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 = [3,4,2,4,0,5,0,9,-3,0,1,5,5,7] 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: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 475)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.24698\) 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)
 
\(f(q)\) \(=\) \(q-0.246980 q^{2} -0.801938 q^{3} -1.93900 q^{4} +0.198062 q^{6} -1.69202 q^{7} +0.972853 q^{8} -2.35690 q^{9} -0.911854 q^{11} +1.55496 q^{12} +1.55496 q^{13} +0.417895 q^{14} +3.63773 q^{16} -5.29590 q^{17} +0.582105 q^{18} +1.35690 q^{21} +0.225209 q^{22} -4.24698 q^{23} -0.780167 q^{24} -0.384043 q^{26} +4.29590 q^{27} +3.28083 q^{28} -5.00969 q^{29} -1.82908 q^{31} -2.84415 q^{32} +0.731250 q^{33} +1.30798 q^{34} +4.57002 q^{36} +6.29590 q^{37} -1.24698 q^{39} -4.18060 q^{41} -0.335126 q^{42} -7.31767 q^{43} +1.76809 q^{44} +1.04892 q^{46} +2.04892 q^{47} -2.91723 q^{48} -4.13706 q^{49} +4.24698 q^{51} -3.01507 q^{52} -2.70171 q^{53} -1.06100 q^{54} -1.64609 q^{56} +1.23729 q^{58} -9.87800 q^{59} +0.542877 q^{61} +0.451747 q^{62} +3.98792 q^{63} -6.57301 q^{64} -0.180604 q^{66} -13.9976 q^{67} +10.2687 q^{68} +3.40581 q^{69} +12.8780 q^{71} -2.29291 q^{72} +2.80731 q^{73} -1.55496 q^{74} +1.54288 q^{77} +0.307979 q^{78} -1.59419 q^{79} +3.62565 q^{81} +1.03252 q^{82} -12.2349 q^{83} -2.63102 q^{84} +1.80731 q^{86} +4.01746 q^{87} -0.887100 q^{88} -2.91723 q^{89} -2.63102 q^{91} +8.23490 q^{92} +1.46681 q^{93} -0.506041 q^{94} +2.28083 q^{96} +1.55496 q^{97} +1.02177 q^{98} +2.14914 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{2} + 2 q^{3} + 4 q^{4} + 5 q^{6} + 9 q^{8} - 3 q^{9} + q^{11} + 5 q^{12} + 5 q^{13} + 7 q^{14} + 18 q^{16} - 2 q^{17} - 4 q^{18} - q^{22} - 8 q^{23} - q^{24} + 9 q^{26} - q^{27} + 21 q^{28}+ \cdots + 20 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.246980 −0.174641 −0.0873205 − 0.996180i \(-0.527830\pi\)
−0.0873205 + 0.996180i \(0.527830\pi\)
\(3\) −0.801938 −0.462999 −0.231499 − 0.972835i \(-0.574363\pi\)
−0.231499 + 0.972835i \(0.574363\pi\)
\(4\) −1.93900 −0.969501
\(5\) 0 0
\(6\) 0.198062 0.0808586
\(7\) −1.69202 −0.639524 −0.319762 − 0.947498i \(-0.603603\pi\)
−0.319762 + 0.947498i \(0.603603\pi\)
\(8\) 0.972853 0.343955
\(9\) −2.35690 −0.785632
\(10\) 0 0
\(11\) −0.911854 −0.274934 −0.137467 − 0.990506i \(-0.543896\pi\)
−0.137467 + 0.990506i \(0.543896\pi\)
\(12\) 1.55496 0.448878
\(13\) 1.55496 0.431268 0.215634 − 0.976474i \(-0.430818\pi\)
0.215634 + 0.976474i \(0.430818\pi\)
\(14\) 0.417895 0.111687
\(15\) 0 0
\(16\) 3.63773 0.909432
\(17\) −5.29590 −1.28444 −0.642222 − 0.766519i \(-0.721987\pi\)
−0.642222 + 0.766519i \(0.721987\pi\)
\(18\) 0.582105 0.137204
\(19\) 0 0
\(20\) 0 0
\(21\) 1.35690 0.296099
\(22\) 0.225209 0.0480148
\(23\) −4.24698 −0.885556 −0.442778 − 0.896631i \(-0.646007\pi\)
−0.442778 + 0.896631i \(0.646007\pi\)
\(24\) −0.780167 −0.159251
\(25\) 0 0
\(26\) −0.384043 −0.0753170
\(27\) 4.29590 0.826746
\(28\) 3.28083 0.620019
\(29\) −5.00969 −0.930276 −0.465138 − 0.885238i \(-0.653995\pi\)
−0.465138 + 0.885238i \(0.653995\pi\)
\(30\) 0 0
\(31\) −1.82908 −0.328513 −0.164257 − 0.986418i \(-0.552523\pi\)
−0.164257 + 0.986418i \(0.552523\pi\)
\(32\) −2.84415 −0.502779
