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Properties

Label 450.5.d.a.251.2
Level $450$
Weight $5$
Character 450.251
Analytic conductor $46.516$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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] = [450,5,Mod(251,450)] 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("450.251"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 450.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-16,0,0,-166] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(46.5164833877\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 251.2
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 450.251
Dual form 450.5.d.a.251.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.82843i q^{2} -8.00000 q^{4} -83.0000 q^{7} -22.6274i q^{8} +80.6102i q^{11} -41.0000 q^{13} -234.759i q^{14} +64.0000 q^{16} -513.360i q^{17} -139.000 q^{19} -228.000 q^{22} +224.860i q^{23} -115.966i q^{26} +664.000 q^{28} +674.580i q^{29} -1051.00 q^{31} +181.019i q^{32} +1452.00 q^{34} +1672.00 q^{37} -393.151i q^{38} +831.558i q^{41} +2515.00 q^{43} -644.881i q^{44} -636.000 q^{46} -2948.64i q^{47} +4488.00 q^{49} +328.000 q^{52} +390.323i q^{53} +1878.08i q^{56} -1908.00 q^{58} +750.947i q^{59} +5825.00 q^{61} -2972.68i q^{62} -512.000 q^{64} -7259.00 q^{67} +4106.88i q^{68} +6907.02i q^{71} +4552.00 q^{73} +4729.13i q^{74} +1112.00 q^{76} -6690.64i q^{77} +9296.00 q^{79} -2352.00 q^{82} -7980.41i q^{83} +7113.49i q^{86} +1824.00 q^{88} +6075.46i q^{89} +3403.00 q^{91} -1798.88i q^{92} +8340.00 q^{94} -1793.00 q^{97} +12694.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{4} - 166 q^{7} - 82 q^{13} + 128 q^{16} - 278 q^{19} - 456 q^{22} + 1328 q^{28} - 2102 q^{31} + 2904 q^{34} + 3344 q^{37} + 5030 q^{43} - 1272 q^{46} + 8976 q^{49} + 656 q^{52} - 3816 q^{58}+ \cdots - 3586 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.82843i 0.707107i
\(3\) 0 0
\(4\) −8.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −83.0000 −1.69388 −0.846939 − 0.531690i \(-0.821557\pi\)
−0.846939 + 0.531690i \(0.821557\pi\)
\(8\) − 22.6274i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 80.6102i 0.666200i 0.942892 + 0.333100i \(0.108095\pi\)
−0.942892 + 0.333100i \(0.891905\pi\)
\(12\) 0 0
\(13\) −41.0000 −0.242604 −0.121302 − 0.992616i \(-0.538707\pi\)
−0.121302 + 0.992616i \(0.538707\pi\)
\(14\) − 234.759i − 1.19775i
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) − 513.360i − 1.77633i −0.459524 − 0.888165i \(-0.651980\pi\)
0.459524 − 0.888165i \(-0.348020\pi\)
\(18\) 0 0
\(19\) −139.000 −0.385042 −0.192521 − 0.981293i \(-0.561666\pi\)
−0.192521 + 0.981293i \(0.561666\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −228.000 −0.471074
\(23\) 224.860i 0.425066i 0.977154 + 0.212533i \(0.0681713\pi\)
−0.977154 + 0.212533i \(0.931829\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) − 115.966i − 0.171547i
\(27\) 0 0
\(28\) 664.000 0.846939
\(29\) 674.580i 0.802116i 0.916053 + 0.401058i \(0.131357\pi\)
−0.916053 + 0.401058i \(0.868643\pi\)
\(30\) 0 0
\(31\) −1051.00 −1.09365 −0.546826 − 0.837246i \(-0.684164\pi\)
