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Properties

Label 369.6.a.e
Level $369$
Weight $6$
Character orbit 369.a
Self dual yes
Analytic conductor $59.182$
Analytic rank $0$
Dimension $10$
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] = [369,6,Mod(1,369)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(369, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("369.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 369 = 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 369.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: \(59.1816295110\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3 x^{9} - 259 x^{8} + 639 x^{7} + 22422 x^{6} - 38356 x^{5} - 735592 x^{4} + 422608 x^{3} + \cdots - 24923264 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 41)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 20) q^{4} + (\beta_{7} - 3) q^{5} + (\beta_{9} - \beta_{8} + \beta_{7} + \cdots + 33) q^{7}+ \cdots + (\beta_{9} + 2 \beta_{8} - 3 \beta_{7} + \cdots - 21) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 20) q^{4} + (\beta_{7} - 3) q^{5} + (\beta_{9} - \beta_{8} + \beta_{7} + \cdots + 33) q^{7}+ \cdots + (79 \beta_{9} - 348 \beta_{8} + \cdots - 27729) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{2} + 207 q^{4} - 32 q^{5} + 342 q^{7} - 249 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{2} + 207 q^{4} - 32 q^{5} + 342 q^{7} - 249 q^{8} + 102 q^{10} - 846 q^{11} + 1504 q^{13} - 3468 q^{14} + 5859 q^{16} - 560 q^{17} + 4240 q^{19} + 6182 q^{20} - 2628 q^{22} + 1508 q^{23} + 11734 q^{25} + 22014 q^{26} - 8662 q^{28} + 124 q^{29} + 10384 q^{31} + 6619 q^{32} + 802 q^{34} + 17890 q^{35} + 5524 q^{37} + 46098 q^{38} - 61738 q^{40} - 16810 q^{41} + 24160 q^{43} + 21594 q^{44} + 42404 q^{46} - 58984 q^{47} + 70326 q^{49} - 6817 q^{50} + 64374 q^{52} - 23456 q^{53} + 96426 q^{55} - 80184 q^{56} - 13378 q^{58} - 52428 q^{59} + 113540 q^{61} + 113008 q^{62} + 37363 q^{64} + 22340 q^{65} + 85506 q^{67} + 71406 q^{68} - 71946 q^{70} - 75236 q^{71} - 85148 q^{73} + 23462 q^{74} + 113376 q^{76} + 172896 q^{77} + 178200 q^{79} + 401850 q^{80} + 5043 q^{82} + 125412 q^{83} - 245912 q^{85} + 18848 q^{86} - 135952 q^{88} + 62696 q^{89} + 30056 q^{91} + 236372 q^{92} + 419014 q^{94} - 28002 q^{95} + 154548 q^{97} - 288367 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3 x^{9} - 259 x^{8} + 639 x^{7} + 22422 x^{6} - 38356 x^{5} - 735592 x^{4} + 422608 x^{3} + \cdots - 24923264 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1786767 \nu^{9} - 21438703 \nu^{8} - 424565695 \nu^{7} + 5464230143 \nu^{6} + \cdots - 31154776964608 ) / 234803267840 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4143687 \nu^{9} - 33647123 \nu^{8} - 908305915 \nu^{7} + 7674998483 \nu^{6} + \cdots - 52427936227328 ) / 234803267840 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2178597 \nu^{9} + 7450813 \nu^{8} + 572371445 \nu^{7} - 1782324453 \nu^{6} + \cdots + 18795027705728 ) / 117401633920 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 272573 \nu^{9} - 859283 \nu^{8} - 65577420 \nu^{7} + 168200372 \nu^{6} + 5032198895 \nu^{5} + \cdots + 1165821278224 ) / 11740163392 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 6706021 \nu^{9} + 41275289 \nu^{8} + 1597715065 \nu^{7} - 8791262049 \nu^{6} + \cdots + 35838881229824 ) / 234803267840 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3072005 \nu^{9} + 26287225 \nu^{8} + 720918549 \nu^{7} - 5818153805 \nu^{6} + \cdots + 23977434144192 ) / 93921307136 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1722879 \nu^{9} + 