Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9025,2,Mod(1,9025)] 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("9025.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,0,10,8,0,-2,0,-10,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(72.0649878242\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1805)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.17557\) of defining polynomial
Character \(\chi\) \(=\) 9025.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.17557 q^{2} +1.17557 q^{3} -0.618034 q^{4} +1.38197 q^{6} -0.236068 q^{7} -3.07768 q^{8} -1.61803 q^{9} +0.854102 q^{11} -0.726543 q^{12} +0.726543 q^{13} -0.277515 q^{14} -2.38197 q^{16} -0.763932 q^{17} -1.90211 q^{18} -0.277515 q^{21} +1.00406 q^{22} +7.09017 q^{23} -3.61803 q^{24} +0.854102 q^{26} -5.42882 q^{27} +0.145898 q^{28} +8.78402 q^{29} +1.17557 q^{31} +3.35520 q^{32} +1.00406 q^{33} -0.898056 q^{34} +1.00000 q^{36} -8.78402 q^{37} +0.854102 q^{39} -1.62460 q^{41} -0.326238 q^{42} -2.61803 q^{43} -0.527864 q^{44} +8.33499 q^{46} -7.47214 q^{47} -2.80017 q^{48} -6.94427 q^{49} -0.898056 q^{51} -0.449028 q^{52} +1.00406 q^{53} -6.38197 q^{54} +0.726543 q^{56} +10.3262 q^{58} +11.3067 q^{59} -13.9443 q^{61} +1.38197 q^{62} +0.381966 q^{63} +8.70820 q^{64} +1.18034 q^{66} -11.5842 q^{67} +0.472136 q^{68} +8.33499 q^{69} -13.0373 q^{71} +4.97980 q^{72} +1.00000 q^{73} -10.3262 q^{74} -0.201626 q^{77} +1.00406 q^{78} -8.50651 q^{79} -1.52786 q^{81} -1.90983 q^{82} +13.2361 q^{83} +0.171513 q^{84} -3.07768 q^{86} +10.3262 q^{87} -2.62866 q^{88} -8.33499 q^{89} -0.171513 q^{91} -4.38197 q^{92} +1.38197 q^{93} -8.78402 q^{94} +3.94427 q^{96} +4.25325 q^{97} -8.16348 q^{98} -1.38197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 10 q^{6} + 8 q^{7} - 2 q^{9} - 10 q^{11} - 14 q^{16} - 12 q^{17} + 6 q^{23} - 10 q^{24} - 10 q^{26} + 14 q^{28} + 4 q^{36} - 10 q^{39} + 30 q^{42} - 6 q^{43} - 20 q^{44} - 12 q^{47} + 8 q^{49}+ \cdots - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.17557 0.831254 0.415627 − 0.909535i \(-0.363562\pi\)
0.415627 + 0.909535i \(0.363562\pi\)
\(3\) 1.17557 0.678716 0.339358 − 0.940657i \(-0.389790\pi\)
0.339358 + 0.940657i \(0.389790\pi\)
\(4\) −0.618034 −0.309017
\(5\) 0 0
\(6\) 1.38197 0.564185
\(7\) −0.236068 −0.0892253 −0.0446127 − 0.999004i \(-0.514205\pi\)
−0.0446127 + 0.999004i \(0.514205\pi\)
\(8\) −3.07768 −1.08813
\(9\) −1.61803 −0.539345
\(10\) 0 0
\(11\) 0.854102 0.257521 0.128761 − 0.991676i \(-0.458900\pi\)
0.128761 + 0.991676i \(0.458900\pi\)
\(12\) −0.726543 −0.209735
\(13\) 0.726543 0.201507 0.100753 − 0.994911i \(-0.467875\pi\)
0.100753 + 0.994911i \(0.467875\pi\)
\(14\) −0.277515 −0.0741689
\(15\) 0 0
\(16\) −2.38197 −0.595492
\(17\) −0.763932 −0.185281 −0.0926404 − 0.995700i \(-0.529531\pi\)
−0.0926404 + 0.995700i \(0.529531\pi\)
\(18\) −1.90211 −0.448332
\(19\) 0 0
\(20\) 0 0
\(21\) −0.277515 −0.0605586
\(22\) 1.00406 0.214066
\(23\) 7.09017 1.47840 0.739201 − 0.673485i \(-0.235203\pi\)
0.739201 + 0.673485i \(0.235203\pi\)
\(24\) −3.61803 −0.738528
\(25\) 0 0
\(26\) 0.854102 0.167503
\(27\) −5.42882 −1.04478
