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Properties

Label 182.3.c.a
Level $182$
Weight $3$
Character orbit 182.c
Analytic conductor $4.959$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [182,3,Mod(181,182)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("182.181"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(182, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 182 = 2 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 182.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.95914081136\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 28x^{14} + 402x^{12} - 2046x^{10} + 9801x^{8} + 25890x^{6} + 157693x^{4} + 753740x^{2} + 1028196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} + \beta_1 q^{3} - 2 q^{4} - \beta_{12} q^{5} - \beta_{9} q^{6} + ( - \beta_{15} + \beta_{12} - \beta_{9}) q^{7} - 2 \beta_{8} q^{8} + (\beta_{2} - 2) q^{9} + (\beta_{7} - \beta_{5} - \beta_1) q^{10}+ \cdots + ( - 4 \beta_{15} + 2 \beta_{14} + \cdots + 21 \beta_{8}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{4} - 32 q^{9} - 12 q^{14} + 64 q^{16} + 24 q^{22} + 44 q^{23} + 12 q^{25} + 20 q^{29} + 112 q^{30} - 148 q^{35} + 64 q^{36} - 240 q^{39} + 176 q^{42} - 196 q^{43} + 76 q^{49} - 384 q^{51} + 524 q^{53}+ \cdots + 532 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 28x^{14} + 402x^{12} - 2046x^{10} + 9801x^{8} + 25890x^{6} + 157693x^{4} + 753740x^{2} + 1028196 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 742533748 \nu^{14} - 18563476350 \nu^{12} + 247980751582 \nu^{10} - 862367262615 \nu^{8} + \cdots + 802725243212854 ) / 819815883749954 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1485067496 \nu^{14} - 37126952700 \nu^{12} + 495961503164 \nu^{10} - 1724734525230 \nu^{8} + \cdots + 44\!\cdots\!47 ) / 409907941874977 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 185961041265 \nu^{14} - 9319433572382 \nu^{12} + 192136546855878 \nu^{10} + \cdots - 10\!\cdots\!30 ) / 42\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 71947090406 \nu^{14} - 1977104855783 \nu^{12} + 27557319831604 \nu^{10} + \cdots + 46\!\cdots\!50 ) / 53\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 726532507609 \nu^{14} - 23102620279854 \nu^{12} + 377194978393130 \nu^{10} + \cdots + 30\!\cdots\!58 ) / 42\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15536021058 \nu^{14} + 477441906849 \nu^{12} - 7062951533964 \nu^{10} + \cdots + 968763259676615 ) / 761257606339243 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 135156744521 \nu^{14} - 4048762065646 \nu^{12} + 62282027617934 \nu^{10} + \cdots + 59\!\cdots\!10 ) / 32\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 825351601861 \nu^{15} - 22356915631636 \nu^{13} + 312967978929222 \nu^{11} + \cdots + 10\!\cdots\!70 \nu ) / 83\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 825351601861 \nu^{15} + 22356915631636 \nu^{13} - 312967978929222 \nu^{11} + \cdots - 26\!\cdots\!14 \nu ) / 41\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 651598704829 \nu^{15} + 18013062165736 \nu^{13} - 254940483059034 \nu^{11} + \cdots - 20\!\cdots\!02 \nu ) / 63\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4477572269803 \nu^{15} + 142064442706555 \nu^{13} + \cdots - 11\!\cdots\!70 \nu ) / 41\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 51318970736 \nu^{15} - 1527431879351 \nu^{13} + 23105872503876 \nu^{11} + \cdots + 18\!\cdots\!80 \nu ) / 45\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 5862433119383 \nu^{15} - 179827485996722 \nu^{13} + \cdots + 16\!\cdots\!62 \nu ) / 47\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3624805195657 \nu^{15} + 99235668145580 \nu^{13} + \cdots - 38\!\cdots\!38 \nu ) / 27\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 89409094293725 \nu^{15} + \cdots + 40\!\cdots\!46 \nu ) / 33\!\cdots\!24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + 2\beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{2} + 4\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{14} - 2\beta_{10} + 5\beta_{9} + 29\beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{6} - 8\beta_{5} + 5\beta_{4} + 8\beta_{3} - 3\beta_{2} + 72\beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 32 \beta_{15} + 55 \beta_{14} - 28 \beta_{13} - 13 \beta_{12} + 10 \beta_{11} + 5 \beta_{10} + \cdots + 403 \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 54\beta_{7} - 6\beta_{6} - 208\beta_{5} - 56\beta_{4} + 130\beta_{3} + 175\beta_{2} + 830\beta _1 - 1407 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 370 \beta_{15} + 539 \beta_{14} - 442 \beta_{13} + 260 \beta_{12} + 196 \beta_{11} + \cdots + 3379 \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 512 \beta_{7} - 659 \beta_{6} - 1552 \beta_{5} - 2411 \beta_{4} + 720 \beta_{3} + 5018 \beta_{2} + \cdots - 32777 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1716 \beta_{15} - 1434 \beta_{14} - 1560 \beta_{13} + 9347 \beta_{12} + 234 \beta_{11} + \cdots - 14020 \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 5798 \beta_{7} - 11851 \beta_{6} + 25556 \beta_{5} - 39001 \beta_{4} - 16950 \beta_{3} + \cdots - 463793 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 131202 \beta_{15} - 173866 \beta_{14} + 89414 \beta_{13} + 116597 \beta_{12} + \cdots - 1202430 \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 259004 \beta_{7} - 86637 \beta_{6} + 987912 \beta_{5} - 274923 \beta_{4} - 588604 \beta_{3} + \cdots - 3049814 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2559650 \beta_{15} - 3814291 \beta_{14} + 2493710 \beta_{13} + 169403 \beta_{12} + \cdots - 25845757 \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 2185961 \beta_{7} + 509406 \beta_{6} + 8225124 \beta_{5} + 1759555 \beta_{4} - 4845007 \beta_{3} + \cdots + 22748338 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 26866418 \beta_{15} - 47046112 \beta_{14} + 35718290 \beta_{13} - 25190074 \beta_{12} + \cdots - 318120166 \beta_{8} ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(15\) \(157\)
