LMFDB - Embedded newform 2415.2.a.q.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2415,2,Mod(1,2415)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2415, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2415.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2415 = 3 \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2415.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:	$19.2838720881$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.15751800.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 8x^{4} + 6x^{3} + 16x^{2} - 7x - 1 $ x^6 - x^5 - 8x^4 + 6x^3 + 16x^2 - 7x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.66208$ of defining polynomial
Character	$\chi$	$=$	2415.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.05680 q^{2} -1.00000 q^{3} +2.23042 q^{4} -1.00000 q^{5} -2.05680 q^{6} +1.00000 q^{7} +0.473937 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.05680 q^{2} -1.00000 q^{3} +2.23042 q^{4} -1.00000 q^{5} -2.05680 q^{6} +1.00000 q^{7} +0.473937 q^{8} +1.00000 q^{9} -2.05680 q^{10} +3.28152 q^{11} -2.23042 q^{12} +4.14348 q^{13} +2.05680 q^{14} +1.00000 q^{15} -3.48606 q^{16} +3.07319 q^{17} +2.05680 q^{18} -7.30174 q^{19} -2.23042 q^{20} -1.00000 q^{21} +6.74943 q^{22} -1.00000 q^{23} -0.473937 q^{24} +1.00000 q^{25} +8.52231 q^{26} -1.00000 q^{27} +2.23042 q^{28} +4.84920 q^{29} +2.05680 q^{30} +4.13382 q^{31} -8.11799 q^{32} -3.28152 q^{33} +6.32094 q^{34} -1.00000 q^{35} +2.23042 q^{36} +9.25670 q^{37} -15.0182 q^{38} -4.14348 q^{39} -0.473937 q^{40} +5.93515 q^{41} -2.05680 q^{42} +4.77427 q^{43} +7.31918 q^{44} -1.00000 q^{45} -2.05680 q^{46} -8.76193 q^{47} +3.48606 q^{48} +1.00000 q^{49} +2.05680 q^{50} -3.07319 q^{51} +9.24172 q^{52} +9.55667 q^{53} -2.05680 q^{54} -3.28152 q^{55} +0.473937 q^{56} +7.30174 q^{57} +9.97382 q^{58} -1.65430 q^{59} +2.23042 q^{60} +12.7382 q^{61} +8.50245 q^{62} +1.00000 q^{63} -9.72497 q^{64} -4.14348 q^{65} -6.74943 q^{66} -7.97142 q^{67} +6.85452 q^{68} +1.00000 q^{69} -2.05680 q^{70} +13.4257 q^{71} +0.473937 q^{72} -1.30705 q^{73} +19.0392 q^{74} -1.00000 q^{75} -16.2860 q^{76} +3.28152 q^{77} -8.52231 q^{78} +6.60879 q^{79} +3.48606 q^{80} +1.00000 q^{81} +12.2074 q^{82} -9.03292 q^{83} -2.23042 q^{84} -3.07319 q^{85} +9.81972 q^{86} -4.84920 q^{87} +1.55523 q^{88} -11.6650 q^{89} -2.05680 q^{90} +4.14348 q^{91} -2.23042 q^{92} -4.13382 q^{93} -18.0215 q^{94} +7.30174 q^{95} +8.11799 q^{96} -16.0074 q^{97} +2.05680 q^{98} +3.28152 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{2} - 6 q^{3} + 5 q^{4} - 6 q^{5} - 3 q^{6} + 6 q^{7} + 9 q^{8} + 6 q^{9}+O(q^{10}) $ 6 * q + 3 * q^2 - 6 * q^3 + 5 * q^4 - 6 * q^5 - 3 * q^6 + 6 * q^7 + 9 * q^8 + 6 * q^9	\( 6 q + 3 q^{2} - 6 q^{3} + 5 q^{4} - 6 q^{5} - 3 q^{6} + 6 q^{7} + 9 q^{8} + 6 q^{9} - 3 q^{10} + 6 q^{11} - 5 q^{12} - 6 q^{13} + 3 q^{14} + 6 q^{15} + 11 q^{16} + 8 q^{17} + 3 q^{18} - 8 q^{19} - 5 q^{20} - 6 q^{21} + 18 q^{22} - 6 q^{23} - 9 q^{24} + 6 q^{25} + 13 q^{26} - 6 q^{27} + 5 q^{28} + 8 q^{29} + 3 q^{30} - 16 q^{31} + 26 q^{32} - 6 q^{33} - 13 q^{34} - 6 q^{35} + 5 q^{36} + 8 q^{37} - 14 q^{38} + 6 q^{39} - 9 q^{40} + 8 q^{41} - 3 q^{42} + 8 q^{43} + 15 q^{44} - 6 q^{45} - 3 q^{46} + 10 q^{47} - 11 q^{48} + 6 q^{49} + 3 q^{50} - 8 q^{51} + 28 q^{53} - 3 q^{54} - 6 q^{55} + 9 q^{56} + 8 q^{57} + 14 q^{58} - 6 q^{59} + 5 q^{60} - 12 q^{61} + 18 q^{62} + 6 q^{63} + 15 q^{64} + 6 q^{65} - 18 q^{66} + 18 q^{67} + 18 q^{68} + 6 q^{69} - 3 q^{70} + 10 q^{71} + 9 q^{72} - 2 q^{73} + q^{74} - 6 q^{75} + 25 q^{76} + 6 q^{77} - 13 q^{78} - 2 q^{79} - 11 q^{80} + 6 q^{81} - 12 q^{82} + 26 q^{83} - 5 q^{84} - 8 q^{85} + 16 q^{86} - 8 q^{87} + 21 q^{88} + 8 q^{89} - 3 q^{90} - 6 q^{91} - 5 q^{92} + 16 q^{93} - 8 q^{94} + 8 q^{95} - 26 q^{96} + 20 q^{97} + 3 q^{98} + 6 q^{99}+O(q^{100}) \) 6 * q + 3 * q^2 - 6 * q^3 + 5 * q^4 - 6 * q^5 - 3 * q^6 + 6 * q^7 + 9 * q^8 + 6 * q^9 - 3 * q^10 + 6 * q^11 - 5 * q^12 - 6 * q^13 + 3 * q^14 + 6 * q^15 + 11 * q^16 + 8 * q^17 + 3 * q^18 - 8 * q^19 - 5 * q^20 - 6 * q^21 + 18 * q^22 - 6 * q^23 - 9 * q^24 + 6 * q^25 + 13 * q^26 - 6 * q^27 + 5 * q^28 + 8 * q^29 + 3 * q^30 - 16 * q^31 + 26 * q^32 - 6 * q^33 - 13 * q^34 - 6 * q^35 + 5 * q^36 + 8 * q^37 - 14 * q^38 + 6 * q^39 - 9 * q^40 + 8 * q^41 - 3 * q^42 + 8 * q^43 + 15 * q^44 - 6 * q^45 - 3 * q^46 + 10 * q^47 - 11 * q^48 + 6 * q^49 + 3 * q^50 - 8 * q^51 + 28 * q^53 - 3 * q^54 - 6 * q^55 + 9 * q^56 + 8 * q^57 + 14 * q^58 - 6 * q^59 + 5 * q^60 - 12 * q^61 + 18 * q^62 + 6 * q^63 + 15 * q^64 + 6 * q^65 - 18 * q^66 + 18 * q^67 + 18 * q^68 + 6 * q^69 - 3 * q^70 + 10 * q^71 + 9 * q^72 - 2 * q^73 + q^74 - 6 * q^75 + 25 * q^76 + 6 * q^77 - 13 * q^78 - 2 * q^79 - 11 * q^80 + 6 * q^81 - 12 * q^82 + 26 * q^83 - 5 * q^84 - 8 * q^85 + 16 * q^86 - 8 * q^87 + 21 * q^88 + 8 * q^89 - 3 * q^90 - 6 * q^91 - 5 * q^92 + 16 * q^93 - 8 * q^94 + 8 * q^95 - 26 * q^96 + 20 * q^97 + 3 * q^98 + 6 * 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 $

$2$	2.05680	1.45438	0.727188	−	0.686438i	$-0.240827\pi$
$2$	2.05680	1.45438	0.727188	+	0.686438i	$0.240827\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.23042	1.11521
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.05680	−0.839685
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.473937	0.167562
$9$	1.00000	0.333333
$10$	−2.05680	−0.650417
$11$	3.28152	0.989415	0.494708	−	0.869059i	$-0.335275\pi$
$11$	3.28152	0.989415	0.494708	+	0.869059i	$0.335275\pi$
$12$	−2.23042	−0.643868
$13$	4.14348	1.14919	0.574597	−	0.818436i	$-0.305159\pi$
$13$	4.14348	1.14919	0.574597	+	0.818436i	$0.305159\pi$
$14$	2.05680	0.549703
$15$	1.00000	0.258199
$16$	−3.48606	−0.871514
$17$	3.07319	0.745358	0.372679	−	0.927960i	$-0.378439\pi$
$17$	3.07319	0.745358	0.372679	+	0.927960i	$0.378439\pi$
$18$	2.05680	0.484792
$19$	−7.30174	−1.67513	−0.837567	−	0.546334i	$-0.816023\pi$
$19$	−7.30174	−1.67513	−0.837567	+	0.546334i	$0.816023\pi$
$20$	−2.23042	−0.498738
$21$	−1.00000	−0.218218
$22$	6.74943	1.43898
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−0.473937	−0.0967420
$25$	1.00000	0.200000
$26$	8.52231	1.67136
$27$	−1.00000	−0.192450
$28$	2.23042	0.421511
