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Properties

Label 471.4.a.b
Level $471$
Weight $4$
Character orbit 471.a
Self dual yes
Analytic conductor $27.790$
Analytic rank $1$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [471,4,Mod(1,471)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(471, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("471.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 471 = 3 \cdot 157 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 471.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: \(27.7898996127\)
Analytic rank: \(1\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 2 x^{16} - 97 x^{15} + 171 x^{14} + 3797 x^{13} - 5785 x^{12} - 76995 x^{11} + 100013 x^{10} + \cdots - 6144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + (\beta_{10} - 2) q^{5} + 3 \beta_1 q^{6} + (\beta_{12} + 1) q^{7} + ( - \beta_{12} - \beta_{11} - 3 \beta_1 - 3) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + (\beta_{10} - 2) q^{5} + 3 \beta_1 q^{6} + (\beta_{12} + 1) q^{7} + ( - \beta_{12} - \beta_{11} - 3 \beta_1 - 3) q^{8} + 9 q^{9} + ( - \beta_{15} - \beta_{14} - \beta_{13} + \cdots - 3) q^{10}+ \cdots + (9 \beta_{15} + 9 \beta_{14} + \cdots - 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q - 2 q^{2} - 51 q^{3} + 62 q^{4} - 28 q^{5} + 6 q^{6} + 10 q^{7} - 45 q^{8} + 153 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q - 2 q^{2} - 51 q^{3} + 62 q^{4} - 28 q^{5} + 6 q^{6} + 10 q^{7} - 45 q^{8} + 153 q^{9} - 47 q^{10} - 71 q^{11} - 186 q^{12} + 4 q^{13} - 63 q^{14} + 84 q^{15} + 166 q^{16} - 236 q^{17} - 18 q^{18} + 32 q^{19} - 193 q^{20} - 30 q^{21} + 54 q^{22} - 269 q^{23} + 135 q^{24} + 271 q^{25} - 209 q^{26} - 459 q^{27} + 85 q^{28} - 410 q^{29} + 141 q^{30} - 411 q^{31} - 522 q^{32} + 213 q^{33} - 303 q^{34} - 198 q^{35} + 558 q^{36} + 146 q^{37} - 176 q^{38} - 12 q^{39} - 792 q^{40} - 1973 q^{41} + 189 q^{42} + 489 q^{43} - 1751 q^{44} - 252 q^{45} - 1056 q^{46} - 620 q^{47} - 498 q^{48} - 547 q^{49} - 3409 q^{50} + 708 q^{51} - 1078 q^{52} - 1542 q^{53} + 54 q^{54} + 578 q^{55} - 3858 q^{56} - 96 q^{57} - 1334 q^{58} - 1478 q^{59} + 579 q^{60} - 2027 q^{61} - 3646 q^{62} + 90 q^{63} - 591 q^{64} - 2772 q^{65} - 162 q^{66} + 1690 q^{67} - 3885 q^{68} + 807 q^{69} - 2081 q^{70} - 2480 q^{71} - 405 q^{72} - 2484 q^{73} - 3682 q^{74} - 813 q^{75} - 3602 q^{76} - 2514 q^{77} + 627 q^{78} + 296 q^{79} - 4125 q^{80} + 1377 q^{81} - 44 q^{82} - 2644 q^{83} - 255 q^{84} - 1780 q^{85} - 4483 q^{86} + 1230 q^{87} - 1048 q^{88} - 2296 q^{89} - 423 q^{90} - 818 q^{91} - 4544 q^{92} + 1233 q^{93} - 4072 q^{94} - 2028 q^{95} + 1566 q^{96} - 768 q^{97} - 445 q^{98} - 639 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 2 x^{16} - 97 x^{15} + 171 x^{14} + 3797 x^{13} - 5785 x^{12} - 76995 x^{11} + 100013 x^{10} + \cdots - 6144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 150542238637961 \nu^{16} + \cdots - 39\!\cdots\!64 ) / 38\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 39\!\cdots\!65 \nu^{16} + \cdots - 86\!\cdots\!08 ) / 30\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10\!\cdots\!53 \nu^{16} + \cdots + 22\!\cdots\!60 ) / 76\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!27 \nu^{16} + \cdots - 17\!\cdots\!16 ) / 61\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13\!\cdots\!59 \nu^{16} + \cdots + 21\!\cdots\!24 ) / 61\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 25\!\cdots\!91 \nu^{16} + \cdots - 10\!\cdots\!76 ) / 61\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18\!\cdots\!56 \nu^{16} - 255418841830635 \nu^{15} + \cdots - 67\!\cdots\!60 ) / 38\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15\!\cdots\!95 \nu^{16} + \cdots + 21\!\cdots\!56 ) / 30\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 35\!\cdots\!17 \nu^{16} + \cdots - 66\!\cdots\!80 ) / 61\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 35\!\cdots\!17 \nu^{16} + \cdots - 11\!\cdots\!24 ) / 61\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 40\!\cdots\!15 \nu^{16} + \cdots + 13\!\cdots\!92 ) / 61\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 44\!\cdots\!63 \nu^{16} + \cdots - 23\!\cdots\!04 ) / 61\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 30\!\cdots\!27 \nu^{16} + \cdots + 34\!\cdots\!52 ) / 30\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 11\!\cdots\!25 \nu^{16} + \cdots - 81\!\cdots\!52 ) / 61\!