LMFDB - Embedded newform 1336.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1336,2,Mod(1,1336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1336, 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("1336.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1336 = 2^{3} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1336.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:	$10.6680137100$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 15 x^{10} + 100 x^{9} + 36 x^{8} - 641 x^{7} + 129 x^{6} + 1804 x^{5} - 433 x^{4} + \cdots + 224 $ x^12 - 5x^11 - 15x^10 + 100x^9 + 36x^8 - 641x^7 + 129x^6 + 1804x^5 - 433x^4 - 2399x^3 + 148x^2 + 1224*x + 224
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.26807$ of defining polynomial
Character	$\chi$	$=$	1336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.26807 q^{3} -0.339536 q^{5} -4.08348 q^{7} -1.39199 q^{9} +O(q^{10})$	$q-1.26807 q^{3} -0.339536 q^{5} -4.08348 q^{7} -1.39199 q^{9} +2.81132 q^{11} -3.28108 q^{13} +0.430557 q^{15} -3.05307 q^{17} +6.09371 q^{19} +5.17815 q^{21} +0.204048 q^{23} -4.88472 q^{25} +5.56937 q^{27} -2.58414 q^{29} +7.00151 q^{31} -3.56496 q^{33} +1.38649 q^{35} +2.47670 q^{37} +4.16065 q^{39} +4.44665 q^{41} +8.73921 q^{43} +0.472631 q^{45} -2.38631 q^{47} +9.67481 q^{49} +3.87152 q^{51} +1.43102 q^{53} -0.954546 q^{55} -7.72727 q^{57} +5.48037 q^{59} -5.24427 q^{61} +5.68416 q^{63} +1.11405 q^{65} -9.65639 q^{67} -0.258748 q^{69} -1.65390 q^{71} +13.8129 q^{73} +6.19418 q^{75} -11.4800 q^{77} +6.14569 q^{79} -2.88639 q^{81} +1.56244 q^{83} +1.03663 q^{85} +3.27689 q^{87} +15.8018 q^{89} +13.3982 q^{91} -8.87842 q^{93} -2.06904 q^{95} +2.47842 q^{97} -3.91333 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9}+O(q^{10}) $ 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9	$ 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9} + 14 q^{11} - 9 q^{13} + 8 q^{15} + 4 q^{17} + 9 q^{19} + 7 q^{21} + 15 q^{23} + 18 q^{25} + 20 q^{27} + 17 q^{29} + 11 q^{31} + 8 q^{33} + 19 q^{35} - 29 q^{37} + 26 q^{39} + 14 q^{41} + 15 q^{43} - 26 q^{45} + 17 q^{47} + 20 q^{49} + 36 q^{51} - 7 q^{53} + 13 q^{55} + 3 q^{57} + 32 q^{59} - 8 q^{61} + 50 q^{63} + 37 q^{65} + 39 q^{67} + q^{69} + 35 q^{71} - 12 q^{73} + 29 q^{75} + 13 q^{77} + 36 q^{79} + 44 q^{81} + 25 q^{83} - 38 q^{85} + 14 q^{87} + 23 q^{89} - 2 q^{91} - 25 q^{93} + 38 q^{95} - 2 q^{97} + 43 q^{99}+O(q^{100}) $ 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9 + 14 * q^11 - 9 * q^13 + 8 * q^15 + 4 * q^17 + 9 * q^19 + 7 * q^21 + 15 * q^23 + 18 * q^25 + 20 * q^27 + 17 * q^29 + 11 * q^31 + 8 * q^33 + 19 * q^35 - 29 * q^37 + 26 * q^39 + 14 * q^41 + 15 * q^43 - 26 * q^45 + 17 * q^47 + 20 * q^49 + 36 * q^51 - 7 * q^53 + 13 * q^55 + 3 * q^57 + 32 * q^59 - 8 * q^61 + 50 * q^63 + 37 * q^65 + 39 * q^67 + q^69 + 35 * q^71 - 12 * q^73 + 29 * q^75 + 13 * q^77 + 36 * q^79 + 44 * q^81 + 25 * q^83 - 38 * q^85 + 14 * q^87 + 23 * q^89 - 2 * q^91 - 25 * q^93 + 38 * q^95 - 2 * q^97 + 43 * 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
$3$	−1.26807	−0.732122	−0.366061	−	0.930591i	$-0.619294\pi$
$3$	−1.26807	−0.732122	−0.366061	+	0.930591i	$0.619294\pi$
$4$	0	0
$5$	−0.339536	−0.151845	−0.0759226	−	0.997114i	$-0.524190\pi$
$5$	−0.339536	−0.151845	−0.0759226	+	0.997114i	$0.524190\pi$
$6$	0	0
$7$	−4.08348	−1.54341	−0.771705	−	0.635981i	$-0.780596\pi$
$7$	−4.08348	−1.54341	−0.771705	+	0.635981i	$0.780596\pi$
$8$	0	0
$9$	−1.39199	−0.463997
$10$	0	0
$11$	2.81132	0.847646	0.423823	−	0.905745i	$-0.360688\pi$
$11$	2.81132	0.847646	0.423823	+	0.905745i	$0.360688\pi$
$12$	0	0
$13$	−3.28108	−0.910008	−0.455004	−	0.890489i	$-0.650362\pi$
$13$	−3.28108	−0.910008	−0.455004	+	0.890489i	$0.650362\pi$
$14$	0	0
$15$	0.430557	0.111169
$16$	0	0
$17$	−3.05307	−0.740479	−0.370240	−	0.928936i	$-0.620724\pi$
$17$	−3.05307	−0.740479	−0.370240	+	0.928936i	$0.620724\pi$
$18$	0	0
$19$	6.09371	1.39799	0.698997	−	0.715125i	$-0.253630\pi$
$19$	6.09371	1.39799	0.698997	+	0.715125i	$0.253630\pi$
$20$	0	0
$21$	5.17815	1.12997
$22$	0	0
$23$	0.204048	0.0425470	0.0212735	−	0.999774i	$-0.493228\pi$
$23$	0.204048	0.0425470	0.0212735	+	0.999774i	$0.493228\pi$
$24$	0	0
$25$	−4.88472	−0.976943
$26$	0	0
$27$	5.56937	1.07182
$28$	0	0
$29$	−2.58414	−0.479864	−0.239932	−	0.970790i	$-0.577125\pi$
$29$	−2.58414	−0.479864	−0.239932	+	0.970790i	$0.577125\pi$
$30$	0	0
$31$	7.00151	1.25751	0.628754	−	0.777604i	$-0.283565\pi$
$31$	7.00151	1.25751	0.628754	+	0.777604i	$0.283565\pi$
$32$	0	0
$33$	−3.56496	−0.620580
$34$	0	0
$35$	1.38649	0.234360
$36$	0	0
$37$	2.47670	0.407166	0.203583	−	0.979058i	$-0.434741\pi$
$37$	2.47670	0.407166	0.203583	+	0.979058i	$0.434741\pi$
$38$	0	0
$39$	4.16065	0.666237
$40$	0	0
$41$	4.44665	0.694450	0.347225	−	0.937782i	$-0.387124\pi$
$41$	4.44665	0.694450	0.347225	+	0.937782i	$0.387124\pi$
