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Properties

Label 798.2.bh.d
Level $798$
Weight $2$
Character orbit 798.bh
Analytic conductor $6.372$
Analytic rank $0$
Dimension $50$
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(65,798)]
 
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, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.bh (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: \(50\)
Relative dimension: \(25\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

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

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1 0.500000 + 0.866025i −1.72453 − 0.161209i −0.500000 + 0.866025i 2.89990 − 1.67426i −0.722655 − 1.57409i −2.58671 − 0.555829i −1.00000 2.94802 + 0.556021i 2.89990 + 1.67426i
65.2 0.500000 + 0.866025i −1.65267 − 0.518341i −0.500000 + 0.866025i −2.58860 + 1.49453i −0.377439 − 1.69043i 2.60274 − 0.475099i −1.00000 2.46264 + 1.71330i −2.58860 − 1.49453i
65.3 0.500000 + 0.866025i −1.63426 − 0.573761i −0.500000 + 0.866025i −2.17971 + 1.25846i −0.320237 − 1.70219i −0.461738 + 2.60515i −1.00000 2.34160 + 1.87535i −2.17971 − 1.25846i
65.4 0.500000 + 0.866025i −1.49560 + 0.873604i −0.500000 + 0.866025i −1.30504 + 0.753464i −1.50436 − 0.858424i 0.837194 − 2.50980i −1.00000 1.47363 − 2.61312i −1.30504 − 0.753464i
65.5 0.500000 + 0.866025i −1.46060 − 0.930938i −0.500000 + 0.866025i −0.120330 + 0.0694726i 0.0759161 − 1.73039i −2.56896 − 0.632805i −1.00000 1.26671 + 2.71946i −0.120330 − 0.0694726i
65.6 0.500000 + 0.866025i −1.45172 − 0.944722i −0.500000 + 0.866025i 2.80375 − 1.61875i 0.0922912 − 1.72959i 2.52933 + 0.776191i −1.00000 1.21500 + 2.74295i 2.80375 + 1.61875i
65.7 0.500000 + 0.866025i −0.898005 + 1.48108i −0.500000 + 0.866025i 0.588436 − 0.339734i −1.73165 − 0.0371575i −2.29134 + 1.32279i −1.00000 −1.38717 − 2.66003i 0.588436 + 0.339734i
65.8 0.500000 + 0.866025i −0.842379 + 1.51341i −0.500000 + 0.866025i 1.13944 − 0.657859i −1.73184 + 0.0271814i 2.50046 + 0.864710i −1.00000 −1.58080 − 2.54972i 1.13944 + 0.657859i
65.9 0.500000 + 0.866025i −0.815916 − 1.52784i −0.500000 + 0.866025i 1.76851 − 1.02105i 0.915186 − 1.47052i 0.305109 + 2.62810i −1.00000 −1.66856 + 2.49317i 1.76851 + 1.02105i
65.10 0.500000 + 0.866025i −0.391050 + 1.68733i −0.500000 + 0.866025i 3.45974 − 1.99748i −1.65680 + 0.505005i 1.01470 − 2.44344i −1.00000 −2.69416 − 1.31966i 3.45974 + 1.99748i
65.11 0.500000 + 0.866025i −0.250812 − 1.71379i −0.500000 + 0.866025i 1.50216 − 0.867275i 1.35878 − 1.07411i −0.675179 − 2.55815i −1.00000 −2.87419 + 0.859681i 1.50216 + 0.867275i
65.12 0.500000 + 0.866025i −0.201798 + 1.72026i −0.500000 + 0.866025i −2.36133 + 1.36331i −1.59068 + 0.685366i −1.73178 − 2.00024i −1.00000 −2.91856 − 0.694287i −2.36133 − 1.36331i
65.13 0.500000 + 0.866025i −0.152116 − 1.72536i −0.500000 + 0.866025i −2.16523 + 1.25010i 1.41815 − 0.994415i 2.58513 + 0.563124i −1.00000 −2.95372 + 0.524910i −2.16523 − 1.25010i
65.14 0.500000 + 0.866025i 0.522227 + 1.65145i −0.500000 + 0.866025i −2.71521 + 1.56763i −1.16908 + 1.27799i −0.151832 + 2.64139i −1.00000 −2.45456 + 1.72486i −2.71521 − 1.56763i
65.15 0.500000 + 0.866025i 0.737374 − 1.56725i −0.500000 + 0.866025i −0.529102 + 0.305477i 1.72597 − 0.145042i −2.63918 + 0.186412i −1.00000 −1.91256 − 2.31130i −0.529102 − 0.305477i
65.16 0.500000 + 0.866025i 0.851478 − 1.50831i −0.500000 + 0.866025i 3.15526 − 1.82169i 1.73197 − 0.0167509i 2.23585 − 1.41457i −1.00000 −1.54997 − 2.56858i 3.15526 + 1.82169i
65.17 0.500000 + 0.866025i 0.859667 + 1.50365i −0.500000 + 0.866025i 2.38391 − 1.37635i −0.872368 + 1.49632i −1.41302 + 2.23683i −1.00000 −1.52195 + 2.58528i 2.38391 + 1.37635i
65.18 0.500000 + 0.866025i 0.902647 + 1.47825i −0.500000 + 0.866025i −1.38374 + 0.798901i −0.828880 + 1.52084i 2.18393 − 1.49348i −1.00000 −1.37046 + 2.66868i −1.38374 − 0.798901i
65.19 0.500000 + 0.866025i 1.15190 − 1.29349i −0.500000 + 0.866025i −3.53113 + 2.03870i 1.69615 + 0.350829i −1.01122 − 2.44488i −1.00000 −0.346248 − 2.97995i −3.53113 − 2.03870i
65.20 0.500000 + 0.866025i 1.24268 − 1.20654i −0.500000 + 0.866025i 1.56409 − 0.903030i 1.66624 + 0.472919i −2.02215 + 1.70614i −1.00000 0.0884990 − 2.99869i 1.56409 + 0.903030i
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.25
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
399.x 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.bh.d yes 50
3.b odd 2 1 798.2.bh.c yes 50
7.c even 3 1 798.2.p.c ✓ 50
19.d odd 6 1 798.2.p.d yes 50
21.h odd 6 1 798.2.p.d yes 50
57.f even 6 1 798.2.p.c ✓ 50
133.n odd 6 1 798.2.bh.c yes 50
399.x even 6 1 inner 798.2.bh.d yes 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.p.c ✓ 50 7.c even 3 1
798.2.p.c ✓ 50 57.f even 6 1
798.2.p.d yes 50 19.d odd 6 1
798.2.p.d yes 50 21.h odd 6 1
798.2.bh.c yes 50 3.b odd 2 1
798.2.bh.c yes 50 133.n odd 6 1
798.2.bh.d yes 50 1.a even 1 1 trivial
798.2.bh.d yes 50 399.x even 6 1 inner

Hecke kernels

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

\( T_{5}^{50} - 77 T_{5}^{48} + 3348 T_{5}^{46} + 66 T_{5}^{45} - 99531 T_{5}^{44} - 4110 T_{5}^{43} + \cdots + 322885837872 \) Copy content Toggle raw display
\( T_{11}^{50} + 3 T_{11}^{49} - 133 T_{11}^{48} - 408 T_{11}^{47} + 10271 T_{11}^{46} + \cdots + 18\!\cdots\!03 \) Copy content Toggle raw display