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Properties

Label 1450.2.a.j.1.2
Level $1450$
Weight $2$
Character 1450.1
Self dual yes
Analytic conductor $11.578$
Analytic rank $0$
Dimension $2$
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] = [1450,2,Mod(1,1450)] 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("1450.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,2,0,0,-4] 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: yes
Analytic conductor: \(11.5783082931\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) 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 \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 1450.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} -4.44949 q^{7} -1.00000 q^{8} +3.00000 q^{9} +2.00000 q^{11} +2.44949 q^{12} +2.44949 q^{13} +4.44949 q^{14} +1.00000 q^{16} +2.00000 q^{17} -3.00000 q^{18} +6.44949 q^{19} -10.8990 q^{21} -2.00000 q^{22} +2.44949 q^{23} -2.44949 q^{24} -2.44949 q^{26} -4.44949 q^{28} -1.00000 q^{29} -3.00000 q^{31} -1.00000 q^{32} +4.89898 q^{33} -2.00000 q^{34} +3.00000 q^{36} +3.44949 q^{37} -6.44949 q^{38} +6.00000 q^{39} +11.3485 q^{41} +10.8990 q^{42} +8.89898 q^{43} +2.00000 q^{44} -2.44949 q^{46} -5.89898 q^{47} +2.44949 q^{48} +12.7980 q^{49} +4.89898 q^{51} +2.44949 q^{52} -10.4495 q^{53} +4.44949 q^{56} +15.7980 q^{57} +1.00000 q^{58} +8.55051 q^{59} +3.44949 q^{61} +3.00000 q^{62} -13.3485 q^{63} +1.00000 q^{64} -4.89898 q^{66} +13.4495 q^{67} +2.00000 q^{68} +6.00000 q^{69} +11.3485 q^{71} -3.00000 q^{72} -13.3485 q^{73} -3.44949 q^{74} +6.44949 q^{76} -8.89898 q^{77} -6.00000 q^{78} -2.89898 q^{79} -9.00000 q^{81} -11.3485 q^{82} +6.00000 q^{83} -10.8990 q^{84} -8.89898 q^{86} -2.44949 q^{87} -2.00000 q^{88} +3.55051 q^{89} -10.8990 q^{91} +2.44949 q^{92} -7.34847 q^{93} +5.89898 q^{94} -2.44949 q^{96} -7.34847 q^{97} -12.7980 q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{7} - 2 q^{8} + 6 q^{9} + 4 q^{11} + 4 q^{14} + 2 q^{16} + 4 q^{17} - 6 q^{18} + 8 q^{19} - 12 q^{21} - 4 q^{22} - 4 q^{28} - 2 q^{29} - 6 q^{31} - 2 q^{32} - 4 q^{34} + 6 q^{36}+ \cdots + 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 2.44949 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −2.44949 −1.00000
\(7\) −4.44949 −1.68175 −0.840875 − 0.541230i \(-0.817959\pi\)
−0.840875 + 0.541230i \(0.817959\pi\)
\(8\) −1.00000 −0.353553
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 2.44949 0.707107
\(13\) 2.44949 0.679366 0.339683 − 0.940540i \(-0.389680\pi\)
0.339683 + 0.940540i \(0.389680\pi\)
\(14\) 4.44949 1.18918
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −3.00000 −0.707107
\(19\) 6.44949 1.47961 0.739807 − 0.672819i \(-0.234917\pi\)
0.739807 + 0.672819i \(0.234917\pi\)
\(20\) 0 0
\(21\) −10.8990 −2.37835
\(22\) −2.00000 −0.426401
\(23\) 2.44949 0.510754 0.255377 − 0.966842i \(-0.417800\pi\)
0.255377 + 0.966842i \(0.417800\pi\)
\(24\) −2.44949 −0.500000
\(25\) 0 0
\(26\) −2.44949 −0.480384
\(27\) 0 0
