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Properties

Label 925.2.y.b.532.7
Level $925$
Weight $2$
Character 925.532
Analytic conductor $7.386$
Analytic rank $0$
Dimension $68$
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] = [925,2,Mod(193,925)] 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("925.193"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([9, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.y (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [68] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(17\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 185)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 532.7
Character \(\chi\) \(=\) 925.532
Dual form 925.2.y.b.193.7

$q$-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+(-0.407096 - 0.235037i) q^{2} +(0.838017 - 3.12752i) q^{3} +(-0.889515 - 1.54069i) q^{4} +(-1.07624 + 1.07624i) q^{6} +(0.968227 - 3.61347i) q^{7} +1.77643i q^{8} +(-6.48105 - 3.74183i) q^{9} +3.20848i q^{11} +(-5.56396 + 1.49086i) q^{12} +(-0.708690 + 0.409162i) q^{13} +(-1.24346 + 1.24346i) q^{14} +(-1.36150 + 2.35819i) q^{16} +(0.330184 - 0.571896i) q^{17} +(1.75894 + 3.04657i) q^{18} +(0.437461 - 1.63263i) q^{19} +(-10.4898 - 6.05630i) q^{21} +(0.754113 - 1.30616i) q^{22} -4.26497i q^{23} +(5.55581 + 1.48867i) q^{24} +0.384673 q^{26} +(-10.2654 + 10.2654i) q^{27} +(-6.42847 + 1.72250i) q^{28} +(-0.937440 + 0.937440i) q^{29} +(3.86165 + 3.86165i) q^{31} +(4.18539 - 2.41643i) q^{32} +(10.0346 + 2.68876i) q^{33} +(-0.268834 + 0.155211i) q^{34} +13.3137i q^{36} +(-2.61665 - 5.49119i) q^{37} +(-0.561817 + 0.561817i) q^{38} +(0.685770 + 2.55933i) q^{39} +(2.21722 - 1.28011i) q^{41} +(2.84691 + 4.93100i) q^{42} -9.49168i q^{43} +(4.94326 - 2.85399i) q^{44} +(-1.00243 + 1.73625i) q^{46} +(6.02730 + 6.02730i) q^{47} +(6.23434 + 6.23434i) q^{48} +(-6.05753 - 3.49732i) q^{49} +(-1.51192 - 1.51192i) q^{51} +(1.26078 + 0.727912i) q^{52} +(2.23181 + 8.32923i) q^{53} +(6.59175 - 1.76625i) q^{54} +(6.41906 + 1.71998i) q^{56} +(-4.73948 - 2.73634i) q^{57} +(0.601962 - 0.161295i) q^{58} +(-2.51108 + 0.672842i) q^{59} +(4.00095 - 14.9318i) q^{61} +(-0.664433 - 2.47970i) q^{62} +(-19.7961 + 19.7961i) q^{63} +3.17421 q^{64} +(-3.45309 - 3.45309i) q^{66} +(5.70346 + 1.52824i) q^{67} -1.17482 q^{68} +(-13.3388 - 3.57412i) q^{69} +(-3.81787 - 6.61274i) q^{71} +(6.64709 - 11.5131i) q^{72} +(-7.41925 - 7.41925i) q^{73} +(-0.225403 + 2.85045i) q^{74} +(-2.90449 + 0.778256i) q^{76} +(11.5938 + 3.10654i) q^{77} +(0.322363 - 1.20307i) q^{78} +(-3.47315 + 12.9620i) q^{79} +(12.2771 + 21.2646i) q^{81} -1.20349 q^{82} +(2.60362 + 9.71686i) q^{83} +21.5487i q^{84} +(-2.23090 + 3.86403i) q^{86} +(2.14627 + 3.71746i) q^{87} -5.69963 q^{88} +(1.41656 + 5.28666i) q^{89} +(0.792323 + 2.95699i) q^{91} +(-6.57098 + 3.79376i) q^{92} +(15.3135 - 8.84127i) q^{93} +(-1.03705 - 3.87033i) q^{94} +(-4.05003 - 15.1149i) q^{96} -3.89183 q^{97} +(1.64400 + 2.84749i) q^{98} +(12.0056 - 20.7943i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 68 q + 6 q^{2} + 4 q^{3} + 30 q^{4} - 8 q^{6} + 2 q^{7} + 10 q^{12} + 6 q^{13} - 26 q^{16} + 10 q^{17} + 8 q^{18} - 4 q^{19} - 12 q^{21} + 14 q^{22} - 24 q^{26} - 68 q^{27} - 14 q^{28} - 14 q^{29} - 24 q^{31}+ \cdots - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.407096 − 0.235037i −0.287861 − 0.166196i 0.349116 − 0.937080i \(-0.386482\pi\)
