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Properties

Label 1575.2.g.d.1574.8
Level $1575$
Weight $2$
Character 1575.1574
Analytic conductor $12.576$
Analytic rank $0$
Dimension $8$
CM discriminant -7
Inner twists $8$

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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] = [1575,2,Mod(1574,1575)] 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("1575.1574"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,16,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(i, \sqrt{2}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1574.8
Root \(1.28897 - 0.581861i\) of defining polynomial
Character \(\chi\) \(=\) 1575.1574
Dual form 1575.2.g.d.1574.7

$q$-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)
 
\(f(q)\) \(=\) \(q+2.57794 q^{2} +4.64575 q^{4} +2.64575i q^{7} +6.82058 q^{8} +0.913230i q^{11} +6.82058i q^{14} +8.29150 q^{16} +2.35425i q^{22} +9.39851 q^{23} +12.2915i q^{28} +6.06910i q^{29} +7.73381 q^{32} -10.5830i q^{37} -5.29150i q^{43} +4.24264i q^{44} +24.2288 q^{46} -7.00000 q^{49} -14.5544 q^{53} +18.0455i q^{56} +15.6458i q^{58} +3.35425 q^{64} -4.00000i q^{67} -7.57205i q^{71} -27.2823i q^{74} -2.41618 q^{77} -8.00000 q^{79} -13.6412i q^{86} +6.22876i q^{88} +43.6631 q^{92} -18.0455 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{4} + 24 q^{16} + 88 q^{46} - 56 q^{49} + 48 q^{64} - 64 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.57794 1.82288 0.911438 − 0.411438i \(-0.134973\pi\)
0.911438 + 0.411438i \(0.134973\pi\)
\(3\) 0 0
\(4\) 4.64575 2.32288
\(5\) 0 0
\(6\) 0 0
\(7\) 2.64575i 1.00000i
\(8\) 6.82058 2.41144
\(9\) 0 0
\(10\) 0 0
\(11\) 0.913230i 0.275349i 0.990478 + 0.137675i \(0.0439628\pi\)
−0.990478 + 0.137675i \(0.956037\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 6.82058i 1.82288i
\(15\) 0 0
\(16\) 8.29150 2.07288
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.35425i 0.501928i
\(23\) 9.39851 1.95973 0.979863 − 0.199673i \(-0.0639880\pi\)
0.979863 + 0.199673i \(0.0639880\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 12.2915i 2.32288i
\(29\) 6.06910i 1.12700i 0.826115 + 0.563502i \(0.190546\pi\)
−0.826115 + 0.563502i \(0.809454\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 7.73381 1.36716
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 10.5830i − 1.73984i −0.493197 − 0.869918i \(-0.664172\pi\)
0.493197 − 0.869918i \(-0.335828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) − 5.29150i − 0.806947i −0.914991 − 0.403473i \(-0.867803\pi\)
0.914991 − 0.403473i \(-0.132197\pi\)
\(44\) 4.24264i 0.639602i
\(45\) 0 0
\(46\) 24.2288 3.57234
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −14.5544 −1.99920 −0.999599 − 0.0283132i \(-0.990986\pi\)
−0.999599 + 0.0283132i \(0.990986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 18.0455i 2.41144i
\(57\) 0 0
\(58\) 15.6458i 2.05439i
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 3.35425 0.419281
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 − 0.244339i \(-0.921429\pi\)
0.969690 − 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 7.57205i − 0.898637i −0.893372 − 0.449319i \(-0.851667\pi\)
0.893372 − 0.449319i \(-0.148333\pi\)
\(72\) 0 0
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) − 27.2823i − 3.17150i
