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Properties

Label 7225.2.a.bq.1.3
Level $7225$
Weight $2$
Character 7225.1
Self dual yes
Analytic conductor $57.692$
Analytic rank $0$
Dimension $12$
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] = [7225,2,Mod(1,7225)] 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("7225.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7225 = 5^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7225.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,4,-8,12,0,8,-16] 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: \(57.6919154604\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 21 x^{8} - 232 x^{7} + 44 x^{6} + 424 x^{5} - 137 x^{4} + \cdots + 17 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.43840\) of defining polynomial
Character \(\chi\) \(=\) 7225.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.43840 q^{2} +0.109907 q^{3} +0.0689897 q^{4} -0.158090 q^{6} -0.695085 q^{7} +2.77756 q^{8} -2.98792 q^{9} +4.85089 q^{11} +0.00758244 q^{12} -5.63906 q^{13} +0.999809 q^{14} -4.13322 q^{16} +4.29782 q^{18} -2.32272 q^{19} -0.0763945 q^{21} -6.97750 q^{22} -4.63686 q^{23} +0.305273 q^{24} +8.11121 q^{26} -0.658113 q^{27} -0.0479537 q^{28} +6.50618 q^{29} +6.63194 q^{31} +0.390093 q^{32} +0.533145 q^{33} -0.206136 q^{36} -0.118625 q^{37} +3.34100 q^{38} -0.619770 q^{39} +1.07877 q^{41} +0.109886 q^{42} +0.641108 q^{43} +0.334661 q^{44} +6.66965 q^{46} -4.93703 q^{47} -0.454269 q^{48} -6.51686 q^{49} -0.389037 q^{52} -11.9864 q^{53} +0.946629 q^{54} -1.93064 q^{56} -0.255283 q^{57} -9.35848 q^{58} -9.91829 q^{59} +1.60292 q^{61} -9.53937 q^{62} +2.07686 q^{63} +7.70533 q^{64} -0.766875 q^{66} +2.99411 q^{67} -0.509622 q^{69} -4.68852 q^{71} -8.29913 q^{72} -5.49911 q^{73} +0.170630 q^{74} -0.160244 q^{76} -3.37178 q^{77} +0.891477 q^{78} +14.8439 q^{79} +8.89143 q^{81} -1.55170 q^{82} -5.03506 q^{83} -0.00527044 q^{84} -0.922169 q^{86} +0.715073 q^{87} +13.4736 q^{88} -2.35657 q^{89} +3.91962 q^{91} -0.319896 q^{92} +0.728895 q^{93} +7.10141 q^{94} +0.0428738 q^{96} +2.70080 q^{97} +9.37384 q^{98} -14.4941 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{2} - 8 q^{3} + 12 q^{4} + 8 q^{6} - 16 q^{7} + 12 q^{8} + 12 q^{9} + 16 q^{11} - 16 q^{12} + 8 q^{13} - 16 q^{14} + 12 q^{16} - 4 q^{18} + 16 q^{21} - 16 q^{22} - 16 q^{23} + 16 q^{26} - 32 q^{27}+ \cdots + 56 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.43840 −1.01710 −0.508551 − 0.861032i \(-0.669819\pi\)
