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Properties

Label 1875.2.b.e.1249.6
Level $1875$
Weight $2$
Character 1875.1249
Analytic conductor $14.972$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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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(1249,1875)] 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("1875.1249"); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.9719503790\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 23x^{10} + 199x^{8} + 794x^{6} + 1399x^{4} + 783x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.6
Root \(-0.364088i\) of defining polynomial
Character \(\chi\) \(=\) 1875.1249
Dual form 1875.2.b.e.1249.7

$q$-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)
 
\(f(q)\) \(=\) \(q-0.364088i q^{2} +1.00000i q^{3} +1.86744 q^{4} +0.364088 q^{6} +2.24941i q^{7} -1.40809i q^{8} -1.00000 q^{9} +4.63962 q^{11} +1.86744i q^{12} -3.75730i q^{13} +0.818981 q^{14} +3.22221 q^{16} -5.36779i q^{17} +0.364088i q^{18} +5.66369 q^{19} -2.24941 q^{21} -1.68923i q^{22} -1.32214i q^{23} +1.40809 q^{24} -1.36799 q^{26} -1.00000i q^{27} +4.20063i q^{28} -8.86744 q^{29} +1.21200 q^{31} -3.98934i q^{32} +4.63962i q^{33} -1.95435 q^{34} -1.86744 q^{36} -2.15975i q^{37} -2.06208i q^{38} +3.75730 q^{39} +3.49965 q^{41} +0.818981i q^{42} -1.16800i q^{43} +8.66420 q^{44} -0.481374 q^{46} -2.36614i q^{47} +3.22221i q^{48} +1.94017 q^{49} +5.36779 q^{51} -7.01653i q^{52} +14.1656i q^{53} -0.364088 q^{54} +3.16736 q^{56} +5.66369i q^{57} +3.22853i q^{58} -0.367089 q^{59} +5.29590 q^{61} -0.441273i q^{62} -2.24941i q^{63} +4.99195 q^{64} +1.68923 q^{66} +8.06723i q^{67} -10.0240i q^{68} +1.32214 q^{69} +11.4185 q^{71} +1.40809i q^{72} +13.7580i q^{73} -0.786340 q^{74} +10.5766 q^{76} +10.4364i q^{77} -1.36799i q^{78} -11.6247 q^{79} +1.00000 q^{81} -1.27418i q^{82} -11.1027i q^{83} -4.20063 q^{84} -0.425253 q^{86} -8.86744i q^{87} -6.53299i q^{88} -16.6526 q^{89} +8.45169 q^{91} -2.46901i q^{92} +1.21200i q^{93} -0.861482 q^{94} +3.98934 q^{96} +14.4790i q^{97} -0.706393i q^{98} -4.63962 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 22 q^{4} - 2 q^{6} - 12 q^{9} + 8 q^{14} + 34 q^{16} + 4 q^{19} + 4 q^{21} + 12 q^{24} + 74 q^{26} - 62 q^{29} - 4 q^{31} - 74 q^{34} + 22 q^{36} + 66 q^{41} + 22 q^{44} - 24 q^{46} - 8 q^{49} - 4 q^{51}+ \cdots - 66 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times\).

