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Properties

Label 245.2.s.a
Level $245$
Weight $2$
Character orbit 245.s
Analytic conductor $1.956$
Analytic rank $0$
Dimension $312$
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] = [245,2,Mod(13,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([21, 22]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 245.s (of order \(28\), degree \(12\), 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: \(1.95633484952\)
Analytic rank: \(0\)
Dimension: \(312\)
Relative dimension: \(26\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 312 q - 10 q^{2} - 14 q^{3} - 14 q^{5} - 56 q^{6} - 18 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 312 q - 10 q^{2} - 14 q^{3} - 14 q^{5} - 56 q^{6} - 18 q^{7} - 6 q^{8} - 14 q^{10} - 28 q^{11} - 14 q^{12} - 14 q^{13} - 42 q^{15} + 4 q^{16} - 28 q^{17} - 44 q^{18} - 14 q^{20} - 8 q^{21} + 18 q^{22} - 26 q^{23} - 6 q^{25} - 14 q^{27} - 14 q^{28} - 20 q^{30} - 78 q^{32} - 14 q^{33} - 34 q^{35} - 80 q^{36} + 2 q^{37} - 42 q^{38} - 14 q^{40} + 78 q^{42} - 6 q^{43} - 126 q^{45} + 48 q^{46} + 28 q^{47} + 224 q^{50} + 20 q^{51} - 14 q^{52} + 40 q^{53} - 56 q^{55} + 72 q^{56} - 102 q^{57} + 94 q^{58} - 10 q^{60} + 28 q^{61} + 14 q^{62} - 6 q^{63} + 2 q^{65} + 168 q^{66} - 16 q^{67} - 162 q^{70} - 108 q^{71} - 156 q^{72} - 14 q^{73} + 70 q^{75} - 140 q^{76} - 10 q^{77} + 102 q^{78} + 164 q^{81} + 126 q^{82} - 56 q^{83} + 6 q^{85} - 68 q^{86} + 14 q^{87} + 132 q^{88} + 70 q^{90} - 188 q^{91} - 22 q^{92} + 70 q^{93} - 56 q^{95} - 280 q^{96} + 236 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −2.25615 − 1.41763i −2.19545 − 0.247368i 2.21277 + 4.59486i 0.182190 − 2.22863i 4.60260 + 3.67045i 1.47954 − 2.19339i 0.924819 − 8.20799i 1.83403 + 0.418606i −3.57044 + 4.76986i
13.2 −2.19512 − 1.37929i 1.36744 + 0.154074i 2.04836 + 4.25347i −0.865428 + 2.06180i −2.78919 − 2.22430i 0.0664125 − 2.64492i 0.789814 − 7.00980i −1.07863 − 0.246190i 4.74354 − 3.33224i
13.3 −2.05537 − 1.29147i 1.44020 + 0.162271i 1.68887 + 3.50697i −1.53927 − 1.62193i −2.75057 − 2.19351i −1.93974 + 1.79928i 0.514340 − 4.56489i −0.876944 − 0.200157i 1.06909 + 5.32158i
13.4 −1.64785 − 1.03541i 2.16061 + 0.243442i 0.775557 + 1.61046i 2.02872 − 0.940365i −3.30829 − 2.63827i 2.63140 + 0.275173i −0.0463088 + 0.411002i 1.68418 + 0.384403i −4.31669 − 0.550983i
13.5 −1.64045 − 1.03076i −1.74153 − 0.196223i 0.760842 + 1.57991i 2.18592 − 0.470927i 2.65464 + 2.11700i −2.44511 + 1.01067i −0.0534575 + 0.474448i 0.0696430 + 0.0158956i −4.07131 − 1.48063i
13.6 −1.43310 − 0.900479i −2.76257 − 0.311267i 0.375159 + 0.779025i 0.0175224 + 2.23600i 3.67876 + 2.93371i 2.33712 − 1.24011i −0.215152 + 1.90953i 4.61011 + 1.05223i 1.98836 − 3.22020i
13.7 −1.37812 − 0.865933i −0.824436 − 0.0928916i 0.281620 + 0.584790i −2.21626 − 0.296941i 1.05574 + 0.841923i 1.96168 + 1.77533i −0.246184 + 2.18495i −2.25372 − 0.514396i 2.79716 + 2.32836i
