LMFDB - Embedded newform 6036.2.a.h.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.a (trivial)

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:	yes
Analytic conductor:	$48.1977026600$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.66295 q^{5} -0.368117 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.66295 q^{5} -0.368117 q^{7} +1.00000 q^{9} +2.19255 q^{11} +0.456235 q^{13} -1.66295 q^{15} -4.55871 q^{17} +1.18276 q^{19} -0.368117 q^{21} +4.16710 q^{23} -2.23459 q^{25} +1.00000 q^{27} -4.37034 q^{29} -3.77530 q^{31} +2.19255 q^{33} +0.612161 q^{35} +7.10906 q^{37} +0.456235 q^{39} +3.74345 q^{41} -0.00494644 q^{43} -1.66295 q^{45} +12.8249 q^{47} -6.86449 q^{49} -4.55871 q^{51} -1.58705 q^{53} -3.64610 q^{55} +1.18276 q^{57} +9.98195 q^{59} +10.0476 q^{61} -0.368117 q^{63} -0.758697 q^{65} +9.92843 q^{67} +4.16710 q^{69} -7.66728 q^{71} -7.07340 q^{73} -2.23459 q^{75} -0.807113 q^{77} -2.49735 q^{79} +1.00000 q^{81} +14.5235 q^{83} +7.58092 q^{85} -4.37034 q^{87} +4.80557 q^{89} -0.167948 q^{91} -3.77530 q^{93} -1.96687 q^{95} -7.92511 q^{97} +2.19255 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9}+O(q^{10}) $ 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9	$ 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9} + 9 q^{11} + 7 q^{13} + 18 q^{15} + 12 q^{17} - q^{19} + 9 q^{21} + 22 q^{23} + 48 q^{25} + 24 q^{27} + 44 q^{29} + 15 q^{31} + 9 q^{33} + 3 q^{35} + 18 q^{37} + 7 q^{39} + 37 q^{41} - 8 q^{43} + 18 q^{45} + 24 q^{47} + 63 q^{49} + 12 q^{51} + 59 q^{53} + 20 q^{55} - q^{57} + 44 q^{59} + 33 q^{61} + 9 q^{63} + 15 q^{65} + 13 q^{67} + 22 q^{69} + 47 q^{71} + 28 q^{73} + 48 q^{75} + 19 q^{77} + 12 q^{79} + 24 q^{81} + 17 q^{83} + 31 q^{85} + 44 q^{87} + 41 q^{89} - q^{91} + 15 q^{93} + 58 q^{95} + 51 q^{97} + 9 q^{99}+O(q^{100}) $ 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9 + 9 * q^11 + 7 * q^13 + 18 * q^15 + 12 * q^17 - q^19 + 9 * q^21 + 22 * q^23 + 48 * q^25 + 24 * q^27 + 44 * q^29 + 15 * q^31 + 9 * q^33 + 3 * q^35 + 18 * q^37 + 7 * q^39 + 37 * q^41 - 8 * q^43 + 18 * q^45 + 24 * q^47 + 63 * q^49 + 12 * q^51 + 59 * q^53 + 20 * q^55 - q^57 + 44 * q^59 + 33 * q^61 + 9 * q^63 + 15 * q^65 + 13 * q^67 + 22 * q^69 + 47 * q^71 + 28 * q^73 + 48 * q^75 + 19 * q^77 + 12 * q^79 + 24 * q^81 + 17 * q^83 + 31 * q^85 + 44 * q^87 + 41 * q^89 - q^91 + 15 * q^93 + 58 * q^95 + 51 * q^97 + 9 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.66295	−0.743695	−0.371847	−	0.928294i	$-0.621276\pi$
$5$	−1.66295	−0.743695	−0.371847	+	0.928294i	$0.621276\pi$
$6$	0	0
$7$	−0.368117	−0.139135	−0.0695675	−	0.997577i	$-0.522162\pi$
$7$	−0.368117	−0.139135	−0.0695675	+	0.997577i	$0.522162\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.19255	0.661078	0.330539	−	0.943792i	$-0.392769\pi$
$11$	2.19255	0.661078	0.330539	+	0.943792i	$0.392769\pi$
$12$	0	0
$13$	0.456235	0.126537	0.0632684	−	0.997997i	$-0.479848\pi$
$13$	0.456235	0.126537	0.0632684	+	0.997997i	$0.479848\pi$
$14$	0	0
$15$	−1.66295	−0.429372
$16$	0	0
$17$	−4.55871	−1.10565	−0.552825	−	0.833298i	$-0.686450\pi$
$17$	−4.55871	−1.10565	−0.552825	+	0.833298i	$0.686450\pi$
$18$	0	0
$19$	1.18276	0.271343	0.135672	−	0.990754i	$-0.456681\pi$
$19$	1.18276	0.271343	0.135672	+	0.990754i	$0.456681\pi$
$20$	0	0
$21$	−0.368117	−0.0803297
$22$	0	0
$23$	4.16710	0.868901	0.434451	−	0.900696i	$-0.356943\pi$
$23$	4.16710	0.868901	0.434451	+	0.900696i	$0.356943\pi$
$24$	0	0
$25$	−2.23459	−0.446918
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.37034	−0.811552	−0.405776	−	0.913973i	$-0.632999\pi$
$29$	−4.37034	−0.811552	−0.405776	+	0.913973i	$0.632999\pi$
$30$	0	0
$31$	−3.77530	−0.678064	−0.339032	−	0.940775i	$-0.610100\pi$
$31$	−3.77530	−0.678064	−0.339032	+	0.940775i	$0.610100\pi$
$32$	0	0
$33$	2.19255	0.381674
$34$	0	0
$35$	0.612161	0.103474
$36$	0	0
$37$	7.10906	1.16872	0.584361	−	0.811494i	$-0.301345\pi$
$37$	7.10906	1.16872	0.584361	+	0.811494i	$0.301345\pi$
$38$	0	0
$39$	0.456235	0.0730560
$40$	0	0
$41$	3.74345	0.584628	0.292314	−	0.956322i	$-0.405575\pi$
$41$	3.74345	0.584628	0.292314	+	0.956322i	$0.405575\pi$
$42$	0	0
$43$	−0.00494644	−0.000754325 0	−0.000377162	−	1.00000i	$-0.500120\pi$
$43$	−0.00494644	−0.000754325 0	−0.000377162		1.00000i	$0.500120\pi$
$44$	0	0
$45$	−1.66295	−0.247898
$46$	0	0
$47$	12.8249	1.87071	0.935355	−	0.353711i	$-0.115080\pi$
$47$	12.8249	1.87071	0.935355	+	0.353711i	$0.115080\pi$
$48$	0	0
$49$	−6.86449	−0.980641
$50$	0	0
$51$	−4.55871	−0.638347
$52$	0	0
$53$	−1.58705	−0.217999	−0.108999	−	0.994042i	$-0.534765\pi$
$53$	−1.58705	−0.217999	−0.108999	+	0.994042i	$0.534765\pi$
$54$	0	0
$55$	−3.64610	−0.491640
$56$	0	0
$57$	1.18276	0.156660
$58$	0	0
$59$	9.98195	1.29954	0.649770	−	0.760131i	$-0.274865\pi$