$42$	0	0
$43$	8.73921	1.33272	0.666359	−	0.745631i	$-0.267852\pi$
$43$	8.73921	1.33272	0.666359	+	0.745631i	$0.267852\pi$
$44$	0	0
$45$	0.472631	0.0704557
$46$	0	0
$47$	−2.38631	−0.348079	−0.174039	−	0.984739i	$-0.555682\pi$
$47$	−2.38631	−0.348079	−0.174039	+	0.984739i	$0.555682\pi$
$48$	0	0
$49$	9.67481	1.38212
$50$	0	0
$51$	3.87152	0.542121
$52$	0	0
$53$	1.43102	0.196566	0.0982831	−	0.995158i	$-0.468665\pi$
$53$	1.43102	0.196566	0.0982831	+	0.995158i	$0.468665\pi$
$54$	0	0
$55$	−0.954546	−0.128711
$56$	0	0
$57$	−7.72727	−1.02350
$58$	0	0
$59$	5.48037	0.713483	0.356742	−	0.934203i	$-0.383888\pi$
$59$	5.48037	0.713483	0.356742	+	0.934203i	$0.383888\pi$
$60$	0	0
$61$	−5.24427	−0.671459	−0.335730	−	0.941958i	$-0.608983\pi$
$61$	−5.24427	−0.671459	−0.335730	+	0.941958i	$0.608983\pi$
$62$	0	0
$63$	5.68416	0.716137
$64$	0	0
$65$	1.11405	0.138180
$66$	0	0
$67$	−9.65639	−1.17972	−0.589858	−	0.807507i	$-0.700816\pi$
$67$	−9.65639	−1.17972	−0.589858	+	0.807507i	$0.700816\pi$
$68$	0	0
$69$	−0.258748	−0.0311496
$70$	0	0
$71$	−1.65390	−0.196282	−0.0981408	−	0.995173i	$-0.531290\pi$
$71$	−1.65390	−0.196282	−0.0981408	+	0.995173i	$0.531290\pi$
$72$	0	0
$73$	13.8129	1.61668	0.808339	−	0.588717i	$-0.200367\pi$
$73$	13.8129	1.61668	0.808339	+	0.588717i	$0.200367\pi$
$74$	0	0
$75$	6.19418	0.715242
$76$	0	0
$77$	−11.4800	−1.30827
$78$	0	0
$79$	6.14569	0.691445	0.345722	−	0.938337i	$-0.387634\pi$
$79$	6.14569	0.691445	0.345722	+	0.938337i	$0.387634\pi$
$80$	0	0
$81$	−2.88639	−0.320710
$82$	0	0
$83$	1.56244	0.171500	0.0857502	−	0.996317i	$-0.472671\pi$
$83$	1.56244	0.171500	0.0857502	+	0.996317i	$0.472671\pi$
$84$	0	0
$85$	1.03663	0.112438
$86$	0	0
$87$	3.27689	0.351319
$88$	0	0
$89$	15.8018	1.67499	0.837494	−	0.546446i	$-0.184020\pi$
$89$	15.8018	1.67499	0.837494	+	0.546446i	$0.184020\pi$
$90$	0	0
$91$	13.3982	1.40452
$92$	0	0
$93$	−8.87842	−0.920650
$94$	0	0
$95$	−2.06904	−0.212279
$96$	0	0
$97$	2.47842	0.251646	0.125823	−	0.992053i	$-0.459843\pi$
$97$	2.47842	0.251646	0.125823	+	0.992053i	$0.459843\pi$
$98$	0	0
$99$	−3.91333	−0.393305
$100$	0	0
$101$	−3.74698	−0.372838	−0.186419	−	0.982470i	$-0.559688\pi$
$101$	−3.74698	−0.372838	−0.186419	+	0.982470i	$0.559688\pi$
$102$	0	0
$103$	−3.95200	−0.389402	−0.194701	−	0.980863i	$-0.562374\pi$
$103$	−3.95200	−0.389402	−0.194701	+	0.980863i	$0.562374\pi$
$104$	0	0
$105$	−1.75817	−0.171580
$106$	0	0
$107$	−9.73495	−0.941113	−0.470557	−	0.882370i	$-0.655947\pi$
$107$	−9.73495	−0.941113	−0.470557	+	0.882370i	$0.655947\pi$
$108$	0	0
$109$	10.7376	1.02847	0.514236	−	0.857649i	$-0.328076\pi$
$109$	10.7376	1.02847	0.514236	+	0.857649i	$0.328076\pi$
$110$	0	0
$111$	−3.14063	−0.298096
$112$	0	0
$113$	3.80794	0.358221	0.179110	−	0.983829i	$-0.442678\pi$
$113$	3.80794	0.358221	0.179110	+	0.983829i	$0.442678\pi$
$114$	0	0
$115$	−0.0692818	−0.00646056
$116$	0	0
$117$	4.56723	0.422241
$118$	0	0
$119$	12.4672	1.14286
$120$	0	0
$121$	−3.09646	−0.281497
$122$	0	0
$123$	−5.63868	−0.508423
$124$	0	0
$125$	3.35622	0.300189
$126$	0	0
$127$	4.16000	0.369140	0.184570	−	0.982819i	$-0.440911\pi$
$127$	4.16000	0.369140	0.184570	+	0.982819i	$0.440911\pi$
$128$	0	0
$129$	−11.0820	−0.975712
$130$	0	0
$131$	−8.25324	−0.721089	−0.360545	−	0.932742i	$-0.617409\pi$
$131$	−8.25324	−0.721089	−0.360545	+	0.932742i	$0.617409\pi$
$132$	0	0
$133$	−24.8836	−2.15768
$134$	0	0
$135$	−1.89100	−0.162752
$136$	0	0
$137$	10.5104	0.897966	0.448983	−	0.893540i	$-0.351786\pi$
$137$	10.5104	0.897966	0.448983	+	0.893540i	$0.351786\pi$
$138$	0	0
$139$	6.53230	0.554063	0.277031	−	0.960861i	$-0.410649\pi$
$139$	6.53230	0.554063	0.277031	+	0.960861i	$0.410649\pi$
$140$	0	0
$141$	3.02601	0.254836
$142$	0	0
$143$	−9.22417	−0.771364
$144$	0	0
$145$	0.877411	0.0728650
$146$	0	0
$147$	−12.2684	−1.01188
$148$	0	0
$149$	18.3752	1.50535	0.752677	−	0.658390i	$-0.228762\pi$
$149$	18.3752	1.50535	0.752677	+	0.658390i	$0.228762\pi$
$150$	0	0
$151$	9.09651	0.740263	0.370132	−	0.928979i	$-0.379313\pi$
$151$	9.09651	0.740263	0.370132	+	0.928979i	$0.379313\pi$
$152$	0	0
$153$	4.24985	0.343580
$154$	0	0
$155$	−2.37727	−0.190947
$156$	0	0
$157$	−11.6017	−0.925914	−0.462957	−	0.886381i	$-0.653212\pi$
$157$	−11.6017	−0.925914	−0.462957	+	0.886381i	$0.653212\pi$
$158$	0	0
$159$	−1.81464	−0.143911
$160$	0	0
$161$	−0.833227	−0.0656675
$162$	0	0
$163$	15.1954	1.19019	0.595097	−	0.803654i	$-0.297114\pi$
$163$	15.1954	1.19019	0.595097	+	0.803654i	$0.297114\pi$
$164$	0	0
$165$	1.21043	0.0942322
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−2.23452	−0.171886
$170$	0	0
$171$	−8.48239	−0.648664
$172$	0	0
$173$	−9.63510	−0.732543	−0.366271	−	0.930508i	$-0.619366\pi$
$173$	−9.63510	−0.732543	−0.366271	+	0.930508i	$0.619366\pi$
$174$	0	0
$175$	19.9466	1.50782
$176$	0	0
$177$	−6.94951	−0.522357
$178$	0	0
$179$	13.4061	1.00202	0.501008	−	0.865443i	$-0.332963\pi$
