# Properties

 Label 350.6.c.f Level $350$ Weight $6$ Character orbit 350.c Analytic conductor $56.134$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,6,Mod(99,350)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("350.99");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 350.c (of order $$2$$, degree $$1$$, 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: $$56.1343369345$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 4 i q^{2} + 8 i q^{3} - 16 q^{4} + 32 q^{6} + 49 i q^{7} + 64 i q^{8} + 179 q^{9} +O(q^{10})$$ q - 4*i * q^2 + 8*i * q^3 - 16 * q^4 + 32 * q^6 + 49*i * q^7 + 64*i * q^8 + 179 * q^9 $$q - 4 i q^{2} + 8 i q^{3} - 16 q^{4} + 32 q^{6} + 49 i q^{7} + 64 i q^{8} + 179 q^{9} - 340 q^{11} - 128 i q^{12} - 294 i q^{13} + 196 q^{14} + 256 q^{16} - 1226 i q^{17} - 716 i q^{18} - 2432 q^{19} - 392 q^{21} + 1360 i q^{22} + 2000 i q^{23} - 512 q^{24} - 1176 q^{26} + 3376 i q^{27} - 784 i q^{28} + 6746 q^{29} + 8856 q^{31} - 1024 i q^{32} - 2720 i q^{33} - 4904 q^{34} - 2864 q^{36} - 9182 i q^{37} + 9728 i q^{38} + 2352 q^{39} - 14574 q^{41} + 1568 i q^{42} + 8108 i q^{43} + 5440 q^{44} + 8000 q^{46} + 312 i q^{47} + 2048 i q^{48} - 2401 q^{49} + 9808 q^{51} + 4704 i q^{52} - 14634 i q^{53} + 13504 q^{54} - 3136 q^{56} - 19456 i q^{57} - 26984 i q^{58} + 27656 q^{59} + 34338 q^{61} - 35424 i q^{62} + 8771 i q^{63} - 4096 q^{64} - 10880 q^{66} - 12316 i q^{67} + 19616 i q^{68} - 16000 q^{69} + 36920 q^{71} + 11456 i q^{72} - 61718 i q^{73} - 36728 q^{74} + 38912 q^{76} - 16660 i q^{77} - 9408 i q^{78} + 64752 q^{79} + 16489 q^{81} + 58296 i q^{82} - 77056 i q^{83} + 6272 q^{84} + 32432 q^{86} + 53968 i q^{87} - 21760 i q^{88} + 8166 q^{89} + 14406 q^{91} - 32000 i q^{92} + 70848 i q^{93} + 1248 q^{94} + 8192 q^{96} - 20650 i q^{97} + 9604 i q^{98} - 60860 q^{99} +O(q^{100})$$ q - 4*i * q^2 + 8*i * q^3 - 16 * q^4 + 32 * q^6 + 49*i * q^7 + 64*i * q^8 + 179 * q^9 - 340 * q^11 - 128*i * q^12 - 294*i * q^13 + 196 * q^14 + 256 * q^16 - 1226*i * q^17 - 716*i * q^18 - 2432 * q^19 - 392 * q^21 + 1360*i * q^22 + 2000*i * q^23 - 512 * q^24 - 1176 * q^26 + 3376*i * q^27 - 784*i * q^28 + 6746 * q^29 + 8856 * q^31 - 1024*i * q^32 - 2720*i * q^33 - 4904 * q^34 - 2864 * q^36 - 9182*i * q^37 + 9728*i * q^38 + 2352 * q^39 - 14574 * q^41 + 1568*i * q^42 + 8108*i * q^43 + 5440 * q^44 + 8000 * q^46 + 312*i * q^47 + 2048*i * q^48 - 2401 * q^49 + 9808 * q^51 + 4704*i * q^52 - 14634*i * q^53 + 13504 * q^54 - 3136 * q^56 - 19456*i * q^57 - 26984*i * q^58 + 27656 * q^59 + 34338 * q^61 - 35424*i * q^62 + 8771*i * q^63 - 4096 * q^64 - 10880 * q^66 - 12316*i * q^67 + 19616*i * q^68 - 16000 * q^69 + 36920 * q^71 + 11456*i * q^72 - 61718*i * q^73 - 36728 * q^74 + 38912 * q^76 - 16660*i * q^77 - 9408*i * q^78 + 64752 * q^79 + 16489 * q^81 + 58296*i * q^82 - 77056*i * q^83 + 6272 * q^84 + 32432 * q^86 + 53968*i * q^87 - 21760*i * q^88 + 8166 * q^89 + 14406 * q^91 - 32000*i * q^92 + 70848*i * q^93 + 1248 * q^94 + 8192 * q^96 - 20650*i * q^97 + 9604*i * q^98 - 60860 