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Properties

Label 693.2.cq.a
Level $693$
Weight $2$
Character orbit 693.cq
Analytic conductor $5.534$
Analytic rank $0$
Dimension $576$
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(29,693)]
 
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([5, 0, 21]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.cq (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: \(576\)
Relative dimension: \(72\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 576 q - 6 q^{3} + 72 q^{4} - 6 q^{5} + 10 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 576 q - 6 q^{3} + 72 q^{4} - 6 q^{5} + 10 q^{6} + 2 q^{9} - 12 q^{11} + 56 q^{12} - 6 q^{15} + 72 q^{16} - 20 q^{18} + 60 q^{19} + 24 q^{20} + 6 q^{22} - 30 q^{24} - 78 q^{25} - 90 q^{29} - 60 q^{30} + 6 q^{31} + 2 q^{33} + 12 q^{34} - 26 q^{36} - 12 q^{37} - 120 q^{38} - 110 q^{39} + 24 q^{45} + 24 q^{47} + 12 q^{48} - 72 q^{49} + 30 q^{51} - 60 q^{57} - 120 q^{59} + 32 q^{60} - 144 q^{64} + 20 q^{66} - 36 q^{67} - 390 q^{68} - 26 q^{69} - 150 q^{72} - 32 q^{75} - 232 q^{78} + 106 q^{81} - 36 q^{82} + 90 q^{83} + 80 q^{84} + 102 q^{86} + 114 q^{88} + 200 q^{90} - 36 q^{91} + 228 q^{92} + 66 q^{93} + 120 q^{95} + 320 q^{96} - 60 q^{97} - 158 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −2.57608 + 1.14694i 1.45021 + 0.947037i 3.98243 − 4.42294i 0.786460 − 1.76642i −4.82206 − 0.776325i −0.207912 + 0.978148i −3.44342 + 10.5977i 1.20624 + 2.74681i 5.45246i
29.2 −2.44704 + 1.08949i −0.771259 − 1.55086i 3.46276 − 3.84578i −1.31888 + 2.96225i 3.57695 + 2.95473i −0.207912 + 0.978148i −2.62808 + 8.08840i −1.81032 + 2.39223i − 8.68565i
29.3 −2.42262 + 1.07862i −0.490008 + 1.66129i 3.36739 − 3.73986i −0.231252 + 0.519400i −0.604799 − 4.55320i 0.207912 − 0.978148i −2.48505 + 7.64820i −2.51978 − 1.62809i − 1.50774i
29.4 −2.34643 + 1.04470i 1.60050 + 0.662127i 3.07609 − 3.41635i −1.47509 + 3.31310i −4.44718 + 0.118401i 0.207912 − 0.978148i −2.06138 + 6.34427i 2.12318 + 2.11946i − 9.31500i
29.5 −2.32607 + 1.03563i 0.937136 − 1.45663i 2.99980 − 3.33162i −0.458233 + 1.02921i −0.671307 + 4.35876i 0.207912 − 0.978148i −1.95378 + 6.01312i −1.24355 − 2.73012i − 2.86857i
29.6 −2.24261 + 0.998476i 1.50527 − 0.856841i 2.69410 − 2.99210i 1.21099 − 2.71992i −2.52019 + 3.42453i 0.207912 − 0.978148i −1.53711 + 4.73073i 1.53165 − 2.57955i 7.30888i
29.7 −2.23348 + 0.994408i 0.561662 − 1.63846i 2.66131 − 2.95568i 0.183538 − 0.412234i 0.374835 + 4.21797i −0.207912 + 0.978148i −1.49382 + 4.59751i −2.36907 − 1.84052i 1.10323i
29.8 −2.11739 + 0.942722i −1.70608 − 0.298812i 2.25635 − 2.50593i −1.18761 + 2.66741i 3.89413 − 0.975658i 0.207912 − 0.978148i −0.982712 + 3.02448i 2.82142 + 1.01960i − 6.76753i
