Properties

Label 100.5.b.e
Level $100$
Weight $5$
Character orbit 100.b
Analytic conductor $10.337$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,5,Mod(51,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.51");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 100.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(10.3369963084\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1816805376000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 6x^{6} + 31x^{4} - 96x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{15} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_1 q^{3} + (\beta_{6} + 6) q^{4} + ( - \beta_{5} - \beta_{4}) q^{6} + (2 \beta_{3} + 6 \beta_{2} + \beta_1) q^{7} + ( - \beta_{7} + 3 \beta_{3} + 7 \beta_{2} + 2 \beta_1) q^{8} + ( - 3 \beta_{6} - 3 \beta_{4} - 39) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_1 q^{3} + (\beta_{6} + 6) q^{4} + ( - \beta_{5} - \beta_{4}) q^{6} + (2 \beta_{3} + 6 \beta_{2} + \beta_1) q^{7} + ( - \beta_{7} + 3 \beta_{3} + 7 \beta_{2} + 2 \beta_1) q^{8} + ( - 3 \beta_{6} - 3 \beta_{4} - 39) q^{9} + 4 \beta_{5} q^{11} + (3 \beta_{7} + 3 \beta_{3} + 3 \beta_{2} + 10 \beta_1) q^{12} + (\beta_{7} - 4 \beta_{3} + 18 \beta_{2}) q^{13} + (8 \beta_{6} - \beta_{5} - \beta_{4} - 80) q^{14} + (12 \beta_{6} - 4 \beta_{5} + 4 \beta_{4} - 104) q^{16} + (2 \beta_{7} - 4 \beta_{3} + 16 \beta_{2}) q^{17} + (6 \beta_{7} + 6 \beta_{3} - 33 \beta_{2} - 12 \beta_1) q^{18} + (24 \beta_{6} + 4 \beta_{5} - 8 \beta_{4}) q^{19} + ( - 15 \beta_{6} - 15 \beta_{4} - 120) q^{21} + ( - 8 \beta_{7} + 8 \beta_{3} - 48 \beta_1) q^{22} + ( - 22 \beta_{3} - 66 \beta_{2} - 13 \beta_1) q^{23} + ( - 4 \beta_{5} - 28 \beta_{4} - 240) q^{24} + (12 \beta_{6} + 2 \beta_{5} - 6 \beta_{4} + 312) q^{26} + ( - 36 \beta_{3} - 108 \beta_{2} - 6 \beta_1) q^{27} + ( - 5 \beta_{7} + 27 \beta_{3} - 69 \beta_{2} + 26 \beta_1) q^{28} + ( - 4 \beta_{6} - 4 \beta_{4} - 338) q^{29} + (24 \beta_{6} - 16 \beta_{5} - 8 \beta_{4}) q^{31} + ( - 8 \beta_{7} + 8 \beta_{3} - 104 \beta_{2} + 80 \beta_1) q^{32} + ( - 12 \beta_{7} - 60 \beta_{3} + 324 \beta_{2}) q^{33} + (8 \beta_{6} + 4 \beta_{5} - 12 \beta_{4} + 272) q^{34} + ( - 39 \beta_{6} + 24 \beta_{5} - 24 \beta_{4} - 714) q^{36} + ( - 21 \beta_{7} - 24 \beta_{3} + 162 \beta_{2}) q^{37} + ( - 24 \beta_{7} + 120 \beta_{3} + 48 \beta_{2} - 16 \beta_1) q^{38} - 24 \beta_{5} q^{39} + (79 \beta_{6} + 79 \beta_{4} + 182) q^{41} + (30 \beta_{7} + 30 \beta_{3} - 90 \beta_{2} - 60 \beta_1) q^{42} + ( - 116 \beta_{3} - 348 \beta_{2} + 77 \beta_1) q^{43} + (24 \beta_{6} + 32 \beta_{5} + 96 \beta_{4} - 240) q^{44} + ( - 88 \beta_{6} + 13 \beta_{5} + 13 \beta_{4} + 880) q^{46} + (34 \beta_{3} + 102 \beta_{2} - 7 \beta_1) q^{47} + (36 \beta_{7} + 132 \beta_{3} - 156 \beta_{2} - 8 \beta_1) q^{48} + (5 \beta_{6} + 5 \beta_{4} + 1001) q^{49} + ( - 24 \beta_{6} - 16 \beta_{5} + 8 \beta_{4}) q^{51} + ( - 10 \beta_{7} + 70 \beta_{3} + 342 \beta_{2} - 12 \beta_1) q^{52} + (59 \beta_{7} - 124 \beta_{3} + 502 \beta_{2}) q^{53} + ( - 144 \beta_{6} + 6 \beta_{5} + 6 \beta_{4} + 1440) q^{54} + ( - 32 \beta_{6} - 36 \beta_{5} + 4 \beta_{4} - 1840) q^{56} + (84 \beta_{7} - 60 \beta_{3} + 132 \beta_{2}) q^{57} + (8 \beta_{7} + 8 \beta_{3} - 330 \beta_{2} - 16 \beta_1) q^{58} + ( - 216 \beta_{6} + 28 \beta_{5} + 72 \beta_{4}) q^{59} + ( - 131 \beta_{6} - 131 \beta_{4} + 822) q^{61} + (16 \beta_{7} + 80 \beta_{3} + 48 \beta_{2} + 224 \beta_1) q^{62} + ( - 18 \beta_{3} - 54 \beta_{2} - 279 \beta_1) q^{63} + ( - 80 \beta_{6} - 96 \beta_{5} - 32 \beta_{4} - 864) q^{64} + (288 \beta_{6} - 24 \beta_{5} + 72 \beta_{4} + 5760) q^{66} + ( - 100 \beta_{3} - 300 \beta_{2} + 139 \beta_1) q^{67} + ( - 4 \beta_{7} + 92 \beta_{3} + 316 \beta_{2} - 56 \beta_1) q^{68} + (171 \beta_{6} + 171 \beta_{4} + 1560) q^{69} + ( - 216 \beta_{6} + 120 \beta_{5} + 72 \beta_{4}) q^{71} + (15 \beta_{7} + 51 \beta_{3} - 681 \beta_{2} - 414 \beta_1) q^{72} + ( - 144 \beta_{7} + 36 \beta_{3} + 108 \beta_{2}) q^{73} + (180 \beta_{6} - 42 \beta_{5} + 126 \beta_{4} + 2952) q^{74} + (216 \beta_{6} - 32 \beta_{5} + 160 \beta_{4} - 6000) q^{76} + ( - 140 \beta_{7} + 20 \beta_{3} + 180 \beta_{2}) q^{77} + (48 \beta_{7} - 48 \beta_{3} + 288 \beta_1) q^{78} + ( - 24 \beta_{6} + 96 \beta_{5} + 8 \beta_{4}) q^{79} + ( - 9 \beta_{6} - 9 \beta_{4} - 2439) q^{81} + ( - 158 \beta_{7} - 158 \beta_{3} + 24 \beta_{2} + 316 \beta_1) q^{82} + (68 \beta_{3} + 204 \beta_{2} + 693 \beta_1) q^{83} + ( - 120 \beta_{6} + 120 \beta_{5} - 120 \beta_{4} - 3120) q^{84} + ( - 464 \beta_{6} - 77 \beta_{5} - 77 \beta_{4} + 4640) q^{86} + ( - 48 \beta_{3} - 144 \beta_{2} - 402 \beta_1) q^{87} + ( - 184 \beta_{7} - 344 \beta_{3} - 504 \beta_{2} - 144 \beta_1) q^{88} + ( - 340 \beta_{6} - 340 \beta_{4} - 3458) q^{89} + (216 \beta_{6} + 8 \beta_{5} - 72 \beta_{4}) q^{91} + (49 \beta_{7} - 303 \beta_{3} + 753 \beta_{2} - 306 \beta_1) q^{92} + (144 \beta_{7} + 240 \beta_{3} - 1488 \beta_{2}) q^{93} + (136 \beta_{6} + 7 \beta_{5} + 7 \beta_{4} - 1360) q^{94} + ( - 96 \beta_{6} + 80 \beta_{5} - 208 \beta_{4} - 9600) q^{96} + (22 \beta_{7} + 236 \beta_{3} - 1224 \beta_{2}) q^{97} + ( - 10 \beta_{7} - 10 \beta_{3} + 991 \beta_{2} + 20 \beta_1) q^{98} + (648 \beta_{6} - 252 \beta_{5} - 216 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 48 q^{4} - 312 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 48 q^{4} - 312 q^{9} - 640 q^{14} - 832 q^{16} - 960 q^{21} - 1920 q^{24} + 2496 q^{26} - 2704 q^{29} + 2176 q^{34} - 5712 q^{36} + 1456 q^{41} - 1920 q^{44} + 7040 q^{46} + 8008 q^{49} + 11520 q^{54} - 14720 q^{56} + 6576 q^{61} - 6912 q^{64} + 46080 q^{66} + 12480 q^{69} + 23616 q^{74} - 48000 q^{76} - 19512 q^{81} - 24960 q^{84} + 37120 q^{86} - 27664 q^{89} - 10880 q^{94} - 76800 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 6x^{6} + 31x^{4} - 96x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 10\nu^{5} - \nu^{3} + 80\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 6\nu^{5} + 31\nu^{3} - 96\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 6\nu^{5} + 31\nu^{3} + 160\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 6\nu^{4} + \nu^{2} + 24 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 2\nu^{4} - \nu^{2} + 28 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 6\nu^{4} - 31\nu^{2} + 72 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} + 10\nu^{5} - 13\nu^{3} + 72\nu ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{4} + 12 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 2\beta_{3} + 8\beta_{2} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{5} + 2\beta_{4} - 26 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{7} - 7\beta_{3} + 11\beta_{2} + 20\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{6} + 12\beta_{5} - 7\beta_{4} - 108 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -19\beta_{7} - 8\beta_{3} - 22\beta_{2} + 58\beta_1 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(77\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
1.88164 0.677813i
1.88164 + 0.677813i
1.39980 1.42849i
1.39980 + 1.42849i
−1.39980 1.42849i
−1.39980 + 1.42849i
−1.88164 0.677813i
−1.88164 + 0.677813i
−3.76328 1.35563i 13.9962i 12.3246 + 10.2032i 0 −18.9737 + 52.6718i 35.6863i −32.5490 55.1050i −114.895 0