$29$	4.84920	0.900473	0.450237	−	0.892909i	$-0.351340\pi$
$29$	4.84920	0.900473	0.450237	+	0.892909i	$0.351340\pi$
$30$	2.05680	0.375519
$31$	4.13382	0.742457	0.371228	−	0.928542i	$-0.378937\pi$
$31$	4.13382	0.742457	0.371228	+	0.928542i	$0.378937\pi$
$32$	−8.11799	−1.43507
$33$	−3.28152	−0.571239
$34$	6.32094	1.08403
$35$	−1.00000	−0.169031
$36$	2.23042	0.371737
$37$	9.25670	1.52179	0.760896	−	0.648874i	$-0.224760\pi$
$37$	9.25670	1.52179	0.760896	+	0.648874i	$0.224760\pi$
$38$	−15.0182	−2.43628
$39$	−4.14348	−0.663488
$40$	−0.473937	−0.0749360
$41$	5.93515	0.926915	0.463457	−	0.886119i	$-0.346609\pi$
$41$	5.93515	0.926915	0.463457	+	0.886119i	$0.346609\pi$
$42$	−2.05680	−0.317371
$43$	4.77427	0.728069	0.364035	−	0.931385i	$-0.381399\pi$
$43$	4.77427	0.728069	0.364035	+	0.931385i	$0.381399\pi$
$44$	7.31918	1.10341
$45$	−1.00000	−0.149071
$46$	−2.05680	−0.303259
$47$	−8.76193	−1.27806	−0.639029	−	0.769182i	$-0.720664\pi$
$47$	−8.76193	−1.27806	−0.639029	+	0.769182i	$0.720664\pi$
$48$	3.48606	0.503169
$49$	1.00000	0.142857
$50$	2.05680	0.290875
$51$	−3.07319	−0.430333
$52$	9.24172	1.28160
$53$	9.55667	1.31271	0.656354	−	0.754453i	$-0.272098\pi$
$53$	9.55667	1.31271	0.656354	+	0.754453i	$0.272098\pi$
$54$	−2.05680	−0.279895
$55$	−3.28152	−0.442480
$56$	0.473937	0.0633325
$57$	7.30174	0.967140
$58$	9.97382	1.30963
$59$	−1.65430	−0.215371	−0.107686	−	0.994185i	$-0.534344\pi$
$59$	−1.65430	−0.215371	−0.107686	+	0.994185i	$0.534344\pi$
$60$	2.23042	0.287947
$61$	12.7382	1.63095	0.815477	−	0.578790i	$-0.196475\pi$
$61$	12.7382	1.63095	0.815477	+	0.578790i	$0.196475\pi$
$62$	8.50245	1.07981
$63$	1.00000	0.125988
$64$	−9.72497	−1.21562
$65$	−4.14348	−0.513935
$66$	−6.74943	−0.830797
$67$	−7.97142	−0.973864	−0.486932	−	0.873440i	$-0.661884\pi$
$67$	−7.97142	−0.973864	−0.486932	+	0.873440i	$0.661884\pi$
$68$	6.85452	0.831233
$69$	1.00000	0.120386
$70$	−2.05680	−0.245835
$71$	13.4257	1.59334	0.796669	−	0.604416i	$-0.206593\pi$
$71$	13.4257	1.59334	0.796669	+	0.604416i	$0.206593\pi$
$72$	0.473937	0.0558540
$73$	−1.30705	−0.152978	−0.0764892	−	0.997070i	$-0.524371\pi$
$73$	−1.30705	−0.152978	−0.0764892	+	0.997070i	$0.524371\pi$
$74$	19.0392	2.21326
$75$	−1.00000	−0.115470
$76$	−16.2860	−1.86813
$77$	3.28152	0.373964
$78$	−8.52231	−0.964961
$79$	6.60879	0.743547	0.371774	−	0.928323i	$-0.378750\pi$
$79$	6.60879	0.743547	0.371774	+	0.928323i	$0.378750\pi$
$80$	3.48606	0.389753
$81$	1.00000	0.111111
$82$	12.2074	1.34808
$83$	−9.03292	−0.991492	−0.495746	−	0.868468i	$-0.665105\pi$
$83$	−9.03292	−0.991492	−0.495746	+	0.868468i	$0.665105\pi$
$84$	−2.23042	−0.243359
$85$	−3.07319	−0.333334
$86$	9.81972	1.05889
$87$	−4.84920	−0.519888
$88$	1.55523	0.165789
$89$	−11.6650	−1.23648	−0.618242	−	0.785988i	$-0.712155\pi$
$89$	−11.6650	−1.23648	−0.618242	+	0.785988i	$0.712155\pi$
$90$	−2.05680	−0.216806
$91$	4.14348	0.434355
$92$	−2.23042	−0.232538
$93$	−4.13382	−0.428658
$94$	−18.0215	−1.85878
$95$	7.30174	0.749143
$96$	8.11799	0.828539
$97$	−16.0074	−1.62531	−0.812653	−	0.582748i	$-0.801978\pi$
$97$	−16.0074	−1.62531	−0.812653	+	0.582748i	$0.801978\pi$
$98$	2.05680	0.207768
$99$	3.28152	0.329805
$100$	2.23042	0.223042
$101$	13.7191	1.36510	0.682551	−	0.730838i	$-0.260870\pi$
$101$	13.7191	1.36510	0.682551	+	0.730838i	$0.260870\pi$
$102$	−6.32094	−0.625866
$103$	4.12085	0.406040	0.203020	−	0.979175i	$-0.434924\pi$
$103$	4.12085	0.406040	0.203020	+	0.979175i	$0.434924\pi$
$104$	1.96375	0.192561
$105$	1.00000	0.0975900
$106$	19.6561	1.90917
$107$	16.7364	1.61797	0.808984	−	0.587830i	$-0.200018\pi$
$107$	16.7364	1.61797	0.808984	+	0.587830i	$0.200018\pi$
$108$	−2.23042	−0.214623
$109$	6.88964	0.659908	0.329954	−	0.943997i	$-0.392967\pi$
$109$	6.88964	0.659908	0.329954	+	0.943997i	$0.392967\pi$
$110$	−6.74943	−0.643533
$111$	−9.25670	−0.878607
$112$	−3.48606	−0.329401
$113$	16.9022	1.59003	0.795013	−	0.606593i	$-0.207464\pi$
$113$	16.9022	1.59003	0.795013	+	0.606593i	$0.207464\pi$
$114$	15.0182	1.40659
$115$	1.00000	0.0932505
$116$	10.8158	1.00422
$117$	4.14348	0.383065
$118$	−3.40256	−0.313231
$119$	3.07319	0.281719
$120$	0.473937	0.0432643
$121$	−0.231630	−0.0210572
$122$	26.1998	2.37202
$123$	−5.93515	−0.535155
$124$	9.22018	0.827997
$125$	−1.00000	−0.0894427
$126$	2.05680	0.183234
$127$	−8.35946	−0.741782	−0.370891	−	0.928676i	$-0.620948\pi$
$127$	−8.35946	−0.741782	−0.370891	+	0.928676i	$0.620948\pi$
$128$	−3.76633	−0.332900
$129$	−4.77427	−0.420351
$130$	−8.52231	−0.747456
$131$	−7.55426	−0.660019	−0.330010	−	0.943978i	$-0.607052\pi$
$131$	−7.55426	−0.660019	−0.330010	+	0.943978i	$0.607052\pi$
$132$	−7.31918	−0.637053
$133$	−7.30174	−0.633141
$134$	−16.3956	−1.41637
$135$	1.00000	0.0860663
$136$	1.45650	0.124894
$137$	−4.09998	−0.350285	−0.175143	−	0.984543i	$-0.556039\pi$
$137$	−4.09998	−0.350285	−0.175143	+	0.984543i	$0.556039\pi$
$138$	2.05680	0.175086
$139$	−9.19530	−0.779935	−0.389967	−	0.920829i	$-0.627514\pi$
$139$	−9.19530	−0.779935	−0.389967	+	0.920829i	$0.627514\pi$
$140$	−2.23042	−0.188505
$141$	8.76193	0.737887
$142$	27.6140	2.31731
$143$	13.5969	1.13703
$144$	−3.48606	−0.290505
$145$	−4.84920	−0.402704
$146$	−2.68833	−0.222488
$147$	−1.00000	−0.0824786
$148$	20.6464	1.69712
$149$	−16.4520	−1.34780	−0.673900	−	0.738823i	$-0.735382\pi$
$149$	−16.4520	−1.34780	−0.673900	+	0.738823i	$0.735382\pi$
$150$	−2.05680	−0.167937
$151$	8.57473	0.697802	0.348901	−	0.937160i	$-0.386555\pi$
$151$	8.57473	0.697802	0.348901	+	0.937160i	$0.386555\pi$
$152$	−3.46057	−0.280689
$153$	3.07319	0.248453
$154$	6.74943	0.543884
$155$	−4.13382	−0.332037
$156$	−9.24172	−0.739930
$157$	−1.11398	−0.0889052	−0.0444526	−	0.999011i	$-0.514154\pi$
$157$	−1.11398	−0.0889052	−0.0444526	+	0.999011i	$0.514154\pi$
$158$	13.5930	1.08140
$159$	−9.55667	−0.757893
$160$	8.11799	0.641784
$161$	−1.00000	−0.0788110
$162$	2.05680	0.161597
$163$	−3.14500	−0.246335	−0.123168	−	0.992386i	$-0.539305\pi$
$163$	−3.14500	−0.246335	−0.123168	+	0.992386i	$0.539305\pi$
$164$	13.2379	1.03371
$165$	3.28152	0.255466
$166$	−18.5789	−1.44200
$167$	19.0105	1.47108	0.735538	−	0.677483i	$-0.236929\pi$
$167$	19.0105	1.47108	0.735538	+	0.677483i	$0.236929\pi$
$168$	−0.473937	−0.0365650
