LMFDB - Embedded newform 6008.2.a.b.1.27

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, 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("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$1$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.27
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.278140 q^{3} +1.58670 q^{5} -2.49171 q^{7} -2.92264 q^{9} +O(q^{10})$	$q+0.278140 q^{3} +1.58670 q^{5} -2.49171 q^{7} -2.92264 q^{9} +5.31839 q^{11} -4.17786 q^{13} +0.441324 q^{15} -3.16226 q^{17} +5.18165 q^{19} -0.693044 q^{21} -2.27106 q^{23} -2.48239 q^{25} -1.64732 q^{27} +4.70849 q^{29} -0.466223 q^{31} +1.47926 q^{33} -3.95359 q^{35} +4.71895 q^{37} -1.16203 q^{39} +4.53789 q^{41} +3.13155 q^{43} -4.63734 q^{45} -1.59850 q^{47} -0.791388 q^{49} -0.879550 q^{51} -3.67829 q^{53} +8.43868 q^{55} +1.44122 q^{57} -3.30428 q^{59} -6.92274 q^{61} +7.28236 q^{63} -6.62900 q^{65} -12.2927 q^{67} -0.631673 q^{69} +12.0893 q^{71} -5.02316 q^{73} -0.690452 q^{75} -13.2519 q^{77} -12.9398 q^{79} +8.30973 q^{81} -12.3149 q^{83} -5.01755 q^{85} +1.30962 q^{87} -15.2492 q^{89} +10.4100 q^{91} -0.129675 q^{93} +8.22171 q^{95} +3.89871 q^{97} -15.5437 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * 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
$3$	0.278140	0.160584	0.0802921	−	0.996771i	$-0.474415\pi$
$3$	0.278140	0.160584	0.0802921	+	0.996771i	$0.474415\pi$
$4$	0	0
$5$	1.58670	0.709593	0.354796	−	0.934944i	$-0.384550\pi$
$5$	1.58670	0.709593	0.354796	+	0.934944i	$0.384550\pi$
$6$	0	0
$7$	−2.49171	−0.941777	−0.470889	−	0.882193i	$-0.656067\pi$
$7$	−2.49171	−0.941777	−0.470889	+	0.882193i	$0.656067\pi$
$8$	0	0
$9$	−2.92264	−0.974213
$10$	0	0
$11$	5.31839	1.60356	0.801778	−	0.597622i	$-0.203888\pi$
$11$	5.31839	1.60356	0.801778	+	0.597622i	$0.203888\pi$
$12$	0	0
$13$	−4.17786	−1.15873	−0.579365	−	0.815069i	$-0.696699\pi$
$13$	−4.17786	−1.15873	−0.579365	+	0.815069i	$0.696699\pi$
$14$	0	0
$15$	0.441324	0.113949
$16$	0	0
$17$	−3.16226	−0.766960	−0.383480	−	0.923549i	$-0.625275\pi$
$17$	−3.16226	−0.766960	−0.383480	+	0.923549i	$0.625275\pi$
$18$	0	0
$19$	5.18165	1.18875	0.594376	−	0.804187i	$-0.297399\pi$
$19$	5.18165	1.18875	0.594376	+	0.804187i	$0.297399\pi$
$20$	0	0
$21$	−0.693044	−0.151235
$22$	0	0
$23$	−2.27106	−0.473550	−0.236775	−	0.971565i	$-0.576090\pi$
$23$	−2.27106	−0.473550	−0.236775	+	0.971565i	$0.576090\pi$
$24$	0	0
$25$	−2.48239	−0.496478
$26$	0	0
$27$	−1.64732	−0.317027
$28$	0	0
$29$	4.70849	0.874345	0.437173	−	0.899378i	$-0.355980\pi$
$29$	4.70849	0.874345	0.437173	+	0.899378i	$0.355980\pi$
$30$	0	0
$31$	−0.466223	−0.0837361	−0.0418680	−	0.999123i	$-0.513331\pi$
$31$	−0.466223	−0.0837361	−0.0418680	+	0.999123i	$0.513331\pi$
$32$	0	0
$33$	1.47926	0.257506
$34$	0	0
$35$	−3.95359	−0.668278
$36$	0	0
$37$	4.71895	0.775791	0.387896	−	0.921703i	$-0.373202\pi$
$37$	4.71895	0.775791	0.387896	+	0.921703i	$0.373202\pi$
$38$	0	0
$39$	−1.16203	−0.186074
$40$	0	0
$41$	4.53789	0.708699	0.354349	−	0.935113i	$-0.384702\pi$
$41$	4.53789	0.708699	0.354349	+	0.935113i	$0.384702\pi$
$42$	0	0
$43$	3.13155	0.477557	0.238778	−	0.971074i	$-0.423253\pi$
$43$	3.13155	0.477557	0.238778	+	0.971074i	$0.423253\pi$
$44$	0	0
$45$	−4.63734	−0.691294
$46$	0	0
$47$	−1.59850	−0.233165	−0.116583	−	0.993181i	$-0.537194\pi$
$47$	−1.59850	−0.233165	−0.116583	+	0.993181i	$0.537194\pi$
$48$	0	0
$49$	−0.791388	−0.113055
$50$	0	0
$51$	−0.879550	−0.123162
$52$	0	0
$53$	−3.67829	−0.505252	−0.252626	−	0.967564i	$-0.581294\pi$
$53$	−3.67829	−0.505252	−0.252626	+	0.967564i	$0.581294\pi$
$54$	0	0
$55$	8.43868	1.13787
$56$	0	0
$57$	1.44122	0.190895
$58$	0	0
$59$	−3.30428	−0.430181	−0.215090	−	0.976594i	$-0.569005\pi$
$59$	−3.30428	−0.430181	−0.215090	+	0.976594i	$0.569005\pi$
$60$	0	0
$61$	−6.92274	−0.886365	−0.443183	−	0.896431i	$-0.646151\pi$
$61$	−6.92274	−0.886365	−0.443183	+	0.896431i	$0.646151\pi$
$62$	0	0
$63$	7.28236	0.917491
$64$	0	0
$65$	−6.62900	−0.822226
$66$	0	0
$67$	−12.2927	−1.50179	−0.750897	−	0.660420i	$-0.770378\pi$
$67$	−12.2927	−1.50179	−0.750897	+	0.660420i	$0.770378\pi$
$68$	0	0
$69$	−0.631673	−0.0760445
$70$	0	0
$71$	12.0893	1.43474	0.717370	−	0.696693i	$-0.245346\pi$
$71$	12.0893	1.43474	0.717370	+	0.696693i	$0.245346\pi$
$72$	0	0
$73$	−5.02316	−0.587917	−0.293958	−	0.955818i	$-0.594973\pi$
$73$	−5.02316	−0.587917	−0.293958	+	0.955818i	$0.594973\pi$
$74$	0	0
$75$	−0.690452	−0.0797265
$76$	0	0
$77$	−13.2519	−1.51019
$78$	0	0
