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Properties

Label 925.2.a.f.1.5
Level $925$
Weight $2$
Character 925.1
Self dual yes
Analytic conductor $7.386$
Analytic rank $1$
Dimension $5$
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 = [5,-2,-3] 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: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.973904.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6 \) 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 185)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.38679\) 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)
 
\(f(q)\) \(=\) \(q+2.47408 q^{2} -2.38679 q^{3} +4.12105 q^{4} -5.90509 q^{6} -4.78404 q^{7} +5.24765 q^{8} +2.69675 q^{9} -3.12105 q^{11} -9.83607 q^{12} +2.81780 q^{13} -11.8361 q^{14} +4.74097 q^{16} -6.37246 q^{17} +6.67196 q^{18} +0.114347 q^{19} +11.4185 q^{21} -7.72172 q^{22} -5.62219 q^{23} -12.5250 q^{24} +6.97145 q^{26} +0.723803 q^{27} -19.7153 q^{28} +2.77357 q^{29} -6.67866 q^{31} +1.23424 q^{32} +7.44929 q^{33} -15.7660 q^{34} +11.1134 q^{36} -1.00000 q^{37} +0.282904 q^{38} -6.72548 q^{39} -3.12105 q^{41} +28.2502 q^{42} +8.57034 q^{43} -12.8620 q^{44} -13.9097 q^{46} -3.40396 q^{47} -11.3157 q^{48} +15.8870 q^{49} +15.2097 q^{51} +11.6123 q^{52} +10.2438 q^{53} +1.79074 q^{54} -25.1049 q^{56} -0.272922 q^{57} +6.86203 q^{58} -9.11059 q^{59} -5.55466 q^{61} -16.5235 q^{62} -12.9013 q^{63} -6.42836 q^{64} +18.4301 q^{66} +7.84948 q^{67} -26.2612 q^{68} +13.4190 q^{69} -4.33996 q^{71} +14.1516 q^{72} -3.22811 q^{73} -2.47408 q^{74} +0.471231 q^{76} +14.9312 q^{77} -16.6394 q^{78} +15.3847 q^{79} -9.81780 q^{81} -7.72172 q^{82} -5.68074 q^{83} +47.0561 q^{84} +21.2037 q^{86} -6.61992 q^{87} -16.3782 q^{88} -9.95042 q^{89} -13.4805 q^{91} -23.1693 q^{92} +15.9405 q^{93} -8.42165 q^{94} -2.94586 q^{96} +5.62970 q^{97} +39.3057 q^{98} -8.41669 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} - 3 q^{3} + 10 q^{4} - 6 q^{6} - 11 q^{7} - 6 q^{8} + 6 q^{9} - 5 q^{11} + 2 q^{12} - 4 q^{13} - 8 q^{14} + 16 q^{16} - 2 q^{18} - 4 q^{19} + 3 q^{21} + 8 q^{22} - 4 q^{23} - 42 q^{24} - 4 q^{26}+ \cdots - 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.47408 1.74944 0.874718 − 0.484632i \(-0.161047\pi\)
0.874718 + 0.484632i \(0.161047\pi\)
\(3\) −2.38679 −1.37801 −0.689006 − 0.724756i \(-0.741953\pi\)
−0.689006 + 0.724756i \(0.741953\pi\)
\(4\) 4.12105 2.06053
\(5\) 0 0
\(6\) −5.90509 −2.41074
\(7\) −4.78404 −1.80820 −0.904098 − 0.427325i \(-0.859456\pi\)
−0.904098 + 0.427325i \(0.859456\pi\)
\(8\) 5.24765 1.85532
\(9\) 2.69675 0.898915
\(10\) 0 0
\(11\) −3.12105 −0.941033 −0.470516 − 0.882391i \(-0.655932\pi\)
−0.470516 + 0.882391i \(0.655932\pi\)
\(12\) −9.83607 −2.83943
\(13\) 2.81780 0.781517 0.390758 − 0.920493i \(-0.372213\pi\)
0.390758 + 0.920493i \(0.372213\pi\)
\(14\) −11.8361 −3.16332
\(15\) 0 0
\(16\) 4.74097 1.18524
\(17\) −6.37246 −1.54555 −0.772774 − 0.634681i \(-0.781132\pi\)
−0.772774 + 0.634681i \(0.781132\pi\)
\(18\) 6.67196 1.57259
\(19\) 0.114347 0.0262330 0.0131165 − 0.999914i \(-0.495825\pi\)
0.0131165 + 0.999914i \(0.495825\pi\)
\(20\) 0 0
\(21\) 11.4185 2.49171
\(22\) −7.72172 −1.64628
\(23\) −5.62219 −1.17231 −0.586153 − 0.810200i \(-0.699358\pi\)
−0.586153 + 0.810200i \(0.699358\pi\)
\(24\) −12.5250 −2.55666
\(25\) 0 0
\(26\) 6.97145 1.36721
