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Properties

Label 361.2.i.a
Level $361$
Weight $2$
Character orbit 361.i
Analytic conductor $2.883$
Analytic rank $0$
Dimension $1080$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [361,2,Mod(7,361)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(361, base_ring=CyclotomicField(114)) chi = DirichletCharacter(H, H._module([50])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("361.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 361.i (of order \(57\), degree \(36\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.88259951297\)
Analytic rank: \(0\)
Dimension: \(1080\)
Relative dimension: \(30\) over \(\Q(\zeta_{57})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{57}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1080 q - 38 q^{2} - 19 q^{3} - 8 q^{4} - 37 q^{5} - 38 q^{6} - 40 q^{7} - 38 q^{8} + 49 q^{9} + 19 q^{10} - 40 q^{11} - 38 q^{12} - 38 q^{13} - 38 q^{15} - 8 q^{16} - 94 q^{17} - 38 q^{18} - 38 q^{19} - 46 q^{20}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.497104 − 2.54490i 1.24599 − 0.137906i −4.37639 + 1.77754i −1.92608 + 1.87372i −0.970344 − 3.10236i −1.47712 + 1.14969i 3.86271 + 5.91232i −1.39391 + 0.312383i 5.72588 + 3.97024i
7.2 −0.491952 − 2.51852i −2.59032 + 0.286697i −4.24794 + 1.72537i −0.768443 + 0.747553i 1.99637 + 6.38274i 0.0293309 − 0.0228291i 3.62809 + 5.55320i 3.70018 − 0.829231i 2.26076 + 1.56758i
7.3 −0.485221 − 2.48406i 0.0922595 − 0.0102113i −4.08212 + 1.65801i 1.31765 − 1.28183i −0.0701317 − 0.224223i 3.34871 − 2.60641i 3.33068 + 5.09798i −2.91898 + 0.654159i −3.82349 − 2.65115i
7.4 −0.409797 − 2.09793i 2.01804 − 0.223357i −2.38040 + 0.966836i 2.44372 − 2.37729i −1.29558 − 4.14219i −1.18549 + 0.922706i 0.665547 + 1.01870i 1.09522 − 0.245444i −5.98883 − 4.15256i
7.5 −0.376595 − 1.92796i −0.174153 + 0.0192752i −1.72221 + 0.699500i 0.192031 − 0.186810i 0.102747 + 0.328500i −3.61536 + 2.81395i −0.151662 − 0.232136i −2.89743 + 0.649330i −0.432480 − 0.299875i
7.6 −0.348190 − 1.78254i 2.54770 − 0.281980i −1.20322 + 0.488707i −1.44340 + 1.40416i −1.38972 − 4.44320i 3.55530 − 2.76720i −0.696676 − 1.06634i 3.48388 − 0.780756i 3.00555 + 2.08400i
7.7 −0.302889 − 1.55062i −2.83012 + 0.313238i −0.459700 + 0.186714i 2.58601 − 2.51571i 1.34293 + 4.29358i −2.10407 + 1.63766i −1.29952 − 1.98906i 4.98409 − 1.11696i −4.68418 − 3.24794i
7.8 −0.301334 − 1.54266i −1.89976 + 0.210265i −0.436011 + 0.177092i −1.83532 + 1.78542i 0.896828 + 2.86732i −0.330256 + 0.257049i −1.31482 − 2.01249i 0.637470 − 0.142860i 3.30734 + 2.29326i
7.9 −0.252437 − 1.29234i −0.391559 + 0.0433377i 0.246574 − 0.100150i −2.30182 + 2.23925i 0.154851 + 0.495086i 1.37783 − 1.07241i −1.63207 − 2.49807i −2.77595 + 0.622105i 3.47493 + 2.40946i
7.10 −0.176813 − 0.905186i 0.734445 − 0.0812884i 1.06489 − 0.432521i 0.951507 − 0.925640i −0.203441 − 0.650437i 1.61292 − 1.25539i −1.58869 − 2.43167i −2.39459 + 0.536640i −1.00612 − 0.697625i
7.11 −0.166828 − 0.854065i 2.73618 − 0.302840i 1.15139 − 0.467655i 0.510455 − 0.496579i −0.715115 − 2.28635i −1.32321 + 1.02989i −1.54341 − 2.36236i 4.46756 − 1.00120i −0.509268 − 0.353119i
7.12 −0.141237 − 0.723052i −3.36489 + 0.372426i 1.35013 − 0.548376i −1.41644 + 1.37793i 0.744529 + 2.38039i 2.60273 − 2.02579i −1.39308 − 2.13227i 8.25640 − 1.85030i 1.19637 + 0.829545i
7.13 −0.141184 − 0.722784i −1.29352 + 0.143167i 1.35050 − 0.548528i 2.25094 − 2.18975i 0.286104 + 0.914726i 1.87275 − 1.45762i −1.39273 − 2.13173i −1.27468 + 0.285662i −1.90051 − 1.31779i
7.14 −0.0831546 − 0.425705i −0.950749 + 0.105229i 1.67868 − 0.681820i −0.486706 + 0.473475i 0.123856 + 0.395989i −3.88236 + 3.02176i −0.904323 − 1.38417i −2.03454 + 0.455951i 0.242033 + 0.167822i
7.15 −0.0671743 − 0.343895i 2.09291 − 0.231643i 1.73924 − 0.706417i −0.792589 + 0.771042i −0.220250 − 0.704179i −2.02753 + 1.57809i −0.743060 − 1.13734i 1.39921 − 0.313570i 0.318399 + 0.220773i
7.16 0.0216586 + 0.110880i −0.0211848 + 0.00234474i 1.84116 − 0.747816i −3.03725 + 2.95468i −0.000718818 − 0.00229819i −0.322988 + 0.251391i 0.246378 + 0.377110i −2.92695 + 0.655944i −0.393398 − 0.272776i
7.17 0.0610592 + 0.312589i −2.00717 + 0.222154i 1.75900 − 0.714446i 0.730176 − 0.710326i −0.191999 − 0.613855i 0.293144 − 0.228163i 0.679133 + 1.03949i 1.05199 − 0.235757i 0.266624 + 0.184873i
7.18 0.152862 + 0.782568i 3.00938 − 0.333078i 1.26394 − 0.513369i −2.74514 + 2.67052i 0.720677 + 2.30413i 0.169613 − 0.132015i 1.46718 + 2.24569i 6.01805 − 1.34868i −2.50949 − 1.74004i
7.19 0.157057 + 0.804046i 1.27603 − 0.141231i 1.23116 − 0.500056i 0.346775 − 0.337348i 0.313967 + 1.00381i 2.33309 − 1.81592i 1.49160 + 2.28306i −1.31907 + 0.295612i 0.325707 + 0.225840i
7.20 0.159102 + 0.814515i 0.592939 − 0.0656265i 1.21487 − 0.493436i 2.54554 − 2.47634i 0.147792 + 0.472517i −1.27640 + 0.993461i 1.50303 + 2.30056i −2.58012 + 0.578218i 2.42202 + 1.67939i
See next 80 embeddings (of 1080 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.30
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
361.i even 57 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.2.i.a ✓ 1080
361.i even 57 1 inner 361.2.i.a ✓ 1080
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
361.2.i.a ✓ 1080 1.a even 1 1 trivial
361.2.i.a ✓ 1080 361.i even 57 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(361, [\chi])\).