LMFDB - Embedded newform 4598.2.a.by.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4598,2,Mod(1,4598)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4598, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4598.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4598 = 2 \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4598.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$36.7152148494$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 16x^{6} - 4x^{5} + 75x^{4} + 32x^{3} - 90x^{2} - 28x - 2 $ x^8 - 16x^6 - 4x^5 + 75x^4 + 32x^3 - 90x^2 - 28x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.84109$ of defining polynomial
Character	$\chi$	$=$	4598.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.67352 q^{3} +1.00000 q^{4} -0.566489 q^{5} -1.67352 q^{6} -3.40033 q^{7} -1.00000 q^{8} -0.199325 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.67352 q^{3} +1.00000 q^{4} -0.566489 q^{5} -1.67352 q^{6} -3.40033 q^{7} -1.00000 q^{8} -0.199325 q^{9} +0.566489 q^{10} +1.67352 q^{12} -1.35442 q^{13} +3.40033 q^{14} -0.948032 q^{15} +1.00000 q^{16} -4.30453 q^{17} +0.199325 q^{18} +1.00000 q^{19} -0.566489 q^{20} -5.69053 q^{21} -2.71065 q^{23} -1.67352 q^{24} -4.67909 q^{25} +1.35442 q^{26} -5.35414 q^{27} -3.40033 q^{28} -5.64462 q^{29} +0.948032 q^{30} +9.62449 q^{31} -1.00000 q^{32} +4.30453 q^{34} +1.92625 q^{35} -0.199325 q^{36} +3.63304 q^{37} -1.00000 q^{38} -2.26664 q^{39} +0.566489 q^{40} +3.64782 q^{41} +5.69053 q^{42} +7.94488 q^{43} +0.112916 q^{45} +2.71065 q^{46} +13.5117 q^{47} +1.67352 q^{48} +4.56226 q^{49} +4.67909 q^{50} -7.20373 q^{51} -1.35442 q^{52} +4.25789 q^{53} +5.35414 q^{54} +3.40033 q^{56} +1.67352 q^{57} +5.64462 q^{58} +7.97498 q^{59} -0.948032 q^{60} -9.37992 q^{61} -9.62449 q^{62} +0.677772 q^{63} +1.00000 q^{64} +0.767262 q^{65} +1.01003 q^{67} -4.30453 q^{68} -4.53633 q^{69} -1.92625 q^{70} +6.30982 q^{71} +0.199325 q^{72} +14.0378 q^{73} -3.63304 q^{74} -7.83056 q^{75} +1.00000 q^{76} +2.26664 q^{78} +7.11185 q^{79} -0.566489 q^{80} -8.36229 q^{81} -3.64782 q^{82} +1.02452 q^{83} -5.69053 q^{84} +2.43847 q^{85} -7.94488 q^{86} -9.44639 q^{87} -15.1613 q^{89} -0.112916 q^{90} +4.60546 q^{91} -2.71065 q^{92} +16.1068 q^{93} -13.5117 q^{94} -0.566489 q^{95} -1.67352 q^{96} +7.04118 q^{97} -4.56226 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^6 - 4 * q^7 - 8 * q^8 + 22 * q^9	$ 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9} + 8 q^{12} + 12 q^{13} + 4 q^{14} + 4 q^{15} + 8 q^{16} + 4 q^{17} - 22 q^{18} + 8 q^{19} + 20 q^{21} + 14 q^{23} - 8 q^{24} + 36 q^{25} - 12 q^{26} + 32 q^{27} - 4 q^{28} + 2 q^{29} - 4 q^{30} - 8 q^{32} - 4 q^{34} - 36 q^{35} + 22 q^{36} + 24 q^{37} - 8 q^{38} - 16 q^{39} - 8 q^{41} - 20 q^{42} - 8 q^{43} + 16 q^{45} - 14 q^{46} - 16 q^{47} + 8 q^{48} + 34 q^{49} - 36 q^{50} - 18 q^{51} + 12 q^{52} + 36 q^{53} - 32 q^{54} + 4 q^{56} + 8 q^{57} - 2 q^{58} - 24 q^{59} + 4 q^{60} - 12 q^{61} - 24 q^{63} + 8 q^{64} - 16 q^{65} + 16 q^{67} + 4 q^{68} + 4 q^{69} + 36 q^{70} + 4 q^{71} - 22 q^{72} + 20 q^{73} - 24 q^{74} + 40 q^{75} + 8 q^{76} + 16 q^{78} + 12 q^{79} + 40 q^{81} + 8 q^{82} - 20 q^{83} + 20 q^{84} - 12 q^{85} + 8 q^{86} + 36 q^{87} + 8 q^{89} - 16 q^{90} - 24 q^{91} + 14 q^{92} + 12 q^{93} + 16 q^{94} - 8 q^{96} + 4 q^{97} - 34 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^6 - 4 * q^7 - 8 * q^8 + 22 * q^9 + 8 * q^12 + 12 * q^13 + 4 * q^14 + 4 * q^15 + 8 * q^16 + 4 * q^17 - 22 * q^18 + 8 * q^19 + 20 * q^21 + 14 * q^23 - 8 * q^24 + 36 * q^25 - 12 * q^26 + 32 * q^27 - 4 * q^28 + 2 * q^29 - 4 * q^30 - 8 * q^32 - 4 * q^34 - 36 * q^35 + 22 * q^36 + 24 * q^37 - 8 * q^38 - 16 * q^39 - 8 * q^41 - 20 * q^42 - 8 * q^43 + 16 * q^45 - 14 * q^46 - 16 * q^47 + 8 * q^48 + 34 * q^49 - 36 * q^50 - 18 * q^51 + 12 * q^52 + 36 * q^53 - 32 * q^54 + 4 * q^56 + 8 * q^57 - 2 * q^58 - 24 * q^59 + 4 * q^60 - 12 * q^61 - 24 * q^63 + 8 * q^64 - 16 * q^65 + 16 * q^67 + 4 * q^68 + 4 * q^69 + 36 * q^70 + 4 * q^71 - 22 * q^72 + 20 * q^73 - 24 * q^74 + 40 * q^75 + 8 * q^76 + 16 * q^78 + 12 * q^79 + 40 * q^81 + 8 * q^82 - 20 * q^83 + 20 * q^84 - 12 * q^85 + 8 * q^86 + 36 * q^87 + 8 * q^89 - 16 * q^90 - 24 * q^91 + 14 * q^92 + 12 * q^93 + 16 * q^94 - 8 * q^96 + 4 * q^97 - 34 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.67352	0.966208	0.483104	−	0.875563i	$-0.339509\pi$
$3$	1.67352	0.966208	0.483104	+	0.875563i	$0.339509\pi$
$4$	1.00000	0.500000
$5$	−0.566489	−0.253342	−0.126671	−	0.991945i	$-0.540429\pi$
$5$	−0.566489	−0.253342	−0.126671	+	0.991945i	$0.540429\pi$
$6$	−1.67352	−0.683212
$7$	−3.40033	−1.28520	−0.642602	−	0.766200i	$-0.722145\pi$
$7$	−3.40033	−1.28520	−0.642602	+	0.766200i	$0.722145\pi$
$8$	−1.00000	−0.353553
$9$	−0.199325	−0.0664418
$10$	0.566489	0.179140
$11$	0	0
$11$	0	0
$12$	1.67352	0.483104
$13$	−1.35442	−0.375647	−0.187824	−	0.982203i	$-0.560143\pi$
$13$	−1.35442	−0.375647	−0.187824	+	0.982203i	$0.560143\pi$
$14$	3.40033	0.908777
$15$	−0.948032	−0.244781
$16$	1.00000	0.250000
$17$	−4.30453	−1.04400	−0.522002	−	0.852945i	$-0.674814\pi$
$17$	−4.30453	−1.04400	−0.522002	+	0.852945i	$0.674814\pi$
$18$	0.199325	0.0469814
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.566489	−0.126671
$21$	−5.69053	−1.24178
$22$	0	0
$23$	−2.71065	−0.565210	−0.282605	−	0.959236i	$-0.591199\pi$
$23$	−2.71065	−0.565210	−0.282605	+	0.959236i	$0.591199\pi$
$24$	−1.67352	−0.341606
$25$	−4.67909	−0.935818
$26$	1.35442	0.265623
$27$	−5.35414	−1.03040
$28$	−3.40033	−0.642602
$29$	−5.64462	−1.04818	−0.524089	−	0.851663i	$-0.675594\pi$
$29$	−5.64462	−1.04818	−0.524089	+	0.851663i	$0.675594\pi$
$30$	0.948032	0.173086
$31$	9.62449	1.72861	0.864305	−	0.502968i	$-0.167759\pi$
$31$	9.62449	1.72861	0.864305	+	0.502968i	$0.167759\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.30453	0.738222
$35$	1.92625	0.325596
$36$	−0.199325	−0.0332209
