LMFDB - Embedded newform 3750.2.c.b.1249.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3750,2,Mod(1249,3750)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3750, 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("3750.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3750 = 2 \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3750.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:	$29.9439007580$
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:	no (minimal twist has level 150)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.2
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	3750.1249
Dual form			3750.2.c.b.1249.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -0.381966i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -0.381966i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.38197 q^{11} +1.00000i q^{12} +2.47214i q^{13} -0.381966 q^{14} +1.00000 q^{16} +3.23607i q^{17} +1.00000i q^{18} -7.70820 q^{19} -0.381966 q^{21} -1.38197i q^{22} -4.47214i q^{23} +1.00000 q^{24} +2.47214 q^{26} +1.00000i q^{27} +0.381966i q^{28} +0.472136 q^{29} +4.38197 q^{31} -1.00000i q^{32} -1.38197i q^{33} +3.23607 q^{34} +1.00000 q^{36} -8.00000i q^{37} +7.70820i q^{38} +2.47214 q^{39} -7.70820 q^{41} +0.381966i q^{42} +5.70820i q^{43} -1.38197 q^{44} -4.47214 q^{46} +11.7082i q^{47} -1.00000i q^{48} +6.85410 q^{49} +3.23607 q^{51} -2.47214i q^{52} +9.09017i q^{53} +1.00000 q^{54} +0.381966 q^{56} +7.70820i q^{57} -0.472136i q^{58} +1.38197 q^{59} +7.23607 q^{61} -4.38197i q^{62} +0.381966i q^{63} -1.00000 q^{64} -1.38197 q^{66} +10.4721i q^{67} -3.23607i q^{68} -4.47214 q^{69} +14.4721 q^{71} -1.00000i q^{72} +12.4721i q^{73} -8.00000 q^{74} +7.70820 q^{76} -0.527864i q^{77} -2.47214i q^{78} +3.38197 q^{79} +1.00000 q^{81} +7.70820i q^{82} +8.85410i q^{83} +0.381966 q^{84} +5.70820 q^{86} -0.472136i q^{87} +1.38197i q^{88} -12.4721 q^{89} +0.944272 q^{91} +4.47214i q^{92} -4.38197i q^{93} +11.7082 q^{94} -1.00000 q^{96} +5.61803i q^{97} -6.85410i q^{98} -1.38197 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 6 q^{14} + 4 q^{16} - 4 q^{19} - 6 q^{21} + 4 q^{24} - 8 q^{26} - 16 q^{29} + 22 q^{31} + 4 q^{34} + 4 q^{36} - 8 q^{39} - 4 q^{41} - 10 q^{44} + 14 q^{49} + 4 q^{51} + 4 q^{54} + 6 q^{56} + 10 q^{59} + 20 q^{61} - 4 q^{64} - 10 q^{66} + 40 q^{71} - 32 q^{74} + 4 q^{76} + 18 q^{79} + 4 q^{81} + 6 q^{84} - 4 q^{86} - 32 q^{89} - 32 q^{91} + 20 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 - 6 * q^14 + 4 * q^16 - 4 * q^19 - 6 * q^21 + 4 * q^24 - 8 * q^26 - 16 * q^29 + 22 * q^31 + 4 * q^34 + 4 * q^36 - 8 * q^39 - 4 * q^41 - 10 * q^44 + 14 * q^49 + 4 * q^51 + 4 * q^54 + 6 * q^56 + 10 * q^59 + 20 * q^61 - 4 * q^64 - 10 * q^66 + 40 * q^71 - 32 * q^74 + 4 * q^76 + 18 * q^79 + 4 * q^81 + 6 * q^84 - 4 * q^86 - 32 * q^89 - 32 * q^91 + 20 * q^94 - 4 * q^96 - 10 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3750\mathbb{Z}\right)^\times$.

$n$	$2501$	$3127$
$\chi(n)$	$1$	$-1$

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
$7$	− 0.381966i	− 0.144370i	−0.997391	−	0.0721848i	$-0.977003\pi$
$7$	− 0.381966i	− 0.144370i	0.997391	−	0.0721848i	$-0.0229971\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.38197	0.416678	0.208339	−	0.978057i	$-0.433194\pi$
$11$	1.38197	0.416678	0.208339	+	0.978057i	$0.433194\pi$
$12$	1.00000i	0.288675i
$13$	2.47214i	0.685647i	0.939400	+	0.342824i	$0.111383\pi$
$13$	2.47214i	0.685647i	−0.939400	+	0.342824i	$0.888617\pi$
$14$	−0.381966	−0.102085
$15$	0	0
$16$	1.00000	0.250000
$17$	3.23607i	0.784862i	0.919781	+	0.392431i	$0.128366\pi$
$17$	3.23607i	0.784862i	−0.919781	+	0.392431i	$0.871634\pi$
$18$	1.00000i	0.235702i
$19$	−7.70820	−1.76838	−0.884192	−	0.467124i	$-0.845290\pi$
$19$	−7.70820	−1.76838	−0.884192	+	0.467124i	$0.845290\pi$
$20$	0	0
$21$	−0.381966	−0.0833518
$22$	− 1.38197i	− 0.294636i
$23$	− 4.47214i	− 0.932505i	−0.884652	−	0.466252i	$-0.845604\pi$
$23$	− 4.47214i	− 0.932505i	0.884652	−	0.466252i	$-0.154396\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	2.47214	0.484826
$27$	1.00000i	0.192450i
$28$	0.381966i	0.0721848i
$29$	0.472136	0.0876734	0.0438367	−	0.999039i	$-0.486042\pi$
$29$	0.472136	0.0876734	0.0438367	+	0.999039i	$0.486042\pi$
$30$	0	0
$31$	4.38197	0.787024	0.393512	−	0.919319i	$-0.371260\pi$
$31$	4.38197	0.787024	0.393512	+	0.919319i	$0.371260\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 1.38197i	− 0.240569i
$34$	3.23607	0.554981
$35$	0	0
$36$	1.00000	0.166667
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	7.70820i	1.25044i
$39$	2.47214	0.395859
$40$	0	0
$41$	−7.70820	−1.20382	−0.601910	−	0.798564i	$-0.705593\pi$
$41$	−7.70820	−1.20382	−0.601910	+	0.798564i	$0.705593\pi$
$42$	0.381966i	0.0589386i
$43$	5.70820i	0.870493i	0.900311	+	0.435246i	$0.143339\pi$
$43$	5.70820i	0.870493i	−0.900311	+	0.435246i	$0.856661\pi$
$44$	−1.38197	−0.208339
$45$	0	0
$46$	−4.47214	−0.659380
$47$	11.7082i	1.70782i	0.520423	+	0.853909i	$0.325774\pi$
$47$	11.7082i	1.70782i	−0.520423	+	0.853909i	$0.674226\pi$
$48$	− 1.00000i	− 0.144338i
$49$	6.85410	0.979157
$50$	0	0
$51$	3.23607	0.453140
$52$	− 2.47214i	− 0.342824i
$53$	9.09017i	1.24863i	0.781172	+	0.624315i	$0.214622\pi$
$53$	9.09017i	1.24863i	−0.781172	+	0.624315i	$0.785378\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0.381966	0.0510424
$57$	7.70820i	1.02098i
$58$	− 0.472136i	− 0.0619945i
$59$	1.38197	0.179917	0.0899583	−	0.995946i	$-0.471327\pi$
$59$	1.38197	0.179917	0.0899583	+	0.995946i	$0.471327\pi$
$60$	0	0
$61$	7.23607	0.926484	0.463242	−	0.886232i	$-0.346686\pi$
$61$	7.23607	0.926484	0.463242	+	0.886232i	$0.346686\pi$
$62$	− 4.38197i	− 0.556510i