$59$	9.98195	1.29954	0.649770	+	0.760131i	$0.274865\pi$
$60$	0	0
$61$	10.0476	1.28646	0.643231	−	0.765672i	$-0.277594\pi$
$61$	10.0476	1.28646	0.643231	+	0.765672i	$0.277594\pi$
$62$	0	0
$63$	−0.368117	−0.0463783
$64$	0	0
$65$	−0.758697	−0.0941048
$66$	0	0
$67$	9.92843	1.21295	0.606475	−	0.795102i	$-0.292583\pi$
$67$	9.92843	1.21295	0.606475	+	0.795102i	$0.292583\pi$
$68$	0	0
$69$	4.16710	0.501660
$70$	0	0
$71$	−7.66728	−0.909939	−0.454969	−	0.890507i	$-0.650350\pi$
$71$	−7.66728	−0.909939	−0.454969	+	0.890507i	$0.650350\pi$
$72$	0	0
$73$	−7.07340	−0.827879	−0.413940	−	0.910304i	$-0.635847\pi$
$73$	−7.07340	−0.827879	−0.413940	+	0.910304i	$0.635847\pi$
$74$	0	0
$75$	−2.23459	−0.258028
$76$	0	0
$77$	−0.807113	−0.0919791
$78$	0	0
$79$	−2.49735	−0.280973	−0.140487	−	0.990083i	$-0.544867\pi$
$79$	−2.49735	−0.280973	−0.140487	+	0.990083i	$0.544867\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.5235	1.59416	0.797081	−	0.603872i	$-0.206376\pi$
$83$	14.5235	1.59416	0.797081	+	0.603872i	$0.206376\pi$
$84$	0	0
$85$	7.58092	0.822266
$86$	0	0
$87$	−4.37034	−0.468550
$88$	0	0
$89$	4.80557	0.509390	0.254695	−	0.967021i	$-0.418025\pi$
$89$	4.80557	0.509390	0.254695	+	0.967021i	$0.418025\pi$
$90$	0	0
$91$	−0.167948	−0.0176057
$92$	0	0
$93$	−3.77530	−0.391480
$94$	0	0
$95$	−1.96687	−0.201797
$96$	0	0
$97$	−7.92511	−0.804673	−0.402337	−	0.915492i	$-0.631802\pi$
$97$	−7.92511	−0.804673	−0.402337	+	0.915492i	$0.631802\pi$
$98$	0	0
$99$	2.19255	0.220359
$100$	0	0
$101$	−4.73353	−0.471004	−0.235502	−	0.971874i	$-0.575673\pi$
$101$	−4.73353	−0.471004	−0.235502	+	0.971874i	$0.575673\pi$
$102$	0	0
$103$	6.69451	0.659630	0.329815	−	0.944046i	$-0.393014\pi$
$103$	6.69451	0.659630	0.329815	+	0.944046i	$0.393014\pi$
$104$	0	0
$105$	0.612161	0.0597408
$106$	0	0
$107$	11.7256	1.13355	0.566777	−	0.823871i	$-0.308190\pi$
$107$	11.7256	1.13355	0.566777	+	0.823871i	$0.308190\pi$
$108$	0	0
$109$	4.40450	0.421875	0.210937	−	0.977500i	$-0.432348\pi$
$109$	4.40450	0.421875	0.210937	+	0.977500i	$0.432348\pi$
$110$	0	0
$111$	7.10906	0.674762
$112$	0	0
$113$	−1.43155	−0.134669	−0.0673347	−	0.997730i	$-0.521450\pi$
$113$	−1.43155	−0.134669	−0.0673347	+	0.997730i	$0.521450\pi$
$114$	0	0
$115$	−6.92970	−0.646198
$116$	0	0
$117$	0.456235	0.0421789
$118$	0	0
$119$	1.67814	0.153835
$120$	0	0
$121$	−6.19273	−0.562976
$122$	0	0
$123$	3.74345	0.337535
$124$	0	0
$125$	12.0308	1.07607
$126$	0	0
$127$	−2.68449	−0.238210	−0.119105	−	0.992882i	$-0.538003\pi$
$127$	−2.68449	−0.238210	−0.119105	+	0.992882i	$0.538003\pi$
$128$	0	0
$129$	−0.00494644	−0.000435510 0
$130$	0	0
$131$	−14.5208	−1.26869	−0.634343	−	0.773052i	$-0.718729\pi$
$131$	−14.5208	−1.26869	−0.634343	+	0.773052i	$0.718729\pi$
$132$	0	0
$133$	−0.435393	−0.0377533
$134$	0	0
$135$	−1.66295	−0.143124
$136$	0	0
$137$	6.45203	0.551234	0.275617	−	0.961267i	$-0.411118\pi$
$137$	6.45203	0.551234	0.275617	+	0.961267i	$0.411118\pi$
$138$	0	0
$139$	−17.2808	−1.46574	−0.732868	−	0.680371i	$-0.761819\pi$
$139$	−17.2808	−1.46574	−0.732868	+	0.680371i	$0.761819\pi$
$140$	0	0
$141$	12.8249	1.08005
$142$	0	0
$143$	1.00032	0.0836507
$144$	0	0
$145$	7.26767	0.603547
$146$	0	0
$147$	−6.86449	−0.566174
$148$	0	0
$149$	4.38049	0.358864	0.179432	−	0.983770i	$-0.442574\pi$
$149$	4.38049	0.358864	0.179432	+	0.983770i	$0.442574\pi$
$150$	0	0
$151$	−6.49785	−0.528787	−0.264394	−	0.964415i	$-0.585172\pi$
$151$	−6.49785	−0.528787	−0.264394	+	0.964415i	$0.585172\pi$
$152$	0	0
$153$	−4.55871	−0.368550
$154$	0	0
$155$	6.27815	0.504273
$156$	0	0
$157$	3.39905	0.271274	0.135637	−	0.990759i	$-0.456692\pi$
$157$	3.39905	0.271274	0.135637	+	0.990759i	$0.456692\pi$
$158$	0	0
$159$	−1.58705	−0.125862
$160$	0	0
$161$	−1.53398	−0.120895
$162$	0	0
$163$	19.6205	1.53680	0.768399	−	0.639970i	$-0.221053\pi$
$163$	19.6205	1.53680	0.768399	+	0.639970i	$0.221053\pi$
$164$	0	0
$165$	−3.64610	−0.283849
$166$	0	0
$167$	22.4576	1.73782	0.868911	−	0.494969i	$-0.164820\pi$
$167$	22.4576	1.73782	0.868911	+	0.494969i	$0.164820\pi$
$168$	0	0
$169$	−12.7918	−0.983988
$170$	0	0
$171$	1.18276	0.0904477
$172$	0	0
$173$	−19.1784	−1.45811	−0.729054	−	0.684457i	$-0.760040\pi$
$173$	−19.1784	−1.45811	−0.729054	+	0.684457i	$0.760040\pi$
$174$	0	0
$175$	0.822590	0.0621819
$176$	0	0
$177$	9.98195	0.750289
$178$	0	0
$179$	23.1979	1.73389	0.866944	−	0.498405i	$-0.166081\pi$
$179$	23.1979	1.73389	0.866944	+	0.498405i	$0.166081\pi$
$180$	0	0
$181$	20.3730	1.51431	0.757157	−	0.653233i	$-0.226588\pi$
$181$	20.3730	1.51431	0.757157	+	0.653233i	$0.226588\pi$
$182$	0	0
$183$	10.0476	0.742739
$184$	0	0
$185$	−11.8220	−0.869172
$186$	0	0
$187$	−9.99519	−0.730920
$188$	0	0
$189$	−0.368117	−0.0267766
$190$	0	0
$191$	2.88469	0.208729	0.104365	−	0.994539i	$-0.466719\pi$
