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Properties

Label 234.2.y.a.149.4
Level $234$
Weight $2$
Character 234.149
Analytic conductor $1.868$
Analytic rank $0$
Dimension $56$
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] = [234,2,Mod(11,234)] 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("234.11"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(234, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([2, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 234.y (of order \(12\), degree \(4\), 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: \(1.86849940730\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 149.4
Character \(\chi\) \(=\) 234.149
Dual form 234.2.y.a.11.4

$q$-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+(-0.707107 - 0.707107i) q^{2} +(0.0694742 + 1.73066i) q^{3} +1.00000i q^{4} +(0.339151 - 0.0908752i) q^{5} +(1.17463 - 1.27288i) q^{6} +(2.97685 - 0.797644i) q^{7} +(0.707107 - 0.707107i) q^{8} +(-2.99035 + 0.240472i) q^{9} +(-0.304074 - 0.175557i) q^{10} +(-2.35360 + 2.35360i) q^{11} +(-1.73066 + 0.0694742i) q^{12} +(2.60734 + 2.49034i) q^{13} +(-2.66897 - 1.54093i) q^{14} +(0.180836 + 0.580640i) q^{15} -1.00000 q^{16} +(3.87756 + 6.71613i) q^{17} +(2.28453 + 1.94445i) q^{18} +(3.67410 + 0.984473i) q^{19} +(0.0908752 + 0.339151i) q^{20} +(1.58726 + 5.09649i) q^{21} +3.32849 q^{22} +(-3.34250 - 5.78938i) q^{23} +(1.27288 + 1.17463i) q^{24} +(-4.22336 + 2.43836i) q^{25} +(-0.0827293 - 3.60460i) q^{26} +(-0.623927 - 5.15856i) q^{27} +(0.797644 + 2.97685i) q^{28} -5.68973i q^{29} +(0.282704 - 0.538445i) q^{30} +(0.293661 + 1.09596i) q^{31} +(0.707107 + 0.707107i) q^{32} +(-4.23679 - 3.90976i) q^{33} +(2.00717 - 7.49087i) q^{34} +(0.937115 - 0.541044i) q^{35} +(-0.240472 - 2.99035i) q^{36} +(8.90068 - 2.38493i) q^{37} +(-1.90186 - 3.29411i) q^{38} +(-4.12878 + 4.68542i) q^{39} +(0.175557 - 0.304074i) q^{40} +(-0.678577 + 2.53248i) q^{41} +(2.48140 - 4.72613i) q^{42} +(-3.68906 - 2.12988i) q^{43} +(-2.35360 - 2.35360i) q^{44} +(-0.992326 + 0.353305i) q^{45} +(-1.73021 + 6.45722i) q^{46} +(-4.17527 - 1.11876i) q^{47} +(-0.0694742 - 1.73066i) q^{48} +(2.16322 - 1.24894i) q^{49} +(4.71055 + 1.26219i) q^{50} +(-11.3539 + 7.17732i) q^{51} +(-2.49034 + 2.60734i) q^{52} -5.65439i q^{53} +(-3.20647 + 4.08883i) q^{54} +(-0.584342 + 1.01211i) q^{55} +(1.54093 - 2.66897i) q^{56} +(-1.44853 + 6.42701i) q^{57} +(-4.02325 + 4.02325i) q^{58} +(7.75962 - 7.75962i) q^{59} +(-0.580640 + 0.180836i) q^{60} +(5.01742 - 8.69042i) q^{61} +(0.567309 - 0.982608i) q^{62} +(-8.71000 + 3.10108i) q^{63} -1.00000i q^{64} +(1.11059 + 0.607659i) q^{65} +(0.231245 + 5.76048i) q^{66} +(-13.3180 - 3.56856i) q^{67} +(-6.71613 + 3.87756i) q^{68} +(9.78722 - 