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Properties

Label 275.4.a.h.1.2
Level $275$
Weight $4$
Character 275.1
Self dual yes
Analytic conductor $16.226$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [275,4,Mod(1,275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("275.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 275.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(16.2255252516\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 38x^{3} + 61x^{2} + 304x - 148 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.75920\) of defining polynomial
Character \(\chi\) \(=\) 275.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.75920 q^{2} -2.30980 q^{3} -0.386826 q^{4} +6.37319 q^{6} +13.2555 q^{7} +23.1409 q^{8} -21.6648 q^{9} -11.0000 q^{11} +0.893490 q^{12} -30.6714 q^{13} -36.5746 q^{14} -60.7558 q^{16} +35.1080 q^{17} +59.7776 q^{18} -134.694 q^{19} -30.6176 q^{21} +30.3512 q^{22} -6.51724 q^{23} -53.4509 q^{24} +84.6285 q^{26} +112.406 q^{27} -5.12758 q^{28} -30.3054 q^{29} +331.803 q^{31} -17.4901 q^{32} +25.4078 q^{33} -96.8700 q^{34} +8.38051 q^{36} +172.426 q^{37} +371.649 q^{38} +70.8448 q^{39} +58.1492 q^{41} +84.4800 q^{42} +210.100 q^{43} +4.25508 q^{44} +17.9823 q^{46} -151.570 q^{47} +140.334 q^{48} -167.291 q^{49} -81.0925 q^{51} +11.8645 q^{52} +318.384 q^{53} -310.150 q^{54} +306.745 q^{56} +311.117 q^{57} +83.6186 q^{58} +414.677 q^{59} +498.088 q^{61} -915.511 q^{62} -287.179 q^{63} +534.305 q^{64} -70.1051 q^{66} +639.464 q^{67} -13.5807 q^{68} +15.0535 q^{69} -267.420 q^{71} -501.344 q^{72} +540.297 q^{73} -475.756 q^{74} +52.1033 q^{76} -145.811 q^{77} -195.475 q^{78} +691.157 q^{79} +325.315 q^{81} -160.445 q^{82} +663.244 q^{83} +11.8437 q^{84} -579.709 q^{86} +69.9994 q^{87} -254.550 q^{88} -119.542 q^{89} -406.565 q^{91} +2.52103 q^{92} -766.399 q^{93} +418.211 q^{94} +40.3987 q^{96} -284.049 q^{97} +461.589 q^{98} +238.313 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + 40 q^{4} - 42 q^{6} + 40 q^{7} + 21 q^{8} + 31 q^{9} - 55 q^{11} + 125 q^{12} + 211 q^{13} - 133 q^{14} + 208 q^{16} + 72 q^{17} + 171 q^{18} + 23 q^{19} + 282 q^{21} - 22 q^{22} + 57 q^{23}+ \cdots - 341 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.75920 −0.975524 −0.487762 − 0.872977i \(-0.662187\pi\)
−0.487762 + 0.872977i \(0.662187\pi\)
\(3\) −2.30980 −0.444521 −0.222260 − 0.974987i \(-0.571344\pi\)
−0.222260 + 0.974987i \(0.571344\pi\)
\(4\) −0.386826 −0.0483532
\(5\) 0 0
\(6\) 6.37319 0.433641
\(7\) 13.2555 0.715731 0.357865 − 0.933773i \(-0.383505\pi\)
0.357865 + 0.933773i \(0.383505\pi\)
\(8\) 23.1409 1.02269
\(9\) −21.6648 −0.802401
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0.893490 0.0214940
\(13\) −30.6714 −0.654363 −0.327181 − 0.944962i \(-0.606099\pi\)
−0.327181 + 0.944962i \(0.606099\pi\)
\(14\) −36.5746 −0.698213
\(15\) 0 0
\(16\) −60.7558 −0.949309
\(17\) 35.1080 0.500879 0.250440 − 0.968132i \(-0.419425\pi\)
0.250440 + 0.968132i \(0.419425\pi\)
\(18\) 59.7776 0.782761
\(19\) −134.694 −1.62637 −0.813185 − 0.582006i \(-0.802268\pi\)
−0.813185 + 0.582006i \(0.802268\pi\)
\(20\) 0 0
\(21\) −30.6176 −0.318157
\(22\) 30.3512 0.294132
\(23\) −6.51724 −0.0590843 −0.0295421 − 0.999564i \(-0.509405\pi\)
−0.0295421 + 0.999564i \(0.509405\pi\)
\(24\) −53.4509 −0.454609
\(25\) 0 0
