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Properties

Label 108.3.f.c
Level $108$
Weight $3$
Character orbit 108.f
Analytic conductor $2.943$
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] = [108,3,Mod(19,108)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(108, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4])) 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("108.19"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 108.f (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.94278685509\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
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: \( x^{16} - 3 x^{15} + 7 x^{14} - 30 x^{13} + 76 x^{12} - 144 x^{11} + 424 x^{10} - 912 x^{9} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - \beta_{2}) q^{4} + ( - \beta_{13} + \beta_{8} + \cdots - \beta_{2}) q^{5} + (\beta_{14} - \beta_{11} + \cdots + \beta_{6}) q^{7} + ( - \beta_{13} - \beta_{11} - \beta_{9} + \cdots + 4) q^{8}+ \cdots + (16 \beta_{15} - \beta_{13} + \cdots - 86) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 3 q^{2} - 5 q^{4} - 6 q^{5} + 54 q^{8} + 20 q^{10} - 46 q^{13} + 12 q^{14} - 17 q^{16} - 12 q^{17} - 36 q^{20} + 33 q^{22} - 30 q^{25} - 72 q^{26} + 12 q^{28} - 42 q^{29} - 87 q^{32} + 11 q^{34}+ \cdots - 1170 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 7 x^{14} - 30 x^{13} + 76 x^{12} - 144 x^{11} + 424 x^{10} - 912 x^{9} + \cdots + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} + 17 \nu^{14} + 83 \nu^{13} - 394 \nu^{12} + 204 \nu^{11} - 2224 \nu^{10} + 6280 \nu^{9} + \cdots - 1228800 ) / 540672 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} + 17 \nu^{14} + 83 \nu^{13} - 394 \nu^{12} + 204 \nu^{11} - 2224 \nu^{10} + 6280 \nu^{9} + \cdots - 1228800 ) / 540672 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13 \nu^{15} + 45 \nu^{14} + 375 \nu^{13} + 158 \nu^{12} - 692 \nu^{11} - 2512 \nu^{10} + \cdots + 786432 ) / 540672 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5 \nu^{15} - 19 \nu^{14} + 91 \nu^{13} - 32 \nu^{12} + 520 \nu^{11} - 1464 \nu^{10} + \cdots + 16384 ) / 135168 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7 \nu^{15} + 97 \nu^{14} - 211 \nu^{13} + 366 \nu^{12} - 1454 \nu^{11} + 3572 \nu^{10} + \cdots + 679936 ) / 135168 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 25 \nu^{15} - 7 \nu^{14} + 15 \nu^{13} - 72 \nu^{12} + 136 \nu^{11} - 280 \nu^{10} + \cdots - 819200 ) / 270336 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 61 \nu^{15} + 371 \nu^{14} - 839 \nu^{13} + 1506 \nu^{12} - 5404 \nu^{11} + 13168 \nu^{10} + \cdots - 16384 ) / 540672 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 67 \nu^{15} + 181 \nu^{14} - 545 \nu^{13} + 2374 \nu^{12} - 5220 \nu^{11} + 11728 \nu^{10} + \cdots + 4472832 ) / 540672 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 7 \nu^{15} - 23 \nu^{14} + 11 \nu^{13} + 118 \nu^{12} + 156 \nu^{11} - 176 \nu^{10} + \cdots + 393216 ) / 49152 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 47 \nu^{15} + 125 \nu^{14} - 73 \nu^{13} + 786 \nu^{12} - 1844 \nu^{11} + 2272 \nu^{10} + \cdots + 892928 ) / 270336 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 25 \nu^{15} + 37 \nu^{14} - 95 \nu^{13} + 610 \nu^{12} - 942 \nu^{11} + 1612 \nu^{10} + \cdots + 712704 ) / 135168 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 26 \nu^{15} - 119 \nu^{14} + 365 \nu^{13} - 1015 \nu^{12} + 2642 \nu^{11} - 6300 \nu^{10} + \cdots - 1220608 ) / 135168 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 135 \nu^{15} + 257 \nu^{14} - 205 \nu^{13} + 2414 \nu^{12} - 4484 \nu^{11} + 4912 \nu^{10} + \cdots + 2965504 ) / 540672 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 137 \nu^{15} + 487 \nu^{14} - 811 \nu^{13} + 3994 \nu^{12} - 8940 \nu^{11} + 