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Properties

Label 234.2.t.a.103.13
Level $234$
Weight $2$
Character 234.103
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 103.13
Character \(\chi\) \(=\) 234.103
Dual form 234.2.t.a.25.13

$q$-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} +(1.21476 - 1.23465i) q^{3} +(0.500000 - 0.866025i) q^{4} +(-2.73536 - 1.57926i) q^{5} +(0.434692 - 1.67662i) q^{6} +(-1.36392 + 0.787458i) q^{7} -1.00000i q^{8} +(-0.0487053 - 2.99960i) q^{9} -3.15852 q^{10} +(3.26194 - 1.88328i) q^{11} +(-0.461854 - 1.66934i) q^{12} +(0.899453 + 3.49156i) q^{13} +(-0.787458 + 1.36392i) q^{14} +(-5.27264 + 1.45878i) q^{15} +(-0.500000 - 0.866025i) q^{16} +7.06565 q^{17} +(-1.54198 - 2.57338i) q^{18} +3.76656i q^{19} +(-2.73536 + 1.57926i) q^{20} +(-0.684603 + 2.64053i) q^{21} +(1.88328 - 3.26194i) q^{22} +(-1.84873 + 3.20209i) q^{23} +(-1.23465 - 1.21476i) q^{24} +(2.48812 + 4.30955i) q^{25} +(2.52473 + 2.57405i) q^{26} +(-3.76262 - 3.58367i) q^{27} +1.57492i q^{28} +(0.109128 + 0.189015i) q^{29} +(-3.83685 + 3.89965i) q^{30} +(2.65792 + 1.53455i) q^{31} +(-0.866025 - 0.500000i) q^{32} +(1.63729 - 6.31508i) q^{33} +(6.11904 - 3.53283i) q^{34} +4.97440 q^{35} +(-2.62209 - 1.45762i) q^{36} +0.292126i q^{37} +(1.88328 + 3.26194i) q^{38} +(5.40346 + 3.13091i) q^{39} +(-1.57926 + 2.73536i) q^{40} +(-6.39272 - 3.69084i) q^{41} +(0.727382 + 2.62907i) q^{42} +(-3.05835 - 5.29722i) q^{43} -3.76656i q^{44} +(-4.60393 + 8.28191i) q^{45} +3.69746i q^{46} +(6.17888 - 3.56738i) q^{47} +(-1.67662 - 0.434692i) q^{48} +(-2.25982 + 3.91412i) q^{49} +(4.30955 + 2.48812i) q^{50} +(8.58309 - 8.72359i) q^{51} +(3.47351 + 0.966830i) q^{52} -14.4175 q^{53} +(-5.05036 - 1.22224i) q^{54} -11.8968 q^{55} +(0.787458 + 1.36392i) q^{56} +(4.65037 + 4.57548i) q^{57} +(0.189015 + 0.109128i) q^{58} +(9.04745 + 5.22355i) q^{59} +(-1.37298 + 5.29562i) q^{60} +(3.00007 + 5.19627i) q^{61} +3.06910 q^{62} +(2.42849 + 4.05286i) q^{63} -1.00000 q^{64} +(3.05375 - 10.9711i) q^{65} +(-1.73960 - 6.28766i) q^{66} +(-6.33583 - 3.65799i) q^{67} +(3.53283 - 6.11904i) q^{68} +(1.70769 + 6.17231i) q^{69} +(4.30796 - 2.48720i) q^{70} +0.772410i q^{71} +(-2.99960 + 0.0487053i) q^{72} +13.5342i q^{73} +(0.146063 + 0.252988i) q^{74} +(8.34324 + 2.16313i) q^{75} +(3.26194 + 1.88328i) q^{76} +(-2.96601 + 5.13728i) q^{77} +(6.24499 + 0.00971288i) q^{78} +(6.34033 + 10.9818i) q^{79} +3.15852i q^{80} +(-8.99526 + 0.292193i) q^{81} -7.38168 q^{82} +(-0.314795 + 0.181747i) q^{83} +(1.94447 + 1.91315i) q^{84} +(-19.3271 - 11.1585i) q^{85} +(-5.29722 - 3.05835i) q^{86} +(0.365931 + 0.0948738i) q^{87} +(-1.88328 - 3.26194i) q^{88} -7.06555i q^{89} +(0.153837 + 9.47431i) q^{90} +(-3.97624 - 4.05392i) q^{91} +(1.84873 + 3.20209i) q^{92} +(5.12337 - 1.41748i) q^{93} +(3.56738 - 6.17888i) q^{94} +(5.94838 - 10.3029i) q^{95} +(-1.66934 + 0.461854i) q^{96} +(-0.535716 + 0.309296i) q^{97} +4.51964i q^{98} +(-5.80797 - 9.69280i) 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 \(\left(\mathbb{Z}/234\mathbb{Z}\right)^\times\).

