LMFDB - Embedded newform 3008.2.a.n.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3008,2,Mod(1,3008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3008, 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("3008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3008 = 2^{6} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3008.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:	$24.0190009280$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4752.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 4x + 1 $ x^4 - 2x^3 - 3x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1504)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.49551$ of defining polynomial
Character	$\chi$	$=$	3008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.49551 q^{3} +2.73205 q^{5} -1.33133 q^{7} -0.763457 q^{9} +O(q^{10})$	$q+1.49551 q^{3} +2.73205 q^{5} -1.33133 q^{7} -0.763457 q^{9} -4.55889 q^{11} -4.62828 q^{13} +4.08580 q^{15} -0.573882 q^{17} -3.35375 q^{19} -1.99102 q^{21} +0.921626 q^{23} +2.46410 q^{25} -5.62828 q^{27} +2.90521 q^{29} -3.82684 q^{31} -6.81785 q^{33} -3.63726 q^{35} +8.60671 q^{37} -6.92163 q^{39} -8.08580 q^{41} -0.715637 q^{43} -2.08580 q^{45} -1.00000 q^{47} -5.22756 q^{49} -0.858244 q^{51} +7.03798 q^{53} -12.4551 q^{55} -5.01556 q^{57} -6.13277 q^{59} -1.89022 q^{61} +1.01641 q^{63} -12.6447 q^{65} +0.669239 q^{67} +1.37830 q^{69} -14.9416 q^{71} +5.24998 q^{73} +3.68508 q^{75} +6.06939 q^{77} +7.57147 q^{79} -6.12676 q^{81} -1.86122 q^{83} -1.56787 q^{85} +4.34477 q^{87} -15.5439 q^{89} +6.16177 q^{91} -5.72307 q^{93} -9.16262 q^{95} -0.438134 q^{97} +3.48052 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 4 q^{5} - 4 q^{7} - 2 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 4 * q^5 - 4 * q^7 - 2 * q^9	$ 4 q - 2 q^{3} + 4 q^{5} - 4 q^{7} - 2 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{15} - 6 q^{17} - 2 q^{19} + 8 q^{21} - 8 q^{23} - 4 q^{25} - 2 q^{27} + 14 q^{29} - 6 q^{31} - 2 q^{33} - 10 q^{35} + 10 q^{37} - 16 q^{39} - 14 q^{41} - 10 q^{43} + 10 q^{45} - 4 q^{47} - 6 q^{49} + 18 q^{53} - 20 q^{55} - 20 q^{57} - 12 q^{59} + 10 q^{61} - 10 q^{63} - 16 q^{65} - 16 q^{67} + 10 q^{69} - 6 q^{71} - 4 q^{73} + 2 q^{75} + 20 q^{77} + 2 q^{79} - 8 q^{81} - 16 q^{83} - 6 q^{85} - 10 q^{87} - 26 q^{89} - 14 q^{91} + 16 q^{95} - 6 q^{97} - 14 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 4 * q^5 - 4 * q^7 - 2 * q^9 - 2 * q^11 + 2 * q^13 - 2 * q^15 - 6 * q^17 - 2 * q^19 + 8 * q^21 - 8 * q^23 - 4 * q^25 - 2 * q^27 + 14 * q^29 - 6 * q^31 - 2 * q^33 - 10 * q^35 + 10 * q^37 - 16 * q^39 - 14 * q^41 - 10 * q^43 + 10 * q^45 - 4 * q^47 - 6 * q^49 + 18 * q^53 - 20 * q^55 - 20 * q^57 - 12 * q^59 + 10 * q^61 - 10 * q^63 - 16 * q^65 - 16 * q^67 + 10 * q^69 - 6 * q^71 - 4 * q^73 + 2 * q^75 + 20 * q^77 + 2 * q^79 - 8 * q^81 - 16 * q^83 - 6 * q^85 - 10 * q^87 - 26 * q^89 - 14 * q^91 + 16 * q^95 - 6 * q^97 - 14 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	0	0
$2$	0	0
$3$	1.49551	0.863432	0.431716	−	0.902010i	$-0.357908\pi$
$3$	1.49551	0.863432	0.431716	+	0.902010i	$0.357908\pi$
$4$	0	0
$5$	2.73205	1.22181	0.610905	−	0.791704i	$-0.290806\pi$
$5$	2.73205	1.22181	0.610905	+	0.791704i	$0.290806\pi$
$6$	0	0
$7$	−1.33133	−0.503196	−0.251598	−	0.967832i	$-0.580956\pi$
$7$	−1.33133	−0.503196	−0.251598	+	0.967832i	$0.580956\pi$
$8$	0	0
$9$	−0.763457	−0.254486
$10$	0	0
$11$	−4.55889	−1.37456	−0.687278	−	0.726394i	$-0.741195\pi$
$11$	−4.55889	−1.37456	−0.687278	+	0.726394i	$0.741195\pi$
$12$	0	0
$13$	−4.62828	−1.28365	−0.641827	−	0.766850i	$-0.721823\pi$
$13$	−4.62828	−1.28365	−0.641827	+	0.766850i	$0.721823\pi$
$14$	0	0
$15$	4.08580	1.05495
$16$	0	0
$17$	−0.573882	−0.139187	−0.0695934	−	0.997575i	$-0.522170\pi$
$17$	−0.573882	−0.139187	−0.0695934	+	0.997575i	$0.522170\pi$
$18$	0	0
$19$	−3.35375	−0.769403	−0.384702	−	0.923041i	$-0.625696\pi$
$19$	−3.35375	−0.769403	−0.384702	+	0.923041i	$0.625696\pi$
$20$	0	0
$21$	−1.99102	−0.434475
$22$	0	0
$23$	0.921626	0.192172	0.0960862	−	0.995373i	$-0.469368\pi$
$23$	0.921626	0.192172	0.0960862	+	0.995373i	$0.469368\pi$
$24$	0	0
$25$	2.46410	0.492820
$26$	0	0
$27$	−5.62828	−1.08316
$28$	0	0
$29$	2.90521	0.539484	0.269742	−	0.962933i	$-0.413062\pi$
$29$	2.90521	0.539484	0.269742	+	0.962933i	$0.413062\pi$
$30$	0	0
$31$	−3.82684	−0.687320	−0.343660	−	0.939094i	$-0.611667\pi$
$31$	−3.82684	−0.687320	−0.343660	+	0.939094i	$0.611667\pi$
$32$	0	0
$33$	−6.81785	−1.18684
$34$	0	0
$35$	−3.63726	−0.614810
$36$	0	0
$37$	8.60671	1.41493	0.707467	−	0.706746i	$-0.249838\pi$
$37$	8.60671	1.41493	0.707467	+	0.706746i	$0.249838\pi$
$38$	0	0
$39$	−6.92163	−1.10835
$40$	0	0
$41$	−8.08580	−1.26279	−0.631395	−	0.775461i	$-0.717517\pi$
$41$	−8.08580	−1.26279	−0.631395	+	0.775461i	$0.717517\pi$
$42$	0	0
$43$	−0.715637	−0.109134	−0.0545668	−	0.998510i	$-0.517378\pi$
