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Properties

Label 3225.2.a.be.1.3
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 1.3
Root \(1.64847\) 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)
 
\(f(q)\) \(=\) \(q-1.64847 q^{2} -1.00000 q^{3} +0.717463 q^{4} +1.64847 q^{6} -3.70598 q^{7} +2.11423 q^{8} +1.00000 q^{9} +3.74249 q^{11} -0.717463 q^{12} -4.69564 q^{13} +6.10921 q^{14} -4.92017 q^{16} -1.77418 q^{17} -1.64847 q^{18} +0.553304 q^{19} +3.70598 q^{21} -6.16940 q^{22} +7.66063 q^{23} -2.11423 q^{24} +7.74063 q^{26} -1.00000 q^{27} -2.65890 q^{28} -5.10440 q^{29} +1.87636 q^{31} +3.88231 q^{32} -3.74249 q^{33} +2.92469 q^{34} +0.717463 q^{36} -3.65420 q^{37} -0.912106 q^{38} +4.69564 q^{39} +1.93308 q^{41} -6.10921 q^{42} +1.00000 q^{43} +2.68510 q^{44} -12.6283 q^{46} +1.10582 q^{47} +4.92017 q^{48} +6.73430 q^{49} +1.77418 q^{51} -3.36894 q^{52} -0.730761 q^{53} +1.64847 q^{54} -7.83529 q^{56} -0.553304 q^{57} +8.41447 q^{58} +10.2249 q^{59} +6.39382 q^{61} -3.09313 q^{62} -3.70598 q^{63} +3.44046 q^{64} +6.16940 q^{66} -2.07793 q^{67} -1.27291 q^{68} -7.66063 q^{69} -11.9514 q^{71} +2.11423 q^{72} +16.5205 q^{73} +6.02385 q^{74} +0.396975 q^{76} -13.8696 q^{77} -7.74063 q^{78} +2.62403 q^{79} +1.00000 q^{81} -3.18663 q^{82} -7.93274 q^{83} +2.65890 q^{84} -1.64847 q^{86} +5.10440 q^{87} +7.91248 q^{88} -9.14369 q^{89} +17.4019 q^{91} +5.49621 q^{92} -1.87636 q^{93} -1.82292 q^{94} -3.88231 q^{96} +0.214169 q^{97} -11.1013 q^{98} +3.74249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{2} - 9 q^{3} + 13 q^{4} + 3 q^{6} - 8 q^{7} - 6 q^{8} + 9 q^{9} + q^{11} - 13 q^{12} - 3 q^{13} + 3 q^{14} + 21 q^{16} - 11 q^{17} - 3 q^{18} + 7 q^{19} + 8 q^{21} - 15 q^{22} - 30 q^{23} + 6 q^{24}+ \cdots + q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.64847 −1.16565 −0.582823 − 0.812599i \(-0.698052\pi\)
−0.582823 + 0.812599i \(0.698052\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.717463 0.358731
\(5\) 0 0
\(6\) 1.64847 0.672986
\(7\) −3.70598 −1.40073 −0.700365 − 0.713785i \(-0.746979\pi\)
−0.700365 + 0.713785i \(0.746979\pi\)
\(8\) 2.11423 0.747492
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.74249 1.12840 0.564202 − 0.825637i \(-0.309184\pi\)
0.564202 + 0.825637i \(0.309184\pi\)
\(12\) −0.717463 −0.207114
\(13\) −4.69564 −1.30234 −0.651168 − 0.758934i \(-0.725721\pi\)
−0.651168 + 0.758934i \(0.725721\pi\)
\(14\) 6.10921 1.63276
\(15\) 0 0
\(16\) −4.92017 −1.23004
\(17\) −1.77418 −0.430302 −0.215151 − 0.976581i \(-0.569024\pi\)
−0.215151 + 0.976581i \(0.569024\pi\)
\(18\) −1.64847 −0.388549
\(19\) 0.553304 0.126937 0.0634683 − 0.997984i \(-0.479784\pi\)
0.0634683 + 0.997984i \(0.479784\pi\)
\(20\) 0 0
\(21\) 3.70598 0.808712
\(22\) −6.16940 −1.31532
\(23\) 7.66063 1.59735 0.798676 − 0.601762i \(-0.205534\pi\)
0.798676 + 0.601762i \(0.205534\pi\)
\(24\) −2.11423 −0.431565
\(25\) 0 0
\(26\) 7.74063 1.51806
\(27\) −1.00000 −0.192450
\(28\) −2.65890 −0.502486
\(29\) −5.10440 −0.947864 −0.473932 − 0.880561i \(-0.657166\pi\)
