LMFDB - Embedded newform 624.6.c.d.337.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [624,6,Mod(337,624)] 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("624.337"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	624.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,54,0,0,0,0,0,486,0,0,0,530] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$100.079503563$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} - 2946x^{3} + 131769x^{2} - 1332936x + 6741792 $ x^6 - 2x^5 + 2x^4 - 2946x^3 + 131769x^2 - 1332936*x + 6741792
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.4
Root			$-15.0768 + 15.0768i$ of defining polynomial
Character	$\chi$	$=$	624.337
Dual form			624.6.c.d.337.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} +9.73803i q^{5} +105.184i q^{7} +81.0000 q^{9} +269.350i q^{11} +(227.464 + 565.290i) q^{13} +87.6423i q^{15} +1663.75 q^{17} -2.82742i q^{19} +946.657i q^{21} -2151.14 q^{23} +3030.17 q^{25} +729.000 q^{27} +220.829 q^{29} -788.106i q^{31} +2424.15i q^{33} -1024.29 q^{35} +980.445i q^{37} +(2047.18 + 5087.61i) q^{39} -14809.7i q^{41} -14142.2 q^{43} +788.781i q^{45} +25181.2i q^{47} +5743.31 q^{49} +14973.7 q^{51} -30900.9 q^{53} -2622.94 q^{55} -25.4468i q^{57} +25094.7i q^{59} +12060.8 q^{61} +8519.91i q^{63} +(-5504.81 + 2215.05i) q^{65} +14450.5i q^{67} -19360.2 q^{69} -33547.5i q^{71} +25805.7i q^{73} +27271.5 q^{75} -28331.4 q^{77} -15620.3 q^{79} +6561.00 q^{81} +2819.78i q^{83} +16201.6i q^{85} +1987.46 q^{87} +16227.2i q^{89} +(-59459.6 + 23925.6i) q^{91} -7092.95i q^{93} +27.5335 q^{95} +126289. i q^{97} +21817.4i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 54 q^{3} + 486 q^{9} + 530 q^{13} - 836 q^{17} + 416 q^{23} + 718 q^{25} + 4374 q^{27} + 18788 q^{29} - 6112 q^{35} + 4770 q^{39} + 24200 q^{43} - 3038 q^{49} - 7524 q^{51} - 42396 q^{53} - 124656 q^{55}+ \cdots - 567168 q^{95}+O(q^{100}) $ 6 * q + 54 * q^3 + 486 * q^9 + 530 * q^13 - 836 * q^17 + 416 * q^23 + 718 * q^25 + 4374 * q^27 + 18788 * q^29 - 6112 * q^35 + 4770 * q^39 + 24200 * q^43 - 3038 * q^49 - 7524 * q^51 - 42396 * q^53 - 124656 * q^55 - 3196 * q^61 - 17168 * q^65 + 3744 * q^69 + 6462 * q^75 - 114024 * q^77 + 169328 * q^79 + 39366 * q^81 + 169092 * q^87 - 236152 * q^91 - 567168 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	9.00000			0.577350
$3$	9.00000			0.577350
$4$		0			0
$5$			9.73803i			0.174199i	0.996200	+	0.0870996i	$0.0277598\pi$
$5$			9.73803i			0.174199i	−0.996200	+	0.0870996i	$0.972240\pi$
$6$		0			0
$7$			105.184i			0.811344i	0.914019	+	0.405672i	$0.132962\pi$
$7$			105.184i			0.811344i	−0.914019	+	0.405672i	$0.867038\pi$
$8$		0			0
$9$	81.0000			0.333333
$10$		0			0
$11$			269.350i			0.671175i	0.942009	+	0.335588i	$0.108935\pi$
$11$			269.350i			0.671175i	−0.942009	+	0.335588i	$0.891065\pi$
$12$		0			0
$13$	227.464	+	565.290i	0.373297	+	0.927712i
$13$	227.464	+	565.290i	0.373297	+	0.927712i
$14$		0			0
$15$			87.6423i			0.100574i
$16$		0			0
$17$	1663.75			1.39625			0.698127	−	0.715974i	$-0.254017\pi$
$17$	1663.75			1.39625			0.698127	+	0.715974i	$0.254017\pi$
$18$		0			0
$19$		−	2.82742i		−	0.00179683i	−1.00000		0.000898415i	$-0.999714\pi$
$19$		−	2.82742i		−	0.00179683i	1.00000		0.000898415i	$-0.000285974\pi$
$20$		0			0
$21$			946.657i			0.468430i
$22$		0			0
$23$	−2151.14			−0.847908			−0.423954	−	0.905684i	$-0.639358\pi$
$23$	−2151.14			−0.847908			−0.423954	+	0.905684i	$0.639358\pi$
$24$		0			0
$25$	3030.17			0.969655
$26$		0			0
$27$	729.000			0.192450
$28$		0			0
$29$	220.829			0.0487598			0.0243799	−	0.999703i	$-0.492239\pi$
$29$	220.829			0.0487598			0.0243799	+	0.999703i	$0.492239\pi$
$30$		0			0
$31$		−	788.106i		−	0.147292i	−0.997284	−	0.0736462i	$-0.976536\pi$
$31$		−	788.106i		−	0.147292i	0.997284	−	0.0736462i	$-0.0234636\pi$
$32$		0			0
$33$			2424.15i			0.387503i
$34$		0			0
$35$	−1024.29			−0.141335
$36$		0			0
$37$			980.445i			0.117739i	0.998266	+	0.0588693i	$0.0187495\pi$
$37$			980.445i			0.117739i	−0.998266	+	0.0588693i	$0.981250\pi$
$38$		0			0
$39$	2047.18	+	5087.61i	0.215523	+	0.535615i
$40$		0			0
$41$		−	14809.7i		−	1.37590i	−0.725759	−	0.687949i	$-0.758511\pi$
$41$		−	14809.7i		−	1.37590i	0.725759	−	0.687949i	$-0.241489\pi$
$42$		0			0
$43$	−14142.2			−1.16640			−0.583198	−	0.812330i	$-0.698199\pi$
$43$	−14142.2			−1.16640			−0.583198	+	0.812330i	$0.698199\pi$
$44$		0			0
$45$			788.781i			0.0580664i
$46$		0			0
$47$			25181.2i			1.66277i	0.555699	+	0.831383i	$0.312451\pi$
$47$			25181.2i			1.66277i	−0.555699	+	0.831383i	$0.687549\pi$
$48$		0			0
$49$	5743.31			0.341721
$50$		0			0
$51$	14973.7			0.806128
$52$		0			0
$53$	−30900.9			−1.51106			−0.755530	−	0.655114i	$-0.772621\pi$
$53$	−30900.9			−1.51106			−0.755530	+	0.655114i	$0.772621\pi$
$54$		0			0
$55$	−2622.94			−0.116918
$56$		0			0
$57$		−	25.4468i		−	0.00103740i
$58$		0			0
$59$			25094.7i			0.938538i	0.883055	+	0.469269i	$0.155483\pi$
$59$			25094.7i			0.938538i	−0.883055	+	0.469269i	$0.844517\pi$
$60$		0			0
$61$	12060.8			0.415003			0.207501	−	0.978235i	$-0.433467\pi$
$61$	12060.8			0.415003			0.207501	+	0.978235i	$0.433467\pi$
$62$		0			0
$63$			8519.91i			0.270448i
$64$		0			0
$65$	−5504.81	+	2215.05i	−0.161607	+	0.0650280i
$66$		0			0
$67$			14450.5i			0.393276i	0.980476	+	0.196638i	$0.0630023\pi$
$67$			14450.5i			0.393276i	−0.980476	+	0.196638i	$0.936998\pi$
$68$		0			0
$69$	−19360.2			−0.489540
$70$		0			0
$71$		−	33547.5i		−	0.789795i	−0.918725	−	0.394897i	$-0.870780\pi$
