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Properties

Label 1575.2.m.d
Level $1575$
Weight $2$
Character orbit 1575.m
Analytic conductor $12.576$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1575,2,Mod(1268,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1575.1268");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.m (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 18x^{10} + 107x^{8} + 240x^{6} + 151x^{4} + 30x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 315)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (2 \beta_{10} - \beta_{4} + \cdots + \beta_1) q^{4}+ \cdots + (\beta_{11} - 2 \beta_{10} + \beta_{9} + \cdots + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + (2 \beta_{10} - \beta_{4} + \cdots + \beta_1) q^{4}+ \cdots - \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 24 q^{8} + 4 q^{13} + 4 q^{14} - 20 q^{16} + 8 q^{17} + 8 q^{22} + 8 q^{23} + 32 q^{29} + 48 q^{32} - 4 q^{37} + 24 q^{38} - 40 q^{43} + 64 q^{44} + 16 q^{46} + 24 q^{47} - 36 q^{52} - 40 q^{53} + 28 q^{58} + 80 q^{59} - 32 q^{61} + 16 q^{62} + 48 q^{67} + 32 q^{68} + 20 q^{73} + 64 q^{74} + 16 q^{76} - 20 q^{82} + 24 q^{83} + 56 q^{89} + 8 q^{92} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 18x^{10} + 107x^{8} + 240x^{6} + 151x^{4} + 30x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{11} - 16\nu^{9} - 70\nu^{7} - 20\nu^{5} + 289\nu^{3} + 82\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 17\nu^{8} + 92\nu^{6} + 174\nu^{4} + 59\nu^{2} + 5 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3 \nu^{11} - 7 \nu^{10} + 53 \nu^{9} - 127 \nu^{8} + 300 \nu^{7} - 760 \nu^{6} + 570 \nu^{5} + \cdots - 71 ) / 40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3 \nu^{11} + 7 \nu^{10} + 53 \nu^{9} + 127 \nu^{8} + 300 \nu^{7} + 760 \nu^{6} + 570 \nu^{5} + \cdots + 71 ) / 40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5 \nu^{11} - 7 \nu^{10} + 85 \nu^{9} - 127 \nu^{8} + 450 \nu^{7} - 750 \nu^{6} + 740 \nu^{5} + \cdots - 41 ) / 40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5 \nu^{11} - 7 \nu^{10} - 85 \nu^{9} - 127 \nu^{8} - 450 \nu^{7} - 750 \nu^{6} - 740 \nu^{5} + \cdots - 41 ) / 40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5 \nu^{11} + 9 \nu^{10} - 85 \nu^{9} + 159 \nu^{8} - 450 \nu^{7} + 910 \nu^{6} - 740 \nu^{5} + \cdots + 57 ) / 40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5 \nu^{11} + 9 \nu^{10} + 85 \nu^{9} + 159 \nu^{8} + 450 \nu^{7} + 910 \nu^{6} + 740 \nu^{5} + \cdots + 57 ) / 40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -6\nu^{10} - 106\nu^{8} - 605\nu^{6} - 1215\nu^{4} - 421\nu^{2} - 3 ) / 10 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -8\nu^{11} - 143\nu^{9} - 840\nu^{7} - 1840\nu^{5} - 1058\nu^{3} - 139\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 29\nu^{11} + 514\nu^{9} + 2960\nu^{7} + 6130\nu^{5} + 2669\nu^{3} + 222\nu ) / 20 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{6} - \beta_{5} + \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{11} + 6\beta_{10} + 7\beta_{8} - 7\beta_{7} + 7\beta_{6} - 7\beta_{5} - 2\beta_{4} - 2\beta_{3} + 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{9} + 4\beta_{8} + 4\beta_{7} + 8\beta_{6} + 8\beta_{5} - \beta_{4} + \beta_{3} - 9\beta_{2} + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 20 \beta_{11} - 62 \beta_{10} - 53 \beta_{8} + 53 \beta_{7} - 51 \beta_{6} + 51 \beta_{5} + \cdots - 46 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 41\beta_{9} - 52\beta_{8} - 52\beta_{7} - 65\beta_{6} - 65\beta_{5} + 15\beta_{4} - 15\beta_{3} + 80\beta_{2} - 126 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 178 \beta_{11} + 560 \beta_{10} + 425 \beta_{8} - 425 \beta_{7} + 395 \beta_{6} - 395 \beta_{5} + \cdots + 442 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 306 \beta_{9} + 518 \beta_{8} + 518 \beta_{7} + 536 \beta_{6} + 536 \beta_{5} - 156 \beta_{4} + \cdots + 957 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1532 \beta_{11} - 4860 \beta_{10} - 3517 \beta_{8} + 3517 \beta_{7} - 3181 \beta_{6} + \cdots - 3976 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2415 \beta_{9} - 4718 \beta_{8} - 4718 \beta_{7} - 4465 \beta_{6} - 4465 \beta_{5} + 1446 \beta_{4} + \cdots - 7637 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 13030 \beta_{11} + 41534 \beta_{10} + 29523 \beta_{8} - 29523 \beta_{7} + 26207 \beta_{6} + \cdots + 34712 \beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(-\beta_{10}\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1268.1
