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Properties

Label 525.2.n.e
Level $525$
Weight $2$
Character orbit 525.n
Analytic conductor $4.192$
Analytic rank $0$
Dimension $32$
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] = [525,2,Mod(106,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.106");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.n (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: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + q^{2} + 8 q^{3} - 3 q^{4} + 7 q^{5} - q^{6} + 32 q^{7} - 3 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + q^{2} + 8 q^{3} - 3 q^{4} + 7 q^{5} - q^{6} + 32 q^{7} - 3 q^{8} - 8 q^{9} + 30 q^{10} - 14 q^{11} + 13 q^{12} + 12 q^{13} + q^{14} + 8 q^{15} - 17 q^{16} + 4 q^{17} - 4 q^{18} - 9 q^{19} - 19 q^{20} + 8 q^{21} - q^{22} - 6 q^{23} + 18 q^{24} - 17 q^{25} + 82 q^{26} + 8 q^{27} - 3 q^{28} - 19 q^{29} + 10 q^{30} - 19 q^{31} - 10 q^{32} - q^{33} + 11 q^{34} + 7 q^{35} - 13 q^{36} - 14 q^{37} + 44 q^{38} - 12 q^{39} - 45 q^{40} + 15 q^{41} - q^{42} - 16 q^{43} - 23 q^{44} + 2 q^{45} + 10 q^{46} + 4 q^{47} + 17 q^{48} + 32 q^{49} - 95 q^{50} - 24 q^{51} + 23 q^{52} + 6 q^{53} - q^{54} - 21 q^{55} - 3 q^{56} - 26 q^{57} - 72 q^{58} - 48 q^{59} - 11 q^{60} - 44 q^{61} - 80 q^{62} - 8 q^{63} - 51 q^{64} - 50 q^{65} - 4 q^{66} + 2 q^{67} + 78 q^{68} + 6 q^{69} + 30 q^{70} - 3 q^{72} + 18 q^{73} + 34 q^{74} + 2 q^{75} + 32 q^{76} - 14 q^{77} + 38 q^{78} - 22 q^{79} + 8 q^{80} - 8 q^{81} + 62 q^{82} - 6 q^{83} + 13 q^{84} - 14 q^{85} - 96 q^{86} - 6 q^{87} + 113 q^{88} - 51 q^{89} + 12 q^{91} - 18 q^{92} - 46 q^{93} + 30 q^{94} + 4 q^{95} + 25 q^{96} + 56 q^{97} + q^{98} + 26 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
106.1 −0.739450 − 2.27579i 0.809017 − 0.587785i −3.01442 + 2.19010i 0.296681 + 2.21630i −1.93591 − 1.40652i 1.00000 3.34142 + 2.42769i 0.309017 − 0.951057i 4.82446 − 2.31403i
106.2 −0.552996 − 1.70195i 0.809017 − 0.587785i −0.972782 + 0.706768i −2.22155 + 0.254352i −1.44776 − 1.05186i 1.00000 −1.15470 − 0.838938i 0.309017 − 0.951057i 1.66140 + 3.64031i
106.3 −0.536523 − 1.65125i 0.809017 − 0.587785i −0.820731 + 0.596296i 1.74179 − 1.40220i −1.40464 − 1.02053i 1.00000 −1.38430 − 1.00575i 0.309017 − 0.951057i −3.24989 − 2.12383i
106.4 −0.0939661 − 0.289198i 0.809017 − 0.587785i 1.54323 − 1.12122i 0.314180 + 2.21389i −0.246007 − 0.178734i 1.00000 −0.961279 − 0.698410i 0.309017 − 0.951057i 0.610729 − 0.298890i
106.5 0.00587616 + 0.0180850i 0.809017 − 0.587785i 1.61774 − 1.17536i 0.200886 − 2.22703i 0.0153840 + 0.0111771i 1.00000 0.0615304 + 0.0447044i 0.309017 − 0.951057i 0.0414561 − 0.00945334i
106.6 0.426116 + 1.31145i 0.809017 − 0.587785i 0.0797057 − 0.0579096i 1.75772 + 1.38218i 1.11559 + 0.810521i 1.00000 2.34108 + 1.70090i 0.309017 − 0.951057i −1.06367 + 2.89413i
