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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,-2] 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: \(\Z[a_1, a_2, a_3]\)
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.a.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +(-1.00000 - 1.00000i) q^{3} -1.00000 q^{4} +(-1.00000 - 1.00000i) q^{6} +(3.00000 + 3.00000i) q^{7} -3.00000 q^{8} -1.00000i q^{9} -2.00000i q^{11} +(1.00000 + 1.00000i) q^{12} -2.00000 q^{13} +(3.00000 + 3.00000i) q^{14} -1.00000 q^{16} -4.00000i q^{17} -1.00000i q^{18} +(3.00000 - 3.00000i) q^{19} -6.00000i q^{21} -2.00000i q^{22} +8.00000 q^{23} +(3.00000 + 3.00000i) q^{24} -2.00000 q^{26} +(-4.00000 + 4.00000i) q^{27} +(-3.00000 - 3.00000i) q^{28} +(-7.00000 - 7.00000i) q^{29} +(3.00000 - 3.00000i) q^{31} +5.00000 q^{32} +(-2.00000 + 2.00000i) q^{33} -4.00000i q^{34} +1.00000i q^{36} +(1.00000 - 6.00000i) q^{37} +(3.00000 - 3.00000i) q^{38} +(2.00000 + 2.00000i) q^{39} -6.00000i q^{42} -12.0000 q^{43} +2.00000i q^{44} +8.00000 q^{46} +(-5.00000 - 5.00000i) q^{47} +(1.00000 + 1.00000i) q^{48} +11.0000i q^{49} +(-4.00000 + 4.00000i) q^{51} +2.00000 q^{52} +(3.00000 - 3.00000i) q^{53} +(-4.00000 + 4.00000i) q^{54} +(-9.00000 - 9.00000i) q^{56} -6.00000 q^{57} +(-7.00000 - 7.00000i) q^{58} +(7.00000 - 7.00000i) q^{59} +(1.00000 - 1.00000i) q^{61} +(3.00000 - 3.00000i) q^{62} +(3.00000 - 3.00000i) q^{63} +7.00000 q^{64} +(-2.00000 + 2.00000i) q^{66} +(-3.00000 + 3.00000i) q^{67} +4.00000i q^{68} +(-8.00000 - 8.00000i) q^{69} +8.00000 q^{71} +3.00000i q^{72} +(-1.00000 - 1.00000i) q^{73} +(1.00000 - 6.00000i) q^{74} +(-3.00000 + 3.00000i) q^{76} +(6.00000 - 6.00000i) q^{77} +(2.00000 + 2.00000i) q^{78} +(3.00000 - 3.00000i) q^{79} +5.00000 q^{81} +(5.00000 - 5.00000i) q^{83} +6.00000i q^{84} -12.0000 q^{86} +14.0000i q^{87} +6.00000i q^{88} +(5.00000 + 5.00000i) q^{89} +(-6.00000 - 6.00000i) q^{91} -8.00000 q^{92} -6.00000 q^{93} +(-5.00000 - 5.00000i) q^{94} +(-5.00000 - 5.00000i) q^{96} +8.00000i q^{97} +11.0000i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{6} + 6 q^{7} - 6 q^{8} + 2 q^{12} - 4 q^{13} + 6 q^{14} - 2 q^{16} + 6 q^{19} + 16 q^{23} + 6 q^{24} - 4 q^{26} - 8 q^{27} - 6 q^{28} - 14 q^{29} + 6 q^{31} + 10 q^{32}+ \cdots - 4 q^{99}+O(q^{100}) \) 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 \(0.384973\pi\)
\(3\) −1.00000 − 1.00000i −0.577350 − 0.577350i 0.356822 − 0.934172i \(-0.383860\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 − 1.00000i −0.408248 − 0.408248i
\(7\) 3.00000 + 3.00000i 1.13389 + 1.13389i 0.989524 + 0.144370i \(0.0461154\pi\)
0.144370 + 0.989524i \(0.453885\pi\)
\(8\) −3.00000 −1.06066
\(9\) − 1.00000i − 0.333333i
\(10\) 0 0
\(11\) − 2.00000i − 0.603023i −0.953463 − 0.301511i \(-0.902509\pi\)
0.953463 − 0.301511i \(-0.0974911\pi\)
\(12\) 1.00000 + 1.00000i 0.288675 + 0.288675i
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 3.00000 + 3.00000i 0.801784 + 0.801784i
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) − 4.00000i − 0.970143i −0.874475 − 0.485071i \(-0.838794\pi\)
0.874475 − 0.485071i \(-0.161206\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 3.00000 − 3.00000i 0.688247 − 0.688247i −0.273597 − 0.961844i \(-0.588214\pi\)
0.961844 + 0.273597i \(0.0882135\pi\)
\(20\) 0 0
\(21\) − 6.00000i − 1.30931i
\(22\) − 2.00000i − 0.426401i
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 3.00000 + 3.00000i 0.612372 + 0.612372i
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) −4.00000 + 4.00000i −0.769800 + 0.769800i
\(28\) −3.00000 − 3.00000i −0.566947 − 0.566947i
\(29\) −7.00000 − 7.00000i −1.29987 − 1.29987i −0.928477 − 0.371391i \(-0.878881\pi\)
−0.371391 − 0.928477i \(-0.621119\pi\)
\(30\) 0 0
