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Properties

Label 75.18.b.e.49.1
Level $75$
Weight $18$
Character 75.49
Analytic conductor $137.417$
Analytic rank $0$
Dimension $6$
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] = [75,18,Mod(49,75)] 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("75.49"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,14174] 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: \(137.416565508\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 364793x^{4} + 33276143056x^{2} + 15375182054400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-416.408i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.18.b.e.49.6

$q$-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)
 
\(f(q)\) \(=\) \(q-500.408i q^{2} -6561.00i q^{3} -119336. q^{4} -3.28318e6 q^{6} +8.90512e6i q^{7} -5.87255e6i q^{8} -4.30467e7 q^{9} +4.48684e8 q^{11} +7.82967e8i q^{12} +3.89738e9i q^{13} +4.45620e9 q^{14} -1.85803e10 q^{16} -5.96526e8i q^{17} +2.15409e10i q^{18} +4.36939e10 q^{19} +5.84265e10 q^{21} -2.24525e11i q^{22} -7.23343e10i q^{23} -3.85298e10 q^{24} +1.95028e12 q^{26} +2.82430e11i q^{27} -1.06271e12i q^{28} -1.82054e12 q^{29} +5.27128e12 q^{31} +8.52803e12i q^{32} -2.94382e12i q^{33} -2.98507e11 q^{34} +5.13704e12 q^{36} -1.62659e13i q^{37} -2.18648e13i q^{38} +2.55707e13 q^{39} +9.78891e12 q^{41} -2.92371e13i q^{42} -1.46155e14i q^{43} -5.35444e13 q^{44} -3.61967e13 q^{46} +2.43226e14i q^{47} +1.21906e14i q^{48} +1.53329e14 q^{49} -3.91381e12 q^{51} -4.65100e14i q^{52} -6.84721e14i q^{53} +1.41330e14 q^{54} +5.22958e13 q^{56} -2.86676e14i q^{57} +9.11015e14i q^{58} -9.83635e14 q^{59} -1.84021e15 q^{61} -2.63779e15i q^{62} -3.83336e14i q^{63} +1.83214e15 q^{64} -1.47311e15 q^{66} -2.46536e15i q^{67} +7.11874e13i q^{68} -4.74585e14 q^{69} +9.43154e14 q^{71} +2.52794e14i q^{72} -1.15522e16i q^{73} -8.13959e15 q^{74} -5.21428e15 q^{76} +3.99559e15i q^{77} -1.27958e16i q^{78} -1.58855e16 q^{79} +1.85302e15 q^{81} -4.89845e15i q^{82} -1.62194e16i q^{83} -6.97241e15 q^{84} -7.31369e16 q^{86} +1.19446e16i q^{87} -2.63492e15i q^{88} +4.03683e16 q^{89} -3.47067e16 q^{91} +8.63212e15i q^{92} -3.45849e16i q^{93} +1.21712e17 q^{94} +5.59524e16 q^{96} -7.85724e16i q^{97} -7.67273e16i q^{98} -1.93144e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 14174 q^{4} - 3319866 q^{6} - 258280326 q^{9} + 1887127360 q^{11} - 3694967976 q^{14} - 43887742718 q^{16} + 156244985992 q^{19} - 56850855048 q^{21} + 212425667586 q^{24} + 5331480861772 q^{26} - 11550537176156 q^{29}+ \cdots - 81\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 500.408i − 1.38220i −0.722761 − 0.691098i \(-0.757127\pi\)
0.722761 − 0.691098i \(-0.242873\pi\)
\(3\) − 6561.00i − 0.577350i
\(4\) −119336. −0.910465
\(5\) 0 0
\(6\) −3.28318e6 −0.798011
\(7\) 8.90512e6i 0.583857i 0.956440 + 0.291929i \(0.0942970\pi\)
−0.956440 + 0.291929i \(0.905703\pi\)
\(8\) − 5.87255e6i − 0.123755i
\(9\) −4.30467e7 −0.333333
\(10\) 0 0
\(11\) 4.48684e8 0.631107 0.315554 − 0.948908i \(-0.397810\pi\)
0.315554 + 0.948908i \(0.397810\pi\)
\(12\) 7.82967e8i 0.525657i
\(13\) 3.89738e9i 1.32512i 0.749010 + 0.662559i \(0.230530\pi\)
−0.749010 + 0.662559i \(0.769470\pi\)
\(14\) 4.45620e9 0.807005
\(15\) 0 0
\(16\) −1.85803e10 −1.08152
\(17\) − 5.96526e8i − 0.0207402i −0.999946 − 0.0103701i \(-0.996699\pi\)
0.999946 − 0.0103701i \(-0.00330097\pi\)
\(18\) 2.15409e10i 0.460732i
\(19\) 4.36939e10 0.590222 0.295111 − 0.955463i \(-0.404643\pi\)
0.295111 + 0.955463i \(0.404643\pi\)
\(20\) 0 0
\(21\) 5.84265e10 0.337090
\(22\) − 2.24525e11i − 0.872314i
\(23\) − 7.23343e10i − 0.192601i −0.995352 − 0.0963003i \(-0.969299\pi\)
