LMFDB - Embedded newform 575.2.b.d.24.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,2,Mod(24,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("575.24");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 575 = 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	575.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$4.59139811622$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 23)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			24.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	575.24
Dual form			575.2.b.d.24.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803i q^{2} -2.23607i q^{3} -0.618034 q^{4} -3.61803 q^{6} -1.23607i q^{7} -2.23607i q^{8} -2.00000 q^{9} +O(q^{10})$	\(q-1.61803i q^{2} -2.23607i q^{3} -0.618034 q^{4} -3.61803 q^{6} -1.23607i q^{7} -2.23607i q^{8} -2.00000 q^{9} -0.763932 q^{11} +1.38197i q^{12} -3.00000i q^{13} -2.00000 q^{14} -4.85410 q^{16} +5.23607i q^{17} +3.23607i q^{18} +2.00000 q^{19} -2.76393 q^{21} +1.23607i q^{22} -1.00000i q^{23} -5.00000 q^{24} -4.85410 q^{26} -2.23607i q^{27} +0.763932i q^{28} +3.00000 q^{29} -6.70820 q^{31} +3.38197i q^{32} +1.70820i q^{33} +8.47214 q^{34} +1.23607 q^{36} +3.23607i q^{37} -3.23607i q^{38} -6.70820 q^{39} +5.47214 q^{41} +4.47214i q^{42} +0.472136 q^{44} -1.61803 q^{46} +2.23607i q^{47} +10.8541i q^{48} +5.47214 q^{49} +11.7082 q^{51} +1.85410i q^{52} +8.47214i q^{53} -3.61803 q^{54} -2.76393 q^{56} -4.47214i q^{57} -4.85410i q^{58} +2.47214 q^{59} +10.9443 q^{61} +10.8541i q^{62} +2.47214i q^{63} -4.23607 q^{64} +2.76393 q^{66} -7.23607i q^{67} -3.23607i q^{68} -2.23607 q^{69} +7.76393 q^{71} +4.47214i q^{72} -15.4721i q^{73} +5.23607 q^{74} -1.23607 q^{76} +0.944272i q^{77} +10.8541i q^{78} -6.94427 q^{79} -11.0000 q^{81} -8.85410i q^{82} +13.2361i q^{83} +1.70820 q^{84} -6.70820i q^{87} +1.70820i q^{88} +1.52786 q^{89} -3.70820 q^{91} +0.618034i q^{92} +15.0000i q^{93} +3.61803 q^{94} +7.56231 q^{96} +4.29180i q^{97} -8.85410i q^{98} +1.52786 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 10 q^{6} - 8 q^{9}+O(q^{10}) $ 4 * q + 2 * q^4 - 10 * q^6 - 8 * q^9	$ 4 q + 2 q^{4} - 10 q^{6} - 8 q^{9} - 12 q^{11} - 8 q^{14} - 6 q^{16} + 8 q^{19} - 20 q^{21} - 20 q^{24} - 6 q^{26} + 12 q^{29} + 16 q^{34} - 4 q^{36} + 4 q^{41} - 16 q^{44} - 2 q^{46} + 4 q^{49} + 20 q^{51} - 10 q^{54} - 20 q^{56} - 8 q^{59} + 8 q^{61} - 8 q^{64} + 20 q^{66} + 40 q^{71} + 12 q^{74} + 4 q^{76} + 8 q^{79} - 44 q^{81} - 20 q^{84} + 24 q^{89} + 12 q^{91} + 10 q^{94} - 10 q^{96} + 24 q^{99}+O(q^{100}) $ 4 * q + 2 * q^4 - 10 * q^6 - 8 * q^9 - 12 * q^11 - 8 * q^14 - 6 * q^16 + 8 * q^19 - 20 * q^21 - 20 * q^24 - 6 * q^26 + 12 * q^29 + 16 * q^34 - 4 * q^36 + 4 * q^41 - 16 * q^44 - 2 * q^46 + 4 * q^49 + 20 * q^51 - 10 * q^54 - 20 * q^56 - 8 * q^59 + 8 * q^61 - 8 * q^64 + 20 * q^66 + 40 * q^71 + 12 * q^74 + 4 * q^76 + 8 * q^79 - 44 * q^81 - 20 * q^84 + 24 * q^89 + 12 * q^91 + 10 * q^94 - 10 * q^96 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/575\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.61803i	− 1.14412i	−0.820211	−	0.572061i	$-0.806144\pi$
$2$	− 1.61803i	− 1.14412i	0.820211	−	0.572061i	$-0.193856\pi$
$3$	− 2.23607i	− 1.29099i	−0.763763	−	0.645497i	$-0.776650\pi$
$3$	− 2.23607i	− 1.29099i	0.763763	−	0.645497i	$-0.223350\pi$
$4$	−0.618034	−0.309017
$5$	0	0
$5$	0	0
$6$	−3.61803	−1.47706
$7$	− 1.23607i	− 0.467190i	−0.972334	−	0.233595i	$-0.924951\pi$
$7$	− 1.23607i	− 0.467190i	0.972334	−	0.233595i	$-0.0750489\pi$
$8$	− 2.23607i	− 0.790569i
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−0.763932	−0.230334	−0.115167	−	0.993346i	$-0.536740\pi$
$11$	−0.763932	−0.230334	−0.115167	+	0.993346i	$0.536740\pi$
$12$	1.38197i	0.398939i
$13$	− 3.00000i	− 0.832050i	−0.909353	−	0.416025i	$-0.863423\pi$
$13$	− 3.00000i	− 0.832050i	0.909353	−	0.416025i	$-0.136577\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−4.85410	−1.21353
$17$	5.23607i	1.26993i	0.772540	+	0.634967i	$0.218986\pi$
$17$	5.23607i	1.26993i	−0.772540	+	0.634967i	$0.781014\pi$
$18$	3.23607i	0.762749i
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−2.76393	−0.603139
$22$	1.23607i	0.263531i
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	−5.00000	−1.02062
$25$	0	0
$26$	−4.85410	−0.951968
$27$	− 2.23607i	− 0.430331i
$28$	0.763932i	0.144370i
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−6.70820	−1.20483	−0.602414	−	0.798183i	$-0.705795\pi$
$31$	−6.70820	−1.20483	−0.602414	+	0.798183i	$0.705795\pi$
$32$	3.38197i	0.597853i
$33$	1.70820i	0.297360i
$34$	8.47214	1.45296
$35$	0	0
$36$	1.23607	0.206011
$37$	3.23607i	0.532006i	0.963972	+	0.266003i	$0.0857032\pi$
$37$	3.23607i	0.532006i	−0.963972	+	0.266003i	$0.914297\pi$
$38$	− 3.23607i	− 0.524960i
$39$	−6.70820	−1.07417
$40$	0	0
$41$	5.47214	0.854604	0.427302	−	0.904109i	$-0.359464\pi$
$41$	5.47214	0.854604	0.427302	+	0.904109i	$0.359464\pi$
$42$	4.47214i	0.690066i
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0.472136	0.0711772
$45$	0	0
$46$	−1.61803	−0.238566
$47$	2.23607i	0.326164i	0.986613	+	0.163082i	$0.0521435\pi$
$47$	2.23607i	0.326164i	−0.986613	+	0.163082i	$0.947856\pi$
$48$	10.8541i	1.56665i
$49$	5.47214	0.781734
$50$	0	0
$51$	11.7082	1.63948
$52$	1.85410i	0.257118i
$53$	8.47214i	1.16374i	0.813283	+	0.581869i	$0.197678\pi$
$53$	8.47214i	1.16374i	−0.813283	+	0.581869i	$0.802322\pi$
$54$	−3.61803	−0.492352
$55$	0	0
$56$	−2.76393	−0.369346
$57$	− 4.47214i	− 0.592349i
$58$	− 4.85410i	− 0.637375i
$59$	2.47214	0.321845	0.160922	−	0.986967i	$-0.448553\pi$
$59$	2.47214	0.321845	0.160922	+	0.986967i	$0.448553\pi$
$60$	0	0
$61$	10.9443	1.40127	0.700635	−	0.713520i	$-0.252900\pi$
$61$	10.9443	1.40127	0.700635	+	0.713520i	$0.252900\pi$
$62$	10.8541i	1.37847i
$63$	2.47214i	0.311460i
