LMFDB - Embedded newform 3174.2.a.bc.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3174,2,Mod(1,3174)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3174, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3174.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3174 = 2 \cdot 3 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3174.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$25.3445176016$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.30972$ of defining polynomial
Character	$\chi$	$=$	3174.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.23648 q^{5} +1.00000 q^{6} -1.47889 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.23648 q^{5} +1.00000 q^{6} -1.47889 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.23648 q^{10} -0.886752 q^{11} +1.00000 q^{12} +0.0673089 q^{13} -1.47889 q^{14} -1.23648 q^{15} +1.00000 q^{16} -4.29177 q^{17} +1.00000 q^{18} -4.66759 q^{19} -1.23648 q^{20} -1.47889 q^{21} -0.886752 q^{22} +1.00000 q^{24} -3.47112 q^{25} +0.0673089 q^{26} +1.00000 q^{27} -1.47889 q^{28} -7.67816 q^{29} -1.23648 q^{30} -1.59145 q^{31} +1.00000 q^{32} -0.886752 q^{33} -4.29177 q^{34} +1.82862 q^{35} +1.00000 q^{36} -5.22871 q^{37} -4.66759 q^{38} +0.0673089 q^{39} -1.23648 q^{40} -0.135933 q^{41} -1.47889 q^{42} +6.57798 q^{43} -0.886752 q^{44} -1.23648 q^{45} +8.31271 q^{47} +1.00000 q^{48} -4.81288 q^{49} -3.47112 q^{50} -4.29177 q^{51} +0.0673089 q^{52} -2.99558 q^{53} +1.00000 q^{54} +1.09645 q^{55} -1.47889 q^{56} -4.66759 q^{57} -7.67816 q^{58} -6.83576 q^{59} -1.23648 q^{60} -4.60015 q^{61} -1.59145 q^{62} -1.47889 q^{63} +1.00000 q^{64} -0.0832260 q^{65} -0.886752 q^{66} +12.4051 q^{67} -4.29177 q^{68} +1.82862 q^{70} -7.31821 q^{71} +1.00000 q^{72} +13.1978 q^{73} -5.22871 q^{74} -3.47112 q^{75} -4.66759 q^{76} +1.31141 q^{77} +0.0673089 q^{78} +7.89446 q^{79} -1.23648 q^{80} +1.00000 q^{81} -0.135933 q^{82} -8.40858 q^{83} -1.47889 q^{84} +5.30668 q^{85} +6.57798 q^{86} -7.67816 q^{87} -0.886752 q^{88} +11.7235 q^{89} -1.23648 q^{90} -0.0995426 q^{91} -1.59145 q^{93} +8.31271 q^{94} +5.77138 q^{95} +1.00000 q^{96} -17.1665 q^{97} -4.81288 q^{98} -0.886752 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{3} + 5 q^{4} - 7 q^{5} + 5 q^{6} - 7 q^{7} + 5 q^{8} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 - 7 * q^5 + 5 * q^6 - 7 * q^7 + 5 * q^8 + 5 * q^9	\( 5 q + 5 q^{2} + 5 q^{3} + 5 q^{4} - 7 q^{5} + 5 q^{6} - 7 q^{7} + 5 q^{8} + 5 q^{9} - 7 q^{10} - 13 q^{11} + 5 q^{12} - 4 q^{13} - 7 q^{14} - 7 q^{15} + 5 q^{16} - 9 q^{17} + 5 q^{18} - 11 q^{19} - 7 q^{20} - 7 q^{21} - 13 q^{22} + 5 q^{24} - 2 q^{25} - 4 q^{26} + 5 q^{27} - 7 q^{28} - 7 q^{29} - 7 q^{30} - 8 q^{31} + 5 q^{32} - 13 q^{33} - 9 q^{34} + q^{35} + 5 q^{36} - 12 q^{37} - 11 q^{38} - 4 q^{39} - 7 q^{40} - 10 q^{41} - 7 q^{42} - 4 q^{43} - 13 q^{44} - 7 q^{45} - 24 q^{47} + 5 q^{48} - 12 q^{49} - 2 q^{50} - 9 q^{51} - 4 q^{52} - 9 q^{53} + 5 q^{54} + 16 q^{55} - 7 q^{56} - 11 q^{57} - 7 q^{58} - 14 q^{59} - 7 q^{60} - 5 q^{61} - 8 q^{62} - 7 q^{63} + 5 q^{64} - 12 q^{65} - 13 q^{66} - 13 q^{67} - 9 q^{68} + q^{70} - 19 q^{71} + 5 q^{72} + 4 q^{73} - 12 q^{74} - 2 q^{75} - 11 q^{76} + 5 q^{77} - 4 q^{78} - 4 q^{79} - 7 q^{80} + 5 q^{81} - 10 q^{82} - 24 q^{83} - 7 q^{84} + 17 q^{85} - 4 q^{86} - 7 q^{87} - 13 q^{88} - 4 q^{89} - 7 q^{90} + 21 q^{91} - 8 q^{93} - 24 q^{94} - 11 q^{95} + 5 q^{96} + 9 q^{97} - 12 q^{98} - 13 q^{99}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 - 7 * q^5 + 5 * q^6 - 7 * q^7 + 5 * q^8 + 5 * q^9 - 7 * q^10 - 13 * q^11 + 5 * q^12 - 4 * q^13 - 7 * q^14 - 7 * q^15 + 5 * q^16 - 9 * q^17 + 5 * q^18 - 11 * q^19 - 7 * q^20 - 7 * q^21 - 13 * q^22 + 5 * q^24 - 2 * q^25 - 4 * q^26 + 5 * q^27 - 7 * q^28 - 7 * q^29 - 7 * q^30 - 8 * q^31 + 5 * q^32 - 13 * q^33 - 9 * q^34 + q^35 + 5 * q^36 - 12 * q^37 - 11 * q^38 - 4 * q^39 - 7 * q^40 - 10 * q^41 - 7 * q^42 - 4 * q^43 - 13 * q^44 - 7 * q^45 - 24 * q^47 + 5 * q^48 - 12 * q^49 - 2 * q^50 - 9 * q^51 - 4 * q^52 - 9 * q^53 + 5 * q^54 + 16 * q^55 - 7 * q^56 - 11 * q^57 - 7 * q^58 - 14 * q^59 - 7 * q^60 - 5 * q^61 - 8 * q^62 - 7 * q^63 + 5 * q^64 - 12 * q^65 - 13 * q^66 - 13 * q^67 - 9 * q^68 + q^70 - 19 * q^71 + 5 * q^72 + 4 * q^73 - 12 * q^74 - 2 * q^75 - 11 * q^76 + 5 * q^77 - 4 * q^78 - 4 * q^79 - 7 * q^80 + 5 * q^81 - 10 * q^82 - 24 * q^83 - 7 * q^84 + 17 * q^85 - 4 * q^86 - 7 * q^87 - 13 * q^88 - 4 * q^89 - 7 * q^90 + 21 * q^91 - 8 * q^93 - 24 * q^94 - 11 * q^95 + 5 * q^96 + 9 * q^97 - 12 * q^98 - 13 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−1.23648	−0.552970	−0.276485	−	0.961018i	$-0.589170\pi$
$5$	−1.23648	−0.552970	−0.276485	+	0.961018i	$0.589170\pi$
$6$	1.00000	0.408248
$7$	−1.47889	−0.558968	−0.279484	−	0.960150i	$-0.590163\pi$
$7$	−1.47889	−0.558968	−0.279484	+	0.960150i	$0.590163\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.23648	−0.391009
$11$	−0.886752	−0.267366	−0.133683	−	0.991024i	$-0.542680\pi$
$11$	−0.886752	−0.267366	−0.133683	+	0.991024i	$0.542680\pi$
$12$	1.00000	0.288675
$13$	0.0673089	0.0186681	0.00933407	−	0.999956i	$-0.497029\pi$
$13$	0.0673089	0.0186681	0.00933407	+	0.999956i	$0.497029\pi$
$14$	−1.47889	−0.395250
$15$	−1.23648	−0.319257
$16$	1.00000	0.250000
$17$	−4.29177	−1.04091	−0.520454	−	0.853890i	$-0.674237\pi$
$17$	−4.29177	−1.04091	−0.520454	+	0.853890i	$0.674237\pi$
$18$	1.00000	0.235702
$19$	−4.66759	−1.07082	−0.535410	−	0.844592i	$-0.679843\pi$
$19$	−4.66759	−1.07082	−0.535410	+	0.844592i	$0.679843\pi$
$20$	−1.23648	−0.276485
$21$	−1.47889	−0.322721
$22$	−0.886752	−0.189056
$23$	0	0
$23$	0	0
$24$	1.00000	0.204124
$25$	−3.47112	−0.694224
$26$	0.0673089	0.0132004
$27$	1.00000	0.192450
$28$	−1.47889	−0.279484
$29$	−7.67816	−1.42580	−0.712899	−	0.701267i	$-0.752618\pi$
$29$	−7.67816	−1.42580	−0.712899	+	0.701267i	$0.752618\pi$
$30$	−1.23648	−0.225749
$31$	−1.59145	−0.285834	−0.142917	−	0.989735i	$-0.545648\pi$
$31$	−1.59145	−0.285834	−0.142917	+	0.989735i	$0.545648\pi$
$32$	1.00000	0.176777
$33$	−0.886752	−0.154364
$34$	−4.29177	−0.736033
$35$	1.82862	0.309093
$36$	1.00000	0.166667
$37$	−5.22871	−0.859594	−0.429797	−	0.902925i	$-0.641415\pi$
