LMFDB - Embedded newform 750.2.c.a.499.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [750,2,Mod(499,750)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(750, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("750.499");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 750 = 2 \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	750.c (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:	$5.98878015160$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			499.4
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	750.499
Dual form			750.2.c.a.499.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.38197i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.38197i q^{7} -1.00000i q^{8} -1.00000 q^{9} -5.85410 q^{11} -1.00000i q^{12} +3.38197i q^{13} +2.38197 q^{14} +1.00000 q^{16} -7.70820i q^{17} -1.00000i q^{18} -4.09017 q^{19} +2.38197 q^{21} -5.85410i q^{22} -3.09017i q^{23} +1.00000 q^{24} -3.38197 q^{26} -1.00000i q^{27} +2.38197i q^{28} -5.70820 q^{29} -6.47214 q^{31} +1.00000i q^{32} -5.85410i q^{33} +7.70820 q^{34} +1.00000 q^{36} +1.61803i q^{37} -4.09017i q^{38} -3.38197 q^{39} +0.381966 q^{41} +2.38197i q^{42} +7.70820i q^{43} +5.85410 q^{44} +3.09017 q^{46} -8.61803i q^{47} +1.00000i q^{48} +1.32624 q^{49} +7.70820 q^{51} -3.38197i q^{52} +0.381966i q^{53} +1.00000 q^{54} -2.38197 q^{56} -4.09017i q^{57} -5.70820i q^{58} +10.8541 q^{59} +11.7082 q^{61} -6.47214i q^{62} +2.38197i q^{63} -1.00000 q^{64} +5.85410 q^{66} -3.23607i q^{67} +7.70820i q^{68} +3.09017 q^{69} -4.47214 q^{71} +1.00000i q^{72} -8.00000i q^{73} -1.61803 q^{74} +4.09017 q^{76} +13.9443i q^{77} -3.38197i q^{78} -12.4721 q^{79} +1.00000 q^{81} +0.381966i q^{82} +2.00000i q^{83} -2.38197 q^{84} -7.70820 q^{86} -5.70820i q^{87} +5.85410i q^{88} -15.5623 q^{89} +8.05573 q^{91} +3.09017i q^{92} -6.47214i q^{93} +8.61803 q^{94} -1.00000 q^{96} +14.1803i q^{97} +1.32624i q^{98} +5.85410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9	$ 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} - 10 q^{11} + 14 q^{14} + 4 q^{16} + 6 q^{19} + 14 q^{21} + 4 q^{24} - 18 q^{26} + 4 q^{29} - 8 q^{31} + 4 q^{34} + 4 q^{36} - 18 q^{39} + 6 q^{41} + 10 q^{44} - 10 q^{46} - 26 q^{49} + 4 q^{51} + 4 q^{54} - 14 q^{56} + 30 q^{59} + 20 q^{61} - 4 q^{64} + 10 q^{66} - 10 q^{69} - 2 q^{74} - 6 q^{76} - 32 q^{79} + 4 q^{81} - 14 q^{84} - 4 q^{86} - 22 q^{89} + 68 q^{91} + 30 q^{94} - 4 q^{96} + 10 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 - 10 * q^11 + 14 * q^14 + 4 * q^16 + 6 * q^19 + 14 * q^21 + 4 * q^24 - 18 * q^26 + 4 * q^29 - 8 * q^31 + 4 * q^34 + 4 * q^36 - 18 * q^39 + 6 * q^41 + 10 * q^44 - 10 * q^46 - 26 * q^49 + 4 * q^51 + 4 * q^54 - 14 * q^56 + 30 * q^59 + 20 * q^61 - 4 * q^64 + 10 * q^66 - 10 * q^69 - 2 * q^74 - 6 * q^76 - 32 * q^79 + 4 * q^81 - 14 * q^84 - 4 * q^86 - 22 * q^89 + 68 * q^91 + 30 * q^94 - 4 * q^96 + 10 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/750\mathbb{Z}\right)^\times$.

$n$	$127$	$251$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	− 2.38197i	− 0.900299i	−0.892953	−	0.450149i	$-0.851371\pi$
$7$	− 2.38197i	− 0.900299i	0.892953	−	0.450149i	$-0.148629\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−5.85410	−1.76508	−0.882539	−	0.470239i	$-0.844168\pi$
$11$	−5.85410	−1.76508	−0.882539	+	0.470239i	$0.844168\pi$
$12$	− 1.00000i	− 0.288675i
$13$	3.38197i	0.937989i	0.883201	+	0.468994i	$0.155384\pi$
$13$	3.38197i	0.937989i	−0.883201	+	0.468994i	$0.844616\pi$
$14$	2.38197	0.636607
$15$	0	0
$16$	1.00000	0.250000
$17$	− 7.70820i	− 1.86951i	−0.355288	−	0.934757i	$-0.615617\pi$
$17$	− 7.70820i	− 1.86951i	0.355288	−	0.934757i	$-0.384383\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−4.09017	−0.938349	−0.469175	−	0.883105i	$-0.655449\pi$
$19$	−4.09017	−0.938349	−0.469175	+	0.883105i	$0.655449\pi$
$20$	0	0
$21$	2.38197	0.519788
$22$	− 5.85410i	− 1.24810i
$23$	− 3.09017i	− 0.644345i	−0.946681	−	0.322172i	$-0.895587\pi$
$23$	− 3.09017i	− 0.644345i	0.946681	−	0.322172i	$-0.104413\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−3.38197	−0.663258
$27$	− 1.00000i	− 0.192450i
$28$	2.38197i	0.450149i
$29$	−5.70820	−1.05999	−0.529993	−	0.848002i	$-0.677806\pi$
$29$	−5.70820	−1.05999	−0.529993	+	0.848002i	$0.677806\pi$
$30$	0	0
$31$	−6.47214	−1.16243	−0.581215	−	0.813750i	$-0.697422\pi$
$31$	−6.47214	−1.16243	−0.581215	+	0.813750i	$0.697422\pi$
$32$	1.00000i	0.176777i
$33$	− 5.85410i	− 1.01907i
$34$	7.70820	1.32195
$35$	0	0
$36$	1.00000	0.166667
$37$	1.61803i	0.266003i	0.991116	+	0.133002i	$0.0424615\pi$
$37$	1.61803i	0.266003i	−0.991116	+	0.133002i	$0.957538\pi$
$38$	− 4.09017i	− 0.663513i
$39$	−3.38197	−0.541548
$40$	0	0
$41$	0.381966	0.0596531	0.0298265	−	0.999555i	$-0.490505\pi$
$41$	0.381966	0.0596531	0.0298265	+	0.999555i	$0.490505\pi$
$42$	2.38197i	0.367545i
$43$	7.70820i	1.17549i	0.809046	+	0.587745i	$0.199984\pi$
$43$	7.70820i	1.17549i	−0.809046	+	0.587745i	$0.800016\pi$
$44$	5.85410	0.882539
$45$	0	0
$46$	3.09017	0.455621
$47$	− 8.61803i	− 1.25707i	−0.777782	−	0.628535i	$-0.783655\pi$
$47$	− 8.61803i	− 1.25707i	0.777782	−	0.628535i	$-0.216345\pi$
$48$	1.00000i	0.144338i
$49$	1.32624	0.189463
$50$	0	0
$51$	7.70820	1.07936
$52$	− 3.38197i	− 0.468994i
$53$	0.381966i	0.0524671i	0.999656	+	0.0262335i	$0.00835135\pi$
$53$	0.381966i	0.0524671i	−0.999656	+	0.0262335i	$0.991649\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−2.38197	−0.318304
$57$	− 4.09017i	− 0.541756i
$58$	− 5.70820i	− 0.749524i
$59$	10.8541	1.41308	0.706542	−	0.707671i	$-0.250254\pi$
$59$	10.8541	1.41308	0.706542	+	0.707671i	$0.250254\pi$
$60$	0	0
