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Properties

Label 798.2.ba.m
Level $798$
Weight $2$
Character orbit 798.ba
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(407,798)]
 
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([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.407");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.ba (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: \(6.37206208130\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + 11 x^{12} - 16 x^{11} + 70 x^{10} - 94 x^{9} + 246 x^{8} - 220 x^{7} + 472 x^{6} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} - 1) q^{2} + \beta_1 q^{3} - \beta_{6} q^{4} - \beta_{12} q^{5} - \beta_{10} q^{6} - q^{7} + q^{8} + ( - \beta_{13} - \beta_{9} + \cdots - \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - 1) q^{2} + \beta_1 q^{3} - \beta_{6} q^{4} - \beta_{12} q^{5} - \beta_{10} q^{6} - q^{7} + q^{8} + ( - \beta_{13} - \beta_{9} + \cdots - \beta_{6}) q^{9}+ \cdots + (2 \beta_{13} + \beta_{12} - \beta_{9} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 7 q^{2} + 5 q^{3} - 7 q^{4} - 4 q^{6} - 14 q^{7} + 14 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 7 q^{2} + 5 q^{3} - 7 q^{4} - 4 q^{6} - 14 q^{7} + 14 q^{8} - 7 q^{9} - q^{12} - 6 q^{13} + 7 q^{14} - 7 q^{16} + 12 q^{17} + 11 q^{18} - 13 q^{19} - 5 q^{21} + 9 q^{22} - 6 q^{23} + 5 q^{24} + 11 q^{25} + 14 q^{27} + 7 q^{28} + 10 q^{29} + 12 q^{30} - 7 q^{32} + 17 q^{33} - 12 q^{34} - 4 q^{36} + 2 q^{38} - 2 q^{39} - 9 q^{41} + 4 q^{42} + 22 q^{43} - 9 q^{44} - 8 q^{45} - 4 q^{48} + 14 q^{49} - 22 q^{50} - 34 q^{51} + 6 q^{52} - 12 q^{53} - 7 q^{54} - 12 q^{55} - 14 q^{56} + 9 q^{57} - 20 q^{58} + 15 q^{59} - 12 q^{60} + 2 q^{61} - 18 q^{62} + 7 q^{63} + 14 q^{64} - 64 q^{65} + 14 q^{66} - 21 q^{67} + 4 q^{69} - 14 q^{71} - 7 q^{72} + 15 q^{73} - 6 q^{74} - 63 q^{75} + 11 q^{76} - 8 q^{78} - 36 q^{79} + 57 q^{81} - 9 q^{82} + q^{84} - 18 q^{85} + 22 q^{86} - 64 q^{87} + 38 q^{89} + 22 q^{90} + 6 q^{91} + 6 q^{92} - 52 q^{93} - 44 q^{95} - q^{96} - 33 q^{97} - 7 q^{98} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 11 x^{12} - 16 x^{11} + 70 x^{10} - 94 x^{9} + 246 x^{8} - 220 x^{7} + 472 x^{6} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 55917240 \nu^{13} - 90133228 \nu^{12} - 132268439 \nu^{11} - 1332762391 \nu^{10} + \cdots - 1751971377 ) / 8434677066 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 40928871 \nu^{13} + 11767079 \nu^{12} + 354479043 \nu^{11} + 305302697 \nu^{10} + \cdots + 9554340477 ) / 4217338533 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 83304926 \nu^{13} - 334390247 \nu^{12} + 1124729052 \nu^{11} - 2730917626 \nu^{10} + \cdots + 19367421618 ) / 8434677066 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 98295277 \nu^{13} + 351623053 \nu^{12} - 1419513560 \nu^{11} + 3105972790 \nu^{10} + \cdots - 12115881717 ) / 8434677066 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 135460287 \nu^{13} + 137376938 \nu^{12} - 1153648476 \nu^{11} + 630183764 \nu^{10} + \cdots - 5342999430 ) / 8434677066 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 135602671 \nu^{13} + 374780944 \nu^{12} - 1683979852 \nu^{11} + 3262538169 \nu^{10} + \cdots + 6919917822 ) / 8434677066 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 142598508 \nu^{13} + 174423582 \nu^{12} - 1459195337 \nu^{11} + 1305307097 \nu^{10} + \cdots + 8401719876 ) / 8434677066 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 242987386 \nu^{13} + 407404107 \nu^{12} - 2434066844 \nu^{11} + 2794855917 \nu^{10} + \cdots + 848896581 ) / 8434677066 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 134009309 \nu^{13} - 395095971 \nu^{12} + 1624967889 \nu^{11} - 3284013057 \nu^{10} + \cdots + 4680958872 ) / 4217338533 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 286652492 \nu^{13} + 319431144 \nu^{12} - 2718157273 \nu^{11} + 1992082874 \nu^{10} + \cdots - 14144893116 ) / 8434677066 