↑

Properties

Label 450.2.q.c
Level $450$
Weight $2$
Character orbit 450.q
Analytic conductor $3.593$
Analytic rank $0$
Dimension $120$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [450,2,Mod(31,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([10, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.q (of order \(15\), 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: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(15\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 15 q^{2} + 2 q^{3} + 15 q^{4} + 4 q^{5} + 4 q^{6} + 16 q^{7} + 30 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 15 q^{2} + 2 q^{3} + 15 q^{4} + 4 q^{5} + 4 q^{6} + 16 q^{7} + 30 q^{8} - 10 q^{9} - 2 q^{10} + 2 q^{11} - 3 q^{12} - 6 q^{13} + 4 q^{14} + 19 q^{15} + 15 q^{16} + 12 q^{17} + 2 q^{18} + 18 q^{19} - q^{20} - 8 q^{21} - 2 q^{22} - 2 q^{24} - 2 q^{25} + 48 q^{26} + 2 q^{27} + 8 q^{28} - 5 q^{30} + 60 q^{32} - q^{33} + 6 q^{34} + 10 q^{35} + 2 q^{36} + 32 q^{37} - 6 q^{38} - 46 q^{39} + q^{40} - 4 q^{41} - 16 q^{42} + 40 q^{43} - 4 q^{44} - 38 q^{45} + 13 q^{47} + 6 q^{48} - 48 q^{49} - 3 q^{50} - 106 q^{51} - 6 q^{52} - 72 q^{53} + 15 q^{54} - 12 q^{55} + 4 q^{56} - 10 q^{57} + 18 q^{59} - 9 q^{60} + 34 q^{61} - 53 q^{63} - 30 q^{64} - 22 q^{65} + 13 q^{66} + 24 q^{68} - 5 q^{69} - 25 q^{70} - 22 q^{71} - 5 q^{72} + 32 q^{73} - 64 q^{74} - 147 q^{75} + 6 q^{76} + 10 q^{77} - 9 q^{78} + 14 q^{79} + 2 q^{80} + 22 q^{81} - 88 q^{82} - 52 q^{83} + 12 q^{84} - 3 q^{85} + 10 q^{86} + 25 q^{87} - 2 q^{88} - 94 q^{89} - 82 q^{90} + 104 q^{91} + 22 q^{93} - 13 q^{94} + 4 q^{95} - 2 q^{96} - 14 q^{97} + 54 q^{98} - 86 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −0.669131 + 0.743145i −1.70489 − 0.305514i −0.104528 − 0.994522i 1.02459 − 1.98752i 1.36784 − 1.06255i −0.359749 − 0.623104i 0.809017 + 0.587785i 2.81332 + 1.04174i 0.791428 + 2.09133i
31.2 −0.669131 + 0.743145i −1.68730 + 0.391172i −0.104528 − 0.994522i 1.26421 + 1.84439i 0.838327 − 1.51565i −1.15080 − 1.99324i 0.809017 + 0.587785i 2.69397 − 1.32005i −2.21657 − 0.294647i
31.3 −0.669131 + 0.743145i −1.68554 + 0.398680i −0.104528 − 0.994522i −1.90572 + 1.16971i 0.831571 − 1.51937i 1.88874 + 3.27139i 0.809017 + 0.587785i 2.68211 − 1.34399i 0.405918 − 2.19892i
31.4 −0.669131 + 0.743145i −1.28015 − 1.16671i −0.104528 − 0.994522i −2.13643 + 0.660051i 1.72362 − 0.170653i −2.25745 − 3.91002i 0.809017 + 0.587785i 0.277564 + 2.98713i 0.939037 − 2.02934i
31.5 −0.669131 + 0.743145i −0.927558 − 1.46275i −0.104528 − 0.994522i 2.23523 − 0.0612255i 1.70769 + 0.289461i 1.34061 + 2.32201i 0.809017 + 0.587785i −1.27927 + 2.71357i −1.45016 + 1.70207i
31.6 −0.669131 + 0.743145i −0.884761 + 1.48903i −0.104528 − 0.994522i 0.0935545 − 2.23411i −0.514541 − 1.65386i 2.10737 + 3.65008i 0.809017 + 0.587785i −1.43439 − 2.63486i 1.59767 + 1.56444i
31.7 −0.669131 + 0.743145i −0.128768 + 1.72726i −0.104528 − 0.994522i 1.10209 + 1.94561i −1.19744 − 1.25145i −0.0474509 − 0.0821874i 0.809017 + 0.587785i −2.96684 − 0.444830i −2.18331 − 0.482855i