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

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
\(3\) 1.21476 − 1.23465i 0.701343 − 0.712824i
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) −2.73536 − 1.57926i −1.22329 − 0.706266i −0.257671 − 0.966233i \(-0.582955\pi\)
−0.965618 + 0.259967i \(0.916288\pi\)
\(6\) 0.434692 − 1.67662i 0.177462 − 0.684476i
\(7\) −1.36392 + 0.787458i −0.515512 + 0.297631i −0.735097 − 0.677962i \(-0.762863\pi\)
0.219584 + 0.975594i \(0.429530\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −0.0487053 − 2.99960i −0.0162351 − 0.999868i
\(10\) −3.15852 −0.998811
\(11\) 3.26194 − 1.88328i 0.983511 − 0.567830i 0.0801828 − 0.996780i \(-0.474450\pi\)
0.903328 + 0.428950i \(0.141116\pi\)
\(12\) −0.461854 − 1.66934i −0.133326 − 0.481896i
\(13\) 0.899453 + 3.49156i 0.249463 + 0.968384i
\(14\) −0.787458 + 1.36392i −0.210457 + 0.364522i
\(15\) −5.27264 + 1.45878i −1.36139 + 0.376654i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 7.06565 1.71367 0.856836 − 0.515588i \(-0.172427\pi\)
0.856836 + 0.515588i \(0.172427\pi\)
\(18\) −1.54198 − 2.57338i −0.363449 − 0.606552i
\(19\) 3.76656i 0.864108i 0.901848 + 0.432054i \(0.142211\pi\)
−0.901848 + 0.432054i \(0.857789\pi\)
\(20\) −2.73536 + 1.57926i −0.611644 + 0.353133i
\(21\) −0.684603 + 2.64053i −0.149393 + 0.576211i
\(22\) 1.88328 − 3.26194i 0.401517 − 0.695447i
\(23\) −1.84873 + 3.20209i −0.385487 + 0.667683i −0.991837 − 0.127515i \(-0.959300\pi\)
0.606350 + 0.795198i \(0.292633\pi\)
\(24\) −1.23465 − 1.21476i −0.252021 − 0.247962i
\(25\) 2.48812 + 4.30955i 0.497624 + 0.861910i
\(26\) 2.52473 + 2.57405i 0.495140 + 0.504813i
\(27\) −3.76262 − 3.58367i −0.724116 − 0.689678i
\(28\) 1.57492i 0.297631i
\(29\) 0.109128 + 0.189015i 0.0202645 + 0.0350992i 0.875980 − 0.482348i \(-0.160216\pi\)
−0.855715 + 0.517447i \(0.826882\pi\)
\(30\) −3.83685 + 3.89965i −0.700510 + 0.711976i
\(31\) 2.65792 + 1.53455i 0.477376 + 0.275613i 0.719322 − 0.694676i \(-0.244452\pi\)
−0.241946 + 0.970290i \(0.577786\pi\)
\(32\) −0.866025 − 0.500000i −0.153093 − 0.0883883i
\(33\) 1.63729 − 6.31508i 0.285016 − 1.09931i
\(34\) 6.11904 − 3.53283i 1.04941 − 0.605875i
\(35\) 4.97440 0.840827
\(36\) −2.62209 − 1.45762i −0.437014 − 0.242937i
\(37\) 0.292126i 0.0480251i 0.999712 + 0.0240126i \(0.00764417\pi\)
−0.999712 + 0.0240126i \(0.992356\pi\)
