↑

Properties

Label 925.2.bq.b.357.15
Level $925$
Weight $2$
Character 925.357
Analytic conductor $7.386$
Analytic rank $0$
Dimension $204$
Inner twists $2$

Related objects

Downloads

Learn more

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(32,925)] 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("925.32"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([9, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.bq (of order \(36\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [204] 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: \(204\)
Relative dimension: \(17\) over \(\Q(\zeta_{36})\)
Twist minimal: no (minimal twist has level 185)
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 357.15
Character \(\chi\) \(=\) 925.357
Dual form 925.2.bq.b.868.15

$q$-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+(1.82148 + 0.321176i) q^{2} +(1.59505 - 2.27797i) q^{3} +(1.33525 + 0.485992i) q^{4} +(3.63698 - 3.63698i) q^{6} +(3.54263 + 0.309940i) q^{7} +(-0.927519 - 0.535504i) q^{8} +(-1.61890 - 4.44788i) q^{9} +(4.83320 + 2.79045i) q^{11} +(3.23687 - 2.26648i) q^{12} +(-1.08466 + 2.98008i) q^{13} +(6.35329 + 1.70236i) q^{14} +(-3.69448 - 3.10004i) q^{16} +(-2.83670 + 1.03247i) q^{17} +(-1.52023 - 8.62168i) q^{18} +(-0.0922365 + 0.131727i) q^{19} +(6.35671 - 7.57564i) q^{21} +(7.90736 + 6.63506i) q^{22} +(-5.58214 + 3.22285i) q^{23} +(-2.69930 + 1.25871i) q^{24} +(-2.93282 + 5.07980i) q^{26} +(-4.65596 - 1.24756i) q^{27} +(4.57968 + 2.13554i) q^{28} +(-2.02173 - 7.54520i) q^{29} +(-3.60337 - 3.60337i) q^{31} +(-4.35691 - 5.19236i) q^{32} +(14.0658 - 6.55898i) q^{33} +(-5.49859 + 0.969550i) q^{34} -6.72581i q^{36} +(-5.82563 + 1.74988i) q^{37} +(-0.210315 + 0.210315i) q^{38} +(5.05845 + 7.22421i) q^{39} +(0.214432 - 0.589148i) q^{41} +(14.0117 - 11.7572i) q^{42} -0.198014i q^{43} +(5.09741 + 6.07486i) q^{44} +(-11.2029 + 4.07751i) q^{46} +(-0.138518 + 0.516958i) q^{47} +(-12.9547 + 3.47120i) q^{48} +(5.56052 + 0.980470i) q^{49} +(-2.17273 + 8.10875i) q^{51} +(-2.89659 + 3.45203i) q^{52} +(2.42823 - 0.212443i) q^{53} +(-8.08006 - 3.76779i) q^{54} +(-3.11989 - 2.18457i) q^{56} +(0.152949 + 0.420224i) q^{57} +(-1.25920 - 14.3928i) q^{58} +(0.179432 + 2.05092i) q^{59} +(-0.187756 - 0.402643i) q^{61} +(-5.40616 - 7.72080i) q^{62} +(-4.35657 - 16.2590i) q^{63} +(-1.44556 - 2.50378i) q^{64} +(27.7271 - 7.42946i) q^{66} +(-1.22935 + 14.0515i) q^{67} -4.28948 q^{68} +(-1.56225 + 17.8566i) q^{69} +(0.219398 + 1.24427i) q^{71} +(-0.880298 + 4.99242i) q^{72} +(6.87081 + 6.87081i) q^{73} +(-11.1733 + 1.31631i) q^{74} +(-0.187177 + 0.131063i) q^{76} +(16.2574 + 11.3835i) q^{77} +(6.89362 + 14.7834i) q^{78} +(6.64942 + 0.581749i) q^{79} +(0.609426 - 0.511369i) q^{81} +(0.579805 - 1.00425i) q^{82} +(-4.10949 + 8.81283i) q^{83} +(12.1695 - 7.02607i) q^{84} +(0.0635975 - 0.360680i) q^{86} +(-20.4125 - 7.42954i) q^{87} +(-2.98859 - 5.17640i) q^{88} +(5.12449 - 0.448334i) q^{89} +(-4.76620 + 10.2212i) q^{91} +(-9.01985 + 1.59044i) q^{92} +(-13.9559 + 2.46081i) q^{93} +(-0.418343 + 0.897140i) q^{94} +(-18.7775 + 1.64282i) q^{96} +(-3.68600 - 6.38433i) q^{97} +(9.81348 + 3.57181i) q^{98} +(4.58714 - 26.0150i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 204 q + 12 q^{2} + 18 q^{3} - 24 q^{6} + 12 q^{7} + 18 q^{8} - 36 q^{11} + 12 q^{12} + 12 q^{13} + 24 q^{14} - 24 q^{16} - 12 q^{17} - 66 q^{18} - 24 q^{21} + 24 q^{22} + 18 q^{23} - 36 q^{24} - 12 q^{26}+ \cdots - 192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(76\) \(852\)
\(\chi(n)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.82148 + 0.321176i 1.28798 + 0.227106i 0.775366 − 0.631512i \(-0.217566\pi\)
0.512616 + 0.858618i \(0.328677\pi\)
\(3\) 1.59505 − 2.27797i 0.920903 − 1.31519i −0.0279962 − 0.999608i \(-0.508913\pi\)
0.948900 − 0.315578i \(-0.102198\pi\)
\(4\) 1.33525 + 0.485992i 0.667626 + 0.242996i
\(5\) 0 0
\(6\) 3.63698 − 3.63698i 1.48479 − 1.48479i
\(7\) 3.54263 + 0.309940i 1.33899 + 0.117146i 0.734034 − 0.679113i \(-0.237635\pi\)
0.604955 + 0.796259i \(0.293191\pi\)
\(8\) −0.927519 − 0.535504i −0.327928 − 0.189329i
\(9\) −1.61890 − 4.44788i −0.539632 − 1.48263i
\(10\) 0 0
\(11\) 4.83320 + 2.79045i 1.45727 + 0.841353i 0.998876 − 0.0473983i \(-0.0150930\pi\)
0.458390 + 0.888751i \(0.348426\pi\)
\(12\) 3.23687 − 2.26648i 0.934404 − 0.654277i
\(13\) −1.08466 + 2.98008i −0.300831 + 0.826526i 0.693525 + 0.720432i \(0.256057\pi\)
−0.994356 + 0.106094i \(0.966166\pi\)
\(14\) 6.35329 + 1.70236i 1.69799 + 0.454974i
\(15\) 0 0
\(16\) −3.69448 − 3.10004i −0.923621 − 0.775010i
\(17\) −2.83670 + 1.03247i −0.688000 + 0.250411i −0.662279 − 0.749258i \(-0.730410\pi\)
−0.0257213 + 0.999669i \(0.508188\pi\)
\(18\) −1.52023 − 8.62168i −0.358323 − 2.03215i
\(19\) −0.0922365 + 0.131727i −0.0211605 + 0.0302203i −0.829593 − 0.558369i \(-0.811427\pi\)
0.808432 + 0.588589i \(0.200316\pi\)
\(20\) 0 0
\(21\) 6.35671 − 7.57564i 1.38715 − 1.65314i
\(22\) 7.90736 + 6.63506i 1.68586 + 1.41460i
