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Properties

Label 1156.4.a.j.1.3
Level $1156$
Weight $4$
Character 1156.1
Self dual yes
Analytic conductor $68.206$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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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] = [1156,4,Mod(1,1156)] 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("1156.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1156, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1156.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,-30] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(68.2062079666\)
Analytic rank: \(1\)
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: \( x^{12} - 216 x^{10} - 74 x^{9} + 17391 x^{8} + 9408 x^{7} - 659646 x^{6} - 424698 x^{5} + \cdots + 168035561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 17^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(5.10659\) of defining polynomial
Character \(\chi\) \(=\) 1156.1

$q$-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)
 
\(f(q)\) \(=\) \(q-5.10659 q^{3} +10.5769 q^{5} +4.18296 q^{7} -0.922756 q^{9} +8.22820 q^{11} -19.4400 q^{13} -54.0121 q^{15} +69.3739 q^{19} -21.3606 q^{21} -74.8678 q^{23} -13.1284 q^{25} +142.590 q^{27} +181.921 q^{29} -285.336 q^{31} -42.0180 q^{33} +44.2429 q^{35} -240.180 q^{37} +99.2721 q^{39} -259.892 q^{41} +12.4851 q^{43} -9.75993 q^{45} -24.7168 q^{47} -325.503 q^{49} +468.130 q^{53} +87.0291 q^{55} -354.264 q^{57} +678.971 q^{59} +717.969 q^{61} -3.85985 q^{63} -205.616 q^{65} +317.490 q^{67} +382.319 q^{69} -891.630 q^{71} -237.892 q^{73} +67.0412 q^{75} +34.4182 q^{77} +358.281 q^{79} -703.234 q^{81} +537.587 q^{83} -928.995 q^{87} +331.539 q^{89} -81.3167 q^{91} +1457.10 q^{93} +733.764 q^{95} -768.911 q^{97} -7.59262 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 30 q^{5} + 18 q^{7} + 108 q^{9} - 66 q^{11} - 72 q^{13} - 138 q^{15} + 138 q^{19} - 42 q^{21} - 132 q^{23} + 444 q^{25} - 222 q^{27} - 564 q^{29} - 54 q^{31} - 390 q^{33} + 678 q^{35} - 474 q^{37}+ \cdots - 6978 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −5.10659 −0.982763 −0.491382 − 0.870944i \(-0.663508\pi\)
−0.491382 + 0.870944i \(0.663508\pi\)
\(4\) 0 0
\(5\) 10.5769 0.946030 0.473015 − 0.881054i \(-0.343166\pi\)
0.473015 + 0.881054i \(0.343166\pi\)
\(6\) 0 0
\(7\) 4.18296 0.225858 0.112929 − 0.993603i \(-0.463977\pi\)
0.112929 + 0.993603i \(0.463977\pi\)
\(8\) 0 0
\(9\) −0.922756 −0.0341761
\(10\) 0 0
\(11\) 8.22820 0.225536 0.112768 − 0.993621i \(-0.464028\pi\)
0.112768 + 0.993621i \(0.464028\pi\)
\(12\) 0 0
\(13\) −19.4400 −0.414745 −0.207373 − 0.978262i \(-0.566491\pi\)
−0.207373 + 0.978262i \(0.566491\pi\)
\(14\) 0 0
\(15\) −54.0121 −0.929724
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) 69.3739 0.837656 0.418828 − 0.908066i \(-0.362441\pi\)
0.418828 + 0.908066i \(0.362441\pi\)
\(20\) 0 0
\(21\) −21.3606 −0.221965
\(22\) 0 0
\(23\) −74.8678 −0.678740 −0.339370 − 0.940653i \(-0.610214\pi\)