6.18694i) q^{69} +(-1.04522 - 0.280065i) q^{70} +(-0.343507 + 1.28199i) q^{71} +(-1.94445 + 2.28453i) q^{72} +(-2.19140 - 2.19140i) q^{73} +(-7.98013 - 4.60733i) q^{74} +(-4.51338 - 7.13979i) q^{75} +(-0.984473 + 3.67410i) q^{76} +(-5.12898 + 8.88365i) q^{77} +(6.23258 - 0.393603i) q^{78} +(-6.22990 - 10.7905i) q^{79} +(-0.339151 + 0.0908752i) q^{80} +(8.88435 - 1.43819i) q^{81} +(2.27056 - 1.31091i) q^{82} +(-1.05166 + 3.92483i) q^{83} +(-5.09649 + 1.58726i) q^{84} +(1.92541 + 1.92541i) q^{85} +(1.10251 + 4.11462i) q^{86} +(9.84697 - 0.395290i) q^{87} +3.32849i q^{88} +(1.43929 + 5.37149i) q^{89} +(0.951504 + 0.451856i) q^{90} +(9.74806 + 5.33364i) q^{91} +(5.78938 - 3.34250i) q^{92} +(-1.87632 + 0.584367i) q^{93} +(2.16128 + 3.74344i) q^{94} +1.33554 q^{95} +(-1.17463 + 1.27288i) q^{96} +(-0.218799 - 0.816569i) q^{97} +(-2.41276 - 0.646496i) q^{98} +(6.47211 - 7.60406i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q + 4 q^{7} + 24 q^{15} - 56 q^{16} - 8 q^{19} + 36 q^{27} - 4 q^{28} + 8 q^{31} - 36 q^{33} + 24 q^{35} - 24 q^{36} - 4 q^{37} - 36 q^{38} - 24 q^{39} - 48 q^{41} + 36 q^{42} + 12 q^{43} - 36 q^{45}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/234\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\)
\(\chi(n)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.707107 − 0.707107i −0.500000 − 0.500000i
\(3\) 0.0694742 + 1.73066i 0.0401110 + 0.999195i
\(4\) 1.00000i 0.500000i
\(5\) 0.339151 − 0.0908752i 0.151673 − 0.0406406i −0.182184 − 0.983265i \(-0.558317\pi\)
0.333857 + 0.942624i \(0.391650\pi\)
\(6\) 1.17463 − 1.27288i 0.479542 − 0.519653i
\(7\) 2.97685 − 0.797644i 1.12514 − 0.301481i 0.352180 − 0.935932i \(-0.385440\pi\)
0.772963 + 0.634451i \(0.218774\pi\)
\(8\) 0.707107 − 0.707107i 0.250000 − 0.250000i
\(9\) −2.99035 + 0.240472i −0.996782 + 0.0801574i
\(10\) −0.304074 − 0.175557i −0.0961568 − 0.0555161i
\(11\) −2.35360 + 2.35360i −0.709637 + 0.709637i −0.966459 − 0.256821i \(-0.917325\pi\)
0.256821 + 0.966459i \(0.417325\pi\)
\(12\) −1.73066 + 0.0694742i −0.499598 + 0.0200555i
\(13\) 2.60734 + 2.49034i 0.723145 + 0.690696i
\(14\) −2.66897 − 1.54093i −0.713312 − 0.411831i
\(15\) 0.180836 + 0.580640i 0.0466917 + 0.149921i
\(16\) −1.00000 −0.250000
\(17\) 3.87756 + 6.71613i 0.940446 + 1.62890i 0.764622 + 0.644479i \(0.222926\pi\)
0.175824 + 0.984422i \(0.443741\pi\)
\(18\) 2.28453 + 1.94445i 0.538470 + 0.458312i
\(19\) 3.67410 + 0.984473i 0.842897 + 0.225854i 0.654332 − 0.756207i \(-0.272950\pi\)
0.188565 + 0.982061i \(0.439616\pi\)
\(20\) 0.0908752 + 0.339151i 0.0203203 + 0.0758364i
\(21\) 1.58726 + 5.09649i 0.346369 + 1.11215i
\(22\) 3.32849 0.709637
\(23\) −3.34250 − 5.78938i −0.696960 − 1.20717i −0.969515 − 0.245030i \(-0.921202\pi\)
0.272556 − 0.962140i \(-0.412131\pi\)
