Properties

Label 350.10.a.j
Level $350$
Weight $10$
Character orbit 350.a
Self dual yes
Analytic conductor $180.263$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(180.262542657\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2305}) \)
Defining polynomial: \(x^{2} - x - 576\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 5\sqrt{2305}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 16 q^{2} + ( 7 + \beta ) q^{3} + 256 q^{4} + ( 112 + 16 \beta ) q^{6} -2401 q^{7} + 4096 q^{8} + ( 37991 + 14 \beta ) q^{9} +O(q^{10})\) \( q + 16 q^{2} + ( 7 + \beta ) q^{3} + 256 q^{4} + ( 112 + 16 \beta ) q^{6} -2401 q^{7} + 4096 q^{8} + ( 37991 + 14 \beta ) q^{9} + ( 22470 - 210 \beta ) q^{11} + ( 1792 + 256 \beta ) q^{12} + ( -50141 - 45 \beta ) q^{13} -38416 q^{14} + 65536 q^{16} + ( 435204 - 318 \beta ) q^{17} + ( 607856 + 224 \beta ) q^{18} + ( 254387 - 2691 \beta ) q^{19} + ( -16807 - 2401 \beta ) q^{21} + ( 359520 - 3360 \beta ) q^{22} + ( -39900 + 1428 \beta ) q^{23} + ( 28672 + 4096 \beta ) q^{24} + ( -802256 - 720 \beta ) q^{26} + ( 934906 + 18406 \beta ) q^{27} -614656 q^{28} + ( 1003164 + 23898 \beta ) q^{29} + ( 1094366 + 7866 \beta ) q^{31} + 1048576 q^{32} + ( -11943960 + 21000 \beta ) q^{33} + ( 6963264 - 5088 \beta ) q^{34} + ( 9725696 + 3584 \beta ) q^{36} + ( 10361788 - 28602 \beta ) q^{37} + ( 4070192 - 43056 \beta ) q^{38} + ( -2944112 - 50456 \beta ) q^{39} + ( 9508296 + 103578 \beta ) q^{41} + ( -268912 - 38416 \beta ) q^{42} + ( -2096858 + 69174 \beta ) q^{43} + ( 5752320 - 53760 \beta ) q^{44} + ( -638400 + 22848 \beta ) q^{46} + ( 37271262 - 40086 \beta ) q^{47} + ( 458752 + 65536 \beta ) q^{48} + 5764801 q^{49} + ( -15278322 + 432978 \beta ) q^{51} + ( -12836096 - 11520 \beta ) q^{52} + ( 1619874 + 92904 \beta ) q^{53} + ( 14958496 + 294496 \beta ) q^{54} -9834496 q^{56} + ( -153288166 + 235550 \beta ) q^{57} + ( 16050624 + 382368 \beta ) q^{58} + ( -66821181 - 223947 \beta ) q^{59} + ( 113900843 - 173241 \beta ) q^{61} + ( 17509856 + 125856 \beta ) q^{62} + ( -91216391 - 33614 \beta ) q^{63} + 16777216 q^{64} + ( -191103360 + 336000 \beta ) q^{66} + ( -166465136 + 330876 \beta ) q^{67} + ( 111412224 - 81408 \beta ) q^{68} + ( 82009200 - 29904 \beta ) q^{69} + ( -83992860 + 1264788 \beta ) q^{71} + ( 155611136 + 57344 \beta ) q^{72} + ( 22342138 - 1208628 \beta ) q^{73} + ( 165788608 - 457632 \beta ) q^{74} + ( 65123072 - 688896 \beta ) q^{76} + ( -53950470 + 504210 \beta ) q^{77} + ( -47105792 - 807296 \beta ) q^{78} + ( 134821388 - 442764 \beta ) q^{79} + ( 319413239 + 788186 \beta ) q^{81} + ( 152132736 + 1657248 \beta ) q^{82} + ( 91552881 + 1215087 \beta ) q^{83} + ( -4302592 - 614656 \beta ) q^{84} + ( -33549728 + 1106784 \beta ) q^{86} + ( 1384144398 + 1170450 \beta ) q^{87} + ( 92037120 - 860160 \beta ) q^{88} + ( 395828874 + 1357008 \beta ) q^{89} + ( 120388541 + 108045 \beta ) q^{91} + ( -10214400 + 365568 \beta ) q^{92} + ( 460938812 + 1149428 \beta ) q^{93} + ( 596340192 - 641376 \beta ) q^{94} + ( 7340032 + 1048576 \beta ) q^{96} + ( 2084740 + 2797938 \beta ) q^{97} + 92236816 q^{98} + ( 684240270 - 7663530 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 32 q^{2} + 14 q^{3} + 512 q^{4} + 224 q^{6} - 4802 q^{7} + 8192 q^{8} + 75982 q^{9} + O(q^{10}) \) \( 2 q + 32 q^{2} + 14 q^{3} + 512 q^{4} + 224 q^{6} - 4802 q^{7} + 