Properties

Label 63.4.i.a
Level $63$
Weight $4$
Character orbit 63.i
Analytic conductor $3.717$
Analytic rank $0$
Dimension $44$
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] = [63,4,Mod(5,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 63.i (of order \(6\), degree \(2\), 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: \(3.71712033036\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 3 q^{3} - 162 q^{4} - 3 q^{5} + 24 q^{6} + 5 q^{7} - 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 3 q^{3} - 162 q^{4} - 3 q^{5} + 24 q^{6} + 5 q^{7} - 45 q^{9} - 6 q^{10} + 9 q^{11} + 186 q^{12} - 36 q^{13} + 54 q^{14} - 141 q^{15} + 526 q^{16} - 72 q^{17} - 54 q^{18} - 6 q^{19} + 24 q^{20} - 81 q^{21} + 14 q^{22} - 285 q^{23} - 114 q^{24} - 349 q^{25} - 96 q^{26} + 432 q^{27} - 156 q^{28} - 132 q^{29} - 447 q^{30} - 3 q^{33} + 24 q^{34} + 765 q^{35} + 1122 q^{36} + 82 q^{37} - 873 q^{38} + 306 q^{39} + 420 q^{40} + 618 q^{41} - 282 q^{42} + 82 q^{43} + 603 q^{44} + 291 q^{45} + 266 q^{46} - 402 q^{47} - 1569 q^{48} - 79 q^{49} - 1845 q^{50} + 453 q^{51} + 189 q^{52} + 564 q^{53} - 2385 q^{54} + 66 q^{56} + 1170 q^{57} + 269 q^{58} + 1494 q^{59} + 2265 q^{60} - 2904 q^{62} - 636 q^{63} - 1144 q^{64} - 372 q^{66} - 590 q^{67} + 3504 q^{68} - 1005 q^{69} - 105 q^{70} + 1830 q^{72} - 6 q^{73} + 1515 q^{74} - 33 q^{75} - 144 q^{76} - 4443 q^{77} - 5985 q^{78} + 1102 q^{79} - 4239 q^{80} - 4017 q^{81} + 18 q^{82} + 1830 q^{83} + 3165 q^{84} - 237 q^{85} + 1209 q^{86} + 2013 q^{87} - 623 q^{88} + 4266 q^{89} + 9993 q^{90} - 1140 q^{91} + 7896 q^{92} - 729 q^{93} + 3975 q^{96} - 792 q^{97} + 5667 q^{98} + 4335 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 5.38106i −0.968958 + 5.10501i −20.9559 −5.57991 9.66469i 27.4704 + 5.21402i −8.46770 + 16.4711i 69.7163i −25.1222 9.89308i −52.0063 + 30.0259i
5.2 5.07706i −4.42649 2.72143i −17.7766 4.21335 + 7.29774i −13.8169 + 22.4736i −4.29909 18.0144i 49.6363i 12.1877 + 24.0928i 37.0511 21.3915i
5.3 4.25176i 5.13732 0.779690i −10.0774 −1.48754 2.57649i −3.31505 21.8426i −13.6257 12.5435i 8.83278i 25.7842 8.01104i −10.9546 + 6.32464i
5.4 4.10714i 1.00877 5.09729i −8.86861 −3.80591 6.59204i −20.9353 4.14318i 13.1152 + 13.0764i 3.56749i −24.9647 10.2840i −27.0744 + 15.6314i
5.5 3.72062i 3.26996 + 4.03824i −5.84303 6.83336 + 11.8357i 15.0248 12.1663i 18.4941 0.984293i 8.02526i −5.61469 + 26.4098i 44.0363 25.4243i
5.6 2.91468i −4.68054 + 2.25667i −0.495360 8.60567 + 14.9055i 6.57748 + 13.6423i −4.20673 + 18.0362i 21.8736i 16.8149 21.1249i 43.4446 25.0828i