$179$	13.4061	1.00202	0.501008	+	0.865443i	$0.332963\pi$
$180$	0	0
$181$	−19.1645	−1.42448	−0.712242	−	0.701934i	$-0.752320\pi$
$181$	−19.1645	−1.42448	−0.712242	+	0.701934i	$0.752320\pi$
$182$	0	0
$183$	6.65011	0.491591
$184$	0	0
$185$	−0.840929	−0.0618263
$186$	0	0
$187$	−8.58318	−0.627664
$188$	0	0
$189$	−22.7424	−1.65427
$190$	0	0
$191$	24.0857	1.74278	0.871389	−	0.490593i	$-0.163220\pi$
$191$	24.0857	1.74278	0.871389	+	0.490593i	$0.163220\pi$
$192$	0	0
$193$	−7.00801	−0.504447	−0.252224	−	0.967669i	$-0.581162\pi$
$193$	−7.00801	−0.504447	−0.252224	+	0.967669i	$0.581162\pi$
$194$	0	0
$195$	−1.41269	−0.101165
$196$	0	0
$197$	16.4779	1.17400	0.587002	−	0.809585i	$-0.300308\pi$
$197$	16.4779	1.17400	0.587002	+	0.809585i	$0.300308\pi$
$198$	0	0
$199$	−1.86255	−0.132032	−0.0660162	−	0.997819i	$-0.521029\pi$
$199$	−1.86255	−0.132032	−0.0660162	+	0.997819i	$0.521029\pi$
$200$	0	0
$201$	12.2450	0.863697
$202$	0	0
$203$	10.5523	0.740626
$204$	0	0
$205$	−1.50980	−0.105449
$206$	0	0
$207$	−0.284033	−0.0197417
$208$	0	0
$209$	17.1314	1.18500
$210$	0	0
$211$	0.154370	0.0106273	0.00531363	−	0.999986i	$-0.498309\pi$
$211$	0.154370	0.0106273	0.00531363	+	0.999986i	$0.498309\pi$
$212$	0	0
$213$	2.09726	0.143702
$214$	0	0
$215$	−2.96728	−0.202367
$216$	0	0
$217$	−28.5905	−1.94085
$218$	0	0
$219$	−17.5158	−1.18361
$220$	0	0
$221$	10.0174	0.673842
$222$	0	0
$223$	−4.14304	−0.277438	−0.138719	−	0.990332i	$-0.544299\pi$
$223$	−4.14304	−0.277438	−0.138719	+	0.990332i	$0.544299\pi$
$224$	0	0
$225$	6.79948	0.453298
$226$	0	0
$227$	−22.3859	−1.48580	−0.742901	−	0.669401i	$-0.766551\pi$
$227$	−22.3859	−1.48580	−0.742901	+	0.669401i	$0.766551\pi$
$228$	0	0
$229$	−15.7127	−1.03832	−0.519161	−	0.854676i	$-0.673756\pi$
$229$	−15.7127	−1.03832	−0.519161	+	0.854676i	$0.673756\pi$
$230$	0	0
$231$	14.5575	0.957810
$232$	0	0
$233$	−8.58198	−0.562224	−0.281112	−	0.959675i	$-0.590703\pi$
$233$	−8.58198	−0.562224	−0.281112	+	0.959675i	$0.590703\pi$
$234$	0	0
$235$	0.810238	0.0528541
$236$	0	0
$237$	−7.79319	−0.506222
$238$	0	0
$239$	−20.3840	−1.31853	−0.659267	−	0.751909i	$-0.729133\pi$
$239$	−20.3840	−1.31853	−0.659267	+	0.751909i	$0.729133\pi$
$240$	0	0
$241$	−7.91994	−0.510168	−0.255084	−	0.966919i	$-0.582103\pi$
$241$	−7.91994	−0.510168	−0.255084	+	0.966919i	$0.582103\pi$
$242$	0	0
$243$	−13.0479	−0.837026
$244$	0	0
$245$	−3.28495	−0.209868
$246$	0	0
$247$	−19.9940	−1.27218
$248$	0	0
$249$	−1.98129	−0.125559
$250$	0	0
$251$	12.5810	0.794108	0.397054	−	0.917795i	$-0.370033\pi$
$251$	12.5810	0.794108	0.397054	+	0.917795i	$0.370033\pi$
$252$	0	0
$253$	0.573646	0.0360648
$254$	0	0
$255$	−1.31452	−0.0823185
$256$	0	0
$257$	−7.89101	−0.492228	−0.246114	−	0.969241i	$-0.579154\pi$
$257$	−7.89101	−0.492228	−0.246114	+	0.969241i	$0.579154\pi$
$258$	0	0
$259$	−10.1135	−0.628425
$260$	0	0
$261$	3.59710	0.222655
$262$	0	0
$263$	22.6589	1.39720	0.698602	−	0.715510i	$-0.253806\pi$
$263$	22.6589	1.39720	0.698602	+	0.715510i	$0.253806\pi$
$264$	0	0
$265$	−0.485884	−0.0298476
$266$	0	0
$267$	−20.0379	−1.22630
$268$	0	0
$269$	−21.6950	−1.32277	−0.661384	−	0.750047i	$-0.730031\pi$
$269$	−21.6950	−1.32277	−0.661384	+	0.750047i	$0.730031\pi$
$270$	0	0
$271$	−2.65542	−0.161305	−0.0806526	−	0.996742i	$-0.525700\pi$
$271$	−2.65542	−0.161305	−0.0806526	+	0.996742i	$0.525700\pi$
$272$	0	0
$273$	−16.9899	−1.02828
$274$	0	0
$275$	−13.7325	−0.828102
$276$	0	0
$277$	−1.04397	−0.0627262	−0.0313631	−	0.999508i	$-0.509985\pi$
$277$	−1.04397	−0.0627262	−0.0313631	+	0.999508i	$0.509985\pi$
$278$	0	0
$279$	−9.74603	−0.583479
$280$	0	0
$281$	−7.63761	−0.455622	−0.227811	−	0.973705i	$-0.573157\pi$
$281$	−7.63761	−0.455622	−0.227811	+	0.973705i	$0.573157\pi$
$282$	0	0
$283$	−31.6025	−1.87857	−0.939287	−	0.343134i	$-0.888512\pi$
$283$	−31.6025	−1.87857	−0.939287	+	0.343134i	$0.888512\pi$
$284$	0	0
$285$	2.62369	0.155414
$286$	0	0
$287$	−18.1578	−1.07182
$288$	0	0
$289$	−7.67874	−0.451691
$290$	0	0
$291$	−3.14282	−0.184235
$292$	0	0
$293$	16.2276	0.948025	0.474012	−	0.880518i	$-0.342805\pi$
$293$	16.2276	0.948025	0.474012	+	0.880518i	$0.342805\pi$
$294$	0	0
$295$	−1.86078	−0.108339
$296$	0	0
$297$	15.6573	0.908528
$298$	0	0
$299$	−0.669499	−0.0387181
$300$	0	0
$301$	−35.6864	−2.05693
$302$	0	0
$303$	4.75144	0.272963
$304$	0	0
$305$	1.78062	0.101958
$306$	0	0
$307$	25.0897	1.43195	0.715974	−	0.698127i	$-0.245983\pi$
$307$	25.0897	1.43195	0.715974	+	0.698127i	$0.245983\pi$
$308$	0	0
$309$	5.01142	0.285090
$310$	0	0
$311$	−4.15439	−0.235574	−0.117787	−	0.993039i	$-0.537580\pi$
$311$	−4.15439	−0.235574	−0.117787	+	0.993039i	$0.537580\pi$
$312$	0	0
$313$	2.01625	0.113965	0.0569827	−	0.998375i	$-0.481852\pi$
$313$	2.01625	0.113965	0.0569827	+	0.998375i	$0.481852\pi$
$314$	0	0
$315$	−1.92998	−0.108742
$316$	0	0
$317$	2.92051	0.164032	0.0820162	−	0.996631i	$-0.473864\pi$
