Properties

Label 1150.4.a.w.1.3
Level $1150$
Weight $4$
Character 1150.1
Self dual yes
Analytic conductor $67.852$
Analytic rank $0$
Dimension $6$
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] = [1150,4,Mod(1,1150)] 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("1150.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-12,5] 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: \(67.8521965066\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 109x^{4} + 94x^{3} + 2808x^{2} + 81x - 9774 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.92662\) of defining polynomial
Character \(\chi\) \(=\) 1150.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -0.926620 q^{3} +4.00000 q^{4} +1.85324 q^{6} -28.2323 q^{7} -8.00000 q^{8} -26.1414 q^{9} -17.5955 q^{11} -3.70648 q^{12} +61.3331 q^{13} +56.4646 q^{14} +16.0000 q^{16} -41.1845 q^{17} +52.2828 q^{18} -162.872 q^{19} +26.1606 q^{21} +35.1911 q^{22} +23.0000 q^{23} +7.41296 q^{24} -122.666 q^{26} +49.2419 q^{27} -112.929 q^{28} -249.549 q^{29} -53.6871 q^{31} -32.0000 q^{32} +16.3044 q^{33} +82.3690 q^{34} -104.566 q^{36} -92.9111 q^{37} +325.744 q^{38} -56.8325 q^{39} -262.366 q^{41} -52.3212 q^{42} -234.613 q^{43} -70.3821 q^{44} -46.0000 q^{46} +297.656 q^{47} -14.8259 q^{48} +454.062 q^{49} +38.1624 q^{51} +245.332 q^{52} +142.966 q^{53} -98.4837 q^{54} +225.858 q^{56} +150.921 q^{57} +499.097 q^{58} -481.025 q^{59} -64.5317 q^{61} +107.374 q^{62} +738.031 q^{63} +64.0000 q^{64} -32.6087 q^{66} -820.850 q^{67} -164.738 q^{68} -21.3123 q^{69} +900.457 q^{71} +209.131 q^{72} +451.981 q^{73} +185.822 q^{74} -651.489 q^{76} +496.762 q^{77} +113.665 q^{78} +156.892 q^{79} +660.189 q^{81} +524.731 q^{82} -399.676 q^{83} +104.642 q^{84} +469.225 q^{86} +231.237 q^{87} +140.764 q^{88} +423.152 q^{89} -1731.57 q^{91} +92.0000 q^{92} +49.7475 q^{93} -595.311 q^{94} +29.6518 q^{96} +626.767 q^{97} -908.124 q^{98} +459.971 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{2} + 5 q^{3} + 24 q^{4} - 10 q^{6} + 42 q^{7} - 48 q^{8} + 61 q^{9} - 49 q^{11} + 20 q^{12} + 16 q^{13} - 84 q^{14} + 96 q^{16} + 175 q^{17} - 122 q^{18} - 229 q^{19} + 92 q^{21} + 98 q^{22}+ \cdots - 142 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) −0.926620 −0.178328 −0.0891640 − 0.996017i \(-0.528420\pi\)
−0.0891640 + 0.996017i \(0.528420\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) 1.85324 0.126097
\(7\) −28.2323 −1.52440 −0.762200 − 0.647341i \(-0.775881\pi\)
−0.762200 + 0.647341i \(0.775881\pi\)
\(8\) −8.00000 −0.353553
\(9\) −26.1414 −0.968199
\(10\) 0 0
\(11\) −17.5955 −0.482296 −0.241148 − 0.970488i \(-0.577524\pi\)
−0.241148 + 0.970488i \(0.577524\pi\)
\(12\) −3.70648 −0.0891640
\(13\) 61.3331 1.30852 0.654259 − 0.756270i \(-0.272980\pi\)
0.654259 + 0.756270i \(0.272980\pi\)
\(14\) 56.4646 1.07791
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −41.1845 −0.587571 −0.293785 − 0.955871i \(-0.594915\pi\)
−0.293785 + 0.955871i \(0.594915\pi\)
\(18\) 52.2828 0.684620
\(19\) −162.872 −1.96660 −0.983301 − 0.181987i \(-0.941747\pi\)
−0.983301 + 0.181987i \(0.941747\pi\)
\(20\) 0 0
\(21\) 26.1606 0.271843
\(22\) 35.1911 0.341034
\(23\) 23.0000 0.208514
\(24\) 7.41296 0.0630485
\(25\) 0 0
