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Properties

Label 880.2.bh.a
Level $880$
Weight $2$
Character orbit 880.bh
Analytic conductor $7.027$
Analytic rank $0$
Dimension $240$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [880,2,Mod(309,880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(880, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.309");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.bh (of order \(4\), degree \(2\), 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: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(120\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 12 q^{10} - 8 q^{14} - 8 q^{16} + 16 q^{19} + 28 q^{20} - 8 q^{24} + 68 q^{30} - 48 q^{31} - 56 q^{34} + 24 q^{35} - 112 q^{36} - 72 q^{40} + 8 q^{44} - 32 q^{46} + 240 q^{49} - 64 q^{50} + 16 q^{51} - 32 q^{54} - 16 q^{59} - 92 q^{60} + 32 q^{61} + 24 q^{64} + 16 q^{65} - 32 q^{69} - 48 q^{70} - 56 q^{75} + 184 q^{76} - 32 q^{79} + 8 q^{80} - 240 q^{81} + 152 q^{84} + 104 q^{86} + 52 q^{90} - 16 q^{91} - 56 q^{94} - 48 q^{95} - 152 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
309.1 −1.41362 − 0.0411133i −1.75781 − 1.75781i 1.99662 + 0.116237i 1.40008 + 1.74350i 2.41260 + 2.55714i 2.70941 −2.81767 − 0.246402i 3.17981i −1.90749 − 2.52219i
309.2 −1.41326 − 0.0518831i 2.19745 + 2.19745i 1.99462 + 0.146649i 2.11042 + 0.738991i −2.99156 − 3.21959i 3.48230 −2.81131 − 0.310740i 6.65759i −2.94424 − 1.15388i
309.3 −1.41246 + 0.0704856i 1.35014 + 1.35014i 1.99006 − 0.199115i −0.174640 + 2.22924i −2.00218 − 1.81184i −2.92492 −2.79684 + 0.421513i 0.645743i 0.0895419 − 3.16101i
309.4 −1.40964 + 0.113668i 0.241370 + 0.241370i 1.97416 − 0.320460i 0.606698 − 2.15219i −0.367680 − 0.312808i −3.70423 −2.74642 + 0.676131i − 2.88348i −0.610591 + 3.10277i
309.5 −1.40791 − 0.133346i −0.791104 − 0.791104i 1.96444 + 0.375479i −2.22023 + 0.265628i 1.00832 + 1.21930i −0.327456 −2.71569 − 0.790591i − 1.74831i 3.16132 − 0.0779211i
309.6 −1.39684 − 0.221014i −2.12980 − 2.12980i 1.90231 + 0.617441i 0.191600 − 2.22784i 2.50427 + 3.44570i −3.36707 −2.52075 − 1.28290i 6.07211i −0.760019 + 3.06959i
309.7 −1.39142 + 0.252888i 1.29286 + 1.29286i 1.87210 − 0.703746i −2.22452 − 0.226952i −2.12586 − 1.47196i −1.90710 −2.42690 + 1.45264i 0.342972i 3.15263 − 0.246769i
309.8 −1.38787 − 0.271693i 0.700741 + 0.700741i 1.85237 + 0.754149i 1.68625 + 1.46852i −0.782151 − 1.16292i −1.44316 −2.36595 − 1.54994i − 2.01792i −1.94131 − 2.49626i
309.9 −1.37493 + 0.330992i 2.27539 + 2.27539i 1.78089 − 0.910184i −1.41815 − 1.72883i −3.88165 − 2.37538i 0.0973471 −2.14734 + 1.84090i 7.35482i 2.52209 + 1.90763i
309.10 −1.36157 + 0.382259i −1.47965 − 1.47965i 1.70776 − 1.04095i 1.78898 − 1.34147i 2.58026 + 1.44904i 0.689816 −1.92732 + 2.07013i 1.37872i −1.92304 + 2.51036i
309.11 −1.35758 + 0.396201i 0.784940 + 0.784940i 1.68605 − 1.07575i 2.07535 − 0.832425i −1.37661 − 0.754625i 1.64919 −1.86274 + 2.12843i − 1.76774i −2.48764 + 1.95234i
309.12 −1.34848 + 0.426152i 0.438085 + 0.438085i 1.63679 − 1.14931i −1.64618 + 1.51330i −0.777440 − 0.404058i 4.03382 −1.71739 + 2.24735i − 2.61616i 1.57494 − 2.74218i
309.13 −1.34667 + 0.431827i −0.561518 − 0.561518i 1.62705 − 1.16306i 0.862844 + 2.06289i 0.998659 + 0.513702i 2.53358 −1.68887 + 2.26886i − 2.36940i −2.05278 − 2.40543i
309.14 −1.34002 − 0.452053i −1.26958 − 1.26958i 1.59130 + 1.21152i 2.16411 − 0.562710i 1.12734 + 2.27518i 0.634467 −1.58470 − 2.34280i 0.223665i −3.15432 − 0.224248i
309.15 −1.32278 − 0.500258i 1.68081 + 1.68081i 1.49948 + 1.32346i −1.32411 + 1.80187i −1.38250 − 3.06418i 3.87870 −1.32141 − 2.50077i 2.65026i 2.65290 − 1.72108i
309.16 −1.29996 − 0.556866i 1.68503 + 1.68503i 1.37980 + 1.44781i 0.0798280 − 2.23464i −1.25214 − 3.12880i −0.237784 −0.987455 − 2.65046i 2.67862i −1.34817 + 2.86050i
309.17 −1.28084 − 0.599532i −0.782530 − 0.782530i 1.28112 + 1.53581i −1.26550 + 1.84351i 0.533148 + 1.47145i −1.91488 −0.720150 − 2.73521i − 1.77529i 2.72614 − 1.60254i
309.18 −1.27879 − 0.603910i −2.25404 − 2.25404i 1.27058 + 1.54454i −2.16203 − 0.570645i 1.52120 + 4.24367i 3.77433 −0.692039 − 2.74246i 7.16140i 2.42015 + 2.03540i
309.19 −1.27480 + 0.612275i −1.74236 − 1.74236i 1.25024 − 1.56106i −1.98320 − 1.03291i 3.28797 + 1.15436i −1.55173 −0.638007 + 2.75553i 3.07163i 3.16062 + 0.102490i
309.20 −1.24787 + 0.665441i −1.02409 − 1.02409i 1.11438 − 1.66077i 1.91036 + 1.16212i 1.95941 + 0.596463i −4.54931 −0.285452 + 2.81399i − 0.902478i −3.15721 − 0.178944i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 309.120
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
16.e even 4 1 inner
80.q even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.bh.a ✓ 240
5.b even 2 1 inner 880.2.bh.a ✓ 240
16.e even 4 1 inner 880.2.bh.a ✓ 240
80.q even 4 1 inner 880.2.bh.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.2.bh.a ✓ 240 1.a even 1 1 trivial
880.2.bh.a ✓ 240 5.b even 2 1 inner
880.2.bh.a ✓ 240 16.e even 4 1 inner
880.2.bh.a ✓ 240 80.q even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(880, [\chi])\).