$71$		−	33547.5i		−	0.789795i	0.918725	−	0.394897i	$-0.129220\pi$
$72$		0			0
$73$			25805.7i			0.566772i	0.959006	+	0.283386i	$0.0914578\pi$
$73$			25805.7i			0.566772i	−0.959006	+	0.283386i	$0.908542\pi$
$74$		0			0
$75$	27271.5			0.559830
$76$		0			0
$77$	−28331.4			−0.544554
$78$		0			0
$79$	−15620.3			−0.281593			−0.140797	−	0.990039i	$-0.544966\pi$
$79$	−15620.3			−0.281593			−0.140797	+	0.990039i	$0.544966\pi$
$80$		0			0
$81$	6561.00			0.111111
$82$		0			0
$83$			2819.78i			0.0449284i	0.999748	+	0.0224642i	$0.00715117\pi$
$83$			2819.78i			0.0449284i	−0.999748	+	0.0224642i	$0.992849\pi$
$84$		0			0
$85$			16201.6i			0.243226i
$86$		0			0
$87$	1987.46			0.0281515
$88$		0			0
$89$			16227.2i			0.217154i	0.994088	+	0.108577i	$0.0346294\pi$
$89$			16227.2i			0.217154i	−0.994088	+	0.108577i	$0.965371\pi$
$90$		0			0
$91$	−59459.6	+	23925.6i	−0.752693	+	0.302872i
$92$		0			0
$93$		−	7092.95i		−	0.0850393i
$94$		0			0
$95$	27.5335			0.000313006
$96$		0			0
$97$			126289.i			1.36281i	0.731906	+	0.681406i	$0.238631\pi$
$97$			126289.i			1.36281i	−0.731906	+	0.681406i	$0.761369\pi$
$98$		0			0
$99$			21817.4i			0.223725i
$100$		0			0
$101$	92398.8			0.901287			0.450643	−	0.892704i	$-0.351195\pi$
$101$	92398.8			0.901287			0.450643	+	0.892704i	$0.351195\pi$
$102$		0			0
$103$	−148091.			−1.37542			−0.687711	−	0.725985i	$-0.741384\pi$
$103$	−148091.			−1.37542			−0.687711	+	0.725985i	$0.741384\pi$
$104$		0			0
$105$	−9218.57			−0.0816001
$106$		0			0
$107$	−88180.1			−0.744580			−0.372290	−	0.928117i	$-0.621427\pi$
$107$	−88180.1			−0.744580			−0.372290	+	0.928117i	$0.621427\pi$
$108$		0			0
$109$			61467.2i			0.495538i	0.968819	+	0.247769i	$0.0796974\pi$
$109$			61467.2i			0.495538i	−0.968819	+	0.247769i	$0.920303\pi$
$110$		0			0
$111$			8824.01i			0.0679764i
$112$		0			0
$113$	−82256.1			−0.605999			−0.303000	−	0.952991i	$-0.597988\pi$
$113$	−82256.1			−0.605999			−0.303000	+	0.952991i	$0.597988\pi$
$114$		0			0
$115$		−	20947.8i		−	0.147705i
$116$		0			0
$117$	18424.6	+	45788.5i	0.124432	+	0.309237i
$118$		0			0
$119$			175000.i			1.13284i
$120$		0			0
$121$	88501.4			0.549524
$122$		0			0
$123$		−	133287.i		−	0.794375i
$124$		0			0
$125$			59939.2i			0.343112i
$126$		0			0
$127$	143150.			0.787556			0.393778	−	0.919206i	$-0.371168\pi$
$127$	143150.			0.787556			0.393778	+	0.919206i	$0.371168\pi$
$128$		0			0
$129$	−127280.			−0.673419
$130$		0			0
$131$	141329.			0.719539			0.359769	−	0.933041i	$-0.382855\pi$
$131$	141329.			0.719539			0.359769	+	0.933041i	$0.382855\pi$
$132$		0			0
$133$	297.400			0.00145785
$134$		0			0
$135$			7099.02i			0.0335247i
$136$		0			0
$137$		−	3958.33i		−	0.0180182i	−0.999959	−	0.00900909i	$-0.997132\pi$
$137$		−	3958.33i		−	0.0180182i	0.999959	−	0.00900909i	$-0.00286772\pi$
$138$		0			0
$139$	445579.			1.95608			0.978042	−	0.208408i	$-0.0668283\pi$
$139$	445579.			1.95608			0.978042	+	0.208408i	$0.0668283\pi$
$140$		0			0
$141$			226631.i			0.959999i
$142$		0			0
$143$	−152261.	+	61267.5i	−0.622657	+	0.250547i
$144$		0			0
$145$			2150.44i			0.00849391i
$146$		0			0
$147$	51689.7			0.197293
$148$		0			0
$149$			49619.3i			0.183098i	0.995801	+	0.0915492i	$0.0291819\pi$
$149$			49619.3i			0.183098i	−0.995801	+	0.0915492i	$0.970818\pi$
$150$		0			0
$151$		−	402323.i		−	1.43593i	−0.696081	−	0.717964i	$-0.745074\pi$
$151$		−	402323.i		−	1.43593i	0.696081	−	0.717964i	$-0.254926\pi$
$152$		0			0
$153$	134763.			0.465418
$154$		0			0
$155$	7674.60			0.0256582
$156$		0			0
$157$	−562429.			−1.82104			−0.910519	−	0.413468i	$-0.864317\pi$
$157$	−562429.			−1.82104			−0.910519	+	0.413468i	$0.864317\pi$
$158$		0			0
$159$	−278108.			−0.872411
$160$		0			0
$161$		−	226265.i		−	0.687945i
$162$		0			0
$163$			313878.i			0.925319i	0.886536	+	0.462659i	$0.153105\pi$
$163$			313878.i			0.925319i	−0.886536	+	0.462659i	$0.846895\pi$
$164$		0			0
$165$	−23606.5			−0.0675027
$166$		0			0
$167$			602228.i			1.67097i	0.549510	+	0.835487i	$0.314814\pi$
$167$			602228.i			1.67097i	−0.549510	+	0.835487i	$0.685186\pi$
$168$		0			0
$169$	−267813.	+	257166.i	−0.721299	+	0.692624i
$170$		0			0
$171$		−	229.021i		−	0.000598943i
$172$		0			0
$173$	304021.			0.772304			0.386152	−	0.922435i	$-0.373804\pi$
$173$	304021.			0.772304			0.386152	+	0.922435i	$0.373804\pi$
$174$		0			0
$175$			318726.i			0.786723i
$176$		0			0
$177$			225852.i			0.541865i
$178$		0			0
$179$	−406190.			−0.947539			−0.473769	−	0.880649i	$-0.657107\pi$
$179$	−406190.			−0.947539			−0.473769	+	0.880649i	$0.657107\pi$
$180$		0			0
$181$	−240967.			−0.546714			−0.273357	−	0.961913i	$-0.588134\pi$
$181$	−240967.			−0.546714			−0.273357	+	0.961913i	$0.588134\pi$
$182$		0			0
$183$	108547.			0.239602
$184$		0			0
$185$	−9547.60			−0.0205100
$186$		0			0
$187$			448130.i			0.937131i
$188$		0			0
$189$			76679.2i			0.156143i
$190$		0			0
$191$	−226140.			−0.448532			−0.224266	−	0.974528i	$-0.571999\pi$
$191$	−226140.			−0.448532			−0.224266	+	0.974528i	$0.571999\pi$
$192$		0			0
$193$		−	406075.i		−	0.784717i	−0.919812	−	0.392359i	$-0.871659\pi$
$193$		−	406075.i		−	0.784717i	0.919812	−	0.392359i	$-0.128341\pi$
$194$		0			0
$195$	−49543.3	+	19935.5i	−0.0933037	+	0.0375439i
$196$		0			0
$197$		−	407390.i		−	0.747902i	−0.927449	−	0.373951i	$-0.878003\pi$
$197$		−	407390.i		−	0.747902i	0.927449	−	0.373951i	$-0.121997\pi$
$198$		0			0
$199$	−202865.			−0.363140			−0.181570	−	0.983378i	$-0.558118\pi$
$199$	−202865.			−0.363140			−0.181570	+	0.983378i	$0.558118\pi$