51.2 −3.76328 + 1.35563i 13.9962i 12.3246 10.2032i 0 −18.9737 52.6718i 35.6863i −32.5490 + 55.1050i −114.895 0
51.3 −2.79959 2.85697i 6.64118i −0.324555 + 15.9967i 0 18.9737 18.5926i 39.0703i 46.6107 43.8570i 36.8947 0
51.4 −2.79959 + 2.85697i 6.64118i −0.324555 15.9967i 0 18.9737 + 18.5926i 39.0703i 46.6107 + 43.8570i 36.8947 0
51.5 2.79959 2.85697i 6.64118i −0.324555 15.9967i 0 18.9737 + 18.5926i 39.0703i −46.6107 43.8570i 36.8947 0
51.6 2.79959 + 2.85697i 6.64118i −0.324555 + 15.9967i 0 18.9737 18.5926i 39.0703i −46.6107 + 43.8570i 36.8947 0
51.7 3.76328 1.35563i 13.9962i 12.3246 10.2032i 0 −18.9737 52.6718i 35.6863i 32.5490 55.1050i −114.895 0
51.8 3.76328 + 1.35563i 13.9962i 12.3246 + 10.2032i 0 −18.9737 + 52.6718i 35.6863i 32.5490 + 55.1050i −114.895 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.5.b.e 8
4.b odd 2 1 inner 100.5.b.e 8
5.b even 2 1 inner 100.5.b.e 8
5.c odd 4 2 20.5.d.c 8
15.e even 4 2 180.5.f.g 8
20.d odd 2 1 inner 100.5.b.e 8
20.e even 4 2 20.5.d.c 8
40.i odd 4 2 320.5.h.f 8
40.k even 4 2 320.5.h.f 8
60.l odd 4 2 180.5.f.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.5.d.c 8 5.c odd 4 2
20.5.d.c 8 20.e even 4 2
100.5.b.e 8 1.a even 1 1 trivial
100.5.b.e 8 4.b odd 2 1 inner
100.5.b.e 8 5.b even 2 1 inner
100.5.b.e 8 20.d odd 2 1 inner
180.5.f.g 8 15.e even 4 2
180.5.f.g 8 60.l odd 4 2
320.5.h.f 8 40.i odd 4 2
320.5.h.f 8 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(100, [\chi])\):

\( T_{3}^{4} + 240T_{3}^{2} + 8640 \) Copy content Toggle raw display
\( T_{13}^{4} - 20928T_{13}^{2} + 82861056 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 24 T^{6} + 496 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( (T^{4} + 240 T^{2} + 8640)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2800 T^{2} + 1944000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 48000 T^{2} + \cdots + 552407040)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 20928 T^{2} + 82861056)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 26368 T^{2} + 16367616)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 447360 T^{2} + \cdots + 10372976640)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 350320 T^{2} + \cdots + 30678152640)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 676 T + 104004)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 1013760 T^{2} + \cdots + 1745879040)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 3008448 T^{2} + \cdots + 126031666176)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 364 T - 3961116)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 10034800 T^{2} + \cdots + 325194264000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 751600 T^{2} + \cdots + 49724936640)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 24538048 T^{2} + \cdots + 20248378776576)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 30276480 T^{2} + \cdots + 225559394979840)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 1644 T - 10307356)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 11037040 T^{2} + \cdots + 7632038946240)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 64765440 T^{2} + \cdots + 13443377725440)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 68179968 T^{2} + \cdots + 11\!\cdots\!16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 27279360 T^{2} + \cdots + 152966937968640)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 118219120 T^{2} + \cdots + 10\!\cdots\!40)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 6916 T - 62026236)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 86769408 T^{2} + \cdots + 13\!\cdots\!76)^{2} \) Copy content Toggle raw display
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