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Properties

Label 12.22.b.a.11.7
Level $12$
Weight $22$
Character 12.11
Analytic conductor $33.537$
Analytic rank $0$
Dimension $40$
Inner twists $4$

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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] = [12,22,Mod(11,12)] 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("12.11"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 12.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(33.5372813144\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 11.7
Character \(\chi\) \(=\) 12.11
Dual form 12.22.b.a.11.8

$q$-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+(-1218.33 - 782.825i) q^{2} +(-54958.0 + 86255.2i) q^{3} +(871523. + 1.90748e6i) q^{4} +3.43031e7i q^{5} +(1.34480e8 - 6.20651e7i) q^{6} -6.03889e8i q^{7} +(4.31420e8 - 3.00620e9i) q^{8} +(-4.41958e9 - 9.48084e9i) q^{9} +(2.68533e10 - 4.17926e10i) q^{10} -1.10950e11 q^{11} +(-2.12428e11 - 2.96582e10i) q^{12} +8.42556e11 q^{13} +(-4.72740e11 + 7.35739e11i) q^{14} +(-2.95882e12 - 1.88523e12i) q^{15} +(-2.87894e12 + 3.32483e12i) q^{16} -9.53568e12i q^{17} +(-2.03731e12 + 1.50106e13i) q^{18} +2.10795e12i q^{19} +(-6.54326e13 + 2.98959e13i) q^{20} +(5.20886e13 + 3.31886e13i) q^{21} +(1.35175e14 + 8.68547e13i) q^{22} -1.21156e14 q^{23} +(2.35591e14 + 2.02427e14i) q^{24} -6.99867e14 q^{25} +(-1.02651e15 - 6.59574e14i) q^{26} +(1.06066e15 + 1.39837e14i) q^{27} +(1.15191e15 - 5.26303e14i) q^{28} -5.55699e14i q^{29} +(2.12903e15 + 4.61308e15i) q^{30} -7.08884e15i q^{31} +(6.11027e15 - 1.79705e15i) q^{32} +(6.09762e15 - 9.57005e15i) q^{33} +(-7.46476e15 + 1.16176e16i) q^{34} +2.07153e16 q^{35} +(1.42328e16 - 1.66930e16i) q^{36} +2.93058e16 q^{37} +(1.65016e15 - 2.56819e15i) q^{38} +(-4.63053e16 + 7.26749e16i) q^{39} +(1.03122e17 + 1.47991e16i) q^{40} +1.67996e16i q^{41} +(-3.74805e16 - 8.12110e16i) q^{42} +1.09563e17i q^{43} +(-9.66958e16 - 2.11636e17i) q^{44} +(3.25222e17 - 1.51605e17i) q^{45} +(1.47609e17 + 9.48443e16i) q^{46} -6.85357e16 q^{47} +(-1.28563e17 - 4.31050e17i) q^{48} +1.93864e17 q^{49} +(8.52671e17 + 5.47873e17i) q^{50} +(8.22502e17 + 5.24062e17i) q^{51} +(7.34307e17 + 1.60716e18i) q^{52} +1.91799e17i q^{53} +(-1.18277e18 - 1.00068e18i) q^{54} -3.80594e18i q^{55} +(-1.81541e18 - 2.60530e17i) q^{56} +(-1.81822e17 - 1.15849e17i) q^{57} +(-4.35015e17 + 6.77026e17i) q^{58} +2.49398e18 q^{59} +(1.01737e18 - 7.28693e18i) q^{60} -6.68065e18 q^{61} +(-5.54932e18 + 8.63657e18i) q^{62} +(-5.72538e18 + 2.66894e18i) q^{63} +(-8.85112e18 - 2.59387e18i) q^{64} +2.89023e19i q^{65} +(-1.49206e19 + 6.88615e18i) q^{66} -7.79677e17i