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 $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.99");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
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

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: \(180.262542657\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{2305})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1153x^{2} + 331776 \) Copy content Toggle raw display
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

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 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.

\(f(q)\) \(=\) \( q + 16 \beta_1 q^{2} + (\beta_{2} - 7 \beta_1) q^{3} - 256 q^{4} + ( - 16 \beta_{3} + 112) q^{6} - 2401 \beta_1 q^{7} - 4096 \beta_1 q^{8} + (14 \beta_{3} - 37991) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 16 \beta_1 q^{2} + (\beta_{2} - 7 \beta_1) q^{3} - 256 q^{4} + ( - 16 \beta_{3} + 112) q^{6} - 2401 \beta_1 q^{7} - 4096 \beta_1 q^{8} + (14 \beta_{3} - 37991) q^{9} + (210 \beta_{3} + 22470) q^{11} + ( - 256 \beta_{2} + 1792 \beta_1) q^{12} + ( - 45 \beta_{2} + 50141 \beta_1) q^{13} + 38416 q^{14} + 65536 q^{16} + (318 \beta_{2} + 435204 \beta_1) q^{17} + (224 \beta_{2} - 607856 \beta_1) q^{18} + ( - 2691 \beta_{3} - 254387) q^{19} + (2401 \beta_{3} - 16807) q^{21} + (3360 \beta_{2} + 359520 \beta_1) q^{22} + (1428 \beta_{2} + 39900 \beta_1) q^{23} + (4096 \beta_{3} - 28672) q^{24} + (720 \beta_{3} - 802256) q^{26} + ( - 18406 \beta_{2} + 934906 \beta_1) q^{27} + 614656 \beta_1 q^{28} + (23898 \beta_{3} - 1003164) q^{29} + ( - 7866 \beta_{3} + 1094366) q^{31} + 1048576 \beta_1 q^{32} + (21000 \beta_{2} + 11943960 \beta_1) q^{33} + ( - 5088 \beta_{3} - 6963264) q^{34} + ( - 3584 \beta_{3} + 9725696) q^{36} + (28602 \beta_{2} + 10361788 \beta_1) q^{37} + ( - 43056 \beta_{2} - 4070192 \beta_1) q^{38} + ( - 50456 \beta_{3} + 2944112) q^{39} + ( - 103578 \beta_{3} + 9508296) q^{41} + (38416 \beta_{2} - 268912 \beta_1) q^{42} + (69174 \beta_{2} + 2096858 \beta_1) q^{43} + ( - 53760 \beta_{3} - 5752320) q^{44} + ( - 22848 \beta_{3} - 638400) q^{46} + (40086 \beta_{2} + 37271262 \beta_1) q^{47} + (65536 \beta_{2} - 458752 \beta_1) q^{48} - 5764801 q^{49} + ( - 432978 \beta_{3} - 15278322) q^{51} + (11520 \beta_{2} - 12836096 \beta_1) q^{52} + (92904 \beta_{2} - 1619874 \beta_1) q^{53} + (294496 \beta_{3} - 14958496) q^{54} - 9834496 q^{56} + ( - 235550 \beta_{2} - 153288166 \beta_1) q^{57} + (382368 \beta_{2} - 16050624 \beta_1) q^{58} + ( - 223947 \beta_{3} + 66821181) q^{59} + (173241 \beta_{3} + 113900843) q^{61} + ( - 125856 \beta_{2} + 17509856 \beta_1) q^{62} + ( - 33614 \beta_{2} + 91216391 \beta_1) q^{63} - 16777216 q^{64} + ( - 336000 \beta_{3} - 191103360) q^{66} + ( - 330876 \beta_{2} - 166465136 \beta_1) q^{67} + ( - 81408 \beta_{2} - 111412224 \beta_1) q^{68} + ( - 29904 \beta_{3} - 82009200) q^{69} + ( - 1264788 \beta_{3} - 83992860) q^{71} + ( - 57344 \beta_{2} + 155611136 \beta_1) q^{72} + ( - 1208628 \beta_{2} - 22342138 \beta_1) q^{73} + ( - 457632 \beta_{3} - 165788608) q^{74} + (688896 \beta_{3} + 65123072) q^{76} + ( - 504210 \beta_{2} - 53950470 \beta_1) q^{77} + ( - 807296 \beta_{2} + 47105792 \beta_1) q^{78} + ( - 442764 \beta_{3} - 134821388) q^{79} + ( - 788186 \beta_{3} + 319413239) q^{81} + ( - 1657248 \beta_{2} + 152132736 \beta_1) q^{82} + (1215087 \beta_{2} - 91552881 \beta_1) q^{83} + ( - 614656 \beta_{3} + 4302592) q^{84} + ( - 1106784 \beta_{3} - 33549728) q^{86} + ( - 1170450 \beta_{2} + 1384144398 \beta_1) q^{87} + ( - 860160 \beta_{2} - 92037120 \beta_1) q^{88} + (1357008 \beta_{3} - 395828874) q^{89} + ( - 