LMFDB - Embedded newform 845.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(1,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, 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("845.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	845.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:	$6.74735897080$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	845.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30278 q^{2} +1.00000 q^{3} +3.30278 q^{4} -1.00000 q^{5} -2.30278 q^{6} -1.00000 q^{7} -3.00000 q^{8} -2.00000 q^{9} +O(q^{10})$	\(q-2.30278 q^{2} +1.00000 q^{3} +3.30278 q^{4} -1.00000 q^{5} -2.30278 q^{6} -1.00000 q^{7} -3.00000 q^{8} -2.00000 q^{9} +2.30278 q^{10} +1.60555 q^{11} +3.30278 q^{12} +2.30278 q^{14} -1.00000 q^{15} +0.302776 q^{16} +7.60555 q^{17} +4.60555 q^{18} -5.60555 q^{19} -3.30278 q^{20} -1.00000 q^{21} -3.69722 q^{22} -3.00000 q^{23} -3.00000 q^{24} +1.00000 q^{25} -5.00000 q^{27} -3.30278 q^{28} -6.21110 q^{29} +2.30278 q^{30} -4.00000 q^{31} +5.30278 q^{32} +1.60555 q^{33} -17.5139 q^{34} +1.00000 q^{35} -6.60555 q^{36} +3.60555 q^{37} +12.9083 q^{38} +3.00000 q^{40} +3.00000 q^{41} +2.30278 q^{42} -10.2111 q^{43} +5.30278 q^{44} +2.00000 q^{45} +6.90833 q^{46} +9.21110 q^{47} +0.302776 q^{48} -6.00000 q^{49} -2.30278 q^{50} +7.60555 q^{51} -3.21110 q^{53} +11.5139 q^{54} -1.60555 q^{55} +3.00000 q^{56} -5.60555 q^{57} +14.3028 q^{58} -10.8167 q^{59} -3.30278 q^{60} -1.00000 q^{61} +9.21110 q^{62} +2.00000 q^{63} -12.8167 q^{64} -3.69722 q^{66} -7.00000 q^{67} +25.1194 q^{68} -3.00000 q^{69} -2.30278 q^{70} +4.81665 q^{71} +6.00000 q^{72} -0.788897 q^{73} -8.30278 q^{74} +1.00000 q^{75} -18.5139 q^{76} -1.60555 q^{77} +5.21110 q^{79} -0.302776 q^{80} +1.00000 q^{81} -6.90833 q^{82} -9.21110 q^{83} -3.30278 q^{84} -7.60555 q^{85} +23.5139 q^{86} -6.21110 q^{87} -4.81665 q^{88} -6.21110 q^{89} -4.60555 q^{90} -9.90833 q^{92} -4.00000 q^{93} -21.2111 q^{94} +5.60555 q^{95} +5.30278 q^{96} -8.39445 q^{97} +13.8167 q^{98} -3.21110 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 2 q^{3} + 3 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9}+O(q^{10}) $ 2 * q - q^2 + 2 * q^3 + 3 * q^4 - 2 * q^5 - q^6 - 2 * q^7 - 6 * q^8 - 4 * q^9	$ 2 q - q^{2} + 2 q^{3} + 3 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9} + q^{10} - 4 q^{11} + 3 q^{12} + q^{14} - 2 q^{15} - 3 q^{16} + 8 q^{17} + 2 q^{18} - 4 q^{19} - 3 q^{20} - 2 q^{21} - 11 q^{22} - 6 q^{23} - 6 q^{24} + 2 q^{25} - 10 q^{27} - 3 q^{28} + 2 q^{29} + q^{30} - 8 q^{31} + 7 q^{32} - 4 q^{33} - 17 q^{34} + 2 q^{35} - 6 q^{36} + 15 q^{38} + 6 q^{40} + 6 q^{41} + q^{42} - 6 q^{43} + 7 q^{44} + 4 q^{45} + 3 q^{46} + 4 q^{47} - 3 q^{48} - 12 q^{49} - q^{50} + 8 q^{51} + 8 q^{53} + 5 q^{54} + 4 q^{55} + 6 q^{56} - 4 q^{57} + 25 q^{58} - 3 q^{60} - 2 q^{61} + 4 q^{62} + 4 q^{63} - 4 q^{64} - 11 q^{66} - 14 q^{67} + 25 q^{68} - 6 q^{69} - q^{70} - 12 q^{71} + 12 q^{72} - 16 q^{73} - 13 q^{74} + 2 q^{75} - 19 q^{76} + 4 q^{77} - 4 q^{79} + 3 q^{80} + 2 q^{81} - 3 q^{82} - 4 q^{83} - 3 q^{84} - 8 q^{85} + 29 q^{86} + 2 q^{87} + 12 q^{88} + 2 q^{89} - 2 q^{90} - 9 q^{92} - 8 q^{93} - 28 q^{94} + 4 q^{95} + 7 q^{96} - 24 q^{97} + 6 q^{98} + 8 q^{99}+O(q^{100}) $ 2 * q - q^2 + 2 * q^3 + 3 * q^4 - 2 * q^5 - q^6 - 2 * q^7 - 6 * q^8 - 4 * q^9 + q^10 - 4 * q^11 + 3 * q^12 + q^14 - 2 * q^15 - 3 * q^16 + 8 * q^17 + 2 * q^18 - 4 * q^19 - 3 * q^20 - 2 * q^21 - 11 * q^22 - 6 * q^23 - 6 * q^24 + 2 * q^25 - 10 * q^27 - 3 * q^28 + 2 * q^29 + q^30 - 8 * q^31 + 7 * q^32 - 4 * q^33 - 17 * q^34 + 2 * q^35 - 6 * q^36 + 15 * q^38 + 6 * q^40 + 6 * q^41 + q^42 - 6 * q^43 + 7 * q^44 + 4 * q^45 + 3 * q^46 + 4 * q^47 - 3 * q^48 - 12 * q^49 - q^50 + 8 * q^51 + 8 * q^53 + 5 * q^54 + 4 * q^55 + 6 * q^56 - 4 * q^57 + 25 * q^58 - 3 * q^60 - 2 * q^61 + 4 * q^62 + 4 * q^63 - 4 * q^64 - 11 * q^66 - 14 * q^67 + 25 * q^68 - 6 * q^69 - q^70 - 12 * q^71 + 12 * q^72 - 16 * q^73 - 13 * q^74 + 2 * q^75 - 19 * q^76 + 4 * q^77 - 4 * q^79 + 3 * q^80 + 2 * q^81 - 3 * q^82 - 4 * q^83 - 3 * q^84 - 8 * q^85 + 29 * q^86 + 2 * q^87 + 12 * q^88 + 2 * q^89 - 2 * q^90 - 9 * q^92 - 8 * q^93 - 28 * q^94 + 4 * q^95 + 7 * q^96 - 24 * q^97 + 6 * q^98 + 8 * 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 $

$2$	−2.30278	−1.62831	−0.814154	−	0.580649i	$-0.802799\pi$
$2$	−2.30278	−1.62831	−0.814154	+	0.580649i	$0.802799\pi$
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	3.30278	1.65139
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.30278	−0.940104
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−3.00000	−1.06066
$9$	−2.00000	−0.666667
$10$	2.30278	0.728202
$11$	1.60555	0.484092	0.242046	−	0.970265i	$-0.422182\pi$
$11$	1.60555	0.484092	0.242046	+	0.970265i	$0.422182\pi$
$12$	3.30278	0.953429
$13$	0	0
$13$	0	0
$14$	2.30278	0.615443
$15$	−1.00000	−0.258199
$16$	0.302776	0.0756939
$17$	7.60555	1.84462	0.922309	−	0.386454i	$-0.126300\pi$
$17$	7.60555	1.84462	0.922309	+	0.386454i	$0.126300\pi$
$18$	4.60555	1.08554
$19$	−5.60555	−1.28600	−0.643001	−	0.765865i	$-0.722311\pi$
$19$	−5.60555	−1.28600	−0.643001	+	0.765865i	$0.722311\pi$
$20$	−3.30278	−0.738523
$21$	−1.00000	−0.218218
$22$	−3.69722	−0.788251
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	−3.00000	−0.612372
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	−3.30278	−0.624166
$29$	−6.21110	−1.15337	−0.576686	−	0.816966i	$-0.695655\pi$
$29$	−6.21110	−1.15337	−0.576686	+	0.816966i	$0.695655\pi$
$30$	2.30278	0.420427
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	5.30278	0.937407
$33$	1.60555	0.279491
$34$	−17.5139	−3.00361
$35$	1.00000	0.169031
$36$	−6.60555	−1.10093
$37$	3.60555	0.592749	0.296374	−	0.955072i	$-0.404222\pi$
$37$	3.60555	0.592749	0.296374	+	0.955072i	$0.404222\pi$
$38$	12.9083	2.09401
$39$	0	0
$40$	3.00000	0.474342
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	2.30278	0.355326
