LMFDB - Embedded newform 845.2.a.g.1.2

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:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
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			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+2.41421 q^{2} -1.41421 q^{3} +3.82843 q^{4} -1.00000 q^{5} -3.41421 q^{6} -4.82843 q^{7} +4.41421 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+2.41421 q^{2} -1.41421 q^{3} +3.82843 q^{4} -1.00000 q^{5} -3.41421 q^{6} -4.82843 q^{7} +4.41421 q^{8} -1.00000 q^{9} -2.41421 q^{10} -3.41421 q^{11} -5.41421 q^{12} -11.6569 q^{14} +1.41421 q^{15} +3.00000 q^{16} +0.828427 q^{17} -2.41421 q^{18} -0.585786 q^{19} -3.82843 q^{20} +6.82843 q^{21} -8.24264 q^{22} +1.41421 q^{23} -6.24264 q^{24} +1.00000 q^{25} +5.65685 q^{27} -18.4853 q^{28} -5.65685 q^{29} +3.41421 q^{30} -1.75736 q^{31} -1.58579 q^{32} +4.82843 q^{33} +2.00000 q^{34} +4.82843 q^{35} -3.82843 q^{36} +8.48528 q^{37} -1.41421 q^{38} -4.41421 q^{40} +3.17157 q^{41} +16.4853 q^{42} -11.0711 q^{43} -13.0711 q^{44} +1.00000 q^{45} +3.41421 q^{46} +4.82843 q^{47} -4.24264 q^{48} +16.3137 q^{49} +2.41421 q^{50} -1.17157 q^{51} +2.48528 q^{53} +13.6569 q^{54} +3.41421 q^{55} -21.3137 q^{56} +0.828427 q^{57} -13.6569 q^{58} -1.75736 q^{59} +5.41421 q^{60} -8.00000 q^{61} -4.24264 q^{62} +4.82843 q^{63} -9.82843 q^{64} +11.6569 q^{66} +2.00000 q^{67} +3.17157 q^{68} -2.00000 q^{69} +11.6569 q^{70} -11.8995 q^{71} -4.41421 q^{72} -8.48528 q^{73} +20.4853 q^{74} -1.41421 q^{75} -2.24264 q^{76} +16.4853 q^{77} -8.48528 q^{79} -3.00000 q^{80} -5.00000 q^{81} +7.65685 q^{82} +3.17157 q^{83} +26.1421 q^{84} -0.828427 q^{85} -26.7279 q^{86} +8.00000 q^{87} -15.0711 q^{88} -6.00000 q^{89} +2.41421 q^{90} +5.41421 q^{92} +2.48528 q^{93} +11.6569 q^{94} +0.585786 q^{95} +2.24264 q^{96} +7.65685 q^{97} +39.3848 q^{98} +3.41421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} + 6 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 4 * q^6 - 4 * q^7 + 6 * q^8 - 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} + 6 q^{8} - 2 q^{9} - 2 q^{10} - 4 q^{11} - 8 q^{12} - 12 q^{14} + 6 q^{16} - 4 q^{17} - 2 q^{18} - 4 q^{19} - 2 q^{20} + 8 q^{21} - 8 q^{22} - 4 q^{24} + 2 q^{25} - 20 q^{28} + 4 q^{30} - 12 q^{31} - 6 q^{32} + 4 q^{33} + 4 q^{34} + 4 q^{35} - 2 q^{36} - 6 q^{40} + 12 q^{41} + 16 q^{42} - 8 q^{43} - 12 q^{44} + 2 q^{45} + 4 q^{46} + 4 q^{47} + 10 q^{49} + 2 q^{50} - 8 q^{51} - 12 q^{53} + 16 q^{54} + 4 q^{55} - 20 q^{56} - 4 q^{57} - 16 q^{58} - 12 q^{59} + 8 q^{60} - 16 q^{61} + 4 q^{63} - 14 q^{64} + 12 q^{66} + 4 q^{67} + 12 q^{68} - 4 q^{69} + 12 q^{70} - 4 q^{71} - 6 q^{72} + 24 q^{74} + 4 q^{76} + 16 q^{77} - 6 q^{80} - 10 q^{81} + 4 q^{82} + 12 q^{83} + 24 q^{84} + 4 q^{85} - 28 q^{86} + 16 q^{87} - 16 q^{88} - 12 q^{89} + 2 q^{90} + 8 q^{92} - 12 q^{93} + 12 q^{94} + 4 q^{95} - 4 q^{96} + 4 q^{97} + 42 q^{98} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 4 * q^6 - 4 * q^7 + 6 * q^8 - 2 * q^9 - 2 * q^10 - 4 * q^11 - 8 * q^12 - 12 * q^14 + 6 * q^16 - 4 * q^17 - 2 * q^18 - 4 * q^19 - 2 * q^20 + 8 * q^21 - 8 * q^22 - 4 * q^24 + 2 * q^25 - 20 * q^28 + 4 * q^30 - 12 * q^31 - 6 * q^32 + 4 * q^33 + 4 * q^34 + 4 * q^35 - 2 * q^36 - 6 * q^40 + 12 * q^41 + 16 * q^42 - 8 * q^43 - 12 * q^44 + 2 * q^45 + 4 * q^46 + 4 * q^47 + 10 * q^49 + 2 * q^50 - 8 * q^51 - 12 * q^53 + 16 * q^54 + 4 * q^55 - 20 * q^56 - 4 * q^57 - 16 * q^58 - 12 * q^59 + 8 * q^60 - 16 * q^61 + 4 * q^63 - 14 * q^64 + 12 * q^66 + 4 * q^67 + 12 * q^68 - 4 * q^69 + 12 * q^70 - 4 * q^71 - 6 * q^72 + 24 * q^74 + 4 * q^76 + 16 * q^77 - 6 * q^80 - 10 * q^81 + 4 * q^82 + 12 * q^83 + 24 * q^84 + 4 * q^85 - 28 * q^86 + 16 * q^87 - 16 * q^88 - 12 * q^89 + 2 * q^90 + 8 * q^92 - 12 * q^93 + 12 * q^94 + 4 * q^95 - 4 * q^96 + 4 * q^97 + 42 * q^98 + 4 * 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 $

$2$	2.41421	1.70711	0.853553	−	0.521005i	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	3.82843	1.91421
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−3.41421	−1.39385
$7$	−4.82843	−1.82497	−0.912487	−	0.409106i	$-0.865841\pi$
$7$	−4.82843	−1.82497	−0.912487	+	0.409106i	$0.865841\pi$
$8$	4.41421	1.56066
$9$	−1.00000	−0.333333
$10$	−2.41421	−0.763441
$11$	−3.41421	−1.02942	−0.514712	−	0.857363i	$-0.672101\pi$
$11$	−3.41421	−1.02942	−0.514712	+	0.857363i	$0.672101\pi$
$12$	−5.41421	−1.56295
$13$	0	0
$13$	0	0
$14$	−11.6569	−3.11543
$15$	1.41421	0.365148
$16$	3.00000	0.750000
$17$	0.828427	0.200923	0.100462	−	0.994941i	$-0.467968\pi$
$17$	0.828427	0.200923	0.100462	+	0.994941i	$0.467968\pi$
$18$	−2.41421	−0.569036
$19$	−0.585786	−0.134389	−0.0671943	−	0.997740i	$-0.521405\pi$
$19$	−0.585786	−0.134389	−0.0671943	+	0.997740i	$0.521405\pi$
$20$	−3.82843	−0.856062
$21$	6.82843	1.49008
$22$	−8.24264	−1.75734
$23$	1.41421	0.294884	0.147442	−	0.989071i	$-0.452896\pi$
$23$	1.41421	0.294884	0.147442	+	0.989071i	$0.452896\pi$
$24$	−6.24264	−1.27427
$25$	1.00000	0.200000
$26$	0	0
$27$	5.65685	1.08866
$28$	−18.4853	−3.49339
$29$	−5.65685	−1.05045	−0.525226	−	0.850963i	$-0.676019\pi$
$29$	−5.65685	−1.05045	−0.525226	+	0.850963i	$0.676019\pi$
$30$	3.41421	0.623347
$31$	−1.75736	−0.315631	−0.157816	−	0.987469i	$-0.550445\pi$
$31$	−1.75736	−0.315631	−0.157816	+	0.987469i	$0.550445\pi$
$32$	−1.58579	−0.280330
$33$	4.82843	0.840521
$34$	2.00000	0.342997
$35$	4.82843	0.816153
$36$	−3.82843	−0.638071
$37$	8.48528	1.39497	0.697486	−	0.716599i	$-0.254302\pi$
$37$	8.48528	1.39497	0.697486	+	0.716599i	$0.254302\pi$
$38$	−1.41421	−0.229416
$39$	0	0
$40$	−4.41421	−0.697948
$41$	3.17157	0.495316	0.247658	−	0.968847i	$-0.420339\pi$
$41$	3.17157	0.495316	0.247658	+	0.968847i	$0.420339\pi$
$42$	16.4853	2.54373
$43$	−11.0711	−1.68832	−0.844161	−	0.536090i	$-0.819901\pi$
$43$	−11.0711	−1.68832	−0.844161	+	0.536090i	$0.819901\pi$
$44$	−13.0711	−1.97054
$45$	1.00000	0.149071
$46$	3.41421	0.503398