q^{67} +(1.81891e19 - 8.31056e18i) q^{68} +(6.65852e18 - 1.04504e19i) q^{69} +(-2.52381e19 - 1.62164e19i) q^{70} +6.90677e18 q^{71} +(-3.04080e19 + 9.19592e18i) q^{72} +5.70164e19 q^{73} +(-3.57042e19 - 2.29413e19i) q^{74} +(3.84633e19 - 6.03672e19i) q^{75} +(-4.02088e18 + 1.83713e18i) q^{76} +6.70018e19i q^{77} +(1.13307e20 - 5.22934e19i) q^{78} +4.06193e19i q^{79} +(-1.14052e20 - 9.87567e19i) q^{80} +(-7.03536e19 + 8.38026e19i) q^{81} +(1.31511e19 - 2.04675e19i) q^{82} +2.07382e20 q^{83} +(-1.79103e19 + 1.28283e20i) q^{84} +3.27103e20 q^{85} +(8.57683e19 - 1.33484e20i) q^{86} +(4.79319e19 + 3.05401e19i) q^{87} +(-4.78663e19 + 3.33539e20i) q^{88} -2.94977e18i q^{89} +(-5.14910e20 - 6.98862e19i) q^{90} -5.08811e20i q^{91} +(-1.05591e20 - 2.31104e20i) q^{92} +(6.11449e20 + 3.89589e20i) q^{93} +(8.34994e19 + 5.36515e19i) q^{94} -7.23093e19 q^{95} +(-1.80804e20 + 6.25805e20i) q^{96} -2.15714e20 q^{97} +(-2.36190e20 - 1.51761e20i) q^{98} +(4.90354e20 + 1.05190e21i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 1212040 q^{4} - 73477608 q^{6} + 4862757768 q^{9} + 70993722608 q^{10} + 230311800264 q^{12} + 532697931824 q^{13} - 3296861950688 q^{16} - 38138435545488 q^{18} + 6345498814320 q^{21} + 345245332418448 q^{22}+ \cdots + 19\!\cdots\!68 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1218.33 − 782.825i −0.841301 − 0.540567i
\(3\) −54958.0 + 86255.2i −0.537351 + 0.843359i
\(4\) 871523. + 1.90748e6i 0.415574 + 0.909559i
\(5\) 3.43031e7i 1.57090i 0.618925 + 0.785450i \(0.287568\pi\)
−0.618925 + 0.785450i \(0.712432\pi\)
\(6\) 1.34480e8 − 6.20651e7i 0.907966 − 0.419044i
\(7\) − 6.03889e8i − 0.808031i −0.914752 − 0.404015i \(-0.867614\pi\)
0.914752 − 0.404015i \(-0.132386\pi\)
\(8\) 4.31420e8 − 3.00620e9i 0.142055 − 0.989859i
\(9\) −4.41958e9 − 9.48084e9i −0.422508 − 0.906359i
\(10\) 2.68533e10 − 4.17926e10i 0.849177 − 1.32160i
\(11\) −1.10950e11 −1.28975 −0.644875 − 0.764288i \(-0.723091\pi\)
−0.644875 + 0.764288i \(0.723091\pi\)
\(12\) −2.12428e11 − 2.96582e10i −0.990394 − 0.138274i
\(13\) 8.42556e11 1.69509 0.847547 − 0.530720i \(-0.178079\pi\)
0.847547 + 0.530720i \(0.178079\pi\)
\(14\) −4.72740e11 + 7.35739e11i −0.436795 + 0.679797i
\(15\) −2.95882e12 − 1.88523e12i −1.32483 − 0.844125i
\(16\) −2.87894e12 + 3.32483e12i −0.654596 + 0.755979i
\(17\) − 9.53568e12i − 1.14720i −0.819137 − 0.573598i \(-0.805547\pi\)
0.819137 − 0.573598i \(-0.194453\pi\)
\(18\) −2.03731e12 + 1.50106e13i −0.134492 + 0.990915i
\(19\) 2.10795e12i 0.0788766i 0.999222 + 0.0394383i \(0.0125569\pi\)
−0.999222 + 0.0394383i \(0.987443\pi\)
\(20\) −6.54326e13 + 2.98959e13i −1.42883 + 0.652826i
