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Properties

Label 400.3.i.b
Level $400$
Weight $3$
Character orbit 400.i
Analytic conductor $10.899$
Analytic rank $0$
Dimension $44$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,3,Mod(93,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.93");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 400.i (of order \(4\), degree \(2\), 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: \(10.8992105744\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 2 q^{2} + 4 q^{4} - 4 q^{6} + 8 q^{8} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 2 q^{2} + 4 q^{4} - 4 q^{6} + 8 q^{8} - 108 q^{9} - 4 q^{11} + 8 q^{12} + 24 q^{16} + 4 q^{17} - 22 q^{18} + 32 q^{19} - 4 q^{21} - 92 q^{22} + 36 q^{24} - 52 q^{26} - 36 q^{28} - 8 q^{31} + 132 q^{32} + 4 q^{33} - 88 q^{34} - 116 q^{36} + 216 q^{38} + 72 q^{39} - 16 q^{42} - 124 q^{43} - 168 q^{44} + 108 q^{46} + 4 q^{47} - 340 q^{48} - 100 q^{51} - 48 q^{52} + 4 q^{53} + 228 q^{54} - 172 q^{56} - 36 q^{57} - 16 q^{58} + 64 q^{59} - 36 q^{61} + 356 q^{62} + 200 q^{63} - 176 q^{64} + 276 q^{66} + 292 q^{67} + 72 q^{68} - 60 q^{69} - 448 q^{72} - 48 q^{73} + 284 q^{74} + 252 q^{76} - 192 q^{77} - 620 q^{78} + 100 q^{81} + 240 q^{82} + 288 q^{84} + 20 q^{86} - 36 q^{87} + 624 q^{88} + 188 q^{91} + 412 q^{92} - 340 q^{94} - 24 q^{96} + 4 q^{97} + 78 q^{98} - 92 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1 −1.97960 − 0.284962i 2.50699i 3.83759 + 1.12822i 0 0.714398 − 4.96283i 7.18571 + 7.18571i −7.27538 − 3.32699i 2.71500 0
93.2 −1.91842 + 0.565399i 1.96075i 3.36065 − 2.16934i 0 −1.10861 − 3.76154i −2.51657 − 2.51657i −5.22058 + 6.06181i 5.15546 0
93.3 −1.86745 + 0.715976i − 4.59062i 2.97476 − 2.67410i 0 3.28677 + 8.57276i 1.15913 + 1.15913i −3.64063 + 7.12361i −12.0738 0
93.4 −1.83264 − 0.800895i − 3.83124i 2.71713 + 2.93550i 0 −3.06842 + 7.02129i 1.69668 + 1.69668i −2.62850 − 7.55586i −5.67842 0
93.5 −1.47324 − 1.35261i 4.50609i 0.340894 + 3.98545i 0 6.09498 − 6.63857i −1.52625 − 1.52625i 4.88854 − 6.33263i −11.3048 0
93.6 −1.30092 + 1.51908i − 1.70661i −0.615196 − 3.95241i 0 2.59247 + 2.22016i 0.332763 + 0.332763i 6.80434 + 4.20725i 6.08749 0
93.7 −1.25977 − 1.55338i 0.119786i −0.825960 + 3.91379i 0 0.186072 − 0.150902i −4.73972 − 4.73972i 7.12012 − 3.64745i 8.98565 0
93.8 −1.12181 + 1.65576i 5.30326i −1.48309 − 3.71489i 0 −8.78093 − 5.94924i −7.26221 − 7.26221i 7.81472 + 1.71176i −19.1246 0
93.9 −0.634321 − 1.89674i − 2.77329i −3.19527 + 2.40629i 0 −5.26021 + 1.75915i 5.39242 + 5.39242i 6.59094 + 4.53426i 1.30888 0
93.10 −0.357454 + 1.96780i 1.24645i −3.74445 − 1.40679i 0 −2.45277 − 0.445550i 3.62600 + 3.62600i 4.10676 − 6.86546i 7.44635 0
