Properties

Label 19.8.a.a
Level $19$
Weight $8$
Character orbit 19.a
Self dual yes
Analytic conductor $5.935$
Analytic rank $1$
Dimension $4$
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] = [19,8,Mod(1,19)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(19, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("19.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 19.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: \(5.93531548420\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 255x^{2} + 475x + 500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 2) q^{2} + ( - \beta_{2} - \beta_1 - 4) q^{3} + ( - \beta_{3} + 4 \beta_{2} - 10 \beta_1 + 8) q^{4} + (6 \beta_{3} + 3 \beta_{2} - 11 \beta_1 - 59) q^{5} + ( - 3 \beta_{3} - 6 \beta_{2} - 18 \beta_1 - 156) q^{6} + ( - 26 \beta_{3} - 4 \beta_{2} - 20 \beta_1 - 311) q^{7} + (27 \beta_{3} - 36 \beta_{2} + 50 \beta_1 - 892) q^{8} + (6 \beta_{3} + 23 \beta_{2} + 77 \beta_1 - 1201) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 2) q^{2} + ( - \beta_{2} - \beta_1 - 4) q^{3} + ( - \beta_{3} + 4 \beta_{2} - 10 \beta_1 + 8) q^{4} + (6 \beta_{3} + 3 \beta_{2} - 11 \beta_1 - 59) q^{5} + ( - 3 \beta_{3} - 6 \beta_{2} - 18 \beta_1 - 156) q^{6} + ( - 26 \beta_{3} - 4 \beta_{2} - 20 \beta_1 - 311) q^{7} + (27 \beta_{3} - 36 \beta_{2} + 50 \beta_1 - 892) q^{8} + (6 \beta_{3} + 23 \beta_{2} + 77 \beta_1 - 1201) q^{9} + (17 \beta_{3} - 14 \beta_{2} + 83 \beta_1 - 1598) q^{10} + (30 \beta_{3} + 87 \beta_{2} + 177 \beta_1 - 2121) q^{11} + ( - 3 \beta_{3} + 32 \beta_{2} - 10 \beta_1 - 1564) q^{12} + ( - 240 \beta_{3} + 123 \beta_{2} - 99 \beta_1 - 1054) q^{13} + (30 \beta_{3} - 192 \beta_{2} - 187 \beta_1 - 586) q^{14} + (150 \beta_{3} + 90 \beta_{2} + 60 \beta_1 - 690) q^{15} + ( - 93 \beta_{3} - 276 \beta_{2} - 858 \beta_1 + 4588) q^{16} + (528 \beta_{3} - 510 \beta_{2} + 778 \beta_1 - 1175) q^{17} + (9 \beta_{3} + 378 \beta_{2} - 1323 \beta_1 + 12942) q^{18} + 6859 q^{19} + ( - 924 \beta_{3} - 12 \beta_{2} - 1196 \beta_1 + 20236) q^{20} + ( - 474 \beta_{3} + 619 \beta_{2} + 1183 \beta_1 + 9100) q^{21} + (141 \beta_{3} + 1002 \beta_{2} - 1683 \beta_1 + 28590) q^{22} + (1428 \beta_{3} - 429 \beta_{2} + 4217 \beta_1 - 7042) q^{23} + (525 \beta_{3} + 780 \beta_{2} + 1530 \beta_1 + 22980) q^{24} + ( - 2062 \beta_{3} - 1931 \beta_{2} + 887 \beta_1 - 5722) q^{25} + (831 \beta_{3} - 1110 \beta_{2} + 2924 \beta_1 + 7376) q^{26} + ( - 174 \beta_{3} + 2495 \beta_{2} + 569 \beta_1 - 17926) q^{27} + (2717 \beta_{3} - 500 \beta_{2} - 814 \beta_1 + 8352) q^{28} + ( - 3510 \beta_{3} - 2907 \beta_{2} + 953 \beta_1 - 63172) q^{29} + (150 \beta_{3} + 1020 \beta_{2} + 510 \beta_1 + 3180) q^{30} + (4306 \beta_{3} - 2428 \beta_{2} - 7694 \beta_1 - 79472) q^{31} + ( - 3609 \beta_{3} + 252 \beta_{2} - 834 \beta_1 - 11508) q^{32} + ( - 162 \beta_{3} - 372 \beta_{2} - 6330 \beta_1 - 92226) q^{33} + ( - 3346 \beta_{3} + 4204 \beta_{2} - 19675 \beta_1 + 57046) q^{34} + (1746 \beta_{3} + 6363 \beta_{2} - 2431 \beta_1 - 140419) q^{35} + (2058 \beta_{3} - 7444 \beta_{2} + 21968 \beta_1 - 35236) q^{36} + (334 \beta_{3} + 9572 \beta_{2} - 18722 \beta_1 - 69986) q^{37} + (6859 \beta_1 - 13718) q^{38} + ( - 5112 \beta_{3} + 1453 \beta_{2} - 1085 \beta_1 - 