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Properties

Label 950.6.a.s
Level $950$
Weight $6$
Character orbit 950.a
Self dual yes
Analytic conductor $152.365$
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] = [950,6,Mod(1,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, 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("950.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 950.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: \(152.364628822\)
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: \( x^{9} - 4 x^{8} - 1619 x^{7} + 7391 x^{6} + 803419 x^{5} - 3168766 x^{4} - 113162562 x^{3} + \cdots + 1738848897 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 5^{3} \)
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.

\(f(q)\) \(=\) \( q - 4 q^{2} - \beta_1 q^{3} + 16 q^{4} + 4 \beta_1 q^{6} + (\beta_{3} - 5) q^{7} - 64 q^{8} + (\beta_{2} - \beta_1 + 119) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} - \beta_1 q^{3} + 16 q^{4} + 4 \beta_1 q^{6} + (\beta_{3} - 5) q^{7} - 64 q^{8} + (\beta_{2} - \beta_1 + 119) q^{9} + ( - \beta_{4} - \beta_{3} + 5 \beta_1 - 41) q^{11} - 16 \beta_1 q^{12} + ( - \beta_{6} + \beta_{3} - \beta_{2} + \cdots - 89) q^{13}+ \cdots + ( - 81 \beta_{8} + 57 \beta_{7} + \cdots - 3160) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 36 q^{2} - 4 q^{3} + 144 q^{4} + 16 q^{6} - 44 q^{7} - 576 q^{8} + 1067 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 36 q^{2} - 4 q^{3} + 144 q^{4} + 16 q^{6} - 44 q^{7} - 576 q^{8} + 1067 q^{9} - 348 q^{11} - 64 q^{12} - 810 q^{13} + 176 q^{14} + 2304 q^{16} + 496 q^{17} - 4268 q^{18} + 3249 q^{19} - 455 q^{21} + 1392 q^{22} - 6360 q^{23} + 256 q^{24} + 3240 q^{26} + 4625 q^{27} - 704 q^{28} + 8796 q^{29} + 9238 q^{31} - 9216 q^{32} - 14858 q^{33} - 1984 q^{34} + 17072 q^{36} + 1919 q^{37} - 12996 q^{38} + 4771 q^{39} + 18212 q^{41} + 1820 q^{42} - 32126 q^{43} - 5568 q^{44} + 25440 q^{46} + 26769 q^{47} - 1024 q^{48} + 21987 q^{49} + 35763 q^{51} - 12960 q^{52} - 10296 q^{53} - 18500 q^{54} + 2816 q^{56} - 1444 q^{57} - 35184 q^{58} + 39826 q^{59} + 52824 q^{61} - 36952 q^{62} - 46202 q^{63} + 36864 q^{64} + 59432 q^{66} - 29154 q^{67} + 7936 q^{68} - 6922 q^{69} + 59762 q^{71} - 68288 q^{72} + 11118 q^{73} - 7676 q^{74} + 51984 q^{76} - 150374 q^{77} - 19084 q^{78} + 44802 q^{79} + 173537 q^{81} - 72848 q^{82} + 35822 q^{83} - 7280 q^{84} + 128504 q^{86} - 157165 q^{87} + 22272 q^{88} - 78584 q^{89} + 261491 q^{91} - 101760 q^{92} + 171366 q^{93} - 107076 q^{94} + 4096 q^{96} + 7620 q^{97} - 87948 q^{98} - 28090 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4 x^{8} - 1619 x^{7} + 7391 x^{6} + 803419 x^{5} - 3168766 x^{4} - 113162562 x^{3} + \cdots + 1738848897 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 362 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 20552829901 \nu^{8} + 198157961338 \nu^{7} + 34643003477602 \nu^{6} + \cdots - 14\!\cdots\!08 ) / 86\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 26312774833 \nu^{8} - 326475433369 \nu^{7} - 44380776842966 \nu^{6} + 522705850244012 \nu^{5} + \cdots + 17\!\cdots\!54 ) / 51\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 148853919751 \nu^{8} - 2435996879488 \nu^{7} - 239583385219727 \nu^{6} + \cdots + 89\!\cdots\!33 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 811066676768 \nu^{8} - 4088208212009 \nu^{7} + \cdots + 60\!\cdots\!19 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 970616045053 \nu^{8} - 6450873399289 \nu^{7} + \cdots + 91\!\cdots\!49 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 375566308151 \nu^{8} - 1920317823063 \nu^{7} - 604726801124577 \nu^{6} + \cdots + 31\!