\(24\) 1.27288 + 1.17463i 0.259827 + 0.239771i
\(25\) −4.22336 + 2.43836i −0.844672 + 0.487672i
\(26\) −0.0827293 − 3.60460i −0.0162246 − 0.706921i
\(27\) −0.623927 − 5.15856i −0.120075 − 0.992765i
\(28\) 0.797644 + 2.97685i 0.150741 + 0.562572i
\(29\) − 5.68973i − 1.05656i −0.849071 − 0.528278i \(-0.822838\pi\)
0.849071 − 0.528278i \(-0.177162\pi\)
\(30\) 0.282704 − 0.538445i 0.0516145 − 0.0983062i
\(31\) 0.293661 + 1.09596i 0.0527430 + 0.196840i 0.987270 − 0.159052i \(-0.0508438\pi\)
−0.934527 + 0.355892i \(0.884177\pi\)
\(32\) 0.707107 + 0.707107i 0.125000 + 0.125000i
\(33\) −4.23679 − 3.90976i −0.737531 − 0.680602i
\(34\) 2.00717 − 7.49087i 0.344227 − 1.28467i
\(35\) 0.937115 − 0.541044i 0.158401 − 0.0914531i
\(36\) −0.240472 − 2.99035i −0.0400787 − 0.498391i
\(37\) 8.90068 − 2.38493i 1.46326 − 0.392080i 0.562647 − 0.826697i \(-0.309783\pi\)
0.900615 + 0.434617i \(0.143116\pi\)
\(38\) −1.90186 − 3.29411i −0.308522 − 0.534375i
\(39\) −4.12878 + 4.68542i −0.661134 + 0.750268i
\(40\) 0.175557 − 0.304074i 0.0277581 − 0.0480784i
\(41\) −0.678577 + 2.53248i −0.105976 + 0.395508i −0.998454 − 0.0555831i \(-0.982298\pi\)
0.892478 + 0.451091i \(0.148965\pi\)
\(42\) 2.48140 − 4.72613i 0.382888 − 0.729257i
\(43\) −3.68906 − 2.12988i −0.562577 − 0.324804i 0.191602 − 0.981473i \(-0.438632\pi\)
−0.754179 + 0.656669i \(0.771965\pi\)
\(44\) −2.35360 − 2.35360i −0.354819 − 0.354819i
\(45\) −0.992326 + 0.353305i −0.147927 + 0.0526676i
\(46\) −1.73021 + 6.45722i −0.255105 + 0.952065i
\(47\) −4.17527 − 1.11876i −0.609026 − 0.163188i −0.0588917 − 0.998264i \(-0.518757\pi\)
−0.550134 + 0.835076i \(0.685423\pi\)
\(48\) −0.0694742 − 1.73066i −0.0100277 − 0.249799i
\(49\) 2.16322 − 1.24894i 0.309031 − 0.178419i
\(50\) 4.71055 + 1.26219i 0.666172 + 0.178500i
\(51\) −11.3539 + 7.17732i −1.58987 + 1.00503i
\(52\) −2.49034 + 2.60734i −0.345348 + 0.361573i
\(53\) − 5.65439i − 0.776690i −0.921514 − 0.388345i \(-0.873047\pi\)
0.921514 − 0.388345i \(-0.126953\pi\)
\(54\) −3.20647 + 4.08883i −0.436345 + 0.556420i
\(55\) −0.584342 + 1.01211i −0.0787927 + 0.136473i
\(56\) 1.54093 − 2.66897i 0.205916 − 0.356656i
\(57\) −1.44853 + 6.42701i −0.191862 + 0.851278i
\(58\) −4.02325 + 4.02325i −0.528278 + 0.528278i
\(59\) 7.75962 − 7.75962i 1.01022 − 1.01022i 0.0102689 − 0.999947i \(-0.496731\pi\)
0.999947 − 0.0102689i \(-0.00326875\pi\)
\(60\) −0.580640 + 0.180836i −0.0749603 + 0.0233458i
\(61\) 5.01742 − 8.69042i 0.642415 − 1.11269i −0.342478 − 0.939526i \(-0.611266\pi\)
0.984892 − 0.173169i \(-0.0554006\pi\)
\(62\) 0.567309 − 0.982608i 0.0720483 − 0.124791i
\(63\) −8.71000 + 3.10108i −1.09736 + 0.390700i
\(64\) − 1.00000i − 0.125000i
\(65\) 1.11059 + 0.607659i 0.137752 + 0.0753708i
\(66\) 0.231245 + 5.76048i 0.0284642 + 0.709066i