\(26\) −122.666 −0.925263
\(27\) 49.2419 0.350985
\(28\) −112.929 −0.762200
\(29\) −249.549 −1.59793 −0.798966 − 0.601377i \(-0.794619\pi\)
−0.798966 + 0.601377i \(0.794619\pi\)
\(30\) 0 0
\(31\) −53.6871 −0.311048 −0.155524 − 0.987832i \(-0.549707\pi\)
−0.155524 + 0.987832i \(0.549707\pi\)
\(32\) −32.0000 −0.176777
\(33\) 16.3044 0.0860068
\(34\) 82.3690 0.415475
\(35\) 0 0
\(36\) −104.566 −0.484100
\(37\) −92.9111 −0.412824 −0.206412 − 0.978465i \(-0.566179\pi\)
−0.206412 + 0.978465i \(0.566179\pi\)
\(38\) 325.744 1.39060
\(39\) −56.8325 −0.233346
\(40\) 0 0
\(41\) −262.366 −0.999382 −0.499691 − 0.866204i \(-0.666553\pi\)
−0.499691 + 0.866204i \(0.666553\pi\)
\(42\) −52.3212 −0.192222
\(43\) −234.613 −0.832048 −0.416024 − 0.909354i \(-0.636577\pi\)
−0.416024 + 0.909354i \(0.636577\pi\)
\(44\) −70.3821 −0.241148
\(45\) 0 0
\(46\) −46.0000 −0.147442
\(47\) 297.656 0.923777 0.461889 − 0.886938i \(-0.347172\pi\)
0.461889 + 0.886938i \(0.347172\pi\)
\(48\) −14.8259 −0.0445820
\(49\) 454.062 1.32380
\(50\) 0 0
\(51\) 38.1624 0.104780
\(52\) 245.332 0.654259
\(53\) 142.966 0.370525 0.185263 − 0.982689i \(-0.440686\pi\)
0.185263 + 0.982689i \(0.440686\pi\)
\(54\) −98.4837 −0.248184
\(55\) 0 0
\(56\) 225.858 0.538957
\(57\) 150.921 0.350700
\(58\) 499.097 1.12991
\(59\) −481.025 −1.06143 −0.530713 − 0.847551i \(-0.678076\pi\)
−0.530713 + 0.847551i \(0.678076\pi\)
\(60\) 0 0
\(61\) −64.5317 −0.135450 −0.0677249 − 0.997704i \(-0.521574\pi\)
−0.0677249 + 0.997704i \(0.521574\pi\)
\(62\) 107.374 0.219944
\(63\) 738.031 1.47592
\(64\) 64.0000 0.125000
\(65\) 0 0
\(66\) −32.6087 −0.0608160
\(67\) −820.850 −1.49676 −0.748379 − 0.663271i \(-0.769168\pi\)
−0.748379 + 0.663271i \(0.769168\pi\)
\(68\) −164.738 −0.293785
\(69\) −21.3123 −0.0371840
\(70\) 0 0
\(71\) 900.457 1.50514 0.752568 − 0.658515i \(-0.228815\pi\)
0.752568 + 0.658515i \(0.228815\pi\)
\(72\) 209.131 0.342310
\(73\) 451.981 0.724663 0.362332 − 0.932049i \(-0.381981\pi\)
0.362332 + 0.932049i \(0.381981\pi\)
\(74\) 185.822 0.291911
\(75\) 0 0
\(76\) −651.489 −0.983301
\(77\) 496.762 0.735212
\(78\) 113.665 0.165000
\(79\) 156.892 0.223439 0.111720 − 0.993740i \(-0.464364\pi\)
0.111720 + 0.993740i \(0.464364\pi\)
\(80\) 0 0
\(81\) 660.189 0.905609
\(82\) 524.731 0.706669
\(83\) −399.676 −0.528556 −0.264278 − 0.964447i \(-0.585134\pi\)
−0.264278 + 0.964447i \(0.585134\pi\)
\(84\) 104.642 0.135922
\(85\) 0 0
\(86\) 469.225 0.588347
\(87\) 231.237 0.284956
\(88\) 140.764 0.170517
\(89\) 423.152 0.503978 0.251989 − 0.967730i \(-0.418915\pi\)
0.251989 + 0.967730i \(0.418915\pi\)
\(90\) 0 0
\(91\) −1731.57 −1.99471
\(92\) 92.0000 0.104257
\(93\) 49.7475 0.0554686
\(94\) −595.311 −0.653209
\(95\) 0 0
\(96\) 29.6518 0.0315242
\(97\) 626.767 0.656067 0.328034 − 0.944666i \(-0.393614\pi\)
0.328034 + 0.944666i \(0.393614\pi\)
\(98\) −908.124 −0.936065
\(99\) 459.971 0.466958
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 1150.4.a.w.1.3 ✓ 6
5.2 odd 4 1150.4.b.s.599.4 12
5.3 odd 4 1150.4.b.s.599.9 12
5.4 even 2 1150.4.a.x.1.4 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.4.a.w.1.3 ✓ 6 1.1 even 1 trivial
1150.4.a.x.1.4 yes 6 5.4 even 2
1150.4.b.s.599.4 12 5.2 odd 4
1150.4.b.s.599.9 12 5.3 odd 4