\(23\) −5.58214 + 3.22285i −1.16396 + 0.672011i −0.952249 − 0.305322i \(-0.901236\pi\)
−0.211708 + 0.977333i \(0.567903\pi\)
\(24\) −2.69930 + 1.25871i −0.550993 + 0.256932i
\(25\) 0 0
\(26\) −2.93282 + 5.07980i −0.575174 + 0.996230i
\(27\) −4.65596 − 1.24756i −0.896040 − 0.240093i
\(28\) 4.57968 + 2.13554i 0.865478 + 0.403579i
\(29\) −2.02173 − 7.54520i −0.375426 − 1.40111i −0.852722 − 0.522365i \(-0.825050\pi\)
0.477296 − 0.878743i \(-0.341617\pi\)
\(30\) 0 0
\(31\) −3.60337 − 3.60337i −0.647185 − 0.647185i 0.305127 − 0.952312i \(-0.401301\pi\)
−0.952312 + 0.305127i \(0.901301\pi\)
\(32\) −4.35691 − 5.19236i −0.770200 − 0.917889i
\(33\) 14.0658 − 6.55898i 2.44854 − 1.14177i
\(34\) −5.49859 + 0.969550i −0.943001 + 0.166276i
\(35\) 0 0
\(36\) − 6.72581i − 1.12097i
\(37\) −5.82563 + 1.74988i −0.957727 + 0.287678i
\(38\) −0.210315 + 0.210315i −0.0341175 + 0.0341175i
\(39\) 5.05845 + 7.22421i 0.810000 + 1.15680i
\(40\) 0 0
\(41\) 0.214432 − 0.589148i 0.0334887 − 0.0920094i −0.921821 − 0.387615i \(-0.873299\pi\)
0.955310 + 0.295605i \(0.0955213\pi\)
\(42\) 14.0117 − 11.7572i 2.16206 − 1.81418i
\(43\) − 0.198014i − 0.0301969i −0.999886 − 0.0150985i \(-0.995194\pi\)
0.999886 − 0.0150985i \(-0.00480617\pi\)
\(44\) 5.09741 + 6.07486i 0.768463 + 0.915819i
\(45\) 0 0
\(46\) −11.2029 + 4.07751i −1.65177 + 0.601196i
\(47\) −0.138518 + 0.516958i −0.0202050 + 0.0754061i −0.975292 − 0.220919i \(-0.929094\pi\)
0.955087 + 0.296325i \(0.0957611\pi\)
\(48\) −12.9547 + 3.47120i −1.86985 + 0.501024i
\(49\) 5.56052 + 0.980470i 0.794360 + 0.140067i
\(50\) 0 0
\(51\) −2.17273 + 8.10875i −0.304244 + 1.13545i
\(52\) −2.89659 + 3.45203i −0.401685 + 0.478710i
\(53\) 2.42823 − 0.212443i 0.333544 − 0.0291813i 0.0808460 − 0.996727i \(-0.474238\pi\)
0.252698 + 0.967545i \(0.418682\pi\)
\(54\) −8.08006 − 3.76779i −1.09956 − 0.512732i
\(55\) 0 0
\(56\) −3.11989 − 2.18457i −0.416912 − 0.291925i
\(57\) 0.152949 + 0.420224i 0.0202586 + 0.0556600i
\(58\) −1.25920 − 14.3928i −0.165341 − 1.88986i
\(59\) 0.179432 + 2.05092i 0.0233601 + 0.267007i 0.998889 + 0.0471341i \(0.0150088\pi\)
−0.975528 + 0.219873i \(0.929436\pi\)
\(60\) 0 0
\(61\) −0.187756 − 0.402643i −0.0240396 − 0.0515532i 0.893921 − 0.448224i \(-0.147943\pi\)
−0.917961 + 0.396671i \(0.870165\pi\)
\(62\) −5.40616 − 7.72080i −0.686583 − 0.980542i
\(63\) −4.35657 − 16.2590i −0.548877 − 2.04844i
\(64\) −1.44556 − 2.50378i −0.180695 − 0.312973i
\(65\) 0 0
\(66\) 27.7271 − 7.42946i 3.41297 − 0.914503i
\(67\) −1.22935 + 14.0515i −0.150189 + 1.71667i 0.430542 + 0.902570i \(0.358322\pi\)
