LMFDB - Embedded newform 8016.2.a.bd.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 26x^{8} - 3x^{7} + 220x^{6} + 42x^{5} - 675x^{4} - 67x^{3} + 628x^{2} - 48x - 2 $ x^10 - 26x^8 - 3x^7 + 220x^6 + 42x^5 - 675x^4 - 67x^3 + 628x^2 - 48x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.15141$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.15141 q^{5} +2.22742 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.15141 q^{5} +2.22742 q^{7} +1.00000 q^{9} +5.26362 q^{11} +3.01189 q^{13} -2.15141 q^{15} +0.872173 q^{17} -5.13559 q^{19} +2.22742 q^{21} -7.30180 q^{23} -0.371439 q^{25} +1.00000 q^{27} -6.24949 q^{29} -10.6516 q^{31} +5.26362 q^{33} -4.79209 q^{35} -5.97452 q^{37} +3.01189 q^{39} +5.74080 q^{41} -8.86244 q^{43} -2.15141 q^{45} -6.77211 q^{47} -2.03860 q^{49} +0.872173 q^{51} -10.3374 q^{53} -11.3242 q^{55} -5.13559 q^{57} +8.89647 q^{59} +4.06058 q^{61} +2.22742 q^{63} -6.47980 q^{65} +6.51933 q^{67} -7.30180 q^{69} +7.06342 q^{71} -4.80957 q^{73} -0.371439 q^{75} +11.7243 q^{77} +1.01320 q^{79} +1.00000 q^{81} -16.0586 q^{83} -1.87640 q^{85} -6.24949 q^{87} -12.2728 q^{89} +6.70874 q^{91} -10.6516 q^{93} +11.0487 q^{95} +18.3889 q^{97} +5.26362 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{3} - 10 q^{5} - q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q + 10 * q^3 - 10 * q^5 - q^7 + 10 * q^9	$ 10 q + 10 q^{3} - 10 q^{5} - q^{7} + 10 q^{9} + q^{11} - 6 q^{13} - 10 q^{15} - 9 q^{17} - 2 q^{19} - q^{21} - 7 q^{23} + 12 q^{25} + 10 q^{27} - 13 q^{29} - 23 q^{31} + q^{33} - q^{35} - 6 q^{37} - 6 q^{39} - 12 q^{41} - 10 q^{45} - 10 q^{47} + 7 q^{49} - 9 q^{51} - 26 q^{53} - 11 q^{55} - 2 q^{57} + 10 q^{59} - 10 q^{61} - q^{63} - 22 q^{65} + 5 q^{67} - 7 q^{69} - 25 q^{71} - 8 q^{73} + 12 q^{75} - 46 q^{77} - 26 q^{79} + 10 q^{81} + 14 q^{83} + 9 q^{85} - 13 q^{87} - 31 q^{89} + 3 q^{91} - 23 q^{93} + 5 q^{95} - 32 q^{97} + q^{99}+O(q^{100}) $ 10 * q + 10 * q^3 - 10 * q^5 - q^7 + 10 * q^9 + q^11 - 6 * q^13 - 10 * q^15 - 9 * q^17 - 2 * q^19 - q^21 - 7 * q^23 + 12 * q^25 + 10 * q^27 - 13 * q^29 - 23 * q^31 + q^33 - q^35 - 6 * q^37 - 6 * q^39 - 12 * q^41 - 10 * q^45 - 10 * q^47 + 7 * q^49 - 9 * q^51 - 26 * q^53 - 11 * q^55 - 2 * q^57 + 10 * q^59 - 10 * q^61 - q^63 - 22 * q^65 + 5 * q^67 - 7 * q^69 - 25 * q^71 - 8 * q^73 + 12 * q^75 - 46 * q^77 - 26 * q^79 + 10 * q^81 + 14 * q^83 + 9 * q^85 - 13 * q^87 - 31 * q^89 + 3 * q^91 - 23 * q^93 + 5 * q^95 - 32 * q^97 + 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.15141	−0.962139	−0.481070	−	0.876682i	$-0.659752\pi$
$5$	−2.15141	−0.962139	−0.481070	+	0.876682i	$0.659752\pi$
$6$	0	0
$7$	2.22742	0.841885	0.420943	−	0.907087i	$-0.361699\pi$
$7$	2.22742	0.841885	0.420943	+	0.907087i	$0.361699\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.26362	1.58704	0.793521	−	0.608543i	$-0.208246\pi$
$11$	5.26362	1.58704	0.793521	+	0.608543i	$0.208246\pi$
$12$	0	0
$13$	3.01189	0.835347	0.417674	−	0.908597i	$-0.362846\pi$
$13$	3.01189	0.835347	0.417674	+	0.908597i	$0.362846\pi$
$14$	0	0
$15$	−2.15141	−0.555491
$16$	0	0
$17$	0.872173	0.211533	0.105767	−	0.994391i	$-0.466270\pi$
$17$	0.872173	0.211533	0.105767	+	0.994391i	$0.466270\pi$
$18$	0	0
$19$	−5.13559	−1.17818	−0.589092	−	0.808066i	$-0.700514\pi$
$19$	−5.13559	−1.17818	−0.589092	+	0.808066i	$0.700514\pi$
$20$	0	0
$21$	2.22742	0.486063
$22$	0	0
$23$	−7.30180	−1.52253	−0.761266	−	0.648440i	$-0.775422\pi$
$23$	−7.30180	−1.52253	−0.761266	+	0.648440i	$0.775422\pi$
$24$	0	0
$25$	−0.371439	−0.0742878
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.24949	−1.16050	−0.580251	−	0.814438i	$-0.697045\pi$
$29$	−6.24949	−1.16050	−0.580251	+	0.814438i	$0.697045\pi$
$30$	0	0
$31$	−10.6516	−1.91309	−0.956544	−	0.291589i	$-0.905816\pi$
$31$	−10.6516	−1.91309	−0.956544	+	0.291589i	$0.905816\pi$
$32$	0	0
$33$	5.26362	0.916279
$34$	0	0
$35$	−4.79209	−0.810011
$36$	0	0
$37$	−5.97452	−0.982206	−0.491103	−	0.871102i	$-0.663406\pi$
$37$	−5.97452	−0.982206	−0.491103	+	0.871102i	$0.663406\pi$
$38$	0	0
$39$	3.01189	0.482288
$40$	0	0
$41$	5.74080	0.896562	0.448281	−	0.893893i	$-0.352036\pi$
$41$	5.74080	0.896562	0.448281	+	0.893893i	$0.352036\pi$
$42$	0	0
$43$	−8.86244	−1.35151	−0.675755	−	0.737127i	$-0.736182\pi$
$43$	−8.86244	−1.35151	−0.675755	+	0.737127i	$0.736182\pi$
$44$	0	0
$45$	−2.15141	−0.320713
$46$	0	0
$47$	−6.77211	−0.987814	−0.493907	−	0.869515i	$-0.664432\pi$
$47$	−6.77211	−0.987814	−0.493907	+	0.869515i	$0.664432\pi$
$48$	0	0
$49$	−2.03860	−0.291229
$50$	0	0
$51$	0.872173	0.122129
$52$	0	0
$53$	−10.3374	−1.41995	−0.709977	−	0.704225i	$-0.751295\pi$
$53$	−10.3374	−1.41995	−0.709977	+	0.704225i	$0.751295\pi$
$54$	0	0
$55$	−11.3242	−1.52696
$56$	0	0
$57$	−5.13559	−0.680225
$58$	0	0
