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Properties

Label 603.2.ba.a
Level $603$
Weight $2$
Character orbit 603.ba
Analytic conductor $4.815$
Analytic rank $0$
Dimension $1320$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [603,2,Mod(49,603)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(603, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([22, 46]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("603.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 603 = 3^{2} \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 603.ba (of order \(33\), degree \(20\), 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: \(4.81497924188\)
Analytic rank: \(0\)
Dimension: \(1320\)
Relative dimension: \(66\) over \(\Q(\zeta_{33})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1320 q - 14 q^{2} - 19 q^{3} + 52 q^{4} - 14 q^{5} - 25 q^{6} - 11 q^{7} - 20 q^{8} - 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1320 q - 14 q^{2} - 19 q^{3} + 52 q^{4} - 14 q^{5} - 25 q^{6} - 11 q^{7} - 20 q^{8} - 15 q^{9} - 44 q^{10} + 3 q^{11} + 21 q^{12} - 11 q^{13} - 7 q^{14} - 22 q^{15} + 46 q^{16} - 78 q^{17} - 24 q^{18} - 44 q^{19} - q^{20} + 32 q^{21} + q^{22} - 45 q^{23} + 83 q^{24} + 46 q^{25} - 46 q^{26} + 2 q^{27} - 32 q^{28} - 8 q^{29} - 22 q^{30} - 11 q^{31} - 48 q^{32} - 8 q^{33} - 11 q^{34} - 47 q^{35} - 61 q^{36} - 22 q^{37} - 266 q^{38} + 36 q^{39} + q^{40} - 16 q^{41} - 46 q^{42} - 8 q^{43} - 34 q^{44} + 79 q^{45} - 44 q^{46} + 21 q^{47} + 17 q^{48} - 107 q^{49} + 231 q^{50} - 40 q^{51} + 45 q^{52} - 8 q^{53} - 356 q^{54} + 16 q^{55} - 7 q^{56} - 69 q^{57} - 93 q^{58} - 94 q^{59} + 7 q^{60} - 59 q^{61} + 10 q^{62} - 18 q^{63} - 128 q^{64} + q^{65} - 169 q^{66} - 11 q^{67} - q^{68} - 30 q^{69} + 88 q^{70} - 48 q^{71} + 113 q^{72} - 198 q^{73} + 79 q^{74} - 68 q^{75} - 102 q^{76} + 207 q^{77} - 25 q^{78} - 27 q^{79} - 170 q^{80} - 83 q^{81} + 22 q^{82} - 2 q^{83} + 61 q^{84} + q^{85} - 137 q^{86} - 42 q^{87} - 53 q^{88} - 168 q^{89} - 15 q^{90} - 2 q^{91} - 19 q^{92} + 106 q^{93} - 11 q^{94} + 129 q^{95} + 243 q^{96} - 4 q^{97} + 9 q^{98} - 88 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 −2.71842 − 0.259578i −1.13727 + 1.30638i 5.35859 + 1.03278i −0.282588 − 0.113131i 3.43068 − 3.25608i −2.01062 − 4.40264i −9.05849 − 2.65981i −0.413248 − 2.97140i 0.738827 + 0.380892i
49.2 −2.71543 − 0.259292i −1.73205 + 0.00164137i 5.34249 + 1.02968i −0.926703 − 0.370996i 4.70369 + 0.444650i 1.95034 + 4.27065i −9.00560 − 2.64428i 2.99999 − 0.00568586i 2.42020 + 1.24770i
49.3 −2.62633 − 0.250784i 0.385488 − 1.68861i 4.87087 + 0.938783i 2.75135 + 1.10147i −1.43589 + 4.33817i 0.419391 + 0.918338i −7.49427 − 2.20052i −2.70280 − 1.30188i −6.94973 − 3.58283i
49.4 −2.57807 − 0.246176i 1.72403 + 0.166500i 4.62200 + 0.890818i −2.27207 − 0.909600i −4.40369 − 0.853663i −1.20005 − 2.62774i −6.72678 − 1.97516i 2.94456 + 0.574101i 5.63364 + 2.90435i
49.5 −2.45903 − 0.234809i 1.17123 + 1.27602i 4.02784 + 0.776303i 3.33791 + 1.33630i −2.58047 − 3.41278i −0.400134 − 0.876170i −4.98199 − 1.46284i −0.256443 + 2.98902i −7.89424 − 4.06976i
49.6 −2.44799 − 0.233755i 1.72339 − 0.173034i 3.97418 + 0.765960i 1.11664 + 0.447036i −4.25929 + 0.0207361i 1.05067 + 2.30064i −4.83068 − 1.41842i 2.94012 − 0.596409i −2.62904 − 1.35536i
