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Properties

Label 51.6.d.a
Level $51$
Weight $6$
Character orbit 51.d
Analytic conductor $8.180$
Analytic rank $0$
Dimension $16$
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] = [51,6,Mod(16,51)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(51, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("51.16");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 51 = 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 51.d (of order \(2\), degree \(1\), 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: \(8.17957481046\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 390 x^{14} + 59223 x^{12} + 4487096 x^{10} + 181150959 x^{8} + 3878696478 x^{6} + \cdots + 440683545600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + \beta_{9} q^{3} + ( - \beta_{3} - \beta_{2} + 16) q^{4} + (\beta_{9} - \beta_{8}) q^{5} + (\beta_{9} - \beta_1) q^{6} + ( - \beta_{12} - 2 \beta_{9} + \beta_{8}) q^{7} + ( - \beta_{5} - 21 \beta_{3} + \cdots + 13) q^{8}+ \cdots - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + \beta_{9} q^{3} + ( - \beta_{3} - \beta_{2} + 16) q^{4} + (\beta_{9} - \beta_{8}) q^{5} + (\beta_{9} - \beta_1) q^{6} + ( - \beta_{12} - 2 \beta_{9} + \beta_{8}) q^{7} + ( - \beta_{5} - 21 \beta_{3} + \cdots + 13) q^{8}+ \cdots + ( - 81 \beta_{14} + 81 \beta_{12} + \cdots - 81 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} + 268 q^{4} + 384 q^{8} - 1296 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{2} + 268 q^{4} + 384 q^{8} - 1296 q^{9} + 566 q^{13} - 702 q^{15} + 7732 q^{16} + 1562 q^{17} - 648 q^{18} - 4654 q^{19} + 1764 q^{21} - 15270 q^{25} - 14348 q^{26} + 14688 q^{30} + 17896 q^{32} + 2502 q^{33} + 9768 q^{34} + 47796 q^{35} - 21708 q^{36} - 19412 q^{38} - 41436 q^{42} - 8938 q^{43} - 21988 q^{47} + 15784 q^{49} - 175124 q^{50} - 10458 q^{51} + 86984 q^{52} + 37368 q^{53} + 72694 q^{55} - 78936 q^{59} + 38772 q^{60} + 311188 q^{64} - 12132 q^{66} - 230752 q^{67} + 278392 q^{68} + 111438 q^{69} - 53760 q^{70} - 31104 q^{72} - 480328 q^{76} - 247348 q^{77} + 104976 q^{81} + 267468 q^{83} - 129420 q^{84} + 319478 q^{85} + 73132 q^{86} + 49140 q^{87} + 199216 q^{89} - 96084 q^{93} - 658344 q^{94} + 906096 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 390 x^{14} + 59223 x^{12} + 4487096 x^{10} + 181150959 x^{8} + 3878696478 x^{6} + \cdots + 440683545600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 9\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 30194471 \nu^{14} + 11586831899 \nu^{12} + 1715556987678 \nu^{10} + 124707599299610 \nu^{8} + \cdots + 21\!\cdots\!56 ) / 233946320965632 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 30194471 \nu^{14} + 11586831899 \nu^{12} + 1715556987678 \nu^{10} + 124707599299610 \nu^{8} + \cdots + 20\!\cdots\!88 ) / 233946320965632 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10780333279 \nu^{14} + 4132007317235 \nu^{12} + 610627214362126 \nu^{10} + \cdots + 70\!\cdots\!04 ) / 42\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1158564651 \nu^{14} - 444528439631 \nu^{12} - 65806669615334 \nu^{10} + \cdots - 80\!\cdots\!96 ) / 116973160482816 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 94599279259 \nu^{14} + 36290292759551 \nu^{12} + \cdots + 65\!\cdots\!64 ) / 42\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 167200887619 \nu^{14} - 64140993516455 \nu^{12} + \cdots - 11\!\cdots\!48 ) / 42\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6614603396347 \nu^{15} + \cdots - 47\!\cdots\!32 \nu ) / 87\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 32934189053 \nu^{15} + 12635538963705 \nu^{13} + \cdots + 22\!\cdots\!08 \nu ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 7092260304884 \nu^{15} + 517859706971655 \nu^{14} + \cdots + 35\!\cdots\!40 ) / 29\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1394108218429 \nu^{15} + 534855551557785 \nu^{13} + \cdots + 96\!\cdots\!84 \nu ) / 80\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 215363264023889 \nu^{15} + \cdots + 14\!\cdots\!04 \nu ) / 87\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 255646315316963 \nu^{15} + \cdots + 17\!\cdots\!68 \nu ) / 87\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 104945753761237 \nu^{15} + \cdots - 72\!\cdots\!12 \nu ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 175712051102603 \nu^{15} + \cdots - 12\!