−0.508551 + 0.861032i \(0.669819\pi\)
\(3\) 0.109907 0.0634547 0.0317274 − 0.999497i \(-0.489899\pi\)
0.0317274 + 0.999497i \(0.489899\pi\)
\(4\) 0.0689897 0.0344949
\(5\) 0 0
\(6\) −0.158090 −0.0645399
\(7\) −0.695085 −0.262717 −0.131359 − 0.991335i \(-0.541934\pi\)
−0.131359 + 0.991335i \(0.541934\pi\)
\(8\) 2.77756 0.982016
\(9\) −2.98792 −0.995974
\(10\) 0 0
\(11\) 4.85089 1.46260 0.731298 − 0.682058i \(-0.238915\pi\)
0.731298 + 0.682058i \(0.238915\pi\)
\(12\) 0.00758244 0.00218886
\(13\) −5.63906 −1.56399 −0.781996 − 0.623283i \(-0.785798\pi\)
−0.781996 + 0.623283i \(0.785798\pi\)
\(14\) 0.999809 0.267210
\(15\) 0 0
\(16\) −4.13322 −1.03330
\(17\) 0 0
\(18\) 4.29782 1.01301
\(19\) −2.32272 −0.532869 −0.266434 − 0.963853i \(-0.585846\pi\)
−0.266434 + 0.963853i \(0.585846\pi\)
\(20\) 0 0
\(21\) −0.0763945 −0.0166706
\(22\) −6.97750 −1.48761
\(23\) −4.63686 −0.966852 −0.483426 − 0.875385i \(-0.660608\pi\)
−0.483426 + 0.875385i \(0.660608\pi\)
\(24\) 0.305273 0.0623136
\(25\) 0 0
\(26\) 8.11121 1.59074
\(27\) −0.658113 −0.126654
\(28\) −0.0479537 −0.00906240
\(29\) 6.50618 1.20817 0.604084 − 0.796921i \(-0.293539\pi\)
0.604084 + 0.796921i \(0.293539\pi\)
\(30\) 0 0
\(31\) 6.63194 1.19113 0.595565 − 0.803307i \(-0.296928\pi\)
0.595565 + 0.803307i \(0.296928\pi\)
\(32\) 0.390093 0.0689593
\(33\) 0.533145 0.0928087
\(34\) 0 0
\(35\) 0 0
\(36\) −0.206136 −0.0343560
\(37\) −0.118625 −0.0195018 −0.00975091 − 0.999952i \(-0.503104\pi\)
−0.00975091 + 0.999952i \(0.503104\pi\)
\(38\) 3.34100 0.541981
\(39\) −0.619770 −0.0992427
\(40\) 0 0
\(41\) 1.07877 0.168475 0.0842375 − 0.996446i \(-0.473155\pi\)
0.0842375 + 0.996446i \(0.473155\pi\)
\(42\) 0.109886 0.0169557
\(43\) 0.641108 0.0977681 0.0488840 − 0.998804i \(-0.484434\pi\)
0.0488840 + 0.998804i \(0.484434\pi\)
\(44\) 0.334661 0.0504521
\(45\) 0 0
\(46\) 6.66965 0.983386
\(47\) −4.93703 −0.720139 −0.360070 − 0.932925i \(-0.617247\pi\)
−0.360070 + 0.932925i \(0.617247\pi\)
\(48\) −0.454269 −0.0655681
\(49\) −6.51686 −0.930980
\(50\) 0 0
\(51\) 0 0
\(52\) −0.389037 −0.0539497
\(53\) −11.9864 −1.64646 −0.823228 − 0.567711i \(-0.807829\pi\)
−0.823228 + 0.567711i \(0.807829\pi\)
\(54\) 0.946629 0.128820
\(55\) 0 0
\(56\) −1.93064 −0.257993
\(57\) −0.255283 −0.0338130
\(58\) −9.35848 −1.22883
\(59\) −9.91829 −1.29125 −0.645626 − 0.763654i \(-0.723403\pi\)
−0.645626 + 0.763654i \(0.723403\pi\)
\(60\) 0 0
\(61\) 1.60292 0.205233 0.102617 − 0.994721i \(-0.467278\pi\)
0.102617 + 0.994721i \(0.467278\pi\)
\(62\) −9.53937 −1.21150
