↑

Properties

Label 5625.2.a.n.1.4
Level $5625$
Weight $2$
Character 5625.1
Self dual yes
Analytic conductor $44.916$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5625,2,Mod(1,5625)] 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("5625.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5625, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5625 = 3^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5625.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2,0,8,0,0,-2,15,0,0,7,0,-1,-16,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.9158511370\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{15 +2 \sqrt{5}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 6x^{2} + 7x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.70636\) of defining polynomial
Character \(\chi\) \(=\) 5625.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.70636 q^{2} +5.32440 q^{4} -0.470294 q^{7} +8.99702 q^{8} +3.18148 q^{11} -0.563444 q^{13} -1.27279 q^{14} +13.7004 q^{16} -1.70636 q^{17} +3.74010 q^{19} +8.61023 q^{22} -2.26981 q^{23} -1.52488 q^{26} -2.50403 q^{28} +8.32440 q^{29} +5.43656 q^{31} +19.0842 q^{32} -4.61803 q^{34} -1.02085 q^{37} +10.1221 q^{38} -1.47214 q^{41} -6.72721 q^{43} +16.9395 q^{44} -6.14292 q^{46} -4.43358 q^{47} -6.77882 q^{49} -3.00000 q^{52} -7.05161 q^{53} -4.23125 q^{56} +22.5288 q^{58} +12.8720 q^{59} +0.126888 q^{61} +14.7133 q^{62} +24.2480 q^{64} +2.79469 q^{67} -9.08535 q^{68} -16.0456 q^{71} +9.78873 q^{73} -2.76279 q^{74} +19.9138 q^{76} -1.49623 q^{77} -4.75499 q^{79} -3.98413 q^{82} +11.6905 q^{83} -18.2063 q^{86} +28.6238 q^{88} +6.20049 q^{89} +0.264985 q^{91} -12.0853 q^{92} -11.9989 q^{94} -8.45443 q^{97} -18.3460 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 8 q^{4} - 2 q^{7} + 15 q^{8} + 7 q^{11} - q^{13} - 16 q^{14} + 4 q^{16} + 2 q^{17} + 5 q^{19} + 6 q^{22} + q^{23} - 3 q^{26} - 9 q^{28} + 20 q^{29} + 23 q^{31} + 12 q^{32} - 14 q^{34}+ \cdots - q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.70636 1.91369 0.956844 − 0.290604i \(-0.0938561\pi\)
0.956844 + 0.290604i \(0.0938561\pi\)
\(3\) 0 0
\(4\) 5.32440 2.66220
\(5\) 0 0
\(6\) 0 0
\(7\) −0.470294 −0.177754 −0.0888772 − 0.996043i \(-0.528328\pi\)
−0.0888772 + 0.996043i \(0.528328\pi\)
\(8\) 8.99702 3.18093
\(9\) 0 0
\(10\) 0 0
\(11\) 3.18148 0.959252 0.479626 − 0.877473i \(-0.340772\pi\)
0.479626 + 0.877473i \(0.340772\pi\)
\(12\) 0 0
\(13\) −0.563444 −0.156271 −0.0781356 − 0.996943i \(-0.524897\pi\)
−0.0781356 + 0.996943i \(0.524897\pi\)
\(14\) −1.27279 −0.340166
\(15\) 0 0
\(16\) 13.7004 3.42510
\(17\) −1.70636 −0.413854 −0.206927 − 0.978356i \(-0.566346\pi\)
−0.206927 + 0.978356i \(0.566346\pi\)
\(18\) 0 0
\(19\) 3.74010 0.858038 0.429019 − 0.903295i \(-0.358859\pi\)
0.429019 + 0.903295i \(0.358859\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 8.61023 1.83571
\(23\) −2.26981 −0.473287 −0.236644 − 0.971597i \(-0.576047\pi\)
−0.236644 + 0.971597i \(0.576047\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.52488 −0.299054
\(27\) 0 0
\(28\) −2.50403 −0.473218
\(29\) 8.32440 1.54580 0.772901 − 0.634527i \(-0.218805\pi\)
0.772901 + 0.634527i \(0.218805\pi\)
\(30\) 0 0
\(31\) 5.43656 0.976434 0.488217 − 0.872722i \(-0.337647\pi\)
0.488217 + 0.872722i \(0.337647\pi\)
\(32\) 19.0842 3.37364
\(33\) 0 0
\(34\) −4.61803 −0.791986
\(35\) 0 0
\(36\) 0 0