66206) q^{39} + ( - 104 \beta_{3} - 6712 \beta_{2} + 20764 \beta_1 + 61256) q^{40} + (13830 \beta_{3} - 6306 \beta_{2} + 17284 \beta_1 - 207854) q^{41} + (1767 \beta_{3} + 4074 \beta_{2} + 14202 \beta_1 + 186204) q^{42} + ( - 2526 \beta_{3} - 12495 \beta_{2} - 15777 \beta_1 + 2927) q^{43} + (1710 \beta_{3} - 15300 \beta_{2} + 41160 \beta_1 + 15756) q^{44} + ( - 10422 \beta_{3} - 7941 \beta_{2} + 17277 \beta_1 + 41613) q^{45} + ( - 7361 \beta_{3} + 21722 \beta_{2} - 53072 \beta_1 + 471320) q^{46} + ( - 14082 \beta_{3} + 33855 \beta_{2} - 30067 \beta_1 + 88385) q^{47} + (1449 \beta_{3} + 5684 \beta_{2} + 28130 \beta_1 + 349652) q^{48} + (14056 \beta_{3} - 15400 \beta_{2} + 29344 \beta_1 - 16918) q^{49} + ( - 6549 \beta_{3} - 8562 \beta_{2} - 51176 \beta_1 + 190456) q^{50} + (10284 \beta_{3} - 1551 \beta_{2} + 7365 \beta_1 + 260712) q^{51} + (22525 \beta_{3} - 2944 \beta_{2} - 29426 \beta_1 + 420748) q^{52} + (7314 \beta_{3} + 6159 \beta_{2} - 61405 \beta_1 + 3542) q^{53} + (9585 \beta_{3} + 6570 \beta_{2} + 32760 \beta_1 + 201240) q^{54} + ( - 29154 \beta_{3} - 23757 \beta_{2} + 33129 \beta_1 + 261861) q^{55} + ( - 7743 \beta_{3} + 31188 \beta_{2} + 22366 \beta_1 - 228164) q^{56} + ( - 6859 \beta_{2} - 6859 \beta_1 - 27436) q^{57} + ( - 9071 \beta_{3} - 16042 \beta_{2} - 127730 \beta_1 + 369716) q^{58} + (9522 \beta_{3} + 10575 \beta_{2} + 122233 \beta_1 - 767012) q^{59} + ( - 15780 \beta_{3} - 6840 \beta_{2} + 13560 \beta_1 + 172920) q^{60} + ( - 27546 \beta_{3} - 13935 \beta_{2} - 30189 \beta_1 + 82493) q^{61} + ( - 6324 \beta_{3} - 18408 \beta_{2} - 79948 \beta_1 - 1192720) q^{62} + (41502 \beta_{3} - 14729 \beta_{2} - 17087 \beta_1 - 24521) q^{63} + (17355 \beta_{3} + 18060 \beta_{2} + 117750 \beta_1 - 449732) q^{64} + ( - 2964 \beta_{3} + 48468 \beta_{2} - 49516 \beta_1 - 1127764) q^{65} + (5004 \beta_{3} - 26712 \beta_{2} - 49446 \beta_1 - 653292) q^{66} + (38388 \beta_{3} + 74733 \beta_{2} + 100737 \beta_1 - 33310) q^{67} + ( - 27747 \beta_{3} - 18396 \beta_{2} + 214042 \beta_1 - 2225504) q^{68} + (18624 \beta_{3} - 27687 \beta_{2} - 68841 \beta_1 - 391902) q^{69} + (26137 \beta_{3} + 9986 \beta_{2} + 15523 \beta_1 + 58802) q^{70} + ( - 53328 \beta_{3} - 85524 \beta_{2} - 70856 \beta_1 - 542294) q^{71} + ( - 54954 \beta_{3} + 32832 \beta_{2} - 209520 \beta_1 + 951984) q^{72} + ( - 38004 \beta_{3} + 23496 \beta_{2} + 204564 \beta_1 - 37495) q^{73} + (56676 \beta_{3} - 54408 \beta_{2} + 289706 \beta_1 - 2045068) q^{74} + ( - 40170 \beta_{3} + 28115 \beta_{2} + 110945 \beta_1 + 1541030) q^{75} + ( - 6859 \beta_{3} + 27436 \beta_{2} - 68590 \beta_1 + 54872) q^{76} + (94302 \beta_{3} - 36297 \beta_{2} - 91371 \beta_1 - 893121) q^{77} + (12009 \beta_{3} - 21882 \beta_{2} - 15336 \beta_1 + 342408) q^{78} + ( - 54354 \beta_{3} + 39378 \beta_{2} - 24108 \beta_1 + 4023872) q^{79} + (70764 \beta_{3} + 70752 \beta_{2} - 99224 \beta_1 - 180416) q^{80} + ( - 25968 \beta_{3} - 63676 \beta_{2} - 292348 \beta_1 + 603053) q^{81} + ( - 56338 \beta_{3} + 111844 \beta_{2} - 512518 \beta_1 + 1665604) q^{82} + (46272 \beta_{3} + 40110 \beta_{2} - 173070 \beta_1 + 2086224) q^{83} + (60999 \beta_{3} - 7208 \beta_{2} + 7258 \beta_1 + 361804) q^{84} + (50402 \beta_{3} - 99089 \beta_{2} + 199973 \beta_1 + 