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Properties

Label 250.4.e.b
Level $250$
Weight $4$
Character orbit 250.e
Analytic conductor $14.750$
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] = [250,4,Mod(49,250)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(250, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([7]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("250.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 250 = 2 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 250.e (of order \(10\), degree \(4\), not 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: \(14.7504775014\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 50)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{4} - 12 q^{6} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{4} - 12 q^{6} + 26 q^{9} - 106 q^{11} - 80 q^{12} + 56 q^{14} - 128 q^{16} - 320 q^{17} + 110 q^{19} - 36 q^{21} + 360 q^{22} + 370 q^{23} - 192 q^{24} + 808 q^{26} + 1200 q^{27} + 120 q^{28} - 10 q^{29} - 486 q^{31} - 2560 q^{33} + 616 q^{34} - 104 q^{36} - 680 q^{37} + 1012 q^{39} - 96 q^{41} + 1020 q^{42} - 136 q^{44} - 832 q^{46} - 1040 q^{47} - 320 q^{48} - 2076 q^{49} + 884 q^{51} + 2550 q^{53} - 120 q^{54} - 224 q^{56} + 2250 q^{59} + 934 q^{61} - 4200 q^{62} - 4660 q^{63} + 512 q^{64} + 16 q^{66} + 3780 q^{67} - 628 q^{69} - 2616 q^{71} + 600 q^{73} - 2584 q^{74} + 800 q^{76} + 4320 q^{77} + 6640 q^{78} - 2800 q^{79} - 5268 q^{81} - 4050 q^{83} + 624 q^{84} - 692 q^{86} - 9390 q^{87} + 1680 q^{88} + 4520 q^{89} + 3764 q^{91} - 1280 q^{92} + 656 q^{94} - 192 q^{96} - 1710 q^{97} - 3280 q^{98} - 2108 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 −1.17557 + 1.61803i −6.75935 + 2.19625i −1.23607 − 3.80423i 0 4.39249 − 13.5187i − 25.2037i 7.60845 + 2.47214i 19.0219 − 13.8202i 0
49.2 −1.17557 + 1.61803i −4.78125 + 1.55352i −1.23607 − 3.80423i 0 3.10704 − 9.56250i 11.4048i 7.60845 + 2.47214i −1.39654 + 1.01465i 0
49.3 −1.17557 + 1.61803i 4.16531 − 1.35339i −1.23607 − 3.80423i 0 −2.70679 + 8.33063i 31.2051i 7.60845 + 2.47214i −6.32528 + 4.59559i 0
49.4 −1.17557 + 1.61803i 6.75819 − 2.19587i −1.23607 − 3.80423i 0 −4.39174 + 13.5164i − 23.3487i 7.60845 + 2.47214i 19.0078 − 13.8100i 0
49.5 1.17557 − 1.61803i −5.96909 + 1.93948i −1.23607 − 3.80423i 0 −3.87895 + 11.9382i 9.80956i −7.60845 − 2.47214i 10.0250 − 7.28362i 0
49.6 1.17557 − 1.61803i −0.234870 + 0.0763140i −1.23607 − 3.80423i 0 −0.152628 + 0.469741i − 19.9138i −7.60845 − 2.47214i −21.7941 + 15.8344i 0
49.7 1.17557 − 1.61803i 1.91394 − 0.621877i −1.23607 − 3.80423i 0 1.24375 − 3.82788i 29.3318i −7.60845 − 2.47214i −18.5670 + 13.4897i 0
