LMFDB - Embedded newform 1248.4.m.a.337.71

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1248,4,Mod(337,1248)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1248.337"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1248, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1248.m (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$73.6343836872$
Analytic rank:	$0$
Dimension:	$84$
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.71
Character	$\chi$	$=$	1248.337
Dual form			1248.4.m.a.337.69

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -3.15976 q^{5} +31.8225i q^{7} -9.00000 q^{9} -30.3274 q^{11} +(42.8769 - 18.9360i) q^{13} -9.47928i q^{15} +33.4761 q^{17} -32.4678 q^{19} -95.4676 q^{21} -92.1829 q^{23} -115.016 q^{25} -27.0000i q^{27} -11.6863i q^{29} +328.498i q^{31} -90.9822i q^{33} -100.551i q^{35} -271.273 q^{37} +(56.8079 + 128.631i) q^{39} -153.109i q^{41} +26.1234i q^{43} +28.4378 q^{45} -72.2968i q^{47} -669.673 q^{49} +100.428i q^{51} +666.675i q^{53} +95.8272 q^{55} -97.4033i q^{57} +512.543 q^{59} -527.725i q^{61} -286.403i q^{63} +(-135.481 + 59.8331i) q^{65} +863.838 q^{67} -276.549i q^{69} -810.528i q^{71} -157.904i q^{73} -345.048i q^{75} -965.094i q^{77} -796.996 q^{79} +81.0000 q^{81} +69.5892 q^{83} -105.776 q^{85} +35.0590 q^{87} -1637.05i q^{89} +(602.590 + 1364.45i) q^{91} -985.493 q^{93} +102.590 q^{95} +904.235i q^{97} +272.946 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 756 q^{9} - 104 q^{17} + 2188 q^{25} - 3396 q^{49} + 1616 q^{55} + 696 q^{65} - 3160 q^{79} + 6804 q^{81} + 2088 q^{87} - 2480 q^{95}+O(q^{100}) $ 84 * q - 756 * q^9 - 104 * q^17 + 2188 * q^25 - 3396 * q^49 + 1616 * q^55 + 696 * q^65 - 3160 * q^79 + 6804 * q^81 + 2088 * q^87 - 2480 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			3.00000i			0.577350i
$3$			3.00000i			0.577350i
$4$		0			0
$5$	−3.15976			−0.282617			−0.141309	−	0.989966i	$-0.545131\pi$
$5$	−3.15976			−0.282617			−0.141309	+	0.989966i	$0.545131\pi$
$6$		0			0
$7$			31.8225i			1.71825i	0.511762	+	0.859127i	$0.328993\pi$
$7$			31.8225i			1.71825i	−0.511762	+	0.859127i	$0.671007\pi$
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	−30.3274			−0.831277			−0.415639	−	0.909530i	$-0.636442\pi$
$11$	−30.3274			−0.831277			−0.415639	+	0.909530i	$0.636442\pi$
$12$		0			0
$13$	42.8769	−	18.9360i	0.914763	−	0.403991i
$13$	42.8769	−	18.9360i	0.914763	−	0.403991i
$14$		0			0
$15$		−	9.47928i		−	0.163169i
$16$		0			0
$17$	33.4761			0.477597			0.238799	−	0.971069i	$-0.423246\pi$
$17$	33.4761			0.477597			0.238799	+	0.971069i	$0.423246\pi$
$18$		0			0
$19$	−32.4678			−0.392032			−0.196016	−	0.980601i	$-0.562800\pi$
$19$	−32.4678			−0.392032			−0.196016	+	0.980601i	$0.562800\pi$
$20$		0			0
$21$	−95.4676			−0.992035
$22$		0			0
$23$	−92.1829			−0.835716			−0.417858	−	0.908512i	$-0.637219\pi$
$23$	−92.1829			−0.835716			−0.417858	+	0.908512i	$0.637219\pi$
$24$		0			0
$25$	−115.016			−0.920127
$26$		0			0
$27$		−	27.0000i		−	0.192450i
$28$		0			0
$29$		−	11.6863i		−	0.0748309i	−0.999300	−	0.0374155i	$-0.988088\pi$
$29$		−	11.6863i		−	0.0748309i	0.999300	−	0.0374155i	$-0.0119125\pi$
$30$		0			0
$31$			328.498i			1.90322i	0.307304	+	0.951611i	$0.400573\pi$
$31$			328.498i			1.90322i	−0.307304	+	0.951611i	$0.599427\pi$
$32$		0			0
$33$		−	90.9822i		−	0.479938i
$34$		0			0
$35$		−	100.551i		−	0.485609i
$36$		0			0
$37$	−271.273			−1.20532			−0.602662	−	0.797996i	$-0.705893\pi$
$37$	−271.273			−1.20532			−0.602662	+	0.797996i	$0.705893\pi$
$38$		0			0
$39$	56.8079	+	128.631i	0.233245	+	0.528139i
$40$		0			0
$41$		−	153.109i		−	0.583209i	−0.956539	−	0.291605i	$-0.905811\pi$
$41$		−	153.109i		−	0.583209i	0.956539	−	0.291605i	$-0.0941892\pi$
$42$		0			0
$43$			26.1234i			0.0926461i	0.998927	+	0.0463230i	$0.0147504\pi$
$43$			26.1234i			0.0926461i	−0.998927	+	0.0463230i	$0.985250\pi$
$44$		0			0
$45$	28.4378			0.0942058
$46$		0			0
$47$		−	72.2968i		−	0.224374i	−0.993687	−	0.112187i	$-0.964214\pi$
$47$		−	72.2968i		−	0.224374i	0.993687	−	0.112187i	$-0.0357855\pi$
$48$		0			0
$49$	−669.673			−1.95240
$50$		0			0
$51$			100.428i			0.275741i
$52$		0			0
$53$			666.675i			1.72783i	0.503641	+	0.863913i	$0.331993\pi$
$53$			666.675i			1.72783i	−0.503641	+	0.863913i	$0.668007\pi$
$54$		0			0
$55$	95.8272			0.234933
$56$		0			0
$57$		−	97.4033i		−	0.226340i
$58$		0			0
$59$	512.543			1.13097			0.565487	−	0.824757i	$-0.308688\pi$
$59$	512.543			1.13097			0.565487	+	0.824757i	$0.308688\pi$
$60$		0			0
$61$		−	527.725i		−	1.10768i	−0.832624	−	0.553838i	$-0.813163\pi$
$61$		−	527.725i		−	1.10768i	0.832624	−	0.553838i	$-0.186837\pi$
$62$		0			0
$63$		−	286.403i		−	0.572752i
$64$		0			0
$65$	−135.481	+	59.8331i	−0.258528	+	0.114175i
$66$		0			0
$67$	863.838			1.57514			0.787572	−	0.616223i	$-0.211338\pi$
$67$	863.838			1.57514			0.787572	+	0.616223i	$0.211338\pi$
$68$		0			0
$69$		−	276.549i		−	0.482501i
$70$		0			0
$71$		−	810.528i		−	1.35482i	−0.735608	−	0.677408i	$-0.763103\pi$
$71$		−	810.528i		−	1.35482i	0.735608	−	0.677408i	$-0.236897\pi$
$72$		0			0
$73$		−	157.904i		−	0.253167i	−0.991956	−	0.126584i	$-0.959599\pi$
$73$		−	157.904i		−	0.253167i	0.991956	−	0.126584i	$-0.0404012\pi$
$74$		0			0
$75$		−	345.048i		−	0.531236i
$76$		0			0
$77$		−	965.094i		−	1.42835i
