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Properties

Label 115.6.a.e
Level $115$
Weight $6$
Character orbit 115.a
Self dual yes
Analytic conductor $18.444$
Analytic rank $0$
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] = [115,6,Mod(1,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 115.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: \(18.4441392785\)
Analytic rank: \(0\)
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: \( x^{12} - 4 x^{11} - 329 x^{10} + 1059 x^{9} + 41059 x^{8} - 99023 x^{7} - 2392947 x^{6} + 3889937 x^{5} + 63665212 x^{4} - 59034365 x^{3} - 601032676 x^{2} + \cdots + 4039776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{4} - \beta_1 + 2) q^{3} + (\beta_{2} - \beta_1 + 25) q^{4} + 25 q^{5} + (\beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 39) q^{6} + ( - \beta_{8} - \beta_{4} - \beta_1 + 2) q^{7} + (\beta_{11} + \beta_{10} - \beta_{6} - 2 \beta_{4} + 2 \beta_{2} - 20 \beta_1 + 63) q^{8} + ( - \beta_{11} - \beta_{10} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - 4 \beta_{4} + 2 \beta_{2} + \cdots + 130) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{4} - \beta_1 + 2) q^{3} + (\beta_{2} - \beta_1 + 25) q^{4} + 25 q^{5} + (\beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 39) q^{6} + ( - \beta_{8} - \beta_{4} - \beta_1 + 2) q^{7} + (\beta_{11} + \beta_{10} - \beta_{6} - 2 \beta_{4} + 2 \beta_{2} - 20 \beta_1 + 63) q^{8} + ( - \beta_{11} - \beta_{10} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - 4 \beta_{4} + 2 \beta_{2} + \cdots + 130) q^{9}+ \cdots + (612 \beta_{11} - 220 \beta_{10} - 392 \beta_{9} + 288 \beta_{8} + \cdots - 63054) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{2} + 22 q^{3} + 294 q^{4} + 300 q^{5} + 454 q^{6} + 16 q^{7} + 675 q^{8} + 1598 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{2} + 22 q^{3} + 294 q^{4} + 300 q^{5} + 454 q^{6} + 16 q^{7} + 675 q^{8} + 1598 q^{9} + 200 q^{10} + 132 q^{11} + 728 q^{12} - 236 q^{13} + 359 q^{14} + 550 q^{15} + 4514 q^{16} + 1666 q^{17} - 3096 q^{18} + 616 q^{19} + 7350 q^{20} + 2732 q^{21} + 305 q^{22} - 6348 q^{23} + 18873 q^{24} + 7500 q^{25} - 4502 q^{26} + 11584 q^{27} + 5407 q^{28} + 23722 q^{29} + 11350 q^{30} + 18446 q^{31} + 35808 q^{32} + 4416 q^{33} + 53123 q^{34} + 400 q^{35} + 68916 q^{36} + 10394 q^{37} + 18681 q^{38} + 27032 q^{39} + 16875 q^{40} + 48232 q^{41} - 18980 q^{42} + 10732 q^{43} - 4765 q^{44} + 39950 q^{45} - 4232 q^{46} - 30448 q^{47} - 2052 q^{48} + 26948 q^{49} + 5000 q^{50} + 1524 q^{51} - 55346 q^{52} + 36494 q^{53} + 55567 q^{54} + 3300 q^{55} - 50981 q^{56} + 37572 q^{57} - 83373 q^{58} - 23870 q^{59} + 18200 q^{60} + 30862 q^{61} + 63582 q^{62} - 49698 q^{63} + 29965 q^{64} - 5900 q^{65} - 235225 q^{66} - 71910 q^{67} - 39371 q^{68} - 11638 q^{69} + 8975 q^{70} + 167158 q^{71} - 296052 q^{72} + 52152 q^{73} - 59356 q^{74} + 13750 q^{75} - 230417 q^{76} + 4808 q^{77} - 469771 q^{78} - 123092 q^{79} + 112850 q^{80} + 159868 q^{81} - 140098 q^{82} + 89322 q^{83} - 488082 q^{84} + 41650 q^{85} - 55318 q^{86} - 334376 q^{87} - 104551 q^{88} - 46184 q^{89} - 77400 q^{90} - 153444 q^{91} - 155526 q^{92} - 16576 q^{93} - 456595 q^{94} + 15400 q^{95} + 330540 q^{96} - 94220 q^{97} + 413841 q^{98} - 740784 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 329 x^{10} + 1059 x^{9} + 41059 x^{8} - 99023 x^{7} - 2392947 x^{6} + 3889937 x^{5} + 63665212 x^{4} - 59034365 x^{3} - 601032676 x^{2} + \cdots + 4039776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4105114832381 \nu^{11} + 201360999557039 \nu^{10} + \cdots - 78\!