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Properties

Label 675.4.a.s.1.3
Level $675$
Weight $4$
Character 675.1
Self dual yes
Analytic conductor $39.826$
Analytic rank $0$
Dimension $3$
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] = [675,4,Mod(1,675)] 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("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,5,0,17,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(39.8262892539\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1772.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 12x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 135)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.654334\) of defining polynomial
Character \(\chi\) \(=\) 675.1

$q$-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)
 
\(f(q)\) \(=\) \(q+5.45876 q^{2} +21.7980 q^{4} +11.8065 q^{7} +75.3201 q^{8} +56.2376 q^{11} -34.5961 q^{13} +64.4489 q^{14} +236.770 q^{16} +39.2675 q^{17} -146.561 q^{19} +306.987 q^{22} -23.5777 q^{23} -188.851 q^{26} +257.359 q^{28} -161.003 q^{29} -29.5465 q^{31} +689.908 q^{32} +214.352 q^{34} +217.688 q^{37} -800.039 q^{38} -142.290 q^{41} +468.030 q^{43} +1225.87 q^{44} -128.705 q^{46} +394.318 q^{47} -203.606 q^{49} -754.126 q^{52} -134.780 q^{53} +889.268 q^{56} -878.875 q^{58} +131.195 q^{59} +259.801 q^{61} -161.287 q^{62} +1871.88 q^{64} -445.244 q^{67} +855.954 q^{68} -560.841 q^{71} +88.6681 q^{73} +1188.30 q^{74} -3194.74 q^{76} +663.970 q^{77} +450.342 q^{79} -776.726 q^{82} +284.295 q^{83} +2554.86 q^{86} +4235.82 q^{88} -625.305 q^{89} -408.459 q^{91} -513.948 q^{92} +2152.49 q^{94} +193.261 q^{97} -1111.44 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 5 q^{2} + 17 q^{4} + 4 q^{7} + 75 q^{8} - 5 q^{11} - 7 q^{13} - 60 q^{14} + 161 q^{16} + 155 q^{17} - 50 q^{19} + 229 q^{22} + 285 q^{23} - 185 q^{26} + 334 q^{28} - 115 q^{29} - 115 q^{31} + 775 q^{32}+ \cdots - 305 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.45876 1.92996 0.964981 − 0.262320i \(-0.0844875\pi\)
0.964981 + 0.262320i \(0.0844875\pi\)
\(3\) 0 0
\(4\) 21.7980 2.72475
\(5\) 0 0
\(6\) 0 0
\(7\) 11.8065 0.637492 0.318746 − 0.947840i \(-0.396738\pi\)
0.318746 + 0.947840i \(0.396738\pi\)
\(8\) 75.3201 3.32871
\(9\) 0 0
\(10\) 0 0
\(11\) 56.2376 1.54148 0.770740 − 0.637150i \(-0.219887\pi\)
0.770740 + 0.637150i \(0.219887\pi\)
\(12\) 0 0
\(13\) −34.5961 −0.738094 −0.369047 − 0.929411i \(-0.620316\pi\)
−0.369047 + 0.929411i \(0.620316\pi\)
\(14\) 64.4489 1.23034
\(15\) 0 0
\(16\) 236.770 3.69953
\(17\) 39.2675 0.560221 0.280111 − 0.959968i \(-0.409629\pi\)
0.280111 + 0.959968i \(0.409629\pi\)
\(18\) 0 0
\(19\) −146.561 −1.76965 −0.884825 − 0.465924i \(-0.845722\pi\)
−0.884825 + 0.465924i \(0.845722\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 306.987 2.97500
\(23\) −23.5777 −0.213752 −0.106876 − 0.994272i \(-0.534085\pi\)
−0.106876 + 0.994272i \(0.534085\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −188.851 −1.42449
\(27\) 0 0
\(28\) 257.359 1.73701
\(29\) −161.003 −1.03095 −0.515473 − 0.856906i \(-0.672384\pi\)
−0.515473 + 0.856906i \(0.672384\pi\)
\(30\) 0 0
\(31\) −29.5465 −0.171184 −0.0855921 − 0.996330i \(-0.527278\pi\)
−0.0855921 + 0.996330i \(0.527278\pi\)
\(32\) 689.908 3.81124