$79$	−12.9398	−1.45584	−0.727922	−	0.685660i	$-0.759514\pi$
$79$	−12.9398	−1.45584	−0.727922	+	0.685660i	$0.759514\pi$
$80$	0	0
$81$	8.30973	0.923303
$82$	0	0
$83$	−12.3149	−1.35173	−0.675866	−	0.737024i	$-0.736230\pi$
$83$	−12.3149	−1.35173	−0.675866	+	0.737024i	$0.736230\pi$
$84$	0	0
$85$	−5.01755	−0.544229
$86$	0	0
$87$	1.30962	0.140406
$88$	0	0
$89$	−15.2492	−1.61641	−0.808205	−	0.588902i	$-0.799561\pi$
$89$	−15.2492	−1.61641	−0.808205	+	0.588902i	$0.799561\pi$
$90$	0	0
$91$	10.4100	1.09126
$92$	0	0
$93$	−0.129675	−0.0134467
$94$	0	0
$95$	8.22171	0.843530
$96$	0	0
$97$	3.89871	0.395854	0.197927	−	0.980217i	$-0.436579\pi$
$97$	3.89871	0.395854	0.197927	+	0.980217i	$0.436579\pi$
$98$	0	0
$99$	−15.5437	−1.56220
$100$	0	0
$101$	2.95625	0.294158	0.147079	−	0.989125i	$-0.453013\pi$
$101$	2.95625	0.294158	0.147079	+	0.989125i	$0.453013\pi$
$102$	0	0
$103$	2.00210	0.197273	0.0986366	−	0.995124i	$-0.468552\pi$
$103$	2.00210	0.197273	0.0986366	+	0.995124i	$0.468552\pi$
$104$	0	0
$105$	−1.09965	−0.107315
$106$	0	0
$107$	−16.8999	−1.63378	−0.816889	−	0.576796i	$-0.804303\pi$
$107$	−16.8999	−1.63378	−0.816889	+	0.576796i	$0.804303\pi$
$108$	0	0
$109$	2.73753	0.262208	0.131104	−	0.991369i	$-0.458148\pi$
$109$	2.73753	0.262208	0.131104	+	0.991369i	$0.458148\pi$
$110$	0	0
$111$	1.31253	0.124580
$112$	0	0
$113$	14.8926	1.40098	0.700491	−	0.713661i	$-0.252964\pi$
$113$	14.8926	1.40098	0.700491	+	0.713661i	$0.252964\pi$
$114$	0	0
$115$	−3.60349	−0.336027
$116$	0	0
$117$	12.2104	1.12885
$118$	0	0
$119$	7.87943	0.722306
$120$	0	0
$121$	17.2853	1.57139
$122$	0	0
$123$	1.26217	0.113806
$124$	0	0
$125$	−11.8723	−1.06189
$126$	0	0
$127$	9.64634	0.855974	0.427987	−	0.903785i	$-0.359223\pi$
$127$	9.64634	0.855974	0.427987	+	0.903785i	$0.359223\pi$
$128$	0	0
$129$	0.871009	0.0766880
$130$	0	0
$131$	−10.3574	−0.904931	−0.452465	−	0.891782i	$-0.649455\pi$
$131$	−10.3574	−0.904931	−0.452465	+	0.891782i	$0.649455\pi$
$132$	0	0
$133$	−12.9112	−1.11954
$134$	0	0
$135$	−2.61380	−0.224960
$136$	0	0
$137$	−20.7335	−1.77138	−0.885692	−	0.464274i	$-0.846315\pi$
$137$	−20.7335	−1.77138	−0.885692	+	0.464274i	$0.846315\pi$
$138$	0	0
$139$	−12.3132	−1.04439	−0.522195	−	0.852826i	$-0.674887\pi$
$139$	−12.3132	−1.04439	−0.522195	+	0.852826i	$0.674887\pi$
$140$	0	0
$141$	−0.444607	−0.0374426
$142$	0	0
$143$	−22.2195	−1.85809
$144$	0	0
$145$	7.47095	0.620429
$146$	0	0
$147$	−0.220117	−0.0181549
$148$	0	0
$149$	0.314134	0.0257349	0.0128674	−	0.999917i	$-0.495904\pi$
$149$	0.314134	0.0257349	0.0128674	+	0.999917i	$0.495904\pi$
$150$	0	0
$151$	−2.56907	−0.209068	−0.104534	−	0.994521i	$-0.533335\pi$
$151$	−2.56907	−0.209068	−0.104534	+	0.994521i	$0.533335\pi$
$152$	0	0
$153$	9.24214	0.747183
$154$	0	0
$155$	−0.739755	−0.0594185
$156$	0	0
$157$	24.6045	1.96366	0.981828	−	0.189772i	$-0.0607749\pi$
$157$	24.6045	1.96366	0.981828	+	0.189772i	$0.0607749\pi$
$158$	0	0
$159$	−1.02308	−0.0811355
$160$	0	0
$161$	5.65883	0.445978
$162$	0	0
$163$	−21.0969	−1.65243	−0.826217	−	0.563352i	$-0.809511\pi$
$163$	−21.0969	−1.65243	−0.826217	+	0.563352i	$0.809511\pi$
$164$	0	0
$165$	2.34713	0.182724
$166$	0	0
$167$	−18.3314	−1.41853	−0.709265	−	0.704942i	$-0.750973\pi$
$167$	−18.3314	−1.41853	−0.709265	+	0.704942i	$0.750973\pi$
$168$	0	0
$169$	4.45449	0.342653
$170$	0	0
$171$	−15.1441	−1.15810
$172$	0	0
$173$	−2.78749	−0.211929	−0.105964	−	0.994370i	$-0.533793\pi$
$173$	−2.78749	−0.211929	−0.105964	+	0.994370i	$0.533793\pi$
$174$	0	0
$175$	6.18540	0.467572
$176$	0	0
$177$	−0.919052	−0.0690802
$178$	0	0
$179$	−5.93477	−0.443585	−0.221793	−	0.975094i	$-0.571191\pi$
$179$	−5.93477	−0.443585	−0.221793	+	0.975094i	$0.571191\pi$
$180$	0	0
$181$	0.676298	0.0502689	0.0251344	−	0.999684i	$-0.491999\pi$
$181$	0.676298	0.0502689	0.0251344	+	0.999684i	$0.491999\pi$
$182$	0	0
$183$	−1.92549	−0.142336
$184$	0	0
$185$	7.48755	0.550496
$186$	0	0
$187$	−16.8181	−1.22986
$188$	0	0
$189$	4.10465	0.298569
$190$	0	0
$191$	4.93976	0.357429	0.178714	−	0.983901i	$-0.442806\pi$
$191$	4.93976	0.357429	0.178714	+	0.983901i	$0.442806\pi$
$192$	0	0
$193$	0.427481	0.0307707	0.0153854	−	0.999882i	$-0.495102\pi$
$193$	0.427481	0.0307707	0.0153854	+	0.999882i	$0.495102\pi$
$194$	0	0
$195$	−1.84379	−0.132036
$196$	0	0
$197$	−11.6733	−0.831691	−0.415845	−	0.909435i	$-0.636514\pi$
$197$	−11.6733	−0.831691	−0.415845	+	0.909435i	$0.636514\pi$
$198$	0	0
$199$	−21.6843	−1.53716	−0.768579	−	0.639755i	$-0.779036\pi$
