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Properties

Label 8325.2.a.ch.1.1
Level $8325$
Weight $2$
Character 8325.1
Self dual yes
Analytic conductor $66.475$
Analytic rank $0$
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] = [8325,2,Mod(1,8325)] 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("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,2,0,10,0,0,-11,6,0,0,5,0,-4] 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: \(66.4754596827\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{7}]\)
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 1.1
Root \(-1.38679\) of defining polynomial
Character \(\chi\) \(=\) 8325.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.47408 q^{2} +4.12105 q^{4} -4.78404 q^{7} -5.24765 q^{8} +3.12105 q^{11} +2.81780 q^{13} +11.8361 q^{14} +4.74097 q^{16} +6.37246 q^{17} +0.114347 q^{19} -7.72172 q^{22} +5.62219 q^{23} -6.97145 q^{26} -19.7153 q^{28} -2.77357 q^{29} -6.67866 q^{31} -1.23424 q^{32} -15.7660 q^{34} -1.00000 q^{37} -0.282904 q^{38} +3.12105 q^{41} +8.57034 q^{43} +12.8620 q^{44} -13.9097 q^{46} +3.40396 q^{47} +15.8870 q^{49} +11.6123 q^{52} -10.2438 q^{53} +25.1049 q^{56} +6.86203 q^{58} +9.11059 q^{59} -5.55466 q^{61} +16.5235 q^{62} -6.42836 q^{64} +7.84948 q^{67} +26.2612 q^{68} +4.33996 q^{71} -3.22811 q^{73} +2.47408 q^{74} +0.471231 q^{76} -14.9312 q^{77} +15.3847 q^{79} -7.72172 q^{82} +5.68074 q^{83} -21.2037 q^{86} -16.3782 q^{88} +9.95042 q^{89} -13.4805 q^{91} +23.1693 q^{92} -8.42165 q^{94} +5.62970 q^{97} -39.3057 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + 10 q^{4} - 11 q^{7} + 6 q^{8} + 5 q^{11} - 4 q^{13} + 8 q^{14} + 16 q^{16} - 4 q^{19} + 8 q^{22} + 4 q^{23} + 4 q^{26} - 28 q^{28} + 4 q^{29} + 8 q^{31} + 14 q^{32} - 32 q^{34} - 5 q^{37}+ \cdots - 18 q^{98}+O(q^{100}) \) 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 \(-0.838953\pi\)
−0.874718 + 0.484632i \(0.838953\pi\)
\(3\) 0 0
\(4\) 4.12105 2.06053
\(5\) 0 0
\(6\) 0 0
\(7\) −4.78404 −1.80820 −0.904098 − 0.427325i \(-0.859456\pi\)
−0.904098 + 0.427325i \(0.859456\pi\)
\(8\) −5.24765 −1.85532
\(9\) 0 0
\(10\) 0 0
\(11\) 3.12105 0.941033 0.470516 − 0.882391i \(-0.344068\pi\)
0.470516 + 0.882391i \(0.344068\pi\)
\(12\) 0 0
\(13\) 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 \(-0.218868\pi\)
0.772774 + 0.634681i \(0.218868\pi\)
\(18\) 0 0
\(19\) 0.114347 0.0262330 0.0131165 − 0.999914i \(-0.495825\pi\)
0.0131165 + 0.999914i \(0.495825\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −7.72172 −1.64628
\(23\) 5.62219 1.17231 0.586153 − 0.810200i \(-0.300642\pi\)
0.586153 + 0.810200i \(0.300642\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −6.97145 −1.36721
\(27\) 0 0
\(28\) −19.7153 −3.72584
\(29\) −2.77357 −0.515039 −0.257520 − 0.966273i \(-0.582905\pi\)
−0.257520 + 0.966273i \(0.582905\pi\)
\(30\) 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
\(33\) 0 0
\(34\) −15.7660 −2.70384
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) −0.282904 −0.0458930
\(39\) 0 0
\(40\) 0 0
\(41\) 3.12105 0.487427 0.243713 − 0.969847i \(-0.421634\pi\)
0.243713 + 0.969847i \(0.421634\pi\)
\(42\) 0 0
\(43\) 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 \(-0.420142\pi\)
0.248259 + 0.968694i \(0.420142\pi\)
\(48\) 0 0
\(49\) 15.8870 2.26957
\(50\) 0 0
\(51\) 0 0
\(52\) 11.6123 1.61034
\(53\) −10.2438 −1.40709 −0.703546 − 0.710650i \(-0.748401\pi\)
−0.703546 + 0.710650i \(0.748401\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 25.1049 3.35479
\(57\) 0 0
\(58\) 6.86203 0.901028
\(59\) 9.11059 1.18610 0.593049 − 0.805167i \(-0.297924\pi\)
0.593049 + 0.805167i \(0.297924\pi\)
\(60\) 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
\(63\) 0 0
\(64\) −6.42836 −0.803544
\(65\) 0 0
\(66\) 0 0
\(67\) 7.84948 0.958967 0.479484 − 0.877551i \(-0.340824\pi\)
0.479484 + 0.877551i \(0.340824\pi\)
\(68\) 26.2612 3.18464
\(69\) 0 0
\(70\) 0 0
\(71\) 4.33996 0.515059 0.257529 − 0.966270i \(-0.417092\pi\)
0.257529 + 0.966270i \(0.417092\pi\)
\(72\) 0 0
\(73\) −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
\(78\) 0 0
\(79\) 15.3847 1.73091 0.865457 − 0.500983i \(-0.167028\pi\)
0.865457 + 0.500983i \(0.167028\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −7.72172 −0.852722
\(83\) 5.68074 0.623542 0.311771 − 0.950157i \(-0.399078\pi\)
0.311771 + 0.950157i \(0.399078\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −21.2037 −2.28645
\(87\) 0 0
\(88\) −16.3782 −1.74592
\(89\) 9.95042 1.05474 0.527371 − 0.849635i \(-0.323178\pi\)
0.527371 + 0.849635i \(0.323178\pi\)
\(90\) 0 0
\(91\) −13.4805 −1.41314
\(92\) 23.1693 2.41557
\(93\) 0 0
\(94\) −8.42165 −0.868627
\(95\) 0 0
\(96\) 0 0
\(97\) 5.62970 0.571610 0.285805 − 0.958288i \(-0.407739\pi\)
0.285805 + 0.958288i \(0.407739\pi\)
\(98\) −39.3057 −3.97047
\(99\) 0 0
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 8325.2.a.ch.1.1 5
3.2 odd 2 925.2.a.f.1.5 5
5.4 even 2 1665.2.a.p.1.5 5
15.2 even 4 925.2.b.f.149.9 10
15.8 even 4 925.2.b.f.149.2 10
15.14 odd 2 185.2.a.e.1.1 ✓ 5
60.59 even 2 2960.2.a.w.1.2 5
105.104 even 2 9065.2.a.k.1.1 5
555.554 odd 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 15.14 odd 2
925.2.a.f.1.5 5 3.2 odd 2
925.2.b.f.149.2 10 15.8 even 4
925.2.b.f.149.9 10 15.2 even 4
1665.2.a.p.1.5 5 5.4 even 2
2960.2.a.w.1.2 5 60.59 even 2
6845.2.a.f.1.5 5 555.554 odd 2
8325.2.a.ch.1.1 5 1.1 even 1 trivial
9065.2.a.k.1.1 5 105.104 even 2