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Properties

Label 755.2.a.k
Level $755$
Weight $2$
Character orbit 755.a
Self dual yes
Analytic conductor $6.029$
Analytic rank $0$
Dimension $18$
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] = [755,2,Mod(1,755)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(755, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("755.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 755 = 5 \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 755.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: \(6.02870535261\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 2 x^{17} - 32 x^{16} + 64 x^{15} + 417 x^{14} - 839 x^{13} - 2829 x^{12} + 5789 x^{11} + \cdots + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{10} q^{3} + (\beta_{2} + 2) q^{4} - q^{5} + (\beta_{14} - \beta_{12} + \beta_{11} + \cdots + 1) q^{6}+ \cdots + (\beta_{15} - \beta_{7} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{10} q^{3} + (\beta_{2} + 2) q^{4} - q^{5} + (\beta_{14} - \beta_{12} + \beta_{11} + \cdots + 1) q^{6}+ \cdots + (3 \beta_{17} - 2 \beta_{16} + \cdots + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{2} + 2 q^{3} + 32 q^{4} - 18 q^{5} + 10 q^{6} + 4 q^{7} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 2 q^{2} + 2 q^{3} + 32 q^{4} - 18 q^{5} + 10 q^{6} + 4 q^{7} + 32 q^{9} + 2 q^{10} + 11 q^{11} + 12 q^{13} + q^{14} - 2 q^{15} + 60 q^{16} - 25 q^{17} - 9 q^{18} + 8 q^{19} - 32 q^{20} + 20 q^{21} + 29 q^{22} + 12 q^{23} + 15 q^{24} + 18 q^{25} + 5 q^{26} + 5 q^{27} + 15 q^{28} + 24 q^{29} - 10 q^{30} + 11 q^{31} - 17 q^{32} - 33 q^{33} + 17 q^{34} - 4 q^{35} + 90 q^{36} + 45 q^{37} - 28 q^{38} + 38 q^{39} + 12 q^{41} - 9 q^{42} + 19 q^{43} + 6 q^{44} - 32 q^{45} - 5 q^{46} - 16 q^{47} - 3 q^{48} + 62 q^{49} - 2 q^{50} + 7 q^{51} + q^{52} - 4 q^{53} - 6 q^{54} - 11 q^{55} + 2 q^{56} + 5 q^{57} + 28 q^{58} - 8 q^{59} + 27 q^{61} - 57 q^{62} + 10 q^{63} + 98 q^{64} - 12 q^{65} - 32 q^{66} + 9 q^{67} - 26 q^{68} + 31 q^{69} - q^{70} + 9 q^{71} - 102 q^{72} + 2 q^{73} - 24 q^{74} + 2 q^{75} - 43 q^{76} + 11 q^{77} + 45 q^{78} + 20 q^{79} - 60 q^{80} + 66 q^{81} - 19 q^{82} - 20 q^{83} - 33 q^{84} + 25 q^{85} + 21 q^{86} - 33 q^{87} + 62 q^{88} + 13 q^{89} + 9 q^{90} + 22 q^{91} - 6 q^{92} + 58 q^{93} + 70 q^{94} - 8 q^{95} - 17 q^{96} - 27 q^{97} - 6 q^{98} + 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 2 x^{17} - 32 x^{16} + 64 x^{15} + 417 x^{14} - 839 x^{13} - 2829 x^{12} + 5789 x^{11} + \cdots + 576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 291189 \nu^{17} - 89672 \nu^{16} - 13773536 \nu^{15} + 3015680 \nu^{14} + 241721301 \nu^{13} + \cdots + 1759381728 ) / 246482048 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6611 \nu^{17} - 102932 \nu^{16} - 147400 \nu^{15} + 3254088 \nu^{14} + 1129363 \nu^{13} + \cdots - 103616736 ) / 3624736 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 352200 \nu^{17} - 1531101 \nu^{16} + 12859718 \nu^{15} + 49775228 \nu^{14} - 191128800 \nu^{13} + \cdots - 1731721392 ) / 61620512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 200609 \nu^{17} - 111305 \nu^{16} + 6840426 \nu^{15} + 2808640 \nu^{14} - 94256757 \nu^{13} + \cdots - 109530144 ) / 30810256 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4207833 \nu^{17} - 3035060 \nu^{16} - 136531352 \nu^{15} + 90560592 \nu^{14} + 1814253113 \nu^{13} + \cdots - 3045233376 ) / 246482048 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 376637 \nu^{17} + 816775 \nu^{16} + 12308969 \nu^{15} - 26268086 \nu^{14} - 165237543 \nu^{13} + \cdots + 1047452304 ) / 15405128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 6655037 \nu^{17} + 7124952 \nu^{16} + 218103744 \nu^{15} - 221147168 \nu^{14} + \cdots + 5512649248 ) / 246482048 