$199$	−21.6843	−1.53716	−0.768579	+	0.639755i	$0.779036\pi$
$200$	0	0
$201$	−3.41909	−0.241164
$202$	0	0
$203$	−11.7322	−0.823439
$204$	0	0
$205$	7.20025	0.502887
$206$	0	0
$207$	6.63750	0.461338
$208$	0	0
$209$	27.5580	1.90623
$210$	0	0
$211$	−17.7958	−1.22511	−0.612557	−	0.790427i	$-0.709859\pi$
$211$	−17.7958	−1.22511	−0.612557	+	0.790427i	$0.709859\pi$
$212$	0	0
$213$	3.36253	0.230396
$214$	0	0
$215$	4.96882	0.338871
$216$	0	0
$217$	1.16169	0.0788608
$218$	0	0
$219$	−1.39714	−0.0944101
$220$	0	0
$221$	13.2115	0.888699
$222$	0	0
$223$	6.56036	0.439314	0.219657	−	0.975577i	$-0.429506\pi$
$223$	6.56036	0.439314	0.219657	+	0.975577i	$0.429506\pi$
$224$	0	0
$225$	7.25513	0.483675
$226$	0	0
$227$	−6.50792	−0.431946	−0.215973	−	0.976399i	$-0.569292\pi$
$227$	−6.50792	−0.431946	−0.215973	+	0.976399i	$0.569292\pi$
$228$	0	0
$229$	12.9817	0.857853	0.428927	−	0.903339i	$-0.358892\pi$
$229$	12.9817	0.857853	0.428927	+	0.903339i	$0.358892\pi$
$230$	0	0
$231$	−3.68588	−0.242513
$232$	0	0
$233$	−9.51412	−0.623291	−0.311646	−	0.950198i	$-0.600880\pi$
$233$	−9.51412	−0.623291	−0.311646	+	0.950198i	$0.600880\pi$
$234$	0	0
$235$	−2.53634	−0.165452
$236$	0	0
$237$	−3.59908	−0.233785
$238$	0	0
$239$	21.4019	1.38437	0.692187	−	0.721718i	$-0.256647\pi$
$239$	21.4019	1.38437	0.692187	+	0.721718i	$0.256647\pi$
$240$	0	0
$241$	−9.72856	−0.626671	−0.313336	−	0.949642i	$-0.601447\pi$
$241$	−9.72856	−0.626671	−0.313336	+	0.949642i	$0.601447\pi$
$242$	0	0
$243$	7.25323	0.465295
$244$	0	0
$245$	−1.25569	−0.0802233
$246$	0	0
$247$	−21.6482	−1.37744
$248$	0	0
$249$	−3.42526	−0.217067
$250$	0	0
$251$	4.75019	0.299830	0.149915	−	0.988699i	$-0.452100\pi$
$251$	4.75019	0.299830	0.149915	+	0.988699i	$0.452100\pi$
$252$	0	0
$253$	−12.0784	−0.759363
$254$	0	0
$255$	−1.39558	−0.0873946
$256$	0	0
$257$	−11.8109	−0.736743	−0.368371	−	0.929679i	$-0.620085\pi$
$257$	−11.8109	−0.736743	−0.368371	+	0.929679i	$0.620085\pi$
$258$	0	0
$259$	−11.7583	−0.730623
$260$	0	0
$261$	−13.7612	−0.851798
$262$	0	0
$263$	−4.75823	−0.293405	−0.146703	−	0.989181i	$-0.546866\pi$
$263$	−4.75823	−0.293405	−0.146703	+	0.989181i	$0.546866\pi$
$264$	0	0
$265$	−5.83634	−0.358523
$266$	0	0
$267$	−4.24140	−0.259570
$268$	0	0
$269$	−10.3181	−0.629103	−0.314551	−	0.949240i	$-0.601854\pi$
$269$	−10.3181	−0.629103	−0.314551	+	0.949240i	$0.601854\pi$
$270$	0	0
$271$	10.9663	0.666153	0.333076	−	0.942900i	$-0.391913\pi$
$271$	10.9663	0.666153	0.333076	+	0.942900i	$0.391913\pi$
$272$	0	0
$273$	2.89544	0.175240
$274$	0	0
$275$	−13.2023	−0.796130
$276$	0	0
$277$	14.3300	0.861007	0.430504	−	0.902589i	$-0.358336\pi$
$277$	14.3300	0.861007	0.430504	+	0.902589i	$0.358336\pi$
$278$	0	0
$279$	1.36260	0.0815768
$280$	0	0
$281$	9.91149	0.591270	0.295635	−	0.955301i	$-0.404469\pi$
$281$	9.91149	0.591270	0.295635	+	0.955301i	$0.404469\pi$
$282$	0	0
$283$	1.90578	0.113287	0.0566434	−	0.998394i	$-0.481960\pi$
$283$	1.90578	0.113287	0.0566434	+	0.998394i	$0.481960\pi$
$284$	0	0
$285$	2.28679	0.135457
$286$	0	0
$287$	−11.3071	−0.667436
$288$	0	0
$289$	−7.00012	−0.411772
$290$	0	0
$291$	1.08439	0.0635679
$292$	0	0
$293$	−10.0845	−0.589142	−0.294571	−	0.955630i	$-0.595177\pi$
$293$	−10.0845	−0.589142	−0.294571	+	0.955630i	$0.595177\pi$
$294$	0	0
$295$	−5.24289	−0.305253
$296$	0	0
$297$	−8.76110	−0.508371
$298$	0	0
$299$	9.48818	0.548716
$300$	0	0
$301$	−7.80291	−0.449752
$302$	0	0
$303$	0.822252	0.0472372
$304$	0	0
$305$	−10.9843	−0.628958
$306$	0	0
$307$	24.8752	1.41970	0.709850	−	0.704353i	$-0.248763\pi$
$307$	24.8752	1.41970	0.709850	+	0.704353i	$0.248763\pi$
$308$	0	0
$309$	0.556865	0.0316789
$310$	0	0
$311$	−24.0379	−1.36306	−0.681531	−	0.731789i	$-0.738686\pi$
$311$	−24.0379	−1.36306	−0.681531	+	0.731789i	$0.738686\pi$
$312$	0	0
$313$	6.12359	0.346126	0.173063	−	0.984911i	$-0.444634\pi$
$313$	6.12359	0.346126	0.173063	+	0.984911i	$0.444634\pi$
$314$	0	0
$315$	11.5549	0.651045
$316$	0	0
$317$	26.9177	1.51185	0.755924	−	0.654659i	$-0.227188\pi$
$317$	26.9177	1.51185	0.755924	+	0.654659i	$0.227188\pi$
$318$	0	0
$319$	25.0416	1.40206
$320$	0	0
$321$	−4.70054	−0.262359
$322$	0	0
$323$	−16.3857	−0.911726
$324$	0	0
$325$	10.3711	0.575284
$326$	0	0
$327$	0.761417	0.0421064
$328$	0	0
$329$	3.98300	0.219590
$330$	0	0
$331$	22.2369	1.22225	0.611125	−	0.791534i	$-0.290717\pi$
$331$	22.2369	1.22225	0.611125	+	0.791534i	$0.290717\pi$
$332$	0	0
$333$	−13.7918	−0.755786
$334$	0	0
$335$	−19.5048	−1.06566
$336$	0	0
$337$	34.9032	1.90130	0.950650	−	0.310265i	$-0.100418\pi$
