↑

Properties

Label 805.2.bb.b
Level $805$
Weight $2$
Character orbit 805.bb
Analytic conductor $6.428$
Analytic rank $0$
Dimension $320$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [805,2,Mod(76,805)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(805, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 11, 19]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("805.76");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 805 = 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 805.bb (of order \(22\), degree \(10\), 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: \(6.42795736271\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(32\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q + 2 q^{2} - 34 q^{4} + 32 q^{5} + 3 q^{7} + 6 q^{8} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q + 2 q^{2} - 34 q^{4} + 32 q^{5} + 3 q^{7} + 6 q^{8} + 28 q^{9} - 2 q^{10} - 6 q^{14} - 24 q^{16} + 2 q^{17} + 6 q^{18} - 8 q^{19} + 34 q^{20} - 17 q^{21} - 8 q^{23} - 32 q^{25} + 10 q^{28} + 6 q^{29} + 12 q^{32} + 20 q^{33} + 16 q^{34} + 8 q^{35} + 15 q^{36} + 58 q^{38} - 24 q^{39} + 16 q^{40} + 44 q^{41} + 50 q^{42} - 44 q^{43} - 11 q^{44} + 324 q^{45} + 16 q^{46} - 176 q^{48} - q^{49} - 9 q^{50} - 44 q^{51} - 154 q^{54} + 125 q^{56} - 123 q^{58} - 8 q^{61} + 120 q^{63} - 62 q^{64} - 112 q^{66} - 48 q^{68} + 12 q^{69} + 6 q^{70} - 10 q^{71} - 218 q^{72} + 11 q^{74} - 212 q^{76} - 96 q^{77} - 200 q^{78} - 44 q^{79} + 46 q^{80} - 72 q^{81} + 22 q^{83} - 100 q^{84} - 2 q^{85} - 22 q^{86} + 165 q^{88} + 12 q^{89} - 6 q^{90} + 92 q^{91} - 258 q^{92} - 44 q^{93} + 484 q^{94} + 8 q^{95} - 12 q^{97} + 7 q^{98} - 264 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1 −1.74334 + 2.01192i −0.284437 − 0.442593i −0.723962 − 5.03527i −0.415415 − 0.909632i 1.38633 + 0.199325i 1.54565 + 2.14732i 6.91357 + 4.44308i 1.13126 − 2.47712i 2.55431 + 0.750014i
76.2 −1.69233 + 1.95306i 1.51516 + 2.35764i −0.665810 − 4.63081i −0.415415 − 0.909632i −7.16877 − 1.03071i 1.17128 − 2.37236i 5.82297 + 3.74220i −2.01650 + 4.41551i 2.47959 + 0.728072i
76.3 −1.61808 + 1.86737i 1.18159 + 1.83859i −0.584240 − 4.06348i −0.415415 − 0.909632i −5.34523 − 0.768528i −2.34723 + 1.22086i 4.37608 + 2.81234i −0.738008 + 1.61601i 2.37079 + 0.696128i
76.4 −1.55872 + 1.79886i −1.27984 − 1.99148i −0.521657 − 3.62820i −0.415415 − 0.909632i 5.57731 + 0.801895i −1.91413 + 1.82650i 3.33499 + 2.14327i −1.08173 + 2.36866i 2.28382 + 0.670589i
76.5 −1.49655 + 1.72711i −0.519871 − 0.808936i −0.458620 − 3.18977i −0.415415 − 0.909632i 2.17514 + 0.312737i 2.57309 − 0.615812i 2.35042 + 1.51052i 0.862134 − 1.88781i 2.19272 + 0.643842i
76.6 −1.35368 + 1.56223i −0.912924 − 1.42054i −0.323483 − 2.24987i −0.415415 − 0.909632i 3.45501 + 0.496756i −0.510857 − 2.59596i 0.474752 + 0.305105i 0.0617479 − 0.135209i 1.98339 + 0.582376i
76.7 −1.11130 + 1.28250i 0.828586 + 1.28930i −0.125208 − 0.870839i −0.415415 − 0.909632i −2.57434 − 0.370135i 2.50140 − 0.861962i −1.59921 − 1.02775i 0.270493 − 0.592296i 1.62826 + 0.478099i
