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Properties

Label 798.2.i.b
Level $798$
Weight $2$
Character orbit 798.i
Analytic conductor $6.372$
Analytic rank $0$
Dimension $14$
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] = [798,2,Mod(163,798)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(798, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.163");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.i (of order \(3\), 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: \(6.37206208130\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 22 x^{12} - 28 x^{11} + 347 x^{10} - 446 x^{9} + 2862 x^{8} - 4007 x^{7} + 16873 x^{6} + \cdots + 21609 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 1) q^{2} + q^{3} + \beta_{3} q^{4} - \beta_1 q^{5} + ( - \beta_{3} - 1) q^{6} + (\beta_{11} - \beta_{3}) q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 1) q^{2} + q^{3} + \beta_{3} q^{4} - \beta_1 q^{5} + ( - \beta_{3} - 1) q^{6} + (\beta_{11} - \beta_{3}) q^{7} + q^{8} + q^{9} + (\beta_{2} + \beta_1) q^{10} + (\beta_{7} - \beta_{6} + \beta_{5} + \cdots + 1) q^{11}+ \cdots + (\beta_{7} - \beta_{6} + \beta_{5} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 7 q^{2} + 14 q^{3} - 7 q^{4} - 7 q^{6} + 6 q^{7} + 14 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 7 q^{2} + 14 q^{3} - 7 q^{4} - 7 q^{6} + 6 q^{7} + 14 q^{8} + 14 q^{9} + 2 q^{11} - 7 q^{12} + q^{13} - 9 q^{14} - 7 q^{16} - 14 q^{17} - 7 q^{18} - 10 q^{19} + 6 q^{21} + 2 q^{22} - 16 q^{23} + 14 q^{24} - 9 q^{25} + q^{26} + 14 q^{27} + 3 q^{28} + 6 q^{29} - 4 q^{31} - 7 q^{32} + 2 q^{33} + 7 q^{34} + 23 q^{35} - 7 q^{36} + 3 q^{37} + 5 q^{38} + q^{39} + 11 q^{41} - 9 q^{42} + 2 q^{43} - 4 q^{44} + 8 q^{46} - 22 q^{47} - 7 q^{48} + 8 q^{49} + 18 q^{50} - 14 q^{51} - 2 q^{52} + 22 q^{53} - 7 q^{54} + 2 q^{55} + 6 q^{56} - 10 q^{57} + 6 q^{58} - 14 q^{59} + 24 q^{61} - 4 q^{62} + 6 q^{63} + 14 q^{64} - 5 q^{65} + 2 q^{66} - 4 q^{67} + 7 q^{68} - 16 q^{69} - 16 q^{70} + 36 q^{71} + 14 q^{72} - 10 q^{73} - 6 q^{74} - 9 q^{75} + 5 q^{76} - 16 q^{77} + q^{78} + 4 q^{79} + 14 q^{81} - 22 q^{82} - 18 q^{83} + 3 q^{84} - 3 q^{85} - 4 q^{86} + 6 q^{87} + 2 q^{88} + 64 q^{89} + 27 q^{91} + 8 q^{92} - 4 q^{93} + 11 q^{94} - 31 q^{95} - 7 q^{96} - 24 q^{97} - q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 22 x^{12} - 28 x^{11} + 347 x^{10} - 446 x^{9} + 2862 x^{8} - 4007 x^{7} + 16873 x^{6} + \cdots + 21609 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 69245462310286 \nu^{13} - 196873453432788 \nu^{12} + \cdots - 61\!\cdots\!21 ) / 79\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 60\!\cdots\!49 \nu^{13} + \cdots - 82\!\cdots\!65 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 65\!\cdots\!89 \nu^{13} + \cdots - 29\!\cdots\!40 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 27\!\cdots\!08 \nu^{13} + \cdots - 86\!\cdots\!14 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 32\!\cdots\!25 \nu^{13} + \cdots + 15\!\cdots\!94 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 61\!\cdots\!16 \nu^{13} + 556254183542996 \nu^{12} + \cdots + 64\!\cdots\!22 ) / 23\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 73\!\cdots\!62 \nu^{13} + \cdots + 59\!\cdots\!08 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10\!\cdots\!73 \nu^{13} + \cdots + 75\!\cdots\!76 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21\!\cdots\!37 \nu^{13} + \cdots - 57\!\cdots\!64 ) / 23\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 26\!\cdots\!59 \nu^{13} + \cdots + 16\!