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Properties

Label 775.2.bs.b
Level $775$
Weight $2$
Character orbit 775.bs
Analytic conductor $6.188$
Analytic rank $0$
Dimension $112$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [775,2,Mod(182,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([5, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.182");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.bs (of order \(20\), degree \(8\), 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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(14\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 155)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 6 q^{2} + 10 q^{3} + 18 q^{7} + 20 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q + 6 q^{2} + 10 q^{3} + 18 q^{7} + 20 q^{8} - 20 q^{11} - 10 q^{12} + 10 q^{13} - 12 q^{16} + 10 q^{17} - 4 q^{18} + 20 q^{21} - 60 q^{22} + 40 q^{27} + 48 q^{28} - 4 q^{31} - 44 q^{32} + 26 q^{33} - 64 q^{36} + 32 q^{38} - 16 q^{41} - 70 q^{42} + 10 q^{43} - 60 q^{46} - 46 q^{47} - 150 q^{48} + 12 q^{51} + 80 q^{52} - 10 q^{53} - 24 q^{56} + 10 q^{58} + 54 q^{62} + 200 q^{63} - 52 q^{66} + 68 q^{67} + 8 q^{71} + 18 q^{72} - 30 q^{73} - 128 q^{76} + 30 q^{77} - 36 q^{78} + 64 q^{81} + 44 q^{82} - 160 q^{83} - 20 q^{86} - 60 q^{91} - 10 q^{93} - 92 q^{97} - 144 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
182.1 −1.16362 + 2.28373i 2.49143 − 1.26945i −2.68586 − 3.69677i 0 7.16691i 4.03395 + 0.638915i 6.50468 − 1.03024i 2.83236 − 3.89841i 0
182.2 −1.07659 + 2.11293i −1.55820 + 0.793944i −2.12984 − 2.93147i 0 − 4.14712i 2.86987 + 0.454543i 3.80255 − 0.602264i 0.0342919 − 0.0471987i 0
182.3 −1.06657 + 2.09327i −0.244956 + 0.124811i −2.06862 − 2.84722i 0 − 0.645878i −3.72361 − 0.589761i 3.52551 − 0.558387i −1.71893 + 2.36590i 0
182.4 −0.679080 + 1.33277i 2.32857 − 1.18647i −0.139553 − 0.192078i 0 3.90916i −1.63626 − 0.259158i −2.60401 + 0.412434i 2.25119 − 3.09850i 0
182.5 −0.664153 + 1.30347i 0.252737 − 0.128776i −0.0823742 − 0.113378i 0 0.414963i −0.00240708 0.000381243i −2.68733 + 0.425631i −1.71606 + 2.36196i 0
182.6 −0.360179 + 0.706891i −1.31115 + 0.668066i 0.805604 + 1.10882i 0 − 1.16747i 2.79496 + 0.442678i −2.64117 + 0.418320i −0.490546 + 0.675179i 0
182.7 0.0652935 − 0.128146i −0.102001 + 0.0519719i 1.16341 + 1.60130i 0 0.0164644i −1.58515 − 0.251064i 0.565264 − 0.0895290i −1.75565 + 2.41645i 0
182.8 0.109757 − 0.215411i 2.34491 − 1.19479i 1.14122 + 1.57075i 0 − 0.636258i 2.38616 + 0.377930i 0.941184 − 0.149069i 2.30773 − 3.17632i 0
182.9 0.352649 − 0.692113i 1.07065 − 0.545523i 0.820912 + 1.12989i 0 − 0.933389i −1.44420 − 0.228739i 2.60593 − 0.412739i −0.914661 + 1.25892i 0
182.10 0.427108 − 0.838246i −2.39620 + 1.22093i 0.655335 + 0.901992i 0 2.53007i 3.20825 + 0.508137i 2.89439 − 0.458427i 2.48776 − 3.42411i 0
182.11 0.654371 − 1.28428i −2.21807 + 1.13016i −0.0455907 − 0.0627502i 0 3.58816i −3.30220 − 0.523017i 2.73684 − 0.433473i 1.87922 − 2.58652i 0
182.12 0.839923 − 1.64844i 2.29637 − 1.17006i −0.836318 − 1.15109i 0 − 4.76820i 1.81446 + 0.287382i 1.05467 − 0.167043i 2.14093 − 2.94674i 0
182.13 0.988338 − 1.93972i −0.625371 + 0.318642i −1.61014 − 2.21617i 0 1.52797i 2.66641 + 0.422318i −1.58972 + 0.251786i −1.47380 + 2.02851i 0
182.14 1.17595 − 2.30794i 1.54939 − 0.789453i −2.76815 − 3.81003i 0 − 4.50425i −1.30001 − 0.205901i −6.93179 + 1.09789i 0.0140116 − 0.0192854i 0
232.1 −2.61621 − 0.414366i 0.365926 + 2.31037i 4.77072 + 1.55010i 0 − 6.19603i 3.43954 − 1.75253i −7.11866 − 3.62714i −2.35073 + 0.763798i 0
232.2 −1.79247 − 0.283900i 0.118876 + 0.750551i 1.23025 + 0.399732i 0 − 1.37909i 2.62202 − 1.33599i 1.14232 + 0.582043i 2.30397 − 0.748607i 0
232.3 −1.65234 − 0.261704i −0.184914 − 1.16750i 0.759614 + 0.246814i 0 1.97750i −1.64328 + 0.837296i 1.79064 + 0.912378i 1.52430 − 0.495275i 0
232.4 −1.30384 − 0.206508i 0.302640 + 1.91079i −0.244762 − 0.0795279i 0 − 2.55386i −1.94164 + 0.989316i 2.65513 + 1.35286i −0.706369 + 0.229513i 0
232.5 −1.01066 − 0.160072i −0.498402 − 3.14678i −0.906307 − 0.294477i 0 3.26010i −0.580348 + 0.295702i 2.69228 + 1.37179i −6.80067 + 2.20967i 0
232.6 −0.186955 − 0.0296108i 0.109767 + 0.693043i −1.86804 − 0.606962i 0 − 0.132818i 0.0773360 − 0.0394047i 0.668577 + 0.340657i 2.38491 − 0.774904i 0
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 182.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
31.f odd 10 1 inner
155.r even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.bs.b 112
5.b even 2 1 155.2.r.a ✓ 112
5.c odd 4 1 155.2.r.a ✓ 112
5.c odd 4 1 inner 775.2.bs.b 112
31.f odd 10 1 inner 775.2.bs.b 112
155.m odd 10 1 155.2.r.a ✓ 112
155.r even 20 1 155.2.r.a ✓ 112
155.r even 20 1 inner 775.2.bs.b 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.r.a ✓ 112 5.b even 2 1
155.2.r.a ✓ 112 5.c odd 4 1
155.2.r.a ✓ 112 155.m odd 10 1
155.2.r.a ✓ 112 155.r even 20 1
775.2.bs.b 112 1.a even 1 1 trivial
775.2.bs.b 112 5.c odd 4 1 inner
775.2.bs.b 112 31.f odd 10 1 inner
775.2.bs.b 112 155.r even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{112} - 6 T_{2}^{111} + 18 T_{2}^{110} - 48 T_{2}^{109} + 55 T_{2}^{108} + 170 T_{2}^{107} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display