$337$	34.9032	1.90130	0.950650	+	0.310265i	$0.100418\pi$
$338$	0	0
$339$	4.14224	0.224976
$340$	0	0
$341$	−2.47956	−0.134275
$342$	0	0
$343$	19.4139	1.04825
$344$	0	0
$345$	−1.00227	−0.0539606
$346$	0	0
$347$	−19.6429	−1.05449	−0.527243	−	0.849714i	$-0.676774\pi$
$347$	−19.6429	−1.05449	−0.527243	+	0.849714i	$0.676774\pi$
$348$	0	0
$349$	−14.5312	−0.777839	−0.388920	−	0.921272i	$-0.627152\pi$
$349$	−14.5312	−0.777839	−0.388920	+	0.921272i	$0.627152\pi$
$350$	0	0
$351$	6.88228	0.367349
$352$	0	0
$353$	−12.8987	−0.686528	−0.343264	−	0.939239i	$-0.611533\pi$
$353$	−12.8987	−0.686528	−0.343264	+	0.939239i	$0.611533\pi$
$354$	0	0
$355$	19.1821	1.01808
$356$	0	0
$357$	2.19158	0.115991
$358$	0	0
$359$	15.9031	0.839333	0.419667	−	0.907678i	$-0.362147\pi$
$359$	15.9031	0.839333	0.419667	+	0.907678i	$0.362147\pi$
$360$	0	0
$361$	7.84949	0.413131
$362$	0	0
$363$	4.80773	0.252340
$364$	0	0
$365$	−7.97024	−0.417181
$366$	0	0
$367$	−8.55160	−0.446390	−0.223195	−	0.974774i	$-0.571649\pi$
$367$	−8.55160	−0.446390	−0.223195	+	0.974774i	$0.571649\pi$
$368$	0	0
$369$	−13.2626	−0.690423
$370$	0	0
$371$	9.16523	0.475835
$372$	0	0
$373$	−11.9876	−0.620693	−0.310346	−	0.950624i	$-0.600445\pi$
$373$	−11.9876	−0.620693	−0.310346	+	0.950624i	$0.600445\pi$
$374$	0	0
$375$	−3.30216	−0.170523
$376$	0	0
$377$	−19.6714	−1.01313
$378$	0	0
$379$	3.65175	0.187578	0.0937888	−	0.995592i	$-0.470102\pi$
$379$	3.65175	0.187578	0.0937888	+	0.995592i	$0.470102\pi$
$380$	0	0
$381$	2.68303	0.137456
$382$	0	0
$383$	−24.6726	−1.26071	−0.630355	−	0.776307i	$-0.717090\pi$
$383$	−24.6726	−1.26071	−0.630355	+	0.776307i	$0.717090\pi$
$384$	0	0
$385$	−21.0267	−1.07162
$386$	0	0
$387$	−9.15239	−0.465242
$388$	0	0
$389$	30.8541	1.56437	0.782183	−	0.623049i	$-0.214106\pi$
$389$	30.8541	1.56437	0.782183	+	0.623049i	$0.214106\pi$
$390$	0	0
$391$	7.18169	0.363194
$392$	0	0
$393$	−2.88081	−0.145318
$394$	0	0
$395$	−20.5316	−1.03306
$396$	0	0
$397$	−6.21276	−0.311809	−0.155905	−	0.987772i	$-0.549829\pi$
$397$	−6.21276	−0.311809	−0.155905	+	0.987772i	$0.549829\pi$
$398$	0	0
$399$	−3.59111	−0.179780
$400$	0	0
$401$	−30.7692	−1.53654	−0.768270	−	0.640126i	$-0.778882\pi$
$401$	−30.7692	−1.53654	−0.768270	+	0.640126i	$0.778882\pi$
$402$	0	0
$403$	1.94781	0.0970275
$404$	0	0
$405$	13.1850	0.655169
$406$	0	0
$407$	25.0972	1.24402
$408$	0	0
$409$	34.8667	1.72405	0.862025	−	0.506866i	$-0.169196\pi$
$409$	34.8667	1.72405	0.862025	+	0.506866i	$0.169196\pi$
$410$	0	0
$411$	−5.76682	−0.284456
$412$	0	0
$413$	8.23330	0.405134
$414$	0	0
$415$	−19.5400	−0.959180
$416$	0	0
$417$	−3.42479	−0.167713
$418$	0	0
$419$	−3.51925	−0.171926	−0.0859632	−	0.996298i	$-0.527397\pi$
$419$	−3.51925	−0.171926	−0.0859632	+	0.996298i	$0.527397\pi$
$420$	0	0
$421$	6.94511	0.338484	0.169242	−	0.985575i	$-0.445868\pi$
$421$	6.94511	0.338484	0.169242	+	0.985575i	$0.445868\pi$
$422$	0	0
$423$	4.67184	0.227153
$424$	0	0
$425$	7.84996	0.380779
$426$	0	0
$427$	17.2494	0.834759
$428$	0	0
$429$	−6.18012	−0.298379
$430$	0	0
$431$	20.4362	0.984376	0.492188	−	0.870489i	$-0.336197\pi$
$431$	20.4362	0.984376	0.492188	+	0.870489i	$0.336197\pi$
$432$	0	0
$433$	9.57364	0.460080	0.230040	−	0.973181i	$-0.426114\pi$
$433$	9.57364	0.460080	0.230040	+	0.973181i	$0.426114\pi$
$434$	0	0
$435$	2.07797	0.0996310
$436$	0	0
$437$	−11.7679	−0.562933
$438$	0	0
$439$	12.0237	0.573862	0.286931	−	0.957951i	$-0.407365\pi$
$439$	12.0237	0.573862	0.286931	+	0.957951i	$0.407365\pi$
$440$	0	0
$441$	2.31294	0.110140
$442$	0	0
$443$	15.7043	0.746135	0.373068	−	0.927804i	$-0.378306\pi$
$443$	15.7043	0.746135	0.373068	+	0.927804i	$0.378306\pi$
$444$	0	0
$445$	−24.1958	−1.14699
$446$	0	0
$447$	0.0873733	0.00413262
$448$	0	0
$449$	−22.1199	−1.04390	−0.521952	−	0.852975i	$-0.674796\pi$
$449$	−22.1199	−1.04390	−0.521952	+	0.852975i	$0.674796\pi$
$450$	0	0
$451$	24.1343	1.13644
$452$	0	0
$453$	−0.714562	−0.0335731
$454$	0	0
$455$	16.5175	0.774353
$456$	0	0
$457$	−1.96675	−0.0920009	−0.0460004	−	0.998941i	$-0.514648\pi$
$457$	−1.96675	−0.0920009	−0.0460004	+	0.998941i	$0.514648\pi$
$458$	0	0
$459$	5.20926	0.243147
$460$	0	0
$461$	−16.8317	−0.783933	−0.391966	−	0.919980i	$-0.628205\pi$
$461$	−16.8317	−0.783933	−0.391966	+	0.919980i	$0.628205\pi$
$462$	0	0
$463$	−12.3927	−0.575936	−0.287968	−	0.957640i	$-0.592980\pi$
$463$	−12.3927	−0.575936	−0.287968	+	0.957640i	$0.592980\pi$
$464$	0	0
$465$	−0.205755	−0.00954167
$466$	0	0
