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Properties

Label 637.2.w.b
Level $637$
Weight $2$
Character orbit 637.w
Analytic conductor $5.086$
Analytic rank $0$
Dimension $174$
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(92,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.92");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.w (of order \(7\), degree \(6\), 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: \(174\)
Relative dimension: \(29\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 174 q - 3 q^{2} - 31 q^{4} - 4 q^{5} - 2 q^{6} + 9 q^{7} - 15 q^{8} - 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 174 q - 3 q^{2} - 31 q^{4} - 4 q^{5} - 2 q^{6} + 9 q^{7} - 15 q^{8} - 31 q^{9} - 10 q^{10} - 5 q^{11} + 25 q^{12} - 29 q^{13} + 15 q^{14} - 10 q^{15} - 51 q^{16} - 9 q^{17} + 44 q^{18} + 24 q^{19} + 63 q^{20} - 28 q^{21} - 8 q^{22} - 13 q^{23} - 48 q^{24} - 49 q^{25} - 3 q^{26} - 9 q^{27} - 44 q^{28} + 2 q^{29} - 22 q^{30} + 10 q^{31} + 24 q^{32} - 26 q^{33} + 118 q^{34} + 5 q^{35} - 55 q^{36} - 32 q^{37} + 16 q^{38} + 42 q^{40} - 14 q^{41} + 4 q^{42} - 50 q^{43} + 35 q^{44} - q^{45} + 4 q^{46} - 24 q^{47} - 116 q^{48} - 25 q^{49} + 156 q^{50} + 12 q^{51} - 31 q^{52} - 30 q^{53} - 78 q^{54} + 25 q^{55} + 3 q^{56} - 63 q^{57} - 12 q^{58} - 4 q^{59} + 128 q^{60} - 42 q^{61} - 38 q^{62} - 85 q^{63} - 105 q^{64} - 4 q^{65} + 15 q^{66} + 94 q^{67} + 214 q^{68} + 32 q^{69} - 57 q^{70} - 29 q^{71} - 64 q^{72} - 66 q^{73} - 90 q^{74} + 131 q^{75} - 21 q^{76} - 82 q^{77} + 19 q^{78} + 6 q^{79} + 22 q^{80} + 49 q^{81} - 50 q^{82} + 25 q^{83} + 89 q^{84} - 86 q^{85} - 28 q^{86} + 24 q^{87} + 48 q^{88} - 50 q^{89} - 155 q^{90} - 5 q^{91} - 98 q^{92} + 89 q^{93} - 28 q^{94} - 130 q^{95} - 105 q^{96} - 42 q^{97} + 195 q^{98} + 438 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
92.1 −1.68715 + 2.11563i −0.820369 − 0.395069i −1.18434 − 5.18892i −0.379531 − 0.182773i 2.21991 − 1.06905i −1.63169 − 2.08269i 8.09995 + 3.90073i −1.35354 − 1.69729i 1.02701 − 0.494580i
92.2 −1.57084 + 1.96978i 2.34650 + 1.13001i −0.967425 − 4.23856i 3.79256 + 1.82640i −5.91185 + 2.84700i 0.107940 − 2.64355i 5.32882 + 2.56623i 2.35865 + 2.95765i −9.55512 + 4.60150i
92.3 −1.44058 + 1.80643i −0.133830 − 0.0644489i −0.742877 − 3.25476i −3.89600 − 1.87621i 0.309215 − 0.148910i −1.21227 + 2.35168i 2.78627 + 1.34180i −1.85671 − 2.32824i 9.00174 − 4.33501i
92.4 −1.42465 + 1.78645i −2.04852 − 0.986513i −0.716746 − 3.14027i 0.342127 + 0.164760i 4.68077 − 2.25414i 2.57961 − 0.587895i 2.51370 + 1.21053i 1.35274 + 1.69628i −0.781746 + 0.376469i
92.5 −1.18201 + 1.48220i −0.0940579 − 0.0452959i −0.354712 − 1.55409i 1.30132 + 0.626680i 0.178315 − 0.0858720i −2.43875 + 1.02591i −0.693369 − 0.333909i −1.86367 − 2.33697i −2.46703 + 1.18806i
92.6 −1.14796 + 1.43949i 1.04469 + 0.503095i −0.309290 − 1.35509i −0.563090 − 0.271170i −1.92345 + 0.926286i 1.79557 − 1.94318i −1.01200 − 0.487355i −1.03220 − 1.29434i 1.03675 − 0.499271i
92.7 −1.12298 + 1.40817i −2.74049 − 1.31975i −0.276823 − 1.21284i −2.53396 − 1.22029i 4.93595 − 2.37703i −1.05285 − 2.42724i −1.22675 − 0.590772i 3.89808 + 4.88803i 4.56397 − 2.19789i
