Properties

Label 103.3.b.b
Level $103$
Weight $3$
Character orbit 103.b
Analytic conductor $2.807$
Analytic rank $0$
Dimension $12$
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] = [103,3,Mod(102,103)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(103, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("103.102");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 103.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: \(2.80654672291\)
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: \( x^{12} + 92x^{10} + 3344x^{8} + 60552x^{6} + 561888x^{4} + 2410584x^{2} + 3274879 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{2} - \beta_1 q^{3} + (\beta_{8} + 1) q^{4} + \beta_{2} q^{5} - \beta_{10} q^{6} + (\beta_{8} + \beta_{6} - \beta_{5}) q^{7} + ( - \beta_{6} + \beta_{5} - \beta_{3} - 2) q^{8} + (\beta_{9} - \beta_{8} + \beta_{5} + 2 \beta_{3} - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{2} - \beta_1 q^{3} + (\beta_{8} + 1) q^{4} + \beta_{2} q^{5} - \beta_{10} q^{6} + (\beta_{8} + \beta_{6} - \beta_{5}) q^{7} + ( - \beta_{6} + \beta_{5} - \beta_{3} - 2) q^{8} + (\beta_{9} - \beta_{8} + \beta_{5} + 2 \beta_{3} - 5) q^{9} + (\beta_{10} + \beta_{7}) q^{10} - \beta_{4} q^{11} + (\beta_{11} - \beta_{2} - 2 \beta_1) q^{12} + ( - 3 \beta_{9} + 2 \beta_{8} + \beta_{6} + \beta_{5} + \beta_{3} + 4) q^{13} + ( - \beta_{9} - 2 \beta_{8} - 3 \beta_{6} + 2 \beta_{5} + \beta_{3} - 4) q^{14} + ( - 2 \beta_{9} + 2 \beta_{8} + 3 \beta_{6} - 6 \beta_{3} + 5) q^{15} + ( - \beta_{9} - 2 \beta_{8} + \beta_{6} - \beta_{5} - 3 \beta_{3} - 5) q^{16} + ( - \beta_{9} - \beta_{8} + 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{3} + 1) q^{17} + (11 \beta_{9} + 3 \beta_{6} - \beta_{5} + 4 \beta_{3} + 5) q^{18} + ( - 2 \beta_{9} + \beta_{8} + 2 \beta_{6} - 4 \beta_{5} - 2 \beta_{3} - 2) q^{19} + ( - \beta_{11} - \beta_{7} + \beta_{4} + \beta_{2} + 2 \beta_1) q^{20} + (\beta_{11} + \beta_{10} + \beta_{7} - \beta_{4} - \beta_{2} - \beta_1) q^{21} + ( - \beta_{11} - \beta_{7} + 2 \beta_1) q^{22} + (2 \beta_{8} - 3 \beta_{6} - 2 \beta_{5} + \beta_{3} + 2) q^{23} + ( - \beta_{10} - \beta_{7} + \beta_{4} - \beta_{2} + \beta_1) q^{24} + ( - 7 \beta_{9} - 2 \beta_{8} - 3 \beta_{6} - 4 \beta_{5} - 7 \beta_{3} - 25) q^{25} + ( - 5 \beta_{9} + 3 \beta_{8} - 3 \beta_{6} + 3 \beta_{5} + 3 \beta_{3} + 16) q^{26} + ( - \beta_{11} + \beta_{10} + \beta_{4} + 2 \beta_{2} + \beta_1) q^{27} + (4 \beta_{9} + 2 \beta_{8} + 5 \beta_{6} - \beta_{5} - 4 \beta_{3} + 16) q^{28} + (8 \beta_{9} - 4 \beta_{8} + 4 \beta_{6} + 2 \beta_{5} + 10) q^{29} + ( - 8 \beta_{9} - \beta_{8} - 14 \beta_{6} + 5 \beta_{5} + \beta_{3} - 12) q^{30} + ( - \beta_{11} + 2 \beta_{10} + \beta_{7} - \beta_{4} + 2 \beta_1) q^{31} + (10 \beta_{9} - \beta_{8} + \beta_{6} - 5 \beta_{5} + 5 \beta_{3} + 6) q^{32} + ( - 2 \beta_{9} + 2 \beta_{8} - 5 \beta_{6} + 12 \beta_{5} + 2 \beta_{3} + 6) q^{33} + (9 \beta_{9} - 3 \beta_{8} + \beta_{5} + 8 \beta_{3} + 12) q^{34} + ( - 2 \beta_{11} - 2 \beta_{10} - \beta_{7} + 2 \beta_{4} + 4 \beta_1) q^{35} + (3 \beta_{9} - 11 \beta_{8} - 2 \beta_{6} - \beta_{5} + 4 \beta_{3} - 