LMFDB - Embedded newform 8649.2.a.bo.1.8

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:	$0$
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} - 19x^{10} + 129x^{8} - 379x^{6} + 473x^{4} - 210x^{2} + 9 $ x^12 - 19x^10 + 129x^8 - 379x^6 + 473x^4 - 210*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 279)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.937929$ 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)

$f(q)$	$=$	$q+0.937929 q^{2} -1.12029 q^{4} +3.32313 q^{5} -3.23052 q^{7} -2.92661 q^{8} +O(q^{10})$	\(q+0.937929 q^{2} -1.12029 q^{4} +3.32313 q^{5} -3.23052 q^{7} -2.92661 q^{8} +3.11686 q^{10} -0.334610 q^{11} +2.81267 q^{13} -3.03000 q^{14} -0.504375 q^{16} +0.999838 q^{17} +2.23052 q^{19} -3.72286 q^{20} -0.313840 q^{22} -5.04233 q^{23} +6.04318 q^{25} +2.63808 q^{26} +3.61911 q^{28} -5.55488 q^{29} +5.38015 q^{32} +0.937777 q^{34} -10.7354 q^{35} -0.688947 q^{37} +2.09207 q^{38} -9.72550 q^{40} -9.91069 q^{41} +12.5341 q^{43} +0.374860 q^{44} -4.72934 q^{46} +9.95339 q^{47} +3.43625 q^{49} +5.66808 q^{50} -3.15100 q^{52} +13.6157 q^{53} -1.11195 q^{55} +9.45447 q^{56} -5.21009 q^{58} +3.14518 q^{59} -10.8984 q^{61} +6.05495 q^{64} +9.34685 q^{65} -4.24648 q^{67} -1.12011 q^{68} -10.0691 q^{70} +4.80108 q^{71} -7.96780 q^{73} -0.646183 q^{74} -2.49883 q^{76} +1.08096 q^{77} +2.26511 q^{79} -1.67610 q^{80} -9.29553 q^{82} +16.3640 q^{83} +3.32259 q^{85} +11.7561 q^{86} +0.979273 q^{88} +12.8544 q^{89} -9.08637 q^{91} +5.64886 q^{92} +9.33557 q^{94} +7.41230 q^{95} +13.6661 q^{97} +3.22296 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 14 q^{4} - 4 q^{7}+O(q^{10}) $ 12 * q + 14 * q^4 - 4 * q^7	$ 12 q + 14 q^{4} - 4 q^{7} - 16 q^{10} + 10 q^{13} + 26 q^{16} - 8 q^{19} - 20 q^{22} + 14 q^{25} - 24 q^{28} + 56 q^{34} + 14 q^{37} - 82 q^{40} + 54 q^{43} + 18 q^{46} + 12 q^{49} - 2 q^{52} + 16 q^{55} + 56 q^{58} + 22 q^{61} + 42 q^{64} - 38 q^{67} + 42 q^{70} + 16 q^{73} + 10 q^{76} + 94 q^{79} - 26 q^{82} - 8 q^{85} - 60 q^{88} + 8 q^{91} + 86 q^{94} + 68 q^{97}+O(q^{100}) $ 12 * q + 14 * q^4 - 4 * q^7 - 16 * q^10 + 10 * q^13 + 26 * q^16 - 8 * q^19 - 20 * q^22 + 14 * q^25 - 24 * q^28 + 56 * q^34 + 14 * q^37 - 82 * q^40 + 54 * q^43 + 18 * q^46 + 12 * q^49 - 2 * q^52 + 16 * q^55 + 56 * q^58 + 22 * q^61 + 42 * q^64 - 38 * q^67 + 42 * q^70 + 16 * q^73 + 10 * q^76 + 94 * q^79 - 26 * q^82 - 8 * q^85 - 60 * q^88 + 8 * q^91 + 86 * q^94 + 68 * q^97

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 $

$2$	0.937929	0.663216	0.331608	−	0.943417i	$-0.392409\pi$
$2$	0.937929	0.663216	0.331608	+	0.943417i	$0.392409\pi$
$3$	0	0
$3$	0	0
$4$	−1.12029	−0.560144
$5$	3.32313	1.48615	0.743074	−	0.669209i	$-0.233367\pi$
$5$	3.32313	1.48615	0.743074	+	0.669209i	$0.233367\pi$
$6$	0	0
$7$	−3.23052	−1.22102	−0.610511	−	0.792008i	$-0.709036\pi$
$7$	−3.23052	−1.22102	−0.610511	+	0.792008i	$0.709036\pi$
$8$	−2.92661	−1.03471
$9$	0	0
$10$	3.11686	0.985637
$11$	−0.334610	−0.100889	−0.0504444	−	0.998727i	$-0.516064\pi$
$11$	−0.334610	−0.100889	−0.0504444	+	0.998727i	$0.516064\pi$
$12$	0	0
$13$	2.81267	0.780093	0.390047	−	0.920795i	$-0.372459\pi$
$13$	2.81267	0.780093	0.390047	+	0.920795i	$0.372459\pi$
$14$	−3.03000	−0.809801
$15$	0	0
$16$	−0.504375	−0.126094
$17$	0.999838	0.242496	0.121248	−	0.992622i	$-0.461310\pi$
$17$	0.999838	0.242496	0.121248	+	0.992622i	$0.461310\pi$
$18$	0	0
$19$	2.23052	0.511716	0.255858	−	0.966714i	$-0.417642\pi$
$19$	2.23052	0.511716	0.255858	+	0.966714i	$0.417642\pi$
$20$	−3.72286	−0.832458
$21$	0	0
$22$	−0.313840	−0.0669110
$23$	−5.04233	−1.05140	−0.525699	−	0.850671i	$-0.676196\pi$
$23$	−5.04233	−1.05140	−0.525699	+	0.850671i	$0.676196\pi$
$24$	0	0
$25$	6.04318	1.20864
$26$	2.63808	0.517370
$27$	0	0
$28$	3.61911	0.683948
$29$	−5.55488	−1.03152	−0.515758	−	0.856734i	$-0.672490\pi$
$29$	−5.55488	−1.03152	−0.515758	+	0.856734i	$0.672490\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	5.38015	0.951085
$33$	0	0
$34$	0.937777	0.160827
$35$	−10.7354	−1.81462
$36$	0	0
$37$	−0.688947	−0.113262	−0.0566311	−	0.998395i	$-0.518036\pi$
$37$	−0.688947	−0.113262	−0.0566311	+	0.998395i	$0.518036\pi$
$38$	2.09207	0.339378
$39$	0	0
$40$	−9.72550	−1.53774
$41$	−9.91069	−1.54779	−0.773895	−	0.633314i	$-0.781694\pi$
$41$	−9.91069	−1.54779	−0.773895	+	0.633314i	$0.781694\pi$
$42$	0	0
$43$	12.5341	1.91143	0.955714	−	0.294298i	$-0.0950857\pi$
$43$	12.5341	1.91143	0.955714	+	0.294298i	$0.0950857\pi$
$44$	0.374860	0.0565123
$45$	0	0
$46$	−4.72934	−0.697304
$47$	9.95339	1.45185	0.725925	−	0.687774i	$-0.241412\pi$
$47$	9.95339	1.45185	0.725925	+	0.687774i	$0.241412\pi$
$48$	0	0
$49$	3.43625	0.490893
$50$	5.66808	0.801587
$51$	0	0
$52$	−3.15100	−0.436965
$53$	13.6157	1.87026	0.935132	−	0.354300i	$-0.115281\pi$
$53$	13.6157	1.87026	0.935132	+	0.354300i	$0.115281\pi$
$54$	0	0
$55$	−1.11195	−0.149936
$56$	9.45447	1.26341
$57$	0	0
$58$	−5.21009	−0.684118
$59$	3.14518	0.409468	0.204734	−	0.978818i	$-0.434367\pi$
$59$	3.14518	0.409468	0.204734	+	0.978818i	$0.434367\pi$
$60$	0	0
