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

Downloads

Learn more

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.

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

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:

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

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