93.11 0.293734 + 1.97831i − 2.88135i −3.82744 + 1.16220i 0 5.70021 − 0.846352i −2.87444 − 2.87444i −3.42344 − 7.23050i 0.697817 0
93.12 0.388354 − 1.96193i 4.95045i −3.69836 − 1.52385i 0 9.71244 + 1.92253i 7.61189 + 7.61189i −4.42596 + 6.66415i −15.5069 0
93.13 0.462923 − 1.94569i − 4.38426i −3.57140 − 1.80141i 0 −8.53040 − 2.02957i −3.84157 − 3.84157i −5.15826 + 6.11493i −10.2217 0
93.14 0.528784 − 1.92883i 2.05195i −3.44078 − 2.03987i 0 3.95786 + 1.08504i −6.87250 − 6.87250i −5.75399 + 5.55803i 4.78950 0
93.15 0.733258 + 1.86073i 3.80597i −2.92466 + 2.72880i 0 −7.08190 + 2.79076i 5.17093 + 5.17093i −7.22210 − 3.44111i −5.48540 0
93.16 1.36329 + 1.46337i − 1.90859i −0.282885 + 3.98998i 0 2.79297 − 2.60196i −8.62025 − 8.62025i −6.22446 + 5.02554i 5.35728 0
93.17 1.59633 − 1.20488i − 0.390820i 1.09653 − 3.84677i 0 −0.470891 − 0.623877i 6.36907 + 6.36907i −2.88447 − 7.46189i 8.84726 0
93.18 1.73567 − 0.993712i 0.616720i 2.02507 − 3.44951i 0 0.612842 + 1.07042i −3.63369 − 3.63369i 0.0870298 − 7.99953i 8.61966 0
93.19 1.79374 + 0.884585i − 3.32036i 2.43502 + 3.17343i 0 2.93714 − 5.95587i 9.08173 + 9.08173i 1.56062 + 7.84630i −2.02480 0
93.20 1.89622 + 0.635873i 1.50709i 3.19133 + 2.41151i 0 −0.958316 + 2.85778i 1.28182 + 1.28182i 4.51807 + 6.60205i 6.72868 0
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 93.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.3.i.b 44
5.b even 2 1 80.3.i.a ✓ 44
5.c odd 4 1 80.3.t.a yes 44
5.c odd 4 1 400.3.t.b 44
16.e even 4 1 400.3.t.b 44
20.d odd 2 1 320.3.i.a 44
20.e even 4 1 320.3.t.a 44
40.e odd 2 1 640.3.i.a 44
40.f even 2 1 640.3.i.b 44
40.i odd 4 1 640.3.t.b 44
40.k even 4 1 640.3.t.a 44
80.i odd 4 1 inner 400.3.i.b 44
80.i odd 4 1 640.3.i.b 44
80.j even 4 1 320.3.i.a 44
80.k odd 4 1 320.3.t.a 44
80.k odd 4 1 640.3.t.a 44
80.q even 4 1 80.3.t.a yes 44
80.q even 4 1 640.3.t.b 44
80.s even 4 1 640.3.i.a 44
80.t odd 4 1 80.3.i.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.3.i.a ✓ 44 5.b even 2 1
80.3.i.a ✓ 44 80.t odd 4 1
80.3.t.a yes 44 5.c odd 4 1
80.3.t.a yes 44 80.q even 4 1
320.3.i.a 44 20.d odd 2 1
320.3.i.a 44 80.j even 4 1
320.3.t.a 44 20.e even 4 1
320.3.t.a 44 80.k odd 4 1
400.3.i.b 44 1.a even 1 1 trivial
400.3.i.b 44 80.i odd 4 1 inner
400.3.t.b 44 5.c odd 4 1
400.3.t.b 44 16.e even 4 1
640.3.i.a 44 40.e odd 2 1
640.3.i.a 44 80.s even 4 1
640.3.i.b 44 40.f even 2 1
640.3.i.b 44 80.i odd 4 1
640.3.t.a 44 40.k even 4 1
640.3.t.a 44 80.k odd 4 1
640.3.t.b 44 40.i odd 4 1
640.3.t.b 44 80.q even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{44} + 252 T_{3}^{42} + 29216 T_{3}^{40} + 2068416 T_{3}^{38} + 100102192 T_{3}^{36} + \cdots + 21\!\cdots\!00 \) acting on \(S_{3}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display