$37$	3.63304	0.597269	0.298634	−	0.954368i	$-0.403469\pi$
$37$	3.63304	0.597269	0.298634	+	0.954368i	$0.403469\pi$
$38$	−1.00000	−0.162221
$39$	−2.26664	−0.362953
$40$	0.566489	0.0895698
$41$	3.64782	0.569694	0.284847	−	0.958573i	$-0.408057\pi$
$41$	3.64782	0.569694	0.284847	+	0.958573i	$0.408057\pi$
$42$	5.69053	0.878068
$43$	7.94488	1.21158	0.605791	−	0.795624i	$-0.292857\pi$
$43$	7.94488	1.21158	0.605791	+	0.795624i	$0.292857\pi$
$44$	0	0
$45$	0.112916	0.0168325
$46$	2.71065	0.399664
$47$	13.5117	1.97088	0.985440	−	0.170023i	$-0.0543841\pi$
$47$	13.5117	1.97088	0.985440	+	0.170023i	$0.0543841\pi$
$48$	1.67352	0.241552
$49$	4.56226	0.651752
$50$	4.67909	0.661723
$51$	−7.20373	−1.00872
$52$	−1.35442	−0.187824
$53$	4.25789	0.584867	0.292433	−	0.956286i	$-0.405535\pi$
$53$	4.25789	0.584867	0.292433	+	0.956286i	$0.405535\pi$
$54$	5.35414	0.728606
$55$	0	0
$56$	3.40033	0.454389
$57$	1.67352	0.221663
$58$	5.64462	0.741174
$59$	7.97498	1.03825	0.519127	−	0.854697i	$-0.326257\pi$
$59$	7.97498	1.03825	0.519127	+	0.854697i	$0.326257\pi$
$60$	−0.948032	−0.122390
$61$	−9.37992	−1.20098	−0.600488	−	0.799634i	$-0.705027\pi$
$61$	−9.37992	−1.20098	−0.600488	+	0.799634i	$0.705027\pi$
$62$	−9.62449	−1.22231
$63$	0.677772	0.0853913
$64$	1.00000	0.125000
$65$	0.767262	0.0951671
$66$	0	0
$67$	1.01003	0.123395	0.0616973	−	0.998095i	$-0.480349\pi$
$67$	1.01003	0.123395	0.0616973	+	0.998095i	$0.480349\pi$
$68$	−4.30453	−0.522002
$69$	−4.53633	−0.546110
$70$	−1.92625	−0.230231
$71$	6.30982	0.748838	0.374419	−	0.927260i	$-0.377842\pi$
$71$	6.30982	0.748838	0.374419	+	0.927260i	$0.377842\pi$
$72$	0.199325	0.0234907
$73$	14.0378	1.64300	0.821498	−	0.570211i	$-0.193139\pi$
$73$	14.0378	1.64300	0.821498	+	0.570211i	$0.193139\pi$
$74$	−3.63304	−0.422333
$75$	−7.83056	−0.904195
$76$	1.00000	0.114708
$77$	0	0
$78$	2.26664	0.256647
$79$	7.11185	0.800145	0.400073	−	0.916483i	$-0.368985\pi$
$79$	7.11185	0.800145	0.400073	+	0.916483i	$0.368985\pi$
$80$	−0.566489	−0.0633354
$81$	−8.36229	−0.929144
$82$	−3.64782	−0.402835
$83$	1.02452	0.112456	0.0562280	−	0.998418i	$-0.482093\pi$
$83$	1.02452	0.112456	0.0562280	+	0.998418i	$0.482093\pi$
$84$	−5.69053	−0.620888
$85$	2.43847	0.264489
$86$	−7.94488	−0.856718
$87$	−9.44639	−1.01276
$88$	0	0
$89$	−15.1613	−1.60710	−0.803548	−	0.595239i	$-0.797057\pi$
$89$	−15.1613	−1.60710	−0.803548	+	0.595239i	$0.797057\pi$
$90$	−0.112916	−0.0119024
$91$	4.60546	0.482784
$92$	−2.71065	−0.282605
$93$	16.1068	1.67020
$94$	−13.5117	−1.39362
$95$	−0.566489	−0.0581206
$96$	−1.67352	−0.170803
$97$	7.04118	0.714923	0.357462	−	0.933928i	$-0.383642\pi$
$97$	7.04118	0.714923	0.357462	+	0.933928i	$0.383642\pi$
$98$	−4.56226	−0.460858
$99$	0	0
$100$	−4.67909	−0.467909
$101$	8.12037	0.808007	0.404003	−	0.914758i	$-0.367618\pi$
$101$	8.12037	0.808007	0.404003	+	0.914758i	$0.367618\pi$
$102$	7.20373	0.713276
$103$	−3.00499	−0.296091	−0.148045	−	0.988981i	$-0.547298\pi$
$103$	−3.00499	−0.296091	−0.148045	+	0.988981i	$0.547298\pi$
$104$	1.35442	0.132811
$105$	3.22362	0.314593
$106$	−4.25789	−0.413563
$107$	7.33334	0.708941	0.354471	−	0.935067i	$-0.384661\pi$
$107$	7.33334	0.708941	0.354471	+	0.935067i	$0.384661\pi$
$108$	−5.35414	−0.515202
$109$	−3.90465	−0.373998	−0.186999	−	0.982360i	$-0.559876\pi$
$109$	−3.90465	−0.373998	−0.186999	+	0.982360i	$0.559876\pi$
$110$	0	0
$111$	6.07998	0.577086
$112$	−3.40033	−0.321301
$113$	−6.36327	−0.598606	−0.299303	−	0.954158i	$-0.596754\pi$
$113$	−6.36327	−0.598606	−0.299303	+	0.954158i	$0.596754\pi$
$114$	−1.67352	−0.156740
$115$	1.53555	0.143191
$116$	−5.64462	−0.524089
$117$	0.269969	0.0249587
$118$	−7.97498	−0.734156
$119$	14.6369	1.34176
$120$	0.948032	0.0865431
$121$	0	0
$122$	9.37992	0.849218
$123$	6.10471	0.550443
$124$	9.62449	0.864305
$125$	5.48310	0.490423
$126$	−0.677772	−0.0603808
$127$	−17.3575	−1.54023	−0.770115	−	0.637905i	$-0.779801\pi$
$127$	−17.3575	−1.54023	−0.770115	+	0.637905i	$0.779801\pi$
$128$	−1.00000	−0.0883883
$129$	13.2959	1.17064
$130$	−0.767262	−0.0672933
$131$	3.39115	0.296286	0.148143	−	0.988966i	$-0.452670\pi$
$131$	3.39115	0.296286	0.148143	+	0.988966i	$0.452670\pi$
$132$	0	0
$133$	−3.40033	−0.294846
$134$	−1.01003	−0.0872531
$135$	3.03306	0.261044
$136$	4.30453	0.369111
$137$	0.0612122	0.00522971	0.00261486	−	0.999997i	$-0.499168\pi$
$137$	0.0612122	0.00522971	0.00261486	+	0.999997i	$0.499168\pi$
$138$	4.53633	0.386158
$139$	16.7756	1.42289	0.711443	−	0.702744i	$-0.248042\pi$
$139$	16.7756	1.42289	0.711443	+	0.702744i	$0.248042\pi$
$140$	1.92625	0.162798
$141$	22.6121	1.90428
$142$	−6.30982	−0.529508
$143$	0	0
$144$	−0.199325	−0.0166104
$145$	3.19761	0.265547
$146$	−14.0378	−1.16177
$147$	7.63504	0.629728
$148$	3.63304	0.298634
$149$	−8.95206	−0.733382	−0.366691	−	0.930343i	$-0.619509\pi$
$149$	−8.95206	−0.733382	−0.366691	+	0.930343i	$0.619509\pi$
$150$	7.83056	0.639362
$151$	8.51404	0.692862	0.346431	−	0.938075i	$-0.387393\pi$
$151$	8.51404	0.692862	0.346431	+	0.938075i	$0.387393\pi$
$152$	−1.00000	−0.0811107
$153$	0.858003	0.0693654
$154$	0	0
$155$	−5.45217	−0.437929
$156$	−2.26664	−0.181477
$157$	12.0876	0.964698	0.482349	−	0.875979i	$-0.339784\pi$
$157$	12.0876	0.964698	0.482349	+	0.875979i	$0.339784\pi$
$158$	−7.11185	−0.565788
$159$	7.12568	0.565103
$160$	0.566489	0.0447849
$161$	9.21711	0.726410
$162$	8.36229	0.657004
$163$	13.6320	1.06774	0.533870	−	0.845566i	$-0.320737\pi$
$163$	13.6320	1.06774	0.533870	+	0.845566i	$0.320737\pi$
$164$	3.64782	0.284847
$165$	0	0
$166$	−1.02452	−0.0795184
$167$	3.77640	0.292226	0.146113	−	0.989268i	$-0.453324\pi$
$167$	3.77640	0.292226	0.146113	+	0.989268i	$0.453324\pi$
$168$	5.69053	0.439034
$169$	−11.1656	−0.858889
$170$	−2.43847	−0.187022
$171$	−0.199325	−0.0152428
$172$	7.94488	0.605791
$173$	1.92088	0.146042	0.0730210	−	0.997330i	$-0.476736\pi$
$173$	1.92088	0.146042	0.0730210	+	0.997330i	$0.476736\pi$
$174$	9.44639	0.716129
$175$	15.9105	1.20272
$176$	0	0
