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 1.3
Root \(1.80194\) 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)
 
\(f(q)\) \(=\) \(q+2.80194 q^{2} +0.554958 q^{3} +5.85086 q^{4} +1.55496 q^{6} +3.04892 q^{7} +10.7899 q^{8} -2.69202 q^{9} -2.93900 q^{11} +3.24698 q^{12} +3.24698 q^{13} +8.54288 q^{14} +18.5308 q^{16} +2.15883 q^{17} -7.54288 q^{18} +1.69202 q^{21} -8.23490 q^{22} -1.19806 q^{23} +5.98792 q^{24} +9.09783 q^{26} -3.15883 q^{27} +17.8388 q^{28} +1.77479 q^{29} +9.34481 q^{31} +30.3424 q^{32} -1.63102 q^{33} +6.04892 q^{34} -15.7506 q^{36} -1.15883 q^{37} +1.80194 q^{39} -8.57002 q^{41} +4.74094 q^{42} -5.27413 q^{43} -17.1957 q^{44} -3.35690 q^{46} -2.35690 q^{47} +10.2838 q^{48} +2.29590 q^{49} +1.19806 q^{51} +18.9976 q^{52} +8.82371 q^{53} -8.85086 q^{54} +32.8974 q^{56} +4.97285 q^{58} +5.70171 q^{59} -9.96077 q^{61} +26.1836 q^{62} -8.20775 q^{63} +47.9560 q^{64} -4.57002 q^{66} +4.98254 q^{67} +12.6310 q^{68} -0.664874 q^{69} -2.70171 q^{71} -29.0465 q^{72} -13.7778 q^{73} -3.24698 q^{74} -8.96077 q^{77} +5.04892 q^{78} -5.66487 q^{79} +6.32304 q^{81} -24.0127 q^{82} +3.00969 q^{83} +9.89977 q^{84} -14.7778 q^{86} +0.984935 q^{87} -31.7114 q^{88} +10.2838 q^{89} +9.89977 q^{91} -7.00969 q^{92} +5.18598 q^{93} -6.60388 q^{94} +16.8388 q^{96} +3.24698 q^{97} +6.43296 q^{98} +7.91185 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{2} + 2 q^{3} + 4 q^{4} + 5 q^{6} + 9 q^{8} - 3 q^{9} + q^{11} + 5 q^{12} + 5 q^{13} + 7 q^{14} + 18 q^{16} - 2 q^{17} - 4 q^{18} - q^{22} - 8 q^{23} - q^{24} + 9 q^{26} - q^{27} + 21 q^{28}+ \cdots + 20 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.80194 1.98127 0.990635 − 0.136540i \(-0.0435982\pi\)
0.990635 + 0.136540i \(0.0435982\pi\)
\(3\) 0.554958 0.320405 0.160203 − 0.987084i \(-0.448785\pi\)
0.160203 + 0.987084i \(0.448785\pi\)
\(4\) 5.85086 2.92543
\(5\) 0 0
\(6\) 1.55496 0.634809
\(7\) 3.04892 1.15238 0.576191 − 0.817315i \(-0.304538\pi\)
0.576191 + 0.817315i \(0.304538\pi\)
\(8\) 10.7899 3.81479
\(9\) −2.69202 −0.897340
\(10\) 0 0
\(11\) −2.93900 −0.886142 −0.443071 − 0.896486i \(-0.646111\pi\)
−0.443071 + 0.896486i \(0.646111\pi\)
\(12\) 3.24698 0.937322
\(13\) 3.24698 0.900550 0.450275 − 0.892890i \(-0.351326\pi\)
0.450275 + 0.892890i \(0.351326\pi\)
\(14\) 8.54288 2.28318
\(15\) 0 0
\(16\) 18.5308 4.63270
\(17\) 2.15883 0.523594 0.261797 − 0.965123i \(-0.415685\pi\)
0.261797 + 0.965123i \(0.415685\pi\)
\(18\) −7.54288 −1.77787
\(19\) 0 0
\(20\) 0 0
\(21\) 1.69202 0.369229
\(22\) −8.23490 −1.75569
\(23\) −1.19806 −0.249813 −0.124907 − 0.992169i \(-0.539863\pi\)
−0.124907 + 0.992169i \(0.539863\pi\)
\(24\) 5.98792 1.22228
\(25\) 0 0
\(26\) 9.09783 1.78423
\(27\) −3.15883 −0.607918
\(28\) 17.8388 3.37121
\(29\) 1.77479 0.329570 0.164785 − 0.986329i \(-0.447307\pi\)
0.164785 + 0.986329i \(0.447307\pi\)
\(30\) 0 0
\(31\) 9.34481 1.67838 0.839189 − 0.543840i \(-0.183030\pi\)
0.839189 + 0.543840i \(0.183030\pi\)
\(32\) 30.3424 5.36383
\(33\) −1.63102 −0.283925
\(34\) 6.04892 1.03738
\(35\) 0 0
\(36\) −15.7506 −2.62510