13.8 −1.22206 − 0.767874i 3.09193 + 0.348377i 0.0360434 + 0.0748448i −0.915744 + 2.03995i −3.51102 − 2.79995i 0.0648196 + 2.64496i −0.309769 + 2.74928i 6.51386 + 1.48675i 2.68553 − 1.78978i
13.9 −1.22098 − 0.767191i −0.0138562 − 0.00156122i 0.0344370 + 0.0715092i 0.649081 + 2.13979i 0.0157204 + 0.0125366i −2.29412 − 1.31797i −0.310092 + 2.75214i −2.92459 − 0.667520i 0.849113 − 3.11060i
13.10 −0.613327 − 0.385379i 2.59016 + 0.291840i −0.640114 − 1.32921i −1.80252 − 1.32322i −1.47614 − 1.17718i −0.138788 − 2.64211i −0.281854 + 2.50152i 3.69895 + 0.844261i 0.595592 + 1.50622i
13.11 −0.524855 − 0.329788i −3.36507 − 0.379152i −0.701055 − 1.45576i −1.73757 − 1.40743i 1.64113 + 1.30876i −2.61648 − 0.392483i −0.250945 + 2.22719i 8.25514 + 1.88418i 0.447816 + 1.31173i
13.12 −0.240927 − 0.151384i −1.09534 − 0.123415i −0.832639 − 1.72899i 1.06661 − 1.96529i 0.245213 + 0.195551i 0.786647 + 2.52610i −0.124854 + 1.10811i −1.74025 − 0.397201i −0.554487 + 0.312022i
13.13 −0.199991 − 0.125663i 0.801645 + 0.0903237i −0.843562 − 1.75168i 1.95541 + 1.08461i −0.148971 − 0.118801i 2.37551 − 1.16489i −0.104306 + 0.925740i −2.29031 − 0.522748i −0.254768 − 0.462634i
13.14 −0.180390 − 0.113347i 0.118994 + 0.0134074i −0.848074 − 1.76104i −1.84970 + 1.25643i −0.0199456 − 0.0159061i −2.64282 + 0.124571i −0.0943311 + 0.837212i −2.91080 − 0.664372i 0.476079 − 0.0169897i
13.15 0.271607 + 0.170662i 2.51922 + 0.283848i −0.823123 − 1.70923i 2.18989 − 0.452085i 0.635796 + 0.507030i −2.52976 + 0.774800i 0.139966 − 1.24223i 3.34112 + 0.762588i 0.671943 + 0.250942i
13.16 0.438293 + 0.275398i −2.22386 − 0.250569i −0.751511 − 1.56053i 0.702447 + 2.12287i −0.905695 − 0.722268i 0.584578 + 2.58036i 0.216297 − 1.91969i 1.95797 + 0.446894i −0.276755 + 1.12389i
13.17 0.855462 + 0.537522i −1.47707 − 0.166426i −0.424883 − 0.882278i −1.97154 + 1.05499i −1.17412 − 0.936332i 1.47269 − 2.19800i 0.337013 − 2.99107i −0.770733 − 0.175915i −2.25366 − 0.157243i
13.18 0.902776 + 0.567252i 1.94463 + 0.219107i −0.374538 − 0.777735i −1.32366 − 1.80220i 1.63127 + 1.30090i 2.20667 + 1.45966i 0.341801 − 3.03357i 0.808787 + 0.184600i −0.172664 − 2.37783i
13.19 0.907025 + 0.569922i −0.517400 − 0.0582969i −0.369884 − 0.768071i 0.252474 − 2.22177i −0.436070 − 0.347754i −1.76494 − 1.97104i 0.342123 − 3.03643i −2.66048 − 0.607237i 1.49523 − 1.87131i
13.20 1.27448 + 0.800808i 1.76594 + 0.198973i 0.115235 + 0.239288i −0.249633 + 2.22209i 2.09131 + 1.66777i 0.671406 + 2.55914i 0.292296 − 2.59420i 0.154165 + 0.0351871i −2.09762 + 2.63210i
See next 80 embeddings (of 312 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
49.f odd 14 1 inner
245.s even 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.2.s.a ✓ 312
5.c odd 4 1 inner 245.2.s.a ✓ 312
49.f odd 14 1 inner 245.2.s.a ✓ 312
245.s even 28 1 inner 245.2.s.a ✓ 312
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.s.a ✓ 312 1.a even 1 1 trivial
245.2.s.a ✓ 312 5.c odd 4 1 inner
245.2.s.a ✓ 312 49.f odd 14 1 inner
245.2.s.a ✓ 312 245.s even 28 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(245, [\chi])\).