$200$		0			0
$201$			130055.i			0.227058i
$202$		0			0
$203$			23227.7i			0.0395609i
$204$		0			0
$205$	144217.			0.239680
$206$		0			0
$207$	−174242.			−0.282636
$208$		0			0
$209$	761.568			0.00120599
$210$		0			0
$211$	−657970.			−1.01742			−0.508710	−	0.860938i	$-0.669877\pi$
$211$	−657970.			−1.01742			−0.508710	+	0.860938i	$0.669877\pi$
$212$		0			0
$213$		−	301927.i		−	0.455988i
$214$		0			0
$215$		−	137717.i		−	0.203185i
$216$		0			0
$217$	82896.2			0.119505
$218$		0			0
$219$			232251.i			0.327226i
$220$		0			0
$221$	378442.	+	940499.i	0.521217	+	1.29532i
$222$		0			0
$223$			978060.i			1.31705i	0.752558	+	0.658527i	$0.228820\pi$
$223$			978060.i			1.31705i	−0.752558	+	0.658527i	$0.771180\pi$
$224$		0			0
$225$	245444.			0.323218
$226$		0			0
$227$		−	812137.i		−	1.04608i	−0.852308	−	0.523040i	$-0.824798\pi$
$227$		−	812137.i		−	1.04608i	0.852308	−	0.523040i	$-0.175202\pi$
$228$		0			0
$229$			1.36329e6i			1.71790i	0.512058	+	0.858951i	$0.328883\pi$
$229$			1.36329e6i			1.71790i	−0.512058	+	0.858951i	$0.671117\pi$
$230$		0			0
$231$	−254982.			−0.314398
$232$		0			0
$233$	931926.			1.12458			0.562292	−	0.826939i	$-0.309920\pi$
$233$	931926.			1.12458			0.562292	+	0.826939i	$0.309920\pi$
$234$		0			0
$235$	−245215.			−0.289653
$236$		0			0
$237$	−140583.			−0.162578
$238$		0			0
$239$			885874.i			1.00318i	0.865107	+	0.501588i	$0.167251\pi$
$239$			885874.i			1.00318i	−0.865107	+	0.501588i	$0.832749\pi$
$240$		0			0
$241$			1.32690e6i			1.47161i	0.677191	+	0.735807i	$0.263197\pi$
$241$			1.32690e6i			1.47161i	−0.677191	+	0.735807i	$0.736803\pi$
$242$		0			0
$243$	59049.0			0.0641500
$244$		0			0
$245$			55928.5i			0.0595275i
$246$		0			0
$247$	1598.32	−	643.137i	0.00166694	−	0.000670751i
$248$		0			0
$249$			25378.1i			0.0259394i
$250$		0			0
$251$	547596.			0.548625			0.274313	−	0.961641i	$-0.411550\pi$
$251$	547596.			0.548625			0.274313	+	0.961641i	$0.411550\pi$
$252$		0			0
$253$		−	579409.i		−	0.569094i
$254$		0			0
$255$			145814.i			0.140427i
$256$		0			0
$257$	−1.65509e6			−1.56311			−0.781554	−	0.623837i	$-0.785573\pi$
$257$	−1.65509e6			−1.56311			−0.781554	+	0.623837i	$0.785573\pi$
$258$		0			0
$259$	−103127.			−0.0955265
$260$		0			0
$261$	17887.2			0.0162533
$262$		0			0
$263$	668188.			0.595675			0.297838	−	0.954617i	$-0.403735\pi$
$263$	668188.			0.595675			0.297838	+	0.954617i	$0.403735\pi$
$264$		0			0
$265$		−	300914.i		−	0.263226i
$266$		0			0
$267$			146045.i			0.125374i
$268$		0			0
$269$	−384942.			−0.324350			−0.162175	−	0.986762i	$-0.551851\pi$
$269$	−384942.			−0.324350			−0.162175	+	0.986762i	$0.551851\pi$
$270$		0			0
$271$			1.05344e6i			0.871334i	0.900108	+	0.435667i	$0.143487\pi$
$271$			1.05344e6i			0.871334i	−0.900108	+	0.435667i	$0.856513\pi$
$272$		0			0
$273$	−535136.	+	215330.i	−0.434568	+	0.174863i
$274$		0			0
$275$			816177.i			0.650808i
$276$		0			0
$277$	−1.62051e6			−1.26897			−0.634485	−	0.772936i	$-0.718788\pi$
$277$	−1.62051e6			−1.26897			−0.634485	+	0.772936i	$0.718788\pi$
$278$		0			0
$279$		−	63836.6i		−	0.0490975i
$280$		0			0
$281$			563009.i			0.425353i	0.977123	+	0.212677i	$0.0682181\pi$
$281$			563009.i			0.425353i	−0.977123	+	0.212677i	$0.931782\pi$
$282$		0			0
$283$	−361892.			−0.268604			−0.134302	−	0.990940i	$-0.542879\pi$
$283$	−361892.			−0.268604			−0.134302	+	0.990940i	$0.542879\pi$
$284$		0			0
$285$	247.802			0.000180714
$286$		0			0
$287$	1.55774e6			1.11633
$288$		0			0
$289$	1.34819e6			0.949526
$290$		0			0
$291$			1.13660e6i			0.786820i
$292$		0			0
$293$			757318.i			0.515358i	0.966231	+	0.257679i	$0.0829577\pi$
$293$			757318.i			0.515358i	−0.966231	+	0.257679i	$0.917042\pi$
$294$		0			0
$295$	−244373.			−0.163493
$296$		0			0
$297$			196356.i			0.129168i
$298$		0			0
$299$	−489306.	−	1.21602e6i	−0.316521	−	0.786614i
$300$		0			0
$301$		−	1.48753e6i		−	0.946348i
$302$		0			0
$303$	831589.			0.520358
$304$		0			0
$305$			117448.i			0.0722932i
$306$		0			0
$307$		−	1.96280e6i		−	1.18858i	−0.804249	−	0.594292i	$-0.797432\pi$
$307$		−	1.96280e6i		−	1.18858i	0.804249	−	0.594292i	$-0.202568\pi$
$308$		0			0
$309$	−1.33282e6			−0.794100
$310$		0			0
$311$	−834419.			−0.489197			−0.244598	−	0.969624i	$-0.578656\pi$
$311$	−834419.			−0.489197			−0.244598	+	0.969624i	$0.578656\pi$
$312$		0			0
$313$	1.61834e6			0.933703			0.466851	−	0.884336i	$-0.345388\pi$
$313$	1.61834e6			0.933703			0.466851	+	0.884336i	$0.345388\pi$
$314$		0			0
$315$	−82967.2			−0.0471118
$316$		0			0
$317$		−	660304.i		−	0.369059i	−0.982827	−	0.184529i	$-0.940924\pi$
$317$		−	660304.i		−	0.369059i	0.982827	−	0.184529i	$-0.0590761\pi$
$318$		0			0
$319$			59480.4i			0.0327263i
$320$		0			0
$321$	−793621.			−0.429883
$322$		0			0
$323$		−	4704.11i		−	0.00250883i
$324$		0			0
$325$	689255.	+	1.71293e6i	0.361969	+	0.899560i
$326$		0			0
$327$			553205.i			0.286099i
$328$		0			0
$329$	−2.64866e6			−1.34908
$330$		0			0
$331$		−	2.73497e6i		−	1.37209i	−0.727558	−	0.686046i	$-0.759345\pi$
$331$		−	2.73497e6i		−	1.37209i	0.727558	−	0.686046i	$-0.240655\pi$
$332$		0			0
$333$			79416.1i			0.0392462i
$334$		0			0
$335$	−140720.			−0.0685083
$336$		0			0
$337$	−880707.			−0.422432			−0.211216	−	0.977439i	$-0.567742\pi$
$337$	−880707.			−0.422432			−0.211216	+	0.977439i	$0.567742\pi$
$338$		0			0
$339$	−740305.			−0.349874
$340$		0			0
$341$	212277.			0.0988590
$342$		0			0
$343$			2.37193e6i			1.08860i
$344$		0			0
$345$		−	188531.i		−	0.0852774i
$346$		0			0
$347$	33079.1			0.0147479			0.00737395	−	0.999973i	$-0.497653\pi$
