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Properties

Label 1440.4.k.c.721.5
Level $1440$
Weight $4$
Character 1440.721
Analytic conductor $84.963$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1440,4,Mod(721,1440)] 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("1440.721"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1440.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] 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: \(84.9627504083\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{30}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 721.5
Root \(1.71681 - 1.02595i\) of defining polynomial
Character \(\chi\) \(=\) 1440.721
Dual form 1440.4.k.c.721.11

$q$-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)
 
\(f(q)\) \(=\) \(q-5.00000i q^{5} +14.6308 q^{7} +14.9969i q^{11} +85.6955i q^{13} +91.9247 q^{17} +60.3737i q^{19} +1.33730 q^{23} -25.0000 q^{25} -25.5118i q^{29} -73.4354 q^{31} -73.1541i q^{35} -211.259i q^{37} -330.839 q^{41} +388.500i q^{43} -550.348 q^{47} -128.939 q^{49} -187.705i q^{53} +74.9847 q^{55} +779.090i q^{59} -358.850i q^{61} +428.477 q^{65} -283.674i q^{67} -534.811 q^{71} +1016.16 q^{73} +219.418i q^{77} -1119.59 q^{79} +1190.49i q^{83} -459.623i q^{85} +398.940 q^{89} +1253.80i q^{91} +301.869 q^{95} +278.022 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 28 q^{7} + 604 q^{23} - 300 q^{25} + 264 q^{31} - 40 q^{41} - 940 q^{47} + 1308 q^{49} - 440 q^{55} - 1592 q^{71} + 432 q^{73} - 2016 q^{79} + 424 q^{89} - 1520 q^{95} - 1584 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 5.00000i − 0.447214i
\(6\) 0 0
\(7\) 14.6308 0.789990 0.394995 − 0.918683i \(-0.370746\pi\)
0.394995 + 0.918683i \(0.370746\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 14.9969i 0.411068i 0.978650 + 0.205534i \(0.0658931\pi\)
−0.978650 + 0.205534i \(0.934107\pi\)
\(12\) 0 0
\(13\) 85.6955i 1.82828i 0.405398 + 0.914140i \(0.367133\pi\)
−0.405398 + 0.914140i \(0.632867\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 91.9247 1.31147 0.655735 − 0.754991i \(-0.272359\pi\)
0.655735 + 0.754991i \(0.272359\pi\)
\(18\) 0 0
\(19\) 60.3737i 0.728983i 0.931207 + 0.364492i \(0.118757\pi\)
−0.931207 + 0.364492i \(0.881243\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.33730 0.0121237 0.00606186 − 0.999982i \(-0.498070\pi\)
0.00606186 + 0.999982i \(0.498070\pi\)
\(24\) 0 0
\(25\) −25.0000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 25.5118i − 0.163360i −0.996659 − 0.0816798i \(-0.973972\pi\)
0.996659 − 0.0816798i \(-0.0260285\pi\)
\(30\) 0 0
\(31\) −73.4354 −0.425464 −0.212732 − 0.977111i \(-0.568236\pi\)
−0.212732 + 0.977111i \(0.568236\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 73.1541i − 0.353294i
\(36\) 0 0
\(37\) − 211.259i − 0.938668i −0.883021 − 0.469334i \(-0.844494\pi\)
0.883021 − 0.469334i \(-0.155506\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −330.839 −1.26020 −0.630102 − 0.776512i \(-0.716987\pi\)
−0.630102 + 0.776512i \(0.716987\pi\)
\(42\) 0 0
\(43\) 388.500i 1.37781i 0.724853 + 0.688904i \(0.241908\pi\)
−0.724853 + 0.688904i \(0.758092\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −550.348 −1.70801 −0.854005 − 0.520265i \(-0.825833\pi\)
−0.854005 + 0.520265i \(0.825833\pi\)
\(48\) 0 0
