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Properties

Label 693.2.db.a
Level $693$
Weight $2$
Character orbit 693.db
Analytic conductor $5.534$
Analytic rank $0$
Dimension $736$
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] = [693,2,Mod(5,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([25, 25, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.db (of order \(30\), degree \(8\), 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: \(5.53363286007\)
Analytic rank: \(0\)
Dimension: \(736\)
Relative dimension: \(92\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 736 q - 9 q^{3} + 170 q^{4} - 9 q^{5} - 12 q^{6} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 736 q - 9 q^{3} + 170 q^{4} - 9 q^{5} - 12 q^{6} - 3 q^{7} + 3 q^{9} - 48 q^{10} - 6 q^{11} - 36 q^{12} - 9 q^{14} - 174 q^{16} - 35 q^{18} - 18 q^{19} + 18 q^{20} - 24 q^{21} - 12 q^{22} - 42 q^{23} - 3 q^{24} + 79 q^{25} - 48 q^{26} - 27 q^{27} - 8 q^{28} - 18 q^{29} - 45 q^{30} - 12 q^{33} - 12 q^{34} - 75 q^{35} + 28 q^{36} - 6 q^{37} - 39 q^{38} + 9 q^{39} - 63 q^{40} + 123 q^{42} - 16 q^{43} + 27 q^{44} - 72 q^{45} + 6 q^{46} - 18 q^{47} - 30 q^{48} - 3 q^{49} - 111 q^{50} + 43 q^{51} + 39 q^{52} + 36 q^{53} - 120 q^{54} - 78 q^{56} - 34 q^{57} - 23 q^{58} + 42 q^{59} - 43 q^{60} - 24 q^{62} + 61 q^{63} + 132 q^{64} + 3 q^{66} - 16 q^{67} - 9 q^{68} - 27 q^{69} - 81 q^{70} - 41 q^{72} - 18 q^{73} - 33 q^{74} - 51 q^{75} - 63 q^{77} + 16 q^{78} + 42 q^{79} + 24 q^{80} - 5 q^{81} - 30 q^{82} - 114 q^{84} - 28 q^{85} + 15 q^{86} + 144 q^{87} + 16 q^{88} - 30 q^{89} - 12 q^{90} + 16 q^{91} - 168 q^{92} - 2 q^{93} + 213 q^{96} - 60 q^{98} - 112 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −2.64708 + 0.860089i 1.71916 − 0.210928i 4.64926 − 3.37788i −3.17850 + 0.675612i −4.36934 + 2.03697i −1.68466 + 2.04008i −6.12971 + 8.43683i 2.91102 − 0.725236i 7.83267 − 4.52220i
5.2 −2.62448 + 0.852746i −1.53036 − 0.811173i 4.54270 − 3.30046i 2.16902 − 0.461040i 4.70812 + 0.823902i −0.337307 + 2.62416i −5.86373 + 8.07073i 1.68400 + 2.48277i −5.29941 + 3.05961i
5.3 −2.53118 + 0.822430i −0.957248 − 1.44349i 4.11245 − 2.98787i −1.30416 + 0.277208i 3.61014 + 2.86648i 0.451073 − 2.60702i −4.82332 + 6.63873i −1.16735 + 2.76356i 3.07309 − 1.77425i
5.4 −2.49755 + 0.811504i 0.845942 − 1.51142i 3.96120 − 2.87798i 3.64232 − 0.774199i −0.886264 + 4.46133i −1.34169 − 2.28032i −4.47067 + 6.15335i −1.56876 − 2.55714i −8.46862 + 4.88936i
5.5 −2.44673 + 0.794990i 0.840753 + 1.51431i 3.73644 − 2.71468i 1.79917 − 0.382426i −3.26096 − 3.03672i −2.20797 − 1.45769i −3.95958 + 5.44990i −1.58627 + 2.54632i −4.09806 + 2.36602i
5.6 −2.37485 + 0.771635i 1.57080 + 0.729795i 3.42645 − 2.48946i 2.85917 − 0.607736i −4.29354 − 0.521071i 2.54263 + 0.731476i −3.28088 + 4.51574i 1.93480 + 2.29272i −6.32115 + 3.64952i
5.7 −2.33942 + 0.760125i 0.697437 − 1.58543i 3.27708 − 2.38094i −1.69336 + 0.359935i −0.426478 + 4.23913i 2.45288 + 0.991658i −2.96499 + 4.08096i −2.02716 − 2.21147i 3.68790 − 2.12921i
5.8 −2.32229 + 0.754559i −1.68533 + 0.399583i 3.20566 − 2.32905i −3.02989 + 0.644024i 3.61232 − 2.19963i −2.13757 − 1.55910i −2.81657 + 3.87667i 2.68067 − 1.34686i 6.55035 − 3.78185i