49.8 1.17557 − 1.61803i 9.37926 − 3.04751i −1.23607 − 3.80423i 0 6.09501 − 18.7585i 2.82989i −7.60845 − 2.47214i 56.8397 − 41.2965i 0
99.1 −1.90211 + 0.618034i −5.49755 − 7.56672i 3.23607 − 2.35114i 0 15.1334 + 10.9951i − 11.0064i −4.70228 + 6.47214i −18.6888 + 57.5183i 0
99.2 −1.90211 + 0.618034i −1.87704 − 2.58353i 3.23607 − 2.35114i 0 5.16706 + 3.75409i 2.29951i −4.70228 + 6.47214i 5.19213 − 15.9797i 0
99.3 −1.90211 + 0.618034i 1.78443 + 2.45606i 3.23607 − 2.35114i 0 −4.91212 − 3.56886i − 1.93842i −4.70228 + 6.47214i 5.49543 − 16.9132i 0
99.4 −1.90211 + 0.618034i 5.11745 + 7.04356i 3.23607 − 2.35114i 0 −14.0871 − 10.2349i 24.3678i −4.70228 + 6.47214i −15.0801 + 46.4116i 0
99.5 1.90211 − 0.618034i −5.72432 − 7.87886i 3.23607 − 2.35114i 0 −15.7577 − 11.4486i 26.8849i 4.70228 − 6.47214i −20.9650 + 64.5237i 0
99.6 1.90211 − 0.618034i −2.86980 − 3.94994i 3.23607 − 2.35114i 0 −7.89989 − 5.73960i − 32.3828i 4.70228 − 6.47214i 0.977169 − 3.00742i 0
99.7 1.90211 − 0.618034i 0.184196 + 0.253524i 3.23607 − 2.35114i 0 0.507047 + 0.368391i 8.77601i 4.70228 − 6.47214i 8.31311 − 25.5851i 0
99.8 1.90211 − 0.618034i 4.41051 + 6.07054i 3.23607 − 2.35114i 0 12.1411 + 8.82101i − 17.5556i 4.70228 − 6.47214i −9.05545 + 27.8698i 0
149.1 −1.90211 − 0.618034i −5.49755 + 7.56672i 3.23607 + 2.35114i 0 15.1334 − 10.9951i 11.0064i −4.70228 − 6.47214i −18.6888 − 57.5183i 0
149.2 −1.90211 − 0.618034i −1.87704 + 2.58353i 3.23607 + 2.35114i 0 5.16706 − 3.75409i − 2.29951i −4.70228 − 6.47214i 5.19213 + 15.9797i 0
149.3 −1.90211 − 0.618034i 1.78443 − 2.45606i 3.23607 + 2.35114i 0 −4.91212 + 3.56886i 1.93842i −4.70228 − 6.47214i 5.49543 + 16.9132i 0
149.4 −1.90211 − 0.618034i 5.11745 − 7.04356i 3.23607 + 2.35114i 0 −14.0871 + 10.2349i − 24.3678i −4.70228 − 6.47214i −15.0801 − 46.4116i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.8
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 250.4.e.b 32
5.b even 2 1 50.4.e.a ✓ 32
5.c odd 4 1 250.4.d.c 32
5.c odd 4 1 250.4.d.d 32
25.d even 5 1 50.4.e.a ✓ 32
25.e even 10 1 inner 250.4.e.b 32
25.f odd 20 1 250.4.d.c 32
25.f odd 20 1 250.4.d.d 32
25.f odd 20 1 1250.4.a.m 16
25.f odd 20 1 1250.4.a.n 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.4.e.a ✓ 32 5.b even 2 1
50.4.e.a ✓ 32 25.d even 5 1
250.4.d.c 32 5.c odd 4 1
250.4.d.c 32 25.f odd 20 1
250.4.d.d 32 5.c odd 4 1
250.4.d.d 32 25.f odd 20 1
250.4.e.b 32 1.a even 1 1 trivial
250.4.e.b 32 25.e even 10 1 inner
1250.4.a.m 16 25.f odd 20 1
1250.4.a.n 16 25.f odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} - 121 T_{3}^{30} - 760 T_{3}^{29} + 15320 T_{3}^{28} + 91960 T_{3}^{27} + \cdots + 91\!\cdots\!76 \) acting on \(S_{4}^{\mathrm{new}}(250, [\chi])\). Copy content Toggle raw display