LMFDB - Embedded newform 600.6.f.f.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,6,Mod(49,600)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("600.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	600.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$96.2302918878$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	600.49
Dual form			600.6.f.f.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000i q^{3} +240.000i q^{7} -81.0000 q^{9} +O(q^{10})$	$q-9.00000i q^{3} +240.000i q^{7} -81.0000 q^{9} -124.000 q^{11} +46.0000i q^{13} -1954.00i q^{17} +1924.00 q^{19} +2160.00 q^{21} +2840.00i q^{23} +729.000i q^{27} +8922.00 q^{29} -4648.00 q^{31} +1116.00i q^{33} +4362.00i q^{37} +414.000 q^{39} -2886.00 q^{41} +11332.0i q^{43} -7008.00i q^{47} -40793.0 q^{49} -17586.0 q^{51} -22594.0i q^{53} -17316.0i q^{57} +28.0000 q^{59} -6386.00 q^{61} -19440.0i q^{63} +39076.0i q^{67} +25560.0 q^{69} -54872.0 q^{71} +21034.0i q^{73} -29760.0i q^{77} -26632.0 q^{79} +6561.00 q^{81} +56188.0i q^{83} -80298.0i q^{87} -64410.0 q^{89} -11040.0 q^{91} +41832.0i q^{93} +116158. i q^{97} +10044.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 162 q^{9}+O(q^{10}) $ 2 * q - 162 * q^9	$ 2 q - 162 q^{9} - 248 q^{11} + 3848 q^{19} + 4320 q^{21} + 17844 q^{29} - 9296 q^{31} + 828 q^{39} - 5772 q^{41} - 81586 q^{49} - 35172 q^{51} + 56 q^{59} - 12772 q^{61} + 51120 q^{69} - 109744 q^{71} - 53264 q^{79} + 13122 q^{81} - 128820 q^{89} - 22080 q^{91} + 20088 q^{99}+O(q^{100}) $ 2 * q - 162 * q^9 - 248 * q^11 + 3848 * q^19 + 4320 * q^21 + 17844 * q^29 - 9296 * q^31 + 828 * q^39 - 5772 * q^41 - 81586 * q^49 - 35172 * q^51 + 56 * q^59 - 12772 * q^61 + 51120 * q^69 - 109744 * q^71 - 53264 * q^79 + 13122 * q^81 - 128820 * q^89 - 22080 * q^91 + 20088 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 9.00000i	− 0.577350i
$3$	− 9.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	240.000i	1.85125i	0.378436	+	0.925627i	$0.376462\pi$
$7$	240.000i	1.85125i	−0.378436	+	0.925627i	$0.623538\pi$
$8$	0	0
$9$	−81.0000	−0.333333
$10$	0	0
$11$	−124.000	−0.308987	−0.154493	−	0.987994i	$-0.549375\pi$
$11$	−124.000	−0.308987	−0.154493	+	0.987994i	$0.549375\pi$
$12$	0	0
$13$	46.0000i	0.0754917i	0.999287	+	0.0377459i	$0.0120177\pi$
$13$	46.0000i	0.0754917i	−0.999287	+	0.0377459i	$0.987982\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1954.00i	− 1.63984i	−0.572476	−	0.819921i	$-0.694017\pi$
$17$	− 1954.00i	− 1.63984i	0.572476	−	0.819921i	$-0.305983\pi$
$18$	0	0
$19$	1924.00	1.22270	0.611352	−	0.791359i	$-0.290626\pi$
$19$	1924.00	1.22270	0.611352	+	0.791359i	$0.290626\pi$
$20$	0	0
$21$	2160.00	1.06882
$22$	0	0
$23$	2840.00i	1.11943i	0.828684	+	0.559717i	$0.189090\pi$
$23$	2840.00i	1.11943i	−0.828684	+	0.559717i	$0.810910\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	729.000i	0.192450i
$28$	0	0
$29$	8922.00	1.97000	0.985002	−	0.172541i	$-0.0551979\pi$
$29$	8922.00	1.97000	0.985002	+	0.172541i	$0.0551979\pi$
$30$	0	0
$31$	−4648.00	−0.868684	−0.434342	−	0.900748i	$-0.643019\pi$
$31$	−4648.00	−0.868684	−0.434342	+	0.900748i	$0.643019\pi$
$32$	0	0
$33$	1116.00i	0.178394i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4362.00i	0.523819i	0.965092	+	0.261910i	$0.0843522\pi$
$37$	4362.00i	0.523819i	−0.965092	+	0.261910i	$0.915648\pi$
$38$	0	0
$39$	414.000	0.0435852
$40$	0	0
$41$	−2886.00	−0.268125	−0.134062	−	0.990973i	$-0.542802\pi$
$41$	−2886.00	−0.268125	−0.134062	+	0.990973i	$0.542802\pi$
$42$	0	0
$43$	11332.0i	0.934621i	0.884093	+	0.467310i	$0.154777\pi$
$43$	11332.0i	0.934621i	−0.884093	+	0.467310i	$0.845223\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 7008.00i	− 0.462753i	−0.972864	−	0.231377i	$-0.925677\pi$
$47$	− 7008.00i	− 0.462753i	0.972864	−	0.231377i	$-0.0743230\pi$
$48$	0	0
$49$	−40793.0	−2.42714
$50$	0	0
$51$	−17586.0	−0.946764
$52$	0	0
$53$	− 22594.0i	− 1.10485i	−0.833562	−	0.552425i	$-0.813703\pi$
$53$	− 22594.0i	− 1.10485i	0.833562	−	0.552425i	$-0.186297\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 17316.0i	− 0.705928i
$58$	0	0
$59$	28.0000	0.00104720	0.000523598	−	1.00000i	$-0.499833\pi$
$59$	28.0000	0.00104720	0.000523598		1.00000i	$0.499833\pi$
$60$	0	0
$61$	−6386.00	−0.219738	−0.109869	−	0.993946i	$-0.535043\pi$
$61$	−6386.00	−0.219738	−0.109869	+	0.993946i	$0.535043\pi$
$62$	0	0
$63$	− 19440.0i	− 0.617085i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	39076.0i	1.06346i	0.846912	+	0.531732i	$0.178459\pi$
$67$	39076.0i	1.06346i	−0.846912	+	0.531732i	$0.821541\pi$
$68$	0	0
$69$	25560.0	0.646306
$70$	0	0
$71$	−54872.0	−1.29183	−0.645914	−	0.763410i	$-0.723524\pi$
$71$	−54872.0	−1.29183	−0.645914	+	0.763410i	$0.723524\pi$
$72$	0	0
$73$	21034.0i	0.461971i	0.972957	+	0.230986i	$0.0741950\pi$
$73$	21034.0i	0.461971i	−0.972957	+	0.230986i	$0.925805\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 29760.0i	− 0.572013i