25) q^{36} + (2 \beta_{11} - 3 \beta_{10} - \beta_{7} - \beta_{4} - \beta_{2} - \beta_1) q^{37} + ( - \beta_{9} - 4 \beta_{8} - 7 \beta_{6} + 3 \beta_{5} - \beta_{3} - 6) q^{38} + (2 \beta_{11} - 2 \beta_{10} + \beta_{7} + \beta_{4} - 3 \beta_{2} - \beta_1) q^{39} + (\beta_{11} + \beta_{10} - \beta_{7} - 2 \beta_{4} - 3 \beta_{2} + \beta_1) q^{40} + (11 \beta_{9} - 2 \beta_{6} + 2 \beta_{5} + 7 \beta_{3} - 2) q^{41} + ( - 2 \beta_{11} - 4 \beta_{10} - 3 \beta_{7} + 2 \beta_{4} + 4 \beta_{2} + 5 \beta_1) q^{42} + (\beta_{11} + 2 \beta_{10} + 2 \beta_{4} + 2 \beta_{2} - 5 \beta_1) q^{43} + (4 \beta_{10} + \beta_{7} + 2 \beta_{4} - 3 \beta_{2} + 3 \beta_1) q^{44} + (2 \beta_{11} + \beta_{10} + 3 \beta_{7} - 2 \beta_{2} - 7 \beta_1) q^{45} + ( - 18 \beta_{9} + \beta_{8} + 5 \beta_{6} - \beta_{5} - 12 \beta_{3} - 5) q^{46} + ( - 3 \beta_{7} + \beta_{4} - 3 \beta_{2} - 3 \beta_1) q^{47} + ( - 2 \beta_{11} + \beta_{7} - \beta_{4} - \beta_{2} + 4 \beta_1) q^{48} + (13 \beta_{9} + 4 \beta_{8} + 8 \beta_{6} - 7 \beta_{3} - 10) q^{49} + (11 \beta_{9} + 6 \beta_{8} + \beta_{6} - 5 \beta_{5} - 18 \beta_{3} + 10) q^{50} + ( - \beta_{11} + \beta_{10} + 2 \beta_{7} - 2 \beta_{4} + 4 \beta_{2} + 3 \beta_1) q^{51} + ( - 16 \beta_{9} + 3 \beta_{8} + 2 \beta_{6} - 4 \beta_{5} - 10 \beta_{3} + \cdots + 18) q^{52}+ \cdots + (2 \beta_{11} - 7 \beta_{10} - 5 \beta_{7} + 3 \beta_{4} - 7 \beta_{2} + 8 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} + 14 q^{4} - 16 q^{8} - 76 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} + 14 q^{4} - 16 q^{8} - 76 q^{9} + 30 q^{13} - 50 q^{14} + 78 q^{15} - 50 q^{16} - 14 q^{17} + 44 q^{18} - 10 q^{19} + 48 q^{23} - 242 q^{25} + 176 q^{26} + 202 q^{28} + 96 q^{29} - 104 q^{30} + 74 q^{32} + 42 q^{33} + 104 q^{34} - 324 q^{36} - 46 q^{38} - 40 q^{41} - 48 q^{46} - 92 q^{49} + 276 q^{50} + 254 q^{52} + 128 q^{55} - 280 q^{56} - 408 q^{58} + 194 q^{59} + 450 q^{60} - 150 q^{61} - 328 q^{63} - 396 q^{64} + 414 q^{66} - 230 q^{68} + 24 q^{72} + 284 q^{76} + 158 q^{79} - 444 q^{81} - 440 q^{82} + 162 q^{83} + 238 q^{91} + 548 q^{92} + 248 q^{93} + 192 q^{97} - 1104 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 92x^{10} + 3344x^{8} + 60552x^{6} + 561888x^{4} + 2410584x^{2} + 3274879 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7411 \nu^{11} - 792209 \nu^{9} - 29272621 \nu^{7} - 467341819 \nu^{5} - 3122794327 \nu^{3} - 6320623191 \nu ) / 191450002 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7411\nu^{10} + 792209\nu^{8} + 29272621\nu^{6} + 467341819\nu^{4} + 3122794327\nu^{2} + 6129173189 ) / 191450002 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10957\nu^{11} + 525433\nu^{9} + 8429909\nu^{7} + 57033155\nu^{5} + 49281519\nu^{3} - 2023136193\nu ) / 191450002 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9184\nu^{10} - 658821\nu^{8} - 18851265\nu^{6} - 262187487\nu^{4} - 1586037923\nu^{2} - 2340193501 ) / 95725001 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 53985\nu^{10} + 2903313\nu^{8} + 50847083\nu^{6} + 314025315\nu^{4} + 476141839\nu^{2} + 584114053 ) / 191450002 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 