$63$	0.381966i	0.0481232i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−1.38197	−0.170108
$67$	10.4721i	1.27938i	0.768635	+	0.639688i	$0.220936\pi$
$67$	10.4721i	1.27938i	−0.768635	+	0.639688i	$0.779064\pi$
$68$	− 3.23607i	− 0.392431i
$69$	−4.47214	−0.538382
$70$	0	0
$71$	14.4721	1.71753	0.858763	−	0.512373i	$-0.171233\pi$
$71$	14.4721	1.71753	0.858763	+	0.512373i	$0.171233\pi$
$72$	− 1.00000i	− 0.117851i
$73$	12.4721i	1.45975i	0.683579	+	0.729877i	$0.260422\pi$
$73$	12.4721i	1.45975i	−0.683579	+	0.729877i	$0.739578\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	7.70820	0.884192
$77$	− 0.527864i	− 0.0601557i
$78$	− 2.47214i	− 0.279914i
$79$	3.38197	0.380501	0.190250	−	0.981736i	$-0.439070\pi$
$79$	3.38197	0.380501	0.190250	+	0.981736i	$0.439070\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	7.70820i	0.851229i
$83$	8.85410i	0.971864i	0.873997	+	0.485932i	$0.161520\pi$
$83$	8.85410i	0.971864i	−0.873997	+	0.485932i	$0.838480\pi$
$84$	0.381966	0.0416759
$85$	0	0
$86$	5.70820	0.615531
$87$	− 0.472136i	− 0.0506183i
$88$	1.38197i	0.147318i
$89$	−12.4721	−1.32204	−0.661022	−	0.750367i	$-0.729877\pi$
$89$	−12.4721	−1.32204	−0.661022	+	0.750367i	$0.729877\pi$
$90$	0	0
$91$	0.944272	0.0989866
$92$	4.47214i	0.466252i
$93$	− 4.38197i	− 0.454389i
$94$	11.7082	1.20761
$95$	0	0
$96$	−1.00000	−0.102062
$97$	5.61803i	0.570425i	0.958464	+	0.285212i	$0.0920642\pi$
$97$	5.61803i	0.570425i	−0.958464	+	0.285212i	$0.907936\pi$
$98$	− 6.85410i	− 0.692369i
$99$	−1.38197	−0.138893
$100$	0	0
$101$	6.61803	0.658519	0.329259	−	0.944239i	$-0.393201\pi$
$101$	6.61803	0.658519	0.329259	+	0.944239i	$0.393201\pi$
$102$	− 3.23607i	− 0.320418i
$103$	− 1.32624i	− 0.130678i	−0.997863	−	0.0653391i	$-0.979187\pi$
$103$	− 1.32624i	− 0.130678i	0.997863	−	0.0653391i	$-0.0208129\pi$
$104$	−2.47214	−0.242413
$105$	0	0
$106$	9.09017	0.882915
$107$	9.38197i	0.906989i	0.891259	+	0.453494i	$0.149823\pi$
$107$	9.38197i	0.906989i	−0.891259	+	0.453494i	$0.850177\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−12.7639	−1.22256	−0.611281	−	0.791413i	$-0.709346\pi$
$109$	−12.7639	−1.22256	−0.611281	+	0.791413i	$0.709346\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	− 0.381966i	− 0.0360924i
$113$	− 14.7639i	− 1.38887i	−0.719554	−	0.694437i	$-0.755654\pi$
$113$	− 14.7639i	− 1.38887i	0.719554	−	0.694437i	$-0.244346\pi$
$114$	7.70820	0.721939
$115$	0	0
$116$	−0.472136	−0.0438367
$117$	− 2.47214i	− 0.228549i
$118$	− 1.38197i	− 0.127220i
$119$	1.23607	0.113310
$120$	0	0
$121$	−9.09017	−0.826379
$122$	− 7.23607i	− 0.655123i
$123$	7.70820i	0.695025i
$124$	−4.38197	−0.393512
$125$	0	0
$126$	0.381966	0.0340282
$127$	− 11.3820i	− 1.00999i	−0.863123	−	0.504993i	$-0.831495\pi$
$127$	− 11.3820i	− 1.00999i	0.863123	−	0.504993i	$-0.168505\pi$
$128$	1.00000i	0.0883883i
$129$	5.70820	0.502579
$130$	0	0
$131$	17.8885	1.56293	0.781465	−	0.623949i	$-0.214473\pi$
$131$	17.8885	1.56293	0.781465	+	0.623949i	$0.214473\pi$
$132$	1.38197i	0.120285i
$133$	2.94427i	0.255301i
$134$	10.4721	0.904655
$135$	0	0
$136$	−3.23607	−0.277491
$137$	− 10.1803i	− 0.869765i	−0.900487	−	0.434883i	$-0.856790\pi$
$137$	− 10.1803i	− 0.869765i	0.900487	−	0.434883i	$-0.143210\pi$
$138$	4.47214i	0.380693i
$139$	−1.52786	−0.129592	−0.0647959	−	0.997899i	$-0.520640\pi$
$139$	−1.52786	−0.129592	−0.0647959	+	0.997899i	$0.520640\pi$
$140$	0	0
$141$	11.7082	0.986009
$142$	− 14.4721i	− 1.21447i
$143$	3.41641i	0.285694i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	12.4721	1.03220
$147$	− 6.85410i	− 0.565317i
$148$	8.00000i	0.657596i
$149$	−10.9098	−0.893768	−0.446884	−	0.894592i	$-0.647466\pi$
$149$	−10.9098	−0.893768	−0.446884	+	0.894592i	$0.647466\pi$
$150$	0	0
$151$	2.67376	0.217588	0.108794	−	0.994064i	$-0.465301\pi$
$151$	2.67376	0.217588	0.108794	+	0.994064i	$0.465301\pi$
$152$	− 7.70820i	− 0.625218i
$153$	− 3.23607i	− 0.261621i
$154$	−0.527864	−0.0425365
$155$	0	0
$156$	−2.47214	−0.197929
$157$	22.6525i	1.80786i	0.427676	+	0.903932i	$0.359332\pi$
$157$	22.6525i	1.80786i	−0.427676	+	0.903932i	$0.640668\pi$
$158$	− 3.38197i	− 0.269055i
$159$	9.09017	0.720897
$160$	0	0
$161$	−1.70820	−0.134625
$162$	− 1.00000i	− 0.0785674i
$163$	8.47214i	0.663589i	0.943352	+	0.331794i	$0.107654\pi$
$163$	8.47214i	0.663589i	−0.943352	+	0.331794i	$0.892346\pi$
$164$	7.70820	0.601910
$165$	0	0
$166$	8.85410	0.687212
$167$	1.70820i	0.132185i	0.997814	+	0.0660924i	$0.0210532\pi$
$167$	1.70820i	0.132185i	−0.997814	+	0.0660924i	$0.978947\pi$
$168$	− 0.381966i	− 0.0294693i
$169$	6.88854	0.529888
$170$	0	0
$171$	7.70820	0.589461
$172$	− 5.70820i	− 0.435246i
$173$	− 6.09017i	− 0.463027i	−0.972832	−	0.231514i	$-0.925632\pi$
$173$	− 6.09017i	− 0.463027i	0.972832	−	0.231514i	$-0.0743678\pi$
$174$	−0.472136	−0.0357925
$175$	0	0
$176$	1.38197	0.104170
$177$	− 1.38197i	− 0.103875i
$178$	12.4721i	0.934826i
$179$	−3.14590	−0.235135	−0.117568	−	0.993065i	$-0.537510\pi$
$179$	−3.14590	−0.235135	−0.117568	+	0.993065i	$0.537510\pi$
$180$	0	0
$181$	24.6525	1.83240	0.916202	−	0.400717i	$-0.131239\pi$
$181$	24.6525	1.83240	0.916202	+	0.400717i	$0.131239\pi$
$182$	− 0.944272i	− 0.0699941i
$183$	− 7.23607i	− 0.534906i
$184$	4.47214	0.329690
$185$	0	0
$186$	−4.38197	−0.321301
$187$	4.47214i	0.327035i
$188$	− 11.7082i	− 0.853909i
$189$	0.381966	0.0277839
$190$	0	0
$191$	17.7082	1.28132	0.640660	−	0.767824i	$-0.278661\pi$
$191$	17.7082	1.28132	0.640660	+	0.767824i	$0.278661\pi$
$192$	1.00000i	0.0721688i
$193$	11.1459i	0.802299i	0.916013	+	0.401150i	$0.131389\pi$
$193$	11.1459i	0.802299i	−0.916013	+	0.401150i	$0.868611\pi$
$194$	5.61803	0.403351
$195$	0	0
$196$	−6.85410	−0.489579
