Properties

Label 49.9.d.c
Level $49$
Weight $9$
Character orbit 49.d
Analytic conductor $19.962$
Analytic rank $0$
Dimension $8$
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] = [49,9,Mod(19,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.19");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 49.d (of order \(6\), degree \(2\), not 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: \(19.9615518930\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 592x^{6} - 1176x^{5} + 336397x^{4} - 348096x^{3} + 8673408x^{2} + 8271396x + 197880489 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_{2} - \beta_1) q^{2} + ( - \beta_{6} + \beta_{5} - 7 \beta_{2} + 7) q^{3} + ( - \beta_{6} + \beta_{4} - 41 \beta_{2} + 3 \beta_1 - 41) q^{4} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} - \beta_{3} + 70 \beta_{2} - \beta_1 + 140) q^{5} + (\beta_{7} + \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 6 \beta_{3} + 12 \beta_1) q^{6} + ( - 6 \beta_{7} - 22 \beta_{6} + 14 \beta_{5} - 38 \beta_{3} + 818) q^{8} + (15 \beta_{7} - 42 \beta_{6} + 84 \beta_{5} - 15 \beta_{4} + 93 \beta_{3} + \cdots - 93 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_{2} - \beta_1) q^{2} + ( - \beta_{6} + \beta_{5} - 7 \beta_{2} + 7) q^{3} + ( - \beta_{6} + \beta_{4} - 41 \beta_{2} + 3 \beta_1 - 41) q^{4} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} - \beta_{3} + 70 \beta_{2} - \beta_1 + 140) q^{5} + (\beta_{7} + \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 6 \beta_{3} + 12 \beta_1) q^{6} + ( - 6 \beta_{7} - 22 \beta_{6} + 14 \beta_{5} - 38 \beta_{3} + 818) q^{8} + (15 \beta_{7} - 42 \beta_{6} + 84 \beta_{5} - 15 \beta_{4} + 93 \beta_{3} + \cdots - 93 \beta_1) q^{9}+ \cdots + (141009 \beta_{7} - 1624701 \beta_{6} + 741846 \beta_{5} + \cdots + 84030501) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 84 q^{3} - 164 q^{4} + 840 q^{5} + 6544 q^{8} + 396 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 84 q^{3} - 164 q^{4} + 840 q^{5} + 6544 q^{8} + 396 q^{9} - 5796 q^{10} + 1784 q^{11} + 40908 q^{12} + 131808 q^{15} - 8584 q^{16} + 141456 q^{17} - 121944 q^{18} + 257544 q^{19} + 1706256 q^{22} - 348940 q^{23} + 895104 q^{24} - 557752 q^{25} + 2913120 q^{26} + 4983176 q^{29} - 551112 q^{30} + 2376696 q^{31} - 332016 q^{32} + 5719140 q^{33} + 13269408 q^{36} + 492740 q^{37} + 7088088 q^{38} - 2850372 q^{39} + 7601832 q^{40} + 4448432 q^{43} - 3678804 q^{44} - 3328164 q^{45} - 226560 q^{46} - 2704128 q^{47} - 15628912 q^{50} - 350856 q^{51} - 11135208 q^{52} + 2281460 q^{53} - 24553368 q^{54} - 43638408 q^{57} + 11442696 q^{58} - 25291140 q^{59} + 17679564 q^{60} - 59368764 q^{61} - 114153056 q^{64} + 16923396 q^{65} - 463428 q^{66} - 107108 q^{67} - 44316972 q^{68} - 82809760 q^{71} + 13297584 q^{72} - 116758404 q^{73} + 72690340 q^{74} - 79832424 q^{75} - 44122512 q^{78} - 50628092 q^{79} + 93591624 q^{80} - 96868872 q^{81} + 91061712 q^{82} + 119231208 q^{85} + 39093088 q^{86} - 26702676 q^{87} + 41392848 q^{88} + 2322516 q^{89} + 253819128 q^{92} - 57693204 q^{93} + 345566088 q^{94} - 172787052 q^{95} + 416455200 q^{96} + 672244008 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 592x^{6} - 1176x^{5} + 336397x^{4} - 348096x^{3} + 8673408x^{2} + 8271396x + 197880489 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 40665038848 \nu^{7} + 933800395536 \nu^{6} - 23107427488768 \nu^{5} + 578443118889849 \nu^{4} + \cdots - 24\!\cdots\!71 ) / 78\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 199147024 \nu^{7} - 206072832 \nu^{6} - 113162941609 \nu^{5} + 117098450112 \nu^{4} - 66871290607312 \nu^{3} + \cdots - 17\!\cdots\!48 ) / 16\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4986016178489 \nu^{7} + 729234428373612 \nu^{6} + \cdots + 58\!\cdots\!33 ) / 21\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 142450897959062 \nu^{7} - 415028421151641 \nu^{6} + \cdots - 22\!\cdots\!49 ) / 10\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 35983628949817 \nu^{7} + 17429303220966 \nu^{6} + \cdots - 87\!\cdots\!11 ) / 21\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 206519207416061 \nu^{7} - 209968840103652 \nu^{6} + \cdots + 29\!\cdots\!97 ) / 10\!\cdots\!50 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{4} + \beta_{3} + 296\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{7} - 25\beta_{6} + 14\beta_{5} - 544\beta_{3} + 441 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -592\beta_{6} + 592\beta_{4} - 161165\beta_{2} + 1180\beta _1 - 161165 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2364\beta_{7} + 8288\beta_{6} - 16576\beta_{5} - 2364\beta_{4} + 308569\beta_{3} + 435120\beta_{2} - 308569\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -338161\beta_{7} + 321697\beta_{6} + 8232\beta_{5} - 1004365\beta_{3} + 91505156 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3004175\beta_{6} + 4709558\beta_{5} + 1705383\beta_{4} - 346152513\beta_{2} + 175714240\beta _1 - 346152513 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-\beta_{2}\)

