Properties

Label 637.2.bp.a
Level $637$
Weight $2$
Character orbit 637.bp
Analytic conductor $5.086$
Analytic rank $0$
Dimension $756$
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] = [637,2,Mod(88,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([34, 35]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.88");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.bp (of order \(42\), degree \(12\), 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: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(756\)
Relative dimension: \(63\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 756 q - 18 q^{2} - 11 q^{3} - 68 q^{4} - 15 q^{6} - 19 q^{7} - 121 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 756 q - 18 q^{2} - 11 q^{3} - 68 q^{4} - 15 q^{6} - 19 q^{7} - 121 q^{9} + 11 q^{10} - 21 q^{11} - 26 q^{12} - 10 q^{13} - 38 q^{14} - 6 q^{15} + 52 q^{16} - 18 q^{17} + 9 q^{18} - 15 q^{20} + 16 q^{21} - q^{22} - 8 q^{23} - 21 q^{24} - 79 q^{25} - 16 q^{26} - 50 q^{27} - 47 q^{28} - 3 q^{29} + 22 q^{30} + 15 q^{31} - 30 q^{32} - 21 q^{33} - 6 q^{35} - 42 q^{36} + 41 q^{37} - 2 q^{38} + 13 q^{39} - 24 q^{40} - 6 q^{41} - 35 q^{42} - 5 q^{43} - 108 q^{44} + 87 q^{45} - 3 q^{46} - 63 q^{47} - 154 q^{48} - 55 q^{49} + 135 q^{50} + 26 q^{51} - 80 q^{52} + 16 q^{53} - 75 q^{54} + 59 q^{55} - 122 q^{56} + 63 q^{57} - 21 q^{58} - 42 q^{59} - 15 q^{60} - 60 q^{61} - 45 q^{62} + 51 q^{63} + 4 q^{64} - 32 q^{65} + 85 q^{66} - 156 q^{68} + 70 q^{69} + 228 q^{70} - 30 q^{71} - 21 q^{72} - 6 q^{73} + 32 q^{74} - 52 q^{75} + 21 q^{76} + 3 q^{77} - 89 q^{78} + 12 q^{79} - 81 q^{81} + 160 q^{82} - 147 q^{83} - 25 q^{84} + 144 q^{85} + 60 q^{86} - 76 q^{87} - 35 q^{88} - 57 q^{89} + 117 q^{90} - 11 q^{91} + 88 q^{92} - 24 q^{93} - 42 q^{94} + 69 q^{95} + 12 q^{96} - 156 q^{97} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
88.1 −2.55146 + 1.00138i −0.303707 0.146257i 4.04110 3.74960i −3.66144 + 0.274387i 0.921355 + 0.0690460i −2.63968 0.179128i −4.17748 + 8.67462i −1.79962 2.25666i 9.06726 4.36656i
88.2 −2.43880 + 0.957159i −2.82876 1.36226i 3.56549 3.30829i −0.302823 + 0.0226935i 8.20268 + 0.614706i 2.57737 + 0.597636i −3.25549 + 6.76010i 4.27566 + 5.36151i 0.716804 0.345195i
88.3 −2.41692 + 0.948570i 1.89451 + 0.912349i 3.47559 3.22488i 1.43933 0.107862i −5.44431 0.407994i 2.21216 + 1.45133i −3.08813 + 6.41257i 0.886329 + 1.11142i −3.37641 + 1.62599i
88.4 −2.40235 + 0.942851i −0.390492 0.188051i 3.41619 3.16976i 1.38128 0.103512i 1.11540 + 0.0835878i 0.480491 2.60175i −2.97878 + 6.18549i −1.75335 2.19863i −3.22071 + 1.55101i
88.5 −2.39358 + 0.939409i −1.57406 0.758028i 3.38061 3.13675i 3.68021 0.275794i 4.47974 + 0.335710i −1.05923 + 2.42447i −2.91375 + 6.05047i 0.0325956 + 0.0408736i −8.54978 + 4.11736i
88.6 −2.24631 + 0.881613i 2.07682 + 1.00014i 2.80258 2.60042i −3.10203 + 0.232465i −5.54693 0.415685i 1.94145 1.79743i −1.90889 + 3.96385i 1.44242 + 1.80874i 6.76318 3.25698i
