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Properties

Label 414.2.a.e.1.2
Level $414$
Weight $2$
Character 414.1
Self dual yes
Analytic conductor $3.306$
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] = [414,2,Mod(1,414)] 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("414.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(414, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 414.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.30580664368\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.64575\) of defining polynomial
Character \(\chi\) \(=\) 414.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.64575 q^{5} +2.00000 q^{7} -1.00000 q^{8} -1.64575 q^{10} -1.64575 q^{11} +5.29150 q^{13} -2.00000 q^{14} +1.00000 q^{16} -3.29150 q^{17} +0.354249 q^{19} +1.64575 q^{20} +1.64575 q^{22} +1.00000 q^{23} -2.29150 q^{25} -5.29150 q^{26} +2.00000 q^{28} +9.29150 q^{29} -1.29150 q^{31} -1.00000 q^{32} +3.29150 q^{34} +3.29150 q^{35} +6.93725 q^{37} -0.354249 q^{38} -1.64575 q^{40} +6.00000 q^{41} +0.354249 q^{43} -1.64575 q^{44} -1.00000 q^{46} +6.00000 q^{47} -3.00000 q^{49} +2.29150 q^{50} +5.29150 q^{52} -1.64575 q^{53} -2.70850 q^{55} -2.00000 q^{56} -9.29150 q^{58} +0.354249 q^{61} +1.29150 q^{62} +1.00000 q^{64} +8.70850 q^{65} -14.9373 q^{67} -3.29150 q^{68} -3.29150 q^{70} -6.00000 q^{71} -7.29150 q^{73} -6.93725 q^{74} +0.354249 q^{76} -3.29150 q^{77} +8.58301 q^{79} +1.64575 q^{80} -6.00000 q^{82} -13.6458 q^{83} -5.41699 q^{85} -0.354249 q^{86} +1.64575 q^{88} +10.5830 q^{91} +1.00000 q^{92} -6.00000 q^{94} +0.583005 q^{95} -1.29150 q^{97} +3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8} + 2 q^{10} + 2 q^{11} - 4 q^{14} + 2 q^{16} + 4 q^{17} + 6 q^{19} - 2 q^{20} - 2 q^{22} + 2 q^{23} + 6 q^{25} + 4 q^{28} + 8 q^{29} + 8 q^{31} - 2 q^{32}+ \cdots + 6 q^{98}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.64575 0.736002 0.368001 − 0.929825i \(-0.380042\pi\)
0.368001 + 0.929825i \(0.380042\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.64575 −0.520432
\(11\) −1.64575 −0.496213 −0.248106 − 0.968733i \(-0.579808\pi\)
−0.248106 + 0.968733i \(0.579808\pi\)
\(12\) 0 0
\(13\) 5.29150 1.46760 0.733799 − 0.679366i \(-0.237745\pi\)
0.733799 + 0.679366i \(0.237745\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.29150 −0.798307 −0.399153 − 0.916884i \(-0.630696\pi\)
−0.399153 + 0.916884i \(0.630696\pi\)
\(18\) 0 0
\(19\) 0.354249 0.0812702 0.0406351 − 0.999174i \(-0.487062\pi\)
0.0406351 + 0.999174i \(0.487062\pi\)
\(20\) 1.64575 0.368001
\(21\) 0 0
\(22\) 1.64575 0.350875
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −2.29150 −0.458301
\(26\) −5.29150 −1.03775
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) 9.29150 1.72539 0.862694 − 0.505726i \(-0.168775\pi\)
0.862694 + 0.505726i \(0.168775\pi\)
\(30\) 0 0
\(31\) −1.29150 −0.231961 −0.115980 − 0.993252i \(-0.537001\pi\)
−0.115980 + 0.993252i \(0.537001\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.29150 0.564488
\(35\) 3.29150 0.556365
\(36\) 0 0
\(37\) 6.93725 1.14048 0.570239 − 0.821479i \(-0.306851\pi\)
0.570239 + 0.821479i \(0.306851\pi\)
\(38\) −0.354249 −0.0574667
\(39\) 0 0
\(40\) −1.64575 −0.260216
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 0.354249 0.0540224 0.0270112 − 0.999635i \(-0.491401\pi\)
0.0270112 + 0.999635i \(0.491401\pi\)
\(44\) −1.64575 −0.248106
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 2.29150 0.324067
\(51\) 0 0
\(52\) 5.29150 0.733799
\(53\) −1.64575 −0.226061 −0.113031 − 0.993592i \(-0.536056\pi\)
−0.113031 + 0.993592i \(0.536056\pi\)
\(54\) 0 0
\(55\) −2.70850 −0.365214
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) −9.29150 −1.22003
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0.354249 0.0453569 0.0226784 − 0.999743i \(-0.492781\pi\)
0.0226784 + 0.999743i \(0.492781\pi\)
\(62\) 1.29150 0.164021
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 8.70850 1.08016
\(66\) 0 0
\(67\) −14.9373 −1.82488 −0.912438 − 0.409215i \(-0.865803\pi\)
−0.912438 + 0.409215i \(0.865803\pi\)
\(68\) −3.29150 −0.399153
\(69\) 0 0
\(70\) −3.29150 −0.393410
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −7.29150 −0.853406 −0.426703 − 0.904392i \(-0.640325\pi\)
−0.426703 + 0.904392i \(0.640325\pi\)
\(74\) −6.93725 −0.806439
\(75\) 0 0
\(76\) 0.354249 0.0406351
\(77\) −3.29150 −0.375102
\(78\) 0 0
\(79\) 8.58301 0.965664 0.482832 − 0.875713i \(-0.339608\pi\)
0.482832 + 0.875713i \(0.339608\pi\)
\(80\) 1.64575 0.184001
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −13.6458 −1.49782 −0.748908 − 0.662674i \(-0.769421\pi\)
−0.748908 + 0.662674i \(0.769421\pi\)
\(84\) 0 0
\(85\) −5.41699 −0.587556
\(86\) −0.354249 −0.0381996
\(87\) 0 0
\(88\) 1.64575 0.175438
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 10.5830 1.10940
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 0.583005 0.0598151
\(96\) 0 0
\(97\) −1.29150 −0.131132 −0.0655661 − 0.997848i \(-0.520885\pi\)
−0.0655661 + 0.997848i \(0.520885\pi\)
\(98\) 3.00000 0.303046
\(99\) 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 414.2.a.e.1.2 ✓ 2
3.2 odd 2 414.2.a.g.1.1 yes 2
4.3 odd 2 3312.2.a.v.1.2 2
12.11 even 2 3312.2.a.z.1.1 2
23.22 odd 2 9522.2.a.bb.1.1 2
69.68 even 2 9522.2.a.bc.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
414.2.a.e.1.2 ✓ 2 1.1 even 1 trivial
414.2.a.g.1.1 yes 2 3.2 odd 2
3312.2.a.v.1.2 2 4.3 odd 2
3312.2.a.z.1.1 2 12.11 even 2
9522.2.a.bb.1.1 2 23.22 odd 2
9522.2.a.bc.1.2 2 69.68 even 2