0.556948i
0.699479i
− 2.15459i
− 0.203482i
2.01185i
− 2.91021i
− 0.556948i
− 0.699479i
2.15459i
0.203482i
− 2.01185i
2.91021i
−1.96418 − 1.96418i 0 5.71600i 0 0 −0.707107 + 0.707107i 7.29890 − 7.29890i 0 0
1268.2 −1.62044 − 1.62044i 0 3.25168i 0 0 0.707107 − 0.707107i 2.02827 − 2.02827i 0 0
1268.3 −0.0241053 − 0.0241053i 0 − 1.99884i 0 0 −0.707107 + 0.707107i −0.0963933 + 0.0963933i 0 0
1268.4 0.876184 + 0.876184i 0 − 0.464602i 0 0 0.707107 − 0.707107i 2.15945 − 2.15945i 0 0
1268.5 1.28118 + 1.28118i 0 1.28283i 0 0 −0.707107 + 0.707107i 0.918816 − 0.918816i 0 0
1268.6 1.45137 + 1.45137i 0 2.21293i 0 0 0.707107 − 0.707107i −0.309035 + 0.309035i 0 0
1457.1 −1.96418 + 1.96418i 0 − 5.71600i 0 0 −0.707107 − 0.707107i 7.29890 + 7.29890i 0 0
1457.2 −1.62044 + 1.62044i 0 − 3.25168i 0 0 0.707107 + 0.707107i 2.02827 + 2.02827i 0 0
1457.3 −0.0241053 + 0.0241053i 0 1.99884i 0 0 −0.707107 − 0.707107i −0.0963933 − 0.0963933i 0 0
1457.4 0.876184 − 0.876184i 0 0.464602i 0 0 0.707107 + 0.707107i 2.15945 + 2.15945i 0 0
1457.5 1.28118 − 1.28118i 0 − 1.28283i 0 0 −0.707107 − 0.707107i 0.918816 + 0.918816i 0 0
1457.6 1.45137 − 1.45137i 0 − 2.21293i 0 0 0.707107 + 0.707107i −0.309035 − 0.309035i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1268.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.2.m.d 12
3.b odd 2 1 1575.2.m.c 12
5.b even 2 1 315.2.m.b yes 12
5.c odd 4 1 315.2.m.a ✓ 12
5.c odd 4 1 1575.2.m.c 12
15.d odd 2 1 315.2.m.a ✓ 12
15.e even 4 1 315.2.m.b yes 12
15.e even 4 1 inner 1575.2.m.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.m.a ✓ 12 5.c odd 4 1
315.2.m.a ✓ 12 15.d odd 2 1
315.2.m.b yes 12 5.b even 2 1
315.2.m.b yes 12 15.e even 4 1
1575.2.m.c 12 3.b odd 2 1
1575.2.m.c 12 5.c odd 4 1
1575.2.m.d 12 1.a even 1 1 trivial
1575.2.m.d 12 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 8 T_{2}^{9} + 59 T_{2}^{8} - 48 T_{2}^{7} + 32 T_{2}^{6} - 192 T_{2}^{5} + 939 T_{2}^{4} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(1575, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 8 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{12} + 80 T^{10} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{12} - 4 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( T^{12} - 8 T^{11} + \cdots + 7311616 \) Copy content Toggle raw display
$19$ \( T^{12} + 144 T^{10} + \cdots + 78287104 \) Copy content Toggle raw display
$23$ \( T^{12} - 8 T^{11} + \cdots + 10137856 \) Copy content Toggle raw display
$29$ \( (T^{6} - 16 T^{5} + \cdots + 6008)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 72 T^{4} + \cdots + 5008)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 4 T^{11} + \cdots + 16711744 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 1248632896 \) Copy content Toggle raw display
$43$ \( T^{12} + 40 T^{11} + \cdots + 15745024 \) Copy content Toggle raw display
$47$ \( T^{12} - 24 T^{11} + \cdots + 65536 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 42054525184 \) Copy content Toggle raw display
$59$ \( (T^{6} - 40 T^{5} + \cdots + 34688)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 16 T^{5} + \cdots - 101248)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 5454708736 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 234947584 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1871081536 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 21143486464 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 1494286336 \) Copy content Toggle raw display
$89$ \( (T^{6} - 28 T^{5} + \cdots + 47800)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 460016704 \) Copy content Toggle raw display
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