Properties

Label 935.2.a.j
Level $935$
Weight $2$
Character orbit 935.a
Self dual yes
Analytic conductor $7.466$
Analytic rank $0$
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] = [935,2,Mod(1,935)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(935, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("935.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 935 = 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 935.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: \(7.46601258899\)
Analytic rank: \(0\)
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: \( x^{11} - x^{10} - 21 x^{9} + 20 x^{8} + 161 x^{7} - 148 x^{6} - 536 x^{5} + 481 x^{4} + 689 x^{3} - 587 x^{2} - 156 x + 61 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + 2) q^{4} + q^{5} + (\beta_{10} - \beta_{9} - \beta_{5}) q^{6} + (\beta_{9} - \beta_{3} - 1) q^{7} + ( - \beta_{7} + \beta_{5} + \beta_{3} - 2 \beta_1 + 1) q^{8} + ( - \beta_{10} - \beta_{6} - \beta_{5} - \beta_{4} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{2} + 2) q^{4} + q^{5} + (\beta_{10} - \beta_{9} - \beta_{5}) q^{6} + (\beta_{9} - \beta_{3} - 1) q^{7} + ( - \beta_{7} + \beta_{5} + \beta_{3} - 2 \beta_1 + 1) q^{8} + ( - \beta_{10} - \beta_{6} - \beta_{5} - \beta_{4} + 2) q^{9} - \beta_1 q^{10} - q^{11} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - 3 \beta_{3} + \beta_1 - 1) q^{12} + ( - \beta_{8} + 1) q^{13} + (\beta_{10} - \beta_{9} - \beta_{7} + \beta_{6} + 2 \beta_{4} + 2 \beta_{3} + 1) q^{14} - \beta_{3} q^{15} + ( - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{5} + \beta_{2} + 3) q^{16} - q^{17} + (\beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{2} - 3 \beta_1 + 2) q^{18} + (\beta_{9} + \beta_{7} - \beta_{6} - \beta_{2}) q^{19} + (\beta_{2} + 2) q^{20} + ( - \beta_{10} + \beta_{9} - 2 \beta_{5} + \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{21} + \beta_1 q^{22} + ( - \beta_{9} - \beta_{7} + \beta_{5} + \beta_{4}) q^{23} + (2 \beta_{10} - 2 \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \cdots - 1) q^{24}+ \cdots + (\beta_{10} + \beta_{6} + \beta_{5} + \beta_{4} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q - q^{2} + 5 q^{3} + 21 q^{4} + 11 q^{5} + q^{6} - 2 q^{7} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q - q^{2} + 5 q^{3} + 21 q^{4} + 11 q^{5} + q^{6} - 2 q^{7} + 26 q^{9} - q^{10} - 11 q^{11} + 8 q^{12} + 13 q^{13} + 5 q^{15} + 29 q^{16} - 11 q^{17} + 3 q^{18} + 4 q^{19} + 21 q^{20} + 34 q^{21} + q^{22} - 7 q^{23} - 29 q^{24} + 11 q^{25} + 4 q^{26} + 14 q^{27} + 8 q^{28} + 9 q^{29} + q^{30} + 20 q^{31} - 26 q^{32} - 5 q^{33} + q^{34} - 2 q^{35} + 62 q^{36} - q^{37} - 6 q^{39} - 9 q^{41} - 74 q^{42} + 5 q^{43} - 21 q^{44} + 26 q^{45} - 10 q^{46} + 24 q^{48} + 43 q^{49} - q^{50} - 5 q^{51} + 18 q^{52} + 14 q^{53} - 9 q^{54} - 11 q^{55} - 6 q^{56} - 32 q^{57} - 10 q^{58} + 13 q^{59} + 8 q^{60} + 11 q^{61} - 26 q^{62} - 40 q^{63} + 24 q^{64} + 13 q^{65} - q^{66} + 26 q^{67} - 21 q^{68} + 56 q^{69} - 8 q^{71} - 26 q^{72} + 42 q^{73} + 16 q^{74} + 5 q^{75} - 70 q^{76} + 2 q^{77} - 56 q^{78} - 27 q^{79} + 29 q^{80} + 71 q^{81} + 57 q^{82} - 41 q^{83} + 76 q^{84} - 11 q^{85} + 47 q^{86} + 26 q^{87} + 37 q^{89} + 3 q^{90} + 30 q^{91} - 47 q^{92} - 8 q^{93} + 94 q^{94} + 4 q^{95} - 154 q^{96} + 29 q^{97} - 69 q^{98} - 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - x^{10} - 21 x^{9} + 20 x^{8} + 161 x^{7} - 148 x^{6} - 536 x^{5} + 481 x^{4} + 689 x^{3} - 587 x^{2} - 156 x + 61 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10 \nu^{10} - 102 \nu^{9} - 87 \nu^{8} + 1544 \nu^{7} - 92 \nu^{6} - 8244 \nu^{5} + 2263 \nu^{4} + 18509 \nu^{3} - 4118 \nu^{2} - 13375 \nu - 820 ) / 1359 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19 \nu^{10} + 78 \nu^{9} - 573 \nu^{8} - 1687 \nu^{7} + 5533 \nu^{6} + 12060 \nu^{5} - 20570 \nu^{4} - 31288 \nu^{3} + 24520 \nu^{2} + 20114 \nu - 199 ) / 1359 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 20 \nu^{10} - 204 \nu^{9} - 174 \nu^{8} + 3088 \nu^{7} - 184 \nu^{6} - 15129 \nu^{5} + 4526 \nu^{4} + 24787 \nu^{3} - 9595 \nu^{2} - 5006 \nu + 1078 ) / 1359 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 28 \nu^{10} + 258 \nu^{9} - 1059 \nu^{8} - 3559 \nu^{7} + 11158 \nu^{6} + 14697 \nu^{5} - 44762 \nu^{4} - 14494 \nu^{3} + 61312 \nu^{2} - 11629 \nu - 9091 ) / 1359 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10 \nu^{10} - 102 \nu^{9} - 87 \nu^{8} + 1544 \nu^{7} - 92 \nu^{6} - 7791 \nu^{5} + 2263 \nu^{4} + 14885 \nu^{3} - 4571 \nu^{2} - 8845 \nu + 539 ) / 453 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 92 \nu^{10} + 123 \nu^{9} + 1344 \nu^{8} - 1702 \nu^{7} - 6764 \nu^{6} + 7623 \nu^{5} + 15058 \nu^{4} - 11008 \nu^{3} - 17018 \nu^{2} + 740 \nu + 6185 ) / 1359 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 169 \nu^{10} + 93 \nu^{9} + 3237 \nu^{8} - 1088 \nu^{7} - 21820 \nu^{6} + 2880 \nu^{5} + 60011 \nu^{4} + 2350 \nu^{3} - 56521 \nu^{2} - 7031 \nu + 4345 ) / 1359 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 241 \nu^{10} + 12 \nu^{9} + 4407 \nu^{8} + 298 \nu^{7} - 28768 \nu^{6} - 4626 \nu^{5} + 78236 \nu^{4} + 16129 \nu^{3} - 73621 \nu^{2} - 11297 \nu + 4813 ) / 1359 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{5} - \beta_{3} + 6\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{10} + \beta_{9} + \beta_{8} + \beta_{5} + 7\beta_{2} + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{7} - 8\beta_{5} - 11\beta_{3} + \beta_{2} + 38\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{10} + 11 \beta_{9} + 9 \beta_{8} + \beta_{7} + \beta_{6} + 11 \beta_{5} + \beta_{4} - 2 \beta_{3} + 47 \beta_{2} + 3 \beta _1 + 143 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{10} + \beta_{9} + \beta_{8} + 68 \beta_{7} + \beta_{6} - 54 \beta_{5} - 2 \beta_{4} - 93 \beta_{3} + 14 \beta_{2} + 248 \beta _1 - 36 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 81 \beta_{10} + 95 \beta_{9} + 65 \beta_{8} + 17 \beta_{7} + 12 \beta_{6} + 88 \beta_{5} + 12 \beta_{4} - 32 \beta_{3} + 315 \beta_{2} + 49 \beta _1 + 920 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 20 \beta_{10} + 20 \beta_{9} + 17 \beta_{8} + 487 \beta_{7} + 14 \beta_{6} - 354 \beta_{5} - 32 \beta_{4} - 719 \beta_{3} + 143 \beta_{2} + 1650 \beta _1 - 142 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 620 \beta_{10} + 751 \beta_{9} + 441 \beta_{8} + 194 \beta_{7} + 102 \beta_{6} + 623 \beta_{5} + 96 \beta_{4} - 353 \beta_{3} + 2122 \beta_{2} + 552 \beta _1 + 6034 \) Copy content Toggle raw display

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
2.70947
2.65249
2.05771
1.45309
1.23394
0.241399
−0.404974
−1.64327
−2.21988
−2.51445
−2.56552
−2.70947 3.24448 5.34122 1.00000 −8.79080 4.32849 −9.05292 7.52662 −2.70947
1.2 −2.65249 −0.210126 5.03568 1.00000 0.557356 −2.01577 −8.05211 −2.95585 −2.65249
1.3 −2.05771 −3.16745 2.23417 1.00000 6.51769 −5.06034 −0.481850 7.03273 −2.05771
1.4 −1.45309 −0.381689 0.111480 1.00000 0.554630 4.39311 2.74420 −2.85431 −1.45309
1.5 −1.23394 1.33781 −0.477390 1.00000 −1.65077 −2.41473 3.05695 −1.21027 −1.23394
