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Properties

Label 950.2.x.a
Level $950$
Weight $2$
Character orbit 950.x
Analytic conductor $7.586$
Analytic rank $0$
Dimension $400$
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] = [950,2,Mod(159,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([21, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.159");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.x (of order \(30\), degree \(8\), 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: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(400\)
Relative dimension: \(50\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 400 q - 50 q^{4} + 2 q^{5} - 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 400 q - 50 q^{4} + 2 q^{5} - 50 q^{9} - 12 q^{11} + 8 q^{14} - 18 q^{15} + 50 q^{16} - 10 q^{17} - 12 q^{19} + 4 q^{20} - 32 q^{21} + 20 q^{22} + 30 q^{23} - 14 q^{25} + 60 q^{27} + 24 q^{29} - 52 q^{30} + 20 q^{33} + 8 q^{34} - 8 q^{35} + 50 q^{36} - 24 q^{39} - 16 q^{41} + 4 q^{44} - 144 q^{45} - 32 q^{46} + 120 q^{47} - 440 q^{49} - 40 q^{50} + 52 q^{51} + 40 q^{53} - 12 q^{54} - 88 q^{55} + 16 q^{56} + 48 q^{59} - 2 q^{60} - 28 q^{61} + 10 q^{63} + 100 q^{64} + 152 q^{65} + 16 q^{66} - 80 q^{67} - 16 q^{69} + 8 q^{70} - 14 q^{71} + 80 q^{73} - 104 q^{75} - 8 q^{76} - 80 q^{77} + 60 q^{78} - 8 q^{79} + 2 q^{80} + 106 q^{81} + 80 q^{83} + 56 q^{84} + 78 q^{85} - 20 q^{86} + 80 q^{87} + 12 q^{89} + 78 q^{90} - 8 q^{91} - 20 q^{92} + 4 q^{95} - 30 q^{97} - 40 q^{98} - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
159.1 −0.994522 − 0.104528i −2.37897 − 2.14203i 0.978148 + 0.207912i 2.19514 + 0.425845i 2.14203 + 2.37897i − 3.35325i −0.951057 − 0.309017i 0.757599 + 7.20807i −2.13861 − 0.652967i
159.2 −0.994522 − 0.104528i −2.37883 − 2.14191i 0.978148 + 0.207912i −2.23592 + 0.0257798i 2.14191 + 2.37883i − 0.582392i −0.951057 − 0.309017i 0.757479 + 7.20693i 2.22637 + 0.208079i
159.3 −0.994522 − 0.104528i −1.73756 − 1.56450i 0.978148 + 0.207912i −0.908640 − 2.04313i 1.56450 + 1.73756i 2.62111i −0.951057 − 0.309017i 0.257848 + 2.45326i 0.690097 + 2.12691i
159.4 −0.994522 − 0.104528i −1.62203 − 1.46048i 0.978148 + 0.207912i 0.144205 + 2.23141i 1.46048 + 1.62203i 0.133313i −0.951057 − 0.309017i 0.184386 + 1.75432i 0.0898316 − 2.23426i
159.5 −0.994522 − 0.104528i −1.46570 − 1.31972i 0.978148 + 0.207912i 2.21506 − 0.305758i 1.31972 + 1.46570i 2.84835i −0.951057 − 0.309017i 0.0930228 + 0.885053i −2.23489 + 0.0725453i
159.6 −0.994522 − 0.104528i −1.44658 − 1.30250i 0.978148 + 0.207912i −1.66215 + 1.49575i 1.30250 + 1.44658i 4.00690i −0.951057 − 0.309017i 0.0824840 + 0.784783i 1.80939 − 1.31382i
159.7 −0.994522 − 0.104528i −1.26268 − 1.13692i 0.978148 + 0.207912i 0.601083 − 2.15376i 1.13692 + 1.26268i − 3.10293i −0.951057 − 0.309017i −0.0118172 − 0.112433i −0.822920 + 2.07913i
