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Properties

Label 109.4.a.a
Level $109$
Weight $4$
Character orbit 109.a
Self dual yes
Analytic conductor $6.431$
Analytic rank $1$
Dimension $12$
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] = [109,4,Mod(1,109)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(109, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("109.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 109 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 109.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: \(6.43120819063\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} - 48 x^{10} + 285 x^{9} + 850 x^{8} - 4937 x^{7} - 6779 x^{6} + 37874 x^{5} + \cdots + 8448 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + ( - \beta_{8} - 2) q^{3} + (\beta_{6} - \beta_{5} + 4) q^{4} + (\beta_{9} + \beta_{8} - \beta_{6} + \cdots - 3) q^{5}+ \cdots + ( - \beta_{11} + 2 \beta_{10} + \cdots + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + ( - \beta_{8} - 2) q^{3} + (\beta_{6} - \beta_{5} + 4) q^{4} + (\beta_{9} + \beta_{8} - \beta_{6} + \cdots - 3) q^{5}+ \cdots + (52 \beta_{11} + 19 \beta_{10} + \cdots - 734) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} - 19 q^{3} + 36 q^{4} - 40 q^{5} - 53 q^{6} - 32 q^{7} - 69 q^{8} + 55 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{2} - 19 q^{3} + 36 q^{4} - 40 q^{5} - 53 q^{6} - 32 q^{7} - 69 q^{8} + 55 q^{9} - 64 q^{10} - 206 q^{11} - 192 q^{12} - 89 q^{13} - 289 q^{14} - 138 q^{15} + 56 q^{16} - 140 q^{17} - 52 q^{18} - 332 q^{19} - 555 q^{20} - 176 q^{21} + 27 q^{22} - 351 q^{23} - 635 q^{24} + 24 q^{25} - 439 q^{26} - 532 q^{27} - 145 q^{28} - 280 q^{29} + 1103 q^{30} - 276 q^{31} + 417 q^{32} + 454 q^{33} + 1247 q^{34} - 12 q^{35} + 2085 q^{36} + 205 q^{37} - 33 q^{38} + 542 q^{39} + 888 q^{40} - 942 q^{41} + 3107 q^{42} + 115 q^{43} - 450 q^{44} - 382 q^{45} + 2325 q^{46} - 543 q^{47} + 1474 q^{48} + 710 q^{49} + 1437 q^{50} - 1462 q^{51} + 1822 q^{52} - 349 q^{53} + 222 q^{54} + 504 q^{55} - 648 q^{56} + 1256 q^{57} + 274 q^{58} - 3014 q^{59} + 2526 q^{60} - 264 q^{61} + 2040 q^{62} - 1486 q^{63} + 19 q^{64} - 1356 q^{65} + 2048 q^{66} + 292 q^{67} + 413 q^{68} - 1130 q^{69} + 2439 q^{70} - 3172 q^{71} + 2309 q^{72} - 429 q^{73} - 2122 q^{74} - 1003 q^{75} - 2173 q^{76} - 838 q^{77} + 509 q^{78} + 57 q^{79} - 3571 q^{80} + 868 q^{81} + 2258 q^{82} - 2061 q^{83} + 404 q^{84} - 276 q^{85} - 2158 q^{86} + 1866 q^{87} + 634 q^{88} - 1573 q^{89} - 3190 q^{90} - 1974 q^{91} - 3794 q^{92} - 1454 q^{93} - 713 q^{94} - 2994 q^{95} - 4784 q^{96} - 1077 q^{97} - 488 q^{98} - 7582 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} - 48 x^{10} + 285 x^{9} + 850 x^{8} - 4937 x^{7} - 6779 x^{6} + 37874 x^{5} + \cdots + 8448 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 459097201 \nu^{11} + 13602333466 \nu^{10} - 62405432280 \nu^{9} - 150758988845 \nu^{8} + \cdots + 292517529054272 ) / 73458444540544 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2771888915 \nu^{11} + 45043455306 \nu^{10} - 597037151752 \nu^{9} - 1758185541177 \nu^{8} + \cdots - 16\!\cdots\!28 ) / 146916889081088 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 245467856 \nu^{11} - 13946509823 \nu^{10} + 111206107294 \nu^{9} + 565233162476 \nu^{8} + \cdots + 57155969927136 ) / 9182305567568 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6928520777 \nu^{11} + 82953363086 \nu^{10} - 962012433912 \nu^{9} - 3895149171547 \nu^{8} + \cdots + 718710330700736 ) / 73458444540544 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6928520777 \nu^{11} + 82953363086 \nu^{10} - 962012433912 \nu^{9} - 3895149171547 \nu^{8} + \cdots - 89332559245248 ) / 73458444540544 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14196954561 \nu^{11} - 151751583534 \nu^{10} + 1688341913240 \nu^{9} + \cdots + 12\!\cdots\!08 ) / 146916889081088 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3721211467 \nu^{11} - 39523549558 \nu^{10} - 74101921896 \nu^{9} + 1814169218591 \nu^{8} + \cdots + 63652271424 ) / 20988127011584 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 9705298821 \nu^{11} + 23446966170 \nu^{10} + 639089100216 \nu^{9} + \cdots - 34257549756096 ) / 20988127011584 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 168674982083 \nu^{11} - 902241494982 \nu^{10} - 8274649275560 \nu^{9} + \cdots - 816645244327872 ) / 146916889081088 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 193133906727 \nu^{11} - 1074736812606 \nu^{10} - 9150529251496 \nu^{9} + \cdots - 718713236053184 ) / 146916889081088 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} + 2\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{11} + 