LMFDB - Embedded newform 1504.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1504,2,Mod(1,1504)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1504, 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("1504.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1504 = 2^{5} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1504.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:	$12.0095004640$
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:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.49551$ of defining polynomial
Character	$\chi$	$=$	1504.1

$q$-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	$-0.642092\pi$
$3$	−1.49551	−0.863432	−0.431716	+	0.902010i	$0.642092\pi$
$4$	0	0
$5$	−2.73205	−1.22181	−0.610905	−	0.791704i	$-0.709194\pi$
$5$	−2.73205	−1.22181	−0.610905	+	0.791704i	$0.709194\pi$
$6$	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	$-0.258805\pi$
$11$	4.55889	1.37456	0.687278	+	0.726394i	$0.258805\pi$
$12$	0	0
$13$	4.62828	1.28365	0.641827	−	0.766850i	$-0.278177\pi$
$13$	4.62828	1.28365	0.641827	+	0.766850i	$0.278177\pi$
$14$	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	$-0.374304\pi$
$19$	3.35375	0.769403	0.384702	+	0.923041i	$0.374304\pi$
$20$	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	$-0.586938\pi$
$29$	−2.90521	−0.539484	−0.269742	+	0.962933i	$0.586938\pi$
$30$	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	$-0.750162\pi$
$37$	−8.60671	−1.41493	−0.707467	+	0.706746i	$0.750162\pi$
$38$	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	$-0.482622\pi$
$43$	0.715637	0.109134	0.0545668	+	0.998510i	$0.482622\pi$
$44$	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	$-0.660588\pi$
$53$	−7.03798	−0.966741	−0.483371	+	0.875416i	$0.660588\pi$
$54$	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	$-0.369285\pi$
$59$	6.13277	0.798419	0.399209	+	0.916860i	$0.369285\pi$
$60$	0	0
$61$	1.89022	0.242018	0.121009	−	0.992651i	$-0.461387\pi$
$61$	1.89022	0.242018	0.121009	+	0.992651i	$0.461387\pi$
$62$	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	$-0.513016\pi$
$67$	−0.669239	−0.0817605	−0.0408803	+	0.999164i	$0.513016\pi$
$68$	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	$-0.467429\pi$
$83$	1.86122	0.204296	0.102148	+	0.994769i	$0.467429\pi$
$84$	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	$-0.689457\pi$
$101$	−11.2694	−1.12134	−0.560672	+	0.828038i	$0.689457\pi$
$102$	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	$-0.453028\pi$
$107$	3.04181	0.294063	0.147032	+	0.989132i	$0.453028\pi$
$108$	0	0
$109$	14.2574	1.36561	0.682806	−	0.730600i	$-0.260759\pi$
$109$	14.2574	1.36561	0.682806	+	0.730600i	$0.260759\pi$
$110$	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	$-0.536063\pi$
$131$	−2.58789	−0.226105	−0.113052	+	0.993589i	$0.536063\pi$
$132$	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	$-0.812392\pi$
$139$	−19.6013	−1.66256	−0.831281	+	0.555852i	$0.812392\pi$
$140$	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	$-0.613976\pi$
$149$	−8.55591	−0.700928	−0.350464	+	0.936576i	$0.613976\pi$
$150$	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	$-0.828455\pi$
$157$	−21.5080	−1.71652	−0.858261	+	0.513214i	$0.828455\pi$
$158$	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	$-0.480853\pi$
$163$	1.53505	0.120234	0.0601171	+	0.998191i	$0.480853\pi$
$164$	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	$-0.720510\pi$
$173$	−16.8005	−1.27731	−0.638657	+	0.769491i	$0.720510\pi$
$174$	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	$-0.630074\pi$
$179$	−10.6327	−0.794724	−0.397362	+	0.917662i	$0.630074\pi$
$180$	0	0
$181$	8.89977	0.661515	0.330757	−	0.943716i	$-0.392696\pi$
$181$	8.89977	0.661515	0.330757	+	0.943716i	$0.392696\pi$
$182$	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	$-0.753895\pi$
$197$	−20.0908	−1.43141	−0.715706	+	0.698402i	$0.753895\pi$
$198$	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	$-0.374174\pi$
$211$	11.1872	0.770156	0.385078	+	0.922884i	$0.374174\pi$
$212$	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	$-0.174481\pi$
$227$	25.7183	1.70698	0.853490	+	0.521109i	$0.174481\pi$
$228$	0	0
$229$	−10.3170	−0.681764	−0.340882	−	0.940106i	$-0.610726\pi$