$61$	11.7082	1.49908	0.749541	−	0.661958i	$-0.230274\pi$
$61$	11.7082	1.49908	0.749541	+	0.661958i	$0.230274\pi$
$62$	− 6.47214i	− 0.821962i
$63$	2.38197i	0.300100i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	5.85410	0.720590
$67$	− 3.23607i	− 0.395349i	−0.980268	−	0.197674i	$-0.936661\pi$
$67$	− 3.23607i	− 0.395349i	0.980268	−	0.197674i	$-0.0633388\pi$
$68$	7.70820i	0.934757i
$69$	3.09017	0.372013
$70$	0	0
$71$	−4.47214	−0.530745	−0.265372	−	0.964146i	$-0.585495\pi$
$71$	−4.47214	−0.530745	−0.265372	+	0.964146i	$0.585495\pi$
$72$	1.00000i	0.117851i
$73$	− 8.00000i	− 0.936329i	−0.883641	−	0.468165i	$-0.844915\pi$
$73$	− 8.00000i	− 0.936329i	0.883641	−	0.468165i	$-0.155085\pi$
$74$	−1.61803	−0.188093
$75$	0	0
$76$	4.09017	0.469175
$77$	13.9443i	1.58910i
$78$	− 3.38197i	− 0.382932i
$79$	−12.4721	−1.40322	−0.701612	−	0.712559i	$-0.747536\pi$
$79$	−12.4721	−1.40322	−0.701612	+	0.712559i	$0.747536\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0.381966i	0.0421811i
$83$	2.00000i	0.219529i	0.993958	+	0.109764i	$0.0350096\pi$
$83$	2.00000i	0.219529i	−0.993958	+	0.109764i	$0.964990\pi$
$84$	−2.38197	−0.259894
$85$	0	0
$86$	−7.70820	−0.831197
$87$	− 5.70820i	− 0.611984i
$88$	5.85410i	0.624049i
$89$	−15.5623	−1.64960	−0.824801	−	0.565424i	$-0.808713\pi$
$89$	−15.5623	−1.64960	−0.824801	+	0.565424i	$0.808713\pi$
$90$	0	0
$91$	8.05573	0.844470
$92$	3.09017i	0.322172i
$93$	− 6.47214i	− 0.671129i
$94$	8.61803	0.888882
$95$	0	0
$96$	−1.00000	−0.102062
$97$	14.1803i	1.43980i	0.694080	+	0.719898i	$0.255811\pi$
$97$	14.1803i	1.43980i	−0.694080	+	0.719898i	$0.744189\pi$
$98$	1.32624i	0.133970i
$99$	5.85410	0.588359
$100$	0	0
$101$	−13.7082	−1.36402	−0.682009	−	0.731344i	$-0.738893\pi$
$101$	−13.7082	−1.36402	−0.682009	+	0.731344i	$0.738893\pi$
$102$	7.70820i	0.763226i
$103$	2.90983i	0.286714i	0.989671	+	0.143357i	$0.0457897\pi$
$103$	2.90983i	0.286714i	−0.989671	+	0.143357i	$0.954210\pi$
$104$	3.38197	0.331629
$105$	0	0
$106$	−0.381966	−0.0370998
$107$	2.00000i	0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$	2.00000i	0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$	1.00000i	0.0962250i
$109$	1.70820	0.163616	0.0818081	−	0.996648i	$-0.473931\pi$
$109$	1.70820	0.163616	0.0818081	+	0.996648i	$0.473931\pi$
$110$	0	0
$111$	−1.61803	−0.153577
$112$	− 2.38197i	− 0.225075i
$113$	3.70820i	0.348838i	0.984671	+	0.174419i	$0.0558048\pi$
$113$	3.70820i	0.348838i	−0.984671	+	0.174419i	$0.944195\pi$
$114$	4.09017	0.383080
$115$	0	0
$116$	5.70820	0.529993
$117$	− 3.38197i	− 0.312663i
$118$	10.8541i	0.999201i
$119$	−18.3607	−1.68312
$120$	0	0
$121$	23.2705	2.11550
$122$	11.7082i	1.06001i
$123$	0.381966i	0.0344407i
$124$	6.47214	0.581215
$125$	0	0
$126$	−2.38197	−0.212202
$127$	5.52786i	0.490519i	0.969458	+	0.245259i	$0.0788731\pi$
$127$	5.52786i	0.490519i	−0.969458	+	0.245259i	$0.921127\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−7.70820	−0.678670
$130$	0	0
$131$	−20.3262	−1.77591	−0.887956	−	0.459929i	$-0.847875\pi$
$131$	−20.3262	−1.77591	−0.887956	+	0.459929i	$0.847875\pi$
$132$	5.85410i	0.509534i
$133$	9.74265i	0.844795i
$134$	3.23607	0.279554
$135$	0	0
$136$	−7.70820	−0.660973
$137$	− 4.94427i	− 0.422418i	−0.977441	−	0.211209i	$-0.932260\pi$
$137$	− 4.94427i	− 0.422418i	0.977441	−	0.211209i	$-0.0677400\pi$
$138$	3.09017i	0.263053i
$139$	2.61803	0.222059	0.111029	−	0.993817i	$-0.464585\pi$
$139$	2.61803	0.222059	0.111029	+	0.993817i	$0.464585\pi$
$140$	0	0
$141$	8.61803	0.725769
$142$	− 4.47214i	− 0.375293i
$143$	− 19.7984i	− 1.65562i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	8.00000	0.662085
$147$	1.32624i	0.109386i
$148$	− 1.61803i	− 0.133002i
$149$	9.41641	0.771422	0.385711	−	0.922620i	$-0.373956\pi$
$149$	9.41641	0.771422	0.385711	+	0.922620i	$0.373956\pi$
$150$	0	0
$151$	−0.291796	−0.0237460	−0.0118730	−	0.999930i	$-0.503779\pi$
$151$	−0.291796	−0.0237460	−0.0118730	+	0.999930i	$0.503779\pi$
$152$	4.09017i	0.331757i
$153$	7.70820i	0.623171i
$154$	−13.9443	−1.12366
$155$	0	0
$156$	3.38197	0.270774
$157$	6.94427i	0.554213i	0.960839	+	0.277107i	$0.0893755\pi$
$157$	6.94427i	0.554213i	−0.960839	+	0.277107i	$0.910624\pi$
$158$	− 12.4721i	− 0.992230i
$159$	−0.381966	−0.0302919
$160$	0	0
$161$	−7.36068	−0.580103
$162$	1.00000i	0.0785674i
$163$	− 24.0000i	− 1.87983i	−0.341415	−	0.939913i	$-0.610906\pi$
$163$	− 24.0000i	− 1.87983i	0.341415	−	0.939913i	$-0.389094\pi$
$164$	−0.381966	−0.0298265
$165$	0	0
$166$	−2.00000	−0.155230
$167$	1.90983i	0.147787i	0.997266	+	0.0738935i	$0.0235425\pi$
$167$	1.90983i	0.147787i	−0.997266	+	0.0738935i	$0.976457\pi$
$168$	− 2.38197i	− 0.183773i
$169$	1.56231	0.120177
$170$	0	0
$171$	4.09017	0.312783
$172$	− 7.70820i	− 0.587745i
$173$	4.38197i	0.333155i	0.986028	+	0.166577i	$0.0532715\pi$
$173$	4.38197i	0.333155i	−0.986028	+	0.166577i	$0.946728\pi$
$174$	5.70820	0.432738
$175$	0	0
$176$	−5.85410	−0.441270
$177$	10.8541i	0.815844i
$178$	− 15.5623i	− 1.16644i
$179$	−3.79837	−0.283904	−0.141952	−	0.989874i	$-0.545338\pi$
$179$	−3.79837	−0.283904	−0.141952	+	0.989874i	$0.545338\pi$
$180$	0	0
$181$	−20.4721	−1.52168	−0.760841	−	0.648938i	$-0.775213\pi$
$181$	−20.4721	−1.52168	−0.760841	+	0.648938i	$0.775213\pi$
$182$	8.05573i	0.597130i
$183$	11.7082i	0.865495i
$184$	−3.09017	−0.227810
$185$	0	0
$186$	6.47214	0.474560
$187$	45.1246i	3.29984i
$188$	8.61803i	0.628535i
$189$	−2.38197	−0.173263
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	9.70820i	0.698812i	0.936971	+	0.349406i	$0.113617\pi$
$193$	9.70820i	0.698812i	−0.936971	+	0.349406i	$0.886383\pi$
$194$	−14.1803	−1.01809
$195$	0	0
$196$	−1.32624	−0.0947313
$197$	14.9443i	1.06474i	0.846513	+	0.532368i	$0.178698\pi$