$78$	0	0
$79$	−26632.0	−0.480105	−0.240052	−	0.970760i	$-0.577165\pi$
$79$	−26632.0	−0.480105	−0.240052	+	0.970760i	$0.577165\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	56188.0i	0.895258i	0.894219	+	0.447629i	$0.147732\pi$
$83$	56188.0i	0.895258i	−0.894219	+	0.447629i	$0.852268\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 80298.0i	− 1.13738i
$88$	0	0
$89$	−64410.0	−0.861942	−0.430971	−	0.902366i	$-0.641829\pi$
$89$	−64410.0	−0.861942	−0.430971	+	0.902366i	$0.641829\pi$
$90$	0	0
$91$	−11040.0	−0.139754
$92$	0	0
$93$	41832.0i	0.501535i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	116158.i	1.25349i	0.779226	+	0.626743i	$0.215613\pi$
$97$	116158.i	1.25349i	−0.779226	+	0.626743i	$0.784387\pi$
$98$	0	0
$99$	10044.0	0.102996
$100$	0	0
$101$	−66834.0	−0.651920	−0.325960	−	0.945384i	$-0.605687\pi$
$101$	−66834.0	−0.651920	−0.325960	+	0.945384i	$0.605687\pi$
$102$	0	0
$103$	64000.0i	0.594411i	0.954814	+	0.297206i	$0.0960547\pi$
$103$	64000.0i	0.594411i	−0.954814	+	0.297206i	$0.903945\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15084.0i	0.127367i	0.997970	+	0.0636835i	$0.0202848\pi$
$107$	15084.0i	0.127367i	−0.997970	+	0.0636835i	$0.979715\pi$
$108$	0	0
$109$	39698.0	0.320039	0.160019	−	0.987114i	$-0.448844\pi$
$109$	39698.0	0.320039	0.160019	+	0.987114i	$0.448844\pi$
$110$	0	0
$111$	39258.0	0.302427
$112$	0	0
$113$	155154.i	1.14305i	0.820583	+	0.571527i	$0.193649\pi$
$113$	155154.i	1.14305i	−0.820583	+	0.571527i	$0.806351\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3726.00i	− 0.0251639i
$118$	0	0
$119$	468960.	3.03577
$120$	0	0
$121$	−145675.	−0.904527
$122$	0	0
$123$	25974.0i	0.154802i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 52072.0i	− 0.286480i	−0.989688	−	0.143240i	$-0.954248\pi$
$127$	− 52072.0i	− 0.286480i	0.989688	−	0.143240i	$-0.0457522\pi$
$128$	0	0
$129$	101988.	0.539604
$130$	0	0
$131$	159964.	0.814412	0.407206	−	0.913336i	$-0.366503\pi$
$131$	159964.	0.814412	0.407206	+	0.913336i	$0.366503\pi$
$132$	0	0
$133$	461760.i	2.26353i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	262278.i	1.19388i	0.802286	+	0.596940i	$0.203617\pi$
$137$	262278.i	1.19388i	−0.802286	+	0.596940i	$0.796383\pi$
$138$	0	0
$139$	−253524.	−1.11297	−0.556483	−	0.830859i	$-0.687850\pi$
$139$	−253524.	−1.11297	−0.556483	+	0.830859i	$0.687850\pi$
$140$	0	0
$141$	−63072.0	−0.267171
$142$	0	0
$143$	− 5704.00i	− 0.0233260i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	367137.i	1.40131i
$148$	0	0
$149$	−355630.	−1.31230	−0.656149	−	0.754631i	$-0.727816\pi$
$149$	−355630.	−1.31230	−0.656149	+	0.754631i	$0.727816\pi$
$150$	0	0
$151$	−1024.00	−0.00365475	−0.00182737	−	0.999998i	$-0.500582\pi$
$151$	−1024.00	−0.00365475	−0.00182737	+	0.999998i	$0.500582\pi$
$152$	0	0
$153$	158274.i	0.546614i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	59954.0i	0.194119i	0.995279	+	0.0970597i	$0.0309438\pi$
$157$	59954.0i	0.194119i	−0.995279	+	0.0970597i	$0.969056\pi$
$158$	0	0
$159$	−203346.	−0.637886
$160$	0	0
$161$	−681600.	−2.07236
$162$	0	0
$163$	− 341556.i	− 1.00692i	−0.864020	−	0.503458i	$-0.832061\pi$
$163$	− 341556.i	− 1.00692i	0.864020	−	0.503458i	$-0.167939\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 5016.00i	− 0.0139177i	−0.999976	−	0.00695883i	$-0.997785\pi$
$167$	− 5016.00i	− 0.0139177i	0.999976	−	0.00695883i	$-0.00221508\pi$
$168$	0	0
$169$	369177.	0.994301
$170$	0	0
$171$	−155844.	−0.407568
$172$	0	0
$173$	− 228666.i	− 0.580880i	−0.956893	−	0.290440i	$-0.906198\pi$
$173$	− 228666.i	− 0.580880i	0.956893	−	0.290440i	$-0.0938016\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 252.000i	0 0.000604599i
$178$	0	0
$179$	−161388.	−0.376477	−0.188239	−	0.982123i	$-0.560278\pi$
$179$	−161388.	−0.376477	−0.188239	+	0.982123i	$0.560278\pi$
$180$	0	0
$181$	−291690.	−0.661797	−0.330899	−	0.943666i	$-0.607352\pi$
$181$	−291690.	−0.661797	−0.330899	+	0.943666i	$0.607352\pi$
$182$	0	0
$183$	57474.0i	0.126866i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	242296.i	0.506690i
$188$	0	0
$189$	−174960.	−0.356274
$190$	0	0
$191$	−55680.0	−0.110437	−0.0552187	−	0.998474i	$-0.517586\pi$
$191$	−55680.0	−0.110437	−0.0552187	+	0.998474i	$0.517586\pi$
$192$	0	0
$193$	− 176254.i	− 0.340601i	−0.985392	−	0.170300i	$-0.945526\pi$
$193$	− 176254.i	− 0.340601i	0.985392	−	0.170300i	$-0.0544738\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	374610.i	0.687723i	0.939020	+	0.343862i	$0.111735\pi$
$197$	374610.i	0.687723i	−0.939020	+	0.343862i	$0.888265\pi$
$198$	0	0
$199$	637760.	1.14163	0.570814	−	0.821079i	$-0.306628\pi$
$199$	637760.	1.14163	0.570814	+	0.821079i	$0.306628\pi$
$200$	0	0
$201$	351684.	0.613992
$202$	0	0
$203$	2.14128e6i	3.64698i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 230040.i	− 0.373145i
$208$	0	0
$209$	−238576.	−0.377799
$210$	0	0
$211$	−904628.	−1.39883	−0.699413	−	0.714717i	$-0.746555\pi$
