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Properties

Label 1875.2.a.h.1.3
Level $1875$
Weight $2$
Character 1875.1
Self dual yes
Analytic conductor $14.972$
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] = [1875,2,Mod(1,1875)] 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("1875.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1875, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2,4] 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: \(14.9719503790\)
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 \(\chi\) \(=\) 1875.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.12233 q^{2} +1.00000 q^{3} +2.50430 q^{4} +2.12233 q^{6} +4.35840 q^{7} +1.07029 q^{8} +1.00000 q^{9} -1.57991 q^{11} +2.50430 q^{12} +1.19794 q^{13} +9.24998 q^{14} -2.73708 q^{16} -1.12233 q^{17} +2.12233 q^{18} +7.67867 q^{19} +4.35840 q^{21} -3.35309 q^{22} -2.32027 q^{23} +1.07029 q^{24} +2.54243 q^{26} +1.00000 q^{27} +10.9147 q^{28} -5.50430 q^{29} +4.80206 q^{31} -7.94959 q^{32} -1.57991 q^{33} -2.38197 q^{34} +2.50430 q^{36} -6.37232 q^{37} +16.2967 q^{38} +1.19794 q^{39} -7.47214 q^{41} +9.24998 q^{42} -1.24998 q^{43} -3.95656 q^{44} -4.92439 q^{46} +4.12765 q^{47} -2.73708 q^{48} +11.9957 q^{49} -1.12233 q^{51} +3.00000 q^{52} +3.74568 q^{53} +2.12233 q^{54} +4.66476 q^{56} +7.67867 q^{57} -11.6820 q^{58} +9.15613 q^{59} +1.39588 q^{61} +10.1916 q^{62} +4.35840 q^{63} -11.3975 q^{64} -3.35309 q^{66} -3.86270 q^{67} -2.81066 q^{68} -2.32027 q^{69} -10.6051 q^{71} +1.07029 q^{72} +4.99672 q^{73} -13.5242 q^{74} +19.2297 q^{76} -6.88586 q^{77} +2.54243 q^{78} +14.5531 q^{79} +1.00000 q^{81} -15.8584 q^{82} -8.73603 q^{83} +10.9147 q^{84} -2.65288 q^{86} -5.50430 q^{87} -1.69096 q^{88} -10.0381 q^{89} +5.22110 q^{91} -5.81066 q^{92} +4.80206 q^{93} +8.76025 q^{94} -7.94959 q^{96} -7.49996 q^{97} +25.4588 q^{98} -1.57991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 4 q^{3} + 8 q^{4} + 2 q^{6} + 2 q^{7} + 15 q^{8} + 4 q^{9} - 7 q^{11} + 8 q^{12} + q^{13} + 16 q^{14} + 4 q^{16} + 2 q^{17} + 2 q^{18} + 5 q^{19} + 2 q^{21} - 6 q^{22} + q^{23} + 15 q^{24}+ \cdots - 7 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.12233 1.50072 0.750358 − 0.661031i \(-0.229881\pi\)
0.750358 + 0.661031i \(0.229881\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.50430 1.25215
\(5\) 0 0
\(6\) 2.12233 0.866439
\(7\) 4.35840 1.64732 0.823660 − 0.567083i \(-0.191928\pi\)
0.823660 + 0.567083i \(0.191928\pi\)
\(8\) 1.07029 0.378405
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.57991 −0.476360 −0.238180 − 0.971221i \(-0.576551\pi\)
−0.238180 + 0.971221i \(0.576551\pi\)
\(12\) 2.50430 0.722929
\(13\) 1.19794 0.332249 0.166124 − 0.986105i \(-0.446875\pi\)
0.166124 + 0.986105i \(0.446875\pi\)
\(14\) 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 \(0.543458\pi\)
\(18\) 2.12233 0.500239
\(19\) 7.67867 1.76161 0.880804 − 0.473480i \(-0.157002\pi\)
0.880804 + 0.473480i \(0.157002\pi\)
\(20\) 0 0
\(21\) 4.35840 0.951081
\(22\) −3.35309 −0.714881
\(23\) −2.32027 −0.483810 −0.241905 − 0.970300i \(-0.577772\pi\)
−0.241905 + 0.970300i \(0.577772\pi\)
\(24\) 1.07029 0.218472
\(25\) 0 0
\(26\) 2.54243 0.498611
\(27\) 1.00000 0.192450
\(28\) 10.9147 2.06269
\(29\) −5.50430 −1.02212 −0.511061 − 0.859544i \(-0.670748\pi\)
−0.511061 + 0.859544i \(0.670748\pi\)
\(30\) 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
\(33\) −1.57991 −0.275026