$197$	14.9443i	1.06474i	−0.846513	+	0.532368i	$0.821302\pi$
$198$	5.85410i	0.416033i
$199$	4.76393	0.337706	0.168853	−	0.985641i	$-0.445994\pi$
$199$	4.76393	0.337706	0.168853	+	0.985641i	$0.445994\pi$
$200$	0	0
$201$	3.23607	0.228255
$202$	− 13.7082i	− 0.964506i
$203$	13.5967i	0.954305i
$204$	−7.70820	−0.539682
$205$	0	0
$206$	−2.90983	−0.202737
$207$	3.09017i	0.214782i
$208$	3.38197i	0.234497i
$209$	23.9443	1.65626
$210$	0	0
$211$	−22.5623	−1.55325	−0.776627	−	0.629961i	$-0.783071\pi$
$211$	−22.5623	−1.55325	−0.776627	+	0.629961i	$0.783071\pi$
$212$	− 0.381966i	− 0.0262335i
$213$	− 4.47214i	− 0.306426i
$214$	−2.00000	−0.136717
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	15.4164i	1.04653i
$218$	1.70820i	0.115694i
$219$	8.00000	0.540590
$220$	0	0
$221$	26.0689	1.75358
$222$	− 1.61803i	− 0.108595i
$223$	− 28.9443i	− 1.93825i	−0.246566	−	0.969126i	$-0.579302\pi$
$223$	− 28.9443i	− 1.93825i	0.246566	−	0.969126i	$-0.420698\pi$
$224$	2.38197	0.159152
$225$	0	0
$226$	−3.70820	−0.246666
$227$	− 10.7639i	− 0.714427i	−0.934023	−	0.357214i	$-0.883727\pi$
$227$	− 10.7639i	− 0.714427i	0.934023	−	0.357214i	$-0.116273\pi$
$228$	4.09017i	0.270878i
$229$	22.7639	1.50428	0.752141	−	0.659002i	$-0.229021\pi$
$229$	22.7639	1.50428	0.752141	+	0.659002i	$0.229021\pi$
$230$	0	0
$231$	−13.9443	−0.917466
$232$	5.70820i	0.374762i
$233$	− 2.47214i	− 0.161955i	−0.996716	−	0.0809775i	$-0.974196\pi$
$233$	− 2.47214i	− 0.161955i	0.996716	−	0.0809775i	$-0.0258042\pi$
$234$	3.38197	0.221086
$235$	0	0
$236$	−10.8541	−0.706542
$237$	− 12.4721i	− 0.810152i
$238$	− 18.3607i	− 1.19015i
$239$	7.05573	0.456397	0.228199	−	0.973615i	$-0.426716\pi$
$239$	7.05573	0.456397	0.228199	+	0.973615i	$0.426716\pi$
$240$	0	0
$241$	15.0902	0.972043	0.486022	−	0.873947i	$-0.338448\pi$
$241$	15.0902	0.972043	0.486022	+	0.873947i	$0.338448\pi$
$242$	23.2705i	1.49589i
$243$	1.00000i	0.0641500i
$244$	−11.7082	−0.749541
$245$	0	0
$246$	−0.381966	−0.0243533
$247$	− 13.8328i	− 0.880161i
$248$	6.47214i	0.410981i
$249$	−2.00000	−0.126745
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	− 2.38197i	− 0.150050i
$253$	18.0902i	1.13732i
$254$	−5.52786	−0.346849
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 17.5279i	− 1.09336i	−0.837342	−	0.546679i	$-0.815892\pi$
$257$	− 17.5279i	− 1.09336i	0.837342	−	0.546679i	$-0.184108\pi$
$258$	− 7.70820i	− 0.479892i
$259$	3.85410	0.239482
$260$	0	0
$261$	5.70820	0.353329
$262$	− 20.3262i	− 1.25576i
$263$	25.3262i	1.56168i	0.624729	+	0.780841i	$0.285209\pi$
$263$	25.3262i	1.56168i	−0.624729	+	0.780841i	$0.714791\pi$
$264$	−5.85410	−0.360295
$265$	0	0
$266$	−9.74265	−0.597360
$267$	− 15.5623i	− 0.952398i
$268$	3.23607i	0.197674i
$269$	9.41641	0.574129	0.287064	−	0.957911i	$-0.407321\pi$
$269$	9.41641	0.574129	0.287064	+	0.957911i	$0.407321\pi$
$270$	0	0
$271$	−9.41641	−0.572006	−0.286003	−	0.958229i	$-0.592327\pi$
$271$	−9.41641	−0.572006	−0.286003	+	0.958229i	$0.592327\pi$
$272$	− 7.70820i	− 0.467379i
$273$	8.05573i	0.487555i
$274$	4.94427	0.298694
$275$	0	0
$276$	−3.09017	−0.186006
$277$	− 27.4508i	− 1.64936i	−0.565598	−	0.824681i	$-0.691355\pi$
$277$	− 27.4508i	− 1.64936i	0.565598	−	0.824681i	$-0.308645\pi$
$278$	2.61803i	0.157019i
$279$	6.47214	0.387477
$280$	0	0
$281$	24.5066	1.46194	0.730970	−	0.682410i	$-0.239068\pi$
$281$	24.5066	1.46194	0.730970	+	0.682410i	$0.239068\pi$
$282$	8.61803i	0.513196i
$283$	8.18034i	0.486271i	0.969992	+	0.243135i	$0.0781759\pi$
$283$	8.18034i	0.486271i	−0.969992	+	0.243135i	$0.921824\pi$
$284$	4.47214	0.265372
$285$	0	0
$286$	19.7984	1.17070
$287$	− 0.909830i	− 0.0537056i
$288$	− 1.00000i	− 0.0589256i
$289$	−42.4164	−2.49508
$290$	0	0
$291$	−14.1803	−0.831266
$292$	8.00000i	0.468165i
$293$	− 20.7984i	− 1.21505i	−0.794299	−	0.607527i	$-0.792162\pi$
$293$	− 20.7984i	− 1.21505i	0.794299	−	0.607527i	$-0.207838\pi$
$294$	−1.32624	−0.0773478
$295$	0	0
$296$	1.61803	0.0940463
$297$	5.85410i	0.339689i
$298$	9.41641i	0.545478i
$299$	10.4508	0.604388
$300$	0	0
$301$	18.3607	1.05829
$302$	− 0.291796i	− 0.0167910i
$303$	− 13.7082i	− 0.787516i
$304$	−4.09017	−0.234587
$305$	0	0
$306$	−7.70820	−0.440649
$307$	6.18034i	0.352731i	0.984325	+	0.176365i	$0.0564340\pi$
$307$	6.18034i	0.352731i	−0.984325	+	0.176365i	$0.943566\pi$
$308$	− 13.9443i	− 0.794549i
$309$	−2.90983	−0.165534
$310$	0	0
$311$	8.94427	0.507183	0.253592	−	0.967311i	$-0.418388\pi$
$311$	8.94427	0.507183	0.253592	+	0.967311i	$0.418388\pi$
$312$	3.38197i	0.191466i
$313$	0.291796i	0.0164933i	0.999966	+	0.00824664i	$0.00262502\pi$
$313$	0.291796i	0.0164933i	−0.999966	+	0.00824664i	$0.997375\pi$
$314$	−6.94427	−0.391888
$315$	0	0
$316$	12.4721	0.701612
$317$	− 23.8541i	− 1.33978i	−0.742460	−	0.669890i	$-0.766341\pi$
$317$	− 23.8541i	− 1.33978i	0.742460	−	0.669890i	$-0.233659\pi$
$318$	− 0.381966i	− 0.0214196i
$319$	33.4164	1.87096
$320$	0	0
$321$	−2.00000	−0.111629
$322$	− 7.36068i	− 0.410195i
$323$	31.5279i	1.75426i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	24.0000	1.32924
$327$	1.70820i	0.0944639i
$328$	− 0.381966i	− 0.0210905i
$329$	−20.5279	−1.13174
$330$	0	0
$331$	6.47214	0.355741	0.177870	−	0.984054i	$-0.443079\pi$
$331$	6.47214	0.355741	0.177870	+	0.984054i	$0.443079\pi$
$332$	− 2.00000i	− 0.109764i
$333$	− 1.61803i	− 0.0886677i
$334$	−1.90983	−0.104501
$335$	0	0
$336$	2.38197	0.129947
$337$	20.8328i	1.13484i	0.823430	+	0.567418i	$0.192058\pi$
$337$	20.8328i	1.13484i	−0.823430	+	0.567418i	$0.807942\pi$
$338$	1.56231i	0.0849782i
$339$	−3.70820	−0.201402
$340$	0	0
$341$	37.8885	2.05178
$342$	4.09017i	0.221171i
$343$	− 19.8328i	− 1.07087i
$344$	7.70820	0.415599
$345$	0	0