$317$	2.92051	0.164032	0.0820162	+	0.996631i	$0.473864\pi$
$318$	0	0
$319$	−7.26487	−0.406754
$320$	0	0
$321$	12.3446	0.689010
$322$	0	0
$323$	−18.6046	−1.03518
$324$	0	0
$325$	16.0271	0.889026
$326$	0	0
$327$	−13.6160	−0.752967
$328$	0	0
$329$	9.74444	0.537228
$330$	0	0
$331$	20.9210	1.14992	0.574962	−	0.818180i	$-0.305017\pi$
$331$	20.9210	1.14992	0.574962	+	0.818180i	$0.305017\pi$
$332$	0	0
$333$	−3.44754	−0.188924
$334$	0	0
$335$	3.27870	0.179134
$336$	0	0
$337$	12.9760	0.706848	0.353424	−	0.935463i	$-0.385017\pi$
$337$	12.9760	0.706848	0.353424	+	0.935463i	$0.385017\pi$
$338$	0	0
$339$	−4.82874	−0.262261
$340$	0	0
$341$	19.6835	1.06592
$342$	0	0
$343$	−10.9225	−0.589760
$344$	0	0
$345$	0.0878544	0.00472992
$346$	0	0
$347$	34.3198	1.84239	0.921193	−	0.389106i	$-0.127216\pi$
$347$	34.3198	1.84239	0.921193	+	0.389106i	$0.127216\pi$
$348$	0	0
$349$	−16.9679	−0.908269	−0.454135	−	0.890933i	$-0.650051\pi$
$349$	−16.9679	−0.908269	−0.454135	+	0.890933i	$0.650051\pi$
$350$	0	0
$351$	−18.2735	−0.975369
$352$	0	0
$353$	30.7542	1.63688	0.818440	−	0.574592i	$-0.194839\pi$
$353$	30.7542	1.63688	0.818440	+	0.574592i	$0.194839\pi$
$354$	0	0
$355$	0.561558	0.0298044
$356$	0	0
$357$	−15.8093	−0.836716
$358$	0	0
$359$	34.1313	1.80138	0.900689	−	0.434463i	$-0.143062\pi$
$359$	34.1313	1.80138	0.900689	+	0.434463i	$0.143062\pi$
$360$	0	0
$361$	18.1333	0.954386
$362$	0	0
$363$	3.92654	0.206090
$364$	0	0
$365$	−4.68998	−0.245485
$366$	0	0
$367$	−33.6092	−1.75438	−0.877192	−	0.480140i	$-0.840586\pi$
$367$	−33.6092	−1.75438	−0.877192	+	0.480140i	$0.840586\pi$
$368$	0	0
$369$	−6.18969	−0.322223
$370$	0	0
$371$	−5.84356	−0.303382
$372$	0	0
$373$	−12.5963	−0.652213	−0.326106	−	0.945333i	$-0.605737\pi$
$373$	−12.5963	−0.652213	−0.326106	+	0.945333i	$0.605737\pi$
$374$	0	0
$375$	−4.25593	−0.219775
$376$	0	0
$377$	8.47878	0.436680
$378$	0	0
$379$	−5.87913	−0.301991	−0.150995	−	0.988534i	$-0.548248\pi$
$379$	−5.87913	−0.301991	−0.150995	+	0.988534i	$0.548248\pi$
$380$	0	0
$381$	−5.27518	−0.270256
$382$	0	0
$383$	28.9190	1.47769	0.738847	−	0.673873i	$-0.235371\pi$
$383$	28.9190	1.47769	0.738847	+	0.673873i	$0.235371\pi$
$384$	0	0
$385$	3.89787	0.198654
$386$	0	0
$387$	−12.1649	−0.618376
$388$	0	0
$389$	14.6655	0.743569	0.371784	−	0.928319i	$-0.378746\pi$
$389$	14.6655	0.743569	0.371784	+	0.928319i	$0.378746\pi$
$390$	0	0
$391$	−0.622974	−0.0315052
$392$	0	0
$393$	10.4657	0.527925
$394$	0	0
$395$	−2.08669	−0.104993
$396$	0	0
$397$	−6.77443	−0.339999	−0.169999	−	0.985444i	$-0.554377\pi$
$397$	−6.77443	−0.339999	−0.169999	+	0.985444i	$0.554377\pi$
$398$	0	0
$399$	31.5542	1.57968
$400$	0	0
$401$	5.12061	0.255711	0.127855	−	0.991793i	$-0.459191\pi$
$401$	5.12061	0.255711	0.127855	+	0.991793i	$0.459191\pi$
$402$	0	0
$403$	−22.9725	−1.14434
$404$	0	0
$405$	0.980035	0.0486983
$406$	0	0
$407$	6.96280	0.345133
$408$	0	0
$409$	−3.30937	−0.163638	−0.0818189	−	0.996647i	$-0.526073\pi$
$409$	−3.30937	−0.163638	−0.0818189	+	0.996647i	$0.526073\pi$
$410$	0	0
$411$	−13.3280	−0.657421
$412$	0	0
$413$	−22.3790	−1.10120
$414$	0	0
$415$	−0.530506	−0.0260415
$416$	0	0
$417$	−8.28344	−0.405642
$418$	0	0
$419$	10.8735	0.531203	0.265601	−	0.964083i	$-0.414429\pi$
$419$	10.8735	0.531203	0.265601	+	0.964083i	$0.414429\pi$
$420$	0	0
$421$	−34.8911	−1.70049	−0.850243	−	0.526390i	$-0.823545\pi$
$421$	−34.8911	−1.70049	−0.850243	+	0.526390i	$0.823545\pi$
$422$	0	0
$423$	3.32172	0.161507
$424$	0	0
$425$	14.9134	0.723406
$426$	0	0
$427$	21.4149	1.03634
$428$	0	0
$429$	11.6969	0.564733
$430$	0	0
$431$	−11.9942	−0.577741	−0.288871	−	0.957368i	$-0.593280\pi$
$431$	−11.9942	−0.577741	−0.288871	+	0.957368i	$0.593280\pi$
$432$	0	0
$433$	36.3972	1.74914	0.874570	−	0.484899i	$-0.161144\pi$
$433$	36.3972	1.74914	0.874570	+	0.484899i	$0.161144\pi$
$434$	0	0
$435$	−1.11262	−0.0533461
$436$	0	0
$437$	1.24341	0.0594805
$438$	0	0
$439$	10.2543	0.489409	0.244705	−	0.969598i	$-0.421309\pi$
$439$	10.2543	0.489409	0.244705	+	0.969598i	$0.421309\pi$
$440$	0	0
$441$	−13.4672	−0.641297
$442$	0	0
$443$	−21.0018	−0.997825	−0.498912	−	0.866652i	$-0.666267\pi$
$443$	−21.0018	−0.997825	−0.498912	+	0.866652i	$0.666267\pi$
$444$	0	0
$445$	−5.36529	−0.254339
$446$	0	0
$447$	−23.3011	−1.10210
$448$	0	0
$449$	17.8067	0.840349	0.420174	−	0.907443i	$-0.361969\pi$
$449$	17.8067	0.840349	0.420174	+	0.907443i	$0.361969\pi$
$450$	0	0
$451$	12.5010	0.588648
$452$	0	0
$453$	−11.5350	−0.541963
$454$	0	0
$455$	−4.54918	−0.213269
$456$	0	0
$457$	17.5032	0.818767	0.409384	−	0.912362i	$-0.365744\pi$
$457$	17.5032	0.818767	0.409384	+	0.912362i	$0.365744\pi$
$458$	0	0
$459$	−17.0037	−0.793664
$460$	0	0
$461$	36.9332	1.72015	0.860076	−	0.510166i	$-0.170416\pi$
$461$	36.9332	1.72015	0.860076	+	0.510166i	$0.170416\pi$
$462$	0	0
$463$	−8.67663	−0.403237	−0.201619	−	0.979464i	$-0.564620\pi$
$463$	−8.67663	−0.403237	−0.201619	+	0.979464i	$0.564620\pi$
