↑

Properties

Label 637.2.bg.a
Level $637$
Weight $2$
Character orbit 637.bg
Analytic conductor $5.086$
Analytic rank $0$
Dimension $372$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [637,2,Mod(64,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([10, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.bg (of order \(14\), degree \(6\), 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: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(372\)
Relative dimension: \(62\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 372 q - 14 q^{3} + 46 q^{4} - 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 372 q - 14 q^{3} + 46 q^{4} - 64 q^{9} - 14 q^{10} + 6 q^{12} - 15 q^{13} - 10 q^{14} - 78 q^{16} - 6 q^{17} - 38 q^{22} - 10 q^{23} + 36 q^{25} - 5 q^{26} - 2 q^{27} - 6 q^{29} - 52 q^{30} - 36 q^{35} + 50 q^{36} + 32 q^{38} - 51 q^{39} + 6 q^{40} - 94 q^{42} + 22 q^{43} + 160 q^{48} + 20 q^{49} + 10 q^{51} - 85 q^{52} + 68 q^{53} - 68 q^{55} - 40 q^{56} + 98 q^{61} + 12 q^{62} - 38 q^{64} + 17 q^{65} - 112 q^{66} + 120 q^{68} + 158 q^{69} - 62 q^{74} - 16 q^{75} + 12 q^{77} + 35 q^{78} - 68 q^{79} + 28 q^{81} - 28 q^{82} - 86 q^{87} + 44 q^{88} - 198 q^{90} + 9 q^{91} - 52 q^{92} - 48 q^{94} + 120 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −2.61088 − 0.595915i 1.16253 − 1.45777i 4.65962 + 2.24396i 2.47973 + 1.97752i −3.90393 + 3.11328i 0.714353 + 2.54749i −6.64097 − 5.29599i −0.106045 − 0.464612i −5.29583 − 6.64076i
64.2 −2.60279 − 0.594070i −1.17016 + 1.46733i 4.61966 + 2.22471i 1.23047 + 0.981265i 3.91737 − 3.12400i −2.53369 − 0.761861i −6.52783 − 5.20577i −0.116227 − 0.509222i −2.61971 − 3.28501i
64.3 −2.52309 − 0.575879i 0.308464 − 0.386802i 4.23241 + 2.03822i −0.723365 − 0.576864i −1.00103 + 0.798298i 2.44959 − 0.999764i −5.45826 − 4.35282i 0.613097 + 2.68615i 1.49291 + 1.87205i
64.4 −2.41957 − 0.552251i −1.38967 + 1.74259i 3.74739 + 1.80465i −2.06602 − 1.64760i 4.32475 − 3.44888i 2.16211 + 1.52488i −4.18977 − 3.34123i −0.437880 − 1.91848i 4.08899 + 5.12743i
64.5 −2.36482 − 0.539755i −0.336282 + 0.421684i 3.49910 + 1.68508i 2.25756 + 1.80035i 1.02285 − 0.815697i 0.108253 − 2.64354i −3.57234 − 2.84885i 0.602831 + 2.64117i −4.36699 − 5.47603i
64.6 −2.35227 − 0.536891i 1.95194 − 2.44765i 3.44299 + 1.65806i 0.0899575 + 0.0717387i −5.90561 + 4.70956i 1.19794 − 2.35901i −3.43590 − 2.74004i −1.51338 − 6.63054i −0.173089 − 0.217046i
64.7 −2.25775 − 0.515318i −1.31430 + 1.64808i 3.02996 + 1.45915i −1.58774 − 1.26618i 3.81666 − 3.04368i −2.13005 + 1.56936i −2.46783 − 1.96803i −0.321224 − 1.40737i 2.93224 + 3.67692i
64.8 −2.18962 − 0.499767i 0.977150 − 1.22531i 2.74274 + 1.32084i −0.753889 − 0.601206i −2.75196 + 2.19461i −2.56305 − 0.656320i −1.83358 − 1.46223i 0.121006 + 0.530163i 1.35027 + 1.69318i
