LMFDB - Embedded newform 1875.2.b.d.1249.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1249,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, 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("1875.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1875.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$14.9719503790$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.324000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 $ x^8 + 9x^6 + 26x^4 + 24*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.7
Root			$1.95630i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.d.1249.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.82709i q^{2} +1.00000i q^{3} -1.33826 q^{4} -1.82709 q^{6} -1.44512i q^{7} +1.20906i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.82709i q^{2} +1.00000i q^{3} -1.33826 q^{4} -1.82709 q^{6} -1.44512i q^{7} +1.20906i q^{8} -1.00000 q^{9} -2.12920 q^{11} -1.33826i q^{12} -5.70353i q^{13} +2.64037 q^{14} -4.88558 q^{16} -4.15622i q^{17} -1.82709i q^{18} -1.70353 q^{19} +1.44512 q^{21} -3.89025i q^{22} -0.323478i q^{23} -1.20906 q^{24} +10.4209 q^{26} -1.00000i q^{27} +1.93395i q^{28} +8.74724 q^{29} -8.45991 q^{31} -6.50828i q^{32} -2.12920i q^{33} +7.59378 q^{34} +1.33826 q^{36} -1.75170i q^{37} -3.11251i q^{38} +5.70353 q^{39} -6.87802 q^{41} +2.64037i q^{42} -11.1411i q^{43} +2.84943 q^{44} +0.591023 q^{46} -12.5982i q^{47} -4.88558i q^{48} +4.91161 q^{49} +4.15622 q^{51} +7.63282i q^{52} +8.34451i q^{53} +1.82709 q^{54} +1.74724 q^{56} -1.70353i q^{57} +15.9820i q^{58} +2.12474 q^{59} -5.38952 q^{61} -15.4570i q^{62} +1.44512i q^{63} +2.12007 q^{64} +3.89025 q^{66} +7.13078i q^{67} +5.56210i q^{68} +0.323478 q^{69} +2.67461 q^{71} -1.20906i q^{72} +6.28253i q^{73} +3.20052 q^{74} +2.27977 q^{76} +3.07697i q^{77} +10.4209i q^{78} -8.37092 q^{79} +1.00000 q^{81} -12.5668i q^{82} -14.5872i q^{83} -1.93395 q^{84} +20.3558 q^{86} +8.74724i q^{87} -2.57433i q^{88} -2.68119 q^{89} -8.24232 q^{91} +0.432897i q^{92} -8.45991i q^{93} +23.0181 q^{94} +6.50828 q^{96} -8.55105i q^{97} +8.97397i q^{98} +2.12920 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{4} - 2 q^{6} - 8 q^{9}+O(q^{10}) $ 8 * q - 2 * q^4 - 2 * q^6 - 8 * q^9	$ 8 q - 2 q^{4} - 2 q^{6} - 8 q^{9} - 12 q^{11} + 20 q^{14} - 18 q^{16} + 18 q^{19} - 10 q^{21} - 6 q^{24} - 4 q^{26} + 56 q^{29} - 20 q^{31} - 14 q^{34} + 2 q^{36} + 14 q^{39} + 18 q^{44} + 10 q^{46} + 6 q^{49} + 14 q^{51} + 2 q^{54} - 8 q^{59} - 86 q^{61} + 14 q^{64} - 12 q^{66} + 20 q^{69} - 54 q^{71} - 10 q^{74} + 18 q^{76} - 20 q^{79} + 8 q^{81} + 10 q^{84} + 48 q^{86} + 18 q^{89} + 10 q^{91} + 44 q^{94} + 12 q^{99}+O(q^{100}) $ 8 * q - 2 * q^4 - 2 * q^6 - 8 * q^9 - 12 * q^11 + 20 * q^14 - 18 * q^16 + 18 * q^19 - 10 * q^21 - 6 * q^24 - 4 * q^26 + 56 * q^29 - 20 * q^31 - 14 * q^34 + 2 * q^36 + 14 * q^39 + 18 * q^44 + 10 * q^46 + 6 * q^49 + 14 * q^51 + 2 * q^54 - 8 * q^59 - 86 * q^61 + 14 * q^64 - 12 * q^66 + 20 * q^69 - 54 * q^71 - 10 * q^74 + 18 * q^76 - 20 * q^79 + 8 * q^81 + 10 * q^84 + 48 * q^86 + 18 * q^89 + 10 * q^91 + 44 * q^94 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times$.

$n$	$626$	$1252$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.82709i	1.29195i	0.763359	+	0.645974i	$0.223549\pi$
$2$	1.82709i	1.29195i	−0.763359	+	0.645974i	$0.776451\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.33826	−0.669131
$5$	0	0
$5$	0	0
$6$	−1.82709	−0.745907
$7$	− 1.44512i	− 0.546206i	−0.961985	−	0.273103i	$-0.911950\pi$
$7$	− 1.44512i	− 0.546206i	0.961985	−	0.273103i	$-0.0880500\pi$
$8$	1.20906i	0.427466i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−2.12920	−0.641979	−0.320990	−	0.947083i	$-0.604015\pi$
$11$	−2.12920	−0.641979	−0.320990	+	0.947083i	$0.604015\pi$
$12$	− 1.33826i	− 0.386323i
$13$	− 5.70353i	− 1.58188i	−0.611897	−	0.790938i	$-0.709593\pi$
$13$	− 5.70353i	− 1.58188i	0.611897	−	0.790938i	$-0.290407\pi$
$14$	2.64037	0.705670
$15$	0	0
$16$	−4.88558	−1.22139
$17$	− 4.15622i	− 1.00803i	−0.863695	−	0.504015i	$-0.831856\pi$
$17$	− 4.15622i	− 1.00803i	0.863695	−	0.504015i	$-0.168144\pi$
$18$	− 1.82709i	− 0.430649i
$19$	−1.70353	−0.390817	−0.195409	−	0.980722i	$-0.562603\pi$
$19$	−1.70353	−0.390817	−0.195409	+	0.980722i	$0.562603\pi$
$20$	0	0
$21$	1.44512	0.315352
$22$	− 3.89025i	− 0.829404i
$23$	− 0.323478i	− 0.0674497i	−0.999431	−	0.0337249i	$-0.989263\pi$
$23$	− 0.323478i	− 0.0674497i	0.999431	−	0.0337249i	$-0.0107370\pi$
$24$	−1.20906	−0.246798
$25$	0	0
$26$	10.4209	2.04370
$27$	− 1.00000i	− 0.192450i
$28$	1.93395i	0.365483i
$29$	8.74724	1.62432	0.812161	−	0.583434i	$-0.198291\pi$
$29$	8.74724	1.62432	0.812161	+	0.583434i	$0.198291\pi$
$30$	0	0
$31$	−8.45991	−1.51944	−0.759722	−	0.650248i	$-0.774665\pi$
$31$	−8.45991	−1.51944	−0.759722	+	0.650248i	$0.774665\pi$
$32$	− 6.50828i	− 1.15051i
$33$	− 2.12920i	− 0.370647i
$34$	7.59378	1.30232
$35$	0	0
$36$	1.33826	0.223044
$37$	− 1.75170i	− 0.287978i	−0.989579	−	0.143989i	$-0.954007\pi$
$37$	− 1.75170i	− 0.287978i	0.989579	−	0.143989i	$-0.0459931\pi$
$38$	− 3.11251i	− 0.504916i
$39$	5.70353	0.913296
$40$	0	0
$41$	−6.87802	−1.07417	−0.537083	−	0.843529i	$-0.680474\pi$
$41$	−6.87802	−1.07417	−0.537083	+	0.843529i	$0.680474\pi$
$42$	2.64037i	0.407419i
$43$	− 11.1411i	− 1.69900i	−0.527587	−	0.849501i	$-0.676903\pi$
$43$	− 11.1411i	− 1.69900i	0.527587	−	0.849501i	$-0.323097\pi$
$44$	2.84943	0.429568
$45$	0	0
$46$	0.591023	0.0871416
$47$	− 12.5982i	− 1.83764i	−0.394673	−	0.918822i	$-0.629142\pi$
$47$	− 12.5982i	− 1.83764i	0.394673	−	0.918822i	$-0.370858\pi$
$48$	− 4.88558i	− 0.705173i
$49$	4.91161	0.701659
$50$	0	0
$51$	4.15622	0.581987
