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Properties

Label 83.6.a.b
Level $83$
Weight $6$
Character orbit 83.a
Self dual yes
Analytic conductor $13.312$
Analytic rank $0$
Dimension $20$
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] = [83,6,Mod(1,83)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(83, 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("83.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 83 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 83.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: \(13.3118570445\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 7 x^{19} - 482 x^{18} + 3222 x^{17} + 96413 x^{16} - 611573 x^{15} - 10357312 x^{14} + \cdots - 7882835431424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{5} + 2) q^{3} + (\beta_{2} + 19) q^{4} + ( - \beta_{4} + 4) q^{5} + (\beta_{8} + \beta_{5} + \beta_{2} + \cdots + 6) q^{6}+ \cdots + ( - \beta_{18} - \beta_{13} + \cdots + 106) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{5} + 2) q^{3} + (\beta_{2} + 19) q^{4} + ( - \beta_{4} + 4) q^{5} + (\beta_{8} + \beta_{5} + \beta_{2} + \cdots + 6) q^{6}+ \cdots + ( - 243 \beta_{19} - 500 \beta_{18} + \cdots + 29585) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 7 q^{2} + 37 q^{3} + 373 q^{4} + 88 q^{5} + 129 q^{6} + 439 q^{7} + 351 q^{8} + 2137 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 7 q^{2} + 37 q^{3} + 373 q^{4} + 88 q^{5} + 129 q^{6} + 439 q^{7} + 351 q^{8} + 2137 q^{9} + 481 q^{10} + 313 q^{11} + 1399 q^{12} + 2804 q^{13} + 1610 q^{15} + 8885 q^{16} + 2381 q^{17} + 3568 q^{18} + 2608 q^{19} + 1647 q^{20} + 5881 q^{21} + 18703 q^{22} + 14802 q^{23} + 34687 q^{24} + 21858 q^{25} + 20301 q^{26} + 17005 q^{27} + 27472 q^{28} + 19515 q^{29} + 22256 q^{30} + 13135 q^{31} + 23009 q^{32} + 17539 q^{33} - 8340 q^{34} - 6310 q^{35} + 20812 q^{36} + 45999 q^{37} - 587 q^{38} - 13794 q^{39} - 18769 q^{40} - 4772 q^{41} - 87449 q^{42} + 16018 q^{43} - 87813 q^{44} + 74 q^{45} + 8347 q^{46} - 9788 q^{47} - 40281 q^{48} + 66263 q^{49} - 127676 q^{50} - 26081 q^{51} - 5633 q^{52} + 21376 q^{53} - 248923 q^{54} - 30582 q^{55} - 169014 q^{56} - 24834 q^{57} - 39633 q^{58} + 40943 q^{59} - 225482 q^{60} + 27191 q^{61} - 213850 q^{62} - 40384 q^{63} + 59353 q^{64} - 27308 q^{65} - 287573 q^{66} + 64688 q^{67} - 270638 q^{68} - 63560 q^{69} - 109632 q^{70} + 63938 q^{71} - 217532 q^{72} + 65234 q^{73} - 428799 q^{74} - 18321 q^{75} - 158945 q^{76} - 183287 q^{77} - 237334 q^{78} - 64074 q^{79} - 201521 q^{80} + 165012 q^{81} - 129233 q^{82} + 137780 q^{83} - 512695 q^{84} + 490786 q^{85} + 2881 q^{86} + 479693 q^{87} + 493355 q^{88} - 88378 q^{89} - 123605 q^{90} + 648104 q^{91} + 385779 q^{92} + 490497 q^{93} - 87006 q^{94} + 225060 q^{95} + 800961 q^{96} + 312918 q^{97} + 565425 q^{98} + 617014 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 7 x^{19} - 482 x^{18} + 3222 x^{17} + 96413 x^{16} - 611573 x^{15} - 10357312 x^{14} + \cdots - 7882835431424 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 51 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 52\!\cdots\!33 \nu^{19} + \cdots - 23\!\cdots\!76 ) / 28\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 26\!\cdots\!11 \nu^{19} + \cdots - 73\!\cdots\!68 ) / 95\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 68\!\cdots\!81 \nu^{19} + \cdots - 79\!\cdots\!48 ) / 14\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23\!\cdots\!49 \nu^{19} + \cdots + 28\!\cdots\!44 ) / 17\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 45\!\cdots\!05 \nu^{19} + \cdots - 16\!\cdots\!56 ) / 28\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 31\!\cdots\!33 \nu^{19} + \cdots + 18\!\cdots\!64 ) / 14\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 71\!\cdots\!31 \nu^{19} + \cdots - 83\!