\(34\) −2.38197 −0.408504
\(35\) 0 0
\(36\) 2.50430 0.417383
\(37\) −6.37232 −1.04760 −0.523801 − 0.851841i \(-0.675486\pi\)
−0.523801 + 0.851841i \(0.675486\pi\)
\(38\) 16.2967 2.64368
\(39\) 1.19794 0.191824
\(40\) 0 0
\(41\) −7.47214 −1.16695 −0.583476 − 0.812131i \(-0.698308\pi\)
−0.583476 + 0.812131i \(0.698308\pi\)
\(42\) 9.24998 1.42730
\(43\) −1.24998 −0.190620 −0.0953102 − 0.995448i \(-0.530384\pi\)
−0.0953102 + 0.995448i \(0.530384\pi\)
\(44\) −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 \(0.402666\pi\)
\(48\) −2.73708 −0.395064
\(49\) 11.9957 1.71367
\(50\) 0 0
\(51\) −1.12233 −0.157158
\(52\) 3.00000 0.416025
\(53\) 3.74568 0.514509 0.257254 − 0.966344i \(-0.417182\pi\)
0.257254 + 0.966344i \(0.417182\pi\)
\(54\) 2.12233 0.288813
\(55\) 0 0
\(56\) 4.66476 0.623355
\(57\) 7.67867 1.01707
\(58\) −11.6820 −1.53392
\(59\) 9.15613 1.19203 0.596013 − 0.802975i \(-0.296751\pi\)
0.596013 + 0.802975i \(0.296751\pi\)
\(60\) 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
\(63\) 4.35840 0.549107
\(64\) −11.3975 −1.42469
\(65\) 0 0
\(66\) −3.35309 −0.412736
\(67\) −3.86270 −0.471904 −0.235952 − 0.971765i \(-0.575821\pi\)
−0.235952 + 0.971765i \(0.575821\pi\)
\(68\) −2.81066 −0.340843
\(69\) −2.32027 −0.279328
\(70\) 0 0
\(71\) −10.6051 −1.25859 −0.629297 − 0.777165i \(-0.716657\pi\)
−0.629297 + 0.777165i \(0.716657\pi\)
\(72\) 1.07029 0.126135
\(73\) 4.99672 0.584821 0.292411 − 0.956293i \(-0.405543\pi\)
0.292411 + 0.956293i \(0.405543\pi\)
\(74\) −13.5242 −1.57215
\(75\) 0 0
\(76\) 19.2297 2.20580
\(77\) −6.88586 −0.784717
\(78\) 2.54243 0.287873
\(79\) 14.5531 1.63735 0.818673 − 0.574259i \(-0.194710\pi\)
0.818673 + 0.574259i \(0.194710\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −15.8584 −1.75126
\(83\) −8.73603 −0.958904 −0.479452 − 0.877568i \(-0.659165\pi\)
−0.479452 + 0.877568i \(0.659165\pi\)
\(84\) 10.9147 1.19090
\(85\) 0 0
\(86\) −2.65288 −0.286067
\(87\) −5.50430 −0.590123
\(88\) −1.69096 −0.180257
\(89\) −10.0381 −1.06404 −0.532020 − 0.846732i \(-0.678567\pi\)
−0.532020 + 0.846732i \(0.678567\pi\)
\(90\) 0 0
\(91\) 5.22110 0.547320
\(92\) −5.81066 −0.605803
\(93\) 4.80206 0.497950
\(94\) 8.76025 0.903550
\(95\) 0 0
\(96\) −7.94959 −0.811351
\(97\) −7.49996 −0.761506 −0.380753 − 0.924677i \(-0.624335\pi\)
−0.380753 + 0.924677i \(0.624335\pi\)
\(98\) 25.4588 2.57173
\(99\) −1.57991 −0.158787
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 1875.2.a.h.1.3 4
3.2 odd 2 5625.2.a.i.1.2 4
5.2 odd 4 1875.2.b.c.1249.7 8
5.3 odd 4 1875.2.b.c.1249.2 8
5.4 even 2 1875.2.a.e.1.2 4
15.14 odd 2 5625.2.a.n.1.3 4
25.2 odd 20 375.2.i.b.274.4 16
25.9 even 10 375.2.g.b.151.1 8
25.11 even 5 75.2.g.b.46.2 yes 8
25.12 odd 20 375.2.i.b.349.1 16
25.13 odd 20 375.2.i.b.349.4 16
25.14 even 10 375.2.g.b.226.1 8
25.16 even 5 75.2.g.b.31.2 ✓ 8
25.23 odd 20 375.2.i.b.274.1 16
75.11 odd 10 225.2.h.c.46.1 8
75.41 odd 10 225.2.h.c.181.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.b.31.2 ✓ 8 25.16 even 5
75.2.g.b.46.2 yes 8 25.11 even 5
225.2.h.c.46.1 8 75.11 odd 10
225.2.h.c.181.1 8 75.41 odd 10
375.2.g.b.151.1 8 25.9 even 10
375.2.g.b.226.1 8 25.14 even 10
375.2.i.b.274.1 16 25.23 odd 20
375.2.i.b.274.4 16 25.2 odd 20
375.2.i.b.349.1 16 25.12 odd 20
375.2.i.b.349.4 16 25.13 odd 20
1875.2.a.e.1.2 4 5.4 even 2
1875.2.a.h.1.3 4 1.1 even 1 trivial
1875.2.b.c.1249.2 8 5.3 odd 4
1875.2.b.c.1249.7 8 5.2 odd 4
5625.2.a.i.1.2 4 3.2 odd 2
5625.2.a.n.1.3 4 15.14 odd 2