$43$	−0.715637	−0.109134	−0.0545668	+	0.998510i	$0.517378\pi$
$44$	0	0
$45$	−2.08580	−0.310933
$46$	0	0
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−5.22756	−0.746794
$50$	0	0
$51$	−0.858244	−0.120178
$52$	0	0
$53$	7.03798	0.966741	0.483371	−	0.875416i	$-0.339412\pi$
$53$	7.03798	0.966741	0.483371	+	0.875416i	$0.339412\pi$
$54$	0	0
$55$	−12.4551	−1.67945
$56$	0	0
$57$	−5.01556	−0.664327
$58$	0	0
$59$	−6.13277	−0.798419	−0.399209	−	0.916860i	$-0.630715\pi$
$59$	−6.13277	−0.798419	−0.399209	+	0.916860i	$0.630715\pi$
$60$	0	0
$61$	−1.89022	−0.242018	−0.121009	−	0.992651i	$-0.538613\pi$
$61$	−1.89022	−0.242018	−0.121009	+	0.992651i	$0.538613\pi$
$62$	0	0
$63$	1.01641	0.128056
$64$	0	0
$65$	−12.6447	−1.56838
$66$	0	0
$67$	0.669239	0.0817605	0.0408803	−	0.999164i	$-0.486984\pi$
$67$	0.669239	0.0817605	0.0408803	+	0.999164i	$0.486984\pi$
$68$	0	0
$69$	1.37830	0.165928
$70$	0	0
$71$	−14.9416	−1.77325	−0.886623	−	0.462492i	$-0.846955\pi$
$71$	−14.9416	−1.77325	−0.886623	+	0.462492i	$0.846955\pi$
$72$	0	0
$73$	5.24998	0.614464	0.307232	−	0.951635i	$-0.400597\pi$
$73$	5.24998	0.614464	0.307232	+	0.951635i	$0.400597\pi$
$74$	0	0
$75$	3.68508	0.425517
$76$	0	0
$77$	6.06939	0.691671
$78$	0	0
$79$	7.57147	0.851857	0.425929	−	0.904757i	$-0.359947\pi$
$79$	7.57147	0.851857	0.425929	+	0.904757i	$0.359947\pi$
$80$	0	0
$81$	−6.12676	−0.680751
$82$	0	0
$83$	−1.86122	−0.204296	−0.102148	−	0.994769i	$-0.532571\pi$
$83$	−1.86122	−0.204296	−0.102148	+	0.994769i	$0.532571\pi$
$84$	0	0
$85$	−1.56787	−0.170060
$86$	0	0
$87$	4.34477	0.465808
$88$	0	0
$89$	−15.5439	−1.64765	−0.823825	−	0.566844i	$-0.808164\pi$
$89$	−15.5439	−1.64765	−0.823825	+	0.566844i	$0.808164\pi$
$90$	0	0
$91$	6.16177	0.645929
$92$	0	0
$93$	−5.72307	−0.593454
$94$	0	0
$95$	−9.16262	−0.940065
$96$	0	0
$97$	−0.438134	−0.0444858	−0.0222429	−	0.999753i	$-0.507081\pi$
$97$	−0.438134	−0.0444858	−0.0222429	+	0.999753i	$0.507081\pi$
$98$	0	0
$99$	3.48052	0.349805
$100$	0	0
$101$	11.2694	1.12134	0.560672	−	0.828038i	$-0.310543\pi$
$101$	11.2694	1.12134	0.560672	+	0.828038i	$0.310543\pi$
$102$	0	0
$103$	3.81185	0.375592	0.187796	−	0.982208i	$-0.439866\pi$
$103$	3.81185	0.375592	0.187796	+	0.982208i	$0.439866\pi$
$104$	0	0
$105$	−5.43955	−0.530846
$106$	0	0
$107$	−3.04181	−0.294063	−0.147032	−	0.989132i	$-0.546972\pi$
$107$	−3.04181	−0.294063	−0.147032	+	0.989132i	$0.546972\pi$
$108$	0	0
$109$	−14.2574	−1.36561	−0.682806	−	0.730600i	$-0.739241\pi$
$109$	−14.2574	−1.36561	−0.682806	+	0.730600i	$0.739241\pi$
$110$	0	0
$111$	12.8714	1.22170
$112$	0	0
$113$	10.1880	0.958408	0.479204	−	0.877703i	$-0.340925\pi$
$113$	10.1880	0.958408	0.479204	+	0.877703i	$0.340925\pi$
$114$	0	0
$115$	2.51793	0.234798
$116$	0	0
$117$	3.53349	0.326671
$118$	0	0
$119$	0.764026	0.0700382
$120$	0	0
$121$	9.78347	0.889406
$122$	0	0
$123$	−12.0924	−1.09033
$124$	0	0
$125$	−6.92820	−0.619677
$126$	0	0
$127$	−5.06040	−0.449038	−0.224519	−	0.974470i	$-0.572081\pi$
$127$	−5.06040	−0.449038	−0.224519	+	0.974470i	$0.572081\pi$
$128$	0	0
$129$	−1.07024	−0.0942295
$130$	0	0
$131$	2.58789	0.226105	0.113052	−	0.993589i	$-0.463937\pi$
$131$	2.58789	0.226105	0.113052	+	0.993589i	$0.463937\pi$
$132$	0	0
$133$	4.46495	0.387161
$134$	0	0
$135$	−15.3767	−1.32342
$136$	0	0
$137$	16.2012	1.38416	0.692080	−	0.721821i	$-0.256695\pi$
$137$	16.2012	1.38416	0.692080	+	0.721821i	$0.256695\pi$
$138$	0	0
$139$	19.6013	1.66256	0.831281	−	0.555852i	$-0.187608\pi$
$139$	19.6013	1.66256	0.831281	+	0.555852i	$0.187608\pi$
$140$	0	0
$141$	−1.49551	−0.125944
$142$	0	0
$143$	21.0998	1.76445
$144$	0	0
$145$	7.93719	0.659148
$146$	0	0
$147$	−7.81785	−0.644806
$148$	0	0
$149$	8.55591	0.700928	0.350464	−	0.936576i	$-0.386024\pi$
$149$	8.55591	0.700928	0.350464	+	0.936576i	$0.386024\pi$
$150$	0	0
$151$	−12.3448	−1.00460	−0.502301	−	0.864693i	$-0.667513\pi$
$151$	−12.3448	−1.00460	−0.502301	+	0.864693i	$0.667513\pi$
$152$	0	0
$153$	0.438134	0.0354210
$154$	0	0
$155$	−10.4551	−0.839775
$156$	0	0
$157$	21.5080	1.71652	0.858261	−	0.513214i	$-0.171545\pi$
$157$	21.5080	1.71652	0.858261	+	0.513214i	$0.171545\pi$
$158$	0	0
$159$	10.5254	0.834715
$160$	0	0
$161$	−1.22699	−0.0967003
$162$	0	0
$163$	−1.53505	−0.120234	−0.0601171	−	0.998191i	$-0.519147\pi$
$163$	−1.53505	−0.120234	−0.0601171	+	0.998191i	$0.519147\pi$
$164$	0	0
$165$	−18.6267	−1.45009
$166$	0	0
$167$	−17.3767	−1.34465	−0.672326	−	0.740255i	$-0.734705\pi$
$167$	−17.3767	−1.34465	−0.672326	+	0.740255i	$0.734705\pi$
$168$	0	0
$169$	8.42096	0.647766
$170$	0	0
$171$	2.56045	0.195802
$172$	0	0
$173$	16.8005	1.27731	0.638657	−	0.769491i	$-0.279490\pi$
$173$	16.8005	1.27731	0.638657	+	0.769491i	$0.279490\pi$
$174$	0	0
$175$	−3.28053	−0.247985
$176$	0	0
$177$	−9.17161	−0.689380
$178$	0	0
$179$	10.6327	0.794724	0.397362	−	0.917662i	$-0.369926\pi$
$179$	10.6327	0.794724	0.397362	+	0.917662i	$0.369926\pi$