\cdots\!52 ) / 28\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 36\!\cdots\!09 \nu^{19} + \cdots + 41\!\cdots\!36 ) / 14\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 76\!\cdots\!57 \nu^{19} + \cdots - 13\!\cdots\!96 ) / 28\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10\!\cdots\!61 \nu^{19} + \cdots - 12\!\cdots\!32 ) / 28\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10\!\cdots\!73 \nu^{19} + \cdots + 43\!\cdots\!36 ) / 14\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 20\!\cdots\!13 \nu^{19} + \cdots + 12\!\cdots\!28 ) / 28\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 23\!\cdots\!13 \nu^{19} + \cdots - 13\!\cdots\!84 ) / 28\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 13\!\cdots\!87 \nu^{19} + \cdots + 88\!\cdots\!24 ) / 14\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 28\!\cdots\!69 \nu^{19} + \cdots - 64\!\cdots\!24 ) / 28\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 18\!\cdots\!03 \nu^{19} + \cdots + 75\!\cdots\!60 ) / 14\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 39\!\cdots\!53 \nu^{19} + \cdots + 19\!\cdots\!96 ) / 28\!\cdots\!56 \) 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
\(\nu^{3}\)\(=\) \( -\beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - \beta_{8} - \beta_{6} + 3\beta_{5} + \beta_{3} + \beta_{2} + 84\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{19} + 2 \beta_{17} - \beta_{16} + \beta_{15} + 5 \beta_{14} + 3 \beta_{13} + 3 \beta_{11} + \cdots + 4318 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 34 \beta_{19} - 2 \beta_{18} - 14 \beta_{17} + \beta_{16} + 17 \beta_{15} - 150 \beta_{14} + 126 \beta_{13} + \cdots + 2629 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 344 \beta_{19} - 112 \beta_{18} + 366 \beta_{17} - 202 \beta_{16} + 148 \beta_{15} + 940 \beta_{14} + \cdots + 413565 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7122 \beta_{19} - 970 \beta_{18} - 3024 \beta_{17} + 900 \beta_{16} + 3322 \beta_{15} - 19687 \beta_{14} + \cdots + 356084 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 47450 \beta_{19} - 27816 \beta_{18} + 50942 \beta_{17} - 29683 \beta_{16} + 18155 \beta_{15} + \cdots + 42482746 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1106214 \beta_{19} - 212530 \beta_{18} - 487798 \beta_{17} + 209725 \beta_{16} + 492133 \beta_{15} + \cdots + 41465379 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 6141628 \beta_{19} - 4813956 \beta_{18} + 6462070 \beta_{17} - 3905088 \beta_{16} + 2094802 \beta_{15} + \cdots + 4582294709 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 154059518 \beta_{19} - 35951478 \beta_{18} - 70190236 \beta_{17} + 36030852 \beta_{16} + \cdots + 4360925858 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 777707314 \beta_{19} - 719119612 \beta_{18} + 792608794 \beta_{17} - 489695853 \beta_{16} + \cdots + 511615778778 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 20364090830 \beta_{19} - 5352846054 \beta_{18} - 9517955406 \beta_{17} + 5416092357 \beta_{16} + \cdots + 412830503729 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 97834545328 \beta_{19} - 99407943984 \beta_{18} + 96334800358 \beta_{17} - 60168279074 \beta_{16} + \cdots + 58507001662177 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2615861606946 \beta_{19} - 739554382346 \beta_{18} - 1246380463872 \beta_{17} + 757647805000 \beta_{16} + \cdots + 33457469768836 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 12304475020098 \beta_{19} - 13118254698352 \beta_{18} + 11726125328038 \beta_{17} + \cdots + 68\!\cdots\!62 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 330365737133142 \beta_{19} - 97377089415834 \beta_{18} - 159673710681102 \beta_{17} + \cdots + 18\!\cdots\!91 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 15\!\cdots\!68 \beta_{19} + \cdots + 79\!\cdots\!93 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 41\!\cdots\!26 \beta_{19} + \cdots - 45\!\cdots\!26 \) 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