$64$	−4.23607	−0.529508
$65$	0	0
$66$	2.76393	0.340217
$67$	− 7.23607i	− 0.884026i	−0.897009	−	0.442013i	$-0.854264\pi$
$67$	− 7.23607i	− 0.884026i	0.897009	−	0.442013i	$-0.145736\pi$
$68$	− 3.23607i	− 0.392431i
$69$	−2.23607	−0.269191
$70$	0	0
$71$	7.76393	0.921409	0.460705	−	0.887554i	$-0.347597\pi$
$71$	7.76393	0.921409	0.460705	+	0.887554i	$0.347597\pi$
$72$	4.47214i	0.527046i
$73$	− 15.4721i	− 1.81088i	−0.424478	−	0.905438i	$-0.639542\pi$
$73$	− 15.4721i	− 1.81088i	0.424478	−	0.905438i	$-0.360458\pi$
$74$	5.23607	0.608681
$75$	0	0
$76$	−1.23607	−0.141787
$77$	0.944272i	0.107610i
$78$	10.8541i	1.22899i
$79$	−6.94427	−0.781292	−0.390646	−	0.920541i	$-0.627748\pi$
$79$	−6.94427	−0.781292	−0.390646	+	0.920541i	$0.627748\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	− 8.85410i	− 0.977772i
$83$	13.2361i	1.45285i	0.687247	+	0.726424i	$0.258819\pi$
$83$	13.2361i	1.45285i	−0.687247	+	0.726424i	$0.741181\pi$
$84$	1.70820	0.186380
$85$	0	0
$86$	0	0
$87$	− 6.70820i	− 0.719195i
$88$	1.70820i	0.182095i
$89$	1.52786	0.161953	0.0809766	−	0.996716i	$-0.474196\pi$
$89$	1.52786	0.161953	0.0809766	+	0.996716i	$0.474196\pi$
$90$	0	0
$91$	−3.70820	−0.388725
$92$	0.618034i	0.0644345i
$93$	15.0000i	1.55543i
$94$	3.61803	0.373172
$95$	0	0
$96$	7.56231	0.771825
$97$	4.29180i	0.435766i	0.975975	+	0.217883i	$0.0699151\pi$
$97$	4.29180i	0.435766i	−0.975975	+	0.217883i	$0.930085\pi$
$98$	− 8.85410i	− 0.894399i
$99$	1.52786	0.153556
$100$	0	0
$101$	−4.47214	−0.444994	−0.222497	−	0.974933i	$-0.571421\pi$
$101$	−4.47214	−0.444994	−0.222497	+	0.974933i	$0.571421\pi$
$102$	− 18.9443i	− 1.87576i
$103$	− 18.1803i	− 1.79136i	−0.444697	−	0.895681i	$-0.646689\pi$
$103$	− 18.1803i	− 1.79136i	0.444697	−	0.895681i	$-0.353311\pi$
$104$	−6.70820	−0.657794
$105$	0	0
$106$	13.7082	1.33146
$107$	− 13.4164i	− 1.29701i	−0.761209	−	0.648507i	$-0.775394\pi$
$107$	− 13.4164i	− 1.29701i	0.761209	−	0.648507i	$-0.224606\pi$
$108$	1.38197i	0.132980i
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	7.23607	0.686817
$112$	6.00000i	0.566947i
$113$	− 13.2361i	− 1.24514i	−0.782562	−	0.622572i	$-0.786088\pi$
$113$	− 13.2361i	− 1.24514i	0.782562	−	0.622572i	$-0.213912\pi$
$114$	−7.23607	−0.677720
$115$	0	0
$116$	−1.85410	−0.172149
$117$	6.00000i	0.554700i
$118$	− 4.00000i	− 0.368230i
$119$	6.47214	0.593300
$120$	0	0
$121$	−10.4164	−0.946946
$122$	− 17.7082i	− 1.60323i
$123$	− 12.2361i	− 1.10329i
$124$	4.14590	0.372313
$125$	0	0
$126$	4.00000	0.356348
$127$	− 20.7082i	− 1.83756i	−0.394775	−	0.918778i	$-0.629177\pi$
$127$	− 20.7082i	− 1.83756i	0.394775	−	0.918778i	$-0.370823\pi$
$128$	13.6180i	1.20368i
$129$	0	0
$130$	0	0
$131$	5.29180	0.462346	0.231173	−	0.972913i	$-0.425744\pi$
$131$	5.29180	0.462346	0.231173	+	0.972913i	$0.425744\pi$
$132$	− 1.05573i	− 0.0918893i
$133$	− 2.47214i	− 0.214361i
$134$	−11.7082	−1.01143
$135$	0	0
$136$	11.7082	1.00397
$137$	13.8885i	1.18658i	0.804989	+	0.593289i	$0.202171\pi$
$137$	13.8885i	1.18658i	−0.804989	+	0.593289i	$0.797829\pi$
$138$	3.61803i	0.307988i
$139$	−2.70820	−0.229707	−0.114853	−	0.993382i	$-0.536640\pi$
$139$	−2.70820	−0.229707	−0.114853	+	0.993382i	$0.536640\pi$
$140$	0	0
$141$	5.00000	0.421076
$142$	− 12.5623i	− 1.05421i
$143$	2.29180i	0.191650i
$144$	9.70820	0.809017
$145$	0	0
$146$	−25.0344	−2.07187
$147$	− 12.2361i	− 1.00921i
$148$	− 2.00000i	− 0.164399i
$149$	11.8885	0.973947	0.486974	−	0.873417i	$-0.338101\pi$
$149$	11.8885	0.973947	0.486974	+	0.873417i	$0.338101\pi$
$150$	0	0
$151$	−0.236068	−0.0192109	−0.00960547	−	0.999954i	$-0.503058\pi$
$151$	−0.236068	−0.0192109	−0.00960547	+	0.999954i	$0.503058\pi$
$152$	− 4.47214i	− 0.362738i
$153$	− 10.4721i	− 0.846622i
$154$	1.52786	0.123119
$155$	0	0
$156$	4.14590	0.331937
$157$	15.4164i	1.23036i	0.788385	+	0.615182i	$0.210917\pi$
$157$	15.4164i	1.23036i	−0.788385	+	0.615182i	$0.789083\pi$
$158$	11.2361i	0.893894i
$159$	18.9443	1.50238
$160$	0	0
$161$	−1.23607	−0.0974158
$162$	17.7984i	1.39837i
$163$	10.2361i	0.801751i	0.916133	+	0.400875i	$0.131294\pi$
$163$	10.2361i	0.801751i	−0.916133	+	0.400875i	$0.868706\pi$
$164$	−3.38197	−0.264087
$165$	0	0
$166$	21.4164	1.66224
$167$	10.4721i	0.810358i	0.914237	+	0.405179i	$0.132791\pi$
$167$	10.4721i	0.810358i	−0.914237	+	0.405179i	$0.867209\pi$
$168$	6.18034i	0.476824i
$169$	4.00000	0.307692
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	− 5.05573i	− 0.384380i	−0.981358	−	0.192190i	$-0.938441\pi$
$173$	− 5.05573i	− 0.384380i	0.981358	−	0.192190i	$-0.0615590\pi$
$174$	−10.8541	−0.822847
$175$	0	0
$176$	3.70820	0.279516
$177$	− 5.52786i	− 0.415500i
$178$	− 2.47214i	− 0.185294i
$179$	12.7082	0.949856	0.474928	−	0.880025i	$-0.342474\pi$
$179$	12.7082	0.949856	0.474928	+	0.880025i	$0.342474\pi$
$180$	0	0
$181$	−14.6525	−1.08911	−0.544555	−	0.838725i	$-0.683301\pi$
$181$	−14.6525	−1.08911	−0.544555	+	0.838725i	$0.683301\pi$
$182$	6.00000i	0.444750i
$183$	− 24.4721i	− 1.80903i
$184$	−2.23607	−0.164845
$185$	0	0
$186$	24.2705	1.77960
$187$	− 4.00000i	− 0.292509i
$188$	− 1.38197i	− 0.100790i
$189$	−2.76393	−0.201046
$190$	0	0
$191$	−3.81966	−0.276381	−0.138190	−	0.990406i	$-0.544129\pi$
$191$	−3.81966	−0.276381	−0.138190	+	0.990406i	$0.544129\pi$
$192$	9.47214i	0.683593i
$193$	7.94427i	0.571841i	0.958253	+	0.285921i	$0.0922994\pi$
$193$	7.94427i	0.571841i	−0.958253	+	0.285921i	$0.907701\pi$
$194$	6.94427	0.498570
$195$	0	0
$196$	−3.38197	−0.241569
$197$	7.47214i	0.532368i	0.963922	+	0.266184i	$0.0857628\pi$
$197$	7.47214i	0.532368i	−0.963922	+	0.266184i	$0.914237\pi$
$198$	− 2.47214i	− 0.175687i
$199$	25.7082	1.82241	0.911203	−	0.411957i	$-0.135155\pi$
$199$	25.7082	1.82241	0.911203	+	0.411957i	$0.135155\pi$
$200$	0	0
$201$	−16.1803	−1.14127
$202$	7.23607i	0.509128i
$203$	− 3.70820i	− 0.260265i
$204$	−7.23607	−0.506626
$205$	0	0
