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Properties

Label 5625.2.a.n.1.3
Level $5625$
Weight $2$
Character 5625.1
Self dual yes
Analytic conductor $44.916$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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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 1.3
Root \(2.12233\) 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)
 
\(f(q)\) \(=\) \(q+2.12233 q^{2} +2.50430 q^{4} -4.35840 q^{7} +1.07029 q^{8} +1.57991 q^{11} -1.19794 q^{13} -9.24998 q^{14} -2.73708 q^{16} -1.12233 q^{17} +7.67867 q^{19} +3.35309 q^{22} -2.32027 q^{23} -2.54243 q^{26} -10.9147 q^{28} +5.50430 q^{29} +4.80206 q^{31} -7.94959 q^{32} -2.38197 q^{34} +6.37232 q^{37} +16.2967 q^{38} +7.47214 q^{41} +1.24998 q^{43} +3.95656 q^{44} -4.92439 q^{46} +4.12765 q^{47} +11.9957 q^{49} -3.00000 q^{52} +3.74568 q^{53} -4.66476 q^{56} +11.6820 q^{58} -9.15613 q^{59} +1.39588 q^{61} +10.1916 q^{62} -11.3975 q^{64} +3.86270 q^{67} -2.81066 q^{68} +10.6051 q^{71} -4.99672 q^{73} +13.5242 q^{74} +19.2297 q^{76} -6.88586 q^{77} +14.5531 q^{79} +15.8584 q^{82} -8.73603 q^{83} +2.65288 q^{86} +1.69096 q^{88} +10.0381 q^{89} +5.22110 q^{91} -5.81066 q^{92} +8.76025 q^{94} +7.49996 q^{97} +25.4588 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 8 q^{4} - 2 q^{7} + 15 q^{8} + 7 q^{11} - q^{13} - 16 q^{14} + 4 q^{16} + 2 q^{17} + 5 q^{19} + 6 q^{22} + q^{23} - 3 q^{26} - 9 q^{28} + 20 q^{29} + 23 q^{31} + 12 q^{32} - 14 q^{34}+ \cdots - q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.12233 1.50072 0.750358 − 0.661031i \(-0.229881\pi\)
0.750358 + 0.661031i \(0.229881\pi\)
\(3\) 0 0
\(4\) 2.50430 1.25215
\(5\) 0 0
\(6\) 0 0
\(7\) −4.35840 −1.64732 −0.823660 − 0.567083i \(-0.808072\pi\)
−0.823660 + 0.567083i \(0.808072\pi\)
\(8\) 1.07029 0.378405
\(9\) 0 0
\(10\) 0 0
\(11\) 1.57991 0.476360 0.238180 − 0.971221i \(-0.423449\pi\)
0.238180 + 0.971221i \(0.423449\pi\)
\(12\) 0 0
\(13\) −1.19794 −0.332249 −0.166124 − 0.986105i \(-0.553125\pi\)
−0.166124 + 0.986105i \(0.553125\pi\)
\(14\) −9.24998 −2.47216
\(15\) 0 0
\(16\) −2.73708 −0.684271
\(17\) −1.12233 −0.272206 −0.136103 − 0.990695i \(-0.543458\pi\)
−0.136103 + 0.990695i \(0.543458\pi\)
\(18\) 0 0
\(19\) 7.67867 1.76161 0.880804 − 0.473480i \(-0.157002\pi\)
0.880804 + 0.473480i \(0.157002\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.35309 0.714881
\(23\) −2.32027 −0.483810 −0.241905 − 0.970300i \(-0.577772\pi\)
−0.241905 + 0.970300i \(0.577772\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.54243 −0.498611
\(27\) 0 0
\(28\) −10.9147 −2.06269
\(29\) 5.50430 1.02212 0.511061 − 0.859544i \(-0.329252\pi\)
0.511061 + 0.859544i \(0.329252\pi\)
\(30\) 0 0
\(31\) 4.80206 0.862475 0.431238 − 0.902238i \(-0.358077\pi\)
0.431238 + 0.902238i \(0.358077\pi\)
\(32\) −7.94959 −1.40530
\(33\) 0 0
\(34\) −2.38197 −0.408504
\(35\) 0 0
\(36\) 0 0
\(37\) 6.37232 1.04760 0.523801 − 0.851841i \(-0.324514\pi\)
0.523801 + 0.851841i \(0.324514\pi\)
\(38\) 16.2967 2.64368
\(39\) 0 0
\(40\) 0 0
\(41\) 7.47214 1.16695 0.583476 − 0.812131i \(-0.301692\pi\)