$467$	−29.4711	−1.36376	−0.681880	−	0.731464i	$-0.738837\pi$
$467$	−29.4711	−1.36376	−0.681880	+	0.731464i	$0.738837\pi$
$468$	0	0
$469$	30.6298	1.41435
$470$	0	0
$471$	6.84351	0.315332
$472$	0	0
$473$	16.6548	0.765789
$474$	0	0
$475$	−12.8629	−0.590190
$476$	0	0
$477$	10.7503	0.492223
$478$	0	0
$479$	26.5295	1.21216	0.606082	−	0.795402i	$-0.292740\pi$
$479$	26.5295	1.21216	0.606082	+	0.795402i	$0.292740\pi$
$480$	0	0
$481$	−19.7151	−0.898932
$482$	0	0
$483$	1.57395	0.0716170
$484$	0	0
$485$	6.18608	0.280895
$486$	0	0
$487$	−13.1334	−0.595133	−0.297567	−	0.954701i	$-0.596175\pi$
$487$	−13.1334	−0.595133	−0.297567	+	0.954701i	$0.596175\pi$
$488$	0	0
$489$	−5.86788	−0.265355
$490$	0	0
$491$	5.43276	0.245177	0.122588	−	0.992458i	$-0.460880\pi$
$491$	5.43276	0.245177	0.122588	+	0.992458i	$0.460880\pi$
$492$	0	0
$493$	−14.8895	−0.670588
$494$	0	0
$495$	−24.6632	−1.10853
$496$	0	0
$497$	−30.1231	−1.35121
$498$	0	0
$499$	−17.9981	−0.805707	−0.402854	−	0.915264i	$-0.631982\pi$
$499$	−17.9981	−0.805707	−0.402854	+	0.915264i	$0.631982\pi$
$500$	0	0
$501$	−5.09871	−0.227793
$502$	0	0
$503$	−20.6584	−0.921112	−0.460556	−	0.887631i	$-0.652350\pi$
$503$	−20.6584	−0.921112	−0.460556	+	0.887631i	$0.652350\pi$
$504$	0	0
$505$	4.69068	0.208733
$506$	0	0
$507$	1.23897	0.0550247
$508$	0	0
$509$	23.3084	1.03313	0.516564	−	0.856249i	$-0.327211\pi$
$509$	23.3084	1.03313	0.516564	+	0.856249i	$0.327211\pi$
$510$	0	0
$511$	12.5163	0.553687
$512$	0	0
$513$	−8.53585	−0.376867
$514$	0	0
$515$	3.17673	0.139984
$516$	0	0
$517$	−8.50145	−0.373893
$518$	0	0
$519$	−0.775311	−0.0340324
$520$	0	0
$521$	22.7892	0.998414	0.499207	−	0.866483i	$-0.333625\pi$
$521$	22.7892	0.998414	0.499207	+	0.866483i	$0.333625\pi$
$522$	0	0
$523$	−15.5606	−0.680416	−0.340208	−	0.940350i	$-0.610497\pi$
$523$	−15.5606	−0.680416	−0.340208	+	0.940350i	$0.610497\pi$
$524$	0	0
$525$	1.72041	0.0750846
$526$	0	0
$527$	1.47432	0.0642223
$528$	0	0
$529$	−17.8423	−0.775751
$530$	0	0
$531$	9.65722	0.419088
$532$	0	0
$533$	−18.9586	−0.821190
$534$	0	0
$535$	−26.8151	−1.15932
$536$	0	0
$537$	−1.65070	−0.0712328
$538$	0	0
$539$	−4.20891	−0.181291
$540$	0	0
$541$	−5.00911	−0.215359	−0.107679	−	0.994186i	$-0.534342\pi$
$541$	−5.00911	−0.215359	−0.107679	+	0.994186i	$0.534342\pi$
$542$	0	0
$543$	0.188106	0.00807238
$544$	0	0
$545$	4.34363	0.186061
$546$	0	0
$547$	6.38513	0.273008	0.136504	−	0.990639i	$-0.456413\pi$
$547$	6.38513	0.273008	0.136504	+	0.990639i	$0.456413\pi$
$548$	0	0
$549$	20.2327	0.863509
$550$	0	0
$551$	24.3978	1.03938
$552$	0	0
$553$	32.2423	1.37108
$554$	0	0
$555$	2.08259	0.0884009
$556$	0	0
$557$	−6.33105	−0.268255	−0.134128	−	0.990964i	$-0.542823\pi$
$557$	−6.33105	−0.268255	−0.134128	+	0.990964i	$0.542823\pi$
$558$	0	0
$559$	−13.0832	−0.553359
$560$	0	0
$561$	−4.67779	−0.197497
$562$	0	0
$563$	−19.9729	−0.841759	−0.420880	−	0.907117i	$-0.638278\pi$
$563$	−19.9729	−0.841759	−0.420880	+	0.907117i	$0.638278\pi$
$564$	0	0
$565$	23.6301	0.994127
$566$	0	0
$567$	−20.7054	−0.869546
$568$	0	0
$569$	11.9790	0.502186	0.251093	−	0.967963i	$-0.419210\pi$
$569$	11.9790	0.502186	0.251093	+	0.967963i	$0.419210\pi$
$570$	0	0
$571$	−19.5625	−0.818666	−0.409333	−	0.912385i	$-0.634239\pi$
$571$	−19.5625	−0.818666	−0.409333	+	0.912385i	$0.634239\pi$
$572$	0	0
$573$	1.37394	0.0573974
$574$	0	0
$575$	5.63767	0.235107
$576$	0	0
$577$	−17.9014	−0.745247	−0.372623	−	0.927983i	$-0.621542\pi$
$577$	−17.9014	−0.745247	−0.372623	+	0.927983i	$0.621542\pi$
$578$	0	0
$579$	0.118899	0.00494129
$580$	0	0
$581$	30.6851	1.27303
$582$	0	0
$583$	−19.5626	−0.810200
$584$	0	0
$585$	19.3742	0.801023
$586$	0	0
$587$	−28.3919	−1.17186	−0.585930	−	0.810362i	$-0.699271\pi$
$587$	−28.3919	−1.17186	−0.585930	+	0.810362i	$0.699271\pi$
$588$	0	0
$589$	−2.41580	−0.0995415
$590$	0	0
$591$	−3.24682	−0.133556
$592$	0	0
$593$	7.44153	0.305587	0.152794	−	0.988258i	$-0.451173\pi$
$593$	7.44153	0.305587	0.152794	+	0.988258i	$0.451173\pi$
$594$	0	0
$595$	12.5023	0.512543
$596$	0	0
$597$	−6.03127	−0.246843
$598$	0	0
$599$	−18.9049	−0.772433	−0.386216	−	0.922408i	$-0.626218\pi$
$599$	−18.9049	−0.772433	−0.386216	+	0.922408i	$0.626218\pi$
$600$	0	0
$601$	−24.6229	−1.00439	−0.502194	−	0.864755i	$-0.667474\pi$
$601$	−24.6229	−1.00439	−0.502194	+	0.864755i	$0.667474\pi$
$602$	0	0
$603$	35.9271	1.46307
$604$	0	0
$605$	27.4265	1.11505
$606$	0	0
$607$	16.4784	0.668836	0.334418	−	0.942425i	$-0.391460\pi$
$607$	16.4784	0.668836	0.334418	+	0.942425i	$0.391460\pi$