\(n\) \(626\) \(1252\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.364088i − 0.257449i −0.991680 − 0.128724i \(-0.958912\pi\)
0.991680 − 0.128724i \(-0.0410883\pi\)
\(3\) 1.00000i 0.577350i
\(4\) 1.86744 0.933720
\(5\) 0 0
\(6\) 0.364088 0.148638
\(7\) 2.24941i 0.850196i 0.905147 + 0.425098i \(0.139760\pi\)
−0.905147 + 0.425098i \(0.860240\pi\)
\(8\) − 1.40809i − 0.497834i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.63962 1.39890 0.699448 − 0.714683i \(-0.253429\pi\)
0.699448 + 0.714683i \(0.253429\pi\)
\(12\) 1.86744i 0.539084i
\(13\) − 3.75730i − 1.04209i −0.853530 − 0.521044i \(-0.825543\pi\)
0.853530 − 0.521044i \(-0.174457\pi\)
\(14\) 0.818981 0.218882
\(15\) 0 0
\(16\) 3.22221 0.805553
\(17\) − 5.36779i − 1.30188i −0.759129 − 0.650940i \(-0.774375\pi\)
0.759129 − 0.650940i \(-0.225625\pi\)
\(18\) 0.364088i 0.0858163i
\(19\) 5.66369 1.29934 0.649669 − 0.760217i \(-0.274907\pi\)
0.649669 + 0.760217i \(0.274907\pi\)
\(20\) 0 0
\(21\) −2.24941 −0.490861
\(22\) − 1.68923i − 0.360144i
\(23\) − 1.32214i − 0.275685i −0.990454 − 0.137842i \(-0.955983\pi\)
0.990454 − 0.137842i \(-0.0440168\pi\)
\(24\) 1.40809 0.287425
\(25\) 0 0
\(26\) −1.36799 −0.268284
\(27\) − 1.00000i − 0.192450i
\(28\) 4.20063i 0.793845i
\(29\) −8.86744 −1.64664 −0.823321 − 0.567576i \(-0.807881\pi\)
−0.823321 + 0.567576i \(0.807881\pi\)
\(30\) 0 0
\(31\) 1.21200 0.217681 0.108841 − 0.994059i \(-0.465286\pi\)
0.108841 + 0.994059i \(0.465286\pi\)
\(32\) − 3.98934i − 0.705223i
\(33\) 4.63962i 0.807653i
\(34\) −1.95435 −0.335168
\(35\) 0 0
\(36\) −1.86744 −0.311240
\(37\) − 2.15975i − 0.355061i −0.984115 − 0.177531i \(-0.943189\pi\)
0.984115 − 0.177531i \(-0.0568109\pi\)
\(38\) − 2.06208i − 0.334513i
\(39\) 3.75730 0.601649
\(40\) 0 0
\(41\) 3.49965 0.546553 0.273277 − 0.961935i \(-0.411893\pi\)
0.273277 + 0.961935i \(0.411893\pi\)
\(42\) 0.818981i 0.126372i
\(43\) − 1.16800i − 0.178118i −0.996026 − 0.0890589i \(-0.971614\pi\)
0.996026 − 0.0890589i \(-0.0283859\pi\)
\(44\) 8.66420 1.30618
\(45\) 0 0
\(46\) −0.481374 −0.0709748
\(47\) − 2.36614i − 0.345137i −0.984998 − 0.172568i \(-0.944793\pi\)
0.984998 − 0.172568i \(-0.0552066\pi\)
\(48\) 3.22221i 0.465086i
\(49\) 1.94017 0.277167
\(50\) 0 0
\(51\) 5.36779 0.751641
\(52\) − 7.01653i − 0.973018i
\(53\) 14.1656i 1.94580i 0.231223 + 0.972901i \(0.425727\pi\)
−0.231223 + 0.972901i \(0.574273\pi\)
\(54\) −0.364088 −0.0495461
\(55\) 0 0
\(56\) 3.16736 0.423256
\(57\) 5.66369i 0.750174i
\(58\) 3.22853i 0.423926i
\(59\) −0.367089 −0.0477909 −0.0238955 − 0.999714i \(-0.507607\pi\)
−0.0238955 + 0.999714i \(0.507607\pi\)
\(60\) 0 0
\(61\) 5.29590 0.678070 0.339035 − 0.940774i \(-0.389899\pi\)
0.339035 + 0.940774i \(0.389899\pi\)
\(62\) − 0.441273i − 0.0560417i
\(63\) − 2.24941i − 0.283399i
\(64\) 4.99195 0.623994
\(65\) 0 0
\(66\) 1.68923 0.207930
\(67\) 8.06723i 0.985570i 0.870151 + 0.492785i \(0.164021\pi\)
−0.870151 + 0.492785i \(0.835979\pi\)
\(68\) − 10.0240i − 1.21559i
\(69\) 1.32214 0.159167
\(70\) 0 0
\(71\) 11.4185 1.35513 0.677563 − 0.735465i \(-0.263036\pi\)
0.677563 + 0.735465i \(0.263036\pi\)
\(72\) 1.40809i 0.165945i
\(73\) 13.7580i 1.61025i 0.593104 + 0.805126i \(0.297902\pi\)
−0.593104 + 0.805126i \(0.702098\pi\)
\(74\) −0.786340 −0.0914101
\(75\) 0 0
\(76\) 10.5766 1.21322
\(77\) 10.4364i 1.18934i
\(78\) − 1.36799i − 0.154894i
\(79\) −11.6247 −1.30789 −0.653943 − 0.756544i \(-0.726886\pi\)
−0.653943 + 0.756544i \(0.726886\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 1.27418i − 0.140710i
\(83\) − 11.1027i − 1.21868i −0.792910 − 0.609338i \(-0.791435\pi\)
0.792910 − 0.609338i \(-0.208565\pi\)
\(84\) −4.20063 −0.458326
\(85\) 0 0
\(86\) −0.425253 −0.0458562
\(87\) − 8.86744i − 0.950689i
\(88\) − 6.53299i − 0.696419i
\(89\) −16.6526 −1.76518 −0.882588 − 0.470148i \(-0.844201\pi\)
−0.882588 + 0.470148i \(0.844201\pi\)
\(90\) 0 0
\(91\) 8.45169 0.885978
\(92\) − 2.46901i − 0.257412i
\(93\) 1.21200i 0.125678i
\(94\) −0.861482 −0.0888551
\(95\) 0 0
\(96\) 3.98934 0.407161
\(97\) 14.4790i 1.47012i 0.678001 + 0.735061i \(0.262847\pi\)
−0.678001 + 0.735061i \(0.737153\pi\)
\(98\) − 0.706393i − 0.0713565i
\(99\) −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.b.e.1249.6 12
5.2 odd 4 1875.2.a.i.1.4 ✓ 6
5.3 odd 4 1875.2.a.l.1.3 yes 6
5.4 even 2 inner 1875.2.b.e.1249.7 12
15.2 even 4 5625.2.a.r.1.3 6
15.8 even 4 5625.2.a.o.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1875.2.a.i.1.4 ✓ 6 5.2 odd 4
1875.2.a.l.1.3 yes 6 5.3 odd 4
1875.2.b.e.1249.6 12 1.1 even 1 trivial
1875.2.b.e.1249.7 12 5.4 even 2 inner
5625.2.a.o.1.4 6 15.8 even 4
5625.2.a.r.1.3 6 15.2 even 4