$43$	−10.2111	−1.55718	−0.778589	−	0.627534i	$-0.784064\pi$
$43$	−10.2111	−1.55718	−0.778589	+	0.627534i	$0.784064\pi$
$44$	5.30278	0.799424
$45$	2.00000	0.298142
$46$	6.90833	1.01858
$47$	9.21110	1.34358	0.671789	−	0.740743i	$-0.265526\pi$
$47$	9.21110	1.34358	0.671789	+	0.740743i	$0.265526\pi$
$48$	0.302776	0.0437019
$49$	−6.00000	−0.857143
$50$	−2.30278	−0.325662
$51$	7.60555	1.06499
$52$	0	0
$53$	−3.21110	−0.441079	−0.220539	−	0.975378i	$-0.570782\pi$
$53$	−3.21110	−0.441079	−0.220539	+	0.975378i	$0.570782\pi$
$54$	11.5139	1.56684
$55$	−1.60555	−0.216492
$56$	3.00000	0.400892
$57$	−5.60555	−0.742473
$58$	14.3028	1.87805
$59$	−10.8167	−1.40821	−0.704104	−	0.710097i	$-0.748651\pi$
$59$	−10.8167	−1.40821	−0.704104	+	0.710097i	$0.748651\pi$
$60$	−3.30278	−0.426387
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	9.21110	1.16981
$63$	2.00000	0.251976
$64$	−12.8167	−1.60208
$65$	0	0
$66$	−3.69722	−0.455097
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	25.1194	3.04618
$69$	−3.00000	−0.361158
$70$	−2.30278	−0.275234
$71$	4.81665	0.571632	0.285816	−	0.958285i	$-0.407735\pi$
$71$	4.81665	0.571632	0.285816	+	0.958285i	$0.407735\pi$
$72$	6.00000	0.707107
$73$	−0.788897	−0.0923335	−0.0461667	−	0.998934i	$-0.514701\pi$
$73$	−0.788897	−0.0923335	−0.0461667	+	0.998934i	$0.514701\pi$
$74$	−8.30278	−0.965178
$75$	1.00000	0.115470
$76$	−18.5139	−2.12369
$77$	−1.60555	−0.182970
$78$	0	0
$79$	5.21110	0.586295	0.293147	−	0.956067i	$-0.405297\pi$
$79$	5.21110	0.586295	0.293147	+	0.956067i	$0.405297\pi$
$80$	−0.302776	−0.0338513
$81$	1.00000	0.111111
$82$	−6.90833	−0.762897
$83$	−9.21110	−1.01105	−0.505525	−	0.862812i	$-0.668701\pi$
$83$	−9.21110	−1.01105	−0.505525	+	0.862812i	$0.668701\pi$
$84$	−3.30278	−0.360362
$85$	−7.60555	−0.824938
$86$	23.5139	2.53557
$87$	−6.21110	−0.665900
$88$	−4.81665	−0.513457
$89$	−6.21110	−0.658376	−0.329188	−	0.944264i	$-0.606775\pi$
$89$	−6.21110	−0.658376	−0.329188	+	0.944264i	$0.606775\pi$
$90$	−4.60555	−0.485468
$91$	0	0
$92$	−9.90833	−1.03301
$93$	−4.00000	−0.414781
$94$	−21.2111	−2.18776
$95$	5.60555	0.575117
$96$	5.30278	0.541212
$97$	−8.39445	−0.852327	−0.426164	−	0.904646i	$-0.640135\pi$
$97$	−8.39445	−0.852327	−0.426164	+	0.904646i	$0.640135\pi$
$98$	13.8167	1.39569
$99$	−3.21110	−0.322728
$100$	3.30278	0.330278
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	−17.5139	−1.73413
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	7.39445	0.718212
$107$	6.21110	0.600450	0.300225	−	0.953868i	$-0.402938\pi$
$107$	6.21110	0.600450	0.300225	+	0.953868i	$0.402938\pi$
$108$	−16.5139	−1.58905
$109$	−19.2111	−1.84009	−0.920045	−	0.391813i	$-0.871848\pi$
$109$	−19.2111	−1.84009	−0.920045	+	0.391813i	$0.871848\pi$
$110$	3.69722	0.352517
$111$	3.60555	0.342224
$112$	−0.302776	−0.0286096
$113$	−1.60555	−0.151038	−0.0755188	−	0.997144i	$-0.524061\pi$
$113$	−1.60555	−0.151038	−0.0755188	+	0.997144i	$0.524061\pi$
$114$	12.9083	1.20898
$115$	3.00000	0.279751
$116$	−20.5139	−1.90467
$117$	0	0
$118$	24.9083	2.29300
$119$	−7.60555	−0.697200
$120$	3.00000	0.273861
$121$	−8.42221	−0.765655
$122$	2.30278	0.208484
$123$	3.00000	0.270501
$124$	−13.2111	−1.18639
$125$	−1.00000	−0.0894427
$126$	−4.60555	−0.410295
$127$	−4.21110	−0.373675	−0.186837	−	0.982391i	$-0.559824\pi$
$127$	−4.21110	−0.373675	−0.186837	+	0.982391i	$0.559824\pi$
$128$	18.9083	1.67128
$129$	−10.2111	−0.899037
$130$	0	0
$131$	−21.2111	−1.85322	−0.926611	−	0.376021i	$-0.877292\pi$
$131$	−21.2111	−1.85322	−0.926611	+	0.376021i	$0.877292\pi$
$132$	5.30278	0.461547
$133$	5.60555	0.486063
$134$	16.1194	1.39251
$135$	5.00000	0.430331
$136$	−22.8167	−1.95651
$137$	−1.60555	−0.137172	−0.0685858	−	0.997645i	$-0.521849\pi$
$137$	−1.60555	−0.137172	−0.0685858	+	0.997645i	$0.521849\pi$
$138$	6.90833	0.588076
$139$	6.39445	0.542370	0.271185	−	0.962527i	$-0.412584\pi$
$139$	6.39445	0.542370	0.271185	+	0.962527i	$0.412584\pi$
$140$	3.30278	0.279135
$141$	9.21110	0.775715
$142$	−11.0917	−0.930793
$143$	0	0
$144$	−0.605551	−0.0504626
$145$	6.21110	0.515804
$146$	1.81665	0.150347
$147$	−6.00000	−0.494872
$148$	11.9083	0.978858
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	−2.30278	−0.188021
$151$	−1.21110	−0.0985581	−0.0492791	−	0.998785i	$-0.515692\pi$
$151$	−1.21110	−0.0985581	−0.0492791	+	0.998785i	$0.515692\pi$
$152$	16.8167	1.36401
$153$	−15.2111	−1.22974
$154$	3.69722	0.297931
$155$	4.00000	0.321288
$156$	0	0
$157$	11.2111	0.894743	0.447372	−	0.894348i	$-0.352360\pi$
$157$	11.2111	0.894743	0.447372	+	0.894348i	$0.352360\pi$
$158$	−12.0000	−0.954669
$159$	−3.21110	−0.254657
$160$	−5.30278	−0.419221
$161$	3.00000	0.236433
$162$	−2.30278	−0.180923
$163$	−3.78890	−0.296769	−0.148385	−	0.988930i	$-0.547407\pi$
$163$	−3.78890	−0.296769	−0.148385	+	0.988930i	$0.547407\pi$
$164$	9.90833	0.773710
$165$	−1.60555	−0.124992
$166$	21.2111	1.64630
$167$	9.00000	0.696441	0.348220	−	0.937413i	$-0.386786\pi$
$167$	9.00000	0.696441	0.348220	+	0.937413i	$0.386786\pi$
$168$	3.00000	0.231455
$169$	0	0
$170$	17.5139	1.34325
$171$	11.2111	0.857334
$172$	−33.7250	−2.57151
$173$	4.81665	0.366203	0.183102	−	0.983094i	$-0.441386\pi$
$173$	4.81665	0.366203	0.183102	+	0.983094i	$0.441386\pi$
$174$	14.3028	1.08429
$175$	−1.00000	−0.0755929
$176$	0.486122	0.0366428
$177$	−10.8167	−0.813029
$178$	14.3028	1.07204
$179$	22.8167	1.70540	0.852698	−	0.522404i	$-0.174965\pi$
$179$	22.8167	1.70540	0.852698	+	0.522404i	$0.174965\pi$
$180$	6.60555	0.492349
$181$	17.6333	1.31067	0.655337	−	0.755337i	$-0.272527\pi$
$181$	17.6333	1.31067	0.655337	+	0.755337i	$0.272527\pi$
$182$	0	0
$183$	−1.00000	−0.0739221
$184$	9.00000	0.663489
$185$	−3.60555	−0.265085
$186$	9.21110	0.675391
$187$	12.2111	0.892964
$188$	30.4222	2.21877
$189$	5.00000	0.363696
$190$	−12.9083	−0.936468
$191$	16.8167	1.21681	0.608405	−	0.793627i	$-0.291810\pi$
$191$	16.8167	1.21681	0.608405	+	0.793627i	$0.291810\pi$
$192$	−12.8167	−0.924962
$193$	15.6056	1.12331	0.561656	−	0.827371i	$-0.310164\pi$
