303$	4.01192e8	0.828520
$304$	−3.15630e8	−0.644349
$305$	1.62328e8	0.327600
$306$	−2.92952e6	−0.00584482
$307$	3.94077e8	0.777315	0.388658	−	0.921382i	$-0.372939\pi$
$307$	3.94077e8	0.777315	0.388658	+	0.921382i	$0.372939\pi$
$308$	7.71572e8	1.50470
$309$	7.63904e8	1.47294
$310$	−6.01794e8	−1.14731
$311$	−1.67240e8	−0.315267	−0.157634	−	0.987498i	$-0.550386\pi$
$311$	−1.67240e8	−0.315267	−0.157634	+	0.987498i	$0.550386\pi$
$312$	0	0
$313$	4.26545e8	0.786249	0.393124	−	0.919485i	$-0.371394\pi$
$313$	4.26545e8	0.786249	0.393124	+	0.919485i	$0.371394\pi$
$314$	5.57292e8	1.01585
$315$	−2.36037e7	−0.0425494
$316$	−3.51568e8	−0.626764
$317$	−6.23487e8	−1.09931	−0.549655	−	0.835392i	$-0.685241\pi$
$317$	−6.23487e8	−1.09931	−0.549655	+	0.835392i	$0.685241\pi$
$318$	−2.45371e8	−0.427886
$319$	6.03824e8	1.04146
$320$	−5.61800e8	−0.958423
$321$	1.27821e8	0.215692
$322$	−2.06497e9	−3.44681
$323$	9.40446e7	0.155283
$324$	−5.12956e8	−0.837862
$325$	0	0
$326$	−1.06110e9	−1.69627
$327$	4.72417e8	0.747150
$328$	−8.06660e7	−0.126221
$329$	−1.96174e9	−3.03707
$330$	−1.13587e9	−1.73992
$331$	−5.24479e8	−0.794932	−0.397466	−	0.917617i	$-0.630110\pi$
$331$	−5.24479e8	−0.794932	−0.397466	+	0.917617i	$0.630110\pi$
$332$	2.30872e7	0.0346249
$333$	−5.73461e6	−0.00851038
$334$	−1.03849e9	−1.52506
$335$	1.26250e9	1.83474
$336$	1.49185e9	2.14554
$337$	−8.13193e8	−1.15741	−0.578707	−	0.815535i	$-0.696443\pi$
$337$	−8.13193e8	−1.15741	−0.578707	+	0.815535i	$0.696443\pi$
$338$	0	0
$339$	−1.79685e7	−0.0250503
$340$	2.34626e8	0.323743
$341$	−4.05697e8	−0.554066
$342$	−9.11139e6	−0.0123167
$343$	2.41990e9	3.23794
$344$	−1.16831e8	−0.154741
$345$	1.39810e9	1.83304
$346$	1.48682e8	0.192971
$347$	6.91267e8	0.888163	0.444081	−	0.895986i	$-0.353530\pi$
$347$	6.91267e8	0.888163	0.444081	+	0.895986i	$0.353530\pi$
$348$	−7.51589e8	−0.955990
$349$	−9.35641e8	−1.17820	−0.589102	−	0.808059i	$-0.700518\pi$
$349$	−9.35641e8	−1.17820	−0.589102	+	0.808059i	$0.700518\pi$
$350$	2.01483e9	2.51188
$351$	0	0
$352$	−1.00213e9	−1.22469
$353$	−1.31459e9	−1.59067	−0.795335	−	0.606170i	$-0.792705\pi$
$353$	−1.31459e9	−1.59067	−0.795335	+	0.606170i	$0.792705\pi$
$354$	−1.75674e9	−2.10472
$355$	1.39353e9	1.65316
$356$	−7.88824e7	−0.0926628
$357$	−4.44508e8	−0.517059
$358$	−1.10407e9	−1.27177
$359$	1.38709e8	0.158225	0.0791124	−	0.996866i	$-0.474791\pi$
$359$	1.38709e8	0.158225	0.0791124	+	0.996866i	$0.474791\pi$
$360$	3.96276e6	0.00447651
$361$	−6.01374e8	−0.672775
$362$	4.03783e8	0.447371
$363$	1.38345e8	0.151806
$364$	0	0
$365$	−1.44274e8	−0.155297
$366$	−2.96171e8	−0.315762
$367$	1.64118e9	1.73311	0.866554	−	0.499084i	$-0.166330\pi$
$367$	1.64118e9	1.73311	0.866554	+	0.499084i	$0.166330\pi$
$368$	1.42073e9	1.48609
$369$	−9.54285e6	−0.00988749
$370$	9.98642e8	1.02495
$371$	5.98597e8	0.608592
$372$	5.04977e8	0.508595
$373$	−1.18561e8	−0.118293	−0.0591466	−	0.998249i	$-0.518838\pi$
$373$	−1.18561e8	−0.118293	−0.0591466	+	0.998249i	$0.518838\pi$
$374$	3.43919e8	0.339942
$375$	5.46892e7	0.0535540
$376$	3.29350e8	0.319522
$377$	0	0
$378$	2.76469e9	2.63284
$379$	−9.89138e8	−0.933297	−0.466648	−	0.884443i	$-0.654539\pi$
$379$	−9.89138e8	−0.933297	−0.466648	+	0.884443i	$0.654539\pi$
$380$	7.29735e8	0.682217
$381$	−1.36226e7	−0.0126190
$382$	9.85159e8	0.904242
$383$	−1.12236e8	−0.102079	−0.0510395	−	0.998697i	$-0.516253\pi$
$383$	−1.12236e8	−0.102079	−0.0510395	+	0.998697i	$0.516253\pi$
$384$	−4.39807e8	−0.396372
$385$	2.77102e9	2.47473
$386$	−2.65569e9	−2.35029
$387$	−1.38213e7	−0.0121216
$388$	−1.42877e9	−1.24180
$389$	−1.22645e9	−1.05640	−0.528199	−	0.849120i	$-0.677133\pi$
$389$	−1.22645e9	−1.05640	−0.528199	+	0.849120i	$0.677133\pi$
$390$	0	0
$391$	−4.23318e8	−0.358136
$392$	−6.47178e8	−0.542653
$393$	1.85704e8	0.154329
$394$	4.03136e8	0.332059
$395$	−1.26262e9	−1.03082
$396$	−1.53243e7	−0.0124008
$397$	−3.43926e8	−0.275866	−0.137933	−	0.990442i	$-0.544046\pi$
$397$	−3.43926e8	−0.275866	−0.137933	+	0.990442i	$0.544046\pi$
