LMFDB - Embedded newform 8649.2.a.be.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8649,2,Mod(1,8649)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8649, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8649.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$8649 = 3^{2} \cdot 31^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	8649.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:	$69.0626127082$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.2051578125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 2x^{7} - 9x^{6} + 19x^{5} + 14x^{4} - 28x^{3} - 11x^{2} + 6x + 1$ x^8 - 2x^7 - 9x^6 + 19x^5 + 14x^4 - 28x^3 - 11x^2 + 6*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.73366$ of defining polynomial
Character	$\chi$	$=$	8649.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.689493 q^{2} -1.52460 q^{4} -3.70752 q^{5} +0.763394 q^{7} -2.43019 q^{8} -2.55631 q^{10} -4.11562 q^{11} +2.90975 q^{13} +0.526355 q^{14} +1.37360 q^{16} +1.30852 q^{17} -3.79954 q^{19} +5.65248 q^{20} -2.83769 q^{22} -3.28296 q^{23} +8.74568 q^{25} +2.00625 q^{26} -1.16387 q^{28} +4.89563 q^{29} +5.80746 q^{32} +0.902218 q^{34} -2.83030 q^{35} +10.4128 q^{37} -2.61976 q^{38} +9.00995 q^{40} -0.755946 q^{41} +7.15689 q^{43} +6.27468 q^{44} -2.26358 q^{46} +0.876067 q^{47} -6.41723 q^{49} +6.03008 q^{50} -4.43621 q^{52} +3.57570 q^{53} +15.2587 q^{55} -1.85519 q^{56} +3.37550 q^{58} -0.927270 q^{59} -2.31704 q^{61} +1.25699 q^{64} -10.7879 q^{65} -2.08690 q^{67} -1.99498 q^{68} -1.95147 q^{70} +7.73608 q^{71} +5.65748 q^{73} +7.17954 q^{74} +5.79278 q^{76} -3.14184 q^{77} +14.0947 q^{79} -5.09266 q^{80} -0.521219 q^{82} -14.1077 q^{83} -4.85137 q^{85} +4.93462 q^{86} +10.0017 q^{88} -4.43481 q^{89} +2.22129 q^{91} +5.00520 q^{92} +0.604042 q^{94} +14.0869 q^{95} -5.27764 q^{97} -4.42463 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 2 q^{2} + 8 q^{4} - 3 q^{5} - 2 q^{7} + 9 q^{8} - 13 q^{10} - 18 q^{11} + 8 q^{13} + 9 q^{14} + 4 q^{16} - 14 q^{17} - 6 q^{19} + 7 q^{20} + 4 q^{22} - 22 q^{23} + 13 q^{25} - 9 q^{26} - 5 q^{28}+ \cdots - 10 q^{98}+O(q^{100})$ 8 * q - 2 * q^2 + 8 * q^4 - 3 * q^5 - 2 * q^7 + 9 * q^8 - 13 * q^10 - 18 * q^11 + 8 * q^13 + 9 * q^14 + 4 * q^16 - 14 * q^17 - 6 * q^19 + 7 * q^20 + 4 * q^22 - 22 * q^23 + 13 * q^25 - 9 * q^26 - 5 * q^28 - 12 * q^29 + 21 * q^32 - 7 * q^34 + 12 * q^35 - 8 * q^37 - 7 * q^38 + 11 * q^40 + 22 * q^41 - 2 * q^43 - 4 * q^44 + 18 * q^46 + 18 * q^47 - 2 * q^49 - 27 * q^50 - q^52 - 6 * q^53 + 28 * q^55 - 30 * q^56 - 5 * q^58 + 4 * q^59 + 30 * q^61 + 9 * q^64 - 3 * q^65 - 13 * q^67 - 30 * q^68 - 39 * q^70 + q^71 + 2 * q^73 - 8 * q^74 - 33 * q^76 + 24 * q^77 + 8 * q^79 - 24 * q^80 - 19 * q^82 - 39 * q^83 - 26 * q^85 + 16 * q^86 - 17 * q^88 - 27 * q^89 + 16 * q^91 - 32 * q^92 + 22 * q^94 + 6 * q^95 + 34 * q^97 - 10 * q^98

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	0.689493	0.487545	0.243772	−	0.969832i	$-0.421615\pi$
$2$	0.689493	0.487545	0.243772	+	0.969832i	$0.421615\pi$
$3$	0	0
$3$	0	0
$4$	−1.52460	−0.762300
$5$	−3.70752	−1.65805	−0.829026	−	0.559210i	$-0.811104\pi$
$5$	−3.70752	−1.65805	−0.829026	+	0.559210i	$0.811104\pi$
$6$	0	0
$7$	0.763394	0.288536	0.144268	−	0.989539i	$-0.453917\pi$
$7$	0.763394	0.288536	0.144268	+	0.989539i	$0.453917\pi$
$8$	−2.43019	−0.859200
$9$	0	0
$10$	−2.55631	−0.808375
$11$	−4.11562	−1.24091	−0.620453	−	0.784243i	$-0.713051\pi$
$11$	−4.11562	−1.24091	−0.620453	+	0.784243i	$0.713051\pi$
$12$	0	0
$13$	2.90975	0.807020	0.403510	−	0.914975i	$-0.367790\pi$
$13$	2.90975	0.807020	0.403510	+	0.914975i	$0.367790\pi$
$14$	0.526355	0.140674
$15$	0	0
$16$	1.37360	0.343401
$17$	1.30852	0.317364	0.158682	−	0.987330i	$-0.449276\pi$
$17$	1.30852	0.317364	0.158682	+	0.987330i	$0.449276\pi$
$18$	0	0
$19$	−3.79954	−0.871675	−0.435837	−	0.900025i	$-0.643548\pi$
$19$	−3.79954	−0.871675	−0.435837	+	0.900025i	$0.643548\pi$
$20$	5.65248	1.26393
$21$	0	0
$22$	−2.83769	−0.604998
$23$	−3.28296	−0.684544	−0.342272	−	0.939601i	$-0.611196\pi$
$23$	−3.28296	−0.684544	−0.342272	+	0.939601i	$0.611196\pi$
$24$	0	0
$25$	8.74568	1.74914
$26$	2.00625	0.393458
$27$	0	0
$28$	−1.16387	−0.219951
$29$	4.89563	0.909096	0.454548	−	0.890722i	$-0.349801\pi$
$29$	4.89563	0.909096	0.454548	+	0.890722i	$0.349801\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	5.80746	1.02662
$33$	0	0
$34$	0.902218	0.154729
$35$	−2.83030	−0.478407
$36$	0	0
$37$	10.4128	1.71185	0.855925	−	0.517099i	$-0.172988\pi$
$37$	10.4128	1.71185	0.855925	+	0.517099i	$0.172988\pi$
$38$	−2.61976	−0.424981
$39$	0	0
$40$	9.00995	1.42460
$41$	−0.755946	−0.118059	−0.0590295	−	0.998256i	$-0.518801\pi$
$41$	−0.755946	−0.118059	−0.0590295	+	0.998256i	$0.518801\pi$
$42$	0	0
$43$	7.15689	1.09142	0.545708	−	0.837976i	$-0.316261\pi$
$43$	7.15689	1.09142	0.545708	+	0.837976i	$0.316261\pi$
$44$	6.27468	0.945943
$45$	0	0
$46$	−2.26358	−0.333746
$47$	0.876067	0.127787	0.0638937	−	0.997957i	$-0.479648\pi$
$47$	0.876067	0.127787	0.0638937	+	0.997957i	$0.479648\pi$
$48$	0	0
$49$	−6.41723	−0.916747
$50$	6.03008	0.852782
$51$	0	0
$52$	−4.43621	−0.615191
$53$	3.57570	0.491161	0.245580	−	0.969376i	$-0.421021\pi$
