Properties

Label 100.9.b.d
Level $100$
Weight $9$
Character orbit 100.b
Analytic conductor $40.738$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 5 x^{15} + 26 x^{14} - 834 x^{13} + 4390 x^{12} - 61783 x^{11} + 466168 x^{10} + \cdots + 206161212459445 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{61}\cdot 5^{16} \)
Twist minimal: no (minimal twist has level 20)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{2} + \beta_1) q^{3} + ( - \beta_{3} - 3) q^{4} + ( - \beta_{8} + \beta_{3} + 5 \beta_{2} + 271) q^{6} + ( - \beta_{12} - \beta_{3} + 3 \beta_{2} + \cdots - 7) q^{7}+ \cdots + ( - \beta_{13} + 2 \beta_{8} + \cdots - 2371) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{2} + \beta_1) q^{3} + ( - \beta_{3} - 3) q^{4} + ( - \beta_{8} + \beta_{3} + 5 \beta_{2} + 271) q^{6} + ( - \beta_{12} - \beta_{3} + 3 \beta_{2} + \cdots - 7) q^{7}+ \cdots + (1088 \beta_{15} + 82 \beta_{14} + \cdots - 446770) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{2} - 52 q^{4} + 4368 q^{6} + 14184 q^{8} - 38800 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{2} - 52 q^{4} + 4368 q^{6} + 14184 q^{8} - 38800 q^{9} + 64040 q^{12} - 51392 q^{13} + 68472 q^{14} - 81424 q^{16} - 27552 q^{17} + 616994 q^{18} + 414496 q^{21} + 389120 q^{22} + 163792 q^{24} + 1037124 q^{26} - 1288520 q^{28} + 2764896 q^{29} + 4379904 q^{32} + 5521600 q^{33} + 3793964 q^{34} - 5468916 q^{36} - 9009472 q^{37} + 3087360 q^{38} - 8576448 q^{41} + 4067400 q^{42} + 16921200 q^{44} - 7974152 q^{46} + 2696640 q^{48} - 32803600 q^{49} - 6679352 q^{52} - 2452032 q^{53} + 8898704 q^{54} + 34134768 q^{56} - 11957760 q^{57} - 52156572 q^{58} + 8371712 q^{61} - 1290000 q^{62} - 47543872 q^{64} + 19358000 q^{66} - 16095192 q^{68} + 7527264 q^{69} + 42242664 q^{72} - 61907232 q^{73} - 138210876 q^{74} + 2570400 q^{76} + 156997440 q^{77} + 104032400 q^{78} + 140586672 q^{81} - 83921012 q^{82} + 69761824 q^{84} - 101724672 q^{86} - 44728480 q^{88} + 106647456 q^{89} + 13876200 q^{92} - 105563840 q^{93} + 55264632 q^{94} - 453389952 q^{96} - 171851232 q^{97} + 285387714 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 5 x^{15} + 26 x^{14} - 834 x^{13} + 4390 x^{12} - 61783 x^{11} + 466168 x^{10} + \cdots + 206161212459445 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 10\!\cdots\!99 \nu^{15} + \cdots + 17\!\cdots\!30 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 71\!\cdots\!31 \nu^{15} + \cdots - 31\!\cdots\!00 ) / 43\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 52\!\cdots\!97 \nu^{15} + \cdots - 31\!\cdots\!55 ) / 91\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 48\!\cdots\!01 \nu^{15} + \cdots + 13\!\cdots\!90 ) / 45\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 26\!\cdots\!88 \nu^{15} + \cdots - 65\!\cdots\!15 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27\!\cdots\!01 \nu^{15} + \cdots + 50\!\cdots\!25 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 33\!\cdots\!98 \nu^{15} + \cdots + 11\!\cdots\!95 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!59 \nu^{15} + \cdots + 96\!\cdots\!20 ) / 43\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 30\!\cdots\!45 \nu^{15} + \cdots - 18\!\cdots\!20 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 51\!\cdots\!21 \nu^{15} + \cdots - 13\!