Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [95,2,Mod(49,95)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("95.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 95.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.758578819202\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 64.2
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 95.64
Dual form 95.2.i.a.49.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.73205 - 1.00000i) q^{2} +(1.00000 - 1.73205i) q^{4} +(-1.23205 + 1.86603i) q^{5} -4.00000i q^{7} +(-1.50000 + 2.59808i) q^{9} +(-0.267949 + 4.46410i) q^{10} -1.00000 q^{11} +(-1.73205 - 1.00000i) q^{13} +(-4.00000 - 6.92820i) q^{14} +(2.00000 + 3.46410i) q^{16} +(-1.73205 + 1.00000i) q^{17} +6.00000i q^{18} +(3.50000 - 2.59808i) q^{19} +(2.00000 + 4.00000i) q^{20} +(-1.73205 + 1.00000i) q^{22} +(5.19615 + 3.00000i) q^{23} +(-1.96410 - 4.59808i) q^{25} -4.00000 q^{26} +(-6.92820 - 4.00000i) q^{28} +(4.50000 - 7.79423i) q^{29} -7.00000 q^{31} +(6.92820 + 4.00000i) q^{32} +(-2.00000 + 3.46410i) q^{34} +(7.46410 + 4.92820i) q^{35} +(3.00000 + 5.19615i) q^{36} +2.00000i q^{37} +(3.46410 - 8.00000i) q^{38} +(-1.00000 - 1.73205i) q^{41} +(-1.73205 + 1.00000i) q^{43} +(-1.00000 + 1.73205i) q^{44} +(-3.00000 - 6.00000i) q^{45} +12.0000 q^{46} +(-5.19615 - 3.00000i) q^{47} -9.00000 q^{49} +(-8.00000 - 6.00000i) q^{50} +(-3.46410 + 2.00000i) q^{52} +(3.46410 + 2.00000i) q^{53} +(1.23205 - 1.86603i) q^{55} -18.0000i q^{58} +(4.50000 + 7.79423i) q^{59} +(3.50000 - 6.06218i) q^{61} +(-12.1244 + 7.00000i) q^{62} +(10.3923 + 6.00000i) q^{63} +8.00000 q^{64} +(4.00000 - 2.00000i) q^{65} +(8.66025 + 5.00000i) q^{67} +4.00000i q^{68} +(17.8564 + 1.07180i) q^{70} +(-0.500000 - 0.866025i) q^{71} +(-8.66025 + 5.00000i) q^{73} +(2.00000 + 3.46410i) q^{74} +(-1.00000 - 8.66025i) q^{76} +4.00000i q^{77} +(0.500000 + 0.866025i) q^{79} +(-8.92820 - 0.535898i) q^{80} +(-4.50000 - 7.79423i) q^{81} +(-3.46410 - 2.00000i) q^{82} +6.00000i q^{83} +(0.267949 - 4.46410i) q^{85} +(-2.00000 + 3.46410i) q^{86} +(-5.50000 + 9.52628i) q^{89} +(-11.1962 - 7.39230i) q^{90} +(-4.00000 + 6.92820i) q^{91} +(10.3923 - 6.00000i) q^{92} -12.0000 q^{94} +(0.535898 + 9.73205i) q^{95} +(-5.19615 + 3.00000i) q^{97} +(-15.5885 + 9.00000i) q^{98} +(1.50000 - 2.59808i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 2 q^{5} - 6 q^{9} - 8 q^{10} - 4 q^{11} - 16 q^{14} + 8 q^{16} + 14 q^{19} + 8 q^{20} + 6 q^{25} - 16 q^{26} + 18 q^{29} - 28 q^{31} - 8 q^{34} + 16 q^{35} + 12 q^{36} - 4 q^{41} - 4 q^{44}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 − 1.00000i 1.22474 − 0.707107i 0.258819 − 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(3\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 1.00000 − 1.73205i 0.500000 − 0.866025i
\(5\) −1.23205 + 1.86603i −0.550990 + 0.834512i
\(6\) 0 0
\(7\) − 4.00000i − 1.51186i −0.654654 − 0.755929i \(-0.727186\pi\)
0.654654 − 0.755929i \(-0.272814\pi\)