\cdots\!68 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} + \beta_{11} + 19\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{16} - 2 \beta_{15} + \beta_{14} - 3 \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{7} + \cdots + 234 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{16} + 2 \beta_{15} - \beta_{14} - 4 \beta_{13} + 28 \beta_{12} + 34 \beta_{11} - \beta_{10} + \cdots + 132 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 82 \beta_{16} - 78 \beta_{15} + 45 \beta_{14} + 12 \beta_{13} + 9 \beta_{12} - 5 \beta_{11} + \cdots + 5231 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 88 \beta_{16} + 84 \beta_{15} - 64 \beta_{14} - 208 \beta_{13} + 696 \beta_{12} + 982 \beta_{11} + \cdots + 4082 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2584 \beta_{16} - 2352 \beta_{15} + 1448 \beta_{14} + 660 \beta_{13} + 464 \beta_{12} - 360 \beta_{11} + \cdots + 125208 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3072 \beta_{16} + 2852 \beta_{15} - 2840 \beta_{14} - 7808 \beta_{13} + 16869 \beta_{12} + 27077 \beta_{11} + \cdots + 107415 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 74634 \beta_{16} - 65578 \beta_{15} + 41405 \beta_{14} + 25140 \beta_{13} + 16176 \beta_{12} + \cdots + 3117374 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 98994 \beta_{16} + 91118 \beta_{15} - 105657 \beta_{14} - 257524 \beta_{13} + 409496 \beta_{12} + \cdots + 2553624 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2081738 \beta_{16} - 1781894 \beta_{15} + 1127945 \beta_{14} + 829808 \beta_{13} + 478265 \beta_{12} + \cdots + 79507123 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3070152 \beta_{16} + 2830384 \beta_{15} - 3560568 \beta_{14} - 7948208 \beta_{13} + 10057908 \beta_{12} + \cdots + 55615318 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 57239408 \beta_{16} - 48105976 \beta_{15} + 30157684 \beta_{14} + 25546376 \beta_{13} + \cdots + 2059366988 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 93214288 \beta_{16} + 86331632 \beta_{15} - 113013232 \beta_{14} - 236131088 \beta_{13} + \cdots + 1088309019 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 1565483938 \beta_{16} - 1299745954 \beta_{15} + 802795465 \beta_{14} + 757732728 \beta_{13} + \cdots + 53902060514 \) 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
5.12331
5.02494
4.83504
3.53567
2.91026
2.41591
1.68957
0.860940
0.000868754 0
−0.577311
−1.19413
−2.03095
−3.51875
−3.64476
−3.90761
−4.23328
−5.28971
−5.12331 −3.00000 18.2483 −5.01489 15.3699 27.0594 −52.5053 9.00000 25.6928
1.2 −5.02494 −3.00000 17.2500 −16.9775 15.0748 −15.6372 −46.4805 9.00000 85.3106
1.3 −4.83504 −3.00000 15.3776 17.4387 14.5051 10.0998 −35.6711 9.00000 −84.3168
1.4 −3.53567 −3.00000 4.50099 10.9478 10.6070 13.4953 12.3714 9.00000 −38.7080
1.5 −2.91026 −3.00000 0.469618 14.8343 8.73078 −22.3381 21.9154 9.00000 −43.1717
1.6 −2.41591 −3.00000 −2.16340 −21.3850 7.24772 3.77349 24.5538 9.00000 51.6642
1.7 −1.68957 −3.00000 −5.14535 −6.98813 5.06871 −31.7624 22.2100 9.00000 11.8069
1.8 −0.860940 −3.00000 −7.25878 −14.2282 2.58282 0.446944 13.1369 9.00000 12.2496
1.9 −0.000868754 0 −3.00000 −8.00000 4.91003 0.00260626 −0.933978 0.0139001 9.00000 −0.00426561
1.10 0.577311 −3.00000 −7.66671 −11.1028 −1.73193 19.2549 −9.04456 9.00000 −6.40974
1.11 1.19413 −3.00000 −6.57405 14.5107 −3.58240 9.27150 −17.4034 9.00000 17.3278
1.12 2.03095 −3.00000 −3.87523 5.58289 −6.09286 −9.48368 −24.1180 9.00000 11.3386
1.13 3.51875 −3.00000 4.38157 −13.8665 −10.5562 30.6530 −12.7323 9.00000 −48.7927
1.14 3.64476 −3.00000 5.28431 −0.937674 −10.9343 18.4043 −9.89806 9.00000 −3.41760
1.15 3.90761 −3.00000 7.26942 4.84061 −11.7228 −16.6234 −2.85484 9.00000 18.9152
1.16 4.23328 −3.00000 9.92067 0.570021 −12.6998 −6.96868 8.13075 9.00000 2.41306
1.17 5.28971 −3.00000 19.9810 −11.1345 −15.8691 −18.7111 63.3761 9.00000 −58.8981
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(157\) \(-1\)

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 471.4.a.b ✓ 17
3.b odd 2 1 1413.4.a.b 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.4.a.b ✓ 17 1.a even 1 1 trivial
1413.4.a.b 17 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} + 2 T_{2}^{16} - 97 T_{2}^{15} - 171 T_{2}^{14} + 3797 T_{2}^{13} + 5785 T_{2}^{12} + \cdots + 6144 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(471))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} + 2 T^{16} + \cdots + 6144 \) Copy content Toggle raw display
$3$ \( (T + 3)^{17} \) Copy content Toggle raw display
$5$ \( T^{17} + \cdots + 904499118408000 \) Copy content Toggle raw display
$7$ \( T^{17} + \cdots - 13\!\cdots\!40 \) Copy content Toggle raw display
$11$ \( T^{17} + \cdots - 99\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{17} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots - 43\!\cdots\!12 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots + 72\!\cdots\!26 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots + 78\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots - 16\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots - 32\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots + 23\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots - 16\!\cdots\!52 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots - 16\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots - 58\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots - 88\!\cdots\!72 \) Copy content Toggle raw display
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