Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [25,22,Mod(1,25)] 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("25.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(25, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,1312] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(69.8693360718\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 54559x - 2833496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(256.626\) of defining polynomial
Character \(\chi\) \(=\) 25.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1592.71 q^{2} +9397.92 q^{3} +439587. q^{4} -1.49682e7 q^{6} +1.90393e8 q^{7} +2.64003e9 q^{8} -1.03720e10 q^{9} +5.63465e9 q^{11} +4.13121e9 q^{12} -9.01426e10 q^{13} -3.03241e11 q^{14} -5.12669e12 q^{16} +9.42995e12 q^{17} +1.65197e13 q^{18} -2.01743e13 q^{19} +1.78930e12 q^{21} -8.97438e12 q^{22} -1.81691e14 q^{23} +2.48108e13 q^{24} +1.43571e14 q^{26} -1.95781e14 q^{27} +8.36942e13 q^{28} +1.06339e15 q^{29} +5.13441e15 q^{31} +2.62882e15 q^{32} +5.29540e13 q^{33} -1.50192e16 q^{34} -4.55941e15 q^{36} +3.84897e16 q^{37} +3.21318e16 q^{38} -8.47153e14 q^{39} +1.47297e17 q^{41} -2.84984e15 q^{42} -8.58814e16 q^{43} +2.47692e15 q^{44} +2.89382e17 q^{46} +4.33330e17 q^{47} -4.81802e16 q^{48} -5.22296e17 q^{49} +8.86220e16 q^{51} -3.96255e16 q^{52} +1.54979e18 q^{53} +3.11823e17 q^{54} +5.02642e17 q^{56} -1.89596e17 q^{57} -1.69367e18 q^{58} -3.53354e18 q^{59} -5.04266e18 q^{61} -8.17765e18 q^{62} -1.97476e18 q^{63} +6.56450e18 q^{64} -8.43405e16 q^{66} -2.20412e19 q^{67} +4.14529e18 q^{68} -1.70752e18 q^{69} -4.58636e19 q^{71} -2.73825e19 q^{72} +2.75815e19 q^{73} -6.13031e19 q^{74} -8.86834e18 q^{76} +1.07280e18 q^{77} +1.34927e18 q^{78} -5.50733e19 q^{79} +1.06655e20 q^{81} -2.34602e20 q^{82} +9.94937e19 q^{83} +7.86551e17 q^{84} +1.36784e20 q^{86} +9.99361e18 q^{87} +1.48756e19 q^{88} -5.03737e20 q^{89} -1.71625e19 q^{91} -7.98692e19 q^{92} +4.82528e19 q^{93} -6.90171e20 q^{94} +2.47054e19 q^{96} -7.08581e20 q^{97} +8.31869e20 q^{98} -5.84427e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 1312 q^{2} - 52194 q^{3} + 991136 q^{4} + 24240576 q^{6} - 684416558 q^{7} - 1119275520 q^{8} + 2005625619 q^{9} + 3188512736 q^{11} + 360872818752 q^{12} + 806175066506 q^{13} - 2283397948608 q^{14}+ \cdots - 13\!\cdots\!72 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1592.71 −1.09982 −0.549912 − 0.835223i \(-0.685339\pi\)
−0.549912 + 0.835223i \(0.685339\pi\)
\(3\) 9397.92 0.0918880 0.0459440 − 0.998944i \(-0.485370\pi\)
0.0459440 + 0.998944i \(0.485370\pi\)
\(4\) 439587. 0.209611
\(5\) 0 0
\(6\) −1.49682e7 −0.101061
\(7\) 1.90393e8 0.254754 0.127377 − 0.991854i \(-0.459344\pi\)
0.127377 + 0.991854i \(0.459344\pi\)
\(8\) 2.64003e9 0.869288
\(9\) −1.03720e10 −0.991557
\(10\) 0 0
\(11\) 5.63465e9 0.0655003 0.0327502 − 0.999464i \(-0.489573\pi\)
0.0327502 + 0.999464i \(0.489573\pi\)
\(12\) 4.13121e9 0.0192608
\(13\) −9.01426e10 −0.181353 −0.0906765 − 0.995880i \(-0.528903\pi\)
−0.0906765 + 0.995880i \(0.528903\pi\)
\(14\) −3.03241e11 −0.280184
\(15\) 0 0
\(16\) −5.12669e12 −1.16567
\(17\) 9.42995e12 1.13448 0.567239 − 0.823553i \(-0.308012\pi\)
0.567239 + 0.823553i \(0.308012\pi\)
\(18\) 1.65197e13 1.09054
\(19\) −2.01743e13 −0.754892 −0.377446 − 0.926032i \(-0.623198\pi\)
−0.377446 + 0.926032i \(0.623198\pi\)
\(20\) 0 0
\(21\) 1.78930e12 0.0234088
\(22\) −8.97438e12 −0.0720388
\(23\) −1.81691e14 −0.914518 −0.457259 − 0.889334i \(-0.651169\pi\)
−0.457259 + 0.889334i \(0.651169\pi\)
\(24\) 2.48108e13 0.0798771