$398$	−8.62468e8	−0.685729
$399$	−1.38251e9	−1.08959
$400$	−1.38623e9	−1.08300
$401$	−2.01823e9	−1.56303	−0.781513	−	0.623889i	$-0.785552\pi$
$401$	−2.01823e9	−1.56303	−0.781513	+	0.623889i	$0.785552\pi$
$402$	−2.30346e9	−1.76843
$403$	0	0
$404$	−9.42565e8	−0.711175
$405$	−1.84223e9	−1.37801
$406$	3.98674e9	2.95649
$407$	6.73230e8	0.494975
$408$	7.46271e7	0.0543984
$409$	5.88275e8	0.425157	0.212578	−	0.977144i	$-0.431814\pi$
$409$	5.88275e8	0.425157	0.212578	+	0.977144i	$0.431814\pi$
$410$	1.66182e9	1.19081
$411$	8.95609e7	0.0636315
$412$	−1.79472e9	−1.26432
$413$	4.28567e9	2.99360
$414$	4.10127e7	0.0284064
$415$	8.29155e7	0.0569465
$416$	0	0
$417$	−2.24622e9	−1.51697
$418$	1.06966e9	0.716353
$419$	−1.20796e9	−0.802236	−0.401118	−	0.916026i	$-0.631378\pi$
$419$	−1.20796e9	−0.802236	−0.401118	+	0.916026i	$0.631378\pi$
$420$	−3.44914e9	−2.27163
$421$	−6.43226e8	−0.420123	−0.210061	−	0.977688i	$-0.567366\pi$
$421$	−6.43226e8	−0.420123	−0.210061	+	0.977688i	$0.567366\pi$
$422$	3.25753e9	2.11006
$423$	3.89624e7	0.0250296
$424$	−1.00497e8	−0.0640283
$425$	4.13040e8	0.260994
$426$	−2.54252e9	−1.59342
$427$	7.22529e8	0.449115
$428$	−3.00303e8	−0.185143
$429$	0	0
$430$	2.40687e9	1.45987
$431$	3.28675e9	1.97741	0.988704	−	0.149884i	$-0.0478901\pi$
$431$	3.28675e9	1.97741	0.988704	+	0.149884i	$0.0478901\pi$
$432$	−1.90215e9	−1.13515
$433$	−6.43138e8	−0.380712	−0.190356	−	0.981715i	$-0.560964\pi$
$433$	−6.43138e8	−0.380712	−0.190356	+	0.981715i	$0.560964\pi$
$434$	−2.67861e9	−1.57288
$435$	−2.69926e9	−1.57229
$436$	−1.10990e9	−0.641330
$437$	−1.31661e9	−0.754693
$438$	2.63231e8	0.149685
$439$	−2.16062e8	−0.121886	−0.0609428	−	0.998141i	$-0.519411\pi$
$439$	−2.16062e8	−0.121886	−0.0609428	+	0.998141i	$0.519411\pi$
$440$	−4.65220e8	−0.260360
$441$	−7.65617e7	−0.0425086
$442$	0	0
$443$	−7.03717e8	−0.384579	−0.192289	−	0.981338i	$-0.561591\pi$
$443$	−7.03717e8	−0.384579	−0.192289	+	0.981338i	$0.561591\pi$
$444$	−8.37980e8	−0.454353
$445$	−2.83299e8	−0.152400
$446$	−3.63060e8	−0.193779
$447$	−2.95973e9	−1.56738
$448$	−2.50060e9	−1.31392
$449$	−2.59771e9	−1.35434	−0.677171	−	0.735826i	$-0.736794\pi$
$449$	−2.59771e9	−1.35434	−0.677171	+	0.735826i	$0.736794\pi$
$450$	−4.00168e7	−0.0207014
$451$	1.12031e9	0.575069
$452$	4.22155e7	0.0215024
$453$	1.94256e9	0.981820
$454$	3.94284e7	0.0197749
$455$	0	0
$456$	2.32105e8	0.114633
$457$	−2.24698e9	−1.10126	−0.550632	−	0.834748i	$-0.685613\pi$
$457$	−2.24698e9	−1.10126	−0.550632	+	0.834748i	$0.685613\pi$
$458$	6.46098e8	0.314246
$459$	5.66762e8	0.273562
$460$	−3.28472e9	−1.57343
$461$	−9.55280e8	−0.454127	−0.227063	−	0.973880i	$-0.572912\pi$
$461$	−9.55280e8	−0.454127	−0.227063	+	0.973880i	$0.572912\pi$
$462$	−5.05580e9	−2.38530
$463$	3.41522e9	1.59914	0.799569	−	0.600575i	$-0.205061\pi$
$463$	3.41522e9	1.59914	0.799569	+	0.600575i	$0.205061\pi$
$464$	−2.74294e9	−1.27469
$465$	1.81358e9	0.836471
$466$	−5.94073e8	−0.271950
$467$	1.28591e9	0.584254	0.292127	−	0.956380i	$-0.405637\pi$
$467$	1.28591e9	0.584254	0.292127	+	0.956380i	$0.405637\pi$
$468$	0	0
$469$	5.61942e9	2.51528
$470$	−6.78503e9	−3.01446
$471$	−1.67946e9	−0.740624
$472$	−7.19509e8	−0.314948
$473$	1.62258e9	0.705006
$474$	2.30368e9	0.993569
$475$	1.28464e9	0.549987
$476$	1.04433e9	0.443827
$477$	−1.18889e7	−0.00501563
$478$	9.40647e8	0.393939
$479$	3.08956e9	1.28446	0.642232	−	0.766510i	$-0.278008\pi$
$479$	3.08956e9	1.28446	0.642232	+	0.766510i	$0.278008\pi$
$480$	4.47981e9	1.84891
$481$	0	0
$482$	2.79138e9	1.13541
$483$	6.22302e9	2.51296
$484$	−3.25029e8	−0.130306
$485$	−5.13128e9	−2.04235
$486$	−1.08965e8	−0.0430585
$487$	−4.40727e9	−1.72909	−0.864546	−	0.502555i	$-0.832394\pi$
$487$	−4.40727e9	−1.72909	−0.864546	+	0.502555i	$0.832394\pi$
$488$	−1.21303e8	−0.0472502
$489$	3.19776e9	1.23670
$490$	1.33327e10	5.11954
$491$	5.27327e8	0.201046	0.100523	−	0.994935i	$-0.467948\pi$
$491$	5.27327e8	0.201046	0.100523	+	0.994935i	$0.467948\pi$
$492$	−1.39447e9	−0.527874
$493$	8.17283e8	0.307191
