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Properties

Label 84.3.g.a
Level $84$
Weight $3$
Character orbit 84.g
Analytic conductor $2.289$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [84,3,Mod(43,84)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(84, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("84.43"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 84.g (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: \(2.28883422063\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.489494783471841.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 7 x^{10} - 11 x^{9} + 18 x^{8} - 22 x^{7} + 33 x^{6} - 44 x^{5} + 72 x^{4} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{2} + \beta_{6} q^{3} + ( - \beta_{6} + \beta_{5}) q^{4} + ( - \beta_{9} - \beta_{8} + \beta_{2} + 1) q^{5} + (\beta_{2} - 1) q^{6} + \beta_1 q^{7} + ( - \beta_{11} - \beta_{10} + \cdots - \beta_1) q^{8}+ \cdots + ( - 3 \beta_{11} - 6 \beta_{10} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} + 2 q^{4} + 8 q^{5} - 12 q^{6} - 10 q^{8} - 36 q^{9} + 28 q^{10} + 24 q^{12} - 24 q^{13} - 14 q^{14} - 14 q^{16} - 40 q^{17} - 6 q^{18} - 20 q^{20} - 88 q^{22} - 36 q^{24} + 180 q^{25} + 100 q^{26}+ \cdots - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 7 x^{10} - 11 x^{9} + 18 x^{8} - 22 x^{7} + 33 x^{6} - 44 x^{5} + 72 x^{4} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 31 \nu^{11} + 23 \nu^{10} + 89 \nu^{9} + 43 \nu^{8} - 66 \nu^{7} + 386 \nu^{6} + 551 \nu^{5} + \cdots + 2608 ) / 1264 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 135 \nu^{11} + 277 \nu^{10} - 673 \nu^{9} + 669 \nu^{8} - 1262 \nu^{7} + 1530 \nu^{6} + \cdots + 5360 ) / 1264 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 135 \nu^{11} - 751 \nu^{10} + 1147 \nu^{9} - 2407 \nu^{8} + 2368 \nu^{7} - 4690 \nu^{6} + \cdots - 20528 ) / 1264 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + \nu^{10} + \nu^{9} + 7\nu^{8} + 16\nu^{6} + 5\nu^{5} + 28\nu^{4} - 2\nu^{3} + 56\nu^{2} - 8\nu + 88 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 177 \nu^{11} - 623 \nu^{10} + 763 \nu^{9} - 1783 \nu^{8} + 1774 \nu^{7} - 3270 \nu^{6} + 2881 \nu^{5} + \cdots - 10960 ) / 1264 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 191 \nu^{11} - 317 \nu^{10} + 793 \nu^{9} - 1101 \nu^{8} + 1734 \nu^{7} - 1954 \nu^{6} + 3191 \nu^{5} + \cdots - 6928 ) / 1264 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 98 \nu^{11} + 9 \nu^{10} + 289 \nu^{9} - 45 \nu^{8} + 431 \nu^{7} - 110 \nu^{6} + 1064 \nu^{5} + \cdots + 416 ) / 632 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 100 \nu^{11} + 331 \nu^{10} - 677 \nu^{9} + 873 \nu^{8} - 1441 \nu^{7} + 1818 \nu^{6} - 2598 \nu^{5} + \cdots + 6592 ) / 632 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 251 \nu^{11} + 405 \nu^{10} - 1057 \nu^{9} + 1293 \nu^{8} - 2330 \nu^{7} + 2002 \nu^{6} + \cdots + 6080 ) / 1264 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 493 \nu^{11} + 781 \nu^{10} - 1869 \nu^{9} + 2257 \nu^{8} - 3828 \nu^{7} + 3586 \nu^{6} + \cdots + 13488 ) / 1264 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 197 \nu^{11} - 468 \nu^{10} + 851 \nu^{9} - 1373 \nu^{8} + 1920 \nu^{7} - 2575 \nu^{6} + 3369 \nu^{5} + \cdots - 9624 ) / 316 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + 2\beta_{6} - 2\beta_{5} + \beta_{4} + \beta_{3} + \beta _1 + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + 3\beta_{10} - 3\beta_{9} + \beta_{8} + \beta_{7} - 2\beta_{6} + \beta_{4} + \beta_{3} + \beta _1 - 6 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{11} + 3 \beta_{10} - \beta_{9} - \beta_{8} + 3 \beta_{7} - 6 \beta_{6} - 2 \beta_{5} + \beta_{4} + \cdots - 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{11} + 3 \beta_{10} + 5 \beta_{9} - 3 \beta_{8} + \beta_{7} + 8 \beta_{6} + 3 \beta_{4} + \beta_{3} + \cdots - 8 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2 \beta_{11} - 3 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + 4 \beta_{6} + 3 \beta_{5} + \cdots - 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3 \beta_{11} - 6 \beta_{10} - 7 \beta_{9} + \beta_{8} - 3 \beta_{7} - 8 \beta_{6} + \beta_{5} + \cdots - 9 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - \beta_{11} - 2 \beta_{10} - 9 \beta_{9} - \beta_{8} - \beta_{7} - 5 \beta_{6} - \beta_{5} - \beta_{4} + \cdots + 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3 \beta_{11} - 5 \beta_{10} + 23 \beta_{9} - 5 \beta_{8} - 3 \beta_{7} - 6 \beta_{5} + \beta_{4} + \cdots + 20 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 10 \beta_{11} + 4 \beta_{10} - 10 \beta_{9} + 4 \beta_{8} + 7 \beta_{7} + 2 \beta_{6} + 4 \beta_{5} + \cdots - 7 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2 \beta_{10} - 8 \beta_{9} + 8 \beta_{8} + 10 \beta_{7} + 15 \beta_{6} - 14 \beta_{5} - 11 \beta_{4} + \cdots + 35 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 45 \beta_{11} + 7 \beta_{10} + 113 \beta_{9} + 33 \beta_{8} + 27 \beta_{7} + 74 \beta_{6} - 20 \beta_{5} + \cdots - 4 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
