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Properties

Label 234.2.t.a.25.7
Level $234$
Weight $2$
Character 234.25
Analytic conductor $1.868$
Analytic rank $0$
Dimension $28$
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] = [234,2,Mod(25,234)] 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("234.25"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(234, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 234.t (of order \(6\), degree \(2\), 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: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 25.7
Character \(\chi\) \(=\) 234.25
Dual form 234.2.t.a.103.7

$q$-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+(-0.866025 - 0.500000i) q^{2} +(1.66410 - 0.480381i) q^{3} +(0.500000 + 0.866025i) q^{4} +(-0.515076 + 0.297379i) q^{5} +(-1.68134 - 0.416029i) q^{6} +(1.45217 + 0.838409i) q^{7} -1.00000i q^{8} +(2.53847 - 1.59880i) q^{9} +0.594758 q^{10} +(0.416337 + 0.240372i) q^{11} +(1.24807 + 1.20096i) q^{12} +(3.56065 - 0.567277i) q^{13} +(-0.838409 - 1.45217i) q^{14} +(-0.714283 + 0.742302i) q^{15} +(-0.500000 + 0.866025i) q^{16} -2.09349 q^{17} +(-2.99778 + 0.115371i) q^{18} -0.480744i q^{19} +(-0.515076 - 0.297379i) q^{20} +(2.81931 + 0.697605i) q^{21} +(-0.240372 - 0.416337i) q^{22} +(-1.83339 - 3.17553i) q^{23} +(-0.480381 - 1.66410i) q^{24} +(-2.32313 + 4.02378i) q^{25} +(-3.36725 - 1.28905i) q^{26} +(3.45624 - 3.88001i) q^{27} +1.67682i q^{28} +(1.23339 - 2.13629i) q^{29} +(0.989738 - 0.285710i) q^{30} +(0.993791 - 0.573765i) q^{31} +(0.866025 - 0.500000i) q^{32} +(0.808297 + 0.200004i) q^{33} +(1.81302 + 1.04675i) q^{34} -0.997301 q^{35} +(2.65384 + 1.39898i) q^{36} -3.65012i q^{37} +(-0.240372 + 0.416337i) q^{38} +(5.65277 - 2.65447i) q^{39} +(0.297379 + 0.515076i) q^{40} +(-8.58235 + 4.95502i) q^{41} +(-2.09279 - 2.01380i) q^{42} +(-3.45822 + 5.98981i) q^{43} +0.480744i q^{44} +(-0.832053 + 1.57839i) q^{45} +3.66679i q^{46} +(5.40488 + 3.12051i) q^{47} +(-0.416029 + 1.68134i) q^{48} +(-2.09414 - 3.62716i) q^{49} +(4.02378 - 2.32313i) q^{50} +(-3.48378 + 1.00567i) q^{51} +(2.27160 + 2.79997i) q^{52} -5.08592 q^{53} +(-4.93319 + 1.63207i) q^{54} -0.285927 q^{55} +(0.838409 - 1.45217i) q^{56} +(-0.230940 - 0.800008i) q^{57} +(-2.13629 + 1.23339i) q^{58} +(-8.13185 + 4.69493i) q^{59} +(-0.999994 - 0.247437i) q^{60} +(-3.90635 + 6.76599i) q^{61} -1.14753 q^{62} +(5.02673 - 0.193456i) q^{63} -1.00000 q^{64} +(-1.66531 + 1.35105i) q^{65} +(-0.600004 - 0.577357i) q^{66} +(-12.4551 + 7.19096i) q^{67} +(-1.04675 - 1.81302i) q^{68} +(-4.57642 - 4.40368i) q^{69} +(0.863688 + 0.498651i) q^{70} +6.51028i q^{71} +(-1.59880 - 2.53847i) q^{72} -5.91514i q^{73} +(-1.82506 + 