$78$		0			0
$79$	−796.996			−1.13505			−0.567526	−	0.823356i	$-0.692099\pi$
$79$	−796.996			−1.13505			−0.567526	+	0.823356i	$0.692099\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	69.5892			0.0920290			0.0460145	−	0.998941i	$-0.485348\pi$
$83$	69.5892			0.0920290			0.0460145	+	0.998941i	$0.485348\pi$
$84$		0			0
$85$	−105.776			−0.134977
$86$		0			0
$87$	35.0590			0.0432036
$88$		0			0
$89$		−	1637.05i		−	1.94975i	−0.222761	−	0.974873i	$-0.571507\pi$
$89$		−	1637.05i		−	1.94975i	0.222761	−	0.974873i	$-0.428493\pi$
$90$		0			0
$91$	602.590	+	1364.45i	0.694160	+	1.57180i
$92$		0			0
$93$	−985.493			−1.09883
$94$		0			0
$95$	102.590			0.110795
$96$		0			0
$97$			904.235i			0.946507i	0.880926	+	0.473253i	$0.156920\pi$
$97$			904.235i			0.946507i	−0.880926	+	0.473253i	$0.843080\pi$
$98$		0			0
$99$	272.946			0.277092
$100$		0			0
$101$		−	571.027i		−	0.562567i	−0.959625	−	0.281283i	$-0.909240\pi$
$101$		−	571.027i		−	0.562567i	0.959625	−	0.281283i	$-0.0907601\pi$
$102$		0			0
$103$	348.693			0.333570			0.166785	−	0.985993i	$-0.446661\pi$
$103$	348.693			0.333570			0.166785	+	0.985993i	$0.446661\pi$
$104$		0			0
$105$	301.654			0.280366
$106$		0			0
$107$		−	224.543i		−	0.202873i	−0.994842	−	0.101436i	$-0.967656\pi$
$107$		−	224.543i		−	0.202873i	0.994842	−	0.101436i	$-0.0323439\pi$
$108$		0			0
$109$	933.844			0.820606			0.410303	−	0.911949i	$-0.365423\pi$
$109$	933.844			0.820606			0.410303	+	0.911949i	$0.365423\pi$
$110$		0			0
$111$		−	813.819i		−	0.695895i
$112$		0			0
$113$	−39.1345			−0.0325793			−0.0162897	−	0.999867i	$-0.505185\pi$
$113$	−39.1345			−0.0325793			−0.0162897	+	0.999867i	$0.505185\pi$
$114$		0			0
$115$	291.276			0.236188
$116$		0			0
$117$	−385.892	+	170.424i	−0.304921	+	0.134664i
$118$		0			0
$119$			1065.29i			0.820634i
$120$		0			0
$121$	−411.250			−0.308978
$122$		0			0
$123$	459.327			0.336716
$124$		0			0
$125$	758.392			0.542661
$126$		0			0
$127$	−2654.51			−1.85472			−0.927360	−	0.374170i	$-0.877928\pi$
$127$	−2654.51			−1.85472			−0.927360	+	0.374170i	$0.877928\pi$
$128$		0			0
$129$	−78.3702			−0.0534892
$130$		0			0
$131$		−	1492.43i		−	0.995378i	−0.867356	−	0.497689i	$-0.834182\pi$
$131$		−	1492.43i		−	0.995378i	0.867356	−	0.497689i	$-0.165818\pi$
$132$		0			0
$133$		−	1033.21i		−	0.673611i
$134$		0			0
$135$			85.3135i			0.0543897i
$136$		0			0
$137$			1648.95i			1.02832i	0.857695	+	0.514158i	$0.171896\pi$
$137$			1648.95i			1.02832i	−0.857695	+	0.514158i	$0.828104\pi$
$138$		0			0
$139$		−	733.234i		−	0.447425i	−0.974655	−	0.223713i	$-0.928182\pi$
$139$		−	733.234i		−	0.447425i	0.974655	−	0.223713i	$-0.0718177\pi$
$140$		0			0
$141$	216.890			0.129542
$142$		0			0
$143$	−1300.34	+	574.278i	−0.760422	+	0.335829i
$144$		0			0
$145$			36.9260i			0.0211485i
$146$		0			0
$147$		−	2009.02i		−	1.12722i
$148$		0			0
$149$	−2501.73			−1.37550			−0.687750	−	0.725948i	$-0.741402\pi$
$149$	−2501.73			−1.37550			−0.687750	+	0.725948i	$0.741402\pi$
$150$		0			0
$151$			398.863i			0.214961i	0.994207	+	0.107480i	$0.0342782\pi$
$151$			398.863i			0.214961i	−0.994207	+	0.107480i	$0.965722\pi$
$152$		0			0
$153$	−301.285			−0.159199
$154$		0			0
$155$		−	1037.97i		−	0.537884i
$156$		0			0
$157$		−	1705.02i		−	0.866720i	−0.901221	−	0.433360i	$-0.857328\pi$
$157$		−	1705.02i		−	0.866720i	0.901221	−	0.433360i	$-0.142672\pi$
$158$		0			0
$159$	−2000.02			−0.997561
$160$		0			0
$161$		−	2933.49i		−	1.43597i
$162$		0			0
$163$	−1303.86			−0.626542			−0.313271	−	0.949664i	$-0.601425\pi$
$163$	−1303.86			−0.626542			−0.313271	+	0.949664i	$0.601425\pi$
$164$		0			0
$165$			287.482i			0.135639i
$166$		0			0
$167$			2448.11i			1.13437i	0.823590	+	0.567186i	$0.191968\pi$
$167$			2448.11i			1.13437i	−0.823590	+	0.567186i	$0.808032\pi$
$168$		0			0
$169$	1479.86	−	1623.83i	0.673582	−	0.739113i
$170$		0			0
$171$	292.210			0.130677
$172$		0			0
$173$		−	2513.15i		−	1.10446i	−0.833692	−	0.552229i	$-0.813777\pi$
$173$		−	2513.15i		−	1.10446i	0.833692	−	0.552229i	$-0.186223\pi$
$174$		0			0
$175$		−	3660.10i		−	1.58101i
$176$		0			0
$177$			1537.63i			0.652968i
$178$		0			0
$179$		−	2275.36i		−	0.950104i	−0.879958	−	0.475052i	$-0.842429\pi$
$179$		−	2275.36i		−	0.950104i	0.879958	−	0.475052i	$-0.157571\pi$
$180$		0			0
$181$		−	3625.71i		−	1.48893i	−0.667661	−	0.744466i	$-0.732704\pi$
$181$		−	3625.71i		−	1.48893i	0.667661	−	0.744466i	$-0.267296\pi$
$182$		0			0
$183$	1583.17			0.639517
$184$		0			0
$185$	857.158			0.340646
$186$		0			0
$187$	−1015.24			−0.397016
$188$		0			0
$189$	859.208			0.330678
$190$		0			0
$191$	−3729.40			−1.41283			−0.706414	−	0.707799i	$-0.749688\pi$
$191$	−3729.40			−1.41283			−0.706414	+	0.707799i	$0.749688\pi$
$192$		0			0
$193$		−	767.818i		−	0.286366i	−0.989696	−	0.143183i	$-0.954266\pi$
$193$		−	767.818i		−	0.286366i	0.989696	−	0.143183i	$-0.0457338\pi$
$194$		0			0
$195$	−179.499	−	406.442i	−0.0659190	−	0.149261i
$196$		0			0
$197$	3005.02			1.08680			0.543398	−	0.839475i	$-0.317137\pi$
$197$	3005.02			1.08680			0.543398	+	0.839475i	$0.317137\pi$
$198$		0			0
$199$	3534.47			1.25906			0.629528	−	0.776978i	$-0.283248\pi$
$199$	3534.47			1.25906			0.629528	+	0.776978i	$0.283248\pi$
$200$		0			0
$201$			2591.51i			0.909409i
$202$		0			0
$203$	371.888			0.128579
$204$		0			0
