LMFDB - Embedded newform 624.6.c.a.337.3

Error: no document with id 240499420 found in table mf_hecke_traces.

Error: table True does not exist

Error: no document with id 275160291 found in table mf_hecke_traces.

Error: table True does not exist

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 = [4,0,-36,0,0,0,0,0,324,0,0,0,312] 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:	$4$
Coefficient field:	4.0.31066572.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 53x^{2} + 300 $ x^4 + 53*x^2 + 300
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.3
Root			$6.82321i$ of defining polynomial
Character	$\chi$	$=$	624.337
Dual form			624.6.c.a.337.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} +33.0580i q^{5} -61.2631i q^{7} +81.0000 q^{9} -164.669i q^{11} +(-443.460 - 417.895i) q^{13} -297.522i q^{15} +336.674 q^{17} -69.6517i q^{19} +551.368i q^{21} -3734.76 q^{23} +2032.17 q^{25} -729.000 q^{27} +4304.83 q^{29} +1850.83i q^{31} +1482.02i q^{33} +2025.24 q^{35} +2739.27i q^{37} +(3991.14 + 3761.05i) q^{39} -12031.5i q^{41} +2161.44 q^{43} +2677.70i q^{45} +10338.2i q^{47} +13053.8 q^{49} -3030.07 q^{51} -696.674 q^{53} +5443.64 q^{55} +626.866i q^{57} -24274.8i q^{59} -26772.3 q^{61} -4962.31i q^{63} +(13814.8 - 14659.9i) q^{65} +68687.0i q^{67} +33612.9 q^{69} -16366.0i q^{71} -83795.1i q^{73} -18289.5 q^{75} -10088.2 q^{77} +16999.8 q^{79} +6561.00 q^{81} -38542.9i q^{83} +11129.8i q^{85} -38743.5 q^{87} +140248. i q^{89} +(-25601.5 + 27167.8i) q^{91} -16657.5i q^{93} +2302.55 q^{95} +87328.3i q^{97} -13338.2i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 36 q^{3} + 324 q^{9} + 312 q^{13} + 384 q^{17} - 2424 q^{23} + 748 q^{25} - 2916 q^{27} - 3960 q^{29} + 20616 q^{35} - 2808 q^{39} - 4832 q^{43} + 31036 q^{49} - 3456 q^{51} - 1824 q^{53} + 10864 q^{55}+ \cdots - 322920 q^{95}+O(q^{100}) $ 4 * q - 36 * q^3 + 324 * q^9 + 312 * q^13 + 384 * q^17 - 2424 * q^23 + 748 * q^25 - 2916 * q^27 - 3960 * q^29 + 20616 * q^35 - 2808 * q^39 - 4832 * q^43 + 31036 * q^49 - 3456 * q^51 - 1824 * q^53 + 10864 * q^55 - 100992 * q^61 + 42744 * q^65 + 21816 * q^69 - 6732 * q^75 - 20136 * q^77 + 205344 * q^79 + 26244 * q^81 + 35640 * q^87 - 77376 * q^91 - 322920 * 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
$5$			33.0580i			0.591360i	0.955287	+	0.295680i	$0.0955462\pi$
$5$			33.0580i			0.591360i	−0.955287	+	0.295680i	$0.904454\pi$
$6$		0			0
$7$		−	61.2631i		−	0.472557i	−0.971685	−	0.236278i	$-0.924072\pi$
$7$		−	61.2631i		−	0.472557i	0.971685	−	0.236278i	$-0.0759277\pi$
$8$		0			0
$9$	81.0000			0.333333
$10$		0			0
$11$		−	164.669i		−	0.410328i	−0.978728	−	0.205164i	$-0.934227\pi$
$11$		−	164.669i		−	0.410328i	0.978728	−	0.205164i	$-0.0657727\pi$
$12$		0			0
$13$	−443.460	−	417.895i	−0.727774	−	0.685817i
$13$	−443.460	−	417.895i	−0.727774	−	0.685817i
$14$		0			0
$15$		−	297.522i		−	0.341422i
$16$		0			0
$17$	336.674			0.282545			0.141272	−	0.989971i	$-0.454881\pi$
$17$	336.674			0.282545			0.141272	+	0.989971i	$0.454881\pi$
$18$		0			0
$19$		−	69.6517i		−	0.0442637i	−0.999755	−	0.0221319i	$-0.992955\pi$
$19$		−	69.6517i		−	0.0442637i	0.999755	−	0.0221319i	$-0.00704537\pi$
$20$		0			0
$21$			551.368i			0.272831i
$22$		0			0
$23$	−3734.76			−1.47212			−0.736060	−	0.676916i	$-0.763316\pi$
$23$	−3734.76			−1.47212			−0.736060	+	0.676916i	$0.763316\pi$
$24$		0			0
$25$	2032.17			0.650294
$26$		0			0
$27$	−729.000			−0.192450
$28$		0			0
$29$	4304.83			0.950519			0.475260	−	0.879846i	$-0.342354\pi$
$29$	4304.83			0.950519			0.475260	+	0.879846i	$0.342354\pi$
$30$		0			0
$31$			1850.83i			0.345910i	0.984930	+	0.172955i	$0.0553315\pi$
$31$			1850.83i			0.345910i	−0.984930	+	0.172955i	$0.944668\pi$
$32$		0			0
$33$			1482.02i			0.236903i
$34$		0			0
$35$	2025.24			0.279451
$36$		0			0
$37$			2739.27i			0.328950i	0.986381	+	0.164475i	$0.0525931\pi$
$37$			2739.27i			0.328950i	−0.986381	+	0.164475i	$0.947407\pi$
$38$		0			0
$39$	3991.14	+	3761.05i	0.420180	+	0.395957i
$40$		0			0
$41$		−	12031.5i		−	1.11779i	−0.829239	−	0.558894i	$-0.811226\pi$
$41$		−	12031.5i		−	1.11779i	0.829239	−	0.558894i	$-0.188774\pi$
$42$		0			0
$43$	2161.44			0.178267			0.0891336	−	0.996020i	$-0.471590\pi$
$43$	2161.44			0.178267			0.0891336	+	0.996020i	$0.471590\pi$
$44$		0			0
$45$			2677.70i			0.197120i
$46$		0			0
$47$			10338.2i			0.682651i	0.939945	+	0.341325i	$0.110876\pi$
$47$			10338.2i			0.682651i	−0.939945	+	0.341325i	$0.889124\pi$
$48$		0			0
$49$	13053.8			0.776690
$50$		0			0
$51$	−3030.07			−0.163127
$52$		0			0
$53$	−696.674			−0.0340675			−0.0170337	−	0.999855i	$-0.505422\pi$
$53$	−696.674			−0.0340675			−0.0170337	+	0.999855i	$0.505422\pi$
$54$		0			0
$55$	5443.64			0.242651
$56$		0			0
$57$			626.866i			0.0255557i
$58$		0			0
$59$		−	24274.8i		−	0.907876i	−0.891033	−	0.453938i	$-0.850019\pi$
$59$		−	24274.8i		−	0.907876i	0.891033	−	0.453938i	$-0.149981\pi$
$60$		0			0
$61$	−26772.3			−0.921214			−0.460607	−	0.887604i	$-0.652368\pi$
$61$	−26772.3			−0.921214			−0.460607	+	0.887604i	$0.652368\pi$
$62$		0			0
$63$		−	4962.31i		−	0.157519i
$64$		0			0
$65$	13814.8	−	14659.9i	0.405565	−	0.430376i
$66$		0			0
$67$			68687.0i			1.86934i	0.355522	+	0.934668i	$0.384303\pi$
$67$			68687.0i			1.86934i	−0.355522	+	0.934668i	$0.615697\pi$
$68$		0			0
$69$	33612.9			0.849929
$70$		0			0
$71$		−	16366.0i		−	0.385298i	−0.981268	−	0.192649i	$-0.938292\pi$
$71$		−	16366.0i		−	0.385298i	0.981268	−	0.192649i	$-0.0617079\pi$