5.7 2.41653i −3.31226 + 4.00362i 2.16037 −5.35965 9.28319i 9.67487 + 8.00418i 3.86176 18.1132i 24.5529i −5.05789 26.5220i −22.4331 + 12.9518i
5.8 1.81805i −3.99842 3.31853i 4.69469 −5.16236 8.94146i −6.03325 + 7.26934i −16.8039 + 7.78656i 23.0796i 4.97478 + 26.5377i −16.2560 + 9.38543i
5.9 1.10690i 0.171156 5.19333i 6.77476 6.23688 + 10.8026i −5.74852 0.189454i 11.7098 14.3485i 16.3542i −26.9414 1.77774i 11.9574 6.90363i
5.10 0.837567i 4.56931 + 2.47415i 7.29848 −10.9584 18.9806i 2.07226 3.82711i 14.8276 + 11.0969i 12.8135i 14.7572 + 22.6103i −15.8975 + 9.17842i
5.11 0.747815i 4.77381 2.05201i 7.44077 4.35110 + 7.53632i −1.53452 3.56993i −11.6200 + 14.4213i 11.5468i 18.5785 19.5918i 5.63577 3.25382i
5.12 0.257625i 1.70621 + 4.90804i 7.93363 3.19386 + 5.53193i −1.26443 + 0.439563i −15.2112 10.5650i 4.10490i −21.1777 + 16.7483i −1.42516 + 0.822818i
5.13 1.15257i −5.00666 1.39045i 6.67158 0.137359 + 0.237913i 1.60260 5.77053i 18.5199 + 0.122959i 16.9100i 23.1333 + 13.9231i −0.274212 + 0.158316i
5.14 1.78786i 2.28100 4.66873i 4.80356 −8.47168 14.6734i 8.34703 + 4.07812i −8.67298 16.3640i 22.8910i −16.5940 21.2988i 26.2340 15.1462i
5.15 1.83815i −1.68860 + 4.91412i 4.62119 0.207277 + 0.359014i −9.03291 3.10391i 5.26194 + 17.7570i 23.1997i −21.2972 16.5960i −0.659923 + 0.381007i
5.16 3.06129i 4.98958 + 1.45054i −1.37153 2.75111 + 4.76507i −4.44054 + 15.2746i 8.35389 16.5291i 20.2917i 22.7918 + 14.4752i −14.5873 + 8.42197i
5.17 3.46625i −2.57609 4.51263i −4.01488 9.02701 + 15.6352i 15.6419 8.92935i −18.1650 + 3.61017i 13.8134i −13.7276 + 23.2498i −54.1956 + 31.2898i
5.18 3.64983i −4.97539 + 1.49850i −5.32127 −6.11568 10.5927i −5.46929 18.1593i −17.8879 4.79841i 9.77690i 22.5090 14.9113i 38.6615 22.3212i
5.19 3.72101i 3.89612 3.43806i −5.84592 1.33006 + 2.30373i 12.7931 + 14.4975i 8.98650 + 16.1939i 8.01534i 3.35949 26.7902i −8.57221 + 4.94917i
5.20 4.92859i 3.60211 + 3.74497i −16.2910 −4.62434 8.00960i −18.4575 + 17.7533i −13.0699 + 13.1217i 40.8632i −1.04964 + 26.9796i 39.4761 22.7915i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.4.i.a 44
3.b odd 2 1 189.4.i.a 44
7.d odd 6 1 63.4.s.a yes 44
9.c even 3 1 189.4.s.a 44
9.d odd 6 1 63.4.s.a yes 44
21.g even 6 1 189.4.s.a 44
63.i even 6 1 inner 63.4.i.a 44
63.t odd 6 1 189.4.i.a 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.i.a 44 1.a even 1 1 trivial
63.4.i.a 44 63.i even 6 1 inner
63.4.s.a yes 44 7.d odd 6 1
63.4.s.a yes 44 9.d odd 6 1
189.4.i.a 44 3.b odd 2 1
189.4.i.a 44 63.t odd 6 1
189.4.s.a 44 9.c even 3 1
189.4.s.a 44 21.g even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(63, [\chi])\).