$177$	13.3463	1.00317
$178$	15.1613	1.13639
$179$	−12.9182	−0.965548	−0.482774	−	0.875745i	$-0.660371\pi$
$179$	−12.9182	−0.965548	−0.482774	+	0.875745i	$0.660371\pi$
$180$	0.112916	0.00841623
$181$	17.7782	1.32144	0.660722	−	0.750630i	$-0.270250\pi$
$181$	17.7782	1.32144	0.660722	+	0.750630i	$0.270250\pi$
$182$	−4.60546	−0.341380
$183$	−15.6975	−1.16039
$184$	2.71065	0.199832
$185$	−2.05808	−0.151313
$186$	−16.1068	−1.18101
$187$	0	0
$188$	13.5117	0.985440
$189$	18.2059	1.32428
$190$	0.566489	0.0410974
$191$	−20.7230	−1.49947	−0.749733	−	0.661741i	$-0.769818\pi$
$191$	−20.7230	−1.49947	−0.749733	+	0.661741i	$0.769818\pi$
$192$	1.67352	0.120776
$193$	−12.1336	−0.873396	−0.436698	−	0.899608i	$-0.643852\pi$
$193$	−12.1336	−0.873396	−0.436698	+	0.899608i	$0.643852\pi$
$194$	−7.04118	−0.505527
$195$	1.28403	0.0919512
$196$	4.56226	0.325876
$197$	14.2097	1.01240	0.506199	−	0.862417i	$-0.331050\pi$
$197$	14.2097	1.01240	0.506199	+	0.862417i	$0.331050\pi$
$198$	0	0
$199$	−0.976956	−0.0692545	−0.0346273	−	0.999400i	$-0.511024\pi$
$199$	−0.976956	−0.0692545	−0.0346273	+	0.999400i	$0.511024\pi$
$200$	4.67909	0.330862
$201$	1.69030	0.119225
$202$	−8.12037	−0.571347
$203$	19.1936	1.34712
$204$	−7.20373	−0.504362
$205$	−2.06645	−0.144327
$206$	3.00499	0.209368
$207$	0.540301	0.0375535
$208$	−1.35442	−0.0939118
$209$	0	0
$210$	−3.22362	−0.222451
$211$	8.44427	0.581327	0.290664	−	0.956825i	$-0.406124\pi$
$211$	8.44427	0.581327	0.290664	+	0.956825i	$0.406124\pi$
$212$	4.25789	0.292433
$213$	10.5596	0.723533
$214$	−7.33334	−0.501297
$215$	−4.50069	−0.306944
$216$	5.35414	0.364303
$217$	−32.7265	−2.22162
$218$	3.90465	0.264456
$219$	23.4925	1.58748
$220$	0	0
$221$	5.83013	0.392177
$222$	−6.07998	−0.408061
$223$	−0.309593	−0.0207319	−0.0103659	−	0.999946i	$-0.503300\pi$
$223$	−0.309593	−0.0207319	−0.0103659	+	0.999946i	$0.503300\pi$
$224$	3.40033	0.227194
$225$	0.932661	0.0621774
$226$	6.36327	0.423278
$227$	−17.8942	−1.18768	−0.593840	−	0.804583i	$-0.702389\pi$
$227$	−17.8942	−1.18768	−0.593840	+	0.804583i	$0.702389\pi$
$228$	1.67352	0.110832
$229$	−4.77162	−0.315318	−0.157659	−	0.987494i	$-0.550395\pi$
$229$	−4.77162	−0.315318	−0.157659	+	0.987494i	$0.550395\pi$
$230$	−1.53555	−0.101251
$231$	0	0
$232$	5.64462	0.370587
$233$	12.5120	0.819686	0.409843	−	0.912156i	$-0.365583\pi$
$233$	12.5120	0.819686	0.409843	+	0.912156i	$0.365583\pi$
$234$	−0.269969	−0.0176484
$235$	−7.65422	−0.499306
$236$	7.97498	0.519127
$237$	11.9018	0.773107
$238$	−14.6369	−0.948766
$239$	−11.2492	−0.727653	−0.363826	−	0.931467i	$-0.618530\pi$
$239$	−11.2492	−0.727653	−0.363826	+	0.931467i	$0.618530\pi$
$240$	−0.948032	−0.0611952
$241$	−23.4105	−1.50801	−0.754003	−	0.656871i	$-0.771879\pi$
$241$	−23.4105	−1.50801	−0.754003	+	0.656871i	$0.771879\pi$
$242$	0	0
$243$	2.06794	0.132658
$244$	−9.37992	−0.600488
$245$	−2.58447	−0.165116
$246$	−6.10471	−0.389222
$247$	−1.35442	−0.0861794
$248$	−9.62449	−0.611156
$249$	1.71456	0.108656
$250$	−5.48310	−0.346782
$251$	−27.2448	−1.71967	−0.859837	−	0.510568i	$-0.829435\pi$
$251$	−27.2448	−1.71967	−0.859837	+	0.510568i	$0.829435\pi$
$252$	0.677772	0.0426956
$253$	0	0
$254$	17.3575	1.08911
$255$	4.08084	0.255552
$256$	1.00000	0.0625000
$257$	18.9406	1.18148	0.590742	−	0.806861i	$-0.298835\pi$
$257$	18.9406	1.18148	0.590742	+	0.806861i	$0.298835\pi$
$258$	−13.2959	−0.827768
$259$	−12.3536	−0.767613
$260$	0.767262	0.0475835
$261$	1.12511	0.0696428
$262$	−3.39115	−0.209506
$263$	28.1414	1.73527	0.867637	−	0.497198i	$-0.165638\pi$
$263$	28.1414	1.73527	0.867637	+	0.497198i	$0.165638\pi$
$264$	0	0
$265$	−2.41205	−0.148171
$266$	3.40033	0.208488
$267$	−25.3728	−1.55279
$268$	1.01003	0.0616973
$269$	−0.144365	−0.00880207	−0.00440104	−	0.999990i	$-0.501401\pi$
$269$	−0.144365	−0.00880207	−0.00440104	+	0.999990i	$0.501401\pi$
$270$	−3.03306	−0.184586
$271$	−7.88292	−0.478853	−0.239427	−	0.970914i	$-0.576959\pi$
$271$	−7.88292	−0.478853	−0.239427	+	0.970914i	$0.576959\pi$
$272$	−4.30453	−0.261001
$273$	7.70734	0.466469
$274$	−0.0612122	−0.00369797
$275$	0	0
$276$	−4.53633	−0.273055
$277$	16.4244	0.986846	0.493423	−	0.869790i	$-0.335746\pi$
$277$	16.4244	0.986846	0.493423	+	0.869790i	$0.335746\pi$
$278$	−16.7756	−1.00613
$279$	−1.91841	−0.114852
$280$	−1.92625	−0.115116
$281$	−1.71799	−0.102487	−0.0512435	−	0.998686i	$-0.516318\pi$
$281$	−1.71799	−0.102487	−0.0512435	+	0.998686i	$0.516318\pi$
$282$	−22.6121	−1.34653
$283$	−5.05430	−0.300447	−0.150223	−	0.988652i	$-0.547999\pi$
$283$	−5.05430	−0.300447	−0.150223	+	0.988652i	$0.547999\pi$
$284$	6.30982	0.374419
$285$	−0.948032	−0.0561566
$286$	0	0
$287$	−12.4038	−0.732174
$288$	0.199325	0.0117454
$289$	1.52902	0.0899424
$290$	−3.19761	−0.187770
$291$	11.7836	0.690765
$292$	14.0378	0.821498
$293$	19.2526	1.12475	0.562376	−	0.826882i	$-0.309888\pi$
$293$	19.2526	1.12475	0.562376	+	0.826882i	$0.309888\pi$
$294$	−7.63504	−0.445285
$295$	−4.51774	−0.263033
$296$	−3.63304	−0.211166
$297$	0	0
$298$	8.95206	0.518579
$299$	3.67135	0.212319
$300$	−7.83056	−0.452097
$301$	−27.0152	−1.55713
$302$	−8.51404	−0.489928
$303$	13.5896	0.780703
$304$	1.00000	0.0573539
$305$	5.31362	0.304257
$306$	−0.858003	−0.0490487
$307$	12.2585	0.699629	0.349814	−	0.936819i	$-0.386245\pi$
$307$	12.2585	0.699629	0.349814	+	0.936819i	$0.386245\pi$
$308$	0	0
$309$	−5.02892	−0.286085
$310$	5.45217	0.309663
$311$	−5.48085	−0.310791	−0.155395	−	0.987852i	$-0.549665\pi$
$311$	−5.48085	−0.310791	−0.155395	+	0.987852i	$0.549665\pi$
$312$	2.26664	0.128323
$313$	8.75492	0.494858	0.247429	−	0.968906i	$-0.420414\pi$
$313$	8.75492	0.494858	0.247429	+	0.968906i	$0.420414\pi$
$314$	−12.0876	−0.682144
$315$	−0.383951	−0.0216332
$316$	7.11185	0.400073
$317$	−18.4394	−1.03566	−0.517829	−	0.855484i	$-0.673260\pi$
$317$	−18.4394	−1.03566	−0.517829	+	0.855484i	$0.673260\pi$
$318$	−7.12568	−0.399588
$319$	0	0
$320$	−0.566489	−0.0316677
$321$	12.2725	0.684985