1093417) q^{85} + ( - 31677 \beta_{3} - 98202 \beta_{2} - 140695 \beta_1 - 2336698) q^{86} + ( - 67848 \beta_{3} + 101565 \beta_{2} + 233283 \beta_1 + 2621958) q^{87} + ( - 122118 \beta_{3} + 12624 \beta_{2} - 438120 \beta_1 + 1149888) q^{88} + ( - 108870 \beta_{3} + 1716 \beta_{2} + 204626 \beta_1 + 803576) q^{89} + ( - 38619 \beta_{3} + 11538 \beta_{2} - 250461 \beta_1 + 2568546) q^{90} + (176582 \beta_{3} - 230993 \beta_{2} + 75005 \beta_1 + 5714162) q^{91} + ( - 35463 \beta_{3} - 143376 \beta_{2} + 848726 \beta_1 - 5910004) q^{92} + (120792 \beta_{3} + 150508 \beta_{2} + 318976 \beta_1 + 3312760) q^{93} + (179569 \beta_{3} - 108886 \beta_{2} + 1101895 \beta_1 - 2217334) q^{94} + (41154 \beta_{3} + 20577 \beta_{2} - 75449 \beta_1 - 404681) q^{95} + ( - 74043 \beta_{3} + 29844 \beta_{2} + 50922 \beta_1 + 167364) q^{96} + ( - 188896 \beta_{3} + 216394 \beta_{2} + 482594 \beta_1 - 969756) q^{97} + ( - 105000 \beta_{3} + 142800 \beta_{2} - 618582 \beta_1 + 2571084) q^{98} + ( - 48906 \beta_{3} - 42663 \beta_{2} - 146937 \beta_1 + 6353907) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 9 q^{2} - 14 q^{3} + 37 q^{4} - 222 q^{5} - 603 q^{6} - 1246 q^{7} - 3555 q^{8} - 4898 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 9 q^{2} - 14 q^{3} + 37 q^{4} - 222 q^{5} - 603 q^{6} - 1246 q^{7} - 3555 q^{8} - 4898 q^{9} - 6444 q^{10} - 8718 q^{11} - 6281 q^{12} - 4480 q^{13} - 1935 q^{14} - 2760 q^{15} + 19393 q^{16} - 4440 q^{17} + 52722 q^{18} + 27436 q^{19} + 81228 q^{20} + 34124 q^{21} + 115182 q^{22} - 30528 q^{23} + 90135 q^{24} - 23906 q^{25} + 28521 q^{26} - 74942 q^{27} + 37439 q^{28} - 254244 q^{29} + 11340 q^{30} - 303460 q^{31} - 49059 q^{32} - 362364 q^{33} + 240309 q^{34} - 563862 q^{35} - 153410 q^{36} - 270460 q^{37} - 61731 q^{38} - 270304 q^{39} + 230868 q^{40} - 828564 q^{41} + 728307 q^{42} + 37454 q^{43} + 38874 q^{44} + 146694 q^{45} + 1909269 q^{46} + 335670 q^{47} + 1366243 q^{48} - 67560 q^{49} + 815013 q^{50} + 1047318 q^{51} + 1737887 q^{52} + 76728 q^{53} + 775215 q^{54} + 1008918 q^{55} - 973953 q^{56} - 96026 q^{57} + 1613565 q^{58} - 3191334 q^{59} + 669180 q^{60} + 346550 q^{61} - 4678848 q^{62} - 24766 q^{63} - 1917383 q^{64} - 4512972 q^{65} - 2532006 q^{66} - 270322 q^{67} - 9125409 q^{68} - 1452456 q^{69} + 235836 q^{70} - 2066124 q^{71} + 3929670 q^{72} - 416044 q^{73} - 8358894 q^{74} + 5984890 q^{75} + 253783 q^{76} - 3350514 q^{77} + 1418859 q^{78} + 16025864 q^{79} - 622428 q^{80} + 2742268 q^{81} + 7006752 q^{82} + 8524128 q^{83} + 1508165 q^{84} + 4323186 q^{85} - 9139572 q^{86} + 10085136 q^{87} + 4902930 q^{88} + 2899092 q^{89} + 10474488 q^{90} + 23189218 q^{91} - 24380829 q^{92} + 12902348 q^{93} - 9682776 q^{94} - 1522698 q^{95} + 514647 q^{96} - 4766908 q^{97} + 10655118 q^{98} + 25556322 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 255x^{2} + 475x + 500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu^{2} - 235\nu - 300 ) / 50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 305\nu - 375 ) / 25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{3} - 8\nu^{2} - 685\nu + 1700 ) / 50 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_{2} + \beta _1 + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -7\beta_{3} + \beta_{2} + 23\beta _1 + 391 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 97\beta_{3} + 154\beta_{2} + 117\beta _1 - 286 ) / 2 \) Copy content Toggle raw display