−0.473932 + 0.880561i \(0.657166\pi\)
\(30\) 0 0
\(31\) 1.87636 0.337005 0.168502 − 0.985701i \(-0.446107\pi\)
0.168502 + 0.985701i \(0.446107\pi\)
\(32\) 3.88231 0.686303
\(33\) −3.74249 −0.651484
\(34\) 2.92469 0.501580
\(35\) 0 0
\(36\) 0.717463 0.119577
\(37\) −3.65420 −0.600747 −0.300373 − 0.953822i \(-0.597111\pi\)
−0.300373 + 0.953822i \(0.597111\pi\)
\(38\) −0.912106 −0.147963
\(39\) 4.69564 0.751904
\(40\) 0 0
\(41\) 1.93308 0.301896 0.150948 − 0.988542i \(-0.451767\pi\)
0.150948 + 0.988542i \(0.451767\pi\)
\(42\) −6.10921 −0.942672
\(43\) 1.00000 0.152499
\(44\) 2.68510 0.404794
\(45\) 0 0
\(46\) −12.6283 −1.86195
\(47\) 1.10582 0.161301 0.0806504 − 0.996742i \(-0.474300\pi\)
0.0806504 + 0.996742i \(0.474300\pi\)
\(48\) 4.92017 0.710166
\(49\) 6.73430 0.962043
\(50\) 0 0
\(51\) 1.77418 0.248435
\(52\) −3.36894 −0.467188
\(53\) −0.730761 −0.100378 −0.0501889 − 0.998740i \(-0.515982\pi\)
−0.0501889 + 0.998740i \(0.515982\pi\)
\(54\) 1.64847 0.224329
\(55\) 0 0
\(56\) −7.83529 −1.04703
\(57\) −0.553304 −0.0732869
\(58\) 8.41447 1.10487
\(59\) 10.2249 1.33117 0.665584 − 0.746323i \(-0.268182\pi\)
0.665584 + 0.746323i \(0.268182\pi\)
\(60\) 0 0
\(61\) 6.39382 0.818645 0.409322 − 0.912390i \(-0.365765\pi\)
0.409322 + 0.912390i \(0.365765\pi\)
\(62\) −3.09313 −0.392828
\(63\) −3.70598 −0.466910
\(64\) 3.44046 0.430057
\(65\) 0 0
\(66\) 6.16940 0.759400
\(67\) −2.07793 −0.253860 −0.126930 − 0.991912i \(-0.540512\pi\)
−0.126930 + 0.991912i \(0.540512\pi\)
\(68\) −1.27291 −0.154363
\(69\) −7.66063 −0.922231
\(70\) 0 0
\(71\) −11.9514 −1.41837 −0.709186 − 0.705022i \(-0.750937\pi\)
−0.709186 + 0.705022i \(0.750937\pi\)
\(72\) 2.11423 0.249164
\(73\) 16.5205 1.93358 0.966791 − 0.255567i \(-0.0822622\pi\)
0.966791 + 0.255567i \(0.0822622\pi\)
\(74\) 6.02385 0.700258
\(75\) 0 0
\(76\) 0.396975 0.0455361
\(77\) −13.8696 −1.58059
\(78\) −7.74063 −0.876454
\(79\) 2.62403 0.295226 0.147613 − 0.989045i \(-0.452841\pi\)
0.147613 + 0.989045i \(0.452841\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −3.18663 −0.351904
\(83\) −7.93274 −0.870731 −0.435365 − 0.900254i \(-0.643381\pi\)
−0.435365 + 0.900254i \(0.643381\pi\)
\(84\) 2.65890 0.290110
\(85\) 0 0
\(86\) −1.64847 −0.177759
\(87\) 5.10440 0.547249
\(88\) 7.91248 0.843473
\(89\) −9.14369 −0.969229 −0.484615 − 0.874728i \(-0.661040\pi\)
−0.484615 + 0.874728i \(0.661040\pi\)
\(90\) 0 0
\(91\) 17.4019 1.82422
\(92\) 5.49621 0.573020
\(93\) −1.87636 −0.194570
\(94\) −1.82292 −0.188020
\(95\) 0 0
\(96\) −3.88231 −0.396237
\(97\) 0.214169 0.0217456 0.0108728 − 0.999941i \(-0.496539\pi\)
0.0108728 + 0.999941i \(0.496539\pi\)
\(98\) −11.1013 −1.12140
\(99\) 3.74249 0.376135
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.1.3 ✓ 9
3.2 odd 2 9675.2.a.ct.1.7 9
5.4 even 2 3225.2.a.bf.1.7 yes 9
15.14 odd 2 9675.2.a.cs.1.3 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3225.2.a.be.1.3 ✓ 9 1.1 even 1 trivial
3225.2.a.bf.1.7 yes 9 5.4 even 2
9675.2.a.cs.1.3 9 15.14 odd 2
9675.2.a.ct.1.7 9 3.2 odd 2