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Properties

Label 245.4.k.a
Level $245$
Weight $4$
Character orbit 245.k
Analytic conductor $14.455$
Analytic rank $0$
Dimension $162$
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] = [245,4,Mod(36,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.36");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.k (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: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(162\)
Relative dimension: \(27\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 162 q + 2 q^{2} - 104 q^{4} - 135 q^{5} + 60 q^{6} - 21 q^{7} + 24 q^{8} - 25 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 162 q + 2 q^{2} - 104 q^{4} - 135 q^{5} + 60 q^{6} - 21 q^{7} + 24 q^{8} - 25 q^{9} + 10 q^{10} + 82 q^{11} - 201 q^{12} - 12 q^{13} - 154 q^{14} - 228 q^{16} + 212 q^{17} - 1902 q^{18} - 166 q^{19} - 520 q^{20} - 14 q^{21} - 110 q^{22} - 88 q^{23} - 1240 q^{24} - 675 q^{25} + 688 q^{26} - 240 q^{27} - 217 q^{28} + 1062 q^{29} - 120 q^{30} + 2372 q^{31} + 344 q^{32} - 912 q^{33} + 278 q^{34} + 350 q^{35} + 1796 q^{36} + 2332 q^{37} + 800 q^{38} + 1560 q^{39} + 120 q^{40} + 1034 q^{41} - 5313 q^{42} + 132 q^{43} + 884 q^{44} - 1525 q^{45} + 610 q^{46} - 1426 q^{47} + 2492 q^{48} + 49 q^{49} - 300 q^{50} + 500 q^{51} - 513 q^{52} + 2177 q^{53} + 3158 q^{54} + 165 q^{55} + 4879 q^{56} + 828 q^{57} + 408 q^{58} + 1599 q^{59} - 795 q^{60} + 242 q^{61} + 1348 q^{62} + 4501 q^{63} - 2192 q^{64} - 130 q^{65} + 849 q^{66} - 11628 q^{67} - 6294 q^{68} - 2488 q^{69} + 105 q^{70} + 2944 q^{71} - 1354 q^{72} - 720 q^{73} + 7190 q^{74} - 5466 q^{76} - 574 q^{77} + 8939 q^{78} - 8284 q^{79} + 10060 q^{80} - 1795 q^{81} - 1361 q^{82} + 330 q^{83} + 441 q^{84} + 10 q^{85} + 3732 q^{86} + 2950 q^{87} + 8209 q^{88} + 2630 q^{89} + 3055 q^{90} - 13601 q^{91} + 5501 q^{92} + 4016 q^{93} - 11551 q^{94} - 235 q^{95} - 32436 q^{96} - 5848 q^{97} + 5467 q^{98} - 16094 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1 −1.14040 − 4.99643i 3.70366 + 4.64424i −16.4561 + 7.92483i 3.11745 + 3.90916i 18.9810 − 23.8014i −16.2661 − 8.85509i 32.7997 + 41.1295i −1.84382 + 8.07832i 15.9767 − 20.0341i
36.2 −1.13782 − 4.98513i −3.38665 − 4.24672i −16.3491 + 7.87331i 3.11745 + 3.90916i −17.3171 + 21.7149i 9.12100 + 16.1185i 32.3470 + 40.5618i −0.557204 + 2.44127i 15.9405 − 19.9888i
36.3 −1.10054 − 4.82179i 0.0876850 + 0.109954i −14.8307 + 7.14210i 3.11745 + 3.90916i 0.433672 − 0.543807i 9.47748 − 15.9115i 26.0903 + 32.7163i 6.00366 − 26.3038i 15.4183 − 19.3339i
36.4 −1.00167 − 4.38860i 4.88383 + 6.12413i −11.0487 + 5.32079i 3.11745 + 3.90916i 21.9844 − 27.5675i 18.5040 − 0.775912i 11.9651 + 15.0037i −7.64510 + 33.4954i 14.0331 − 17.5969i
36.5 −0.841324 − 3.68608i 0.839767 + 1.05303i −5.67160 + 2.73130i 3.11745 + 3.90916i 3.17505 − 3.98139i −11.4148 + 14.5843i −4.01924 − 5.03997i 5.60439 − 24.5544i 11.7867 − 14.7800i
36.6 −0.803608 − 3.52084i −4.83758 − 6.06614i −4.54277 + 2.18768i 3.11745 + 3.90916i −17.4704 + 21.9071i −18.1539 − 3.66561i −6.66021 − 8.35164i −7.38775 + 32.3678i 11.2583 − 14.1175i
