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 = [4,0,0,4,0,0,0,0,0,0,-14] 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: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{11})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.52434\) 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)
 
\(f(q)\) \(=\) \(q+1.73205 q^{2} +1.00000 q^{4} -5.04868 q^{7} -1.73205 q^{8} -0.627719 q^{11} -4.25639 q^{13} -8.74456 q^{14} -5.00000 q^{16} -1.58457 q^{17} +4.00000 q^{19} -1.08724 q^{22} +3.46410 q^{23} -7.37228 q^{26} -5.04868 q^{28} +1.00000 q^{29} -3.37228 q^{31} -5.19615 q^{32} -2.74456 q^{34} -3.16915 q^{37} +6.92820 q^{38} +4.74456 q^{41} +10.8896 q^{43} -0.627719 q^{44} +6.00000 q^{46} -10.8896 q^{47} +18.4891 q^{49} -4.25639 q^{52} +4.25639 q^{53} +8.74456 q^{56} +1.73205 q^{58} +10.7446 q^{59} +6.00000 q^{61} -5.84096 q^{62} +1.00000 q^{64} +1.87953 q^{67} -1.58457 q^{68} -6.74456 q^{71} +6.92820 q^{73} -5.48913 q^{74} +4.00000 q^{76} +3.16915 q^{77} +11.3723 q^{79} +8.21782 q^{82} -9.80240 q^{83} +18.8614 q^{86} +1.08724 q^{88} +0.744563 q^{89} +21.4891 q^{91} +3.46410 q^{92} -18.8614 q^{94} +6.92820 q^{97} +32.0241 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 14 q^{11} - 12 q^{14} - 20 q^{16} + 16 q^{19} - 18 q^{26} + 4 q^{29} - 2 q^{31} + 12 q^{34} - 4 q^{41} - 14 q^{44} + 24 q^{46} + 28 q^{49} + 12 q^{56} + 20 q^{59} + 24 q^{61} + 4 q^{64}+ \cdots - 18 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 − 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −5.04868 −1.90822 −0.954110 − 0.299456i \(-0.903195\pi\)
−0.954110 + 0.299456i \(0.903195\pi\)
\(8\) −1.73205 −0.612372
\(9\) 0 0
\(10\) 0 0
\(11\) −0.627719 −0.189264 −0.0946322 − 0.995512i \(-0.530167\pi\)
−0.0946322 + 0.995512i \(0.530167\pi\)
\(12\) 0 0
\(13\) −4.25639 −1.18051 −0.590255 − 0.807217i \(-0.700973\pi\)
−0.590255 + 0.807217i \(0.700973\pi\)
\(14\) −8.74456 −2.33708
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) −1.58457 −0.384316 −0.192158 − 0.981364i \(-0.561549\pi\)
−0.192158 + 0.981364i \(0.561549\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.08724 −0.231800
\(23\) 3.46410 0.722315 0.361158 − 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −7.37228 −1.44582
\(27\) 0 0
\(28\) −5.04868 −0.954110
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −3.37228 −0.605680 −0.302840 − 0.953041i \(-0.597935\pi\)
−0.302840 + 0.953041i \(0.597935\pi\)
\(32\) −5.19615 −0.918559
\(33\) 0 0
\(34\) −2.74456 −0.470689
\(35\) 0 0
\(36\) 0 0
\(37\) −3.16915 −0.521005 −0.260502 − 0.965473i \(-0.583888\pi\)
−0.260502 + 0.965473i \(0.583888\pi\)
\(38\) 6.92820 1.12390
\(39\) 0 0
\(40\) 0 0
\(41\) 4.74456 0.740976 0.370488 − 0.928837i \(-0.379190\pi\)
0.370488 + 0.928837i \(0.379190\pi\)
\(42\) 0 0
\(43\) 10.8896 1.66065 0.830327 − 0.557276i \(-0.188154\pi\)
0.830327 + 0.557276i \(0.188154\pi\)
\(44\) −0.627719 −0.0946322
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −10.8896 −1.58842 −0.794208 − 0.607645i \(-0.792114\pi\)
−0.794208 + 0.607645i \(0.792114\pi\)
\(48\) 0 0
\(49\) 18.4891 2.64130
\(50\) 0 0
\(51\) 0 0
\(52\) −4.25639 −0.590255
\(53\) 4.25639 0.584660 0.292330 − 0.956318i \(-0.405569\pi\)
0.292330 + 0.956318i \(0.405569\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 8.74456 1.16854
\(57\) 0 0
\(58\) 1.73205 0.227429
\(59\) 10.7446 1.39882 0.699411 − 0.714719i \(-0.253446\pi\)
0.699411 + 0.714719i \(0.253446\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −5.84096 −0.741803
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 1.87953 0.229621 0.114810 − 0.993387i \(-0.463374\pi\)
0.114810 + 0.993387i \(0.463374\pi\)
\(68\) −1.58457 −0.192158
\(69\) 0 0
\(70\) 0 0
\(71\) −6.74456 −0.800432 −0.400216 − 0.916421i \(-0.631065\pi\)
−0.400216 + 0.916421i \(0.631065\pi\)
\(72\) 0 0
\(73\) 6.92820 0.810885 0.405442 − 0.914121i \(-0.367117\pi\)
0.405442 + 0.914121i \(0.367117\pi\)
\(74\) −5.48913 −0.638098
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 3.16915 0.361158
\(78\) 0 0
\(79\) 11.3723 1.27948 0.639741 − 0.768591i \(-0.279042\pi\)
0.639741 + 0.768591i \(0.279042\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 8.21782 0.907507
\(83\) −9.80240 −1.07595 −0.537976 − 0.842960i \(-0.680811\pi\)
−0.537976 + 0.842960i \(0.680811\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 18.8614 2.03388
\(87\) 0 0
\(88\) 1.08724 0.115900
\(89\) 0.744563 0.0789235 0.0394617 − 0.999221i \(-0.487436\pi\)
0.0394617 + 0.999221i \(0.487436\pi\)
\(90\) 0 0
\(91\) 21.4891 2.25267
\(92\) 3.46410 0.361158
\(93\) 0 0
\(94\) −18.8614 −1.94541
\(95\) 0 0
\(96\) 0 0
\(97\) 6.92820 0.703452 0.351726 − 0.936103i \(-0.385595\pi\)
0.351726 + 0.936103i \(0.385595\pi\)
\(98\) 32.0241 3.23492
\(99\) 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.bk.1.3 4
3.2 odd 2 725.2.a.g.1.1 4
5.2 odd 4 1305.2.c.e.784.3 4
5.3 odd 4 1305.2.c.e.784.1 4
5.4 even 2 inner 6525.2.a.bk.1.2 4
15.2 even 4 145.2.b.a.59.2 ✓ 4
15.8 even 4 145.2.b.a.59.3 yes 4
15.14 odd 2 725.2.a.g.1.4 4
60.23 odd 4 2320.2.d.c.929.3 4
60.47 odd 4 2320.2.d.c.929.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.b.a.59.2 ✓ 4 15.2 even 4
145.2.b.a.59.3 yes 4 15.8 even 4
725.2.a.g.1.1 4 3.2 odd 2
725.2.a.g.1.4 4 15.14 odd 2
1305.2.c.e.784.1 4 5.3 odd 4
1305.2.c.e.784.3 4 5.2 odd 4
2320.2.d.c.929.2 4 60.47 odd 4
2320.2.d.c.929.3 4 60.23 odd 4
6525.2.a.bk.1.2 4 5.4 even 2 inner
6525.2.a.bk.1.3 4 1.1 even 1 trivial