$37$	−5.22871	−0.859594	−0.429797	+	0.902925i	$0.641415\pi$
$38$	−4.66759	−0.757184
$39$	0.0673089	0.0107781
$40$	−1.23648	−0.195504
$41$	−0.135933	−0.0212291	−0.0106145	−	0.999944i	$-0.503379\pi$
$41$	−0.135933	−0.0212291	−0.0106145	+	0.999944i	$0.503379\pi$
$42$	−1.47889	−0.228198
$43$	6.57798	1.00313	0.501567	−	0.865119i	$-0.332757\pi$
$43$	6.57798	1.00313	0.501567	+	0.865119i	$0.332757\pi$
$44$	−0.886752	−0.133683
$45$	−1.23648	−0.184323
$46$	0	0
$47$	8.31271	1.21253	0.606266	−	0.795262i	$-0.292666\pi$
$47$	8.31271	1.21253	0.606266	+	0.795262i	$0.292666\pi$
$48$	1.00000	0.144338
$49$	−4.81288	−0.687554
$50$	−3.47112	−0.490890
$51$	−4.29177	−0.600968
$52$	0.0673089	0.00933407
$53$	−2.99558	−0.411474	−0.205737	−	0.978607i	$-0.565959\pi$
$53$	−2.99558	−0.411474	−0.205737	+	0.978607i	$0.565959\pi$
$54$	1.00000	0.136083
$55$	1.09645	0.147845
$56$	−1.47889	−0.197625
$57$	−4.66759	−0.618238
$58$	−7.67816	−1.00819
$59$	−6.83576	−0.889940	−0.444970	−	0.895545i	$-0.646786\pi$
$59$	−6.83576	−0.889940	−0.444970	+	0.895545i	$0.646786\pi$
$60$	−1.23648	−0.159629
$61$	−4.60015	−0.588988	−0.294494	−	0.955653i	$-0.595151\pi$
$61$	−4.60015	−0.588988	−0.294494	+	0.955653i	$0.595151\pi$
$62$	−1.59145	−0.202115
$63$	−1.47889	−0.186323
$64$	1.00000	0.125000
$65$	−0.0832260	−0.0103229
$66$	−0.886752	−0.109152
$67$	12.4051	1.51553	0.757763	−	0.652530i	$-0.226293\pi$
$67$	12.4051	1.51553	0.757763	+	0.652530i	$0.226293\pi$
$68$	−4.29177	−0.520454
$69$	0	0
$70$	1.82862	0.218562
$71$	−7.31821	−0.868512	−0.434256	−	0.900790i	$-0.642989\pi$
$71$	−7.31821	−0.868512	−0.434256	+	0.900790i	$0.642989\pi$
$72$	1.00000	0.117851
$73$	13.1978	1.54469	0.772343	−	0.635206i	$-0.219085\pi$
$73$	13.1978	1.54469	0.772343	+	0.635206i	$0.219085\pi$
$74$	−5.22871	−0.607825
$75$	−3.47112	−0.400810
$76$	−4.66759	−0.535410
$77$	1.31141	0.149449
$78$	0.0673089	0.00762123
$79$	7.89446	0.888197	0.444098	−	0.895978i	$-0.353524\pi$
$79$	7.89446	0.888197	0.444098	+	0.895978i	$0.353524\pi$
$80$	−1.23648	−0.138243
$81$	1.00000	0.111111
$82$	−0.135933	−0.0150112
$83$	−8.40858	−0.922961	−0.461481	−	0.887150i	$-0.652682\pi$
$83$	−8.40858	−0.922961	−0.461481	+	0.887150i	$0.652682\pi$
$84$	−1.47889	−0.161360
$85$	5.30668	0.575591
$86$	6.57798	0.709322
$87$	−7.67816	−0.823185
$88$	−0.886752	−0.0945281
$89$	11.7235	1.24268	0.621342	−	0.783539i	$-0.286588\pi$
$89$	11.7235	1.24268	0.621342	+	0.783539i	$0.286588\pi$
$90$	−1.23648	−0.130336
$91$	−0.0995426	−0.0104349
$92$	0	0
$93$	−1.59145	−0.165026
$94$	8.31271	0.857390
$95$	5.77138	0.592131
$96$	1.00000	0.102062
$97$	−17.1665	−1.74299	−0.871496	−	0.490402i	$-0.836850\pi$
$97$	−17.1665	−1.74299	−0.871496	+	0.490402i	$0.836850\pi$
$98$	−4.81288	−0.486174
$99$	−0.886752	−0.0891220
$100$	−3.47112	−0.347112
$101$	5.98712	0.595741	0.297870	−	0.954606i	$-0.403724\pi$
$101$	5.98712	0.595741	0.297870	+	0.954606i	$0.403724\pi$
$102$	−4.29177	−0.424949
$103$	−15.6415	−1.54120	−0.770602	−	0.637317i	$-0.780044\pi$
$103$	−15.6415	−1.54120	−0.770602	+	0.637317i	$0.780044\pi$
$104$	0.0673089	0.00660018
$105$	1.82862	0.178455
$106$	−2.99558	−0.290956
$107$	−13.8185	−1.33589	−0.667944	−	0.744212i	$-0.732825\pi$
$107$	−13.8185	−1.33589	−0.667944	+	0.744212i	$0.732825\pi$
$108$	1.00000	0.0962250
$109$	10.7865	1.03316	0.516581	−	0.856238i	$-0.327204\pi$
$109$	10.7865	1.03316	0.516581	+	0.856238i	$0.327204\pi$
$110$	1.09645	0.104542
$111$	−5.22871	−0.496287
$112$	−1.47889	−0.139742
$113$	−4.57032	−0.429939	−0.214970	−	0.976621i	$-0.568965\pi$
$113$	−4.57032	−0.429939	−0.214970	+	0.976621i	$0.568965\pi$
$114$	−4.66759	−0.437160
$115$	0	0
$116$	−7.67816	−0.712899
$117$	0.0673089	0.00622271
$118$	−6.83576	−0.629283
$119$	6.34706	0.581834
$120$	−1.23648	−0.112875
$121$	−10.2137	−0.928515
$122$	−4.60015	−0.416478
$123$	−0.135933	−0.0122566
$124$	−1.59145	−0.142917
$125$	10.4744	0.936855
$126$	−1.47889	−0.131750
$127$	−14.4076	−1.27846	−0.639232	−	0.769014i	$-0.720748\pi$
$127$	−14.4076	−1.27846	−0.639232	+	0.769014i	$0.720748\pi$
$128$	1.00000	0.0883883
$129$	6.57798	0.579159
$130$	−0.0832260	−0.00729941
$131$	−4.57998	−0.400154	−0.200077	−	0.979780i	$-0.564119\pi$
$131$	−4.57998	−0.400154	−0.200077	+	0.979780i	$0.564119\pi$
$132$	−0.886752	−0.0771819
$133$	6.90286	0.598554
$134$	12.4051	1.07164
$135$	−1.23648	−0.106419
$136$	−4.29177	−0.368016
$137$	−9.22788	−0.788391	−0.394196	−	0.919027i	$-0.628977\pi$
$137$	−9.22788	−0.788391	−0.394196	+	0.919027i	$0.628977\pi$
$138$	0	0
$139$	7.82143	0.663405	0.331702	−	0.943384i	$-0.392377\pi$
$139$	7.82143	0.663405	0.331702	+	0.943384i	$0.392377\pi$
$140$	1.82862	0.154546
$141$	8.31271	0.700056
$142$	−7.31821	−0.614131
$143$	−0.0596863	−0.00499122
$144$	1.00000	0.0833333
$145$	9.49388	0.788424
$146$	13.1978	1.09226
$147$	−4.81288	−0.396960
$148$	−5.22871	−0.429797
$149$	−4.95242	−0.405718	−0.202859	−	0.979208i	$-0.565023\pi$
$149$	−4.95242	−0.405718	−0.202859	+	0.979208i	$0.565023\pi$
$150$	−3.47112	−0.283416
$151$	15.3636	1.25028	0.625138	−	0.780514i	$-0.285043\pi$
$151$	15.3636	1.25028	0.625138	+	0.780514i	$0.285043\pi$
$152$	−4.66759	−0.378592
$153$	−4.29177	−0.346969
$154$	1.31141	0.105676
$155$	1.96780	0.158057
$156$	0.0673089	0.00538903
$157$	12.5073	0.998188	0.499094	−	0.866548i	$-0.333666\pi$
$157$	12.5073	0.998188	0.499094	+	0.866548i	$0.333666\pi$
$158$	7.89446	0.628050
$159$	−2.99558	−0.237565
$160$	−1.23648	−0.0977522
$161$	0	0
$162$	1.00000	0.0785674
$163$	−10.4300	−0.816943	−0.408472	−	0.912771i	$-0.633938\pi$
$163$	−10.4300	−0.816943	−0.408472	+	0.912771i	$0.633938\pi$
$164$	−0.135933	−0.0106145
$165$	1.09645	0.0853586
$166$	−8.40858	−0.652632
$167$	−25.3684	−1.96306	−0.981532	−	0.191299i	$-0.938730\pi$
$167$	−25.3684	−1.96306	−0.981532	+	0.191299i	$0.938730\pi$
$168$	−1.47889	−0.114099
$169$	−12.9955	−0.999652
$170$	5.30668	0.407004
$171$	−4.66759	−0.356940
$172$	6.57798	0.501567
$173$	17.1947	1.30729	0.653644	−	0.756802i	$-0.273239\pi$
$173$	17.1947	1.30729	0.653644	+	0.756802i	$0.273239\pi$
$174$	−7.67816	−0.582079
$175$	5.13341	0.388049
$176$	−0.886752	−0.0668415
$177$	−6.83576	−0.513807
$178$	11.7235	0.878711
