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Properties

Label 525.2.m.b
Level $525$
Weight $2$
Character orbit 525.m
Analytic conductor $4.192$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,2,Mod(118,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.118");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.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: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 105)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + \beta_{9} q^{3} + ( - \beta_{12} - \beta_{11} + \cdots + \beta_{2}) q^{4}+ \cdots + \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + \beta_{9} q^{3} + ( - \beta_{12} - \beta_{11} + \cdots + \beta_{2}) q^{4}+ \cdots + (\beta_{15} - \beta_{7} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{7} - 24 q^{8} - 16 q^{11} - 48 q^{16} + 8 q^{21} + 16 q^{22} + 40 q^{23} - 24 q^{28} - 48 q^{32} - 16 q^{36} - 32 q^{37} + 16 q^{42} + 16 q^{43} + 64 q^{46} - 16 q^{51} - 24 q^{53} + 24 q^{56} - 8 q^{57} - 32 q^{58} - 8 q^{63} + 32 q^{67} + 64 q^{71} - 24 q^{72} + 24 q^{77} + 8 q^{78} - 16 q^{81} + 64 q^{86} + 64 q^{88} - 48 q^{91} + 40 q^{92} - 24 q^{93} + 96 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{14} - 16\nu^{12} + 6\nu^{10} - 20\nu^{8} + 81\nu^{6} - 60\nu^{4} + 176\nu^{2} - 832 ) / 256 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 17\nu^{14} - 72\nu^{12} + 6\nu^{10} - 36\nu^{8} + 513\nu^{6} - 1332\nu^{4} + 3696\nu^{2} - 4544 ) / 3840 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{15} - 3\nu^{13} + 24\nu^{11} + 6\nu^{9} - 78\nu^{7} + 117\nu^{5} - 216\nu^{3} - 16\nu ) / 960 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -29\nu^{15} + 144\nu^{13} + 18\nu^{11} + 132\nu^{9} - 621\nu^{7} + 204\nu^{5} - 1392\nu^{3} + 4928\nu ) / 7680 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 43\nu^{14} - 48\nu^{12} - 126\nu^{10} + 36\nu^{8} + 27\nu^{6} + 492\nu^{4} + 144\nu^{2} - 1216 ) / 3840 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{15} - 4\nu^{13} + 6\nu^{11} - 12\nu^{9} + 33\nu^{7} - 48\nu^{5} + 96\nu^{3} - 128\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -43\nu^{14} + 108\nu^{12} - 114\nu^{10} + 324\nu^{8} - 747\nu^{6} + 528\nu^{4} - 3024\nu^{2} + 6016 ) / 1920 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{15} + 27 \nu^{14} - 72 \nu^{12} - 6 \nu^{11} + 66 \nu^{10} - 12 \nu^{9} - 300 \nu^{8} + \cdots - 4416 ) / 768 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 149\nu^{15} - 264\nu^{13} + 222\nu^{11} - 1332\nu^{9} + 2181\nu^{7} - 1764\nu^{5} + 8112\nu^{3} - 14528\nu ) / 7680 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14 \nu^{15} + 17 \nu^{14} - 24 \nu^{13} - 12 \nu^{12} + 12 \nu^{11} + 6 \nu^{10} - 72 \nu^{9} + \cdots - 1664 ) / 960 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14 \nu^{15} - 17 \nu^{14} - 24 \nu^{13} + 12 \nu^{12} + 12 \nu^{11} - 6 \nu^{10} - 72 \nu^{9} + \cdots + 1664 ) / 960 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 56 \nu^{15} - 145 \nu^{14} + 96 \nu^{13} + 360 \nu^{12} - 48 \nu^{11} - 390 \nu^{10} + \cdots + 22720 ) / 3840 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 13 \nu^{15} + 18 \nu^{14} + 16 \nu^{13} - 48 \nu^{12} - 14 \nu^{11} + 44 \nu^{10} + 68 \nu^{9} + \cdots - 2944 ) / 512 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 215 \nu^{15} + 274 \nu^{14} - 480 \nu^{13} - 384 \nu^{12} + 330 \nu^{11} + 492 \nu^{10} + \cdots - 28288 ) / 7680 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -401\nu^{14} + 816\nu^{12} - 678\nu^{10} + 3348\nu^{8} - 6369\nu^{6} + 5916\nu^{4} - 25968\nu^{2} + 49472 ) / 3840 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{14} + \beta_{9} + \beta_{8} + \beta_{6} - \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{12} + \beta_{10} - \beta_{7} - \beta_{5} + 