Properties

Label 8.14.b.b
Level $8$
Weight $14$
Character orbit 8.b
Analytic conductor $8.578$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8,14,Mod(5,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.5");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 8.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: \(8.57847431615\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 5 x^{9} + 752 x^{8} + 708 x^{7} - 743866 x^{6} + 96647426 x^{5} + 2540283092 x^{4} + \cdots + 31\!\cdots\!68 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 11) q^{2} + (\beta_{3} + 3 \beta_1) q^{3} + (\beta_{4} - 2 \beta_{3} + 12 \beta_1 - 472) q^{4} + (10 \beta_{3} + \beta_{2} + 53 \beta_1) q^{5} + (\beta_{6} + \beta_{5} + 5 \beta_{4} + \cdots - 26769) q^{6}+ \cdots + (\beta_{9} + 2 \beta_{8} + \cdots + 201402) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 11) q^{2} + (\beta_{3} + 3 \beta_1) q^{3} + (\beta_{4} - 2 \beta_{3} + 12 \beta_1 - 472) q^{4} + (10 \beta_{3} + \beta_{2} + 53 \beta_1) q^{5} + (\beta_{6} + \beta_{5} + 5 \beta_{4} + \cdots - 26769) q^{6}+ \cdots + (1778304 \beta_{9} - 21666480 \beta_{7} + \cdots + 77120736) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 110 q^{2} - 4716 q^{4} - 267668 q^{6} + 586960 q^{7} - 270712 q^{8} + 2014054 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 110 q^{2} - 4716 q^{4} - 267668 q^{6} + 586960 q^{7} - 270712 q^{8} + 2014054 q^{9} - 4542088 q^{10} + 27987880 q^{12} + 1408688 q^{14} - 145914416 q^{15} + 56624912 q^{16} + 217326004 q^{17} - 147615262 q^{18} + 21655184 q^{20} - 177987876 q^{22} - 78679952 q^{23} + 320199056 q^{24} - 3076402574 q^{25} + 3734872040 q^{26} - 1653812448 q^{28} + 6338232752 q^{30} + 648233792 q^{31} - 11298380000 q^{32} + 15484079688 q^{33} - 6096822724 q^{34} + 4004708940 q^{36} - 18764968628 q^{38} - 63497510288 q^{39} + 7466802592 q^{40} + 59324640356 q^{41} + 53897620960 q^{42} + 13325704392 q^{44} - 55046867440 q^{46} - 10176534816 q^{47} - 301841943264 q^{48} + 182708552058 q^{49} + 326454435302 q^{50} - 53296499536 q^{52} + 35449773752 q^{54} - 123010753008 q^{55} - 462152447680 q^{56} - 511372324504 q^{57} + 766482705096 q^{58} + 1813082440992 q^{60} - 1665308528960 q^{62} - 898991123792 q^{63} - 2180548996032 q^{64} + 1577231990240 q^{65} + 2269525079448 q^{66} + 2338280915304 q^{68} - 6070110714688 q^{70} + 726361179984 q^{71} - 3600753685960 q^{72} - 633240365532 q^{73} + 7528513982264 q^{74} + 10338420845032 q^{76} - 8252024440816 q^{78} + 5445103565344 q^{79} - 15406871881920 q^{80} - 9674575380574 q^{81} + 12273334206796 q^{82} + 20362643366464 q^{84} - 26794541719396 q^{86} + 7632221772720 q^{87} - 27677491769136 q^{88} + 5506344808004 q^{89} + 31454099524040 q^{90} + 33971694298464 q^{92} - 45356008560096 q^{94} - 14214732035504 q^{95} - 35398666935232 q^{96} + 1361133320788 q^{97} + 54325451514942 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 5 x^{9} + 752 x^{8} + 708 x^{7} - 743866 x^{6} + 96647426 x^{5} + 2540283092 x^{4} + \cdots + 31\!\cdots\!68 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4673 \nu^{9} + 7354742 \nu^{8} + 521346978 \nu^{7} - 35423339110 \nu^{6} + \cdots - 24\!\cdots\!60 ) / 17\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 53727 \nu^{9} - 600394 \nu^{8} + 135456610 \nu^{7} + 4741322330 \nu^{6} + \cdots + 17\!\cdots\!44 ) / 69\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 53727 \nu^{9} - 600394 \nu^{8} + 135456610 \nu^{7} + 4741322330 \nu^{6} + \cdots + 38\!\cdots\!56 ) / 34\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1098741 \nu^{9} + 90806382 \nu^{8} + 2440766858 \nu^{7} + 54449976802 \nu^{6} + \cdots + 45\!\cdots\!48 ) / 17\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 393119 \nu^{9} + 2793526 \nu^{8} - 136735774 \nu^{7} + 5861994714 \nu^{6} + \cdots + 28\!\cdots\!68 ) / 58\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 712491 \nu^{9} - 83201106 \nu^{8} - 2967283894 \nu^{7} + 68951707810 \nu^{6} + \cdots + 80\!\cdots\!76 ) / 87\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1412391 \nu^{9} - 53876346 \nu^{8} - 6991130158 \nu^{7} - 262411250934 \nu^{6} + \cdots + 77\!\cdots\!44 ) / 87\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12461681 \nu^{9} - 378837910 \nu^{8} - 5471783426 \nu^{7} - 1181809514106 \nu^{6} + \cdots + 11\!\cdots\!72 ) / 34\!