$47$	4.82843	0.704298	0.352149	−	0.935944i	$-0.385451\pi$
$47$	4.82843	0.704298	0.352149	+	0.935944i	$0.385451\pi$
$48$	−4.24264	−0.612372
$49$	16.3137	2.33053
$50$	2.41421	0.341421
$51$	−1.17157	−0.164053
$52$	0	0
$53$	2.48528	0.341380	0.170690	−	0.985325i	$-0.445400\pi$
$53$	2.48528	0.341380	0.170690	+	0.985325i	$0.445400\pi$
$54$	13.6569	1.85846
$55$	3.41421	0.460372
$56$	−21.3137	−2.84816
$57$	0.828427	0.109728
$58$	−13.6569	−1.79323
$59$	−1.75736	−0.228789	−0.114394	−	0.993435i	$-0.536493\pi$
$59$	−1.75736	−0.228789	−0.114394	+	0.993435i	$0.536493\pi$
$60$	5.41421	0.698972
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	−4.24264	−0.538816
$63$	4.82843	0.608325
$64$	−9.82843	−1.22855
$65$	0	0
$66$	11.6569	1.43486
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	3.17157	0.384610
$69$	−2.00000	−0.240772
$70$	11.6569	1.39326
$71$	−11.8995	−1.41221	−0.706105	−	0.708107i	$-0.749549\pi$
$71$	−11.8995	−1.41221	−0.706105	+	0.708107i	$0.749549\pi$
$72$	−4.41421	−0.520220
$73$	−8.48528	−0.993127	−0.496564	−	0.868000i	$-0.665405\pi$
$73$	−8.48528	−0.993127	−0.496564	+	0.868000i	$0.665405\pi$
$74$	20.4853	2.38137
$75$	−1.41421	−0.163299
$76$	−2.24264	−0.257249
$77$	16.4853	1.87867
$78$	0	0
$79$	−8.48528	−0.954669	−0.477334	−	0.878722i	$-0.658397\pi$
$79$	−8.48528	−0.954669	−0.477334	+	0.878722i	$0.658397\pi$
$80$	−3.00000	−0.335410
$81$	−5.00000	−0.555556
$82$	7.65685	0.845558
$83$	3.17157	0.348125	0.174063	−	0.984735i	$-0.444310\pi$
$83$	3.17157	0.348125	0.174063	+	0.984735i	$0.444310\pi$
$84$	26.1421	2.85234
$85$	−0.828427	−0.0898555
$86$	−26.7279	−2.88215
$87$	8.00000	0.857690
$88$	−15.0711	−1.60658
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	2.41421	0.254480
$91$	0	0
$92$	5.41421	0.564471
$93$	2.48528	0.257712
$94$	11.6569	1.20231
$95$	0.585786	0.0601004
$96$	2.24264	0.228889
$97$	7.65685	0.777436	0.388718	−	0.921357i	$-0.372918\pi$
$97$	7.65685	0.777436	0.388718	+	0.921357i	$0.372918\pi$
$98$	39.3848	3.97846
$99$	3.41421	0.343141
$100$	3.82843	0.382843
$101$	−3.65685	−0.363871	−0.181935	−	0.983311i	$-0.558236\pi$
$101$	−3.65685	−0.363871	−0.181935	+	0.983311i	$0.558236\pi$
$102$	−2.82843	−0.280056
$103$	14.5858	1.43718	0.718590	−	0.695434i	$-0.244788\pi$
$103$	14.5858	1.43718	0.718590	+	0.695434i	$0.244788\pi$
$104$	0	0
$105$	−6.82843	−0.666386
$106$	6.00000	0.582772
$107$	−9.41421	−0.910106	−0.455053	−	0.890464i	$-0.650380\pi$
$107$	−9.41421	−0.910106	−0.455053	+	0.890464i	$0.650380\pi$
$108$	21.6569	2.08393
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	8.24264	0.785905
$111$	−12.0000	−1.13899
$112$	−14.4853	−1.36873
$113$	−8.82843	−0.830509	−0.415254	−	0.909705i	$-0.636307\pi$
$113$	−8.82843	−0.830509	−0.415254	+	0.909705i	$0.636307\pi$
$114$	2.00000	0.187317
$115$	−1.41421	−0.131876
$116$	−21.6569	−2.01079
$117$	0	0
$118$	−4.24264	−0.390567
$119$	−4.00000	−0.366679
$120$	6.24264	0.569873
$121$	0.656854	0.0597140
$122$	−19.3137	−1.74858
$123$	−4.48528	−0.404424
$124$	−6.72792	−0.604185
$125$	−1.00000	−0.0894427
$126$	11.6569	1.03848
$127$	−6.58579	−0.584394	−0.292197	−	0.956358i	$-0.594386\pi$
$127$	−6.58579	−0.584394	−0.292197	+	0.956358i	$0.594386\pi$
$128$	−20.5563	−1.81694
$129$	15.6569	1.37851
$130$	0	0
$131$	−16.9706	−1.48272	−0.741362	−	0.671105i	$-0.765820\pi$
$131$	−16.9706	−1.48272	−0.741362	+	0.671105i	$0.765820\pi$
$132$	18.4853	1.60894
$133$	2.82843	0.245256
$134$	4.82843	0.417113
$135$	−5.65685	−0.486864
$136$	3.65685	0.313573
$137$	17.3137	1.47921	0.739605	−	0.673041i	$-0.235012\pi$
$137$	17.3137	1.47921	0.739605	+	0.673041i	$0.235012\pi$
$138$	−4.82843	−0.411023
$139$	4.48528	0.380437	0.190218	−	0.981742i	$-0.439080\pi$
$139$	4.48528	0.380437	0.190218	+	0.981742i	$0.439080\pi$
$140$	18.4853	1.56229
$141$	−6.82843	−0.575057
$142$	−28.7279	−2.41079
$143$	0	0
$144$	−3.00000	−0.250000
$145$	5.65685	0.469776
$146$	−20.4853	−1.69537
$147$	−23.0711	−1.90287
$148$	32.4853	2.67027
$149$	11.6569	0.954967	0.477483	−	0.878641i	$-0.341549\pi$
$149$	11.6569	0.954967	0.477483	+	0.878641i	$0.341549\pi$
$150$	−3.41421	−0.278769
$151$	−9.75736	−0.794043	−0.397021	−	0.917809i	$-0.629956\pi$
$151$	−9.75736	−0.794043	−0.397021	+	0.917809i	$0.629956\pi$
$152$	−2.58579	−0.209735
$153$	−0.828427	−0.0669744
$154$	39.7990	3.20709
$155$	1.75736	0.141154
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	−20.4853	−1.62972
$159$	−3.51472	−0.278735
$160$	1.58579	0.125367
$161$	−6.82843	−0.538155
$162$	−12.0711	−0.948393
$163$	−18.9706	−1.48589	−0.742945	−	0.669353i	$-0.766571\pi$
$163$	−18.9706	−1.48589	−0.742945	+	0.669353i	$0.766571\pi$
$164$	12.1421	0.948141
$165$	−4.82843	−0.375893
$166$	7.65685	0.594287
$167$	3.17157	0.245424	0.122712	−	0.992442i	$-0.460841\pi$
$167$	3.17157	0.245424	0.122712	+	0.992442i	$0.460841\pi$
$168$	30.1421	2.32552
$169$	0	0
$170$	−2.00000	−0.153393
$171$	0.585786	0.0447962
$172$	−42.3848	−3.23181
$173$	16.8284	1.27944	0.639721	−	0.768607i	$-0.279050\pi$
$173$	16.8284	1.27944	0.639721	+	0.768607i	$0.279050\pi$
$174$	19.3137	1.46417
$175$	−4.82843	−0.364995
$176$	−10.2426	−0.772068
$177$	2.48528	0.186805
$178$	−14.4853	−1.08572
$179$	5.65685	0.422813	0.211407	−	0.977398i	$-0.432196\pi$
$179$	5.65685	0.422813	0.211407	+	0.977398i	$0.432196\pi$
$180$	3.82843	0.285354
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	11.3137	0.836333
$184$	6.24264	0.460214
$185$	−8.48528	−0.623850
$186$	6.00000	0.439941
$187$	−2.82843	−0.206835
$188$	18.4853	1.34818
$189$	−27.3137	−1.98678
$190$	1.41421	0.102598
$191$	2.34315	0.169544	0.0847720	−	0.996400i	$-0.472984\pi$
$191$	2.34315	0.169544	0.0847720	+	0.996400i	$0.472984\pi$
$192$	13.8995	1.00311
$193$	−4.34315	−0.312626	−0.156313	−	0.987708i	$-0.549961\pi$
$193$	−4.34315	−0.312626	−0.156313	+	0.987708i	$0.549961\pi$
$194$	18.4853	1.32717
$195$	0	0
$196$	62.4558	4.46113
$197$	−10.9706	−0.781620	−0.390810	−	0.920471i	$-0.627805\pi$
$197$	−10.9706	−0.781620	−0.390810	+	0.920471i	$0.627805\pi$
