# Properties

 Label 350.6.c.d 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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} + 10 i q^{3} - 16 q^{4} - 40 q^{6} - 49 i q^{7} - 64 i q^{8} + 143 q^{9} +O(q^{10})$$ q + 4*i * q^2 + 10*i * q^3 - 16 * q^4 - 40 * q^6 - 49*i * q^7 - 64*i * q^8 + 143 * q^9 $$q + 4 i q^{2} + 10 i q^{3} - 16 q^{4} - 40 q^{6} - 49 i q^{7} - 64 i q^{8} + 143 q^{9} - 336 q^{11} - 160 i q^{12} + 584 i q^{13} + 196 q^{14} + 256 q^{16} + 1458 i q^{17} + 572 i q^{18} - 470 q^{19} + 490 q^{21} - 1344 i q^{22} - 4200 i q^{23} + 640 q^{24} - 2336 q^{26} + 3860 i q^{27} + 784 i q^{28} - 4866 q^{29} - 7372 q^{31} + 1024 i q^{32} - 3360 i q^{33} - 5832 q^{34} - 2288 q^{36} - 14330 i q^{37} - 1880 i q^{38} - 5840 q^{39} + 6222 q^{41} + 1960 i q^{42} + 3704 i q^{43} + 5376 q^{44} + 16800 q^{46} + 1812 i q^{47} + 2560 i q^{48} - 2401 q^{49} - 14580 q^{51} - 9344 i q^{52} - 37242 i q^{53} - 15440 q^{54} - 3136 q^{56} - 4700 i q^{57} - 19464 i q^{58} - 34302 q^{59} + 24476 q^{61} - 29488 i q^{62} - 7007 i q^{63} - 4096 q^{64} + 13440 q^{66} + 17452 i q^{67} - 23328 i q^{68} + 42000 q^{69} + 28224 q^{71} - 9152 i q^{72} + 3602 i q^{73} + 57320 q^{74} + 7520 q^{76} + 16464 i q^{77} - 23360 i q^{78} - 42872 q^{79} - 3851 q^{81} + 24888 i q^{82} - 35202 i q^{83} - 7840 q^{84} - 14816 q^{86} - 48660 i q^{87} + 21504 i q^{88} - 26730 q^{89} + 28616 q^{91} + 67200 i q^{92} - 73720 i q^{93} - 7248 q^{94} - 10240 q^{96} + 16978 i q^{97} - 9604 i q^{98} - 48048 q^{99} +O(q^{100})$$ q + 4*i * q^2 + 10*i * q^3 - 16 * q^4 - 40 * q^6 - 49*i * q^7 - 64*i * q^8 + 143 * q^9 - 336 * q^11 - 160*i * q^12 + 584*i * q^13 + 196 * q^14 + 256 * q^16 + 1458*i * q^17 + 572*i * q^18 - 470 * q^19 + 490 * q^21 - 1344*i * q^22 - 4200*i * q^23 + 640 * q^24 - 2336 * q^26 + 3860*i * q^27 + 784*i * q^28 - 4866 * q^29 - 7372 * q^31 + 1024*i * q^32 - 3360*i * q^33 - 5832 * q^34 - 2288 * q^36 - 14330*i * q^37 - 1880*i * q^38 - 5840 * q^39 + 6222 * q^41 + 1960*i * q^42 + 3704*i * q^43 + 5376 * q^44 + 16800 * q^46 + 1812*i * q^47 + 2560*i * q^48 - 2401 * q^49 - 14580 * q^51 - 9344*i * q^52 - 37242*i * q^53 - 15440 * q^54 - 3136 * q^56 - 4700*i * q^57 - 19464*i * q^58 - 34302 * q^59 + 24476 * q^61 - 29488*i * q^62 - 7007*i * q^63 - 4096 * q^64 + 13440 * q^66 + 17452*i * q^67 - 23328*i * q^68 + 42000 * q^69 + 28224 * q^71 - 9152*i * q^72 + 3602*i * q^73 + 57320 * q^74 + 7520 * q^76 + 16464*i * q^77 - 23360*i * q^78 - 42872 * q^79 - 3851 * q^81 + 24888*i * q^82 - 35202*i * q^83 - 7840 * q^84 - 14816 * q^86 - 48660*i * q^87 + 21504*i * q^88 - 26730 * q^89 + 28616 * q^91 + 67200*i * q^92 - 73720*i * q^93 - 7248 * q^94 - 10240 * q^96 + 16978*i * q^97 - 9604*i * q^98 - 48048 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 32 q^{4} - 80 q^{6} + 286 q^{9}+O(q^{10})$$ 2 * q - 32 * q^4 - 80 * q^6 + 286 * q^9 $$2 q - 32 q^{4} - 80 q^{6} + 286 q^{9} - 672 