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Properties

Label 49.6.e.a
Level $49$
Weight $6$
Character orbit 49.e
Analytic conductor $7.859$
Analytic rank $0$
Dimension $138$
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] = [49,6,Mod(8,49)]
 
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([12]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.8");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.e (of order \(7\), 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: \(7.85880717084\)
Analytic rank: \(0\)
Dimension: \(138\)
Relative dimension: \(23\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 138 q - 5 q^{2} + 13 q^{3} - 389 q^{4} + 67 q^{5} + 331 q^{6} - 56 q^{7} + 305 q^{8} - 918 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 138 q - 5 q^{2} + 13 q^{3} - 389 q^{4} + 67 q^{5} + 331 q^{6} - 56 q^{7} + 305 q^{8} - 918 q^{9} - 771 q^{10} + 410 q^{11} - 1316 q^{12} + 483 q^{13} + 2597 q^{14} + 2557 q^{15} - 7357 q^{16} - 1422 q^{17} - 2480 q^{18} - 6758 q^{19} - 3647 q^{20} - 399 q^{21} - 857 q^{22} - 2392 q^{23} - 3571 q^{24} - 8066 q^{25} - 343 q^{26} + 16396 q^{27} + 32102 q^{28} + 30341 q^{29} + 13716 q^{30} - 32712 q^{31} - 26993 q^{32} - 16775 q^{33} + 35163 q^{34} - 12019 q^{35} + 35461 q^{36} - 34872 q^{37} + 13383 q^{38} - 20671 q^{39} + 165169 q^{40} + 51233 q^{41} - 13741 q^{42} - 17203 q^{43} - 4267 q^{44} + 30777 q^{45} - 74611 q^{46} + 64264 q^{47} - 247084 q^{48} - 70350 q^{49} - 263036 q^{50} - 57453 q^{51} - 36043 q^{52} + 124590 q^{53} + 144840 q^{54} + 90977 q^{55} + 404404 q^{56} - 103899 q^{57} + 24861 q^{58} - 8079 q^{59} + 72065 q^{60} - 236659 q^{61} - 173227 q^{62} - 119805 q^{63} + 105381 q^{64} - 33383 q^{65} + 536577 q^{66} + 19716 q^{67} - 256340 q^{68} - 202795 q^{69} + 194803 q^{70} + 61778 q^{71} - 392108 q^{72} - 157760 q^{73} - 328319 q^{74} + 355167 q^{75} - 61334 q^{76} - 24402 q^{77} + 497217 q^{78} - 73876 q^{79} - 1340760 q^{80} - 815116 q^{81} + 248976 q^{82} + 303107 q^{83} + 1250354 q^{84} + 205487 q^{85} + 1083219 q^{86} + 197027 q^{87} + 1052915 q^{88} + 219783 q^{89} + 82170 q^{90} - 837235 q^{91} + 35936 q^{92} + 138365 q^{93} - 571908 q^{94} - 610016 q^{95} - 1054494 q^{96} + 762020 q^{97} - 2160767 q^{98} - 1090772 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −6.65295 − 8.34253i −18.5736 + 8.94459i −18.2155 + 79.8071i 81.4863 − 39.2418i 198.190 + 95.4433i −120.860 + 46.9014i 479.339 − 230.837i 113.466 − 142.282i −869.500 − 418.729i
8.2 −6.59749 − 8.27300i 25.4427 − 12.2525i −17.7949 + 77.9644i 45.4620 − 21.8934i −269.223 − 129.651i 124.236 − 37.0458i 457.324 − 220.236i 345.696 − 433.489i −481.059 − 231.666i
8.3 −6.16552 − 7.73132i −10.0578 + 4.84359i −14.6390 + 64.1376i −72.0665 + 34.7054i 99.4591 + 47.8970i 83.2368 − 99.3913i 301.024 − 144.965i −73.8086 + 92.5531i 712.646 + 343.192i
8.4 −4.98029 − 6.24508i 6.83688 − 3.29247i −7.07712 + 31.0069i 10.7661 − 5.18467i −54.6114 − 26.2994i −98.9047 − 83.8145i −1.40854 + 0.678317i −115.605 + 144.965i −85.9969 − 41.4139i
8.5 −4.94829 − 6.20496i 4.98390 − 2.40012i −6.89529 + 30.2102i −5.69000 + 2.74016i −39.5545 − 19.0484i 19.6454 + 128.145i −7.24231 + 3.48771i −132.429 + 166.061i 45.1584 + 21.7471i
