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Properties

Label 418.2.v.a
Level $418$
Weight $2$
Character orbit 418.v
Analytic conductor $3.338$
Analytic rank $0$
Dimension $240$
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] = [418,2,Mod(13,418)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(418, base_ring=CyclotomicField(90))
 
chi = DirichletCharacter(H, H._module([9, 25]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("418.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.v (of order \(90\), degree \(24\), 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: \(3.33774680449\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(10\) over \(\Q(\zeta_{90})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{90}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 3 q^{3} - 12 q^{6} + 33 q^{7} - 30 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 3 q^{3} - 12 q^{6} + 33 q^{7} - 30 q^{8} + 3 q^{9} - 3 q^{11} - 6 q^{13} - 18 q^{14} + 21 q^{15} + 21 q^{17} - 60 q^{18} + 45 q^{19} + 12 q^{20} + 48 q^{21} - 12 q^{22} + 12 q^{24} - 18 q^{25} - 96 q^{26} - 9 q^{27} - 6 q^{28} + 18 q^{29} + 9 q^{31} + 87 q^{33} - 24 q^{34} - 36 q^{35} - 12 q^{36} - 36 q^{38} + 60 q^{41} - 6 q^{42} - 12 q^{43} + 18 q^{44} + 48 q^{45} + 12 q^{46} - 54 q^{47} + 6 q^{48} - 81 q^{49} - 21 q^{50} - 75 q^{51} + 3 q^{52} - 39 q^{53} + 27 q^{54} - 126 q^{55} - 90 q^{57} - 24 q^{58} + 69 q^{59} - 42 q^{60} + 66 q^{61} - 45 q^{62} - 9 q^{63} + 30 q^{64} - 12 q^{66} - 9 q^{67} + 12 q^{68} - 54 q^{69} + 9 q^{70} + 48 q^{71} + 6 q^{72} - 12 q^{74} + 72 q^{77} - 36 q^{79} - 27 q^{81} + 45 q^{82} + 36 q^{83} + 36 q^{84} - 210 q^{85} + 3 q^{86} + 216 q^{87} + 3 q^{88} + 18 q^{89} - 96 q^{90} - 108 q^{91} - 30 q^{92} + 147 q^{93} - 18 q^{94} + 66 q^{95} - 9 q^{97} - 12 q^{98} - 42 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 0.374607 + 0.927184i −0.866547 − 3.02201i −0.719340 + 0.694658i −2.04892 + 1.08943i 2.47734 − 1.93551i 1.73061 + 3.88702i −0.913545 − 0.406737i −5.83749 + 3.64767i −1.77764 − 1.49162i
13.2 0.374607 + 0.927184i −0.630231 − 2.19788i −0.719340 + 0.694658i 1.48095 − 0.787437i 1.80175 − 1.40768i −1.06207 − 2.38544i −0.913545 − 0.406737i −1.88932 + 1.18058i 1.28487 + 1.07814i
13.3 0.374607 + 0.927184i −0.301053 − 1.04990i −0.719340 + 0.694658i −0.793784 + 0.422062i 0.860670 − 0.672429i −0.868426 − 1.95052i −0.913545 − 0.406737i 1.53250 − 0.957610i −0.688686 − 0.577876i
13.4 0.374607 + 0.927184i −0.185288 − 0.646175i −0.719340 + 0.694658i −3.75384 + 1.99595i 0.529713 − 0.413857i −0.225011 − 0.505382i −0.913545 − 0.406737i 2.16093 − 1.35030i −3.25683 − 2.73280i
13.5 0.374607 + 0.927184i −0.0175505 − 0.0612060i −0.719340 + 0.694658i 2.65042 − 1.40925i 0.0501746 − 0.0392007i 1.63684 + 3.67640i −0.913545 − 0.406737i 2.54071 − 1.58761i 2.29950 + 1.92951i
13.6 0.374607 + 0.927184i −0.0174768 − 0.0609489i −0.719340 + 0.694658i −0.867424 + 0.461218i 0.0499639 − 0.0390361i 0.714147 + 1.60400i −0.913545 − 0.406737i 2.54073 − 1.58763i −0.752576 − 0.631487i
13.7 0.374607 + 0.927184i 0.195526 + 0.681879i −0.719340 + 0.694658i 2.12313 − 1.12889i −0.558982 + 0.436725i −0.266302 − 0.598123i −0.913545 − 0.406737i 2.11742 − 1.32311i 1.84203 + 1.54564i