$197$	5.90983i	0.421058i	0.977588	+	0.210529i	$0.0675186\pi$
$197$	5.90983i	0.421058i	−0.977588	+	0.210529i	$0.932481\pi$
$198$	1.38197i	0.0982120i
$199$	18.5066	1.31190	0.655948	−	0.754806i	$-0.272269\pi$
$199$	18.5066	1.31190	0.655948	+	0.754806i	$0.272269\pi$
$200$	0	0
$201$	10.4721	0.738648
$202$	− 6.61803i	− 0.465643i
$203$	− 0.180340i	− 0.0126574i
$204$	−3.23607	−0.226570
$205$	0	0
$206$	−1.32624	−0.0924034
$207$	4.47214i	0.310835i
$208$	2.47214i	0.171412i
$209$	−10.6525	−0.736847
$210$	0	0
$211$	23.4164	1.61205	0.806026	−	0.591880i	$-0.201614\pi$
$211$	23.4164	1.61205	0.806026	+	0.591880i	$0.201614\pi$
$212$	− 9.09017i	− 0.624315i
$213$	− 14.4721i	− 0.991614i
$214$	9.38197	0.641338
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	− 1.67376i	− 0.113622i
$218$	12.7639i	0.864483i
$219$	12.4721	0.842789
$220$	0	0
$221$	−8.00000	−0.538138
$222$	8.00000i	0.536925i
$223$	3.09017i	0.206933i	0.994633	+	0.103467i	$0.0329935\pi$
$223$	3.09017i	0.206933i	−0.994633	+	0.103467i	$0.967007\pi$
$224$	−0.381966	−0.0255212
$225$	0	0
$226$	−14.7639	−0.982082
$227$	− 11.2705i	− 0.748050i	−0.927419	−	0.374025i	$-0.877977\pi$
$227$	− 11.2705i	− 0.748050i	0.927419	−	0.374025i	$-0.122023\pi$
$228$	− 7.70820i	− 0.510488i
$229$	14.4721	0.956346	0.478173	−	0.878266i	$-0.341299\pi$
$229$	14.4721	0.956346	0.478173	+	0.878266i	$0.341299\pi$
$230$	0	0
$231$	−0.527864	−0.0347309
$232$	0.472136i	0.0309972i
$233$	− 12.6525i	− 0.828891i	−0.910074	−	0.414446i	$-0.863976\pi$
$233$	− 12.6525i	− 0.828891i	0.910074	−	0.414446i	$-0.136024\pi$
$234$	−2.47214	−0.161609
$235$	0	0
$236$	−1.38197	−0.0899583
$237$	− 3.38197i	− 0.219682i
$238$	− 1.23607i	− 0.0801224i
$239$	−25.7082	−1.66293	−0.831463	−	0.555581i	$-0.812496\pi$
$239$	−25.7082	−1.66293	−0.831463	+	0.555581i	$0.812496\pi$
$240$	0	0
$241$	−10.5623	−0.680378	−0.340189	−	0.940357i	$-0.610491\pi$
$241$	−10.5623	−0.680378	−0.340189	+	0.940357i	$0.610491\pi$
$242$	9.09017i	0.584338i
$243$	− 1.00000i	− 0.0641500i
$244$	−7.23607	−0.463242
$245$	0	0
$246$	7.70820	0.491457
$247$	− 19.0557i	− 1.21249i
$248$	4.38197i	0.278255i
$249$	8.85410	0.561106
$250$	0	0
$251$	−6.56231	−0.414209	−0.207105	−	0.978319i	$-0.566404\pi$
$251$	−6.56231	−0.414209	−0.207105	+	0.978319i	$0.566404\pi$
$252$	− 0.381966i	− 0.0240616i
$253$	− 6.18034i	− 0.388555i
$254$	−11.3820	−0.714168
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.0000i	1.37232i	0.727450	+	0.686161i	$0.240706\pi$
$257$	22.0000i	1.37232i	−0.727450	+	0.686161i	$0.759294\pi$
$258$	− 5.70820i	− 0.355377i
$259$	−3.05573	−0.189874
$260$	0	0
$261$	−0.472136	−0.0292245
$262$	− 17.8885i	− 1.10516i
$263$	1.70820i	0.105332i	0.998612	+	0.0526662i	$0.0167719\pi$
$263$	1.70820i	0.105332i	−0.998612	+	0.0526662i	$0.983228\pi$
$264$	1.38197	0.0850541
$265$	0	0
$266$	2.94427	0.180525
$267$	12.4721i	0.763282i
$268$	− 10.4721i	− 0.639688i
$269$	−2.09017	−0.127440	−0.0637200	−	0.997968i	$-0.520296\pi$
$269$	−2.09017	−0.127440	−0.0637200	+	0.997968i	$0.520296\pi$
$270$	0	0
$271$	11.0344	0.670295	0.335147	−	0.942166i	$-0.391214\pi$
$271$	11.0344	0.670295	0.335147	+	0.942166i	$0.391214\pi$
$272$	3.23607i	0.196215i
$273$	− 0.944272i	− 0.0571499i
$274$	−10.1803	−0.615017
$275$	0	0
$276$	4.47214	0.269191
$277$	0.944272i	0.0567358i	0.999598	+	0.0283679i	$0.00903099\pi$
$277$	0.944272i	0.0567358i	−0.999598	+	0.0283679i	$0.990969\pi$
$278$	1.52786i	0.0916352i
$279$	−4.38197	−0.262341
$280$	0	0
$281$	−29.8885	−1.78300	−0.891501	−	0.453020i	$-0.850347\pi$
$281$	−29.8885	−1.78300	−0.891501	+	0.453020i	$0.850347\pi$
$282$	− 11.7082i	− 0.697213i
$283$	31.4164i	1.86751i	0.357911	+	0.933756i	$0.383489\pi$
$283$	31.4164i	1.86751i	−0.357911	+	0.933756i	$0.616511\pi$
$284$	−14.4721	−0.858763
$285$	0	0
$286$	3.41641	0.202016
$287$	2.94427i	0.173795i
$288$	1.00000i	0.0589256i
$289$	6.52786	0.383992
$290$	0	0
$291$	5.61803	0.329335
$292$	− 12.4721i	− 0.729877i
$293$	− 10.9098i	− 0.637359i	−0.947863	−	0.318680i	$-0.896761\pi$
$293$	− 10.9098i	− 0.637359i	0.947863	−	0.318680i	$-0.103239\pi$
$294$	−6.85410	−0.399739
$295$	0	0
$296$	8.00000	0.464991
$297$	1.38197i	0.0801898i
$298$	10.9098i	0.631989i
$299$	11.0557	0.639369
$300$	0	0
$301$	2.18034	0.125673
$302$	− 2.67376i	− 0.153858i
$303$	− 6.61803i	− 0.380196i
$304$	−7.70820	−0.442096
$305$	0	0
$306$	−3.23607	−0.184994
$307$	− 10.0000i	− 0.570730i	−0.958419	−	0.285365i	$-0.907885\pi$
$307$	− 10.0000i	− 0.570730i	0.958419	−	0.285365i	$-0.0921148\pi$
$308$	0.527864i	0.0300778i
$309$	−1.32624	−0.0754470
$310$	0	0
$311$	−18.9443	−1.07423	−0.537116	−	0.843509i	$-0.680486\pi$
$311$	−18.9443	−1.07423	−0.537116	+	0.843509i	$0.680486\pi$
$312$	2.47214i	0.139957i
$313$	17.2705i	0.976187i	0.872791	+	0.488093i	$0.162307\pi$
$313$	17.2705i	0.976187i	−0.872791	+	0.488093i	$0.837693\pi$
$314$	22.6525	1.27835
$315$	0	0
$316$	−3.38197	−0.190250
$317$	0.437694i	0.0245833i	0.999924	+	0.0122917i	$0.00391266\pi$
$317$	0.437694i	0.0245833i	−0.999924	+	0.0122917i	$0.996087\pi$
$318$	− 9.09017i	− 0.509751i
$319$	0.652476	0.0365316
$320$	0	0
$321$	9.38197	0.523650
$322$	1.70820i	0.0951945i
$323$	− 24.9443i	− 1.38794i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	8.47214	0.469228
$327$	12.7639i	0.705847i
$328$	− 7.70820i	− 0.425614i
$329$	4.47214	0.246557
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	− 8.85410i	− 0.485932i
$333$	8.00000i	0.438397i
$334$	1.70820	0.0934688
$335$	0	0
$336$	−0.381966	−0.0208380
$337$	− 12.0902i	− 0.658594i	−0.944226	−	0.329297i	$-0.893188\pi$
$337$	− 12.0902i	− 0.658594i	0.944226	−	0.329297i	$-0.106812\pi$
$338$	− 6.88854i	− 0.374687i
$339$	−14.7639	−0.801867
$340$	0	0
$341$	6.05573	0.327936
$342$	− 7.70820i	− 0.416812i
