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Properties

Label 352.2.bf.a
Level $352$
Weight $2$
Character orbit 352.bf
Analytic conductor $2.811$
Analytic rank $0$
Dimension $736$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [352,2,Mod(5,352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(352, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([0, 5, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("352.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 352 = 2^{5} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 352.bf (of order \(40\), degree \(16\), 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: \(2.81073415115\)
Analytic rank: \(0\)
Dimension: \(736\)
Relative dimension: \(46\) over \(\Q(\zeta_{40})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{40}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 736 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 12 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 736 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 12 q^{7} - 12 q^{8} - 12 q^{9} - 32 q^{10} - 16 q^{11} - 64 q^{12} - 12 q^{13} - 28 q^{14} - 52 q^{16} + 48 q^{18} - 12 q^{19} - 28 q^{20} - 32 q^{21} - 28 q^{22} - 48 q^{23} - 12 q^{24} - 12 q^{25} - 72 q^{26} - 12 q^{27} - 12 q^{28} - 12 q^{29} - 28 q^{30} - 24 q^{31} + 8 q^{32} - 32 q^{33} - 8 q^{34} - 12 q^{35} - 12 q^{36} - 12 q^{37} - 12 q^{38} - 12 q^{39} + 28 q^{40} - 12 q^{41} - 52 q^{42} - 48 q^{43} - 56 q^{44} - 56 q^{45} - 76 q^{46} - 12 q^{48} - 12 q^{50} - 36 q^{51} - 28 q^{52} + 36 q^{53} - 120 q^{54} + 16 q^{55} + 184 q^{56} - 12 q^{57} - 12 q^{58} + 20 q^{59} + 4 q^{60} - 12 q^{61} + 4 q^{62} + 56 q^{63} - 156 q^{64} - 64 q^{65} + 8 q^{66} + 48 q^{67} - 100 q^{68} - 12 q^{69} + 36 q^{70} + 20 q^{71} + 168 q^{72} - 12 q^{73} - 44 q^{74} - 148 q^{75} - 32 q^{76} - 32 q^{77} + 16 q^{78} - 148 q^{80} - 52 q^{82} - 12 q^{83} + 100 q^{84} - 52 q^{85} + 92 q^{86} - 32 q^{87} - 20 q^{88} - 32 q^{89} - 180 q^{90} + 132 q^{91} + 132 q^{92} + 12 q^{93} + 8 q^{94} - 152 q^{95} - 76 q^{96} - 24 q^{97} - 48 q^{98} - 68 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −1.41182 − 0.0823192i 0.738774 − 3.07722i 1.98645 + 0.232439i 0.695843 − 0.814727i −1.29633 + 4.28365i −2.15580 − 0.341445i −2.78536 − 0.491684i −6.25046 − 3.18477i −1.04947 + 1.09296i
5.2 −1.41113 − 0.0932770i −0.649566 + 2.70564i 1.98260 + 0.263253i −2.56267 + 3.00050i 1.16900 − 3.75743i −1.80767 − 0.286306i −2.77316 − 0.556416i −4.22552 − 2.15301i 3.89614 − 3.99507i
5.3 −1.40637 − 0.148716i 0.504695 − 2.10221i 1.95577 + 0.418299i −2.24806 + 2.63214i −1.02242 + 2.88143i 1.59514 + 0.252645i −2.68833 − 0.879137i −1.49153 − 0.759974i 3.55305 − 3.36745i
5.4 −1.39808 + 0.212978i 0.118797 − 0.494823i 1.90928 − 0.595522i 1.66537 − 1.94989i −0.0607012 + 0.717106i 1.41232 + 0.223690i −2.54250 + 1.23922i 2.44228 + 1.24440i −1.91304 + 3.08080i
5.5 −1.35615 + 0.401056i −0.427613 + 1.78114i 1.67831 − 1.08779i 0.798011 − 0.934351i −0.134425 − 2.58699i −1.05423 − 0.166974i −1.83978 + 2.14830i −0.316571 − 0.161301i −0.707500 + 1.58717i
5.6 −1.34285 − 0.443581i −0.229576 + 0.956251i 1.60647 + 1.19132i −0.0398970 + 0.0467134i 0.732459 − 1.18226i 3.01590 + 0.477672i −1.62880 − 2.31236i 1.81131 + 0.922908i 0.0742968 − 0.0450314i
