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Properties

Label 261.2.q.a
Level $261$
Weight $2$
Character orbit 261.q
Analytic conductor $2.084$
Analytic rank $0$
Dimension $336$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [261,2,Mod(7,261)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(261, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([28, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("261.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 261 = 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 261.q (of order \(21\), degree \(12\), 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: \(2.08409549276\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(28\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q - 5 q^{2} - 10 q^{3} + 21 q^{4} - 9 q^{5} - 40 q^{6} - 5 q^{7} + 2 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 336 q - 5 q^{2} - 10 q^{3} + 21 q^{4} - 9 q^{5} - 40 q^{6} - 5 q^{7} + 2 q^{8} - 6 q^{9} - 28 q^{10} - q^{11} - 22 q^{12} - 5 q^{13} - 9 q^{14} - 26 q^{15} + 21 q^{16} - 60 q^{17} - 90 q^{18} - 20 q^{19} - 15 q^{20} - 2 q^{21} - 13 q^{22} - 32 q^{23} + 44 q^{24} + 15 q^{25} - 4 q^{26} - 43 q^{27} - 72 q^{28} - q^{29} - 8 q^{30} - 5 q^{31} + 7 q^{32} - 37 q^{33} - 15 q^{34} + 16 q^{35} - 104 q^{36} - 20 q^{37} + 63 q^{38} + 38 q^{39} + 5 q^{40} - 20 q^{41} + 11 q^{42} - 5 q^{43} - 8 q^{44} + 30 q^{45} - 80 q^{46} + 5 q^{47} + 12 q^{48} - 19 q^{49} - 3 q^{50} - 62 q^{51} + q^{52} - 4 q^{53} + 26 q^{54} - 100 q^{55} - 5 q^{56} - 50 q^{57} - 7 q^{58} + 154 q^{59} + 23 q^{60} + 7 q^{61} - 4 q^{62} - 144 q^{63} + 18 q^{64} - 65 q^{65} + 44 q^{66} - 5 q^{67} - 9 q^{68} + 22 q^{69} - 14 q^{70} + 18 q^{71} + 37 q^{72} - 20 q^{73} - 77 q^{74} + 16 q^{75} - 5 q^{76} + 39 q^{77} + 52 q^{78} - 5 q^{79} - 104 q^{80} + 54 q^{81} + 12 q^{82} - 23 q^{83} + 175 q^{84} - 42 q^{85} + 128 q^{86} + 42 q^{87} - 8 q^{88} + 34 q^{89} + 222 q^{90} + 12 q^{91} - 115 q^{92} + 64 q^{93} - 25 q^{94} + 89 q^{95} - 49 q^{96} + 55 q^{97} - 88 q^{98} + 196 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.44742 − 0.754930i 1.52271 − 0.825436i 3.76749 + 2.56863i −1.43861 + 1.33483i −4.34987 + 0.870650i 1.12701 − 0.768385i −4.08772 − 5.12584i 1.63731 − 2.51381i 4.52859 − 2.18085i
7.2 −2.44589 − 0.754458i 1.09572 + 1.34141i 3.76071 + 2.56401i 2.97490 − 2.76030i −1.66799 − 4.10763i 2.64157 − 1.80099i −4.07208 − 5.10622i −0.598775 + 2.93964i −9.35882 + 4.50697i
7.3 −2.41664 − 0.745434i −1.72562 + 0.149092i 3.63198 + 2.47624i −0.623182 + 0.578228i 4.28134 + 0.926035i 0.501374 − 0.341831i −3.77770 − 4.73709i 2.95554 − 0.514554i 1.93704 − 0.932827i
7.4 −2.27130 − 0.700603i 0.0581640 − 1.73107i 3.01548 + 2.05592i 1.42240 − 1.31979i −1.34490 + 3.89104i −4.04895 + 2.76053i −2.44473 − 3.06559i −2.99323 − 0.201372i −4.15534 + 2.00111i
7.5 −1.73892 − 0.536384i −0.330340 + 1.70026i 1.08364 + 0.738814i −1.02914 + 0.954898i 1.48643 − 2.77941i 1.84077 − 1.25502i 0.781135 + 0.979512i −2.78175 − 1.12333i 2.30177 − 1.10847i
7.6 −1.64561 − 0.507603i −1.09912 + 1.33864i 0.797889 + 0.543992i 2.10385 − 1.95209i 2.48821 − 1.64496i −2.18752 + 1.49143i 1.11056 + 1.39260i −0.583889 − 2.94263i −4.45301 + 2.14446i
7.7 −1.58794 − 0.489814i −0.840358 − 1.51453i 0.629152 + 0.428949i −2.01805 + 1.87248i 0.592598 + 2.81660i 0.587584 − 0.400608i 1.28324 + 1.60913i −1.58760 + 2.54549i 4.12171 − 1.98491i
