Properties

Label 209.4.a.d
Level $209$
Weight $4$
Character orbit 209.a
Self dual yes
Analytic conductor $12.331$
Analytic rank $0$
Dimension $15$
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] = [209,4,Mod(1,209)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(209, 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("209.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 209.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: \(12.3313991912\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 4 x^{14} - 89 x^{13} + 352 x^{12} + 2997 x^{11} - 11764 x^{10} - 47311 x^{9} + 186264 x^{8} + 347582 x^{7} - 1421856 x^{6} - 959244 x^{5} + 4697632 x^{4} + \cdots + 1892352 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 5) q^{4} + (\beta_{6} + 1) q^{5} + ( - \beta_{5} + \beta_{2} + 4) q^{6} + ( - \beta_{9} + 5) q^{7} + (\beta_{4} + \beta_{3} + 5 \beta_1 - 1) q^{8} + (\beta_{13} + \beta_{7} - \beta_{4} - \beta_{3} + 11) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 5) q^{4} + (\beta_{6} + 1) q^{5} + ( - \beta_{5} + \beta_{2} + 4) q^{6} + ( - \beta_{9} + 5) q^{7} + (\beta_{4} + \beta_{3} + 5 \beta_1 - 1) q^{8} + (\beta_{13} + \beta_{7} - \beta_{4} - \beta_{3} + 11) q^{9} + ( - \beta_{13} + \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{10} + 11 q^{11} + (\beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + 4 \beta_{3} - \beta_{2} + 5 \beta_1 - 6) q^{12} + ( - \beta_{14} + \beta_{12} - \beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \cdots + 3) q^{13}+ \cdots + (11 \beta_{13} + 11 \beta_{7} - 11 \beta_{4} - 11 \beta_{3} + 121) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 4 q^{2} + 3 q^{3} + 74 q^{4} + 10 q^{5} + 61 q^{6} + 73 q^{7} + 12 q^{8} + 160 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 4 q^{2} + 3 q^{3} + 74 q^{4} + 10 q^{5} + 61 q^{6} + 73 q^{7} + 12 q^{8} + 160 q^{9} + 165 q^{11} - 63 q^{12} + 43 q^{13} + 107 q^{14} - 16 q^{15} + 478 q^{16} + 73 q^{17} - q^{18} + 285 q^{19} + 30 q^{20} + 207 q^{21} + 44 q^{22} + 355 q^{23} + 769 q^{24} + 409 q^{25} + 25 q^{26} - 363 q^{27} + 1039 q^{28} + 385 q^{29} + 764 q^{30} + 478 q^{31} + 340 q^{32} + 33 q^{33} + 629 q^{34} + 466 q^{35} + 349 q^{36} - 2 q^{37} + 76 q^{38} + 1083 q^{39} + 40 q^{40} + 366 q^{41} + 1571 q^{42} + 732 q^{43} + 814 q^{44} + 676 q^{45} + 709 q^{46} + 332 q^{47} - 1467 q^{48} + 1420 q^{49} - 1020 q^{50} + 297 q^{51} + 2729 q^{52} - 555 q^{53} + 1049 q^{54} + 110 q^{55} + 279 q^{56} + 57 q^{57} - 2307 q^{58} - 653 q^{59} - 2638 q^{60} + 884 q^{61} - 3728 q^{62} + 762 q^{63} + 2110 q^{64} + 560 q^{65} + 671 q^{66} + 665 q^{67} - 5445 q^{68} - 2861 q^{69} - 4538 q^{70} + 380 q^{71} - 7873 q^{72} + 1263 q^{73} + 224 q^{74} - 365 q^{75} + 1406 q^{76} + 803 q^{77} + 169 q^{78} + 3604 q^{79} - 5990 q^{80} + 3811 q^{81} - 2818 q^{82} - 1420 q^{83} - 7939 q^{84} + 3178 q^{85} - 6696 q^{86} - 1549 q^{87} + 132 q^{88} - 3856 q^{89} - 6178 q^{90} + 2483 q^{91} - 3281 q^{92} - 1394 q^{93} - 4060 q^{94} + 190 q^{95} + 1833 q^{96} - 2472 q^{97} + 1447 q^{98} + 1760 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 4 x^{14} - 89 x^{13} + 352 x^{12} + 2997 x^{11} - 11764 x^{10} - 47311 x^{9} + 186264 x^{8} + 347582 x^{7} - 1421856 x^{6} - 959244 x^{5} + 4697632 x^{4} + \cdots + 1892352 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 380293821 \nu^{14} - 4908486504 \nu^{13} - 20446292901 \nu^{12} + 422741663876 \nu^{11} - 40066613855 \nu^{10} + \cdots + 92\!