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Properties

Label 297.3.b.d
Level $297$
Weight $3$
Character orbit 297.b
Analytic conductor $8.093$
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] = [297,3,Mod(188,297)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(297, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("297.188");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 297.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: \(8.09266385150\)
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: \( x^{12} - 2x^{11} + x^{10} - 10x^{8} + 74x^{7} - 103x^{6} + 6x^{5} + 300x^{4} - 364x^{3} + 383x^{2} - 214x + 53 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{10} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( - \beta_1 - 1) q^{4} + ( - \beta_{9} + \beta_{5}) q^{5} + (\beta_{2} - 1) q^{7} + ( - \beta_{11} - \beta_{7} - \beta_{5}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + ( - \beta_1 - 1) q^{4} + ( - \beta_{9} + \beta_{5}) q^{5} + (\beta_{2} - 1) q^{7} + ( - \beta_{11} - \beta_{7} - \beta_{5}) q^{8} + ( - \beta_{6} - \beta_{2} - 3) q^{10} - \beta_{7} q^{11} + ( - \beta_{4} - \beta_{3} - \beta_1 + 3) q^{13} + (\beta_{10} - \beta_{9} + \cdots - \beta_{5}) q^{14}+ \cdots + ( - \beta_{11} + 8 \beta_{10} + \cdots + 28 \beta_{5}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{4} - 12 q^{7} - 36 q^{10} + 36 q^{13} + 12 q^{16} + 84 q^{19} - 84 q^{25} - 72 q^{31} + 204 q^{34} - 204 q^{37} - 108 q^{40} - 72 q^{43} + 96 q^{46} + 192 q^{49} + 12 q^{52} + 228 q^{58} + 276 q^{61} - 156 q^{64} + 120 q^{67} - 600 q^{70} - 96 q^{73} - 516 q^{76} - 96 q^{79} + 684 q^{82} - 372 q^{85} - 132 q^{88} + 240 q^{91} + 228 q^{94} + 444 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{11} + x^{10} - 10x^{8} + 74x^{7} - 103x^{6} + 6x^{5} + 300x^{4} - 364x^{3} + 383x^{2} - 214x + 53 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 5475532 \nu^{11} + 22200465 \nu^{10} - 16348702 \nu^{9} - 17234720 \nu^{8} + \cdots - 2258043449 ) / 1540812045 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12323176 \nu^{11} - 40801770 \nu^{10} - 14343275 \nu^{9} + 10821224 \nu^{8} - 95878962 \nu^{7} + \cdots - 8352112039 ) / 1540812045 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8232427 \nu^{11} + 32424030 \nu^{10} - 29813561 \nu^{9} + 749826 \nu^{8} + 102593689 \nu^{7} + \cdots + 5078140382 ) / 513604015 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 54806650 \nu^{11} + 224726790 \nu^{10} - 215283394 \nu^{9} - 91443869 \nu^{8} + \cdots + 16450760971 ) / 1540812045 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 60025061 \nu^{11} - 85604157 \nu^{10} - 18603595 \nu^{9} + 67579396 \nu^{8} - 690524193 \nu^{7} + \cdots - 4706261477 ) / 1540812045 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 79520083 \nu^{11} + 307726500 \nu^{10} - 208924477 \nu^{9} - 92121359 \nu^{8} + \cdots + 35384751835 ) / 1540812045 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 665184 \nu^{11} - 1068652 \nu^{10} + 555676 \nu^{9} - 830406 \nu^{8} - 5456264 \nu^{7} + \cdots - 66135663 ) / 10927745 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 115755245 \nu^{11} - 228730644 \nu^{10} + 178698569 \nu^{9} - 329628407 \nu^{8} + \cdots - 4840544387 ) / 1540812045 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 157963067 \nu^{11} + 255572919 \nu^{10} - 101783918 \nu^{9} + 60853070 \nu^{8} + \cdots + 18609560636 ) / 1540812045 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 465289681 \nu^{11} + 752109066 \nu^{10} - 143198791 \nu^{9} - 46551731 \nu^{8} + \cdots + 32405171587 ) / 1540812045 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 156148673 \nu^{11} + 226891692 \nu^{10} + 13615864 \nu^{9} - 101215360 \nu^{8} + \cdots + 12350555819 ) / 513604015 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{9} - 3\beta_{7} + \beta_{6} - 3\beta_{5} - \beta_{4} + \beta_{3} + 3\beta_{2} + 3 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3 \beta_{11} - \beta_{10} - 3 \beta_{9} - 3 \beta_{7} + \beta_{6} + 13 \beta_{5} + 2 \beta_{4} + \cdots + 3 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{11} - 3 \beta_{9} + 3 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} + 39 \beta_{5} - 5 \beta_{4} + \cdots + 3 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 4 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} + 6 \beta_{8} - 15 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} + \cdots + 21 