LMFDB - Embedded newform 4008.2.a.j.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, 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("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.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:	$32.0040411301$
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:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.15141$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-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	$-0.638301\pi$
$7$	−2.22742	−0.841885	−0.420943	+	0.907087i	$0.638301\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.26362	−1.58704	−0.793521	−	0.608543i	$-0.791754\pi$
$11$	−5.26362	−1.58704	−0.793521	+	0.608543i	$0.791754\pi$
$12$	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	$-0.299486\pi$
$19$	5.13559	1.17818	0.589092	+	0.808066i	$0.299486\pi$
$20$	0	0
$21$	2.22742	0.486063
$22$	0	0
$23$	7.30180	1.52253	0.761266	−	0.648440i	$-0.224578\pi$
$23$	7.30180	1.52253	0.761266	+	0.648440i	$0.224578\pi$
$24$	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	$-0.0941841\pi$
$31$	10.6516	1.91309	0.956544	+	0.291589i	$0.0941841\pi$
$32$	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	$-0.263818\pi$
$43$	8.86244	1.35151	0.675755	+	0.737127i	$0.263818\pi$
$44$	0	0
$45$	−2.15141	−0.320713
$46$	0	0
$47$	6.77211	0.987814	0.493907	−	0.869515i	$-0.335568\pi$
$47$	6.77211	0.987814	0.493907	+	0.869515i	$0.335568\pi$
$48$	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	$-0.696600\pi$
$59$	−8.89647	−1.15822	−0.579111	+	0.815249i	$0.696600\pi$
$60$	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	$-0.630376\pi$
$67$	−6.51933	−0.796463	−0.398231	+	0.917285i	$0.630376\pi$
$68$	0	0
$69$	−7.30180	−0.879034
$70$	0	0
$71$	−7.06342	−0.838273	−0.419137	−	0.907923i	$-0.637667\pi$
$71$	−7.06342	−0.838273	−0.419137	+	0.907923i	$0.637667\pi$
$72$	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	$-0.518152\pi$
$79$	−1.01320	−0.113993	−0.0569967	+	0.998374i	$0.518152\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	16.0586	1.76266	0.881330	−	0.472501i	$-0.156649\pi$
$83$	16.0586	1.76266	0.881330	+	0.472501i	$0.156649\pi$
$84$	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	$-0.683243\pi$
$103$	−11.0501	−1.08880	−0.544401	+	0.838825i	$0.683243\pi$
$104$	0	0
$105$	−4.79209	−0.467660
$106$	0	0
$107$	−2.02820	−0.196073	−0.0980367	−	0.995183i	$-0.531256\pi$
$107$	−2.02820	−0.196073	−0.0980367	+	0.995183i	$0.531256\pi$
$108$	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	$-0.493450\pi$
$127$	0.463781	0.0411539	0.0205770	+	0.999788i	$0.493450\pi$
$128$	0	0
$129$	−8.86244	−0.780294
$130$	0	0
$131$	6.15679	0.537922	0.268961	−	0.963151i	$-0.413320\pi$
$131$	6.15679	0.537922	0.268961	+	0.963151i	$0.413320\pi$
$132$	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	$-0.637256\pi$
$139$	−9.85541	−0.835925	−0.417963	+	0.908464i	$0.637256\pi$
$140$	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	$-0.571257\pi$
$151$	−5.45582	−0.443988	−0.221994	+	0.975048i	$0.571257\pi$
$152$	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	$-0.744645\pi$
$163$	−17.7492	−1.39023	−0.695113	+	0.718901i	$0.744645\pi$
$164$	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	$-0.338950\pi$
$179$	12.9681	0.969281	0.484641	+	0.874713i	$0.338950\pi$
$180$	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	$-0.420655\pi$
$191$	6.81882	0.493392	0.246696	+	0.969093i	$0.420655\pi$
$192$	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	$-0.339623\pi$
$199$	13.6212	0.965585	0.482792	+	0.875735i	$0.339623\pi$
$200$	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	$-0.360948\pi$
$211$	12.2913	0.846167	0.423084	+	0.906091i	$0.360948\pi$
$212$	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	$-0.980164\pi$
$223$	−29.8084	−1.99612	−0.998059	+	0.0622749i	$0.980164\pi$
$224$	0	0
$225$	−0.371439	−0.0247626
$226$	0	0
$227$	5.90639	0.392021	0.196010	−	0.980602i	$-0.437201\pi$
$227$	5.90639	0.392021	0.196010	+	0.980602i	$0.437201\pi$
$228$	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	$-0.531691\pi$
$239$	−3.07320	−0.198789	−0.0993943	+	0.995048i	$0.531691\pi$
$240$	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	$-0.537307\pi$
$251$	−3.70516	−0.233867	−0.116934	+	0.993140i	$0.537307\pi$