$72$		0			0
$73$		−	83795.1i		−	1.84040i	−0.391451	−	0.920199i	$-0.628027\pi$
$73$		−	83795.1i		−	1.84040i	0.391451	−	0.920199i	$-0.371973\pi$
$74$		0			0
$75$	−18289.5			−0.375447
$76$		0			0
$77$	−10088.2			−0.193903
$78$		0			0
$79$	16999.8			0.306462			0.153231	−	0.988190i	$-0.451032\pi$
$79$	16999.8			0.306462			0.153231	+	0.988190i	$0.451032\pi$
$80$		0			0
$81$	6561.00			0.111111
$82$		0			0
$83$		−	38542.9i		−	0.614114i	−0.951691	−	0.307057i	$-0.900656\pi$
$83$		−	38542.9i		−	0.614114i	0.951691	−	0.307057i	$-0.0993442\pi$
$84$		0			0
$85$			11129.8i			0.167086i
$86$		0			0
$87$	−38743.5			−0.548783
$88$		0			0
$89$			140248.i			1.87681i	0.345531	+	0.938407i	$0.387699\pi$
$89$			140248.i			1.87681i	−0.345531	+	0.938407i	$0.612301\pi$
$90$		0			0
$91$	−25601.5	+	27167.8i	−0.324088	+	0.343915i
$92$		0			0
$93$		−	16657.5i		−	0.199711i
$94$		0			0
$95$	2302.55			0.0261758
$96$		0			0
$97$			87328.3i			0.942379i	0.882032	+	0.471189i	$0.156175\pi$
$97$			87328.3i			0.942379i	−0.882032	+	0.471189i	$0.843825\pi$
$98$		0			0
$99$		−	13338.2i		−	0.136776i
$100$		0			0
$101$	−128464.			−1.25308			−0.626539	−	0.779390i	$-0.715529\pi$
$101$	−128464.			−1.25308			−0.626539	+	0.779390i	$0.715529\pi$
$102$		0			0
$103$	63628.0			0.590956			0.295478	−	0.955350i	$-0.404521\pi$
$103$	63628.0			0.590956			0.295478	+	0.955350i	$0.404521\pi$
$104$		0			0
$105$	−18227.1			−0.161341
$106$		0			0
$107$	−125275.			−1.05780			−0.528900	−	0.848684i	$-0.677395\pi$
$107$	−125275.			−1.05780			−0.528900	+	0.848684i	$0.677395\pi$
$108$		0			0
$109$			56877.0i			0.458533i	0.973364	+	0.229267i	$0.0736327\pi$
$109$			56877.0i			0.458533i	−0.973364	+	0.229267i	$0.926367\pi$
$110$		0			0
$111$		−	24653.4i		−	0.189920i
$112$		0			0
$113$	−235161.			−1.73249			−0.866243	−	0.499623i	$-0.833472\pi$
$113$	−235161.			−1.73249			−0.866243	+	0.499623i	$0.833472\pi$
$114$		0			0
$115$		−	123464.i		−	0.870553i
$116$		0			0
$117$	−35920.3	−	33849.5i	−0.242591	−	0.228606i
$118$		0			0
$119$		−	20625.7i		−	0.133518i
$120$		0			0
$121$	133935.			0.831631
$122$		0			0
$123$			108283.i			0.645356i
$124$		0			0
$125$			170486.i			0.975917i
$126$		0			0
$127$	−109597.			−0.602961			−0.301481	−	0.953472i	$-0.597481\pi$
$127$	−109597.			−0.602961			−0.301481	+	0.953472i	$0.597481\pi$
$128$		0			0
$129$	−19452.9			−0.102923
$130$		0			0
$131$	219695.			1.11851			0.559257	−	0.828994i	$-0.311087\pi$
$131$	219695.			1.11851			0.559257	+	0.828994i	$0.311087\pi$
$132$		0			0
$133$	−4267.08			−0.0209171
$134$		0			0
$135$		−	24099.3i		−	0.113807i
$136$		0			0
$137$			9145.44i			0.0416297i	0.999783	+	0.0208148i	$0.00662605\pi$
$137$			9145.44i			0.0416297i	−0.999783	+	0.0208148i	$0.993374\pi$
$138$		0			0
$139$	−146786.			−0.644390			−0.322195	−	0.946673i	$-0.604421\pi$
$139$	−146786.			−0.644390			−0.322195	+	0.946673i	$0.604421\pi$
$140$		0			0
$141$		−	93043.4i		−	0.394129i
$142$		0			0
$143$	−68814.4	+	73024.3i	−0.281410	+	0.298626i
$144$		0			0
$145$			142309.i			0.562099i
$146$		0			0
$147$	−117484.			−0.448422
$148$		0			0
$149$		−	205420.i		−	0.758012i	−0.925394	−	0.379006i	$-0.876266\pi$
$149$		−	205420.i		−	0.758012i	0.925394	−	0.379006i	$-0.123734\pi$
$150$		0			0
$151$			325694.i			1.16243i	0.813749	+	0.581216i	$0.197423\pi$
$151$			325694.i			1.16243i	−0.813749	+	0.581216i	$0.802577\pi$
$152$		0			0
$153$	27270.6			0.0941816
$154$		0			0
$155$	−61184.9			−0.204557
$156$		0			0
$157$	−557622.			−1.80547			−0.902735	−	0.430196i	$-0.858444\pi$
$157$	−557622.			−1.80547			−0.902735	+	0.430196i	$0.858444\pi$
$158$		0			0
$159$	6270.07			0.0196689
$160$		0			0
$161$			228803.i			0.695661i
$162$		0			0
$163$			635012.i			1.87203i	0.351958	+	0.936016i	$0.385516\pi$
$163$			635012.i			1.87203i	−0.351958	+	0.936016i	$0.614484\pi$
$164$		0			0
$165$	−48992.8			−0.140095
$166$		0			0
$167$			480228.i			1.33247i	0.745743	+	0.666234i	$0.232095\pi$
$167$			480228.i			1.33247i	−0.745743	+	0.666234i	$0.767905\pi$
$168$		0			0
$169$	22021.3	+	370639.i	0.0593099	+	0.998240i
$170$		0			0
$171$		−	5641.79i		−	0.0147546i
$172$		0			0
$173$	−337273.			−0.856774			−0.428387	−	0.903595i	$-0.640918\pi$
$173$	−337273.			−0.856774			−0.428387	+	0.903595i	$0.640918\pi$
$174$		0			0
$175$		−	124497.i		−	0.307301i
$176$		0			0
$177$			218474.i			0.524162i
$178$		0			0
$179$	−570180.			−1.33009			−0.665043	−	0.746805i	$-0.731587\pi$
$179$	−570180.			−1.33009			−0.665043	+	0.746805i	$0.731587\pi$
$180$		0			0
$181$	192663.			0.437121			0.218560	−	0.975823i	$-0.429864\pi$
$181$	192663.			0.437121			0.218560	+	0.975823i	$0.429864\pi$
$182$		0			0
$183$	240950.			0.531863
$184$		0			0
$185$	−90554.8			−0.194528
$186$		0			0
$187$		−	55439.9i		−	0.115936i
$188$		0			0
$189$			44660.8i			0.0909436i
$190$		0			0
$191$	320668.			0.636023			0.318011	−	0.948087i	$-0.396985\pi$
$191$	320668.			0.636023			0.318011	+	0.948087i	$0.396985\pi$
$192$		0			0
$193$			386084.i			0.746086i	0.927814	+	0.373043i	$0.121686\pi$
$193$			386084.i			0.746086i	−0.927814	+	0.373043i	$0.878314\pi$
$194$		0			0
$195$	−124333.	+	131939.i	−0.234153	+	0.248478i
$196$		0			0
$197$			891618.i			1.63687i	0.574602	+	0.818433i	$0.305157\pi$
$197$			891618.i			1.63687i	−0.574602	+	0.818433i	$0.694843\pi$
$198$		0			0
$199$	−700392.			−1.25374			−0.626872	−	0.779123i	$-0.715665\pi$
$199$	−700392.			−1.25374			−0.626872	+	0.779123i	$0.715665\pi$
$200$		0			0
$201$		−	618183.i		−	1.07926i
