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Properties

Label 275.2.t.a
Level $275$
Weight $2$
Character orbit 275.t
Analytic conductor $2.196$
Analytic rank $0$
Dimension $112$
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] = [275,2,Mod(14,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([3, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.14");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.t (of order \(10\), degree \(4\), 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: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(28\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 5 q^{2} + 25 q^{4} - 7 q^{5} - 3 q^{6} - 15 q^{7} - 5 q^{8} - 100 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q - 5 q^{2} + 25 q^{4} - 7 q^{5} - 3 q^{6} - 15 q^{7} - 5 q^{8} - 100 q^{9} - 9 q^{14} + 24 q^{15} - 27 q^{16} - 10 q^{17} - 5 q^{18} + 5 q^{19} + 27 q^{20} - 11 q^{21} - 50 q^{22} - 10 q^{23} + 22 q^{24} - 21 q^{25} + 11 q^{26} - 15 q^{28} + 11 q^{29} + 16 q^{30} + 10 q^{31} + 65 q^{33} - 4 q^{34} - 3 q^{35} - 36 q^{36} - 35 q^{38} - 82 q^{39} - 33 q^{40} - 14 q^{41} + 45 q^{42} + 23 q^{44} + 12 q^{45} - 33 q^{46} + 10 q^{48} - q^{49} + 48 q^{50} - 19 q^{51} + 35 q^{52} - 20 q^{53} + 13 q^{54} + 32 q^{55} - q^{56} + 50 q^{57} + 30 q^{58} - 7 q^{59} + 13 q^{60} + 22 q^{61} + 50 q^{62} + 25 q^{63} - 13 q^{64} + 60 q^{65} - 35 q^{66} - 15 q^{67} - 65 q^{68} - 40 q^{69} - 19 q^{70} - 28 q^{71} - 30 q^{72} - 20 q^{73} + 5 q^{74} + 60 q^{75} + 50 q^{76} - 50 q^{77} + 85 q^{78} + 11 q^{79} + 39 q^{80} + 32 q^{81} - 80 q^{82} - 55 q^{83} + 50 q^{84} + 68 q^{85} - 13 q^{86} - 10 q^{87} - 170 q^{88} + 4 q^{89} - 227 q^{90} + 18 q^{91} + 55 q^{93} + 74 q^{94} - 32 q^{95} + 238 q^{96} - 5 q^{97} - 40 q^{98} - 45 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −2.59478 − 0.843096i − 0.767711i 4.40405 + 3.19973i 1.66576 − 1.49172i −0.647254 + 1.99204i 0.839042 + 0.272621i −5.52254 − 7.60112i 2.41062 −5.57995 + 2.46630i
14.2 −2.55745 − 0.830966i 3.13796i 4.23202 + 3.07474i −1.63877 − 1.52132i 2.60754 − 8.02518i −1.57548 − 0.511905i −5.10699 − 7.02917i −6.84678 2.92691 + 5.25247i
14.3 −2.43242 − 0.790342i − 1.91347i 3.67401 + 2.66933i −0.635186 + 2.14395i −1.51230 + 4.65438i 0.179903 + 0.0584539i −3.82043 − 5.25837i −0.661378 3.23950 − 4.71299i
14.4 −2.11080 − 0.685842i 1.62840i 2.36708 + 1.71978i 0.729523 + 2.11372i 1.11682 − 3.43723i −0.541206 − 0.175848i −1.20784 − 1.66246i 0.348325 −0.0902048 − 4.96198i
14.5 −1.86879 − 0.607208i − 3.22322i 1.50565 + 1.09392i −1.02548 − 1.98706i −1.95716 + 6.02353i −2.52840 − 0.821528i 0.160436 + 0.220822i −7.38913 0.709846 + 4.33608i
14.6 −1.72777 − 0.561386i − 0.954902i 1.05199 + 0.764317i −1.05602 − 1.97099i −0.536068 + 1.64985i 4.67375 + 1.51859i 0.747117 + 1.02832i 2.08816 0.718070 + 3.99825i
