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Properties

Label 798.2.i.d
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: \( x^{14} - 3 x^{13} + 19 x^{12} - 22 x^{11} + 141 x^{10} - 94 x^{9} + 768 x^{8} + 4 x^{7} + 2293 x^{6} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{2} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + 1) q^{2} + q^{3} - \beta_{6} q^{4} - \beta_{7} q^{5} + ( - \beta_{6} + 1) q^{6} + (\beta_{11} + \beta_{4}) q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + 1) q^{2} + q^{3} - \beta_{6} q^{4} - \beta_{7} q^{5} + ( - \beta_{6} + 1) q^{6} + (\beta_{11} + \beta_{4}) q^{7} - q^{8} + q^{9} + ( - \beta_{7} - \beta_1) q^{10} + (\beta_{13} + \beta_{4} - \beta_{2}) q^{11} - \beta_{6} q^{12} + (\beta_{12} - \beta_{9}) q^{13} + \beta_{11} q^{14} - \beta_{7} q^{15} + (\beta_{6} - 1) q^{16} + (\beta_{12} + \beta_{11} - \beta_{10} + \cdots + 2) q^{17}+ \cdots + (\beta_{13} + \beta_{4} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 7 q^{2} + 14 q^{3} - 7 q^{4} - 2 q^{5} + 7 q^{6} - 6 q^{7} - 14 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 7 q^{2} + 14 q^{3} - 7 q^{4} - 2 q^{5} + 7 q^{6} - 6 q^{7} - 14 q^{8} + 14 q^{9} + 2 q^{10} + 2 q^{11} - 7 q^{12} + q^{13} - 3 q^{14} - 2 q^{15} - 7 q^{16} + 2 q^{17} + 7 q^{18} + 6 q^{19} + 4 q^{20} - 6 q^{21} - 2 q^{22} + 8 q^{23} - 14 q^{24} - 17 q^{25} - q^{26} + 14 q^{27} + 3 q^{28} - 8 q^{29} + 2 q^{30} + 4 q^{31} + 7 q^{32} + 2 q^{33} + q^{34} + 7 q^{35} - 7 q^{36} - 7 q^{37} + 9 q^{38} + q^{39} + 2 q^{40} + 5 q^{41} - 3 q^{42} + 16 q^{43} - 4 q^{44} - 2 q^{45} + 4 q^{46} + 14 q^{47} - 7 q^{48} + 8 q^{49} - 34 q^{50} + 2 q^{51} - 2 q^{52} - 6 q^{53} + 7 q^{54} + 8 q^{55} + 6 q^{56} + 6 q^{57} + 8 q^{58} - 14 q^{59} + 4 q^{60} - 4 q^{61} - 4 q^{62} - 6 q^{63} + 14 q^{64} - 31 q^{65} - 2 q^{66} - 14 q^{67} - q^{68} + 8 q^{69} + 14 q^{70} - 6 q^{71} - 14 q^{72} + 22 q^{73} - 14 q^{74} - 17 q^{75} + 3 q^{76} - 8 q^{77} - q^{78} - 28 q^{79} - 2 q^{80} + 14 q^{81} + 10 q^{82} + 10 q^{83} + 3 q^{84} + 3 q^{85} + 32 q^{86} - 8 q^{87} - 2 q^{88} + 16 q^{89} + 2 q^{90} - 3 q^{91} - 4 q^{92} + 4 q^{93} + 7 q^{94} - 47 q^{95} + 7 q^{96} + 16 q^{97} - 11 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} + 19 x^{12} - 22 x^{11} + 141 x^{10} - 94 x^{9} + 768 x^{8} + 4 x^{7} + 2293 x^{6} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 11821453292272 \nu^{13} + 55916451882485 \nu^{12} - 290006846871284 \nu^{11} + \cdots - 17\!\cdots\!38 ) / 13\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 370390223145424 \nu^{13} + \cdots + 45\!\cdots\!70 ) / 37\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 691158106318048 \nu^{13} + \cdots - 10\!\cdots\!77 ) / 37\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 777104280324898 \nu^{13} + \cdots - 98\!\cdots\!16 ) / 37\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 490263616664596 \nu^{13} + \cdots + 15\!\cdots\!12 ) / 12\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 260482600330581 \nu^{13} - 790997375881393 \nu^{12} + \cdots + 44\!\cdots\!13 ) / 41\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 272827411742178 \nu^{13} + 816270716253835 \nu^{12} + \cdots - 26\!\cdots\!50 ) / 41\!\cdots\!19 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 32\!\cdots\!22 \nu^{13} + \cdots + 97\!\cdots\!38 ) / 37\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 40\!\cdots\!27 \nu^{13} + \cdots + 43\!\cdots\!47 ) / 37\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 67\!\cdots\!27 \nu^{13} + \cdots + 10\!\cdots\!04 ) / 37\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 67\!