$52$	7.63282i	1.05848i
$53$	8.34451i	1.14621i	0.819483	+	0.573103i	$0.194261\pi$
$53$	8.34451i	1.14621i	−0.819483	+	0.573103i	$0.805739\pi$
$54$	1.82709	0.248636
$55$	0	0
$56$	1.74724	0.233485
$57$	− 1.70353i	− 0.225639i
$58$	15.9820i	2.09854i
$59$	2.12474	0.276617	0.138309	−	0.990389i	$-0.455833\pi$
$59$	2.12474	0.276617	0.138309	+	0.990389i	$0.455833\pi$
$60$	0	0
$61$	−5.38952	−0.690058	−0.345029	−	0.938592i	$-0.612131\pi$
$61$	−5.38952	−0.690058	−0.345029	+	0.938592i	$0.612131\pi$
$62$	− 15.4570i	− 1.96304i
$63$	1.44512i	0.182069i
$64$	2.12007	0.265008
$65$	0	0
$66$	3.89025	0.478857
$67$	7.13078i	0.871164i	0.900149	+	0.435582i	$0.143457\pi$
$67$	7.13078i	0.871164i	−0.900149	+	0.435582i	$0.856543\pi$
$68$	5.56210i	0.674504i
$69$	0.323478	0.0389421
$70$	0	0
$71$	2.67461	0.317418	0.158709	−	0.987325i	$-0.449267\pi$
$71$	2.67461	0.317418	0.158709	+	0.987325i	$0.449267\pi$
$72$	− 1.20906i	− 0.142489i
$73$	6.28253i	0.735315i	0.929961	+	0.367657i	$0.119840\pi$
$73$	6.28253i	0.735315i	−0.929961	+	0.367657i	$0.880160\pi$
$74$	3.20052	0.372053
$75$	0	0
$76$	2.27977	0.261508
$77$	3.07697i	0.350653i
$78$	10.4209i	1.17993i
$79$	−8.37092	−0.941802	−0.470901	−	0.882186i	$-0.656071\pi$
$79$	−8.37092	−0.941802	−0.470901	+	0.882186i	$0.656071\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 12.5668i	− 1.38777i
$83$	− 14.5872i	− 1.60115i	−0.599230	−	0.800577i	$-0.704527\pi$
$83$	− 14.5872i	− 1.60115i	0.599230	−	0.800577i	$-0.295473\pi$
$84$	−1.93395	−0.211012
$85$	0	0
$86$	20.3558	2.19502
$87$	8.74724i	0.937802i
$88$	− 2.57433i	− 0.274424i
$89$	−2.68119	−0.284206	−0.142103	−	0.989852i	$-0.545386\pi$
$89$	−2.68119	−0.284206	−0.142103	+	0.989852i	$0.545386\pi$
$90$	0	0
$91$	−8.24232	−0.864030
$92$	0.432897i	0.0451327i
$93$	− 8.45991i	− 0.877252i
$94$	23.0181	2.37414
$95$	0	0
$96$	6.50828	0.664249
$97$	− 8.55105i	− 0.868228i	−0.900858	−	0.434114i	$-0.857061\pi$
$97$	− 8.55105i	− 0.868228i	0.900858	−	0.434114i	$-0.142939\pi$
$98$	8.97397i	0.906507i
$99$	2.12920	0.213993
$100$	0	0
$101$	9.85377	0.980487	0.490243	−	0.871586i	$-0.336908\pi$
$101$	9.85377	0.980487	0.490243	+	0.871586i	$0.336908\pi$
$102$	7.59378i	0.751897i
$103$	− 16.1279i	− 1.58913i	−0.607180	−	0.794564i	$-0.707699\pi$
$103$	− 16.1279i	− 1.58913i	0.607180	−	0.794564i	$-0.292301\pi$
$104$	6.89590	0.676198
$105$	0	0
$106$	−15.2462	−1.48084
$107$	1.81198i	0.175170i	0.996157	+	0.0875852i	$0.0279150\pi$
$107$	1.81198i	0.175170i	−0.996157	+	0.0875852i	$0.972085\pi$
$108$	1.33826i	0.128774i
$109$	18.4646	1.76859	0.884293	−	0.466933i	$-0.154641\pi$
$109$	18.4646	1.76859	0.884293	+	0.466933i	$0.154641\pi$
$110$	0	0
$111$	1.75170	0.166264
$112$	7.06027i	0.667133i
$113$	13.3653i	1.25730i	0.777689	+	0.628650i	$0.216392\pi$
$113$	13.3653i	1.25730i	−0.777689	+	0.628650i	$0.783608\pi$
$114$	3.11251	0.291513
$115$	0	0
$116$	−11.7061	−1.08688
$117$	5.70353i	0.527292i
$118$	3.88209i	0.357375i
$119$	−6.00625	−0.550592
$120$	0	0
$121$	−6.46649	−0.587863
$122$	− 9.84715i	− 0.891519i
$123$	− 6.87802i	− 0.620170i
$124$	11.3216	1.01671
$125$	0	0
$126$	−2.64037	−0.235223
$127$	− 1.01381i	− 0.0899608i	−0.998988	−	0.0449804i	$-0.985677\pi$
$127$	− 1.01381i	− 0.0899608i	0.998988	−	0.0449804i	$-0.0143225\pi$
$128$	− 9.14301i	− 0.808136i
$129$	11.1411	0.980919
$130$	0	0
$131$	2.08550	0.182211	0.0911055	−	0.995841i	$-0.470960\pi$
$131$	2.08550	0.182211	0.0911055	+	0.995841i	$0.470960\pi$
$132$	2.84943i	0.248011i
$133$	2.46182i	0.213467i
$134$	−13.0286	−1.12550
$135$	0	0
$136$	5.02510	0.430899
$137$	16.9365i	1.44698i	0.690333	+	0.723492i	$0.257464\pi$
$137$	16.9365i	1.44698i	−0.690333	+	0.723492i	$0.742536\pi$
$138$	0.591023i	0.0503112i
$139$	−14.8883	−1.26281	−0.631406	−	0.775452i	$-0.717522\pi$
$139$	−14.8883	−1.26281	−0.631406	+	0.775452i	$0.717522\pi$
$140$	0	0
$141$	12.5982	1.06096
$142$	4.88676i	0.410088i
$143$	12.1440i	1.01553i
$144$	4.88558	0.407132
$145$	0	0
$146$	−11.4788	−0.949989
$147$	4.91161i	0.405103i
$148$	2.34424i	0.192695i
$149$	4.70991	0.385851	0.192925	−	0.981213i	$-0.438202\pi$
$149$	4.70991	0.385851	0.192925	+	0.981213i	$0.438202\pi$
$150$	0	0
$151$	−9.63091	−0.783752	−0.391876	−	0.920018i	$-0.628174\pi$
$151$	−9.63091	−0.783752	−0.391876	+	0.920018i	$0.628174\pi$
$152$	− 2.05967i	− 0.167061i
$153$	4.15622i	0.336010i
$154$	−5.62190	−0.453025
$155$	0	0
$156$	−7.63282	−0.611114
$157$	18.5557i	1.48091i	0.672107	+	0.740454i	$0.265389\pi$
$157$	18.5557i	1.48091i	−0.672107	+	0.740454i	$0.734611\pi$
$158$	− 15.2944i	− 1.21676i
$159$	−8.34451	−0.661763
$160$	0	0
$161$	−0.467465	−0.0368414
$162$	1.82709i	0.143550i
$163$	− 0.451705i	− 0.0353803i	−0.999844	−	0.0176902i	$-0.994369\pi$
$163$	− 0.451705i	− 0.0353803i	0.999844	−	0.0176902i	$-0.00563124\pi$
$164$	9.20459	0.718758
$165$	0	0
$166$	26.6521	2.06861
$167$	4.72648i	0.365746i	0.983137	+	0.182873i	$0.0585397\pi$
$167$	4.72648i	0.365746i	−0.983137	+	0.182873i	$0.941460\pi$
$168$	1.74724i	0.134802i
$169$	−19.5303	−1.50233
$170$	0	0
$171$	1.70353	0.130272
$172$	14.9097i	1.13685i
$173$	− 2.29489i	− 0.174477i	−0.996187	−	0.0872385i	$-0.972196\pi$
$173$	− 2.29489i	− 0.174477i	0.996187	−	0.0872385i	$-0.0278042\pi$
$174$	−15.9820	−1.21159
$175$	0	0
$176$	10.4024	0.784110
$177$	2.12474i	0.159705i
$178$	− 4.89878i	− 0.367179i
$179$	−2.68757	−0.200878	−0.100439	−	0.994943i	$-0.532025\pi$
$179$	−2.68757	−0.200878	−0.100439	+	0.994943i	$0.532025\pi$
$180$	0	0
$181$	−23.9088	−1.77712	−0.888562	−	0.458756i	$-0.848295\pi$
$181$	−23.9088	−1.77712	−0.888562	+	0.458756i	$0.848295\pi$
$182$	− 15.0595i	− 1.11628i
$183$	− 5.38952i	− 0.398405i
$184$	0.391103	0.0288325
$185$	0	0
$186$	15.4570	1.13336
$187$	8.84943i	0.647135i
