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Properties

Label 451.6.a.d
Level $451$
Weight $6$
Character orbit 451.a
Self dual yes
Analytic conductor $72.333$
Analytic rank $0$
Dimension $46$
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] = [451,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("451.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 451 = 11 \cdot 41 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(72.3331027357\)
Analytic rank: \(0\)
Dimension: \(46\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 46 q + 12 q^{2} + 24 q^{3} + 848 q^{4} + 286 q^{5} + 126 q^{6} - 243 q^{7} + 102 q^{8} + 4688 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 46 q + 12 q^{2} + 24 q^{3} + 848 q^{4} + 286 q^{5} + 126 q^{6} - 243 q^{7} + 102 q^{8} + 4688 q^{9} + 250 q^{10} - 5566 q^{11} - 2644 q^{12} + 405 q^{13} + 509 q^{14} + 853 q^{15} + 16732 q^{16} + 1771 q^{17} + 1601 q^{18} - 1985 q^{19} + 16889 q^{20} + 8921 q^{21} - 1452 q^{22} + 12817 q^{23} + 15896 q^{24} + 40674 q^{25} + 35117 q^{26} - 5136 q^{27} - 13528 q^{28} + 24380 q^{29} + 67916 q^{30} + 21147 q^{31} + 19601 q^{32} - 2904 q^{33} + 25132 q^{34} + 35309 q^{35} + 107514 q^{36} + 27752 q^{37} + 31715 q^{38} - 8416 q^{39} + 25247 q^{40} + 77326 q^{41} + 50945 q^{42} - 6262 q^{43} - 102608 q^{44} + 102140 q^{45} + 37222 q^{46} - 47958 q^{47} + 17059 q^{48} + 160347 q^{49} + 26802 q^{50} - 44639 q^{51} + 16431 q^{52} + 168642 q^{53} + 145293 q^{54} - 34606 q^{55} + 61477 q^{56} - 2180 q^{57} - 13400 q^{58} + 97987 q^{59} - 72300 q^{60} - 2647 q^{61} + 130547 q^{62} + 78780 q^{63} + 491864 q^{64} + 178685 q^{65} - 15246 q^{66} - 4374 q^{67} + 49475 q^{68} + 267572 q^{69} + 261692 q^{70} + 154953 q^{71} - 21828 q^{72} + 109425 q^{73} + 295415 q^{74} - 38514 q^{75} - 105973 q^{76} + 29403 q^{77} + 271193 q^{78} - 152786 q^{79} + 577596 q^{80} + 893778 q^{81} + 20172 q^{82} + 131500 q^{83} + 123892 q^{84} + 131881 q^{85} + 159256 q^{86} + 513191 q^{87} - 12342 q^{88} + 65366 q^{89} + 186979 q^{90} + 339246 q^{91} + 655758 q^{92} + 249609 q^{93} - 532348 q^{94} + 243812 q^{95} + 321210 q^{96} - 161871 q^{97} + 441772 q^{98} - 567248 q^{99}+O(q^{100}) \) 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 −11.1613 −30.2184 92.5743 99.5043 337.276 −76.1996 −676.088 670.151 −1110.60
1.2 −11.1583 22.6808 92.5070 −49.9654 −253.079 −231.738 −675.153 271.420 557.528
1.3 −10.0575 −4.61544 69.1540 −9.34657 46.4200 48.9200 −373.678 −221.698 94.0035
1.4 −9.84036 −20.4036 64.8328 −22.9434 200.779 −33.3459 −323.086 173.309 225.771
1.5 −9.49737 21.1979 58.2000 35.0854 −201.324 110.828 −248.831 206.352 −333.219
1.6 −9.17130 2.12952 52.1127 −52.8112 −19.5304 195.233 −184.459 −238.465 484.347
1.7 −8.92678 −4.72444 47.6874 102.150 42.1740 199.130 −140.038 −220.680 −911.870
1.8 −8.17193 15.9332 34.7805 −49.4891 −130.205 −176.219 −22.7222 10.8654 404.421
1.9 −8.13236 −16.1223 34.1353 92.1940 131.112 −18.7350 −17.3651 16.9276 −749.755
1.10 −7.67985 −18.9460 26.9801 −50.9871 145.502 −21.5291 38.5518 115.949 391.574
1.11 −7.38725 −0.447022 22.5714 67.1748 3.30226 −191.457 69.6511 −242.800 −496.237
1.12 −6.45430 22.3308 9.65802 19.2076 −144.130 207.280 144.202 255.666 −123.972
1.13 −5.30125 −29.2973 −3.89674 −97.9021 155.312 −28.7082 190.298 615.332 519.004
1.14 −5.09012 17.3449 −6.09066 −61.4068 −88.2877 −4.74787 193.886 57.8458 312.568
1.15 −5.04796 −2.19319 −6.51812 2.75012 11.0711 −63.3151 194.438 −238.190 −13.8825
1.16 −4.19356 26.0136 −14.4140 62.7817 −109.090 −214.149 194.640 433.709 −263.279
1.17 −3.01681 7.31575 −22.8989 80.3837 −22.0702 101.674 165.619 −189.480 −242.502
1.18 −2.96546 −1.02280 −23.2060 −84.4498 3.03307 117.573 163.711 −241.954 250.432
1.19 −1.80596 −26.8562 −28.7385 42.2255 48.5011 −199.702 109.691 478.255 −76.2573
1.20 −1.75438 −17.1280 −28.9221 46.5933 30.0491 58.6018 106.881 50.3692 −81.7425
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.46
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 451.6.a.d ✓ 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
451.6.a.d ✓ 46 1.a even 1 1 trivial