$179$	−6.10800	−0.456533	−0.228267	−	0.973599i	$-0.573306\pi$
$179$	−6.10800	−0.456533	−0.228267	+	0.973599i	$0.573306\pi$
$180$	−1.23648	−0.0921617
$181$	21.4901	1.59734	0.798672	−	0.601766i	$-0.205536\pi$
$181$	21.4901	1.59734	0.798672	+	0.601766i	$0.205536\pi$
$182$	−0.0995426	−0.00737859
$183$	−4.60015	−0.340053
$184$	0	0
$185$	6.46519	0.475330
$186$	−1.59145	−0.116691
$187$	3.80574	0.278303
$188$	8.31271	0.606266
$189$	−1.47889	−0.107574
$190$	5.77138	0.418700
$191$	−2.83473	−0.205114	−0.102557	−	0.994727i	$-0.532702\pi$
$191$	−2.83473	−0.205114	−0.102557	+	0.994727i	$0.532702\pi$
$192$	1.00000	0.0721688
$193$	−24.0877	−1.73387	−0.866936	−	0.498420i	$-0.833914\pi$
$193$	−24.0877	−1.73387	−0.866936	+	0.498420i	$0.833914\pi$
$194$	−17.1665	−1.23248
$195$	−0.0832260	−0.00595994
$196$	−4.81288	−0.343777
$197$	−24.6054	−1.75306	−0.876532	−	0.481343i	$-0.840149\pi$
$197$	−24.6054	−1.75306	−0.876532	+	0.481343i	$0.840149\pi$
$198$	−0.886752	−0.0630187
$199$	6.27783	0.445024	0.222512	−	0.974930i	$-0.428574\pi$
$199$	6.27783	0.445024	0.222512	+	0.974930i	$0.428574\pi$
$200$	−3.47112	−0.245445
$201$	12.4051	0.874989
$202$	5.98712	0.421252
$203$	11.3552	0.796976
$204$	−4.29177	−0.300484
$205$	0.168078	0.0117391
$206$	−15.6415	−1.08980
$207$	0	0
$208$	0.0673089	0.00466703
$209$	4.13900	0.286301
$210$	1.82862	0.126187
$211$	19.3895	1.33483	0.667414	−	0.744686i	$-0.267401\pi$
$211$	19.3895	1.33483	0.667414	+	0.744686i	$0.267401\pi$
$212$	−2.99558	−0.205737
$213$	−7.31821	−0.501436
$214$	−13.8185	−0.944615
$215$	−8.13354	−0.554703
$216$	1.00000	0.0680414
$217$	2.35359	0.159772
$218$	10.7865	0.730556
$219$	13.1978	0.891825
$220$	1.09645	0.0739227
$221$	−0.288874	−0.0194318
$222$	−5.22871	−0.350928
$223$	20.5735	1.37770	0.688852	−	0.724902i	$-0.258115\pi$
$223$	20.5735	1.37770	0.688852	+	0.724902i	$0.258115\pi$
$224$	−1.47889	−0.0988126
$225$	−3.47112	−0.231408
$226$	−4.57032	−0.304013
$227$	−23.0232	−1.52811	−0.764053	−	0.645154i	$-0.776793\pi$
$227$	−23.0232	−1.52811	−0.764053	+	0.645154i	$0.776793\pi$
$228$	−4.66759	−0.309119
$229$	7.50367	0.495857	0.247928	−	0.968778i	$-0.420250\pi$
$229$	7.50367	0.495857	0.247928	+	0.968778i	$0.420250\pi$
$230$	0	0
$231$	1.31141	0.0862845
$232$	−7.67816	−0.504096
$233$	24.3682	1.59642	0.798208	−	0.602382i	$-0.205782\pi$
$233$	24.3682	1.59642	0.798208	+	0.602382i	$0.205782\pi$
$234$	0.0673089	0.00440012
$235$	−10.2785	−0.670495
$236$	−6.83576	−0.444970
$237$	7.89446	0.512801
$238$	6.34706	0.411419
$239$	8.69645	0.562527	0.281263	−	0.959631i	$-0.409247\pi$
$239$	8.69645	0.562527	0.281263	+	0.959631i	$0.409247\pi$
$240$	−1.23648	−0.0798144
$241$	−5.67714	−0.365697	−0.182848	−	0.983141i	$-0.558532\pi$
$241$	−5.67714	−0.365697	−0.182848	+	0.983141i	$0.558532\pi$
$242$	−10.2137	−0.656560
$243$	1.00000	0.0641500
$244$	−4.60015	−0.294494
$245$	5.95102	0.380197
$246$	−0.135933	−0.00866674
$247$	−0.314171	−0.0199902
$248$	−1.59145	−0.101057
$249$	−8.40858	−0.532872
$250$	10.4744	0.662457
$251$	5.98412	0.377714	0.188857	−	0.982005i	$-0.439522\pi$
$251$	5.98412	0.377714	0.188857	+	0.982005i	$0.439522\pi$
$252$	−1.47889	−0.0931614
$253$	0	0
$254$	−14.4076	−0.904010
$255$	5.30668	0.332317
$256$	1.00000	0.0625000
$257$	15.5582	0.970496	0.485248	−	0.874376i	$-0.338729\pi$
$257$	15.5582	0.970496	0.485248	+	0.874376i	$0.338729\pi$
$258$	6.57798	0.409527
$259$	7.73269	0.480486
$260$	−0.0832260	−0.00516146
$261$	−7.67816	−0.475266
$262$	−4.57998	−0.282952
$263$	10.2667	0.633069	0.316535	−	0.948581i	$-0.397481\pi$
$263$	10.2667	0.633069	0.316535	+	0.948581i	$0.397481\pi$
$264$	−0.886752	−0.0545758
$265$	3.70397	0.227533
$266$	6.90286	0.423242
$267$	11.7235	0.717464
$268$	12.4051	0.757763
$269$	−3.24314	−0.197738	−0.0988688	−	0.995100i	$-0.531522\pi$
$269$	−3.24314	−0.197738	−0.0988688	+	0.995100i	$0.531522\pi$
$270$	−1.23648	−0.0752497
$271$	15.5064	0.941944	0.470972	−	0.882148i	$-0.343903\pi$
$271$	15.5064	0.941944	0.470972	+	0.882148i	$0.343903\pi$
$272$	−4.29177	−0.260227
$273$	−0.0995426	−0.00602459
$274$	−9.22788	−0.557477
$275$	3.07802	0.185612
$276$	0	0
$277$	2.16313	0.129970	0.0649849	−	0.997886i	$-0.479300\pi$
$277$	2.16313	0.129970	0.0649849	+	0.997886i	$0.479300\pi$
$278$	7.82143	0.469098
$279$	−1.59145	−0.0952779
$280$	1.82862	0.109281
$281$	20.5561	1.22627	0.613137	−	0.789977i	$-0.289907\pi$
$281$	20.5561	1.22627	0.613137	+	0.789977i	$0.289907\pi$
$282$	8.31271	0.495015
$283$	16.9363	1.00676	0.503381	−	0.864065i	$-0.332089\pi$
$283$	16.9363	1.00676	0.503381	+	0.864065i	$0.332089\pi$
$284$	−7.31821	−0.434256
$285$	5.77138	0.341867
$286$	−0.0596863	−0.00352933
$287$	0.201029	0.0118664
$288$	1.00000	0.0589256
$289$	1.41930	0.0834884
$290$	9.49388	0.557500
$291$	−17.1665	−1.00632
$292$	13.1978	0.772343
$293$	2.06234	0.120483	0.0602416	−	0.998184i	$-0.480813\pi$
$293$	2.06234	0.120483	0.0602416	+	0.998184i	$0.480813\pi$
$294$	−4.81288	−0.280693
$295$	8.45227	0.492110
$296$	−5.22871	−0.303912
$297$	−0.886752	−0.0514546
$298$	−4.95242	−0.286886
$299$	0	0
$300$	−3.47112	−0.200405
$301$	−9.72812	−0.560720
$302$	15.3636	0.884079
$303$	5.98712	0.343951
$304$	−4.66759	−0.267705
$305$	5.68798	0.325693
$306$	−4.29177	−0.245344
$307$	−4.08297	−0.233027	−0.116514	−	0.993189i	$-0.537172\pi$
$307$	−4.08297	−0.233027	−0.116514	+	0.993189i	$0.537172\pi$
$308$	1.31141	0.0747245
$309$	−15.6415	−0.889814
$310$	1.96780	0.111763
$311$	−12.7493	−0.722945	−0.361472	−	0.932383i	$-0.617726\pi$
$311$	−12.7493	−0.722945	−0.361472	+	0.932383i	$0.617726\pi$
$312$	0.0673089	0.00381062
$313$	−7.78228	−0.439881	−0.219940	−	0.975513i	$-0.570586\pi$
$313$	−7.78228	−0.439881	−0.219940	+	0.975513i	$0.570586\pi$
$314$	12.5073	0.705825
$315$	1.82862	0.103031
$316$	7.89446	0.444098
$317$	19.7507	1.10931	0.554655	−	0.832080i	$-0.312850\pi$
$317$	19.7507	1.10931	0.554655	+	0.832080i	$0.312850\pi$
$318$	−2.99558	−0.167984
$319$	6.80862	0.381210
$320$	−1.23648	−0.0691213
$321$	−13.8185	−0.771275
$322$	0	0
$323$	20.0322	1.11462
$324$	1.00000	0.0555556
$325$	−0.233637	−0.0129599
$326$	−10.4300	−0.577666
$327$	10.7865	0.596496
$328$	−0.135933	−0.00750562
$329$	−12.2936	−0.677768
$330$	1.09645	0.0603576