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 12 \beta_{11} + 22 \beta_{10} - 21 \beta_{9} + 33 \beta_{8} - 48 \beta_{7} + 18 \beta_{6} + \cdots - 402 ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 69 \beta_{11} + 7 \beta_{10} + 33 \beta_{9} + 3 \beta_{8} + 114 \beta_{7} - 40 \beta_{6} + 494 \beta_{5} + \cdots - 132 ) / 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 39 \beta_{11} + 20 \beta_{10} - 39 \beta_{9} + 42 \beta_{8} - 102 \beta_{7} - 129 \beta_{6} + \cdots + 192 ) / 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 126 \beta_{11} + 14 \beta_{10} - 66 \beta_{9} + 12 \beta_{8} - 225 \beta_{7} - 30 \beta_{6} + \cdots - 795 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 699 \beta_{11} - 525 \beta_{10} + 879 \beta_{9} - 765 \beta_{8} + 2265 \beta_{7} + 280 \beta_{6} + \cdots - 9339 ) / 18 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1125 \beta_{11} - 1171 \beta_{10} + 2157 \beta_{9} - 2061 \beta_{8} + 5037 \beta_{7} - 1454 \beta_{6} + \cdots + 13053 ) / 18 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 5811 \beta_{11} - 312 \beta_{10} - 615 \beta_{9} - 798 \beta_{8} - 5313 \beta_{7} - 313 \beta_{6} + \cdots + 7923 ) / 18 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/297\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(244\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
188.1
−0.604421 + 2.29308i
2.01816 − 0.933356i
−1.95098 + 0.0270721i
0.432813 − 0.247971i
0.0735781 + 0.876088i
1.03084 + 1.24132i
1.03084 − 1.24132i
0.0735781 − 0.876088i
0.432813 + 0.247971i
−1.95098 − 0.0270721i
2.01816 + 0.933356i
−0.604421 − 2.29308i
− 3.62355i 0 −9.13011 3.19477i 0 −5.88599 18.5892i 0 11.5764
188.2 − 2.94414i 0 −4.66795 − 8.17178i 0 12.2734 1.96653i 0 −24.0589
188.3 − 1.80705i 0 0.734588 − 0.135033i 0 −9.55014 − 8.55561i 0 −0.244011
188.4 − 1.75118i 0 0.933359 − 8.30682i 0 −5.72331 − 8.63921i 0 −14.5468
188.5 − 1.22706i 0 2.49433 6.11904i 0 −4.74537 − 7.96891i 0 7.50841
188.6 − 0.603507i 0 3.63578 2.92427i 0 7.63136 − 4.60825i 0 1.76482
188.7 0.603507i 0 3.63578 − 2.92427i 0 7.63136 4.60825i 0 1.76482
188.8 1.22706i 0 2.49433 − 6.11904i 0 −4.74537 7.96891i 0 7.50841
188.9 1.75118i 0 0.933359 8.30682i 0 −5.72331 8.63921i 0 −14.5468
188.10 1.80705i 0 0.734588 0.135033i 0 −9.55014 8.55561i 0 −0.244011
188.11 2.94414i 0 −4.66795 8.17178i 0 12.2734 − 1.96653i 0 −24.0589
188.12 3.62355i 0 −9.13011 − 3.19477i 0 −5.88599 − 18.5892i 0 11.5764
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 188.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.3.b.d ✓ 12
3.b odd 2 1 inner 297.3.b.d ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.3.b.d ✓ 12 1.a even 1 1 trivial
297.3.b.d ✓ 12 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 30T_{2}^{10} + 315T_{2}^{8} + 1444T_{2}^{6} + 3039T_{2}^{4} + 2646T_{2}^{2} + 625 \) acting on \(S_{3}^{\mathrm{new}}(297, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 30 T^{10} + \cdots + 625 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 192 T^{10} + \cdots + 274576 \) Copy content Toggle raw display
$7$ \( (T^{6} + 6 T^{5} + \cdots + 142993)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 11)^{6} \) Copy content Toggle raw display
$13$ \( (T^{6} - 18 T^{5} + \cdots + 70756)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 218733480908881 \) Copy content Toggle raw display
$19$ \( (T^{6} - 42 T^{5} + \cdots + 2965572)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 5388224775025 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 945571484025 \) Copy content Toggle raw display
$31$ \( (T^{6} + 36 T^{5} + \cdots - 2956700)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 102 T^{5} + \cdots + 5552691021)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( (T^{6} + 36 T^{5} + \cdots - 1782944379)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 33\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 153410692212649 \) Copy content Toggle raw display
$61$ \( (T^{6} - 138 T^{5} + \cdots + 124608534084)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 60 T^{5} + \cdots - 254827052)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} + 48 T^{5} + \cdots + 10531209808)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 48 T^{5} + \cdots - 2751804035)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{6} - 222 T^{5} + \cdots - 9431342615)^{2} \) Copy content Toggle raw display
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