Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7406,2,Mod(1,7406)] 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("7406.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7406, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7406 = 2 \cdot 7 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7406.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,6,2,6,4,2,6] 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: \(59.1372077370\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.84770496.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 10x^{4} + 24x^{3} + 6x^{2} - 28x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.0360320\) of defining polynomial
Character \(\chi\) \(=\) 7406.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.0360320 q^{3} +1.00000 q^{4} -2.45350 q^{5} +0.0360320 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.99870 q^{9} -2.45350 q^{10} +2.96397 q^{11} +0.0360320 q^{12} -2.61396 q^{13} +1.00000 q^{14} -0.0884043 q^{15} +1.00000 q^{16} -6.94560 q^{17} -2.99870 q^{18} +7.92725 q^{19} -2.45350 q^{20} +0.0360320 q^{21} +2.96397 q^{22} +0.0360320 q^{24} +1.01964 q^{25} -2.61396 q^{26} -0.216145 q^{27} +1.00000 q^{28} -1.09041 q^{29} -0.0884043 q^{30} +5.77313 q^{31} +1.00000 q^{32} +0.106798 q^{33} -6.94560 q^{34} -2.45350 q^{35} -2.99870 q^{36} +1.32294 q^{37} +7.92725 q^{38} -0.0941863 q^{39} -2.45350 q^{40} +6.72272 q^{41} +0.0360320 q^{42} +3.65999 q^{43} +2.96397 q^{44} +7.35730 q^{45} -6.27495 q^{47} +0.0360320 q^{48} +1.00000 q^{49} +1.01964 q^{50} -0.250264 q^{51} -2.61396 q^{52} -7.89380 q^{53} -0.216145 q^{54} -7.27209 q^{55} +1.00000 q^{56} +0.285635 q^{57} -1.09041 q^{58} -6.68669 q^{59} -0.0884043 q^{60} +1.94725 q^{61} +5.77313 q^{62} -2.99870 q^{63} +1.00000 q^{64} +6.41335 q^{65} +0.106798 q^{66} +2.03542 q^{67} -6.94560 q^{68} -2.45350 q^{70} -14.8102 q^{71} -2.99870 q^{72} -2.21615 q^{73} +1.32294 q^{74} +0.0367398 q^{75} +7.92725 q^{76} +2.96397 q^{77} -0.0941863 q^{78} +13.1042 q^{79} -2.45350 q^{80} +8.98832 q^{81} +6.72272 q^{82} +7.38367 q^{83} +0.0360320 q^{84} +17.0410 q^{85} +3.65999 q^{86} -0.0392897 q^{87} +2.96397 q^{88} -6.09008 q^{89} +7.35730 q^{90} -2.61396 q^{91} +0.208018 q^{93} -6.27495 q^{94} -19.4495 q^{95} +0.0360320 q^{96} +2.05080 q^{97} +1.00000 q^{98} -8.88806 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 4 q^{5} + 2 q^{6} + 6 q^{7} + 6 q^{8} + 6 q^{9} + 4 q^{10} + 16 q^{11} + 2 q^{12} + 6 q^{14} + 6 q^{16} + 2 q^{17} + 6 q^{18} + 18 q^{19} + 4 q^{20} + 2 q^{21} + 16 q^{22}+ \cdots + 28 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0.0360320 0.0208031 0.0104015 − 0.999946i \(-0.496689\pi\)
0.0104015 + 0.999946i \(0.496689\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.45350 −1.09724 −0.548618 − 0.836073i \(-0.684846\pi\)
−0.548618 + 0.836073i \(0.684846\pi\)
\(6\) 0.0360320 0.0147100
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −2.99870 −0.999567
\(10\) −2.45350 −0.775864
\(11\) 2.96397 0.893670 0.446835 − 0.894616i \(-0.352551\pi\)
0.446835 + 0.894616i \(0.352551\pi\)
\(12\) 0.0360320 0.0104015
\(13\) −2.61396 −0.724983 −0.362492 − 0.931987i \(-0.618074\pi\)
−0.362492 + 0.931987i \(0.618074\pi\)
\(14\) 1.00000 0.267261
\(15\) −0.0884043 −0.0228259
\(16\) 1.00000 0.250000
\(17\) −6.94560 −1.68456 −0.842278 − 0.539044i \(-0.818786\pi\)
−0.842278 + 0.539044i \(0.818786\pi\)
\(18\) −2.99870 −0.706801
\(19\) 7.92725 1.81864 0.909318 − 0.416101i \(-0.136604\pi\)
