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Properties

Label 1156.2.h.d.733.1
Level $1156$
Weight $2$
Character 1156.733
Analytic conductor $9.231$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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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] = [1156,2,Mod(733,1156)] 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("1156.733"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1156, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1156.h (of order \(8\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(33)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.23070647366\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{16})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 68)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 733.1
Root \(-0.382683 + 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 1156.733
Dual form 1156.2.h.d.757.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.30656 + 0.541196i) q^{3} +(1.08239 + 2.61313i) q^{5} +(1.62359 - 3.91969i) q^{7} +(-0.707107 + 0.707107i) q^{9} +(1.30656 + 0.541196i) q^{11} +4.00000i q^{13} +(-2.82843 - 2.82843i) q^{15} +(2.82843 + 2.82843i) q^{19} +6.00000i q^{21} +(-1.30656 - 0.541196i) q^{23} +(-2.12132 + 2.12132i) q^{25} +(2.16478 - 5.22625i) q^{27} +(-1.08239 - 2.61313i) q^{29} +(3.91969 - 1.62359i) q^{31} -2.00000 q^{33} +12.0000 q^{35} +(7.83938 - 3.24718i) q^{37} +(-2.16478 - 5.22625i) q^{39} +(-4.32957 + 10.4525i) q^{41} +(-5.65685 + 5.65685i) q^{43} +(-2.61313 - 1.08239i) q^{45} +12.0000i q^{47} +(-7.77817 - 7.77817i) q^{49} +(4.24264 + 4.24264i) q^{53} +4.00000i q^{55} +(-5.22625 - 2.16478i) q^{57} +(-3.24718 + 7.83938i) q^{61} +(1.62359 + 3.91969i) q^{63} +(-10.4525 + 4.32957i) q^{65} +4.00000 q^{67} +2.00000 q^{69} +(-6.53281 + 2.70598i) q^{71} +(1.62359 - 3.91969i) q^{75} +(4.24264 - 4.24264i) q^{77} +(-3.91969 - 1.62359i) q^{79} +5.00000i q^{81} +(2.82843 + 2.82843i) q^{87} +12.0000i q^{89} +(15.6788 + 6.49435i) q^{91} +(-4.24264 + 4.24264i) q^{93} +(-4.32957 + 10.4525i) q^{95} +(-1.30656 + 0.541196i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{33} + 96 q^{35} + 32 q^{67} + 16 q^{69}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times\).

\(n\) \(579\) \(581\)
\(\chi(n)\) \(1\) \(e\left(\frac{7}{8}\right)\)

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
\(3\) −1.30656 + 0.541196i −0.754344 + 0.312460i −0.726513 − 0.687153i \(-0.758860\pi\)
−0.0278317 + 0.999613i \(0.508860\pi\)
\(4\) 0 0
\(5\) 1.08239 + 2.61313i 0.484061 + 1.16863i 0.957664 + 0.287887i \(0.0929529\pi\)
−0.473604 + 0.880738i \(0.657047\pi\)
\(6\) 0 0
\(7\) 1.62359 − 3.91969i 0.613659 − 1.48150i −0.245294 − 0.969449i \(-0.578885\pi\)
0.858953 − 0.512054i \(-0.171115\pi\)
\(8\) 0 0
\(9\) −0.707107 + 0.707107i −0.235702 + 0.235702i
\(10\) 0 0
\(11\) 1.30656 + 0.541196i 0.393944 + 0.163177i 0.570857 − 0.821050i \(-0.306611\pi\)