$206$	−29.4164	−2.04954
$207$	2.00000i	0.139010i
$208$	14.5623i	1.00971i
$209$	−1.52786	−0.105685
$210$	0	0
$211$	3.41641	0.235195	0.117598	−	0.993061i	$-0.462481\pi$
$211$	3.41641	0.235195	0.117598	+	0.993061i	$0.462481\pi$
$212$	− 5.23607i	− 0.359615i
$213$	− 17.3607i	− 1.18953i
$214$	−21.7082	−1.48394
$215$	0	0
$216$	−5.00000	−0.340207
$217$	8.29180i	0.562884i
$218$	0	0
$219$	−34.5967	−2.33783
$220$	0	0
$221$	15.7082	1.05665
$222$	− 11.7082i	− 0.785803i
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	4.18034	0.279311
$225$	0	0
$226$	−21.4164	−1.42460
$227$	10.1803i	0.675693i	0.941201	+	0.337846i	$0.109698\pi$
$227$	10.1803i	0.675693i	−0.941201	+	0.337846i	$0.890302\pi$
$228$	2.76393i	0.183046i
$229$	12.0000	0.792982	0.396491	−	0.918039i	$-0.370228\pi$
$229$	12.0000	0.792982	0.396491	+	0.918039i	$0.370228\pi$
$230$	0	0
$231$	2.11146	0.138924
$232$	− 6.70820i	− 0.440415i
$233$	15.4721i	1.01361i	0.862060	+	0.506807i	$0.169174\pi$
$233$	15.4721i	1.01361i	−0.862060	+	0.506807i	$0.830826\pi$
$234$	9.70820	0.634645
$235$	0	0
$236$	−1.52786	−0.0994555
$237$	15.5279i	1.00864i
$238$	− 10.4721i	− 0.678808i
$239$	−18.2361	−1.17959	−0.589797	−	0.807552i	$-0.700792\pi$
$239$	−18.2361	−1.17959	−0.589797	+	0.807552i	$0.700792\pi$
$240$	0	0
$241$	17.1246	1.10309	0.551547	−	0.834144i	$-0.314038\pi$
$241$	17.1246	1.10309	0.551547	+	0.834144i	$0.314038\pi$
$242$	16.8541i	1.08342i
$243$	17.8885i	1.14755i
$244$	−6.76393	−0.433016
$245$	0	0
$246$	−19.7984	−1.26230
$247$	− 6.00000i	− 0.381771i
$248$	15.0000i	0.952501i
$249$	29.5967	1.87562
$250$	0	0
$251$	15.7082	0.991493	0.495747	−	0.868467i	$-0.334895\pi$
$251$	15.7082	0.991493	0.495747	+	0.868467i	$0.334895\pi$
$252$	− 1.52786i	− 0.0962464i
$253$	0.763932i	0.0480280i
$254$	−33.5066	−2.10239
$255$	0	0
$256$	13.5623	0.847644
$257$	1.47214i	0.0918293i	0.998945	+	0.0459147i	$0.0146202\pi$
$257$	1.47214i	0.0918293i	−0.998945	+	0.0459147i	$0.985380\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	−6.00000	−0.371391
$262$	− 8.56231i	− 0.528981i
$263$	14.9443i	0.921503i	0.887529	+	0.460752i	$0.152420\pi$
$263$	14.9443i	0.921503i	−0.887529	+	0.460752i	$0.847580\pi$
$264$	3.81966	0.235084
$265$	0	0
$266$	−4.00000	−0.245256
$267$	− 3.41641i	− 0.209081i
$268$	4.47214i	0.273179i
$269$	−9.94427	−0.606313	−0.303156	−	0.952941i	$-0.598040\pi$
$269$	−9.94427	−0.606313	−0.303156	+	0.952941i	$0.598040\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	− 25.4164i	− 1.54110i
$273$	8.29180i	0.501842i
$274$	22.4721	1.35759
$275$	0	0
$276$	1.38197	0.0831846
$277$	6.52786i	0.392221i	0.980582	+	0.196111i	$0.0628312\pi$
$277$	6.52786i	0.392221i	−0.980582	+	0.196111i	$0.937169\pi$
$278$	4.38197i	0.262813i
$279$	13.4164	0.803219
$280$	0	0
$281$	−13.2361	−0.789598	−0.394799	−	0.918768i	$-0.629186\pi$
$281$	−13.2361	−0.789598	−0.394799	+	0.918768i	$0.629186\pi$
$282$	− 8.09017i	− 0.481763i
$283$	− 14.2918i	− 0.849559i	−0.905297	−	0.424780i	$-0.860352\pi$
$283$	− 14.2918i	− 0.849559i	0.905297	−	0.424780i	$-0.139648\pi$
$284$	−4.79837	−0.284731
$285$	0	0
$286$	3.70820	0.219271
$287$	− 6.76393i	− 0.399262i
$288$	− 6.76393i	− 0.398569i
$289$	−10.4164	−0.612730
$290$	0	0
$291$	9.59675	0.562571
$292$	9.56231i	0.559592i
$293$	10.4721i	0.611789i	0.952065	+	0.305894i	$0.0989554\pi$
$293$	10.4721i	0.611789i	−0.952065	+	0.305894i	$0.901045\pi$
$294$	−19.7984	−1.15466
$295$	0	0
$296$	7.23607	0.420588
$297$	1.70820i	0.0991200i
$298$	− 19.2361i	− 1.11432i
$299$	−3.00000	−0.173494
$300$	0	0
$301$	0	0
$302$	0.381966i	0.0219797i
$303$	10.0000i	0.574485i
$304$	−9.70820	−0.556804
$305$	0	0
$306$	−16.9443	−0.968640
$307$	18.4721i	1.05426i	0.849785	+	0.527130i	$0.176732\pi$
$307$	18.4721i	1.05426i	−0.849785	+	0.527130i	$0.823268\pi$
$308$	− 0.583592i	− 0.0332532i
$309$	−40.6525	−2.31264
$310$	0	0
$311$	−9.18034	−0.520569	−0.260285	−	0.965532i	$-0.583816\pi$
$311$	−9.18034	−0.520569	−0.260285	+	0.965532i	$0.583816\pi$
$312$	15.0000i	0.849208i
$313$	20.3607i	1.15085i	0.817853	+	0.575427i	$0.195164\pi$
$313$	20.3607i	1.15085i	−0.817853	+	0.575427i	$0.804836\pi$
$314$	24.9443	1.40769
$315$	0	0
$316$	4.29180	0.241432
$317$	− 1.41641i	− 0.0795534i	−0.999209	−	0.0397767i	$-0.987335\pi$
$317$	− 1.41641i	− 0.0795534i	0.999209	−	0.0397767i	$-0.0126647\pi$
$318$	− 30.6525i	− 1.71891i
$319$	−2.29180	−0.128316
$320$	0	0
$321$	−30.0000	−1.67444
$322$	2.00000i	0.111456i
$323$	10.4721i	0.582685i
$324$	6.79837	0.377687
$325$	0	0
$326$	16.5623	0.917301
$327$	0	0
$328$	− 12.2361i	− 0.675624i
$329$	2.76393	0.152381
$330$	0	0
$331$	11.6525	0.640478	0.320239	−	0.947337i	$-0.396237\pi$
$331$	11.6525	0.640478	0.320239	+	0.947337i	$0.396237\pi$
$332$	− 8.18034i	− 0.448954i
$333$	− 6.47214i	− 0.354671i
$334$	16.9443	0.927149
$335$	0	0
$336$	13.4164	0.731925
$337$	− 3.41641i	− 0.186104i	−0.995661	−	0.0930518i	$-0.970338\pi$
$337$	− 3.41641i	− 0.186104i	0.995661	−	0.0930518i	$-0.0296622\pi$
$338$	− 6.47214i	− 0.352038i
$339$	−29.5967	−1.60747
$340$	0	0
$341$	5.12461	0.277513
$342$	6.47214i	0.349973i
$343$	− 15.4164i	− 0.832408i
$344$	0	0
$345$	0	0
$346$	−8.18034	−0.439778
$347$	25.8885i	1.38977i	0.719121	+	0.694885i	$0.244545\pi$
$347$	25.8885i	1.38977i	−0.719121	+	0.694885i	$0.755455\pi$
$348$	4.14590i	0.222243i
$349$	2.41641	0.129347	0.0646737	−	0.997906i	$-0.479399\pi$
$349$	2.41641	0.129347	0.0646737	+	0.997906i	$0.479399\pi$
$350$	0	0
$351$	−6.70820	−0.358057
$352$	− 2.58359i	− 0.137706i
$353$	35.3607i	1.88206i	0.338324	+	0.941030i	$0.390140\pi$
$353$	35.3607i	1.88206i	−0.338324	+	0.941030i	$0.609860\pi$
$354$	−8.94427	−0.475383
$355$	0	0
$356$	−0.944272	−0.0500463
$357$	− 14.4721i	− 0.765947i
$358$	− 20.5623i	− 1.08675i
$359$	−15.8885	−0.838565	−0.419283	−	0.907856i	$-0.637718\pi$
