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Properties

Label 234.2.t.a.25.12
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 25.12
Character \(\chi\) \(=\) 234.25
Dual form 234.2.t.a.103.12

$q$-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} +(0.523143 - 1.65116i) q^{3} +(0.500000 + 0.866025i) q^{4} +(-0.419378 + 0.242128i) q^{5} +(1.27863 - 1.16837i) q^{6} +(4.37722 + 2.52719i) q^{7} +1.00000i q^{8} +(-2.45264 - 1.72758i) q^{9} -0.484256 q^{10} +(-2.78126 - 1.60576i) q^{11} +(1.69152 - 0.372524i) q^{12} +(-0.722307 - 3.53246i) q^{13} +(2.52719 + 4.37722i) q^{14} +(0.180397 + 0.819127i) q^{15} +(-0.500000 + 0.866025i) q^{16} +4.20245 q^{17} +(-1.26026 - 2.72245i) q^{18} +3.21153i q^{19} +(-0.419378 - 0.242128i) q^{20} +(6.46270 - 5.90540i) q^{21} +(-1.60576 - 2.78126i) q^{22} +(-3.13608 - 5.43186i) q^{23} +(1.65116 + 0.523143i) q^{24} +(-2.38275 + 4.12704i) q^{25} +(1.14069 - 3.42035i) q^{26} +(-4.13559 + 3.14593i) q^{27} +5.05438i q^{28} +(-2.29627 + 3.97725i) q^{29} +(-0.253335 + 0.799583i) q^{30} +(-5.61504 + 3.24185i) q^{31} +(-0.866025 + 0.500000i) q^{32} +(-4.10637 + 3.75226i) q^{33} +(3.63943 + 2.10123i) q^{34} -2.44761 q^{35} +(0.269808 - 2.98784i) q^{36} -2.08429i q^{37} +(-1.60576 + 2.78126i) q^{38} +(-6.21052 - 0.655337i) q^{39} +(-0.242128 - 0.419378i) q^{40} +(-9.57301 + 5.52698i) q^{41} +(8.54956 - 1.88288i) q^{42} +(4.73367 - 8.19896i) q^{43} -3.21153i q^{44} +(1.44688 + 0.130656i) q^{45} -6.27217i q^{46} +(-4.57369 - 2.64062i) q^{47} +(1.16837 + 1.27863i) q^{48} +(9.27337 + 16.0619i) q^{49} +(-4.12704 + 2.38275i) q^{50} +(2.19848 - 6.93891i) q^{51} +(2.69805 - 2.39177i) q^{52} +6.41990 q^{53} +(-5.15449 + 0.656660i) q^{54} +1.55520 q^{55} +(-2.52719 + 4.37722i) q^{56} +(5.30274 + 1.68009i) q^{57} +(-3.97725 + 2.29627i) q^{58} +(3.13771 - 1.81156i) q^{59} +(-0.619186 + 0.565792i) q^{60} +(0.500864 - 0.867521i) q^{61} -6.48369 q^{62} +(-6.36984 - 13.7603i) q^{63} -1.00000 q^{64} +(1.15823 + 1.30655i) q^{65} +(-5.43235 + 1.19637i) q^{66} +(0.936987 - 0.540970i) q^{67} +(2.10123 + 3.63943i) q^{68} +(-10.6095 + 2.33653i) q^{69} +(-2.11969 - 1.22381i) q^{70} +4.63041i q^{71} +(1.72758 - 2.45264i) q^{72} -0.325525i q^{73} +(1.04214 - 1.80504i) q^{74} +(5.56788 + 6.09332i) q^{75} +(-2.78126 + 1.60576i) q^{76} +(-8.11614 - 14.0576i) q^{77} +(-5.05080 - 3.67280i) q^{78} +(3.91818 - 6.78649i) q^{79} -0.484256i q^{80} +(3.03092 + 8.47428i) q^{81} -11.0540 q^{82} +(-5.08022 - 2.93306i) q^{83} +(8.34557 + 2.64416i) q^{84} +(-1.76242 + 1.01753i) q^{85} +(8.19896 - 4.73367i) q^{86} +(5.36580 + 5.87217i) q^{87} +(1.60576 - 2.78126i) q^{88} -8.42912i q^{89} +(1.18771 + 0.836592i) q^{90} +(5.76550 - 17.2878i) q^{91} +(3.13608 - 5.43186i) q^{92} +(2.41533 + 10.9673i) q^{93} +(-2.64062 - 4.57369i) q^{94} +(-0.777601 - 1.34684i) q^{95} +(0.372524 + 1.69152i) q^{96} +(11.3021 + 6.52525i) q^{97} +18.5467i q^{98} +(4.04736 + 8.74323i) 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
\(3\) 0.523143 − 1.65116i 0.302036 − 0.953296i
\(4\) 0.500000 + 0.866025i 0.250000 + 0.433013i
\(5\) −0.419378 + 0.242128i −0.187552 + 0.108283i −0.590836 − 0.806792i \(-0.701202\pi\)
0.403284 + 0.915075i \(0.367869\pi\)
\(6\) 1.27863 − 1.16837i 0.522000 − 0.476986i
\(7\) 4.37722 + 2.52719i 1.65443 + 0.955188i 0.975217 + 0.221249i \(0.0710134\pi\)
0.679216 + 0.733938i \(0.262320\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −2.45264 − 1.72758i −0.817548 − 0.575861i
\(10\) −0.484256 −0.153135
\(11\) −2.78126 − 1.60576i −0.838583 − 0.484156i 0.0181994 − 0.999834i \(-0.494207\pi\)
−0.856782 + 0.515678i \(0.827540\pi\)
\(12\) 1.69152 − 0.372524i 0.488299 − 0.107538i
\(13\) −0.722307 − 3.53246i −0.200332 − 0.979728i
\(14\) 2.52719 + 4.37722i 0.675420 + 1.16986i
\(15\) 0.180397 + 0.819127i 0.0465783 + 0.211498i
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 4.20245 1.01924 0.509622 − 0.860398i \(-0.329785\pi\)
0.509622 + 0.860398i \(0.329785\pi\)
\(18\) −1.26026 − 2.72245i −0.297046 − 0.641688i
\(19\) 3.21153i 0.736775i 0.929672 + 0.368388i \(0.120090\pi\)
−0.929672 + 0.368388i \(0.879910\pi\)
\(20\) −0.419378 − 0.242128i −0.0937758 − 0.0541415i
\(21\) 6.46270 − 5.90540i 1.41028 − 1.28866i
\(22\) −1.60576 − 2.78126i −0.342350 − 0.592968i
\(23\) −3.13608 − 5.43186i −0.653919 − 1.13262i −0.982164 − 0.188028i \(-0.939791\pi\)
0.328245 − 0.944593i \(-0.393543\pi\)
\(24\) 1.65116 + 0.523143i 0.337041 + 0.106786i
\(25\) −2.38275 + 4.12704i −0.476550 + 0.825408i
\(26\) 1.14069 − 3.42035i 0.223708 − 0.670787i
\(27\) −4.13559 + 3.14593i −0.795895 + 0.605435i
\(28\) 5.05438i 0.955188i
\(29\) −2.29627 + 3.97725i −0.426406 + 0.738557i −0.996551 − 0.0829873i \(-0.973554\pi\)
0.570144 + 0.821544i \(0.306887\pi\)
\(30\) −0.253335 + 0.799583i −0.0462524 + 0.145983i
\(31\) −5.61504 + 3.24185i −1.00849 + 0.582253i −0.910750 − 0.412959i \(-0.864495\pi\)
−0.0977420 + 0.995212i \(0.531162\pi\)
\(32\) −0.866025 + 0.500000i −0.153093 + 0.0883883i
