LMFDB - Embedded newform 575.2.b.d.24.3

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			24.3
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	575.24
Dual form			575.2.b.d.24.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034i q^{2} +2.23607i q^{3} +1.61803 q^{4} -1.38197 q^{6} +3.23607i q^{7} +2.23607i q^{8} -2.00000 q^{9} +O(q^{10})$	\(q+0.618034i q^{2} +2.23607i q^{3} +1.61803 q^{4} -1.38197 q^{6} +3.23607i q^{7} +2.23607i q^{8} -2.00000 q^{9} -5.23607 q^{11} +3.61803i q^{12} -3.00000i q^{13} -2.00000 q^{14} +1.85410 q^{16} +0.763932i q^{17} -1.23607i q^{18} +2.00000 q^{19} -7.23607 q^{21} -3.23607i q^{22} -1.00000i q^{23} -5.00000 q^{24} +1.85410 q^{26} +2.23607i q^{27} +5.23607i q^{28} +3.00000 q^{29} +6.70820 q^{31} +5.61803i q^{32} -11.7082i q^{33} -0.472136 q^{34} -3.23607 q^{36} -1.23607i q^{37} +1.23607i q^{38} +6.70820 q^{39} -3.47214 q^{41} -4.47214i q^{42} -8.47214 q^{44} +0.618034 q^{46} -2.23607i q^{47} +4.14590i q^{48} -3.47214 q^{49} -1.70820 q^{51} -4.85410i q^{52} -0.472136i q^{53} -1.38197 q^{54} -7.23607 q^{56} +4.47214i q^{57} +1.85410i q^{58} -6.47214 q^{59} -6.94427 q^{61} +4.14590i q^{62} -6.47214i q^{63} +0.236068 q^{64} +7.23607 q^{66} -2.76393i q^{67} +1.23607i q^{68} +2.23607 q^{69} +12.2361 q^{71} -4.47214i q^{72} -6.52786i q^{73} +0.763932 q^{74} +3.23607 q^{76} -16.9443i q^{77} +4.14590i q^{78} +10.9443 q^{79} -11.0000 q^{81} -2.14590i q^{82} +8.76393i q^{83} -11.7082 q^{84} +6.70820i q^{87} -11.7082i q^{88} +10.4721 q^{89} +9.70820 q^{91} -1.61803i q^{92} +15.0000i q^{93} +1.38197 q^{94} -12.5623 q^{96} +17.7082i q^{97} -2.14590i q^{98} +10.4721 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.618034i	0.437016i	0.975835	+	0.218508i	$0.0701190\pi$
$2$	0.618034i	0.437016i	−0.975835	+	0.218508i	$0.929881\pi$
$3$	2.23607i	1.29099i	0.763763	+	0.645497i	$0.223350\pi$
$3$	2.23607i	1.29099i	−0.763763	+	0.645497i	$0.776650\pi$
$4$	1.61803	0.809017
$5$	0	0
$5$	0	0
$6$	−1.38197	−0.564185
$7$	3.23607i	1.22312i	0.791199	+	0.611559i	$0.209457\pi$
$7$	3.23607i	1.22312i	−0.791199	+	0.611559i	$0.790543\pi$
$8$	2.23607i	0.790569i
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−5.23607	−1.57873	−0.789367	−	0.613922i	$-0.789591\pi$
$11$	−5.23607	−1.57873	−0.789367	+	0.613922i	$0.789591\pi$
$12$	3.61803i	1.04444i
$13$	− 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
$16$	1.85410	0.463525
$17$	0.763932i	0.185281i	0.995700	+	0.0926404i	$0.0295307\pi$
$17$	0.763932i	0.185281i	−0.995700	+	0.0926404i	$0.970469\pi$
$18$	− 1.23607i	− 0.291344i
$19$	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
$21$	−7.23607	−1.57904
$22$	− 3.23607i	− 0.689932i
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	−5.00000	−1.02062
$25$	0	0
$26$	1.85410	0.363619
$27$	2.23607i	0.430331i
$28$	5.23607i	0.989524i
$29$	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	$-0.294205\pi$
$31$	6.70820	1.20483	0.602414	+	0.798183i	$0.294205\pi$
$32$	5.61803i	0.993137i
$33$	− 11.7082i	− 2.03814i
$34$	−0.472136	−0.0809706
$35$	0	0
$36$	−3.23607	−0.539345
$37$	− 1.23607i	− 0.203208i	−0.994825	−	0.101604i	$-0.967602\pi$
$37$	− 1.23607i	− 0.203208i	0.994825	−	0.101604i	$-0.0323975\pi$
$38$	1.23607i	0.200517i
$39$	6.70820	1.07417
$40$	0	0
$41$	−3.47214	−0.542257	−0.271128	−	0.962543i	$-0.587397\pi$
$41$	−3.47214	−0.542257	−0.271128	+	0.962543i	$0.587397\pi$
$42$	− 4.47214i	− 0.690066i
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	−8.47214	−1.27722
$45$	0	0
$46$	0.618034	0.0911241
$47$	− 2.23607i	− 0.326164i	−0.986613	−	0.163082i	$-0.947856\pi$
$47$	− 2.23607i	− 0.326164i	0.986613	−	0.163082i	$-0.0521435\pi$
$48$	4.14590i	0.598409i
$49$	−3.47214	−0.496019
$50$	0	0
$51$	−1.70820	−0.239196
$52$	− 4.85410i	− 0.673143i
$53$	− 0.472136i	− 0.0648529i	−0.999474	−	0.0324264i	$-0.989677\pi$
$53$	− 0.472136i	− 0.0648529i	0.999474	−	0.0324264i	$-0.0103235\pi$
$54$	−1.38197	−0.188062
$55$	0	0
$56$	−7.23607	−0.966960
$57$	4.47214i	0.592349i
$58$	1.85410i	0.243456i
$59$	−6.47214	−0.842600	−0.421300	−	0.906921i	$-0.638426\pi$
$59$	−6.47214	−0.842600	−0.421300	+	0.906921i	$0.638426\pi$
$60$	0	0
$61$	−6.94427	−0.889123	−0.444561	−	0.895748i	$-0.646640\pi$
$61$	−6.94427	−0.889123	−0.444561	+	0.895748i	$0.646640\pi$
$62$	4.14590i	0.526530i
$63$	− 6.47214i	− 0.815412i
$64$	0.236068	0.0295085
$65$	0	0
$66$	7.23607	0.890698
$67$	− 2.76393i	− 0.337668i	−0.985644	−	0.168834i	$-0.946000\pi$
$67$	− 2.76393i	− 0.337668i	0.985644	−	0.168834i	$-0.0540002\pi$
$68$	1.23607i	0.149895i
$69$	2.23607	0.269191
$70$	0	0
$71$	12.2361	1.45215	0.726077	−	0.687613i	$-0.241342\pi$
$71$	12.2361	1.45215	0.726077	+	0.687613i	$0.241342\pi$
$72$	− 4.47214i	− 0.527046i
$73$	− 6.52786i	− 0.764029i	−0.924156	−	0.382014i	$-0.875230\pi$
$73$	− 6.52786i	− 0.764029i	0.924156	−	0.382014i	$-0.124770\pi$
$74$	0.763932	0.0888053
$75$	0	0
$76$	3.23607	0.371202
$77$	− 16.9443i	− 1.93098i
$78$	4.14590i	0.469431i
$79$	10.9443	1.23133	0.615663	−	0.788009i	$-0.288888\pi$
$79$	10.9443	1.23133	0.615663	+	0.788009i	$0.288888\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	− 2.14590i	− 0.236975i
$83$	8.76393i	0.961967i	0.876730	+	0.480983i	$0.159720\pi$
$83$	8.76393i	0.961967i	−0.876730	+	0.480983i	$0.840280\pi$
$84$	−11.7082	−1.27747
$85$	0	0
$86$	0	0
$87$	6.70820i	0.719195i
$88$	− 11.7082i	− 1.24810i
$89$	10.4721	1.11004	0.555022	−	0.831836i	$-0.312710\pi$
$89$	10.4721	1.11004	0.555022	+	0.831836i	$0.312710\pi$
$90$	0	0
$91$	9.70820	1.01770
