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Properties

Label 6845.2.a.f.1.4
Level $6845$
Weight $2$
Character 6845.1
Self dual yes
Analytic conductor $54.658$
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] = [6845,2,Mod(1,6845)] 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("6845.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6845, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6845 = 5 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6845.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,-2,3,10,5,6,11] 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: \(54.6576001836\)
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: \(\Z[a_1, a_2, a_3]\)
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 1.4
Root \(2.10563\) of defining polynomial
Character \(\chi\) \(=\) 6845.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.13359 q^{2} -1.10563 q^{3} -0.714970 q^{4} +1.00000 q^{5} -1.25333 q^{6} +2.46164 q^{7} -3.07767 q^{8} -1.77758 q^{9} +1.13359 q^{10} +1.71497 q^{11} +0.790492 q^{12} -6.49255 q^{13} +2.79049 q^{14} -1.10563 q^{15} -2.05888 q^{16} -3.32980 q^{17} -2.01505 q^{18} -0.734568 q^{19} -0.714970 q^{20} -2.72166 q^{21} +1.94408 q^{22} +2.08603 q^{23} +3.40276 q^{24} +1.00000 q^{25} -7.35990 q^{26} +5.28224 q^{27} -1.76000 q^{28} +4.21126 q^{29} -1.25333 q^{30} -7.46459 q^{31} +3.82141 q^{32} -1.89612 q^{33} -3.77463 q^{34} +2.46164 q^{35} +1.27092 q^{36} -0.832700 q^{38} +7.17836 q^{39} -3.07767 q^{40} +1.71497 q^{41} -3.08525 q^{42} -1.81885 q^{43} -1.22615 q^{44} -1.77758 q^{45} +2.36471 q^{46} -0.882270 q^{47} +2.27636 q^{48} -0.940340 q^{49} +1.13359 q^{50} +3.68152 q^{51} +4.64198 q^{52} -7.03066 q^{53} +5.98790 q^{54} +1.71497 q^{55} -7.57610 q^{56} +0.812159 q^{57} +4.77385 q^{58} -0.387867 q^{59} +0.790492 q^{60} +11.8224 q^{61} -8.46180 q^{62} -4.37577 q^{63} +8.44967 q^{64} -6.49255 q^{65} -2.14943 q^{66} +12.1086 q^{67} +2.38070 q^{68} -2.30638 q^{69} +2.79049 q^{70} +13.7486 q^{71} +5.47081 q^{72} +16.6719 q^{73} -1.10563 q^{75} +0.525194 q^{76} +4.22163 q^{77} +8.13733 q^{78} -8.23253 q^{79} -2.05888 q^{80} -0.507447 q^{81} +1.94408 q^{82} -4.80275 q^{83} +1.94590 q^{84} -3.32980 q^{85} -2.06183 q^{86} -4.65609 q^{87} -5.27811 q^{88} +1.52506 q^{89} -2.01505 q^{90} -15.9823 q^{91} -1.49145 q^{92} +8.25307 q^{93} -1.00013 q^{94} -0.734568 q^{95} -4.22506 q^{96} +18.1588 q^{97} -1.06596 q^{98} -3.04850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 3 q^{3} + 10 q^{4} + 5 q^{5} + 6 q^{6} + 11 q^{7} - 6 q^{8} + 6 q^{9} - 2 q^{10} - 5 q^{11} - 2 q^{12} - 4 q^{13} + 8 q^{14} + 3 q^{15} + 16 q^{16} - 2 q^{18} + 4 q^{19} + 10 q^{20} + 3 q^{21}+ \cdots - 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.13359 0.801570 0.400785 − 0.916172i \(-0.368737\pi\)
0.400785 + 0.916172i \(0.368737\pi\)
\(3\) −1.10563 −0.638335 −0.319168 − 0.947698i \(-0.603403\pi\)
−0.319168 + 0.947698i \(0.603403\pi\)
\(4\) −0.714970 −0.357485
\(5\) 1.00000 0.447214
\(6\) −1.25333 −0.511671
\(7\) 2.46164 0.930412 0.465206 − 0.885203i \(-0.345980\pi\)
0.465206 + 0.885203i \(0.345980\pi\)
\(8\) −3.07767 −1.08812
\(9\) −1.77758 −0.592528
\(10\) 1.13359 0.358473
\(11\) 1.71497 0.517083 0.258541 − 0.966000i \(-0.416758\pi\)
0.258541 + 0.966000i \(0.416758\pi\)
\(12\) 0.790492 0.228195
\(13\) −6.49255 −1.80071 −0.900355 − 0.435156i \(-0.856693\pi\)
−0.900355 + 0.435156i \(0.856693\pi\)
\(14\) 2.79049 0.745790
\(15\) −1.10563 −0.285472
\(16\) −2.05888 −0.514719
\(17\) −3.32980 −0.807594 −0.403797 − 0.914849i \(-0.632310\pi\)
−0.403797 + 0.914849i \(0.632310\pi\)
\(18\) −2.01505 −0.474953
\(19\) −0.734568 −0.168521 −0.0842607 − 0.996444i \(-0.526853\pi\)
−0.0842607 + 0.996444i \(0.526853\pi\)
\(20\) −0.714970 −0.159872
\(21\) −2.72166 −0.593915
\(22\) 1.94408 0.414478
