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Properties

Label 109.4.a.b
Level $109$
Weight $4$
Character orbit 109.a
Self dual yes
Analytic conductor $6.431$
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] = [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: \(0\)
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: \( x^{15} - 4 x^{14} - 88 x^{13} + 335 x^{12} + 2994 x^{11} - 10431 x^{10} - 50374 x^{9} + 148793 x^{8} + \cdots - 2560 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{8} + 1) q^{3} + (\beta_{2} + 5) q^{4} + (\beta_{11} + 3) q^{5} + ( - \beta_{11} - \beta_{8} + \beta_{5} + \cdots + 1) q^{6}+ \cdots + ( - \beta_{12} - \beta_{11} + \cdots + 12) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{8} + 1) q^{3} + (\beta_{2} + 5) q^{4} + (\beta_{11} + 3) q^{5} + ( - \beta_{11} - \beta_{8} + \beta_{5} + \cdots + 1) q^{6}+ \cdots + ( - 43 \beta_{14} - 3 \beta_{13} + \cdots + 213) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 4 q^{2} + 17 q^{3} + 72 q^{4} + 40 q^{5} + 31 q^{6} + 24 q^{7} + 51 q^{8} + 190 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 4 q^{2} + 17 q^{3} + 72 q^{4} + 40 q^{5} + 31 q^{6} + 24 q^{7} + 51 q^{8} + 190 q^{9} - 4 q^{10} + 212 q^{11} + 96 q^{12} + 15 q^{13} + 47 q^{14} + 102 q^{15} + 392 q^{16} + 166 q^{17} + 218 q^{18} + 238 q^{19} + 405 q^{20} + 160 q^{21} + 247 q^{22} + 293 q^{23} + 373 q^{24} + 399 q^{25} + 653 q^{26} + 764 q^{27} - 557 q^{28} + 212 q^{29} - 1539 q^{30} + 92 q^{31} - 773 q^{32} - 82 q^{33} - 1469 q^{34} + 764 q^{35} - 383 q^{36} - 503 q^{37} - 181 q^{38} - 30 q^{39} - 1428 q^{40} - 144 q^{41} - 3691 q^{42} - 517 q^{43} + 944 q^{44} - 782 q^{45} - 2139 q^{46} + 1033 q^{47} - 1590 q^{48} + 1073 q^{49} - 825 q^{50} - 610 q^{51} - 3428 q^{52} + 139 q^{53} + 56 q^{54} + 104 q^{55} - 644 q^{56} - 724 q^{57} - 1742 q^{58} + 3212 q^{59} - 2718 q^{60} - 92 q^{61} + 478 q^{62} - 622 q^{63} - 881 q^{64} + 16 q^{65} + 32 q^{66} + 454 q^{67} + 1379 q^{68} - 646 q^{69} - 2011 q^{70} + 1444 q^{71} - 149 q^{72} + 747 q^{73} - 1378 q^{74} + 3665 q^{75} - 517 q^{76} + 282 q^{77} - 1899 q^{78} + 1009 q^{79} + 3533 q^{80} + 3207 q^{81} + 1798 q^{82} + 4515 q^{83} + 736 q^{84} + 804 q^{85} + 738 q^{86} + 3154 q^{87} - 2 q^{88} + 4791 q^{89} - 4784 q^{90} + 3494 q^{91} + 1286 q^{92} + 946 q^{93} + 547 q^{94} + 1034 q^{95} + 2154 q^{96} + 679 q^{97} + 3020 q^{98} + 3344 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 4 x^{14} - 88 x^{13} + 335 x^{12} + 2994 x^{11} - 10431 x^{10} - 50374 x^{9} + 148793 x^{8} + \cdots - 2560 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 119723576611525 \nu^{14} - 935110355878260 \nu^{13} + \cdots - 10\!\cdots\!32 ) / 19\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 201433656926377 \nu^{14} + \cdots - 37\!\cdots\!92 ) / 19\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 20245914951998 \nu^{14} + 54662936317677 \nu^{13} + \cdots + 11\!\cdots\!76 ) / 15\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 172470902377657 \nu^{14} - 584804298913900 \nu^{13} + \cdots - 71\!\cdots\!64 ) / 63\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 320320054972711 \nu^{14} + 737628888337782 \nu^{13} + \cdots - 19\!\cdots\!24 ) / 95\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 125232311046595 \nu^{14} - 622894933278886 \nu^{13} + \cdots + 11\!\cdots\!84 ) / 31\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 266234796202525 \nu^{14} + \cdots - 59\!\cdots\!60 ) / 47\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 16\!\cdots\!85 \nu^{14} + \cdots - 38\!\cdots\!40 ) / 19\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 19\!\cdots\!59 \nu^{14} + \cdots + 28\!\cdots\!84 ) / 19\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 21\!\cdots\!21 \nu^{14} + \cdots + 33\!\cdots\!12 ) / 19\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 565392491042735 \nu^{14} + \cdots + 73\!\cdots\!16 ) / 47\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 23\!\cdots\!75 \nu^{14} + \cdots + 36\!\cdots\!84 ) / 19\!