\cdots\!56 ) / 191245670030336 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 380293821 \nu^{14} + 4908486504 \nu^{13} + 20446292901 \nu^{12} - 422741663876 \nu^{11} + 40066613855 \nu^{10} + \cdots - 90\!\cdots\!20 ) / 191245670030336 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 846827805 \nu^{14} - 3349964292 \nu^{13} - 72219559721 \nu^{12} + 294951798848 \nu^{11} + 2290870065937 \nu^{10} + \cdots - 250390314379008 ) / 47811417507584 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5432385381 \nu^{14} + 1518603268 \nu^{13} + 554713706389 \nu^{12} - 96477229832 \nu^{11} - 22478641474713 \nu^{10} + \cdots + 40\!\cdots\!44 ) / 191245670030336 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13253258539 \nu^{14} - 11985921096 \nu^{13} + 1381445027931 \nu^{12} + 1175032606428 \nu^{11} - 57103164345047 \nu^{10} + \cdots + 10\!\cdots\!60 ) / 191245670030336 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 7788051871 \nu^{14} + 23575675464 \nu^{13} + 718531648515 \nu^{12} - 2025127928720 \nu^{11} - 25498441507163 \nu^{10} + \cdots + 12\!\cdots\!36 ) / 95622835015168 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9784419435 \nu^{14} - 41832632276 \nu^{13} - 860736438067 \nu^{12} + 3655921059216 \nu^{11} + 28446907113983 \nu^{10} + \cdots + 27\!\cdots\!92 ) / 95622835015168 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5721887747 \nu^{14} - 23283421280 \nu^{13} - 489951876929 \nu^{12} + 2023392642758 \nu^{11} + 15451457720175 \nu^{10} + \cdots + 10\!\cdots\!40 ) / 47811417507584 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 25790001789 \nu^{14} + 98910769148 \nu^{13} + 2300202479581 \nu^{12} - 8629420333984 \nu^{11} - 77673532577825 \nu^{10} + \cdots - 26\!\cdots\!60 ) / 191245670030336 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 8376263217 \nu^{14} - 32508155010 \nu^{13} - 740736899457 \nu^{12} + 2822971755806 \nu^{11} + 24720137446005 \nu^{10} + \cdots + 213227978333184 ) / 47811417507584 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 38217519565 \nu^{14} - 105576373016 \nu^{13} - 3522469617117 \nu^{12} + 9129819470172 \nu^{11} + 124913173709841 \nu^{10} + \cdots - 65\!\cdots\!08 ) / 191245670030336 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 10495297293 \nu^{14} - 34294170950 \nu^{13} - 950355451183 \nu^{12} + 2971433743968 \nu^{11} + 32855068475965 \nu^{10} + \cdots - 98\!\cdots\!04 ) / 47811417507584 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + 21\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{14} - \beta_{13} - 2 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - 2 \beta_{6} - 3 \beta_{3} + 27 \beta_{2} + \beta _1 + 279 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{13} - 4 \beta_{12} - 4 \beta_{9} - 10 \beta_{8} + 4 \beta_{7} + 10 \beta_{5} + 33 \beta_{4} + 31 \beta_{3} - 8 \beta_{2} + 499 \beta _1 - 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 31 \beta_{14} - 39 \beta_{13} - 8 \beta_{12} - 78 \beta_{11} + 47 \beta_{10} - 59 \beta_{9} - 59 \beta_{8} + 20 \beta_{7} - 102 \beta_{6} - 4 \beta_{5} + 4 \beta_{4} - 173 \beta_{3} + 703 \beta_{2} + 59 \beta _1 + 6831 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 32 \beta_{14} + 94 \beta_{13} - 236 \beta_{12} - 16 \beta_{11} + 8 \beta_{10} - 236 \beta_{9} - 510 \beta_{8} + 236 \beta_{7} - 80 \beta_{6} + 518 \beta_{5} + 965 \beta_{4} + 935 \beta_{3} - 384 \beta_{2} + 12607 \beta _1 + 323 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 695 \beta_{14} - 1175 \beta_{13} - 488 \beta_{12} - 2518 \beta_{11} + 1639 \beta_{10} - 2611 \beta_{9} - 2467 \beta_{8} + 1244 \beta_{7} - 3758 \beta_{6} - 164 \beta_{5} + 268 \beta_{4} - 6893 \beta_{3} + \cdots + 178079 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2464 \beta_{14} + 2998 \beta_{13} - 10012 \beta_{12} - 1168 \beta_{11} + 576 \beta_{10} - 9980 \beta_{9} - 19534 \beta_{8} + 9804 \beta_{7} - 5072 \beta_{6} + 19950 \beta_{5} + 27757 \beta_{4} + 27671 \beta_{3} + \cdots + 27611 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 11799 \beta_{14} - 32623 \beta_{13} - 20696 \beta_{12} - 77438 \beta_{11} + 51399 \beta_{10} - 101739 \beta_{9} - 90635 \beta_{8} + 54492 \beta_{7} - 123718 \beta_{6} - 3564 \beta_{5} + \cdots + 4825751 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 123424 \beta_{14} + 81646 \beta_{13} - 373868 \beta_{12} - 58992 \beta_{11} + 28152 \beta_{10} - 371244 \beta_{9} - 674430 \beta_{8} + 357100 \beta_{7} - 223280 \beta_{6} + 690486 \beta_{5} + \cdots + 1427675 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 80943 \beta_{14} - 878591 \beta_{13} - 765256 \beta_{12} - 2341718 \beta_{11} + 1542367 \beta_{10} - 3683291 \beta_{9} - 3123947 \beta_{8} + 2075164 \beta_{7} - 3884814 \beta_{6} + \cdots + 134270847 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 5132448 \beta_{14} + 2028630 \beta_{13} - 13071132 \beta_{12} - 2529040 \beta_{11} + 1178544 \beta_{10} - 12971260 \beta_{9} - 22181278 \beta_{8} + 12211436 \beta_{7} + \cdots + 62924891 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 5199849 \beta_{14} - 23432799 \beta_{13} - 26580216 \beta_{12} - 70390798 \beta_{11} + 45484103 \beta_{10} - 127217915 \beta_{9} - 103884283 \beta_{8} + 73493308 \beta_{7} + \cdots + 3807550375 \) 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
−5.35570
−5.02576
−3.99447
−3.50782
−2.28978
−0.882088
−0.803379
1.01142
1.14693
1.95379
2.76521
4.06083
4.47117
4.87155
5.57810
−5.35570 −9.24103 20.6835 6.18620 49.4922 25.8904 −67.9293 58.3966 −33.1314
1.2 −5.02576 1.82026 17.2582 −17.8848 −9.14820 −17.9233 −46.5296 −23.6866 89.8846
1.3 −3.99447 6.47217 7.95577 7.38287 −25.8529 21.5415 0.176689 14.8890 −29.4906
1.4 −3.50782 −5.87430 4.30483 18.0882 20.6060 8.74967 12.9620 7.50737 −63.4501
1.5 −2.28978 −1.98101 −2.75691 −0.461460 4.53608 −28.2040 24.6310 −23.0756 1.05664
1.6 −0.882088 1.30340 −7.22192 −20.0123 −1.14972 22.5444 13.4271 −25.3011 17.6526
1.7 −0.803379 8.56898 −7.35458 6.36703 −6.88414 −13.3731 12.3356 46.4274 −5.11514
1.8 1.01142 1.23042 −6.97704 20.5649 1.24447 20.3274 −15.1480 −25.4861 20.7997
1.9 1.14693 −4.66405 −6.68456 −7.91994 −5.34933 −5.62137 −16.8421 −5.24664 −9.08360
1.10 1.95379 −10.1846 −4.18272 −9.12890 −19.8985 −23.2297 −23.8024 76.7254 −17.8359
