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Properties

Label 353.2.a.c
Level $353$
Weight $2$
Character orbit 353.a
Self dual yes
Analytic conductor $2.819$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [353,2,Mod(1,353)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(353, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("353.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 353 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 353.a (trivial)

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: yes
Analytic conductor: \(2.81871919135\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 5x^{10} - x^{9} + 36x^{8} - 28x^{7} - 82x^{6} + 87x^{5} + 65x^{4} - 71x^{3} - 21x^{2} + 14x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{10} - \beta_{8} - \beta_{5} + \cdots - 1) q^{3}+ \cdots + (\beta_{10} - \beta_{9} + \cdots + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{10} - \beta_{8} - \beta_{5} + \cdots - 1) q^{3}+ \cdots + (3 \beta_{10} - 5 \beta_{9} + 3 \beta_{8} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q - 5 q^{2} - 5 q^{3} + 5 q^{4} - 4 q^{5} - 2 q^{6} - 25 q^{7} - 12 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q - 5 q^{2} - 5 q^{3} + 5 q^{4} - 4 q^{5} - 2 q^{6} - 25 q^{7} - 12 q^{8} + 6 q^{9} - 7 q^{10} - 4 q^{11} - 12 q^{12} - 13 q^{13} + 10 q^{14} - 19 q^{15} + q^{16} - 2 q^{17} - 14 q^{18} - 18 q^{19} + 7 q^{20} + 3 q^{21} - 11 q^{22} - 19 q^{23} + 15 q^{24} - 3 q^{25} + 20 q^{26} - 8 q^{27} - 25 q^{28} + 6 q^{29} + 17 q^{30} - 18 q^{31} - 9 q^{32} - 11 q^{33} - 6 q^{34} + 2 q^{35} + 23 q^{36} - 17 q^{37} + 22 q^{38} - 25 q^{39} - 12 q^{40} + 4 q^{41} + 16 q^{42} - 45 q^{43} + 17 q^{44} - 8 q^{45} - q^{46} - 11 q^{47} - q^{48} + 30 q^{49} + 15 q^{50} - 11 q^{51} - 15 q^{52} + 2 q^{53} + 20 q^{54} - 14 q^{55} + 71 q^{56} + 12 q^{57} - 3 q^{58} + 11 q^{59} + 24 q^{60} + 15 q^{61} + 35 q^{62} - 29 q^{63} + 6 q^{64} + 14 q^{65} + 64 q^{66} - 36 q^{67} + 29 q^{68} + 27 q^{69} + 22 q^{70} + 4 q^{71} + 18 q^{72} - 2 q^{73} + 37 q^{74} + 55 q^{75} - 6 q^{76} + 12 q^{77} + 55 q^{78} - 33 q^{79} + 52 q^{80} + 35 q^{81} + 18 q^{82} - 2 q^{83} + 48 q^{84} - 22 q^{85} + 32 q^{86} + 6 q^{87} - 17 q^{88} + 23 q^{89} + 27 q^{90} + 23 q^{91} + 19 q^{92} - 2 q^{93} + 2 q^{94} - 2 q^{95} + 47 q^{96} - 13 q^{97} - 35 q^{98} - 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 5x^{10} - x^{9} + 36x^{8} - 28x^{7} - 82x^{6} + 87x^{5} + 65x^{4} - 71x^{3} - 21x^{2} + 14x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} - 3\nu^{9} - 9\nu^{8} + 26\nu^{7} + 32\nu^{6} - 74\nu^{5} - 55\nu^{4} + 73\nu^{3} + 39\nu^{2} - 15\nu - 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} - 5\nu^{9} - \nu^{8} + 36\nu^{7} - 28\nu^{6} - 80\nu^{5} + 83\nu^{4} + 55\nu^{3} - 55\nu^{2} - 9\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{10} + 5\nu^{9} + \nu^{8} - 36\nu^{7} + 28\nu^{6} + 82\nu^{5} - 87\nu^{4} - 63\nu^{3} + 69\nu^{2} + 13\nu - 