92.8 −1.10143 + 1.38115i 3.05365 + 1.47056i −0.249384 − 1.09262i 0.775588 + 0.373504i −5.39445 + 2.59783i 0.705731 + 2.54989i −1.39947 − 0.673949i 5.29177 + 6.63567i −1.37012 + 0.659815i
92.9 −0.807700 + 1.01282i −2.74330 − 1.32111i 0.0716088 + 0.313739i 2.02464 + 0.975017i 3.55382 − 1.71143i 0.327305 + 2.62543i −2.70992 − 1.30503i 3.90993 + 4.90289i −2.62283 + 1.26309i
92.10 −0.487175 + 0.610898i 1.39678 + 0.672652i 0.309185 + 1.35463i −2.44805 − 1.17892i −1.09140 + 0.525588i −2.52492 − 0.790430i −2.38614 − 1.14911i −0.371947 − 0.466407i 1.91283 − 0.921169i
92.11 −0.440322 + 0.552147i −1.61690 − 0.778657i 0.334060 + 1.46361i 2.37681 + 1.14461i 1.14189 − 0.549905i 2.31524 − 1.28050i −2.22779 − 1.07285i 0.137583 + 0.172524i −1.67856 + 0.808350i
92.12 −0.347630 + 0.435914i −0.809113 − 0.389648i 0.375867 + 1.64678i −1.77186 − 0.853282i 0.451125 − 0.217250i 2.01500 + 1.71458i −1.85320 − 0.892452i −1.36763 − 1.71495i 0.987907 − 0.475751i
92.13 −0.165915 + 0.208051i 2.56994 + 1.23762i 0.429285 + 1.88082i 0.602735 + 0.290262i −0.683878 + 0.329338i 1.20359 − 2.35614i −0.942039 − 0.453662i 3.20241 + 4.01570i −0.160392 + 0.0772407i
92.14 −0.118827 + 0.149004i 1.11947 + 0.539107i 0.436959 + 1.91444i 0.156007 + 0.0751292i −0.213352 + 0.102745i 0.0412747 + 2.64543i −0.680602 − 0.327760i −0.907899 − 1.13847i −0.0297324 + 0.0143184i
92.15 0.153433 − 0.192399i −1.67003 − 0.804242i 0.431566 + 1.89082i 0.217813 + 0.104893i −0.410972 + 0.197914i −2.59813 − 0.499745i 0.873441 + 0.420627i 0.271713 + 0.340717i 0.0536009 − 0.0258129i
92.16 0.252803 − 0.317004i 1.07472 + 0.517556i 0.408459 + 1.78958i 3.73621 + 1.79926i 0.435758 − 0.209850i −1.42709 + 2.22787i 1.40118 + 0.674774i −0.983320 − 1.23304i 1.51490 − 0.729536i
92.17 0.311250 − 0.390295i −1.89343 − 0.911826i 0.389588 + 1.70690i −2.41457 − 1.16280i −0.945210 + 0.455189i 0.718518 − 2.54632i 1.68699 + 0.812413i 0.883165 + 1.10745i −1.20537 + 0.580476i
92.18 0.529450 − 0.663910i 2.82603 + 1.36094i 0.284583 + 1.24684i −2.32207 − 1.11825i 2.39979 − 1.15568i −1.55674 + 2.13929i 2.50862 + 1.20809i 4.26380 + 5.34664i −1.97184 + 0.949586i
92.19 0.708170 − 0.888017i 0.627836 + 0.302350i 0.157972 + 0.692122i −3.97243 − 1.91302i 0.713107 − 0.343414i 2.60413 + 0.467470i 2.77316 + 1.33548i −1.56771 − 1.96584i −4.51195 + 2.17284i
92.20 0.743539 − 0.932368i 1.30142 + 0.626732i 0.128582 + 0.563353i 0.955511 + 0.460150i 1.55200 − 0.747405i 2.35959 − 1.19679i 2.76975 + 1.33384i −0.569562 − 0.714208i 1.13949 − 0.548748i
See next 80 embeddings (of 174 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 92.29
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.w.b ✓ 174
49.e even 7 1 inner 637.2.w.b ✓ 174
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.w.b ✓ 174 1.a even 1 1 trivial
637.2.w.b ✓ 174 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{174} + 3 T_{2}^{173} + 49 T_{2}^{172} + 147 T_{2}^{171} + 1338 T_{2}^{170} + \cdots + 84\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\). Copy content Toggle raw display