1.6 −0.241399 2.96345 −1.94173 1.00000 −0.715376 1.66096 0.951530 5.78206 −0.241399
1.7 0.404974 −2.08927 −1.83600 1.00000 −0.846099 −3.63572 −1.55348 1.36504 0.404974
1.8 1.64327 2.10175 0.700339 1.00000 3.45374 1.99073 −2.13569 1.41733 1.64327
1.9 2.21988 3.25200 2.92788 1.00000 7.21906 −3.88294 2.05978 7.57551 2.21988
1.10 2.51445 0.737750 4.32248 1.00000 1.85504 3.31449 5.83977 −2.45573 2.51445
1.11 2.56552 −2.78870 4.58187 1.00000 −7.15446 −0.678270 6.62382 4.77686 2.56552
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(1\)
\(17\) \(1\)

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 935.2.a.j 11
3.b odd 2 1 8415.2.a.by 11
5.b even 2 1 4675.2.a.bl 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
935.2.a.j 11 1.a even 1 1 trivial
4675.2.a.bl 11 5.b even 2 1
8415.2.a.by 11 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(935))\):

\( T_{2}^{11} + T_{2}^{10} - 21 T_{2}^{9} - 20 T_{2}^{8} + 161 T_{2}^{7} + 148 T_{2}^{6} - 536 T_{2}^{5} - 481 T_{2}^{4} + 689 T_{2}^{3} + 587 T_{2}^{2} - 156 T_{2} - 61 \) Copy content Toggle raw display
\( T_{7}^{11} + 2 T_{7}^{10} - 58 T_{7}^{9} - 104 T_{7}^{8} + 1216 T_{7}^{7} + 1952 T_{7}^{6} - 11208 T_{7}^{5} - 15616 T_{7}^{4} + 44160 T_{7}^{3} + 49664 T_{7}^{2} - 61440 T_{7} - 49152 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} + T^{10} - 21 T^{9} - 20 T^{8} + \cdots - 61 \) Copy content Toggle raw display
$3$ \( T^{11} - 5 T^{10} - 17 T^{9} + 112 T^{8} + \cdots + 96 \) Copy content Toggle raw display
$5$ \( (T - 1)^{11} \) Copy content Toggle raw display
$7$ \( T^{11} + 2 T^{10} - 58 T^{9} + \cdots - 49152 \) Copy content Toggle raw display
$11$ \( (T + 1)^{11} \) Copy content Toggle raw display
$13$ \( T^{11} - 13 T^{10} + T^{9} + \cdots + 103172 \) Copy content Toggle raw display
$17$ \( (T + 1)^{11} \) Copy content Toggle raw display
$19$ \( T^{11} - 4 T^{10} - 166 T^{9} + \cdots + 78316544 \) Copy content Toggle raw display
$23$ \( T^{11} + 7 T^{10} - 115 T^{9} + \cdots + 146992 \) Copy content Toggle raw display
$29$ \( T^{11} - 9 T^{10} - 137 T^{9} + \cdots - 6391716 \) Copy content Toggle raw display
$31$ \( T^{11} - 20 T^{10} - 44 T^{9} + \cdots + 5378048 \) Copy content Toggle raw display
$37$ \( T^{11} + T^{10} - 205 T^{9} + \cdots - 19954876 \) Copy content Toggle raw display
$41$ \( T^{11} + 9 T^{10} - 223 T^{9} + \cdots - 8169524 \) Copy content Toggle raw display
$43$ \( T^{11} - 5 T^{10} - 317 T^{9} + \cdots + 352464976 \) Copy content Toggle raw display
$47$ \( T^{11} - 374 T^{9} + \cdots - 109016064 \) Copy content Toggle raw display
$53$ \( T^{11} - 14 T^{10} - 238 T^{9} + \cdots + 49237760 \) Copy content Toggle raw display
$59$ \( T^{11} - 13 T^{10} + \cdots + 4778628112 \) Copy content Toggle raw display
$61$ \( T^{11} - 11 T^{10} - 225 T^{9} + \cdots - 82231988 \) Copy content Toggle raw display
$67$ \( T^{11} - 26 T^{10} + 74 T^{9} + \cdots - 26957824 \) Copy content Toggle raw display
$71$ \( T^{11} + 8 T^{10} - 320 T^{9} + \cdots + 75526144 \) Copy content Toggle raw display
$73$ \( T^{11} - 42 T^{10} + \cdots - 148165888 \) Copy content Toggle raw display
$79$ \( T^{11} + 27 T^{10} + 139 T^{9} + \cdots + 53056 \) Copy content Toggle raw display
$83$ \( T^{11} + 41 T^{10} + \cdots - 6681430544 \) Copy content Toggle raw display
$89$ \( T^{11} - 37 T^{10} + \cdots + 2613838716 \) Copy content Toggle raw display
$97$ \( T^{11} - 29 T^{10} - 37 T^{9} + \cdots - 1056116 \) Copy content Toggle raw display
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