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{10} - 3\nu^{9} - 9\nu^{8} + 26\nu^{7} + 34\nu^{6} - 78\nu^{5} - 67\nu^{4} + 93\nu^{3} + 57\nu^{2} - 37\nu - 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{9} + 4\nu^{8} + 5\nu^{7} - 31\nu^{6} - 2\nu^{5} + 77\nu^{4} - 14\nu^{3} - 64\nu^{2} + 9\nu + 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{10} + 4\nu^{9} + 5\nu^{8} - 31\nu^{7} - 2\nu^{6} + 77\nu^{5} - 14\nu^{4} - 64\nu^{3} + 9\nu^{2} + 12\nu \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{10} + 7 \nu^{9} - 7 \nu^{8} - 44 \nu^{7} + 86 \nu^{6} + 72 \nu^{5} - 219 \nu^{4} - 5 \nu^{3} + \cdots - 30 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3 \nu^{10} - 17 \nu^{9} + 7 \nu^{8} + 110 \nu^{7} - 154 \nu^{6} - 194 \nu^{5} + 409 \nu^{4} + 47 \nu^{3} + \cdots + 46 ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( 2 \nu^{10} - 11 \nu^{9} + 3 \nu^{8} + 73 \nu^{7} - 91 \nu^{6} - 137 \nu^{5} + 247 \nu^{4} + 49 \nu^{3} + \cdots + 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{3} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 3\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - \beta_{9} + 6\beta_{7} + 2\beta_{6} + 2\beta_{5} + 2\beta_{4} + 5\beta_{3} + 6\beta_{2} + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{10} - 2\beta_{9} + 9\beta_{7} + 8\beta_{6} + 8\beta_{5} + 9\beta_{4} + 8\beta_{3} + 9\beta_{2} + 10\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10 \beta_{10} - 10 \beta_{9} + 35 \beta_{7} + 18 \beta_{6} + 19 \beta_{5} + 20 \beta_{4} + 27 \beta_{3} + \cdots + 40 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 23 \beta_{10} - 23 \beta_{9} + \beta_{8} + 66 \beta_{7} + 56 \beta_{6} + 57 \beta_{5} + 65 \beta_{4} + \cdots + 66 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 79 \beta_{10} - 78 \beta_{9} + 4 \beta_{8} + 210 \beta_{7} + 134 \beta_{6} + 143 \beta_{5} + 154 \beta_{4} + \cdots + 223 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 194 \beta_{10} - 190 \beta_{9} + 21 \beta_{8} + 451 \beta_{7} + 381 \beta_{6} + 392 \beta_{5} + \cdots + 450 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 578 \beta_{10} - 557 \beta_{9} + 73 \beta_{8} + 1291 \beta_{7} + 946 \beta_{6} + 1002 \beta_{5} + \cdots + 1324 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.56511
2.48645
1.91829
1.28831
1.17586
0.645288
−0.324006
−0.376422
−0.922632
−1.60888
−1.84737
−2.56511 −2.77268 4.57978 0.586311 7.11223 −4.25313 −6.61741 4.68776 −1.50395
1.2 −2.48645 1.15542 4.18244 2.08307 −2.87289 −4.23388 −5.42654 −1.66501 −5.17944
1.3 −1.91829 −0.105720 1.67984 −1.21033 0.202803 0.607461 0.614168 −2.98882 2.32177
1.4 −1.28831 3.08439 −0.340262 −3.66467 −3.97365 −3.89556 3.01498 6.51346 4.72122
1.5 −1.17586 −3.00119 −0.617362 1.15807 3.52897 0.707132 3.07764 6.00716 −1.36172
1.6 −0.645288 −0.108267 −1.58360 −0.0627156 0.0698636 1.18221 2.31246 −2.98828 0.0404697
1.7 0.324006 −1.14497 −1.89502 3.74094 −0.370976 −4.94712 −1.26201 −1.68905 1.21209