88.7 −2.21641 + 0.869875i 1.15945 + 0.558362i 2.68967 2.49565i 1.79264 0.134340i −3.05552 0.228980i −2.36417 + 1.18772i −1.72435 + 3.58064i −0.837911 1.05071i −3.85637 + 1.85713i
88.8 −2.18175 + 0.856274i 2.75338 + 1.32596i 2.56073 2.37601i −1.56425 + 0.117224i −7.14257 0.535261i −2.17276 + 1.50967i −1.51852 + 3.15323i 3.95246 + 4.95622i 3.31243 1.59518i
88.9 −2.12012 + 0.832087i −2.53379 1.22021i 2.33645 2.16791i −1.25010 + 0.0936823i 6.38726 + 0.478659i −2.64572 0.0132775i −1.17328 + 2.43635i 3.06070 + 3.83799i 2.57242 1.23881i
88.10 −2.06002 + 0.808498i −1.04079 0.501218i 2.12391 1.97070i −2.71907 + 0.203766i 2.54928 + 0.191042i 0.772418 + 2.53049i −0.861620 + 1.78917i −1.03845 1.30217i 5.43659 2.61812i
88.11 −2.01195 + 0.789633i −0.739417 0.356085i 1.95833 1.81706i 1.06565 0.0798593i 1.76885 + 0.132557i 0.267626 2.63218i −0.629693 + 1.30757i −1.45053 1.81890i −2.08097 + 1.00214i
88.12 −1.75260 + 0.687843i 1.74124 + 0.838539i 1.13236 1.05068i 3.79954 0.284736i −3.62848 0.271917i 2.60964 0.435647i 0.371913 0.772285i 0.458315 + 0.574709i −6.46320 + 3.11251i
88.13 −1.69388 + 0.664798i 1.46656 + 0.706257i 0.961158 0.891824i −0.587608 + 0.0440351i −2.95369 0.221348i −1.88488 1.85668i 0.543844 1.12930i −0.218477 0.273962i 0.966061 0.465230i
88.14 −1.58862 + 0.623486i 0.0795416 + 0.0383052i 0.668864 0.620615i −3.18924 + 0.239001i −0.150244 0.0112592i 2.60068 0.486269i 0.805296 1.67222i −1.86561 2.33940i 4.91747 2.36813i
88.15 −1.56862 + 0.615640i −1.77186 0.853281i 0.615466 0.571069i 0.139416 0.0104478i 3.30469 + 0.247652i 2.63298 + 0.259694i 0.848421 1.76177i 0.540916 + 0.678287i −0.212259 + 0.102218i
88.16 −1.43231 + 0.562140i −1.25604 0.604875i 0.269405 0.249971i 2.05445 0.153960i 2.13905 + 0.160300i −0.494528 + 2.59912i 1.08986 2.26311i −0.658718 0.826007i −2.85606 + 1.37541i
88.17 −1.39678 + 0.548196i −2.79574 1.34636i 0.184376 0.171076i 4.02490 0.301624i 4.64310 + 0.347952i 0.853845 2.50419i 1.13834 2.36379i 4.13300 + 5.18262i −5.45655 + 2.62774i
88.18 −1.39588 + 0.547841i 2.88642 + 1.39003i 0.182234 0.169089i 2.44886 0.183517i −4.79059 0.359005i −0.897086 2.48902i 1.13950 2.36620i 4.52878 + 5.67890i −3.31777 + 1.59775i
88.19 −1.39074 + 0.545824i 1.60197 + 0.771470i 0.170123 0.157851i −0.583615 + 0.0437359i −2.64901 0.198516i 0.511124 + 2.59591i 1.14602 2.37973i 0.100684 + 0.126254i 0.787784 0.379377i
88.20 −1.32370 + 0.519514i −1.03568 0.498757i 0.0161854 0.0150178i 0.809944 0.0606969i 1.63004 + 0.122155i −2.62057 0.364129i 1.22034 2.53406i −1.04660 1.31239i −1.04059 + 0.501122i
See next 80 embeddings (of 756 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 88.63
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
637.bp even 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.bp.a 756
13.e even 6 1 637.2.bz.a yes 756
49.g even 21 1 637.2.bz.a yes 756
637.bp even 42 1 inner 637.2.bp.a 756
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.bp.a 756 1.a even 1 1 trivial
637.2.bp.a 756 637.bp even 42 1 inner
637.2.bz.a yes 756 13.e even 6 1
637.2.bz.a yes 756 49.g even 21 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(637, [\chi])\).