LMFDB - Embedded newform 8009.2.a.b.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8009,2,Mod(1,8009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8009, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8009 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8009.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:	$63.9521869788$
Analytic rank:	$0$
Dimension:	$361$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	8009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69966 q^{2} +0.961365 q^{3} +5.28818 q^{4} +0.760429 q^{5} -2.59536 q^{6} -4.15994 q^{7} -8.87697 q^{8} -2.07578 q^{9} +O(q^{10})$	\(q-2.69966 q^{2} +0.961365 q^{3} +5.28818 q^{4} +0.760429 q^{5} -2.59536 q^{6} -4.15994 q^{7} -8.87697 q^{8} -2.07578 q^{9} -2.05290 q^{10} +1.40672 q^{11} +5.08387 q^{12} -5.02835 q^{13} +11.2304 q^{14} +0.731050 q^{15} +13.3885 q^{16} +3.13519 q^{17} +5.60390 q^{18} -5.59467 q^{19} +4.02129 q^{20} -3.99922 q^{21} -3.79767 q^{22} +0.397352 q^{23} -8.53401 q^{24} -4.42175 q^{25} +13.5748 q^{26} -4.87967 q^{27} -21.9985 q^{28} -0.0786862 q^{29} -1.97359 q^{30} -5.10988 q^{31} -18.3904 q^{32} +1.35237 q^{33} -8.46395 q^{34} -3.16334 q^{35} -10.9771 q^{36} -6.84271 q^{37} +15.1037 q^{38} -4.83408 q^{39} -6.75031 q^{40} +0.453162 q^{41} +10.7965 q^{42} +6.93489 q^{43} +7.43899 q^{44} -1.57848 q^{45} -1.07272 q^{46} -1.21566 q^{47} +12.8712 q^{48} +10.3051 q^{49} +11.9372 q^{50} +3.01406 q^{51} -26.5908 q^{52} -1.68351 q^{53} +13.1735 q^{54} +1.06971 q^{55} +36.9276 q^{56} -5.37852 q^{57} +0.212426 q^{58} -3.61128 q^{59} +3.86592 q^{60} +3.10320 q^{61} +13.7949 q^{62} +8.63510 q^{63} +22.8710 q^{64} -3.82371 q^{65} -3.65095 q^{66} -10.7034 q^{67} +16.5794 q^{68} +0.382001 q^{69} +8.53995 q^{70} -8.50214 q^{71} +18.4266 q^{72} +7.48300 q^{73} +18.4730 q^{74} -4.25091 q^{75} -29.5856 q^{76} -5.85187 q^{77} +13.0504 q^{78} +0.587347 q^{79} +10.1810 q^{80} +1.53618 q^{81} -1.22338 q^{82} +5.77193 q^{83} -21.1486 q^{84} +2.38409 q^{85} -18.7219 q^{86} -0.0756461 q^{87} -12.4874 q^{88} -12.7033 q^{89} +4.26137 q^{90} +20.9176 q^{91} +2.10127 q^{92} -4.91246 q^{93} +3.28187 q^{94} -4.25435 q^{95} -17.6799 q^{96} -7.77601 q^{97} -27.8202 q^{98} -2.92004 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9}+O(q^{10}) $ 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9	\( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9} + 65 q^{10} + 33 q^{11} + 52 q^{12} + 89 q^{13} + 32 q^{14} + 55 q^{15} + 512 q^{16} + 42 q^{17} + 34 q^{18} + 191 q^{19} + 48 q^{20} + 53 q^{21} + 61 q^{22} + 52 q^{23} + 139 q^{24} + 458 q^{25} + 57 q^{26} + 80 q^{27} + 194 q^{28} + 47 q^{29} + 32 q^{30} + 254 q^{31} + 55 q^{32} + 40 q^{33} + 122 q^{34} + 93 q^{35} + 519 q^{36} + 43 q^{37} + 25 q^{38} + 210 q^{39} + 184 q^{40} + 54 q^{41} + 48 q^{42} + 151 q^{43} + 56 q^{44} + 82 q^{45} + 101 q^{46} + 117 q^{47} + 77 q^{48} + 563 q^{49} + 38 q^{50} + 143 q^{51} + 241 q^{52} + 14 q^{53} + 164 q^{54} + 452 q^{55} + 52 q^{56} + 21 q^{57} + 55 q^{58} + 125 q^{59} + 39 q^{60} + 227 q^{61} + 58 q^{62} + 292 q^{63} + 710 q^{64} + 15 q^{65} + 105 q^{66} + 120 q^{67} + 125 q^{68} + 136 q^{69} + 88 q^{70} + 105 q^{71} + 78 q^{72} + 108 q^{73} + 41 q^{74} + 128 q^{75} + 461 q^{76} + 28 q^{77} + 13 q^{78} + 400 q^{79} + 59 q^{80} + 485 q^{81} + 175 q^{82} + 97 q^{83} + 76 q^{84} + 144 q^{85} - 14 q^{86} + 327 q^{87} + 145 q^{88} + 52 q^{89} + 60 q^{90} + 192 q^{91} + 11 q^{92} + 32 q^{93} + 366 q^{94} + 182 q^{95} + 275 q^{96} + 117 q^{97} + 42 q^{98} + 111 q^{99}+O(q^{100}) \) 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9 + 65 * q^10 + 33 * q^11 + 52 * q^12 + 89 * q^13 + 32 * q^14 + 55 * q^15 + 512 * q^16 + 42 * q^17 + 34 * q^18 + 191 * q^19 + 48 * q^20 + 53 * q^21 + 61 * q^22 + 52 * q^23 + 139 * q^24 + 458 * q^25 + 57 * q^26 + 80 * q^27 + 194 * q^28 + 47 * q^29 + 32 * q^30 + 254 * q^31 + 55 * q^32 + 40 * q^33 + 122 * q^34 + 93 * q^35 + 519 * q^36 + 43 * q^37 + 25 * q^38 + 210 * q^39 + 184 * q^40 + 54 * q^41 + 48 * q^42 + 151 * q^43 + 56 * q^44 + 82 * q^45 + 101 * q^46 + 117 * q^47 + 77 * q^48 + 563 * q^49 + 38 * q^50 + 143 * q^51 + 241 * q^52 + 14 * q^53 + 164 * q^54 + 452 * q^55 + 52 * q^56 + 21 * q^57 + 55 * q^58 + 125 * q^59 + 39 * q^60 + 227 * q^61 + 58 * q^62 + 292 * q^63 + 710 * q^64 + 15 * q^65 + 105 * q^66 + 120 * q^67 + 125 * q^68 + 136 * q^69 + 88 * q^70 + 105 * q^71 + 78 * q^72 + 108 * q^73 + 41 * q^74 + 128 * q^75 + 461 * q^76 + 28 * q^77 + 13 * q^78 + 400 * q^79 + 59 * q^80 + 485 * q^81 + 175 * q^82 + 97 * q^83 + 76 * q^84 + 144 * q^85 - 14 * q^86 + 327 * q^87 + 145 * q^88 + 52 * q^89 + 60 * q^90 + 192 * q^91 + 11 * q^92 + 32 * q^93 + 366 * q^94 + 182 * q^95 + 275 * q^96 + 117 * q^97 + 42 * q^98 + 111 * 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 $

$2$	−2.69966	−1.90895	−0.954475	−	0.298291i	$-0.903583\pi$
$2$	−2.69966	−1.90895	−0.954475	+	0.298291i	$0.903583\pi$
$3$	0.961365	0.555044	0.277522	−	0.960719i	$-0.410487\pi$
$3$	0.961365	0.555044	0.277522	+	0.960719i	$0.410487\pi$
$4$	5.28818	2.64409
$5$	0.760429	0.340074	0.170037	−	0.985438i	$-0.445611\pi$
$5$	0.760429	0.340074	0.170037	+	0.985438i	$0.445611\pi$
$6$	−2.59536	−1.05955
$7$	−4.15994	−1.57231	−0.786154	−	0.618030i	$-0.787931\pi$
$7$	−4.15994	−1.57231	−0.786154	+	0.618030i	$0.787931\pi$
$8$	−8.87697	−3.13848
$9$	−2.07578	−0.691926
$10$	−2.05290	−0.649185
$11$	1.40672	0.424142	0.212071	−	0.977254i	$-0.431979\pi$
$11$	1.40672	0.424142	0.212071	+	0.977254i	$0.431979\pi$
$12$	5.08387	1.46759
$13$	−5.02835	−1.39461	−0.697307	−	0.716773i	$-0.745618\pi$
$13$	−5.02835	−1.39461	−0.697307	+	0.716773i	$0.745618\pi$
$14$	11.2304	3.00146
$15$	0.731050	0.188756
$16$	13.3885	3.34712
$17$	3.13519	0.760395	0.380198	−	0.924905i	$-0.375856\pi$
$17$	3.13519	0.760395	0.380198	+	0.924905i	$0.375856\pi$
$18$	5.60390	1.32085
$19$	−5.59467	−1.28351	−0.641753	−	0.766911i	$-0.721793\pi$
$19$	−5.59467	−1.28351	−0.641753	+	0.766911i	$0.721793\pi$
$20$	4.02129	0.899187
$21$	−3.99922	−0.872701
$22$	−3.79767	−0.809666
$23$	0.397352	0.0828537	0.0414268	−	0.999142i	$-0.486810\pi$
$23$	0.397352	0.0828537	0.0414268	+	0.999142i	$0.486810\pi$
$24$	−8.53401	−1.74200
$25$	−4.42175	−0.884349
$26$	13.5748	2.66225
