LMFDB - Embedded newform 1216.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.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:	$9.70980888579$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 608)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.69353$ of defining polynomial
Character	$\chi$	$=$	1216.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155 q^{3} -2.64453 q^{5} -0.180969 q^{7} +3.56155 q^{9} +O(q^{10})$	$q-2.56155 q^{3} -2.64453 q^{5} -0.180969 q^{7} +3.56155 q^{9} +0.644529 q^{11} +3.94860 q^{13} +6.77410 q^{15} +5.56802 q^{17} -1.00000 q^{19} +0.463560 q^{21} -4.46356 q^{23} +1.99353 q^{25} -1.43845 q^{27} +3.94860 q^{29} -5.48504 q^{31} -1.65100 q^{33} +0.478577 q^{35} -7.48504 q^{37} -10.1146 q^{39} +8.41216 q^{41} +5.76763 q^{43} -9.41863 q^{45} -11.2527 q^{47} -6.96725 q^{49} -14.2628 q^{51} -7.58667 q^{53} -1.70448 q^{55} +2.56155 q^{57} -11.8506 q^{59} +2.47858 q^{61} -0.644529 q^{63} -10.4422 q^{65} +8.97372 q^{67} +11.4336 q^{69} -7.63806 q^{71} +10.6911 q^{73} -5.10654 q^{75} -0.116639 q^{77} +10.6081 q^{79} -7.00000 q^{81} -5.65100 q^{83} -14.7248 q^{85} -10.1146 q^{87} -17.1733 q^{89} -0.714573 q^{91} +14.0502 q^{93} +2.64453 q^{95} +3.28906 q^{97} +2.29552 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - q^{5} - q^{7} + 6 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - q^5 - q^7 + 6 * q^9	$ 4 q - 2 q^{3} - q^{5} - q^{7} + 6 q^{9} - 7 q^{11} - 10 q^{13} - 8 q^{15} + 5 q^{17} - 4 q^{19} - 8 q^{21} - 8 q^{23} + 17 q^{25} - 14 q^{27} - 10 q^{29} - 6 q^{31} + 12 q^{33} - 5 q^{35} - 14 q^{37} - 12 q^{39} - 2 q^{41} - 3 q^{43} + 7 q^{45} - 3 q^{47} + 7 q^{49} + 6 q^{51} - 4 q^{53} - 35 q^{55} + 2 q^{57} - 20 q^{59} + 3 q^{61} + 7 q^{63} - 12 q^{65} - 8 q^{67} + 4 q^{69} - 30 q^{71} + 9 q^{73} - 34 q^{75} + 7 q^{77} + 10 q^{79} - 28 q^{81} - 4 q^{83} - 19 q^{85} - 12 q^{87} - 16 q^{89} - 10 q^{91} + 20 q^{93} + q^{95} - 6 q^{97} - 19 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - q^5 - q^7 + 6 * q^9 - 7 * q^11 - 10 * q^13 - 8 * q^15 + 5 * q^17 - 4 * q^19 - 8 * q^21 - 8 * q^23 + 17 * q^25 - 14 * q^27 - 10 * q^29 - 6 * q^31 + 12 * q^33 - 5 * q^35 - 14 * q^37 - 12 * q^39 - 2 * q^41 - 3 * q^43 + 7 * q^45 - 3 * q^47 + 7 * q^49 + 6 * q^51 - 4 * q^53 - 35 * q^55 + 2 * q^57 - 20 * q^59 + 3 * q^61 + 7 * q^63 - 12 * q^65 - 8 * q^67 + 4 * q^69 - 30 * q^71 + 9 * q^73 - 34 * q^75 + 7 * q^77 + 10 * q^79 - 28 * q^81 - 4 * q^83 - 19 * q^85 - 12 * q^87 - 16 * q^89 - 10 * q^91 + 20 * q^93 + q^95 - 6 * q^97 - 19 * 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
$3$	−2.56155	−1.47891	−0.739457	−	0.673204i	$-0.764917\pi$
$3$	−2.56155	−1.47891	−0.739457	+	0.673204i	$0.764917\pi$
$4$	0	0
$5$	−2.64453	−1.18267	−0.591335	−	0.806426i	$-0.701399\pi$
$5$	−2.64453	−1.18267	−0.591335	+	0.806426i	$0.701399\pi$
$6$	0	0
$7$	−0.180969	−0.0683997	−0.0341998	−	0.999415i	$-0.510888\pi$
$7$	−0.180969	−0.0683997	−0.0341998	+	0.999415i	$0.510888\pi$
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	0.644529	0.194333	0.0971664	−	0.995268i	$-0.469022\pi$
$11$	0.644529	0.194333	0.0971664	+	0.995268i	$0.469022\pi$
$12$	0	0
$13$	3.94860	1.09515	0.547573	−	0.836758i	$-0.315552\pi$
$13$	3.94860	1.09515	0.547573	+	0.836758i	$0.315552\pi$
$14$	0	0
$15$	6.77410	1.74907
$16$	0	0
$17$	5.56802	1.35044	0.675221	−	0.737615i	$-0.264048\pi$
$17$	5.56802	1.35044	0.675221	+	0.737615i	$0.264048\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.463560	0.101157
$22$	0	0
$23$	−4.46356	−0.930717	−0.465358	−	0.885122i	$-0.654075\pi$
$23$	−4.46356	−0.930717	−0.465358	+	0.885122i	$0.654075\pi$
$24$	0	0
$25$	1.99353	0.398707
$26$	0	0
$27$	−1.43845	−0.276829
$28$	0	0
$29$	3.94860	0.733237	0.366619	−	0.930371i	$-0.380515\pi$
$29$	3.94860	0.733237	0.366619	+	0.930371i	$0.380515\pi$
$30$	0	0
$31$	−5.48504	−0.985143	−0.492571	−	0.870272i	$-0.663943\pi$
$31$	−5.48504	−0.985143	−0.492571	+	0.870272i	$0.663943\pi$
$32$	0	0
$33$	−1.65100	−0.287401
$34$	0	0
$35$	0.478577	0.0808942
$36$	0	0
$37$	−7.48504	−1.23053	−0.615267	−	0.788319i	$-0.710952\pi$
$37$	−7.48504	−1.23053	−0.615267	+	0.788319i	$0.710952\pi$
$38$	0	0
$39$	−10.1146	−1.61963
$40$	0	0
$41$	8.41216	1.31376	0.656880	−	0.753995i	$-0.271876\pi$
$41$	8.41216	1.31376	0.656880	+	0.753995i	$0.271876\pi$
$42$	0	0
$43$	5.76763	0.879556	0.439778	−	0.898107i	$-0.355057\pi$
$43$	5.76763	0.879556	0.439778	+	0.898107i	$0.355057\pi$
$44$	0	0
$45$	−9.41863	−1.40405
$46$	0	0
$47$	−11.2527	−1.64137	−0.820686	−	0.571380i	$-0.806408\pi$
$47$	−11.2527	−1.64137	−0.820686	+	0.571380i	$0.806408\pi$
$48$	0	0
$49$	−6.96725	−0.995321
$50$	0	0
$51$	−14.2628	−1.99719
$52$	0	0
$53$	−7.58667	−1.04211	−0.521054	−	0.853523i	$-0.674461\pi$
$53$	−7.58667	−1.04211	−0.521054	+	0.853523i	$0.674461\pi$
$54$	0	0
$55$	−1.70448	−0.229831