$347$	33079.1			0.0147479			0.00737395	+	0.999973i	$0.497653\pi$
$348$		0			0
$349$			4.06448e6i			1.78624i	0.449814	+	0.893122i	$0.351490\pi$
$349$			4.06448e6i			1.78624i	−0.449814	+	0.893122i	$0.648510\pi$
$350$		0			0
$351$	165821.	+	412097.i	0.0718410	+	0.178538i
$352$		0			0
$353$		−	3.57770e6i		−	1.52815i	−0.645125	−	0.764077i	$-0.723195\pi$
$353$		−	3.57770e6i		−	1.52815i	0.645125	−	0.764077i	$-0.276805\pi$
$354$		0			0
$355$	326686.			0.137582
$356$		0			0
$357$			1.57500e6i			0.654047i
$358$		0			0
$359$		−	3.31455e6i		−	1.35734i	−0.734444	−	0.678669i	$-0.762557\pi$
$359$		−	3.31455e6i		−	1.35734i	0.734444	−	0.678669i	$-0.237443\pi$
$360$		0			0
$361$	2.47609e6			0.999997
$362$		0			0
$363$	796513.			0.317268
$364$		0			0
$365$	−251297.			−0.0987313
$366$		0			0
$367$	−192406.			−0.0745681			−0.0372840	−	0.999305i	$-0.511871\pi$
$367$	−192406.			−0.0745681			−0.0372840	+	0.999305i	$0.511871\pi$
$368$		0			0
$369$		−	1.19958e6i		−	0.458632i
$370$		0			0
$371$		−	3.25029e6i		−	1.22599i
$372$		0			0
$373$	1.60161e6			0.596053			0.298027	−	0.954558i	$-0.403672\pi$
$373$	1.60161e6			0.596053			0.298027	+	0.954558i	$0.403672\pi$
$374$		0			0
$375$			539453.i			0.198096i
$376$		0			0
$377$	50230.7	+	124833.i	0.0182019	+	0.0452350i
$378$		0			0
$379$			3.76006e6i			1.34461i	0.740274	+	0.672305i	$0.234696\pi$
$379$			3.76006e6i			1.34461i	−0.740274	+	0.672305i	$0.765304\pi$
$380$		0			0
$381$	1.28835e6			0.454695
$382$		0			0
$383$		−	4.69693e6i		−	1.63613i	−0.575128	−	0.818063i	$-0.695048\pi$
$383$		−	4.69693e6i		−	1.63613i	0.575128	−	0.818063i	$-0.304952\pi$
$384$		0			0
$385$		−	275892.i		−	0.0948608i
$386$		0			0
$387$	−1.14552e6			−0.388799
$388$		0			0
$389$	4.85881e6			1.62801			0.814003	−	0.580860i	$-0.197284\pi$
$389$	4.85881e6			1.62801			0.814003	+	0.580860i	$0.197284\pi$
$390$		0			0
$391$	−3.57894e6			−1.18389
$392$		0			0
$393$	1.27196e6			0.415426
$394$		0			0
$395$		−	152111.i		−	0.0490533i
$396$		0			0
$397$		−	3.91926e6i		−	1.24804i	−0.781409	−	0.624019i	$-0.785499\pi$
$397$		−	3.91926e6i		−	1.24804i	0.781409	−	0.624019i	$-0.214501\pi$
$398$		0			0
$399$	2676.60			0.000841688
$400$		0			0
$401$		−	5.01898e6i		−	1.55867i	−0.626606	−	0.779336i	$-0.715557\pi$
$401$		−	5.01898e6i		−	1.55867i	0.626606	−	0.779336i	$-0.284443\pi$
$402$		0			0
$403$	445509.	−	179266.i	0.136645	−	0.0549838i
$404$		0			0
$405$			63891.2i			0.0193555i
$406$		0			0
$407$	−264083.			−0.0790232
$408$		0			0
$409$		−	2.97886e6i		−	0.880526i	−0.897869	−	0.440263i	$-0.854885\pi$
$409$		−	2.97886e6i		−	0.880526i	0.897869	−	0.440263i	$-0.145115\pi$
$410$		0			0
$411$		−	35625.0i		−	0.0104028i
$412$		0			0
$413$	−2.63956e6			−0.761477
$414$		0			0
$415$	−27459.1			−0.00782648
$416$		0			0
$417$	4.01021e6			1.12935
$418$		0			0
$419$	4.76887e6			1.32703			0.663515	−	0.748163i	$-0.269064\pi$
$419$	4.76887e6			1.32703			0.663515	+	0.748163i	$0.269064\pi$
$420$		0			0
$421$			651294.i			0.179090i	0.995983	+	0.0895451i	$0.0285413\pi$
$421$			651294.i			0.179090i	−0.995983	+	0.0895451i	$0.971459\pi$
$422$		0			0
$423$			2.03968e6i			0.554256i
$424$		0			0
$425$	5.04143e6			1.35388
$426$		0			0
$427$			1.26860e6i			0.336710i
$428$		0			0
$429$	−1.37035e6	+	551407.i	−0.359491	+	0.144654i
$430$		0			0
$431$		−	5.22621e6i		−	1.35517i	−0.735444	−	0.677585i	$-0.763027\pi$
$431$		−	5.22621e6i		−	1.35517i	0.735444	−	0.677585i	$-0.236973\pi$
$432$		0			0
$433$	−6.33506e6			−1.62379			−0.811897	−	0.583801i	$-0.801565\pi$
$433$	−6.33506e6			−1.62379			−0.811897	+	0.583801i	$0.801565\pi$
$434$		0			0
$435$			19354.0i			0.00490396i
$436$		0			0
$437$			6082.18i			0.00152355i
$438$		0			0
$439$	7.44321e6			1.84331			0.921657	−	0.388006i	$-0.126836\pi$
$439$	7.44321e6			1.84331			0.921657	+	0.388006i	$0.126836\pi$
$440$		0			0
$441$	465208.			0.113907
$442$		0			0
$443$	−327031.			−0.0791734			−0.0395867	−	0.999216i	$-0.512604\pi$
$443$	−327031.			−0.0791734			−0.0395867	+	0.999216i	$0.512604\pi$
$444$		0			0
$445$	−158021.			−0.0378281
$446$		0			0
$447$			446574.i			0.105712i
$448$		0			0
$449$		−	5.82118e6i		−	1.36268i	−0.731965	−	0.681342i	$-0.761396\pi$
$449$		−	5.82118e6i		−	1.36268i	0.731965	−	0.681342i	$-0.238604\pi$
$450$		0			0
$451$	3.98899e6			0.923468
$452$		0			0
$453$		−	3.62091e6i		−	0.829033i
$454$		0			0
$455$	−232988.	−	579019.i	−0.0527601	−	0.131119i
$456$		0			0
$457$		−	8.15788e6i		−	1.82720i	−0.406612	−	0.913601i	$-0.633290\pi$
$457$		−	8.15788e6i		−	1.82720i	0.406612	−	0.913601i	$-0.366710\pi$
$458$		0			0
$459$	1.21287e6			0.268709
$460$		0			0
$461$			3.15011e6i			0.690357i	0.938537	+	0.345179i	$0.112182\pi$
$461$			3.15011e6i			0.690357i	−0.938537	+	0.345179i	$0.887818\pi$
$462$		0			0
$463$		−	202622.i		−	0.0439273i	−0.999759	−	0.0219637i	$-0.993008\pi$
$463$		−	202622.i		−	0.0439273i	0.999759	−	0.0219637i	$-0.00699182\pi$
$464$		0			0
$465$	69071.4			0.0148138
$466$		0			0
$467$	−91749.8			−0.0194676			−0.00973382	−	0.999953i	$-0.503098\pi$
$467$	−91749.8			−0.0194676			−0.00973382	+	0.999953i	$0.503098\pi$
$468$		0			0
$469$	−1.51997e6			−0.319082
$470$		0			0
$471$	−5.06186e6			−1.05138
$472$		0			0
$473$		−	3.80921e6i		−	0.782856i
$474$		0			0
$475$		−	8567.58i		−	0.00174230i
$476$		0			0
$477$	−2.50298e6			−0.503687
$478$		0			0
$479$		−	2.82910e6i		−	0.563390i	−0.959504	−	0.281695i	$-0.909103\pi$
$479$		−	2.82910e6i		−	0.563390i	0.959504	−	0.281695i	$-0.0908967\pi$
$480$		0			0
$481$	−554236.	+	223016.i	−0.109228	+	0.0439514i
