# Properties

 Label 350.10.c.b Level $350$ Weight $10$ Character orbit 350.c Analytic conductor $180.263$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 350.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$180.262542657$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ 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

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -16 i q^{2} + 6 i q^{3} -256 q^{4} + 96 q^{6} -2401 i q^{7} + 4096 i q^{8} + 19647 q^{9} +O(q^{10})$$ $$q -16 i q^{2} + 6 i q^{3} -256 q^{4} + 96 q^{6} -2401 i q^{7} + 4096 i q^{8} + 19647 q^{9} -54152 q^{11} -1536 i q^{12} + 113172 i q^{13} -38416 q^{14} + 65536 q^{16} + 6262 i q^{17} -314352 i q^{18} -257078 q^{19} + 14406 q^{21} + 866432 i q^{22} + 266000 i q^{23} -24576 q^{24} + 1810752 q^{26} + 235980 i q^{27} + 614656 i q^{28} -1574714 q^{29} -4637484 q^{31} -1048576 i q^{32} -324912 i q^{33} + 100192 q^{34} -5029632 q^{36} -11946238 i q^{37} + 4113248 i q^{38} -679032 q^{39} + 21909126 q^{41} -230496 i q^{42} -27520592 i q^{43} + 13862912 q^{44} + 4256000 q^{46} + 52927836 i q^{47} + 393216 i q^{48} -5764801 q^{49} -37572 q^{51} -28972032 i q^{52} -16221222 i q^{53} + 3775680 q^{54} + 9834496 q^{56} -1542468 i q^{57} + 25195424 i q^{58} + 140509618 q^{59} -202963560 q^{61} + 74199744 i q^{62} -47172447 i q^{63} -16777216 q^{64} -5198592 q^{66} + 153734572 i q^{67} -1603072 i q^{68} -1596000 q^{69} + 279655936 q^{71} + 80474112 i q^{72} + 404022830 i q^{73} -191139808 q^{74} + 65811968 q^{76} + 130018952 i q^{77} + 10864512 i q^{78} + 130689816 q^{79} + 385296021 q^{81} -350546016 i q^{82} -420134014 i q^{83} -3687936 q^{84} -440329472 q^{86} -9448284 i q^{87} -221806592 i q^{88} + 469542390 q^{89} + 271725972 q^{91} -68096000 i q^{92} -27824904 i q^{93} + 846845376 q^{94} + 6291456 q^{96} -872501690 i q^{97} + 92236816 i q^{98} -1063924344 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 512 q^{4} + 192 q^{6} + 39294 q^{9} + O(q^{10})$$ $$2 q - 512 q^{4} + 192 q^{6} + 39294 q^{9} - 108304 q^{11} - 76832 q^{14} + 131072 q^{16} - 514156 q^{19} + 28812 q^{21} - 49152 q^{24} + 3621504 q^{26} - 3149428 q^{29} - 9274968 q^{31} + 200384 q^{34} - 10059264 q^{36} - 1358064 q^{39} + 43818252 q^{41} + 27725824 q^{44} + 8512000 q^{46} - 11529602 q^{49} - 75144 q^{51} + 7551360 q^{54} + 19668992 q^{56} + 281019236 q^{59} - 405927120 q^{61} - 33554432 q^{64} - 10397184 q^{66} - 3192000 q^{69} + 559311872 q^{71} - 382279616 q^{74} + 131623936 q^{76} + 261379632 q^{79} + 770592042 q^{81} - 7375872 q^{84} - 880658944 q^{86} + 939084780 q^{89} + 543451944 q^{91} + 1693690752 q^{94} + 12582912 q^{96} - 2127848688 q^{99} + O(q^{100})$$

## 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.

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
16.0000i 6.00000i −256.000 0 96.0000 2401.00i 4096.00i 19647.0 0
99.2 16.0000i 6.00000i −256.000 0 96.0000 2401.00i 4096.00i 19647.0 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 350.10.c.b 2
5.b even 2 1 inner 350.10.c.b 2
5.c odd 4 1 14.10.a.a 1
5.c odd 4 1 350.10.a.c 1
15.e even 4 1 126.10.a.e 1
20.e even 4 1 112.10.a.b 1
35.f even 4 1 98.10.a.a 1
35.k even 12 2 98.10.c.e 2
35.l odd 12 2 98.10.c.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.a 1 5.c odd 4 1
98.10.a.a 1 35.f even 4 1
98.10.c.e 2 35.k even 12 2
98.10.c.f 2 35.l odd 12 2
112.10.a.b 1 20.e even 4 1
126.10.a.e 1 15.e even 4 1
350.10.a.c 1 5.c odd 4 1
350.10.c.b 2 1.a even 1 1 trivial
350.10.c.b 2 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 36$$ acting on $$S_{10}^{\mathrm{new}}(350, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256 + T^{2}$$
$3$ $$36 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$5764801 + T^{2}$$
$11$ $$( 54152 + T )^{2}$$
$13$ $$12807901584 + T^{2}$$
$17$ $$39212644 + T^{2}$$
$19$ $$( 257078 + T )^{2}$$
$23$ $$70756000000 + T^{2}$$
$29$ $$( 1574714 + T )^{2}$$
$31$ $$( 4637484 + T )^{2}$$
$37$ $$142712602352644 + T^{2}$$
$41$ $$( -21909126 + T )^{2}$$
$43$ $$757382984030464 + T^{2}$$
$47$ $$2801355823642896 + T^{2}$$
$53$ $$263128043173284 + T^{2}$$
$59$ $$( -140509618 + T )^{2}$$
$61$ $$( 202963560 + T )^{2}$$
$67$ $$23634318628023184 + T^{2}$$
$71$ $$( -279655936 + T )^{2}$$
$73$ $$163234447161208900 + T^{2}$$
$79$ $$( -130689816 + T )^{2}$$
$83$ $$176512589719752196 + T^{2}$$
$89$ $$( -469542390 + T )^{2}$$
$97$ $$761259199052856100 + T^{2}$$