$92$	− 1.61803i	− 0.168692i
$93$	15.0000i	1.55543i
$94$	1.38197	0.142539
$95$	0	0
$96$	−12.5623	−1.28213
$97$	17.7082i	1.79800i	0.437953	+	0.898998i	$0.355704\pi$
$97$	17.7082i	1.79800i	−0.437953	+	0.898998i	$0.644296\pi$
$98$	− 2.14590i	− 0.216768i
$99$	10.4721	1.05249
$100$	0	0
$101$	4.47214	0.444994	0.222497	−	0.974933i	$-0.428579\pi$
$101$	4.47214	0.444994	0.222497	+	0.974933i	$0.428579\pi$
$102$	− 1.05573i	− 0.104533i
$103$	4.18034i	0.411901i	0.978562	+	0.205951i	$0.0660286\pi$
$103$	4.18034i	0.411901i	−0.978562	+	0.205951i	$0.933971\pi$
$104$	6.70820	0.657794
$105$	0	0
$106$	0.291796	0.0283417
$107$	13.4164i	1.29701i	0.761209	+	0.648507i	$0.224606\pi$
$107$	13.4164i	1.29701i	−0.761209	+	0.648507i	$0.775394\pi$
$108$	3.61803i	0.348145i
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	2.76393	0.262341
$112$	6.00000i	0.566947i
$113$	− 8.76393i	− 0.824441i	−0.911084	−	0.412221i	$-0.864753\pi$
$113$	− 8.76393i	− 0.824441i	0.911084	−	0.412221i	$-0.135247\pi$
$114$	−2.76393	−0.258866
$115$	0	0
$116$	4.85410	0.450692
$117$	6.00000i	0.554700i
$118$	− 4.00000i	− 0.368230i
$119$	−2.47214	−0.226620
$120$	0	0
$121$	16.4164	1.49240
$122$	− 4.29180i	− 0.388561i
$123$	− 7.76393i	− 0.700050i
$124$	10.8541	0.974727
$125$	0	0
$126$	4.00000	0.356348
$127$	− 7.29180i	− 0.647042i	−0.946221	−	0.323521i	$-0.895133\pi$
$127$	− 7.29180i	− 0.647042i	0.946221	−	0.323521i	$-0.104867\pi$
$128$	11.3820i	1.00603i
$129$	0	0
$130$	0	0
$131$	18.7082	1.63454	0.817272	−	0.576253i	$-0.195486\pi$
$131$	18.7082	1.63454	0.817272	+	0.576253i	$0.195486\pi$
$132$	− 18.9443i	− 1.64889i
$133$	6.47214i	0.561205i
$134$	1.70820	0.147566
$135$	0	0
$136$	−1.70820	−0.146477
$137$	− 21.8885i	− 1.87006i	−0.354563	−	0.935032i	$-0.615370\pi$
$137$	− 21.8885i	− 1.87006i	0.354563	−	0.935032i	$-0.384630\pi$
$138$	1.38197i	0.117641i
$139$	10.7082	0.908258	0.454129	−	0.890936i	$-0.349951\pi$
$139$	10.7082	0.908258	0.454129	+	0.890936i	$0.349951\pi$
$140$	0	0
$141$	5.00000	0.421076
$142$	7.56231i	0.634615i
$143$	15.7082i	1.31359i
$144$	−3.70820	−0.309017
$145$	0	0
$146$	4.03444	0.333893
$147$	− 7.76393i	− 0.640358i
$148$	− 2.00000i	− 0.164399i
$149$	−23.8885	−1.95703	−0.978513	−	0.206186i	$-0.933895\pi$
$149$	−23.8885	−1.95703	−0.978513	+	0.206186i	$0.933895\pi$
$150$	0	0
$151$	4.23607	0.344726	0.172363	−	0.985033i	$-0.444860\pi$
$151$	4.23607	0.344726	0.172363	+	0.985033i	$0.444860\pi$
$152$	4.47214i	0.362738i
$153$	− 1.52786i	− 0.123520i
$154$	10.4721	0.843869
$155$	0	0
$156$	10.8541	0.869024
$157$	− 11.4164i	− 0.911129i	−0.890203	−	0.455564i	$-0.849438\pi$
$157$	− 11.4164i	− 0.911129i	0.890203	−	0.455564i	$-0.150562\pi$
$158$	6.76393i	0.538110i
$159$	1.05573	0.0837247
$160$	0	0
$161$	3.23607	0.255038
$162$	− 6.79837i	− 0.534131i
$163$	5.76393i	0.451466i	0.974189	+	0.225733i	$0.0724777\pi$
$163$	5.76393i	0.451466i	−0.974189	+	0.225733i	$0.927522\pi$
$164$	−5.61803	−0.438695
$165$	0	0
$166$	−5.41641	−0.420395
$167$	1.52786i	0.118230i	0.998251	+	0.0591148i	$0.0188278\pi$
$167$	1.52786i	0.118230i	−0.998251	+	0.0591148i	$0.981172\pi$
$168$	− 16.1803i	− 1.24834i
$169$	4.00000	0.307692
$170$	0	0
$171$	−4.00000	−0.305888
$172$	0	0
$173$	− 22.9443i	− 1.74442i	−0.489131	−	0.872210i	$-0.662686\pi$
$173$	− 22.9443i	− 1.74442i	0.489131	−	0.872210i	$-0.337314\pi$
$174$	−4.14590	−0.314300
$175$	0	0
$176$	−9.70820	−0.731783
$177$	− 14.4721i	− 1.08779i
$178$	6.47214i	0.485107i
$179$	−0.708204	−0.0529336	−0.0264668	−	0.999650i	$-0.508426\pi$
$179$	−0.708204	−0.0529336	−0.0264668	+	0.999650i	$0.508426\pi$
$180$	0	0
$181$	16.6525	1.23777	0.618884	−	0.785482i	$-0.287585\pi$
$181$	16.6525	1.23777	0.618884	+	0.785482i	$0.287585\pi$
$182$	6.00000i	0.444750i
$183$	− 15.5279i	− 1.14785i
$184$	2.23607	0.164845
$185$	0	0
$186$	−9.27051	−0.679747
$187$	− 4.00000i	− 0.292509i
$188$	− 3.61803i	− 0.263872i
$189$	−7.23607	−0.526346
$190$	0	0
$191$	−26.1803	−1.89434	−0.947171	−	0.320728i	$-0.896073\pi$
$191$	−26.1803	−1.89434	−0.947171	+	0.320728i	$0.896073\pi$
$192$	0.527864i	0.0380953i
$193$	− 9.94427i	− 0.715804i	−0.933759	−	0.357902i	$-0.883492\pi$
$193$	− 9.94427i	− 0.715804i	0.933759	−	0.357902i	$-0.116508\pi$
$194$	−10.9443	−0.785753
$195$	0	0
$196$	−5.61803	−0.401288
$197$	− 1.47214i	− 0.104885i	−0.998624	−	0.0524427i	$-0.983299\pi$
$197$	− 1.47214i	− 0.104885i	0.998624	−	0.0524427i	$-0.0167007\pi$
$198$	6.47214i	0.459955i
$199$	12.2918	0.871342	0.435671	−	0.900106i	$-0.356511\pi$
$199$	12.2918	0.871342	0.435671	+	0.900106i	$0.356511\pi$
$200$	0	0
$201$	6.18034	0.435928
$202$	2.76393i	0.194470i
$203$	9.70820i	0.681382i
$204$	−2.76393	−0.193514
$205$	0	0
$206$	−2.58359	−0.180007
$207$	2.00000i	0.139010i
$208$	− 5.56231i	− 0.385677i
$209$	−10.4721	−0.724373
$210$	0	0
$211$	−23.4164	−1.61205	−0.806026	−	0.591880i	$-0.798386\pi$
$211$	−23.4164	−1.61205	−0.806026	+	0.591880i	$0.798386\pi$
$212$	− 0.763932i	− 0.0524671i
$213$	27.3607i	1.87472i
$214$	−8.29180	−0.566816
$215$	0	0
$216$	−5.00000	−0.340207
$217$	21.7082i	1.47365i
$218$	0	0
$219$	14.5967	0.986357
$220$	0	0
$221$	2.29180	0.154163
$222$	1.70820i	0.114647i
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	−18.1803	−1.21473
$225$	0	0
$226$	5.41641	0.360294
$227$	− 12.1803i	− 0.808438i	−0.914662	−	0.404219i	$-0.867543\pi$
$227$	− 12.1803i	− 0.808438i	0.914662	−	0.404219i	$-0.132457\pi$
$228$	7.23607i	0.479220i
$229$	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
$231$	37.8885	2.49288
$232$	6.70820i	0.440415i