$56$	0	0
$57$	2.56155	0.339286
$58$	0	0
$59$	−11.8506	−1.54282	−0.771409	−	0.636340i	$-0.780448\pi$
$59$	−11.8506	−1.54282	−0.771409	+	0.636340i	$0.780448\pi$
$60$	0	0
$61$	2.47858	0.317349	0.158675	−	0.987331i	$-0.449278\pi$
$61$	2.47858	0.317349	0.158675	+	0.987331i	$0.449278\pi$
$62$	0	0
$63$	−0.644529	−0.0812030
$64$	0	0
$65$	−10.4422	−1.29519
$66$	0	0
$67$	8.97372	1.09631	0.548157	−	0.836375i	$-0.315330\pi$
$67$	8.97372	1.09631	0.548157	+	0.836375i	$0.315330\pi$
$68$	0	0
$69$	11.4336	1.37645
$70$	0	0
$71$	−7.63806	−0.906471	−0.453236	−	0.891391i	$-0.649730\pi$
$71$	−7.63806	−0.906471	−0.453236	+	0.891391i	$0.649730\pi$
$72$	0	0
$73$	10.6911	1.25130	0.625651	−	0.780103i	$-0.284834\pi$
$73$	10.6911	1.25130	0.625651	+	0.780103i	$0.284834\pi$
$74$	0	0
$75$	−5.10654	−0.589653
$76$	0	0
$77$	−0.116639	−0.0132923
$78$	0	0
$79$	10.6081	1.19351	0.596755	−	0.802424i	$-0.296456\pi$
$79$	10.6081	1.19351	0.596755	+	0.802424i	$0.296456\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−5.65100	−0.620277	−0.310139	−	0.950691i	$-0.600375\pi$
$83$	−5.65100	−0.620277	−0.310139	+	0.950691i	$0.600375\pi$
$84$	0	0
$85$	−14.7248	−1.59713
$86$	0	0
$87$	−10.1146	−1.08439
$88$	0	0
$89$	−17.1733	−1.82037	−0.910185	−	0.414202i	$-0.864061\pi$
$89$	−17.1733	−1.82037	−0.910185	+	0.414202i	$0.864061\pi$
$90$	0	0
$91$	−0.714573	−0.0749076
$92$	0	0
$93$	14.0502	1.45694
$94$	0	0
$95$	2.64453	0.271323
$96$	0	0
$97$	3.28906	0.333953	0.166977	−	0.985961i	$-0.446600\pi$
$97$	3.28906	0.333953	0.166977	+	0.985961i	$0.446600\pi$
$98$	0	0
$99$	2.29552	0.230709
$100$	0	0
$101$	−0.246211	−0.0244989	−0.0122495	−	0.999925i	$-0.503899\pi$
$101$	−0.246211	−0.0244989	−0.0122495	+	0.999925i	$0.503899\pi$
$102$	0	0
$103$	14.4122	1.42007	0.710036	−	0.704165i	$-0.248678\pi$
$103$	14.4122	1.42007	0.710036	+	0.704165i	$0.248678\pi$
$104$	0	0
$105$	−1.22590	−0.119636
$106$	0	0
$107$	−9.69759	−0.937501	−0.468751	−	0.883330i	$-0.655296\pi$
$107$	−9.69759	−0.937501	−0.468751	+	0.883330i	$0.655296\pi$
$108$	0	0
$109$	−19.4839	−1.86622	−0.933108	−	0.359596i	$-0.882915\pi$
$109$	−19.4839	−1.86622	−0.933108	+	0.359596i	$0.882915\pi$
$110$	0	0
$111$	19.1733	1.81985
$112$	0	0
$113$	−7.28906	−0.685697	−0.342848	−	0.939391i	$-0.611392\pi$
$113$	−7.28906	−0.685697	−0.342848	+	0.939391i	$0.611392\pi$
$114$	0	0
$115$	11.8040	1.10073
$116$	0	0
$117$	14.0632	1.30014
$118$	0	0
$119$	−1.00764	−0.0923699
$120$	0	0
$121$	−10.5846	−0.962235
$122$	0	0
$123$	−21.5482	−1.94294
$124$	0	0
$125$	7.95069	0.711131
$126$	0	0
$127$	−0.595216	−0.0528169	−0.0264084	−	0.999651i	$-0.508407\pi$
$127$	−0.595216	−0.0528169	−0.0264084	+	0.999651i	$0.508407\pi$
$128$	0	0
$129$	−14.7741	−1.30079
$130$	0	0
$131$	15.7806	1.37875	0.689377	−	0.724403i	$-0.257884\pi$
$131$	15.7806	1.37875	0.689377	+	0.724403i	$0.257884\pi$
$132$	0	0
$133$	0.180969	0.0156920
$134$	0	0
$135$	3.80402	0.327398
$136$	0	0
$137$	−15.9802	−1.36528	−0.682640	−	0.730755i	$-0.739168\pi$
$137$	−15.9802	−1.36528	−0.682640	+	0.730755i	$0.739168\pi$
$138$	0	0
$139$	−4.47858	−0.379868	−0.189934	−	0.981797i	$-0.560827\pi$
$139$	−4.47858	−0.379868	−0.189934	+	0.981797i	$0.560827\pi$
$140$	0	0
$141$	28.8243	2.42745
$142$	0	0
$143$	2.54499	0.212823
$144$	0	0
$145$	−10.4422	−0.867177
$146$	0	0
$147$	17.8470	1.47199
$148$	0	0
$149$	−10.3758	−0.850017	−0.425009	−	0.905189i	$-0.639729\pi$
$149$	−10.3758	−0.850017	−0.425009	+	0.905189i	$0.639729\pi$
$150$	0	0
$151$	−16.6584	−1.35564	−0.677820	−	0.735228i	$-0.737075\pi$
$151$	−16.6584	−1.35564	−0.677820	+	0.735228i	$0.737075\pi$
$152$	0	0
$153$	19.8308	1.60322
$154$	0	0
$155$	14.5054	1.16510
$156$	0	0
$157$	11.0203	0.879517	0.439758	−	0.898116i	$-0.355064\pi$
$157$	11.0203	0.879517	0.439758	+	0.898116i	$0.355064\pi$
$158$	0	0
$159$	19.4336	1.54119
$160$	0	0
$161$	0.807764	0.0636607
$162$	0	0
$163$	7.80402	0.611258	0.305629	−	0.952151i	$-0.401133\pi$
$163$	7.80402	0.611258	0.305629	+	0.952151i	$0.401133\pi$
$164$	0	0
$165$	4.36610	0.339901
$166$	0	0
$167$	−10.2462	−0.792876	−0.396438	−	0.918062i	$-0.629754\pi$
$167$	−10.2462	−0.792876	−0.396438	+	0.918062i	$0.629754\pi$
$168$	0	0
$169$	2.59147	0.199344
$170$	0	0
$171$	−3.56155	−0.272359
$172$	0	0
$173$	1.80402	0.137157	0.0685784	−	0.997646i	$-0.478154\pi$
$173$	1.80402	0.137157	0.0685784	+	0.997646i	$0.478154\pi$
$174$	0	0
$175$	−0.360767	−0.0272714
$176$	0	0
$177$	30.3560	2.28169
$178$	0	0
$179$	−1.75379	−0.131084	−0.0655422	−	0.997850i	$-0.520878\pi$
$179$	−1.75379	−0.131084	−0.0655422	+	0.997850i	$0.520878\pi$
$180$	0	0
$181$	−6.52789	−0.485214	−0.242607	−	0.970125i	$-0.578003\pi$
$181$	−6.52789	−0.485214	−0.242607	+	0.970125i	$0.578003\pi$
$182$	0	0
$183$	−6.34900	−0.469332
$184$	0	0
$185$	19.7944	1.45531
$186$	0	0
$187$	3.58875	0.262435
$188$	0	0
$189$	0.260314	0.0189350
$190$	0	0
