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Properties

Label 6975.2.a.ba.1.3
Level $6975$
Weight $2$
Character 6975.1
Self dual yes
Analytic conductor $55.696$
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] = [6975,2,Mod(1,6975)] 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("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6975.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-2,0,2,0,0,8,-6,0,0,-3,0,4,-10,0,4,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(55.6956554098\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 6975.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.17009 q^{2} -0.630898 q^{4} +0.290725 q^{7} -3.07838 q^{8} -0.460811 q^{11} -1.80098 q^{13} +0.340173 q^{14} -2.34017 q^{16} -3.09171 q^{17} +3.00000 q^{19} -0.539189 q^{22} -2.46081 q^{23} -2.10731 q^{26} -0.183417 q^{28} -9.58864 q^{29} +1.00000 q^{31} +3.41855 q^{32} -3.61757 q^{34} +6.87936 q^{37} +3.51026 q^{38} +3.32684 q^{41} +10.8371 q^{43} +0.290725 q^{44} -2.87936 q^{46} +10.3402 q^{47} -6.91548 q^{49} +1.13624 q^{52} -1.92881 q^{53} -0.894960 q^{56} -11.2195 q^{58} -0.986669 q^{59} -6.68035 q^{61} +1.17009 q^{62} +8.68035 q^{64} +0.0289294 q^{67} +1.95055 q^{68} +13.9421 q^{71} -9.91548 q^{73} +8.04945 q^{74} -1.89269 q^{76} -0.133969 q^{77} +16.6381 q^{79} +3.89269 q^{82} -10.0628 q^{83} +12.6803 q^{86} +1.41855 q^{88} +5.53919 q^{89} -0.523590 q^{91} +1.55252 q^{92} +12.0989 q^{94} +2.55252 q^{97} -8.09171 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 2 q^{4} + 8 q^{7} - 6 q^{8} - 3 q^{11} + 4 q^{13} - 10 q^{14} + 4 q^{16} - 7 q^{17} + 9 q^{19} - 9 q^{23} - 18 q^{26} + 4 q^{28} - 9 q^{29} + 3 q^{31} - 4 q^{32} - 6 q^{34} + 8 q^{37} - 6 q^{38}+ \cdots - 22 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.17009 0.827376 0.413688 − 0.910419i \(-0.364240\pi\)
0.413688 + 0.910419i \(0.364240\pi\)
\(3\) 0 0
\(4\) −0.630898 −0.315449
\(5\) 0 0
\(6\) 0 0
\(7\) 0.290725 0.109884 0.0549418 − 0.998490i \(-0.482503\pi\)
0.0549418 + 0.998490i \(0.482503\pi\)
\(8\) −3.07838 −1.08837
\(9\) 0 0
\(10\) 0 0
\(11\) −0.460811 −0.138940 −0.0694699 − 0.997584i \(-0.522131\pi\)
−0.0694699 + 0.997584i \(0.522131\pi\)
\(12\) 0 0
\(13\) −1.80098 −0.499503 −0.249752 − 0.968310i \(-0.580349\pi\)
−0.249752 + 0.968310i \(0.580349\pi\)
\(14\) 0.340173 0.0909151
\(15\) 0 0
\(16\) −2.34017 −0.585043
\(17\) −3.09171 −0.749850 −0.374925 − 0.927055i \(-0.622331\pi\)
−0.374925 + 0.927055i \(0.622331\pi\)
\(18\) 0 0
\(19\) 3.00000 0.688247 0.344124 − 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.539189 −0.114955
\(23\) −2.46081 −0.513115 −0.256557 − 0.966529i \(-0.582588\pi\)
−0.256557 + 0.966529i \(0.582588\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.10731 −0.413277
\(27\) 0 0
\(28\) −0.183417 −0.0346626
\(29\) −9.58864 −1.78057 −0.890283 − 0.455408i \(-0.849493\pi\)
