Properties

Label 525.2.i.j
Level $525$
Weight $2$
Character orbit 525.i
Analytic conductor $4.192$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,2,Mod(151,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.i (of order \(3\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 8x^{6} - 3x^{5} + 50x^{4} - 27x^{3} + 53x^{2} + 20x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{7} + 2 \beta_{5} - \beta_{3} + \beta_{2}) q^{4} - \beta_{3} q^{6} + ( - \beta_{4} + \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{7} - \beta_{6} - 3 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{8} + ( - \beta_{5} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{7} + 2 \beta_{5} - \beta_{3} + \beta_{2}) q^{4} - \beta_{3} q^{6} + ( - \beta_{4} + \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{7} - \beta_{6} - 3 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{8} + ( - \beta_{5} - 1) q^{9} + ( - \beta_{6} + 3 \beta_{5} - \beta_{4} + \beta_{3} - \beta_1) q^{11} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 - 2) q^{12} + ( - \beta_{7} - \beta_{4} + 2 \beta_{3} + 1) q^{13} + (\beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{14} + ( - 2 \beta_{7} - \beta_{6} - 5 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 - 5) q^{16} + (\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{17} + (\beta_{3} - \beta_1) q^{18} + (\beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} - 3 \beta_1) q^{19} + ( - \beta_{3} + \beta_{2}) q^{21} + ( - \beta_{7} - \beta_{6} - 4 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{22} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{23} + (\beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3} - 2 \beta_1) q^{24} + (4 \beta_{5} + 3 \beta_1 + 4) q^{26} + q^{27} + ( - \beta_{6} + 5 \beta_{5} - 4 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{28} + 4 q^{29} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} + 3 \beta_1) q^{31} + ( - 2 \beta_{7} - 4 \beta_{5} + 5 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{32} + ( - \beta_{7} - 3 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{33} + (\beta_{7} + 3 \beta_{6} - 2 \beta_{4} + 4 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 5) q^{34} + (\beta_{6} - \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 + 2) q^{36} + (2 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 4 \beta_1 + 3) q^{37} + ( - 2 \beta_{7} - \beta_{6} - 7 \beta_{5} - \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{38} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1) q^{39} + ( - \beta_{7} + \beta_{6} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{41} + ( - 2 \beta_{7} - \beta_{6} - 3 \beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 2) q^{42} + (\beta_{7} + 2 \beta_{6} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{43} + ( - 3 \beta_{7} - 4 \beta_{6} - 7 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} - \beta_{2} + \cdots - 7) q^{44}+ \cdots + (\beta_{7} + \beta_{6} - \beta_{2} - \beta_1 + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - 4 q^{3} - 7 q^{4} - 2 q^{6} - q^{7} - 6 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} - 4 q^{3} - 7 q^{4} - 2 q^{6} - q^{7} - 6 q^{8} - 4 q^{9} - 8 q^{11} - 7 q^{12} + 14 q^{13} - 12 q^{14} - 17 q^{16} - 6 q^{17} + q^{18} - 