8192 q^{8} + 75982 q^{9} + 44940 q^{11} + 3584 q^{12} - 100282 q^{13} - 76832 q^{14} + 131072 q^{16} + 870408 q^{17} + 1215712 q^{18} + 508774 q^{19} - 33614 q^{21} + 719040 q^{22} - 79800 q^{23} + 57344 q^{24} - 1604512 q^{26} + 1869812 q^{27} - 1229312 q^{28} + 2006328 q^{29} + 2188732 q^{31} + 2097152 q^{32} - 23887920 q^{33} + 13926528 q^{34} + 19451392 q^{36} + 20723576 q^{37} + 8140384 q^{38} - 5888224 q^{39} + 19016592 q^{41} - 537824 q^{42} - 4193716 q^{43} + 11504640 q^{44} - 1276800 q^{46} + 74542524 q^{47} + 917504 q^{48} + 11529602 q^{49} - 30556644 q^{51} - 25672192 q^{52} + 3239748 q^{53} + 29916992 q^{54} - 19668992 q^{56} - 306576332 q^{57} + 32101248 q^{58} - 133642362 q^{59} + 227801686 q^{61} + 35019712 q^{62} - 182432782 q^{63} + 33554432 q^{64} - 382206720 q^{66} - 332930272 q^{67} + 222824448 q^{68} + 164018400 q^{69} - 167985720 q^{71} + 311222272 q^{72} + 44684276 q^{73} + 331577216 q^{74} + 130246144 q^{76} - 107900940 q^{77} - 94211584 q^{78} + 269642776 q^{79} + 638826478 q^{81} + 304265472 q^{82} + 183105762 q^{83} - 8605184 q^{84} - 67099456 q^{86} + 2768288796 q^{87} + 184074240 q^{88} + 791657748 q^{89} + 240777082 q^{91} - 20428800 q^{92} + 921877624 q^{93} + 1192680384 q^{94} + 14680064 q^{96} + 4169480 q^{97} + 184473632 q^{98} + 1368480540 q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−23.5052
24.5052
16.0000 −233.052 256.000 0 −3728.83 −2401.00 4096.00 34630.3 0
1.2 16.0000 247.052 256.000 0 3952.83 −2401.00 4096.00 41351.7 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.10.a.j 2
5.b even 2 1 14.10.a.c 2
5.c odd 4 2 350.10.c.j 4
15.d odd 2 1 126.10.a.o 2
20.d odd 2 1 112.10.a.c 2
35.c odd 2 1 98.10.a.e 2
35.i odd 6 2 98.10.c.h 4
35.j even 6 2 98.10.c.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.c 2 5.b even 2 1
98.10.a.e 2 35.c odd 2 1
98.10.c.h 4 35.i odd 6 2
98.10.c.j 4 35.j even 6 2
112.10.a.c 2 20.d odd 2 1
126.10.a.o 2 15.d odd 2 1
350.10.a.j 2 1.a even 1 1 trivial
350.10.c.j 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 14 T_{3} - 57576 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(350))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -16 + T )^{2} \)
$3$ \( -57576 - 14 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 2401 + T )^{2} \)
$11$ \( -2036361600 - 44940 T + T^{2} \)
$13$ \( 2397429256 + 100282 T + T^{2} \)
$17$ \( 183575251116 - 870408 T + T^{2} \)
$19$ \( -352577596856 - 508774 T + T^{2} \)
$23$ \( -115915968000 + 79800 T + T^{2} \)
$29$ \( -31904129519604 - 2006328 T + T^{2} \)
$31$ \( -2367849772544 - 2188732 T + T^{2} \)
$37$ \( 60225113026444 - 20723576 T + T^{2} \)
$41$ \( -527816477266884 - 19016592 T + T^{2} \)
$43$ \( -271341247682336 + 4193716 T + T^{2} \)
$47$ \( 1296550084878144 - 74542524 T + T^{2} \)
$53$ \( -494746212296124 - 3239748 T + T^{2} \)
$59$ \( 1575046316366136 + 133642362 T + T^{2} \)
$61$ \( 11243934945943024 - 227801686 T + T^{2} \)
$67$ \( 21401918313456496 + 332930272 T + T^{2} \)
$71$ \( -85127259938918400 + 167985720 T + T^{2} \)
$73$ \( -83678371011966956 - 44684276 T + T^{2} \)
$79$ \( 6880003984764544 - 269642776 T + T^{2} \)
$83$ \( -76697718543013464 - 183105762 T + T^{2} \)
$89$ \( 50565747709419876 - 791657748 T + T^{2} \)
$97$ \( -451110491471642900 - 4169480 T + T^{2} \)
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