0.995352 − 0.0963003i \(-0.0307009\pi\)
\(24\) −3.85298e10 −0.0714499
\(25\) 0 0
\(26\) 1.95028e12 1.83157
\(27\) 2.82430e11i 0.192450i
\(28\) − 1.06271e12i − 0.531582i
\(29\) −1.82054e12 −0.675800 −0.337900 − 0.941182i \(-0.609716\pi\)
−0.337900 + 0.941182i \(0.609716\pi\)
\(30\) 0 0
\(31\) 5.27128e12 1.11005 0.555024 − 0.831835i \(-0.312709\pi\)
0.555024 + 0.831835i \(0.312709\pi\)
\(32\) 8.52803e12i 1.37112i
\(33\) − 2.94382e12i − 0.364370i
\(34\) −2.98507e11 −0.0286671
\(35\) 0 0
\(36\) 5.13704e12 0.303488
\(37\) − 1.62659e13i − 0.761313i −0.924717 − 0.380656i \(-0.875698\pi\)
0.924717 − 0.380656i \(-0.124302\pi\)
\(38\) − 2.18648e13i − 0.815803i
\(39\) 2.55707e13 0.765057
\(40\) 0 0
\(41\) 9.78891e12 0.191457 0.0957285 − 0.995407i \(-0.469482\pi\)
0.0957285 + 0.995407i \(0.469482\pi\)
\(42\) − 2.92371e13i − 0.465924i
\(43\) − 1.46155e14i − 1.90691i −0.301534 − 0.953455i \(-0.597499\pi\)
0.301534 − 0.953455i \(-0.402501\pi\)
\(44\) −5.35444e13 −0.574601
\(45\) 0 0
\(46\) −3.61967e13 −0.266212
\(47\) 2.43226e14i 1.48997i 0.667079 + 0.744987i \(0.267544\pi\)
−0.667079 + 0.744987i \(0.732456\pi\)
\(48\) 1.21906e14i 0.624415i
\(49\) 1.53329e14 0.659111
\(50\) 0 0
\(51\) −3.91381e12 −0.0119744
\(52\) − 4.65100e14i − 1.20647i
\(53\) − 6.84721e14i − 1.51067i −0.655340 − 0.755334i \(-0.727475\pi\)
0.655340 − 0.755334i \(-0.272525\pi\)
\(54\) 1.41330e14 0.266004
\(55\) 0 0
\(56\) 5.22958e13 0.0722551
\(57\) − 2.86676e14i − 0.340765i
\(58\) 9.11015e14i 0.934087i
\(59\) −9.83635e14 −0.872151 −0.436076 − 0.899910i \(-0.643632\pi\)
−0.436076 + 0.899910i \(0.643632\pi\)
\(60\) 0 0
\(61\) −1.84021e15 −1.22903 −0.614516 − 0.788904i \(-0.710649\pi\)
−0.614516 + 0.788904i \(0.710649\pi\)
\(62\) − 2.63779e15i − 1.53430i
\(63\) − 3.83336e14i − 0.194619i
\(64\) 1.83214e15 0.813631
\(65\) 0 0
\(66\) −1.47311e15 −0.503631
\(67\) − 2.46536e15i − 0.741729i −0.928687 − 0.370864i \(-0.879062\pi\)
0.928687 − 0.370864i \(-0.120938\pi\)
\(68\) 7.11874e13i 0.0188833i
\(69\) −4.74585e14 −0.111198
\(70\) 0 0
\(71\) 9.43154e14 0.173335 0.0866676 − 0.996237i \(-0.472378\pi\)
0.0866676 + 0.996237i \(0.472378\pi\)
\(72\) 2.52794e14i 0.0412516i
\(73\) − 1.15522e16i − 1.67657i −0.545231 − 0.838286i \(-0.683558\pi\)
0.545231 − 0.838286i \(-0.316442\pi\)
\(74\) −8.13959e15 −1.05228
\(75\) 0 0
\(76\) −5.21428e15 −0.537377
\(77\) 3.99559e15i 0.368477i
\(78\) − 1.27958e16i − 1.05746i
\(79\) −1.58855e16 −1.17807 −0.589035 − 0.808108i \(-0.700492\pi\)
−0.589035 + 0.808108i \(0.700492\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) − 4.89845e15i − 0.264631i
\(83\) − 1.62194e16i − 0.790442i −0.918586 − 0.395221i \(-0.870668\pi\)
0.918586 − 0.395221i \(-0.129332\pi\)
\(84\) −6.97241e15 −0.306909
\(85\) 0 0
\(86\) −7.31369e16 −2.63572
\(87\) 1.19446e16i 0.390173i
\(88\) − 2.63492e15i − 0.0781026i
\(89\) 4.03683e16 1.08699 0.543495 − 0.839413i \(-0.317101\pi\)
0.543495 + 0.839413i \(0.317101\pi\)
\(90\) 0 0
\(91\) −3.47067e16 −0.773679
\(92\) 8.63212e15i 0.175356i
\(93\) − 3.45849e16i − 0.640886i
\(94\) 1.21712e17 2.05944
\(95\) 0 0
\(96\) 5.59524e16 0.791614
\(97\) − 7.85724e16i − 1.01791i −0.860792 − 0.508956i \(-0.830031\pi\)
0.860792 − 0.508956i \(-0.169969\pi\)
\(98\) − 7.67273e16i − 0.911020i
\(99\) −1.93144e16 −0.210369
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 75.18.b.e.49.1 6
5.2 odd 4 75.18.a.d.1.3 3
5.3 odd 4 15.18.a.c.1.1 ✓ 3
5.4 even 2 inner 75.18.b.e.49.6 6
15.8 even 4 45.18.a.d.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.c.1.1 ✓ 3 5.3 odd 4
45.18.a.d.1.3 3 15.8 even 4
75.18.a.d.1.3 3 5.2 odd 4
75.18.b.e.49.1 6 1.1 even 1 trivial
75.18.b.e.49.6 6 5.4 even 2 inner