Properties

Label 925.2.a.j.1.6
Level $925$
Weight $2$
Character 925.1
Self dual yes
Analytic conductor $7.386$
Analytic rank $0$
Dimension $7$
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] = [925,2,Mod(1,925)] 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("925.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,-1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 10x^{5} + 9x^{4} + 26x^{3} - 23x^{2} - 9x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-2.13289\) of defining polynomial
Character \(\chi\) \(=\) 925.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.13289 q^{2} +2.92598 q^{3} +2.54923 q^{4} +6.24079 q^{6} -3.01925 q^{7} +1.17144 q^{8} +5.56135 q^{9} +5.51375 q^{11} +7.45898 q^{12} -0.501630 q^{13} -6.43973 q^{14} -2.59990 q^{16} -6.61915 q^{17} +11.8618 q^{18} +4.42510 q^{19} -8.83426 q^{21} +11.7602 q^{22} -1.67307 q^{23} +3.42761 q^{24} -1.06992 q^{26} +7.49445 q^{27} -7.69675 q^{28} -1.51711 q^{29} -9.00821 q^{31} -7.88818 q^{32} +16.1331 q^{33} -14.1179 q^{34} +14.1771 q^{36} -1.00000 q^{37} +9.43826 q^{38} -1.46776 q^{39} +3.32768 q^{41} -18.8425 q^{42} +3.05695 q^{43} +14.0558 q^{44} -3.56848 q^{46} -3.80128 q^{47} -7.60725 q^{48} +2.11587 q^{49} -19.3675 q^{51} -1.27877 q^{52} +6.03086 q^{53} +15.9849 q^{54} -3.53687 q^{56} +12.9477 q^{57} -3.23583 q^{58} +13.4763 q^{59} +11.9483 q^{61} -19.2135 q^{62} -16.7911 q^{63} -11.6248 q^{64} +34.4102 q^{66} -10.8898 q^{67} -16.8737 q^{68} -4.89537 q^{69} -2.46808 q^{71} +6.51479 q^{72} +13.2652 q^{73} -2.13289 q^{74} +11.2806 q^{76} -16.6474 q^{77} -3.13057 q^{78} -11.4110 q^{79} +5.24456 q^{81} +7.09758 q^{82} -8.36622 q^{83} -22.5205 q^{84} +6.52014 q^{86} -4.43904 q^{87} +6.45903 q^{88} -7.41023 q^{89} +1.51454 q^{91} -4.26503 q^{92} -26.3578 q^{93} -8.10773 q^{94} -23.0806 q^{96} +5.75342 q^{97} +4.51291 q^{98} +30.6639 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - q^{2} + q^{3} + 7 q^{4} + 6 q^{6} + 10 q^{9} + 16 q^{11} - 11 q^{12} + q^{13} - 3 q^{14} + 3 q^{16} - 4 q^{17} + 23 q^{18} + 9 q^{19} + 2 q^{21} - q^{22} + q^{23} + 24 q^{26} + q^{27} - 7 q^{28}+ \cdots + 39 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.13289 1.50818 0.754091 − 0.656770i \(-0.228078\pi\)
0.754091 + 0.656770i \(0.228078\pi\)
\(3\) 2.92598 1.68931 0.844657 − 0.535308i \(-0.179804\pi\)
0.844657 + 0.535308i \(0.179804\pi\)
\(4\) 2.54923 1.27461
\(5\) 0 0
\(6\) 6.24079 2.54779
\(7\) −3.01925 −1.14117 −0.570585 − 0.821239i \(-0.693283\pi\)
−0.570585 + 0.821239i \(0.693283\pi\)
\(8\) 1.17144 0.414167
\(9\) 5.56135 1.85378
\(10\) 0 0
\(11\) 5.51375 1.66246 0.831230 − 0.555929i \(-0.187637\pi\)
0.831230 + 0.555929i \(0.187637\pi\)
\(12\) 7.45898 2.15322
\(13\) −0.501630 −0.139127 −0.0695635 − 0.997578i \(-0.522161\pi\)
−0.0695635 + 0.997578i \(0.522161\pi\)
\(14\) −6.43973 −1.72109
\(15\) 0 0
\(16\) −2.59990 −0.649975
\(17\) −6.61915 −1.60538 −0.802690 − 0.596397i \(-0.796598\pi\)
−0.802690 + 0.596397i \(0.796598\pi\)
\(18\) 11.8618 2.79584
\(19\) 4.42510 1.01519 0.507594 − 0.861597i \(-0.330535\pi\)
0.507594 + 0.861597i \(0.330535\pi\)
\(20\) 0 0
\(21\) −8.83426 −1.92779
\(22\) 11.7602 2.50729
\(23\) −1.67307 −0.348859 −0.174430 − 0.984670i \(-0.555808\pi\)
−0.174430 + 0.984670i \(0.555808\pi\)
\(24\) 3.42761 0.699658