$202$		0			0
$203$		−	263727.i		−	0.449174i
$204$		0			0
$205$	397737.			0.661015
$206$		0			0
$207$	−302516.			−0.490707
$208$		0			0
$209$	−11469.5			−0.0181626
$210$		0			0
$211$	−598648.			−0.925689			−0.462845	−	0.886439i	$-0.653171\pi$
$211$	−598648.			−0.925689			−0.462845	+	0.886439i	$0.653171\pi$
$212$		0			0
$213$			147294.i			0.222452i
$214$		0			0
$215$			71452.8i			0.105420i
$216$		0			0
$217$	113388.			0.163462
$218$		0			0
$219$			754156.i			1.06255i
$220$		0			0
$221$	−149302.	−	140694.i	−0.205629	−	0.193774i
$222$		0			0
$223$		−	694921.i		−	0.935778i	−0.883787	−	0.467889i	$-0.845015\pi$
$223$		−	694921.i		−	0.935778i	0.883787	−	0.467889i	$-0.154985\pi$
$224$		0			0
$225$	164606.			0.216765
$226$		0			0
$227$			1.06606e6i			1.37315i	0.727059	+	0.686575i	$0.240887\pi$
$227$			1.06606e6i			1.37315i	−0.727059	+	0.686575i	$0.759113\pi$
$228$		0			0
$229$		−	985635.i		−	1.24202i	−0.783804	−	0.621008i	$-0.786723\pi$
$229$		−	985635.i		−	1.24202i	0.783804	−	0.621008i	$-0.213277\pi$
$230$		0			0
$231$	90793.4			0.111950
$232$		0			0
$233$	−1.01282e6			−1.22220			−0.611098	−	0.791555i	$-0.709272\pi$
$233$	−1.01282e6			−1.22220			−0.611098	+	0.791555i	$0.709272\pi$
$234$		0			0
$235$	−341759.			−0.403692
$236$		0			0
$237$	−152999.			−0.176936
$238$		0			0
$239$		−	112715.i		−	0.127640i	−0.997961	−	0.0638201i	$-0.979672\pi$
$239$		−	112715.i		−	0.127640i	0.997961	−	0.0638201i	$-0.0203284\pi$
$240$		0			0
$241$		−	940022.i		−	1.04255i	−0.853390	−	0.521273i	$-0.825457\pi$
$241$		−	940022.i		−	1.04255i	0.853390	−	0.521273i	$-0.174543\pi$
$242$		0			0
$243$	−59049.0			−0.0641500
$244$		0			0
$245$			431534.i			0.459303i
$246$		0			0
$247$	−29107.1	+	30887.8i	−0.0303568	+	0.0322140i
$248$		0			0
$249$			346886.i			0.354559i
$250$		0			0
$251$	−1.22654e6			−1.22885			−0.614423	−	0.788977i	$-0.710611\pi$
$251$	−1.22654e6			−1.22885			−0.614423	+	0.788977i	$0.710611\pi$
$252$		0			0
$253$			615001.i			0.604052i
$254$		0			0
$255$		−	100168.i		−	0.0964669i
$256$		0			0
$257$	810365.			0.765328			0.382664	−	0.923888i	$-0.375007\pi$
$257$	810365.			0.765328			0.382664	+	0.923888i	$0.375007\pi$
$258$		0			0
$259$	167816.			0.155448
$260$		0			0
$261$	348691.			0.316840
$262$		0			0
$263$	25213.2			0.0224770			0.0112385	−	0.999937i	$-0.496423\pi$
$263$	25213.2			0.0224770			0.0112385	+	0.999937i	$0.496423\pi$
$264$		0			0
$265$		−	23030.7i		−	0.0201461i
$266$		0			0
$267$		−	1.26223e6i		−	1.08358i
$268$		0			0
$269$	−498829.			−0.420311			−0.210156	−	0.977668i	$-0.567397\pi$
$269$	−498829.			−0.420311			−0.210156	+	0.977668i	$0.567397\pi$
$270$		0			0
$271$		−	541761.i		−	0.448110i	−0.974577	−	0.224055i	$-0.928070\pi$
$271$		−	541761.i		−	0.448110i	0.974577	−	0.224055i	$-0.0719295\pi$
$272$		0			0
$273$	230414.	−	244510.i	0.187112	−	0.198559i
$274$		0			0
$275$		−	334636.i		−	0.266834i
$276$		0			0
$277$	−1.42654e6			−1.11708			−0.558540	−	0.829478i	$-0.688638\pi$
$277$	−1.42654e6			−1.11708			−0.558540	+	0.829478i	$0.688638\pi$
$278$		0			0
$279$			149917.i			0.115303i
$280$		0			0
$281$		−	186175.i		−	0.140655i	−0.997524	−	0.0703274i	$-0.977596\pi$
$281$		−	186175.i		−	0.140655i	0.997524	−	0.0703274i	$-0.0224044\pi$
$282$		0			0
$283$	1.64391e6			1.22015			0.610073	−	0.792345i	$-0.291140\pi$
$283$	1.64391e6			1.22015			0.610073	+	0.792345i	$0.291140\pi$
$284$		0			0
$285$	−20722.9			−0.0151126
$286$		0			0
$287$	−737086.			−0.528219
$288$		0			0
$289$	−1.30651e6			−0.920168
$290$		0			0
$291$		−	785954.i		−	0.544083i
$292$		0			0
$293$			2.04281e6i			1.39014i	0.718940	+	0.695072i	$0.244627\pi$
$293$			2.04281e6i			1.39014i	−0.718940	+	0.695072i	$0.755373\pi$
$294$		0			0
$295$	802478.			0.536881
$296$		0			0
$297$			120044.i			0.0789676i
$298$		0			0
$299$	1.65622e6	+	1.56074e6i	1.07137	+	1.00961i
$300$		0			0
$301$		−	132416.i		−	0.0842414i
$302$		0			0
$303$	1.15618e6			0.723465
$304$		0			0
$305$		−	885038.i		−	0.544769i
$306$		0			0
$307$			3.15457e6i			1.91026i	0.296179	+	0.955132i	$0.404287\pi$
$307$			3.15457e6i			1.91026i	−0.296179	+	0.955132i	$0.595713\pi$
$308$		0			0
$309$	−572652.			−0.341189
$310$		0			0
$311$	270180.			0.158399			0.0791994	−	0.996859i	$-0.474764\pi$
$311$	270180.			0.158399			0.0791994	+	0.996859i	$0.474764\pi$
$312$		0			0
$313$	−1.05582e6			−0.609157			−0.304578	−	0.952487i	$-0.598516\pi$
$313$	−1.05582e6			−0.609157			−0.304578	+	0.952487i	$0.598516\pi$
$314$		0			0
$315$	164044.			0.0931504
$316$		0			0
$317$		−	165844.i		−	0.0926938i	−0.998925	−	0.0463469i	$-0.985242\pi$
$317$		−	165844.i		−	0.0926938i	0.998925	−	0.0463469i	$-0.0147580\pi$
$318$		0			0
$319$		−	708873.i		−	0.390024i
$320$		0			0
$321$	1.12747e6			0.610722
$322$		0			0
$323$		−	23449.9i		−	0.0125065i
$324$		0			0
$325$	−901186.	−	849232.i	−0.473267	−	0.445983i
$326$		0			0
$327$		−	511893.i		−	0.264734i
$328$		0			0
$329$	633348.			0.322591
$330$		0			0
$331$		−	2.29011e6i		−	1.14891i	−0.818535	−	0.574456i	$-0.805213\pi$
$331$		−	2.29011e6i		−	1.14891i	0.818535	−	0.574456i	$-0.194787\pi$
$332$		0			0
$333$			221881.i			0.109650i
$334$		0			0
$335$	−2.27065e6			−1.10545
$336$		0			0
$337$	1.89805e6			0.910402			0.455201	−	0.890389i	$-0.349567\pi$
$337$	1.89805e6			0.910402			0.455201	+	0.890389i	$0.349567\pi$
$338$		0			0
$339$	2.11645e6			1.00025
$340$		0			0
$341$	304775.			0.141936
$342$		0			0
$343$		−	1.82937e6i		−	0.839587i
$344$		0			0
$345$			1.11117e6i			0.502614i
$346$		0			0
