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Properties

Label 209.4.a.c
Level $209$
Weight $4$
Character orbit 209.a
Self dual yes
Analytic conductor $12.331$
Analytic rank $0$
Dimension $13$
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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 2 x^{12} - 91 x^{11} + 176 x^{10} + 3117 x^{9} - 5786 x^{8} - 49725 x^{7} + 87196 x^{6} + \cdots - 86016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{5} + 1) q^{3} + (\beta_{2} + 6) q^{4} + ( - \beta_{10} + \beta_{5} - \beta_1 + 1) q^{5} + ( - \beta_{12} + \beta_{10} + \beta_{7} + \cdots + 2) q^{6}+ \cdots + (\beta_{12} - \beta_{10} - \beta_{7} + \cdots + 11) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{5} + 1) q^{3} + (\beta_{2} + 6) q^{4} + ( - \beta_{10} + \beta_{5} - \beta_1 + 1) q^{5} + ( - \beta_{12} + \beta_{10} + \beta_{7} + \cdots + 2) q^{6}+ \cdots + ( - 11 \beta_{12} + 11 \beta_{10} + \cdots - 121) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 2 q^{2} + 11 q^{3} + 82 q^{4} + 8 q^{5} + 13 q^{6} + 39 q^{7} + 6 q^{8} + 156 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 2 q^{2} + 11 q^{3} + 82 q^{4} + 8 q^{5} + 13 q^{6} + 39 q^{7} + 6 q^{8} + 156 q^{9} + 124 q^{10} - 143 q^{11} + 247 q^{12} - 23 q^{13} + 47 q^{14} + 278 q^{15} + 526 q^{16} + 73 q^{17} - 165 q^{18} - 247 q^{19} + 142 q^{20} + 77 q^{21} + 22 q^{22} + 185 q^{23} - 251 q^{24} + 641 q^{25} - 47 q^{26} + 1109 q^{27} + 289 q^{28} - 41 q^{29} + 520 q^{30} + 930 q^{31} - 194 q^{32} - 121 q^{33} - 333 q^{34} + 122 q^{35} + 1469 q^{36} + 1094 q^{37} + 38 q^{38} + 543 q^{39} + 2084 q^{40} - 766 q^{41} - 1277 q^{42} + 1024 q^{43} - 902 q^{44} - 348 q^{45} + 1927 q^{46} + 516 q^{47} + 3067 q^{48} + 1966 q^{49} + 3066 q^{50} + 1975 q^{51} - 553 q^{52} + 1017 q^{53} - 2231 q^{54} - 88 q^{55} - 1793 q^{56} - 209 q^{57} - 159 q^{58} + 15 q^{59} - 3158 q^{60} - 862 q^{61} - 3298 q^{62} + 552 q^{63} + 3694 q^{64} - 1640 q^{65} - 143 q^{66} + 621 q^{67} - 1533 q^{68} + 335 q^{69} - 3250 q^{70} + 632 q^{71} - 8349 q^{72} - 629 q^{73} - 4086 q^{74} - 2759 q^{75} - 1558 q^{76} - 429 q^{77} - 5813 q^{78} + 2444 q^{79} + 2810 q^{80} + 2265 q^{81} - 846 q^{82} + 732 q^{83} - 761 q^{84} - 3786 q^{85} - 4664 q^{86} - 1185 q^{87} - 66 q^{88} + 3178 q^{89} - 6174 q^{90} + 1349 q^{91} - 1677 q^{92} + 1328 q^{93} + 3156 q^{94} - 152 q^{95} - 13811 q^{96} + 1410 q^{97} - 6273 q^{98} - 1716 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 2 x^{12} - 91 x^{11} + 176 x^{10} + 3117 x^{9} - 5786 x^{8} - 49725 x^{7} + 87196 x^{6} + \cdots - 86016 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 179351 \nu^{12} - 6111956 \nu^{11} + 22627657 \nu^{10} + 624540444 \nu^{9} + \cdots + 665042042368 ) / 142316773888 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 632365 \nu^{12} - 12143320 \nu^{11} + 16549089 \nu^{10} + 1022191756 \nu^{9} + \cdots + 4814040741376 ) / 142316773888 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1490925 \nu^{12} - 3374106 \nu^{11} - 117105595 \nu^{10} + 266779852 \nu^{9} + \cdots - 863686138368 ) / 142316773888 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1017601 \nu^{12} - 276553 \nu^{11} - 80542386 \nu^{10} - 32232160 \nu^{9} + \cdots - 637257679104 ) / 71158386944 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1920205 \nu^{12} - 11132819 \nu^{11} - 167383848 \nu^{10} + 911265656 \nu^{9} + \cdots + 1253807937024 ) / 71158386944 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 139235 \nu^{12} - 445304 \nu^{11} - 12795239 \nu^{10} + 39284284 \nu^{9} + 442283335 \nu^{8} + \cdots + 4926684672 ) / 3309692416 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10489149 \nu^{12} - 29313602 \nu^{11} + 912896051 \nu^{10} + 2404111664 \nu^{9} + \cdots - 463328483328 ) / 142316773888 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7125793 \nu^{12} + 3639059 \nu^{11} - 649932012 \nu^{10} - 291399086 \nu^{9} + \cdots + 456899993088 ) / 71158386944 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 16394151 \nu^{12} + 20435252 \nu^{11} + 1462858283 \nu^{10} - 1790208568 \nu^{9} + \cdots - 802773986816 ) / 142316773888 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 17020531 \nu^{12} + 12986530 \nu^{11} - 1507070505 \nu^{10} - 1046131624 \nu^{9} + \cdots + 548980252160 ) / 142316773888 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} + \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} + 23\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{12} - 4 \beta_{10} - 2 \beta_{9} - 2 \beta_{6} + 6 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + \cdots + 310 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 