Properties

 Label 275.4.b.c Level $275$ Weight $4$ Character orbit 275.b Analytic conductor $16.226$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Learn more

Newspace parameters

 Level: $$N$$ $$=$$ $$275 = 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 275.b (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$16.2255252516$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 11) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + ( 4 - 8 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} + ( 4 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{4} + ( -13 + 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{6} + ( -4 + 8 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{7} + ( 10 - 20 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{8} + ( -22 + 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + ( 4 - 8 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} + ( 4 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{4} + ( -13 + 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{6} + ( -4 + 8 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{7} + ( 10 - 20 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{8} + ( -22 + 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{9} -11 q^{11} + ( 14 - 28 \zeta_{12}^{2} - 20 \zeta_{12}^{3} ) q^{12} + ( 20 - 40 \zeta_{12}^{2} - 40 \zeta_{12}^{3} ) q^{13} + ( 2 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{14} + ( -4 + 64 \zeta_{12} - 32 \zeta_{12}^{3} ) q^{16} + ( 12 - 24 \zeta_{12}^{2} - 62 \zeta_{12}^{3} ) q^{17} + ( -30 + 60 \zeta_{12}^{2} - 46 \zeta_{12}^{3} ) q^{18} + ( -36 + 120 \zeta_{12} - 60 \zeta_{12}^{3} ) q^{19} + ( 38 + 72 \zeta_{12} - 36 \zeta_{12}^{3} ) q^{21} + ( -11 + 22 \zeta_{12}^{2} - 11 \zeta_{12}^{3} ) q^{22} + ( 36 - 72 \zeta_{12}^{2} + 49 \zeta_{12}^{3} ) q^{23} + ( -126 + 68 \zeta_{12} - 34 \zeta_{12}^{3} ) q^{24} + ( -20 - 40 \zeta_{12} + 20 \zeta_{12}^{3} ) q^{26} + ( 12 - 24 \zeta_{12}^{2} - 91 \zeta_{12}^{3} ) q^{27} + ( -36 + 72 \zeta_{12}^{2} + 64 \zeta_{12}^{3} ) q^{28} + ( -72 - 112 \zeta_{12} + 56 \zeta_{12}^{3} ) q^{29} + ( -17 - 56 \zeta_{12} + 28 \zeta_{12}^{3} ) q^{31} + ( 44 - 88 \zeta_{12}^{2} - 52 \zeta_{12}^{3} ) q^{32} + ( -44 + 88 \zeta_{12}^{2} - 11 \zeta_{12}^{3} ) q^{33} + ( 26 - 100 \zeta_{12} + 50 \zeta_{12}^{3} ) q^{34} + ( -40 - 24 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{36} + ( -8 + 16 \zeta_{12}^{2} + 27 \zeta_{12}^{3} ) q^{37} + ( -96 + 192 \zeta_{12}^{2} - 216 \zeta_{12}^{3} ) q^{38} + ( -200 - 280 \zeta_{12} + 140 \zeta_{12}^{3} ) q^{39} + ( 268 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{41} + ( 2 - 4 \zeta_{12}^{2} - 70 \zeta_{12}^{3} ) q^{42} + ( 16 - 32 \zeta_{12}^{2} + 30 \zeta_{12}^{3} ) q^{43} + ( -44 - 44 \zeta_{12} + 22 \zeta_{12}^{3} ) q^{44} + ( -157 + 170 \zeta_{12} - 85 \zeta_{12}^{3} ) q^{46} + ( -120 + 240 \zeta_{12}^{2} - 136 \zeta_{12}^{3} ) q^{47} + ( -48 + 96 \zeta_{12}^{2} - 388 \zeta_{12}^{3} ) q^{48} + ( 195 - 160 \zeta_{12} + 80 \zeta_{12}^{3} ) q^{49} + ( -82 - 472 \zeta_{12} + 236 \zeta_{12}^{3} ) q^{51} + ( 160 - 320 \zeta_{12}^{2} - 280 \zeta_{12}^{3} ) q^{52} + ( 56 - 112 \zeta_{12}^{2} + 246 \zeta_{12}^{3} ) q^{53} + ( 55 - 158 \zeta_{12} + 79 \zeta_{12}^{3} ) q^{54} + ( 60 + 152 \zeta_{12} - 76 \zeta_{12}^{3} ) q^{56} + ( -204 + 408 \zeta_{12}^{2} - 756 \zeta_{12}^{3} ) q^{57} + ( -16 + 32 \zeta_{12}^{2} + 96 \zeta_{12}^{3} ) q^{58} + ( -317 - 264 \zeta_{12} + 132 \zeta_{12}^{3} ) q^{59} + ( 420 - 368 \zeta_{12} + 184 \zeta_{12}^{3} ) q^{61} + ( 11 - 22 \zeta_{12}^{2} + 67 \zeta_{12}^{3} ) q^{62} + ( 8 - 16 \zeta_{12}^{2} - 124 \zeta_{12}^{3} ) q^{63} + ( -112 + 496 \zeta_{12} - 248 \zeta_{12}^{3} ) q^{64} + ( 143 - 110 \zeta_{12} + 55 \zeta_{12}^{3} ) q^{66} + ( -20 + 40 \zeta_{12}^{2} + 377 \zeta_{12}^{3} ) q^{67} + ( 172 - 344 \zeta_{12}^{2} - 320 \zeta_{12}^{3} ) q^{68} + ( -481 + 464 \zeta_{12} - 232 \zeta_{12}^{3} ) q^{69} + ( -339 - 152 \zeta_{12} + 76 \zeta_{12}^{3} ) q^{71} + ( -268 + 536 \zeta_{12}^{2} - 372 \zeta_{12}^{3} ) q^{72} + ( 468 - 936 \zeta_{12}^{2} + 200 \zeta_{12}^{3} ) q^{73} + ( -3 + 38 \zeta_{12} - 19 \zeta_{12}^{3} ) q^{74} + ( 216 + 336 \zeta_{12} - 168 \zeta_{12}^{3} ) q^{76} + ( 44 - 88 \zeta_{12}^{2} - 110 \zeta_{12}^{3} ) q^{77} + ( -60 + 120 \zeta_{12}^{2} + 220 \zeta_{12}^{3} ) q^{78} + ( -158 + 1312 \zeta_{12} - 656 \zeta_{12}^{3} ) q^{79} + ( -647 - 272 \zeta_{12} + 136 \zeta_{12}^{3} ) q^{81} + ( 264 - 528 \zeta_{12}^{2} + 256 \zeta_{12}^{3} ) q^{82} + ( -120 + 240 \zeta_{12}^{2} - 234 \zeta_{12}^{3} ) q^{83} + ( 368 + 440 \zeta_{12} - 220 \zeta_{12}^{3} ) q^{84} + ( -78 + 92 \zeta_{12} - 46 \zeta_{12}^{3} ) q^{86} + ( -232 + 464 \zeta_{12}^{2} + 600 \zeta_{12}^{3} ) q^{87} + ( -110 + 220 \zeta_{12}^{2} - 66 \zeta_{12}^{3} ) q^{88} + ( 921 - 656 \zeta_{12} + 328 \zeta_{12}^{3} ) q^{89} + ( 640 + 720 \zeta_{12} - 360 \zeta_{12}^{3} ) q^{91} + ( 46 - 92 \zeta_{12}^{2} - 20 \zeta_{12}^{3} ) q^{92} + ( -40 + 80 \zeta_{12}^{2} + 319 \zeta_{12}^{3} ) q^{93} + ( 496 - 512 \zeta_{12} + 256 \zeta_{12}^{3} ) q^{94} + ( -476 - 328 \zeta_{12} + 164 \zeta_{12}^{3} ) q^{96} + ( 144 - 288 \zeta_{12}^{2} + 1097 \zeta_{12}^{3} ) q^{97} + ( 275 - 550 \zeta_{12}^{2} + 435 \zeta_{12}^{3} ) q^{98} + ( 242 - 176 \zeta_{12} + 88 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 16q^{4} - 52q^{6} - 88q^{9} + O(q^{10})$$ $$4q + 16q^{4} - 52q^{6} - 88q^{9} - 44q^{11} + 8q^{14} - 16q^{16} - 144q^{19} + 152q^{21} - 504q^{24} - 80q^{26} - 288q^{29} - 68q^{31} + 104q^{34} - 160q^{36} - 800q^{39} + 1072q^{41} - 176q^{44} - 628q^{46} + 780q^{49} - 328q^{51} + 220q^{54} + 240q^{56} - 1268q^{59} + 1680q^{61} - 448q^{64} + 572q^{66} - 1924q^{69} - 1356q^{71} - 12q^{74} + 864q^{76} - 632q^{79} - 2588q^{81} + 1472q^{84} - 312q^{86} + 3684q^{89} + 2560q^{91} + 1984q^{94} - 1904q^{96} + 968q^{99} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$\chi(n)$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