\(8\) 0 0
\(9\) −1.50000 + 2.59808i −0.500000 + 0.866025i
\(10\) −0.267949 + 4.46410i −0.0847330 + 1.41167i
\(11\) −1.00000 −0.301511 −0.150756 − 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) −1.73205 − 1.00000i −0.480384 − 0.277350i 0.240192 − 0.970725i \(-0.422790\pi\)
−0.720577 + 0.693375i \(0.756123\pi\)
\(14\) −4.00000 − 6.92820i −1.06904 − 1.85164i
\(15\) 0 0
\(16\) 2.00000 + 3.46410i 0.500000 + 0.866025i
\(17\) −1.73205 + 1.00000i −0.420084 + 0.242536i −0.695113 − 0.718900i \(-0.744646\pi\)
0.275029 + 0.961436i \(0.411312\pi\)
\(18\) 6.00000i 1.41421i
\(19\) 3.50000 − 2.59808i 0.802955 − 0.596040i
\(20\) 2.00000 + 4.00000i 0.447214 + 0.894427i
\(21\) 0 0
\(22\) −1.73205 + 1.00000i −0.369274 + 0.213201i
\(23\) 5.19615 + 3.00000i 1.08347 + 0.625543i 0.931831 − 0.362892i \(-0.118211\pi\)
0.151642 + 0.988436i \(0.451544\pi\)
\(24\) 0 0
\(25\) −1.96410 − 4.59808i −0.392820 − 0.919615i
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) −6.92820 − 4.00000i −1.30931 − 0.755929i
\(29\) 4.50000 − 7.79423i 0.835629 − 1.44735i −0.0578882 − 0.998323i \(-0.518437\pi\)
0.893517 − 0.449029i \(-0.148230\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 − 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 6.92820 + 4.00000i 1.22474 + 0.707107i
\(33\) 0 0
\(34\) −2.00000 + 3.46410i −0.342997 + 0.594089i
\(35\) 7.46410 + 4.92820i 1.26166 + 0.833018i
\(36\) 3.00000 + 5.19615i 0.500000 + 0.866025i
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 3.46410 − 8.00000i 0.561951 − 1.29777i
\(39\) 0 0
\(40\) 0 0
\(41\) −1.00000 − 1.73205i −0.156174 − 0.270501i 0.777312 − 0.629115i \(-0.216583\pi\)
−0.933486 + 0.358614i \(0.883249\pi\)
\(42\) 0 0
\(43\) −1.73205 + 1.00000i −0.264135 + 0.152499i −0.626219 − 0.779647i \(-0.715399\pi\)
0.362084 + 0.932145i \(0.382065\pi\)
\(44\) −1.00000 + 1.73205i −0.150756 + 0.261116i
\(45\) −3.00000 − 6.00000i −0.447214 − 0.894427i
\(46\) 12.0000 1.76930
\(47\) −5.19615 − 3.00000i −0.757937 − 0.437595i 0.0706177 − 0.997503i \(-0.477503\pi\)
−0.828554 + 0.559908i \(0.810836\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) −8.00000 − 6.00000i −1.13137 − 0.848528i
\(51\) 0 0
\(52\) −3.46410 + 2.00000i −0.480384 + 0.277350i
\(53\) 3.46410 + 2.00000i 0.475831 + 0.274721i 0.718677 − 0.695344i \(-0.244748\pi\)
−0.242846 + 0.970065i \(0.578081\pi\)
\(54\) 0 0
\(55\) 1.23205 − 1.86603i 0.166130 − 0.251615i
\(56\) 0 0
\(57\) 0 0
\(58\) − 18.0000i − 2.36352i
\(59\) 4.50000 + 7.79423i 0.585850 + 1.01472i 0.994769 + 0.102151i \(0.0325726\pi\)
−0.408919 + 0.912571i \(0.634094\pi\)
\(60\) 0 0
\(61\) 3.50000 − 6.06218i 0.448129 − 0.776182i −0.550135 − 0.835076i \(-0.685424\pi\)
0.998264 + 0.0588933i \(0.0187572\pi\)
\(62\) −12.1244 + 7.00000i −1.53979 + 0.889001i
\(63\) 10.3923 + 6.00000i 1.30931 + 0.755929i
\(64\) 8.00000 1.00000
\(65\) 4.00000 − 2.00000i 0.496139 − 0.248069i
\(66\) 0 0
\(67\) 8.66025 + 5.00000i 1.05802 + 0.610847i 0.924883 − 0.380251i \(-0.124162\pi\)