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
−12.1698 21.0787i
−2.23583 3.87257i
2.77916 + 4.81365i
11.6264 + 20.1376i
−12.1698 + 21.0787i
−2.23583 + 3.87257i
2.77916 4.81365i
11.6264 20.1376i
−12.6698 + 21.9447i 7.68317 4.43588i −193.046 334.366i 538.352 + 310.818i 224.806i 0 3296.47 −3241.15 + 5613.83i −13641.6 + 7875.98i
19.2 −2.73583 + 4.73860i −58.8837 + 33.9965i 113.030 + 195.774i −586.541 338.640i 372.035i 0 −2637.67 −968.977 + 1678.32i 3209.35 1852.92i
19.3 2.27916 3.94762i 124.416 71.8315i 117.611 + 203.708i 163.006 + 94.1113i 654.862i 0 2239.15 7039.03 12192.0i 743.031 428.989i
19.4 11.1264 19.2716i −31.2153 + 18.0222i −119.595 207.145i 305.183 + 176.198i 802.090i 0 374.058 −2630.90 + 4556.86i 6791.21 3920.91i
31.1 −12.6698 21.9447i 7.68317 + 4.43588i −193.046 + 334.366i 538.352 310.818i 224.806i 0 3296.47 −3241.15 5613.83i −13641.6 7875.98i
31.2 −2.73583 4.73860i −58.8837 33.9965i 113.030 195.774i −586.541 + 338.640i 372.035i 0 −2637.67 −968.977 1678.32i 3209.35 + 1852.92i
31.3 2.27916 + 3.94762i 124.416 + 71.8315i 117.611 203.708i 163.006 94.1113i 654.862i 0 2239.15 7039.03 + 12192.0i 743.031 + 428.989i
31.4 11.1264 + 19.2716i −31.2153 18.0222i −119.595 + 207.145i 305.183 176.198i 802.090i 0 374.058 −2630.90 4556.86i 6791.21 + 3920.91i
\(n\): 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
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.9.d.c 8
7.b odd 2 1 7.9.d.a 8
7.c even 3 1 7.9.d.a 8
7.c even 3 1 49.9.b.a 8
7.d odd 6 1 49.9.b.a 8
7.d odd 6 1 inner 49.9.d.c 8
21.c even 2 1 63.9.m.b 8
21.h odd 6 1 63.9.m.b 8
28.d even 2 1 112.9.s.a 8
28.g odd 6 1 112.9.s.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.9.d.a 8 7.b odd 2 1
7.9.d.a 8 7.c even 3 1
49.9.b.a 8 7.c even 3 1
49.9.b.a 8 7.d odd 6 1
49.9.d.c 8 1.a even 1 1 trivial
49.9.d.c 8 7.d odd 6 1 inner
63.9.m.b 8 21.c even 2 1
63.9.m.b 8 21.h odd 6 1
112.9.s.a 8 28.d even 2 1
112.9.s.a 8 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 4 T_{2}^{7} + 602 T_{2}^{6} - 1160 T_{2}^{5} + 331700 T_{2}^{4} + 234400 T_{2}^{3} + 8591968 T_{2}^{2} - 8325888 T_{2} + 197796096 \) acting on \(S_{9}^{\mathrm{new}}(49, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4 T^{7} + 602 T^{6} + \cdots + 197796096 \) Copy content Toggle raw display
$3$ \( T^{8} - 84 T^{7} + \cdots + 9756895701609 \) Copy content Toggle raw display
$5$ \( T^{8} - 840 T^{7} + \cdots + 77\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 1784 T^{7} + \cdots + 39\!\cdots\!29 \) Copy content Toggle raw display
$13$ \( T^{8} + 2926140840 T^{6} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{8} - 141456 T^{7} + \cdots + 21\!\cdots\!69 \) Copy content Toggle raw display
$19$ \( T^{8} - 257544 T^{7} + \cdots + 42\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{8} + 348940 T^{7} + \cdots + 98\!\cdots\!21 \) Copy content Toggle raw display
$29$ \( (T^{4} - 2491588 T^{3} + \cdots - 15\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 2376696 T^{7} + \cdots + 38\!\cdots\!09 \) Copy content Toggle raw display
$37$ \( T^{8} - 492740 T^{7} + \cdots + 47\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( T^{8} + 26277784311912 T^{6} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} - 2224216 T^{3} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 2704128 T^{7} + \cdots + 20\!\cdots\!89 \) Copy content Toggle raw display
$53$ \( T^{8} - 2281460 T^{7} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{8} + 25291140 T^{7} + \cdots + 20\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{8} + 59368764 T^{7} + \cdots + 25\!\cdots\!81 \) Copy content Toggle raw display
$67$ \( T^{8} + 107108 T^{7} + \cdots + 40\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( (T^{4} + 41404880 T^{3} + \cdots + 63\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 116758404 T^{7} + \cdots + 63\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{8} + 50628092 T^{7} + \cdots + 13\!\cdots\!89 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 44\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{8} - 2322516 T^{7} + \cdots + 31\!\cdots\!49 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 20\!\cdots\!64 \) Copy content Toggle raw display
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