187891 \nu^{11} + 15073221 \nu^{9} + 441088073 \nu^{7} + 5635079717 \nu^{5} + 28678839689 \nu^{3} + 34276772639 \nu ) / 574350006 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 361441 \nu^{10} + 26960115 \nu^{8} + 743975321 \nu^{6} + 9210107111 \nu^{4} + 48121836605 \nu^{2} + 70832662427 ) / 574350006 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 372079 \nu^{10} + 26159787 \nu^{8} + 681447185 \nu^{6} + 7979181119 \nu^{4} + 39475648187 \nu^{2} + 56139684383 ) / 574350006 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 372079 \nu^{11} - 26159787 \nu^{9} - 681447185 \nu^{7} - 7979181119 \nu^{5} - 39475648187 \nu^{3} - 56139684383 \nu ) / 574350006 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 191837 \nu^{11} - 14668371 \nu^{9} - 415896592 \nu^{7} - 5306066284 \nu^{5} - 28745109793 \nu^{3} - 44610090997 \nu ) / 287175003 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_{5} + 2\beta_{3} - 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{10} - \beta_{4} - 2\beta_{2} - 19\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -40\beta_{9} + 37\beta_{8} + 7\beta_{6} - 27\beta_{5} - 50\beta_{3} + 266 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -37\beta_{11} + 33\beta_{10} - 7\beta_{7} + 27\beta_{4} + 67\beta_{2} + 393\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1180\beta_{9} - 1072\beta_{8} - 277\beta_{6} + 601\beta_{5} + 1187\beta_{3} - 5596 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1072\beta_{11} - 903\beta_{10} + 277\beta_{7} - 601\beta_{4} - 1935\beta_{2} - 8503\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -31483\beta_{9} + 28600\beta_{8} + 8076\beta_{6} - 12704\beta_{5} - 28383\beta_{3} + 123607 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -28600\beta_{11} + 23407\beta_{10} - 8076\beta_{7} + 12704\beta_{4} + 52355\beta_{2} + 189846\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 805600\beta_{9} - 734824\beta_{8} - 210600\beta_{6} + 265391\beta_{5} + 681638\beta_{3} - 2811492 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 734824\beta_{11} - 595000\beta_{10} + 210600\beta_{7} - 265391\beta_{4} - 1361671\beta_{2} - 4337536\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/103\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
102.1
4.87595i
4.87595i
3.83937i
3.83937i
3.02467i
3.02467i
4.18237i
4.18237i
1.54521i
1.54521i
4.94529i
4.94529i
−3.18549 4.87595i 6.14732 8.33804i 15.5323i 12.6976 −6.84026 −14.7749 26.5607i
102.2 −3.18549 4.87595i 6.14732 8.33804i 15.5323i 12.6976 −6.84026 −14.7749 26.5607i
102.3 −2.64285 3.83937i 2.98465 6.46444i 10.1469i −3.38248 2.68341 −5.74077 17.0845i
102.4 −2.64285 3.83937i 2.98465 6.46444i 10.1469i −3.38248 2.68341 −5.74077 17.0845i
102.5 −0.771370 3.02467i −3.40499 7.47643i 2.33314i −8.64515 5.71198 −0.148612 5.76709i
102.6 −0.771370 3.02467i −3.40499 7.47643i 2.33314i −8.64515 5.71198 −0.148612 5.76709i
102.7 0.671610 4.18237i −3.54894 1.09651i 2.80892i −0.741172 −5.06994 −8.49219 0.736424i
102.8 0.671610 4.18237i −3.54894 1.09651i 2.80892i −0.741172 −5.06994 −8.49219 0.736424i
102.9 1.88143 1.54521i −0.460236 9.32562i 2.90720i −0.103817 −8.39160 6.61233 17.5455i