$59$	8.89647	1.15822	0.579111	−	0.815249i	$-0.303400\pi$
$59$	8.89647	1.15822	0.579111	+	0.815249i	$0.303400\pi$
$60$	0	0
$61$	4.06058	0.519904	0.259952	−	0.965622i	$-0.416293\pi$
$61$	4.06058	0.519904	0.259952	+	0.965622i	$0.416293\pi$
$62$	0	0
$63$	2.22742	0.280628
$64$	0	0
$65$	−6.47980	−0.803721
$66$	0	0
$67$	6.51933	0.796463	0.398231	−	0.917285i	$-0.369624\pi$
$67$	6.51933	0.796463	0.398231	+	0.917285i	$0.369624\pi$
$68$	0	0
$69$	−7.30180	−0.879034
$70$	0	0
$71$	7.06342	0.838273	0.419137	−	0.907923i	$-0.362333\pi$
$71$	7.06342	0.838273	0.419137	+	0.907923i	$0.362333\pi$
$72$	0	0
$73$	−4.80957	−0.562917	−0.281459	−	0.959573i	$-0.590818\pi$
$73$	−4.80957	−0.562917	−0.281459	+	0.959573i	$0.590818\pi$
$74$	0	0
$75$	−0.371439	−0.0428901
$76$	0	0
$77$	11.7243	1.33611
$78$	0	0
$79$	1.01320	0.113993	0.0569967	−	0.998374i	$-0.481848\pi$
$79$	1.01320	0.113993	0.0569967	+	0.998374i	$0.481848\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−16.0586	−1.76266	−0.881330	−	0.472501i	$-0.843351\pi$
$83$	−16.0586	−1.76266	−0.881330	+	0.472501i	$0.843351\pi$
$84$	0	0
$85$	−1.87640	−0.203524
$86$	0	0
$87$	−6.24949	−0.670016
$88$	0	0
$89$	−12.2728	−1.30091	−0.650456	−	0.759544i	$-0.725422\pi$
$89$	−12.2728	−1.30091	−0.650456	+	0.759544i	$0.725422\pi$
$90$	0	0
$91$	6.70874	0.703267
$92$	0	0
$93$	−10.6516	−1.10452
$94$	0	0
$95$	11.0487	1.13358
$96$	0	0
$97$	18.3889	1.86711	0.933553	−	0.358441i	$-0.116691\pi$
$97$	18.3889	1.86711	0.933553	+	0.358441i	$0.116691\pi$
$98$	0	0
$99$	5.26362	0.529014
$100$	0	0
$101$	−16.8050	−1.67216	−0.836080	−	0.548607i	$-0.815158\pi$
$101$	−16.8050	−1.67216	−0.836080	+	0.548607i	$0.815158\pi$
$102$	0	0
$103$	11.0501	1.08880	0.544401	−	0.838825i	$-0.316757\pi$
$103$	11.0501	1.08880	0.544401	+	0.838825i	$0.316757\pi$
$104$	0	0
$105$	−4.79209	−0.467660
$106$	0	0
$107$	2.02820	0.196073	0.0980367	−	0.995183i	$-0.468744\pi$
$107$	2.02820	0.196073	0.0980367	+	0.995183i	$0.468744\pi$
$108$	0	0
$109$	−2.32393	−0.222592	−0.111296	−	0.993787i	$-0.535500\pi$
$109$	−2.32393	−0.222592	−0.111296	+	0.993787i	$0.535500\pi$
$110$	0	0
$111$	−5.97452	−0.567077
$112$	0	0
$113$	−7.09277	−0.667232	−0.333616	−	0.942709i	$-0.608269\pi$
$113$	−7.09277	−0.667232	−0.333616	+	0.942709i	$0.608269\pi$
$114$	0	0
$115$	15.7092	1.46489
$116$	0	0
$117$	3.01189	0.278449
$118$	0	0
$119$	1.94269	0.178087
$120$	0	0
$121$	16.7057	1.51870
$122$	0	0
$123$	5.74080	0.517630
$124$	0	0
$125$	11.5562	1.03361
$126$	0	0
$127$	−0.463781	−0.0411539	−0.0205770	−	0.999788i	$-0.506550\pi$
$127$	−0.463781	−0.0411539	−0.0205770	+	0.999788i	$0.506550\pi$
$128$	0	0
$129$	−8.86244	−0.780294
$130$	0	0
$131$	−6.15679	−0.537922	−0.268961	−	0.963151i	$-0.586680\pi$
$131$	−6.15679	−0.537922	−0.268961	+	0.963151i	$0.586680\pi$
$132$	0	0
$133$	−11.4391	−0.991896
$134$	0	0
$135$	−2.15141	−0.185164
$136$	0	0
$137$	−15.6011	−1.33289	−0.666446	−	0.745553i	$-0.732185\pi$
$137$	−15.6011	−1.33289	−0.666446	+	0.745553i	$0.732185\pi$
$138$	0	0
$139$	9.85541	0.835925	0.417963	−	0.908464i	$-0.362744\pi$
$139$	9.85541	0.835925	0.417963	+	0.908464i	$0.362744\pi$
$140$	0	0
$141$	−6.77211	−0.570315
$142$	0	0
$143$	15.8534	1.32573
$144$	0	0
$145$	13.4452	1.11656
$146$	0	0
$147$	−2.03860	−0.168141
$148$	0	0
$149$	−10.8616	−0.889818	−0.444909	−	0.895576i	$-0.646764\pi$
$149$	−10.8616	−0.889818	−0.444909	+	0.895576i	$0.646764\pi$
$150$	0	0
$151$	5.45582	0.443988	0.221994	−	0.975048i	$-0.428743\pi$
$151$	5.45582	0.443988	0.221994	+	0.975048i	$0.428743\pi$
$152$	0	0
$153$	0.872173	0.0705110
$154$	0	0
$155$	22.9160	1.84066
$156$	0	0
$157$	−9.40270	−0.750417	−0.375209	−	0.926940i	$-0.622429\pi$
$157$	−9.40270	−0.750417	−0.375209	+	0.926940i	$0.622429\pi$
$158$	0	0
$159$	−10.3374	−0.819811
$160$	0	0
$161$	−16.2642	−1.28180
$162$	0	0
$163$	17.7492	1.39023	0.695113	−	0.718901i	$-0.255355\pi$
$163$	17.7492	1.39023	0.695113	+	0.718901i	$0.255355\pi$
$164$	0	0
$165$	−11.3242	−0.881588
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−3.92853	−0.302195
$170$	0	0
$171$	−5.13559	−0.392728
$172$	0	0
$173$	23.1988	1.76377	0.881886	−	0.471463i	$-0.156274\pi$
$173$	23.1988	1.76377	0.881886	+	0.471463i	$0.156274\pi$
$174$	0	0
$175$	−0.827350	−0.0625418
$176$	0	0
$177$	8.89647	0.668700
$178$	0	0
$179$	−12.9681	−0.969281	−0.484641	−	0.874713i	$-0.661050\pi$
$179$	−12.9681	−0.969281	−0.484641	+	0.874713i	$0.661050\pi$
$180$	0	0
$181$	5.51396	0.409849	0.204925	−	0.978778i	$-0.434305\pi$
$181$	5.51396	0.409849	0.204925	+	0.978778i	$0.434305\pi$
$182$	0	0
$183$	4.06058	0.300167
$184$	0	0
$185$	12.8536	0.945019
$186$	0	0
$187$	4.59079	0.335712
$188$	0	0
$189$	2.22742	0.162021
$190$	0	0
$191$	−6.81882	−0.493392	−0.246696	−	0.969093i	$-0.579345\pi$
$191$	−6.81882	−0.493392	−0.246696	+	0.969093i	$0.579345\pi$