$494$	0	0
$495$	−5.50358e7	−0.0203952
$496$	1.84293e9	0.678145
$497$	6.20264e9	2.26636
$498$	−1.51281e8	−0.0548886
$499$	4.53287e9	1.63313	0.816566	−	0.577252i	$-0.195875\pi$
$499$	4.53287e9	1.63313	0.816566	+	0.577252i	$0.195875\pi$
$500$	−1.28487e8	−0.0459691
$501$	3.12960e9	1.11188
$502$	−6.74737e9	−2.38052
$503$	−1.82521e9	−0.639478	−0.319739	−	0.947506i	$-0.603595\pi$
$503$	−1.82521e9	−0.639478	−0.319739	+	0.947506i	$0.603595\pi$
$504$	1.76384e7	0.00613695
$505$	−3.38513e9	−1.16965
$506$	−4.81479e9	−1.65216
$507$	0	0
$508$	3.20052e7	0.0108317
$509$	2.42206e9	0.814089	0.407045	−	0.913408i	$-0.366559\pi$
$509$	2.42206e9	0.814089	0.407045	+	0.913408i	$0.366559\pi$
$510$	−1.53741e9	−0.513210
$511$	−6.42170e8	−0.212901
$512$	3.86130e9	1.27142
$513$	1.76274e9	0.576472
$514$	−1.11530e9	−0.362259
$515$	−6.44558e9	−2.07939
$516$	−2.01966e9	−0.647148
$517$	−4.57410e9	−1.45576
$518$	4.44499e9	1.40513
$519$	−4.48071e8	−0.140689
$520$	0	0
$521$	2.47043e9	0.765315	0.382658	−	0.923890i	$-0.375009\pi$
$521$	2.47043e9	0.765315	0.382658	+	0.923890i	$0.375009\pi$
$522$	−7.91814e7	−0.0243656
$523$	1.46575e9	0.448027	0.224013	−	0.974586i	$-0.428084\pi$
$523$	1.46575e9	0.448027	0.224013	+	0.974586i	$0.428084\pi$
$524$	−4.36295e8	−0.132471
$525$	−6.07191e9	−1.83134
$526$	1.63904e9	0.491064
$527$	−5.49116e8	−0.163428
$528$	3.47847e9	1.02842
$529$	2.52155e9	0.740580
$530$	2.07036e9	0.604061
$531$	−8.51185e7	−0.0246713
$532$	3.24808e9	0.935268
$533$	0	0
$534$	5.16885e8	0.146892
$535$	−1.07851e9	−0.304499
$536$	−9.43429e8	−0.264626
$537$	3.32726e9	0.927206
$538$	−5.38032e9	−1.48960
$539$	8.98817e9	2.47235
$540$	4.39777e9	1.20186
$541$	−2.22699e9	−0.604683	−0.302342	−	0.953200i	$-0.597768\pi$
$541$	−2.22699e9	−0.604683	−0.302342	+	0.953200i	$0.597768\pi$
$542$	−7.35203e9	−1.98340
$543$	−1.21685e9	−0.326165
$544$	−1.35640e9	−0.361236
$545$	−3.98610e9	−1.05478
$546$	0	0
$547$	−1.77912e9	−0.464781	−0.232391	−	0.972623i	$-0.574655\pi$
$547$	−1.77912e9	−0.464781	−0.232391	+	0.972623i	$0.574655\pi$
$548$	−2.10415e8	−0.0546193
$549$	−1.43503e7	−0.00370132
$550$	4.69788e9	1.20402
$551$	2.54191e9	0.647336
$552$	−1.04476e9	−0.264382
$553$	−5.61997e9	−1.41318
$554$	3.54860e9	0.886692
$555$	−3.00952e9	−0.747261
$556$	5.27731e9	1.30212
$557$	3.58268e9	0.878445	0.439222	−	0.898378i	$-0.355254\pi$
$557$	3.58268e9	0.878445	0.439222	+	0.898378i	$0.355254\pi$
$558$	5.32004e7	0.0129627
$559$	0	0
$560$	−1.25877e10	−3.02893
$561$	−1.03644e9	−0.247842
$562$	−1.62557e9	−0.386302
$563$	5.14381e9	1.21480	0.607401	−	0.794396i	$-0.292212\pi$
$563$	5.14381e9	1.21480	0.607401	+	0.794396i	$0.292212\pi$
$564$	5.69346e9	1.33629
$565$	1.51613e8	0.0353644
$566$	2.58854e9	0.600064
$567$	−8.19985e9	−1.88914
$568$	−1.04134e9	−0.238438
$569$	−7.75397e9	−1.76454	−0.882269	−	0.470745i	$-0.843985\pi$
$569$	−7.75397e9	−1.76454	−0.882269	+	0.470745i	$0.843985\pi$
$570$	−4.78166e9	−1.08148
$571$	−1.59816e9	−0.359249	−0.179624	−	0.983735i	$-0.557488\pi$
$571$	−1.59816e9	−0.359249	−0.179624	+	0.983735i	$0.557488\pi$
$572$	0	0
$573$	−2.96889e9	−0.659255
$574$	7.39683e9	1.63250
$575$	−5.78247e9	−1.26846
$576$	4.96648e7	0.0108285
$577$	3.75139e9	0.812974	0.406487	−	0.913657i	$-0.366754\pi$
$577$	3.75139e9	0.812974	0.406487	+	0.913657i	$0.366754\pi$
$578$	−5.85156e9	−1.26045
$579$	8.00324e9	1.71353
$580$	6.34167e9	1.34960
$581$	3.69060e8	0.0780693
$582$	9.36214e9	1.96854
$583$	1.39572e9	0.291716
$584$	1.07812e8	0.0223987
$585$	0	0
$586$	−5.35487e9	−1.09928
$587$	−6.89634e9	−1.40730	−0.703648	−	0.710548i	$-0.748447\pi$
$587$	−6.89634e9	−1.40730	−0.703648	+	0.710548i	$0.748447\pi$
$588$	−1.11877e10	−2.26945
$589$	−1.70786e9	−0.344389
$590$	1.48228e10	2.97131
$591$	−1.21490e9	−0.242094
$592$	−3.05823e9	−0.605821
$593$	4.03921e9	0.795435	0.397718	−	0.917508i	$-0.369802\pi$
$593$	4.03921e9	0.795435	0.397718	+	0.917508i	$0.369802\pi$
$594$	6.44631e9	1.26200
$595$	3.75062e9	0.729950
$596$	6.95362e9	1.34539
$597$	2.59915e9	0.499944
$598$	0	0