−0.546826 + 0.837246i \(0.684164\pi\)
\(32\) 181.019i 0.176777i
\(33\) 0 0
\(34\) 1452.00 1.25606
\(35\) 0 0
\(36\) 0 0
\(37\) 1672.00 1.22133 0.610665 − 0.791889i \(-0.290902\pi\)
0.610665 + 0.791889i \(0.290902\pi\)
\(38\) − 393.151i − 0.272265i
\(39\) 0 0
\(40\) 0 0
\(41\) 831.558i 0.494680i 0.968929 + 0.247340i \(0.0795565\pi\)
−0.968929 + 0.247340i \(0.920443\pi\)
\(42\) 0 0
\(43\) 2515.00 1.36019 0.680097 − 0.733122i \(-0.261937\pi\)
0.680097 + 0.733122i \(0.261937\pi\)
\(44\) − 644.881i − 0.333100i
\(45\) 0 0
\(46\) −636.000 −0.300567
\(47\) − 2948.64i − 1.33483i −0.744687 − 0.667414i \(-0.767401\pi\)
0.744687 − 0.667414i \(-0.232599\pi\)
\(48\) 0 0
\(49\) 4488.00 1.86922
\(50\) 0 0
\(51\) 0 0
\(52\) 328.000 0.121302
\(53\) 390.323i 0.138954i 0.997584 + 0.0694772i \(0.0221331\pi\)
−0.997584 + 0.0694772i \(0.977867\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1878.08i 0.598876i
\(57\) 0 0
\(58\) −1908.00 −0.567182
\(59\) 750.947i 0.215727i 0.994166 + 0.107864i \(0.0344010\pi\)
−0.994166 + 0.107864i \(0.965599\pi\)
\(60\) 0 0
\(61\) 5825.00 1.56544 0.782720 − 0.622374i \(-0.213832\pi\)
0.782720 + 0.622374i \(0.213832\pi\)
\(62\) − 2972.68i − 0.773329i
\(63\) 0 0
\(64\) −512.000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −7259.00 −1.61706 −0.808532 − 0.588452i \(-0.799737\pi\)
−0.808532 + 0.588452i \(0.799737\pi\)
\(68\) 4106.88i 0.888165i
\(69\) 0 0
\(70\) 0 0
\(71\) 6907.02i 1.37017i 0.728464 + 0.685084i \(0.240235\pi\)
−0.728464 + 0.685084i \(0.759765\pi\)
\(72\) 0 0
\(73\) 4552.00 0.854194 0.427097 − 0.904206i \(-0.359536\pi\)
0.427097 + 0.904206i \(0.359536\pi\)
\(74\) 4729.13i 0.863610i
\(75\) 0 0
\(76\) 1112.00 0.192521
\(77\) − 6690.64i − 1.12846i
\(78\) 0 0
\(79\) 9296.00 1.48950 0.744752 − 0.667341i \(-0.232568\pi\)
0.744752 + 0.667341i \(0.232568\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −2352.00 −0.349792
\(83\) − 7980.41i − 1.15843i −0.815176 − 0.579214i \(-0.803360\pi\)
0.815176 − 0.579214i \(-0.196640\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 7113.49i 0.961803i
\(87\) 0 0
\(88\) 1824.00 0.235537
\(89\) 6075.46i 0.767007i 0.923539 + 0.383503i \(0.125283\pi\)
−0.923539 + 0.383503i \(0.874717\pi\)
\(90\) 0 0
\(91\) 3403.00 0.410941
\(92\) − 1798.88i − 0.212533i
\(93\) 0 0
\(94\) 8340.00 0.943866
\(95\) 0 0
\(96\) 0 0
\(97\) −1793.00 −0.190562 −0.0952811 − 0.995450i \(-0.530375\pi\)
−0.0952811 + 0.995450i \(0.530375\pi\)
\(98\) 12694.0i 1.32174i
\(99\) 0 0
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 450.5.d.a.251.2 yes 2
3.2 odd 2 inner 450.5.d.a.251.1 ✓ 2
5.2 odd 4 450.5.b.b.449.1 4
5.3 odd 4 450.5.b.b.449.4 4
5.4 even 2 450.5.d.d.251.1 yes 2
15.2 even 4 450.5.b.b.449.3 4
15.8 even 4 450.5.b.b.449.2 4
15.14 odd 2 450.5.d.d.251.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
450.5.b.b.449.1 4 5.2 odd 4
450.5.b.b.449.2 4 15.8 even 4
450.5.b.b.449.3 4 15.2 even 4
450.5.b.b.449.4 4 5.3 odd 4
450.5.d.a.251.1 ✓ 2 3.2 odd 2 inner
450.5.d.a.251.2 yes 2 1.1 even 1 trivial
450.5.d.d.251.1 yes 2 5.4 even 2
450.5.d.d.251.2 yes 2 15.14 odd 2