\(49\) −128.939 −0.375915
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 187.705i − 0.486477i −0.969966 − 0.243239i \(-0.921790\pi\)
0.969966 − 0.243239i \(-0.0782098\pi\)
\(54\) 0 0
\(55\) 74.9847 0.183835
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 779.090i 1.71913i 0.511023 + 0.859567i \(0.329267\pi\)
−0.511023 + 0.859567i \(0.670733\pi\)
\(60\) 0 0
\(61\) − 358.850i − 0.753215i −0.926373 − 0.376607i \(-0.877091\pi\)
0.926373 − 0.376607i \(-0.122909\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 428.477 0.817632
\(66\) 0 0
\(67\) − 283.674i − 0.517258i −0.965977 − 0.258629i \(-0.916729\pi\)
0.965977 − 0.258629i \(-0.0832706\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −534.811 −0.893949 −0.446975 − 0.894547i \(-0.647499\pi\)
−0.446975 + 0.894547i \(0.647499\pi\)
\(72\) 0 0
\(73\) 1016.16 1.62921 0.814607 − 0.580014i \(-0.196953\pi\)
0.814607 + 0.580014i \(0.196953\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 219.418i 0.324740i
\(78\) 0 0
\(79\) −1119.59 −1.59447 −0.797237 − 0.603666i \(-0.793706\pi\)
−0.797237 + 0.603666i \(0.793706\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1190.49i 1.57437i 0.616716 + 0.787186i \(0.288463\pi\)
−0.616716 + 0.787186i \(0.711537\pi\)
\(84\) 0 0
\(85\) − 459.623i − 0.586508i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 398.940 0.475141 0.237571 − 0.971370i \(-0.423649\pi\)
0.237571 + 0.971370i \(0.423649\pi\)
\(90\) 0 0
\(91\) 1253.80i 1.44432i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 301.869 0.326011
\(96\) 0 0
\(97\) 278.022 0.291019 0.145510 − 0.989357i \(-0.453518\pi\)
0.145510 + 0.989357i \(0.453518\pi\)
\(98\) 0 0
\(99\) 0 0
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 1440.4.k.c.721.5 12
3.2 odd 2 160.4.d.a.81.9 12
4.3 odd 2 360.4.k.c.181.2 12
8.3 odd 2 360.4.k.c.181.1 12
8.5 even 2 inner 1440.4.k.c.721.11 12
12.11 even 2 40.4.d.a.21.11 ✓ 12
15.2 even 4 800.4.f.b.49.10 12
15.8 even 4 800.4.f.c.49.3 12
15.14 odd 2 800.4.d.d.401.4 12
24.5 odd 2 160.4.d.a.81.4 12
24.11 even 2 40.4.d.a.21.12 yes 12
48.5 odd 4 1280.4.a.ba.1.5 6
48.11 even 4 1280.4.a.bc.1.2 6
48.29 odd 4 1280.4.a.bd.1.2 6
48.35 even 4 1280.4.a.bb.1.5 6
60.23 odd 4 200.4.f.b.149.7 12
60.47 odd 4 200.4.f.c.149.6 12
60.59 even 2 200.4.d.b.101.2 12
120.29 odd 2 800.4.d.d.401.9 12
120.53 even 4 800.4.f.b.49.9 12
120.59 even 2 200.4.d.b.101.1 12
120.77 even 4 800.4.f.c.49.4 12
120.83 odd 4 200.4.f.c.149.5 12
120.107 odd 4 200.4.f.b.149.8 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.d.a.21.11 ✓ 12 12.11 even 2
40.4.d.a.21.12 yes 12 24.11 even 2
160.4.d.a.81.4 12 24.5 odd 2
160.4.d.a.81.9 12 3.2 odd 2
200.4.d.b.101.1 12 120.59 even 2
200.4.d.b.101.2 12 60.59 even 2
200.4.f.b.149.7 12 60.23 odd 4
200.4.f.b.149.8 12 120.107 odd 4
200.4.f.c.149.5 12 120.83 odd 4
200.4.f.c.149.6 12 60.47 odd 4
360.4.k.c.181.1 12 8.3 odd 2
360.4.k.c.181.2 12 4.3 odd 2
800.4.d.d.401.4 12 15.14 odd 2
800.4.d.d.401.9 12 120.29 odd 2
800.4.f.b.49.9 12 120.53 even 4
800.4.f.b.49.10 12 15.2 even 4
800.4.f.c.49.3 12 15.8 even 4
800.4.f.c.49.4 12 120.77 even 4
1280.4.a.ba.1.5 6 48.5 odd 4
1280.4.a.bb.1.5 6 48.35 even 4
1280.4.a.bc.1.2 6 48.11 even 4
1280.4.a.bd.1.2 6 48.29 odd 4
1440.4.k.c.721.5 12 1.1 even 1 trivial
1440.4.k.c.721.11 12 8.5 even 2 inner