Properties

 Label 350.4.c.g Level $350$ Weight $4$ Character orbit 350.c Analytic conductor $20.651$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 350.c (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$20.6506685020$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} + 2 i q^{3} -4 q^{4} -4 q^{6} + 7 i q^{7} -8 i q^{8} + 23 q^{9} +O(q^{10})$$ $$q + 2 i q^{2} + 2 i q^{3} -4 q^{4} -4 q^{6} + 7 i q^{7} -8 i q^{8} + 23 q^{9} + 48 q^{11} -8 i q^{12} -56 i q^{13} -14 q^{14} + 16 q^{16} -114 i q^{17} + 46 i q^{18} -2 q^{19} -14 q^{21} + 96 i q^{22} + 120 i q^{23} + 16 q^{24} + 112 q^{26} + 100 i q^{27} -28 i q^{28} + 54 q^{29} + 236 q^{31} + 32 i q^{32} + 96 i q^{33} + 228 q^{34} -92 q^{36} + 146 i q^{37} -4 i q^{38} + 112 q^{39} + 126 q^{41} -28 i q^{42} + 376 i q^{43} -192 q^{44} -240 q^{46} -12 i q^{47} + 32 i q^{48} -49 q^{49} + 228 q^{51} + 224 i q^{52} -174 i q^{53} -200 q^{54} + 56 q^{56} -4 i q^{57} + 108 i q^{58} -138 q^{59} + 380 q^{61} + 472 i q^{62} + 161 i q^{63} -64 q^{64} -192 q^{66} -484 i q^{67} + 456 i q^{68} -240 q^{69} + 576 q^{71} -184 i q^{72} + 1150 i q^{73} -292 q^{74} + 8 q^{76} + 336 i q^{77} + 224 i q^{78} -776 q^{79} + 421 q^{81} + 252 i q^{82} -378 i q^{83} + 56 q^{84} -752 q^{86} + 108 i q^{87} -384 i q^{88} + 390 q^{89} + 392 q^{91} -480 i q^{92} + 472 i q^{93} + 24 q^{94} -64 q^{96} -1330 i q^{97} -98 i q^{98} + 1104 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{4} - 8q^{6} + 46q^{9} + O(q^{10})$$ $$2q - 8q^{4} - 8q^{6} + 46q^{9} + 96q^{11} - 28q^{14} + 32q^{16} - 4q^{19} - 28q^{21} + 32q^{24} + 224q^{26} + 108q^{29} + 472q^{31} + 456q^{34} - 184q^{36} + 224q^{39} + 252q^{41} - 384q^{44} - 480q^{46} - 98q^{49} + 456q^{51} - 400q^{54} + 112q^{56} - 276q^{59} + 760q^{61} - 128q^{64} - 384q^{66} - 480q^{69} + 1152q^{71} - 584q^{74} + 16q^{76} - 1552q^{79} + 842q^{81} + 112q^{84} - 1504q^{86} + 780q^{89} + 784q^{91} + 48q^{94} - 128q^{96} + 2208q^{99} + O(q^{100})$$

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
2.00000i 2.00000i −4.00000 0 −4.00000 7.00000i 8.00000i 23.0000 0
99.2 2.00000i 2.00000i −4.00000 0 −4.00000 7.00000i 8.00000i 23.0000 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 350.4.c.g 2
5.b even 2 1 inner 350.4.c.g 2
5.c odd 4 1 14.4.a.b 1
5.c odd 4 1 350.4.a.f 1
15.e even 4 1 126.4.a.d 1
20.e even 4 1 112.4.a.e 1
35.f even 4 1 98.4.a.e 1
35.f even 4 1 2450.4.a.i 1
35.k even 12 2 98.4.c.b 2
35.l odd 12 2 98.4.c.c 2
40.i odd 4 1 448.4.a.k 1
40.k even 4 1 448.4.a.g 1
55.e even 4 1 1694.4.a.b 1
60.l odd 4 1 1008.4.a.r 1
65.h odd 4 1 2366.4.a.c 1
105.k odd 4 1 882.4.a.b 1
105.w odd 12 2 882.4.g.v 2
105.x even 12 2 882.4.g.p 2
140.j odd 4 1 784.4.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.b 1 5.c odd 4 1
98.4.a.e 1 35.f even 4 1
98.4.c.b 2 35.k even 12 2
98.4.c.c 2 35.l odd 12 2
112.4.a.e 1 20.e even 4 1
126.4.a.d 1 15.e even 4 1
350.4.a.f 1 5.c odd 4 1
350.4.c.g 2 1.a even 1 1 trivial
350.4.c.g 2 5.b even 2 1 inner
448.4.a.g 1 40.k even 4 1
448.4.a.k 1 40.i odd 4 1
784.4.a.h 1 140.j odd 4 1
882.4.a.b 1 105.k odd 4 1
882.4.g.p 2 105.x even 12 2
882.4.g.v 2 105.w odd 12 2
1008.4.a.r 1 60.l odd 4 1
1694.4.a.b 1 55.e even 4 1
2366.4.a.c 1 65.h odd 4 1
2450.4.a.i 1 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(350, [\chi])$$:

 $$T_{3}^{2} + 4$$ $$T_{11} - 48$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T^{2}$$
$3$ $$4 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$49 + T^{2}$$
$11$ $$( -48 + T )^{2}$$
$13$ $$3136 + T^{2}$$
$17$ $$12996 + T^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$14400 + T^{2}$$
$29$ $$( -54 + T )^{2}$$
$31$ $$( -236 + T )^{2}$$
$37$ $$21316 + T^{2}$$
$41$ $$( -126 + T )^{2}$$
$43$ $$141376 + T^{2}$$
$47$ $$144 + T^{2}$$
$53$ $$30276 + T^{2}$$
$59$ $$( 138 + T )^{2}$$
$61$ $$( -380 + T )^{2}$$
$67$ $$234256 + T^{2}$$
$71$ $$( -576 + T )^{2}$$
$73$ $$1322500 + T^{2}$$
$79$ $$( 776 + T )^{2}$$
$83$ $$142884 + T^{2}$$
$89$ $$( -390 + T )^{2}$$
$97$ $$1768900 + T^{2}$$