\cdots\!68 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} - 2\beta_{3} - 8\beta _1 - 592 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{9} - 2\beta_{7} + 2\beta_{6} - 6\beta_{5} - 20\beta_{4} + 50\beta_{3} + \beta_{2} - 502\beta _1 - 10617 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 19 \beta_{9} + 48 \beta_{8} + 66 \beta_{7} - 108 \beta_{6} + 134 \beta_{5} + 344 \beta_{4} + \cdots + 3214105 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 565 \beta_{9} + 720 \beta_{8} - 1278 \beta_{7} - 3928 \beta_{6} + 1798 \beta_{5} + 14726 \beta_{4} + \cdots - 179362061 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 45887 \beta_{9} - 57600 \beta_{8} + 89730 \beta_{7} + 299902 \beta_{6} - 29882 \beta_{5} + \cdots - 8831822803 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2487037 \beta_{9} - 3107424 \beta_{8} - 5870294 \beta_{7} - 24474454 \beta_{6} - 5731714 \beta_{5} + \cdots + 1283839484109 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 51318481 \beta_{9} + 98556528 \beta_{8} - 12162394 \beta_{7} + 1717394936 \beta_{6} + \cdots - 69547145106093 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 488837006 \beta_{9} - 167230080 \beta_{8} + 194693340 \beta_{7} - 24404269452 \beta_{6} + \cdots + 912576881846255 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−46.7129 17.5509i
−46.7129 + 17.5509i
−17.5296 43.4857i
−17.5296 + 43.4857i
1.33949 44.8086i
1.33949 + 44.8086i
27.9424 31.0290i
27.9424 + 31.0290i
37.4606 15.6556i
37.4606 + 15.6556i
−83.4258 35.1018i 622.159i 5727.73 + 5856.79i 26675.4i −21838.9 + 51904.1i 50480.5 −272257. 689661.i 1.20724e6 936352. 2.22541e6i
5.2 −83.4258 + 35.1018i 622.159i 5727.73 5856.79i 26675.4i −21838.9 51904.1i 50480.5 −272257. + 689661.i 1.20724e6 936352. + 2.22541e6i
5.3 −25.0592 86.9715i 1746.24i −6936.08 + 4358.86i 64905.5i −151873. + 43759.4i 201238. 552909. + 494011.i −1.45504e6 −5.64492e6 + 1.62648e6i
5.4 −25.0592 + 86.9715i 1746.24i −6936.08 4358.86i 64905.5i −151873. 43759.4i 201238. 552909. 494011.i −1.45504e6 −5.64492e6 1.62648e6i
5.5 12.6790 89.6172i 86.8898i −7870.49 2272.51i 45531.7i 7786.82 + 1101.67i −249036. −303446. + 676518.i 1.58677e6 4.08042e6 + 577295.i
5.6 12.6790 + 89.6172i 86.8898i −7870.49 + 2272.51i 45531.7i 7786.82 1101.67i −249036. −303446. 676518.i 1.58677e6 4.08042e6 577295.i
5.7 65.8848 62.0580i 1231.41i 489.620 8177.36i 25270.7i 76418.8 + 81131.3i 608245. −475211. 569148.i 77950.5 −1.56825e6 1.66495e6i
5.8 65.8848 + 62.0580i 1231.41i 489.620 + 8177.36i 25270.7i 76418.8 81131.3i 608245. −475211. + 569148.i 77950.5 −1.56825e6 + 1.66495e6i
5.9 84.9212 31.3112i 1415.71i 6231.21 5317.98i 2384.10i −44327.5 120223.i −317448. 362649. 646716.i −409898. −74649.0 202460.i
5.10 84.9212 + 31.3112i 1415.71i 6231.21 + 5317.98i 2384.10i −44327.5 + 120223.i −317448. 362649. + 646716.i −409898. −74649.0 + 202460.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.14.b.b 10
3.b odd 2 1 72.14.d.c 10
4.b odd 2 1 32.14.b.b 10
8.b even 2 1 inner 8.14.b.b 10
8.d odd 2 1 32.14.b.b 10
24.h odd 2 1 72.14.d.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.14.b.b 10 1.a even 1 1 trivial
8.14.b.b 10 8.b even 2 1 inner
32.14.b.b 10 4.b odd 2 1
32.14.b.b 10 8.d odd 2 1
72.14.d.c 10 3.b odd 2 1
72.14.d.c 10 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 6964588 T_{3}^{8} + 16370347169952 T_{3}^{6} + \cdots + 27\!\cdots\!16 \) acting on \(S_{14}^{\mathrm{new}}(8, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 36\!\cdots\!32 \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{5} + \cdots - 48\!\cdots\!64)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{5} + \cdots - 45\!\cdots\!36)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T^{5} + \cdots - 19\!\cdots\!36)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 12\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 92\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 24\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( (T^{5} + \cdots - 55\!\cdots\!44)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 98\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 38\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 48\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 21\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 23\!\cdots\!44)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 69\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 59\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 17\!\cdots\!56)^{2} \) Copy content Toggle raw display
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