3.16110i) q^{74} +(-1.93298 + 7.81197i) q^{75} +(0.416337 - 0.240372i) q^{76} +(0.403061 + 0.698121i) q^{77} +(-6.22268 - 0.527544i) q^{78} +(1.02895 - 1.78219i) q^{79} -0.594758i q^{80} +(3.88765 - 8.11703i) q^{81} +9.91005 q^{82} +(9.57834 + 5.53006i) q^{83} +(0.805511 + 2.79040i) q^{84} +(1.07831 - 0.622560i) q^{85} +(5.98981 - 3.45822i) q^{86} +(1.02625 - 4.14750i) q^{87} +(0.240372 - 0.416337i) q^{88} -9.48720i q^{89} +(1.50978 - 0.950902i) q^{90} +(5.64626 + 2.16150i) q^{91} +(1.83339 - 3.17553i) q^{92} +(1.37814 - 1.43220i) q^{93} +(-3.12051 - 5.40488i) q^{94} +(0.142963 + 0.247620i) q^{95} +(1.20096 - 1.24807i) q^{96} +(-8.41374 - 4.85767i) q^{97} +4.18828i q^{98} +(1.44117 - 0.0554640i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + 14 q^{4} - 16 q^{9} + 2 q^{13} + 8 q^{14} - 14 q^{16} + 16 q^{17} - 8 q^{23} + 14 q^{25} + 8 q^{26} + 18 q^{27} - 16 q^{29} - 8 q^{30} - 68 q^{35} - 8 q^{36} - 8 q^{39} - 10 q^{42} - 4 q^{43} + 10 q^{49}+ \cdots + 8 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(145\) \(209\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 − 0.500000i −0.612372 − 0.353553i
\(3\) 1.66410 − 0.480381i 0.960770 − 0.277348i
\(4\) 0.500000 + 0.866025i 0.250000 + 0.433013i
\(5\) −0.515076 + 0.297379i −0.230349 + 0.132992i −0.610733 − 0.791837i \(-0.709125\pi\)
0.380384 + 0.924829i \(0.375792\pi\)
\(6\) −1.68134 − 0.416029i −0.686406 − 0.169843i
\(7\) 1.45217 + 0.838409i 0.548868 + 0.316889i 0.748665 − 0.662948i \(-0.230695\pi\)
−0.199798 + 0.979837i \(0.564028\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 2.53847 − 1.59880i 0.846156 − 0.532935i
\(10\) 0.594758 0.188079
\(11\) 0.416337 + 0.240372i 0.125530 + 0.0724750i 0.561450 − 0.827510i \(-0.310244\pi\)
−0.435920 + 0.899985i \(0.643577\pi\)
\(12\) 1.24807 + 1.20096i 0.360288 + 0.346688i
\(13\) 3.56065 − 0.567277i 0.987545 − 0.157334i
\(14\) −0.838409 − 1.45217i −0.224074 − 0.388108i
\(15\) −0.714283 + 0.742302i −0.184427 + 0.191661i
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −2.09349 −0.507746 −0.253873 − 0.967238i \(-0.581705\pi\)
−0.253873 + 0.967238i \(0.581705\pi\)
\(18\) −2.99778 + 0.115371i −0.706584 + 0.0271932i
\(19\) − 0.480744i − 0.110290i −0.998478 − 0.0551452i \(-0.982438\pi\)
0.998478 − 0.0551452i \(-0.0175622\pi\)
\(20\) −0.515076 − 0.297379i −0.115174 − 0.0664960i
\(21\) 2.81931 + 0.697605i 0.615224 + 0.152230i
\(22\) −0.240372 − 0.416337i −0.0512475 − 0.0887633i
\(23\) −1.83339 − 3.17553i −0.382289 − 0.662144i 0.609100 − 0.793093i \(-0.291531\pi\)
−0.991389 + 0.130950i \(0.958197\pi\)
\(24\) −0.480381 − 1.66410i −0.0980573 − 0.339683i
\(25\) −2.32313 + 4.02378i −0.464626 + 0.804756i
\(26\) −3.36725 − 1.28905i −0.660372 − 0.252803i