7.8 −1.37970 − 0.425582i 1.72230 − 0.183558i 0.0699795 + 0.0477112i 0.492252 − 0.456743i −2.45437 − 0.479723i 0.163360 − 0.111377i 1.72420 + 2.16208i 2.93261 − 0.632282i −0.873542 + 0.420676i
7.9 −1.24479 − 0.383966i 0.798366 − 1.53708i −0.250415 − 0.170730i 2.23188 − 2.07088i −1.58398 + 1.60679i 2.93424 − 2.00053i 1.87055 + 2.34559i −1.72522 − 2.45430i −3.57336 + 1.72084i
7.10 −1.11367 − 0.343520i 1.26067 + 1.18773i −0.530233 − 0.361507i 0.196428 − 0.182259i −0.995961 − 1.75580i −2.10964 + 1.43833i 1.91960 + 2.40710i 0.178601 + 2.99468i −0.281365 + 0.135498i
7.11 −0.798432 − 0.246284i −1.49975 − 0.866458i −1.07564 − 0.733359i 1.73661 − 1.61133i 0.984054 + 1.06117i 1.64257 − 1.11988i 1.72013 + 2.15697i 1.49850 + 2.59894i −1.78341 + 0.858843i
7.12 −0.578865 − 0.178556i 1.24199 − 1.20725i −1.34928 − 0.919921i −2.14683 + 1.99197i −0.934508 + 0.477068i −3.38240 + 2.30608i 1.37218 + 1.72066i 0.0850969 − 2.99879i 1.59840 − 0.769750i
7.13 −0.231669 − 0.0714605i −1.66097 + 0.491106i −1.60391 − 1.09353i −1.73028 + 1.60547i 0.419890 + 0.00491935i 3.03807 − 2.07132i 0.595752 + 0.747049i 2.51763 − 1.63142i 0.515580 − 0.248290i
7.14 −0.189018 − 0.0583043i −1.70982 − 0.276636i −1.62015 − 1.10460i 0.827004 − 0.767348i 0.307057 + 0.151979i −3.35721 + 2.28891i 0.488494 + 0.612552i 2.84694 + 0.945995i −0.201058 + 0.0968245i
7.15 −0.00356936 − 0.00110100i −0.353602 + 1.69557i −1.65247 − 1.12663i −0.638623 + 0.592556i 0.00312896 − 0.00566279i −1.40581 + 0.958464i 0.00931567 + 0.0116815i −2.74993 − 1.19912i 0.00293188 − 0.00141192i
7.16 0.510660 + 0.157518i 1.70190 + 0.321795i −1.41652 − 0.965764i 1.10132 − 1.02188i 0.818402 + 0.432407i 1.15558 − 0.787864i −1.23762 − 1.55193i 2.79290 + 1.09532i 0.723367 − 0.348355i
7.17 0.593900 + 0.183194i 0.918881 − 1.46822i −1.33332 − 0.909042i −1.38781 + 1.28770i 0.814692 − 0.703640i 2.77469 − 1.89175i −1.40034 − 1.75597i −1.31132 − 2.69823i −1.06012 + 0.510525i
7.18 0.697178 + 0.215051i 1.12130 + 1.32011i −1.21267 − 0.826782i −3.10978 + 2.88546i 0.497853 + 1.16149i −0.507723 + 0.346159i −1.57743 − 1.97804i −0.485385 + 2.96047i −2.78859 + 1.34292i
7.19 0.771348 + 0.237929i −0.832788 + 1.51870i −1.11411 − 0.759588i 2.79791 − 2.59608i −1.00371 + 0.973304i 1.85057 − 1.26169i −1.68521 − 2.11319i −1.61293 − 2.52952i 2.77584 − 1.33677i
7.20 0.985541 + 0.303999i −0.426367 − 1.67875i −0.773602 − 0.527433i 1.85682 − 1.72287i 0.0901373 − 1.78409i −0.903456 + 0.615966i −1.88816 − 2.36768i −2.63642 + 1.43153i 2.35372 − 1.13349i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
29.d even 7 1 inner
261.q even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 261.2.q.a ✓ 336
3.b odd 2 1 783.2.u.a 336
9.c even 3 1 inner 261.2.q.a ✓ 336
9.d odd 6 1 783.2.u.a 336
29.d even 7 1 inner 261.2.q.a ✓ 336
87.j odd 14 1 783.2.u.a 336
261.q even 21 1 inner 261.2.q.a ✓ 336
261.t odd 42 1 783.2.u.a 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.2.q.a ✓ 336 1.a even 1 1 trivial
261.2.q.a ✓ 336 9.c even 3 1 inner
261.2.q.a ✓ 336 29.d even 7 1 inner
261.2.q.a ✓ 336 261.q even 21 1 inner
783.2.u.a 336 3.b odd 2 1
783.2.u.a 336 9.d odd 6 1
783.2.u.a 336 87.j odd 14 1
783.2.u.a 336 261.t odd 42 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(261, [\chi])\).