$599$	−6.23761e9	−1.18584	−0.592918	−	0.805263i	$-0.702024\pi$
$599$	−6.23761e9	−1.18584	−0.592918	+	0.805263i	$0.702024\pi$
$600$	1.01940e9	0.192670
$601$	5.89985e9	1.10861	0.554307	−	0.832312i	$-0.312983\pi$
$601$	5.89985e9	1.10861	0.554307	+	0.832312i	$0.312983\pi$
$602$	1.07131e10	2.00137
$603$	−1.11608e8	−0.0207294
$604$	−4.56389e9	−0.842763
$605$	−1.16731e9	−0.214310
$606$	6.17625e9	1.12738
$607$	8.39238e8	0.152309	0.0761543	−	0.997096i	$-0.475736\pi$
$607$	8.39238e8	0.152309	0.0761543	+	0.997096i	$0.475736\pi$
$608$	−4.21867e9	−0.761226
$609$	−1.20145e10	−2.15549
$610$	2.49900e9	0.445771
$611$	0	0
$612$	−2.07417e7	−0.00365775
$613$	−3.62141e9	−0.634988	−0.317494	−	0.948260i	$-0.602841\pi$
$613$	−3.62141e9	−0.634988	−0.317494	+	0.948260i	$0.602841\pi$
$614$	6.06672e9	1.05771
$615$	−5.00809e9	−0.868180
$616$	−2.07071e9	−0.356933
$617$	−6.19739e8	−0.106221	−0.0531105	−	0.998589i	$-0.516914\pi$
$617$	−6.19739e8	−0.106221	−0.0531105	+	0.998589i	$0.516914\pi$
$618$	1.17601e10	2.00425
$619$	4.55108e9	0.771254	0.385627	−	0.922655i	$-0.373985\pi$
$619$	4.55108e9	0.771254	0.385627	+	0.922655i	$0.373985\pi$
$620$	−4.26084e9	−0.718000
$621$	−7.93455e9	−1.32954
$622$	−2.57462e9	−0.428989
$623$	−1.26097e9	−0.208929
$624$	0	0
$625$	−6.32971e9	−1.03706
$626$	6.56656e9	1.06986
$627$	−3.22354e9	−0.522271
$628$	3.94575e9	0.635728
$629$	9.11225e8	0.145998
$630$	−3.63374e8	−0.0578977
$631$	8.13531e9	1.28905	0.644527	−	0.764581i	$-0.277054\pi$
$631$	8.13531e9	1.28905	0.644527	+	0.764581i	$0.277054\pi$
$632$	9.43522e8	0.148676
$633$	−9.81695e9	−1.53838
$634$	−9.59843e9	−1.49585
$635$	1.14944e8	0.0178146
$636$	−1.73728e9	−0.267775
$637$	0	0
$638$	9.29571e9	1.41713
$639$	−1.23192e8	−0.0186779
$640$	3.71095e9	0.559571
$641$	1.58796e9	0.238142	0.119071	−	0.992886i	$-0.462008\pi$
$641$	1.58796e9	0.238142	0.119071	+	0.992886i	$0.462008\pi$
$642$	1.96777e9	0.293495
$643$	1.12354e10	1.66667	0.833335	−	0.552768i	$-0.186428\pi$
$643$	1.12354e10	1.66667	0.833335	+	0.552768i	$0.186428\pi$
$644$	−1.46204e10	−2.15705
$645$	−7.25340e9	−1.06435
$646$	1.44779e9	0.211297
$647$	−9.93285e9	−1.44181	−0.720906	−	0.693033i	$-0.756274\pi$
$647$	−9.93285e9	−1.44181	−0.720906	+	0.693033i	$0.756274\pi$
$648$	1.37665e9	0.198752
$649$	9.99271e9	1.43492
$650$	0	0
$651$	8.07231e9	1.14674
$652$	−7.51285e9	−1.06154
$653$	6.28507e8	0.0883311	0.0441656	−	0.999024i	$-0.485937\pi$
$653$	6.28507e8	0.0883311	0.0441656	+	0.999024i	$0.485937\pi$
$654$	7.27274e9	1.01666
$655$	−1.56691e9	−0.217871
$656$	−5.08915e9	−0.703852
$657$	1.27543e7	0.00175459
$658$	−3.02004e10	−4.13259
$659$	1.23032e10	1.67463	0.837315	−	0.546721i	$-0.184124\pi$
$659$	1.23032e10	1.67463	0.837315	+	0.546721i	$0.184124\pi$
$660$	−8.04222e9	−1.08886
$661$	1.18989e10	1.60252	0.801259	−	0.598318i	$-0.204164\pi$
$661$	1.18989e10	1.60252	0.801259	+	0.598318i	$0.204164\pi$
$662$	−8.07422e9	−1.08168
$663$	0	0
$664$	−6.19604e7	−0.00821346
$665$	1.16652e10	1.53821
$666$	−8.82829e7	−0.0115802
$667$	−1.14418e10	−1.49298
$668$	−7.35272e9	−0.954401
$669$	1.09412e9	0.141278
$670$	1.94358e10	2.49656
$671$	1.68469e9	0.215274
$672$	1.99398e10	2.53471
$673$	−1.44651e9	−0.182923	−0.0914613	−	0.995809i	$-0.529154\pi$
$673$	−1.44651e9	−0.182923	−0.0914613	+	0.995809i	$0.529154\pi$
$674$	−1.25189e10	−1.57491
$675$	7.74189e9	0.968911
$676$	0	0
$677$	−6.60976e9	−0.818701	−0.409351	−	0.912377i	$-0.634245\pi$
$677$	−6.60976e9	−0.818701	−0.409351	+	0.912377i	$0.634245\pi$
$678$	−2.76621e8	−0.0340864
$679$	−2.28395e10	−2.79990
$680$	−6.29680e8	−0.0767961
$681$	−1.18822e8	−0.0144172
$682$	−6.24560e9	−0.753926
$683$	2.92558e9	0.351349	0.175675	−	0.984448i	$-0.443789\pi$
$683$	2.92558e9	0.351349	0.175675	+	0.984448i	$0.443789\pi$
$684$	−6.45108e7	−0.00770789
$685$	−7.55687e8	−0.0898307
$686$	3.72538e10	4.40592
$687$	−1.94709e9	−0.229107
$688$	−7.37079e9	−0.862888
$689$	0	0
$690$	2.15235e10	2.49425
$691$	1.15973e9	0.133716	0.0668579	−	0.997763i	$-0.478703\pi$
