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Properties

Label 3225.2.a.be.1.4
Level $3225$
Weight $2$
Character 3225.1
Self dual yes
Analytic conductor $25.752$
Analytic rank $1$
Dimension $9$
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] = [3225,2,Mod(1,3225)] 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("3225.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3225 = 3 \cdot 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3225.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,-3,-9,13,0,3,-8,-6,9,0,1,-13,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(25.7517546519\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 11x^{7} + 36x^{6} + 29x^{5} - 120x^{4} - 13x^{3} + 127x^{2} - 4x - 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.52360\) of defining polynomial
Character \(\chi\) \(=\) 3225.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.52360 q^{2} -1.00000 q^{3} +0.321366 q^{4} +1.52360 q^{6} +2.34301 q^{7} +2.55757 q^{8} +1.00000 q^{9} +2.08395 q^{11} -0.321366 q^{12} +2.66696 q^{13} -3.56982 q^{14} -4.53946 q^{16} -7.74555 q^{17} -1.52360 q^{18} +2.53046 q^{19} -2.34301 q^{21} -3.17511 q^{22} -3.96722 q^{23} -2.55757 q^{24} -4.06339 q^{26} -1.00000 q^{27} +0.752963 q^{28} +6.99393 q^{29} -5.64286 q^{31} +1.80118 q^{32} -2.08395 q^{33} +11.8011 q^{34} +0.321366 q^{36} -5.09401 q^{37} -3.85542 q^{38} -2.66696 q^{39} +0.781322 q^{41} +3.56982 q^{42} +1.00000 q^{43} +0.669710 q^{44} +6.04446 q^{46} -12.6244 q^{47} +4.53946 q^{48} -1.51031 q^{49} +7.74555 q^{51} +0.857069 q^{52} -12.3269 q^{53} +1.52360 q^{54} +5.99242 q^{56} -2.53046 q^{57} -10.6560 q^{58} -12.5209 q^{59} -7.15946 q^{61} +8.59748 q^{62} +2.34301 q^{63} +6.33462 q^{64} +3.17511 q^{66} -3.32474 q^{67} -2.48915 q^{68} +3.96722 q^{69} +15.8379 q^{71} +2.55757 q^{72} +9.85036 q^{73} +7.76125 q^{74} +0.813204 q^{76} +4.88272 q^{77} +4.06339 q^{78} +15.2699 q^{79} +1.00000 q^{81} -1.19042 q^{82} +8.77190 q^{83} -0.752963 q^{84} -1.52360 q^{86} -6.99393 q^{87} +5.32986 q^{88} +12.3329 q^{89} +6.24871 q^{91} -1.27493 q^{92} +5.64286 q^{93} +19.2345 q^{94} -1.80118 q^{96} +3.52186 q^{97} +2.30111 q^{98} +2.08395 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{2} - 9 q^{3} + 13 q^{4} + 3 q^{6} - 8 q^{7} - 6 q^{8} + 9 q^{9} + q^{11} - 13 q^{12} - 3 q^{13} + 3 q^{14} + 21 q^{16} - 11 q^{17} - 3 q^{18} + 7 q^{19} + 8 q^{21} - 15 q^{22} - 30 q^{23} + 6 q^{24}+ \cdots + q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.52360 −1.07735 −0.538675 − 0.842514i \(-0.681075\pi\)
−0.538675 + 0.842514i \(0.681075\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.321366 0.160683
\(5\) 0 0
\(6\) 1.52360 0.622008
\(7\) 2.34301 0.885574 0.442787 − 0.896627i \(-0.353990\pi\)
0.442787 + 0.896627i \(0.353990\pi\)
\(8\) 2.55757 0.904238
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.08395 0.628335 0.314168 − 0.949368i \(-0.398275\pi\)
0.314168 + 0.949368i \(0.398275\pi\)
\(12\) −0.321366 −0.0927703
\(13\) 2.66696 0.739681 0.369841 − 0.929095i \(-0.379412\pi\)
0.369841 + 0.929095i \(0.379412\pi\)
\(14\) −3.56982 −0.954074
\(15\) 0 0
\(16\) −4.53946 −1.13486
\(17\) −7.74555 −1.87857 −0.939286 − 0.343135i \(-0.888511\pi\)
−0.939286 + 0.343135i \(0.888511\pi\)
\(18\) −1.52360 −0.359117
\(19\) 2.53046 0.580528 0.290264 − 0.956947i \(-0.406257\pi\)
0.290264 + 0.956947i \(0.406257\pi\)
