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Properties

Label 531.2.m.a
Level $531$
Weight $2$
Character orbit 531.m
Analytic conductor $4.240$
Analytic rank $0$
Dimension $3248$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [531,2,Mod(4,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, base_ring=CyclotomicField(174))
 
chi = DirichletCharacter(H, H._module([58, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("531.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 531.m (of order \(87\), degree \(56\), 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: \(4.24005634733\)
Analytic rank: \(0\)
Dimension: \(3248\)
Relative dimension: \(58\) over \(\Q(\zeta_{87})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{87}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 3248 q - 29 q^{2} - 58 q^{3} + 29 q^{4} - 25 q^{5} - 66 q^{6} - 27 q^{7} - 116 q^{8} - 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 3248 q - 29 q^{2} - 58 q^{3} + 29 q^{4} - 25 q^{5} - 66 q^{6} - 27 q^{7} - 116 q^{8} - 62 q^{9} - 116 q^{10} - 33 q^{11} - 58 q^{12} - 27 q^{13} - 33 q^{14} - 51 q^{15} + 29 q^{16} - 126 q^{17} - 78 q^{18} - 120 q^{19} - 31 q^{20} - 60 q^{21} - 29 q^{22} - 29 q^{23} - 16 q^{24} + 29 q^{25} - 92 q^{26} - 55 q^{27} - 132 q^{28} - 27 q^{29} - 64 q^{30} - 21 q^{31} - 49 q^{32} - 32 q^{33} - 17 q^{34} - 136 q^{35} - 70 q^{36} - 120 q^{37} - 33 q^{38} - 80 q^{39} - 17 q^{40} - 17 q^{41} - 68 q^{42} - 21 q^{43} - 104 q^{44} - 58 q^{45} - 116 q^{46} - 17 q^{47} + 106 q^{48} + 19 q^{49} - 5 q^{50} - 250 q^{51} - 21 q^{52} - 108 q^{53} + 175 q^{54} - 116 q^{55} - q^{56} + 63 q^{57} - 58 q^{58} - 17 q^{59} - 816 q^{60} - 27 q^{61} - 128 q^{62} + 95 q^{63} - 256 q^{64} - 59 q^{65} + 67 q^{66} - 15 q^{67} - 43 q^{68} - 302 q^{69} - 41 q^{70} - 134 q^{71} + 210 q^{72} - 144 q^{73} - 51 q^{74} - 47 q^{75} - 9 q^{76} - 85 q^{77} + 36 q^{78} - 39 q^{79} - 128 q^{80} - 22 q^{81} - 164 q^{82} - 35 q^{83} - 190 q^{84} - 17 q^{85} + 31 q^{86} - 75 q^{87} - 29 q^{88} - 36 q^{89} - 86 q^{90} - 136 q^{91} - 31 q^{92} - 41 q^{94} - 13 q^{95} - 30 q^{96} - 3 q^{97} - 156 q^{98} - 86 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −1.11737 − 2.53423i 0.109412 + 1.72859i −3.82487 + 4.18674i 2.16943 − 0.910101i 4.25840 − 2.20874i −1.18066 + 0.171731i 9.63467 + 3.24630i −2.97606 + 0.378256i −4.73045 − 4.48092i
4.2 −1.08034 − 2.45026i 0.836206 − 1.51683i −3.48769 + 3.81766i −3.50876 + 1.47197i −4.62001 − 0.410233i −1.37358 + 0.199792i 8.04683 + 2.71129i −1.60152 − 2.53676i 7.39737 + 7.00716i
4.3 −1.07888 − 2.44694i −0.114188 − 1.72828i −3.47456 + 3.80329i 2.48452 − 1.04229i −4.10580 + 2.14401i 4.47379 − 0.650731i 7.98656 + 2.69099i −2.97392 + 0.394700i −5.23090 − 4.95498i
4.4 −1.03633 − 2.35043i −1.73198 + 0.0156715i −3.10161 + 3.39505i −1.79241 + 0.751936i 1.83173 + 4.05466i −2.03940 + 0.296638i 6.32554 + 2.13132i 2.99951 − 0.0542854i 3.62489 + 3.43368i
4.5 −0.992901 − 2.25194i 1.08539 + 1.34979i −2.73643 + 2.99533i −3.27551 + 1.37412i 1.96197 − 3.78444i 3.11928 − 0.453711i 4.79775 + 1.61655i −0.643876 + 2.93009i 6.34669 + 6.01190i
4.6 −0.986609 − 2.23767i −1.69894 + 0.337037i −2.68482 + 2.93883i 1.19347 − 0.500676i 2.43037 + 3.46915i 1.19400 − 0.173672i 4.58999 + 1.54655i 2.77281 − 1.14521i −2.29784 − 2.17663i