$193$	15.6056	1.12331	0.561656	+	0.827371i	$0.310164\pi$
$194$	19.3305	1.38785
$195$	0	0
$196$	−19.8167	−1.41548
$197$	1.18335	0.0843099	0.0421550	−	0.999111i	$-0.486578\pi$
$197$	1.18335	0.0843099	0.0421550	+	0.999111i	$0.486578\pi$
$198$	7.39445	0.525501
$199$	12.8167	0.908549	0.454274	−	0.890862i	$-0.349899\pi$
$199$	12.8167	0.908549	0.454274	+	0.890862i	$0.349899\pi$
$200$	−3.00000	−0.212132
$201$	−7.00000	−0.493742
$202$	20.7250	1.45820
$203$	6.21110	0.435934
$204$	25.1194	1.75871
$205$	−3.00000	−0.209529
$206$	9.21110	0.641768
$207$	6.00000	0.417029
$208$	0	0
$209$	−9.00000	−0.622543
$210$	−2.30278	−0.158907
$211$	−23.6056	−1.62507	−0.812537	−	0.582910i	$-0.801914\pi$
$211$	−23.6056	−1.62507	−0.812537	+	0.582910i	$0.801914\pi$
$212$	−10.6056	−0.728392
$213$	4.81665	0.330032
$214$	−14.3028	−0.977718
$215$	10.2111	0.696391
$216$	15.0000	1.02062
$217$	4.00000	0.271538
$218$	44.2389	2.99623
$219$	−0.788897	−0.0533087
$220$	−5.30278	−0.357513
$221$	0	0
$222$	−8.30278	−0.557246
$223$	−4.21110	−0.281996	−0.140998	−	0.990010i	$-0.545031\pi$
$223$	−4.21110	−0.281996	−0.140998	+	0.990010i	$0.545031\pi$
$224$	−5.30278	−0.354307
$225$	−2.00000	−0.133333
$226$	3.69722	0.245936
$227$	−27.4222	−1.82008	−0.910038	−	0.414525i	$-0.863948\pi$
$227$	−27.4222	−1.82008	−0.910038	+	0.414525i	$0.863948\pi$
$228$	−18.5139	−1.22611
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	−6.90833	−0.455522
$231$	−1.60555	−0.105638
$232$	18.6333	1.22334
$233$	15.2111	0.996512	0.498256	−	0.867030i	$-0.333974\pi$
$233$	15.2111	0.996512	0.498256	+	0.867030i	$0.333974\pi$
$234$	0	0
$235$	−9.21110	−0.600866
$236$	−35.7250	−2.32550
$237$	5.21110	0.338497
$238$	17.5139	1.13526
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	−0.302776	−0.0195441
$241$	1.78890	0.115233	0.0576165	−	0.998339i	$-0.481650\pi$
$241$	1.78890	0.115233	0.0576165	+	0.998339i	$0.481650\pi$
$242$	19.3944	1.24672
$243$	16.0000	1.02640
$244$	−3.30278	−0.211439
$245$	6.00000	0.383326
$246$	−6.90833	−0.440459
$247$	0	0
$248$	12.0000	0.762001
$249$	−9.21110	−0.583730
$250$	2.30278	0.145640
$251$	−7.18335	−0.453409	−0.226704	−	0.973964i	$-0.572795\pi$
$251$	−7.18335	−0.453409	−0.226704	+	0.973964i	$0.572795\pi$
$252$	6.60555	0.416111
$253$	−4.81665	−0.302820
$254$	9.69722	0.608458
$255$	−7.60555	−0.476278
$256$	−17.9083	−1.11927
$257$	−16.3944	−1.02266	−0.511329	−	0.859385i	$-0.670847\pi$
$257$	−16.3944	−1.02266	−0.511329	+	0.859385i	$0.670847\pi$
$258$	23.5139	1.46391
$259$	−3.60555	−0.224038
$260$	0	0
$261$	12.4222	0.768915
$262$	48.8444	3.01762
$263$	11.7889	0.726935	0.363467	−	0.931607i	$-0.381593\pi$
$263$	11.7889	0.726935	0.363467	+	0.931607i	$0.381593\pi$
$264$	−4.81665	−0.296445
$265$	3.21110	0.197256
$266$	−12.9083	−0.791460
$267$	−6.21110	−0.380113
$268$	−23.1194	−1.41224
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	−11.5139	−0.700712
$271$	−20.8167	−1.26452	−0.632261	−	0.774756i	$-0.717873\pi$
$271$	−20.8167	−1.26452	−0.632261	+	0.774756i	$0.717873\pi$
$272$	2.30278	0.139626
$273$	0	0
$274$	3.69722	0.223357
$275$	1.60555	0.0968184
$276$	−9.90833	−0.596411
$277$	27.6056	1.65866	0.829328	−	0.558761i	$-0.188723\pi$
$277$	27.6056	1.65866	0.829328	+	0.558761i	$0.188723\pi$
$278$	−14.7250	−0.883146
$279$	8.00000	0.478947
$280$	−3.00000	−0.179284
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	−21.2111	−1.26310
$283$	5.00000	0.297219	0.148610	−	0.988896i	$-0.452520\pi$
$283$	5.00000	0.297219	0.148610	+	0.988896i	$0.452520\pi$
$284$	15.9083	0.943986
$285$	5.60555	0.332044
$286$	0	0
$287$	−3.00000	−0.177084
$288$	−10.6056	−0.624938
$289$	40.8444	2.40261
$290$	−14.3028	−0.839888
$291$	−8.39445	−0.492091
$292$	−2.60555	−0.152478
$293$	10.3944	0.607250	0.303625	−	0.952792i	$-0.401803\pi$
$293$	10.3944	0.607250	0.303625	+	0.952792i	$0.401803\pi$
$294$	13.8167	0.805804
$295$	10.8167	0.629770
$296$	−10.8167	−0.628705
$297$	−8.02776	−0.465818
$298$	−6.90833	−0.400189
$299$	0	0
$300$	3.30278	0.190686
$301$	10.2111	0.588558
$302$	2.78890	0.160483
$303$	−9.00000	−0.517036
$304$	−1.69722	−0.0973425
$305$	1.00000	0.0572598
$306$	35.0278	2.00240
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	−5.30278	−0.302154
$309$	−4.00000	−0.227552
$310$	−9.21110	−0.523155
$311$	−9.21110	−0.522314	−0.261157	−	0.965296i	$-0.584104\pi$
$311$	−9.21110	−0.522314	−0.261157	+	0.965296i	$0.584104\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	−25.8167	−1.45692
$315$	−2.00000	−0.112687
$316$	17.2111	0.968200
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	7.39445	0.414660
$319$	−9.97224	−0.558338
$320$	12.8167	0.716473
$321$	6.21110	0.346670
$322$	−6.90833	−0.384986
$323$	−42.6333	−2.37218
$324$	3.30278	0.183488
$325$	0	0
$326$	8.72498	0.483232
$327$	−19.2111	−1.06238
$328$	−9.00000	−0.496942
$329$	−9.21110	−0.507825
$330$	3.69722	0.203526
$331$	10.0278	0.551175	0.275588	−	0.961276i	$-0.411128\pi$
$331$	10.0278	0.551175	0.275588	+	0.961276i	$0.411128\pi$
$332$	−30.4222	−1.66964
$333$	−7.21110	−0.395166
$334$	−20.7250	−1.13402
$335$	7.00000	0.382451
$336$	−0.302776	−0.0165178
$337$	−25.6333	−1.39634	−0.698168	−	0.715934i	$-0.746001\pi$
$337$	−25.6333	−1.39634	−0.698168	+	0.715934i	$0.746001\pi$
$338$	0	0
$339$	−1.60555	−0.0872016
$340$	−25.1194	−1.36229
$341$	−6.42221	−0.347782
$342$	−25.8167	−1.39600
$343$	13.0000	0.701934
$344$	30.6333	1.65164
$345$	3.00000	0.161515
$346$	−11.0917	−0.596292
$347$	5.78890	0.310764	0.155382	−	0.987854i	$-0.450339\pi$
$347$	5.78890	0.310764	0.155382	+	0.987854i	$0.450339\pi$
$348$	−20.5139	−1.09966
$349$	−3.78890	−0.202815	−0.101408	−	0.994845i	$-0.532335\pi$
$349$	−3.78890	−0.202815	−0.101408	+	0.994845i	$0.532335\pi$
$350$	2.30278	0.123089
$351$	0	0
$352$	8.51388	0.453791
$353$	16.8167	0.895060	0.447530	−	0.894269i	$-0.352304\pi$
$353$	16.8167	0.895060	0.447530	+	0.894269i	$0.352304\pi$
$354$	24.9083	1.32386
$355$	−4.81665	−0.255641
$356$	−20.5139	−1.08723