$608$	0	0
$609$	−3.26319	−0.132231
$610$	0	0
$611$	6.67831	0.270175
$612$	0	0
$613$	−13.5886	−0.548839	−0.274419	−	0.961610i	$-0.588486\pi$
$613$	−13.5886	−0.548839	−0.274419	+	0.961610i	$0.588486\pi$
$614$	0	0
$615$	2.00268	0.0807557
$616$	0	0
$617$	6.83640	0.275223	0.137612	−	0.990486i	$-0.456057\pi$
$617$	6.83640	0.275223	0.137612	+	0.990486i	$0.456057\pi$
$618$	0	0
$619$	33.8366	1.36001	0.680003	−	0.733209i	$-0.261978\pi$
$619$	33.8366	1.36001	0.680003	+	0.733209i	$0.261978\pi$
$620$	0	0
$621$	3.74117	0.150128
$622$	0	0
$623$	37.9965	1.52230
$624$	0	0
$625$	−6.42578	−0.257031
$626$	0	0
$627$	7.66499	0.306110
$628$	0	0
$629$	−14.9226	−0.595001
$630$	0	0
$631$	49.3139	1.96316	0.981578	−	0.191060i	$-0.0611924\pi$
$631$	49.3139	1.96316	0.981578	+	0.191060i	$0.0611924\pi$
$632$	0	0
$633$	−4.94972	−0.196734
$634$	0	0
$635$	15.3058	0.607393
$636$	0	0
$637$	3.30631	0.131001
$638$	0	0
$639$	−35.3327	−1.39774
$640$	0	0
$641$	17.2970	0.683191	0.341596	−	0.939847i	$-0.389033\pi$
$641$	17.2970	0.683191	0.341596	+	0.939847i	$0.389033\pi$
$642$	0	0
$643$	20.3460	0.802366	0.401183	−	0.915998i	$-0.368599\pi$
$643$	20.3460	0.802366	0.401183	+	0.915998i	$0.368599\pi$
$644$	0	0
$645$	1.38203	0.0544173
$646$	0	0
$647$	−21.0241	−0.826541	−0.413270	−	0.910608i	$-0.635614\pi$
$647$	−21.0241	−0.826541	−0.413270	+	0.910608i	$0.635614\pi$
$648$	0	0
$649$	−17.5735	−0.689819
$650$	0	0
$651$	0.323113	0.0126638
$652$	0	0
$653$	−24.4175	−0.955530	−0.477765	−	0.878488i	$-0.658553\pi$
$653$	−24.4175	−0.955530	−0.477765	+	0.878488i	$0.658553\pi$
$654$	0	0
$655$	−16.4341	−0.642132
$656$	0	0
$657$	14.6809	0.572756
$658$	0	0
$659$	−26.5985	−1.03613	−0.518064	−	0.855342i	$-0.673347\pi$
$659$	−26.5985	−1.03613	−0.518064	+	0.855342i	$0.673347\pi$
$660$	0	0
$661$	−39.9783	−1.55497	−0.777487	−	0.628899i	$-0.783506\pi$
$661$	−39.9783	−1.55497	−0.777487	+	0.628899i	$0.783506\pi$
$662$	0	0
$663$	3.67464	0.142711
$664$	0	0
$665$	−20.4861	−0.794417
$666$	0	0
$667$	−10.6933	−0.414046
$668$	0	0
$669$	1.82470	0.0705469
$670$	0	0
$671$	−36.8178	−1.42134
$672$	0	0
$673$	1.23630	0.0476561	0.0238280	−	0.999716i	$-0.492415\pi$
$673$	1.23630	0.0476561	0.0238280	+	0.999716i	$0.492415\pi$
$674$	0	0
$675$	4.08930	0.157397
$676$	0	0
$677$	27.8016	1.06850	0.534251	−	0.845326i	$-0.320594\pi$
$677$	27.8016	1.06850	0.534251	+	0.845326i	$0.320594\pi$
$678$	0	0
$679$	−9.71446	−0.372807
$680$	0	0
$681$	−1.81011	−0.0693637
$682$	0	0
$683$	−0.659163	−0.0252222	−0.0126111	−	0.999920i	$-0.504014\pi$
$683$	−0.659163	−0.0252222	−0.0126111	+	0.999920i	$0.504014\pi$
$684$	0	0
$685$	−32.8978	−1.25696
$686$	0	0
$687$	3.61072	0.137758
$688$	0	0
$689$	15.3674	0.585451
$690$	0	0
$691$	29.0771	1.10615	0.553073	−	0.833133i	$-0.313455\pi$
$691$	29.0771	1.10615	0.553073	+	0.833133i	$0.313455\pi$
$692$	0	0
$693$	38.7305	1.47125
$694$	0	0
$695$	−19.5373	−0.741092
$696$	0	0
$697$	−14.3500	−0.543544
$698$	0	0
$699$	−2.64626	−0.100091
$700$	0	0
$701$	10.4430	0.394427	0.197213	−	0.980361i	$-0.436811\pi$
$701$	10.4430	0.394427	0.197213	+	0.980361i	$0.436811\pi$
$702$	0	0
$703$	24.4520	0.922224
$704$	0	0
$705$	−0.705456	−0.0265690
$706$	0	0
$707$	−7.36613	−0.277032
$708$	0	0
$709$	−46.7017	−1.75392	−0.876960	−	0.480564i	$-0.840432\pi$
$709$	−46.7017	−1.75392	−0.876960	+	0.480564i	$0.840432\pi$
$710$	0	0
$711$	37.8184	1.41830
$712$	0	0
$713$	1.05882	0.0396532
$714$	0	0
$715$	−35.2556	−1.31848
$716$	0	0
$717$	5.95272	0.222308
$718$	0	0
$719$	−46.3746	−1.72948	−0.864741	−	0.502218i	$-0.832517\pi$
$719$	−46.3746	−1.72948	−0.864741	+	0.502218i	$0.832517\pi$
$720$	0	0
$721$	−4.98866	−0.185787
$722$	0	0
$723$	−2.70590	−0.100634
$724$	0	0
$725$	−11.6883	−0.434093
$726$	0	0
$727$	−0.373294	−0.0138447	−0.00692235	−	0.999976i	$-0.502203\pi$
$727$	−0.373294	−0.0138447	−0.00692235	+	0.999976i	$0.502203\pi$
$728$	0	0
$729$	−22.9118	−0.848584
$730$	0	0
$731$	−9.90277	−0.366267
$732$	0	0
$733$	−24.0591	−0.888642	−0.444321	−	0.895868i	$-0.646555\pi$
$733$	−24.0591	−0.888642	−0.444321	+	0.895868i	$0.646555\pi$
$734$	0	0
$735$	−0.349258	−0.0128826
$736$	0	0
$737$	−65.3774	−2.40821
$738$	0	0
$739$	−3.57096	−0.131360	−0.0656800	−	0.997841i	$-0.520922\pi$
$739$	−3.57096	−0.131360	−0.0656800	+	0.997841i	$0.520922\pi$
$740$	0	0
$741$	−6.02123	−0.221195
$742$	0	0
$743$	5.89842	0.216392	0.108196	−	0.994130i	$-0.465493\pi$
$743$	5.89842	0.216392	0.108196	+	0.994130i	$0.465493\pi$
$744$	0	0
$745$	0.498436	0.0182613
$746$	0	0
$747$	35.9919	1.31688