14.7 −1.68518 − 0.547549i − 0.0850558i 0.922000 + 0.669872i −2.12259 + 0.703300i −0.0465722 + 0.143335i −3.52655 − 1.14585i 0.896050 + 1.23331i 2.99277 3.96204 − 0.0229685i
14.8 −1.59820 − 0.519288i 1.05092i 0.666560 + 0.484285i 1.38553 − 1.75508i 0.545732 − 1.67959i −4.24096 − 1.37797i 1.16167 + 1.59890i 1.89556 −3.12575 + 2.08548i
14.9 −1.13256 − 0.367991i − 1.49036i −0.470760 − 0.342027i 2.22482 + 0.223989i −0.548441 + 1.68793i 1.42538 + 0.463133i 1.80722 + 2.48743i 0.778813 −2.43732 − 1.07239i
14.10 −1.06046 − 0.344564i 2.25353i −0.612185 − 0.444778i −0.456645 − 2.18894i 0.776485 − 2.38978i 1.81830 + 0.590801i 1.80674 + 2.48677i −2.07839 −0.269978 + 2.47863i
14.11 −0.865077 − 0.281080i 2.89076i −0.948683 − 0.689258i −1.29223 + 1.82487i 0.812536 − 2.50073i 2.22372 + 0.722531i 1.69624 + 2.33467i −5.35649 1.63081 − 1.21543i
14.12 −0.811974 − 0.263826i − 0.366438i −1.02834 − 0.747130i 0.951890 + 2.02334i −0.0966761 + 0.297538i 0.115861 + 0.0376455i 1.64152 + 2.25937i 2.86572 −0.239100 − 1.89403i
14.13 −0.450191 − 0.146276i − 1.58953i −1.43676 − 1.04387i −2.23095 − 0.151255i −0.232510 + 0.715592i −0.490730 − 0.159448i 1.05059 + 1.44601i 0.473391 0.982227 + 0.394427i
14.14 −0.0103576 − 0.00336538i − 3.04074i −1.61794 − 1.17550i 0.102131 + 2.23373i −0.0102332 + 0.0314946i −2.92404 − 0.950078i 0.0256045 + 0.0352416i −6.24610 0.00645952 − 0.0234797i
14.15 0.00945998 + 0.00307373i 1.69155i −1.61795 − 1.17551i 2.05080 − 0.891184i −0.00519936 + 0.0160020i 1.10602 + 0.359369i −0.0233858 − 0.0321878i 0.138673 0.0221398 − 0.00212696i
14.16 0.377314 + 0.122597i − 2.95651i −1.49070 − 1.08306i 1.39438 − 1.74806i 0.362458 − 1.11553i 3.83523 + 1.24614i −0.896068 − 1.23333i −5.74093 0.740424 − 0.488622i
14.17 0.586551 + 0.190582i − 0.0710651i −1.31031 − 0.951998i −1.15414 + 1.91519i 0.0135437 − 0.0416833i 3.96127 + 1.28709i −1.31215 − 1.80602i 2.99495 −1.04196 + 0.903400i
14.18 0.604363 + 0.196370i 0.471833i −1.29134 − 0.938213i −0.900442 − 2.04675i −0.0926537 + 0.285159i −0.423214 − 0.137511i −1.34324 − 1.84881i 2.77737 −0.142274 − 1.41380i
14.19 0.773806 + 0.251425i 2.52599i −1.08247 − 0.786462i −2.20690 − 0.359974i −0.635096 + 1.95463i −3.68312 − 1.19672i −1.59636 − 2.19721i −3.38062 −1.61721 − 0.833420i
14.20 0.931499 + 0.302662i − 1.26928i −0.841948 − 0.611711i 2.01089 − 0.977916i 0.384162 − 1.18233i −4.02744 − 1.30859i −1.75053 − 2.40940i 1.38894 2.16912 − 0.302307i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
275.t even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.2.t.a yes 112
11.c even 5 1 275.2.n.a ✓ 112
25.e even 10 1 275.2.n.a ✓ 112
275.t even 10 1 inner 275.2.t.a yes 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.n.a ✓ 112 11.c even 5 1
275.2.n.a ✓ 112 25.e even 10 1
275.2.t.a yes 112 1.a even 1 1 trivial
275.2.t.a yes 112 275.t even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(275, [\chi])\).