$211$	−904628.	−1.39883	−0.699413	+	0.714717i	$0.746555\pi$
$212$	0	0
$213$	493848.i	0.745838i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1.11552e6i	− 1.60816i
$218$	0	0
$219$	189306.	0.266719
$220$	0	0
$221$	89884.0	0.123795
$222$	0	0
$223$	619048.i	0.833609i	0.908996	+	0.416804i	$0.136850\pi$
$223$	619048.i	0.833609i	−0.908996	+	0.416804i	$0.863150\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.46975e6i	1.89312i	0.322527	+	0.946560i	$0.395468\pi$
$227$	1.46975e6i	1.89312i	−0.322527	+	0.946560i	$0.604532\pi$
$228$	0	0
$229$	3290.00	0.00414579	0.00207289	−	0.999998i	$-0.499340\pi$
$229$	3290.00	0.00414579	0.00207289	+	0.999998i	$0.499340\pi$
$230$	0	0
$231$	−267840.	−0.330252
$232$	0	0
$233$	935402.i	1.12878i	0.825509	+	0.564389i	$0.190888\pi$
$233$	935402.i	1.12878i	−0.825509	+	0.564389i	$0.809112\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	239688.i	0.277189i
$238$	0	0
$239$	875600.	0.991542	0.495771	−	0.868453i	$-0.334886\pi$
$239$	875600.	0.991542	0.495771	+	0.868453i	$0.334886\pi$
$240$	0	0
$241$	−959214.	−1.06383	−0.531916	−	0.846797i	$-0.678528\pi$
$241$	−959214.	−1.06383	−0.531916	+	0.846797i	$0.678528\pi$
$242$	0	0
$243$	− 59049.0i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	88504.0i	0.0923040i
$248$	0	0
$249$	505692.	0.516878
$250$	0	0
$251$	318868.	0.319467	0.159734	−	0.987160i	$-0.448936\pi$
$251$	318868.	0.319467	0.159734	+	0.987160i	$0.448936\pi$
$252$	0	0
$253$	− 352160.i	− 0.345891i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.71469e6i	− 1.61940i	−0.586847	−	0.809698i	$-0.699631\pi$
$257$	− 1.71469e6i	− 1.61940i	0.586847	−	0.809698i	$-0.300369\pi$
$258$	0	0
$259$	−1.04688e6	−0.969723
$260$	0	0
$261$	−722682.	−0.656668
$262$	0	0
$263$	− 1.11028e6i	− 0.989790i	−0.868953	−	0.494895i	$-0.835206\pi$
$263$	− 1.11028e6i	− 0.989790i	0.868953	−	0.494895i	$-0.164794\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	579690.i	0.497643i
$268$	0	0
$269$	398378.	0.335672	0.167836	−	0.985815i	$-0.446322\pi$
$269$	398378.	0.335672	0.167836	+	0.985815i	$0.446322\pi$
$270$	0	0
$271$	1.44198e6	1.19271	0.596355	−	0.802721i	$-0.296615\pi$
$271$	1.44198e6	1.19271	0.596355	+	0.802721i	$0.296615\pi$
$272$	0	0
$273$	99360.0i	0.0806873i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 117238.i	− 0.0918056i	−0.998946	−	0.0459028i	$-0.985384\pi$
$277$	− 117238.i	− 0.0918056i	0.998946	−	0.0459028i	$-0.0146164\pi$
$278$	0	0
$279$	376488.	0.289561
$280$	0	0
$281$	−1.67514e6	−1.26557	−0.632784	−	0.774328i	$-0.718088\pi$
$281$	−1.67514e6	−1.26557	−0.632784	+	0.774328i	$0.718088\pi$
$282$	0	0
$283$	1.92468e6i	1.42854i	0.699872	+	0.714269i	$0.253240\pi$
$283$	1.92468e6i	1.42854i	−0.699872	+	0.714269i	$0.746760\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 692640.i	− 0.496367i
$288$	0	0
$289$	−2.39826e6	−1.68908
$290$	0	0
$291$	1.04542e6	0.723701
$292$	0	0
$293$	1.28062e6i	0.871469i	0.900075	+	0.435734i	$0.143511\pi$
$293$	1.28062e6i	0.871469i	−0.900075	+	0.435734i	$0.856489\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 90396.0i	− 0.0594645i
$298$	0	0
$299$	−130640.	−0.0845081
$300$	0	0
$301$	−2.71968e6	−1.73022
$302$	0	0
$303$	601506.i	0.376386i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.26319e6i	1.37049i	0.728314	+	0.685243i	$0.240304\pi$
$307$	2.26319e6i	1.37049i	−0.728314	+	0.685243i	$0.759696\pi$
$308$	0	0
$309$	576000.	0.343183
$310$	0	0
$311$	247848.	0.145306	0.0726532	−	0.997357i	$-0.476853\pi$
$311$	247848.	0.145306	0.0726532	+	0.997357i	$0.476853\pi$
$312$	0	0
$313$	− 1.82391e6i	− 1.05231i	−0.850390	−	0.526154i	$-0.823634\pi$
$313$	− 1.82391e6i	− 1.05231i	0.850390	−	0.526154i	$-0.176366\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.85629e6i	− 1.59645i	−0.602361	−	0.798224i	$-0.705773\pi$
$317$	− 2.85629e6i	− 1.59645i	0.602361	−	0.798224i	$-0.294227\pi$
$318$	0	0
$319$	−1.10633e6	−0.608705
$320$	0	0
$321$	135756.	0.0735354
$322$	0	0
$323$	− 3.75950e6i	− 2.00504i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 357282.i	− 0.184774i
$328$	0	0
$329$	1.68192e6	0.856674
$330$	0	0
$331$	−147148.	−0.0738218	−0.0369109	−	0.999319i	$-0.511752\pi$
$331$	−147148.	−0.0738218	−0.0369109	+	0.999319i	$0.511752\pi$
$332$	0	0
$333$	− 353322.i	− 0.174606i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.24728e6i	1.55756i	0.627297	+	0.778780i	$0.284161\pi$
$337$	3.24728e6i	1.55756i	−0.627297	+	0.778780i	$0.715839\pi$
$338$	0	0
$339$	1.39639e6	0.659943
$340$	0	0
$341$	576352.	0.268412
$342$	0	0
$343$	− 5.75664e6i	− 2.64201i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.55675e6i	0.694056i	0.937855	+	0.347028i	$0.112809\pi$
$347$	1.55675e6i	0.694056i	−0.937855	+	0.347028i	$0.887191\pi$
$348$	0	0
$349$	−4.03217e6	−1.77205	−0.886024	−	0.463639i	$-0.846544\pi$
$349$	−4.03217e6	−1.77205	−0.886024	+	0.463639i	$0.846544\pi$
$350$	0	0
$351$	−33534.0	−0.0145284