−0.636977 + 0.770883i \(0.719815\pi\)
\(3\) 0.838017 − 3.12752i 0.483829 − 1.80568i −0.101446 − 0.994841i \(-0.532347\pi\)
0.585275 − 0.810835i \(-0.300986\pi\)
\(4\) −0.889515 − 1.54069i −0.444757 − 0.770343i
\(5\) 0 0
\(6\) −1.07624 + 1.07624i −0.439372 + 0.439372i
\(7\) 0.968227 − 3.61347i 0.365955 − 1.36576i −0.500166 − 0.865930i \(-0.666728\pi\)
0.866121 − 0.499834i \(-0.166606\pi\)
\(8\) 1.77643i 0.628061i
\(9\) −6.48105 − 3.74183i −2.16035 − 1.24728i
\(10\) 0 0
\(11\) 3.20848i 0.967394i 0.875235 + 0.483697i \(0.160706\pi\)
−0.875235 + 0.483697i \(0.839294\pi\)
\(12\) −5.56396 + 1.49086i −1.60618 + 0.430373i
\(13\) −0.708690 + 0.409162i −0.196555 + 0.113481i −0.595048 − 0.803690i \(-0.702867\pi\)
0.398492 + 0.917172i \(0.369533\pi\)
\(14\) −1.24346 + 1.24346i −0.332329 + 0.332329i
\(15\) 0 0
\(16\) −1.36150 + 2.35819i −0.340376 + 0.589548i
\(17\) 0.330184 − 0.571896i 0.0800815 − 0.138705i −0.823203 − 0.567747i \(-0.807815\pi\)
0.903285 + 0.429042i \(0.141149\pi\)
\(18\) 1.75894 + 3.04657i 0.414586 + 0.718084i
\(19\) 0.437461 − 1.63263i 0.100360 − 0.374550i −0.897417 − 0.441183i \(-0.854559\pi\)
0.997778 + 0.0666328i \(0.0212256\pi\)
\(20\) 0 0
\(21\) −10.4898 − 6.05630i −2.28907 − 1.32159i
\(22\) 0.754113 − 1.30616i 0.160777 − 0.278475i
\(23\) − 4.26497i − 0.889308i −0.895702 − 0.444654i \(-0.853327\pi\)
0.895702 − 0.444654i \(-0.146673\pi\)
\(24\) 5.55581 + 1.48867i 1.13408 + 0.303874i
\(25\) 0 0
\(26\) 0.384673 0.0754407
\(27\) −10.2654 + 10.2654i −1.97557 + 1.97557i
\(28\) −6.42847 + 1.72250i −1.21487 + 0.325523i
\(29\) −0.937440 + 0.937440i −0.174078 + 0.174078i −0.788769 − 0.614690i \(-0.789281\pi\)
0.614690 + 0.788769i \(0.289281\pi\)
\(30\) 0 0
\(31\) 3.86165 + 3.86165i 0.693573 + 0.693573i 0.963016 − 0.269443i \(-0.0868396\pi\)
−0.269443 + 0.963016i \(0.586840\pi\)
\(32\) 4.18539 − 2.41643i 0.739879 − 0.427169i
\(33\) 10.0346 + 2.68876i 1.74680 + 0.468054i
\(34\) −0.268834 + 0.155211i −0.0461046 + 0.0266185i
\(35\) 0 0
\(36\) 13.3137i 2.21894i
\(37\) −2.61665 − 5.49119i −0.430175 − 0.902746i
\(38\) −0.561817 + 0.561817i −0.0911388 + 0.0911388i
\(39\) 0.685770 + 2.55933i 0.109811 + 0.409820i
\(40\) 0 0
\(41\) 2.21722 − 1.28011i 0.346271 − 0.199920i −0.316771 − 0.948502i \(-0.602599\pi\)
0.663042 + 0.748583i \(0.269265\pi\)
\(42\) 2.84691 + 4.93100i 0.439288 + 0.760869i
\(43\) − 9.49168i − 1.44747i −0.690079 − 0.723734i \(-0.742424\pi\)
0.690079 − 0.723734i \(-0.257576\pi\)
\(44\) 4.94326 − 2.85399i 0.745225 − 0.430256i
\(45\) 0 0
\(46\) −1.00243 + 1.73625i −0.147800 + 0.255997i
\(47\) 6.02730 + 6.02730i 0.879172 + 0.879172i 0.993449 − 0.114277i \(-0.0364552\pi\)
−0.114277 + 0.993449i \(0.536455\pi\)
\(48\) 6.23434 + 6.23434i 0.899849 + 0.899849i
\(49\) −6.05753 − 3.49732i −0.865362 − 0.499617i
\(50\) 0 0
\(51\) −1.51192 − 1.51192i −0.211711 − 0.211711i
\(52\) 1.26078 + 0.727912i 0.174839 + 0.100943i