$27$	−4.87967	−0.939094
$28$	−21.9985	−4.15732
$29$	−0.0786862	−0.0146117	−0.00730583	−	0.999973i	$-0.502326\pi$
$29$	−0.0786862	−0.0146117	−0.00730583	+	0.999973i	$0.502326\pi$
$30$	−1.97359	−0.360326
$31$	−5.10988	−0.917761	−0.458881	−	0.888498i	$-0.651749\pi$
$31$	−5.10988	−0.917761	−0.458881	+	0.888498i	$0.651749\pi$
$32$	−18.3904	−3.25100
$33$	1.35237	0.235418
$34$	−8.46395	−1.45156
$35$	−3.16334	−0.534702
$36$	−10.9771	−1.82951
$37$	−6.84271	−1.12493	−0.562467	−	0.826820i	$-0.690148\pi$
$37$	−6.84271	−1.12493	−0.562467	+	0.826820i	$0.690148\pi$
$38$	15.1037	2.45015
$39$	−4.83408	−0.774072
$40$	−6.75031	−1.06732
$41$	0.453162	0.0707720	0.0353860	−	0.999374i	$-0.488734\pi$
$41$	0.453162	0.0707720	0.0353860	+	0.999374i	$0.488734\pi$
$42$	10.7965	1.66594
$43$	6.93489	1.05756	0.528780	−	0.848759i	$-0.322650\pi$
$43$	6.93489	1.05756	0.528780	+	0.848759i	$0.322650\pi$
$44$	7.43899	1.12147
$45$	−1.57848	−0.235306
$46$	−1.07272	−0.158164
$47$	−1.21566	−0.177322	−0.0886610	−	0.996062i	$-0.528259\pi$
$47$	−1.21566	−0.177322	−0.0886610	+	0.996062i	$0.528259\pi$
$48$	12.8712	1.85780
$49$	10.3051	1.47215
$50$	11.9372	1.68818
$51$	3.01406	0.422053
$52$	−26.5908	−3.68748
$53$	−1.68351	−0.231248	−0.115624	−	0.993293i	$-0.536887\pi$
$53$	−1.68351	−0.231248	−0.115624	+	0.993293i	$0.536887\pi$
$54$	13.1735	1.79268
$55$	1.06971	0.144240
$56$	36.9276	4.93466
$57$	−5.37852	−0.712403
$58$	0.212426	0.0278929
$59$	−3.61128	−0.470148	−0.235074	−	0.971977i	$-0.575533\pi$
$59$	−3.61128	−0.470148	−0.235074	+	0.971977i	$0.575533\pi$
$60$	3.86592	0.499089
$61$	3.10320	0.397325	0.198662	−	0.980068i	$-0.436340\pi$
$61$	3.10320	0.397325	0.198662	+	0.980068i	$0.436340\pi$
$62$	13.7949	1.75196
$63$	8.63510	1.08792
$64$	22.8710	2.85887
$65$	−3.82371	−0.474272
$66$	−3.65095	−0.449400
$67$	−10.7034	−1.30763	−0.653816	−	0.756653i	$-0.726833\pi$
$67$	−10.7034	−1.30763	−0.653816	+	0.756653i	$0.726833\pi$
$68$	16.5794	2.01055
$69$	0.382001	0.0459875
$70$	8.53995	1.02072
$71$	−8.50214	−1.00902	−0.504509	−	0.863406i	$-0.668326\pi$
$71$	−8.50214	−1.00902	−0.504509	+	0.863406i	$0.668326\pi$
$72$	18.4266	2.17160
$73$	7.48300	0.875818	0.437909	−	0.899019i	$-0.355719\pi$
$73$	7.48300	0.875818	0.437909	+	0.899019i	$0.355719\pi$
$74$	18.4730	2.14744
$75$	−4.25091	−0.490853
$76$	−29.5856	−3.39370
$77$	−5.85187	−0.666882
$78$	13.0504	1.47767
$79$	0.587347	0.0660817	0.0330409	−	0.999454i	$-0.489481\pi$
$79$	0.587347	0.0660817	0.0330409	+	0.999454i	$0.489481\pi$
$80$	10.1810	1.13827
$81$	1.53618	0.170687
$82$	−1.22338	−0.135100
$83$	5.77193	0.633551	0.316776	−	0.948501i	$-0.397400\pi$
$83$	5.77193	0.633551	0.316776	+	0.948501i	$0.397400\pi$
$84$	−21.1486	−2.30750
$85$	2.38409	0.258591
$86$	−18.7219	−2.01883
$87$	−0.0756461	−0.00811012
$88$	−12.4874	−1.33116
$89$	−12.7033	−1.34655	−0.673274	−	0.739393i	$-0.735112\pi$
$89$	−12.7033	−1.34655	−0.673274	+	0.739393i	$0.735112\pi$
$90$	4.26137	0.449188
$91$	20.9176	2.19276
$92$	2.10127	0.219072
$93$	−4.91246	−0.509398
$94$	3.28187	0.338499
$95$	−4.25435	−0.436488
$96$	−17.6799	−1.80445
$97$	−7.77601	−0.789534	−0.394767	−	0.918781i	$-0.629175\pi$
$97$	−7.77601	−0.789534	−0.394767	+	0.918781i	$0.629175\pi$
$98$	−27.8202	−2.81027
$99$	−2.92004	−0.293475
$100$	−23.3830	−2.33830
$101$	2.70624	0.269281	0.134640	−	0.990895i	$-0.457012\pi$
$101$	2.70624	0.269281	0.134640	+	0.990895i	$0.457012\pi$
$102$	−8.13695	−0.805678
$103$	0.0573895	0.00565476	0.00282738	−	0.999996i	$-0.499100\pi$
$103$	0.0573895	0.00565476	0.00282738	+	0.999996i	$0.499100\pi$
$104$	44.6365	4.37697
$105$	−3.04112	−0.296783
$106$	4.54491	0.441441
$107$	7.02038	0.678686	0.339343	−	0.940663i	$-0.389795\pi$
$107$	7.02038	0.678686	0.339343	+	0.940663i	$0.389795\pi$
$108$	−25.8046	−2.48305
$109$	−1.54124	−0.147624	−0.0738122	−	0.997272i	$-0.523517\pi$
$109$	−1.54124	−0.147624	−0.0738122	+	0.997272i	$0.523517\pi$
$110$	−2.88786	−0.275347
$111$	−6.57834	−0.624388
$112$	−55.6952	−5.26270
$113$	−0.141995	−0.0133577	−0.00667886	−	0.999978i	$-0.502126\pi$
$113$	−0.141995	−0.0133577	−0.00667886	+	0.999978i	$0.502126\pi$
$114$	14.5202	1.35994
$115$	0.302158	0.0281764
$116$	−0.416107	−0.0386345
$117$	10.4377	0.964969
$118$	9.74923	0.897489
$119$	−13.0422	−1.19558
$120$	−6.48951	−0.592409
$121$	−9.02114	−0.820104
$122$	−8.37761	−0.758473
$123$	0.435654	0.0392816
$124$	−27.0219	−2.42664
$125$	−7.16457	−0.640819
$126$	−23.3119	−2.07679
$127$	−5.06800	−0.449712	−0.224856	−	0.974392i	$-0.572191\pi$
$127$	−5.06800	−0.449712	−0.224856	+	0.974392i	$0.572191\pi$
$128$	−24.9630	−2.20644
$129$	6.66696	0.586993
$130$	10.3227	0.905362
$131$	−9.01506	−0.787649	−0.393825	−	0.919186i	$-0.628848\pi$
$131$	−9.01506	−0.787649	−0.393825	+	0.919186i	$0.628848\pi$
$132$	7.15158	0.622465
$133$	23.2735	2.01807
$134$	28.8957	2.49621
$135$	−3.71065	−0.319362
$136$	−27.8310	−2.38649
$137$	−13.4902	−1.15254	−0.576272	−	0.817258i	$-0.695493\pi$
$137$	−13.4902	−1.15254	−0.576272	+	0.817258i	$0.695493\pi$
$138$	−1.03127	−0.0877878
$139$	9.84268	0.834845	0.417423	−	0.908712i	$-0.362933\pi$
$139$	9.84268	0.834845	0.417423	+	0.908712i	$0.362933\pi$
$140$	−16.7283	−1.41380
$141$	−1.16869	−0.0984216
$142$	22.9529	1.92616
$143$	−7.07348	−0.591514
$144$	−27.7915	−2.31596
$145$	−0.0598353	−0.00496905
$146$	−20.2016	−1.67189
$147$	9.90695	0.817111
$148$	−36.1854	−2.97443
$149$	12.6957	1.04007	0.520037	−	0.854144i	$-0.325918\pi$
$149$	12.6957	1.04007	0.520037	+	0.854144i	$0.325918\pi$
$150$	11.4760	0.937014
$151$	0.317612	0.0258469	0.0129235	−	0.999916i	$-0.495886\pi$
$151$	0.317612	0.0258469	0.0129235	+	0.999916i	$0.495886\pi$
$152$	49.6638	4.02826
$153$	−6.50795	−0.526137
$154$	15.7981	1.27304
$155$	−3.88570	−0.312107
$156$	−25.5635	−2.04672
$157$	−24.6482	−1.96714	−0.983571	−	0.180520i	$-0.942222\pi$
$157$	−24.6482	−1.96714	−0.983571	+	0.180520i	$0.942222\pi$
$158$	−1.58564	−0.126147
$159$	−1.61847	−0.128353
$160$	−13.9846	−1.10558
$161$	−1.65296	−0.130272
$162$	−4.14717	−0.325833
$163$	−13.1271	−1.02819	−0.514096	−	0.857733i	$-0.671873\pi$
$163$	−13.1271	−1.02819	−0.514096	+	0.857733i	$0.671873\pi$
$164$	2.39640	0.187128
$165$	1.02838	0.0800595
$166$	−15.5823	−1.20942
$167$	−2.42966	−0.188013	−0.0940064	−	0.995572i	$-0.529967\pi$