$343$	− 5.29180i	− 0.285730i
$344$	−5.70820	−0.307766
$345$	0	0
$346$	−6.09017	−0.327410
$347$	− 0.437694i	− 0.0234967i	−0.999931	−	0.0117483i	$-0.996260\pi$
$347$	− 0.437694i	− 0.0234967i	0.999931	−	0.0117483i	$-0.00373969\pi$
$348$	0.472136i	0.0253091i
$349$	−7.88854	−0.422264	−0.211132	−	0.977458i	$-0.567715\pi$
$349$	−7.88854	−0.422264	−0.211132	+	0.977458i	$0.567715\pi$
$350$	0	0
$351$	−2.47214	−0.131953
$352$	− 1.38197i	− 0.0736590i
$353$	− 8.76393i	− 0.466457i	−0.972422	−	0.233229i	$-0.925071\pi$
$353$	− 8.76393i	− 0.466457i	0.972422	−	0.233229i	$-0.0749290\pi$
$354$	−1.38197	−0.0734507
$355$	0	0
$356$	12.4721	0.661022
$357$	− 1.23607i	− 0.0654197i
$358$	3.14590i	0.166266i
$359$	0.180340	0.00951798	0.00475899	−	0.999989i	$-0.498485\pi$
$359$	0.180340	0.00951798	0.00475899	+	0.999989i	$0.498485\pi$
$360$	0	0
$361$	40.4164	2.12718
$362$	− 24.6525i	− 1.29571i
$363$	9.09017i	0.477110i
$364$	−0.944272	−0.0494933
$365$	0	0
$366$	−7.23607	−0.378235
$367$	28.3820i	1.48153i	0.671766	+	0.740763i	$0.265536\pi$
$367$	28.3820i	1.48153i	−0.671766	+	0.740763i	$0.734464\pi$
$368$	− 4.47214i	− 0.233126i
$369$	7.70820	0.401273
$370$	0	0
$371$	3.47214	0.180264
$372$	4.38197i	0.227194i
$373$	− 20.4721i	− 1.06001i	−0.847995	−	0.530004i	$-0.822191\pi$
$373$	− 20.4721i	− 1.06001i	0.847995	−	0.530004i	$-0.177809\pi$
$374$	4.47214	0.231249
$375$	0	0
$376$	−11.7082	−0.603805
$377$	1.16718i	0.0601130i
$378$	− 0.381966i	− 0.0196462i
$379$	−2.18034	−0.111997	−0.0559983	−	0.998431i	$-0.517834\pi$
$379$	−2.18034	−0.111997	−0.0559983	+	0.998431i	$0.517834\pi$
$380$	0	0
$381$	−11.3820	−0.583116
$382$	− 17.7082i	− 0.906031i
$383$	− 20.0000i	− 1.02195i	−0.859595	−	0.510976i	$-0.829284\pi$
$383$	− 20.0000i	− 1.02195i	0.859595	−	0.510976i	$-0.170716\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	11.1459	0.567311
$387$	− 5.70820i	− 0.290164i
$388$	− 5.61803i	− 0.285212i
$389$	14.6180	0.741164	0.370582	−	0.928800i	$-0.379158\pi$
$389$	14.6180	0.741164	0.370582	+	0.928800i	$0.379158\pi$
$390$	0	0
$391$	14.4721	0.731887
$392$	6.85410i	0.346184i
$393$	− 17.8885i	− 0.902358i
$394$	5.90983	0.297733
$395$	0	0
$396$	1.38197	0.0694464
$397$	35.2361i	1.76845i	0.467063	+	0.884224i	$0.345312\pi$
$397$	35.2361i	1.76845i	−0.467063	+	0.884224i	$0.654688\pi$
$398$	− 18.5066i	− 0.927651i
$399$	2.94427	0.147398
$400$	0	0
$401$	12.2918	0.613823	0.306912	−	0.951738i	$-0.400704\pi$
$401$	12.2918	0.613823	0.306912	+	0.951738i	$0.400704\pi$
$402$	− 10.4721i	− 0.522303i
$403$	10.8328i	0.539621i
$404$	−6.61803	−0.329259
$405$	0	0
$406$	−0.180340	−0.00895012
$407$	− 11.0557i	− 0.548012i
$408$	3.23607i	0.160209i
$409$	18.7984	0.929520	0.464760	−	0.885437i	$-0.346141\pi$
$409$	18.7984	0.929520	0.464760	+	0.885437i	$0.346141\pi$
$410$	0	0
$411$	−10.1803	−0.502159
$412$	1.32624i	0.0653391i
$413$	− 0.527864i	− 0.0259745i
$414$	4.47214	0.219793
$415$	0	0
$416$	2.47214	0.121206
$417$	1.52786i	0.0748198i
$418$	10.6525i	0.521030i
$419$	−8.09017	−0.395231	−0.197615	−	0.980280i	$-0.563320\pi$
$419$	−8.09017	−0.395231	−0.197615	+	0.980280i	$0.563320\pi$
$420$	0	0
$421$	−27.2361	−1.32740	−0.663702	−	0.747997i	$-0.731016\pi$
$421$	−27.2361	−1.32740	−0.663702	+	0.747997i	$0.731016\pi$
$422$	− 23.4164i	− 1.13989i
$423$	− 11.7082i	− 0.569272i
$424$	−9.09017	−0.441458
$425$	0	0
$426$	−14.4721	−0.701177
$427$	− 2.76393i	− 0.133756i
$428$	− 9.38197i	− 0.453494i
$429$	3.41641	0.164946
$430$	0	0
$431$	−36.8328	−1.77417	−0.887087	−	0.461602i	$-0.847275\pi$
$431$	−36.8328	−1.77417	−0.887087	+	0.461602i	$0.847275\pi$
$432$	1.00000i	0.0481125i
$433$	− 22.5066i	− 1.08160i	−0.841152	−	0.540799i	$-0.818122\pi$
$433$	− 22.5066i	− 1.08160i	0.841152	−	0.540799i	$-0.181878\pi$
$434$	−1.67376	−0.0803432
$435$	0	0
$436$	12.7639	0.611281
$437$	34.4721i	1.64903i
$438$	− 12.4721i	− 0.595942i
$439$	−14.5066	−0.692361	−0.346181	−	0.938168i	$-0.612522\pi$
$439$	−14.5066	−0.692361	−0.346181	+	0.938168i	$0.612522\pi$
$440$	0	0
$441$	−6.85410	−0.326386
$442$	8.00000i	0.380521i
$443$	14.3820i	0.683308i	0.939826	+	0.341654i	$0.110987\pi$
$443$	14.3820i	0.683308i	−0.939826	+	0.341654i	$0.889013\pi$
$444$	8.00000	0.379663
$445$	0	0
$446$	3.09017	0.146324
$447$	10.9098i	0.516017i
$448$	0.381966i	0.0180462i
$449$	14.2918	0.674472	0.337236	−	0.941420i	$-0.390508\pi$
$449$	14.2918	0.674472	0.337236	+	0.941420i	$0.390508\pi$
$450$	0	0
$451$	−10.6525	−0.501605
$452$	14.7639i	0.694437i
$453$	− 2.67376i	− 0.125624i
$454$	−11.2705	−0.528951
$455$	0	0
$456$	−7.70820	−0.360970
$457$	− 7.09017i	− 0.331664i	−0.986154	−	0.165832i	$-0.946969\pi$
$457$	− 7.09017i	− 0.331664i	0.986154	−	0.165832i	$-0.0530310\pi$
$458$	− 14.4721i	− 0.676239i
$459$	−3.23607	−0.151047
$460$	0	0
$461$	−12.5623	−0.585085	−0.292542	−	0.956253i	$-0.594501\pi$
$461$	−12.5623	−0.585085	−0.292542	+	0.956253i	$0.594501\pi$
$462$	0.527864i	0.0245585i
$463$	30.8328i	1.43292i	0.697627	+	0.716461i	$0.254239\pi$
$463$	30.8328i	1.43292i	−0.697627	+	0.716461i	$0.745761\pi$
$464$	0.472136	0.0219184
$465$	0	0
$466$	−12.6525	−0.586115
$467$	− 26.3262i	− 1.21823i	−0.793081	−	0.609117i	$-0.791524\pi$
$467$	− 26.3262i	− 1.21823i	0.793081	−	0.609117i	$-0.208476\pi$
$468$	2.47214i	0.114275i
$469$	4.00000	0.184703
$470$	0	0
$471$	22.6525	1.04377
$472$	1.38197i	0.0636101i
$473$	7.88854i	0.362716i
$474$	−3.38197	−0.155339
$475$	0	0
$476$	−1.23607	−0.0566551
$477$	− 9.09017i	− 0.416210i
$478$	25.7082i	1.17587i
$479$	−11.1246	−0.508296	−0.254148	−	0.967165i	$-0.581795\pi$
$479$	−11.1246	−0.508296	−0.254148	+	0.967165i	$0.581795\pi$
$480$	0	0
$481$	19.7771	0.901758
$482$	10.5623i	0.481100i
$483$	1.70820i	0.0777260i
$484$	9.09017	0.413190
