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Properties

Label 182.2.e.d
Level $182$
Weight $2$
Character orbit 182.e
Analytic conductor $1.453$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [182,2,Mod(107,182)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(182, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("182.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 182 = 2 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 182.e (of order \(3\), degree \(2\), minimal)

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: no
Analytic conductor: \(1.45327731679\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.23207289578928.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 3x^{8} + 13x^{7} + x^{6} - 39x^{5} + 3x^{4} + 117x^{3} - 81x^{2} - 162x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + q^{2} - \beta_{5} q^{3} + q^{4} + \beta_{6} q^{5} - \beta_{5} q^{6} + ( - \beta_{8} - \beta_{7} + \cdots + \beta_{2}) q^{7}+ \cdots + ( - \beta_{8} + \beta_{5} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_{5} q^{3} + q^{4} + \beta_{6} q^{5} - \beta_{5} q^{6} + ( - \beta_{8} - \beta_{7} + \cdots + \beta_{2}) q^{7}+ \cdots + (\beta_{8} + \beta_{6} - 3 \beta_{5} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} - q^{3} + 10 q^{4} - 2 q^{5} - q^{6} - 3 q^{7} + 10 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} - q^{3} + 10 q^{4} - 2 q^{5} - q^{6} - 3 q^{7} + 10 q^{8} - 4 q^{9} - 2 q^{10} - 4 q^{11} - q^{12} - 6 q^{13} - 3 q^{14} + 3 q^{15} + 10 q^{16} + 6 q^{17} - 4 q^{18} - 5 q^{19} - 2 q^{20} - 16 q^{21} - 4 q^{22} - 8 q^{23} - q^{24} - 3 q^{25} - 6 q^{26} + 8 q^{27} - 3 q^{28} + q^{29} + 3 q^{30} - 21 q^{31} + 10 q^{32} + 6 q^{34} + 26 q^{35} - 4 q^{36} - 20 q^{37} - 5 q^{38} + 11 q^{39} - 2 q^{40} - 9 q^{41} - 16 q^{42} + 9 q^{43} - 4 q^{44} - 6 q^{45} - 8 q^{46} - 13 q^{47} - q^{48} + 13 q^{49} - 3 q^{50} - 6 q^{51} - 6 q^{52} - 15 q^{53} + 8 q^{54} + 7 q^{55} - 3 q^{56} - 26 q^{57} + q^{58} - 2 q^{59} + 3 q^{60} - 15 q^{61} - 21 q^{62} + 14 q^{63} + 10 q^{64} - 19 q^{65} + 6 q^{68} - 18 q^{69} + 26 q^{70} - q^{71} - 4 q^{72} - 4 q^{73} - 20 q^{74} + 50 q^{75} - 5 q^{76} + 16 q^{77} + 11 q^{78} - q^{79} - 2 q^{80} + 23 q^{81} - 9 q^{82} + 96 q^{83} - 16 q^{84} + 12 q^{85} + 9 q^{86} + 54 q^{87} - 4 q^{88} + 18 q^{89} - 6 q^{90} + 4 q^{91} - 8 q^{92} + 4 q^{93} - 13 q^{94} - 12 q^{95} - q^{96} - 19 q^{97} + 13 q^{98} + 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 3x^{8} + 13x^{7} + x^{6} - 39x^{5} + 3x^{4} + 117x^{3} - 81x^{2} - 162x + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{9} - \nu^{8} - 21\nu^{7} + 35\nu^{6} + 68\nu^{5} - 111\nu^{4} - 201\nu^{3} + 351\nu^{2} + 405\nu - 891 ) / 81 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{9} + 2\nu^{8} + 45\nu^{7} - 28\nu^{6} - 136\nu^{5} + 99\nu^{4} + 393\nu^{3} - 342\nu^{2} - 675\nu + 567 ) / 81 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} + 2\nu^{7} - 8\nu^{6} - 5\nu^{5} + 26\nu^{4} + 13\nu^{3} - 69\nu^{2} - 24\nu + 162 ) / 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{8} - 2\nu^{7} + 8\nu^{6} + 5\nu^{5} - 26\nu^{4} - 13\nu^{3} + 78\nu^{2} + 33\nu - 171 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{9} + 20\nu^{8} + 15\nu^{7} - 88\nu^{6} - 37\nu^{5} + 276\nu^{4} + 87\nu^{3} - 783\nu^{2} + 108\nu + 972 ) / 81 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8\nu^{9} + 5\nu^{8} - 30\nu^{7} - 31\nu^{6} + 92\nu^{5} + 96\nu^{4} - 273\nu^{3} - 243\nu^{2} + 459\nu + 729 ) / 81 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{9} - 4\nu^{8} - 57\nu^{7} + 59\nu^{6} + 164\nu^{5} - 174\nu^{4} - 480\nu^{3} + 567\nu^{2} + 891\nu - 1215 ) / 81 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -6\nu^{9} + 2\nu^{8} + 35\nu^{7} - 24\nu^{6} - 109\nu^{5} + 77\nu^{4} + 306\nu^{3} - 273\nu^{2} - 531\nu + 432 ) / 27 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 23 \nu^{9} + 37 \nu^{8} + 141 \nu^{7} - 245 \nu^{6} - 410 \nu^{5} + 780 \nu^{4} + 1119 \nu^{3} + \cdots + 3969 ) / 81 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} + 2\beta_{4} + \beta_{2} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{9} - 4 \beta_{8} - 3 \beta_{7} + \beta_{6} + 5 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + \cdots + 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{9} - 2\beta_{8} + 9\beta_{7} - 10\beta_{6} + \beta_{5} - 2\beta_{4} + 9\beta_{3} + 5\beta_{2} - 6\beta _1 - 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2 \beta_{9} - 19 \beta_{8} - 12 \beta_{7} - 11 \beta_{6} + 8 \beta_{5} - 7 \beta_{4} - 9 \beta_{3} + \cdots + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{9} - 8\beta_{8} + 24\beta_{7} - 13\beta_{6} + \beta_{5} - 17\beta_{4} + 35\beta_{2} - 3\beta _1 + 41 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 19 \beta_{9} - 28 \beta_{8} - 21 \beta_{7} + \beta_{6} - 31 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + \cdots - 23 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 49 \beta_{9} + 46 \beta_{8} - 21 \beta_{7} + 41 \beta_{6} + 64 \beta_{5} - 32 \beta_{4} - 12 \beta_{3} + \cdots + 185 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 23 \beta_{9} + 29 \beta_{8} - 45 \beta_{7} + 151 \beta_{6} + 2 \beta_{5} + 50 \beta_{4} + 3 \beta_{3} + \cdots - 104 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/182\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(157\)
\(\chi(n)\) \(\beta_{2}\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−1.73100 + 0.0603688i
1.16412 − 1.28251i
−1.24530 + 1.20384i
1.67396 + 0.444815i
1.13823 + 1.30554i
−1.73100 − 0.0603688i
1.16412 + 1.28251i
−1.24530 − 1.20384i
1.67396 − 0.444815i
1.13823 − 1.30554i
1.00000 −1.31322 + 2.27456i 1.00000 −0.917780 + 1.58964i −1.31322 + 2.27456i −2.62858 + 0.300964i 1.00000 −1.94908 − 3.37591i −0.917780 + 1.58964i
107.2 1.00000 −1.02863 + 1.78164i 1.00000 1.69274 − 2.93192i −1.02863 + 1.78164i 1.70214 + 2.02552i 1.00000 −0.616155 − 1.06721i 1.69274 − 2.93192i
107.3 1.00000 −0.0800993 + 0.138736i 1.00000 −1.66520 + 2.88422i −0.0800993 + 0.138736i 2.46571 − 0.959314i 1.00000 1.48717 + 2.57585i −1.66520 + 2.88422i
107.4 1.00000 0.722201 − 1.25089i 1.00000 0.451759 − 0.782469i 0.722201 − 1.25089i −1.86963 + 1.87203i 1.00000 0.456853 + 0.791292i 0.451759 − 0.782469i
107.5 1.00000 1.19975 − 2.07802i 1.00000 −0.561520 + 0.972581i 1.19975 − 2.07802i −1.16964 − 2.37317i 1.00000 −1.37878 − 2.38812i −0.561520 + 0.972581i
165.1 1.00000 −1.31322 − 2.27456i 1.00000 −0.917780 − 1.58964i −1.31322 − 2.27456i −2.62858 − 0.300964i 1.00000 −1.94908 + 3.37591i −0.917780 − 1.58964i
165.2 1.00000 −1.02863 − 1.78164i 1.00000 1.69274 + 2.93192i −1.02863 − 1.78164i 1.70214 − 2.02552i 1.00000 −0.616155 + 1.06721i 1.69274 + 2.93192i
165.3 1.00000 −0.0800993 − 0.138736i 1.00000 −1.66520 − 2.88422i −0.0800993 − 0.138736i 2.46571 + 0.959314i 1.00000 1.48717 − 2.57585i −1.66520 − 2.88422i
165.4 1.00000 0.722201 + 1.25089i 1.00000 0.451759 + 0.782469i 0.722201 + 1.25089i −1.86963 − 1.87203i 1.00000 0.456853 − 0.791292i 0.451759 + 0.782469i
