Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [425,2,Mod(84,425)] 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("425.84"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([7, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.q (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(44\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 84.20
Character \(\chi\) \(=\) 425.84
Dual form 425.2.q.a.339.20

$q$-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+(-0.453067 + 0.623593i) q^{2} +(0.240491 + 0.740156i) q^{3} +(0.434435 + 1.33705i) q^{4} +(-2.23529 + 0.0588141i) q^{5} +(-0.570514 - 0.185371i) q^{6} -2.18212 q^{7} +(-2.49676 - 0.811247i) q^{8} +(1.93706 - 1.40735i) q^{9} +(0.976061 - 1.42056i) q^{10} +(-2.32047 + 3.19385i) q^{11} +(-0.885151 + 0.643100i) q^{12} +(-2.37819 - 3.27329i) q^{13} +(0.988648 - 1.36076i) q^{14} +(-0.581100 - 1.64032i) q^{15} +(-0.637648 + 0.463278i) q^{16} +(0.465129 - 4.09679i) q^{17} +1.84556i q^{18} +(-1.20332 + 3.70344i) q^{19} +(-1.04973 - 2.96316i) q^{20} +(-0.524782 - 1.61511i) q^{21} +(-0.940336 - 2.89406i) q^{22} +(-1.21493 - 0.882696i) q^{23} -2.04309i q^{24} +(4.99308 - 0.262933i) q^{25} +3.11868 q^{26} +(3.39635 + 2.46759i) q^{27} +(-0.947992 - 2.91762i) q^{28} +(-8.67033 + 2.81716i) q^{29} +(1.28617 + 0.380805i) q^{30} +(1.85861 + 0.603899i) q^{31} -5.85803i q^{32} +(-2.92200 - 0.949416i) q^{33} +(2.34399 + 2.14617i) q^{34} +(4.87769 - 0.128340i) q^{35} +(2.72324 + 1.97855i) q^{36} +(0.531257 - 0.385981i) q^{37} +(-1.76425 - 2.42829i) q^{38} +(1.85081 - 2.54743i) q^{39} +(5.62871 + 1.66653i) q^{40} +(-2.25941 - 3.10981i) q^{41} +(1.24493 + 0.404503i) q^{42} +10.0827i q^{43} +(-5.27845 - 1.71507i) q^{44} +(-4.24712 + 3.25978i) q^{45} +(1.10088 - 0.357699i) q^{46} +(4.67828 - 1.52006i) q^{47} +(-0.496247 - 0.360544i) q^{48} -2.23833 q^{49} +(-2.09824 + 3.23278i) q^{50} +(3.14412 - 0.640973i) q^{51} +(3.34340 - 4.60180i) q^{52} +(-9.85596 + 3.20240i) q^{53} +(-3.07754 + 0.999955i) q^{54} +(4.99909 - 7.27568i) q^{55} +(5.44824 + 1.77024i) q^{56} -3.03051 q^{57} +(2.17148 - 6.68312i) q^{58} +(-11.9770 + 8.70178i) q^{59} +(1.94075 - 1.48958i) q^{60} +(1.64612 - 2.26569i) q^{61} +(-1.21866 + 0.885409i) q^{62} +(-4.22690 + 3.07102i) q^{63} +(2.37773 + 1.72752i) q^{64} +(5.50846 + 7.17690i) q^{65} +(1.91591 - 1.39199i) q^{66} +(-4.45821 - 1.44856i) q^{67} +(5.67970 - 1.15789i) q^{68} +(0.361153 - 1.11152i) q^{69} +(-2.12989 + 3.09984i) q^{70} +(-7.01974 + 2.28085i) q^{71} +(-5.97808 + 1.94239i) q^{72} +(3.73499 + 2.71363i) q^{73} +0.506163i q^{74} +(1.39540 + 