\cdots\!52 ) / 49\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4485970522399 \nu^{11} - 13077055877179 \nu^{10} + \cdots + 42\!\cdots\!52 ) / 49\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1306807350427 \nu^{11} + 24571239904313 \nu^{10} + \cdots + 38\!\cdots\!36 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12427339108299 \nu^{11} - 7270386628201 \nu^{10} + \cdots - 14\!\cdots\!72 ) / 24\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 36432839572533 \nu^{11} - 115989961322647 \nu^{10} + \cdots - 43\!\cdots\!84 ) / 49\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 32723488673401 \nu^{11} + 19321474848899 \nu^{10} + \cdots + 50\!\cdots\!48 ) / 24\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6009477378277 \nu^{11} - 21549568159577 \nu^{10} + \cdots + 11\!\cdots\!96 ) / 38\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 100656069415571 \nu^{11} - 543196049549311 \nu^{10} + \cdots + 59\!\cdots\!28 ) / 49\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 116538806587371 \nu^{11} + 502501164538551 \nu^{10} + \cdots - 87\!\cdots\!48 ) / 49\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 56 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - \beta_{10} + \beta_{6} + 2\beta_{4} + \beta_{2} + 84\beta _1 + 42 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} - 2 \beta_{9} + 4 \beta_{8} + 4 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + 21 \beta_{4} - \beta_{3} + 114 \beta_{2} + 166 \beta _1 + 4683 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 147 \beta_{11} - 157 \beta_{10} + 2 \beta_{9} + 124 \beta_{6} - 3 \beta_{5} + 377 \beta_{4} - 19 \beta_{3} + 167 \beta_{2} + 7886 \beta _1 + 7309 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 164 \beta_{11} + 42 \beta_{10} - 262 \beta_{9} + 750 \beta_{8} + 688 \beta_{7} + 375 \beta_{6} + 345 \beta_{5} + 4441 \beta_{4} - 261 \beta_{3} + 12147 \beta_{2} + 22211 \beta _1 + 436251 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 17202 \beta_{11} - 19728 \beta_{10} + 850 \beta_{9} + 476 \beta_{8} + 928 \beta_{7} + 13386 \beta_{6} - 474 \beta_{5} + 59574 \beta_{4} - 3598 \beta_{3} + 23652 \beta_{2} + 783431 \beta _1 + 996504 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 19130 \beta_{11} - 11362 \beta_{10} - 23402 \beta_{9} + 107278 \beta_{8} + 94540 \beta_{7} + 55570 \beta_{6} + 33768 \beta_{5} + 672664 \beta_{4} - 45862 \beta_{3} + 1285313 \beta_{2} + \cdots + 42918402 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1882579 \beta_{11} - 2322275 \beta_{10} + 181432 \beta_{9} + 147218 \beta_{8} + 266624 \beta_{7} + 1429391 \beta_{6} - 54402 \beta_{5} + 8505182 \beta_{4} - 506724 \beta_{3} + \cdots + 125078532 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1861399 \beta_{11} - 3471513 \beta_{10} - 1533108 \beta_{9} + 14006586 \beta_{8} + 12065488 \beta_{7} + 7544088 \beta_{6} + 3180243 \beta_{5} + 90854665 \beta_{4} + \cdots + 4356449237 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 201720793 \beta_{11} - 267429131 \beta_{10} + 30247242 \beta_{9} + 30042510 \beta_{8} + 50327464 \beta_{7} + 154434086 \beta_{6} - 5900985 \beta_{5} + \cdots + 15167057835 \) 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
10.8081
9.88408
9.44196
6.41367
4.85960
0.286598
−0.0230442
−4.18883
−6.78525
−7.26134
−9.31650
−10.1191
−9.80811 −29.7028 64.1990 25.0000 291.328 −68.7900 −315.812 639.254 −245.203
1.2 −8.88408 −9.77577 46.9269 25.0000 86.8487 148.975 −132.612 −147.434 −222.102