165.5 1.00000 1.19975 + 2.07802i 1.00000 −0.561520 − 0.972581i 1.19975 + 2.07802i −1.16964 + 2.37317i 1.00000 −1.37878 + 2.38812i −0.561520 − 0.972581i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 182.2.e.d ✓ 10
3.b odd 2 1 1638.2.m.j 10
7.b odd 2 1 1274.2.e.s 10
7.c even 3 1 182.2.h.d yes 10
7.c even 3 1 1274.2.g.p 10
7.d odd 6 1 1274.2.g.q 10
7.d odd 6 1 1274.2.h.s 10
13.c even 3 1 182.2.h.d yes 10
21.h odd 6 1 1638.2.p.k 10
39.i odd 6 1 1638.2.p.k 10
91.g even 3 1 1274.2.g.p 10
91.h even 3 1 inner 182.2.e.d ✓ 10
91.m odd 6 1 1274.2.g.q 10
91.n odd 6 1 1274.2.h.s 10
91.v odd 6 1 1274.2.e.s 10
273.s odd 6 1 1638.2.m.j 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.e.d ✓ 10 1.a even 1 1 trivial
182.2.e.d ✓ 10 91.h even 3 1 inner
182.2.h.d yes 10 7.c even 3 1
182.2.h.d yes 10 13.c even 3 1
1274.2.e.s 10 7.b odd 2 1
1274.2.e.s 10 91.v odd 6 1
1274.2.g.p 10 7.c even 3 1
1274.2.g.p 10 91.g even 3 1
1274.2.g.q 10 7.d odd 6 1
1274.2.g.q 10 91.m odd 6 1
1274.2.h.s 10 7.d odd 6 1
1274.2.h.s 10 91.n odd 6 1
1638.2.m.j 10 3.b odd 2 1
1638.2.m.j 10 273.s odd 6 1
1638.2.p.k 10 21.h odd 6 1
1638.2.p.k 10 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(182, [\chi])\):

\( T_{3}^{10} + T_{3}^{9} + 10T_{3}^{8} + 3T_{3}^{7} + 69T_{3}^{6} + 21T_{3}^{5} + 195T_{3}^{4} - 54T_{3}^{3} + 342T_{3}^{2} + 54T_{3} + 9 \) Copy content Toggle raw display
\( T_{5}^{10} + 2 T_{5}^{9} + 16 T_{5}^{8} + 26 T_{5}^{7} + 187 T_{5}^{6} + 293 T_{5}^{5} + 667 T_{5}^{4} + \cdots + 441 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + T^{9} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{10} + 2 T^{9} + \cdots + 441 \) Copy content Toggle raw display
$7$ \( T^{10} + 3 T^{9} + \cdots + 16807 \) Copy content Toggle raw display
$11$ \( T^{10} + 4 T^{9} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{10} + 6 T^{9} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( (T^{5} - 3 T^{4} + \cdots - 2592)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 5 T^{9} + \cdots + 50253921 \) Copy content Toggle raw display
$23$ \( (T^{5} + 4 T^{4} - 3 T^{3} + \cdots + 12)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} - T^{9} + \cdots + 103041 \) Copy content Toggle raw display
$31$ \( T^{10} + 21 T^{9} + \cdots + 32228329 \) Copy content Toggle raw display
$37$ \( (T^{5} + 10 T^{4} + \cdots + 12)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + 9 T^{9} + \cdots + 7733961 \) Copy content Toggle raw display
$43$ \( T^{10} - 9 T^{9} + \cdots + 6538249 \) Copy content Toggle raw display
$47$ \( T^{10} + 13 T^{9} + \cdots + 9529569 \) Copy content Toggle raw display
$53$ \( T^{10} + 15 T^{9} + \cdots + 5948721 \) Copy content Toggle raw display
$59$ \( (T^{5} + T^{4} - 129 T^{3} + \cdots + 7212)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 493417369 \) Copy content Toggle raw display
$67$ \( T^{10} + 220 T^{8} + \cdots + 22118209 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 1069878681 \) Copy content Toggle raw display
$73$ \( T^{10} + 4 T^{9} + \cdots + 41744521 \) Copy content Toggle raw display
$79$ \( T^{10} + T^{9} + \cdots + 66049 \) Copy content Toggle raw display
$83$ \( (T^{5} - 48 T^{4} + \cdots - 20412)^{2} \) Copy content Toggle raw display
$89$ \( (T^{5} - 9 T^{4} + \cdots - 6696)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 19 T^{9} + \cdots + 42523441 \) Copy content Toggle raw display
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