\(33\) 0.731250 0.127294
\(34\) 1.30798 0.224316
\(35\) 0 0
\(36\) 4.57002 0.761671
\(37\) 6.29590 1.03504 0.517520 − 0.855671i \(-0.326855\pi\)
0.517520 + 0.855671i \(0.326855\pi\)
\(38\) 0 0
\(39\) −1.24698 −0.199677
\(40\) 0 0
\(41\) −4.18060 −0.652901 −0.326450 − 0.945214i \(-0.605853\pi\)
−0.326450 + 0.945214i \(0.605853\pi\)
\(42\) −0.335126 −0.0517110
\(43\) −7.31767 −1.11593 −0.557967 − 0.829863i \(-0.688418\pi\)
−0.557967 + 0.829863i \(0.688418\pi\)
\(44\) 1.76809 0.266549
\(45\) 0 0
\(46\) 1.04892 0.154654
\(47\) 2.04892 0.298865 0.149433 − 0.988772i \(-0.452255\pi\)
0.149433 + 0.988772i \(0.452255\pi\)
\(48\) −2.91723 −0.421066
\(49\) −4.13706 −0.591009
\(50\) 0 0
\(51\) 4.24698 0.594696
\(52\) −3.01507 −0.418114
\(53\) −2.70171 −0.371108 −0.185554 − 0.982634i \(-0.559408\pi\)
−0.185554 + 0.982634i \(0.559408\pi\)
\(54\) −1.06100 −0.144384
\(55\) 0 0
\(56\) −1.64609 −0.219968
\(57\) 0 0
\(58\) 1.23729 0.162464
\(59\) −9.87800 −1.28601 −0.643003 − 0.765864i \(-0.722312\pi\)
−0.643003 + 0.765864i \(0.722312\pi\)
\(60\) 0 0
\(61\) 0.542877 0.0695082 0.0347541 − 0.999396i \(-0.488935\pi\)
0.0347541 + 0.999396i \(0.488935\pi\)
\(62\) 0.451747 0.0573719
\(63\) 3.98792 0.502430
\(64\) −6.57301 −0.821626
\(65\) 0 0
\(66\) −0.180604 −0.0222308
\(67\) −13.9976 −1.71008 −0.855040 − 0.518562i \(-0.826467\pi\)
−0.855040 + 0.518562i \(0.826467\pi\)
\(68\) 10.2687 1.24527
\(69\) 3.40581 0.410012
\(70\) 0 0
\(71\) 12.8780 1.52834 0.764169 − 0.645016i \(-0.223149\pi\)
0.764169 + 0.645016i \(0.223149\pi\)
\(72\) −2.29291 −0.270222
\(73\) 2.80731 0.328571 0.164286 − 0.986413i \(-0.447468\pi\)
0.164286 + 0.986413i \(0.447468\pi\)
\(74\) −1.55496 −0.180760
\(75\) 0 0
\(76\) 0 0
\(77\) 1.54288 0.175827
\(78\) 0.307979 0.0348717
\(79\) −1.59419 −0.179360 −0.0896800 − 0.995971i \(-0.528584\pi\)
−0.0896800 + 0.995971i \(0.528584\pi\)
\(80\) 0 0
\(81\) 3.62565 0.402850
\(82\) 1.03252 0.114023
\(83\) −12.2349 −1.34295 −0.671477 − 0.741025i \(-0.734340\pi\)
−0.671477 + 0.741025i \(0.734340\pi\)
\(84\) −2.63102 −0.287068
\(85\) 0 0
\(86\) 1.80731 0.194888
\(87\) 4.01746 0.430717
\(88\) −0.887100 −0.0945652
\(89\) −2.91723 −0.309226 −0.154613 − 0.987975i \(-0.549413\pi\)
−0.154613 + 0.987975i \(0.549413\pi\)
\(90\) 0 0
\(91\) −2.63102 −0.275806
\(92\) 8.23490 0.858547
\(93\) 1.46681 0.152101
\(94\) −0.506041 −0.0521941
\(95\) 0 0
\(96\) 2.28083 0.232786
\(97\) 1.55496 0.157882 0.0789410 − 0.996879i \(-0.474846\pi\)
0.0789410 + 0.996879i \(0.474846\pi\)
\(98\) 1.02177 0.103214
\(99\) 2.14914 0.215997
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.be.1.1 3
5.4 even 2 9025.2.a.w.1.3 3
19.18 odd 2 475.2.a.d.1.3 ✓ 3
57.56 even 2 4275.2.a.bn.1.1 3
76.75 even 2 7600.2.a.bw.1.1 3
95.18 even 4 475.2.b.c.324.3 6
95.37 even 4 475.2.b.c.324.4 6
95.94 odd 2 475.2.a.h.1.1 yes 3
285.284 even 2 4275.2.a.z.1.3 3
380.379 even 2 7600.2.a.bn.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.d.1.3 ✓ 3 19.18 odd 2
475.2.a.h.1.1 yes 3 95.94 odd 2
475.2.b.c.324.3 6 95.18 even 4
475.2.b.c.324.4 6 95.37 even 4
4275.2.a.z.1.3 3 285.284 even 2
4275.2.a.bn.1.1 3 57.56 even 2
7600.2.a.bn.1.3 3 380.379 even 2
7600.2.a.bw.1.1 3 76.75 even 2
9025.2.a.w.1.3 3 5.4 even 2
9025.2.a.be.1.1 3 1.1 even 1 trivial