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Properties

Label 6525.2.a.cf.1.8
Level $6525$
Weight $2$
Character 6525.1
Self dual yes
Analytic conductor $52.102$
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] = [6525,2,Mod(1,6525)] 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("6525.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6525.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,16,0,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(52.1023873189\)
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: \( x^{12} - 20x^{10} + 148x^{8} - 502x^{6} + 792x^{4} - 496x^{2} + 45 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1305)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(1.27263\) of defining polynomial
Character \(\chi\) \(=\) 6525.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.27263 q^{2} -0.380419 q^{4} +0.255813 q^{7} -3.02939 q^{8} -4.63446 q^{11} +5.02700 q^{13} +0.325555 q^{14} -3.09444 q^{16} -0.336444 q^{17} -2.91437 q^{19} -5.89795 q^{22} +8.65656 q^{23} +6.39750 q^{26} -0.0973162 q^{28} -1.00000 q^{29} -3.26943 q^{31} +2.12070 q^{32} -0.428167 q^{34} -3.86954 q^{37} -3.70891 q^{38} -5.71649 q^{41} +6.98619 q^{43} +1.76304 q^{44} +11.0166 q^{46} -0.336444 q^{47} -6.93456 q^{49} -1.91237 q^{52} +6.01484 q^{53} -0.774958 q^{56} -1.27263 q^{58} +13.2799 q^{59} -7.77792 q^{61} -4.16077 q^{62} +8.88775 q^{64} -11.2605 q^{67} +0.127989 q^{68} +13.9668 q^{71} +8.46426 q^{73} -4.92448 q^{74} +1.10868 q^{76} -1.18556 q^{77} -15.3102 q^{79} -7.27496 q^{82} +7.60521 q^{83} +8.89082 q^{86} +14.0396 q^{88} +13.0383 q^{89} +1.28597 q^{91} -3.29312 q^{92} -0.428167 q^{94} +11.5445 q^{97} -8.82511 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16 q^{4} + 12 q^{11} + 16 q^{14} + 16 q^{16} - 20 q^{19} + 56 q^{26} - 12 q^{29} - 16 q^{31} + 4 q^{34} + 32 q^{41} + 68 q^{44} + 20 q^{46} - 4 q^{49} + 76 q^{56} + 44 q^{59} - 52 q^{61} + 36 q^{64}+ \cdots + 4 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.27263 0.899884 0.449942 − 0.893058i \(-0.351445\pi\)
0.449942 + 0.893058i \(0.351445\pi\)
\(3\) 0 0
\(4\) −0.380419 −0.190209
\(5\) 0 0
\(6\) 0 0
\(7\) 0.255813 0.0966884 0.0483442 − 0.998831i \(-0.484606\pi\)
0.0483442 + 0.998831i \(0.484606\pi\)
\(8\) −3.02939 −1.07105
\(9\) 0 0
\(10\) 0 0
\(11\) −4.63446 −1.39734 −0.698672 − 0.715442i \(-0.746225\pi\)
−0.698672 + 0.715442i \(0.746225\pi\)
\(12\) 0 0
\(13\) 5.02700 1.39424 0.697120 − 0.716955i \(-0.254465\pi\)
0.697120 + 0.716955i \(0.254465\pi\)
\(14\) 0.325555 0.0870083
\(15\) 0 0
\(16\) −3.09444 −0.773611
\(17\) −0.336444 −0.0815995 −0.0407998 − 0.999167i \(-0.512991\pi\)
−0.0407998 + 0.999167i \(0.512991\pi\)
\(18\) 0 0
\(19\) −2.91437 −0.668602 −0.334301 − 0.942466i \(-0.608500\pi\)
−0.334301 + 0.942466i \(0.608500\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −5.89795 −1.25745
\(23\) 8.65656 1.80502 0.902509 − 0.430671i \(-0.141723\pi\)
0.902509 + 0.430671i \(0.141723\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 6.39750 1.25465
\(27\) 0 0
\(28\) −0.0973162 −0.0183910