$53$	3.57570	0.491161	0.245580	+	0.969376i	$0.421021\pi$
$54$	0	0
$55$	15.2587	2.05749
$56$	−1.85519	−0.247910
$57$	0	0
$58$	3.37550	0.443225
$59$	−0.927270	−0.120720	−0.0603601	−	0.998177i	$-0.519225\pi$
$59$	−0.927270	−0.120720	−0.0603601	+	0.998177i	$0.519225\pi$
$60$	0	0
$61$	−2.31704	−0.296666	−0.148333	−	0.988937i	$-0.547391\pi$
$61$	−2.31704	−0.296666	−0.148333	+	0.988937i	$0.547391\pi$
$62$	0	0
$63$	0	0
$64$	1.25699	0.157124
$65$	−10.7879	−1.33808
$66$	0	0
$67$	−2.08690	−0.254955	−0.127478	−	0.991841i	$-0.540688\pi$
$67$	−2.08690	−0.254955	−0.127478	+	0.991841i	$0.540688\pi$
$68$	−1.99498	−0.241926
$69$	0	0
$70$	−1.95147	−0.233245
$71$	7.73608	0.918104	0.459052	−	0.888409i	$-0.348189\pi$
$71$	7.73608	0.918104	0.459052	+	0.888409i	$0.348189\pi$
$72$	0	0
$73$	5.65748	0.662158	0.331079	−	0.943603i	$-0.392587\pi$
$73$	5.65748	0.662158	0.331079	+	0.943603i	$0.392587\pi$
$74$	7.17954	0.834604
$75$	0	0
$76$	5.79278	0.664477
$77$	−3.14184	−0.358046
$78$	0	0
$79$	14.0947	1.58578	0.792890	−	0.609365i	$-0.208576\pi$
$79$	14.0947	1.58578	0.792890	+	0.609365i	$0.208576\pi$
$80$	−5.09266	−0.569377
$81$	0	0
$82$	−0.521219	−0.0575590
$83$	−14.1077	−1.54852	−0.774261	−	0.632866i	$-0.781878\pi$
$83$	−14.1077	−1.54852	−0.774261	+	0.632866i	$0.781878\pi$
$84$	0	0
$85$	−4.85137	−0.526205
$86$	4.93462	0.532114
$87$	0	0
$88$	10.0017	1.06619
$89$	−4.43481	−0.470089	−0.235045	−	0.971985i	$-0.575524\pi$
$89$	−4.43481	−0.470089	−0.235045	+	0.971985i	$0.575524\pi$
$90$	0	0
$91$	2.22129	0.232854
$92$	5.00520	0.521828
$93$	0	0
$94$	0.604042	0.0623021
$95$	14.0869	1.44528
$96$	0	0
$97$	−5.27764	−0.535863	−0.267932	−	0.963438i	$-0.586340\pi$
$97$	−5.27764	−0.535863	−0.267932	+	0.963438i	$0.586340\pi$
$98$	−4.42463	−0.446955
$99$	0	0
$100$	−13.3337	−1.33337
$101$	−3.80743	−0.378853	−0.189427	−	0.981895i	$-0.560663\pi$
$101$	−3.80743	−0.378853	−0.189427	+	0.981895i	$0.560663\pi$
$102$	0	0
$103$	−6.24569	−0.615406	−0.307703	−	0.951482i	$-0.599560\pi$
$103$	−6.24569	−0.615406	−0.307703	+	0.951482i	$0.599560\pi$
$104$	−7.07124	−0.693392
$105$	0	0
$106$	2.46542	0.239463
$107$	−2.32738	−0.224997	−0.112498	−	0.993652i	$-0.535885\pi$
$107$	−2.32738	−0.224997	−0.112498	+	0.993652i	$0.535885\pi$
$108$	0	0
$109$	1.47683	0.141454	0.0707272	−	0.997496i	$-0.477468\pi$
$109$	1.47683	0.141454	0.0707272	+	0.997496i	$0.477468\pi$
$110$	10.5208	1.00312
$111$	0	0
$112$	1.04860	0.0990835
$113$	−6.38055	−0.600231	−0.300116	−	0.953903i	$-0.597025\pi$
$113$	−6.38055	−0.600231	−0.300116	+	0.953903i	$0.597025\pi$
$114$	0	0
$115$	12.1716	1.13501
$116$	−7.46388	−0.693004
$117$	0	0
$118$	−0.639346	−0.0588566
$119$	0.998919	0.0915708
$120$	0	0
$121$	5.93835	0.539850
$122$	−1.59758	−0.144638
$123$	0	0
$124$	0	0
$125$	−13.8872	−1.24211
$126$	0	0
$127$	1.43201	0.127071	0.0635354	−	0.997980i	$-0.479762\pi$
$127$	1.43201	0.127071	0.0635354	+	0.997980i	$0.479762\pi$
$128$	−10.7482	−0.950019
$129$	0	0
$130$	−7.43821	−0.652374
$131$	−6.29290	−0.549813	−0.274907	−	0.961471i	$-0.588647\pi$
$131$	−6.29290	−0.549813	−0.274907	+	0.961471i	$0.588647\pi$
$132$	0	0
$133$	−2.90055	−0.251509
$134$	−1.43890	−0.124302
$135$	0	0
$136$	−3.17996	−0.272679
$137$	12.7499	1.08930	0.544649	−	0.838664i	$-0.316663\pi$
$137$	12.7499	1.08930	0.544649	+	0.838664i	$0.316663\pi$
$138$	0	0
$139$	−18.7137	−1.58728	−0.793640	−	0.608388i	$-0.791816\pi$
$139$	−18.7137	−1.58728	−0.793640	+	0.608388i	$0.791816\pi$
$140$	4.31507	0.364690
$141$	0	0
$142$	5.33397	0.447617
$143$	−11.9754	−1.00144
$144$	0	0
$145$	−18.1506	−1.50733
$146$	3.90079	0.322832
$147$	0	0
$148$	−15.8753	−1.30494
$149$	20.1353	1.64955	0.824774	−	0.565462i	$-0.191302\pi$
$149$	20.1353	1.64955	0.824774	+	0.565462i	$0.191302\pi$
$150$	0	0
$151$	−15.3725	−1.25099	−0.625497	−	0.780227i	$-0.715104\pi$
$151$	−15.3725	−1.25099	−0.625497	+	0.780227i	$0.715104\pi$
$152$	9.23359	0.748943
$153$	0	0
$154$	−2.16628	−0.174564
$155$	0	0
$156$	0	0
$157$	15.1473	1.20888	0.604442	−	0.796649i	$-0.293396\pi$
$157$	15.1473	1.20888	0.604442	+	0.796649i	$0.293396\pi$
$158$	9.71821	0.773139
$159$	0	0
$160$	−21.5313	−1.70220
$161$	−2.50619	−0.197515
$162$	0	0
$163$	−1.04487	−0.0818405	−0.0409202	−	0.999162i	$-0.513029\pi$
$163$	−1.04487	−0.0818405	−0.0409202	+	0.999162i	$0.513029\pi$
$164$	1.15251	0.0899963
$165$	0	0
$166$	−9.72717	−0.754975
$167$	−11.2894	−0.873600	−0.436800	−	0.899559i	$-0.643888\pi$
$167$	−11.2894	−0.873600	−0.436800	+	0.899559i	$0.643888\pi$
$168$	0	0
$169$	−4.53335	−0.348719
$170$	−3.34499	−0.256549
$171$	0	0
$172$	−10.9114	−0.831986
$173$	20.6527	1.57019	0.785097	−	0.619373i	$-0.212613\pi$
$173$	20.6527	1.57019	0.785097	+	0.619373i	$0.212613\pi$
$174$	0	0
$175$	6.67640	0.504688
$176$	−5.65324	−0.426129
$177$	0	0
$178$	−3.05777	−0.229190
$179$	−0.131662	−0.00984091	−0.00492045	−	0.999988i	$-0.501566\pi$
$179$	−0.131662	−0.00984091	−0.00492045	+	0.999988i	$0.501566\pi$
$180$	0	0
$181$	−7.73199	−0.574714	−0.287357	−	0.957824i	$-0.592777\pi$
$181$	−7.73199	−0.574714	−0.287357	+	0.957824i	$0.592777\pi$
$182$	1.53156	0.113527
$183$	0	0
$184$	7.97820	0.588161
$185$	−38.6056	−2.83834
$186$	0	0
$187$	−5.38539	−0.393819
$188$	−1.33565	−0.0974124
$189$	0	0