$167$	−2.42966	−0.188013	−0.0940064	+	0.995572i	$0.529967\pi$
$168$	35.5009	2.73896
$169$	12.2843	0.944946
$170$	−6.43624	−0.493637
$171$	11.6133	0.888091
$172$	36.6729	2.79628
$173$	−16.5323	−1.25693	−0.628463	−	0.777839i	$-0.716316\pi$
$173$	−16.5323	−1.25693	−0.628463	+	0.777839i	$0.716316\pi$
$174$	0.204219	0.0154818
$175$	18.3942	1.39047
$176$	18.8338	1.41965
$177$	−3.47175	−0.260953
$178$	34.2946	2.57049
$179$	−3.30223	−0.246820	−0.123410	−	0.992356i	$-0.539383\pi$
$179$	−3.30223	−0.246820	−0.123410	+	0.992356i	$0.539383\pi$
$180$	−8.34729	−0.622171
$181$	25.7026	1.91046	0.955231	−	0.295861i	$-0.0956066\pi$
$181$	25.7026	1.91046	0.955231	+	0.295861i	$0.0956066\pi$
$182$	−56.4705	−4.18587
$183$	2.98331	0.220533
$184$	−3.52728	−0.260035
$185$	−5.20339	−0.382561
$186$	13.2620	0.972416
$187$	4.41033	0.322516
$188$	−6.42862	−0.468855
$189$	20.2991	1.47655
$190$	11.4853	0.833233
$191$	24.2259	1.75293	0.876463	−	0.481470i	$-0.159897\pi$
$191$	24.2259	1.75293	0.876463	+	0.481470i	$0.159897\pi$
$192$	21.9873	1.58680
$193$	−15.4434	−1.11164	−0.555821	−	0.831302i	$-0.687596\pi$
$193$	−15.4434	−1.11164	−0.555821	+	0.831302i	$0.687596\pi$
$194$	20.9926	1.50718
$195$	−3.67598	−0.263242
$196$	54.4951	3.89251
$197$	22.6253	1.61199	0.805995	−	0.591923i	$-0.201631\pi$
$197$	22.6253	1.61199	0.805995	+	0.591923i	$0.201631\pi$
$198$	7.88312	0.560229
$199$	23.8023	1.68730	0.843651	−	0.536892i	$-0.180402\pi$
$199$	23.8023	1.68730	0.843651	+	0.536892i	$0.180402\pi$
$200$	39.2517	2.77552
$201$	−10.2899	−0.725794
$202$	−7.30593	−0.514043
$203$	0.327330	0.0229740
$204$	15.9389	1.11595
$205$	0.344598	0.0240678
$206$	−0.154932	−0.0107946
$207$	−0.824815	−0.0573286
$208$	−67.3219	−4.66793
$209$	−7.87014	−0.544389
$210$	8.21001	0.566544
$211$	23.0936	1.58983	0.794913	−	0.606723i	$-0.207516\pi$
$211$	23.0936	1.58983	0.794913	+	0.606723i	$0.207516\pi$
$212$	−8.90271	−0.611441
$213$	−8.17366	−0.560050
$214$	−18.9527	−1.29558
$215$	5.27349	0.359649
$216$	43.3167	2.94733
$217$	21.2568	1.44300
$218$	4.16084	0.281807
$219$	7.19389	0.486118
$220$	5.65682	0.381383
$221$	−15.7648	−1.06046
$222$	17.7593	1.19193
$223$	12.4039	0.830626	0.415313	−	0.909679i	$-0.363672\pi$
$223$	12.4039	0.830626	0.415313	+	0.909679i	$0.363672\pi$
$224$	76.5030	5.11157
$225$	9.17856	0.611904
$226$	0.383338	0.0254992
$227$	−15.8167	−1.04979	−0.524896	−	0.851166i	$-0.675896\pi$
$227$	−15.8167	−1.04979	−0.524896	+	0.851166i	$0.675896\pi$
$228$	−28.4426	−1.88366
$229$	26.4036	1.74480	0.872400	−	0.488793i	$-0.162563\pi$
$229$	26.4036	1.74480	0.872400	+	0.488793i	$0.162563\pi$
$230$	−0.815726	−0.0537874
$231$	−5.62578	−0.370149
$232$	0.698495	0.0458584
$233$	4.42636	0.289981	0.144990	−	0.989433i	$-0.453685\pi$
$233$	4.42636	0.289981	0.144990	+	0.989433i	$0.453685\pi$
$234$	−28.1784	−1.84208
$235$	−0.924422	−0.0603027
$236$	−19.0971	−1.24311
$237$	0.564655	0.0366783
$238$	35.2095	2.28229
$239$	−19.6279	−1.26963	−0.634813	−	0.772666i	$-0.718923\pi$
$239$	−19.6279	−1.26963	−0.634813	+	0.772666i	$0.718923\pi$
$240$	9.78764	0.631790
$241$	21.1324	1.36126	0.680629	−	0.732628i	$-0.261707\pi$
$241$	21.1324	1.36126	0.680629	+	0.732628i	$0.261707\pi$
$242$	24.3540	1.56554
$243$	16.1159	1.03383
$244$	16.4103	1.05056
$245$	7.83629	0.500642
$246$	−1.17612	−0.0749866
$247$	28.1320	1.79000
$248$	45.3602	2.88038
$249$	5.54893	0.351649
$250$	19.3419	1.22329
$251$	−5.84691	−0.369054	−0.184527	−	0.982827i	$-0.559075\pi$
$251$	−5.84691	−0.369054	−0.184527	+	0.982827i	$0.559075\pi$
$252$	45.6640	2.87656
$253$	0.558963	0.0351417
$254$	13.6819	0.858478
$255$	2.29198	0.143529
$256$	21.6499	1.35312
$257$	2.04635	0.127648	0.0638239	−	0.997961i	$-0.479670\pi$
$257$	2.04635	0.127648	0.0638239	+	0.997961i	$0.479670\pi$
$258$	−17.9985	−1.12054
$259$	28.4652	1.76874
$260$	−20.2204	−1.25402
$261$	0.163335	0.0101102
$262$	24.3376	1.50358
$263$	1.20237	0.0741411	0.0370705	−	0.999313i	$-0.488197\pi$
$263$	1.20237	0.0741411	0.0370705	+	0.999313i	$0.488197\pi$
$264$	−12.0050	−0.738854
$265$	−1.28019	−0.0786416
$266$	−62.8306	−3.85239
$267$	−12.2125	−0.747393
$268$	−56.6017	−3.45750
$269$	−9.53501	−0.581360	−0.290680	−	0.956820i	$-0.593881\pi$
$269$	−9.53501	−0.581360	−0.290680	+	0.956820i	$0.593881\pi$
$270$	10.0175	0.609646
$271$	−17.9686	−1.09152	−0.545758	−	0.837943i	$-0.683758\pi$
$271$	−17.9686	−1.09152	−0.545758	+	0.837943i	$0.683758\pi$
$272$	41.9754	2.54513
$273$	20.1095	1.21708
$274$	36.4190	2.20015
$275$	−6.22016	−0.375090
$276$	2.02009	0.121595
$277$	2.04488	0.122865	0.0614326	−	0.998111i	$-0.480433\pi$
$277$	2.04488	0.122865	0.0614326	+	0.998111i	$0.480433\pi$
$278$	−26.5719	−1.59368
$279$	10.6070	0.635023
$280$	28.0809	1.67815
$281$	−0.615372	−0.0367100	−0.0183550	−	0.999832i	$-0.505843\pi$
$281$	−0.615372	−0.0367100	−0.0183550	+	0.999832i	$0.505843\pi$
$282$	3.15507	0.187882
$283$	1.35803	0.0807264	0.0403632	−	0.999185i	$-0.487149\pi$
$283$	1.35803	0.0807264	0.0403632	+	0.999185i	$0.487149\pi$
$284$	−44.9608	−2.66793
$285$	−4.08999	−0.242270
$286$	19.0960	1.12917
$287$	−1.88513	−0.111275
$288$	38.1744	2.24945
$289$	−7.17059	−0.421799
$290$	0.161535	0.00948567
$291$	−7.47558	−0.438227
$292$	39.5714	2.31574
$293$	0.0542595	0.00316987	0.00158494	−	0.999999i	$-0.499495\pi$
$293$	0.0542595	0.00316987	0.00158494	+	0.999999i	$0.499495\pi$
$294$	−26.7454	−1.55982
$295$	−2.74612	−0.159885
$296$	60.7425	3.53059
$297$	−6.86434	−0.398309
$298$	−34.2742	−1.98545
$299$	−1.99803	−0.115549
$300$	−22.4796	−1.29786
$301$	−28.8487	−1.66281
$302$	−0.857446	−0.0493405
$303$	2.60168	0.149463
$304$	−74.9041	−4.29605
$305$	2.35977	0.135120
$306$	17.5693	1.00437
$307$	−14.9063	−0.850750	−0.425375	−	0.905017i	$-0.639858\pi$
$307$	−14.9063	−0.850750	−0.425375	+	0.905017i	$0.639858\pi$
$308$	−30.9457	−1.76330
$309$	0.0551723	0.00313864
$310$	10.4901	0.595797
$311$	28.0025	1.58787	0.793937	−	0.608000i	$-0.208028\pi$
$311$	28.0025	1.58787	0.793937	+	0.608000i	$0.208028\pi$
$312$	42.9120	2.42941
$313$	1.06094	0.0599676	0.0299838	−	0.999550i	$-0.490454\pi$
$313$	1.06094	0.0599676	0.0299838	+	0.999550i	$0.490454\pi$
$314$	66.5419	3.75518
$315$	6.56639	0.369974
$316$	3.10600	0.174726
$317$	−10.8010	−0.606646	−0.303323	−	0.952888i	$-0.598096\pi$