2.73205i 7.92820i 0.535898 0 −21.6603 3.07180i 23.3205i −35.8564 0
199.2 0.732051i 5.92820i 7.46410 0 −4.33975 16.9282i 11.3205i −8.14359 0
199.3 0.732051i 5.92820i 7.46410 0 −4.33975 16.9282i 11.3205i −8.14359 0
199.4 2.73205i 7.92820i 0.535898 0 −21.6603 3.07180i 23.3205i −35.8564 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 275.4.b.c 4
5.b even 2 1 inner 275.4.b.c 4
5.c odd 4 1 11.4.a.a 2
5.c odd 4 1 275.4.a.b 2
15.e even 4 1 99.4.a.c 2
15.e even 4 1 2475.4.a.q 2
20.e even 4 1 176.4.a.i 2
35.f even 4 1 539.4.a.e 2
40.i odd 4 1 704.4.a.p 2
40.k even 4 1 704.4.a.n 2
55.e even 4 1 121.4.a.c 2
55.k odd 20 4 121.4.c.c 8
55.l even 20 4 121.4.c.f 8
60.l odd 4 1 1584.4.a.bc 2
65.h odd 4 1 1859.4.a.a 2
165.l odd 4 1 1089.4.a.v 2
220.i odd 4 1 1936.4.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 5.c odd 4 1
99.4.a.c 2 15.e even 4 1
121.4.a.c 2 55.e even 4 1
121.4.c.c 8 55.k odd 20 4
121.4.c.f 8 55.l even 20 4
176.4.a.i 2 20.e even 4 1
275.4.a.b 2 5.c odd 4 1
275.4.b.c 4 1.a even 1 1 trivial
275.4.b.c 4 5.b even 2 1 inner
539.4.a.e 2 35.f even 4 1
704.4.a.n 2 40.k even 4 1
704.4.a.p 2 40.i odd 4 1
1089.4.a.v 2 165.l odd 4 1
1584.4.a.bc 2 60.l odd 4 1
1859.4.a.a 2 65.h odd 4 1
1936.4.a.w 2 220.i odd 4 1
2475.4.a.q 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 8 T_{2}^{2} + 4$$ acting on $$S_{4}^{\mathrm{new}}(275, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 8 T^{2} + T^{4}$$
$3$ $$2209 + 98 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$2704 + 296 T^{2} + T^{4}$$
$11$ $$( 11 + T )^{4}$$
$13$ $$160000 + 5600 T^{2} + T^{4}$$
$17$ $$11641744 + 8552 T^{2} + T^{4}$$
$19$ $$( -9504 + 72 T + T^{2} )^{2}$$
$23$ $$2211169 + 12578 T^{2} + T^{4}$$
$29$ $$( -4224 + 144 T + T^{2} )^{2}$$
$31$ $$( -2063 + 34 T + T^{2} )^{2}$$
$37$ $$288369 + 1842 T^{2} + T^{4}$$
$41$ $$( 71776 - 536 T + T^{2} )^{2}$$
$43$ $$17424 + 3336 T^{2} + T^{4}$$
$47$ $$610287616 + 123392 T^{2} + T^{4}$$
$53$ $$2612027664 + 139848 T^{2} + T^{4}$$
$59$ $$( 48217 + 634 T + T^{2} )^{2}$$
$61$ $$( 74832 - 840 T + T^{2} )^{2}$$
$67$ $$19860983041 + 286658 T^{2} + T^{4}$$
$71$ $$( 97593 + 678 T + T^{2} )^{2}$$
$73$ $$380777853184 + 1394144 T^{2} + T^{4}$$
$79$ $$( -1266044 + 316 T + T^{2} )^{2}$$
$83$ $$133541136 + 195912 T^{2} + T^{4}$$
$89$ $$( 525489 - 1842 T + T^{2} )^{2}$$
$97$ $$1302339722401 + 2531234 T^{2} + T^{4}$$
show more
show less