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Properties

Label 925.2.bb.e.176.16
Level $925$
Weight $2$
Character 925.176
Analytic conductor $7.386$
Analytic rank $0$
Dimension $96$
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] = [925,2,Mod(151,925)] 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("925.151"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.bb (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [96,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 185)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 176.16
Character \(\chi\) \(=\) 925.176
Dual form 925.2.bb.e.226.16

$q$-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+(2.43972 - 0.430188i) q^{2} +(0.198906 - 1.12805i) q^{3} +(3.88777 - 1.41503i) q^{4} -2.83769i q^{6} +(2.28969 + 1.92128i) q^{7} +(4.58542 - 2.64739i) q^{8} +(1.58614 + 0.577309i) q^{9} +(-0.267671 - 0.463619i) q^{11} +(-0.822927 - 4.66705i) q^{12} +(0.877498 + 2.41091i) q^{13} +(6.41272 + 3.70238i) q^{14} +(3.70957 - 3.11270i) q^{16} +(-0.391407 + 1.07538i) q^{17} +(4.11809 + 0.726131i) q^{18} +(-6.95889 - 1.22704i) q^{19} +(2.62274 - 2.20074i) q^{21} +(-0.852484 - 1.01595i) q^{22} +(-6.63166 - 3.82879i) q^{23} +(-2.07432 - 5.69916i) q^{24} +(3.17799 + 5.50444i) q^{26} +(2.68491 - 4.65039i) q^{27} +(11.6205 + 4.22951i) q^{28} +(-4.54393 + 2.62344i) q^{29} +2.99528i q^{31} +(0.904401 - 1.07782i) q^{32} +(-0.576227 + 0.209729i) q^{33} +(-0.492305 + 2.79200i) q^{34} +6.98347 q^{36} +(-2.22697 - 5.66044i) q^{37} -17.5056 q^{38} +(2.89416 - 0.510319i) q^{39} +(6.61853 - 2.40895i) q^{41} +(5.45200 - 6.49744i) q^{42} -10.0569i q^{43} +(-1.69668 - 1.42368i) q^{44} +(-17.8265 - 6.48830i) q^{46} +(0.583692 - 1.01098i) q^{47} +(-2.77342 - 4.80371i) q^{48} +(0.335840 + 1.90465i) q^{49} +(1.13523 + 0.655426i) q^{51} +(6.82301 + 8.13135i) q^{52} +(-3.60832 + 3.02774i) q^{53} +(4.54986 - 12.5006i) q^{54} +(15.5856 + 2.74816i) q^{56} +(-2.76833 + 7.60592i) q^{57} +(-9.95732 + 8.35518i) q^{58} +(3.09121 + 3.68396i) q^{59} +(-0.833377 - 2.28968i) q^{61} +(1.28853 + 7.30764i) q^{62} +(2.52261 + 4.36929i) q^{63} +(-3.09968 + 5.36880i) q^{64} +(-1.31561 + 0.759566i) q^{66} +(-9.96765 - 8.36385i) q^{67} +4.73468i q^{68} +(-5.63814 + 6.71927i) q^{69} +(-1.13206 + 6.42024i) q^{71} +(8.80150 - 1.55194i) q^{72} +10.5389 q^{73} +(-7.86823 - 12.8519i) q^{74} +(-28.7909 + 5.07661i) q^{76} +(0.277859 - 1.57582i) q^{77} +(6.84140 - 2.49007i) q^{78} +(-3.98028 + 4.74352i) q^{79} +(-0.832722 - 0.698737i) q^{81} +(15.1110 - 8.72436i) q^{82} +(4.78589 + 1.74192i) q^{83} +(7.08247 - 12.2672i) q^{84} +(-4.32636 - 24.5360i) q^{86} +(2.05556 + 5.64759i) q^{87} +(-2.45476 - 1.41726i) q^{88} +(5.26842 + 6.27865i) q^{89} +(-2.62283 + 7.20616i) q^{91} +(-31.2002 - 5.50144i) q^{92} +(3.37883 + 0.595779i) q^{93} +(0.989130 - 2.71761i) q^{94} +(-1.03595 - 1.23459i) q^{96} +(4.12441 + 2.38123i) q^{97} +(1.63871 + 4.50232i) q^{98} +(-0.156913 - 0.889896i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 12 q^{4} + 6 q^{9} - 30 q^{11} + 36 q^{14} + 18 q^{19} - 24 q^{21} - 96 q^{24} + 48 q^{26} + 18 q^{29} + 54 q^{34} + 24 q^{36} + 36 q^{39} + 72 q^{41} + 84 q^{44} - 18 q^{46} + 6 q^{49} - 18 q^{51}+ \cdots + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(76\) \(852\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.43972 − 0.430188i 1.72514 − 0.304189i 0.778778 − 0.627299i \(-0.215840\pi\)