$357$	−7.60555	−0.402528
$358$	−52.5416	−2.77691
$359$	18.4222	0.972287	0.486143	−	0.873879i	$-0.338403\pi$
$359$	18.4222	0.972287	0.486143	+	0.873879i	$0.338403\pi$
$360$	−6.00000	−0.316228
$361$	12.4222	0.653800
$362$	−40.6056	−2.13418
$363$	−8.42221	−0.442051
$364$	0	0
$365$	0.788897	0.0412928
$366$	2.30278	0.120368
$367$	11.4222	0.596234	0.298117	−	0.954529i	$-0.403641\pi$
$367$	11.4222	0.596234	0.298117	+	0.954529i	$0.403641\pi$
$368$	−0.908327	−0.0473498
$369$	−6.00000	−0.312348
$370$	8.30278	0.431641
$371$	3.21110	0.166712
$372$	−13.2111	−0.684964
$373$	−20.3944	−1.05598	−0.527992	−	0.849249i	$-0.677055\pi$
$373$	−20.3944	−1.05598	−0.527992	+	0.849249i	$0.677055\pi$
$374$	−28.1194	−1.45402
$375$	−1.00000	−0.0516398
$376$	−27.6333	−1.42508
$377$	0	0
$378$	−11.5139	−0.592210
$379$	9.60555	0.493404	0.246702	−	0.969091i	$-0.420653\pi$
$379$	9.60555	0.493404	0.246702	+	0.969091i	$0.420653\pi$
$380$	18.5139	0.949742
$381$	−4.21110	−0.215741
$382$	−38.7250	−1.98134
$383$	−24.6333	−1.25870	−0.629352	−	0.777121i	$-0.716679\pi$
$383$	−24.6333	−1.25870	−0.629352	+	0.777121i	$0.716679\pi$
$384$	18.9083	0.964912
$385$	1.60555	0.0818265
$386$	−35.9361	−1.82910
$387$	20.4222	1.03812
$388$	−27.7250	−1.40752
$389$	−15.2111	−0.771234	−0.385617	−	0.922659i	$-0.626011\pi$
$389$	−15.2111	−0.771234	−0.385617	+	0.922659i	$0.626011\pi$
$390$	0	0
$391$	−22.8167	−1.15389
$392$	18.0000	0.909137
$393$	−21.2111	−1.06996
$394$	−2.72498	−0.137283
$395$	−5.21110	−0.262199
$396$	−10.6056	−0.532949
$397$	22.0278	1.10554	0.552771	−	0.833333i	$-0.313571\pi$
$397$	22.0278	1.10554	0.552771	+	0.833333i	$0.313571\pi$
$398$	−29.5139	−1.47940
$399$	5.60555	0.280629
$400$	0.302776	0.0151388
$401$	12.2111	0.609793	0.304897	−	0.952385i	$-0.401378\pi$
$401$	12.2111	0.609793	0.304897	+	0.952385i	$0.401378\pi$
$402$	16.1194	0.803964
$403$	0	0
$404$	−29.7250	−1.47887
$405$	−1.00000	−0.0496904
$406$	−14.3028	−0.709835
$407$	5.78890	0.286945
$408$	−22.8167	−1.12959
$409$	8.21110	0.406013	0.203006	−	0.979177i	$-0.434929\pi$
$409$	8.21110	0.406013	0.203006	+	0.979177i	$0.434929\pi$
$410$	6.90833	0.341178
$411$	−1.60555	−0.0791960
$412$	−13.2111	−0.650864
$413$	10.8167	0.532253
$414$	−13.8167	−0.679051
$415$	9.21110	0.452155
$416$	0	0
$417$	6.39445	0.313138
$418$	20.7250	1.01369
$419$	17.2389	0.842173	0.421087	−	0.907020i	$-0.361649\pi$
$419$	17.2389	0.842173	0.421087	+	0.907020i	$0.361649\pi$
$420$	3.30278	0.161159
$421$	32.4222	1.58016	0.790081	−	0.613003i	$-0.210039\pi$
$421$	32.4222	1.58016	0.790081	+	0.613003i	$0.210039\pi$
$422$	54.3583	2.64612
$423$	−18.4222	−0.895718
$424$	9.63331	0.467835
$425$	7.60555	0.368923
$426$	−11.0917	−0.537393
$427$	1.00000	0.0483934
$428$	20.5139	0.991576
$429$	0	0
$430$	−23.5139	−1.13394
$431$	29.2389	1.40839	0.704193	−	0.710008i	$-0.251309\pi$
$431$	29.2389	1.40839	0.704193	+	0.710008i	$0.251309\pi$
$432$	−1.51388	−0.0728365
$433$	3.60555	0.173272	0.0866359	−	0.996240i	$-0.472388\pi$
$433$	3.60555	0.173272	0.0866359	+	0.996240i	$0.472388\pi$
$434$	−9.21110	−0.442147
$435$	6.21110	0.297800
$436$	−63.4500	−3.03870
$437$	16.8167	0.804450
$438$	1.81665	0.0868031
$439$	−27.2389	−1.30004	−0.650020	−	0.759917i	$-0.725239\pi$
$439$	−27.2389	−1.30004	−0.650020	+	0.759917i	$0.725239\pi$
$440$	4.81665	0.229625
$441$	12.0000	0.571429
$442$	0	0
$443$	6.42221	0.305128	0.152564	−	0.988294i	$-0.451247\pi$
$443$	6.42221	0.305128	0.152564	+	0.988294i	$0.451247\pi$
$444$	11.9083	0.565144
$445$	6.21110	0.294434
$446$	9.69722	0.459177
$447$	3.00000	0.141895
$448$	12.8167	0.605530
$449$	30.6333	1.44568	0.722838	−	0.691018i	$-0.242837\pi$
$449$	30.6333	1.44568	0.722838	+	0.691018i	$0.242837\pi$
$450$	4.60555	0.217108
$451$	4.81665	0.226807
$452$	−5.30278	−0.249422
$453$	−1.21110	−0.0569026
$454$	63.1472	2.96364
$455$	0	0
$456$	16.8167	0.787512
$457$	−26.8167	−1.25443	−0.627215	−	0.778846i	$-0.715805\pi$
$457$	−26.8167	−1.25443	−0.627215	+	0.778846i	$0.715805\pi$
$458$	−32.2389	−1.50642
$459$	−38.0278	−1.77498
$460$	9.90833	0.461978
$461$	36.2111	1.68652	0.843260	−	0.537507i	$-0.180634\pi$
$461$	36.2111	1.68652	0.843260	+	0.537507i	$0.180634\pi$
$462$	3.69722	0.172010
$463$	−34.4222	−1.59974	−0.799868	−	0.600176i	$-0.795097\pi$
$463$	−34.4222	−1.59974	−0.799868	+	0.600176i	$0.795097\pi$
$464$	−1.88057	−0.0873033
$465$	4.00000	0.185496
$466$	−35.0278	−1.62263
$467$	2.78890	0.129055	0.0645274	−	0.997916i	$-0.479446\pi$
$467$	2.78890	0.129055	0.0645274	+	0.997916i	$0.479446\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	21.2111	0.978395
$471$	11.2111	0.516580
$472$	32.4500	1.49363
$473$	−16.3944	−0.753818
$474$	−12.0000	−0.551178
$475$	−5.60555	−0.257200
$476$	−25.1194	−1.15135
$477$	6.42221	0.294053
$478$	0	0
$479$	−28.8167	−1.31667	−0.658333	−	0.752727i	$-0.728738\pi$
$479$	−28.8167	−1.31667	−0.658333	+	0.752727i	$0.728738\pi$
$480$	−5.30278	−0.242037
$481$	0	0
$482$	−4.11943	−0.187635
$483$	3.00000	0.136505
$484$	−27.8167	−1.26439
$485$	8.39445	0.381172
$486$	−36.8444	−1.67130
$487$	−1.00000	−0.0453143	−0.0226572	−	0.999743i	$-0.507213\pi$
$487$	−1.00000	−0.0453143	−0.0226572	+	0.999743i	$0.507213\pi$
$488$	3.00000	0.135804
$489$	−3.78890	−0.171340
$490$	−13.8167	−0.624173
$491$	−16.8167	−0.758925	−0.379462	−	0.925207i	$-0.623891\pi$
$491$	−16.8167	−0.758925	−0.379462	+	0.925207i	$0.623891\pi$
$492$	9.90833	0.446702
$493$	−47.2389	−2.12753
$494$	0	0
$495$	3.21110	0.144328
$496$	−1.21110	−0.0543801
$497$	−4.81665	−0.216056
$498$	21.2111	0.950492
$499$	2.42221	0.108433	0.0542164	−	0.998529i	$-0.482734\pi$
$499$	2.42221	0.108433	0.0542164	+	0.998529i	$0.482734\pi$
$500$	−3.30278	−0.147705
$501$	9.00000	0.402090
$502$	16.5416	0.738289
$503$	3.00000	0.133763	0.0668817	−	0.997761i	$-0.478695\pi$
$503$	3.00000	0.133763	0.0668817	+	0.997761i	$0.478695\pi$
$504$	−6.00000	−0.267261
$505$	9.00000	0.400495
$506$	11.0917	0.493085
$507$	0	0
$508$	−13.9083	−0.617082