\cdots\!36 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 35\!\cdots\!76 \nu^{13} + \cdots + 29\!\cdots\!48 ) / 16\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 37\!\cdots\!92 \nu^{13} + \cdots + 34\!\cdots\!33 ) / 79\!\cdots\!29 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{6} + 7\beta_{3} - \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - 2\beta_{8} + 2\beta_{7} + 2\beta_{5} + 9\beta_{2} + 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{12} + 3 \beta_{11} - \beta_{10} - 3 \beta_{9} - 14 \beta_{7} + 14 \beta_{6} - 14 \beta_{5} + \cdots - 72 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 21 \beta_{13} + 12 \beta_{12} - 25 \beta_{11} - 13 \beta_{10} + 21 \beta_{9} + 45 \beta_{8} + \cdots - 101 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 74 \beta_{13} - 100 \beta_{12} + 20 \beta_{11} + 100 \beta_{10} - 208 \beta_{8} + 187 \beta_{7} + \cdots + 884 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 328 \beta_{12} + 328 \beta_{11} - 141 \beta_{10} - 367 \beta_{9} - 576 \beta_{7} + 576 \beta_{6} + \cdots - 2701 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1373 \beta_{13} + 333 \beta_{12} - 1864 \beta_{11} - 1531 \beta_{10} + 1373 \beta_{9} + \cdots - 4187 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5993 \beta_{13} - 7915 \beta_{12} + 1774 \beta_{11} + 7915 \beta_{10} - 12338 \beta_{8} + 8921 \beta_{7} + \cdots + 41815 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 26042 \beta_{12} + 26042 \beta_{11} - 5225 \beta_{10} - 22977 \beta_{9} - 37363 \beta_{7} + \cdots - 171646 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 94672 \beta_{13} + 23848 \beta_{12} - 127111 \beta_{11} - 103263 \beta_{10} + 94672 \beta_{9} + \cdots - 248181 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 366612 \beta_{13} - 499669 \beta_{12} + 79951 \beta_{11} + 499669 \beta_{10} - 749189 \beta_{8} + \cdots + 2523630 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1652611 \beta_{12} + 1652611 \beta_{11} - 336288 \beta_{10} - 1468683 \beta_{9} - 2067330 \beta_{7} + \cdots - 9587753 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/798\mathbb{Z}\right)^\times\).

\(n\) \(115\) \(211\) \(533\)
\(\chi(n)\) \(\beta_{3}\) \(-1 - \beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
1.33643 + 2.31477i
1.17085 + 2.02797i
1.16367 + 2.01553i
0.362765 + 0.628327i
−0.600600 − 1.04027i
−1.48907 − 2.57915i
−1.94404 − 3.36718i
1.33643 − 2.31477i
1.17085 − 2.02797i
1.16367 − 2.01553i
0.362765 − 0.628327i
−0.600600 + 1.04027i
−1.48907 + 2.57915i
−1.94404 + 3.36718i
−0.500000 − 0.866025i 1.00000 −0.500000 + 0.866025i −1.33643 − 2.31477i −0.500000 − 0.866025i −2.63378 + 0.251374i 1.00000 1.00000 −1.33643 + 2.31477i
163.2 −0.500000 − 0.866025i 1.00000 −0.500000 + 0.866025i −1.17085 − 2.02797i −0.500000 − 0.866025i 0.736832 + 2.54108i 1.00000 1.00000 −1.17085 + 2.02797i
163.3 −0.500000 − 0.866025i 1.00000 −0.500000 + 0.866025i −1.16367 − 2.01553i −0.500000 − 0.866025i 2.23730 − 1.41226i 1.00000 1.00000 −1.16367 + 2.01553i
163.4 −0.500000 − 0.866025i 1.00000 −0.500000 + 0.866025i −0.362765 − 0.628327i −0.500000 − 0.866025i 2.15579 − 1.53381i 1.00000 1.00000 −0.362765 + 0.628327i
163.5 −0.500000 − 0.866025i 1.00000 −0.500000 + 0.866025i 0.600600 + 1.04027i −0.500000 − 0.866025i −1.92214 − 1.81807i 1.00000 1.00000 0.600600 − 1.04027i
163.6 −0.500000 − 0.866025i 1.00000 −0.500000 + 0.866025i 1.48907 + 2.57915i −0.500000 − 0.866025i 2.38132 + 1.15296i 1.00000 1.00000 1.48907 − 2.57915i
163.7 −0.500000 − 0.866025i 1.00000 −0.500000 + 0.866025i 1.94404 + 3.36718i −0.500000 − 0.866025i 0.0446759 − 2.64537i 1.00000 1.00000 1.94404 − 3.36718i