\(67\) −13.3180 − 3.56856i −1.62706 − 0.435969i −0.673994 − 0.738737i \(-0.735423\pi\)
−0.953064 + 0.302768i \(0.902089\pi\)
\(68\) −6.71613 + 3.87756i −0.814450 + 0.470223i
\(69\) 9.78722 − 6.18694i 1.17824 − 0.744820i
\(70\) −1.04522 − 0.280065i −0.124927 − 0.0334741i
\(71\) −0.343507 + 1.28199i −0.0407668 + 0.152144i −0.983309 − 0.181943i \(-0.941761\pi\)
0.942542 + 0.334087i \(0.108428\pi\)
\(72\) −1.94445 + 2.28453i −0.229156 + 0.269235i
\(73\) −2.19140 − 2.19140i −0.256484 − 0.256484i 0.567138 − 0.823622i \(-0.308050\pi\)
−0.823622 + 0.567138i \(0.808050\pi\)
\(74\) −7.98013 − 4.60733i −0.927671 − 0.535591i
\(75\) −4.51338 − 7.13979i −0.521160 − 0.824432i
\(76\) −0.984473 + 3.67410i −0.112927 + 0.421448i
\(77\) −5.12898 + 8.88365i −0.584502 + 1.01239i
\(78\) 6.23258 − 0.393603i 0.705701 − 0.0445668i
\(79\) −6.22990 − 10.7905i −0.700919 − 1.21403i −0.968144 − 0.250393i \(-0.919440\pi\)
0.267225 − 0.963634i \(-0.413893\pi\)
\(80\) −0.339151 + 0.0908752i −0.0379182 + 0.0101602i
\(81\) 8.88435 − 1.43819i 0.987150 − 0.159799i
\(82\) 2.27056 − 1.31091i 0.250742 − 0.144766i
\(83\) −1.05166 + 3.92483i −0.115434 + 0.430806i −0.999319 − 0.0368984i \(-0.988252\pi\)
0.883885 + 0.467705i \(0.154919\pi\)
\(84\) −5.09649 + 1.58726i −0.556073 + 0.173185i
\(85\) 1.92541 + 1.92541i 0.208840 + 0.208840i
\(86\) 1.10251 + 4.11462i 0.118887 + 0.443691i
\(87\) 9.84697 − 0.395290i 1.05571 − 0.0423795i
\(88\) 3.32849i 0.354819i
\(89\) 1.43929 + 5.37149i 0.152564 + 0.569377i 0.999302 + 0.0373662i \(0.0118968\pi\)
−0.846737 + 0.532011i \(0.821437\pi\)
\(90\) 0.951504 + 0.451856i 0.100297 + 0.0476298i
\(91\) 9.74806 + 5.33364i 1.02187 + 0.559117i
\(92\) 5.78938 − 3.34250i 0.603585 − 0.348480i
\(93\) −1.87632 + 0.584367i −0.194566 + 0.0605960i
\(94\) 2.16128 + 3.74344i 0.222919 + 0.386107i
\(95\) 1.33554 0.137023
\(96\) −1.17463 + 1.27288i −0.119886 + 0.129913i
\(97\) −0.218799 − 0.816569i −0.0222157 − 0.0829101i 0.953928 − 0.300036i \(-0.0969986\pi\)
−0.976144 + 0.217126i \(0.930332\pi\)
\(98\) −2.41276 − 0.646496i −0.243725 − 0.0653060i
\(99\) 6.47211 − 7.60406i 0.650471 − 0.764237i
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 234.2.y.a.149.4 yes 56
3.2 odd 2 702.2.bb.a.71.12 56
9.2 odd 6 234.2.z.a.227.8 yes 56
9.7 even 3 702.2.bc.a.305.5 56
13.11 odd 12 234.2.z.a.167.8 yes 56
39.11 even 12 702.2.bc.a.557.5 56
117.11 even 12 inner 234.2.y.a.11.4 ✓ 56
117.115 odd 12 702.2.bb.a.89.12 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
234.2.y.a.11.4 ✓ 56 117.11 even 12 inner
234.2.y.a.149.4 yes 56 1.1 even 1 trivial
234.2.z.a.167.8 yes 56 13.11 odd 12
234.2.z.a.227.8 yes 56 9.2 odd 6
702.2.bb.a.71.12 56 3.2 odd 2
702.2.bb.a.89.12 56 117.115 odd 12
702.2.bc.a.305.5 56 9.7 even 3
702.2.bc.a.557.5 56 39.11 even 12