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Properties

Label 213.4.e.a
Level $213$
Weight $4$
Character orbit 213.e
Analytic conductor $12.567$
Analytic rank $0$
Dimension $72$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [213,4,Mod(25,213)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(213, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("213.25");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 213 = 3 \cdot 71 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 213.e (of order \(5\), degree \(4\), 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: \(12.5674068312\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 3 q^{2} - 54 q^{3} - 53 q^{4} + 4 q^{5} + 6 q^{6} + 28 q^{7} + 4 q^{8} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 3 q^{2} - 54 q^{3} - 53 q^{4} + 4 q^{5} + 6 q^{6} + 28 q^{7} + 4 q^{8} - 162 q^{9} - 14 q^{10} - 54 q^{11} - 159 q^{12} + 102 q^{13} - 501 q^{14} - 18 q^{15} - 657 q^{16} + 120 q^{17} + 18 q^{18} + 182 q^{19} + 650 q^{20} + 84 q^{21} - 663 q^{22} - 660 q^{23} - 48 q^{24} + 6 q^{25} + 592 q^{26} - 486 q^{27} + 37 q^{28} + 672 q^{29} - 102 q^{30} + 580 q^{31} + 784 q^{32} + 348 q^{33} - 2414 q^{34} - 74 q^{35} - 702 q^{36} - 212 q^{37} + 559 q^{38} - 864 q^{39} - 638 q^{40} + 152 q^{41} - 1503 q^{42} - 364 q^{43} + 1885 q^{44} + 36 q^{45} + 313 q^{46} - 1006 q^{47} + 3834 q^{48} - 626 q^{49} + 3569 q^{50} - 540 q^{51} - 1773 q^{52} - 406 q^{53} - 81 q^{54} - 1618 q^{55} + 3822 q^{56} - 384 q^{57} - 1400 q^{58} + 2144 q^{59} - 1140 q^{60} + 582 q^{61} - 2054 q^{62} + 252 q^{63} - 3290 q^{64} + 1654 q^{65} - 1989 q^{66} - 816 q^{67} - 452 q^{68} + 1530 q^{69} + 8734 q^{70} + 548 q^{71} + 216 q^{72} - 164 q^{73} + 4281 q^{74} - 2712 q^{75} - 1220 q^{76} + 1344 q^{77} - 504 q^{78} + 1516 q^{79} - 6385 q^{80} - 1458 q^{81} + 1238 q^{82} - 2522 q^{83} + 2091 q^{84} - 2704 q^{85} - 3043 q^{86} - 1374 q^{87} + 8575 q^{88} + 540 q^{89} - 126 q^{90} - 2408 q^{91} + 551 q^{92} + 1560 q^{93} - 2860 q^{94} - 684 q^{95} - 1278 q^{96} - 2988 q^{97} + 3050 q^{98} + 1044 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −4.45185 + 3.23446i −2.42705 + 1.76336i 6.88511 − 21.1902i −4.08590 − 12.5751i 5.10137 − 15.7004i 20.4940 − 14.8897i 24.2837 + 74.7375i 2.78115 − 8.55951i 58.8635 + 42.7668i
25.2 −4.14902 + 3.01444i −2.42705 + 1.76336i 5.65537 − 17.4054i 5.22278 + 16.0741i 4.75435 − 14.6324i −8.14459 + 5.91739i 16.3251 + 50.2435i 2.78115 − 8.55951i −70.1237 − 50.9478i
25.3 −3.52463 + 2.56079i −2.42705 + 1.76336i 3.39322 − 10.4432i 3.55946 + 10.9549i 4.03887 − 12.4304i 23.8904 − 17.3574i 4.01285 + 12.3503i 2.78115 − 8.55951i −40.5990 − 29.4969i
25.4 −3.07942 + 2.23733i −2.42705 + 1.76336i 2.00505 − 6.17090i −3.00712 − 9.25498i 3.52870 − 10.8602i −5.10617 + 3.70985i −1.77792 − 5.47186i 2.78115 − 8.55951i 29.9666 + 21.7720i
25.5 −2.74605 + 1.99512i −2.42705 + 1.76336i 1.08815 − 3.34897i 2.19350 + 6.75089i 3.14670 − 9.68453i −4.17536 + 3.03358i −4.69768 − 14.4580i 2.78115 − 8.55951i −19.4923 − 14.1620i
25.6 −1.65087 + 1.19943i −2.42705 + 1.76336i −1.18539 + 3.64827i −6.53529 − 20.1136i 1.89173 − 5.82213i 5.37927 − 3.90827i −7.46350 − 22.9703i 2.78115 − 8.55951i 34.9136 + 25.3662i