$359$	−15.8885	−0.838565	−0.419283	+	0.907856i	$0.637718\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	23.7082i	1.24608i
$363$	23.2918i	1.22250i
$364$	2.29180	0.120123
$365$	0	0
$366$	−39.5967	−2.06976
$367$	18.1803i	0.949006i	0.880254	+	0.474503i	$0.157372\pi$
$367$	18.1803i	0.949006i	−0.880254	+	0.474503i	$0.842628\pi$
$368$	4.85410i	0.253038i
$369$	−10.9443	−0.569736
$370$	0	0
$371$	10.4721	0.543686
$372$	− 9.27051i	− 0.480654i
$373$	5.70820i	0.295560i	0.989020	+	0.147780i	$0.0472127\pi$
$373$	5.70820i	0.295560i	−0.989020	+	0.147780i	$0.952787\pi$
$374$	−6.47214	−0.334666
$375$	0	0
$376$	5.00000	0.257855
$377$	− 9.00000i	− 0.463524i
$378$	4.47214i	0.230022i
$379$	20.3607	1.04586	0.522929	−	0.852376i	$-0.324839\pi$
$379$	20.3607	1.04586	0.522929	+	0.852376i	$0.324839\pi$
$380$	0	0
$381$	−46.3050	−2.37227
$382$	6.18034i	0.316214i
$383$	− 24.9443i	− 1.27459i	−0.770619	−	0.637296i	$-0.780053\pi$
$383$	− 24.9443i	− 1.27459i	0.770619	−	0.637296i	$-0.219947\pi$
$384$	30.4508	1.55394
$385$	0	0
$386$	12.8541	0.654257
$387$	0	0
$388$	− 2.65248i	− 0.134659i
$389$	−34.4721	−1.74781	−0.873903	−	0.486100i	$-0.838419\pi$
$389$	−34.4721	−1.74781	−0.873903	+	0.486100i	$0.838419\pi$
$390$	0	0
$391$	5.23607	0.264799
$392$	− 12.2361i	− 0.618015i
$393$	− 11.8328i	− 0.596887i
$394$	12.0902	0.609094
$395$	0	0
$396$	−0.944272	−0.0474514
$397$	2.41641i	0.121276i	0.998160	+	0.0606380i	$0.0193135\pi$
$397$	2.41641i	0.121276i	−0.998160	+	0.0606380i	$0.980686\pi$
$398$	− 41.5967i	− 2.08506i
$399$	−5.52786	−0.276739
$400$	0	0
$401$	8.18034	0.408507	0.204253	−	0.978918i	$-0.434523\pi$
$401$	8.18034	0.408507	0.204253	+	0.978918i	$0.434523\pi$
$402$	26.1803i	1.30576i
$403$	20.1246i	1.00248i
$404$	2.76393	0.137511
$405$	0	0
$406$	−6.00000	−0.297775
$407$	− 2.47214i	− 0.122539i
$408$	− 26.1803i	− 1.29612i
$409$	23.3607	1.15511	0.577556	−	0.816351i	$-0.304007\pi$
$409$	23.3607	1.15511	0.577556	+	0.816351i	$0.304007\pi$
$410$	0	0
$411$	31.0557	1.53187
$412$	11.2361i	0.553561i
$413$	− 3.05573i	− 0.150363i
$414$	3.23607	0.159044
$415$	0	0
$416$	10.1459	0.497444
$417$	6.05573i	0.296550i
$418$	2.47214i	0.120916i
$419$	31.4164	1.53479	0.767396	−	0.641173i	$-0.221552\pi$
$419$	31.4164	1.53479	0.767396	+	0.641173i	$0.221552\pi$
$420$	0	0
$421$	−23.7082	−1.15547	−0.577734	−	0.816225i	$-0.696063\pi$
$421$	−23.7082	−1.15547	−0.577734	+	0.816225i	$0.696063\pi$
$422$	− 5.52786i	− 0.269092i
$423$	− 4.47214i	− 0.217443i
$424$	18.9443	0.920015
$425$	0	0
$426$	−28.0902	−1.36097
$427$	− 13.5279i	− 0.654659i
$428$	8.29180i	0.400799i
$429$	5.12461	0.247419
$430$	0	0
$431$	−26.4721	−1.27512	−0.637559	−	0.770402i	$-0.720056\pi$
$431$	−26.4721	−1.27512	−0.637559	+	0.770402i	$0.720056\pi$
$432$	10.8541i	0.522218i
$433$	− 40.1803i	− 1.93094i	−0.260507	−	0.965472i	$-0.583890\pi$
$433$	− 40.1803i	− 1.93094i	0.260507	−	0.965472i	$-0.416110\pi$
$434$	13.4164	0.644008
$435$	0	0
$436$	0	0
$437$	− 2.00000i	− 0.0956730i
$438$	55.9787i	2.67477i
$439$	5.29180	0.252564	0.126282	−	0.991994i	$-0.459696\pi$
$439$	5.29180	0.252564	0.126282	+	0.991994i	$0.459696\pi$
$440$	0	0
$441$	−10.9443	−0.521156
$442$	− 25.4164i	− 1.20894i
$443$	2.12461i	0.100943i	0.998725	+	0.0504717i	$0.0160725\pi$
$443$	2.12461i	0.100943i	−0.998725	+	0.0504717i	$0.983928\pi$
$444$	−4.47214	−0.212238
$445$	0	0
$446$	−6.47214	−0.306465
$447$	− 26.5836i	− 1.25736i
$448$	5.23607i	0.247381i
$449$	−2.94427	−0.138949	−0.0694744	−	0.997584i	$-0.522132\pi$
$449$	−2.94427	−0.138949	−0.0694744	+	0.997584i	$0.522132\pi$
$450$	0	0
$451$	−4.18034	−0.196845
$452$	8.18034i	0.384771i
$453$	0.527864i	0.0248012i
$454$	16.4721	0.773076
$455$	0	0
$456$	−10.0000	−0.468293
$457$	35.1246i	1.64306i	0.570165	+	0.821530i	$0.306879\pi$
$457$	35.1246i	1.64306i	−0.570165	+	0.821530i	$0.693121\pi$
$458$	− 19.4164i	− 0.907269i
$459$	11.7082	0.546492
$460$	0	0
$461$	7.47214	0.348012	0.174006	−	0.984745i	$-0.444329\pi$
$461$	7.47214	0.348012	0.174006	+	0.984745i	$0.444329\pi$
$462$	− 3.41641i	− 0.158946i
$463$	20.0000i	0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$	20.0000i	0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$	−14.5623	−0.676038
$465$	0	0
$466$	25.0344	1.15970
$467$	− 30.9443i	− 1.43193i	−0.698136	−	0.715965i	$-0.745987\pi$
$467$	− 30.9443i	− 1.43193i	0.698136	−	0.715965i	$-0.254013\pi$
$468$	− 3.70820i	− 0.171412i
$469$	−8.94427	−0.413008
$470$	0	0
$471$	34.4721	1.58839
$472$	− 5.52786i	− 0.254441i
$473$	0	0
$474$	25.1246	1.15401
$475$	0	0
$476$	−4.00000	−0.183340
$477$	− 16.9443i	− 0.775825i
$478$	29.5066i	1.34960i
$479$	17.5967	0.804016	0.402008	−	0.915636i	$-0.368312\pi$
$479$	17.5967	0.804016	0.402008	+	0.915636i	$0.368312\pi$
$480$	0	0
$481$	9.70820	0.442656
$482$	− 27.7082i	− 1.26207i
$483$	2.76393i	0.125763i
$484$	6.43769	0.292622
$485$	0	0
$486$	28.9443	1.31294
$487$	− 1.29180i	− 0.0585369i	−0.999572	−	0.0292684i	$-0.990682\pi$
$487$	− 1.29180i	− 0.0585369i	0.999572	−	0.0292684i	$-0.00931776\pi$
$488$	− 24.4721i	− 1.10780i
$489$	22.8885	1.03506
$490$	0	0
$491$	39.6525	1.78949	0.894746	−	0.446576i	$-0.147357\pi$
$491$	39.6525	1.78949	0.894746	+	0.446576i	$0.147357\pi$
$492$	7.56231i	0.340935i
$493$	15.7082i	0.707462i
$494$	−9.70820	−0.436793
$495$	0	0
$496$	32.5623	1.46209
$497$	− 9.59675i	− 0.430473i
$498$	− 47.8885i	− 2.14594i
$499$	−32.7082	−1.46422	−0.732110	−	0.681186i	$-0.761464\pi$
$499$	−32.7082	−1.46422	−0.732110	+	0.681186i	$0.761464\pi$
$500$	0	0
$501$	23.4164	1.04617
$502$	− 25.4164i	− 1.13439i
$503$	9.05573i	0.403775i	0.979409	+	0.201887i	$0.0647075\pi$
$503$	9.05573i	0.403775i	−0.979409	+	0.201887i	$0.935292\pi$
$504$	5.52786	0.246231
$505$	0	0
$506$	1.23607	0.0549499
$507$	− 8.94427i	− 0.397229i
$508$	12.7984i	0.567836i
$509$	−34.3050	−1.52054	−0.760270	−	0.649607i	$-0.774933\pi$
$509$	−34.3050	−1.52054	−0.760270	+	0.649607i	$0.774933\pi$