\(33\) 0 0
\(34\) 214.352 1.08121
\(35\) 0 0
\(36\) 0 0
\(37\) 217.688 0.967233 0.483617 − 0.875280i \(-0.339323\pi\)
0.483617 + 0.875280i \(0.339323\pi\)
\(38\) −800.039 −3.41536
\(39\) 0 0
\(40\) 0 0
\(41\) −142.290 −0.541999 −0.270999 − 0.962580i \(-0.587354\pi\)
−0.270999 + 0.962580i \(0.587354\pi\)
\(42\) 0 0
\(43\) 468.030 1.65986 0.829929 − 0.557869i \(-0.188381\pi\)
0.829929 + 0.557869i \(0.188381\pi\)
\(44\) 1225.87 4.20015
\(45\) 0 0
\(46\) −128.705 −0.412533
\(47\) 394.318 1.22377 0.611886 − 0.790946i \(-0.290411\pi\)
0.611886 + 0.790946i \(0.290411\pi\)
\(48\) 0 0
\(49\) −203.606 −0.593604
\(50\) 0 0
\(51\) 0 0
\(52\) −754.126 −2.01112
\(53\) −134.780 −0.349311 −0.174655 − 0.984630i \(-0.555881\pi\)
−0.174655 + 0.984630i \(0.555881\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 889.268 2.12203
\(57\) 0 0
\(58\) −878.875 −1.98969
\(59\) 131.195 0.289495 0.144747 − 0.989469i \(-0.453763\pi\)
0.144747 + 0.989469i \(0.453763\pi\)
\(60\) 0 0
\(61\) 259.801 0.545313 0.272657 − 0.962111i \(-0.412098\pi\)
0.272657 + 0.962111i \(0.412098\pi\)
\(62\) −161.287 −0.330379
\(63\) 0 0
\(64\) 1871.88 3.65602
\(65\) 0 0
\(66\) 0 0
\(67\) −445.244 −0.811869 −0.405935 − 0.913902i \(-0.633054\pi\)
−0.405935 + 0.913902i \(0.633054\pi\)
\(68\) 855.954 1.52647
\(69\) 0 0
\(70\) 0 0
\(71\) −560.841 −0.937459 −0.468729 − 0.883342i \(-0.655288\pi\)
−0.468729 + 0.883342i \(0.655288\pi\)
\(72\) 0 0
\(73\) 88.6681 0.142162 0.0710809 − 0.997471i \(-0.477355\pi\)
0.0710809 + 0.997471i \(0.477355\pi\)
\(74\) 1188.30 1.86672
\(75\) 0 0
\(76\) −3194.74 −4.82186
\(77\) 663.970 0.982681
\(78\) 0 0
\(79\) 450.342 0.641360 0.320680 − 0.947188i \(-0.396089\pi\)
0.320680 + 0.947188i \(0.396089\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −776.726 −1.04604
\(83\) 284.295 0.375969 0.187985 − 0.982172i \(-0.439804\pi\)
0.187985 + 0.982172i \(0.439804\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2554.86 3.20346
\(87\) 0 0
\(88\) 4235.82 5.13114
\(89\) −625.305 −0.744744 −0.372372 − 0.928083i \(-0.621455\pi\)
−0.372372 + 0.928083i \(0.621455\pi\)
\(90\) 0 0
\(91\) −408.459 −0.470529
\(92\) −513.948 −0.582421
\(93\) 0 0
\(94\) 2152.49 2.36183
\(95\) 0 0
\(96\) 0 0
\(97\) 193.261 0.202296 0.101148 − 0.994871i \(-0.467748\pi\)
0.101148 + 0.994871i \(0.467748\pi\)
\(98\) −1111.44 −1.14563
\(99\) 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 675.4.a.s.1.3 3
3.2 odd 2 675.4.a.p.1.1 3
5.2 odd 4 675.4.b.m.649.6 6
5.3 odd 4 675.4.b.m.649.1 6
5.4 even 2 135.4.a.e.1.1 ✓ 3
15.2 even 4 675.4.b.n.649.1 6
15.8 even 4 675.4.b.n.649.6 6
15.14 odd 2 135.4.a.h.1.3 yes 3
20.19 odd 2 2160.4.a.bi.1.2 3
45.4 even 6 405.4.e.v.136.3 6
45.14 odd 6 405.4.e.q.136.1 6
45.29 odd 6 405.4.e.q.271.1 6
45.34 even 6 405.4.e.v.271.3 6
60.59 even 2 2160.4.a.bq.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.e.1.1 ✓ 3 5.4 even 2
135.4.a.h.1.3 yes 3 15.14 odd 2
405.4.e.q.136.1 6 45.14 odd 6
405.4.e.q.271.1 6 45.29 odd 6
405.4.e.v.136.3 6 45.4 even 6
405.4.e.v.271.3 6 45.34 even 6
675.4.a.p.1.1 3 3.2 odd 2
675.4.a.s.1.3 3 1.1 even 1 trivial
675.4.b.m.649.1 6 5.3 odd 4
675.4.b.m.649.6 6 5.2 odd 4
675.4.b.n.649.1 6 15.2 even 4
675.4.b.n.649.6 6 15.8 even 4
2160.4.a.bi.1.2 3 20.19 odd 2
2160.4.a.bq.1.2 3 60.59 even 2