Properties

Label 798.2.p.d
Level $798$
Weight $2$
Character orbit 798.p
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(107,798)]
 
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, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.p (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.

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

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 1.00000 −1.73110 0.0573132i 1.00000 2.04210i −1.73110 0.0573132i 0.305109 + 2.62810i 1.00000 2.99343 + 0.198430i 2.04210i
107.2 1.00000 −1.60960 0.639688i 1.00000 1.73455i −1.60960 0.639688i −0.675179 2.55815i 1.00000 2.18160 + 2.05928i 1.73455i
107.3 1.00000 −1.57026 0.730943i 1.00000 2.50019i −1.57026 0.730943i 2.58513 + 0.563124i 1.00000 1.93145 + 2.29554i 2.50019i
107.4 1.00000 −1.54401 + 0.784869i 1.00000 3.23749i −1.54401 + 0.784869i 2.52933 + 0.776191i 1.00000 1.76796 2.42370i 3.23749i
107.5 1.00000 −1.53652 + 0.799448i 1.00000 0.138945i −1.53652 + 0.799448i −2.56896 0.632805i 1.00000 1.72177 2.45673i 0.138945i
107.6 1.00000 −1.31402 + 1.12843i 1.00000 2.51691i −1.31402 + 1.12843i −0.461738 + 2.60515i 1.00000 0.453299 2.96556i 2.51691i
107.7 1.00000 −1.27523 + 1.17208i 1.00000 2.98905i −1.27523 + 1.17208i 2.60274 0.475099i 1.00000 0.252435 2.98936i 2.98905i
107.8 1.00000 −1.00188 + 1.41288i 1.00000 3.34851i −1.00188 + 1.41288i −2.58671 0.555829i 1.00000 −0.992484 2.83107i 3.34851i
107.9 1.00000 −0.988593 1.42221i 1.00000 0.610954i −0.988593 1.42221i −2.63918 + 0.186412i 1.00000 −1.04537 + 2.81198i 0.610954i
107.10 1.00000 −0.880492 1.49155i 1.00000 3.64338i −0.880492 1.49155i 2.23585 1.41457i 1.00000 −1.44947 + 2.62660i 3.64338i
107.11 1.00000 −0.544247 1.64432i 1.00000 4.07740i −0.544247 1.64432i −1.01122 2.44488i 1.00000 −2.40759 + 1.78984i 4.07740i
107.12 1.00000 −0.423559 1.67946i 1.00000 1.80606i −0.423559 1.67946i −2.02215 + 1.70614i 1.00000 −2.64119 + 1.42270i 1.80606i
107.13 1.00000 0.00876401 + 1.73203i 1.00000 1.50693i 0.00876401 + 1.73203i 0.837194 2.50980i 1.00000 −2.99985 + 0.0303590i 1.50693i
107.14 1.00000 0.309702 1.70414i 1.00000 0.0905266i 0.309702 1.70414i 2.19691 1.47431i 1.00000 −2.80817 1.05555i 0.0905266i
107.15 1.00000 0.584344 1.63050i 1.00000 3.26298i 0.584344 1.63050i 0.0400578 + 2.64545i 1.00000 −2.31708 1.90555i 3.26298i
107.16 1.00000 0.833647 + 1.51823i 1.00000 0.679467i 0.833647 + 1.51823i −2.29134 + 1.32279i 1.00000 −1.61007 + 2.53134i 0.679467i
107.17 1.00000 0.889459 + 1.48622i 1.00000 1.31572i 0.889459 + 1.48622i 2.50046 + 0.864710i 1.00000 −1.41773 + 2.64387i 1.31572i
107.18 1.00000 1.25084 1.19809i 1.00000 1.95673i 1.25084 1.19809i 1.98940 + 1.74422i 1.00000 0.129182 2.99722i 1.95673i
107.19 1.00000 1.26574 + 1.18232i 1.00000 3.99497i 1.26574 + 1.18232i 1.01470 2.44344i 1.00000 0.204220 + 2.99304i 3.99497i
107.20 1.00000 1.32297 1.11792i 1.00000 2.52686i 1.32297 1.11792i −2.32425 1.26406i 1.00000 0.500522 2.95795i 2.52686i
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.25
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{5}^{50} + 154 T_{5}^{48} + 11091 T_{5}^{46} + 496680 T_{5}^{44} + 15511693 T_{5}^{42} + 359151444 T_{5}^{40} + 6397835009 T_{5}^{38} + 89814995056 T_{5}^{36} + 1009611897357 T_{5}^{34} + \cdots + 322885837872 \) Copy content Toggle raw display
\( T_{11}^{50} + 3 T_{11}^{49} - 133 T_{11}^{48} - 408 T_{11}^{47} + 10271 T_{11}^{46} + 29751 T_{11}^{45} - 539483 T_{11}^{44} - 1454952 T_{11}^{43} + 21385223 T_{11}^{42} + 52190625 T_{11}^{41} + \cdots + 18\!\cdots\!03 \) Copy content Toggle raw display