$482$		0			0
$483$		−	2.03639e6i		−	0.397185i
$484$		0			0
$485$	−1.22981e6			−0.237401
$486$		0			0
$487$			1.30176e6i			0.248719i	0.992237	+	0.124359i	$0.0396876\pi$
$487$			1.30176e6i			0.248719i	−0.992237	+	0.124359i	$0.960312\pi$
$488$		0			0
$489$			2.82490e6i			0.534233i
$490$		0			0
$491$	6.07696e6			1.13758			0.568791	−	0.822482i	$-0.307412\pi$
$491$	6.07696e6			1.13758			0.568791	+	0.822482i	$0.307412\pi$
$492$		0			0
$493$	367404.			0.0680810
$494$		0			0
$495$	−212458.			−0.0389727
$496$		0			0
$497$	3.52866e6			0.640795
$498$		0			0
$499$		−	3.78708e6i		−	0.680853i	−0.940271	−	0.340426i	$-0.889429\pi$
$499$		−	3.78708e6i		−	0.680853i	0.940271	−	0.340426i	$-0.110571\pi$
$500$		0			0
$501$			5.42005e6i			0.964737i
$502$		0			0
$503$	−2.28832e6			−0.403271			−0.201636	−	0.979461i	$-0.564626\pi$
$503$	−2.28832e6			−0.403271			−0.201636	+	0.979461i	$0.564626\pi$
$504$		0			0
$505$			899783.i			0.157003i
$506$		0			0
$507$	−2.41032e6	+	2.31450e6i	−0.416442	+	0.399886i
$508$		0			0
$509$		−	3.70029e6i		−	0.633055i	−0.948583	−	0.316528i	$-0.897483\pi$
$509$		−	3.70029e6i		−	0.633055i	0.948583	−	0.316528i	$-0.102517\pi$
$510$		0			0
$511$	−2.71435e6			−0.459847
$512$		0			0
$513$		−	2061.19i		−	0.000345800i
$514$		0			0
$515$		−	1.44212e6i		−	0.239597i
$516$		0			0
$517$	−6.78256e6			−1.11601
$518$		0			0
$519$	2.73619e6			0.445890
$520$		0			0
$521$	6.13756e6			0.990608			0.495304	−	0.868720i	$-0.335057\pi$
$521$	6.13756e6			0.990608			0.495304	+	0.868720i	$0.335057\pi$
$522$		0			0
$523$	−4.10572e6			−0.656349			−0.328175	−	0.944617i	$-0.606433\pi$
$523$	−4.10572e6			−0.656349			−0.328175	+	0.944617i	$0.606433\pi$
$524$		0			0
$525$			2.86853e6i			0.454215i
$526$		0			0
$527$		−	1.31121e6i		−	0.205658i
$528$		0			0
$529$	−1.80895e6			−0.281053
$530$		0			0
$531$			2.03267e6i			0.312846i
$532$		0			0
$533$	8.37177e6	−	3.36867e6i	1.27644	−	0.513618i
$534$		0			0
$535$		−	858701.i		−	0.129705i
$536$		0			0
$537$	−3.65571e6			−0.547062
$538$		0			0
$539$			1.54696e6i			0.229355i
$540$		0			0
$541$			2.51266e6i			0.369098i	0.982823	+	0.184549i	$0.0590824\pi$
$541$			2.51266e6i			0.369098i	−0.982823	+	0.184549i	$0.940918\pi$
$542$		0			0
$543$	−2.16870e6			−0.315646
$544$		0			0
$545$	−598569.			−0.0863223
$546$		0			0
$547$	2.29454e6			0.327889			0.163945	−	0.986470i	$-0.447578\pi$
$547$	2.29454e6			0.327889			0.163945	+	0.986470i	$0.447578\pi$
$548$		0			0
$549$	976924.			0.138334
$550$		0			0
$551$		−	624.378i		−	8.76130e-5i
$552$		0			0
$553$		−	1.64301e6i		−	0.228469i
$554$		0			0
$555$	−85928.4			−0.0118414
$556$		0			0
$557$		−	5.96324e6i		−	0.814412i	−0.913336	−	0.407206i	$-0.866503\pi$
$557$		−	5.96324e6i		−	0.814412i	0.913336	−	0.407206i	$-0.133497\pi$
$558$		0			0
$559$	−3.21684e6	−	7.99445e6i	−0.435412	−	1.08208i
$560$		0			0
$561$			4.03317e6i			0.541053i
$562$		0			0
$563$	7.37754e6			0.980935			0.490468	−	0.871459i	$-0.336826\pi$
$563$	7.37754e6			0.980935			0.490468	+	0.871459i	$0.336826\pi$
$564$		0			0
$565$		−	801012.i		−	0.105565i
$566$		0			0
$567$			690113.i			0.0901493i
$568$		0			0
$569$	2.60845e6			0.337756			0.168878	−	0.985637i	$-0.445986\pi$
$569$	2.60845e6			0.337756			0.168878	+	0.985637i	$0.445986\pi$
$570$		0			0
$571$	−1.09867e7			−1.41019			−0.705095	−	0.709113i	$-0.749095\pi$
$571$	−1.09867e7			−1.41019			−0.705095	+	0.709113i	$0.749095\pi$
$572$		0			0
$573$	−2.03526e6			−0.258960
$574$		0			0
$575$	−6.51831e6			−0.822177
$576$		0			0
$577$			1.10048e7i			1.37607i	0.725677	+	0.688036i	$0.241527\pi$
$577$			1.10048e7i			1.37607i	−0.725677	+	0.688036i	$0.758473\pi$
$578$		0			0
$579$		−	3.65468e6i		−	0.453057i
$580$		0			0
$581$	−296596.			−0.0364523
$582$		0			0
$583$		−	8.32318e6i		−	1.01419i
$584$		0			0
$585$	−445890.	+	179419.i	−0.0538689	+	0.0216760i
$586$		0			0
$587$		−	1.09851e7i		−	1.31586i	−0.753080	−	0.657929i	$-0.771433\pi$
$587$		−	1.09851e7i		−	1.31586i	0.753080	−	0.657929i	$-0.228567\pi$
$588$		0			0
$589$	−2228.31			−0.000264659
$590$		0			0
$591$		−	3.66651e6i		−	0.431801i
$592$		0			0
$593$		−	3.90620e6i		−	0.456161i	−0.973642	−	0.228081i	$-0.926755\pi$
$593$		−	3.90620e6i		−	0.456161i	0.973642	−	0.228081i	$-0.0732449\pi$
$594$		0			0
$595$	−1.70415e6			−0.197340
$596$		0			0
$597$	−1.82578e6			−0.209659
$598$		0			0
$599$	1.69973e7			1.93559			0.967797	−	0.251733i	$-0.0810005\pi$
$599$	1.69973e7			1.93559			0.967797	+	0.251733i	$0.0810005\pi$
$600$		0			0
$601$	1.07099e7			1.20948			0.604739	−	0.796424i	$-0.293278\pi$
$601$	1.07099e7			1.20948			0.604739	+	0.796424i	$0.293278\pi$
$602$		0			0
$603$			1.17049e6i			0.131092i
$604$		0			0
$605$			861829.i			0.0957267i
$606$		0			0
$607$	1.43102e7			1.57642			0.788212	−	0.615404i	$-0.211007\pi$
$607$	1.43102e7			1.57642			0.788212	+	0.615404i	$0.211007\pi$
$608$		0			0
$609$			209050.i			0.0228405i
$610$		0			0
$611$	−1.42347e7	+	5.72781e6i	−1.54257	+	0.620705i
$612$		0			0
$613$			1.67594e7i			1.80139i	0.434450	+	0.900696i	$0.356943\pi$
$613$			1.67594e7i			1.80139i	−0.434450	+	0.900696i	$0.643057\pi$
$614$		0			0
$615$	1.29795e6			0.138379
$616$		0			0
$617$		−	1.26793e7i		−	1.34085i	−0.741976	−	0.670427i	$-0.766111\pi$
$617$		−	1.26793e7i		−	1.34085i	0.741976	−	0.670427i	$-0.233889\pi$
$618$		0			0
$619$			5.43928e6i			0.570577i	0.958442	+	0.285289i	$0.0920895\pi$
$619$			5.43928e6i			0.570577i	−0.958442	+	0.285289i	$0.907911\pi$
$620$		0			0
$621$	−1.56818e6			−0.163180
$622$		0			0
$623$	−1.70684e6			−0.176187
