Properties

Label 1575.4.a.q
Level 1575
Weight 4
Character orbit 1575.a
Self dual yes
Analytic conductor 92.928
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + 2 \beta ) q^{2} + ( 1 - 4 \beta ) q^{4} + 7 q^{7} + ( -9 - 10 \beta ) q^{8} +O(q^{10})\) \( q + ( -1 + 2 \beta ) q^{2} + ( 1 - 4 \beta ) q^{4} + 7 q^{7} + ( -9 - 10 \beta ) q^{8} + ( 8 + 40 \beta ) q^{11} + ( 38 - 4 \beta ) q^{13} + ( -7 + 14 \beta ) q^{14} + ( -39 + 24 \beta ) q^{16} + ( -62 + 4 \beta ) q^{17} + ( -48 + 32 \beta ) q^{19} + ( 152 - 24 \beta ) q^{22} + ( -8 + 68 \beta ) q^{23} + ( -54 + 80 \beta ) q^{26} + ( 7 - 28 \beta ) q^{28} + ( -94 - 108 \beta ) q^{29} + ( -60 - 36 \beta ) q^{31} + ( 207 - 22 \beta ) q^{32} + ( 78 - 128 \beta ) q^{34} + ( 66 - 132 \beta ) q^{37} + ( 176 - 128 \beta ) q^{38} + ( -50 + 160 \beta ) q^{41} + ( 268 + 124 \beta ) q^{43} + ( -312 + 8 \beta ) q^{44} + ( 280 - 84 \beta ) q^{46} + ( -464 + 84 \beta ) q^{47} + 49 q^{49} + ( 70 - 156 \beta ) q^{52} + ( 442 - 128 \beta ) q^{53} + ( -63 - 70 \beta ) q^{56} + ( -338 - 80 \beta ) q^{58} + ( -52 - 408 \beta ) q^{59} + ( -234 + 84 \beta ) q^{61} + ( -84 - 84 \beta ) q^{62} + ( 17 + 244 \beta ) q^{64} + ( 844 + 76 \beta ) q^{67} + ( -94 + 252 \beta ) q^{68} + ( 68 - 300 \beta ) q^{71} + ( -254 + 644 \beta ) q^{73} + ( -594 + 264 \beta ) q^{74} + ( -304 + 224 \beta ) q^{76} + ( 56 + 280 \beta ) q^{77} + ( -216 + 464 \beta ) q^{79} + ( 690 - 260 \beta ) q^{82} + ( -292 + 168 \beta ) q^{83} + ( 228 + 412 \beta ) q^{86} + ( -872 - 440 \beta ) q^{88} + ( 702 + 224 \beta ) q^{89} + ( 266 - 28 \beta ) q^{91} + ( -552 + 100 \beta ) q^{92} + ( 800 - 1012 \beta ) q^{94} + ( 594 + 92 \beta ) q^{97} + ( -49 + 98 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + 14q^{7} - 18q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + 14q^{7} - 18q^{8} + 16q^{11} + 76q^{13} - 14q^{14} - 78q^{16} - 124q^{17} - 96q^{19} + 304q^{22} - 16q^{23} - 108q^{26} + 14q^{28} - 188q^{29} - 120q^{31} + 414q^{32} + 156q^{34} + 132q^{37} + 352q^{38} - 100q^{41} + 536q^{43} - 624q^{44} + 560q^{46} - 928q^{47} + 98q^{49} + 140q^{52} + 884q^{53} - 126q^{56} - 676q^{58} - 104q^{59} - 468q^{61} - 168q^{62} + 34q^{64} + 1688q^{67} - 188q^{68} + 136q^{71} - 508q^{73} - 1188q^{74} - 608q^{76} + 112q^{77} - 432q^{79} + 1380q^{82} - 584q^{83} + 456q^{86} - 1744q^{88} + 1404q^{89} + 532q^{91} - 1104q^{92} + 1600q^{94} + 1188q^{97} - 98q^{98} + O(q^{100}) \)

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
−1.41421
1.41421
−3.82843 0 6.65685 0 0 7.00000 5.14214 0 0
1.2 1.82843 0 −4.65685 0 0 7.00000 −23.1421 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 1575.4.a.q 2
3.b odd 2 1 525.4.a.l 2
5.b even 2 1 315.4.a.k 2
15.d odd 2 1 105.4.a.e 2
15.e even 4 2 525.4.d.l 4
35.c odd 2 1 2205.4.a.bb 2
60.h even 2 1 1680.4.a.bo 2
105.g even 2 1 735.4.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.e 2 15.d odd 2 1
315.4.a.k 2 5.b even 2 1
525.4.a.l 2 3.b odd 2 1
525.4.d.l 4 15.e even 4 2
735.4.a.o 2 105.g even 2 1
1575.4.a.q 2 1.a even 1 1 trivial
1680.4.a.bo 2 60.h even 2 1
2205.4.a.bb 2 35.c odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{2} + 2 T_{2} - 7 \)
\( T_{11}^{2} - 16 T_{11} - 3136 \)
\( T_{13}^{2} - 76 T_{13} + 1412 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 9 T^{2} + 16 T^{3} + 64 T^{4} \)
$3$ 1
$5$ 1
$7$ \( ( 1 - 7 T )^{2} \)
$11$ \( 1 - 16 T - 474 T^{2} - 21296 T^{3} + 1771561 T^{4} \)
$13$ \( 1 - 76 T + 5806 T^{2} - 166972 T^{3} + 4826809 T^{4} \)
$17$ \( 1 + 124 T + 13638 T^{2} + 609212 T^{3} + 24137569 T^{4} \)
$19$ \( 1 + 96 T + 13974 T^{2} + 658464 T^{3} + 47045881 T^{4} \)
$23$ \( 1 + 16 T + 15150 T^{2} + 194672 T^{3} + 148035889 T^{4} \)
$29$ \( 1 + 188 T + 34286 T^{2} + 4585132 T^{3} + 594823321 T^{4} \)
$31$ \( 1 + 120 T + 60590 T^{2} + 3574920 T^{3} + 887503681 T^{4} \)
$37$ \( 1 - 132 T + 70814 T^{2} - 6686196 T^{3} + 2565726409 T^{4} \)
$41$ \( 1 + 100 T + 89142 T^{2} + 6892100 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 - 536 T + 200086 T^{2} - 42615752 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 + 928 T + 408830 T^{2} + 96347744 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 - 884 T + 460350 T^{2} - 131607268 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 + 104 T + 80534 T^{2} + 21359416 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 468 T + 494606 T^{2} + 106227108 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 - 1688 T + 1302310 T^{2} - 507687944 T^{3} + 90458382169 T^{4} \)
$71$ \( 1 - 136 T + 540446 T^{2} - 48675896 T^{3} + 128100283921 T^{4} \)
$73$ \( 1 + 508 T + 13078 T^{2} + 197620636 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 + 432 T + 602142 T^{2} + 212992848 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 + 584 T + 1172390 T^{2} + 333923608 T^{3} + 326940373369 T^{4} \)
$89$ \( 1 - 1404 T + 1802390 T^{2} - 989776476 T^{3} + 496981290961 T^{4} \)
$97$ \( 1 - 1188 T + 2161254 T^{2} - 1084255524 T^{3} + 832972004929 T^{4} \)
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