$464$	0	0
$465$	3.01455	0.139796
$466$	0	0
$467$	−22.4427	−1.03852	−0.519261	−	0.854616i	$-0.673793\pi$
$467$	−22.4427	−1.03852	−0.519261	+	0.854616i	$0.673793\pi$
$468$	0	0
$469$	39.4317	1.82079
$470$	0	0
$471$	14.7118	0.677882
$472$	0	0
$473$	24.5687	1.12967
$474$	0	0
$475$	−29.7661	−1.36576
$476$	0	0
$477$	−1.99197	−0.0912061
$478$	0	0
$479$	16.1267	0.736849	0.368425	−	0.929658i	$-0.379897\pi$
$479$	16.1267	0.736849	0.368425	+	0.929658i	$0.379897\pi$
$480$	0	0
$481$	−8.12624	−0.370525
$482$	0	0
$483$	1.05659	0.0480766
$484$	0	0
$485$	−0.841515	−0.0382112
$486$	0	0
$487$	32.4310	1.46959	0.734795	−	0.678289i	$-0.237278\pi$
$487$	32.4310	1.46959	0.734795	+	0.678289i	$0.237278\pi$
$488$	0	0
$489$	−19.2689	−0.871368
$490$	0	0
$491$	11.2092	0.505863	0.252932	−	0.967484i	$-0.418605\pi$
$491$	11.2092	0.505863	0.252932	+	0.967484i	$0.418605\pi$
$492$	0	0
$493$	7.88958	0.355329
$494$	0	0
$495$	1.32872	0.0597215
$496$	0	0
$497$	6.75366	0.302943
$498$	0	0
$499$	25.3022	1.13268	0.566341	−	0.824171i	$-0.308358\pi$
$499$	25.3022	1.13268	0.566341	+	0.824171i	$0.308358\pi$
$500$	0	0
$501$	−1.26807	−0.0566533
$502$	0	0
$503$	35.4969	1.58273	0.791364	−	0.611345i	$-0.209371\pi$
$503$	35.4969	1.58273	0.791364	+	0.611345i	$0.209371\pi$
$504$	0	0
$505$	1.27224	0.0566137
$506$	0	0
$507$	2.83354	0.125842
$508$	0	0
$509$	−4.85286	−0.215099	−0.107550	−	0.994200i	$-0.534300\pi$
$509$	−4.85286	−0.215099	−0.107550	+	0.994200i	$0.534300\pi$
$510$	0	0
$511$	−56.4047	−2.49520
$512$	0	0
$513$	33.9381	1.49840
$514$	0	0
$515$	1.34185	0.0591288
$516$	0	0
$517$	−6.70868	−0.295047
$518$	0	0
$519$	12.2180	0.536311
$520$	0	0
$521$	−35.2582	−1.54469	−0.772346	−	0.635202i	$-0.780917\pi$
$521$	−35.2582	−1.54469	−0.772346	+	0.635202i	$0.780917\pi$
$522$	0	0
$523$	−12.3986	−0.542151	−0.271076	−	0.962558i	$-0.587379\pi$
$523$	−12.3986	−0.542151	−0.271076	+	0.962558i	$0.587379\pi$
$524$	0	0
$525$	−25.2938	−1.10391
$526$	0	0
$527$	−21.3761	−0.931158
$528$	0	0
$529$	−22.9584	−0.998190
$530$	0	0
$531$	−7.62862	−0.331054
$532$	0	0
$533$	−14.5898	−0.631955
$534$	0	0
$535$	3.30537	0.142904
$536$	0	0
$537$	−16.9999	−0.733598
$538$	0	0
$539$	27.1990	1.17154
$540$	0	0
$541$	−12.2209	−0.525417	−0.262708	−	0.964875i	$-0.584616\pi$
$541$	−12.2209	−0.525417	−0.262708	+	0.964875i	$0.584616\pi$
$542$	0	0
$543$	24.3020	1.04290
$544$	0	0
$545$	−3.64579	−0.156169
$546$	0	0
$547$	22.1595	0.947473	0.473737	−	0.880667i	$-0.342905\pi$
$547$	22.1595	0.947473	0.473737	+	0.880667i	$0.342905\pi$
$548$	0	0
$549$	7.29997	0.311555
$550$	0	0
$551$	−15.7470	−0.670846
$552$	0	0
$553$	−25.0958	−1.06718
$554$	0	0
$555$	1.06636	0.0452644
$556$	0	0
$557$	−28.1364	−1.19218	−0.596089	−	0.802918i	$-0.703280\pi$
$557$	−28.1364	−1.19218	−0.596089	+	0.802918i	$0.703280\pi$
$558$	0	0
$559$	−28.6740	−1.21278
$560$	0	0
$561$	10.8841	0.459527
$562$	0	0
$563$	8.73478	0.368127	0.184064	−	0.982914i	$-0.441075\pi$
$563$	8.73478	0.368127	0.184064	+	0.982914i	$0.441075\pi$
$564$	0	0
$565$	−1.29293	−0.0543941
$566$	0	0
$567$	11.7865	0.494988
$568$	0	0
$569$	11.1529	0.467553	0.233776	−	0.972290i	$-0.424892\pi$
$569$	11.1529	0.467553	0.233776	+	0.972290i	$0.424892\pi$
$570$	0	0
$571$	37.0009	1.54844	0.774220	−	0.632917i	$-0.218143\pi$
$571$	37.0009	1.54844	0.774220	+	0.632917i	$0.218143\pi$
$572$	0	0
$573$	−30.5424	−1.27593
$574$	0	0
$575$	−0.996718	−0.0415660
$576$	0	0
$577$	47.3480	1.97112	0.985561	−	0.169323i	$-0.0541580\pi$
$577$	47.3480	1.97112	0.985561	+	0.169323i	$0.0541580\pi$
$578$	0	0
$579$	8.88667	0.369317
$580$	0	0
$581$	−6.38021	−0.264696
$582$	0	0
$583$	4.02307	0.166619
$584$	0	0
$585$	−1.55074	−0.0641152
$586$	0	0
$587$	−8.16219	−0.336890	−0.168445	−	0.985711i	$-0.553874\pi$
$587$	−8.16219	−0.336890	−0.168445	+	0.985711i	$0.553874\pi$
$588$	0	0
$589$	42.6652	1.75799
$590$	0	0
$591$	−20.8952	−0.859515
$592$	0	0
$593$	−11.4939	−0.471999	−0.236000	−	0.971753i	$-0.575836\pi$
$593$	−11.4939	−0.471999	−0.236000	+	0.971753i	$0.575836\pi$
$594$	0	0
$595$	−4.23305	−0.173538
$596$	0	0
$597$	2.36185	0.0966639
$598$	0	0
$599$	18.7338	0.765440	0.382720	−	0.923864i	$-0.374987\pi$
$599$	18.7338	0.765440	0.382720	+	0.923864i	$0.374987\pi$
$600$	0	0
$601$	−8.66239	−0.353346	−0.176673	−	0.984270i	$-0.556534\pi$
$601$	−8.66239	−0.353346	−0.176673	+	0.984270i	$0.556534\pi$
$602$	0	0
$603$	13.4416	0.547384
$604$	0	0
$605$	1.05136	0.0427439
$606$	0	0
$607$	17.3746	0.705214	0.352607	−	0.935772i	$-0.385295\pi$
$607$	17.3746	0.705214	0.352607	+	0.935772i	$0.385295\pi$
$608$	0	0
$609$	−13.3811	−0.542229
$610$	0	0
$611$	7.82966	0.316754
$612$	0	0
$613$	6.03241	0.243647	0.121823	−	0.992552i	$-0.461126\pi$
$613$	6.03241	0.243647	0.121823	+	0.992552i	$0.461126\pi$
$614$	0	0
$615$	1.91454	0.0772016
$616$	0	0
$617$	47.4933	1.91201	0.956004	−	0.293353i	$-0.0947711\pi$
$617$	47.4933	1.91201	0.956004	+	0.293353i	$0.0947711\pi$
