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Properties

Label 71.6.a.b
Level $71$
Weight $6$
Character orbit 71.a
Self dual yes
Analytic conductor $11.387$
Analytic rank $0$
Dimension $18$
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] = [71,6,Mod(1,71)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(71, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("71.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 71.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: \(11.3872512067\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 4 x^{17} - 453 x^{16} + 1684 x^{15} + 83700 x^{14} - 280456 x^{13} - 8137672 x^{12} + \cdots + 1095970392576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 1) q^{3} + (\beta_{2} + 19) q^{4} + (\beta_{9} + \beta_{2} + \beta_1 + 7) q^{5} + (\beta_{4} + \beta_{3} + \beta_{2} + \cdots + 7) q^{6}+ \cdots + (\beta_{17} - \beta_{16} - \beta_{15} + \cdots + 97) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + 1) q^{3} + (\beta_{2} + 19) q^{4} + (\beta_{9} + \beta_{2} + \beta_1 + 7) q^{5} + (\beta_{4} + \beta_{3} + \beta_{2} + \cdots + 7) q^{6}+ \cdots + ( - 188 \beta_{17} - 479 \beta_{16} + \cdots + 19788) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 4 q^{2} + 16 q^{3} + 346 q^{4} + 128 q^{5} + 126 q^{6} + 244 q^{7} + 192 q^{8} + 1802 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 4 q^{2} + 16 q^{3} + 346 q^{4} + 128 q^{5} + 126 q^{6} + 244 q^{7} + 192 q^{8} + 1802 q^{9} + 750 q^{10} + 854 q^{11} + 768 q^{12} + 1258 q^{13} + 304 q^{14} + 1152 q^{15} + 7842 q^{16} + 2592 q^{17} + 8567 q^{18} + 8712 q^{19} + 19723 q^{20} + 12472 q^{21} + 13072 q^{22} + 5304 q^{23} + 20933 q^{24} + 21034 q^{25} + 9734 q^{26} + 8752 q^{27} + 11600 q^{28} + 1804 q^{29} + 1789 q^{30} + 12940 q^{31} - 15592 q^{32} + 752 q^{33} - 2818 q^{34} - 4380 q^{35} - 24948 q^{36} + 29304 q^{37} - 36799 q^{38} - 12476 q^{39} - 36874 q^{40} + 18528 q^{41} - 85136 q^{42} - 10336 q^{43} - 39522 q^{44} - 28044 q^{45} - 4870 q^{46} - 28520 q^{47} - 129189 q^{48} + 71878 q^{49} - 130768 q^{50} + 17276 q^{51} - 17358 q^{52} - 31226 q^{53} - 212998 q^{54} + 29764 q^{55} - 165366 q^{56} - 15940 q^{57} - 80234 q^{58} - 12794 q^{59} - 186813 q^{60} + 105870 q^{61} - 171004 q^{62} - 93576 q^{63} + 45970 q^{64} - 59100 q^{65} - 229136 q^{66} + 32514 q^{67} - 264178 q^{68} + 58968 q^{69} - 246938 q^{70} + 90738 q^{71} + 42848 q^{72} + 178648 q^{73} - 122089 q^{74} + 220892 q^{75} + 377676 q^{76} + 371028 q^{77} - 277210 q^{78} + 199868 q^{79} + 307806 q^{80} + 540714 q^{81} + 320684 q^{82} + 260564 q^{83} + 73436 q^{84} + 106112 q^{85} - 157186 q^{86} + 446884 q^{87} + 13240 q^{88} + 239728 q^{89} + 238645 q^{90} + 578488 q^{91} - 22950 q^{92} + 159264 q^{93} + 309116 q^{94} + 427992 q^{95} + 102774 q^{96} + 459920 q^{97} - 532804 q^{98} + 364890 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 4 x^{17} - 453 x^{16} + 1684 x^{15} + 83700 x^{14} - 280456 x^{13} - 8137672 x^{12} + \cdots + 1095970392576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 16\!\cdots\!93 \nu^{17} + \cdots + 17\!\cdots\!56 ) / 23\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 31\!\cdots\!53 \nu^{17} + \cdots - 71\!\cdots\!04 ) / 23\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12\!\cdots\!13 \nu^{17} + \cdots - 21\!\cdots\!56 ) / 78\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20\!\cdots\!51 \nu^{17} + \cdots + 95\!\cdots\!96 ) / 59\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21\!\cdots\!11 \nu^{17} + \cdots - 18\!\cdots\!28 ) / 59\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!57 \nu^{17} + \cdots + 50\!\cdots\!84 ) / 23\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 23\!\cdots\!13 \nu^{17} + \cdots + 30\!\cdots\!16 ) / 39\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 26\!\cdots\!53 \nu^{17} + \cdots + 32\!\cdots\!56 ) / 39\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 64\!\cdots\!87 \nu^{17} + \cdots - 77\!\cdots\!56 ) / 59\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!13 \nu^{17} + \cdots + 37\!\cdots\!24 ) / 11\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14\!\cdots\!03 \nu^{17} + \cdots + 96\!\cdots\!20 ) / 78\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 47\!\cdots\!33 \nu^{17} + \cdots + 17\!\cdots\!52 ) / 23\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 10\!\cdots\!01 \nu^{17} + \cdots + 38\!\cdots\!48 ) / 39\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 22\!\cdots\!85 \nu^{17} + \cdots - 19\!\cdots\!80 ) / 63\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 98\!\cdots\!99 \nu^{17} + \cdots + 19\!\cdots\!68 ) / 23\!