1.3 −8.44196 28.1750 39.2667 25.0000 −237.853 −83.6202 −61.3452 550.833 −211.049
1.4 −5.41367 6.74608 −2.69220 25.0000 −36.5211 167.896 187.812 −197.490 −135.342
1.5 −3.85960 −12.6293 −17.1035 25.0000 48.7440 −113.441 189.520 −83.5011 −96.4901
1.6 0.713402 22.8326 −31.4911 25.0000 16.2889 23.6618 −45.2947 278.329 17.8351
1.7 1.02304 −8.89333 −30.9534 25.0000 −9.09827 −228.319 −64.4041 −163.909 25.5761
1.8 5.18883 −22.3016 −5.07606 25.0000 −115.719 6.29344 −192.381 254.359 129.721
1.9 7.78525 27.6988 28.6101 25.0000 215.642 167.772 −26.3911 524.222 194.631
1.10 8.26134 12.5950 36.2497 25.0000 104.052 63.2337 35.1081 −84.3653 206.533
1.11 10.3165 −11.9831 74.4302 25.0000 −123.623 148.504 437.732 −99.4063 257.913
1.12 11.1191 19.2382 91.6335 25.0000 213.911 −216.165 663.068 127.109 277.976
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(23\) \(1\)

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 115.6.a.e ✓ 12
3.b odd 2 1 1035.6.a.m 12
5.b even 2 1 575.6.a.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.6.a.e ✓ 12 1.a even 1 1 trivial
575.6.a.g 12 5.b even 2 1
1035.6.a.m 12 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 8 T_{2}^{11} - 307 T_{2}^{10} + 2231 T_{2}^{9} + 35620 T_{2}^{8} - 227565 T_{2}^{7} - 1917514 T_{2}^{6} + 10198454 T_{2}^{5} + 46692536 T_{2}^{4} - 185549448 T_{2}^{3} + \cdots - 429434688 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(115))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 8 T^{11} - 307 T^{10} + \cdots - 429434688 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 253857988051200 \) Copy content Toggle raw display
$5$ \( (T - 25)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 16 T^{11} + \cdots - 18\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{12} - 132 T^{11} + \cdots + 25\!\cdots\!80 \) Copy content Toggle raw display
$13$ \( T^{12} + 236 T^{11} + \cdots - 11\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{12} - 1666 T^{11} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} - 616 T^{11} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T + 529)^{12} \) Copy content Toggle raw display
$29$ \( T^{12} - 23722 T^{11} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} - 18446 T^{11} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} - 10394 T^{11} + \cdots + 23\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( T^{12} - 48232 T^{11} + \cdots - 44\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{12} - 10732 T^{11} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + 30448 T^{11} + \cdots - 34\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{12} - 36494 T^{11} + \cdots - 38\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{12} + 23870 T^{11} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} - 30862 T^{11} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{12} + 71910 T^{11} + \cdots + 35\!\cdots\!20 \) Copy content Toggle raw display
$71$ \( T^{12} - 167158 T^{11} + \cdots - 37\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{12} - 52152 T^{11} + \cdots - 11\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{12} + 123092 T^{11} + \cdots + 12\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{12} - 89322 T^{11} + \cdots + 47\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{12} + 46184 T^{11} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + 94220 T^{11} + \cdots + 31\!\cdots\!96 \) Copy content Toggle raw display
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