$191$	−7.45709	−0.539576	−0.269788	−	0.962920i	$-0.586954\pi$
$191$	−7.45709	−0.539576	−0.269788	+	0.962920i	$0.586954\pi$
$192$	0	0
$193$	19.0203	1.36911	0.684556	−	0.728960i	$-0.259996\pi$
$193$	19.0203	1.36911	0.684556	+	0.728960i	$0.259996\pi$
$194$	0	0
$195$	26.7482	1.91548
$196$	0	0
$197$	−0.153020	−0.0109022	−0.00545112	−	0.999985i	$-0.501735\pi$
$197$	−0.153020	−0.0109022	−0.00545112	+	0.999985i	$0.501735\pi$
$198$	0	0
$199$	21.3672	1.51468	0.757341	−	0.653019i	$-0.226498\pi$
$199$	21.3672	1.51468	0.757341	+	0.653019i	$0.226498\pi$
$200$	0	0
$201$	−22.9866	−1.62135
$202$	0	0
$203$	−0.714573	−0.0501532
$204$	0	0
$205$	−22.2462	−1.55374
$206$	0	0
$207$	−15.8972	−1.10493
$208$	0	0
$209$	−0.644529	−0.0445830
$210$	0	0
$211$	−6.82070	−0.469556	−0.234778	−	0.972049i	$-0.575436\pi$
$211$	−6.82070	−0.469556	−0.234778	+	0.972049i	$0.575436\pi$
$212$	0	0
$213$	19.5653	1.34059
$214$	0	0
$215$	−15.2527	−1.04022
$216$	0	0
$217$	0.992620	0.0673835
$218$	0	0
$219$	−27.3859	−1.85057
$220$	0	0
$221$	21.9859	1.47893
$222$	0	0
$223$	−8.42510	−0.564186	−0.282093	−	0.959387i	$-0.591029\pi$
$223$	−8.42510	−0.564186	−0.282093	+	0.959387i	$0.591029\pi$
$224$	0	0
$225$	7.10008	0.473338
$226$	0	0
$227$	−23.3859	−1.55218	−0.776088	−	0.630625i	$-0.782799\pi$
$227$	−23.3859	−1.55218	−0.776088	+	0.630625i	$0.782799\pi$
$228$	0	0
$229$	3.37567	0.223070	0.111535	−	0.993760i	$-0.464423\pi$
$229$	3.37567	0.223070	0.111535	+	0.993760i	$0.464423\pi$
$230$	0	0
$231$	0.298778	0.0196582
$232$	0	0
$233$	−15.8308	−1.03711	−0.518555	−	0.855044i	$-0.673530\pi$
$233$	−15.8308	−1.03711	−0.518555	+	0.855044i	$0.673530\pi$
$234$	0	0
$235$	29.7580	1.94120
$236$	0	0
$237$	−27.1733	−1.76510
$238$	0	0
$239$	2.39715	0.155059	0.0775293	−	0.996990i	$-0.475297\pi$
$239$	2.39715	0.155059	0.0775293	+	0.996990i	$0.475297\pi$
$240$	0	0
$241$	5.44208	0.350555	0.175278	−	0.984519i	$-0.443918\pi$
$241$	5.44208	0.350555	0.175278	+	0.984519i	$0.443918\pi$
$242$	0	0
$243$	22.2462	1.42710
$244$	0	0
$245$	18.4251	1.17714
$246$	0	0
$247$	−3.94860	−0.251244
$248$	0	0
$249$	14.4753	0.917336
$250$	0	0
$251$	−12.8405	−0.810486	−0.405243	−	0.914209i	$-0.632813\pi$
$251$	−12.8405	−0.810486	−0.405243	+	0.914209i	$0.632813\pi$
$252$	0	0
$253$	−2.87689	−0.180869
$254$	0	0
$255$	37.7183	2.36201
$256$	0	0
$257$	−19.1863	−1.19681	−0.598403	−	0.801195i	$-0.704198\pi$
$257$	−19.1863	−1.19681	−0.598403	+	0.801195i	$0.704198\pi$
$258$	0	0
$259$	1.35456	0.0841681
$260$	0	0
$261$	14.0632	0.870488
$262$	0	0
$263$	20.2098	1.24619	0.623096	−	0.782146i	$-0.285875\pi$
$263$	20.2098	1.24619	0.623096	+	0.782146i	$0.285875\pi$
$264$	0	0
$265$	20.0632	1.23247
$266$	0	0
$267$	43.9904	2.69217
$268$	0	0
$269$	−16.4422	−1.00250	−0.501249	−	0.865303i	$-0.667126\pi$
$269$	−16.4422	−1.00250	−0.501249	+	0.865303i	$0.667126\pi$
$270$	0	0
$271$	8.02886	0.487719	0.243859	−	0.969811i	$-0.421586\pi$
$271$	8.02886	0.487719	0.243859	+	0.969811i	$0.421586\pi$
$272$	0	0
$273$	1.83042	0.110782
$274$	0	0
$275$	1.28489	0.0774818
$276$	0	0
$277$	−9.68738	−0.582058	−0.291029	−	0.956714i	$-0.593998\pi$
$277$	−9.68738	−0.582058	−0.291029	+	0.956714i	$0.593998\pi$
$278$	0	0
$279$	−19.5353	−1.16955
$280$	0	0
$281$	−8.15302	−0.486368	−0.243184	−	0.969980i	$-0.578192\pi$
$281$	−8.15302	−0.486368	−0.243184	+	0.969980i	$0.578192\pi$
$282$	0	0
$283$	−27.9742	−1.66289	−0.831447	−	0.555604i	$-0.812487\pi$
$283$	−27.9742	−1.66289	−0.831447	+	0.555604i	$0.812487\pi$
$284$	0	0
$285$	−6.77410	−0.401263
$286$	0	0
$287$	−1.52234	−0.0898607
$288$	0	0
$289$	14.0028	0.823696
$290$	0	0
$291$	−8.42510	−0.493888
$292$	0	0
$293$	12.5438	0.732818	0.366409	−	0.930454i	$-0.380587\pi$
$293$	12.5438	0.732818	0.366409	+	0.930454i	$0.380587\pi$
$294$	0	0
$295$	31.3393	1.82464
$296$	0	0
$297$	−0.927121	−0.0537970
$298$	0	0
$299$	−17.6248	−1.01927
$300$	0	0
$301$	−1.04376	−0.0601614
$302$	0	0
$303$	0.630683	0.0362318
$304$	0	0
$305$	−6.55467	−0.375319
$306$	0	0
$307$	12.9572	0.739504	0.369752	−	0.929131i	$-0.379443\pi$
$307$	12.9572	0.739504	0.369752	+	0.929131i	$0.379443\pi$
$308$	0	0
$309$	−36.9175	−2.10016
$310$	0	0
$311$	31.8467	1.80586	0.902931	−	0.429786i	$-0.141411\pi$
$311$	31.8467	1.80586	0.902931	+	0.429786i	$0.141411\pi$
$312$	0	0
$313$	16.2628	0.919226	0.459613	−	0.888119i	$-0.347988\pi$
$313$	16.2628	0.919226	0.459613	+	0.888119i	$0.347988\pi$
$314$	0	0
$315$	1.70448	0.0960363
$316$	0	0
$317$	6.88866	0.386905	0.193453	−	0.981110i	$-0.438031\pi$
$317$	6.88866	0.386905	0.193453	+	0.981110i	$0.438031\pi$
$318$	0	0
$319$	2.54499	0.142492
$320$	0	0
$321$	24.8409	1.38648
$322$	0	0
$323$	−5.56802	−0.309813
$324$	0	0
$325$	7.87167	0.436642
$326$	0	0
$327$	49.9090	2.75997
$328$	0	0
$329$	2.03638	0.112269
$330$	0	0
$331$	−25.6976	−1.41247	−0.706234	−	0.707979i	$-0.749607\pi$