$190$	9.71279	0.704640
$191$	21.1027	1.52694	0.763468	−	0.645846i	$-0.223495\pi$
$191$	21.1027	1.52694	0.763468	+	0.645846i	$0.223495\pi$
$192$	0	0
$193$	5.66324	0.407649	0.203824	−	0.979007i	$-0.434663\pi$
$193$	5.66324	0.407649	0.203824	+	0.979007i	$0.434663\pi$
$194$	−3.63889	−0.261257
$195$	0	0
$196$	9.78371	0.698836
$197$	15.9148	1.13388	0.566941	−	0.823758i	$-0.308127\pi$
$197$	15.9148	1.13388	0.566941	+	0.823758i	$0.308127\pi$
$198$	0	0
$199$	4.26145	0.302086	0.151043	−	0.988527i	$-0.451737\pi$
$199$	4.26145	0.302086	0.151043	+	0.988527i	$0.451737\pi$
$200$	−21.2536	−1.50286
$201$	0	0
$202$	−2.62520	−0.184708
$203$	3.73730	0.262307
$204$	0	0
$205$	2.80268	0.195748
$206$	−4.30636	−0.300038
$207$	0	0
$208$	3.99685	0.277131
$209$	15.6375	1.08167
$210$	0	0
$211$	−11.5083	−0.792263	−0.396131	−	0.918194i	$-0.629648\pi$
$211$	−11.5083	−0.792263	−0.396131	+	0.918194i	$0.629648\pi$
$212$	−5.45152	−0.374412
$213$	0	0
$214$	−1.60471	−0.109696
$215$	−26.5343	−1.80962
$216$	0	0
$217$	0	0
$218$	1.01826	0.0689654
$219$	0	0
$220$	−23.2635	−1.56842
$221$	3.80748	0.256119
$222$	0	0
$223$	−9.42391	−0.631072	−0.315536	−	0.948914i	$-0.602184\pi$
$223$	−9.42391	−0.631072	−0.315536	+	0.948914i	$0.602184\pi$
$224$	4.43338	0.296218
$225$	0	0
$226$	−4.39934	−0.292640
$227$	−21.7630	−1.44446	−0.722231	−	0.691652i	$-0.756883\pi$
$227$	−21.7630	−1.44446	−0.722231	+	0.691652i	$0.756883\pi$
$228$	0	0
$229$	−16.9559	−1.12048	−0.560238	−	0.828331i	$-0.689291\pi$
$229$	−16.9559	−1.12048	−0.560238	+	0.828331i	$0.689291\pi$
$230$	8.39224	0.553368
$231$	0	0
$232$	−11.8973	−0.781096
$233$	9.83611	0.644385	0.322192	−	0.946674i	$-0.395580\pi$
$233$	9.83611	0.644385	0.322192	+	0.946674i	$0.395580\pi$
$234$	0	0
$235$	−3.24803	−0.211878
$236$	1.41372	0.0920250
$237$	0	0
$238$	0.688748	0.0446449
$239$	0.590635	0.0382050	0.0191025	−	0.999818i	$-0.493919\pi$
$239$	0.590635	0.0382050	0.0191025	+	0.999818i	$0.493919\pi$
$240$	0	0
$241$	−15.3704	−0.990097	−0.495049	−	0.868865i	$-0.664850\pi$
$241$	−15.3704	−0.990097	−0.495049	+	0.868865i	$0.664850\pi$
$242$	4.09445	0.263201
$243$	0	0
$244$	3.53256	0.226149
$245$	23.7920	1.52001
$246$	0	0
$247$	−11.0557	−0.703459
$248$	0	0
$249$	0	0
$250$	−9.57510	−0.605582
$251$	−6.05150	−0.381967	−0.190984	−	0.981593i	$-0.561168\pi$
$251$	−6.05150	−0.381967	−0.190984	+	0.981593i	$0.561168\pi$
$252$	0	0
$253$	13.5114	0.849455
$254$	0.987363	0.0619527
$255$	0	0
$256$	−9.92482	−0.620301
$257$	4.52784	0.282439	0.141219	−	0.989978i	$-0.454898\pi$
$257$	4.52784	0.282439	0.141219	+	0.989978i	$0.454898\pi$
$258$	0	0
$259$	7.94906	0.493930
$260$	16.4473	1.02002
$261$	0	0
$262$	−4.33891	−0.268059
$263$	−2.50886	−0.154703	−0.0773516	−	0.997004i	$-0.524646\pi$
$263$	−2.50886	−0.154703	−0.0773516	+	0.997004i	$0.524646\pi$
$264$	0	0
$265$	−13.2570	−0.814370
$266$	−1.99991	−0.122622
$267$	0	0
$268$	3.18168	0.194352
$269$	13.3398	0.813340	0.406670	−	0.913575i	$-0.366690\pi$
$269$	13.3398	0.813340	0.406670	+	0.913575i	$0.366690\pi$
$270$	0	0
$271$	−13.8242	−0.839759	−0.419879	−	0.907580i	$-0.637928\pi$
$271$	−13.8242	−0.839759	−0.419879	+	0.907580i	$0.637928\pi$
$272$	1.79739	0.108983
$273$	0	0
$274$	8.79097	0.531082
$275$	−35.9939	−2.17051
$276$	0	0
$277$	−11.5707	−0.695219	−0.347609	−	0.937639i	$-0.613006\pi$
$277$	−11.5707	−0.695219	−0.347609	+	0.937639i	$0.613006\pi$
$278$	−12.9030	−0.773870
$279$	0	0
$280$	6.87814	0.411048
$281$	15.3993	0.918648	0.459324	−	0.888269i	$-0.348092\pi$
$281$	15.3993	0.918648	0.459324	+	0.888269i	$0.348092\pi$
$282$	0	0
$283$	−22.6957	−1.34912	−0.674560	−	0.738220i	$-0.735667\pi$
$283$	−22.6957	−1.34912	−0.674560	+	0.738220i	$0.735667\pi$
$284$	−11.7944	−0.699871
$285$	0	0
$286$	−8.25698	−0.488245
$287$	−0.577084	−0.0340642
$288$	0	0
$289$	−15.2878	−0.899280
$290$	−12.5147	−0.734890
$291$	0	0
$292$	−8.62539	−0.504763
$293$	−12.4009	−0.724466	−0.362233	−	0.932088i	$-0.617986\pi$
$293$	−12.4009	−0.724466	−0.362233	+	0.932088i	$0.617986\pi$
$294$	0	0
$295$	3.43787	0.200160
$296$	−25.3050	−1.47082
$297$	0	0
$298$	13.8831	0.804229
$299$	−9.55259	−0.552441
$300$	0	0
$301$	5.46353	0.314912
$302$	−10.5992	−0.609915
$303$	0	0
$304$	−5.21907	−0.299334
$305$	8.59046	0.491888
$306$	0	0
$307$	30.7200	1.75328	0.876641	−	0.481145i	$-0.159779\pi$
$307$	30.7200	1.75328	0.876641	+	0.481145i	$0.159779\pi$
$308$	4.79005	0.272938
$309$	0	0
$310$	0	0
$311$	−2.38141	−0.135037	−0.0675187	−	0.997718i	$-0.521508\pi$
$311$	−2.38141	−0.135037	−0.0675187	+	0.997718i	$0.521508\pi$
$312$	0	0
$313$	−10.7456	−0.607377	−0.303688	−	0.952771i	$-0.598218\pi$
$313$	−10.7456	−0.607377	−0.303688	+	0.952771i	$0.598218\pi$
$314$	10.4439	0.589385
$315$	0	0
$316$	−21.4888	−1.20884
$317$	−8.74635	−0.491244	−0.245622	−	0.969366i	$-0.578992\pi$
$317$	−8.74635	−0.491244	−0.245622	+	0.969366i	$0.578992\pi$
$318$	0	0
$319$	−20.1486	−1.12810
$320$	−4.66033	−0.260520
$321$	0	0
$322$	−1.72800	−0.0962977
$323$	−4.97179	−0.276638
$324$	0	0
$325$	25.4477	1.41159
$326$	−0.720430	−0.0399009
$327$	0	0
$328$	1.83709	0.101436
$329$	0.668784	0.0368713
$330$	0	0
$331$	32.3691	1.77917	0.889583	−	0.456773i	$-0.150995\pi$
$331$	32.3691	1.77917	0.889583	+	0.456773i	$0.150995\pi$
