# 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 $2$

# Related objects

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: $$x^{2} - x + 10$$ x^2 - x + 10 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.

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

## 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
 0.5 + 3.12250i 0.5 − 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{9}^{\mathrm{new}}(100, [\chi])$$:

 $$T_{3}^{2} + 9984$$ T3^2 + 9984 $$T_{13} - 5470$$ T13 - 5470

## Hecke characteristic polynomials

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