3.63243i) q^{75} -5.47447 q^{76} +(5.06356 - 6.96939i) q^{77} +(0.750015 + 2.30831i) q^{78} +(9.21871 - 2.99534i) q^{79} +(1.39808 - 1.07307i) q^{80} +(1.21006 - 3.72418i) q^{81} +2.96291 q^{82} +(-7.65726 - 2.48800i) q^{83} +(1.93151 - 1.40332i) q^{84} +(-0.798752 + 9.18488i) q^{85} +(-6.28753 - 4.56816i) q^{86} +(-4.17028 - 5.73989i) q^{87} +(8.38466 - 6.09181i) q^{88} +(13.3639 + 9.70947i) q^{89} +(-0.108545 - 4.12537i) q^{90} +(5.18950 + 7.14274i) q^{91} +(0.652405 - 2.00790i) q^{92} +1.52089i q^{93} +(-1.17167 + 3.60603i) q^{94} +(2.47196 - 8.34905i) q^{95} +(4.33585 - 1.40880i) q^{96} +(4.91324 + 15.1214i) q^{97} +(1.01411 - 1.39581i) q^{98} +9.45240i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 176 q - 10 q^{2} + 38 q^{4} - 10 q^{8} - 50 q^{9} - 10 q^{13} + 6 q^{15} - 58 q^{16} - 8 q^{19} - 44 q^{21} - 20 q^{25} - 64 q^{26} + 78 q^{30} - 20 q^{33} - 23 q^{34} - 46 q^{35} + 52 q^{36} - 10 q^{38}+ \cdots - 250 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(e\left(\frac{7}{10}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.453067 + 0.623593i −0.320366 + 0.440947i −0.938579 − 0.345064i \(-0.887857\pi\)
0.618213 + 0.786011i \(0.287857\pi\)
\(3\) 0.240491 + 0.740156i 0.138848 + 0.427329i 0.996169 − 0.0874531i \(-0.0278728\pi\)
−0.857321 + 0.514782i \(0.827873\pi\)
\(4\) 0.434435 + 1.33705i 0.217218 + 0.668527i
\(5\) −2.23529 + 0.0588141i −0.999654 + 0.0263024i
\(6\) −0.570514 − 0.185371i −0.232911 − 0.0756775i
\(7\) −2.18212 −0.824766 −0.412383 − 0.911011i \(-0.635303\pi\)
−0.412383 + 0.911011i \(0.635303\pi\)
\(8\) −2.49676 − 0.811247i −0.882738 − 0.286819i
\(9\) 1.93706 − 1.40735i 0.645685 − 0.469118i
\(10\) 0.976061 − 1.42056i 0.308658 − 0.449220i
\(11\) −2.32047 + 3.19385i −0.699648 + 0.962983i 0.300310 + 0.953842i \(0.402910\pi\)
−0.999958 + 0.00914152i \(0.997090\pi\)
\(12\) −0.885151 + 0.643100i −0.255521 + 0.185647i
\(13\) −2.37819 − 3.27329i −0.659590 − 0.907848i 0.339877 − 0.940470i \(-0.389614\pi\)
−0.999468 + 0.0326214i \(0.989614\pi\)
\(14\) 0.988648 − 1.36076i 0.264227 − 0.363678i
\(15\) −0.581100 − 1.64032i −0.150039 − 0.423529i
\(16\) −0.637648 + 0.463278i −0.159412 + 0.115820i
\(17\) 0.465129 − 4.09679i 0.112810 − 0.993617i
\(18\) 1.84556i 0.435003i
\(19\) −1.20332 + 3.70344i −0.276061 + 0.849628i 0.712876 + 0.701290i \(0.247392\pi\)
−0.988937 + 0.148338i \(0.952608\pi\)
\(20\) −1.04973 − 2.96316i −0.234727 − 0.662583i
\(21\) −0.524782 − 1.61511i −0.114517 − 0.352446i
\(22\) −0.940336 − 2.89406i −0.200480 − 0.617015i
\(23\) −1.21493 − 0.882696i −0.253330 − 0.184055i 0.453872 − 0.891067i \(-0.350042\pi\)