$322$	−9.21711	−0.513650
$323$	−4.30453	−0.239511
$324$	−8.36229	−0.464572
$325$	6.33743	0.351537
$326$	−13.6320	−0.755007
$327$	−6.53452	−0.361360
$328$	−3.64782	−0.201417
$329$	−45.9442	−2.53298
$330$	0	0
$331$	6.57096	0.361172	0.180586	−	0.983559i	$-0.442201\pi$
$331$	6.57096	0.361172	0.180586	+	0.983559i	$0.442201\pi$
$332$	1.02452	0.0562280
$333$	−0.724158	−0.0396836
$334$	−3.77640	−0.206635
$335$	−0.572170	−0.0312610
$336$	−5.69053	−0.310444
$337$	23.5185	1.28114	0.640568	−	0.767902i	$-0.278699\pi$
$337$	23.5185	1.28114	0.640568	+	0.767902i	$0.278699\pi$
$338$	11.1656	0.607326
$339$	−10.6491	−0.578378
$340$	2.43847	0.132245
$341$	0	0
$342$	0.199325	0.0107783
$343$	8.28912	0.447571
$344$	−7.94488	−0.428359
$345$	2.56978	0.138352
$346$	−1.92088	−0.103267
$347$	6.22613	0.334236	0.167118	−	0.985937i	$-0.446554\pi$
$347$	6.22613	0.334236	0.167118	+	0.985937i	$0.446554\pi$
$348$	−9.44639	−0.506379
$349$	−22.3550	−1.19664	−0.598319	−	0.801258i	$-0.704164\pi$
$349$	−22.3550	−1.19664	−0.598319	+	0.801258i	$0.704164\pi$
$350$	−15.9105	−0.850450
$351$	7.25173	0.387069
$352$	0	0
$353$	36.4241	1.93866	0.969329	−	0.245768i	$-0.0790402\pi$
$353$	36.4241	1.93866	0.969329	+	0.245768i	$0.0790402\pi$
$354$	−13.3463	−0.709348
$355$	−3.57444	−0.189712
$356$	−15.1613	−0.803548
$357$	24.4951	1.29642
$358$	12.9182	0.682746
$359$	−7.54684	−0.398307	−0.199153	−	0.979968i	$-0.563819\pi$
$359$	−7.54684	−0.398307	−0.199153	+	0.979968i	$0.563819\pi$
$360$	−0.112916	−0.00595118
$361$	1.00000	0.0526316
$362$	−17.7782	−0.934403
$363$	0	0
$364$	4.60546	0.241392
$365$	−7.95224	−0.416240
$366$	15.6975	0.820521
$367$	25.7229	1.34273	0.671363	−	0.741129i	$-0.265709\pi$
$367$	25.7229	1.34273	0.671363	+	0.741129i	$0.265709\pi$
$368$	−2.71065	−0.141302
$369$	−0.727103	−0.0378515
$370$	2.05808	0.106995
$371$	−14.4783	−0.751674
$372$	16.1068	0.835099
$373$	7.13854	0.369619	0.184810	−	0.982774i	$-0.440833\pi$
$373$	7.13854	0.369619	0.184810	+	0.982774i	$0.440833\pi$
$374$	0	0
$375$	9.17609	0.473851
$376$	−13.5117	−0.696811
$377$	7.64515	0.393745
$378$	−18.2059	−0.936408
$379$	−27.1836	−1.39633	−0.698165	−	0.715937i	$-0.746000\pi$
$379$	−27.1836	−1.39633	−0.698165	+	0.715937i	$0.746000\pi$
$380$	−0.566489	−0.0290603
$381$	−29.0482	−1.48818
$382$	20.7230	1.06028
$383$	−30.3425	−1.55043	−0.775215	−	0.631698i	$-0.782358\pi$
$383$	−30.3425	−1.55043	−0.775215	+	0.631698i	$0.782358\pi$
$384$	−1.67352	−0.0854015
$385$	0	0
$386$	12.1336	0.617584
$387$	−1.58362	−0.0804997
$388$	7.04118	0.357462
$389$	5.82380	0.295278	0.147639	−	0.989041i	$-0.452833\pi$
$389$	5.82380	0.295278	0.147639	+	0.989041i	$0.452833\pi$
$390$	−1.28403	−0.0650193
$391$	11.6681	0.590081
$392$	−4.56226	−0.230429
$393$	5.67516	0.286274
$394$	−14.2097	−0.715874
$395$	−4.02878	−0.202710
$396$	0	0
$397$	2.13644	0.107225	0.0536124	−	0.998562i	$-0.482926\pi$
$397$	2.13644	0.107225	0.0536124	+	0.998562i	$0.482926\pi$
$398$	0.976956	0.0489704
$399$	−5.69053	−0.284883
$400$	−4.67909	−0.233954
$401$	3.52506	0.176033	0.0880164	−	0.996119i	$-0.471947\pi$
$401$	3.52506	0.176033	0.0880164	+	0.996119i	$0.471947\pi$
$402$	−1.69030	−0.0843047
$403$	−13.0356	−0.649347
$404$	8.12037	0.404003
$405$	4.73715	0.235391
$406$	−19.1936	−0.952561
$407$	0	0
$408$	7.20373	0.356638
$409$	−13.0620	−0.645874	−0.322937	−	0.946421i	$-0.604670\pi$
$409$	−13.0620	−0.645874	−0.322937	+	0.946421i	$0.604670\pi$
$410$	2.06645	0.102055
$411$	0.102440	0.00505299
$412$	−3.00499	−0.148045
$413$	−27.1176	−1.33437
$414$	−0.540301	−0.0265544
$415$	−0.580381	−0.0284898
$416$	1.35442	0.0664057
$417$	28.0743	1.37480
$418$	0	0
$419$	24.4288	1.19342	0.596712	−	0.802456i	$-0.296474\pi$
$419$	24.4288	1.19342	0.596712	+	0.802456i	$0.296474\pi$
$420$	3.22362	0.157297
$421$	−11.0083	−0.536514	−0.268257	−	0.963347i	$-0.586448\pi$
$421$	−11.0083	−0.536514	−0.268257	+	0.963347i	$0.586448\pi$
$422$	−8.44427	−0.411061
$423$	−2.69322	−0.130949
$424$	−4.25789	−0.206782
$425$	20.1413	0.976997
$426$	−10.5596	−0.511615
$427$	31.8948	1.54350
$428$	7.33334	0.354471
$429$	0	0
$430$	4.50069	0.217042
$431$	13.5147	0.650978	0.325489	−	0.945546i	$-0.394471\pi$
$431$	13.5147	0.650978	0.325489	+	0.945546i	$0.394471\pi$
$432$	−5.35414	−0.257601
$433$	22.8978	1.10040	0.550199	−	0.835033i	$-0.314552\pi$
$433$	22.8978	1.10040	0.550199	+	0.835033i	$0.314552\pi$
$434$	32.7265	1.57092
$435$	5.35128	0.256574
$436$	−3.90465	−0.186999
$437$	−2.71065	−0.129668
$438$	−23.4925	−1.12252
$439$	4.19717	0.200320	0.100160	−	0.994971i	$-0.468065\pi$
$439$	4.19717	0.200320	0.100160	+	0.994971i	$0.468065\pi$
$440$	0	0
$441$	−0.909374	−0.0433035
$442$	−5.83013	−0.277311
$443$	−31.0222	−1.47391	−0.736954	−	0.675943i	$-0.763737\pi$
$443$	−31.0222	−1.47391	−0.736954	+	0.675943i	$0.763737\pi$
$444$	6.07998	0.288543
$445$	8.58873	0.407145
$446$	0.309593	0.0146596
$447$	−14.9815	−0.708599
$448$	−3.40033	−0.160651
$449$	−30.0246	−1.41695	−0.708474	−	0.705737i	$-0.750616\pi$
$449$	−30.0246	−1.41695	−0.708474	+	0.705737i	$0.750616\pi$
$450$	−0.932661	−0.0439661
$451$	0	0
$452$	−6.36327	−0.299303
$453$	14.2484	0.669449
$454$	17.8942	0.839817
$455$	−2.60894	−0.122309
$456$	−1.67352	−0.0783698
$457$	9.16793	0.428858	0.214429	−	0.976740i	$-0.431211\pi$
$457$	9.16793	0.428858	0.214429	+	0.976740i	$0.431211\pi$
$458$	4.77162	0.222963
$459$	23.0471	1.07575
$460$	1.53555	0.0715956
$461$	30.4549	1.41843	0.709213	−	0.704995i	$-0.249051\pi$
$461$	30.4549	1.41843	0.709213	+	0.704995i	$0.249051\pi$
$462$	0	0
$463$	16.4686	0.765359	0.382679	−	0.923881i	$-0.375001\pi$
$463$	16.4686	0.765359	0.382679	+	0.923881i	$0.375001\pi$
$464$	−5.64462	−0.262045
$465$	−9.12433	−0.423131
$466$	−12.5120	−0.579606
$467$	−36.8201	−1.70383	−0.851916	−	0.523679i	$-0.824559\pi$
$467$	−36.8201	−1.70383	−0.851916	+	0.523679i	$0.824559\pi$
$468$	0.269969	0.0124793
$469$	−3.43443	−0.158587
$470$	7.65422	0.353063
$471$	20.2289	0.932099
$472$	−7.97498	−0.367078
$473$	0	0