$332$	21.5086	1.18044
$333$	0	0
$334$	−7.78396	−0.425919
$335$	7.73721	0.422729
$336$	0	0
$337$	−27.7338	−1.51076	−0.755378	−	0.655289i	$-0.772547\pi$
$337$	−27.7338	−1.51076	−0.755378	+	0.655289i	$0.772547\pi$
$338$	−3.12571	−0.170016
$339$	0	0
$340$	7.39640	0.401126
$341$	0	0
$342$	0	0
$343$	−10.2426	−0.553050
$344$	−17.3926	−0.937745
$345$	0	0
$346$	14.2399	0.765540
$347$	−25.9161	−1.39125	−0.695624	−	0.718407i	$-0.744872\pi$
$347$	−25.9161	−1.39125	−0.695624	+	0.718407i	$0.744872\pi$
$348$	0	0
$349$	5.59393	0.299436	0.149718	−	0.988729i	$-0.452163\pi$
$349$	5.59393	0.299436	0.149718	+	0.988729i	$0.452163\pi$
$350$	4.60333	0.246058
$351$	0	0
$352$	−23.9013	−1.27394
$353$	11.6690	0.621077	0.310538	−	0.950561i	$-0.399491\pi$
$353$	11.6690	0.621077	0.310538	+	0.950561i	$0.399491\pi$
$354$	0	0
$355$	−28.6817	−1.52226
$356$	6.76131	0.358349
$357$	0	0
$358$	−0.0907803	−0.00479789
$359$	−26.1998	−1.38278	−0.691388	−	0.722484i	$-0.743000\pi$
$359$	−26.1998	−1.38278	−0.691388	+	0.722484i	$0.743000\pi$
$360$	0	0
$361$	−4.56348	−0.240183
$362$	−5.33115	−0.280199
$363$	0	0
$364$	−3.38657	−0.177505
$365$	−20.9752	−1.09789
$366$	0	0
$367$	−27.0146	−1.41015	−0.705076	−	0.709132i	$-0.749087\pi$
$367$	−27.0146	−1.41015	−0.705076	+	0.709132i	$0.749087\pi$
$368$	−4.50948	−0.235073
$369$	0	0
$370$	−26.6183	−1.38382
$371$	2.72967	0.141717
$372$	0	0
$373$	−12.4058	−0.642351	−0.321175	−	0.947020i	$-0.604078\pi$
$373$	−12.4058	−0.642351	−0.321175	+	0.947020i	$0.604078\pi$
$374$	−3.71319	−0.192004
$375$	0	0
$376$	−2.12900	−0.109795
$377$	14.2451	0.733658
$378$	0	0
$379$	−4.63901	−0.238290	−0.119145	−	0.992877i	$-0.538015\pi$
$379$	−4.63901	−0.238290	−0.119145	+	0.992877i	$0.538015\pi$
$380$	−21.4768	−1.10174
$381$	0	0
$382$	14.5501	0.744450
$383$	−5.02133	−0.256578	−0.128289	−	0.991737i	$-0.540949\pi$
$383$	−5.02133	−0.256578	−0.128289	+	0.991737i	$0.540949\pi$
$384$	0	0
$385$	11.6484	0.593659
$386$	3.90476	0.198747
$387$	0	0
$388$	8.04629	0.408488
$389$	18.6018	0.943151	0.471575	−	0.881826i	$-0.343686\pi$
$389$	18.6018	0.943151	0.471575	+	0.881826i	$0.343686\pi$
$390$	0	0
$391$	−4.29583	−0.217249
$392$	15.5951	0.787670
$393$	0	0
$394$	10.9731	0.552819
$395$	−52.2564	−2.62930
$396$	0	0
$397$	32.5588	1.63408	0.817040	−	0.576581i	$-0.195614\pi$
$397$	32.5588	1.63408	0.817040	+	0.576581i	$0.195614\pi$
$398$	2.93824	0.147281
$399$	0	0
$400$	12.0131	0.600655
$401$	−12.3903	−0.618740	−0.309370	−	0.950942i	$-0.600118\pi$
$401$	−12.3903	−0.618740	−0.309370	+	0.950942i	$0.600118\pi$
$402$	0	0
$403$	0	0
$404$	5.80481	0.288800
$405$	0	0
$406$	2.57684	0.127886
$407$	−42.8551	−2.12425
$408$	0	0
$409$	3.44808	0.170497	0.0852484	−	0.996360i	$-0.472832\pi$
$409$	3.44808	0.170497	0.0852484	+	0.996360i	$0.472832\pi$
$410$	1.93243	0.0954358
$411$	0	0
$412$	9.52218	0.469124
$413$	−0.707872	−0.0348321
$414$	0	0
$415$	52.3046	2.56753
$416$	16.8983	0.828506
$417$	0	0
$418$	10.7819	0.527361
$419$	40.0471	1.95643	0.978216	−	0.207592i	$-0.0665626\pi$
$419$	40.0471	1.95643	0.978216	+	0.207592i	$0.0665626\pi$
$420$	0	0
$421$	21.6844	1.05683	0.528417	−	0.848985i	$-0.322786\pi$
$421$	21.6844	1.05683	0.528417	+	0.848985i	$0.322786\pi$
$422$	−7.93488	−0.386264
$423$	0	0
$424$	−8.68963	−0.422006
$425$	11.4439	0.555112
$426$	0	0
$427$	−1.76881	−0.0855989
$428$	3.54833	0.171515
$429$	0	0
$430$	−18.2952	−0.882273
$431$	11.6207	0.559748	0.279874	−	0.960037i	$-0.409707\pi$
$431$	11.6207	0.559748	0.279874	+	0.960037i	$0.409707\pi$
$432$	0	0
$433$	−24.5964	−1.18203	−0.591015	−	0.806661i	$-0.701272\pi$
$433$	−24.5964	−1.18203	−0.591015	+	0.806661i	$0.701272\pi$
$434$	0	0
$435$	0	0
$436$	−2.25157	−0.107831
$437$	12.4737	0.596700
$438$	0	0
$439$	18.1932	0.868316	0.434158	−	0.900837i	$-0.357046\pi$
$439$	18.1932	0.868316	0.434158	+	0.900837i	$0.357046\pi$
$440$	−37.0816	−1.76779
$441$	0	0
$442$	2.62523	0.124869
$443$	−17.7603	−0.843819	−0.421909	−	0.906638i	$-0.638640\pi$
$443$	−17.7603	−0.843819	−0.421909	+	0.906638i	$0.638640\pi$
$444$	0	0
$445$	16.4421	0.779432
$446$	−6.49772	−0.307676
$447$	0	0
$448$	0.959582	0.0453360
$449$	−9.19857	−0.434107	−0.217054	−	0.976160i	$-0.569645\pi$
$449$	−9.19857	−0.434107	−0.217054	+	0.976160i	$0.569645\pi$
$450$	0	0
$451$	3.11119	0.146500
$452$	9.72778	0.457556
$453$	0	0
$454$	−15.0054	−0.704241
$455$	−8.23545	−0.386084
$456$	0	0
$457$	31.1999	1.45947	0.729735	−	0.683730i	$-0.239644\pi$
$457$	31.1999	1.45947	0.729735	+	0.683730i	$0.239644\pi$
$458$	−11.6910	−0.546283
$459$	0	0
$460$	−18.5568	−0.865218
$461$	−6.66847	−0.310582	−0.155291	−	0.987869i	$-0.549631\pi$
$461$	−6.66847	−0.310582	−0.155291	+	0.987869i	$0.549631\pi$
$462$	0	0
$463$	−35.3185	−1.64139	−0.820694	−	0.571368i	$-0.806413\pi$
$463$	−35.3185	−1.64139	−0.820694	+	0.571368i	$0.806413\pi$
$464$	6.72466	0.312184
$465$	0	0
$466$	6.78192	0.314167
$467$	−35.9434	−1.66326	−0.831632	−	0.555327i	$-0.812593\pi$
$467$	−35.9434	−1.66326	−0.831632	+	0.555327i	$0.812593\pi$
$468$	0	0
$469$	−1.59312	−0.0735637
$470$	−2.23949	−0.103300
$471$	0	0
$472$	2.25344	0.103723
$473$	−29.4551	−1.35434
$474$	0	0
$475$	−33.2296	−1.52468
$476$	−1.52295	−0.0698044
$477$	0	0
$478$	0.407239	0.0186267