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	6.79837i	0.308064i	0.988066	+	0.154032i	$0.0492258\pi$
$487$	6.79837i	0.308064i	−0.988066	+	0.154032i	$0.950774\pi$
$488$	7.23607i	0.327561i
$489$	8.47214	0.383123
$490$	0	0
$491$	8.27051	0.373243	0.186621	−	0.982432i	$-0.440246\pi$
$491$	8.27051	0.373243	0.186621	+	0.982432i	$0.440246\pi$
$492$	− 7.70820i	− 0.347513i
$493$	1.52786i	0.0688115i
$494$	−19.0557	−0.857358
$495$	0	0
$496$	4.38197	0.196756
$497$	− 5.52786i	− 0.247959i
$498$	− 8.85410i	− 0.396762i
$499$	13.5967	0.608674	0.304337	−	0.952564i	$-0.401565\pi$
$499$	13.5967	0.608674	0.304337	+	0.952564i	$0.401565\pi$
$500$	0	0
$501$	1.70820	0.0763169
$502$	6.56231i	0.292890i
$503$	− 11.8885i	− 0.530084i	−0.964237	−	0.265042i	$-0.914614\pi$
$503$	− 11.8885i	− 0.530084i	0.964237	−	0.265042i	$-0.0853858\pi$
$504$	−0.381966	−0.0170141
$505$	0	0
$506$	−6.18034	−0.274750
$507$	− 6.88854i	− 0.305931i
$508$	11.3820i	0.504993i
$509$	−20.5066	−0.908938	−0.454469	−	0.890763i	$-0.650171\pi$
$509$	−20.5066	−0.908938	−0.454469	+	0.890763i	$0.650171\pi$
$510$	0	0
$511$	4.76393	0.210744
$512$	− 1.00000i	− 0.0441942i
$513$	− 7.70820i	− 0.340326i
$514$	22.0000	0.970378
$515$	0	0
$516$	−5.70820	−0.251290
$517$	16.1803i	0.711611i
$518$	3.05573i	0.134261i
$519$	−6.09017	−0.267329
$520$	0	0
$521$	−8.18034	−0.358387	−0.179194	−	0.983814i	$-0.557349\pi$
$521$	−8.18034	−0.358387	−0.179194	+	0.983814i	$0.557349\pi$
$522$	0.472136i	0.0206648i
$523$	28.9443i	1.26564i	0.774297	+	0.632822i	$0.218104\pi$
$523$	28.9443i	1.26564i	−0.774297	+	0.632822i	$0.781896\pi$
$524$	−17.8885	−0.781465
$525$	0	0
$526$	1.70820	0.0744812
$527$	14.1803i	0.617705i
$528$	− 1.38197i	− 0.0601424i
$529$	3.00000	0.130435
$530$	0	0
$531$	−1.38197	−0.0599722
$532$	− 2.94427i	− 0.127650i
$533$	− 19.0557i	− 0.825395i
$534$	12.4721	0.539722
$535$	0	0
$536$	−10.4721	−0.452327
$537$	3.14590i	0.135756i
$538$	2.09017i	0.0901136i
$539$	9.47214	0.407994
$540$	0	0
$541$	−3.81966	−0.164220	−0.0821100	−	0.996623i	$-0.526166\pi$
$541$	−3.81966	−0.164220	−0.0821100	+	0.996623i	$0.526166\pi$
$542$	− 11.0344i	− 0.473970i
$543$	− 24.6525i	− 1.05794i
$544$	3.23607	0.138745
$545$	0	0
$546$	−0.944272	−0.0404111
$547$	3.70820i	0.158551i	0.996853	+	0.0792757i	$0.0252607\pi$
$547$	3.70820i	0.158551i	−0.996853	+	0.0792757i	$0.974739\pi$
$548$	10.1803i	0.434883i
$549$	−7.23607	−0.308828
$550$	0	0
$551$	−3.63932	−0.155040
$552$	− 4.47214i	− 0.190347i
$553$	− 1.29180i	− 0.0549328i
$554$	0.944272	0.0401183
$555$	0	0
$556$	1.52786	0.0647959
$557$	2.67376i	0.113291i	0.998394	+	0.0566455i	$0.0180405\pi$
$557$	2.67376i	0.113291i	−0.998394	+	0.0566455i	$0.981960\pi$
$558$	4.38197i	0.185503i
$559$	−14.1115	−0.596851
$560$	0	0
$561$	4.47214	0.188814
$562$	29.8885i	1.26077i
$563$	− 12.2705i	− 0.517140i	−0.965992	−	0.258570i	$-0.916749\pi$
$563$	− 12.2705i	− 0.517140i	0.965992	−	0.258570i	$-0.0832513\pi$
$564$	−11.7082	−0.493004
$565$	0	0
$566$	31.4164	1.32053
$567$	− 0.381966i	− 0.0160411i
$568$	14.4721i	0.607237i
$569$	−43.7771	−1.83523	−0.917615	−	0.397469i	$-0.869889\pi$
$569$	−43.7771	−1.83523	−0.917615	+	0.397469i	$0.869889\pi$
$570$	0	0
$571$	−27.8885	−1.16710	−0.583550	−	0.812077i	$-0.698337\pi$
$571$	−27.8885	−1.16710	−0.583550	+	0.812077i	$0.698337\pi$
$572$	− 3.41641i	− 0.142847i
$573$	− 17.7082i	− 0.739771i
$574$	2.94427	0.122892
$575$	0	0
$576$	1.00000	0.0416667
$577$	37.9787i	1.58107i	0.612414	+	0.790537i	$0.290199\pi$
$577$	37.9787i	1.58107i	−0.612414	+	0.790537i	$0.709801\pi$
$578$	− 6.52786i	− 0.271523i
$579$	11.1459	0.463208
$580$	0	0
$581$	3.38197	0.140308
$582$	− 5.61803i	− 0.232875i
$583$	12.5623i	0.520278i
$584$	−12.4721	−0.516101
$585$	0	0
$586$	−10.9098	−0.450681
$587$	30.0344i	1.23965i	0.784738	+	0.619827i	$0.212797\pi$
$587$	30.0344i	1.23965i	−0.784738	+	0.619827i	$0.787203\pi$
$588$	6.85410i	0.282658i
$589$	−33.7771	−1.39176
$590$	0	0
$591$	5.90983	0.243098
$592$	− 8.00000i	− 0.328798i
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	1.38197	0.0567028
$595$	0	0
$596$	10.9098	0.446884
$597$	− 18.5066i	− 0.757424i
$598$	− 11.0557i	− 0.452102i
$599$	8.47214	0.346162	0.173081	−	0.984908i	$-0.444628\pi$
$599$	8.47214	0.346162	0.173081	+	0.984908i	$0.444628\pi$
$600$	0	0
$601$	−41.2705	−1.68346	−0.841730	−	0.539899i	$-0.818462\pi$
$601$	−41.2705	−1.68346	−0.841730	+	0.539899i	$0.818462\pi$
$602$	− 2.18034i	− 0.0888640i
$603$	− 10.4721i	− 0.426458i
$604$	−2.67376	−0.108794
$605$	0	0
$606$	−6.61803	−0.268839
$607$	13.5623i	0.550477i	0.961376	+	0.275239i	$0.0887568\pi$
$607$	13.5623i	0.550477i	−0.961376	+	0.275239i	$0.911243\pi$
$608$	7.70820i	0.312609i
$609$	−0.180340	−0.00730774
$610$	0	0
$611$	−28.9443	−1.17096
$612$	3.23607i	0.130810i
$613$	− 32.2492i	− 1.30253i	−0.758849	−	0.651267i	$-0.774238\pi$
$613$	− 32.2492i	− 1.30253i	0.758849	−	0.651267i	$-0.225762\pi$
$614$	−10.0000	−0.403567
$615$	0	0
$616$	0.527864	0.0212682
$617$	− 37.8885i	− 1.52534i	−0.646791	−	0.762668i	$-0.723889\pi$
$617$	− 37.8885i	− 1.52534i	0.646791	−	0.762668i	$-0.276111\pi$
$618$	1.32624i	0.0533491i
$619$	−17.4164	−0.700025	−0.350012	−	0.936745i	$-0.613823\pi$
$619$	−17.4164	−0.700025	−0.350012	+	0.936745i	$0.613823\pi$
$620$	0	0
$621$	4.47214	0.179461
$622$	18.9443i	0.759596i
$623$	4.76393i	0.190863i
$624$	2.47214	0.0989646
$625$	0	0
$626$	17.2705	0.690268
$627$	10.6525i	0.425419i
$628$	− 22.6525i	− 0.903932i
$629$	25.8885	1.03224
$630$	0	0
$631$	37.5279	1.49396	0.746980	−	0.664846i	$-0.231503\pi$
$631$	37.5279	1.49396	0.746980	+	0.664846i	$0.231503\pi$
$632$	3.38197i	0.134527i
$633$	− 23.4164i	− 0.930719i
$634$	0.437694	0.0173831
$635$	0	0
$636$	−9.09017	−0.360449
$637$	16.9443i	0.671356i
$638$	− 0.652476i	− 0.0258318i
