Properties

Label 1859.4.a.h
Level $1859$
Weight $4$
Character orbit 1859.a
Self dual yes
Analytic conductor $109.685$
Analytic rank $0$
Dimension $17$
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] = [1859,4,Mod(1,1859)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1859, 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("1859.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1859 = 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1859.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: \(109.684550701\)
Analytic rank: \(0\)
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: \( x^{17} - 93 x^{15} - 7 x^{14} + 3449 x^{13} + 406 x^{12} - 65242 x^{11} - 7942 x^{10} + 669163 x^{9} + 59532 x^{8} - 3663297 x^{7} - 79027 x^{6} + 9967603 x^{5} + \cdots - 2210688 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 143)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} + 3) q^{4} + ( - \beta_{7} + 1) q^{5} + ( - \beta_{5} - \beta_1 - 1) q^{6} + (\beta_{10} + 4) q^{7} + (\beta_{3} + 3 \beta_1 + 1) q^{8} + (\beta_{14} - \beta_{7} + \beta_{6} - \beta_{2} + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} + 3) q^{4} + ( - \beta_{7} + 1) q^{5} + ( - \beta_{5} - \beta_1 - 1) q^{6} + (\beta_{10} + 4) q^{7} + (\beta_{3} + 3 \beta_1 + 1) q^{8} + (\beta_{14} - \beta_{7} + \beta_{6} - \beta_{2} + 7) q^{9} + ( - \beta_{11} - 2 \beta_{6} + \beta_{2} + 2 \beta_1 + 1) q^{10} - 11 q^{11} + (\beta_{13} - \beta_{11} + \beta_{8} - \beta_{6} - 2 \beta_1 - 7) q^{12} + (\beta_{15} + \beta_{13} + \beta_{12} + \beta_{11} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} + \cdots + 1) q^{14}+ \cdots + ( - 11 \beta_{14} + 11 \beta_{7} - 11 \beta_{6} + 11 \beta_{2} - 77) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q - 6 q^{3} + 50 q^{4} + 24 q^{5} - 16 q^{6} + 62 q^{7} + 21 q^{8} + 135 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q - 6 q^{3} + 50 q^{4} + 24 q^{5} - 16 q^{6} + 62 q^{7} + 21 q^{8} + 135 q^{9} + 2 q^{10} - 187 q^{11} - 127 q^{12} + 148 q^{15} + 126 q^{16} - 74 q^{17} - 90 q^{18} + 159 q^{19} + 222 q^{20} + 184 q^{21} - 215 q^{23} - 214 q^{24} + 95 q^{25} - 192 q^{27} + 358 q^{28} - 157 q^{29} + 829 q^{30} + 394 q^{31} + 553 q^{32} + 66 q^{33} + 702 q^{34} + 58 q^{35} - 700 q^{36} - 88 q^{37} - 1318 q^{38} + 733 q^{40} + 512 q^{41} + 337 q^{42} + 927 q^{43} - 550 q^{44} + 1482 q^{45} + 1361 q^{46} + 143 q^{47} - 178 q^{48} + 1835 q^{49} + 583 q^{50} - 568 q^{51} + 106 q^{53} + 67 q^{54} - 264 q^{55} + 2059 q^{56} - 1298 q^{57} + 1690 q^{58} + 266 q^{59} - 37 q^{60} - 624 q^{61} + 643 q^{62} + 2360 q^{63} - 1589 q^{64} + 176 q^{66} + 676 q^{67} - 413 q^{68} + 764 q^{69} + 1061 q^{70} + 763 q^{71} + 1366 q^{72} + 2374 q^{73} - 1649 q^{74} + 2420 q^{75} + 2101 q^{76} - 682 q^{77} + 2164 q^{79} + 1013 q^{80} + 537 q^{81} + 3152 q^{82} - 777 q^{83} + 3381 q^{84} + 1690 q^{85} - 2894 q^{86} - 4200 q^{87} - 231 q^{88} + 1687 q^{89} - 5399 q^{90} + 5542 q^{92} + 4310 q^{93} + 1777 q^{94} + 1124 q^{95} + 3465 q^{96} + 2047 q^{97} - 1553 q^{98} - 1485 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 93 x^{15} - 7 x^{14} + 3449 x^{13} + 406 x^{12} - 65242 x^{11} - 7942 x^{10} + 669163 x^{9} + 59532 x^{8} - 3663297 x^{7} - 79027 x^{6} + 9967603 x^{5} + \cdots - 2210688 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 19\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 41310869995 \nu^{16} + 278727206397 \nu^{15} - 4357449870192 \nu^{14} - 22798802169445 \nu^{13} + 178373934891776 \nu^{12} + \cdots + 19\!