$61$	−10.8984	−1.39539	−0.697696	−	0.716394i	$-0.745792\pi$
$61$	−10.8984	−1.39539	−0.697696	+	0.716394i	$0.745792\pi$
$62$	0	0
$63$	0	0
$64$	6.05495	0.756869
$65$	9.34685	1.15933
$66$	0	0
$67$	−4.24648	−0.518790	−0.259395	−	0.965771i	$-0.583523\pi$
$67$	−4.24648	−0.518790	−0.259395	+	0.965771i	$0.583523\pi$
$68$	−1.12011	−0.135833
$69$	0	0
$70$	−10.0691	−1.20348
$71$	4.80108	0.569784	0.284892	−	0.958560i	$-0.408042\pi$
$71$	4.80108	0.569784	0.284892	+	0.958560i	$0.408042\pi$
$72$	0	0
$73$	−7.96780	−0.932561	−0.466280	−	0.884637i	$-0.654406\pi$
$73$	−7.96780	−0.932561	−0.466280	+	0.884637i	$0.654406\pi$
$74$	−0.646183	−0.0751173
$75$	0	0
$76$	−2.49883	−0.286635
$77$	1.08096	0.123187
$78$	0	0
$79$	2.26511	0.254844	0.127422	−	0.991849i	$-0.459330\pi$
$79$	2.26511	0.254844	0.127422	+	0.991849i	$0.459330\pi$
$80$	−1.67610	−0.187394
$81$	0	0
$82$	−9.29553	−1.02652
$83$	16.3640	1.79618	0.898091	−	0.439810i	$-0.144954\pi$
$83$	16.3640	1.79618	0.898091	+	0.439810i	$0.144954\pi$
$84$	0	0
$85$	3.32259	0.360385
$86$	11.7561	1.26769
$87$	0	0
$88$	0.979273	0.104391
$89$	12.8544	1.36256	0.681280	−	0.732023i	$-0.261424\pi$
$89$	12.8544	1.36256	0.681280	+	0.732023i	$0.261424\pi$
$90$	0	0
$91$	−9.08637	−0.952510
$92$	5.64886	0.588934
$93$	0	0
$94$	9.33557	0.962891
$95$	7.41230	0.760486
$96$	0	0
$97$	13.6661	1.38758	0.693792	−	0.720175i	$-0.255939\pi$
$97$	13.6661	1.38758	0.693792	+	0.720175i	$0.255939\pi$
$98$	3.22296	0.325568
$99$	0	0
$100$	−6.77011	−0.677011
$101$	−17.5233	−1.74364	−0.871818	−	0.489829i	$-0.837059\pi$
$101$	−17.5233	−1.74364	−0.871818	+	0.489829i	$0.837059\pi$
$102$	0	0
$103$	6.66916	0.657132	0.328566	−	0.944481i	$-0.393435\pi$
$103$	6.66916	0.657132	0.328566	+	0.944481i	$0.393435\pi$
$104$	−8.23158	−0.807172
$105$	0	0
$106$	12.7706	1.24039
$107$	−4.80853	−0.464858	−0.232429	−	0.972613i	$-0.574667\pi$
$107$	−4.80853	−0.464858	−0.232429	+	0.972613i	$0.574667\pi$
$108$	0	0
$109$	0.323585	0.0309939	0.0154969	−	0.999880i	$-0.495067\pi$
$109$	0.323585	0.0309939	0.0154969	+	0.999880i	$0.495067\pi$
$110$	−1.04293	−0.0994397
$111$	0	0
$112$	1.62939	0.153963
$113$	−10.2403	−0.963329	−0.481664	−	0.876356i	$-0.659968\pi$
$113$	−10.2403	−0.963329	−0.481664	+	0.876356i	$0.659968\pi$
$114$	0	0
$115$	−16.7563	−1.56253
$116$	6.22307	0.577798
$117$	0	0
$118$	2.94996	0.271565
$119$	−3.22999	−0.296093
$120$	0	0
$121$	−10.8880	−0.989821
$122$	−10.2219	−0.925447
$123$	0	0
$124$	0	0
$125$	3.46663	0.310065
$126$	0	0
$127$	1.84316	0.163554	0.0817769	−	0.996651i	$-0.473941\pi$
$127$	1.84316	0.163554	0.0817769	+	0.996651i	$0.473941\pi$
$128$	−5.08119	−0.449118
$129$	0	0
$130$	8.76668	0.768889
$131$	9.89286	0.864343	0.432172	−	0.901791i	$-0.357747\pi$
$131$	9.89286	0.864343	0.432172	+	0.901791i	$0.357747\pi$
$132$	0	0
$133$	−7.20573	−0.624816
$134$	−3.98290	−0.344070
$135$	0	0
$136$	−2.92614	−0.250914
$137$	16.7143	1.42800	0.713999	−	0.700146i	$-0.246882\pi$
$137$	16.7143	1.42800	0.713999	+	0.700146i	$0.246882\pi$
$138$	0	0
$139$	15.9838	1.35573	0.677864	−	0.735187i	$-0.262906\pi$
$139$	15.9838	1.35573	0.677864	+	0.735187i	$0.262906\pi$
$140$	12.0268	1.01645
$141$	0	0
$142$	4.50308	0.377890
$143$	−0.941146	−0.0787026
$144$	0	0
$145$	−18.4596	−1.53299
$146$	−7.47323	−0.618489
$147$	0	0
$148$	0.771820	0.0634432
$149$	−9.35389	−0.766301	−0.383150	−	0.923686i	$-0.625161\pi$
$149$	−9.35389	−0.766301	−0.383150	+	0.923686i	$0.625161\pi$
$150$	0	0
$151$	22.5609	1.83598	0.917990	−	0.396604i	$-0.129811\pi$
$151$	22.5609	1.83598	0.917990	+	0.396604i	$0.129811\pi$
$152$	−6.52786	−0.529479
$153$	0	0
$154$	1.01387	0.0816998
$155$	0	0
$156$	0	0
$157$	16.4691	1.31438	0.657190	−	0.753725i	$-0.271745\pi$
$157$	16.4691	1.31438	0.657190	+	0.753725i	$0.271745\pi$
$158$	2.12451	0.169017
$159$	0	0
$160$	17.8789	1.41345
$161$	16.2893	1.28378
$162$	0	0
$163$	−17.2308	−1.34962	−0.674811	−	0.737991i	$-0.735775\pi$
$163$	−17.2308	−1.34962	−0.674811	+	0.737991i	$0.735775\pi$
$164$	11.1028	0.866986
$165$	0	0
$166$	15.3483	1.19126
$167$	12.7718	0.988310	0.494155	−	0.869374i	$-0.335478\pi$
$167$	12.7718	0.988310	0.494155	+	0.869374i	$0.335478\pi$
$168$	0	0
$169$	−5.08891	−0.391455
$170$	3.11635	0.239013
$171$	0	0
$172$	−14.0418	−1.07068
$173$	2.43050	0.184787	0.0923937	−	0.995723i	$-0.470548\pi$
$173$	2.43050	0.184787	0.0923937	+	0.995723i	$0.470548\pi$
$174$	0	0
$175$	−19.5226	−1.47577
$176$	0.168769	0.0127214
$177$	0	0
$178$	12.0565	0.903671
$179$	2.14995	0.160694	0.0803472	−	0.996767i	$-0.474397\pi$
$179$	2.14995	0.160694	0.0803472	+	0.996767i	$0.474397\pi$
$180$	0	0
$181$	3.62790	0.269660	0.134830	−	0.990869i	$-0.456951\pi$
$181$	3.62790	0.269660	0.134830	+	0.990869i	$0.456951\pi$
$182$	−8.52237	−0.631720
$183$	0	0
$184$	14.7569	1.08789
$185$	−2.28946	−0.168324
$186$	0	0
$187$	−0.334556	−0.0244651
$188$	−11.1507	−0.813246
$189$	0	0
$190$	6.95221	0.504366
$191$	−8.66318	−0.626846	−0.313423	−	0.949614i	$-0.601476\pi$
$191$	−8.66318	−0.626846	−0.313423	+	0.949614i	$0.601476\pi$
$192$	0	0
$193$	16.3715	1.17845	0.589224	−	0.807970i	$-0.299434\pi$