0.583476 + 0.812131i \(0.301692\pi\)
\(42\) 0 0
\(43\) 1.24998 0.190620 0.0953102 − 0.995448i \(-0.469616\pi\)
0.0953102 + 0.995448i \(0.469616\pi\)
\(44\) 3.95656 0.596473
\(45\) 0 0
\(46\) −4.92439 −0.726062
\(47\) 4.12765 0.602079 0.301040 − 0.953612i \(-0.402666\pi\)
0.301040 + 0.953612i \(0.402666\pi\)
\(48\) 0 0
\(49\) 11.9957 1.71367
\(50\) 0 0
\(51\) 0 0
\(52\) −3.00000 −0.416025
\(53\) 3.74568 0.514509 0.257254 − 0.966344i \(-0.417182\pi\)
0.257254 + 0.966344i \(0.417182\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −4.66476 −0.623355
\(57\) 0 0
\(58\) 11.6820 1.53392
\(59\) −9.15613 −1.19203 −0.596013 − 0.802975i \(-0.703249\pi\)
−0.596013 + 0.802975i \(0.703249\pi\)
\(60\) 0 0
\(61\) 1.39588 0.178724 0.0893620 − 0.995999i \(-0.471517\pi\)
0.0893620 + 0.995999i \(0.471517\pi\)
\(62\) 10.1916 1.29433
\(63\) 0 0
\(64\) −11.3975 −1.42469
\(65\) 0 0
\(66\) 0 0
\(67\) 3.86270 0.471904 0.235952 − 0.971765i \(-0.424179\pi\)
0.235952 + 0.971765i \(0.424179\pi\)
\(68\) −2.81066 −0.340843
\(69\) 0 0
\(70\) 0 0
\(71\) 10.6051 1.25859 0.629297 − 0.777165i \(-0.283343\pi\)
0.629297 + 0.777165i \(0.283343\pi\)
\(72\) 0 0
\(73\) −4.99672 −0.584821 −0.292411 − 0.956293i \(-0.594457\pi\)
−0.292411 + 0.956293i \(0.594457\pi\)
\(74\) 13.5242 1.57215
\(75\) 0 0
\(76\) 19.2297 2.20580
\(77\) −6.88586 −0.784717
\(78\) 0 0
\(79\) 14.5531 1.63735 0.818673 − 0.574259i \(-0.194710\pi\)
0.818673 + 0.574259i \(0.194710\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 15.8584 1.75126
\(83\) −8.73603 −0.958904 −0.479452 − 0.877568i \(-0.659165\pi\)
−0.479452 + 0.877568i \(0.659165\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.65288 0.286067
\(87\) 0 0
\(88\) 1.69096 0.180257
\(89\) 10.0381 1.06404 0.532020 − 0.846732i \(-0.321433\pi\)
0.532020 + 0.846732i \(0.321433\pi\)
\(90\) 0 0
\(91\) 5.22110 0.547320
\(92\) −5.81066 −0.605803
\(93\) 0 0
\(94\) 8.76025 0.903550
\(95\) 0 0
\(96\) 0 0
\(97\) 7.49996 0.761506 0.380753 − 0.924677i \(-0.375665\pi\)
0.380753 + 0.924677i \(0.375665\pi\)
\(98\) 25.4588 2.57173
\(99\) 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.1.3 4
3.2 odd 2 1875.2.a.e.1.2 4
5.4 even 2 5625.2.a.i.1.2 4
15.2 even 4 1875.2.b.c.1249.2 8
15.8 even 4 1875.2.b.c.1249.7 8
15.14 odd 2 1875.2.a.h.1.3 4
25.9 even 10 225.2.h.c.181.1 8
25.14 even 10 225.2.h.c.46.1 8
75.2 even 20 375.2.i.b.274.1 16
75.11 odd 10 375.2.g.b.226.1 8
75.14 odd 10 75.2.g.b.46.2 yes 8
75.23 even 20 375.2.i.b.274.4 16
75.38 even 20 375.2.i.b.349.1 16
75.41 odd 10 375.2.g.b.151.1 8
75.59 odd 10 75.2.g.b.31.2 ✓ 8
75.62 even 20 375.2.i.b.349.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.b.31.2 ✓ 8 75.59 odd 10
75.2.g.b.46.2 yes 8 75.14 odd 10
225.2.h.c.46.1 8 25.14 even 10
225.2.h.c.181.1 8 25.9 even 10
375.2.g.b.151.1 8 75.41 odd 10
375.2.g.b.226.1 8 75.11 odd 10
375.2.i.b.274.1 16 75.2 even 20
375.2.i.b.274.4 16 75.23 even 20
375.2.i.b.349.1 16 75.38 even 20
375.2.i.b.349.4 16 75.62 even 20
1875.2.a.e.1.2 4 3.2 odd 2
1875.2.a.h.1.3 4 15.14 odd 2
1875.2.b.c.1249.2 8 15.2 even 4
1875.2.b.c.1249.7 8 15.8 even 4
5625.2.a.i.1.2 4 5.4 even 2
5625.2.a.n.1.3 4 1.1 even 1 trivial