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Properties

Label 102.4.h.a
Level $102$
Weight $4$
Character orbit 102.h
Analytic conductor $6.018$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [102,4,Mod(19,102)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(102, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("102.19");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 102 = 2 \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 102.h (of order \(8\), degree \(4\), 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: \(6.01819482059\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 380 x^{14} + 61646 x^{12} + 5539124 x^{10} + 298284737 x^{8} + 9669948280 x^{6} + \cdots + 2353192816144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{3} q^{2} + \beta_{8} q^{3} + 4 \beta_{9} q^{4} + ( - \beta_{15} - \beta_{12} - 2 \beta_{9} + \cdots - 4) q^{5}+ \cdots - 9 \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{3} q^{2} + \beta_{8} q^{3} + 4 \beta_{9} q^{4} + ( - \beta_{15} - \beta_{12} - 2 \beta_{9} + \cdots - 4) q^{5}+ \cdots + (18 \beta_{14} - 36 \beta_{13} + \cdots - 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 64 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 64 q^{5} - 128 q^{10} - 40 q^{11} + 48 q^{14} + 24 q^{15} - 256 q^{16} + 64 q^{17} - 288 q^{18} - 168 q^{19} + 96 q^{20} - 272 q^{22} - 136 q^{23} + 56 q^{25} - 448 q^{26} - 96 q^{28} + 576 q^{29} - 24 q^{31} + 120 q^{33} - 272 q^{34} + 1648 q^{35} + 360 q^{37} + 360 q^{39} - 192 q^{40} + 616 q^{41} - 48 q^{42} - 536 q^{43} - 544 q^{44} - 216 q^{45} + 816 q^{46} + 2648 q^{49} + 240 q^{50} - 264 q^{51} - 32 q^{52} + 1672 q^{53} - 1608 q^{57} + 1152 q^{58} + 2032 q^{59} - 96 q^{60} - 864 q^{61} + 48 q^{62} + 256 q^{65} + 240 q^{66} - 1888 q^{67} - 480 q^{68} - 1224 q^{69} - 1088 q^{70} - 2544 q^{71} - 3368 q^{73} - 720 q^{74} - 96 q^{75} + 672 q^{76} - 1792 q^{77} + 720 q^{78} - 96 q^{79} + 1024 q^{80} + 1920 q^{82} - 2368 q^{83} + 1824 q^{85} + 3024 q^{86} + 864 q^{87} - 320 q^{88} + 1152 q^{90} - 3200 q^{91} + 1632 q^{92} - 264 q^{93} - 544 q^{94} - 3976 q^{95} - 3208 q^{97} - 360 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 380 x^{14} + 61646 x^{12} + 5539124 x^{10} + 298284737 x^{8} + 9669948280 x^{6} + \cdots + 2353192816144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6576 \nu^{14} - 2096990 \nu^{12} - 277364663 \nu^{10} - 19511195811 \nu^{8} + \cdots - 254958186452180 ) / 3233649424 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1366677711 \nu^{15} + 57858804343 \nu^{14} + 472335018997 \nu^{13} + 18729689947245 \nu^{12} + \cdots + 25\!\cdots\!08 ) / 31\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1366677711 \nu^{15} - 57858804343 \nu^{14} + 472335018997 \nu^{13} - 18729689947245 \nu^{12} + \cdots - 25\!\cdots\!08 ) / 31\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 32103697150 \nu^{15} - 676719580635 \nu^{14} - 9772118554250 \nu^{13} + \cdots - 30\!\cdots\!68 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 32103697150 \nu^{15} + 676719580635 \nu^{14} - 9772118554250 \nu^{13} + \cdots + 30\!\cdots\!68 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 34884988529 \nu^{15} - 588718117262 \nu^{14} - 9964597734803 \nu^{13} + \cdots - 19\!\cdots\!84 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 34884988529 \nu^{15} - 588718117262 \nu^{14} + 9964597734803 \nu^{13} + \cdots - 19\!\cdots\!84 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 166203515 \nu^{15} + 53069672788 \nu^{13} + 7028974061810 \nu^{11} + \cdots + 64\!\cdots\!44 \nu ) / 49\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 256102821704 \nu^{15} + 793103356591 \nu^{14} + 82809101249464 \nu^{13} + \cdots + 30\!\cdots\!12 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 256102821704 \nu^{15} + 793103356591 \nu^{14} - 82809101249464 \nu^{13} + \cdots + 30\!