\(33\) −4.10637 + 3.75226i −0.714827 + 0.653185i
\(34\) 3.63943 + 2.10123i 0.624157 + 0.360357i
\(35\) −2.44761 −0.413722
\(36\) 0.269808 − 2.98784i 0.0449679 − 0.497974i
\(37\) − 2.08429i − 0.342654i −0.985214 − 0.171327i \(-0.945194\pi\)
0.985214 − 0.171327i \(-0.0548055\pi\)
\(38\) −1.60576 + 2.78126i −0.260489 + 0.451181i
\(39\) −6.21052 − 0.655337i −0.994479 − 0.104938i
\(40\) −0.242128 − 0.419378i −0.0382838 − 0.0663095i
\(41\) −9.57301 + 5.52698i −1.49505 + 0.863169i −0.999984 − 0.00568392i \(-0.998191\pi\)
−0.495070 + 0.868853i \(0.664857\pi\)
\(42\) 8.54956 − 1.88288i 1.31923 − 0.290534i
\(43\) 4.73367 − 8.19896i 0.721879 − 1.25033i −0.238367 − 0.971175i \(-0.576612\pi\)
0.960246 − 0.279155i \(-0.0900544\pi\)
\(44\) − 3.21153i − 0.484156i
\(45\) 1.44688 + 0.130656i 0.215688 + 0.0194770i
\(46\) − 6.27217i − 0.924780i
\(47\) −4.57369 − 2.64062i −0.667141 − 0.385174i 0.127851 − 0.991793i \(-0.459192\pi\)
−0.794992 + 0.606619i \(0.792525\pi\)
\(48\) 1.16837 + 1.27863i 0.168640 + 0.184555i
\(49\) 9.27337 + 16.0619i 1.32477 + 2.29456i
\(50\) −4.12704 + 2.38275i −0.583652 + 0.336971i
\(51\) 2.19848 − 6.93891i 0.307849 − 0.971642i
\(52\) 2.69805 − 2.39177i 0.374152 − 0.331678i
\(53\) 6.41990 0.881841 0.440921 − 0.897546i \(-0.354652\pi\)
0.440921 + 0.897546i \(0.354652\pi\)
\(54\) −5.15449 + 0.656660i −0.701438 + 0.0893601i
\(55\) 1.55520 0.209703
\(56\) −2.52719 + 4.37722i −0.337710 + 0.584931i
\(57\) 5.30274 + 1.68009i 0.702365 + 0.222533i
\(58\) −3.97725 + 2.29627i −0.522239 + 0.301515i
\(59\) 3.13771 − 1.81156i 0.408495 − 0.235844i −0.281648 − 0.959518i \(-0.590881\pi\)
0.690143 + 0.723673i \(0.257548\pi\)
\(60\) −0.619186 + 0.565792i −0.0799366 + 0.0730434i
\(61\) 0.500864 − 0.867521i 0.0641290 − 0.111075i −0.832178 − 0.554508i \(-0.812906\pi\)
0.896307 + 0.443433i \(0.146240\pi\)
\(62\) −6.48369 −0.823430
\(63\) −6.36984 − 13.7603i −0.802524 − 1.73363i
\(64\) −1.00000 −0.125000
\(65\) 1.15823 + 1.30655i 0.143660 + 0.162057i
\(66\) −5.43235 + 1.19637i −0.668676 + 0.147263i
\(67\) 0.936987 − 0.540970i 0.114471 − 0.0660900i −0.441671 − 0.897177i \(-0.645614\pi\)
0.556142 + 0.831087i \(0.312281\pi\)
\(68\) 2.10123 + 3.63943i 0.254811 + 0.441346i
\(69\) −10.6095 + 2.33653i −1.27723 + 0.281286i
\(70\) −2.11969 − 1.22381i −0.253352 − 0.146273i
\(71\) 4.63041i 0.549529i 0.961512 + 0.274764i \(0.0885999\pi\)
−0.961512 + 0.274764i \(0.911400\pi\)
\(72\) 1.72758 − 2.45264i 0.203597 − 0.289047i