$198$	8.24264	0.585779
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	4.41421	0.312132
$201$	−2.82843	−0.199502
$202$	−8.82843	−0.621166
$203$	27.3137	1.91705
$204$	−4.48528	−0.314033
$205$	−3.17157	−0.221512
$206$	35.2132	2.45342
$207$	−1.41421	−0.0982946
$208$	0	0
$209$	2.00000	0.138343
$210$	−16.4853	−1.13759
$211$	3.31371	0.228125	0.114063	−	0.993474i	$-0.463614\pi$
$211$	3.31371	0.228125	0.114063	+	0.993474i	$0.463614\pi$
$212$	9.51472	0.653474
$213$	16.8284	1.15306
$214$	−22.7279	−1.55365
$215$	11.0711	0.755041
$216$	24.9706	1.69903
$217$	8.48528	0.576018
$218$	4.82843	0.327022
$219$	12.0000	0.810885
$220$	13.0711	0.881251
$221$	0	0
$222$	−28.9706	−1.94438
$223$	−9.51472	−0.637153	−0.318576	−	0.947897i	$-0.603205\pi$
$223$	−9.51472	−0.637153	−0.318576	+	0.947897i	$0.603205\pi$
$224$	7.65685	0.511595
$225$	−1.00000	−0.0666667
$226$	−21.3137	−1.41777
$227$	16.3431	1.08473	0.542366	−	0.840142i	$-0.317528\pi$
$227$	16.3431	1.08473	0.542366	+	0.840142i	$0.317528\pi$
$228$	3.17157	0.210043
$229$	4.82843	0.319071	0.159536	−	0.987192i	$-0.449000\pi$
$229$	4.82843	0.319071	0.159536	+	0.987192i	$0.449000\pi$
$230$	−3.41421	−0.225127
$231$	−23.3137	−1.53393
$232$	−24.9706	−1.63940
$233$	−20.6274	−1.35135	−0.675674	−	0.737201i	$-0.736147\pi$
$233$	−20.6274	−1.35135	−0.675674	+	0.737201i	$0.736147\pi$
$234$	0	0
$235$	−4.82843	−0.314972
$236$	−6.72792	−0.437950
$237$	12.0000	0.779484
$238$	−9.65685	−0.625961
$239$	3.41421	0.220847	0.110424	−	0.993885i	$-0.464779\pi$
$239$	3.41421	0.220847	0.110424	+	0.993885i	$0.464779\pi$
$240$	4.24264	0.273861
$241$	14.4853	0.933079	0.466539	−	0.884500i	$-0.345501\pi$
$241$	14.4853	0.933079	0.466539	+	0.884500i	$0.345501\pi$
$242$	1.58579	0.101938
$243$	−9.89949	−0.635053
$244$	−30.6274	−1.96072
$245$	−16.3137	−1.04224
$246$	−10.8284	−0.690395
$247$	0	0
$248$	−7.75736	−0.492593
$249$	−4.48528	−0.284243
$250$	−2.41421	−0.152688
$251$	19.7990	1.24970	0.624851	−	0.780744i	$-0.285160\pi$
$251$	19.7990	1.24970	0.624851	+	0.780744i	$0.285160\pi$
$252$	18.4853	1.16446
$253$	−4.82843	−0.303561
$254$	−15.8995	−0.997623
$255$	1.17157	0.0733667
$256$	−29.9706	−1.87316
$257$	27.6569	1.72519	0.862594	−	0.505898i	$-0.168839\pi$
$257$	27.6569	1.72519	0.862594	+	0.505898i	$0.168839\pi$
$258$	37.7990	2.35326
$259$	−40.9706	−2.54579
$260$	0	0
$261$	5.65685	0.350150
$262$	−40.9706	−2.53117
$263$	−10.5858	−0.652748	−0.326374	−	0.945241i	$-0.605827\pi$
$263$	−10.5858	−0.652748	−0.326374	+	0.945241i	$0.605827\pi$
$264$	21.3137	1.31177
$265$	−2.48528	−0.152670
$266$	6.82843	0.418678
$267$	8.48528	0.519291
$268$	7.65685	0.467717
$269$	−25.3137	−1.54340	−0.771702	−	0.635984i	$-0.780594\pi$
$269$	−25.3137	−1.54340	−0.771702	+	0.635984i	$0.780594\pi$
$270$	−13.6569	−0.831130
$271$	−26.7279	−1.62361	−0.811803	−	0.583932i	$-0.801514\pi$
$271$	−26.7279	−1.62361	−0.811803	+	0.583932i	$0.801514\pi$
$272$	2.48528	0.150692
$273$	0	0
$274$	41.7990	2.52517
$275$	−3.41421	−0.205885
$276$	−7.65685	−0.460888
$277$	−12.8284	−0.770785	−0.385393	−	0.922753i	$-0.625934\pi$
$277$	−12.8284	−0.770785	−0.385393	+	0.922753i	$0.625934\pi$
$278$	10.8284	0.649446
$279$	1.75736	0.105210
$280$	21.3137	1.27374
$281$	−21.7990	−1.30042	−0.650209	−	0.759755i	$-0.725319\pi$
$281$	−21.7990	−1.30042	−0.650209	+	0.759755i	$0.725319\pi$
$282$	−16.4853	−0.981684
$283$	−16.7279	−0.994372	−0.497186	−	0.867644i	$-0.665633\pi$
$283$	−16.7279	−0.994372	−0.497186	+	0.867644i	$0.665633\pi$
$284$	−45.5563	−2.70327
$285$	−0.828427	−0.0490718
$286$	0	0
$287$	−15.3137	−0.903940
$288$	1.58579	0.0934434
$289$	−16.3137	−0.959630
$290$	13.6569	0.801958
$291$	−10.8284	−0.634774
$292$	−32.4853	−1.90106
$293$	−26.1421	−1.52724	−0.763620	−	0.645666i	$-0.776580\pi$
$293$	−26.1421	−1.52724	−0.763620	+	0.645666i	$0.776580\pi$
$294$	−55.6985	−3.24840
$295$	1.75736	0.102317
$296$	37.4558	2.17708
$297$	−19.3137	−1.12070
$298$	28.1421	1.63023
$299$	0	0
$300$	−5.41421	−0.312590
$301$	53.4558	3.08114
$302$	−23.5563	−1.35552
$303$	5.17157	0.297099
$304$	−1.75736	−0.100791
$305$	8.00000	0.458079
$306$	−2.00000	−0.114332
$307$	−24.8284	−1.41703	−0.708517	−	0.705694i	$-0.750635\pi$
$307$	−24.8284	−1.41703	−0.708517	+	0.705694i	$0.750635\pi$
$308$	63.1127	3.59618
$309$	−20.6274	−1.17345
$310$	4.24264	0.240966
$311$	8.48528	0.481156	0.240578	−	0.970630i	$-0.422663\pi$
$311$	8.48528	0.481156	0.240578	+	0.970630i	$0.422663\pi$
$312$	0	0
$313$	−4.82843	−0.272919	−0.136459	−	0.990646i	$-0.543572\pi$
$313$	−4.82843	−0.272919	−0.136459	+	0.990646i	$0.543572\pi$
$314$	43.4558	2.45236
$315$	−4.82843	−0.272051
$316$	−32.4853	−1.82744
$317$	2.14214	0.120314	0.0601572	−	0.998189i	$-0.480840\pi$
$317$	2.14214	0.120314	0.0601572	+	0.998189i	$0.480840\pi$
$318$	−8.48528	−0.475831
$319$	19.3137	1.08136
$320$	9.82843	0.549426
$321$	13.3137	0.743099
$322$	−16.4853	−0.918689
$323$	−0.485281	−0.0270018
$324$	−19.1421	−1.06345
$325$	0	0
$326$	−45.7990	−2.53657
$327$	−2.82843	−0.156412
$328$	14.0000	0.773021
$329$	−23.3137	−1.28533
$330$	−11.6569	−0.641689
$331$	26.0416	1.43138	0.715689	−	0.698419i	$-0.246113\pi$
$331$	26.0416	1.43138	0.715689	+	0.698419i	$0.246113\pi$
$332$	12.1421	0.666386
$333$	−8.48528	−0.464991
$334$	7.65685	0.418964
$335$	−2.00000	−0.109272
$336$	20.4853	1.11756
$337$	12.8284	0.698809	0.349404	−	0.936972i	$-0.386384\pi$
$337$	12.8284	0.698809	0.349404	+	0.936972i	$0.386384\pi$
$338$	0	0
$339$	12.4853	0.678107
$340$	−3.17157	−0.172003
$341$	6.00000	0.324918
$342$	1.41421	0.0764719
$343$	−44.9706	−2.42818
$344$	−48.8701	−2.63490
$345$	2.00000	0.107676
$346$	40.6274	2.18414
$347$	−4.24264	−0.227757	−0.113878	−	0.993495i	$-0.536327\pi$
$347$	−4.24264	−0.227757	−0.113878	+	0.993495i	$0.536327\pi$
$348$	30.6274	1.64180
$349$	−18.4853	−0.989494	−0.494747	−	0.869037i	$-0.664739\pi$
$349$	−18.4853	−0.989494	−0.494747	+	0.869037i	$0.664739\pi$
$350$	−11.6569	−0.623085
$351$	0	0
$352$	5.41421	0.288579
$353$	−14.8284	−0.789238	−0.394619	−	0.918845i	$-0.629123\pi$