\(21\) 5.20886e13 + 3.31886e13i 0.681460 + 0.434196i
\(22\) 1.35175e14 + 8.68547e13i 1.08507 + 0.697197i
\(23\) −1.21156e14 −0.609824 −0.304912 − 0.952381i \(-0.598627\pi\)
−0.304912 + 0.952381i \(0.598627\pi\)
\(24\) 2.35591e14 + 2.02427e14i 0.758473 + 0.651705i
\(25\) −6.99867e14 −1.46773
\(26\) −1.02651e15 − 6.59574e14i −1.42608 − 0.916312i
\(27\) 1.06066e15 + 1.39837e14i 0.991421 + 0.130708i
\(28\) 1.15191e15 − 5.26303e14i 0.734952 − 0.335797i
\(29\) − 5.55699e14i − 0.245279i −0.992451 − 0.122639i \(-0.960864\pi\)
0.992451 − 0.122639i \(-0.0391359\pi\)
\(30\) 2.12903e15 + 4.61308e15i 0.658276 + 1.42632i
\(31\) − 7.08884e15i − 1.55338i −0.629884 − 0.776689i \(-0.716898\pi\)
0.629884 − 0.776689i \(-0.283102\pi\)
\(32\) 6.11027e15 − 1.79705e15i 0.959370 − 0.282153i
\(33\) 6.09762e15 − 9.57005e15i 0.693049 − 1.08772i
\(34\) −7.46476e15 + 1.16176e16i −0.620137 + 0.965137i
\(35\) 2.07153e16 1.26934
\(36\) 1.42328e16 − 1.66930e16i 0.648804 − 0.760955i
\(37\) 2.93058e16 1.00192 0.500962 − 0.865469i \(-0.332980\pi\)
0.500962 + 0.865469i \(0.332980\pi\)
\(38\) 1.65016e15 − 2.56819e15i 0.0426381 − 0.0663589i
\(39\) −4.63053e16 + 7.26749e16i −0.910861 + 1.42957i
\(40\) 1.03122e17 + 1.47991e16i 1.55497 + 0.223154i
\(41\) 1.67996e16i 0.195464i 0.995213 + 0.0977321i \(0.0311588\pi\)
−0.995213 + 0.0977321i \(0.968841\pi\)
\(42\) −3.74805e16 − 8.12110e16i −0.338601 − 0.733664i
\(43\) 1.09563e17i 0.773113i 0.922266 + 0.386557i \(0.126336\pi\)
−0.922266 + 0.386557i \(0.873664\pi\)
\(44\) −9.66958e16 − 2.11636e17i −0.535987 − 1.17310i
\(45\) 3.25222e17 − 1.51605e17i 1.42380 − 0.663717i
\(46\) 1.47609e17 + 9.48443e16i 0.513045 + 0.329651i
\(47\) −6.85357e16 −0.190059 −0.0950297 − 0.995474i \(-0.530295\pi\)
−0.0950297 + 0.995474i \(0.530295\pi\)
\(48\) −1.28563e17 − 4.31050e17i −0.285814 − 0.958285i
\(49\) 1.93864e17 0.347086
\(50\) 8.52671e17 + 5.47873e17i 1.23480 + 0.793405i
\(51\) 8.22502e17 + 5.24062e17i 0.967498 + 0.616447i
\(52\) 7.34307e17 + 1.60716e18i 0.704438 + 1.54179i
\(53\) 1.91799e17i 0.150643i 0.997159 + 0.0753217i \(0.0239984\pi\)
−0.997159 + 0.0753217i \(0.976002\pi\)
\(54\) −1.18277e18 − 1.00068e18i −0.763427 − 0.645894i
\(55\) − 3.80594e18i − 2.02607i
\(56\) −1.81541e18 − 2.60530e17i −0.799836 − 0.114785i
\(57\) −1.81822e17 − 1.15849e17i −0.0665212 − 0.0423844i
\(58\) −4.35015e17 + 6.77026e17i −0.132590 + 0.206353i
\(59\) 2.49398e18 0.635252 0.317626 − 0.948216i \(-0.397114\pi\)
0.317626 + 0.948216i \(0.397114\pi\)
\(60\) 1.01737e18 − 7.28693e18i 0.217215 − 1.55581i
\(61\) −6.68065e18 −1.19910 −0.599550 − 0.800337i \(-0.704654\pi\)
−0.599550 + 0.800337i \(0.704654\pi\)
\(62\) −5.54932e18 + 8.63657e18i −0.839705 + 1.30686i