$169$	4.16842	0.320648
$170$	−6.32094	−0.484794
$171$	−7.30174	−0.558378
$172$	10.6486	0.811952
$173$	−8.65577	−0.658086	−0.329043	−	0.944315i	$-0.606726\pi$
$173$	−8.65577	−0.658086	−0.329043	+	0.944315i	$0.606726\pi$
$174$	−9.97382	−0.756114
$175$	1.00000	0.0755929
$176$	−11.4396	−0.862289
$177$	1.65430	0.124345
$178$	−23.9925	−1.79831
$179$	−23.5515	−1.76032	−0.880161	−	0.474674i	$-0.842566\pi$
$179$	−23.5515	−1.76032	−0.880161	+	0.474674i	$0.842566\pi$
$180$	−2.23042	−0.166246
$181$	−8.83819	−0.656937	−0.328469	−	0.944515i	$-0.606533\pi$
$181$	−8.83819	−0.656937	−0.328469	+	0.944515i	$0.606533\pi$
$182$	8.52231	0.631715
$183$	−12.7382	−0.941631
$184$	−0.473937	−0.0349391
$185$	−9.25670	−0.680566
$186$	−8.50245	−0.623430
$187$	10.0847	0.737469
$188$	−19.5428	−1.42531
$189$	−1.00000	−0.0727393
$190$	15.0182	1.08954
$191$	−10.3537	−0.749165	−0.374582	−	0.927194i	$-0.622214\pi$
$191$	−10.3537	−0.749165	−0.374582	+	0.927194i	$0.622214\pi$
$192$	9.72497	0.701839
$193$	−15.5438	−1.11887	−0.559435	−	0.828874i	$-0.688982\pi$
$193$	−15.5438	−1.11887	−0.559435	+	0.828874i	$0.688982\pi$
$194$	−32.9240	−2.36381
$195$	4.14348	0.296721
$196$	2.23042	0.159316
$197$	−11.1054	−0.791229	−0.395615	−	0.918417i	$-0.629468\pi$
$197$	−11.1054	−0.791229	−0.395615	+	0.918417i	$0.629468\pi$
$198$	6.74943	0.479661
$199$	−8.77322	−0.621917	−0.310959	−	0.950423i	$-0.600650\pi$
$199$	−8.77322	−0.621917	−0.310959	+	0.950423i	$0.600650\pi$
$200$	0.473937	0.0335124
$201$	7.97142	0.562261
$202$	28.2175	1.98537
$203$	4.84920	0.340347
$204$	−6.85452	−0.479912
$205$	−5.93515	−0.414529
$206$	8.47577	0.590535
$207$	−1.00000	−0.0695048
$208$	−14.4444	−1.00154
$209$	−23.9608	−1.65740
$210$	2.05680	0.141933
$211$	−25.6533	−1.76605	−0.883023	−	0.469329i	$-0.844496\pi$
$211$	−25.6533	−1.76605	−0.883023	+	0.469329i	$0.844496\pi$
$212$	21.3154	1.46395
$213$	−13.4257	−0.919914
$214$	34.4234	2.35314
$215$	−4.77427	−0.325603
$216$	−0.473937	−0.0322473
$217$	4.13382	0.280622
$218$	14.1706	0.959755
$219$	1.30705	0.0883221
$220$	−7.31918	−0.493459
$221$	12.7337	0.856562
$222$	−19.0392	−1.27783
$223$	−5.74216	−0.384524	−0.192262	−	0.981344i	$-0.561582\pi$
$223$	−5.74216	−0.384524	−0.192262	+	0.981344i	$0.561582\pi$
$224$	−8.11799	−0.542406
$225$	1.00000	0.0666667
$226$	34.7644	2.31250
$227$	27.0123	1.79287	0.896433	−	0.443179i	$-0.146149\pi$
$227$	27.0123	1.79287	0.896433	+	0.443179i	$0.146149\pi$
$228$	16.2860	1.07857
$229$	−29.6456	−1.95904	−0.979520	−	0.201349i	$-0.935467\pi$
$229$	−29.6456	−1.95904	−0.979520	+	0.201349i	$0.935467\pi$
$230$	2.05680	0.135621
$231$	−3.28152	−0.215908
$232$	2.29821	0.150885
$233$	−13.7589	−0.901372	−0.450686	−	0.892682i	$-0.648821\pi$
$233$	−13.7589	−0.901372	−0.450686	+	0.892682i	$0.648821\pi$
$234$	8.52231	0.557121
$235$	8.76193	0.571565
$236$	−3.68979	−0.240185
$237$	−6.60879	−0.429287
$238$	6.32094	0.409726
$239$	15.4591	0.999968	0.499984	−	0.866035i	$-0.333339\pi$
$239$	15.4591	0.999968	0.499984	+	0.866035i	$0.333339\pi$
$240$	−3.48606	−0.225024
$241$	−12.8275	−0.826291	−0.413145	−	0.910665i	$-0.635570\pi$
$241$	−12.8275	−0.826291	−0.413145	+	0.910665i	$0.635570\pi$
$242$	−0.476416	−0.0306252
$243$	−1.00000	−0.0641500
$244$	28.4115	1.81886
$245$	−1.00000	−0.0638877
$246$	−12.2074	−0.778316
$247$	−30.2546	−1.92506
$248$	1.95917	0.124408
$249$	9.03292	0.572438
$250$	−2.05680	−0.130083
$251$	−9.54991	−0.602785	−0.301393	−	0.953500i	$-0.597451\pi$
$251$	−9.54991	−0.602785	−0.301393	+	0.953500i	$0.597451\pi$
$252$	2.23042	0.140504
$253$	−3.28152	−0.206307
$254$	−17.1937	−1.07883
$255$	3.07319	0.192451
$256$	11.7033	0.731459
$257$	−2.17492	−0.135668	−0.0678339	−	0.997697i	$-0.521609\pi$
$257$	−2.17492	−0.135668	−0.0678339	+	0.997697i	$0.521609\pi$
$258$	−9.81972	−0.611349
$259$	9.25670	0.575183
$260$	−9.24172	−0.573147
$261$	4.84920	0.300158
$262$	−15.5376	−0.959917
$263$	12.3196	0.759661	0.379830	−	0.925056i	$-0.375982\pi$
$263$	12.3196	0.759661	0.379830	+	0.925056i	$0.375982\pi$
$264$	−1.55523	−0.0957180
$265$	−9.55667	−0.587061
$266$	−15.0182	−0.920826
$267$	11.6650	0.713884
$268$	−17.7797	−1.08607
$269$	14.6898	0.895654	0.447827	−	0.894120i	$-0.352198\pi$
$269$	14.6898	0.895654	0.447827	+	0.894120i	$0.352198\pi$
$270$	2.05680	0.125173
$271$	−8.65889	−0.525990	−0.262995	−	0.964797i	$-0.584710\pi$
$271$	−8.65889	−0.525990	−0.262995	+	0.964797i	$0.584710\pi$
$272$	−10.7133	−0.649590
$273$	−4.14348	−0.250775
$274$	−8.43284	−0.509446
$275$	3.28152	0.197883
$276$	2.23042	0.134256
$277$	18.5542	1.11481	0.557407	−	0.830240i	$-0.311796\pi$
$277$	18.5542	1.11481	0.557407	+	0.830240i	$0.311796\pi$
$278$	−18.9129	−1.13432
$279$	4.13382	0.247486
$280$	−0.473937	−0.0283232
$281$	17.0301	1.01593	0.507965	−	0.861378i	$-0.330398\pi$
$281$	17.0301	1.01593	0.507965	+	0.861378i	$0.330398\pi$
$282$	18.0215	1.07317
$283$	11.5793	0.688320	0.344160	−	0.938911i	$-0.388164\pi$
$283$	11.5793	0.688320	0.344160	+	0.938911i	$0.388164\pi$
$284$	29.9450	1.77691
$285$	−7.30174	−0.432518
$286$	27.9661	1.65367
$287$	5.93515	0.350341
$288$	−8.11799	−0.478357
$289$	−7.55550	−0.444441
$290$	−9.97382	−0.585683
$291$	16.0074	0.938371
$292$	−2.91527	−0.170603
$293$	−23.5087	−1.37339	−0.686696	−	0.726945i	$-0.740940\pi$
$293$	−23.5087	−1.37339	−0.686696	+	0.726945i	$0.740940\pi$
$294$	−2.05680	−0.119955
$295$	1.65430	0.0963169
$296$	4.38709	0.254995
$297$	−3.28152	−0.190413
$298$	−33.8385	−1.96021
$299$	−4.14348	−0.239624
$300$	−2.23042	−0.128774
$301$	4.77427	0.275184
$302$	17.6365	1.01487
$303$	−13.7191	−0.788142
$304$	25.4543	1.45990
$305$	−12.7382	−0.729384
$306$	6.32094	0.361344
$307$	2.23100	0.127330	0.0636648	−	0.997971i	$-0.479721\pi$
$307$	2.23100	0.127330	0.0636648	+	0.997971i	$0.479721\pi$
$308$	7.31918	0.417049
$309$	−4.12085	−0.234427
$310$	−8.50245	−0.482907
$311$	−12.2995	−0.697442	−0.348721	−	0.937227i	$-0.613384\pi$
$311$	−12.2995	−0.697442	−0.348721	+	0.937227i	$0.613384\pi$
$312$	−1.96375	−0.111175
$313$	10.9900	0.621194	0.310597	−	0.950542i	$-0.399471\pi$
$313$	10.9900	0.621194	0.310597	+	0.950542i	$0.399471\pi$
$314$	−2.29123	−0.129302
$315$	−1.00000	−0.0563436
$316$	14.7404	0.829213
$317$	17.4233	0.978590	0.489295	−	0.872118i	$-0.337254\pi$