\(28\) −4.44949 −0.840875
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 − 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.89898 0.852803
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) 3.44949 0.567093 0.283546 − 0.958959i \(-0.408489\pi\)
0.283546 + 0.958959i \(0.408489\pi\)
\(38\) −6.44949 −1.04625
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) 11.3485 1.77233 0.886167 − 0.463367i \(-0.153359\pi\)
0.886167 + 0.463367i \(0.153359\pi\)
\(42\) 10.8990 1.68175
\(43\) 8.89898 1.35708 0.678541 − 0.734563i \(-0.262613\pi\)
0.678541 + 0.734563i \(0.262613\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −2.44949 −0.361158
\(47\) −5.89898 −0.860455 −0.430227 − 0.902721i \(-0.641567\pi\)
−0.430227 + 0.902721i \(0.641567\pi\)
\(48\) 2.44949 0.353553
\(49\) 12.7980 1.82828
\(50\) 0 0
\(51\) 4.89898 0.685994
\(52\) 2.44949 0.339683
\(53\) −10.4495 −1.43535 −0.717674 − 0.696379i \(-0.754793\pi\)
−0.717674 + 0.696379i \(0.754793\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.44949 0.594588
\(57\) 15.7980 2.09249
\(58\) 1.00000 0.131306
\(59\) 8.55051 1.11318 0.556591 − 0.830787i \(-0.312109\pi\)
0.556591 + 0.830787i \(0.312109\pi\)
\(60\) 0 0
\(61\) 3.44949 0.441662 0.220831 − 0.975312i \(-0.429123\pi\)
0.220831 + 0.975312i \(0.429123\pi\)
\(62\) 3.00000 0.381000
\(63\) −13.3485 −1.68175
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.89898 −0.603023
\(67\) 13.4495 1.64312 0.821558 − 0.570124i \(-0.193105\pi\)
0.821558 + 0.570124i \(0.193105\pi\)
\(68\) 2.00000 0.242536
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 11.3485 1.34682 0.673408 − 0.739271i \(-0.264830\pi\)
0.673408 + 0.739271i \(0.264830\pi\)
\(72\) −3.00000 −0.353553
\(73\) −13.3485 −1.56232 −0.781160 − 0.624331i \(-0.785372\pi\)
−0.781160 + 0.624331i \(0.785372\pi\)
\(74\) −3.44949 −0.400995
\(75\) 0 0
\(76\) 6.44949 0.739807
\(77\) −8.89898 −1.01413
\(78\) −6.00000 −0.679366
\(79\) −2.89898 −0.326161 −0.163080 − 0.986613i \(-0.552143\pi\)
−0.163080 + 0.986613i \(0.552143\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) −11.3485 −1.25323
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) −10.8990 −1.18918
\(85\) 0 0
\(86\) −8.89898 −0.959602
\(87\) −2.44949 −0.262613
\(88\) −2.00000 −0.213201
\(89\) 3.55051 0.376353 0.188177 − 0.982135i \(-0.439742\pi\)
0.188177 + 0.982135i \(0.439742\pi\)
\(90\) 0 0
\(91\) −10.8990 −1.14252
\(92\) 2.44949 0.255377
\(93\) −7.34847 −0.762001
\(94\) 5.89898 0.608433
\(95\) 0 0
\(96\) −2.44949 −0.250000
\(97\) −7.34847 −0.746124 −0.373062 − 0.927806i \(-0.621692\pi\)
−0.373062 + 0.927806i \(0.621692\pi\)
\(98\) −12.7980 −1.29279
\(99\) 6.00000 0.603023
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 1450.2.a.j.1.2 ✓ 2
5.2 odd 4 1450.2.b.i.349.1 4
5.3 odd 4 1450.2.b.i.349.4 4
5.4 even 2 1450.2.a.o.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1450.2.a.j.1.2 ✓ 2 1.1 even 1 trivial
1450.2.a.o.1.1 yes 2 5.4 even 2
1450.2.b.i.349.1 4 5.2 odd 4
1450.2.b.i.349.4 4 5.3 odd 4