\(38\) 1.88328 + 3.26194i 0.305508 + 0.529156i
\(39\) 5.40346 + 3.13091i 0.865247 + 0.501346i
\(40\) −1.57926 + 2.73536i −0.249703 + 0.432498i
\(41\) −6.39272 − 3.69084i −0.998375 − 0.576412i −0.0906081 − 0.995887i \(-0.528881\pi\)
−0.907767 + 0.419474i \(0.862214\pi\)
\(42\) 0.727382 + 2.62907i 0.112237 + 0.405674i
\(43\) −3.05835 − 5.29722i −0.466395 − 0.807819i 0.532869 − 0.846198i \(-0.321114\pi\)
−0.999263 + 0.0383789i \(0.987781\pi\)
\(44\) − 3.76656i − 0.567830i
\(45\) −4.60393 + 8.28191i −0.686313 + 1.23459i
\(46\) 3.69746i 0.545161i
\(47\) 6.17888 − 3.56738i 0.901282 − 0.520356i 0.0236662 − 0.999720i \(-0.492466\pi\)
0.877616 + 0.479364i \(0.159133\pi\)
\(48\) −1.67662 − 0.434692i −0.241999 − 0.0627423i
\(49\) −2.25982 + 3.91412i −0.322831 + 0.559160i
\(50\) 4.30955 + 2.48812i 0.609462 + 0.351873i
\(51\) 8.58309 − 8.72359i 1.20187 − 1.22155i
\(52\) 3.47351 + 0.966830i 0.481689 + 0.134075i
\(53\) −14.4175 −1.98040 −0.990201 − 0.139649i \(-0.955403\pi\)
−0.990201 + 0.139649i \(0.955403\pi\)
\(54\) −5.05036 − 1.22224i −0.687267 − 0.166326i
\(55\) −11.8968 −1.60416
\(56\) 0.787458 + 1.36392i 0.105229 + 0.182261i
\(57\) 4.65037 + 4.57548i 0.615957 + 0.606037i
\(58\) 0.189015 + 0.109128i 0.0248188 + 0.0143292i
\(59\) 9.04745 + 5.22355i 1.17788 + 0.680048i 0.955523 − 0.294918i \(-0.0952924\pi\)
0.222355 + 0.974966i \(0.428626\pi\)
\(60\) −1.37298 + 5.29562i −0.177251 + 0.683662i
\(61\) 3.00007 + 5.19627i 0.384119 + 0.665314i 0.991647 − 0.128985i \(-0.0411718\pi\)
−0.607527 + 0.794299i \(0.707838\pi\)
\(62\) 3.06910 0.389776
\(63\) 2.42849 + 4.05286i 0.305961 + 0.510612i
\(64\) −1.00000 −0.125000
\(65\) 3.05375 − 10.9711i 0.378771 − 1.36080i
\(66\) −1.73960 − 6.28766i −0.214130 − 0.773958i
\(67\) −6.33583 − 3.65799i −0.774045 − 0.446895i 0.0602708 − 0.998182i \(-0.480804\pi\)
−0.834316 + 0.551287i \(0.814137\pi\)
\(68\) 3.53283 − 6.11904i 0.428418 − 0.742042i
\(69\) 1.70769 + 6.17231i 0.205581 + 0.743059i
\(70\) 4.30796 − 2.48720i 0.514900 − 0.297277i
\(71\) 0.772410i 0.0916682i 0.998949 + 0.0458341i \(0.0145946\pi\)
−0.998949 + 0.0458341i \(0.985405\pi\)
\(72\) −2.99960 + 0.0487053i −0.353507 + 0.00573997i
\(73\) 13.5342i 1.58406i 0.610480 + 0.792032i \(0.290976\pi\)
−0.610480 + 0.792032i \(0.709024\pi\)
\(74\) 0.146063 + 0.252988i 0.0169795 + 0.0294093i