$331$	−25.6976	−1.41247	−0.706234	+	0.707979i	$0.749607\pi$
$332$	0	0
$333$	−26.6584	−1.46087
$334$	0	0
$335$	−23.7313	−1.29658
$336$	0	0
$337$	7.78382	0.424012	0.212006	−	0.977268i	$-0.432000\pi$
$337$	7.78382	0.424012	0.212006	+	0.977268i	$0.432000\pi$
$338$	0	0
$339$	18.6713	1.01409
$340$	0	0
$341$	−3.53527	−0.191446
$342$	0	0
$343$	2.52763	0.136479
$344$	0	0
$345$	−30.2366	−1.62788
$346$	0	0
$347$	32.3758	1.73802	0.869012	−	0.494792i	$-0.164756\pi$
$347$	32.3758	1.73802	0.869012	+	0.494792i	$0.164756\pi$
$348$	0	0
$349$	−25.0567	−1.34125	−0.670627	−	0.741795i	$-0.733975\pi$
$349$	−25.0567	−1.34125	−0.670627	+	0.741795i	$0.733975\pi$
$350$	0	0
$351$	−5.67986	−0.303168
$352$	0	0
$353$	−2.70731	−0.144096	−0.0720478	−	0.997401i	$-0.522953\pi$
$353$	−2.70731	−0.144096	−0.0720478	+	0.997401i	$0.522953\pi$
$354$	0	0
$355$	20.1991	1.07206
$356$	0	0
$357$	2.58111	0.136607
$358$	0	0
$359$	−27.6862	−1.46122	−0.730611	−	0.682794i	$-0.760765\pi$
$359$	−27.6862	−1.46122	−0.730611	+	0.682794i	$0.760765\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	27.1130	1.42306
$364$	0	0
$365$	−28.2730	−1.47988
$366$	0	0
$367$	15.3020	0.798757	0.399379	−	0.916786i	$-0.369226\pi$
$367$	15.3020	0.798757	0.399379	+	0.916786i	$0.369226\pi$
$368$	0	0
$369$	29.9604	1.55967
$370$	0	0
$371$	1.37295	0.0712799
$372$	0	0
$373$	−27.2506	−1.41098	−0.705491	−	0.708719i	$-0.749274\pi$
$373$	−27.2506	−1.41098	−0.705491	+	0.708719i	$0.749274\pi$
$374$	0	0
$375$	−20.3661	−1.05170
$376$	0	0
$377$	15.5915	0.803001
$378$	0	0
$379$	−30.0968	−1.54597	−0.772985	−	0.634424i	$-0.781237\pi$
$379$	−30.0968	−1.54597	−0.772985	+	0.634424i	$0.781237\pi$
$380$	0	0
$381$	1.52468	0.0781116
$382$	0	0
$383$	37.8146	1.93224	0.966118	−	0.258100i	$-0.0830966\pi$
$383$	37.8146	1.93224	0.966118	+	0.258100i	$0.0830966\pi$
$384$	0	0
$385$	0.308457	0.0157204
$386$	0	0
$387$	20.5417	1.04420
$388$	0	0
$389$	−2.44128	−0.123778	−0.0618890	−	0.998083i	$-0.519712\pi$
$389$	−2.44128	−0.123778	−0.0618890	+	0.998083i	$0.519712\pi$
$390$	0	0
$391$	−24.8532	−1.25688
$392$	0	0
$393$	−40.4228	−2.03906
$394$	0	0
$395$	−28.0536	−1.41153
$396$	0	0
$397$	−16.4260	−0.824398	−0.412199	−	0.911094i	$-0.635239\pi$
$397$	−16.4260	−0.824398	−0.412199	+	0.911094i	$0.635239\pi$
$398$	0	0
$399$	−0.463560	−0.0232071
$400$	0	0
$401$	−30.0932	−1.50278	−0.751391	−	0.659857i	$-0.770617\pi$
$401$	−30.0932	−1.50278	−0.751391	+	0.659857i	$0.770617\pi$
$402$	0	0
$403$	−21.6583	−1.07887
$404$	0	0
$405$	18.5117	0.919854
$406$	0	0
$407$	−4.82433	−0.239133
$408$	0	0
$409$	−39.4454	−1.95045	−0.975225	−	0.221216i	$-0.928998\pi$
$409$	−39.4454	−1.95045	−0.975225	+	0.221216i	$0.928998\pi$
$410$	0	0
$411$	40.9341	2.01913
$412$	0	0
$413$	2.14459	0.105528
$414$	0	0
$415$	14.9442	0.733583
$416$	0	0
$417$	11.4721	0.561792
$418$	0	0
$419$	−13.3522	−0.652298	−0.326149	−	0.945318i	$-0.605751\pi$
$419$	−13.3522	−0.652298	−0.326149	+	0.945318i	$0.605751\pi$
$420$	0	0
$421$	−21.8587	−1.06533	−0.532665	−	0.846326i	$-0.678809\pi$
$421$	−21.8587	−1.06533	−0.532665	+	0.846326i	$0.678809\pi$
$422$	0	0
$423$	−40.0770	−1.94861
$424$	0	0
$425$	11.1000	0.538431
$426$	0	0
$427$	−0.448544	−0.0217066
$428$	0	0
$429$	−6.51912	−0.314746
$430$	0	0
$431$	−14.3490	−0.691167	−0.345584	−	0.938388i	$-0.612319\pi$
$431$	−14.3490	−0.691167	−0.345584	+	0.938388i	$0.612319\pi$
$432$	0	0
$433$	19.2536	0.925269	0.462634	−	0.886549i	$-0.346904\pi$
$433$	19.2536	0.925269	0.462634	+	0.886549i	$0.346904\pi$
$434$	0	0
$435$	26.7482	1.28248
$436$	0	0
$437$	4.46356	0.213521
$438$	0	0
$439$	−19.3020	−0.921234	−0.460617	−	0.887599i	$-0.652372\pi$
$439$	−19.3020	−0.921234	−0.460617	+	0.887599i	$0.652372\pi$
$440$	0	0
$441$	−24.8142	−1.18163
$442$	0	0
$443$	−4.61866	−0.219439	−0.109720	−	0.993963i	$-0.534995\pi$
$443$	−4.61866	−0.219439	−0.109720	+	0.993963i	$0.534995\pi$
$444$	0	0
$445$	45.4154	2.15290
$446$	0	0
$447$	26.5781	1.25710
$448$	0	0
$449$	8.54082	0.403066	0.201533	−	0.979482i	$-0.435408\pi$
$449$	8.54082	0.403066	0.201533	+	0.979482i	$0.435408\pi$
$450$	0	0
$451$	5.42188	0.255307
$452$	0	0
$453$	42.6713	2.00487
$454$	0	0
$455$	1.88971	0.0885909
$456$	0	0
$457$	25.7340	1.20378	0.601892	−	0.798577i	$-0.294414\pi$
$457$	25.7340	1.20378	0.601892	+	0.798577i	$0.294414\pi$
$458$	0	0
$459$	−8.00930	−0.373842
$460$	0	0
$461$	12.3629	0.575795	0.287898	−	0.957661i	$-0.407044\pi$
$461$	12.3629	0.575795	0.287898	+	0.957661i	$0.407044\pi$
$462$	0	0
$463$	−21.1069	−0.980922	−0.490461	−	0.871463i	$-0.663172\pi$
$463$	−21.1069	−0.980922	−0.490461	+	0.871463i	$0.663172\pi$
$464$	0	0
$465$	−37.1562	−1.72308
$466$	0	0
$467$	−25.2729	−1.16949	−0.584745	−	0.811218i	$-0.698805\pi$
$467$	−25.2729	−1.16949	−0.584745	+	0.811218i	$0.698805\pi$
$468$	0	0
$469$	−1.62396	−0.0749875
$470$	0	0
$471$	−28.2291	−1.30073
$472$	0	0
$473$	3.71741	0.170927