$509$	3.00000	0.132973	0.0664863	−	0.997787i	$-0.478821\pi$
$509$	3.00000	0.132973	0.0664863	+	0.997787i	$0.478821\pi$
$510$	17.5139	0.775528
$511$	0.788897	0.0348988
$512$	3.42221	0.151242
$513$	28.0278	1.23746
$514$	37.7527	1.66520
$515$	4.00000	0.176261
$516$	−33.7250	−1.48466
$517$	14.7889	0.650415
$518$	8.30278	0.364803
$519$	4.81665	0.211428
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	−28.6056	−1.25203
$523$	−1.42221	−0.0621887	−0.0310943	−	0.999516i	$-0.509899\pi$
$523$	−1.42221	−0.0621887	−0.0310943	+	0.999516i	$0.509899\pi$
$524$	−70.0555	−3.06039
$525$	−1.00000	−0.0436436
$526$	−27.1472	−1.18367
$527$	−30.4222	−1.32521
$528$	0.486122	0.0211557
$529$	−14.0000	−0.608696
$530$	−7.39445	−0.321194
$531$	21.6333	0.938806
$532$	18.5139	0.802678
$533$	0	0
$534$	14.3028	0.618942
$535$	−6.21110	−0.268529
$536$	21.0000	0.907062
$537$	22.8167	0.984611
$538$	20.7250	0.893517
$539$	−9.63331	−0.414936
$540$	16.5139	0.710644
$541$	−25.6333	−1.10206	−0.551031	−	0.834485i	$-0.685765\pi$
$541$	−25.6333	−1.10206	−0.551031	+	0.834485i	$0.685765\pi$
$542$	47.9361	2.05903
$543$	17.6333	0.756718
$544$	40.3305	1.72916
$545$	19.2111	0.822913
$546$	0	0
$547$	32.8444	1.40433	0.702163	−	0.712016i	$-0.252218\pi$
$547$	32.8444	1.40433	0.702163	+	0.712016i	$0.252218\pi$
$548$	−5.30278	−0.226523
$549$	2.00000	0.0853579
$550$	−3.69722	−0.157650
$551$	34.8167	1.48324
$552$	9.00000	0.383065
$553$	−5.21110	−0.221599
$554$	−63.5694	−2.70080
$555$	−3.60555	−0.153047
$556$	21.1194	0.895663
$557$	−1.60555	−0.0680294	−0.0340147	−	0.999421i	$-0.510829\pi$
$557$	−1.60555	−0.0680294	−0.0340147	+	0.999421i	$0.510829\pi$
$558$	−18.4222	−0.779874
$559$	0	0
$560$	0.302776	0.0127946
$561$	12.2111	0.515553
$562$	13.8167	0.582820
$563$	9.42221	0.397099	0.198549	−	0.980091i	$-0.436377\pi$
$563$	9.42221	0.397099	0.198549	+	0.980091i	$0.436377\pi$
$564$	30.4222	1.28101
$565$	1.60555	0.0675460
$566$	−11.5139	−0.483964
$567$	−1.00000	−0.0419961
$568$	−14.4500	−0.606307
$569$	−27.4222	−1.14960	−0.574799	−	0.818294i	$-0.694920\pi$
$569$	−27.4222	−1.14960	−0.574799	+	0.818294i	$0.694920\pi$
$570$	−12.9083	−0.540670
$571$	20.8444	0.872311	0.436156	−	0.899871i	$-0.356340\pi$
$571$	20.8444	0.872311	0.436156	+	0.899871i	$0.356340\pi$
$572$	0	0
$573$	16.8167	0.702526
$574$	6.90833	0.288348
$575$	−3.00000	−0.125109
$576$	25.6333	1.06805
$577$	−13.6333	−0.567562	−0.283781	−	0.958889i	$-0.591589\pi$
$577$	−13.6333	−0.567562	−0.283781	+	0.958889i	$0.591589\pi$
$578$	−94.0555	−3.91219
$579$	15.6056	0.648545
$580$	20.5139	0.851792
$581$	9.21110	0.382141
$582$	19.3305	0.801276
$583$	−5.15559	−0.213523
$584$	2.36669	0.0979344
$585$	0	0
$586$	−23.9361	−0.988790
$587$	−33.4222	−1.37948	−0.689741	−	0.724056i	$-0.742276\pi$
$587$	−33.4222	−1.37948	−0.689741	+	0.724056i	$0.742276\pi$
$588$	−19.8167	−0.817225
$589$	22.4222	0.923891
$590$	−24.9083	−1.02546
$591$	1.18335	0.0486764
$592$	1.09167	0.0448675
$593$	20.7889	0.853698	0.426849	−	0.904323i	$-0.359624\pi$
$593$	20.7889	0.853698	0.426849	+	0.904323i	$0.359624\pi$
$594$	18.4861	0.758495
$595$	7.60555	0.311797
$596$	9.90833	0.405861
$597$	12.8167	0.524551
$598$	0	0
$599$	−21.2111	−0.866662	−0.433331	−	0.901235i	$-0.642662\pi$
$599$	−21.2111	−0.866662	−0.433331	+	0.901235i	$0.642662\pi$
$600$	−3.00000	−0.122474
$601$	13.7889	0.562461	0.281230	−	0.959640i	$-0.409257\pi$
$601$	13.7889	0.562461	0.281230	+	0.959640i	$0.409257\pi$
$602$	−23.5139	−0.958354
$603$	14.0000	0.570124
$604$	−4.00000	−0.162758
$605$	8.42221	0.342411
$606$	20.7250	0.841895
$607$	−34.2111	−1.38859	−0.694293	−	0.719693i	$-0.744283\pi$
$607$	−34.2111	−1.38859	−0.694293	+	0.719693i	$0.744283\pi$
$608$	−29.7250	−1.20551
$609$	6.21110	0.251687
$610$	−2.30278	−0.0932367
$611$	0	0
$612$	−50.2389	−2.03079
$613$	−5.60555	−0.226406	−0.113203	−	0.993572i	$-0.536111\pi$
$613$	−5.60555	−0.226406	−0.113203	+	0.993572i	$0.536111\pi$
$614$	36.8444	1.48692
$615$	−3.00000	−0.120972
$616$	4.81665	0.194069
$617$	−38.4500	−1.54794	−0.773969	−	0.633224i	$-0.781731\pi$
$617$	−38.4500	−1.54794	−0.773969	+	0.633224i	$0.781731\pi$
$618$	9.21110	0.370525
$619$	14.4222	0.579677	0.289839	−	0.957076i	$-0.406398\pi$
$619$	14.4222	0.579677	0.289839	+	0.957076i	$0.406398\pi$
$620$	13.2111	0.530571
$621$	15.0000	0.601929
$622$	21.2111	0.850488
$623$	6.21110	0.248843
$624$	0	0
$625$	1.00000	0.0400000
$626$	−32.2389	−1.28852
$627$	−9.00000	−0.359425
$628$	37.0278	1.47757
$629$	27.4222	1.09339
$630$	4.60555	0.183490
$631$	−36.0278	−1.43424	−0.717121	−	0.696949i	$-0.754541\pi$
$631$	−36.0278	−1.43424	−0.717121	+	0.696949i	$0.754541\pi$
$632$	−15.6333	−0.621860
$633$	−23.6056	−0.938236
$634$	−13.8167	−0.548729
$635$	4.21110	0.167113
$636$	−10.6056	−0.420537
$637$	0	0
$638$	22.9638	0.909147
$639$	−9.63331	−0.381088
$640$	−18.9083	−0.747417
$641$	9.42221	0.372155	0.186077	−	0.982535i	$-0.440422\pi$
$641$	9.42221	0.372155	0.186077	+	0.982535i	$0.440422\pi$
$642$	−14.3028	−0.564486
$643$	2.63331	0.103848	0.0519238	−	0.998651i	$-0.483465\pi$
$643$	2.63331	0.103848	0.0519238	+	0.998651i	$0.483465\pi$
$644$	9.90833	0.390443
$645$	10.2111	0.402062
$646$	98.1749	3.86264
$647$	39.4222	1.54985	0.774923	−	0.632055i	$-0.217788\pi$
$647$	39.4222	1.54985	0.774923	+	0.632055i	$0.217788\pi$
$648$	−3.00000	−0.117851
$649$	−17.3667	−0.681702
$650$	0	0
$651$	4.00000	0.156772
$652$	−12.5139	−0.490081
$653$	−7.18335	−0.281106	−0.140553	−	0.990073i	$-0.544888\pi$
$653$	−7.18335	−0.281106	−0.140553	+	0.990073i	$0.544888\pi$
$654$	44.2389	1.72988
$655$	21.2111	0.828786
$656$	0.908327	0.0354642
$657$	1.57779	0.0615556
$658$	21.2111	0.826895
$659$	−34.8167	−1.35626	−0.678132	−	0.734940i	$-0.737210\pi$
$659$	−34.8167	−1.35626	−0.678132	+	0.734940i	$0.737210\pi$
$660$	−5.30278	−0.206410
$661$	−4.63331	−0.180215	−0.0901074	−	0.995932i	$-0.528721\pi$
$661$	−4.63331	−0.180215	−0.0901074	+	0.995932i	$0.528721\pi$
$662$	−23.0917	−0.897483
$663$	0	0
$664$	27.6333	1.07238
$665$	−5.60555	−0.217374
