Properties

Label 4.9.b.b
Level $4$
Weight $9$
Character orbit 4.b
Analytic conductor $1.630$
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] = [4,9,Mod(3,4)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 4.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: \(1.62951444024\)
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: 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 \(\beta = 2\sqrt{-39}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 10) q^{2} - 8 \beta q^{3} + (20 \beta - 56) q^{4} + 610 q^{5} + (80 \beta - 1248) q^{6} - 112 \beta q^{7} + ( - 144 \beta + 3680) q^{8} - 3423 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 10) q^{2} - 8 \beta q^{3} + (20 \beta - 56) q^{4} + 610 q^{5} + (80 \beta - 1248) q^{6} - 112 \beta q^{7} + ( - 144 \beta + 3680) q^{8} - 3423 q^{9} + ( - 610 \beta - 6100) q^{10} + 1480 \beta q^{11} + (448 \beta + 24960) q^{12} - 5470 q^{13} + (1120 \beta - 17472) q^{14} - 4880 \beta q^{15} + ( - 2240 \beta - 59264) q^{16} + 73090 q^{17} + (3423 \beta + 34230) q^{18} + 1560 \beta q^{19} + (12200 \beta - 34160) q^{20} - 139776 q^{21} + ( - 14800 \beta + 230880) q^{22} + 18992 \beta q^{23} + ( - 29440 \beta - 179712) q^{24} - 18525 q^{25} + (5470 \beta + 54700) q^{26} - 25104 \beta q^{27} + (6272 \beta + 349440) q^{28} - 128222 q^{29} + (48800 \beta - 761280) q^{30} + 5440 \beta q^{31} + (81664 \beta + 243200) q^{32} + 1847040 q^{33} + ( - 73090 \beta - 730900) q^{34} - 68320 \beta q^{35} + ( - 68460 \beta + 191688) q^{36} - 3472030 q^{37} + ( - 15600 \beta + 243360) q^{38} + 43760 \beta q^{39} + ( - 87840 \beta + 2244800) q^{40} + 2146882 q^{41} + (139776 \beta + 1397760) q^{42} + 474632 \beta q^{43} + ( - 82880 \beta - 4617600) q^{44} - 2088030 q^{45} + ( - 189920 \beta + 2962752) q^{46} - 610592 \beta q^{47} + (474112 \beta - 2795520) q^{48} + 3807937 q^{49} + (18525 \beta + 185250) q^{50} - 584720 \beta q^{51} + ( - 109400 \beta + 306320) q^{52} + 824290 q^{53} + (251040 \beta - 3916224) q^{54} + 902800 \beta q^{55} + ( - 412160 \beta - 2515968) q^{56} + 1946880 q^{57} + (128222 \beta + 1282220) q^{58} + 298280 \beta q^{59} + (273280 \beta + 15225600) q^{60} - 14746078 q^{61} + ( - 54400 \beta + 848640) q^{62} + 383376 \beta q^{63} + ( - 1059840 \beta + 10307584) q^{64} - 3336700 q^{65} + ( - 1847040 \beta - 18470400) q^{66} - 1221512 \beta q^{67} + (1461800 \beta - 4093040) q^{68} + 23702016 q^{69} + (683200 \beta - 10657920) q^{70} + 95760 \beta q^{71} + (492912 \beta - 12596640) q^{72} - 5725630 q^{73} + (3472030 \beta + 34720300) q^{74} + 148200 \beta q^{75} + ( - 87360 \beta - 4867200) q^{76} + 25858560 q^{77} + ( - 437600 \beta + 6826560) q^{78} - 2875360 \beta q^{79} + ( - 1366400 \beta - 36151040) q^{80} - 53788095 q^{81} + ( - 2146882 \beta - 21468820) q^{82} + 4160152 \beta q^{83} + ( - 2795520 \beta + 7827456) q^{84} + 44584900 q^{85} + ( - 4746320 \beta + 74042592) q^{86} + 1025776 \beta q^{87} + (5446400 \beta + 33246720) q^{88} - 83324222 q^{89} + (2088030 \beta + 20880300) q^{90} + 612640 \beta q^{91} + ( - 1063552 \beta - 59255040) q^{92} + 6789120 q^{93} + (6105920 \beta - 95252352) q^{94} + 951600 \beta q^{95} + ( - 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