$191$	2.88469	0.208729	0.104365	+	0.994539i	$0.466719\pi$
$192$	0	0
$193$	12.3677	0.890243	0.445121	−	0.895470i	$-0.353160\pi$
$193$	12.3677	0.890243	0.445121	+	0.895470i	$0.353160\pi$
$194$	0	0
$195$	−0.758697	−0.0543314
$196$	0	0
$197$	−20.2887	−1.44551	−0.722754	−	0.691106i	$-0.757124\pi$
$197$	−20.2887	−1.44551	−0.722754	+	0.691106i	$0.757124\pi$
$198$	0	0
$199$	4.12642	0.292514	0.146257	−	0.989247i	$-0.453277\pi$
$199$	4.12642	0.292514	0.146257	+	0.989247i	$0.453277\pi$
$200$	0	0
$201$	9.92843	0.700297
$202$	0	0
$203$	1.60880	0.112915
$204$	0	0
$205$	−6.22518	−0.434785
$206$	0	0
$207$	4.16710	0.289634
$208$	0	0
$209$	2.59325	0.179379
$210$	0	0
$211$	−9.40963	−0.647785	−0.323893	−	0.946094i	$-0.604992\pi$
$211$	−9.40963	−0.647785	−0.323893	+	0.946094i	$0.604992\pi$
$212$	0	0
$213$	−7.66728	−0.525353
$214$	0	0
$215$	0.00822569	0.000560988 0
$216$	0	0
$217$	1.38975	0.0943425
$218$	0	0
$219$	−7.07340	−0.477976
$220$	0	0
$221$	−2.07984	−0.139905
$222$	0	0
$223$	20.5388	1.37538	0.687690	−	0.726004i	$-0.258625\pi$
$223$	20.5388	1.37538	0.687690	+	0.726004i	$0.258625\pi$
$224$	0	0
$225$	−2.23459	−0.148973
$226$	0	0
$227$	−9.97482	−0.662052	−0.331026	−	0.943622i	$-0.607395\pi$
$227$	−9.97482	−0.662052	−0.331026	+	0.943622i	$0.607395\pi$
$228$	0	0
$229$	24.6372	1.62807	0.814037	−	0.580813i	$-0.197265\pi$
$229$	24.6372	1.62807	0.814037	+	0.580813i	$0.197265\pi$
$230$	0	0
$231$	−0.807113	−0.0531042
$232$	0	0
$233$	15.7881	1.03431	0.517156	−	0.855891i	$-0.326991\pi$
$233$	15.7881	1.03431	0.517156	+	0.855891i	$0.326991\pi$
$234$	0	0
$235$	−21.3273	−1.39124
$236$	0	0
$237$	−2.49735	−0.162220
$238$	0	0
$239$	1.61623	0.104545	0.0522727	−	0.998633i	$-0.483353\pi$
$239$	1.61623	0.104545	0.0522727	+	0.998633i	$0.483353\pi$
$240$	0	0
$241$	4.84691	0.312217	0.156108	−	0.987740i	$-0.450105\pi$
$241$	4.84691	0.312217	0.156108	+	0.987740i	$0.450105\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	11.4153	0.729298
$246$	0	0
$247$	0.539615	0.0343349
$248$	0	0
$249$	14.5235	0.920390
$250$	0	0
$251$	7.29225	0.460283	0.230141	−	0.973157i	$-0.426081\pi$
$251$	7.29225	0.460283	0.230141	+	0.973157i	$0.426081\pi$
$252$	0	0
$253$	9.13658	0.574412
$254$	0	0
$255$	7.58092	0.474735
$256$	0	0
$257$	16.0497	1.00115	0.500576	−	0.865693i	$-0.333122\pi$
$257$	16.0497	1.00115	0.500576	+	0.865693i	$0.333122\pi$
$258$	0	0
$259$	−2.61696	−0.162610
$260$	0	0
$261$	−4.37034	−0.270517
$262$	0	0
$263$	16.4973	1.01727	0.508634	−	0.860983i	$-0.330151\pi$
$263$	16.4973	1.01727	0.508634	+	0.860983i	$0.330151\pi$
$264$	0	0
$265$	2.63920	0.162125
$266$	0	0
$267$	4.80557	0.294096
$268$	0	0
$269$	5.44385	0.331918	0.165959	−	0.986133i	$-0.446928\pi$
$269$	5.44385	0.331918	0.165959	+	0.986133i	$0.446928\pi$
$270$	0	0
$271$	1.72563	0.104824	0.0524122	−	0.998626i	$-0.483309\pi$
$271$	1.72563	0.104824	0.0524122	+	0.998626i	$0.483309\pi$
$272$	0	0
$273$	−0.167948	−0.0101647
$274$	0	0
$275$	−4.89944	−0.295448
$276$	0	0
$277$	−6.46661	−0.388541	−0.194270	−	0.980948i	$-0.562234\pi$
$277$	−6.46661	−0.388541	−0.194270	+	0.980948i	$0.562234\pi$
$278$	0	0
$279$	−3.77530	−0.226021
$280$	0	0
$281$	29.8668	1.78170	0.890852	−	0.454293i	$-0.150108\pi$
$281$	29.8668	1.78170	0.890852	+	0.454293i	$0.150108\pi$
$282$	0	0
$283$	−14.4565	−0.859348	−0.429674	−	0.902984i	$-0.641372\pi$
$283$	−14.4565	−0.859348	−0.429674	+	0.902984i	$0.641372\pi$
$284$	0	0
$285$	−1.96687	−0.116507
$286$	0	0
$287$	−1.37803	−0.0813423
$288$	0	0
$289$	3.78183	0.222460
$290$	0	0
$291$	−7.92511	−0.464578
$292$	0	0
$293$	9.36992	0.547397	0.273698	−	0.961816i	$-0.411753\pi$
$293$	9.36992	0.547397	0.273698	+	0.961816i	$0.411753\pi$
$294$	0	0
$295$	−16.5995	−0.966461
$296$	0	0
$297$	2.19255	0.127225
$298$	0	0
$299$	1.90118	0.109948
$300$	0	0
$301$	0.00182087	0.000104953 0
$302$	0	0
$303$	−4.73353	−0.271934
$304$	0	0
$305$	−16.7087	−0.956735
$306$	0	0
$307$	27.5209	1.57070	0.785351	−	0.619051i	$-0.212483\pi$
$307$	27.5209	1.57070	0.785351	+	0.619051i	$0.212483\pi$
$308$	0	0
$309$	6.69451	0.380837
$310$	0	0
$311$	7.23684	0.410364	0.205182	−	0.978724i	$-0.434221\pi$
$311$	7.23684	0.410364	0.205182	+	0.978724i	$0.434221\pi$
$312$	0	0
$313$	−17.7165	−1.00140	−0.500699	−	0.865621i	$-0.666924\pi$
$313$	−17.7165	−1.00140	−0.500699	+	0.865621i	$0.666924\pi$
$314$	0	0
$315$	0.612161	0.0344913
$316$	0	0
$317$	20.9788	1.17829	0.589144	−	0.808028i	$-0.299465\pi$
$317$	20.9788	1.17829	0.589144	+	0.808028i	$0.299465\pi$
$318$	0	0
$319$	−9.58218	−0.536499
$320$	0	0
$321$	11.7256	0.654457
$322$	0	0
$323$	−5.39185	−0.300010
$324$	0	0
$325$	−1.01950	−0.0565515
$326$	0	0
$327$	4.40450	0.243569
$328$	0	0
$329$	−4.72107	−0.260281
$330$	0	0
$331$	23.1690	1.27348	0.636740	−	0.771078i	$-0.280282\pi$
$331$	23.1690	1.27348	0.636740	+	0.771078i	$0.280282\pi$