−0.339370 + 0.940653i \(0.610214\pi\)
\(24\) 0 0
\(25\) −13.1284 −0.105027
\(26\) 0 0
\(27\) 142.590 1.01635
\(28\) 0 0
\(29\) 181.921 1.16489 0.582446 − 0.812870i \(-0.302096\pi\)
0.582446 + 0.812870i \(0.302096\pi\)
\(30\) 0 0
\(31\) −285.336 −1.65316 −0.826579 − 0.562820i \(-0.809716\pi\)
−0.826579 + 0.562820i \(0.809716\pi\)
\(32\) 0 0
\(33\) −42.0180 −0.221648
\(34\) 0 0
\(35\) 44.2429 0.213669
\(36\) 0 0
\(37\) −240.180 −1.06717 −0.533586 − 0.845746i \(-0.679156\pi\)
−0.533586 + 0.845746i \(0.679156\pi\)
\(38\) 0 0
\(39\) 99.2721 0.407596
\(40\) 0 0
\(41\) −259.892 −0.989959 −0.494979 − 0.868905i \(-0.664824\pi\)
−0.494979 + 0.868905i \(0.664824\pi\)
\(42\) 0 0
\(43\) 12.4851 0.0442783 0.0221391 − 0.999755i \(-0.492952\pi\)
0.0221391 + 0.999755i \(0.492952\pi\)
\(44\) 0 0
\(45\) −9.75993 −0.0323317
\(46\) 0 0
\(47\) −24.7168 −0.0767089 −0.0383545 − 0.999264i \(-0.512212\pi\)
−0.0383545 + 0.999264i \(0.512212\pi\)
\(48\) 0 0
\(49\) −325.503 −0.948988
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 468.130 1.21326 0.606628 − 0.794986i \(-0.292522\pi\)
0.606628 + 0.794986i \(0.292522\pi\)
\(54\) 0 0
\(55\) 87.0291 0.213364
\(56\) 0 0
\(57\) −354.264 −0.823218
\(58\) 0 0
\(59\) 678.971 1.49821 0.749106 − 0.662450i \(-0.230483\pi\)
0.749106 + 0.662450i \(0.230483\pi\)
\(60\) 0 0
\(61\) 717.969 1.50699 0.753496 − 0.657452i \(-0.228366\pi\)
0.753496 + 0.657452i \(0.228366\pi\)
\(62\) 0 0
\(63\) −3.85985 −0.00771897
\(64\) 0 0
\(65\) −205.616 −0.392361
\(66\) 0 0
\(67\) 317.490 0.578919 0.289459 − 0.957190i \(-0.406525\pi\)
0.289459 + 0.957190i \(0.406525\pi\)
\(68\) 0 0
\(69\) 382.319 0.667041
\(70\) 0 0
\(71\) −891.630 −1.49038 −0.745190 − 0.666852i \(-0.767641\pi\)
−0.745190 + 0.666852i \(0.767641\pi\)
\(72\) 0 0
\(73\) −237.892 −0.381413 −0.190707 − 0.981647i \(-0.561078\pi\)
−0.190707 + 0.981647i \(0.561078\pi\)
\(74\) 0 0
\(75\) 67.0412 0.103217
\(76\) 0 0
\(77\) 34.4182 0.0509392
\(78\) 0 0
\(79\) 358.281 0.510250 0.255125 − 0.966908i \(-0.417883\pi\)
0.255125 + 0.966908i \(0.417883\pi\)
\(80\) 0 0
\(81\) −703.234 −0.964656
\(82\) 0 0
\(83\) 537.587 0.710938 0.355469 − 0.934688i \(-0.384321\pi\)
0.355469 + 0.934688i \(0.384321\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −928.995 −1.14481
\(88\) 0 0
\(89\) 331.539 0.394865 0.197433 − 0.980316i \(-0.436740\pi\)
0.197433 + 0.980316i \(0.436740\pi\)
\(90\) 0 0
\(91\) −81.3167 −0.0936737
\(92\) 0 0
\(93\) 1457.10 1.62466
\(94\) 0 0
\(95\) 733.764 0.792448
\(96\) 0 0
\(97\) −768.911 −0.804857 −0.402428 − 0.915451i \(-0.631834\pi\)
−0.402428 + 0.915451i \(0.631834\pi\)
\(98\) 0 0
\(99\) −7.59262 −0.00770795
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 1156.4.a.j.1.3 ✓ 12
17.4 even 4 1156.4.b.h.577.19 24
17.13 even 4 1156.4.b.h.577.6 24
17.16 even 2 1156.4.a.k.1.10 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1156.4.a.j.1.3 ✓ 12 1.1 even 1 trivial
1156.4.a.k.1.10 yes 12 17.16 even 2
1156.4.b.h.577.6 24 17.13 even 4
1156.4.b.h.577.19 24 17.4 even 4