LMFDB - Embedded newform 6008.2.a.b.1.24

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			1.24
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)

$f(q)$	$=$	$q-0.252021 q^{3} -1.14323 q^{5} -2.95394 q^{7} -2.93649 q^{9} +O(q^{10})$	$q-0.252021 q^{3} -1.14323 q^{5} -2.95394 q^{7} -2.93649 q^{9} +4.60695 q^{11} -2.71345 q^{13} +0.288117 q^{15} +6.70585 q^{17} -3.81186 q^{19} +0.744454 q^{21} +8.43814 q^{23} -3.69304 q^{25} +1.49612 q^{27} -8.76408 q^{29} +5.25299 q^{31} -1.16105 q^{33} +3.37702 q^{35} +5.29340 q^{37} +0.683845 q^{39} -7.12412 q^{41} +10.1901 q^{43} +3.35707 q^{45} -4.85654 q^{47} +1.72574 q^{49} -1.69001 q^{51} +9.41547 q^{53} -5.26679 q^{55} +0.960668 q^{57} -12.5427 q^{59} +12.5093 q^{61} +8.67419 q^{63} +3.10208 q^{65} +0.810453 q^{67} -2.12659 q^{69} -9.99792 q^{71} -3.66059 q^{73} +0.930722 q^{75} -13.6086 q^{77} -5.78439 q^{79} +8.43240 q^{81} -2.78063 q^{83} -7.66630 q^{85} +2.20873 q^{87} +5.17695 q^{89} +8.01535 q^{91} -1.32386 q^{93} +4.35782 q^{95} +0.683095 q^{97} -13.5282 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.252021	−0.145504	−0.0727522	−	0.997350i	$-0.523178\pi$
$3$	−0.252021	−0.145504	−0.0727522	+	0.997350i	$0.523178\pi$
$4$	0	0
$5$	−1.14323	−0.511266	−0.255633	−	0.966774i	$-0.582284\pi$
$5$	−1.14323	−0.511266	−0.255633	+	0.966774i	$0.582284\pi$
$6$	0	0
$7$	−2.95394	−1.11648	−0.558242	−	0.829678i	$-0.688524\pi$
$7$	−2.95394	−1.11648	−0.558242	+	0.829678i	$0.688524\pi$
$8$	0	0
$9$	−2.93649	−0.978828
$10$	0	0
$11$	4.60695	1.38905	0.694524	−	0.719469i	$-0.255615\pi$
$11$	4.60695	1.38905	0.694524	+	0.719469i	$0.255615\pi$
$12$	0	0
$13$	−2.71345	−0.752575	−0.376287	−	0.926503i	$-0.622799\pi$
$13$	−2.71345	−0.752575	−0.376287	+	0.926503i	$0.622799\pi$
$14$	0	0
$15$	0.288117	0.0743914
$16$	0	0
$17$	6.70585	1.62641	0.813204	−	0.581979i	$-0.197721\pi$
$17$	6.70585	1.62641	0.813204	+	0.581979i	$0.197721\pi$
$18$	0	0
$19$	−3.81186	−0.874501	−0.437250	−	0.899340i	$-0.644048\pi$
$19$	−3.81186	−0.874501	−0.437250	+	0.899340i	$0.644048\pi$
$20$	0	0
$21$	0.744454	0.162453
$22$	0	0
$23$	8.43814	1.75947	0.879737	−	0.475460i	$-0.157718\pi$
$23$	8.43814	1.75947	0.879737	+	0.475460i	$0.157718\pi$
$24$	0	0
$25$	−3.69304	−0.738607
$26$	0	0
$27$	1.49612	0.287928
$28$	0	0
$29$	−8.76408	−1.62745	−0.813725	−	0.581251i	$-0.802564\pi$
$29$	−8.76408	−1.62745	−0.813725	+	0.581251i	$0.802564\pi$
$30$	0	0
$31$	5.25299	0.943464	0.471732	−	0.881742i	$-0.343629\pi$
$31$	5.25299	0.943464	0.471732	+	0.881742i	$0.343629\pi$
$32$	0	0
$33$	−1.16105	−0.202113
$34$	0	0