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
2.64844
−0.751230
−16.3077
15.4104
−19.5150 −3.33349 252.835 170.463 65.0531 31.4807 −2436.16 −2175.89 −3326.58
1.2 −4.43255 22.5580 −108.352 160.438 −99.9896 −1314.43 1047.64 −1678.14 −711.151
1.3 3.18411 20.6576 −117.861 −477.630 65.7761 883.696 −782.850 −1760.26 −1520.83
1.4 11.7634 −53.8822 10.3786 −75.2709 −633.840 −846.751 −1383.63 716.286 −885.446
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(19\) \(-1\)

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 19.8.a.a 4
3.b odd 2 1 171.8.a.d 4
4.b odd 2 1 304.8.a.f 4
5.b even 2 1 475.8.a.a 4
19.b odd 2 1 361.8.a.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.8.a.a 4 1.a even 1 1 trivial
171.8.a.d 4 3.b odd 2 1
304.8.a.f 4 4.b odd 2 1
361.8.a.b 4 19.b odd 2 1
475.8.a.a 4 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 9T_{2}^{3} - 234T_{2}^{2} - 396T_{2} + 3240 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(19))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 9 T^{3} - 234 T^{2} + \cdots + 3240 \) Copy content Toggle raw display
$3$ \( T^{4} + 14 T^{3} - 1827 T^{2} + \cdots + 83700 \) Copy content Toggle raw display
$5$ \( T^{4} + 222 T^{3} + \cdots + 983232000 \) Copy content Toggle raw display
$7$ \( T^{4} + 1246 T^{3} + \cdots + 30962663047 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots - 235640376975204 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots - 256119966125456 \) Copy content Toggle raw display
$17$ \( T^{4} + 4440 T^{3} + \cdots - 30\!\cdots\!31 \) Copy content Toggle raw display
$19$ \( (T - 6859)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 30528 T^{3} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} + 254244 T^{3} + \cdots - 58\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{4} + 303460 T^{3} + \cdots + 52\!\cdots\!32 \) Copy content Toggle raw display
$37$ \( T^{4} + 270460 T^{3} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{4} + 828564 T^{3} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{4} - 37454 T^{3} + \cdots - 47\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{4} - 335670 T^{3} + \cdots + 57\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{4} - 76728 T^{3} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{4} + 3191334 T^{3} + \cdots - 29\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} - 346550 T^{3} + \cdots + 48\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{4} + 270322 T^{3} + \cdots - 35\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{4} + 2066124 T^{3} + \cdots - 80\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{4} + 416044 T^{3} + \cdots + 61\!\cdots\!53 \) Copy content Toggle raw display
$79$ \( T^{4} - 16025864 T^{3} + \cdots + 96\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{4} - 8524128 T^{3} + \cdots - 58\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{4} - 2899092 T^{3} + \cdots + 24\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{4} + 4766908 T^{3} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
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