$639$	−14.4721	−0.572509
$640$	0	0
$641$	−36.1803	−1.42904	−0.714519	−	0.699616i	$-0.753354\pi$
$641$	−36.1803	−1.42904	−0.714519	+	0.699616i	$0.753354\pi$
$642$	− 9.38197i	− 0.370277i
$643$	− 13.8885i	− 0.547711i	−0.961771	−	0.273855i	$-0.911701\pi$
$643$	− 13.8885i	− 0.547711i	0.961771	−	0.273855i	$-0.0882990\pi$
$644$	1.70820	0.0673127
$645$	0	0
$646$	−24.9443	−0.981419
$647$	5.05573i	0.198761i	0.995049	+	0.0993806i	$0.0316861\pi$
$647$	5.05573i	0.198761i	−0.995049	+	0.0993806i	$0.968314\pi$
$648$	1.00000i	0.0392837i
$649$	1.90983	0.0749674
$650$	0	0
$651$	−1.67376	−0.0655999
$652$	− 8.47214i	− 0.331794i
$653$	2.85410i	0.111690i	0.998439	+	0.0558448i	$0.0177852\pi$
$653$	2.85410i	0.111690i	−0.998439	+	0.0558448i	$0.982215\pi$
$654$	12.7639	0.499109
$655$	0	0
$656$	−7.70820	−0.300955
$657$	− 12.4721i	− 0.486584i
$658$	− 4.47214i	− 0.174342i
$659$	18.3820	0.716060	0.358030	−	0.933710i	$-0.383449\pi$
$659$	18.3820	0.716060	0.358030	+	0.933710i	$0.383449\pi$
$660$	0	0
$661$	−21.5279	−0.837337	−0.418668	−	0.908139i	$-0.637503\pi$
$661$	−21.5279	−0.837337	−0.418668	+	0.908139i	$0.637503\pi$
$662$	18.0000i	0.699590i
$663$	8.00000i	0.310694i
$664$	−8.85410	−0.343606
$665$	0	0
$666$	8.00000	0.309994
$667$	− 2.11146i	− 0.0817559i
$668$	− 1.70820i	− 0.0660924i
$669$	3.09017	0.119473
$670$	0	0
$671$	10.0000	0.386046
$672$	0.381966i	0.0147347i
$673$	24.4508i	0.942511i	0.881997	+	0.471255i	$0.156199\pi$
$673$	24.4508i	0.942511i	−0.881997	+	0.471255i	$0.843801\pi$
$674$	−12.0902	−0.465696
$675$	0	0
$676$	−6.88854	−0.264944
$677$	− 17.9098i	− 0.688331i	−0.938909	−	0.344165i	$-0.888162\pi$
$677$	− 17.9098i	− 0.688331i	0.938909	−	0.344165i	$-0.111838\pi$
$678$	14.7639i	0.567005i
$679$	2.14590	0.0823520
$680$	0	0
$681$	−11.2705	−0.431887
$682$	− 6.05573i	− 0.231886i
$683$	− 4.50658i	− 0.172439i	−0.996276	−	0.0862197i	$-0.972521\pi$
$683$	− 4.50658i	− 0.172439i	0.996276	−	0.0862197i	$-0.0274787\pi$
$684$	−7.70820	−0.294731
$685$	0	0
$686$	−5.29180	−0.202042
$687$	− 14.4721i	− 0.552146i
$688$	5.70820i	0.217623i
$689$	−22.4721	−0.856120
$690$	0	0
$691$	6.76393	0.257312	0.128656	−	0.991689i	$-0.458934\pi$
$691$	6.76393	0.257312	0.128656	+	0.991689i	$0.458934\pi$
$692$	6.09017i	0.231514i
$693$	0.527864i	0.0200519i
$694$	−0.437694	−0.0166146
$695$	0	0
$696$	0.472136	0.0178963
$697$	− 24.9443i	− 0.944832i
$698$	7.88854i	0.298586i
$699$	−12.6525	−0.478561
$700$	0	0
$701$	−12.8328	−0.484689	−0.242344	−	0.970190i	$-0.577916\pi$
$701$	−12.8328	−0.484689	−0.242344	+	0.970190i	$0.577916\pi$
$702$	2.47214i	0.0933048i
$703$	61.6656i	2.32576i
$704$	−1.38197	−0.0520848
$705$	0	0
$706$	−8.76393	−0.329835
$707$	− 2.52786i	− 0.0950701i
$708$	1.38197i	0.0519375i
$709$	30.2918	1.13763	0.568816	−	0.822465i	$-0.307402\pi$
$709$	30.2918	1.13763	0.568816	+	0.822465i	$0.307402\pi$
$710$	0	0
$711$	−3.38197	−0.126834
$712$	− 12.4721i	− 0.467413i
$713$	− 19.5967i	− 0.733904i
$714$	−1.23607	−0.0462587
$715$	0	0
$716$	3.14590	0.117568
$717$	25.7082i	0.960090i
$718$	− 0.180340i	− 0.00673022i
$719$	43.4164	1.61916	0.809579	−	0.587010i	$-0.199695\pi$
$719$	43.4164	1.61916	0.809579	+	0.587010i	$0.199695\pi$
$720$	0	0
$721$	−0.506578	−0.0188659
$722$	− 40.4164i	− 1.50414i
$723$	10.5623i	0.392816i
$724$	−24.6525	−0.916202
$725$	0	0
$726$	9.09017	0.337368
$727$	35.4164i	1.31352i	0.754099	+	0.656761i	$0.228074\pi$
$727$	35.4164i	1.31352i	−0.754099	+	0.656761i	$0.771926\pi$
$728$	0.944272i	0.0349970i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−18.4721	−0.683217
$732$	7.23607i	0.267453i
$733$	− 0.111456i	− 0.00411673i	−0.999998	−	0.00205836i	$-0.999345\pi$
$733$	− 0.111456i	− 0.00411673i	0.999998	−	0.00205836i	$-0.000655198\pi$
$734$	28.3820	1.04760
$735$	0	0
$736$	−4.47214	−0.164845
$737$	14.4721i	0.533088i
$738$	− 7.70820i	− 0.283743i
$739$	25.8197	0.949792	0.474896	−	0.880042i	$-0.342486\pi$
$739$	25.8197	0.949792	0.474896	+	0.880042i	$0.342486\pi$
$740$	0	0
$741$	−19.0557	−0.700030
$742$	− 3.47214i	− 0.127466i
$743$	37.0132i	1.35788i	0.734193	+	0.678940i	$0.237561\pi$
$743$	37.0132i	1.35788i	−0.734193	+	0.678940i	$0.762439\pi$
$744$	4.38197	0.160651
$745$	0	0
$746$	−20.4721	−0.749538
$747$	− 8.85410i	− 0.323955i
$748$	− 4.47214i	− 0.163517i
$749$	3.58359	0.130942
$750$	0	0
$751$	0.201626	0.00735744	0.00367872	−	0.999993i	$-0.498829\pi$
$751$	0.201626	0.00735744	0.00367872	+	0.999993i	$0.498829\pi$
$752$	11.7082i	0.426954i
$753$	6.56231i	0.239144i
$754$	1.16718	0.0425063
$755$	0	0
$756$	−0.381966	−0.0138920
$757$	− 23.1246i	− 0.840478i	−0.907413	−	0.420239i	$-0.861946\pi$
$757$	− 23.1246i	− 0.840478i	0.907413	−	0.420239i	$-0.138054\pi$
$758$	2.18034i	0.0791935i
$759$	−6.18034	−0.224332
$760$	0	0
$761$	−25.4164	−0.921344	−0.460672	−	0.887570i	$-0.652392\pi$
$761$	−25.4164	−0.921344	−0.460672	+	0.887570i	$0.652392\pi$
$762$	11.3820i	0.412325i
$763$	4.87539i	0.176501i
$764$	−17.7082	−0.640660
$765$	0	0
$766$	−20.0000	−0.722629
$767$	3.41641i	0.123359i
$768$	− 1.00000i	− 0.0360844i
$769$	35.6869	1.28690	0.643452	−	0.765487i	$-0.277502\pi$
$769$	35.6869	1.28690	0.643452	+	0.765487i	$0.277502\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	− 11.1459i	− 0.401150i
$773$	− 7.85410i	− 0.282492i	−0.989974	−	0.141246i	$-0.954889\pi$
$773$	− 7.85410i	− 0.282492i	0.989974	−	0.141246i	$-0.0451109\pi$
$774$	−5.70820	−0.205177
$775$	0	0
$776$	−5.61803	−0.201676
$777$	3.05573i	0.109624i
$778$	− 14.6180i	− 0.524082i
$779$	59.4164	2.12881
$780$	0	0
$781$	20.0000	0.715656
$782$	− 14.4721i	− 0.517523i
$783$	0.472136i	0.0168728i
$784$	6.85410	0.244789
$785$	0	0
$786$	−17.8885	−0.638063
$787$	30.0689i	1.07184i	0.844269	+	0.535920i	$0.180035\pi$