$35$	3.37702	0.570820
$36$	0	0
$37$	5.29340	0.870230	0.435115	−	0.900375i	$-0.356708\pi$
$37$	5.29340	0.870230	0.435115	+	0.900375i	$0.356708\pi$
$38$	0	0
$39$	0.683845	0.109503
$40$	0	0
$41$	−7.12412	−1.11260	−0.556301	−	0.830981i	$-0.687780\pi$
$41$	−7.12412	−1.11260	−0.556301	+	0.830981i	$0.687780\pi$
$42$	0	0
$43$	10.1901	1.55397	0.776987	−	0.629517i	$-0.216747\pi$
$43$	10.1901	1.55397	0.776987	+	0.629517i	$0.216747\pi$
$44$	0	0
$45$	3.35707	0.500442
$46$	0	0
$47$	−4.85654	−0.708399	−0.354199	−	0.935170i	$-0.615247\pi$
$47$	−4.85654	−0.708399	−0.354199	+	0.935170i	$0.615247\pi$
$48$	0	0
$49$	1.72574	0.246535
$50$	0	0
$51$	−1.69001	−0.236649
$52$	0	0
$53$	9.41547	1.29331	0.646657	−	0.762781i	$-0.276167\pi$
$53$	9.41547	1.29331	0.646657	+	0.762781i	$0.276167\pi$
$54$	0	0
$55$	−5.26679	−0.710173
$56$	0	0
$57$	0.960668	0.127244
$58$	0	0
$59$	−12.5427	−1.63291	−0.816457	−	0.577406i	$-0.804065\pi$
$59$	−12.5427	−1.63291	−0.816457	+	0.577406i	$0.804065\pi$
$60$	0	0
$61$	12.5093	1.60165	0.800823	−	0.598901i	$-0.204396\pi$
$61$	12.5093	1.60165	0.800823	+	0.598901i	$0.204396\pi$
$62$	0	0
$63$	8.67419	1.09285
$64$	0	0
$65$	3.10208	0.384766
$66$	0	0
$67$	0.810453	0.0990126	0.0495063	−	0.998774i	$-0.484235\pi$
$67$	0.810453	0.0990126	0.0495063	+	0.998774i	$0.484235\pi$
$68$	0	0
$69$	−2.12659	−0.256011
$70$	0	0
$71$	−9.99792	−1.18654	−0.593268	−	0.805005i	$-0.702162\pi$
$71$	−9.99792	−1.18654	−0.593268	+	0.805005i	$0.702162\pi$
$72$	0	0
$73$	−3.66059	−0.428439	−0.214220	−	0.976786i	$-0.568721\pi$
$73$	−3.66059	−0.428439	−0.214220	+	0.976786i	$0.568721\pi$
$74$	0	0
$75$	0.930722	0.107471
$76$	0	0
$77$	−13.6086	−1.55085
$78$	0	0
$79$	−5.78439	−0.650794	−0.325397	−	0.945577i	$-0.605498\pi$
$79$	−5.78439	−0.650794	−0.325397	+	0.945577i	$0.605498\pi$
$80$	0	0
$81$	8.43240	0.936934
$82$	0	0
$83$	−2.78063	−0.305214	−0.152607	−	0.988287i	$-0.548767\pi$
$83$	−2.78063	−0.305214	−0.152607	+	0.988287i	$0.548767\pi$
$84$	0	0
$85$	−7.66630	−0.831527
$86$	0	0
$87$	2.20873	0.236801
$88$	0	0
$89$	5.17695	0.548756	0.274378	−	0.961622i	$-0.411528\pi$
$89$	5.17695	0.548756	0.274378	+	0.961622i	$0.411528\pi$
$90$	0	0
$91$	8.01535	0.840237
$92$	0	0
$93$	−1.32386	−0.137278
$94$	0	0
$95$	4.35782	0.447102
$96$	0	0
$97$	0.683095	0.0693578	0.0346789	−	0.999399i	$-0.488959\pi$
$97$	0.683095	0.0693578	0.0346789	+	0.999399i	$0.488959\pi$
$98$	0	0
$99$	−13.5282	−1.35964
$100$	0	0
$101$	17.7475	1.76594	0.882969	−	0.469431i	$-0.155541\pi$
$101$	17.7475	1.76594	0.882969	+	0.469431i	$0.155541\pi$
$102$	0	0
$103$	11.1758	1.10119	0.550595	−	0.834773i	$-0.314401\pi$