$691$	1.15973e9	0.133716	0.0668579	+	0.997763i	$0.478703\pi$
$692$	1.05270e9	0.120763
$693$	−2.44967e8	−0.0279602
$694$	1.06419e10	1.20854
$695$	1.89529e10	2.14156
$696$	2.01708e9	0.226773
$697$	1.51635e9	0.169623
$698$	−1.44040e10	−1.60320
$699$	1.79031e9	0.198270
$700$	1.42654e10	1.57196
$701$	5.69383e8	0.0624297	0.0312149	−	0.999513i	$-0.490062\pi$
$701$	5.69383e8	0.0624297	0.0312149	+	0.999513i	$0.490062\pi$
$702$	0	0
$703$	2.83409e9	0.307659
$704$	−5.83054e9	−0.629802
$705$	2.04475e10	2.19775
$706$	−2.02378e10	−2.16445
$707$	−1.50673e10	−1.60350
$708$	−1.24381e10	−1.31716
$709$	−1.83393e10	−1.93250	−0.966252	−	0.257600i	$-0.917068\pi$
$709$	−1.83393e10	−1.93250	−0.966252	+	0.257600i	$0.917068\pi$
$710$	2.14530e10	2.24949
$711$	1.11619e8	0.0116465
$712$	2.11701e8	0.0219808
$713$	7.68751e9	0.794277
$714$	−6.84309e9	−0.703571
$715$	0	0
$716$	−7.81710e9	−0.795884
$717$	−2.83475e9	−0.287209
$718$	2.13539e9	0.215299
$719$	−2.83967e9	−0.284916	−0.142458	−	0.989801i	$-0.545501\pi$
$719$	−2.83967e9	−0.284916	−0.142458	+	0.989801i	$0.545501\pi$
$720$	2.50007e8	0.0249625
$721$	−2.86895e10	−2.85069
$722$	−9.25801e9	−0.915456
$723$	−8.41214e9	−0.827794
$724$	2.85888e9	0.279969
$725$	1.11640e10	1.08802
$726$	2.12978e9	0.206565
$727$	−6.58137e9	−0.635251	−0.317626	−	0.948216i	$-0.602886\pi$
$727$	−6.58137e9	−0.635251	−0.317626	+	0.948216i	$0.602886\pi$
$728$	0	0
$729$	1.06206e10	1.01532
$730$	−2.22106e9	−0.211315
$731$	2.19619e9	0.207950
$732$	−2.09696e9	−0.197607
$733$	−2.01542e10	−1.89017	−0.945087	−	0.326818i	$-0.894023\pi$
$733$	−2.01542e10	−1.89017	−0.945087	+	0.326818i	$0.894023\pi$
$734$	2.52656e10	2.35827
$735$	−4.01796e10	−3.73250
$736$	1.89893e10	1.75565
$737$	1.31026e10	1.20565
$738$	−1.46910e8	−0.0134541
$739$	−1.77114e10	−1.61435	−0.807176	−	0.590311i	$-0.799005\pi$
$739$	−1.77114e10	−1.61435	−0.807176	+	0.590311i	$0.799005\pi$
$740$	7.07061e9	0.641425
$741$	0	0
$742$	9.21526e9	0.828121
$743$	1.00196e10	0.896164	0.448082	−	0.893992i	$-0.352107\pi$
$743$	1.00196e10	0.896164	0.448082	+	0.893992i	$0.352107\pi$
$744$	−1.35524e9	−0.120645
$745$	2.49733e10	2.21273
$746$	−1.82521e9	−0.160963
$747$	−7.32997e6	−0.000643399 0
$748$	2.43502e9	0.212739
$749$	−4.80049e9	−0.417445
$750$	8.41926e8	0.0728718
$751$	2.67720e9	0.230643	0.115322	−	0.993328i	$-0.463210\pi$
$751$	2.67720e9	0.230643	0.115322	+	0.993328i	$0.463210\pi$
$752$	2.07784e10	1.78176
$753$	2.03340e10	1.73556
$754$	0	0
$755$	−1.63908e10	−1.38607
$756$	1.95746e10	1.64766
$757$	−2.31024e10	−1.93563	−0.967815	−	0.251662i	$-0.919023\pi$
$757$	−2.31024e10	−1.93563	−0.967815	+	0.251662i	$0.919023\pi$
$758$	−1.52275e10	−1.26995
$759$	1.45099e10	1.20454
$760$	−1.95843e9	−0.161831
$761$	−9.32276e9	−0.766829	−0.383414	−	0.923576i	$-0.625252\pi$
$761$	−9.32276e9	−0.766829	−0.383414	+	0.923576i	$0.625252\pi$
$762$	−2.09717e8	−0.0171709
$763$	−1.77423e10	−1.44602
$764$	6.97515e9	0.565883
$765$	−7.44917e7	−0.00601579
$766$	−1.72784e9	−0.138901
$767$	0	0
$768$	−1.52933e10	−1.21825
$769$	−1.29575e10	−1.02749	−0.513745	−	0.857943i	$-0.671742\pi$
$769$	−1.29575e10	−1.02749	−0.513745	+	0.857943i	$0.671742\pi$
$770$	4.26592e10	3.36741
$771$	3.36108e9	0.264112
$772$	−1.88029e10	−1.47084
$773$	1.43468e10	1.11719	0.558594	−	0.829441i	$-0.311341\pi$
$773$	1.43468e10	1.11719	0.558594	+	0.829441i	$0.311341\pi$
$774$	−2.12775e8	−0.0164940
$775$	−7.50084e9	−0.578834
$776$	3.83447e9	0.294570
$777$	−1.33955e10	−1.02444
$778$	−1.88810e10	−1.43746
$779$	4.71616e9	0.357443
$780$	0	0
$781$	1.44624e10	1.08633
$782$	−6.51688e9	−0.487322
$783$	1.53189e10	1.14041
$784$	−4.08299e10	−3.02602
$785$	1.41708e10	1.04556
$786$	2.85887e9	0.209998
$787$	2.67966e7	0.00195960	0.000979800	−	1.00000i	$-0.499688\pi$
$787$	2.67966e7	0.00195960	0.000979800		1.00000i	$0.499688\pi$
$788$	2.85430e9	0.207806
$789$	−4.93943e9	−0.358020
$790$	−1.94377e10	−1.40265
$791$	6.74834e8	0.0484819
$792$	4.11268e7	0.00294162
$793$	0	0
$794$	−5.29466e9	−0.375376
$795$	−6.23928e9	−0.440402
$796$	−6.10648e9	−0.429136
