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Properties

Label 213.2.m.a
Level $213$
Weight $2$
Character orbit 213.m
Analytic conductor $1.701$
Analytic rank $0$
Dimension $144$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [213,2,Mod(4,213)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(213, base_ring=CyclotomicField(70))
 
chi = DirichletCharacter(H, H._module([0, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("213.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 213 = 3 \cdot 71 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 213.m (of order \(35\), degree \(24\), 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: \(1.70081356305\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(6\) over \(\Q(\zeta_{35})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{35}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 3 q^{2} - 6 q^{3} + 13 q^{4} - 2 q^{5} - 2 q^{6} - 8 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 3 q^{2} - 6 q^{3} + 13 q^{4} - 2 q^{5} - 2 q^{6} - 8 q^{8} + 6 q^{9} - 9 q^{10} + 3 q^{11} - 27 q^{12} - 19 q^{14} - 10 q^{15} + 3 q^{16} + 10 q^{17} - 5 q^{18} + 10 q^{19} - 6 q^{20} + q^{22} + 20 q^{23} + 33 q^{24} - 30 q^{25} - 46 q^{26} - 6 q^{27} - 11 q^{28} - 115 q^{29} - 16 q^{30} - 13 q^{31} - 19 q^{32} + 102 q^{34} + 8 q^{35} + 4 q^{36} - 4 q^{37} + 7 q^{38} - 10 q^{39} - 114 q^{40} + 19 q^{41} - 100 q^{42} - 47 q^{43} - 29 q^{44} - 2 q^{45} - 107 q^{46} - 31 q^{47} + 37 q^{48} + 24 q^{49} + 61 q^{50} - 5 q^{51} - 28 q^{52} - 56 q^{53} + 3 q^{54} - 223 q^{55} - 41 q^{56} - 39 q^{57} + 131 q^{58} + 20 q^{59} + 33 q^{60} + q^{61} + 47 q^{62} + 244 q^{64} + 117 q^{65} - 15 q^{66} + 76 q^{67} - 258 q^{68} - 5 q^{69} - 10 q^{70} + 95 q^{71} + 12 q^{72} - 8 q^{73} + 119 q^{74} - 63 q^{75} + 64 q^{76} + 29 q^{77} + 12 q^{78} + 122 q^{79} + 137 q^{80} + 6 q^{81} + 167 q^{82} - 68 q^{83} - 2 q^{84} + 3 q^{85} + 34 q^{86} - 26 q^{87} + 79 q^{88} - 36 q^{89} + 26 q^{90} - 6 q^{91} + 9 q^{92} - 25 q^{93} - 96 q^{94} + 170 q^{95} - 6 q^{96} - 61 q^{97} - 319 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.71771 − 0.244599i −0.134233 − 0.990950i 5.35828 + 0.972385i 2.67542 − 1.94381i 0.122423 + 2.72595i 2.63499 + 1.57433i −9.06370 − 2.50142i −0.963963 + 0.266037i −7.74649 + 4.62831i
4.2 −1.52037 − 0.136836i −0.134233 − 0.990950i 0.324936 + 0.0589673i −0.391278 + 0.284280i 0.0684868 + 1.52498i 3.04628 + 1.82007i 2.45705 + 0.678103i −0.963963 + 0.266037i 0.633787 − 0.378670i
4.3 0.304291 + 0.0273867i −0.134233 − 0.990950i −1.87602 − 0.340447i 2.20367 − 1.60106i −0.0137071 − 0.305213i −2.45611 − 1.46746i −1.15055 − 0.317532i −0.963963 + 0.266037i 0.714405 − 0.426837i
4.4 0.649238 + 0.0584325i −0.134233 − 0.990950i −1.54976 − 0.281241i −3.45259 + 2.50845i −0.0292457 − 0.651206i −1.79414 − 1.07195i −2.24647 − 0.619987i −0.963963 + 0.266037i −2.38813 + 1.42684i
4.5 1.96539 + 0.176889i −0.134233 − 0.990950i 1.86362 + 0.338197i 0.334155 − 0.242778i −0.0885334 − 1.97135i 2.68871 + 1.60643i −0.201527 − 0.0556179i −0.963963 + 0.266037i 0.699690 − 0.418046i
4.6 2.69572 + 0.242619i −0.134233 − 0.990950i 5.24019 + 0.950955i −1.15218 + 0.837111i −0.121432 − 2.70389i −4.11971 − 2.46141i 8.67722 + 2.39476i −0.963963 + 0.266037i −3.30907 + 1.97708i