$233$	6.52786i	0.427655i	0.976871	+	0.213827i	$0.0685930\pi$
$233$	6.52786i	0.427655i	−0.976871	+	0.213827i	$0.931407\pi$
$234$	−3.70820	−0.242413
$235$	0	0
$236$	−10.4721	−0.681678
$237$	24.4721i	1.58964i
$238$	− 1.52786i	− 0.0990367i
$239$	−13.7639	−0.890315	−0.445157	−	0.895452i	$-0.646852\pi$
$239$	−13.7639	−0.890315	−0.445157	+	0.895452i	$0.646852\pi$
$240$	0	0
$241$	−23.1246	−1.48959	−0.744794	−	0.667295i	$-0.767452\pi$
$241$	−23.1246	−1.48959	−0.744794	+	0.667295i	$0.767452\pi$
$242$	10.1459i	0.652203i
$243$	− 17.8885i	− 1.14755i
$244$	−11.2361	−0.719316
$245$	0	0
$246$	4.79837	0.305933
$247$	− 6.00000i	− 0.381771i
$248$	15.0000i	0.952501i
$249$	−19.5967	−1.24189
$250$	0	0
$251$	2.29180	0.144657	0.0723284	−	0.997381i	$-0.476957\pi$
$251$	2.29180	0.144657	0.0723284	+	0.997381i	$0.476957\pi$
$252$	− 10.4721i	− 0.659683i
$253$	5.23607i	0.329189i
$254$	4.50658	0.282768
$255$	0	0
$256$	−6.56231	−0.410144
$257$	− 7.47214i	− 0.466099i	−0.972465	−	0.233050i	$-0.925130\pi$
$257$	− 7.47214i	− 0.466099i	0.972465	−	0.233050i	$-0.0748704\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	−6.00000	−0.371391
$262$	11.5623i	0.714322i
$263$	− 2.94427i	− 0.181552i	−0.995871	−	0.0907758i	$-0.971065\pi$
$263$	− 2.94427i	− 0.181552i	0.995871	−	0.0907758i	$-0.0289347\pi$
$264$	26.1803	1.61129
$265$	0	0
$266$	−4.00000	−0.245256
$267$	23.4164i	1.43306i
$268$	− 4.47214i	− 0.273179i
$269$	7.94427	0.484371	0.242185	−	0.970230i	$-0.422136\pi$
$269$	7.94427	0.484371	0.242185	+	0.970230i	$0.422136\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	1.41641i	0.0858823i
$273$	21.7082i	1.31384i
$274$	13.5279	0.817248
$275$	0	0
$276$	3.61803	0.217780
$277$	15.4721i	0.929631i	0.885408	+	0.464815i	$0.153879\pi$
$277$	15.4721i	0.929631i	−0.885408	+	0.464815i	$0.846121\pi$
$278$	6.61803i	0.396923i
$279$	−13.4164	−0.803219
$280$	0	0
$281$	−8.76393	−0.522812	−0.261406	−	0.965229i	$-0.584186\pi$
$281$	−8.76393	−0.522812	−0.261406	+	0.965229i	$0.584186\pi$
$282$	3.09017i	0.184017i
$283$	− 27.7082i	− 1.64708i	−0.567257	−	0.823541i	$-0.691995\pi$
$283$	− 27.7082i	− 1.64708i	0.567257	−	0.823541i	$-0.308005\pi$
$284$	19.7984	1.17482
$285$	0	0
$286$	−9.70820	−0.574058
$287$	− 11.2361i	− 0.663244i
$288$	− 11.2361i	− 0.662092i
$289$	16.4164	0.965671
$290$	0	0
$291$	−39.5967	−2.32120
$292$	− 10.5623i	− 0.618112i
$293$	1.52786i	0.0892588i	0.999004	+	0.0446294i	$0.0142107\pi$
$293$	1.52786i	0.0892588i	−0.999004	+	0.0446294i	$0.985789\pi$
$294$	4.79837	0.279847
$295$	0	0
$296$	2.76393	0.160650
$297$	− 11.7082i	− 0.679379i
$298$	− 14.7639i	− 0.855252i
$299$	−3.00000	−0.173494
$300$	0	0
$301$	0	0
$302$	2.61803i	0.150651i
$303$	10.0000i	0.574485i
$304$	3.70820	0.212680
$305$	0	0
$306$	0.944272	0.0539804
$307$	9.52786i	0.543784i	0.962328	+	0.271892i	$0.0876493\pi$
$307$	9.52786i	0.543784i	−0.962328	+	0.271892i	$0.912351\pi$
$308$	− 27.4164i	− 1.56219i
$309$	−9.34752	−0.531762
$310$	0	0
$311$	13.1803	0.747389	0.373694	−	0.927552i	$-0.378091\pi$
$311$	13.1803	0.747389	0.373694	+	0.927552i	$0.378091\pi$
$312$	15.0000i	0.849208i
$313$	− 24.3607i	− 1.37695i	−0.725261	−	0.688474i	$-0.758281\pi$
$313$	− 24.3607i	− 1.37695i	0.725261	−	0.688474i	$-0.241719\pi$
$314$	7.05573	0.398178
$315$	0	0
$316$	17.7082	0.996164
$317$	25.4164i	1.42753i	0.700386	+	0.713764i	$0.253011\pi$
$317$	25.4164i	1.42753i	−0.700386	+	0.713764i	$0.746989\pi$
$318$	0.652476i	0.0365890i
$319$	−15.7082	−0.879491
$320$	0	0
$321$	−30.0000	−1.67444
$322$	2.00000i	0.111456i
$323$	1.52786i	0.0850126i
$324$	−17.7984	−0.988799
$325$	0	0
$326$	−3.56231	−0.197298
$327$	0	0
$328$	− 7.76393i	− 0.428691i
$329$	7.23607	0.398937
$330$	0	0
$331$	−19.6525	−1.08020	−0.540099	−	0.841602i	$-0.681613\pi$
$331$	−19.6525	−1.08020	−0.540099	+	0.841602i	$0.681613\pi$
$332$	14.1803i	0.778247i
$333$	2.47214i	0.135472i
$334$	−0.944272	−0.0516683
$335$	0	0
$336$	−13.4164	−0.731925
$337$	23.4164i	1.27557i	0.770213	+	0.637787i	$0.220150\pi$
$337$	23.4164i	1.27557i	−0.770213	+	0.637787i	$0.779850\pi$
$338$	2.47214i	0.134466i
$339$	19.5967	1.06435
$340$	0	0
$341$	−35.1246	−1.90210
$342$	− 2.47214i	− 0.133678i
$343$	11.4164i	0.616428i
$344$	0	0
$345$	0	0
$346$	14.1803	0.762340
$347$	− 9.88854i	− 0.530845i	−0.964132	−	0.265422i	$-0.914489\pi$
$347$	− 9.88854i	− 0.530845i	0.964132	−	0.265422i	$-0.0855114\pi$
$348$	10.8541i	0.581841i
$349$	−24.4164	−1.30698	−0.653490	−	0.756935i	$-0.726696\pi$
$349$	−24.4164	−1.30698	−0.653490	+	0.756935i	$0.726696\pi$
$350$	0	0
$351$	6.70820	0.358057
$352$	− 29.4164i	− 1.56790i
$353$	− 9.36068i	− 0.498219i	−0.968475	−	0.249109i	$-0.919862\pi$
$353$	− 9.36068i	− 0.498219i	0.968475	−	0.249109i	$-0.0801379\pi$
$354$	8.94427	0.475383
$355$	0	0
$356$	16.9443	0.898045
$357$	− 5.52786i	− 0.292566i
$358$	− 0.437694i	− 0.0231329i
$359$	19.8885	1.04968	0.524839	−	0.851202i	$-0.324126\pi$
$359$	19.8885	1.04968	0.524839	+	0.851202i	$0.324126\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	10.2918i	0.540925i
$363$	36.7082i	1.92668i
$364$	15.7082	0.823334
$365$	0	0
$366$	9.59675	0.501630
$367$	− 4.18034i	− 0.218212i	−0.994030	−	0.109106i	$-0.965201\pi$
$367$	− 4.18034i	− 0.218212i	0.994030	−	0.109106i	$-0.0347988\pi$
$368$	− 1.85410i	− 0.0966517i
$369$	6.94427	0.361504
$370$	0	0
$371$	1.52786	0.0793227
$372$	24.2705i	1.25837i
$373$	− 7.70820i	− 0.399116i	−0.979886	−	0.199558i	$-0.936049\pi$
$373$	− 7.70820i	− 0.399116i	0.979886	−	0.199558i	$-0.0639506\pi$
$374$	2.47214	0.127831
$375$	0	0
$376$	5.00000	0.257855