0.133135 + 0.991098i \(0.457496\pi\)
\(68\) 4.00000i 0.485071i
\(69\) 0 0
\(70\) 17.8564 + 1.07180i 2.13425 + 0.128104i
\(71\) −0.500000 − 0.866025i −0.0593391 − 0.102778i 0.834830 − 0.550508i \(-0.185566\pi\)
−0.894169 + 0.447730i \(0.852233\pi\)
\(72\) 0 0
\(73\) −8.66025 + 5.00000i −1.01361 + 0.585206i −0.912245 − 0.409644i \(-0.865653\pi\)
−0.101361 + 0.994850i \(0.532320\pi\)
\(74\) 2.00000 + 3.46410i 0.232495 + 0.402694i
\(75\) 0 0
\(76\) −1.00000 − 8.66025i −0.114708 − 0.993399i
\(77\) 4.00000i 0.455842i
\(78\) 0 0
\(79\) 0.500000 + 0.866025i 0.0562544 + 0.0974355i 0.892781 − 0.450490i \(-0.148751\pi\)
−0.836527 + 0.547926i \(0.815418\pi\)
\(80\) −8.92820 − 0.535898i −0.998203 − 0.0599153i
\(81\) −4.50000 − 7.79423i −0.500000 − 0.866025i
\(82\) −3.46410 − 2.00000i −0.382546 − 0.220863i
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 0 0
\(85\) 0.267949 − 4.46410i 0.0290632 − 0.484200i
\(86\) −2.00000 + 3.46410i −0.215666 + 0.373544i
\(87\) 0 0
\(88\) 0 0
\(89\) −5.50000 + 9.52628i −0.582999 + 1.00978i 0.412123 + 0.911128i \(0.364787\pi\)
−0.995122 + 0.0986553i \(0.968546\pi\)
\(90\) −11.1962 − 7.39230i −1.18018 − 0.779217i
\(91\) −4.00000 + 6.92820i −0.419314 + 0.726273i
\(92\) 10.3923 − 6.00000i 1.08347 − 0.625543i
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0.535898 + 9.73205i 0.0549820 + 0.998487i
\(96\) 0 0
\(97\) −5.19615 + 3.00000i −0.527589 + 0.304604i −0.740034 − 0.672569i \(-0.765191\pi\)
0.212445 + 0.977173i \(0.431857\pi\)
\(98\) −15.5885 + 9.00000i −1.57467 + 0.909137i
\(99\) 1.50000 − 2.59808i 0.150756 − 0.261116i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 95.2.i.a.64.2 yes 4
3.2 odd 2 855.2.be.a.64.1 4
5.2 odd 4 475.2.e.c.26.1 2
5.3 odd 4 475.2.e.a.26.1 2
5.4 even 2 inner 95.2.i.a.64.1 yes 4
15.14 odd 2 855.2.be.a.64.2 4
19.7 even 3 1805.2.b.b.1084.2 2
19.11 even 3 inner 95.2.i.a.49.1 ✓ 4
19.12 odd 6 1805.2.b.a.1084.1 2
57.11 odd 6 855.2.be.a.334.2 4
95.7 odd 12 9025.2.a.b.1.1 1
95.12 even 12 9025.2.a.j.1.1 1
95.49 even 6 inner 95.2.i.a.49.2 yes 4
95.64 even 6 1805.2.b.b.1084.1 2
95.68 odd 12 475.2.e.a.201.1 2
95.69 odd 6 1805.2.b.a.1084.2 2
95.83 odd 12 9025.2.a.i.1.1 1
95.87 odd 12 475.2.e.c.201.1 2
95.88 even 12 9025.2.a.a.1.1 1
285.239 odd 6 855.2.be.a.334.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.i.a.49.1 ✓ 4 19.11 even 3 inner
95.2.i.a.49.2 yes 4 95.49 even 6 inner
95.2.i.a.64.1 yes 4 5.4 even 2 inner
95.2.i.a.64.2 yes 4 1.1 even 1 trivial
475.2.e.a.26.1 2 5.3 odd 4
475.2.e.a.201.1 2 95.68 odd 12
475.2.e.c.26.1 2 5.2 odd 4
475.2.e.c.201.1 2 95.87 odd 12
855.2.be.a.64.1 4 3.2 odd 2
855.2.be.a.64.2 4 15.14 odd 2
855.2.be.a.334.1 4 285.239 odd 6
855.2.be.a.334.2 4 57.11 odd 6
1805.2.b.a.1084.1 2 19.12 odd 6
1805.2.b.a.1084.2 2 95.69 odd 6
1805.2.b.b.1084.1 2 95.64 even 6
1805.2.b.b.1084.2 2 19.7 even 3
9025.2.a.a.1.1 1 95.88 even 12
9025.2.a.b.1.1 1 95.7 odd 12
9025.2.a.i.1.1 1 95.83 odd 12
9025.2.a.j.1.1 1 95.12 even 12