$332$	0	0
$333$	7.10906	0.389574
$334$	0	0
$335$	−16.5105	−0.902065
$336$	0	0
$337$	13.8083	0.752188	0.376094	−	0.926581i	$-0.377267\pi$
$337$	13.8083	0.752188	0.376094	+	0.926581i	$0.377267\pi$
$338$	0	0
$339$	−1.43155	−0.0777514
$340$	0	0
$341$	−8.27753	−0.448253
$342$	0	0
$343$	5.10375	0.275577
$344$	0	0
$345$	−6.92970	−0.373082
$346$	0	0
$347$	−13.2438	−0.710966	−0.355483	−	0.934683i	$-0.615684\pi$
$347$	−13.2438	−0.710966	−0.355483	+	0.934683i	$0.615684\pi$
$348$	0	0
$349$	−27.5930	−1.47702	−0.738510	−	0.674243i	$-0.764470\pi$
$349$	−27.5930	−1.47702	−0.738510	+	0.674243i	$0.764470\pi$
$350$	0	0
$351$	0.456235	0.0243520
$352$	0	0
$353$	9.31765	0.495928	0.247964	−	0.968769i	$-0.420238\pi$
$353$	9.31765	0.495928	0.247964	+	0.968769i	$0.420238\pi$
$354$	0	0
$355$	12.7503	0.676717
$356$	0	0
$357$	1.67814	0.0888164
$358$	0	0
$359$	29.8035	1.57297	0.786484	−	0.617610i	$-0.211899\pi$
$359$	29.8035	1.57297	0.786484	+	0.617610i	$0.211899\pi$
$360$	0	0
$361$	−17.6011	−0.926373
$362$	0	0
$363$	−6.19273	−0.325034
$364$	0	0
$365$	11.7627	0.615689
$366$	0	0
$367$	−28.2583	−1.47507	−0.737536	−	0.675308i	$-0.764011\pi$
$367$	−28.2583	−1.47507	−0.737536	+	0.675308i	$0.764011\pi$
$368$	0	0
$369$	3.74345	0.194876
$370$	0	0
$371$	0.584221	0.0303313
$372$	0	0
$373$	13.3949	0.693561	0.346781	−	0.937946i	$-0.387275\pi$
$373$	13.3949	0.693561	0.346781	+	0.937946i	$0.387275\pi$
$374$	0	0
$375$	12.0308	0.621267
$376$	0	0
$377$	−1.99390	−0.102691
$378$	0	0
$379$	14.6211	0.751035	0.375517	−	0.926815i	$-0.377465\pi$
$379$	14.6211	0.751035	0.375517	+	0.926815i	$0.377465\pi$
$380$	0	0
$381$	−2.68449	−0.137531
$382$	0	0
$383$	21.7466	1.11120	0.555600	−	0.831450i	$-0.312489\pi$
$383$	21.7466	1.11120	0.555600	+	0.831450i	$0.312489\pi$
$384$	0	0
$385$	1.34219	0.0684044
$386$	0	0
$387$	−0.00494644	−0.000251442 0
$388$	0	0
$389$	9.54245	0.483821	0.241911	−	0.970299i	$-0.422226\pi$
$389$	9.54245	0.483821	0.241911	+	0.970299i	$0.422226\pi$
$390$	0	0
$391$	−18.9966	−0.960700
$392$	0	0
$393$	−14.5208	−0.732476
$394$	0	0
$395$	4.15297	0.208958
$396$	0	0
$397$	−22.2301	−1.11570	−0.557849	−	0.829943i	$-0.688373\pi$
$397$	−22.2301	−1.11570	−0.557849	+	0.829943i	$0.688373\pi$
$398$	0	0
$399$	−0.435393	−0.0217969
$400$	0	0
$401$	0.631121	0.0315167	0.0157583	−	0.999876i	$-0.494984\pi$
$401$	0.631121	0.0315167	0.0157583	+	0.999876i	$0.494984\pi$
$402$	0	0
$403$	−1.72242	−0.0858000
$404$	0	0
$405$	−1.66295	−0.0826328
$406$	0	0
$407$	15.5869	0.772616
$408$	0	0
$409$	4.28633	0.211946	0.105973	−	0.994369i	$-0.466204\pi$
$409$	4.28633	0.211946	0.105973	+	0.994369i	$0.466204\pi$
$410$	0	0
$411$	6.45203	0.318255
$412$	0	0
$413$	−3.67452	−0.180811
$414$	0	0
$415$	−24.1519	−1.18557
$416$	0	0
$417$	−17.2808	−0.846243
$418$	0	0
$419$	−26.2678	−1.28327	−0.641634	−	0.767011i	$-0.721743\pi$
$419$	−26.2678	−1.28327	−0.641634	+	0.767011i	$0.721743\pi$
$420$	0	0
$421$	32.2789	1.57318	0.786589	−	0.617476i	$-0.211845\pi$
$421$	32.2789	1.57318	0.786589	+	0.617476i	$0.211845\pi$
$422$	0	0
$423$	12.8249	0.623570
$424$	0	0
$425$	10.1868	0.494134
$426$	0	0
$427$	−3.69868	−0.178992
$428$	0	0
$429$	1.00032	0.0482957
$430$	0	0
$431$	23.6864	1.14093	0.570467	−	0.821321i	$-0.306762\pi$
$431$	23.6864	1.14093	0.570467	+	0.821321i	$0.306762\pi$
$432$	0	0
$433$	−2.22438	−0.106897	−0.0534484	−	0.998571i	$-0.517021\pi$
$433$	−2.22438	−0.106897	−0.0534484	+	0.998571i	$0.517021\pi$
$434$	0	0
$435$	7.26767	0.348458
$436$	0	0
$437$	4.92867	0.235770
$438$	0	0
$439$	21.0605	1.00516	0.502581	−	0.864530i	$-0.332384\pi$
$439$	21.0605	1.00516	0.502581	+	0.864530i	$0.332384\pi$
$440$	0	0
$441$	−6.86449	−0.326880
$442$	0	0
$443$	28.8937	1.37278	0.686390	−	0.727234i	$-0.259195\pi$
$443$	28.8937	1.37278	0.686390	+	0.727234i	$0.259195\pi$
$444$	0	0
$445$	−7.99144	−0.378830
$446$	0	0
$447$	4.38049	0.207190
$448$	0	0
$449$	−10.4086	−0.491214	−0.245607	−	0.969369i	$-0.578987\pi$
$449$	−10.4086	−0.491214	−0.245607	+	0.969369i	$0.578987\pi$
$450$	0	0
$451$	8.20769	0.386485
$452$	0	0
$453$	−6.49785	−0.305296
$454$	0	0
$455$	0.279289	0.0130933
$456$	0	0
$457$	−32.5910	−1.52454	−0.762272	−	0.647257i	$-0.775916\pi$
$457$	−32.5910	−1.52454	−0.762272	+	0.647257i	$0.775916\pi$
$458$	0	0
$459$	−4.55871	−0.212782
$460$	0	0
$461$	38.8968	1.81160	0.905802	−	0.423700i	$-0.139269\pi$
$461$	38.8968	1.81160	0.905802	+	0.423700i	$0.139269\pi$
$462$	0	0
$463$	5.36768	0.249457	0.124729	−	0.992191i	$-0.460194\pi$
$463$	5.36768	0.249457	0.124729	+	0.992191i	$0.460194\pi$
$464$	0	0
$465$	6.27815	0.291142
$466$	0	0
$467$	−35.9372	−1.66298	−0.831489	−	0.555542i	$-0.812511\pi$
$467$	−35.9372	−1.66298	−0.831489	+	0.555542i	$0.812511\pi$
$468$	0	0
$469$	−3.65482	−0.168764
$470$	0	0
$471$	3.39905	0.156620
$472$	0	0