$474$	−11.9018	−0.546669
$475$	−4.67909	−0.214691
$476$	14.6369	0.670879
$477$	−0.848706	−0.0388596
$478$	11.2492	0.514528
$479$	31.1691	1.42415	0.712077	−	0.702102i	$-0.247755\pi$
$479$	31.1691	1.42415	0.712077	+	0.702102i	$0.247755\pi$
$480$	0.948032	0.0432715
$481$	−4.92065	−0.224362
$482$	23.4105	1.06632
$483$	15.4250	0.701864
$484$	0	0
$485$	−3.98875	−0.181120
$486$	−2.06794	−0.0938037
$487$	40.4109	1.83119	0.915597	−	0.402097i	$-0.131718\pi$
$487$	40.4109	1.83119	0.915597	+	0.402097i	$0.131718\pi$
$488$	9.37992	0.424609
$489$	22.8134	1.03166
$490$	2.58447	0.116755
$491$	9.86715	0.445298	0.222649	−	0.974899i	$-0.428530\pi$
$491$	9.86715	0.445298	0.222649	+	0.974899i	$0.428530\pi$
$492$	6.10471	0.275222
$493$	24.2974	1.09430
$494$	1.35442	0.0609380
$495$	0	0
$496$	9.62449	0.432153
$497$	−21.4555	−0.962410
$498$	−1.71456	−0.0768313
$499$	−13.4498	−0.602094	−0.301047	−	0.953609i	$-0.597336\pi$
$499$	−13.4498	−0.602094	−0.301047	+	0.953609i	$0.597336\pi$
$500$	5.48310	0.245212
$501$	6.31988	0.282352
$502$	27.2448	1.21599
$503$	22.5378	1.00491	0.502456	−	0.864603i	$-0.332430\pi$
$503$	22.5378	1.00491	0.502456	+	0.864603i	$0.332430\pi$
$504$	−0.677772	−0.0301904
$505$	−4.60010	−0.204702
$506$	0	0
$507$	−18.6858	−0.829866
$508$	−17.3575	−0.770115
$509$	4.87076	0.215893	0.107946	−	0.994157i	$-0.465573\pi$
$509$	4.87076	0.215893	0.107946	+	0.994157i	$0.465573\pi$
$510$	−4.08084	−0.180702
$511$	−47.7331	−2.11159
$512$	−1.00000	−0.0441942
$513$	−5.35414	−0.236391
$514$	−18.9406	−0.835435
$515$	1.70230	0.0750121
$516$	13.2959	0.585321
$517$	0	0
$518$	12.3536	0.542784
$519$	3.21464	0.141107
$520$	−0.767262	−0.0336466
$521$	6.08484	0.266582	0.133291	−	0.991077i	$-0.457446\pi$
$521$	6.08484	0.266582	0.133291	+	0.991077i	$0.457446\pi$
$522$	−1.12511	−0.0492449
$523$	−40.9388	−1.79013	−0.895065	−	0.445936i	$-0.852871\pi$
$523$	−40.9388	−1.79013	−0.895065	+	0.445936i	$0.852871\pi$
$524$	3.39115	0.148143
$525$	26.6265	1.16208
$526$	−28.1414	−1.22702
$527$	−41.4290	−1.80467
$528$	0	0
$529$	−15.6524	−0.680538
$530$	2.41205	0.104773
$531$	−1.58961	−0.0689834
$532$	−3.40033	−0.147423
$533$	−4.94067	−0.214004
$534$	25.3728	1.09799
$535$	−4.15426	−0.179604
$536$	−1.01003	−0.0436266
$537$	−21.6188	−0.932921
$538$	0.144365	0.00622400
$539$	0	0
$540$	3.03306	0.130522
$541$	−20.1046	−0.864366	−0.432183	−	0.901786i	$-0.642257\pi$
$541$	−20.1046	−0.864366	−0.432183	+	0.901786i	$0.642257\pi$
$542$	7.88292	0.338601
$543$	29.7522	1.27679
$544$	4.30453	0.184555
$545$	2.21194	0.0947493
$546$	−7.70734	−0.329844
$547$	−30.7418	−1.31442	−0.657212	−	0.753706i	$-0.728264\pi$
$547$	−30.7418	−1.31442	−0.657212	+	0.753706i	$0.728264\pi$
$548$	0.0612122	0.00261486
$549$	1.86965	0.0797949
$550$	0	0
$551$	−5.64462	−0.240469
$552$	4.53633	0.193079
$553$	−24.1826	−1.02835
$554$	−16.4244	−0.697805
$555$	−3.44424	−0.146200
$556$	16.7756	0.711443
$557$	−9.53569	−0.404040	−0.202020	−	0.979381i	$-0.564751\pi$
$557$	−9.53569	−0.404040	−0.202020	+	0.979381i	$0.564751\pi$
$558$	1.91841	0.0812126
$559$	−10.7607	−0.455128
$560$	1.92625	0.0813990
$561$	0	0
$562$	1.71799	0.0724692
$563$	38.1065	1.60600	0.802998	−	0.595981i	$-0.203237\pi$
$563$	38.1065	1.60600	0.802998	+	0.595981i	$0.203237\pi$
$564$	22.6121	0.952140
$565$	3.60472	0.151652
$566$	5.05430	0.212448
$567$	28.4346	1.19414
$568$	−6.30982	−0.264754
$569$	22.1572	0.928879	0.464439	−	0.885605i	$-0.346256\pi$
$569$	22.1572	0.928879	0.464439	+	0.885605i	$0.346256\pi$
$570$	0.948032	0.0397087
$571$	19.3268	0.808801	0.404400	−	0.914582i	$-0.367480\pi$
$571$	19.3268	0.808801	0.404400	+	0.914582i	$0.367480\pi$
$572$	0	0
$573$	−34.6804	−1.44880
$574$	12.4038	0.517725
$575$	12.6834	0.528933
$576$	−0.199325	−0.00830522
$577$	26.9783	1.12312	0.561560	−	0.827436i	$-0.310201\pi$
$577$	26.9783	1.12312	0.561560	+	0.827436i	$0.310201\pi$
$578$	−1.52902	−0.0635989
$579$	−20.3058	−0.843882
$580$	3.19761	0.132774
$581$	−3.48372	−0.144529
$582$	−11.7836	−0.488444
$583$	0	0
$584$	−14.0378	−0.580887
$585$	−0.152935	−0.00632307
$586$	−19.2526	−0.795319
$587$	−27.5285	−1.13622	−0.568112	−	0.822951i	$-0.692326\pi$
$587$	−27.5285	−1.13622	−0.568112	+	0.822951i	$0.692326\pi$
$588$	7.63504	0.314864
$589$	9.62449	0.396570
$590$	4.51774	0.185992
$591$	23.7802	0.978188
$592$	3.63304	0.149317
$593$	0.523811	0.0215103	0.0107552	−	0.999942i	$-0.496576\pi$
$593$	0.523811	0.0215103	0.0107552	+	0.999942i	$0.496576\pi$
$594$	0	0
$595$	−8.29162	−0.339923
$596$	−8.95206	−0.366691
$597$	−1.63496	−0.0669143
$598$	−3.67135	−0.150133
$599$	−2.50378	−0.102302	−0.0511508	−	0.998691i	$-0.516289\pi$
$599$	−2.50378	−0.102302	−0.0511508	+	0.998691i	$0.516289\pi$
$600$	7.83056	0.319681
$601$	19.0143	0.775608	0.387804	−	0.921742i	$-0.373234\pi$
$601$	19.0143	0.775608	0.387804	+	0.921742i	$0.373234\pi$
$602$	27.0152	1.10106
$603$	−0.201324	−0.00819855
$604$	8.51404	0.346431
$605$	0	0
$606$	−13.5896	−0.552040
$607$	37.9481	1.54026	0.770132	−	0.637884i	$-0.220190\pi$
$607$	37.9481	1.54026	0.770132	+	0.637884i	$0.220190\pi$
$608$	−1.00000	−0.0405554
$609$	32.1209	1.30160
$610$	−5.31362	−0.215142
$611$	−18.3004	−0.740356
$612$	0.858003	0.0346827
$613$	17.9322	0.724274	0.362137	−	0.932125i	$-0.382047\pi$
$613$	17.9322	0.724274	0.362137	+	0.932125i	$0.382047\pi$
$614$	−12.2585	−0.494712
$615$	−3.45825	−0.139450
$616$	0	0
$617$	−40.0578	−1.61267	−0.806333	−	0.591462i	$-0.798551\pi$
$617$	−40.0578	−1.61267	−0.806333	+	0.591462i	$0.798551\pi$
$618$	5.02892	0.202293
$619$	−24.2014	−0.972737	−0.486368	−	0.873754i	$-0.661679\pi$
$619$	−24.2014	−0.972737	−0.486368	+	0.873754i	$0.661679\pi$
$620$	−5.45217	−0.218965
$621$	14.5132	0.582395
$622$	5.48085	0.219762
$623$	51.5535	2.06545
$624$	−2.26664	−0.0907383
$625$	20.2893	0.811573
$626$	−8.75492	−0.349917
$627$	0	0
$628$	12.0876	0.482349
$629$	−15.6386	−0.623550
$630$	0.383951	0.0152970
$631$	48.8748	1.94567	0.972837	−	0.231489i	$-0.0743599\pi$