\(27\) 3.45624 − 3.88001i 0.665153 − 0.746707i
\(28\) 1.67682i 0.316889i
\(29\) 1.23339 − 2.13629i 0.229035 − 0.396700i −0.728488 − 0.685059i \(-0.759776\pi\)
0.957522 + 0.288359i \(0.0931098\pi\)
\(30\) 0.989738 − 0.285710i 0.180701 − 0.0521633i
\(31\) 0.993791 − 0.573765i 0.178490 − 0.103051i −0.408093 − 0.912940i \(-0.633806\pi\)
0.586583 + 0.809889i \(0.300473\pi\)
\(32\) 0.866025 − 0.500000i 0.153093 − 0.0883883i
\(33\) 0.808297 + 0.200004i 0.140706 + 0.0348162i
\(34\) 1.81302 + 1.04675i 0.310930 + 0.179515i
\(35\) −0.997301 −0.168575
\(36\) 2.65384 + 1.39898i 0.442307 + 0.233163i
\(37\) − 3.65012i − 0.600076i −0.953927 − 0.300038i \(-0.903001\pi\)
0.953927 − 0.300038i \(-0.0969994\pi\)
\(38\) −0.240372 + 0.416337i −0.0389935 + 0.0675388i
\(39\) 5.65277 − 2.65447i 0.905167 − 0.425056i
\(40\) 0.297379 + 0.515076i 0.0470198 + 0.0814406i
\(41\) −8.58235 + 4.95502i −1.34034 + 0.773845i −0.986857 − 0.161597i \(-0.948336\pi\)
−0.353481 + 0.935442i \(0.615002\pi\)
\(42\) −2.09279 − 2.01380i −0.322925 − 0.310736i
\(43\) −3.45822 + 5.98981i −0.527374 + 0.913438i 0.472117 + 0.881536i \(0.343490\pi\)
−0.999491 + 0.0319023i \(0.989843\pi\)
\(44\) 0.480744i 0.0724750i
\(45\) −0.832053 + 1.57839i −0.124035 + 0.235293i
\(46\) 3.66679i 0.540638i
\(47\) 5.40488 + 3.12051i 0.788383 + 0.455173i 0.839393 − 0.543525i \(-0.182911\pi\)
−0.0510099 + 0.998698i \(0.516244\pi\)
\(48\) −0.416029 + 1.68134i −0.0600486 + 0.242681i
\(49\) −2.09414 − 3.62716i −0.299163 − 0.518165i
\(50\) 4.02378 − 2.32313i 0.569049 − 0.328540i
\(51\) −3.48378 + 1.00567i −0.487827 + 0.140822i
\(52\) 2.27160 + 2.79997i 0.315014 + 0.388286i
\(53\) −5.08592 −0.698605 −0.349302 − 0.937010i \(-0.613581\pi\)
−0.349302 + 0.937010i \(0.613581\pi\)
\(54\) −4.93319 + 1.63207i −0.671322 + 0.222096i
\(55\) −0.285927 −0.0385544
\(56\) 0.838409 − 1.45217i 0.112037 − 0.194054i
\(57\) −0.230940 − 0.800008i −0.0305888 − 0.105964i
\(58\) −2.13629 + 1.23339i −0.280509 + 0.161952i
\(59\) −8.13185 + 4.69493i −1.05868 + 0.611227i −0.925066 − 0.379807i \(-0.875991\pi\)
−0.133611 + 0.991034i \(0.542657\pi\)
\(60\) −0.999994 − 0.247437i −0.129099 − 0.0319439i
\(61\) −3.90635 + 6.76599i −0.500157 + 0.866297i 0.499843 + 0.866116i \(0.333391\pi\)
−1.00000 0.000180927i \(0.999942\pi\)
\(62\) −1.14753 −0.145737
\(63\) 5.02673 − 0.193456i 0.633309 − 0.0243732i
\(64\) −1.00000 −0.125000
\(65\) −1.66531 + 1.35105i −0.206556 + 0.167577i
\(66\) −0.600004 − 0.577357i −0.0738554 − 0.0710677i
\(67\) −12.4551 + 7.19096i −1.52163 + 0.878516i −0.521960 + 0.852970i \(0.674799\pi\)
−0.999674 + 0.0255454i \(0.991868\pi\)
\(68\) −1.04675 − 1.81302i −0.126936 − 0.219860i
\(69\) −4.57642 − 4.40368i −0.550936 − 0.530141i