$510$	0	0
$511$	−19.1246	−0.846023
$512$	5.29180i	0.233867i
$513$	− 4.47214i	− 0.197450i
$514$	2.38197	0.105064
$515$	0	0
$516$	0	0
$517$	− 1.70820i	− 0.0751267i
$518$	− 6.47214i	− 0.284369i
$519$	−11.3050	−0.496232
$520$	0	0
$521$	4.58359	0.200811	0.100405	−	0.994947i	$-0.467986\pi$
$521$	4.58359	0.200811	0.100405	+	0.994947i	$0.467986\pi$
$522$	9.70820i	0.424917i
$523$	− 0.875388i	− 0.0382781i	−0.999817	−	0.0191390i	$-0.993907\pi$
$523$	− 0.875388i	− 0.0382781i	0.999817	−	0.0191390i	$-0.00609251\pi$
$524$	−3.27051	−0.142873
$525$	0	0
$526$	24.1803	1.05431
$527$	− 35.1246i	− 1.53005i
$528$	− 8.29180i	− 0.360854i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−4.94427	−0.214563
$532$	1.52786i	0.0662413i
$533$	− 16.4164i	− 0.711074i
$534$	−5.52786	−0.239214
$535$	0	0
$536$	−16.1803	−0.698884
$537$	− 28.4164i	− 1.22626i
$538$	16.0902i	0.693696i
$539$	−4.18034	−0.180060
$540$	0	0
$541$	−7.58359	−0.326044	−0.163022	−	0.986622i	$-0.552124\pi$
$541$	−7.58359	−0.326044	−0.163022	+	0.986622i	$0.552124\pi$
$542$	− 12.9443i	− 0.556004i
$543$	32.7639i	1.40603i
$544$	−17.7082	−0.759233
$545$	0	0
$546$	13.4164	0.574169
$547$	37.5410i	1.60514i	0.596559	+	0.802569i	$0.296534\pi$
$547$	37.5410i	1.60514i	−0.596559	+	0.802569i	$0.703466\pi$
$548$	− 8.58359i	− 0.366673i
$549$	−21.8885	−0.934180
$550$	0	0
$551$	6.00000	0.255609
$552$	5.00000i	0.212814i
$553$	8.58359i	0.365011i
$554$	10.5623	0.448749
$555$	0	0
$556$	1.67376	0.0709833
$557$	19.4164i	0.822700i	0.911478	+	0.411350i	$0.134943\pi$
$557$	19.4164i	0.822700i	−0.911478	+	0.411350i	$0.865057\pi$
$558$	− 21.7082i	− 0.918982i
$559$	0	0
$560$	0	0
$561$	−8.94427	−0.377627
$562$	21.4164i	0.903397i
$563$	15.0557i	0.634523i	0.948338	+	0.317262i	$0.102763\pi$
$563$	15.0557i	0.634523i	−0.948338	+	0.317262i	$0.897237\pi$
$564$	−3.09017	−0.130120
$565$	0	0
$566$	−23.1246	−0.972000
$567$	13.5967i	0.571010i
$568$	− 17.3607i	− 0.728438i
$569$	−0.180340	−0.00756024	−0.00378012	−	0.999993i	$-0.501203\pi$
$569$	−0.180340	−0.00756024	−0.00378012	+	0.999993i	$0.501203\pi$
$570$	0	0
$571$	−27.7082	−1.15955	−0.579776	−	0.814776i	$-0.696860\pi$
$571$	−27.7082	−1.15955	−0.579776	+	0.814776i	$0.696860\pi$
$572$	− 1.41641i	− 0.0592230i
$573$	8.54102i	0.356806i
$574$	−10.9443	−0.456805
$575$	0	0
$576$	8.47214	0.353006
$577$	− 12.8885i	− 0.536557i	−0.963341	−	0.268279i	$-0.913545\pi$
$577$	− 12.8885i	− 0.536557i	0.963341	−	0.268279i	$-0.0864548\pi$
$578$	16.8541i	0.701038i
$579$	17.7639	0.738244
$580$	0	0
$581$	16.3607	0.678755
$582$	− 15.5279i	− 0.643651i
$583$	− 6.47214i	− 0.268048i
$584$	−34.5967	−1.43162
$585$	0	0
$586$	16.9443	0.699961
$587$	− 11.2918i	− 0.466062i	−0.972469	−	0.233031i	$-0.925136\pi$
$587$	− 11.2918i	− 0.466062i	0.972469	−	0.233031i	$-0.0748644\pi$
$588$	7.56231i	0.311864i
$589$	−13.4164	−0.552813
$590$	0	0
$591$	16.7082	0.687284
$592$	− 15.7082i	− 0.645603i
$593$	− 14.9443i	− 0.613688i	−0.951760	−	0.306844i	$-0.900727\pi$
$593$	− 14.9443i	− 0.613688i	0.951760	−	0.306844i	$-0.0992729\pi$
$594$	2.76393	0.113406
$595$	0	0
$596$	−7.34752	−0.300966
$597$	− 57.4853i	− 2.35272i
$598$	4.85410i	0.198499i
$599$	1.88854	0.0771638	0.0385819	−	0.999255i	$-0.487716\pi$
$599$	1.88854	0.0771638	0.0385819	+	0.999255i	$0.487716\pi$
$600$	0	0
$601$	11.1115	0.453246	0.226623	−	0.973983i	$-0.427232\pi$
$601$	11.1115	0.453246	0.226623	+	0.973983i	$0.427232\pi$
$602$	0	0
$603$	14.4721i	0.589351i
$604$	0.145898	0.00593651
$605$	0	0
$606$	16.1803	0.657281
$607$	17.5279i	0.711434i	0.934594	+	0.355717i	$0.115763\pi$
$607$	17.5279i	0.711434i	−0.934594	+	0.355717i	$0.884237\pi$
$608$	6.76393i	0.274314i
$609$	−8.29180	−0.336001
$610$	0	0
$611$	6.70820	0.271385
$612$	6.47214i	0.261621i
$613$	7.70820i	0.311331i	0.987810	+	0.155666i	$0.0497523\pi$
$613$	7.70820i	0.311331i	−0.987810	+	0.155666i	$0.950248\pi$
$614$	29.8885	1.20620
$615$	0	0
$616$	2.11146	0.0850730
$617$	− 16.4721i	− 0.663143i	−0.943430	−	0.331572i	$-0.892421\pi$
$617$	− 16.4721i	− 0.663143i	0.943430	−	0.331572i	$-0.107579\pi$
$618$	65.7771i	2.64594i
$619$	7.41641	0.298091	0.149045	−	0.988830i	$-0.452380\pi$
$619$	7.41641	0.298091	0.149045	+	0.988830i	$0.452380\pi$
$620$	0	0
$621$	−2.23607	−0.0897303
$622$	14.8541i	0.595595i
$623$	− 1.88854i	− 0.0756629i
$624$	32.5623	1.30354
$625$	0	0
$626$	32.9443	1.31672
$627$	3.41641i	0.136438i
$628$	− 9.52786i	− 0.380203i
$629$	−16.9443	−0.675612
$630$	0	0
$631$	−32.3607	−1.28826	−0.644129	−	0.764917i	$-0.722780\pi$
$631$	−32.3607	−1.28826	−0.644129	+	0.764917i	$0.722780\pi$
$632$	15.5279i	0.617665i
$633$	− 7.63932i	− 0.303636i
$634$	−2.29180	−0.0910188
$635$	0	0
$636$	−11.7082	−0.464260
$637$	− 16.4164i	− 0.650442i
$638$	3.70820i	0.146809i
$639$	−15.5279	−0.614273
$640$	0	0
$641$	45.3050	1.78944	0.894719	−	0.446629i	$-0.147376\pi$
$641$	45.3050	1.78944	0.894719	+	0.446629i	$0.147376\pi$
$642$	48.5410i	1.91576i
$643$	− 19.5967i	− 0.772820i	−0.922327	−	0.386410i	$-0.873715\pi$
$643$	− 19.5967i	− 0.772820i	0.922327	−	0.386410i	$-0.126285\pi$
$644$	0.763932	0.0301031
$645$	0	0
$646$	16.9443	0.666663
$647$	− 6.70820i	− 0.263727i	−0.991268	−	0.131863i	$-0.957904\pi$
$647$	− 6.70820i	− 0.263727i	0.991268	−	0.131863i	$-0.0420960\pi$
$648$	24.5967i	0.966252i
$649$	−1.88854	−0.0741318
$650$	0	0
$651$	18.5410	0.726680
$652$	− 6.32624i	− 0.247755i
$653$	− 24.3050i	− 0.951126i	−0.879682	−	0.475563i	$-0.842244\pi$
$653$	− 24.3050i	− 0.951126i	0.879682	−	0.475563i	$-0.157756\pi$
$654$	0	0
$655$	0	0
$656$	−26.5623	−1.03708
$657$	30.9443i	1.20725i
$658$	− 4.47214i	− 0.174342i
$659$	−20.6525	−0.804506	−0.402253	−	0.915528i	$-0.631773\pi$
$659$	−20.6525	−0.804506	−0.402253	+	0.915528i	$0.631773\pi$
$660$	0	0
$661$	−5.05573	−0.196645	−0.0983225	−	0.995155i	$-0.531348\pi$
$661$	−5.05573	−0.196645	−0.0983225	+	0.995155i	$0.531348\pi$