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 7671053 \nu^{17} - 3019592 \nu^{16} - 252716832 \nu^{15} + 90268544 \nu^{14} + 3409619053 \nu^{13} + \cdots - 2534526752 ) / 246482048 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2012075 \nu^{17} - 240291 \nu^{16} - 66473910 \nu^{15} + 4644444 \nu^{14} + 901092691 \nu^{13} + \cdots + 402009264 ) / 61620512 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1080450 \nu^{17} - 770753 \nu^{16} - 35433032 \nu^{15} + 24019784 \nu^{14} + 475927878 \nu^{13} + \cdots - 857558448 ) / 30810256 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2667907 \nu^{17} + 32580 \nu^{16} - 87752232 \nu^{15} - 4559040 \nu^{14} + 1182540091 \nu^{13} + \cdots + 516806112 ) / 61620512 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 5514599 \nu^{17} + 1603132 \nu^{16} + 181517160 \nu^{15} - 43215552 \nu^{14} + \cdots - 176155936 ) / 123241024 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 19147913 \nu^{17} - 20921528 \nu^{16} - 620919424 \nu^{15} + 652093696 \nu^{14} + \cdots - 18920576800 ) / 246482048 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 5467460 \nu^{17} - 2351091 \nu^{16} - 180215470 \nu^{15} + 71663140 \nu^{14} + 2434217780 \nu^{13} + \cdots - 1937384048 ) / 61620512 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + \beta_{13} + \beta_{11} + \beta_{9} + \beta_{5} + \beta_{4} + 8\beta_{2} - \beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + 10\beta_{3} + 38\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{17} + 10 \beta_{15} - \beta_{14} + 12 \beta_{13} + 2 \beta_{12} + 10 \beta_{11} + \beta_{10} + \cdots + 168 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{16} - \beta_{15} + \beta_{14} - 3 \beta_{13} + 19 \beta_{11} + 2 \beta_{10} - \beta_{9} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 14 \beta_{17} + 78 \beta_{15} - 20 \beta_{14} + 109 \beta_{13} + 32 \beta_{12} + 82 \beta_{11} + \cdots + 1160 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4 \beta_{17} - 14 \beta_{16} - 19 \beta_{15} + 16 \beta_{14} - 51 \beta_{13} + 231 \beta_{11} + \cdots + 181 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 138 \beta_{17} - 6 \beta_{16} + 559 \beta_{15} - 254 \beta_{14} + 905 \beta_{13} + 350 \beta_{12} + \cdots + 8140 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 81 \beta_{17} - 144 \beta_{16} - 247 \beta_{15} + 181 \beta_{14} - 587 \beta_{13} - 2 \beta_{12} + \cdots + 1774 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1185 \beta_{17} - 137 \beta_{16} + 3835 \beta_{15} - 2662 \beta_{14} + 7237 \beta_{13} + 3268 \beta_{12} + \cdots + 57823 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1081 \beta_{17} - 1319 \beta_{16} - 2721 \beta_{15} + 1786 \beta_{14} - 5742 \beta_{13} + \cdots + 16118 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 9476 \beta_{17} - 1993 \beta_{16} + 25488 \beta_{15} - 25193 \beta_{14} + 56809 \beta_{13} + \cdots + 414993 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 12018 \beta_{17} - 11392 \beta_{16} - 27312 \beta_{15} + 16418 \beta_{14} - 51517 \beta_{13} + \cdots + 140033 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 72575 \beta_{17} - 23670 \beta_{16} + 164341 \beta_{15} - 224285 \beta_{14} + 441297 \beta_{13} + \cdots + 3005281 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 120606 \beta_{17} - 95053 \beta_{16} - 258403 \beta_{15} + 144563 \beta_{14} - 438785 \beta_{13} + \cdots + 1182797 \) 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
2.78505
2.67329
2.53901
2.45014
1.76982
1.47536
1.43785
0.899027
0.348479
0.305904
−0.335211
−1.01684
−1.43913
−1.63907
−2.41658
−2.50213
−2.58952
−2.74545
−2.78505 −1.67120 5.75652 −1.00000 4.65438 −4.20569 −10.4621 −0.207087 2.78505
1.2 −2.67329 3.37901 5.14647 −1.00000 −9.03307 0.692364 −8.41144 8.41770 2.67329