$192$	0	0
$193$	3.94100	0.283679	0.141840	−	0.989890i	$-0.454698\pi$
$193$	3.94100	0.283679	0.141840	+	0.989890i	$0.454698\pi$
$194$	0	0
$195$	−6.47980	−0.464028
$196$	0	0
$197$	10.7619	0.766753	0.383377	−	0.923592i	$-0.374761\pi$
$197$	10.7619	0.766753	0.383377	+	0.923592i	$0.374761\pi$
$198$	0	0
$199$	−13.6212	−0.965585	−0.482792	−	0.875735i	$-0.660377\pi$
$199$	−13.6212	−0.965585	−0.482792	+	0.875735i	$0.660377\pi$
$200$	0	0
$201$	6.51933	0.459838
$202$	0	0
$203$	−13.9202	−0.977009
$204$	0	0
$205$	−12.3508	−0.862618
$206$	0	0
$207$	−7.30180	−0.507510
$208$	0	0
$209$	−27.0318	−1.86983
$210$	0	0
$211$	−12.2913	−0.846167	−0.423084	−	0.906091i	$-0.639052\pi$
$211$	−12.2913	−0.846167	−0.423084	+	0.906091i	$0.639052\pi$
$212$	0	0
$213$	7.06342	0.483977
$214$	0	0
$215$	19.0667	1.30034
$216$	0	0
$217$	−23.7256	−1.61060
$218$	0	0
$219$	−4.80957	−0.325000
$220$	0	0
$221$	2.62689	0.176704
$222$	0	0
$223$	29.8084	1.99612	0.998059	−	0.0622749i	$-0.0198355\pi$
$223$	29.8084	1.99612	0.998059	+	0.0622749i	$0.0198355\pi$
$224$	0	0
$225$	−0.371439	−0.0247626
$226$	0	0
$227$	−5.90639	−0.392021	−0.196010	−	0.980602i	$-0.562799\pi$
$227$	−5.90639	−0.392021	−0.196010	+	0.980602i	$0.562799\pi$
$228$	0	0
$229$	21.3820	1.41297	0.706483	−	0.707730i	$-0.250281\pi$
$229$	21.3820	1.41297	0.706483	+	0.707730i	$0.250281\pi$
$230$	0	0
$231$	11.7243	0.771402
$232$	0	0
$233$	−16.8878	−1.10636	−0.553178	−	0.833063i	$-0.686585\pi$
$233$	−16.8878	−1.10636	−0.553178	+	0.833063i	$0.686585\pi$
$234$	0	0
$235$	14.5696	0.950415
$236$	0	0
$237$	1.01320	0.0658141
$238$	0	0
$239$	3.07320	0.198789	0.0993943	−	0.995048i	$-0.468309\pi$
$239$	3.07320	0.198789	0.0993943	+	0.995048i	$0.468309\pi$
$240$	0	0
$241$	−0.156195	−0.0100614	−0.00503070	−	0.999987i	$-0.501601\pi$
$241$	−0.156195	−0.0100614	−0.00503070	+	0.999987i	$0.501601\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	4.38587	0.280203
$246$	0	0
$247$	−15.4678	−0.984193
$248$	0	0
$249$	−16.0586	−1.01767
$250$	0	0
$251$	3.70516	0.233867	0.116934	−	0.993140i	$-0.462693\pi$
$251$	3.70516	0.233867	0.116934	+	0.993140i	$0.462693\pi$
$252$	0	0
$253$	−38.4339	−2.41632
$254$	0	0
$255$	−1.87640	−0.117505
$256$	0	0
$257$	−2.46075	−0.153497	−0.0767487	−	0.997050i	$-0.524454\pi$
$257$	−2.46075	−0.153497	−0.0767487	+	0.997050i	$0.524454\pi$
$258$	0	0
$259$	−13.3078	−0.826904
$260$	0	0
$261$	−6.24949	−0.386834
$262$	0	0
$263$	18.5170	1.14181	0.570904	−	0.821017i	$-0.306593\pi$
$263$	18.5170	1.14181	0.570904	+	0.821017i	$0.306593\pi$
$264$	0	0
$265$	22.2400	1.36619
$266$	0	0
$267$	−12.2728	−0.751082
$268$	0	0
$269$	−31.3755	−1.91300	−0.956500	−	0.291731i	$-0.905769\pi$
$269$	−31.3755	−1.91300	−0.956500	+	0.291731i	$0.905769\pi$
$270$	0	0
$271$	−6.93351	−0.421180	−0.210590	−	0.977574i	$-0.567539\pi$
$271$	−6.93351	−0.421180	−0.210590	+	0.977574i	$0.567539\pi$
$272$	0	0
$273$	6.70874	0.406031
$274$	0	0
$275$	−1.95511	−0.117898
$276$	0	0
$277$	28.2322	1.69631	0.848154	−	0.529750i	$-0.177714\pi$
$277$	28.2322	1.69631	0.848154	+	0.529750i	$0.177714\pi$
$278$	0	0
$279$	−10.6516	−0.637696
$280$	0	0
$281$	26.6487	1.58973	0.794864	−	0.606787i	$-0.207542\pi$
$281$	26.6487	1.58973	0.794864	+	0.606787i	$0.207542\pi$
$282$	0	0
$283$	25.5145	1.51668	0.758341	−	0.651858i	$-0.226010\pi$
$283$	25.5145	1.51668	0.758341	+	0.651858i	$0.226010\pi$
$284$	0	0
$285$	11.0487	0.654471
$286$	0	0
$287$	12.7872	0.754802
$288$	0	0
$289$	−16.2393	−0.955254
$290$	0	0
$291$	18.3889	1.07797
$292$	0	0
$293$	−22.3309	−1.30459	−0.652293	−	0.757967i	$-0.726193\pi$
$293$	−22.3309	−1.30459	−0.652293	+	0.757967i	$0.726193\pi$
$294$	0	0
$295$	−19.1400	−1.11437
$296$	0	0
$297$	5.26362	0.305426
$298$	0	0
$299$	−21.9922	−1.27184
$300$	0	0
$301$	−19.7404	−1.13782
$302$	0	0
$303$	−16.8050	−0.965422
$304$	0	0
$305$	−8.73597	−0.500220
$306$	0	0
$307$	19.6253	1.12008	0.560038	−	0.828467i	$-0.310786\pi$
$307$	19.6253	1.12008	0.560038	+	0.828467i	$0.310786\pi$
$308$	0	0
$309$	11.0501	0.628620
$310$	0	0
$311$	−23.4421	−1.32928	−0.664640	−	0.747164i	$-0.731415\pi$
$311$	−23.4421	−1.32928	−0.664640	+	0.747164i	$0.731415\pi$
$312$	0	0
$313$	−20.7190	−1.17111	−0.585553	−	0.810634i	$-0.699123\pi$
$313$	−20.7190	−1.17111	−0.585553	+	0.810634i	$0.699123\pi$
$314$	0	0
$315$	−4.79209	−0.270004
$316$	0	0
$317$	−24.8655	−1.39658	−0.698292	−	0.715813i	$-0.746056\pi$
$317$	−24.8655	−1.39658	−0.698292	+	0.715813i	$0.746056\pi$
$318$	0	0
$319$	−32.8950	−1.84176
$320$	0	0
$321$	2.02820	0.113203
$322$	0	0
$323$	−4.47912	−0.249225
$324$	0	0
$325$	−1.11873	−0.0620561
$326$	0	0
$327$	−2.32393	−0.128514
$328$	0	0
$329$	−15.0843	−0.831626
$330$	0	0
$331$	22.5680	1.24045	0.620225	−	0.784424i	$-0.287041\pi$
$331$	22.5680	1.24045	0.620225	+	0.784424i	$0.287041\pi$