$618$	0	0
$619$	−9.05320	−0.363879	−0.181939	−	0.983310i	$-0.558237\pi$
$619$	−9.05320	−0.363879	−0.181939	+	0.983310i	$0.558237\pi$
$620$	0	0
$621$	1.13642	0.0456029
$622$	0	0
$623$	−64.5264	−2.58519
$624$	0	0
$625$	23.2840	0.931361
$626$	0	0
$627$	−21.7239	−0.867568
$628$	0	0
$629$	−7.56154	−0.301498
$630$	0	0
$631$	13.3038	0.529616	0.264808	−	0.964301i	$-0.414691\pi$
$631$	13.3038	0.529616	0.264808	+	0.964301i	$0.414691\pi$
$632$	0	0
$633$	−0.195752	−0.00778045
$634$	0	0
$635$	−1.41247	−0.0560522
$636$	0	0
$637$	−31.7438	−1.25774
$638$	0	0
$639$	2.30221	0.0910740
$640$	0	0
$641$	−6.50825	−0.257060	−0.128530	−	0.991706i	$-0.541026\pi$
$641$	−6.50825	−0.257060	−0.128530	+	0.991706i	$0.541026\pi$
$642$	0	0
$643$	41.5717	1.63943	0.819713	−	0.572774i	$-0.194133\pi$
$643$	41.5717	1.63943	0.819713	+	0.572774i	$0.194133\pi$
$644$	0	0
$645$	3.76273	0.148157
$646$	0	0
$647$	−47.0603	−1.85013	−0.925066	−	0.379807i	$-0.875990\pi$
$647$	−47.0603	−1.85013	−0.925066	+	0.379807i	$0.875990\pi$
$648$	0	0
$649$	15.4071	0.604781
$650$	0	0
$651$	36.2549	1.42094
$652$	0	0
$653$	48.4752	1.89698	0.948490	−	0.316808i	$-0.102611\pi$
$653$	48.4752	1.89698	0.948490	+	0.316808i	$0.102611\pi$
$654$	0	0
$655$	2.80227	0.109494
$656$	0	0
$657$	−19.2274	−0.750133
$658$	0	0
$659$	−17.2186	−0.670740	−0.335370	−	0.942087i	$-0.608861\pi$
$659$	−17.2186	−0.670740	−0.335370	+	0.942087i	$0.608861\pi$
$660$	0	0
$661$	8.04694	0.312990	0.156495	−	0.987679i	$-0.449981\pi$
$661$	8.04694	0.312990	0.156495	+	0.987679i	$0.449981\pi$
$662$	0	0
$663$	−12.7028	−0.493335
$664$	0	0
$665$	8.44887	0.327633
$666$	0	0
$667$	−0.527290	−0.0204168
$668$	0	0
$669$	5.25367	0.203119
$670$	0	0
$671$	−14.7433	−0.569160
$672$	0	0
$673$	18.9163	0.729171	0.364585	−	0.931170i	$-0.381211\pi$
$673$	18.9163	0.729171	0.364585	+	0.931170i	$0.381211\pi$
$674$	0	0
$675$	−27.2048	−1.04711
$676$	0	0
$677$	27.8057	1.06866	0.534330	−	0.845276i	$-0.320564\pi$
$677$	27.8057	1.06866	0.534330	+	0.845276i	$0.320564\pi$
$678$	0	0
$679$	−10.1206	−0.388393
$680$	0	0
$681$	28.3869	1.08779
$682$	0	0
$683$	35.4005	1.35456	0.677280	−	0.735725i	$-0.263158\pi$
$683$	35.4005	1.35456	0.677280	+	0.735725i	$0.263158\pi$
$684$	0	0
$685$	−3.56867	−0.136352
$686$	0	0
$687$	19.9248	0.760179
$688$	0	0
$689$	−4.69530	−0.178877
$690$	0	0
$691$	24.9001	0.947244	0.473622	−	0.880728i	$-0.342946\pi$
$691$	24.9001	0.947244	0.473622	+	0.880728i	$0.342946\pi$
$692$	0	0
$693$	15.9800	0.607031
$694$	0	0
$695$	−2.21795	−0.0841318
$696$	0	0
$697$	−13.5760	−0.514226
$698$	0	0
$699$	10.8826	0.411617
$700$	0	0
$701$	−12.7962	−0.483306	−0.241653	−	0.970363i	$-0.577690\pi$
$701$	−12.7962	−0.483306	−0.241653	+	0.970363i	$0.577690\pi$
$702$	0	0
$703$	15.0923	0.569216
$704$	0	0
$705$	−1.02744	−0.0386957
$706$	0	0
$707$	15.3007	0.575443
$708$	0	0
$709$	−15.8228	−0.594236	−0.297118	−	0.954841i	$-0.596025\pi$
$709$	−15.8228	−0.594236	−0.297118	+	0.954841i	$0.596025\pi$
$710$	0	0
$711$	−8.55475	−0.320828
$712$	0	0
$713$	1.42865	0.0535032
$714$	0	0
$715$	3.13194	0.117128
$716$	0	0
$717$	25.8485	0.965329
$718$	0	0
$719$	43.5601	1.62452	0.812259	−	0.583298i	$-0.198238\pi$
$719$	43.5601	1.62452	0.812259	+	0.583298i	$0.198238\pi$
$720$	0	0
$721$	16.1379	0.601007
$722$	0	0
$723$	10.0431	0.373506
$724$	0	0
$725$	12.6228	0.468799
$726$	0	0
$727$	−48.8752	−1.81268	−0.906340	−	0.422549i	$-0.861135\pi$
$727$	−48.8752	−1.81268	−0.906340	+	0.422549i	$0.861135\pi$
$728$	0	0
$729$	25.2049	0.933516
$730$	0	0
$731$	−26.6815	−0.986849
$732$	0	0
$733$	5.61109	0.207250	0.103625	−	0.994616i	$-0.466956\pi$
$733$	5.61109	0.207250	0.103625	+	0.994616i	$0.466956\pi$
$734$	0	0
$735$	4.16555	0.153649
$736$	0	0
$737$	−27.1472	−0.999981
$738$	0	0
$739$	0.778753	0.0286469	0.0143235	−	0.999897i	$-0.495441\pi$
$739$	0.778753	0.0286469	0.0143235	+	0.999897i	$0.495441\pi$
$740$	0	0
$741$	25.3538	0.931395
$742$	0	0
$743$	−26.2378	−0.962571	−0.481285	−	0.876564i	$-0.659830\pi$
$743$	−26.2378	−0.962571	−0.481285	+	0.876564i	$0.659830\pi$
$744$	0	0
$745$	−6.23904	−0.228581
$746$	0	0
$747$	−2.17491	−0.0795756
$748$	0	0
$749$	39.7525	1.45252
$750$	0	0
$751$	−45.6386	−1.66538	−0.832688	−	0.553742i	$-0.813200\pi$
$751$	−45.6386	−1.66538	−0.832688	+	0.553742i	$0.813200\pi$
$752$	0	0
$753$	−15.9537	−0.581385
$754$	0	0
$755$	−3.08859	−0.112405
$756$	0	0
$757$	28.3332	1.02979	0.514893	−	0.857254i	$-0.327832\pi$
$757$	28.3332	1.02979	0.514893	+	0.857254i	$0.327832\pi$
$758$	0	0
$759$	−0.727425	−0.0264038
$760$	0	0
$761$	9.80438	0.355408	0.177704	−	0.984084i	$-0.443133\pi$
$761$	9.80438	0.355408	0.177704	+	0.984084i	$0.443133\pi$
$762$	0	0
$763$	−43.8466	−1.58735
$764$	0	0
$765$	−1.44298	−0.0521710
$766$	0	0
$767$	−17.9815	−0.649275
$768$	0	0
$769$	32.2777	1.16396	0.581981	−	0.813202i	$-0.302278\pi$
$769$	32.2777	1.16396	0.581981	+	0.813202i	$0.302278\pi$