$479$	−33.2444	−1.51898	−0.759488	−	0.650521i	$-0.774550\pi$
$479$	−33.2444	−1.51898	−0.759488	+	0.650521i	$0.774550\pi$
$480$	0	0
$481$	30.2986	1.38150
$482$	−10.5978	−0.482717
$483$	0	0
$484$	−9.05361	−0.411528
$485$	19.5669	0.888489
$486$	0	0
$487$	−28.5154	−1.29215	−0.646077	−	0.763272i	$-0.723592\pi$
$487$	−28.5154	−1.29215	−0.646077	+	0.763272i	$0.723592\pi$
$488$	5.63084	0.254896
$489$	0	0
$490$	16.4044	0.741075
$491$	−27.9636	−1.26198	−0.630991	−	0.775790i	$-0.717351\pi$
$491$	−27.9636	−1.26198	−0.630991	+	0.775790i	$0.717351\pi$
$492$	0	0
$493$	6.40605	0.288514
$494$	−7.62284	−0.342968
$495$	0	0
$496$	0	0
$497$	5.90568	0.264906
$498$	0	0
$499$	41.3050	1.84906	0.924532	−	0.381104i	$-0.124456\pi$
$499$	41.3050	1.84906	0.924532	+	0.381104i	$0.124456\pi$
$500$	21.1724	0.946857
$501$	0	0
$502$	−4.17247	−0.186226
$503$	25.5860	1.14082	0.570412	−	0.821358i	$-0.306783\pi$
$503$	25.5860	1.14082	0.570412	+	0.821358i	$0.306783\pi$
$504$	0	0
$505$	14.1161	0.628159
$506$	9.31602	0.414148
$507$	0	0
$508$	−2.18325	−0.0968660
$509$	−24.4678	−1.08452	−0.542258	−	0.840212i	$-0.682430\pi$
$509$	−24.4678	−1.08452	−0.542258	+	0.840212i	$0.682430\pi$
$510$	0	0
$511$	4.31888	0.191056
$512$	14.6534	0.647594
$513$	0	0
$514$	3.12191	0.137702
$515$	23.1560	1.02038
$516$	0	0
$517$	−3.60556	−0.158572
$518$	5.48082	0.240813
$519$	0	0
$520$	26.2167	1.14968
$521$	−14.9656	−0.655654	−0.327827	−	0.944738i	$-0.606316\pi$
$521$	−14.9656	−0.655654	−0.327827	+	0.944738i	$0.606316\pi$
$522$	0	0
$523$	20.9656	0.916763	0.458381	−	0.888756i	$-0.348429\pi$
$523$	20.9656	0.916763	0.458381	+	0.888756i	$0.348429\pi$
$524$	9.59415	0.419123
$525$	0	0
$526$	−1.72984	−0.0754247
$527$	0	0
$528$	0	0
$529$	−12.2222	−0.531399
$530$	−9.14059	−0.397042
$531$	0	0
$532$	4.42217	0.191726
$533$	−2.19961	−0.0952759
$534$	0	0
$535$	8.62882	0.373056
$536$	5.07155	0.219058
$537$	0	0
$538$	9.19768	0.396540
$539$	26.4109	1.13760
$540$	0	0
$541$	1.73130	0.0744345	0.0372172	−	0.999307i	$-0.488151\pi$
$541$	1.73130	0.0744345	0.0372172	+	0.999307i	$0.488151\pi$
$542$	−9.53167	−0.409420
$543$	0	0
$544$	7.59920	0.325813
$545$	−5.47536	−0.234539
$546$	0	0
$547$	−13.1214	−0.561032	−0.280516	−	0.959849i	$-0.590506\pi$
$547$	−13.1214	−0.561032	−0.280516	+	0.959849i	$0.590506\pi$
$548$	−19.4385	−0.830372
$549$	0	0
$550$	−24.8175	−1.05822
$551$	−18.6012	−0.792436
$552$	0	0
$553$	10.7598	0.457554
$554$	−7.97795	−0.338950
$555$	0	0
$556$	28.5310	1.20998
$557$	−28.0246	−1.18744	−0.593721	−	0.804671i	$-0.702342\pi$
$557$	−28.0246	−1.18744	−0.593721	+	0.804671i	$0.702342\pi$
$558$	0	0
$559$	20.8248	0.880794
$560$	−3.88770	−0.164286
$561$	0	0
$562$	10.6177	0.447882
$563$	−22.6519	−0.954662	−0.477331	−	0.878723i	$-0.658396\pi$
$563$	−22.6519	−0.954662	−0.477331	+	0.878723i	$0.658396\pi$
$564$	0	0
$565$	23.6560	0.995215
$566$	−15.6485	−0.657757
$567$	0	0
$568$	−18.8001	−0.788836
$569$	−46.4420	−1.94695	−0.973475	−	0.228792i	$-0.926522\pi$
$569$	−46.4420	−1.94695	−0.973475	+	0.228792i	$0.926522\pi$
$570$	0	0
$571$	12.4984	0.523040	0.261520	−	0.965198i	$-0.415776\pi$
$571$	12.4984	0.523040	0.261520	+	0.965198i	$0.415776\pi$
$572$	18.2577	0.763395
$573$	0	0
$574$	−0.397896	−0.0166078
$575$	−28.7117	−1.19736
$576$	0	0
$577$	−30.4843	−1.26908	−0.634539	−	0.772891i	$-0.718810\pi$
$577$	−30.4843	−1.26908	−0.634539	+	0.772891i	$0.718810\pi$
$578$	−10.5408	−0.438440
$579$	0	0
$580$	27.6725	1.14904
$581$	−10.7697	−0.446804
$582$	0	0
$583$	−14.7162	−0.609485
$584$	−13.7487	−0.568926
$585$	0	0
$586$	−8.55030	−0.353210
$587$	33.6581	1.38922	0.694609	−	0.719387i	$-0.255577\pi$
$587$	33.6581	1.38922	0.694609	+	0.719387i	$0.255577\pi$
$588$	0	0
$589$	0	0
$590$	2.37039	0.0975872
$591$	0	0
$592$	14.3030	0.587851
$593$	44.8879	1.84333	0.921663	−	0.387992i	$-0.126831\pi$
$593$	44.8879	1.84333	0.921663	+	0.387992i	$0.126831\pi$
$594$	0	0
$595$	−3.70351	−0.151829
$596$	−30.6983	−1.25745
$597$	0	0
$598$	−6.58644	−0.269340
$599$	19.5846	0.800206	0.400103	−	0.916470i	$-0.368974\pi$
$599$	19.5846	0.800206	0.400103	+	0.916470i	$0.368974\pi$
$600$	0	0
$601$	−30.4595	−1.24247	−0.621234	−	0.783625i	$-0.713368\pi$
$601$	−30.4595	−1.24247	−0.621234	+	0.783625i	$0.713368\pi$
$602$	3.76706	0.153534
$603$	0	0
$604$	23.4368	0.953632
$605$	−22.0165	−0.895099
$606$	0	0
$607$	−13.3746	−0.542857	−0.271428	−	0.962459i	$-0.587496\pi$
$607$	−13.3746	−0.542857	−0.271428	+	0.962459i	$0.587496\pi$
$608$	−22.0657	−0.894882
$609$	0	0
$610$	5.92306	0.239818
$611$	2.54914	0.103127
$612$	0	0
$613$	5.19521	0.209832	0.104916	−	0.994481i	$-0.466543\pi$
$613$	5.19521	0.209832	0.104916	+	0.994481i	$0.466543\pi$
$614$	21.1812	0.854804
$615$	0	0
$616$	7.63526	0.307633
$617$	30.0131	1.20828	0.604140	−	0.796878i	$-0.293517\pi$
$617$	30.0131	1.20828	0.604140	+	0.796878i	$0.293517\pi$
$618$	0	0
$619$	−18.3260	−0.736584	−0.368292	−	0.929710i	$-0.620057\pi$
$619$	−18.3260	−0.736584	−0.368292	+	0.929710i	$0.620057\pi$
$620$	0	0
$621$	0	0
$622$	−1.64197	−0.0658369
$623$	−3.38551	−0.135638
$624$	0	0
$625$	7.75849	0.310339
$626$	−7.40901	−0.296123
$627$	0	0
$628$	−23.0935	−0.921532
$629$	13.6254	0.543279
$630$	0	0
$631$	−10.3352	−0.411438	−0.205719	−	0.978611i	$-0.565953\pi$
