Properties

Label 825.2.p.b
Level $825$
Weight $2$
Character orbit 825.p
Analytic conductor $6.588$
Analytic rank $0$
Dimension $116$
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] = [825,2,Mod(181,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.p (of order \(5\), 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: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(116\)
Relative dimension: \(29\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 116 q + 3 q^{2} + 116 q^{3} - 25 q^{4} + 9 q^{5} + 3 q^{6} - 3 q^{7} - 11 q^{8} + 116 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 116 q + 3 q^{2} + 116 q^{3} - 25 q^{4} + 9 q^{5} + 3 q^{6} - 3 q^{7} - 11 q^{8} + 116 q^{9} - 14 q^{10} + 9 q^{11} - 25 q^{12} + 52 q^{13} - 9 q^{14} + 9 q^{15} - 45 q^{16} - 8 q^{17} + 3 q^{18} - 5 q^{19} + 12 q^{20} - 3 q^{21} + 26 q^{22} - 4 q^{23} - 11 q^{24} + 15 q^{25} - 37 q^{26} + 116 q^{27} + 7 q^{28} - 6 q^{29} - 14 q^{30} + 50 q^{31} + 86 q^{32} + 9 q^{33} - 31 q^{34} - 21 q^{35} - 25 q^{36} + 4 q^{37} + 55 q^{38} + 52 q^{39} - 71 q^{40} + 13 q^{41} - 9 q^{42} + 28 q^{43} - 5 q^{44} + 9 q^{45} - 45 q^{46} - 42 q^{47} - 45 q^{48} - 6 q^{49} - 64 q^{50} - 8 q^{51} - 48 q^{52} + 7 q^{53} + 3 q^{54} - 25 q^{55} + 7 q^{56} - 5 q^{57} - 29 q^{58} + 12 q^{60} - 6 q^{61} - 63 q^{62} - 3 q^{63} - 75 q^{64} - 49 q^{65} + 26 q^{66} + 11 q^{67} - 50 q^{68} - 4 q^{69} - 78 q^{70} + 11 q^{71} - 11 q^{72} + 9 q^{73} + 88 q^{74} + 15 q^{75} - 4 q^{76} + 11 q^{77} - 37 q^{78} - 33 q^{79} - 98 q^{80} + 116 q^{81} + q^{82} - 68 q^{83} + 7 q^{84} - 23 q^{85} + 33 q^{86} - 6 q^{87} + 2 q^{88} - 2 q^{89} - 14 q^{90} + 28 q^{91} - 52 q^{92} + 50 q^{93} - 9 q^{94} + 26 q^{95} + 86 q^{96} - 31 q^{97} - 91 q^{98} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1 −2.24910 + 1.63407i 1.00000 1.77024 5.44824i 0.914795 2.04038i −2.24910 + 1.63407i 0.0784489 0.0569965i 3.20318 + 9.85837i 1.00000 1.27665 + 6.08385i
181.2 −2.21592 + 1.60996i 1.00000 1.70029 5.23297i 0.903122 + 2.04557i −2.21592 + 1.60996i −3.85379 + 2.79994i 2.96434 + 9.12330i 1.00000 −5.29454 3.07884i
181.3 −2.01843 + 1.46648i 1.00000 1.30548 4.01784i −1.24961 + 1.85431i −2.01843 + 1.46648i 3.43159 2.49320i 1.71511 + 5.27857i 1.00000 −0.197063 5.57532i
181.4 −1.85827 + 1.35011i 1.00000 1.01233 3.11564i 2.13700 + 0.658209i −1.85827 + 1.35011i 3.74931 2.72403i 0.905688 + 2.78742i 1.00000 −4.85978 + 1.66206i
181.5 −1.85228 + 1.34576i 1.00000 1.00183 3.08331i −2.15795 0.585878i −1.85228 + 1.34576i −0.321352 + 0.233476i 0.878713 + 2.70440i 1.00000 4.78557 1.81887i
181.6 −1.52829 + 1.11037i 1.00000 0.484719 1.49181i −0.921953 + 2.03716i −1.52829 + 1.11037i 0.173367 0.125958i −0.251841 0.775087i 1.00000 −0.852979 4.13707i
181.7 −1.46778 + 1.06640i 1.00000 0.399122 1.22837i 1.84597 + 1.26190i −1.46778 + 1.06640i −2.35600 + 1.71173i −0.397167 1.22235i 1.00000 −4.05517 + 0.116353i