13.8 0.374607 + 0.927184i 0.682065 + 2.37864i −0.719340 + 0.694658i −1.17948 + 0.627143i −1.94993 + 1.52345i 1.45204 + 3.26133i −0.913545 − 0.406737i −2.64858 + 1.65502i −1.02332 − 0.858667i
13.9 0.374607 + 0.927184i 0.765882 + 2.67095i −0.719340 + 0.694658i −2.13408 + 1.13471i −2.18956 + 1.71067i −1.72666 − 3.87814i −0.913545 − 0.406737i −4.00325 + 2.50151i −1.85152 − 1.55361i
13.10 0.374607 + 0.927184i 0.885806 + 3.08917i −0.719340 + 0.694658i 3.09438 − 1.64531i −2.53240 + 1.97853i −0.131685 − 0.295769i −0.913545 − 0.406737i −6.21419 + 3.88306i 2.68468 + 2.25272i
29.1 −0.438371 + 0.898794i −2.91493 + 1.17771i −0.615661 − 0.788011i −0.450149 + 0.129078i 0.219304 − 3.13619i 0.395058 − 1.85860i 0.978148 − 0.207912i 4.95179 − 4.78189i 0.0813176 − 0.461175i
29.2 −0.438371 + 0.898794i −2.15354 + 0.870087i −0.615661 − 0.788011i −2.44299 + 0.700517i 0.162021 − 2.31701i −0.805999 + 3.79193i 0.978148 − 0.207912i 1.72267 − 1.66356i 0.441317 − 2.50284i
29.3 −0.438371 + 0.898794i −0.914225 + 0.369371i −0.615661 − 0.788011i 0.610535 − 0.175068i 0.0687815 − 0.983622i 0.321044 − 1.51039i 0.978148 − 0.207912i −1.45865 + 1.40860i −0.110291 + 0.625490i
29.4 −0.438371 + 0.898794i −0.913816 + 0.369206i −0.615661 − 0.788011i 0.341675 − 0.0979736i 0.0687508 − 0.983182i −0.672800 + 3.16528i 0.978148 − 0.207912i −1.45927 + 1.40920i −0.0617222 + 0.350044i
29.5 −0.438371 + 0.898794i −0.559221 + 0.225940i −0.615661 − 0.788011i −0.807264 + 0.231479i 0.0420729 − 0.601670i 0.675252 − 3.17681i 0.978148 − 0.207912i −1.89634 + 1.83127i 0.145829 − 0.827038i
29.6 −0.438371 + 0.898794i 0.467532 − 0.188895i −0.615661 − 0.788011i 3.20340 − 0.918561i −0.0351747 + 0.503021i −0.476423 + 2.24139i 0.978148 − 0.207912i −1.97511 + 1.90735i −0.578682 + 3.28187i
29.7 −0.438371 + 0.898794i 1.43643 − 0.580356i −0.615661 − 0.788011i −3.49602 + 1.00247i −0.108070 + 1.54547i 0.232262 − 1.09270i 0.978148 − 0.207912i −0.431498 + 0.416692i 0.631543 − 3.58166i
29.8 −0.438371 + 0.898794i 2.01881 − 0.815653i −0.615661 − 0.788011i 3.65522 − 1.04812i −0.151885 + 2.17205i 0.300599 − 1.41421i 0.978148 − 0.207912i 1.25229 − 1.20932i −0.660301 + 3.74475i
29.9 −0.438371 + 0.898794i 2.12470 − 0.858435i −0.615661 − 0.788011i −0.690733 + 0.198064i −0.159852 + 2.28598i 0.729217 − 3.43070i 0.978148 − 0.207912i 1.61943 − 1.56386i 0.124778 − 0.707652i
29.10 −0.438371 + 0.898794i 3.07495 − 1.24236i −0.615661 − 0.788011i −0.517758 + 0.148465i −0.231343 + 3.30836i −0.797517 + 3.75202i 0.978148 − 0.207912i 5.75383 − 5.55641i 0.0935310 − 0.530441i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
209.w even 90 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 418.2.v.a ✓ 240
11.d odd 10 1 418.2.v.b yes 240
19.f odd 18 1 418.2.v.b yes 240
209.w even 90 1 inner 418.2.v.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.v.a ✓ 240 1.a even 1 1 trivial
418.2.v.a ✓ 240 209.w even 90 1 inner
418.2.v.b yes 240 11.d odd 10 1
418.2.v.b yes 240 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{240} - 3 T_{3}^{239} + 3 T_{3}^{238} + 3 T_{3}^{237} - 18 T_{3}^{236} + 3 T_{3}^{235} + \cdots + 14\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\). Copy content Toggle raw display