$180$	0	0
$181$	−8.89977	−0.661515	−0.330757	−	0.943716i	$-0.607304\pi$
$181$	−8.89977	−0.661515	−0.330757	+	0.943716i	$0.607304\pi$
$182$	0	0
$183$	−2.82684	−0.208966
$184$	0	0
$185$	23.5140	1.72878
$186$	0	0
$187$	2.61626	0.191320
$188$	0	0
$189$	7.49310	0.545043
$190$	0	0
$191$	−21.7756	−1.57563	−0.787814	−	0.615913i	$-0.788787\pi$
$191$	−21.7756	−1.57563	−0.787814	+	0.615913i	$0.788787\pi$
$192$	0	0
$193$	−10.0908	−0.726353	−0.363177	−	0.931720i	$-0.618308\pi$
$193$	−10.0908	−0.726353	−0.363177	+	0.931720i	$0.618308\pi$
$194$	0	0
$195$	−18.9102	−1.35419
$196$	0	0
$197$	20.0908	1.43141	0.715706	−	0.698402i	$-0.246105\pi$
$197$	20.0908	1.43141	0.715706	+	0.698402i	$0.246105\pi$
$198$	0	0
$199$	12.6617	0.897566	0.448783	−	0.893641i	$-0.351858\pi$
$199$	12.6617	0.897566	0.448783	+	0.893641i	$0.351858\pi$
$200$	0	0
$201$	1.00085	0.0705946
$202$	0	0
$203$	−3.86780	−0.271466
$204$	0	0
$205$	−22.0908	−1.54289
$206$	0	0
$207$	−0.703622	−0.0489051
$208$	0	0
$209$	15.2894	1.05759
$210$	0	0
$211$	−11.1872	−0.770156	−0.385078	−	0.922884i	$-0.625826\pi$
$211$	−11.1872	−0.770156	−0.385078	+	0.922884i	$0.625826\pi$
$212$	0	0
$213$	−22.3453	−1.53108
$214$	0	0
$215$	−1.95516	−0.133341
$216$	0	0
$217$	5.09479	0.345857
$218$	0	0
$219$	7.85139	0.530547
$220$	0	0
$221$	2.65608	0.178668
$222$	0	0
$223$	−4.51135	−0.302102	−0.151051	−	0.988526i	$-0.548266\pi$
$223$	−4.51135	−0.302102	−0.151051	+	0.988526i	$0.548266\pi$
$224$	0	0
$225$	−1.88124	−0.125416
$226$	0	0
$227$	−25.7183	−1.70698	−0.853490	−	0.521109i	$-0.825519\pi$
$227$	−25.7183	−1.70698	−0.853490	+	0.521109i	$0.825519\pi$
$228$	0	0
$229$	10.3170	0.681764	0.340882	−	0.940106i	$-0.389274\pi$
$229$	10.3170	0.681764	0.340882	+	0.940106i	$0.389274\pi$
$230$	0	0
$231$	9.07682	0.597211
$232$	0	0
$233$	−5.74543	−0.376396	−0.188198	−	0.982131i	$-0.560265\pi$
$233$	−5.74543	−0.376396	−0.188198	+	0.982131i	$0.560265\pi$
$234$	0	0
$235$	−2.73205	−0.178219
$236$	0	0
$237$	11.3232	0.735521
$238$	0	0
$239$	−14.9924	−0.969780	−0.484890	−	0.874575i	$-0.661140\pi$
$239$	−14.9924	−0.969780	−0.484890	+	0.874575i	$0.661140\pi$
$240$	0	0
$241$	28.8035	1.85540	0.927698	−	0.373332i	$-0.121785\pi$
$241$	28.8035	1.85540	0.927698	+	0.373332i	$0.121785\pi$
$242$	0	0
$243$	7.72221	0.495380
$244$	0	0
$245$	−14.2820	−0.912441
$246$	0	0
$247$	15.5221	0.987647
$248$	0	0
$249$	−2.78347	−0.176395
$250$	0	0
$251$	29.4596	1.85947	0.929736	−	0.368227i	$-0.120035\pi$
$251$	29.4596	1.85947	0.929736	+	0.368227i	$0.120035\pi$
$252$	0	0
$253$	−4.20159	−0.264152
$254$	0	0
$255$	−2.34477	−0.146835
$256$	0	0
$257$	−5.38488	−0.335899	−0.167950	−	0.985796i	$-0.553715\pi$
$257$	−5.38488	−0.335899	−0.167950	+	0.985796i	$0.553715\pi$
$258$	0	0
$259$	−11.4584	−0.711989
$260$	0	0
$261$	−2.21800	−0.137291
$262$	0	0
$263$	−3.91037	−0.241124	−0.120562	−	0.992706i	$-0.538470\pi$
$263$	−3.91037	−0.241124	−0.120562	+	0.992706i	$0.538470\pi$
$264$	0	0
$265$	19.2281	1.18117
$266$	0	0
$267$	−23.2460	−1.42263
$268$	0	0
$269$	−6.32240	−0.385484	−0.192742	−	0.981250i	$-0.561738\pi$
$269$	−6.32240	−0.385484	−0.192742	+	0.981250i	$0.561738\pi$
$270$	0	0
$271$	−7.67913	−0.466474	−0.233237	−	0.972420i	$-0.574932\pi$
$271$	−7.67913	−0.466474	−0.233237	+	0.972420i	$0.574932\pi$
$272$	0	0
$273$	9.21497	0.557716
$274$	0	0
$275$	−11.2336	−0.677410
$276$	0	0
$277$	7.47210	0.448955	0.224477	−	0.974479i	$-0.427932\pi$
$277$	7.47210	0.448955	0.224477	+	0.974479i	$0.427932\pi$
$278$	0	0
$279$	2.92163	0.174913
$280$	0	0
$281$	−31.9586	−1.90649	−0.953246	−	0.302196i	$-0.902280\pi$
$281$	−31.9586	−1.90649	−0.953246	+	0.302196i	$0.902280\pi$
$282$	0	0
$283$	−31.3282	−1.86227	−0.931135	−	0.364676i	$-0.881180\pi$
$283$	−31.3282	−1.86227	−0.931135	+	0.364676i	$0.881180\pi$
$284$	0	0
$285$	−13.7028	−0.811682
$286$	0	0
$287$	10.7649	0.635431
$288$	0	0
$289$	−16.6707	−0.980627
$290$	0	0
$291$	−0.655233	−0.0384104
$292$	0	0
$293$	−11.8558	−0.692622	−0.346311	−	0.938120i	$-0.612566\pi$
$293$	−11.8558	−0.692622	−0.346311	+	0.938120i	$0.612566\pi$
$294$	0	0
$295$	−16.7550	−0.975516
$296$	0	0
$297$	25.6587	1.48887
$298$	0	0
$299$	−4.26554	−0.246683
$300$	0	0
$301$	0.952750	0.0549156
$302$	0	0
$303$	16.8534	0.968204
$304$	0	0
$305$	−5.16418	−0.295700
$306$	0	0
$307$	−13.3071	−0.759475	−0.379737	−	0.925094i	$-0.623986\pi$
$307$	−13.3071	−0.759475	−0.379737	+	0.925094i	$0.623986\pi$
$308$	0	0
$309$	5.70064	0.324298
$310$	0	0
$311$	17.2467	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$311$	17.2467	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$312$	0	0
$313$	0.293347	0.0165810	0.00829049	−	0.999966i	$-0.497361\pi$
$313$	0.293347	0.0165810	0.00829049	+	0.999966i	$0.497361\pi$
$314$	0	0
$315$	2.77689	0.156460
$316$	0	0
$317$	−0.774714	−0.0435123	−0.0217561	−	0.999763i	$-0.506926\pi$
$317$	−0.774714	−0.0435123	−0.0217561	+	0.999763i	$0.506926\pi$
