Properties

Label 13.8.c.a
Level $13$
Weight $8$
Character orbit 13.c
Analytic conductor $4.061$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [13,8,Mod(3,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.3");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 13.c (of order \(3\), degree \(2\), 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: \(4.06100533129\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + 796 x^{14} - 475 x^{13} + 449889 x^{12} - 92038 x^{11} + 116806037 x^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{12}\cdot 3\cdot 13^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1 - 1) q^{2} + ( - \beta_{8} + \beta_{6} - 4 \beta_{2} - 4) q^{3} + (\beta_{5} + 2 \beta_{3} + 72 \beta_{2}) q^{4} + (\beta_{9} - 4 \beta_{3} + 4 \beta_1 + 23) q^{5} + (\beta_{12} + 5 \beta_{8} + \cdots - 41 \beta_{2}) q^{6}+ \cdots + ( - \beta_{13} + \beta_{12} + \cdots + 747 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1 - 1) q^{2} + ( - \beta_{8} + \beta_{6} - 4 \beta_{2} - 4) q^{3} + (\beta_{5} + 2 \beta_{3} + 72 \beta_{2}) q^{4} + (\beta_{9} - 4 \beta_{3} + 4 \beta_1 + 23) q^{5} + (\beta_{12} + 5 \beta_{8} + \cdots - 41 \beta_{2}) q^{6}+ \cdots + ( - 4750 \beta_{15} - 1325 \beta_{14} + \cdots + 1732779) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 9 q^{2} - 28 q^{3} - 577 q^{4} + 384 q^{5} + 342 q^{6} + 196 q^{7} + 5922 q^{8} - 5988 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 9 q^{2} - 28 q^{3} - 577 q^{4} + 384 q^{5} + 342 q^{6} + 196 q^{7} + 5922 q^{8} - 5988 q^{9} - 6813 q^{10} + 5052 q^{11} - 15816 q^{12} + 17064 q^{13} + 14484 q^{14} - 184 q^{15} - 61377 q^{16} - 22824 q^{17} + 226318 q^{18} + 63692 q^{19} - 121971 q^{20} - 271240 q^{21} - 73726 q^{22} - 72468 q^{23} + 224952 q^{24} + 58488 q^{25} + 154443 q^{26} + 358616 q^{27} - 172816 q^{28} - 221772 q^{29} - 258116 q^{30} + 58432 q^{31} + 77823 q^{32} - 352372 q^{33} + 643334 q^{34} + 31560 q^{35} - 1162993 q^{36} - 368884 q^{37} + 2364492 q^{38} - 672996 q^{39} + 766030 q^{40} - 385824 q^{41} + 1560954 q^{42} + 391492 q^{43} - 6149616 q^{44} + 232096 q^{45} - 2599078 q^{46} + 1089984 q^{47} + 4585712 q^{48} - 1108780 q^{49} + 1626894 q^{50} + 8040712 q^{51} - 1312478 q^{52} + 3329328 q^{53} - 5616702 q^{54} - 418216 q^{55} - 3609156 q^{56} - 12068664 q^{57} + 3685081 q^{58} - 6318924 q^{59} + 1448784 q^{60} + 1763548 q^{61} - 12998388 q^{62} + 6641128 q^{63} + 32227074 q^{64} + 9500508 q^{65} + 16245108 q^{66} - 8607964 q^{67} - 10118091 q^{68} - 4618492 q^{69} - 25381840 q^{70} + 9473220 q^{71} - 35174445 q^{72} + 5020888 q^{73} + 2206221 q^{74} - 21834988 q^{75} + 18702876 q^{76} + 35263992 q^{77} + 28395902 q^{78} + 12644416 q^{79} - 24671247 q^{80} - 9316272 q^{81} - 9347751 q^{82} - 22158432 q^{83} + 37532636 q^{84} - 36911824 q^{85} - 30103956 q^{86} + 5199988 q^{87} - 36676404 q^{88} + 18469860 q^{89} + 132263562 q^{90} - 9614540 q^{91} + 64127424 q^{92} - 26340696 q^{93} - 6026128 q^{94} - 16267080 q^{95} - 164216320 q^{96} + 19660916 q^{97} - 51546651 q^{98} + 27386736 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 796 x^{14} - 475 x^{13} + 449889 x^{12} - 92038 x^{11} + 116806037 x^{10} + \cdots + 11\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 29\!\cdots\!41 \nu^{15} + \cdots + 12\!\cdots\!60 ) / 25\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\!\cdots\!48 \nu^{15} + \cdots - 57\!\cdots\!00 ) / 44\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\!\cdots\!84 \nu^{15} + \cdots - 69\!\cdots\!45 ) / 44\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 58\!\cdots\!59 \nu^{15} + \cdots - 24\!\cdots\!40 ) / 25\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 76\!\cdots\!37 \nu^{15} + \cdots - 58\!