102.10 1.88143 1.54521i −0.460236 9.32562i 2.90720i −0.103817 −8.39160 6.61233 17.5455i
102.11 3.04667 4.94529i 5.28219 3.95241i 15.0667i 0.175006 3.90641 −15.4559 12.0417i
102.12 3.04667 4.94529i 5.28219 3.95241i 15.0667i 0.175006 3.90641 −15.4559 12.0417i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 102.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 103.3.b.b 12
3.b odd 2 1 927.3.d.c 12
103.b odd 2 1 inner 103.3.b.b 12
309.c even 2 1 927.3.d.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.3.b.b 12 1.a even 1 1 trivial
103.3.b.b 12 103.b odd 2 1 inner
927.3.d.c 12 3.b odd 2 1
927.3.d.c 12 309.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + T_{2}^{5} - 15T_{2}^{4} - 10T_{2}^{3} + 55T_{2}^{2} + 9T_{2} - 25 \) acting on \(S_{3}^{\mathrm{new}}(103, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{5} - 15 T^{4} - 10 T^{3} + 55 T^{2} + \cdots - 25)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + 92 T^{10} + 3344 T^{8} + \cdots + 3274879 \) Copy content Toggle raw display
$5$ \( T^{12} + 271 T^{10} + \cdots + 265265199 \) Copy content Toggle raw display
$7$ \( (T^{6} - 124 T^{4} - 454 T^{3} - 240 T^{2} + \cdots + 5)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 1001 T^{10} + \cdots + 1697697273600 \) Copy content Toggle raw display
$13$ \( (T^{6} - 15 T^{5} - 307 T^{4} + \cdots - 262480)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 7 T^{5} - 562 T^{4} + \cdots + 226885)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 5 T^{5} - 747 T^{4} + \cdots - 2887376)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 24 T^{5} - 1127 T^{4} + \cdots - 2308060)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 48 T^{5} - 2212 T^{4} + \cdots + 56410112)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 5924 T^{10} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + 10164 T^{10} + \cdots + 67\!\cdots\!91 \) Copy content Toggle raw display
$41$ \( (T^{6} + 20 T^{5} - 1841 T^{4} + \cdots + 3565468)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 10812 T^{10} + \cdots + 13\!\cdots\!79 \) Copy content Toggle raw display
$47$ \( T^{12} + 13319 T^{10} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{12} + 15497 T^{10} + \cdots + 14\!\cdots\!39 \) Copy content Toggle raw display
$59$ \( (T^{6} - 97 T^{5} - 7281 T^{4} + \cdots - 18095168848)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 75 T^{5} - 7235 T^{4} + \cdots - 19383425104)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 39292 T^{10} + \cdots + 48\!\cdots\!71 \) Copy content Toggle raw display
$71$ \( T^{12} + 40631 T^{10} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{12} + 42187 T^{10} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{6} - 79 T^{5} + \cdots - 205105099109)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 81 T^{5} + \cdots - 167129829040)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 28828 T^{10} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{6} - 96 T^{5} - 16423 T^{4} + \cdots - 28741698880)^{2} \) Copy content Toggle raw display
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