$317$	−10.8010	−0.606646	−0.303323	+	0.952888i	$0.598096\pi$
$318$	4.36932	0.245019
$319$	−0.110689	−0.00619742
$320$	17.3917	0.972228
$321$	6.74915	0.376701
$322$	4.46244	0.248682
$323$	−17.5404	−0.975972
$324$	8.12361	0.451311
$325$	22.2341	1.23333
$326$	35.4387	1.96277
$327$	−1.48170	−0.0819380
$328$	−4.02271	−0.222117
$329$	5.05706	0.278805
$330$	−2.77629	−0.152830
$331$	0.200865	0.0110405	0.00552026	−	0.999985i	$-0.498243\pi$
$331$	0.200865	0.0110405	0.00552026	+	0.999985i	$0.498243\pi$
$332$	30.5230	1.67517
$333$	14.2039	0.778371
$334$	6.55927	0.358907
$335$	−8.13921	−0.444692
$336$	−53.5434	−2.92103
$337$	−19.2335	−1.04772	−0.523859	−	0.851805i	$-0.675508\pi$
$337$	−19.2335	−1.04772	−0.523859	+	0.851805i	$0.675508\pi$
$338$	−33.1635	−1.80386
$339$	−0.136509	−0.00741413
$340$	12.6075	0.683737
$341$	−7.18817	−0.389261
$342$	−31.3520	−1.69532
$343$	−13.7489	−0.742373
$344$	−61.5608	−3.31913
$345$	0.290484	0.0156392
$346$	44.6316	2.39941
$347$	13.3770	0.718112	0.359056	−	0.933316i	$-0.383099\pi$
$347$	13.3770	0.718112	0.359056	+	0.933316i	$0.383099\pi$
$348$	−0.400030	−0.0214439
$349$	27.7583	1.48587	0.742933	−	0.669366i	$-0.233434\pi$
$349$	27.7583	1.48587	0.742933	+	0.669366i	$0.233434\pi$
$350$	−49.6581	−2.65434
$351$	24.5367	1.30967
$352$	−25.8702	−1.37888
$353$	−8.91840	−0.474679	−0.237339	−	0.971427i	$-0.576275\pi$
$353$	−8.91840	−0.474679	−0.237339	+	0.971427i	$0.576275\pi$
$354$	9.37257	0.498146
$355$	−6.46528	−0.343141
$356$	−67.1773	−3.56039
$357$	−12.5383	−0.663598
$358$	8.91489	0.471167
$359$	24.1159	1.27279	0.636395	−	0.771364i	$-0.280425\pi$
$359$	24.1159	1.27279	0.636395	+	0.771364i	$0.280425\pi$
$360$	14.0121	0.738505
$361$	12.3004	0.647388
$362$	−69.3884	−3.64698
$363$	−8.67261	−0.455194
$364$	110.616	5.79786
$365$	5.69029	0.297843
$366$	−8.05394	−0.420986
$367$	−10.4384	−0.544879	−0.272440	−	0.962173i	$-0.587830\pi$
$367$	−10.4384	−0.544879	−0.272440	+	0.962173i	$0.587830\pi$
$368$	5.31994	0.277321
$369$	−0.940664	−0.0489690
$370$	14.0474	0.730290
$371$	7.00330	0.363593
$372$	−25.9780	−1.34689
$373$	19.0513	0.986436	0.493218	−	0.869906i	$-0.335820\pi$
$373$	19.0513	0.986436	0.493218	+	0.869906i	$0.335820\pi$
$374$	−11.9064	−0.615666
$375$	−6.88777	−0.355683
$376$	10.7914	0.556522
$377$	0.395662	0.0203776
$378$	−54.8008	−2.81865
$379$	6.79771	0.349175	0.174587	−	0.984642i	$-0.444141\pi$
$379$	6.79771	0.349175	0.174587	+	0.984642i	$0.444141\pi$
$380$	−22.4978	−1.15411
$381$	−4.87220	−0.249610
$382$	−65.4018	−3.34625
$383$	2.81472	0.143826	0.0719129	−	0.997411i	$-0.477090\pi$
$383$	2.81472	0.143826	0.0719129	+	0.997411i	$0.477090\pi$
$384$	−23.9986	−1.22467
$385$	−4.44993	−0.226790
$386$	41.6921	2.12207
$387$	−14.3953	−0.731753
$388$	−41.1209	−2.08760
$389$	5.18167	0.262721	0.131361	−	0.991335i	$-0.458065\pi$
$389$	5.18167	0.262721	0.131361	+	0.991335i	$0.458065\pi$
$390$	9.92390	0.502516
$391$	1.24577	0.0630015
$392$	−91.4779	−4.62033
$393$	−8.66676	−0.437180
$394$	−61.0808	−3.07721
$395$	0.446636	0.0224727
$396$	−15.4417	−0.775973
$397$	−8.27275	−0.415197	−0.207599	−	0.978214i	$-0.566565\pi$
$397$	−8.27275	−0.415197	−0.207599	+	0.978214i	$0.566565\pi$
$398$	−64.2582	−3.22097
$399$	22.3743	1.12012
$400$	−59.2004	−2.96002
$401$	16.7138	0.834646	0.417323	−	0.908758i	$-0.362968\pi$
$401$	16.7138	0.834646	0.417323	+	0.908758i	$0.362968\pi$
$402$	27.7793	1.38550
$403$	25.6943	1.27992
$404$	14.3111	0.712002
$405$	1.16816	0.0580463
$406$	−0.883680	−0.0438563
$407$	−9.62577	−0.477132
$408$	−26.7557	−1.32461
$409$	−23.6183	−1.16785	−0.583924	−	0.811808i	$-0.698483\pi$
$409$	−23.6183	−1.16785	−0.583924	+	0.811808i	$0.698483\pi$
$410$	−0.930298	−0.0459441
$411$	−12.9690	−0.639713
$412$	0.303486	0.0149517
$413$	15.0227	0.739218
$414$	2.22672	0.109437
$415$	4.38914	0.215455
$416$	92.4734	4.53388
$417$	9.46241	0.463376
$418$	21.2467	1.03921
$419$	7.38440	0.360752	0.180376	−	0.983598i	$-0.442269\pi$
$419$	7.38440	0.360752	0.180376	+	0.983598i	$0.442269\pi$
$420$	−16.0820	−0.784721
$421$	25.1072	1.22365	0.611825	−	0.790993i	$-0.290436\pi$
$421$	25.1072	1.22365	0.611825	+	0.790993i	$0.290436\pi$
$422$	−62.3448	−3.03490
$423$	2.52344	0.122694
$424$	14.9445	0.725768
$425$	−13.8630	−0.672455
$426$	22.0661	1.06911
$427$	−12.9091	−0.624717
$428$	37.1250	1.79451
$429$	−6.80020	−0.328317
$430$	−14.2366	−0.686552
$431$	9.20800	0.443534	0.221767	−	0.975100i	$-0.428818\pi$
$431$	9.20800	0.443534	0.221767	+	0.975100i	$0.428818\pi$
$432$	−65.3314	−3.14326
$433$	−14.0469	−0.675050	−0.337525	−	0.941317i	$-0.609590\pi$
$433$	−14.0469	−0.675050	−0.337525	+	0.941317i	$0.609590\pi$
$434$	−57.3861	−2.75462
$435$	−0.0575236	−0.00275804
$436$	−8.15037	−0.390332
$437$	−2.22306	−0.106343
$438$	−19.4211	−0.927975
$439$	12.1979	0.582173	0.291087	−	0.956697i	$-0.405983\pi$
$439$	12.1979	0.582173	0.291087	+	0.956697i	$0.405983\pi$
$440$	−9.49580	−0.452694
$441$	−21.3911	−1.01862
$442$	42.5597	2.02436
$443$	−39.3857	−1.87127	−0.935634	−	0.352971i	$-0.885172\pi$
$443$	−39.3857	−1.87127	−0.935634	+	0.352971i	$0.885172\pi$
$444$	−34.7874	−1.65094
$445$	−9.65996	−0.457926
$446$	−33.4863	−1.58562
$447$	12.2052	0.577288
$448$	−95.1418	−4.49503
$449$	−10.1634	−0.479642	−0.239821	−	0.970817i	$-0.577089\pi$
$449$	−10.1634	−0.479642	−0.239821	+	0.970817i	$0.577089\pi$
$450$	−24.7790	−1.16809
$451$	0.637472	0.0300174
$452$	−0.750893	−0.0353190
$453$	0.305341	0.0143462
$454$	42.6998	2.00400
$455$	15.9064	0.745702
$456$	47.7450	2.23586
$457$	−16.8894	−0.790051	−0.395026	−	0.918670i	$-0.629264\pi$
$457$	−16.8894	−0.790051	−0.395026	+	0.918670i	$0.629264\pi$
$458$	−71.2808	−3.33073
$459$	−15.2987	−0.714082
$460$	1.59787	0.0745009
$461$	20.2728	0.944196	0.472098	−	0.881546i	$-0.343497\pi$
$461$	20.2728	0.944196	0.472098	+	0.881546i	$0.343497\pi$
$462$	15.1877	0.706596
$463$	22.1406	1.02896	0.514481	−	0.857502i	$-0.327985\pi$
$463$	22.1406	1.02896	0.514481	+	0.857502i	$0.327985\pi$
$464$	−1.05349	−0.0489069
$465$	−3.73558	−0.173233
$466$	−11.9497	−0.553558
$467$	17.2656	0.798954	0.399477	−	0.916743i	$-0.369192\pi$
$467$	17.2656	0.798954	0.399477	+	0.916743i	$0.369192\pi$
$468$	55.1966	2.55146
$469$	44.5256	2.05600
$470$	2.49563	0.115115
$471$	−23.6959	−1.09185
$472$	32.0572	1.47555
