Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(1,925)] 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("925.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,-5,-8] 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: yes
Analytic conductor: \(7.38616218697\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 6x^{7} + 30x^{6} + 15x^{5} - 70x^{4} - 22x^{3} + 44x^{2} + 4x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 185)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(1.97415\) of defining polynomial
Character \(\chi\) \(=\) 925.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.974151 q^{2} +1.62921 q^{3} -1.05103 q^{4} +1.58709 q^{6} -4.15169 q^{7} -2.97216 q^{8} -0.345685 q^{9} -1.12356 q^{11} -1.71234 q^{12} -0.328108 q^{13} -4.04438 q^{14} -0.793280 q^{16} +1.55144 q^{17} -0.336750 q^{18} -4.32598 q^{19} -6.76396 q^{21} -1.09452 q^{22} -1.73985 q^{23} -4.84227 q^{24} -0.319627 q^{26} -5.45081 q^{27} +4.36355 q^{28} +8.20254 q^{29} -10.1080 q^{31} +5.17155 q^{32} -1.83051 q^{33} +1.51134 q^{34} +0.363325 q^{36} +1.00000 q^{37} -4.21416 q^{38} -0.534556 q^{39} -1.91727 q^{41} -6.58913 q^{42} -4.22272 q^{43} +1.18089 q^{44} -1.69488 q^{46} -10.7452 q^{47} -1.29242 q^{48} +10.2365 q^{49} +2.52762 q^{51} +0.344851 q^{52} +2.11437 q^{53} -5.30992 q^{54} +12.3395 q^{56} -7.04792 q^{57} +7.99051 q^{58} +11.5917 q^{59} +9.72982 q^{61} -9.84669 q^{62} +1.43518 q^{63} +6.62444 q^{64} -1.78319 q^{66} +5.29915 q^{67} -1.63061 q^{68} -2.83458 q^{69} +9.90260 q^{71} +1.02743 q^{72} -8.12992 q^{73} +0.974151 q^{74} +4.54673 q^{76} +4.66467 q^{77} -0.520738 q^{78} -0.0298662 q^{79} -7.84345 q^{81} -1.86771 q^{82} -13.8190 q^{83} +7.10912 q^{84} -4.11357 q^{86} +13.3636 q^{87} +3.33940 q^{88} -9.37865 q^{89} +1.36220 q^{91} +1.82864 q^{92} -16.4680 q^{93} -10.4674 q^{94} +8.42553 q^{96} -10.3113 q^{97} +9.97194 q^{98} +0.388398 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 5 q^{2} - 8 q^{3} + 11 q^{4} + 2 q^{6} - 8 q^{7} - 15 q^{8} + 13 q^{9} - 16 q^{12} - 6 q^{13} - 4 q^{14} + 11 q^{16} - 18 q^{17} + 3 q^{18} - 4 q^{19} + 4 q^{21} - 6 q^{22} - 16 q^{23} + 6 q^{24}+ \cdots + 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.974151 0.688829 0.344415 − 0.938818i \(-0.388077\pi\)
0.344415 + 0.938818i \(0.388077\pi\)
\(3\) 1.62921 0.940623 0.470311 − 0.882500i \(-0.344142\pi\)
0.470311 + 0.882500i \(0.344142\pi\)
\(4\) −1.05103 −0.525515
\(5\) 0 0
\(6\) 1.58709 0.647928
\(7\) −4.15169 −1.56919 −0.784596 − 0.620007i \(-0.787130\pi\)
−0.784596 + 0.620007i \(0.787130\pi\)
\(8\) −2.97216 −1.05082
\(9\) −0.345685 −0.115228
\(10\) 0 0
\(11\) −1.12356 −0.338766 −0.169383 − 0.985550i \(-0.554177\pi\)
−0.169383 + 0.985550i \(0.554177\pi\)
\(12\) −1.71234 −0.494311
\(13\) −0.328108 −0.0910008 −0.0455004 − 0.998964i \(-0.514488\pi\)
−0.0455004 + 0.998964i \(0.514488\pi\)
\(14\) −4.04438 −1.08091
\(15\) 0 0
\(16\) −0.793280 −0.198320
\(17\) 1.55144 0.376280 0.188140 − 0.982142i \(-0.439754\pi\)
0.188140 + 0.982142i \(0.439754\pi\)
\(18\) −0.336750 −0.0793727
\(19\) −4.32598 −0.992449 −0.496224 − 0.868194i \(-0.665281\pi\)
−0.496224 + 0.868194i \(0.665281\pi\)
\(20\) 0 0
\(21\) −6.76396 −1.47602
\(22\) −1.09452 −0.233352
\(23\) −1.73985 −0.362785 −0.181392 − 0.983411i \(-0.558060\pi\)
−0.181392 + 0.983411i \(0.558060\pi\)
\(24\) −4.84227 −0.988424
\(25\) 0 0
\(26\) −0.319627 −0.0626840