$205$			483.787i			0.164825i
$206$		0			0
$207$	829.646			0.278572
$208$		0			0
$209$	984.662			0.325888
$210$		0			0
$211$			2159.43i			0.704556i	0.935895	+	0.352278i	$0.114593\pi$
$211$			2159.43i			0.704556i	−0.935895	+	0.352278i	$0.885407\pi$
$212$		0			0
$213$	2431.58			0.782203
$214$		0			0
$215$		−	82.5437i		−	0.0261834i
$216$		0			0
$217$	−10453.6			−3.27022
$218$		0			0
$219$	473.711			0.146166
$220$		0			0
$221$	1435.35	−	633.902i	0.436888	−	0.192945i
$222$		0			0
$223$		−	3677.94i		−	1.10445i	−0.833694	−	0.552227i	$-0.813778\pi$
$223$		−	3677.94i		−	1.10445i	0.833694	−	0.552227i	$-0.186222\pi$
$224$		0			0
$225$	1035.14			0.306709
$226$		0			0
$227$	2985.71			0.872990			0.436495	−	0.899707i	$-0.356220\pi$
$227$	2985.71			0.872990			0.436495	+	0.899707i	$0.356220\pi$
$228$		0			0
$229$	111.331			0.0321264			0.0160632	−	0.999871i	$-0.494887\pi$
$229$	111.331			0.0321264			0.0160632	+	0.999871i	$0.494887\pi$
$230$		0			0
$231$	2895.28			0.824656
$232$		0			0
$233$	4372.15			1.22931			0.614654	−	0.788797i	$-0.289296\pi$
$233$	4372.15			1.22931			0.614654	+	0.788797i	$0.289296\pi$
$234$		0			0
$235$			228.440i			0.0634120i
$236$		0			0
$237$		−	2390.99i		−	0.655322i
$238$		0			0
$239$			3323.79i			0.899573i	0.893136	+	0.449787i	$0.148500\pi$
$239$			3323.79i			0.899573i	−0.893136	+	0.449787i	$0.851500\pi$
$240$		0			0
$241$			1479.10i			0.395342i	0.980268	+	0.197671i	$0.0633377\pi$
$241$			1479.10i			0.395342i	−0.980268	+	0.197671i	$0.936662\pi$
$242$		0			0
$243$			243.000i			0.0641500i
$244$		0			0
$245$	2116.00			0.551782
$246$		0			0
$247$	−1392.12	+	614.808i	−0.358617	+	0.158378i
$248$		0			0
$249$			208.768i			0.0531330i
$250$		0			0
$251$		−	3963.27i		−	0.996651i	−0.866990	−	0.498325i	$-0.833948\pi$
$251$		−	3963.27i		−	0.996651i	0.866990	−	0.498325i	$-0.166052\pi$
$252$		0			0
$253$	2795.67			0.694712
$254$		0			0
$255$		−	317.329i		−	0.0779292i
$256$		0			0
$257$	−4738.68			−1.15016			−0.575080	−	0.818097i	$-0.695029\pi$
$257$	−4738.68			−1.15016			−0.575080	+	0.818097i	$0.695029\pi$
$258$		0			0
$259$		−	8632.59i		−	2.07106i
$260$		0			0
$261$			105.177i			0.0249436i
$262$		0			0
$263$	−4228.54			−0.991417			−0.495709	−	0.868489i	$-0.665092\pi$
$263$	−4228.54			−0.991417			−0.495709	+	0.868489i	$0.665092\pi$
$264$		0			0
$265$		−	2106.53i		−	0.488314i
$266$		0			0
$267$	4911.16			1.12569
$268$		0			0
$269$			6051.15i			1.37154i	0.727817	+	0.685771i	$0.240535\pi$
$269$			6051.15i			1.37154i	−0.727817	+	0.685771i	$0.759465\pi$
$270$		0			0
$271$		−	3361.97i		−	0.753598i	−0.926295	−	0.376799i	$-0.877025\pi$
$271$		−	3361.97i		−	0.753598i	0.926295	−	0.376799i	$-0.122975\pi$
$272$		0			0
$273$	−4093.35	+	1807.77i	−0.907476	+	0.400774i
$274$		0			0
$275$	3488.13			0.764881
$276$		0			0
$277$			5266.13i			1.14228i	0.820853	+	0.571139i	$0.193498\pi$
$277$			5266.13i			1.14228i	−0.820853	+	0.571139i	$0.806502\pi$
$278$		0			0
$279$		−	2956.48i		−	0.634408i
$280$		0			0
$281$		−	5935.42i		−	1.26006i	−0.776570	−	0.630031i	$-0.783042\pi$
$281$		−	5935.42i		−	1.26006i	0.776570	−	0.630031i	$-0.216958\pi$
$282$		0			0
$283$		−	173.311i		−	0.0364038i	−0.999834	−	0.0182019i	$-0.994206\pi$
$283$		−	173.311i		−	0.0364038i	0.999834	−	0.0182019i	$-0.00579417\pi$
$284$		0			0
$285$			307.771i			0.0639676i
$286$		0			0
$287$	4872.31			1.00210
$288$		0			0
$289$	−3792.35			−0.771901
$290$		0			0
$291$	−2712.70			−0.546466
$292$		0			0
$293$	8551.32			1.70503			0.852515	−	0.522703i	$-0.175077\pi$
$293$	8551.32			1.70503			0.852515	+	0.522703i	$0.175077\pi$
$294$		0			0
$295$	−1619.51			−0.319633
$296$		0			0
$297$			818.839i			0.159979i
$298$		0			0
$299$	−3952.52	+	1745.57i	−0.764482	+	0.337622i
$300$		0			0
$301$	−831.313			−0.159190
$302$		0			0
$303$	1713.08			0.324798
$304$		0			0
$305$			1667.48i			0.313049i
$306$		0			0
$307$	−8547.36			−1.58900			−0.794501	−	0.607263i	$-0.792267\pi$
$307$	−8547.36			−1.58900			−0.794501	+	0.607263i	$0.792267\pi$
$308$		0			0
$309$			1046.08i			0.192587i
$310$		0			0
$311$	5897.36			1.07527			0.537634	−	0.843178i	$-0.319318\pi$
$311$	5897.36			1.07527			0.537634	+	0.843178i	$0.319318\pi$
$312$		0			0
$313$	2734.59			0.493829			0.246914	−	0.969037i	$-0.420583\pi$
$313$	2734.59			0.493829			0.246914	+	0.969037i	$0.420583\pi$
$314$		0			0
$315$			904.963i			0.161870i
$316$		0			0
$317$	−571.593			−0.101274			−0.0506370	−	0.998717i	$-0.516125\pi$
$317$	−571.593			−0.101274			−0.0506370	+	0.998717i	$0.516125\pi$
$318$		0			0
$319$			354.416i			0.0622052i
$320$		0			0
$321$	673.629			0.117129
$322$		0			0
$323$	−1086.89			−0.187234
$324$		0			0
$325$	−4931.53	+	2177.94i	−0.841698	+	0.371724i
$326$		0			0
$327$			2801.53i			0.473777i
$328$		0			0
$329$	2300.67			0.385531
$330$		0			0
$331$	−606.517			−0.100717			−0.0503583	−	0.998731i	$-0.516036\pi$
$331$	−606.517			−0.100717			−0.0503583	+	0.998731i	$0.516036\pi$
$332$		0			0
$333$	2441.46			0.401775
$334$		0			0
$335$	−2729.52			−0.445163
$336$		0			0
$337$	−2.45017			−0.000396051			−0.000198026	−	1.00000i	$-0.500063\pi$
$337$	−2.45017			−0.000396051			−0.000198026		1.00000i	$0.500063\pi$
$338$		0			0
$339$		−	117.404i		−	0.0188097i
$340$		0			0
$341$		−	9962.47i		−	1.58211i
$342$		0			0
$343$		−	10395.6i		−	1.63646i
$344$		0			0
$345$			873.827i			0.136363i