$193$	16.3715	1.17845	0.589224	+	0.807970i	$0.299434\pi$
$194$	12.8179	0.920268
$195$	0	0
$196$	−3.84959	−0.274971
$197$	3.61105	0.257277	0.128639	−	0.991692i	$-0.458939\pi$
$197$	3.61105	0.257277	0.128639	+	0.991692i	$0.458939\pi$
$198$	0	0
$199$	2.70200	0.191540	0.0957698	−	0.995404i	$-0.469469\pi$
$199$	2.70200	0.191540	0.0957698	+	0.995404i	$0.469469\pi$
$200$	−17.6860	−1.25059
$201$	0	0
$202$	−16.4356	−1.15641
$203$	17.9452	1.25950
$204$	0	0
$205$	−32.9345	−2.30025
$206$	6.25520	0.435820
$207$	0	0
$208$	−1.41864	−0.0983648
$209$	−0.746354	−0.0516264
$210$	0	0
$211$	−7.21680	−0.496825	−0.248412	−	0.968654i	$-0.579909\pi$
$211$	−7.21680	−0.496825	−0.248412	+	0.968654i	$0.579909\pi$
$212$	−15.2535	−1.04762
$213$	0	0
$214$	−4.51006	−0.308301
$215$	41.6523	2.84066
$216$	0	0
$217$	0	0
$218$	0.303500	0.0205556
$219$	0	0
$220$	1.24571	0.0839856
$221$	2.81221	0.189170
$222$	0	0
$223$	11.8817	0.795661	0.397830	−	0.917459i	$-0.369763\pi$
$223$	11.8817	0.795661	0.397830	+	0.917459i	$0.369763\pi$
$224$	−17.3807	−1.16130
$225$	0	0
$226$	−9.60470	−0.638895
$227$	4.85328	0.322124	0.161062	−	0.986944i	$-0.448508\pi$
$227$	4.85328	0.322124	0.161062	+	0.986944i	$0.448508\pi$
$228$	0	0
$229$	13.0135	0.859956	0.429978	−	0.902839i	$-0.358521\pi$
$229$	13.0135	0.859956	0.429978	+	0.902839i	$0.358521\pi$
$230$	−15.7162	−1.03630
$231$	0	0
$232$	16.2570	1.06732
$233$	4.53957	0.297397	0.148698	−	0.988883i	$-0.452492\pi$
$233$	4.53957	0.297397	0.148698	+	0.988883i	$0.452492\pi$
$234$	0	0
$235$	33.0764	2.15767
$236$	−3.52351	−0.229361
$237$	0	0
$238$	−3.02951	−0.196374
$239$	3.58117	0.231647	0.115823	−	0.993270i	$-0.463049\pi$
$239$	3.58117	0.231647	0.115823	+	0.993270i	$0.463049\pi$
$240$	0	0
$241$	7.09725	0.457174	0.228587	−	0.973523i	$-0.426589\pi$
$241$	7.09725	0.457174	0.228587	+	0.973523i	$0.426589\pi$
$242$	−10.2122	−0.656465
$243$	0	0
$244$	12.2093	0.781622
$245$	11.4191	0.729539
$246$	0	0
$247$	6.27370	0.399186
$248$	0	0
$249$	0	0
$250$	3.25146	0.205640
$251$	14.4295	0.910780	0.455390	−	0.890292i	$-0.349500\pi$
$251$	14.4295	0.910780	0.455390	+	0.890292i	$0.349500\pi$
$252$	0	0
$253$	1.68721	0.106074
$254$	1.72875	0.108471
$255$	0	0
$256$	−16.8757	−1.05473
$257$	11.1555	0.695859	0.347929	−	0.937521i	$-0.386885\pi$
$257$	11.1555	0.695859	0.347929	+	0.937521i	$0.386885\pi$
$258$	0	0
$259$	2.22566	0.138295
$260$	−10.4712	−0.649395
$261$	0	0
$262$	9.27880	0.573246
$263$	−4.02975	−0.248485	−0.124243	−	0.992252i	$-0.539650\pi$
$263$	−4.02975	−0.248485	−0.124243	+	0.992252i	$0.539650\pi$
$264$	0	0
$265$	45.2468	2.77949
$266$	−6.75846	−0.414388
$267$	0	0
$268$	4.75729	0.290597
$269$	−20.4854	−1.24902	−0.624510	−	0.781017i	$-0.714701\pi$
$269$	−20.4854	−1.24902	−0.624510	+	0.781017i	$0.714701\pi$
$270$	0	0
$271$	7.68981	0.467123	0.233561	−	0.972342i	$-0.424962\pi$
$271$	7.68981	0.467123	0.233561	+	0.972342i	$0.424962\pi$
$272$	−0.504293	−0.0305773
$273$	0	0
$274$	15.6768	0.947072
$275$	−2.02211	−0.121938
$276$	0	0
$277$	−3.29137	−0.197759	−0.0988797	−	0.995099i	$-0.531526\pi$
$277$	−3.29137	−0.197759	−0.0988797	+	0.995099i	$0.531526\pi$
$278$	14.9917	0.899141
$279$	0	0
$280$	31.4184	1.87761
$281$	26.2518	1.56605	0.783024	−	0.621991i	$-0.213676\pi$
$281$	26.2518	1.56605	0.783024	+	0.621991i	$0.213676\pi$
$282$	0	0
$283$	18.6328	1.10761	0.553803	−	0.832648i	$-0.313176\pi$
$283$	18.6328	1.10761	0.553803	+	0.832648i	$0.313176\pi$
$284$	−5.37860	−0.319161
$285$	0	0
$286$	−0.882728	−0.0521968
$287$	32.0167	1.88988
$288$	0	0
$289$	−16.0003	−0.941196
$290$	−17.3138	−1.01670
$291$	0	0
$292$	8.92624	0.522369
$293$	22.8158	1.33291	0.666457	−	0.745544i	$-0.267810\pi$
$293$	22.8158	1.33291	0.666457	+	0.745544i	$0.267810\pi$
$294$	0	0
$295$	10.4518	0.608530
$296$	2.01628	0.117194
$297$	0	0
$298$	−8.77329	−0.508223
$299$	−14.1824	−0.820188
$300$	0	0
$301$	−40.4915	−2.33389
$302$	21.1605	1.21765
$303$	0	0
$304$	−1.12502	−0.0645242
$305$	−36.2167	−2.07376
$306$	0	0
$307$	−7.73615	−0.441525	−0.220763	−	0.975328i	$-0.570855\pi$
$307$	−7.73615	−0.441525	−0.220763	+	0.975328i	$0.570855\pi$
$308$	−1.21099	−0.0690027
$309$	0	0
$310$	0	0
$311$	−11.3500	−0.643601	−0.321800	−	0.946808i	$-0.604288\pi$
$311$	−11.3500	−0.643601	−0.321800	+	0.946808i	$0.604288\pi$
$312$	0	0
$313$	−21.6477	−1.22360	−0.611801	−	0.791011i	$-0.709555\pi$
$313$	−21.6477	−1.22360	−0.611801	+	0.791011i	$0.709555\pi$
$314$	15.4469	0.871718
$315$	0	0
$316$	−2.53757	−0.142750
$317$	15.6499	0.878987	0.439493	−	0.898246i	$-0.355158\pi$
$317$	15.6499	0.878987	0.439493	+	0.898246i	$0.355158\pi$
$318$	0	0
$319$	1.85872	0.104068
$320$	20.1214	1.12482
$321$	0	0
$322$	15.2782	0.851423
$323$	2.23016	0.124089
$324$	0	0
$325$	16.9975	0.942849
$326$	−16.1613	−0.895091
$327$	0	0
$328$	29.0047	1.60152
$329$	−32.1546	−1.77274
$330$	0	0
$331$	−19.9903	−1.09877	−0.549384	−	0.835570i	$-0.685138\pi$
$331$	−19.9903	−1.09877	−0.549384	+	0.835570i	$0.685138\pi$
$332$	−18.3324	−1.00612
$333$	0	0
$334$	11.9790	0.655463
$335$	−14.1116	−0.770999
$336$	0	0
$337$	−2.54853	−0.138827	−0.0694136	−	0.997588i	$-0.522113\pi$