159.8 −0.994522 − 0.104528i −1.10634 − 0.996156i 0.978148 + 0.207912i 2.09315 + 0.786590i 0.996156 + 1.10634i 0.326434i −0.951057 − 0.309017i −0.0819168 − 0.779387i −1.99946 − 1.00107i
159.9 −0.994522 − 0.104528i −1.07277 − 0.965928i 0.978148 + 0.207912i −1.93765 − 1.11603i 0.965928 + 1.07277i − 2.14291i −0.951057 − 0.309017i −0.0957633 − 0.911127i 1.81038 + 1.31246i
159.10 −0.994522 − 0.104528i −0.852655 − 0.767734i 0.978148 + 0.207912i −0.257892 + 2.22115i 0.767734 + 0.852655i − 4.58392i −0.951057 − 0.309017i −0.175980 − 1.67434i 0.488652 − 2.18202i
159.11 −0.994522 − 0.104528i −0.455776 − 0.410383i 0.978148 + 0.207912i −0.905416 − 2.04456i 0.410383 + 0.455776i 3.69355i −0.951057 − 0.309017i −0.274267 − 2.60948i 0.686741 + 2.12800i
159.12 −0.994522 − 0.104528i −0.305771 − 0.275318i 0.978148 + 0.207912i 1.48246 − 1.67401i 0.275318 + 0.305771i − 4.36047i −0.951057 − 0.309017i −0.295889 − 2.81520i −1.64932 + 1.50988i
159.13 −0.994522 − 0.104528i 0.0825043 + 0.0742872i 0.978148 + 0.207912i 0.751196 + 2.10611i −0.0742872 − 0.0825043i − 0.0772685i −0.951057 − 0.309017i −0.312297 − 2.97131i −0.526932 − 2.17310i
159.14 −0.994522 − 0.104528i 0.353923 + 0.318674i 0.978148 + 0.207912i −2.22902 − 0.177361i −0.318674 − 0.353923i − 2.02267i −0.951057 − 0.309017i −0.289877 − 2.75799i 2.19827 + 0.409386i
159.15 −0.994522 − 0.104528i 0.415578 + 0.374188i 0.978148 + 0.207912i −1.68108 + 1.47444i −0.374188 − 0.415578i − 0.399754i −0.951057 − 0.309017i −0.280897 − 2.67256i 1.82599 − 1.29065i
159.16 −0.994522 − 0.104528i 0.650894 + 0.586067i 0.978148 + 0.207912i −2.14939 − 0.616527i −0.586067 − 0.650894i 3.13390i −0.951057 − 0.309017i −0.233398 − 2.22063i 2.07317 + 0.837823i
159.17 −0.994522 − 0.104528i 0.792127 + 0.713235i 0.978148 + 0.207912i 1.59520 − 1.56695i −0.713235 − 0.792127i 0.0485779i −0.951057 − 0.309017i −0.194823 − 1.85362i −1.75025 + 1.39162i
159.18 −0.994522 − 0.104528i 1.14808 + 1.03373i 0.978148 + 0.207912i 1.84118 + 1.26888i −1.03373 − 1.14808i 2.15604i −0.951057 − 0.309017i −0.0641093 − 0.609959i −1.69846 − 1.45439i
159.19 −0.994522 − 0.104528i 1.25394 + 1.12906i 0.978148 + 0.207912i −0.311300 − 2.21429i −1.12906 − 1.25394i − 1.86474i −0.951057 − 0.309017i −0.0159781 − 0.152021i 0.0781379 + 2.23470i
159.20 −0.994522 − 0.104528i 1.37103 + 1.23448i 0.978148 + 0.207912i 2.22915 + 0.175792i −1.23448 − 1.37103i − 1.39428i −0.951057 − 0.309017i 0.0421934 + 0.401443i −2.19856 − 0.407839i
See next 80 embeddings (of 400 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 159.50
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner
25.e even 10 1 inner
475.x even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.x.a ✓ 400
19.c even 3 1 inner 950.2.x.a ✓ 400
25.e even 10 1 inner 950.2.x.a ✓ 400
475.x even 30 1 inner 950.2.x.a ✓ 400
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.x.a ✓ 400 1.a even 1 1 trivial
950.2.x.a ✓ 400 19.c even 3 1 inner
950.2.x.a ✓ 400 25.e even 10 1 inner
950.2.x.a ✓ 400 475.x even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(950, [\chi])\).