$318$	0	0
$319$	−13.2445	−0.741552
$320$	0	0
$321$	−4.54905	−0.253903
$322$	0	0
$323$	1.92466	0.107091
$324$	0	0
$325$	−11.4045	−0.632611
$326$	0	0
$327$	−21.3221	−1.17911
$328$	0	0
$329$	1.33133	0.0733986
$330$	0	0
$331$	19.3282	1.06238	0.531188	−	0.847254i	$-0.321746\pi$
$331$	19.3282	1.06238	0.531188	+	0.847254i	$0.321746\pi$
$332$	0	0
$333$	−6.57085	−0.360080
$334$	0	0
$335$	1.82839	0.0998958
$336$	0	0
$337$	−2.99484	−0.163140	−0.0815698	−	0.996668i	$-0.525993\pi$
$337$	−2.99484	−0.163140	−0.0815698	+	0.996668i	$0.525993\pi$
$338$	0	0
$339$	15.2363	0.827520
$340$	0	0
$341$	17.4461	0.944761
$342$	0	0
$343$	16.2789	0.878979
$344$	0	0
$345$	3.76558	0.202732
$346$	0	0
$347$	16.4117	0.881026	0.440513	−	0.897746i	$-0.354797\pi$
$347$	16.4117	0.881026	0.440513	+	0.897746i	$0.354797\pi$
$348$	0	0
$349$	−1.95904	−0.104865	−0.0524325	−	0.998624i	$-0.516697\pi$
$349$	−1.95904	−0.104865	−0.0524325	+	0.998624i	$0.516697\pi$
$350$	0	0
$351$	26.0492	1.39041
$352$	0	0
$353$	−5.58874	−0.297459	−0.148729	−	0.988878i	$-0.547518\pi$
$353$	−5.58874	−0.297459	−0.148729	+	0.988878i	$0.547518\pi$
$354$	0	0
$355$	−40.8213	−2.16657
$356$	0	0
$357$	1.14261	0.0604732
$358$	0	0
$359$	−13.1642	−0.694779	−0.347389	−	0.937721i	$-0.612932\pi$
$359$	−13.1642	−0.694779	−0.347389	+	0.937721i	$0.612932\pi$
$360$	0	0
$361$	−7.75235	−0.408018
$362$	0	0
$363$	14.6313	0.767942
$364$	0	0
$365$	14.3432	0.750758
$366$	0	0
$367$	16.8043	0.877176	0.438588	−	0.898688i	$-0.355479\pi$
$367$	16.8043	0.877176	0.438588	+	0.898688i	$0.355479\pi$
$368$	0	0
$369$	6.17316	0.321362
$370$	0	0
$371$	−9.36988	−0.486460
$372$	0	0
$373$	−11.2030	−0.580067	−0.290034	−	0.957016i	$-0.593666\pi$
$373$	−11.2030	−0.580067	−0.290034	+	0.957016i	$0.593666\pi$
$374$	0	0
$375$	−10.3612	−0.535049
$376$	0	0
$377$	−13.4461	−0.692511
$378$	0	0
$379$	−1.06579	−0.0547459	−0.0273730	−	0.999625i	$-0.508714\pi$
$379$	−1.06579	−0.0547459	−0.0273730	+	0.999625i	$0.508714\pi$
$380$	0	0
$381$	−7.56787	−0.387714
$382$	0	0
$383$	2.11939	0.108296	0.0541478	−	0.998533i	$-0.482756\pi$
$383$	2.11939	0.108296	0.0541478	+	0.998533i	$0.482756\pi$
$384$	0	0
$385$	16.5819	0.845091
$386$	0	0
$387$	0.546358	0.0277729
$388$	0	0
$389$	−33.5555	−1.70133	−0.850667	−	0.525705i	$-0.823802\pi$
$389$	−33.5555	−1.70133	−0.850667	+	0.525705i	$0.823802\pi$
$390$	0	0
$391$	−0.528904	−0.0267478
$392$	0	0
$393$	3.87021	0.195226
$394$	0	0
$395$	20.6857	1.04081
$396$	0	0
$397$	−25.4661	−1.27811	−0.639055	−	0.769161i	$-0.720674\pi$
$397$	−25.4661	−1.27811	−0.639055	+	0.769161i	$0.720674\pi$
$398$	0	0
$399$	6.67737	0.334287
$400$	0	0
$401$	16.9631	0.847096	0.423548	−	0.905874i	$-0.360785\pi$
$401$	16.9631	0.847096	0.423548	+	0.905874i	$0.360785\pi$
$402$	0	0
$403$	17.7117	0.882281
$404$	0	0
$405$	−16.7386	−0.831749
$406$	0	0
$407$	−39.2370	−1.94491
$408$	0	0
$409$	21.9820	1.08694	0.543471	−	0.839428i	$-0.317110\pi$
$409$	21.9820	1.08694	0.543471	+	0.839428i	$0.317110\pi$
$410$	0	0
$411$	24.2290	1.19513
$412$	0	0
$413$	8.16475	0.401761
$414$	0	0
$415$	−5.08495	−0.249610
$416$	0	0
$417$	29.3139	1.43551
$418$	0	0
$419$	−27.5418	−1.34550	−0.672752	−	0.739868i	$-0.734888\pi$
$419$	−27.5418	−1.34550	−0.672752	+	0.739868i	$0.734888\pi$
$420$	0	0
$421$	31.8794	1.55371	0.776853	−	0.629682i	$-0.216815\pi$
$421$	31.8794	1.55371	0.776853	+	0.629682i	$0.216815\pi$
$422$	0	0
$423$	0.763457	0.0371205
$424$	0	0
$425$	−1.41410	−0.0685941
$426$	0	0
$427$	2.51651	0.121782
$428$	0	0
$429$	31.5549	1.52349
$430$	0	0
$431$	−14.9353	−0.719410	−0.359705	−	0.933066i	$-0.617123\pi$
$431$	−14.9353	−0.719410	−0.359705	+	0.933066i	$0.617123\pi$
$432$	0	0
$433$	25.8887	1.24413	0.622066	−	0.782965i	$-0.286293\pi$
$433$	25.8887	1.24413	0.622066	+	0.782965i	$0.286293\pi$
$434$	0	0
$435$	11.8701	0.569129
$436$	0	0
$437$	−3.09091	−0.147858
$438$	0	0
$439$	−35.9520	−1.71590	−0.857949	−	0.513735i	$-0.828261\pi$
$439$	−35.9520	−1.71590	−0.857949	+	0.513735i	$0.828261\pi$
$440$	0	0
$441$	3.99102	0.190048
$442$	0	0
$443$	10.3753	0.492944	0.246472	−	0.969150i	$-0.420729\pi$
$443$	10.3753	0.492944	0.246472	+	0.969150i	$0.420729\pi$
$444$	0	0
$445$	−42.4667	−2.01312
$446$	0	0
$447$	12.7954	0.605203
$448$	0	0
$449$	7.42159	0.350246	0.175123	−	0.984547i	$-0.443968\pi$
$449$	7.42159	0.350246	0.175123	+	0.984547i	$0.443968\pi$
$450$	0	0
$451$	36.8623	1.73578
$452$	0	0
$453$	−18.4617	−0.867406
$454$	0	0
$455$	16.8343	0.789203
$456$	0	0
$457$	−11.8992	−0.556621	−0.278311	−	0.960491i	$-0.589774\pi$
$457$	−11.8992	−0.556621	−0.278311	+	0.960491i	$0.589774\pi$
$458$	0	0
$459$	3.22997	0.150762
$460$	0	0
$461$	−9.51008	−0.442929	−0.221464	−	0.975168i	$-0.571084\pi$
$461$	−9.51008	−0.442929	−0.221464	+	0.975168i	$0.571084\pi$
$462$	0	0
$463$	−8.90480	−0.413841	−0.206920	−	0.978358i	$-0.566344\pi$
$463$	−8.90480	−0.413841	−0.206920	+	0.978358i	$0.566344\pi$