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 356897242 \nu^{13} - 603300349 \nu^{12} + 3903377163 \nu^{11} - 4746939296 \nu^{10} + \cdots + 3353280258 ) / 8434677066 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 442671019 \nu^{13} + 762723114 \nu^{12} - 4428449131 \nu^{11} + 5605068932 \nu^{10} + \cdots - 7700761110 ) / 8434677066 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 476858332 \nu^{13} - 1037169449 \nu^{12} + 5257848552 \nu^{11} - 8349622654 \nu^{10} + \cdots + 6447761616 ) / 8434677066 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{10} - \beta_{7} + 4\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 2\beta_{2} + \beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{13} + 2 \beta_{12} - \beta_{11} - 5 \beta_{10} - 5 \beta_{9} - 4 \beta_{8} + 4 \beta_{3} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{13} + 7 \beta_{10} - 5 \beta_{9} + 5 \beta_{8} + 3 \beta_{7} - 14 \beta_{6} - 5 \beta_{5} + \cdots - 10 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2 \beta_{13} - 7 \beta_{12} + 14 \beta_{11} + 7 \beta_{10} + 5 \beta_{9} + 26 \beta_{7} + 14 \beta_{6} + \cdots - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{10} + 11\beta_{9} - 12\beta_{8} + 8\beta_{7} + 24\beta_{4} - 12\beta_{3} + 11\beta_{2} + 20\beta _1 + 29 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 38 \beta_{13} - 40 \beta_{12} - 40 \beta_{11} + 79 \beta_{10} + 77 \beta_{9} + 95 \beta_{8} + \cdots + 36 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 78 \beta_{13} - 230 \beta_{10} + 20 \beta_{9} - 113 \beta_{7} + 264 \beta_{6} + 117 \beta_{5} + \cdots - 264 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 289 \beta_{13} + 426 \beta_{12} - 213 \beta_{11} - 555 \beta_{10} - 457 \beta_{9} - 482 \beta_{8} + \cdots + 430 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 382 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 1103 \beta_{10} - 551 \beta_{9} + 581 \beta_{8} + \cdots - 1108 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 548 \beta_{13} - 1099 \beta_{12} + 2198 \beta_{11} + 767 \beta_{10} + 417 \beta_{9} + 3218 \beta_{7} + \cdots - 2254 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 98 \beta_{13} + 32 \beta_{12} - 16 \beta_{11} - 136 \beta_{10} + 896 \beta_{9} - 1464 \beta_{8} + \cdots + 3119 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 4538 \beta_{13} - 5582 \beta_{12} - 5582 \beta_{11} + 9793 \beta_{10} + 8397 \beta_{9} + \cdots + 3434 \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(115\) \(211\) \(533\)
\(\chi(n)\) \(1\) \(\beta_{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
407.1
0.724695 + 1.25521i
−0.135255 − 0.234268i
−0.752869 − 1.30401i
0.908218 + 1.57308i
−1.12786 − 1.95351i
1.10162 + 1.90806i
0.281454 + 0.487493i
0.724695 − 1.25521i
−0.135255 + 0.234268i
−0.752869 + 1.30401i
0.908218 − 1.57308i
−1.12786 + 1.95351i
1.10162 − 1.90806i
0.281454 − 0.487493i
−0.500000 + 0.866025i −1.45018 + 0.947086i −0.500000 − 0.866025i −0.558987 − 0.322731i −0.0951095 − 1.72944i −1.00000 1.00000 1.20606 − 2.74689i 0.558987 − 0.322731i
407.2 −0.500000 + 0.866025i −0.311581 − 1.70379i −0.500000 − 0.866025i 1.75774 + 1.01483i 1.63132 + 0.582060i −1.00000 1.00000 −2.80583 + 1.06174i −1.75774 + 1.01483i
407.3 −0.500000 + 0.866025i −0.220161 + 1.71800i −0.500000 − 0.866025i 3.13494 + 1.80996i −1.37775 − 1.04967i −1.00000 1.00000 −2.90306 − 0.756474i −3.13494 + 1.80996i
407.4 −0.500000 + 0.866025i 0.333099 + 1.69972i −0.500000 − 0.866025i −3.60141 − 2.07928i −1.63855 − 0.561387i −1.00000 1.00000 −2.77809 + 1.13235i 3.60141 − 2.07928i
407.5 −0.500000 + 0.866025i 0.723433 − 1.57374i −0.500000 − 0.866025i 0.0595041 + 0.0343547i 1.00118 + 1.41338i −1.00000 1.00000 −1.95329 − 2.27698i −0.0595041 + 0.0343547i
407.6 −0.500000 + 0.866025i 1.69766 − 0.343451i −0.500000 − 0.866025i 1.60196 + 0.924891i −0.551391 + 1.64194i −1.00000 1.00000 2.76408 − 1.16612i −1.60196 + 0.924891i
