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Properties

Label 57.3.h.b
Level $57$
Weight $3$
Character orbit 57.h
Analytic conductor $1.553$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [57,3,Mod(11,57)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(57, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("57.11");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 57.h (of order \(6\), 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: \(1.55313750685\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 44x^{14} + 686x^{12} + 4668x^{10} + 13913x^{8} + 18672x^{6} + 10976x^{4} + 2816x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_{6} q^{3} + ( - 2 \beta_{9} - 3 \beta_{8} - 2 \beta_1) q^{4} + (\beta_{15} + \beta_{14} + \beta_{13}) q^{5} + ( - \beta_{13} + \beta_{12} + \cdots + 2 \beta_1) q^{6}+ \cdots + (\beta_{12} + 3 \beta_{9} + \cdots + 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_{6} q^{3} + ( - 2 \beta_{9} - 3 \beta_{8} - 2 \beta_1) q^{4} + (\beta_{15} + \beta_{14} + \beta_{13}) q^{5} + ( - \beta_{13} + \beta_{12} + \cdots + 2 \beta_1) q^{6}+ \cdots + (2 \beta_{13} + 27 \beta_{12} + \cdots + 11 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{3} + 24 q^{4} - 17 q^{6} - 68 q^{7} + 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + q^{3} + 24 q^{4} - 17 q^{6} - 68 q^{7} + 25 q^{9} - 16 q^{10} + 86 q^{12} - 74 q^{13} + 10 q^{15} - 72 q^{16} + 34 q^{18} + 66 q^{19} - 12 q^{21} + 18 q^{22} + 123 q^{24} + 96 q^{25} + 4 q^{27} - 110 q^{28} - 416 q^{30} - 76 q^{31} - 123 q^{33} + 240 q^{34} + 53 q^{36} - 52 q^{37} + 144 q^{39} + 264 q^{40} - 84 q^{42} - 202 q^{43} + 304 q^{45} - 184 q^{46} + 245 q^{48} + 100 q^{49} - 42 q^{51} + 166 q^{52} - 278 q^{54} + 168 q^{55} - 28 q^{57} + 280 q^{58} + 26 q^{60} + 126 q^{61} - 108 q^{63} - 560 q^{64} + 87 q^{66} - 124 q^{67} - 116 q^{69} - 156 q^{70} - 597 q^{72} + 228 q^{73} - 406 q^{75} - 152 q^{76} - 426 q^{78} - 62 q^{79} + 313 q^{81} + 146 q^{82} + 144 q^{84} - 252 q^{85} - 16 q^{87} + 924 q^{88} + 46 q^{90} - 10 q^{91} - 226 q^{93} + 480 q^{94} + 962 q^{96} + 318 q^{97} + 183 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 44x^{14} + 686x^{12} + 4668x^{10} + 13913x^{8} + 18672x^{6} + 10976x^{4} + 2816x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{14} - 44\nu^{12} - 686\nu^{10} - 4668\nu^{8} - 13913\nu^{6} - 18672\nu^{4} - 10912\nu^{2} - 2112 ) / 128 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3623 \nu^{15} - 15546 \nu^{14} - 155232 \nu^{13} - 678080 \nu^{12} - 2304130 \nu^{11} + \cdots - 15888256 ) / 806912 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 197 \nu^{15} + 1781 \nu^{14} + 8668 \nu^{13} + 77986 \nu^{12} + 135142 \nu^{11} + 1205166 \nu^{10} + \cdots + 1909088 ) / 50432 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3623 \nu^{15} + 15546 \nu^{14} - 155232 \nu^{13} + 678080 \nu^{12} - 2304130 \nu^{11} + \cdots + 15888256 ) / 806912 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 197 \nu^{15} + 1781 \nu^{14} - 8668 \nu^{13} + 77986 \nu^{12} - 135142 \nu^{11} + 1205166 \nu^{10} + \cdots + 1909088 ) / 50432 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3869 \nu^{15} - 77038 \nu^{14} - 168912 \nu^{13} - 3363296 \nu^{12} - 2594806 \nu^{11} + \cdots - 58398848 ) / 806912 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3869 \nu^{15} + 77038 \nu^{14} - 168912 \nu^{13} + 3363296 \nu^{12} - 2594806 \nu^{11} + \cdots + 58398848 ) / 806912 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 987 \nu^{15} - 43235 \nu^{13} - 668598 \nu^{11} - 4475286 \nu^{9} - 12837431 \nu^{7} + \cdots - 25216 ) / 50432 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 24899 \nu^{15} + 1576 \nu^{14} - 1088224 \nu^{13} + 69344 \nu^{12} - 16760778 \nu^{11} + \cdots + 3328512 ) / 403456 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 55887 \nu^{15} - 62462 \nu^{14} - 2442752 \nu^{13} - 2724320 \nu^{12} - 37626034 \nu^{11} + \cdots - 42512512 ) / 806912 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 333 \nu^{15} + 250 \nu^{14} + 14496 \nu^{13} + 10912 \nu^{12} + 221654 \nu^{11} + 167660 \nu^{10} + \cdots + 194944 ) / 4096 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 34735 \nu^{15} - 13894 \nu^{14} - 1512312 \nu^{13} - 606816 \nu^{12} - 23130322 \nu^{11} + \cdots - 9997440 ) / 403456 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 96229 \nu^{15} - 30942 \nu^{14} - 4203136 \nu^{13} - 1343744 \nu^{12} - 64660774 \nu^{11} + \cdots - 10538624 ) / 806912 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 96229 \nu^{15} - 30942 \nu^{14} + 4203136 \nu^{13} - 1343744 \nu^{12} + 64660774 \nu^{11} + \cdots - 10538624 ) / 806912 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 173 \nu^{15} - 40 \nu^{14} + 7556 \nu^{13} - 1752 \nu^{12} + 116230 \nu^{11} - 27088 \nu^{10} + \cdots - 40064 ) / 1024 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{14} + 2\beta_{13} - \beta_{10} - 2\beta_{9} - \beta_{8} + 3\beta_{5} - 3\beta_{3} - \beta _1 - 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} + \beta_{14} + 2 \beta_{13} + \beta_{10} - \beta_{8} + \beta_{5} + 6 \beta_{4} + \beta_{3} + \cdots - 33 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 9 \beta_{15} + 9 \beta_{14} - 18 \beta_{13} - 4 \beta_{12} + 4 \beta_{11} + 9 \beta_{10} + 26 \beta_{9} + \cdots + 17 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 27 \beta_{15} - 3 \beta_{14} - 30 \beta_{13} - 4 \beta_{12} - 4 \beta_{11} - 27 \beta_{10} + \cdots + 435 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 111 \beta_{15} - 135 \beta_{14} + 246 \beta_{13} + 80 \beta_{12} - 80 \beta_{11} - 111 \beta_{10} + \cdots - 463 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 629 \beta_{15} - 163 \beta_{14} + 466 \beta_{13} + 40 \beta_{12} + 40 \beta_{11} + 629 \beta_{10} + \cdots - 6861 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1457 \beta_{15} + 2393 \beta_{14} - 3850 \beta_{13} - 1404 \beta_{12} + 1404 \beta_{11} + \cdots + 10553 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 12939 \beta_{15} + 5445 \beta_{14} - 7494 \beta_{13} + 580 \beta_{12} + 580 \beta_{11} + \cdots + 115779 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 20143 \beta_{15} - 44623 \beta_{14} + 64766 \beta_{13} + 24504 \beta_{12} - 24504 \beta_{11} + \cdots - 217807 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 251509 \beta_{15} - 127163 \beta_{14} + 124346 \beta_{13} - 34448 \beta_{12} - 34448 \beta_{11} + \cdots - 2029677 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 294993 \beta_{15} + 842241 \beta_{14} - 1137234 \beta_{13} - 433588 \beta_{12} + 433588 \beta_{11} + \cdots + 4284377 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 1589593 \beta_{15} + 881375 \beta_{14} - 708218 \beta_{13} + 328900 \beta_{12} + 328900 \beta_{11} + \cdots + 12140993 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 4576719 \beta_{15} - 15897687 \beta_{14} + 20474406 \beta_{13} + 7783520 \beta_{12} - 7783520 \beta_{11} + \cdots - 82223983 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 89432309 \beta_{15} - 52212019 \beta_{14} + 37220290 \beta_{13} - 22972744 \beta_{12} - 22972744 \beta_{11} + \cdots - 663026829 ) / 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 74743217 \beta_{15} + 299151881 \beta_{14} - 373895098 \beta_{13} - 141334188 \beta_{12} + \cdots + 1557133049 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(40\)
\(\chi(n)\) \(-1\) \(\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.463724i
− 3.37248i
− 2.98614i
1.27007i
1.57472i
− 0.669760i
− 0.593035i
4.31291i
− 0.463724i
3.37248i
2.98614i
− 1.27007i
− 1.57472i
0.669760i
0.593035i
− 4.31291i
−3.33349 − 1.92459i 2.86714 − 0.882896i 5.40812 + 9.36713i 5.16366 + 2.98124i −11.2568 − 2.57495i −0.122988 − 26.2369i 7.44099 − 5.06277i −11.4753 − 19.8759i
11.2 −2.40707 − 1.38972i 0.837465 + 2.88074i 1.86266 + 3.22622i −7.11641 − 4.10866i 1.98759 − 8.09799i −6.71636 0.763449i −7.59731 + 4.82503i 11.4198 + 19.7797i
11.3 −2.00605 − 1.15819i −1.67757 − 2.48712i 0.682817 + 1.18267i 2.24545 + 1.29641i 0.484722 + 6.93222i −12.9625 6.10220i −3.37152 + 8.34463i −3.00298 − 5.20132i
11.4 −0.263834 − 0.152324i 1.99941 + 2.23660i −1.95359 − 3.38372i 5.35269 + 3.09037i −0.186823 − 0.894649i 2.80180 2.40892i −1.00472 + 8.94374i −0.941479 − 1.63069i
