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Properties

Label 145.4.l.a
Level $145$
Weight $4$
Character orbit 145.l
Analytic conductor $8.555$
Analytic rank $0$
Dimension $264$
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] = [145,4,Mod(4,145)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(145, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("145.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 145.l (of order \(14\), 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: \(8.55527695083\)
Analytic rank: \(0\)
Dimension: \(264\)
Relative dimension: \(44\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 264 q - 186 q^{4} - 13 q^{5} + 26 q^{6} - 346 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 264 q - 186 q^{4} - 13 q^{5} + 26 q^{6} - 346 q^{9} - 7 q^{10} - 14 q^{11} - 14 q^{14} + 287 q^{15} - 682 q^{16} - 14 q^{19} + 245 q^{20} + 196 q^{21} + 294 q^{24} + 833 q^{25} + 238 q^{26} - 658 q^{29} - 558 q^{30} + 294 q^{31} + 1482 q^{34} + 985 q^{35} - 2048 q^{36} + 1134 q^{39} - 175 q^{40} + 434 q^{44} + 189 q^{45} + 4036 q^{49} - 448 q^{50} + 792 q^{51} + 2058 q^{54} - 2863 q^{55} - 490 q^{56} - 9424 q^{59} - 8638 q^{60} + 2002 q^{61} - 2586 q^{64} + 577 q^{65} - 4802 q^{66} + 2954 q^{69} + 5822 q^{71} + 7858 q^{74} + 2674 q^{76} - 14 q^{79} + 4874 q^{80} - 10402 q^{81} - 1750 q^{84} + 3654 q^{85} - 668 q^{86} - 14 q^{89} + 5362 q^{90} - 4194 q^{91} + 5484 q^{94} + 5593 q^{95} - 1418 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −1.22949 + 5.38673i −1.51372 + 0.728971i −20.2975 − 9.77477i 5.99780 − 9.43538i −2.06567 − 9.05028i −0.590313 − 1.22580i 50.0501 − 62.7608i −15.0743 + 18.9025i 43.4516 + 43.9093i
4.2 −1.17319 + 5.14007i 9.31451 − 4.48563i −17.8362 − 8.58946i 8.33518 + 7.45150i 12.1288 + 53.1397i 4.89419 + 10.1629i 38.7780 − 48.6261i 49.8049 − 62.4534i −48.0799 + 34.1014i
4.3 −1.13150 + 4.95740i 1.23699 − 0.595705i −16.0878 − 7.74749i −7.11073 + 8.62772i 1.55350 + 6.80632i −7.00612 − 14.5484i 31.2477 − 39.1834i −15.6589 + 19.6357i −34.7253 − 45.0130i
4.4 −1.11824 + 4.89933i −7.44754 + 3.58654i −15.5452 − 7.48619i −11.1714 − 0.447354i −9.24353 − 40.4986i 3.62003 + 7.51706i 28.9947 − 36.3582i 25.7683 − 32.3124i 14.6840 − 54.2321i
4.5 −1.03621 + 4.53993i 4.18291 − 2.01438i −12.3295 − 5.93756i −6.10001 − 9.36963i 4.81079 + 21.0774i 12.6431 + 26.2537i 16.5049 − 20.6965i −3.39522 + 4.25747i 48.8583 − 17.9847i
4.6 −0.963461 + 4.22120i 1.49755 − 0.721183i −9.68250 − 4.66285i 10.7489 + 3.07603i 1.60142 + 7.01629i −10.9823 − 22.8051i 7.41506 − 9.29819i −15.1117 + 18.9494i −23.3406 + 42.4094i
4.7 −0.961410 + 4.21221i −8.59612 + 4.13967i −9.61067 − 4.62825i 11.1309 − 1.04985i −9.17279 − 40.1886i −5.62912 − 11.6890i 7.18450 − 9.00908i 39.9222 − 50.0608i −6.27919 + 47.8952i
