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Properties

Label 925.2.k.b.68.1
Level $925$
Weight $2$
Character 925.68
Analytic conductor $7.386$
Analytic rank $0$
Dimension $2$
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(68,925)] 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("925.68"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 185)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 68.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 925.68
Dual form 925.2.k.b.857.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +(2.00000 - 2.00000i) q^{3} -1.00000 q^{4} +(2.00000 - 2.00000i) q^{6} -3.00000 q^{8} -5.00000i q^{9} -4.00000i q^{11} +(-2.00000 + 2.00000i) q^{12} +4.00000 q^{13} -1.00000 q^{16} -2.00000i q^{17} -5.00000i q^{18} -4.00000i q^{22} -4.00000 q^{23} +(-6.00000 + 6.00000i) q^{24} +4.00000 q^{26} +(-4.00000 - 4.00000i) q^{27} +(-1.00000 + 1.00000i) q^{29} +(-6.00000 - 6.00000i) q^{31} +5.00000 q^{32} +(-8.00000 - 8.00000i) q^{33} -2.00000i q^{34} +5.00000i q^{36} +(1.00000 - 6.00000i) q^{37} +(8.00000 - 8.00000i) q^{39} +12.0000 q^{43} +4.00000i q^{44} -4.00000 q^{46} +(-8.00000 + 8.00000i) q^{47} +(-2.00000 + 2.00000i) q^{48} +7.00000i q^{49} +(-4.00000 - 4.00000i) q^{51} -4.00000 q^{52} +(9.00000 + 9.00000i) q^{53} +(-4.00000 - 4.00000i) q^{54} +(-1.00000 + 1.00000i) q^{58} +(4.00000 + 4.00000i) q^{59} +(1.00000 + 1.00000i) q^{61} +(-6.00000 - 6.00000i) q^{62} +7.00000 q^{64} +(-8.00000 - 8.00000i) q^{66} +(6.00000 + 6.00000i) q^{67} +2.00000i q^{68} +(-8.00000 + 8.00000i) q^{69} -4.00000 q^{71} +15.0000i q^{72} +(11.0000 - 11.0000i) q^{73} +(1.00000 - 6.00000i) q^{74} +(8.00000 - 8.00000i) q^{78} +(-6.00000 - 6.00000i) q^{79} -1.00000 q^{81} +(2.00000 + 2.00000i) q^{83} +12.0000 q^{86} +4.00000i q^{87} +12.0000i q^{88} +(-1.00000 + 1.00000i) q^{89} +4.00000 q^{92} -24.0000 q^{93} +(-8.00000 + 8.00000i) q^{94} +(10.0000 - 10.0000i) q^{96} +4.00000i q^{97} +7.00000i q^{98} -20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{3} - 2 q^{4} + 4 q^{6} - 6 q^{8} - 4 q^{12} + 8 q^{13} - 2 q^{16} - 8 q^{23} - 12 q^{24} + 8 q^{26} - 8 q^{27} - 2 q^{29} - 12 q^{31} + 10 q^{32} - 16 q^{33} + 2 q^{37} + 16 q^{39}+ \cdots - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(76\) \(852\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 − 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 2.00000 − 2.00000i 1.15470 − 1.15470i 0.169102 − 0.985599i \(-0.445913\pi\)
0.985599 − 0.169102i \(-0.0540867\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.00000 − 2.00000i 0.816497 − 0.816497i
\(7\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) −3.00000 −1.06066
\(9\) − 5.00000i − 1.66667i
\(10\) 0 0
\(11\) − 4.00000i − 1.20605i −0.797724 − 0.603023i \(-0.793963\pi\)
0.797724 − 0.603023i \(-0.206037\pi\)
\(12\) −2.00000 + 2.00000i −0.577350 + 0.577350i
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 − 0.242536i \(-0.922021\pi\)
0.970143 − 0.242536i \(-0.0779791\pi\)
\(18\) − 5.00000i − 1.17851i
\(19\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 4.00000i − 0.852803i
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −6.00000 + 6.00000i −1.22474 + 1.22474i
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) −4.00000 − 4.00000i −0.769800 − 0.769800i
\(28\) 0 0
\(29\) −1.00000 + 1.00000i −0.185695 + 0.185695i −0.793832 − 0.608137i \(-0.791917\pi\)
0.608137 + 0.793832i \(0.291917\pi\)
\(30\) 0 0