$188$	16.8597i	1.22962i
$189$	−1.44512	−0.105117
$190$	0	0
$191$	−3.91824	−0.283514	−0.141757	−	0.989902i	$-0.545275\pi$
$191$	−3.91824	−0.283514	−0.141757	+	0.989902i	$0.545275\pi$
$192$	2.12007i	0.153003i
$193$	8.54752i	0.615264i	0.951505	+	0.307632i	$0.0995366\pi$
$193$	8.54752i	0.615264i	−0.951505	+	0.307632i	$0.900463\pi$
$194$	15.6236	1.12171
$195$	0	0
$196$	−6.57302	−0.469502
$197$	2.45235i	0.174723i	0.996177	+	0.0873614i	$0.0278435\pi$
$197$	2.45235i	0.174723i	−0.996177	+	0.0873614i	$0.972157\pi$
$198$	3.89025i	0.276468i
$199$	10.4345	0.739680	0.369840	−	0.929095i	$-0.379412\pi$
$199$	10.4345	0.739680	0.369840	+	0.929095i	$0.379412\pi$
$200$	0	0
$201$	−7.13078	−0.502967
$202$	18.0037i	1.26674i
$203$	− 12.6409i	− 0.887214i
$204$	−5.56210	−0.389425
$205$	0	0
$206$	29.4671	2.05307
$207$	0.323478i	0.0224832i
$208$	27.8651i	1.93209i
$209$	3.62717	0.250897
$210$	0	0
$211$	19.2618	1.32604	0.663018	−	0.748604i	$-0.269275\pi$
$211$	19.2618	1.32604	0.663018	+	0.748604i	$0.269275\pi$
$212$	− 11.1671i	− 0.766962i
$213$	2.67461i	0.183261i
$214$	−3.31065	−0.226311
$215$	0	0
$216$	1.20906	0.0822659
$217$	12.2256i	0.829929i
$218$	33.7365i	2.28492i
$219$	−6.28253	−0.424534
$220$	0	0
$221$	−23.7051	−1.59458
$222$	3.20052i	0.214805i
$223$	− 7.61706i	− 0.510076i	−0.966931	−	0.255038i	$-0.917912\pi$
$223$	− 7.61706i	− 0.510076i	0.966931	−	0.255038i	$-0.0820880\pi$
$224$	−9.40528	−0.628417
$225$	0	0
$226$	−24.4196	−1.62437
$227$	− 1.31803i	− 0.0874810i	−0.999043	−	0.0437405i	$-0.986073\pi$
$227$	− 1.31803i	− 0.0874810i	0.999043	−	0.0437405i	$-0.0139275\pi$
$228$	2.27977i	0.150982i
$229$	−1.12632	−0.0744292	−0.0372146	−	0.999307i	$-0.511849\pi$
$229$	−1.12632	−0.0744292	−0.0372146	+	0.999307i	$0.511849\pi$
$230$	0	0
$231$	−3.07697	−0.202450
$232$	10.5759i	0.694342i
$233$	− 26.6071i	− 1.74309i	−0.490319	−	0.871543i	$-0.663120\pi$
$233$	− 26.6071i	− 1.74309i	0.490319	−	0.871543i	$-0.336880\pi$
$234$	−10.4209	−0.681234
$235$	0	0
$236$	−2.84345	−0.185093
$237$	− 8.37092i	− 0.543750i
$238$	− 10.9740i	− 0.711337i
$239$	24.8692	1.60865	0.804326	−	0.594188i	$-0.202527\pi$
$239$	24.8692	1.60865	0.804326	+	0.594188i	$0.202527\pi$
$240$	0	0
$241$	−12.3532	−0.795743	−0.397871	−	0.917441i	$-0.630251\pi$
$241$	−12.3532	−0.795743	−0.397871	+	0.917441i	$0.630251\pi$
$242$	− 11.8149i	− 0.759488i
$243$	1.00000i	0.0641500i
$244$	7.21259	0.461739
$245$	0	0
$246$	12.5668	0.801228
$247$	9.71616i	0.618224i
$248$	− 10.2285i	− 0.649511i
$249$	14.5872	0.924426
$250$	0	0
$251$	−29.0986	−1.83669	−0.918343	−	0.395786i	$-0.870472\pi$
$251$	−29.0986	−1.83669	−0.918343	+	0.395786i	$0.870472\pi$
$252$	− 1.93395i	− 0.121828i
$253$	0.688750i	0.0433013i
$254$	1.85232	0.116225
$255$	0	0
$256$	20.9452	1.30908
$257$	− 12.0500i	− 0.751656i	−0.926690	−	0.375828i	$-0.877358\pi$
$257$	− 12.0500i	− 0.751656i	0.926690	−	0.375828i	$-0.122642\pi$
$258$	20.3558i	1.26730i
$259$	−2.53143	−0.157296
$260$	0	0
$261$	−8.74724	−0.541440
$262$	3.81040i	0.235407i
$263$	1.02892i	0.0634460i	0.999497	+	0.0317230i	$0.0100994\pi$
$263$	1.02892i	0.0634460i	−0.999497	+	0.0317230i	$0.989901\pi$
$264$	2.57433	0.158439
$265$	0	0
$266$	−4.49797	−0.275788
$267$	− 2.68119i	− 0.164086i
$268$	− 9.54285i	− 0.582922i
$269$	−23.9072	−1.45765	−0.728824	−	0.684701i	$-0.759933\pi$
$269$	−23.9072	−1.45765	−0.728824	+	0.684701i	$0.759933\pi$
$270$	0	0
$271$	3.72214	0.226104	0.113052	−	0.993589i	$-0.463937\pi$
$271$	3.72214	0.226104	0.113052	+	0.993589i	$0.463937\pi$
$272$	20.3055i	1.23120i
$273$	− 8.24232i	− 0.498848i
$274$	−30.9445	−1.86943
$275$	0	0
$276$	−0.432897	−0.0260574
$277$	− 17.5802i	− 1.05629i	−0.849153	−	0.528147i	$-0.822887\pi$
$277$	− 17.5802i	− 1.05629i	0.849153	−	0.528147i	$-0.177113\pi$
$278$	− 27.2024i	− 1.63149i
$279$	8.45991	0.506481
$280$	0	0
$281$	13.2536	0.790644	0.395322	−	0.918543i	$-0.370633\pi$
$281$	13.2536	0.790644	0.395322	+	0.918543i	$0.370633\pi$
$282$	23.0181i	1.37071i
$283$	− 12.0996i	− 0.719249i	−0.933097	−	0.359624i	$-0.882905\pi$
$283$	− 12.0996i	− 0.719249i	0.933097	−	0.359624i	$-0.117095\pi$
$284$	−3.57933	−0.212394
$285$	0	0
$286$	−22.1882	−1.31201
$287$	9.93960i	0.586716i
$288$	6.50828i	0.383504i
$289$	−0.274126	−0.0161251
$290$	0	0
$291$	8.55105	0.501272
$292$	− 8.40767i	− 0.492022i
$293$	− 3.61673i	− 0.211291i	−0.994404	−	0.105646i	$-0.966309\pi$
$293$	− 3.61673i	− 0.211291i	0.994404	−	0.105646i	$-0.0336910\pi$
$294$	−8.97397	−0.523372
$295$	0	0
$296$	2.11791	0.123101
$297$	2.12920i	0.123549i
$298$	8.60543i	0.498499i
$299$	−1.84497	−0.106697
$300$	0	0
$301$	−16.1003	−0.928005
$302$	− 17.5965i	− 1.01257i
$303$	9.85377i	0.566084i
$304$	8.32275	0.477342
$305$	0	0
$306$	−7.59378	−0.434108
$307$	14.8273i	0.846238i	0.906074	+	0.423119i	$0.139065\pi$
$307$	14.8273i	0.846238i	−0.906074	+	0.423119i	$0.860935\pi$
$308$	− 4.11778i	− 0.234633i
$309$	16.1279	0.917484
$310$	0	0
$311$	−6.61579	−0.375147	−0.187574	−	0.982251i	$-0.560062\pi$
$311$	−6.61579	−0.375147	−0.187574	+	0.982251i	$0.560062\pi$
$312$	6.89590i	0.390403i
$313$	7.17823i	0.405737i	0.979206	+	0.202869i	$0.0650264\pi$
$313$	7.17823i	0.405737i	−0.979206	+	0.202869i	$0.934974\pi$
$314$	−33.9030	−1.91326
$315$	0	0
$316$	11.2025	0.630189
$317$	− 9.75255i	− 0.547758i	−0.961764	−	0.273879i	$-0.911693\pi$
$317$	− 9.75255i	− 0.547758i	0.961764	−	0.273879i	$-0.0883068\pi$
$318$	− 15.2462i	− 0.854963i
$319$	−18.6247	−1.04278
$320$	0	0
$321$	−1.81198	−0.101135
$322$	− 0.854102i	− 0.0475972i
$323$	7.08025i	0.393956i
$324$	−1.33826	−0.0743478
$325$	0	0
$326$	0.825307	0.0457095
$327$	18.4646i	1.02109i
$328$	− 8.31592i	− 0.459170i
$329$	−18.2060	−1.00373
$330$	0	0