$464$	0	0
$465$	−15.6357	−0.725089
$466$	0	0
$467$	34.2504	1.58492	0.792460	−	0.609924i	$-0.208800\pi$
$467$	34.2504	1.58492	0.792460	+	0.609924i	$0.208800\pi$
$468$	0	0
$469$	−0.890978	−0.0411415
$470$	0	0
$471$	32.1653	1.48210
$472$	0	0
$473$	3.26251	0.150010
$474$	0	0
$475$	−8.26399	−0.379178
$476$	0	0
$477$	−5.37320	−0.246022
$478$	0	0
$479$	39.8773	1.82204	0.911020	−	0.412363i	$-0.135297\pi$
$479$	39.8773	1.82204	0.911020	+	0.412363i	$0.135297\pi$
$480$	0	0
$481$	−39.8342	−1.81629
$482$	0	0
$483$	−1.83497	−0.0834941
$484$	0	0
$485$	−1.19700	−0.0543532
$486$	0	0
$487$	−9.24340	−0.418859	−0.209429	−	0.977824i	$-0.567161\pi$
$487$	−9.24340	−0.418859	−0.209429	+	0.977824i	$0.567161\pi$
$488$	0	0
$489$	−2.29567	−0.103814
$490$	0	0
$491$	39.4030	1.77823	0.889117	−	0.457679i	$-0.151319\pi$
$491$	39.4030	1.77823	0.889117	+	0.457679i	$0.151319\pi$
$492$	0	0
$493$	−1.66725	−0.0750891
$494$	0	0
$495$	9.50894	0.427395
$496$	0	0
$497$	19.8923	0.892290
$498$	0	0
$499$	−23.0654	−1.03255	−0.516275	−	0.856423i	$-0.672682\pi$
$499$	−23.0654	−1.03255	−0.516275	+	0.856423i	$0.672682\pi$
$500$	0	0
$501$	−25.9871	−1.16102
$502$	0	0
$503$	23.0702	1.02865	0.514326	−	0.857595i	$-0.328042\pi$
$503$	23.0702	1.02865	0.514326	+	0.857595i	$0.328042\pi$
$504$	0	0
$505$	30.7885	1.37007
$506$	0	0
$507$	12.5936	0.559302
$508$	0	0
$509$	−20.0608	−0.889181	−0.444591	−	0.895734i	$-0.646651\pi$
$509$	−20.0608	−0.889181	−0.444591	+	0.895734i	$0.646651\pi$
$510$	0	0
$511$	−6.98946	−0.309195
$512$	0	0
$513$	18.8759	0.833389
$514$	0	0
$515$	10.4142	0.458903
$516$	0	0
$517$	4.55889	0.200500
$518$	0	0
$519$	25.1252	1.10287
$520$	0	0
$521$	−30.3346	−1.32898	−0.664491	−	0.747296i	$-0.731352\pi$
$521$	−30.3346	−1.32898	−0.664491	+	0.747296i	$0.731352\pi$
$522$	0	0
$523$	−32.5771	−1.42450	−0.712248	−	0.701928i	$-0.752323\pi$
$523$	−32.5771	−1.42450	−0.712248	+	0.701928i	$0.752323\pi$
$524$	0	0
$525$	−4.90606	−0.214118
$526$	0	0
$527$	2.19615	0.0956659
$528$	0	0
$529$	−22.1506	−0.963070
$530$	0	0
$531$	4.68211	0.203186
$532$	0	0
$533$	37.4233	1.62099
$534$	0	0
$535$	−8.31038	−0.359289
$536$	0	0
$537$	15.9012	0.686190
$538$	0	0
$539$	23.8319	1.02651
$540$	0	0
$541$	10.9292	0.469883	0.234941	−	0.972010i	$-0.424510\pi$
$541$	10.9292	0.469883	0.234941	+	0.972010i	$0.424510\pi$
$542$	0	0
$543$	−13.3097	−0.571173
$544$	0	0
$545$	−38.9520	−1.66852
$546$	0	0
$547$	−24.1079	−1.03078	−0.515389	−	0.856956i	$-0.672353\pi$
$547$	−24.1079	−1.03078	−0.515389	+	0.856956i	$0.672353\pi$
$548$	0	0
$549$	1.44310	0.0615901
$550$	0	0
$551$	−9.74336	−0.415081
$552$	0	0
$553$	−10.0801	−0.428651
$554$	0	0
$555$	35.1653	1.49268
$556$	0	0
$557$	2.56920	0.108861	0.0544303	−	0.998518i	$-0.482666\pi$
$557$	2.56920	0.108861	0.0544303	+	0.998518i	$0.482666\pi$
$558$	0	0
$559$	3.31217	0.140090
$560$	0	0
$561$	3.91264	0.165192
$562$	0	0
$563$	1.69837	0.0715778	0.0357889	−	0.999359i	$-0.488606\pi$
$563$	1.69837	0.0715778	0.0357889	+	0.999359i	$0.488606\pi$
$564$	0	0
$565$	27.8342	1.17099
$566$	0	0
$567$	8.15675	0.342551
$568$	0	0
$569$	−27.6133	−1.15761	−0.578806	−	0.815465i	$-0.696481\pi$
$569$	−27.6133	−1.15761	−0.578806	+	0.815465i	$0.696481\pi$
$570$	0	0
$571$	−8.99839	−0.376571	−0.188285	−	0.982114i	$-0.560293\pi$
$571$	−8.99839	−0.376571	−0.188285	+	0.982114i	$0.560293\pi$
$572$	0	0
$573$	−32.5656	−1.36045
$574$	0	0
$575$	2.27098	0.0947064
$576$	0	0
$577$	−3.93217	−0.163698	−0.0818491	−	0.996645i	$-0.526083\pi$
$577$	−3.93217	−0.163698	−0.0818491	+	0.996645i	$0.526083\pi$
$578$	0	0
$579$	−15.0909	−0.627157
$580$	0	0
$581$	2.47790	0.102801
$582$	0	0
$583$	−32.0854	−1.32884
$584$	0	0
$585$	9.65368	0.399130
$586$	0	0
$587$	−12.0100	−0.495708	−0.247854	−	0.968797i	$-0.579725\pi$
$587$	−12.0100	−0.495708	−0.247854	+	0.968797i	$0.579725\pi$
$588$	0	0
$589$	12.8343	0.528827
$590$	0	0
$591$	30.0460	1.23593
$592$	0	0
$593$	−36.2379	−1.48811	−0.744056	−	0.668118i	$-0.767100\pi$
$593$	−36.2379	−1.48811	−0.744056	+	0.668118i	$0.767100\pi$
$594$	0	0
$595$	2.08736	0.0855734
$596$	0	0
$597$	18.9357	0.774987
$598$	0	0
$599$	19.9221	0.813993	0.406997	−	0.913430i	$-0.366576\pi$
$599$	19.9221	0.813993	0.406997	+	0.913430i	$0.366576\pi$
$600$	0	0
$601$	−31.3304	−1.27799	−0.638997	−	0.769209i	$-0.720650\pi$
$601$	−31.3304	−1.27799	−0.638997	+	0.769209i	$0.720650\pi$
$602$	0	0
$603$	−0.510935	−0.0208069
$604$	0	0
$605$	26.7289	1.08669
$606$	0	0
$607$	21.9871	0.892427	0.446213	−	0.894927i	$-0.352772\pi$
$607$	21.9871	0.892427	0.446213	+	0.894927i	$0.352772\pi$
$608$	0	0
$609$	−5.78432	−0.234393
$610$	0	0
$611$	4.62828	0.187240
$612$	0	0
$613$	−22.2454	−0.898485	−0.449243	−	0.893410i	$-0.648306\pi$
$613$	−22.2454	−0.898485	−0.449243	+	0.893410i	$0.648306\pi$
$614$	0	0
$615$	−33.0370	−1.33218
$616$	0	0
$617$	−2.95899	−0.119124	−0.0595621	−	0.998225i	$-0.518970\pi$
$617$	−2.95899	−0.119124	−0.0595621	+	0.998225i	$0.518970\pi$