$631$	−10.3352	−0.411438	−0.205719	+	0.978611i	$0.565953\pi$
$632$	−34.2528	−1.36250
$633$	0	0
$634$	−6.03054	−0.239503
$635$	−5.30922	−0.210690
$636$	0	0
$637$	−18.6725	−0.739833
$638$	−13.8923	−0.550001
$639$	0	0
$640$	39.8493	1.57518
$641$	0.101252	0.00399921	0.00199961	−	0.999998i	$-0.499364\pi$
$641$	0.101252	0.00399921	0.00199961	+	0.999998i	$0.499364\pi$
$642$	0	0
$643$	13.7118	0.540742	0.270371	−	0.962756i	$-0.412854\pi$
$643$	13.7118	0.540742	0.270371	+	0.962756i	$0.412854\pi$
$644$	3.82094	0.150566
$645$	0	0
$646$	−3.42801	−0.134873
$647$	−10.8344	−0.425943	−0.212971	−	0.977058i	$-0.568314\pi$
$647$	−10.8344	−0.425943	−0.212971	+	0.977058i	$0.568314\pi$
$648$	0	0
$649$	3.81629	0.149803
$650$	17.5460	0.688212
$651$	0	0
$652$	1.59301	0.0623870
$653$	24.2522	0.949061	0.474531	−	0.880239i	$-0.342618\pi$
$653$	24.2522	0.949061	0.474531	+	0.880239i	$0.342618\pi$
$654$	0	0
$655$	23.3310	0.911619
$656$	−1.03837	−0.0405415
$657$	0	0
$658$	0.461122	0.0179764
$659$	8.53349	0.332418	0.166209	−	0.986091i	$-0.446847\pi$
$659$	8.53349	0.332418	0.166209	+	0.986091i	$0.446847\pi$
$660$	0	0
$661$	37.8842	1.47352	0.736762	−	0.676152i	$-0.236354\pi$
$661$	37.8842	1.47352	0.736762	+	0.676152i	$0.236354\pi$
$662$	22.3183	0.867424
$663$	0	0
$664$	34.2844	1.33049
$665$	10.7538	0.417015
$666$	0	0
$667$	−16.0722	−0.622316
$668$	17.2118	0.665945
$669$	0	0
$670$	5.33475	0.206099
$671$	9.53606	0.368136
$672$	0	0
$673$	−7.12011	−0.274460	−0.137230	−	0.990539i	$-0.543820\pi$
$673$	−7.12011	−0.274460	−0.137230	+	0.990539i	$0.543820\pi$
$674$	−19.1223	−0.736562
$675$	0	0
$676$	6.91154	0.265829
$677$	48.0610	1.84713	0.923567	−	0.383438i	$-0.125260\pi$
$677$	48.0610	1.84713	0.923567	+	0.383438i	$0.125260\pi$
$678$	0	0
$679$	−4.02892	−0.154616
$680$	11.7897	0.452116
$681$	0	0
$682$	0	0
$683$	−32.5731	−1.24638	−0.623188	−	0.782072i	$-0.714163\pi$
$683$	−32.5731	−1.24638	−0.623188	+	0.782072i	$0.714163\pi$
$684$	0	0
$685$	−47.2705	−1.80611
$686$	−7.06222	−0.269637
$687$	0	0
$688$	9.83073	0.374793
$689$	10.4044	0.396376
$690$	0	0
$691$	27.2531	1.03676	0.518378	−	0.855152i	$-0.326536\pi$
$691$	27.2531	1.03676	0.518378	+	0.855152i	$0.326536\pi$
$692$	−31.4871	−1.19696
$693$	0	0
$694$	−17.8689	−0.678296
$695$	69.3815	2.63179
$696$	0	0
$697$	−0.989173	−0.0374676
$698$	3.85698	0.145989
$699$	0	0
$700$	−10.1788	−0.384724
$701$	−8.48915	−0.320631	−0.160315	−	0.987066i	$-0.551251\pi$
$701$	−8.48915	−0.320631	−0.160315	+	0.987066i	$0.551251\pi$
$702$	0	0
$703$	−39.5638	−1.49218
$704$	−5.17332	−0.194977
$705$	0	0
$706$	8.04567	0.302803
$707$	−2.90657	−0.109313
$708$	0	0
$709$	23.3861	0.878283	0.439142	−	0.898418i	$-0.355283\pi$
$709$	23.3861	0.878283	0.439142	+	0.898418i	$0.355283\pi$
$710$	−19.7758	−0.742172
$711$	0	0
$712$	10.7774	0.403901
$713$	0	0
$714$	0	0
$715$	44.3991	1.66043
$716$	0.200732	0.00750172
$717$	0	0
$718$	−18.0646	−0.674165
$719$	12.9769	0.483956	0.241978	−	0.970282i	$-0.422204\pi$
$719$	12.9769	0.483956	0.241978	+	0.970282i	$0.422204\pi$
$720$	0	0
$721$	−4.76792	−0.177567
$722$	−3.14649	−0.117100
$723$	0	0
$724$	11.7882	0.438104
$725$	42.8156	1.59013
$726$	0	0
$727$	−12.4230	−0.460744	−0.230372	−	0.973103i	$-0.573994\pi$
$727$	−12.4230	−0.460744	−0.230372	+	0.973103i	$0.573994\pi$
$728$	−5.39814	−0.200068
$729$	0	0
$730$	−14.4622	−0.535272
$731$	9.36496	0.346376
$732$	0	0
$733$	−34.1573	−1.26163	−0.630815	−	0.775933i	$-0.717279\pi$
$733$	−34.1573	−1.26163	−0.630815	+	0.775933i	$0.717279\pi$
$734$	−18.6264	−0.687512
$735$	0	0
$736$	−19.0657	−0.702769
$737$	8.58888	0.316376
$738$	0	0
$739$	12.3749	0.455219	0.227610	−	0.973752i	$-0.426909\pi$
$739$	12.3749	0.455219	0.227610	+	0.973752i	$0.426909\pi$
$740$	58.8580	2.16366
$741$	0	0
$742$	1.88209	0.0690936
$743$	−16.2455	−0.595990	−0.297995	−	0.954567i	$-0.596318\pi$
$743$	−16.2455	−0.595990	−0.297995	+	0.954567i	$0.596318\pi$
$744$	0	0
$745$	−74.6520	−2.73504
$746$	−8.55374	−0.313175
$747$	0	0
$748$	8.21057	0.300208
$749$	−1.77671	−0.0649196
$750$	0	0
$751$	−10.7883	−0.393670	−0.196835	−	0.980437i	$-0.563066\pi$
$751$	−10.7883	−0.393670	−0.196835	+	0.980437i	$0.563066\pi$
$752$	1.20337	0.0438823
$753$	0	0
$754$	9.82187	0.357691
$755$	56.9936	2.07421
$756$	0	0
$757$	−41.1017	−1.49387	−0.746933	−	0.664900i	$-0.768474\pi$
$757$	−41.1017	−1.49387	−0.746933	+	0.664900i	$0.768474\pi$
$758$	−3.19857	−0.116177
$759$	0	0
$760$	−34.2337	−1.24179
$761$	19.6605	0.712693	0.356347	−	0.934354i	$-0.384022\pi$
$761$	19.6605	0.712693	0.356347	+	0.934354i	$0.384022\pi$
$762$	0	0
$763$	1.12740	0.0408146
$764$	−32.1731	−1.16398
$765$	0	0
$766$	−3.46217	−0.125093
$767$	−2.69812	−0.0974236
$768$	0	0
$769$	3.78834	0.136611	0.0683055	−	0.997664i	$-0.478241\pi$
$769$	3.78834	0.136611	0.0683055	+	0.997664i	$0.478241\pi$
$770$	8.03151	0.289435
$771$	0	0
$772$	−8.63417	−0.310750
$773$	−25.3696	−0.912480	−0.456240	−	0.889857i	$-0.650804\pi$
$773$	−25.3696	−0.912480	−0.456240	+	0.889857i	$0.650804\pi$
$774$	0	0
$775$	0	0
$776$	12.8256	0.460414
$777$	0	0
$778$	12.8258	0.459828
$779$	2.87225	0.102909
$780$	0	0
$781$	−31.8388	−1.13928
$782$	−2.96194	−0.105919
$783$	0	0
$784$	−8.81473	−0.314812
$785$	−56.1588	−2.00439
$786$	0	0