$317$	17.4233	0.978590	0.489295	+	0.872118i	$0.337254\pi$
$318$	−19.6561	−1.10226
$319$	15.9127	0.890942
$320$	9.72497	0.543642
$321$	−16.7364	−0.934135
$322$	−2.05680	−0.114621
$323$	−22.4397	−1.24858
$324$	2.23042	0.123912
$325$	4.14348	0.229839
$326$	−6.46863	−0.358265
$327$	−6.88964	−0.380998
$328$	2.81289	0.155316
$329$	−8.76193	−0.483061
$330$	6.74943	0.371544
$331$	−24.1162	−1.32554	−0.662772	−	0.748821i	$-0.730620\pi$
$331$	−24.1162	−1.32554	−0.662772	+	0.748821i	$0.730620\pi$
$332$	−20.1472	−1.10572
$333$	9.25670	0.507264
$334$	39.1008	2.13950
$335$	7.97142	0.435525
$336$	3.48606	0.190180
$337$	11.0479	0.601817	0.300908	−	0.953653i	$-0.402710\pi$
$337$	11.0479	0.601817	0.300908	+	0.953653i	$0.402710\pi$
$338$	8.57361	0.466343
$339$	−16.9022	−0.918002
$340$	−6.85452	−0.371739
$341$	13.5652	0.734598
$342$	−15.0182	−0.812093
$343$	1.00000	0.0539949
$344$	2.26270	0.121997
$345$	−1.00000	−0.0538382
$346$	−17.8032	−0.957105
$347$	−14.7716	−0.792982	−0.396491	−	0.918039i	$-0.629772\pi$
$347$	−14.7716	−0.792982	−0.396491	+	0.918039i	$0.629772\pi$
$348$	−10.8158	−0.579786
$349$	8.45803	0.452748	0.226374	−	0.974040i	$-0.427313\pi$
$349$	8.45803	0.452748	0.226374	+	0.974040i	$0.427313\pi$
$350$	2.05680	0.109941
$351$	−4.14348	−0.221163
$352$	−26.6393	−1.41988
$353$	14.5398	0.773873	0.386937	−	0.922106i	$-0.373533\pi$
$353$	14.5398	0.773873	0.386937	+	0.922106i	$0.373533\pi$
$354$	3.40256	0.180844
$355$	−13.4257	−0.712562
$356$	−26.0178	−1.37894
$357$	−3.07319	−0.162651
$358$	−48.4407	−2.56017
$359$	−6.77822	−0.357741	−0.178870	−	0.983873i	$-0.557244\pi$
$359$	−6.77822	−0.357741	−0.178870	+	0.983873i	$0.557244\pi$
$360$	−0.473937	−0.0249787
$361$	34.3155	1.80608
$362$	−18.1784	−0.955434
$363$	0.231630	0.0121574
$364$	9.24172	0.484398
$365$	1.30705	0.0684140
$366$	−26.1998	−1.36949
$367$	−15.4057	−0.804171	−0.402086	−	0.915602i	$-0.631715\pi$
$367$	−15.4057	−0.804171	−0.402086	+	0.915602i	$0.631715\pi$
$368$	3.48606	0.181723
$369$	5.93515	0.308972
$370$	−19.0392	−0.989799
$371$	9.55667	0.496157
$372$	−9.22018	−0.478044
$373$	25.1856	1.30406	0.652032	−	0.758192i	$-0.273917\pi$
$373$	25.1856	1.30406	0.652032	+	0.758192i	$0.273917\pi$
$374$	20.7423	1.07256
$375$	1.00000	0.0516398
$376$	−4.15260	−0.214154
$377$	20.0925	1.03482
$378$	−2.05680	−0.105790
$379$	−15.5459	−0.798541	−0.399271	−	0.916833i	$-0.630737\pi$
$379$	−15.5459	−0.798541	−0.399271	+	0.916833i	$0.630737\pi$
$380$	16.2860	0.835454
$381$	8.35946	0.428268
$382$	−21.2954	−1.08957
$383$	10.7481	0.549201	0.274600	−	0.961558i	$-0.411454\pi$
$383$	10.7481	0.549201	0.274600	+	0.961558i	$0.411454\pi$
$384$	3.76633	0.192200
$385$	−3.28152	−0.167242
$386$	−31.9706	−1.62726
$387$	4.77427	0.242690
$388$	−35.7033	−1.81256
$389$	−17.2969	−0.876989	−0.438494	−	0.898734i	$-0.644488\pi$
$389$	−17.2969	−0.876989	−0.438494	+	0.898734i	$0.644488\pi$
$390$	8.52231	0.431544
$391$	−3.07319	−0.155418
$392$	0.473937	0.0239374
$393$	7.55426	0.381062
$394$	−22.8417	−1.15075
$395$	−6.60879	−0.332524
$396$	7.31918	0.367803
$397$	−16.1628	−0.811190	−0.405595	−	0.914053i	$-0.632936\pi$
$397$	−16.1628	−0.811190	−0.405595	+	0.914053i	$0.632936\pi$
$398$	−18.0448	−0.904502
$399$	7.30174	0.365544
$400$	−3.48606	−0.174303
$401$	11.6421	0.581381	0.290690	−	0.956817i	$-0.406115\pi$
$401$	11.6421	0.581381	0.290690	+	0.956817i	$0.406115\pi$
$402$	16.3956	0.817739
$403$	17.1284	0.853227
$404$	30.5994	1.52238
$405$	−1.00000	−0.0496904
$406$	9.97382	0.494993
$407$	30.3760	1.50568
$408$	−1.45650	−0.0721075
$409$	23.4913	1.16157	0.580786	−	0.814056i	$-0.302745\pi$
$409$	23.4913	1.16157	0.580786	+	0.814056i	$0.302745\pi$
$410$	−12.2074	−0.602881
$411$	4.09998	0.202237
$412$	9.19125	0.452821
$413$	−1.65430	−0.0814027
$414$	−2.05680	−0.101086
$415$	9.03292	0.443408
$416$	−33.6367	−1.64918
$417$	9.19530	0.450296
$418$	−49.2826	−2.41049
$419$	18.1840	0.888347	0.444173	−	0.895941i	$-0.353497\pi$
$419$	18.1840	0.888347	0.444173	+	0.895941i	$0.353497\pi$
$420$	2.23042	0.108834
$421$	−24.3109	−1.18484	−0.592420	−	0.805629i	$-0.701827\pi$
$421$	−24.3109	−1.18484	−0.592420	+	0.805629i	$0.701827\pi$
$422$	−52.7637	−2.56850
$423$	−8.76193	−0.426019
$424$	4.52926	0.219960
$425$	3.07319	0.149072
$426$	−27.6140	−1.33790
$427$	12.7382	0.616442
$428$	37.3293	1.80438
$429$	−13.5969	−0.656465
$430$	−9.81972	−0.473549
$431$	2.52590	0.121669	0.0608343	−	0.998148i	$-0.480624\pi$
$431$	2.52590	0.121669	0.0608343	+	0.998148i	$0.480624\pi$
$432$	3.48606	0.167723
$433$	24.5767	1.18108	0.590541	−	0.807007i	$-0.298914\pi$
$433$	24.5767	1.18108	0.590541	+	0.807007i	$0.298914\pi$
$434$	8.50245	0.408130
$435$	4.84920	0.232501
$436$	15.3668	0.735938
$437$	7.30174	0.349290
$438$	2.68833	0.128454
$439$	15.3459	0.732419	0.366209	−	0.930532i	$-0.380655\pi$
$439$	15.3459	0.732419	0.366209	+	0.930532i	$0.380655\pi$
$440$	−1.55523	−0.0741429
$441$	1.00000	0.0476190
$442$	26.1907	1.24576
$443$	23.8220	1.13182	0.565908	−	0.824468i	$-0.308526\pi$
$443$	23.8220	1.13182	0.565908	+	0.824468i	$0.308526\pi$
$444$	−20.6464	−0.979833
$445$	11.6650	0.552972
$446$	−11.8105	−0.559242
$447$	16.4520	0.778153
$448$	−9.72497	−0.459462
$449$	−40.3021	−1.90197	−0.950986	−	0.309233i	$-0.899928\pi$
$449$	−40.3021	−1.90197	−0.950986	+	0.309233i	$0.899928\pi$
$450$	2.05680	0.0969585
$451$	19.4763	0.917104
$452$	37.6991	1.77322
$453$	−8.57473	−0.402876
$454$	55.5588	2.60750
$455$	−4.14348	−0.194249
$456$	3.46057	0.162056
$457$	28.8916	1.35149	0.675746	−	0.737135i	$-0.263822\pi$
$457$	28.8916	1.35149	0.675746	+	0.737135i	$0.263822\pi$
$458$	−60.9751	−2.84918
$459$	−3.07319	−0.143444
$460$	2.23042	0.103994
$461$	2.33103	0.108567	0.0542834	−	0.998526i	$-0.482713\pi$
$461$	2.33103	0.108567	0.0542834	+	0.998526i	$0.482713\pi$
$462$	−6.74943	−0.314012
$463$	−35.1421	−1.63319	−0.816596	−	0.577209i	$-0.804142\pi$
$463$	−35.1421	−1.63319	−0.816596	+	0.577209i	$0.804142\pi$
$464$	−16.9046	−0.784775
$465$	4.13382	0.191701
$466$	−28.2992	−1.31094
$467$	24.0426	1.11256	0.556280	−	0.830995i	$-0.312228\pi$
$467$	24.0426	1.11256	0.556280	+	0.830995i	$0.312228\pi$
$468$	9.24172	0.427199
$469$	−7.97142	−0.368086
$470$	18.0215	0.831271
$471$	1.11398	0.0513295
