Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1725,2,Mod(1174,1725)] 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("1725.1174"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1725, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1725 = 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1725.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-14,0,-2,0,0,-4,0,-2,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.7741943487\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{21})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 11x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1174.4
Root \(2.79129i\) of defining polynomial
Character \(\chi\) \(=\) 1725.1174
Dual form 1725.2.b.n.1174.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.79129i q^{2} +1.00000i q^{3} -5.79129 q^{4} -2.79129 q^{6} -1.79129i q^{7} -10.5826i q^{8} -1.00000 q^{9} +1.79129 q^{11} -5.79129i q^{12} -2.20871i q^{13} +5.00000 q^{14} +17.9564 q^{16} +5.00000i q^{17} -2.79129i q^{18} +5.58258 q^{19} +1.79129 q^{21} +5.00000i q^{22} -1.00000i q^{23} +10.5826 q^{24} +6.16515 q^{26} -1.00000i q^{27} +10.3739i q^{28} +5.58258 q^{29} +0.208712 q^{31} +28.9564i q^{32} +1.79129i q^{33} -13.9564 q^{34} +5.79129 q^{36} -5.00000i q^{37} +15.5826i q^{38} +2.20871 q^{39} -7.00000 q^{41} +5.00000i q^{42} -11.0000i q^{43} -10.3739 q^{44} +2.79129 q^{46} +12.1652i q^{47} +17.9564i q^{48} +3.79129 q^{49} -5.00000 q^{51} +12.7913i q^{52} -5.37386i q^{53} +2.79129 q^{54} -18.9564 q^{56} +5.58258i q^{57} +15.5826i q^{58} -0.626136 q^{59} -2.20871 q^{61} +0.582576i q^{62} +1.79129i q^{63} -44.9129 q^{64} -5.00000 q^{66} +6.37386i q^{67} -28.9564i q^{68} +1.00000 q^{69} +16.5390 q^{71} +10.5826i q^{72} +7.00000i q^{73} +13.9564 q^{74} -32.3303 q^{76} -3.20871i q^{77} +6.16515i q^{78} +8.16515 q^{79} +1.00000 q^{81} -19.5390i q^{82} +12.1652i q^{83} -10.3739 q^{84} +30.7042 q^{86} +5.58258i q^{87} -18.9564i q^{88} +14.5826 q^{89} -3.95644 q^{91} +5.79129i q^{92} +0.208712i q^{93} -33.9564 q^{94} -28.9564 q^{96} -14.9564i q^{97} +10.5826i q^{98} -1.79129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{4} - 2 q^{6} - 4 q^{9} - 2 q^{11} + 20 q^{14} + 26 q^{16} + 4 q^{19} - 2 q^{21} + 24 q^{24} - 12 q^{26} + 4 q^{29} + 10 q^{31} - 10 q^{34} + 14 q^{36} + 18 q^{39} - 28 q^{41} - 14 q^{44} + 2 q^{46}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1725\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.79129i 1.97374i 0.161521 + 0.986869i \(0.448360\pi\)
−0.161521 + 0.986869i \(0.551640\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −5.79129 −2.89564
\(5\) 0 0
\(6\) −2.79129 −1.13954
\(7\) − 1.79129i − 0.677043i −0.940959 − 0.338522i \(-0.890073\pi\)
0.940959 − 0.338522i \(-0.109927\pi\)
\(8\) − 10.5826i − 3.74151i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.79129 0.540094 0.270047 − 0.962847i \(-0.412961\pi\)
0.270047 + 0.962847i \(0.412961\pi\)
\(12\) − 5.79129i − 1.67180i
\(13\) − 2.20871i − 0.612587i −0.951937 − 0.306293i \(-0.900911\pi\)
0.951937 − 0.306293i \(-0.0990888\pi\)
\(14\) 5.00000 1.33631
\(15\) 0 0
\(16\) 17.9564 4.48911
\(17\) 5.00000i 1.21268i 0.795206 + 0.606339i \(0.207363\pi\)
−0.795206 + 0.606339i \(0.792637\pi\)
\(18\) − 2.79129i − 0.657913i
\(19\) 5.58258 1.28073 0.640365 − 0.768070i \(-0.278783\pi\)
0.640365 + 0.768070i \(0.278783\pi\)
\(20\) 0 0
\(21\) 1.79129 0.390891
\(22\) 5.00000i 1.06600i
\(23\) − 1.00000i − 0.208514i
