Properties

Label 22.4.c.a
Level $22$
Weight $4$
Character orbit 22.c
Analytic conductor $1.298$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 22.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.29804202013\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \zeta_{10} q^{2} + (8 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{3} + 4 \zeta_{10}^{2} q^{4} + ( - \zeta_{10}^{3} + 1) q^{5} + ( - 16 \zeta_{10}^{3} + 10 \zeta_{10}^{2} - 10 \zeta_{10} + 16) q^{6} + (11 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 11 \zeta_{10}) q^{7} - 8 \zeta_{10}^{3} q^{8} + ( - 39 \zeta_{10}^{2} - 7 \zeta_{10} - 39) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \zeta_{10} q^{2} + (8 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{3} + 4 \zeta_{10}^{2} q^{4} + ( - \zeta_{10}^{3} + 1) q^{5} + ( - 16 \zeta_{10}^{3} + 10 \zeta_{10}^{2} - 10 \zeta_{10} + 16) q^{6} + (11 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 11 \zeta_{10}) q^{7} - 8 \zeta_{10}^{3} q^{8} + ( - 39 \zeta_{10}^{2} - 7 \zeta_{10} - 39) q^{9} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2) q^{10} + ( - 22 \zeta_{10}^{3} + 22 \zeta_{10} + 11) q^{11} + (12 \zeta_{10}^{3} - 12 \zeta_{10}^{2} - 32) q^{12} + (29 \zeta_{10}^{2} + 4 \zeta_{10} + 29) q^{13} + ( - 16 \zeta_{10}^{3} - 22 \zeta_{10} + 22) q^{14} + (8 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 8 \zeta_{10}) q^{15} + (16 \zeta_{10}^{3} - 16 \zeta_{10}^{2} + 16 \zeta_{10} - 16) q^{16} + ( - 77 \zeta_{10}^{3} + 104 \zeta_{10}^{2} - 104 \zeta_{10} + 77) q^{17} + (78 \zeta_{10}^{3} + 14 \zeta_{10}^{2} + 78 \zeta_{10}) q^{18} + (32 \zeta_{10}^{3} - 9 \zeta_{10} + 9) q^{19} + (4 \zeta_{10}^{2} + 4) q^{20} + (79 \zeta_{10}^{3} - 79 \zeta_{10}^{2} - 97) q^{21} + (44 \zeta_{10}^{3} - 88 \zeta_{10}^{2} + 22 \zeta_{10} - 44) q^{22} + ( - 24 \zeta_{10}^{3} + 24 \zeta_{10}^{2} - 44) q^{23} + (24 \zeta_{10}^{2} + 40 \zeta_{10} + 24) q^{24} + (123 \zeta_{10}^{3} - \zeta_{10} + 1) q^{25} + ( - 58 \zeta_{10}^{3} - 8 \zeta_{10}^{2} - 58 \zeta_{10}) q^{26} + ( - 269 \zeta_{10}^{3} + 17 \zeta_{10}^{2} - 17 \zeta_{10} + 269) q^{27} + (32 \zeta_{10}^{3} + 12 \zeta_{10}^{2} - 12 \zeta_{10} - 32) q^{28} + ( - 159 \zeta_{10}^{3} + 107 \zeta_{10}^{2} - 159 \zeta_{10}) q^{29} + ( - 22 \zeta_{10}^{3} - 16 \zeta_{10} + 16) q^{30} + ( - 143 \zeta_{10}^{2} + 202 \zeta_{10} - 143) q^{31} + 32 q^{32} + (264 \zeta_{10}^{3} - 44 \zeta_{10}^{2} + 253 \zeta_{10} - 143) q^{33} + ( - 54 \zeta_{10}^{3} + 54 \zeta_{10}^{2} - 154) q^{34} + (8 \zeta_{10}^{2} + 11 \zeta_{10} + 8) q^{35} + ( - 184 \zeta_{10}^{3} - 156 \zeta_{10} + 156) q^{36} + ( - 79 \zeta_{10}^{3} - 97 \zeta_{10}^{2} - 79 \zeta_{10}) q^{37} + ( - 64 \zeta_{10}^{3} + 82 \zeta_{10}^{2} - 82 \zeta_{10} + 64) q^{38} + (351 \zeta_{10}^{3} - 107 \zeta_{10}^{2} + 107 \zeta_{10} - 351) q^{39} + ( - 8 \zeta_{10}^{3} - 8 \zeta_{10}) q^{40} + ( - 40 \zeta_{10}^{3} - 129 \zeta_{10} + 129) q^{41} + (158 \zeta_{10}^{2} + 36 \zeta_{10} + 158) q^{42} + (68 \zeta_{10}^{3} - 68 \zeta_{10}^{2} + 256) q^{43} + (88 \zeta_{10}^{3} + 44 \zeta_{10}^{2} + 88) q^{44} + (46 \zeta_{10}^{3} - 46 \zeta_{10}^{2} - 85) q^{45} + ( - 48 \zeta_{10}^{2} + 136 \zeta_{10} - 48) q^{46} + (188 \zeta_{10}^{3} - 21 \zeta_{10} + 21) q^{47} + ( - 48 \zeta_{10}^{3} - 80 \zeta_{10}^{2} - 48 \zeta_{10}) q^{48} + ( - 158 \zeta_{10}^{3} + 213 \zeta_{10}^{2} - 213 \zeta_{10} + 158) q^{49} + ( - 246 \zeta_{10}^{3} + 248 \zeta_{10}^{2} - 248 \zeta_{10} + 246) q^{50} + (96 \zeta_{10}^{3} + 439 \zeta_{10}^{2} + 96 \zeta_{10}) q^{51} + (132 \zeta_{10}^{3} + 116 \zeta_{10} - 116) q^{52} + ( - 223 \zeta_{10}^{2} + 424 \zeta_{10} - 223) q^{53} + (504 \zeta_{10}^{3} - 504 \zeta_{10}^{2} - 538) q^{54} + ( - 55 \zeta_{10}^{3} + 22 \zeta_{10}^{2} - 22 \zeta_{10} + 33) q^{55} + ( - 88 \zeta_{10}^{3} + 88 \zeta_{10}^{2} + 64) q^{56} + ( - 51 \zeta_{10}^{2} - 178 \zeta_{10} - 51) q^{57} + (104 \zeta_{10}^{3} + 318 \zeta_{10} - 318) q^{58} + ( - 301 \zeta_{10}^{3} + 225 \zeta_{10}^{2} - 301 \zeta_{10}) q^{59} + (44 \zeta_{10}^{3} - 12 \zeta_{10}^{2} + 12 \zeta_{10} - 44) q^{60} + ( - 429 \zeta_{10}^{3} - 24 \zeta_{10}^{2} + 24 \zeta_{10} + 429) q^{61} + (286 \zeta_{10}^{3} - 404 \zeta_{10}^{2} + 286 \zeta_{10}) q^{62} + ( - 797 \zeta_{10}^{3} - 389 \zeta_{10} + 389) q^{63} - 64 \zeta_{10} q^{64} + ( - 33 \zeta_{10}^{3} + 33 \zeta_{10}^{2} + 62) q^{65} + ( - 440 \zeta_{10}^{3} + 22 \zeta_{10}^{2} - 242 \zeta_{10} + 528) q^{66} + (116 \zeta_{10}^{3} - 116 \zeta_{10}^{2} - 80) q^{67} + ( - 108 \zeta_{10}^{2} + 416 \zeta_{10} - 108) q^{68} + ( - 280 \zeta_{10}^{3} - 12 \zeta_{10} + 12) q^{69} + ( - 16 \zeta_{10}^{3} - 22 \zeta_{10}^{2} - 16 \zeta_{10}) q^{70} + (783 \zeta_{10}^{3} - 342 \zeta_{10}^{2} + 342 \zeta_{10} - 783) q^{71} + (368 \zeta_{10}^{3} - 56 \zeta_{10}^{2} + 56 \zeta_{10} - 368) q^{72} + ( - 51 \zeta_{10}^{3} + 51 \zeta_{10}^{2} - 51 \zeta_{10}) q^{73} + (352 \zeta_{10}^{3} + 158 \zeta_{10} - 158) q^{74} + ( - 364 \zeta_{10}^{2} - 617 \zeta_{10} - 364) q^{75} + ( - 36 \zeta_{10}^{3} + 36 \zeta_{10}^{2} - 128) q^{76} + (55 \zeta_{10}^{3} + 209 \zeta_{10}^{2} + 363 \zeta_{10} - 66) q^{77} + ( - 488 \zeta_{10}^{3} + 488 \zeta_{10}^{2} + 702) q^{78} + (537 \zeta_{10}^{2} - 934 \zeta_{10} + 537) q^{79} + (16 \zeta_{10}^{3} + 16 \zeta_{10} - 16) q^{80} + (1014 \zeta_{10}^{3} + 652 \zeta_{10}^{2} + 1014 \zeta_{10}) q^{81} + (80 \zeta_{10}^{3} + 178 \zeta_{10}^{2} - 178 \zeta_{10} - 80) q^{82} + (63 \zeta_{10}^{3} - 538 \zeta_{10}^{2} + 538 \zeta_{10} - 63) q^{83} + ( - 316 \zeta_{10}^{3} - 72 \zeta_{10}^{2} - 316 \zeta_{10}) q^{84} + ( - 50 \zeta_{10}^{3} - 77 \zeta_{10} + 77) q^{85} + (136 \zeta_{10}^{2} - 648 \zeta_{10} + 136) q^{86} + ( - 951 \zeta_{10}^{3} + 951 \zeta_{10}^{2} + 893) q^{87} + ( - 264 \zeta_{10}^{3} + 176 \zeta_{10}^{2} - 352 \zeta_{10} + 176) q^{88} + (940 \zeta_{10}^{3} - 940 \zeta_{10}^{2} - 38) q^{89} + (92 \zeta_{10}^{2} + 78 \zeta_{10} + 92) q^{90} + (583 \zeta_{10}^{3} + 276 \zeta_{10} - 276) q^{91} + (96 \zeta_{10}^{3} - 272 \zeta_{10}^{2} + 96 \zeta_{10}) q^{92} + (43 \zeta_{10}^{3} - 581 \zeta_{10}^{2} + 581 \zeta_{10} - 43) q^{93} + ( - 376 \zeta_{10}^{3} + 418 \zeta_{10}^{2} - 418 \zeta_{10} + 376) q^{94} + (32 \zeta_{10}^{3} - 9 \zeta_{10}^{2} + 32 \zeta_{10}) q^{95} + (256 \zeta_{10}^{3} + 96 \zeta_{10} - 96) q^{96} + (665 \zeta_{10}^{2} + 24 \zeta_{10} + 665) q^{97} + ( - 110 \zeta_{10}^{3} + 110 \zeta_{10}^{2} - 316) q^{98} + (154 \zeta_{10}^{3} - 737 \zeta_{10}^{2} - 781 \zeta_{10} - 1441) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - q^{3} - 4 q^{4} + 3 q^{5} + 28 q^{6} + 25 q^{7} - 8 q^{8} - 124 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - q^{3} - 4 q^{4} + 3 q^{5} + 28 q^{6} + 25 q^{7} - 8 q^{8} - 124 q^{9} - 4 q^{10} + 44 q^{11} - 104 q^{12} + 91 q^{13} + 50 q^{14} + 13 q^{15} - 16 q^{16} + 23 q^{17} + 142 q^{18} + 59 q^{19} + 12 q^{20} - 230 q^{21} - 22 q^{22} - 224 q^{23} + 112 q^{24} + 126 q^{25} - 108 q^{26} + 773 q^{27} - 120 q^{28} - 425 q^{29} + 26 q^{30} - 227 q^{31} + 128 q^{32} - 11 q^{33} - 724 q^{34} + 35 q^{35} + 284 q^{36} - 61 q^{37} + 28 q^{38} - 839 q^{39} - 16 q^{40} + 347 q^{41} + 510 q^{42} + 1160 q^{43} + 396 q^{44} - 248 q^{45} - 8 q^{46} + 251 q^{47} - 16 q^{48} + 48 q^{49} + 242 q^{50} - 247 q^{51} - 216 q^{52} - 245 q^{53} - 1144 q^{54} + 33 q^{55} + 80 q^{56} - 331 q^{57} - 850 q^{58} - 827 q^{59} - 108 q^{60} + 1335 q^{61} + 976 q^{62} + 370 q^{63} - 64 q^{64} + 182 q^{65} + 1408 q^{66} - 88 q^{67} + 92 q^{68} - 244 q^{69} - 10 q^{70} - 1665 q^{71} - 992 q^{72} - 153 q^{73} - 122 q^{74} - 1709 q^{75} - 584 q^{76} - 55 q^{77} + 1832 q^{78} + 677 q^{79} - 32 q^{80} + 1376 q^{81} - 596 q^{82} + 887 q^{83} - 560 q^{84} + 181 q^{85} - 240 q^{86} + 1670 q^{87} - 88 q^{88} + 1728 q^{89} + 354 q^{90} - 245 q^{91} + 464 q^{92} + 1033 q^{93} + 292 q^{94} + 73 q^{95} - 32 q^{96} + 2019 q^{97} - 1484 q^{98} - 5654 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).