\(75\) 8.34324 + 2.16313i 0.963395 + 0.249777i
\(76\) 3.26194 + 1.88328i 0.374170 + 0.216027i
\(77\) −2.96601 + 5.13728i −0.338008 + 0.585447i
\(78\) 6.24499 + 0.00971288i 0.707106 + 0.00109977i
\(79\) 6.34033 + 10.9818i 0.713343 + 1.23555i 0.963595 + 0.267366i \(0.0861532\pi\)
−0.250252 + 0.968181i \(0.580513\pi\)
\(80\) 3.15852i 0.353133i
\(81\) −8.99526 + 0.292193i −0.999473 + 0.0324659i
\(82\) −7.38168 −0.815170
\(83\) −0.314795 + 0.181747i −0.0345533 + 0.0199494i −0.517177 − 0.855878i \(-0.673017\pi\)
0.482624 + 0.875828i \(0.339684\pi\)
\(84\) 1.94447 + 1.91315i 0.212159 + 0.208742i
\(85\) −19.3271 − 11.1585i −2.09632 − 1.21031i
\(86\) −5.29722 − 3.05835i −0.571214 − 0.329791i
\(87\) 0.365931 + 0.0948738i 0.0392319 + 0.0101715i
\(88\) −1.88328 − 3.26194i −0.200758 − 0.347724i
\(89\) − 7.06555i − 0.748947i −0.927238 − 0.374473i \(-0.877824\pi\)
0.927238 − 0.374473i \(-0.122176\pi\)
\(90\) 0.153837 + 9.47431i 0.0162158 + 0.998680i
\(91\) −3.97624 − 4.05392i −0.416823 − 0.424966i
\(92\) 1.84873 + 3.20209i 0.192743 + 0.333841i
\(93\) 5.12337 − 1.41748i 0.531268 − 0.146986i
\(94\) 3.56738 − 6.17888i 0.367947 − 0.637303i
\(95\) 5.94838 − 10.3029i 0.610290 − 1.05705i
\(96\) −1.66934 + 0.461854i −0.170376 + 0.0471378i
\(97\) −0.535716 + 0.309296i −0.0543937 + 0.0314042i −0.526950 − 0.849896i \(-0.676665\pi\)
0.472557 + 0.881300i \(0.343331\pi\)
\(98\) 4.51964i 0.456552i
\(99\) −5.80797 − 9.69280i −0.583723 − 0.974163i
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.103.13 yes 28
3.2 odd 2 702.2.t.a.415.7 28
9.2 odd 6 702.2.t.a.181.8 28
9.4 even 3 2106.2.b.c.649.14 14
9.5 odd 6 2106.2.b.d.649.1 14
9.7 even 3 inner 234.2.t.a.25.6 ✓ 28
13.12 even 2 inner 234.2.t.a.103.6 yes 28
39.38 odd 2 702.2.t.a.415.8 28
117.25 even 6 inner 234.2.t.a.25.13 yes 28
117.38 odd 6 702.2.t.a.181.7 28
117.77 odd 6 2106.2.b.d.649.14 14
117.103 even 6 2106.2.b.c.649.1 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
234.2.t.a.25.6 ✓ 28 9.7 even 3 inner
234.2.t.a.25.13 yes 28 117.25 even 6 inner
234.2.t.a.103.6 yes 28 13.12 even 2 inner
234.2.t.a.103.13 yes 28 1.1 even 1 trivial
702.2.t.a.181.7 28 117.38 odd 6
702.2.t.a.181.8 28 9.2 odd 6
702.2.t.a.415.7 28 3.2 odd 2
702.2.t.a.415.8 28 39.38 odd 2
2106.2.b.c.649.1 14 117.103 even 6
2106.2.b.c.649.14 14 9.4 even 3
2106.2.b.d.649.1 14 9.5 odd 6
2106.2.b.d.649.14 14 117.77 odd 6