$377$	− 9.00000i	− 0.463524i
$378$	− 4.47214i	− 0.230022i
$379$	−24.3607	−1.25132	−0.625662	−	0.780094i	$-0.715171\pi$
$379$	−24.3607	−1.25132	−0.625662	+	0.780094i	$0.715171\pi$
$380$	0	0
$381$	16.3050	0.835328
$382$	− 16.1803i	− 0.827858i
$383$	− 7.05573i	− 0.360531i	−0.983618	−	0.180265i	$-0.942304\pi$
$383$	− 7.05573i	− 0.360531i	0.983618	−	0.180265i	$-0.0576957\pi$
$384$	−25.4508	−1.29878
$385$	0	0
$386$	6.14590	0.312818
$387$	0	0
$388$	28.6525i	1.45461i
$389$	−25.5279	−1.29431	−0.647157	−	0.762357i	$-0.724042\pi$
$389$	−25.5279	−1.29431	−0.647157	+	0.762357i	$0.724042\pi$
$390$	0	0
$391$	0.763932	0.0386337
$392$	− 7.76393i	− 0.392138i
$393$	41.8328i	2.11019i
$394$	0.909830	0.0458366
$395$	0	0
$396$	16.9443	0.851482
$397$	− 24.4164i	− 1.22542i	−0.790306	−	0.612712i	$-0.790078\pi$
$397$	− 24.4164i	− 1.22542i	0.790306	−	0.612712i	$-0.209922\pi$
$398$	7.59675i	0.380791i
$399$	−14.4721	−0.724513
$400$	0	0
$401$	−14.1803	−0.708132	−0.354066	−	0.935220i	$-0.615201\pi$
$401$	−14.1803	−0.708132	−0.354066	+	0.935220i	$0.615201\pi$
$402$	3.81966i	0.190507i
$403$	− 20.1246i	− 1.00248i
$404$	7.23607	0.360008
$405$	0	0
$406$	−6.00000	−0.297775
$407$	6.47214i	0.320812i
$408$	− 3.81966i	− 0.189101i
$409$	−21.3607	−1.05622	−0.528109	−	0.849177i	$-0.677099\pi$
$409$	−21.3607	−1.05622	−0.528109	+	0.849177i	$0.677099\pi$
$410$	0	0
$411$	48.9443	2.41424
$412$	6.76393i	0.333235i
$413$	− 20.9443i	− 1.03060i
$414$	−1.23607	−0.0607494
$415$	0	0
$416$	16.8541	0.826340
$417$	23.9443i	1.17256i
$418$	− 6.47214i	− 0.316563i
$419$	4.58359	0.223923	0.111962	−	0.993713i	$-0.464287\pi$
$419$	4.58359	0.223923	0.111962	+	0.993713i	$0.464287\pi$
$420$	0	0
$421$	−10.2918	−0.501591	−0.250796	−	0.968040i	$-0.580692\pi$
$421$	−10.2918	−0.501591	−0.250796	+	0.968040i	$0.580692\pi$
$422$	− 14.4721i	− 0.704493i
$423$	4.47214i	0.217443i
$424$	1.05573	0.0512707
$425$	0	0
$426$	−16.9098	−0.819284
$427$	− 22.4721i	− 1.08750i
$428$	21.7082i	1.04931i
$429$	−35.1246	−1.69583
$430$	0	0
$431$	−17.5279	−0.844288	−0.422144	−	0.906529i	$-0.638722\pi$
$431$	−17.5279	−0.844288	−0.422144	+	0.906529i	$0.638722\pi$
$432$	4.14590i	0.199470i
$433$	− 17.8197i	− 0.856358i	−0.903694	−	0.428179i	$-0.859155\pi$
$433$	− 17.8197i	− 0.856358i	0.903694	−	0.428179i	$-0.140845\pi$
$434$	−13.4164	−0.644008
$435$	0	0
$436$	0	0
$437$	− 2.00000i	− 0.0956730i
$438$	9.02129i	0.431054i
$439$	18.7082	0.892894	0.446447	−	0.894810i	$-0.352689\pi$
$439$	18.7082	0.892894	0.446447	+	0.894810i	$0.352689\pi$
$440$	0	0
$441$	6.94427	0.330680
$442$	1.41641i	0.0673717i
$443$	− 38.1246i	− 1.81135i	−0.423967	−	0.905677i	$-0.639363\pi$
$443$	− 38.1246i	− 1.81135i	0.423967	−	0.905677i	$-0.360637\pi$
$444$	4.47214	0.212238
$445$	0	0
$446$	2.47214	0.117059
$447$	− 53.4164i	− 2.52651i
$448$	0.763932i	0.0360924i
$449$	14.9443	0.705264	0.352632	−	0.935762i	$-0.385287\pi$
$449$	14.9443	0.705264	0.352632	+	0.935762i	$0.385287\pi$
$450$	0	0
$451$	18.1803	0.856079
$452$	− 14.1803i	− 0.666987i
$453$	9.47214i	0.445040i
$454$	7.52786	0.353300
$455$	0	0
$456$	−10.0000	−0.468293
$457$	− 5.12461i	− 0.239719i	−0.992791	−	0.119860i	$-0.961756\pi$
$457$	− 5.12461i	− 0.239719i	0.992791	−	0.119860i	$-0.0382444\pi$
$458$	7.41641i	0.346546i
$459$	−1.70820	−0.0797321
$460$	0	0
$461$	−1.47214	−0.0685642	−0.0342821	−	0.999412i	$-0.510914\pi$
$461$	−1.47214	−0.0685642	−0.0342821	+	0.999412i	$0.510914\pi$
$462$	23.4164i	1.08943i
$463$	20.0000i	0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$	20.0000i	0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$	5.56231	0.258224
$465$	0	0
$466$	−4.03444	−0.186892
$467$	− 13.0557i	− 0.604147i	−0.953285	−	0.302074i	$-0.902321\pi$
$467$	− 13.0557i	− 0.604147i	0.953285	−	0.302074i	$-0.0976788\pi$
$468$	9.70820i	0.448762i
$469$	8.94427	0.413008
$470$	0	0
$471$	25.5279	1.17626
$472$	− 14.4721i	− 0.666134i
$473$	0	0
$474$	−15.1246	−0.694696
$475$	0	0
$476$	−4.00000	−0.183340
$477$	0.944272i	0.0432352i
$478$	− 8.50658i	− 0.389082i
$479$	−31.5967	−1.44369	−0.721846	−	0.692054i	$-0.756706\pi$
$479$	−31.5967	−1.44369	−0.721846	+	0.692054i	$0.756706\pi$
$480$	0	0
$481$	−3.70820	−0.169080
$482$	− 14.2918i	− 0.650973i
$483$	7.23607i	0.329252i
$484$	26.5623	1.20738
$485$	0	0
$486$	11.0557	0.501498
$487$	− 14.7082i	− 0.666492i	−0.942840	−	0.333246i	$-0.891856\pi$
$487$	− 14.7082i	− 0.666492i	0.942840	−	0.333246i	$-0.108144\pi$
$488$	− 15.5279i	− 0.702913i
$489$	−12.8885	−0.582840
$490$	0	0
$491$	8.34752	0.376718	0.188359	−	0.982100i	$-0.439683\pi$
$491$	8.34752	0.376718	0.188359	+	0.982100i	$0.439683\pi$
$492$	− 12.5623i	− 0.566352i
$493$	2.29180i	0.103217i
$494$	3.70820	0.166840
$495$	0	0
$496$	12.4377	0.558469
$497$	39.5967i	1.77616i
$498$	− 12.1115i	− 0.542727i
$499$	−19.2918	−0.863619	−0.431810	−	0.901965i	$-0.642125\pi$
$499$	−19.2918	−0.863619	−0.431810	+	0.901965i	$0.642125\pi$
$500$	0	0
$501$	−3.41641	−0.152634
$502$	1.41641i	0.0632174i
$503$	26.9443i	1.20139i	0.799480	+	0.600693i	$0.205109\pi$
$503$	26.9443i	1.20139i	−0.799480	+	0.600693i	$0.794891\pi$
$504$	14.4721	0.644640
$505$	0	0
$506$	−3.23607	−0.143861
$507$	8.94427i	0.397229i
$508$	− 11.7984i	− 0.523468i
$509$	28.3050	1.25459	0.627297	−	0.778780i	$-0.284161\pi$
$509$	28.3050	1.25459	0.627297	+	0.778780i	$0.284161\pi$
$510$	0	0
$511$	21.1246	0.934498
$512$	18.7082i	0.826794i
$513$	4.47214i	0.197450i
$514$	4.61803	0.203693
$515$	0	0
$516$	0	0
$517$	11.7082i	0.514926i
$518$	2.47214i	0.108619i
$519$	51.3050	2.25204