$797$	2.23570e10	1.56426	0.782130	−	0.623116i	$-0.214133\pi$
$797$	2.23570e10	1.56426	0.782130	+	0.623116i	$0.214133\pi$
$798$	−2.12834e10	−1.48262
$799$	−6.19110e9	−0.429392
$800$	−1.85282e10	−1.27944
$801$	2.50444e7	0.00172186
$802$	−3.10702e10	−2.12684
$803$	−1.49732e9	−0.102049
$804$	−1.63090e10	−1.10670
$805$	−5.25079e10	−3.54763
$806$	0	0
$807$	1.62142e10	1.08602
$808$	2.52961e9	0.168700
$809$	2.09973e10	1.39426	0.697130	−	0.716944i	$-0.254460\pi$
$809$	2.09973e10	1.39426	0.697130	+	0.716944i	$0.254460\pi$
$810$	−2.83607e10	−1.87508
$811$	−1.13973e10	−0.750292	−0.375146	−	0.926966i	$-0.622407\pi$
$811$	−1.13973e10	−0.750292	−0.375146	+	0.926966i	$0.622407\pi$
$812$	2.82270e10	1.85020
$813$	2.21562e10	1.44603
$814$	1.03642e10	0.673520
$815$	−2.69817e10	−1.74589
$816$	4.70816e9	0.303344
$817$	6.83058e9	0.438208
$818$	9.05636e9	0.578518
$819$	0	0
$820$	1.17661e10	0.745218
$821$	−2.52022e10	−1.58941	−0.794707	−	0.606993i	$-0.792376\pi$
$821$	−2.52022e10	−1.58941	−0.794707	+	0.606993i	$0.792376\pi$
$822$	1.37877e9	0.0865844
$823$	2.03342e10	1.27153	0.635765	−	0.771883i	$-0.280685\pi$
$823$	2.03342e10	1.27153	0.635765	+	0.771883i	$0.280685\pi$
$824$	4.81660e9	0.299913
$825$	−1.41576e10	−0.877812
$826$	6.59768e10	4.07344
$827$	2.63091e10	1.61747	0.808735	−	0.588174i	$-0.200153\pi$
$827$	2.63091e10	1.61747	0.808735	+	0.588174i	$0.200153\pi$
$828$	2.90379e8	0.0177770
$829$	2.51842e10	1.53528	0.767641	−	0.640880i	$-0.221431\pi$
$829$	2.51842e10	1.53528	0.767641	+	0.640880i	$0.221431\pi$
$830$	1.27646e9	0.0774881
$831$	−1.06941e10	−0.646460
$832$	0	0
$833$	1.21656e10	0.729249
$834$	−3.45801e10	−2.06417
$835$	−2.64066e10	−1.56968
$836$	7.57341e9	0.448301
$837$	−1.02925e10	−0.606708
$838$	−1.85962e10	−1.09162
$839$	1.42680e10	0.834055	0.417028	−	0.908894i	$-0.363072\pi$
$839$	1.42680e10	0.834055	0.417028	+	0.908894i	$0.363072\pi$
$840$	9.25665e9	0.538861
$841$	4.84030e9	0.280599
$842$	−9.90230e9	−0.571668
$843$	4.89883e9	0.281641
$844$	2.30641e10	1.32050
$845$	0	0
$846$	5.99817e8	0.0340583
$847$	−5.19574e9	−0.293803
$848$	−6.34025e9	−0.357044
$849$	−7.80088e9	−0.437488
$850$	6.35864e9	0.355139
$851$	−1.27570e10	−0.709568
$852$	−1.80016e10	−0.997179
$853$	−3.11924e10	−1.72078	−0.860392	−	0.509632i	$-0.829781\pi$
$853$	−3.11924e10	−1.72078	−0.860392	+	0.509632i	$0.829781\pi$
$854$	1.11231e10	0.611118
$855$	−2.31684e8	−0.0126770
$856$	8.05941e8	0.0439182
$857$	2.06191e10	1.11902	0.559509	−	0.828824i	$-0.310990\pi$
$857$	2.06191e10	1.11902	0.559509	+	0.828824i	$0.310990\pi$
$858$	0	0
$859$	1.79442e10	0.965935	0.482967	−	0.875638i	$-0.339559\pi$
$859$	1.79442e10	0.965935	0.482967	+	0.875638i	$0.339559\pi$
$860$	1.70412e10	0.913600
$861$	−2.22912e10	−1.19021
$862$	5.05987e10	2.69069
$863$	1.86934e10	0.990036	0.495018	−	0.868883i	$-0.335161\pi$
$863$	1.86934e10	0.990036	0.495018	+	0.868883i	$0.335161\pi$
$864$	−2.54239e10	−1.34105
$865$	3.78068e9	0.198616
$866$	−9.90095e9	−0.518041
$867$	1.76344e10	0.918953
$868$	−1.89652e10	−0.984323
$869$	−1.31039e10	−0.677376
$870$	−4.15544e10	−2.13944
$871$	0	0
$872$	2.97870e9	0.152132
$873$	4.53620e8	0.0230751
$874$	−2.02688e10	−1.02692
$875$	−2.05393e9	−0.103647
$876$	1.86374e9	0.0936744
$877$	2.23792e10	1.12033	0.560164	−	0.828382i	$-0.310738\pi$
$877$	2.23792e10	1.12033	0.560164	+	0.828382i	$0.310738\pi$
$878$	−3.32622e9	−0.165852
$879$	1.61375e10	0.801449
$880$	−2.93503e10	−1.45185
$881$	−1.28714e10	−0.634174	−0.317087	−	0.948396i	$-0.602705\pi$
$881$	−1.28714e10	−0.634174	−0.317087	+	0.948396i	$0.602705\pi$
$882$	−1.17865e9	−0.0578421
$883$	2.70734e10	1.32337	0.661685	−	0.749782i	$-0.269842\pi$
$883$	2.70734e10	1.32337	0.661685	+	0.749782i	$0.269842\pi$
$884$	0	0
$885$	−4.46702e10	−2.16629
$886$	−1.08336e10	−0.523303
$887$	5.45686e9	0.262549	0.131274	−	0.991346i	$-0.458093\pi$
$887$	5.45686e9	0.262549	0.131274	+	0.991346i	$0.458093\pi$
$888$	2.24893e9	0.107778
$889$	5.11619e8	0.0244225
$890$	−4.36131e9	−0.207373
$891$	−1.91192e10	−0.905521