6556601 \nu^{8} + 418654030 \nu^{7} - 1446935606 \nu^{6} + \cdots + 6996626226656 ) / 29350408480 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 52 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - 2\beta_{8} + 3\beta_{7} - \beta_{6} - 2\beta_{3} - 2\beta_{2} + 87\beta _1 + 21 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4 \beta_{9} - 8 \beta_{8} + 16 \beta_{7} + 12 \beta_{6} - 6 \beta_{5} + 4 \beta_{4} - 4 \beta_{3} + \cdots + 4550 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 85 \beta_{9} - 298 \beta_{8} + 483 \beta_{7} - 53 \beta_{6} - 78 \beta_{5} + 68 \beta_{4} + \cdots + 1939 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 660 \beta_{9} - 1528 \beta_{8} + 3316 \beta_{7} + 2036 \beta_{6} - 1316 \beta_{5} + 1040 \beta_{4} + \cdots + 445344 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 6153 \beta_{9} - 37474 \beta_{8} + 65107 \beta_{7} + 1487 \beta_{6} - 14572 \beta_{5} + 15536 \beta_{4} + \cdots + 205585 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 83344 \beta_{9} - 216336 \beta_{8} + 505604 \beta_{7} + 277832 \beta_{6} - 201594 \beta_{5} + \cdots + 45168886 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 412801 \beta_{9} - 4451426 \beta_{8} + 8212751 \beta_{7} + 883503 \beta_{6} - 2070682 \beta_{5} + \cdots + 28534771 \) 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
10.7652
9.20306
8.15557
3.67603
2.00833
−2.52657
−3.43528
−4.78711
−9.84411
−10.2151
−10.7652 0 83.8885 106.130 0 150.634 −558.588 0 −1142.50
1.2 −9.20306 0 52.6962 −60.0346 0 92.4544 −190.469 0 552.502
1.3 −8.15557 0 34.5133 −37.0474 0 −182.955 −20.4971 0 302.143
1.4 −3.67603 0 −18.4868 −17.9916 0 214.269 185.591 0 66.1377
1.5 −2.00833 0 −27.9666 −74.6642 0 126.297 120.433 0 149.950
1.6 2.52657 0 −25.6165 62.7567 0 238.318 −145.572 0 158.559
1.7 3.43528 0 −20.1989 −41.7210 0 27.9809 −179.318 0 −143.323
1.8 4.78711 0 −9.08361 34.1675 0 −209.673 −196.672 0 163.563
1.9 9.84411 0 64.9065 −85.4409 0 −100.276 323.935 0 −841.090
1.10 10.2151 0 72.3478 81.8458 0 −15.0493 412.156 0 836.061
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(41\) \(1\)

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 369.6.a.e 10
3.b odd 2 1 41.6.a.b ✓ 10
12.b even 2 1 656.6.a.g 10
15.d odd 2 1 1025.6.a.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
41.6.a.b ✓ 10 3.b odd 2 1
369.6.a.e 10 1.a even 1 1 trivial
656.6.a.g 10 12.b even 2 1
1025.6.a.b 10 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 3 T_{2}^{9} - 259 T_{2}^{8} - 639 T_{2}^{7} + 22422 T_{2}^{6} + 38356 T_{2}^{5} + \cdots - 24923264 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(369))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 3 T^{9} + \cdots - 24923264 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 11\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 84\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots - 16\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 59\!\cdots\!08 \) Copy content Toggle raw display
$41$ \( (T + 1681)^{10} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 75\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 35\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 47\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 16\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 71\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 94\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 45\!\cdots\!20 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 75\!\cdots\!60 \) Copy content Toggle raw display
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