$520$	0	0
$521$	31.4164	1.37638	0.688189	−	0.725532i	$-0.258406\pi$
$521$	31.4164	1.37638	0.688189	+	0.725532i	$0.258406\pi$
$522$	− 3.70820i	− 0.162304i
$523$	− 41.1246i	− 1.79825i	−0.437688	−	0.899127i	$-0.644203\pi$
$523$	− 41.1246i	− 1.79825i	0.437688	−	0.899127i	$-0.355797\pi$
$524$	30.2705	1.32237
$525$	0	0
$526$	1.81966	0.0793410
$527$	5.12461i	0.223232i
$528$	− 21.7082i	− 0.944728i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	12.9443	0.561734
$532$	10.4721i	0.454025i
$533$	10.4164i	0.451185i
$534$	−14.4721	−0.626271
$535$	0	0
$536$	6.18034	0.266950
$537$	− 1.58359i	− 0.0683370i
$538$	4.90983i	0.211678i
$539$	18.1803	0.783083
$540$	0	0
$541$	−34.4164	−1.47968	−0.739838	−	0.672785i	$-0.765098\pi$
$541$	−34.4164	−1.47968	−0.739838	+	0.672785i	$0.765098\pi$
$542$	4.94427i	0.212375i
$543$	37.2361i	1.59795i
$544$	−4.29180	−0.184009
$545$	0	0
$546$	−13.4164	−0.574169
$547$	− 29.5410i	− 1.26308i	−0.775342	−	0.631541i	$-0.782423\pi$
$547$	− 29.5410i	− 1.26308i	0.775342	−	0.631541i	$-0.217577\pi$
$548$	− 35.4164i	− 1.51291i
$549$	13.8885	0.592749
$550$	0	0
$551$	6.00000	0.255609
$552$	5.00000i	0.212814i
$553$	35.4164i	1.50606i
$554$	−9.56231	−0.406263
$555$	0	0
$556$	17.3262	0.734796
$557$	− 7.41641i	− 0.314243i	−0.987579	−	0.157122i	$-0.949779\pi$
$557$	− 7.41641i	− 0.314243i	0.987579	−	0.157122i	$-0.0502215\pi$
$558$	− 8.29180i	− 0.351020i
$559$	0	0
$560$	0	0
$561$	8.94427	0.377627
$562$	− 5.41641i	− 0.228477i
$563$	32.9443i	1.38844i	0.719765	+	0.694218i	$0.244250\pi$
$563$	32.9443i	1.38844i	−0.719765	+	0.694218i	$0.755750\pi$
$564$	8.09017	0.340658
$565$	0	0
$566$	17.1246	0.719801
$567$	− 35.5967i	− 1.49492i
$568$	27.3607i	1.14803i
$569$	22.1803	0.929848	0.464924	−	0.885351i	$-0.346082\pi$
$569$	22.1803	0.929848	0.464924	+	0.885351i	$0.346082\pi$
$570$	0	0
$571$	−14.2918	−0.598093	−0.299047	−	0.954239i	$-0.596669\pi$
$571$	−14.2918	−0.598093	−0.299047	+	0.954239i	$0.596669\pi$
$572$	25.4164i	1.06271i
$573$	− 58.5410i	− 2.44559i
$574$	6.94427	0.289848
$575$	0	0
$576$	−0.472136	−0.0196723
$577$	22.8885i	0.952863i	0.879212	+	0.476431i	$0.158070\pi$
$577$	22.8885i	0.952863i	−0.879212	+	0.476431i	$0.841930\pi$
$578$	10.1459i	0.422014i
$579$	22.2361	0.924099
$580$	0	0
$581$	−28.3607	−1.17660
$582$	− 24.4721i	− 1.01440i
$583$	2.47214i	0.102385i
$584$	14.5967	0.604018
$585$	0	0
$586$	−0.944272	−0.0390075
$587$	− 24.7082i	− 1.01982i	−0.860229	−	0.509908i	$-0.829679\pi$
$587$	− 24.7082i	− 1.01982i	0.860229	−	0.509908i	$-0.170321\pi$
$588$	− 12.5623i	− 0.518061i
$589$	13.4164	0.552813
$590$	0	0
$591$	3.29180	0.135406
$592$	− 2.29180i	− 0.0941922i
$593$	2.94427i	0.120907i	0.998171	+	0.0604534i	$0.0192546\pi$
$593$	2.94427i	0.120907i	−0.998171	+	0.0604534i	$0.980745\pi$
$594$	7.23607	0.296899
$595$	0	0
$596$	−38.6525	−1.58327
$597$	27.4853i	1.12490i
$598$	− 1.85410i	− 0.0758199i
$599$	−33.8885	−1.38465	−0.692324	−	0.721587i	$-0.743413\pi$
$599$	−33.8885	−1.38465	−0.692324	+	0.721587i	$0.743413\pi$
$600$	0	0
$601$	46.8885	1.91262	0.956312	−	0.292349i	$-0.0944368\pi$
$601$	46.8885	1.91262	0.956312	+	0.292349i	$0.0944368\pi$
$602$	0	0
$603$	5.52786i	0.225112i
$604$	6.85410	0.278889
$605$	0	0
$606$	−6.18034	−0.251059
$607$	26.4721i	1.07447i	0.843432	+	0.537235i	$0.180531\pi$
$607$	26.4721i	1.07447i	−0.843432	+	0.537235i	$0.819469\pi$
$608$	11.2361i	0.455683i
$609$	−21.7082	−0.879661
$610$	0	0
$611$	−6.70820	−0.271385
$612$	− 2.47214i	− 0.0999302i
$613$	− 5.70820i	− 0.230552i	−0.993333	−	0.115276i	$-0.963225\pi$
$613$	− 5.70820i	− 0.230552i	0.993333	−	0.115276i	$-0.0367753\pi$
$614$	−5.88854	−0.237642
$615$	0	0
$616$	37.8885	1.52657
$617$	− 7.52786i	− 0.303060i	−0.988453	−	0.151530i	$-0.951580\pi$
$617$	− 7.52786i	− 0.303060i	0.988453	−	0.151530i	$-0.0484201\pi$
$618$	− 5.77709i	− 0.232389i
$619$	−19.4164	−0.780411	−0.390206	−	0.920728i	$-0.627596\pi$
$619$	−19.4164	−0.780411	−0.390206	+	0.920728i	$0.627596\pi$
$620$	0	0
$621$	2.23607	0.0897303
$622$	8.14590i	0.326621i
$623$	33.8885i	1.35772i
$624$	12.4377	0.497906
$625$	0	0
$626$	15.0557	0.601748
$627$	− 23.4164i	− 0.935161i
$628$	− 18.4721i	− 0.737118i
$629$	0.944272	0.0376506
$630$	0	0
$631$	12.3607	0.492071	0.246035	−	0.969261i	$-0.420872\pi$
$631$	12.3607	0.492071	0.246035	+	0.969261i	$0.420872\pi$
$632$	24.4721i	0.973449i
$633$	− 52.3607i	− 2.08115i
$634$	−15.7082	−0.623852
$635$	0	0
$636$	1.70820	0.0677347
$637$	10.4164i	0.412713i
$638$	− 9.70820i	− 0.384351i
$639$	−24.4721	−0.968103
$640$	0	0
$641$	−17.3050	−0.683504	−0.341752	−	0.939790i	$-0.611020\pi$
$641$	−17.3050	−0.683504	−0.341752	+	0.939790i	$0.611020\pi$
$642$	− 18.5410i	− 0.731756i
$643$	29.5967i	1.16718i	0.812048	+	0.583591i	$0.198353\pi$
$643$	29.5967i	1.16718i	−0.812048	+	0.583591i	$0.801647\pi$
$644$	5.23607	0.206330
$645$	0	0
$646$	−0.944272	−0.0371519
$647$	6.70820i	0.263727i	0.991268	+	0.131863i	$0.0420960\pi$
$647$	6.70820i	0.263727i	−0.991268	+	0.131863i	$0.957904\pi$
$648$	− 24.5967i	− 0.966252i
$649$	33.8885	1.33024
$650$	0	0
$651$	−48.5410	−1.90247
$652$	9.32624i	0.365244i
$653$	38.3050i	1.49899i	0.662011	+	0.749494i	$0.269703\pi$
$653$	38.3050i	1.49899i	−0.662011	+	0.749494i	$0.730297\pi$
$654$	0	0
$655$	0	0
$656$	−6.43769	−0.251350
$657$	13.0557i	0.509352i
$658$	4.47214i	0.174342i
$659$	10.6525	0.414962	0.207481	−	0.978239i	$-0.433474\pi$
$659$	10.6525	0.414962	0.207481	+	0.978239i	$0.433474\pi$
$660$	0	0
$661$	−22.9443	−0.892429	−0.446214	−	0.894926i	$-0.647228\pi$