$352$	0	0
$353$	1.79399e6i	0.766271i	0.923692	+	0.383135i	$0.125156\pi$
$353$	1.79399e6i	0.766271i	−0.923692	+	0.383135i	$0.874844\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 4.22064e6i	− 1.75270i
$358$	0	0
$359$	−1.55278e6	−0.635876	−0.317938	−	0.948111i	$-0.602990\pi$
$359$	−1.55278e6	−0.635876	−0.317938	+	0.948111i	$0.602990\pi$
$360$	0	0
$361$	1.22568e6	0.495003
$362$	0	0
$363$	1.31108e6i	0.522229i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.11545e6i	1.20741i	0.797207	+	0.603706i	$0.206310\pi$
$367$	3.11545e6i	1.20741i	−0.797207	+	0.603706i	$0.793690\pi$
$368$	0	0
$369$	233766.	0.0893749
$370$	0	0
$371$	5.42256e6	2.04536
$372$	0	0
$373$	− 630682.i	− 0.234714i	−0.993090	−	0.117357i	$-0.962558\pi$
$373$	− 630682.i	− 0.234714i	0.993090	−	0.117357i	$-0.0374421\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	410412.i	0.148719i
$378$	0	0
$379$	−48404.0	−0.0173094	−0.00865472	−	0.999963i	$-0.502755\pi$
$379$	−48404.0	−0.0173094	−0.00865472	+	0.999963i	$0.502755\pi$
$380$	0	0
$381$	−468648.	−0.165400
$382$	0	0
$383$	1.74182e6i	0.606747i	0.952872	+	0.303373i	$0.0981129\pi$
$383$	1.74182e6i	0.606747i	−0.952872	+	0.303373i	$0.901887\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 917892.i	− 0.311540i
$388$	0	0
$389$	3.06819e6	1.02804	0.514019	−	0.857779i	$-0.328156\pi$
$389$	3.06819e6	1.02804	0.514019	+	0.857779i	$0.328156\pi$
$390$	0	0
$391$	5.54936e6	1.83570
$392$	0	0
$393$	− 1.43968e6i	− 0.470201i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 5.35984e6i	− 1.70677i	−0.521280	−	0.853386i	$-0.674545\pi$
$397$	− 5.35984e6i	− 1.70677i	0.521280	−	0.853386i	$-0.325455\pi$
$398$	0	0
$399$	4.15584e6	1.30685
$400$	0	0
$401$	−2.76473e6	−0.858603	−0.429302	−	0.903161i	$-0.641240\pi$
$401$	−2.76473e6	−0.858603	−0.429302	+	0.903161i	$0.641240\pi$
$402$	0	0
$403$	− 213808.i	− 0.0655785i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 540888.i	− 0.161853i
$408$	0	0
$409$	1.20893e6	0.357350	0.178675	−	0.983908i	$-0.442819\pi$
$409$	1.20893e6	0.357350	0.178675	+	0.983908i	$0.442819\pi$
$410$	0	0
$411$	2.36050e6	0.689287
$412$	0	0
$413$	6720.00i	0.00193863i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.28172e6i	0.642571i
$418$	0	0
$419$	4.38008e6	1.21884	0.609421	−	0.792847i	$-0.291402\pi$
$419$	4.38008e6	1.21884	0.609421	+	0.792847i	$0.291402\pi$
$420$	0	0
$421$	−922810.	−0.253751	−0.126875	−	0.991919i	$-0.540495\pi$
$421$	−922810.	−0.253751	−0.126875	+	0.991919i	$0.540495\pi$
$422$	0	0
$423$	567648.i	0.154251i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1.53264e6i	− 0.406790i
$428$	0	0
$429$	−51336.0	−0.0134672
$430$	0	0
$431$	6.12678e6	1.58869	0.794345	−	0.607466i	$-0.207814\pi$
$431$	6.12678e6	1.58869	0.794345	+	0.607466i	$0.207814\pi$
$432$	0	0
$433$	− 1.76315e6i	− 0.451928i	−0.974136	−	0.225964i	$-0.927447\pi$
$433$	− 1.76315e6i	− 0.451928i	0.974136	−	0.225964i	$-0.0725532\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.46416e6i	1.36874i
$438$	0	0
$439$	−3.85906e6	−0.955696	−0.477848	−	0.878443i	$-0.658583\pi$
$439$	−3.85906e6	−0.955696	−0.477848	+	0.878443i	$0.658583\pi$
$440$	0	0
$441$	3.30423e6	0.809048
$442$	0	0
$443$	− 4.39396e6i	− 1.06377i	−0.846817	−	0.531884i	$-0.821484\pi$
$443$	− 4.39396e6i	− 1.06377i	0.846817	−	0.531884i	$-0.178516\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.20067e6i	0.757656i
$448$	0	0
$449$	793390.	0.185725	0.0928626	−	0.995679i	$-0.470398\pi$
$449$	793390.	0.185725	0.0928626	+	0.995679i	$0.470398\pi$
$450$	0	0
$451$	357864.	0.0828470
$452$	0	0
$453$	9216.00i	0.00211007i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 7.04302e6i	− 1.57750i	−0.614717	−	0.788748i	$-0.710730\pi$
$457$	− 7.04302e6i	− 1.57750i	0.614717	−	0.788748i	$-0.289270\pi$
$458$	0	0
$459$	1.42447e6	0.315588
$460$	0	0
$461$	7.43005e6	1.62832	0.814160	−	0.580641i	$-0.197198\pi$
$461$	7.43005e6	1.62832	0.814160	+	0.580641i	$0.197198\pi$
$462$	0	0
$463$	− 4.10567e6i	− 0.890086i	−0.895509	−	0.445043i	$-0.853188\pi$
$463$	− 4.10567e6i	− 0.890086i	0.895509	−	0.445043i	$-0.146812\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.39817e6i	− 0.721030i	−0.932753	−	0.360515i	$-0.882601\pi$
$467$	− 3.39817e6i	− 0.721030i	0.932753	−	0.360515i	$-0.117399\pi$
$468$	0	0
$469$	−9.37824e6	−1.96874
$470$	0	0
$471$	539586.	0.112075
$472$	0	0
$473$	− 1.40517e6i	− 0.288786i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.83011e6i	0.368283i
$478$	0	0
$479$	−2.78133e6	−0.553877	−0.276939	−	0.960888i	$-0.589320\pi$
$479$	−2.78133e6	−0.553877	−0.276939	+	0.960888i	$0.589320\pi$
$480$	0	0
$481$	−200652.	−0.0395440
$482$	0	0
$483$	6.13440e6i	1.19648i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.06734e6i	0.394994i	0.980304	+	0.197497i	$0.0632812\pi$
$487$	2.06734e6i	0.394994i	−0.980304	+	0.197497i	$0.936719\pi$
$488$	0	0
$489$	−3.07400e6	−0.581343
$490$	0	0
$491$	−7.65976e6	−1.43387	−0.716937	−	0.697138i	$-0.754457\pi$