$353$	−14.8284	−0.789238	−0.394619	+	0.918845i	$0.629123\pi$
$354$	6.00000	0.318896
$355$	11.8995	0.631560
$356$	−22.9706	−1.21744
$357$	5.65685	0.299392
$358$	13.6569	0.721787
$359$	8.10051	0.427528	0.213764	−	0.976885i	$-0.431428\pi$
$359$	8.10051	0.427528	0.213764	+	0.976885i	$0.431428\pi$
$360$	4.41421	0.232649
$361$	−18.6569	−0.981940
$362$	0	0
$363$	−0.928932	−0.0487563
$364$	0	0
$365$	8.48528	0.444140
$366$	27.3137	1.42771
$367$	35.5563	1.85603	0.928013	−	0.372547i	$-0.121516\pi$
$367$	35.5563	1.85603	0.928013	+	0.372547i	$0.121516\pi$
$368$	4.24264	0.221163
$369$	−3.17157	−0.165105
$370$	−20.4853	−1.06498
$371$	−12.0000	−0.623009
$372$	9.51472	0.493315
$373$	−2.68629	−0.139091	−0.0695455	−	0.997579i	$-0.522155\pi$
$373$	−2.68629	−0.139091	−0.0695455	+	0.997579i	$0.522155\pi$
$374$	−6.82843	−0.353090
$375$	1.41421	0.0730297
$376$	21.3137	1.09917
$377$	0	0
$378$	−65.9411	−3.39165
$379$	−29.0711	−1.49328	−0.746640	−	0.665228i	$-0.768334\pi$
$379$	−29.0711	−1.49328	−0.746640	+	0.665228i	$0.768334\pi$
$380$	2.24264	0.115045
$381$	9.31371	0.477156
$382$	5.65685	0.289430
$383$	29.1127	1.48759	0.743795	−	0.668408i	$-0.233024\pi$
$383$	29.1127	1.48759	0.743795	+	0.668408i	$0.233024\pi$
$384$	29.0711	1.48353
$385$	−16.4853	−0.840168
$386$	−10.4853	−0.533687
$387$	11.0711	0.562774
$388$	29.3137	1.48818
$389$	28.6274	1.45147	0.725734	−	0.687976i	$-0.241500\pi$
$389$	28.6274	1.45147	0.725734	+	0.687976i	$0.241500\pi$
$390$	0	0
$391$	1.17157	0.0592490
$392$	72.0122	3.63717
$393$	24.0000	1.21064
$394$	−26.4853	−1.33431
$395$	8.48528	0.426941
$396$	13.0711	0.656846
$397$	−11.7990	−0.592174	−0.296087	−	0.955161i	$-0.595682\pi$
$397$	−11.7990	−0.592174	−0.296087	+	0.955161i	$0.595682\pi$
$398$	9.65685	0.484054
$399$	−4.00000	−0.200250
$400$	3.00000	0.150000
$401$	5.31371	0.265354	0.132677	−	0.991159i	$-0.457643\pi$
$401$	5.31371	0.265354	0.132677	+	0.991159i	$0.457643\pi$
$402$	−6.82843	−0.340571
$403$	0	0
$404$	−14.0000	−0.696526
$405$	5.00000	0.248452
$406$	65.9411	3.27260
$407$	−28.9706	−1.43602
$408$	−5.17157	−0.256031
$409$	−7.17157	−0.354611	−0.177306	−	0.984156i	$-0.556738\pi$
$409$	−7.17157	−0.354611	−0.177306	+	0.984156i	$0.556738\pi$
$410$	−7.65685	−0.378145
$411$	−24.4853	−1.20777
$412$	55.8406	2.75107
$413$	8.48528	0.417533
$414$	−3.41421	−0.167799
$415$	−3.17157	−0.155686
$416$	0	0
$417$	−6.34315	−0.310625
$418$	4.82843	0.236166
$419$	10.8284	0.529003	0.264502	−	0.964385i	$-0.414793\pi$
$419$	10.8284	0.529003	0.264502	+	0.964385i	$0.414793\pi$
$420$	−26.1421	−1.27561
$421$	34.9706	1.70436	0.852180	−	0.523248i	$-0.175280\pi$
$421$	34.9706	1.70436	0.852180	+	0.523248i	$0.175280\pi$
$422$	8.00000	0.389434
$423$	−4.82843	−0.234766
$424$	10.9706	0.532778
$425$	0.828427	0.0401846
$426$	40.6274	1.96840
$427$	38.6274	1.86931
$428$	−36.0416	−1.74214
$429$	0	0
$430$	26.7279	1.28893
$431$	−40.3848	−1.94527	−0.972633	−	0.232346i	$-0.925360\pi$
$431$	−40.3848	−1.94527	−0.972633	+	0.232346i	$0.925360\pi$
$432$	16.9706	0.816497
$433$	7.65685	0.367965	0.183982	−	0.982930i	$-0.441101\pi$
$433$	7.65685	0.367965	0.183982	+	0.982930i	$0.441101\pi$
$434$	20.4853	0.983325
$435$	−8.00000	−0.383571
$436$	7.65685	0.366697
$437$	−0.828427	−0.0396290
$438$	28.9706	1.38427
$439$	0.970563	0.0463224	0.0231612	−	0.999732i	$-0.492627\pi$
$439$	0.970563	0.0463224	0.0231612	+	0.999732i	$0.492627\pi$
$440$	15.0711	0.718485
$441$	−16.3137	−0.776843
$442$	0	0
$443$	−9.41421	−0.447283	−0.223641	−	0.974671i	$-0.571794\pi$
$443$	−9.41421	−0.447283	−0.223641	+	0.974671i	$0.571794\pi$
$444$	−45.9411	−2.18027
$445$	6.00000	0.284427
$446$	−22.9706	−1.08769
$447$	−16.4853	−0.779727
$448$	47.4558	2.24208
$449$	33.1127	1.56268	0.781342	−	0.624103i	$-0.214535\pi$
$449$	33.1127	1.56268	0.781342	+	0.624103i	$0.214535\pi$
$450$	−2.41421	−0.113807
$451$	−10.8284	−0.509891
$452$	−33.7990	−1.58977
$453$	13.7990	0.648333
$454$	39.4558	1.85175
$455$	0	0
$456$	3.65685	0.171248
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	11.6569	0.544689
$459$	4.68629	0.218737
$460$	−5.41421	−0.252439
$461$	−9.51472	−0.443145	−0.221572	−	0.975144i	$-0.571119\pi$
$461$	−9.51472	−0.443145	−0.221572	+	0.975144i	$0.571119\pi$
$462$	−56.2843	−2.61858
$463$	4.34315	0.201843	0.100922	−	0.994894i	$-0.467821\pi$
$463$	4.34315	0.201843	0.100922	+	0.994894i	$0.467821\pi$
$464$	−16.9706	−0.787839
$465$	−2.48528	−0.115252
$466$	−49.7990	−2.30689
$467$	−13.4142	−0.620736	−0.310368	−	0.950617i	$-0.600452\pi$
$467$	−13.4142	−0.620736	−0.310368	+	0.950617i	$0.600452\pi$
$468$	0	0
$469$	−9.65685	−0.445912
$470$	−11.6569	−0.537691
$471$	−25.4558	−1.17294
$472$	−7.75736	−0.357061
$473$	37.7990	1.73800
$474$	28.9706	1.33066
$475$	−0.585786	−0.0268777
$476$	−15.3137	−0.701903
$477$	−2.48528	−0.113793
$478$	8.24264	0.377010
$479$	30.7279	1.40399	0.701997	−	0.712180i	$-0.252292\pi$
$479$	30.7279	1.40399	0.701997	+	0.712180i	$0.252292\pi$
$480$	−2.24264	−0.102362
$481$	0	0
$482$	34.9706	1.59287
$483$	9.65685	0.439402
$484$	2.51472	0.114305
$485$	−7.65685	−0.347680
$486$	−23.8995	−1.08410
$487$	10.9706	0.497124	0.248562	−	0.968616i	$-0.420042\pi$
$487$	10.9706	0.497124	0.248562	+	0.968616i	$0.420042\pi$
$488$	−35.3137	−1.59858
$489$	26.8284	1.21322
$490$	−39.3848	−1.77922
$491$	5.17157	0.233390	0.116695	−	0.993168i	$-0.462770\pi$
$491$	5.17157	0.233390	0.116695	+	0.993168i	$0.462770\pi$
$492$	−17.1716	−0.774154
$493$	−4.68629	−0.211060
$494$	0	0
$495$	−3.41421	−0.153457
$496$	−5.27208	−0.236723
$497$	57.4558	2.57725
$498$	−10.8284	−0.485233
$499$	−41.5563	−1.86032	−0.930159	−	0.367157i	$-0.880331\pi$
$499$	−41.5563	−1.86032	−0.930159	+	0.367157i	$0.880331\pi$
$500$	−3.82843	−0.171212
$501$	−4.48528	−0.200388
$502$	47.7990	2.13337
$503$	37.8995	1.68985	0.844927	−	0.534881i	$-0.179644\pi$
$503$	37.8995	1.68985	0.844927	+	0.534881i	$0.179644\pi$
$504$	21.3137	0.949388
$505$	3.65685	0.162728
$506$	−11.6569	−0.518210
$507$	0	0