$474$	0	0
$475$	−1.99353	−0.0914696
$476$	0	0
$477$	−27.0203	−1.23718
$478$	0	0
$479$	3.27613	0.149690	0.0748450	−	0.997195i	$-0.476154\pi$
$479$	3.27613	0.149690	0.0748450	+	0.997195i	$0.476154\pi$
$480$	0	0
$481$	−29.5555	−1.34761
$482$	0	0
$483$	−2.06913	−0.0941487
$484$	0	0
$485$	−8.69801	−0.394956
$486$	0	0
$487$	−14.0202	−0.635316	−0.317658	−	0.948205i	$-0.602896\pi$
$487$	−14.0202	−0.635316	−0.317658	+	0.948205i	$0.602896\pi$
$488$	0	0
$489$	−19.9904	−0.903997
$490$	0	0
$491$	21.6487	0.976990	0.488495	−	0.872567i	$-0.337546\pi$
$491$	21.6487	0.976990	0.488495	+	0.872567i	$0.337546\pi$
$492$	0	0
$493$	21.9859	0.990195
$494$	0	0
$495$	−6.07058	−0.272852
$496$	0	0
$497$	1.38225	0.0620023
$498$	0	0
$499$	34.4187	1.54079	0.770397	−	0.637564i	$-0.220058\pi$
$499$	34.4187	1.54079	0.770397	+	0.637564i	$0.220058\pi$
$500$	0	0
$501$	26.2462	1.17259
$502$	0	0
$503$	−14.3178	−0.638399	−0.319200	−	0.947687i	$-0.603414\pi$
$503$	−14.3178	−0.638399	−0.319200	+	0.947687i	$0.603414\pi$
$504$	0	0
$505$	0.651113	0.0289741
$506$	0	0
$507$	−6.63818	−0.294812
$508$	0	0
$509$	−17.3393	−0.768550	−0.384275	−	0.923219i	$-0.625549\pi$
$509$	−17.3393	−0.768550	−0.384275	+	0.923219i	$0.625549\pi$
$510$	0	0
$511$	−1.93476	−0.0855886
$512$	0	0
$513$	1.43845	0.0635090
$514$	0	0
$515$	−38.1134	−1.67948
$516$	0	0
$517$	−7.25268	−0.318972
$518$	0	0
$519$	−4.62108	−0.202843
$520$	0	0
$521$	30.5856	1.33998	0.669990	−	0.742370i	$-0.266298\pi$
$521$	30.5856	1.33998	0.669990	+	0.742370i	$0.266298\pi$
$522$	0	0
$523$	−22.7275	−0.993804	−0.496902	−	0.867807i	$-0.665529\pi$
$523$	−22.7275	−0.993804	−0.496902	+	0.867807i	$0.665529\pi$
$524$	0	0
$525$	0.924124	0.0403321
$526$	0	0
$527$	−30.5408	−1.33038
$528$	0	0
$529$	−3.07663	−0.133766
$530$	0	0
$531$	−42.2066	−1.83161
$532$	0	0
$533$	33.2163	1.43876
$534$	0	0
$535$	25.6456	1.10875
$536$	0	0
$537$	4.49242	0.193862
$538$	0	0
$539$	−4.49060	−0.193424
$540$	0	0
$541$	31.9913	1.37541	0.687707	−	0.725988i	$-0.258617\pi$
$541$	31.9913	1.37541	0.687707	+	0.725988i	$0.258617\pi$
$542$	0	0
$543$	16.7215	0.717590
$544$	0	0
$545$	51.5257	2.20712
$546$	0	0
$547$	16.1005	0.688406	0.344203	−	0.938895i	$-0.388149\pi$
$547$	16.1005	0.688406	0.344203	+	0.938895i	$0.388149\pi$
$548$	0	0
$549$	8.82758	0.376752
$550$	0	0
$551$	−3.94860	−0.168216
$552$	0	0
$553$	−1.91974	−0.0816357
$554$	0	0
$555$	−50.7044	−2.15228
$556$	0	0
$557$	14.3126	0.606445	0.303223	−	0.952920i	$-0.401937\pi$
$557$	14.3126	0.606445	0.303223	+	0.952920i	$0.401937\pi$
$558$	0	0
$559$	22.7741	0.963242
$560$	0	0
$561$	−9.19277	−0.388119
$562$	0	0
$563$	−13.9143	−0.586418	−0.293209	−	0.956048i	$-0.594723\pi$
$563$	−13.9143	−0.586418	−0.293209	+	0.956048i	$0.594723\pi$
$564$	0	0
$565$	19.2761	0.810953
$566$	0	0
$567$	1.26678	0.0531998
$568$	0	0
$569$	−7.71415	−0.323394	−0.161697	−	0.986840i	$-0.551697\pi$
$569$	−7.71415	−0.323394	−0.161697	+	0.986840i	$0.551697\pi$
$570$	0	0
$571$	9.42188	0.394294	0.197147	−	0.980374i	$-0.436832\pi$
$571$	9.42188	0.394294	0.197147	+	0.980374i	$0.436832\pi$
$572$	0	0
$573$	19.1017	0.797987
$574$	0	0
$575$	−8.89826	−0.371083
$576$	0	0
$577$	−10.3851	−0.432337	−0.216168	−	0.976356i	$-0.569356\pi$
$577$	−10.3851	−0.432337	−0.216168	+	0.976356i	$0.569356\pi$
$578$	0	0
$579$	−48.7215	−2.02480
$580$	0	0
$581$	1.02265	0.0424268
$582$	0	0
$583$	−4.88983	−0.202516
$584$	0	0
$585$	−37.1904	−1.53764
$586$	0	0
$587$	38.8866	1.60502	0.802510	−	0.596638i	$-0.203497\pi$
$587$	38.8866	1.60502	0.802510	+	0.596638i	$0.203497\pi$
$588$	0	0
$589$	5.48504	0.226007
$590$	0	0
$591$	0.391969	0.0161235
$592$	0	0
$593$	19.1304	0.785590	0.392795	−	0.919626i	$-0.371508\pi$
$593$	19.1304	0.785590	0.392795	+	0.919626i	$0.371508\pi$
$594$	0	0
$595$	2.66472	0.109243
$596$	0	0
$597$	−54.7333	−2.24008
$598$	0	0
$599$	−27.5353	−1.12506	−0.562530	−	0.826777i	$-0.690172\pi$
$599$	−27.5353	−1.12506	−0.562530	+	0.826777i	$0.690172\pi$
$600$	0	0
$601$	−5.73669	−0.234004	−0.117002	−	0.993132i	$-0.537328\pi$
$601$	−5.73669	−0.234004	−0.117002	+	0.993132i	$0.537328\pi$
$602$	0	0
$603$	31.9604	1.30153
$604$	0	0
$605$	27.9912	1.13801
$606$	0	0
$607$	25.3191	1.02767	0.513835	−	0.857889i	$-0.328224\pi$
$607$	25.3191	1.02767	0.513835	+	0.857889i	$0.328224\pi$
$608$	0	0
$609$	1.83042	0.0741722
$610$	0	0
$611$	−44.4324	−1.79754
$612$	0	0
$613$	−1.55872	−0.0629560	−0.0314780	−	0.999504i	$-0.510021\pi$
$613$	−1.55872	−0.0629560	−0.0314780	+	0.999504i	$0.510021\pi$
$614$	0	0
$615$	56.9848	2.29785
$616$	0	0
$617$	−11.6317	−0.468275	−0.234138	−	0.972203i	$-0.575227\pi$
$617$	−11.6317	−0.468275	−0.234138	+	0.972203i	$0.575227\pi$
$618$	0	0
$619$	−15.6754	−0.630046	−0.315023	−	0.949084i	$-0.602012\pi$
$619$	−15.6754	−0.630046	−0.315023	+	0.949084i	$0.602012\pi$
$620$	0	0
$621$	6.42060	0.257650
$622$	0	0
$623$	3.10783	0.124513