$472$	−0.784033	−0.0360880
$473$	15.6669	0.720363
$474$	−13.5930	−0.624345
$475$	−7.30174	−0.335027
$476$	6.85452	0.314176
$477$	9.55667	0.437570
$478$	31.7963	1.45433
$479$	−22.9522	−1.04871	−0.524355	−	0.851500i	$-0.675694\pi$
$479$	−22.9522	−1.04871	−0.524355	+	0.851500i	$0.675694\pi$
$480$	−8.11799	−0.370534
$481$	38.3549	1.74883
$482$	−26.3836	−1.20174
$483$	1.00000	0.0455016
$484$	−0.516632	−0.0234833
$485$	16.0074	0.726859
$486$	−2.05680	−0.0932983
$487$	1.18259	0.0535880	0.0267940	−	0.999641i	$-0.491470\pi$
$487$	1.18259	0.0535880	0.0267940	+	0.999641i	$0.491470\pi$
$488$	6.03708	0.273286
$489$	3.14500	0.142222
$490$	−2.05680	−0.0929167
$491$	8.18683	0.369466	0.184733	−	0.982789i	$-0.440858\pi$
$491$	8.18683	0.369466	0.184733	+	0.982789i	$0.440858\pi$
$492$	−13.2379	−0.596811
$493$	14.9025	0.671175
$494$	−62.2277	−2.79976
$495$	−3.28152	−0.147493
$496$	−14.4107	−0.647061
$497$	13.4257	0.602225
$498$	18.5789	0.832540
$499$	−33.5961	−1.50397	−0.751983	−	0.659183i	$-0.770902\pi$
$499$	−33.5961	−1.50397	−0.751983	+	0.659183i	$0.770902\pi$
$500$	−2.23042	−0.0997476
$501$	−19.0105	−0.849326
$502$	−19.6423	−0.876677
$503$	26.8856	1.19877	0.599384	−	0.800462i	$-0.295412\pi$
$503$	26.8856	1.19877	0.599384	+	0.800462i	$0.295412\pi$
$504$	0.473937	0.0211108
$505$	−13.7191	−0.610492
$506$	−6.74943	−0.300049
$507$	−4.16842	−0.185126
$508$	−18.6451	−0.827245
$509$	−14.3506	−0.636078	−0.318039	−	0.948078i	$-0.603024\pi$
$509$	−14.3506	−0.636078	−0.318039	+	0.948078i	$0.603024\pi$
$510$	6.32094	0.279896
$511$	−1.30705	−0.0578204
$512$	31.6041	1.39672
$513$	7.30174	0.322380
$514$	−4.47338	−0.197312
$515$	−4.12085	−0.181587
$516$	−10.6486	−0.468781
$517$	−28.7524	−1.26453
$518$	19.0392	0.836533
$519$	8.65577	0.379946
$520$	−1.96375	−0.0861161
$521$	2.33511	0.102303	0.0511516	−	0.998691i	$-0.483711\pi$
$521$	2.33511	0.102303	0.0511516	+	0.998691i	$0.483711\pi$
$522$	9.97382	0.436542
$523$	−29.7285	−1.29994	−0.649968	−	0.759961i	$-0.725218\pi$
$523$	−29.7285	−1.29994	−0.649968	+	0.759961i	$0.725218\pi$
$524$	−16.8492	−0.736061
$525$	−1.00000	−0.0436436
$526$	25.3390	1.10483
$527$	12.7040	0.553396
$528$	11.4396	0.497843
$529$	1.00000	0.0434783
$530$	−19.6561	−0.853808
$531$	−1.65430	−0.0717904
$532$	−16.2860	−0.706087
$533$	24.5922	1.06521
$534$	23.9925	1.03826
$535$	−16.7364	−0.723578
$536$	−3.77795	−0.163183
$537$	23.5515	1.01632
$538$	30.2140	1.30262
$539$	3.28152	0.141345
$540$	2.23042	0.0959822
$541$	−26.1068	−1.12242	−0.561209	−	0.827674i	$-0.689664\pi$
$541$	−26.1068	−1.12242	−0.561209	+	0.827674i	$0.689664\pi$
$542$	−17.8096	−0.764988
$543$	8.83819	0.379283
$544$	−24.9481	−1.06964
$545$	−6.88964	−0.295120
$546$	−8.52231	−0.364721
$547$	−18.1614	−0.776524	−0.388262	−	0.921549i	$-0.626924\pi$
$547$	−18.1614	−0.776524	−0.388262	+	0.921549i	$0.626924\pi$
$548$	−9.14470	−0.390642
$549$	12.7382	0.543651
$550$	6.74943	0.287797
$551$	−35.4076	−1.50841
$552$	0.473937	0.0201721
$553$	6.60879	0.281034
$554$	38.1623	1.62136
$555$	9.25670	0.392925
$556$	−20.5094	−0.869793
$557$	22.5546	0.955669	0.477835	−	0.878450i	$-0.341422\pi$
$557$	22.5546	0.955669	0.477835	+	0.878450i	$0.341422\pi$
$558$	8.50245	0.359937
$559$	19.7821	0.836693
$560$	3.48606	0.147313
$561$	−10.0847	−0.425778
$562$	35.0275	1.47755
$563$	−21.6888	−0.914073	−0.457037	−	0.889448i	$-0.651089\pi$
$563$	−21.6888	−0.914073	−0.457037	+	0.889448i	$0.651089\pi$
$564$	19.5428	0.822901
$565$	−16.9022	−0.711081
$566$	23.8164	1.00108
$567$	1.00000	0.0419961
$568$	6.36294	0.266983
$569$	28.6284	1.20016	0.600082	−	0.799939i	$-0.295135\pi$
$569$	28.6284	1.20016	0.600082	+	0.799939i	$0.295135\pi$
$570$	−15.0182	−0.629044
$571$	−18.0017	−0.753348	−0.376674	−	0.926346i	$-0.622932\pi$
$571$	−18.0017	−0.753348	−0.376674	+	0.926346i	$0.622932\pi$
$572$	30.3269	1.26803
$573$	10.3537	0.432531
$574$	12.2074	0.509528
$575$	−1.00000	−0.0417029
$576$	−9.72497	−0.405207
$577$	−13.1022	−0.545451	−0.272725	−	0.962092i	$-0.587925\pi$
$577$	−13.1022	−0.545451	−0.272725	+	0.962092i	$0.587925\pi$
$578$	−15.5401	−0.646385
$579$	15.5438	0.645980
$580$	−10.8158	−0.449100
$581$	−9.03292	−0.374749
$582$	32.9240	1.36474
$583$	31.3604	1.29881
$584$	−0.619458	−0.0256334
$585$	−4.14348	−0.171312
$586$	−48.3527	−1.99743
$587$	−22.1968	−0.916160	−0.458080	−	0.888911i	$-0.651463\pi$
$587$	−22.1968	−0.916160	−0.458080	+	0.888911i	$0.651463\pi$
$588$	−2.23042	−0.0919812
$589$	−30.1841	−1.24372
$590$	3.40256	0.140081
$591$	11.1054	0.456816
$592$	−32.2694	−1.32626
$593$	−26.8869	−1.10411	−0.552056	−	0.833807i	$-0.686156\pi$
$593$	−26.8869	−1.10411	−0.552056	+	0.833807i	$0.686156\pi$
$594$	−6.74943	−0.276932
$595$	−3.07319	−0.125989
$596$	−36.6949	−1.50308
$597$	8.77322	0.359064
$598$	−8.52231	−0.348503
$599$	22.0512	0.900988	0.450494	−	0.892779i	$-0.351248\pi$
$599$	22.0512	0.900988	0.450494	+	0.892779i	$0.351248\pi$
$600$	−0.473937	−0.0193484
$601$	−8.90835	−0.363379	−0.181690	−	0.983356i	$-0.558157\pi$
$601$	−8.90835	−0.363379	−0.181690	+	0.983356i	$0.558157\pi$
$602$	9.81972	0.400222
$603$	−7.97142	−0.324621
$604$	19.1253	0.778197
$605$	0.231630	0.00941708
$606$	−28.2175	−1.14626
$607$	−15.7124	−0.637746	−0.318873	−	0.947797i	$-0.603304\pi$
$607$	−15.7124	−0.637746	−0.318873	+	0.947797i	$0.603304\pi$
$608$	59.2755	2.40394
$609$	−4.84920	−0.196499
$610$	−26.1998	−1.06080
$611$	−36.3049	−1.46874
$612$	6.85452	0.277078
$613$	−15.6991	−0.634079	−0.317040	−	0.948412i	$-0.602689\pi$
$613$	−15.6991	−0.634079	−0.317040	+	0.948412i	$0.602689\pi$
$614$	4.58871	0.185185
$615$	5.93515	0.239328
$616$	1.55523	0.0626622
$617$	−38.4455	−1.54776	−0.773879	−	0.633333i	$-0.781686\pi$
$617$	−38.4455	−1.54776	−0.773879	+	0.633333i	$0.781686\pi$
$618$	−8.47577	−0.340945
$619$	36.8139	1.47968	0.739838	−	0.672785i	$-0.234902\pi$
$619$	36.8139	1.47968	0.739838	+	0.672785i	$0.234902\pi$
$620$	−9.22018	−0.370291
$621$	1.00000	0.0401286
$622$	−25.2976	−1.01434
$623$	−11.6650	−0.467347
$624$	14.4444	0.578239
$625$	1.00000	0.0400000
$626$	22.6043	0.903450
$627$	23.9608	0.956903
$628$	−2.48465	−0.0991482
$629$	28.4476	1.13428
$630$	−2.05680	−0.0819449
$631$	−39.3316	−1.56577	−0.782883	−	0.622169i	$-0.786251\pi$