$661$	−22.9443	−0.892429	−0.446214	+	0.894926i	$0.647228\pi$
$662$	− 12.1459i	− 0.472064i
$663$	5.12461i	0.199023i
$664$	−19.5967	−0.760501
$665$	0	0
$666$	−1.52786	−0.0592035
$667$	− 3.00000i	− 0.116160i
$668$	2.47214i	0.0956498i
$669$	8.94427	0.345806
$670$	0	0
$671$	36.3607	1.40369
$672$	− 40.6525i	− 1.56820i
$673$	− 3.00000i	− 0.115642i	−0.998327	−	0.0578208i	$-0.981585\pi$
$673$	− 3.00000i	− 0.115642i	0.998327	−	0.0578208i	$-0.0184152\pi$
$674$	−14.4721	−0.557446
$675$	0	0
$676$	6.47214	0.248928
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	12.1115i	0.465138i
$679$	−57.3050	−2.19916
$680$	0	0
$681$	27.2361	1.04369
$682$	− 21.7082i	− 0.831250i
$683$	− 26.5967i	− 1.01770i	−0.860856	−	0.508848i	$-0.830071\pi$
$683$	− 26.5967i	− 1.01770i	0.860856	−	0.508848i	$-0.169929\pi$
$684$	−6.47214	−0.247468
$685$	0	0
$686$	−7.05573	−0.269389
$687$	26.8328i	1.02374i
$688$	0	0
$689$	−1.41641	−0.0539608
$690$	0	0
$691$	7.05573	0.268413	0.134206	−	0.990953i	$-0.457152\pi$
$691$	7.05573	0.268413	0.134206	+	0.990953i	$0.457152\pi$
$692$	− 37.1246i	− 1.41127i
$693$	33.8885i	1.28732i
$694$	6.11146	0.231988
$695$	0	0
$696$	−15.0000	−0.568574
$697$	− 2.65248i	− 0.100470i
$698$	− 15.0902i	− 0.571171i
$699$	−14.5967	−0.552100
$700$	0	0
$701$	−3.81966	−0.144267	−0.0721333	−	0.997395i	$-0.522981\pi$
$701$	−3.81966	−0.144267	−0.0721333	+	0.997395i	$0.522981\pi$
$702$	4.14590i	0.156477i
$703$	− 2.47214i	− 0.0932384i
$704$	−1.23607	−0.0465861
$705$	0	0
$706$	5.78522	0.217730
$707$	14.4721i	0.544281i
$708$	− 23.4164i	− 0.880042i
$709$	42.0689	1.57993	0.789965	−	0.613152i	$-0.210099\pi$
$709$	42.0689	1.57993	0.789965	+	0.613152i	$0.210099\pi$
$710$	0	0
$711$	−21.8885	−0.820885
$712$	23.4164i	0.877567i
$713$	− 6.70820i	− 0.251224i
$714$	3.41641	0.127856
$715$	0	0
$716$	−1.14590	−0.0428242
$717$	− 30.7771i	− 1.14939i
$718$	12.2918i	0.458726i
$719$	3.05573	0.113959	0.0569797	−	0.998375i	$-0.481853\pi$
$719$	3.05573	0.113959	0.0569797	+	0.998375i	$0.481853\pi$
$720$	0	0
$721$	−13.5279	−0.503804
$722$	− 9.27051i	− 0.345013i
$723$	− 51.7082i	− 1.92305i
$724$	26.9443	1.00138
$725$	0	0
$726$	−22.6869	−0.841990
$727$	− 27.7082i	− 1.02764i	−0.857898	−	0.513820i	$-0.828230\pi$
$727$	− 27.7082i	− 1.02764i	0.857898	−	0.513820i	$-0.171770\pi$
$728$	21.7082i	0.804560i
$729$	7.00000	0.259259
$730$	0	0
$731$	0	0
$732$	− 25.1246i	− 0.928632i
$733$	31.2361i	1.15373i	0.816839	+	0.576865i	$0.195724\pi$
$733$	31.2361i	1.15373i	−0.816839	+	0.576865i	$0.804276\pi$
$734$	2.58359	0.0953621
$735$	0	0
$736$	5.61803	0.207083
$737$	14.4721i	0.533088i
$738$	4.29180i	0.157983i
$739$	−26.8197	−0.986577	−0.493289	−	0.869866i	$-0.664205\pi$
$739$	−26.8197	−0.986577	−0.493289	+	0.869866i	$0.664205\pi$
$740$	0	0
$741$	13.4164	0.492864
$742$	0.944272i	0.0346653i
$743$	− 41.1246i	− 1.50872i	−0.656463	−	0.754358i	$-0.727948\pi$
$743$	− 41.1246i	− 1.50872i	0.656463	−	0.754358i	$-0.272052\pi$
$744$	−33.5410	−1.22967
$745$	0	0
$746$	4.76393	0.174420
$747$	− 17.5279i	− 0.641311i
$748$	− 6.47214i	− 0.236645i
$749$	−43.4164	−1.58640
$750$	0	0
$751$	0.360680	0.0131614	0.00658070	−	0.999978i	$-0.497905\pi$
$751$	0.360680	0.0131614	0.00658070	+	0.999978i	$0.497905\pi$
$752$	− 4.14590i	− 0.151185i
$753$	5.12461i	0.186751i
$754$	5.56231	0.202567
$755$	0	0
$756$	−11.7082	−0.425823
$757$	1.59675i	0.0580348i	0.999579	+	0.0290174i	$0.00923782\pi$
$757$	1.59675i	0.0580348i	−0.999579	+	0.0290174i	$0.990762\pi$
$758$	− 15.0557i	− 0.546849i
$759$	−11.7082	−0.424981
$760$	0	0
$761$	46.3050	1.67855	0.839277	−	0.543705i	$-0.182979\pi$
$761$	46.3050	1.67855	0.839277	+	0.543705i	$0.182979\pi$
$762$	10.0770i	0.365052i
$763$	0	0
$764$	−42.3607	−1.53256
$765$	0	0
$766$	4.36068	0.157558
$767$	19.4164i	0.701086i
$768$	− 14.6738i	− 0.529494i
$769$	23.1246	0.833895	0.416947	−	0.908931i	$-0.363100\pi$
$769$	23.1246	0.833895	0.416947	+	0.908931i	$0.363100\pi$
$770$	0	0
$771$	16.7082	0.601731
$772$	− 16.0902i	− 0.579098i
$773$	5.52786i	0.198823i	0.995046	+	0.0994117i	$0.0316961\pi$
$773$	5.52786i	0.198823i	−0.995046	+	0.0994117i	$0.968304\pi$
$774$	0	0
$775$	0	0
$776$	−39.5967	−1.42144
$777$	8.94427i	0.320874i
$778$	− 15.7771i	− 0.565636i
$779$	−6.94427	−0.248804
$780$	0	0
$781$	−64.0689	−2.29256
$782$	0.472136i	0.0168835i
$783$	6.70820i	0.239732i
$784$	−6.43769	−0.229918
$785$	0	0
$786$	−25.8541	−0.922185
$787$	24.5836i	0.876310i	0.898899	+	0.438155i	$0.144368\pi$
$787$	24.5836i	0.876310i	−0.898899	+	0.438155i	$0.855632\pi$
$788$	− 2.38197i	− 0.0848540i
$789$	6.58359	0.234382
$790$	0	0
$791$	28.3607	1.00839
$792$	23.4164i	0.832066i
$793$	20.8328i	0.739795i
$794$	15.0902	0.535530
$795$	0	0
$796$	19.8885	0.704931
$797$	− 34.3607i	− 1.21712i	−0.793509	−	0.608559i	$-0.791748\pi$
$797$	− 34.3607i	− 1.21712i	0.793509	−	0.608559i	$-0.208252\pi$
$798$	− 8.94427i	− 0.316624i
$799$	1.70820	0.0604319
$800$	0	0
$801$	−20.9443	−0.740029
$802$	− 8.76393i	− 0.309465i
$803$	34.1803i	1.20620i
$804$	10.0000	0.352673
$805$	0	0
$806$	12.4377	0.438099
$807$	17.7639i	0.625320i
$808$	10.0000i	0.351799i
$809$	−12.1115	−0.425816	−0.212908	−	0.977072i	$-0.568293\pi$
$809$	−12.1115	−0.425816	−0.212908	+	0.977072i	$0.568293\pi$
$810$	0	0
$811$	−24.3475	−0.854957	−0.427479	−	0.904025i	$-0.640598\pi$
$811$	−24.3475	−0.854957	−0.427479	+	0.904025i	$0.640598\pi$
$812$	15.7082i	0.551250i
$813$	17.8885i	0.627379i
$814$	−4.00000	−0.140200
$815$	0	0
$816$	−3.16718	−0.110874
$817$	0	0
$818$	− 13.2016i	− 0.461584i
$819$	−19.4164	−0.678464
$820$	0	0
