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Properties

Label 644.2.w.a
Level $644$
Weight $2$
Character orbit 644.w
Analytic conductor $5.142$
Analytic rank $0$
Dimension $360$
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] = [644,2,Mod(15,644)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(644, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 0, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("644.15");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 644.w (of order \(22\), degree \(10\), 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: \(5.14236589017\)
Analytic rank: \(0\)
Dimension: \(360\)
Relative dimension: \(36\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 360 q - q^{2} + 3 q^{4} - q^{6} - 36 q^{7} + 2 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 360 q - q^{2} + 3 q^{4} - q^{6} - 36 q^{7} + 2 q^{8} + 36 q^{9} + 8 q^{11} + 3 q^{12} - q^{14} + 8 q^{15} - 9 q^{16} + 8 q^{18} + 8 q^{19} + 63 q^{20} + 24 q^{22} - 8 q^{23} + 4 q^{24} + 36 q^{25} - 7 q^{26} - 19 q^{28} + 8 q^{29} + 9 q^{32} - 8 q^{34} + 52 q^{36} + 220 q^{39} - 88 q^{40} - q^{42} - 40 q^{43} - 28 q^{44} - 63 q^{46} - 49 q^{48} - 36 q^{49} - 34 q^{50} - 25 q^{52} - 55 q^{54} - 220 q^{55} + 2 q^{56} + 36 q^{58} + 54 q^{60} + 5 q^{62} + 36 q^{63} - 12 q^{64} - 225 q^{66} + 8 q^{67} - 60 q^{68} - 28 q^{69} + 44 q^{71} + 36 q^{72} + 146 q^{74} + 55 q^{76} - 36 q^{77} - 17 q^{78} + 40 q^{79} - 50 q^{80} - 72 q^{81} - 71 q^{82} + 80 q^{84} + 24 q^{85} + 94 q^{86} - 50 q^{88} - 44 q^{89} + 309 q^{90} + 21 q^{92} - 272 q^{93} - 35 q^{94} + 294 q^{96} - 44 q^{97} - q^{98} - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1 −1.40577 + 0.154330i −2.06068 + 0.941080i 1.95236 − 0.433904i 0.418011 − 0.362209i 2.75160 − 1.64096i 0.841254 + 0.540641i −2.67761 + 0.911277i 1.39618 − 1.61128i −0.531727 + 0.573693i
15.2 −1.39145 − 0.252703i 1.27073 − 0.580322i 1.87228 + 0.703250i 2.20448 − 1.91020i −1.91481 + 0.486373i 0.841254 + 0.540641i −2.42748 − 1.45167i −0.686608 + 0.792387i −3.55015 + 2.10087i
15.3 −1.37669 − 0.323619i 1.19717 − 0.546731i 1.79054 + 0.891044i −1.44044 + 1.24815i −1.82507 + 0.365251i 0.841254 + 0.540641i −2.17666 − 1.80614i −0.830270 + 0.958183i 2.38695 − 1.25215i
15.4 −1.31412 + 0.522583i 1.85000 − 0.844869i 1.45381 − 1.37347i −2.09624 + 1.81640i −1.98961 + 2.07704i 0.841254 + 0.540641i −1.19273 + 2.56464i 0.744132 − 0.858775i 1.80549 − 3.48243i
15.5 −1.28374 + 0.593300i −1.34266 + 0.613171i 1.29599 − 1.52329i −2.17137 + 1.88151i 1.35983 − 1.58375i 0.841254 + 0.540641i −0.759951 + 2.72442i −0.537831 + 0.620690i 1.67119 − 3.70365i
15.6 −1.27324 − 0.615516i −0.941519 + 0.429977i 1.24228 + 1.56740i 2.17631 − 1.88578i 1.46344 + 0.0320562i 0.841254 + 0.540641i −0.616959 − 2.76032i −1.26300 + 1.45758i −3.93169 + 1.06150i
15.7 −1.10844 − 0.878271i 3.05941 − 1.39718i 0.457282 + 1.94702i −0.519837 + 0.450441i −4.61828 − 1.13829i 0.841254 + 0.540641i 1.20314 − 2.55978i 5.44327 − 6.28187i 0.971817 − 0.0427300i