$331$	−4.29179	−0.235898	−0.117949	−	0.993020i	$-0.537632\pi$
$331$	−4.29179	−0.235898	−0.117949	+	0.993020i	$0.537632\pi$
$332$	−8.40858	−0.461481
$333$	−5.22871	−0.286531
$334$	−25.3684	−1.38810
$335$	−15.3387	−0.838040
$336$	−1.47889	−0.0806801
$337$	28.9585	1.57747	0.788735	−	0.614733i	$-0.210736\pi$
$337$	28.9585	1.57747	0.788735	+	0.614733i	$0.210736\pi$
$338$	−12.9955	−0.706860
$339$	−4.57032	−0.248226
$340$	5.30668	0.287795
$341$	1.41123	0.0764221
$342$	−4.66759	−0.252395
$343$	17.4700	0.943290
$344$	6.57798	0.354661
$345$	0	0
$346$	17.1947	0.924392
$347$	−18.7893	−1.00866	−0.504332	−	0.863510i	$-0.668261\pi$
$347$	−18.7893	−1.00866	−0.504332	+	0.863510i	$0.668261\pi$
$348$	−7.67816	−0.411592
$349$	−15.5823	−0.834102	−0.417051	−	0.908883i	$-0.636936\pi$
$349$	−15.5823	−0.834102	−0.417051	+	0.908883i	$0.636936\pi$
$350$	5.13341	0.274392
$351$	0.0673089	0.00359268
$352$	−0.886752	−0.0472641
$353$	−23.2408	−1.23698	−0.618492	−	0.785791i	$-0.712256\pi$
$353$	−23.2408	−1.23698	−0.618492	+	0.785791i	$0.712256\pi$
$354$	−6.83576	−0.363317
$355$	9.04881	0.480261
$356$	11.7235	0.621342
$357$	6.34706	0.335922
$358$	−6.10800	−0.322818
$359$	14.4495	0.762614	0.381307	−	0.924449i	$-0.375474\pi$
$359$	14.4495	0.762614	0.381307	+	0.924449i	$0.375474\pi$
$360$	−1.23648	−0.0651682
$361$	2.78643	0.146654
$362$	21.4901	1.12949
$363$	−10.2137	−0.536079
$364$	−0.0995426	−0.00521745
$365$	−16.3188	−0.854165
$366$	−4.60015	−0.240453
$367$	26.6056	1.38880	0.694400	−	0.719589i	$-0.255670\pi$
$367$	26.6056	1.38880	0.694400	+	0.719589i	$0.255670\pi$
$368$	0	0
$369$	−0.135933	−0.00707636
$370$	6.46519	0.336109
$371$	4.43013	0.230001
$372$	−1.59145	−0.0825130
$373$	−4.88055	−0.252705	−0.126353	−	0.991985i	$-0.540327\pi$
$373$	−4.88055	−0.252705	−0.126353	+	0.991985i	$0.540327\pi$
$374$	3.80574	0.196790
$375$	10.4744	0.540894
$376$	8.31271	0.428695
$377$	−0.516808	−0.0266170
$378$	−1.47889	−0.0760660
$379$	3.41084	0.175203	0.0876015	−	0.996156i	$-0.472080\pi$
$379$	3.41084	0.175203	0.0876015	+	0.996156i	$0.472080\pi$
$380$	5.77138	0.296066
$381$	−14.4076	−0.738121
$382$	−2.83473	−0.145037
$383$	−28.3725	−1.44977	−0.724883	−	0.688872i	$-0.758106\pi$
$383$	−28.3725	−1.44977	−0.724883	+	0.688872i	$0.758106\pi$
$384$	1.00000	0.0510310
$385$	−1.62153	−0.0826409
$386$	−24.0877	−1.22603
$387$	6.57798	0.334378
$388$	−17.1665	−0.871496
$389$	−10.6537	−0.540166	−0.270083	−	0.962837i	$-0.587051\pi$
$389$	−10.6537	−0.540166	−0.270083	+	0.962837i	$0.587051\pi$
$390$	−0.0832260	−0.00421431
$391$	0	0
$392$	−4.81288	−0.243087
$393$	−4.57998	−0.231029
$394$	−24.6054	−1.23960
$395$	−9.76134	−0.491146
$396$	−0.886752	−0.0445610
$397$	20.9156	1.04972	0.524862	−	0.851187i	$-0.324117\pi$
$397$	20.9156	1.04972	0.524862	+	0.851187i	$0.324117\pi$
$398$	6.27783	0.314679
$399$	6.90286	0.345575
$400$	−3.47112	−0.173556
$401$	4.05114	0.202304	0.101152	−	0.994871i	$-0.467747\pi$
$401$	4.05114	0.202304	0.101152	+	0.994871i	$0.467747\pi$
$402$	12.4051	0.618711
$403$	−0.107119	−0.00533598
$404$	5.98712	0.297870
$405$	−1.23648	−0.0614411
$406$	11.3552	0.563547
$407$	4.63657	0.229826
$408$	−4.29177	−0.212474
$409$	24.3805	1.20554	0.602770	−	0.797915i	$-0.294064\pi$
$409$	24.3805	1.20554	0.602770	+	0.797915i	$0.294064\pi$
$410$	0.168078	0.00830077
$411$	−9.22788	−0.455178
$412$	−15.6415	−0.770602
$413$	10.1093	0.497448
$414$	0	0
$415$	10.3970	0.510370
$416$	0.0673089	0.00330009
$417$	7.82143	0.383017
$418$	4.13900	0.202445
$419$	14.1181	0.689715	0.344858	−	0.938655i	$-0.387927\pi$
$419$	14.1181	0.689715	0.344858	+	0.938655i	$0.387927\pi$
$420$	1.82862	0.0892274
$421$	22.3142	1.08753	0.543763	−	0.839239i	$-0.316999\pi$
$421$	22.3142	1.08753	0.543763	+	0.839239i	$0.316999\pi$
$422$	19.3895	0.943867
$423$	8.31271	0.404178
$424$	−2.99558	−0.145478
$425$	14.8973	0.722623
$426$	−7.31821	−0.354568
$427$	6.80312	0.329226
$428$	−13.8185	−0.667944
$429$	−0.0596863	−0.00288168
$430$	−8.13354	−0.392234
$431$	−18.0169	−0.867842	−0.433921	−	0.900951i	$-0.642870\pi$
$431$	−18.0169	−0.867842	−0.433921	+	0.900951i	$0.642870\pi$
$432$	1.00000	0.0481125
$433$	20.7161	0.995554	0.497777	−	0.867305i	$-0.334150\pi$
$433$	20.7161	0.995554	0.497777	+	0.867305i	$0.334150\pi$
$434$	2.35359	0.112976
$435$	9.49388	0.455197
$436$	10.7865	0.516581
$437$	0	0
$438$	13.1978	0.630615
$439$	−13.5915	−0.648686	−0.324343	−	0.945940i	$-0.605143\pi$
$439$	−13.5915	−0.648686	−0.324343	+	0.945940i	$0.605143\pi$
$440$	1.09645	0.0522712
$441$	−4.81288	−0.229185
$442$	−0.288874	−0.0137404
$443$	10.4025	0.494238	0.247119	−	0.968985i	$-0.420516\pi$
$443$	10.4025	0.494238	0.247119	+	0.968985i	$0.420516\pi$
$444$	−5.22871	−0.248143
$445$	−14.4958	−0.687168
$446$	20.5735	0.974184
$447$	−4.95242	−0.234241
$448$	−1.47889	−0.0698711
$449$	10.7318	0.506465	0.253232	−	0.967405i	$-0.418506\pi$
$449$	10.7318	0.506465	0.253232	+	0.967405i	$0.418506\pi$
$450$	−3.47112	−0.163630
$451$	0.120538	0.00567593
$452$	−4.57032	−0.214970
$453$	15.3636	0.721847
$454$	−23.0232	−1.08053
$455$	0.123082	0.00577019
$456$	−4.66759	−0.218580
$457$	0.845263	0.0395397	0.0197699	−	0.999805i	$-0.493707\pi$
$457$	0.845263	0.0395397	0.0197699	+	0.999805i	$0.493707\pi$
$458$	7.50367	0.350624
$459$	−4.29177	−0.200323
$460$	0	0
$461$	−17.2385	−0.802875	−0.401437	−	0.915886i	$-0.631489\pi$
$461$	−17.2385	−0.802875	−0.401437	+	0.915886i	$0.631489\pi$
$462$	1.31141	0.0610123
$463$	−1.81253	−0.0842354	−0.0421177	−	0.999113i	$-0.513410\pi$
$463$	−1.81253	−0.0842354	−0.0421177	+	0.999113i	$0.513410\pi$
$464$	−7.67816	−0.356449
$465$	1.96780	0.0912545
$466$	24.3682	1.12884
$467$	−32.8225	−1.51884	−0.759422	−	0.650598i	$-0.774518\pi$
$467$	−32.8225	−1.51884	−0.759422	+	0.650598i	$0.774518\pi$
$468$	0.0673089	0.00311136
$469$	−18.3458	−0.847131
$470$	−10.2785	−0.474111
$471$	12.5073	0.576304
$472$	−6.83576	−0.314641
$473$	−5.83304	−0.268204
$474$	7.89446	0.362605
$475$	16.2018	0.743389
$476$	6.34706	0.290917
$477$	−2.99558	−0.137158
$478$	8.69645	0.397766
$479$	−20.2241	−0.924062	−0.462031	−	0.886864i	$-0.652879\pi$
$479$	−20.2241	−0.924062	−0.462031	+	0.886864i	$0.652879\pi$
$480$	−1.23648	−0.0564373
$481$	−0.351939	−0.0160470