$892$	−2.57055e9	−0.121269
$893$	−1.92556e10	−0.904849
$894$	−4.55643e10	−2.13277
$895$	−2.80743e10	−1.30897
$896$	1.65176e10	0.767130
$897$	0	0
$898$	−3.99911e10	−1.84288
$899$	−1.48419e10	−0.681290
$900$	−2.83328e8	−0.0129551
$901$	1.88913e9	0.0860449
$902$	1.72469e10	0.782506
$903$	−3.22852e10	−1.45914
$904$	−1.13296e8	−0.00510065
$905$	1.02674e10	0.460458
$906$	2.99053e10	1.33598
$907$	1.25119e10	0.556796	0.278398	−	0.960466i	$-0.410197\pi$
$907$	1.25119e10	0.556796	0.278398	+	0.960466i	$0.410197\pi$
$908$	2.79162e8	0.0123753
$909$	2.99255e8	0.0132150
$910$	0	0
$911$	2.67041e10	1.17021	0.585106	−	0.810957i	$-0.301053\pi$
$911$	2.67041e10	1.17021	0.585106	+	0.810957i	$0.301053\pi$
$912$	1.46433e10	0.639231
$913$	8.60522e8	0.0374209
$914$	−3.45916e10	−1.49851
$915$	−7.53103e9	−0.324998
$916$	4.57453e9	0.196658
$917$	−6.97438e9	−0.298685
$918$	8.72516e9	0.372241
$919$	3.23480e9	0.137481	0.0687406	−	0.997635i	$-0.478102\pi$
$919$	3.23480e9	0.137481	0.0687406	+	0.997635i	$0.478102\pi$
$920$	8.81540e9	0.373237
$921$	−1.82828e10	−0.771141
$922$	−1.47063e10	−0.617938
$923$	0	0
$924$	−3.57962e10	−1.49274
$925$	1.24472e10	0.517101
$926$	5.25765e10	2.17597
$927$	5.69808e8	0.0234936
$928$	−3.66619e10	−1.50590
$929$	3.49201e10	1.42896	0.714480	−	0.699656i	$-0.246663\pi$
$929$	3.49201e10	1.42896	0.714480	+	0.699656i	$0.246663\pi$
$930$	2.79196e10	1.13820
$931$	3.78374e10	1.53673
$932$	−4.20617e9	−0.170189
$933$	7.75891e9	0.312763
$934$	1.97963e10	0.795004
$935$	8.74515e9	0.349886
$936$	0	0
$937$	−3.59978e10	−1.42951	−0.714756	−	0.699374i	$-0.753462\pi$
$937$	−3.59978e10	−1.42951	−0.714756	+	0.699374i	$0.753462\pi$
$938$	8.65096e10	3.42259
$939$	−1.97891e10	−0.780003
$940$	−4.80396e10	−1.88648
$941$	3.00932e10	1.17735	0.588674	−	0.808371i	$-0.299650\pi$
$941$	3.00932e10	1.17735	0.588674	+	0.808371i	$0.299650\pi$
$942$	−2.58549e10	−1.00778
$943$	−2.12286e10	−0.824387
$944$	−4.53931e10	−1.75626
$945$	7.03004e10	2.70986
$946$	2.49793e10	0.959314
$947$	−1.15710e10	−0.442737	−0.221368	−	0.975190i	$-0.571052\pi$
$947$	−1.15710e10	−0.442737	−0.221368	+	0.975190i	$0.571052\pi$
$948$	1.63106e10	0.621785
$949$	0	0
$950$	1.97767e10	0.748377
$951$	2.89260e10	1.09058
$952$	−2.80273e9	−0.105282
$953$	2.16596e10	0.810636	0.405318	−	0.914176i	$-0.367161\pi$
$953$	2.16596e10	0.810636	0.405318	+	0.914176i	$0.367161\pi$
$954$	−1.83026e8	−0.00682486
$955$	2.50506e10	0.930692
$956$	6.66000e9	0.246531
$957$	−2.80137e10	−1.03319
$958$	4.75630e10	1.74779
$959$	−3.36359e9	−0.123151
$960$	2.60641e10	0.950810
$961$	−1.75406e10	−0.637548
$962$	0	0
$963$	9.53435e7	0.00344032
$964$	1.97636e10	0.710552
$965$	−6.75288e10	−2.41904
$966$	9.58018e10	3.41943
$967$	1.28792e10	0.458031	0.229016	−	0.973423i	$-0.426449\pi$
$967$	1.28792e10	0.458031	0.229016	+	0.973423i	$0.426449\pi$
$968$	8.72299e8	0.0309102
$969$	−4.36310e9	−0.154050
$970$	−7.89948e10	−2.77906
$971$	−2.88983e10	−1.01299	−0.506495	−	0.862243i	$-0.669059\pi$
$971$	−2.88983e10	−1.01299	−0.506495	+	0.862243i	$0.669059\pi$
$972$	−7.71495e8	−0.0269464
$973$	8.43603e10	2.93591
$974$	−6.78488e10	−2.35280
$975$	0	0
$976$	−7.65291e9	−0.263483
$977$	−3.58124e10	−1.22858	−0.614288	−	0.789082i	$-0.710557\pi$
$977$	−3.58124e10	−1.22858	−0.614288	+	0.789082i	$0.710557\pi$
$978$	4.92287e10	1.68280
$979$	−2.94016e9	−0.100145
$980$	9.43984e10	3.20386
$981$	3.52383e8	0.0119172
$982$	8.11806e9	0.273566
$983$	1.35420e10	0.454723	0.227361	−	0.973810i	$-0.426990\pi$
$983$	1.35420e10	0.454723	0.227361	+	0.973810i	$0.426990\pi$
$984$	3.74241e9	0.125219
$985$	1.02509e10	0.341772
$986$	1.25819e10	0.417999
$987$	9.10126e10	3.01295
$988$	0	0
$989$	−3.07462e10	−1.01066
$990$	−8.47263e8	−0.0277521
$991$	1.54107e10	0.502997	0.251498	−	0.967858i	$-0.419077\pi$
$991$	1.54107e10	0.502997	0.251498	+	0.967858i	$0.419077\pi$
$992$	2.46324e10	0.801152
$993$	2.43326e10	0.788618
$994$	9.54880e10	3.08387
$995$	−2.19308e10	−0.705788
$996$	−1.07111e9	−0.0343498
