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Properties

Label 869.2.z.a
Level $869$
Weight $2$
Character orbit 869.z
Analytic conductor $6.939$
Analytic rank $0$
Dimension $3744$
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] = [869,2,Mod(17,869)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(869, base_ring=CyclotomicField(130))
 
chi = DirichletCharacter(H, H._module([117, 35]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("869.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 869 = 11 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 869.z (of order \(130\), degree \(48\), 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: \(6.93899993565\)
Analytic rank: \(0\)
Dimension: \(3744\)
Relative dimension: \(78\) over \(\Q(\zeta_{130})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{130}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 3744 q - 55 q^{2} - 39 q^{3} - 113 q^{4} - 33 q^{5} - 65 q^{6} - 65 q^{7} + 55 q^{8} - 95 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 3744 q - 55 q^{2} - 39 q^{3} - 113 q^{4} - 33 q^{5} - 65 q^{6} - 65 q^{7} + 55 q^{8} - 95 q^{9} - 70 q^{11} - 104 q^{12} - 45 q^{13} - 39 q^{14} - 117 q^{15} - 9 q^{16} - 65 q^{17} - 5 q^{18} - 55 q^{19} - 76 q^{20} - 57 q^{22} - 204 q^{23} - 95 q^{25} - 34 q^{26} - 39 q^{27} - 65 q^{28} - 65 q^{29} - 65 q^{30} + 53 q^{31} - 52 q^{33} - 234 q^{34} - 65 q^{35} + 61 q^{36} - 91 q^{37} - 31 q^{38} + 455 q^{39} - 155 q^{40} - 65 q^{41} - 97 q^{42} - 40 q^{44} - 40 q^{45} - 135 q^{46} - 65 q^{47} + 13 q^{48} + 39 q^{49} - 105 q^{50} - 95 q^{51} - 325 q^{52} - 195 q^{53} - 154 q^{55} - 65 q^{57} - 39 q^{58} - 39 q^{59} + 507 q^{60} - 65 q^{61} - 55 q^{62} - 65 q^{63} - 197 q^{64} + 338 q^{66} - 28 q^{67} - 65 q^{68} - 52 q^{69} - 13 q^{70} - 39 q^{71} - 45 q^{72} + 45 q^{73} - 65 q^{74} - 39 q^{75} + 247 q^{77} - 145 q^{79} + 266 q^{80} - 87 q^{81} - 39 q^{82} - 135 q^{83} - 205 q^{84} - 65 q^{85} - 221 q^{86} + 264 q^{88} - 132 q^{89} - 755 q^{90} - 52 q^{91} - 114 q^{92} - 143 q^{93} - 65 q^{94} - 5 q^{95} - 65 q^{96} + 25 q^{97} - 69 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −2.74364 + 0.199259i −0.00821958 − 0.0847672i 5.50886 − 0.804411i −1.28215 − 1.67792i 0.0394422 + 0.230933i 0.0622405 − 0.281643i −9.58192 + 2.11751i 2.93699 − 0.574987i 3.85210 + 4.34813i
17.2 −2.70123 + 0.196179i 0.152894 + 1.57677i 5.27916 − 0.770870i 1.32703 + 1.73665i −0.722333 − 4.22924i −0.515825 + 2.33415i −8.81992 + 1.94912i 0.481269 − 0.0942199i −3.92532 − 4.43076i
17.3 −2.67036 + 0.193937i −0.167725 − 1.72972i 5.11421 − 0.746784i 2.44819 + 3.20388i 0.783344 + 4.58646i 0.169331 − 0.766238i −8.28333 + 1.83054i −0.0197018 + 0.00385709i −7.15889 − 8.08072i
17.4 −2.56537 + 0.186312i −0.299464 − 3.08832i 4.56741 − 0.666939i −0.810957 − 1.06128i 1.34363 + 7.86690i 0.720928 − 3.26226i −6.56977 + 1.45186i −6.50395 + 1.27330i 2.27814 + 2.57148i
17.5 −2.52841 + 0.183627i −0.168399 − 1.73667i 4.38014 − 0.639594i −1.40983 − 1.84500i 0.744684 + 4.36010i −0.535463 + 2.42301i −6.00666 + 1.32741i −0.0435633 + 0.00852856i 3.90342 + 4.40605i
17.6 −2.48165 + 0.180231i 0.254943 + 2.62918i 4.14708 − 0.605562i −1.88465 − 2.46640i −1.10654 − 6.47875i 0.0231386 − 0.104704i −5.32332 + 1.17640i −3.90349 + 0.764202i 5.12157 + 5.78105i
