# Properties

 Label 350.10.c.j Level $350$ Weight $10$ Character orbit 350.c Analytic conductor $180.263$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{2305})$$ Defining polynomial: $$x^{4} + 1153 x^{2} + 331776$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 5^{2}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 16 \beta_{1} q^{2} + ( -7 \beta_{1} + \beta_{2} ) q^{3} -256 q^{4} + ( 112 - 16 \beta_{3} ) q^{6} -2401 \beta_{1} q^{7} -4096 \beta_{1} q^{8} + ( -37991 + 14 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + 16 \beta_{1} q^{2} + ( -7 \beta_{1} + \beta_{2} ) q^{3} -256 q^{4} + ( 112 - 16 \beta_{3} ) q^{6} -2401 \beta_{1} q^{7} -4096 \beta_{1} q^{8} + ( -37991 + 14 \beta_{3} ) q^{9} + ( 22470 + 210 \beta_{3} ) q^{11} + ( 1792 \beta_{1} - 256 \beta_{2} ) q^{12} + ( 50141 \beta_{1} - 45 \beta_{2} ) q^{13} + 38416 q^{14} + 65536 q^{16} + ( 435204 \beta_{1} + 318 \beta_{2} ) q^{17} + ( -607856 \beta_{1} + 224 \beta_{2} ) q^{18} + ( -254387 - 2691 \beta_{3} ) q^{19} + ( -16807 + 2401 \beta_{3} ) q^{21} + ( 359520 \beta_{1} + 3360 \beta_{2} ) q^{22} + ( 39900 \beta_{1} + 1428 \beta_{2} ) q^{23} + ( -28672 + 4096 \beta_{3} ) q^{24} + ( -802256 + 720 \beta_{3} ) q^{26} + ( 934906 \beta_{1} - 18406 \beta_{2} ) q^{27} + 614656 \beta_{1} q^{28} + ( -1003164 + 23898 \beta_{3} ) q^{29} + ( 1094366 - 7866 \beta_{3} ) q^{31} + 1048576 \beta_{1} q^{32} + ( 11943960 \beta_{1} + 21000 \beta_{2} ) q^{33} + ( -6963264 - 5088 \beta_{3} ) q^{34} + ( 9725696 - 3584 \beta_{3} ) q^{36} + ( 10361788 \beta_{1} + 28602 \beta_{2} ) q^{37} + ( -4070192 \beta_{1} - 43056 \beta_{2} ) q^{38} + ( 2944112 - 50456 \beta_{3} ) q^{39} + ( 9508296 - 103578 \beta_{3} ) q^{41} + ( -268912 \beta_{1} + 38416 \beta_{2} ) q^{42} + ( 2096858 \beta_{1} + 69174 \beta_{2} ) q^{43} + ( -5752320 - 53760 \beta_{3} ) q^{44} + ( -638400 - 22848 \beta_{3} ) q^{46} + ( 37271262 \beta_{1} + 40086 \beta_{2} ) q^{47} + ( -458752 \beta_{1} + 65536 \beta_{2} ) q^{48} -5764801 q^{49} + ( -15278322 - 432978 \beta_{3} ) q^{51} + ( -12836096 \beta_{1} + 11520 \beta_{2} ) q^{52} + ( -1619874 \beta_{1} + 92904 \beta_{2} ) q^{53} + ( -14958496 + 294496 \beta_{3} ) q^{54} -9834496 q^{56} + ( -153288166 \beta_{1} - 235550 \beta_{2} ) q^{57} + ( -16050624 \beta_{1} + 382368 \beta_{2} ) q^{58} + ( 66821181 - 223947 \beta_{3} ) q^{59} + ( 113900843 + 173241 \beta_{3} ) q^{61} + ( 17509856 \beta_{1} - 125856 \beta_{2} ) q^{62} + ( 91216391 \beta_{1} - 33614 \beta_{2} ) q^{63} -16777216 q^{64} + ( -191103360 - 336000 \beta_{3} ) q^{66} + ( -166465136 \beta_{1} - 330876 \beta_{2} ) q^{67} + ( -111412224 \beta_{1} - 81408 \beta_{2} ) q^{68} + ( -82009200 - 29904 \beta_{3} ) q^{69} + ( -83992860 - 1264788 \beta_{3} ) q^{71} + ( 155611136 \beta_{1} - 57344 \beta_{2} ) q^{72} + ( -22342138 \beta_{1} - 1208628 \beta_{2} ) q^{73} + ( -165788608 - 457632 \beta_{3} ) q^{74} + ( 65123072 + 688896 \beta_{3} ) q^{76} + ( -53950470 \beta_{1} - 504210 \beta_{2} ) q^{77} + ( 47105792 \beta_{1} - 807296 \beta_{2} ) q^{78} + ( -134821388 - 442764 \beta_{3} ) q^{79} + ( 319413239 - 788186 \beta_{3} ) q^{81} + ( 152132736 \beta_{1} - 1657248 \beta_{2} ) q^{82} + ( -91552881 \beta_{1} + 1215087 \beta_{2} ) q^{83} + ( 4302592 - 614656 \beta_{3} ) q^{84} + ( -33549728 - 1106784 \beta_{3} ) q^{86} + ( 1384144398 \beta_{1} - 1170450 \beta_{2} ) q^{87} + ( -92037120 \beta_{1} - 860160 \beta_{2} ) q^{88} + ( -395828874 + 1357008 \beta_{3} ) q^{89} + ( 120388541 - 108045 \beta_{3} ) q^{91} + ( -10214400 \beta_{1} - 365568 \beta_{2} ) q^{92} + ( -460938812 \beta_{1} + 1149428 \beta_{2} ) q^{93} + ( -596340192 - 641376 \beta_{3} ) q^{94} + ( 7340032 - 1048576 \beta_{3} ) q^{96} + ( 2084740 \beta_{1} - 2797938 \beta_{2} ) q^{97} -92236816 \beta_{1} q^{98} + ( -684240270 - 7663530 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 1024 q^{4} + 448 q^{6} - 151964 q^{9} + O(q^{10})$$ $$4 q - 1024 q^{4} + 448 q^{6} - 151964 q^{9} + 89880 q^{11} + 153664 q^{14} + 262144 q^{16} - 1017548 q^{19} - 67228 q^{21} - 114688 q^{24} - 3209024 q^{26} - 4012656 q^{29} + 4377464 q^{31} - 27853056 q^{34} + 38902784 q^{36} + 11776448 q^{39} + 38033184 q^{41} - 23009280 q^{44} - 2553600 q^{46} - 23059204 q^{49} - 61113288 q^{51} - 59833984 q^{54} - 39337984 q^{56} + 267284724 q^{59} + 455603372 q^{61} - 67108864 q^{64} - 764413440 q^{66} - 328036800 q^{69} - 335971440 q^{71} - 663154432 q^{74} + 260492288 q^{76} - 539285552 q^{79} + 1277652956 q^{81} + 17210368 q^{84} - 134198912 q^{86} - 1583315496 q^{89} + 481554164 q^{91} - 2385360768 q^{94} + 29360128 q^{96} - 2736961080 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 1153 x^{2} + 331776$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 577 \nu$$$$)/576$$ $$\beta_{2}$$ $$=$$ $$($$$$5 \nu^{3} + 8645 \nu$$$$)/576$$ $$\beta_{3}$$ $$=$$ $$10 \nu^{2} + 5765$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - 5 \beta_{1}$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 5765$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$-577 \beta_{2} + 8645 \beta_{1}$$$$)/10$$

## 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
 − 23.5052i 24.5052i − 24.5052i 23.5052i
16.0000i 233.052i −256.000 0 −3728.83 2401.00i 4096.00i −34630.3 0
99.2 16.0000i 247.052i −256.000 0 3952.83 2401.00i 4096.00i −41351.7 0
99.3 16.0000i 247.052i −256.000 0 3952.83 2401.00i 4096.00i −41351.7 0
99.4 16.0000i 233.052i −256.000 0 −3728.83 2401.00i 4096.00i −34630.3 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.j 4
5.b even 2 1 inner 350.10.c.j 4
5.c odd 4 1 14.10.a.c 2
5.c odd 4 1 350.10.a.j 2
15.e even 4 1 126.10.a.o 2
20.e even 4 1 112.10.a.c 2
35.f even 4 1 98.10.a.e 2
35.k even 12 2 98.10.c.h 4
35.l odd 12 2 98.10.c.j 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 115348 T_{3}^{2} + 3314995776$$ acting on $$S_{10}^{\mathrm{new}}(350, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 256 + T^{2} )^{2}$$
$3$ $$3314995776 + 115348 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 5764801 + T^{2} )^{2}$$
$11$ $$( -2036361600 - 44940 T + T^{2} )^{2}$$
$13$ $$5747667037524713536 + 5261621012 T^{2} + T^{4}$$
$17$ $$33\!\cdots\!56$$$$+ 390459584232 T^{2} + T^{4}$$
$19$ $$( -352577596856 + 508774 T + T^{2} )^{2}$$
$23$ $$13\!\cdots\!00$$$$+ 238199976000 T^{2} + T^{4}$$
$29$ $$( -31904129519604 + 2006328 T + T^{2} )^{2}$$
$31$ $$( -2367849772544 - 2188732 T + T^{2} )^{2}$$
$37$ $$36\!\cdots\!36$$$$+ 309016376174888 T^{2} + T^{4}$$
$41$ $$( -527816477266884 - 19016592 T + T^{2} )^{2}$$
$43$ $$73\!\cdots\!96$$$$+ 560269749253328 T^{2} + T^{4}$$
$47$ $$16\!\cdots\!36$$$$+ 2963487714534288 T^{2} + T^{4}$$
$53$ $$24\!\cdots\!76$$$$+ 999988391695752 T^{2} + T^{4}$$
$59$ $$( 1575046316366136 - 133642362 T + T^{2} )^{2}$$
$61$ $$( 11243934945943024 - 227801686 T + T^{2} )^{2}$$
$67$ $$45\!\cdots\!16$$$$+ 68038729387080992 T^{2} + T^{4}$$
$71$ $$( -85127259938918400 + 167985720 T + T^{2} )^{2}$$
$73$ $$70\!\cdots\!36$$$$+ 169353426545578088 T^{2} + T^{4}$$
$79$ $$( 6880003984764544 + 269642776 T + T^{2} )^{2}$$
$83$ $$58\!\cdots\!96$$$$+ 186923157163627572 T^{2} + T^{4}$$
$89$ $$( 50565747709419876 + 791657748 T + T^{2} )^{2}$$
$97$ $$20\!\cdots\!00$$$$+ 902238367506756200 T^{2} + T^{4}$$