$787$	30.0689i	1.07184i	−0.844269	+	0.535920i	$0.819965\pi$
$788$	− 5.90983i	− 0.210529i
$789$	1.70820	0.0608137
$790$	0	0
$791$	−5.63932	−0.200511
$792$	− 1.38197i	− 0.0491060i
$793$	17.8885i	0.635241i
$794$	35.2361	1.25048
$795$	0	0
$796$	−18.5066	−0.655948
$797$	36.5066i	1.29313i	0.762859	+	0.646565i	$0.223795\pi$
$797$	36.5066i	1.29313i	−0.762859	+	0.646565i	$0.776205\pi$
$798$	− 2.94427i	− 0.104226i
$799$	−37.8885	−1.34040
$800$	0	0
$801$	12.4721	0.440681
$802$	− 12.2918i	− 0.434038i
$803$	17.2361i	0.608248i
$804$	−10.4721	−0.369324
$805$	0	0
$806$	10.8328	0.381570
$807$	2.09017i	0.0735775i
$808$	6.61803i	0.232822i
$809$	−22.8328	−0.802759	−0.401380	−	0.915912i	$-0.631469\pi$
$809$	−22.8328	−0.802759	−0.401380	+	0.915912i	$0.631469\pi$
$810$	0	0
$811$	7.70820	0.270672	0.135336	−	0.990800i	$-0.456789\pi$
$811$	7.70820	0.270672	0.135336	+	0.990800i	$0.456789\pi$
$812$	0.180340i	0.00632869i
$813$	− 11.0344i	− 0.386995i
$814$	−11.0557	−0.387503
$815$	0	0
$816$	3.23607	0.113285
$817$	− 44.0000i	− 1.53937i
$818$	− 18.7984i	− 0.657270i
$819$	−0.944272	−0.0329955
$820$	0	0
$821$	−33.9098	−1.18346	−0.591731	−	0.806136i	$-0.701555\pi$
$821$	−33.9098	−1.18346	−0.591731	+	0.806136i	$0.701555\pi$
$822$	10.1803i	0.355080i
$823$	32.0344i	1.11665i	0.829622	+	0.558325i	$0.188556\pi$
$823$	32.0344i	1.11665i	−0.829622	+	0.558325i	$0.811444\pi$
$824$	1.32624	0.0462017
$825$	0	0
$826$	−0.527864	−0.0183667
$827$	43.1033i	1.49885i	0.662090	+	0.749425i	$0.269670\pi$
$827$	43.1033i	1.49885i	−0.662090	+	0.749425i	$0.730330\pi$
$828$	− 4.47214i	− 0.155417i
$829$	9.88854	0.343443	0.171722	−	0.985146i	$-0.445067\pi$
$829$	9.88854	0.343443	0.171722	+	0.985146i	$0.445067\pi$
$830$	0	0
$831$	0.944272	0.0327564
$832$	− 2.47214i	− 0.0857059i
$833$	22.1803i	0.768503i
$834$	1.52786	0.0529056
$835$	0	0
$836$	10.6525	0.368424
$837$	4.38197i	0.151463i
$838$	8.09017i	0.279470i
$839$	38.3607	1.32436	0.662179	−	0.749346i	$-0.269632\pi$
$839$	38.3607	1.32436	0.662179	+	0.749346i	$0.269632\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	27.2361i	0.938617i
$843$	29.8885i	1.02942i
$844$	−23.4164	−0.806026
$845$	0	0
$846$	−11.7082	−0.402536
$847$	3.47214i	0.119304i
$848$	9.09017i	0.312158i
$849$	31.4164	1.07821
$850$	0	0
$851$	−35.7771	−1.22642
$852$	14.4721i	0.495807i
$853$	5.59675i	0.191629i	0.995399	+	0.0958145i	$0.0305456\pi$
$853$	5.59675i	0.191629i	−0.995399	+	0.0958145i	$0.969454\pi$
$854$	−2.76393	−0.0945798
$855$	0	0
$856$	−9.38197	−0.320669
$857$	− 42.0689i	− 1.43705i	−0.695503	−	0.718523i	$-0.744819\pi$
$857$	− 42.0689i	− 1.43705i	0.695503	−	0.718523i	$-0.255181\pi$
$858$	− 3.41641i	− 0.116634i
$859$	15.7082	0.535957	0.267979	−	0.963425i	$-0.413644\pi$
$859$	15.7082	0.535957	0.267979	+	0.963425i	$0.413644\pi$
$860$	0	0
$861$	2.94427	0.100341
$862$	36.8328i	1.25453i
$863$	12.1803i	0.414624i	0.978275	+	0.207312i	$0.0664715\pi$
$863$	12.1803i	0.414624i	−0.978275	+	0.207312i	$0.933529\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−22.5066	−0.764805
$867$	− 6.52786i	− 0.221698i
$868$	1.67376i	0.0568112i
$869$	4.67376	0.158547
$870$	0	0
$871$	−25.8885	−0.877200
$872$	− 12.7639i	− 0.432241i
$873$	− 5.61803i	− 0.190142i
$874$	34.4721	1.16604
$875$	0	0
$876$	−12.4721	−0.421394
$877$	− 3.12461i	− 0.105511i	−0.998607	−	0.0527553i	$-0.983200\pi$
$877$	− 3.12461i	− 0.105511i	0.998607	−	0.0527553i	$-0.0168003\pi$
$878$	14.5066i	0.489573i
$879$	−10.9098	−0.367979
$880$	0	0
$881$	25.5967	0.862376	0.431188	−	0.902262i	$-0.358095\pi$
$881$	25.5967	0.862376	0.431188	+	0.902262i	$0.358095\pi$
$882$	6.85410i	0.230790i
$883$	28.0689i	0.944593i	0.881440	+	0.472297i	$0.156575\pi$
$883$	28.0689i	0.944593i	−0.881440	+	0.472297i	$0.843425\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	14.3820	0.483172
$887$	3.41641i	0.114712i	0.998354	+	0.0573559i	$0.0182670\pi$
$887$	3.41641i	0.114712i	−0.998354	+	0.0573559i	$0.981733\pi$
$888$	− 8.00000i	− 0.268462i
$889$	−4.34752	−0.145811
$890$	0	0
$891$	1.38197	0.0462976
$892$	− 3.09017i	− 0.103467i
$893$	− 90.2492i	− 3.02008i
$894$	10.9098	0.364879
$895$	0	0
$896$	0.381966	0.0127606
$897$	− 11.0557i	− 0.369140i
$898$	− 14.2918i	− 0.476923i
$899$	2.06888	0.0690011
$900$	0	0
$901$	−29.4164	−0.980003
$902$	10.6525i	0.354689i
$903$	− 2.18034i	− 0.0725572i
$904$	14.7639	0.491041
$905$	0	0
$906$	−2.67376	−0.0888298
$907$	− 21.5279i	− 0.714821i	−0.933947	−	0.357410i	$-0.883660\pi$
$907$	− 21.5279i	− 0.714821i	0.933947	−	0.357410i	$-0.116340\pi$
$908$	11.2705i	0.374025i
$909$	−6.61803	−0.219506
$910$	0	0
$911$	−4.18034	−0.138501	−0.0692504	−	0.997599i	$-0.522061\pi$
$911$	−4.18034	−0.138501	−0.0692504	+	0.997599i	$0.522061\pi$
$912$	7.70820i	0.255244i
$913$	12.2361i	0.404955i
$914$	−7.09017	−0.234522
$915$	0	0
$916$	−14.4721	−0.478173
$917$	− 6.83282i	− 0.225639i
$918$	3.23607i	0.106806i
$919$	−48.7214	−1.60717	−0.803585	−	0.595190i	$-0.797077\pi$
$919$	−48.7214	−1.60717	−0.803585	+	0.595190i	$0.797077\pi$
$920$	0	0
$921$	−10.0000	−0.329511
$922$	12.5623i	0.413718i
$923$	35.7771i	1.17762i
$924$	0.527864	0.0173655
$925$	0	0
$926$	30.8328	1.01323
$927$	1.32624i	0.0435594i
$928$	− 0.472136i	− 0.0154986i
$929$	−57.0132	−1.87054	−0.935270	−	0.353934i	$-0.884844\pi$
$929$	−57.0132	−1.87054	−0.935270	+	0.353934i	$0.884844\pi$
$930$	0	0
$931$	−52.8328	−1.73153
$932$	12.6525i	0.414446i
$933$	18.9443i	0.620208i
$934$	−26.3262	−0.861421
$935$	0	0
$936$	2.47214	0.0808043
$937$	− 51.6869i	− 1.68854i	−0.535920	−	0.844269i	$-0.680035\pi$
$937$	− 51.6869i	− 1.68854i	0.535920	−	0.844269i	$-0.319965\pi$
$938$	− 4.00000i	− 0.130605i
$939$	17.2705	0.563602
$940$	0	0
$941$	−24.5623	−0.800708	−0.400354	−	0.916360i	$-0.631113\pi$