1.3 −2.53901 −3.30364 4.44659 −1.00000 8.38798 3.67389 −6.21194 7.91401 2.53901
1.4 −2.45014 −0.0247552 4.00319 −1.00000 0.0606537 1.10013 −4.90809 −2.99939 2.45014
1.5 −1.76982 2.26705 1.13225 −1.00000 −4.01226 3.82507 1.53576 2.13951 1.76982
1.6 −1.47536 −2.78489 0.176690 −1.00000 4.10872 −0.173538 2.69004 4.75562 1.47536
1.7 −1.43785 −0.254287 0.0674170 −1.00000 0.365627 −3.46667 2.77877 −2.93534 1.43785
1.8 −0.899027 −0.0342120 −1.19175 −1.00000 0.0307575 4.96999 2.86947 −2.99883 0.899027
1.9 −0.348479 1.65774 −1.87856 −1.00000 −0.577689 −0.779308 1.35160 −0.251882 0.348479
1.10 −0.305904 −0.840395 −1.90642 −1.00000 0.257080 −4.91901 1.19499 −2.29374 0.305904
1.11 0.335211 2.61395 −1.88763 −1.00000 0.876225 −0.699553 −1.30318 3.83275 −0.335211
1.12 1.01684 −3.08977 −0.966037 −1.00000 −3.14180 −3.86206 −3.01598 6.54668 −1.01684
1.13 1.43913 2.89473 0.0711026 −1.00000 4.16590 4.37243 −2.77594 5.37946 −1.43913
1.14 1.63907 −1.56822 0.686561 −1.00000 −2.57043 3.96416 −2.15282 −0.540673 −1.63907
1.15 2.41658 0.530330 3.83984 −1.00000 1.28158 1.66759 4.44611 −2.71875 −2.41658
1.16 2.50213 2.91576 4.26066 −1.00000 7.29562 −4.35529 5.65646 5.50167 −2.50213
1.17 2.58952 1.68398 4.70561 −1.00000 4.36069 2.60507 7.00623 −0.164218 −2.58952
1.18 2.74545 −2.37118 5.53751 −1.00000 −6.50997 −0.409554 9.71206 2.62250 −2.74545
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(151\) \( -1 \)

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 755.2.a.k ✓ 18
3.b odd 2 1 6795.2.a.bi 18
5.b even 2 1 3775.2.a.r 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
755.2.a.k ✓ 18 1.a even 1 1 trivial
3775.2.a.r 18 5.b even 2 1
6795.2.a.bi 18 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(755))\):

\( T_{2}^{18} + 2 T_{2}^{17} - 32 T_{2}^{16} - 64 T_{2}^{15} + 417 T_{2}^{14} + 839 T_{2}^{13} - 2829 T_{2}^{12} + \cdots + 576 \) Copy content Toggle raw display
\( T_{3}^{18} - 2 T_{3}^{17} - 41 T_{3}^{16} + 79 T_{3}^{15} + 681 T_{3}^{14} - 1245 T_{3}^{13} - 5903 T_{3}^{12} + \cdots + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 2 T^{17} + \cdots + 576 \) Copy content Toggle raw display
$3$ \( T^{18} - 2 T^{17} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( (T + 1)^{18} \) Copy content Toggle raw display
$7$ \( T^{18} - 4 T^{17} + \cdots - 187232 \) Copy content Toggle raw display
$11$ \( T^{18} - 11 T^{17} + \cdots + 15945984 \) Copy content Toggle raw display
$13$ \( T^{18} - 12 T^{17} + \cdots + 6024352 \) Copy content Toggle raw display
$17$ \( T^{18} + 25 T^{17} + \cdots + 8306688 \) Copy content Toggle raw display
$19$ \( T^{18} - 8 T^{17} + \cdots - 324288 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 143258712 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 3481410264 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 12920205824 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 27363905232896 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 789639168 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 123008974848 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 268399804416 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 41277800448 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots - 322195609536 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 49054901641216 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots - 22497872398488 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 190086356140032 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots - 14\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 40860420133992 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 309732808310784 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 70\!\cdots\!28 \) Copy content Toggle raw display
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