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Properties

Label 1035.4.a.r
Level $1035$
Weight $4$
Character orbit 1035.a
Self dual yes
Analytic conductor $61.067$
Analytic rank $1$
Dimension $8$
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] = [1035,4,Mod(1,1035)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1035, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1035.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1035 = 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1035.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: \(61.0669768559\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 49x^{6} + 31x^{5} + 750x^{4} + 249x^{3} - 2892x^{2} - 620x + 2400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 115)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} + 5) q^{4} - 5 q^{5} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{6} + \beta_{5} - \beta_{4} + \cdots - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} + 5) q^{4} - 5 q^{5} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 1) q^{7}+ \cdots + (25 \beta_{7} + 48 \beta_{6} + \cdots - 202) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{2} + 42 q^{4} - 40 q^{5} + 11 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{2} + 42 q^{4} - 40 q^{5} + 11 q^{7} - 3 q^{8} + 30 q^{10} - 41 q^{11} + 28 q^{13} - 161 q^{14} + 98 q^{16} - 71 q^{17} + 177 q^{19} - 210 q^{20} + 171 q^{22} - 184 q^{23} + 200 q^{25} - 396 q^{26} + 179 q^{28} - 225 q^{29} - 36 q^{31} + 360 q^{32} - 1051 q^{34} - 55 q^{35} - 348 q^{37} + 309 q^{38} + 15 q^{40} - 620 q^{41} - 390 q^{43} + 141 q^{44} + 138 q^{46} - 123 q^{47} + 881 q^{49} - 150 q^{50} - 1890 q^{52} - 1406 q^{53} + 205 q^{55} - 1031 q^{56} - 129 q^{58} + 676 q^{59} + 1447 q^{61} + 1556 q^{62} - 2883 q^{64} - 140 q^{65} - 1582 q^{67} - 953 q^{68} + 805 q^{70} - 1396 q^{71} + 17 q^{73} + 964 q^{74} + 2271 q^{76} - 488 q^{77} + 708 q^{79} - 490 q^{80} - 3332 q^{82} - 1486 q^{83} + 355 q^{85} - 3570 q^{86} - 1031 q^{88} - 1360 q^{89} + 2693 q^{91} - 966 q^{92} - 3207 q^{94} - 885 q^{95} - 855 q^{97} - 1817 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 49x^{6} + 31x^{5} + 750x^{4} + 249x^{3} - 2892x^{2} - 620x + 2400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -21\nu^{7} + 372\nu^{6} - 611\nu^{5} - 10501\nu^{4} + 19940\nu^{3} + 88331\nu^{2} - 77178\nu - 104240 ) / 3680 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -43\nu^{7} + 236\nu^{6} + 1027\nu^{5} - 3803\nu^{4} - 10340\nu^{3} + 373\nu^{2} + 31226\nu + 17520 ) / 3680 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 57\nu^{7} - 484\nu^{6} - 1233\nu^{5} + 15097\nu^{4} + 6860\nu^{3} - 135927\nu^{2} - 25774\nu + 211440 ) / 3680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 79\nu^{7} - 348\nu^{6} - 2871\nu^{5} + 8399\nu^{4} + 33460\nu^{3} - 36929\nu^{2} - 71618\nu + 30800 ) / 3680 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 233\nu^{7} - 1236\nu^{6} - 6977\nu^{5} + 29593\nu^{4} + 63260\nu^{3} - 146183\nu^{2} - 44766\nu + 109520 ) / 3680 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 3\beta_{2} + 23\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} - 9\beta_{6} + 5\beta_{5} - 2\beta_{4} + 6\beta_{3} + 30\beta_{2} + 87\beta _1 + 268 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14\beta_{7} - 79\beta_{6} + 45\beta_{5} - 36\beta_{4} + 54\beta_{3} + 133\beta_{2} + 652\beta _1 + 956 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 128\beta_{7} - 548\beta_{6} + 266\beta_{5} - 133\beta_{4} + 353\beta_{3} + 951\beta_{2} + 3189\beta _1 + 7538 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 860 \beta_{7} - 3858 \beta_{6} + 1852 \beta_{5} - 1258 \beta_{4} + 2456 \beta_{3} + 5030 \beta_{2} + \cdots + 36204 \) 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
−3.81977
−3.57019
−3.05234
−1.16661
0.954235
1.76920
4.85207
6.03341
−4.81977 0 15.2302 −5.00000 0 13.5587 −34.8481 0 24.0989
1.2 −4.57019 0 12.8867 −5.00000 0 −18.4259 −22.3330 0 22.8510
1.3 −4.05234 0 8.42146 −5.00000 0 32.7645 −1.70790 0 20.2617
1.4 −2.16661 0 −3.30580 −5.00000 0 9.86011 24.4953 0 10.8331
1.5 −0.0457645 0 −7.99791 −5.00000 0 −33.8594 0.732137 0 0.228823
1.6 0.769202 0 −7.40833 −5.00000 0 10.0918 −11.8521 0 −3.84601
1.7 3.85207 0 6.83843 −5.00000 0 16.9190 −4.47444 0 −19.2603
1.8 5.03341 0 17.3352 −5.00000 0 −19.9088 46.9881 0 −25.1671
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(23\) \(1\)

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 1035.4.a.r 8
3.b odd 2 1 115.4.a.f ✓ 8
12.b even 2 1 1840.4.a.v 8
15.d odd 2 1 575.4.a.n 8
15.e even 4 2 575.4.b.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.a.f ✓ 8 3.b odd 2 1
575.4.a.n 8 15.d odd 2 1
575.4.b.k 16 15.e even 4 2
1035.4.a.r 8 1.a even 1 1 trivial
1840.4.a.v 8 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1035))\):

\( T_{2}^{8} + 6T_{2}^{7} - 35T_{2}^{6} - 249T_{2}^{5} + 170T_{2}^{4} + 2565T_{2}^{3} + 1916T_{2}^{2} - 2802T_{2} - 132 \) Copy content Toggle raw display
\( T_{7}^{8} - 11 T_{7}^{7} - 1752 T_{7}^{6} + 22539 T_{7}^{5} + 783793 T_{7}^{4} - 12717810 T_{7}^{3} + \cdots - 9289584128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 6 T^{7} + \cdots - 132 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T + 5)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots - 9289584128 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots - 11345758080 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots - 5251923133176 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 52834929586560 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots - 54772847726720 \) Copy content Toggle raw display
$23$ \( (T + 23)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots - 15\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots - 18\!\cdots\!22 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots - 30\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots - 62\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots - 94\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 27\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 30\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots - 59\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 11\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 84\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 13\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
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