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 20 q^{2} - 112 q^{4} + 1220 q^{5} - 2496 q^{6} + 7360 q^{8} - 6846 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 20 q^{2} - 112 q^{4} + 1220 q^{5} - 2496 q^{6} + 7360 q^{8} - 6846 q^{9} - 12200 q^{10} + 49920 q^{12} - 10940 q^{13} - 34944 q^{14} - 118528 q^{16} + 146180 q^{17} + 68460 q^{18} - 68320 q^{20} - 279552 q^{21} + 461760 q^{22} - 359424 q^{24} - 37050 q^{25} + 109400 q^{26} + 698880 q^{28} - 256444 q^{29} - 1522560 q^{30} + 486400 q^{32} + 3694080 q^{33} - 1461800 q^{34} + 383376 q^{36} - 6944060 q^{37} + 486720 q^{38} + 4489600 q^{40} + 4293764 q^{41} + 2795520 q^{42} - 9235200 q^{44} - 4176060 q^{45} + 5925504 q^{46} - 5591040 q^{48} + 7615874 q^{49} + 370500 q^{50} + 612640 q^{52} + 1648580 q^{53} - 7832448 q^{54} - 5031936 q^{56} + 3893760 q^{57} + 2564440 q^{58} + 30451200 q^{60} - 29492156 q^{61} + 1697280 q^{62} + 20615168 q^{64} - 6673400 q^{65} - 36940800 q^{66} - 8186080 q^{68} + 47404032 q^{69} - 21315840 q^{70} - 25193280 q^{72} - 11451260 q^{73} + 69440600 q^{74} - 9734400 q^{76} + 51717120 q^{77} + 13653120 q^{78} - 72302080 q^{80} - 107576190 q^{81} - 42937640 q^{82} + 15654912 q^{84} + 89169800 q^{85} + 148085184 q^{86} + 66493440 q^{88} - 166648444 q^{89} + 41760600 q^{90} - 118510080 q^{92} + 13578240 q^{93} - 190504704 q^{94} + 203833344 q^{96} + 241238020 q^{97} - 76158740 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.500000 + 3.12250i
0.500000 3.12250i
−10.0000 12.4900i 99.9200i −56.0000 + 249.800i 610.000 −1248.00 + 999.200i 1398.88i 3680.00 1798.56i −3423.00 −6100.00 7618.90i
3.2 −10.0000 + 12.4900i 99.9200i −56.0000 249.800i 610.000 −1248.00 999.200i 1398.88i 3680.00 + 1798.56i −3423.00 −6100.00 + 7618.90i
\(n\): 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 4.9.b.b 2
3.b odd 2 1 36.9.d.b 2
4.b odd 2 1 inner 4.9.b.b 2
5.b even 2 1 100.9.b.c 2
5.c odd 4 2 100.9.d.b 4
8.b even 2 1 64.9.c.b 2
8.d odd 2 1 64.9.c.b 2
12.b even 2 1 36.9.d.b 2
16.e even 4 2 256.9.d.e 4
16.f odd 4 2 256.9.d.e 4
20.d odd 2 1 100.9.b.c 2
20.e even 4 2 100.9.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.9.b.b 2 1.a even 1 1 trivial
4.9.b.b 2 4.b odd 2 1 inner
36.9.d.b 2 3.b odd 2 1
36.9.d.b 2 12.b even 2 1
64.9.c.b 2 8.b even 2 1
64.9.c.b 2 8.d odd 2 1
100.9.b.c 2 5.b even 2 1
100.9.b.c 2 20.d odd 2 1
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 16.e even 4 2
256.9.d.e 4 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 9984 \) acting on \(S_{9}^{\mathrm{new}}(4, [\chi])\). 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
$5$ \( (T - 610)^{2} \) 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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