$662$	− 18.8541i	− 0.732785i
$663$	− 35.1246i	− 1.36413i
$664$	29.5967	1.14858
$665$	0	0
$666$	−10.4721	−0.405787
$667$	− 3.00000i	− 0.116160i
$668$	− 6.47214i	− 0.250414i
$669$	−8.94427	−0.345806
$670$	0	0
$671$	−8.36068	−0.322760
$672$	− 9.34752i	− 0.360589i
$673$	− 3.00000i	− 0.115642i	−0.998327	−	0.0578208i	$-0.981585\pi$
$673$	− 3.00000i	− 0.115642i	0.998327	−	0.0578208i	$-0.0184152\pi$
$674$	−5.52786	−0.212925
$675$	0	0
$676$	−2.47214	−0.0950822
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	47.8885i	1.83915i
$679$	5.30495	0.203585
$680$	0	0
$681$	22.7639	0.872316
$682$	− 8.29180i	− 0.317509i
$683$	22.5967i	0.864641i	0.901720	+	0.432320i	$0.142305\pi$
$683$	22.5967i	0.864641i	−0.901720	+	0.432320i	$0.857695\pi$
$684$	2.47214	0.0945245
$685$	0	0
$686$	−24.9443	−0.952377
$687$	− 26.8328i	− 1.02374i
$688$	0	0
$689$	25.4164	0.968288
$690$	0	0
$691$	24.9443	0.948925	0.474462	−	0.880276i	$-0.342642\pi$
$691$	24.9443	0.948925	0.474462	+	0.880276i	$0.342642\pi$
$692$	3.12461i	0.118780i
$693$	− 1.88854i	− 0.0717398i
$694$	41.8885	1.59007
$695$	0	0
$696$	−15.0000	−0.568574
$697$	28.6525i	1.08529i
$698$	− 3.90983i	− 0.147989i
$699$	34.5967	1.30857
$700$	0	0
$701$	−26.1803	−0.988818	−0.494409	−	0.869229i	$-0.664615\pi$
$701$	−26.1803	−0.988818	−0.494409	+	0.869229i	$0.664615\pi$
$702$	10.8541i	0.409662i
$703$	6.47214i	0.244101i
$704$	3.23607	0.121964
$705$	0	0
$706$	57.2148	2.15331
$707$	5.52786i	0.207897i
$708$	3.41641i	0.128396i
$709$	−16.0689	−0.603480	−0.301740	−	0.953390i	$-0.597567\pi$
$709$	−16.0689	−0.603480	−0.301740	+	0.953390i	$0.597567\pi$
$710$	0	0
$711$	13.8885	0.520861
$712$	− 3.41641i	− 0.128035i
$713$	6.70820i	0.251224i
$714$	−23.4164	−0.876337
$715$	0	0
$716$	−7.85410	−0.293522
$717$	40.7771i	1.52285i
$718$	25.7082i	0.959422i
$719$	20.9443	0.781090	0.390545	−	0.920584i	$-0.372287\pi$
$719$	20.9443	0.781090	0.390545	+	0.920584i	$0.372287\pi$
$720$	0	0
$721$	−22.4721	−0.836906
$722$	24.2705i	0.903255i
$723$	− 38.2918i	− 1.42409i
$724$	9.05573	0.336553
$725$	0	0
$726$	37.6869	1.39869
$727$	− 14.2918i	− 0.530053i	−0.964241	−	0.265027i	$-0.914619\pi$
$727$	− 14.2918i	− 0.530053i	0.964241	−	0.265027i	$-0.0853808\pi$
$728$	8.29180i	0.307314i
$729$	7.00000	0.259259
$730$	0	0
$731$	0	0
$732$	15.1246i	0.559022i
$733$	26.7639i	0.988548i	0.869306	+	0.494274i	$0.164566\pi$
$733$	26.7639i	0.988548i	−0.869306	+	0.494274i	$0.835434\pi$
$734$	29.4164	1.08578
$735$	0	0
$736$	3.38197	0.124661
$737$	5.52786i	0.203621i
$738$	17.7082i	0.651848i
$739$	−49.1803	−1.80913	−0.904564	−	0.426338i	$-0.859803\pi$
$739$	−49.1803	−1.80913	−0.904564	+	0.426338i	$0.859803\pi$
$740$	0	0
$741$	−13.4164	−0.492864
$742$	− 16.9443i	− 0.622044i
$743$	− 0.875388i	− 0.0321149i	−0.999871	−	0.0160574i	$-0.994889\pi$
$743$	− 0.875388i	− 0.0321149i	0.999871	−	0.0160574i	$-0.00511146\pi$
$744$	33.5410	1.22967
$745$	0	0
$746$	9.23607	0.338156
$747$	− 26.4721i	− 0.968565i
$748$	2.47214i	0.0903902i
$749$	−16.5836	−0.605951
$750$	0	0
$751$	−44.3607	−1.61874	−0.809372	−	0.587296i	$-0.800192\pi$
$751$	−44.3607	−1.61874	−0.809372	+	0.587296i	$0.800192\pi$
$752$	− 10.8541i	− 0.395808i
$753$	− 35.1246i	− 1.28001i
$754$	−14.5623	−0.530328
$755$	0	0
$756$	1.70820	0.0621268
$757$	− 47.5967i	− 1.72993i	−0.501829	−	0.864967i	$-0.667339\pi$
$757$	− 47.5967i	− 1.72993i	0.501829	−	0.864967i	$-0.332661\pi$
$758$	− 32.9443i	− 1.19659i
$759$	1.70820	0.0620039
$760$	0	0
$761$	−16.3050	−0.591054	−0.295527	−	0.955334i	$-0.595495\pi$
$761$	−16.3050	−0.591054	−0.295527	+	0.955334i	$0.595495\pi$
$762$	74.9230i	2.71417i
$763$	0	0
$764$	2.36068	0.0854064
$765$	0	0
$766$	−40.3607	−1.45829
$767$	− 7.41641i	− 0.267791i
$768$	− 30.3262i	− 1.09430i
$769$	−17.1246	−0.617529	−0.308765	−	0.951138i	$-0.599916\pi$
$769$	−17.1246	−0.617529	−0.308765	+	0.951138i	$0.599916\pi$
$770$	0	0
$771$	3.29180	0.118551
$772$	− 4.90983i	− 0.176709i
$773$	14.4721i	0.520527i	0.965538	+	0.260263i	$0.0838094\pi$
$773$	14.4721i	0.520527i	−0.965538	+	0.260263i	$0.916191\pi$
$774$	0	0
$775$	0	0
$776$	9.59675	0.344503
$777$	− 8.94427i	− 0.320874i
$778$	55.7771i	1.99971i
$779$	10.9443	0.392119
$780$	0	0
$781$	−5.93112	−0.212232
$782$	− 8.47214i	− 0.302963i
$783$	− 6.70820i	− 0.239732i
$784$	−26.5623	−0.948654
$785$	0	0
$786$	−19.1459	−0.682912
$787$	51.4164i	1.83280i	0.400267	+	0.916399i	$0.368917\pi$
$787$	51.4164i	1.83280i	−0.400267	+	0.916399i	$0.631083\pi$
$788$	− 4.61803i	− 0.164511i
$789$	33.4164	1.18966
$790$	0	0
$791$	−16.3607	−0.581719
$792$	− 3.41641i	− 0.121397i
$793$	− 32.8328i	− 1.16593i
$794$	3.90983	0.138755
$795$	0	0
$796$	−15.8885	−0.563155
$797$	10.3607i	0.366994i	0.983020	+	0.183497i	$0.0587418\pi$
$797$	10.3607i	0.366994i	−0.983020	+	0.183497i	$0.941258\pi$
$798$	8.94427i	0.316624i
$799$	−11.7082	−0.414206
$800$	0	0
$801$	−3.05573	−0.107969
$802$	− 13.2361i	− 0.467382i
$803$	11.8197i	0.417107i
$804$	10.0000	0.352673
$805$	0	0
$806$	32.5623	1.14696
$807$	22.2361i	0.782747i
$808$	10.0000i	0.351799i
$809$	−47.8885	−1.68367	−0.841836	−	0.539734i	$-0.818525\pi$
$809$	−47.8885	−1.68367	−0.841836	+	0.539734i	$0.818525\pi$
$810$	0	0
$811$	−55.6525	−1.95422	−0.977111	−	0.212728i	$-0.931765\pi$
$811$	−55.6525	−1.95422	−0.977111	+	0.212728i	$0.931765\pi$
$812$	2.29180i	0.0804263i
$813$	− 17.8885i	− 0.627379i
$814$	−4.00000	−0.140200
$815$	0	0
$816$	−56.8328	−1.98955
$817$	0	0
$818$	− 37.7984i	− 1.32159i
$819$	7.41641	0.259150
$820$	0	0
$821$	−21.0557	−0.734850	−0.367425	−	0.930053i	$-0.619761\pi$
$821$	−21.0557	−0.734850	−0.367425	+	0.930053i	$0.619761\pi$
$822$	− 50.2492i	− 1.75264i
$823$	− 27.5410i	− 0.960020i	−0.877263	−	0.480010i	$-0.840633\pi$
$823$	− 27.5410i	− 0.960020i	0.877263	−	0.480010i	$-0.159367\pi$
$824$	−40.6525	−1.41620
$825$	0	0
$826$	−4.94427	−0.172033