$331$	6.68822	0.367618	0.183809	−	0.982962i	$-0.441157\pi$
$331$	6.68822	0.367618	0.183809	+	0.982962i	$0.441157\pi$
$332$	19.5215i	1.07138i
$333$	1.75170i	0.0959928i
$334$	−8.63570	−0.472525
$335$	0	0
$336$	−7.06027	−0.385169
$337$	− 20.8191i	− 1.13409i	−0.823687	−	0.567045i	$-0.808086\pi$
$337$	− 20.8191i	− 1.13409i	0.823687	−	0.567045i	$-0.191914\pi$
$338$	− 35.6836i	− 1.94093i
$339$	−13.3653	−0.725902
$340$	0	0
$341$	18.0129	0.975452
$342$	3.11251i	0.168305i
$343$	− 17.2138i	− 0.929456i
$344$	13.4702	0.726266
$345$	0	0
$346$	4.19297	0.225415
$347$	− 30.9907i	− 1.66367i	−0.555023	−	0.831835i	$-0.687291\pi$
$347$	− 30.9907i	− 1.66367i	0.555023	−	0.831835i	$-0.312709\pi$
$348$	− 11.7061i	− 0.627512i
$349$	7.45484	0.399048	0.199524	−	0.979893i	$-0.436060\pi$
$349$	7.45484	0.399048	0.199524	+	0.979893i	$0.436060\pi$
$350$	0	0
$351$	−5.70353	−0.304432
$352$	13.8575i	0.738605i
$353$	3.68948i	0.196371i	0.995168	+	0.0981856i	$0.0313039\pi$
$353$	3.68948i	0.196371i	−0.995168	+	0.0981856i	$0.968696\pi$
$354$	−3.88209	−0.206331
$355$	0	0
$356$	3.58814	0.190171
$357$	− 6.00625i	− 0.317884i
$358$	− 4.91043i	− 0.259524i
$359$	9.92989	0.524079	0.262040	−	0.965057i	$-0.415605\pi$
$359$	9.92989	0.524079	0.262040	+	0.965057i	$0.415605\pi$
$360$	0	0
$361$	−16.0980	−0.847262
$362$	− 43.6835i	− 2.29595i
$363$	− 6.46649i	− 0.339403i
$364$	11.0304	0.578149
$365$	0	0
$366$	9.84715	0.514719
$367$	− 12.5507i	− 0.655142i	−0.944826	−	0.327571i	$-0.893770\pi$
$367$	− 12.5507i	− 0.655142i	0.944826	−	0.327571i	$-0.106230\pi$
$368$	1.58038i	0.0823828i
$369$	6.87802	0.358056
$370$	0	0
$371$	12.0589	0.626065
$372$	11.3216i	0.586996i
$373$	− 14.8576i	− 0.769299i	−0.923063	−	0.384650i	$-0.874322\pi$
$373$	− 14.8576i	− 0.769299i	0.923063	−	0.384650i	$-0.125678\pi$
$374$	−16.1687	−0.836064
$375$	0	0
$376$	15.2320	0.785530
$377$	− 49.8902i	− 2.56947i
$378$	− 2.64037i	− 0.135806i
$379$	−11.2957	−0.580223	−0.290112	−	0.956993i	$-0.593692\pi$
$379$	−11.2957	−0.580223	−0.290112	+	0.956993i	$0.593692\pi$
$380$	0	0
$381$	1.01381	0.0519389
$382$	− 7.15898i	− 0.366285i
$383$	3.04968i	0.155831i	0.996960	+	0.0779157i	$0.0248265\pi$
$383$	3.04968i	0.155831i	−0.996960	+	0.0779157i	$0.975173\pi$
$384$	9.14301	0.466577
$385$	0	0
$386$	−15.6171	−0.794889
$387$	11.1411i	0.566334i
$388$	11.4435i	0.580958i
$389$	21.3939	1.08472	0.542358	−	0.840148i	$-0.317532\pi$
$389$	21.3939	1.08472	0.542358	+	0.840148i	$0.317532\pi$
$390$	0	0
$391$	−1.34444	−0.0679914
$392$	5.93842i	0.299936i
$393$	2.08550i	0.105200i
$394$	−4.48067	−0.225733
$395$	0	0
$396$	−2.84943	−0.143189
$397$	34.4305i	1.72802i	0.503476	+	0.864009i	$0.332054\pi$
$397$	34.4305i	1.72802i	−0.503476	+	0.864009i	$0.667946\pi$
$398$	19.0647i	0.955629i
$399$	−2.46182	−0.123245
$400$	0	0
$401$	36.2976	1.81262	0.906309	−	0.422616i	$-0.138888\pi$
$401$	36.2976	1.81262	0.906309	+	0.422616i	$0.138888\pi$
$402$	− 13.0286i	− 0.649807i
$403$	48.2514i	2.40357i
$404$	−13.1869	−0.656074
$405$	0	0
$406$	23.0960	1.14623
$407$	3.72974i	0.184876i
$408$	5.02510i	0.248780i
$409$	−24.9896	−1.23566	−0.617828	−	0.786313i	$-0.711987\pi$
$409$	−24.9896	−1.23566	−0.617828	+	0.786313i	$0.711987\pi$
$410$	0	0
$411$	−16.9365	−0.835416
$412$	21.5833i	1.06333i
$413$	− 3.07051i	− 0.151090i
$414$	−0.591023	−0.0290472
$415$	0	0
$416$	−37.1202	−1.81997
$417$	− 14.8883i	− 0.729085i
$418$	6.62717i	0.324146i
$419$	−2.70313	−0.132057	−0.0660284	−	0.997818i	$-0.521033\pi$
$419$	−2.70313	−0.132057	−0.0660284	+	0.997818i	$0.521033\pi$
$420$	0	0
$421$	26.6496	1.29882	0.649411	−	0.760438i	$-0.275016\pi$
$421$	26.6496	1.29882	0.649411	+	0.760438i	$0.275016\pi$
$422$	35.1930i	1.71317i
$423$	12.5982i	0.612548i
$424$	−10.0890	−0.489965
$425$	0	0
$426$	−4.88676	−0.236764
$427$	7.78853i	0.376914i
$428$	− 2.42490i	− 0.117212i
$429$	−12.1440	−0.586317
$430$	0	0
$431$	33.6242	1.61962	0.809811	−	0.586691i	$-0.199570\pi$
$431$	33.6242	1.61962	0.809811	+	0.586691i	$0.199570\pi$
$432$	4.88558i	0.235058i
$433$	− 5.34260i	− 0.256749i	−0.991726	−	0.128375i	$-0.959024\pi$
$433$	− 5.34260i	− 0.256749i	0.991726	−	0.128375i	$-0.0409760\pi$
$434$	−22.3373	−1.07223
$435$	0	0
$436$	−24.7104	−1.18341
$437$	0.551055i	0.0263605i
$438$	− 11.4788i	− 0.548476i
$439$	−26.9690	−1.28716	−0.643580	−	0.765379i	$-0.722552\pi$
$439$	−26.9690	−1.28716	−0.643580	+	0.765379i	$0.722552\pi$
$440$	0	0
$441$	−4.91161	−0.233886
$442$	− 43.3114i	− 2.06011i
$443$	− 16.8670i	− 0.801374i	−0.916215	−	0.400687i	$-0.868771\pi$
$443$	− 16.8670i	− 0.801374i	0.916215	−	0.400687i	$-0.131229\pi$
$444$	−2.34424	−0.111253
$445$	0	0
$446$	13.9171	0.658992
$447$	4.70991i	0.222771i
$448$	− 3.06376i	− 0.144749i
$449$	−21.5942	−1.01909	−0.509546	−	0.860443i	$-0.670187\pi$
$449$	−21.5942	−1.01909	−0.509546	+	0.860443i	$0.670187\pi$
$450$	0	0
$451$	14.6447	0.689593
$452$	− 17.8862i	− 0.841297i
$453$	− 9.63091i	− 0.452500i
$454$	2.40817	0.113021
$455$	0	0
$456$	2.05967	0.0964528
$457$	− 5.60699i	− 0.262284i	−0.991364	−	0.131142i	$-0.958136\pi$
$457$	− 5.60699i	− 0.262284i	0.991364	−	0.131142i	$-0.0418644\pi$
$458$	− 2.05788i	− 0.0961587i
$459$	−4.15622	−0.193996
$460$	0	0
$461$	−9.54504	−0.444557	−0.222278	−	0.974983i	$-0.571349\pi$
$461$	−9.54504	−0.444557	−0.222278	+	0.974983i	$0.571349\pi$
$462$	− 5.62190i	− 0.261554i
$463$	23.5262i	1.09336i	0.837343	+	0.546678i	$0.184108\pi$
$463$	23.5262i	1.09336i	−0.837343	+	0.546678i	$0.815892\pi$
$464$	−42.7353	−1.98394
$465$	0	0
$466$	48.6135	2.25198
$467$	− 36.5834i	− 1.69288i	−0.532486	−	0.846439i	$-0.678742\pi$
$467$	− 36.5834i	− 1.69288i	0.532486	−	0.846439i	$-0.321258\pi$
$468$	− 7.63282i	− 0.352827i
$469$	10.3049	0.475835
$470$	0	0
$471$	−18.5557	−0.855003