0.946361 + 0.323110i \(0.104729\pi\)
\(3\) 0.198906 − 1.12805i 0.114838 − 0.651280i −0.871992 − 0.489520i \(-0.837172\pi\)
0.986830 − 0.161760i \(-0.0517170\pi\)
\(4\) 3.88777 − 1.41503i 1.94388 − 0.707516i
\(5\) 0 0
\(6\) − 2.83769i − 1.15848i
\(7\) 2.28969 + 1.92128i 0.865423 + 0.726176i 0.963129 − 0.269039i \(-0.0867060\pi\)
−0.0977060 + 0.995215i \(0.531150\pi\)
\(8\) 4.58542 − 2.64739i 1.62119 − 0.935994i
\(9\) 1.58614 + 0.577309i 0.528715 + 0.192436i
\(10\) 0 0
\(11\) −0.267671 − 0.463619i −0.0807058 − 0.139786i 0.822848 − 0.568262i \(-0.192384\pi\)
−0.903553 + 0.428476i \(0.859051\pi\)
\(12\) −0.822927 − 4.66705i −0.237559 − 1.34726i
\(13\) 0.877498 + 2.41091i 0.243374 + 0.668665i 0.999892 + 0.0146920i \(0.00467678\pi\)
−0.756518 + 0.653973i \(0.773101\pi\)
\(14\) 6.41272 + 3.70238i 1.71387 + 0.989504i
\(15\) 0 0
\(16\) 3.70957 − 3.11270i 0.927391 − 0.778174i
\(17\) −0.391407 + 1.07538i −0.0949301 + 0.260818i −0.978065 − 0.208301i \(-0.933206\pi\)
0.883135 + 0.469120i \(0.155429\pi\)
\(18\) 4.11809 + 0.726131i 0.970644 + 0.171151i
\(19\) −6.95889 − 1.22704i −1.59648 − 0.281502i −0.696540 − 0.717517i \(-0.745278\pi\)
−0.899939 + 0.436015i \(0.856389\pi\)
\(20\) 0 0
\(21\) 2.62274 − 2.20074i 0.572328 − 0.480240i
\(22\) −0.852484 − 1.01595i −0.181750 − 0.216601i
\(23\) −6.63166 − 3.82879i −1.38280 − 0.798358i −0.390306 − 0.920685i \(-0.627631\pi\)
−0.992490 + 0.122328i \(0.960964\pi\)
\(24\) −2.07432 − 5.69916i −0.423420 − 1.16334i
\(25\) 0 0
\(26\) 3.17799 + 5.50444i 0.623255 + 1.07951i
\(27\) 2.68491 − 4.65039i 0.516710 − 0.894968i
\(28\) 11.6205 + 4.22951i 2.19606 + 0.799302i
\(29\) −4.54393 + 2.62344i −0.843786 + 0.487160i −0.858549 − 0.512731i \(-0.828634\pi\)
0.0147634 + 0.999891i \(0.495300\pi\)
\(30\) 0 0
\(31\) 2.99528i 0.537969i 0.963145 + 0.268984i \(0.0866880\pi\)
−0.963145 + 0.268984i \(0.913312\pi\)
\(32\) 0.904401 − 1.07782i 0.159877 − 0.190534i
\(33\) −0.576227 + 0.209729i −0.100308 + 0.0365092i
\(34\) −0.492305 + 2.79200i −0.0844297 + 0.478824i
\(35\) 0 0
\(36\) 6.98347 1.16391
\(37\) −2.22697 − 5.66044i −0.366112 − 0.930571i
\(38\) −17.5056 −2.83978
\(39\) 2.89416 − 0.510319i 0.463437 − 0.0817164i
\(40\) 0 0
\(41\) 6.61853 − 2.40895i 1.03364 − 0.376215i 0.231175 − 0.972912i \(-0.425743\pi\)
0.802466 + 0.596698i \(0.203521\pi\)
\(42\) 5.45200 − 6.49744i 0.841262 − 1.00258i
\(43\) − 10.0569i − 1.53367i −0.641847 − 0.766833i \(-0.721831\pi\)
0.641847 − 0.766833i \(-0.278169\pi\)
\(44\) −1.69668 − 1.42368i −0.255784 − 0.214628i
\(45\) 0 0
\(46\) −17.8265 − 6.48830i −2.62837 − 0.956647i
\(47\) 0.583692 − 1.01098i 0.0851402 − 0.147467i −0.820311 − 0.571918i \(-0.806200\pi\)
0.905451 + 0.424451i \(0.139533\pi\)
\(48\) −2.77342 − 4.80371i −0.400309 − 0.693356i
\(49\) 0.335840 + 1.90465i 0.0479772 + 0.272092i
\(50\) 0 0
\(51\) 1.13523 + 0.655426i 0.158964 + 0.0917780i
\(52\) 6.82301 + 8.13135i 0.946182 + 1.12762i
\(53\) −3.60832 + 3.02774i −0.495641 + 0.415893i −0.856043 − 0.516905i \(-0.827084\pi\)
0.360401 + 0.932797i \(0.382640\pi\)