$346$	−4.38197	−0.235576
$347$	14.1803i	0.761241i	0.924731	+	0.380620i	$0.124289\pi$
$347$	14.1803i	0.761241i	−0.924731	+	0.380620i	$0.875711\pi$
$348$	5.70820i	0.305992i
$349$	20.0000	1.07058	0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000	1.07058	0.535288	+	0.844670i	$0.320203\pi$
$350$	0	0
$351$	3.38197	0.180516
$352$	− 5.85410i	− 0.312025i
$353$	− 9.12461i	− 0.485654i	−0.970070	−	0.242827i	$-0.921925\pi$
$353$	− 9.12461i	− 0.485654i	0.970070	−	0.242827i	$-0.0780747\pi$
$354$	−10.8541	−0.576889
$355$	0	0
$356$	15.5623	0.824801
$357$	− 18.3607i	− 0.971750i
$358$	− 3.79837i	− 0.200750i
$359$	17.4164	0.919203	0.459601	−	0.888125i	$-0.347992\pi$
$359$	17.4164	0.919203	0.459601	+	0.888125i	$0.347992\pi$
$360$	0	0
$361$	−2.27051	−0.119501
$362$	− 20.4721i	− 1.07599i
$363$	23.2705i	1.22139i
$364$	−8.05573	−0.422235
$365$	0	0
$366$	−11.7082	−0.611998
$367$	− 15.4164i	− 0.804730i	−0.915479	−	0.402365i	$-0.868188\pi$
$367$	− 15.4164i	− 0.804730i	0.915479	−	0.402365i	$-0.131812\pi$
$368$	− 3.09017i	− 0.161086i
$369$	−0.381966	−0.0198844
$370$	0	0
$371$	0.909830	0.0472360
$372$	6.47214i	0.335565i
$373$	− 19.3262i	− 1.00067i	−0.865831	−	0.500337i	$-0.833209\pi$
$373$	− 19.3262i	− 1.00067i	0.865831	−	0.500337i	$-0.166791\pi$
$374$	−45.1246	−2.33334
$375$	0	0
$376$	−8.61803	−0.444441
$377$	− 19.3050i	− 0.994256i
$378$	− 2.38197i	− 0.122515i
$379$	−35.9230	−1.84524	−0.922620	−	0.385710i	$-0.873956\pi$
$379$	−35.9230	−1.84524	−0.922620	+	0.385710i	$0.873956\pi$
$380$	0	0
$381$	−5.52786	−0.283201
$382$	− 4.00000i	− 0.204658i
$383$	27.0344i	1.38140i	0.723144	+	0.690698i	$0.242696\pi$
$383$	27.0344i	1.38140i	−0.723144	+	0.690698i	$0.757304\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−9.70820	−0.494135
$387$	− 7.70820i	− 0.391830i
$388$	− 14.1803i	− 0.719898i
$389$	34.5410	1.75130	0.875650	−	0.482947i	$-0.160434\pi$
$389$	34.5410	1.75130	0.875650	+	0.482947i	$0.160434\pi$
$390$	0	0
$391$	−23.8197	−1.20461
$392$	− 1.32624i	− 0.0669851i
$393$	− 20.3262i	− 1.02532i
$394$	−14.9443	−0.752882
$395$	0	0
$396$	−5.85410	−0.294180
$397$	− 23.3262i	− 1.17071i	−0.810777	−	0.585355i	$-0.800955\pi$
$397$	− 23.3262i	− 1.17071i	0.810777	−	0.585355i	$-0.199045\pi$
$398$	4.76393i	0.238794i
$399$	−9.74265	−0.487742
$400$	0	0
$401$	−1.32624	−0.0662292	−0.0331146	−	0.999452i	$-0.510543\pi$
$401$	−1.32624	−0.0662292	−0.0331146	+	0.999452i	$0.510543\pi$
$402$	3.23607i	0.161400i
$403$	− 21.8885i	− 1.09035i
$404$	13.7082	0.682009
$405$	0	0
$406$	−13.5967	−0.674795
$407$	− 9.47214i	− 0.469516i
$408$	− 7.70820i	− 0.381613i
$409$	4.85410	0.240020	0.120010	−	0.992773i	$-0.461707\pi$
$409$	4.85410	0.240020	0.120010	+	0.992773i	$0.461707\pi$
$410$	0	0
$411$	4.94427	0.243883
$412$	− 2.90983i	− 0.143357i
$413$	− 25.8541i	− 1.27220i
$414$	−3.09017	−0.151874
$415$	0	0
$416$	−3.38197	−0.165815
$417$	2.61803i	0.128206i
$418$	23.9443i	1.17115i
$419$	−17.8885	−0.873913	−0.436956	−	0.899483i	$-0.643944\pi$
$419$	−17.8885	−0.873913	−0.436956	+	0.899483i	$0.643944\pi$
$420$	0	0
$421$	−25.1246	−1.22450	−0.612249	−	0.790665i	$-0.709735\pi$
$421$	−25.1246	−1.22450	−0.612249	+	0.790665i	$0.709735\pi$
$422$	− 22.5623i	− 1.09832i
$423$	8.61803i	0.419023i
$424$	0.381966	0.0185499
$425$	0	0
$426$	4.47214	0.216676
$427$	− 27.8885i	− 1.34962i
$428$	− 2.00000i	− 0.0966736i
$429$	19.7984	0.955874
$430$	0	0
$431$	17.2361	0.830232	0.415116	−	0.909768i	$-0.363741\pi$
$431$	17.2361	0.830232	0.415116	+	0.909768i	$0.363741\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	38.3607i	1.84350i	0.387789	+	0.921748i	$0.373239\pi$
$433$	38.3607i	1.84350i	−0.387789	+	0.921748i	$0.626761\pi$
$434$	−15.4164	−0.740011
$435$	0	0
$436$	−1.70820	−0.0818081
$437$	12.6393i	0.604621i
$438$	8.00000i	0.382255i
$439$	32.6525	1.55842	0.779209	−	0.626764i	$-0.215621\pi$
$439$	32.6525	1.55842	0.779209	+	0.626764i	$0.215621\pi$
$440$	0	0
$441$	−1.32624	−0.0631542
$442$	26.0689i	1.23997i
$443$	− 13.1246i	− 0.623569i	−0.950153	−	0.311785i	$-0.899073\pi$
$443$	− 13.1246i	− 0.623569i	0.950153	−	0.311785i	$-0.100927\pi$
$444$	1.61803	0.0767885
$445$	0	0
$446$	28.9443	1.37055
$447$	9.41641i	0.445381i
$448$	2.38197i	0.112537i
$449$	−16.5623	−0.781624	−0.390812	−	0.920471i	$-0.627806\pi$
$449$	−16.5623	−0.781624	−0.390812	+	0.920471i	$0.627806\pi$
$450$	0	0
$451$	−2.23607	−0.105292
$452$	− 3.70820i	− 0.174419i
$453$	− 0.291796i	− 0.0137098i
$454$	10.7639	0.505176
$455$	0	0
$456$	−4.09017	−0.191540
$457$	− 33.8885i	− 1.58524i	−0.609716	−	0.792620i	$-0.708717\pi$
$457$	− 33.8885i	− 1.58524i	0.609716	−	0.792620i	$-0.291283\pi$
$458$	22.7639i	1.06369i
$459$	−7.70820	−0.359788
$460$	0	0
$461$	19.5967	0.912712	0.456356	−	0.889797i	$-0.349154\pi$
$461$	19.5967	0.912712	0.456356	+	0.889797i	$0.349154\pi$
$462$	− 13.9443i	− 0.648746i
$463$	− 15.0557i	− 0.699699i	−0.936806	−	0.349850i	$-0.886233\pi$
$463$	− 15.0557i	− 0.699699i	0.936806	−	0.349850i	$-0.113767\pi$
$464$	−5.70820	−0.264997
$465$	0	0
$466$	2.47214	0.114519
$467$	− 21.2361i	− 0.982688i	−0.870966	−	0.491344i	$-0.836506\pi$
$467$	− 21.2361i	− 0.982688i	0.870966	−	0.491344i	$-0.163494\pi$
$468$	3.38197i	0.156331i
$469$	−7.70820	−0.355932
$470$	0	0
$471$	−6.94427	−0.319975
$472$	− 10.8541i	− 0.499601i
$473$	− 45.1246i	− 2.07483i
$474$	12.4721	0.572864
$475$	0	0
$476$	18.3607	0.841560
$477$	− 0.381966i	− 0.0174890i
$478$	7.05573i	0.322721i
$479$	9.12461	0.416914	0.208457	−	0.978032i	$-0.433156\pi$
$479$	9.12461	0.416914	0.208457	+	0.978032i	$0.433156\pi$
$480$	0	0
$481$	−5.47214	−0.249508
$482$	15.0902i	0.687338i
$483$	− 7.36068i	− 0.334923i
$484$	−23.2705	−1.05775
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	12.7984i	0.579950i	0.957034	+	0.289975i	$0.0936469\pi$