$482$	−5.67714	−0.258587
$483$	0	0
$484$	−10.2137	−0.464258
$485$	21.2260	0.963823
$486$	1.00000	0.0453609
$487$	−39.5598	−1.79262	−0.896312	−	0.443424i	$-0.853764\pi$
$487$	−39.5598	−1.79262	−0.896312	+	0.443424i	$0.853764\pi$
$488$	−4.60015	−0.208239
$489$	−10.4300	−0.471662
$490$	5.95102	0.268840
$491$	24.3279	1.09790	0.548951	−	0.835854i	$-0.315027\pi$
$491$	24.3279	1.09790	0.548951	+	0.835854i	$0.315027\pi$
$492$	−0.135933	−0.00612831
$493$	32.9529	1.48412
$494$	−0.314171	−0.0141352
$495$	1.09645	0.0492818
$496$	−1.59145	−0.0714584
$497$	10.8228	0.485471
$498$	−8.40858	−0.376797
$499$	−15.8756	−0.710689	−0.355344	−	0.934735i	$-0.615636\pi$
$499$	−15.8756	−0.710689	−0.355344	+	0.934735i	$0.615636\pi$
$500$	10.4744	0.468428
$501$	−25.3684	−1.13338
$502$	5.98412	0.267084
$503$	6.54382	0.291774	0.145887	−	0.989301i	$-0.453396\pi$
$503$	6.54382	0.291774	0.145887	+	0.989301i	$0.453396\pi$
$504$	−1.47889	−0.0658751
$505$	−7.40295	−0.329427
$506$	0	0
$507$	−12.9955	−0.577149
$508$	−14.4076	−0.639232
$509$	−17.8626	−0.791745	−0.395873	−	0.918305i	$-0.629558\pi$
$509$	−17.8626	−0.791745	−0.395873	+	0.918305i	$0.629558\pi$
$510$	5.30668	0.234984
$511$	−19.5181	−0.863431
$512$	1.00000	0.0441942
$513$	−4.66759	−0.206079
$514$	15.5582	0.686244
$515$	19.3404	0.852240
$516$	6.57798	0.289580
$517$	−7.37131	−0.324190
$518$	7.73269	0.339755
$519$	17.1947	0.754763
$520$	−0.0832260	−0.00364970
$521$	11.8337	0.518445	0.259222	−	0.965818i	$-0.416534\pi$
$521$	11.8337	0.518445	0.259222	+	0.965818i	$0.416534\pi$
$522$	−7.67816	−0.336064
$523$	−16.0125	−0.700177	−0.350088	−	0.936717i	$-0.613848\pi$
$523$	−16.0125	−0.700177	−0.350088	+	0.936717i	$0.613848\pi$
$524$	−4.57998	−0.200077
$525$	5.13341	0.224040
$526$	10.2667	0.447648
$527$	6.83016	0.297526
$528$	−0.886752	−0.0385909
$529$	0	0
$530$	3.70397	0.160890
$531$	−6.83576	−0.296647
$532$	6.90286	0.299277
$533$	−0.00914947	−0.000396307 0
$534$	11.7235	0.507324
$535$	17.0863	0.738706
$536$	12.4051	0.535819
$537$	−6.10800	−0.263580
$538$	−3.24314	−0.139822
$539$	4.26783	0.183829
$540$	−1.23648	−0.0532096
$541$	−19.5524	−0.840621	−0.420311	−	0.907380i	$-0.638079\pi$
$541$	−19.5524	−0.840621	−0.420311	+	0.907380i	$0.638079\pi$
$542$	15.5064	0.666055
$543$	21.4901	0.922228
$544$	−4.29177	−0.184008
$545$	−13.3373	−0.571308
$546$	−0.0995426	−0.00426003
$547$	−1.13521	−0.0485380	−0.0242690	−	0.999705i	$-0.507726\pi$
$547$	−1.13521	−0.0485380	−0.0242690	+	0.999705i	$0.507726\pi$
$548$	−9.22788	−0.394196
$549$	−4.60015	−0.196329
$550$	3.07802	0.131247
$551$	35.8385	1.52677
$552$	0	0
$553$	−11.6751	−0.496474
$554$	2.16313	0.0919026
$555$	6.46519	0.274432
$556$	7.82143	0.331702
$557$	41.2506	1.74784	0.873922	−	0.486066i	$-0.161569\pi$
$557$	41.2506	1.74784	0.873922	+	0.486066i	$0.161569\pi$
$558$	−1.59145	−0.0673716
$559$	0.442757	0.0187266
$560$	1.82862	0.0772732
$561$	3.80574	0.160678
$562$	20.5561	0.867106
$563$	−28.2407	−1.19020	−0.595101	−	0.803651i	$-0.702888\pi$
$563$	−28.2407	−1.19020	−0.595101	+	0.803651i	$0.702888\pi$
$564$	8.31271	0.350028
$565$	5.65110	0.237744
$566$	16.9363	0.711888
$567$	−1.47889	−0.0621076
$568$	−7.31821	−0.307065
$569$	−41.5315	−1.74109	−0.870546	−	0.492086i	$-0.836234\pi$
$569$	−41.5315	−1.74109	−0.870546	+	0.492086i	$0.836234\pi$
$570$	5.77138	0.241737
$571$	−31.5189	−1.31902	−0.659512	−	0.751694i	$-0.729237\pi$
$571$	−31.5189	−1.31902	−0.659512	+	0.751694i	$0.729237\pi$
$572$	−0.0596863	−0.00249561
$573$	−2.83473	−0.118423
$574$	0.201029	0.00839081
$575$	0	0
$576$	1.00000	0.0416667
$577$	20.5890	0.857132	0.428566	−	0.903511i	$-0.359019\pi$
$577$	20.5890	0.857132	0.428566	+	0.903511i	$0.359019\pi$
$578$	1.41930	0.0590352
$579$	−24.0877	−1.00105
$580$	9.49388	0.394212
$581$	12.4354	0.515906
$582$	−17.1665	−0.711574
$583$	2.65633	0.110014
$584$	13.1978	0.546129
$585$	−0.0832260	−0.00344097
$586$	2.06234	0.0851945
$587$	8.51084	0.351280	0.175640	−	0.984454i	$-0.443801\pi$
$587$	8.51084	0.351280	0.175640	+	0.984454i	$0.443801\pi$
$588$	−4.81288	−0.198480
$589$	7.42826	0.306076
$590$	8.45227	0.347975
$591$	−24.6054	−1.01213
$592$	−5.22871	−0.214899
$593$	−9.47438	−0.389066	−0.194533	−	0.980896i	$-0.562319\pi$
$593$	−9.47438	−0.389066	−0.194533	+	0.980896i	$0.562319\pi$
$594$	−0.886752	−0.0363839
$595$	−7.84801	−0.321737
$596$	−4.95242	−0.202859
$597$	6.27783	0.256935
$598$	0	0
$599$	−2.52422	−0.103137	−0.0515684	−	0.998669i	$-0.516422\pi$
$599$	−2.52422	−0.103137	−0.0515684	+	0.998669i	$0.516422\pi$
$600$	−3.47112	−0.141708
$601$	−18.8422	−0.768588	−0.384294	−	0.923211i	$-0.625555\pi$
$601$	−18.8422	−0.768588	−0.384294	+	0.923211i	$0.625555\pi$
$602$	−9.72812	−0.396489
$603$	12.4051	0.505175
$604$	15.3636	0.625138
$605$	12.6290	0.513441
$606$	5.98712	0.243210
$607$	3.25339	0.132051	0.0660256	−	0.997818i	$-0.478968\pi$
$607$	3.25339	0.132051	0.0660256	+	0.997818i	$0.478968\pi$
$608$	−4.66759	−0.189296
$609$	11.3552	0.460134
$610$	5.68798	0.230300
$611$	0.559519	0.0226357
$612$	−4.29177	−0.173485
$613$	16.1279	0.651398	0.325699	−	0.945474i	$-0.394400\pi$
$613$	16.1279	0.651398	0.325699	+	0.945474i	$0.394400\pi$
$614$	−4.08297	−0.164775
$615$	0.168078	0.00677755
$616$	1.31141	0.0528382
$617$	2.31927	0.0933702	0.0466851	−	0.998910i	$-0.485134\pi$
$617$	2.31927	0.0933702	0.0466851	+	0.998910i	$0.485134\pi$
$618$	−15.6415	−0.629194
$619$	32.9947	1.32617	0.663085	−	0.748544i	$-0.269247\pi$
$619$	32.9947	1.32617	0.663085	+	0.748544i	$0.269247\pi$
$620$	1.96780	0.0790287
$621$	0	0
$622$	−12.7493	−0.511199
$623$	−17.3377	−0.694622
$624$	0.0673089	0.00269451
$625$	4.40427	0.176171
$626$	−7.78228	−0.311043
$627$	4.13900	0.165296
$628$	12.5073	0.499094
$629$	22.4404	0.894758
$630$	1.82862	0.0728539
$631$	−30.4452	−1.21200	−0.606001	−	0.795464i	$-0.707227\pi$
$631$	−30.4452	−1.21200	−0.606001	+	0.795464i	$0.707227\pi$
$632$	7.89446	0.314025
$633$	19.3895	0.770664
$634$	19.7507	0.784401
$635$	17.8146	0.706952
$636$	−2.99558	−0.118782
$637$	−0.323950	−0.0128354
$638$	6.80862	0.269556
$639$	−7.31821	−0.289504
$640$	−1.23648	−0.0488761
$641$	−44.7383	−1.76706	−0.883528	−	0.468378i	$-0.844839\pi$
$641$	−44.7383	−1.76706	−0.883528	+	0.468378i	$0.844839\pi$