$473$	9.75544	0.448556
$474$	−1.52438	−0.0700170
$475$	24.7382	1.13507
$476$	−68.9694	−3.16121
$477$	3.49460	0.160007
$478$	52.9888	2.42365
$479$	42.0792	1.92265	0.961324	−	0.275418i	$-0.0888164\pi$
$479$	42.0792	1.92265	0.961324	+	0.275418i	$0.0888164\pi$
$480$	−13.4443	−0.613646
$481$	34.4075	1.56885
$482$	−57.0504	−2.59857
$483$	−1.58910	−0.0723065
$484$	−47.7054	−2.16843
$485$	−5.91311	−0.268500
$486$	−43.5074	−1.97353
$487$	−2.04064	−0.0924700	−0.0462350	−	0.998931i	$-0.514722\pi$
$487$	−2.04064	−0.0924700	−0.0462350	+	0.998931i	$0.514722\pi$
$488$	−27.5471	−1.24700
$489$	−12.6199	−0.570692
$490$	−21.1553	−0.955701
$491$	−0.695284	−0.0313777	−0.0156889	−	0.999877i	$-0.504994\pi$
$491$	−0.695284	−0.0313777	−0.0156889	+	0.999877i	$0.504994\pi$
$492$	2.30382	0.103864
$493$	−0.246696	−0.0111106
$494$	−75.9469	−3.41701
$495$	−2.22048	−0.0998033
$496$	−68.4134	−3.07185
$497$	35.3684	1.58649
$498$	−14.9802	−0.671280
$499$	4.27946	0.191575	0.0957874	−	0.995402i	$-0.469463\pi$
$499$	4.27946	0.191575	0.0957874	+	0.995402i	$0.469463\pi$
$500$	−37.8875	−1.69438
$501$	−2.33579	−0.104356
$502$	15.7847	0.704505
$503$	−14.7789	−0.658957	−0.329478	−	0.944163i	$-0.606873\pi$
$503$	−14.7789	−0.658957	−0.329478	+	0.944163i	$0.606873\pi$
$504$	−76.6536	−3.41442
$505$	2.05790	0.0915755
$506$	−1.50901	−0.0670838
$507$	11.8097	0.524487
$508$	−26.8005	−1.18908
$509$	−41.4975	−1.83935	−0.919673	−	0.392686i	$-0.871546\pi$
$509$	−41.4975	−1.83935	−0.919673	+	0.392686i	$0.871546\pi$
$510$	−6.18757	−0.273990
$511$	−31.1288	−1.37706
$512$	−8.52131	−0.376592
$513$	27.3002	1.20533
$514$	−5.52446	−0.243673
$515$	0.0436407	0.00192304
$516$	35.2561	1.55206
$517$	−1.71009	−0.0752097
$518$	−76.8465	−3.37644
$519$	−15.8936	−0.697650
$520$	33.9429	1.48850
$521$	−4.30866	−0.188766	−0.0943830	−	0.995536i	$-0.530088\pi$
$521$	−4.30866	−0.188766	−0.0943830	+	0.995536i	$0.530088\pi$
$522$	−0.440949	−0.0192998
$523$	2.47787	0.108350	0.0541749	−	0.998531i	$-0.482747\pi$
$523$	2.47787	0.108350	0.0541749	+	0.998531i	$0.482747\pi$
$524$	−47.6732	−2.08261
$525$	17.6835	0.771773
$526$	−3.24598	−0.141532
$527$	−16.0204	−0.697861
$528$	18.1062	0.787971
$529$	−22.8421	−0.993135
$530$	3.45609	0.150123
$531$	7.49620	0.325308
$532$	123.074	5.33595
$533$	−2.27866	−0.0986996
$534$	32.9697	1.42674
$535$	5.33850	0.230804
$536$	95.0141	4.10398
$537$	−3.17464	−0.136996
$538$	25.7413	1.10979
$539$	14.4964	0.624403
$540$	−19.6226	−0.844421
$541$	11.3178	0.486588	0.243294	−	0.969953i	$-0.421772\pi$
$541$	11.3178	0.486588	0.243294	+	0.969953i	$0.421772\pi$
$542$	48.5093	2.08365
$543$	24.7096	1.06039
$544$	−57.6574	−2.47204
$545$	−1.17201	−0.0502032
$546$	−54.2888	−2.32335
$547$	−0.866194	−0.0370358	−0.0185179	−	0.999829i	$-0.505895\pi$
$547$	−0.866194	−0.0370358	−0.0185179	+	0.999829i	$0.505895\pi$
$548$	−71.3385	−3.04743
$549$	−6.44156	−0.274919
$550$	16.7923	0.716028
$551$	0.440224	0.0187542
$552$	−3.39101	−0.144331
$553$	−2.44333	−0.103901
$554$	−5.52050	−0.234543
$555$	−5.00236	−0.212338
$556$	52.0499	2.20741
$557$	33.2859	1.41037	0.705184	−	0.709024i	$-0.250864\pi$
$557$	33.2859	1.41037	0.705184	+	0.709024i	$0.250864\pi$
$558$	−28.6352	−1.21223
$559$	−34.8710	−1.47489
$560$	−42.3523	−1.78971
$561$	4.23994	0.179010
$562$	1.66130	0.0700775
$563$	−11.4687	−0.483348	−0.241674	−	0.970358i	$-0.577696\pi$
$563$	−11.4687	−0.483348	−0.241674	+	0.970358i	$0.577696\pi$
$564$	−6.18025	−0.260235
$565$	−0.107977	−0.00454262
$566$	−3.66622	−0.154103
$567$	−6.39042	−0.268373
$568$	75.4732	3.16679
$569$	−35.5751	−1.49139	−0.745693	−	0.666290i	$-0.767881\pi$
$569$	−35.5751	−1.49139	−0.745693	+	0.666290i	$0.767881\pi$
$570$	11.0416	0.462481
$571$	29.8016	1.24716	0.623579	−	0.781761i	$-0.285678\pi$
$571$	29.8016	1.24716	0.623579	+	0.781761i	$0.285678\pi$
$572$	−37.4058	−1.56402
$573$	23.2899	0.972951
$574$	5.08921	0.212419
$575$	−1.75699	−0.0732716
$576$	−47.4750	−1.97813
$577$	35.8291	1.49159	0.745793	−	0.666178i	$-0.232071\pi$
$577$	35.8291	1.49159	0.745793	+	0.666178i	$0.232071\pi$
$578$	19.3582	0.805194
$579$	−14.8468	−0.617011
$580$	−0.316420	−0.0131386
$581$	−24.0109	−0.996138
$582$	20.1816	0.836552
$583$	−2.36823	−0.0980821
$584$	−66.4263	−2.74874
$585$	7.93716	0.328161
$586$	−0.146482	−0.00605113
$587$	−10.9463	−0.451802	−0.225901	−	0.974150i	$-0.572533\pi$
$587$	−10.9463	−0.451802	−0.225901	+	0.974150i	$0.572533\pi$
$588$	52.3897	2.16051
$589$	28.5881	1.17795
$590$	7.41360	0.305213
$591$	21.7512	0.894725
$592$	−91.6134	−3.76529
$593$	−31.4035	−1.28959	−0.644794	−	0.764356i	$-0.723057\pi$
$593$	−31.4035	−1.28959	−0.644794	+	0.764356i	$0.723057\pi$
$594$	18.5314	0.760352
$595$	−9.91767	−0.406585
$596$	67.1373	2.75005
$597$	22.8827	0.936527
$598$	5.39400	0.220577
$599$	14.0647	0.574667	0.287334	−	0.957831i	$-0.407231\pi$
$599$	14.0647	0.574667	0.287334	+	0.957831i	$0.407231\pi$
$600$	37.7352	1.54053
$601$	19.7498	0.805610	0.402805	−	0.915286i	$-0.368035\pi$
$601$	19.7498	0.805610	0.402805	+	0.915286i	$0.368035\pi$
$602$	77.8817	3.17422
$603$	22.2179	0.904785
$604$	1.67959	0.0683416
$605$	−6.85994	−0.278896
$606$	−7.02366	−0.285317
$607$	0.765652	0.0310768	0.0155384	−	0.999879i	$-0.495054\pi$
$607$	0.765652	0.0310768	0.0155384	+	0.999879i	$0.495054\pi$
$608$	102.888	4.17267
$609$	0.314683	0.0127516
$610$	−6.37058	−0.257937
$611$	6.11275	0.247296
$612$	−34.4152	−1.39115
$613$	9.10082	0.367579	0.183789	−	0.982966i	$-0.441164\pi$
$613$	9.10082	0.367579	0.183789	+	0.982966i	$0.441164\pi$
$614$	40.2421	1.62404
$615$	0.331284	0.0133587
$616$	51.9469	2.09300
$617$	27.8116	1.11965	0.559826	−	0.828610i	$-0.310868\pi$
$617$	27.8116	1.11965	0.559826	+	0.828610i	$0.310868\pi$
$618$	−0.148946	−0.00599151
$619$	−13.1064	−0.526789	−0.263395	−	0.964688i	$-0.584842\pi$
$619$	−13.1064	−0.526789	−0.263395	+	0.964688i	$0.584842\pi$
$620$	−20.5483	−0.825239
$621$	−1.93895	−0.0778074
$622$	−75.5972	−3.03117
$623$	52.8449	2.11719
$624$	−64.7209	−2.59091
$625$	16.6606	0.666423
$626$	−2.86417	−0.114475
$627$	−7.56608	−0.302160
$628$	−130.344	−5.20130
$629$	−21.4532	−0.855394
$630$	−17.7270	−0.706262
$631$	−12.4132	−0.494163	−0.247082	−	0.968995i	$-0.579472\pi$
$631$	−12.4132	−0.494163	−0.247082	+	0.968995i	$0.579472\pi$
$632$	−5.21386	−0.207396