\cdots\!24 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22792255353 \nu^{16} - 173935419925 \nu^{15} - 1445507747716 \nu^{14} + 13203532405837 \nu^{13} + 28731625134576 \nu^{12} + \cdots - 92\!\cdots\!40 ) / 424284949467392 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 124885018663 \nu^{16} + 68376766059 \nu^{15} + 11092500475884 \nu^{14} - 3462328112507 \nu^{13} - 391117832151176 \nu^{12} + \cdots - 31\!\cdots\!20 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 294827222141 \nu^{16} + 219464620161 \nu^{15} + 25860702212988 \nu^{14} - 12725541872401 \nu^{13} + \cdots - 63\!\cdots\!08 ) / 25\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 177480472921 \nu^{16} - 398292066701 \nu^{15} - 14709832798652 \nu^{14} + 27780697951405 \nu^{13} + 478644757930352 \nu^{12} + \cdots + 13\!\cdots\!48 ) / 848569898934784 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 183146367977 \nu^{16} - 411803762979 \nu^{15} + 17660657765996 \nu^{14} + 35199004200787 \nu^{13} + \cdots - 68\!\cdots\!80 ) / 848569898934784 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 278509815379 \nu^{16} + 240283192845 \nu^{15} - 25773301577160 \nu^{14} - 23681279993365 \nu^{13} + 946899009302480 \nu^{12} + \cdots + 10\!\cdots\!16 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 719004694813 \nu^{16} - 1831736510361 \nu^{15} - 59159334330924 \nu^{14} + 132140276331041 \nu^{13} + \cdots + 60\!\cdots\!52 ) / 25\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 483443617781 \nu^{16} - 896950422627 \nu^{15} + 47322682896696 \nu^{14} + 76549836795275 \nu^{13} + \cdots - 21\!\cdots\!52 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 508877270167 \nu^{16} - 1725555260937 \nu^{15} + 52228351735548 \nu^{14} + 142874728959613 \nu^{13} + \cdots - 29\!\cdots\!36 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 675317839955 \nu^{16} + 243395485401 \nu^{15} - 61841470872860 \nu^{14} - 30140401950649 \nu^{13} + \cdots + 20\!\cdots\!52 ) / 848569898934784 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1065195885323 \nu^{16} + 1645100590137 \nu^{15} - 102145188288576 \nu^{14} - 145383659542541 \nu^{13} + \cdots + 42\!\cdots\!80 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 545813418179 \nu^{16} + 175570637925 \nu^{15} - 49939980916872 \nu^{14} - 23109325952993 \nu^{13} + \cdots + 18\!\cdots\!36 ) / 636427424201088 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 19\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} + \beta_{12} - \beta_{10} - \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4} + 26\beta_{2} + 4\beta _1 + 207 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{16} + 2 \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + 4 \beta_{9} - \beta_{7} - 8 \beta_{6} + 4 \beta_{4} + 32 \beta_{3} + 2 \beta_{2} + 411 \beta _1 + 63 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{15} + 36 \beta_{14} + \beta_{13} + 40 \beta_{12} + 8 \beta_{11} - 37 \beta_{10} - 8 \beta_{9} - 37 \beta_{8} + 7 \beta_{7} - 30 \beta_{6} - 50 \beta_{5} - 46 \beta_{4} + 3 \beta_{3} + 648 \beta_{2} + 191 \beta _1 + 4479 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 88 \beta_{16} + 16 \beta_{15} + 77 \beta_{14} - 28 \beta_{13} + 43 \beta_{12} - 44 \beta_{11} + 55 \beta_{10} + 176 \beta_{9} + \beta_{8} + 