$618$	0	0
$619$	47.6368	1.91469	0.957343	−	0.288953i	$-0.0933072\pi$
$619$	47.6368	1.91469	0.957343	+	0.288953i	$0.0933072\pi$
$620$	0	0
$621$	−5.18717	−0.208154
$622$	0	0
$623$	20.6941	0.829090
$624$	0	0
$625$	−31.2487	−1.24995
$626$	0	0
$627$	22.8654	0.913156
$628$	0	0
$629$	−4.93923	−0.196940
$630$	0	0
$631$	27.5871	1.09823	0.549113	−	0.835748i	$-0.314966\pi$
$631$	27.5871	1.09823	0.549113	+	0.835748i	$0.314966\pi$
$632$	0	0
$633$	−16.7305	−0.664977
$634$	0	0
$635$	−13.8253	−0.548640
$636$	0	0
$637$	24.1946	0.958625
$638$	0	0
$639$	11.4073	0.451266
$640$	0	0
$641$	−31.2086	−1.23267	−0.616333	−	0.787486i	$-0.711382\pi$
$641$	−31.2086	−1.23267	−0.616333	+	0.787486i	$0.711382\pi$
$642$	0	0
$643$	47.7306	1.88231	0.941155	−	0.337976i	$-0.109742\pi$
$643$	47.7306	1.88231	0.941155	+	0.337976i	$0.109742\pi$
$644$	0	0
$645$	−2.92395	−0.115131
$646$	0	0
$647$	38.3427	1.50741	0.753704	−	0.657215i	$-0.228265\pi$
$647$	38.3427	1.50741	0.753704	+	0.657215i	$0.228265\pi$
$648$	0	0
$649$	27.9586	1.09747
$650$	0	0
$651$	7.61929	0.298624
$652$	0	0
$653$	−13.6737	−0.535093	−0.267546	−	0.963545i	$-0.586213\pi$
$653$	−13.6737	−0.535093	−0.267546	+	0.963545i	$0.586213\pi$
$654$	0	0
$655$	7.07024	0.276257
$656$	0	0
$657$	−4.00813	−0.156372
$658$	0	0
$659$	−47.0756	−1.83381	−0.916903	−	0.399111i	$-0.869319\pi$
$659$	−47.0756	−1.83381	−0.916903	+	0.399111i	$0.869319\pi$
$660$	0	0
$661$	−37.9738	−1.47701	−0.738504	−	0.674249i	$-0.764467\pi$
$661$	−37.9738	−1.47701	−0.738504	+	0.674249i	$0.764467\pi$
$662$	0	0
$663$	3.97219	0.154267
$664$	0	0
$665$	12.1985	0.473037
$666$	0	0
$667$	2.67752	0.103674
$668$	0	0
$669$	−6.74676	−0.260845
$670$	0	0
$671$	8.61730	0.332667
$672$	0	0
$673$	9.86298	0.380190	0.190095	−	0.981766i	$-0.439120\pi$
$673$	9.86298	0.380190	0.190095	+	0.981766i	$0.439120\pi$
$674$	0	0
$675$	−13.8687	−0.533805
$676$	0	0
$677$	8.08078	0.310570	0.155285	−	0.987870i	$-0.450370\pi$
$677$	8.08078	0.310570	0.155285	+	0.987870i	$0.450370\pi$
$678$	0	0
$679$	0.583301	0.0223850
$680$	0	0
$681$	−38.4618	−1.47386
$682$	0	0
$683$	33.6100	1.28605	0.643026	−	0.765844i	$-0.277679\pi$
$683$	33.6100	1.28605	0.643026	+	0.765844i	$0.277679\pi$
$684$	0	0
$685$	44.2624	1.69118
$686$	0	0
$687$	15.4291	0.588657
$688$	0	0
$689$	−32.5737	−1.24096
$690$	0	0
$691$	−14.7081	−0.559524	−0.279762	−	0.960069i	$-0.590255\pi$
$691$	−14.7081	−0.559524	−0.279762	+	0.960069i	$0.590255\pi$
$692$	0	0
$693$	−4.63372	−0.176020
$694$	0	0
$695$	53.5518	2.03134
$696$	0	0
$697$	4.64029	0.175764
$698$	0	0
$699$	−8.59234	−0.324992
$700$	0	0
$701$	40.1957	1.51817	0.759086	−	0.650990i	$-0.225646\pi$
$701$	40.1957	1.51817	0.759086	+	0.650990i	$0.225646\pi$
$702$	0	0
$703$	−28.8648	−1.08866
$704$	0	0
$705$	−4.08580	−0.153880
$706$	0	0
$707$	−15.0033	−0.564256
$708$	0	0
$709$	28.3083	1.06314	0.531569	−	0.847015i	$-0.321602\pi$
$709$	28.3083	1.06314	0.531569	+	0.847015i	$0.321602\pi$
$710$	0	0
$711$	−5.78049	−0.216785
$712$	0	0
$713$	−3.52691	−0.132084
$714$	0	0
$715$	57.6458	2.15583
$716$	0	0
$717$	−22.4213	−0.837339
$718$	0	0
$719$	12.0671	0.450028	0.225014	−	0.974356i	$-0.427757\pi$
$719$	12.0671	0.450028	0.225014	+	0.974356i	$0.427757\pi$
$720$	0	0
$721$	−5.07483	−0.188996
$722$	0	0
$723$	43.0758	1.60201
$724$	0	0
$725$	7.15874	0.265869
$726$	0	0
$727$	42.8793	1.59031	0.795153	−	0.606409i	$-0.207391\pi$
$727$	42.8793	1.59031	0.795153	+	0.606409i	$0.207391\pi$
$728$	0	0
$729$	29.9289	1.10848
$730$	0	0
$731$	0.410691	0.0151900
$732$	0	0
$733$	−11.7604	−0.434381	−0.217191	−	0.976129i	$-0.569689\pi$
$733$	−11.7604	−0.434381	−0.217191	+	0.976129i	$0.569689\pi$
$734$	0	0
$735$	−21.3588	−0.787830
$736$	0	0
$737$	−3.05099	−0.112384
$738$	0	0
$739$	−31.7177	−1.16675	−0.583377	−	0.812202i	$-0.698269\pi$
$739$	−31.7177	−1.16675	−0.583377	+	0.812202i	$0.698269\pi$
$740$	0	0
$741$	23.2134	0.852766
$742$	0	0
$743$	−20.9387	−0.768165	−0.384083	−	0.923299i	$-0.625482\pi$
$743$	−20.9387	−0.768165	−0.384083	+	0.923299i	$0.625482\pi$
$744$	0	0
$745$	23.3752	0.856401
$746$	0	0
$747$	1.42096	0.0519903
$748$	0	0
$749$	4.04966	0.147971
$750$	0	0
$751$	−1.36846	−0.0499359	−0.0249680	−	0.999688i	$-0.507948\pi$
$751$	−1.36846	−0.0499359	−0.0249680	+	0.999688i	$0.507948\pi$
$752$	0	0
$753$	44.0570	1.60553
$754$	0	0
$755$	−33.7265	−1.22743
$756$	0	0
$757$	34.8208	1.26558	0.632792	−	0.774322i	$-0.281909\pi$
$757$	34.8208	1.26558	0.632792	+	0.774322i	$0.281909\pi$
$758$	0	0
$759$	−6.28351	−0.228077
$760$	0	0
$761$	−2.88818	−0.104696	−0.0523481	−	0.998629i	$-0.516671\pi$
$761$	−2.88818	−0.104696	−0.0523481	+	0.998629i	$0.516671\pi$
$762$	0	0
$763$	18.9813	0.687170
$764$	0	0
$765$	1.19700	0.0432778
$766$	0	0
$767$	28.3842	1.02489
$768$	0	0
$769$	4.96505	0.179044	0.0895221	−	0.995985i	$-0.471466\pi$
$769$	4.96505	0.179044	0.0895221	+	0.995985i	$0.471466\pi$
$770$	0	0