64.9 −2.00613 − 0.457887i 1.28345 − 1.60940i 2.01298 + 0.969400i −3.18000 − 2.53597i −3.31170 + 2.64099i 1.11355 + 2.40000i −0.376841 − 0.300521i −0.275350 − 1.20639i 5.21832 + 6.54357i
64.10 −1.88499 − 0.430237i 1.08343 − 1.35858i 1.56615 + 0.754218i 3.17017 + 2.52812i −2.62677 + 2.09478i −2.64573 + 0.0106413i 0.395605 + 0.315485i −0.00435563 − 0.0190833i −4.88805 − 6.12942i
64.11 −1.88222 − 0.429605i −1.05130 + 1.31828i 1.55626 + 0.749455i −2.59166 − 2.06678i 2.54512 − 2.02966i −0.135831 − 2.64226i 0.411595 + 0.328236i 0.0349133 + 0.152965i 3.99018 + 5.00353i
64.12 −1.85833 − 0.424153i −1.93405 + 2.42522i 1.47156 + 0.708667i 0.570338 + 0.454830i 4.62277 − 3.68653i 1.87952 − 1.86209i 0.546465 + 0.435791i −1.47358 − 6.45618i −0.866962 − 1.08714i
64.13 −1.78946 − 0.408433i −0.0524771 + 0.0658042i 1.23342 + 0.593985i −0.502763 − 0.400940i 0.120782 − 0.0963208i −0.780674 + 2.52795i 0.905514 + 0.722123i 0.665986 + 2.91788i 0.735919 + 0.922814i
64.14 −1.70350 − 0.388812i −0.586579 + 0.735547i 0.948790 + 0.456913i 2.01042 + 1.60326i 1.28523 − 1.02493i 0.688104 + 2.55470i 1.29359 + 1.03160i 0.470608 + 2.06187i −2.80138 − 3.51282i
64.15 −1.65183 − 0.377020i 1.58163 − 1.98330i 0.784472 + 0.377782i 0.373783 + 0.298082i −3.36034 + 2.67978i 2.26261 + 1.37135i 1.49595 + 1.19298i −0.764369 − 3.34892i −0.505044 − 0.633305i
64.16 −1.61405 − 0.368396i 0.414662 − 0.519970i 0.667497 + 0.321450i −1.00464 − 0.801173i −0.860839 + 0.686496i −1.55883 − 2.13777i 1.62978 + 1.29971i 0.569139 + 2.49356i 1.32639 + 1.66324i
64.17 −1.54111 − 0.351748i −0.155761 + 0.195319i 0.449345 + 0.216393i 1.53910 + 1.22739i 0.308748 − 0.246218i 2.61776 + 0.383848i 1.85537 + 1.47961i 0.653675 + 2.86394i −1.94018 − 2.43291i
64.18 −1.51840 − 0.346565i −1.97447 + 2.47591i 0.383492 + 0.184680i 1.58125 + 1.26101i 3.85610 − 3.07513i −2.63761 + 0.207343i 1.91703 + 1.52878i −1.56402 − 6.85243i −1.96395 − 2.46272i
64.19 −1.13406 − 0.258842i 1.54451 − 1.93676i −0.582843 − 0.280682i 2.60535 + 2.07770i −2.25288 + 1.79661i 1.54254 − 2.14955i 2.40722 + 1.91969i −0.697945 − 3.05790i −2.41683 − 3.03061i
64.20 −0.916322 − 0.209144i −0.289551 + 0.363086i −1.00603 − 0.484480i −2.03025 − 1.61907i 0.341260 − 0.272146i 2.40685 − 1.09867i 2.29019 + 1.82637i 0.619571 + 2.71452i 1.52175 + 1.90821i
See next 80 embeddings (of 372 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.62
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
49.e even 7 1 inner
637.bg even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.bg.a ✓ 372
13.b even 2 1 inner 637.2.bg.a ✓ 372
49.e even 7 1 inner 637.2.bg.a ✓ 372
637.bg even 14 1 inner 637.2.bg.a ✓ 372
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.bg.a ✓ 372 1.a even 1 1 trivial
637.2.bg.a ✓ 372 13.b even 2 1 inner
637.2.bg.a ✓ 372 49.e even 7 1 inner
637.2.bg.a ✓ 372 637.bg even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(637, [\chi])\).