$229$	−10.3170	−0.681764	−0.340882	+	0.940106i	$0.610726\pi$
$230$	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	$-0.879965\pi$
$251$	−29.4596	−1.85947	−0.929736	+	0.368227i	$0.879965\pi$
$252$	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	$-0.438262\pi$
$269$	6.32240	0.385484	0.192742	+	0.981250i	$0.438262\pi$
$270$	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	$-0.572068\pi$
$277$	−7.47210	−0.448955	−0.224477	+	0.974479i	$0.572068\pi$
$278$	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	$-0.118820\pi$
$283$	31.3282	1.86227	0.931135	+	0.364676i	$0.118820\pi$
$284$	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	$-0.387434\pi$
$293$	11.8558	0.692622	0.346311	+	0.938120i	$0.387434\pi$
$294$	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	$-0.376014\pi$
$307$	13.3071	0.759475	0.379737	+	0.925094i	$0.376014\pi$
$308$	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	$-0.493074\pi$
$317$	0.774714	0.0435123	0.0217561	+	0.999763i	$0.493074\pi$
$318$	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	$-0.678254\pi$
$331$	−19.3282	−1.06238	−0.531188	+	0.847254i	$0.678254\pi$
$332$	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	$-0.645203\pi$
$347$	−16.4117	−0.881026	−0.440513	+	0.897746i	$0.645203\pi$
$348$	0	0
$349$	1.95904	0.104865	0.0524325	−	0.998624i	$-0.483303\pi$
$349$	1.95904	0.104865	0.0524325	+	0.998624i	$0.483303\pi$
$350$	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	$-0.406334\pi$
$373$	11.2030	0.580067	0.290034	+	0.957016i	$0.406334\pi$
$374$	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	$-0.491286\pi$
$379$	1.06579	0.0547459	0.0273730	+	0.999625i	$0.491286\pi$
$380$	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	$-0.176198\pi$
$389$	33.5555	1.70133	0.850667	+	0.525705i	$0.176198\pi$
$390$	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	$-0.279326\pi$
$397$	25.4661	1.27811	0.639055	+	0.769161i	$0.279326\pi$
$398$	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	$-0.265112\pi$
$419$	27.5418	1.34550	0.672752	+	0.739868i	$0.265112\pi$
$420$	0	0
$421$	−31.8794	−1.55371	−0.776853	−	0.629682i	$-0.783185\pi$
$421$	−31.8794	−1.55371	−0.776853	+	0.629682i	$0.783185\pi$
$422$	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	$-0.579271\pi$
$443$	−10.3753	−0.492944	−0.246472	+	0.969150i	$0.579271\pi$
$444$	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	$-0.428916\pi$
$461$	9.51008	0.442929	0.221464	+	0.975168i	$0.428916\pi$
$462$	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	$-0.791200\pi$
$467$	−34.2504	−1.58492	−0.792460	+	0.609924i	$0.791200\pi$
$468$	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	$-0.848681\pi$
$491$	−39.4030	−1.77823	−0.889117	+	0.457679i	$0.848681\pi$
$492$	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	$-0.327318\pi$
$499$	23.0654	1.03255	0.516275	+	0.856423i	$0.327318\pi$
$500$	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	$-0.353349\pi$
$509$	20.0608	0.889181	0.444591	+	0.895734i	$0.353349\pi$
$510$	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	$-0.247677\pi$
$523$	32.5771	1.42450	0.712248	+	0.701928i	$0.247677\pi$
$524$	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	$-0.575490\pi$
$541$	−10.9292	−0.469883	−0.234941	+	0.972010i	$0.575490\pi$
$542$	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	$-0.327647\pi$
$547$	24.1079	1.03078	0.515389	+	0.856956i	$0.327647\pi$
$548$	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	$-0.517334\pi$
$557$	−2.56920	−0.108861	−0.0544303	+	0.998518i	$0.517334\pi$
$558$	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	$-0.511394\pi$
$563$	−1.69837	−0.0715778	−0.0357889	+	0.999359i	$0.511394\pi$
$564$	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	$-0.439707\pi$
$571$	8.99839	0.376571	0.188285	+	0.982114i	$0.439707\pi$
$572$	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	$-0.420275\pi$
$587$	12.0100	0.495708	0.247854	+	0.968797i	$0.420275\pi$
$588$	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	$-0.351694\pi$
$613$	22.2454	0.898485	0.449243	+	0.893410i	$0.351694\pi$
$614$	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	$-0.906693\pi$
$619$	−47.6368	−1.91469	−0.957343	+	0.288953i	$0.906693\pi$