$346$		0			0
$347$		−	902.500i		−	0.139622i	−0.997560	−	0.0698109i	$-0.977760\pi$
$347$		−	902.500i		−	0.139622i	0.997560	−	0.0698109i	$-0.0222396\pi$
$348$		0			0
$349$	−4953.37			−0.759735			−0.379868	−	0.925041i	$-0.624030\pi$
$349$	−4953.37			−0.759735			−0.379868	+	0.925041i	$0.624030\pi$
$350$		0			0
$351$	−511.271	−	1157.68i	−0.0777482	−	0.176046i
$352$		0			0
$353$		−	4514.62i		−	0.680705i	−0.940298	−	0.340352i	$-0.889454\pi$
$353$		−	4514.62i		−	0.680705i	0.940298	−	0.340352i	$-0.110546\pi$
$354$		0			0
$355$			2561.07i			0.382895i
$356$		0			0
$357$	−3195.88			−0.473793
$358$		0			0
$359$			3303.93i			0.485724i	0.970061	+	0.242862i	$0.0780863\pi$
$359$			3303.93i			0.485724i	−0.970061	+	0.242862i	$0.921914\pi$
$360$		0			0
$361$	−5804.85			−0.846311
$362$		0			0
$363$		−	1233.75i		−	0.178388i
$364$		0			0
$365$			498.937i			0.0715495i
$366$		0			0
$367$	−12769.7			−1.81627			−0.908134	−	0.418679i	$-0.862493\pi$
$367$	−12769.7			−1.81627			−0.908134	+	0.418679i	$0.862493\pi$
$368$		0			0
$369$			1377.98i			0.194403i
$370$		0			0
$371$	−21215.3			−2.96885
$372$		0			0
$373$			8952.31i			1.24272i	0.783527	+	0.621358i	$0.213419\pi$
$373$			8952.31i			1.24272i	−0.783527	+	0.621358i	$0.786581\pi$
$374$		0			0
$375$			2275.18i			0.313306i
$376$		0			0
$377$	−221.292	−	501.074i	−0.0302310	−	0.0684525i
$378$		0			0
$379$	−2688.52			−0.364380			−0.182190	−	0.983263i	$-0.558319\pi$
$379$	−2688.52			−0.364380			−0.182190	+	0.983263i	$0.558319\pi$
$380$		0			0
$381$		−	7963.52i		−	1.07082i
$382$		0			0
$383$			12013.3i			1.60274i	0.598168	+	0.801371i	$0.295896\pi$
$383$			12013.3i			1.60274i	−0.598168	+	0.801371i	$0.704104\pi$
$384$		0			0
$385$			3049.46i			0.403676i
$386$		0			0
$387$		−	235.111i		−	0.0308820i
$388$		0			0
$389$		−	11974.3i		−	1.56073i	−0.625325	−	0.780364i	$-0.715034\pi$
$389$		−	11974.3i		−	1.56073i	0.625325	−	0.780364i	$-0.284966\pi$
$390$		0			0
$391$	−3085.93			−0.399136
$392$		0			0
$393$	4477.30			0.574681
$394$		0			0
$395$	2518.31			0.320785
$396$		0			0
$397$	3566.63			0.450891			0.225446	−	0.974256i	$-0.427616\pi$
$397$	3566.63			0.450891			0.225446	+	0.974256i	$0.427616\pi$
$398$		0			0
$399$	3099.62			0.388910
$400$		0			0
$401$			7221.88i			0.899361i	0.893190	+	0.449680i	$0.148462\pi$
$401$			7221.88i			0.899361i	−0.893190	+	0.449680i	$0.851538\pi$
$402$		0			0
$403$	6220.42	+	14085.0i	0.768886	+	1.74100i
$404$		0			0
$405$	−255.940			−0.0314019
$406$		0			0
$407$	8227.00			1.00196
$408$		0			0
$409$		−	4995.74i		−	0.603969i	−0.953313	−	0.301984i	$-0.902351\pi$
$409$		−	4995.74i		−	0.603969i	0.953313	−	0.301984i	$-0.0976491\pi$
$410$		0			0
$411$	−4946.85			−0.593699
$412$		0			0
$413$			16310.4i			1.94330i
$414$		0			0
$415$	−219.885			−0.0260090
$416$		0			0
$417$	2199.70			0.258321
$418$		0			0
$419$		−	4651.46i		−	0.542336i	−0.962532	−	0.271168i	$-0.912590\pi$
$419$		−	4651.46i		−	0.542336i	0.962532	−	0.271168i	$-0.0874099\pi$
$420$		0			0
$421$	−4982.78			−0.576831			−0.288416	−	0.957505i	$-0.593129\pi$
$421$	−4982.78			−0.576831			−0.288416	+	0.957505i	$0.593129\pi$
$422$		0			0
$423$			650.671i			0.0747913i
$424$		0			0
$425$	−3850.29			−0.439450
$426$		0			0
$427$	16793.5			1.90327
$428$		0			0
$429$	−1722.83	−	3901.03i	−0.193891	−	0.439030i
$430$		0			0
$431$		−	1566.50i		−	0.175071i	−0.996161	−	0.0875354i	$-0.972101\pi$
$431$		−	1566.50i		−	0.175071i	0.996161	−	0.0875354i	$-0.0278991\pi$
$432$		0			0
$433$	6832.62			0.758325			0.379163	−	0.925330i	$-0.376212\pi$
$433$	6832.62			0.758325			0.379163	+	0.925330i	$0.376212\pi$
$434$		0			0
$435$	−110.778			−0.0122101
$436$		0			0
$437$	2992.97			0.327628
$438$		0			0
$439$	−11087.7			−1.20543			−0.602717	−	0.797955i	$-0.705915\pi$
$439$	−11087.7			−1.20543			−0.602717	+	0.797955i	$0.705915\pi$
$440$		0			0
$441$	6027.06			0.650800
$442$		0			0
$443$		−	6400.66i		−	0.686467i	−0.939250	−	0.343233i	$-0.888478\pi$
$443$		−	6400.66i		−	0.686467i	0.939250	−	0.343233i	$-0.111522\pi$
$444$		0			0
$445$			5172.70i			0.551032i
$446$		0			0
$447$		−	7505.18i		−	0.794145i
$448$		0			0
$449$		−	1185.40i		−	0.124593i	−0.998058	−	0.0622965i	$-0.980158\pi$
$449$		−	1185.40i		−	0.124593i	0.998058	−	0.0622965i	$-0.0198424\pi$
$450$		0			0
$451$			4643.39i			0.484809i
$452$		0			0
$453$	−1196.59			−0.124108
$454$		0			0
$455$	−1904.04	−	4311.34i	−0.196182	−	0.444217i
$456$		0			0
$457$			10739.6i			1.09930i	0.835396	+	0.549648i	$0.185238\pi$
$457$			10739.6i			1.09930i	−0.835396	+	0.549648i	$0.814762\pi$
$458$		0			0
$459$		−	903.855i		−	0.0919136i
$460$		0			0
$461$	−18672.2			−1.88645			−0.943223	−	0.332159i	$-0.892223\pi$
$461$	−18672.2			−1.88645			−0.943223	+	0.332159i	$0.892223\pi$
$462$		0			0
$463$			6961.59i			0.698775i	0.936978	+	0.349387i	$0.113610\pi$
$463$			6961.59i			0.698775i	−0.936978	+	0.349387i	$0.886390\pi$
$464$		0			0
$465$	3113.92			0.310547
$466$		0			0
$467$			8319.54i			0.824374i	0.911099	+	0.412187i	$0.135235\pi$
$467$			8319.54i			0.824374i	−0.911099	+	0.412187i	$0.864765\pi$
$468$		0			0
$469$			27489.5i			2.70650i
$470$		0			0
$471$	5115.05			0.500401
$472$		0			0
$473$		−	792.255i		−	0.0770146i
$474$		0			0
$475$	3734.31			0.360720
$476$		0			0
$477$		−	6000.07i		−	0.575942i
$478$		0			0
$479$		−	10801.1i		−	1.03030i	−0.857100	−	0.515151i	$-0.827736\pi$