76.8 −1.09870 + 1.26796i −1.64073 − 2.55302i −0.115967 − 0.806570i −0.415415 − 0.909632i 5.03980 + 0.724614i −0.0736811 − 2.64473i −1.67272 − 1.07499i −2.57969 + 5.64873i 1.60979 + 0.472678i
76.9 −1.05312 + 1.21537i 0.141570 + 0.220287i −0.0834240 − 0.580227i −0.415415 − 0.909632i −0.416820 − 0.0599297i 0.0706956 + 2.64481i −1.91270 − 1.22922i 1.21776 − 2.66652i 1.54302 + 0.453072i
76.10 −0.767643 + 0.885908i 1.72318 + 2.68132i 0.0890736 + 0.619520i −0.415415 − 0.909632i −3.69819 − 0.531720i −2.63992 − 0.175566i −2.58949 − 1.66416i −2.97389 + 6.51191i 1.12474 + 0.330254i
76.11 −0.603281 + 0.696223i 0.737851 + 1.14812i 0.163851 + 1.13961i −0.415415 − 0.909632i −1.24448 − 0.178929i −0.318147 − 2.62655i −2.44225 − 1.56954i 0.472493 − 1.03462i 0.883919 + 0.259542i
76.12 −0.573328 + 0.661656i −1.07837 − 1.67798i 0.175546 + 1.22095i −0.415415 − 0.909632i 1.72851 + 0.248522i 2.03843 + 1.68665i −2.38152 − 1.53051i −0.406491 + 0.890091i 0.840033 + 0.246656i
76.13 −0.455668 + 0.525869i −0.640871 − 0.997215i 0.215725 + 1.50040i −0.415415 − 0.909632i 0.816429 + 0.117385i −0.919500 + 2.48083i −2.05804 − 1.32262i 0.662523 − 1.45072i 0.667639 + 0.196036i
76.14 −0.405797 + 0.468315i 1.70223 + 2.64873i 0.229982 + 1.59956i −0.415415 − 0.909632i −1.93120 − 0.277665i 2.42316 + 1.06222i −1.88502 − 1.21143i −2.87192 + 6.28862i 0.594568 + 0.174581i
76.15 −0.222337 + 0.256591i −1.54604 − 2.40568i 0.268225 + 1.86554i −0.415415 − 0.909632i 0.961019 + 0.138174i −2.59109 − 0.535035i −1.10956 − 0.713071i −2.15083 + 4.70966i 0.325766 + 0.0956534i
76.16 −0.100332 + 0.115789i 0.359063 + 0.558713i 0.281289 + 1.95641i −0.415415 − 0.909632i −0.100718 − 0.0144811i −0.822876 − 2.51453i −0.512531 − 0.329384i 1.06301 − 2.32767i 0.147005 + 0.0431645i
76.17 0.0706938 − 0.0815850i −0.141975 − 0.220917i 0.282971 + 1.96811i −0.415415 − 0.909632i −0.0280602 − 0.00403445i −2.60143 − 0.482234i 0.362203 + 0.232774i 1.21760 − 2.66617i −0.103580 − 0.0304137i
76.18 0.136706 − 0.157767i 1.11625 + 1.73691i 0.278428 + 1.93651i −0.415415 − 0.909632i 0.426626 + 0.0613395i −1.76097 + 1.97458i 0.694813 + 0.446529i −0.524619 + 1.14876i −0.200300 − 0.0588133i
76.19 0.353457 − 0.407911i 0.273369 + 0.425370i 0.243170 + 1.69128i −0.415415 − 0.909632i 0.270137 + 0.0388399i 2.52204 + 0.799578i 1.68397 + 1.08222i 1.14004 − 2.49633i −0.517881 − 0.152063i
76.20 0.476258 − 0.549631i −1.26437 − 1.96739i 0.209357 + 1.45611i −0.415415 − 0.909632i −1.68351 − 0.242052i 1.08048 − 2.41507i 2.12366 + 1.36480i −1.02577 + 2.24612i −0.697807 − 0.204895i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
161.k even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 805.2.bb.b yes 320
7.b odd 2 1 805.2.bb.a ✓ 320
23.d odd 22 1 805.2.bb.a ✓ 320
161.k even 22 1 inner 805.2.bb.b yes 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.bb.a ✓ 320 7.b odd 2 1
805.2.bb.a ✓ 320 23.d odd 22 1
805.2.bb.b yes 320 1.a even 1 1 trivial
805.2.bb.b yes 320 161.k even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{320} - 62 T_{3}^{318} + 2147 T_{3}^{316} - 22 T_{3}^{315} - 54527 T_{3}^{314} + \cdots + 26\!\cdots\!16 \) acting on \(S_{2}^{\mathrm{new}}(805, [\chi])\). Copy content Toggle raw display