−0.707201 + 0.707012i \(0.750042\pi\)
\(24\) − 2.04309i − 0.417044i
\(25\) 4.99308 − 0.262933i 0.998616 − 0.0525867i
\(26\) 3.11868 0.611623
\(27\) 3.39635 + 2.46759i 0.653628 + 0.474888i
\(28\) −0.947992 − 2.91762i −0.179154 − 0.551378i
\(29\) −8.67033 + 2.81716i −1.61004 + 0.523134i −0.969563 − 0.244843i \(-0.921264\pi\)
−0.640478 + 0.767977i \(0.721264\pi\)
\(30\) 1.28617 + 0.380805i 0.234821 + 0.0695252i
\(31\) 1.85861 + 0.603899i 0.333816 + 0.108463i 0.471129 − 0.882064i \(-0.343847\pi\)
−0.137313 + 0.990528i \(0.543847\pi\)
\(32\) − 5.85803i − 1.03556i
\(33\) −2.92200 − 0.949416i −0.508655 − 0.165272i
\(34\) 2.34399 + 2.14617i 0.401991 + 0.368065i
\(35\) 4.87769 − 0.128340i 0.824480 − 0.0216934i
\(36\) 2.72324 + 1.97855i 0.453873 + 0.329758i
\(37\) 0.531257 − 0.385981i 0.0873381 − 0.0634548i −0.543259 − 0.839565i \(-0.682810\pi\)
0.630597 + 0.776110i \(0.282810\pi\)
\(38\) −1.76425 − 2.42829i −0.286200 − 0.393920i
\(39\) 1.85081 − 2.54743i 0.296367 − 0.407915i
\(40\) 5.62871 + 1.66653i 0.889977 + 0.263502i
\(41\) −2.25941 − 3.10981i −0.352860 − 0.485670i 0.595282 − 0.803517i \(-0.297040\pi\)
−0.948142 + 0.317847i \(0.897040\pi\)
\(42\) 1.24493 + 0.404503i 0.192097 + 0.0624162i
\(43\) 10.0827i 1.53760i 0.639487 + 0.768802i \(0.279147\pi\)
−0.639487 + 0.768802i \(0.720853\pi\)
\(44\) −5.27845 − 1.71507i −0.795757 − 0.258557i
\(45\) −4.24712 + 3.25978i −0.633123 + 0.485939i
\(46\) 1.10088 − 0.357699i 0.162317 − 0.0527399i
\(47\) 4.67828 − 1.52006i 0.682397 − 0.221724i 0.0527523 − 0.998608i \(-0.483201\pi\)
0.629644 + 0.776884i \(0.283201\pi\)
\(48\) −0.496247 − 0.360544i −0.0716270 − 0.0520401i
\(49\) −2.23833 −0.319762
\(50\) −2.09824 + 3.23278i −0.296735 + 0.457184i
\(51\) 3.14412 − 0.640973i 0.440265 − 0.0897541i
\(52\) 3.34340 − 4.60180i 0.463647 − 0.638155i
\(53\) −9.85596 + 3.20240i −1.35382 + 0.439883i −0.893975 − 0.448116i \(-0.852095\pi\)
−0.459845 + 0.887999i \(0.652095\pi\)
\(54\) −3.07754 + 0.999955i −0.418801 + 0.136077i
\(55\) 4.99909 − 7.27568i 0.674077 − 0.981052i
\(56\) 5.44824 + 1.77024i 0.728052 + 0.236558i
\(57\) −3.03051 −0.401401
\(58\) 2.17148 − 6.68312i 0.285129 − 0.877536i
\(59\) −11.9770 + 8.70178i −1.55927 + 1.13287i −0.622662 + 0.782491i \(0.713949\pi\)
−0.936606 + 0.350384i \(0.886051\pi\)
\(60\) 1.94075 − 1.48958i 0.250550 − 0.192304i
\(61\) 1.64612 − 2.26569i 0.210764 − 0.290092i −0.690526 − 0.723307i \(-0.742621\pi\)
0.901290 + 0.433215i \(0.142621\pi\)
\(62\) −1.21866 + 0.885409i −0.154770 + 0.112447i
\(63\) −4.22690 + 3.07102i −0.532539 + 0.386912i
