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Properties

Label 275.4.a.i.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 = [5,2,12] 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: \( x^{5} - 2x^{4} - 24x^{3} + 31x^{2} + 108x - 84 \) 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 \(-2.28165\) 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)
 
\(f(q)\) \(=\) \(q-2.28165 q^{2} +2.58679 q^{3} -2.79408 q^{4} -5.90215 q^{6} -27.1421 q^{7} +24.6283 q^{8} -20.3085 q^{9} +11.0000 q^{11} -7.22769 q^{12} -9.09567 q^{13} +61.9287 q^{14} -33.8405 q^{16} +91.1457 q^{17} +46.3369 q^{18} -80.9261 q^{19} -70.2109 q^{21} -25.0981 q^{22} +208.001 q^{23} +63.7082 q^{24} +20.7531 q^{26} -122.377 q^{27} +75.8371 q^{28} +136.309 q^{29} -213.838 q^{31} -119.814 q^{32} +28.4547 q^{33} -207.963 q^{34} +56.7436 q^{36} +351.681 q^{37} +184.645 q^{38} -23.5286 q^{39} -1.21191 q^{41} +160.197 q^{42} +231.165 q^{43} -30.7348 q^{44} -474.586 q^{46} +283.227 q^{47} -87.5383 q^{48} +393.693 q^{49} +235.775 q^{51} +25.4140 q^{52} -238.248 q^{53} +279.222 q^{54} -668.463 q^{56} -209.339 q^{57} -311.010 q^{58} +740.562 q^{59} +446.493 q^{61} +487.904 q^{62} +551.216 q^{63} +544.098 q^{64} -64.9236 q^{66} +56.2007 q^{67} -254.668 q^{68} +538.056 q^{69} +684.245 q^{71} -500.164 q^{72} +428.030 q^{73} -802.412 q^{74} +226.114 q^{76} -298.563 q^{77} +53.6840 q^{78} -1262.60 q^{79} +231.766 q^{81} +2.76516 q^{82} +147.237 q^{83} +196.175 q^{84} -527.438 q^{86} +352.603 q^{87} +270.911 q^{88} +305.209 q^{89} +246.875 q^{91} -581.172 q^{92} -553.155 q^{93} -646.225 q^{94} -309.934 q^{96} +205.445 q^{97} -898.269 q^{98} -223.394 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + 12 q^{3} + 12 q^{4} + 8 q^{6} + 24 q^{7} + 27 q^{8} + 31 q^{9} + 55 q^{11} + 3 q^{12} + 111 q^{13} + 47 q^{14} - 56 q^{16} + 40 q^{17} + 217 q^{18} + 205 q^{19} - 94 q^{21} + 22 q^{22} + 287 q^{23}+ \cdots + 341 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.28165 −0.806685 −0.403342 − 0.915049i \(-0.632152\pi\)
−0.403342 + 0.915049i \(0.632152\pi\)
\(3\) 2.58679 0.497828 0.248914 − 0.968526i \(-0.419926\pi\)
0.248914 + 0.968526i \(0.419926\pi\)
\(4\) −2.79408 −0.349260
\(5\) 0 0
\(6\) −5.90215 −0.401590
\(7\) −27.1421 −1.46553 −0.732767 − 0.680479i \(-0.761772\pi\)
−0.732767 + 0.680479i \(0.761772\pi\)
\(8\) 24.6283 1.08843
\(9\) −20.3085 −0.752167
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) −7.22769 −0.173871
\(13\) −9.09567 −0.194053 −0.0970263 − 0.995282i \(-0.530933\pi\)
−0.0970263 + 0.995282i \(0.530933\pi\)
\(14\) 61.9287 1.18222
\(15\) 0 0
\(16\) −33.8405 −0.528758
\(17\) 91.1457 1.30036 0.650179 − 0.759781i \(-0.274694\pi\)
0.650179 + 0.759781i \(0.274694\pi\)
\(18\) 46.3369 0.606762
\(19\) −80.9261 −0.977143 −0.488571 − 0.872524i \(-0.662482\pi\)
−0.488571 + 0.872524i \(0.662482\pi\)
\(20\) 0 0
\(21\) −70.2109 −0.729584
\(22\) −25.0981 −0.243225
\(23\) 208.001 1.88571 0.942854 − 0.333207i \(-0.108131\pi\)
0.942854 + 0.333207i \(0.108131\pi\)
\(24\) 63.7082 0.541849
\(25\) 0 0
\(26\) 20.7531 0.156539
\(27\) −122.377 −0.872278
\(28\) 75.8371 0.511852
\(29\) 136.309 0.872827 0.436414 − 0.899746i \(-0.356248\pi\)