\cdots\!10 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 19\!\cdots\!49 \nu^{15} + \cdots - 13\!\cdots\!15 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 73\!\cdots\!49 \nu^{15} + \cdots + 30\!\cdots\!35 ) / 72\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 51\!\cdots\!09 \nu^{15} + \cdots - 16\!\cdots\!60 ) / 43\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 58\!\cdots\!04 \nu^{15} + \cdots + 45\!\cdots\!65 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11\!\cdots\!87 \nu^{15} + \cdots - 30\!\cdots\!60 ) / 43\!\cdots\!60 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 125\beta_{5} + 32\beta_{4} - 125\beta_{3} - 250\beta_{2} + 128\beta _1 + 5093 ) / 16000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 20 \beta_{15} + 12 \beta_{14} + 123 \beta_{13} - 115 \beta_{11} - 135 \beta_{10} + 20 \beta_{9} + \cdots - 19897 ) / 16000 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 350 \beta_{15} - 494 \beta_{14} + 299 \beta_{13} - 720 \beta_{12} - 415 \beta_{11} + 275 \beta_{10} + \cdots + 2242245 ) / 16000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 790 \beta_{15} + 642 \beta_{14} - 1352 \beta_{13} - 3280 \beta_{12} + 6360 \beta_{11} + \cdots - 955271 ) / 16000 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1575 \beta_{15} + 2457 \beta_{14} + 2243 \beta_{13} - 7520 \beta_{12} + 1405 \beta_{11} + \cdots + 19995941 ) / 1600 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 83480 \beta_{15} + 121160 \beta_{14} - 313370 \beta_{13} - 795280 \beta_{12} + 198110 \beta_{11} + \cdots + 512868853 ) / 16000 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 558550 \beta_{15} + 1267282 \beta_{14} - 1464947 \beta_{13} - 1564400 \beta_{12} + \cdots - 10155933897 ) / 16000 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 7880500 \beta_{15} - 3097684 \beta_{14} - 5067471 \beta_{13} - 50600000 \beta_{12} + \cdots - 111520988635 ) / 16000 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 98172710 \beta_{15} + 86068282 \beta_{14} - 499335522 \beta_{13} - 333381120 \beta_{12} + \cdots - 1075416741271 ) / 16000 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 103616980 \beta_{15} + 177985124 \beta_{14} - 408489084 \beta_{13} - 258440720 \beta_{12} + \cdots - 236808055548 ) / 1600 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 4387774530 \beta_{15} + 15735417730 \beta_{14} - 34049618190 \beta_{13} - 44479097680 \beta_{12} + \cdots - 112607212297717 ) / 16000 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 27582252350 \beta_{15} + 146548866662 \beta_{14} - 306234377467 \beta_{13} + \cdots - 15\!\cdots\!47 ) / 16000 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 89308249690 \beta_{15} + 87341593326 \beta_{14} - 1952057514441 \beta_{13} + 530274756720 \beta_{12} + \cdots - 93\!\cdots\!95 ) / 16000 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 5578949131920 \beta_{15} + 3079412068112 \beta_{14} - 24707521165142 \beta_{13} + \cdots - 11\!\cdots\!61 ) / 16000 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 1448295323885 \beta_{15} + 4931541114483 \beta_{14} - 7401580619303 \beta_{13} + \cdots - 44\!\cdots\!06 ) / 800 \) 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
−5.64565 + 3.35245i
−5.64565 3.35245i
−4.16577 + 5.52698i
−4.16577 5.52698i
−6.29294 + 5.63875i
−6.29294 5.63875i