29.9 −2.05287 + 0.913996i 0.0948586 + 1.72945i 2.04062 − 2.26633i 1.50243 − 3.37451i −1.77544 − 3.46363i 0.207912 − 0.978148i −0.728885 + 2.24328i −2.98200 + 0.328107i 8.30064i
29.10 −2.03260 + 0.904971i −1.32256 + 1.11841i 1.97422 − 2.19260i 0.544572 − 1.22313i 1.67611 − 3.47015i −0.207912 + 0.978148i −0.653470 + 2.01117i 0.498333 − 2.95832i 2.97895i
29.11 −1.97441 + 0.879066i −1.35151 − 1.08324i 1.78729 − 1.98499i 0.890532 − 2.00017i 3.62069 + 0.950707i −0.207912 + 0.978148i −0.448186 + 1.37937i 0.653162 + 2.92803i 4.73200i
29.12 −1.95377 + 0.869875i −1.05969 − 1.37006i 1.72228 − 1.91279i 0.700122 − 1.57250i 3.26217 + 1.75498i 0.207912 − 0.978148i −0.379288 + 1.16733i −0.754112 + 2.90367i 3.68133i
29.13 −1.91396 + 0.852151i 1.72697 + 0.132579i 1.59883 − 1.77568i 0.424640 − 0.953756i −3.41833 + 1.21789i −0.207912 + 0.978148i −0.252115 + 0.775929i 2.96485 + 0.457921i 2.18731i
29.14 −1.88769 + 0.840455i 1.67168 − 0.453303i 1.51876 − 1.68675i −1.47945 + 3.32290i −2.77464 + 2.26067i −0.207912 + 0.978148i −0.172243 + 0.530108i 2.58903 − 1.51555i − 7.51603i
29.15 −1.84553 + 0.821683i 0.354972 + 1.69529i 1.39255 − 1.54659i −0.00639606 + 0.0143658i −2.04810 − 2.83703i −0.207912 + 0.978148i −0.0506535 + 0.155896i −2.74799 + 1.20356i − 0.0317680i
29.16 −1.43049 + 0.636897i −1.65919 + 0.497078i 0.302414 − 0.335865i −1.42359 + 3.19745i 2.05687 − 1.76780i −0.207912 + 0.978148i 0.749071 − 2.30540i 2.50583 − 1.64949i − 5.48061i
29.17 −1.39631 + 0.621678i −1.19012 + 1.25842i 0.224941 − 0.249823i 0.140763 − 0.316158i 0.879445 − 2.49702i 0.207912 − 0.978148i 0.785857 − 2.41862i −0.167236 − 2.99534i 0.528965i
29.18 −1.39084 + 0.619244i 1.52131 + 0.828013i 0.212724 − 0.236254i −0.497370 + 1.11711i −2.62865 − 0.209573i 0.207912 − 0.978148i 0.791370 − 2.43559i 1.62879 + 2.51933i − 1.86172i
29.19 −1.37697 + 0.613064i 1.15539 − 1.29038i 0.181924 − 0.202047i −0.651759 + 1.46387i −0.799849 + 2.48513i −0.207912 + 0.978148i 0.804913 − 2.47727i −0.330145 − 2.98178i − 2.41527i
29.20 −1.35909 + 0.605104i −1.72528 + 0.153020i 0.142704 − 0.158489i 1.67642 − 3.76529i 2.25221 − 1.25194i 0.207912 − 0.978148i 0.821408 − 2.52803i 2.95317 − 0.528003i 6.13177i
See next 80 embeddings (of 576 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
11.d odd 10 1 inner
99.p 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.cq.a ✓ 576
9.d odd 6 1 inner 693.2.cq.a ✓ 576
11.d odd 10 1 inner 693.2.cq.a ✓ 576
99.p even 30 1 inner 693.2.cq.a ✓ 576
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.cq.a ✓ 576 1.a even 1 1 trivial
693.2.cq.a ✓ 576 9.d odd 6 1 inner
693.2.cq.a ✓ 576 11.d odd 10 1 inner
693.2.cq.a ✓ 576 99.p even 30 1 inner

Hecke kernels

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