−0.176913 + 0.984226i \(0.556611\pi\)
\(12\) 0 0
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) −2.82843 − 2.82843i −0.730297 − 0.730297i
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) 2.82843 + 2.82843i 0.648886 + 0.648886i 0.952724 − 0.303838i \(-0.0982682\pi\)
−0.303838 + 0.952724i \(0.598268\pi\)
\(20\) 0 0
\(21\) 6.00000i 1.30931i
\(22\) 0 0
\(23\) −1.30656 − 0.541196i −0.272437 − 0.112847i 0.242282 − 0.970206i \(-0.422104\pi\)
−0.514720 + 0.857359i \(0.672104\pi\)
\(24\) 0 0
\(25\) −2.12132 + 2.12132i −0.424264 + 0.424264i
\(26\) 0 0
\(27\) 2.16478 − 5.22625i 0.416613 − 1.00579i
\(28\) 0 0
\(29\) −1.08239 − 2.61313i −0.200995 − 0.485245i 0.790955 − 0.611874i \(-0.209584\pi\)
−0.991950 + 0.126629i \(0.959584\pi\)
\(30\) 0 0
\(31\) 3.91969 − 1.62359i 0.703997 − 0.291605i −0.00182103 − 0.999998i \(-0.500580\pi\)
0.705818 + 0.708393i \(0.250580\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 12.0000 2.02837
\(36\) 0 0
\(37\) 7.83938 − 3.24718i 1.28879 − 0.533833i 0.370162 − 0.928967i \(-0.379302\pi\)
0.918623 + 0.395134i \(0.129302\pi\)
\(38\) 0 0
\(39\) −2.16478 − 5.22625i −0.346643 − 0.836870i
\(40\) 0 0
\(41\) −4.32957 + 10.4525i −0.676165 + 1.63241i 0.0947747 + 0.995499i \(0.469787\pi\)
−0.770940 + 0.636908i \(0.780213\pi\)
\(42\) 0 0
\(43\) −5.65685 + 5.65685i −0.862662 + 0.862662i −0.991647 − 0.128984i \(-0.958828\pi\)
0.128984 + 0.991647i \(0.458828\pi\)
\(44\) 0 0
\(45\) −2.61313 − 1.08239i −0.389542 − 0.161354i
\(46\) 0 0
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 0 0
\(49\) −7.77817 − 7.77817i −1.11117 − 1.11117i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.24264 + 4.24264i 0.582772 + 0.582772i 0.935664 − 0.352892i \(-0.114802\pi\)
−0.352892 + 0.935664i \(0.614802\pi\)
\(54\) 0 0
\(55\) 4.00000i 0.539360i
\(56\) 0 0
\(57\) −5.22625 − 2.16478i −0.692234 − 0.286733i
\(58\) 0 0
\(59\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(60\) 0 0
\(61\) −3.24718 + 7.83938i −0.415758 + 1.00373i 0.567805 + 0.823163i \(0.307793\pi\)
−0.983563 + 0.180566i \(0.942207\pi\)
\(62\) 0 0
\(63\) 1.62359 + 3.91969i 0.204553 + 0.493834i
\(64\) 0 0
\(65\) −10.4525 + 4.32957i −1.29647 + 0.537017i
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) −6.53281 + 2.70598i −0.775302 + 0.321141i −0.735018 − 0.678047i \(-0.762826\pi\)
−0.0402844 + 0.999188i \(0.512826\pi\)
\(72\) 0 0
\(73\) 0 0 0.923880 − 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(74\) 0 0
\(75\) 1.62359 − 3.91969i 0.187476 − 0.452607i
\(76\) 0 0
\(77\) 4.24264 − 4.24264i 0.483494 − 0.483494i
\(78\) 0 0
\(79\) −3.91969 − 1.62359i −0.440999 − 0.182668i 0.151125 − 0.988515i \(-0.451710\pi\)
−0.592124 + 0.805847i \(0.701710\pi\)
\(80\) 0 0
\(81\) 5.00000i 0.555556i
\(82\) 0 0
\(83\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.82843 + 2.82843i 0.303239 + 0.303239i
\(88\) 0 0