$337$	−2.54853	−0.138827	−0.0694136	+	0.997588i	$0.522113\pi$
$338$	−4.77304	−0.259619
$339$	0	0
$340$	−3.72226	−0.201868
$341$	0	0
$342$	0	0
$343$	11.5128	0.621631
$344$	−36.6823	−1.97778
$345$	0	0
$346$	2.27963	0.122554
$347$	26.3242	1.41316	0.706578	−	0.707635i	$-0.250238\pi$
$347$	26.3242	1.41316	0.706578	+	0.707635i	$0.250238\pi$
$348$	0	0
$349$	13.9172	0.744971	0.372486	−	0.928038i	$-0.378506\pi$
$349$	13.9172	0.744971	0.372486	+	0.928038i	$0.378506\pi$
$350$	−18.3108	−0.978755
$351$	0	0
$352$	−1.80025	−0.0959538
$353$	−2.74298	−0.145994	−0.0729970	−	0.997332i	$-0.523256\pi$
$353$	−2.74298	−0.145994	−0.0729970	+	0.997332i	$0.523256\pi$
$354$	0	0
$355$	15.9546	0.846783
$356$	−14.4006	−0.763230
$357$	0	0
$358$	2.01650	0.106575
$359$	8.93111	0.471366	0.235683	−	0.971830i	$-0.424267\pi$
$359$	8.93111	0.471366	0.235683	+	0.971830i	$0.424267\pi$
$360$	0	0
$361$	−14.0248	−0.738147
$362$	3.40271	0.178843
$363$	0	0
$364$	10.1794	0.533543
$365$	−26.4780	−1.38592
$366$	0	0
$367$	16.5480	0.863800	0.431900	−	0.901922i	$-0.357843\pi$
$367$	16.5480	0.863800	0.431900	+	0.901922i	$0.357843\pi$
$368$	2.54322	0.132575
$369$	0	0
$370$	−2.14735	−0.111635
$371$	−43.9859	−2.28363
$372$	0	0
$373$	17.8477	0.924121	0.462060	−	0.886849i	$-0.347110\pi$
$373$	17.8477	0.924121	0.462060	+	0.886849i	$0.347110\pi$
$374$	−0.313790	−0.0162257
$375$	0	0
$376$	−29.1297	−1.50225
$377$	−15.6240	−0.804678
$378$	0	0
$379$	−15.3086	−0.786350	−0.393175	−	0.919464i	$-0.628623\pi$
$379$	−15.3086	−0.786350	−0.393175	+	0.919464i	$0.628623\pi$
$380$	−8.30392	−0.425982
$381$	0	0
$382$	−8.12545	−0.415734
$383$	4.68977	0.239636	0.119818	−	0.992796i	$-0.461769\pi$
$383$	4.68977	0.239636	0.119818	+	0.992796i	$0.461769\pi$
$384$	0	0
$385$	3.59218	0.183075
$386$	15.3553	0.781566
$387$	0	0
$388$	−15.3100	−0.777248
$389$	−30.3909	−1.54088	−0.770441	−	0.637511i	$-0.779964\pi$
$389$	−30.3909	−1.54088	−0.770441	+	0.637511i	$0.779964\pi$
$390$	0	0
$391$	−5.04151	−0.254960
$392$	−10.0566	−0.507933
$393$	0	0
$394$	3.38691	0.170630
$395$	7.52724	0.378737
$396$	0	0
$397$	24.6017	1.23472	0.617361	−	0.786680i	$-0.288202\pi$
$397$	24.6017	1.23472	0.617361	+	0.786680i	$0.288202\pi$
$398$	2.53428	0.127032
$399$	0	0
$400$	−3.04803	−0.152401
$401$	17.1414	0.856002	0.428001	−	0.903778i	$-0.359218\pi$
$401$	17.1414	0.856002	0.428001	+	0.903778i	$0.359218\pi$
$402$	0	0
$403$	0	0
$404$	19.6312	0.976689
$405$	0	0
$406$	16.8313	0.835322
$407$	0.230529	0.0114269
$408$	0	0
$409$	28.8218	1.42515	0.712573	−	0.701598i	$-0.247530\pi$
$409$	28.8218	1.42515	0.712573	+	0.701598i	$0.247530\pi$
$410$	−30.8902	−1.52556
$411$	0	0
$412$	−7.47138	−0.368089
$413$	−10.1606	−0.499969
$414$	0	0
$415$	54.3796	2.66939
$416$	15.1326	0.741935
$417$	0	0
$418$	−0.700027	−0.0342394
$419$	−13.1340	−0.641638	−0.320819	−	0.947141i	$-0.603958\pi$
$419$	−13.1340	−0.641638	−0.320819	+	0.947141i	$0.603958\pi$
$420$	0	0
$421$	−0.884246	−0.0430955	−0.0215478	−	0.999768i	$-0.506859\pi$
$421$	−0.884246	−0.0430955	−0.0215478	+	0.999768i	$0.506859\pi$
$422$	−6.76885	−0.329502
$423$	0	0
$424$	−39.8479	−1.93519
$425$	6.04220	0.293090
$426$	0	0
$427$	35.2074	1.70380
$428$	5.38694	0.260388
$429$	0	0
$430$	39.0669	1.88397
$431$	24.6917	1.18936	0.594679	−	0.803963i	$-0.297279\pi$
$431$	24.6917	1.18936	0.594679	+	0.803963i	$0.297279\pi$
$432$	0	0
$433$	−22.5897	−1.08559	−0.542796	−	0.839864i	$-0.682635\pi$
$433$	−22.5897	−1.08559	−0.542796	+	0.839864i	$0.682635\pi$
$434$	0	0
$435$	0	0
$436$	−0.362509	−0.0173610
$437$	−11.2470	−0.538017
$438$	0	0
$439$	−21.6637	−1.03395	−0.516977	−	0.855999i	$-0.672943\pi$
$439$	−21.6637	−1.03395	−0.516977	+	0.855999i	$0.672943\pi$
$440$	3.25425	0.155140
$441$	0	0
$442$	2.63765	0.125460
$443$	3.45850	0.164318	0.0821592	−	0.996619i	$-0.473818\pi$
$443$	3.45850	0.164318	0.0821592	+	0.996619i	$0.473818\pi$
$444$	0	0
$445$	42.7167	2.02497
$446$	11.1442	0.527695
$447$	0	0
$448$	−19.5606	−0.924153
$449$	−22.7874	−1.07541	−0.537703	−	0.843134i	$-0.680708\pi$
$449$	−22.7874	−1.07541	−0.537703	+	0.843134i	$0.680708\pi$
$450$	0	0
$451$	3.31622	0.156155
$452$	11.4721	0.539603
$453$	0	0
$454$	4.55204	0.213638
$455$	−30.1952	−1.41557
$456$	0	0
$457$	0.754779	0.0353070	0.0176535	−	0.999844i	$-0.494380\pi$
$457$	0.754779	0.0353070	0.0176535	+	0.999844i	$0.494380\pi$
$458$	12.2057	0.570336
$459$	0	0
$460$	18.7719	0.875244
$461$	2.35672	0.109763	0.0548817	−	0.998493i	$-0.482522\pi$
$461$	2.35672	0.109763	0.0548817	+	0.998493i	$0.482522\pi$
$462$	0	0
$463$	30.1479	1.40109	0.700547	−	0.713606i	$-0.252939\pi$
$463$	30.1479	1.40109	0.700547	+	0.713606i	$0.252939\pi$
$464$	2.80174	0.130068
$465$	0	0
$466$	4.25779	0.197238
$467$	−27.5707	−1.27582	−0.637910	−	0.770111i	$-0.720201\pi$
$467$	−27.5707	−1.27582	−0.637910	+	0.770111i	$0.720201\pi$
$468$	0	0
$469$	13.7183	0.633454
$470$	31.0233	1.43100
$471$	0	0
$472$	−9.20472	−0.423681
$473$	−4.19403	−0.192841
$474$	0	0
$475$	13.4794	0.618479
$476$	3.61853	0.165855
$477$	0	0
$478$	3.35889	0.153632
$479$	15.2674	0.697587	0.348793	−	0.937200i	$-0.386592\pi$