$487$	12.7984i	0.579950i	−0.957034	+	0.289975i	$0.906353\pi$
$488$	− 11.7082i	− 0.530005i
$489$	24.0000	1.08532
$490$	0	0
$491$	11.0344	0.497977	0.248989	−	0.968506i	$-0.419902\pi$
$491$	11.0344	0.497977	0.248989	+	0.968506i	$0.419902\pi$
$492$	− 0.381966i	− 0.0172204i
$493$	44.0000i	1.98166i
$494$	13.8328	0.622368
$495$	0	0
$496$	−6.47214	−0.290607
$497$	10.6525i	0.477829i
$498$	− 2.00000i	− 0.0896221i
$499$	5.38197	0.240930	0.120465	−	0.992718i	$-0.461561\pi$
$499$	5.38197	0.240930	0.120465	+	0.992718i	$0.461561\pi$
$500$	0	0
$501$	−1.90983	−0.0853249
$502$	− 24.0000i	− 1.07117i
$503$	− 30.1459i	− 1.34414i	−0.740488	−	0.672070i	$-0.765406\pi$
$503$	− 30.1459i	− 1.34414i	0.740488	−	0.672070i	$-0.234594\pi$
$504$	2.38197	0.106101
$505$	0	0
$506$	−18.0902	−0.804206
$507$	1.56231i	0.0693844i
$508$	− 5.52786i	− 0.245259i
$509$	−12.2918	−0.544824	−0.272412	−	0.962181i	$-0.587821\pi$
$509$	−12.2918	−0.544824	−0.272412	+	0.962181i	$0.587821\pi$
$510$	0	0
$511$	−19.0557	−0.842976
$512$	1.00000i	0.0441942i
$513$	4.09017i	0.180585i
$514$	17.5279	0.773121
$515$	0	0
$516$	7.70820	0.339335
$517$	50.4508i	2.21883i
$518$	3.85410i	0.169340i
$519$	−4.38197	−0.192347
$520$	0	0
$521$	1.09017	0.0477612	0.0238806	−	0.999715i	$-0.492398\pi$
$521$	1.09017	0.0477612	0.0238806	+	0.999715i	$0.492398\pi$
$522$	5.70820i	0.249841i
$523$	32.3607i	1.41503i	0.706696	+	0.707517i	$0.250185\pi$
$523$	32.3607i	1.41503i	−0.706696	+	0.707517i	$0.749815\pi$
$524$	20.3262	0.887956
$525$	0	0
$526$	−25.3262	−1.10428
$527$	49.8885i	2.17318i
$528$	− 5.85410i	− 0.254767i
$529$	13.4508	0.584820
$530$	0	0
$531$	−10.8541	−0.471028
$532$	− 9.74265i	− 0.422397i
$533$	1.29180i	0.0559539i
$534$	15.5623	0.673447
$535$	0	0
$536$	−3.23607	−0.139777
$537$	− 3.79837i	− 0.163912i
$538$	9.41641i	0.405970i
$539$	−7.76393	−0.334416
$540$	0	0
$541$	−13.4164	−0.576816	−0.288408	−	0.957508i	$-0.593126\pi$
$541$	−13.4164	−0.576816	−0.288408	+	0.957508i	$0.593126\pi$
$542$	− 9.41641i	− 0.404469i
$543$	− 20.4721i	− 0.878543i
$544$	7.70820	0.330487
$545$	0	0
$546$	−8.05573	−0.344753
$547$	0.763932i	0.0326634i	0.999867	+	0.0163317i	$0.00519877\pi$
$547$	0.763932i	0.0326634i	−0.999867	+	0.0163317i	$0.994801\pi$
$548$	4.94427i	0.211209i
$549$	−11.7082	−0.499694
$550$	0	0
$551$	23.3475	0.994638
$552$	− 3.09017i	− 0.131526i
$553$	29.7082i	1.26332i
$554$	27.4508	1.16627
$555$	0	0
$556$	−2.61803	−0.111029
$557$	− 13.4508i	− 0.569931i	−0.958538	−	0.284965i	$-0.908018\pi$
$557$	− 13.4508i	− 0.569931i	0.958538	−	0.284965i	$-0.0919821\pi$
$558$	6.47214i	0.273987i
$559$	−26.0689	−1.10260
$560$	0	0
$561$	−45.1246	−1.90516
$562$	24.5066i	1.03375i
$563$	6.94427i	0.292666i	0.989235	+	0.146333i	$0.0467471\pi$
$563$	6.94427i	0.292666i	−0.989235	+	0.146333i	$0.953253\pi$
$564$	−8.61803	−0.362885
$565$	0	0
$566$	−8.18034	−0.343845
$567$	− 2.38197i	− 0.100033i
$568$	4.47214i	0.187647i
$569$	−27.9230	−1.17059	−0.585296	−	0.810820i	$-0.699022\pi$
$569$	−27.9230	−1.17059	−0.585296	+	0.810820i	$0.699022\pi$
$570$	0	0
$571$	4.27051	0.178715	0.0893576	−	0.996000i	$-0.471519\pi$
$571$	4.27051	0.178715	0.0893576	+	0.996000i	$0.471519\pi$
$572$	19.7984i	0.827812i
$573$	− 4.00000i	− 0.167102i
$574$	0.909830	0.0379756
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 9.23607i	− 0.384503i	−0.981346	−	0.192251i	$-0.938421\pi$
$577$	− 9.23607i	− 0.384503i	0.981346	−	0.192251i	$-0.0615789\pi$
$578$	− 42.4164i	− 1.76429i
$579$	−9.70820	−0.403459
$580$	0	0
$581$	4.76393	0.197641
$582$	− 14.1803i	− 0.587794i
$583$	− 2.23607i	− 0.0926085i
$584$	−8.00000	−0.331042
$585$	0	0
$586$	20.7984	0.859173
$587$	− 25.2361i	− 1.04160i	−0.853678	−	0.520802i	$-0.825633\pi$
$587$	− 25.2361i	− 1.04160i	0.853678	−	0.520802i	$-0.174367\pi$
$588$	− 1.32624i	− 0.0546931i
$589$	26.4721	1.09077
$590$	0	0
$591$	−14.9443	−0.614725
$592$	1.61803i	0.0665008i
$593$	31.3050i	1.28554i	0.766059	+	0.642770i	$0.222215\pi$
$593$	31.3050i	1.28554i	−0.766059	+	0.642770i	$0.777785\pi$
$594$	−5.85410	−0.240197
$595$	0	0
$596$	−9.41641	−0.385711
$597$	4.76393i	0.194975i
$598$	10.4508i	0.427367i
$599$	6.76393	0.276367	0.138183	−	0.990407i	$-0.455874\pi$
$599$	6.76393	0.276367	0.138183	+	0.990407i	$0.455874\pi$
$600$	0	0
$601$	10.4377	0.425762	0.212881	−	0.977078i	$-0.431715\pi$
$601$	10.4377	0.425762	0.212881	+	0.977078i	$0.431715\pi$
$602$	18.3607i	0.748325i
$603$	3.23607i	0.131783i
$604$	0.291796	0.0118730
$605$	0	0
$606$	13.7082	0.556858
$607$	9.97871i	0.405023i	0.979280	+	0.202512i	$0.0649104\pi$
$607$	9.97871i	0.405023i	−0.979280	+	0.202512i	$0.935090\pi$
$608$	− 4.09017i	− 0.165878i
$609$	−13.5967	−0.550968
$610$	0	0
$611$	29.1459	1.17912
$612$	− 7.70820i	− 0.311586i
$613$	− 31.7426i	− 1.28207i	−0.767510	−	0.641037i	$-0.778505\pi$
$613$	− 31.7426i	− 1.28207i	0.767510	−	0.641037i	$-0.221495\pi$
$614$	−6.18034	−0.249418
$615$	0	0
$616$	13.9443	0.561831
$617$	− 5.52786i	− 0.222543i	−0.993790	−	0.111272i	$-0.964508\pi$
$617$	− 5.52786i	− 0.222543i	0.993790	−	0.111272i	$-0.0354924\pi$
$618$	− 2.90983i	− 0.117051i
$619$	11.8541	0.476457	0.238228	−	0.971209i	$-0.423433\pi$
$619$	11.8541	0.476457	0.238228	+	0.971209i	$0.423433\pi$
$620$	0	0
$621$	−3.09017	−0.124004
$622$	8.94427i	0.358633i
$623$	37.0689i	1.48513i
$624$	−3.38197	−0.135387
$625$	0	0
$626$	−0.291796	−0.0116625
$627$	23.9443i	0.956242i
$628$	− 6.94427i	− 0.277107i
$629$	12.4721	0.497297
$630$	0	0
$631$	33.7082	1.34190	0.670951	−	0.741502i	$-0.265886\pi$
$631$	33.7082	1.34190	0.670951	+	0.741502i	$0.265886\pi$
$632$	12.4721i	0.496115i
$633$	− 22.5623i	− 0.896771i
$634$	23.8541	0.947367
$635$	0	0
$636$	0.381966	0.0151459