14 \beta_{7} - 407 \beta_{6} + \beta_{5} + 177 \beta_{4} + 892 \beta_{3} + 113 \beta_{2} + 9485 \beta _1 + 2488 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{16} + 70 \beta_{15} + 1024 \beta_{14} + 65 \beta_{13} + 1243 \beta_{12} + 411 \beta_{11} - 1055 \beta_{10} - 398 \beta_{9} - 1096 \beta_{8} + 315 \beta_{7} - 636 \beta_{6} - 1750 \beta_{5} - 1570 \beta_{4} + \cdots + 103475 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2892 \beta_{16} + 887 \beta_{15} + 2231 \beta_{14} - 603 \beta_{13} + 1393 \beta_{12} - 1450 \beta_{11} + 2076 \beta_{10} + 5790 \beta_{9} + 90 \beta_{8} + 1557 \beta_{7} - 14487 \beta_{6} + 107 \beta_{5} + \cdots + 82167 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 154 \beta_{16} + 2932 \beta_{15} + 27330 \beta_{14} + 2713 \beta_{13} + 35465 \beta_{12} + 14693 \beta_{11} - 28161 \beta_{10} - 14078 \beta_{9} - 30310 \beta_{8} + 10457 \beta_{7} - 11940 \beta_{6} + \cdots + 2487570 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 85404 \beta_{16} + 33111 \beta_{15} + 59472 \beta_{14} - 11917 \beta_{13} + 40444 \beta_{12} - 43356 \beta_{11} + 66825 \beta_{10} + 171188 \beta_{9} + 4549 \beta_{8} + 65085 \beta_{7} + \cdots + 2465723 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7496 \beta_{16} + 99188 \beta_{15} + 714934 \beta_{14} + 93336 \beta_{13} + 973360 \beta_{12} + 454612 \beta_{11} - 739022 \beta_{10} - 435808 \beta_{9} - 814256 \beta_{8} + 309022 \beta_{7} + \cdots + 61381628 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2395480 \beta_{16} + 1052466 \beta_{15} + 1542287 \beta_{14} - 225820 \beta_{13} + 1112321 \beta_{12} - 1241998 \beta_{11} + 1978649 \beta_{10} + 4807826 \beta_{9} + \cdots + 69948387 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 295642 \beta_{16} + 3011076 \beta_{15} + 18586881 \beta_{14} + 2905893 \beta_{13} + 26170874 \beta_{12} + 13077169 \beta_{11} - 19335396 \beta_{10} - 12623450 \beta_{9} + \cdots + 1541335383 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 65313734 \beta_{16} + 30847469 \beta_{15} + 39648325 \beta_{14} - 4120830 \beta_{13} + 29698754 \beta_{12} - 34787907 \beta_{11} + 55835459 \beta_{10} + 131277070 \beta_{9} + \cdots + 1916892151 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 10358698 \beta_{16} + 86056501 \beta_{15} + 482470073 \beta_{14} + 85537486 \beta_{13} + 695333740 \beta_{12} + 361031457 \beta_{11} - 506175581 \beta_{10} + \cdots + 39163943496 \) 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
−5.11519
−4.19083
−4.15571
−4.13824
−2.78419
−1.88904
−1.41378
−0.991815
0.631226
0.664163
1.06843
1.71387
3.12700
3.17505
4.11212
5.08131
5.10562
−5.11519 −3.58195 18.1651 2.16776 18.3223 −20.9807 −51.9966 −14.1696 −11.0885
1.2 −4.19083 4.70544 9.56305 −14.4259 −19.7197 34.7012 −6.55048 −4.85885 60.4566
1.3 −4.15571 7.17608 9.26990 11.1721 −29.8217 7.78423 −5.27734 24.4962 −46.4282
1.4 −4.13824 −5.47866 9.12507 3.23279 22.6720 −3.07367 −4.65582 3.01573 −13.3781
1.5 −2.78419 −8.94641 −0.248293 17.0433 24.9085 30.9353 22.9648 53.0382 −47.4517
1.6 −1.88904 1.56640 −4.43152 −2.20557 −2.95900 1.80472 23.4837 −24.5464 4.16640
1.7 −1.41378 5.76511 −6.00123 4.58681 −8.15059 −36.4239 19.7947 6.23646 −6.48474
1.8 −0.991815 −5.36099 −7.01630 −18.5970 5.31711 13.4049 14.8934 1.74021 18.4447
1.9 0.631226 9.24271 −7.60155 19.0524 5.83424 25.4436 −9.84810 58.4278 12.0263