−11.1404
−8.89409
−8.70507
−8.33679
−8.28078
−5.43020
−3.99687
−3.32870
−0.174138
0.785291
1.31467
2.02651
2.21749
4.18473
7.72261
7.85145
7.96539
9.64299
10.7613
10.8147
−11.1404 −18.7873 92.1092 15.4532 209.299 140.462 −669.643 109.964 −172.156
1.2 −8.89409 27.5180 47.1049 −65.6247 −244.748 200.127 −134.344 514.241 583.672
1.3 −8.70507 −5.26468 43.7782 −95.8353 45.8294 −53.7456 −102.530 −215.283 834.253
1.4 −8.33679 3.89204 37.5020 69.7041 −32.4471 167.155 −45.8693 −227.852 −581.108
1.5 −8.28078 −15.4255 36.5714 89.0687 127.735 −154.526 −37.8546 −5.05530 −737.559
1.6 −5.43020 29.4882 −2.51296 68.7972 −160.127 −38.8283 187.412 626.553 −373.582
1.7 −3.99687 15.8248 −16.0251 −94.0136 −63.2496 −85.4591 191.950 7.42412 375.760
1.8 −3.32870 −6.83509 −20.9197 −7.83055 22.7520 −154.436 176.154 −196.282 26.0656
1.9 −0.174138 12.2955 −31.9697 106.874 −2.14112 130.197 11.1396 −91.8201 −18.6108
1.10 0.785291 −12.8980 −31.3833 −70.4730 −10.1287 91.0057 −49.7743 −76.6417 −55.3418
1.11 1.31467 −22.4625 −30.2717 −72.9331 −29.5307 −127.699 −81.8664 261.565 −95.8826
1.12 2.02651 −2.42555 −27.8933 16.9209 −4.91539 133.064 −121.374 −237.117 34.2903
1.13 2.21749 26.0548 −27.0827 15.9428 57.7764 42.0115 −131.015 435.855 35.3530
1.14 4.18473 −27.6750 −14.4881 67.6710 −115.812 −156.986 −194.540 522.903 283.184
1.15 7.72261 22.5661 27.6387 −12.0780 174.269 116.539 −33.6807 266.230 −93.2735
1.16 7.85145 −26.6393 29.6453 −51.0117 −209.157 205.205 −18.4879 466.654 −400.516
1.17 7.96539 18.6234 31.4475 81.3109 148.343 10.0531 −4.40090 103.831 647.673
1.18 9.64299 −7.99410 60.9873 80.3213 −77.0871 −4.21826 279.525 −179.094 774.538
1.19 10.7613 22.7515 83.8049 −20.3300 244.835 −248.756 557.487 274.631 −218.776
1.20 10.8147 4.39262 84.9571 −33.9339 47.5047 227.835 572.713 −223.705 −366.984
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(83\) \(-1\)

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 83.6.a.b ✓ 20
3.b odd 2 1 747.6.a.f 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
83.6.a.b ✓ 20 1.a even 1 1 trivial
747.6.a.f 20 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 7 T_{2}^{19} - 482 T_{2}^{18} + 3222 T_{2}^{17} + 96413 T_{2}^{16} - 611573 T_{2}^{15} + \cdots - 7882835431424 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(83))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots - 7882835431424 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 29\!\cdots\!52 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 85\!\cdots\!92 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots - 15\!\cdots\!32 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 77\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots - 17\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots - 39\!\cdots\!50 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 80\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots - 11\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 10\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots - 81\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 38\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots - 41\!\cdots\!20 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots - 59\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots - 48\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots - 43\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 79\!\cdots\!60 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( (T - 6889)^{20} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots - 14\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 38\!\cdots\!44 \) Copy content Toggle raw display
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