1.11 2.76521 9.50454 −0.353606 −8.85840 26.2821 30.2159 −23.0995 63.3363 −24.4953
1.12 4.06083 6.11046 8.49033 12.0550 24.8135 7.80092 1.99116 10.3377 48.9531
1.13 4.47117 −5.94903 11.9913 11.1099 −26.5991 15.8968 17.8460 8.39093 49.6743
1.14 4.87155 5.53765 15.7320 5.16356 26.9770 −21.3437 37.6671 3.66558 25.1546
1.15 5.57810 0.346090 23.1152 −12.6518 1.93052 29.7281 84.3144 −26.8802 −70.5733
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(19\) \(-1\)

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 209.4.a.d 15
3.b odd 2 1 1881.4.a.l 15
11.b odd 2 1 2299.4.a.l 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.4.a.d 15 1.a even 1 1 trivial
1881.4.a.l 15 3.b odd 2 1
2299.4.a.l 15 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} - 4 T_{2}^{14} - 89 T_{2}^{13} + 352 T_{2}^{12} + 2997 T_{2}^{11} - 11764 T_{2}^{10} - 47311 T_{2}^{9} + 186264 T_{2}^{8} + 347582 T_{2}^{7} - 1421856 T_{2}^{6} - 959244 T_{2}^{5} + 4697632 T_{2}^{4} + \cdots + 1892352 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(209))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} - 4 T^{14} - 89 T^{13} + \cdots + 1892352 \) Copy content Toggle raw display
$3$ \( T^{15} - 3 T^{14} - 278 T^{13} + \cdots - 547620616 \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots + 100114696008744 \) Copy content Toggle raw display
$7$ \( T^{15} - 73 T^{14} + \cdots - 46\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( (T - 11)^{15} \) Copy content Toggle raw display
$13$ \( T^{15} - 43 T^{14} + \cdots + 12\!\cdots\!60 \) Copy content Toggle raw display
$17$ \( T^{15} - 73 T^{14} + \cdots - 70\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( (T - 19)^{15} \) Copy content Toggle raw display
$23$ \( T^{15} - 355 T^{14} + \cdots - 14\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{15} - 385 T^{14} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{15} - 478 T^{14} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{15} + 2 T^{14} + \cdots + 63\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( T^{15} - 366 T^{14} + \cdots - 93\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{15} - 732 T^{14} + \cdots + 31\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{15} - 332 T^{14} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{15} + 555 T^{14} + \cdots + 54\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{15} + 653 T^{14} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{15} - 884 T^{14} + \cdots - 11\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{15} - 665 T^{14} + \cdots + 54\!\cdots\!48 \) Copy content Toggle raw display
$71$ \( T^{15} - 380 T^{14} + \cdots - 13\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{15} - 1263 T^{14} + \cdots + 61\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{15} - 3604 T^{14} + \cdots + 34\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{15} + 1420 T^{14} + \cdots - 16\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( T^{15} + 3856 T^{14} + \cdots + 18\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{15} + 2472 T^{14} + \cdots + 21\!\cdots\!88 \) Copy content Toggle raw display
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