$479$		−	10801.1i		−	1.03030i	0.857100	−	0.515151i	$-0.172264\pi$
$480$		0			0
$481$	−11631.4	+	5136.82i	−1.10259	+	0.486941i
$482$		0			0
$483$	8800.48			0.829059
$484$		0			0
$485$		−	2857.16i		−	0.267499i
$486$		0			0
$487$			11483.4i			1.06851i	0.845324	+	0.534255i	$0.179408\pi$
$487$			11483.4i			1.06851i	−0.845324	+	0.534255i	$0.820592\pi$
$488$		0			0
$489$		−	3911.59i		−	0.361734i
$490$		0			0
$491$			18363.8i			1.68787i	0.536443	+	0.843936i	$0.319768\pi$
$491$			18363.8i			1.68787i	−0.536443	+	0.843936i	$0.680232\pi$
$492$		0			0
$493$		−	391.213i		−	0.0357390i
$494$		0			0
$495$	−862.445			−0.0783112
$496$		0			0
$497$	25793.0			2.32792
$498$		0			0
$499$	−76.2986			−0.00684488			−0.00342244	−	0.999994i	$-0.501089\pi$
$499$	−76.2986			−0.00684488			−0.00342244	+	0.999994i	$0.501089\pi$
$500$		0			0
$501$	−7344.32			−0.654930
$502$		0			0
$503$	−2934.86			−0.260157			−0.130079	−	0.991504i	$-0.541523\pi$
$503$	−2934.86			−0.260157			−0.130079	+	0.991504i	$0.541523\pi$
$504$		0			0
$505$			1804.31i			0.158991i
$506$		0			0
$507$	4871.49	+	4439.58i	0.426727	+	0.388893i
$508$		0			0
$509$	13001.0			1.13214			0.566069	−	0.824358i	$-0.308463\pi$
$509$	13001.0			1.13214			0.566069	+	0.824358i	$0.308463\pi$
$510$		0			0
$511$	5024.89			0.435006
$512$		0			0
$513$			876.629i			0.0754466i
$514$		0			0
$515$	−1101.78			−0.0942727
$516$		0			0
$517$			2192.57i			0.186517i
$518$		0			0
$519$	7539.45			0.637659
$520$		0			0
$521$	11843.0			0.995878			0.497939	−	0.867212i	$-0.334090\pi$
$521$	11843.0			0.995878			0.497939	+	0.867212i	$0.334090\pi$
$522$		0			0
$523$			5094.18i			0.425914i	0.977062	+	0.212957i	$0.0683094\pi$
$523$			5094.18i			0.425914i	−0.977062	+	0.212957i	$0.931691\pi$
$524$		0			0
$525$	10980.3			0.912798
$526$		0			0
$527$			10996.8i			0.908974i
$528$		0			0
$529$	−3669.31			−0.301579
$530$		0			0
$531$	−4612.89			−0.376991
$532$		0			0
$533$	−2899.26	−	6564.83i	−0.235612	−	0.533498i
$534$		0			0
$535$			709.502i			0.0573354i
$536$		0			0
$537$	6826.09			0.548543
$538$		0			0
$539$	20309.4			1.62299
$540$		0			0
$541$	−4124.52			−0.327776			−0.163888	−	0.986479i	$-0.552404\pi$
$541$	−4124.52			−0.327776			−0.163888	+	0.986479i	$0.552404\pi$
$542$		0			0
$543$	10877.1			0.859635
$544$		0			0
$545$	−2950.72			−0.231917
$546$		0			0
$547$			6416.94i			0.501588i	0.968040	+	0.250794i	$0.0806917\pi$
$547$			6416.94i			0.501588i	−0.968040	+	0.250794i	$0.919308\pi$
$548$		0			0
$549$			4749.52i			0.369225i
$550$		0			0
$551$			379.429i			0.0293361i
$552$		0			0
$553$		−	25362.4i		−	1.95031i
$554$		0			0
$555$			2571.47i			0.196672i
$556$		0			0
$557$	−14142.7			−1.07585			−0.537924	−	0.842994i	$-0.680791\pi$
$557$	−14142.7			−1.07585			−0.537924	+	0.842994i	$0.680791\pi$
$558$		0			0
$559$	494.672	+	1120.09i	0.0374282	+	0.0847492i
$560$		0			0
$561$		−	3045.73i		−	0.229217i
$562$		0			0
$563$			3830.12i			0.286714i	0.989671	+	0.143357i	$0.0457898\pi$
$563$			3830.12i			0.286714i	−0.989671	+	0.143357i	$0.954210\pi$
$564$		0			0
$565$	123.656			0.00920749
$566$		0			0
$567$			2577.62i			0.190917i
$568$		0			0
$569$	22320.1			1.64448			0.822238	−	0.569144i	$-0.192725\pi$
$569$	22320.1			1.64448			0.822238	+	0.569144i	$0.192725\pi$
$570$		0			0
$571$			19368.3i			1.41951i	0.704449	+	0.709754i	$0.251194\pi$
$571$			19368.3i			1.41951i	−0.704449	+	0.709754i	$0.748806\pi$
$572$		0			0
$573$		−	11188.2i		−	0.815696i
$574$		0			0
$575$	10602.5			0.768965
$576$		0			0
$577$			3965.92i			0.286141i	0.989712	+	0.143071i	$0.0456976\pi$
$577$			3965.92i			0.286141i	−0.989712	+	0.143071i	$0.954302\pi$
$578$		0			0
$579$	2303.45			0.165334
$580$		0			0
$581$			2214.50i			0.158129i
$582$		0			0
$583$		−	20218.5i		−	1.43630i
$584$		0			0
$585$	1219.33	−	538.497i	0.0861760	−	0.0380583i
$586$		0			0
$587$	−1149.02			−0.0807926			−0.0403963	−	0.999184i	$-0.512862\pi$
$587$	−1149.02			−0.0807926			−0.0403963	+	0.999184i	$0.512862\pi$
$588$		0			0
$589$		−	10665.6i		−	0.746125i
$590$		0			0
$591$			9015.07i			0.627463i
$592$		0			0
$593$			6509.04i			0.450749i	0.974272	+	0.225374i	$0.0723606\pi$
$593$			6509.04i			0.450749i	−0.974272	+	0.225374i	$0.927639\pi$
$594$		0			0
$595$		−	3366.07i		−	0.231925i
$596$		0			0
$597$			10603.4i			0.726917i
$598$		0			0
$599$	−11194.9			−0.763622			−0.381811	−	0.924240i	$-0.624699\pi$
$599$	−11194.9			−0.763622			−0.381811	+	0.924240i	$0.624699\pi$
$600$		0			0
$601$	−2151.98			−0.146058			−0.0730292	−	0.997330i	$-0.523267\pi$
$601$	−2151.98			−0.146058			−0.0730292	+	0.997330i	$0.523267\pi$
$602$		0			0
$603$	−7774.54			−0.525048
$604$		0			0
$605$	1299.45			0.0873225
$606$		0			0
$607$	7666.01			0.512609			0.256305	−	0.966596i	$-0.417495\pi$
$607$	7666.01			0.512609			0.256305	+	0.966596i	$0.417495\pi$
$608$		0			0
$609$			1115.67i			0.0742349i
$610$		0			0
$611$	−1369.01	−	3099.86i	−0.0906451	−	0.205249i
$612$		0			0
$613$	3102.78			0.204437			0.102219	−	0.994762i	$-0.467406\pi$
$613$	3102.78			0.204437			0.102219	+	0.994762i	$0.467406\pi$
$614$		0			0
$615$	−1451.36			−0.0951618
$616$		0			0
$617$			3290.36i			0.214692i	0.994222	+	0.107346i	$0.0342353\pi$
$617$			3290.36i			0.214692i	−0.994222	+	0.107346i	$0.965765\pi$
$618$		0			0
$619$	−28037.4			−1.82054			−0.910272	−	0.414011i	$-0.864127\pi$