\cdots\!33 ) / 86\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 362 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} - \beta_{7} + 2\beta_{6} - 2\beta_{5} + 4\beta_{4} - 4\beta_{3} - 5\beta_{2} + 619\beta _1 - 573 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 8 \beta_{8} + 43 \beta_{7} - 27 \beta_{6} - 10 \beta_{5} - 18 \beta_{4} + 113 \beta_{3} + \cdots + 224904 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 774 \beta_{8} - 762 \beta_{7} + 1571 \beta_{6} - 1803 \beta_{5} + 4129 \beta_{4} - 1016 \beta_{3} + \cdots - 1037252 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 14194 \beta_{8} + 46904 \beta_{7} - 21840 \beta_{6} - 9026 \beta_{5} - 23418 \beta_{4} + \cdots + 148755133 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 438209 \beta_{8} - 655715 \beta_{7} + 1129850 \beta_{6} - 1453156 \beta_{5} + 3487462 \beta_{4} + \cdots - 1130584963 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 15598218 \beta_{8} + 41259729 \beta_{7} - 14362377 \beta_{6} - 5718564 \beta_{5} + \cdots + 101273684126 \) 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
25.4696
23.6730
20.6219
3.89701
−1.70950
−2.55326
−11.2051
−26.3986
−27.7949
−4.00000 −25.4696 16.0000 0 101.878 150.352 −64.0000 405.701 0
1.2 −4.00000 −23.6730 16.0000 0 94.6919 −232.821 −64.0000 317.409 0
1.3 −4.00000 −20.6219 16.0000 0 82.4874 −19.0798 −64.0000 182.261 0
1.4 −4.00000 −3.89701 16.0000 0 15.5881 253.939 −64.0000 −227.813 0
1.5 −4.00000 1.70950 16.0000 0 −6.83800 −83.7281 −64.0000 −240.078 0
1.6 −4.00000 2.55326 16.0000 0 −10.2130 6.81059 −64.0000 −236.481 0
1.7 −4.00000 11.2051 16.0000 0 −44.8205 −120.847 −64.0000 −117.445 0
1.8 −4.00000 26.3986 16.0000 0 −105.595 71.1419 −64.0000 453.889 0
1.9 −4.00000 27.7949 16.0000 0 −111.180 −69.7683 −64.0000 529.557 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(19\) \(-1\)

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 950.6.a.s ✓ 9
5.b even 2 1 950.6.a.u yes 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.6.a.s ✓ 9 1.a even 1 1 trivial
950.6.a.u yes 9 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{9} + 4 T_{3}^{8} - 1619 T_{3}^{7} - 7391 T_{3}^{6} + 803419 T_{3}^{5} + 3168766 T_{3}^{4} + \cdots - 1738848897 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(950))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 4)^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + \cdots - 1738848897 \) Copy content Toggle raw display
$5$ \( T^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots + 58\!\cdots\!95 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots + 50\!\cdots\!60 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots + 46\!\cdots\!75 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots - 35\!\cdots\!37 \) Copy content Toggle raw display
$19$ \( (T - 361)^{9} \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots - 25\!\cdots\!29 \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots + 45\!\cdots\!85 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots - 28\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots - 12\!\cdots\!75 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots - 72\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots - 83\!\cdots\!88 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots + 34\!\cdots\!75 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots + 32\!\cdots\!79 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots - 77\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots - 95\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots - 39\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots - 52\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots + 42\!\cdots\!33 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots - 98\!\cdots\!60 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots - 72\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots - 44\!\cdots\!00 \) Copy content Toggle raw display
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