$347$	−3.22565e6			−1.43812			−0.719058	−	0.694950i	$-0.755426\pi$
$347$	−3.22565e6			−1.43812			−0.719058	+	0.694950i	$0.755426\pi$
$348$		0			0
$349$			2.95721e6i			1.29963i	0.760094	+	0.649813i	$0.225153\pi$
$349$			2.95721e6i			1.29963i	−0.760094	+	0.649813i	$0.774847\pi$
$350$		0			0
$351$	323283.	+	304645.i	0.140060	+	0.131986i
$352$		0			0
$353$			101605.i			0.0433987i	0.999765	+	0.0216993i	$0.00690766\pi$
$353$			101605.i			0.0433987i	−0.999765	+	0.0216993i	$0.993092\pi$
$354$		0			0
$355$	541028.			0.227850
$356$		0			0
$357$			185631.i			0.0770869i
$358$		0			0
$359$			3.01953e6i			1.23653i	0.785971	+	0.618263i	$0.212163\pi$
$359$			3.01953e6i			1.23653i	−0.785971	+	0.618263i	$0.787837\pi$
$360$		0			0
$361$	2.47125e6			0.998041
$362$		0			0
$363$	−1.20542e6			−0.480142
$364$		0			0
$365$	2.77010e6			1.08834
$366$		0			0
$367$	3.66414e6			1.42006			0.710031	−	0.704170i	$-0.248681\pi$
$367$	3.66414e6			1.42006			0.710031	+	0.704170i	$0.248681\pi$
$368$		0			0
$369$		−	974550.i		−	0.372596i
$370$		0			0
$371$			42680.4i			0.0160988i
$372$		0			0
$373$	2.15700e6			0.802746			0.401373	−	0.915915i	$-0.368533\pi$
$373$	2.15700e6			0.802746			0.401373	+	0.915915i	$0.368533\pi$
$374$		0			0
$375$		−	1.53437e6i		−	0.563446i
$376$		0			0
$377$	−1.90902e6	−	1.79896e6i	−0.691763	−	0.651882i
$378$		0			0
$379$		−	474934.i		−	0.169838i	−0.996388	−	0.0849191i	$-0.972937\pi$
$379$		−	474934.i		−	0.169838i	0.996388	−	0.0849191i	$-0.0270632\pi$
$380$		0			0
$381$	986373.			0.348120
$382$		0			0
$383$		−	1.52732e6i		−	0.532025i	−0.963970	−	0.266012i	$-0.914294\pi$
$383$		−	1.52732e6i		−	0.532025i	0.963970	−	0.266012i	$-0.0857062\pi$
$384$		0			0
$385$		−	333494.i		−	0.114667i
$386$		0			0
$387$	175076.			0.0594224
$388$		0			0
$389$	−2.38936e6			−0.800585			−0.400293	−	0.916387i	$-0.631092\pi$
$389$	−2.38936e6			−0.800585			−0.400293	+	0.916387i	$0.631092\pi$
$390$		0			0
$391$	−1.25740e6			−0.415940
$392$		0			0
$393$	−1.97725e6			−0.645774
$394$		0			0
$395$			561981.i			0.181229i
$396$		0			0
$397$		−	2.68613e6i		−	0.855363i	−0.903930	−	0.427681i	$-0.859331\pi$
$397$		−	2.68613e6i		−	0.855363i	0.903930	−	0.427681i	$-0.140669\pi$
$398$		0			0
$399$	38403.8			0.0120765
$400$		0			0
$401$			2.14312e6i			0.665557i	0.943005	+	0.332779i	$0.107986\pi$
$401$			2.14312e6i			0.665557i	−0.943005	+	0.332779i	$0.892014\pi$
$402$		0			0
$403$	773453.	−	820771.i	0.237231	−	0.251744i
$404$		0			0
$405$			216894.i			0.0657066i
$406$		0			0
$407$	451074.			0.134977
$408$		0			0
$409$		−	3.97614e6i		−	1.17531i	−0.809110	−	0.587657i	$-0.800051\pi$
$409$		−	3.97614e6i		−	1.17531i	0.809110	−	0.587657i	$-0.199949\pi$
$410$		0			0
$411$		−	82309.0i		−	0.0240349i
$412$		0			0
$413$	−1.48715e6			−0.429023
$414$		0			0
$415$	1.27415e6			0.363162
$416$		0			0
$417$	1.32108e6			0.372039
$418$		0			0
$419$	−2.59228e6			−0.721352			−0.360676	−	0.932691i	$-0.617454\pi$
$419$	−2.59228e6			−0.721352			−0.360676	+	0.932691i	$0.617454\pi$
$420$		0			0
$421$			2.81076e6i			0.772890i	0.922312	+	0.386445i	$0.126297\pi$
$421$			2.81076e6i			0.772890i	−0.922312	+	0.386445i	$0.873703\pi$
$422$		0			0
$423$			837391.i			0.227550i
$424$		0			0
$425$	684178.			0.183737
$426$		0			0
$427$			1.64015e6i			0.435326i
$428$		0			0
$429$	619330.	−	657219.i	0.162472	−	0.172412i
$430$		0			0
$431$		−	1.22819e6i		−	0.318472i	−0.987241	−	0.159236i	$-0.949097\pi$
$431$		−	1.22819e6i		−	0.318472i	0.987241	−	0.159236i	$-0.0509030\pi$
$432$		0			0
$433$	5.44083e6			1.39459			0.697294	−	0.716785i	$-0.254387\pi$
$433$	5.44083e6			1.39459			0.697294	+	0.716785i	$0.254387\pi$
$434$		0			0
$435$		−	1.28078e6i		−	0.324528i
$436$		0			0
$437$			260133.i			0.0651616i
$438$		0			0
$439$	−211253.			−0.0523169			−0.0261585	−	0.999658i	$-0.508327\pi$
$439$	−211253.			−0.0523169			−0.0261585	+	0.999658i	$0.508327\pi$
$440$		0			0
$441$	1.05736e6			0.258897
$442$		0			0
$443$	−2.23978e6			−0.542246			−0.271123	−	0.962545i	$-0.587395\pi$
$443$	−2.23978e6			−0.542246			−0.271123	+	0.962545i	$0.587395\pi$
$444$		0			0
$445$	−4.63632e6			−1.10987
$446$		0			0
$447$			1.84878e6i			0.437639i
$448$		0			0
$449$		−	5.59942e6i		−	1.31077i	−0.755294	−	0.655386i	$-0.772506\pi$
$449$		−	5.59942e6i		−	1.31077i	0.755294	−	0.655386i	$-0.227494\pi$
$450$		0			0
$451$	−1.98122e6			−0.458660
$452$		0			0
$453$		−	2.93125e6i		−	0.671131i
$454$		0			0
$455$	−898113.	−	846336.i	−0.203377	−	0.191652i
$456$		0			0
$457$		−	2.28019e6i		−	0.510718i	−0.966846	−	0.255359i	$-0.917806\pi$
$457$		−	2.28019e6i		−	0.510718i	0.966846	−	0.255359i	$-0.0821937\pi$
$458$		0			0
$459$	−245435.			−0.0543758
$460$		0			0
$461$		−	1.16004e6i		−	0.254227i	−0.991888	−	0.127114i	$-0.959429\pi$
$461$		−	1.16004e6i		−	0.254227i	0.991888	−	0.127114i	$-0.0405713\pi$
$462$		0			0
$463$		−	5.43708e6i		−	1.17873i	−0.807868	−	0.589364i	$-0.799379\pi$
$463$		−	5.43708e6i		−	1.17873i	0.807868	−	0.589364i	$-0.200621\pi$
$464$		0			0
$465$	550664.			0.118101
$466$		0			0
$467$	6.30219e6			1.33721			0.668605	−	0.743618i	$-0.266892\pi$
$467$	6.30219e6			1.33721			0.668605	+	0.743618i	$0.266892\pi$
$468$		0			0
$469$	4.20798e6			0.883368
$470$		0			0
$471$	5.01859e6			1.04239
$472$		0			0
$473$		−	355922.i		−	0.0731480i
$474$		0			0
$475$		−	141544.i		−	0.0287844i
$476$		0			0
$477$	−56430.6			−0.0113558
$478$		0			0
$479$			6.02572e6i			1.19997i	0.800011	+	0.599985i	$0.204827\pi$
$479$			6.02572e6i			1.19997i	−0.800011	+	0.599985i	$0.795173\pi$