31.8 −0.669131 + 0.743145i −0.0822897 − 1.73009i −0.104528 − 0.994522i −1.35994 − 1.77499i 1.34077 + 1.09651i 0.132744 + 0.229920i 0.809017 + 0.587785i −2.98646 + 0.284738i 2.22905 + 0.177068i
31.9 −0.669131 + 0.743145i 0.606701 − 1.62232i −0.104528 − 0.994522i −0.202828 + 2.22685i 0.799655 + 1.53641i 0.989724 + 1.71425i 0.809017 + 0.587785i −2.26383 − 1.96852i −1.51915 − 1.64078i
31.10 −0.669131 + 0.743145i 0.689762 + 1.58878i −0.104528 − 0.994522i −1.44248 − 1.70858i −1.64224 − 0.550509i −1.01564 − 1.75914i 0.809017 + 0.587785i −2.04846 + 2.19176i 2.23493 + 0.0712903i
31.11 −0.669131 + 0.743145i 0.882941 + 1.49011i −0.104528 − 0.994522i 2.11952 − 0.712480i −1.69817 − 0.340922i −1.46519 − 2.53778i 0.809017 + 0.587785i −1.44083 + 2.63135i −0.888761 + 2.05185i
31.12 −0.669131 + 0.743145i 1.21095 − 1.23839i −0.104528 − 0.994522i −1.55164 + 1.61010i 0.110020 + 1.72855i −0.0993321 − 0.172048i 0.809017 + 0.587785i −0.0672111 − 2.99925i −0.158283 − 2.23046i
31.13 −0.669131 + 0.743145i 1.52380 − 0.823428i −0.104528 − 0.994522i 1.26095 − 1.84662i −0.407695 + 1.68339i −1.10262 − 1.90980i 0.809017 + 0.587785i 1.64393 − 2.50948i 0.528561 + 2.17270i
31.14 −0.669131 + 0.743145i 1.69823 + 0.340598i −0.104528 − 0.994522i −2.07884 − 0.823674i −1.38945 + 1.03413i 0.746133 + 1.29234i 0.809017 + 0.587785i 2.76799 + 1.15683i 2.00312 − 0.993731i
31.15 −0.669131 + 0.743145i 1.70428 + 0.308935i −0.104528 − 0.994522i 2.07773 + 0.826460i −1.36997 + 1.05981i 2.29291 + 3.97143i 0.809017 + 0.587785i 2.80912 + 1.05302i −2.00445 + 0.991045i
61.1 0.104528 + 0.994522i −1.66231 − 0.486533i −0.978148 + 0.207912i −1.22161 − 1.87288i 0.310109 − 1.70406i 0.0160288 − 0.0277626i −0.309017 − 0.951057i 2.52657 + 1.61754i 1.73493 − 1.41069i
61.2 0.104528 + 0.994522i −1.47041 + 0.915364i −0.978148 + 0.207912i 2.03309 − 0.930893i −1.06405 − 1.36667i 0.687693 − 1.19112i −0.309017 − 0.951057i 1.32422 − 2.69192i 1.13831 + 1.92464i
61.3 0.104528 + 0.994522i −1.38806 + 1.03600i −0.978148 + 0.207912i −2.16518 + 0.558560i −1.17542 − 1.27216i 2.56839 − 4.44858i −0.309017 − 0.951057i 0.853398 − 2.87606i −0.781824 − 2.09493i
61.4 0.104528 + 0.994522i −1.21293 − 1.23645i −0.978148 + 0.207912i −1.30029 + 1.81914i 1.10289 − 1.33553i 0.172767 − 0.299242i −0.309017 − 0.951057i −0.0576240 + 2.99945i −1.94509 − 1.10301i
61.5 0.104528 + 0.994522i −0.793716 − 1.53949i −0.978148 + 0.207912i 2.05812 + 0.874155i 1.44809 − 0.950288i −1.13930 + 1.97333i −0.309017 − 0.951057i −1.74003 + 2.44383i −0.654234 + 2.13822i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
25.d even 5 1 inner
225.q even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.q.c ✓ 120
9.c even 3 1 inner 450.2.q.c ✓ 120
25.d even 5 1 inner 450.2.q.c ✓ 120
225.q even 15 1 inner 450.2.q.c ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.q.c ✓ 120 1.a even 1 1 trivial
450.2.q.c ✓ 120 9.c even 3 1 inner
450.2.q.c ✓ 120 25.d even 5 1 inner
450.2.q.c ✓ 120 225.q even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{60} - 8 T_{7}^{59} + 149 T_{7}^{58} - 888 T_{7}^{57} + 10684 T_{7}^{56} - 54003 T_{7}^{55} + \cdots + 855036081 \) acting on \(S_{2}^{\mathrm{new}}(450, [\chi])\). Copy content Toggle raw display