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Properties

Label 6525.2.a.cf.1.6
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 1.6
Root \(-0.328889\) 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)
 
\(f(q)\) \(=\) \(q-0.328889 q^{2} -1.89183 q^{4} +1.86588 q^{7} +1.27998 q^{8} +0.564087 q^{11} +4.95986 q^{13} -0.613668 q^{14} +3.36269 q^{16} +7.06879 q^{17} -3.32971 q^{19} -0.185522 q^{22} +2.36471 q^{23} -1.63124 q^{26} -3.52994 q^{28} -1.00000 q^{29} +4.87352 q^{31} -3.66591 q^{32} -2.32485 q^{34} +8.50474 q^{37} +1.09510 q^{38} +7.86030 q^{41} -4.39693 q^{43} -1.06716 q^{44} -0.777727 q^{46} +7.06879 q^{47} -3.51848 q^{49} -9.38323 q^{52} +4.18169 q^{53} +2.38829 q^{56} +0.328889 q^{58} -9.01198 q^{59} -1.26059 q^{61} -1.60285 q^{62} -5.51971 q^{64} +6.61254 q^{67} -13.3730 q^{68} -5.46480 q^{71} -5.77659 q^{73} -2.79711 q^{74} +6.29925 q^{76} +1.05252 q^{77} +12.1018 q^{79} -2.58516 q^{82} +9.41830 q^{83} +1.44610 q^{86} +0.722020 q^{88} -0.622547 q^{89} +9.25453 q^{91} -4.47364 q^{92} -2.32485 q^{94} -19.2638 q^{97} +1.15719 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 16 q^{4} + 12 q^{11} + 16 q^{14} + 16 q^{16} - 20 q^{19} + 56 q^{26} - 12 q^{29} - 16 q^{31} + 4 q^{34} + 32 q^{41} + 68 q^{44} + 20 q^{46} - 4 q^{49} + 76 q^{56} + 44 q^{59} - 52 q^{61} + 36 q^{64}+ \cdots + 4 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.328889 −0.232560 −0.116280 − 0.993216i \(-0.537097\pi\)
−0.116280 + 0.993216i \(0.537097\pi\)
\(3\) 0 0
\(4\) −1.89183 −0.945916
\(5\) 0 0
\(6\) 0 0
\(7\) 1.86588 0.705237 0.352619 − 0.935767i \(-0.385291\pi\)
0.352619 + 0.935767i \(0.385291\pi\)
\(8\) 1.27998 0.452541
\(9\) 0 0
\(10\) 0 0
\(11\) 0.564087 0.170079 0.0850393 − 0.996378i \(-0.472898\pi\)
0.0850393 + 0.996378i \(0.472898\pi\)
\(12\) 0 0
\(13\) 4.95986 1.37562 0.687809 − 0.725891i \(-0.258572\pi\)
0.687809 + 0.725891i \(0.258572\pi\)
\(14\) −0.613668 −0.164010
\(15\) 0 0
\(16\) 3.36269 0.840673
\(17\) 7.06879 1.71443 0.857217 − 0.514956i \(-0.172192\pi\)
0.857217 + 0.514956i \(0.172192\pi\)
\(18\) 0 0
\(19\) −3.32971 −0.763887 −0.381944 − 0.924186i \(-0.624745\pi\)
−0.381944 + 0.924186i \(0.624745\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.185522 −0.0395534
\(23\) 2.36471 0.493076 0.246538 − 0.969133i \(-0.420707\pi\)
0.246538 + 0.969133i \(0.420707\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.63124 −0.319913
\(27\) 0 0
\(28\) −3.52994 −0.667095
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 4.87352 0.875310 0.437655 − 0.899143i \(-0.355809\pi\)
0.437655 + 0.899143i \(0.355809\pi\)
\(32\) −3.66591 −0.648048
\(33\) 0 0
\(34\) −2.32485 −0.398708
\(35\) 0 0
\(36\) 0 0
\(37\) 8.50474 1.39817 0.699085 − 0.715038i \(-0.253591\pi\)
0.699085 + 0.715038i \(0.253591\pi\)
\(38\) 1.09510 0.177649
