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Properties

Label 71.10.d.a
Level $71$
Weight $10$
Character orbit 71.d
Analytic conductor $36.568$
Analytic rank $0$
Dimension $318$
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] = [71,10,Mod(20,71)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(71, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("71.20");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 71.d (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: \(36.5675443676\)
Analytic rank: \(0\)
Dimension: \(318\)
Relative dimension: \(53\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 318 q - 39 q^{2} + 365 q^{3} - 13147 q^{4} + 1580 q^{5} - 1171 q^{6} + 4797 q^{7} - 24583 q^{8} - 282600 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 318 q - 39 q^{2} + 365 q^{3} - 13147 q^{4} + 1580 q^{5} - 1171 q^{6} + 4797 q^{7} - 24583 q^{8} - 282600 q^{9} + 53418 q^{10} + 93595 q^{11} - 264318 q^{12} + 93261 q^{13} - 1012814 q^{14} + 394544 q^{15} - 5390759 q^{16} - 2512272 q^{17} + 1928859 q^{18} + 172239 q^{19} - 321340 q^{20} - 1797959 q^{21} - 662377 q^{22} + 2484777 q^{23} + 2757450 q^{24} + 115099986 q^{25} + 16197322 q^{26} - 5408419 q^{27} + 4226763 q^{28} + 2200387 q^{29} - 25817075 q^{30} + 16026151 q^{31} + 27754513 q^{32} + 20400413 q^{33} - 5425559 q^{34} - 19121996 q^{35} - 112798447 q^{36} + 839243 q^{37} - 17533488 q^{38} - 52072083 q^{39} + 129858690 q^{40} - 10552613 q^{41} - 15263970 q^{42} + 134839057 q^{43} - 85249581 q^{44} - 126033474 q^{45} - 41358830 q^{46} - 175584667 q^{47} - 284955639 q^{48} - 338745896 q^{49} - 320457874 q^{50} - 388384918 q^{51} + 916523873 q^{52} - 49839055 q^{53} - 159905650 q^{54} - 86861648 q^{55} + 80863215 q^{56} + 1134490414 q^{57} + 469329720 q^{58} + 152695965 q^{59} - 2436904680 q^{60} + 93720841 q^{61} - 391776376 q^{62} + 787923301 q^{63} + 209320005 q^{64} + 23115566 q^{65} - 96052564 q^{66} - 430249881 q^{67} - 236548009 q^{68} + 563281657 q^{69} - 672265124 q^{70} + 566361418 q^{71} - 2690414940 q^{72} + 579246267 q^{73} - 1214424359 q^{74} - 238619677 q^{75} - 3551600894 q^{76} + 4813344341 q^{77} + 2777103971 q^{78} - 92545825 q^{79} + 2406742418 q^{80} - 2142018252 q^{81} + 3428427960 q^{82} + 169330275 q^{83} + 2853151198 q^{84} - 1240809976 q^{85} + 1892781751 q^{86} + 4064767249 q^{87} - 1905142488 q^{88} + 2792991379 q^{89} - 4030568949 q^{90} + 419255617 q^{91} + 1709869451 q^{92} + 4590816657 q^{93} - 5253433315 q^{94} + 2394084038 q^{95} + 1399467994 q^{96} + 2405915905 q^{97} + 1821354325 q^{98} + 8253584575 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −9.53258 − 41.7650i 10.8423 − 5.22136i −1192.15 + 574.108i 1339.82 −321.425 − 403.054i 2729.79 − 11960.0i 21666.5 + 27168.9i −12181.9 + 15275.6i −12771.9 − 55957.6i
20.2 −9.26041 − 40.5725i −58.9595 + 28.3934i −1099.08 + 529.288i 298.347 1697.98 + 2129.20i −1636.32 + 7169.18i 18367.6 + 23032.2i −9602.11 + 12040.7i −2762.81 − 12104.7i
20.3 −9.07254 − 39.7494i 215.637 − 103.845i −1036.41 + 499.107i 1388.67 −6084.17 − 7629.31i −1197.66 + 5247.31i 16226.6 + 20347.5i 23443.4 − 29397.1i −12598.8 − 55198.9i