1.19375 + 0.758257i
1.19375 − 0.758257i
0.0311486 − 1.41387i
0.0311486 + 1.41387i
1.10978 − 0.876576i
1.10978 + 0.876576i
−0.434363 + 1.34586i
−0.434363 − 1.34586i
−1.15503 + 0.816025i
−1.15503 − 0.816025i
0.754714 − 1.19600i
0.754714 + 1.19600i
−1.83926 − 0.785573i − 1.73205i 2.76575 + 2.88975i −8.17539 −1.36065 + 3.18569i 2.64575i −2.81683 − 7.48769i −3.00000 15.0367 + 6.42236i
43.2 −1.83926 + 0.785573i 1.73205i 2.76575 − 2.88975i −8.17539 −1.36065 − 3.18569i − 2.64575i −2.81683 + 7.48769i −3.00000 15.0367 − 6.42236i
43.3 −1.51951 − 1.30042i − 1.73205i 0.617841 + 3.95200i 7.86764 −2.25238 + 2.63187i − 2.64575i 4.20042 − 6.80856i −3.00000 −11.9550 − 10.2312i
43.4 −1.51951 + 1.30042i 1.73205i 0.617841 − 3.95200i 7.86764 −2.25238 − 2.63187i 2.64575i 4.20042 + 6.80856i −3.00000 −11.9550 + 10.2312i
43.5 0.0345996 − 1.99970i − 1.73205i −3.99761 − 0.138378i −6.29204 −3.46358 − 0.0599283i − 2.64575i −0.415030 + 7.98923i −3.00000 −0.217702 + 12.5822i
43.6 0.0345996 + 1.99970i 1.73205i −3.99761 + 0.138378i −6.29204 −3.46358 + 0.0599283i 2.64575i −0.415030 − 7.98923i −3.00000 −0.217702 − 12.5822i
43.7 0.913644 − 1.77912i 1.73205i −2.33051 − 3.25096i 8.22808 3.08152 + 1.58248i − 2.64575i −7.91309 + 1.17604i −3.00000 7.51753 − 14.6387i
43.8 0.913644 + 1.77912i − 1.73205i −2.33051 + 3.25096i 8.22808 3.08152 − 1.58248i 2.64575i −7.91309 − 1.17604i −3.00000 7.51753 + 14.6387i
43.9 1.42562 − 1.40272i − 1.73205i 0.0647610 − 3.99948i 1.94731 −2.42958 − 2.46924i 2.64575i −5.51781 − 5.79256i −3.00000 2.77611 − 2.73152i
43.10 1.42562 + 1.40272i 1.73205i 0.0647610 + 3.99948i 1.94731 −2.42958 + 2.46924i − 2.64575i −5.51781 + 5.79256i −3.00000 2.77611 + 2.73152i
43.11 1.98491 − 0.245189i 1.73205i 3.87976 − 0.973359i 0.424396 0.424680 + 3.43797i 2.64575i 7.46234 − 2.88331i −3.00000 0.842389 − 0.104057i
43.12 1.98491 + 0.245189i − 1.73205i 3.87976 + 0.973359i 0.424396 0.424680 − 3.43797i − 2.64575i 7.46234 + 2.88331i −3.00000 0.842389 + 0.104057i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.3.g.a ✓ 12
3.b odd 2 1 252.3.g.b 12
4.b odd 2 1 inner 84.3.g.a ✓ 12
7.b odd 2 1 588.3.g.d 12
8.b even 2 1 1344.3.m.e 12
8.d odd 2 1 1344.3.m.e 12
12.b even 2 1 252.3.g.b 12
28.d even 2 1 588.3.g.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.3.g.a ✓ 12 1.a even 1 1 trivial
84.3.g.a ✓ 12 4.b odd 2 1 inner
252.3.g.b 12 3.b odd 2 1
252.3.g.b 12 12.b even 2 1
588.3.g.d 12 7.b odd 2 1
588.3.g.d 12 28.d even 2 1
1344.3.m.e 12 8.b even 2 1
1344.3.m.e 12 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(84, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} - 4 T^{5} + \cdots + 2752)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 8305770496 \) Copy content Toggle raw display
$13$ \( (T^{6} + 12 T^{5} + \cdots - 1102016)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 20 T^{5} + \cdots + 39448000)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 4294967296 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{6} - 36 T^{5} + \cdots - 12566720)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 128544076201984 \) Copy content Toggle raw display
$37$ \( (T^{6} + 44 T^{5} + \cdots - 155225024)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 100 T^{5} + \cdots + 8374720)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 49\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{6} - 52 T^{5} + \cdots - 11239563200)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 90\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{6} - 52 T^{5} + \cdots + 44445760)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} - 156 T^{5} + \cdots + 11256899392)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 84\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{6} + 276 T^{5} + \cdots + 733352728000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 132 T^{5} + \cdots - 290193606848)^{2} \) Copy content Toggle raw display
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