$472$	2.56893i	0.118245i
$473$	23.7217i	1.09072i
$474$	15.2944	0.702496
$475$	0	0
$476$	8.03793	0.368418
$477$	− 8.34451i	− 0.382069i
$478$	45.4382i	2.07830i
$479$	−29.2750	−1.33761	−0.668804	−	0.743439i	$-0.733193\pi$
$479$	−29.2750	−1.33761	−0.668804	+	0.743439i	$0.733193\pi$
$480$	0	0
$481$	−9.99091	−0.455546
$482$	− 22.5705i	− 1.02806i
$483$	− 0.467465i	− 0.0212704i
$484$	8.65385	0.393357
$485$	0	0
$486$	−1.82709	−0.0828785
$487$	− 21.2869i	− 0.964600i	−0.876006	−	0.482300i	$-0.839801\pi$
$487$	− 21.2869i	− 0.964600i	0.876006	−	0.482300i	$-0.160199\pi$
$488$	− 6.51624i	− 0.294976i
$489$	0.451705	0.0204268
$490$	0	0
$491$	−33.1940	−1.49803	−0.749013	−	0.662556i	$-0.769472\pi$
$491$	−33.1940	−1.49803	−0.749013	+	0.662556i	$0.769472\pi$
$492$	9.20459i	0.414975i
$493$	− 36.3554i	− 1.63737i
$494$	−17.7523	−0.798714
$495$	0	0
$496$	41.3316	1.85584
$497$	− 3.86515i	− 0.173376i
$498$	26.6521i	1.19431i
$499$	−4.72872	−0.211686	−0.105843	−	0.994383i	$-0.533754\pi$
$499$	−4.72872	−0.211686	−0.105843	+	0.994383i	$0.533754\pi$
$500$	0	0
$501$	−4.72648	−0.211163
$502$	− 53.1657i	− 2.37290i
$503$	0.400444i	0.0178549i	0.999960	+	0.00892745i	$0.00284173\pi$
$503$	0.400444i	0.0178549i	−0.999960	+	0.00892745i	$0.997158\pi$
$504$	−1.74724	−0.0778282
$505$	0	0
$506$	−1.25841	−0.0559431
$507$	− 19.5303i	− 0.867371i
$508$	1.35674i	0.0601956i
$509$	27.6104	1.22381	0.611904	−	0.790932i	$-0.290404\pi$
$509$	27.6104	1.22381	0.611904	+	0.790932i	$0.290404\pi$
$510$	0	0
$511$	9.07905	0.401633
$512$	19.9828i	0.883126i
$513$	1.70353i	0.0752128i
$514$	22.0164	0.971100
$515$	0	0
$516$	−14.9097	−0.656363
$517$	26.8242i	1.17973i
$518$	− 4.62516i	− 0.203218i
$519$	2.29489	0.100734
$520$	0	0
$521$	−28.7374	−1.25901	−0.629505	−	0.776996i	$-0.716742\pi$
$521$	−28.7374	−1.25901	−0.629505	+	0.776996i	$0.716742\pi$
$522$	− 15.9820i	− 0.699513i
$523$	− 18.0660i	− 0.789969i	−0.918688	−	0.394985i	$-0.870750\pi$
$523$	− 18.0660i	− 0.789969i	0.918688	−	0.394985i	$-0.129250\pi$
$524$	−2.79094	−0.121923
$525$	0	0
$526$	−1.87993	−0.0819690
$527$	35.1612i	1.53165i
$528$	10.4024i	0.452706i
$529$	22.8954	0.995451
$530$	0	0
$531$	−2.12474	−0.0922058
$532$	− 3.29456i	− 0.142837i
$533$	39.2290i	1.69920i
$534$	4.89878	0.211991
$535$	0	0
$536$	−8.62152	−0.372393
$537$	− 2.68757i	− 0.115977i
$538$	− 43.6807i	− 1.88321i
$539$	−10.4578	−0.450451
$540$	0	0
$541$	−5.26032	−0.226159	−0.113079	−	0.993586i	$-0.536071\pi$
$541$	−5.26032	−0.226159	−0.113079	+	0.993586i	$0.536071\pi$
$542$	6.80068i	0.292114i
$543$	− 23.9088i	− 1.02602i
$544$	−27.0498	−1.15975
$545$	0	0
$546$	15.0595	0.644486
$547$	− 36.7888i	− 1.57297i	−0.617607	−	0.786487i	$-0.711898\pi$
$547$	− 36.7888i	− 1.57297i	0.617607	−	0.786487i	$-0.288102\pi$
$548$	− 22.6655i	− 0.968221i
$549$	5.38952	0.230019
$550$	0	0
$551$	−14.9012	−0.634813
$552$	0.391103i	0.0166464i
$553$	12.0970i	0.514418i
$554$	32.1207	1.36468
$555$	0	0
$556$	19.9245	0.844986
$557$	23.3475i	0.989266i	0.869102	+	0.494633i	$0.164698\pi$
$557$	23.3475i	0.989266i	−0.869102	+	0.494633i	$0.835302\pi$
$558$	15.4570i	0.654348i
$559$	−63.5436	−2.68761
$560$	0	0
$561$	−8.84943	−0.373623
$562$	24.2156i	1.02147i
$563$	29.2369i	1.23219i	0.787672	+	0.616095i	$0.211286\pi$
$563$	29.2369i	1.23219i	−0.787672	+	0.616095i	$0.788714\pi$
$564$	−16.8597	−0.709923
$565$	0	0
$566$	22.1071	0.929232
$567$	− 1.44512i	− 0.0606895i
$568$	3.23376i	0.135685i
$569$	20.2530	0.849051	0.424526	−	0.905416i	$-0.360441\pi$
$569$	20.2530	0.849051	0.424526	+	0.905416i	$0.360441\pi$
$570$	0	0
$571$	−4.57564	−0.191484	−0.0957422	−	0.995406i	$-0.530522\pi$
$571$	−4.57564	−0.191484	−0.0957422	+	0.995406i	$0.530522\pi$
$572$	− 16.2518i	− 0.679523i
$573$	− 3.91824i	− 0.163687i
$574$	−18.1606	−0.758007
$575$	0	0
$576$	−2.12007	−0.0883361
$577$	− 31.3502i	− 1.30513i	−0.757734	−	0.652564i	$-0.773694\pi$
$577$	− 31.3502i	− 1.30513i	0.757734	−	0.652564i	$-0.226306\pi$
$578$	− 0.500853i	− 0.0208327i
$579$	−8.54752	−0.355223
$580$	0	0
$581$	−21.0803	−0.874559
$582$	15.6236i	0.647617i
$583$	− 17.7672i	− 0.735841i
$584$	−7.59594	−0.314322
$585$	0	0
$586$	6.60809	0.272978
$587$	− 4.67218i	− 0.192842i	−0.995341	−	0.0964208i	$-0.969261\pi$
$587$	− 4.67218i	− 0.192842i	0.995341	−	0.0964208i	$-0.0307395\pi$
$588$	− 6.57302i	− 0.271067i
$589$	14.4117	0.593825
$590$	0	0
$591$	−2.45235	−0.100876
$592$	8.55809i	0.351735i
$593$	7.37978i	0.303051i	0.988453	+	0.151526i	$0.0484186\pi$
$593$	7.37978i	0.303051i	−0.988453	+	0.151526i	$0.951581\pi$
$594$	−3.89025	−0.159619
$595$	0	0
$596$	−6.30309	−0.258185
$597$	10.4345i	0.427055i
$598$	− 3.37092i	− 0.137847i
$599$	46.7505	1.91018	0.955088	−	0.296323i	$-0.0957605\pi$
$599$	46.7505	1.91018	0.955088	+	0.296323i	$0.0957605\pi$
$600$	0	0
$601$	32.3225	1.31846	0.659230	−	0.751941i	$-0.270882\pi$
$601$	32.3225	1.31846	0.659230	+	0.751941i	$0.270882\pi$
$602$	− 29.4167i	− 1.19893i
$603$	− 7.13078i	− 0.290388i
$604$	12.8887	0.524433
$605$	0	0
$606$	−18.0037	−0.731352
$607$	− 14.0891i	− 0.571858i	−0.958251	−	0.285929i	$-0.907698\pi$
$607$	− 14.0891i	− 0.571858i	0.958251	−	0.285929i	$-0.0923021\pi$
$608$	11.0871i	0.449640i
$609$	12.6409	0.512233
$610$	0	0
$611$	−71.8545	−2.90692
$612$	− 5.56210i	− 0.224835i
$613$	12.2959i	0.496625i	0.968680	+	0.248313i	$0.0798760\pi$
$613$	12.2959i	0.496625i	−0.968680	+	0.248313i	$0.920124\pi$
$614$	−27.0908	−1.09330
$615$	0	0
$616$	−3.72023	−0.149892
$617$	− 1.96969i	− 0.0792969i	−0.999214	−	0.0396485i	$-0.987376\pi$
$617$	− 1.96969i	− 0.0792969i	0.999214	−	0.0396485i	$-0.0126238\pi$
$618$	29.4671i	1.18534i
$619$	−14.9714	−0.601749	−0.300875	−	0.953664i	$-0.597279\pi$
$619$	−14.9714	−0.601749	−0.300875	+	0.953664i	$0.597279\pi$