$479$	15.2674	0.697587	0.348793	+	0.937200i	$0.386592\pi$
$480$	0	0
$481$	−1.93778	−0.0883550
$482$	6.65672	0.303205
$483$	0	0
$484$	12.1977	0.554443
$485$	45.4143	2.06216
$486$	0	0
$487$	−6.84600	−0.310222	−0.155111	−	0.987897i	$-0.549574\pi$
$487$	−6.84600	−0.310222	−0.155111	+	0.987897i	$0.549574\pi$
$488$	31.8953	1.44383
$489$	0	0
$490$	10.7103	0.483842
$491$	30.1317	1.35982	0.679912	−	0.733294i	$-0.262018\pi$
$491$	30.1317	1.35982	0.679912	+	0.733294i	$0.262018\pi$
$492$	0	0
$493$	−5.55398	−0.250139
$494$	5.88429	0.264747
$495$	0	0
$496$	0	0
$497$	−15.5100	−0.695718
$498$	0	0
$499$	37.1887	1.66480	0.832398	−	0.554178i	$-0.186967\pi$
$499$	37.1887	1.66480	0.832398	+	0.554178i	$0.186967\pi$
$500$	−3.88363	−0.173681
$501$	0	0
$502$	13.5338	0.604044
$503$	−15.4086	−0.687035	−0.343517	−	0.939146i	$-0.611618\pi$
$503$	−15.4086	−0.687035	−0.343517	+	0.939146i	$0.611618\pi$
$504$	0	0
$505$	−58.2323	−2.59130
$506$	1.58249	0.0703501
$507$	0	0
$508$	−2.06487	−0.0916137
$509$	−16.3722	−0.725685	−0.362843	−	0.931850i	$-0.618194\pi$
$509$	−16.3722	−0.725685	−0.362843	+	0.931850i	$0.618194\pi$
$510$	0	0
$511$	25.7401	1.13868
$512$	−5.66583	−0.250397
$513$	0	0
$514$	10.4630	0.461505
$515$	22.1625	0.976595
$516$	0	0
$517$	−3.33050	−0.146475
$518$	2.08751	0.0917198
$519$	0	0
$520$	−27.3546	−1.19958
$521$	−27.8806	−1.22147	−0.610735	−	0.791835i	$-0.709126\pi$
$521$	−27.8806	−1.22147	−0.610735	+	0.791835i	$0.709126\pi$
$522$	0	0
$523$	−14.3134	−0.625880	−0.312940	−	0.949773i	$-0.601314\pi$
$523$	−14.3134	−0.625880	−0.312940	+	0.949773i	$0.601314\pi$
$524$	−11.0829	−0.484157
$525$	0	0
$526$	−3.77962	−0.164799
$527$	0	0
$528$	0	0
$529$	2.42504	0.105437
$530$	42.4383	1.84340
$531$	0	0
$532$	8.07250	0.349987
$533$	−27.8755	−1.20742
$534$	0	0
$535$	−15.9794	−0.690848
$536$	12.4278	0.536799
$537$	0	0
$538$	−19.2139	−0.828370
$539$	−1.14980	−0.0495255
$540$	0	0
$541$	−10.6556	−0.458121	−0.229061	−	0.973412i	$-0.573565\pi$
$541$	−10.6556	−0.458121	−0.229061	+	0.973412i	$0.573565\pi$
$542$	7.21249	0.309803
$543$	0	0
$544$	5.37928	0.230635
$545$	1.07532	0.0460615
$546$	0	0
$547$	10.3669	0.443257	0.221629	−	0.975131i	$-0.428863\pi$
$547$	10.3669	0.443257	0.221629	+	0.975131i	$0.428863\pi$
$548$	−18.7248	−0.799886
$549$	0	0
$550$	−1.89660	−0.0808711
$551$	−12.3903	−0.527843
$552$	0	0
$553$	−7.31747	−0.311170
$554$	−3.08707	−0.131157
$555$	0	0
$556$	−17.9065	−0.759404
$557$	21.0838	0.893347	0.446674	−	0.894697i	$-0.352608\pi$
$557$	21.0838	0.893347	0.446674	+	0.894697i	$0.352608\pi$
$558$	0	0
$559$	35.2541	1.49109
$560$	5.41468	0.228812
$561$	0	0
$562$	24.6223	1.03863
$563$	−12.4385	−0.524221	−0.262110	−	0.965038i	$-0.584418\pi$
$563$	−12.4385	−0.524221	−0.262110	+	0.965038i	$0.584418\pi$
$564$	0	0
$565$	−34.0299	−1.43165
$566$	17.4763	0.734582
$567$	0	0
$568$	−14.0509	−0.589562
$569$	−16.0051	−0.670967	−0.335483	−	0.942046i	$-0.608900\pi$
$569$	−16.0051	−0.670967	−0.335483	+	0.942046i	$0.608900\pi$
$570$	0	0
$571$	34.7111	1.45261	0.726307	−	0.687371i	$-0.241235\pi$
$571$	34.7111	1.45261	0.726307	+	0.687371i	$0.241235\pi$
$572$	1.05436	0.0440848
$573$	0	0
$574$	30.0294	1.25340
$575$	−30.4717	−1.27076
$576$	0	0
$577$	12.2404	0.509574	0.254787	−	0.966997i	$-0.417995\pi$
$577$	12.2404	0.509574	0.254787	+	0.966997i	$0.417995\pi$
$578$	−15.0072	−0.624216
$579$	0	0
$580$	20.6801	0.858693
$581$	−52.8642	−2.19318
$582$	0	0
$583$	−4.55596	−0.188689
$584$	23.3186	0.964932
$585$	0	0
$586$	21.3996	0.884010
$587$	−24.6103	−1.01577	−0.507887	−	0.861424i	$-0.669573\pi$
$587$	−24.6103	−1.01577	−0.507887	+	0.861424i	$0.669573\pi$
$588$	0	0
$589$	0	0
$590$	9.80308	0.403587
$591$	0	0
$592$	0.347487	0.0142816
$593$	−6.92065	−0.284197	−0.142099	−	0.989853i	$-0.545385\pi$
$593$	−6.92065	−0.284197	−0.142099	+	0.989853i	$0.545385\pi$
$594$	0	0
$595$	−10.7337	−0.440038
$596$	10.4791	0.429239
$597$	0	0
$598$	−13.3021	−0.543962
$599$	−28.8675	−1.17950	−0.589748	−	0.807588i	$-0.700773\pi$
$599$	−28.8675	−1.17950	−0.589748	+	0.807588i	$0.700773\pi$
$600$	0	0
$601$	17.5571	0.716169	0.358085	−	0.933689i	$-0.383430\pi$
$601$	17.5571	0.716169	0.358085	+	0.933689i	$0.383430\pi$
$602$	−37.9782	−1.54788
$603$	0	0
$604$	−25.2747	−1.02841
$605$	−36.1823	−1.47102
$606$	0	0
$607$	−4.52381	−0.183616	−0.0918078	−	0.995777i	$-0.529265\pi$
$607$	−4.52381	−0.183616	−0.0918078	+	0.995777i	$0.529265\pi$
$608$	12.0005	0.486686
$609$	0	0
$610$	−33.9687	−1.37535
$611$	27.9955	1.13258
$612$	0	0
$613$	−20.5091	−0.828353	−0.414177	−	0.910197i	$-0.635930\pi$
$613$	−20.5091	−0.828353	−0.414177	+	0.910197i	$0.635930\pi$
$614$	−7.25596	−0.292827
$615$	0	0
$616$	−3.16356	−0.127463
$617$	−33.1358	−1.33400	−0.666999	−	0.745059i	$-0.732421\pi$
$617$	−33.1358	−1.33400	−0.666999	+	0.745059i	$0.732421\pi$
$618$	0	0
$619$	10.0674	0.404644	0.202322	−	0.979319i	$-0.435151\pi$
$619$	10.0674	0.404644	0.202322	+	0.979319i	$0.435151\pi$
$620$	0	0
$621$	0	0
$622$	−10.6455	−0.426846
$623$	−41.5262	−1.66371
$624$	0	0
$625$	−18.6958	−0.747834
$626$	−20.3040	−0.811513
$627$	0	0
$628$	−18.4502	−0.736243
$629$	−0.688835	−0.0274657