2\beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} + \beta_{14} + \beta_{11} + \beta_{10} - 3 \beta_{9} + \beta_{8} - \beta_{7} + \cdots - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{15} + \beta_{12} - \beta_{11} + 2\beta_{10} - 4\beta_{7} - \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2 \beta_{15} + \beta_{14} + 4 \beta_{13} + 4 \beta_{11} + 4 \beta_{10} - \beta_{9} - \beta_{8} + \cdots - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4\beta_{15} - 7\beta_{12} - 6\beta_{11} - \beta_{10} - \beta_{7} + 3\beta_{5} + 2\beta_{2} + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( \beta_{15} + \beta_{14} + 8 \beta_{13} + 7 \beta_{11} + 7 \beta_{10} - 3 \beta_{9} - 7 \beta_{8} + \cdots - 9 \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 9\beta_{15} - 7\beta_{12} - 3\beta_{11} - 4\beta_{10} - 24\beta_{7} + 12\beta_{5} + 11\beta_{2} - 7\beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 10 \beta_{15} + \beta_{14} - 8 \beta_{13} + 10 \beta_{11} + 10 \beta_{10} - 33 \beta_{9} + \cdots + 11 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 8\beta_{15} - 3\beta_{12} + 4\beta_{11} - 7\beta_{10} - 53\beta_{7} - 13\beta_{5} - 14\beta_{2} - 16\beta _1 - 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 7 \beta_{15} + \beta_{14} + 13 \beta_{11} + 13 \beta_{10} - 19 \beta_{9} - 15 \beta_{8} + \cdots + 8 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 9\beta_{15} - 23\beta_{12} - 21\beta_{11} - 2\beta_{10} + 4\beta_{7} - 8\beta_{5} + 15\beta_{2} - 39\beta _1 - 71 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 6 \beta_{15} + 25 \beta_{14} + 12 \beta_{13} + 24 \beta_{11} + 24 \beta_{10} - 25 \beta_{9} + \cdots + 31 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 12 \beta_{15} - 39 \beta_{12} + 6 \beta_{11} - 45 \beta_{10} - 81 \beta_{7} + 123 \beta_{5} - 30 \beta_{2} + \cdots - 67 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 135 \beta_{15} - 103 \beta_{14} - 120 \beta_{13} + 3 \beta_{11} + 3 \beta_{10} + 85 \beta_{9} + \cdots + 32 \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(\beta_{7}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
118.1
−0.944649 − 1.05244i
0.944649 + 1.05244i
−1.36166 − 0.381939i
1.36166 + 0.381939i
1.40927 + 0.118126i
−1.40927 − 0.118126i
−0.517174 + 1.31626i
0.517174 − 1.31626i
−0.944649 + 1.05244i
0.944649 − 1.05244i
−1.36166 + 0.381939i
1.36166 − 0.381939i
1.40927 − 0.118126i
−1.40927 + 0.118126i
−0.517174 − 1.31626i
0.517174 + 1.31626i
−1.48838 + 1.48838i −0.707107 + 0.707107i − 2.43055i 0 − 2.10489i 1.97552 − 1.75993i 0.640825 + 0.640825i − 1.00000i 0
118.2 −1.48838 + 1.48838i 0.707107 − 0.707107i − 2.43055i 0 2.10489i 1.75993 − 1.97552i 0.640825 + 0.640825i − 1.00000i 0
118.3 −0.540143 + 0.540143i −0.707107 + 0.707107i 1.41649i 0 − 0.763878i −2.57351 + 0.614060i −1.84539 − 1.84539i − 1.00000i 0
118.4 −0.540143 + 0.540143i 0.707107 − 0.707107i 1.41649i 0 0.763878i −0.614060 + 2.57351i −1.84539 − 1.84539i − 1.00000i 0
118.5 0.167056 − 0.167056i −0.707107 + 0.707107i 1.94418i 0 0.236253i 2.64501 − 0.0627175i 0.658899 + 0.658899i − 1.00000i 0
118.6 0.167056 − 0.167056i 0.707107 − 0.707107i 1.94418i 0 − 0.236253i 0.0627175 − 2.64501i 0.658899 + 0.658899i − 1.00000i 0
118.7 1.86147 − 1.86147i −0.707107 + 0.707107i − 4.93012i 0 2.63251i −1.46123 − 2.20563i −5.45433 − 5.45433i − 1.00000i 0
118.8 1.86147 − 1.86147i 0.707107 − 0.707107i − 4.93012i 0 − 2.63251i 2.20563 + 1.46123i −5.45433 − 5.45433i − 1.00000i 0
307.1 −1.48838 − 1.48838i −0.707107 − 0.707107i 2.43055i 0 2.10489i 1.97552 + 1.75993i 0.640825 − 0.640825i 1.00000i 0
307.2 −1.48838 − 1.48838i 0.707107 + 0.707107i 2.43055i 0 − 2.10489i 1.75993 + 1.97552i 0.640825 − 0.640825i 1.00000i 0
307.3 −0.540143 − 0.540143i −0.707107 − 0.707107i − 1.41649i 0 0.763878i −2.57351 − 0.614060i −1.84539 + 1.84539i 1.00000i 0