\(26\) 84.6285 0.638347
\(27\) 112.406 0.801205
\(28\) −5.12758 −0.0346079
\(29\) −30.3054 −0.194054 −0.0970271 − 0.995282i \(-0.530933\pi\)
−0.0970271 + 0.995282i \(0.530933\pi\)
\(30\) 0 0
\(31\) 331.803 1.92237 0.961187 − 0.275897i \(-0.0889748\pi\)
0.961187 + 0.275897i \(0.0889748\pi\)
\(32\) −17.4901 −0.0966202
\(33\) 25.4078 0.134028
\(34\) −96.8700 −0.488620
\(35\) 0 0
\(36\) 8.38051 0.0387987
\(37\) 172.426 0.766124 0.383062 − 0.923723i \(-0.374870\pi\)
0.383062 + 0.923723i \(0.374870\pi\)
\(38\) 371.649 1.58656
\(39\) 70.8448 0.290878
\(40\) 0 0
\(41\) 58.1492 0.221497 0.110749 − 0.993848i \(-0.464675\pi\)
0.110749 + 0.993848i \(0.464675\pi\)
\(42\) 84.4800 0.310370
\(43\) 210.100 0.745116 0.372558 − 0.928009i \(-0.378481\pi\)
0.372558 + 0.928009i \(0.378481\pi\)
\(44\) 4.25508 0.0145790
\(45\) 0 0
\(46\) 17.9823 0.0576381
\(47\) −151.570 −0.470398 −0.235199 − 0.971947i \(-0.575574\pi\)
−0.235199 + 0.971947i \(0.575574\pi\)
\(48\) 140.334 0.421988
\(49\) −167.291 −0.487729
\(50\) 0 0
\(51\) −81.0925 −0.222651
\(52\) 11.8645 0.0316406
\(53\) 318.384 0.825159 0.412579 − 0.910922i \(-0.364628\pi\)
0.412579 + 0.910922i \(0.364628\pi\)
\(54\) −310.150 −0.781595
\(55\) 0 0
\(56\) 306.745 0.731973
\(57\) 311.117 0.722955
\(58\) 83.6186 0.189304
\(59\) 414.677 0.915023 0.457512 − 0.889204i \(-0.348741\pi\)
0.457512 + 0.889204i \(0.348741\pi\)
\(60\) 0 0
\(61\) 498.088 1.04547 0.522735 − 0.852495i \(-0.324912\pi\)
0.522735 + 0.852495i \(0.324912\pi\)
\(62\) −915.511 −1.87532
\(63\) −287.179 −0.574303
\(64\) 534.305 1.04356
\(65\) 0 0
\(66\) −70.1051 −0.130748
\(67\) 639.464 1.16601 0.583007 − 0.812467i \(-0.301876\pi\)
0.583007 + 0.812467i \(0.301876\pi\)
\(68\) −13.5807 −0.0242191
\(69\) 15.0535 0.0262642
\(70\) 0 0
\(71\) −267.420 −0.446998 −0.223499 − 0.974704i \(-0.571748\pi\)
−0.223499 + 0.974704i \(0.571748\pi\)
\(72\) −501.344 −0.820610
\(73\) 540.297 0.866260 0.433130 − 0.901331i \(-0.357409\pi\)
0.433130 + 0.901331i \(0.357409\pi\)
\(74\) −475.756 −0.747373
\(75\) 0 0
\(76\) 52.1033 0.0786402
\(77\) −145.811 −0.215801
\(78\) −195.475 −0.283758
\(79\) 691.157 0.984319 0.492160 − 0.870505i \(-0.336208\pi\)
0.492160 + 0.870505i \(0.336208\pi\)
\(80\) 0 0
\(81\) 325.315 0.446249
\(82\) −160.445 −0.216076
\(83\) 663.244 0.877115 0.438557 − 0.898703i \(-0.355490\pi\)
0.438557 + 0.898703i \(0.355490\pi\)
\(84\) 11.8437 0.0153839
\(85\) 0 0
\(86\) −579.709 −0.726879
\(87\) 69.9994 0.0862612
\(88\) −254.550 −0.308354
\(89\) −119.542 −0.142376 −0.0711880 − 0.997463i \(-0.522679\pi\)
−0.0711880 + 0.997463i \(0.522679\pi\)
\(90\) 0 0
\(91\) −406.565 −0.468348
\(92\) 2.52103 0.00285691
\(93\) −766.399 −0.854536
\(94\) 418.211 0.458884
\(95\) 0 0
\(96\) 40.3987 0.0429497
\(97\) −284.049 −0.297328 −0.148664 − 0.988888i \(-0.547497\pi\)
−0.148664 + 0.988888i \(0.547497\pi\)
\(98\) 461.589 0.475792
\(99\) 238.313 0.241933
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 275.4.a.h.1.2 yes 5
3.2 odd 2 2475.4.a.bh.1.4 5
5.2 odd 4 275.4.b.f.199.4 10
5.3 odd 4 275.4.b.f.199.7 10
5.4 even 2 275.4.a.g.1.4 ✓ 5
15.14 odd 2 2475.4.a.bl.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.4.a.g.1.4 ✓ 5 5.4 even 2
275.4.a.h.1.2 yes 5 1.1 even 1 trivial
275.4.b.f.199.4 10 5.2 odd 4
275.4.b.f.199.7 10 5.3 odd 4
2475.4.a.bh.1.4 5 3.2 odd 2
2475.4.a.bl.1.2 5 15.14 odd 2