$997$	2.02192e10	0.646147	0.323074	−	0.946374i	$-0.395284\pi$
$997$	2.02192e10	0.646147	0.323074	+	0.946374i	$0.395284\pi$
$998$	6.97824e10	2.22223
$999$	1.70797e10	0.542003

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.8.a.i.1.17	yes	21
13.5	odd	4		169.8.b.f.168.8		42
13.8	odd	4		169.8.b.f.168.35		42
13.12	even	2		169.8.a.h.1.5	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.8.a.h.1.5	✓	21	13.12	even	2
169.8.a.i.1.17	yes	21	1.1	even	1	trivial
169.8.b.f.168.8		42	13.5	odd	4
169.8.b.f.168.35		42	13.8	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

\(f(q)\)	\(=\)	\(q+15.3948 q^{2} -46.3939 q^{3} +108.998 q^{4} +391.457 q^{5} -714.223 q^{6} +1742.39 q^{7} -292.525 q^{8} -34.6060 q^{9} +6026.39 q^{10} +4062.66 q^{11} -5056.86 q^{12} +26823.7 q^{14} -18161.2 q^{15} -18455.1 q^{16} +5498.86 q^{17} -532.750 q^{18} +17102.6 q^{19} +42668.2 q^{20} -80836.3 q^{21} +62543.6 q^{22} -76982.9 q^{23} +13571.4 q^{24} +75113.7 q^{25} +103069. q^{27} +189918. q^{28} +148628. q^{29} -279588. q^{30} -99859.9 q^{31} -246669. q^{32} -188483. q^{33} +84653.6 q^{34} +682071. q^{35} -3771.99 q^{36} +165712. q^{37} +263290. q^{38} -114511. q^{40} +275758. q^{41} -1.24446e6 q^{42} +399389. q^{43} +442823. q^{44} -13546.7 q^{45} -1.18513e6 q^{46} -1.12589e6 q^{47} +856206. q^{48} +2.21238e6 q^{49} +1.15636e6 q^{50} -255114. q^{51} +343549. q^{53} +1.58672e6 q^{54} +1.59036e6 q^{55} -509693. q^{56} -793454. q^{57} +2.28809e6 q^{58} +2.45965e6 q^{59} -1.97954e6 q^{60} +414676. q^{61} -1.53732e6 q^{62} -60297.1 q^{63} -1.43515e6 q^{64} -2.90164e6 q^{66} +3.22512e6 q^{67} +599367. q^{68} +3.57154e6 q^{69} +1.05003e7 q^{70} +3.55984e6 q^{71} +10123.1 q^{72} -368557. q^{73} +2.55109e6 q^{74} -3.48482e6 q^{75} +1.86415e6 q^{76} +7.07874e6 q^{77} -3.22544e6 q^{79} -7.22440e6 q^{80} -4.70609e6 q^{81} +4.24522e6 q^{82} +211812. q^{83} -8.81103e6 q^{84} +2.15257e6 q^{85} +6.14850e6 q^{86} -6.89542e6 q^{87} -1.18843e6 q^{88} -723703. q^{89} -208549. q^{90} -8.39102e6 q^{92} +4.63289e6 q^{93} -1.73328e7 q^{94} +6.69492e6 q^{95} +1.14439e7 q^{96} -1.31082e7 q^{97} +3.40591e7 q^{98} -140592. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 31 q^{2} - 26 q^{3} + 1409 q^{4} + 680 q^{5} + 1470 q^{6} + 2929 q^{7} + 4716 q^{8} + 15465 q^{9} - 5167 q^{10} + 14824 q^{11} + 21795 q^{12} - 179 q^{14} + 36398 q^{15} + 113205 q^{16} + 45016 q^{17}+ \cdots + 37605493 q^{99}+O(q^{100}) \) 21 * q + 31 * q^2 - 26 * q^3 + 1409 * q^4 + 680 * q^5 + 1470 * q^6 + 2929 * q^7 + 4716 * q^8 + 15465 * q^9 - 5167 * q^10 + 14824 * q^11 + 21795 * q^12 - 179 * q^14 + 36398 * q^15 + 113205 * q^16 + 45016 * q^17 + 72967 * q^18 + 119316 * q^19 + 131130 * q^20 + 94178 * q^21 - 274174 * q^22 - 128105 * q^23 + 316902 * q^24 + 366537 * q^25 - 106643 * q^27 + 170352 * q^28 - 62329 * q^29 + 9761 * q^30 + 467367 * q^31 + 677464 * q^32 + 587690 * q^33 + 578614 * q^34 + 848860 * q^35 - 274245 * q^36 + 697064 * q^37 + 225831 * q^38 - 2288827 * q^40 + 2351315 * q^41 + 2736836 * q^42 + 894489 * q^43 - 263025 * q^44 - 1938345 * q^45 - 3170392 * q^46 + 1070934 * q^47 + 6302388 * q^48 + 131530 * q^49 + 3970474 * q^50 - 1184341 * q^51 - 4550027 * q^53 + 7916233 * q^54 + 5524030 * q^55 + 2334386 * q^56 + 5590305 * q^57 + 7510959 * q^58 + 10746233 * q^59 + 15186203 * q^60 + 5717012 * q^61 - 13593653 * q^62 + 22347508 * q^63 + 9767924 * q^64 - 17806977 * q^66 + 11305923 * q^67 + 12252492 * q^68 + 2772206 * q^69 + 28670666 * q^70 + 23267025 * q^71 + 49774322 * q^72 + 4763727 * q^73 + 9625078 * q^74 - 43212397 * q^75 + 63359948 * q^76 + 9446036 * q^77 - 18613281 * q^79 + 57904093 * q^80 + 41055449 * q^81 + 24190327 * q^82 + 21455789 * q^83 + 60768429 * q^84 + 26216249 * q^85 + 35632347 * q^86 + 9027894 * q^87 - 75678180 * q^88 + 54037507 * q^89 + 3335560 * q^90 - 69339390 * q^92 + 20085986 * q^93 + 28955713 * q^94 + 17660031 * q^95 + 66667956 * q^96 + 10916609 * q^97 + 41738840 * q^98 + 37605493 * q^99

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