↑

Properties

Label 925.2.b.b.149.2
Level $925$
Weight $2$
Character 925.149
Analytic conductor $7.386$
Analytic rank $1$
Dimension $2$
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(149,925)] 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("925.149"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-4,0,12,0,0,-12,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 37)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 925.149
Dual form 925.2.b.b.149.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000i q^{2} -3.00000i q^{3} -2.00000 q^{4} +6.00000 q^{6} +1.00000i q^{7} -6.00000 q^{9} -5.00000 q^{11} +6.00000i q^{12} -2.00000i q^{13} -2.00000 q^{14} -4.00000 q^{16} -12.0000i q^{18} +3.00000 q^{21} -10.0000i q^{22} +2.00000i q^{23} +4.00000 q^{26} +9.00000i q^{27} -2.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} -8.00000i q^{32} +15.0000i q^{33} +12.0000 q^{36} +1.00000i q^{37} -6.00000 q^{39} -9.00000 q^{41} +6.00000i q^{42} +2.00000i q^{43} +10.0000 q^{44} -4.00000 q^{46} +9.00000i q^{47} +12.0000i q^{48} +6.00000 q^{49} +4.00000i q^{52} +1.00000i q^{53} -18.0000 q^{54} -12.0000i q^{58} -8.00000 q^{59} -8.00000 q^{61} -8.00000i q^{62} -6.00000i q^{63} +8.00000 q^{64} -30.0000 q^{66} -8.00000i q^{67} +6.00000 q^{69} +9.00000 q^{71} -1.00000i q^{73} -2.00000 q^{74} -5.00000i q^{77} -12.0000i q^{78} -4.00000 q^{79} +9.00000 q^{81} -18.0000i q^{82} -15.0000i q^{83} -6.00000 q^{84} -4.00000 q^{86} +18.0000i q^{87} -4.00000 q^{89} +2.00000 q^{91} -4.00000i q^{92} +12.0000i q^{93} -18.0000 q^{94} -24.0000 q^{96} -4.00000i q^{97} +12.0000i q^{98} +30.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} + 12 q^{6} - 12 q^{9} - 10 q^{11} - 4 q^{14} - 8 q^{16} + 6 q^{21} + 8 q^{26} - 12 q^{29} - 8 q^{31} + 24 q^{36} - 12 q^{39} - 18 q^{41} + 20 q^{44} - 8 q^{46} + 12 q^{49} - 36 q^{54} - 16 q^{59}+ \cdots + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(76\) \(852\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) − 3.00000i − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) 6.00000 2.44949
\(7\) 1.00000i 0.377964i 0.981981 + 0.188982i \(0.0605189\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) 0 0
\(9\) −6.00000 −2.00000
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 − 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 6.00000i 1.73205i
\(13\) − 2.00000i − 0.554700i −0.960769 − 0.277350i \(-0.910544\pi\)
0.960769 − 0.277350i \(-0.0894562\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) − 12.0000i − 2.82843i
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) − 10.0000i − 2.13201i
\(23\) 2.00000i 0.417029i 0.978019 + 0.208514i \(0.0668628\pi\)
−0.978019 + 0.208514i \(0.933137\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) 9.00000i 1.73205i
\(28\) − 2.00000i − 0.377964i
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) − 8.00000i − 1.41421i
\(33\) 15.0000i 2.61116i
\(34\) 0 0
\(35\) 0 0
\(36\) 12.0000 2.00000
\(37\) 1.00000i 0.164399i
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 − 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 6.00000i 0.925820i
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) 10.0000 1.50756
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 9.00000i 1.31278i 0.754420 + 0.656392i \(0.227918\pi\)
−0.754420 + 0.656392i \(0.772082\pi\)
\(48\) 12.0000i 1.73205i
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 4.00000i 0.554700i
\(53\) 1.00000i 0.137361i 0.997639 + 0.0686803i \(0.0218788\pi\)
−0.997639 + 0.0686803i \(0.978121\pi\)
\(54\) −18.0000 −2.44949
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) − 12.0000i − 1.57568i
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) − 8.00000i − 1.01600i
\(63\) − 6.00000i − 0.755929i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) −30.0000 −3.69274
\(67\) − 8.00000i − 0.977356i −0.872464 − 0.488678i \(-0.837479\pi\)
0.872464 − 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 9.00000 1.06810 0.534052 − 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) − 1.00000i − 0.117041i −0.998286 − 0.0585206i \(-0.981362\pi\)
0.998286 − 0.0585206i \(-0.0186383\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 0 0
\(77\) − 5.00000i − 0.569803i
\(78\) − 12.0000i − 1.35873i
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) − 18.0000i − 1.98777i
\(83\) − 15.0000i − 1.64646i −0.567705 − 0.823232i \(-0.692169\pi\)
0.567705 − 0.823232i \(-0.307831\pi\)
\(84\) −6.00000 −0.654654
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 18.0000i 1.92980i
\(88\) 0 0
\(89\) −4.00000 −0.423999 −0.212000 − 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) − 4.00000i − 0.417029i
\(93\) 12.0000i 1.24434i
\(94\) −18.0000 −1.85656
\(95\) 0 0
\(96\) −24.0000 −2.44949
\(97\) − 4.00000i − 0.406138i −0.979164 − 0.203069i \(-0.934908\pi\)
0.979164 − 0.203069i \(-0.0650917\pi\)
\(98\) 12.0000i 1.21218i
\(99\) 30.0000 3.01511
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.b.b.149.2 2
5.2 odd 4 37.2.a.a.1.1 ✓ 1
5.3 odd 4 925.2.a.e.1.1 1
5.4 even 2 inner 925.2.b.b.149.1 2
15.2 even 4 333.2.a.d.1.1 1
15.8 even 4 8325.2.a.e.1.1 1
20.7 even 4 592.2.a.e.1.1 1
35.27 even 4 1813.2.a.a.1.1 1
40.27 even 4 2368.2.a.b.1.1 1
40.37 odd 4 2368.2.a.q.1.1 1
55.32 even 4 4477.2.a.b.1.1 1
60.47 odd 4 5328.2.a.r.1.1 1
65.12 odd 4 6253.2.a.c.1.1 1
185.117 even 4 1369.2.b.c.1368.2 2
185.142 even 4 1369.2.b.c.1368.1 2
185.147 odd 4 1369.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
37.2.a.a.1.1 ✓ 1 5.2 odd 4
333.2.a.d.1.1 1 15.2 even 4
592.2.a.e.1.1 1 20.7 even 4
925.2.a.e.1.1 1 5.3 odd 4
925.2.b.b.149.1 2 5.4 even 2 inner
925.2.b.b.149.2 2 1.1 even 1 trivial
1369.2.a.e.1.1 1 185.147 odd 4
1369.2.b.c.1368.1 2 185.142 even 4
1369.2.b.c.1368.2 2 185.117 even 4
1813.2.a.a.1.1 1 35.27 even 4
2368.2.a.b.1.1 1 40.27 even 4
2368.2.a.q.1.1 1 40.37 odd 4
4477.2.a.b.1.1 1 55.32 even 4
5328.2.a.r.1.1 1 60.47 odd 4
6253.2.a.c.1.1 1 65.12 odd 4
8325.2.a.e.1.1 1 15.8 even 4