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Properties

Label 471.2.b.b
Level $471$
Weight $2$
Character orbit 471.b
Analytic conductor $3.761$
Analytic rank $0$
Dimension $14$
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] = [471,2,Mod(313,471)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(471, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("471.313");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 471.b (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: \(3.76095393520\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 24x^{12} + 224x^{10} + 1027x^{8} + 2399x^{6} + 2652x^{4} + 1094x^{2} + 147 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} - 1) q^{4} + \beta_{4} q^{5} + \beta_1 q^{6} + \beta_{12} q^{7} + (\beta_{3} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} - 1) q^{4} + \beta_{4} q^{5} + \beta_1 q^{6} + \beta_{12} q^{7} + (\beta_{3} - \beta_1) q^{8} + q^{9} + ( - \beta_{7} + \beta_{5} - \beta_{2}) q^{10} + \beta_{5} q^{11} + (\beta_{2} - 1) q^{12} + ( - \beta_{11} - \beta_{2}) q^{13} + (\beta_{11} - 2 \beta_{9} - \beta_{2}) q^{14} + \beta_{4} q^{15} + ( - \beta_{10} - \beta_{9} - 2 \beta_{2} + 1) q^{16} + (\beta_{11} + \beta_{10} - \beta_{7} + 1) q^{17} + \beta_1 q^{18} + (\beta_{9} - \beta_{5} + \beta_{2} - 1) q^{19} + ( - \beta_{13} - 2 \beta_{4} + \beta_1) q^{20} + \beta_{12} q^{21} + (\beta_{12} - \beta_{8} + \beta_{6} + \cdots - \beta_1) q^{22}+ \cdots + \beta_{5} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 14 q^{3} - 20 q^{4} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 14 q^{3} - 20 q^{4} + 14 q^{9} + 6 q^{10} - 2 q^{11} - 20 q^{12} + 24 q^{16} + 18 q^{17} - 12 q^{19} - 18 q^{25} + 14 q^{27} + 6 q^{30} - 14 q^{31} - 2 q^{33} + 16 q^{35} - 20 q^{36} - 14 q^{37} - 36 q^{40} + 24 q^{44} - 8 q^{46} + 22 q^{47} + 24 q^{48} - 48 q^{49} + 18 q^{51} - 50 q^{52} - 62 q^{56} - 12 q^{57} + 20 q^{58} - 34 q^{64} + 42 q^{67} - 56 q^{68} + 38 q^{71} - 18 q^{75} + 52 q^{76} + 14 q^{81} + 10 q^{82} + 34 q^{86} - 48 q^{89} + 6 q^{90} - 14 q^{93} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 24x^{12} + 224x^{10} + 1027x^{8} + 2399x^{6} + 2652x^{4} + 1094x^{2} + 147 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25\nu^{13} + 593\nu^{11} + 5418\nu^{9} + 23876\nu^{7} + 51582\nu^{5} + 47750\nu^{3} + 10522\nu ) / 133 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{12} + 52\nu^{10} + 533\nu^{8} + 2702\nu^{6} + 6858\nu^{4} + 7468\nu^{2} + 1791 ) / 19 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 39\nu^{13} + 957\nu^{11} + 9149\nu^{9} + 42790\nu^{7} + 99721\nu^{5} + 101223\nu^{3} + 25187\nu ) / 266 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{12} + 78\nu^{10} + 790\nu^{8} + 3901\nu^{6} + 9508\nu^{4} + 9853\nu^{2} + 2259 ) / 19 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7\nu^{13} + 163\nu^{11} + 1457\nu^{9} + 6265\nu^{7} + 13230\nu^{5} + 12192\nu^{3} + 2953\nu ) / 19 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7\nu^{12} + 163\nu^{10} + 1457\nu^{8} + 6265\nu^{6} + 13211\nu^{4} + 12021\nu^{2} + 2687 ) / 19 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -7\nu^{12} - 163\nu^{10} - 1457\nu^{8} - 6265\nu^{6} - 13230\nu^{4} - 12173\nu^{2} - 2858 ) / 19 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 21\nu^{12} + 489\nu^{10} + 4371\nu^{8} + 18776\nu^{6} + 39443\nu^{4} + 35645\nu^{2} + 7985 ) / 38 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 137\nu^{13} + 3239\nu^{11} + 29547\nu^{9} + 130500\nu^{7} + 284675\nu^{5} + 269517\nu^{3} + 62539\nu ) / 266 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -93\nu^{13} - 2190\nu^{11} - 19873\nu^{9} - 87111\nu^{7} - 187778\nu^{5} - 174172\nu^{3} - 38280\nu ) / 133 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{10} - \beta_{9} - 8\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{12} + \beta_{8} + \beta_{6} - 9\beta_{3} + 30\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{11} + 10\beta_{10} + 13\beta_{9} + 58\beta_{2} - 88 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{13} + 15\beta_{12} - 12\beta_{8} - 13\beta_{6} + 68\beta_{3} - 195\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 32\beta_{11} - 78\beta_{10} - 126\beta_{9} - 2\beta_{7} + 3\beta_{5} - 414\beta_{2} + 559 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -34\beta_{13} - 157\beta_{12} + 109\beta_{8} + 127\beta_{6} - 9\beta_{4} - 490\beta_{3} + 1322\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -352\beta_{11} + 565\beta_{10} + 1092\beta_{9} + 43\beta_{7} - 61\beta_{5} + 2958\beta_{2} - 3714 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 395\beta_{13} + 1426\beta_{12} - 899\beta_{8} - 1110\beta_{6} + 190\beta_{4} + 3480\beta_{3} - 9181\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3326\beta_{11} - 3984\beta_{10} - 8947\beta_{9} - 585\beta_{7} + 796\beta_{5} - 21237\beta_{2} + 25350 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -3911\beta_{13} - 12062\beta_{12} + 7099\beta_{8} + 9158\beta_{6} - 2551\beta_{4} - 24636\beta_{3} + 64733\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/471\mathbb{Z}\right)^\times\).