\(23\) 2.08603 0.434968 0.217484 − 0.976064i \(-0.430215\pi\)
0.217484 + 0.976064i \(0.430215\pi\)
\(24\) 3.40276 0.694585
\(25\) 1.00000 0.200000
\(26\) −7.35990 −1.44340
\(27\) 5.28224 1.01657
\(28\) −1.76000 −0.332608
\(29\) 4.21126 0.782011 0.391006 − 0.920388i \(-0.372127\pi\)
0.391006 + 0.920388i \(0.372127\pi\)
\(30\) −1.25333 −0.228826
\(31\) −7.46459 −1.34068 −0.670340 − 0.742054i \(-0.733852\pi\)
−0.670340 + 0.742054i \(0.733852\pi\)
\(32\) 3.82141 0.675536
\(33\) −1.89612 −0.330072
\(34\) −3.77463 −0.647344
\(35\) 2.46164 0.416093
\(36\) 1.27092 0.211820
\(37\) 0 0
\(38\) −0.832700 −0.135082
\(39\) 7.17836 1.14946
\(40\) −3.07767 −0.486622
\(41\) 1.71497 0.267833 0.133917 − 0.990993i \(-0.457245\pi\)
0.133917 + 0.990993i \(0.457245\pi\)
\(42\) −3.08525 −0.476064
\(43\) −1.81885 −0.277372 −0.138686 − 0.990336i \(-0.544288\pi\)
−0.138686 + 0.990336i \(0.544288\pi\)
\(44\) −1.22615 −0.184849
\(45\) −1.77758 −0.264986
\(46\) 2.36471 0.348657
\(47\) −0.882270 −0.128692 −0.0643462 − 0.997928i \(-0.520496\pi\)
−0.0643462 + 0.997928i \(0.520496\pi\)
\(48\) 2.27636 0.328564
\(49\) −0.940340 −0.134334
\(50\) 1.13359 0.160314
\(51\) 3.68152 0.515516
\(52\) 4.64198 0.643727
\(53\) −7.03066 −0.965735 −0.482867 − 0.875693i \(-0.660405\pi\)
−0.482867 + 0.875693i \(0.660405\pi\)
\(54\) 5.98790 0.814850
\(55\) 1.71497 0.231247
\(56\) −7.57610 −1.01240
\(57\) 0.812159 0.107573
\(58\) 4.77385 0.626837
\(59\) −0.387867 −0.0504959 −0.0252480 − 0.999681i \(-0.508038\pi\)
−0.0252480 + 0.999681i \(0.508038\pi\)
\(60\) 0.790492 0.102052
\(61\) 11.8224 1.51370 0.756848 − 0.653590i \(-0.226738\pi\)
0.756848 + 0.653590i \(0.226738\pi\)
\(62\) −8.46180 −1.07465
\(63\) −4.37577 −0.551295
\(64\) 8.44967 1.05621
\(65\) −6.49255 −0.805302
\(66\) −2.14943 −0.264576
\(67\) 12.1086 1.47930 0.739649 − 0.672992i \(-0.234991\pi\)
0.739649 + 0.672992i \(0.234991\pi\)
\(68\) 2.38070 0.288703
\(69\) −2.30638 −0.277655
\(70\) 2.79049 0.333528
\(71\) 13.7486 1.63166 0.815828 − 0.578295i \(-0.196282\pi\)
0.815828 + 0.578295i \(0.196282\pi\)
\(72\) 5.47081 0.644741
\(73\) 16.6719 1.95129 0.975646 − 0.219349i \(-0.0703933\pi\)
0.975646 + 0.219349i \(0.0703933\pi\)
\(74\) 0 0
\(75\) −1.10563 −0.127667
\(76\) 0.525194 0.0602439
\(77\) 4.22163 0.481100
\(78\) 8.13733 0.921371
\(79\) −8.23253 −0.926232 −0.463116 − 0.886298i \(-0.653269\pi\)
−0.463116 + 0.886298i \(0.653269\pi\)
\(80\) −2.05888 −0.230190
\(81\) −0.507447 −0.0563829
\(82\) 1.94408 0.214687
\(83\) −4.80275 −0.527171 −0.263585 − 0.964636i \(-0.584905\pi\)
−0.263585 + 0.964636i \(0.584905\pi\)
\(84\) 1.94590 0.212316
\(85\) −3.32980 −0.361167
\(86\) −2.06183 −0.222333
\(87\) −4.65609 −0.499185
\(88\) −5.27811 −0.562648
\(89\) 1.52506 0.161656 0.0808280 − 0.996728i \(-0.474244\pi\)
0.0808280 + 0.996728i \(0.474244\pi\)
\(90\) −2.01505 −0.212405
\(91\) −15.9823 −1.67540
\(92\) −1.49145 −0.155494
\(93\) 8.25307 0.855804
\(94\) −1.00013 −0.103156
\(95\) −0.734568 −0.0753650
\(96\) −4.22506 −0.431218
\(97\) 18.1588 1.84375 0.921875 − 0.387487i \(-0.126657\pi\)
0.921875 + 0.387487i \(0.126657\pi\)
\(98\) −1.06596 −0.107678
\(99\) −3.04850 −0.306386
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 6845.2.a.f.1.4 5
37.36 even 2 185.2.a.e.1.2 ✓ 5
111.110 odd 2 1665.2.a.p.1.4 5
148.147 odd 2 2960.2.a.w.1.4 5
185.73 odd 4 925.2.b.f.149.7 10
185.147 odd 4 925.2.b.f.149.4 10
185.184 even 2 925.2.a.f.1.4 5
259.258 odd 2 9065.2.a.k.1.2 5
555.554 odd 2 8325.2.a.ch.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.a.e.1.2 ✓ 5 37.36 even 2
925.2.a.f.1.4 5 185.184 even 2
925.2.b.f.149.4 10 185.147 odd 4
925.2.b.f.149.7 10 185.73 odd 4
1665.2.a.p.1.4 5 111.110 odd 2
2960.2.a.w.1.4 5 148.147 odd 2
6845.2.a.f.1.4 5 1.1 even 1 trivial
8325.2.a.ch.1.2 5 555.554 odd 2
9065.2.a.k.1.2 5 259.258 odd 2