49.7 −2.43805 − 0.232805i 0.450542 + 1.67243i 3.92603 + 0.756680i −2.14823 − 0.860023i −0.709094 − 4.18235i 0.864113 + 1.89214i −4.69582 − 1.37882i −2.59402 + 1.50700i 5.03728 + 2.59690i
49.8 −2.35395 − 0.224775i −1.18551 − 1.26276i 3.52670 + 0.679717i −1.42982 − 0.572413i 2.50679 + 3.23896i 0.387590 + 0.848703i −3.61116 − 1.06033i −0.189144 + 2.99403i 3.23706 + 1.66882i
49.9 −2.05110 − 0.195856i 1.06165 − 1.36854i 2.20478 + 0.424937i 0.842759 + 0.337390i −2.44559 + 2.59907i −1.51801 − 3.32398i −0.485068 − 0.142429i −0.745788 − 2.90582i −1.66250 − 0.857079i
49.10 −2.02320 − 0.193192i −1.66738 + 0.468857i 2.09214 + 0.403227i 2.06142 + 0.825270i 3.46403 − 0.626465i −0.582972 − 1.27653i −0.254765 − 0.0748057i 2.56035 − 1.56353i −4.01123 − 2.06793i
49.11 −2.02264 − 0.193139i 1.26304 − 1.18522i 2.08993 + 0.402801i −3.46105 − 1.38560i −2.78359 + 2.15333i 1.52793 + 3.34571i −0.250306 − 0.0734965i 0.190524 − 2.99394i 6.73286 + 3.47103i
49.12 −2.00448 − 0.191404i −0.976934 − 1.43024i 2.01743 + 0.388828i −1.96620 − 0.787146i 1.68449 + 3.05388i −1.33020 − 2.91274i −0.105408 − 0.0309507i −1.09120 + 2.79451i 3.79053 + 1.95415i
49.13 −1.93935 − 0.185186i −0.396611 + 1.68603i 1.76293 + 0.339777i 1.01307 + 0.405573i 1.08140 − 3.19636i 0.207046 + 0.453367i 0.382492 + 0.112310i −2.68540 − 1.33740i −1.88960 − 0.974154i
49.14 −1.92863 − 0.184162i −1.69060 + 0.376659i 1.72184 + 0.331858i −3.48970 − 1.39706i 3.32991 − 0.415091i −0.257998 − 0.564937i 0.458166 + 0.134530i 2.71626 − 1.27356i 6.47305 + 3.33709i
49.15 −1.78698 − 0.170636i −0.0855694 − 1.72994i 1.20034 + 0.231346i 0.651405 + 0.260783i −0.142279 + 3.10597i 1.72721 + 3.78205i 1.33929 + 0.393251i −2.98536 + 0.296059i −1.11955 − 0.577169i
49.16 −1.56186 − 0.149139i 0.783495 + 1.54471i 0.453293 + 0.0873650i 1.43975 + 0.576391i −0.993329 − 2.52947i −1.26641 − 2.77305i 2.31586 + 0.679999i −1.77227 + 2.42055i −2.16273 − 1.11496i
49.17 −1.51640 − 0.144799i −1.68275 − 0.410311i 0.314646 + 0.0606431i 3.21816 + 1.28836i 2.49231 + 0.865855i 1.28735 + 2.81890i 2.45484 + 0.720806i 2.66329 + 1.38090i −4.69347 − 2.41965i
49.18 −1.40241 − 0.133914i 1.63155 + 0.581409i −0.0150359 − 0.00289794i −1.10256 − 0.441399i −2.21025 − 1.03386i 0.0381937 + 0.0836326i 2.72415 + 0.799881i 2.32393 + 1.89720i 1.48713 + 0.766670i
49.19 −1.39961 − 0.133647i −0.901498 + 1.47895i −0.0228096 − 0.00439619i −0.379360 − 0.151873i 1.45940 − 1.94948i 0.856511 + 1.87550i 2.72939 + 0.801420i −1.37460 − 2.66655i 0.510659 + 0.263263i
49.20 −1.24619 − 0.118996i −0.302508 − 1.70543i −0.425036 − 0.0819190i 1.39292 + 0.557642i 0.174042 + 2.16128i 0.330480 + 0.723650i 2.92222 + 0.858041i −2.81698 + 1.03181i −1.66949 − 0.860679i
See next 80 embeddings (of 1320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.66
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
603.ba even 33 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 603.2.ba.a yes 1320
9.c even 3 1 603.2.y.a ✓ 1320
67.g even 33 1 603.2.y.a ✓ 1320
603.ba even 33 1 inner 603.2.ba.a yes 1320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
603.2.y.a ✓ 1320 9.c even 3 1
603.2.y.a ✓ 1320 67.g even 33 1
603.2.ba.a yes 1320 1.a even 1 1 trivial
603.2.ba.a yes 1320 603.ba even 33 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(603, [\chi])\).