1.10 0.664163 −0.846704 −7.55889 8.92433 −0.562349 −3.53056 −10.3336 −26.2831 5.92721
1.11 1.06843 −9.85108 −6.85846 −2.35929 −10.5252 −12.2506 −15.8752 70.0437 −2.52073
1.12 1.71387 7.58218 −5.06264 −12.0244 12.9949 −1.14865 −22.3877 30.4895 −20.6082
1.13 3.12700 −0.537604 1.77811 0.516265 −1.68109 34.6419 −19.4558 −26.7110 1.61436
1.14 3.17505 −0.600459 2.08094 −12.4679 −1.90649 −10.6254 −18.7933 −26.6394 −39.5863
1.15 4.11212 −4.01551 8.90957 12.3424 −16.5123 −29.6370 3.74025 −10.8757 50.7537
1.16 5.08131 −7.39360 17.8197 −8.39690 −37.5692 14.1925 49.8972 27.6653 −42.6673
1.17 5.10562 4.57504 18.0674 15.4387 23.3584 16.7622 51.4001 −6.06900 78.8244
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(1\)
\(13\) \(1\)

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 1859.4.a.h 17
13.b even 2 1 1859.4.a.g 17
13.e even 6 2 143.4.e.b 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.e.b 34 13.e even 6 2
1859.4.a.g 17 13.b even 2 1
1859.4.a.h 17 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} - 93 T_{2}^{15} - 7 T_{2}^{14} + 3449 T_{2}^{13} + 406 T_{2}^{12} - 65242 T_{2}^{11} - 7942 T_{2}^{10} + 669163 T_{2}^{9} + 59532 T_{2}^{8} - 3663297 T_{2}^{7} - 79027 T_{2}^{6} + 9967603 T_{2}^{5} + \cdots - 2210688 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1859))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} - 93 T^{15} - 7 T^{14} + \cdots - 2210688 \) Copy content Toggle raw display
$3$ \( T^{17} + 6 T^{16} + \cdots - 7355831456 \) Copy content Toggle raw display
$5$ \( T^{17} + \cdots + 179912814666900 \) Copy content Toggle raw display
$7$ \( T^{17} - 62 T^{16} + \cdots - 15\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( (T + 11)^{17} \) Copy content Toggle raw display
$13$ \( T^{17} \) Copy content Toggle raw display
$17$ \( T^{17} + 74 T^{16} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{17} - 159 T^{16} + \cdots - 20\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{17} + 215 T^{16} + \cdots + 38\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{17} + 157 T^{16} + \cdots + 17\!\cdots\!77 \) Copy content Toggle raw display
$31$ \( T^{17} - 394 T^{16} + \cdots - 11\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{17} + 88 T^{16} + \cdots - 60\!\cdots\!98 \) Copy content Toggle raw display
$41$ \( T^{17} - 512 T^{16} + \cdots - 56\!\cdots\!50 \) Copy content Toggle raw display
$43$ \( T^{17} - 927 T^{16} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{17} - 143 T^{16} + \cdots + 24\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{17} - 106 T^{16} + \cdots - 60\!\cdots\!50 \) Copy content Toggle raw display
$59$ \( T^{17} - 266 T^{16} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{17} + 624 T^{16} + \cdots - 28\!\cdots\!14 \) Copy content Toggle raw display
$67$ \( T^{17} - 676 T^{16} + \cdots + 56\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{17} - 763 T^{16} + \cdots + 60\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{17} - 2374 T^{16} + \cdots + 66\!\cdots\!32 \) Copy content Toggle raw display
$79$ \( T^{17} - 2164 T^{16} + \cdots - 82\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{17} + 777 T^{16} + \cdots + 65\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{17} - 1687 T^{16} + \cdots - 19\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{17} - 2047 T^{16} + \cdots + 46\!\cdots\!84 \) Copy content Toggle raw display
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