$631$	48.8748	1.94567	0.972837	+	0.231489i	$0.0743599\pi$
$632$	−7.11185	−0.282894
$633$	14.1317	0.561683
$634$	18.4394	0.732320
$635$	9.83284	0.390204
$636$	7.12568	0.282552
$637$	−6.17920	−0.244829
$638$	0	0
$639$	−1.25771	−0.0497541
$640$	0.566489	0.0223925
$641$	−13.6718	−0.540002	−0.270001	−	0.962860i	$-0.587024\pi$
$641$	−13.6718	−0.540002	−0.270001	+	0.962860i	$0.587024\pi$
$642$	−12.2725	−0.484357
$643$	35.5605	1.40237	0.701185	−	0.712980i	$-0.252655\pi$
$643$	35.5605	1.40237	0.701185	+	0.712980i	$0.252655\pi$
$644$	9.21711	0.363205
$645$	−7.53200	−0.296572
$646$	4.30453	0.169360
$647$	−34.6366	−1.36170	−0.680852	−	0.732421i	$-0.738391\pi$
$647$	−34.6366	−1.36170	−0.680852	+	0.732421i	$0.738391\pi$
$648$	8.36229	0.328502
$649$	0	0
$650$	−6.33743	−0.248574
$651$	−54.7685	−2.14655
$652$	13.6320	0.533870
$653$	7.70803	0.301638	0.150819	−	0.988561i	$-0.451809\pi$
$653$	7.70803	0.301638	0.150819	+	0.988561i	$0.451809\pi$
$654$	6.53452	0.255520
$655$	−1.92105	−0.0750617
$656$	3.64782	0.142424
$657$	−2.79808	−0.109164
$658$	45.9442	1.79109
$659$	2.66717	0.103898	0.0519491	−	0.998650i	$-0.483457\pi$
$659$	2.66717	0.103898	0.0519491	+	0.998650i	$0.483457\pi$
$660$	0	0
$661$	36.9320	1.43649	0.718244	−	0.695792i	$-0.244946\pi$
$661$	36.9320	1.43649	0.718244	+	0.695792i	$0.244946\pi$
$662$	−6.57096	−0.255387
$663$	9.75684	0.378924
$664$	−1.02452	−0.0397592
$665$	1.92625	0.0746968
$666$	0.724158	0.0280605
$667$	15.3006	0.592441
$668$	3.77640	0.146113
$669$	−0.518110	−0.0200313
$670$	0.572170	0.0221049
$671$	0	0
$672$	5.69053	0.219517
$673$	49.1070	1.89293	0.946467	−	0.322799i	$-0.104624\pi$
$673$	49.1070	1.89293	0.946467	+	0.322799i	$0.104624\pi$
$674$	−23.5185	−0.905900
$675$	25.0525	0.964271
$676$	−11.1656	−0.429445
$677$	3.66330	0.140792	0.0703961	−	0.997519i	$-0.477574\pi$
$677$	3.66330	0.140792	0.0703961	+	0.997519i	$0.477574\pi$
$678$	10.6491	0.408975
$679$	−23.9423	−0.918823
$680$	−2.43847	−0.0935112
$681$	−29.9463	−1.14755
$682$	0	0
$683$	−23.1273	−0.884941	−0.442471	−	0.896783i	$-0.645898\pi$
$683$	−23.1273	−0.884941	−0.442471	+	0.896783i	$0.645898\pi$
$684$	−0.199325	−0.00762139
$685$	−0.0346761	−0.00132490
$686$	−8.28912	−0.316480
$687$	−7.98541	−0.304662
$688$	7.94488	0.302896
$689$	−5.76696	−0.219704
$690$	−2.56978	−0.0978300
$691$	46.5289	1.77004	0.885021	−	0.465552i	$-0.154144\pi$
$691$	46.5289	1.77004	0.885021	+	0.465552i	$0.154144\pi$
$692$	1.92088	0.0730210
$693$	0	0
$694$	−6.22613	−0.236341
$695$	−9.50319	−0.360476
$696$	9.44639	0.358064
$697$	−15.7022	−0.594763
$698$	22.3550	0.846151
$699$	20.9390	0.791987
$700$	15.9105	0.601359
$701$	15.3125	0.578344	0.289172	−	0.957277i	$-0.406620\pi$
$701$	15.3125	0.578344	0.289172	+	0.957277i	$0.406620\pi$
$702$	−7.25173	−0.273699
$703$	3.63304	0.137023
$704$	0	0
$705$	−12.8095	−0.482434
$706$	−36.4241	−1.37084
$707$	−27.6119	−1.03845
$708$	13.3463	0.501584
$709$	35.5773	1.33613	0.668067	−	0.744101i	$-0.267122\pi$
$709$	35.5773	1.33613	0.668067	+	0.744101i	$0.267122\pi$
$710$	3.57444	0.134147
$711$	−1.41757	−0.0531631
$712$	15.1613	0.568195
$713$	−26.0886	−0.977027
$714$	−24.4951	−0.916706
$715$	0	0
$716$	−12.9182	−0.482774
$717$	−18.8258	−0.703064
$718$	7.54684	0.281645
$719$	41.8647	1.56129	0.780646	−	0.624974i	$-0.214890\pi$
$719$	41.8647	1.56129	0.780646	+	0.624974i	$0.214890\pi$
$720$	0.112916	0.00420812
$721$	10.2180	0.380537
$722$	−1.00000	−0.0372161
$723$	−39.1780	−1.45705
$724$	17.7782	0.660722
$725$	26.4117	0.980904
$726$	0	0
$727$	17.6492	0.654574	0.327287	−	0.944925i	$-0.393866\pi$
$727$	17.6492	0.654574	0.327287	+	0.944925i	$0.393866\pi$
$728$	−4.60546	−0.170690
$729$	28.5476	1.05732
$730$	7.95224	0.294326
$731$	−34.1990	−1.26490
$732$	−15.6975	−0.580196
$733$	42.7110	1.57756	0.788782	−	0.614672i	$-0.210712\pi$
$733$	42.7110	1.57756	0.788782	+	0.614672i	$0.210712\pi$
$734$	−25.7229	−0.949450
$735$	−4.32517	−0.159536
$736$	2.71065	0.0999159
$737$	0	0
$738$	0.727103	0.0267650
$739$	5.03814	0.185331	0.0926655	−	0.995697i	$-0.470461\pi$
$739$	5.03814	0.185331	0.0926655	+	0.995697i	$0.470461\pi$
$740$	−2.05808	−0.0756565
$741$	−2.26664	−0.0832672
$742$	14.4783	0.531514
$743$	−22.7649	−0.835165	−0.417583	−	0.908639i	$-0.637123\pi$
$743$	−22.7649	−0.835165	−0.417583	+	0.908639i	$0.637123\pi$
$744$	−16.1068	−0.590504
$745$	5.07125	0.185796
$746$	−7.13854	−0.261360
$747$	−0.204213	−0.00747177
$748$	0	0
$749$	−24.9358	−0.911134
$750$	−9.17609	−0.335063
$751$	−10.7708	−0.393031	−0.196515	−	0.980501i	$-0.562963\pi$
$751$	−10.7708	−0.393031	−0.196515	+	0.980501i	$0.562963\pi$
$752$	13.5117	0.492720
$753$	−45.5947	−1.66156
$754$	−7.64515	−0.278420
$755$	−4.82311	−0.175531
$756$	18.2059	0.662141
$757$	−39.7209	−1.44368	−0.721841	−	0.692059i	$-0.756704\pi$
$757$	−39.7209	−1.44368	−0.721841	+	0.692059i	$0.756704\pi$
$758$	27.1836	0.987355
$759$	0	0
$760$	0.566489	0.0205487
$761$	−50.5954	−1.83408	−0.917041	−	0.398794i	$-0.869429\pi$
$761$	−50.5954	−1.83408	−0.917041	+	0.398794i	$0.869429\pi$
$762$	29.0482	1.05230
$763$	13.2771	0.480664
$764$	−20.7230	−0.749733
$765$	−0.486049	−0.0175731
$766$	30.3425	1.09632
$767$	−10.8014	−0.390017
$768$	1.67352	0.0603880
$769$	21.0614	0.759495	0.379748	−	0.925090i	$-0.376011\pi$
$769$	21.0614	0.759495	0.379748	+	0.925090i	$0.376011\pi$
$770$	0	0
$771$	31.6976	1.14156
$772$	−12.1336	−0.436698
$773$	−6.82724	−0.245559	−0.122779	−	0.992434i	$-0.539181\pi$
$773$	−6.82724	−0.245559	−0.122779	+	0.992434i	$0.539181\pi$
$774$	1.58362	0.0569219
$775$	−45.0339	−1.61766
$776$	−7.04118	−0.252764
$777$	−20.6739	−0.741674
$778$	−5.82380	−0.208793
$779$	3.64782	0.130697
$780$	1.28403	0.0459756
$781$	0	0
$782$	−11.6681	−0.417250
$783$	30.2221	1.08005
$784$	4.56226	0.162938
$785$	−6.84751	−0.244398
$786$	−5.67516	−0.202426
$787$	45.8465	1.63425	0.817125	−	0.576461i	$-0.195567\pi$
$787$	45.8465	1.63425	0.817125	+	0.576461i	$0.195567\pi$
$788$	14.2097	0.506199
$789$	47.0953	1.67664