$620$	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	$-0.890258\pi$
$643$	−47.7306	−1.88231	−0.941155	+	0.337976i	$0.890258\pi$
$644$	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	$-0.413787\pi$
$653$	13.6737	0.535093	0.267546	+	0.963545i	$0.413787\pi$
$654$	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	$-0.130681\pi$
$659$	47.0756	1.83381	0.916903	+	0.399111i	$0.130681\pi$
$660$	0	0
$661$	37.9738	1.47701	0.738504	−	0.674249i	$-0.235533\pi$
$661$	37.9738	1.47701	0.738504	+	0.674249i	$0.235533\pi$
$662$	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	$-0.549630\pi$
$677$	−8.08078	−0.310570	−0.155285	+	0.987870i	$0.549630\pi$
$678$	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	$-0.722321\pi$
$683$	−33.6100	−1.28605	−0.643026	+	0.765844i	$0.722321\pi$
$684$	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	$-0.409745\pi$
$691$	14.7081	0.559524	0.279762	+	0.960069i	$0.409745\pi$
$692$	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	$-0.774354\pi$
$701$	−40.1957	−1.51817	−0.759086	+	0.650990i	$0.774354\pi$
$702$	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	$-0.678398\pi$
$709$	−28.3083	−1.06314	−0.531569	+	0.847015i	$0.678398\pi$
$710$	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	$-0.430311\pi$
$733$	11.7604	0.434381	0.217191	+	0.976129i	$0.430311\pi$
$734$	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	$-0.301731\pi$
$739$	31.7177	1.16675	0.583377	+	0.812202i	$0.301731\pi$
$740$	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	$-0.718091\pi$
$757$	−34.8208	−1.26558	−0.632792	+	0.774322i	$0.718091\pi$
$758$	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	$-0.621116\pi$
$773$	−20.6510	−0.742764	−0.371382	+	0.928480i	$0.621116\pi$
$774$	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	$-0.607002\pi$
$787$	−18.5075	−0.659720	−0.329860	+	0.944030i	$0.607002\pi$
$788$	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	$-0.613842\pi$
$797$	−19.7657	−0.700138	−0.350069	+	0.936724i	$0.613842\pi$
$798$	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	$-0.239281\pi$
$811$	41.6071	1.46102	0.730512	+	0.682900i	$0.239281\pi$
$812$	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	$-0.645352\pi$
$821$	−25.2681	−0.881863	−0.440931	+	0.897541i	$0.645352\pi$
$822$	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	$-0.493426\pi$
$827$	1.18774	0.0413017	0.0206508	+	0.999787i	$0.493426\pi$
$828$	0	0
$829$	1.94702	0.0676229	0.0338115	−	0.999428i	$-0.489235\pi$
$829$	1.94702	0.0676229	0.0338115	+	0.999428i	$0.489235\pi$
$830$	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	$-0.282694\pi$
$853$	36.8512	1.26176	0.630880	+	0.775880i	$0.282694\pi$
$854$	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	$-0.155745\pi$
$859$	51.7397	1.76534	0.882668	+	0.469998i	$0.155745\pi$
$860$	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	$-0.509911\pi$
$877$	−1.84388	−0.0622632	−0.0311316	+	0.999515i	$0.509911\pi$
$878$	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	$-0.547469\pi$
$883$	−8.82996	−0.297152	−0.148576	+	0.988901i	$0.547469\pi$
$884$	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	$-0.189982\pi$
$907$	49.8193	1.65422	0.827112	+	0.562037i	$0.189982\pi$
$908$	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	$-0.812727\pi$
$941$	−51.0361	−1.66373	−0.831865	+	0.554978i	$0.812727\pi$
$942$	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	$-0.408237\pi$
$947$	17.4981	0.568611	0.284305	+	0.958734i	$0.408237\pi$
$948$	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	$-0.842680\pi$
$971$	−54.8638	−1.76066	−0.880332	+	0.474358i	$0.842680\pi$
$972$	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	$-0.276828\pi$
$997$	40.7367	1.29014	0.645071	+	0.764122i	$0.276828\pi$
$998$	0	0
$999$	−48.4410	−1.53260

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1504.2.a.a.1.4	✓	4	4.3	odd	2
1504.2.a.b.1.1	yes	4	1.1	even	1	trivial
3008.2.a.n.1.4		4	8.5	even	2
3008.2.a.s.1.1		4	8.3	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:	\( N \)	\(=\)	\( 1504 = 2^{5} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1504.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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