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Properties

Label 475.3.c.g
Level $475$
Weight $3$
Character orbit 475.c
Analytic conductor $12.943$
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] = [475,3,Mod(151,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("475.151");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 475.c (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: \(12.9428125571\)
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: \( x^{12} + 30x^{10} + 329x^{8} + 1620x^{6} + 3479x^{4} + 2470x^{2} + 55 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 95)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{9} q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{3} + \beta_{2} - 2) q^{6} + ( - \beta_{5} - 2) q^{7} + ( - \beta_{9} + \beta_{7}) q^{8} + ( - \beta_{6} - \beta_{2} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{9} q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{3} + \beta_{2} - 2) q^{6} + ( - \beta_{5} - 2) q^{7} + ( - \beta_{9} + \beta_{7}) q^{8} + ( - \beta_{6} - \beta_{2} - 4) q^{9} + (\beta_{6} - \beta_{5} - 2 \beta_{3} + \cdots + 3) q^{11}+ \cdots + ( - 5 \beta_{6} + 5 \beta_{5} + \cdots - 29) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4} - 28 q^{6} - 20 q^{7} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{4} - 28 q^{6} - 20 q^{7} - 48 q^{9} + 32 q^{11} - 44 q^{16} + 44 q^{17} + 8 q^{19} - 36 q^{23} + 100 q^{24} + 108 q^{26} + 36 q^{28} - 80 q^{36} + 44 q^{38} + 76 q^{39} - 100 q^{42} - 320 q^{43} - 256 q^{44} + 56 q^{47} + 72 q^{49} - 76 q^{54} - 60 q^{57} - 68 q^{58} - 296 q^{61} + 376 q^{62} + 96 q^{63} + 188 q^{64} + 152 q^{66} + 340 q^{68} + 244 q^{73} + 136 q^{74} + 248 q^{76} + 200 q^{77} - 372 q^{81} - 424 q^{82} + 160 q^{83} - 444 q^{87} - 716 q^{92} - 296 q^{93} - 44 q^{96} - 312 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 30x^{10} + 329x^{8} + 1620x^{6} + 3479x^{4} + 2470x^{2} + 55 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{10} + 125\nu^{8} + 308\nu^{6} - 3088\nu^{4} - 8767\nu^{2} + 2743 ) / 1336 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25\nu^{10} + 685\nu^{8} + 6444\nu^{6} + 24280\nu^{4} + 31863\nu^{2} + 6695 ) / 1336 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -37\nu^{10} - 947\nu^{8} - 8308\nu^{6} - 30056\nu^{4} - 42027\nu^{2} - 12113 ) / 1336 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -16\nu^{10} - 405\nu^{8} - 3376\nu^{6} - 9928\nu^{4} - 4200\nu^{2} + 5301 ) / 668 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -23\nu^{11} - 697\nu^{9} - 7692\nu^{7} - 37568\nu^{5} - 75593\nu^{3} - 36019\nu ) / 1336 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -16\nu^{11} - 405\nu^{9} - 3376\nu^{7} - 9928\nu^{5} - 2864\nu^{3} + 15321\nu ) / 668 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -23\nu^{11} - 697\nu^{9} - 7692\nu^{7} - 37568\nu^{5} - 76929\nu^{3} - 46707\nu ) / 1336 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -14\nu^{11} - 417\nu^{9} - 4624\nu^{7} - 23884\nu^{5} - 56614\nu^{3} - 45399\nu ) / 668 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -53\nu^{11} - 1519\nu^{9} - 15692\nu^{7} - 72048\nu^{5} - 146427\nu^{3} - 108181\nu ) / 1336 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{7} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + \beta_{4} + \beta_{3} - 11\beta_{2} + 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{11} + \beta_{10} + 17\beta_{9} + \beta_{8} - 15\beta_{7} + 73\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -15\beta_{6} - 5\beta_{5} - 21\beta_{4} - 20\beta_{3} + 118\beta_{2} - 370 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 41\beta_{11} - 26\beta_{10} - 224\beta_{9} - 20\beta_{8} + 189\beta_{7} - 724\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 188\beta_{6} + 112\beta_{5} + 328\beta_{4} + 280\beta_{3} - 1280\beta_{2} + 3705 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -608\beta_{11} + 440\beta_{10} + 2712\beta_{9} + 300\beta_{8} - 2264\beta_{7} + 7565\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -2256\beta_{6} - 1780\beta_{5} - 4492\beta_{4} - 3488\beta_{3} + 14065\beta_{2} - 38889 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 7980\beta_{11} - 6272\beta_{10} - 31753\beta_{9} - 4036\beta_{8} + 26557\beta_{7} - 81632\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