$620$	0	0
$621$	−0.323478	−0.0129807
$622$	− 12.0877i	− 0.484671i
$623$	3.87466i	0.155235i
$624$	−27.8651	−1.11550
$625$	0	0
$626$	−13.1153	−0.524192
$627$	3.62717i	0.144855i
$628$	− 24.8324i	− 0.990921i
$629$	−7.28046	−0.290291
$630$	0	0
$631$	18.0729	0.719472	0.359736	−	0.933054i	$-0.382867\pi$
$631$	18.0729	0.719472	0.359736	+	0.933054i	$0.382867\pi$
$632$	− 10.1209i	− 0.402588i
$633$	19.2618i	0.765587i
$634$	17.8188	0.707675
$635$	0	0
$636$	11.1671	0.442806
$637$	− 28.0136i	− 1.10994i
$638$	− 34.0289i	− 1.34722i
$639$	−2.67461	−0.105806
$640$	0	0
$641$	12.2381	0.483374	0.241687	−	0.970354i	$-0.422299\pi$
$641$	12.2381	0.483374	0.241687	+	0.970354i	$0.422299\pi$
$642$	− 3.31065i	− 0.130661i
$643$	16.4453i	0.648540i	0.945964	+	0.324270i	$0.105119\pi$
$643$	16.4453i	0.648540i	−0.945964	+	0.324270i	$0.894881\pi$
$644$	0.625591	0.0246517
$645$	0	0
$646$	−12.9363	−0.508971
$647$	17.3991i	0.684028i	0.939695	+	0.342014i	$0.111109\pi$
$647$	17.3991i	0.684028i	−0.939695	+	0.342014i	$0.888891\pi$
$648$	1.20906i	0.0474962i
$649$	−4.52400	−0.177583
$650$	0	0
$651$	−12.2256	−0.479160
$652$	0.604500i	0.0236740i
$653$	28.4645i	1.11390i	0.830546	+	0.556950i	$0.188029\pi$
$653$	28.4645i	1.11390i	−0.830546	+	0.556950i	$0.811971\pi$
$654$	−33.7365	−1.31920
$655$	0	0
$656$	33.6031	1.31198
$657$	− 6.28253i	− 0.245105i
$658$	− 33.2641i	− 1.29677i
$659$	1.51333	0.0589509	0.0294754	−	0.999566i	$-0.490616\pi$
$659$	1.51333	0.0589509	0.0294754	+	0.999566i	$0.490616\pi$
$660$	0	0
$661$	26.7842	1.04179	0.520893	−	0.853622i	$-0.325599\pi$
$661$	26.7842	1.04179	0.520893	+	0.853622i	$0.325599\pi$
$662$	12.2200i	0.474943i
$663$	− 23.7051i	− 0.920630i
$664$	17.6368	0.684439
$665$	0	0
$666$	−3.20052	−0.124018
$667$	− 2.82954i	− 0.109560i
$668$	− 6.32526i	− 0.244732i
$669$	7.61706	0.294492
$670$	0	0
$671$	11.4754	0.443003
$672$	− 9.40528i	− 0.362817i
$673$	− 9.31141i	− 0.358929i	−0.983764	−	0.179464i	$-0.942564\pi$
$673$	− 9.31141i	− 0.358929i	0.983764	−	0.179464i	$-0.0574364\pi$
$674$	38.0385	1.46519
$675$	0	0
$676$	26.1366	1.00526
$677$	24.7854i	0.952579i	0.879288	+	0.476290i	$0.158019\pi$
$677$	24.7854i	0.952579i	−0.879288	+	0.476290i	$0.841981\pi$
$678$	− 24.4196i	− 0.937828i
$679$	−12.3573	−0.474231
$680$	0	0
$681$	1.31803	0.0505072
$682$	32.9112i	1.26023i
$683$	37.3864i	1.43055i	0.698842	+	0.715276i	$0.253699\pi$
$683$	37.3864i	1.43055i	−0.698842	+	0.715276i	$0.746301\pi$
$684$	−2.27977	−0.0871693
$685$	0	0
$686$	31.4511	1.20081
$687$	− 1.12632i	− 0.0429717i
$688$	54.4307i	2.07515i
$689$	47.5932	1.81316
$690$	0	0
$691$	−32.8909	−1.25123	−0.625615	−	0.780132i	$-0.715152\pi$
$691$	−32.8909	−1.25123	−0.625615	+	0.780132i	$0.715152\pi$
$692$	3.07116i	0.116748i
$693$	− 3.07697i	− 0.116884i
$694$	56.6229	2.14938
$695$	0	0
$696$	−10.5759	−0.400879
$697$	28.5865i	1.08279i
$698$	13.6207i	0.515550i
$699$	26.6071	1.00637
$700$	0	0
$701$	−42.3336	−1.59892	−0.799459	−	0.600721i	$-0.794880\pi$
$701$	−42.3336	−1.59892	−0.799459	+	0.600721i	$0.794880\pi$
$702$	− 10.4209i	− 0.393311i
$703$	2.98409i	0.112547i
$704$	−4.51406	−0.170130
$705$	0	0
$706$	−6.74101	−0.253701
$707$	− 14.2399i	− 0.535548i
$708$	− 2.84345i	− 0.106864i
$709$	−18.3593	−0.689498	−0.344749	−	0.938695i	$-0.612036\pi$
$709$	−18.3593	−0.689498	−0.344749	+	0.938695i	$0.612036\pi$
$710$	0	0
$711$	8.37092	0.313934
$712$	− 3.24171i	− 0.121488i
$713$	2.73659i	0.102486i
$714$	10.9740	0.410690
$715$	0	0
$716$	3.59667	0.134414
$717$	24.8692i	0.928756i
$718$	18.1428i	0.677084i
$719$	19.2706	0.718671	0.359336	−	0.933208i	$-0.383003\pi$
$719$	19.2706	0.718671	0.359336	+	0.933208i	$0.383003\pi$
$720$	0	0
$721$	−23.3068	−0.867992
$722$	− 29.4125i	− 1.09462i
$723$	− 12.3532i	− 0.459422i
$724$	31.9962	1.18913
$725$	0	0
$726$	11.8149	0.438491
$727$	17.4238i	0.646214i	0.946362	+	0.323107i	$0.104727\pi$
$727$	17.4238i	0.646214i	−0.946362	+	0.323107i	$0.895273\pi$
$728$	− 9.96543i	− 0.369343i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−46.3048	−1.71265
$732$	7.21259i	0.266585i
$733$	− 24.2010i	− 0.893883i	−0.894563	−	0.446942i	$-0.852513\pi$
$733$	− 24.2010i	− 0.893883i	0.894563	−	0.446942i	$-0.147487\pi$
$734$	22.9313	0.846410
$735$	0	0
$736$	−2.10528	−0.0776018
$737$	− 15.1829i	− 0.559269i
$738$	12.5668i	0.462589i
$739$	38.5798	1.41918	0.709591	−	0.704613i	$-0.248880\pi$
$739$	38.5798	1.41918	0.709591	+	0.704613i	$0.248880\pi$
$740$	0	0
$741$	−9.71616	−0.356932
$742$	22.0326i	0.808844i
$743$	33.8580i	1.24213i	0.783759	+	0.621065i	$0.213300\pi$
$743$	33.8580i	1.24213i	−0.783759	+	0.621065i	$0.786700\pi$
$744$	10.2285	0.374995
$745$	0	0
$746$	27.1462	0.993895
$747$	14.5872i	0.533718i
$748$	− 11.8429i	− 0.433018i
$749$	2.61853	0.0956791
$750$	0	0
$751$	−45.9138	−1.67542	−0.837710	−	0.546115i	$-0.816106\pi$
$751$	−45.9138	−1.67542	−0.837710	+	0.546115i	$0.816106\pi$
$752$	61.5497i	2.24449i
$753$	− 29.0986i	− 1.06041i
$754$	91.1539	3.31963
$755$	0	0
$756$	1.93395	0.0703372
$757$	10.3561i	0.376400i	0.982131	+	0.188200i	$0.0602653\pi$
$757$	10.3561i	0.376400i	−0.982131	+	0.188200i	$0.939735\pi$
$758$	− 20.6383i	− 0.749618i
$759$	−0.688750	−0.0250000
$760$	0	0
$761$	49.2652	1.78586	0.892931	−	0.450193i	$-0.148645\pi$
$761$	49.2652	1.78586	0.892931	+	0.450193i	$0.148645\pi$
$762$	1.85232i	0.0671024i
$763$	− 26.6836i	− 0.966012i
$764$	5.24362	0.189708
$765$	0	0
$766$	−5.57205	−0.201326
$767$	− 12.1185i	− 0.437574i
$768$	20.9452i	0.755797i
$769$	1.26208	0.0455116	0.0227558	−	0.999741i	$-0.492756\pi$
$769$	1.26208	0.0455116	0.0227558	+	0.999741i	$0.492756\pi$
$770$	0	0
$771$	12.0500	0.433969
$772$	− 11.4388i	− 0.411692i
$773$	26.9903i	0.970772i	0.874300	+	0.485386i	$0.161321\pi$
