Properties

Label 49.5.f.a
Level $49$
Weight $5$
Character orbit 49.f
Analytic conductor $5.065$
Analytic rank $0$
Dimension $102$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [49,5,Mod(6,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([9]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.6");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 49.f (of order \(14\), degree \(6\), 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: \(5.06512819111\)
Analytic rank: \(0\)
Dimension: \(102\)
Relative dimension: \(17\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 102 q - 5 q^{2} - 7 q^{3} - 117 q^{4} - 7 q^{5} + 217 q^{6} - 56 q^{7} + 117 q^{8} + 266 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 102 q - 5 q^{2} - 7 q^{3} - 117 q^{4} - 7 q^{5} + 217 q^{6} - 56 q^{7} + 117 q^{8} + 266 q^{9} - 7 q^{10} + 64 q^{11} + 798 q^{12} - 7 q^{13} - 2051 q^{14} - 353 q^{15} - 1037 q^{16} + 728 q^{17} - 180 q^{18} - 2023 q^{20} - 7 q^{21} + 1951 q^{22} + 2620 q^{23} + 4907 q^{24} + 1666 q^{25} - 1351 q^{26} - 3052 q^{27} - 5782 q^{28} + 3991 q^{29} - 128 q^{30} - 6541 q^{32} - 7 q^{33} + 8757 q^{34} + 4193 q^{35} + 197 q^{36} + 646 q^{37} + 3465 q^{38} - 8421 q^{39} - 17535 q^{40} - 3241 q^{41} + 12313 q^{42} + 2465 q^{43} - 5095 q^{44} - 15043 q^{45} - 8467 q^{46} - 9310 q^{47} + 308 q^{49} - 10380 q^{50} + 10799 q^{51} + 31577 q^{52} + 4516 q^{53} + 17570 q^{54} + 21329 q^{55} + 37170 q^{56} - 12155 q^{57} - 18147 q^{58} - 15757 q^{59} + 33621 q^{60} + 17857 q^{61} - 14007 q^{62} - 20419 q^{63} + 25381 q^{64} - 3871 q^{65} - 65415 q^{66} - 3100 q^{67} - 22519 q^{69} - 11361 q^{70} - 14216 q^{71} + 43614 q^{72} + 8624 q^{73} + 22389 q^{74} + 30737 q^{75} - 64890 q^{76} - 3458 q^{77} + 22869 q^{78} + 7156 q^{79} - 6488 q^{81} + 36568 q^{82} + 64043 q^{83} - 56126 q^{84} - 52657 q^{85} - 6849 q^{86} - 3647 q^{87} - 48491 q^{88} - 69181 q^{89} - 75838 q^{90} + 56161 q^{91} + 34526 q^{92} - 93991 q^{93} - 266 q^{94} - 22086 q^{95} + 24878 q^{96} + 18025 q^{98} - 7496 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −4.72332 + 5.92286i 3.57449 7.42250i −9.21016 40.3523i −7.63324 + 15.8506i 27.0790 + 56.2301i 48.8460 3.88170i 173.297 + 83.4557i 8.18611 + 10.2651i −57.8265 120.078i
6.2 −4.23981 + 5.31655i −3.45640 + 7.17729i −6.72942 29.4835i 10.4167 21.6306i −23.5040 48.8065i −33.0265 36.1974i 87.2550 + 42.0198i 10.9359 + 13.7132i 70.8352 + 147.091i
6.3 −3.61910 + 4.53821i −4.63704 + 9.62891i −3.93710 17.2496i −13.6463 + 28.3369i −26.9161 55.8918i −0.846585 + 48.9927i 8.85487 + 4.26428i −20.7111 25.9709i −79.2113 164.484i
6.4 −3.02455 + 3.79266i 2.80761 5.83006i −1.67606 7.34331i −2.45023 + 5.08795i 13.6197 + 28.2816i −31.7325 + 37.3369i −37.0095 17.8228i 24.3958 + 30.5913i −11.8860 24.6816i
6.5 −2.51536 + 3.15417i 6.48273 13.4615i −0.0613763 0.268907i 0.0863095 0.179223i 26.1535 + 54.3083i −13.2950 47.1619i −57.1543 27.5241i −88.6843 111.207i 0.348201 + 0.723047i