$480$		0			0
$481$	1.14473e6	−	1.21476e6i	0.225600	−	0.239402i
$482$		0			0
$483$		−	2.05923e6i		−	0.401640i
$484$		0			0
$485$	−2.88690e6			−0.557285
$486$		0			0
$487$			6.09211e6i			1.16398i	0.813196	+	0.581990i	$0.197726\pi$
$487$			6.09211e6i			1.16398i	−0.813196	+	0.581990i	$0.802274\pi$
$488$		0			0
$489$		−	5.71511e6i		−	1.08082i
$490$		0			0
$491$	−1.65104e6			−0.309068			−0.154534	−	0.987987i	$-0.549388\pi$
$491$	−1.65104e6			−0.309068			−0.154534	+	0.987987i	$0.549388\pi$
$492$		0			0
$493$	1.44932e6			0.268564
$494$		0			0
$495$	440935.			0.0808838
$496$		0			0
$497$	−1.00263e6			−0.182075
$498$		0			0
$499$		−	1.06216e7i		−	1.90958i	−0.297279	−	0.954791i	$-0.596079\pi$
$499$		−	1.06216e7i		−	1.90958i	0.297279	−	0.954791i	$-0.403921\pi$
$500$		0			0
$501$		−	4.32205e6i		−	0.769300i
$502$		0			0
$503$	4.17356e6			0.735506			0.367753	−	0.929923i	$-0.380127\pi$
$503$	4.17356e6			0.735506			0.367753	+	0.929923i	$0.380127\pi$
$504$		0			0
$505$		−	4.24677e6i		−	0.741020i
$506$		0			0
$507$	−198192.	−	3.33575e6i	−0.0342426	−	0.576334i
$508$		0			0
$509$			6.50979e6i			1.11371i	0.830610	+	0.556855i	$0.187992\pi$
$509$			6.50979e6i			1.11371i	−0.830610	+	0.556855i	$0.812008\pi$
$510$		0			0
$511$	−5.13355e6			−0.869692
$512$		0			0
$513$			50776.1i			0.00851856i
$514$		0			0
$515$			2.10341e6i			0.349468i
$516$		0			0
$517$	1.70238e6			0.280111
$518$		0			0
$519$	3.03546e6			0.494659
$520$		0			0
$521$	5.00448e6			0.807727			0.403864	−	0.914819i	$-0.367667\pi$
$521$	5.00448e6			0.807727			0.403864	+	0.914819i	$0.367667\pi$
$522$		0			0
$523$	5.64842e6			0.902969			0.451484	−	0.892279i	$-0.350895\pi$
$523$	5.64842e6			0.902969			0.451484	+	0.892279i	$0.350895\pi$
$524$		0			0
$525$			1.12047e6i			0.177420i
$526$		0			0
$527$			623127.i			0.0977350i
$528$		0			0
$529$	7.51211e6			1.16714
$530$		0			0
$531$		−	1.96626e6i		−	0.302625i
$532$		0			0
$533$	−5.02789e6	+	5.33549e6i	−0.766598	+	0.813497i
$534$		0			0
$535$		−	4.14133e6i		−	0.625541i
$536$		0			0
$537$	5.13162e6			0.767925
$538$		0			0
$539$		−	2.14956e6i		−	0.318697i
$540$		0			0
$541$			1.27506e7i			1.87300i	0.350670	+	0.936499i	$0.385954\pi$
$541$			1.27506e7i			1.87300i	−0.350670	+	0.936499i	$0.614046\pi$
$542$		0			0
$543$	−1.73396e6			−0.252372
$544$		0			0
$545$	−1.88024e6			−0.271158
$546$		0			0
$547$	1.22119e7			1.74507			0.872537	−	0.488547i	$-0.162473\pi$
$547$	1.22119e7			1.74507			0.872537	+	0.488547i	$0.162473\pi$
$548$		0			0
$549$	−2.16855e6			−0.307071
$550$		0			0
$551$		−	299839.i		−	0.0420735i
$552$		0			0
$553$		−	1.04146e6i		−	0.144821i
$554$		0			0
$555$	814993.			0.112311
$556$		0			0
$557$			3.33060e6i			0.454867i	0.973794	+	0.227433i	$0.0730334\pi$
$557$			3.33060e6i			0.454867i	−0.973794	+	0.227433i	$0.926967\pi$
$558$		0			0
$559$	−958512.	−	903253.i	−0.129738	−	0.122259i
$560$		0			0
$561$			498959.i			0.0669357i
$562$		0			0
$563$	−5.83292e6			−0.775559			−0.387779	−	0.921752i	$-0.626758\pi$
$563$	−5.83292e6			−0.775559			−0.387779	+	0.921752i	$0.626758\pi$
$564$		0			0
$565$		−	7.77396e6i		−	1.02452i
$566$		0			0
$567$		−	401947.i		−	0.0525063i
$568$		0			0
$569$	−237693.			−0.0307777			−0.0153888	−	0.999882i	$-0.504899\pi$
$569$	−237693.			−0.0307777			−0.0153888	+	0.999882i	$0.504899\pi$
$570$		0			0
$571$	7.42085e6			0.952496			0.476248	−	0.879311i	$-0.341996\pi$
$571$	7.42085e6			0.952496			0.476248	+	0.879311i	$0.341996\pi$
$572$		0			0
$573$	−2.88602e6			−0.367208
$574$		0			0
$575$	−7.58966e6			−0.957311
$576$		0			0
$577$		−	6.17238e6i		−	0.771814i	−0.922538	−	0.385907i	$-0.873888\pi$
$577$		−	6.17238e6i		−	0.771814i	0.922538	−	0.385907i	$-0.126112\pi$
$578$		0			0
$579$		−	3.47476e6i		−	0.430753i
$580$		0			0
$581$	−2.36126e6			−0.290204
$582$		0			0
$583$			114721.i			0.0139788i
$584$		0			0
$585$	1.11900e6	−	1.18745e6i	0.135188	−	0.143459i
$586$		0			0
$587$			4.27561e6i			0.512157i	0.966656	+	0.256078i	$0.0824305\pi$
$587$			4.27561e6i			0.512157i	−0.966656	+	0.256078i	$0.917569\pi$
$588$		0			0
$589$	128914.			0.0153113
$590$		0			0
$591$		−	8.02456e6i		−	0.945045i
$592$		0			0
$593$			1.00584e6i			0.117460i	0.998274	+	0.0587302i	$0.0187052\pi$
$593$			1.00584e6i			0.117460i	−0.998274	+	0.0587302i	$0.981295\pi$
$594$		0			0
$595$	681845.			0.0789575
$596$		0			0
$597$	6.30353e6			0.723849
$598$		0			0
$599$	2.06786e6			0.235480			0.117740	−	0.993044i	$-0.462435\pi$
$599$	2.06786e6			0.235480			0.117740	+	0.993044i	$0.462435\pi$
$600$		0			0
$601$	4.96901e6			0.561157			0.280578	−	0.959831i	$-0.409474\pi$
$601$	4.96901e6			0.561157			0.280578	+	0.959831i	$0.409474\pi$
$602$		0			0
$603$			5.56365e6i			0.623112i
$604$		0			0
$605$			4.42763e6i			0.491793i
$606$		0			0
$607$	7.68015e6			0.846054			0.423027	−	0.906117i	$-0.360968\pi$
$607$	7.68015e6			0.846054			0.423027	+	0.906117i	$0.360968\pi$
$608$		0			0
$609$			2.37355e6i			0.259331i
$610$		0			0
$611$	4.32026e6	−	4.58456e6i	0.468174	−	0.496815i
$612$		0			0
$613$		−	1.42866e7i		−	1.53560i	−0.640689	−	0.767801i	$-0.721351\pi$
$613$		−	1.42866e7i		−	1.53560i	0.640689	−	0.767801i	$-0.278649\pi$
$614$		0			0
$615$	−3.57963e6			−0.381637
$616$		0			0
$617$			1.20205e7i			1.27119i	0.772022	+	0.635596i	$0.219246\pi$
$617$			1.20205e7i			1.27119i	−0.772022	+	0.635596i	$0.780754\pi$
$618$		0			0
$619$		−	6.90208e6i		−	0.724025i	−0.932173	−	0.362013i	$-0.882090\pi$
$619$		−	6.90208e6i		−	0.724025i	0.932173	−	0.362013i	$-0.117910\pi$
$620$		0			0