$491$	−7.65976e6	−1.43387	−0.716937	+	0.697138i	$0.754457\pi$
$492$	0	0
$493$	− 1.74336e7i	− 3.23050i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.31693e7i	− 2.39150i
$498$	0	0
$499$	386580.	0.0695005	0.0347503	−	0.999396i	$-0.488936\pi$
$499$	386580.	0.0695005	0.0347503	+	0.999396i	$0.488936\pi$
$500$	0	0
$501$	−45144.0	−0.00803537
$502$	0	0
$503$	− 2.57326e6i	− 0.453485i	−0.973955	−	0.226743i	$-0.927192\pi$
$503$	− 2.57326e6i	− 0.453485i	0.973955	−	0.226743i	$-0.0728076\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 3.32259e6i	− 0.574060i
$508$	0	0
$509$	−360678.	−0.0617057	−0.0308528	−	0.999524i	$-0.509822\pi$
$509$	−360678.	−0.0617057	−0.0308528	+	0.999524i	$0.509822\pi$
$510$	0	0
$511$	−5.04816e6	−0.855226
$512$	0	0
$513$	1.40260e6i	0.235309i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	868992.i	0.142985i
$518$	0	0
$519$	−2.05799e6	−0.335371
$520$	0	0
$521$	−1.55908e6	−0.251636	−0.125818	−	0.992053i	$-0.540156\pi$
$521$	−1.55908e6	−0.251636	−0.125818	+	0.992053i	$0.540156\pi$
$522$	0	0
$523$	− 9.18220e6i	− 1.46789i	−0.679210	−	0.733944i	$-0.737678\pi$
$523$	− 9.18220e6i	− 1.46789i	0.679210	−	0.733944i	$-0.262322\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.08219e6i	1.42451i
$528$	0	0
$529$	−1.62926e6	−0.253134
$530$	0	0
$531$	−2268.00	−0.000349065 0
$532$	0	0
$533$	− 132756.i	− 0.0202412i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.45249e6i	0.217359i
$538$	0	0
$539$	5.05833e6	0.749955
$540$	0	0
$541$	−6.67773e6	−0.980925	−0.490462	−	0.871462i	$-0.663172\pi$
$541$	−6.67773e6	−0.980925	−0.490462	+	0.871462i	$0.663172\pi$
$542$	0	0
$543$	2.62521e6i	0.382089i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 8.89656e6i	− 1.27132i	−0.771971	−	0.635658i	$-0.780729\pi$
$547$	− 8.89656e6i	− 1.27132i	0.771971	−	0.635658i	$-0.219271\pi$
$548$	0	0
$549$	517266.	0.0732459
$550$	0	0
$551$	1.71659e7	2.40873
$552$	0	0
$553$	− 6.39168e6i	− 0.888796i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.46070e6i	0.609207i	0.952479	+	0.304603i	$0.0985239\pi$
$557$	4.46070e6i	0.609207i	−0.952479	+	0.304603i	$0.901476\pi$
$558$	0	0
$559$	−521272.	−0.0705562
$560$	0	0
$561$	2.18066e6	0.292538
$562$	0	0
$563$	6.37660e6i	0.847849i	0.905698	+	0.423924i	$0.139348\pi$
$563$	6.37660e6i	0.847849i	−0.905698	+	0.423924i	$0.860652\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.57464e6i	0.205695i
$568$	0	0
$569$	−5.51143e6	−0.713648	−0.356824	−	0.934172i	$-0.616140\pi$
$569$	−5.51143e6	−0.713648	−0.356824	+	0.934172i	$0.616140\pi$
$570$	0	0
$571$	1.35431e6	0.173831	0.0869155	−	0.996216i	$-0.472299\pi$
$571$	1.35431e6	0.173831	0.0869155	+	0.996216i	$0.472299\pi$
$572$	0	0
$573$	501120.i	0.0637610i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5.00736e6i	0.626137i	0.949731	+	0.313068i	$0.101357\pi$
$577$	5.00736e6i	0.626137i	−0.949731	+	0.313068i	$0.898643\pi$
$578$	0	0
$579$	−1.58629e6	−0.196646
$580$	0	0
$581$	−1.34851e7	−1.65735
$582$	0	0
$583$	2.80166e6i	0.341384i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.69964e6i	− 0.323378i	−0.986842	−	0.161689i	$-0.948306\pi$
$587$	− 2.69964e6i	− 0.323378i	0.986842	−	0.161689i	$-0.0516941\pi$
$588$	0	0
$589$	−8.94275e6	−1.06214
$590$	0	0
$591$	3.37149e6	0.397057
$592$	0	0
$593$	1.31035e7i	1.53021i	0.643908	+	0.765103i	$0.277312\pi$
$593$	1.31035e7i	1.53021i	−0.643908	+	0.765103i	$0.722688\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 5.73984e6i	− 0.659119i
$598$	0	0
$599$	5.22804e6	0.595349	0.297675	−	0.954667i	$-0.403789\pi$
$599$	5.22804e6	0.595349	0.297675	+	0.954667i	$0.403789\pi$
$600$	0	0
$601$	1.02248e7	1.15470	0.577351	−	0.816496i	$-0.304087\pi$
$601$	1.02248e7	1.15470	0.577351	+	0.816496i	$0.304087\pi$
$602$	0	0
$603$	− 3.16516e6i	− 0.354488i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 8.81684e6i	− 0.971273i	−0.874161	−	0.485636i	$-0.838588\pi$
$607$	− 8.81684e6i	− 0.971273i	0.874161	−	0.485636i	$-0.161412\pi$
$608$	0	0
$609$	1.92715e7	2.10558
$610$	0	0
$611$	322368.	0.0349340
$612$	0	0
$613$	1.13600e7i	1.22103i	0.792006	+	0.610514i	$0.209037\pi$
$613$	1.13600e7i	1.22103i	−0.792006	+	0.610514i	$0.790963\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.77356e6i	0.504812i	0.967621	+	0.252406i	$0.0812218\pi$
$617$	4.77356e6i	0.504812i	−0.967621	+	0.252406i	$0.918778\pi$
$618$	0	0
$619$	2.55931e6	0.268470	0.134235	−	0.990950i	$-0.457142\pi$
$619$	2.55931e6	0.268470	0.134235	+	0.990950i	$0.457142\pi$
$620$	0	0
$621$	−2.07036e6	−0.215435
$622$	0	0
$623$	− 1.54584e7i	− 1.59567i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.14718e6i	0.218122i
$628$	0	0
$629$	8.52335e6	0.858981
$630$	0	0
$631$	−8.41981e6	−0.841839	−0.420919	−	0.907098i	$-0.638292\pi$
$631$	−8.41981e6	−0.841839	−0.420919	+	0.907098i	$0.638292\pi$
$632$	0	0
$633$	8.14165e6i	0.807613i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 1.87648e6i	− 0.183229i