6.6 −2.24615 + 2.81658i 0.457590 0.950195i 0.672393 + 2.94595i 17.5653 36.4748i 1.64849 + 3.42312i 45.9934 + 16.9000i −61.7402 29.7325i 49.8092 + 62.4588i 63.2798 + 131.402i
6.7 −1.53378 + 1.92330i −4.33690 + 9.00566i 2.21373 + 9.69900i −8.18548 + 16.9973i −10.6688 22.1539i 24.7221 42.3063i −57.5115 27.6961i −11.7906 14.7850i −20.1362 41.8133i
6.8 −0.190677 + 0.239101i −7.10131 + 14.7460i 3.53952 + 15.5077i 11.9420 24.7979i −2.17173 4.50965i −32.6258 + 36.5590i −8.79138 4.23370i −116.514 146.104i 3.65213 + 7.58373i
6.9 0.101848 0.127714i 0.716533 1.48790i 3.55440 + 15.5728i −5.00109 + 10.3849i −0.117047 0.243050i −48.8363 + 4.00153i 4.70567 + 2.26613i 48.8023 + 61.1961i 0.816937 + 1.69639i
6.10 0.238140 0.298618i 3.88212 8.06131i 3.52787 + 15.4566i −18.4576 + 38.3276i −1.48276 3.07899i 46.2093 + 16.3002i 10.9617 + 5.27888i 0.588851 + 0.738396i 7.04983 + 14.6391i
6.11 1.25074 1.56837i 7.29580 15.1499i 2.66488 + 11.6756i 13.2800 27.5763i −14.6355 30.3910i 1.44197 + 48.9788i 50.5626 + 24.3497i −125.788 157.733i −26.6401 55.3187i
6.12 1.54183 1.93339i 0.771738 1.60253i 2.19957 + 9.63692i 14.3069 29.7086i −1.90843 3.96290i −14.7592 46.7244i 57.6714 + 27.7731i 48.5302 + 60.8549i −35.3796 73.4666i
6.13 2.38635 2.99238i −3.28786 + 6.82732i 0.300624 + 1.31712i 0.646610 1.34270i 12.5840 + 26.1309i 43.0574 + 23.3893i 59.8326 + 28.8139i 14.7005 + 18.4338i −2.47484 5.13905i
6.14 3.33676 4.18417i −4.73280 + 9.82775i −2.81295 12.3243i −18.3794 + 38.1652i 25.3288 + 52.5957i −48.9209 2.78244i 16.1949 + 7.79907i −23.6827 29.6972i 98.3620 + 204.251i
6.15 3.56270 4.46749i 4.66338 9.68361i −3.70526 16.2338i −6.40559 + 13.3014i −26.6472 55.3334i 12.2647 47.4402i −3.35304 1.61474i −21.5224 26.9883i 36.6024 + 76.0057i
6.16 4.29229 5.38236i 2.34281 4.86490i −6.98570 30.6064i 3.49106 7.24926i −16.1286 33.4914i −15.3426 + 46.5360i −95.4785 45.9800i 32.3242 + 40.5332i −24.0335 49.9061i
6.17 4.60462 5.77401i −5.81900 + 12.0833i −8.57632 37.5753i 15.6615 32.5215i 42.9747 + 89.2379i 8.91373 48.1824i −149.989 72.2309i −61.6423 77.2970i −115.664 240.179i
13.1 −1.74700 7.65410i −7.57880 + 6.04389i −41.1178 + 19.8013i 5.79433 4.62082i 59.5007 + 47.4502i 15.8681 46.3595i 145.074 + 181.917i 2.88536 12.6416i −45.4909 36.2778i
13.2 −1.44765 6.34256i 9.73419 7.76276i −23.7169 + 11.4215i −17.9952 + 14.3507i −63.3275 50.5020i −38.0892 30.8256i 41.8754 + 52.5101i 16.4699 72.1592i 117.071 + 93.3611i
13.3 −1.26018 5.52122i −2.76841 + 2.20774i −14.4803 + 6.97334i −15.1005 + 12.0422i 15.6781 + 12.5029i 3.85271 + 48.8483i 0.253896 + 0.318375i −15.2342 + 66.7453i 85.5170 + 68.1975i
See next 80 embeddings (of 102 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.f odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.5.f.a 102
49.f odd 14 1 inner 49.5.f.a 102
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.5.f.a 102 1.a even 1 1 trivial
49.5.f.a 102 49.f odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(49, [\chi])\).