\cdots\!20 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 256633258937 \nu^{15} + 102694049837 \nu^{14} + 81154510193483 \nu^{13} + \cdots - 14\!\cdots\!04 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 256633258937 \nu^{15} + 102694049837 \nu^{14} - 81154510193483 \nu^{13} + \cdots - 14\!\cdots\!96 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 5731887709 \nu^{15} - 1357036645 \nu^{14} - 1857611983727 \nu^{13} + \cdots + 88\!\cdots\!68 ) / 39\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2564777 \nu^{15} - 2083537 \nu^{14} - 830253555 \nu^{13} - 722987339 \nu^{12} + \cdots - 14\!\cdots\!76 ) / 14072842293248 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} + 3 \beta_{8} + \cdots - 47 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} - \beta_{14} + 7 \beta_{13} - 7 \beta_{12} - 3 \beta_{11} + 3 \beta_{10} + 17 \beta_{9} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 49 \beta_{15} + 49 \beta_{14} - 75 \beta_{13} - 75 \beta_{12} - 35 \beta_{11} - 35 \beta_{10} + \cdots + 2625 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{15} + 3 \beta_{14} - 633 \beta_{13} + 633 \beta_{12} + 297 \beta_{11} - 297 \beta_{10} + \cdots - 333 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2333 \beta_{15} - 2333 \beta_{14} + 5151 \beta_{13} + 5151 \beta_{12} + 999 \beta_{11} + 999 \beta_{10} + \cdots - 151625 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3897 \beta_{15} + 3897 \beta_{14} + 47893 \beta_{13} - 47893 \beta_{12} - 24437 \beta_{11} + \cdots + 27353 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 111749 \beta_{15} + 111749 \beta_{14} - 341023 \beta_{13} - 341023 \beta_{12} - 19095 \beta_{11} + \cdots + 8966489 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 500185 \beta_{15} - 500185 \beta_{14} - 3372597 \beta_{13} + 3372597 \beta_{12} + 1848821 \beta_{11} + \cdots - 2023961 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5351757 \beta_{15} - 5351757 \beta_{14} + 22099767 \beta_{13} + 22099767 \beta_{12} - 187137 \beta_{11} + \cdots - 540279105 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 47129377 \beta_{15} + 47129377 \beta_{14} + 229095213 \beta_{13} - 229095213 \beta_{12} + \cdots + 142953169 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 252743941 \beta_{15} + 252743941 \beta_{14} - 1413672351 \beta_{13} - 1413672351 \beta_{12} + \cdots + 33038726633 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3921591321 \beta_{15} - 3921591321 \beta_{14} - 15264175381 \beta_{13} + 15264175381 \beta_{12} + \cdots - 9870324537 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 11479180605 \beta_{15} - 11479180605 \beta_{14} + 89723495015 \beta_{13} + 89723495015 \beta_{12} + \cdots - 2043706907313 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 305143538481 \beta_{15} + 305143538481 \beta_{14} + 1006368846813 \beta_{13} - 1006368846813 \beta_{12} + \cdots + 673788764673 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/102\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(37\)
\(\chi(n)\) \(1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
− 8.14846i
7.94955i
− 1.41058i
6.43792i
6.86964i
− 7.53781i
− 6.38753i
7.88413i
8.14846i
− 7.94955i
1.41058i
− 6.43792i
− 6.86964i
7.53781i
6.38753i
− 7.88413i
−1.41421 − 1.41421i −2.77164 + 1.14805i 4.00000i −3.77473 − 9.11299i 5.54328 + 2.29610i −6.00068 + 14.4869i 5.65685 − 5.65685i 6.36396 − 6.36396i −7.54945 + 18.2260i
19.2 −1.41421 − 1.41421i −2.77164 + 1.14805i 4.00000i 1.44785 + 3.49541i 5.54328 + 2.29610i 3.33817 − 8.05905i 5.65685 − 5.65685i 6.36396 − 6.36396i 2.89569 − 6.99082i
19.3 −1.41421 − 1.41421i 2.77164 − 1.14805i 4.00000i −7.57764 − 18.2940i −5.54328 − 2.29610i −7.31420 + 17.6580i 5.65685 − 5.65685i 6.36396 − 6.36396i −15.1553 + 36.5881i
19.4 −1.41421 − 1.41421i 2.77164 − 1.14805i 4.00000i 0.975586 + 2.35527i −5.54328 − 2.29610i 5.73407 − 13.8433i 5.65685 − 5.65685i 6.36396 − 6.36396i 1.95117 − 4.71055i