\(70\) 0.863688 + 0.498651i 0.103231 + 0.0596002i
\(71\) 6.51028i 0.772628i 0.922367 + 0.386314i \(0.126252\pi\)
−0.922367 + 0.386314i \(0.873748\pi\)
\(72\) −1.59880 − 2.53847i −0.188421 − 0.299161i
\(73\) − 5.91514i − 0.692315i −0.938176 − 0.346157i \(-0.887486\pi\)
0.938176 − 0.346157i \(-0.112514\pi\)
\(74\) −1.82506 + 3.16110i −0.212159 + 0.367470i
\(75\) −1.93298 + 7.81197i −0.223201 + 0.902048i
\(76\) 0.416337 − 0.240372i 0.0477571 − 0.0275726i
\(77\) 0.403061 + 0.698121i 0.0459330 + 0.0795583i
\(78\) −6.22268 − 0.527544i −0.704579 − 0.0597326i
\(79\) 1.02895 − 1.78219i 0.115766 − 0.200512i −0.802320 − 0.596894i \(-0.796401\pi\)
0.918086 + 0.396382i \(0.129734\pi\)
\(80\) − 0.594758i − 0.0664960i
\(81\) 3.88765 − 8.11703i 0.431961 − 0.901892i
\(82\) 9.91005 1.09438
\(83\) 9.57834 + 5.53006i 1.05136 + 0.607003i 0.923030 − 0.384729i \(-0.125705\pi\)
0.128330 + 0.991732i \(0.459038\pi\)
\(84\) 0.805511 + 2.79040i 0.0878885 + 0.304457i
\(85\) 1.07831 − 0.622560i 0.116959 − 0.0675261i
\(86\) 5.98981 − 3.45822i 0.645898 − 0.372909i
\(87\) 1.02625 − 4.14750i 0.110026 − 0.444659i
\(88\) 0.240372 − 0.416337i 0.0256238 − 0.0443817i
\(89\) − 9.48720i − 1.00564i −0.864391 − 0.502821i \(-0.832296\pi\)
0.864391 − 0.502821i \(-0.167704\pi\)
\(90\) 1.50978 − 0.950902i 0.159144 − 0.100234i
\(91\) 5.64626 + 2.16150i 0.591889 + 0.226586i
\(92\) 1.83339 − 3.17553i 0.191144 − 0.331072i
\(93\) 1.37814 − 1.43220i 0.142907 − 0.148512i
\(94\) −3.12051 − 5.40488i −0.321856 − 0.557471i
\(95\) 0.142963 + 0.247620i 0.0146677 + 0.0254053i
\(96\) 1.20096 − 1.24807i 0.122573 − 0.127381i
\(97\) −8.41374 − 4.85767i −0.854286 − 0.493222i 0.00780887 − 0.999970i \(-0.497514\pi\)
−0.862095 + 0.506747i \(0.830848\pi\)
\(98\) 4.18828i 0.423080i
\(99\) 1.44117 − 0.0554640i 0.144843 − 0.00557435i
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.t.a.25.7 ✓ 28
3.2 odd 2 702.2.t.a.181.12 28
9.2 odd 6 2106.2.b.d.649.5 14
9.4 even 3 inner 234.2.t.a.103.14 yes 28
9.5 odd 6 702.2.t.a.415.3 28
9.7 even 3 2106.2.b.c.649.10 14
13.12 even 2 inner 234.2.t.a.25.14 yes 28
39.38 odd 2 702.2.t.a.181.3 28
117.25 even 6 2106.2.b.c.649.5 14
117.38 odd 6 2106.2.b.d.649.10 14
117.77 odd 6 702.2.t.a.415.12 28
117.103 even 6 inner 234.2.t.a.103.7 yes 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
234.2.t.a.25.7 ✓ 28 1.1 even 1 trivial
234.2.t.a.25.14 yes 28 13.12 even 2 inner
234.2.t.a.103.7 yes 28 117.103 even 6 inner
234.2.t.a.103.14 yes 28 9.4 even 3 inner
702.2.t.a.181.3 28 39.38 odd 2
702.2.t.a.181.12 28 3.2 odd 2
702.2.t.a.415.3 28 9.5 odd 6
702.2.t.a.415.12 28 117.77 odd 6
2106.2.b.c.649.5 14 117.25 even 6
2106.2.b.c.649.10 14 9.7 even 3
2106.2.b.d.649.5 14 9.2 odd 6
2106.2.b.d.649.10 14 117.38 odd 6