q^{11} + 392 q^{14} + 512 q^{16} - 940 q^{19} + 980 q^{21} + 1280 q^{24} - 4672 q^{26} - 9732 q^{29} - 14744 q^{31} - 11664 q^{34} - 4576 q^{36} - 11680 q^{39} + 12444 q^{41} + 10752 q^{44} + 33600 q^{46} - 4802 q^{49} - 29160 q^{51} - 30880 q^{54} - 6272 q^{56} - 68604 q^{59} + 48952 q^{61} - 8192 q^{64} + 26880 q^{66} + 84000 q^{69} + 56448 q^{71} + 114640 q^{74} + 15040 q^{76} - 85744 q^{79} - 7702 q^{81} - 15680 q^{84} - 29632 q^{86} - 53460 q^{89} + 57232 q^{91} - 14496 q^{94} - 20480 q^{96} - 96096 q^{99}+O(q^{100})$$ 2 * q - 32 * q^4 - 80 * q^6 + 286 * q^9 - 672 * q^11 + 392 * q^14 + 512 * q^16 - 940 * q^19 + 980 * q^21 + 1280 * q^24 - 4672 * q^26 - 9732 * q^29 - 14744 * q^31 - 11664 * q^34 - 4576 * q^36 - 11680 * q^39 + 12444 * q^41 + 10752 * q^44 + 33600 * q^46 - 4802 * q^49 - 29160 * q^51 - 30880 * q^54 - 6272 * q^56 - 68604 * q^59 + 48952 * q^61 - 8192 * q^64 + 26880 * q^66 + 84000 * q^69 + 56448 * q^71 + 114640 * q^74 + 15040 * q^76 - 85744 * q^79 - 7702 * q^81 - 15680 * q^84 - 29632 * q^86 - 53460 * q^89 + 57232 * q^91 - 14496 * q^94 - 20480 * q^96 - 96096 * 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 10.0000i −16.0000 0 −40.0000 49.0000i 64.0000i 143.000 0
99.2 4.00000i 10.0000i −16.0000 0 −40.0000 49.0000i 64.0000i 143.000 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.d 2
5.b even 2 1 inner 350.6.c.d 2
5.c odd 4 1 14.6.a.a 1
5.c odd 4 1 350.6.a.i 1
15.e even 4 1 126.6.a.f 1
20.e even 4 1 112.6.a.c 1
35.f even 4 1 98.6.a.a 1
35.k even 12 2 98.6.c.d 2
35.l odd 12 2 98.6.c.c 2
40.i odd 4 1 448.6.a.e 1
40.k even 4 1 448.6.a.l 1
60.l odd 4 1 1008.6.a.b 1
105.k odd 4 1 882.6.a.x 1
140.j odd 4 1 784.6.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.a.a 1 5.c odd 4 1
98.6.a.a 1 35.f even 4 1
98.6.c.c 2 35.l odd 12 2
98.6.c.d 2 35.k even 12 2
112.6.a.c 1 20.e even 4 1
126.6.a.f 1 15.e even 4 1
350.6.a.i 1 5.c odd 4 1
350.6.c.d 2 1.a even 1 1 trivial
350.6.c.d 2 5.b even 2 1 inner
448.6.a.e 1 40.i odd 4 1
448.6.a.l 1 40.k even 4 1
784.6.a.i 1 140.j odd 4 1
882.6.a.x 1 105.k odd 4 1
1008.6.a.b 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 $$S_{6}^{\mathrm{new}}(350, [\chi])$$:

 $$T_{3}^{2} + 100$$ T3^2 + 100 $$T_{11} + 336$$ T11 + 336

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 16$$
$3$ $$T^{2} + 100$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2401$$
$11$ $$(T + 336)^{2}$$
$13$ $$T^{2} + 341056$$
$17$ $$T^{2} + 2125764$$
$19$ $$(T + 470)^{2}$$
$23$ $$T^{2} + 17640000$$
$29$ $$(T + 4866)^{2}$$
$31$ $$(T + 7372)^{2}$$
$37$ $$T^{2} + 205348900$$
$41$ $$(T - 6222)^{2}$$
$43$ $$T^{2} + 13719616$$
$47$ $$T^{2} + 3283344$$
$53$ $$T^{2} + 1386966564$$
$59$ $$(T + 34302)^{2}$$
$61$ $$(T - 24476)^{2}$$
$67$ $$T^{2} + 304572304$$
$71$ $$(T - 28224)^{2}$$
$73$ $$T^{2} + 12974404$$
$79$ $$(T + 42872)^{2}$$
$83$ $$T^{2} + 1239180804$$
$89$ $$(T + 26730)^{2}$$
$97$ $$T^{2} + 288252484$$