\cdots\!77 \nu^{13} + \cdots + 10\!\cdots\!65 ) / 37\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 93\!\cdots\!72 \nu^{13} + \cdots + 55\!\cdots\!87 ) / 37\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 96\!\cdots\!77 \nu^{13} + \cdots + 15\!\cdots\!48 ) / 37\!\cdots\!71 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{10} + \beta_{4} - \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{12} + 3 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - 13 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \cdots + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{12} + 13 \beta_{11} - 12 \beta_{10} + \beta_{9} + 2 \beta_{8} - 2 \beta_{6} + \beta_{5} + \cdots - 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3 \beta_{13} - 6 \beta_{12} + \beta_{11} - 10 \beta_{10} + 15 \beta_{9} + 9 \beta_{8} - 6 \beta_{7} + \cdots - 63 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6 \beta_{13} - 29 \beta_{12} - 68 \beta_{11} + 42 \beta_{10} + 13 \beta_{9} - 13 \beta_{8} - 36 \beta_{7} + \cdots - 16 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 72 \beta_{12} - 223 \beta_{11} + 172 \beta_{10} - 72 \beta_{9} - 123 \beta_{8} + 123 \beta_{6} + \cdots + 339 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 108 \beta_{13} + 146 \beta_{12} - 137 \beta_{11} + 286 \beta_{10} - 307 \beta_{9} - 161 \beta_{8} + \cdots + 784 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 477 \beta_{13} + 1064 \beta_{12} + 1822 \beta_{11} - 656 \beta_{10} - 499 \beta_{9} + 499 \beta_{8} + \cdots + 565 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1403 \beta_{12} + 5943 \beta_{11} - 4399 \beta_{10} + 1403 \beta_{9} + 2947 \beta_{8} - 2947 \beta_{6} + \cdots - 5414 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 5004 \beta_{13} - 5000 \beta_{12} + 645 \beta_{11} - 6413 \beta_{10} + 9403 \beta_{9} + 4403 \beta_{8} + \cdots - 22645 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 14955 \beta_{13} - 27184 \beta_{12} - 45992 \beta_{11} + 15340 \beta_{10} + 15653 \beta_{9} + \cdots - 11531 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 34264 \beta_{12} - 142141 \beta_{11} + 92769 \beta_{10} - 34264 \beta_{9} - 83636 \beta_{8} + \cdots + 147787 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 50715 \beta_{13} + 51215 \beta_{12} - 6363 \beta_{11} + 53115 \beta_{10} - 82023 \beta_{9} + \cdots + 166145 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
−0.389110 − 0.673959i
1.48109 + 2.56532i
−0.0290551 − 0.0503249i
−0.929463 − 1.60988i
1.33601 + 2.31404i
−1.04653 − 1.81265i
1.07706 + 1.86552i
−0.389110 + 0.673959i
1.48109 − 2.56532i
−0.0290551 + 0.0503249i
−0.929463 + 1.60988i
1.33601 − 2.31404i
−1.04653 + 1.81265i
1.07706 − 1.86552i
0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −1.90708 − 3.30316i 0.500000 + 0.866025i 1.86290 + 1.87872i −1.00000 1.00000 1.90708 − 3.30316i
163.2 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −1.70314 − 2.94993i 0.500000 + 0.866025i −2.54533 + 0.722006i −1.00000 1.00000 1.70314 − 2.94993i
163.3 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.643489 − 1.11456i 0.500000 + 0.866025i −1.27905 − 2.31604i −1.00000 1.00000 0.643489 − 1.11456i
163.4 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.207656 − 0.359670i 0.500000 + 0.866025i 0.344397 − 2.62324i −1.00000 1.00000 0.207656 − 0.359670i
163.5 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 0.211160 + 0.365740i 0.500000 + 0.866025i −1.36269 + 2.26783i −1.00000 1.00000 −0.211160 + 0.365740i
163.6 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 1.04259 + 1.80582i 0.500000 + 0.866025i 2.53343 + 0.762724i −1.00000 1.00000 −1.04259 + 1.80582i