\(20\) 0 0
\(21\) −2.34301 −0.511287
\(22\) −3.17511 −0.676937
\(23\) −3.96722 −0.827222 −0.413611 − 0.910454i \(-0.635733\pi\)
−0.413611 + 0.910454i \(0.635733\pi\)
\(24\) −2.55757 −0.522062
\(25\) 0 0
\(26\) −4.06339 −0.796896
\(27\) −1.00000 −0.192450
\(28\) 0.752963 0.142297
\(29\) 6.99393 1.29874 0.649370 − 0.760472i \(-0.275032\pi\)
0.649370 + 0.760472i \(0.275032\pi\)
\(30\) 0 0
\(31\) −5.64286 −1.01349 −0.506744 − 0.862097i \(-0.669151\pi\)
−0.506744 + 0.862097i \(0.669151\pi\)
\(32\) 1.80118 0.318407
\(33\) −2.08395 −0.362769
\(34\) 11.8011 2.02388
\(35\) 0 0
\(36\) 0.321366 0.0535609
\(37\) −5.09401 −0.837451 −0.418725 − 0.908113i \(-0.637523\pi\)
−0.418725 + 0.908113i \(0.637523\pi\)
\(38\) −3.85542 −0.625432
\(39\) −2.66696 −0.427055
\(40\) 0 0
\(41\) 0.781322 0.122022 0.0610110 − 0.998137i \(-0.480568\pi\)
0.0610110 + 0.998137i \(0.480568\pi\)
\(42\) 3.56982 0.550835
\(43\) 1.00000 0.152499
\(44\) 0.669710 0.100963
\(45\) 0 0
\(46\) 6.04446 0.891207
\(47\) −12.6244 −1.84146 −0.920728 − 0.390206i \(-0.872404\pi\)
−0.920728 + 0.390206i \(0.872404\pi\)
\(48\) 4.53946 0.655214
\(49\) −1.51031 −0.215758
\(50\) 0 0
\(51\) 7.74555 1.08459
\(52\) 0.857069 0.118854
\(53\) −12.3269 −1.69323 −0.846617 − 0.532203i \(-0.821364\pi\)
−0.846617 + 0.532203i \(0.821364\pi\)
\(54\) 1.52360 0.207336
\(55\) 0 0
\(56\) 5.99242 0.800770
\(57\) −2.53046 −0.335168
\(58\) −10.6560 −1.39920
\(59\) −12.5209 −1.63009 −0.815043 − 0.579401i \(-0.803287\pi\)
−0.815043 + 0.579401i \(0.803287\pi\)
\(60\) 0 0
\(61\) −7.15946 −0.916675 −0.458338 − 0.888778i \(-0.651555\pi\)
−0.458338 + 0.888778i \(0.651555\pi\)
\(62\) 8.59748 1.09188
\(63\) 2.34301 0.295191
\(64\) 6.33462 0.791828
\(65\) 0 0
\(66\) 3.17511 0.390830
\(67\) −3.32474 −0.406182 −0.203091 − 0.979160i \(-0.565099\pi\)
−0.203091 + 0.979160i \(0.565099\pi\)
\(68\) −2.48915 −0.301854
\(69\) 3.96722 0.477597
\(70\) 0 0
\(71\) 15.8379 1.87961 0.939804 − 0.341714i \(-0.111008\pi\)
0.939804 + 0.341714i \(0.111008\pi\)
\(72\) 2.55757 0.301413
\(73\) 9.85036 1.15290 0.576448 − 0.817134i \(-0.304438\pi\)
0.576448 + 0.817134i \(0.304438\pi\)
\(74\) 7.76125 0.902227
\(75\) 0 0
\(76\) 0.813204 0.0932809
\(77\) 4.88272 0.556437
\(78\) 4.06339 0.460088
\(79\) 15.2699 1.71800 0.859000 − 0.511975i \(-0.171086\pi\)
0.859000 + 0.511975i \(0.171086\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −1.19042 −0.131460
\(83\) 8.77190 0.962841 0.481421 − 0.876490i \(-0.340121\pi\)
0.481421 + 0.876490i \(0.340121\pi\)
\(84\) −0.752963 −0.0821550
\(85\) 0 0
\(86\) −1.52360 −0.164294
\(87\) −6.99393 −0.749828
\(88\) 5.32986 0.568165
\(89\) 12.3329 1.30728 0.653641 − 0.756804i \(-0.273240\pi\)
0.653641 + 0.756804i \(0.273240\pi\)
\(90\) 0 0
\(91\) 6.24871 0.655043
\(92\) −1.27493 −0.132920
\(93\) 5.64286 0.585138
\(94\) 19.2345 1.98389
\(95\) 0 0
\(96\) −1.80118 −0.183832
\(97\) 3.52186 0.357591 0.178796 − 0.983886i \(-0.442780\pi\)
0.178796 + 0.983886i \(0.442780\pi\)
\(98\) 2.30111 0.232447
\(99\) 2.08395 0.209445
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 3225.2.a.be.1.4 ✓ 9
3.2 odd 2 9675.2.a.ct.1.6 9
5.4 even 2 3225.2.a.bf.1.6 yes 9
15.14 odd 2 9675.2.a.cs.1.4 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3225.2.a.be.1.4 ✓ 9 1.1 even 1 trivial
3225.2.a.bf.1.6 yes 9 5.4 even 2
9675.2.a.cs.1.4 9 15.14 odd 2
9675.2.a.ct.1.6 9 3.2 odd 2