−0.890283 + 0.455408i \(0.849493\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 3.41855 0.604320
\(33\) 0 0
\(34\) −3.61757 −0.620408
\(35\) 0 0
\(36\) 0 0
\(37\) 6.87936 1.13096 0.565480 − 0.824762i \(-0.308691\pi\)
0.565480 + 0.824762i \(0.308691\pi\)
\(38\) 3.51026 0.569439
\(39\) 0 0
\(40\) 0 0
\(41\) 3.32684 0.519565 0.259783 − 0.965667i \(-0.416349\pi\)
0.259783 + 0.965667i \(0.416349\pi\)
\(42\) 0 0
\(43\) 10.8371 1.65264 0.826321 − 0.563199i \(-0.190430\pi\)
0.826321 + 0.563199i \(0.190430\pi\)
\(44\) 0.290725 0.0438284
\(45\) 0 0
\(46\) −2.87936 −0.424539
\(47\) 10.3402 1.50827 0.754135 − 0.656720i \(-0.228057\pi\)
0.754135 + 0.656720i \(0.228057\pi\)
\(48\) 0 0
\(49\) −6.91548 −0.987926
\(50\) 0 0
\(51\) 0 0
\(52\) 1.13624 0.157568
\(53\) −1.92881 −0.264942 −0.132471 − 0.991187i \(-0.542291\pi\)
−0.132471 + 0.991187i \(0.542291\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.894960 −0.119594
\(57\) 0 0
\(58\) −11.2195 −1.47320
\(59\) −0.986669 −0.128453 −0.0642267 − 0.997935i \(-0.520458\pi\)
−0.0642267 + 0.997935i \(0.520458\pi\)
\(60\) 0 0
\(61\) −6.68035 −0.855331 −0.427665 − 0.903937i \(-0.640664\pi\)
−0.427665 + 0.903937i \(0.640664\pi\)
\(62\) 1.17009 0.148601
\(63\) 0 0
\(64\) 8.68035 1.08504
\(65\) 0 0
\(66\) 0 0
\(67\) 0.0289294 0.00353429 0.00176715 − 0.999998i \(-0.499437\pi\)
0.00176715 + 0.999998i \(0.499437\pi\)
\(68\) 1.95055 0.236539
\(69\) 0 0
\(70\) 0 0
\(71\) 13.9421 1.65463 0.827314 − 0.561740i \(-0.189868\pi\)
0.827314 + 0.561740i \(0.189868\pi\)
\(72\) 0 0
\(73\) −9.91548 −1.16052 −0.580260 − 0.814432i \(-0.697049\pi\)
−0.580260 + 0.814432i \(0.697049\pi\)
\(74\) 8.04945 0.935729
\(75\) 0 0
\(76\) −1.89269 −0.217107
\(77\) −0.133969 −0.0152672
\(78\) 0 0
\(79\) 16.6381 1.87193 0.935965 − 0.352092i \(-0.114530\pi\)
0.935965 + 0.352092i \(0.114530\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.89269 0.429876
\(83\) −10.0628 −1.10453 −0.552267 − 0.833667i \(-0.686237\pi\)
−0.552267 + 0.833667i \(0.686237\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 12.6803 1.36736
\(87\) 0 0
\(88\) 1.41855 0.151218
\(89\) 5.53919 0.587153 0.293576 − 0.955936i \(-0.405154\pi\)
0.293576 + 0.955936i \(0.405154\pi\)
\(90\) 0 0
\(91\) −0.523590 −0.0548872
\(92\) 1.55252 0.161861
\(93\) 0 0
\(94\) 12.0989 1.24791
\(95\) 0 0
\(96\) 0 0
\(97\) 2.55252 0.259169 0.129585 − 0.991568i \(-0.458636\pi\)
0.129585 + 0.991568i \(0.458636\pi\)
\(98\) −8.09171 −0.817386
\(99\) 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 6975.2.a.ba.1.3 yes 3
3.2 odd 2 6975.2.a.bh.1.1 yes 3
5.4 even 2 6975.2.a.bg.1.1 yes 3
15.14 odd 2 6975.2.a.z.1.3 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6975.2.a.z.1.3 ✓ 3 15.14 odd 2
6975.2.a.ba.1.3 yes 3 1.1 even 1 trivial
6975.2.a.bg.1.1 yes 3 5.4 even 2
6975.2.a.bh.1.1 yes 3 3.2 odd 2