106.7 0.434585 + 1.33751i 0.809017 − 0.587785i 0.0179543 − 0.0130446i −0.889598 − 2.05149i 1.13776 + 0.826629i 1.00000 2.30076 + 1.67160i 0.309017 − 0.951057i 2.35729 − 2.08140i
106.8 0.747341 + 2.30008i 0.809017 − 0.587785i −3.11382 + 2.26232i 2.22695 + 0.201777i 1.95657 + 1.42153i 1.00000 −3.61747 − 2.62825i 0.309017 − 0.951057i 1.20019 + 5.27295i
211.1 −1.99857 + 1.45205i −0.309017 + 0.951057i 1.26781 − 3.90193i −2.13808 − 0.654681i −0.763386 − 2.34946i 1.00000 1.60519 + 4.94028i −0.809017 − 0.587785i 5.22374 − 1.79617i
211.2 −1.45586 + 1.05774i −0.309017 + 0.951057i 0.382665 − 1.17772i 2.23051 + 0.157617i −0.556087 − 1.71146i 1.00000 −0.423555 − 1.30357i −0.809017 − 0.587785i −3.41401 + 2.12983i
211.3 −0.939939 + 0.682906i −0.309017 + 0.951057i −0.200909 + 0.618333i 0.388818 − 2.20200i −0.359025 − 1.10496i 1.00000 −0.951471 − 2.92833i −0.809017 − 0.587785i 1.13830 + 2.33528i
211.4 −0.581451 + 0.422449i −0.309017 + 0.951057i −0.458412 + 1.41085i −1.95398 + 1.08719i −0.222095 − 0.683537i 1.00000 −0.773656 − 2.38107i −0.809017 − 0.587785i 0.676862 − 1.45760i
211.5 0.631204 − 0.458597i −0.309017 + 0.951057i −0.429926 + 1.32318i −0.785185 − 2.09368i 0.241099 + 0.742025i 1.00000 0.817630 + 2.51641i −0.809017 − 0.587785i −1.45577 − 0.961455i
211.6 1.12690 − 0.818742i −0.309017 + 0.951057i −0.0184645 + 0.0568279i 1.08970 + 1.95258i 0.430438 + 1.32475i 1.00000 0.886596 + 2.72866i −0.809017 − 0.587785i 2.82664 + 1.30818i
211.7 1.83906 − 1.33616i −0.309017 + 0.951057i 0.978804 − 3.01245i −0.524744 + 2.17362i 0.702460 + 2.16195i 1.00000 −0.820105 − 2.52402i −0.809017 − 0.587785i 1.93927 + 4.69858i
211.8 2.18766 − 1.58943i −0.309017 + 0.951057i 1.64155 − 5.05217i 1.76591 − 1.37170i 0.835613 + 2.57175i 1.00000 −2.76769 − 8.51806i −0.809017 − 0.587785i 1.68301 − 5.80761i
316.1 −1.99857 − 1.45205i −0.309017 − 0.951057i 1.26781 + 3.90193i −2.13808 + 0.654681i −0.763386 + 2.34946i 1.00000 1.60519 − 4.94028i −0.809017 + 0.587785i 5.22374 + 1.79617i
316.2 −1.45586 − 1.05774i −0.309017 − 0.951057i 0.382665 + 1.17772i 2.23051 − 0.157617i −0.556087 + 1.71146i 1.00000 −0.423555 + 1.30357i −0.809017 + 0.587785i −3.41401 − 2.12983i
316.3 −0.939939 − 0.682906i −0.309017 − 0.951057i −0.200909 − 0.618333i 0.388818 + 2.20200i −0.359025 + 1.10496i 1.00000 −0.951471 + 2.92833i −0.809017 + 0.587785i 1.13830 − 2.33528i
316.4 −0.581451 − 0.422449i −0.309017 − 0.951057i −0.458412 − 1.41085i −1.95398 − 1.08719i −0.222095 + 0.683537i 1.00000 −0.773656 + 2.38107i −0.809017 + 0.587785i 0.676862 + 1.45760i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 106.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.n.e ✓ 32
25.d even 5 1 inner 525.2.n.e ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.n.e ✓ 32 1.a even 1 1 trivial
525.2.n.e ✓ 32 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - T_{2}^{31} + 10 T_{2}^{30} - 10 T_{2}^{29} + 90 T_{2}^{28} - 43 T_{2}^{27} + 688 T_{2}^{26} + \cdots + 25 \) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display