$787$	25.9466	0.924896	0.462448	−	0.886646i	$-0.346971\pi$
$787$	25.9466	0.924896	0.462448	+	0.886646i	$0.346971\pi$
$788$	−24.2637	−0.864359
$789$	0	0
$790$	−36.0304	−1.28190
$791$	−4.87087	−0.173188
$792$	0	0
$793$	−6.74201	−0.239416
$794$	22.4491	0.796687
$795$	0	0
$796$	−6.49701	−0.230280
$797$	−18.5026	−0.655396	−0.327698	−	0.944782i	$-0.606273\pi$
$797$	−18.5026	−0.655396	−0.327698	+	0.944782i	$0.606273\pi$
$798$	0	0
$799$	1.14635	0.0405551
$800$	50.7902	1.79570
$801$	0	0
$802$	−8.54300	−0.301664
$803$	−23.2840	−0.821676
$804$	0	0
$805$	9.29174	0.327491
$806$	0	0
$807$	0	0
$808$	9.25276	0.325511
$809$	36.7478	1.29199	0.645993	−	0.763344i	$-0.276444\pi$
$809$	36.7478	1.29199	0.645993	+	0.763344i	$0.276444\pi$
$810$	0	0
$811$	−39.4301	−1.38458	−0.692289	−	0.721620i	$-0.743398\pi$
$811$	−39.4301	−1.38458	−0.692289	+	0.721620i	$0.743398\pi$
$812$	−5.69788	−0.199956
$813$	0	0
$814$	−29.5483	−1.03567
$815$	3.87387	0.135696
$816$	0	0
$817$	−27.1929	−0.951359
$818$	2.37743	0.0831248
$819$	0	0
$820$	−4.27297	−0.149218
$821$	1.48058	0.0516727	0.0258363	−	0.999666i	$-0.491775\pi$
$821$	1.48058	0.0516727	0.0258363	+	0.999666i	$0.491775\pi$
$822$	0	0
$823$	38.8652	1.35476	0.677378	−	0.735635i	$-0.263116\pi$
$823$	38.8652	1.35476	0.677378	+	0.735635i	$0.263116\pi$
$824$	15.1782	0.528758
$825$	0	0
$826$	−0.488073	−0.0169822
$827$	33.9363	1.18008	0.590040	−	0.807374i	$-0.299112\pi$
$827$	33.9363	1.18008	0.590040	+	0.807374i	$0.299112\pi$
$828$	0	0
$829$	−16.5307	−0.574135	−0.287067	−	0.957910i	$-0.592680\pi$
$829$	−16.5307	−0.574135	−0.287067	+	0.957910i	$0.592680\pi$
$830$	36.0636	1.25179
$831$	0	0
$832$	3.65754	0.126802
$833$	−8.39710	−0.290942
$834$	0	0
$835$	41.8556	1.44847
$836$	−23.8409	−0.824555
$837$	0	0
$838$	27.6122	0.953848
$839$	54.0781	1.86698	0.933491	−	0.358600i	$-0.116746\pi$
$839$	54.0781	1.86698	0.933491	+	0.358600i	$0.116746\pi$
$840$	0	0
$841$	−5.03279	−0.173545
$842$	14.9512	0.515254
$843$	0	0
$844$	17.5455	0.603942
$845$	16.8075	0.578194
$846$	0	0
$847$	4.53330	0.155766
$848$	4.91160	0.168665
$849$	0	0
$850$	7.89051	0.270642
$851$	−34.1847	−1.17184
$852$	0	0
$853$	39.5762	1.35506	0.677532	−	0.735493i	$-0.263050\pi$
$853$	39.5762	1.35506	0.677532	+	0.735493i	$0.263050\pi$
$854$	−1.21958	−0.0417333
$855$	0	0
$856$	5.65598	0.193317
$857$	−12.7250	−0.434679	−0.217340	−	0.976096i	$-0.569738\pi$
$857$	−12.7250	−0.434679	−0.217340	+	0.976096i	$0.569738\pi$
$858$	0	0
$859$	21.1543	0.721777	0.360888	−	0.932609i	$-0.382474\pi$
$859$	21.1543	0.721777	0.360888	+	0.932609i	$0.382474\pi$
$860$	40.4542	1.37948
$861$	0	0
$862$	8.01237	0.272902
$863$	9.33974	0.317928	0.158964	−	0.987284i	$-0.449185\pi$
$863$	9.33974	0.317928	0.158964	+	0.987284i	$0.449185\pi$
$864$	0	0
$865$	−76.5702	−2.60346
$866$	−16.9591	−0.576292
$867$	0	0
$868$	0	0
$869$	−58.0085	−1.96780
$870$	0	0
$871$	−6.07235	−0.205754
$872$	−3.58896	−0.121538
$873$	0	0
$874$	8.60055	0.290918
$875$	−10.6014	−0.358392
$876$	0	0
$877$	22.4353	0.757585	0.378792	−	0.925482i	$-0.376339\pi$
$877$	22.4353	0.757585	0.378792	+	0.925482i	$0.376339\pi$
$878$	12.5441	0.423343
$879$	0	0
$880$	20.9595	0.706543
$881$	−46.4315	−1.56432	−0.782158	−	0.623080i	$-0.785881\pi$
$881$	−46.4315	−1.56432	−0.782158	+	0.623080i	$0.785881\pi$
$882$	0	0
$883$	−36.5396	−1.22965	−0.614827	−	0.788662i	$-0.710774\pi$
$883$	−36.5396	−1.22965	−0.614827	+	0.788662i	$0.710774\pi$
$884$	−5.80488	−0.195239
$885$	0	0
$886$	−12.2456	−0.411400
$887$	−28.6850	−0.963150	−0.481575	−	0.876405i	$-0.659935\pi$
$887$	−28.6850	−0.963150	−0.481575	+	0.876405i	$0.659935\pi$
$888$	0	0
$889$	1.09319	0.0366644
$890$	11.3367	0.380008
$891$	0	0
$892$	14.3677	0.481066
$893$	−3.32865	−0.111389
$894$	0	0
$895$	0.488140	0.0163167
$896$	−8.20514	−0.274114
$897$	0	0
$898$	−6.34235	−0.211647
$899$	0	0
$900$	0	0
$901$	4.67889	0.155877
$902$	2.14514	0.0714254
$903$	0	0
$904$	15.5059	0.515719
$905$	28.6665	0.952906
$906$	0	0
$907$	−22.1896	−0.736794	−0.368397	−	0.929669i	$-0.620093\pi$
$907$	−22.1896	−0.736794	−0.368397	+	0.929669i	$0.620093\pi$
$908$	33.1799	1.10111
$909$	0	0
$910$	−5.67829	−0.188233
$911$	−28.8324	−0.955261	−0.477631	−	0.878561i	$-0.658504\pi$
$911$	−28.8324	−0.955261	−0.477631	+	0.878561i	$0.658504\pi$
$912$	0	0
$913$	58.0620	1.92157
$914$	21.5121	0.711557
$915$	0	0
$916$	25.8510	0.854139
$917$	−4.80396	−0.158641
$918$	0	0
$919$	48.0623	1.58543	0.792715	−	0.609592i	$-0.208667\pi$
$919$	48.0623	1.58543	0.792715	+	0.609592i	$0.208667\pi$
$920$	−29.5793	−0.975201
$921$	0	0
$922$	−4.59786	−0.151423
$923$	22.5101	0.740928
$924$	0	0
$925$	91.0668	2.99426
$926$	−24.3518	−0.800251
$927$	0	0
$928$	28.4312	0.933300
$929$	7.01617	0.230193	0.115097	−	0.993354i	$-0.463282\pi$
$929$	7.01617	0.230193	0.115097	+	0.993354i	$0.463282\pi$
$930$	0	0
$931$	24.3825	0.799105
$932$	−14.9961	−0.491214
$933$	0	0
$934$	−24.7827	−0.810916
$935$	19.9664	0.652972
$936$	0	0
$937$	−12.8807	−0.420793	−0.210396	−	0.977616i	$-0.567475\pi$
$937$	−12.8807	−0.420793	−0.210396	+	0.977616i	$0.567475\pi$
$938$	−1.09845	−0.0358656
$939$	0	0
$940$	4.95195	0.161515
$941$	−50.0392	−1.63123	−0.815616	−	0.578593i	$-0.803602\pi$
$941$	−50.0392	−1.63123	−0.815616	+	0.578593i	$0.803602\pi$