$619$	−28037.4			−1.82054			−0.910272	+	0.414011i	$0.864127\pi$
$620$		0			0
$621$			2488.94i			0.160834i
$622$		0			0
$623$	52095.2			3.35016
$624$		0			0
$625$	11980.7			0.766762
$626$		0			0
$627$			2953.99i			0.188151i
$628$		0			0
$629$	−9081.17			−0.575660
$630$		0			0
$631$			19986.5i			1.26094i	0.776216	+	0.630468i	$0.217137\pi$
$631$			19986.5i			1.26094i	−0.776216	+	0.630468i	$0.782863\pi$
$632$		0			0
$633$	−6478.29			−0.406775
$634$		0			0
$635$	8387.61			0.524176
$636$		0			0
$637$	−28713.5	+	12680.9i	−1.78598	+	0.788752i
$638$		0			0
$639$			7294.75i			0.451605i
$640$		0			0
$641$	−20297.4			−1.25070			−0.625350	−	0.780344i	$-0.715044\pi$
$641$	−20297.4			−1.25070			−0.625350	+	0.780344i	$0.715044\pi$
$642$		0			0
$643$	−9815.56			−0.602003			−0.301002	−	0.953624i	$-0.597321\pi$
$643$	−9815.56			−0.602003			−0.301002	+	0.953624i	$0.597321\pi$
$644$		0			0
$645$	247.631			0.0151170
$646$		0			0
$647$	4770.57			0.289877			0.144938	−	0.989441i	$-0.453702\pi$
$647$	4770.57			0.289877			0.144938	+	0.989441i	$0.453702\pi$
$648$		0			0
$649$	−15544.1			−0.940152
$650$		0			0
$651$		−	31360.9i		−	1.88806i
$652$		0			0
$653$			23058.8i			1.38187i	0.722918	+	0.690933i	$0.242800\pi$
$653$			23058.8i			1.38187i	−0.722918	+	0.690933i	$0.757200\pi$
$654$		0			0
$655$			4715.73i			0.281311i
$656$		0			0
$657$			1421.13i			0.0843891i
$658$		0			0
$659$		−	8574.94i		−	0.506878i	−0.967351	−	0.253439i	$-0.918438\pi$
$659$		−	8574.94i		−	0.506878i	0.967351	−	0.253439i	$-0.0815616\pi$
$660$		0			0
$661$	−23616.3			−1.38967			−0.694833	−	0.719171i	$-0.744522\pi$
$661$	−23616.3			−1.38967			−0.694833	+	0.719171i	$0.744522\pi$
$662$		0			0
$663$	1901.71	+	4306.06i	0.111397	+	0.252237i
$664$		0			0
$665$			3264.68i			0.190374i
$666$		0			0
$667$			1077.28i			0.0625374i
$668$		0			0
$669$	11033.8			0.637657
$670$		0			0
$671$			16004.5i			0.920786i
$672$		0			0
$673$	−14768.2			−0.845874			−0.422937	−	0.906159i	$-0.639001\pi$
$673$	−14768.2			−0.845874			−0.422937	+	0.906159i	$0.639001\pi$
$674$		0			0
$675$			3105.43i			0.177079i
$676$		0			0
$677$		−	23267.6i		−	1.32089i	−0.750872	−	0.660447i	$-0.770367\pi$
$677$		−	23267.6i		−	1.32089i	0.750872	−	0.660447i	$-0.229633\pi$
$678$		0			0
$679$	−28775.0			−1.62634
$680$		0			0
$681$			8957.14i			0.504021i
$682$		0			0
$683$	13908.6			0.779206			0.389603	−	0.920983i	$-0.372612\pi$
$683$	13908.6			0.779206			0.389603	+	0.920983i	$0.372612\pi$
$684$		0			0
$685$		−	5210.28i		−	0.290620i
$686$		0			0
$687$			333.993i			0.0185482i
$688$		0			0
$689$	12624.1	+	28584.9i	0.698027	+	1.58055i
$690$		0			0
$691$	13598.7			0.748651			0.374326	−	0.927297i	$-0.377874\pi$
$691$	13598.7			0.748651			0.374326	+	0.927297i	$0.377874\pi$
$692$		0			0
$693$			8685.85i			0.476115i
$694$		0			0
$695$			2316.84i			0.126450i
$696$		0			0
$697$		−	5125.49i		−	0.278539i
$698$		0			0
$699$			13116.4i			0.709741i
$700$		0			0
$701$			12703.4i			0.684453i	0.939617	+	0.342227i	$0.111181\pi$
$701$			12703.4i			0.684453i	−0.939617	+	0.342227i	$0.888819\pi$
$702$		0			0
$703$	8807.63			0.472526
$704$		0			0
$705$	−685.321			−0.0366109
$706$		0			0
$707$	18171.5			0.966633
$708$		0			0
$709$	−3667.61			−0.194273			−0.0971367	−	0.995271i	$-0.530968\pi$
$709$	−3667.61			−0.194273			−0.0971367	+	0.995271i	$0.530968\pi$
$710$		0			0
$711$	7172.96			0.378350
$712$		0			0
$713$		−	30281.9i		−	1.59055i
$714$		0			0
$715$	4108.78	−	1814.58i	0.214908	−	0.0949111i
$716$		0			0
$717$	−9971.37			−0.519369
$718$		0			0
$719$	−13537.6			−0.702181			−0.351090	−	0.936342i	$-0.614189\pi$
$719$	−13537.6			−0.702181			−0.351090	+	0.936342i	$0.614189\pi$
$720$		0			0
$721$			11096.3i			0.573158i
$722$		0			0
$723$	−4437.31			−0.228251
$724$		0			0
$725$			1344.11i			0.0688540i
$726$		0			0
$727$	−11439.0			−0.583561			−0.291780	−	0.956485i	$-0.594248\pi$
$727$	−11439.0			−0.583561			−0.291780	+	0.956485i	$0.594248\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$			874.510i			0.0442475i
$732$		0			0
$733$	29164.6			1.46960			0.734802	−	0.678281i	$-0.237275\pi$
$733$	29164.6			1.46960			0.734802	+	0.678281i	$0.237275\pi$
$734$		0			0
$735$			6348.01i			0.318571i
$736$		0			0
$737$	−26197.9			−1.30938
$738$		0			0
$739$	−31446.7			−1.56534			−0.782671	−	0.622436i	$-0.786143\pi$
$739$	−31446.7			−1.56534			−0.782671	+	0.622436i	$0.786143\pi$
$740$		0			0
$741$	−1844.42	−	4176.35i	−0.0914394	−	0.207047i
$742$		0			0
$743$		−	574.677i		−	0.0283753i	−0.999899	−	0.0141876i	$-0.995484\pi$
$743$		−	574.677i		−	0.0283753i	0.999899	−	0.0141876i	$-0.00451622\pi$
$744$		0			0
$745$	7904.86			0.388740
$746$		0			0
$747$	−626.303			−0.0306763
$748$		0			0
$749$	7145.53			0.348587
$750$		0			0
$751$	10250.8			0.498080			0.249040	−	0.968493i	$-0.419885\pi$
$751$	10250.8			0.498080			0.249040	+	0.968493i	$0.419885\pi$
$752$		0			0
$753$	11889.8			0.575417
$754$		0			0
$755$		−	1260.31i		−	0.0607516i
$756$		0			0
$757$			873.168i			0.0419232i	0.999780	+	0.0209616i	$0.00667277\pi$
$757$			873.168i			0.0419232i	−0.999780	+	0.0209616i	$0.993327\pi$
$758$		0			0
$759$			8387.00i			0.401092i
$760$		0			0
$761$			25191.5i			1.19999i	0.800004	+	0.599995i	$0.204831\pi$
$761$			25191.5i			1.19999i	−0.800004	+	0.599995i	$0.795169\pi$
$762$		0			0
