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Properties

Label 1875.2.a.i.1.4
Level $1875$
Weight $2$
Character 1875.1
Self dual yes
Analytic conductor $14.972$
Analytic rank $0$
Dimension $6$
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 = [6,-1,6] 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: \(6\)
Coefficient field: 6.6.46840000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.364088\) 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)
 
\(f(q)\) \(=\) \(q+0.364088 q^{2} +1.00000 q^{3} -1.86744 q^{4} +0.364088 q^{6} -2.24941 q^{7} -1.40809 q^{8} +1.00000 q^{9} +4.63962 q^{11} -1.86744 q^{12} -3.75730 q^{13} -0.818981 q^{14} +3.22221 q^{16} +5.36779 q^{17} +0.364088 q^{18} -5.66369 q^{19} -2.24941 q^{21} +1.68923 q^{22} -1.32214 q^{23} -1.40809 q^{24} -1.36799 q^{26} +1.00000 q^{27} +4.20063 q^{28} +8.86744 q^{29} +1.21200 q^{31} +3.98934 q^{32} +4.63962 q^{33} +1.95435 q^{34} -1.86744 q^{36} +2.15975 q^{37} -2.06208 q^{38} -3.75730 q^{39} +3.49965 q^{41} -0.818981 q^{42} -1.16800 q^{43} -8.66420 q^{44} -0.481374 q^{46} +2.36614 q^{47} +3.22221 q^{48} -1.94017 q^{49} +5.36779 q^{51} +7.01653 q^{52} +14.1656 q^{53} +0.364088 q^{54} +3.16736 q^{56} -5.66369 q^{57} +3.22853 q^{58} +0.367089 q^{59} +5.29590 q^{61} +0.441273 q^{62} -2.24941 q^{63} -4.99195 q^{64} +1.68923 q^{66} -8.06723 q^{67} -10.0240 q^{68} -1.32214 q^{69} +11.4185 q^{71} -1.40809 q^{72} +13.7580 q^{73} +0.786340 q^{74} +10.5766 q^{76} -10.4364 q^{77} -1.36799 q^{78} +11.6247 q^{79} +1.00000 q^{81} +1.27418 q^{82} -11.1027 q^{83} +4.20063 q^{84} -0.425253 q^{86} +8.86744 q^{87} -6.53299 q^{88} +16.6526 q^{89} +8.45169 q^{91} +2.46901 q^{92} +1.21200 q^{93} +0.861482 q^{94} +3.98934 q^{96} -14.4790 q^{97} -0.706393 q^{98} +4.63962 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} + 6 q^{3} + 11 q^{4} - q^{6} + 2 q^{7} - 6 q^{8} + 6 q^{9} + 11 q^{12} - 4 q^{14} + 17 q^{16} - 2 q^{17} - q^{18} - 2 q^{19} + 2 q^{21} - 9 q^{22} + q^{23} - 6 q^{24} + 37 q^{26} + 6 q^{27}+ \cdots - 15 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.364088 0.257449 0.128724 − 0.991680i \(-0.458912\pi\)
0.128724 + 0.991680i \(0.458912\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.86744 −0.933720
\(5\) 0 0
\(6\) 0.364088 0.148638
\(7\) −2.24941 −0.850196 −0.425098 − 0.905147i \(-0.639760\pi\)
−0.425098 + 0.905147i \(0.639760\pi\)
\(8\) −1.40809 −0.497834
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.63962 1.39890 0.699448 − 0.714683i \(-0.253429\pi\)
0.699448 + 0.714683i \(0.253429\pi\)
\(12\) −1.86744 −0.539084
\(13\) −3.75730 −1.04209 −0.521044 − 0.853530i \(-0.674457\pi\)
−0.521044 + 0.853530i \(0.674457\pi\)
\(14\) −0.818981 −0.218882
\(15\) 0 0
\(16\) 3.22221 0.805553
\(17\) 5.36779 1.30188 0.650940 − 0.759129i \(-0.274375\pi\)
0.650940 + 0.759129i \(0.274375\pi\)
\(18\) 0.364088 0.0858163
\(19\) −5.66369 −1.29934 −0.649669 − 0.760217i \(-0.725093\pi\)
−0.649669 + 0.760217i \(0.725093\pi\)
\(20\) 0 0
\(21\) −2.24941 −0.490861
\(22\) 1.68923 0.360144
\(23\) −1.32214 −0.275685 −0.137842 − 0.990454i \(-0.544017\pi\)
−0.137842 + 0.990454i \(0.544017\pi\)
\(24\) −1.40809 −0.287425
\(25\) 0 0
\(26\) −1.36799 −0.268284
\(27\) 1.00000 0.192450
\(28\) 4.20063 0.793845