15.8 −1.05338 + 0.943610i 1.04632 − 0.477837i 0.219199 − 1.98795i 0.642420 − 0.556660i −0.651272 + 1.49066i 0.841254 + 0.540641i 1.64495 + 2.30090i −1.09813 + 1.26731i −0.151439 + 1.19257i
15.9 −0.910112 + 1.08245i −3.06454 + 1.39953i −0.343393 − 1.97030i 2.22841 − 1.93093i 1.27415 − 4.59093i 0.841254 + 0.540641i 2.44528 + 1.42149i 5.46812 − 6.31055i 0.0620299 + 4.16951i
15.10 −0.868816 − 1.11587i −2.46864 + 1.12739i −0.490317 + 1.93897i −1.18240 + 1.02456i 3.40281 + 1.77518i 0.841254 + 0.540641i 2.58962 − 1.13748i 2.85860 − 3.29900i 2.17056 + 0.429251i
15.11 −0.851344 − 1.12925i −0.540348 + 0.246769i −0.550426 + 1.92277i 1.49922 − 1.29908i 0.738686 + 0.400105i 0.841254 + 0.540641i 2.63989 − 1.01537i −1.73350 + 2.00057i −2.74334 − 0.587032i
15.12 −0.665880 + 1.24764i 2.56801 − 1.17277i −1.11321 − 1.66156i 1.82272 − 1.57939i −0.246790 + 3.98488i 0.841254 + 0.540641i 2.81429 − 0.282488i 3.25471 − 3.75613i 0.756804 + 3.32578i
15.13 −0.623251 + 1.26947i 0.395610 − 0.180669i −1.22312 − 1.58240i −2.58184 + 2.23717i −0.0172101 + 0.614817i 0.841254 + 0.540641i 2.77112 − 0.566480i −1.84072 + 2.12430i −1.23090 − 4.67189i
15.14 −0.503844 − 1.32142i −0.540570 + 0.246870i −1.49228 + 1.33157i −2.49828 + 2.16477i 0.598581 + 0.589934i 0.841254 + 0.540641i 2.51144 + 1.30102i −1.73331 + 2.00035i 4.11931 + 2.21056i
15.15 −0.454375 + 1.33923i −0.488724 + 0.223193i −1.58709 − 1.21703i 0.500943 − 0.434070i −0.0768432 − 0.755929i 0.841254 + 0.540641i 2.35101 − 1.57249i −1.77555 + 2.04909i 0.353704 + 0.868109i
15.16 −0.309401 − 1.37995i 2.23791 − 1.02202i −1.80854 + 0.853918i 1.81601 − 1.57358i −2.10275 − 2.77199i 0.841254 + 0.540641i 1.73793 + 2.23150i 1.99912 − 2.30711i −2.73334 − 2.01914i
15.17 −0.132461 − 1.40800i 1.93748 − 0.884818i −1.96491 + 0.373009i −2.70459 + 2.34354i −1.50246 − 2.61076i 0.841254 + 0.540641i 0.785469 + 2.71717i 1.00635 − 1.16139i 3.65795 + 3.49763i
15.18 −0.0536018 − 1.41320i −1.91376 + 0.873985i −1.99425 + 0.151500i 2.92462 − 2.53420i 1.33769 + 2.65767i 0.841254 + 0.540641i 0.320995 + 2.81015i 0.934048 − 1.07795i −3.73809 − 3.99723i
15.19 −0.0185458 + 1.41409i −2.33313 + 1.06550i −1.99931 − 0.0524511i −0.461405 + 0.399809i −1.46345 − 3.31902i 0.841254 + 0.540641i 0.111250 − 2.82624i 2.34360 − 2.70466i −0.556810 − 0.659883i
15.20 0.176746 − 1.40313i −2.95704 + 1.35043i −1.93752 − 0.495994i −1.30360 + 1.12957i 1.37218 + 4.38778i 0.841254 + 0.540641i −1.03839 + 2.63092i 4.95582 − 5.71932i 1.35453 + 2.02876i
See next 80 embeddings (of 360 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
92.h even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 644.2.w.a ✓ 360
4.b odd 2 1 644.2.w.b yes 360
23.d odd 22 1 644.2.w.b yes 360
92.h even 22 1 inner 644.2.w.a ✓ 360
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
644.2.w.a ✓ 360 1.a even 1 1 trivial
644.2.w.a ✓ 360 92.h even 22 1 inner
644.2.w.b yes 360 4.b odd 2 1
644.2.w.b yes 360 23.d odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{360} - 72 T_{3}^{358} + 2853 T_{3}^{356} - 82336 T_{3}^{354} + 660 T_{3}^{353} + \cdots + 19\!\cdots\!24 \) acting on \(S_{2}^{\mathrm{new}}(644, [\chi])\). Copy content Toggle raw display