$790$	4.02878	0.143338
$791$	21.6372	0.769331
$792$	0	0
$793$	12.7043	0.451143
$794$	−2.13644	−0.0758194
$795$	−4.03662	−0.143164
$796$	−0.976956	−0.0346273
$797$	−39.1554	−1.38696	−0.693478	−	0.720478i	$-0.743923\pi$
$797$	−39.1554	−1.38696	−0.693478	+	0.720478i	$0.743923\pi$
$798$	5.69053	0.201443
$799$	−58.1615	−2.05761
$800$	4.67909	0.165431
$801$	3.02204	0.106778
$802$	−3.52506	−0.124474
$803$	0	0
$804$	1.69030	0.0596124
$805$	−5.22140	−0.184030
$806$	13.0356	0.459158
$807$	−0.241597	−0.00850463
$808$	−8.12037	−0.285674
$809$	−29.0753	−1.02223	−0.511116	−	0.859512i	$-0.670768\pi$
$809$	−29.0753	−1.02223	−0.511116	+	0.859512i	$0.670768\pi$
$810$	−4.73715	−0.166446
$811$	20.3401	0.714238	0.357119	−	0.934059i	$-0.383759\pi$
$811$	20.3401	0.714238	0.357119	+	0.934059i	$0.383759\pi$
$812$	19.1936	0.673562
$813$	−13.1922	−0.462672
$814$	0	0
$815$	−7.72238	−0.270503
$816$	−7.20373	−0.252181
$817$	7.94488	0.277956
$818$	13.0620	0.456702
$819$	−0.917985	−0.0320770
$820$	−2.06645	−0.0721636
$821$	10.1130	0.352947	0.176474	−	0.984305i	$-0.443531\pi$
$821$	10.1130	0.352947	0.176474	+	0.984305i	$0.443531\pi$
$822$	−0.102440	−0.00357301
$823$	17.1457	0.597662	0.298831	−	0.954306i	$-0.403403\pi$
$823$	17.1457	0.597662	0.298831	+	0.954306i	$0.403403\pi$
$824$	3.00499	0.104684
$825$	0	0
$826$	27.1176	0.943541
$827$	−46.5796	−1.61973	−0.809866	−	0.586615i	$-0.800460\pi$
$827$	−46.5796	−1.61973	−0.809866	+	0.586615i	$0.800460\pi$
$828$	0.540301	0.0187768
$829$	−32.4293	−1.12632	−0.563158	−	0.826350i	$-0.690414\pi$
$829$	−32.4293	−1.12632	−0.563158	+	0.826350i	$0.690414\pi$
$830$	0.580381	0.0201453
$831$	27.4866	0.953498
$832$	−1.35442	−0.0469559
$833$	−19.6384	−0.680431
$834$	−28.0743	−0.972134
$835$	−2.13929	−0.0740331
$836$	0	0
$837$	−51.5309	−1.78117
$838$	−24.4288	−0.843878
$839$	44.9176	1.55073	0.775363	−	0.631516i	$-0.217567\pi$
$839$	44.9176	1.55073	0.775363	+	0.631516i	$0.217567\pi$
$840$	−3.22362	−0.111226
$841$	2.86168	0.0986786
$842$	11.0083	0.379373
$843$	−2.87510	−0.0990237
$844$	8.44427	0.290664
$845$	6.32517	0.217592
$846$	2.69322	0.0925948
$847$	0	0
$848$	4.25789	0.146217
$849$	−8.45847	−0.290294
$850$	−20.1413	−0.690841
$851$	−9.84791	−0.337582
$852$	10.5596	0.361767
$853$	41.7985	1.43115	0.715577	−	0.698534i	$-0.246164\pi$
$853$	41.7985	1.43115	0.715577	+	0.698534i	$0.246164\pi$
$854$	−31.8948	−1.09142
$855$	0.112916	0.00386163
$856$	−7.33334	−0.250649
$857$	−20.6115	−0.704076	−0.352038	−	0.935986i	$-0.614511\pi$
$857$	−20.6115	−0.704076	−0.352038	+	0.935986i	$0.614511\pi$
$858$	0	0
$859$	49.0170	1.67244	0.836220	−	0.548394i	$-0.184761\pi$
$859$	49.0170	1.67244	0.836220	+	0.548394i	$0.184761\pi$
$860$	−4.50069	−0.153472
$861$	−20.7580	−0.707432
$862$	−13.5147	−0.460311
$863$	−10.0126	−0.340833	−0.170416	−	0.985372i	$-0.554511\pi$
$863$	−10.0126	−0.340833	−0.170416	+	0.985372i	$0.554511\pi$
$864$	5.35414	0.182152
$865$	−1.08816	−0.0369985
$866$	−22.8978	−0.778099
$867$	2.55885	0.0869031
$868$	−32.7265	−1.11081
$869$	0	0
$870$	−5.35128	−0.181425
$871$	−1.36800	−0.0463528
$872$	3.90465	0.132228
$873$	−1.40348	−0.0475008
$874$	2.71065	0.0916891
$875$	−18.6444	−0.630295
$876$	23.4925	0.793738
$877$	−40.9542	−1.38293	−0.691463	−	0.722412i	$-0.743033\pi$
$877$	−40.9542	−1.38293	−0.691463	+	0.722412i	$0.743033\pi$
$878$	−4.19717	−0.141648
$879$	32.2197	1.08674
$880$	0	0
$881$	−47.6017	−1.60374	−0.801870	−	0.597498i	$-0.796161\pi$
$881$	−47.6017	−1.60374	−0.801870	+	0.597498i	$0.796161\pi$
$882$	0.909374	0.0306202
$883$	−18.8140	−0.633142	−0.316571	−	0.948569i	$-0.602532\pi$
$883$	−18.8140	−0.633142	−0.316571	+	0.948569i	$0.602532\pi$
$884$	5.83013	0.196088
$885$	−7.56053	−0.254145
$886$	31.0222	1.04221
$887$	36.2917	1.21856	0.609278	−	0.792957i	$-0.291459\pi$
$887$	36.2917	1.21856	0.609278	+	0.792957i	$0.291459\pi$
$888$	−6.07998	−0.204031
$889$	59.0213	1.97951
$890$	−8.58873	−0.287895
$891$	0	0
$892$	−0.309593	−0.0103659
$893$	13.5117	0.452151
$894$	14.9815	0.501055
$895$	7.31800	0.244614
$896$	3.40033	0.113597
$897$	6.14408	0.205145
$898$	30.0246	1.00193
$899$	−54.3266	−1.81189
$900$	0.932661	0.0310887
$901$	−18.3283	−0.610603
$902$	0	0
$903$	−45.2106	−1.50451
$904$	6.36327	0.211639
$905$	−10.0712	−0.334777
$906$	−14.2484	−0.473372
$907$	−44.4140	−1.47474	−0.737371	−	0.675488i	$-0.763933\pi$
$907$	−44.4140	−1.47474	−0.737371	+	0.675488i	$0.763933\pi$
$908$	−17.8942	−0.593840
$909$	−1.61859	−0.0536854
$910$	2.60894	0.0864857
$911$	14.7495	0.488674	0.244337	−	0.969690i	$-0.421430\pi$
$911$	14.7495	0.488674	0.244337	+	0.969690i	$0.421430\pi$
$912$	1.67352	0.0554158
$913$	0	0
$914$	−9.16793	−0.303248
$915$	8.89246	0.293976
$916$	−4.77162	−0.157659
$917$	−11.5310	−0.380789
$918$	−23.0471	−0.760667
$919$	0.209240	0.00690220	0.00345110	−	0.999994i	$-0.498901\pi$
$919$	0.209240	0.00690220	0.00345110	+	0.999994i	$0.498901\pi$
$920$	−1.53555	−0.0506257
$921$	20.5148	0.675987
$922$	−30.4549	−1.00298
$923$	−8.54611	−0.281299
$924$	0	0
$925$	−16.9993	−0.558935
$926$	−16.4686	−0.541190
$927$	0.598971	0.0196728
$928$	5.64462	0.185294
$929$	12.5178	0.410695	0.205348	−	0.978689i	$-0.434168\pi$
$929$	12.5178	0.410695	0.205348	+	0.978689i	$0.434168\pi$
$930$	9.12433	0.299199
$931$	4.56226	0.149522
$932$	12.5120	0.409843
$933$	−9.17233	−0.300288
$934$	36.8201	1.20479
$935$	0	0
$936$	−0.269969	−0.00882422
$937$	−38.2958	−1.25107	−0.625534	−	0.780197i	$-0.715119\pi$
$937$	−38.2958	−1.25107	−0.625534	+	0.780197i	$0.715119\pi$
$938$	3.43443	0.112138
$939$	14.6516	0.478135
$940$	−7.65422	−0.249653
$941$	−58.5062	−1.90725	−0.953623	−	0.301002i	$-0.902679\pi$
$941$	−58.5062	−1.90725	−0.953623	+	0.301002i	$0.902679\pi$
$942$	−20.2289	−0.659094
$943$	−9.88797	−0.321997
$944$	7.97498	0.259563
$945$	−10.3134	−0.335496
$946$	0	0
$947$	−34.2392	−1.11262	−0.556312	−	0.830974i	$-0.687784\pi$
$947$	−34.2392	−1.11262	−0.556312	+	0.830974i	$0.687784\pi$
$948$	11.9018	0.386553