$621$	2.72264e6			0.283310
$622$		0			0
$623$	8.59202e6			0.886901
$624$		0			0
$625$	714605.			0.0731755
$626$		0			0
$627$	103226.			0.0104862
$628$		0			0
$629$			922241.i			0.0929432i
$630$		0			0
$631$		−	8.31995e6i		−	0.831855i	−0.909398	−	0.415927i	$-0.863457\pi$
$631$		−	8.31995e6i		−	0.831855i	0.909398	−	0.415927i	$-0.136543\pi$
$632$		0			0
$633$	5.38783e6			0.534447
$634$		0			0
$635$		−	3.62306e6i		−	0.356567i
$636$		0			0
$637$	−5.78886e6	−	5.45512e6i	−0.565255	−	0.532667i
$638$		0			0
$639$		−	1.32565e6i		−	0.128433i
$640$		0			0
$641$	5.62851e6			0.541064			0.270532	−	0.962711i	$-0.412800\pi$
$641$	5.62851e6			0.541064			0.270532	+	0.962711i	$0.412800\pi$
$642$		0			0
$643$			1.34363e7i			1.28160i	0.767708	+	0.640799i	$0.221397\pi$
$643$			1.34363e7i			1.28160i	−0.767708	+	0.640799i	$0.778603\pi$
$644$		0			0
$645$		−	643075.i		−	0.0608643i
$646$		0			0
$647$	−1.14169e7			−1.07223			−0.536115	−	0.844145i	$-0.680109\pi$
$647$	−1.14169e7			−1.07223			−0.536115	+	0.844145i	$0.680109\pi$
$648$		0			0
$649$	−3.99732e6			−0.372527
$650$		0			0
$651$	−1.02049e6			−0.0943749
$652$		0			0
$653$	−1.01592e7			−0.932345			−0.466173	−	0.884694i	$-0.654367\pi$
$653$	−1.01592e7			−0.932345			−0.466173	+	0.884694i	$0.654367\pi$
$654$		0			0
$655$			7.26267e6i			0.661444i
$656$		0			0
$657$		−	6.78741e6i		−	0.613466i
$658$		0			0
$659$	7.04782e6			0.632181			0.316090	−	0.948729i	$-0.397630\pi$
$659$	7.04782e6			0.632181			0.316090	+	0.948729i	$0.397630\pi$
$660$		0			0
$661$			1.08823e7i			0.968762i	0.874857	+	0.484381i	$0.160955\pi$
$661$			1.08823e7i			0.968762i	−0.874857	+	0.484381i	$0.839045\pi$
$662$		0			0
$663$	1.34371e6	+	1.26625e6i	0.118720	+	0.111876i
$664$		0			0
$665$		−	141061.i		−	0.0123695i
$666$		0			0
$667$	−1.60775e7			−1.39928
$668$		0			0
$669$			6.25429e6i			0.540272i
$670$		0			0
$671$			4.40857e6i			0.378000i
$672$		0			0
$673$	2.82794e6			0.240676			0.120338	−	0.992733i	$-0.461602\pi$
$673$	2.82794e6			0.240676			0.120338	+	0.992733i	$0.461602\pi$
$674$		0			0
$675$	−1.48145e6			−0.125149
$676$		0			0
$677$	−2.19071e7			−1.83702			−0.918508	−	0.395401i	$-0.870605\pi$
$677$	−2.19071e7			−1.83702			−0.918508	+	0.395401i	$0.870605\pi$
$678$		0			0
$679$	5.35000e6			0.445328
$680$		0			0
$681$		−	9.59456e6i		−	0.792788i
$682$		0			0
$683$		−	7.16197e6i		−	0.587463i	−0.955888	−	0.293732i	$-0.905103\pi$
$683$		−	7.16197e6i		−	0.587463i	0.955888	−	0.293732i	$-0.0948972\pi$
$684$		0			0
$685$	−302330.			−0.0246181
$686$		0			0
$687$			8.87071e6i			0.717078i
$688$		0			0
$689$	308947.	+	291136.i	0.0247934	+	0.0233641i
$690$		0			0
$691$			1.89454e7i			1.50941i	0.656062	+	0.754707i	$0.272221\pi$
$691$			1.89454e7i			1.50941i	−0.656062	+	0.754707i	$0.727779\pi$
$692$		0			0
$693$	−817141.			−0.0646344
$694$		0			0
$695$		−	4.85247e6i		−	0.381066i
$696$		0			0
$697$		−	4.05069e6i		−	0.315825i
$698$		0			0
$699$	9.11534e6			0.705635
$700$		0			0
$701$	−2.38204e7			−1.83085			−0.915426	−	0.402487i	$-0.868146\pi$
$701$	−2.38204e7			−1.83085			−0.915426	+	0.402487i	$0.868146\pi$
$702$		0			0
$703$	190795.			0.0145606
$704$		0			0
$705$	3.07583e6			0.233072
$706$		0			0
$707$			7.87011e6i			0.592151i
$708$		0			0
$709$			1.15066e7i			0.859666i	0.902908	+	0.429833i	$0.141428\pi$
$709$			1.15066e7i			0.859666i	−0.902908	+	0.429833i	$0.858572\pi$
$710$		0			0
$711$	1.37699e6			0.102154
$712$		0			0
$713$		−	6.91242e6i		−	0.509221i
$714$		0			0
$715$	−2.41404e6	−	2.27487e6i	−0.176595	−	0.166414i
$716$		0			0
$717$			1.01444e6i			0.0736931i
$718$		0			0
$719$	−2.25090e7			−1.62381			−0.811903	−	0.583792i	$-0.801568\pi$
$719$	−2.25090e7			−1.62381			−0.811903	+	0.583792i	$0.801568\pi$
$720$		0			0
$721$		−	3.89805e6i		−	0.279260i
$722$		0			0
$723$			8.46020e6i			0.601915i
$724$		0			0
$725$	8.74813e6			0.618117
$726$		0			0
$727$	−1.44393e7			−1.01323			−0.506617	−	0.862171i	$-0.669104\pi$
$727$	−1.44393e7			−1.01323			−0.506617	+	0.862171i	$0.669104\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	727700.			0.0503685
$732$		0			0
$733$		−	1.27813e6i		−	0.0878648i	−0.999035	−	0.0439324i	$-0.986011\pi$
$733$		−	1.27813e6i		−	0.0878648i	0.999035	−	0.0439324i	$-0.0139886\pi$
$734$		0			0
$735$		−	3.88380e6i		−	0.265179i
$736$		0			0
$737$	1.13106e7			0.767040
$738$		0			0
$739$		−	8.59162e6i		−	0.578714i	−0.957221	−	0.289357i	$-0.906559\pi$
$739$		−	8.59162e6i		−	0.578714i	0.957221	−	0.289357i	$-0.0934414\pi$
$740$		0			0
$741$	261964.	−	277990.i	0.0175265	−	0.0185988i
$742$		0			0
$743$			1.97043e7i			1.30945i	0.755866	+	0.654727i	$0.227216\pi$
$743$			1.97043e7i			1.30945i	−0.755866	+	0.654727i	$0.772784\pi$
$744$		0			0
$745$	6.79076e6			0.448258
$746$		0			0
$747$		−	3.12197e6i		−	0.204705i
$748$		0			0
$749$			7.67472e6i			0.499871i
$750$		0			0
$751$	−1.40159e7			−0.906818			−0.453409	−	0.891302i	$-0.649792\pi$
$751$	−1.40159e7			−0.906818			−0.453409	+	0.891302i	$0.649792\pi$
$752$		0			0
$753$	1.10389e7			0.709474
$754$		0			0
$755$	−1.07668e7			−0.687416
$756$		0			0
$757$	−1.63210e7			−1.03516			−0.517581	−	0.855634i	$-0.673167\pi$
$757$	−1.63210e7			−1.03516			−0.517581	+	0.855634i	$0.673167\pi$
$758$		0			0
$759$		−	5.53501e6i		−	0.348750i
$760$		0			0
$761$			1.94245e7i			1.21587i	0.793987	+	0.607935i	$0.208002\pi$
$761$			1.94245e7i			1.21587i	−0.793987	+	0.607935i	$0.791998\pi$
$762$		0			0
$763$	3.48446e6			0.216683
$764$		0			0
$765$			901512.i			0.0556952i
$766$		0			0