4.8 −0.933184 + 4.08854i −3.09067 + 1.48839i −8.63761 − 4.15966i 6.13723 + 9.34529i −3.20118 − 14.0253i 13.0757 + 27.1519i 4.14964 − 5.20348i −9.49727 + 11.9092i −43.9358 + 16.3715i
4.9 −0.876009 + 3.83805i 7.21876 − 3.47637i −6.75546 − 3.25326i −3.79214 − 10.5176i 7.01878 + 30.7513i −13.1196 − 27.2431i −1.23219 + 1.54511i 23.1911 − 29.0808i 43.6889 − 5.34091i
4.10 −0.852663 + 3.73576i 3.85280 − 1.85541i −6.02111 − 2.89962i −8.79598 + 6.90150i 3.64623 + 15.9752i 2.52847 + 5.25043i −3.14662 + 3.94573i −5.43271 + 6.81240i −18.2824 − 38.7443i
4.11 −0.813494 + 3.56415i −4.04306 + 1.94703i −4.83363 − 2.32775i −8.46672 − 7.30169i −3.65052 − 15.9940i −3.02198 − 6.27520i −6.00628 + 7.53164i −4.27884 + 5.36550i 32.9119 − 24.2368i
4.12 −0.613303 + 2.68706i 4.58454 − 2.20780i 0.363625 + 0.175112i 8.91258 − 6.75025i 3.12076 + 13.6730i 1.96614 + 4.08273i −14.4410 + 18.1085i −0.690605 + 0.865992i 12.6722 + 28.0885i
4.13 −0.607251 + 2.66054i −6.07887 + 2.92743i 0.498029 + 0.239838i −3.21496 + 10.7081i −4.09715 − 17.9508i −12.3167 − 25.5759i −14.5524 + 18.2481i 11.5486 − 14.4815i −26.5371 − 15.0560i
4.14 −0.580541 + 2.54352i −3.75365 + 1.80766i 1.07530 + 0.517839i 5.31821 − 9.83446i −2.41867 − 10.5969i 3.97503 + 8.25423i −14.9545 + 18.7524i −6.01199 + 7.53880i 21.9267 + 19.2363i
4.15 −0.509945 + 2.23422i 7.41750 − 3.57208i 2.47608 + 1.19242i −8.01020 + 7.79978i 4.19828 + 18.3939i 7.17849 + 14.9063i −15.3575 + 19.2576i 25.4254 − 31.8824i −13.3416 − 21.8740i
4.16 −0.448563 + 1.96528i 5.11010 − 2.46090i 3.54662 + 1.70796i 9.33192 + 6.15754i 2.54415 + 11.1467i 1.93728 + 4.02281i −15.0023 + 18.8123i 3.22291 − 4.04140i −16.2873 + 15.5778i
4.17 −0.289803 + 1.26971i 0.615601 − 0.296458i 5.67958 + 2.73514i −10.3366 − 4.26085i 0.198012 + 0.867547i −7.30075 − 15.1602i −11.6149 + 14.5646i −16.5431 + 20.7445i 8.40560 − 11.8897i
4.18 −0.283968 + 1.24415i −0.0343415 + 0.0165380i 5.74049 + 2.76447i 2.29220 + 10.9428i −0.0108238 − 0.0474221i −2.17231 − 4.51085i −11.4348 + 14.3388i −16.8333 + 21.1083i −14.2654 − 0.255591i
4.19 −0.280430 + 1.22864i −6.72549 + 3.23883i 5.77682 + 2.78197i −9.41942 + 6.02284i −2.09334 − 9.17150i 12.9023 + 26.7919i −11.3240 + 14.1999i 17.9080 − 22.4560i −4.75843 − 13.2621i
4.20 −0.0402191 + 0.176211i −7.09917 + 3.41878i 7.17832 + 3.45690i 10.3547 + 4.21658i −0.316905 − 1.38845i 4.97906 + 10.3391i −1.79938 + 2.25635i 21.8759 − 27.4315i −1.15947 + 1.65503i
See next 80 embeddings (of 264 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
29.e even 14 1 inner
145.l even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 145.4.l.a ✓ 264
5.b even 2 1 inner 145.4.l.a ✓ 264
29.e even 14 1 inner 145.4.l.a ✓ 264
145.l even 14 1 inner 145.4.l.a ✓ 264
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.4.l.a ✓ 264 1.a even 1 1 trivial
145.4.l.a ✓ 264 5.b even 2 1 inner
145.4.l.a ✓ 264 29.e even 14 1 inner
145.4.l.a ✓ 264 145.l even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(145, [\chi])\).