\(28\) 0.145898 0.0275721
\(29\) 8.78402 1.63115 0.815576 − 0.578650i \(-0.196420\pi\)
0.815576 + 0.578650i \(0.196420\pi\)
\(30\) 0 0
\(31\) 1.17557 0.211139 0.105569 − 0.994412i \(-0.466333\pi\)
0.105569 + 0.994412i \(0.466333\pi\)
\(32\) 3.35520 0.593121
\(33\) 1.00406 0.174784
\(34\) −0.898056 −0.154015
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −8.78402 −1.44408 −0.722042 − 0.691849i \(-0.756796\pi\)
−0.722042 + 0.691849i \(0.756796\pi\)
\(38\) 0 0
\(39\) 0.854102 0.136766
\(40\) 0 0
\(41\) −1.62460 −0.253720 −0.126860 − 0.991921i \(-0.540490\pi\)
−0.126860 + 0.991921i \(0.540490\pi\)
\(42\) −0.326238 −0.0503396
\(43\) −2.61803 −0.399246 −0.199623 − 0.979873i \(-0.563972\pi\)
−0.199623 + 0.979873i \(0.563972\pi\)
\(44\) −0.527864 −0.0795785
\(45\) 0 0
\(46\) 8.33499 1.22893
\(47\) −7.47214 −1.08992 −0.544962 − 0.838461i \(-0.683456\pi\)
−0.544962 + 0.838461i \(0.683456\pi\)
\(48\) −2.80017 −0.404170
\(49\) −6.94427 −0.992039
\(50\) 0 0
\(51\) −0.898056 −0.125753
\(52\) −0.449028 −0.0622690
\(53\) 1.00406 0.137918 0.0689589 − 0.997620i \(-0.478032\pi\)
0.0689589 + 0.997620i \(0.478032\pi\)
\(54\) −6.38197 −0.868476
\(55\) 0 0
\(56\) 0.726543 0.0970883
\(57\) 0 0
\(58\) 10.3262 1.35590
\(59\) 11.3067 1.47200 0.736002 − 0.676979i \(-0.236711\pi\)
0.736002 + 0.676979i \(0.236711\pi\)
\(60\) 0 0
\(61\) −13.9443 −1.78538 −0.892691 − 0.450670i \(-0.851185\pi\)
−0.892691 + 0.450670i \(0.851185\pi\)
\(62\) 1.38197 0.175510
\(63\) 0.381966 0.0481232
\(64\) 8.70820 1.08853
\(65\) 0 0
\(66\) 1.18034 0.145290
\(67\) −11.5842 −1.41523 −0.707617 − 0.706596i \(-0.750230\pi\)
−0.707617 + 0.706596i \(0.750230\pi\)
\(68\) 0.472136 0.0572549
\(69\) 8.33499 1.00342
\(70\) 0 0
\(71\) −13.0373 −1.54724 −0.773620 − 0.633650i \(-0.781556\pi\)
−0.773620 + 0.633650i \(0.781556\pi\)
\(72\) 4.97980 0.586875
\(73\) 1.00000 0.117041 0.0585206 − 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) −10.3262 −1.20040
\(75\) 0 0
\(76\) 0 0
\(77\) −0.201626 −0.0229774
\(78\) 1.00406 0.113687
\(79\) −8.50651 −0.957057 −0.478528 − 0.878072i \(-0.658830\pi\)
−0.478528 + 0.878072i \(0.658830\pi\)
\(80\) 0 0
\(81\) −1.52786 −0.169763
\(82\) −1.90983 −0.210905
\(83\) 13.2361 1.45285 0.726424 − 0.687247i \(-0.241181\pi\)
0.726424 + 0.687247i \(0.241181\pi\)
\(84\) 0.171513 0.0187136
\(85\) 0 0
\(86\) −3.07768 −0.331875
\(87\) 10.3262 1.10709
\(88\) −2.62866 −0.280216
\(89\) −8.33499 −0.883508 −0.441754 − 0.897136i \(-0.645644\pi\)
−0.441754 + 0.897136i \(0.645644\pi\)
\(90\) 0 0
\(91\) −0.171513 −0.0179795
\(92\) −4.38197 −0.456852
\(93\) 1.38197 0.143303
\(94\) −8.78402 −0.906003
\(95\) 0 0
\(96\) 3.94427 0.402561
\(97\) 4.25325 0.431853 0.215926 − 0.976410i \(-0.430723\pi\)
0.215926 + 0.976410i \(0.430723\pi\)
\(98\) −8.16348 −0.824636
\(99\) −1.38197 −0.138893
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 9025.2.a.bl.1.3 4
5.4 even 2 1805.2.a.l.1.2 ✓ 4
19.18 odd 2 inner 9025.2.a.bl.1.2 4
95.94 odd 2 1805.2.a.l.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.a.l.1.2 ✓ 4 5.4 even 2
1805.2.a.l.1.3 yes 4 95.94 odd 2
9025.2.a.bl.1.2 4 19.18 odd 2 inner
9025.2.a.bl.1.3 4 1.1 even 1 trivial