8.6 −3.57517 − 4.48312i −26.8141 + 12.9130i −0.195868 + 0.858155i −65.6815 + 31.6305i 153.756 + 74.0448i −61.0537 + 114.365i −160.773 + 77.4243i 400.744 − 502.517i 376.626 + 181.373i
8.7 −3.42301 − 4.29231i 25.2791 − 12.1737i 0.413685 − 1.81247i −87.2410 + 42.0130i −138.784 − 66.8348i −126.240 − 29.5041i −167.480 + 80.6542i 339.322 − 425.497i 478.959 + 230.655i
8.8 −3.16704 − 3.97134i −14.8323 + 7.14284i 1.37924 − 6.04286i 68.5349 − 33.0046i 75.3410 + 36.2823i 129.221 − 10.4405i −174.815 + 84.1863i 17.4677 − 21.9038i −348.126 − 167.648i
8.9 −2.27658 − 2.85474i −14.5247 + 6.99475i 4.15395 − 18.1996i 0.00127143 0.000612289i 53.0349 + 25.5402i −61.2225 − 114.275i −166.684 + 80.2708i 10.5335 − 13.2086i −0.00464244 − 0.00223568i
8.10 −1.52548 − 1.91289i 14.5313 − 6.99791i 5.78861 − 25.3616i 13.8983 − 6.69309i −35.5534 − 17.1216i 106.641 − 73.7197i −127.884 + 61.5859i 10.6801 − 13.3924i −34.0047 − 16.3758i
8.11 −1.16235 − 1.45754i 17.4856 − 8.42061i 6.34730 − 27.8093i 97.8327 − 47.1137i −32.5978 − 15.6983i −91.6882 + 91.6530i −101.660 + 48.9568i 83.3305 − 104.493i −182.386 − 87.8326i
8.12 −0.469190 − 0.588346i 2.32193 − 1.11818i 6.99466 − 30.6456i −89.5344 + 43.1175i −1.74731 − 0.841459i 102.032 + 79.9786i −43.0080 + 20.7116i −147.367 + 184.792i 67.3767 + 32.4469i
8.13 0.700150 + 0.877961i −3.63552 + 1.75077i 6.84007 − 29.9683i −6.35053 + 3.05825i −4.08252 − 1.96604i −127.348 + 24.2804i 63.4759 − 30.5684i −141.356 + 177.255i −7.13135 − 3.43428i
8.14 1.63882 + 2.05502i −16.7655 + 8.07386i 5.58330 − 24.4621i 39.0863 − 18.8230i −44.0677 − 21.2219i 49.2203 + 119.935i 135.202 − 65.1096i 64.3879 − 80.7399i 102.737 + 49.4756i
8.15 2.68695 + 3.36932i 16.1005 − 7.75358i 2.98801 − 13.0913i −9.63295 + 4.63899i 69.3854 + 33.4142i 31.3108 − 125.804i 176.385 − 84.9428i 47.5990 − 59.6873i −41.5135 − 19.9918i
8.16 2.96579 + 3.71898i −21.3031 + 10.2591i 2.08576 − 9.13830i −47.1674 + 22.7146i −101.334 − 48.7998i 76.1422 − 104.925i 177.313 − 85.3894i 197.068 − 247.115i −224.364 − 108.048i
8.17 3.11301 + 3.90359i 24.6500 − 11.8708i 1.57347 − 6.89382i −13.5648 + 6.53245i 123.075 + 59.2697i −7.10586 + 129.447i 175.759 − 84.6410i 315.200 − 395.248i −67.7273 − 32.6157i
8.18 3.97480 + 4.98424i −2.41848 + 1.16468i −1.92294 + 8.42495i 98.2899 − 47.3339i −15.4180 − 7.42491i −15.6636 − 128.692i 134.165 − 64.6103i −147.015 + 184.352i 626.606 + 301.757i
8.19 4.98133 + 6.24639i 2.75495 − 1.32671i −7.08306 + 31.0329i −79.8726 + 38.4646i 22.0105 + 10.5997i −129.023 − 12.6496i 1.21670 − 0.585929i −145.678 + 182.675i −638.137 − 307.310i
8.20 4.98712 + 6.25365i 7.50846 − 3.61589i −7.11609 + 31.1776i 29.9404 − 14.4185i 60.0580 + 28.9224i 75.7713 + 105.194i 0.148300 − 0.0714174i −108.206 + 135.686i 239.485 + 115.330i
See next 80 embeddings (of 138 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.23
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.6.e.a ✓ 138
49.e even 7 1 inner 49.6.e.a ✓ 138
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.6.e.a ✓ 138 1.a even 1 1 trivial
49.6.e.a ✓ 138 49.e even 7 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(49, [\chi])\).