307.4 −0.540143 − 0.540143i 0.707107 + 0.707107i − 1.41649i 0 − 0.763878i −0.614060 − 2.57351i −1.84539 + 1.84539i 1.00000i 0
307.5 0.167056 + 0.167056i −0.707107 − 0.707107i − 1.94418i 0 − 0.236253i 2.64501 + 0.0627175i 0.658899 − 0.658899i 1.00000i 0
307.6 0.167056 + 0.167056i 0.707107 + 0.707107i − 1.94418i 0 0.236253i 0.0627175 + 2.64501i 0.658899 − 0.658899i 1.00000i 0
307.7 1.86147 + 1.86147i −0.707107 − 0.707107i 4.93012i 0 − 2.63251i −1.46123 + 2.20563i −5.45433 + 5.45433i 1.00000i 0
307.8 1.86147 + 1.86147i 0.707107 + 0.707107i 4.93012i 0 2.63251i 2.20563 − 1.46123i −5.45433 + 5.45433i 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 118.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.m.b 16
5.b even 2 1 105.2.m.a ✓ 16
5.c odd 4 1 105.2.m.a ✓ 16
5.c odd 4 1 inner 525.2.m.b 16
7.b odd 2 1 inner 525.2.m.b 16
15.d odd 2 1 315.2.p.e 16
15.e even 4 1 315.2.p.e 16
20.d odd 2 1 1680.2.cz.d 16
20.e even 4 1 1680.2.cz.d 16
35.c odd 2 1 105.2.m.a ✓ 16
35.f even 4 1 105.2.m.a ✓ 16
35.f even 4 1 inner 525.2.m.b 16
35.i odd 6 2 735.2.v.a 32
35.j even 6 2 735.2.v.a 32
35.k even 12 2 735.2.v.a 32
35.l odd 12 2 735.2.v.a 32
105.g even 2 1 315.2.p.e 16
105.k odd 4 1 315.2.p.e 16
140.c even 2 1 1680.2.cz.d 16
140.j odd 4 1 1680.2.cz.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.m.a ✓ 16 5.b even 2 1
105.2.m.a ✓ 16 5.c odd 4 1
105.2.m.a ✓ 16 35.c odd 2 1
105.2.m.a ✓ 16 35.f even 4 1
315.2.p.e 16 15.d odd 2 1
315.2.p.e 16 15.e even 4 1
315.2.p.e 16 105.g even 2 1
315.2.p.e 16 105.k odd 4 1
525.2.m.b 16 1.a even 1 1 trivial
525.2.m.b 16 5.c odd 4 1 inner
525.2.m.b 16 7.b odd 2 1 inner
525.2.m.b 16 35.f even 4 1 inner
735.2.v.a 32 35.i odd 6 2
735.2.v.a 32 35.j even 6 2
735.2.v.a 32 35.k even 12 2
735.2.v.a 32 35.l odd 12 2
1680.2.cz.d 16 20.d odd 2 1
1680.2.cz.d 16 20.e even 4 1
1680.2.cz.d 16 140.c even 2 1
1680.2.cz.d 16 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 4T_{2}^{5} + 34T_{2}^{4} + 24T_{2}^{3} + 8T_{2}^{2} - 4T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 4 T^{5} + 34 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 8 T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{4} + 4 T^{3} - 12 T^{2} + \cdots - 60)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} + 736 T^{12} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 100000000 \) Copy content Toggle raw display
$19$ \( (T^{8} - 104 T^{6} + \cdots + 144)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 20 T^{7} + \cdots + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 48 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 120 T^{6} + \cdots + 274576)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 16 T^{7} + \cdots + 144)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 184 T^{6} + \cdots + 129600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 8 T^{7} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 25216 T^{12} + \cdots + 1048576 \) Copy content Toggle raw display
$53$ \( (T^{8} + 12 T^{7} + \cdots + 129600)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 160 T^{6} + \cdots + 1183744)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 288 T^{6} + \cdots + 640000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 16 T^{7} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 16 T^{3} + \cdots - 1132)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 169459576016896 \) Copy content Toggle raw display
$79$ \( (T^{8} + 320 T^{6} + \cdots + 18939904)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 45137758519296 \) Copy content Toggle raw display
$89$ \( (T^{8} - 136 T^{6} + \cdots + 107584)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 60\!\cdots\!56 \) Copy content Toggle raw display
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