Properties

Label 100.9.b.c
Level $100$
Weight $9$
Character orbit 100.b
Analytic conductor $40.738$
Analytic rank $0$
Dimension $2$
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] = [100,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.51");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
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: \(40.7378610061\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-39}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 4)
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 \(\beta = 2\sqrt{-39}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 10) q^{2} - 8 \beta q^{3} + ( - 20 \beta - 56) q^{4} + ( - 80 \beta - 1248) q^{6} - 112 \beta q^{7} + ( - 144 \beta - 3680) q^{8} - 3423 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 10) q^{2} - 8 \beta q^{3} + ( - 20 \beta - 56) q^{4} + ( - 80 \beta - 1248) q^{6} - 112 \beta q^{7} + ( - 144 \beta - 3680) q^{8} - 3423 q^{9} - 1480 \beta q^{11} + (448 \beta - 24960) q^{12} + 5470 q^{13} + ( - 1120 \beta - 17472) q^{14} + (2240 \beta - 59264) q^{16} - 73090 q^{17} + (3423 \beta - 34230) q^{18} - 1560 \beta q^{19} - 139776 q^{21} + ( - 14800 \beta - 230880) q^{22} + 18992 \beta q^{23} + (29440 \beta - 179712) q^{24} + ( - 5470 \beta + 54700) q^{26} - 25104 \beta q^{27} + (6272 \beta - 349440) q^{28} - 128222 q^{29} - 5440 \beta q^{31} + (81664 \beta - 243200) q^{32} - 1847040 q^{33} + (73090 \beta - 730900) q^{34} + (68460 \beta + 191688) q^{36} + 3472030 q^{37} + ( - 15600 \beta - 243360) q^{38} - 43760 \beta q^{39} + 2146882 q^{41} + (139776 \beta - 1397760) q^{42} + 474632 \beta q^{43} + (82880 \beta - 4617600) q^{44} + (189920 \beta + 2962752) q^{46} - 610592 \beta q^{47} + (474112 \beta + 2795520) q^{48} + 3807937 q^{49} + 584720 \beta q^{51} + ( - 109400 \beta - 306320) q^{52} - 824290 q^{53} + ( - 251040 \beta - 3916224) q^{54} + (412160 \beta - 2515968) q^{56} - 1946880 q^{57} + (128222 \beta - 1282220) q^{58} - 298280 \beta q^{59} - 14746078 q^{61} + ( - 54400 \beta - 848640) q^{62} + 383376 \beta q^{63} + (1059840 \beta + 10307584) q^{64} + (1847040 \beta - 18470400) q^{66} - 1221512 \beta q^{67} + (1461800 \beta + 4093040) q^{68} + 23702016 q^{69} - 95760 \beta q^{71} + (492912 \beta + 12596640) q^{72} + 5725630 q^{73} + ( - 3472030 \beta + 34720300) q^{74} + (87360 \beta - 4867200) q^{76} - 25858560 q^{77} + ( - 437600 \beta - 6826560) q^{78} + 2875360 \beta q^{79} - 53788095 q^{81} + ( - 2146882 \beta + 21468820) q^{82} + 4160152 \beta q^{83} + (2795520 \beta + 7827456) q^{84} + (4746320 \beta + 74042592) q^{86} + 1025776 \beta q^{87} + (5446400 \beta - 33246720) q^{88} - 83324222 q^{89} - 612640 \beta q^{91} + ( - 1063552 \beta + 59255040) q^{92} - 6789120 q^{93} + ( - 6105920 \beta - 95252352) q^{94} + (1945600 \beta + 101916672) q^{96} - 120619010 q^{97} + ( - 3807937 \beta + 38079370) q^{98} + 5066040 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 20 q^{2} - 112 q^{4} - 2496 q^{6} - 7360 q^{8} - 6846 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 20 q^{2} - 112 q^{4} - 2496 q^{6} - 7360 q^{8} - 6846 q^{9} - 49920 q^{12} + 10940 q^{13} - 34944 q^{14} - 118528 q^{16} - 146180 q^{17} - 68460 q^{18} - 279552 q^{21} - 461760 q^{22} - 359424 q^{24} + 109400 q^{26} - 698880 q^{28} - 256444 q^{29} - 486400 q^{32} - 3694080 q^{33} - 1461800 