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Properties

Label 935.2.a.i
Level $935$
Weight $2$
Character orbit 935.a
Self dual yes
Analytic conductor $7.466$
Analytic rank $0$
Dimension $9$
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 10x^{7} + 31x^{6} + 29x^{5} - 97x^{4} - 19x^{3} + 94x^{2} - 10x - 9 \) 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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 10x^{7} + 31x^{6} + 29x^{5} - 97x^{4} - 19x^{3} + 94x^{2} - 10x - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9\nu^{8} + 26\nu^{7} + 79\nu^{6} - 173\nu^{5} - 183\nu^{4} + 61\nu^{3} - 100\nu^{2} + 504\nu + 396 ) / 125 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -11\nu^{8} + 4\nu^{7} + 166\nu^{6} - 17\nu^{5} - 807\nu^{4} - 106\nu^{3} + 1350\nu^{2} + 366\nu - 516 ) / 125 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{8} - 4\nu^{7} - 166\nu^{6} + 17\nu^{5} + 807\nu^{4} + 106\nu^{3} - 1225\nu^{2} - 366\nu + 141 ) / 125 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\nu^{8} - 52\nu^{7} - 158\nu^{6} + 471\nu^{5} + 366\nu^{4} - 1122\nu^{3} - 175\nu^{2} + 492\nu - 42 ) / 125 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4\nu^{8} + 6\nu^{7} + 49\nu^{6} - 63\nu^{5} - 173\nu^{4} + 191\nu^{3} + 125\nu^{2} - 151\nu + 76 ) / 25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -28\nu^{8} + 67\nu^{7} + 343\nu^{6} - 691\nu^{5} - 1361\nu^{4} + 2037\nu^{3} + 1675\nu^{2} - 1682\nu - 143 ) / 125 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 29\nu^{8} - 56\nu^{7} - 324\nu^{6} + 488\nu^{5} + 1173\nu^{4} - 1141\nu^{3} - 1400\nu^{2} + 751\nu + 224 ) / 125 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{5} + \beta_{4} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + \beta_{5} + 8\beta_{4} + 7\beta_{3} + \beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{8} + 9\beta_{5} + 11\beta_{4} + 3\beta_{3} + 2\beta_{2} + 28\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{8} + 2\beta_{7} + 8\beta_{6} + 13\beta_{5} + 57\beta_{4} + 47\beta_{3} + 11\beta_{2} + 15\beta _1 + 79 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -52\beta_{8} + 5\beta_{7} - \beta_{6} + 68\beta_{5} + 98\beta_{4} + 40\beta_{3} + 26\beta_{2} + 171\beta _1 + 97 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 12 \beta_{8} + 32 \beta_{7} + 47 \beta_{6} + 124 \beta_{5} + 405 \beta_{4} + 317 \beta_{3} + \cdots + 495 \) 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
−2.30518
−1.82213
−1.27485
−0.263215
0.434208
1.25326
1.67199
2.51099
2.79493
−2.30518 2.11406 3.31387 −1.00000 −4.87331 1.72738 −3.02871 1.46927 2.30518
1.2 −1.82213 −2.88880 1.32016 −1.00000 5.26377 5.06716 1.23876 5.34517 1.82213
1.3 −1.27485 −0.573451 −0.374758 −1.00000 0.731064 1.11173 3.02746 −2.67115 1.27485
1.4 −0.263215 −0.359599 −1.93072 −1.00000 0.0946517 −1.72139 1.03462 −2.87069 0.263215
1.5 0.434208 1.91729 −1.81146 −1.00000 0.832505 4.14473 −1.65497 0.676011 −0.434208
1.6 1.25326 −1.82448 −0.429335 −1.00000 −2.28655 −3.29038 −3.04459 0.328712 −1.25326
1.7 1.67199 3.38712 0.795535 −1.00000 5.66322 0.816098 −2.01385 8.47260 −1.67199
1.8 2.51099 −0.503277 4.30510 −1.00000 −1.26373 4.45748 5.78808 −2.74671 −2.51099
1.9 2.79493 1.73112 5.81162 −1.00000 4.83836 −4.31281 10.6532 −0.00321109 −2.79493
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.i ✓ 9
3.b odd 2 1 8415.2.a.bt 9
5.b even 2 1 4675.2.a.bi 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
935.2.a.i ✓ 9 1.a even 1 1 trivial
4675.2.a.bi 9 5.b even 2 1
8415.2.a.bt 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(935))\):

\( T_{2}^{9} - 3T_{2}^{8} - 10T_{2}^{7} + 31T_{2}^{6} + 29T_{2}^{5} - 97T_{2}^{4} - 19T_{2}^{3} + 94T_{2}^{2} - 10T_{2} - 9 \) Copy content Toggle raw display
\( T_{7}^{9} - 8T_{7}^{8} - 18T_{7}^{7} + 256T_{7}^{6} - 168T_{7}^{5} - 2176T_{7}^{4} + 3352T_{7}^{3} + 3200T_{7}^{2} - 8064T_{7} + 3584 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - 3 T^{8} + \cdots - 9 \) Copy content Toggle raw display
$3$ \( T^{9} - 3 T^{8} + \cdots + 13 \) Copy content Toggle raw display
$5$ \( (T + 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - 8 T^{8} + \cdots + 3584 \) Copy content Toggle raw display
$11$ \( (T - 1)^{9} \) Copy content Toggle raw display
$13$ \( T^{9} - 23 T^{8} + \cdots + 9253 \) Copy content Toggle raw display
$17$ \( (T + 1)^{9} \) Copy content Toggle raw display
$19$ \( T^{9} - 12 T^{8} + \cdots - 814912 \) Copy content Toggle raw display
$23$ \( T^{9} + T^{8} + \cdots + 221277 \) Copy content Toggle raw display
$29$ \( T^{9} + 19 T^{8} + \cdots - 2015229 \) Copy content Toggle raw display
$31$ \( T^{9} - 14 T^{8} + \cdots + 7616 \) Copy content Toggle raw display
$37$ \( T^{9} - 15 T^{8} + \cdots - 4459 \) Copy content Toggle raw display
$41$ \( T^{9} + 7 T^{8} + \cdots - 8401809 \) Copy content Toggle raw display
$43$ \( T^{9} - 33 T^{8} + \cdots - 97661 \) Copy content Toggle raw display
$47$ \( T^{9} + 10 T^{8} + \cdots - 55872 \) Copy content Toggle raw display
$53$ \( T^{9} - 18 T^{8} + \cdots + 49430976 \) Copy content Toggle raw display
$59$ \( T^{9} + 7 T^{8} + \cdots - 88495959 \) Copy content Toggle raw display
$61$ \( T^{9} - 13 T^{8} + \cdots - 84097973 \) Copy content Toggle raw display
$67$ \( T^{9} - 8 T^{8} + \cdots - 473152 \) Copy content Toggle raw display
$71$ \( T^{9} + 16 T^{8} + \cdots - 122636736 \) Copy content Toggle raw display
$73$ \( T^{9} - 30 T^{8} + \cdots - 3846848 \) Copy content Toggle raw display
$79$ \( T^{9} + 21 T^{8} + \cdots + 4749787 \) Copy content Toggle raw display
$83$ \( T^{9} - 33 T^{8} + \cdots - 948279 \) Copy content Toggle raw display
$89$ \( T^{9} + 11 T^{8} + \cdots + 42921507 \) Copy content Toggle raw display
$97$ \( T^{9} - 27 T^{8} + \cdots - 28201033 \) Copy content Toggle raw display
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