$508$	−25.2132	−1.11866
$509$	−41.1127	−1.82229	−0.911144	−	0.412088i	$-0.864800\pi$
$509$	−41.1127	−1.82229	−0.911144	+	0.412088i	$0.864800\pi$
$510$	2.82843	0.125245
$511$	40.9706	1.81243
$512$	−31.2426	−1.38074
$513$	−3.31371	−0.146304
$514$	66.7696	2.94508
$515$	−14.5858	−0.642727
$516$	59.9411	2.63876
$517$	−16.4853	−0.725022
$518$	−98.9117	−4.34593
$519$	−23.7990	−1.04466
$520$	0	0
$521$	−17.6569	−0.773561	−0.386780	−	0.922172i	$-0.626413\pi$
$521$	−17.6569	−0.773561	−0.386780	+	0.922172i	$0.626413\pi$
$522$	13.6569	0.597744
$523$	−19.7574	−0.863929	−0.431965	−	0.901891i	$-0.642179\pi$
$523$	−19.7574	−0.863929	−0.431965	+	0.901891i	$0.642179\pi$
$524$	−64.9706	−2.83825
$525$	6.82843	0.298017
$526$	−25.5563	−1.11431
$527$	−1.45584	−0.0634176
$528$	14.4853	0.630391
$529$	−21.0000	−0.913043
$530$	−6.00000	−0.260623
$531$	1.75736	0.0762629
$532$	10.8284	0.469472
$533$	0	0
$534$	20.4853	0.886485
$535$	9.41421	0.407012
$536$	8.82843	0.381330
$537$	−8.00000	−0.345225
$538$	−61.1127	−2.63476
$539$	−55.6985	−2.39910
$540$	−21.6569	−0.931963
$541$	7.17157	0.308330	0.154165	−	0.988045i	$-0.450731\pi$
$541$	7.17157	0.308330	0.154165	+	0.988045i	$0.450731\pi$
$542$	−64.5269	−2.77167
$543$	0	0
$544$	−1.31371	−0.0563248
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	13.2132	0.564956	0.282478	−	0.959274i	$-0.408844\pi$
$547$	13.2132	0.564956	0.282478	+	0.959274i	$0.408844\pi$
$548$	66.2843	2.83152
$549$	8.00000	0.341432
$550$	−8.24264	−0.351467
$551$	3.31371	0.141169
$552$	−8.82843	−0.375763
$553$	40.9706	1.74225
$554$	−30.9706	−1.31581
$555$	12.0000	0.509372
$556$	17.1716	0.728237
$557$	35.7990	1.51685	0.758426	−	0.651759i	$-0.225969\pi$
$557$	35.7990	1.51685	0.758426	+	0.651759i	$0.225969\pi$
$558$	4.24264	0.179605
$559$	0	0
$560$	14.4853	0.612115
$561$	4.00000	0.168880
$562$	−52.6274	−2.21995
$563$	−7.75736	−0.326934	−0.163467	−	0.986549i	$-0.552268\pi$
$563$	−7.75736	−0.326934	−0.163467	+	0.986549i	$0.552268\pi$
$564$	−26.1421	−1.10078
$565$	8.82843	0.371415
$566$	−40.3848	−1.69750
$567$	24.1421	1.01387
$568$	−52.5269	−2.20398
$569$	−10.3431	−0.433607	−0.216804	−	0.976215i	$-0.569563\pi$
$569$	−10.3431	−0.433607	−0.216804	+	0.976215i	$0.569563\pi$
$570$	−2.00000	−0.0837708
$571$	−11.5147	−0.481876	−0.240938	−	0.970541i	$-0.577455\pi$
$571$	−11.5147	−0.481876	−0.240938	+	0.970541i	$0.577455\pi$
$572$	0	0
$573$	−3.31371	−0.138432
$574$	−36.9706	−1.54312
$575$	1.41421	0.0589768
$576$	9.82843	0.409518
$577$	34.8284	1.44993	0.724963	−	0.688788i	$-0.241857\pi$
$577$	34.8284	1.44993	0.724963	+	0.688788i	$0.241857\pi$
$578$	−39.3848	−1.63819
$579$	6.14214	0.255258
$580$	21.6569	0.899252
$581$	−15.3137	−0.635320
$582$	−26.1421	−1.08363
$583$	−8.48528	−0.351424
$584$	−37.4558	−1.54993
$585$	0	0
$586$	−63.1127	−2.60716
$587$	−20.3431	−0.839651	−0.419826	−	0.907605i	$-0.637909\pi$
$587$	−20.3431	−0.839651	−0.419826	+	0.907605i	$0.637909\pi$
$588$	−88.3259	−3.64250
$589$	1.02944	0.0424172
$590$	4.24264	0.174667
$591$	15.5147	0.638190
$592$	25.4558	1.04623
$593$	24.6274	1.01133	0.505663	−	0.862731i	$-0.331248\pi$
$593$	24.6274	1.01133	0.505663	+	0.862731i	$0.331248\pi$
$594$	−46.6274	−1.91315
$595$	4.00000	0.163984
$596$	44.6274	1.82801
$597$	−5.65685	−0.231520
$598$	0	0
$599$	25.4558	1.04010	0.520049	−	0.854137i	$-0.325914\pi$
$599$	25.4558	1.04010	0.520049	+	0.854137i	$0.325914\pi$
$600$	−6.24264	−0.254855
$601$	−44.6274	−1.82039	−0.910195	−	0.414180i	$-0.864069\pi$
$601$	−44.6274	−1.82039	−0.910195	+	0.414180i	$0.864069\pi$
$602$	129.054	5.25984
$603$	−2.00000	−0.0814463
$604$	−37.3553	−1.51997
$605$	−0.656854	−0.0267049
$606$	12.4853	0.507180
$607$	31.7574	1.28899	0.644496	−	0.764608i	$-0.277067\pi$
$607$	31.7574	1.28899	0.644496	+	0.764608i	$0.277067\pi$
$608$	0.928932	0.0376732
$609$	−38.6274	−1.56526
$610$	19.3137	0.781989
$611$	0	0
$612$	−3.17157	−0.128203
$613$	14.6863	0.593174	0.296587	−	0.955006i	$-0.404152\pi$
$613$	14.6863	0.593174	0.296587	+	0.955006i	$0.404152\pi$
$614$	−59.9411	−2.41903
$615$	4.48528	0.180864
$616$	72.7696	2.93197
$617$	−10.9706	−0.441658	−0.220829	−	0.975313i	$-0.570876\pi$
$617$	−10.9706	−0.441658	−0.220829	+	0.975313i	$0.570876\pi$
$618$	−49.7990	−2.00321
$619$	−1.75736	−0.0706342	−0.0353171	−	0.999376i	$-0.511244\pi$
$619$	−1.75736	−0.0706342	−0.0353171	+	0.999376i	$0.511244\pi$
$620$	6.72792	0.270200
$621$	8.00000	0.321029
$622$	20.4853	0.821385
$623$	28.9706	1.16068
$624$	0	0
$625$	1.00000	0.0400000
$626$	−11.6569	−0.465902
$627$	−2.82843	−0.112956
$628$	68.9117	2.74988
$629$	7.02944	0.280282
$630$	−11.6569	−0.464420
$631$	9.75736	0.388434	0.194217	−	0.980959i	$-0.437783\pi$
$631$	9.75736	0.388434	0.194217	+	0.980959i	$0.437783\pi$
$632$	−37.4558	−1.48991
$633$	−4.68629	−0.186263
$634$	5.17157	0.205389
$635$	6.58579	0.261349
$636$	−13.4558	−0.533559
$637$	0	0
$638$	46.6274	1.84600
$639$	11.8995	0.470737
$640$	20.5563	0.812561
$641$	47.6569	1.88233	0.941166	−	0.337944i	$-0.109731\pi$
$641$	47.6569	1.88233	0.941166	+	0.337944i	$0.109731\pi$
$642$	32.1421	1.26855
$643$	−9.51472	−0.375224	−0.187612	−	0.982243i	$-0.560075\pi$
$643$	−9.51472	−0.375224	−0.187612	+	0.982243i	$0.560075\pi$
$644$	−26.1421	−1.03014
$645$	−15.6569	−0.616488
$646$	−1.17157	−0.0460949
$647$	9.41421	0.370111	0.185055	−	0.982728i	$-0.440754\pi$
$647$	9.41421	0.370111	0.185055	+	0.982728i	$0.440754\pi$
$648$	−22.0711	−0.867033
$649$	6.00000	0.235521
$650$	0	0
$651$	−12.0000	−0.470317
$652$	−72.6274	−2.84431
$653$	46.9706	1.83810	0.919050	−	0.394141i	$-0.128958\pi$
$653$	46.9706	1.83810	0.919050	+	0.394141i	$0.128958\pi$
$654$	−6.82843	−0.267013
$655$	16.9706	0.663095
$656$	9.51472	0.371487
$657$	8.48528	0.331042
$658$	−56.2843	−2.19419
$659$	17.8579	0.695644	0.347822	−	0.937561i	$-0.386921\pi$
$659$	17.8579	0.695644	0.347822	+	0.937561i	$0.386921\pi$
$660$	−18.4853	−0.719539
$661$	−29.5980	−1.15123	−0.575614	−	0.817722i	$-0.695237\pi$
$661$	−29.5980	−1.15123	−0.575614	+	0.817722i	$0.695237\pi$