q^{34} + 383376 q^{36} + 6944060 q^{37} - 486720 q^{38} + 4293764 q^{41} - 2795520 q^{42} - 9235200 q^{44} + 5925504 q^{46} + 5591040 q^{48} + 7615874 q^{49} - 612640 q^{52} - 1648580 q^{53} - 7832448 q^{54} - 5031936 q^{56} - 3893760 q^{57} - 2564440 q^{58} - 29492156 q^{61} - 1697280 q^{62} + 20615168 q^{64} - 36940800 q^{66} + 8186080 q^{68} + 47404032 q^{69} + 25193280 q^{72} + 11451260 q^{73} + 69440600 q^{74} - 9734400 q^{76} - 51717120 q^{77} - 13653120 q^{78} - 107576190 q^{81} + 42937640 q^{82} + 15654912 q^{84} + 148085184 q^{86} - 66493440 q^{88} - 166648444 q^{89} + 118510080 q^{92} - 13578240 q^{93} - 190504704 q^{94} + 203833344 q^{96} - 241238020 q^{97} + 76158740 q^{98}+O(q^{100}) \) 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
0.500000 + 3.12250i
0.500000 3.12250i
10.0000 12.4900i 99.9200i −56.0000 249.800i 0 −1248.00 999.200i 1398.88i −3680.00 1798.56i −3423.00 0
51.2 10.0000 + 12.4900i 99.9200i −56.0000 + 249.800i 0 −1248.00 + 999.200i 1398.88i −3680.00 + 1798.56i −3423.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.9.b.c 2
4.b odd 2 1 inner 100.9.b.c 2
5.b even 2 1 4.9.b.b 2
5.c odd 4 2 100.9.d.b 4
15.d odd 2 1 36.9.d.b 2
20.d odd 2 1 4.9.b.b 2
20.e even 4 2 100.9.d.b 4
40.e odd 2 1 64.9.c.b 2
40.f even 2 1 64.9.c.b 2
60.h even 2 1 36.9.d.b 2
80.k odd 4 2 256.9.d.e 4
80.q even 4 2 256.9.d.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.9.b.b 2 5.b even 2 1
4.9.b.b 2 20.d odd 2 1
36.9.d.b 2 15.d odd 2 1
36.9.d.b 2 60.h even 2 1
64.9.c.b 2 40.e odd 2 1
64.9.c.b 2 40.f even 2 1
100.9.b.c 2 1.a even 1 1 trivial
100.9.b.c 2 4.b odd 2 1 inner
100.9.d.b 4 5.c odd 4 2
100.9.d.b 4 20.e even 4 2
256.9.d.e 4 80.k odd 4 2
256.9.d.e 4 80.q even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(100, [\chi])\):

\( T_{3}^{2} + 9984 \) Copy content Toggle raw display
\( T_{13} - 5470 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 20T + 256 \) Copy content Toggle raw display
$3$ \( T^{2} + 9984 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1956864 \) Copy content Toggle raw display
$11$ \( T^{2} + 341702400 \) Copy content Toggle raw display
$13$ \( (T - 5470)^{2} \) Copy content Toggle raw display
$17$ \( (T + 73090)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 379641600 \) Copy content Toggle raw display
$23$ \( T^{2} + 56268585984 \) Copy content Toggle raw display
$29$ \( (T + 128222)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4616601600 \) Copy content Toggle raw display
$37$ \( (T - 3472030)^{2} \) Copy content Toggle raw display
$41$ \( (T - 2146882)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 35142983526144 \) Copy content Toggle raw display
$47$ \( T^{2} + 58160324112384 \) Copy content Toggle raw display
$53$ \( (T + 824290)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 13879469510400 \) Copy content Toggle raw display
$61$ \( (T + 14746078)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 232766284318464 \) Copy content Toggle raw display
$71$ \( T^{2} + 1430516505600 \) Copy content Toggle raw display
$73$ \( (T - 5725630)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + 26\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T + 83324222)^{2} \) Copy content Toggle raw display
$97$ \( (T + 120619010)^{2} \) Copy content Toggle raw display
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