$252$	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	$-0.693407\pi$
$263$	−18.5170	−1.14181	−0.570904	+	0.821017i	$0.693407\pi$
$264$	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	$-0.432461\pi$
$271$	6.93351	0.421180	0.210590	+	0.977574i	$0.432461\pi$
$272$	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	$-0.773990\pi$
$283$	−25.5145	−1.51668	−0.758341	+	0.651858i	$0.773990\pi$
$284$	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	$-0.689214\pi$
$307$	−19.6253	−1.12008	−0.560038	+	0.828467i	$0.689214\pi$
$308$	0	0
$309$	11.0501	0.628620
$310$	0	0
$311$	23.4421	1.32928	0.664640	−	0.747164i	$-0.268585\pi$
$311$	23.4421	1.32928	0.664640	+	0.747164i	$0.268585\pi$
$312$	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	$-0.712959\pi$
$331$	−22.5680	−1.24045	−0.620225	+	0.784424i	$0.712959\pi$
$332$	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	$-0.666740\pi$
$347$	−18.6354	−1.00040	−0.500200	+	0.865910i	$0.666740\pi$
$348$	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	$-0.657730\pi$
$359$	−18.0185	−0.950982	−0.475491	+	0.879721i	$0.657730\pi$
$360$	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	$-0.289881\pi$
$367$	23.4946	1.22641	0.613203	+	0.789926i	$0.289881\pi$
$368$	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	$-0.771514\pi$
$379$	−29.3283	−1.50650	−0.753248	+	0.657737i	$0.771514\pi$
$380$	0	0
$381$	−0.463781	−0.0237602
$382$	0	0
$383$	21.0500	1.07561	0.537803	−	0.843070i	$-0.319254\pi$
$383$	21.0500	1.07561	0.537803	+	0.843070i	$0.319254\pi$
$384$	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	$-0.714644\pi$
$419$	−25.5611	−1.24874	−0.624370	+	0.781129i	$0.714644\pi$
$420$	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	$-0.533435\pi$
$431$	−4.35336	−0.209694	−0.104847	+	0.994488i	$0.533435\pi$
$432$	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	$-0.246033\pi$
$439$	29.9980	1.43173	0.715863	+	0.698240i	$0.246033\pi$
$440$	0	0
$441$	−2.03860	−0.0970764
$442$	0	0
$443$	6.80439	0.323286	0.161643	−	0.986849i	$-0.448321\pi$
$443$	6.80439	0.323286	0.161643	+	0.986849i	$0.448321\pi$
$444$	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	$-0.310638\pi$
$463$	24.1178	1.12085	0.560425	+	0.828205i	$0.310638\pi$
$464$	0	0
$465$	22.9160	1.06270
$466$	0	0
$467$	−41.3413	−1.91305	−0.956523	−	0.291658i	$-0.905793\pi$
$467$	−41.3413	−1.91305	−0.956523	+	0.291658i	$0.905793\pi$
$468$	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	$-0.315345\pi$
$479$	23.9922	1.09623	0.548116	+	0.836402i	$0.315345\pi$
$480$	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	$-0.686223\pi$
$487$	−24.3733	−1.10446	−0.552230	+	0.833692i	$0.686223\pi$
$488$	0	0
$489$	17.7492	0.802647
$490$	0	0
$491$	−9.67535	−0.436642	−0.218321	−	0.975877i	$-0.570058\pi$
$491$	−9.67535	−0.436642	−0.218321	+	0.975877i	$0.570058\pi$
$492$	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	$-0.467077\pi$
$499$	4.61271	0.206493	0.103247	+	0.994656i	$0.467077\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	40.1374	1.78964	0.894818	−	0.446431i	$-0.147305\pi$
$503$	40.1374	1.78964	0.894818	+	0.446431i	$0.147305\pi$
$504$	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	$-0.679739\pi$
$523$	−24.4762	−1.07027	−0.535135	+	0.844767i	$0.679739\pi$
$524$	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	$-0.334880\pi$
$547$	23.1909	0.991570	0.495785	+	0.868445i	$0.334880\pi$
$548$	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	$-0.592645\pi$
$563$	−13.6177	−0.573919	−0.286960	+	0.957943i	$0.592645\pi$
$564$	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	$-0.520563\pi$
$571$	−3.08523	−0.129113	−0.0645565	+	0.997914i	$0.520563\pi$
$572$	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	$-0.914943\pi$
$587$	−46.7364	−1.92902	−0.964510	+	0.264047i	$0.914943\pi$
$588$	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	$-0.543692\pi$
$599$	−6.69774	−0.273662	−0.136831	+	0.990594i	$0.543692\pi$
$600$	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	$-0.199519\pi$
$607$	39.9079	1.61981	0.809905	+	0.586561i	$0.199519\pi$
$608$	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	$-0.406480\pi$
$619$	14.4099	0.579184	0.289592	+	0.957150i	$0.406480\pi$
$620$	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	$-0.432092\pi$