$624$	0	0
$625$	−30.9935	−1.23974
$626$	0	0
$627$	1.65100	0.0659344
$628$	0	0
$629$	−41.6769	−1.66177
$630$	0	0
$631$	−9.63171	−0.383432	−0.191716	−	0.981450i	$-0.561405\pi$
$631$	−9.63171	−0.383432	−0.191716	+	0.981450i	$0.561405\pi$
$632$	0	0
$633$	17.4716	0.694433
$634$	0	0
$635$	1.57407	0.0624649
$636$	0	0
$637$	−27.5109	−1.09002
$638$	0	0
$639$	−27.2034	−1.07615
$640$	0	0
$641$	−16.5150	−0.652302	−0.326151	−	0.945318i	$-0.605752\pi$
$641$	−16.5150	−0.652302	−0.326151	+	0.945318i	$0.605752\pi$
$642$	0	0
$643$	18.3887	0.725180	0.362590	−	0.931949i	$-0.381893\pi$
$643$	18.3887	0.725180	0.362590	+	0.931949i	$0.381893\pi$
$644$	0	0
$645$	39.0705	1.53840
$646$	0	0
$647$	17.3414	0.681760	0.340880	−	0.940107i	$-0.389275\pi$
$647$	17.3414	0.681760	0.340880	+	0.940107i	$0.389275\pi$
$648$	0	0
$649$	−7.63806	−0.299820
$650$	0	0
$651$	−2.54265	−0.0996543
$652$	0	0
$653$	3.46886	0.135747	0.0678734	−	0.997694i	$-0.478379\pi$
$653$	3.46886	0.135747	0.0678734	+	0.997694i	$0.478379\pi$
$654$	0	0
$655$	−41.7322	−1.63061
$656$	0	0
$657$	38.0770	1.48553
$658$	0	0
$659$	−36.6677	−1.42837	−0.714185	−	0.699957i	$-0.753202\pi$
$659$	−36.6677	−1.42837	−0.714185	+	0.699957i	$0.753202\pi$
$660$	0	0
$661$	48.6960	1.89405	0.947027	−	0.321153i	$-0.104070\pi$
$661$	48.6960	1.89405	0.947027	+	0.321153i	$0.104070\pi$
$662$	0	0
$663$	−56.3180	−2.18721
$664$	0	0
$665$	−0.478577	−0.0185584
$666$	0	0
$667$	−17.6248	−0.682436
$668$	0	0
$669$	21.5813	0.834382
$670$	0	0
$671$	1.59751	0.0616714
$672$	0	0
$673$	24.2535	0.934903	0.467451	−	0.884019i	$-0.345172\pi$
$673$	24.2535	0.934903	0.467451	+	0.884019i	$0.345172\pi$
$674$	0	0
$675$	−2.86759	−0.110374
$676$	0	0
$677$	−16.7471	−0.643642	−0.321821	−	0.946801i	$-0.604295\pi$
$677$	−16.7471	−0.643642	−0.321821	+	0.946801i	$0.604295\pi$
$678$	0	0
$679$	−0.595216	−0.0228423
$680$	0	0
$681$	59.9042	2.29553
$682$	0	0
$683$	−17.0299	−0.651632	−0.325816	−	0.945433i	$-0.605639\pi$
$683$	−17.0299	−0.651632	−0.325816	+	0.945433i	$0.605639\pi$
$684$	0	0
$685$	42.2601	1.61467
$686$	0	0
$687$	−8.64695	−0.329902
$688$	0	0
$689$	−29.9567	−1.14126
$690$	0	0
$691$	−38.8811	−1.47911	−0.739554	−	0.673097i	$-0.764964\pi$
$691$	−38.8811	−1.47911	−0.739554	+	0.673097i	$0.764964\pi$
$692$	0	0
$693$	−0.415418	−0.0157804
$694$	0	0
$695$	11.8437	0.449258
$696$	0	0
$697$	46.8391	1.77416
$698$	0	0
$699$	40.5514	1.53380
$700$	0	0
$701$	8.17888	0.308912	0.154456	−	0.988000i	$-0.450637\pi$
$701$	8.17888	0.308912	0.154456	+	0.988000i	$0.450637\pi$
$702$	0	0
$703$	7.48504	0.282304
$704$	0	0
$705$	−76.2268	−2.87087
$706$	0	0
$707$	0.0445565	0.00167572
$708$	0	0
$709$	−23.3522	−0.877011	−0.438505	−	0.898729i	$-0.644492\pi$
$709$	−23.3522	−0.877011	−0.438505	+	0.898729i	$0.644492\pi$
$710$	0	0
$711$	37.7815	1.41692
$712$	0	0
$713$	24.4828	0.916889
$714$	0	0
$715$	−6.73030	−0.251699
$716$	0	0
$717$	−6.14042	−0.229318
$718$	0	0
$719$	−24.1810	−0.901798	−0.450899	−	0.892575i	$-0.648897\pi$
$719$	−24.1810	−0.901798	−0.450899	+	0.892575i	$0.648897\pi$
$720$	0	0
$721$	−2.60815	−0.0971325
$722$	0	0
$723$	−13.9402	−0.518441
$724$	0	0
$725$	7.87167	0.292347
$726$	0	0
$727$	10.3503	0.383870	0.191935	−	0.981408i	$-0.438524\pi$
$727$	10.3503	0.383870	0.191935	+	0.981408i	$0.438524\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	32.1143	1.18779
$732$	0	0
$733$	1.81128	0.0669011	0.0334505	−	0.999440i	$-0.489350\pi$
$733$	1.81128	0.0669011	0.0334505	+	0.999440i	$0.489350\pi$
$734$	0	0
$735$	−47.1969	−1.74088
$736$	0	0
$737$	5.78382	0.213050
$738$	0	0
$739$	17.0342	0.626612	0.313306	−	0.949652i	$-0.398564\pi$
$739$	17.0342	0.626612	0.313306	+	0.949652i	$0.398564\pi$
$740$	0	0
$741$	10.1146	0.371567
$742$	0	0
$743$	0.992620	0.0364157	0.0182079	−	0.999834i	$-0.494204\pi$
$743$	0.992620	0.0364157	0.0182079	+	0.999834i	$0.494204\pi$
$744$	0	0
$745$	27.4391	1.00529
$746$	0	0
$747$	−20.1263	−0.736383
$748$	0	0
$749$	1.75496	0.0641248
$750$	0	0
$751$	40.1263	1.46423	0.732115	−	0.681181i	$-0.238533\pi$
$751$	40.1263	1.46423	0.732115	+	0.681181i	$0.238533\pi$
$752$	0	0
$753$	32.8917	1.19864
$754$	0	0
$755$	44.0536	1.60327
$756$	0	0
$757$	41.8810	1.52219	0.761096	−	0.648640i	$-0.224662\pi$
$757$	41.8810	1.52219	0.761096	+	0.648640i	$0.224662\pi$
$758$	0	0
$759$	7.36932	0.267489
$760$	0	0
$761$	23.8142	0.863265	0.431633	−	0.902050i	$-0.357938\pi$
$761$	23.8142	0.863265	0.431633	+	0.902050i	$0.357938\pi$
$762$	0	0
$763$	3.52597	0.127649
$764$	0	0
$765$	−52.4431	−1.89608
$766$	0	0
$767$	−46.7934	−1.68961
$768$	0	0
$769$	−47.5828	−1.71588	−0.857939	−	0.513751i	$-0.828256\pi$
$769$	−47.5828	−1.71588	−0.857939	+	0.513751i	$0.828256\pi$
$770$	0	0
$771$	49.1466	1.76997
$772$	0	0
$773$	−33.4240	−1.20218	−0.601090	−	0.799182i	$-0.705267\pi$
$773$	−33.4240	−1.20218	−0.601090	+	0.799182i	$0.705267\pi$