$773$	26.9903i	0.970772i	−0.874300	+	0.485386i	$0.838679\pi$
$774$	−20.3558	−0.731674
$775$	0	0
$776$	10.3387	0.371138
$777$	− 2.53143i	− 0.0908146i
$778$	39.0887i	1.40140i
$779$	11.7169	0.419803
$780$	0	0
$781$	−5.69480	−0.203776
$782$	− 2.45642i	− 0.0878413i
$783$	− 8.74724i	− 0.312601i
$784$	−23.9961	−0.857003
$785$	0	0
$786$	−3.81040	−0.135912
$787$	7.62303i	0.271732i	0.990727	+	0.135866i	$0.0433816\pi$
$787$	7.62303i	0.271732i	−0.990727	+	0.135866i	$0.956618\pi$
$788$	− 3.28189i	− 0.116912i
$789$	−1.02892	−0.0366306
$790$	0	0
$791$	19.3145	0.686744
$792$	2.57433i	0.0914748i
$793$	30.7393i	1.09159i
$794$	−62.9077	−2.23251
$795$	0	0
$796$	−13.9641	−0.494943
$797$	4.98266i	0.176495i	0.996099	+	0.0882474i	$0.0281266\pi$
$797$	4.98266i	0.176495i	−0.996099	+	0.0882474i	$0.971873\pi$
$798$	− 4.49797i	− 0.159226i
$799$	−52.3610	−1.85240
$800$	0	0
$801$	2.68119	0.0947353
$802$	66.3191i	2.34181i
$803$	− 13.3768i	− 0.472057i
$804$	9.54285	0.336550
$805$	0	0
$806$	−88.1596	−3.10529
$807$	− 23.9072i	− 0.841574i
$808$	11.9138i	0.419125i
$809$	−5.74271	−0.201903	−0.100952	−	0.994891i	$-0.532189\pi$
$809$	−5.74271	−0.201903	−0.100952	+	0.994891i	$0.532189\pi$
$810$	0	0
$811$	38.8021	1.36253	0.681264	−	0.732038i	$-0.261431\pi$
$811$	38.8021	1.36253	0.681264	+	0.732038i	$0.261431\pi$
$812$	16.9168i	0.593662i
$813$	3.72214i	0.130541i
$814$	−6.81457	−0.238851
$815$	0	0
$816$	−20.3055	−0.710835
$817$	18.9792i	0.664000i
$818$	− 45.6583i	− 1.59640i
$819$	8.24232	0.288010
$820$	0	0
$821$	−41.8919	−1.46204	−0.731019	−	0.682357i	$-0.760955\pi$
$821$	−41.8919	−1.46204	−0.731019	+	0.682357i	$0.760955\pi$
$822$	− 30.9445i	− 1.07931i
$823$	29.2212i	1.01859i	0.860593	+	0.509294i	$0.170093\pi$
$823$	29.2212i	1.01859i	−0.860593	+	0.509294i	$0.829907\pi$
$824$	19.4995	0.679299
$825$	0	0
$826$	5.61010	0.195200
$827$	7.41934i	0.257996i	0.991645	+	0.128998i	$0.0411760\pi$
$827$	7.41934i	0.257996i	−0.991645	+	0.128998i	$0.958824\pi$
$828$	− 0.432897i	− 0.0150442i
$829$	−29.7586	−1.03356	−0.516779	−	0.856119i	$-0.672869\pi$
$829$	−29.7586	−1.03356	−0.516779	+	0.856119i	$0.672869\pi$
$830$	0	0
$831$	17.5802	0.609852
$832$	− 12.0919i	− 0.419210i
$833$	− 20.4137i	− 0.707294i
$834$	27.2024	0.941940
$835$	0	0
$836$	−4.85410	−0.167883
$837$	8.45991i	0.292417i
$838$	− 4.93887i	− 0.170610i
$839$	−29.5793	−1.02119	−0.510595	−	0.859821i	$-0.670575\pi$
$839$	−29.5793	−1.02119	−0.510595	+	0.859821i	$0.670575\pi$
$840$	0	0
$841$	47.5142	1.63842
$842$	48.6912i	1.67801i
$843$	13.2536i	0.456479i
$844$	−25.7773	−0.887291
$845$	0	0
$846$	−23.0181	−0.791380
$847$	9.34488i	0.321094i
$848$	− 40.7678i	− 1.39997i
$849$	12.0996	0.415258
$850$	0	0
$851$	−0.566637	−0.0194241
$852$	− 3.57933i	− 0.122626i
$853$	− 37.5298i	− 1.28499i	−0.766288	−	0.642497i	$-0.777898\pi$
$853$	− 37.5298i	− 1.28499i	0.766288	−	0.642497i	$-0.222102\pi$
$854$	−14.2304	−0.486953
$855$	0	0
$856$	−2.19078	−0.0748794
$857$	49.4333i	1.68861i	0.535864	+	0.844305i	$0.319986\pi$
$857$	49.4333i	1.68861i	−0.535864	+	0.844305i	$0.680014\pi$
$858$	− 22.1882i	− 0.757492i
$859$	−2.82651	−0.0964394	−0.0482197	−	0.998837i	$-0.515355\pi$
$859$	−2.82651	−0.0964394	−0.0482197	+	0.998837i	$0.515355\pi$
$860$	0	0
$861$	−9.93960	−0.338741
$862$	61.4345i	2.09247i
$863$	33.3977i	1.13687i	0.822727	+	0.568436i	$0.192451\pi$
$863$	33.3977i	1.13687i	−0.822727	+	0.568436i	$0.807549\pi$
$864$	−6.50828	−0.221416
$865$	0	0
$866$	9.76142	0.331707
$867$	− 0.274126i	− 0.00930980i
$868$	− 16.3611i	− 0.555331i
$869$	17.8234	0.604617
$870$	0	0
$871$	40.6707	1.37807
$872$	22.3247i	0.756011i
$873$	8.55105i	0.289409i
$874$	−1.00683	−0.0340564
$875$	0	0
$876$	8.40767	0.284069
$877$	9.28115i	0.313402i	0.987646	+	0.156701i	$0.0500859\pi$
$877$	9.28115i	0.313402i	−0.987646	+	0.156701i	$0.949914\pi$
$878$	− 49.2748i	− 1.66294i
$879$	3.61673	0.121989
$880$	0	0
$881$	6.58736	0.221934	0.110967	−	0.993824i	$-0.464605\pi$
$881$	6.58736	0.221934	0.110967	+	0.993824i	$0.464605\pi$
$882$	− 8.97397i	− 0.302169i
$883$	4.18883i	0.140965i	0.997513	+	0.0704827i	$0.0224540\pi$
$883$	4.18883i	0.140965i	−0.997513	+	0.0704827i	$0.977546\pi$
$884$	31.7236	1.06698
$885$	0	0
$886$	30.8175	1.03533
$887$	26.8716i	0.902260i	0.892458	+	0.451130i	$0.148979\pi$
$887$	26.8716i	0.902260i	−0.892458	+	0.451130i	$0.851021\pi$
$888$	2.11791i	0.0710724i
$889$	−1.46508	−0.0491371
$890$	0	0
$891$	−2.12920	−0.0713310
$892$	10.1936i	0.341307i
$893$	21.4615i	0.718183i
$894$	−8.60543	−0.287809
$895$	0	0
$896$	−13.2128	−0.441408
$897$	− 1.84497i	− 0.0616016i
$898$	− 39.4545i	− 1.31661i
$899$	−74.0008	−2.46807
$900$	0	0
$901$	34.6816	1.15541
$902$	26.7572i	0.890918i
$903$	− 16.1003i	− 0.535784i
$904$	−16.1594	−0.537453
$905$	0	0
$906$	17.5965	0.584606
$907$	− 39.1821i	− 1.30102i	−0.759497	−	0.650511i	$-0.774555\pi$
$907$	− 39.1821i	− 1.30102i	0.759497	−	0.650511i	$-0.225445\pi$
$908$	1.76387i	0.0585362i
$909$	−9.85377	−0.326829
$910$	0	0
$911$	54.2926	1.79880	0.899398	−	0.437132i	$-0.144006\pi$
$911$	54.2926	1.79880	0.899398	+	0.437132i	$0.144006\pi$
$912$	8.32275i	0.275594i
$913$	31.0591i	1.02791i
$914$	10.2445	0.338857
$915$	0	0
$916$	1.50731	0.0498028
$917$	− 3.01381i	− 0.0995247i
$918$	− 7.59378i	− 0.250632i
$919$	28.0959	0.926798	0.463399	−	0.886150i	$-0.346630\pi$
$919$	28.0959	0.926798	0.463399	+	0.886150i	$0.346630\pi$
$920$	0	0
$921$	−14.8273	−0.488576
$922$	− 17.4396i	− 0.574344i
$923$	− 15.2547i	− 0.502116i
$924$	4.11778	0.135465
$925$	0	0
$926$	−42.9846	−1.41256
$927$	16.1279i	0.529710i
$928$	− 56.9295i	− 1.86880i
$929$	31.3772	1.02945	0.514727	−	0.857354i	$-0.327893\pi$
$929$	31.3772	1.02945	0.514727	+	0.857354i	$0.327893\pi$