20.4 −9.05154 − 39.6574i −193.935 + 93.3940i −1029.48 + 495.772i −1620.75 5459.17 + 6845.58i 170.492 − 746.974i 15994.2 + 20056.0i 16616.0 − 20835.8i 14670.2 + 64274.5i
20.5 −8.95526 − 39.2356i 130.752 − 62.9667i −997.936 + 480.581i −2122.55 −3641.45 − 4566.23i 706.772 − 3096.57i 14945.5 + 18741.0i 859.053 − 1077.22i 19008.0 + 83279.3i
20.6 −8.34140 − 36.5461i 154.740 − 74.5190i −804.740 + 387.543i −439.915 −4014.13 − 5033.56i −434.079 + 1901.83i 8909.31 + 11171.9i 6119.35 − 7673.42i 3669.51 + 16077.2i
20.7 −8.01298 − 35.1072i −218.285 + 105.120i −707.008 + 340.477i 1948.26 5439.59 + 6821.03i 549.604 − 2407.97i 6123.07 + 7678.09i 24325.8 − 30503.5i −15611.3 − 68397.7i
20.8 −7.04268 − 30.8560i −1.87390 + 0.902422i −441.198 + 212.470i 2492.71 41.0424 + 51.4656i −562.725 + 2465.46i −440.182 − 551.971i −12269.5 + 15385.4i −17555.4 − 76915.2i
20.9 −6.96659 − 30.5226i −84.4198 + 40.6544i −421.800 + 203.128i −646.807 1829.00 + 2293.49i 900.660 − 3946.05i −855.696 − 1073.01i −6798.23 + 8524.71i 4506.04 + 19742.2i
20.10 −6.50612 − 28.5052i 201.921 − 97.2399i −308.920 + 148.768i 2033.50 −4085.56 − 5123.14i 2312.96 − 10133.7i −3083.11 − 3866.09i 19044.3 − 23880.8i −13230.2 − 57965.4i
20.11 −6.38096 − 27.9568i 37.3781 − 18.0003i −279.570 + 134.634i −447.492 −741.740 − 930.113i 935.407 − 4098.29i −3606.22 − 4522.06i −11199.0 + 14043.2i 2855.42 + 12510.4i
20.12 −6.28815 − 27.5502i −17.6259 + 8.48819i −258.176 + 124.331i −2555.71 344.686 + 432.222i −2791.48 + 12230.3i −3972.14 − 4980.90i −12033.5 + 15089.6i 16070.7 + 70410.3i
20.13 −6.19074 − 27.1234i −120.693 + 58.1228i −236.057 + 113.679i −1780.94 2323.67 + 2913.79i 2085.18 − 9135.76i −4336.45 − 5437.73i −1083.54 + 1358.72i 11025.3 + 48305.2i
20.14 −6.12328 − 26.8279i 126.386 − 60.8645i −220.944 + 106.401i 599.216 −2406.76 − 3017.99i −1357.49 + 5947.55i −4577.01 − 5739.39i −3.11736 + 3.90905i −3669.17 − 16075.7i
20.15 −5.20545 − 22.8066i −175.548 + 84.5397i −31.7465 + 15.2883i 521.154 2841.87 + 3563.59i −868.251 + 3804.05i −6953.77 − 8719.75i 11398.1 − 14292.8i −2712.84 − 11885.7i
20.16 −4.27579 − 18.7335i 239.317 − 115.249i 128.636 − 61.9476i −1677.84 −3182.29 − 3990.46i −385.918 + 1690.82i −7844.53 − 9836.73i 31718.3 − 39773.5i 7174.11 + 31431.8i
20.17 −4.12206 − 18.0599i 140.813 − 67.8117i 152.127 − 73.2604i −395.061 −1805.11 − 2263.54i 2164.06 − 9481.38i −7863.62 − 9860.67i 2957.59 − 3708.70i 1628.47 + 7134.78i
20.18 −3.84066 − 16.8270i −186.399 + 89.7649i 192.898 − 92.8948i −301.856 2226.37 + 2791.78i −2471.18 + 10826.9i −7813.78 − 9798.17i 14414.6 − 18075.3i 1159.33 + 5079.34i
20.19 −3.36585 − 14.7467i −6.95901 + 3.35128i 255.159 − 122.878i 1794.39 72.8435 + 91.3428i −430.346 + 1885.47i −7499.49 − 9404.06i −12235.0 + 15342.1i −6039.65 − 26461.5i
20.20 −2.37790 − 10.4182i −231.887 + 111.671i 358.411 − 172.601i −2485.15 1714.81 + 2150.31i 503.090 − 2204.18i −6061.78 − 7601.23i 29028.9 − 36401.1i 5909.44 + 25890.9i
See next 80 embeddings (of 318 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.53
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 71.10.d.a ✓ 318
71.d even 7 1 inner 71.10.d.a ✓ 318
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.10.d.a ✓ 318 1.a even 1 1 trivial
71.10.d.a ✓ 318 71.d even 7 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(71, [\chi])\).