$642$	−13.8185	−0.545374
$643$	26.8514	1.05892	0.529459	−	0.848336i	$-0.322395\pi$
$643$	26.8514	1.05892	0.529459	+	0.848336i	$0.322395\pi$
$644$	0	0
$645$	−8.13354	−0.320258
$646$	20.0322	0.788158
$647$	−22.9140	−0.900841	−0.450420	−	0.892817i	$-0.648726\pi$
$647$	−22.9140	−0.900841	−0.450420	+	0.892817i	$0.648726\pi$
$648$	1.00000	0.0392837
$649$	6.06163	0.237940
$650$	−0.233637	−0.00916401
$651$	2.35359	0.0922444
$652$	−10.4300	−0.408472
$653$	−46.0369	−1.80156	−0.900782	−	0.434271i	$-0.857006\pi$
$653$	−46.0369	−1.80156	−0.900782	+	0.434271i	$0.857006\pi$
$654$	10.7865	0.421787
$655$	5.66304	0.221273
$656$	−0.135933	−0.00530727
$657$	13.1978	0.514895
$658$	−12.2936	−0.479254
$659$	−27.6386	−1.07665	−0.538323	−	0.842739i	$-0.680942\pi$
$659$	−27.6386	−1.07665	−0.538323	+	0.842739i	$0.680942\pi$
$660$	1.09645	0.0426793
$661$	5.29014	0.205763	0.102881	−	0.994694i	$-0.467194\pi$
$661$	5.29014	0.205763	0.102881	+	0.994694i	$0.467194\pi$
$662$	−4.29179	−0.166805
$663$	−0.288874	−0.0112190
$664$	−8.40858	−0.326316
$665$	−8.53525	−0.330983
$666$	−5.22871	−0.202608
$667$	0	0
$668$	−25.3684	−0.981532
$669$	20.5735	0.795418
$670$	−15.3387	−0.592584
$671$	4.07919	0.157475
$672$	−1.47889	−0.0570495
$673$	−40.6460	−1.56679	−0.783394	−	0.621525i	$-0.786513\pi$
$673$	−40.6460	−1.56679	−0.783394	+	0.621525i	$0.786513\pi$
$674$	28.9585	1.11544
$675$	−3.47112	−0.133603
$676$	−12.9955	−0.499826
$677$	−15.4320	−0.593100	−0.296550	−	0.955017i	$-0.595836\pi$
$677$	−15.4320	−0.593100	−0.296550	+	0.955017i	$0.595836\pi$
$678$	−4.57032	−0.175522
$679$	25.3874	0.974278
$680$	5.30668	0.203502
$681$	−23.0232	−0.882252
$682$	1.41123	0.0540386
$683$	0.323786	0.0123893	0.00619467	−	0.999981i	$-0.498028\pi$
$683$	0.323786	0.0123893	0.00619467	+	0.999981i	$0.498028\pi$
$684$	−4.66759	−0.178470
$685$	11.4101	0.435957
$686$	17.4700	0.667006
$687$	7.50367	0.286283
$688$	6.57798	0.250783
$689$	−0.201629	−0.00768145
$690$	0	0
$691$	−11.8762	−0.451791	−0.225896	−	0.974152i	$-0.572531\pi$
$691$	−11.8762	−0.451791	−0.225896	+	0.974152i	$0.572531\pi$
$692$	17.1947	0.653644
$693$	1.31141	0.0498164
$694$	−18.7893	−0.713233
$695$	−9.67103	−0.366843
$696$	−7.67816	−0.291040
$697$	0.583391	0.0220975
$698$	−15.5823	−0.589799
$699$	24.3682	0.921691
$700$	5.13341	0.194025
$701$	29.1388	1.10056	0.550278	−	0.834981i	$-0.314522\pi$
$701$	29.1388	1.10056	0.550278	+	0.834981i	$0.314522\pi$
$702$	0.0673089	0.00254041
$703$	24.4055	0.920470
$704$	−0.886752	−0.0334207
$705$	−10.2785	−0.387110
$706$	−23.2408	−0.874680
$707$	−8.85430	−0.333000
$708$	−6.83576	−0.256904
$709$	−13.3204	−0.500259	−0.250130	−	0.968212i	$-0.580473\pi$
$709$	−13.3204	−0.500259	−0.250130	+	0.968212i	$0.580473\pi$
$710$	9.04881	0.339596
$711$	7.89446	0.296066
$712$	11.7235	0.439355
$713$	0	0
$714$	6.34706	0.237533
$715$	0.0738009	0.00276000
$716$	−6.10800	−0.228267
$717$	8.69645	0.324775
$718$	14.4495	0.539249
$719$	40.7753	1.52066	0.760330	−	0.649537i	$-0.225037\pi$
$719$	40.7753	1.52066	0.760330	+	0.649537i	$0.225037\pi$
$720$	−1.23648	−0.0460808
$721$	23.1321	0.861484
$722$	2.78643	0.103700
$723$	−5.67714	−0.211135
$724$	21.4901	0.798672
$725$	26.6518	0.989823
$726$	−10.2137	−0.379065
$727$	−32.8678	−1.21900	−0.609500	−	0.792786i	$-0.708630\pi$
$727$	−32.8678	−1.21900	−0.609500	+	0.792786i	$0.708630\pi$
$728$	−0.0995426	−0.00368929
$729$	1.00000	0.0370370
$730$	−16.3188	−0.603986
$731$	−28.2312	−1.04417
$732$	−4.60015	−0.170026
$733$	49.1083	1.81386	0.906929	−	0.421284i	$-0.138421\pi$
$733$	49.1083	1.81386	0.906929	+	0.421284i	$0.138421\pi$
$734$	26.6056	0.982030
$735$	5.95102	0.219507
$736$	0	0
$737$	−11.0003	−0.405200
$738$	−0.135933	−0.00500374
$739$	−42.3548	−1.55805	−0.779023	−	0.626995i	$-0.784285\pi$
$739$	−42.3548	−1.55805	−0.779023	+	0.626995i	$0.784285\pi$
$740$	6.46519	0.237665
$741$	−0.314171	−0.0115413
$742$	4.43013	0.162635
$743$	−2.66106	−0.0976250	−0.0488125	−	0.998808i	$-0.515544\pi$
$743$	−2.66106	−0.0976250	−0.0488125	+	0.998808i	$0.515544\pi$
$744$	−1.59145	−0.0583455
$745$	6.12356	0.224350
$746$	−4.88055	−0.178689
$747$	−8.40858	−0.307654
$748$	3.80574	0.139152
$749$	20.4361	0.746719
$750$	10.4744	0.382470
$751$	37.1251	1.35471	0.677357	−	0.735654i	$-0.263125\pi$
$751$	37.1251	1.35471	0.677357	+	0.735654i	$0.263125\pi$
$752$	8.31271	0.303133
$753$	5.98412	0.218073
$754$	−0.516808	−0.0188210
$755$	−18.9968	−0.691365
$756$	−1.47889	−0.0537868
$757$	−23.2476	−0.844950	−0.422475	−	0.906375i	$-0.638838\pi$
$757$	−23.2476	−0.844950	−0.422475	+	0.906375i	$0.638838\pi$
$758$	3.41084	0.123887
$759$	0	0
$760$	5.77138	0.209350
$761$	17.6451	0.639633	0.319816	−	0.947480i	$-0.396379\pi$
$761$	17.6451	0.639633	0.319816	+	0.947480i	$0.396379\pi$
$762$	−14.4076	−0.521931
$763$	−15.9521	−0.577505
$764$	−2.83473	−0.102557
$765$	5.30668	0.191864
$766$	−28.3725	−1.02514
$767$	−0.460108	−0.0166135
$768$	1.00000	0.0360844
$769$	36.5665	1.31862	0.659311	−	0.751871i	$-0.270848\pi$
$769$	36.5665	1.31862	0.659311	+	0.751871i	$0.270848\pi$
$770$	−1.62153	−0.0584359
$771$	15.5582	0.560316
$772$	−24.0877	−0.866936
$773$	9.85872	0.354594	0.177297	−	0.984157i	$-0.443265\pi$
$773$	9.85872	0.354594	0.177297	+	0.984157i	$0.443265\pi$
$774$	6.57798	0.236441
$775$	5.52413	0.198433
$776$	−17.1665	−0.616241
$777$	7.73269	0.277409
$778$	−10.6537	−0.381955
$779$	0.634478	0.0227325
$780$	−0.0832260	−0.00297997
$781$	6.48944	0.232210
$782$	0	0
$783$	−7.67816	−0.274395
$784$	−4.81288	−0.171889
$785$	−15.4650	−0.551968
$786$	−4.57998	−0.163362
$787$	38.9210	1.38738	0.693692	−	0.720272i	$-0.255983\pi$
$787$	38.9210	1.38738	0.693692	+	0.720272i	$0.255983\pi$
$788$	−24.6054	−0.876532
$789$	10.2667	0.365503
$790$	−9.76134	−0.347293
$791$	6.75901	0.240323
$792$	−0.886752	−0.0315094
$793$	−0.309631	−0.0109953
$794$	20.9156	0.742268
$795$	3.70397	0.131366
$796$	6.27783	0.222512
$797$	1.36928	0.0485024	0.0242512	−	0.999706i	$-0.492280\pi$
$797$	1.36928	0.0485024	0.0242512	+	0.999706i	$0.492280\pi$
$798$	6.90286	0.244359
$799$	−35.6762	−1.26213
$800$	−3.47112	−0.122723
$801$	11.7235	0.414228
$802$	4.05114	0.143051
$803$	−11.7032	−0.412996
$804$	12.4051	0.437494