$821$	−38.9443	−1.35916	−0.679582	−	0.733599i	$-0.737839\pi$
$821$	−38.9443	−1.35916	−0.679582	+	0.733599i	$0.737839\pi$
$822$	30.2492i	1.05506i
$823$	39.5410i	1.37831i	0.724612	+	0.689157i	$0.242019\pi$
$823$	39.5410i	1.37831i	−0.724612	+	0.689157i	$0.757981\pi$
$824$	−9.34752	−0.325636
$825$	0	0
$826$	12.9443	0.450389
$827$	1.52786i	0.0531290i	0.999647	+	0.0265645i	$0.00845674\pi$
$827$	1.52786i	0.0531290i	−0.999647	+	0.0265645i	$0.991543\pi$
$828$	3.23607i	0.112461i
$829$	−40.2492	−1.39791	−0.698957	−	0.715164i	$-0.746352\pi$
$829$	−40.2492	−1.39791	−0.698957	+	0.715164i	$0.746352\pi$
$830$	0	0
$831$	−34.5967	−1.20015
$832$	− 0.708204i	− 0.0245526i
$833$	− 2.65248i	− 0.0919028i
$834$	−14.7984	−0.512426
$835$	0	0
$836$	−16.9443	−0.586030
$837$	15.0000i	0.518476i
$838$	2.83282i	0.0978580i
$839$	41.1246	1.41978	0.709890	−	0.704313i	$-0.248745\pi$
$839$	41.1246	1.41978	0.709890	+	0.704313i	$0.248745\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	− 6.36068i	− 0.219204i
$843$	− 19.5967i	− 0.674948i
$844$	−37.8885	−1.30418
$845$	0	0
$846$	−2.76393	−0.0950259
$847$	53.1246i	1.82538i
$848$	− 0.875388i	− 0.0300610i
$849$	61.9574	2.12637
$850$	0	0
$851$	−1.23607	−0.0423719
$852$	44.2705i	1.51668i
$853$	10.5836i	0.362375i	0.983449	+	0.181188i	$0.0579941\pi$
$853$	10.5836i	0.362375i	−0.983449	+	0.181188i	$0.942006\pi$
$854$	13.8885	0.475256
$855$	0	0
$856$	−30.0000	−1.02538
$857$	1.47214i	0.0502872i	0.999684	+	0.0251436i	$0.00800430\pi$
$857$	1.47214i	0.0502872i	−0.999684	+	0.0251436i	$0.991996\pi$
$858$	− 21.7082i	− 0.741106i
$859$	16.7082	0.570077	0.285038	−	0.958516i	$-0.407994\pi$
$859$	16.7082	0.570077	0.285038	+	0.958516i	$0.407994\pi$
$860$	0	0
$861$	25.1246	0.856244
$862$	− 10.8328i	− 0.368967i
$863$	21.5410i	0.733265i	0.930366	+	0.366632i	$0.119489\pi$
$863$	21.5410i	0.733265i	−0.930366	+	0.366632i	$0.880511\pi$
$864$	−12.5623	−0.427378
$865$	0	0
$866$	11.0132	0.374242
$867$	36.7082i	1.24668i
$868$	35.1246i	1.19221i
$869$	−57.3050	−1.94394
$870$	0	0
$871$	−8.29180	−0.280957
$872$	0	0
$873$	− 35.4164i	− 1.19866i
$874$	1.23607	0.0418106
$875$	0	0
$876$	23.6180	0.797979
$877$	− 36.4721i	− 1.23158i	−0.787912	−	0.615788i	$-0.788838\pi$
$877$	− 36.4721i	− 1.23158i	0.787912	−	0.615788i	$-0.211162\pi$
$878$	11.5623i	0.390209i
$879$	−3.41641	−0.115233
$880$	0	0
$881$	44.1803	1.48847	0.744237	−	0.667916i	$-0.232813\pi$
$881$	44.1803	1.48847	0.744237	+	0.667916i	$0.232813\pi$
$882$	4.29180i	0.144512i
$883$	− 4.00000i	− 0.134611i	−0.997732	−	0.0673054i	$-0.978560\pi$
$883$	− 4.00000i	− 0.134611i	0.997732	−	0.0673054i	$-0.0214402\pi$
$884$	3.70820	0.124720
$885$	0	0
$886$	23.5623	0.791591
$887$	23.0689i	0.774577i	0.921959	+	0.387289i	$0.126588\pi$
$887$	23.0689i	0.774577i	−0.921959	+	0.387289i	$0.873412\pi$
$888$	6.18034i	0.207399i
$889$	23.5967	0.791410
$890$	0	0
$891$	57.5967	1.92956
$892$	− 6.47214i	− 0.216703i
$893$	− 4.47214i	− 0.149654i
$894$	33.0132	1.10413
$895$	0	0
$896$	−36.8328	−1.23050
$897$	− 6.70820i	− 0.223980i
$898$	9.23607i	0.308212i
$899$	20.1246	0.671193
$900$	0	0
$901$	0.360680	0.0120160
$902$	11.2361i	0.374120i
$903$	0	0
$904$	19.5967	0.651778
$905$	0	0
$906$	−5.85410	−0.194490
$907$	− 40.2492i	− 1.33645i	−0.743958	−	0.668227i	$-0.767054\pi$
$907$	− 40.2492i	− 1.33645i	0.743958	−	0.668227i	$-0.232946\pi$
$908$	− 19.7082i	− 0.654040i
$909$	−8.94427	−0.296663
$910$	0	0
$911$	31.3050	1.03718	0.518590	−	0.855023i	$-0.326457\pi$
$911$	31.3050	1.03718	0.518590	+	0.855023i	$0.326457\pi$
$912$	8.29180i	0.274569i
$913$	− 45.8885i	− 1.51869i
$914$	3.16718	0.104761
$915$	0	0
$916$	19.4164	0.641536
$917$	60.5410i	1.99924i
$918$	− 1.05573i	− 0.0348442i
$919$	−41.1246	−1.35658	−0.678288	−	0.734796i	$-0.737278\pi$
$919$	−41.1246	−1.35658	−0.678288	+	0.734796i	$0.737278\pi$
$920$	0	0
$921$	−21.3050	−0.702022
$922$	− 0.909830i	− 0.0299637i
$923$	− 36.7082i	− 1.20827i
$924$	61.3050	2.01678
$925$	0	0
$926$	−12.3607	−0.406197
$927$	− 8.36068i	− 0.274601i
$928$	16.8541i	0.553263i
$929$	24.0557	0.789243	0.394621	−	0.918844i	$-0.370876\pi$
$929$	24.0557	0.789243	0.394621	+	0.918844i	$0.370876\pi$
$930$	0	0
$931$	−6.94427	−0.227589
$932$	10.5623i	0.345980i
$933$	29.4721i	0.964874i
$934$	8.06888	0.264022
$935$	0	0
$936$	−13.4164	−0.438529
$937$	34.1803i	1.11662i	0.829631	+	0.558312i	$0.188551\pi$
$937$	34.1803i	1.11662i	−0.829631	+	0.558312i	$0.811449\pi$
$938$	5.52786i	0.180491i
$939$	54.4721	1.77763
$940$	0	0
$941$	6.65248	0.216865	0.108432	−	0.994104i	$-0.465417\pi$
$941$	6.65248	0.216865	0.108432	+	0.994104i	$0.465417\pi$
$942$	15.7771i	0.514045i
$943$	3.47214i	0.113068i
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	− 10.8197i	− 0.351592i	−0.984427	−	0.175796i	$-0.943750\pi$
$947$	− 10.8197i	− 0.351592i	0.984427	−	0.175796i	$-0.0562499\pi$
$948$	39.5967i	1.28604i
$949$	−19.5836	−0.635710
$950$	0	0
$951$	−56.8328	−1.84293
$952$	− 5.52786i	− 0.179159i
$953$	− 20.4721i	− 0.663158i	−0.943428	−	0.331579i	$-0.892419\pi$
$953$	− 20.4721i	− 0.663158i	0.943428	−	0.331579i	$-0.107581\pi$
$954$	−0.583592	−0.0188945
$955$	0	0
$956$	−22.2705	−0.720280
$957$	− 35.1246i	− 1.13542i
$958$	− 19.5279i	− 0.630917i
$959$	70.8328	2.28731
$960$	0	0
$961$	14.0000	0.451613
$962$	− 2.29180i	− 0.0738905i
$963$	− 26.8328i	− 0.864675i
$964$	−37.4164	−1.20510
$965$	0	0
$966$	−4.47214	−0.143889
$967$	27.5410i	0.885659i	0.896606	+	0.442830i	$0.146025\pi$
$967$	27.5410i	0.885659i	−0.896606	+	0.442830i	$0.853975\pi$
$968$	36.7082i	1.17985i