$633$	22.2013	0.882424
$634$	29.1591	1.15806
$635$	−3.85385	−0.152936
$636$	−8.55875	−0.339377
$637$	−51.8176	−2.05309
$638$	0.298824	0.0118306
$639$	17.6485	0.698166
$640$	−18.9826	−0.750355
$641$	−7.37676	−0.291365	−0.145682	−	0.989331i	$-0.546538\pi$
$641$	−7.37676	−0.291365	−0.145682	+	0.989331i	$0.546538\pi$
$642$	−18.2204	−0.719103
$643$	33.8088	1.33329	0.666644	−	0.745376i	$-0.267730\pi$
$643$	33.8088	1.33329	0.666644	+	0.745376i	$0.267730\pi$
$644$	−8.74115	−0.344450
$645$	5.06975	0.199621
$646$	47.3531	1.86308
$647$	−15.3324	−0.602777	−0.301389	−	0.953501i	$-0.597450\pi$
$647$	−15.3324	−0.602777	−0.301389	+	0.953501i	$0.597450\pi$
$648$	−13.6366	−0.535698
$649$	−5.08005	−0.199410
$650$	−60.0245	−2.35436
$651$	20.4355	0.800931
$652$	−69.4183	−2.71863
$653$	−17.1568	−0.671399	−0.335699	−	0.941969i	$-0.608973\pi$
$653$	−17.1568	−0.671399	−0.335699	+	0.941969i	$0.608973\pi$
$654$	4.00008	0.156416
$655$	−6.85531	−0.267859
$656$	6.06715	0.236882
$657$	−15.5330	−0.606001
$658$	−13.6524	−0.532224
$659$	−38.0356	−1.48166	−0.740829	−	0.671694i	$-0.765567\pi$
$659$	−38.0356	−1.48166	−0.740829	+	0.671694i	$0.765567\pi$
$660$	5.43827	0.211684
$661$	39.0923	1.52052	0.760258	−	0.649622i	$-0.225073\pi$
$661$	39.0923	1.52052	0.760258	+	0.649622i	$0.225073\pi$
$662$	−0.542267	−0.0210758
$663$	−15.1558	−0.588601
$664$	−51.2372	−1.98839
$665$	17.6979	0.686293
$666$	−38.3458	−1.48587
$667$	−0.0312661	−0.00121063
$668$	−12.8485	−0.497123
$669$	11.9247	0.461034
$670$	21.9731	0.848895
$671$	4.36534	0.168522
$672$	73.5473	2.83715
$673$	−15.7237	−0.606105	−0.303052	−	0.952974i	$-0.598006\pi$
$673$	−15.7237	−0.606105	−0.303052	+	0.952974i	$0.598006\pi$
$674$	51.9241	2.00004
$675$	21.5767	0.830487
$676$	64.9616	2.49852
$677$	−40.0437	−1.53900	−0.769502	−	0.638645i	$-0.779495\pi$
$677$	−40.0437	−1.53900	−0.769502	+	0.638645i	$0.779495\pi$
$678$	0.368527	0.0141532
$679$	32.3477	1.24139
$680$	−21.1635	−0.811583
$681$	−15.2056	−0.582681
$682$	19.4056	0.743080
$683$	−26.8455	−1.02721	−0.513607	−	0.858026i	$-0.671691\pi$
$683$	−26.8455	−1.02721	−0.513607	+	0.858026i	$0.671691\pi$
$684$	61.4132	2.34819
$685$	−10.2583	−0.391951
$686$	37.1175	1.41715
$687$	25.3835	0.968441
$688$	92.8475	3.53978
$689$	8.46529	0.322502
$690$	−0.784210	−0.0298544
$691$	−9.70651	−0.369253	−0.184626	−	0.982809i	$-0.559108\pi$
$691$	−9.70651	−0.369253	−0.184626	+	0.982809i	$0.559108\pi$
$692$	−87.4256	−3.32342
$693$	12.1472	0.461433
$694$	−36.1133	−1.37084
$695$	7.48467	0.283910
$696$	0.671509	0.0254535
$697$	1.42075	0.0538147
$698$	−74.9379	−2.83644
$699$	4.25535	0.160952
$700$	97.2718	3.67653
$701$	−20.3521	−0.768689	−0.384344	−	0.923190i	$-0.625572\pi$
$701$	−20.3521	−0.768689	−0.384344	+	0.923190i	$0.625572\pi$
$702$	−66.2408	−2.50010
$703$	38.2827	1.44386
$704$	32.1730	1.21257
$705$	−0.888707	−0.0334706
$706$	24.0767	0.906137
$707$	−11.2578	−0.423392
$708$	−18.3593	−0.689983
$709$	17.4034	0.653600	0.326800	−	0.945093i	$-0.394030\pi$
$709$	17.4034	0.653600	0.326800	+	0.945093i	$0.394030\pi$
$710$	17.4541	0.655039
$711$	−1.21920	−0.0457236
$712$	112.767	4.22612
$713$	−2.03042	−0.0760399
$714$	33.8492	1.26677
$715$	−5.37888	−0.201159
$716$	−17.4628	−0.652614
$717$	−18.8696	−0.704699
$718$	−65.1048	−2.42969
$719$	−45.3724	−1.69211	−0.846053	−	0.533099i	$-0.821027\pi$
$719$	−45.3724	−1.69211	−0.846053	+	0.533099i	$0.821027\pi$
$720$	−21.1335	−0.787597
$721$	−0.238737	−0.00889102
$722$	−33.2069	−1.23583
$723$	20.3160	0.755559
$724$	135.920	5.05143
$725$	0.347930	0.0129218
$726$	23.4131	0.868942
$727$	−19.6863	−0.730123	−0.365062	−	0.930983i	$-0.618952\pi$
$727$	−19.6863	−0.730123	−0.365062	+	0.930983i	$0.618952\pi$
$728$	−185.685	−6.88195
$729$	10.8847	0.403136
$730$	−15.3619	−0.568568
$731$	21.7422	0.804163
$732$	15.7763	0.583108
$733$	−6.14743	−0.227060	−0.113530	−	0.993535i	$-0.536216\pi$
$733$	−6.14743	−0.227060	−0.113530	+	0.993535i	$0.536216\pi$
$734$	28.1801	1.04015
$735$	7.53353	0.277879
$736$	−7.30747	−0.269357
$737$	−15.0567	−0.554622
$738$	2.53947	0.0934793
$739$	−0.0317700	−0.00116868	−0.000584339	−	1.00000i	$-0.500186\pi$
$739$	−0.0317700	−0.00116868	−0.000584339		1.00000i	$0.500186\pi$
$740$	−27.5165	−1.01153
$741$	27.0451	0.993527
$742$	−18.9066	−0.694082
$743$	14.7019	0.539359	0.269679	−	0.962950i	$-0.413082\pi$
$743$	14.7019	0.539359	0.269679	+	0.962950i	$0.413082\pi$
$744$	43.6077	1.59874
$745$	9.65421	0.353703
$746$	−51.4319	−1.88306
$747$	−11.9812	−0.438370
$748$	23.3226	0.852760
$749$	−29.2043	−1.06710
$750$	18.5947	0.678981
$751$	−6.06488	−0.221311	−0.110655	−	0.993859i	$-0.535295\pi$
$751$	−6.06488	−0.221311	−0.110655	+	0.993859i	$0.535295\pi$
$752$	−16.2758	−0.593517
$753$	−5.62101	−0.204841
$754$	−1.06815	−0.0388998
$755$	0.241522	0.00878987
$756$	107.345	3.90412
$757$	−29.1305	−1.05876	−0.529382	−	0.848383i	$-0.677576\pi$
$757$	−29.1305	−1.05876	−0.529382	+	0.848383i	$0.677576\pi$
$758$	−18.3515	−0.666557
$759$	0.537368	0.0195052
$760$	37.7658	1.36991
$761$	−6.54175	−0.237138	−0.118569	−	0.992946i	$-0.537831\pi$
$761$	−6.54175	−0.237138	−0.118569	+	0.992946i	$0.537831\pi$
$762$	13.1533	0.476493
$763$	6.41148	0.232111
$764$	128.111	4.63489
$765$	−4.94884	−0.178926
$766$	−7.59881	−0.274556
$767$	18.1588	0.655675
$768$	20.8135	0.751041
$769$	−7.07681	−0.255196	−0.127598	−	0.991826i	$-0.540727\pi$
$769$	−7.07681	−0.255196	−0.127598	+	0.991826i	$0.540727\pi$
$770$	12.0133	0.432930
$771$	1.96729	0.0708502
$772$	−81.6676	−2.93928
$773$	49.0532	1.76432	0.882161	−	0.470948i	$-0.156088\pi$
$773$	49.0532	1.76432	0.882161	+	0.470948i	$0.156088\pi$
$774$	38.8624	1.39688
$775$	22.5946	0.811622
$776$	69.0274	2.47794
$777$	27.3655	0.981731
$778$	−13.9888	−0.501522
$779$	−2.53529	−0.0908364
$780$	−19.4392	−0.696036
$781$	−11.9601	−0.427967
$782$	−3.36317	−0.120267
$783$	0.383963	0.0137217
$784$	137.969	4.92747
$785$	−18.7432	−0.668975
$786$	23.3973	0.834555
$787$	−40.6045	−1.44739	−0.723697	−	0.690117i	$-0.757559\pi$
$787$	−40.6045	−1.44739	−0.723697	+	0.690117i	$0.757559\pi$
$788$	119.647	4.26224
$789$	1.15591	0.0411516
$790$	−1.20577	−0.0428993
$791$	0.590689	0.0210025
$792$	25.9211	0.921066
$793$	−15.6040	−0.554114
$794$	22.3336	0.792591
$795$	−1.23073	−0.0436496