\cdots\!28 \nu ) / 29\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} - 49 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{13} - \beta_{12} + 8\beta_{11} - 45\beta_{9} + 2\beta_{8} - 86\beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 3\beta_{5} - \beta_{4} + 199\beta_{3} - 123\beta_{2} + 4276 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 72 \beta_{15} + 45 \beta_{14} - 13 \beta_{13} + 509 \beta_{12} - 1354 \beta_{11} + 9390 \beta_{9} + \cdots + 8649 \beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16 \beta_{15} - 16 \beta_{14} + 16 \beta_{12} - 16 \beta_{11} - 32 \beta_{10} - 170 \beta_{6} + \cdots - 449225 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 12348 \beta_{15} - 11160 \beta_{14} + 19155 \beta_{13} - 91185 \beta_{12} + 192900 \beta_{11} + \cdots - 955942 \beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3644 \beta_{15} + 3644 \beta_{14} - 3644 \beta_{12} + 3644 \beta_{11} + 7288 \beta_{10} + \cdots + 51245860 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1461348 \beta_{15} + 1947105 \beta_{14} - 4377629 \beta_{13} + 13086025 \beta_{12} + \cdots + 111776537 \beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 594476 \beta_{15} - 594476 \beta_{14} + 594476 \beta_{12} - 594476 \beta_{11} - 1188952 \beta_{10} + \cdots - 6116578489 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 146781576 \beta_{15} - 296509644 \beta_{14} + 764292787 \beta_{13} - 1736641757 \beta_{12} + \cdots - 13521578262 \beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 86182248 \beta_{15} + 86182248 \beta_{14} - 86182248 \beta_{12} + 86182248 \beta_{11} + \cdots + 749895470900 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 13140350208 \beta_{15} + 42299910405 \beta_{14} - 118959889245 \beta_{13} + 223684184901 \beta_{12} + \cdots + 1670621546281 \beta_1 ) / 9 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 11863221832 \beta_{15} - 11863221832 \beta_{14} + 11863221832 \beta_{12} - 11863221832 \beta_{11} + \cdots - 93486361397737 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 1031418582804 \beta_{15} - 5826297048768 \beta_{14} + 17395118599379 \beta_{13} + \cdots - 209251753519814 \beta_1 ) / 9 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/51\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(37\)
\(\chi(n)\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
− 11.3826i
11.3826i
− 7.85248i
7.85248i
− 5.09078i
5.09078i
− 2.52066i
2.52066i
2.52419i
− 2.52419i
3.20181i
− 3.20181i
7.27046i
− 7.27046i
9.85003i
− 9.85003i
−10.3826 − 9.00000i 75.7977 − 30.3750i 93.4431i 146.432i −454.732 −81.0000 315.370i
16.2 −10.3826 9.00000i 75.7977 30.3750i − 93.4431i − 146.432i −454.732 −81.0000 − 315.370i
16.3 −6.85248 − 9.00000i 14.9565 − 17.1775i 61.6723i − 167.890i 116.790 −81.0000 117.709i
16.4 −6.85248 9.00000i 14.9565 17.1775i − 61.6723i 167.890i 116.790 −81.0000 − 117.709i
16.5 −4.09078 − 9.00000i −15.2655 46.7156i 36.8170i 40.2932i 193.353 −81.0000 − 191.103i
16.6 −4.09078 9.00000i −15.2655 − 46.7156i − 36.8170i − 40.2932i 193.353 −81.0000 191.103i
16.7 −1.52066 − 9.00000i −29.6876 − 104.386i 13.6860i 239.950i 93.8060 −81.0000 158.736i
16.8 −1.52066 9.00000i −29.6876 104.386i − 13.6860i − 239.950i 93.8060 −81.0000 − 158.736i
16.9 3.52419 − 9.00000i −19.5801 78.5067i − 31.7177i 44.7605i −181.778 −81.0000 276.672i
16.10 3.52419 9.00000i −19.5801 − 78.5067i 31.7177i − 44.7605i −181.778 −81.0000 − 276.672i
16.11 4.20181 − 9.00000i −14.3448 − 10.3936i − 37.8163i − 87.0287i −194.732 −81.0000 − 43.6720i
16.12 4.20181 9.00000i −14.3448 10.3936i 37.8163i 87.0287i −194.732 −81.0000 43.6720i
16.13 8.27046 − 9.00000i 36.4005 − 78.6153i − 74.4341i − 35.4435i 36.3943 −81.0000 − 650.184i
16.14 8.27046 9.00000i 36.4005 78.6153i 74.4341i 35.4435i 36.3943 −81.0000 650.184i
16.15 10.8500 − 9.00000i 85.7232 76.7253i − 97.6503i − 83.0733i 582.899 −81.0000 832.472i
16.16 10.8500 9.00000i 85.7232 − 76.7253i 97.6503i 83.0733i 582.899 −81.0000 − 832.472i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 51.6.d.a ✓ 16
3.b odd 2 1 153.6.d.d 16
4.b odd 2 1 816.6.c.e 16
17.b even 2 1 inner 51.6.d.a ✓ 16
17.c even 4 1 867.6.a.j 8
17.c even 4 1 867.6.a.k 8
51.c odd 2 1 153.6.d.d 16
68.d odd 2 1 816.6.c.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.6.d.a ✓ 16 1.a even 1 1 trivial
51.6.d.a ✓ 16 17.b even 2 1 inner
153.6.d.d 16 3.b odd 2 1
153.6.d.d 16 51.c odd 2 1
816.6.c.e 16 4.b odd 2 1
816.6.c.e 16 68.d odd 2 1
867.6.a.j 8 17.c even 4 1
867.6.a.k 8 17.c even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(51, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 4 T^{7} + \cdots + 588096)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 81)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 15\!\cdots\!80)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 59\!\cdots\!60)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 11\!\cdots\!16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 63\!\cdots\!84)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 16\!\cdots\!80)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 10\!\cdots\!76)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
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