2 \beta_{10} + \beta_{8} + 2 \beta_{7} + 6 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 5 \beta_{11} + 6 \beta_{10} - 4 \beta_{8} + 12 \beta_{7} + 47 \beta_{6} - 28 \beta_{5} + 12 \beta_{4} + \cdots + 212 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 70 \beta_{11} + 72 \beta_{10} - 2 \beta_{9} + 13 \beta_{8} + 100 \beta_{7} + 286 \beta_{6} + \cdots + 620 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 277 \beta_{11} + 336 \beta_{10} + 34 \beta_{9} - 183 \beta_{8} + 620 \beta_{7} + 1878 \beta_{6} + \cdots + 5459 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2317 \beta_{11} + 2514 \beta_{10} + 74 \beta_{9} - 202 \beta_{8} + 4118 \beta_{7} + 11383 \beta_{6} + \cdots + 23588 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 11472 \beta_{11} + 13866 \beta_{10} + 1930 \beta_{9} - 6550 \beta_{8} + 25356 \beta_{7} + \cdots + 166931 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 79356 \beta_{11} + 89894 \beta_{10} + 8012 \beta_{9} - 21120 \beta_{8} + 158142 \beta_{7} + \cdots + 867899 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 437582 \beta_{11} + 526688 \beta_{10} + 82576 \beta_{9} - 227177 \beta_{8} + 965506 \beta_{7} + \cdots + 5602244 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2799135 \beta_{11} + 3256566 \beta_{10} + 419238 \beta_{9} - 1024716 \beta_{8} + 5897348 \beta_{7} + \cdots + 31676272 \) 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
−4.11420
−3.73577
−3.20174
−1.86974
−1.69540
−0.0581710
1.63639
1.94532
2.96276
3.89854
4.19220
6.03981
−5.11420 7.39483 18.1550 −9.81201 −37.8187 −16.5055 −51.9350 27.6835 50.1806
1.2 −4.73577 −1.46046 14.4275 −10.3187 6.91639 22.5325 −30.4393 −24.8671 48.8671
1.3 −4.20174 −9.34760 9.65463 −2.24647 39.2762 24.4855 −6.95233 60.3777 9.43910
1.4 −2.86974 −6.55632 0.235431 17.1471 18.8150 −8.08509 22.2823 15.9854 −49.2079
1.5 −2.69540 3.75940 −0.734830 1.45307 −10.1331 −13.2092 23.5438 −12.8669 −3.91660
1.6 −1.05817 −0.146709 −6.88027 0.972968 0.155243 31.8116 15.7459 −26.9785 −1.02957
1.7 0.636386 3.31977 −7.59501 −0.524144 2.11266 −33.0680 −9.92445 −15.9791 −0.333558
1.8 0.945321 5.17257 −7.10637 −18.7353 4.88974 0.205070 −14.2804 −0.244564 −17.7109
1.9 1.96276 −5.58098 −4.14756 14.7197 −10.9541 −5.04083 −23.8428 4.14731 28.8913
1.10 2.89854 −3.25136 0.401553 −2.96333 −9.42420 −15.8978 −22.0244 −16.4287 −8.58934
1.11 3.19220 −2.79842 2.19013 −15.7264 −8.93312 8.24565 −18.5462 −19.1688 −50.2019
1.12 5.03981 −9.50472 17.3997 −13.9664 −47.9020 −27.4738 47.3728 63.3397 −70.3883
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(109\) \(-1\)

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 109.4.a.a ✓ 12
3.b odd 2 1 981.4.a.b 12
4.b odd 2 1 1744.4.a.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
109.4.a.a ✓ 12 1.a even 1 1 trivial
981.4.a.b 12 3.b odd 2 1
1744.4.a.f 12 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 6 T_{2}^{11} - 48 T_{2}^{10} - 305 T_{2}^{9} + 760 T_{2}^{8} + 5175 T_{2}^{7} + \cdots + 45864 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(109))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 6 T^{11} + \cdots + 45864 \) Copy content Toggle raw display
$3$ \( T^{12} + 19 T^{11} + \cdots + 3025488 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 518760828 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots - 3808782019904 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 14\!\cdots\!12 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 35\!\cdots\!88 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots - 16\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots - 32\!\cdots\!67 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots - 12\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 99\!\cdots\!28 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 23\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots - 26\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 14\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots - 11\!\cdots\!57 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 51\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 47\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 26\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 44\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 57\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 26\!\cdots\!61 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 15\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 95\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 22\!\cdots\!41 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 14\!\cdots\!09 \) Copy content Toggle raw display
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