$637$	4.48529i	0.177714i
$638$	33.4164i	1.32297i
$639$	4.47214	0.176915
$640$	0	0
$641$	−29.1459	−1.15119	−0.575597	−	0.817734i	$-0.695230\pi$
$641$	−29.1459	−1.15119	−0.575597	+	0.817734i	$0.695230\pi$
$642$	− 2.00000i	− 0.0789337i
$643$	− 15.0557i	− 0.593740i	−0.954918	−	0.296870i	$-0.904057\pi$
$643$	− 15.0557i	− 0.593740i	0.954918	−	0.296870i	$-0.0959428\pi$
$644$	7.36068	0.290051
$645$	0	0
$646$	−31.5279	−1.24045
$647$	− 14.8541i	− 0.583975i	−0.956422	−	0.291988i	$-0.905683\pi$
$647$	− 14.8541i	− 0.583975i	0.956422	−	0.291988i	$-0.0943166\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	−63.5410	−2.49420
$650$	0	0
$651$	−15.4164	−0.604217
$652$	24.0000i	0.939913i
$653$	12.1459i	0.475306i	0.971350	+	0.237653i	$0.0763781\pi$
$653$	12.1459i	0.475306i	−0.971350	+	0.237653i	$0.923622\pi$
$654$	−1.70820	−0.0667961
$655$	0	0
$656$	0.381966	0.0149133
$657$	8.00000i	0.312110i
$658$	− 20.5279i	− 0.800259i
$659$	−14.5066	−0.565096	−0.282548	−	0.959253i	$-0.591180\pi$
$659$	−14.5066	−0.565096	−0.282548	+	0.959253i	$0.591180\pi$
$660$	0	0
$661$	−0.875388	−0.0340487	−0.0170243	−	0.999855i	$-0.505419\pi$
$661$	−0.875388	−0.0340487	−0.0170243	+	0.999855i	$0.505419\pi$
$662$	6.47214i	0.251547i
$663$	26.0689i	1.01243i
$664$	2.00000	0.0776151
$665$	0	0
$666$	1.61803	0.0626975
$667$	17.6393i	0.682997i
$668$	− 1.90983i	− 0.0738935i
$669$	28.9443	1.11905
$670$	0	0
$671$	−68.5410	−2.64600
$672$	2.38197i	0.0918863i
$673$	− 9.52786i	− 0.367272i	−0.982994	−	0.183636i	$-0.941213\pi$
$673$	− 9.52786i	− 0.367272i	0.982994	−	0.183636i	$-0.0587868\pi$
$674$	−20.8328	−0.802450
$675$	0	0
$676$	−1.56231	−0.0600887
$677$	24.7426i	0.950937i	0.879733	+	0.475469i	$0.157721\pi$
$677$	24.7426i	0.950937i	−0.879733	+	0.475469i	$0.842279\pi$
$678$	− 3.70820i	− 0.142413i
$679$	33.7771	1.29625
$680$	0	0
$681$	10.7639	0.412475
$682$	37.8885i	1.45083i
$683$	− 29.8885i	− 1.14365i	−0.820374	−	0.571827i	$-0.806235\pi$
$683$	− 29.8885i	− 1.14365i	0.820374	−	0.571827i	$-0.193765\pi$
$684$	−4.09017	−0.156392
$685$	0	0
$686$	19.8328	0.757220
$687$	22.7639i	0.868498i
$688$	7.70820i	0.293873i
$689$	−1.29180	−0.0492135
$690$	0	0
$691$	−33.8885	−1.28918	−0.644590	−	0.764528i	$-0.722972\pi$
$691$	−33.8885	−1.28918	−0.644590	+	0.764528i	$0.722972\pi$
$692$	− 4.38197i	− 0.166577i
$693$	− 13.9443i	− 0.529699i
$694$	−14.1803	−0.538278
$695$	0	0
$696$	−5.70820	−0.216369
$697$	− 2.94427i	− 0.111522i
$698$	20.0000i	0.757011i
$699$	2.47214	0.0935048
$700$	0	0
$701$	−17.7082	−0.668830	−0.334415	−	0.942426i	$-0.608539\pi$
$701$	−17.7082	−0.668830	−0.334415	+	0.942426i	$0.608539\pi$
$702$	3.38197i	0.127644i
$703$	− 6.61803i	− 0.249604i
$704$	5.85410	0.220635
$705$	0	0
$706$	9.12461	0.343409
$707$	32.6525i	1.22802i
$708$	− 10.8541i	− 0.407922i
$709$	−18.6525	−0.700508	−0.350254	−	0.936655i	$-0.613905\pi$
$709$	−18.6525	−0.700508	−0.350254	+	0.936655i	$0.613905\pi$
$710$	0	0
$711$	12.4721	0.467742
$712$	15.5623i	0.583222i
$713$	20.0000i	0.749006i
$714$	18.3607	0.687131
$715$	0	0
$716$	3.79837	0.141952
$717$	7.05573i	0.263501i
$718$	17.4164i	0.649975i
$719$	−12.1115	−0.451681	−0.225841	−	0.974164i	$-0.572513\pi$
$719$	−12.1115	−0.451681	−0.225841	+	0.974164i	$0.572513\pi$
$720$	0	0
$721$	6.93112	0.258128
$722$	− 2.27051i	− 0.0844996i
$723$	15.0902i	0.561209i
$724$	20.4721	0.760841
$725$	0	0
$726$	−23.2705	−0.863650
$727$	12.6738i	0.470044i	0.971990	+	0.235022i	$0.0755162\pi$
$727$	12.6738i	0.470044i	−0.971990	+	0.235022i	$0.924484\pi$
$728$	− 8.05573i	− 0.298565i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	59.4164	2.19760
$732$	− 11.7082i	− 0.432748i
$733$	17.9230i	0.662001i	0.943631	+	0.331000i	$0.107386\pi$
$733$	17.9230i	0.662001i	−0.943631	+	0.331000i	$0.892614\pi$
$734$	15.4164	0.569030
$735$	0	0
$736$	3.09017	0.113905
$737$	18.9443i	0.697821i
$738$	− 0.381966i	− 0.0140604i
$739$	−23.3262	−0.858070	−0.429035	−	0.903288i	$-0.641146\pi$
$739$	−23.3262	−0.858070	−0.429035	+	0.903288i	$0.641146\pi$
$740$	0	0
$741$	13.8328	0.508161
$742$	0.909830i	0.0334009i
$743$	− 38.1459i	− 1.39944i	−0.714419	−	0.699719i	$-0.753309\pi$
$743$	− 38.1459i	− 1.39944i	0.714419	−	0.699719i	$-0.246691\pi$
$744$	−6.47214	−0.237280
$745$	0	0
$746$	19.3262	0.707584
$747$	− 2.00000i	− 0.0731762i
$748$	− 45.1246i	− 1.64992i
$749$	4.76393	0.174070
$750$	0	0
$751$	−4.06888	−0.148476	−0.0742378	−	0.997241i	$-0.523652\pi$
$751$	−4.06888	−0.148476	−0.0742378	+	0.997241i	$0.523652\pi$
$752$	− 8.61803i	− 0.314267i
$753$	− 24.0000i	− 0.874609i
$754$	19.3050	0.703045
$755$	0	0
$756$	2.38197	0.0866313
$757$	− 11.6738i	− 0.424290i	−0.977238	−	0.212145i	$-0.931955\pi$
$757$	− 11.6738i	− 0.424290i	0.977238	−	0.212145i	$-0.0680449\pi$
$758$	− 35.9230i	− 1.30478i
$759$	−18.0902	−0.656632
$760$	0	0
$761$	55.0344	1.99500	0.997498	−	0.0706879i	$-0.0225194\pi$
$761$	55.0344	1.99500	0.997498	+	0.0706879i	$0.0225194\pi$
$762$	− 5.52786i	− 0.200253i
$763$	− 4.06888i	− 0.147303i
$764$	4.00000	0.144715
$765$	0	0
$766$	−27.0344	−0.976794
$767$	36.7082i	1.32546i
$768$	1.00000i	0.0360844i
$769$	−11.6738	−0.420967	−0.210483	−	0.977597i	$-0.567504\pi$
$769$	−11.6738	−0.420967	−0.210483	+	0.977597i	$0.567504\pi$
$770$	0	0
$771$	17.5279	0.631251
$772$	− 9.70820i	− 0.349406i
$773$	− 26.9443i	− 0.969118i	−0.874759	−	0.484559i	$-0.838980\pi$
$773$	− 26.9443i	− 0.969118i	0.874759	−	0.484559i	$-0.161020\pi$
$774$	7.70820	0.277066
$775$	0	0
$776$	14.1803	0.509045
$777$	3.85410i	0.138265i
$778$	34.5410i	1.23836i
$779$	−1.56231	−0.0559754
$780$	0	0
$781$	26.1803	0.936806
$782$	− 23.8197i	− 0.851789i
$783$	5.70820i	0.203995i
$784$	1.32624	0.0473656
$785$	0	0
$786$	20.3262	0.725013