25.7 −1.49268 + 1.08450i −2.42705 + 1.76336i −1.42017 + 4.37082i 0.977219 + 3.00757i 1.71046 − 5.26427i −25.7679 + 18.7215i −7.18153 − 22.1025i 2.78115 − 8.55951i −4.72039 − 3.42956i
25.8 −0.874305 + 0.635220i −2.42705 + 1.76336i −2.11123 + 6.49770i 2.65444 + 8.16952i 1.00186 − 3.08342i 24.8129 − 18.0276i −4.95325 − 15.2445i 2.78115 − 8.55951i −7.51023 − 5.45650i
25.9 −0.650145 + 0.472358i −2.42705 + 1.76336i −2.27257 + 6.99425i −1.23490 − 3.80064i 0.745000 − 2.29287i 19.7303 − 14.3349i −3.81296 − 11.7351i 2.78115 − 8.55951i 2.59813 + 1.88765i
25.10 0.0761761 − 0.0553452i −2.42705 + 1.76336i −2.46940 + 7.60002i 2.73364 + 8.41328i −0.0872900 + 0.268651i −16.8996 + 12.2783i 0.465289 + 1.43201i 2.78115 − 8.55951i 0.673873 + 0.489597i
25.11 0.967439 − 0.702885i −2.42705 + 1.76336i −2.03025 + 6.24846i −4.75190 − 14.6249i −1.10859 + 3.41188i −15.9550 + 11.5920i 5.38404 + 16.5704i 2.78115 − 8.55951i −14.8768 − 10.8086i
25.12 1.18313 − 0.859597i −2.42705 + 1.76336i −1.81124 + 5.57442i 4.95557 + 15.2517i −1.35575 + 4.17257i 1.07203 − 0.778879i 6.26415 + 19.2791i 2.78115 − 8.55951i 18.9734 + 13.7850i
25.13 1.65993 − 1.20601i −2.42705 + 1.76336i −1.17123 + 3.60468i −3.66716 − 11.2863i −1.90211 + 5.85408i 7.24017 − 5.26029i 7.47540 + 23.0069i 2.78115 − 8.55951i −19.6986 − 14.3119i
25.14 2.36924 − 1.72135i −2.42705 + 1.76336i 0.178096 − 0.548123i −1.22638 − 3.77442i −2.71490 + 8.35561i 6.97372 − 5.06671i 6.71818 + 20.6764i 2.78115 − 8.55951i −9.40270 − 6.83146i
25.15 3.31511 − 2.40857i −2.42705 + 1.76336i 2.71661 − 8.36087i 0.580947 + 1.78797i −3.79878 + 11.6914i 12.9289 − 9.39341i −1.00179 − 3.08320i 2.78115 − 8.55951i 6.23235 + 4.52807i
25.16 3.49781 − 2.54131i −2.42705 + 1.76336i 3.30429 − 10.1696i −2.85407 − 8.78392i −4.00813 + 12.3358i −27.2990 + 19.8339i −3.59786 − 11.0731i 2.78115 − 8.55951i −32.3056 − 23.4714i
25.17 3.75658 − 2.72931i −2.42705 + 1.76336i 4.19058 − 12.8973i 6.09511 + 18.7588i −4.30465 + 13.2484i −0.0113511 + 0.00824702i −7.97941 − 24.5581i 2.78115 − 8.55951i 74.0955 + 53.8335i
25.18 4.48455 − 3.25821i −2.42705 + 1.76336i 7.02307 − 21.6148i −0.609934 − 1.87718i −5.13883 + 15.8157i 3.48979 − 2.53548i −25.2267 − 77.6399i 2.78115 − 8.55951i −8.85154 − 6.43102i
76.1 −1.63584 − 5.03459i 0.927051 + 2.85317i −16.1990 + 11.7693i −16.0098 − 11.6318i 12.8480 − 9.33465i −8.64946 − 26.6203i 51.4911 + 37.4105i −7.28115 + 5.29007i −32.3719 + 99.6303i
76.2 −1.50162 − 4.62152i 0.927051 + 2.85317i −12.6314 + 9.17727i 3.04861 + 2.21494i 11.7939 − 8.56877i 0.213623 + 0.657465i 29.9302 + 21.7455i −7.28115 + 5.29007i 5.65854 − 17.4152i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 213.4.e.a ✓ 72
71.c even 5 1 inner 213.4.e.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
213.4.e.a ✓ 72 1.a even 1 1 trivial
213.4.e.a ✓ 72 71.c even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{72} + 3 T_{2}^{71} + 103 T_{2}^{70} + 288 T_{2}^{69} + 6470 T_{2}^{68} + 16949 T_{2}^{67} + \cdots + 59\!\cdots\!16 \) acting on \(S_{4}^{\mathrm{new}}(213, [\chi])\). Copy content Toggle raw display