10.1 −2.22642 + 0.614452i 0.393025 − 0.919528i 2.86248 − 1.71025i 1.02098 − 3.14225i −0.310032 + 2.28875i 0.0174822 + 0.389272i −2.12998 + 2.22778i −0.691063 − 0.722795i −0.342363 + 7.62331i
10.2 −1.83896 + 0.507519i 0.393025 − 0.919528i 1.40728 − 0.840812i −1.27048 + 3.91013i −0.256077 + 1.89044i −0.0436465 − 0.971864i 0.475483 − 0.497316i −0.691063 − 0.722795i 0.351886 − 7.83535i
10.3 −1.13322 + 0.312750i 0.393025 − 0.919528i −0.530516 + 0.316969i 0.127778 − 0.393260i −0.157803 + 1.16495i 0.113837 + 2.53478i 2.12687 − 2.22453i −0.691063 − 0.722795i −0.0218089 + 0.485614i
10.4 −0.234477 + 0.0647116i 0.393025 − 0.919528i −1.66611 + 0.995452i 0.362611 − 1.11600i −0.0326513 + 0.241042i −0.170382 − 3.79385i 0.662439 − 0.692857i −0.691063 − 0.722795i −0.0128058 + 0.285142i
10.5 1.67458 − 0.462156i 0.393025 − 0.919528i 0.873746 − 0.522039i 0.462116 − 1.42225i 0.233188 − 1.72147i −0.0124413 − 0.277028i −1.17911 + 1.23325i −0.691063 − 0.722795i 0.116553 − 2.59524i
10.6 2.30662 − 0.636587i 0.393025 − 0.919528i 3.19836 − 1.91093i −0.460104 + 1.41606i 0.321200 − 2.37120i 0.0951507 + 2.11870i 2.85371 − 2.98475i −0.691063 − 0.722795i −0.159844 + 3.55920i
16.1 −2.41222 − 0.437753i 0.963963 − 0.266037i 3.75469 + 1.40916i 0.985648 − 3.03351i −2.44175 + 0.219761i −0.223673 − 0.415654i −4.23109 − 2.52796i 0.858449 − 0.512899i −3.70552 + 6.88602i
16.2 −1.03873 − 0.188501i 0.963963 − 0.266037i −0.829046 − 0.311146i 0.460164 − 1.41624i −1.05144 + 0.0946317i 2.02247 + 3.75838i 2.61502 + 1.56240i 0.858449 − 0.512899i −0.744948 + 1.38434i
16.3 −0.0752058 − 0.0136478i 0.963963 − 0.266037i −1.86700 − 0.700697i −1.30978 + 4.03109i −0.0761264 + 0.00685150i 1.01476 + 1.88574i 0.262076 + 0.156583i 0.858449 − 0.512899i 0.153519 − 0.285286i
16.4 0.0242924 + 0.00440842i 0.963963 − 0.266037i −1.87190 − 0.702535i 0.139526 − 0.429417i 0.0245898 − 0.00221312i −2.10788 − 3.91709i −0.0847646 − 0.0506445i 0.858449 − 0.512899i 0.00528248 − 0.00981649i
16.5 1.28666 + 0.233495i 0.963963 − 0.266037i −0.271491 − 0.101892i 1.03158 − 3.17488i 1.30241 − 0.117219i 0.673083 + 1.25080i −2.57067 − 1.53591i 0.858449 − 0.512899i 2.06862 − 3.84413i
16.6 2.12691 + 0.385977i 0.963963 − 0.266037i 2.50229 + 0.939126i −0.711377 + 2.18939i 2.15294 − 0.193769i −1.37876 − 2.56217i 1.24834 + 0.745847i 0.858449 − 0.512899i −2.35809 + 4.38206i
19.1 −1.07636 − 2.51828i −0.983930 + 0.178557i −3.80104 + 3.97558i 0.786441 + 2.42042i 1.50872 + 2.28562i −1.99719 − 1.74489i 8.97486 + 3.36832i 0.936235 − 0.351375i 5.24879 − 4.58573i
19.2 −0.715969 − 1.67509i −0.983930 + 0.178557i −0.911197 + 0.953037i −0.620477 − 1.90963i 1.00356 + 1.52033i −0.289947 − 0.253319i −1.16224 − 0.436196i 0.936235 − 0.351375i −2.75457 + 2.40659i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.g even 35 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 213.2.m.a ✓ 144
3.b odd 2 1 639.2.v.c 144
71.g even 35 1 inner 213.2.m.a ✓ 144
213.o odd 70 1 639.2.v.c 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
213.2.m.a ✓ 144 1.a even 1 1 trivial
213.2.m.a ✓ 144 71.g even 35 1 inner
639.2.v.c 144 3.b odd 2 1
639.2.v.c 144 213.o odd 70 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{144} + 3 T_{2}^{143} - 8 T_{2}^{142} - 33 T_{2}^{141} + 17 T_{2}^{140} + 216 T_{2}^{139} + \cdots + 44453083921 \) acting on \(S_{2}^{\mathrm{new}}(213, [\chi])\). Copy content Toggle raw display