−2.93811 + 7.65619i
−2.93811 7.65619i
2.32463 + 7.96873i
2.32463 7.96873i
3.05707 + 7.10588i
3.05707 7.10588i
6.86496 + 2.82928i
6.86496 2.82928i
9.29581 + 2.24761i
9.29581 2.24761i
−14.5274 6.70489i 150.211i 166.089 + 194.809i 0 −1007.15 + 2182.17i 2626.96i −1106.66 3943.67i −16002.3 0
51.2 −14.5274 + 6.70489i 150.211i 166.089 194.809i 0 −1007.15 2182.17i 2626.96i −1106.66 + 3943.67i −16002.3 0
51.3 −11.5676 11.0540i 27.2434i 11.6196 + 255.736i 0 301.148 315.141i 3325.58i 2692.49 3086.70i 5818.80 0
51.4 −11.5676 + 11.0540i 27.2434i 11.6196 255.736i 0 301.148 + 315.141i 3325.58i 2692.49 + 3086.70i 5818.80 0
51.5 −11.3498 11.2775i 137.297i 1.63618 + 255.995i 0 1548.37 1558.29i 3940.57i 2868.41 2923.94i −12289.4 0
51.6 −11.3498 + 11.2775i 137.297i 1.63618 255.995i 0 1548.37 + 1558.29i 3940.57i 2868.41 + 2923.94i −12289.4 0
51.7 −4.64016 15.3124i 75.7492i −212.938 + 142.104i 0 −1159.90 + 351.488i 210.345i 3164.01 + 2601.20i 823.060 0
51.8 −4.64016 + 15.3124i 75.7492i −212.938 142.104i 0 −1159.90 351.488i 210.345i 3164.01 2601.20i 823.060 0
51.9 1.41320 15.9375i 39.9624i −252.006 45.0455i 0 636.899 + 56.4746i 2633.20i −1074.04 + 3952.68i 4964.01 0
51.10 1.41320 + 15.9375i 39.9624i −252.006 + 45.0455i 0 636.899 56.4746i 2633.20i −1074.04 3952.68i 4964.01 0
51.11 7.35022 14.2118i 110.171i −147.949 208.919i 0 1565.72 + 809.778i 3540.70i −4056.56 + 567.011i −5576.56 0
51.12 7.35022 + 14.2118i 110.171i −147.949 + 208.919i 0 1565.72 809.778i 3540.70i −4056.56 567.011i −5576.56 0
51.13 14.9660 5.65855i 25.1248i 191.962 169.372i 0 −142.170 376.017i 2973.76i 1914.50 3621.04i 5929.74 0
51.14 14.9660 + 5.65855i 25.1248i 191.962 + 169.372i 0 −142.170 + 376.017i 2973.76i 1914.50 + 3621.04i 5929.74 0
51.15 15.3556 4.49522i 98.1237i 215.586 138.053i 0 441.088 + 1506.74i 820.952i 2689.86 3088.99i −3067.27 0
51.16 15.3556 + 4.49522i 98.1237i 215.586 + 138.053i 0 441.088 1506.74i 820.952i 2689.86 + 3088.99i −3067.27 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.16
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.d 16
4.b odd 2 1 inner 100.9.b.d 16
5.b even 2 1 20.9.b.a 16
5.c odd 4 2 100.9.d.c 32
15.d odd 2 1 180.9.c.a 16
20.d odd 2 1 20.9.b.a 16
20.e even 4 2 100.9.d.c 32
40.e odd 2 1 320.9.b.d 16
40.f even 2 1 320.9.b.d 16
60.h even 2 1 180.9.c.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.9.b.a 16 5.b even 2 1
20.9.b.a 16 20.d odd 2 1
100.9.b.d 16 1.a even 1 1 trivial
100.9.b.d 16 4.b odd 2 1 inner
100.9.d.c 32 5.c odd 4 2
100.9.d.c 32 20.e even 4 2
180.9.c.a 16 15.d odd 2 1
180.9.c.a 16 60.h even 2 1
320.9.b.d 16 40.e odd 2 1
320.9.b.d 16 40.f even 2 1

Hecke kernels

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

\( T_{3}^{16} + 71888 T_{3}^{14} + 2013496736 T_{3}^{12} + 27929868057600 T_{3}^{10} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
\( T_{13}^{8} + 25696 T_{13}^{7} - 3918908944 T_{13}^{6} - 65251903733888 T_{13}^{5} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 92\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 61\!\cdots\!04)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 56\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 83\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 36\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
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