11.5 0.263834 + 0.152324i −2.93665 − 0.613242i −1.95359 − 3.38372i −5.35269 − 3.09037i −0.681377 − 0.609118i 2.80180 − 2.40892i 8.24787 + 3.60176i −0.941479 − 1.63069i
11.6 2.00605 + 1.15819i 2.99269 + 0.209258i 0.682817 + 1.18267i −2.24545 − 1.29641i 5.76112 + 3.88589i −12.9625 − 6.10220i 8.91242 + 1.25249i −3.00298 − 5.20132i
11.7 2.40707 + 1.38972i −2.91352 + 0.715104i 1.86266 + 3.22622i 7.11641 + 4.10866i −8.00686 − 2.32769i −6.71636 − 0.763449i 7.97725 − 4.16694i 11.4198 + 19.7797i
11.8 3.33349 + 1.92459i −0.668960 − 2.92446i 5.40812 + 9.36713i −5.16366 − 2.98124i 3.39843 − 11.0362i −0.122988 26.2369i −8.10498 + 3.91270i −11.4753 − 19.8759i
26.1 −3.33349 + 1.92459i 2.86714 + 0.882896i 5.40812 − 9.36713i 5.16366 − 2.98124i −11.2568 + 2.57495i −0.122988 26.2369i 7.44099 + 5.06277i −11.4753 + 19.8759i
26.2 −2.40707 + 1.38972i 0.837465 − 2.88074i 1.86266 − 3.22622i −7.11641 + 4.10866i 1.98759 + 8.09799i −6.71636 − 0.763449i −7.59731 − 4.82503i 11.4198 − 19.7797i
26.3 −2.00605 + 1.15819i −1.67757 + 2.48712i 0.682817 − 1.18267i 2.24545 − 1.29641i 0.484722 − 6.93222i −12.9625 − 6.10220i −3.37152 − 8.34463i −3.00298 + 5.20132i
26.4 −0.263834 + 0.152324i 1.99941 − 2.23660i −1.95359 + 3.38372i 5.35269 − 3.09037i −0.186823 + 0.894649i 2.80180 − 2.40892i −1.00472 − 8.94374i −0.941479 + 1.63069i
26.5 0.263834 − 0.152324i −2.93665 + 0.613242i −1.95359 + 3.38372i −5.35269 + 3.09037i −0.681377 + 0.609118i 2.80180 2.40892i 8.24787 − 3.60176i −0.941479 + 1.63069i
26.6 2.00605 − 1.15819i 2.99269 − 0.209258i 0.682817 − 1.18267i −2.24545 + 1.29641i 5.76112 − 3.88589i −12.9625 6.10220i 8.91242 − 1.25249i −3.00298 + 5.20132i
26.7 2.40707 − 1.38972i −2.91352 − 0.715104i 1.86266 − 3.22622i 7.11641 − 4.10866i −8.00686 + 2.32769i −6.71636 0.763449i 7.97725 + 4.16694i 11.4198 − 19.7797i
26.8 3.33349 − 1.92459i −0.668960 + 2.92446i 5.40812 − 9.36713i −5.16366 + 2.98124i 3.39843 + 11.0362i −0.122988 − 26.2369i −8.10498 − 3.91270i −11.4753 + 19.8759i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.c even 3 1 inner
57.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.3.h.b ✓ 16
3.b odd 2 1 inner 57.3.h.b ✓ 16
19.c even 3 1 inner 57.3.h.b ✓ 16
57.h odd 6 1 inner 57.3.h.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.h.b ✓ 16 1.a even 1 1 trivial
57.3.h.b ✓ 16 3.b odd 2 1 inner
57.3.h.b ✓ 16 19.c even 3 1 inner
57.3.h.b ✓ 16 57.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 28 T_{2}^{14} + 546 T_{2}^{12} - 5392 T_{2}^{10} + 38779 T_{2}^{8} - 148176 T_{2}^{6} + \cdots + 3249 \) acting on \(S_{3}^{\mathrm{new}}(57, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 28 T^{14} + \cdots + 3249 \) Copy content Toggle raw display
$3$ \( T^{16} - T^{15} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 380087046144 \) Copy content Toggle raw display
$7$ \( (T^{4} + 17 T^{3} + \cdots - 30)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} + 309 T^{6} + \cdots + 1846800)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 37 T^{7} + \cdots + 152769600)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 91\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( (T^{8} - 33 T^{7} + \cdots + 16983563041)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 46\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + 19 T^{3} + \cdots - 6914)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 13 T^{3} + \cdots - 79070)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{8} + 101 T^{7} + \cdots + 41404110400)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 35\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 8273458355044)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 178714519612225)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 103465413522025)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 733563856659456)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 2171161248768)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 398212800772900)^{2} \) Copy content Toggle raw display
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