$763$			29717.3i			1.41001i
$764$		0			0
$765$	951.988			0.0449924
$766$		0			0
$767$	21976.3	−	9705.49i	1.03457	−	0.456903i
$768$		0			0
$769$		−	21012.5i		−	0.985347i	−0.870214	−	0.492673i	$-0.836020\pi$
$769$		−	21012.5i		−	0.985347i	0.870214	−	0.492673i	$-0.163980\pi$
$770$		0			0
$771$		−	14216.1i		−	0.664045i
$772$		0			0
$773$	−12118.6			−0.563874			−0.281937	−	0.959433i	$-0.590977\pi$
$773$	−12118.6			−0.563874			−0.281937	+	0.959433i	$0.590977\pi$
$774$		0			0
$775$		−	37782.5i		−	1.75121i
$776$		0			0
$777$	25897.8			1.19572
$778$		0			0
$779$			4971.10i			0.228637i
$780$		0			0
$781$			24581.2i			1.12623i
$782$		0			0
$783$	−315.531			−0.0144012
$784$		0			0
$785$			5387.44i			0.244950i
$786$		0			0
$787$	−37757.5			−1.71018			−0.855090	−	0.518480i	$-0.826498\pi$
$787$	−37757.5			−1.71018			−0.855090	+	0.518480i	$0.826498\pi$
$788$		0			0
$789$		−	12685.6i		−	0.572395i
$790$		0			0
$791$		−	1245.36i		−	0.0559796i
$792$		0			0
$793$	−9992.98	−	22627.2i	−0.447492	−	1.01326i
$794$		0			0
$795$	6319.59			0.281928
$796$		0			0
$797$		−	10568.9i		−	0.469724i	−0.972029	−	0.234862i	$-0.924536\pi$
$797$		−	10568.9i		−	0.469724i	0.972029	−	0.234862i	$-0.0754639\pi$
$798$		0			0
$799$		−	2420.22i		−	0.107160i
$800$		0			0
$801$			14733.5i			0.649915i
$802$		0			0
$803$			4788.81i			0.210452i
$804$		0			0
$805$			9269.13i			0.405831i
$806$		0			0
$807$	−18153.4			−0.791861
$808$		0			0
$809$	19569.5			0.850464			0.425232	−	0.905084i	$-0.360192\pi$
$809$	19569.5			0.850464			0.425232	+	0.905084i	$0.360192\pi$
$810$		0			0
$811$	−2364.54			−0.102380			−0.0511900	−	0.998689i	$-0.516301\pi$
$811$	−2364.54			−0.102380			−0.0511900	+	0.998689i	$0.516301\pi$
$812$		0			0
$813$	10085.9			0.435090
$814$		0			0
$815$	4119.89			0.177072
$816$		0			0
$817$		−	848.168i		−	0.0363203i
$818$		0			0
$819$	−5423.31	−	12280.1i	−0.231387	−	0.523932i
$820$		0			0
$821$	7292.36			0.309994			0.154997	−	0.987915i	$-0.450463\pi$
$821$	7292.36			0.309994			0.154997	+	0.987915i	$0.450463\pi$
$822$		0			0
$823$	−34525.2			−1.46230			−0.731151	−	0.682216i	$-0.761016\pi$
$823$	−34525.2			−1.46230			−0.731151	+	0.682216i	$0.761016\pi$
$824$		0			0
$825$			10464.4i			0.441604i
$826$		0			0
$827$	−45368.3			−1.90763			−0.953815	−	0.300394i	$-0.902882\pi$
$827$	−45368.3			−1.90763			−0.953815	+	0.300394i	$0.902882\pi$
$828$		0			0
$829$		−	31898.5i		−	1.33641i	−0.743979	−	0.668203i	$-0.767064\pi$
$829$		−	31898.5i		−	1.33641i	0.743979	−	0.668203i	$-0.232936\pi$
$830$		0			0
$831$	−15798.4			−0.659494
$832$		0			0
$833$	−22418.1			−0.932460
$834$		0			0
$835$		−	7735.42i		−	0.320593i
$836$		0			0
$837$	8869.43			0.366275
$838$		0			0
$839$		−	29527.6i		−	1.21502i	−0.794311	−	0.607511i	$-0.792168\pi$
$839$		−	29527.6i		−	1.21502i	0.794311	−	0.607511i	$-0.207832\pi$
$840$		0			0
$841$	24252.4			0.994400
$842$		0			0
$843$	17806.3			0.727497
$844$		0			0
$845$	−4676.00	+	5130.91i	−0.190366	+	0.208886i
$846$		0			0
$847$		−	13087.0i		−	0.530903i
$848$		0			0
$849$	519.933			0.0210177
$850$		0			0
$851$	25006.7			1.00731
$852$		0			0
$853$	39673.4			1.59249			0.796243	−	0.604977i	$-0.206818\pi$
$853$	39673.4			1.59249			0.796243	+	0.604977i	$0.206818\pi$
$854$		0			0
$855$	−923.312			−0.0369317
$856$		0			0
$857$	−38078.7			−1.51779			−0.758893	−	0.651216i	$-0.774259\pi$
$857$	−38078.7			−1.51779			−0.758893	+	0.651216i	$0.774259\pi$
$858$		0			0
$859$			44320.6i			1.76042i	0.474587	+	0.880209i	$0.342597\pi$
$859$			44320.6i			1.76042i	−0.474587	+	0.880209i	$0.657403\pi$
$860$		0			0
$861$			14616.9i			0.578564i
$862$		0			0
$863$		−	38185.8i		−	1.50621i	−0.657900	−	0.753106i	$-0.728555\pi$
$863$		−	38185.8i		−	1.50621i	0.657900	−	0.753106i	$-0.271445\pi$
$864$		0			0
$865$			7940.95i			0.312139i
$866$		0			0
$867$		−	11377.0i		−	0.445657i
$868$		0			0
$869$	24170.8			0.943542
$870$		0			0
$871$	37038.7	−	16357.6i	1.44088	−	0.636344i
$872$		0			0
$873$		−	8138.11i		−	0.315502i
$874$		0			0
$875$			24134.0i			0.932431i
$876$		0			0
$877$	−14880.5			−0.572953			−0.286476	−	0.958087i	$-0.592484\pi$
$877$	−14880.5			−0.572953			−0.286476	+	0.958087i	$0.592484\pi$
$878$		0			0
$879$			25654.0i			0.984399i
$880$		0			0
$881$	46557.3			1.78043			0.890213	−	0.455545i	$-0.150555\pi$
$881$	46557.3			1.78043			0.890213	+	0.455545i	$0.150555\pi$
$882$		0			0
$883$		−	36222.1i		−	1.38049i	−0.723577	−	0.690244i	$-0.757503\pi$
$883$		−	36222.1i		−	1.38049i	0.723577	−	0.690244i	$-0.242497\pi$
$884$		0			0
$885$		−	4858.54i		−	0.184540i
$886$		0			0
$887$	7013.25			0.265481			0.132741	−	0.991151i	$-0.457622\pi$
$887$	7013.25			0.265481			0.132741	+	0.991151i	$0.457622\pi$
$888$		0			0
$889$		−	84473.1i		−	3.18688i
$890$		0			0
$891$	−2456.52			−0.0923642
$892$		0			0
$893$			2347.31i			0.0879618i
$894$		0			0
$895$			7189.60i			0.268516i
$896$		0			0
$897$	−5236.71	−	11857.6i	−0.194926	−	0.441374i
$898$		0			0
$899$	3838.93			0.142420
$900$		0			0
$901$			22317.7i			0.825205i
$902$		0			0
$903$		−	2493.94i		−	0.0919081i
$904$		0			0
$905$			11456.4i			0.420798i
$906$		0			0
$907$		−	43949.2i		−	1.60894i	−0.593992	−	0.804471i	$-0.702449\pi$
$907$		−	43949.2i		−	1.60894i	0.593992	−	0.804471i	$-0.297551\pi$
$908$		0			0
$909$			5139.24i			0.187522i
$910$		0			0
$911$	−46103.2			−1.67669			−0.838346	−	0.545139i	$-0.816477\pi$