$624$		0			0
$625$	8.88559e6			0.909885
$626$		0			0
$627$	6854.11			0.000696277
$628$		0			0
$629$			1.63121e6i			0.164393i
$630$		0			0
$631$			5.28439e6i			0.528350i	0.964475	+	0.264175i	$0.0850996\pi$
$631$			5.28439e6i			0.528350i	−0.964475	+	0.264175i	$0.914900\pi$
$632$		0			0
$633$	−5.92173e6			−0.587407
$634$		0			0
$635$			1.39400e6i			0.137192i
$636$		0			0
$637$	1.30639e6	+	3.24664e6i	0.127563	+	0.317019i
$638$		0			0
$639$		−	2.71735e6i		−	0.263265i
$640$		0			0
$641$	8.01855e6			0.770816			0.385408	−	0.922746i	$-0.374061\pi$
$641$	8.01855e6			0.770816			0.385408	+	0.922746i	$0.374061\pi$
$642$		0			0
$643$			1.39227e7i			1.32799i	0.747735	+	0.663997i	$0.231141\pi$
$643$			1.39227e7i			1.32799i	−0.747735	+	0.663997i	$0.768859\pi$
$644$		0			0
$645$		−	1.23945e6i		−	0.117309i
$646$		0			0
$647$	9.55517e6			0.897383			0.448692	−	0.893687i	$-0.351890\pi$
$647$	9.55517e6			0.897383			0.448692	+	0.893687i	$0.351890\pi$
$648$		0			0
$649$	−6.75926e6			−0.629923
$650$		0			0
$651$	746066.			0.0689961
$652$		0			0
$653$	−1.87340e7			−1.71928			−0.859640	−	0.510901i	$-0.829312\pi$
$653$	−1.87340e7			−1.71928			−0.859640	+	0.510901i	$0.829312\pi$
$654$		0			0
$655$			1.37627e6i			0.125343i
$656$		0			0
$657$			2.09026e6i			0.188924i
$658$		0			0
$659$	3.39787e6			0.304785			0.152392	−	0.988320i	$-0.451302\pi$
$659$	3.39787e6			0.304785			0.152392	+	0.988320i	$0.451302\pi$
$660$		0			0
$661$			1.81823e7i			1.61862i	0.587382	+	0.809310i	$0.300159\pi$
$661$			1.81823e7i			1.61862i	−0.587382	+	0.809310i	$0.699841\pi$
$662$		0			0
$663$	3.40598e6	+	8.46449e6i	0.300925	+	0.747854i
$664$		0			0
$665$			2896.09i			0.000253956i
$666$		0			0
$667$	−475034.			−0.0413438
$668$		0			0
$669$			8.80254e6i			0.760401i
$670$		0			0
$671$			3.24858e6i			0.278540i
$672$		0			0
$673$	686844.			0.0584548			0.0292274	−	0.999573i	$-0.490695\pi$
$673$	686844.			0.0584548			0.0292274	+	0.999573i	$0.490695\pi$
$674$		0			0
$675$	2.20899e6			0.186610
$676$		0			0
$677$	−6.64858e6			−0.557516			−0.278758	−	0.960361i	$-0.589923\pi$
$677$	−6.64858e6			−0.557516			−0.278758	+	0.960361i	$0.589923\pi$
$678$		0			0
$679$	−1.32836e7			−1.10571
$680$		0			0
$681$		−	7.30923e6i		−	0.603954i
$682$		0			0
$683$		−	1.66853e7i		−	1.36862i	−0.729193	−	0.684308i	$-0.760104\pi$
$683$		−	1.66853e7i		−	1.36862i	0.729193	−	0.684308i	$-0.239896\pi$
$684$		0			0
$685$	38546.4			0.00313875
$686$		0			0
$687$			1.22696e7i			0.991831i
$688$		0			0
$689$	−7.02885e6	−	1.74680e7i	−0.564074	−	1.40183i
$690$		0			0
$691$			1.75290e7i			1.39657i	0.715822	+	0.698283i	$0.246052\pi$
$691$			1.75290e7i			1.39657i	−0.715822	+	0.698283i	$0.753948\pi$
$692$		0			0
$693$	−2.29484e6			−0.181518
$694$		0			0
$695$			4.33906e6i			0.340748i
$696$		0			0
$697$		−	2.46395e7i		−	1.92110i
$698$		0			0
$699$	8.38734e6			0.649279
$700$		0			0
$701$	7.06379e6			0.542929			0.271464	−	0.962449i	$-0.412492\pi$
$701$	7.06379e6			0.542929			0.271464	+	0.962449i	$0.412492\pi$
$702$		0			0
$703$	2772.13			0.000211556
$704$		0			0
$705$	−2.20694e6			−0.167231
$706$		0			0
$707$			9.71889e6i			0.731253i
$708$		0			0
$709$			9.07731e6i			0.678175i	0.940755	+	0.339087i	$0.110118\pi$
$709$			9.07731e6i			0.678175i	−0.940755	+	0.339087i	$0.889882\pi$
$710$		0			0
$711$	−1.26525e6			−0.0938644
$712$		0			0
$713$			1.69532e6i			0.124890i
$714$		0			0
$715$	−596625.	−	1.48272e6i	−0.0436452	−	0.108466i
$716$		0			0
$717$			7.97287e6i			0.579184i
$718$		0			0
$719$	1.03755e7			0.748492			0.374246	−	0.927330i	$-0.377902\pi$
$719$	1.03755e7			0.748492			0.374246	+	0.927330i	$0.377902\pi$
$720$		0			0
$721$		−	1.55768e7i		−	1.11594i
$722$		0			0
$723$			1.19421e7i			0.849637i
$724$		0			0
$725$	669150.			0.0472801
$726$		0			0
$727$	2.94915e6			0.206948			0.103474	−	0.994632i	$-0.467004\pi$
$727$	2.94915e6			0.206948			0.103474	+	0.994632i	$0.467004\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	−2.35290e7			−1.62858
$732$		0			0
$733$			1.71605e7i			1.17970i	0.807513	+	0.589849i	$0.200813\pi$
$733$			1.71605e7i			1.17970i	−0.807513	+	0.589849i	$0.799187\pi$
$734$		0			0
$735$			503356.i			0.0343682i
$736$		0			0
$737$	−3.89226e6			−0.263957
$738$		0			0
$739$		−	2.20849e7i		−	1.48760i	−0.668405	−	0.743798i	$-0.733023\pi$
$739$		−	2.20849e7i		−	1.48760i	0.668405	−	0.743798i	$-0.266977\pi$
$740$		0			0
$741$	14384.8	−	5788.23i	0.000962409	−	0.000387258i
$742$		0			0
$743$		−	2.21459e7i		−	1.47171i	−0.677140	−	0.735854i	$-0.736781\pi$
$743$		−	2.21459e7i		−	1.47171i	0.677140	−	0.735854i	$-0.263219\pi$
$744$		0			0
$745$	−483194.			−0.0318956
$746$		0			0
$747$			228402.i			0.0149761i
$748$		0			0
$749$		−	9.27514e6i		−	0.604110i
$750$		0			0
$751$	−6.38710e6			−0.413242			−0.206621	−	0.978421i	$-0.566247\pi$
$751$	−6.38710e6			−0.413242			−0.206621	+	0.978421i	$0.566247\pi$
$752$		0			0
$753$	4.92836e6			0.316749
$754$		0			0
$755$	3.91783e6			0.250137
$756$		0			0
$757$	−2.47266e6			−0.156828			−0.0784141	−	0.996921i	$-0.524986\pi$
$757$	−2.47266e6			−0.156828			−0.0784141	+	0.996921i	$0.524986\pi$
$758$		0			0
$759$		−	5.21469e6i		−	0.328567i
$760$		0			0
$761$		−	1.59009e7i		−	0.995315i	−0.867374	−	0.497657i	$-0.834194\pi$
$761$		−	1.59009e7i		−	0.995315i	0.867374	−	0.497657i	$-0.165806\pi$
$762$		0			0
$763$	−6.46537e6			−0.402052
$764$		0			0
$765$			1.31233e6i			0.0810755i
$766$		0			0
$767$	−1.41858e7	+	5.70814e6i	−0.870693	+	0.350353i
$768$		0			0
$769$			4.80537e6i			0.293029i	0.989209	+	0.146515i	$0.0468055\pi$