1.8 0.376422 0.834323 −1.85831 −0.943004 0.314057 −2.01249 −1.45235 −2.30391 −0.354967
1.9 0.922632 0.942426 −1.14875 −3.31962 0.869512 −3.67648 −2.90514 −2.11183 −3.06279
1.10 1.60888 −1.23587 0.588487 −2.27254 −1.98837 −1.38010 −2.27095 −1.47262 −3.65624
1.11 1.84737 −2.64785 1.41276 −0.0955050 −4.89156 −3.09805 −1.08485 4.01114 −0.176433
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(353\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 353.2.a.c ✓ 11
3.b odd 2 1 3177.2.a.e 11
4.b odd 2 1 5648.2.a.o 11
5.b even 2 1 8825.2.a.h 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
353.2.a.c ✓ 11 1.a even 1 1 trivial
3177.2.a.e 11 3.b odd 2 1
5648.2.a.o 11 4.b odd 2 1
8825.2.a.h 11 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{11} + 5 T_{2}^{10} - T_{2}^{9} - 36 T_{2}^{8} - 28 T_{2}^{7} + 82 T_{2}^{6} + 87 T_{2}^{5} + \cdots - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(353))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} + 5 T^{10} + \cdots - 4 \) Copy content Toggle raw display
$3$ \( T^{11} + 5 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{11} + 4 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{11} + 25 T^{10} + \cdots - 5575 \) Copy content Toggle raw display
$11$ \( T^{11} + 4 T^{10} + \cdots - 14697 \) Copy content Toggle raw display
$13$ \( T^{11} + 13 T^{10} + \cdots + 32157 \) Copy content Toggle raw display
$17$ \( T^{11} + 2 T^{10} + \cdots - 181723 \) Copy content Toggle raw display
$19$ \( T^{11} + 18 T^{10} + \cdots - 25369 \) Copy content Toggle raw display
$23$ \( T^{11} + 19 T^{10} + \cdots + 70145 \) Copy content Toggle raw display
$29$ \( T^{11} - 6 T^{10} + \cdots - 4538213 \) Copy content Toggle raw display
$31$ \( T^{11} + 18 T^{10} + \cdots + 541771 \) Copy content Toggle raw display
$37$ \( T^{11} + 17 T^{10} + \cdots + 1397891 \) Copy content Toggle raw display
$41$ \( T^{11} - 4 T^{10} + \cdots - 545 \) Copy content Toggle raw display
$43$ \( T^{11} + 45 T^{10} + \cdots - 89666496 \) Copy content Toggle raw display
$47$ \( T^{11} + 11 T^{10} + \cdots - 24920991 \) Copy content Toggle raw display
$53$ \( T^{11} + \cdots + 1074428021 \) Copy content Toggle raw display
$59$ \( T^{11} - 11 T^{10} + \cdots - 5289017 \) Copy content Toggle raw display
$61$ \( T^{11} - 15 T^{10} + \cdots + 1724720 \) Copy content Toggle raw display
$67$ \( T^{11} + 36 T^{10} + \cdots + 87917885 \) Copy content Toggle raw display
$71$ \( T^{11} - 4 T^{10} + \cdots - 15597383 \) Copy content Toggle raw display
$73$ \( T^{11} + 2 T^{10} + \cdots + 61406435 \) Copy content Toggle raw display
$79$ \( T^{11} + 33 T^{10} + \cdots - 29961895 \) Copy content Toggle raw display
$83$ \( T^{11} + 2 T^{10} + \cdots - 11080247 \) Copy content Toggle raw display
$89$ \( T^{11} - 23 T^{10} + \cdots - 25222653 \) Copy content Toggle raw display
$97$ \( T^{11} + 13 T^{10} + \cdots + 5563451 \) Copy content Toggle raw display
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