$767$	−1.01443e7	+	1.07649e7i	−0.622637	+	0.660728i
$768$		0			0
$769$		−	1.44865e7i		−	0.883378i	−0.897168	−	0.441689i	$-0.854380\pi$
$769$		−	1.44865e7i		−	0.883378i	0.897168	−	0.441689i	$-0.145620\pi$
$770$		0			0
$771$	−7.29328e6			−0.441862
$772$		0			0
$773$			1.93608e7i			1.16540i	0.812687	+	0.582700i	$0.198004\pi$
$773$			1.93608e7i			1.16540i	−0.812687	+	0.582700i	$0.801996\pi$
$774$		0			0
$775$			3.76120e6i			0.224943i
$776$		0			0
$777$	−1.51035e6			−0.0897478
$778$		0			0
$779$	−838014.			−0.0494775
$780$		0			0
$781$	−2.69498e6			−0.158099
$782$		0			0
$783$	−3.13822e6			−0.182928
$784$		0			0
$785$		−	1.84339e7i		−	1.06768i
$786$		0			0
$787$			1.82561e6i			0.105068i	0.998619	+	0.0525341i	$0.0167298\pi$
$787$			1.82561e6i			0.105068i	−0.998619	+	0.0525341i	$0.983270\pi$
$788$		0			0
$789$	−226919.			−0.0129771
$790$		0			0
$791$			1.44067e7i			0.818698i
$792$		0			0
$793$	1.18724e7	+	1.11880e7i	0.670436	+	0.631784i
$794$		0			0
$795$			207276.i			0.0116314i
$796$		0			0
$797$	−4.34190e6			−0.242122			−0.121061	−	0.992645i	$-0.538630\pi$
$797$	−4.34190e6			−0.242122			−0.121061	+	0.992645i	$0.538630\pi$
$798$		0			0
$799$			3.48059e6i			0.192879i
$800$		0			0
$801$			1.13601e7i			0.625605i
$802$		0			0
$803$	−1.37985e7			−0.755166
$804$		0			0
$805$	−7.56378e6			−0.411386
$806$		0			0
$807$	4.48946e6			0.242667
$808$		0			0
$809$	−2.31393e7			−1.24302			−0.621510	−	0.783406i	$-0.713480\pi$
$809$	−2.31393e7			−1.24302			−0.621510	+	0.783406i	$0.713480\pi$
$810$		0			0
$811$		−	1.93460e6i		−	0.103285i	−0.998666	−	0.0516427i	$-0.983554\pi$
$811$		−	1.93460e6i		−	0.103285i	0.998666	−	0.0516427i	$-0.0164457\pi$
$812$		0			0
$813$			4.87585e6i			0.258716i
$814$		0			0
$815$	−2.09922e7			−1.10704
$816$		0			0
$817$		−	150548.i		−	0.00789077i
$818$		0			0
$819$	−2.07372e6	+	2.20059e6i	−0.108029	+	0.114638i
$820$		0			0
$821$			1.93584e7i			1.00233i	0.865352	+	0.501165i	$0.167095\pi$
$821$			1.93584e7i			1.00233i	−0.865352	+	0.501165i	$0.832905\pi$
$822$		0			0
$823$	1.38012e6			0.0710260			0.0355130	−	0.999369i	$-0.488693\pi$
$823$	1.38012e6			0.0710260			0.0355130	+	0.999369i	$0.488693\pi$
$824$		0			0
$825$			3.01172e6i			0.154056i
$826$		0			0
$827$		−	595025.i		−	0.0302532i	−0.999886	−	0.0151266i	$-0.995185\pi$
$827$		−	595025.i		−	0.0302532i	0.999886	−	0.0151266i	$-0.00481513\pi$
$828$		0			0
$829$	−1.58534e7			−0.801193			−0.400597	−	0.916254i	$-0.631197\pi$
$829$	−1.58534e7			−0.801193			−0.400597	+	0.916254i	$0.631197\pi$
$830$		0			0
$831$	1.28388e7			0.644946
$832$		0			0
$833$	4.39489e6			0.219450
$834$		0			0
$835$	−1.58754e7			−0.787967
$836$		0			0
$837$		−	1.34926e6i		−	0.0665704i
$838$		0			0
$839$			3.11440e7i			1.52746i	0.645538	+	0.763728i	$0.276633\pi$
$839$			3.11440e7i			1.52746i	−0.645538	+	0.763728i	$0.723367\pi$
$840$		0			0
$841$	−1.97959e6			−0.0965131
$842$		0			0
$843$			1.67557e6i			0.0812071i
$844$		0			0
$845$	−1.22526e7	+	727982.i	−0.590319	+	0.0350735i
$846$		0			0
$847$		−	8.20528e6i		−	0.392993i
$848$		0			0
$849$	−1.47952e7			−0.704452
$850$		0			0
$851$		−	1.02305e7i		−	0.484255i
$852$		0			0
$853$			6.07749e6i			0.285991i	0.989723	+	0.142995i	$0.0456734\pi$
$853$			6.07749e6i			0.285991i	−0.989723	+	0.142995i	$0.954327\pi$
$854$		0			0
$855$	186506.			0.00872526
$856$		0			0
$857$	1.18639e7			0.551790			0.275895	−	0.961188i	$-0.411026\pi$
$857$	1.18639e7			0.551790			0.275895	+	0.961188i	$0.411026\pi$
$858$		0			0
$859$	8.22653e6			0.380394			0.190197	−	0.981746i	$-0.439087\pi$
$859$	8.22653e6			0.380394			0.190197	+	0.981746i	$0.439087\pi$
$860$		0			0
$861$	6.63378e6			0.304967
$862$		0			0
$863$			1.04409e7i			0.477212i	0.971116	+	0.238606i	$0.0766905\pi$
$863$			1.04409e7i			0.477212i	−0.971116	+	0.238606i	$0.923310\pi$
$864$		0			0
$865$		−	1.11496e7i		−	0.506662i
$866$		0			0
$867$	1.17586e7			0.531259
$868$		0			0
$869$		−	2.79935e6i		−	0.125750i
$870$		0			0
$871$	2.87039e7	−	3.04600e7i	1.28202	−	1.36045i
$872$		0			0
$873$			7.07359e6i			0.314126i
$874$		0			0
$875$	1.04445e7			0.461176
$876$		0			0
$877$			3.73473e7i			1.63968i	0.572590	+	0.819842i	$0.305939\pi$
$877$			3.73473e7i			1.63968i	−0.572590	+	0.819842i	$0.694061\pi$
$878$		0			0
$879$		−	1.83853e7i		−	0.802600i
$880$		0			0
$881$	1.87870e7			0.815489			0.407745	−	0.913096i	$-0.366315\pi$
$881$	1.87870e7			0.815489			0.407745	+	0.913096i	$0.366315\pi$
$882$		0			0
$883$	671977.			0.0290036			0.0145018	−	0.999895i	$-0.495384\pi$
$883$	671977.			0.0290036			0.0145018	+	0.999895i	$0.495384\pi$
$884$		0			0
$885$	−7.22230e6			−0.309968
$886$		0			0
$887$	1.74761e6			0.0745823			0.0372912	−	0.999304i	$-0.488127\pi$
$887$	1.74761e6			0.0745823			0.0372912	+	0.999304i	$0.488127\pi$
$888$		0			0
$889$			6.71426e6i			0.284934i
$890$		0			0
$891$		−	1.08040e6i		−	0.0455920i
$892$		0			0
$893$	720071.			0.0302167
$894$		0			0
$895$		−	1.88490e7i		−	0.786559i
$896$		0			0
$897$	−1.49060e7	−	1.40466e7i	−0.618556	−	0.582896i
$898$		0			0
$899$			7.96752e6i			0.328794i
$900$		0			0
$901$	−234552.			−0.00962559
$902$		0			0
$903$			1.19175e6i			0.0486368i
$904$		0			0
$905$			6.36905e6i			0.258496i
$906$		0			0
$907$	−3.53654e7			−1.42745			−0.713723	−	0.700428i	$-0.752993\pi$
$907$	−3.53654e7			−1.42745			−0.713723	+	0.700428i	$0.752993\pi$
$908$		0			0
$909$	−1.04056e7			−0.417693
$910$		0			0
$911$	8.75487e6			0.349505			0.174753	−	0.984612i	$-0.444087\pi$
$911$	8.75487e6			0.349505			0.174753	+	0.984612i	$0.444087\pi$
