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Properties

Label 1008.2.df.c
Level $1008$
Weight $2$
Character orbit 1008.df
Analytic conductor $8.049$
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] = [1008,2,Mod(689,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.689");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.df (of order \(6\), degree \(2\), not 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: \(8.04892052375\)
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: \( x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 126)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} - \beta_{11} q^{5} + ( - \beta_{10} - \beta_{8} + \beta_{4} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{14} + \beta_{13} - \beta_{11} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} - \beta_{11} q^{5} + ( - \beta_{10} - \beta_{8} + \beta_{4} + \cdots + 1) q^{7}+ \cdots + (\beta_{15} - 3 \beta_{14} + 2 \beta_{13} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{7} - 6 q^{9} - 6 q^{13} + 18 q^{15} + 18 q^{17} - 18 q^{21} + 16 q^{25} + 36 q^{27} + 6 q^{29} - 6 q^{31} + 18 q^{33} + 30 q^{35} - 2 q^{37} + 30 q^{39} + 6 q^{41} + 2 q^{43} + 12 q^{45} + 18 q^{47} + 10 q^{49} + 36 q^{53} + 6 q^{57} - 30 q^{59} - 60 q^{61} - 42 q^{63} + 42 q^{65} - 14 q^{67} + 42 q^{69} - 30 q^{75} - 18 q^{77} + 16 q^{79} + 54 q^{81} - 12 q^{85} + 48 q^{87} + 24 q^{89} + 12 q^{91} + 30 q^{93} + 66 q^{95} - 6 q^{97} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 154 \nu^{15} + 1325 \nu^{14} - 3608 \nu^{13} + 224 \nu^{12} + 22478 \nu^{11} - 55022 \nu^{10} + \cdots + 1285227 ) / 47385 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1342 \nu^{15} - 9134 \nu^{14} + 18833 \nu^{13} + 17821 \nu^{12} - 164858 \nu^{11} + 301448 \nu^{10} + \cdots - 6445089 ) / 142155 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2846 \nu^{15} - 22369 \nu^{14} + 55246 \nu^{13} + 17972 \nu^{12} - 402586 \nu^{11} + \cdots - 20783061 ) / 142155 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2782 \nu^{15} - 15918 \nu^{14} + 26947 \nu^{13} + 42629 \nu^{12} - 270897 \nu^{11} + 425335 \nu^{10} + \cdots - 7405182 ) / 47385 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5666 \nu^{15} + 35256 \nu^{14} - 67793 \nu^{13} - 74626 \nu^{12} + 609708 \nu^{11} + \cdots + 21264930 ) / 47385 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 432 \nu^{15} + 2816 \nu^{14} - 5740 \nu^{13} - 5150 \nu^{12} + 49010 \nu^{11} - 90874 \nu^{10} + \cdots + 1853118 ) / 3645 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4120 \nu^{15} - 25571 \nu^{14} + 48788 \nu^{13} + 55006 \nu^{12} - 441224 \nu^{11} + 771188 \nu^{10} + \cdots - 14963454 ) / 28431 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20555 \nu^{15} - 129232 \nu^{14} + 250807 \nu^{13} + 267434 \nu^{12} - 2233957 \nu^{11} + \cdots - 77795964 ) / 142155 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 21860 \nu^{15} + 133486 \nu^{14} - 249901 \nu^{13} - 298607 \nu^{12} + 2298061 \nu^{11} + \cdots + 75541167 ) / 142155 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2015 \nu^{15} - 12538 \nu^{14} + 24088 \nu^{13} + 26576 \nu^{12} - 216643 \nu^{11} + 381184 \nu^{10} + \cdots - 7453296 ) / 10935 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 29357 \nu^{15} - 190006 \nu^{14} + 382390 \nu^{13} + 360905 \nu^{12} - 3304405 \nu^{11} + \cdots - 122535423 ) / 142155 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11745 \nu^{15} + 73063 \nu^{14} - 139508 \nu^{13} - 157621 \nu^{12} + 1262908 \nu^{11} + \cdots + 42448941 ) / 47385 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 45758 \nu^{15} + 290392 \nu^{14} - 571483 \nu^{13} - 583571 \nu^{12} + 5039353 \nu^{11} + \cdots + 181431333 ) / 142155 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 17267 \nu^{15} + 108242 \nu^{14} - 209241 \nu^{13} - 227007 \nu^{12} + 1874096 \nu^{11} + \cdots + 65023155 ) / 47385 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 56402 \nu^{15} + 348301 \nu^{14} - 661000 \nu^{13} - 757115 \nu^{12} + 6004885 \nu^{11} + \cdots + 200862828 ) / 142155 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{14} - \beta_{10} + \beta_{9} - \beta_{7} + 