$473$	−0.0108453	−0.000498668 0
$474$	0	0
$475$	−2.64298	−0.121268
$476$	0	0
$477$	−1.58705	−0.0726662
$478$	0	0
$479$	−29.9898	−1.37027	−0.685134	−	0.728417i	$-0.740256\pi$
$479$	−29.9898	−1.37027	−0.685134	+	0.728417i	$0.740256\pi$
$480$	0	0
$481$	3.24340	0.147886
$482$	0	0
$483$	−1.53398	−0.0697985
$484$	0	0
$485$	13.1791	0.598432
$486$	0	0
$487$	−27.1921	−1.23219	−0.616096	−	0.787671i	$-0.711287\pi$
$487$	−27.1921	−1.23219	−0.616096	+	0.787671i	$0.711287\pi$
$488$	0	0
$489$	19.6205	0.887271
$490$	0	0
$491$	−7.88618	−0.355898	−0.177949	−	0.984040i	$-0.556946\pi$
$491$	−7.88618	−0.355898	−0.177949	+	0.984040i	$0.556946\pi$
$492$	0	0
$493$	19.9231	0.897292
$494$	0	0
$495$	−3.64610	−0.163880
$496$	0	0
$497$	2.82245	0.126604
$498$	0	0
$499$	−28.4236	−1.27242	−0.636208	−	0.771517i	$-0.719498\pi$
$499$	−28.4236	−1.27242	−0.636208	+	0.771517i	$0.719498\pi$
$500$	0	0
$501$	22.4576	1.00333
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	7.87163	0.350283
$506$	0	0
$507$	−12.7918	−0.568106
$508$	0	0
$509$	−19.6883	−0.872667	−0.436334	−	0.899785i	$-0.643723\pi$
$509$	−19.6883	−0.872667	−0.436334	+	0.899785i	$0.643723\pi$
$510$	0	0
$511$	2.60384	0.115187
$512$	0	0
$513$	1.18276	0.0522200
$514$	0	0
$515$	−11.1327	−0.490563
$516$	0	0
$517$	28.1193	1.23668
$518$	0	0
$519$	−19.1784	−0.841839
$520$	0	0
$521$	−12.7066	−0.556688	−0.278344	−	0.960481i	$-0.589785\pi$
$521$	−12.7066	−0.556688	−0.278344	+	0.960481i	$0.589785\pi$
$522$	0	0
$523$	15.1500	0.662464	0.331232	−	0.943549i	$-0.392536\pi$
$523$	15.1500	0.662464	0.331232	+	0.943549i	$0.392536\pi$
$524$	0	0
$525$	0.822590	0.0359008
$526$	0	0
$527$	17.2105	0.749701
$528$	0	0
$529$	−5.63524	−0.245010
$530$	0	0
$531$	9.98195	0.433180
$532$	0	0
$533$	1.70789	0.0739770
$534$	0	0
$535$	−19.4991	−0.843018
$536$	0	0
$537$	23.1979	1.00106
$538$	0	0
$539$	−15.0507	−0.648281
$540$	0	0
$541$	−1.89852	−0.0816236	−0.0408118	−	0.999167i	$-0.512994\pi$
$541$	−1.89852	−0.0816236	−0.0408118	+	0.999167i	$0.512994\pi$
$542$	0	0
$543$	20.3730	0.874290
$544$	0	0
$545$	−7.32448	−0.313746
$546$	0	0
$547$	−43.9992	−1.88127	−0.940635	−	0.339420i	$-0.889769\pi$
$547$	−43.9992	−1.88127	−0.940635	+	0.339420i	$0.889769\pi$
$548$	0	0
$549$	10.0476	0.428820
$550$	0	0
$551$	−5.16906	−0.220209
$552$	0	0
$553$	0.919315	0.0390932
$554$	0	0
$555$	−11.8220	−0.501817
$556$	0	0
$557$	−25.8827	−1.09668	−0.548342	−	0.836254i	$-0.684741\pi$
$557$	−25.8827	−1.09668	−0.548342	+	0.836254i	$0.684741\pi$
$558$	0	0
$559$	−0.00225674	−9.54498e−5 0
$560$	0	0
$561$	−9.99519	−0.421997
$562$	0	0
$563$	15.4335	0.650445	0.325223	−	0.945637i	$-0.394561\pi$
$563$	15.4335	0.650445	0.325223	+	0.945637i	$0.394561\pi$
$564$	0	0
$565$	2.38061	0.100153
$566$	0	0
$567$	−0.368117	−0.0154594
$568$	0	0
$569$	−46.0889	−1.93215	−0.966074	−	0.258265i	$-0.916849\pi$
$569$	−46.0889	−1.93215	−0.966074	+	0.258265i	$0.916849\pi$
$570$	0	0
$571$	−40.2916	−1.68615	−0.843076	−	0.537794i	$-0.819258\pi$
$571$	−40.2916	−1.68615	−0.843076	+	0.537794i	$0.819258\pi$
$572$	0	0
$573$	2.88469	0.120510
$574$	0	0
$575$	−9.31177	−0.388327
$576$	0	0
$577$	−46.6938	−1.94389	−0.971945	−	0.235209i	$-0.924423\pi$
$577$	−46.6938	−1.94389	−0.971945	+	0.235209i	$0.924423\pi$
$578$	0	0
$579$	12.3677	0.513982
$580$	0	0
$581$	−5.34635	−0.221804
$582$	0	0
$583$	−3.47969	−0.144114
$584$	0	0
$585$	−0.758697	−0.0313683
$586$	0	0
$587$	21.3721	0.882119	0.441060	−	0.897478i	$-0.354603\pi$
$587$	21.3721	0.882119	0.441060	+	0.897478i	$0.354603\pi$
$588$	0	0
$589$	−4.46527	−0.183988
$590$	0	0
$591$	−20.2887	−0.834564
$592$	0	0
$593$	−42.7576	−1.75584	−0.877921	−	0.478805i	$-0.841070\pi$
$593$	−42.7576	−1.75584	−0.877921	+	0.478805i	$0.841070\pi$
$594$	0	0
$595$	−2.79066	−0.114406
$596$	0	0
$597$	4.12642	0.168883
$598$	0	0
$599$	2.69047	0.109930	0.0549649	−	0.998488i	$-0.482495\pi$
$599$	2.69047	0.109930	0.0549649	+	0.998488i	$0.482495\pi$
$600$	0	0
$601$	8.38143	0.341886	0.170943	−	0.985281i	$-0.445319\pi$
$601$	8.38143	0.341886	0.170943	+	0.985281i	$0.445319\pi$
$602$	0	0
$603$	9.92843	0.404317
$604$	0	0
$605$	10.2982	0.418682
$606$	0	0
$607$	2.31357	0.0939049	0.0469525	−	0.998897i	$-0.485049\pi$
$607$	2.31357	0.0939049	0.0469525	+	0.998897i	$0.485049\pi$
$608$	0	0
$609$	1.60880	0.0651917
$610$	0	0
$611$	5.85118	0.236714
$612$	0	0
$613$	−43.3663	−1.75155	−0.875775	−	0.482720i	$-0.839649\pi$
$613$	−43.3663	−1.75155	−0.875775	+	0.482720i	$0.839649\pi$
$614$	0	0
$615$	−6.22518	−0.251023
$616$	0	0
$617$	−4.89148	−0.196923	−0.0984617	−	0.995141i	$-0.531392\pi$
$617$	−4.89148	−0.196923	−0.0984617	+	0.995141i	$0.531392\pi$
$618$	0	0
$619$	25.2350	1.01428	0.507139	−	0.861864i	$-0.330703\pi$
$619$	25.2350	1.01428	0.507139	+	0.861864i	$0.330703\pi$
$620$	0	0
$621$	4.16710	0.167220
$622$	0	0