$969$	−3.41641	−0.109751
$970$	0	0
$971$	16.4721	0.528616	0.264308	−	0.964438i	$-0.414856\pi$
$971$	16.4721	0.528616	0.264308	+	0.964438i	$0.414856\pi$
$972$	− 28.9443i	− 0.928388i
$973$	34.6525i	1.11091i
$974$	9.09017	0.291268
$975$	0	0
$976$	−12.8754	−0.412131
$977$	− 23.3475i	− 0.746953i	−0.927640	−	0.373477i	$-0.878166\pi$
$977$	− 23.3475i	− 0.746953i	0.927640	−	0.373477i	$-0.121834\pi$
$978$	− 7.96556i	− 0.254710i
$979$	−54.8328	−1.75246
$980$	0	0
$981$	0	0
$982$	5.15905i	0.164632i
$983$	40.4721i	1.29086i	0.763819	+	0.645430i	$0.223322\pi$
$983$	40.4721i	1.29086i	−0.763819	+	0.645430i	$0.776678\pi$
$984$	17.3607	0.553438
$985$	0	0
$986$	−1.41641	−0.0451076
$987$	16.1803i	0.515026i
$988$	− 9.70820i	− 0.308859i
$989$	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	$0.375513\pi$
$992$	37.6869i	1.19656i
$993$	− 43.9443i	− 1.39453i
$994$	−24.4721	−0.776209
$995$	0	0
$996$	−31.7082	−1.00471
$997$	16.8328i	0.533101i	0.963821	+	0.266550i	$0.0858838\pi$
$997$	16.8328i	0.533101i	−0.963821	+	0.266550i	$0.914116\pi$
$998$	− 11.9230i	− 0.377416i
$999$	2.76393	0.0874469

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	575.2.b.d.24.3		4
5.2	odd	4		575.2.a.f.1.1		2
5.3	odd	4		23.2.a.a.1.2	✓	2
5.4	even	2	inner	575.2.b.d.24.2		4
15.2	even	4		5175.2.a.be.1.2		2
15.8	even	4		207.2.a.d.1.1		2
20.3	even	4		368.2.a.h.1.2		2
20.7	even	4		9200.2.a.bt.1.1		2
35.13	even	4		1127.2.a.c.1.2		2
40.3	even	4		1472.2.a.s.1.1		2
40.13	odd	4		1472.2.a.t.1.2		2
55.43	even	4		2783.2.a.c.1.1		2
60.23	odd	4		3312.2.a.ba.1.1		2
65.38	odd	4		3887.2.a.i.1.1		2
85.33	odd	4		6647.2.a.b.1.2		2
95.18	even	4		8303.2.a.e.1.1		2
115.3	odd	44		529.2.c.o.170.1		20
115.8	odd	44		529.2.c.o.501.1		20
115.13	odd	44		529.2.c.o.399.1		20
115.18	odd	44		529.2.c.o.255.2		20
115.28	even	44		529.2.c.n.255.2		20
115.33	even	44		529.2.c.n.399.1		20
115.38	even	44		529.2.c.n.501.1		20
115.43	even	44		529.2.c.n.170.1		20
115.48	odd	44		529.2.c.o.487.2		20
115.53	even	44		529.2.c.n.118.1		20
115.58	odd	44		529.2.c.o.466.2		20
115.63	even	44		529.2.c.n.266.1		20
115.68	even	4		529.2.a.a.1.2		2
115.73	odd	44		529.2.c.o.177.1		20
115.78	odd	44		529.2.c.o.334.2		20
115.83	even	44		529.2.c.n.334.2		20
115.88	even	44		529.2.c.n.177.1		20
115.98	odd	44		529.2.c.o.266.1		20
115.103	even	44		529.2.c.n.466.2		20
115.108	odd	44		529.2.c.o.118.1		20
115.113	even	44		529.2.c.n.487.2		20
345.68	odd	4		4761.2.a.w.1.1		2
460.183	odd	4		8464.2.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.2.a.a.1.2	✓	2	5.3	odd	4
207.2.a.d.1.1		2	15.8	even	4
368.2.a.h.1.2		2	20.3	even	4
529.2.a.a.1.2		2	115.68	even	4
529.2.c.n.118.1		20	115.53	even	44
529.2.c.n.170.1		20	115.43	even	44
529.2.c.n.177.1		20	115.88	even	44
529.2.c.n.255.2		20	115.28	even	44
529.2.c.n.266.1		20	115.63	even	44
529.2.c.n.334.2		20	115.83	even	44
529.2.c.n.399.1		20	115.33	even	44
529.2.c.n.466.2		20	115.103	even	44
529.2.c.n.487.2		20	115.113	even	44
529.2.c.n.501.1		20	115.38	even	44
529.2.c.o.118.1		20	115.108	odd	44
529.2.c.o.170.1		20	115.3	odd	44
529.2.c.o.177.1		20	115.73	odd	44
529.2.c.o.255.2		20	115.18	odd	44
529.2.c.o.266.1		20	115.98	odd	44
529.2.c.o.334.2		20	115.78	odd	44
529.2.c.o.399.1		20	115.13	odd	44
529.2.c.o.466.2		20	115.58	odd	44
529.2.c.o.487.2		20	115.48	odd	44
529.2.c.o.501.1		20	115.8	odd	44
575.2.a.f.1.1		2	5.2	odd	4
575.2.b.d.24.2		4	5.4	even	2	inner
575.2.b.d.24.3		4	1.1	even	1	trivial
1127.2.a.c.1.2		2	35.13	even	4
1472.2.a.s.1.1		2	40.3	even	4
1472.2.a.t.1.2		2	40.13	odd	4
2783.2.a.c.1.1		2	55.43	even	4
3312.2.a.ba.1.1		2	60.23	odd	4
3887.2.a.i.1.1		2	65.38	odd	4
4761.2.a.w.1.1		2	345.68	odd	4
5175.2.a.be.1.2		2	15.2	even	4
6647.2.a.b.1.2		2	85.33	odd	4
8303.2.a.e.1.1		2	95.18	even	4
8464.2.a.bb.1.2		2	460.183	odd	4
9200.2.a.bt.1.1		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)

\(f(q)\)	\(=\)	\(q+0.618034i q^{2} +2.23607i q^{3} +1.61803 q^{4} -1.38197 q^{6} +3.23607i q^{7} +2.23607i q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q+0.618034i q^{2} +2.23607i q^{3} +1.61803 q^{4} -1.38197 q^{6} +3.23607i q^{7} +2.23607i q^{8} -2.00000 q^{9} -5.23607 q^{11} +3.61803i q^{12} -3.00000i q^{13} -2.00000 q^{14} +1.85410 q^{16} +0.763932i q^{17} -1.23607i q^{18} +2.00000 q^{19} -7.23607 q^{21} -3.23607i q^{22} -1.00000i q^{23} -5.00000 q^{24} +1.85410 q^{26} +2.23607i q^{27} +5.23607i q^{28} +3.00000 q^{29} +6.70820 q^{31} +5.61803i q^{32} -11.7082i q^{33} -0.472136 q^{34} -3.23607 q^{36} -1.23607i q^{37} +1.23607i q^{38} +6.70820 q^{39} -3.47214 q^{41} -4.47214i q^{42} -8.47214 q^{44} +0.618034 q^{46} -2.23607i q^{47} +4.14590i q^{48} -3.47214 q^{49} -1.70820 q^{51} -4.85410i q^{52} -0.472136i q^{53} -1.38197 q^{54} -7.23607 q^{56} +4.47214i q^{57} +1.85410i q^{58} -6.47214 q^{59} -6.94427 q^{61} +4.14590i q^{62} -6.47214i q^{63} +0.236068 q^{64} +7.23607 q^{66} -2.76393i q^{67} +1.23607i q^{68} +2.23607 q^{69} +12.2361 q^{71} -4.47214i q^{72} -6.52786i q^{73} +0.763932 q^{74} +3.23607 q^{76} -16.9443i q^{77} +4.14590i q^{78} +10.9443 q^{79} -11.0000 q^{81} -2.14590i q^{82} +8.76393i q^{83} -11.7082 q^{84} +6.70820i q^{87} -11.7082i q^{88} +10.4721 q^{89} +9.70820 q^{91} -1.61803i q^{92} +15.0000i q^{93} +1.38197 q^{94} -12.5623 q^{96} +17.7082i q^{97} -2.14590i q^{98} +10.4721 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 10 q^{6} - 8 q^{9}+O(q^{10}) \) 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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