$827$	10.4721i	0.364152i	0.983284	+	0.182076i	$0.0582817\pi$
$827$	10.4721i	0.364152i	−0.983284	+	0.182076i	$0.941718\pi$
$828$	− 1.23607i	− 0.0429563i
$829$	40.2492	1.39791	0.698957	−	0.715164i	$-0.253648\pi$
$829$	40.2492	1.39791	0.698957	+	0.715164i	$0.253648\pi$
$830$	0	0
$831$	14.5967	0.506356
$832$	12.7082i	0.440578i
$833$	28.6525i	0.992749i
$834$	9.79837	0.339290
$835$	0	0
$836$	0.944272	0.0326583
$837$	15.0000i	0.518476i
$838$	− 50.8328i	− 1.75599i
$839$	0.875388	0.0302218	0.0151109	−	0.999886i	$-0.495190\pi$
$839$	0.875388	0.0302218	0.0151109	+	0.999886i	$0.495190\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	38.3607i	1.32200i
$843$	29.5967i	1.01937i
$844$	−2.11146	−0.0726793
$845$	0	0
$846$	−7.23607	−0.248781
$847$	12.8754i	0.442404i
$848$	− 41.1246i	− 1.41222i
$849$	−31.9574	−1.09678
$850$	0	0
$851$	3.23607	0.110931
$852$	10.7295i	0.367586i
$853$	37.4164i	1.28111i	0.767911	+	0.640557i	$0.221296\pi$
$853$	37.4164i	1.28111i	−0.767911	+	0.640557i	$0.778704\pi$
$854$	−21.8885	−0.749011
$855$	0	0
$856$	−30.0000	−1.02538
$857$	− 7.47214i	− 0.255243i	−0.991823	−	0.127622i	$-0.959266\pi$
$857$	− 7.47214i	− 0.255243i	0.991823	−	0.127622i	$-0.0407343\pi$
$858$	− 8.29180i	− 0.283077i
$859$	3.29180	0.112315	0.0561573	−	0.998422i	$-0.482115\pi$
$859$	3.29180	0.112315	0.0561573	+	0.998422i	$0.482115\pi$
$860$	0	0
$861$	−15.1246	−0.515445
$862$	42.8328i	1.45889i
$863$	− 45.5410i	− 1.55023i	−0.631818	−	0.775117i	$-0.717691\pi$
$863$	− 45.5410i	− 1.55023i	0.631818	−	0.775117i	$-0.282309\pi$
$864$	7.56231	0.257275
$865$	0	0
$866$	−65.0132	−2.20924
$867$	23.2918i	0.791031i
$868$	− 5.12461i	− 0.173941i
$869$	5.30495	0.179958
$870$	0	0
$871$	−21.7082	−0.735554
$872$	0	0
$873$	− 8.58359i	− 0.290511i
$874$	−3.23607	−0.109462
$875$	0	0
$876$	21.3820	0.722430
$877$	− 27.5279i	− 0.929550i	−0.885429	−	0.464775i	$-0.846135\pi$
$877$	− 27.5279i	− 0.929550i	0.885429	−	0.464775i	$-0.153865\pi$
$878$	− 8.56231i	− 0.288964i
$879$	23.4164	0.789816
$880$	0	0
$881$	21.8197	0.735123	0.367562	−	0.929999i	$-0.380193\pi$
$881$	21.8197	0.735123	0.367562	+	0.929999i	$0.380193\pi$
$882$	17.7082i	0.596266i
$883$	− 4.00000i	− 0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$	− 4.00000i	− 0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	−9.70820	−0.326522
$885$	0	0
$886$	3.43769	0.115492
$887$	− 35.0689i	− 1.17750i	−0.808316	−	0.588749i	$-0.799621\pi$
$887$	− 35.0689i	− 1.17750i	0.808316	−	0.588749i	$-0.200379\pi$
$888$	− 16.1803i	− 0.542977i
$889$	−25.5967	−0.858487
$890$	0	0
$891$	8.40325	0.281520
$892$	2.47214i	0.0827732i
$893$	4.47214i	0.149654i
$894$	−43.0132	−1.43858
$895$	0	0
$896$	16.8328	0.562345
$897$	6.70820i	0.223980i
$898$	4.76393i	0.158974i
$899$	−20.1246	−0.671193
$900$	0	0
$901$	−44.3607	−1.47787
$902$	6.76393i	0.225214i
$903$	0	0
$904$	−29.5967	−0.984373
$905$	0	0
$906$	0.854102	0.0283756
$907$	40.2492i	1.33645i	0.743958	+	0.668227i	$0.232946\pi$
$907$	40.2492i	1.33645i	−0.743958	+	0.668227i	$0.767054\pi$
$908$	− 6.29180i	− 0.208801i
$909$	8.94427	0.296663
$910$	0	0
$911$	−31.3050	−1.03718	−0.518590	−	0.855023i	$-0.673543\pi$
$911$	−31.3050	−1.03718	−0.518590	+	0.855023i	$0.673543\pi$
$912$	21.7082i	0.718830i
$913$	− 10.1115i	− 0.334640i
$914$	56.8328	1.87986
$915$	0	0
$916$	−7.41641	−0.245045
$917$	− 6.54102i	− 0.216003i
$918$	− 18.9443i	− 0.625254i
$919$	−0.875388	−0.0288764	−0.0144382	−	0.999896i	$-0.504596\pi$
$919$	−0.875388	−0.0288764	−0.0144382	+	0.999896i	$0.504596\pi$
$920$	0	0
$921$	41.3050	1.36104
$922$	− 12.0902i	− 0.398169i
$923$	− 23.2918i	− 0.766659i
$924$	−1.30495	−0.0429298
$925$	0	0
$926$	32.3607	1.06344
$927$	36.3607i	1.19424i
$928$	10.1459i	0.333055i
$929$	41.9443	1.37615	0.688073	−	0.725641i	$-0.258457\pi$
$929$	41.9443	1.37615	0.688073	+	0.725641i	$0.258457\pi$
$930$	0	0
$931$	10.9443	0.358684
$932$	− 9.56231i	− 0.313224i
$933$	20.5279i	0.672052i
$934$	−50.0689	−1.63830
$935$	0	0
$936$	13.4164	0.438529
$937$	11.8197i	0.386131i	0.981186	+	0.193066i	$0.0618431\pi$
$937$	11.8197i	0.386131i	−0.981186	+	0.193066i	$0.938157\pi$
$938$	14.4721i	0.472532i
$939$	45.5279	1.48575
$940$	0	0
$941$	−24.6525	−0.803648	−0.401824	−	0.915717i	$-0.631624\pi$
$941$	−24.6525	−0.803648	−0.401824	+	0.915717i	$0.631624\pi$
$942$	− 55.7771i	− 1.81732i
$943$	− 5.47214i	− 0.178197i
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	− 33.1803i	− 1.07822i	−0.842237	−	0.539108i	$-0.818761\pi$
$947$	− 33.1803i	− 1.07822i	0.842237	−	0.539108i	$-0.181239\pi$
$948$	− 9.59675i	− 0.311688i
$949$	−46.4164	−1.50674
$950$	0	0
$951$	−3.16718	−0.102703
$952$	− 14.4721i	− 0.469045i
$953$	− 11.5279i	− 0.373424i	−0.982415	−	0.186712i	$-0.940217\pi$
$953$	− 11.5279i	− 0.373424i	0.982415	−	0.186712i	$-0.0597831\pi$
$954$	−27.4164	−0.887639
$955$	0	0
$956$	11.2705	0.364514
$957$	5.12461i	0.165655i
$958$	− 28.4721i	− 0.919893i
$959$	17.1672	0.554357
$960$	0	0
$961$	14.0000	0.451613
$962$	− 15.7082i	− 0.506453i
$963$	26.8328i	0.864675i
$964$	−10.5836	−0.340875
$965$	0	0
$966$	4.47214	0.143889
$967$	− 39.5410i	− 1.27155i	−0.771873	−	0.635777i	$-0.780680\pi$
$967$	− 39.5410i	− 1.27155i	0.771873	−	0.635777i	$-0.219320\pi$
$968$	23.2918i	0.748627i
$969$	23.4164	0.752243
$970$	0	0
$971$	7.52786	0.241581	0.120790	−	0.992678i	$-0.461457\pi$
$971$	7.52786	0.241581	0.120790	+	0.992678i	$0.461457\pi$
$972$	− 11.0557i	− 0.354613i
$973$	3.34752i	0.107317i
$974$	−2.09017	−0.0669734
$975$	0	0
$976$	−53.1246	−1.70048
$977$	− 54.6525i	− 1.74849i	−0.485487	−	0.874244i	$-0.661358\pi$
$977$	− 54.6525i	− 1.74849i	0.485487	−	0.874244i	$-0.338642\pi$
$978$	− 37.0344i	− 1.18423i
$979$	−1.16718	−0.0373034
$980$	0	0
$981$	0	0
$982$	− 64.1591i	− 2.04740i
$983$	31.5279i	1.00558i	0.864408	+	0.502791i	$0.167694\pi$
$983$	31.5279i	1.00558i	−0.864408	+	0.502791i	$0.832306\pi$