\cdots\!60 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10\!\cdots\!07 \nu^{15} + \cdots - 11\!\cdots\!80 ) / 14\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 65\!\cdots\!33 \nu^{15} + \cdots - 13\!\cdots\!40 ) / 89\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 17\!\cdots\!03 \nu^{15} + \cdots - 64\!\cdots\!60 ) / 39\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 69\!\cdots\!75 \nu^{15} + \cdots + 49\!\cdots\!60 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!77 \nu^{15} + \cdots - 14\!\cdots\!80 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14\!\cdots\!95 \nu^{15} + \cdots + 21\!\cdots\!40 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 49\!\cdots\!41 \nu^{15} + \cdots - 53\!\cdots\!00 ) / 59\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 19\!\cdots\!29 \nu^{15} + \cdots - 13\!\cdots\!00 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 25\!\cdots\!17 \nu^{15} + \cdots - 96\!\cdots\!00 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 199\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} - \beta_{13} + \beta_{11} + \beta_{9} - 8\beta_{6} + 3\beta_{4} + 334\beta_{3} - 334\beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5 \beta_{14} + 11 \beta_{12} + 11 \beta_{11} - 17 \beta_{9} + 124 \beta_{8} + 17 \beta_{7} + \cdots - 66756 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 542 \beta_{13} + 630 \beta_{12} + 24 \beta_{10} + 6576 \beta_{8} - 910 \beta_{7} + \cdots + 84118 \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 616 \beta_{15} - 3774 \beta_{14} - 3774 \beta_{13} - 7546 \beta_{11} - 616 \beta_{10} + 9510 \beta_{9} + \cdots + 27466237 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 8392 \beta_{15} + 258295 \beta_{14} - 324287 \beta_{12} - 324287 \beta_{11} - 527559 \beta_{9} + \cdots - 87425103 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2219595 \beta_{13} - 3865581 \beta_{12} + 496568 \beta_{10} - 47530580 \beta_{8} + \cdots + 12143482322 \beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2179728 \beta_{15} - 121169452 \beta_{14} - 121169452 \beta_{13} + 155690300 \beta_{11} + \cdots + 60937775228 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 286467696 \beta_{15} + 1208461516 \beta_{14} + 1780323076 \beta_{12} + 1780323076 \beta_{11} + \cdots - 5531905738659 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 56972432477 \beta_{13} + 72333241677 \beta_{12} + 604393968 \beta_{10} + 868403841640 \beta_{8} + \cdots + 36688056060797 \beta_{2} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 146903709456 \beta_{15} - 636281294433 \beta_{14} - 636281294433 \beta_{13} - 777016867935 \beta_{11} + \cdots + 25\!\cdots\!40 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 249537409896 \beta_{15} + 26911761090858 \beta_{14} - 33048714152786 \beta_{12} + \cdots - 20\!\cdots\!38 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 328967591306026 \beta_{13} - 327829774832030 \beta_{12} + 71587381350424 \beta_{10} + \cdots + 11\!\cdots\!37 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 148314289835896 \beta_{15} + \cdots + 11\!\cdots\!07 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
10.3972 + 18.0085i
8.10170 + 14.0326i
4.62264 + 8.00665i
2.89297 + 5.01078i
−3.54173 6.13445i
−5.36984 9.30084i
−5.55707 9.62513i
−11.0459 19.1321i
10.3972 18.0085i
8.10170 14.0326i
4.62264 8.00665i
2.89297 5.01078i
−3.54173 + 6.13445i
−5.36984 + 9.30084i
−5.55707 + 9.62513i
−11.0459 + 19.1321i
−10.8972 18.8745i 33.9160 + 58.7442i −173.499 + 300.509i 157.804 739.180 1280.30i −421.054 + 729.287i 4772.94 −1207.09 + 2090.74i −1719.62 2978.47i
3.2 −8.60170 14.8986i −43.0086 74.4931i −83.9785 + 145.455i 289.904 −739.895 + 1281.54i 484.274 838.787i 687.398 −2605.99 + 4513.70i −2493.67 4319.16i