\cdots\!68 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + 21\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + 4 \beta_{8} + \cdots + 274 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 8 \beta_{14} - 4 \beta_{13} + 6 \beta_{12} + 8 \beta_{11} - \beta_{10} + 3 \beta_{9} + 37 \beta_{7} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 66 \beta_{14} - 20 \beta_{13} - 32 \beta_{12} - 72 \beta_{11} - 68 \beta_{10} - 60 \beta_{9} + \cdots + 6433 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 381 \beta_{14} - 223 \beta_{13} + 253 \beta_{12} + 421 \beta_{11} - 46 \beta_{10} + 154 \beta_{9} + \cdots - 1131 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2745 \beta_{14} + 83 \beta_{13} - 983 \beta_{12} - 3085 \beta_{11} - 1817 \beta_{10} - 1437 \beta_{9} + \cdots + 159878 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 13261 \beta_{14} - 8969 \beta_{13} + 7837 \beta_{12} + 15799 \beta_{11} - 1591 \beta_{10} + \cdots - 67285 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 96567 \beta_{14} + 24351 \beta_{13} - 31263 \beta_{12} - 110817 \beta_{11} - 44991 \beta_{10} + \cdots + 4116386 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 415072 \beta_{14} - 319510 \beta_{13} + 218068 \beta_{12} + 523338 \beta_{11} - 51409 \beta_{10} + \cdots - 2784472 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3138641 \beta_{14} + 1313053 \beta_{13} - 1000103 \beta_{12} - 3677155 \beta_{11} - 1082960 \beta_{10} + \cdots + 108581860 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 12471391 \beta_{14} - 10724029 \beta_{13} + 5810813 \beta_{12} + 16379619 \beta_{11} + \cdots - 101553233 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 97771154 \beta_{14} + 53686668 \beta_{13} - 31632534 \beta_{12} - 117056840 \beta_{11} + \cdots + 2915162345 \) 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
−5.42909
−4.83296
−3.98415
−2.72174
−2.57704
−0.951721
−0.476779
−0.00914468
0.209278
2.26836
3.67025
4.06788
4.54729
5.05201
5.16756
−5.42909 −5.14163 21.4750 2.68542 27.9144 −26.8991 −73.1569 −0.563608 −14.5794
1.2 −4.83296 6.14390 15.3575 19.9133 −29.6933 29.4875 −35.5587 10.7475 −96.2404
1.3 −3.98415 0.273863 7.87342 7.57215 −1.09111 −5.90031 0.504316 −26.9250 −30.1685
1.4 −2.72174 7.88108 −0.592158 −12.7286 −21.4502 14.4486 23.3856 35.1114 34.6440
1.5 −2.57704 −4.81088 −1.35884 −18.3413 12.3979 −19.0628 24.1182 −3.85540 47.2664
1.6 −0.951721 9.89918 −7.09423 14.1549 −9.42126 −6.21685 14.3655 70.9938 −13.4716
1.7 −0.476779 −4.03144 −7.77268 −6.79446 1.92211 18.1436 7.52009 −10.7475 3.23946
1.8 −0.00914468 0.820286 −7.99992 18.0810 −0.00750126 6.95666 0.146314 −26.3271 −0.165345
1.9 0.209278 −9.15419 −7.95620 −1.27734 −1.91577 −7.55784 −3.33928 56.7992 −0.267319
1.10 2.26836 6.79191 −2.85452 0.198399 15.4065 27.2196 −24.6220 19.1300 0.450042
1.11 3.67025 5.68443 5.47071 11.2905 20.8633 −1.29290 −9.28311 5.31279 41.4390
1.12 4.06788 −7.69739 8.54763 9.12850 −31.3120 32.9791 2.22770 32.2498 37.1336
1.13 4.54729 10.0864 12.6778 −14.0830 45.8657 −29.9321 21.2715 74.7351 −64.0396
1.14 5.05201 −1.78412 17.5228 16.8944 −9.01341 −23.9782 48.1091 −23.8169 85.3508
1.15 5.16756 2.03862 18.7036 −6.69390 10.5347 15.6050 55.3117 −22.8440 −34.5911
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.b ✓ 15
3.b odd 2 1 981.4.a.e 15
4.b odd 2 1 1744.4.a.g 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
109.4.a.b ✓ 15 1.a even 1 1 trivial
981.4.a.e 15 3.b odd 2 1
1744.4.a.g 15 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} - 4 T_{2}^{14} - 88 T_{2}^{13} + 335 T_{2}^{12} + 2994 T_{2}^{11} - 10431 T_{2}^{10} + \cdots - 2560 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(109))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} - 4 T^{14} + \cdots - 2560 \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots - 1071653120 \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots - 6838377112424 \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots - 99\!\cdots\!20 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 36\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots - 45\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots + 14\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 24\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 95\!\cdots\!30 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 54\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 21\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 95\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 55\!\cdots\!80 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 35\!\cdots\!06 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 16\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 10\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 23\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 18\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 94\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 41\!\cdots\!14 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 19\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 68\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 22\!\cdots\!50 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 36\!\cdots\!30 \) Copy content Toggle raw display
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