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Properties

Label 525.2.z.a
Level $525$
Weight $2$
Character orbit 525.z
Analytic conductor $4.192$
Analytic rank $0$
Dimension $56$
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(64,525)]
 
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, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.z (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: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 12 q^{4} - 2 q^{5} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 12 q^{4} - 2 q^{5} + 14 q^{9} + 36 q^{10} + 12 q^{11} + 20 q^{13} - 4 q^{15} + 4 q^{16} - 22 q^{19} - 22 q^{20} + 14 q^{21} + 30 q^{22} - 10 q^{23} + 16 q^{25} - 60 q^{26} - 20 q^{28} - 18 q^{29} - 18 q^{30} + 16 q^{31} - 10 q^{33} + 6 q^{35} - 12 q^{36} + 10 q^{37} + 100 q^{38} - 22 q^{40} + 8 q^{41} - 66 q^{44} + 2 q^{45} + 40 q^{46} - 100 q^{47} - 56 q^{49} + 62 q^{50} + 32 q^{51} + 80 q^{52} - 30 q^{53} - 58 q^{55} - 40 q^{58} - 20 q^{59} + 66 q^{60} + 28 q^{61} - 50 q^{62} - 12 q^{64} + 78 q^{65} - 20 q^{67} + 4 q^{69} - 18 q^{70} + 40 q^{71} - 20 q^{73} + 100 q^{74} - 8 q^{75} - 164 q^{76} + 20 q^{77} - 90 q^{78} - 4 q^{79} - 158 q^{80} - 14 q^{81} + 30 q^{83} - 12 q^{84} + 46 q^{85} + 80 q^{86} - 40 q^{87} + 130 q^{88} - 38 q^{89} + 4 q^{90} - 100 q^{92} - 10 q^{94} + 8 q^{95} + 10 q^{96} - 130 q^{97} + 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1 −1.58398 − 2.18016i 0.951057 + 0.309017i −1.62607 + 5.00454i −0.548304 − 2.16780i −0.832747 − 2.56293i 1.00000i 8.36050 − 2.71649i 0.809017 + 0.587785i −3.85765 + 4.62914i
64.2 −1.23567 − 1.70075i −0.951057 − 0.309017i −0.747648 + 2.30102i 2.01518 + 0.969039i 0.649630 + 1.99936i − 1.00000i 0.838609 − 0.272480i 0.809017 + 0.587785i −0.842003 − 4.62474i
64.3 −1.17586 − 1.61843i −0.951057 − 0.309017i −0.618636 + 1.90397i −0.258970 + 2.22102i 0.618185 + 1.90258i − 1.00000i 0.00370838 − 0.00120493i 0.809017 + 0.587785i 3.89907 − 2.19248i
64.4 −1.00461 − 1.38273i 0.951057 + 0.309017i −0.284658 + 0.876087i −2.14464 + 0.632860i −0.528155 − 1.62549i 1.00000i −1.75363 + 0.569787i 0.809017 + 0.587785i 3.02960 + 2.32968i
64.5 −0.894373 − 1.23100i 0.951057 + 0.309017i −0.0974215 + 0.299833i 1.95748 + 1.08087i −0.470200 − 1.44713i 1.00000i −2.43803 + 0.792163i 0.809017 + 0.587785i −0.420165 − 3.37636i
64.6 −0.292589 − 0.402714i −0.951057 − 0.309017i 0.541463 − 1.66645i −1.58524 + 1.57703i 0.153823 + 0.473419i − 1.00000i −1.77637 + 0.577177i 0.809017 + 0.587785i 1.09892 + 0.176977i
64.7 0.158974 + 0.218810i −0.951057 − 0.309017i 0.595429 − 1.83254i −0.756459 − 2.10423i −0.0835778 − 0.257226i − 1.00000i 1.01009 − 0.328197i 0.809017 + 0.587785i 0.340167 − 0.500039i