39\beta_{12} + 39\beta_{9} - 39\beta_{8} - 8\beta_{7} - 39\beta_{6} + 63\beta_{5} + 8\beta_{4} + 595\beta _1 - 62 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 126 \beta_{12} + 8 \beta_{11} - 212 \beta_{10} - 78 \beta_{9} + 24 \beta_{8} - 32 \beta_{7} + \cdots + 7782 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1275 \beta_{12} + 72 \beta_{11} + 64 \beta_{10} + 1283 \beta_{9} - 1251 \beta_{8} - 416 \beta_{7} + \cdots - 1566 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5526 \beta_{12} + 480 \beta_{11} - 8460 \beta_{10} - 2454 \beta_{9} + 1392 \beta_{8} - 2016 \beta_{7} + \cdots + 208294 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 40243 \beta_{12} + 4864 \beta_{11} + 3776 \beta_{10} + 40307 \beta_{9} - 37843 \beta_{8} - 16536 \beta_{7} + \cdots - 32310 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 210454 \beta_{12} + 19096 \beta_{11} - 304948 \beta_{10} - 73094 \beta_{9} + 55848 \beta_{8} + \cdots + 5793542 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1263587 \beta_{12} + 225368 \beta_{11} + 150720 \beta_{10} + 1243259 \beta_{9} - 1122331 \beta_{8} + \cdots - 421022 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7470214 \beta_{12} + 645520 \beta_{11} - 10444476 \beta_{10} - 2139558 \beta_{9} + 1929504 \beta_{8} + \cdots + 165422118 \) 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
5.62918
4.82823
4.48413
3.21121
2.98386
1.33472
0.224790
−0.264443
−2.57872
−3.02090
−4.33968
−4.99897
−5.49342
−5.62918 10.0382 23.6877 −2.17343 −56.5070 1.58350 −88.3091 73.7660 12.2346
1.2 −4.82823 −1.78306 15.3118 −12.0513 8.60905 32.7048 −35.3032 −23.8207 58.1866
1.3 −4.48413 −7.61802 12.1074 0.924111 34.1602 −24.8241 −18.4180 31.0342 −4.14383
1.4 −3.21121 −2.40039 2.31189 14.8396 7.70816 20.3294 18.2657 −21.2381 −47.6532
1.5 −2.98386 3.69270 0.903438 −11.2324 −11.0185 −31.8562 21.1752 −13.3640 33.5161
1.6 −1.33472 8.27114 −6.21852 12.3033 −11.0397 31.4819 18.9777 41.4117 −16.4215
1.7 −0.224790 −4.32964 −7.94947 −17.8280 0.973257 −19.8188 3.58527 −8.25425 4.00756
1.8 0.264443 −4.63433 −7.93007 7.68704 −1.22552 −7.09559 −4.21260 −5.52296 2.03279
1.9 2.57872 8.66678 −1.35022 16.9434 22.3492 −6.89893 −24.1116 48.1130 43.6921
1.10 3.02090 −8.02896 1.12583 −20.1089 −24.2547 33.1653 −20.7662 37.4642 −60.7468
1.11 4.33968 2.40523 10.8328 6.68433 10.4379 24.5435 12.2935 −21.2149 29.0079
1.12 4.99897 8.31216 16.9897 −8.69474 41.5522 −3.84543 44.9392 42.0920 −43.4648
1.13 5.49342 −1.59182 22.1777 20.7070 −8.74456 −10.4694 77.8840 −24.4661 113.752
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
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.c ✓ 13
3.b odd 2 1 1881.4.a.k 13
11.b odd 2 1 2299.4.a.k 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.4.a.c ✓ 13 1.a even 1 1 trivial
1881.4.a.k 13 3.b odd 2 1
2299.4.a.k 13 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} + 2 T_{2}^{12} - 91 T_{2}^{11} - 176 T_{2}^{10} + 3117 T_{2}^{9} + 5786 T_{2}^{8} + \cdots + 86016 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(209))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} + 2 T^{12} + \cdots + 86016 \) Copy content Toggle raw display
$3$ \( T^{13} + \cdots + 444198064 \) Copy content Toggle raw display
$5$ \( T^{13} + \cdots - 2789379165656 \) Copy content Toggle raw display
$7$ \( T^{13} + \cdots + 833325521817216 \) Copy content Toggle raw display
$11$ \( (T + 11)^{13} \) Copy content Toggle raw display
$13$ \( T^{13} + \cdots + 49\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{13} + \cdots + 90\!\cdots\!88 \) Copy content Toggle raw display
$19$ \( (T + 19)^{13} \) Copy content Toggle raw display
$23$ \( T^{13} + \cdots - 27\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots - 10\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots + 19\!\cdots\!88 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots - 29\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots + 69\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots - 13\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots - 96\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots - 93\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots + 89\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots + 16\!\cdots\!52 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 37\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots - 71\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots - 24\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 26\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 88\!\cdots\!40 \) Copy content Toggle raw display
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