\(\chi(n)\) \(-1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
3.75057 − 1.41421i
2.46367 − 1.41421i
1.34606 − 1.41421i
0.227406 − 1.41421i
−0.227406 − 1.41421i
−1.34606 − 1.41421i
−2.46367 − 1.41421i
−3.75057 − 1.41421i
−3.75057 + 1.41421i
−2.46367 + 1.41421i
−1.34606 + 1.41421i
−0.227406 + 1.41421i
0.227406 + 1.41421i
1.34606 + 1.41421i
2.46367 + 1.41421i
3.75057 + 1.41421i
− 1.41421i − 5.30410i −2.00000 −0.294416 −7.50113 −6.67347 + 2.11302i 2.82843i −19.1335 0.416366i
181.2 − 1.41421i − 3.48415i −2.00000 −8.04266 −4.92734 3.92712 − 5.79463i 2.82843i −3.13933 11.3740i
181.3 − 1.41421i − 1.90361i −2.00000 4.40738 −2.69211 −5.61366 − 4.18172i 2.82843i 5.37627 − 6.23297i
181.4 − 1.41421i − 0.321601i −2.00000 4.33635 −0.454813 4.00366 + 5.74201i 2.82843i 8.89657 − 6.13252i
181.5 − 1.41421i 0.321601i −2.00000 −4.33635 0.454813 −4.00366 + 5.74201i 2.82843i 8.89657 6.13252i
181.6 − 1.41421i 1.90361i −2.00000 −4.40738 2.69211 5.61366 − 4.18172i 2.82843i 5.37627 6.23297i
181.7 − 1.41421i 3.48415i −2.00000 8.04266 4.92734 −3.92712 − 5.79463i 2.82843i −3.13933 − 11.3740i
181.8 − 1.41421i 5.30410i −2.00000 0.294416 7.50113 6.67347 + 2.11302i 2.82843i −19.1335 − 0.416366i
181.9 1.41421i − 5.30410i −2.00000 0.294416 7.50113 6.67347 − 2.11302i − 2.82843i −19.1335 0.416366i
181.10 1.41421i − 3.48415i −2.00000 8.04266 4.92734 −3.92712 + 5.79463i − 2.82843i −3.13933 11.3740i
181.11 1.41421i − 1.90361i −2.00000 −4.40738 2.69211 5.61366 + 4.18172i − 2.82843i 5.37627 − 6.23297i
181.12 1.41421i − 0.321601i −2.00000 −4.33635 0.454813 −4.00366 − 5.74201i − 2.82843i 8.89657 − 6.13252i
181.13 1.41421i 0.321601i −2.00000 4.33635 −0.454813 4.00366 − 5.74201i − 2.82843i 8.89657 6.13252i
181.14 1.41421i 1.90361i −2.00000 4.40738 −2.69211 −5.61366 + 4.18172i − 2.82843i 5.37627 6.23297i
181.15 1.41421i 3.48415i −2.00000 −8.04266 −4.92734 3.92712 + 5.79463i − 2.82843i −3.13933 − 11.3740i
181.16 1.41421i 5.30410i −2.00000 −0.294416 −7.50113 −6.67347 − 2.11302i − 2.82843i −19.1335 − 0.416366i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
13.b even 2 1 inner
91.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 182.3.c.a ✓ 16
3.b odd 2 1 1638.3.f.a 16
7.b odd 2 1 inner 182.3.c.a ✓ 16
13.b even 2 1 inner 182.3.c.a ✓ 16
21.c even 2 1 1638.3.f.a 16
39.d odd 2 1 1638.3.f.a 16
91.b odd 2 1 inner 182.3.c.a ✓ 16
273.g even 2 1 1638.3.f.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.3.c.a ✓ 16 1.a even 1 1 trivial
182.3.c.a ✓ 16 7.b odd 2 1 inner
182.3.c.a ✓ 16 13.b even 2 1 inner
182.3.c.a ✓ 16 91.b odd 2 1 inner
1638.3.f.a 16 3.b odd 2 1
1638.3.f.a 16 21.c even 2 1
1638.3.f.a 16 39.d odd 2 1
1638.3.f.a 16 273.g even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(182, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{8} \) Copy content Toggle raw display
$3$ \( (T^{8} + 44 T^{6} + \cdots + 128)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 103 T^{6} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{8} + 630 T^{6} + \cdots + 32764176)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 66\!\cdots\!41 \) Copy content Toggle raw display
$17$ \( (T^{8} + 1104 T^{6} + \cdots + 23328)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 625 T^{6} + \cdots + 192119202)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 11 T^{3} + \cdots - 11984)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 5 T^{3} + \cdots + 864)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 3241 T^{6} + \cdots + 7213686498)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 7068 T^{6} + \cdots + 785307902976)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 1671435517952)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 49 T^{3} + \cdots - 43676)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 2689 T^{6} + \cdots + 18899179362)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 131 T^{3} + \cdots - 5593778)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 2907135619200)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 83692402900992)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 10302 T^{6} + \cdots + 19601120016)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 13210416807200)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 11 T^{3} + \cdots + 132007736)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 30601 T^{6} + \cdots + 142867574882)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 12\!\cdots\!48)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 26815390738208)^{2} \) Copy content Toggle raw display
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