\(29\) 8.86744 1.64664 0.823321 − 0.567576i \(-0.192119\pi\)
0.823321 + 0.567576i \(0.192119\pi\)
\(30\) 0 0
\(31\) 1.21200 0.217681 0.108841 − 0.994059i \(-0.465286\pi\)
0.108841 + 0.994059i \(0.465286\pi\)
\(32\) 3.98934 0.705223
\(33\) 4.63962 0.807653
\(34\) 1.95435 0.335168
\(35\) 0 0
\(36\) −1.86744 −0.311240
\(37\) 2.15975 0.355061 0.177531 − 0.984115i \(-0.443189\pi\)
0.177531 + 0.984115i \(0.443189\pi\)
\(38\) −2.06208 −0.334513
\(39\) −3.75730 −0.601649
\(40\) 0 0
\(41\) 3.49965 0.546553 0.273277 − 0.961935i \(-0.411893\pi\)
0.273277 + 0.961935i \(0.411893\pi\)
\(42\) −0.818981 −0.126372
\(43\) −1.16800 −0.178118 −0.0890589 − 0.996026i \(-0.528386\pi\)
−0.0890589 + 0.996026i \(0.528386\pi\)
\(44\) −8.66420 −1.30618
\(45\) 0 0
\(46\) −0.481374 −0.0709748
\(47\) 2.36614 0.345137 0.172568 − 0.984998i \(-0.444793\pi\)
0.172568 + 0.984998i \(0.444793\pi\)
\(48\) 3.22221 0.465086
\(49\) −1.94017 −0.277167
\(50\) 0 0
\(51\) 5.36779 0.751641
\(52\) 7.01653 0.973018
\(53\) 14.1656 1.94580 0.972901 − 0.231223i \(-0.0742728\pi\)
0.972901 + 0.231223i \(0.0742728\pi\)
\(54\) 0.364088 0.0495461
\(55\) 0 0
\(56\) 3.16736 0.423256
\(57\) −5.66369 −0.750174
\(58\) 3.22853 0.423926
\(59\) 0.367089 0.0477909 0.0238955 − 0.999714i \(-0.492393\pi\)
0.0238955 + 0.999714i \(0.492393\pi\)
\(60\) 0 0
\(61\) 5.29590 0.678070 0.339035 − 0.940774i \(-0.389899\pi\)
0.339035 + 0.940774i \(0.389899\pi\)
\(62\) 0.441273 0.0560417
\(63\) −2.24941 −0.283399
\(64\) −4.99195 −0.623994
\(65\) 0 0
\(66\) 1.68923 0.207930
\(67\) −8.06723 −0.985570 −0.492785 − 0.870151i \(-0.664021\pi\)
−0.492785 + 0.870151i \(0.664021\pi\)
\(68\) −10.0240 −1.21559
\(69\) −1.32214 −0.159167
\(70\) 0 0
\(71\) 11.4185 1.35513 0.677563 − 0.735465i \(-0.263036\pi\)
0.677563 + 0.735465i \(0.263036\pi\)
\(72\) −1.40809 −0.165945
\(73\) 13.7580 1.61025 0.805126 − 0.593104i \(-0.202098\pi\)
0.805126 + 0.593104i \(0.202098\pi\)
\(74\) 0.786340 0.0914101
\(75\) 0 0
\(76\) 10.5766 1.21322
\(77\) −10.4364 −1.18934
\(78\) −1.36799 −0.154894
\(79\) 11.6247 1.30789 0.653943 − 0.756544i \(-0.273114\pi\)
0.653943 + 0.756544i \(0.273114\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 1.27418 0.140710
\(83\) −11.1027 −1.21868 −0.609338 − 0.792910i \(-0.708565\pi\)
−0.609338 + 0.792910i \(0.708565\pi\)
\(84\) 4.20063 0.458326
\(85\) 0 0
\(86\) −0.425253 −0.0458562
\(87\) 8.86744 0.950689
\(88\) −6.53299 −0.696419
\(89\) 16.6526 1.76518 0.882588 − 0.470148i \(-0.155799\pi\)
0.882588 + 0.470148i \(0.155799\pi\)
\(90\) 0 0
\(91\) 8.45169 0.885978
\(92\) 2.46901 0.257412
\(93\) 1.21200 0.125678
\(94\) 0.861482 0.0888551
\(95\) 0 0
\(96\) 3.98934 0.407161
\(97\) −14.4790 −1.47012 −0.735061 − 0.678001i \(-0.762847\pi\)
−0.735061 + 0.678001i \(0.762847\pi\)
\(98\) −0.706393 −0.0713565
\(99\) 4.63962 0.466299
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.i.1.4 ✓ 6
3.2 odd 2 5625.2.a.r.1.3 6
5.2 odd 4 1875.2.b.e.1249.7 12
5.3 odd 4 1875.2.b.e.1249.6 12
5.4 even 2 1875.2.a.l.1.3 yes 6
15.14 odd 2 5625.2.a.o.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.i.1.4 ✓ 6 1.1 even 1 trivial
1875.2.a.l.1.3 yes 6 5.4 even 2
1875.2.b.e.1249.6 12 5.3 odd 4
1875.2.b.e.1249.7 12 5.2 odd 4
5625.2.a.o.1.4 6 15.14 odd 2
5625.2.a.r.1.3 6 3.2 odd 2