36.7 −0.766504 − 3.35827i −3.47540 − 4.35801i −3.48273 + 1.67719i 3.11745 + 3.90916i −11.9715 + 15.0118i 12.3632 − 13.7896i −8.87957 − 11.1346i −0.905803 + 3.96858i 10.7385 − 13.4656i
36.8 −0.587416 − 2.57364i 3.55417 + 4.45679i 0.929193 − 0.447476i 3.11745 + 3.90916i 9.38238 − 11.7651i −9.23134 − 16.0556i −14.8647 − 18.6397i −1.22276 + 5.35728i 8.22952 − 10.3195i
36.9 −0.556963 − 2.44022i −0.274452 − 0.344152i 1.56330 − 0.752847i 3.11745 + 3.90916i −0.686946 + 0.861403i 14.2696 + 11.8059i −15.1924 − 19.0507i 5.96495 − 26.1341i 7.80288 − 9.78451i
36.10 −0.345536 − 1.51389i 5.17615 + 6.49068i 5.03528 − 2.42486i 3.11745 + 3.90916i 8.03765 − 10.0789i −16.2481 + 8.88807i −13.1562 − 16.4974i −9.32840 + 40.8704i 4.84085 − 6.07023i
36.11 −0.249114 − 1.09144i 2.10425 + 2.63864i 6.07857 − 2.92728i 3.11745 + 3.90916i 2.35573 − 2.95399i 17.2008 + 6.86539i −10.2932 − 12.9073i 3.47348 − 15.2183i 3.49001 − 4.37634i
36.12 −0.179866 − 0.788044i −4.00110 − 5.01722i 6.61909 − 3.18759i 3.11745 + 3.90916i −3.23413 + 4.05547i −3.32819 + 18.2188i −7.73429 − 9.69849i −3.15563 + 13.8257i 2.51986 − 3.15981i
36.13 −0.0705601 − 0.309144i −4.10148 − 5.14310i 7.11716 − 3.42744i 3.11745 + 3.90916i −1.30056 + 1.63085i −1.32902 − 18.4725i −3.14340 − 3.94170i −3.62122 + 15.8656i 0.988526 − 1.23957i
36.14 0.0749341 + 0.328308i 5.15803 + 6.46796i 7.10558 − 3.42187i 3.11745 + 3.90916i −1.73697 + 2.17809i 16.6577 − 8.09445i 3.33556 + 4.18266i −9.22122 + 40.4008i −1.04980 + 1.31641i
36.15 0.137340 + 0.601728i −5.59226 − 7.01248i 6.86454 − 3.30579i 3.11745 + 3.90916i 3.45156 − 4.32812i 18.4628 + 1.45746i 6.01052 + 7.53695i −11.8933 + 52.1082i −1.92410 + 2.41274i
36.16 0.143755 + 0.629830i 0.474877 + 0.595477i 6.83173 − 3.28999i 3.11745 + 3.90916i −0.306784 + 0.384695i −14.9979 − 10.8657i 6.27656 + 7.87055i 5.87898 − 25.7575i −2.01396 + 2.52542i
36.17 0.195114 + 0.854850i −0.858110 − 1.07604i 6.51505 − 3.13748i 3.11745 + 3.90916i 0.752420 − 0.943505i −17.2758 + 6.67421i 8.32683 + 10.4415i 5.58656 − 24.4763i −2.73348 + 3.42768i
36.18 0.495821 + 2.17233i −0.603627 − 0.756925i 2.73455 − 1.31689i 3.11745 + 3.90916i 1.34500 − 1.68658i 18.3840 − 2.24261i 15.3307 + 19.2240i 5.79950 − 25.4093i −6.94630 + 8.71038i
36.19 0.554391 + 2.42895i 4.29512 + 5.38591i 1.61532 − 0.777898i 3.11745 + 3.90916i −10.7009 + 13.4185i −6.25205 + 17.4331i 15.2119 + 19.0752i −4.55189 + 19.9431i −7.76685 + 9.73932i
36.20 0.688303 + 3.01565i −5.93423 − 7.44129i −1.41266 + 0.680299i 3.11745 + 3.90916i 18.3558 − 23.0175i −17.4332 + 6.25165i 12.4048 + 15.5551i −14.1497 + 61.9937i −9.64291 + 12.0918i
See next 80 embeddings (of 162 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.27
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 245.4.k.a ✓ 162
49.e even 7 1 inner 245.4.k.a ✓ 162
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.k.a ✓ 162 1.a even 1 1 trivial
245.4.k.a ✓ 162 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{162} - 2 T_{2}^{161} + 162 T_{2}^{160} - 340 T_{2}^{159} + 14619 T_{2}^{158} + \cdots + 86\!\cdots\!76 \) acting on \(S_{4}^{\mathrm{new}}(245, [\chi])\). Copy content Toggle raw display