\(53\) 2.23181 + 8.32923i 0.306563 + 1.14411i 0.931592 + 0.363506i \(0.118420\pi\)
−0.625029 + 0.780602i \(0.714913\pi\)
\(54\) 6.59175 − 1.76625i 0.897023 − 0.240357i
\(55\) 0 0
\(56\) 6.41906 + 1.71998i 0.857783 + 0.229842i
\(57\) −4.73948 − 2.73634i −0.627759 − 0.362437i
\(58\) 0.601962 − 0.161295i 0.0790415 − 0.0211791i
\(59\) −2.51108 + 0.672842i −0.326915 + 0.0875966i −0.418544 − 0.908197i \(-0.637459\pi\)
0.0916288 + 0.995793i \(0.470793\pi\)
\(60\) 0 0
\(61\) 4.00095 − 14.9318i 0.512269 − 1.91182i 0.117378 − 0.993087i \(-0.462551\pi\)
0.394891 − 0.918728i \(-0.370782\pi\)
\(62\) −0.664433 − 2.47970i −0.0843831 − 0.314922i
\(63\) −19.7961 + 19.7961i −2.49408 + 2.49408i
\(64\) 3.17421 0.396776
\(65\) 0 0
\(66\) −3.45309 − 3.45309i −0.425046 − 0.425046i
\(67\) 5.70346 + 1.52824i 0.696789 + 0.186704i 0.589792 − 0.807555i \(-0.299210\pi\)
0.106997 + 0.994259i \(0.465877\pi\)
\(68\) −1.17482 −0.142467
\(69\) −13.3388 − 3.57412i −1.60580 − 0.430273i
\(70\) 0 0
\(71\) −3.81787 − 6.61274i −0.453098 − 0.784788i 0.545479 − 0.838125i \(-0.316348\pi\)
−0.998577 + 0.0533363i \(0.983014\pi\)
\(72\) 6.64709 − 11.5131i 0.783367 − 1.35683i
\(73\) −7.41925 − 7.41925i −0.868357 − 0.868357i 0.123933 − 0.992291i \(-0.460449\pi\)
−0.992291 + 0.123933i \(0.960449\pi\)
\(74\) −0.225403 + 2.85045i −0.0262026 + 0.331358i
\(75\) 0 0
\(76\) −2.90449 + 0.778256i −0.333168 + 0.0892721i
\(77\) 11.5938 + 3.10654i 1.32123 + 0.354023i
\(78\) 0.322363 − 1.20307i 0.0365004 − 0.136221i
\(79\) −3.47315 + 12.9620i −0.390760 + 1.45833i 0.438124 + 0.898914i \(0.355643\pi\)
−0.828884 + 0.559421i \(0.811024\pi\)
\(80\) 0 0
\(81\) 12.2771 + 21.2646i 1.36413 + 2.36274i
\(82\) −1.20349 −0.132904
\(83\) 2.60362 + 9.71686i 0.285785 + 1.06656i 0.948264 + 0.317483i \(0.102838\pi\)
−0.662479 + 0.749081i \(0.730496\pi\)
\(84\) 21.5487i 2.35115i
\(85\) 0 0
\(86\) −2.23090 + 3.86403i −0.240564 + 0.416669i
\(87\) 2.14627 + 3.71746i 0.230105 + 0.398553i
\(88\) −5.69963 −0.607583
\(89\) 1.41656 + 5.28666i 0.150155 + 0.560384i 0.999472 + 0.0325015i \(0.0103474\pi\)
−0.849317 + 0.527883i \(0.822986\pi\)
\(90\) 0 0
\(91\) 0.792323 + 2.95699i 0.0830581 + 0.309977i
\(92\) −6.57098 + 3.79376i −0.685072 + 0.395526i
\(93\) 15.3135 − 8.84127i 1.58794 − 0.916797i
\(94\) −1.03705 − 3.87033i −0.106964 − 0.399194i
\(95\) 0 0
\(96\) −4.05003 − 15.1149i −0.413354 − 1.54266i
\(97\) −3.89183 −0.395156 −0.197578 − 0.980287i \(-0.563308\pi\)
−0.197578 + 0.980287i \(0.563308\pi\)
\(98\) 1.64400 + 2.84749i 0.166069 + 0.287640i
\(99\) 12.0056 − 20.7943i 1.20661 − 2.08991i
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 925.2.y.b.532.7 68
5.2 odd 4 185.2.p.a.88.11 yes 68
5.3 odd 4 925.2.t.b.643.7 68
5.4 even 2 185.2.u.a.162.11 yes 68
37.8 odd 12 925.2.t.b.82.7 68
185.8 even 12 inner 925.2.y.b.193.7 68
185.82 even 12 185.2.u.a.8.11 yes 68
185.119 odd 12 185.2.p.a.82.11 ✓ 68
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.p.a.82.11 ✓ 68 185.119 odd 12
185.2.p.a.88.11 yes 68 5.2 odd 4
185.2.u.a.8.11 yes 68 185.82 even 12
185.2.u.a.162.11 yes 68 5.4 even 2
925.2.t.b.82.7 68 37.8 odd 12
925.2.t.b.643.7 68 5.3 odd 4
925.2.y.b.193.7 68 185.8 even 12 inner
925.2.y.b.532.7 68 1.1 even 1 trivial