$787$	10.5836i	0.377264i	0.982048	+	0.188632i	$0.0604054\pi$
$787$	10.5836i	0.377264i	−0.982048	+	0.188632i	$0.939595\pi$
$788$	− 14.9443i	− 0.532368i
$789$	−25.3262	−0.901638
$790$	0	0
$791$	8.83282	0.314059
$792$	− 5.85410i	− 0.208016i
$793$	39.5967i	1.40612i
$794$	23.3262	0.827817
$795$	0	0
$796$	−4.76393	−0.168853
$797$	33.8673i	1.19964i	0.800135	+	0.599820i	$0.204761\pi$
$797$	33.8673i	1.19964i	−0.800135	+	0.599820i	$0.795239\pi$
$798$	− 9.74265i	− 0.344886i
$799$	−66.4296	−2.35011
$800$	0	0
$801$	15.5623	0.549867
$802$	− 1.32624i	− 0.0468311i
$803$	46.8328i	1.65269i
$804$	−3.23607	−0.114127
$805$	0	0
$806$	21.8885	0.770991
$807$	9.41641i	0.331473i
$808$	13.7082i	0.482253i
$809$	−46.3262	−1.62874	−0.814372	−	0.580343i	$-0.802918\pi$
$809$	−46.3262	−1.62874	−0.814372	+	0.580343i	$0.802918\pi$
$810$	0	0
$811$	−3.27051	−0.114843	−0.0574216	−	0.998350i	$-0.518288\pi$
$811$	−3.27051	−0.114843	−0.0574216	+	0.998350i	$0.518288\pi$
$812$	− 13.5967i	− 0.477152i
$813$	− 9.41641i	− 0.330248i
$814$	9.47214	0.331998
$815$	0	0
$816$	7.70820	0.269841
$817$	− 31.5279i	− 1.10302i
$818$	4.85410i	0.169720i
$819$	−8.05573	−0.281490
$820$	0	0
$821$	−29.8885	−1.04312	−0.521559	−	0.853215i	$-0.674649\pi$
$821$	−29.8885	−1.04312	−0.521559	+	0.853215i	$0.674649\pi$
$822$	4.94427i	0.172451i
$823$	− 34.9230i	− 1.21734i	−0.793424	−	0.608669i	$-0.791704\pi$
$823$	− 34.9230i	− 1.21734i	0.793424	−	0.608669i	$-0.208296\pi$
$824$	2.90983	0.101369
$825$	0	0
$826$	25.8541	0.899579
$827$	− 5.81966i	− 0.202369i	−0.994868	−	0.101185i	$-0.967737\pi$
$827$	− 5.81966i	− 0.202369i	0.994868	−	0.101185i	$-0.0322633\pi$
$828$	− 3.09017i	− 0.107391i
$829$	−5.63932	−0.195862	−0.0979308	−	0.995193i	$-0.531222\pi$
$829$	−5.63932	−0.195862	−0.0979308	+	0.995193i	$0.531222\pi$
$830$	0	0
$831$	27.4508	0.952259
$832$	− 3.38197i	− 0.117249i
$833$	− 10.2229i	− 0.354203i
$834$	−2.61803	−0.0906551
$835$	0	0
$836$	−23.9443	−0.828130
$837$	6.47214i	0.223710i
$838$	− 17.8885i	− 0.617949i
$839$	−23.3475	−0.806046	−0.403023	−	0.915190i	$-0.632041\pi$
$839$	−23.3475	−0.806046	−0.403023	+	0.915190i	$0.632041\pi$
$840$	0	0
$841$	3.58359	0.123572
$842$	− 25.1246i	− 0.865851i
$843$	24.5066i	0.844051i
$844$	22.5623	0.776627
$845$	0	0
$846$	−8.61803	−0.296294
$847$	− 55.4296i	− 1.90458i
$848$	0.381966i	0.0131168i
$849$	−8.18034	−0.280749
$850$	0	0
$851$	5.00000	0.171398
$852$	4.47214i	0.153213i
$853$	48.2705i	1.65275i	0.563120	+	0.826375i	$0.309601\pi$
$853$	48.2705i	1.65275i	−0.563120	+	0.826375i	$0.690399\pi$
$854$	27.8885	0.954326
$855$	0	0
$856$	2.00000	0.0683586
$857$	− 3.70820i	− 0.126670i	−0.997992	−	0.0633349i	$-0.979826\pi$
$857$	− 3.70820i	− 0.126670i	0.997992	−	0.0633349i	$-0.0201736\pi$
$858$	19.7984i	0.675905i
$859$	1.03444	0.0352947	0.0176474	−	0.999844i	$-0.494382\pi$
$859$	1.03444	0.0352947	0.0176474	+	0.999844i	$0.494382\pi$
$860$	0	0
$861$	0.909830	0.0310069
$862$	17.2361i	0.587063i
$863$	11.0344i	0.375617i	0.982206	+	0.187808i	$0.0601384\pi$
$863$	11.0344i	0.375617i	−0.982206	+	0.187808i	$0.939862\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−38.3607	−1.30355
$867$	− 42.4164i	− 1.44054i
$868$	− 15.4164i	− 0.523267i
$869$	73.0132	2.47680
$870$	0	0
$871$	10.9443	0.370833
$872$	− 1.70820i	− 0.0578471i
$873$	− 14.1803i	− 0.479932i
$874$	−12.6393	−0.427531
$875$	0	0
$876$	−8.00000	−0.270295
$877$	8.97871i	0.303190i	0.988443	+	0.151595i	$0.0484409\pi$
$877$	8.97871i	0.303190i	−0.988443	+	0.151595i	$0.951559\pi$
$878$	32.6525i	1.10197i
$879$	20.7984	0.701512
$880$	0	0
$881$	−16.1591	−0.544412	−0.272206	−	0.962239i	$-0.587753\pi$
$881$	−16.1591	−0.544412	−0.272206	+	0.962239i	$0.587753\pi$
$882$	− 1.32624i	− 0.0446568i
$883$	− 7.41641i	− 0.249582i	−0.992183	−	0.124791i	$-0.960174\pi$
$883$	− 7.41641i	− 0.249582i	0.992183	−	0.124791i	$-0.0398260\pi$
$884$	−26.0689	−0.876791
$885$	0	0
$886$	13.1246	0.440930
$887$	27.0344i	0.907728i	0.891071	+	0.453864i	$0.149955\pi$
$887$	27.0344i	0.907728i	−0.891071	+	0.453864i	$0.850045\pi$
$888$	1.61803i	0.0542977i
$889$	13.1672	0.441613
$890$	0	0
$891$	−5.85410	−0.196120
$892$	28.9443i	0.969126i
$893$	35.2492i	1.17957i
$894$	−9.41641	−0.314932
$895$	0	0
$896$	−2.38197	−0.0795759
$897$	10.4508i	0.348944i
$898$	− 16.5623i	− 0.552691i
$899$	36.9443	1.23216
$900$	0	0
$901$	2.94427	0.0980879
$902$	− 2.23607i	− 0.0744529i
$903$	18.3607i	0.611005i
$904$	3.70820	0.123333
$905$	0	0
$906$	0.291796	0.00969428
$907$	16.6525i	0.552936i	0.961023	+	0.276468i	$0.0891640\pi$
$907$	16.6525i	0.552936i	−0.961023	+	0.276468i	$0.910836\pi$
$908$	10.7639i	0.357214i
$909$	13.7082	0.454672
$910$	0	0
$911$	−29.0557	−0.962659	−0.481330	−	0.876540i	$-0.659846\pi$
$911$	−29.0557	−0.962659	−0.481330	+	0.876540i	$0.659846\pi$
$912$	− 4.09017i	− 0.135439i
$913$	− 11.7082i	− 0.387485i
$914$	33.8885	1.12093
$915$	0	0
$916$	−22.7639	−0.752141
$917$	48.4164i	1.59885i
$918$	− 7.70820i	− 0.254409i
$919$	−36.7639	−1.21273	−0.606365	−	0.795186i	$-0.707373\pi$
$919$	−36.7639	−1.21273	−0.606365	+	0.795186i	$0.707373\pi$
$920$	0	0
$921$	−6.18034	−0.203649
$922$	19.5967i	0.645385i
$923$	− 15.1246i	− 0.497833i
$924$	13.9443	0.458733
$925$	0	0
$926$	15.0557	0.494762
$927$	− 2.90983i	− 0.0955714i
$928$	− 5.70820i	− 0.187381i
$929$	−41.8115	−1.37179	−0.685896	−	0.727700i	$-0.740589\pi$
$929$	−41.8115	−1.37179	−0.685896	+	0.727700i	$0.740589\pi$
$930$	0	0
$931$	−5.42454	−0.177782
$932$	2.47214i	0.0809775i
$933$	8.94427i	0.292822i
$934$	21.2361	0.694865
$935$	0	0
$936$	−3.38197	−0.110543
$937$	− 6.65248i	− 0.217327i	−0.994079	−	0.108663i	$-0.965343\pi$
$937$	− 6.65248i	− 0.217327i	0.994079	−	0.108663i	$-0.0346571\pi$
$938$	− 7.70820i	− 0.251682i