\(89\) 12.0000i 1.27200i 0.771690 + 0.635999i \(0.219412\pi\)
−0.771690 + 0.635999i \(0.780588\pi\)
\(90\) 0 0
\(91\) 15.6788 + 6.49435i 1.64358 + 0.680793i
\(92\) 0 0
\(93\) −4.24264 + 4.24264i −0.439941 + 0.439941i
\(94\) 0 0
\(95\) −4.32957 + 10.4525i −0.444204 + 1.07240i
\(96\) 0 0
\(97\) 0 0 0.923880 − 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(98\) 0 0
\(99\) −1.30656 + 0.541196i −0.131315 + 0.0543923i
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 1156.2.h.d.733.1 8
17.2 even 8 inner 1156.2.h.d.977.1 8
17.3 odd 16 1156.2.a.c.1.1 2
17.4 even 4 inner 1156.2.h.d.1001.2 8
17.5 odd 16 68.2.b.a.33.1 ✓ 2
17.6 odd 16 1156.2.e.a.905.1 2
17.7 odd 16 1156.2.e.b.829.1 2
17.8 even 8 inner 1156.2.h.d.757.2 8
17.9 even 8 inner 1156.2.h.d.757.1 8
17.10 odd 16 1156.2.e.a.829.1 2
17.11 odd 16 1156.2.e.b.905.1 2
17.12 odd 16 68.2.b.a.33.2 yes 2
17.13 even 4 inner 1156.2.h.d.1001.1 8
17.14 odd 16 1156.2.a.c.1.2 2
17.15 even 8 inner 1156.2.h.d.977.2 8
17.16 even 2 inner 1156.2.h.d.733.2 8
51.5 even 16 612.2.b.a.577.2 2
51.29 even 16 612.2.b.a.577.1 2
68.3 even 16 4624.2.a.n.1.2 2
68.31 even 16 4624.2.a.n.1.1 2
68.39 even 16 272.2.b.c.33.2 2
68.63 even 16 272.2.b.c.33.1 2
85.12 even 16 1700.2.g.a.849.3 4
85.22 even 16 1700.2.g.a.849.2 4
85.29 odd 16 1700.2.c.a.101.1 2
85.39 odd 16 1700.2.c.a.101.2 2
85.63 even 16 1700.2.g.a.849.1 4
85.73 even 16 1700.2.g.a.849.4 4
119.90 even 16 3332.2.b.a.2549.2 2
119.97 even 16 3332.2.b.a.2549.1 2
136.5 odd 16 1088.2.b.e.577.2 2
136.29 odd 16 1088.2.b.e.577.1 2
136.107 even 16 1088.2.b.f.577.1 2
136.131 even 16 1088.2.b.f.577.2 2
204.107 odd 16 2448.2.c.d.577.2 2
204.131 odd 16 2448.2.c.d.577.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
68.2.b.a.33.1 ✓ 2 17.5 odd 16
68.2.b.a.33.2 yes 2 17.12 odd 16
272.2.b.c.33.1 2 68.63 even 16
272.2.b.c.33.2 2 68.39 even 16
612.2.b.a.577.1 2 51.29 even 16
612.2.b.a.577.2 2 51.5 even 16
1088.2.b.e.577.1 2 136.29 odd 16
1088.2.b.e.577.2 2 136.5 odd 16
1088.2.b.f.577.1 2 136.107 even 16
1088.2.b.f.577.2 2 136.131 even 16
1156.2.a.c.1.1 2 17.3 odd 16
1156.2.a.c.1.2 2 17.14 odd 16
1156.2.e.a.829.1 2 17.10 odd 16
1156.2.e.a.905.1 2 17.6 odd 16
1156.2.e.b.829.1 2 17.7 odd 16
1156.2.e.b.905.1 2 17.11 odd 16
1156.2.h.d.733.1 8 1.1 even 1 trivial
1156.2.h.d.733.2 8 17.16 even 2 inner
1156.2.h.d.757.1 8 17.9 even 8 inner
1156.2.h.d.757.2 8 17.8 even 8 inner
1156.2.h.d.977.1 8 17.2 even 8 inner
1156.2.h.d.977.2 8 17.15 even 8 inner
1156.2.h.d.1001.1 8 17.13 even 4 inner
1156.2.h.d.1001.2 8 17.4 even 4 inner
1700.2.c.a.101.1 2 85.29 odd 16
1700.2.c.a.101.2 2 85.39 odd 16
1700.2.g.a.849.1 4 85.63 even 16
1700.2.g.a.849.2 4 85.22 even 16
1700.2.g.a.849.3 4 85.12 even 16
1700.2.g.a.849.4 4 85.73 even 16
2448.2.c.d.577.1 2 204.131 odd 16
2448.2.c.d.577.2 2 204.107 odd 16
3332.2.b.a.2549.1 2 119.97 even 16
3332.2.b.a.2549.2 2 119.90 even 16
4624.2.a.n.1.1 2 68.31 even 16
4624.2.a.n.1.2 2 68.3 even 16