\(73\) − 0.325525i − 0.0380998i −0.999819 − 0.0190499i \(-0.993936\pi\)
0.999819 − 0.0190499i \(-0.00606414\pi\)
\(74\) 1.04214 − 1.80504i 0.121147 − 0.209832i
\(75\) 5.56788 + 6.09332i 0.642923 + 0.703596i
\(76\) −2.78126 + 1.60576i −0.319033 + 0.184194i
\(77\) −8.11614 − 14.0576i −0.924920 − 1.60201i
\(78\) −5.05080 − 3.67280i −0.571890 − 0.415862i
\(79\) 3.91818 − 6.78649i 0.440830 − 0.763540i −0.556921 − 0.830565i \(-0.688017\pi\)
0.997751 + 0.0670253i \(0.0213508\pi\)
\(80\) − 0.484256i − 0.0541415i
\(81\) 3.03092 + 8.47428i 0.336769 + 0.941587i
\(82\) −11.0540 −1.22071
\(83\) −5.08022 − 2.93306i −0.557626 − 0.321946i 0.194566 − 0.980889i \(-0.437670\pi\)
−0.752192 + 0.658944i \(0.771003\pi\)
\(84\) 8.34557 + 2.64416i 0.910577 + 0.288501i
\(85\) −1.76242 + 1.01753i −0.191161 + 0.110367i
\(86\) 8.19896 − 4.73367i 0.884117 − 0.510445i
\(87\) 5.36580 + 5.87217i 0.575274 + 0.629563i
\(88\) 1.60576 − 2.78126i 0.171175 − 0.296484i
\(89\) − 8.42912i − 0.893485i −0.894662 − 0.446743i \(-0.852584\pi\)
0.894662 − 0.446743i \(-0.147416\pi\)
\(90\) 1.18771 + 0.836592i 0.125195 + 0.0881845i
\(91\) 5.76550 − 17.2878i 0.604388 − 1.81225i
\(92\) 3.13608 − 5.43186i 0.326959 − 0.566310i
\(93\) 2.41533 + 10.9673i 0.250458 + 1.13725i
\(94\) −2.64062 − 4.57369i −0.272359 − 0.471740i
\(95\) −0.777601 − 1.34684i −0.0797802 − 0.138183i
\(96\) 0.372524 + 1.69152i 0.0380206 + 0.172640i
\(97\) 11.3021 + 6.52525i 1.14755 + 0.662539i 0.948289 − 0.317407i \(-0.102812\pi\)
0.199262 + 0.979946i \(0.436145\pi\)
\(98\) 18.5467i 1.87350i
\(99\) 4.04736 + 8.74323i 0.406775 + 0.878728i
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.25.12 yes 28
3.2 odd 2 702.2.t.a.181.4 28
9.2 odd 6 2106.2.b.d.649.11 14
9.4 even 3 inner 234.2.t.a.103.5 yes 28
9.5 odd 6 702.2.t.a.415.11 28
9.7 even 3 2106.2.b.c.649.4 14
13.12 even 2 inner 234.2.t.a.25.5 ✓ 28
39.38 odd 2 702.2.t.a.181.11 28
117.25 even 6 2106.2.b.c.649.11 14
117.38 odd 6 2106.2.b.d.649.4 14
117.77 odd 6 702.2.t.a.415.4 28
117.103 even 6 inner 234.2.t.a.103.12 yes 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
234.2.t.a.25.5 ✓ 28 13.12 even 2 inner
234.2.t.a.25.12 yes 28 1.1 even 1 trivial
234.2.t.a.103.5 yes 28 9.4 even 3 inner
234.2.t.a.103.12 yes 28 117.103 even 6 inner
702.2.t.a.181.4 28 3.2 odd 2
702.2.t.a.181.11 28 39.38 odd 2
702.2.t.a.415.4 28 117.77 odd 6
702.2.t.a.415.11 28 9.5 odd 6
2106.2.b.c.649.4 14 9.7 even 3
2106.2.b.c.649.11 14 117.25 even 6
2106.2.b.d.649.4 14 117.38 odd 6
2106.2.b.d.649.11 14 9.2 odd 6