163.7 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 2.20762 + 3.82370i 0.500000 + 0.866025i −2.55365 − 0.692010i −1.00000 1.00000 −2.20762 + 3.82370i
235.1 0.500000 − 0.866025i 1.00000 −0.500000 − 0.866025i −1.90708 + 3.30316i 0.500000 − 0.866025i 1.86290 − 1.87872i −1.00000 1.00000 1.90708 + 3.30316i
235.2 0.500000 − 0.866025i 1.00000 −0.500000 − 0.866025i −1.70314 + 2.94993i 0.500000 − 0.866025i −2.54533 − 0.722006i −1.00000 1.00000 1.70314 + 2.94993i
235.3 0.500000 − 0.866025i 1.00000 −0.500000 − 0.866025i −0.643489 + 1.11456i 0.500000 − 0.866025i −1.27905 + 2.31604i −1.00000 1.00000 0.643489 + 1.11456i
235.4 0.500000 − 0.866025i 1.00000 −0.500000 − 0.866025i −0.207656 + 0.359670i 0.500000 − 0.866025i 0.344397 + 2.62324i −1.00000 1.00000 0.207656 + 0.359670i
235.5 0.500000 − 0.866025i 1.00000 −0.500000 − 0.866025i 0.211160 − 0.365740i 0.500000 − 0.866025i −1.36269 − 2.26783i −1.00000 1.00000 −0.211160 − 0.365740i
235.6 0.500000 − 0.866025i 1.00000 −0.500000 − 0.866025i 1.04259 − 1.80582i 0.500000 − 0.866025i 2.53343 − 0.762724i −1.00000 1.00000 −1.04259 − 1.80582i
235.7 0.500000 − 0.866025i 1.00000 −0.500000 − 0.866025i 2.20762 − 3.82370i 0.500000 − 0.866025i −2.55365 + 0.692010i −1.00000 1.00000 −2.20762 − 3.82370i
\(n\): 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.d ✓ 14
7.c even 3 1 798.2.l.a yes 14
19.c even 3 1 798.2.l.a yes 14
133.g even 3 1 inner 798.2.i.d ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.i.d ✓ 14 1.a even 1 1 trivial
798.2.i.d ✓ 14 133.g even 3 1 inner
798.2.l.a yes 14 7.c even 3 1
798.2.l.a yes 14 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} + 2 T_{5}^{13} + 28 T_{5}^{12} + 52 T_{5}^{11} + 575 T_{5}^{10} + 958 T_{5}^{9} + 4564 T_{5}^{8} + \cdots + 729 \) 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
$5$ \( T^{14} + 2 T^{13} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( T^{14} + 6 T^{13} + \cdots + 823543 \) Copy content Toggle raw display
$11$ \( T^{14} - 2 T^{13} + \cdots + 6561 \) Copy content Toggle raw display
$13$ \( T^{14} - T^{13} + \cdots + 2259009 \) Copy content Toggle raw display
$17$ \( (T^{7} - T^{6} - 78 T^{5} + \cdots - 27)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 893871739 \) Copy content Toggle raw display
$23$ \( (T^{7} - 4 T^{6} + \cdots + 243)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + 8 T^{13} + \cdots + 321489 \) Copy content Toggle raw display
$31$ \( T^{14} - 4 T^{13} + \cdots + 4084441 \) Copy content Toggle raw display
$37$ \( T^{14} + 7 T^{13} + \cdots + 6561 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 141776649 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 168974001 \) Copy content Toggle raw display
$47$ \( (T^{7} - 7 T^{6} + \cdots + 345789)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 503418969 \) Copy content Toggle raw display
$59$ \( (T^{7} + 7 T^{6} + \cdots + 6561)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} + 2 T^{6} + \cdots - 170097)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 166291868521 \) Copy content Toggle raw display
$71$ \( T^{14} + 6 T^{13} + \cdots + 4782969 \) Copy content Toggle raw display
$73$ \( (T^{7} - 11 T^{6} + \cdots + 485253)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 216300336561 \) Copy content Toggle raw display
$83$ \( (T^{7} - 5 T^{6} + \cdots + 177147)^{2} \) Copy content Toggle raw display
$89$ \( (T^{7} - 8 T^{6} + \cdots - 55323)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 5978215711521 \) Copy content Toggle raw display
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