$631$	10.6369	0.423447	0.211724	+	0.977330i	$0.432092\pi$
$632$	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	$-0.458372\pi$
$643$	6.61351	0.260811	0.130406	+	0.991461i	$0.458372\pi$
$644$	0	0
$645$	19.0667	0.750752
$646$	0	0
$647$	−36.7460	−1.44463	−0.722316	−	0.691563i	$-0.756923\pi$
$647$	−36.7460	−1.44463	−0.722316	+	0.691563i	$0.756923\pi$
$648$	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	$-0.707386\pi$
$659$	−31.1337	−1.21279	−0.606397	+	0.795162i	$0.707386\pi$
$660$	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	$-0.460638\pi$
$683$	6.44700	0.246688	0.123344	+	0.992364i	$0.460638\pi$
$684$	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	$-0.675805\pi$
$691$	−27.5831	−1.04931	−0.524654	+	0.851315i	$0.675805\pi$
$692$	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	$-0.465312\pi$
$719$	5.83258	0.217519	0.108759	+	0.994068i	$0.465312\pi$
$720$	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	$-0.607453\pi$
$727$	−17.8601	−0.662394	−0.331197	+	0.943562i	$0.607453\pi$
$728$	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	$-0.779670\pi$
$739$	−41.8562	−1.53970	−0.769852	+	0.638222i	$0.779670\pi$
$740$	0	0
$741$	−15.4678	−0.568224
$742$	0	0
$743$	−40.6051	−1.48966	−0.744828	−	0.667256i	$-0.767469\pi$
$743$	−40.6051	−1.48966	−0.744828	+	0.667256i	$0.767469\pi$
$744$	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	$-0.567018\pi$
$751$	−11.4546	−0.417985	−0.208992	+	0.977917i	$0.567018\pi$
$752$	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	$-0.372435\pi$
$787$	21.8883	0.780234	0.390117	+	0.920765i	$0.372435\pi$
$788$	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	$-0.408334\pi$
$811$	16.1763	0.568027	0.284013	+	0.958820i	$0.408334\pi$
$812$	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	$-0.450091\pi$
$823$	8.95939	0.312305	0.156152	+	0.987733i	$0.450091\pi$
$824$	0	0
$825$	−1.95511	−0.0680683
$826$	0	0
$827$	22.6162	0.786444	0.393222	−	0.919444i	$-0.371360\pi$
$827$	22.6162	0.786444	0.393222	+	0.919444i	$0.371360\pi$
$828$	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	$-0.773504\pi$
$839$	−43.8738	−1.51469	−0.757346	+	0.653014i	$0.773504\pi$
$840$	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	$-0.537229\pi$
$859$	−6.84019	−0.233384	−0.116692	+	0.993168i	$0.537229\pi$
$860$	0	0
$861$	12.7872	0.435785
$862$	0	0
$863$	45.2413	1.54003	0.770016	−	0.638024i	$-0.220248\pi$
$863$	45.2413	1.54003	0.770016	+	0.638024i	$0.220248\pi$
$864$	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	$-0.801729\pi$
$883$	−48.2694	−1.62440	−0.812198	+	0.583382i	$0.801729\pi$
$884$	0	0
$885$	−19.1400	−0.643383
$886$	0	0
$887$	−25.5333	−0.857323	−0.428661	−	0.903465i	$-0.641015\pi$
$887$	−25.5333	−0.857323	−0.428661	+	0.903465i	$0.641015\pi$
$888$	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	$-0.276851\pi$
$907$	38.8512	1.29003	0.645016	+	0.764169i	$0.276851\pi$
$908$	0	0
$909$	−16.8050	−0.557387
$910$	0	0
$911$	13.4762	0.446486	0.223243	−	0.974763i	$-0.428336\pi$
$911$	13.4762	0.446486	0.223243	+	0.974763i	$0.428336\pi$
$912$	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	$-0.381260\pi$
$919$	22.0962	0.728885	0.364443	+	0.931226i	$0.381260\pi$
$920$	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	$-0.291044\pi$
$947$	37.5627	1.22062	0.610312	+	0.792161i	$0.291044\pi$
$948$	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	$-0.148641\pi$
$967$	55.5346	1.78587	0.892936	+	0.450183i	$0.148641\pi$
$968$	0	0
$969$	−4.47912	−0.143890
$970$	0	0
$971$	−3.02146	−0.0969634	−0.0484817	−	0.998824i	$-0.515438\pi$
$971$	−3.02146	−0.0969634	−0.0484817	+	0.998824i	$0.515438\pi$
$972$	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	$-0.767755\pi$
$983$	−46.7425	−1.49086	−0.745428	+	0.666586i	$0.767755\pi$
$984$	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	$-0.446354\pi$
$991$	10.5609	0.335477	0.167739	+	0.985832i	$0.446354\pi$
$992$	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	4008.2.a.j.1.4	✓	10
4.3	odd	2		8016.2.a.bd.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.j.1.4	✓	10	1.1	even	1	trivial
8016.2.a.bd.1.4		10	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:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.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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