3.3 −5.12264 8.87268i −4.84859 8.39800i 11.5171 19.9481i −294.418 −49.6752 + 86.0399i −715.408 + 1239.12i −1547.39 1046.48 1812.56i 1508.20 + 2612.27i
3.4 −3.39297 5.87680i 15.0885 + 26.1340i 40.9755 70.9716i 93.2351 102.389 177.344i 809.227 1401.62i −1424.72 638.177 1105.35i −316.344 547.924i
3.5 3.04173 + 5.26843i −12.8141 22.1946i 45.4958 78.8010i 442.310 77.9538 135.020i −380.773 + 659.518i 1332.23 765.099 1325.19i 1345.39 + 2330.28i
3.6 4.86984 + 8.43482i −34.7632 60.2115i 16.5693 28.6988i −459.709 338.582 586.442i 430.545 745.725i 1569.44 −1323.45 + 2292.29i −2238.71 3877.56i
3.7 5.05707 + 8.75910i 35.2053 + 60.9774i 12.8521 22.2605i −163.930 −356.071 + 616.734i −123.469 + 213.855i 1554.59 −1385.33 + 2399.46i −829.005 1435.88i
3.8 10.5459 + 18.2660i −2.77530 4.80695i −158.432 + 274.412i 126.804 58.5360 101.387i 14.6580 25.3884i −3983.49 1078.10 1867.32i 1337.27 + 2316.22i
9.1 −10.8972 + 18.8745i 33.9160 58.7442i −173.499 300.509i 157.804 739.180 + 1280.30i −421.054 729.287i 4772.94 −1207.09 2090.74i −1719.62 + 2978.47i
9.2 −8.60170 + 14.8986i −43.0086 + 74.4931i −83.9785 145.455i 289.904 −739.895 1281.54i 484.274 + 838.787i 687.398 −2605.99 4513.70i −2493.67 + 4319.16i
9.3 −5.12264 + 8.87268i −4.84859 + 8.39800i 11.5171 + 19.9481i −294.418 −49.6752 86.0399i −715.408 1239.12i −1547.39 1046.48 + 1812.56i 1508.20 2612.27i
9.4 −3.39297 + 5.87680i 15.0885 26.1340i 40.9755 + 70.9716i 93.2351 102.389 + 177.344i 809.227 + 1401.62i −1424.72 638.177 + 1105.35i −316.344 + 547.924i
9.5 3.04173 5.26843i −12.8141 + 22.1946i 45.4958 + 78.8010i 442.310 77.9538 + 135.020i −380.773 659.518i 1332.23 765.099 + 1325.19i 1345.39 2330.28i
9.6 4.86984 8.43482i −34.7632 + 60.2115i 16.5693 + 28.6988i −459.709 338.582 + 586.442i 430.545 + 745.725i 1569.44 −1323.45 2292.29i −2238.71 + 3877.56i
9.7 5.05707 8.75910i 35.2053 60.9774i 12.8521 + 22.2605i −163.930 −356.071 616.734i −123.469 213.855i 1554.59 −1385.33 2399.46i −829.005 + 1435.88i
9.8 10.5459 18.2660i −2.77530 + 4.80695i −158.432 274.412i 126.804 58.5360 + 101.387i 14.6580 + 25.3884i −3983.49 1078.10 + 1867.32i 1337.27 2316.22i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.8.c.a 16
3.b odd 2 1 117.8.g.d 16
4.b odd 2 1 208.8.i.d 16
13.c even 3 1 inner 13.8.c.a 16
13.c even 3 1 169.8.a.f 8
13.e even 6 1 169.8.a.e 8
13.f odd 12 2 169.8.b.e 16
39.i odd 6 1 117.8.g.d 16
52.j odd 6 1 208.8.i.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.8.c.a 16 1.a even 1 1 trivial
13.8.c.a 16 13.c even 3 1 inner
117.8.g.d 16 3.b odd 2 1
117.8.g.d 16 39.i odd 6 1
169.8.a.e 8 13.e even 6 1
169.8.a.f 8 13.c even 3 1
169.8.b.e 16 13.f odd 12 2
208.8.i.d 16 4.b odd 2 1
208.8.i.d 16 52.j odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots - 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 24\!\cdots\!41 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 33\!\cdots\!89 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 44\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 25\!\cdots\!49 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 85\!\cdots\!69 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 50\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 66\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 57\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 62\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 45\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 29\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 72\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 92\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 16\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
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