$332$	0	0
$333$	−5.97452	−0.327402
$334$	0	0
$335$	−14.0257	−0.766308
$336$	0	0
$337$	9.50371	0.517700	0.258850	−	0.965917i	$-0.416656\pi$
$337$	9.50371	0.517700	0.258850	+	0.965917i	$0.416656\pi$
$338$	0	0
$339$	−7.09277	−0.385226
$340$	0	0
$341$	−56.0661	−3.03615
$342$	0	0
$343$	−20.1328	−1.08707
$344$	0	0
$345$	15.7092	0.845753
$346$	0	0
$347$	18.6354	1.00040	0.500200	−	0.865910i	$-0.333260\pi$
$347$	18.6354	1.00040	0.500200	+	0.865910i	$0.333260\pi$
$348$	0	0
$349$	6.27961	0.336140	0.168070	−	0.985775i	$-0.446247\pi$
$349$	6.27961	0.336140	0.168070	+	0.985775i	$0.446247\pi$
$350$	0	0
$351$	3.01189	0.160763
$352$	0	0
$353$	−35.8067	−1.90580	−0.952899	−	0.303289i	$-0.901915\pi$
$353$	−35.8067	−1.90580	−0.952899	+	0.303289i	$0.901915\pi$
$354$	0	0
$355$	−15.1963	−0.806536
$356$	0	0
$357$	1.94269	0.102818
$358$	0	0
$359$	18.0185	0.950982	0.475491	−	0.879721i	$-0.342270\pi$
$359$	18.0185	0.950982	0.475491	+	0.879721i	$0.342270\pi$
$360$	0	0
$361$	7.37424	0.388118
$362$	0	0
$363$	16.7057	0.876822
$364$	0	0
$365$	10.3473	0.541605
$366$	0	0
$367$	−23.4946	−1.22641	−0.613203	−	0.789926i	$-0.710119\pi$
$367$	−23.4946	−1.22641	−0.613203	+	0.789926i	$0.710119\pi$
$368$	0	0
$369$	5.74080	0.298854
$370$	0	0
$371$	−23.0258	−1.19544
$372$	0	0
$373$	−4.96876	−0.257273	−0.128636	−	0.991692i	$-0.541060\pi$
$373$	−4.96876	−0.257273	−0.128636	+	0.991692i	$0.541060\pi$
$374$	0	0
$375$	11.5562	0.596758
$376$	0	0
$377$	−18.8228	−0.969422
$378$	0	0
$379$	29.3283	1.50650	0.753248	−	0.657737i	$-0.228486\pi$
$379$	29.3283	1.50650	0.753248	+	0.657737i	$0.228486\pi$
$380$	0	0
$381$	−0.463781	−0.0237602
$382$	0	0
$383$	−21.0500	−1.07561	−0.537803	−	0.843070i	$-0.680746\pi$
$383$	−21.0500	−1.07561	−0.537803	+	0.843070i	$0.680746\pi$
$384$	0	0
$385$	−25.2237	−1.28552
$386$	0	0
$387$	−8.86244	−0.450503
$388$	0	0
$389$	−25.8912	−1.31274	−0.656368	−	0.754441i	$-0.727908\pi$
$389$	−25.8912	−1.31274	−0.656368	+	0.754441i	$0.727908\pi$
$390$	0	0
$391$	−6.36844	−0.322066
$392$	0	0
$393$	−6.15679	−0.310569
$394$	0	0
$395$	−2.17980	−0.109677
$396$	0	0
$397$	9.44154	0.473857	0.236929	−	0.971527i	$-0.423859\pi$
$397$	9.44154	0.473857	0.236929	+	0.971527i	$0.423859\pi$
$398$	0	0
$399$	−11.4391	−0.572671
$400$	0	0
$401$	−3.79399	−0.189463	−0.0947315	−	0.995503i	$-0.530199\pi$
$401$	−3.79399	−0.189463	−0.0947315	+	0.995503i	$0.530199\pi$
$402$	0	0
$403$	−32.0815	−1.59809
$404$	0	0
$405$	−2.15141	−0.106904
$406$	0	0
$407$	−31.4476	−1.55880
$408$	0	0
$409$	7.08701	0.350430	0.175215	−	0.984530i	$-0.443938\pi$
$409$	7.08701	0.350430	0.175215	+	0.984530i	$0.443938\pi$
$410$	0	0
$411$	−15.6011	−0.769545
$412$	0	0
$413$	19.8162	0.975090
$414$	0	0
$415$	34.5486	1.69593
$416$	0	0
$417$	9.85541	0.482622
$418$	0	0
$419$	25.5611	1.24874	0.624370	−	0.781129i	$-0.285356\pi$
$419$	25.5611	1.24874	0.624370	+	0.781129i	$0.285356\pi$
$420$	0	0
$421$	7.98682	0.389254	0.194627	−	0.980877i	$-0.437650\pi$
$421$	7.98682	0.389254	0.194627	+	0.980877i	$0.437650\pi$
$422$	0	0
$423$	−6.77211	−0.329271
$424$	0	0
$425$	−0.323959	−0.0157143
$426$	0	0
$427$	9.04462	0.437700
$428$	0	0
$429$	15.8534	0.765411
$430$	0	0
$431$	4.35336	0.209694	0.104847	−	0.994488i	$-0.466565\pi$
$431$	4.35336	0.209694	0.104847	+	0.994488i	$0.466565\pi$
$432$	0	0
$433$	−9.60449	−0.461562	−0.230781	−	0.973006i	$-0.574128\pi$
$433$	−9.60449	−0.461562	−0.230781	+	0.973006i	$0.574128\pi$
$434$	0	0
$435$	13.4452	0.644649
$436$	0	0
$437$	37.4990	1.79382
$438$	0	0
$439$	−29.9980	−1.43173	−0.715863	−	0.698240i	$-0.753967\pi$
$439$	−29.9980	−1.43173	−0.715863	+	0.698240i	$0.753967\pi$
$440$	0	0
$441$	−2.03860	−0.0970764
$442$	0	0
$443$	−6.80439	−0.323286	−0.161643	−	0.986849i	$-0.551679\pi$
$443$	−6.80439	−0.323286	−0.161643	+	0.986849i	$0.551679\pi$
$444$	0	0
$445$	26.4038	1.25166
$446$	0	0
$447$	−10.8616	−0.513736
$448$	0	0
$449$	26.9990	1.27416	0.637081	−	0.770796i	$-0.280141\pi$
$449$	26.9990	1.27416	0.637081	+	0.770796i	$0.280141\pi$
$450$	0	0
$451$	30.2174	1.42288
$452$	0	0
$453$	5.45582	0.256337
$454$	0	0
$455$	−14.4332	−0.676640
$456$	0	0
$457$	−20.9589	−0.980417	−0.490209	−	0.871605i	$-0.663079\pi$
$457$	−20.9589	−0.980417	−0.490209	+	0.871605i	$0.663079\pi$
$458$	0	0
$459$	0.872173	0.0407095
$460$	0	0
$461$	4.94509	0.230316	0.115158	−	0.993347i	$-0.463263\pi$
$461$	4.94509	0.230316	0.115158	+	0.993347i	$0.463263\pi$
$462$	0	0
$463$	−24.1178	−1.12085	−0.560425	−	0.828205i	$-0.689362\pi$
$463$	−24.1178	−1.12085	−0.560425	+	0.828205i	$0.689362\pi$
$464$	0	0
$465$	22.9160	1.06270
$466$	0	0
$467$	41.3413	1.91305	0.956523	−	0.291658i	$-0.0942070\pi$
$467$	41.3413	1.91305	0.956523	+	0.291658i	$0.0942070\pi$
$468$	0	0
$469$	14.5213	0.670530
$470$	0	0
$471$	−9.40270	−0.433254
$472$	0	0