\(25\) 0 0
\(26\) 1.43571e14 0.199456
\(27\) −1.95781e14 −0.183000
\(28\) 8.36942e13 0.0533993
\(29\) 1.06339e15 0.469366 0.234683 − 0.972072i \(-0.424595\pi\)
0.234683 + 0.972072i \(0.424595\pi\)
\(30\) 0 0
\(31\) 5.13441e15 1.12510 0.562552 − 0.826762i \(-0.309819\pi\)
0.562552 + 0.826762i \(0.309819\pi\)
\(32\) 2.62882e15 0.412748
\(33\) 5.29540e13 0.00601869
\(34\) −1.50192e16 −1.24772
\(35\) 0 0
\(36\) −4.55941e15 −0.207842
\(37\) 3.84897e16 1.31591 0.657955 − 0.753057i \(-0.271422\pi\)
0.657955 + 0.753057i \(0.271422\pi\)
\(38\) 3.21318e16 0.830248
\(39\) −8.47153e14 −0.0166642
\(40\) 0 0
\(41\) 1.47297e17 1.71381 0.856907 − 0.515472i \(-0.172383\pi\)
0.856907 + 0.515472i \(0.172383\pi\)
\(42\) −2.84984e15 −0.0257456
\(43\) −8.58814e16 −0.606010 −0.303005 − 0.952989i \(-0.597990\pi\)
−0.303005 + 0.952989i \(0.597990\pi\)
\(44\) 2.47692e15 0.0137296
\(45\) 0 0
\(46\) 2.89382e17 1.00581
\(47\) 4.33330e17 1.20169 0.600843 − 0.799367i \(-0.294832\pi\)
0.600843 + 0.799367i \(0.294832\pi\)
\(48\) −4.81802e16 −0.107111
\(49\) −5.22296e17 −0.935100
\(50\) 0 0
\(51\) 8.86220e16 0.104245
\(52\) −3.96255e16 −0.0380137
\(53\) 1.54979e18 1.21724 0.608621 − 0.793461i \(-0.291723\pi\)
0.608621 + 0.793461i \(0.291723\pi\)
\(54\) 3.11823e17 0.201268
\(55\) 0 0
\(56\) 5.02642e17 0.221455
\(57\) −1.89596e17 −0.0693655
\(58\) −1.69367e18 −0.516220
\(59\) −3.53354e18 −0.900045 −0.450023 − 0.893017i \(-0.648584\pi\)
−0.450023 + 0.893017i \(0.648584\pi\)
\(60\) 0 0
\(61\) −5.04266e18 −0.905101 −0.452550 − 0.891739i \(-0.649486\pi\)
−0.452550 + 0.891739i \(0.649486\pi\)
\(62\) −8.17765e18 −1.23742
\(63\) −1.97476e18 −0.252603
\(64\) 6.56450e18 0.711724
\(65\) 0 0
\(66\) −8.43405e16 −0.00661950
\(67\) −2.20412e19 −1.47723 −0.738617 − 0.674125i \(-0.764521\pi\)
−0.738617 + 0.674125i \(0.764521\pi\)
\(68\) 4.14529e18 0.237799
\(69\) −1.70752e18 −0.0840332
\(70\) 0 0
\(71\) −4.58636e19 −1.67207 −0.836037 − 0.548673i \(-0.815133\pi\)
−0.836037 + 0.548673i \(0.815133\pi\)
\(72\) −2.73825e19 −0.861948
\(73\) 2.75815e19 0.751153 0.375577 − 0.926791i \(-0.377445\pi\)
0.375577 + 0.926791i \(0.377445\pi\)
\(74\) −6.13031e19 −1.44727
\(75\) 0 0
\(76\) −8.86834e18 −0.158234
\(77\) 1.07280e18 0.0166865
\(78\) 1.34927e18 0.0183276
\(79\) −5.50733e19 −0.654421 −0.327210 − 0.944952i \(-0.606109\pi\)
−0.327210 + 0.944952i \(0.606109\pi\)
\(80\) 0 0
\(81\) 1.06655e20 0.974741
\(82\) −2.34602e20 −1.88489
\(83\) 9.94937e19 0.703842 0.351921 − 0.936030i \(-0.385529\pi\)
0.351921 + 0.936030i \(0.385529\pi\)
\(84\) 7.86551e17 0.00490676
\(85\) 0 0
\(86\) 1.36784e20 0.666504
\(87\) 9.99361e18 0.0431291
\(88\) 1.48756e19 0.0569386
\(89\) −5.03737e20 −1.71241 −0.856207 − 0.516633i \(-0.827185\pi\)
−0.856207 + 0.516633i \(0.827185\pi\)
\(90\) 0 0
\(91\) −1.71625e19 −0.0462004
\(92\) −7.98692e19 −0.191693
\(93\) 4.82528e19 0.103384
\(94\) −6.90171e20 −1.32164
\(95\) 0 0
\(96\) 2.47054e19 0.0379266
\(97\) −7.08581e20 −0.975633 −0.487816 − 0.872946i \(-0.662206\pi\)
−0.487816 + 0.872946i \(0.662206\pi\)
\(98\) 8.31869e20 1.02845
\(99\) −5.84427e19 −0.0649473
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 25.22.a.b.1.1 3
5.2 odd 4 25.22.b.b.24.2 6
5.3 odd 4 25.22.b.b.24.5 6
5.4 even 2 5.22.a.a.1.3 ✓ 3
15.14 odd 2 45.22.a.d.1.1 3
20.19 odd 2 80.22.a.e.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.22.a.a.1.3 ✓ 3 5.4 even 2
25.22.a.b.1.1 3 1.1 even 1 trivial
25.22.b.b.24.2 6 5.2 odd 4
25.22.b.b.24.5 6 5.3 odd 4
45.22.a.d.1.1 3 15.14 odd 2
80.22.a.e.1.2 3 20.19 odd 2