↑

Properties

Label 925.2.k.b.857.1
Level $925$
Weight $2$
Character 925.857
Analytic conductor $7.386$
Analytic rank $0$
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(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 857.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 925.857
Dual form 925.2.k.b.68.1

$q$-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 \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 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 0.985599 + 0.169102i \(0.0540867\pi\)
0.169102 + 0.985599i \(0.445913\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.00000 + 2.00000i 0.816497 + 0.816497i
\(7\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) −3.00000 −1.06066
\(9\) 5.00000i 1.66667i
\(10\) 0 0
\(11\) 4.00000i 1.20605i 0.797724 + 0.603023i \(0.206037\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) −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 \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 5.00000i 1.17851i
\(19\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(20\) 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 0.608137 − 0.793832i \(-0.291917\pi\)
−0.793832 + 0.608137i \(0.791917\pi\)
\(30\) 0 0
\(31\) −6.00000 + 6.00000i −1.07763 + 1.07763i −0.0809104 + 0.996721i \(0.525783\pi\)
−0.996721 + 0.0809104i \(0.974217\pi\)
\(32\) 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 −0.982928 − 0.183992i \(-0.941098\pi\)
−0.183992 − 0.982928i \(-0.558902\pi\)
\(48\) −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 0.274721 − 0.961524i \(-0.411414\pi\)
0.961524 − 0.274721i \(-0.0885855\pi\)
\(54\) −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 −0.397044 − 0.917800i \(-0.629964\pi\)
0.917800 + 0.397044i \(0.129964\pi\)
\(60\) 0 0
\(61\) 1.00000 − 1.00000i 0.128037 − 0.128037i −0.640184 − 0.768221i \(-0.721142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) −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 −0.238200 − 0.971216i \(-0.576557\pi\)
0.971216 + 0.238200i \(0.0765572\pi\)
\(68\) − 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 0.936329 + 0.351123i \(0.114200\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 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 −0.958876 − 0.283824i \(-0.908397\pi\)
0.283824 + 0.958876i \(0.408397\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) 2.00000 − 2.00000i 0.219529 − 0.219529i −0.588771 − 0.808300i \(-0.700388\pi\)
0.808300 + 0.588771i \(0.200388\pi\)
\(84\) 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 0.652118 − 0.758118i \(-0.273881\pi\)
−0.758118 + 0.652118i \(0.773881\pi\)
\(90\) 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 \(-0.934908\pi\)
0.979164 − 0.203069i \(-0.0650917\pi\)
\(98\) − 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.857.1 2
5.2 odd 4 185.2.f.b.43.1 ✓ 2
5.3 odd 4 925.2.f.a.43.1 2
5.4 even 2 185.2.k.a.117.1 yes 2
37.31 odd 4 925.2.f.a.882.1 2
185.68 even 4 inner 925.2.k.b.68.1 2
185.142 even 4 185.2.k.a.68.1 yes 2
185.179 odd 4 185.2.f.b.142.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.f.b.43.1 ✓ 2 5.2 odd 4
185.2.f.b.142.1 yes 2 185.179 odd 4
185.2.k.a.68.1 yes 2 185.142 even 4
185.2.k.a.117.1 yes 2 5.4 even 2
925.2.f.a.43.1 2 5.3 odd 4
925.2.f.a.882.1 2 37.31 odd 4
925.2.k.b.68.1 2 185.68 even 4 inner
925.2.k.b.857.1 2 1.1 even 1 trivial