$631$	−39.3316	−1.56577	−0.782883	+	0.622169i	$0.786251\pi$
$632$	3.13215	0.124590
$633$	25.6533	1.01963
$634$	35.8362	1.42324
$635$	8.35946	0.331735
$636$	−21.3154	−0.845211
$637$	4.14348	0.164171
$638$	32.7293	1.29577
$639$	13.4257	0.531113
$640$	3.76633	0.148877
$641$	25.9396	1.02455	0.512276	−	0.858821i	$-0.328802\pi$
$641$	25.9396	1.02455	0.512276	+	0.858821i	$0.328802\pi$
$642$	−34.4234	−1.35858
$643$	−23.3184	−0.919586	−0.459793	−	0.888026i	$-0.652076\pi$
$643$	−23.3184	−0.919586	−0.459793	+	0.888026i	$0.652076\pi$
$644$	−2.23042	−0.0878910
$645$	4.77427	0.187987
$646$	−46.1539	−1.81590
$647$	43.2282	1.69948	0.849738	−	0.527206i	$-0.176760\pi$
$647$	43.2282	1.69948	0.849738	+	0.527206i	$0.176760\pi$
$648$	0.473937	0.0186180
$649$	−5.42861	−0.213092
$650$	8.52231	0.334272
$651$	−4.13382	−0.162017
$652$	−7.01468	−0.274716
$653$	−10.2180	−0.399862	−0.199931	−	0.979810i	$-0.564072\pi$
$653$	−10.2180	−0.399862	−0.199931	+	0.979810i	$0.564072\pi$
$654$	−14.1706	−0.554115
$655$	7.55426	0.295170
$656$	−20.6903	−0.807819
$657$	−1.30705	−0.0509928
$658$	−18.0215	−0.702552
$659$	29.0316	1.13091	0.565455	−	0.824779i	$-0.308700\pi$
$659$	29.0316	1.13091	0.565455	+	0.824779i	$0.308700\pi$
$660$	7.31918	0.284899
$661$	−24.2973	−0.945055	−0.472528	−	0.881316i	$-0.656658\pi$
$661$	−24.2973	−0.945055	−0.472528	+	0.881316i	$0.656658\pi$
$662$	−49.6021	−1.92784
$663$	−12.7337	−0.494536
$664$	−4.28104	−0.166136
$665$	7.30174	0.283149
$666$	19.0392	0.737753
$667$	−4.84920	−0.187762
$668$	42.4015	1.64056
$669$	5.74216	0.222005
$670$	16.3956	0.633418
$671$	41.8005	1.61369
$672$	8.11799	0.313158
$673$	−7.05708	−0.272031	−0.136015	−	0.990707i	$-0.543430\pi$
$673$	−7.05708	−0.272031	−0.136015	+	0.990707i	$0.543430\pi$
$674$	22.7233	0.875269
$675$	−1.00000	−0.0384900
$676$	9.29735	0.357590
$677$	−37.5332	−1.44252	−0.721260	−	0.692665i	$-0.756437\pi$
$677$	−37.5332	−1.44252	−0.721260	+	0.692665i	$0.756437\pi$
$678$	−34.7644	−1.33512
$679$	−16.0074	−0.614308
$680$	−1.45650	−0.0558542
$681$	−27.0123	−1.03511
$682$	27.9009	1.06838
$683$	−14.3432	−0.548829	−0.274414	−	0.961612i	$-0.588484\pi$
$683$	−14.3432	−0.548829	−0.274414	+	0.961612i	$0.588484\pi$
$684$	−16.2860	−0.622710
$685$	4.09998	0.156652
$686$	2.05680	0.0785290
$687$	29.6456	1.13105
$688$	−16.6434	−0.634523
$689$	39.5978	1.50856
$690$	−2.05680	−0.0783010
$691$	−2.30218	−0.0875792	−0.0437896	−	0.999041i	$-0.513943\pi$
$691$	−2.30218	−0.0875792	−0.0437896	+	0.999041i	$0.513943\pi$
$692$	−19.3060	−0.733905
$693$	3.28152	0.124655
$694$	−30.3823	−1.15330
$695$	9.19530	0.348797
$696$	−2.29821	−0.0871136
$697$	18.2398	0.690884
$698$	17.3965	0.658466
$699$	13.7589	0.520408
$700$	2.23042	0.0843021
$701$	−23.6985	−0.895079	−0.447540	−	0.894264i	$-0.647700\pi$
$701$	−23.6985	−0.895079	−0.447540	+	0.894264i	$0.647700\pi$
$702$	−8.52231	−0.321654
$703$	−67.5900	−2.54921
$704$	−31.9127	−1.20275
$705$	−8.76193	−0.329993
$706$	29.9054	1.12550
$707$	13.7191	0.515960
$708$	3.68979	0.138671
$709$	−43.6383	−1.63887	−0.819435	−	0.573173i	$-0.805712\pi$
$709$	−43.6383	−1.63887	−0.819435	+	0.573173i	$0.805712\pi$
$710$	−27.6140	−1.03633
$711$	6.60879	0.247849
$712$	−5.52846	−0.207188
$713$	−4.13382	−0.154813
$714$	−6.32094	−0.236555
$715$	−13.5969	−0.508496
$716$	−52.5299	−1.96313
$717$	−15.4591	−0.577332
$718$	−13.9414	−0.520290
$719$	6.42348	0.239555	0.119778	−	0.992801i	$-0.461782\pi$
$719$	6.42348	0.239555	0.119778	+	0.992801i	$0.461782\pi$
$720$	3.48606	0.129918
$721$	4.12085	0.153469
$722$	70.5800	2.62672
$723$	12.8275	0.477059
$724$	−19.7129	−0.732624
$725$	4.84920	0.180095
$726$	0.476416	0.0176814
$727$	47.0618	1.74542	0.872712	−	0.488235i	$-0.162359\pi$
$727$	47.0618	1.74542	0.872712	+	0.488235i	$0.162359\pi$
$728$	1.96375	0.0727814
$729$	1.00000	0.0370370
$730$	2.68833	0.0994997
$731$	14.6722	0.542672
$732$	−28.4115	−1.05012
$733$	42.2680	1.56121	0.780603	−	0.625027i	$-0.214912\pi$
$733$	42.2680	1.56121	0.780603	+	0.625027i	$0.214912\pi$
$734$	−31.6865	−1.16957
$735$	1.00000	0.0368856
$736$	8.11799	0.299233
$737$	−26.1584	−0.963556
$738$	12.2074	0.449361
$739$	23.9479	0.880937	0.440469	−	0.897768i	$-0.354812\pi$
$739$	23.9479	0.880937	0.440469	+	0.897768i	$0.354812\pi$
$740$	−20.6464	−0.758976
$741$	30.2546	1.11143
$742$	19.6561	0.721600
$743$	48.1651	1.76701	0.883503	−	0.468426i	$-0.155179\pi$
$743$	48.1651	1.76701	0.883503	+	0.468426i	$0.155179\pi$
$744$	−1.95917	−0.0718267
$745$	16.4520	0.602754
$746$	51.8018	1.89660
$747$	−9.03292	−0.330497
$748$	22.4932	0.822434
$749$	16.7364	0.611535
$750$	2.05680	0.0751037
$751$	44.7153	1.63169	0.815843	−	0.578274i	$-0.196273\pi$
$751$	44.7153	1.63169	0.815843	+	0.578274i	$0.196273\pi$
$752$	30.5446	1.11385
$753$	9.54991	0.348018
$754$	41.3263	1.50502
$755$	−8.57473	−0.312067
$756$	−2.23042	−0.0811198
$757$	8.03463	0.292024	0.146012	−	0.989283i	$-0.453356\pi$
$757$	8.03463	0.292024	0.146012	+	0.989283i	$0.453356\pi$
$758$	−31.9749	−1.16138
$759$	3.28152	0.119112
$760$	3.46057	0.125528
$761$	−34.9071	−1.26538	−0.632691	−	0.774404i	$-0.718050\pi$
$761$	−34.9071	−1.26538	−0.632691	+	0.774404i	$0.718050\pi$
$762$	17.1937	0.622863
$763$	6.88964	0.249422
$764$	−23.0931	−0.835478
$765$	−3.07319	−0.111111
$766$	22.1066	0.798745
$767$	−6.85455	−0.247503
$768$	−11.7033	−0.422308
$769$	54.0448	1.94891	0.974453	−	0.224590i	$-0.0721041\pi$
$769$	54.0448	1.94891	0.974453	+	0.224590i	$0.0721041\pi$
$770$	−6.74943	−0.243233
$771$	2.17492	0.0783278
$772$	−34.6694	−1.24778
$773$	42.6344	1.53345	0.766727	−	0.641973i	$-0.221884\pi$
$773$	42.6344	1.53345	0.766727	+	0.641973i	$0.221884\pi$
$774$	9.81972	0.352962
$775$	4.13382	0.148491
$776$	−7.58651	−0.272340
$777$	−9.25670	−0.332082
$778$	−35.5763	−1.27547
$779$	−43.3369	−1.55271
$780$	9.24172	0.330907
$781$	44.0567	1.57647
$782$	−6.32094	−0.226036
$783$	−4.84920	−0.173296
$784$	−3.48606	−0.124502
$785$	1.11398	0.0397596
$786$	15.5376	0.554208
$787$	−19.4354	−0.692799	−0.346399	−	0.938087i	$-0.612596\pi$
$787$	−19.4354	−0.692799	−0.346399	+	0.938087i	$0.612596\pi$
$788$	−24.7698	−0.882389
$789$	−12.3196	−0.438590
$790$	−13.5930	−0.483616
$791$	16.9022	0.600973
$792$	1.55523	0.0552628
$793$	52.7803	1.87428
$794$	−33.2437	−1.17978