$623$	−1.76901	−0.0708739
$624$	0	0
$625$	−8.83367	−0.353347
$626$	0	0
$627$	2.59325	0.103565
$628$	0	0
$629$	−32.4081	−1.29220
$630$	0	0
$631$	−9.09545	−0.362084	−0.181042	−	0.983475i	$-0.557947\pi$
$631$	−9.09545	−0.362084	−0.181042	+	0.983475i	$0.557947\pi$
$632$	0	0
$633$	−9.40963	−0.373999
$634$	0	0
$635$	4.46418	0.177156
$636$	0	0
$637$	−3.13182	−0.124087
$638$	0	0
$639$	−7.66728	−0.303313
$640$	0	0
$641$	33.3278	1.31637	0.658185	−	0.752856i	$-0.271324\pi$
$641$	33.3278	1.31637	0.658185	+	0.752856i	$0.271324\pi$
$642$	0	0
$643$	−40.6480	−1.60300	−0.801501	−	0.597993i	$-0.795965\pi$
$643$	−40.6480	−1.60300	−0.801501	+	0.597993i	$0.795965\pi$
$644$	0	0
$645$	0.00822569	0.000323886 0
$646$	0	0
$647$	−31.0752	−1.22169	−0.610845	−	0.791750i	$-0.709170\pi$
$647$	−31.0752	−1.22169	−0.610845	+	0.791750i	$0.709170\pi$
$648$	0	0
$649$	21.8859	0.859097
$650$	0	0
$651$	1.38975	0.0544687
$652$	0	0
$653$	−45.9073	−1.79649	−0.898246	−	0.439492i	$-0.855158\pi$
$653$	−45.9073	−1.79649	−0.898246	+	0.439492i	$0.855158\pi$
$654$	0	0
$655$	24.1473	0.943515
$656$	0	0
$657$	−7.07340	−0.275960
$658$	0	0
$659$	−7.74964	−0.301883	−0.150942	−	0.988543i	$-0.548231\pi$
$659$	−7.74964	−0.301883	−0.150942	+	0.988543i	$0.548231\pi$
$660$	0	0
$661$	−23.5760	−0.917000	−0.458500	−	0.888694i	$-0.651613\pi$
$661$	−23.5760	−0.917000	−0.458500	+	0.888694i	$0.651613\pi$
$662$	0	0
$663$	−2.07984	−0.0807744
$664$	0	0
$665$	0.724038	0.0280770
$666$	0	0
$667$	−18.2117	−0.705159
$668$	0	0
$669$	20.5388	0.794076
$670$	0	0
$671$	22.0298	0.850451
$672$	0	0
$673$	15.9703	0.615610	0.307805	−	0.951449i	$-0.400406\pi$
$673$	15.9703	0.615610	0.307805	+	0.951449i	$0.400406\pi$
$674$	0	0
$675$	−2.23459	−0.0860094
$676$	0	0
$677$	9.21248	0.354064	0.177032	−	0.984205i	$-0.443350\pi$
$677$	9.21248	0.354064	0.177032	+	0.984205i	$0.443350\pi$
$678$	0	0
$679$	2.91737	0.111958
$680$	0	0
$681$	−9.97482	−0.382236
$682$	0	0
$683$	−32.3227	−1.23679	−0.618396	−	0.785867i	$-0.712217\pi$
$683$	−32.3227	−1.23679	−0.618396	+	0.785867i	$0.712217\pi$
$684$	0	0
$685$	−10.7294	−0.409950
$686$	0	0
$687$	24.6372	0.939969
$688$	0	0
$689$	−0.724069	−0.0275848
$690$	0	0
$691$	16.3396	0.621589	0.310795	−	0.950477i	$-0.399405\pi$
$691$	16.3396	0.621589	0.310795	+	0.950477i	$0.399405\pi$
$692$	0	0
$693$	−0.807113	−0.0306597
$694$	0	0
$695$	28.7371	1.09006
$696$	0	0
$697$	−17.0653	−0.646394
$698$	0	0
$699$	15.7881	0.597160
$700$	0	0
$701$	26.7561	1.01056	0.505282	−	0.862954i	$-0.331388\pi$
$701$	26.7561	1.01056	0.505282	+	0.862954i	$0.331388\pi$
$702$	0	0
$703$	8.40829	0.317125
$704$	0	0
$705$	−21.3273	−0.803231
$706$	0	0
$707$	1.74249	0.0655331
$708$	0	0
$709$	14.2573	0.535446	0.267723	−	0.963496i	$-0.413729\pi$
$709$	14.2573	0.535446	0.267723	+	0.963496i	$0.413729\pi$
$710$	0	0
$711$	−2.49735	−0.0936578
$712$	0	0
$713$	−15.7321	−0.589171
$714$	0	0
$715$	−1.66348	−0.0622106
$716$	0	0
$717$	1.61623	0.0603593
$718$	0	0
$719$	50.7047	1.89097	0.945484	−	0.325669i	$-0.105590\pi$
$719$	50.7047	1.89097	0.945484	+	0.325669i	$0.105590\pi$
$720$	0	0
$721$	−2.46436	−0.0917776
$722$	0	0
$723$	4.84691	0.180259
$724$	0	0
$725$	9.76592	0.362697
$726$	0	0
$727$	4.55518	0.168942	0.0844711	−	0.996426i	$-0.473080\pi$
$727$	4.55518	0.168942	0.0844711	+	0.996426i	$0.473080\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.0225494	0.000834019 0
$732$	0	0
$733$	12.4934	0.461455	0.230727	−	0.973018i	$-0.425889\pi$
$733$	12.4934	0.461455	0.230727	+	0.973018i	$0.425889\pi$
$734$	0	0
$735$	11.4153	0.421060
$736$	0	0
$737$	21.7686	0.801855
$738$	0	0
$739$	17.6252	0.648355	0.324177	−	0.945996i	$-0.394913\pi$
$739$	17.6252	0.648355	0.324177	+	0.945996i	$0.394913\pi$
$740$	0	0
$741$	0.539615	0.0198233
$742$	0	0
$743$	35.7215	1.31049	0.655247	−	0.755415i	$-0.272565\pi$
$743$	35.7215	1.31049	0.655247	+	0.755415i	$0.272565\pi$
$744$	0	0
$745$	−7.28454	−0.266885
$746$	0	0
$747$	14.5235	0.531388
$748$	0	0
$749$	−4.31638	−0.157717
$750$	0	0
$751$	33.9560	1.23907	0.619537	−	0.784968i	$-0.287321\pi$
$751$	33.9560	1.23907	0.619537	+	0.784968i	$0.287321\pi$
$752$	0	0
$753$	7.29225	0.265744
$754$	0	0
$755$	10.8056	0.393257
$756$	0	0
$757$	26.2452	0.953899	0.476949	−	0.878931i	$-0.341742\pi$
$757$	26.2452	0.953899	0.476949	+	0.878931i	$0.341742\pi$
$758$	0	0
$759$	9.13658	0.331637
$760$	0	0
$761$	13.9276	0.504876	0.252438	−	0.967613i	$-0.418768\pi$
$761$	13.9276	0.504876	0.252438	+	0.967613i	$0.418768\pi$
$762$	0	0
$763$	−1.62137	−0.0586976
$764$	0	0
$765$	7.58092	0.274089
$766$	0	0
$767$	4.55411	0.164439
$768$	0	0
$769$	−40.8045	−1.47145	−0.735723	−	0.677282i	$-0.763158\pi$
$769$	−40.8045	−1.47145	−0.735723	+	0.677282i	$0.763158\pi$
$770$	0	0
$771$	16.0497	0.578015
$772$	0	0
$773$	11.7416	0.422316	0.211158	−	0.977452i	$-0.432277\pi$