3 q^{19} + 2 q^{21} + 24 q^{22} + 2 q^{23} + 3 q^{24} + 19 q^{26} + 8 q^{27} - 15 q^{28} + 32 q^{29} - 9 q^{31} + 17 q^{32} - 8 q^{33} + 28 q^{34} + 14 q^{36} + 8 q^{37} + 27 q^{38} - 7 q^{39} + 8 q^{41} - 3 q^{42} - 10 q^{43} - 26 q^{44} + 6 q^{46} + 6 q^{47} + 34 q^{48} - 21 q^{49} - 6 q^{51} - 35 q^{52} - 6 q^{53} + q^{54} - 21 q^{56} + 6 q^{57} + 4 q^{58} - 10 q^{59} + 20 q^{61} - 88 q^{62} - q^{63} + 42 q^{64} - 12 q^{66} + q^{67} + 8 q^{68} - 4 q^{69} + 44 q^{71} + 3 q^{72} + 4 q^{73} + 21 q^{74} - 46 q^{76} - 32 q^{77} - 38 q^{78} - 8 q^{79} - 4 q^{81} - 8 q^{82} - 4 q^{83} - 18 q^{84} - 12 q^{86} - 16 q^{87} + 28 q^{88} + 10 q^{89} + 21 q^{91} + 132 q^{92} - 9 q^{93} - 22 q^{94} + 17 q^{96} + 24 q^{97} - 67 q^{98} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 8x^{6} - 3x^{5} + 50x^{4} - 27x^{3} + 53x^{2} + 20x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 689\nu^{7} - 7573\nu^{6} + 14556\nu^{5} - 51899\nu^{4} + 65994\nu^{3} - 301543\nu^{2} + 311369\nu - 57732 ) / 75828 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -329\nu^{7} + 21\nu^{6} - 2209\nu^{5} - 1081\nu^{4} - 17462\nu^{3} - 3431\nu^{2} - 1692\nu - 8164 ) / 18957 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1653 \nu^{7} + 3659 \nu^{6} + 10196 \nu^{5} + 37929 \nu^{4} + 67606 \nu^{3} + 217641 \nu^{2} - 19483 \nu + 208424 ) / 75828 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2041 \nu^{7} + 3357 \nu^{6} - 16412 \nu^{5} + 14959 \nu^{4} - 97726 \nu^{3} + 124955 \nu^{2} - 94449 \nu - 34052 ) / 75828 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1871\nu^{7} - 1195\nu^{6} + 17076\nu^{5} - 4685\nu^{4} + 112020\nu^{3} - 34651\nu^{2} + 187457\nu - 33126 ) / 37914 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6159 \nu^{7} - 5771 \nu^{6} + 42256 \nu^{5} - 12261 \nu^{4} + 255062 \nu^{3} - 136173 \nu^{2} + 59659 \nu + 161284 ) / 75828 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 4\beta_{5} - \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - \beta_{6} - 7\beta_{3} + \beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{7} - 7\beta_{6} - 25\beta_{5} + 8\beta_{4} - 8\beta_{3} + \beta_{2} + 7\beta _1 - 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{7} + 8\beta_{6} - 12\beta_{5} + 8\beta_{4} + 41\beta_{3} - 2\beta_{2} - 39\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{7} + 57\beta_{6} - 47\beta_{4} + 120\beta_{3} - 57\beta_{2} - 57\beta _1 + 166 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 83\beta_{7} + 26\beta_{6} + 121\beta_{5} - 83\beta_{4} + 83\beta_{3} - 57\beta_{2} + 177\beta _1 + 121 \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
−1.21868 2.11082i
−0.247087 0.427967i
0.614340 + 1.06407i
1.35143 + 2.34074i
−1.21868 + 2.11082i
−0.247087 + 0.427967i
0.614340 1.06407i
1.35143 2.34074i
−1.21868 2.11082i −0.500000 + 0.866025i −1.97036 + 3.41277i 0 2.43736 0.970361 2.46138i 4.73024 −0.500000 0.866025i 0
151.2 −0.247087 0.427967i −0.500000 + 0.866025i 0.877896 1.52056i 0 0.494173 −1.87790 + 1.86373i −1.85601 −0.500000 0.866025i 0
151.3 0.614340 + 1.06407i −0.500000 + 0.866025i 0.245174 0.424653i 0 −1.22868 −1.24517 2.33443i 3.05984 −0.500000 0.866025i 0
151.4 1.35143 + 2.34074i −0.500000 + 0.866025i −2.65271 + 4.59463i 0 −2.70285 1.65271 + 2.06605i −8.93406 −0.500000 0.866025i 0