$949$	−19.0130	−0.617187
$950$	4.67909	0.151810
$951$	−30.8587	−1.00066
$952$	−14.6369	−0.474383
$953$	24.3715	0.789471	0.394736	−	0.918795i	$-0.370836\pi$
$953$	24.3715	0.789471	0.394736	+	0.918795i	$0.370836\pi$
$954$	0.848706	0.0274779
$955$	11.7394	0.379877
$956$	−11.2492	−0.363826
$957$	0	0
$958$	−31.1691	−1.00703
$959$	−0.208142	−0.00672126
$960$	−0.948032	−0.0305976
$961$	61.6309	1.98809
$962$	4.92065	0.158648
$963$	−1.46172	−0.0471033
$964$	−23.4105	−0.754003
$965$	6.87355	0.221268
$966$	−15.4250	−0.496293
$967$	−44.3499	−1.42620	−0.713099	−	0.701063i	$-0.752709\pi$
$967$	−44.3499	−1.42620	−0.713099	+	0.701063i	$0.752709\pi$
$968$	0	0
$969$	−7.20373	−0.231417
$970$	3.98875	0.128071
$971$	−19.7687	−0.634408	−0.317204	−	0.948357i	$-0.602744\pi$
$971$	−19.7687	−0.634408	−0.317204	+	0.948357i	$0.602744\pi$
$972$	2.06794	0.0663292
$973$	−57.0426	−1.82870
$974$	−40.4109	−1.29485
$975$	10.6058	0.339658
$976$	−9.37992	−0.300244
$977$	18.2277	0.583155	0.291577	−	0.956547i	$-0.405820\pi$
$977$	18.2277	0.583155	0.291577	+	0.956547i	$0.405820\pi$
$978$	−22.8134	−0.729494
$979$	0	0
$980$	−2.58447	−0.0825579
$981$	0.778296	0.0248491
$982$	−9.86715	−0.314873
$983$	56.2431	1.79388	0.896939	−	0.442155i	$-0.145786\pi$
$983$	56.2431	1.79388	0.896939	+	0.442155i	$0.145786\pi$
$984$	−6.10471	−0.194611
$985$	−8.04964	−0.256483
$986$	−24.2974	−0.773788
$987$	−76.8886	−2.44739
$988$	−1.35442	−0.0430897
$989$	−21.5358	−0.684798
$990$	0	0
$991$	48.6909	1.54672	0.773358	−	0.633970i	$-0.218576\pi$
$991$	48.6909	1.54672	0.773358	+	0.633970i	$0.218576\pi$
$992$	−9.62449	−0.305578
$993$	10.9966	0.348968
$994$	21.4555	0.680527
$995$	0.553435	0.0175451
$996$	1.71456	0.0543279
$997$	50.2387	1.59107	0.795537	−	0.605905i	$-0.207189\pi$
$997$	50.2387	1.59107	0.795537	+	0.605905i	$0.207189\pi$
$998$	13.4498	0.425745
$999$	−19.4518	−0.615429

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4598.2.a.by.1.5	✓	8
11.10	odd	2		4598.2.a.cb.1.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4598.2.a.by.1.5	✓	8	1.1	even	1	trivial
4598.2.a.cb.1.5	yes	8	11.10	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:	\( N \)	\(=\)	\( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4598.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.67352 q^{3} +1.00000 q^{4} -0.566489 q^{5} -1.67352 q^{6} -3.40033 q^{7} -1.00000 q^{8} -0.199325 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.67352 q^{3} +1.00000 q^{4} -0.566489 q^{5} -1.67352 q^{6} -3.40033 q^{7} -1.00000 q^{8} -0.199325 q^{9} +0.566489 q^{10} +1.67352 q^{12} -1.35442 q^{13} +3.40033 q^{14} -0.948032 q^{15} +1.00000 q^{16} -4.30453 q^{17} +0.199325 q^{18} +1.00000 q^{19} -0.566489 q^{20} -5.69053 q^{21} -2.71065 q^{23} -1.67352 q^{24} -4.67909 q^{25} +1.35442 q^{26} -5.35414 q^{27} -3.40033 q^{28} -5.64462 q^{29} +0.948032 q^{30} +9.62449 q^{31} -1.00000 q^{32} +4.30453 q^{34} +1.92625 q^{35} -0.199325 q^{36} +3.63304 q^{37} -1.00000 q^{38} -2.26664 q^{39} +0.566489 q^{40} +3.64782 q^{41} +5.69053 q^{42} +7.94488 q^{43} +0.112916 q^{45} +2.71065 q^{46} +13.5117 q^{47} +1.67352 q^{48} +4.56226 q^{49} +4.67909 q^{50} -7.20373 q^{51} -1.35442 q^{52} +4.25789 q^{53} +5.35414 q^{54} +3.40033 q^{56} +1.67352 q^{57} +5.64462 q^{58} +7.97498 q^{59} -0.948032 q^{60} -9.37992 q^{61} -9.62449 q^{62} +0.677772 q^{63} +1.00000 q^{64} +0.767262 q^{65} +1.01003 q^{67} -4.30453 q^{68} -4.53633 q^{69} -1.92625 q^{70} +6.30982 q^{71} +0.199325 q^{72} +14.0378 q^{73} -3.63304 q^{74} -7.83056 q^{75} +1.00000 q^{76} +2.26664 q^{78} +7.11185 q^{79} -0.566489 q^{80} -8.36229 q^{81} -3.64782 q^{82} +1.02452 q^{83} -5.69053 q^{84} +2.43847 q^{85} -7.94488 q^{86} -9.44639 q^{87} -15.1613 q^{89} -0.112916 q^{90} +4.60546 q^{91} -2.71065 q^{92} +16.1068 q^{93} -13.5117 q^{94} -0.566489 q^{95} -1.67352 q^{96} +7.04118 q^{97} -4.56226 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^6 - 4 * q^7 - 8 * q^8 + 22 * q^9	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9} + 8 q^{12} + 12 q^{13} + 4 q^{14} + 4 q^{15} + 8 q^{16} + 4 q^{17} - 22 q^{18} + 8 q^{19} + 20 q^{21} + 14 q^{23} - 8 q^{24} + 36 q^{25} - 12 q^{26} + 32 q^{27} - 4 q^{28} + 2 q^{29} - 4 q^{30} - 8 q^{32} - 4 q^{34} - 36 q^{35} + 22 q^{36} + 24 q^{37} - 8 q^{38} - 16 q^{39} - 8 q^{41} - 20 q^{42} - 8 q^{43} + 16 q^{45} - 14 q^{46} - 16 q^{47} + 8 q^{48} + 34 q^{49} - 36 q^{50} - 18 q^{51} + 12 q^{52} + 36 q^{53} - 32 q^{54} + 4 q^{56} + 8 q^{57} - 2 q^{58} - 24 q^{59} + 4 q^{60} - 12 q^{61} - 24 q^{63} + 8 q^{64} - 16 q^{65} + 16 q^{67} + 4 q^{68} + 4 q^{69} + 36 q^{70} + 4 q^{71} - 22 q^{72} + 20 q^{73} - 24 q^{74} + 40 q^{75} + 8 q^{76} + 16 q^{78} + 12 q^{79} + 40 q^{81} + 8 q^{82} - 20 q^{83} + 20 q^{84} - 12 q^{85} + 8 q^{86} + 36 q^{87} + 8 q^{89} - 16 q^{90} - 24 q^{91} + 14 q^{92} + 12 q^{93} + 16 q^{94} - 8 q^{96} + 4 q^{97} - 34 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^6 - 4 * q^7 - 8 * q^8 + 22 * q^9 + 8 * q^12 + 12 * q^13 + 4 * q^14 + 4 * q^15 + 8 * q^16 + 4 * q^17 - 22 * q^18 + 8 * q^19 + 20 * q^21 + 14 * q^23 - 8 * q^24 + 36 * q^25 - 12 * q^26 + 32 * q^27 - 4 * q^28 + 2 * q^29 - 4 * q^30 - 8 * q^32 - 4 * q^34 - 36 * q^35 + 22 * q^36 + 24 * q^37 - 8 * q^38 - 16 * q^39 - 8 * q^41 - 20 * q^42 - 8 * q^43 + 16 * q^45 - 14 * q^46 - 16 * q^47 + 8 * q^48 + 34 * q^49 - 36 * q^50 - 18 * q^51 + 12 * q^52 + 36 * q^53 - 32 * q^54 + 4 * q^56 + 8 * q^57 - 2 * q^58 - 24 * q^59 + 4 * q^60 - 12 * q^61 - 24 * q^63 + 8 * q^64 - 16 * q^65 + 16 * q^67 + 4 * q^68 + 4 * q^69 + 36 * q^70 + 4 * q^71 - 22 * q^72 + 20 * q^73 - 24 * q^74 + 40 * q^75 + 8 * q^76 + 16 * q^78 + 12 * q^79 + 40 * q^81 + 8 * q^82 - 20 * q^83 + 20 * q^84 - 12 * q^85 + 8 * q^86 + 36 * q^87 + 8 * q^89 - 16 * q^90 - 24 * q^91 + 14 * q^92 + 12 * q^93 + 16 * q^94 - 8 * q^96 + 4 * q^97 - 34 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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