\(64\) 2.37773 + 1.72752i 0.297216 + 0.215940i
\(65\) 5.50846 + 7.17690i 0.683241 + 0.890185i
\(66\) 1.91591 − 1.39199i 0.235832 − 0.171342i
\(67\) −4.45821 − 1.44856i −0.544657 − 0.176970i 0.0237490 − 0.999718i \(-0.492440\pi\)
−0.568406 + 0.822748i \(0.692440\pi\)
\(68\) 5.67970 − 1.15789i 0.688764 − 0.140414i
\(69\) 0.361153 − 1.11152i 0.0434777 − 0.133811i
\(70\) −2.12989 + 3.09984i −0.254570 + 0.370502i
\(71\) −7.01974 + 2.28085i −0.833090 + 0.270687i −0.694346 − 0.719641i \(-0.744306\pi\)
−0.138743 + 0.990328i \(0.544306\pi\)
\(72\) −5.97808 + 1.94239i −0.704523 + 0.228913i
\(73\) 3.73499 + 2.71363i 0.437147 + 0.317606i 0.784500 − 0.620128i \(-0.212919\pi\)
−0.347353 + 0.937734i \(0.612919\pi\)
\(74\) 0.506163i 0.0588402i
\(75\) 1.39540 + 3.63243i 0.161127 + 0.419436i
\(76\) −5.47447 −0.627965
\(77\) 5.06356 − 6.96939i 0.577046 − 0.794235i
\(78\) 0.750015 + 2.30831i 0.0849225 + 0.261364i
\(79\) 9.21871 − 2.99534i 1.03719 − 0.337002i 0.259558 − 0.965727i \(-0.416423\pi\)
0.777627 + 0.628725i \(0.216423\pi\)
\(80\) 1.39808 − 1.07307i 0.156310 − 0.119972i
\(81\) 1.21006 − 3.72418i 0.134451 − 0.413798i
\(82\) 2.96291 0.327199
\(83\) −7.65726 − 2.48800i −0.840494 − 0.273093i −0.143035 − 0.989718i \(-0.545686\pi\)
−0.697459 + 0.716625i \(0.745686\pi\)
\(84\) 1.93151 − 1.40332i 0.210745 − 0.153115i
\(85\) −0.798752 + 9.18488i −0.0866369 + 0.996240i
\(86\) −6.28753 − 4.56816i −0.678001 − 0.492597i
\(87\) −4.17028 − 5.73989i −0.447101 − 0.615381i
\(88\) 8.38466 − 6.09181i 0.893808 − 0.649390i
\(89\) 13.3639 + 9.70947i 1.41657 + 1.02920i 0.992325 + 0.123656i \(0.0394618\pi\)
0.424249 + 0.905546i \(0.360538\pi\)
\(90\) −0.108545 − 4.12537i −0.0114416 − 0.434852i
\(91\) 5.18950 + 7.14274i 0.544008 + 0.748762i
\(92\) 0.652405 − 2.00790i 0.0680180 − 0.209338i
\(93\) 1.52089i 0.157709i
\(94\) −1.17167 + 3.60603i −0.120849 + 0.371933i
\(95\) 2.47196 − 8.34905i 0.253618 − 0.856595i
\(96\) 4.33585 − 1.40880i 0.442526 − 0.143785i
\(97\) 4.91324 + 15.1214i 0.498864 + 1.53534i 0.810848 + 0.585257i \(0.199006\pi\)
−0.311984 + 0.950087i \(0.600994\pi\)
\(98\) 1.01411 − 1.39581i 0.102441 − 0.140998i
\(99\) 9.45240i 0.950002i
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 425.2.q.a.84.20 yes 176
17.16 even 2 inner 425.2.q.a.84.19 ✓ 176
25.14 even 10 inner 425.2.q.a.339.19 yes 176
425.339 even 10 inner 425.2.q.a.339.20 yes 176
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
425.2.q.a.84.19 ✓ 176 17.16 even 2 inner
425.2.q.a.84.20 yes 176 1.1 even 1 trivial
425.2.q.a.339.19 yes 176 25.14 even 10 inner
425.2.q.a.339.20 yes 176 425.339 even 10 inner