$795$	9.55667	0.338940
$796$	−19.5680	−0.693570
$797$	6.54344	0.231780	0.115890	−	0.993262i	$-0.463028\pi$
$797$	6.54344	0.231780	0.115890	+	0.993262i	$0.463028\pi$
$798$	15.0182	0.531639
$799$	−26.9271	−0.952611
$800$	−8.11799	−0.287014
$801$	−11.6650	−0.412161
$802$	23.9455	0.845547
$803$	−4.28910	−0.151359
$804$	17.7797	0.627040
$805$	1.00000	0.0352454
$806$	35.2297	1.24091
$807$	−14.6898	−0.517106
$808$	6.50199	0.228739
$809$	2.45868	0.0864424	0.0432212	−	0.999066i	$-0.486238\pi$
$809$	2.45868	0.0864424	0.0432212	+	0.999066i	$0.486238\pi$
$810$	−2.05680	−0.0722686
$811$	−13.9114	−0.488496	−0.244248	−	0.969713i	$-0.578541\pi$
$811$	−13.9114	−0.488496	−0.244248	+	0.969713i	$0.578541\pi$
$812$	10.8158	0.379559
$813$	8.65889	0.303680
$814$	62.4774	2.18983
$815$	3.14500	0.110165
$816$	10.7133	0.375041
$817$	−34.8605	−1.21961
$818$	48.3170	1.68936
$819$	4.14348	0.144785
$820$	−13.2379	−0.462288
$821$	8.44715	0.294807	0.147404	−	0.989076i	$-0.452908\pi$
$821$	8.44715	0.294807	0.147404	+	0.989076i	$0.452908\pi$
$822$	8.43284	0.294129
$823$	−52.9454	−1.84556	−0.922780	−	0.385326i	$-0.874089\pi$
$823$	−52.9454	−1.84556	−0.922780	+	0.385326i	$0.874089\pi$
$824$	1.95303	0.0680369
$825$	−3.28152	−0.114248
$826$	−3.40256	−0.118390
$827$	44.8264	1.55877	0.779383	−	0.626548i	$-0.215533\pi$
$827$	44.8264	1.55877	0.779383	+	0.626548i	$0.215533\pi$
$828$	−2.23042	−0.0775126
$829$	27.8579	0.967545	0.483773	−	0.875194i	$-0.339266\pi$
$829$	27.8579	0.967545	0.483773	+	0.875194i	$0.339266\pi$
$830$	18.5789	0.644883
$831$	−18.5542	−0.643638
$832$	−40.2952	−1.39699
$833$	3.07319	0.106480
$834$	18.9129	0.654900
$835$	−19.0105	−0.657885
$836$	−53.4428	−1.84836
$837$	−4.13382	−0.142886
$838$	37.4009	1.29199
$839$	−31.2719	−1.07963	−0.539813	−	0.841785i	$-0.681505\pi$
$839$	−31.2719	−1.07963	−0.539813	+	0.841785i	$0.681505\pi$
$840$	0.473937	0.0163524
$841$	−5.48530	−0.189148
$842$	−50.0026	−1.72320
$843$	−17.0301	−0.586548
$844$	−57.2178	−1.96952
$845$	−4.16842	−0.143398
$846$	−18.0215	−0.619593
$847$	−0.231630	−0.00795889
$848$	−33.3151	−1.14404
$849$	−11.5793	−0.397402
$850$	6.32094	0.216806
$851$	−9.25670	−0.317316
$852$	−29.9450	−1.02590
$853$	−19.6653	−0.673329	−0.336664	−	0.941625i	$-0.609299\pi$
$853$	−19.6653	−0.673329	−0.336664	+	0.941625i	$0.609299\pi$
$854$	26.1998	0.896540
$855$	7.30174	0.249714
$856$	7.93200	0.271110
$857$	41.6049	1.42120	0.710598	−	0.703599i	$-0.248425\pi$
$857$	41.6049	1.42120	0.710598	+	0.703599i	$0.248425\pi$
$858$	−27.9661	−0.954747
$859$	−1.06574	−0.0363627	−0.0181814	−	0.999835i	$-0.505788\pi$
$859$	−1.06574	−0.0363627	−0.0181814	+	0.999835i	$0.505788\pi$
$860$	−10.6486	−0.363116
$861$	−5.93515	−0.202269
$862$	5.19528	0.176952
$863$	9.99294	0.340164	0.170082	−	0.985430i	$-0.445597\pi$
$863$	9.99294	0.340164	0.170082	+	0.985430i	$0.445597\pi$
$864$	8.11799	0.276180
$865$	8.65577	0.294305
$866$	50.5494	1.71774
$867$	7.55550	0.256598
$868$	9.22018	0.312953
$869$	21.6869	0.735677
$870$	9.97382	0.338144
$871$	−33.0294	−1.11916
$872$	3.26526	0.110576
$873$	−16.0074	−0.541769
$874$	15.0182	0.507999
$875$	−1.00000	−0.0338062
$876$	2.91527	0.0984979
$877$	19.6497	0.663523	0.331761	−	0.943363i	$-0.392357\pi$
$877$	19.6497	0.663523	0.331761	+	0.943363i	$0.392357\pi$
$878$	31.5634	1.06521
$879$	23.5087	0.792928
$880$	11.4396	0.385627
$881$	41.7493	1.40657	0.703285	−	0.710908i	$-0.251716\pi$
$881$	41.7493	1.40657	0.703285	+	0.710908i	$0.251716\pi$
$882$	2.05680	0.0692560
$883$	−33.7044	−1.13425	−0.567123	−	0.823633i	$-0.691943\pi$
$883$	−33.7044	−1.13425	−0.567123	+	0.823633i	$0.691943\pi$
$884$	28.4016	0.955248
$885$	−1.65430	−0.0556086
$886$	48.9971	1.64609
$887$	31.8674	1.07000	0.535001	−	0.844852i	$-0.320311\pi$
$887$	31.8674	1.07000	0.535001	+	0.844852i	$0.320311\pi$
$888$	−4.38709	−0.147221
$889$	−8.35946	−0.280367
$890$	23.9925	0.804230
$891$	3.28152	0.109935
$892$	−12.8075	−0.428825
$893$	63.9773	2.14092
$894$	33.8385	1.13173
$895$	23.5515	0.787240
$896$	−3.76633	−0.125824
$897$	4.14348	0.138347
$898$	−82.8933	−2.76618
$899$	20.0457	0.668562
$900$	2.23042	0.0743475
$901$	29.3695	0.978438
$902$	40.0589	1.33381
$903$	−4.77427	−0.158878
$904$	8.01058	0.266428
$905$	8.83819	0.293791
$906$	−17.6365	−0.585934
$907$	28.2260	0.937229	0.468615	−	0.883403i	$-0.344753\pi$
$907$	28.2260	0.937229	0.468615	+	0.883403i	$0.344753\pi$
$908$	60.2488	1.99943
$909$	13.7191	0.455034
$910$	−8.52231	−0.282512
$911$	−32.9759	−1.09254	−0.546271	−	0.837609i	$-0.683953\pi$
$911$	−32.9759	−1.09254	−0.546271	+	0.837609i	$0.683953\pi$
$912$	−25.4543	−0.842876
$913$	−29.6417	−0.980997
$914$	59.4242	1.96558
$915$	12.7382	0.421110
$916$	−66.1224	−2.18474
$917$	−7.55426	−0.249464
$918$	−6.32094	−0.208622
$919$	−28.6030	−0.943527	−0.471763	−	0.881725i	$-0.656382\pi$
$919$	−28.6030	−0.943527	−0.471763	+	0.881725i	$0.656382\pi$
$920$	0.473937	0.0156252
$921$	−2.23100	−0.0735138
$922$	4.79446	0.157897
$923$	55.6291	1.83106
$924$	−7.31918	−0.240783
$925$	9.25670	0.304358
$926$	−72.2803	−2.37528
$927$	4.12085	0.135347
$928$	−39.3657	−1.29224
$929$	−3.31729	−0.108837	−0.0544184	−	0.998518i	$-0.517330\pi$
$929$	−3.31729	−0.108837	−0.0544184	+	0.998518i	$0.517330\pi$
$930$	8.50245	0.278806
$931$	−7.30174	−0.239305
$932$	−30.6881	−1.00522
$933$	12.2995	0.402668
$934$	49.4508	1.61808
$935$	−10.0847	−0.329806
$936$	1.96375	0.0641871
$937$	52.0307	1.69977	0.849885	−	0.526968i	$-0.176671\pi$
$937$	52.0307	1.69977	0.849885	+	0.526968i	$0.176671\pi$
$938$	−16.3956	−0.535336
$939$	−10.9900	−0.358647
$940$	19.5428	0.637416
$941$	−10.6552	−0.347349	−0.173675	−	0.984803i	$-0.555564\pi$
$941$	−10.6552	−0.347349	−0.173675	+	0.984803i	$0.555564\pi$
$942$	2.29123	0.0746524
$943$	−5.93515	−0.193275
$944$	5.76697	0.187699
$945$	1.00000	0.0325300
$946$	32.2236	1.04768
$947$	−24.8645	−0.807987	−0.403993	−	0.914762i	$-0.632378\pi$
$947$	−24.8645	−0.807987	−0.403993	+	0.914762i	$0.632378\pi$
$948$	−14.7404	−0.478746
$949$	−5.41572	−0.175802
$950$	−15.0182	−0.487256
$951$	−17.4233	−0.564989
$952$	1.45650	0.0472054
$953$	−49.8418	−1.61453	−0.807267	−	0.590186i	$-0.799054\pi$
$953$	−49.8418	−1.61453	−0.807267	+	0.590186i	$0.799054\pi$
$954$	19.6561	0.636391