\(39\) 0 0
\(40\) 0 0
\(41\) 7.86030 1.22757 0.613786 − 0.789473i \(-0.289646\pi\)
0.613786 + 0.789473i \(0.289646\pi\)
\(42\) 0 0
\(43\) −4.39693 −0.670526 −0.335263 − 0.942125i \(-0.608825\pi\)
−0.335263 + 0.942125i \(0.608825\pi\)
\(44\) −1.06716 −0.160880
\(45\) 0 0
\(46\) −0.777727 −0.114670
\(47\) 7.06879 1.03109 0.515544 − 0.856863i \(-0.327590\pi\)
0.515544 + 0.856863i \(0.327590\pi\)
\(48\) 0 0
\(49\) −3.51848 −0.502640
\(50\) 0 0
\(51\) 0 0
\(52\) −9.38323 −1.30122
\(53\) 4.18169 0.574400 0.287200 − 0.957871i \(-0.407276\pi\)
0.287200 + 0.957871i \(0.407276\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.38829 0.319149
\(57\) 0 0
\(58\) 0.328889 0.0431852
\(59\) −9.01198 −1.17326 −0.586630 − 0.809855i \(-0.699546\pi\)
−0.586630 + 0.809855i \(0.699546\pi\)
\(60\) 0 0
\(61\) −1.26059 −0.161402 −0.0807009 − 0.996738i \(-0.525716\pi\)
−0.0807009 + 0.996738i \(0.525716\pi\)
\(62\) −1.60285 −0.203562
\(63\) 0 0
\(64\) −5.51971 −0.689964
\(65\) 0 0
\(66\) 0 0
\(67\) 6.61254 0.807851 0.403925 − 0.914792i \(-0.367646\pi\)
0.403925 + 0.914792i \(0.367646\pi\)
\(68\) −13.3730 −1.62171
\(69\) 0 0
\(70\) 0 0
\(71\) −5.46480 −0.648552 −0.324276 − 0.945962i \(-0.605121\pi\)
−0.324276 + 0.945962i \(0.605121\pi\)
\(72\) 0 0
\(73\) −5.77659 −0.676099 −0.338049 − 0.941128i \(-0.609767\pi\)
−0.338049 + 0.941128i \(0.609767\pi\)
\(74\) −2.79711 −0.325158
\(75\) 0 0
\(76\) 6.29925 0.722573
\(77\) 1.05252 0.119946
\(78\) 0 0
\(79\) 12.1018 1.36156 0.680782 − 0.732486i \(-0.261640\pi\)
0.680782 + 0.732486i \(0.261640\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −2.58516 −0.285484
\(83\) 9.41830 1.03379 0.516896 − 0.856048i \(-0.327087\pi\)
0.516896 + 0.856048i \(0.327087\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.44610 0.155937
\(87\) 0 0
\(88\) 0.722020 0.0769676
\(89\) −0.622547 −0.0659899 −0.0329949 − 0.999456i \(-0.510505\pi\)
−0.0329949 + 0.999456i \(0.510505\pi\)
\(90\) 0 0
\(91\) 9.25453 0.970138
\(92\) −4.47364 −0.466409
\(93\) 0 0
\(94\) −2.32485 −0.239790
\(95\) 0 0
\(96\) 0 0
\(97\) −19.2638 −1.95594 −0.977970 − 0.208743i \(-0.933063\pi\)
−0.977970 + 0.208743i \(0.933063\pi\)
\(98\) 1.15719 0.116894
\(99\) 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.1.6 12
3.2 odd 2 6525.2.a.ce.1.7 12
5.2 odd 4 1305.2.c.l.784.6 yes 12
5.3 odd 4 1305.2.c.l.784.7 yes 12
5.4 even 2 inner 6525.2.a.cf.1.7 12
15.2 even 4 1305.2.c.k.784.7 yes 12
15.8 even 4 1305.2.c.k.784.6 ✓ 12
15.14 odd 2 6525.2.a.ce.1.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.c.k.784.6 ✓ 12 15.8 even 4
1305.2.c.k.784.7 yes 12 15.2 even 4
1305.2.c.l.784.6 yes 12 5.2 odd 4
1305.2.c.l.784.7 yes 12 5.3 odd 4
6525.2.a.ce.1.6 12 15.14 odd 2
6525.2.a.ce.1.7 12 3.2 odd 2
6525.2.a.cf.1.6 12 1.1 even 1 trivial
6525.2.a.cf.1.7 12 5.4 even 2 inner