\(37\) −1.02085 −0.167827 −0.0839135 − 0.996473i \(-0.526742\pi\)
−0.0839135 + 0.996473i \(0.526742\pi\)
\(38\) 10.1221 1.64202
\(39\) 0 0
\(40\) 0 0
\(41\) −1.47214 −0.229909 −0.114955 − 0.993371i \(-0.536672\pi\)
−0.114955 + 0.993371i \(0.536672\pi\)
\(42\) 0 0
\(43\) −6.72721 −1.02589 −0.512945 − 0.858421i \(-0.671446\pi\)
−0.512945 + 0.858421i \(0.671446\pi\)
\(44\) 16.9395 2.55372
\(45\) 0 0
\(46\) −6.14292 −0.905724
\(47\) −4.43358 −0.646703 −0.323352 − 0.946279i \(-0.604810\pi\)
−0.323352 + 0.946279i \(0.604810\pi\)
\(48\) 0 0
\(49\) −6.77882 −0.968403
\(50\) 0 0
\(51\) 0 0
\(52\) −3.00000 −0.416025
\(53\) −7.05161 −0.968613 −0.484307 − 0.874898i \(-0.660928\pi\)
−0.484307 + 0.874898i \(0.660928\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −4.23125 −0.565424
\(57\) 0 0
\(58\) 22.5288 2.95818
\(59\) 12.8720 1.67579 0.837894 − 0.545833i \(-0.183787\pi\)
0.837894 + 0.545833i \(0.183787\pi\)
\(60\) 0 0
\(61\) 0.126888 0.0162464 0.00812319 − 0.999967i \(-0.497414\pi\)
0.00812319 + 0.999967i \(0.497414\pi\)
\(62\) 14.7133 1.86859
\(63\) 0 0
\(64\) 24.2480 3.03100
\(65\) 0 0
\(66\) 0 0
\(67\) 2.79469 0.341426 0.170713 − 0.985321i \(-0.445393\pi\)
0.170713 + 0.985321i \(0.445393\pi\)
\(68\) −9.08535 −1.10176
\(69\) 0 0
\(70\) 0 0
\(71\) −16.0456 −1.90427 −0.952134 − 0.305681i \(-0.901116\pi\)
−0.952134 + 0.305681i \(0.901116\pi\)
\(72\) 0 0
\(73\) 9.78873 1.14568 0.572842 − 0.819666i \(-0.305841\pi\)
0.572842 + 0.819666i \(0.305841\pi\)
\(74\) −2.76279 −0.321168
\(75\) 0 0
\(76\) 19.9138 2.28427
\(77\) −1.49623 −0.170511
\(78\) 0 0
\(79\) −4.75499 −0.534978 −0.267489 − 0.963561i \(-0.586194\pi\)
−0.267489 + 0.963561i \(0.586194\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −3.98413 −0.439974
\(83\) 11.6905 1.28320 0.641599 − 0.767040i \(-0.278271\pi\)
0.641599 + 0.767040i \(0.278271\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −18.2063 −1.96323
\(87\) 0 0
\(88\) 28.6238 3.05131
\(89\) 6.20049 0.657250 0.328625 − 0.944460i \(-0.393415\pi\)
0.328625 + 0.944460i \(0.393415\pi\)
\(90\) 0 0
\(91\) 0.264985 0.0277779
\(92\) −12.0853 −1.25998
\(93\) 0 0
\(94\) −11.9989 −1.23759
\(95\) 0 0
\(96\) 0 0
\(97\) −8.45443 −0.858417 −0.429209 − 0.903205i \(-0.641207\pi\)
−0.429209 + 0.903205i \(0.641207\pi\)
\(98\) −18.3460 −1.85322
\(99\) 0 0
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 5625.2.a.n.1.4 4
3.2 odd 2 1875.2.a.e.1.1 4
5.4 even 2 5625.2.a.i.1.1 4
15.2 even 4 1875.2.b.c.1249.1 8
15.8 even 4 1875.2.b.c.1249.8 8
15.14 odd 2 1875.2.a.h.1.4 4
25.4 even 10 225.2.h.c.91.2 8
25.19 even 10 225.2.h.c.136.2 8
75.8 even 20 375.2.i.b.199.1 16
75.17 even 20 375.2.i.b.199.4 16
75.29 odd 10 75.2.g.b.16.1 ✓ 8
75.44 odd 10 75.2.g.b.61.1 yes 8
75.47 even 20 375.2.i.b.49.1 16
75.53 even 20 375.2.i.b.49.4 16
75.56 odd 10 375.2.g.b.301.2 8
75.71 odd 10 375.2.g.b.76.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.b.16.1 ✓ 8 75.29 odd 10
75.2.g.b.61.1 yes 8 75.44 odd 10
225.2.h.c.91.2 8 25.4 even 10
225.2.h.c.136.2 8 25.19 even 10
375.2.g.b.76.2 8 75.71 odd 10
375.2.g.b.301.2 8 75.56 odd 10
375.2.i.b.49.1 16 75.47 even 20
375.2.i.b.49.4 16 75.53 even 20
375.2.i.b.199.1 16 75.8 even 20
375.2.i.b.199.4 16 75.17 even 20
1875.2.a.e.1.1 4 3.2 odd 2
1875.2.a.h.1.4 4 15.14 odd 2
1875.2.b.c.1249.1 8 15.2 even 4
1875.2.b.c.1249.8 8 15.8 even 4
5625.2.a.i.1.1 4 5.4 even 2
5625.2.a.n.1.4 4 1.1 even 1 trivial