$638$	0	0
$639$	4.44463e6	0.430610
$640$	0	0
$641$	−1.21494e7	−1.16791	−0.583957	−	0.811785i	$-0.698496\pi$
$641$	−1.21494e7	−1.16791	−0.583957	+	0.811785i	$0.698496\pi$
$642$	0	0
$643$	− 1.08968e7i	− 1.03937i	−0.854358	−	0.519685i	$-0.826049\pi$
$643$	− 1.08968e7i	− 1.03937i	0.854358	−	0.519685i	$-0.173951\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.32166e7i	− 1.24124i	−0.784110	−	0.620622i	$-0.786880\pi$
$647$	− 1.32166e7i	− 1.24124i	0.784110	−	0.620622i	$-0.213120\pi$
$648$	0	0
$649$	−3472.00	−0.000323570 0
$650$	0	0
$651$	−1.00397e7	−0.928469
$652$	0	0
$653$	1.65915e7i	1.52266i	0.648365	+	0.761329i	$0.275453\pi$
$653$	1.65915e7i	1.52266i	−0.648365	+	0.761329i	$0.724547\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 1.70375e6i	− 0.153990i
$658$	0	0
$659$	2.29372e6	0.205743	0.102872	−	0.994695i	$-0.467197\pi$
$659$	2.29372e6	0.205743	0.102872	+	0.994695i	$0.467197\pi$
$660$	0	0
$661$	−719194.	−0.0640239	−0.0320120	−	0.999487i	$-0.510191\pi$
$661$	−719194.	−0.0640239	−0.0320120	+	0.999487i	$0.510191\pi$
$662$	0	0
$663$	− 808956.i	− 0.0714728i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.53385e7i	2.20529i
$668$	0	0
$669$	5.57143e6	0.481284
$670$	0	0
$671$	791864.	0.0678960
$672$	0	0
$673$	8.64695e6i	0.735911i	0.929843	+	0.367955i	$0.119942\pi$
$673$	8.64695e6i	0.735911i	−0.929843	+	0.367955i	$0.880058\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.69592e7i	1.42211i	0.703135	+	0.711056i	$0.251783\pi$
$677$	1.69592e7i	1.42211i	−0.703135	+	0.711056i	$0.748217\pi$
$678$	0	0
$679$	−2.78779e7	−2.32052
$680$	0	0
$681$	1.32277e7	1.09299
$682$	0	0
$683$	1.87105e7i	1.53473i	0.641209	+	0.767367i	$0.278433\pi$
$683$	1.87105e7i	1.53473i	−0.641209	+	0.767367i	$0.721567\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 29610.0i	− 0.00239357i
$688$	0	0
$689$	1.03932e6	0.0834071
$690$	0	0
$691$	−1.16204e7	−0.925820	−0.462910	−	0.886405i	$-0.653195\pi$
$691$	−1.16204e7	−0.925820	−0.462910	+	0.886405i	$0.653195\pi$
$692$	0	0
$693$	2.41056e6i	0.190671i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5.63924e6i	0.439682i
$698$	0	0
$699$	8.41862e6	0.651700
$700$	0	0
$701$	2.23497e7	1.71781	0.858907	−	0.512132i	$-0.171144\pi$
$701$	2.23497e7	1.71781	0.858907	+	0.512132i	$0.171144\pi$
$702$	0	0
$703$	8.39249e6i	0.640475i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.60402e7i	− 1.20687i
$708$	0	0
$709$	−1.02353e7	−0.764687	−0.382344	−	0.924020i	$-0.624883\pi$
$709$	−1.02353e7	−0.764687	−0.382344	+	0.924020i	$0.624883\pi$
$710$	0	0
$711$	2.15719e6	0.160035
$712$	0	0
$713$	− 1.32003e7i	− 0.972435i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 7.88040e6i	− 0.572467i
$718$	0	0
$719$	1.70339e7	1.22883	0.614416	−	0.788982i	$-0.289392\pi$
$719$	1.70339e7	1.22883	0.614416	+	0.788982i	$0.289392\pi$
$720$	0	0
$721$	−1.53600e7	−1.10041
$722$	0	0
$723$	8.63293e6i	0.614203i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.62280e7i	1.13875i	0.822077	+	0.569377i	$0.192815\pi$
$727$	1.62280e7i	1.13875i	−0.822077	+	0.569377i	$0.807185\pi$
$728$	0	0
$729$	−531441.	−0.0370370
$730$	0	0
$731$	2.21427e7	1.53263
$732$	0	0
$733$	− 2.17495e7i	− 1.49517i	−0.664168	−	0.747583i	$-0.731214\pi$
$733$	− 2.17495e7i	− 1.49517i	0.664168	−	0.747583i	$-0.268786\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 4.84542e6i	− 0.328597i
$738$	0	0
$739$	−1.96200e7	−1.32156	−0.660781	−	0.750578i	$-0.729775\pi$
$739$	−1.96200e7	−1.32156	−0.660781	+	0.750578i	$0.729775\pi$
$740$	0	0
$741$	796536.	0.0532917
$742$	0	0
$743$	1.74018e7i	1.15644i	0.815882	+	0.578218i	$0.196252\pi$
$743$	1.74018e7i	1.15644i	−0.815882	+	0.578218i	$0.803748\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 4.55123e6i	− 0.298419i
$748$	0	0
$749$	−3.62016e6	−0.235789
$750$	0	0
$751$	−2.62693e7	−1.69961	−0.849803	−	0.527101i	$-0.823279\pi$
$751$	−2.62693e7	−1.69961	−0.849803	+	0.527101i	$0.823279\pi$
$752$	0	0
$753$	− 2.86981e6i	− 0.184445i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5.70356e6i	0.361748i	0.983506	+	0.180874i	$0.0578927\pi$
$757$	5.70356e6i	0.361748i	−0.983506	+	0.180874i	$0.942107\pi$
$758$	0	0
$759$	−3.16944e6	−0.199700
$760$	0	0
$761$	−2.13762e7	−1.33804	−0.669020	−	0.743244i	$-0.733286\pi$
$761$	−2.13762e7	−1.33804	−0.669020	+	0.743244i	$0.733286\pi$
$762$	0	0
$763$	9.52752e6i	0.592473i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1288.00i	0 7.90547e-5i
$768$	0	0
$769$	2.01523e6	0.122888	0.0614439	−	0.998111i	$-0.480429\pi$
$769$	2.01523e6	0.122888	0.0614439	+	0.998111i	$0.480429\pi$
$770$	0	0
$771$	−1.54322e7	−0.934958
$772$	0	0
$773$	− 1.27674e7i	− 0.768520i	−0.923225	−	0.384260i	$-0.874457\pi$
$773$	− 1.27674e7i	− 0.768520i	0.923225	−	0.384260i	$-0.125543\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	9.42192e6i	0.559870i
$778$	0	0
$779$	−5.55266e6	−0.327837
$780$	0	0
$781$	6.80413e6	0.399158
$782$	0	0