\(63\) −5.72538e18 + 2.66894e18i −0.732366 + 0.341399i
\(64\) −8.85112e18 − 2.59387e18i −0.959641 − 0.281228i
\(65\) 2.89023e19i 2.66282i
\(66\) −1.49206e19 + 6.88615e18i −1.17105 + 0.540462i
\(67\) − 7.79677e17i − 0.0522552i −0.999659 − 0.0261276i \(-0.991682\pi\)
0.999659 − 0.0261276i \(-0.00831761\pi\)
\(68\) 1.81891e19 − 8.31056e18i 1.04344 − 0.476745i
\(69\) 6.65852e18 − 1.04504e19i 0.327690 − 0.514300i
\(70\) −2.52381e19 − 1.62164e19i −1.06789 − 0.686161i
\(71\) 6.90677e18 0.251804 0.125902 − 0.992043i \(-0.459818\pi\)
0.125902 + 0.992043i \(0.459818\pi\)
\(72\) −3.04080e19 + 9.19592e18i −0.957187 + 0.289470i
\(73\) 5.70164e19 1.55278 0.776389 − 0.630254i \(-0.217049\pi\)
0.776389 + 0.630254i \(0.217049\pi\)
\(74\) −3.57042e19 − 2.29413e19i −0.842920 − 0.541607i
\(75\) 3.84633e19 − 6.03672e19i 0.788685 − 1.23782i
\(76\) −4.02088e18 + 1.83713e18i −0.0717429 + 0.0327791i
\(77\) 6.70018e19i 1.04216i
\(78\) 1.13307e20 − 5.22934e19i 1.53909 − 0.710319i
\(79\) 4.06193e19i 0.482668i 0.970442 + 0.241334i \(0.0775849\pi\)
−0.970442 + 0.241334i \(0.922415\pi\)
\(80\) −1.14052e20 − 9.87567e19i −1.18757 − 1.02830i
\(81\) −7.03536e19 + 8.38026e19i −0.642975 + 0.765887i
\(82\) 1.31511e19 − 2.04675e19i 0.105662 − 0.164444i
\(83\) 2.07382e20 1.46707 0.733537 − 0.679650i \(-0.237868\pi\)
0.733537 + 0.679650i \(0.237868\pi\)
\(84\) −1.79103e19 + 1.28283e20i −0.111730 + 0.800269i
\(85\) 3.27103e20 1.80213
\(86\) 8.57683e19 − 1.33484e20i 0.417920 − 0.650421i
\(87\) 4.79319e19 + 3.05401e19i 0.206858 + 0.131801i
\(88\) −4.78663e19 + 3.33539e20i −0.183215 + 1.27667i
\(89\) − 2.94977e18i − 0.0100275i −0.999987 − 0.00501376i \(-0.998404\pi\)
0.999987 − 0.00501376i \(-0.00159594\pi\)
\(90\) −5.14910e20 − 6.98862e19i −1.55663 − 0.211274i
\(91\) − 5.08811e20i − 1.36969i
\(92\) −1.05591e20 − 2.31104e20i −0.253427 − 0.554671i
\(93\) 6.11449e20 + 3.89589e20i 1.31006 + 0.834710i
\(94\) 8.34994e19 + 5.36515e19i 0.159897 + 0.102740i
\(95\) −7.23093e19 −0.123907
\(96\) −1.80804e20 + 6.25805e20i −0.277562 + 0.960708i
\(97\) −2.15714e20 −0.297012 −0.148506 − 0.988911i \(-0.547446\pi\)
−0.148506 + 0.988911i \(0.547446\pi\)
\(98\) −2.36190e20 − 1.51761e20i −0.292004 − 0.187623i
\(99\) 4.90354e20 + 1.05190e21i 0.544929 + 1.16898i
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 12.22.b.a.11.7 ✓ 40
3.2 odd 2 inner 12.22.b.a.11.34 yes 40
4.3 odd 2 inner 12.22.b.a.11.33 yes 40
12.11 even 2 inner 12.22.b.a.11.8 yes 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.22.b.a.11.7 ✓ 40 1.1 even 1 trivial
12.22.b.a.11.8 yes 40 12.11 even 2 inner
12.22.b.a.11.33 yes 40 4.3 odd 2 inner
12.22.b.a.11.34 yes 40 3.2 odd 2 inner