$662$	62.8701	2.44351
$663$	0	0
$664$	14.0000	0.543305
$665$	−2.82843	−0.109682
$666$	−20.4853	−0.793789
$667$	−8.00000	−0.309761
$668$	12.1421	0.469793
$669$	13.4558	0.520233
$670$	−4.82843	−0.186538
$671$	27.3137	1.05443
$672$	−10.8284	−0.417716
$673$	6.48528	0.249989	0.124995	−	0.992157i	$-0.460109\pi$
$673$	6.48528	0.249989	0.124995	+	0.992157i	$0.460109\pi$
$674$	30.9706	1.19294
$675$	5.65685	0.217732
$676$	0	0
$677$	−20.1421	−0.774125	−0.387063	−	0.922053i	$-0.626510\pi$
$677$	−20.1421	−0.774125	−0.387063	+	0.922053i	$0.626510\pi$
$678$	30.1421	1.15760
$679$	−36.9706	−1.41880
$680$	−3.65685	−0.140234
$681$	−23.1127	−0.885681
$682$	14.4853	0.554670
$683$	−10.6863	−0.408900	−0.204450	−	0.978877i	$-0.565541\pi$
$683$	−10.6863	−0.408900	−0.204450	+	0.978877i	$0.565541\pi$
$684$	2.24264	0.0857495
$685$	−17.3137	−0.661523
$686$	−108.569	−4.14517
$687$	−6.82843	−0.260521
$688$	−33.2132	−1.26624
$689$	0	0
$690$	4.82843	0.183815
$691$	−6.92893	−0.263589	−0.131795	−	0.991277i	$-0.542074\pi$
$691$	−6.92893	−0.263589	−0.131795	+	0.991277i	$0.542074\pi$
$692$	64.4264	2.44912
$693$	−16.4853	−0.626224
$694$	−10.2426	−0.388805
$695$	−4.48528	−0.170136
$696$	35.3137	1.33856
$697$	2.62742	0.0995205
$698$	−44.6274	−1.68917
$699$	29.1716	1.10337
$700$	−18.4853	−0.698678
$701$	14.6863	0.554694	0.277347	−	0.960770i	$-0.410545\pi$
$701$	14.6863	0.554694	0.277347	+	0.960770i	$0.410545\pi$
$702$	0	0
$703$	−4.97056	−0.187468
$704$	33.5563	1.26470
$705$	6.82843	0.257173
$706$	−35.7990	−1.34731
$707$	17.6569	0.664054
$708$	9.51472	0.357585
$709$	−45.1127	−1.69424	−0.847121	−	0.531399i	$-0.821666\pi$
$709$	−45.1127	−1.69424	−0.847121	+	0.531399i	$0.821666\pi$
$710$	28.7279	1.07814
$711$	8.48528	0.318223
$712$	−26.4853	−0.992578
$713$	−2.48528	−0.0930745
$714$	13.6569	0.511095
$715$	0	0
$716$	21.6569	0.809355
$717$	−4.82843	−0.180321
$718$	19.5563	0.729836
$719$	28.9706	1.08042	0.540210	−	0.841530i	$-0.318345\pi$
$719$	28.9706	1.08042	0.540210	+	0.841530i	$0.318345\pi$
$720$	3.00000	0.111803
$721$	−70.4264	−2.62282
$722$	−45.0416	−1.67628
$723$	−20.4853	−0.761856
$724$	0	0
$725$	−5.65685	−0.210090
$726$	−2.24264	−0.0832322
$727$	−51.3553	−1.90466	−0.952332	−	0.305063i	$-0.901322\pi$
$727$	−51.3553	−1.90466	−0.952332	+	0.305063i	$0.901322\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	20.4853	0.758194
$731$	−9.17157	−0.339223
$732$	43.3137	1.60092
$733$	−21.3137	−0.787240	−0.393620	−	0.919273i	$-0.628777\pi$
$733$	−21.3137	−0.787240	−0.393620	+	0.919273i	$0.628777\pi$
$734$	85.8406	3.16844
$735$	23.0711	0.850989
$736$	−2.24264	−0.0826648
$737$	−6.82843	−0.251528
$738$	−7.65685	−0.281853
$739$	−5.27208	−0.193937	−0.0969683	−	0.995287i	$-0.530915\pi$
$739$	−5.27208	−0.193937	−0.0969683	+	0.995287i	$0.530915\pi$
$740$	−32.4853	−1.19418
$741$	0	0
$742$	−28.9706	−1.06354
$743$	21.5147	0.789298	0.394649	−	0.918832i	$-0.370866\pi$
$743$	21.5147	0.789298	0.394649	+	0.918832i	$0.370866\pi$
$744$	10.9706	0.402200
$745$	−11.6569	−0.427074
$746$	−6.48528	−0.237443
$747$	−3.17157	−0.116042
$748$	−10.8284	−0.395927
$749$	45.4558	1.66092
$750$	3.41421	0.124669
$751$	−27.5147	−1.00403	−0.502013	−	0.864860i	$-0.667407\pi$
$751$	−27.5147	−1.00403	−0.502013	+	0.864860i	$0.667407\pi$
$752$	14.4853	0.528224
$753$	−28.0000	−1.02038
$754$	0	0
$755$	9.75736	0.355107
$756$	−104.569	−3.80312
$757$	−24.1421	−0.877461	−0.438730	−	0.898619i	$-0.644572\pi$
$757$	−24.1421	−0.877461	−0.438730	+	0.898619i	$0.644572\pi$
$758$	−70.1838	−2.54919
$759$	6.82843	0.247856
$760$	2.58579	0.0937963
$761$	−8.62742	−0.312744	−0.156372	−	0.987698i	$-0.549980\pi$
$761$	−8.62742	−0.312744	−0.156372	+	0.987698i	$0.549980\pi$
$762$	22.4853	0.814556
$763$	−9.65685	−0.349602
$764$	8.97056	0.324544
$765$	0.828427	0.0299518
$766$	70.2843	2.53947
$767$	0	0
$768$	42.3848	1.52943
$769$	22.9706	0.828340	0.414170	−	0.910200i	$-0.364072\pi$
$769$	22.9706	0.828340	0.414170	+	0.910200i	$0.364072\pi$
$770$	−39.7990	−1.43426
$771$	−39.1127	−1.40861
$772$	−16.6274	−0.598434
$773$	22.1421	0.796397	0.398199	−	0.917299i	$-0.369635\pi$
$773$	22.1421	0.796397	0.398199	+	0.917299i	$0.369635\pi$
$774$	26.7279	0.960715
$775$	−1.75736	−0.0631262
$776$	33.7990	1.21331
$777$	57.9411	2.07863
$778$	69.1127	2.47781
$779$	−1.85786	−0.0665649
$780$	0	0
$781$	40.6274	1.45376
$782$	2.82843	0.101144
$783$	−32.0000	−1.14359
$784$	48.9411	1.74790
$785$	−18.0000	−0.642448
$786$	57.9411	2.06669
$787$	−22.4853	−0.801514	−0.400757	−	0.916184i	$-0.631253\pi$
$787$	−22.4853	−0.801514	−0.400757	+	0.916184i	$0.631253\pi$
$788$	−42.0000	−1.49619
$789$	14.9706	0.532966
$790$	20.4853	0.728834
$791$	42.6274	1.51566
$792$	15.0711	0.535527
$793$	0	0
$794$	−28.4853	−1.01090
$795$	3.51472	0.124654
$796$	15.3137	0.542780
$797$	−22.9706	−0.813659	−0.406830	−	0.913504i	$-0.633366\pi$
$797$	−22.9706	−0.813659	−0.406830	+	0.913504i	$0.633366\pi$
$798$	−9.65685	−0.341849
$799$	4.00000	0.141510
$800$	−1.58579	−0.0560660
$801$	6.00000	0.212000
$802$	12.8284	0.452988
$803$	28.9706	1.02235
$804$	−10.8284	−0.381889
$805$	6.82843	0.240670
$806$	0	0
$807$	35.7990	1.26018
$808$	−16.1421	−0.567878
$809$	45.2548	1.59108	0.795538	−	0.605904i	$-0.207189\pi$
$809$	45.2548	1.59108	0.795538	+	0.605904i	$0.207189\pi$
$810$	12.0711	0.424134
$811$	28.3848	0.996724	0.498362	−	0.866969i	$-0.333935\pi$
$811$	28.3848	0.996724	0.498362	+	0.866969i	$0.333935\pi$
$812$	104.569	3.66964
$813$	37.7990	1.32567
$814$	−69.9411	−2.45144
$815$	18.9706	0.664510
$816$	−3.51472	−0.123040
$817$	6.48528	0.226891
$818$	−17.3137	−0.605360
$819$	0	0
$820$	−12.1421	−0.424022
$821$	51.2548	1.78881	0.894403	−	0.447262i	$-0.147601\pi$
$821$	51.2548	1.78881	0.894403	+	0.447262i	$0.147601\pi$
$822$	−59.1127	−2.06179
$823$	−2.38478	−0.0831281	−0.0415640	−	0.999136i	$-0.513234\pi$
$823$	−2.38478	−0.0831281	−0.0415640	+	0.999136i	$0.513234\pi$
$824$	64.3848	2.24295
$825$	4.82843	0.168104
$826$	20.4853	0.712774