64.8 0.341242 + 0.469679i 0.951057 + 0.309017i 0.513881 − 1.58156i 2.23604 + 0.0116437i 0.179402 + 0.552141i 1.00000i 2.02247 − 0.657140i 0.809017 + 0.587785i 0.757561 + 1.05419i
64.9 0.342442 + 0.471330i −0.951057 − 0.309017i 0.513148 − 1.57931i 2.09675 + 0.776951i −0.180032 − 0.554082i − 1.00000i 2.02826 − 0.659022i 0.809017 + 0.587785i 0.351813 + 1.25432i
64.10 0.507633 + 0.698697i 0.951057 + 0.309017i 0.387548 − 1.19275i −2.23580 + 0.0345861i 0.266878 + 0.821367i 1.00000i 2.67284 − 0.868457i 0.809017 + 0.587785i −1.15913 − 1.54459i
64.11 0.975068 + 1.34207i −0.951057 − 0.309017i −0.232349 + 0.715098i −2.23159 − 0.141462i −0.512624 − 1.57769i − 1.00000i 1.96912 − 0.639806i 0.809017 + 0.587785i −1.98610 − 3.13287i
64.12 1.14001 + 1.56909i 0.951057 + 0.309017i −0.544387 + 1.67545i −1.00833 + 1.99581i 0.599339 + 1.84458i 1.00000i 0.439612 − 0.142839i 0.809017 + 0.587785i −4.28112 + 0.693093i
64.13 1.22763 + 1.68969i −0.951057 − 0.309017i −0.729938 + 2.24652i 1.98040 − 1.03828i −0.645404 − 1.98635i − 1.00000i −0.719317 + 0.233720i 0.809017 + 0.587785i 4.18557 + 2.07164i
64.14 1.49408 + 2.05642i 0.951057 + 0.309017i −1.37856 + 4.24278i 1.10152 − 1.94593i 0.785482 + 2.41746i 1.00000i −5.94967 + 1.93316i 0.809017 + 0.587785i 5.64741 − 0.642187i
169.1 −2.50474 + 0.813838i 0.587785 + 0.809017i 3.99333 − 2.90133i 0.517719 − 2.17531i −2.13065 − 1.54801i − 1.00000i −4.54501 + 6.25567i −0.309017 + 0.951057i 0.473598 + 5.86991i
169.2 −2.47318 + 0.803584i −0.587785 − 0.809017i 3.85283 − 2.79924i −2.08479 − 0.808499i 2.10381 + 1.52851i 1.00000i −4.22228 + 5.81147i −0.309017 + 0.951057i 5.80574 + 0.324262i
169.3 −1.98360 + 0.644512i 0.587785 + 0.809017i 1.90126 − 1.38134i −1.68150 + 1.47397i −1.68735 − 1.22593i − 1.00000i −0.429178 + 0.590713i −0.309017 + 0.951057i 2.38543 − 4.00751i
169.4 −1.32106 + 0.429238i −0.587785 − 0.809017i −0.0570843 + 0.0414741i 0.747101 − 2.10757i 1.12376 + 0.816459i 1.00000i 1.69053 − 2.32681i −0.309017 + 0.951057i −0.0823162 + 3.10490i
169.5 −0.752353 + 0.244454i −0.587785 − 0.809017i −1.11176 + 0.807738i 2.10831 + 0.745003i 0.639990 + 0.464980i 1.00000i 1.56894 − 2.15946i −0.309017 + 0.951057i −1.76831 − 0.0451201i
169.6 −0.748945 + 0.243347i 0.587785 + 0.809017i −1.11633 + 0.811064i −1.45716 − 1.69608i −0.637090 − 0.462873i − 1.00000i 1.56445 − 2.15328i −0.309017 + 0.951057i 1.50407 + 0.915677i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.z.a ✓ 56
25.e even 10 1 inner 525.2.z.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.z.a ✓ 56 1.a even 1 1 trivial
525.2.z.a ✓ 56 25.e even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} - 20 T_{2}^{54} + 245 T_{2}^{52} - 2390 T_{2}^{50} - 70 T_{2}^{49} + 20680 T_{2}^{48} + \cdots + 9025 \) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display