$939$	−0.291796	−0.00952240
$940$	0	0
$941$	−18.1803	−0.592662	−0.296331	−	0.955085i	$-0.595763\pi$
$941$	−18.1803	−0.592662	−0.296331	+	0.955085i	$0.595763\pi$
$942$	− 6.94427i	− 0.226257i
$943$	− 1.18034i	− 0.0384372i
$944$	10.8541	0.353271
$945$	0	0
$946$	45.1246	1.46713
$947$	− 13.8885i	− 0.451317i	−0.974206	−	0.225659i	$-0.927547\pi$
$947$	− 13.8885i	− 0.451317i	0.974206	−	0.225659i	$-0.0724534\pi$
$948$	12.4721i	0.405076i
$949$	27.0557	0.878266
$950$	0	0
$951$	23.8541	0.773522
$952$	18.3607i	0.595073i
$953$	− 22.5410i	− 0.730175i	−0.930973	−	0.365088i	$-0.881039\pi$
$953$	− 22.5410i	− 0.730175i	0.930973	−	0.365088i	$-0.118961\pi$
$954$	0.381966	0.0123666
$955$	0	0
$956$	−7.05573	−0.228199
$957$	33.4164i	1.08020i
$958$	9.12461i	0.294803i
$959$	−11.7771	−0.380302
$960$	0	0
$961$	10.8885	0.351243
$962$	− 5.47214i	− 0.176429i
$963$	− 2.00000i	− 0.0644491i
$964$	−15.0902	−0.486022
$965$	0	0
$966$	7.36068	0.236826
$967$	− 58.6869i	− 1.88724i	−0.331025	−	0.943622i	$-0.607394\pi$
$967$	− 58.6869i	− 1.88724i	0.331025	−	0.943622i	$-0.392606\pi$
$968$	− 23.2705i	− 0.747943i
$969$	−31.5279	−1.01282
$970$	0	0
$971$	−6.72949	−0.215960	−0.107980	−	0.994153i	$-0.534438\pi$
$971$	−6.72949	−0.215960	−0.107980	+	0.994153i	$0.534438\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 6.23607i	− 0.199919i
$974$	−12.7984	−0.410086
$975$	0	0
$976$	11.7082	0.374770
$977$	− 9.30495i	− 0.297692i	−0.988860	−	0.148846i	$-0.952444\pi$
$977$	− 9.30495i	− 0.297692i	0.988860	−	0.148846i	$-0.0475558\pi$
$978$	24.0000i	0.767435i
$979$	91.1033	2.91167
$980$	0	0
$981$	−1.70820	−0.0545388
$982$	11.0344i	0.352123i
$983$	− 22.2148i	− 0.708541i	−0.935143	−	0.354271i	$-0.884729\pi$
$983$	− 22.2148i	− 0.708541i	0.935143	−	0.354271i	$-0.115271\pi$
$984$	0.381966	0.0121766
$985$	0	0
$986$	−44.0000	−1.40125
$987$	− 20.5279i	− 0.653409i
$988$	13.8328i	0.440080i
$989$	23.8197	0.757421
$990$	0	0
$991$	16.1115	0.511797	0.255899	−	0.966704i	$-0.417629\pi$
$991$	16.1115	0.511797	0.255899	+	0.966704i	$0.417629\pi$
$992$	− 6.47214i	− 0.205491i
$993$	6.47214i	0.205387i
$994$	−10.6525	−0.337876
$995$	0	0
$996$	2.00000	0.0633724
$997$	− 4.09017i	− 0.129537i	−0.997900	−	0.0647685i	$-0.979369\pi$
$997$	− 4.09017i	− 0.129537i	0.997900	−	0.0647685i	$-0.0206309\pi$
$998$	5.38197i	0.170363i
$999$	1.61803	0.0511923

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	750.2.c.a.499.4		4
3.2	odd	2		2250.2.c.g.1999.2		4
4.3	odd	2		6000.2.f.k.1249.1		4
5.2	odd	4		750.2.a.d.1.1	✓	2
5.3	odd	4		750.2.a.e.1.2	yes	2
5.4	even	2	inner	750.2.c.a.499.1		4
15.2	even	4		2250.2.a.p.1.1		2
15.8	even	4		2250.2.a.a.1.2		2
15.14	odd	2		2250.2.c.g.1999.3		4
20.3	even	4		6000.2.a.bb.1.1		2
20.7	even	4		6000.2.a.a.1.2		2
20.19	odd	2		6000.2.f.k.1249.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
750.2.a.d.1.1	✓	2	5.2	odd	4
750.2.a.e.1.2	yes	2	5.3	odd	4
750.2.c.a.499.1		4	5.4	even	2	inner
750.2.c.a.499.4		4	1.1	even	1	trivial
2250.2.a.a.1.2		2	15.8	even	4
2250.2.a.p.1.1		2	15.2	even	4
2250.2.c.g.1999.2		4	3.2	odd	2
2250.2.c.g.1999.3		4	15.14	odd	2
6000.2.a.a.1.2		2	20.7	even	4
6000.2.a.bb.1.1		2	20.3	even	4
6000.2.f.k.1249.1		4	4.3	odd	2
6000.2.f.k.1249.4		4	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	750.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.38197i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -2.38197i q^{7} -1.00000i q^{8} -1.00000 q^{9} -5.85410 q^{11} -1.00000i q^{12} +3.38197i q^{13} +2.38197 q^{14} +1.00000 q^{16} -7.70820i q^{17} -1.00000i q^{18} -4.09017 q^{19} +2.38197 q^{21} -5.85410i q^{22} -3.09017i q^{23} +1.00000 q^{24} -3.38197 q^{26} -1.00000i q^{27} +2.38197i q^{28} -5.70820 q^{29} -6.47214 q^{31} +1.00000i q^{32} -5.85410i q^{33} +7.70820 q^{34} +1.00000 q^{36} +1.61803i q^{37} -4.09017i q^{38} -3.38197 q^{39} +0.381966 q^{41} +2.38197i q^{42} +7.70820i q^{43} +5.85410 q^{44} +3.09017 q^{46} -8.61803i q^{47} +1.00000i q^{48} +1.32624 q^{49} +7.70820 q^{51} -3.38197i q^{52} +0.381966i q^{53} +1.00000 q^{54} -2.38197 q^{56} -4.09017i q^{57} -5.70820i q^{58} +10.8541 q^{59} +11.7082 q^{61} -6.47214i q^{62} +2.38197i q^{63} -1.00000 q^{64} +5.85410 q^{66} -3.23607i q^{67} +7.70820i q^{68} +3.09017 q^{69} -4.47214 q^{71} +1.00000i q^{72} -8.00000i q^{73} -1.61803 q^{74} +4.09017 q^{76} +13.9443i q^{77} -3.38197i q^{78} -12.4721 q^{79} +1.00000 q^{81} +0.381966i q^{82} +2.00000i q^{83} -2.38197 q^{84} -7.70820 q^{86} -5.70820i q^{87} +5.85410i q^{88} -15.5623 q^{89} +8.05573 q^{91} +3.09017i q^{92} -6.47214i q^{93} +8.61803 q^{94} -1.00000 q^{96} +14.1803i q^{97} +1.32624i q^{98} +5.85410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9	\( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} - 10 q^{11} + 14 q^{14} + 4 q^{16} + 6 q^{19} + 14 q^{21} + 4 q^{24} - 18 q^{26} + 4 q^{29} - 8 q^{31} + 4 q^{34} + 4 q^{36} - 18 q^{39} + 6 q^{41} + 10 q^{44} - 10 q^{46} - 26 q^{49} + 4 q^{51} + 4 q^{54} - 14 q^{56} + 30 q^{59} + 20 q^{61} - 4 q^{64} + 10 q^{66} - 10 q^{69} - 2 q^{74} - 6 q^{76} - 32 q^{79} + 4 q^{81} - 14 q^{84} - 4 q^{86} - 22 q^{89} + 68 q^{91} + 30 q^{94} - 4 q^{96} + 10 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 - 10 * q^11 + 14 * q^14 + 4 * q^16 + 6 * q^19 + 14 * q^21 + 4 * q^24 - 18 * q^26 + 4 * q^29 - 8 * q^31 + 4 * q^34 + 4 * q^36 - 18 * q^39 + 6 * q^41 + 10 * q^44 - 10 * q^46 - 26 * q^49 + 4 * q^51 + 4 * q^54 - 14 * q^56 + 30 * q^59 + 20 * q^61 - 4 * q^64 + 10 * q^66 - 10 * q^69 - 2 * q^74 - 6 * q^76 - 32 * q^79 + 4 * q^81 - 14 * q^84 - 4 * q^86 - 22 * q^89 + 68 * q^91 + 30 * q^94 - 4 * q^96 + 10 * q^99

Introduction

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