\(54\) 4.54986 − 12.5006i 0.619158 − 1.70112i
\(55\) 0 0
\(56\) 15.5856 + 2.74816i 2.08271 + 0.367238i
\(57\) −2.76833 + 7.60592i −0.366674 + 1.00743i
\(58\) −9.95732 + 8.35518i −1.30746 + 1.09709i
\(59\) 3.09121 + 3.68396i 0.402441 + 0.479611i 0.928763 − 0.370675i \(-0.120874\pi\)
−0.526321 + 0.850286i \(0.676429\pi\)
\(60\) 0 0
\(61\) −0.833377 − 2.28968i −0.106703 − 0.293164i 0.874838 − 0.484415i \(-0.160968\pi\)
−0.981541 + 0.191251i \(0.938745\pi\)
\(62\) 1.28853 + 7.30764i 0.163644 + 0.928071i
\(63\) 2.52261 + 4.36929i 0.317819 + 0.550479i
\(64\) −3.09968 + 5.36880i −0.387460 + 0.671100i
\(65\) 0 0
\(66\) −1.31561 + 0.759566i −0.161940 + 0.0934961i
\(67\) −9.96765 − 8.36385i −1.21774 − 1.02181i −0.998940 − 0.0460382i \(-0.985340\pi\)
−0.218803 − 0.975769i \(-0.570215\pi\)
\(68\) 4.73468i 0.574165i
\(69\) −5.63814 + 6.71927i −0.678752 + 0.808905i
\(70\) 0 0
\(71\) −1.13206 + 6.42024i −0.134351 + 0.761943i 0.840958 + 0.541100i \(0.181992\pi\)
−0.975309 + 0.220843i \(0.929119\pi\)
\(72\) 8.80150 − 1.55194i 1.03727 − 0.182898i
\(73\) 10.5389 1.23348 0.616742 − 0.787166i \(-0.288452\pi\)
0.616742 + 0.787166i \(0.288452\pi\)
\(74\) −7.86823 − 12.8519i −0.914663 − 1.49400i
\(75\) 0 0
\(76\) −28.7909 + 5.07661i −3.30254 + 0.582327i
\(77\) 0.277859 − 1.57582i 0.0316650 − 0.179581i
\(78\) 6.84140 − 2.49007i 0.774636 − 0.281944i
\(79\) −3.98028 + 4.74352i −0.447817 + 0.533687i −0.941974 − 0.335685i \(-0.891032\pi\)
0.494158 + 0.869372i \(0.335477\pi\)
\(80\) 0 0
\(81\) −0.832722 − 0.698737i −0.0925247 − 0.0776374i
\(82\) 15.1110 − 8.72436i 1.66873 − 0.963444i
\(83\) 4.78589 + 1.74192i 0.525319 + 0.191201i 0.591047 − 0.806637i \(-0.298715\pi\)
−0.0657278 + 0.997838i \(0.520937\pi\)
\(84\) 7.08247 − 12.2672i 0.772761 − 1.33846i
\(85\) 0 0
\(86\) −4.32636 − 24.5360i −0.466524 − 2.64579i
\(87\) 2.05556 + 5.64759i 0.220379 + 0.605486i
\(88\) −2.45476 − 1.41726i −0.261679 − 0.151080i
\(89\) 5.26842 + 6.27865i 0.558451 + 0.665536i 0.969218 − 0.246204i \(-0.0791835\pi\)
−0.410767 + 0.911740i \(0.634739\pi\)
\(90\) 0 0
\(91\) −2.62283 + 7.20616i −0.274947 + 0.755411i
\(92\) −31.2002 − 5.50144i −3.25284 − 0.573564i
\(93\) 3.37883 + 0.595779i 0.350368 + 0.0617794i
\(94\) 0.989130 − 2.71761i 0.102021 − 0.280300i
\(95\) 0 0
\(96\) −1.03595 − 1.23459i −0.105731 − 0.126005i
\(97\) 4.12441 + 2.38123i 0.418770 + 0.241777i 0.694551 − 0.719443i \(-0.255603\pi\)
−0.275781 + 0.961221i \(0.588936\pi\)
\(98\) 1.63871 + 4.50232i 0.165535 + 0.454803i
\(99\) −0.156913 − 0.889896i −0.0157703 − 0.0894379i
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.bb.e.176.16 96
5.2 odd 4 185.2.v.a.139.16 yes 96
5.3 odd 4 185.2.v.a.139.1 yes 96
5.4 even 2 inner 925.2.bb.e.176.1 96
37.4 even 18 inner 925.2.bb.e.226.16 96
185.4 even 18 inner 925.2.bb.e.226.1 96
185.78 odd 36 185.2.v.a.4.16 yes 96
185.152 odd 36 185.2.v.a.4.1 ✓ 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.v.a.4.1 ✓ 96 185.152 odd 36
185.2.v.a.4.16 yes 96 185.78 odd 36
185.2.v.a.139.1 yes 96 5.3 odd 4
185.2.v.a.139.16 yes 96 5.2 odd 4
925.2.bb.e.176.1 96 5.4 even 2 inner
925.2.bb.e.176.16 96 1.1 even 1 trivial
925.2.bb.e.226.1 96 185.4 even 18 inner
925.2.bb.e.226.16 96 37.4 even 18 inner