$769$			4.80537e6i			0.293029i	−0.989209	+	0.146515i	$0.953194\pi$
$770$		0			0
$771$	−1.48958e7			−0.902461
$772$		0			0
$773$			2.22008e7i			1.33635i	0.744004	+	0.668175i	$0.232924\pi$
$773$			2.22008e7i			1.33635i	−0.744004	+	0.668175i	$0.767076\pi$
$774$		0			0
$775$		−	2.38810e6i		−	0.142823i
$776$		0			0
$777$	−928145.			−0.0551523
$778$		0			0
$779$	−41873.2			−0.00247225
$780$		0			0
$781$	9.03603e6			0.530090
$782$		0			0
$783$	160985.			0.00938382
$784$		0			0
$785$		−	5.47695e6i		−	0.317223i
$786$		0			0
$787$			2.85397e7i			1.64253i	0.570548	+	0.821264i	$0.306731\pi$
$787$			2.85397e7i			1.64253i	−0.570548	+	0.821264i	$0.693269\pi$
$788$		0			0
$789$	6.01370e6			0.343913
$790$		0			0
$791$		−	8.65203e6i		−	0.491674i
$792$		0			0
$793$	2.74339e6	+	6.81785e6i	0.154919	+	0.385003i
$794$		0			0
$795$		−	2.70823e6i		−	0.151973i
$796$		0			0
$797$	−2.10013e7			−1.17112			−0.585558	−	0.810630i	$-0.699125\pi$
$797$	−2.10013e7			−1.17112			−0.585558	+	0.810630i	$0.699125\pi$
$798$		0			0
$799$			4.18951e7i			2.32165i
$800$		0			0
$801$			1.31440e6i			0.0723847i
$802$		0			0
$803$	−6.95078e6			−0.380403
$804$		0			0
$805$	2.20338e6			0.119839
$806$		0			0
$807$	−3.46448e6			−0.187264
$808$		0			0
$809$	3.59907e7			1.93339			0.966694	−	0.255935i	$-0.0823834\pi$
$809$	3.59907e7			1.93339			0.966694	+	0.255935i	$0.0823834\pi$
$810$		0			0
$811$		−	3.20415e7i		−	1.71065i	−0.518093	−	0.855324i	$-0.673358\pi$
$811$		−	3.20415e7i		−	1.71065i	0.518093	−	0.855324i	$-0.326642\pi$
$812$		0			0
$813$			9.48092e6i			0.503065i
$814$		0			0
$815$	−3.05655e6			−0.161190
$816$		0			0
$817$			39986.0i			0.00209581i
$818$		0			0
$819$	−4.81622e6	+	1.93797e6i	−0.250898	+	0.100957i
$820$		0			0
$821$			2.92849e7i			1.51630i	0.652079	+	0.758151i	$0.273897\pi$
$821$			2.92849e7i			1.51630i	−0.652079	+	0.758151i	$0.726103\pi$
$822$		0			0
$823$	1.27788e6			0.0657646			0.0328823	−	0.999459i	$-0.489531\pi$
$823$	1.27788e6			0.0657646			0.0328823	+	0.999459i	$0.489531\pi$
$824$		0			0
$825$			7.34560e6i			0.375744i
$826$		0			0
$827$			3.34508e7i			1.70076i	0.526169	+	0.850380i	$0.323628\pi$
$827$			3.34508e7i			1.70076i	−0.526169	+	0.850380i	$0.676372\pi$
$828$		0			0
$829$	−1.65727e7			−0.837541			−0.418770	−	0.908092i	$-0.637539\pi$
$829$	−1.65727e7			−0.837541			−0.418770	+	0.908092i	$0.637539\pi$
$830$		0			0
$831$	−1.45845e7			−0.732640
$832$		0			0
$833$	9.55540e6			0.477129
$834$		0			0
$835$	−5.86451e6			−0.291082
$836$		0			0
$837$		−	574529.i		−	0.0283464i
$838$		0			0
$839$			7.23411e6i			0.354797i	0.984139	+	0.177399i	$0.0567682\pi$
$839$			7.23411e6i			0.354797i	−0.984139	+	0.177399i	$0.943232\pi$
$840$		0			0
$841$	−2.04624e7			−0.997622
$842$		0			0
$843$			5.06708e6i			0.245578i
$844$		0			0
$845$	−2.50429e6	−	2.60797e6i	−0.120654	−	0.125650i
$846$		0			0
$847$			9.30894e6i			0.445853i
$848$		0			0
$849$	−3.25703e6			−0.155079
$850$		0			0
$851$		−	2.10907e6i		−	0.0998315i
$852$		0			0
$853$			3.08382e6i			0.145116i	0.997364	+	0.0725582i	$0.0231163\pi$
$853$			3.08382e6i			0.145116i	−0.997364	+	0.0725582i	$0.976884\pi$
$854$		0			0
$855$	2230.22			0.000104335
$856$		0			0
$857$	9.15615e6			0.425854			0.212927	−	0.977068i	$-0.431700\pi$
$857$	9.15615e6			0.425854			0.212927	+	0.977068i	$0.431700\pi$
$858$		0			0
$859$	3.40397e7			1.57399			0.786996	−	0.616958i	$-0.211635\pi$
$859$	3.40397e7			1.57399			0.786996	+	0.616958i	$0.211635\pi$
$860$		0			0
$861$	1.40197e7			0.644511
$862$		0			0
$863$		−	1.92683e7i		−	0.880676i	−0.897832	−	0.440338i	$-0.854859\pi$
$863$		−	1.92683e7i		−	0.880676i	0.897832	−	0.440338i	$-0.145141\pi$
$864$		0			0
$865$			2.96057e6i			0.134535i
$866$		0			0
$867$	1.21337e7			0.548209
$868$		0			0
$869$		−	4.20734e6i		−	0.188998i
$870$		0			0
$871$	−8.16875e6	+	3.28698e6i	−0.364846	+	0.146808i
$872$		0			0
$873$			1.02294e7i			0.454270i
$874$		0			0
$875$	−6.30466e6			−0.278382
$876$		0			0
$877$		−	2.63479e7i		−	1.15677i	−0.815765	−	0.578384i	$-0.803684\pi$
$877$		−	2.63479e7i		−	1.15677i	0.815765	−	0.578384i	$-0.196316\pi$
$878$		0			0
$879$			6.81586e6i			0.297542i
$880$		0			0
$881$	4.12657e6			0.179122			0.0895610	−	0.995981i	$-0.471454\pi$
$881$	4.12657e6			0.179122			0.0895610	+	0.995981i	$0.471454\pi$
$882$		0			0
$883$	−1.36506e6			−0.0589182			−0.0294591	−	0.999566i	$-0.509378\pi$
$883$	−1.36506e6			−0.0589182			−0.0294591	+	0.999566i	$0.509378\pi$
$884$		0			0
$885$	−2.19936e6			−0.0943925
$886$		0			0
$887$	2.77026e7			1.18225			0.591127	−	0.806578i	$-0.298683\pi$
$887$	2.77026e7			1.18225			0.591127	+	0.806578i	$0.298683\pi$
$888$		0			0
$889$			1.50571e7i			0.638978i
$890$		0			0
$891$			1.76721e6i			0.0745750i
$892$		0			0
$893$	71197.9			0.00298771
$894$		0			0
$895$		−	3.95549e6i		−	0.165060i
$896$		0			0
$897$	−4.40376e6	−	1.09442e7i	−0.182744	−	0.454152i
$898$		0			0
$899$		−	174037.i		−	0.00718194i
$900$		0			0
$901$	−5.14113e7			−2.10983
$902$		0			0
$903$		−	1.33878e7i		−	0.546374i
$904$		0			0
$905$		−	2.34654e6i		−	0.0952372i
$906$		0			0
$907$	2.08574e7			0.841863			0.420931	−	0.907093i	$-0.361703\pi$
$907$	2.08574e7			0.841863			0.420931	+	0.907093i	$0.361703\pi$
$908$		0			0
$909$	7.48430e6			0.300429
$910$		0			0
$911$	−3.95317e6			−0.157815			−0.0789077	−	0.996882i	$-0.525143\pi$
$911$	−3.95317e6			−0.157815			−0.0789077	+	0.996882i	$0.525143\pi$
$912$		0			0
$913$	−759510.			−0.0301548
$914$		0			0
$915$			1.05703e6i			0.0417385i
$916$		0			0
$917$			1.48656e7i			0.583793i