$796$	125.871	4.46138
$797$	−23.7513	−0.841314	−0.420657	−	0.907220i	$-0.638200\pi$
$797$	−23.7513	−0.841314	−0.420657	+	0.907220i	$0.638200\pi$
$798$	−60.4031	−2.13825
$799$	−3.81132	−0.134835
$800$	81.3177	2.87502
$801$	26.3692	0.931711
$802$	−45.1215	−1.59330
$803$	10.5265	0.371471
$804$	−54.4149	−1.91906
$805$	−1.25696	−0.0443020
$806$	−69.3658	−2.44331
$807$	−9.16662	−0.322680
$808$	−24.0232	−0.845133
$809$	36.9412	1.29878	0.649391	−	0.760455i	$-0.275024\pi$
$809$	36.9412	1.29878	0.649391	+	0.760455i	$0.275024\pi$
$810$	−3.15363	−0.110807
$811$	−21.8401	−0.766910	−0.383455	−	0.923560i	$-0.625266\pi$
$811$	−21.8401	−0.766910	−0.383455	+	0.923560i	$0.625266\pi$
$812$	1.73098	0.0607454
$813$	−17.2744	−0.605840
$814$	25.9863	0.910821
$815$	−9.98222	−0.349662
$816$	40.3537	1.41266
$817$	−38.7984	−1.35739
$818$	63.7613	2.22936
$819$	−43.4203	−1.51723
$820$	1.82229	0.0636373
$821$	9.66497	0.337310	0.168655	−	0.985675i	$-0.446058\pi$
$821$	9.66497	0.337310	0.168655	+	0.985675i	$0.446058\pi$
$822$	35.0119	1.22118
$823$	46.4664	1.61972	0.809858	−	0.586626i	$-0.199544\pi$
$823$	46.4664	1.61972	0.809858	+	0.586626i	$0.199544\pi$
$824$	−0.509445	−0.0177474
$825$	−5.97984	−0.208191
$826$	−40.5562	−1.41113
$827$	−15.3333	−0.533192	−0.266596	−	0.963808i	$-0.585899\pi$
$827$	−15.3333	−0.533192	−0.266596	+	0.963808i	$0.585899\pi$
$828$	−4.36177	−0.151582
$829$	11.5602	0.401501	0.200750	−	0.979642i	$-0.435662\pi$
$829$	11.5602	0.401501	0.200750	+	0.979642i	$0.435662\pi$
$830$	−11.8492	−0.411292
$831$	1.96588	0.0681956
$832$	−115.003	−3.98702
$833$	32.3084	1.11942
$834$	−25.5453	−0.884562
$835$	−1.84759	−0.0639384
$836$	−41.6187	−1.43941
$837$	24.9345	0.861864
$838$	−19.9354	−0.688656
$839$	35.6785	1.23176	0.615880	−	0.787840i	$-0.288801\pi$
$839$	35.6785	1.23176	0.615880	+	0.787840i	$0.288801\pi$
$840$	26.9960	0.931449
$841$	−28.9938	−0.999786
$842$	−67.7809	−2.33589
$843$	−0.591597	−0.0203757
$844$	122.123	4.20364
$845$	9.34135	0.321352
$846$	−6.81242	−0.234216
$847$	37.5274	1.28946
$848$	−22.5396	−0.774015
$849$	1.30556	0.0448067
$850$	37.4255	1.28368
$851$	−2.71896	−0.0932049
$852$	−43.2238	−1.48082
$853$	40.9401	1.40176	0.700881	−	0.713278i	$-0.252790\pi$
$853$	40.9401	1.40176	0.700881	+	0.713278i	$0.252790\pi$
$854$	34.8503	1.19255
$855$	8.83109	0.302017
$856$	−62.3197	−2.13004
$857$	54.7171	1.86910	0.934550	−	0.355833i	$-0.115803\pi$
$857$	54.7171	1.86910	0.934550	+	0.355833i	$0.115803\pi$
$858$	18.3582	0.626740
$859$	−31.8550	−1.08688	−0.543439	−	0.839449i	$-0.682878\pi$
$859$	−31.8550	−1.08688	−0.543439	+	0.839449i	$0.682878\pi$
$860$	27.8872	0.950944
$861$	−1.81229	−0.0617628
$862$	−24.8585	−0.846684
$863$	48.0079	1.63421	0.817104	−	0.576490i	$-0.195578\pi$
$863$	48.0079	1.63421	0.817104	+	0.576490i	$0.195578\pi$
$864$	89.7392	3.05299
$865$	−12.5716	−0.427448
$866$	37.9218	1.28864
$867$	−6.89355	−0.234117
$868$	112.410	3.81543
$869$	0.826233	0.0280280
$870$	0.155294	0.00526497
$871$	53.8206	1.82364
$872$	13.6816	0.463316
$873$	16.1413	0.546299
$874$	6.00150	0.203004
$875$	29.8042	1.00757
$876$	38.0426	1.28534
$877$	−9.68280	−0.326965	−0.163482	−	0.986546i	$-0.552273\pi$
$877$	−9.68280	−0.326965	−0.163482	+	0.986546i	$0.552273\pi$
$878$	−32.9302	−1.11134
$879$	0.0521632	0.00175942
$880$	14.3218	0.482788
$881$	49.4598	1.66634	0.833172	−	0.553015i	$-0.186523\pi$
$881$	49.4598	1.66634	0.833172	+	0.553015i	$0.186523\pi$
$882$	57.7486	1.94450
$883$	−6.21168	−0.209040	−0.104520	−	0.994523i	$-0.533331\pi$
$883$	−6.21168	−0.209040	−0.104520	+	0.994523i	$0.533331\pi$
$884$	−83.3672	−2.80394
$885$	−2.64002	−0.0887434
$886$	106.328	3.57216
$887$	−43.3545	−1.45570	−0.727851	−	0.685735i	$-0.759481\pi$
$887$	−43.3545	−1.45570	−0.727851	+	0.685735i	$0.759481\pi$
$888$	58.3957	1.95963
$889$	21.0826	0.707086
$890$	26.0786	0.874158
$891$	2.16098	0.0723955
$892$	65.5940	2.19625
$893$	6.80121	0.227594
$894$	−32.9500	−1.10201
$895$	−2.51111	−0.0839371
$896$	103.845	3.46921
$897$	−1.92083	−0.0641347
$898$	27.4378	0.915613
$899$	0.402077	0.0134100
$900$	48.5379	1.61793
$901$	−5.27813	−0.175840
$902$	−1.72096	−0.0573017
$903$	−27.7341	−0.922934
$904$	1.26048	0.0419230
$905$	19.5450	0.649699
$906$	−0.824319	−0.0273861
$907$	−14.1331	−0.469280	−0.234640	−	0.972082i	$-0.575391\pi$
$907$	−14.1331	−0.469280	−0.234640	+	0.972082i	$0.575391\pi$
$908$	−83.6416	−2.77574
$909$	−5.61755	−0.186322
$910$	−42.9418	−1.42351
$911$	−10.4430	−0.345992	−0.172996	−	0.984923i	$-0.555345\pi$
$911$	−10.4430	−0.345992	−0.172996	+	0.984923i	$0.555345\pi$
$912$	−72.0102	−2.38450
$913$	8.11948	0.268716
$914$	45.5956	1.50817
$915$	2.26860	0.0749976
$916$	139.627	4.61340
$917$	37.5021	1.23843
$918$	41.3013	1.36315
$919$	22.5122	0.742608	0.371304	−	0.928511i	$-0.378911\pi$
$919$	22.5122	0.742608	0.371304	+	0.928511i	$0.378911\pi$
$920$	−2.68225	−0.0884312
$921$	−14.3304	−0.472204
$922$	−54.7296	−1.80242
$923$	42.7517	1.40719
$924$	−29.7501	−0.978708
$925$	30.2567	0.994835
$926$	−59.7722	−1.96424
$927$	−0.119128	−0.00391267
$928$	1.44707	0.0475024
$929$	44.3517	1.45513	0.727566	−	0.686038i	$-0.240652\pi$
$929$	44.3517	1.45513	0.727566	+	0.686038i	$0.240652\pi$
$930$	10.0848	0.330694
$931$	−57.6536	−1.88952
$932$	23.4074	0.766734
$933$	26.9206	0.881341
$934$	−46.6112	−1.52516
$935$	3.35375	0.109679
$936$	−92.6555	−3.02854
$937$	53.9986	1.76406	0.882029	−	0.471196i	$-0.156177\pi$
$937$	53.9986	1.76406	0.882029	+	0.471196i	$0.156177\pi$
$938$	−120.204	−3.92481
$939$	1.01995	0.0332847
$940$	−4.88851	−0.159446
$941$	35.4855	1.15679	0.578397	−	0.815755i	$-0.303678\pi$
$941$	35.4855	1.15679	0.578397	+	0.815755i	$0.303678\pi$
$942$	63.9710	2.08429
$943$	0.180065	0.00586372
$944$	−48.3495	−1.57364
$945$	15.4361	0.502135
$946$	−26.3364	−0.856270
$947$	29.2719	0.951208	0.475604	−	0.879660i	$-0.342230\pi$
$947$	29.2719	0.951208	0.475604	+	0.879660i	$0.342230\pi$
$948$	2.98600	0.0969807
$949$	−37.6271	−1.22143
$950$	−66.7849	−2.16679
$951$	−10.3837	−0.336716
$952$	115.775	3.75229
$953$	3.13741	0.101631	0.0508153	−	0.998708i	$-0.483818\pi$
$953$	3.13741	0.101631	0.0508153	+	0.998708i	$0.483818\pi$
$954$	−9.43423	−0.305444
$955$	18.4221	0.596125
$956$	−103.796	−3.35700
$957$	−0.106413	−0.00343984
$958$	−113.600	−3.67024