\(37\) −1.15883 −0.190511 −0.0952555 − 0.995453i \(-0.530367\pi\)
−0.0952555 + 0.995453i \(0.530367\pi\)
\(38\) 0 0
\(39\) 1.80194 0.288541
\(40\) 0 0
\(41\) −8.57002 −1.33841 −0.669206 − 0.743077i \(-0.733366\pi\)
−0.669206 + 0.743077i \(0.733366\pi\)
\(42\) 4.74094 0.731543
\(43\) −5.27413 −0.804297 −0.402148 − 0.915575i \(-0.631736\pi\)
−0.402148 + 0.915575i \(0.631736\pi\)
\(44\) −17.1957 −2.59234
\(45\) 0 0
\(46\) −3.35690 −0.494947
\(47\) −2.35690 −0.343789 −0.171894 − 0.985115i \(-0.554989\pi\)
−0.171894 + 0.985115i \(0.554989\pi\)
\(48\) 10.2838 1.48434
\(49\) 2.29590 0.327985
\(50\) 0 0
\(51\) 1.19806 0.167762
\(52\) 18.9976 2.63449
\(53\) 8.82371 1.21203 0.606015 − 0.795453i \(-0.292767\pi\)
0.606015 + 0.795453i \(0.292767\pi\)
\(54\) −8.85086 −1.20445
\(55\) 0 0
\(56\) 32.8974 4.39610
\(57\) 0 0
\(58\) 4.97285 0.652968
\(59\) 5.70171 0.742299 0.371150 − 0.928573i \(-0.378964\pi\)
0.371150 + 0.928573i \(0.378964\pi\)
\(60\) 0 0
\(61\) −9.96077 −1.27535 −0.637673 − 0.770307i \(-0.720103\pi\)
−0.637673 + 0.770307i \(0.720103\pi\)
\(62\) 26.1836 3.32532
\(63\) −8.20775 −1.03408
\(64\) 47.9560 5.99450
\(65\) 0 0
\(66\) −4.57002 −0.562531
\(67\) 4.98254 0.608714 0.304357 − 0.952558i \(-0.401558\pi\)
0.304357 + 0.952558i \(0.401558\pi\)
\(68\) 12.6310 1.53174
\(69\) −0.664874 −0.0800415
\(70\) 0 0
\(71\) −2.70171 −0.320634 −0.160317 − 0.987066i \(-0.551252\pi\)
−0.160317 + 0.987066i \(0.551252\pi\)
\(72\) −29.0465 −3.42317
\(73\) −13.7778 −1.61257 −0.806283 − 0.591530i \(-0.798524\pi\)
−0.806283 + 0.591530i \(0.798524\pi\)
\(74\) −3.24698 −0.377454
\(75\) 0 0
\(76\) 0 0
\(77\) −8.96077 −1.02117
\(78\) 5.04892 0.571677
\(79\) −5.66487 −0.637348 −0.318674 − 0.947864i \(-0.603238\pi\)
−0.318674 + 0.947864i \(0.603238\pi\)
\(80\) 0 0
\(81\) 6.32304 0.702560
\(82\) −24.0127 −2.65176
\(83\) 3.00969 0.330356 0.165178 − 0.986264i \(-0.447180\pi\)
0.165178 + 0.986264i \(0.447180\pi\)
\(84\) 9.89977 1.08015
\(85\) 0 0
\(86\) −14.7778 −1.59353
\(87\) 0.984935 0.105596
\(88\) −31.7114 −3.38045
\(89\) 10.2838 1.09008 0.545041 − 0.838409i \(-0.316514\pi\)
0.545041 + 0.838409i \(0.316514\pi\)
\(90\) 0 0
\(91\) 9.89977 1.03778
\(92\) −7.00969 −0.730811
\(93\) 5.18598 0.537761
\(94\) −6.60388 −0.681138
\(95\) 0 0
\(96\) 16.8388 1.71860
\(97\) 3.24698 0.329681 0.164840 − 0.986320i \(-0.447289\pi\)
0.164840 + 0.986320i \(0.447289\pi\)
\(98\) 6.43296 0.649827
\(99\) 7.91185 0.795171
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.1.3 3
5.4 even 2 9025.2.a.w.1.1 3
19.18 odd 2 475.2.a.d.1.1 ✓ 3
57.56 even 2 4275.2.a.bn.1.3 3
76.75 even 2 7600.2.a.bw.1.2 3
95.18 even 4 475.2.b.c.324.6 6
95.37 even 4 475.2.b.c.324.1 6
95.94 odd 2 475.2.a.h.1.3 yes 3
285.284 even 2 4275.2.a.z.1.1 3
380.379 even 2 7600.2.a.bn.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.d.1.1 ✓ 3 19.18 odd 2
475.2.a.h.1.3 yes 3 95.94 odd 2
475.2.b.c.324.1 6 95.37 even 4
475.2.b.c.324.6 6 95.18 even 4
4275.2.a.z.1.1 3 285.284 even 2
4275.2.a.bn.1.3 3 57.56 even 2
7600.2.a.bn.1.2 3 380.379 even 2
7600.2.a.bw.1.2 3 76.75 even 2
9025.2.a.w.1.1 3 5.4 even 2
9025.2.a.be.1.3 3 1.1 even 1 trivial