181.8 −1.05047 + 0.763209i 1.00000 −0.0970409 + 0.298661i 2.18286 0.484899i −1.05047 + 0.763209i −0.171244 + 0.124416i −0.928488 2.85759i 1.00000 −1.92294 + 2.17535i
181.9 −1.02297 + 0.743233i 1.00000 −0.123957 + 0.381499i −2.20077 + 0.395720i −1.02297 + 0.743233i −2.88045 + 2.09277i −0.938220 2.88754i 1.00000 1.95722 2.04050i
181.10 −0.877840 + 0.637788i 1.00000 −0.254204 + 0.782361i −0.399046 2.20017i −0.877840 + 0.637788i −2.74356 + 1.99331i −0.946440 2.91284i 1.00000 1.75354 + 1.67689i
181.11 −0.814826 + 0.592006i 1.00000 −0.304563 + 0.937350i −2.23603 0.0137857i −0.814826 + 0.592006i 2.67438 1.94305i −0.929222 2.85985i 1.00000 1.83013 1.31251i
181.12 −0.492147 + 0.357566i 1.00000 −0.503678 + 1.55016i 0.646593 + 2.14054i −0.492147 + 0.357566i 2.83888 2.06257i −0.682368 2.10011i 1.00000 −1.08360 0.822262i
181.13 −0.427030 + 0.310255i 1.00000 −0.531938 + 1.63714i 2.19795 0.411090i −0.427030 + 0.310255i −1.87664 + 1.36346i −0.606999 1.86815i 1.00000 −0.811049 + 0.857475i
181.14 −0.187290 + 0.136074i 1.00000 −0.601473 + 1.85114i −0.345638 2.20919i −0.187290 + 0.136074i −1.94890 + 1.41596i −0.282319 0.868889i 1.00000 0.365348 + 0.366727i
181.15 −0.0469946 + 0.0341436i 1.00000 −0.616991 + 1.89890i −1.58251 1.57976i −0.0469946 + 0.0341436i 2.70882 1.96807i −0.0717408 0.220795i 1.00000 0.128308 + 0.0202077i
181.16 0.0949068 0.0689538i 1.00000 −0.613781 + 1.88902i 1.85498 1.24861i 0.0949068 0.0689538i 1.41611 1.02886i 0.144506 + 0.444743i 1.00000 0.0899537 0.246410i
181.17 0.280424 0.203740i 1.00000 −0.580906 + 1.78785i −1.32433 + 1.80171i 0.280424 0.203740i −1.31728 + 0.957058i 0.415581 + 1.27903i 1.00000 −0.00429482 + 0.775062i
181.18 0.646368 0.469614i 1.00000 −0.420780 + 1.29503i 1.72428 + 1.42368i 0.646368 0.469614i 0.903501 0.656432i 0.829965 + 2.55437i 1.00000 1.78310 + 0.110476i
181.19 0.878025 0.637923i 1.00000 −0.254051 + 0.781888i −1.04939 + 1.97453i 0.878025 0.637923i 2.61952 1.90319i 0.946473 + 2.91294i 1.00000 0.338212 + 2.40312i
181.20 1.13803 0.826829i 1.00000 −0.00656158 + 0.0201945i 2.21577 + 0.300584i 1.13803 0.826829i −2.29890 + 1.67025i 0.878610 + 2.70408i 1.00000 2.77015 1.48999i
See next 80 embeddings (of 116 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.29
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
275.k even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.p.b 116
11.c even 5 1 825.2.r.b yes 116
25.d even 5 1 825.2.r.b yes 116
275.k even 5 1 inner 825.2.p.b 116
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.p.b 116 1.a even 1 1 trivial
825.2.p.b 116 275.k even 5 1 inner
825.2.r.b yes 116 11.c even 5 1
825.2.r.b yes 116 25.d even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{116} - 3 T_{2}^{115} + 46 T_{2}^{114} - 116 T_{2}^{113} + 1134 T_{2}^{112} - 2627 T_{2}^{111} + 20690 T_{2}^{110} - 45716 T_{2}^{109} + 312863 T_{2}^{108} - 669700 T_{2}^{107} + 4086294 T_{2}^{106} - 8492312 T_{2}^{105} + \cdots + 262278025 \) acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\). Copy content Toggle raw display