\(31\) 3.00000 − 3.00000i 0.538816 − 0.538816i −0.384365 − 0.923181i \(-0.625580\pi\)
0.923181 + 0.384365i \(0.125580\pi\)
\(32\) 5.00000 0.883883
\(33\) −2.00000 + 2.00000i −0.348155 + 0.348155i
\(34\) − 4.00000i − 0.685994i
\(35\) 0 0
\(36\) 1.00000i 0.166667i
\(37\) 1.00000 − 6.00000i 0.164399 − 0.986394i
\(38\) 3.00000 − 3.00000i 0.486664 − 0.486664i
\(39\) 2.00000 + 2.00000i 0.320256 + 0.320256i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) − 6.00000i − 0.925820i
\(43\) −12.0000 −1.82998 −0.914991 − 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 2.00000i 0.301511i
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) −5.00000 − 5.00000i −0.729325 − 0.729325i 0.241160 − 0.970485i \(-0.422472\pi\)
−0.970485 + 0.241160i \(0.922472\pi\)
\(48\) 1.00000 + 1.00000i 0.144338 + 0.144338i
\(49\) 11.0000i 1.57143i
\(50\) 0 0
\(51\) −4.00000 + 4.00000i −0.560112 + 0.560112i
\(52\) 2.00000 0.277350
\(53\) 3.00000 − 3.00000i 0.412082 − 0.412082i −0.470381 − 0.882463i \(-0.655884\pi\)
0.882463 + 0.470381i \(0.155884\pi\)
\(54\) −4.00000 + 4.00000i −0.544331 + 0.544331i
\(55\) 0 0
\(56\) −9.00000 − 9.00000i −1.20268 − 1.20268i
\(57\) −6.00000 −0.794719
\(58\) −7.00000 − 7.00000i −0.919145 − 0.919145i
\(59\) 7.00000 − 7.00000i 0.911322 − 0.911322i −0.0850540 − 0.996376i \(-0.527106\pi\)
0.996376 + 0.0850540i \(0.0271063\pi\)
\(60\) 0 0
\(61\) 1.00000 − 1.00000i 0.128037 − 0.128037i −0.640184 − 0.768221i \(-0.721142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 3.00000 − 3.00000i 0.381000 − 0.381000i
\(63\) 3.00000 − 3.00000i 0.377964 − 0.377964i
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) −2.00000 + 2.00000i −0.246183 + 0.246183i
\(67\) −3.00000 + 3.00000i −0.366508 + 0.366508i −0.866202 − 0.499694i \(-0.833446\pi\)
0.499694 + 0.866202i \(0.333446\pi\)
\(68\) 4.00000i 0.485071i
\(69\) −8.00000 − 8.00000i −0.963087 − 0.963087i
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 3.00000i 0.353553i
\(73\) −1.00000 − 1.00000i −0.117041 − 0.117041i 0.646160 − 0.763202i \(-0.276374\pi\)
−0.763202 + 0.646160i \(0.776374\pi\)
\(74\) 1.00000 − 6.00000i 0.116248 − 0.697486i
\(75\) 0 0
\(76\) −3.00000 + 3.00000i −0.344124 + 0.344124i
\(77\) 6.00000 − 6.00000i 0.683763 − 0.683763i
\(78\) 2.00000 + 2.00000i 0.226455 + 0.226455i
\(79\) 3.00000 − 3.00000i 0.337526 − 0.337526i −0.517909 − 0.855436i \(-0.673290\pi\)
0.855436 + 0.517909i \(0.173290\pi\)
\(80\) 0 0
\(81\) 5.00000 0.555556
\(82\) 0 0
\(83\) 5.00000 − 5.00000i 0.548821 − 0.548821i −0.377279 − 0.926100i \(-0.623140\pi\)
0.926100 + 0.377279i \(0.123140\pi\)
\(84\) 6.00000i 0.654654i
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) 14.0000i 1.50096i
\(88\) 6.00000i 0.639602i
\(89\) 5.00000 + 5.00000i 0.529999 + 0.529999i 0.920572 − 0.390573i \(-0.127723\pi\)
−0.390573 + 0.920572i \(0.627723\pi\)
\(90\) 0 0
\(91\) −6.00000 − 6.00000i −0.628971 − 0.628971i
\(92\) −8.00000 −0.834058
\(93\) −6.00000 −0.622171
\(94\) −5.00000 − 5.00000i −0.515711 − 0.515711i
\(95\) 0 0
\(96\) −5.00000 − 5.00000i −0.510310 − 0.510310i
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 11.0000i 1.11117i
\(99\) −2.00000 −0.201008
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.a.857.1 2
5.2 odd 4 185.2.f.a.43.1 ✓ 2
5.3 odd 4 925.2.f.b.43.1 2
5.4 even 2 185.2.k.b.117.1 yes 2
37.31 odd 4 925.2.f.b.882.1 2
185.68 even 4 inner 925.2.k.a.68.1 2
185.142 even 4 185.2.k.b.68.1 yes 2
185.179 odd 4 185.2.f.a.142.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.f.a.43.1 ✓ 2 5.2 odd 4
185.2.f.a.142.1 yes 2 185.179 odd 4
185.2.k.b.68.1 yes 2 185.142 even 4
185.2.k.b.117.1 yes 2 5.4 even 2
925.2.f.b.43.1 2 5.3 odd 4
925.2.f.b.882.1 2 37.31 odd 4
925.2.k.a.68.1 2 185.68 even 4 inner
925.2.k.a.857.1 2 1.1 even 1 trivial