$770$	0	0
$771$	10.0064	0.360371
$772$	0	0
$773$	−28.9014	−1.03951	−0.519755	−	0.854315i	$-0.673977\pi$
$773$	−28.9014	−1.03951	−0.519755	+	0.854315i	$0.673977\pi$
$774$	0	0
$775$	−34.2004	−1.22851
$776$	0	0
$777$	12.8247	0.460084
$778$	0	0
$779$	27.0966	0.970837
$780$	0	0
$781$	−4.64964	−0.166377
$782$	0	0
$783$	−14.3920	−0.514330
$784$	0	0
$785$	3.93919	0.140596
$786$	0	0
$787$	−37.9186	−1.35165	−0.675827	−	0.737060i	$-0.736213\pi$
$787$	−37.9186	−1.35165	−0.675827	+	0.737060i	$0.736213\pi$
$788$	0	0
$789$	−28.7331	−1.02292
$790$	0	0
$791$	−15.5496	−0.552881
$792$	0	0
$793$	17.2069	0.611033
$794$	0	0
$795$	0.616137	0.0218521
$796$	0	0
$797$	−3.20227	−0.113430	−0.0567150	−	0.998390i	$-0.518063\pi$
$797$	−3.20227	−0.113430	−0.0567150	+	0.998390i	$0.518063\pi$
$798$	0	0
$799$	7.28557	0.257745
$800$	0	0
$801$	−21.9960	−0.777189
$802$	0	0
$803$	38.8325	1.37037
$804$	0	0
$805$	0.282911	0.00997130
$806$	0	0
$807$	27.5109	0.968428
$808$	0	0
$809$	13.1070	0.460819	0.230410	−	0.973094i	$-0.425993\pi$
$809$	13.1070	0.460819	0.230410	+	0.973094i	$0.425993\pi$
$810$	0	0
$811$	−4.81554	−0.169097	−0.0845483	−	0.996419i	$-0.526945\pi$
$811$	−4.81554	−0.169097	−0.0845483	+	0.996419i	$0.526945\pi$
$812$	0	0
$813$	3.36727	0.118095
$814$	0	0
$815$	−5.15938	−0.180725
$816$	0	0
$817$	53.2542	1.86313
$818$	0	0
$819$	−18.6502	−0.651690
$820$	0	0
$821$	14.5024	0.506139	0.253069	−	0.967448i	$-0.418560\pi$
$821$	14.5024	0.506139	0.253069	+	0.967448i	$0.418560\pi$
$822$	0	0
$823$	−41.2620	−1.43830	−0.719152	−	0.694853i	$-0.755469\pi$
$823$	−41.2620	−1.43830	−0.719152	+	0.694853i	$0.755469\pi$
$824$	0	0
$825$	17.4138	0.606272
$826$	0	0
$827$	34.5334	1.20084	0.600422	−	0.799684i	$-0.294999\pi$
$827$	34.5334	1.20084	0.600422	+	0.799684i	$0.294999\pi$
$828$	0	0
$829$	−39.6825	−1.37823	−0.689115	−	0.724652i	$-0.742001\pi$
$829$	−39.6825	−1.37823	−0.689115	+	0.724652i	$0.742001\pi$
$830$	0	0
$831$	1.32383	0.0459232
$832$	0	0
$833$	−29.5379	−1.02343
$834$	0	0
$835$	−0.339536	−0.0117501
$836$	0	0
$837$	38.9939	1.34783
$838$	0	0
$839$	5.33133	0.184058	0.0920290	−	0.995756i	$-0.470665\pi$
$839$	5.33133	0.184058	0.0920290	+	0.995756i	$0.470665\pi$
$840$	0	0
$841$	−22.3222	−0.769731
$842$	0	0
$843$	9.68505	0.333571
$844$	0	0
$845$	0.758701	0.0261001
$846$	0	0
$847$	12.6443	0.434465
$848$	0	0
$849$	40.0743	1.37535
$850$	0	0
$851$	0.505366	0.0173237
$852$	0	0
$853$	7.06747	0.241986	0.120993	−	0.992653i	$-0.461392\pi$
$853$	7.06747	0.241986	0.120993	+	0.992653i	$0.461392\pi$
$854$	0	0
$855$	2.88008	0.0984966
$856$	0	0
$857$	−57.2790	−1.95661	−0.978307	−	0.207162i	$-0.933577\pi$
$857$	−57.2790	−1.95661	−0.978307	+	0.207162i	$0.933577\pi$
$858$	0	0
$859$	−7.80660	−0.266358	−0.133179	−	0.991092i	$-0.542518\pi$
$859$	−7.80660	−0.266358	−0.133179	+	0.991092i	$0.542518\pi$
$860$	0	0
$861$	23.0254	0.784705
$862$	0	0
$863$	41.5422	1.41411	0.707057	−	0.707157i	$-0.250022\pi$
$863$	41.5422	1.41411	0.707057	+	0.707157i	$0.250022\pi$
$864$	0	0
$865$	3.27147	0.111233
$866$	0	0
$867$	9.73721	0.330693
$868$	0	0
$869$	17.2775	0.586100
$870$	0	0
$871$	31.6834	1.07355
$872$	0	0
$873$	−3.44994	−0.116763
$874$	0	0
$875$	−13.7051	−0.463315
$876$	0	0
$877$	26.0074	0.878207	0.439103	−	0.898437i	$-0.355296\pi$
$877$	26.0074	0.878207	0.439103	+	0.898437i	$0.355296\pi$
$878$	0	0
$879$	−20.5778	−0.694070
$880$	0	0
$881$	30.5891	1.03057	0.515286	−	0.857018i	$-0.327686\pi$
$881$	30.5891	1.03057	0.515286	+	0.857018i	$0.327686\pi$
$882$	0	0
$883$	−31.4371	−1.05794	−0.528971	−	0.848640i	$-0.677422\pi$
$883$	−31.4371	−1.05794	−0.528971	+	0.848640i	$0.677422\pi$
$884$	0	0
$885$	2.35961	0.0793174
$886$	0	0
$887$	−27.6686	−0.929020	−0.464510	−	0.885568i	$-0.653770\pi$
$887$	−27.6686	−0.929020	−0.464510	+	0.885568i	$0.653770\pi$
$888$	0	0
$889$	−16.9873	−0.569735
$890$	0	0
$891$	−8.11458	−0.271849
$892$	0	0
$893$	−14.5415	−0.486612
$894$	0	0
$895$	−4.55184	−0.152151
$896$	0	0
$897$	0.848973	0.0283464
$898$	0	0
$899$	−18.0929	−0.603432
$900$	0	0
$901$	−4.36902	−0.145553
$902$	0	0
$903$	45.2530	1.50592
$904$	0	0
$905$	6.50703	0.216301
$906$	0	0
$907$	48.7131	1.61749	0.808746	−	0.588159i	$-0.200147\pi$
$907$	48.7131	1.61749	0.808746	+	0.588159i	$0.200147\pi$
$908$	0	0
$909$	5.21576	0.172996
$910$	0	0
$911$	−21.6411	−0.717002	−0.358501	−	0.933529i	$-0.616712\pi$
$911$	−21.6411	−0.717002	−0.358501	+	0.933529i	$0.616712\pi$
$912$	0	0
$913$	4.39253	0.145372
$914$	0	0
$915$	−2.25796	−0.0746457
$916$	0	0
$917$	33.7019	1.11294
$918$	0	0
$919$	23.0283	0.759634	0.379817	−	0.925062i	$-0.375987\pi$
$919$	23.0283	0.759634	0.379817	+	0.925062i	$0.375987\pi$
$920$	0	0
$921$	−31.8156	−1.04836
$922$	0	0
$923$	5.42657	0.178618
$924$	0	0
$925$	−12.0980	−0.397778
$926$	0	0
$927$	5.50114	0.180681
$928$	0	0
$929$	23.5264	0.771877	0.385939	−	0.922524i	$-0.373878\pi$
$929$	23.5264	0.771877	0.385939	+	0.922524i	$0.373878\pi$