5.9 −2.28116 + 0.741192i −1.02769 + 1.39422i 3.03627 − 2.20598i 1.95147 − 0.414797i 1.31094 − 3.94215i −2.31689 + 1.27750i −2.47149 + 3.40171i −0.887697 − 2.86566i −4.14416 + 2.39263i
5.10 −2.26202 + 0.734976i −0.547140 + 1.64336i 2.95853 − 2.14950i −2.37089 + 0.503949i 0.0298131 − 4.11946i 0.154772 + 2.64122i −2.31642 + 3.18828i −2.40127 − 1.79830i 4.99263 − 2.88250i
5.11 −2.24868 + 0.730641i −1.61319 + 0.630558i 2.90470 − 2.11039i −1.14834 + 0.244087i 3.16685 − 2.59659i 2.63731 − 0.211177i −2.21028 + 3.04219i 2.20479 − 2.03442i 2.40390 − 1.38790i
5.12 −2.21462 + 0.719574i 1.73163 + 0.0380337i 2.76872 − 2.01159i −0.344883 + 0.0733072i −3.86228 + 1.16181i 1.71938 − 2.01090i −1.94675 + 2.67948i 2.99711 + 0.131721i 0.711035 − 0.410516i
5.13 −2.01345 + 0.654211i −0.137264 − 1.72660i 2.00797 − 1.45888i 1.09272 − 0.232264i 1.40594 + 3.38664i −0.257905 + 2.63315i −0.599780 + 0.825527i −2.96232 + 0.474002i −2.04819 + 1.18252i
5.14 −1.96414 + 0.638188i 1.32193 − 1.11915i 1.83253 − 1.33141i 0.807298 − 0.171597i −1.88224 + 3.04180i −2.63246 + 0.264899i −0.321846 + 0.442983i 0.495023 − 2.95888i −1.47614 + 0.852248i
5.15 −1.93410 + 0.628429i −1.70677 − 0.294867i 1.72780 − 1.25532i 2.20682 − 0.469074i 3.48637 − 0.502278i 2.55795 + 0.675927i −0.162187 + 0.223231i 2.82611 + 1.00654i −3.97344 + 2.29406i
5.16 −1.80602 + 0.586811i −1.18802 − 1.26040i 1.29932 − 0.944013i −3.77091 + 0.801532i 2.88521 + 1.57916i −0.609191 + 2.57466i 0.439718 − 0.605220i −0.177207 + 2.99476i 6.33999 − 3.66040i
5.17 −1.79123 + 0.582004i 1.38620 + 1.03848i 1.25173 − 0.909432i −0.701482 + 0.149105i −3.08740 − 1.05337i −2.58578 + 0.560143i 0.501247 − 0.689907i 0.843128 + 2.87909i 1.16973 − 0.675345i
5.18 −1.79046 + 0.581756i 0.870345 + 1.49750i 1.24928 − 0.907653i −3.45177 + 0.733696i −2.42950 − 2.17488i 2.60320 + 0.472612i 0.504381 − 0.694221i −1.48500 + 2.60668i 5.75342 − 3.32174i
5.19 −1.78970 + 0.581508i 0.00461531 + 1.73204i 1.24683 − 0.905875i 3.79286 − 0.806198i −1.01546 − 3.09715i 2.34297 + 1.22902i 0.507513 − 0.698531i −2.99996 + 0.0159878i −6.31926 + 3.64843i
5.20 −1.70106 + 0.552708i −1.09179 − 1.34462i 0.970083 − 0.704807i 0.674586 − 0.143388i 2.60038 + 1.68383i −2.18958 − 1.48517i 0.842008 − 1.15892i −0.615987 + 2.93608i −1.06826 + 0.616760i
See next 80 embeddings (of 736 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.92
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner
63.i even 6 1 inner
693.db even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.db.a yes 736
7.d odd 6 1 693.2.ce.a ✓ 736
9.d odd 6 1 693.2.ce.a ✓ 736
11.c even 5 1 inner 693.2.db.a yes 736
63.i even 6 1 inner 693.2.db.a yes 736
77.p odd 30 1 693.2.ce.a ✓ 736
99.n odd 30 1 693.2.ce.a ✓ 736
693.db even 30 1 inner 693.2.db.a yes 736
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.ce.a ✓ 736 7.d odd 6 1
693.2.ce.a ✓ 736 9.d odd 6 1
693.2.ce.a ✓ 736 77.p odd 30 1
693.2.ce.a ✓ 736 99.n odd 30 1
693.2.db.a yes 736 1.a even 1 1 trivial
693.2.db.a yes 736 11.c even 5 1 inner
693.2.db.a yes 736 63.i even 6 1 inner
693.2.db.a yes 736 693.db even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(693, [\chi])\).