\(n\) \(158\) \(319\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
313.1
− 2.72109i
− 2.49123i
− 2.14537i
− 1.66261i
− 1.53977i
− 0.580612i
− 0.560879i
0.560879i
0.580612i
1.53977i
1.66261i
2.14537i
2.49123i
2.72109i
− 2.72109i 1.00000 −5.40431 1.84852i − 2.72109i 5.01715i 9.26343i 1.00000 5.02998
313.2 − 2.49123i 1.00000 −4.20622 − 1.87721i − 2.49123i − 2.20605i 5.49621i 1.00000 −4.67656
313.3 − 2.14537i 1.00000 −2.60262 4.18337i − 2.14537i − 4.37069i 1.29283i 1.00000 8.97487
313.4 − 1.66261i 1.00000 −0.764270 − 1.85555i − 1.66261i 0.948177i − 2.05454i 1.00000 −3.08505
313.5 − 1.53977i 1.00000 −0.370884 − 3.16196i − 1.53977i 0.921702i − 2.50846i 1.00000 −4.86868
313.6 − 0.580612i 1.00000 1.66289 2.44616i − 0.580612i 1.30605i − 2.12672i 1.00000 1.42027
313.7 − 0.560879i 1.00000 1.68542 0.365796i − 0.560879i − 4.51711i − 2.06707i 1.00000 0.205167
313.8 0.560879i 1.00000 1.68542 − 0.365796i 0.560879i 4.51711i 2.06707i 1.00000 0.205167
313.9 0.580612i 1.00000 1.66289 − 2.44616i 0.580612i − 1.30605i 2.12672i 1.00000 1.42027
313.10 1.53977i 1.00000 −0.370884 3.16196i 1.53977i − 0.921702i 2.50846i 1.00000 −4.86868
313.11 1.66261i 1.00000 −0.764270 1.85555i 1.66261i − 0.948177i 2.05454i 1.00000 −3.08505
313.12 2.14537i 1.00000 −2.60262 − 4.18337i 2.14537i 4.37069i − 1.29283i 1.00000 8.97487
313.13 2.49123i 1.00000 −4.20622 1.87721i 2.49123i 2.20605i − 5.49621i 1.00000 −4.67656
313.14 2.72109i 1.00000 −5.40431 − 1.84852i 2.72109i − 5.01715i − 9.26343i 1.00000 5.02998
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 313.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 471.2.b.b ✓ 14
3.b odd 2 1 1413.2.b.e 14
157.b even 2 1 inner 471.2.b.b ✓ 14
471.d odd 2 1 1413.2.b.e 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.2.b.b ✓ 14 1.a even 1 1 trivial
471.2.b.b ✓ 14 157.b even 2 1 inner
1413.2.b.e 14 3.b odd 2 1
1413.2.b.e 14 471.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 24T_{2}^{12} + 224T_{2}^{10} + 1027T_{2}^{8} + 2399T_{2}^{6} + 2652T_{2}^{4} + 1094T_{2}^{2} + 147 \) acting on \(S_{2}^{\mathrm{new}}(471, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 24 T^{12} + \cdots + 147 \) Copy content Toggle raw display
$3$ \( (T - 1)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 44 T^{12} + \cdots + 5808 \) Copy content Toggle raw display
$7$ \( T^{14} + 73 T^{12} + \cdots + 62208 \) Copy content Toggle raw display
$11$ \( (T^{7} + T^{6} - 35 T^{5} + \cdots - 1152)^{2} \) Copy content Toggle raw display
$13$ \( (T^{7} - 52 T^{5} + \cdots + 2464)^{2} \) Copy content Toggle raw display
$17$ \( (T^{7} - 9 T^{6} + \cdots - 1296)^{2} \) Copy content Toggle raw display
$19$ \( (T^{7} + 6 T^{6} - 48 T^{5} + \cdots - 96)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + 87 T^{12} + \cdots + 4747692 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 181398528 \) Copy content Toggle raw display
$31$ \( (T^{7} + 7 T^{6} + \cdots - 144)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} + 7 T^{6} + \cdots - 1296)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 4026589488 \) Copy content Toggle raw display
$43$ \( T^{14} + 243 T^{12} + \cdots + 110592 \) Copy content Toggle raw display
$47$ \( (T^{7} - 11 T^{6} + \cdots + 27648)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 1063932672 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 20635140288 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 73638346752 \) Copy content Toggle raw display
$67$ \( (T^{7} - 21 T^{6} + \cdots + 36352)^{2} \) Copy content Toggle raw display
$71$ \( (T^{7} - 19 T^{6} + \cdots - 28224)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 429632335872 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 19554905088 \) Copy content Toggle raw display
$83$ \( T^{14} + 533 T^{12} + \cdots + 30950832 \) Copy content Toggle raw display
$89$ \( (T^{7} + 24 T^{6} + \cdots - 240048)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 134547222528 \) Copy content Toggle raw display
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