$930$	0	0
$931$	58.9555	1.93219
$932$	0	0
$933$	5.26807	0.172469
$934$	0	0
$935$	2.91430	0.0953078
$936$	0	0
$937$	−40.6086	−1.32662	−0.663312	−	0.748343i	$-0.730850\pi$
$937$	−40.6086	−1.32662	−0.663312	+	0.748343i	$0.730850\pi$
$938$	0	0
$939$	−2.55676	−0.0834366
$940$	0	0
$941$	32.6490	1.06433	0.532164	−	0.846641i	$-0.321379\pi$
$941$	32.6490	1.06433	0.532164	+	0.846641i	$0.321379\pi$
$942$	0	0
$943$	0.907332	0.0295468
$944$	0	0
$945$	7.72187	0.251192
$946$	0	0
$947$	−34.3652	−1.11672	−0.558359	−	0.829599i	$-0.688569\pi$
$947$	−34.3652	−1.11672	−0.558359	+	0.829599i	$0.688569\pi$
$948$	0	0
$949$	−45.3212	−1.47119
$950$	0	0
$951$	−3.70342	−0.120092
$952$	0	0
$953$	−5.47688	−0.177413	−0.0887067	−	0.996058i	$-0.528273\pi$
$953$	−5.47688	−0.177413	−0.0887067	+	0.996058i	$0.528273\pi$
$954$	0	0
$955$	−8.17796	−0.264633
$956$	0	0
$957$	9.21238	0.297794
$958$	0	0
$959$	−42.9191	−1.38593
$960$	0	0
$961$	18.0211	0.581325
$962$	0	0
$963$	13.5510	0.436673
$964$	0	0
$965$	2.37947	0.0765979
$966$	0	0
$967$	1.38197	0.0444413	0.0222206	−	0.999753i	$-0.492926\pi$
$967$	1.38197	0.0444413	0.0222206	+	0.999753i	$0.492926\pi$
$968$	0	0
$969$	23.5919	0.757882
$970$	0	0
$971$	26.5046	0.850573	0.425286	−	0.905059i	$-0.360173\pi$
$971$	26.5046	0.850573	0.425286	+	0.905059i	$0.360173\pi$
$972$	0	0
$973$	−26.6745	−0.855146
$974$	0	0
$975$	−20.3236	−0.650876
$976$	0	0
$977$	−61.3580	−1.96302	−0.981508	−	0.191420i	$-0.938691\pi$
$977$	−61.3580	−1.96302	−0.981508	+	0.191420i	$0.938691\pi$
$978$	0	0
$979$	44.4240	1.41980
$980$	0	0
$981$	−14.9466	−0.477208
$982$	0	0
$983$	17.6436	0.562743	0.281371	−	0.959599i	$-0.409211\pi$
$983$	17.6436	0.562743	0.281371	+	0.959599i	$0.409211\pi$
$984$	0	0
$985$	−5.59486	−0.178267
$986$	0	0
$987$	−12.3567	−0.393317
$988$	0	0
$989$	1.78322	0.0567031
$990$	0	0
$991$	9.48199	0.301205	0.150603	−	0.988594i	$-0.451879\pi$
$991$	9.48199	0.301205	0.150603	+	0.988594i	$0.451879\pi$
$992$	0	0
$993$	−26.5294	−0.841885
$994$	0	0
$995$	0.632402	0.0200485
$996$	0	0
$997$	−39.0000	−1.23514	−0.617571	−	0.786515i	$-0.711883\pi$
$997$	−39.0000	−1.23514	−0.617571	+	0.786515i	$0.711883\pi$
$998$	0	0
$999$	13.7936	0.436411

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1336.2.a.e.1.3	✓	12
4.3	odd	2		2672.2.a.n.1.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.e.1.3	✓	12	1.1	even	1	trivial
2672.2.a.n.1.10		12	4.3	odd	2

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:	\( N \)	\(=\)	\( 1336 = 2^{3} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1336.a (trivial)

\(f(q)\)	\(=\)	\(q-1.26807 q^{3} -0.339536 q^{5} -4.08348 q^{7} -1.39199 q^{9} +O(q^{10})\)	\(q-1.26807 q^{3} -0.339536 q^{5} -4.08348 q^{7} -1.39199 q^{9} +2.81132 q^{11} -3.28108 q^{13} +0.430557 q^{15} -3.05307 q^{17} +6.09371 q^{19} +5.17815 q^{21} +0.204048 q^{23} -4.88472 q^{25} +5.56937 q^{27} -2.58414 q^{29} +7.00151 q^{31} -3.56496 q^{33} +1.38649 q^{35} +2.47670 q^{37} +4.16065 q^{39} +4.44665 q^{41} +8.73921 q^{43} +0.472631 q^{45} -2.38631 q^{47} +9.67481 q^{49} +3.87152 q^{51} +1.43102 q^{53} -0.954546 q^{55} -7.72727 q^{57} +5.48037 q^{59} -5.24427 q^{61} +5.68416 q^{63} +1.11405 q^{65} -9.65639 q^{67} -0.258748 q^{69} -1.65390 q^{71} +13.8129 q^{73} +6.19418 q^{75} -11.4800 q^{77} +6.14569 q^{79} -2.88639 q^{81} +1.56244 q^{83} +1.03663 q^{85} +3.27689 q^{87} +15.8018 q^{89} +13.3982 q^{91} -8.87842 q^{93} -2.06904 q^{95} +2.47842 q^{97} -3.91333 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9}+O(q^{10}) \) 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9	\( 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9} + 14 q^{11} - 9 q^{13} + 8 q^{15} + 4 q^{17} + 9 q^{19} + 7 q^{21} + 15 q^{23} + 18 q^{25} + 20 q^{27} + 17 q^{29} + 11 q^{31} + 8 q^{33} + 19 q^{35} - 29 q^{37} + 26 q^{39} + 14 q^{41} + 15 q^{43} - 26 q^{45} + 17 q^{47} + 20 q^{49} + 36 q^{51} - 7 q^{53} + 13 q^{55} + 3 q^{57} + 32 q^{59} - 8 q^{61} + 50 q^{63} + 37 q^{65} + 39 q^{67} + q^{69} + 35 q^{71} - 12 q^{73} + 29 q^{75} + 13 q^{77} + 36 q^{79} + 44 q^{81} + 25 q^{83} - 38 q^{85} + 14 q^{87} + 23 q^{89} - 2 q^{91} - 25 q^{93} + 38 q^{95} - 2 q^{97} + 43 q^{99}+O(q^{100}) \) 12 * q + 5 * q^3 - 2 * q^5 + 10 * q^7 + 19 * q^9 + 14 * q^11 - 9 * q^13 + 8 * q^15 + 4 * q^17 + 9 * q^19 + 7 * q^21 + 15 * q^23 + 18 * q^25 + 20 * q^27 + 17 * q^29 + 11 * q^31 + 8 * q^33 + 19 * q^35 - 29 * q^37 + 26 * q^39 + 14 * q^41 + 15 * q^43 - 26 * q^45 + 17 * q^47 + 20 * q^49 + 36 * q^51 - 7 * q^53 + 13 * q^55 + 3 * q^57 + 32 * q^59 - 8 * q^61 + 50 * q^63 + 37 * q^65 + 39 * q^67 + q^69 + 35 * q^71 - 12 * q^73 + 29 * q^75 + 13 * q^77 + 36 * q^79 + 44 * q^81 + 25 * q^83 - 38 * q^85 + 14 * q^87 + 23 * q^89 - 2 * q^91 - 25 * q^93 + 38 * q^95 - 2 * q^97 + 43 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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