$827$	−56.1421	−1.95225	−0.976127	−	0.217202i	$-0.930307\pi$
$827$	−56.1421	−1.95225	−0.976127	+	0.217202i	$0.930307\pi$
$828$	−5.41421	−0.188157
$829$	40.9706	1.42297	0.711483	−	0.702703i	$-0.248024\pi$
$829$	40.9706	1.42297	0.711483	+	0.702703i	$0.248024\pi$
$830$	−7.65685	−0.265773
$831$	18.1421	0.629344
$832$	0	0
$833$	13.5147	0.468257
$834$	−15.3137	−0.530270
$835$	−3.17157	−0.109757
$836$	7.65685	0.264818
$837$	−9.94113	−0.343616
$838$	26.1421	0.903065
$839$	−6.72792	−0.232274	−0.116137	−	0.993233i	$-0.537051\pi$
$839$	−6.72792	−0.232274	−0.116137	+	0.993233i	$0.537051\pi$
$840$	−30.1421	−1.04000
$841$	3.00000	0.103448
$842$	84.4264	2.90953
$843$	30.8284	1.06179
$844$	12.6863	0.436680
$845$	0	0
$846$	−11.6569	−0.400771
$847$	−3.17157	−0.108977
$848$	7.45584	0.256035
$849$	23.6569	0.811901
$850$	2.00000	0.0685994
$851$	12.0000	0.411355
$852$	64.4264	2.20721
$853$	13.4558	0.460719	0.230360	−	0.973106i	$-0.426010\pi$
$853$	13.4558	0.460719	0.230360	+	0.973106i	$0.426010\pi$
$854$	93.2548	3.19111
$855$	−0.585786	−0.0200335
$856$	−41.5563	−1.42037
$857$	11.6569	0.398191	0.199095	−	0.979980i	$-0.436200\pi$
$857$	11.6569	0.398191	0.199095	+	0.979980i	$0.436200\pi$
$858$	0	0
$859$	−27.7990	−0.948489	−0.474245	−	0.880393i	$-0.657279\pi$
$859$	−27.7990	−0.948489	−0.474245	+	0.880393i	$0.657279\pi$
$860$	42.3848	1.44531
$861$	21.6569	0.738064
$862$	−97.4975	−3.32078
$863$	31.4558	1.07077	0.535385	−	0.844608i	$-0.320167\pi$
$863$	31.4558	1.07077	0.535385	+	0.844608i	$0.320167\pi$
$864$	−8.97056	−0.305185
$865$	−16.8284	−0.572184
$866$	18.4853	0.628155
$867$	23.0711	0.783535
$868$	32.4853	1.10262
$869$	28.9706	0.982759
$870$	−19.3137	−0.654796
$871$	0	0
$872$	8.82843	0.298968
$873$	−7.65685	−0.259145
$874$	−2.00000	−0.0676510
$875$	4.82843	0.163231
$876$	45.9411	1.55221
$877$	−25.3137	−0.854783	−0.427392	−	0.904067i	$-0.640567\pi$
$877$	−25.3137	−0.854783	−0.427392	+	0.904067i	$0.640567\pi$
$878$	2.34315	0.0790773
$879$	36.9706	1.24699
$880$	10.2426	0.345279
$881$	19.0294	0.641118	0.320559	−	0.947229i	$-0.396129\pi$
$881$	19.0294	0.641118	0.320559	+	0.947229i	$0.396129\pi$
$882$	−39.3848	−1.32615
$883$	−23.7574	−0.799499	−0.399749	−	0.916624i	$-0.630903\pi$
$883$	−23.7574	−0.799499	−0.399749	+	0.916624i	$0.630903\pi$
$884$	0	0
$885$	−2.48528	−0.0835418
$886$	−22.7279	−0.763559
$887$	−22.3848	−0.751607	−0.375804	−	0.926699i	$-0.622633\pi$
$887$	−22.3848	−0.751607	−0.375804	+	0.926699i	$0.622633\pi$
$888$	−52.9706	−1.77758
$889$	31.7990	1.06650
$890$	14.4853	0.485548
$891$	17.0711	0.571902
$892$	−36.4264	−1.21965
$893$	−2.82843	−0.0946497
$894$	−39.7990	−1.33108
$895$	−5.65685	−0.189088
$896$	99.2548	3.31587
$897$	0	0
$898$	79.9411	2.66767
$899$	9.94113	0.331555
$900$	−3.82843	−0.127614
$901$	2.05887	0.0685911
$902$	−26.1421	−0.870438
$903$	−75.5980	−2.51574
$904$	−38.9706	−1.29614
$905$	0	0
$906$	33.3137	1.10677
$907$	9.21320	0.305919	0.152960	−	0.988232i	$-0.451120\pi$
$907$	9.21320	0.305919	0.152960	+	0.988232i	$0.451120\pi$
$908$	62.5685	2.07641
$909$	3.65685	0.121290
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	2.48528	0.0822959
$913$	−10.8284	−0.358369
$914$	43.4558	1.43739
$915$	−11.3137	−0.374020
$916$	18.4853	0.610771
$917$	81.9411	2.70593
$918$	11.3137	0.373408
$919$	0.485281	0.0160080	0.00800398	−	0.999968i	$-0.497452\pi$
$919$	0.485281	0.0160080	0.00800398	+	0.999968i	$0.497452\pi$
$920$	−6.24264	−0.205814
$921$	35.1127	1.15700
$922$	−22.9706	−0.756495
$923$	0	0
$924$	−89.2548	−2.93627
$925$	8.48528	0.278994
$926$	10.4853	0.344568
$927$	−14.5858	−0.479060
$928$	8.97056	0.294473
$929$	−16.8284	−0.552123	−0.276061	−	0.961140i	$-0.589029\pi$
$929$	−16.8284	−0.552123	−0.276061	+	0.961140i	$0.589029\pi$
$930$	−6.00000	−0.196748
$931$	−9.55635	−0.313197
$932$	−78.9706	−2.58677
$933$	−12.0000	−0.392862
$934$	−32.3848	−1.05966
$935$	2.82843	0.0924995
$936$	0	0
$937$	−22.9706	−0.750416	−0.375208	−	0.926941i	$-0.622429\pi$
$937$	−22.9706	−0.750416	−0.375208	+	0.926941i	$0.622429\pi$
$938$	−23.3137	−0.761220
$939$	6.82843	0.222837
$940$	−18.4853	−0.602923
$941$	−18.7696	−0.611870	−0.305935	−	0.952052i	$-0.598969\pi$
$941$	−18.7696	−0.611870	−0.305935	+	0.952052i	$0.598969\pi$
$942$	−61.4558	−2.00234
$943$	4.48528	0.146061
$944$	−5.27208	−0.171592
$945$	27.3137	0.888515
$946$	91.2548	2.96695
$947$	−17.1127	−0.556088	−0.278044	−	0.960568i	$-0.589686\pi$
$947$	−17.1127	−0.556088	−0.278044	+	0.960568i	$0.589686\pi$
$948$	45.9411	1.49210
$949$	0	0
$950$	−1.41421	−0.0458831
$951$	−3.02944	−0.0982362
$952$	−17.6569	−0.572262
$953$	35.2548	1.14202	0.571008	−	0.820944i	$-0.306553\pi$
$953$	35.2548	1.14202	0.571008	+	0.820944i	$0.306553\pi$
$954$	−6.00000	−0.194257
$955$	−2.34315	−0.0758224
$956$	13.0711	0.422749
$957$	−27.3137	−0.882927
$958$	74.1838	2.39677
$959$	−83.5980	−2.69952
$960$	−13.8995	−0.448604
$961$	−27.9117	−0.900377
$962$	0	0
$963$	9.41421	0.303369
$964$	55.4558	1.78611
$965$	4.34315	0.139811
$966$	23.3137	0.750106
$967$	−47.9411	−1.54168	−0.770841	−	0.637027i	$-0.780164\pi$
$967$	−47.9411	−1.54168	−0.770841	+	0.637027i	$0.780164\pi$
$968$	2.89949	0.0931933
$969$	0.686292	0.0220469
$970$	−18.4853	−0.593527
$971$	44.2843	1.42115	0.710575	−	0.703622i	$-0.248435\pi$
$971$	44.2843	1.42115	0.710575	+	0.703622i	$0.248435\pi$
$972$	−37.8995	−1.21563
$973$	−21.6569	−0.694287
$974$	26.4853	0.848643
$975$	0	0
$976$	−24.0000	−0.768221
$977$	39.5147	1.26419	0.632094	−	0.774892i	$-0.282196\pi$
$977$	39.5147	1.26419	0.632094	+	0.774892i	$0.282196\pi$
$978$	64.7696	2.07110
$979$	20.4853	0.654712
$980$	−62.4558	−1.99508
$981$	−2.00000	−0.0638551
$982$	12.4853	0.398421
$983$	1.02944	0.0328339	0.0164170	−	0.999865i	$-0.494774\pi$
$983$	1.02944	0.0328339	0.0164170	+	0.999865i	$0.494774\pi$
$984$	−19.7990	−0.631169
$985$	10.9706	0.349551
$986$	−11.3137	−0.360302
$987$	32.9706	1.04946
$988$	0	0
$989$	−15.6569	−0.497859
$990$	−8.24264	−0.261968
$991$	48.9706	1.55560	0.777801	−	0.628511i	$-0.216335\pi$
