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Properties

Label 555.2.bx.a
Level $555$
Weight $2$
Character orbit 555.bx
Analytic conductor $4.432$
Analytic rank $0$
Dimension $456$
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] = [555,2,Mod(22,555)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(555, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([0, 9, 31]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("555.22");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 555 = 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 555.bx (of order \(36\), 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: \(4.43169731218\)
Analytic rank: \(0\)
Dimension: \(456\)
Relative dimension: \(38\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 456 q + 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 456 q + 36 q^{8} + 48 q^{14} + 36 q^{17} + 60 q^{20} + 36 q^{25} - 48 q^{30} - 24 q^{35} + 24 q^{37} - 168 q^{38} - 48 q^{39} - 60 q^{41} + 360 q^{44} - 48 q^{48} - 48 q^{49} - 204 q^{50} - 60 q^{52} + 36 q^{53} + 72 q^{57} - 48 q^{58} + 72 q^{60} - 144 q^{61} + 48 q^{62} - 228 q^{64} - 36 q^{65} - 24 q^{67} + 120 q^{70} - 96 q^{71} - 180 q^{73} - 60 q^{74} - 120 q^{76} + 36 q^{77} + 72 q^{78} + 96 q^{79} - 264 q^{80} + 120 q^{83} - 240 q^{86} - 216 q^{88} + 24 q^{89} + 48 q^{91} - 72 q^{93} - 72 q^{94} + 12 q^{95} - 72 q^{97} + 180 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1 −2.51502 − 0.915392i −0.906308 − 0.422618i 3.95529 + 3.31888i 0.895573 + 2.04889i 1.89252 + 1.89252i −1.77292 + 1.24141i −4.23313 − 7.33199i 0.642788 + 0.766044i −0.376845 − 5.97280i
22.2 −2.46345 − 0.896623i 0.906308 + 0.422618i 3.73258 + 3.13200i −0.220417 − 2.22518i −1.85372 − 1.85372i 2.78240 − 1.94826i −3.76525 − 6.52160i 0.642788 + 0.766044i −1.45216 + 5.67925i
22.3 −2.33139 − 0.848557i −0.906308 − 0.422618i 3.18325 + 2.67106i −2.21197 + 0.327389i 1.75434 + 1.75434i 1.84444 − 1.29149i −2.67384 − 4.63123i 0.642788 + 0.766044i 5.43478 + 1.11371i
22.4 −2.20959 − 0.804224i 0.906308 + 0.422618i 2.70341 + 2.26843i −2.23607 − 0.00271234i −1.66269 − 1.66269i −3.86520 + 2.70644i −1.79769 − 3.11370i 0.642788 + 0.766044i 4.93860 + 1.80429i
22.5 −2.19026 − 0.797188i 0.906308 + 0.422618i 2.62962 + 2.20652i 1.90522 + 1.17052i −1.64814 − 1.64814i −2.02508 + 1.41798i −1.66972 − 2.89204i 0.642788 + 0.766044i −3.23980 − 4.08256i
22.6 −2.06031 − 0.749890i −0.906308 − 0.422618i 2.15043 + 1.80443i 1.40626 − 1.73851i 1.55035 + 1.55035i −1.87633 + 1.31382i −0.884896 − 1.53269i 0.642788 + 0.766044i −4.20102 + 2.52732i
22.7 −1.77883 − 0.647441i −0.906308 − 0.422618i 1.21296 + 1.01780i 2.14093 − 0.645316i 1.33855 + 1.33855i 3.27641 − 2.29416i 0.394297 + 0.682943i 0.642788 + 0.766044i −4.22615 − 0.238217i
22.8 −1.65574 − 0.602640i 0.906308 + 0.422618i 0.846213 + 0.710057i −1.52095 − 1.63912i −1.24592 − 1.24592i 1.52495 − 1.06778i 0.788802 + 1.36625i 0.642788 + 0.766044i 1.53050 + 3.63054i