$666$	16.6056	0.643452
$667$	18.6333	0.721485
$668$	29.7250	1.15009
$669$	−4.21110	−0.162811
$670$	−16.1194	−0.622748
$671$	−1.60555	−0.0619816
$672$	−5.30278	−0.204559
$673$	−17.6056	−0.678644	−0.339322	−	0.940670i	$-0.610198\pi$
$673$	−17.6056	−0.678644	−0.339322	+	0.940670i	$0.610198\pi$
$674$	59.0278	2.27366
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−9.63331	−0.370238	−0.185119	−	0.982716i	$-0.559267\pi$
$677$	−9.63331	−0.370238	−0.185119	+	0.982716i	$0.559267\pi$
$678$	3.69722	0.141991
$679$	8.39445	0.322149
$680$	22.8167	0.874979
$681$	−27.4222	−1.05082
$682$	14.7889	0.566296
$683$	36.2111	1.38558	0.692790	−	0.721140i	$-0.256381\pi$
$683$	36.2111	1.38558	0.692790	+	0.721140i	$0.256381\pi$
$684$	37.0278	1.41579
$685$	1.60555	0.0613450
$686$	−29.9361	−1.14296
$687$	14.0000	0.534133
$688$	−3.09167	−0.117869
$689$	0	0
$690$	−6.90833	−0.262996
$691$	−30.0278	−1.14231	−0.571155	−	0.820842i	$-0.693504\pi$
$691$	−30.0278	−1.14231	−0.571155	+	0.820842i	$0.693504\pi$
$692$	15.9083	0.604744
$693$	3.21110	0.121980
$694$	−13.3305	−0.506020
$695$	−6.39445	−0.242555
$696$	18.6333	0.706294
$697$	22.8167	0.864242
$698$	8.72498	0.330245
$699$	15.2111	0.575337
$700$	−3.30278	−0.124833
$701$	−36.4222	−1.37565	−0.687824	−	0.725878i	$-0.741434\pi$
$701$	−36.4222	−1.37565	−0.687824	+	0.725878i	$0.741434\pi$
$702$	0	0
$703$	−20.2111	−0.762276
$704$	−20.5778	−0.775555
$705$	−9.21110	−0.346910
$706$	−38.7250	−1.45743
$707$	9.00000	0.338480
$708$	−35.7250	−1.34263
$709$	−13.8444	−0.519938	−0.259969	−	0.965617i	$-0.583712\pi$
$709$	−13.8444	−0.519938	−0.259969	+	0.965617i	$0.583712\pi$
$710$	11.0917	0.416263
$711$	−10.4222	−0.390863
$712$	18.6333	0.698313
$713$	12.0000	0.449404
$714$	17.5139	0.655440
$715$	0	0
$716$	75.3583	2.81627
$717$	0	0
$718$	−42.4222	−1.58318
$719$	−25.6056	−0.954926	−0.477463	−	0.878652i	$-0.658444\pi$
$719$	−25.6056	−0.954926	−0.477463	+	0.878652i	$0.658444\pi$
$720$	0.605551	0.0225676
$721$	4.00000	0.148968
$722$	−28.6056	−1.06459
$723$	1.78890	0.0665298
$724$	58.2389	2.16443
$725$	−6.21110	−0.230675
$726$	19.3944	0.719796
$727$	13.5778	0.503573	0.251786	−	0.967783i	$-0.418982\pi$
$727$	13.5778	0.503573	0.251786	+	0.967783i	$0.418982\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	−1.81665	−0.0672374
$731$	−77.6611	−2.87240
$732$	−3.30278	−0.122074
$733$	−46.8444	−1.73024	−0.865119	−	0.501567i	$-0.832757\pi$
$733$	−46.8444	−1.73024	−0.865119	+	0.501567i	$0.832757\pi$
$734$	−26.3028	−0.970853
$735$	6.00000	0.221313
$736$	−15.9083	−0.586389
$737$	−11.2389	−0.413989
$738$	13.8167	0.508598
$739$	−35.6056	−1.30977	−0.654886	−	0.755728i	$-0.727283\pi$
$739$	−35.6056	−1.30977	−0.654886	+	0.755728i	$0.727283\pi$
$740$	−11.9083	−0.437759
$741$	0	0
$742$	−7.39445	−0.271459
$743$	36.6333	1.34395	0.671973	−	0.740576i	$-0.265447\pi$
$743$	36.6333	1.34395	0.671973	+	0.740576i	$0.265447\pi$
$744$	12.0000	0.439941
$745$	−3.00000	−0.109911
$746$	46.9638	1.71947
$747$	18.4222	0.674033
$748$	40.3305	1.47463
$749$	−6.21110	−0.226949
$750$	2.30278	0.0840855
$751$	46.4500	1.69498	0.847492	−	0.530809i	$-0.178112\pi$
$751$	46.4500	1.69498	0.847492	+	0.530809i	$0.178112\pi$
$752$	2.78890	0.101701
$753$	−7.18335	−0.261776
$754$	0	0
$755$	1.21110	0.0440765
$756$	16.5139	0.600604
$757$	0.816654	0.0296818	0.0148409	−	0.999890i	$-0.495276\pi$
$757$	0.816654	0.0296818	0.0148409	+	0.999890i	$0.495276\pi$
$758$	−22.1194	−0.803414
$759$	−4.81665	−0.174833
$760$	−16.8167	−0.610004
$761$	18.6333	0.675457	0.337728	−	0.941244i	$-0.390341\pi$
$761$	18.6333	0.675457	0.337728	+	0.941244i	$0.390341\pi$
$762$	9.69722	0.351293
$763$	19.2111	0.695489
$764$	55.5416	2.00943
$765$	15.2111	0.549959
$766$	56.7250	2.04956
$767$	0	0
$768$	−17.9083	−0.646211
$769$	11.0000	0.396670	0.198335	−	0.980134i	$-0.436447\pi$
$769$	11.0000	0.396670	0.198335	+	0.980134i	$0.436447\pi$
$770$	−3.69722	−0.133239
$771$	−16.3944	−0.590432
$772$	51.5416	1.85502
$773$	22.3944	0.805472	0.402736	−	0.915316i	$-0.368059\pi$
$773$	22.3944	0.805472	0.402736	+	0.915316i	$0.368059\pi$
$774$	−47.0278	−1.69038
$775$	−4.00000	−0.143684
$776$	25.1833	0.904029
$777$	−3.60555	−0.129348
$778$	35.0278	1.25581
$779$	−16.8167	−0.602519
$780$	0	0
$781$	7.73338	0.276722
$782$	52.5416	1.87889
$783$	31.0555	1.10983
$784$	−1.81665	−0.0648805
$785$	−11.2111	−0.400141
$786$	48.8444	1.74222
$787$	14.6333	0.521621	0.260811	−	0.965390i	$-0.416010\pi$
$787$	14.6333	0.521621	0.260811	+	0.965390i	$0.416010\pi$
$788$	3.90833	0.139228
$789$	11.7889	0.419696
$790$	12.0000	0.426941
$791$	1.60555	0.0570868
$792$	9.63331	0.342305
$793$	0	0
$794$	−50.7250	−1.80016
$795$	3.21110	0.113886
$796$	42.3305	1.50037
$797$	−14.4500	−0.511844	−0.255922	−	0.966697i	$-0.582379\pi$
$797$	−14.4500	−0.511844	−0.255922	+	0.966697i	$0.582379\pi$
$798$	−12.9083	−0.456950
$799$	70.0555	2.47839
$800$	5.30278	0.187481
$801$	12.4222	0.438917
$802$	−28.1194	−0.992932
$803$	−1.26662	−0.0446979
$804$	−23.1194	−0.815359
$805$	−3.00000	−0.105736
$806$	0	0
$807$	−9.00000	−0.316815
$808$	27.0000	0.949857
$809$	−55.0555	−1.93565	−0.967824	−	0.251627i	$-0.919034\pi$
$809$	−55.0555	−1.93565	−0.967824	+	0.251627i	$0.919034\pi$
$810$	2.30278	0.0809113
$811$	−46.4222	−1.63010	−0.815052	−	0.579388i	$-0.803292\pi$
$811$	−46.4222	−1.63010	−0.815052	+	0.579388i	$0.803292\pi$
$812$	20.5139	0.719896
$813$	−20.8167	−0.730072
$814$	−13.3305	−0.467235
$815$	3.78890	0.132719
$816$	2.30278	0.0806133
$817$	57.2389	2.00253
$818$	−18.9083	−0.661114
$819$	0	0
$820$	−9.90833	−0.346014
$821$	21.4222	0.747640	0.373820	−	0.927501i	$-0.378048\pi$
$821$	21.4222	0.747640	0.373820	+	0.927501i	$0.378048\pi$
$822$	3.69722	0.128956
$823$	−16.6333	−0.579801	−0.289900	−	0.957057i	$-0.593622\pi$
$823$	−16.6333	−0.579801	−0.289900	+	0.957057i	$0.593622\pi$
$824$	12.0000	0.418040
$825$	1.60555	0.0558981
$826$	−24.9083	−0.866672
$827$	−42.4222	−1.47516	−0.737582	−	0.675257i	$-0.764033\pi$
