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Properties

Label 650.2.bf.b
Level $650$
Weight $2$
Character orbit 650.bf
Analytic conductor $5.190$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [650,2,Mod(69,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([27, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.69");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.bf (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: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 18 q^{2} + 18 q^{4} - 3 q^{5} + 16 q^{7} - 36 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 18 q^{2} + 18 q^{4} - 3 q^{5} + 16 q^{7} - 36 q^{8} + 10 q^{9} - 3 q^{13} + 8 q^{14} + 2 q^{15} + 18 q^{16} + 15 q^{17} - 160 q^{18} - 15 q^{19} - 12 q^{20} + 10 q^{23} + 15 q^{25} - 3 q^{26} - 4 q^{28} - 9 q^{29} - 10 q^{30} - 72 q^{32} - 22 q^{33} + 5 q^{34} - 27 q^{35} + 10 q^{36} + 23 q^{37} - 26 q^{39} - 3 q^{40} + 36 q^{41} - 18 q^{43} + 9 q^{45} + 10 q^{46} - 88 q^{49} - 9 q^{50} - 24 q^{51} + 6 q^{52} + 5 q^{53} - 18 q^{54} + 38 q^{55} - 4 q^{56} + 56 q^{57} - 9 q^{58} + 2 q^{59} + 8 q^{60} + 11 q^{61} - 10 q^{63} - 36 q^{64} - 54 q^{65} - 16 q^{66} + 14 q^{67} - 15 q^{68} - 57 q^{69} + 12 q^{70} + 15 q^{71} - 20 q^{72} - 14 q^{73} + 33 q^{74} + 108 q^{75} - 40 q^{77} - 48 q^{78} - 8 q^{79} + 16 q^{81} + 9 q^{82} + 36 q^{83} + 15 q^{84} + 54 q^{85} + 20 q^{86} + 40 q^{87} - 90 q^{89} - 3 q^{90} + 104 q^{91} + 20 q^{92} + 48 q^{93} + 75 q^{94} - 9 q^{95} + 14 q^{97} + 82 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1 −0.978148 − 0.207912i −3.11164 + 0.327047i 0.913545 + 0.406737i −2.17482 + 0.519748i 3.11164 + 0.327047i 1.97509 + 3.42096i −0.809017 − 0.587785i 6.64091 − 1.41157i 2.23536 − 0.0562186i
69.2 −0.978148 − 0.207912i −3.04735 + 0.320290i 0.913545 + 0.406737i −0.908637 − 2.04313i 3.04735 + 0.320290i −2.64268 − 4.57726i −0.809017 − 0.587785i 6.24934 − 1.32834i 0.463991 + 2.18740i
69.3 −0.978148 − 0.207912i −2.94881 + 0.309933i 0.913545 + 0.406737i 1.83439 − 1.27868i 2.94881 + 0.309933i 1.57011 + 2.71951i −0.809017 − 0.587785i 5.66499 − 1.20413i −2.06016 + 0.869343i
69.4 −0.978148 − 0.207912i −2.22859 + 0.234234i 0.913545 + 0.406737i −1.11362 + 1.93903i 2.22859 + 0.234234i −1.32317 − 2.29180i −0.809017 − 0.587785i 1.97729 − 0.420287i 1.49244 − 1.66512i
69.5 −0.978148 − 0.207912i −1.64135 + 0.172513i 0.913545 + 0.406737i 1.54993 + 1.61174i 1.64135 + 0.172513i −0.186368 − 0.322799i −0.809017 − 0.587785i −0.270171 + 0.0574266i −1.18096 − 1.89877i
69.6 −0.978148 − 0.207912i −1.27274 + 0.133770i 0.913545 + 0.406737i 1.51547 + 1.64419i 1.27274 + 0.133770i 1.69150 + 2.92977i −0.809017 − 0.587785i −1.33247 + 0.283225i −1.14051 − 1.92334i
69.7 −0.978148 − 0.207912i −1.18698 + 0.124757i 0.913545 + 0.406737i 1.97587 − 1.04687i 1.18698 + 0.124757i −2.02152 − 3.50138i −0.809017 − 0.587785i −1.54108 + 0.327566i −2.15035 + 0.613182i
69.8 −0.978148 − 0.207912i −0.635705 + 0.0668153i 0.913545 + 0.406737i −2.21226 + 0.325412i 0.635705 + 0.0668153i −0.239980 − 0.415657i −0.809017 − 0.587785i −2.53479 + 0.538785i 2.23158 + 0.141655i