0.909318 + 0.416101i \(0.136604\pi\)
\(20\) −2.45350 −0.548618
\(21\) 0.0360320 0.00786282
\(22\) 2.96397 0.631920
\(23\) 0 0
\(24\) 0.0360320 0.00735500
\(25\) 1.01964 0.203929
\(26\) −2.61396 −0.512641
\(27\) −0.216145 −0.0415971
\(28\) 1.00000 0.188982
\(29\) −1.09041 −0.202484 −0.101242 − 0.994862i \(-0.532282\pi\)
−0.101242 + 0.994862i \(0.532282\pi\)
\(30\) −0.0884043 −0.0161404
\(31\) 5.77313 1.03689 0.518443 − 0.855112i \(-0.326512\pi\)
0.518443 + 0.855112i \(0.326512\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.106798 0.0185911
\(34\) −6.94560 −1.19116
\(35\) −2.45350 −0.414717
\(36\) −2.99870 −0.499784
\(37\) 1.32294 0.217490 0.108745 − 0.994070i \(-0.465317\pi\)
0.108745 + 0.994070i \(0.465317\pi\)
\(38\) 7.92725 1.28597
\(39\) −0.0941863 −0.0150819
\(40\) −2.45350 −0.387932
\(41\) 6.72272 1.04991 0.524956 − 0.851129i \(-0.324082\pi\)
0.524956 + 0.851129i \(0.324082\pi\)
\(42\) 0.0360320 0.00555986
\(43\) 3.65999 0.558143 0.279071 − 0.960270i \(-0.409973\pi\)
0.279071 + 0.960270i \(0.409973\pi\)
\(44\) 2.96397 0.446835
\(45\) 7.35730 1.09676
\(46\) 0 0
\(47\) −6.27495 −0.915295 −0.457648 − 0.889134i \(-0.651308\pi\)
−0.457648 + 0.889134i \(0.651308\pi\)
\(48\) 0.0360320 0.00520077
\(49\) 1.00000 0.142857
\(50\) 1.01964 0.144200
\(51\) −0.250264 −0.0350439
\(52\) −2.61396 −0.362492
\(53\) −7.89380 −1.08430 −0.542148 − 0.840283i \(-0.682389\pi\)
−0.542148 + 0.840283i \(0.682389\pi\)
\(54\) −0.216145 −0.0294136
\(55\) −7.27209 −0.980568
\(56\) 1.00000 0.133631
\(57\) 0.285635 0.0378332
\(58\) −1.09041 −0.143178
\(59\) −6.68669 −0.870533 −0.435266 − 0.900302i \(-0.643346\pi\)
−0.435266 + 0.900302i \(0.643346\pi\)
\(60\) −0.0884043 −0.0114130
\(61\) 1.94725 0.249319 0.124660 − 0.992200i \(-0.460216\pi\)
0.124660 + 0.992200i \(0.460216\pi\)
\(62\) 5.77313 0.733189
\(63\) −2.99870 −0.377801
\(64\) 1.00000 0.125000
\(65\) 6.41335 0.795479
\(66\) 0.106798 0.0131459
\(67\) 2.03542 0.248666 0.124333 − 0.992241i \(-0.460321\pi\)
0.124333 + 0.992241i \(0.460321\pi\)
\(68\) −6.94560 −0.842278
\(69\) 0 0
\(70\) −2.45350 −0.293249
\(71\) −14.8102 −1.75765 −0.878824 − 0.477146i \(-0.841671\pi\)
−0.878824 + 0.477146i \(0.841671\pi\)
\(72\) −2.99870 −0.353400
\(73\) −2.21615 −0.259380 −0.129690 − 0.991555i \(-0.541398\pi\)
−0.129690 + 0.991555i \(0.541398\pi\)
\(74\) 1.32294 0.153789
\(75\) 0.0367398 0.00424235
\(76\) 7.92725 0.909318
\(77\) 2.96397 0.337776
\(78\) −0.0941863 −0.0106645
\(79\) 13.1042 1.47433 0.737166 − 0.675711i \(-0.236163\pi\)
0.737166 + 0.675711i \(0.236163\pi\)
\(80\) −2.45350 −0.274309
\(81\) 8.98832 0.998702
\(82\) 6.72272 0.742400
\(83\) 7.38367 0.810463 0.405232 − 0.914214i \(-0.367191\pi\)
0.405232 + 0.914214i \(0.367191\pi\)
\(84\) 0.0360320 0.00393141
\(85\) 17.0410 1.84836
\(86\) 3.65999 0.394667
\(87\) −0.0392897 −0.00421229
\(88\) 2.96397 0.315960
\(89\) −6.09008 −0.645548 −0.322774 − 0.946476i \(-0.604615\pi\)
−0.322774 + 0.946476i \(0.604615\pi\)
\(90\) 7.35730 0.775528
\(91\) −2.61396 −0.274018
\(92\) 0 0
\(93\) 0.208018 0.0215704
\(94\) −6.27495 −0.647212
\(95\) −19.4495 −1.99548
\(96\) 0.0360320 0.00367750
\(97\) 2.05080 0.208227 0.104113 − 0.994565i \(-0.466800\pi\)
0.104113 + 0.994565i \(0.466800\pi\)
\(98\) 1.00000 0.101015
\(99\) −8.88806 −0.893283
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 7406.2.a.bl.1.3 yes 6
23.22 odd 2 7406.2.a.bk.1.3 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.bk.1.3 ✓ 6 23.22 odd 2
7406.2.a.bl.1.3 yes 6 1.1 even 1 trivial