$473$	−46.6485	−2.14490
$474$	0	0
$475$	1.90756	0.0875247
$476$	0	0
$477$	−10.3374	−0.473318
$478$	0	0
$479$	−23.9922	−1.09623	−0.548116	−	0.836402i	$-0.684655\pi$
$479$	−23.9922	−1.09623	−0.548116	+	0.836402i	$0.684655\pi$
$480$	0	0
$481$	−17.9946	−0.820483
$482$	0	0
$483$	−16.2642	−0.740046
$484$	0	0
$485$	−39.5619	−1.79642
$486$	0	0
$487$	24.3733	1.10446	0.552230	−	0.833692i	$-0.313777\pi$
$487$	24.3733	1.10446	0.552230	+	0.833692i	$0.313777\pi$
$488$	0	0
$489$	17.7492	0.802647
$490$	0	0
$491$	9.67535	0.436642	0.218321	−	0.975877i	$-0.429942\pi$
$491$	9.67535	0.436642	0.218321	+	0.975877i	$0.429942\pi$
$492$	0	0
$493$	−5.45064	−0.245484
$494$	0	0
$495$	−11.3242	−0.508985
$496$	0	0
$497$	15.7332	0.705730
$498$	0	0
$499$	−4.61271	−0.206493	−0.103247	−	0.994656i	$-0.532923\pi$
$499$	−4.61271	−0.206493	−0.103247	+	0.994656i	$0.532923\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−40.1374	−1.78964	−0.894818	−	0.446431i	$-0.852695\pi$
$503$	−40.1374	−1.78964	−0.894818	+	0.446431i	$0.852695\pi$
$504$	0	0
$505$	36.1544	1.60885
$506$	0	0
$507$	−3.92853	−0.174472
$508$	0	0
$509$	6.02612	0.267103	0.133551	−	0.991042i	$-0.457362\pi$
$509$	6.02612	0.267103	0.133551	+	0.991042i	$0.457362\pi$
$510$	0	0
$511$	−10.7129	−0.473912
$512$	0	0
$513$	−5.13559	−0.226742
$514$	0	0
$515$	−23.7734	−1.04758
$516$	0	0
$517$	−35.6458	−1.56770
$518$	0	0
$519$	23.1988	1.01831
$520$	0	0
$521$	−23.7073	−1.03863	−0.519317	−	0.854581i	$-0.673814\pi$
$521$	−23.7073	−1.03863	−0.519317	+	0.854581i	$0.673814\pi$
$522$	0	0
$523$	24.4762	1.07027	0.535135	−	0.844767i	$-0.320261\pi$
$523$	24.4762	1.07027	0.535135	+	0.844767i	$0.320261\pi$
$524$	0	0
$525$	−0.827350	−0.0361085
$526$	0	0
$527$	−9.29005	−0.404681
$528$	0	0
$529$	30.3163	1.31810
$530$	0	0
$531$	8.89647	0.386074
$532$	0	0
$533$	17.2906	0.748941
$534$	0	0
$535$	−4.36349	−0.188650
$536$	0	0
$537$	−12.9681	−0.559615
$538$	0	0
$539$	−10.7304	−0.462193
$540$	0	0
$541$	6.31761	0.271615	0.135808	−	0.990735i	$-0.456637\pi$
$541$	6.31761	0.271615	0.135808	+	0.990735i	$0.456637\pi$
$542$	0	0
$543$	5.51396	0.236626
$544$	0	0
$545$	4.99973	0.214165
$546$	0	0
$547$	−23.1909	−0.991570	−0.495785	−	0.868445i	$-0.665120\pi$
$547$	−23.1909	−0.991570	−0.495785	+	0.868445i	$0.665120\pi$
$548$	0	0
$549$	4.06058	0.173301
$550$	0	0
$551$	32.0948	1.36728
$552$	0	0
$553$	2.25681	0.0959693
$554$	0	0
$555$	12.8536	0.545607
$556$	0	0
$557$	12.7445	0.540002	0.270001	−	0.962860i	$-0.412976\pi$
$557$	12.7445	0.540002	0.270001	+	0.962860i	$0.412976\pi$
$558$	0	0
$559$	−26.6927	−1.12898
$560$	0	0
$561$	4.59079	0.193823
$562$	0	0
$563$	13.6177	0.573919	0.286960	−	0.957943i	$-0.407355\pi$
$563$	13.6177	0.573919	0.286960	+	0.957943i	$0.407355\pi$
$564$	0	0
$565$	15.2594	0.641970
$566$	0	0
$567$	2.22742	0.0935428
$568$	0	0
$569$	−4.58265	−0.192115	−0.0960574	−	0.995376i	$-0.530623\pi$
$569$	−4.58265	−0.192115	−0.0960574	+	0.995376i	$0.530623\pi$
$570$	0	0
$571$	3.08523	0.129113	0.0645565	−	0.997914i	$-0.479437\pi$
$571$	3.08523	0.129113	0.0645565	+	0.997914i	$0.479437\pi$
$572$	0	0
$573$	−6.81882	−0.284860
$574$	0	0
$575$	2.71217	0.113105
$576$	0	0
$577$	29.3635	1.22242	0.611210	−	0.791468i	$-0.290683\pi$
$577$	29.3635	1.22242	0.611210	+	0.791468i	$0.290683\pi$
$578$	0	0
$579$	3.94100	0.163782
$580$	0	0
$581$	−35.7692	−1.48396
$582$	0	0
$583$	−54.4123	−2.25353
$584$	0	0
$585$	−6.47980	−0.267907
$586$	0	0
$587$	46.7364	1.92902	0.964510	−	0.264047i	$-0.0850574\pi$
$587$	46.7364	1.92902	0.964510	+	0.264047i	$0.0850574\pi$
$588$	0	0
$589$	54.7023	2.25397
$590$	0	0
$591$	10.7619	0.442685
$592$	0	0
$593$	−3.52374	−0.144703	−0.0723514	−	0.997379i	$-0.523050\pi$
$593$	−3.52374	−0.144703	−0.0723514	+	0.997379i	$0.523050\pi$
$594$	0	0
$595$	−4.17953	−0.171344
$596$	0	0
$597$	−13.6212	−0.557481
$598$	0	0
$599$	6.69774	0.273662	0.136831	−	0.990594i	$-0.456308\pi$
$599$	6.69774	0.273662	0.136831	+	0.990594i	$0.456308\pi$
$600$	0	0
$601$	−11.3079	−0.461258	−0.230629	−	0.973042i	$-0.574078\pi$
$601$	−11.3079	−0.461258	−0.230629	+	0.973042i	$0.574078\pi$
$602$	0	0
$603$	6.51933	0.265488
$604$	0	0
$605$	−35.9408	−1.46120
$606$	0	0
$607$	−39.9079	−1.61981	−0.809905	−	0.586561i	$-0.800481\pi$
$607$	−39.9079	−1.61981	−0.809905	+	0.586561i	$0.800481\pi$
$608$	0	0
$609$	−13.9202	−0.564077
$610$	0	0
$611$	−20.3968	−0.825167
$612$	0	0
$613$	36.3501	1.46817	0.734084	−	0.679059i	$-0.237612\pi$
$613$	36.3501	1.46817	0.734084	+	0.679059i	$0.237612\pi$
$614$	0	0
$615$	−12.3508	−0.498032
$616$	0	0
$617$	−7.14711	−0.287732	−0.143866	−	0.989597i	$-0.545953\pi$
$617$	−7.14711	−0.287732	−0.143866	+	0.989597i	$0.545953\pi$
$618$	0	0
$619$	−14.4099	−0.579184	−0.289592	−	0.957150i	$-0.593520\pi$
$619$	−14.4099	−0.579184	−0.289592	+	0.957150i	$0.593520\pi$