$941$	−24.5623	−0.800708	−0.400354	+	0.916360i	$0.631113\pi$
$942$	− 22.6525i	− 0.738058i
$943$	34.4721i	1.12257i
$944$	1.38197	0.0449792
$945$	0	0
$946$	7.88854	0.256479
$947$	13.4377i	0.436666i	0.975874	+	0.218333i	$0.0700620\pi$
$947$	13.4377i	0.436666i	−0.975874	+	0.218333i	$0.929938\pi$
$948$	3.38197i	0.109841i
$949$	−30.8328	−1.00088
$950$	0	0
$951$	0.437694	0.0141932
$952$	1.23607i	0.0400612i
$953$	− 57.3050i	− 1.85629i	−0.372219	−	0.928145i	$-0.621403\pi$
$953$	− 57.3050i	− 1.85629i	0.372219	−	0.928145i	$-0.378597\pi$
$954$	−9.09017	−0.294305
$955$	0	0
$956$	25.7082	0.831463
$957$	− 0.652476i	− 0.0210915i
$958$	11.1246i	0.359420i
$959$	−3.88854	−0.125568
$960$	0	0
$961$	−11.7984	−0.380593
$962$	− 19.7771i	− 0.637639i
$963$	− 9.38197i	− 0.302330i
$964$	10.5623	0.340189
$965$	0	0
$966$	1.70820	0.0549606
$967$	− 22.0902i	− 0.710372i	−0.934796	−	0.355186i	$-0.884418\pi$
$967$	− 22.0902i	− 0.710372i	0.934796	−	0.355186i	$-0.115582\pi$
$968$	− 9.09017i	− 0.292169i
$969$	−24.9443	−0.801325
$970$	0	0
$971$	37.2148	1.19428	0.597140	−	0.802137i	$-0.296304\pi$
$971$	37.2148	1.19428	0.597140	+	0.802137i	$0.296304\pi$
$972$	1.00000i	0.0320750i
$973$	0.583592i	0.0187091i
$974$	6.79837	0.217834
$975$	0	0
$976$	7.23607	0.231621
$977$	5.23607i	0.167517i	0.996486	+	0.0837583i	$0.0266924\pi$
$977$	5.23607i	0.167517i	−0.996486	+	0.0837583i	$0.973308\pi$
$978$	− 8.47214i	− 0.270909i
$979$	−17.2361	−0.550867
$980$	0	0
$981$	12.7639	0.407521
$982$	− 8.27051i	− 0.263923i
$983$	17.8197i	0.568359i	0.958771	+	0.284179i	$0.0917211\pi$
$983$	17.8197i	0.568359i	−0.958771	+	0.284179i	$0.908279\pi$
$984$	−7.70820	−0.245729
$985$	0	0
$986$	1.52786	0.0486571
$987$	− 4.47214i	− 0.142350i
$988$	19.0557i	0.606243i
$989$	25.5279	0.811739
$990$	0	0
$991$	−26.4508	−0.840239	−0.420119	−	0.907469i	$-0.638012\pi$
$991$	−26.4508	−0.840239	−0.420119	+	0.907469i	$0.638012\pi$
$992$	− 4.38197i	− 0.139128i
$993$	18.0000i	0.571213i
$994$	−5.52786	−0.175333
$995$	0	0
$996$	−8.85410	−0.280553
$997$	− 43.5967i	− 1.38072i	−0.723465	−	0.690361i	$-0.757452\pi$
$997$	− 43.5967i	− 1.38072i	0.723465	−	0.690361i	$-0.242548\pi$
$998$	− 13.5967i	− 0.430398i
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3750.2.c.b.1249.2		4
5.2	odd	4		3750.2.a.f.1.1		2
5.3	odd	4		3750.2.a.d.1.2		2
5.4	even	2	inner	3750.2.c.b.1249.3		4
25.2	odd	20		150.2.g.a.121.1	yes	4
25.9	even	10		750.2.h.b.349.1		8
25.11	even	5		750.2.h.b.649.1		8
25.12	odd	20		150.2.g.a.31.1	✓	4
25.13	odd	20		750.2.g.b.151.1		4
25.14	even	10		750.2.h.b.649.2		8
25.16	even	5		750.2.h.b.349.2		8
25.23	odd	20		750.2.g.b.601.1		4
75.2	even	20		450.2.h.c.271.1		4
75.62	even	20		450.2.h.c.181.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.2.g.a.31.1	✓	4	25.12	odd	20
150.2.g.a.121.1	yes	4	25.2	odd	20
450.2.h.c.181.1		4	75.62	even	20
450.2.h.c.271.1		4	75.2	even	20
750.2.g.b.151.1		4	25.13	odd	20
750.2.g.b.601.1		4	25.23	odd	20
750.2.h.b.349.1		8	25.9	even	10
750.2.h.b.349.2		8	25.16	even	5
750.2.h.b.649.1		8	25.11	even	5
750.2.h.b.649.2		8	25.14	even	10
3750.2.a.d.1.2		2	5.3	odd	4
3750.2.a.f.1.1		2	5.2	odd	4
3750.2.c.b.1249.2		4	1.1	even	1	trivial
3750.2.c.b.1249.3		4	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 3750 = 2 \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3750.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -0.381966i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -0.381966i q^{7} +1.00000i q^{8} -1.00000 q^{9} +1.38197 q^{11} +1.00000i q^{12} +2.47214i q^{13} -0.381966 q^{14} +1.00000 q^{16} +3.23607i q^{17} +1.00000i q^{18} -7.70820 q^{19} -0.381966 q^{21} -1.38197i q^{22} -4.47214i q^{23} +1.00000 q^{24} +2.47214 q^{26} +1.00000i q^{27} +0.381966i q^{28} +0.472136 q^{29} +4.38197 q^{31} -1.00000i q^{32} -1.38197i q^{33} +3.23607 q^{34} +1.00000 q^{36} -8.00000i q^{37} +7.70820i q^{38} +2.47214 q^{39} -7.70820 q^{41} +0.381966i q^{42} +5.70820i q^{43} -1.38197 q^{44} -4.47214 q^{46} +11.7082i q^{47} -1.00000i q^{48} +6.85410 q^{49} +3.23607 q^{51} -2.47214i q^{52} +9.09017i q^{53} +1.00000 q^{54} +0.381966 q^{56} +7.70820i q^{57} -0.472136i q^{58} +1.38197 q^{59} +7.23607 q^{61} -4.38197i q^{62} +0.381966i q^{63} -1.00000 q^{64} -1.38197 q^{66} +10.4721i q^{67} -3.23607i q^{68} -4.47214 q^{69} +14.4721 q^{71} -1.00000i q^{72} +12.4721i q^{73} -8.00000 q^{74} +7.70820 q^{76} -0.527864i q^{77} -2.47214i q^{78} +3.38197 q^{79} +1.00000 q^{81} +7.70820i q^{82} +8.85410i q^{83} +0.381966 q^{84} +5.70820 q^{86} -0.472136i q^{87} +1.38197i q^{88} -12.4721 q^{89} +0.944272 q^{91} +4.47214i q^{92} -4.38197i q^{93} +11.7082 q^{94} -1.00000 q^{96} +5.61803i q^{97} -6.85410i q^{98} -1.38197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9	\( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 10 q^{11} - 6 q^{14} + 4 q^{16} - 4 q^{19} - 6 q^{21} + 4 q^{24} - 8 q^{26} - 16 q^{29} + 22 q^{31} + 4 q^{34} + 4 q^{36} - 8 q^{39} - 4 q^{41} - 10 q^{44} + 14 q^{49} + 4 q^{51} + 4 q^{54} + 6 q^{56} + 10 q^{59} + 20 q^{61} - 4 q^{64} - 10 q^{66} + 40 q^{71} - 32 q^{74} + 4 q^{76} + 18 q^{79} + 4 q^{81} + 6 q^{84} - 4 q^{86} - 32 q^{89} - 32 q^{91} + 20 q^{94} - 4 q^{96} - 10 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^6 - 4 * q^9 + 10 * q^11 - 6 * q^14 + 4 * q^16 - 4 * q^19 - 6 * q^21 + 4 * q^24 - 8 * q^26 - 16 * q^29 + 22 * q^31 + 4 * q^34 + 4 * q^36 - 8 * q^39 - 4 * q^41 - 10 * q^44 + 14 * q^49 + 4 * q^51 + 4 * q^54 + 6 * q^56 + 10 * q^59 + 20 * q^61 - 4 * q^64 - 10 * q^66 + 40 * q^71 - 32 * q^74 + 4 * q^76 + 18 * q^79 + 4 * q^81 + 6 * q^84 - 4 * q^86 - 32 * q^89 - 32 * q^91 + 20 * q^94 - 4 * q^96 - 10 * q^99

Introduction

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