69.9 −0.978148 − 0.207912i −0.570472 + 0.0599590i 0.913545 + 0.406737i −1.55855 − 1.60341i 0.570472 + 0.0599590i −0.100796 − 0.174584i −0.809017 − 0.587785i −2.61260 + 0.555325i 1.19113 + 1.89241i
69.10 −0.978148 − 0.207912i −0.285296 + 0.0299858i 0.913545 + 0.406737i −0.324735 − 2.21236i 0.285296 + 0.0299858i 1.21478 + 2.10407i −0.809017 − 0.587785i −2.85395 + 0.606625i −0.142337 + 2.23153i
69.11 −0.978148 − 0.207912i 1.12633 − 0.118382i 0.913545 + 0.406737i 1.33810 − 1.79150i −1.12633 − 0.118382i 0.825503 + 1.42981i −0.809017 − 0.587785i −1.67985 + 0.357063i −1.68134 + 1.47415i
69.12 −0.978148 − 0.207912i 1.25765 − 0.132184i 0.913545 + 0.406737i −1.59228 + 1.56992i −1.25765 − 0.132184i −1.01333 − 1.75514i −0.809017 − 0.587785i −1.37023 + 0.291252i 1.88389 − 1.20456i
69.13 −0.978148 − 0.207912i 1.60387 − 0.168574i 0.913545 + 0.406737i −0.433633 + 2.19362i −1.60387 − 0.168574i 2.21888 + 3.84321i −0.809017 − 0.587785i −0.390460 + 0.0829947i 0.880236 − 2.05553i
69.14 −0.978148 − 0.207912i 1.83484 − 0.192849i 0.913545 + 0.406737i 1.47312 + 1.68224i −1.83484 − 0.192849i −0.288231 − 0.499231i −0.809017 − 0.587785i 0.395000 − 0.0839599i −1.09117 − 1.95175i
69.15 −0.978148 − 0.207912i 2.14241 − 0.225177i 0.913545 + 0.406737i −2.13010 − 0.680210i −2.14241 − 0.225177i 2.10849 + 3.65201i −0.809017 − 0.587785i 1.60479 − 0.341108i 1.94213 + 1.10822i
69.16 −0.978148 − 0.207912i 2.49179 − 0.261898i 0.913545 + 0.406737i 0.816723 − 2.08158i −2.49179 − 0.261898i −1.32821 − 2.30052i −0.809017 − 0.587785i 3.20598 − 0.681453i −1.23166 + 1.86628i
69.17 −0.978148 − 0.207912i 3.15857 − 0.331979i 0.913545 + 0.406737i 2.23408 + 0.0943403i −3.15857 − 0.331979i 1.11854 + 1.93737i −0.809017 − 0.587785i 6.93192 − 1.47343i −2.16564 − 0.556769i
69.18 −0.978148 − 0.207912i 3.31348 − 0.348261i 0.913545 + 0.406737i −2.01161 + 0.976444i −3.31348 − 0.348261i −1.57862 − 2.73425i −0.809017 − 0.587785i 7.92344 − 1.68418i 2.17066 − 0.536870i
179.1 −0.978148 + 0.207912i −3.11164 − 0.327047i 0.913545 − 0.406737i −2.17482 − 0.519748i 3.11164 − 0.327047i 1.97509 − 3.42096i −0.809017 + 0.587785i 6.64091 + 1.41157i 2.23536 + 0.0562186i
179.2 −0.978148 + 0.207912i −3.04735 − 0.320290i 0.913545 − 0.406737i −0.908637 + 2.04313i 3.04735 − 0.320290i −2.64268 + 4.57726i −0.809017 + 0.587785i 6.24934 + 1.32834i 0.463991 − 2.18740i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 69.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
325.bh even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.bf.b yes 144
13.e even 6 1 650.2.bf.a ✓ 144
25.e even 10 1 650.2.bf.a ✓ 144
325.bh even 30 1 inner 650.2.bf.b yes 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.bf.a ✓ 144 13.e even 6 1
650.2.bf.a ✓ 144 25.e even 10 1
650.2.bf.b yes 144 1.a even 1 1 trivial
650.2.bf.b yes 144 325.bh even 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{144} + 22 T_{3}^{142} + 112 T_{3}^{140} + 54 T_{3}^{139} - 2511 T_{3}^{138} + \cdots + 38\!\cdots\!96 \) acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\). Copy content Toggle raw display