$620$	0	0
$621$	−7.30180	−0.293011
$622$	0	0
$623$	−27.3366	−1.09522
$624$	0	0
$625$	−23.0048	−0.920194
$626$	0	0
$627$	−27.0318	−1.07955
$628$	0	0
$629$	−5.21082	−0.207769
$630$	0	0
$631$	−10.6369	−0.423447	−0.211724	−	0.977330i	$-0.567908\pi$
$631$	−10.6369	−0.423447	−0.211724	+	0.977330i	$0.567908\pi$
$632$	0	0
$633$	−12.2913	−0.488535
$634$	0	0
$635$	0.997783	0.0395958
$636$	0	0
$637$	−6.14005	−0.243278
$638$	0	0
$639$	7.06342	0.279424
$640$	0	0
$641$	−15.2423	−0.602036	−0.301018	−	0.953618i	$-0.597326\pi$
$641$	−15.2423	−0.602036	−0.301018	+	0.953618i	$0.597326\pi$
$642$	0	0
$643$	−6.61351	−0.260811	−0.130406	−	0.991461i	$-0.541628\pi$
$643$	−6.61351	−0.260811	−0.130406	+	0.991461i	$0.541628\pi$
$644$	0	0
$645$	19.0667	0.750752
$646$	0	0
$647$	36.7460	1.44463	0.722316	−	0.691563i	$-0.243077\pi$
$647$	36.7460	1.44463	0.722316	+	0.691563i	$0.243077\pi$
$648$	0	0
$649$	46.8277	1.83815
$650$	0	0
$651$	−23.7256	−0.929880
$652$	0	0
$653$	−21.7995	−0.853080	−0.426540	−	0.904469i	$-0.640268\pi$
$653$	−21.7995	−0.853080	−0.426540	+	0.904469i	$0.640268\pi$
$654$	0	0
$655$	13.2458	0.517556
$656$	0	0
$657$	−4.80957	−0.187639
$658$	0	0
$659$	31.1337	1.21279	0.606397	−	0.795162i	$-0.292614\pi$
$659$	31.1337	1.21279	0.606397	+	0.795162i	$0.292614\pi$
$660$	0	0
$661$	0.809186	0.0314737	0.0157369	−	0.999876i	$-0.494991\pi$
$661$	0.809186	0.0314737	0.0157369	+	0.999876i	$0.494991\pi$
$662$	0	0
$663$	2.62689	0.102020
$664$	0	0
$665$	24.6102	0.954342
$666$	0	0
$667$	45.6326	1.76690
$668$	0	0
$669$	29.8084	1.15246
$670$	0	0
$671$	21.3734	0.825110
$672$	0	0
$673$	−0.361915	−0.0139508	−0.00697540	−	0.999976i	$-0.502220\pi$
$673$	−0.361915	−0.0139508	−0.00697540	+	0.999976i	$0.502220\pi$
$674$	0	0
$675$	−0.371439	−0.0142967
$676$	0	0
$677$	21.2802	0.817863	0.408932	−	0.912565i	$-0.365901\pi$
$677$	21.2802	0.817863	0.408932	+	0.912565i	$0.365901\pi$
$678$	0	0
$679$	40.9597	1.57189
$680$	0	0
$681$	−5.90639	−0.226333
$682$	0	0
$683$	−6.44700	−0.246688	−0.123344	−	0.992364i	$-0.539362\pi$
$683$	−6.44700	−0.246688	−0.123344	+	0.992364i	$0.539362\pi$
$684$	0	0
$685$	33.5643	1.28243
$686$	0	0
$687$	21.3820	0.815776
$688$	0	0
$689$	−31.1351	−1.18615
$690$	0	0
$691$	27.5831	1.04931	0.524654	−	0.851315i	$-0.324195\pi$
$691$	27.5831	1.04931	0.524654	+	0.851315i	$0.324195\pi$
$692$	0	0
$693$	11.7243	0.445369
$694$	0	0
$695$	−21.2030	−0.804277
$696$	0	0
$697$	5.00697	0.189652
$698$	0	0
$699$	−16.8878	−0.638755
$700$	0	0
$701$	25.6133	0.967399	0.483700	−	0.875234i	$-0.339293\pi$
$701$	25.6133	0.967399	0.483700	+	0.875234i	$0.339293\pi$
$702$	0	0
$703$	30.6827	1.15722
$704$	0	0
$705$	14.5696	0.548722
$706$	0	0
$707$	−37.4318	−1.40777
$708$	0	0
$709$	−39.6608	−1.48949	−0.744747	−	0.667347i	$-0.767430\pi$
$709$	−39.6608	−1.48949	−0.744747	+	0.667347i	$0.767430\pi$
$710$	0	0
$711$	1.01320	0.0379978
$712$	0	0
$713$	77.7760	2.91274
$714$	0	0
$715$	−34.1072	−1.27554
$716$	0	0
$717$	3.07320	0.114771
$718$	0	0
$719$	−5.83258	−0.217519	−0.108759	−	0.994068i	$-0.534688\pi$
$719$	−5.83258	−0.217519	−0.108759	+	0.994068i	$0.534688\pi$
$720$	0	0
$721$	24.6133	0.916646
$722$	0	0
$723$	−0.156195	−0.00580895
$724$	0	0
$725$	2.32130	0.0862111
$726$	0	0
$727$	17.8601	0.662394	0.331197	−	0.943562i	$-0.392547\pi$
$727$	17.8601	0.662394	0.331197	+	0.943562i	$0.392547\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−7.72958	−0.285889
$732$	0	0
$733$	−51.5311	−1.90335	−0.951673	−	0.307113i	$-0.900637\pi$
$733$	−51.5311	−1.90335	−0.951673	+	0.307113i	$0.900637\pi$
$734$	0	0
$735$	4.38587	0.161775
$736$	0	0
$737$	34.3153	1.26402
$738$	0	0
$739$	41.8562	1.53970	0.769852	−	0.638222i	$-0.220330\pi$
$739$	41.8562	1.53970	0.769852	+	0.638222i	$0.220330\pi$
$740$	0	0
$741$	−15.4678	−0.568224
$742$	0	0
$743$	40.6051	1.48966	0.744828	−	0.667256i	$-0.232531\pi$
$743$	40.6051	1.48966	0.744828	+	0.667256i	$0.232531\pi$
$744$	0	0
$745$	23.3678	0.856129
$746$	0	0
$747$	−16.0586	−0.587554
$748$	0	0
$749$	4.51765	0.165071
$750$	0	0
$751$	11.4546	0.417985	0.208992	−	0.977917i	$-0.432982\pi$
$751$	11.4546	0.417985	0.208992	+	0.977917i	$0.432982\pi$
$752$	0	0
$753$	3.70516	0.135023
$754$	0	0
$755$	−11.7377	−0.427179
$756$	0	0
$757$	−42.2945	−1.53722	−0.768610	−	0.639718i	$-0.779051\pi$
$757$	−42.2945	−1.53722	−0.768610	+	0.639718i	$0.779051\pi$
$758$	0	0
$759$	−38.4339	−1.39506
$760$	0	0
$761$	4.64853	0.168509	0.0842546	−	0.996444i	$-0.473149\pi$
$761$	4.64853	0.168509	0.0842546	+	0.996444i	$0.473149\pi$
$762$	0	0
$763$	−5.17637	−0.187397
$764$	0	0
$765$	−1.87640	−0.0678414
$766$	0	0
$767$	26.7952	0.967518
$768$	0	0
$769$	−1.64981	−0.0594937	−0.0297469	−	0.999557i	$-0.509470\pi$
$769$	−1.64981	−0.0594937	−0.0297469	+	0.999557i	$0.509470\pi$
$770$	0	0