17.7 −2.44218 + 0.177365i 0.286047 + 2.94995i 3.95376 − 0.577334i 0.791545 + 1.03587i −1.22180 − 7.15358i 0.660089 − 2.98696i −4.77155 + 1.05447i −5.67630 + 1.11127i −2.11682 − 2.38940i
17.8 −2.39340 + 0.173822i 0.140419 + 1.44812i 3.71913 − 0.543072i 0.712101 + 0.931908i −0.587794 − 3.44152i 0.888675 − 4.02133i −4.12062 + 0.910618i 0.866780 − 0.169693i −1.86633 − 2.10665i
17.9 −2.28416 + 0.165889i −0.181019 − 1.86682i 3.21087 − 0.468855i −0.216942 − 0.283907i 0.723160 + 4.23409i −0.797130 + 3.60708i −2.78391 + 0.615219i −0.508130 + 0.0994786i 0.542629 + 0.612501i
17.10 −2.22795 + 0.161806i −0.0491564 − 0.506942i 2.95856 − 0.432013i 1.43019 + 1.87166i 0.191544 + 1.12149i 0.152498 − 0.690068i −2.15924 + 0.477172i 2.68954 − 0.526541i −3.48924 − 3.93854i
17.11 −2.15092 + 0.156212i −0.314534 − 3.24373i 2.62305 − 0.383021i 1.31228 + 1.71734i 1.18325 + 6.92788i −0.659341 + 2.98357i −1.37058 + 0.302885i −7.47876 + 1.46415i −3.09087 − 3.48887i
17.12 −2.09037 + 0.151814i −0.205124 − 2.11541i 2.36758 − 0.345718i −1.29230 − 1.69119i 0.749934 + 4.39084i 0.994381 − 4.49965i −0.803645 + 0.177598i −1.48876 + 0.291461i 2.95813 + 3.33903i
17.13 −2.07744 + 0.150875i 0.0181292 + 0.186963i 2.31399 − 0.337892i −2.03944 − 2.66896i −0.0658704 − 0.385670i 0.479325 − 2.16898i −0.688513 + 0.152155i 2.90948 − 0.569601i 4.63950 + 5.23692i
17.14 −2.01114 + 0.146060i 0.152138 + 1.56897i 2.04434 − 0.298518i −1.04489 − 1.36742i −0.535135 − 3.13320i −0.701460 + 3.17416i −0.129994 + 0.0287275i 0.505590 − 0.0989813i 2.30115 + 2.59747i
17.15 −1.99485 + 0.144877i 0.181897 + 1.87588i 1.97944 − 0.289041i 2.58539 + 3.38343i −0.634631 − 3.71575i −0.0811295 + 0.367118i −0.000849869 0 0.000187813i −0.541717 + 0.106054i −5.64766 − 6.37489i
17.16 −1.98498 + 0.144161i 0.0331807 + 0.342187i 1.94037 − 0.283335i 0.164563 + 0.215360i −0.115193 − 0.674453i 0.00806547 − 0.0364969i 0.0758942 − 0.0167719i 2.82812 − 0.553672i −0.357702 − 0.403763i
17.17 −1.74571 + 0.126783i 0.315735 + 3.25612i 1.05242 − 0.153676i −0.0671366 − 0.0878599i −0.964002 − 5.64421i −0.713674 + 3.22943i 1.60041 − 0.353675i −7.55850 + 1.47976i 0.128340 + 0.144866i
17.18 −1.69819 + 0.123332i −0.0709470 − 0.731664i 0.889614 − 0.129903i 1.13981 + 1.49164i 0.210719 + 1.23375i 0.703043 − 3.18133i 1.83038 − 0.404497i 2.41381 − 0.472561i −2.11957 − 2.39250i
17.19 −1.56303 + 0.113516i 0.0162189 + 0.167263i 0.451157 − 0.0658786i −2.53773 − 3.32106i −0.0443376 − 0.259595i −0.937577 + 4.24261i 2.36276 − 0.522147i 2.91640 − 0.570955i 4.34353 + 4.90283i
17.20 −1.52476 + 0.110736i 0.0799102 + 0.824101i 0.333614 − 0.0487148i 0.150456 + 0.196898i −0.213102 − 1.24771i −0.414915 + 1.87753i 2.48223 − 0.548549i 2.27135 − 0.444672i −0.251213 − 0.283561i
See next 80 embeddings (of 3744 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.78
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
79.f odd 26 1 inner
869.z even 130 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 869.2.z.a ✓ 3744
11.d odd 10 1 inner 869.2.z.a ✓ 3744
79.f odd 26 1 inner 869.2.z.a ✓ 3744
869.z even 130 1 inner 869.2.z.a ✓ 3744
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
869.2.z.a ✓ 3744 1.a even 1 1 trivial
869.2.z.a ✓ 3744 11.d odd 10 1 inner
869.2.z.a ✓ 3744 79.f odd 26 1 inner
869.2.z.a ✓ 3744 869.z even 130 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(869, [\chi])\).