$918$		0			0
$919$	−1.66130e7			−0.648873			−0.324436	−	0.945907i	$-0.605175\pi$
$919$	−1.66130e7			−0.648873			−0.324436	+	0.945907i	$0.605175\pi$
$920$		0			0
$921$		−	1.76652e7i		−	0.686230i
$922$		0			0
$923$	1.89641e7	−	7.63084e6i	0.732702	−	0.294828i
$924$		0			0
$925$			2.97092e6i			0.114166i
$926$		0			0
$927$	−1.19954e7			−0.458474
$928$		0			0
$929$			4.33406e7i			1.64762i	0.566869	+	0.823808i	$0.308155\pi$
$929$			4.33406e7i			1.64762i	−0.566869	+	0.823808i	$0.691845\pi$
$930$		0			0
$931$		−	16238.8i		−	0.000614014i
$932$		0			0
$933$	−7.50978e6			−0.282438
$934$		0			0
$935$	−4.36391e6			−0.163247
$936$		0			0
$937$	−5.84680e6			−0.217555			−0.108778	−	0.994066i	$-0.534694\pi$
$937$	−5.84680e6			−0.217555			−0.108778	+	0.994066i	$0.534694\pi$
$938$		0			0
$939$	1.45650e7			0.539073
$940$		0			0
$941$			3.23340e7i			1.19038i	0.803585	+	0.595190i	$0.202923\pi$
$941$			3.23340e7i			1.19038i	−0.803585	+	0.595190i	$0.797077\pi$
$942$		0			0
$943$			3.18577e7i			1.16663i
$944$		0			0
$945$	−746705.			−0.0272000
$946$		0			0
$947$			3.74163e6i			0.135577i	0.997700	+	0.0677884i	$0.0215943\pi$
$947$			3.74163e6i			0.135577i	−0.997700	+	0.0677884i	$0.978406\pi$
$948$		0			0
$949$	−1.45877e7	+	5.86987e6i	−0.525801	+	0.211574i
$950$		0			0
$951$		−	5.94273e6i		−	0.213076i
$952$		0			0
$953$	−5.30640e7			−1.89264			−0.946319	−	0.323234i	$-0.895230\pi$
$953$	−5.30640e7			−1.89264			−0.946319	+	0.323234i	$0.895230\pi$
$954$		0			0
$955$		−	2.20216e6i		−	0.0781340i
$956$		0			0
$957$			535324.i			0.0188946i
$958$		0			0
$959$	416354.			0.0146189
$960$		0			0
$961$	2.80080e7			0.978305
$962$		0			0
$963$	−7.14259e6			−0.248193
$964$		0			0
$965$	3.95437e6			0.136697
$966$		0			0
$967$		−	888502.i		−	0.0305557i	−0.999883	−	0.0152779i	$-0.995137\pi$
$967$		−	888502.i		−	0.0305557i	0.999883	−	0.0152779i	$-0.00486328\pi$
$968$		0			0
$969$		−	42337.0i		−	0.00144847i
$970$		0			0
$971$	1.92989e7			0.656878			0.328439	−	0.944525i	$-0.393477\pi$
$971$	1.92989e7			0.656878			0.328439	+	0.944525i	$0.393477\pi$
$972$		0			0
$973$			4.68678e7i			1.58706i
$974$		0			0
$975$	6.20329e6	+	1.54163e7i	0.208983	+	0.519361i
$976$		0			0
$977$			2.73358e7i			0.916212i	0.888898	+	0.458106i	$0.151472\pi$
$977$			2.73358e7i			0.916212i	−0.888898	+	0.458106i	$0.848528\pi$
$978$		0			0
$979$	−4.37079e6			−0.145748
$980$		0			0
$981$			4.97884e6i			0.165179i
$982$		0			0
$983$		−	2.36204e7i		−	0.779656i	−0.920888	−	0.389828i	$-0.872534\pi$
$983$		−	2.36204e7i		−	0.779656i	0.920888	−	0.389828i	$-0.127466\pi$
$984$		0			0
$985$	3.96717e6			0.130284
$986$		0			0
$987$	−2.38379e7			−0.778889
$988$		0			0
$989$	3.04218e7			0.988996
$990$		0			0
$991$	3.49750e7			1.13129			0.565644	−	0.824649i	$-0.308628\pi$
$991$	3.49750e7			1.13129			0.565644	+	0.824649i	$0.308628\pi$
$992$		0			0
$993$		−	2.46148e7i		−	0.792177i
$994$		0			0
$995$		−	1.97550e6i		−	0.0632587i
$996$		0			0
$997$	1.96013e7			0.624521			0.312261	−	0.949996i	$-0.398914\pi$
$997$	1.96013e7			0.624521			0.312261	+	0.949996i	$0.398914\pi$
$998$		0			0
$999$			714744.i			0.0226588i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.6.c.d.337.4		6
4.3	odd	2		78.6.b.a.25.5	yes	6
12.11	even	2		234.6.b.c.181.2		6
13.12	even	2	inner	624.6.c.d.337.3		6
52.31	even	4		1014.6.a.q.1.2		3
52.47	even	4		1014.6.a.o.1.2		3
52.51	odd	2		78.6.b.a.25.2	✓	6
156.155	even	2		234.6.b.c.181.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.6.b.a.25.2	✓	6	52.51	odd	2
78.6.b.a.25.5	yes	6	4.3	odd	2
234.6.b.c.181.2		6	12.11	even	2
234.6.b.c.181.5		6	156.155	even	2
624.6.c.d.337.3		6	13.12	even	2	inner
624.6.c.d.337.4		6	1.1	even	1	trivial
1014.6.a.o.1.2		3	52.47	even	4
1014.6.a.q.1.2		3	52.31	even	4

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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	624.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +9.73803i q^{5} +105.184i q^{7} +81.0000 q^{9} +269.350i q^{11} +(227.464 + 565.290i) q^{13} +87.6423i q^{15} +1663.75 q^{17} -2.82742i q^{19} +946.657i q^{21} -2151.14 q^{23} +3030.17 q^{25} +729.000 q^{27} +220.829 q^{29} -788.106i q^{31} +2424.15i q^{33} -1024.29 q^{35} +980.445i q^{37} +(2047.18 + 5087.61i) q^{39} -14809.7i q^{41} -14142.2 q^{43} +788.781i q^{45} +25181.2i q^{47} +5743.31 q^{49} +14973.7 q^{51} -30900.9 q^{53} -2622.94 q^{55} -25.4468i q^{57} +25094.7i q^{59} +12060.8 q^{61} +8519.91i q^{63} +(-5504.81 + 2215.05i) q^{65} +14450.5i q^{67} -19360.2 q^{69} -33547.5i q^{71} +25805.7i q^{73} +27271.5 q^{75} -28331.4 q^{77} -15620.3 q^{79} +6561.00 q^{81} +2819.78i q^{83} +16201.6i q^{85} +1987.46 q^{87} +16227.2i q^{89} +(-59459.6 + 23925.6i) q^{91} -7092.95i q^{93} +27.5335 q^{95} +126289. i q^{97} +21817.4i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 54 q^{3} + 486 q^{9} + 530 q^{13} - 836 q^{17} + 416 q^{23} + 718 q^{25} + 4374 q^{27} + 18788 q^{29} - 6112 q^{35} + 4770 q^{39} + 24200 q^{43} - 3038 q^{49} - 7524 q^{51} - 42396 q^{53} - 124656 q^{55}+ \cdots - 567168 q^{95}+O(q^{100}) \) 6 * q + 54 * q^3 + 486 * q^9 + 530 * q^13 - 836 * q^17 + 416 * q^23 + 718 * q^25 + 4374 * q^27 + 18788 * q^29 - 6112 * q^35 + 4770 * q^39 + 24200 * q^43 - 3038 * q^49 - 7524 * q^51 - 42396 * q^53 - 124656 * q^55 - 3196 * q^61 - 17168 * q^65 + 3744 * q^69 + 6462 * q^75 - 114024 * q^77 + 169328 * q^79 + 39366 * q^81 + 169092 * q^87 - 236152 * q^91 - 567168 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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