5.7 −1.32175 + 0.502961i 0.0699635 − 0.291419i 1.49406 − 1.32958i −1.39403 + 1.63220i 0.0540980 + 0.420372i −2.54848 − 0.403640i −1.30605 + 2.50883i 2.59299 + 1.32119i 1.02163 − 2.85850i
5.8 −1.23195 − 0.694480i −0.575954 + 2.39902i 1.03539 + 1.71113i 2.29907 − 2.69186i 2.37562 − 2.55548i −4.37967 − 0.693671i −0.0872092 − 2.82708i −2.75055 − 1.40148i −4.70178 + 1.71958i
5.9 −1.17766 − 0.783017i 0.173703 − 0.723525i 0.773769 + 1.84426i −0.596836 + 0.698806i −0.771096 + 0.716055i −4.47404 − 0.708619i 0.532846 − 2.77778i 2.17970 + 1.11061i 1.25005 − 0.355623i
5.10 −1.13292 + 0.846462i 0.508744 − 2.11907i 0.567005 − 1.91794i 1.09532 − 1.28245i 1.21735 + 2.83136i 5.18941 + 0.821922i 0.981095 + 2.65282i −1.55862 − 0.794156i −0.155358 + 2.38006i
5.11 −0.956907 − 1.04131i −0.684146 + 2.84967i −0.168659 + 1.99288i −0.167757 + 0.196418i 3.62206 − 2.01446i 3.70182 + 0.586310i 2.23659 − 1.73137i −4.97956 − 2.53721i 0.365059 − 0.0132666i
5.12 −0.918365 − 1.07546i 0.558117 − 2.32472i −0.313211 + 1.97532i 0.499422 − 0.584748i −3.01269 + 1.53471i −0.373062 − 0.0590872i 2.41201 − 1.47722i −2.41982 − 1.23296i −1.08752 9.41241e-5i
5.13 −0.911046 + 1.08166i −0.361228 + 1.50462i −0.339990 − 1.97089i 1.40625 − 1.64651i −1.29840 − 1.76151i −1.17467 − 0.186049i 2.44159 + 1.42782i 0.539615 + 0.274948i 0.499807 + 3.02113i
5.14 −0.886713 + 1.10170i −0.228515 + 0.951834i −0.427481 − 1.95378i −2.05225 + 2.40288i −0.846007 − 1.09576i 3.51420 + 0.556595i 2.53153 + 1.26149i 1.81925 + 0.926955i −0.827491 − 4.39163i
5.15 −0.853219 + 1.12784i 0.535652 − 2.23115i −0.544034 − 1.92458i 2.05057 − 2.40091i 2.05935 + 2.50779i −4.64073 − 0.735020i 2.63480 + 1.02851i −2.01809 − 1.02827i 0.958249 + 4.36121i
5.16 −0.750928 − 1.19838i −0.000523823 0.00218188i −0.872215 + 1.79979i 2.82960 − 3.31303i 0.00300807 − 0.00101070i 1.32527 + 0.209902i 2.81180 − 0.306269i 2.67302 + 1.36197i −6.09509 − 0.903076i
5.17 −0.745829 − 1.20156i 0.408333 − 1.70083i −0.887477 + 1.79231i −1.89197 + 2.21522i −2.34819 + 0.777894i 4.39034 + 0.695362i 2.81547 − 0.270406i −0.0530682 − 0.0270396i 4.07280 + 0.621140i
5.18 −0.591259 + 1.28468i 0.205717 − 0.856871i −1.30082 − 1.51916i −0.548568 + 0.642290i 0.979176 + 0.770914i 0.362741 + 0.0574525i 2.72077 − 0.772928i 1.98111 + 1.00943i −0.500794 − 1.08450i
5.19 −0.397914 + 1.35708i 0.756597 − 3.15145i −1.68333 − 1.08000i −2.26684 + 2.65413i 3.97571 + 2.28077i −0.688584 − 0.109061i 2.13547 − 1.85466i −6.68620 − 3.40679i −2.69986 − 4.13241i
5.20 −0.358537 − 1.36801i −0.130545 + 0.543760i −1.74290 + 0.980964i −0.262273 + 0.307082i 0.790675 − 0.0163708i −2.17329 − 0.344215i 1.96686 + 2.03260i 2.39439 + 1.22000i 0.514126 + 0.248692i
See next 80 embeddings (of 736 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
32.g even 8 1 inner
352.bf even 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 352.2.bf.a ✓ 736
11.c even 5 1 inner 352.2.bf.a ✓ 736
32.g even 8 1 inner 352.2.bf.a ✓ 736
352.bf even 40 1 inner 352.2.bf.a ✓ 736
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
352.2.bf.a ✓ 736 1.a even 1 1 trivial
352.2.bf.a ✓ 736 11.c even 5 1 inner
352.2.bf.a ✓ 736 32.g even 8 1 inner
352.2.bf.a ✓ 736 352.bf even 40 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(352, [\chi])\).