\(24\) 10.5826 2.16016
\(25\) 0 0
\(26\) 6.16515 1.20909
\(27\) − 1.00000i − 0.192450i
\(28\) 10.3739i 1.96048i
\(29\) 5.58258 1.03666 0.518329 − 0.855181i \(-0.326554\pi\)
0.518329 + 0.855181i \(0.326554\pi\)
\(30\) 0 0
\(31\) 0.208712 0.0374858 0.0187429 − 0.999824i \(-0.494034\pi\)
0.0187429 + 0.999824i \(0.494034\pi\)
\(32\) 28.9564i 5.11882i
\(33\) 1.79129i 0.311823i
\(34\) −13.9564 −2.39351
\(35\) 0 0
\(36\) 5.79129 0.965215
\(37\) − 5.00000i − 0.821995i −0.911636 − 0.410997i \(-0.865181\pi\)
0.911636 − 0.410997i \(-0.134819\pi\)
\(38\) 15.5826i 2.52783i
\(39\) 2.20871 0.353677
\(40\) 0 0
\(41\) −7.00000 −1.09322 −0.546608 − 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 5.00000i 0.771517i
\(43\) − 11.0000i − 1.67748i −0.544529 − 0.838742i \(-0.683292\pi\)
0.544529 − 0.838742i \(-0.316708\pi\)
\(44\) −10.3739 −1.56392
\(45\) 0 0
\(46\) 2.79129 0.411553
\(47\) 12.1652i 1.77447i 0.461318 + 0.887235i \(0.347377\pi\)
−0.461318 + 0.887235i \(0.652623\pi\)
\(48\) 17.9564i 2.59179i
\(49\) 3.79129 0.541613
\(50\) 0 0
\(51\) −5.00000 −0.700140
\(52\) 12.7913i 1.77383i
\(53\) − 5.37386i − 0.738157i −0.929398 − 0.369078i \(-0.879673\pi\)
0.929398 − 0.369078i \(-0.120327\pi\)
\(54\) 2.79129 0.379846
\(55\) 0 0
\(56\) −18.9564 −2.53316
\(57\) 5.58258i 0.739430i
\(58\) 15.5826i 2.04609i
\(59\) −0.626136 −0.0815160 −0.0407580 − 0.999169i \(-0.512977\pi\)
−0.0407580 + 0.999169i \(0.512977\pi\)
\(60\) 0 0
\(61\) −2.20871 −0.282797 −0.141398 − 0.989953i \(-0.545160\pi\)
−0.141398 + 0.989953i \(0.545160\pi\)
\(62\) 0.582576i 0.0739872i
\(63\) 1.79129i 0.225681i
\(64\) −44.9129 −5.61411
\(65\) 0 0
\(66\) −5.00000 −0.615457
\(67\) 6.37386i 0.778691i 0.921092 + 0.389346i \(0.127299\pi\)
−0.921092 + 0.389346i \(0.872701\pi\)
\(68\) − 28.9564i − 3.51148i
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 16.5390 1.96282 0.981410 − 0.191923i \(-0.0614724\pi\)
0.981410 + 0.191923i \(0.0614724\pi\)
\(72\) 10.5826i 1.24717i
\(73\) 7.00000i 0.819288i 0.912245 + 0.409644i \(0.134347\pi\)
−0.912245 + 0.409644i \(0.865653\pi\)
\(74\) 13.9564 1.62240
\(75\) 0 0
\(76\) −32.3303 −3.70854
\(77\) − 3.20871i − 0.365667i
\(78\) 6.16515i 0.698066i
\(79\) 8.16515 0.918651 0.459326 − 0.888268i \(-0.348091\pi\)
0.459326 + 0.888268i \(0.348091\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 19.5390i − 2.15772i
\(83\) 12.1652i 1.33530i 0.744476 + 0.667649i \(0.232699\pi\)
−0.744476 + 0.667649i \(0.767301\pi\)
\(84\) −10.3739 −1.13188
\(85\) 0 0
\(86\) 30.7042 3.31092
\(87\) 5.58258i 0.598515i
\(88\) − 18.9564i − 2.02076i
\(89\) 14.5826 1.54575 0.772875 − 0.634558i \(-0.218818\pi\)
0.772875 + 0.634558i \(0.218818\pi\)
\(90\) 0 0
\(91\) −3.95644 −0.414748
\(92\) 5.79129i 0.603783i
\(93\) 0.208712i 0.0216424i
\(94\) −33.9564 −3.50234
\(95\) 0 0
\(96\) −28.9564 −2.95535
\(97\) − 14.9564i − 1.51860i −0.650743 − 0.759298i \(-0.725542\pi\)
0.650743 − 0.759298i \(-0.274458\pi\)
\(98\) 10.5826i 1.06900i
\(99\) −1.79129 −0.180031
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 1725.2.b.n.1174.4 4
5.2 odd 4 1725.2.a.x.1.1 ✓ 2
5.3 odd 4 1725.2.a.bb.1.2 yes 2
5.4 even 2 inner 1725.2.b.n.1174.1 4
15.2 even 4 5175.2.a.bn.1.2 2
15.8 even 4 5175.2.a.bg.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1725.2.a.x.1.1 ✓ 2 5.2 odd 4
1725.2.a.bb.1.2 yes 2 5.3 odd 4
1725.2.b.n.1174.1 4 5.4 even 2 inner
1725.2.b.n.1174.4 4 1.1 even 1 trivial
5175.2.a.bg.1.1 2 15.8 even 4
5175.2.a.bn.1.2 2 15.2 even 4