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −3.26943 −0.587207 −0.293603 − 0.955927i \(-0.594854\pi\)
−0.293603 + 0.955927i \(0.594854\pi\)
\(32\) 2.12070 0.374890
\(33\) 0 0
\(34\) −0.428167 −0.0734301
\(35\) 0 0
\(36\) 0 0
\(37\) −3.86954 −0.636148 −0.318074 − 0.948066i \(-0.603036\pi\)
−0.318074 + 0.948066i \(0.603036\pi\)
\(38\) −3.70891 −0.601664
\(39\) 0 0
\(40\) 0 0
\(41\) −5.71649 −0.892765 −0.446383 − 0.894842i \(-0.647288\pi\)
−0.446383 + 0.894842i \(0.647288\pi\)
\(42\) 0 0
\(43\) 6.98619 1.06538 0.532692 − 0.846309i \(-0.321180\pi\)
0.532692 + 0.846309i \(0.321180\pi\)
\(44\) 1.76304 0.265788
\(45\) 0 0
\(46\) 11.0166 1.62431
\(47\) −0.336444 −0.0490753 −0.0245377 − 0.999699i \(-0.507811\pi\)
−0.0245377 + 0.999699i \(0.507811\pi\)
\(48\) 0 0
\(49\) −6.93456 −0.990651
\(50\) 0 0
\(51\) 0 0
\(52\) −1.91237 −0.265197
\(53\) 6.01484 0.826202 0.413101 − 0.910685i \(-0.364446\pi\)
0.413101 + 0.910685i \(0.364446\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.774958 −0.103558
\(57\) 0 0
\(58\) −1.27263 −0.167104
\(59\) 13.2799 1.72890 0.864451 − 0.502718i \(-0.167666\pi\)
0.864451 + 0.502718i \(0.167666\pi\)
\(60\) 0 0
\(61\) −7.77792 −0.995861 −0.497930 − 0.867217i \(-0.665906\pi\)
−0.497930 + 0.867217i \(0.665906\pi\)
\(62\) −4.16077 −0.528418
\(63\) 0 0
\(64\) 8.88775 1.11097
\(65\) 0 0
\(66\) 0 0
\(67\) −11.2605 −1.37569 −0.687847 − 0.725856i \(-0.741444\pi\)
−0.687847 + 0.725856i \(0.741444\pi\)
\(68\) 0.127989 0.0155210
\(69\) 0 0
\(70\) 0 0
\(71\) 13.9668 1.65756 0.828778 − 0.559578i \(-0.189037\pi\)
0.828778 + 0.559578i \(0.189037\pi\)
\(72\) 0 0
\(73\) 8.46426 0.990667 0.495333 − 0.868703i \(-0.335046\pi\)
0.495333 + 0.868703i \(0.335046\pi\)
\(74\) −4.92448 −0.572459
\(75\) 0 0
\(76\) 1.10868 0.127174
\(77\) −1.18556 −0.135107
\(78\) 0 0
\(79\) −15.3102 −1.72253 −0.861267 − 0.508153i \(-0.830328\pi\)
−0.861267 + 0.508153i \(0.830328\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −7.27496 −0.803385
\(83\) 7.60521 0.834781 0.417390 − 0.908727i \(-0.362945\pi\)
0.417390 + 0.908727i \(0.362945\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 8.89082 0.958722
\(87\) 0 0
\(88\) 14.0396 1.49662
\(89\) 13.0383 1.38206 0.691029 − 0.722827i \(-0.257157\pi\)
0.691029 + 0.722827i \(0.257157\pi\)
\(90\) 0 0
\(91\) 1.28597 0.134807
\(92\) −3.29312 −0.343331
\(93\) 0 0
\(94\) −0.428167 −0.0441621
\(95\) 0 0
\(96\) 0 0
\(97\) 11.5445 1.17216 0.586081 − 0.810253i \(-0.300670\pi\)
0.586081 + 0.810253i \(0.300670\pi\)
\(98\) −8.82511 −0.891471
\(99\) 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 6525.2.a.cf.1.8 12
3.2 odd 2 6525.2.a.ce.1.5 12
5.2 odd 4 1305.2.c.l.784.8 yes 12
5.3 odd 4 1305.2.c.l.784.5 yes 12
5.4 even 2 inner 6525.2.a.cf.1.5 12
15.2 even 4 1305.2.c.k.784.5 ✓ 12
15.8 even 4 1305.2.c.k.784.8 yes 12
15.14 odd 2 6525.2.a.ce.1.8 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.c.k.784.5 ✓ 12 15.2 even 4
1305.2.c.k.784.8 yes 12 15.8 even 4
1305.2.c.l.784.5 yes 12 5.3 odd 4
1305.2.c.l.784.8 yes 12 5.2 odd 4
6525.2.a.ce.1.5 12 3.2 odd 2
6525.2.a.ce.1.8 12 15.14 odd 2
6525.2.a.cf.1.5 12 5.4 even 2 inner
6525.2.a.cf.1.8 12 1.1 even 1 trivial