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Properties

Label 145.3.p.a
Level $145$
Weight $3$
Character orbit 145.p
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(7,145)]
 
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([7, 12]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("145.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 145.p (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.

\(\operatorname{Tr}(f)(q) = \) \( 336 q - 14 q^{2} - 10 q^{3} - 18 q^{5} - 60 q^{6} - 26 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 336 q - 14 q^{2} - 10 q^{3} - 18 q^{5} - 60 q^{6} - 26 q^{7} - 2 q^{8} - 30 q^{10} - 44 q^{11} - 44 q^{12} + 14 q^{13} - 118 q^{15} + 156 q^{16} + 20 q^{18} + 66 q^{20} + 48 q^{21} + 70 q^{22} - 18 q^{23} + 74 q^{25} - 120 q^{26} + 164 q^{27} - 40 q^{28} + 76 q^{30} + 72 q^{31} + 54 q^{32} + 2 q^{33} - 106 q^{35} + 180 q^{36} - 184 q^{37} - 692 q^{38} + 226 q^{40} - 56 q^{41} + 18 q^{42} + 186 q^{43} - 408 q^{45} - 272 q^{46} - 34 q^{47} - 714 q^{48} - 2 q^{50} - 380 q^{51} - 320 q^{52} - 488 q^{53} - 242 q^{55} - 700 q^{56} + 4 q^{57} + 154 q^{58} + 322 q^{60} + 28 q^{61} + 816 q^{62} + 502 q^{63} + 464 q^{65} + 884 q^{66} + 134 q^{67} + 640 q^{68} - 1216 q^{70} + 260 q^{71} - 1602 q^{72} + 1160 q^{73} - 1680 q^{75} + 92 q^{76} - 108 q^{77} + 354 q^{78} + 1176 q^{80} + 656 q^{81} - 22 q^{82} - 26 q^{83} - 450 q^{85} - 808 q^{86} + 2044 q^{87} - 760 q^{88} + 1074 q^{90} - 488 q^{91} + 150 q^{92} - 54 q^{93} + 722 q^{95} + 196 q^{96} - 1392 q^{97} - 546 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.92961 + 3.07095i 0.540452 − 0.189112i −3.97184 − 8.24761i 4.93292 − 0.816269i −0.462105 + 2.02462i 1.93718 + 5.53615i 18.5759 + 2.09300i −6.78016 + 5.40700i −7.01188 + 16.7239i
7.2 −1.91194 + 3.04284i −3.50423 + 1.22618i −3.86783 − 8.03162i −4.86846 − 1.13934i 2.96881 − 13.0072i 0.436797 + 1.24829i 17.5498 + 1.97738i 3.73960 − 2.98223i 12.7751 − 12.6356i
7.3 −1.77351 + 2.82253i −0.565136 + 0.197749i −3.08578 − 6.40768i 0.275211 + 4.99242i 0.444121 − 1.94582i −3.96947 − 11.3441i 10.3085 + 1.16149i −6.75621 + 5.38790i −14.5793 − 8.07732i
7.4 −1.73403 + 2.75969i 4.06680 − 1.42303i −2.87350 − 5.96688i −1.15747 − 4.86418i −3.12481 + 13.6907i −1.52180 − 4.34904i 8.49441 + 0.957090i 7.47734 − 5.96298i 15.4307 + 5.24036i
7.5 −1.34504 + 2.14061i 2.76872 − 0.968817i −1.03757 − 2.15454i −4.52204 + 2.13335i −1.65017 + 7.22985i 0.288884 + 0.825582i −4.04126 − 0.455341i −0.309286 + 0.246647i 1.51562 − 12.5494i
7.6 −1.31668 + 2.09549i −4.60175 + 1.61022i −0.921882 − 1.91431i 3.58496 − 3.48541i 2.68484 − 11.7631i −1.74887 − 4.99798i −4.61179 − 0.519624i 11.5468 − 9.20827i 2.58337 + 12.1014i
7.7 −1.14970 + 1.82974i −3.07636 + 1.07647i −0.290597 − 0.603431i 1.15733 + 4.86421i 1.56725 − 6.86655i 2.90442 + 8.30037i −7.15128 − 0.805755i 1.26874 − 1.01179i −10.2308 − 3.47477i