$773$	11.7416	0.422316	0.211158	+	0.977452i	$0.432277\pi$
$774$	0	0
$775$	8.43625	0.303039
$776$	0	0
$777$	−2.61696	−0.0938830
$778$	0	0
$779$	4.42759	0.158635
$780$	0	0
$781$	−16.8109	−0.601540
$782$	0	0
$783$	−4.37034	−0.156183
$784$	0	0
$785$	−5.65246	−0.201745
$786$	0	0
$787$	−36.6852	−1.30769	−0.653843	−	0.756630i	$-0.726844\pi$
$787$	−36.6852	−1.30769	−0.653843	+	0.756630i	$0.726844\pi$
$788$	0	0
$789$	16.4973	0.587320
$790$	0	0
$791$	0.526979	0.0187372
$792$	0	0
$793$	4.58406	0.162785
$794$	0	0
$795$	2.63920	0.0936026
$796$	0	0
$797$	50.2870	1.78126	0.890629	−	0.454731i	$-0.150265\pi$
$797$	50.2870	1.78126	0.890629	+	0.454731i	$0.150265\pi$
$798$	0	0
$799$	−58.4652	−2.06835
$800$	0	0
$801$	4.80557	0.169797
$802$	0	0
$803$	−15.5088	−0.547293
$804$	0	0
$805$	2.55094	0.0899087
$806$	0	0
$807$	5.44385	0.191633
$808$	0	0
$809$	28.5894	1.00515	0.502574	−	0.864534i	$-0.332386\pi$
$809$	28.5894	1.00515	0.502574	+	0.864534i	$0.332386\pi$
$810$	0	0
$811$	−29.2950	−1.02869	−0.514343	−	0.857585i	$-0.671964\pi$
$811$	−29.2950	−1.02869	−0.514343	+	0.857585i	$0.671964\pi$
$812$	0	0
$813$	1.72563	0.0605204
$814$	0	0
$815$	−32.6280	−1.14291
$816$	0	0
$817$	−0.00585044	−0.000204681 0
$818$	0	0
$819$	−0.167948	−0.00586857
$820$	0	0
$821$	40.7726	1.42297	0.711487	−	0.702699i	$-0.248022\pi$
$821$	40.7726	1.42297	0.711487	+	0.702699i	$0.248022\pi$
$822$	0	0
$823$	14.4265	0.502877	0.251439	−	0.967873i	$-0.419096\pi$
$823$	14.4265	0.502877	0.251439	+	0.967873i	$0.419096\pi$
$824$	0	0
$825$	−4.89944	−0.170577
$826$	0	0
$827$	−41.4957	−1.44295	−0.721473	−	0.692442i	$-0.756535\pi$
$827$	−41.4957	−1.44295	−0.721473	+	0.692442i	$0.756535\pi$
$828$	0	0
$829$	27.6006	0.958610	0.479305	−	0.877648i	$-0.340889\pi$
$829$	27.6006	0.958610	0.479305	+	0.877648i	$0.340889\pi$
$830$	0	0
$831$	−6.46661	−0.224324
$832$	0	0
$833$	31.2932	1.08425
$834$	0	0
$835$	−37.3459	−1.29241
$836$	0	0
$837$	−3.77530	−0.130493
$838$	0	0
$839$	−34.2188	−1.18136	−0.590681	−	0.806905i	$-0.701141\pi$
$839$	−34.2188	−1.18136	−0.590681	+	0.806905i	$0.701141\pi$
$840$	0	0
$841$	−9.90011	−0.341383
$842$	0	0
$843$	29.8668	1.02867
$844$	0	0
$845$	21.2722	0.731787
$846$	0	0
$847$	2.27965	0.0783297
$848$	0	0
$849$	−14.4565	−0.496145
$850$	0	0
$851$	29.6242	1.01550
$852$	0	0
$853$	54.2407	1.85717	0.928583	−	0.371126i	$-0.121028\pi$
$853$	54.2407	1.85717	0.928583	+	0.371126i	$0.121028\pi$
$854$	0	0
$855$	−1.96687	−0.0672655
$856$	0	0
$857$	−35.5888	−1.21569	−0.607845	−	0.794056i	$-0.707966\pi$
$857$	−35.5888	−1.21569	−0.607845	+	0.794056i	$0.707966\pi$
$858$	0	0
$859$	−28.7810	−0.981995	−0.490997	−	0.871161i	$-0.663368\pi$
$859$	−28.7810	−0.981995	−0.490997	+	0.871161i	$0.663368\pi$
$860$	0	0
$861$	−1.37803	−0.0469630
$862$	0	0
$863$	−50.4000	−1.71564	−0.857819	−	0.513952i	$-0.828181\pi$
$863$	−50.4000	−1.71564	−0.857819	+	0.513952i	$0.828181\pi$
$864$	0	0
$865$	31.8928	1.08439
$866$	0	0
$867$	3.78183	0.128438
$868$	0	0
$869$	−5.47555	−0.185745
$870$	0	0
$871$	4.52969	0.153483
$872$	0	0
$873$	−7.92511	−0.268224
$874$	0	0
$875$	−4.42873	−0.149718
$876$	0	0
$877$	−27.7699	−0.937722	−0.468861	−	0.883272i	$-0.655335\pi$
$877$	−27.7699	−0.937722	−0.468861	+	0.883272i	$0.655335\pi$
$878$	0	0
$879$	9.36992	0.316040
$880$	0	0
$881$	−14.1616	−0.477117	−0.238558	−	0.971128i	$-0.576675\pi$
$881$	−14.1616	−0.477117	−0.238558	+	0.971128i	$0.576675\pi$
$882$	0	0
$883$	−26.5922	−0.894900	−0.447450	−	0.894309i	$-0.647668\pi$
$883$	−26.5922	−0.894900	−0.447450	+	0.894309i	$0.647668\pi$
$884$	0	0
$885$	−16.5995	−0.557986
$886$	0	0
$887$	−43.2589	−1.45249	−0.726246	−	0.687434i	$-0.758737\pi$
$887$	−43.2589	−1.45249	−0.726246	+	0.687434i	$0.758737\pi$
$888$	0	0
$889$	0.988206	0.0331434
$890$	0	0
$891$	2.19255	0.0734531
$892$	0	0
$893$	15.1688	0.507604
$894$	0	0
$895$	−38.5769	−1.28948
$896$	0	0
$897$	1.90118	0.0634785
$898$	0	0
$899$	16.4994	0.550284
$900$	0	0
$901$	7.23492	0.241030
$902$	0	0
$903$	0.00182087	6.05947e−5 0
$904$	0	0
$905$	−33.8794	−1.12619
$906$	0	0
$907$	22.9336	0.761499	0.380749	−	0.924678i	$-0.375666\pi$
$907$	22.9336	0.761499	0.380749	+	0.924678i	$0.375666\pi$
$908$	0	0
$909$	−4.73353	−0.157001
$910$	0	0
$911$	31.0071	1.02731	0.513655	−	0.857997i	$-0.328291\pi$
$911$	31.0071	1.02731	0.513655	+	0.857997i	$0.328291\pi$
$912$	0	0
$913$	31.8435	1.05387
$914$	0	0
$915$	−16.7087	−0.552371
$916$	0	0
$917$	5.34534	0.176519
$918$	0	0
$919$	−42.7615	−1.41057	−0.705285	−	0.708924i	$-0.749181\pi$
$919$	−42.7615	−1.41057	−0.705285	+	0.708924i	$0.749181\pi$
$920$	0	0
$921$	27.5209	0.906845
$922$	0	0
$923$	−3.49808	−0.115141
$924$	0	0
$925$	−15.8858	−0.522322
$926$	0	0
$927$	6.69451	0.219877
$928$	0	0
$929$	22.2717	0.730712	0.365356	−	0.930868i	$-0.380947\pi$