$630$	0	0
$631$	−20.5723	−0.818972	−0.409486	−	0.912316i	$-0.634292\pi$
$631$	−20.5723	−0.818972	−0.409486	+	0.912316i	$0.634292\pi$
$632$	−6.62908	−0.263691
$633$	0	0
$634$	14.6785	0.582958
$635$	6.12505	0.243065
$636$	0	0
$637$	9.66502	0.382942
$638$	1.74335	0.0690198
$639$	0	0
$640$	−16.8854	−0.667456
$641$	25.3655	1.00188	0.500938	−	0.865483i	$-0.332989\pi$
$641$	25.3655	1.00188	0.500938	+	0.865483i	$0.332989\pi$
$642$	0	0
$643$	31.5622	1.24469	0.622345	−	0.782743i	$-0.286180\pi$
$643$	31.5622	1.24469	0.622345	+	0.782743i	$0.286180\pi$
$644$	−18.2487	−0.719101
$645$	0	0
$646$	2.09173	0.0822980
$647$	−8.43852	−0.331753	−0.165876	−	0.986147i	$-0.553045\pi$
$647$	−8.43852	−0.331753	−0.165876	+	0.986147i	$0.553045\pi$
$648$	0	0
$649$	−1.05241	−0.0413107
$650$	15.9424	0.625313
$651$	0	0
$652$	19.3035	0.755983
$653$	2.26014	0.0884461	0.0442231	−	0.999022i	$-0.485919\pi$
$653$	2.26014	0.0884461	0.0442231	+	0.999022i	$0.485919\pi$
$654$	0	0
$655$	32.8753	1.28454
$656$	4.99870	0.195167
$657$	0	0
$658$	−30.1587	−1.17571
$659$	−39.1765	−1.52610	−0.763050	−	0.646340i	$-0.776299\pi$
$659$	−39.1765	−1.52610	−0.763050	+	0.646340i	$0.776299\pi$
$660$	0	0
$661$	27.2343	1.05929	0.529645	−	0.848219i	$-0.322325\pi$
$661$	27.2343	1.05929	0.529645	+	0.848219i	$0.322325\pi$
$662$	−18.7495	−0.728720
$663$	0	0
$664$	−47.8910	−1.85853
$665$	−23.9456	−0.928569
$666$	0	0
$667$	28.0095	1.08453
$668$	−14.3081	−0.553596
$669$	0	0
$670$	−13.2357	−0.511339
$671$	3.64670	0.140779
$672$	0	0
$673$	−29.3279	−1.13051	−0.565253	−	0.824918i	$-0.691221\pi$
$673$	−29.3279	−1.13051	−0.565253	+	0.824918i	$0.691221\pi$
$674$	−2.39034	−0.0920724
$675$	0	0
$676$	5.70105	0.219271
$677$	1.27454	0.0489844	0.0244922	−	0.999700i	$-0.492203\pi$
$677$	1.27454	0.0489844	0.0244922	+	0.999700i	$0.492203\pi$
$678$	0	0
$679$	−44.1487	−1.69427
$680$	−9.72392	−0.372895
$681$	0	0
$682$	0	0
$683$	19.6193	0.750713	0.375357	−	0.926881i	$-0.377520\pi$
$683$	19.6193	0.750713	0.375357	+	0.926881i	$0.377520\pi$
$684$	0	0
$685$	55.5438	2.12222
$686$	10.7982	0.412275
$687$	0	0
$688$	−6.32187	−0.241019
$689$	38.2965	1.45898
$690$	0	0
$691$	−0.366769	−0.0139526	−0.00697628	−	0.999976i	$-0.502221\pi$
$691$	−0.366769	−0.0139526	−0.00697628	+	0.999976i	$0.502221\pi$
$692$	−2.72286	−0.103508
$693$	0	0
$694$	24.6902	0.937227
$695$	53.1163	2.01481
$696$	0	0
$697$	−9.90909	−0.375333
$698$	13.0534	0.494077
$699$	0	0
$700$	21.8710	0.826645
$701$	15.1616	0.572647	0.286324	−	0.958133i	$-0.407567\pi$
$701$	15.1616	0.572647	0.286324	+	0.958133i	$0.407567\pi$
$702$	0	0
$703$	−1.53671	−0.0579581
$704$	−2.02605	−0.0763595
$705$	0	0
$706$	−2.57272	−0.0968255
$707$	56.6095	2.12902
$708$	0	0
$709$	15.7219	0.590448	0.295224	−	0.955428i	$-0.404606\pi$
$709$	15.7219	0.590448	0.295224	+	0.955428i	$0.404606\pi$
$710$	14.9643	0.561600
$711$	0	0
$712$	−37.6197	−1.40986
$713$	0	0
$714$	0	0
$715$	−3.12755	−0.116964
$716$	−2.40856	−0.0900121
$717$	0	0
$718$	8.37675	0.312618
$719$	−25.5678	−0.953518	−0.476759	−	0.879034i	$-0.658189\pi$
$719$	−25.5678	−0.953518	−0.476759	+	0.879034i	$0.658189\pi$
$720$	0	0
$721$	−21.5448	−0.802372
$722$	−13.1543	−0.489551
$723$	0	0
$724$	−4.06430	−0.151048
$725$	−33.5692	−1.24673
$726$	0	0
$727$	−15.9397	−0.591170	−0.295585	−	0.955317i	$-0.595514\pi$
$727$	−15.9397	−0.591170	−0.295585	+	0.955317i	$0.595514\pi$
$728$	26.5923	0.985575
$729$	0	0
$730$	−24.8345	−0.919167
$731$	12.5320	0.463514
$732$	0	0
$733$	−33.7453	−1.24641	−0.623205	−	0.782059i	$-0.714170\pi$
$733$	−33.7453	−1.24641	−0.623205	+	0.782059i	$0.714170\pi$
$734$	15.5209	0.572886
$735$	0	0
$736$	−27.1285	−0.999969
$737$	1.42091	0.0523401
$738$	0	0
$739$	−43.4900	−1.59981	−0.799903	−	0.600129i	$-0.795116\pi$
$739$	−43.4900	−1.59981	−0.799903	+	0.600129i	$0.795116\pi$
$740$	2.56486	0.0942860
$741$	0	0
$742$	−41.2556	−1.51454
$743$	−40.8299	−1.49790	−0.748952	−	0.662624i	$-0.769443\pi$
$743$	−40.8299	−1.49790	−0.748952	+	0.662624i	$0.769443\pi$
$744$	0	0
$745$	−31.0842	−1.13884
$746$	16.7399	0.612892
$747$	0	0
$748$	0.374799	0.0137040
$749$	15.5340	0.567602
$750$	0	0
$751$	47.5276	1.73431	0.867153	−	0.498041i	$-0.165947\pi$
$751$	47.5276	1.73431	0.867153	+	0.498041i	$0.165947\pi$
$752$	−5.02024	−0.183069
$753$	0	0
$754$	−14.6542	−0.533676
$755$	74.9728	2.72854
$756$	0	0
$757$	−0.123210	−0.00447814	−0.00223907	−	0.999997i	$-0.500713\pi$
$757$	−0.123210	−0.00447814	−0.00223907	+	0.999997i	$0.500713\pi$
$758$	−14.3584	−0.521520
$759$	0	0
$760$	−21.6929	−0.786885
$761$	32.1088	1.16394	0.581972	−	0.813209i	$-0.302281\pi$
$761$	32.1088	1.16394	0.581972	+	0.813209i	$0.302281\pi$
$762$	0	0
$763$	−1.04535	−0.0378442
$764$	9.70526	0.351124
$765$	0	0
$766$	4.39867	0.158930
$767$	8.84634	0.319423
$768$	0	0
$769$	−6.79859	−0.245164	−0.122582	−	0.992458i	$-0.539117\pi$
$769$	−6.79859	−0.245164	−0.122582	+	0.992458i	$0.539117\pi$
$770$	3.36921	0.121418
$771$	0	0
$772$	−18.3408	−0.660101
$773$	−39.1251	−1.40723	−0.703615	−	0.710581i	$-0.748432\pi$
$773$	−39.1251	−1.40723	−0.703615	+	0.710581i	$0.748432\pi$
$774$	0	0