7.8 −1.09604 + 1.74434i −0.618244 + 0.216333i −0.105878 − 0.219857i −0.609100 − 4.96276i 0.300263 − 1.31554i 2.24749 + 6.42295i −7.68905 − 0.866347i −6.70106 + 5.34391i 9.32434 + 4.37691i
7.9 −1.05534 + 1.67957i 4.39940 − 1.53942i 0.0283324 + 0.0588327i 3.67837 + 3.38667i −2.05732 + 9.01370i 0.564288 + 1.61264i −8.01326 − 0.902877i 9.94842 − 7.93360i −9.57010 + 2.60398i
7.10 −0.674184 + 1.07296i −0.496495 + 0.173731i 1.03882 + 2.15714i −4.12637 − 2.82367i 0.148323 − 0.649844i −3.70941 − 10.6009i −8.05174 − 0.907214i −6.82016 + 5.43890i 5.81160 − 2.52375i
7.11 −0.318847 + 0.507441i 1.14204 − 0.399619i 1.57970 + 3.28028i 4.15304 − 2.78429i −0.161354 + 0.706937i 1.90356 + 5.44005i −4.55036 − 0.512702i −5.89191 + 4.69864i 0.0886804 + 2.99518i
7.12 −0.228563 + 0.363755i 0.850754 − 0.297692i 1.65546 + 3.43759i −3.51429 + 3.55665i −0.0861636 + 0.377508i 0.213584 + 0.610387i −3.33643 − 0.375925i −6.40132 + 5.10488i −0.490517 − 2.09126i
7.13 −0.142603 + 0.226951i −5.24368 + 1.83484i 1.70436 + 3.53915i −4.93254 + 0.818554i 0.331344 − 1.45171i 0.145447 + 0.415662i −2.11166 − 0.237927i 17.0930 − 13.6312i 0.517623 − 1.23617i
7.14 −0.0740409 + 0.117835i −2.56051 + 0.895963i 1.72713 + 3.58643i 4.02352 + 2.96838i 0.0840067 − 0.368057i −1.97198 − 5.63561i −1.10365 − 0.124352i −1.28300 + 1.02316i −0.647685 + 0.254332i
7.15 0.0856163 − 0.136258i 4.01715 − 1.40566i 1.72430 + 3.58054i 2.74601 − 4.17845i 0.152402 − 0.667715i −3.27858 − 9.36964i 1.27515 + 0.143675i 7.12512 − 5.68210i −0.334243 − 0.731908i
7.16 0.153849 − 0.244849i 4.78367 − 1.67388i 1.69925 + 3.52854i −4.45588 − 2.26830i 0.326115 − 1.42880i 4.02607 + 11.5058i 2.27480 + 0.256309i 13.0451 − 10.4031i −1.24092 + 0.742043i
7.17 0.512981 − 0.816405i −2.75716 + 0.964772i 1.33217 + 2.76627i −1.66600 − 4.71428i −0.626727 + 2.74587i 2.19499 + 6.27291i 6.77430 + 0.763280i −0.365331 + 0.291342i −4.70339 − 1.05821i
7.18 0.746161 − 1.18751i 3.67291 − 1.28521i 0.882116 + 1.83173i −0.308330 + 4.99048i 1.21439 − 5.32058i −2.71564 − 7.76086i 8.40801 + 0.947356i 4.80203 − 3.82949i 5.69618 + 4.08985i
7.19 0.960438 − 1.52853i 0.835952 − 0.292512i 0.321576 + 0.667760i 4.62944 + 1.88900i 0.355767 − 1.55872i 1.67317 + 4.78164i 8.50505 + 0.958289i −6.42323 + 5.12236i 7.33368 − 5.26196i
7.20 0.972841 − 1.54827i −3.15075 + 1.10249i 0.284822 + 0.591440i 2.27517 − 4.45237i −1.35822 + 5.95076i −2.42197 − 6.92160i 8.46096 + 0.953321i 1.67524 − 1.33596i −4.68009 − 7.85402i
See next 80 embeddings (of 336 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
29.d even 7 1 inner
145.p 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.p.a ✓ 336
5.c odd 4 1 inner 145.3.p.a ✓ 336
29.d even 7 1 inner 145.3.p.a ✓ 336
145.p odd 28 1 inner 145.3.p.a ✓ 336
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.3.p.a ✓ 336 1.a even 1 1 trivial
145.3.p.a ✓ 336 5.c odd 4 1 inner
145.3.p.a ✓ 336 29.d even 7 1 inner
145.3.p.a ✓ 336 145.p odd 28 1 inner

Hecke kernels

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