$774$	0	0
$775$	−10.9346	−0.392783
$776$	0	0
$777$	−3.46977	−0.124477
$778$	0	0
$779$	−8.41216	−0.301397
$780$	0	0
$781$	−4.92295	−0.176157
$782$	0	0
$783$	−5.67986	−0.202982
$784$	0	0
$785$	−29.1435	−1.04018
$786$	0	0
$787$	−0.315342	−0.0112407	−0.00562036	−	0.999984i	$-0.501789\pi$
$787$	−0.315342	−0.0112407	−0.00562036	+	0.999984i	$0.501789\pi$
$788$	0	0
$789$	−51.7685	−1.84301
$790$	0	0
$791$	1.31909	0.0469015
$792$	0	0
$793$	9.78692	0.347544
$794$	0	0
$795$	−51.3928	−1.82272
$796$	0	0
$797$	−11.0942	−0.392978	−0.196489	−	0.980506i	$-0.562954\pi$
$797$	−11.0942	−0.392978	−0.196489	+	0.980506i	$0.562954\pi$
$798$	0	0
$799$	−62.6551	−2.21658
$800$	0	0
$801$	−61.1637	−2.16111
$802$	0	0
$803$	6.89074	0.243169
$804$	0	0
$805$	−2.13616	−0.0752896
$806$	0	0
$807$	42.1176	1.48261
$808$	0	0
$809$	17.4021	0.611824	0.305912	−	0.952060i	$-0.401039\pi$
$809$	17.4021	0.611824	0.305912	+	0.952060i	$0.401039\pi$
$810$	0	0
$811$	44.2703	1.55454	0.777270	−	0.629167i	$-0.216604\pi$
$811$	44.2703	1.55454	0.777270	+	0.629167i	$0.216604\pi$
$812$	0	0
$813$	−20.5664	−0.721294
$814$	0	0
$815$	−20.6379	−0.722916
$816$	0	0
$817$	−5.76763	−0.201784
$818$	0	0
$819$	−2.54499	−0.0889291
$820$	0	0
$821$	−5.12402	−0.178830	−0.0894148	−	0.995994i	$-0.528500\pi$
$821$	−5.12402	−0.178830	−0.0894148	+	0.995994i	$0.528500\pi$
$822$	0	0
$823$	6.29435	0.219407	0.109704	−	0.993964i	$-0.465010\pi$
$823$	6.29435	0.219407	0.109704	+	0.993964i	$0.465010\pi$
$824$	0	0
$825$	−3.29131	−0.114589
$826$	0	0
$827$	28.9996	1.00841	0.504207	−	0.863583i	$-0.331785\pi$
$827$	28.9996	1.00841	0.504207	+	0.863583i	$0.331785\pi$
$828$	0	0
$829$	11.0215	0.382792	0.191396	−	0.981513i	$-0.438699\pi$
$829$	11.0215	0.382792	0.191396	+	0.981513i	$0.438699\pi$
$830$	0	0
$831$	24.8147	0.860813
$832$	0	0
$833$	−38.7938	−1.34412
$834$	0	0
$835$	27.0964	0.937710
$836$	0	0
$837$	7.88994	0.272716
$838$	0	0
$839$	14.8727	0.513464	0.256732	−	0.966483i	$-0.417354\pi$
$839$	14.8727	0.513464	0.256732	+	0.966483i	$0.417354\pi$
$840$	0	0
$841$	−13.4085	−0.462363
$842$	0	0
$843$	20.8844	0.719297
$844$	0	0
$845$	−6.85321	−0.235758
$846$	0	0
$847$	1.91548	0.0658166
$848$	0	0
$849$	71.6574	2.45928
$850$	0	0
$851$	33.4099	1.14528
$852$	0	0
$853$	−19.8801	−0.680682	−0.340341	−	0.940302i	$-0.610542\pi$
$853$	−19.8801	−0.680682	−0.340341	+	0.940302i	$0.610542\pi$
$854$	0	0
$855$	9.41863	0.322110
$856$	0	0
$857$	−23.9474	−0.818029	−0.409014	−	0.912528i	$-0.634127\pi$
$857$	−23.9474	−0.818029	−0.409014	+	0.912528i	$0.634127\pi$
$858$	0	0
$859$	44.5394	1.51966	0.759832	−	0.650119i	$-0.225281\pi$
$859$	44.5394	1.51966	0.759832	+	0.650119i	$0.225281\pi$
$860$	0	0
$861$	3.89955	0.132896
$862$	0	0
$863$	−7.47937	−0.254601	−0.127300	−	0.991864i	$-0.540631\pi$
$863$	−7.47937	−0.254601	−0.127300	+	0.991864i	$0.540631\pi$
$864$	0	0
$865$	−4.77077	−0.162211
$866$	0	0
$867$	−35.8690	−1.21818
$868$	0	0
$869$	6.83726	0.231938
$870$	0	0
$871$	35.4336	1.20062
$872$	0	0
$873$	11.7142	0.396464
$874$	0	0
$875$	−1.43882	−0.0486411
$876$	0	0
$877$	−43.0919	−1.45511	−0.727555	−	0.686049i	$-0.759343\pi$
$877$	−43.0919	−1.45511	−0.727555	+	0.686049i	$0.759343\pi$
$878$	0	0
$879$	−32.1317	−1.08377
$880$	0	0
$881$	12.4283	0.418722	0.209361	−	0.977838i	$-0.432862\pi$
$881$	12.4283	0.418722	0.209361	+	0.977838i	$0.432862\pi$
$882$	0	0
$883$	−34.0244	−1.14501	−0.572507	−	0.819900i	$-0.694029\pi$
$883$	−34.0244	−1.14501	−0.572507	+	0.819900i	$0.694029\pi$
$884$	0	0
$885$	−80.2772	−2.69849
$886$	0	0
$887$	45.2609	1.51971	0.759855	−	0.650092i	$-0.225270\pi$
$887$	45.2609	1.51971	0.759855	+	0.650092i	$0.225270\pi$
$888$	0	0
$889$	0.107715	0.00361266
$890$	0	0
$891$	−4.51170	−0.151148
$892$	0	0
$893$	11.2527	0.376556
$894$	0	0
$895$	4.63795	0.155029
$896$	0	0
$897$	45.1469	1.50741
$898$	0	0
$899$	−21.6583	−0.722343
$900$	0	0
$901$	−42.2427	−1.40731
$902$	0	0
$903$	2.67365	0.0889734
$904$	0	0
$905$	17.2632	0.573848
$906$	0	0
$907$	−24.7681	−0.822412	−0.411206	−	0.911542i	$-0.634892\pi$
$907$	−24.7681	−0.822412	−0.411206	+	0.911542i	$0.634892\pi$
$908$	0	0
$909$	−0.876894	−0.0290848
$910$	0	0
$911$	12.8875	0.426981	0.213491	−	0.976945i	$-0.431517\pi$
$911$	12.8875	0.426981	0.213491	+	0.976945i	$0.431517\pi$
$912$	0	0
$913$	−3.64223	−0.120540
$914$	0	0
$915$	16.7901	0.555064
$916$	0	0
$917$	−2.85579	−0.0943064
$918$	0	0
$919$	−33.0846	−1.09136	−0.545681	−	0.837993i	$-0.683729\pi$
$919$	−33.0846	−1.09136	−0.545681	+	0.837993i	$0.683729\pi$
$920$	0	0
$921$	−33.1904	−1.09366
$922$	0	0
$923$	−30.1597	−0.992718
$924$	0	0
$925$	−14.9217	−0.490622
$926$	0	0
$927$	51.3297	1.68589
$928$	0	0
$929$	48.0572	1.57671	0.788353	−	0.615224i	$-0.210934\pi$
$929$	48.0572	1.57671	0.788353	+	0.615224i	$0.210934\pi$
$930$	0	0
$931$	6.96725	0.228342
$932$	0	0
$933$	−81.5771	−2.67071
$934$	0	0