$930$	0	0
$931$	−8.36710	−0.274221
$932$	35.6072i	1.16635i
$933$	− 6.61579i	− 0.216591i
$934$	66.8412	2.18711
$935$	0	0
$936$	−6.89590	−0.225399
$937$	− 49.7286i	− 1.62456i	−0.583266	−	0.812281i	$-0.698226\pi$
$937$	− 49.7286i	− 1.62456i	0.583266	−	0.812281i	$-0.301774\pi$
$938$	18.8279i	0.614754i
$939$	−7.17823	−0.234253
$940$	0	0
$941$	−4.49598	−0.146565	−0.0732824	−	0.997311i	$-0.523347\pi$
$941$	−4.49598	−0.146565	−0.0732824	+	0.997311i	$0.523347\pi$
$942$	− 33.9030i	− 1.10462i
$943$	2.22489i	0.0724523i
$944$	−10.3806	−0.337859
$945$	0	0
$946$	−43.3417	−1.40916
$947$	− 28.8993i	− 0.939101i	−0.882906	−	0.469551i	$-0.844416\pi$
$947$	− 28.8993i	− 0.939101i	0.882906	−	0.469551i	$-0.155584\pi$
$948$	11.2025i	0.363840i
$949$	35.8326	1.16318
$950$	0	0
$951$	9.75255	0.316248
$952$	− 7.26190i	− 0.235359i
$953$	46.1922i	1.49631i	0.663523	+	0.748156i	$0.269060\pi$
$953$	46.1922i	1.49631i	−0.663523	+	0.748156i	$0.730940\pi$
$954$	15.2462	0.493613
$955$	0	0
$956$	−33.2814	−1.07640
$957$	− 18.6247i	− 0.602050i
$958$	− 53.4880i	− 1.72812i
$959$	24.4754	0.790351
$960$	0	0
$961$	40.5701	1.30871
$962$	− 18.2543i	− 0.588542i
$963$	− 1.81198i	− 0.0583901i
$964$	16.5319	0.532456
$965$	0	0
$966$	0.854102	0.0274803
$967$	8.41820i	0.270711i	0.990797	+	0.135356i	$0.0432177\pi$
$967$	8.41820i	0.270711i	−0.990797	+	0.135356i	$0.956782\pi$
$968$	− 7.81835i	− 0.251291i
$969$	−7.08025	−0.227450
$970$	0	0
$971$	26.4421	0.848567	0.424284	−	0.905529i	$-0.360526\pi$
$971$	26.4421	0.848567	0.424284	+	0.905529i	$0.360526\pi$
$972$	− 1.33826i	− 0.0429247i
$973$	21.5155i	0.689756i
$974$	38.8931	1.24621
$975$	0	0
$976$	26.3309	0.842833
$977$	39.8235i	1.27407i	0.770836	+	0.637034i	$0.219839\pi$
$977$	39.8235i	1.27407i	−0.770836	+	0.637034i	$0.780161\pi$
$978$	0.825307i	0.0263904i
$979$	5.70881	0.182454
$980$	0	0
$981$	−18.4646	−0.589529
$982$	− 60.6485i	− 1.93537i
$983$	24.1179i	0.769242i	0.923075	+	0.384621i	$0.125668\pi$
$983$	24.1179i	0.769242i	−0.923075	+	0.384621i	$0.874332\pi$
$984$	8.31592	0.265102
$985$	0	0
$986$	66.4246	2.11539
$987$	− 18.2060i	− 0.579505i
$988$	− 13.0028i	− 0.413673i
$989$	−3.60390	−0.114597
$990$	0	0
$991$	−10.4044	−0.330506	−0.165253	−	0.986251i	$-0.552844\pi$
$991$	−10.4044	−0.330506	−0.165253	+	0.986251i	$0.552844\pi$
$992$	55.0595i	1.74814i
$993$	6.68822i	0.212244i
$994$	7.06198	0.223992
$995$	0	0
$996$	−19.5215	−0.618562
$997$	− 13.3642i	− 0.423249i	−0.977351	−	0.211624i	$-0.932125\pi$
$997$	− 13.3642i	− 0.423249i	0.977351	−	0.211624i	$-0.0678754\pi$
$998$	− 8.63980i	− 0.273488i
$999$	−1.75170	−0.0554215

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.d.1249.7		8
5.2	odd	4		1875.2.a.f.1.1	✓	4
5.3	odd	4		1875.2.a.g.1.4	yes	4
5.4	even	2	inner	1875.2.b.d.1249.2		8
15.2	even	4		5625.2.a.m.1.4		4
15.8	even	4		5625.2.a.j.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.f.1.1	✓	4	5.2	odd	4
1875.2.a.g.1.4	yes	4	5.3	odd	4
1875.2.b.d.1249.2		8	5.4	even	2	inner
1875.2.b.d.1249.7		8	1.1	even	1	trivial
5625.2.a.j.1.1		4	15.8	even	4
5625.2.a.m.1.4		4	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.82709i q^{2} +1.00000i q^{3} -1.33826 q^{4} -1.82709 q^{6} -1.44512i q^{7} +1.20906i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.82709i q^{2} +1.00000i q^{3} -1.33826 q^{4} -1.82709 q^{6} -1.44512i q^{7} +1.20906i q^{8} -1.00000 q^{9} -2.12920 q^{11} -1.33826i q^{12} -5.70353i q^{13} +2.64037 q^{14} -4.88558 q^{16} -4.15622i q^{17} -1.82709i q^{18} -1.70353 q^{19} +1.44512 q^{21} -3.89025i q^{22} -0.323478i q^{23} -1.20906 q^{24} +10.4209 q^{26} -1.00000i q^{27} +1.93395i q^{28} +8.74724 q^{29} -8.45991 q^{31} -6.50828i q^{32} -2.12920i q^{33} +7.59378 q^{34} +1.33826 q^{36} -1.75170i q^{37} -3.11251i q^{38} +5.70353 q^{39} -6.87802 q^{41} +2.64037i q^{42} -11.1411i q^{43} +2.84943 q^{44} +0.591023 q^{46} -12.5982i q^{47} -4.88558i q^{48} +4.91161 q^{49} +4.15622 q^{51} +7.63282i q^{52} +8.34451i q^{53} +1.82709 q^{54} +1.74724 q^{56} -1.70353i q^{57} +15.9820i q^{58} +2.12474 q^{59} -5.38952 q^{61} -15.4570i q^{62} +1.44512i q^{63} +2.12007 q^{64} +3.89025 q^{66} +7.13078i q^{67} +5.56210i q^{68} +0.323478 q^{69} +2.67461 q^{71} -1.20906i q^{72} +6.28253i q^{73} +3.20052 q^{74} +2.27977 q^{76} +3.07697i q^{77} +10.4209i q^{78} -8.37092 q^{79} +1.00000 q^{81} -12.5668i q^{82} -14.5872i q^{83} -1.93395 q^{84} +20.3558 q^{86} +8.74724i q^{87} -2.57433i q^{88} -2.68119 q^{89} -8.24232 q^{91} +0.432897i q^{92} -8.45991i q^{93} +23.0181 q^{94} +6.50828 q^{96} -8.55105i q^{97} +8.97397i q^{98} +2.12920 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{4} - 2 q^{6} - 8 q^{9}+O(q^{10}) \) 8 * q - 2 * q^4 - 2 * q^6 - 8 * q^9	\( 8 q - 2 q^{4} - 2 q^{6} - 8 q^{9} - 12 q^{11} + 20 q^{14} - 18 q^{16} + 18 q^{19} - 10 q^{21} - 6 q^{24} - 4 q^{26} + 56 q^{29} - 20 q^{31} - 14 q^{34} + 2 q^{36} + 14 q^{39} + 18 q^{44} + 10 q^{46} + 6 q^{49} + 14 q^{51} + 2 q^{54} - 8 q^{59} - 86 q^{61} + 14 q^{64} - 12 q^{66} + 20 q^{69} - 54 q^{71} - 10 q^{74} + 18 q^{76} - 20 q^{79} + 8 q^{81} + 10 q^{84} + 48 q^{86} + 18 q^{89} + 10 q^{91} + 44 q^{94} + 12 q^{99}+O(q^{100}) \) 8 * q - 2 * q^4 - 2 * q^6 - 8 * q^9 - 12 * q^11 + 20 * q^14 - 18 * q^16 + 18 * q^19 - 10 * q^21 - 6 * q^24 - 4 * q^26 + 56 * q^29 - 20 * q^31 - 14 * q^34 + 2 * q^36 + 14 * q^39 + 18 * q^44 + 10 * q^46 + 6 * q^49 + 14 * q^51 + 2 * q^54 - 8 * q^59 - 86 * q^61 + 14 * q^64 - 12 * q^66 + 20 * q^69 - 54 * q^71 - 10 * q^74 + 18 * q^76 - 20 * q^79 + 8 * q^81 + 10 * q^84 + 48 * q^86 + 18 * q^89 + 10 * q^91 + 44 * q^94 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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