\(63\) 2.07686 0.261659
\(64\) 7.70533 0.963166
\(65\) 0 0
\(66\) −0.766875 −0.0943958
\(67\) 2.99411 0.365789 0.182894 − 0.983133i \(-0.441453\pi\)
0.182894 + 0.983133i \(0.441453\pi\)
\(68\) 0 0
\(69\) −0.509622 −0.0613513
\(70\) 0 0
\(71\) −4.68852 −0.556425 −0.278213 − 0.960520i \(-0.589742\pi\)
−0.278213 + 0.960520i \(0.589742\pi\)
\(72\) −8.29913 −0.978062
\(73\) −5.49911 −0.643622 −0.321811 − 0.946804i \(-0.604292\pi\)
−0.321811 + 0.946804i \(0.604292\pi\)
\(74\) 0.170630 0.0198353
\(75\) 0 0
\(76\) −0.160244 −0.0183812
\(77\) −3.37178 −0.384250
\(78\) 0.891477 0.100940
\(79\) 14.8439 1.67007 0.835037 − 0.550193i \(-0.185446\pi\)
0.835037 + 0.550193i \(0.185446\pi\)
\(80\) 0 0
\(81\) 8.89143 0.987937
\(82\) −1.55170 −0.171356
\(83\) −5.03506 −0.552670 −0.276335 − 0.961061i \(-0.589120\pi\)
−0.276335 + 0.961061i \(0.589120\pi\)
\(84\) −0.00527044 −0.000575052 0
\(85\) 0 0
\(86\) −0.922169 −0.0994400
\(87\) 0.715073 0.0766639
\(88\) 13.4736 1.43629
\(89\) −2.35657 −0.249796 −0.124898 − 0.992170i \(-0.539860\pi\)
−0.124898 + 0.992170i \(0.539860\pi\)
\(90\) 0 0
\(91\) 3.91962 0.410888
\(92\) −0.319896 −0.0333514
\(93\) 0.728895 0.0755829
\(94\) 7.10141 0.732454
\(95\) 0 0
\(96\) 0.0428738 0.00437579
\(97\) 2.70080 0.274225 0.137113 − 0.990555i \(-0.456218\pi\)
0.137113 + 0.990555i \(0.456218\pi\)
\(98\) 9.37384 0.946900
\(99\) −14.4941 −1.45671
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 7225.2.a.bq.1.3 12
5.4 even 2 1445.2.a.q.1.10 12
17.3 odd 16 425.2.m.b.26.2 24
17.6 odd 16 425.2.m.b.376.2 24
17.16 even 2 7225.2.a.bs.1.3 12
85.3 even 16 425.2.n.f.349.5 24
85.4 even 4 1445.2.d.j.866.6 24
85.23 even 16 425.2.n.c.274.2 24
85.37 even 16 425.2.n.c.349.2 24
85.54 odd 16 85.2.l.a.26.5 ✓ 24
85.57 even 16 425.2.n.f.274.5 24
85.64 even 4 1445.2.d.j.866.5 24
85.74 odd 16 85.2.l.a.36.5 yes 24
85.84 even 2 1445.2.a.p.1.10 12
255.74 even 16 765.2.be.b.631.2 24
255.224 even 16 765.2.be.b.451.2 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.l.a.26.5 ✓ 24 85.54 odd 16
85.2.l.a.36.5 yes 24 85.74 odd 16
425.2.m.b.26.2 24 17.3 odd 16
425.2.m.b.376.2 24 17.6 odd 16
425.2.n.c.274.2 24 85.23 even 16
425.2.n.c.349.2 24 85.37 even 16
425.2.n.f.274.5 24 85.57 even 16
425.2.n.f.349.5 24 85.3 even 16
765.2.be.b.451.2 24 255.224 even 16
765.2.be.b.631.2 24 255.74 even 16
1445.2.a.p.1.10 12 85.84 even 2
1445.2.a.q.1.10 12 5.4 even 2
1445.2.d.j.866.5 24 85.64 even 4
1445.2.d.j.866.6 24 85.4 even 4
7225.2.a.bq.1.3 12 1.1 even 1 trivial
7225.2.a.bs.1.3 12 17.16 even 2