Properties

Label 13.7.f.a
Level $13$
Weight $7$
Character orbit 13.f
Analytic conductor $2.991$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [13,7,Mod(2,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.2");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 13.f (of order \(12\), 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: \(2.99070308706\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{2} - 2 q^{3} - 6 q^{4} - 114 q^{5} + 634 q^{6} + 316 q^{7} - 1398 q^{8} - 1946 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{2} - 2 q^{3} - 6 q^{4} - 114 q^{5} + 634 q^{6} + 316 q^{7} - 1398 q^{8} - 1946 q^{9} + 4674 q^{10} + 228 q^{11} - 1552 q^{13} - 14616 q^{14} + 7486 q^{15} - 2138 q^{16} + 20898 q^{17} - 5630 q^{18} - 14036 q^{19} - 14634 q^{20} - 314 q^{21} + 37852 q^{22} + 29634 q^{23} + 47154 q^{24} + 510 q^{26} - 107060 q^{27} - 40572 q^{28} + 1026 q^{29} - 247734 q^{30} - 118116 q^{31} + 180876 q^{32} + 306046 q^{33} + 101898 q^{34} + 104862 q^{35} + 105066 q^{36} - 31470 q^{37} + 28510 q^{39} - 275328 q^{40} - 357216 q^{41} - 25636 q^{42} - 480486 q^{43} - 797712 q^{44} + 464248 q^{45} + 1012050 q^{46} + 140028 q^{47} + 676102 q^{48} + 854562 q^{49} + 756714 q^{50} - 771704 q^{52} - 1397076 q^{53} - 1271048 q^{54} - 205490 q^{55} - 1461108 q^{56} - 1483394 q^{57} + 1133806 q^{58} + 1568052 q^{59} + 886808 q^{60} + 768676 q^{61} + 3741966 q^{62} + 1336264 q^{63} - 946968 q^{65} - 5076928 q^{66} - 1489892 q^{67} - 569052 q^{68} - 1475766 q^{69} - 2626556 q^{70} + 2136948 q^{71} + 4787052 q^{72} + 326326 q^{73} + 2159718 q^{74} + 3466008 q^{75} + 1522158 q^{76} - 1573544 q^{78} - 1734344 q^{79} - 5214300 q^{80} - 773072 q^{81} - 3540426 q^{82} - 3642132 q^{83} + 2952364 q^{84} + 3509916 q^{85} - 132324 q^{86} + 531698 q^{87} + 4478052 q^{88} + 632754 q^{89} - 2946788 q^{91} - 1656060 q^{92} - 295514 q^{93} + 1552210 q^{94} - 2753718 q^{95} - 3979208 q^{96} + 2258214 q^{97} + 4984308 q^{98} + 4336480 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −14.0703 3.77014i −17.8024 30.8347i 128.335 + 74.0942i −91.2022 + 91.2022i 134.235 + 500.973i 284.589 76.2554i −867.158 867.158i −269.354 + 466.535i 1627.09 939.401i
2.2 −9.37184 2.51118i 10.4091 + 18.0291i 26.0998 + 15.0687i 160.035 160.035i −52.2782 195.105i 162.036 43.4174i 232.320 + 232.320i 147.801 255.999i −1901.70 + 1097.95i
2.3 −4.77368 1.27910i 4.54257 + 7.86796i −34.2737 19.7879i −109.630 + 109.630i −11.6208 43.3696i −504.391 + 135.151i 361.954 + 361.954i 323.230 559.851i 663.567 383.110i
2.4 2.48669 + 0.666306i −16.9072 29.2841i −49.6860 28.6862i 34.0715 34.0715i −22.5307 84.0857i 190.342 51.0019i −220.944 220.944i −207.205 + 358.889i 107.427 62.0232i
2.5 5.89025 + 1.57829i 20.5701 + 35.6285i −23.2216 13.4070i −37.1940 + 37.1940i 64.9312 + 242.326i 588.562 157.705i −391.586 391.586i −481.762 + 834.436i −277.785 + 160.379i