\(n\) \(13\)
\(\chi(n)\) \(\zeta_{10}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 + 0.587785i
−1.61803 + 1.17557i −3.04508 9.37181i 1.23607 3.80423i 1.30902 + 0.951057i 15.9443 + 11.5842i 4.57295 14.0741i 2.47214 + 7.60845i −56.7148 + 41.2057i −3.23607
5.1 0.618034 + 1.90211i 2.54508 + 1.84911i −3.23607 + 2.35114i 0.190983 0.587785i −1.94427 + 5.98385i 7.92705 5.75934i −6.47214 4.70228i −5.28522 16.2662i 1.23607
9.1 0.618034 1.90211i 2.54508 1.84911i −3.23607 2.35114i 0.190983 + 0.587785i −1.94427 5.98385i 7.92705 + 5.75934i −6.47214 + 4.70228i −5.28522 + 16.2662i 1.23607
15.1 −1.61803 1.17557i −3.04508 + 9.37181i 1.23607 + 3.80423i 1.30902 0.951057i 15.9443 11.5842i 4.57295 + 14.0741i 2.47214 7.60845i −56.7148 41.2057i −3.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 22.4.c.a 4
3.b odd 2 1 198.4.f.b 4
4.b odd 2 1 176.4.m.a 4
11.b odd 2 1 242.4.c.j 4
11.c even 5 1 inner 22.4.c.a 4
11.c even 5 1 242.4.a.k 2
11.c even 5 2 242.4.c.f 4
11.d odd 10 1 242.4.a.h 2
11.d odd 10 1 242.4.c.j 4
11.d odd 10 2 242.4.c.m 4
33.f even 10 1 2178.4.a.bi 2
33.h odd 10 1 198.4.f.b 4
33.h odd 10 1 2178.4.a.z 2
44.g even 10 1 1936.4.a.bc 2
44.h odd 10 1 176.4.m.a 4
44.h odd 10 1 1936.4.a.bb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.4.c.a 4 1.a even 1 1 trivial
22.4.c.a 4 11.c even 5 1 inner
176.4.m.a 4 4.b odd 2 1
176.4.m.a 4 44.h odd 10 1
198.4.f.b 4 3.b odd 2 1
198.4.f.b 4 33.h odd 10 1
242.4.a.h 2 11.d odd 10 1
242.4.a.k 2 11.c even 5 1
242.4.c.f 4 11.c even 5 2
242.4.c.j 4 11.b odd 2 1
242.4.c.j 4 11.d odd 10 1
242.4.c.m 4 11.d odd 10 2
1936.4.a.bb 2 44.h odd 10 1
1936.4.a.bc 2 44.g even 10 1
2178.4.a.z 2 33.h odd 10 1
2178.4.a.bi 2 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + T_{3}^{3} + 76T_{3}^{2} - 434T_{3} + 961 \) acting on \(S_{4}^{\mathrm{new}}(22, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} + 76 T^{2} - 434 T + 961 \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} - 25 T^{3} + 460 T^{2} + \cdots + 21025 \) Copy content Toggle raw display
$11$ \( T^{4} - 44 T^{3} + 726 T^{2} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( T^{4} - 91 T^{3} + 3496 T^{2} + \cdots + 502681 \) Copy content Toggle raw display
$17$ \( T^{4} - 23 T^{3} + 10924 T^{2} + \cdots + 52983841 \) Copy content Toggle raw display
$19$ \( T^{4} - 59 T^{3} + 1636 T^{2} + \cdots + 1515361 \) Copy content Toggle raw display
$23$ \( (T^{2} + 112 T + 2416)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 425 T^{3} + \cdots + 951414025 \) Copy content Toggle raw display
$31$ \( T^{4} + 227 T^{3} + \cdots + 72777961 \) Copy content Toggle raw display
$37$ \( T^{4} + 61 T^{3} + \cdots + 117310561 \) Copy content Toggle raw display
$41$ \( T^{4} - 347 T^{3} + \cdots + 408080401 \) Copy content Toggle raw display
$43$ \( (T^{2} - 580 T + 78320)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 251 T^{3} + \cdots + 1509400201 \) Copy content Toggle raw display
$53$ \( T^{4} + 245 T^{3} + \cdots + 1259895025 \) Copy content Toggle raw display
$59$ \( T^{4} + 827 T^{3} + \cdots + 11599505401 \) Copy content Toggle raw display
$61$ \( T^{4} - 1335 T^{3} + \cdots + 46442405025 \) Copy content Toggle raw display
$67$ \( (T^{2} + 44 T - 16336)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 1665 T^{3} + \cdots + 5373623025 \) Copy content Toggle raw display
$73$ \( T^{4} + 153 T^{3} + 10404 T^{2} + \cdots + 6765201 \) Copy content Toggle raw display
$79$ \( T^{4} - 677 T^{3} + \cdots + 6794540041 \) Copy content Toggle raw display
$83$ \( T^{4} - 887 T^{3} + \cdots + 36760776361 \) Copy content Toggle raw display
$89$ \( (T^{2} - 864 T - 917876)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 2019 T^{3} + \cdots + 181211124721 \) Copy content Toggle raw display
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