$805$	0	0
$806$	−0.107119	−0.00377311
$807$	−3.24314	−0.114164
$808$	5.98712	0.210626
$809$	−43.2994	−1.52233	−0.761163	−	0.648561i	$-0.775371\pi$
$809$	−43.2994	−1.52233	−0.761163	+	0.648561i	$0.775371\pi$
$810$	−1.23648	−0.0434454
$811$	9.57980	0.336392	0.168196	−	0.985754i	$-0.446206\pi$
$811$	9.57980	0.336392	0.168196	+	0.985754i	$0.446206\pi$
$812$	11.3552	0.398488
$813$	15.5064	0.543832
$814$	4.63657	0.162512
$815$	12.8965	0.451745
$816$	−4.29177	−0.150242
$817$	−30.7034	−1.07417
$818$	24.3805	0.852445
$819$	−0.0995426	−0.00347830
$820$	0.168078	0.00586953
$821$	23.4537	0.818541	0.409271	−	0.912413i	$-0.365783\pi$
$821$	23.4537	0.818541	0.409271	+	0.912413i	$0.365783\pi$
$822$	−9.22788	−0.321859
$823$	−9.33665	−0.325455	−0.162728	−	0.986671i	$-0.552029\pi$
$823$	−9.33665	−0.325455	−0.162728	+	0.986671i	$0.552029\pi$
$824$	−15.6415	−0.544898
$825$	3.07802	0.107163
$826$	10.1093	0.351749
$827$	−54.7053	−1.90229	−0.951145	−	0.308745i	$-0.900091\pi$
$827$	−54.7053	−1.90229	−0.951145	+	0.308745i	$0.900091\pi$
$828$	0	0
$829$	−55.3720	−1.92315	−0.961574	−	0.274545i	$-0.911473\pi$
$829$	−55.3720	−1.92315	−0.961574	+	0.274545i	$0.911473\pi$
$830$	10.3970	0.360886
$831$	2.16313	0.0750381
$832$	0.0673089	0.00233352
$833$	20.6558	0.715680
$834$	7.82143	0.270834
$835$	31.3675	1.08552
$836$	4.13900	0.143150
$837$	−1.59145	−0.0550087
$838$	14.1181	0.487702
$839$	11.6657	0.402744	0.201372	−	0.979515i	$-0.435460\pi$
$839$	11.6657	0.402744	0.201372	+	0.979515i	$0.435460\pi$
$840$	1.82862	0.0630933
$841$	29.9541	1.03290
$842$	22.3142	0.768998
$843$	20.5561	0.707989
$844$	19.3895	0.667414
$845$	16.0686	0.552777
$846$	8.31271	0.285797
$847$	15.1049	0.519011
$848$	−2.99558	−0.102869
$849$	16.9363	0.581254
$850$	14.8973	0.510972
$851$	0	0
$852$	−7.31821	−0.250718
$853$	39.0443	1.33685	0.668425	−	0.743779i	$-0.266969\pi$
$853$	39.0443	1.33685	0.668425	+	0.743779i	$0.266969\pi$
$854$	6.80312	0.232798
$855$	5.77138	0.197377
$856$	−13.8185	−0.472308
$857$	36.8064	1.25728	0.628641	−	0.777696i	$-0.283612\pi$
$857$	36.8064	1.25728	0.628641	+	0.777696i	$0.283612\pi$
$858$	−0.0596863	−0.00203766
$859$	−3.04908	−0.104033	−0.0520167	−	0.998646i	$-0.516565\pi$
$859$	−3.04908	−0.104033	−0.0520167	+	0.998646i	$0.516565\pi$
$860$	−8.13354	−0.277351
$861$	0.201029	0.00685106
$862$	−18.0169	−0.613657
$863$	−13.7177	−0.466958	−0.233479	−	0.972362i	$-0.575011\pi$
$863$	−13.7177	−0.466958	−0.233479	+	0.972362i	$0.575011\pi$
$864$	1.00000	0.0340207
$865$	−21.2609	−0.722891
$866$	20.7161	0.703963
$867$	1.41930	0.0482021
$868$	2.35359	0.0798860
$869$	−7.00043	−0.237473
$870$	9.49388	0.321873
$871$	0.834974	0.0282920
$872$	10.7865	0.365278
$873$	−17.1665	−0.580998
$874$	0	0
$875$	−15.4904	−0.523673
$876$	13.1978	0.445912
$877$	21.0860	0.712023	0.356012	−	0.934481i	$-0.384136\pi$
$877$	21.0860	0.712023	0.356012	+	0.934481i	$0.384136\pi$
$878$	−13.5915	−0.458690
$879$	2.06234	0.0695610
$880$	1.09645	0.0369613
$881$	−9.28725	−0.312895	−0.156448	−	0.987686i	$-0.550004\pi$
$881$	−9.28725	−0.312895	−0.156448	+	0.987686i	$0.550004\pi$
$882$	−4.81288	−0.162058
$883$	45.4416	1.52923	0.764616	−	0.644486i	$-0.222929\pi$
$883$	45.4416	1.52923	0.764616	+	0.644486i	$0.222929\pi$
$884$	−0.288874	−0.00971590
$885$	8.45227	0.284120
$886$	10.4025	0.349479
$887$	5.72147	0.192108	0.0960540	−	0.995376i	$-0.469378\pi$
$887$	5.72147	0.192108	0.0960540	+	0.995376i	$0.469378\pi$
$888$	−5.22871	−0.175464
$889$	21.3072	0.714621
$890$	−14.4958	−0.485901
$891$	−0.886752	−0.0297073
$892$	20.5735	0.688852
$893$	−38.8003	−1.29840
$894$	−4.95242	−0.165634
$895$	7.55241	0.252449
$896$	−1.47889	−0.0494063
$897$	0	0
$898$	10.7318	0.358125
$899$	12.2194	0.407541
$900$	−3.47112	−0.115704
$901$	12.8563	0.428306
$902$	0.120538	0.00401349
$903$	−9.72812	−0.323732
$904$	−4.57032	−0.152007
$905$	−26.5720	−0.883284
$906$	15.3636	0.510423
$907$	−26.2917	−0.873003	−0.436502	−	0.899704i	$-0.643783\pi$
$907$	−26.2917	−0.873003	−0.436502	+	0.899704i	$0.643783\pi$
$908$	−23.0232	−0.764053
$909$	5.98712	0.198580
$910$	0.123082	0.00408014
$911$	−49.3931	−1.63647	−0.818233	−	0.574887i	$-0.805046\pi$
$911$	−49.3931	−1.63647	−0.818233	+	0.574887i	$0.805046\pi$
$912$	−4.66759	−0.154559
$913$	7.45633	0.246768
$914$	0.845263	0.0279588
$915$	5.68798	0.188039
$916$	7.50367	0.247928
$917$	6.77329	0.223674
$918$	−4.29177	−0.141650
$919$	28.9737	0.955753	0.477876	−	0.878427i	$-0.341407\pi$
$919$	28.9737	0.955753	0.477876	+	0.878427i	$0.341407\pi$
$920$	0	0
$921$	−4.08297	−0.134538
$922$	−17.2385	−0.567718
$923$	−0.492581	−0.0162135
$924$	1.31141	0.0431422
$925$	18.1495	0.596751
$926$	−1.81253	−0.0595634
$927$	−15.6415	−0.513735
$928$	−7.67816	−0.252048
$929$	−48.9460	−1.60587	−0.802934	−	0.596068i	$-0.796729\pi$
$929$	−48.9460	−1.60587	−0.802934	+	0.596068i	$0.796729\pi$
$930$	1.96780	0.0645267
$931$	22.4646	0.736247
$932$	24.3682	0.798208
$933$	−12.7493	−0.417392
$934$	−32.8225	−1.07399
$935$	−4.70571	−0.153893
$936$	0.0673089	0.00220006
$937$	45.9165	1.50003	0.750014	−	0.661422i	$-0.230047\pi$
$937$	45.9165	1.50003	0.750014	+	0.661422i	$0.230047\pi$
$938$	−18.3458	−0.599012
$939$	−7.78228	−0.253965
$940$	−10.2785	−0.335247
$941$	−37.6436	−1.22715	−0.613573	−	0.789638i	$-0.710268\pi$
$941$	−37.6436	−1.22715	−0.613573	+	0.789638i	$0.710268\pi$
$942$	12.5073	0.407508
$943$	0	0
$944$	−6.83576	−0.222485
$945$	1.82862	0.0594849
$946$	−5.83304	−0.189649
$947$	−8.57147	−0.278535	−0.139268	−	0.990255i	$-0.544475\pi$
$947$	−8.57147	−0.278535	−0.139268	+	0.990255i	$0.544475\pi$
$948$	7.89446	0.256400
$949$	0.888330	0.0288364
$950$	16.2018	0.525655
$951$	19.7507	0.640461
$952$	6.34706	0.205710
$953$	38.8498	1.25847	0.629234	−	0.777216i	$-0.283369\pi$
$953$	38.8498	1.25847	0.629234	+	0.777216i	$0.283369\pi$
$954$	−2.99558	−0.0969854
$955$	3.50508	0.113422
$956$	8.69645	0.281263
$957$	6.80862	0.220091
$958$	−20.2241	−0.653410
$959$	13.6470	0.440686
$960$	−1.23648	−0.0399072
$961$	−28.4673	−0.918299
$962$	−0.351939	−0.0113470
$963$	−13.8185	−0.445296
$964$	−5.67714	−0.182848
$965$	29.7840	0.958779
$966$	0	0
$967$	17.0473	0.548205	0.274103	−	0.961700i	$-0.411619\pi$