2.6 12.5088 + 3.35172i −1.31220 2.27280i 89.8106 + 51.8522i 24.9454 24.9454i −8.79628 32.8282i −400.516 + 107.318i 363.576 + 363.576i 361.056 625.368i 395.648 228.427i
6.1 −2.92596 10.9198i 14.1081 24.4359i −55.2556 + 31.9018i −47.2751 + 47.2751i −308.115 82.5592i 27.7639 103.616i −1.56966 1.56966i −33.5751 58.1539i 654.560 + 377.911i
6.2 −2.74273 10.2360i −22.3986 + 38.7956i −41.8275 + 24.1491i −35.0169 + 35.0169i 458.545 + 122.867i −112.597 + 420.218i −117.658 117.658i −638.899 1106.61i 454.475 + 262.391i
6.3 −0.397723 1.48432i −0.503519 + 0.872121i 53.3806 30.8193i 84.5480 84.5480i 1.49477 + 0.400522i 41.0022 153.022i −136.519 136.519i 363.993 + 630.454i −159.123 91.8697i
6.4 1.76668 + 6.59335i −10.7542 + 18.6268i 15.0745 8.70327i −144.857 + 144.857i −141.812 37.9984i 49.5658 184.982i 392.922 + 392.922i 133.195 + 230.701i −1211.01 699.177i
6.5 1.99444 + 7.44337i 25.6304 44.3931i 3.99974 2.30925i −23.8066 + 23.8066i 381.553 + 102.237i −75.6362 + 282.278i 373.897 + 373.897i −949.333 1644.29i −224.682 129.720i
6.6 3.63541 + 13.5675i −6.58210 + 11.4005i −115.436 + 66.6469i 128.381 128.381i −178.606 47.8572i −92.7197 + 346.035i −688.234 688.234i 277.852 + 481.253i 2208.54 + 1275.10i
7.1 −14.0703 + 3.77014i −17.8024 + 30.8347i 128.335 74.0942i −91.2022 91.2022i 134.235 500.973i 284.589 + 76.2554i −867.158 + 867.158i −269.354 466.535i 1627.09 + 939.401i
7.2 −9.37184 + 2.51118i 10.4091 18.0291i 26.0998 15.0687i 160.035 + 160.035i −52.2782 + 195.105i 162.036 + 43.4174i 232.320 232.320i 147.801 + 255.999i −1901.70 1097.95i
7.3 −4.77368 + 1.27910i 4.54257 7.86796i −34.2737 + 19.7879i −109.630 109.630i −11.6208 + 43.3696i −504.391 135.151i 361.954 361.954i 323.230 + 559.851i 663.567 + 383.110i
7.4 2.48669 0.666306i −16.9072 + 29.2841i −49.6860 + 28.6862i 34.0715 + 34.0715i −22.5307 + 84.0857i 190.342 + 51.0019i −220.944 + 220.944i −207.205 358.889i 107.427 + 62.0232i
7.5 5.89025 1.57829i 20.5701 35.6285i −23.2216 + 13.4070i −37.1940 37.1940i 64.9312 242.326i 588.562 + 157.705i −391.586 + 391.586i −481.762 834.436i −277.785 160.379i
7.6 12.5088 3.35172i −1.31220 + 2.27280i 89.8106 51.8522i 24.9454 + 24.9454i −8.79628 + 32.8282i −400.516 107.318i 363.576 363.576i 361.056 + 625.368i 395.648 + 228.427i
11.1 −2.92596 + 10.9198i 14.1081 + 24.4359i −55.2556 31.9018i −47.2751 47.2751i −308.115 + 82.5592i 27.7639 + 103.616i −1.56966 + 1.56966i −33.5751 + 58.1539i 654.560 377.911i
11.2 −2.74273 + 10.2360i −22.3986 38.7956i −41.8275 24.1491i −35.0169 35.0169i 458.545 122.867i −112.597 420.218i −117.658 + 117.658i −638.899 + 1106.61i 454.475 262.391i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.7.f.a 24
13.f odd 12 1 inner 13.7.f.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.7.f.a 24 1.a even 1 1 trivial
13.7.f.a 24 13.f odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(13, [\chi])\).