407.7 −0.500000 + 0.866025i 1.72773 + 0.122201i −0.500000 − 0.866025i −2.39375 − 1.38203i −0.969696 + 1.43516i −1.00000 1.00000 2.97013 + 0.422261i 2.39375 − 1.38203i
449.1 −0.500000 − 0.866025i −1.45018 − 0.947086i −0.500000 + 0.866025i −0.558987 + 0.322731i −0.0951095 + 1.72944i −1.00000 1.00000 1.20606 + 2.74689i 0.558987 + 0.322731i
449.2 −0.500000 − 0.866025i −0.311581 + 1.70379i −0.500000 + 0.866025i 1.75774 − 1.01483i 1.63132 − 0.582060i −1.00000 1.00000 −2.80583 − 1.06174i −1.75774 − 1.01483i
449.3 −0.500000 − 0.866025i −0.220161 − 1.71800i −0.500000 + 0.866025i 3.13494 − 1.80996i −1.37775 + 1.04967i −1.00000 1.00000 −2.90306 + 0.756474i −3.13494 − 1.80996i
449.4 −0.500000 − 0.866025i 0.333099 − 1.69972i −0.500000 + 0.866025i −3.60141 + 2.07928i −1.63855 + 0.561387i −1.00000 1.00000 −2.77809 − 1.13235i 3.60141 + 2.07928i
449.5 −0.500000 − 0.866025i 0.723433 + 1.57374i −0.500000 + 0.866025i 0.0595041 − 0.0343547i 1.00118 − 1.41338i −1.00000 1.00000 −1.95329 + 2.27698i −0.0595041 − 0.0343547i
449.6 −0.500000 − 0.866025i 1.69766 + 0.343451i −0.500000 + 0.866025i 1.60196 − 0.924891i −0.551391 − 1.64194i −1.00000 1.00000 2.76408 + 1.16612i −1.60196 − 0.924891i
449.7 −0.500000 − 0.866025i 1.72773 − 0.122201i −0.500000 + 0.866025i −2.39375 + 1.38203i −0.969696 − 1.43516i −1.00000 1.00000 2.97013 − 0.422261i 2.39375 + 1.38203i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 407.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.ba.m ✓ 14
3.b odd 2 1 798.2.ba.n yes 14
19.d odd 6 1 798.2.ba.n yes 14
57.f even 6 1 inner 798.2.ba.m ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.ba.m ✓ 14 1.a even 1 1 trivial
798.2.ba.m ✓ 14 57.f even 6 1 inner
798.2.ba.n yes 14 3.b odd 2 1
798.2.ba.n yes 14 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\):

\( T_{5}^{14} - 23 T_{5}^{12} + 404 T_{5}^{10} - 252 T_{5}^{9} - 2577 T_{5}^{8} + 2238 T_{5}^{7} + \cdots + 48 \) Copy content Toggle raw display
\( T_{11}^{14} + 31T_{11}^{12} + 380T_{11}^{10} + 2352T_{11}^{8} + 7728T_{11}^{6} + 12800T_{11}^{4} + 8704T_{11}^{2} + 768 \) Copy content Toggle raw display
\( T_{13}^{14} + 6 T_{13}^{13} - 21 T_{13}^{12} - 198 T_{13}^{11} + 576 T_{13}^{10} + 3900 T_{13}^{9} + \cdots + 108 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{14} - 5 T^{13} + \cdots + 2187 \) Copy content Toggle raw display
$5$ \( T^{14} - 23 T^{12} + \cdots + 48 \) Copy content Toggle raw display
$7$ \( (T + 1)^{14} \) Copy content Toggle raw display
$11$ \( T^{14} + 31 T^{12} + \cdots + 768 \) Copy content Toggle raw display
$13$ \( T^{14} + 6 T^{13} + \cdots + 108 \) Copy content Toggle raw display
$17$ \( T^{14} - 12 T^{13} + \cdots + 110592 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 893871739 \) Copy content Toggle raw display
$23$ \( T^{14} + 6 T^{13} + \cdots + 112908 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 119771136 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 9920130048 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 2045509632 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 33850112256 \) Copy content Toggle raw display
$43$ \( T^{14} - 22 T^{13} + \cdots + 2985984 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 18833129472 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 6642902016 \) Copy content Toggle raw display
$59$ \( T^{14} - 15 T^{13} + \cdots + 59049 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 225120016 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 7342436352 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 3029181444 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 16302472167424 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 603489042432 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 18091828947 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 110841053184 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 3143467414272 \) Copy content Toggle raw display
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