$929$	22.2717	0.730712	0.365356	+	0.930868i	$0.380947\pi$
$930$	0	0
$931$	−8.11903	−0.266090
$932$	0	0
$933$	7.23684	0.236923
$934$	0	0
$935$	16.6215	0.543582
$936$	0	0
$937$	38.6188	1.26162	0.630811	−	0.775937i	$-0.282722\pi$
$937$	38.6188	1.26162	0.630811	+	0.775937i	$0.282722\pi$
$938$	0	0
$939$	−17.7165	−0.578157
$940$	0	0
$941$	33.9395	1.10639	0.553197	−	0.833050i	$-0.313408\pi$
$941$	33.9395	1.10639	0.553197	+	0.833050i	$0.313408\pi$
$942$	0	0
$943$	15.5993	0.507984
$944$	0	0
$945$	0.612161	0.0199136
$946$	0	0
$947$	−41.7224	−1.35580	−0.677899	−	0.735155i	$-0.737109\pi$
$947$	−41.7224	−1.35580	−0.677899	+	0.735155i	$0.737109\pi$
$948$	0	0
$949$	−3.22713	−0.104757
$950$	0	0
$951$	20.9788	0.680285
$952$	0	0
$953$	−10.8556	−0.351648	−0.175824	−	0.984422i	$-0.556259\pi$
$953$	−10.8556	−0.351648	−0.175824	+	0.984422i	$0.556259\pi$
$954$	0	0
$955$	−4.79711	−0.155231
$956$	0	0
$957$	−9.58218	−0.309748
$958$	0	0
$959$	−2.37510	−0.0766960
$960$	0	0
$961$	−16.7471	−0.540229
$962$	0	0
$963$	11.7256	0.377851
$964$	0	0
$965$	−20.5668	−0.662069
$966$	0	0
$967$	−11.9595	−0.384593	−0.192296	−	0.981337i	$-0.561594\pi$
$967$	−11.9595	−0.384593	−0.192296	+	0.981337i	$0.561594\pi$
$968$	0	0
$969$	−5.39185	−0.173211
$970$	0	0
$971$	24.6461	0.790931	0.395466	−	0.918481i	$-0.370583\pi$
$971$	24.6461	0.790931	0.395466	+	0.918481i	$0.370583\pi$
$972$	0	0
$973$	6.36134	0.203935
$974$	0	0
$975$	−1.01950	−0.0326500
$976$	0	0
$977$	21.5985	0.690996	0.345498	−	0.938419i	$-0.387710\pi$
$977$	21.5985	0.690996	0.345498	+	0.938419i	$0.387710\pi$
$978$	0	0
$979$	10.5364	0.336746
$980$	0	0
$981$	4.40450	0.140625
$982$	0	0
$983$	−27.1426	−0.865713	−0.432856	−	0.901463i	$-0.642494\pi$
$983$	−27.1426	−0.865713	−0.432856	+	0.901463i	$0.642494\pi$
$984$	0	0
$985$	33.7391	1.07502
$986$	0	0
$987$	−4.72107	−0.150273
$988$	0	0
$989$	−0.0206123	−0.000655434 0
$990$	0	0
$991$	13.4523	0.427327	0.213664	−	0.976907i	$-0.431460\pi$
$991$	13.4523	0.427327	0.213664	+	0.976907i	$0.431460\pi$
$992$	0	0
$993$	23.1690	0.735245
$994$	0	0
$995$	−6.86205	−0.217542
$996$	0	0
$997$	11.3277	0.358752	0.179376	−	0.983781i	$-0.442592\pi$
$997$	11.3277	0.358752	0.179376	+	0.983781i	$0.442592\pi$
$998$	0	0
$999$	7.10906	0.224921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.h.1.6	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.h.1.6	✓	24	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.66295 q^{5} -0.368117 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.66295 q^{5} -0.368117 q^{7} +1.00000 q^{9} +2.19255 q^{11} +0.456235 q^{13} -1.66295 q^{15} -4.55871 q^{17} +1.18276 q^{19} -0.368117 q^{21} +4.16710 q^{23} -2.23459 q^{25} +1.00000 q^{27} -4.37034 q^{29} -3.77530 q^{31} +2.19255 q^{33} +0.612161 q^{35} +7.10906 q^{37} +0.456235 q^{39} +3.74345 q^{41} -0.00494644 q^{43} -1.66295 q^{45} +12.8249 q^{47} -6.86449 q^{49} -4.55871 q^{51} -1.58705 q^{53} -3.64610 q^{55} +1.18276 q^{57} +9.98195 q^{59} +10.0476 q^{61} -0.368117 q^{63} -0.758697 q^{65} +9.92843 q^{67} +4.16710 q^{69} -7.66728 q^{71} -7.07340 q^{73} -2.23459 q^{75} -0.807113 q^{77} -2.49735 q^{79} +1.00000 q^{81} +14.5235 q^{83} +7.58092 q^{85} -4.37034 q^{87} +4.80557 q^{89} -0.167948 q^{91} -3.77530 q^{93} -1.96687 q^{95} -7.92511 q^{97} +2.19255 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9}+O(q^{10}) \) 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9	\( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9} + 9 q^{11} + 7 q^{13} + 18 q^{15} + 12 q^{17} - q^{19} + 9 q^{21} + 22 q^{23} + 48 q^{25} + 24 q^{27} + 44 q^{29} + 15 q^{31} + 9 q^{33} + 3 q^{35} + 18 q^{37} + 7 q^{39} + 37 q^{41} - 8 q^{43} + 18 q^{45} + 24 q^{47} + 63 q^{49} + 12 q^{51} + 59 q^{53} + 20 q^{55} - q^{57} + 44 q^{59} + 33 q^{61} + 9 q^{63} + 15 q^{65} + 13 q^{67} + 22 q^{69} + 47 q^{71} + 28 q^{73} + 48 q^{75} + 19 q^{77} + 12 q^{79} + 24 q^{81} + 17 q^{83} + 31 q^{85} + 44 q^{87} + 41 q^{89} - q^{91} + 15 q^{93} + 58 q^{95} + 51 q^{97} + 9 q^{99}+O(q^{100}) \) 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9 + 9 * q^11 + 7 * q^13 + 18 * q^15 + 12 * q^17 - q^19 + 9 * q^21 + 22 * q^23 + 48 * q^25 + 24 * q^27 + 44 * q^29 + 15 * q^31 + 9 * q^33 + 3 * q^35 + 18 * q^37 + 7 * q^39 + 37 * q^41 - 8 * q^43 + 18 * q^45 + 24 * q^47 + 63 * q^49 + 12 * q^51 + 59 * q^53 + 20 * q^55 - q^57 + 44 * q^59 + 33 * q^61 + 9 * q^63 + 15 * q^65 + 13 * q^67 + 22 * q^69 + 47 * q^71 + 28 * q^73 + 48 * q^75 + 19 * q^77 + 12 * q^79 + 24 * q^81 + 17 * q^83 + 31 * q^85 + 44 * q^87 + 41 * q^89 - q^91 + 15 * q^93 + 58 * q^95 + 51 * q^97 + 9 * q^99

Introduction

L-functions

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Varieties

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Database

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