$984$	−27.3607	−0.872227
$985$	0	0
$986$	25.4164	0.809423
$987$	− 6.18034i	− 0.196722i
$988$	3.70820i	0.117974i
$989$	0	0
$990$	0	0
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	− 22.6869i	− 0.720310i
$993$	− 26.0557i	− 0.826854i
$994$	−15.5279	−0.492514
$995$	0	0
$996$	−18.2918	−0.579598
$997$	− 36.8328i	− 1.16651i	−0.812290	−	0.583253i	$-0.801779\pi$
$997$	− 36.8328i	− 1.16651i	0.812290	−	0.583253i	$-0.198221\pi$
$998$	52.9230i	1.67525i
$999$	7.23607	0.228939

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	575.2.b.d.24.1		4
5.2	odd	4		575.2.a.f.1.2		2
5.3	odd	4		23.2.a.a.1.1	✓	2
5.4	even	2	inner	575.2.b.d.24.4		4
15.2	even	4		5175.2.a.be.1.1		2
15.8	even	4		207.2.a.d.1.2		2
20.3	even	4		368.2.a.h.1.1		2
20.7	even	4		9200.2.a.bt.1.2		2
35.13	even	4		1127.2.a.c.1.1		2
40.3	even	4		1472.2.a.s.1.2		2
40.13	odd	4		1472.2.a.t.1.1		2
55.43	even	4		2783.2.a.c.1.2		2
60.23	odd	4		3312.2.a.ba.1.2		2
65.38	odd	4		3887.2.a.i.1.2		2
85.33	odd	4		6647.2.a.b.1.1		2
95.18	even	4		8303.2.a.e.1.2		2
115.3	odd	44		529.2.c.o.170.2		20
115.8	odd	44		529.2.c.o.501.2		20
115.13	odd	44		529.2.c.o.399.2		20
115.18	odd	44		529.2.c.o.255.1		20
115.28	even	44		529.2.c.n.255.1		20
115.33	even	44		529.2.c.n.399.2		20
115.38	even	44		529.2.c.n.501.2		20
115.43	even	44		529.2.c.n.170.2		20
115.48	odd	44		529.2.c.o.487.1		20
115.53	even	44		529.2.c.n.118.2		20
115.58	odd	44		529.2.c.o.466.1		20
115.63	even	44		529.2.c.n.266.2		20
115.68	even	4		529.2.a.a.1.1		2
115.73	odd	44		529.2.c.o.177.2		20
115.78	odd	44		529.2.c.o.334.1		20
115.83	even	44		529.2.c.n.334.1		20
115.88	even	44		529.2.c.n.177.2		20
115.98	odd	44		529.2.c.o.266.2		20
115.103	even	44		529.2.c.n.466.1		20
115.108	odd	44		529.2.c.o.118.2		20
115.113	even	44		529.2.c.n.487.1		20
345.68	odd	4		4761.2.a.w.1.2		2
460.183	odd	4		8464.2.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.2.a.a.1.1	✓	2	5.3	odd	4
207.2.a.d.1.2		2	15.8	even	4
368.2.a.h.1.1		2	20.3	even	4
529.2.a.a.1.1		2	115.68	even	4
529.2.c.n.118.2		20	115.53	even	44
529.2.c.n.170.2		20	115.43	even	44
529.2.c.n.177.2		20	115.88	even	44
529.2.c.n.255.1		20	115.28	even	44
529.2.c.n.266.2		20	115.63	even	44
529.2.c.n.334.1		20	115.83	even	44
529.2.c.n.399.2		20	115.33	even	44
529.2.c.n.466.1		20	115.103	even	44
529.2.c.n.487.1		20	115.113	even	44
529.2.c.n.501.2		20	115.38	even	44
529.2.c.o.118.2		20	115.108	odd	44
529.2.c.o.170.2		20	115.3	odd	44
529.2.c.o.177.2		20	115.73	odd	44
529.2.c.o.255.1		20	115.18	odd	44
529.2.c.o.266.2		20	115.98	odd	44
529.2.c.o.334.1		20	115.78	odd	44
529.2.c.o.399.2		20	115.13	odd	44
529.2.c.o.466.1		20	115.58	odd	44
529.2.c.o.487.1		20	115.48	odd	44
529.2.c.o.501.2		20	115.8	odd	44
575.2.a.f.1.2		2	5.2	odd	4
575.2.b.d.24.1		4	1.1	even	1	trivial
575.2.b.d.24.4		4	5.4	even	2	inner
1127.2.a.c.1.1		2	35.13	even	4
1472.2.a.s.1.2		2	40.3	even	4
1472.2.a.t.1.1		2	40.13	odd	4
2783.2.a.c.1.2		2	55.43	even	4
3312.2.a.ba.1.2		2	60.23	odd	4
3887.2.a.i.1.2		2	65.38	odd	4
4761.2.a.w.1.2		2	345.68	odd	4
5175.2.a.be.1.1		2	15.2	even	4
6647.2.a.b.1.1		2	85.33	odd	4
8303.2.a.e.1.2		2	95.18	even	4
8464.2.a.bb.1.1		2	460.183	odd	4
9200.2.a.bt.1.2		2	20.7	even	4

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:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	575.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.61803i q^{2} -2.23607i q^{3} -0.618034 q^{4} -3.61803 q^{6} -1.23607i q^{7} -2.23607i q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q-1.61803i q^{2} -2.23607i q^{3} -0.618034 q^{4} -3.61803 q^{6} -1.23607i q^{7} -2.23607i q^{8} -2.00000 q^{9} -0.763932 q^{11} +1.38197i q^{12} -3.00000i q^{13} -2.00000 q^{14} -4.85410 q^{16} +5.23607i q^{17} +3.23607i q^{18} +2.00000 q^{19} -2.76393 q^{21} +1.23607i q^{22} -1.00000i q^{23} -5.00000 q^{24} -4.85410 q^{26} -2.23607i q^{27} +0.763932i q^{28} +3.00000 q^{29} -6.70820 q^{31} +3.38197i q^{32} +1.70820i q^{33} +8.47214 q^{34} +1.23607 q^{36} +3.23607i q^{37} -3.23607i q^{38} -6.70820 q^{39} +5.47214 q^{41} +4.47214i q^{42} +0.472136 q^{44} -1.61803 q^{46} +2.23607i q^{47} +10.8541i q^{48} +5.47214 q^{49} +11.7082 q^{51} +1.85410i q^{52} +8.47214i q^{53} -3.61803 q^{54} -2.76393 q^{56} -4.47214i q^{57} -4.85410i q^{58} +2.47214 q^{59} +10.9443 q^{61} +10.8541i q^{62} +2.47214i q^{63} -4.23607 q^{64} +2.76393 q^{66} -7.23607i q^{67} -3.23607i q^{68} -2.23607 q^{69} +7.76393 q^{71} +4.47214i q^{72} -15.4721i q^{73} +5.23607 q^{74} -1.23607 q^{76} +0.944272i q^{77} +10.8541i q^{78} -6.94427 q^{79} -11.0000 q^{81} -8.85410i q^{82} +13.2361i q^{83} +1.70820 q^{84} -6.70820i q^{87} +1.70820i q^{88} +1.52786 q^{89} -3.70820 q^{91} +0.618034i q^{92} +15.0000i q^{93} +3.61803 q^{94} +7.56231 q^{96} +4.29180i q^{97} -8.85410i q^{98} +1.52786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 10 q^{6} - 8 q^{9}+O(q^{10}) \) 4 * q + 2 * q^4 - 10 * q^6 - 8 * q^9	\( 4 q + 2 q^{4} - 10 q^{6} - 8 q^{9} - 12 q^{11} - 8 q^{14} - 6 q^{16} + 8 q^{19} - 20 q^{21} - 20 q^{24} - 6 q^{26} + 12 q^{29} + 16 q^{34} - 4 q^{36} + 4 q^{41} - 16 q^{44} - 2 q^{46} + 4 q^{49} + 20 q^{51} - 10 q^{54} - 20 q^{56} - 8 q^{59} + 8 q^{61} - 8 q^{64} + 20 q^{66} + 40 q^{71} + 12 q^{74} + 4 q^{76} + 8 q^{79} - 44 q^{81} - 20 q^{84} + 24 q^{89} + 12 q^{91} + 10 q^{94} - 10 q^{96} + 24 q^{99}+O(q^{100}) \) 4 * q + 2 * q^4 - 10 * q^6 - 8 * q^9 - 12 * q^11 - 8 * q^14 - 6 * q^16 + 8 * q^19 - 20 * q^21 - 20 * q^24 - 6 * q^26 + 12 * q^29 + 16 * q^34 - 4 * q^36 + 4 * q^41 - 16 * q^44 - 2 * q^46 + 4 * q^49 + 20 * q^51 - 10 * q^54 - 20 * q^56 - 8 * q^59 + 8 * q^61 - 8 * q^64 + 20 * q^66 + 40 * q^71 + 12 * q^74 + 4 * q^76 + 8 * q^79 - 44 * q^81 - 20 * q^84 + 24 * q^89 + 12 * q^91 + 10 * q^94 - 10 * q^96 + 24 * q^99

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