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Properties

Label 525.3.h.d
Level $525$
Weight $3$
Character orbit 525.h
Analytic conductor $14.305$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,3,Mod(76,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.76");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.h (of order \(2\), degree \(1\), 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: \(14.3052138789\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 21 x^{10} - 26 x^{9} + 295 x^{8} - 372 x^{7} + 1704 x^{6} - 1074 x^{5} + 5613 x^{4} + \cdots + 441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - \beta_{3} q^{3} + (\beta_1 + 4) q^{4} - \beta_{9} q^{6} + (\beta_{11} + \beta_{3} + 1) q^{7} + (\beta_{7} + 3 \beta_{5} + \beta_1 - 1) q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} - \beta_{3} q^{3} + (\beta_1 + 4) q^{4} - \beta_{9} q^{6} + (\beta_{11} + \beta_{3} + 1) q^{7} + (\beta_{7} + 3 \beta_{5} + \beta_1 - 1) q^{8} - 3 q^{9} + (\beta_{8} + \beta_{7} + 2 \beta_{5} + \cdots - 1) q^{11}+ \cdots + ( - 3 \beta_{8} - 3 \beta_{7} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{2} + 44 q^{4} + 8 q^{7} - 4 q^{8} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{2} + 44 q^{4} + 8 q^{7} - 4 q^{8} - 36 q^{9} - 16 q^{11} - 40 q^{14} + 92 q^{16} - 12 q^{18} + 36 q^{21} + 88 q^{22} + 64 q^{23} - 88 q^{28} + 104 q^{29} + 228 q^{32} - 132 q^{36} - 32 q^{37} - 24 q^{39} + 60 q^{42} - 152 q^{43} + 192 q^{44} + 200 q^{46} + 60 q^{49} + 24 q^{51} - 176 q^{53} - 368 q^{56} + 240 q^{57} + 400 q^{58} - 24 q^{63} - 20 q^{64} - 168 q^{67} + 32 q^{71} + 12 q^{72} + 184 q^{74} - 8 q^{77} - 456 q^{78} + 120 q^{79} + 108 q^{81} + 108 q^{84} + 400 q^{86} + 536 q^{88} + 24 q^{91} - 192 q^{92} - 48 q^{93} - 884 q^{98} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 21 x^{10} - 26 x^{9} + 295 x^{8} - 372 x^{7} + 1704 x^{6} - 1074 x^{5} + 5613 x^{4} + \cdots + 441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1163071964 \nu^{11} - 179172244 \nu^{10} - 20630949354 \nu^{9} - 16671982336 \nu^{8} + \cdots - 2151653525805 ) / 5604734688861 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6667044052 \nu^{11} + 14676332312 \nu^{10} - 143980659226 \nu^{9} + 199624049398 \nu^{8} + \cdots + 4796809862817 ) / 5604734688861 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7830116016 \nu^{11} + 14497160068 \nu^{10} - 164611608580 \nu^{9} + 182952067062 \nu^{8} + \cdots + 8249891025873 ) / 5604734688861 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 28650310720 \nu^{11} + 90431027955 \nu^{10} - 729823436526 \nu^{9} + \cdots + 259629670498401 ) / 16814204066583 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5593516534 \nu^{11} - 2605315656 \nu^{10} + 102990651177 \nu^{9} + 27516350446 \nu^{8} + \cdots + 10342332760284 ) / 2402029152369 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 96862294048 \nu^{11} + 204220441760 \nu^{10} - 2007177917898 \nu^{9} + \cdots + 46133280083394 ) / 16814204066583 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 840872 \nu^{11} + 322002 \nu^{10} - 15493812 \nu^{9} - 5194574 \nu^{8} - 224506950 \nu^{7} + \cdots - 1589299452 ) / 133884909 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 119594038472 \nu^{11} + 165327704385 \nu^{10} - 2428788089634 \nu^{9} + \cdots - 55476225402885 ) / 16814204066583 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 63482984056 \nu^{11} + 122784179216 \nu^{10} - 1310387571065 \nu^{9} + \cdots + 46611711121164 ) / 5604734688861 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 202706987158 \nu^{11} + 408710094497 \nu^{10} - 4200036380223 \nu^{9} + \cdots + 91244674865799 ) / 16814204066583 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 708381238 \nu^{11} - 1256290715 \nu^{10} + 14498890029 \nu^{9} - 15120183251 \nu^{8} + \cdots - 774397903797 ) / 52709103657 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} + \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{11} - 2 \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{4} - 13 \beta_{3} + \cdots - 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{8} - \beta_{7} - 6\beta_{5} + \beta_{4} - 11\beta _1 - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{11} - 26 \beta_{10} + 34 \beta_{9} - 11 \beta_{8} - 15 \beta_{7} - 17 \beta_{6} - 32 \beta_{5} + \cdots - 138 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 35 \beta_{11} - 41 \beta_{10} + 76 \beta_{9} + 20 \beta_{8} + 21 \beta_{7} + 13 \beta_{6} + 110 \beta_{5} + \cdots + 117 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 144\beta_{11} + 144\beta_{10} - 13\beta_{8} + 213\beta_{7} + 476\beta_{5} + 13\beta_{4} + 194\beta _1 + 1644 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 641 \beta_{11} + 551 \beta_{10} - 1216 \beta_{9} + 194 \beta_{8} + 357 \beta_{7} - 121 \beta_{6} + \cdots + 1644 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4610 \beta_{11} + 1052 \beta_{10} - 7120 \beta_{9} + 1900 \beta_{8} - 2952 \beta_{7} + 3194 \beta_{6} + \cdots - 20703 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 429 \beta_{11} + 429 \beta_{10} - 3194 \beta_{8} - 5598 \beta_{7} - 24704 \beta_{5} + 3194 \beta_{4} + \cdots - 24291 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 17228 \beta_{11} - 62174 \beta_{10} + 99298 \beta_{9} - 21683 \beta_{8} - 40491 \beta_{7} + \cdots - 268665 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 124562 \beta_{11} - 128546 \beta_{10} + 268264 \beta_{9} + 44063 \beta_{8} + 84483 \beta_{7} + \cdots + 366186 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
1.31896 − 2.28450i
1.31896 + 2.28450i
0.378061 − 0.654821i
0.378061 + 0.654821i
−1.74681 + 3.02556i
−1.74681 − 3.02556i
−1.01714 + 1.76174i
−1.01714 − 1.76174i
0.198184 − 0.343264i
0.198184 + 0.343264i
1.86875 − 3.23677i
1.86875 + 3.23677i
−3.50369 − 1.73205i 8.27584 0 6.06857i 6.69736 − 2.03600i −14.9812 −3.00000 0
76.2 −3.50369 1.73205i 8.27584 0 − 6.06857i 6.69736 + 2.03600i −14.9812 −3.00000 0
76.3 −2.91758 − 1.73205i 4.51225 0 5.05339i −6.13981 + 3.36195i −1.49451 −3.00000 0
76.4 −2.91758 1.73205i 4.51225 0 − 5.05339i −6.13981 − 3.36195i −1.49451 −3.00000 0
76.5 0.112974 − 1.73205i −3.98724 0 − 0.195676i 6.71303 + 1.98374i −0.902349 −3.00000 0
76.6 0.112974 1.73205i −3.98724 0 0.195676i 6.71303 − 1.98374i −0.902349 −3.00000 0
76.7 1.71214 − 1.73205i −1.06857 0 − 2.96552i 3.33344 + 6.15534i −8.67811 −3.00000 0
76.8 1.71214 1.73205i −1.06857 0 2.96552i 3.33344 − 6.15534i −8.67811 −3.00000 0
76.9 2.79155 − 1.73205i 3.79273 0 − 4.83510i −4.15782 − 5.63139i −0.578591 −3.00000 0
76.10 2.79155 1.73205i 3.79273 0 4.83510i −4.15782 + 5.63139i −0.578591 −3.00000 0
76.11 3.80460 − 1.73205i 10.4750 0 − 6.58976i −2.44621 + 6.55866i 24.6348 −3.00000 0
76.12 3.80460 1.73205i 10.4750 0 6.58976i −2.44621 − 6.55866i 24.6348 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.3.h.d 12
5.b even 2 1 105.3.h.a ✓ 12
5.c odd 4 2 525.3.e.c 24
7.b odd 2 1 inner 525.3.h.d 12
15.d odd 2 1 315.3.h.d 12
20.d odd 2 1 1680.3.s.c 12
35.c odd 2 1 105.3.h.a ✓ 12
35.f even 4 2 525.3.e.c 24
105.g even 2 1 315.3.h.d 12
140.c even 2 1 1680.3.s.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.h.a ✓ 12 5.b even 2 1
105.3.h.a ✓ 12 35.c odd 2 1
315.3.h.d 12 15.d odd 2 1
315.3.h.d 12 105.g even 2 1
525.3.e.c 24 5.c odd 4 2
525.3.e.c 24 35.f even 4 2
525.3.h.d 12 1.a even 1 1 trivial
525.3.h.d 12 7.b odd 2 1 inner
1680.3.s.c 12 20.d odd 2 1
1680.3.s.c 12 140.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 2T_{2}^{5} - 21T_{2}^{4} + 40T_{2}^{3} + 103T_{2}^{2} - 198T_{2} + 21 \) acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 2 T^{5} - 21 T^{4} + \cdots + 21)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( (T^{6} + 8 T^{5} + \cdots + 388416)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 1316818944 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 867491057664 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 44\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T^{6} - 32 T^{5} + \cdots - 11126976)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 52 T^{5} + \cdots + 172225344)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( (T^{6} + 16 T^{5} + \cdots + 82379584)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( (T^{6} + 76 T^{5} + \cdots + 44197696)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 57\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{6} + 88 T^{5} + \cdots + 19593854784)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 46\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( (T^{6} + 84 T^{5} + \cdots + 35588736064)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 16 T^{5} + \cdots - 12730697664)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{6} - 60 T^{5} + \cdots - 595422656)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 74\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 32\!\cdots\!44 \) Copy content Toggle raw display
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