108045 \beta_{3} + 120388541) q^{91} + ( - 365568 \beta_{2} - 10214400 \beta_1) q^{92} + (1149428 \beta_{2} - 460938812 \beta_1) q^{93} + ( - 641376 \beta_{3} - 596340192) q^{94} + ( - 1048576 \beta_{3} + 7340032) q^{96} + ( - 2797938 \beta_{2} + 2084740 \beta_1) q^{97} - 92236816 \beta_1 q^{98} + ( - 7663530 \beta_{3} - 684240270) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 1024 q^{4} + 448 q^{6} - 151964 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 1024 q^{4} + 448 q^{6} - 151964 q^{9} + 89880 q^{11} + 153664 q^{14} + 262144 q^{16} - 1017548 q^{19} - 67228 q^{21} - 114688 q^{24} - 3209024 q^{26} - 4012656 q^{29} + 4377464 q^{31} - 27853056 q^{34} + 38902784 q^{36} + 11776448 q^{39} + 38033184 q^{41} - 23009280 q^{44} - 2553600 q^{46} - 23059204 q^{49} - 61113288 q^{51} - 59833984 q^{54} - 39337984 q^{56} + 267284724 q^{59} + 455603372 q^{61} - 67108864 q^{64} - 764413440 q^{66} - 328036800 q^{69} - 335971440 q^{71} - 663154432 q^{74} + 260492288 q^{76} - 539285552 q^{79} + 1277652956 q^{81} + 17210368 q^{84} - 134198912 q^{86} - 1583315496 q^{89} + 481554164 q^{91} - 2385360768 q^{94} + 29360128 q^{96} - 2736961080 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 1153x^{2} + 331776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 577\nu ) / 576 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{3} + 8645\nu ) / 576 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 10\nu^{2} + 5765 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 5\beta_1 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 5765 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -577\beta_{2} + 8645\beta_1 ) / 10 \) Copy content Toggle raw display

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
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:

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 \( T_{3}^{4} + 115348T_{3}^{2} + 3314995776 \) acting on \(S_{10}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 256)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 115348 T^{2} + \cdots + 3314995776 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 5764801)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 44940 T - 2036361600)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 5261621012 T^{2} + \cdots + 57\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{4} + 390459584232 T^{2} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{2} + 508774 T - 352577596856)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 238199976000 T^{2} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{2} + 2006328 T - 31904129519604)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2188732 T - 2367849772544)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 309016376174888 T^{2} + \cdots + 36\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{2} - 19016592 T - 527816477266884)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 560269749253328 T^{2} + \cdots + 73\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{4} + 999988391695752 T^{2} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{2} - 133642362 T + 15\!\cdots\!36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 227801686 T + 11\!\cdots\!24)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{2} + 167985720 T - 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 70\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{2} + 269642776 T + 68\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 58\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{2} + 791657748 T + 50\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
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