$775$	0	0
$776$	−39.9954	−1.43575
$777$	0	0
$778$	−28.5046	−1.02194
$779$	−22.1060	−0.792029
$780$	0	0
$781$	−1.60649	−0.0574847
$782$	−4.72858	−0.169094
$783$	0	0
$784$	−1.73316	−0.0618985
$785$	54.7291	1.95336
$786$	0	0
$787$	−7.45432	−0.265718	−0.132859	−	0.991135i	$-0.542416\pi$
$787$	−7.45432	−0.265718	−0.132859	+	0.991135i	$0.542416\pi$
$788$	−4.04542	−0.144112
$789$	0	0
$790$	7.06002	0.251184
$791$	33.0816	1.17624
$792$	0	0
$793$	−30.6535	−1.08854
$794$	23.0746	0.818887
$795$	0	0
$796$	−3.02702	−0.107290
$797$	−50.1923	−1.77790	−0.888951	−	0.458002i	$-0.848565\pi$
$797$	−50.1923	−1.77790	−0.888951	+	0.458002i	$0.848565\pi$
$798$	0	0
$799$	9.95177	0.352068
$800$	32.5132	1.14952
$801$	0	0
$802$	16.0774	0.567714
$803$	2.66611	0.0940848
$804$	0	0
$805$	54.1315	1.90789
$806$	0	0
$807$	0	0
$808$	51.2840	1.80416
$809$	26.9576	0.947780	0.473890	−	0.880584i	$-0.342849\pi$
$809$	26.9576	0.947780	0.473890	+	0.880584i	$0.342849\pi$
$810$	0	0
$811$	−18.2946	−0.642412	−0.321206	−	0.947009i	$-0.604088\pi$
$811$	−18.2946	−0.642412	−0.321206	+	0.947009i	$0.604088\pi$
$812$	−20.1038	−0.705504
$813$	0	0
$814$	0.216219	0.00757849
$815$	−57.2602	−2.00574
$816$	0	0
$817$	27.9575	0.978108
$818$	27.0328	0.945179
$819$	0	0
$820$	36.8962	1.28847
$821$	19.1193	0.667267	0.333634	−	0.942703i	$-0.391725\pi$
$821$	19.1193	0.667267	0.333634	+	0.942703i	$0.391725\pi$
$822$	0	0
$823$	−9.06118	−0.315853	−0.157926	−	0.987451i	$-0.550481\pi$
$823$	−9.06118	−0.315853	−0.157926	+	0.987451i	$0.550481\pi$
$824$	−19.5180	−0.679943
$825$	0	0
$826$	−9.52989	−0.331587
$827$	21.9892	0.764640	0.382320	−	0.924030i	$-0.375125\pi$
$827$	21.9892	0.764640	0.382320	+	0.924030i	$0.375125\pi$
$828$	0	0
$829$	−44.7043	−1.55265	−0.776323	−	0.630336i	$-0.782917\pi$
$829$	−44.7043	−1.55265	−0.776323	+	0.630336i	$0.782917\pi$
$830$	51.0043	1.77038
$831$	0	0
$832$	17.0306	0.590428
$833$	3.43569	0.119040
$834$	0	0
$835$	42.4423	1.46878
$836$	0.836132	0.0289182
$837$	0	0
$838$	−12.3188	−0.425544
$839$	−26.9212	−0.929423	−0.464712	−	0.885462i	$-0.653842\pi$
$839$	−26.9212	−0.929423	−0.464712	+	0.885462i	$0.653842\pi$
$840$	0	0
$841$	1.85673	0.0640251
$842$	−0.829360	−0.0285816
$843$	0	0
$844$	8.08490	0.278294
$845$	−16.9111	−0.581760
$846$	0	0
$847$	35.1740	1.20859
$848$	−6.86743	−0.235828
$849$	0	0
$850$	5.66716	0.194382
$851$	3.47389	0.119084
$852$	0	0
$853$	−29.1937	−0.999572	−0.499786	−	0.866149i	$-0.666588\pi$
$853$	−29.1937	−0.999572	−0.499786	+	0.866149i	$0.666588\pi$
$854$	33.0220	1.12999
$855$	0	0
$856$	14.0727	0.480995
$857$	−12.5597	−0.429032	−0.214516	−	0.976720i	$-0.568817\pi$
$857$	−12.5597	−0.429032	−0.214516	+	0.976720i	$0.568817\pi$
$858$	0	0
$859$	−12.0549	−0.411307	−0.205654	−	0.978625i	$-0.565932\pi$
$859$	−12.0549	−0.411307	−0.205654	+	0.978625i	$0.565932\pi$
$860$	−46.6626	−1.59118
$861$	0	0
$862$	23.1591	0.788801
$863$	50.0546	1.70388	0.851940	−	0.523639i	$-0.175426\pi$
$863$	50.0546	1.70388	0.851940	+	0.523639i	$0.175426\pi$
$864$	0	0
$865$	8.07686	0.274621
$866$	−21.1876	−0.719983
$867$	0	0
$868$	0	0
$869$	−0.757927	−0.0257109
$870$	0	0
$871$	−11.9439	−0.404705
$872$	−0.947008	−0.0320698
$873$	0	0
$874$	−10.5489	−0.356821
$875$	−11.1990	−0.378596
$876$	0	0
$877$	7.84583	0.264935	0.132467	−	0.991187i	$-0.457710\pi$
$877$	7.84583	0.264935	0.132467	+	0.991187i	$0.457710\pi$
$878$	−20.3190	−0.685735
$879$	0	0
$880$	0.560841	0.0189059
$881$	−31.9914	−1.07782	−0.538909	−	0.842364i	$-0.681163\pi$
$881$	−31.9914	−1.07782	−0.538909	+	0.842364i	$0.681163\pi$
$882$	0	0
$883$	−22.0077	−0.740619	−0.370309	−	0.928908i	$-0.620748\pi$
$883$	−22.0077	−0.740619	−0.370309	+	0.928908i	$0.620748\pi$
$884$	−3.15049	−0.105962
$885$	0	0
$886$	3.24383	0.108979
$887$	−8.28596	−0.278215	−0.139108	−	0.990277i	$-0.544423\pi$
$887$	−8.28596	−0.278215	−0.139108	+	0.990277i	$0.544423\pi$
$888$	0	0
$889$	−5.95435	−0.199703
$890$	40.0652	1.34299
$891$	0	0
$892$	−13.3110	−0.445685
$893$	22.2012	0.742935
$894$	0	0
$895$	7.14454	0.238816
$896$	16.4149	0.548383
$897$	0	0
$898$	−21.3730	−0.713227
$899$	0	0
$900$	0	0
$901$	13.6135	0.453532
$902$	3.11038	0.103564
$903$	0	0
$904$	29.9694	0.996769
$905$	12.0560	0.400754
$906$	0	0
$907$	−8.67318	−0.287988	−0.143994	−	0.989579i	$-0.545995\pi$
$907$	−8.67318	−0.287988	−0.143994	+	0.989579i	$0.545995\pi$
$908$	−5.43708	−0.180436
$909$	0	0
$910$	−28.3209	−0.938830
$911$	23.0473	0.763593	0.381796	−	0.924247i	$-0.375306\pi$
$911$	23.0473	0.763593	0.381796	+	0.924247i	$0.375306\pi$
$912$	0	0
$913$	−5.47555	−0.181214
$914$	0.707929	0.0234162
$915$	0	0
$916$	−14.5789	−0.481699
$917$	−31.9591	−1.05538
$918$	0	0
$919$	54.5467	1.79933	0.899664	−	0.436582i	$-0.143811\pi$
$919$	54.5467	1.79933	0.899664	+	0.436582i	$0.143811\pi$
$920$	49.0391	1.61677
$921$	0	0
$922$	2.21044	0.0727968
$923$	13.5038	0.444484
$924$	0	0
$925$	−4.16343	−0.136893
$926$	28.2766	0.929228
$927$	0	0
$928$	−29.8861	−0.981060
$929$	42.1963	1.38442	0.692208	−	0.721698i	$-0.256638\pi$
$929$	42.1963	1.38442	0.692208	+	0.721698i	$0.256638\pi$
$930$	0	0
$931$	7.66462	0.251198