$771$	−8.05312	−0.290026
$772$	0	0
$773$	20.6510	0.742764	0.371382	−	0.928480i	$-0.378884\pi$
$773$	20.6510	0.742764	0.371382	+	0.928480i	$0.378884\pi$
$774$	0	0
$775$	−9.42972	−0.338725
$776$	0	0
$777$	−17.1361	−0.614754
$778$	0	0
$779$	27.1178	0.971595
$780$	0	0
$781$	68.1173	2.43743
$782$	0	0
$783$	−16.3513	−0.584349
$784$	0	0
$785$	58.7608	2.09726
$786$	0	0
$787$	18.5075	0.659720	0.329860	−	0.944030i	$-0.392998\pi$
$787$	18.5075	0.659720	0.329860	+	0.944030i	$0.392998\pi$
$788$	0	0
$789$	−5.84799	−0.208194
$790$	0	0
$791$	−13.5636	−0.482267
$792$	0	0
$793$	8.74846	0.310667
$794$	0	0
$795$	28.7558	1.01986
$796$	0	0
$797$	19.7657	0.700138	0.350069	−	0.936724i	$-0.386158\pi$
$797$	19.7657	0.700138	0.350069	+	0.936724i	$0.386158\pi$
$798$	0	0
$799$	0.573882	0.0203025
$800$	0	0
$801$	11.8671	0.419303
$802$	0	0
$803$	−23.9341	−0.844615
$804$	0	0
$805$	−3.35220	−0.118149
$806$	0	0
$807$	−9.45520	−0.332839
$808$	0	0
$809$	−31.7574	−1.11653	−0.558266	−	0.829662i	$-0.688533\pi$
$809$	−31.7574	−1.11653	−0.558266	+	0.829662i	$0.688533\pi$
$810$	0	0
$811$	−41.6071	−1.46102	−0.730512	−	0.682900i	$-0.760719\pi$
$811$	−41.6071	−1.46102	−0.730512	+	0.682900i	$0.760719\pi$
$812$	0	0
$813$	−11.4842	−0.402768
$814$	0	0
$815$	−4.19383	−0.146903
$816$	0	0
$817$	2.40007	0.0839678
$818$	0	0
$819$	−4.70425	−0.164380
$820$	0	0
$821$	25.2681	0.881863	0.440931	−	0.897541i	$-0.354648\pi$
$821$	25.2681	0.881863	0.440931	+	0.897541i	$0.354648\pi$
$822$	0	0
$823$	−20.4328	−0.712242	−0.356121	−	0.934440i	$-0.615901\pi$
$823$	−20.4328	−0.712242	−0.356121	+	0.934440i	$0.615901\pi$
$824$	0	0
$825$	−16.7999	−0.584897
$826$	0	0
$827$	−1.18774	−0.0413017	−0.0206508	−	0.999787i	$-0.506574\pi$
$827$	−1.18774	−0.0413017	−0.0206508	+	0.999787i	$0.506574\pi$
$828$	0	0
$829$	−1.94702	−0.0676229	−0.0338115	−	0.999428i	$-0.510765\pi$
$829$	−1.94702	−0.0676229	−0.0338115	+	0.999428i	$0.510765\pi$
$830$	0	0
$831$	11.1746	0.387642
$832$	0	0
$833$	3.00000	0.103944
$834$	0	0
$835$	−47.4741	−1.64291
$836$	0	0
$837$	21.5385	0.744480
$838$	0	0
$839$	−6.43171	−0.222047	−0.111024	−	0.993818i	$-0.535413\pi$
$839$	−6.43171	−0.222047	−0.111024	+	0.993818i	$0.535413\pi$
$840$	0	0
$841$	−20.5597	−0.708957
$842$	0	0
$843$	−47.7944	−1.64613
$844$	0	0
$845$	23.0065	0.791448
$846$	0	0
$847$	−13.0250	−0.447546
$848$	0	0
$849$	−46.8516	−1.60794
$850$	0	0
$851$	7.93217	0.271911
$852$	0	0
$853$	−36.8512	−1.26176	−0.630880	−	0.775880i	$-0.717306\pi$
$853$	−36.8512	−1.26176	−0.630880	+	0.775880i	$0.717306\pi$
$854$	0	0
$855$	6.99527	0.239233
$856$	0	0
$857$	13.7520	0.469760	0.234880	−	0.972024i	$-0.424530\pi$
$857$	13.7520	0.469760	0.234880	+	0.972024i	$0.424530\pi$
$858$	0	0
$859$	−51.7397	−1.76534	−0.882668	−	0.469998i	$-0.844255\pi$
$859$	−51.7397	−1.76534	−0.882668	+	0.469998i	$0.844255\pi$
$860$	0	0
$861$	16.0990	0.548651
$862$	0	0
$863$	28.4882	0.969750	0.484875	−	0.874583i	$-0.338865\pi$
$863$	28.4882	0.969750	0.484875	+	0.874583i	$0.338865\pi$
$864$	0	0
$865$	45.8997	1.56064
$866$	0	0
$867$	−24.9311	−0.846705
$868$	0	0
$869$	−34.5175	−1.17093
$870$	0	0
$871$	−3.09742	−0.104952
$872$	0	0
$873$	0.334496	0.0113210
$874$	0	0
$875$	9.22373	0.311819
$876$	0	0
$877$	1.84388	0.0622632	0.0311316	−	0.999515i	$-0.490089\pi$
$877$	1.84388	0.0622632	0.0311316	+	0.999515i	$0.490089\pi$
$878$	0	0
$879$	−17.7304	−0.598032
$880$	0	0
$881$	44.2342	1.49029	0.745144	−	0.666903i	$-0.232381\pi$
$881$	44.2342	1.49029	0.745144	+	0.666903i	$0.232381\pi$
$882$	0	0
$883$	8.82996	0.297152	0.148576	−	0.988901i	$-0.452531\pi$
$883$	8.82996	0.297152	0.148576	+	0.988901i	$0.452531\pi$
$884$	0	0
$885$	−25.0573	−0.842292
$886$	0	0
$887$	−11.6029	−0.389587	−0.194793	−	0.980844i	$-0.562404\pi$
$887$	−11.6029	−0.389587	−0.194793	+	0.980844i	$0.562404\pi$
$888$	0	0
$889$	6.73707	0.225954
$890$	0	0
$891$	27.9312	0.935732
$892$	0	0
$893$	3.35375	0.112229
$894$	0	0
$895$	29.0490	0.971001
$896$	0	0
$897$	−6.37915	−0.212994
$898$	0	0
$899$	−11.1178	−0.370799
$900$	0	0
$901$	−4.03897	−0.134558
$902$	0	0
$903$	1.42484	0.0474159
$904$	0	0
$905$	−24.3146	−0.808246
$906$	0	0
$907$	−49.8193	−1.65422	−0.827112	−	0.562037i	$-0.810018\pi$
$907$	−49.8193	−1.65422	−0.827112	+	0.562037i	$0.810018\pi$
$908$	0	0
$909$	−8.60368	−0.285366
$910$	0	0
$911$	16.1716	0.535787	0.267894	−	0.963448i	$-0.413672\pi$
$911$	16.1716	0.535787	0.267894	+	0.963448i	$0.413672\pi$
$912$	0	0
$913$	8.48510	0.280816
$914$	0	0
$915$	−7.72307	−0.255317
$916$	0	0
$917$	−3.44533	−0.113775
$918$	0	0
$919$	−9.21767	−0.304063	−0.152031	−	0.988376i	$-0.548581\pi$
$919$	−9.21767	−0.304063	−0.152031	+	0.988376i	$0.548581\pi$
$920$	0	0
$921$	−19.9008	−0.655755
$922$	0	0
$923$	69.1541	2.27623
$924$	0	0
$925$	21.2078	0.697308
$926$	0	0
$927$	−2.91018	−0.0955828
$928$	0	0
$929$	6.60501	0.216703	0.108352	−	0.994113i	$-0.465443\pi$
$929$	6.60501	0.216703	0.108352	+	0.994113i	$0.465443\pi$