0.436414 + 0.899746i \(0.356248\pi\)
\(30\) 0 0
\(31\) −213.838 −1.23892 −0.619459 − 0.785029i \(-0.712648\pi\)
−0.619459 + 0.785029i \(0.712648\pi\)
\(32\) −119.814 −0.661886
\(33\) 28.4547 0.150101
\(34\) −207.963 −1.04898
\(35\) 0 0
\(36\) 56.7436 0.262702
\(37\) 351.681 1.56259 0.781297 − 0.624159i \(-0.214558\pi\)
0.781297 + 0.624159i \(0.214558\pi\)
\(38\) 184.645 0.788246
\(39\) −23.5286 −0.0966048
\(40\) 0 0
\(41\) −1.21191 −0.00461632 −0.00230816 − 0.999997i \(-0.500735\pi\)
−0.00230816 + 0.999997i \(0.500735\pi\)
\(42\) 160.197 0.588544
\(43\) 231.165 0.819823 0.409911 − 0.912125i \(-0.365560\pi\)
0.409911 + 0.912125i \(0.365560\pi\)
\(44\) −30.7348 −0.105306
\(45\) 0 0
\(46\) −474.586 −1.52117
\(47\) 283.227 0.878999 0.439499 − 0.898243i \(-0.355156\pi\)
0.439499 + 0.898243i \(0.355156\pi\)
\(48\) −87.5383 −0.263231
\(49\) 393.693 1.14779
\(50\) 0 0
\(51\) 235.775 0.647354
\(52\) 25.4140 0.0677747
\(53\) −238.248 −0.617470 −0.308735 − 0.951148i \(-0.599906\pi\)
−0.308735 + 0.951148i \(0.599906\pi\)
\(54\) 279.222 0.703653
\(55\) 0 0
\(56\) −668.463 −1.59513
\(57\) −209.339 −0.486449
\(58\) −311.010 −0.704097
\(59\) 740.562 1.63412 0.817059 − 0.576554i \(-0.195603\pi\)
0.817059 + 0.576554i \(0.195603\pi\)
\(60\) 0 0
\(61\) 446.493 0.937172 0.468586 − 0.883418i \(-0.344764\pi\)
0.468586 + 0.883418i \(0.344764\pi\)
\(62\) 487.904 0.999417
\(63\) 551.216 1.10233
\(64\) 544.098 1.06269
\(65\) 0 0
\(66\) −64.9236 −0.121084
\(67\) 56.2007 0.102478 0.0512389 − 0.998686i \(-0.483683\pi\)
0.0512389 + 0.998686i \(0.483683\pi\)
\(68\) −254.668 −0.454162
\(69\) 538.056 0.938758
\(70\) 0 0
\(71\) 684.245 1.14373 0.571866 − 0.820347i \(-0.306220\pi\)
0.571866 + 0.820347i \(0.306220\pi\)
\(72\) −500.164 −0.818679
\(73\) 428.030 0.686262 0.343131 − 0.939287i \(-0.388512\pi\)
0.343131 + 0.939287i \(0.388512\pi\)
\(74\) −802.412 −1.26052
\(75\) 0 0
\(76\) 226.114 0.341276
\(77\) −298.563 −0.441875
\(78\) 53.6840 0.0779296
\(79\) −1262.60 −1.79815 −0.899076 − 0.437793i \(-0.855760\pi\)
−0.899076 + 0.437793i \(0.855760\pi\)
\(80\) 0 0
\(81\) 231.766 0.317923
\(82\) 2.76516 0.00372391
\(83\) 147.237 0.194715 0.0973575 − 0.995249i \(-0.468961\pi\)
0.0973575 + 0.995249i \(0.468961\pi\)
\(84\) 196.175 0.254814
\(85\) 0 0
\(86\) −527.438 −0.661338
\(87\) 352.603 0.434518
\(88\) 270.911 0.328173
\(89\) 305.209 0.363506 0.181753 − 0.983344i \(-0.441823\pi\)
0.181753 + 0.983344i \(0.441823\pi\)
\(90\) 0 0
\(91\) 246.875 0.284391
\(92\) −581.172 −0.658601
\(93\) −553.155 −0.616768
\(94\) −646.225 −0.709075
\(95\) 0 0
\(96\) −309.934 −0.329505
\(97\) 205.445 0.215049 0.107524 − 0.994202i \(-0.465708\pi\)
0.107524 + 0.994202i \(0.465708\pi\)
\(98\) −898.269 −0.925907
\(99\) −223.394 −0.226787
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.i.1.2 yes 5
3.2 odd 2 2475.4.a.bg.1.4 5
5.2 odd 4 275.4.b.g.199.4 10
5.3 odd 4 275.4.b.g.199.7 10
5.4 even 2 275.4.a.f.1.4 ✓ 5
15.14 odd 2 2475.4.a.bk.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.4.a.f.1.4 ✓ 5 5.4 even 2
275.4.a.i.1.2 yes 5 1.1 even 1 trivial
275.4.b.g.199.4 10 5.2 odd 4
275.4.b.g.199.7 10 5.3 odd 4
2475.4.a.bg.1.4 5 3.2 odd 2
2475.4.a.bk.1.2 5 15.14 odd 2