\(27\) 0.723803 0.139296
\(28\) −19.7153 −3.72584
\(29\) 2.77357 0.515039 0.257520 − 0.966273i \(-0.417095\pi\)
0.257520 + 0.966273i \(0.417095\pi\)
\(30\) 0 0
\(31\) −6.67866 −1.19952 −0.599761 − 0.800179i \(-0.704738\pi\)
−0.599761 + 0.800179i \(0.704738\pi\)
\(32\) 1.23424 0.218184
\(33\) 7.44929 1.29675
\(34\) −15.7660 −2.70384
\(35\) 0 0
\(36\) 11.1134 1.85224
\(37\) −1.00000 −0.164399
\(38\) 0.282904 0.0458930
\(39\) −6.72548 −1.07694
\(40\) 0 0
\(41\) −3.12105 −0.487427 −0.243713 − 0.969847i \(-0.578366\pi\)
−0.243713 + 0.969847i \(0.578366\pi\)
\(42\) 28.2502 4.35910
\(43\) 8.57034 1.30696 0.653482 − 0.756942i \(-0.273307\pi\)
0.653482 + 0.756942i \(0.273307\pi\)
\(44\) −12.8620 −1.93902
\(45\) 0 0
\(46\) −13.9097 −2.05088
\(47\) −3.40396 −0.496518 −0.248259 − 0.968694i \(-0.579858\pi\)
−0.248259 + 0.968694i \(0.579858\pi\)
\(48\) −11.3157 −1.63328
\(49\) 15.8870 2.26957
\(50\) 0 0
\(51\) 15.2097 2.12978
\(52\) 11.6123 1.61034
\(53\) 10.2438 1.40709 0.703546 − 0.710650i \(-0.251599\pi\)
0.703546 + 0.710650i \(0.251599\pi\)
\(54\) 1.79074 0.243689
\(55\) 0 0
\(56\) −25.1049 −3.35479
\(57\) −0.272922 −0.0361494
\(58\) 6.86203 0.901028
\(59\) −9.11059 −1.18610 −0.593049 − 0.805167i \(-0.702076\pi\)
−0.593049 + 0.805167i \(0.702076\pi\)
\(60\) 0 0
\(61\) −5.55466 −0.711201 −0.355601 − 0.934638i \(-0.615724\pi\)
−0.355601 + 0.934638i \(0.615724\pi\)
\(62\) −16.5235 −2.09849
\(63\) −12.9013 −1.62541
\(64\) −6.42836 −0.803544
\(65\) 0 0
\(66\) 18.4301 2.26859
\(67\) 7.84948 0.958967 0.479484 − 0.877551i \(-0.340824\pi\)
0.479484 + 0.877551i \(0.340824\pi\)
\(68\) −26.2612 −3.18464
\(69\) 13.4190 1.61545
\(70\) 0 0
\(71\) −4.33996 −0.515059 −0.257529 − 0.966270i \(-0.582908\pi\)
−0.257529 + 0.966270i \(0.582908\pi\)
\(72\) 14.1516 1.66778
\(73\) −3.22811 −0.377822 −0.188911 − 0.981994i \(-0.560496\pi\)
−0.188911 + 0.981994i \(0.560496\pi\)
\(74\) −2.47408 −0.287606
\(75\) 0 0
\(76\) 0.471231 0.0540539
\(77\) 14.9312 1.70157
\(78\) −16.6394 −1.88404
\(79\) 15.3847 1.73091 0.865457 − 0.500983i \(-0.167028\pi\)
0.865457 + 0.500983i \(0.167028\pi\)
\(80\) 0 0
\(81\) −9.81780 −1.09087
\(82\) −7.72172 −0.852722
\(83\) −5.68074 −0.623542 −0.311771 − 0.950157i \(-0.600922\pi\)
−0.311771 + 0.950157i \(0.600922\pi\)
\(84\) 47.0561 5.13424
\(85\) 0 0
\(86\) 21.2037 2.28645
\(87\) −6.61992 −0.709730
\(88\) −16.3782 −1.74592
\(89\) −9.95042 −1.05474 −0.527371 − 0.849635i \(-0.676822\pi\)
−0.527371 + 0.849635i \(0.676822\pi\)
\(90\) 0 0
\(91\) −13.4805 −1.41314
\(92\) −23.1693 −2.41557
\(93\) 15.9405 1.65296
\(94\) −8.42165 −0.868627
\(95\) 0 0
\(96\) −2.94586 −0.300660
\(97\) 5.62970 0.571610 0.285805 − 0.958288i \(-0.407739\pi\)
0.285805 + 0.958288i \(0.407739\pi\)
\(98\) 39.3057 3.97047
\(99\) −8.41669 −0.845909
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.f.1.5 5
3.2 odd 2 8325.2.a.ch.1.1 5
5.2 odd 4 925.2.b.f.149.9 10
5.3 odd 4 925.2.b.f.149.2 10
5.4 even 2 185.2.a.e.1.1 ✓ 5
15.14 odd 2 1665.2.a.p.1.5 5
20.19 odd 2 2960.2.a.w.1.2 5
35.34 odd 2 9065.2.a.k.1.1 5
185.184 even 2 6845.2.a.f.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.1 ✓ 5 5.4 even 2
925.2.a.f.1.5 5 1.1 even 1 trivial
925.2.b.f.149.2 10 5.3 odd 4
925.2.b.f.149.9 10 5.2 odd 4
1665.2.a.p.1.5 5 15.14 odd 2
2960.2.a.w.1.2 5 20.19 odd 2
6845.2.a.f.1.5 5 185.184 even 2
8325.2.a.ch.1.1 5 3.2 odd 2
9065.2.a.k.1.1 5 35.34 odd 2