$783$	6.50414e6i	0.379128i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	2.72384e7i	1.56764i	0.620990	+	0.783818i	$0.286731\pi$
$787$	2.72384e7i	1.56764i	−0.620990	+	0.783818i	$0.713269\pi$
$788$	0	0
$789$	−9.99252e6	−0.571456
$790$	0	0
$791$	−3.72370e7	−2.11608
$792$	0	0
$793$	− 293756.i	− 0.0165884i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.66724e6i	0.427556i	0.976882	+	0.213778i	$0.0685770\pi$
$797$	7.66724e6i	0.427556i	−0.976882	+	0.213778i	$0.931423\pi$
$798$	0	0
$799$	−1.36936e7	−0.758843
$800$	0	0
$801$	5.21721e6	0.287314
$802$	0	0
$803$	− 2.60822e6i	− 0.142743i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 3.58540e6i	− 0.193800i
$808$	0	0
$809$	1.05541e7	0.566956	0.283478	−	0.958979i	$-0.408512\pi$
$809$	1.05541e7	0.566956	0.283478	+	0.958979i	$0.408512\pi$
$810$	0	0
$811$	−1.32883e6	−0.0709442	−0.0354721	−	0.999371i	$-0.511293\pi$
$811$	−1.32883e6	−0.0709442	−0.0354721	+	0.999371i	$0.511293\pi$
$812$	0	0
$813$	− 1.29778e7i	− 0.688611i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.18028e7i	1.14276i
$818$	0	0
$819$	894240.	0.0465848
$820$	0	0
$821$	−6.15933e6	−0.318915	−0.159458	−	0.987205i	$-0.550975\pi$
$821$	−6.15933e6	−0.318915	−0.159458	+	0.987205i	$0.550975\pi$
$822$	0	0
$823$	1.00734e7i	0.518414i	0.965822	+	0.259207i	$0.0834612\pi$
$823$	1.00734e7i	0.518414i	−0.965822	+	0.259207i	$0.916539\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.49152e6i	0.330052i	0.986289	+	0.165026i	$0.0527708\pi$
$827$	6.49152e6i	0.330052i	−0.986289	+	0.165026i	$0.947229\pi$
$828$	0	0
$829$	1.93536e7	0.978082	0.489041	−	0.872261i	$-0.337347\pi$
$829$	1.93536e7	0.978082	0.489041	+	0.872261i	$0.337347\pi$
$830$	0	0
$831$	−1.05514e6	−0.0530040
$832$	0	0
$833$	7.97095e7i	3.98013i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3.38839e6i	− 0.167178i
$838$	0	0
$839$	2.78622e7	1.36650	0.683251	−	0.730183i	$-0.260565\pi$
$839$	2.78622e7	1.36650	0.683251	+	0.730183i	$0.260565\pi$
$840$	0	0
$841$	5.90909e7	2.88092
$842$	0	0
$843$	1.50763e7i	0.730677i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 3.49620e7i	− 1.67451i
$848$	0	0
$849$	1.73221e7	0.824766
$850$	0	0
$851$	−1.23881e7	−0.586381
$852$	0	0
$853$	1.07651e7i	0.506577i	0.967391	+	0.253288i	$0.0815121\pi$
$853$	1.07651e7i	0.506577i	−0.967391	+	0.253288i	$0.918488\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.22439e7i	− 0.569465i	−0.958607	−	0.284733i	$-0.908095\pi$
$857$	− 1.22439e7i	− 0.569465i	0.958607	−	0.284733i	$-0.0919048\pi$
$858$	0	0
$859$	1.38664e6	0.0641179	0.0320590	−	0.999486i	$-0.489794\pi$
$859$	1.38664e6	0.0641179	0.0320590	+	0.999486i	$0.489794\pi$
$860$	0	0
$861$	−6.23376e6	−0.286578
$862$	0	0
$863$	− 1.09856e7i	− 0.502109i	−0.967973	−	0.251055i	$-0.919223\pi$
$863$	− 1.09856e7i	− 0.502109i	0.967973	−	0.251055i	$-0.0807773\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.15843e7i	0.975194i
$868$	0	0
$869$	3.30237e6	0.148346
$870$	0	0
$871$	−1.79750e6	−0.0802828
$872$	0	0
$873$	− 9.40880e6i	− 0.417829i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 8.17798e6i	− 0.359044i	−0.983754	−	0.179522i	$-0.942545\pi$
$877$	− 8.17798e6i	− 0.359044i	0.983754	−	0.179522i	$-0.0574550\pi$
$878$	0	0
$879$	1.15256e7	0.503143
$880$	0	0
$881$	4.66520e6	0.202503	0.101251	−	0.994861i	$-0.467715\pi$
$881$	4.66520e6	0.202503	0.101251	+	0.994861i	$0.467715\pi$
$882$	0	0
$883$	3.82201e7i	1.64964i	0.565393	+	0.824822i	$0.308724\pi$
$883$	3.82201e7i	1.64964i	−0.565393	+	0.824822i	$0.691276\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.72172e6i	0.329538i	0.986332	+	0.164769i	$0.0526878\pi$
$887$	7.72172e6i	0.329538i	−0.986332	+	0.164769i	$0.947312\pi$
$888$	0	0
$889$	1.24973e7	0.530348
$890$	0	0
$891$	−813564.	−0.0343319
$892$	0	0
$893$	− 1.34834e7i	− 0.565810i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.17576e6i	0.0487908i
$898$	0	0
$899$	−4.14695e7	−1.71131
$900$	0	0
$901$	−4.41487e7	−1.81178
$902$	0	0
$903$	2.44771e7i	0.998944i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 4.33137e7i	− 1.74826i	−0.485689	−	0.874131i	$-0.661431\pi$
$907$	− 4.33137e7i	− 1.74826i	0.485689	−	0.874131i	$-0.338569\pi$
$908$	0	0
$909$	5.41355e6	0.217307
$910$	0	0
$911$	3.44456e6	0.137511	0.0687556	−	0.997634i	$-0.478097\pi$
$911$	3.44456e6	0.137511	0.0687556	+	0.997634i	$0.478097\pi$
$912$	0	0
$913$	− 6.96731e6i	− 0.276623i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.83914e7i	1.50768i
$918$	0	0
$919$	4.37073e7	1.70712	0.853562	−	0.520991i	$-0.174437\pi$
$919$	4.37073e7	1.70712	0.853562	+	0.520991i	$0.174437\pi$
$920$	0	0
$921$	2.03687e7	0.791251
$922$	0	0
$923$	− 2.52411e6i	− 0.0975224i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 5.18400e6i	− 0.198137i
$928$	0	0
$929$	4.13022e7	1.57012	0.785062	−	0.619418i	$-0.212631\pi$
$929$	4.13022e7	1.57012	0.785062	+	0.619418i	$0.212631\pi$
$930$	0	0
$931$	−7.84857e7	−2.96768
$932$	0	0