$935$	−9.49055	−0.310374
$936$	0	0
$937$	−4.71132	−0.153912	−0.0769560	−	0.997034i	$-0.524520\pi$
$937$	−4.71132	−0.153912	−0.0769560	+	0.997034i	$0.524520\pi$
$938$	0	0
$939$	−41.6580	−1.35946
$940$	0	0
$941$	55.0727	1.79532	0.897659	−	0.440690i	$-0.145266\pi$
$941$	55.0727	1.79532	0.897659	+	0.440690i	$0.145266\pi$
$942$	0	0
$943$	−37.5482	−1.22274
$944$	0	0
$945$	−0.688407	−0.0223939
$946$	0	0
$947$	−23.5984	−0.766846	−0.383423	−	0.923573i	$-0.625255\pi$
$947$	−23.5984	−0.766846	−0.383423	+	0.923573i	$0.625255\pi$
$948$	0	0
$949$	42.2150	1.37036
$950$	0	0
$951$	−17.6457	−0.572200
$952$	0	0
$953$	35.4123	1.14712	0.573558	−	0.819165i	$-0.305563\pi$
$953$	35.4123	1.14712	0.573558	+	0.819165i	$0.305563\pi$
$954$	0	0
$955$	19.7205	0.638140
$956$	0	0
$957$	−6.51912	−0.210733
$958$	0	0
$959$	2.89191	0.0933847
$960$	0	0
$961$	−0.914306	−0.0294938
$962$	0	0
$963$	−34.5385	−1.11299
$964$	0	0
$965$	−50.2998	−1.61921
$966$	0	0
$967$	58.6786	1.88698	0.943488	−	0.331407i	$-0.107523\pi$
$967$	58.6786	1.88698	0.943488	+	0.331407i	$0.107523\pi$
$968$	0	0
$969$	14.2628	0.458186
$970$	0	0
$971$	20.7839	0.666988	0.333494	−	0.942752i	$-0.391772\pi$
$971$	20.7839	0.666988	0.333494	+	0.942752i	$0.391772\pi$
$972$	0	0
$973$	0.810482	0.0259828
$974$	0	0
$975$	−20.1637	−0.645756
$976$	0	0
$977$	11.7111	0.374670	0.187335	−	0.982296i	$-0.440015\pi$
$977$	11.7111	0.374670	0.187335	+	0.982296i	$0.440015\pi$
$978$	0	0
$979$	−11.0687	−0.353758
$980$	0	0
$981$	−69.3928	−2.21554
$982$	0	0
$983$	−55.7921	−1.77949	−0.889745	−	0.456457i	$-0.849118\pi$
$983$	−55.7921	−1.77949	−0.889745	+	0.456457i	$0.849118\pi$
$984$	0	0
$985$	0.404666	0.0128937
$986$	0	0
$987$	−5.21630	−0.166037
$988$	0	0
$989$	−25.7442	−0.818617
$990$	0	0
$991$	−52.1167	−1.65554	−0.827771	−	0.561066i	$-0.810391\pi$
$991$	−52.1167	−1.65554	−0.827771	+	0.561066i	$0.810391\pi$
$992$	0	0
$993$	65.8257	2.08892
$994$	0	0
$995$	−56.5063	−1.79137
$996$	0	0
$997$	35.9871	1.13972	0.569862	−	0.821740i	$-0.306997\pi$
$997$	35.9871	1.13972	0.569862	+	0.821740i	$0.306997\pi$
$998$	0	0
$999$	10.7668	0.340648

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.a.w.1.1		4
4.3	odd	2		1216.2.a.x.1.3		4
8.3	odd	2		608.2.a.i.1.2	✓	4
8.5	even	2		608.2.a.j.1.4	yes	4
24.5	odd	2		5472.2.a.bs.1.2		4
24.11	even	2		5472.2.a.bt.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.2.a.i.1.2	✓	4	8.3	odd	2
608.2.a.j.1.4	yes	4	8.5	even	2
1216.2.a.w.1.1		4	1.1	even	1	trivial
1216.2.a.x.1.3		4	4.3	odd	2
5472.2.a.bs.1.2		4	24.5	odd	2
5472.2.a.bt.1.2		4	24.11	even	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56155 q^{3} -2.64453 q^{5} -0.180969 q^{7} +3.56155 q^{9} +O(q^{10})\)	\(q-2.56155 q^{3} -2.64453 q^{5} -0.180969 q^{7} +3.56155 q^{9} +0.644529 q^{11} +3.94860 q^{13} +6.77410 q^{15} +5.56802 q^{17} -1.00000 q^{19} +0.463560 q^{21} -4.46356 q^{23} +1.99353 q^{25} -1.43845 q^{27} +3.94860 q^{29} -5.48504 q^{31} -1.65100 q^{33} +0.478577 q^{35} -7.48504 q^{37} -10.1146 q^{39} +8.41216 q^{41} +5.76763 q^{43} -9.41863 q^{45} -11.2527 q^{47} -6.96725 q^{49} -14.2628 q^{51} -7.58667 q^{53} -1.70448 q^{55} +2.56155 q^{57} -11.8506 q^{59} +2.47858 q^{61} -0.644529 q^{63} -10.4422 q^{65} +8.97372 q^{67} +11.4336 q^{69} -7.63806 q^{71} +10.6911 q^{73} -5.10654 q^{75} -0.116639 q^{77} +10.6081 q^{79} -7.00000 q^{81} -5.65100 q^{83} -14.7248 q^{85} -10.1146 q^{87} -17.1733 q^{89} -0.714573 q^{91} +14.0502 q^{93} +2.64453 q^{95} +3.28906 q^{97} +2.29552 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - q^{5} - q^{7} + 6 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - q^5 - q^7 + 6 * q^9	\( 4 q - 2 q^{3} - q^{5} - q^{7} + 6 q^{9} - 7 q^{11} - 10 q^{13} - 8 q^{15} + 5 q^{17} - 4 q^{19} - 8 q^{21} - 8 q^{23} + 17 q^{25} - 14 q^{27} - 10 q^{29} - 6 q^{31} + 12 q^{33} - 5 q^{35} - 14 q^{37} - 12 q^{39} - 2 q^{41} - 3 q^{43} + 7 q^{45} - 3 q^{47} + 7 q^{49} + 6 q^{51} - 4 q^{53} - 35 q^{55} + 2 q^{57} - 20 q^{59} + 3 q^{61} + 7 q^{63} - 12 q^{65} - 8 q^{67} + 4 q^{69} - 30 q^{71} + 9 q^{73} - 34 q^{75} + 7 q^{77} + 10 q^{79} - 28 q^{81} - 4 q^{83} - 19 q^{85} - 12 q^{87} - 16 q^{89} - 10 q^{91} + 20 q^{93} + q^{95} - 6 q^{97} - 19 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - q^5 - q^7 + 6 * q^9 - 7 * q^11 - 10 * q^13 - 8 * q^15 + 5 * q^17 - 4 * q^19 - 8 * q^21 - 8 * q^23 + 17 * q^25 - 14 * q^27 - 10 * q^29 - 6 * q^31 + 12 * q^33 - 5 * q^35 - 14 * q^37 - 12 * q^39 - 2 * q^41 - 3 * q^43 + 7 * q^45 - 3 * q^47 + 7 * q^49 + 6 * q^51 - 4 * q^53 - 35 * q^55 + 2 * q^57 - 20 * q^59 + 3 * q^61 + 7 * q^63 - 12 * q^65 - 8 * q^67 + 4 * q^69 - 30 * q^71 + 9 * q^73 - 34 * q^75 + 7 * q^77 + 10 * q^79 - 28 * q^81 - 4 * q^83 - 19 * q^85 - 12 * q^87 - 16 * q^89 - 10 * q^91 + 20 * q^93 + q^95 - 6 * q^97 - 19 * q^99

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