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Properties

Label 451.4.a.a
Level $451$
Weight $4$
Character orbit 451.a
Self dual yes
Analytic conductor $26.610$
Analytic rank $1$
Dimension $21$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [451,4,Mod(1,451)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(451, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("451.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 451 = 11 \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 451.a (trivial)

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: yes
Analytic conductor: \(26.6098614126\)
Analytic rank: \(1\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 21 q - 6 q^{2} - q^{3} + 80 q^{4} - 53 q^{5} - 12 q^{6} + 18 q^{7} - 54 q^{8} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q - 6 q^{2} - q^{3} + 80 q^{4} - 53 q^{5} - 12 q^{6} + 18 q^{7} - 54 q^{8} + 60 q^{9} - 50 q^{10} - 231 q^{11} - 160 q^{12} - 64 q^{13} - 105 q^{14} - 38 q^{15} + 156 q^{16} - 132 q^{17} - 157 q^{18} + 71 q^{19} - 679 q^{20} - 452 q^{21} + 66 q^{22} - 548 q^{23} - 536 q^{24} + 198 q^{25} - 595 q^{26} + 170 q^{27} + 268 q^{28} - 1003 q^{29} - 228 q^{30} - 360 q^{31} - 863 q^{32} + 11 q^{33} + 50 q^{34} - 391 q^{35} - 322 q^{36} - 703 q^{37} - 1089 q^{38} - 160 q^{39} + 453 q^{40} + 861 q^{41} - 819 q^{42} - 27 q^{43} - 880 q^{44} - 1193 q^{45} - 582 q^{46} - 1188 q^{47} - 1085 q^{48} - 577 q^{49} - 1492 q^{50} - 1636 q^{51} - 631 q^{52} - 1185 q^{53} - 171 q^{54} + 583 q^{55} - 1195 q^{56} + 406 q^{57} - 1196 q^{58} - 2759 q^{59} - 1716 q^{60} - 1343 q^{61} - 2755 q^{62} - 1765 q^{63} - 892 q^{64} - 1114 q^{65} + 132 q^{66} - 235 q^{67} - 1355 q^{68} - 3248 q^{69} - 2416 q^{70} - 1812 q^{71} - 1814 q^{72} - 1894 q^{73} - 2091 q^{74} - 195 q^{75} + 151 q^{76} - 198 q^{77} + 3173 q^{78} + 1603 q^{79} - 2012 q^{80} - 2183 q^{81} - 246 q^{82} - 1069 q^{83} - 114 q^{84} - 1035 q^{85} - 1966 q^{86} - 2206 q^{87} + 594 q^{88} + 1142 q^{89} - 2719 q^{90} - 4754 q^{91} - 6294 q^{92} - 4925 q^{93} - 2250 q^{94} - 4591 q^{95} + 476 q^{96} - 2744 q^{97} + 2852 q^{98} - 660 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.31114 6.64481 20.2082 −10.8110 −35.2915 0.577015 −64.8395 17.1535 57.4187
1.2 −5.06302 −4.32412 17.6342 −19.1824 21.8931 −17.5501 −48.7783 −8.30200 97.1208
1.3 −5.03352 −6.67153 17.3364 9.32981 33.5813 25.1399 −46.9947 17.5094 −46.9618
1.4 −4.18775 4.04312 9.53725 10.3287 −16.9316 0.863556 −6.43762 −10.6532 −43.2540
1.5 −3.53707 5.04400 4.51087 0.693162 −17.8410 18.7401 12.3413 −1.55806 −2.45176
1.6 −3.25163 −5.77324 2.57311 12.7781 18.7725 −6.61988 17.6463 6.33034 −41.5497
1.7 −3.03635 −0.987385 1.21941 −20.8169 2.99805 26.8815 20.5882 −26.0251 63.2074
1.8 −2.13529 −6.02417 −3.44053 −9.71475 12.8634 −17.9314 24.4289 9.29057 20.7438
1.9 −1.96679 7.91730 −4.13174 −2.72225 −15.5717 −9.53743 23.8606 35.6837 5.35409
1.10 −0.793595 −1.63234 −7.37021 2.82947 1.29542 3.55742 12.1977 −24.3355 −2.24545
1.11 −0.114517 7.25979 −7.98689 7.98094 −0.831371 −32.9298 1.83077 25.7045 −0.913955
1.12 −0.0847912 0.306225 −7.99281 18.9107 −0.0259652 4.55997 1.35605 −26.9062 −1.60346
1.13 0.165018 −9.17167 −7.97277 −10.6382 −1.51349 3.75561 −2.63580 57.1195 −1.75550
1.14 2.09407 4.65137 −3.61486 −0.984610 9.74031 10.1998 −24.3224 −5.36474 −2.06185
1.15 2.13999 −2.68137 −3.42045 −4.37250 −5.73810 34.4983 −24.4396 −19.8103 −9.35711
1.16 2.64686 9.11767 −0.994124 −20.2754 24.1332 −3.77950 −23.8062 56.1318 −53.6661
1.17 3.37967 3.57211 3.42216 2.41621 12.0725 −22.9935 −15.4716 −14.2400 8.16600
1.18 3.87133 −3.98549 6.98716 10.7826 −15.4291 −6.29148 −3.92104 −11.1159 41.7430
1.19 4.47283 −0.804126 12.0062 −14.8722 −3.59672 22.8156 17.9190 −26.3534 −66.5208
1.20 4.86952 0.469039 15.7122 −10.5808 2.28400 −23.7618 37.5549 −26.7800 −51.5233
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \( +1 \)
\(41\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 451.4.a.a ✓ 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
451.4.a.a ✓ 21 1.a even 1 1 trivial