$827$	−42.4222	−1.47516	−0.737582	+	0.675257i	$0.764033\pi$
$828$	19.8167	0.688676
$829$	29.4222	1.02188	0.510938	−	0.859618i	$-0.329298\pi$
$829$	29.4222	1.02188	0.510938	+	0.859618i	$0.329298\pi$
$830$	−21.2111	−0.736248
$831$	27.6056	0.957626
$832$	0	0
$833$	−45.6333	−1.58110
$834$	−14.7250	−0.509884
$835$	−9.00000	−0.311458
$836$	−29.7250	−1.02806
$837$	20.0000	0.691301
$838$	−39.6972	−1.37132
$839$	−20.0278	−0.691435	−0.345717	−	0.938339i	$-0.612364\pi$
$839$	−20.0278	−0.691435	−0.345717	+	0.938339i	$0.612364\pi$
$840$	−3.00000	−0.103510
$841$	9.57779	0.330269
$842$	−74.6611	−2.57299
$843$	−6.00000	−0.206651
$844$	−77.9638	−2.68363
$845$	0	0
$846$	42.4222	1.45851
$847$	8.42221	0.289390
$848$	−0.972244	−0.0333870
$849$	5.00000	0.171600
$850$	−17.5139	−0.600721
$851$	−10.8167	−0.370790
$852$	15.9083	0.545010
$853$	47.2111	1.61648	0.808239	−	0.588855i	$-0.200421\pi$
$853$	47.2111	1.61648	0.808239	+	0.588855i	$0.200421\pi$
$854$	−2.30278	−0.0787994
$855$	−11.2111	−0.383412
$856$	−18.6333	−0.636873
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	10.7889	0.368112	0.184056	−	0.982916i	$-0.441077\pi$
$859$	10.7889	0.368112	0.184056	+	0.982916i	$0.441077\pi$
$860$	33.7250	1.15001
$861$	−3.00000	−0.102240
$862$	−67.3305	−2.29329
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	−26.5139	−0.902020
$865$	−4.81665	−0.163771
$866$	−8.30278	−0.282140
$867$	40.8444	1.38715
$868$	13.2111	0.448414
$869$	8.36669	0.283821
$870$	−14.3028	−0.484910
$871$	0	0
$872$	57.6333	1.95171
$873$	16.7889	0.568218
$874$	−38.7250	−1.30989
$875$	1.00000	0.0338062
$876$	−2.60555	−0.0880334
$877$	−1.97224	−0.0665979	−0.0332990	−	0.999445i	$-0.510601\pi$
$877$	−1.97224	−0.0665979	−0.0332990	+	0.999445i	$0.510601\pi$
$878$	62.7250	2.11687
$879$	10.3944	0.350596
$880$	−0.486122	−0.0163872
$881$	−21.8444	−0.735957	−0.367978	−	0.929834i	$-0.619950\pi$
$881$	−21.8444	−0.735957	−0.367978	+	0.929834i	$0.619950\pi$
$882$	−27.6333	−0.930462
$883$	11.6333	0.391492	0.195746	−	0.980655i	$-0.437287\pi$
$883$	11.6333	0.391492	0.195746	+	0.980655i	$0.437287\pi$
$884$	0	0
$885$	10.8167	0.363598
$886$	−14.7889	−0.496843
$887$	−37.0555	−1.24420	−0.622101	−	0.782937i	$-0.713721\pi$
$887$	−37.0555	−1.24420	−0.622101	+	0.782937i	$0.713721\pi$
$888$	−10.8167	−0.362983
$889$	4.21110	0.141236
$890$	−14.3028	−0.479430
$891$	1.60555	0.0537880
$892$	−13.9083	−0.465685
$893$	−51.6333	−1.72784
$894$	−6.90833	−0.231049
$895$	−22.8167	−0.762677
$896$	−18.9083	−0.631683
$897$	0	0
$898$	−70.5416	−2.35400
$899$	24.8444	0.828607
$900$	−6.60555	−0.220185
$901$	−24.4222	−0.813622
$902$	−11.0917	−0.369312
$903$	10.2111	0.339804
$904$	4.81665	0.160200
$905$	−17.6333	−0.586151
$906$	2.78890	0.0926549
$907$	−38.2666	−1.27062	−0.635311	−	0.772256i	$-0.719128\pi$
$907$	−38.2666	−1.27062	−0.635311	+	0.772256i	$0.719128\pi$
$908$	−90.5694	−3.00565
$909$	18.0000	0.597022
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	−1.69722	−0.0562007
$913$	−14.7889	−0.489441
$914$	61.7527	2.04260
$915$	1.00000	0.0330590
$916$	46.2389	1.52777
$917$	21.2111	0.700452
$918$	87.5694	2.89022
$919$	−38.8167	−1.28044	−0.640222	−	0.768190i	$-0.721157\pi$
$919$	−38.8167	−1.28044	−0.640222	+	0.768190i	$0.721157\pi$
$920$	−9.00000	−0.296721
$921$	−16.0000	−0.527218
$922$	−83.3860	−2.74617
$923$	0	0
$924$	−5.30278	−0.174449
$925$	3.60555	0.118550
$926$	79.2666	2.60486
$927$	8.00000	0.262754
$928$	−32.9361	−1.08118
$929$	−15.4222	−0.505986	−0.252993	−	0.967468i	$-0.581415\pi$
$929$	−15.4222	−0.505986	−0.252993	+	0.967468i	$0.581415\pi$
$930$	−9.21110	−0.302044
$931$	33.6333	1.10229
$932$	50.2389	1.64563
$933$	−9.21110	−0.301558
$934$	−6.42221	−0.210141
$935$	−12.2111	−0.399346
$936$	0	0
$937$	54.4777	1.77971	0.889855	−	0.456244i	$-0.150806\pi$
$937$	54.4777	1.77971	0.889855	+	0.456244i	$0.150806\pi$
$938$	−16.1194	−0.526318
$939$	14.0000	0.456873
$940$	−30.4222	−0.992263
$941$	−9.63331	−0.314037	−0.157018	−	0.987596i	$-0.550188\pi$
$941$	−9.63331	−0.314037	−0.157018	+	0.987596i	$0.550188\pi$
$942$	−25.8167	−0.841152
$943$	−9.00000	−0.293080
$944$	−3.27502	−0.106593
$945$	−5.00000	−0.162650
$946$	37.7527	1.22745
$947$	−18.6333	−0.605501	−0.302751	−	0.953070i	$-0.597905\pi$
$947$	−18.6333	−0.605501	−0.302751	+	0.953070i	$0.597905\pi$
$948$	17.2111	0.558991
$949$	0	0
$950$	12.9083	0.418801
$951$	6.00000	0.194563
$952$	22.8167	0.739492
$953$	−14.4500	−0.468080	−0.234040	−	0.972227i	$-0.575195\pi$
$953$	−14.4500	−0.468080	−0.234040	+	0.972227i	$0.575195\pi$
$954$	−14.7889	−0.478808
$955$	−16.8167	−0.544174
$956$	0	0
$957$	−9.97224	−0.322357
$958$	66.3583	2.14394
$959$	1.60555	0.0518460
$960$	12.8167	0.413656
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−12.4222	−0.400300
$964$	5.90833	0.190294
$965$	−15.6056	−0.502360
$966$	−6.90833	−0.222272
$967$	−44.4777	−1.43031	−0.715153	−	0.698967i	$-0.753643\pi$
$967$	−44.4777	−1.43031	−0.715153	+	0.698967i	$0.753643\pi$
$968$	25.2666	0.812100
$969$	−42.6333	−1.36958
$970$	−19.3305	−0.620666
$971$	−44.0278	−1.41292	−0.706459	−	0.707754i	$-0.749709\pi$
$971$	−44.0278	−1.41292	−0.706459	+	0.707754i	$0.749709\pi$
$972$	52.8444	1.69499
$973$	−6.39445	−0.204997
$974$	2.30278	0.0737857
$975$	0	0
$976$	−0.302776	−0.00969161
$977$	28.8167	0.921926	0.460963	−	0.887419i	$-0.347504\pi$
$977$	28.8167	0.921926	0.460963	+	0.887419i	$0.347504\pi$
$978$	8.72498	0.278994
$979$	−9.97224	−0.318714
$980$	19.8167	0.633020
$981$	38.4222	1.22673
$982$	38.7250	1.23576
$983$	18.4222	0.587577	0.293789	−	0.955870i	$-0.405084\pi$
$983$	18.4222	0.587577	0.293789	+	0.955870i	$0.405084\pi$
$984$	−9.00000	−0.286910
$985$	−1.18335	−0.0377045
$986$	108.780	3.46428
$987$	−9.21110	−0.293193
$988$	0	0
$989$	30.6333	0.974083
$990$	−7.39445	−0.235011
$991$	40.0278	1.27152	0.635762	−	0.771885i	$-0.280686\pi$
$991$	40.0278	1.27152	0.635762	+	0.771885i	$0.280686\pi$
$992$	−21.2111	−0.673453
$993$	10.0278	0.318221
$994$	11.0917	0.351807