$942$	0	0
$943$	2.48174	0.0808165
$944$	−1.27370	−0.0414555
$945$	0	0
$946$	−20.3090	−0.660304
$947$	51.8679	1.68548	0.842741	−	0.538320i	$-0.180941\pi$
$947$	51.8679	1.68548	0.842741	+	0.538320i	$0.180941\pi$
$948$	0	0
$949$	16.4619	0.534374
$950$	−22.9115	−0.743349
$951$	0	0
$952$	−2.42756	−0.0786777
$953$	18.7352	0.606893	0.303447	−	0.952848i	$-0.401863\pi$
$953$	18.7352	0.606893	0.303447	+	0.952848i	$0.401863\pi$
$954$	0	0
$955$	−78.2385	−2.53174
$956$	−0.900482	−0.0291237
$957$	0	0
$958$	−22.9218	−0.740569
$959$	9.73320	0.314301
$960$	0	0
$961$	0	0
$962$	20.8907	0.673542
$963$	0	0
$964$	23.4338	0.754751
$965$	−20.9965	−0.675902
$966$	0	0
$967$	47.7846	1.53665	0.768324	−	0.640061i	$-0.221091\pi$
$967$	47.7846	1.53665	0.768324	+	0.640061i	$0.221091\pi$
$968$	−14.4313	−0.463839
$969$	0	0
$970$	13.4913	0.433178
$971$	31.1007	0.998068	0.499034	−	0.866583i	$-0.333688\pi$
$971$	31.1007	0.998068	0.499034	+	0.866583i	$0.333688\pi$
$972$	0	0
$973$	−14.2860	−0.457987
$974$	−19.6611	−0.629983
$975$	0	0
$976$	−3.18269	−0.101876
$977$	22.0791	0.706375	0.353187	−	0.935553i	$-0.385098\pi$
$977$	22.0791	0.706375	0.353187	+	0.935553i	$0.385098\pi$
$978$	0	0
$979$	18.2520	0.583337
$980$	−36.2733	−1.15871
$981$	0	0
$982$	−19.2807	−0.615273
$983$	−37.5769	−1.19852	−0.599258	−	0.800556i	$-0.704538\pi$
$983$	−37.5769	−1.19852	−0.599258	+	0.800556i	$0.704538\pi$
$984$	0	0
$985$	−59.0044	−1.88004
$986$	4.41693	0.140664
$987$	0	0
$988$	16.8555	0.536246
$989$	−23.4958	−0.747122
$990$	0	0
$991$	6.77397	0.215182	0.107591	−	0.994195i	$-0.465686\pi$
$991$	6.77397	0.215182	0.107591	+	0.994195i	$0.465686\pi$
$992$	0	0
$993$	0	0
$994$	4.07192	0.129154
$995$	−15.7994	−0.500875
$996$	0	0
$997$	−29.5733	−0.936595	−0.468298	−	0.883571i	$-0.655132\pi$
$997$	−29.5733	−0.936595	−0.468298	+	0.883571i	$0.655132\pi$
$998$	28.4795	0.901502
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.be.1.6		8
3.2	odd	2		961.2.a.j.1.3		8
31.22	odd	30		279.2.y.c.19.1		16
31.24	odd	30		279.2.y.c.235.1		16
31.30	odd	2		8649.2.a.bf.1.6		8
93.2	odd	10		961.2.d.q.531.1		16
93.5	odd	6		961.2.c.i.521.3		16
93.8	odd	10		961.2.d.n.374.4		16
93.11	even	30		961.2.g.k.338.2		16
93.14	odd	30		961.2.g.j.816.2		16
93.17	even	30		961.2.g.k.816.2		16
93.20	odd	30		961.2.g.j.338.2		16
93.23	even	10		961.2.d.o.374.4		16
93.26	even	6		961.2.c.j.521.3		16
93.29	even	10		961.2.d.p.531.1		16
93.35	odd	10		961.2.d.n.388.4		16
93.38	odd	30		961.2.g.l.235.2		16
93.41	odd	30		961.2.g.m.844.1		16
93.44	even	30		961.2.g.t.448.1		16
93.47	odd	10		961.2.d.q.628.1		16
93.50	odd	30		961.2.g.n.547.1		16
93.53	even	30		31.2.g.a.19.2	yes	16
93.56	odd	6		961.2.c.i.439.3		16
93.59	odd	30		961.2.g.m.846.1		16
93.65	even	30		961.2.g.s.846.1		16
93.68	even	6		961.2.c.j.439.3		16
93.71	odd	30		961.2.g.l.732.2		16
93.74	even	30		961.2.g.t.547.1		16
93.77	even	10		961.2.d.p.628.1		16
93.80	odd	30		961.2.g.n.448.1		16
93.83	even	30		961.2.g.s.844.1		16
93.86	even	30		31.2.g.a.18.2	✓	16
93.89	even	10		961.2.d.o.388.4		16
93.92	even	2		961.2.a.i.1.3		8
372.179	odd	30		496.2.bg.c.49.1		16
372.239	odd	30		496.2.bg.c.81.1		16
465.53	odd	60		775.2.ck.a.174.3		32
465.179	even	30		775.2.bl.a.576.1		16
465.239	even	30		775.2.bl.a.701.1		16
465.272	odd	60		775.2.ck.a.49.3		32
465.332	odd	60		775.2.ck.a.174.2		32
465.458	odd	60		775.2.ck.a.49.2		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.2.g.a.18.2	✓	16	93.86	even	30
31.2.g.a.19.2	yes	16	93.53	even	30
279.2.y.c.19.1		16	31.22	odd	30
279.2.y.c.235.1		16	31.24	odd	30
496.2.bg.c.49.1		16	372.179	odd	30
496.2.bg.c.81.1		16	372.239	odd	30
775.2.bl.a.576.1		16	465.179	even	30
775.2.bl.a.701.1		16	465.239	even	30
775.2.ck.a.49.2		32	465.458	odd	60
775.2.ck.a.49.3		32	465.272	odd	60
775.2.ck.a.174.2		32	465.332	odd	60
775.2.ck.a.174.3		32	465.53	odd	60
961.2.a.i.1.3		8	93.92	even	2
961.2.a.j.1.3		8	3.2	odd	2
961.2.c.i.439.3		16	93.56	odd	6
961.2.c.i.521.3		16	93.5	odd	6
961.2.c.j.439.3		16	93.68	even	6
961.2.c.j.521.3		16	93.26	even	6
961.2.d.n.374.4		16	93.8	odd	10
961.2.d.n.388.4		16	93.35	odd	10
961.2.d.o.374.4		16	93.23	even	10
961.2.d.o.388.4		16	93.89	even	10
961.2.d.p.531.1		16	93.29	even	10
961.2.d.p.628.1		16	93.77	even	10
961.2.d.q.531.1		16	93.2	odd	10
961.2.d.q.628.1		16	93.47	odd	10
961.2.g.j.338.2		16	93.20	odd	30
961.2.g.j.816.2		16	93.14	odd	30
961.2.g.k.338.2		16	93.11	even	30
961.2.g.k.816.2		16	93.17	even	30
961.2.g.l.235.2		16	93.38	odd	30
961.2.g.l.732.2		16	93.71	odd	30
961.2.g.m.844.1		16	93.41	odd	30
961.2.g.m.846.1		16	93.59	odd	30
961.2.g.n.448.1		16	93.80	odd	30
961.2.g.n.547.1		16	93.50	odd	30
961.2.g.s.844.1		16	93.83	even	30
961.2.g.s.846.1		16	93.65	even	30
961.2.g.t.448.1		16	93.44	even	30
961.2.g.t.547.1		16	93.74	even	30
8649.2.a.be.1.6		8	1.1	even	1	trivial
8649.2.a.bf.1.6		8	31.30	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 $\F_{q}$

Introduction

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Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	8649.2.a.be.1.6

Level	$8649$
Weight	$2$
Character	8649.1
Self dual	yes
Analytic conductor	$69.063$
Analytic rank	$1$
Dimension	$8$
CM	no
Inner twists	$1$