2\beta_{4} + \beta_{3} - 2\beta_{2} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{15} - \beta_{12} - \beta_{10} - \beta_{9} + 2 \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{5} + \cdots + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 4 \beta_{12} + \beta_{11} - 2 \beta_{10} - \beta_{9} + \cdots - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3 \beta_{15} + 10 \beta_{14} - 4 \beta_{13} - 6 \beta_{12} + 5 \beta_{11} + 4 \beta_{10} - 3 \beta_{9} + \cdots + 11 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3 \beta_{15} - 2 \beta_{14} - 6 \beta_{13} - 6 \beta_{12} + 3 \beta_{11} - 8 \beta_{10} - 4 \beta_{9} + \cdots - 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 7 \beta_{15} - 3 \beta_{14} + 4 \beta_{13} - 4 \beta_{12} - 2 \beta_{11} - 10 \beta_{10} + 7 \beta_{9} + \cdots - 31 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6 \beta_{15} - 35 \beta_{14} + \beta_{13} - 18 \beta_{12} - 38 \beta_{11} - 59 \beta_{10} + \beta_{9} + \cdots - 85 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 10 \beta_{15} + 35 \beta_{14} - 18 \beta_{13} - 40 \beta_{12} - 48 \beta_{11} + 7 \beta_{10} + \cdots - 36 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 32 \beta_{14} - 126 \beta_{13} - 72 \beta_{12} - 90 \beta_{11} - 46 \beta_{10} - 20 \beta_{9} + \cdots + 64 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 90 \beta_{15} + 171 \beta_{14} - 167 \beta_{13} - 36 \beta_{12} - 5 \beta_{11} + 153 \beta_{10} + \cdots - 225 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 100 \beta_{15} + 83 \beta_{14} - 141 \beta_{13} - 113 \beta_{12} - 90 \beta_{11} - 27 \beta_{10} + \cdots - 325 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 251 \beta_{15} + 97 \beta_{14} + 298 \beta_{13} - 319 \beta_{12} - 128 \beta_{11} + 867 \beta_{10} + \cdots - 845 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1440 \beta_{15} - 170 \beta_{14} + 597 \beta_{13} - 1254 \beta_{12} - 999 \beta_{11} + 1288 \beta_{10} + \cdots + 1901 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 835 \beta_{15} - 918 \beta_{14} + 882 \beta_{13} - 1877 \beta_{12} - 1842 \beta_{11} + 2872 \beta_{10} + \cdots - 1013 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1012 \beta_{15} + 838 \beta_{14} + 431 \beta_{13} - 2534 \beta_{12} - 3259 \beta_{11} + 1145 \beta_{10} + \cdots + 561 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1 + \beta_{7}\) \(1\) \(-\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
689.1
0.765614 + 1.55365i
0.320287 − 1.70218i
1.27866 + 1.16834i
1.58110 − 0.707199i
1.71298 − 0.256290i
−1.68301 − 0.409224i
−1.70672 + 0.295146i
1.73109 + 0.0577511i
0.765614 − 1.55365i
0.320287 + 1.70218i
1.27866 − 1.16834i
1.58110 + 0.707199i
1.71298 + 0.256290i
−1.68301 + 0.409224i
−1.70672 − 0.295146i
1.73109 − 0.0577511i
0 −1.47073 − 0.914855i 0 −3.64414 0 −2.62749 + 0.310282i 0 1.32608 + 2.69100i 0
689.2 0 −1.33318 + 1.10572i 0 −0.0676069 0 2.64192 + 0.142361i 0 0.554753 − 2.94826i 0
689.3 0 −0.626615 + 1.61473i 0 3.55225 0 1.49384 − 2.18368i 0 −2.21471 − 2.02363i 0
689.4 0 −0.361565 − 1.69389i 0 −0.900258 0 −1.05755 − 2.42520i 0 −2.73854 + 1.22490i 0
689.5 0 0.128499 + 1.72728i 0 −3.61932 0 −0.266972 + 2.63225i 0 −2.96698 + 0.443907i 0
689.6 0 0.206076 − 1.71975i 0 1.42985 0 2.43739 + 1.02913i 0 −2.91507 − 0.708796i 0
689.7 0 1.72571 − 0.148116i 0 0.967324 0 −2.40137 + 1.11060i 0 2.95612 − 0.511208i 0
689.8 0 1.73181 + 0.0288796i 0 2.28190 0 −1.21977 − 2.34780i 0 2.99833 + 0.100028i 0
929.1 0 −1.47073 + 0.914855i 0 −3.64414 0 −2.62749 − 0.310282i 0 1.32608 − 2.69100i 0
929.2 0 −1.33318 − 1.10572i 0 −0.0676069 0 2.64192 − 0.142361i 0 0.554753 + 2.94826i 0
929.3 0 −0.626615 − 1.61473i 0 3.55225 0 1.49384 + 2.18368i 0 −2.21471 + 2.02363i 0
929.4 0 −0.361565 + 1.69389i 0 −0.900258 0 −1.05755 + 2.42520i 0 −2.73854 − 1.22490i 0
929.5 0 0.128499 − 1.72728i 0 −3.61932 0 −0.266972 − 2.63225i 0 −2.96698 − 0.443907i 0
929.6 0 0.206076 + 1.71975i 0 1.42985 0 2.43739 − 1.02913i 0 −2.91507 + 0.708796i 0
929.7 0 1.72571 + 0.148116i 0 0.967324 0 −2.40137 − 1.11060i 0 2.95612 + 0.511208i 0