226.1 −1.21868 + 2.11082i −0.500000 0.866025i −1.97036 3.41277i 0 2.43736 0.970361 + 2.46138i 4.73024 −0.500000 + 0.866025i 0
226.2 −0.247087 + 0.427967i −0.500000 0.866025i 0.877896 + 1.52056i 0 0.494173 −1.87790 1.86373i −1.85601 −0.500000 + 0.866025i 0
226.3 0.614340 1.06407i −0.500000 0.866025i 0.245174 + 0.424653i 0 −1.22868 −1.24517 + 2.33443i 3.05984 −0.500000 + 0.866025i 0
226.4 1.35143 2.34074i −0.500000 0.866025i −2.65271 4.59463i 0 −2.70285 1.65271 2.06605i −8.93406 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.i.j yes 8
5.b even 2 1 525.2.i.i 8
5.c odd 4 2 525.2.r.h 16
7.c even 3 1 inner 525.2.i.j yes 8
7.c even 3 1 3675.2.a.br 4
7.d odd 6 1 3675.2.a.bq 4
35.i odd 6 1 3675.2.a.bx 4
35.j even 6 1 525.2.i.i 8
35.j even 6 1 3675.2.a.bw 4
35.l odd 12 2 525.2.r.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.i.i 8 5.b even 2 1
525.2.i.i 8 35.j even 6 1
525.2.i.j yes 8 1.a even 1 1 trivial
525.2.i.j yes 8 7.c even 3 1 inner
525.2.r.h 16 5.c odd 4 2
525.2.r.h 16 35.l odd 12 2
3675.2.a.bq 4 7.d odd 6 1
3675.2.a.br 4 7.c even 3 1
3675.2.a.bw 4 35.j even 6 1
3675.2.a.bx 4 35.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - T_{2}^{7} + 8T_{2}^{6} - 3T_{2}^{5} + 50T_{2}^{4} - 27T_{2}^{3} + 53T_{2}^{2} + 20T_{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + 8 T^{6} - 3 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + T^{7} + 11 T^{6} + 12 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 8 T^{7} + 72 T^{6} + \cdots + 107584 \) Copy content Toggle raw display
$13$ \( (T^{4} - 7 T^{3} - 19 T^{2} + 179 T - 194)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 6 T^{7} + 56 T^{6} + \cdots + 1600 \) Copy content Toggle raw display
$19$ \( T^{8} + 3 T^{7} + 46 T^{6} + \cdots + 38416 \) Copy content Toggle raw display
$23$ \( T^{8} - 2 T^{7} + 72 T^{6} + \cdots + 921600 \) Copy content Toggle raw display
$29$ \( (T - 4)^{8} \) Copy content Toggle raw display
$31$ \( T^{8} + 9 T^{7} + 137 T^{6} + \cdots + 589824 \) Copy content Toggle raw display
$37$ \( T^{8} - 8 T^{7} + 126 T^{6} + \cdots + 597529 \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} - 60 T^{2} + 236 T - 200)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 5 T^{3} - 52 T^{2} + 32 T + 160)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 6 T^{7} + 100 T^{6} + \cdots + 732736 \) Copy content Toggle raw display
$53$ \( T^{8} + 6 T^{7} + 112 T^{6} + \cdots + 1236544 \) Copy content Toggle raw display
$59$ \( T^{8} + 10 T^{7} + 140 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$61$ \( (T^{2} - 5 T + 25)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - T^{7} + 18 T^{6} - T^{5} + \cdots + 3600 \) Copy content Toggle raw display
$71$ \( (T^{4} - 22 T^{3} - 32 T^{2} + 2880 T - 12736)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 4 T^{7} + 138 T^{6} + \cdots + 2455489 \) Copy content Toggle raw display
$79$ \( T^{8} + 8 T^{7} + 134 T^{6} + \cdots + 23409 \) Copy content Toggle raw display
$83$ \( (T^{4} + 2 T^{3} - 112 T^{2} + 396 T - 312)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 10 T^{7} + 96 T^{6} + \cdots + 576 \) Copy content Toggle raw display
$97$ \( (T^{4} - 12 T^{3} - 58 T^{2} + 196 T - 79)^{2} \) Copy content Toggle raw display
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