$911$	−46103.2			−1.67669			−0.838346	+	0.545139i	$0.816477\pi$
$912$		0			0
$913$	−2110.46			−0.0765016
$914$		0			0
$915$	−5002.45			−0.180739
$916$		0			0
$917$	47493.0			1.71031
$918$		0			0
$919$	6268.85			0.225017			0.112508	−	0.993651i	$-0.464111\pi$
$919$	6268.85			0.225017			0.112508	+	0.993651i	$0.464111\pi$
$920$		0			0
$921$		−	25642.1i		−	0.917410i
$922$		0			0
$923$	−15348.1	−	34752.9i	−0.547334	−	1.23934i
$924$		0			0
$925$	31200.7			1.10905
$926$		0			0
$927$	−3138.23			−0.111190
$928$		0			0
$929$			50758.1i			1.79259i	0.443456	+	0.896296i	$0.353752\pi$
$929$			50758.1i			1.79259i	−0.443456	+	0.896296i	$0.646248\pi$
$930$		0			0
$931$	21742.8			0.765403
$932$		0			0
$933$			17692.1i			0.620807i
$934$		0			0
$935$	3207.92			0.112204
$936$		0			0
$937$	25483.1			0.888470			0.444235	−	0.895910i	$-0.353476\pi$
$937$	25483.1			0.888470			0.444235	+	0.895910i	$0.353476\pi$
$938$		0			0
$939$			8203.78i			0.285112i
$940$		0			0
$941$	−13135.7			−0.455062			−0.227531	−	0.973771i	$-0.573065\pi$
$941$	−13135.7			−0.455062			−0.227531	+	0.973771i	$0.573065\pi$
$942$		0			0
$943$			14114.0i			0.487397i
$944$		0			0
$945$	−2714.89			−0.0934554
$946$		0			0
$947$	18483.8			0.634258			0.317129	−	0.948382i	$-0.397281\pi$
$947$	18483.8			0.634258			0.317129	+	0.948382i	$0.397281\pi$
$948$		0			0
$949$	−2990.06	−	6770.42i	−0.102277	−	0.231588i
$950$		0			0
$951$		−	1714.78i		−	0.0584706i
$952$		0			0
$953$	−34098.4			−1.15903			−0.579514	−	0.814962i	$-0.696758\pi$
$953$	−34098.4			−1.15903			−0.579514	+	0.814962i	$0.696758\pi$
$954$		0			0
$955$	11784.0			0.399290
$956$		0			0
$957$	−1063.25			−0.0359142
$958$		0			0
$959$	−52473.7			−1.76691
$960$		0			0
$961$	−78119.7			−2.62226
$962$		0			0
$963$			2020.89i			0.0676243i
$964$		0			0
$965$			2426.12i			0.0809322i
$966$		0			0
$967$			37439.7i			1.24507i	0.782593	+	0.622533i	$0.213896\pi$
$967$			37439.7i			1.24507i	−0.782593	+	0.622533i	$0.786104\pi$
$968$		0			0
$969$		−	3260.68i		−	0.108099i
$970$		0			0
$971$		−	51229.6i		−	1.69314i	−0.532279	−	0.846569i	$-0.678664\pi$
$971$		−	51229.6i		−	1.69314i	0.532279	−	0.846569i	$-0.321336\pi$
$972$		0			0
$973$	23333.4			0.768791
$974$		0			0
$975$	−6533.81	−	14794.6i	−0.214615	−	0.485955i
$976$		0			0
$977$			43308.5i			1.41818i	0.705118	+	0.709090i	$0.250894\pi$
$977$			43308.5i			1.41818i	−0.705118	+	0.709090i	$0.749106\pi$
$978$		0			0
$979$			49647.6i			1.62078i
$980$		0			0
$981$	−8404.59			−0.273535
$982$		0			0
$983$			6203.56i			0.201285i	0.994923	+	0.100642i	$0.0320898\pi$
$983$			6203.56i			0.201285i	−0.994923	+	0.100642i	$0.967910\pi$
$984$		0			0
$985$	−9495.15			−0.307148
$986$		0			0
$987$			6902.00i			0.222587i
$988$		0			0
$989$		−	2408.13i		−	0.0774258i
$990$		0			0
$991$	−41109.0			−1.31773			−0.658866	−	0.752261i	$-0.728963\pi$
$991$	−41109.0			−1.31773			−0.658866	+	0.752261i	$0.728963\pi$
$992$		0			0
$993$		−	1819.55i		−	0.0581488i
$994$		0			0
$995$	−11168.1			−0.355831
$996$		0			0
$997$		−	52762.0i		−	1.67602i	−0.545656	−	0.838009i	$-0.683720\pi$
$997$		−	52762.0i		−	1.67602i	0.545656	−	0.838009i	$-0.316280\pi$
$998$		0			0
$999$			7324.37i			0.231965i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1248.4.m.a.337.71		84
4.3	odd	2		312.4.m.a.181.17	✓	84
8.3	odd	2		312.4.m.a.181.67	yes	84
8.5	even	2	inner	1248.4.m.a.337.70		84
13.12	even	2	inner	1248.4.m.a.337.72		84
52.51	odd	2		312.4.m.a.181.68	yes	84
104.51	odd	2		312.4.m.a.181.18	yes	84
104.77	even	2	inner	1248.4.m.a.337.69		84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.m.a.181.17	✓	84	4.3	odd	2
312.4.m.a.181.18	yes	84	104.51	odd	2
312.4.m.a.181.67	yes	84	8.3	odd	2
312.4.m.a.181.68	yes	84	52.51	odd	2
1248.4.m.a.337.69		84	104.77	even	2	inner
1248.4.m.a.337.70		84	8.5	even	2	inner
1248.4.m.a.337.71		84	1.1	even	1	trivial
1248.4.m.a.337.72		84	13.12	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1248.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} -3.15976 q^{5} +31.8225i q^{7} -9.00000 q^{9} -30.3274 q^{11} +(42.8769 - 18.9360i) q^{13} -9.47928i q^{15} +33.4761 q^{17} -32.4678 q^{19} -95.4676 q^{21} -92.1829 q^{23} -115.016 q^{25} -27.0000i q^{27} -11.6863i q^{29} +328.498i q^{31} -90.9822i q^{33} -100.551i q^{35} -271.273 q^{37} +(56.8079 + 128.631i) q^{39} -153.109i q^{41} +26.1234i q^{43} +28.4378 q^{45} -72.2968i q^{47} -669.673 q^{49} +100.428i q^{51} +666.675i q^{53} +95.8272 q^{55} -97.4033i q^{57} +512.543 q^{59} -527.725i q^{61} -286.403i q^{63} +(-135.481 + 59.8331i) q^{65} +863.838 q^{67} -276.549i q^{69} -810.528i q^{71} -157.904i q^{73} -345.048i q^{75} -965.094i q^{77} -796.996 q^{79} +81.0000 q^{81} +69.5892 q^{83} -105.776 q^{85} +35.0590 q^{87} -1637.05i q^{89} +(602.590 + 1364.45i) q^{91} -985.493 q^{93} +102.590 q^{95} +904.235i q^{97} +272.946 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 756 q^{9} - 104 q^{17} + 2188 q^{25} - 3396 q^{49} + 1616 q^{55} + 696 q^{65} - 3160 q^{79} + 6804 q^{81} + 2088 q^{87} - 2480 q^{95}+O(q^{100}) \) 84 * q - 756 * q^9 - 104 * q^17 + 2188 * q^25 - 3396 * q^49 + 1616 * q^55 + 696 * q^65 - 3160 * q^79 + 6804 * q^81 + 2088 * q^87 - 2480 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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