$933$	− 2.23063e6i	− 0.0838926i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 9.57460e6i	− 0.356264i	−0.984007	−	0.178132i	$-0.942995\pi$
$937$	− 9.57460e6i	− 0.356264i	0.984007	−	0.178132i	$-0.0570054\pi$
$938$	0	0
$939$	−1.64152e7	−0.607550
$940$	0	0
$941$	8.71623e6	0.320889	0.160444	−	0.987045i	$-0.448707\pi$
$941$	8.71623e6	0.320889	0.160444	+	0.987045i	$0.448707\pi$
$942$	0	0
$943$	− 8.19624e6i	− 0.300148i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.30605e7i	0.473244i	0.971602	+	0.236622i	$0.0760403\pi$
$947$	1.30605e7i	0.473244i	−0.971602	+	0.236622i	$0.923960\pi$
$948$	0	0
$949$	−967564.	−0.0348750
$950$	0	0
$951$	−2.57066e7	−0.921710
$952$	0	0
$953$	1.13875e7i	0.406158i	0.979162	+	0.203079i	$0.0650948\pi$
$953$	1.13875e7i	0.406158i	−0.979162	+	0.203079i	$0.934905\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9.95695e6i	0.351436i
$958$	0	0
$959$	−6.29467e7	−2.21017
$960$	0	0
$961$	−7.02525e6	−0.245388
$962$	0	0
$963$	− 1.22180e6i	− 0.0424557i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 4.62711e7i	− 1.59127i	−0.605778	−	0.795634i	$-0.707138\pi$
$967$	− 4.62711e7i	− 1.59127i	0.605778	−	0.795634i	$-0.292862\pi$
$968$	0	0
$969$	−3.38355e7	−1.15761
$970$	0	0
$971$	1.63206e7	0.555506	0.277753	−	0.960653i	$-0.410410\pi$
$971$	1.63206e7	0.555506	0.277753	+	0.960653i	$0.410410\pi$
$972$	0	0
$973$	− 6.08458e7i	− 2.06038i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.95213e7i	0.654294i	0.944973	+	0.327147i	$0.106087\pi$
$977$	1.95213e7i	0.654294i	−0.944973	+	0.327147i	$0.893913\pi$
$978$	0	0
$979$	7.98684e6	0.266329
$980$	0	0
$981$	−3.21554e6	−0.106680
$982$	0	0
$983$	− 4.33962e7i	− 1.43241i	−0.697888	−	0.716207i	$-0.745877\pi$
$983$	− 4.33962e7i	− 1.43241i	0.697888	−	0.716207i	$-0.254123\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 1.51373e7i	− 0.494601i
$988$	0	0
$989$	−3.21829e7	−1.04625
$990$	0	0
$991$	3.83518e7	1.24051	0.620257	−	0.784399i	$-0.287028\pi$
$991$	3.83518e7	1.24051	0.620257	+	0.784399i	$0.287028\pi$
$992$	0	0
$993$	1.32433e6i	0.0426210i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	7.82206e6i	0.249220i	0.992206	+	0.124610i	$0.0397680\pi$
$997$	7.82206e6i	0.249220i	−0.992206	+	0.124610i	$0.960232\pi$
$998$	0	0
$999$	−3.17990e6	−0.100809

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.6.f.f.49.1		2
5.2	odd	4		24.6.a.a.1.1	✓	1
5.3	odd	4		600.6.a.i.1.1		1
5.4	even	2	inner	600.6.f.f.49.2		2
15.2	even	4		72.6.a.e.1.1		1
20.7	even	4		48.6.a.d.1.1		1
40.27	even	4		192.6.a.f.1.1		1
40.37	odd	4		192.6.a.n.1.1		1
60.47	odd	4		144.6.a.i.1.1		1
80.27	even	4		768.6.d.a.385.1		2
80.37	odd	4		768.6.d.r.385.2		2
80.67	even	4		768.6.d.a.385.2		2
80.77	odd	4		768.6.d.r.385.1		2
120.77	even	4		576.6.a.k.1.1		1
120.107	odd	4		576.6.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.6.a.a.1.1	✓	1	5.2	odd	4
48.6.a.d.1.1		1	20.7	even	4
72.6.a.e.1.1		1	15.2	even	4
144.6.a.i.1.1		1	60.47	odd	4
192.6.a.f.1.1		1	40.27	even	4
192.6.a.n.1.1		1	40.37	odd	4
576.6.a.k.1.1		1	120.77	even	4
576.6.a.l.1.1		1	120.107	odd	4
600.6.a.i.1.1		1	5.3	odd	4
600.6.f.f.49.1		2	1.1	even	1	trivial
600.6.f.f.49.2		2	5.4	even	2	inner
768.6.d.a.385.1		2	80.27	even	4
768.6.d.a.385.2		2	80.67	even	4
768.6.d.r.385.1		2	80.77	odd	4
768.6.d.r.385.2		2	80.37	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	600.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-9.00000i q^{3} +240.000i q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q-9.00000i q^{3} +240.000i q^{7} -81.0000 q^{9} -124.000 q^{11} +46.0000i q^{13} -1954.00i q^{17} +1924.00 q^{19} +2160.00 q^{21} +2840.00i q^{23} +729.000i q^{27} +8922.00 q^{29} -4648.00 q^{31} +1116.00i q^{33} +4362.00i q^{37} +414.000 q^{39} -2886.00 q^{41} +11332.0i q^{43} -7008.00i q^{47} -40793.0 q^{49} -17586.0 q^{51} -22594.0i q^{53} -17316.0i q^{57} +28.0000 q^{59} -6386.00 q^{61} -19440.0i q^{63} +39076.0i q^{67} +25560.0 q^{69} -54872.0 q^{71} +21034.0i q^{73} -29760.0i q^{77} -26632.0 q^{79} +6561.00 q^{81} +56188.0i q^{83} -80298.0i q^{87} -64410.0 q^{89} -11040.0 q^{91} +41832.0i q^{93} +116158. i q^{97} +10044.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 162 q^{9}+O(q^{10}) \) 2 * q - 162 * q^9	\( 2 q - 162 q^{9} - 248 q^{11} + 3848 q^{19} + 4320 q^{21} + 17844 q^{29} - 9296 q^{31} + 828 q^{39} - 5772 q^{41} - 81586 q^{49} - 35172 q^{51} + 56 q^{59} - 12772 q^{61} + 51120 q^{69} - 109744 q^{71} - 53264 q^{79} + 13122 q^{81} - 128820 q^{89} - 22080 q^{91} + 20088 q^{99}+O(q^{100}) \) 2 * q - 162 * q^9 - 248 * q^11 + 3848 * q^19 + 4320 * q^21 + 17844 * q^29 - 9296 * q^31 + 828 * q^39 - 5772 * q^41 - 81586 * q^49 - 35172 * q^51 + 56 * q^59 - 12772 * q^61 + 51120 * q^69 - 109744 * q^71 - 53264 * q^79 + 13122 * q^81 - 128820 * q^89 - 22080 * q^91 + 20088 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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