$748$	0	0
$749$	42.1097	1.53865
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	1.32122	0.0481479
$754$	0	0
$755$	−4.07634	−0.148353
$756$	0	0
$757$	−8.88934	−0.323089	−0.161544	−	0.986865i	$-0.551647\pi$
$757$	−8.88934	−0.323089	−0.161544	+	0.986865i	$0.551647\pi$
$758$	0	0
$759$	−3.35949	−0.121942
$760$	0	0
$761$	41.0345	1.48750	0.743750	−	0.668458i	$-0.233045\pi$
$761$	41.0345	1.48750	0.743750	+	0.668458i	$0.233045\pi$
$762$	0	0
$763$	−6.82113	−0.246941
$764$	0	0
$765$	14.6645	0.530195
$766$	0	0
$767$	13.8048	0.498463
$768$	0	0
$769$	20.0124	0.721667	0.360834	−	0.932630i	$-0.382492\pi$
$769$	20.0124	0.721667	0.360834	+	0.932630i	$0.382492\pi$
$770$	0	0
$771$	−3.28508	−0.118309
$772$	0	0
$773$	48.4597	1.74297	0.871487	−	0.490419i	$-0.163156\pi$
$773$	48.4597	1.74297	0.871487	+	0.490419i	$0.163156\pi$
$774$	0	0
$775$	1.15735	0.0415732
$776$	0	0
$777$	−3.27044	−0.117326
$778$	0	0
$779$	23.5137	0.842467
$780$	0	0
$781$	64.2958	2.30068
$782$	0	0
$783$	−7.75640	−0.277191
$784$	0	0
$785$	39.0400	1.39340
$786$	0	0
$787$	−50.8456	−1.81245	−0.906226	−	0.422794i	$-0.861049\pi$
$787$	−50.8456	−1.81245	−0.906226	+	0.422794i	$0.861049\pi$
$788$	0	0
$789$	−1.32345	−0.0471162
$790$	0	0
$791$	−37.1081	−1.31941
$792$	0	0
$793$	28.9222	1.02706
$794$	0	0
$795$	−1.62332	−0.0575732
$796$	0	0
$797$	−8.19747	−0.290369	−0.145185	−	0.989405i	$-0.546378\pi$
$797$	−8.19747	−0.290369	−0.145185	+	0.989405i	$0.546378\pi$
$798$	0	0
$799$	5.05487	0.178829
$800$	0	0
$801$	44.5678	1.57473
$802$	0	0
$803$	−26.7151	−0.942757
$804$	0	0
$805$	8.97885	0.316463
$806$	0	0
$807$	−2.86986	−0.101024
$808$	0	0
$809$	−1.90765	−0.0670693	−0.0335347	−	0.999438i	$-0.510676\pi$
$809$	−1.90765	−0.0670693	−0.0335347	+	0.999438i	$0.510676\pi$
$810$	0	0
$811$	16.8725	0.592472	0.296236	−	0.955115i	$-0.404269\pi$
$811$	16.8725	0.592472	0.296236	+	0.955115i	$0.404269\pi$
$812$	0	0
$813$	3.05015	0.106974
$814$	0	0
$815$	−33.4743	−1.17255
$816$	0	0
$817$	16.2266	0.567697
$818$	0	0
$819$	−30.4247	−1.06312
$820$	0	0
$821$	34.7184	1.21168	0.605841	−	0.795586i	$-0.292837\pi$
$821$	34.7184	1.21168	0.605841	+	0.795586i	$0.292837\pi$
$822$	0	0
$823$	−20.4354	−0.712332	−0.356166	−	0.934423i	$-0.615916\pi$
$823$	−20.4354	−0.712332	−0.356166	+	0.934423i	$0.615916\pi$
$824$	0	0
$825$	−3.67209	−0.127846
$826$	0	0
$827$	23.3289	0.811225	0.405612	−	0.914045i	$-0.367058\pi$
$827$	23.3289	0.811225	0.405612	+	0.914045i	$0.367058\pi$
$828$	0	0
$829$	−22.6949	−0.788227	−0.394114	−	0.919062i	$-0.628948\pi$
$829$	−22.6949	−0.788227	−0.394114	+	0.919062i	$0.628948\pi$
$830$	0	0
$831$	3.98575	0.138264
$832$	0	0
$833$	2.50257	0.0867091
$834$	0	0
$835$	−29.0865	−1.00658
$836$	0	0
$837$	0.768019	0.0265466
$838$	0	0
$839$	17.4196	0.601393	0.300696	−	0.953720i	$-0.402781\pi$
$839$	17.4196	0.601393	0.300696	+	0.953720i	$0.402781\pi$
$840$	0	0
$841$	−6.83009	−0.235520
$842$	0	0
$843$	2.75678	0.0949485
$844$	0	0
$845$	7.06793	0.243144
$846$	0	0
$847$	−43.0699	−1.47990
$848$	0	0
$849$	0.530073	0.0181921
$850$	0	0
$851$	−10.7170	−0.367376
$852$	0	0
$853$	−31.0103	−1.06177	−0.530887	−	0.847443i	$-0.678141\pi$
$853$	−31.0103	−1.06177	−0.530887	+	0.847443i	$0.678141\pi$
$854$	0	0
$855$	−24.0291	−0.821777
$856$	0	0
$857$	−25.3182	−0.864855	−0.432427	−	0.901669i	$-0.642343\pi$
$857$	−25.3182	−0.864855	−0.432427	+	0.901669i	$0.642343\pi$
$858$	0	0
$859$	−21.4794	−0.732869	−0.366434	−	0.930444i	$-0.619422\pi$
$859$	−21.4794	−0.732869	−0.366434	+	0.930444i	$0.619422\pi$
$860$	0	0
$861$	−3.14495	−0.107180
$862$	0	0
$863$	18.5454	0.631291	0.315646	−	0.948877i	$-0.397779\pi$
$863$	18.5454	0.631291	0.315646	+	0.948877i	$0.397779\pi$
$864$	0	0
$865$	−4.42290	−0.150383
$866$	0	0
$867$	−1.94701	−0.0661240
$868$	0	0
$869$	−68.8190	−2.33452
$870$	0	0
$871$	51.3572	1.74017
$872$	0	0
$873$	−11.3945	−0.385646
$874$	0	0
$875$	29.5823	1.00006
$876$	0	0
$877$	−35.2153	−1.18914	−0.594569	−	0.804045i	$-0.702677\pi$
$877$	−35.2153	−1.18914	−0.594569	+	0.804045i	$0.702677\pi$
$878$	0	0
$879$	−2.80490	−0.0946069
$880$	0	0
$881$	7.75356	0.261224	0.130612	−	0.991434i	$-0.458306\pi$
$881$	7.75356	0.261224	0.130612	+	0.991434i	$0.458306\pi$
$882$	0	0
$883$	−28.7798	−0.968518	−0.484259	−	0.874925i	$-0.660911\pi$
$883$	−28.7798	−0.968518	−0.484259	+	0.874925i	$0.660911\pi$
$884$	0	0
$885$	−1.45826	−0.0490188
$886$	0	0
$887$	−11.5055	−0.386318	−0.193159	−	0.981167i	$-0.561873\pi$
$887$	−11.5055	−0.386318	−0.193159	+	0.981167i	$0.561873\pi$