$967$	17.0473	0.548205	0.274103	+	0.961700i	$0.411619\pi$
$968$	−10.2137	−0.328280
$969$	20.0322	0.643528
$970$	21.2260	0.681526
$971$	−46.5548	−1.49402	−0.747008	−	0.664815i	$-0.768510\pi$
$971$	−46.5548	−1.49402	−0.747008	+	0.664815i	$0.768510\pi$
$972$	1.00000	0.0320750
$973$	−11.5670	−0.370822
$974$	−39.5598	−1.26758
$975$	−0.233637	−0.00748238
$976$	−4.60015	−0.147247
$977$	−4.79148	−0.153293	−0.0766465	−	0.997058i	$-0.524421\pi$
$977$	−4.79148	−0.153293	−0.0766465	+	0.997058i	$0.524421\pi$
$978$	−10.4300	−0.333516
$979$	−10.3958	−0.332252
$980$	5.95102	0.190099
$981$	10.7865	0.344387
$982$	24.3279	0.776334
$983$	−25.8138	−0.823332	−0.411666	−	0.911335i	$-0.635053\pi$
$983$	−25.8138	−0.823332	−0.411666	+	0.911335i	$0.635053\pi$
$984$	−0.135933	−0.00433337
$985$	30.4241	0.969392
$986$	32.9529	1.04943
$987$	−12.2936	−0.391309
$988$	−0.314171	−0.00999510
$989$	0	0
$990$	1.09645	0.0348475
$991$	18.1853	0.577674	0.288837	−	0.957378i	$-0.406731\pi$
$991$	18.1853	0.577674	0.288837	+	0.957378i	$0.406731\pi$
$992$	−1.59145	−0.0505287
$993$	−4.29179	−0.136196
$994$	10.8228	0.343280
$995$	−7.76241	−0.246085
$996$	−8.40858	−0.266436
$997$	4.31292	0.136592	0.0682958	−	0.997665i	$-0.478244\pi$
$997$	4.31292	0.136592	0.0682958	+	0.997665i	$0.478244\pi$
$998$	−15.8756	−0.502533
$999$	−5.22871	−0.165429

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3174.2.a.bc.1.4		5
3.2	odd	2		9522.2.a.bt.1.2		5
23.9	even	11		138.2.e.a.127.1	yes	10
23.18	even	11		138.2.e.a.25.1	✓	10
23.22	odd	2		3174.2.a.bd.1.2		5
69.32	odd	22		414.2.i.d.127.1		10
69.41	odd	22		414.2.i.d.163.1		10
69.68	even	2		9522.2.a.bq.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.2.e.a.25.1	✓	10	23.18	even	11
138.2.e.a.127.1	yes	10	23.9	even	11
414.2.i.d.127.1		10	69.32	odd	22
414.2.i.d.163.1		10	69.41	odd	22
3174.2.a.bc.1.4		5	1.1	even	1	trivial
3174.2.a.bd.1.2		5	23.22	odd	2
9522.2.a.bq.1.4		5	69.68	even	2
9522.2.a.bt.1.2		5	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:	\( N \)	\(=\)	\( 3174 = 2 \cdot 3 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3174.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.23648 q^{5} +1.00000 q^{6} -1.47889 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.23648 q^{5} +1.00000 q^{6} -1.47889 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.23648 q^{10} -0.886752 q^{11} +1.00000 q^{12} +0.0673089 q^{13} -1.47889 q^{14} -1.23648 q^{15} +1.00000 q^{16} -4.29177 q^{17} +1.00000 q^{18} -4.66759 q^{19} -1.23648 q^{20} -1.47889 q^{21} -0.886752 q^{22} +1.00000 q^{24} -3.47112 q^{25} +0.0673089 q^{26} +1.00000 q^{27} -1.47889 q^{28} -7.67816 q^{29} -1.23648 q^{30} -1.59145 q^{31} +1.00000 q^{32} -0.886752 q^{33} -4.29177 q^{34} +1.82862 q^{35} +1.00000 q^{36} -5.22871 q^{37} -4.66759 q^{38} +0.0673089 q^{39} -1.23648 q^{40} -0.135933 q^{41} -1.47889 q^{42} +6.57798 q^{43} -0.886752 q^{44} -1.23648 q^{45} +8.31271 q^{47} +1.00000 q^{48} -4.81288 q^{49} -3.47112 q^{50} -4.29177 q^{51} +0.0673089 q^{52} -2.99558 q^{53} +1.00000 q^{54} +1.09645 q^{55} -1.47889 q^{56} -4.66759 q^{57} -7.67816 q^{58} -6.83576 q^{59} -1.23648 q^{60} -4.60015 q^{61} -1.59145 q^{62} -1.47889 q^{63} +1.00000 q^{64} -0.0832260 q^{65} -0.886752 q^{66} +12.4051 q^{67} -4.29177 q^{68} +1.82862 q^{70} -7.31821 q^{71} +1.00000 q^{72} +13.1978 q^{73} -5.22871 q^{74} -3.47112 q^{75} -4.66759 q^{76} +1.31141 q^{77} +0.0673089 q^{78} +7.89446 q^{79} -1.23648 q^{80} +1.00000 q^{81} -0.135933 q^{82} -8.40858 q^{83} -1.47889 q^{84} +5.30668 q^{85} +6.57798 q^{86} -7.67816 q^{87} -0.886752 q^{88} +11.7235 q^{89} -1.23648 q^{90} -0.0995426 q^{91} -1.59145 q^{93} +8.31271 q^{94} +5.77138 q^{95} +1.00000 q^{96} -17.1665 q^{97} -4.81288 q^{98} -0.886752 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{3} + 5 q^{4} - 7 q^{5} + 5 q^{6} - 7 q^{7} + 5 q^{8} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 - 7 * q^5 + 5 * q^6 - 7 * q^7 + 5 * q^8 + 5 * q^9	\( 5 q + 5 q^{2} + 5 q^{3} + 5 q^{4} - 7 q^{5} + 5 q^{6} - 7 q^{7} + 5 q^{8} + 5 q^{9} - 7 q^{10} - 13 q^{11} + 5 q^{12} - 4 q^{13} - 7 q^{14} - 7 q^{15} + 5 q^{16} - 9 q^{17} + 5 q^{18} - 11 q^{19} - 7 q^{20} - 7 q^{21} - 13 q^{22} + 5 q^{24} - 2 q^{25} - 4 q^{26} + 5 q^{27} - 7 q^{28} - 7 q^{29} - 7 q^{30} - 8 q^{31} + 5 q^{32} - 13 q^{33} - 9 q^{34} + q^{35} + 5 q^{36} - 12 q^{37} - 11 q^{38} - 4 q^{39} - 7 q^{40} - 10 q^{41} - 7 q^{42} - 4 q^{43} - 13 q^{44} - 7 q^{45} - 24 q^{47} + 5 q^{48} - 12 q^{49} - 2 q^{50} - 9 q^{51} - 4 q^{52} - 9 q^{53} + 5 q^{54} + 16 q^{55} - 7 q^{56} - 11 q^{57} - 7 q^{58} - 14 q^{59} - 7 q^{60} - 5 q^{61} - 8 q^{62} - 7 q^{63} + 5 q^{64} - 12 q^{65} - 13 q^{66} - 13 q^{67} - 9 q^{68} + q^{70} - 19 q^{71} + 5 q^{72} + 4 q^{73} - 12 q^{74} - 2 q^{75} - 11 q^{76} + 5 q^{77} - 4 q^{78} - 4 q^{79} - 7 q^{80} + 5 q^{81} - 10 q^{82} - 24 q^{83} - 7 q^{84} + 17 q^{85} - 4 q^{86} - 7 q^{87} - 13 q^{88} - 4 q^{89} - 7 q^{90} + 21 q^{91} - 8 q^{93} - 24 q^{94} - 11 q^{95} + 5 q^{96} + 9 q^{97} - 12 q^{98} - 13 q^{99}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^3 + 5 * q^4 - 7 * q^5 + 5 * q^6 - 7 * q^7 + 5 * q^8 + 5 * q^9 - 7 * q^10 - 13 * q^11 + 5 * q^12 - 4 * q^13 - 7 * q^14 - 7 * q^15 + 5 * q^16 - 9 * q^17 + 5 * q^18 - 11 * q^19 - 7 * q^20 - 7 * q^21 - 13 * q^22 + 5 * q^24 - 2 * q^25 - 4 * q^26 + 5 * q^27 - 7 * q^28 - 7 * q^29 - 7 * q^30 - 8 * q^31 + 5 * q^32 - 13 * q^33 - 9 * q^34 + q^35 + 5 * q^36 - 12 * q^37 - 11 * q^38 - 4 * q^39 - 7 * q^40 - 10 * q^41 - 7 * q^42 - 4 * q^43 - 13 * q^44 - 7 * q^45 - 24 * q^47 + 5 * q^48 - 12 * q^49 - 2 * q^50 - 9 * q^51 - 4 * q^52 - 9 * q^53 + 5 * q^54 + 16 * q^55 - 7 * q^56 - 11 * q^57 - 7 * q^58 - 14 * q^59 - 7 * q^60 - 5 * q^61 - 8 * q^62 - 7 * q^63 + 5 * q^64 - 12 * q^65 - 13 * q^66 - 13 * q^67 - 9 * q^68 + q^70 - 19 * q^71 + 5 * q^72 + 4 * q^73 - 12 * q^74 - 2 * q^75 - 11 * q^76 + 5 * q^77 - 4 * q^78 - 4 * q^79 - 7 * q^80 + 5 * q^81 - 10 * q^82 - 24 * q^83 - 7 * q^84 + 17 * q^85 - 4 * q^86 - 7 * q^87 - 13 * q^88 - 4 * q^89 - 7 * q^90 + 21 * q^91 - 8 * q^93 - 24 * q^94 - 11 * q^95 + 5 * q^96 + 9 * q^97 - 12 * q^98 - 13 * q^99

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