4.7 −0.967951 − 2.19536i 1.72737 − 0.127256i −2.53369 + 2.77341i 0.812833 − 0.340993i −1.95138 − 3.66901i 3.32533 − 0.483682i 3.99375 + 1.34565i 2.96761 − 0.439635i −1.53538 − 1.45439i
4.8 −0.917418 − 2.08074i 1.04953 − 1.37785i −2.13887 + 2.34123i 3.27750 − 1.37495i −3.82982 − 0.919745i −4.96812 + 0.722633i 2.52379 + 0.850364i −0.796955 − 2.89221i −5.86775 − 5.55823i
4.9 −0.914097 − 2.07321i 1.47791 + 0.903206i −2.11367 + 2.31364i −0.362328 + 0.152001i 0.521583 − 3.88964i −3.18656 + 0.463498i 2.43442 + 0.820250i 1.36844 + 2.66971i 0.646333 + 0.612240i
4.10 −0.864970 − 1.96179i −0.788647 − 1.54209i −1.75148 + 1.91719i −0.481587 + 0.202031i −2.34309 + 2.88102i −1.07926 + 0.156983i 1.21254 + 0.408551i −1.75607 + 2.43233i 0.812901 + 0.770021i
4.11 −0.774002 − 1.75547i −0.188020 + 1.72182i −1.13364 + 1.24089i 0.0443068 − 0.0185872i 3.16812 − 1.00263i −0.953357 + 0.138670i −0.580418 − 0.195566i −2.92930 − 0.647471i −0.0669229 − 0.0633927i
4.12 −0.738184 − 1.67423i −1.32405 − 1.11665i −0.909179 + 0.995197i −2.29772 + 0.963923i −0.892133 + 3.04105i 3.68113 − 0.535435i −1.13060 − 0.380943i 0.506207 + 2.95698i 3.30997 + 3.13537i
4.13 −0.721710 − 1.63687i −1.14463 + 1.29993i −0.809512 + 0.886101i 3.57669 − 1.50047i 2.95390 + 0.935442i 1.52572 − 0.221922i −1.35587 − 0.456846i −0.379630 − 2.97588i −5.03740 − 4.77168i
4.14 −0.713364 − 1.61794i 1.54698 + 0.779016i −0.759884 + 0.831777i 3.58957 − 1.50587i 0.156844 − 3.05864i 0.869170 − 0.126424i −1.46349 − 0.493107i 1.78627 + 2.41024i −4.99708 − 4.73348i
4.15 −0.700893 − 1.58966i −1.19780 + 1.25111i −0.686795 + 0.751772i −2.23191 + 0.936314i 2.82836 + 1.02720i 2.79042 − 0.405878i −1.61631 − 0.544599i −0.130538 − 2.99716i 3.05275 + 2.89172i
4.16 −0.585309 − 1.32751i 1.25099 − 1.19792i −0.0707233 + 0.0774144i −2.01564 + 0.845586i −2.32246 − 0.959547i 3.08503 − 0.448730i −2.60557 − 0.877919i 0.129969 − 2.99718i 2.30229 + 2.18085i
4.17 −0.584975 − 1.32675i 1.38185 − 1.04426i −0.0691071 + 0.0756453i 1.01737 − 0.426799i −2.19382 − 1.22251i 1.15974 − 0.168689i −2.60738 − 0.878528i 0.819038 − 2.88603i −1.16139 − 1.10013i
4.18 −0.548422 − 1.24385i 1.72799 + 0.118560i 0.102576 − 0.112281i −2.33072 + 0.977763i −0.800197 − 2.21437i −3.60762 + 0.524743i −2.77236 − 0.934117i 2.97189 + 0.409740i 2.49440 + 2.36282i
4.19 −0.478377 − 1.08498i −1.62569 + 0.597611i 0.400626 − 0.438530i 0.172397 − 0.0723227i 1.42609 + 1.47795i −4.30271 + 0.625846i −2.91482 − 0.982117i 2.28572 − 1.94306i −0.160939 − 0.152450i
4.20 −0.470938 − 1.06811i −1.40021 − 1.01952i 0.429888 − 0.470560i 2.30323 − 0.966231i −0.429549 + 1.97570i −1.82284 + 0.265139i −2.91749 − 0.983017i 0.921150 + 2.85508i −2.11672 − 2.00506i
See next 80 embeddings (of 3248 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
59.c even 29 1 inner
531.m even 87 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 531.2.m.a ✓ 3248
9.c even 3 1 inner 531.2.m.a ✓ 3248
59.c even 29 1 inner 531.2.m.a ✓ 3248
531.m even 87 1 inner 531.2.m.a ✓ 3248
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
531.2.m.a ✓ 3248 1.a even 1 1 trivial
531.2.m.a ✓ 3248 9.c even 3 1 inner
531.2.m.a ✓ 3248 59.c even 29 1 inner
531.2.m.a ✓ 3248 531.m even 87 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(531, [\chi])\).