$932$	−5.08563	−0.166585
$933$	0	0
$934$	−25.8594	−0.846145
$935$	−1.11177	−0.0363588
$936$	0	0
$937$	−7.68306	−0.250995	−0.125497	−	0.992094i	$-0.540053\pi$
$937$	−7.68306	−0.250995	−0.125497	+	0.992094i	$0.540053\pi$
$938$	12.8668	0.420117
$939$	0	0
$940$	−37.0551	−1.20860
$941$	49.0904	1.60030	0.800151	−	0.599799i	$-0.204753\pi$
$941$	49.0904	1.60030	0.800151	+	0.599799i	$0.204753\pi$
$942$	0	0
$943$	49.9729	1.62734
$944$	−1.58635	−0.0516313
$945$	0	0
$946$	−3.93370	−0.127896
$947$	31.2317	1.01489	0.507447	−	0.861683i	$-0.330589\pi$
$947$	31.2317	1.01489	0.507447	+	0.861683i	$0.330589\pi$
$948$	0	0
$949$	−22.4108	−0.727484
$950$	12.6428	0.410185
$951$	0	0
$952$	9.45293	0.306371
$953$	−46.5281	−1.50719	−0.753596	−	0.657338i	$-0.771682\pi$
$953$	−46.5281	−1.50719	−0.753596	+	0.657338i	$0.771682\pi$
$954$	0	0
$955$	−28.7889	−0.931586
$956$	−4.01195	−0.129756
$957$	0	0
$958$	14.3198	0.462651
$959$	−53.9958	−1.74362
$960$	0	0
$961$	0	0
$962$	−1.81750	−0.0585985
$963$	0	0
$964$	−7.95097	−0.256083
$965$	54.4047	1.75135
$966$	0	0
$967$	49.4544	1.59035	0.795174	−	0.606382i	$-0.207380\pi$
$967$	49.4544	1.59035	0.795174	+	0.606382i	$0.207380\pi$
$968$	31.8650	1.02418
$969$	0	0
$970$	42.5954	1.36766
$971$	−8.88234	−0.285048	−0.142524	−	0.989791i	$-0.545522\pi$
$971$	−8.88234	−0.285048	−0.142524	+	0.989791i	$0.545522\pi$
$972$	0	0
$973$	−51.6360	−1.65537
$974$	−6.42106	−0.205744
$975$	0	0
$976$	5.49686	0.175950
$977$	54.4355	1.74155	0.870773	−	0.491685i	$-0.163619\pi$
$977$	54.4355	1.74155	0.870773	+	0.491685i	$0.163619\pi$
$978$	0	0
$979$	−4.30120	−0.137467
$980$	−12.7927	−0.408647
$981$	0	0
$982$	28.2614	0.901857
$983$	−41.7509	−1.33165	−0.665823	−	0.746110i	$-0.731919\pi$
$983$	−41.7509	−1.33165	−0.665823	+	0.746110i	$0.731919\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	−5.20924	−0.165896
$987$	0	0
$988$	−7.02836	−0.223602
$989$	−63.2009	−2.00967
$990$	0	0
$991$	22.9011	0.727478	0.363739	−	0.931501i	$-0.381500\pi$
$991$	22.9011	0.727478	0.363739	+	0.931501i	$0.381500\pi$
$992$	0	0
$993$	0	0
$994$	−14.5473	−0.461411
$995$	8.97909	0.284656
$996$	0	0
$997$	28.4025	0.899517	0.449759	−	0.893150i	$-0.351510\pi$
$997$	28.4025	0.899517	0.449759	+	0.893150i	$0.351510\pi$
$998$	34.8804	1.10412
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.bo.1.8		12
3.2	odd	2	inner	8649.2.a.bo.1.5		12
31.23	odd	10		279.2.i.d.64.3	✓	24
31.27	odd	10		279.2.i.d.109.3	yes	24
31.30	odd	2		8649.2.a.bn.1.8		12
93.23	even	10		279.2.i.d.64.4	yes	24
93.89	even	10		279.2.i.d.109.4	yes	24
93.92	even	2		8649.2.a.bn.1.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
279.2.i.d.64.3	✓	24	31.23	odd	10
279.2.i.d.64.4	yes	24	93.23	even	10
279.2.i.d.109.3	yes	24	31.27	odd	10
279.2.i.d.109.4	yes	24	93.89	even	10
8649.2.a.bn.1.5		12	93.92	even	2
8649.2.a.bn.1.8		12	31.30	odd	2
8649.2.a.bo.1.5		12	3.2	odd	2	inner
8649.2.a.bo.1.8		12	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:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q+0.937929 q^{2} -1.12029 q^{4} +3.32313 q^{5} -3.23052 q^{7} -2.92661 q^{8} +O(q^{10})\)	\(q+0.937929 q^{2} -1.12029 q^{4} +3.32313 q^{5} -3.23052 q^{7} -2.92661 q^{8} +3.11686 q^{10} -0.334610 q^{11} +2.81267 q^{13} -3.03000 q^{14} -0.504375 q^{16} +0.999838 q^{17} +2.23052 q^{19} -3.72286 q^{20} -0.313840 q^{22} -5.04233 q^{23} +6.04318 q^{25} +2.63808 q^{26} +3.61911 q^{28} -5.55488 q^{29} +5.38015 q^{32} +0.937777 q^{34} -10.7354 q^{35} -0.688947 q^{37} +2.09207 q^{38} -9.72550 q^{40} -9.91069 q^{41} +12.5341 q^{43} +0.374860 q^{44} -4.72934 q^{46} +9.95339 q^{47} +3.43625 q^{49} +5.66808 q^{50} -3.15100 q^{52} +13.6157 q^{53} -1.11195 q^{55} +9.45447 q^{56} -5.21009 q^{58} +3.14518 q^{59} -10.8984 q^{61} +6.05495 q^{64} +9.34685 q^{65} -4.24648 q^{67} -1.12011 q^{68} -10.0691 q^{70} +4.80108 q^{71} -7.96780 q^{73} -0.646183 q^{74} -2.49883 q^{76} +1.08096 q^{77} +2.26511 q^{79} -1.67610 q^{80} -9.29553 q^{82} +16.3640 q^{83} +3.32259 q^{85} +11.7561 q^{86} +0.979273 q^{88} +12.8544 q^{89} -9.08637 q^{91} +5.64886 q^{92} +9.33557 q^{94} +7.41230 q^{95} +13.6661 q^{97} +3.22296 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 14 q^{4} - 4 q^{7}+O(q^{10}) \) 12 * q + 14 * q^4 - 4 * q^7	\( 12 q + 14 q^{4} - 4 q^{7} - 16 q^{10} + 10 q^{13} + 26 q^{16} - 8 q^{19} - 20 q^{22} + 14 q^{25} - 24 q^{28} + 56 q^{34} + 14 q^{37} - 82 q^{40} + 54 q^{43} + 18 q^{46} + 12 q^{49} - 2 q^{52} + 16 q^{55} + 56 q^{58} + 22 q^{61} + 42 q^{64} - 38 q^{67} + 42 q^{70} + 16 q^{73} + 10 q^{76} + 94 q^{79} - 26 q^{82} - 8 q^{85} - 60 q^{88} + 8 q^{91} + 86 q^{94} + 68 q^{97}+O(q^{100}) \) 12 * q + 14 * q^4 - 4 * q^7 - 16 * q^10 + 10 * q^13 + 26 * q^16 - 8 * q^19 - 20 * q^22 + 14 * q^25 - 24 * q^28 + 56 * q^34 + 14 * q^37 - 82 * q^40 + 54 * q^43 + 18 * q^46 + 12 * q^49 - 2 * q^52 + 16 * q^55 + 56 * q^58 + 22 * q^61 + 42 * q^64 - 38 * q^67 + 42 * q^70 + 16 * q^73 + 10 * q^76 + 94 * q^79 - 26 * q^82 - 8 * q^85 - 60 * q^88 + 8 * q^91 + 86 * q^94 + 68 * q^97

Introduction

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