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32 q^{4} + 64 q^{6} + 358 q^{9}+O(q^{10})$$ 2 * q - 32 * q^4 + 64 * q^6 + 358 * q^9 $$2 q - 32 q^{4} + 64 q^{6} + 358 q^{9} - 680 q^{11} + 392 q^{14} + 512 q^{16} - 4864 q^{19} - 784 q^{21} - 1024 q^{24} - 2352 q^{26} + 13492 q^{29} + 17712 q^{31} - 9808 q^{34} - 5728 q^{36} + 4704 q^{39} - 29148 q^{41} + 10880 q^{44} + 16000 q^{46} - 4802 q^{49} + 19616 q^{51} + 27008 q^{54} - 6272 q^{56} + 55312 q^{59} + 68676 q^{61} - 8192 q^{64} - 21760 q^{66} - 32000 q^{69} + 73840 q^{71} - 73456 q^{74} + 77824 q^{76} + 129504 q^{79} + 32978 q^{81} + 12544 q^{84} + 64864 q^{86} + 16332 q^{89} + 28812 q^{91} + 2496 q^{94} + 16384 q^{96} - 121720 q^{99}+O(q^{100})$$ 2 * q - 32 * q^4 + 64 * q^6 + 358 * q^9 - 680 * q^11 + 392 * q^14 + 512 * q^16 - 4864 * q^19 - 784 * q^21 - 1024 * q^24 - 2352 * q^26 + 13492 * q^29 + 17712 * q^31 - 9808 * q^34 - 5728 * q^36 + 4704 * q^39 - 29148 * q^41 + 10880 * q^44 + 16000 * q^46 - 4802 * q^49 + 19616 * q^51 + 27008 * q^54 - 6272 * q^56 + 55312 * q^59 + 68676 * q^61 - 8192 * q^64 - 21760 * q^66 - 32000 * q^69 + 73840 * q^71 - 73456 * q^74 + 77824 * q^76 + 129504 * q^79 + 32978 * q^81 + 12544 * q^84 + 64864 * q^86 + 16332 * q^89 + 28812 * q^91 + 2496 * q^94 + 16384 * q^96 - 121720 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 1.00000i − 1.00000i
4.00000i 8.00000i −16.0000 0 32.0000 49.0000i 64.0000i 179.000 0
99.2 4.00000i 8.00000i −16.0000 0 32.0000 49.0000i 64.0000i 179.000 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.6.c.f 2
5.b even 2 1 inner 350.6.c.f 2
5.c odd 4 1 14.6.a.b 1
5.c odd 4 1 350.6.a.b 1
15.e even 4 1 126.6.a.c 1
20.e even 4 1 112.6.a.d 1
35.f even 4 1 98.6.a.b 1
35.k even 12 2 98.6.c.b 2
35.l odd 12 2 98.6.c.a 2
40.i odd 4 1 448.6.a.f 1
40.k even 4 1 448.6.a.k 1
60.l odd 4 1 1008.6.a.n 1
105.k odd 4 1 882.6.a.g 1
140.j odd 4 1 784.6.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.b 1 5.c odd 4 1
98.6.a.b 1 35.f even 4 1
98.6.c.a 2 35.l odd 12 2
98.6.c.b 2 35.k even 12 2
112.6.a.d 1 20.e even 4 1
126.6.a.c 1 15.e even 4 1
350.6.a.b 1 5.c odd 4 1
350.6.c.f 2 1.a even 1 1 trivial
350.6.c.f 2 5.b even 2 1 inner
448.6.a.f 1 40.i odd 4 1
448.6.a.k 1 40.k even 4 1
784.6.a.h 1 140.j odd 4 1
882.6.a.g 1 105.k odd 4 1
1008.6.a.n 1 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(350, [\chi])$$:

 $$T_{3}^{2} + 64$$ T3^2 + 64 $$T_{11} + 340$$ T11 + 340

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 16$$
$3$ $$T^{2} + 64$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2401$$
$11$ $$(T + 340)^{2}$$
$13$ $$T^{2} + 86436$$
$17$ $$T^{2} + 1503076$$
$19$ $$(T + 2432)^{2}$$
$23$ $$T^{2} + 4000000$$
$29$ $$(T - 6746)^{2}$$
$31$ $$(T - 8856)^{2}$$
$37$ $$T^{2} + 84309124$$
$41$ $$(T + 14574)^{2}$$
$43$ $$T^{2} + 65739664$$
$47$ $$T^{2} + 97344$$
$53$ $$T^{2} + 214153956$$
$59$ $$(T - 27656)^{2}$$
$61$ $$(T - 34338)^{2}$$
$67$ $$T^{2} + 151683856$$
$71$ $$(T - 36920)^{2}$$
$73$ $$T^{2} + 3809111524$$
$79$ $$(T - 64752)^{2}$$
$83$ $$T^{2} + 5937627136$$
$89$ $$(T - 8166)^{2}$$
$97$ $$T^{2} + 426422500$$