$959$	56.1183	1.81216
$960$	16.7198	0.539630
$961$	−4.88915	−0.157714
$962$	−92.8887	−2.99485
$963$	−14.5727	−0.469600
$964$	111.752	3.59929
$965$	−11.7436	−0.378041
$966$	4.29003	0.138029
$967$	−28.1283	−0.904544	−0.452272	−	0.891880i	$-0.649386\pi$
$967$	−28.1283	−0.904544	−0.452272	+	0.891880i	$0.649386\pi$
$968$	80.0804	2.57388
$969$	−16.8627	−0.541708
$970$	15.9634	0.512554
$971$	6.10664	0.195972	0.0979858	−	0.995188i	$-0.468760\pi$
$971$	6.10664	0.195972	0.0979858	+	0.995188i	$0.468760\pi$
$972$	85.2235	2.73355
$973$	−40.9450	−1.31263
$974$	5.50903	0.176521
$975$	21.3751	0.684550
$976$	41.5472	1.32989
$977$	51.7975	1.65715	0.828574	−	0.559880i	$-0.189153\pi$
$977$	51.7975	1.65715	0.828574	+	0.559880i	$0.189153\pi$
$978$	34.0695	1.08942
$979$	−17.8700	−0.571127
$980$	41.4397	1.32374
$981$	3.19928	0.102145
$982$	1.87703	0.0598985
$983$	−24.1757	−0.771087	−0.385543	−	0.922690i	$-0.625986\pi$
$983$	−24.1757	−0.771087	−0.385543	+	0.922690i	$0.625986\pi$
$984$	−3.86729	−0.123285
$985$	17.2050	0.548196
$986$	0.665996	0.0212096
$987$	4.86168	0.154749
$988$	148.767	4.73291
$989$	2.75559	0.0876227
$990$	5.99455	0.190519
$991$	−7.05575	−0.224133	−0.112067	−	0.993701i	$-0.535747\pi$
$991$	−7.05575	−0.224133	−0.112067	+	0.993701i	$0.535747\pi$
$992$	93.9727	2.98364
$993$	0.193104	0.00612798
$994$	−95.4826	−3.02853
$995$	18.1000	0.573808
$996$	29.3437	0.929791
$997$	−24.5820	−0.778521	−0.389261	−	0.921128i	$-0.627269\pi$
$997$	−24.5820	−0.778521	−0.389261	+	0.921128i	$0.627269\pi$
$998$	−11.5531	−0.365707
$999$	33.3902	1.05642

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8009.2.a.b.1.12	✓	361

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8009.2.a.b.1.12	✓	361	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8009 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69966 q^{2} +0.961365 q^{3} +5.28818 q^{4} +0.760429 q^{5} -2.59536 q^{6} -4.15994 q^{7} -8.87697 q^{8} -2.07578 q^{9} +O(q^{10})\)	\(q-2.69966 q^{2} +0.961365 q^{3} +5.28818 q^{4} +0.760429 q^{5} -2.59536 q^{6} -4.15994 q^{7} -8.87697 q^{8} -2.07578 q^{9} -2.05290 q^{10} +1.40672 q^{11} +5.08387 q^{12} -5.02835 q^{13} +11.2304 q^{14} +0.731050 q^{15} +13.3885 q^{16} +3.13519 q^{17} +5.60390 q^{18} -5.59467 q^{19} +4.02129 q^{20} -3.99922 q^{21} -3.79767 q^{22} +0.397352 q^{23} -8.53401 q^{24} -4.42175 q^{25} +13.5748 q^{26} -4.87967 q^{27} -21.9985 q^{28} -0.0786862 q^{29} -1.97359 q^{30} -5.10988 q^{31} -18.3904 q^{32} +1.35237 q^{33} -8.46395 q^{34} -3.16334 q^{35} -10.9771 q^{36} -6.84271 q^{37} +15.1037 q^{38} -4.83408 q^{39} -6.75031 q^{40} +0.453162 q^{41} +10.7965 q^{42} +6.93489 q^{43} +7.43899 q^{44} -1.57848 q^{45} -1.07272 q^{46} -1.21566 q^{47} +12.8712 q^{48} +10.3051 q^{49} +11.9372 q^{50} +3.01406 q^{51} -26.5908 q^{52} -1.68351 q^{53} +13.1735 q^{54} +1.06971 q^{55} +36.9276 q^{56} -5.37852 q^{57} +0.212426 q^{58} -3.61128 q^{59} +3.86592 q^{60} +3.10320 q^{61} +13.7949 q^{62} +8.63510 q^{63} +22.8710 q^{64} -3.82371 q^{65} -3.65095 q^{66} -10.7034 q^{67} +16.5794 q^{68} +0.382001 q^{69} +8.53995 q^{70} -8.50214 q^{71} +18.4266 q^{72} +7.48300 q^{73} +18.4730 q^{74} -4.25091 q^{75} -29.5856 q^{76} -5.85187 q^{77} +13.0504 q^{78} +0.587347 q^{79} +10.1810 q^{80} +1.53618 q^{81} -1.22338 q^{82} +5.77193 q^{83} -21.1486 q^{84} +2.38409 q^{85} -18.7219 q^{86} -0.0756461 q^{87} -12.4874 q^{88} -12.7033 q^{89} +4.26137 q^{90} +20.9176 q^{91} +2.10127 q^{92} -4.91246 q^{93} +3.28187 q^{94} -4.25435 q^{95} -17.6799 q^{96} -7.77601 q^{97} -27.8202 q^{98} -2.92004 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9}+O(q^{10}) \) 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9	\( 361 q + 10 q^{2} + 23 q^{3} + 414 q^{4} + 21 q^{5} + 49 q^{6} + 106 q^{7} + 30 q^{8} + 406 q^{9} + 65 q^{10} + 33 q^{11} + 52 q^{12} + 89 q^{13} + 32 q^{14} + 55 q^{15} + 512 q^{16} + 42 q^{17} + 34 q^{18} + 191 q^{19} + 48 q^{20} + 53 q^{21} + 61 q^{22} + 52 q^{23} + 139 q^{24} + 458 q^{25} + 57 q^{26} + 80 q^{27} + 194 q^{28} + 47 q^{29} + 32 q^{30} + 254 q^{31} + 55 q^{32} + 40 q^{33} + 122 q^{34} + 93 q^{35} + 519 q^{36} + 43 q^{37} + 25 q^{38} + 210 q^{39} + 184 q^{40} + 54 q^{41} + 48 q^{42} + 151 q^{43} + 56 q^{44} + 82 q^{45} + 101 q^{46} + 117 q^{47} + 77 q^{48} + 563 q^{49} + 38 q^{50} + 143 q^{51} + 241 q^{52} + 14 q^{53} + 164 q^{54} + 452 q^{55} + 52 q^{56} + 21 q^{57} + 55 q^{58} + 125 q^{59} + 39 q^{60} + 227 q^{61} + 58 q^{62} + 292 q^{63} + 710 q^{64} + 15 q^{65} + 105 q^{66} + 120 q^{67} + 125 q^{68} + 136 q^{69} + 88 q^{70} + 105 q^{71} + 78 q^{72} + 108 q^{73} + 41 q^{74} + 128 q^{75} + 461 q^{76} + 28 q^{77} + 13 q^{78} + 400 q^{79} + 59 q^{80} + 485 q^{81} + 175 q^{82} + 97 q^{83} + 76 q^{84} + 144 q^{85} - 14 q^{86} + 327 q^{87} + 145 q^{88} + 52 q^{89} + 60 q^{90} + 192 q^{91} + 11 q^{92} + 32 q^{93} + 366 q^{94} + 182 q^{95} + 275 q^{96} + 117 q^{97} + 42 q^{98} + 111 q^{99}+O(q^{100}) \) 361 * q + 10 * q^2 + 23 * q^3 + 414 * q^4 + 21 * q^5 + 49 * q^6 + 106 * q^7 + 30 * q^8 + 406 * q^9 + 65 * q^10 + 33 * q^11 + 52 * q^12 + 89 * q^13 + 32 * q^14 + 55 * q^15 + 512 * q^16 + 42 * q^17 + 34 * q^18 + 191 * q^19 + 48 * q^20 + 53 * q^21 + 61 * q^22 + 52 * q^23 + 139 * q^24 + 458 * q^25 + 57 * q^26 + 80 * q^27 + 194 * q^28 + 47 * q^29 + 32 * q^30 + 254 * q^31 + 55 * q^32 + 40 * q^33 + 122 * q^34 + 93 * q^35 + 519 * q^36 + 43 * q^37 + 25 * q^38 + 210 * q^39 + 184 * q^40 + 54 * q^41 + 48 * q^42 + 151 * q^43 + 56 * q^44 + 82 * q^45 + 101 * q^46 + 117 * q^47 + 77 * q^48 + 563 * q^49 + 38 * q^50 + 143 * q^51 + 241 * q^52 + 14 * q^53 + 164 * q^54 + 452 * q^55 + 52 * q^56 + 21 * q^57 + 55 * q^58 + 125 * q^59 + 39 * q^60 + 227 * q^61 + 58 * q^62 + 292 * q^63 + 710 * q^64 + 15 * q^65 + 105 * q^66 + 120 * q^67 + 125 * q^68 + 136 * q^69 + 88 * q^70 + 105 * q^71 + 78 * q^72 + 108 * q^73 + 41 * q^74 + 128 * q^75 + 461 * q^76 + 28 * q^77 + 13 * q^78 + 400 * q^79 + 59 * q^80 + 485 * q^81 + 175 * q^82 + 97 * q^83 + 76 * q^84 + 144 * q^85 - 14 * q^86 + 327 * q^87 + 145 * q^88 + 52 * q^89 + 60 * q^90 + 192 * q^91 + 11 * q^92 + 32 * q^93 + 366 * q^94 + 182 * q^95 + 275 * q^96 + 117 * q^97 + 42 * q^98 + 111 * q^99

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