17456 \nu^{10} + \cdots + 3080192 ) / 540672 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} - \beta_{11} - \beta_{9} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + \beta_{14} - 3\beta_{12} - \beta_{11} + 2\beta_{10} + \beta_{6} + 4\beta_{2} + 3\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5 \beta_{14} + \beta_{13} + \beta_{12} - 6 \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{6} + \cdots - 9 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 7 \beta_{15} - 11 \beta_{13} + \beta_{11} - 2 \beta_{10} - 6 \beta_{9} - 2 \beta_{8} + 5 \beta_{7} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5 \beta_{15} - 17 \beta_{14} - 13 \beta_{12} + 17 \beta_{11} + 22 \beta_{10} + 11 \beta_{6} + 2 \beta_{4} + \cdots - 48 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 23 \beta_{14} + 35 \beta_{13} + 43 \beta_{12} - 30 \beta_{9} - 22 \beta_{8} + 43 \beta_{7} + \cdots - 83 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 5 \beta_{15} - 5 \beta_{13} + 15 \beta_{11} - 70 \beta_{10} + 82 \beta_{9} - 70 \beta_{8} + 3 \beta_{7} + \cdots - 80 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 83 \beta_{15} - 247 \beta_{14} - 43 \beta_{12} + 247 \beta_{11} + 10 \beta_{10} - 67 \beta_{6} + \cdots + 176 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 95 \beta_{14} + 245 \beta_{13} + 269 \beta_{12} - 98 \beta_{9} + 134 \beta_{8} + 269 \beta_{7} + \cdots + 123 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 637 \beta_{15} + 365 \beta_{13} - 583 \beta_{11} - 490 \beta_{10} + 942 \beta_{9} - 490 \beta_{8} + \cdots - 944 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 949 \beta_{15} - 337 \beta_{14} - 893 \beta_{12} + 337 \beta_{11} - 730 \beta_{10} - 1253 \beta_{6} + \cdots + 3408 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1111 \beta_{14} + 739 \beta_{13} - 1525 \beta_{12} - 654 \beta_{9} + 2506 \beta_{8} - 1525 \beta_{7} + \cdots + 3245 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2459 \beta_{15} + 1131 \beta_{13} - 3137 \beta_{11} - 1478 \beta_{10} + 1698 \beta_{9} - 1478 \beta_{8} + \cdots - 13136 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(55\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−1.59523 − 1.20633i
−1.26364 + 1.55023i
−0.710719 + 1.86946i
−0.523926 − 1.93016i
0.186266 − 1.99131i
1.63139 − 1.15696i
1.84233 + 0.778342i
1.93353 − 0.511345i
−1.59523 + 1.20633i
−1.26364 − 1.55023i
−0.710719 − 1.86946i
−0.523926 + 1.93016i
0.186266 + 1.99131i
1.63139 + 1.15696i
1.84233 − 0.778342i
1.93353 + 0.511345i
−1.59523 − 1.20633i 0 1.08951 + 3.84876i −1.10093 − 1.90686i 0 −7.23844 − 4.17912i 2.90487 − 7.45397i 0 −0.544081 + 4.36996i
19.2 −1.26364 + 1.55023i 0 −0.806428 − 3.91787i −1.35609 − 2.34881i 0 10.0431 + 5.79837i 7.09263 + 3.70062i 0 5.35481 + 0.865806i
19.3 −0.710719 + 1.86946i 0 −2.98976 − 2.65732i −1.35609 − 2.34881i 0 −10.0431 − 5.79837i 7.09263 − 3.70062i 0 5.35481 − 0.865806i
19.4 −0.523926 − 1.93016i 0 −3.45100 + 2.02252i 4.03104 + 6.98197i 0 3.90254 + 2.25313i 5.71184 + 5.60133i 0 11.3643 − 11.4386i
19.5 0.186266 − 1.99131i 0 −3.93061 − 0.741826i −3.07403 − 5.32438i 0 0.511543 + 0.295340i −2.20934 + 7.68888i 0 −11.1751 + 5.12959i
19.6 1.63139 − 1.15696i 0 1.32286 − 3.77492i −3.07403 − 5.32438i 0 −0.511543 − 0.295340i −2.20934 − 7.68888i 0 −11.1751 − 5.12959i
19.7 1.84233 + 0.778342i 0 2.78837 + 2.86793i −1.10093 − 1.90686i 0 7.23844 + 4.17912i 2.90487 + 7.45397i 0 −0.544081 − 4.36996i
19.8 1.93353 − 0.511345i 0 3.47705 − 1.97740i 4.03104 + 6.98197i 0 −3.90254 − 2.25313i 5.71184 − 5.60133i 0 11.3643 + 11.4386i
91.1 −1.59523 + 1.20633i 0 1.08951 − 3.84876i −1.10093 + 1.90686i 0 −7.23844 + 4.17912i 2.90487 + 7.45397i 0 −0.544081 − 4.36996i
91.2 −1.26364 − 1.55023i 0 −0.806428 + 3.91787i −1.35609 + 2.34881i 0 10.0431 − 5.79837i 7.09263 − 3.70062i 0 5.35481 − 0.865806i