$888$	0	0
$889$	−24.0359	−0.806137
$890$	0	0
$891$	44.1944	1.48057
$892$	0	0
$893$	−8.28287	−0.277176
$894$	0	0
$895$	−9.41668	−0.314765
$896$	0	0
$897$	2.63904	0.0881150
$898$	0	0
$899$	−2.19521	−0.0732143
$900$	0	0
$901$	11.6317	0.387509
$902$	0	0
$903$	−2.17030	−0.0722231
$904$	0	0
$905$	1.07308	0.0356704
$906$	0	0
$907$	−6.38439	−0.211990	−0.105995	−	0.994367i	$-0.533803\pi$
$907$	−6.38439	−0.211990	−0.105995	+	0.994367i	$0.533803\pi$
$908$	0	0
$909$	−8.64006	−0.286573
$910$	0	0
$911$	−23.6633	−0.784001	−0.392001	−	0.919965i	$-0.628217\pi$
$911$	−23.6633	−0.784001	−0.392001	+	0.919965i	$0.628217\pi$
$912$	0	0
$913$	−65.4953	−2.16758
$914$	0	0
$915$	−3.05517	−0.101001
$916$	0	0
$917$	25.8076	0.852243
$918$	0	0
$919$	36.2900	1.19710	0.598548	−	0.801087i	$-0.295745\pi$
$919$	36.2900	1.19710	0.598548	+	0.801087i	$0.295745\pi$
$920$	0	0
$921$	6.91878	0.227981
$922$	0	0
$923$	−50.5075	−1.66247
$924$	0	0
$925$	−11.7143	−0.385164
$926$	0	0
$927$	−5.85143	−0.192186
$928$	0	0
$929$	36.8539	1.20914	0.604569	−	0.796553i	$-0.293345\pi$
$929$	36.8539	1.20914	0.604569	+	0.796553i	$0.293345\pi$
$930$	0	0
$931$	−4.10070	−0.134395
$932$	0	0
$933$	−6.68589	−0.218886
$934$	0	0
$935$	−26.6853	−0.872702
$936$	0	0
$937$	52.5859	1.71791	0.858954	−	0.512053i	$-0.171115\pi$
$937$	52.5859	1.71791	0.858954	+	0.512053i	$0.171115\pi$
$938$	0	0
$939$	1.70321	0.0555823
$940$	0	0
$941$	15.0363	0.490170	0.245085	−	0.969502i	$-0.421184\pi$
$941$	15.0363	0.490170	0.245085	+	0.969502i	$0.421184\pi$
$942$	0	0
$943$	−10.3058	−0.335604
$944$	0	0
$945$	6.51283	0.211862
$946$	0	0
$947$	48.1553	1.56484	0.782418	−	0.622753i	$-0.213986\pi$
$947$	48.1553	1.56484	0.782418	+	0.622753i	$0.213986\pi$
$948$	0	0
$949$	20.9861	0.681236
$950$	0	0
$951$	7.48689	0.242779
$952$	0	0
$953$	37.1339	1.20288	0.601442	−	0.798916i	$-0.294593\pi$
$953$	37.1339	1.20288	0.601442	+	0.798916i	$0.294593\pi$
$954$	0	0
$955$	7.83790	0.253629
$956$	0	0
$957$	6.96507	0.225149
$958$	0	0
$959$	51.6619	1.66825
$960$	0	0
$961$	−30.7826	−0.992988
$962$	0	0
$963$	49.3924	1.59165
$964$	0	0
$965$	0.678282	0.0218347
$966$	0	0
$967$	−12.7318	−0.409428	−0.204714	−	0.978822i	$-0.565626\pi$
$967$	−12.7318	−0.409428	−0.204714	+	0.978822i	$0.565626\pi$
$968$	0	0
$969$	−4.55752	−0.146409
$970$	0	0
$971$	53.5608	1.71885	0.859423	−	0.511264i	$-0.170823\pi$
$971$	53.5608	1.71885	0.859423	+	0.511264i	$0.170823\pi$
$972$	0	0
$973$	30.6809	0.983583
$974$	0	0
$975$	2.88461	0.0923815
$976$	0	0
$977$	32.6257	1.04379	0.521895	−	0.853010i	$-0.325225\pi$
$977$	32.6257	1.04379	0.521895	+	0.853010i	$0.325225\pi$
$978$	0	0
$979$	−81.1011	−2.59200
$980$	0	0
$981$	−8.00081	−0.255446
$982$	0	0
$983$	−29.6145	−0.944556	−0.472278	−	0.881450i	$-0.656568\pi$
$983$	−29.6145	−0.944556	−0.472278	+	0.881450i	$0.656568\pi$
$984$	0	0
$985$	−18.5221	−0.590162
$986$	0	0
$987$	1.10783	0.0352626
$988$	0	0
$989$	−7.11195	−0.226147
$990$	0	0
$991$	32.7361	1.03990	0.519949	−	0.854197i	$-0.325951\pi$
$991$	32.7361	1.03990	0.519949	+	0.854197i	$0.325951\pi$
$992$	0	0
$993$	6.18496	0.196274
$994$	0	0
$995$	−34.4064	−1.09076
$996$	0	0
$997$	33.1839	1.05094	0.525472	−	0.850811i	$-0.323889\pi$
$997$	33.1839	1.05094	0.525472	+	0.850811i	$0.323889\pi$
$998$	0	0
$999$	−7.77364	−0.245947

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.27	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.27	✓	44	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:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q+0.278140 q^{3} +1.58670 q^{5} -2.49171 q^{7} -2.92264 q^{9} +O(q^{10})\)	\(q+0.278140 q^{3} +1.58670 q^{5} -2.49171 q^{7} -2.92264 q^{9} +5.31839 q^{11} -4.17786 q^{13} +0.441324 q^{15} -3.16226 q^{17} +5.18165 q^{19} -0.693044 q^{21} -2.27106 q^{23} -2.48239 q^{25} -1.64732 q^{27} +4.70849 q^{29} -0.466223 q^{31} +1.47926 q^{33} -3.95359 q^{35} +4.71895 q^{37} -1.16203 q^{39} +4.53789 q^{41} +3.13155 q^{43} -4.63734 q^{45} -1.59850 q^{47} -0.791388 q^{49} -0.879550 q^{51} -3.67829 q^{53} +8.43868 q^{55} +1.44122 q^{57} -3.30428 q^{59} -6.92274 q^{61} +7.28236 q^{63} -6.62900 q^{65} -12.2927 q^{67} -0.631673 q^{69} +12.0893 q^{71} -5.02316 q^{73} -0.690452 q^{75} -13.2519 q^{77} -12.9398 q^{79} +8.30973 q^{81} -12.3149 q^{83} -5.01755 q^{85} +1.30962 q^{87} -15.2492 q^{89} +10.4100 q^{91} -0.129675 q^{93} +8.22171 q^{95} +3.89871 q^{97} -15.5437 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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