\(31\) −6.00000 − 6.00000i −1.07763 − 1.07763i −0.996721 − 0.0809104i \(-0.974217\pi\)
−0.0809104 − 0.996721i \(-0.525783\pi\)
\(32\) 5.00000 0.883883
\(33\) −8.00000 − 8.00000i −1.39262 − 1.39262i
\(34\) − 2.00000i − 0.342997i
\(35\) 0 0
\(36\) 5.00000i 0.833333i
\(37\) 1.00000 − 6.00000i 0.164399 − 0.986394i
\(38\) 0 0
\(39\) 8.00000 − 8.00000i 1.28103 − 1.28103i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 − 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 4.00000i 0.603023i
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −8.00000 + 8.00000i −1.16692 + 1.16692i −0.183992 + 0.982928i \(0.558902\pi\)
−0.982928 + 0.183992i \(0.941098\pi\)
\(48\) −2.00000 + 2.00000i −0.288675 + 0.288675i
\(49\) 7.00000i 1.00000i
\(50\) 0 0
\(51\) −4.00000 − 4.00000i −0.560112 − 0.560112i
\(52\) −4.00000 −0.554700
\(53\) 9.00000 + 9.00000i 1.23625 + 1.23625i 0.961524 + 0.274721i \(0.0885855\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) −4.00000 − 4.00000i −0.544331 − 0.544331i
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −1.00000 + 1.00000i −0.131306 + 0.131306i
\(59\) 4.00000 + 4.00000i 0.520756 + 0.520756i 0.917800 − 0.397044i \(-0.129964\pi\)
−0.397044 + 0.917800i \(0.629964\pi\)
\(60\) 0 0
\(61\) 1.00000 + 1.00000i 0.128037 + 0.128037i 0.768221 − 0.640184i \(-0.221142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −6.00000 − 6.00000i −0.762001 − 0.762001i
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) −8.00000 − 8.00000i −0.984732 − 0.984732i
\(67\) 6.00000 + 6.00000i 0.733017 + 0.733017i 0.971216 − 0.238200i \(-0.0765572\pi\)
−0.238200 + 0.971216i \(0.576557\pi\)
\(68\) 2.00000i 0.242536i
\(69\) −8.00000 + 8.00000i −0.963087 + 0.963087i
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 15.0000i 1.76777i
\(73\) 11.0000 − 11.0000i 1.28745 − 1.28745i 0.351123 − 0.936329i \(-0.385800\pi\)
0.936329 − 0.351123i \(-0.114200\pi\)
\(74\) 1.00000 − 6.00000i 0.116248 − 0.697486i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 8.00000 − 8.00000i 0.905822 − 0.905822i
\(79\) −6.00000 − 6.00000i −0.675053 − 0.675053i 0.283824 − 0.958876i \(-0.408397\pi\)
−0.958876 + 0.283824i \(0.908397\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) 2.00000 + 2.00000i 0.219529 + 0.219529i 0.808300 − 0.588771i \(-0.200388\pi\)
−0.588771 + 0.808300i \(0.700388\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 4.00000i 0.428845i
\(88\) 12.0000i 1.27920i
\(89\) −1.00000 + 1.00000i −0.106000 + 0.106000i −0.758118 − 0.652118i \(-0.773881\pi\)
0.652118 + 0.758118i \(0.273881\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) −24.0000 −2.48868
\(94\) −8.00000 + 8.00000i −0.825137 + 0.825137i
\(95\) 0 0
\(96\) 10.0000 − 10.0000i 1.02062 − 1.02062i
\(97\) 4.00000i 0.406138i 0.979164 + 0.203069i \(0.0650917\pi\)
−0.979164 + 0.203069i \(0.934908\pi\)
\(98\) 7.00000i 0.707107i
\(99\) −20.0000 −2.01008
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.k.b.68.1 2
5.2 odd 4 925.2.f.a.882.1 2
5.3 odd 4 185.2.f.b.142.1 yes 2
5.4 even 2 185.2.k.a.68.1 yes 2
37.6 odd 4 925.2.f.a.43.1 2
185.43 even 4 185.2.k.a.117.1 yes 2
185.117 even 4 inner 925.2.k.b.857.1 2
185.154 odd 4 185.2.f.b.43.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.f.b.43.1 ✓ 2 185.154 odd 4
185.2.f.b.142.1 yes 2 5.3 odd 4
185.2.k.a.68.1 yes 2 5.4 even 2
185.2.k.a.117.1 yes 2 185.43 even 4
925.2.f.a.43.1 2 37.6 odd 4
925.2.f.a.882.1 2 5.2 odd 4
925.2.k.b.68.1 2 1.1 even 1 trivial
925.2.k.b.857.1 2 185.117 even 4 inner