\(25\) 0 0
\(26\) −1.06992 −0.209829
\(27\) 7.49445 1.44231
\(28\) −7.69675 −1.45455
\(29\) −1.51711 −0.281721 −0.140860 − 0.990029i \(-0.544987\pi\)
−0.140860 + 0.990029i \(0.544987\pi\)
\(30\) 0 0
\(31\) −9.00821 −1.61792 −0.808961 − 0.587862i \(-0.799970\pi\)
−0.808961 + 0.587862i \(0.799970\pi\)
\(32\) −7.88818 −1.39445
\(33\) 16.1331 2.80842
\(34\) −14.1179 −2.42120
\(35\) 0 0
\(36\) 14.1771 2.36286
\(37\) −1.00000 −0.164399
\(38\) 9.43826 1.53109
\(39\) −1.46776 −0.235029
\(40\) 0 0
\(41\) 3.32768 0.519696 0.259848 − 0.965649i \(-0.416327\pi\)
0.259848 + 0.965649i \(0.416327\pi\)
\(42\) −18.8425 −2.90746
\(43\) 3.05695 0.466180 0.233090 − 0.972455i \(-0.425116\pi\)
0.233090 + 0.972455i \(0.425116\pi\)
\(44\) 14.0558 2.11899
\(45\) 0 0
\(46\) −3.56848 −0.526143
\(47\) −3.80128 −0.554474 −0.277237 − 0.960802i \(-0.589419\pi\)
−0.277237 + 0.960802i \(0.589419\pi\)
\(48\) −7.60725 −1.09801
\(49\) 2.11587 0.302267
\(50\) 0 0
\(51\) −19.3675 −2.71199
\(52\) −1.27877 −0.177333
\(53\) 6.03086 0.828403 0.414202 − 0.910185i \(-0.364061\pi\)
0.414202 + 0.910185i \(0.364061\pi\)
\(54\) 15.9849 2.17526
\(55\) 0 0
\(56\) −3.53687 −0.472634
\(57\) 12.9477 1.71497
\(58\) −3.23583 −0.424886
\(59\) 13.4763 1.75446 0.877231 − 0.480069i \(-0.159388\pi\)
0.877231 + 0.480069i \(0.159388\pi\)
\(60\) 0 0
\(61\) 11.9483 1.52982 0.764909 − 0.644139i \(-0.222784\pi\)
0.764909 + 0.644139i \(0.222784\pi\)
\(62\) −19.2135 −2.44012
\(63\) −16.7911 −2.11548
\(64\) −11.6248 −1.45310
\(65\) 0 0
\(66\) 34.4102 4.23560
\(67\) −10.8898 −1.33040 −0.665198 − 0.746667i \(-0.731653\pi\)
−0.665198 + 0.746667i \(0.731653\pi\)
\(68\) −16.8737 −2.04624
\(69\) −4.89537 −0.589333
\(70\) 0 0
\(71\) −2.46808 −0.292907 −0.146454 − 0.989218i \(-0.546786\pi\)
−0.146454 + 0.989218i \(0.546786\pi\)
\(72\) 6.51479 0.767775
\(73\) 13.2652 1.55258 0.776289 − 0.630378i \(-0.217100\pi\)
0.776289 + 0.630378i \(0.217100\pi\)
\(74\) −2.13289 −0.247944
\(75\) 0 0
\(76\) 11.2806 1.29397
\(77\) −16.6474 −1.89715
\(78\) −3.13057 −0.354467
\(79\) −11.4110 −1.28384 −0.641918 − 0.766773i \(-0.721861\pi\)
−0.641918 + 0.766773i \(0.721861\pi\)
\(80\) 0 0
\(81\) 5.24456 0.582729
\(82\) 7.09758 0.783797
\(83\) −8.36622 −0.918312 −0.459156 − 0.888356i \(-0.651848\pi\)
−0.459156 + 0.888356i \(0.651848\pi\)
\(84\) −22.5205 −2.45719
\(85\) 0 0
\(86\) 6.52014 0.703085
\(87\) −4.43904 −0.475915
\(88\) 6.45903 0.688535
\(89\) −7.41023 −0.785483 −0.392741 − 0.919649i \(-0.628473\pi\)
−0.392741 + 0.919649i \(0.628473\pi\)
\(90\) 0 0
\(91\) 1.51454 0.158767
\(92\) −4.26503 −0.444660
\(93\) −26.3578 −2.73318
\(94\) −8.10773 −0.836248
\(95\) 0 0
\(96\) −23.0806 −2.35566
\(97\) 5.75342 0.584171 0.292085 − 0.956392i \(-0.405651\pi\)
0.292085 + 0.956392i \(0.405651\pi\)
\(98\) 4.51291 0.455873
\(99\) 30.6639 3.08184
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 925.2.a.j.1.6 ✓ 7
3.2 odd 2 8325.2.a.cn.1.2 7
5.2 odd 4 925.2.b.i.149.12 14
5.3 odd 4 925.2.b.i.149.3 14
5.4 even 2 925.2.a.k.1.2 yes 7
15.14 odd 2 8325.2.a.cm.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
925.2.a.j.1.6 ✓ 7 1.1 even 1 trivial
925.2.a.k.1.2 yes 7 5.4 even 2
925.2.b.i.149.3 14 5.3 odd 4
925.2.b.i.149.12 14 5.2 odd 4
8325.2.a.cm.1.6 7 15.14 odd 2
8325.2.a.cn.1.2 7 3.2 odd 2