$912$		0			0
$913$	−6.34683e6			−0.251988
$914$		0			0
$915$			7.96534e6i			0.314522i
$916$		0			0
$917$		−	1.34592e7i		−	0.528561i
$918$		0			0
$919$	4.34862e6			0.169849			0.0849243	−	0.996387i	$-0.472935\pi$
$919$	4.34862e6			0.169849			0.0849243	+	0.996387i	$0.472935\pi$
$920$		0			0
$921$		−	2.83911e7i		−	1.10289i
$922$		0			0
$923$	−6.83927e6	+	7.25768e6i	−0.264244	+	0.280410i
$924$		0			0
$925$			5.56666e6i			0.213914i
$926$		0			0
$927$	5.15387e6			0.196985
$928$		0			0
$929$		−	3.42017e7i		−	1.30019i	−0.759851	−	0.650097i	$-0.774728\pi$
$929$		−	3.42017e7i		−	1.30019i	0.759851	−	0.650097i	$-0.225272\pi$
$930$		0			0
$931$		−	909222.i		−	0.0343792i
$932$		0			0
$933$	−2.43162e6			−0.0914515
$934$		0			0
$935$	1.83273e6			0.0685599
$936$		0			0
$937$	2.46362e7			0.916697			0.458348	−	0.888773i	$-0.348441\pi$
$937$	2.46362e7			0.916697			0.458348	+	0.888773i	$0.348441\pi$
$938$		0			0
$939$	9.50238e6			0.351697
$940$		0			0
$941$			1.57679e7i			0.580496i	0.956951	+	0.290248i	$0.0937378\pi$
$941$			1.57679e7i			0.580496i	−0.956951	+	0.290248i	$0.906262\pi$
$942$		0			0
$943$			4.49347e7i			1.64552i
$944$		0			0
$945$	−1.47640e6			−0.0537804
$946$		0			0
$947$			4.99665e7i			1.81052i	0.424857	+	0.905261i	$0.360325\pi$
$947$			4.99665e7i			1.81052i	−0.424857	+	0.905261i	$0.639675\pi$
$948$		0			0
$949$	−3.50175e7	+	3.71598e7i	−1.26218	+	1.33939i
$950$		0			0
$951$			1.49259e6i			0.0535168i
$952$		0			0
$953$	−2.99714e7			−1.06899			−0.534496	−	0.845171i	$-0.679499\pi$
$953$	−2.99714e7			−1.06899			−0.534496	+	0.845171i	$0.679499\pi$
$954$		0			0
$955$			1.06007e7i			0.376118i
$956$		0			0
$957$			6.37986e6i			0.225181i
$958$		0			0
$959$	560278.			0.0196724
$960$		0			0
$961$	2.52036e7			0.880346
$962$		0			0
$963$	−1.01472e7			−0.352600
$964$		0			0
$965$	−1.27632e7			−0.441205
$966$		0			0
$967$			3.53755e6i			0.121657i	0.998148	+	0.0608284i	$0.0193743\pi$
$967$			3.53755e6i			0.121657i	−0.998148	+	0.0608284i	$0.980626\pi$
$968$		0			0
$969$			211049.i			0.00722062i
$970$		0			0
$971$	4.50145e6			0.153216			0.0766081	−	0.997061i	$-0.475591\pi$
$971$	4.50145e6			0.153216			0.0766081	+	0.997061i	$0.475591\pi$
$972$		0			0
$973$			8.99259e6i			0.304511i
$974$		0			0
$975$	8.11067e6	+	7.64309e6i	0.273241	+	0.257488i
$976$		0			0
$977$			3.17740e6i			0.106497i	0.998581	+	0.0532483i	$0.0169575\pi$
$977$			3.17740e6i			0.106497i	−0.998581	+	0.0532483i	$0.983043\pi$
$978$		0			0
$979$	2.30945e7			0.770109
$980$		0			0
$981$			4.60704e6i			0.152844i
$982$		0			0
$983$		−	4.62115e7i		−	1.52534i	−0.646789	−	0.762669i	$-0.723888\pi$
$983$		−	4.62115e7i		−	1.52534i	0.646789	−	0.762669i	$-0.276112\pi$
$984$		0			0
$985$	−2.94751e7			−0.967977
$986$		0			0
$987$	−5.70013e6			−0.186248
$988$		0			0
$989$	−8.07245e6			−0.262431
$990$		0			0
$991$	−2.68291e7			−0.867803			−0.433902	−	0.900960i	$-0.642863\pi$
$991$	−2.68291e7			−0.867803			−0.433902	+	0.900960i	$0.642863\pi$
$992$		0			0
$993$			2.06110e7i			0.663325i
$994$		0			0
$995$		−	2.31536e7i		−	0.741413i
$996$		0			0
$997$	2.27086e7			0.723524			0.361762	−	0.932271i	$-0.382175\pi$
$997$	2.27086e7			0.723524			0.361762	+	0.932271i	$0.382175\pi$
$998$		0			0
$999$		−	1.99693e6i		−	0.0633065i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.6.c.a.337.3		4
4.3	odd	2		39.6.b.a.25.4	yes	4
12.11	even	2		117.6.b.b.64.1		4
13.12	even	2	inner	624.6.c.a.337.2		4
52.31	even	4		507.6.a.f.1.4		4
52.47	even	4		507.6.a.f.1.1		4
52.51	odd	2		39.6.b.a.25.1	✓	4
156.155	even	2		117.6.b.b.64.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.6.b.a.25.1	✓	4	52.51	odd	2
39.6.b.a.25.4	yes	4	4.3	odd	2
117.6.b.b.64.1		4	12.11	even	2
117.6.b.b.64.4		4	156.155	even	2
507.6.a.f.1.1		4	52.47	even	4
507.6.a.f.1.4		4	52.31	even	4
624.6.c.a.337.2		4	13.12	even	2	inner
624.6.c.a.337.3		4	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +33.0580i q^{5} -61.2631i q^{7} +81.0000 q^{9} -164.669i q^{11} +(-443.460 - 417.895i) q^{13} -297.522i q^{15} +336.674 q^{17} -69.6517i q^{19} +551.368i q^{21} -3734.76 q^{23} +2032.17 q^{25} -729.000 q^{27} +4304.83 q^{29} +1850.83i q^{31} +1482.02i q^{33} +2025.24 q^{35} +2739.27i q^{37} +(3991.14 + 3761.05i) q^{39} -12031.5i q^{41} +2161.44 q^{43} +2677.70i q^{45} +10338.2i q^{47} +13053.8 q^{49} -3030.07 q^{51} -696.674 q^{53} +5443.64 q^{55} +626.866i q^{57} -24274.8i q^{59} -26772.3 q^{61} -4962.31i q^{63} +(13814.8 - 14659.9i) q^{65} +68687.0i q^{67} +33612.9 q^{69} -16366.0i q^{71} -83795.1i q^{73} -18289.5 q^{75} -10088.2 q^{77} +16999.8 q^{79} +6561.00 q^{81} -38542.9i q^{83} +11129.8i q^{85} -38743.5 q^{87} +140248. i q^{89} +(-25601.5 + 27167.8i) q^{91} -16657.5i q^{93} +2302.55 q^{95} +87328.3i q^{97} -13338.2i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 36 q^{3} + 324 q^{9} + 312 q^{13} + 384 q^{17} - 2424 q^{23} + 748 q^{25} - 2916 q^{27} - 3960 q^{29} + 20616 q^{35} - 2808 q^{39} - 4832 q^{43} + 31036 q^{49} - 3456 q^{51} - 1824 q^{53} + 10864 q^{55}+ \cdots - 322920 q^{95}+O(q^{100}) \) 4 * q - 36 * q^3 + 324 * q^9 + 312 * q^13 + 384 * q^17 - 2424 * q^23 + 748 * q^25 - 2916 * q^27 - 3960 * q^29 + 20616 * q^35 - 2808 * q^39 - 4832 * q^43 + 31036 * q^49 - 3456 * q^51 - 1824 * q^53 + 10864 * q^55 - 100992 * q^61 + 42744 * q^65 + 21816 * q^69 - 6732 * q^75 - 20136 * q^77 + 205344 * q^79 + 26244 * q^81 + 35640 * q^87 - 77376 * q^91 - 322920 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more