$930$	0	0
$931$	17.5319	0.574586
$932$	0	0
$933$	25.7926	0.844412
$934$	0	0
$935$	7.14776	0.233757
$936$	0	0
$937$	−52.0041	−1.69890	−0.849450	−	0.527668i	$-0.823066\pi$
$937$	−52.0041	−1.69890	−0.849450	+	0.527668i	$0.823066\pi$
$938$	0	0
$939$	0.438703	0.0143165
$940$	0	0
$941$	51.0361	1.66373	0.831865	−	0.554978i	$-0.187273\pi$
$941$	51.0361	1.66373	0.831865	+	0.554978i	$0.187273\pi$
$942$	0	0
$943$	−7.45209	−0.242673
$944$	0	0
$945$	20.4715	0.665939
$946$	0	0
$947$	−17.4981	−0.568611	−0.284305	−	0.958734i	$-0.591763\pi$
$947$	−17.4981	−0.568611	−0.284305	+	0.958734i	$0.591763\pi$
$948$	0	0
$949$	−24.2984	−0.788758
$950$	0	0
$951$	−1.15859	−0.0375699
$952$	0	0
$953$	−14.5337	−0.470793	−0.235397	−	0.971899i	$-0.575639\pi$
$953$	−14.5337	−0.470793	−0.235397	+	0.971899i	$0.575639\pi$
$954$	0	0
$955$	−59.4921	−1.92512
$956$	0	0
$957$	−19.8073	−0.640280
$958$	0	0
$959$	−21.5691	−0.696503
$960$	0	0
$961$	−16.3553	−0.527591
$962$	0	0
$963$	2.32229	0.0748348
$964$	0	0
$965$	−27.5686	−0.887466
$966$	0	0
$967$	−26.9042	−0.865182	−0.432591	−	0.901590i	$-0.642401\pi$
$967$	−26.9042	−0.865182	−0.432591	+	0.901590i	$0.642401\pi$
$968$	0	0
$969$	2.87834	0.0924656
$970$	0	0
$971$	54.8638	1.76066	0.880332	−	0.474358i	$-0.157320\pi$
$971$	54.8638	1.76066	0.880332	+	0.474358i	$0.157320\pi$
$972$	0	0
$973$	−26.0958	−0.836594
$974$	0	0
$975$	−17.0556	−0.546216
$976$	0	0
$977$	24.0891	0.770679	0.385340	−	0.922775i	$-0.374084\pi$
$977$	24.0891	0.770679	0.385340	+	0.922775i	$0.374084\pi$
$978$	0	0
$979$	70.8629	2.26479
$980$	0	0
$981$	10.8849	0.347529
$982$	0	0
$983$	−16.8207	−0.536497	−0.268248	−	0.963350i	$-0.586445\pi$
$983$	−16.8207	−0.536497	−0.268248	+	0.963350i	$0.586445\pi$
$984$	0	0
$985$	54.8892	1.74891
$986$	0	0
$987$	1.99102	0.0633747
$988$	0	0
$989$	−0.659550	−0.0209725
$990$	0	0
$991$	−2.95104	−0.0937429	−0.0468715	−	0.998901i	$-0.514925\pi$
$991$	−2.95104	−0.0937429	−0.0468715	+	0.998901i	$0.514925\pi$
$992$	0	0
$993$	28.9055	0.917288
$994$	0	0
$995$	34.5925	1.09666
$996$	0	0
$997$	−40.7367	−1.29014	−0.645071	−	0.764122i	$-0.723172\pi$
$997$	−40.7367	−1.29014	−0.645071	+	0.764122i	$0.723172\pi$
$998$	0	0
$999$	−48.4410	−1.53260

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3008.2.a.n.1.4		4
4.3	odd	2		3008.2.a.s.1.1		4
8.3	odd	2		1504.2.a.a.1.4	✓	4
8.5	even	2		1504.2.a.b.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1504.2.a.a.1.4	✓	4	8.3	odd	2
1504.2.a.b.1.1	yes	4	8.5	even	2
3008.2.a.n.1.4		4	1.1	even	1	trivial
3008.2.a.s.1.1		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3008 = 2^{6} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.49551 q^{3} +2.73205 q^{5} -1.33133 q^{7} -0.763457 q^{9} +O(q^{10})\)	\(q+1.49551 q^{3} +2.73205 q^{5} -1.33133 q^{7} -0.763457 q^{9} -4.55889 q^{11} -4.62828 q^{13} +4.08580 q^{15} -0.573882 q^{17} -3.35375 q^{19} -1.99102 q^{21} +0.921626 q^{23} +2.46410 q^{25} -5.62828 q^{27} +2.90521 q^{29} -3.82684 q^{31} -6.81785 q^{33} -3.63726 q^{35} +8.60671 q^{37} -6.92163 q^{39} -8.08580 q^{41} -0.715637 q^{43} -2.08580 q^{45} -1.00000 q^{47} -5.22756 q^{49} -0.858244 q^{51} +7.03798 q^{53} -12.4551 q^{55} -5.01556 q^{57} -6.13277 q^{59} -1.89022 q^{61} +1.01641 q^{63} -12.6447 q^{65} +0.669239 q^{67} +1.37830 q^{69} -14.9416 q^{71} +5.24998 q^{73} +3.68508 q^{75} +6.06939 q^{77} +7.57147 q^{79} -6.12676 q^{81} -1.86122 q^{83} -1.56787 q^{85} +4.34477 q^{87} -15.5439 q^{89} +6.16177 q^{91} -5.72307 q^{93} -9.16262 q^{95} -0.438134 q^{97} +3.48052 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 4 q^{5} - 4 q^{7} - 2 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 4 * q^5 - 4 * q^7 - 2 * q^9	\( 4 q - 2 q^{3} + 4 q^{5} - 4 q^{7} - 2 q^{9} - 2 q^{11} + 2 q^{13} - 2 q^{15} - 6 q^{17} - 2 q^{19} + 8 q^{21} - 8 q^{23} - 4 q^{25} - 2 q^{27} + 14 q^{29} - 6 q^{31} - 2 q^{33} - 10 q^{35} + 10 q^{37} - 16 q^{39} - 14 q^{41} - 10 q^{43} + 10 q^{45} - 4 q^{47} - 6 q^{49} + 18 q^{53} - 20 q^{55} - 20 q^{57} - 12 q^{59} + 10 q^{61} - 10 q^{63} - 16 q^{65} - 16 q^{67} + 10 q^{69} - 6 q^{71} - 4 q^{73} + 2 q^{75} + 20 q^{77} + 2 q^{79} - 8 q^{81} - 16 q^{83} - 6 q^{85} - 10 q^{87} - 26 q^{89} - 14 q^{91} + 16 q^{95} - 6 q^{97} - 14 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 4 * q^5 - 4 * q^7 - 2 * q^9 - 2 * q^11 + 2 * q^13 - 2 * q^15 - 6 * q^17 - 2 * q^19 + 8 * q^21 - 8 * q^23 - 4 * q^25 - 2 * q^27 + 14 * q^29 - 6 * q^31 - 2 * q^33 - 10 * q^35 + 10 * q^37 - 16 * q^39 - 14 * q^41 - 10 * q^43 + 10 * q^45 - 4 * q^47 - 6 * q^49 + 18 * q^53 - 20 * q^55 - 20 * q^57 - 12 * q^59 + 10 * q^61 - 10 * q^63 - 16 * q^65 - 16 * q^67 + 10 * q^69 - 6 * q^71 - 4 * q^73 + 2 * q^75 + 20 * q^77 + 2 * q^79 - 8 * q^81 - 16 * q^83 - 6 * q^85 - 10 * q^87 - 26 * q^89 - 14 * q^91 + 16 * q^95 - 6 * q^97 - 14 * q^99

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