− 3.38208i
− 2.84623i
− 2.36559i
− 1.88109i
− 1.14149i
− 0.151673i
0.151673i
1.14149i
1.88109i
2.36559i
2.84623i
3.38208i
− 3.38208i − 4.08851i −7.43849 0 −13.8277 −4.56923 11.6293i −7.71590 0
151.2 − 2.84623i − 2.18419i −4.10101 0 −6.21669 9.15076 0.287487i 4.22932 0
151.3 − 2.36559i 3.55563i −1.59600 0 8.41115 −11.0785 − 5.68687i −3.64250 0
151.4 − 1.88109i 0.840697i 0.461500 0 1.58143 −2.31342 − 8.39248i 8.29323 0
151.5 − 1.14149i − 4.13699i 2.69701 0 −4.72232 −7.54444 − 7.64455i −8.11468 0
151.6 − 0.151673i 5.10387i 3.97700 0 0.774118 6.35478 − 1.20989i −17.0495 0
151.7 0.151673i − 5.10387i 3.97700 0 0.774118 6.35478 1.20989i −17.0495 0
151.8 1.14149i 4.13699i 2.69701 0 −4.72232 −7.54444 7.64455i −8.11468 0
151.9 1.88109i − 0.840697i 0.461500 0 1.58143 −2.31342 8.39248i 8.29323 0
151.10 2.36559i − 3.55563i −1.59600 0 8.41115 −11.0785 5.68687i −3.64250 0
151.11 2.84623i 2.18419i −4.10101 0 −6.21669 9.15076 − 0.287487i 4.22932 0
151.12 3.38208i 4.08851i −7.43849 0 −13.8277 −4.56923 − 11.6293i −7.71590 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.3.c.g 12
5.b even 2 1 95.3.c.a ✓ 12
5.c odd 4 2 475.3.d.c 24
15.d odd 2 1 855.3.e.a 12
19.b odd 2 1 inner 475.3.c.g 12
20.d odd 2 1 1520.3.h.a 12
95.d odd 2 1 95.3.c.a ✓ 12
95.g even 4 2 475.3.d.c 24
285.b even 2 1 855.3.e.a 12
380.d even 2 1 1520.3.h.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.3.c.a ✓ 12 5.b even 2 1
95.3.c.a ✓ 12 95.d odd 2 1
475.3.c.g 12 1.a even 1 1 trivial
475.3.c.g 12 19.b odd 2 1 inner
475.3.d.c 24 5.c odd 4 2
475.3.d.c 24 95.g even 4 2
855.3.e.a 12 15.d odd 2 1
855.3.e.a 12 285.b even 2 1
1520.3.h.a 12 20.d odd 2 1
1520.3.h.a 12 380.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(475, [\chi])\):

\( T_{2}^{12} + 30T_{2}^{10} + 329T_{2}^{8} + 1620T_{2}^{6} + 3479T_{2}^{4} + 2470T_{2}^{2} + 55 \) Copy content Toggle raw display
\( T_{7}^{6} + 10T_{7}^{5} - 115T_{7}^{4} - 1192T_{7}^{3} + 1840T_{7}^{2} + 31200T_{7} + 51376 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 30 T^{10} + \cdots + 55 \) Copy content Toggle raw display
$3$ \( T^{12} + 78 T^{10} + \cdots + 317680 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + 10 T^{5} + \cdots + 51376)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 16 T^{5} + \cdots + 1609984)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 1391784557680 \) Copy content Toggle raw display
$17$ \( (T^{6} - 22 T^{5} + \cdots + 1301904)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 22\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( (T^{6} + 18 T^{5} + \cdots - 97445584)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 63\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 20\!\cdots\!80 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 19\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 20\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( (T^{6} + 160 T^{5} + \cdots + 10954816)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 28 T^{5} + \cdots - 253813824)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 41\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 89\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( (T^{6} + 148 T^{5} + \cdots - 76754624)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 86\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 17\!\cdots\!80 \) Copy content Toggle raw display
$73$ \( (T^{6} - 122 T^{5} + \cdots + 5877225616)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 11\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( (T^{6} - 80 T^{5} + \cdots + 5457804544)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 52\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
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