22.9 −1.64320 − 0.598075i 0.906308 + 0.422618i 0.810314 + 0.679934i −0.662392 + 2.13571i −1.23648 − 1.23648i 0.676135 − 0.473435i 0.823799 + 1.42686i 0.642788 + 0.766044i 2.36575 − 3.11322i
22.10 −1.57067 − 0.571678i −0.906308 − 0.422618i 0.608106 + 0.510262i −1.63654 + 1.52372i 1.18191 + 1.18191i −3.48670 + 2.44141i 1.00804 + 1.74598i 0.642788 + 0.766044i 3.44155 − 1.45769i
22.11 −1.23603 − 0.449879i 0.906308 + 0.422618i −0.206703 − 0.173445i 0.625681 − 2.14675i −0.930099 − 0.930099i −0.189486 + 0.132680i 1.49282 + 2.58564i 0.642788 + 0.766044i −1.73914 + 2.37197i
22.12 −1.19480 − 0.434873i −0.906308 − 0.422618i −0.293649 − 0.246401i −1.09090 − 1.95190i 0.899074 + 0.899074i −0.617647 + 0.432481i 1.51518 + 2.62437i 0.642788 + 0.766044i 0.454585 + 2.80654i
22.13 −0.913220 − 0.332385i 0.906308 + 0.422618i −0.808599 − 0.678495i 2.21097 + 0.334055i −0.687186 − 0.687186i 3.20606 − 2.24491i 1.48473 + 2.57164i 0.642788 + 0.766044i −1.90807 − 1.03996i
22.14 −0.648426 − 0.236008i −0.906308 − 0.422618i −1.16733 − 0.979508i −0.608457 − 2.15169i 0.487933 + 0.487933i 3.67283 − 2.57175i 1.21580 + 2.10582i 0.642788 + 0.766044i −0.113276 + 1.53881i
22.15 −0.614395 − 0.223621i −0.906308 − 0.422618i −1.20461 − 1.01079i −1.42120 + 1.72633i 0.462324 + 0.462324i 3.23227 − 2.26326i 1.16790 + 2.02286i 0.642788 + 0.766044i 1.25922 − 0.742835i
22.16 −0.488876 − 0.177936i 0.906308 + 0.422618i −1.32475 − 1.11160i 1.96602 − 1.06526i −0.367873 − 0.367873i −4.20483 + 2.94425i 0.970096 + 1.68026i 0.642788 + 0.766044i −1.15069 + 0.170954i
22.17 −0.377581 − 0.137428i −0.906308 − 0.422618i −1.40841 − 1.18179i 2.19473 + 0.427975i 0.284125 + 0.284125i −2.46740 + 1.72770i 0.771189 + 1.33574i 0.642788 + 0.766044i −0.769872 − 0.463213i
22.18 −0.372244 − 0.135486i 0.906308 + 0.422618i −1.41188 − 1.18471i −1.99098 − 1.01783i −0.280109 − 0.280109i −2.54761 + 1.78385i 0.761187 + 1.31841i 0.642788 + 0.766044i 0.603231 + 0.648632i
22.19 −0.142814 − 0.0519801i 0.906308 + 0.422618i −1.51439 − 1.27073i 1.35364 + 1.77980i −0.107466 − 0.107466i 1.14891 − 0.804474i 0.302204 + 0.523433i 0.642788 + 0.766044i −0.100804 − 0.324542i
22.20 0.216485 + 0.0787943i −0.906308 − 0.422618i −1.49143 − 1.25146i 0.111562 + 2.23328i −0.162903 − 0.162903i 0.736167 − 0.515470i −0.454644 − 0.787467i 0.642788 + 0.766044i −0.151818 + 0.492264i
See next 80 embeddings (of 456 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.z even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 555.2.bx.a ✓ 456
5.c odd 4 1 555.2.ch.a yes 456
37.i odd 36 1 555.2.ch.a yes 456
185.z even 36 1 inner 555.2.bx.a ✓ 456
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.bx.a ✓ 456 1.a even 1 1 trivial
555.2.bx.a ✓ 456 185.z even 36 1 inner
555.2.ch.a yes 456 5.c odd 4 1
555.2.ch.a yes 456 37.i odd 36 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(555, [\chi])\).