\cdots\!36 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 51 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + \beta_{16} - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} - \beta_{10} - \beta_{9} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{16} + 5 \beta_{15} + \beta_{14} - 7 \beta_{13} + 5 \beta_{11} - 4 \beta_{10} + 3 \beta_{9} + \cdots + 4307 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 127 \beta_{17} + 139 \beta_{16} - 138 \beta_{15} - 160 \beta_{14} + 161 \beta_{13} + 18 \beta_{12} + \cdots - 145 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 60 \beta_{17} - 51 \beta_{16} + 847 \beta_{15} + 292 \beta_{14} - 1122 \beta_{13} - 73 \beta_{12} + \cdots + 411683 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14831 \beta_{17} + 15837 \beta_{16} - 17072 \beta_{15} - 20125 \beta_{14} + 19422 \beta_{13} + \cdots - 159338 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 6361 \beta_{17} + 1608 \beta_{16} + 118443 \beta_{15} + 52827 \beta_{14} - 146742 \beta_{13} + \cdots + 41598632 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1737132 \beta_{17} + 1716054 \beta_{16} - 2092572 \beta_{15} - 2360210 \beta_{14} + 2188663 \beta_{13} + \cdots - 34635376 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 26471 \beta_{17} + 681023 \beta_{16} + 15710611 \beta_{15} + 8073093 \beta_{14} - 18162558 \beta_{13} + \cdots + 4343658685 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 206221558 \beta_{17} + 184635829 \beta_{16} - 257236751 \beta_{15} - 271340217 \beta_{14} + \cdots - 5723955528 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 141147017 \beta_{17} + 95288761 \beta_{16} + 2040014396 \beta_{15} + 1139702244 \beta_{14} + \cdots + 464121173536 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 24772431810 \beta_{17} + 20031086151 \beta_{16} - 31700162301 \beta_{15} - 31133234156 \beta_{14} + \cdots - 842534839204 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 35716679929 \beta_{17} + 9349885255 \beta_{16} + 261768334218 \beta_{15} + 153904363801 \beta_{14} + \cdots + 50487479401277 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2999898267753 \beta_{17} + 2202286536200 \beta_{16} - 3908946012449 \beta_{15} - 3588294639363 \beta_{14} + \cdots - 116603895260361 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 6604469410262 \beta_{17} + 595362830822 \beta_{16} + 33305110253486 \beta_{15} + \cdots + 55\!\cdots\!75 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 364955245909749 \beta_{17} + 245524500322873 \beta_{16} - 481625922844585 \beta_{15} + \cdots - 15\!\cdots\!30 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−10.9970
−10.0791
−9.20558
−7.05077
−6.47643
−4.27080
−4.12518
−2.20292
0.380427
0.603680
2.11263
3.40360
5.61750
7.20268
9.35702
9.41870
9.69208
10.6195
−10.9970 −4.66494 88.9340 88.1527 51.3003 218.123 −626.103 −221.238 −969.415
1.2 −10.0791 −18.6697 69.5887 19.0730 188.174 −242.873 −378.862 105.556 −192.240
1.3 −9.20558 2.25110 52.7426 −72.9411 −20.7227 141.786 −190.948 −237.933 671.465
1.4 −7.05077 26.7973 17.7134 68.8957 −188.942 125.874 100.731 475.094 −485.768
1.5 −6.47643 −0.000816341 0 9.94414 92.7550 0.00528697 −110.372 142.843 −243.000 −600.721
1.6 −4.27080 27.7302 −13.7602 −106.460 −118.430 63.2068 195.433 525.963 454.670
1.7 −4.12518 −3.85735 −14.9829 −58.9919 15.9123 −130.388 193.813 −228.121 243.352
1.8 −2.20292 −14.4527 −27.1471 −87.2333 31.8381 −108.923 130.297 −34.1208 192.168
1.9 0.380427 19.4431 −31.8553 31.5209 7.39668 75.0968 −24.2923 135.033 11.9914
1.10 0.603680 −10.4088 −31.6356 24.6673 −6.28361 19.8097 −38.4155 −134.656 14.8911
1.11 2.11263 −30.6752 −27.5368 −10.4067 −64.8055 −170.804 −125.779 697.968 −21.9855
1.12 3.40360 −20.1757 −20.4155 −73.4452 −68.6698 232.957 −178.401 164.058 −249.978
1.13 5.61750 14.3304 −0.443638 93.9728 80.5011 121.192 −182.252 −37.6395 527.893
1.14 7.20268 29.9731 19.8786 11.2373 215.887 −78.5152 −87.3063 655.385 80.9387
1.15 9.35702 −27.1298 55.5538 69.5231 −253.854 −11.7298 220.394 493.027 650.529
1.16 9.41870 −1.97104 56.7119 18.7949 −18.5647 169.933 232.754 −239.115 177.023
1.17 9.69208 17.8720 61.9364 −48.1274 173.217 114.866 290.146 76.4078 −466.455
1.18 10.6195 9.60889 80.7734 67.0129 102.041 −185.239 517.948 −150.669 711.642
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(71\) \(-1\)

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 71.6.a.b ✓ 18
3.b odd 2 1 639.6.a.f 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.6.a.b ✓ 18 1.a even 1 1 trivial
639.6.a.f 18 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 4 T_{2}^{17} - 453 T_{2}^{16} + 1684 T_{2}^{15} + 83700 T_{2}^{14} - 280456 T_{2}^{13} + \cdots + 1095970392576 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(71))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + \cdots + 1095970392576 \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 32\!\cdots\!84 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots - 11\!\cdots\!80 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 22\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 19\!\cdots\!48 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots - 11\!\cdots\!48 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots - 41\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots - 81\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 12\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 49\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots - 48\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 10\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 69\!\cdots\!20 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 39\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 60\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots - 33\!\cdots\!60 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots - 37\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( (T - 5041)^{18} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 53\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 12\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 19\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 43\!\cdots\!00 \) Copy content Toggle raw display
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