$995$	−12.8167	−0.406315
$996$	−30.4222	−0.963964
$997$	−18.4500	−0.584316	−0.292158	−	0.956370i	$-0.594373\pi$
$997$	−18.4500	−0.584316	−0.292158	+	0.956370i	$0.594373\pi$
$998$	−5.57779	−0.176562
$999$	−18.0278	−0.570373

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.a.c.1.1		2
3.2	odd	2		7605.2.a.bg.1.2		2
5.4	even	2		4225.2.a.x.1.2		2
13.2	odd	12		845.2.m.d.316.1		8
13.3	even	3		65.2.e.b.61.2	yes	4
13.4	even	6		845.2.e.d.146.1		4
13.5	odd	4		845.2.c.d.506.4		4
13.6	odd	12		845.2.m.d.361.4		8
13.7	odd	12		845.2.m.d.361.1		8
13.8	odd	4		845.2.c.d.506.1		4
13.9	even	3		65.2.e.b.16.2	✓	4
13.10	even	6		845.2.e.d.191.1		4
13.11	odd	12		845.2.m.d.316.4		8
13.12	even	2		845.2.a.f.1.2		2
39.29	odd	6		585.2.j.d.451.1		4
39.35	odd	6		585.2.j.d.406.1		4
39.38	odd	2		7605.2.a.bb.1.1		2
52.3	odd	6		1040.2.q.o.321.2		4
52.35	odd	6		1040.2.q.o.81.2		4
65.3	odd	12		325.2.o.b.74.1		8
65.9	even	6		325.2.e.a.276.1		4
65.22	odd	12		325.2.o.b.224.1		8
65.29	even	6		325.2.e.a.126.1		4
65.42	odd	12		325.2.o.b.74.4		8
65.48	odd	12		325.2.o.b.224.4		8
65.64	even	2		4225.2.a.t.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.e.b.16.2	✓	4	13.9	even	3
65.2.e.b.61.2	yes	4	13.3	even	3
325.2.e.a.126.1		4	65.29	even	6
325.2.e.a.276.1		4	65.9	even	6
325.2.o.b.74.1		8	65.3	odd	12
325.2.o.b.74.4		8	65.42	odd	12
325.2.o.b.224.1		8	65.22	odd	12
325.2.o.b.224.4		8	65.48	odd	12
585.2.j.d.406.1		4	39.35	odd	6
585.2.j.d.451.1		4	39.29	odd	6
845.2.a.c.1.1		2	1.1	even	1	trivial
845.2.a.f.1.2		2	13.12	even	2
845.2.c.d.506.1		4	13.8	odd	4
845.2.c.d.506.4		4	13.5	odd	4
845.2.e.d.146.1		4	13.4	even	6
845.2.e.d.191.1		4	13.10	even	6
845.2.m.d.316.1		8	13.2	odd	12
845.2.m.d.316.4		8	13.11	odd	12
845.2.m.d.361.1		8	13.7	odd	12
845.2.m.d.361.4		8	13.6	odd	12
1040.2.q.o.81.2		4	52.35	odd	6
1040.2.q.o.321.2		4	52.3	odd	6
4225.2.a.t.1.1		2	65.64	even	2
4225.2.a.x.1.2		2	5.4	even	2
7605.2.a.bb.1.1		2	39.38	odd	2
7605.2.a.bg.1.2		2	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:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30278 q^{2} +1.00000 q^{3} +3.30278 q^{4} -1.00000 q^{5} -2.30278 q^{6} -1.00000 q^{7} -3.00000 q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q-2.30278 q^{2} +1.00000 q^{3} +3.30278 q^{4} -1.00000 q^{5} -2.30278 q^{6} -1.00000 q^{7} -3.00000 q^{8} -2.00000 q^{9} +2.30278 q^{10} +1.60555 q^{11} +3.30278 q^{12} +2.30278 q^{14} -1.00000 q^{15} +0.302776 q^{16} +7.60555 q^{17} +4.60555 q^{18} -5.60555 q^{19} -3.30278 q^{20} -1.00000 q^{21} -3.69722 q^{22} -3.00000 q^{23} -3.00000 q^{24} +1.00000 q^{25} -5.00000 q^{27} -3.30278 q^{28} -6.21110 q^{29} +2.30278 q^{30} -4.00000 q^{31} +5.30278 q^{32} +1.60555 q^{33} -17.5139 q^{34} +1.00000 q^{35} -6.60555 q^{36} +3.60555 q^{37} +12.9083 q^{38} +3.00000 q^{40} +3.00000 q^{41} +2.30278 q^{42} -10.2111 q^{43} +5.30278 q^{44} +2.00000 q^{45} +6.90833 q^{46} +9.21110 q^{47} +0.302776 q^{48} -6.00000 q^{49} -2.30278 q^{50} +7.60555 q^{51} -3.21110 q^{53} +11.5139 q^{54} -1.60555 q^{55} +3.00000 q^{56} -5.60555 q^{57} +14.3028 q^{58} -10.8167 q^{59} -3.30278 q^{60} -1.00000 q^{61} +9.21110 q^{62} +2.00000 q^{63} -12.8167 q^{64} -3.69722 q^{66} -7.00000 q^{67} +25.1194 q^{68} -3.00000 q^{69} -2.30278 q^{70} +4.81665 q^{71} +6.00000 q^{72} -0.788897 q^{73} -8.30278 q^{74} +1.00000 q^{75} -18.5139 q^{76} -1.60555 q^{77} +5.21110 q^{79} -0.302776 q^{80} +1.00000 q^{81} -6.90833 q^{82} -9.21110 q^{83} -3.30278 q^{84} -7.60555 q^{85} +23.5139 q^{86} -6.21110 q^{87} -4.81665 q^{88} -6.21110 q^{89} -4.60555 q^{90} -9.90833 q^{92} -4.00000 q^{93} -21.2111 q^{94} +5.60555 q^{95} +5.30278 q^{96} -8.39445 q^{97} +13.8167 q^{98} -3.21110 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 2 q^{3} + 3 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9}+O(q^{10}) \) 2 * q - q^2 + 2 * q^3 + 3 * q^4 - 2 * q^5 - q^6 - 2 * q^7 - 6 * q^8 - 4 * q^9	\( 2 q - q^{2} + 2 q^{3} + 3 q^{4} - 2 q^{5} - q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9} + q^{10} - 4 q^{11} + 3 q^{12} + q^{14} - 2 q^{15} - 3 q^{16} + 8 q^{17} + 2 q^{18} - 4 q^{19} - 3 q^{20} - 2 q^{21} - 11 q^{22} - 6 q^{23} - 6 q^{24} + 2 q^{25} - 10 q^{27} - 3 q^{28} + 2 q^{29} + q^{30} - 8 q^{31} + 7 q^{32} - 4 q^{33} - 17 q^{34} + 2 q^{35} - 6 q^{36} + 15 q^{38} + 6 q^{40} + 6 q^{41} + q^{42} - 6 q^{43} + 7 q^{44} + 4 q^{45} + 3 q^{46} + 4 q^{47} - 3 q^{48} - 12 q^{49} - q^{50} + 8 q^{51} + 8 q^{53} + 5 q^{54} + 4 q^{55} + 6 q^{56} - 4 q^{57} + 25 q^{58} - 3 q^{60} - 2 q^{61} + 4 q^{62} + 4 q^{63} - 4 q^{64} - 11 q^{66} - 14 q^{67} + 25 q^{68} - 6 q^{69} - q^{70} - 12 q^{71} + 12 q^{72} - 16 q^{73} - 13 q^{74} + 2 q^{75} - 19 q^{76} + 4 q^{77} - 4 q^{79} + 3 q^{80} + 2 q^{81} - 3 q^{82} - 4 q^{83} - 3 q^{84} - 8 q^{85} + 29 q^{86} + 2 q^{87} + 12 q^{88} + 2 q^{89} - 2 q^{90} - 9 q^{92} - 8 q^{93} - 28 q^{94} + 4 q^{95} + 7 q^{96} - 24 q^{97} + 6 q^{98} + 8 q^{99}+O(q^{100}) \) 2 * q - q^2 + 2 * q^3 + 3 * q^4 - 2 * q^5 - q^6 - 2 * q^7 - 6 * q^8 - 4 * q^9 + q^10 - 4 * q^11 + 3 * q^12 + q^14 - 2 * q^15 - 3 * q^16 + 8 * q^17 + 2 * q^18 - 4 * q^19 - 3 * q^20 - 2 * q^21 - 11 * q^22 - 6 * q^23 - 6 * q^24 + 2 * q^25 - 10 * q^27 - 3 * q^28 + 2 * q^29 + q^30 - 8 * q^31 + 7 * q^32 - 4 * q^33 - 17 * q^34 + 2 * q^35 - 6 * q^36 + 15 * q^38 + 6 * q^40 + 6 * q^41 + q^42 - 6 * q^43 + 7 * q^44 + 4 * q^45 + 3 * q^46 + 4 * q^47 - 3 * q^48 - 12 * q^49 - q^50 + 8 * q^51 + 8 * q^53 + 5 * q^54 + 4 * q^55 + 6 * q^56 - 4 * q^57 + 25 * q^58 - 3 * q^60 - 2 * q^61 + 4 * q^62 + 4 * q^63 - 4 * q^64 - 11 * q^66 - 14 * q^67 + 25 * q^68 - 6 * q^69 - q^70 - 12 * q^71 + 12 * q^72 - 16 * q^73 - 13 * q^74 + 2 * q^75 - 19 * q^76 + 4 * q^77 - 4 * q^79 + 3 * q^80 + 2 * q^81 - 3 * q^82 - 4 * q^83 - 3 * q^84 - 8 * q^85 + 29 * q^86 + 2 * q^87 + 12 * q^88 + 2 * q^89 - 2 * q^90 - 9 * q^92 - 8 * q^93 - 28 * q^94 + 4 * q^95 + 7 * q^96 - 24 * q^97 + 6 * q^98 + 8 * q^99

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