$771$	−2.46075	−0.0886218
$772$	0	0
$773$	8.80820	0.316809	0.158404	−	0.987374i	$-0.449365\pi$
$773$	8.80820	0.316809	0.158404	+	0.987374i	$0.449365\pi$
$774$	0	0
$775$	3.95642	0.142119
$776$	0	0
$777$	−13.3078	−0.477413
$778$	0	0
$779$	−29.4824	−1.05632
$780$	0	0
$781$	37.1791	1.33037
$782$	0	0
$783$	−6.24949	−0.223339
$784$	0	0
$785$	20.2291	0.722006
$786$	0	0
$787$	−21.8883	−0.780234	−0.390117	−	0.920765i	$-0.627565\pi$
$787$	−21.8883	−0.780234	−0.390117	+	0.920765i	$0.627565\pi$
$788$	0	0
$789$	18.5170	0.659223
$790$	0	0
$791$	−15.7986	−0.561732
$792$	0	0
$793$	12.2300	0.434301
$794$	0	0
$795$	22.2400	0.788772
$796$	0	0
$797$	35.2309	1.24794	0.623971	−	0.781448i	$-0.285518\pi$
$797$	35.2309	1.24794	0.623971	+	0.781448i	$0.285518\pi$
$798$	0	0
$799$	−5.90645	−0.208955
$800$	0	0
$801$	−12.2728	−0.433637
$802$	0	0
$803$	−25.3157	−0.893373
$804$	0	0
$805$	34.9909	1.23327
$806$	0	0
$807$	−31.3755	−1.10447
$808$	0	0
$809$	12.7891	0.449641	0.224820	−	0.974400i	$-0.427820\pi$
$809$	12.7891	0.449641	0.224820	+	0.974400i	$0.427820\pi$
$810$	0	0
$811$	−16.1763	−0.568027	−0.284013	−	0.958820i	$-0.591666\pi$
$811$	−16.1763	−0.568027	−0.284013	+	0.958820i	$0.591666\pi$
$812$	0	0
$813$	−6.93351	−0.243169
$814$	0	0
$815$	−38.1858	−1.33759
$816$	0	0
$817$	45.5138	1.59233
$818$	0	0
$819$	6.70874	0.234422
$820$	0	0
$821$	−14.5505	−0.507815	−0.253908	−	0.967228i	$-0.581716\pi$
$821$	−14.5505	−0.507815	−0.253908	+	0.967228i	$0.581716\pi$
$822$	0	0
$823$	−8.95939	−0.312305	−0.156152	−	0.987733i	$-0.549909\pi$
$823$	−8.95939	−0.312305	−0.156152	+	0.987733i	$0.549909\pi$
$824$	0	0
$825$	−1.95511	−0.0680683
$826$	0	0
$827$	−22.6162	−0.786444	−0.393222	−	0.919444i	$-0.628640\pi$
$827$	−22.6162	−0.786444	−0.393222	+	0.919444i	$0.628640\pi$
$828$	0	0
$829$	3.17997	0.110445	0.0552225	−	0.998474i	$-0.482413\pi$
$829$	3.17997	0.110445	0.0552225	+	0.998474i	$0.482413\pi$
$830$	0	0
$831$	28.2322	0.979364
$832$	0	0
$833$	−1.77802	−0.0616046
$834$	0	0
$835$	−2.15141	−0.0744526
$836$	0	0
$837$	−10.6516	−0.368174
$838$	0	0
$839$	43.8738	1.51469	0.757346	−	0.653014i	$-0.226496\pi$
$839$	43.8738	1.51469	0.757346	+	0.653014i	$0.226496\pi$
$840$	0	0
$841$	10.0562	0.346764
$842$	0	0
$843$	26.6487	0.917830
$844$	0	0
$845$	8.45189	0.290754
$846$	0	0
$847$	37.2106	1.27857
$848$	0	0
$849$	25.5145	0.875656
$850$	0	0
$851$	43.6248	1.49544
$852$	0	0
$853$	−29.3229	−1.00400	−0.501999	−	0.864868i	$-0.667402\pi$
$853$	−29.3229	−1.00400	−0.501999	+	0.864868i	$0.667402\pi$
$854$	0	0
$855$	11.0487	0.377859
$856$	0	0
$857$	0.0318938	0.00108947	0.000544737	−	1.00000i	$-0.499827\pi$
$857$	0.0318938	0.00108947	0.000544737		1.00000i	$0.499827\pi$
$858$	0	0
$859$	6.84019	0.233384	0.116692	−	0.993168i	$-0.462771\pi$
$859$	6.84019	0.233384	0.116692	+	0.993168i	$0.462771\pi$
$860$	0	0
$861$	12.7872	0.435785
$862$	0	0
$863$	−45.2413	−1.54003	−0.770016	−	0.638024i	$-0.779752\pi$
$863$	−45.2413	−1.54003	−0.770016	+	0.638024i	$0.779752\pi$
$864$	0	0
$865$	−49.9101	−1.69699
$866$	0	0
$867$	−16.2393	−0.551516
$868$	0	0
$869$	5.33307	0.180912
$870$	0	0
$871$	19.6355	0.665323
$872$	0	0
$873$	18.3889	0.622368
$874$	0	0
$875$	25.7404	0.870185
$876$	0	0
$877$	−37.3692	−1.26187	−0.630935	−	0.775836i	$-0.717328\pi$
$877$	−37.3692	−1.26187	−0.630935	+	0.775836i	$0.717328\pi$
$878$	0	0
$879$	−22.3309	−0.753203
$880$	0	0
$881$	38.6654	1.30267	0.651336	−	0.758789i	$-0.274209\pi$
$881$	38.6654	1.30267	0.651336	+	0.758789i	$0.274209\pi$
$882$	0	0
$883$	48.2694	1.62440	0.812198	−	0.583382i	$-0.198271\pi$
$883$	48.2694	1.62440	0.812198	+	0.583382i	$0.198271\pi$
$884$	0	0
$885$	−19.1400	−0.643383
$886$	0	0
$887$	25.5333	0.857323	0.428661	−	0.903465i	$-0.358985\pi$
$887$	25.5333	0.857323	0.428661	+	0.903465i	$0.358985\pi$
$888$	0	0
$889$	−1.03304	−0.0346469
$890$	0	0
$891$	5.26362	0.176338
$892$	0	0
$893$	34.7788	1.16383
$894$	0	0
$895$	27.8997	0.932584
$896$	0	0
$897$	−21.9922	−0.734299
$898$	0	0
$899$	66.5672	2.22014
$900$	0	0
$901$	−9.01602	−0.300367
$902$	0	0
$903$	−19.7404	−0.656918
$904$	0	0
$905$	−11.8628	−0.394332
$906$	0	0
$907$	−38.8512	−1.29003	−0.645016	−	0.764169i	$-0.723149\pi$
$907$	−38.8512	−1.29003	−0.645016	+	0.764169i	$0.723149\pi$
$908$	0	0
$909$	−16.8050	−0.557387
$910$	0	0
$911$	−13.4762	−0.446486	−0.223243	−	0.974763i	$-0.571664\pi$
$911$	−13.4762	−0.446486	−0.223243	+	0.974763i	$0.571664\pi$
$912$	0	0
$913$	−84.5264	−2.79742
$914$	0	0
$915$	−8.73597	−0.288802
$916$	0	0
$917$	−13.7138	−0.452868
$918$	0	0
$919$	−22.0962	−0.728885	−0.364443	−	0.931226i	$-0.618740\pi$
$919$	−22.0962	−0.728885	−0.364443	+	0.931226i	$0.618740\pi$
$920$	0	0
$921$	19.6253	0.646677
$922$	0	0
$923$	21.2742	0.700249
$924$	0	0
$925$	2.21917	0.0729659
$926$	0	0
$927$	11.0501	0.362934
$928$	0	0