\(27\) −5.45081 −1.04901
\(28\) 4.36355 0.824633
\(29\) 8.20254 1.52317 0.761587 − 0.648063i \(-0.224421\pi\)
0.761587 + 0.648063i \(0.224421\pi\)
\(30\) 0 0
\(31\) −10.1080 −1.81545 −0.907723 − 0.419571i \(-0.862181\pi\)
−0.907723 + 0.419571i \(0.862181\pi\)
\(32\) 5.17155 0.914210
\(33\) −1.83051 −0.318651
\(34\) 1.51134 0.259193
\(35\) 0 0
\(36\) 0.363325 0.0605542
\(37\) 1.00000 0.164399
\(38\) −4.21416 −0.683627
\(39\) −0.534556 −0.0855974
\(40\) 0 0
\(41\) −1.91727 −0.299428 −0.149714 − 0.988729i \(-0.547835\pi\)
−0.149714 + 0.988729i \(0.547835\pi\)
\(42\) −6.58913 −1.01672
\(43\) −4.22272 −0.643959 −0.321979 − 0.946747i \(-0.604348\pi\)
−0.321979 + 0.946747i \(0.604348\pi\)
\(44\) 1.18089 0.178026
\(45\) 0 0
\(46\) −1.69488 −0.249897
\(47\) −10.7452 −1.56735 −0.783673 − 0.621174i \(-0.786656\pi\)
−0.783673 + 0.621174i \(0.786656\pi\)
\(48\) −1.29242 −0.186544
\(49\) 10.2365 1.46236
\(50\) 0 0
\(51\) 2.52762 0.353938
\(52\) 0.344851 0.0478222
\(53\) 2.11437 0.290431 0.145215 − 0.989400i \(-0.453612\pi\)
0.145215 + 0.989400i \(0.453612\pi\)
\(54\) −5.30992 −0.722588
\(55\) 0 0
\(56\) 12.3395 1.64894
\(57\) −7.04792 −0.933520
\(58\) 7.99051 1.04921
\(59\) 11.5917 1.50911 0.754557 − 0.656234i \(-0.227852\pi\)
0.754557 + 0.656234i \(0.227852\pi\)
\(60\) 0 0
\(61\) 9.72982 1.24578 0.622888 − 0.782311i \(-0.285959\pi\)
0.622888 + 0.782311i \(0.285959\pi\)
\(62\) −9.84669 −1.25053
\(63\) 1.43518 0.180816
\(64\) 6.62444 0.828054
\(65\) 0 0
\(66\) −1.78319 −0.219496
\(67\) 5.29915 0.647395 0.323697 − 0.946161i \(-0.395074\pi\)
0.323697 + 0.946161i \(0.395074\pi\)
\(68\) −1.63061 −0.197741
\(69\) −2.83458 −0.341243
\(70\) 0 0
\(71\) 9.90260 1.17522 0.587611 − 0.809144i \(-0.300069\pi\)
0.587611 + 0.809144i \(0.300069\pi\)
\(72\) 1.02743 0.121084
\(73\) −8.12992 −0.951535 −0.475767 − 0.879571i \(-0.657830\pi\)
−0.475767 + 0.879571i \(0.657830\pi\)
\(74\) 0.974151 0.113243
\(75\) 0 0
\(76\) 4.54673 0.521546
\(77\) 4.66467 0.531588
\(78\) −0.520738 −0.0589620
\(79\) −0.0298662 −0.00336021 −0.00168011 − 0.999999i \(-0.500535\pi\)
−0.00168011 + 0.999999i \(0.500535\pi\)
\(80\) 0 0
\(81\) −7.84345 −0.871494
\(82\) −1.86771 −0.206255
\(83\) −13.8190 −1.51683 −0.758416 − 0.651770i \(-0.774027\pi\)
−0.758416 + 0.651770i \(0.774027\pi\)
\(84\) 7.10912 0.775669
\(85\) 0 0
\(86\) −4.11357 −0.443577
\(87\) 13.3636 1.43273
\(88\) 3.33940 0.355981
\(89\) −9.37865 −0.994135 −0.497067 − 0.867712i \(-0.665590\pi\)
−0.497067 + 0.867712i \(0.665590\pi\)
\(90\) 0 0
\(91\) 1.36220 0.142798
\(92\) 1.82864 0.190649
\(93\) −16.4680 −1.70765
\(94\) −10.4674 −1.07963
\(95\) 0 0
\(96\) 8.42553 0.859927
\(97\) −10.3113 −1.04695 −0.523475 − 0.852041i \(-0.675365\pi\)
−0.523475 + 0.852041i \(0.675365\pi\)
\(98\) 9.97194 1.00732
\(99\) 0.388398 0.0390354
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.a.l.1.7 9
3.2 odd 2 8325.2.a.cr.1.3 9
5.2 odd 4 185.2.b.a.149.12 yes 18
5.3 odd 4 185.2.b.a.149.7 ✓ 18
5.4 even 2 925.2.a.m.1.3 9
15.2 even 4 1665.2.c.e.334.7 18
15.8 even 4 1665.2.c.e.334.12 18
15.14 odd 2 8325.2.a.cq.1.7 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.b.a.149.7 ✓ 18 5.3 odd 4
185.2.b.a.149.12 yes 18 5.2 odd 4
925.2.a.l.1.7 9 1.1 even 1 trivial
925.2.a.m.1.3 9 5.4 even 2
1665.2.c.e.334.7 18 15.2 even 4
1665.2.c.e.334.12 18 15.8 even 4
8325.2.a.cq.1.7 9 15.14 odd 2
8325.2.a.cr.1.3 9 3.2 odd 2