$955$	10.3537	0.335037
$956$	34.4804	1.11518
$957$	−15.9127	−0.514386
$958$	−47.2080	−1.52522
$959$	−4.09998	−0.132395
$960$	−9.72497	−0.313872
$961$	−13.9115	−0.448758
$962$	78.8884	2.54346
$963$	16.7364	0.539323
$964$	−28.6107	−0.921490
$965$	15.5438	0.500374
$966$	2.05680	0.0661764
$967$	6.02535	0.193762	0.0968810	−	0.995296i	$-0.469113\pi$
$967$	6.02535	0.193762	0.0968810	+	0.995296i	$0.469113\pi$
$968$	−0.109778	−0.00352839
$969$	22.4397	0.720865
$970$	32.9240	1.05713
$971$	32.0748	1.02933	0.514665	−	0.857392i	$-0.327916\pi$
$971$	32.0748	1.02933	0.514665	+	0.857392i	$0.327916\pi$
$972$	−2.23042	−0.0715409
$973$	−9.19530	−0.294788
$974$	2.43234	0.0779372
$975$	−4.14348	−0.132698
$976$	−44.4059	−1.42140
$977$	37.0518	1.18539	0.592696	−	0.805426i	$-0.298064\pi$
$977$	37.0518	1.18539	0.592696	+	0.805426i	$0.298064\pi$
$978$	6.46863	0.206844
$979$	−38.2788	−1.22340
$980$	−2.23042	−0.0712483
$981$	6.88964	0.219969
$982$	16.8387	0.537344
$983$	44.6877	1.42532	0.712658	−	0.701512i	$-0.247491\pi$
$983$	44.6877	1.42532	0.712658	+	0.701512i	$0.247491\pi$
$984$	−2.81289	−0.0896716
$985$	11.1054	0.353849
$986$	30.6515	0.976141
$987$	8.76193	0.278895
$988$	−67.4807	−2.14685
$989$	−4.77427	−0.151813
$990$	−6.74943	−0.214511
$991$	32.8090	1.04221	0.521106	−	0.853492i	$-0.325520\pi$
$991$	32.8090	1.04221	0.521106	+	0.853492i	$0.325520\pi$
$992$	−33.5583	−1.06548
$993$	24.1162	0.765303
$994$	27.6140	0.875862
$995$	8.77322	0.278130
$996$	20.1472	0.638390
$997$	−56.7509	−1.79732	−0.898660	−	0.438646i	$-0.855458\pi$
$997$	−56.7509	−1.79732	−0.898660	+	0.438646i	$0.855458\pi$
$998$	−69.1003	−2.18733
$999$	−9.25670	−0.292869

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2415.2.a.q.1.5	✓	6
3.2	odd	2		7245.2.a.bj.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2415.2.a.q.1.5	✓	6	1.1	even	1	trivial
7245.2.a.bj.1.2		6	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2415.a (trivial)

\(f(q)\)	\(=\)	\(q+2.05680 q^{2} -1.00000 q^{3} +2.23042 q^{4} -1.00000 q^{5} -2.05680 q^{6} +1.00000 q^{7} +0.473937 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.05680 q^{2} -1.00000 q^{3} +2.23042 q^{4} -1.00000 q^{5} -2.05680 q^{6} +1.00000 q^{7} +0.473937 q^{8} +1.00000 q^{9} -2.05680 q^{10} +3.28152 q^{11} -2.23042 q^{12} +4.14348 q^{13} +2.05680 q^{14} +1.00000 q^{15} -3.48606 q^{16} +3.07319 q^{17} +2.05680 q^{18} -7.30174 q^{19} -2.23042 q^{20} -1.00000 q^{21} +6.74943 q^{22} -1.00000 q^{23} -0.473937 q^{24} +1.00000 q^{25} +8.52231 q^{26} -1.00000 q^{27} +2.23042 q^{28} +4.84920 q^{29} +2.05680 q^{30} +4.13382 q^{31} -8.11799 q^{32} -3.28152 q^{33} +6.32094 q^{34} -1.00000 q^{35} +2.23042 q^{36} +9.25670 q^{37} -15.0182 q^{38} -4.14348 q^{39} -0.473937 q^{40} +5.93515 q^{41} -2.05680 q^{42} +4.77427 q^{43} +7.31918 q^{44} -1.00000 q^{45} -2.05680 q^{46} -8.76193 q^{47} +3.48606 q^{48} +1.00000 q^{49} +2.05680 q^{50} -3.07319 q^{51} +9.24172 q^{52} +9.55667 q^{53} -2.05680 q^{54} -3.28152 q^{55} +0.473937 q^{56} +7.30174 q^{57} +9.97382 q^{58} -1.65430 q^{59} +2.23042 q^{60} +12.7382 q^{61} +8.50245 q^{62} +1.00000 q^{63} -9.72497 q^{64} -4.14348 q^{65} -6.74943 q^{66} -7.97142 q^{67} +6.85452 q^{68} +1.00000 q^{69} -2.05680 q^{70} +13.4257 q^{71} +0.473937 q^{72} -1.30705 q^{73} +19.0392 q^{74} -1.00000 q^{75} -16.2860 q^{76} +3.28152 q^{77} -8.52231 q^{78} +6.60879 q^{79} +3.48606 q^{80} +1.00000 q^{81} +12.2074 q^{82} -9.03292 q^{83} -2.23042 q^{84} -3.07319 q^{85} +9.81972 q^{86} -4.84920 q^{87} +1.55523 q^{88} -11.6650 q^{89} -2.05680 q^{90} +4.14348 q^{91} -2.23042 q^{92} -4.13382 q^{93} -18.0215 q^{94} +7.30174 q^{95} +8.11799 q^{96} -16.0074 q^{97} +2.05680 q^{98} +3.28152 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} - 6 q^{3} + 5 q^{4} - 6 q^{5} - 3 q^{6} + 6 q^{7} + 9 q^{8} + 6 q^{9}+O(q^{10}) \) 6 * q + 3 * q^2 - 6 * q^3 + 5 * q^4 - 6 * q^5 - 3 * q^6 + 6 * q^7 + 9 * q^8 + 6 * q^9	\( 6 q + 3 q^{2} - 6 q^{3} + 5 q^{4} - 6 q^{5} - 3 q^{6} + 6 q^{7} + 9 q^{8} + 6 q^{9} - 3 q^{10} + 6 q^{11} - 5 q^{12} - 6 q^{13} + 3 q^{14} + 6 q^{15} + 11 q^{16} + 8 q^{17} + 3 q^{18} - 8 q^{19} - 5 q^{20} - 6 q^{21} + 18 q^{22} - 6 q^{23} - 9 q^{24} + 6 q^{25} + 13 q^{26} - 6 q^{27} + 5 q^{28} + 8 q^{29} + 3 q^{30} - 16 q^{31} + 26 q^{32} - 6 q^{33} - 13 q^{34} - 6 q^{35} + 5 q^{36} + 8 q^{37} - 14 q^{38} + 6 q^{39} - 9 q^{40} + 8 q^{41} - 3 q^{42} + 8 q^{43} + 15 q^{44} - 6 q^{45} - 3 q^{46} + 10 q^{47} - 11 q^{48} + 6 q^{49} + 3 q^{50} - 8 q^{51} + 28 q^{53} - 3 q^{54} - 6 q^{55} + 9 q^{56} + 8 q^{57} + 14 q^{58} - 6 q^{59} + 5 q^{60} - 12 q^{61} + 18 q^{62} + 6 q^{63} + 15 q^{64} + 6 q^{65} - 18 q^{66} + 18 q^{67} + 18 q^{68} + 6 q^{69} - 3 q^{70} + 10 q^{71} + 9 q^{72} - 2 q^{73} + q^{74} - 6 q^{75} + 25 q^{76} + 6 q^{77} - 13 q^{78} - 2 q^{79} - 11 q^{80} + 6 q^{81} - 12 q^{82} + 26 q^{83} - 5 q^{84} - 8 q^{85} + 16 q^{86} - 8 q^{87} + 21 q^{88} + 8 q^{89} - 3 q^{90} - 6 q^{91} - 5 q^{92} + 16 q^{93} - 8 q^{94} + 8 q^{95} - 26 q^{96} + 20 q^{97} + 3 q^{98} + 6 q^{99}+O(q^{100}) \) 6 * q + 3 * q^2 - 6 * q^3 + 5 * q^4 - 6 * q^5 - 3 * q^6 + 6 * q^7 + 9 * q^8 + 6 * q^9 - 3 * q^10 + 6 * q^11 - 5 * q^12 - 6 * q^13 + 3 * q^14 + 6 * q^15 + 11 * q^16 + 8 * q^17 + 3 * q^18 - 8 * q^19 - 5 * q^20 - 6 * q^21 + 18 * q^22 - 6 * q^23 - 9 * q^24 + 6 * q^25 + 13 * q^26 - 6 * q^27 + 5 * q^28 + 8 * q^29 + 3 * q^30 - 16 * q^31 + 26 * q^32 - 6 * q^33 - 13 * q^34 - 6 * q^35 + 5 * q^36 + 8 * q^37 - 14 * q^38 + 6 * q^39 - 9 * q^40 + 8 * q^41 - 3 * q^42 + 8 * q^43 + 15 * q^44 - 6 * q^45 - 3 * q^46 + 10 * q^47 - 11 * q^48 + 6 * q^49 + 3 * q^50 - 8 * q^51 + 28 * q^53 - 3 * q^54 - 6 * q^55 + 9 * q^56 + 8 * q^57 + 14 * q^58 - 6 * q^59 + 5 * q^60 - 12 * q^61 + 18 * q^62 + 6 * q^63 + 15 * q^64 + 6 * q^65 - 18 * q^66 + 18 * q^67 + 18 * q^68 + 6 * q^69 - 3 * q^70 + 10 * q^71 + 9 * q^72 - 2 * q^73 + q^74 - 6 * q^75 + 25 * q^76 + 6 * q^77 - 13 * q^78 - 2 * q^79 - 11 * q^80 + 6 * q^81 - 12 * q^82 + 26 * q^83 - 5 * q^84 - 8 * q^85 + 16 * q^86 - 8 * q^87 + 21 * q^88 + 8 * q^89 - 3 * q^90 - 6 * q^91 - 5 * q^92 + 16 * q^93 - 8 * q^94 + 8 * q^95 - 26 * q^96 + 20 * q^97 + 3 * q^98 + 6 * q^99

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