$929$	7.99169	0.262199	0.131099	−	0.991369i	$-0.458149\pi$
$929$	7.99169	0.262199	0.131099	+	0.991369i	$0.458149\pi$
$930$	0	0
$931$	10.4694	0.343122
$932$	0	0
$933$	−23.4421	−0.767460
$934$	0	0
$935$	−9.87666	−0.323001
$936$	0	0
$937$	−5.16626	−0.168774	−0.0843871	−	0.996433i	$-0.526893\pi$
$937$	−5.16626	−0.168774	−0.0843871	+	0.996433i	$0.526893\pi$
$938$	0	0
$939$	−20.7190	−0.676139
$940$	0	0
$941$	−29.0921	−0.948376	−0.474188	−	0.880424i	$-0.657258\pi$
$941$	−29.0921	−0.948376	−0.474188	+	0.880424i	$0.657258\pi$
$942$	0	0
$943$	−41.9182	−1.36504
$944$	0	0
$945$	−4.79209	−0.155887
$946$	0	0
$947$	−37.5627	−1.22062	−0.610312	−	0.792161i	$-0.708956\pi$
$947$	−37.5627	−1.22062	−0.610312	+	0.792161i	$0.708956\pi$
$948$	0	0
$949$	−14.4859	−0.470231
$950$	0	0
$951$	−24.8655	−0.806318
$952$	0	0
$953$	−39.5701	−1.28180	−0.640900	−	0.767624i	$-0.721439\pi$
$953$	−39.5701	−1.28180	−0.640900	+	0.767624i	$0.721439\pi$
$954$	0	0
$955$	14.6701	0.474712
$956$	0	0
$957$	−32.8950	−1.06334
$958$	0	0
$959$	−34.7502	−1.12214
$960$	0	0
$961$	82.4570	2.65990
$962$	0	0
$963$	2.02820	0.0653578
$964$	0	0
$965$	−8.47869	−0.272939
$966$	0	0
$967$	−55.5346	−1.78587	−0.892936	−	0.450183i	$-0.851359\pi$
$967$	−55.5346	−1.78587	−0.892936	+	0.450183i	$0.851359\pi$
$968$	0	0
$969$	−4.47912	−0.143890
$970$	0	0
$971$	3.02146	0.0969634	0.0484817	−	0.998824i	$-0.484562\pi$
$971$	3.02146	0.0969634	0.0484817	+	0.998824i	$0.484562\pi$
$972$	0	0
$973$	21.9521	0.703753
$974$	0	0
$975$	−1.11873	−0.0358281
$976$	0	0
$977$	44.5274	1.42456	0.712279	−	0.701896i	$-0.247663\pi$
$977$	44.5274	1.42456	0.712279	+	0.701896i	$0.247663\pi$
$978$	0	0
$979$	−64.5992	−2.06460
$980$	0	0
$981$	−2.32393	−0.0741975
$982$	0	0
$983$	46.7425	1.49086	0.745428	−	0.666586i	$-0.232245\pi$
$983$	46.7425	1.49086	0.745428	+	0.666586i	$0.232245\pi$
$984$	0	0
$985$	−23.1532	−0.737723
$986$	0	0
$987$	−15.0843	−0.480139
$988$	0	0
$989$	64.7118	2.05771
$990$	0	0
$991$	−10.5609	−0.335477	−0.167739	−	0.985832i	$-0.553646\pi$
$991$	−10.5609	−0.335477	−0.167739	+	0.985832i	$0.553646\pi$
$992$	0	0
$993$	22.5680	0.716175
$994$	0	0
$995$	29.3049	0.929027
$996$	0	0
$997$	−26.5039	−0.839388	−0.419694	−	0.907666i	$-0.637863\pi$
$997$	−26.5039	−0.839388	−0.419694	+	0.907666i	$0.637863\pi$
$998$	0	0
$999$	−5.97452	−0.189026

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.bd.1.4		10
4.3	odd	2		4008.2.a.j.1.4	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.j.1.4	✓	10	4.3	odd	2
8016.2.a.bd.1.4		10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.15141 q^{5} +2.22742 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.15141 q^{5} +2.22742 q^{7} +1.00000 q^{9} +5.26362 q^{11} +3.01189 q^{13} -2.15141 q^{15} +0.872173 q^{17} -5.13559 q^{19} +2.22742 q^{21} -7.30180 q^{23} -0.371439 q^{25} +1.00000 q^{27} -6.24949 q^{29} -10.6516 q^{31} +5.26362 q^{33} -4.79209 q^{35} -5.97452 q^{37} +3.01189 q^{39} +5.74080 q^{41} -8.86244 q^{43} -2.15141 q^{45} -6.77211 q^{47} -2.03860 q^{49} +0.872173 q^{51} -10.3374 q^{53} -11.3242 q^{55} -5.13559 q^{57} +8.89647 q^{59} +4.06058 q^{61} +2.22742 q^{63} -6.47980 q^{65} +6.51933 q^{67} -7.30180 q^{69} +7.06342 q^{71} -4.80957 q^{73} -0.371439 q^{75} +11.7243 q^{77} +1.01320 q^{79} +1.00000 q^{81} -16.0586 q^{83} -1.87640 q^{85} -6.24949 q^{87} -12.2728 q^{89} +6.70874 q^{91} -10.6516 q^{93} +11.0487 q^{95} +18.3889 q^{97} +5.26362 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{3} - 10 q^{5} - q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q + 10 * q^3 - 10 * q^5 - q^7 + 10 * q^9	\( 10 q + 10 q^{3} - 10 q^{5} - q^{7} + 10 q^{9} + q^{11} - 6 q^{13} - 10 q^{15} - 9 q^{17} - 2 q^{19} - q^{21} - 7 q^{23} + 12 q^{25} + 10 q^{27} - 13 q^{29} - 23 q^{31} + q^{33} - q^{35} - 6 q^{37} - 6 q^{39} - 12 q^{41} - 10 q^{45} - 10 q^{47} + 7 q^{49} - 9 q^{51} - 26 q^{53} - 11 q^{55} - 2 q^{57} + 10 q^{59} - 10 q^{61} - q^{63} - 22 q^{65} + 5 q^{67} - 7 q^{69} - 25 q^{71} - 8 q^{73} + 12 q^{75} - 46 q^{77} - 26 q^{79} + 10 q^{81} + 14 q^{83} + 9 q^{85} - 13 q^{87} - 31 q^{89} + 3 q^{91} - 23 q^{93} + 5 q^{95} - 32 q^{97} + q^{99}+O(q^{100}) \) 10 * q + 10 * q^3 - 10 * q^5 - q^7 + 10 * q^9 + q^11 - 6 * q^13 - 10 * q^15 - 9 * q^17 - 2 * q^19 - q^21 - 7 * q^23 + 12 * q^25 + 10 * q^27 - 13 * q^29 - 23 * q^31 + q^33 - q^35 - 6 * q^37 - 6 * q^39 - 12 * q^41 - 10 * q^45 - 10 * q^47 + 7 * q^49 - 9 * q^51 - 26 * q^53 - 11 * q^55 - 2 * q^57 + 10 * q^59 - 10 * q^61 - q^63 - 22 * q^65 + 5 * q^67 - 7 * q^69 - 25 * q^71 - 8 * q^73 + 12 * q^75 - 46 * q^77 - 26 * q^79 + 10 * q^81 + 14 * q^83 + 9 * q^85 - 13 * q^87 - 31 * q^89 + 3 * q^91 - 23 * q^93 + 5 * q^95 - 32 * q^97 + q^99

Introduction

L-functions

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