25.1 1.41421 − 1.41421i −1.14805 + 2.77164i − 4.00000i −11.2162 − 4.64590i 2.29610 + 5.54328i 32.3046 − 13.3810i −5.65685 − 5.65685i −6.36396 − 6.36396i −22.4324 + 9.29180i
25.2 1.41421 − 1.41421i −1.14805 + 2.77164i − 4.00000i −8.09306 − 3.35225i 2.29610 + 5.54328i −31.4899 + 13.0435i −5.65685 − 5.65685i −6.36396 − 6.36396i −16.1861 + 6.70451i
25.3 1.41421 − 1.41421i 1.14805 − 2.77164i − 4.00000i −13.5193 − 5.59990i −2.29610 − 5.54328i −11.2359 + 4.65404i −5.65685 − 5.65685i −6.36396 − 6.36396i −27.0387 + 11.1998i
25.4 1.41421 − 1.41421i 1.14805 − 2.77164i − 4.00000i 9.75753 + 4.04170i −2.29610 − 5.54328i 14.6637 − 6.07392i −5.65685 − 5.65685i −6.36396 − 6.36396i 19.5151 − 8.08340i
43.1 −1.41421 + 1.41421i −2.77164 − 1.14805i − 4.00000i −3.77473 + 9.11299i 5.54328 − 2.29610i −6.00068 − 14.4869i 5.65685 + 5.65685i 6.36396 + 6.36396i −7.54945 − 18.2260i
43.2 −1.41421 + 1.41421i −2.77164 − 1.14805i − 4.00000i 1.44785 − 3.49541i 5.54328 − 2.29610i 3.33817 + 8.05905i 5.65685 + 5.65685i 6.36396 + 6.36396i 2.89569 + 6.99082i
43.3 −1.41421 + 1.41421i 2.77164 + 1.14805i − 4.00000i −7.57764 + 18.2940i −5.54328 + 2.29610i −7.31420 − 17.6580i 5.65685 + 5.65685i 6.36396 + 6.36396i −15.1553 − 36.5881i
43.4 −1.41421 + 1.41421i 2.77164 + 1.14805i − 4.00000i 0.975586 − 2.35527i −5.54328 + 2.29610i 5.73407 + 13.8433i 5.65685 + 5.65685i 6.36396 + 6.36396i 1.95117 + 4.71055i
49.1 1.41421 + 1.41421i −1.14805 − 2.77164i 4.00000i −11.2162 + 4.64590i 2.29610 − 5.54328i 32.3046 + 13.3810i −5.65685 + 5.65685i −6.36396 + 6.36396i −22.4324 − 9.29180i
49.2 1.41421 + 1.41421i −1.14805 − 2.77164i 4.00000i −8.09306 + 3.35225i 2.29610 − 5.54328i −31.4899 − 13.0435i −5.65685 + 5.65685i −6.36396 + 6.36396i −16.1861 − 6.70451i
49.3 1.41421 + 1.41421i 1.14805 + 2.77164i 4.00000i −13.5193 + 5.59990i −2.29610 + 5.54328i −11.2359 − 4.65404i −5.65685 + 5.65685i −6.36396 + 6.36396i −27.0387 − 11.1998i
49.4 1.41421 + 1.41421i 1.14805 + 2.77164i 4.00000i 9.75753 − 4.04170i −2.29610 + 5.54328i 14.6637 + 6.07392i −5.65685 + 5.65685i −6.36396 + 6.36396i 19.5151 + 8.08340i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 102.4.h.a ✓ 16
3.b odd 2 1 306.4.l.d 16
17.d even 8 1 inner 102.4.h.a ✓ 16
17.e odd 16 1 1734.4.a.bf 8
17.e odd 16 1 1734.4.a.bg 8
51.g odd 8 1 306.4.l.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.4.h.a ✓ 16 1.a even 1 1 trivial
102.4.h.a ✓ 16 17.d even 8 1 inner
306.4.l.d 16 3.b odd 2 1
306.4.l.d 16 51.g odd 8 1
1734.4.a.bf 8 17.e odd 16 1
1734.4.a.bg 8 17.e odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 64 T_{5}^{15} + 2020 T_{5}^{14} + 37696 T_{5}^{13} + 373128 T_{5}^{12} + \cdots + 958700061745924 \) acting on \(S_{4}^{\mathrm{new}}(102, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 16)^{4} \) Copy content Toggle raw display
$3$ \( (T^{8} + 6561)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 958700061745924 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 81\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 82\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 33\!\cdots\!21 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 71\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 36\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 61\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 36\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 49\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 82\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 57\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 43\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 96\!\cdots\!12)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 93\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 69\!\cdots\!36 \) Copy content Toggle raw display
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