$991$	48.9706	1.55560	0.777801	+	0.628511i	$0.216335\pi$
$992$	2.78680	0.0884809
$993$	−36.8284	−1.16871
$994$	138.711	4.39964
$995$	−4.00000	−0.126809
$996$	−17.1716	−0.544102
$997$	28.8284	0.913005	0.456503	−	0.889722i	$-0.349102\pi$
$997$	28.8284	0.913005	0.456503	+	0.889722i	$0.349102\pi$
$998$	−100.326	−3.17576
$999$	48.0000	1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.a.g.1.2		2
3.2	odd	2		7605.2.a.x.1.1		2
5.4	even	2		4225.2.a.r.1.1		2
13.2	odd	12		845.2.m.f.316.4		8
13.3	even	3		845.2.e.c.191.1		4
13.4	even	6		845.2.e.h.146.2		4
13.5	odd	4		845.2.c.b.506.1		4
13.6	odd	12		845.2.m.f.361.1		8
13.7	odd	12		845.2.m.f.361.4		8
13.8	odd	4		845.2.c.b.506.4		4
13.9	even	3		845.2.e.c.146.1		4
13.10	even	6		845.2.e.h.191.2		4
13.11	odd	12		845.2.m.f.316.1		8
13.12	even	2		65.2.a.b.1.1	✓	2
39.38	odd	2		585.2.a.m.1.2		2
52.51	odd	2		1040.2.a.j.1.2		2
65.12	odd	4		325.2.b.f.274.1		4
65.38	odd	4		325.2.b.f.274.4		4
65.64	even	2		325.2.a.i.1.2		2
91.90	odd	2		3185.2.a.j.1.1		2
104.51	odd	2		4160.2.a.z.1.1		2
104.77	even	2		4160.2.a.bf.1.2		2
143.142	odd	2		7865.2.a.j.1.2		2
156.155	even	2		9360.2.a.cd.1.1		2
195.38	even	4		2925.2.c.r.2224.1		4
195.77	even	4		2925.2.c.r.2224.4		4
195.194	odd	2		2925.2.a.u.1.1		2
260.259	odd	2		5200.2.a.bu.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.b.1.1	✓	2	13.12	even	2
325.2.a.i.1.2		2	65.64	even	2
325.2.b.f.274.1		4	65.12	odd	4
325.2.b.f.274.4		4	65.38	odd	4
585.2.a.m.1.2		2	39.38	odd	2
845.2.a.g.1.2		2	1.1	even	1	trivial
845.2.c.b.506.1		4	13.5	odd	4
845.2.c.b.506.4		4	13.8	odd	4
845.2.e.c.146.1		4	13.9	even	3
845.2.e.c.191.1		4	13.3	even	3
845.2.e.h.146.2		4	13.4	even	6
845.2.e.h.191.2		4	13.10	even	6
845.2.m.f.316.1		8	13.11	odd	12
845.2.m.f.316.4		8	13.2	odd	12
845.2.m.f.361.1		8	13.6	odd	12
845.2.m.f.361.4		8	13.7	odd	12
1040.2.a.j.1.2		2	52.51	odd	2
2925.2.a.u.1.1		2	195.194	odd	2
2925.2.c.r.2224.1		4	195.38	even	4
2925.2.c.r.2224.4		4	195.77	even	4
3185.2.a.j.1.1		2	91.90	odd	2
4160.2.a.z.1.1		2	104.51	odd	2
4160.2.a.bf.1.2		2	104.77	even	2
4225.2.a.r.1.1		2	5.4	even	2
5200.2.a.bu.1.1		2	260.259	odd	2
7605.2.a.x.1.1		2	3.2	odd	2
7865.2.a.j.1.2		2	143.142	odd	2
9360.2.a.cd.1.1		2	156.155	even	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)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} -1.41421 q^{3} +3.82843 q^{4} -1.00000 q^{5} -3.41421 q^{6} -4.82843 q^{7} +4.41421 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+2.41421 q^{2} -1.41421 q^{3} +3.82843 q^{4} -1.00000 q^{5} -3.41421 q^{6} -4.82843 q^{7} +4.41421 q^{8} -1.00000 q^{9} -2.41421 q^{10} -3.41421 q^{11} -5.41421 q^{12} -11.6569 q^{14} +1.41421 q^{15} +3.00000 q^{16} +0.828427 q^{17} -2.41421 q^{18} -0.585786 q^{19} -3.82843 q^{20} +6.82843 q^{21} -8.24264 q^{22} +1.41421 q^{23} -6.24264 q^{24} +1.00000 q^{25} +5.65685 q^{27} -18.4853 q^{28} -5.65685 q^{29} +3.41421 q^{30} -1.75736 q^{31} -1.58579 q^{32} +4.82843 q^{33} +2.00000 q^{34} +4.82843 q^{35} -3.82843 q^{36} +8.48528 q^{37} -1.41421 q^{38} -4.41421 q^{40} +3.17157 q^{41} +16.4853 q^{42} -11.0711 q^{43} -13.0711 q^{44} +1.00000 q^{45} +3.41421 q^{46} +4.82843 q^{47} -4.24264 q^{48} +16.3137 q^{49} +2.41421 q^{50} -1.17157 q^{51} +2.48528 q^{53} +13.6569 q^{54} +3.41421 q^{55} -21.3137 q^{56} +0.828427 q^{57} -13.6569 q^{58} -1.75736 q^{59} +5.41421 q^{60} -8.00000 q^{61} -4.24264 q^{62} +4.82843 q^{63} -9.82843 q^{64} +11.6569 q^{66} +2.00000 q^{67} +3.17157 q^{68} -2.00000 q^{69} +11.6569 q^{70} -11.8995 q^{71} -4.41421 q^{72} -8.48528 q^{73} +20.4853 q^{74} -1.41421 q^{75} -2.24264 q^{76} +16.4853 q^{77} -8.48528 q^{79} -3.00000 q^{80} -5.00000 q^{81} +7.65685 q^{82} +3.17157 q^{83} +26.1421 q^{84} -0.828427 q^{85} -26.7279 q^{86} +8.00000 q^{87} -15.0711 q^{88} -6.00000 q^{89} +2.41421 q^{90} +5.41421 q^{92} +2.48528 q^{93} +11.6569 q^{94} +0.585786 q^{95} +2.24264 q^{96} +7.65685 q^{97} +39.3848 q^{98} +3.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} + 6 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 4 * q^6 - 4 * q^7 + 6 * q^8 - 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} + 6 q^{8} - 2 q^{9} - 2 q^{10} - 4 q^{11} - 8 q^{12} - 12 q^{14} + 6 q^{16} - 4 q^{17} - 2 q^{18} - 4 q^{19} - 2 q^{20} + 8 q^{21} - 8 q^{22} - 4 q^{24} + 2 q^{25} - 20 q^{28} + 4 q^{30} - 12 q^{31} - 6 q^{32} + 4 q^{33} + 4 q^{34} + 4 q^{35} - 2 q^{36} - 6 q^{40} + 12 q^{41} + 16 q^{42} - 8 q^{43} - 12 q^{44} + 2 q^{45} + 4 q^{46} + 4 q^{47} + 10 q^{49} + 2 q^{50} - 8 q^{51} - 12 q^{53} + 16 q^{54} + 4 q^{55} - 20 q^{56} - 4 q^{57} - 16 q^{58} - 12 q^{59} + 8 q^{60} - 16 q^{61} + 4 q^{63} - 14 q^{64} + 12 q^{66} + 4 q^{67} + 12 q^{68} - 4 q^{69} + 12 q^{70} - 4 q^{71} - 6 q^{72} + 24 q^{74} + 4 q^{76} + 16 q^{77} - 6 q^{80} - 10 q^{81} + 4 q^{82} + 12 q^{83} + 24 q^{84} + 4 q^{85} - 28 q^{86} + 16 q^{87} - 16 q^{88} - 12 q^{89} + 2 q^{90} + 8 q^{92} - 12 q^{93} + 12 q^{94} + 4 q^{95} - 4 q^{96} + 4 q^{97} + 42 q^{98} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 4 * q^6 - 4 * q^7 + 6 * q^8 - 2 * q^9 - 2 * q^10 - 4 * q^11 - 8 * q^12 - 12 * q^14 + 6 * q^16 - 4 * q^17 - 2 * q^18 - 4 * q^19 - 2 * q^20 + 8 * q^21 - 8 * q^22 - 4 * q^24 + 2 * q^25 - 20 * q^28 + 4 * q^30 - 12 * q^31 - 6 * q^32 + 4 * q^33 + 4 * q^34 + 4 * q^35 - 2 * q^36 - 6 * q^40 + 12 * q^41 + 16 * q^42 - 8 * q^43 - 12 * q^44 + 2 * q^45 + 4 * q^46 + 4 * q^47 + 10 * q^49 + 2 * q^50 - 8 * q^51 - 12 * q^53 + 16 * q^54 + 4 * q^55 - 20 * q^56 - 4 * q^57 - 16 * q^58 - 12 * q^59 + 8 * q^60 - 16 * q^61 + 4 * q^63 - 14 * q^64 + 12 * q^66 + 4 * q^67 + 12 * q^68 - 4 * q^69 + 12 * q^70 - 4 * q^71 - 6 * q^72 + 24 * q^74 + 4 * q^76 + 16 * q^77 - 6 * q^80 - 10 * q^81 + 4 * q^82 + 12 * q^83 + 24 * q^84 + 4 * q^85 - 28 * q^86 + 16 * q^87 - 16 * q^88 - 12 * q^89 + 2 * q^90 + 8 * q^92 - 12 * q^93 + 12 * q^94 + 4 * q^95 - 4 * q^96 + 4 * q^97 + 42 * q^98 + 4 * q^99

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