−0.580731 + 0.814095i \(0.697233\pi\)
\(68\) −4.28948 −0.520176
\(69\) −1.56225 + 17.8566i −0.188072 + 2.14968i
\(70\) 0 0
\(71\) 0.219398 + 1.24427i 0.0260378 + 0.147668i 0.995055 − 0.0993256i \(-0.0316685\pi\)
−0.969017 + 0.246993i \(0.920557\pi\)
\(72\) −0.880298 + 4.99242i −0.103744 + 0.588362i
\(73\) 6.87081 + 6.87081i 0.804168 + 0.804168i 0.983744 − 0.179576i \(-0.0574727\pi\)
−0.179576 + 0.983744i \(0.557473\pi\)
\(74\) −11.1733 + 1.31631i −1.29887 + 0.153018i
\(75\) 0 0
\(76\) −0.187177 + 0.131063i −0.0214707 + 0.0150340i
\(77\) 16.2574 + 11.3835i 1.85270 + 1.29728i
\(78\) 6.89362 + 14.7834i 0.780549 + 1.67389i
\(79\) 6.64942 + 0.581749i 0.748118 + 0.0654519i 0.454837 − 0.890574i \(-0.349697\pi\)
0.293281 + 0.956026i \(0.405253\pi\)
\(80\) 0 0
\(81\) 0.609426 − 0.511369i 0.0677140 − 0.0568188i
\(82\) 0.579805 − 1.00425i 0.0640287 − 0.110901i
\(83\) −4.10949 + 8.81283i −0.451075 + 0.967333i 0.540922 + 0.841073i \(0.318075\pi\)
−0.991997 + 0.126261i \(0.959702\pi\)
\(84\) 12.1695 − 7.02607i 1.32780 − 0.766607i
\(85\) 0 0
\(86\) 0.0635975 − 0.360680i 0.00685790 − 0.0388931i
\(87\) −20.4125 − 7.42954i −2.18845 − 0.796530i
\(88\) −2.98859 − 5.17640i −0.318585 − 0.551806i
\(89\) 5.12449 − 0.448334i 0.543194 − 0.0475234i 0.187743 − 0.982218i \(-0.439883\pi\)
0.355451 + 0.934695i \(0.384327\pi\)
\(90\) 0 0
\(91\) −4.76620 + 10.2212i −0.499634 + 1.07147i
\(92\) −9.01985 + 1.59044i −0.940384 + 0.165815i
\(93\) −13.9559 + 2.46081i −1.44716 + 0.255174i
\(94\) −0.418343 + 0.897140i −0.0431488 + 0.0925329i
\(95\) 0 0
\(96\) −18.7775 + 1.64282i −1.91647 + 0.167670i
\(97\) −3.68600 − 6.38433i −0.374256 − 0.648231i 0.615959 − 0.787778i \(-0.288769\pi\)
−0.990215 + 0.139547i \(0.955435\pi\)
\(98\) 9.81348 + 3.57181i 0.991311 + 0.360808i
\(99\) 4.58714 − 26.0150i 0.461025 − 2.61460i
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.bq.b.357.15 204
5.2 odd 4 185.2.z.a.98.3 yes 204
5.3 odd 4 925.2.bn.b.468.15 204
5.4 even 2 185.2.bc.a.172.3 yes 204
37.17 odd 36 925.2.bn.b.757.15 204
185.17 even 36 185.2.bc.a.128.3 yes 204
185.54 odd 36 185.2.z.a.17.3 ✓ 204
185.128 even 36 inner 925.2.bq.b.868.15 204
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.z.a.17.3 ✓ 204 185.54 odd 36
185.2.z.a.98.3 yes 204 5.2 odd 4
185.2.bc.a.128.3 yes 204 185.17 even 36
185.2.bc.a.172.3 yes 204 5.4 even 2
925.2.bn.b.468.15 204 5.3 odd 4
925.2.bn.b.757.15 204 37.17 odd 36
925.2.bq.b.357.15 204 1.1 even 1 trivial
925.2.bq.b.868.15 204 185.128 even 36 inner