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Properties

Label 145.3.q.a
Level $145$
Weight $3$
Character orbit 145.q
Analytic conductor $3.951$
Analytic rank $0$
Dimension $336$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [145,3,Mod(13,145)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(145, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([21, 18]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("145.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 145.q (of order \(28\), 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: \(3.95096383322\)
Analytic rank: \(0\)
Dimension: \(336\)
Relative dimension: \(28\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 336 q - 14 q^{2} - 14 q^{3} - 2 q^{5} + 20 q^{6} + 6 q^{7} - 14 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 336 q - 14 q^{2} - 14 q^{3} - 2 q^{5} + 20 q^{6} + 6 q^{7} - 14 q^{8} - 14 q^{10} - 28 q^{11} + 46 q^{13} + 154 q^{15} + 156 q^{16} - 112 q^{18} - 214 q^{20} - 448 q^{21} + 38 q^{22} - 2 q^{23} + 74 q^{25} - 168 q^{26} - 308 q^{27} - 8 q^{28} + 76 q^{30} + 56 q^{31} - 14 q^{32} + 122 q^{33} - 130 q^{35} + 52 q^{36} + 112 q^{37} + 616 q^{38} - 434 q^{40} + 250 q^{42} - 322 q^{43} + 228 q^{45} - 14 q^{47} - 1246 q^{48} - 14 q^{50} - 172 q^{51} - 292 q^{52} + 108 q^{53} - 14 q^{55} + 1148 q^{56} - 52 q^{57} + 382 q^{58} - 14 q^{60} - 28 q^{61} - 476 q^{62} + 46 q^{63} - 156 q^{65} - 532 q^{66} - 378 q^{67} + 840 q^{68} - 508 q^{71} + 490 q^{72} - 84 q^{73} - 28 q^{76} + 532 q^{77} + 1114 q^{78} - 2804 q^{80} + 808 q^{81} - 14 q^{82} - 26 q^{83} - 98 q^{85} + 424 q^{86} - 2032 q^{87} - 472 q^{88} + 546 q^{90} + 352 q^{91} + 510 q^{92} + 322 q^{93} + 994 q^{95} + 1092 q^{96} + 56 q^{97} + 434 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.28147 − 3.66223i −0.338319 − 3.00266i −8.64244 + 6.89211i −4.82185 − 1.32280i −10.5629 + 5.08682i −0.985062 − 8.74267i 23.1745 + 14.5615i −0.127168 + 0.0290253i 1.33465 + 19.3538i
13.2 −1.13047 − 3.23069i −0.00195180 − 0.0173227i −6.03206 + 4.81040i 1.52140 + 4.76291i −0.0537578 + 0.0258884i 0.793340 + 7.04109i 10.7674 + 6.76561i 8.77405 − 2.00262i 13.6676 − 10.2995i
13.3 −1.11210 − 3.17819i 0.431794 + 3.83228i −5.73680 + 4.57495i −0.779017 − 4.93894i 11.6995 − 5.63418i 0.954256 + 8.46925i 9.51576 + 5.97915i −5.72554 + 1.30682i −14.8305 + 7.96844i
13.4 −1.07802 − 3.08081i 0.125573 + 1.11449i −5.20194 + 4.14841i 4.48081 − 2.21864i 3.29816 − 1.58831i −0.724028 − 6.42593i 7.33353 + 4.60796i 7.54803 − 1.72279i −11.6656 − 11.4128i
13.5 −0.865592 − 2.47372i 0.424401 + 3.76666i −2.24272 + 1.78851i −4.96550 + 0.586321i 8.95030 − 4.31024i −0.274288 − 2.43438i −2.51080 − 1.57764i −5.23325 + 1.19446i 5.74850 + 11.7758i
13.6 −0.846451 − 2.41902i −0.555048 − 4.92619i −2.00785 + 1.60120i 4.84592 + 1.23170i −11.4467 + 5.51245i −1.07453 − 9.53672i −3.10717 − 1.95237i −15.1849 + 3.46586i −1.12232 − 12.7649i
13.7 −0.818451 − 2.33900i −0.545620 − 4.84251i −1.67373 + 1.33475i −3.03939 + 3.97015i −10.8801 + 5.23956i 0.916870 + 8.13745i −3.90109 − 2.45121i −14.3778 + 3.28165i 11.7738 + 3.85975i