\(75\) 0 0
\(76\) 0 0
\(77\) −2.41618 −0.275349
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) − 13.6412i − 1.47096i
\(87\) 0 0
\(88\) 6.22876i 0.663988i
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 43.6631 4.55220
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) −18.0455 −1.82288
\(99\) 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 1575.2.g.d.1574.8 8
3.2 odd 2 inner 1575.2.g.d.1574.2 8
5.2 odd 4 1575.2.b.a.251.4 4
5.3 odd 4 63.2.c.a.62.1 ✓ 4
5.4 even 2 inner 1575.2.g.d.1574.1 8
7.6 odd 2 CM 1575.2.g.d.1574.8 8
15.2 even 4 1575.2.b.a.251.1 4
15.8 even 4 63.2.c.a.62.4 yes 4
15.14 odd 2 inner 1575.2.g.d.1574.7 8
20.3 even 4 1008.2.k.a.881.1 4
21.20 even 2 inner 1575.2.g.d.1574.2 8
35.3 even 12 441.2.p.b.215.4 8
35.13 even 4 63.2.c.a.62.1 ✓ 4
35.18 odd 12 441.2.p.b.215.4 8
35.23 odd 12 441.2.p.b.80.1 8
35.27 even 4 1575.2.b.a.251.4 4
35.33 even 12 441.2.p.b.80.1 8
35.34 odd 2 inner 1575.2.g.d.1574.1 8
40.3 even 4 4032.2.k.b.3905.2 4
40.13 odd 4 4032.2.k.c.3905.3 4
45.13 odd 12 567.2.o.f.188.4 8
45.23 even 12 567.2.o.f.188.1 8
45.38 even 12 567.2.o.f.377.4 8
45.43 odd 12 567.2.o.f.377.1 8
60.23 odd 4 1008.2.k.a.881.2 4
105.23 even 12 441.2.p.b.80.4 8
105.38 odd 12 441.2.p.b.215.1 8
105.53 even 12 441.2.p.b.215.1 8
105.62 odd 4 1575.2.b.a.251.1 4
105.68 odd 12 441.2.p.b.80.4 8
105.83 odd 4 63.2.c.a.62.4 yes 4
105.104 even 2 inner 1575.2.g.d.1574.7 8
120.53 even 4 4032.2.k.c.3905.4 4
120.83 odd 4 4032.2.k.b.3905.1 4
140.83 odd 4 1008.2.k.a.881.1 4
280.13 even 4 4032.2.k.c.3905.3 4
280.83 odd 4 4032.2.k.b.3905.2 4
315.13 even 12 567.2.o.f.188.4 8
315.83 odd 12 567.2.o.f.377.4 8
315.223 even 12 567.2.o.f.377.1 8
315.293 odd 12 567.2.o.f.188.1 8
420.83 even 4 1008.2.k.a.881.2 4
840.83 even 4 4032.2.k.b.3905.1 4
840.293 odd 4 4032.2.k.c.3905.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.c.a.62.1 ✓ 4 5.3 odd 4
63.2.c.a.62.1 ✓ 4 35.13 even 4
63.2.c.a.62.4 yes 4 15.8 even 4
63.2.c.a.62.4 yes 4 105.83 odd 4
441.2.p.b.80.1 8 35.23 odd 12
441.2.p.b.80.1 8 35.33 even 12
441.2.p.b.80.4 8 105.23 even 12
441.2.p.b.80.4 8 105.68 odd 12
441.2.p.b.215.1 8 105.38 odd 12
441.2.p.b.215.1 8 105.53 even 12
441.2.p.b.215.4 8 35.3 even 12
441.2.p.b.215.4 8 35.18 odd 12
567.2.o.f.188.1 8 45.23 even 12
567.2.o.f.188.1 8 315.293 odd 12
567.2.o.f.188.4 8 45.13 odd 12
567.2.o.f.188.4 8 315.13 even 12
567.2.o.f.377.1 8 45.43 odd 12
567.2.o.f.377.1 8 315.223 even 12
567.2.o.f.377.4 8 45.38 even 12
567.2.o.f.377.4 8 315.83 odd 12
1008.2.k.a.881.1 4 20.3 even 4
1008.2.k.a.881.1 4 140.83 odd 4
1008.2.k.a.881.2 4 60.23 odd 4
1008.2.k.a.881.2 4 420.83 even 4
1575.2.b.a.251.1 4 15.2 even 4
1575.2.b.a.251.1 4 105.62 odd 4
1575.2.b.a.251.4 4 5.2 odd 4
1575.2.b.a.251.4 4 35.27 even 4
1575.2.g.d.1574.1 8 5.4 even 2 inner
1575.2.g.d.1574.1 8 35.34 odd 2 inner
1575.2.g.d.1574.2 8 3.2 odd 2 inner
1575.2.g.d.1574.2 8 21.20 even 2 inner
1575.2.g.d.1574.7 8 15.14 odd 2 inner
1575.2.g.d.1574.7 8 105.104 even 2 inner
1575.2.g.d.1574.8 8 1.1 even 1 trivial
1575.2.g.d.1574.8 8 7.6 odd 2 CM
4032.2.k.b.3905.1 4 120.83 odd 4
4032.2.k.b.3905.1 4 840.83 even 4
4032.2.k.b.3905.2 4 40.3 even 4
4032.2.k.b.3905.2 4 280.83 odd 4
4032.2.k.c.3905.3 4 40.13 odd 4
4032.2.k.c.3905.3 4 280.13 even 4
4032.2.k.c.3905.4 4 120.53 even 4
4032.2.k.c.3905.4 4 840.293 odd 4