91.3 −0.710719 − 1.86946i 0 −2.98976 + 2.65732i −1.35609 + 2.34881i 0 −10.0431 + 5.79837i 7.09263 + 3.70062i 0 5.35481 + 0.865806i
91.4 −0.523926 + 1.93016i 0 −3.45100 − 2.02252i 4.03104 − 6.98197i 0 3.90254 − 2.25313i 5.71184 − 5.60133i 0 11.3643 + 11.4386i
91.5 0.186266 + 1.99131i 0 −3.93061 + 0.741826i −3.07403 + 5.32438i 0 0.511543 − 0.295340i −2.20934 − 7.68888i 0 −11.1751 − 5.12959i
91.6 1.63139 + 1.15696i 0 1.32286 + 3.77492i −3.07403 + 5.32438i 0 −0.511543 + 0.295340i −2.20934 + 7.68888i 0 −11.1751 + 5.12959i
91.7 1.84233 − 0.778342i 0 2.78837 − 2.86793i −1.10093 + 1.90686i 0 7.23844 − 4.17912i 2.90487 − 7.45397i 0 −0.544081 + 4.36996i
91.8 1.93353 + 0.511345i 0 3.47705 + 1.97740i 4.03104 − 6.98197i 0 −3.90254 + 2.25313i 5.71184 + 5.60133i 0 11.3643 − 11.4386i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.3.f.c 16
3.b odd 2 1 36.3.f.c ✓ 16
4.b odd 2 1 inner 108.3.f.c 16
8.b even 2 1 1728.3.o.g 16
8.d odd 2 1 1728.3.o.g 16
9.c even 3 1 inner 108.3.f.c 16
9.c even 3 1 324.3.d.g 8
9.d odd 6 1 36.3.f.c ✓ 16
9.d odd 6 1 324.3.d.i 8
12.b even 2 1 36.3.f.c ✓ 16
24.f even 2 1 576.3.o.g 16
24.h odd 2 1 576.3.o.g 16
36.f odd 6 1 inner 108.3.f.c 16
36.f odd 6 1 324.3.d.g 8
36.h even 6 1 36.3.f.c ✓ 16
36.h even 6 1 324.3.d.i 8
72.j odd 6 1 576.3.o.g 16
72.l even 6 1 576.3.o.g 16
72.n even 6 1 1728.3.o.g 16
72.p odd 6 1 1728.3.o.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.f.c ✓ 16 3.b odd 2 1
36.3.f.c ✓ 16 9.d odd 6 1
36.3.f.c ✓ 16 12.b even 2 1
36.3.f.c ✓ 16 36.h even 6 1
108.3.f.c 16 1.a even 1 1 trivial
108.3.f.c 16 4.b odd 2 1 inner
108.3.f.c 16 9.c even 3 1 inner
108.3.f.c 16 36.f odd 6 1 inner
324.3.d.g 8 9.c even 3 1
324.3.d.g 8 36.f odd 6 1
324.3.d.i 8 9.d odd 6 1
324.3.d.i 8 36.h even 6 1
576.3.o.g 16 24.f even 2 1
576.3.o.g 16 24.h odd 2 1
576.3.o.g 16 72.j odd 6 1
576.3.o.g 16 72.l even 6 1
1728.3.o.g 16 8.b even 2 1
1728.3.o.g 16 8.d odd 2 1
1728.3.o.g 16 72.n even 6 1
1728.3.o.g 16 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(108, [\chi])\):

\( T_{5}^{8} + 3T_{5}^{7} + 62T_{5}^{6} + 351T_{5}^{5} + 3870T_{5}^{4} + 15291T_{5}^{3} + 49337T_{5}^{2} + 75480T_{5} + 87616 \) Copy content Toggle raw display
\( T_{7}^{16} - 225 T_{7}^{14} + 37002 T_{7}^{12} - 2674161 T_{7}^{10} + 141530490 T_{7}^{8} + \cdots + 4430766096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 3 T^{15} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 3 T^{7} + \cdots + 87616)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 4430766096 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 134421415700625 \) Copy content Toggle raw display
$13$ \( (T^{8} + 23 T^{7} + \cdots + 12100)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 3 T^{3} + \cdots + 2200)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1215 T^{6} + \cdots + 7464960000)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{8} + 21 T^{7} + \cdots + 3658072324)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{4} - 14 T^{3} + \cdots + 5920)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 42 T^{7} + \cdots + 12391919761)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 14\!\cdots\!41 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{4} - 18 T^{3} + \cdots - 16160)^{4} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 933144135518464)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 15\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 29 T^{3} + \cdots + 20112040)^{4} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 36\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} - 96 T^{3} + \cdots - 4957424)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 74 T^{7} + \cdots + 4182855625)^{2} \) Copy content Toggle raw display
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