13.8 −0.666780 − 1.90555i −0.261601 − 2.32177i −0.0591936 + 0.0472054i −1.84977 − 4.64525i −4.24982 + 2.04660i 0.205067 + 1.82002i −6.70818 − 4.21503i 3.45216 − 0.787933i −7.61836 + 6.62218i
13.9 −0.608272 − 1.73834i 0.172620 + 1.53205i 0.475491 − 0.379191i −1.35828 + 4.81197i 2.55822 − 1.23197i −1.10132 − 9.77449i −7.18601 − 4.51527i 6.45699 − 1.47376i 9.19105 − 0.565830i
13.10 −0.508777 − 1.45400i 0.598187 + 5.30906i 1.27206 − 1.01443i 3.52961 + 3.54145i 7.41504 − 3.57089i 0.795334 + 7.05878i −7.33952 − 4.61172i −19.0539 + 4.34894i 3.35349 − 6.93386i
13.11 −0.361773 − 1.03389i −0.0794260 − 0.704926i 2.18928 − 1.74589i 4.94849 − 0.715879i −0.700080 + 0.337141i 0.554326 + 4.91978i −6.30694 − 3.96291i 8.28374 − 1.89071i −2.53037 − 4.85719i
13.12 −0.227468 − 0.650066i 0.434535 + 3.85660i 2.75648 − 2.19822i 2.19950 − 4.49023i 2.40820 − 1.15973i −0.906656 − 8.04679i −4.38860 − 2.75754i −5.91023 + 1.34897i −3.41926 − 0.408437i
13.13 −0.0732720 − 0.209399i 0.106491 + 0.945134i 3.08885 − 2.46327i −4.45921 + 2.26174i 0.190108 − 0.0915510i 1.49265 + 13.2476i −1.49351 − 0.938436i 7.89241 − 1.80139i 0.800342 + 0.768032i
13.14 −0.00824722 − 0.0235692i −0.449508 − 3.98949i 3.12684 − 2.49357i −4.36005 + 2.44745i −0.0903220 + 0.0434968i −0.858846 − 7.62247i −0.169132 − 0.106272i −6.93965 + 1.58393i 0.0936426 + 0.0825782i
13.15 0.0957374 + 0.273602i 0.0828588 + 0.735392i 3.06163 − 2.44157i −4.23070 − 2.66481i −0.193272 + 0.0930749i −0.833399 − 7.39662i 1.94288 + 1.22080i 8.24041 − 1.88082i 0.324059 − 1.41265i
13.16 0.162530 + 0.464483i −0.347597 − 3.08501i 2.93800 − 2.34297i 3.60636 + 3.46326i 1.37644 − 0.662858i 0.0259599 + 0.230400i 3.23247 + 2.03109i −0.622095 + 0.141989i −1.02248 + 2.23798i
13.17 0.245491 + 0.701572i −0.621944 − 5.51990i 2.69539 − 2.14950i 1.64112 − 4.72300i 3.71993 − 1.79142i 0.645332 + 5.72748i 4.68714 + 2.94513i −21.3082 + 4.86345i 3.71640 − 0.00808755i
13.18 0.370622 + 1.05918i 0.297229 + 2.63798i 2.14283 − 1.70885i 2.70675 + 4.20399i −2.68393 + 1.29251i −0.476748 − 4.23126i 6.40475 + 4.02437i 1.90374 − 0.434517i −3.44959 + 4.42501i
13.19 0.521392 + 1.49005i 0.669518 + 5.94214i 1.17892 − 0.940154i −4.97656 + 0.483551i −8.50502 + 4.09580i −0.171378 − 1.52102i 7.36225 + 4.62601i −26.0864 + 5.95404i −3.31526 − 7.16323i
13.20 0.555400 + 1.58724i 0.347446 + 3.08367i 0.916459 − 0.730851i 2.60347 − 4.26872i −4.70155 + 2.26415i 1.02805 + 9.12415i 7.36447 + 4.62740i −0.613932 + 0.140126i 8.22145 + 1.76149i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
29.e even 14 1 inner
145.q odd 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 145.3.q.a ✓ 336
5.c odd 4 1 inner 145.3.q.a ✓ 336
29.e even 14 1 inner 145.3.q.a ✓ 336
145.q odd 28 1 inner 145.3.q.a ✓ 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.3.q.a ✓ 336 1.a even 1 1 trivial
145.3.q.a ✓ 336 5.c odd 4 1 inner
145.3.q.a ✓ 336 29.e even 14 1 inner
145.3.q.a ✓ 336 145.q odd 28 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(145, [\chi])\).