929.8 0 1.73181 − 0.0288796i 0 2.28190 0 −1.21977 + 2.34780i 0 2.99833 − 0.100028i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 689.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.df.c 16
3.b odd 2 1 3024.2.df.c 16
4.b odd 2 1 126.2.t.a yes 16
7.d odd 6 1 1008.2.ca.c 16
9.c even 3 1 3024.2.ca.c 16
9.d odd 6 1 1008.2.ca.c 16
12.b even 2 1 378.2.t.a 16
21.g even 6 1 3024.2.ca.c 16
28.d even 2 1 882.2.t.a 16
28.f even 6 1 126.2.l.a ✓ 16
28.f even 6 1 882.2.m.b 16
28.g odd 6 1 882.2.l.b 16
28.g odd 6 1 882.2.m.a 16
36.f odd 6 1 378.2.l.a 16
36.f odd 6 1 1134.2.k.b 16
36.h even 6 1 126.2.l.a ✓ 16
36.h even 6 1 1134.2.k.a 16
63.k odd 6 1 3024.2.df.c 16
63.s even 6 1 inner 1008.2.df.c 16
84.h odd 2 1 2646.2.t.b 16
84.j odd 6 1 378.2.l.a 16
84.j odd 6 1 2646.2.m.b 16
84.n even 6 1 2646.2.l.a 16
84.n even 6 1 2646.2.m.a 16
252.n even 6 1 378.2.t.a 16
252.o even 6 1 882.2.t.a 16
252.r odd 6 1 882.2.m.a 16
252.r odd 6 1 1134.2.k.b 16
252.s odd 6 1 882.2.l.b 16
252.u odd 6 1 2646.2.m.b 16
252.bb even 6 1 882.2.m.b 16
252.bi even 6 1 2646.2.l.a 16
252.bj even 6 1 1134.2.k.a 16
252.bj even 6 1 2646.2.m.a 16
252.bl odd 6 1 2646.2.t.b 16
252.bn odd 6 1 126.2.t.a yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.l.a ✓ 16 28.f even 6 1
126.2.l.a ✓ 16 36.h even 6 1
126.2.t.a yes 16 4.b odd 2 1
126.2.t.a yes 16 252.bn odd 6 1
378.2.l.a 16 36.f odd 6 1
378.2.l.a 16 84.j odd 6 1
378.2.t.a 16 12.b even 2 1
378.2.t.a 16 252.n even 6 1
882.2.l.b 16 28.g odd 6 1
882.2.l.b 16 252.s odd 6 1
882.2.m.a 16 28.g odd 6 1
882.2.m.a 16 252.r odd 6 1
882.2.m.b 16 28.f even 6 1
882.2.m.b 16 252.bb even 6 1
882.2.t.a 16 28.d even 2 1
882.2.t.a 16 252.o even 6 1
1008.2.ca.c 16 7.d odd 6 1
1008.2.ca.c 16 9.d odd 6 1
1008.2.df.c 16 1.a even 1 1 trivial
1008.2.df.c 16 63.s even 6 1 inner
1134.2.k.a 16 36.h even 6 1
1134.2.k.a 16 252.bj even 6 1
1134.2.k.b 16 36.f odd 6 1
1134.2.k.b 16 252.r odd 6 1
2646.2.l.a 16 84.n even 6 1
2646.2.l.a 16 252.bi even 6 1
2646.2.m.a 16 84.n even 6 1
2646.2.m.a 16 252.bj even 6 1
2646.2.m.b 16 84.j odd 6 1
2646.2.m.b 16 252.u odd 6 1
2646.2.t.b 16 84.h odd 2 1
2646.2.t.b 16 252.bl odd 6 1
3024.2.ca.c 16 9.c even 3 1
3024.2.ca.c 16 21.g even 6 1
3024.2.df.c 16 3.b odd 2 1
3024.2.df.c 16 63.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 24T_{5}^{6} + 12T_{5}^{5} + 153T_{5}^{4} - 162T_{5}^{3} - 117T_{5}^{2} + 126T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 3 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T^{8} - 24 T^{6} + 12 T^{5} + \cdots + 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 2 T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} + 108 T^{14} + \cdots + 61732449 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 390971529 \) Copy content Toggle raw display
$17$ \( T^{16} - 18 T^{15} + \cdots + 56070144 \) Copy content Toggle raw display
$19$ \( T^{16} - 72 T^{14} + \cdots + 9199089 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 187388721 \) Copy content Toggle raw display
$29$ \( T^{16} - 6 T^{15} + \cdots + 1108809 \) Copy content Toggle raw display
$31$ \( T^{16} + 6 T^{15} + \cdots + 65610000 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 32746159681 \) Copy content Toggle raw display
$41$ \( T^{16} - 6 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 2999643361 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 588203099136 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 36759242529 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 216504090000 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 547560000 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 2603856784 \) Copy content Toggle raw display
$71$ \( T^{16} + 486 T^{14} + \cdots + 65610000 \) Copy content Toggle raw display
$73$ \( T^{16} - 150 T^{14} + \cdots + 71115489 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 970422010000 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 953512641 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 131145120363321 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 9120206721024 \) Copy content Toggle raw display
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