235.1 −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i −1.33643 + 2.31477i −0.500000 + 0.866025i −2.63378 − 0.251374i 1.00000 1.00000 −1.33643 − 2.31477i
235.2 −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i −1.17085 + 2.02797i −0.500000 + 0.866025i 0.736832 − 2.54108i 1.00000 1.00000 −1.17085 − 2.02797i
235.3 −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i −1.16367 + 2.01553i −0.500000 + 0.866025i 2.23730 + 1.41226i 1.00000 1.00000 −1.16367 − 2.01553i
235.4 −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i −0.362765 + 0.628327i −0.500000 + 0.866025i 2.15579 + 1.53381i 1.00000 1.00000 −0.362765 − 0.628327i
235.5 −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i 0.600600 − 1.04027i −0.500000 + 0.866025i −1.92214 + 1.81807i 1.00000 1.00000 0.600600 + 1.04027i
235.6 −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i 1.48907 − 2.57915i −0.500000 + 0.866025i 2.38132 − 1.15296i 1.00000 1.00000 1.48907 + 2.57915i
235.7 −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i 1.94404 − 3.36718i −0.500000 + 0.866025i 0.0446759 + 2.64537i 1.00000 1.00000 1.94404 + 3.36718i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.i.b ✓ 14
7.c even 3 1 798.2.l.c yes 14
19.c even 3 1 798.2.l.c yes 14
133.g even 3 1 inner 798.2.i.b ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.i.b ✓ 14 1.a even 1 1 trivial
798.2.i.b ✓ 14 133.g even 3 1 inner
798.2.l.c yes 14 7.c even 3 1
798.2.l.c yes 14 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} + 22 T_{5}^{12} + 28 T_{5}^{11} + 347 T_{5}^{10} + 446 T_{5}^{9} + 2862 T_{5}^{8} + \cdots + 21609 \) acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{7} \) Copy content Toggle raw display
$3$ \( (T - 1)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 22 T^{12} + \cdots + 21609 \) Copy content Toggle raw display
$7$ \( T^{14} - 6 T^{13} + \cdots + 823543 \) Copy content Toggle raw display
$11$ \( T^{14} - 2 T^{13} + \cdots + 35721 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 572214241 \) Copy content Toggle raw display
$17$ \( (T^{7} + 7 T^{6} + \cdots - 2535)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 893871739 \) Copy content Toggle raw display
$23$ \( (T^{7} + 8 T^{6} + \cdots + 1131)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} - 6 T^{13} + \cdots + 11025 \) Copy content Toggle raw display
$31$ \( T^{14} + 4 T^{13} + \cdots + 56205009 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 5192359755625 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 771895089 \) Copy content Toggle raw display
$43$ \( T^{14} - 2 T^{13} + \cdots + 505521 \) Copy content Toggle raw display
$47$ \( (T^{7} + 11 T^{6} + \cdots + 231417)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 4455162009 \) Copy content Toggle raw display
$59$ \( (T^{7} + 7 T^{6} + \cdots - 45927)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 12 T^{6} + \cdots + 196007)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 2124011569 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 1659729773025 \) Copy content Toggle raw display
$73$ \( (T^{7} + 5 T^{6} + \cdots - 8379)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 166783208881 \) Copy content Toggle raw display
$83$ \( (T^{7} + 9 T^{6} + \cdots - 62685)^{2} \) Copy content Toggle raw display
$89$ \( (T^{7} - 32 T^{6} + \cdots + 7548381)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 598829841 \) Copy content Toggle raw display
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