$103$	11.1758	1.10119	0.550595	+	0.834773i	$0.314401\pi$
$104$	0	0
$105$	−0.851079	−0.0830568
$106$	0	0
$107$	−4.88596	−0.472343	−0.236172	−	0.971711i	$-0.575893\pi$
$107$	−4.88596	−0.472343	−0.236172	+	0.971711i	$0.575893\pi$
$108$	0	0
$109$	−8.12888	−0.778606	−0.389303	−	0.921110i	$-0.627284\pi$
$109$	−8.12888	−0.778606	−0.389303	+	0.921110i	$0.627284\pi$
$110$	0	0
$111$	−1.33405	−0.126622
$112$	0	0
$113$	−17.7705	−1.67170	−0.835852	−	0.548955i	$-0.815026\pi$
$113$	−17.7705	−1.67170	−0.835852	+	0.548955i	$0.815026\pi$
$114$	0	0
$115$	−9.64670	−0.899560
$116$	0	0
$117$	7.96800	0.736642
$118$	0	0
$119$	−19.8087	−1.81586
$120$	0	0
$121$	10.2240	0.929455
$122$	0	0
$123$	1.79543	0.161888
$124$	0	0
$125$	9.93810	0.888891
$126$	0	0
$127$	−2.12777	−0.188809	−0.0944046	−	0.995534i	$-0.530095\pi$
$127$	−2.12777	−0.188809	−0.0944046	+	0.995534i	$0.530095\pi$
$128$	0	0
$129$	−2.56811	−0.226110
$130$	0	0
$131$	−2.22280	−0.194207	−0.0971036	−	0.995274i	$-0.530958\pi$
$131$	−2.22280	−0.194207	−0.0971036	+	0.995274i	$0.530958\pi$
$132$	0	0
$133$	11.2600	0.976365
$134$	0	0
$135$	−1.71040	−0.147208
$136$	0	0
$137$	4.00853	0.342472	0.171236	−	0.985230i	$-0.445224\pi$
$137$	4.00853	0.342472	0.171236	+	0.985230i	$0.445224\pi$
$138$	0	0
$139$	−18.0644	−1.53220	−0.766102	−	0.642719i	$-0.777806\pi$
$139$	−18.0644	−1.53220	−0.766102	+	0.642719i	$0.777806\pi$
$140$	0	0
$141$	1.22395	0.103075
$142$	0	0
$143$	−12.5007	−1.04536
$144$	0	0
$145$	10.0193	0.832060
$146$	0	0
$147$	−0.434923	−0.0358719
$148$	0	0
$149$	−0.282976	−0.0231823	−0.0115911	−	0.999933i	$-0.503690\pi$
$149$	−0.282976	−0.0231823	−0.0115911	+	0.999933i	$0.503690\pi$
$150$	0	0
$151$	17.5934	1.43173	0.715865	−	0.698238i	$-0.246032\pi$
$151$	17.5934	1.43173	0.715865	+	0.698238i	$0.246032\pi$
$152$	0	0
$153$	−19.6916	−1.59197
$154$	0	0
$155$	−6.00535	−0.482361
$156$	0	0
$157$	−3.06030	−0.244239	−0.122119	−	0.992515i	$-0.538969\pi$
$157$	−3.06030	−0.244239	−0.122119	+	0.992515i	$0.538969\pi$
$158$	0	0
$159$	−2.37289	−0.188183
$160$	0	0
$161$	−24.9257	−1.96442
$162$	0	0
$163$	−22.2746	−1.74468	−0.872340	−	0.488899i	$-0.837399\pi$
$163$	−22.2746	−1.74468	−0.872340	+	0.488899i	$0.837399\pi$
$164$	0	0
$165$	1.32734	0.103333
$166$	0	0
$167$	−13.4126	−1.03790	−0.518949	−	0.854805i	$-0.673677\pi$
$167$	−13.4126	−1.03790	−0.518949	+	0.854805i	$0.673677\pi$
$168$	0	0
$169$	−5.63721	−0.433631
$170$	0	0
$171$	11.1935	0.855986
$172$	0	0
$173$	−6.74954	−0.513158	−0.256579	−	0.966523i	$-0.582595\pi$
$173$	−6.74954	−0.513158	−0.256579	+	0.966523i	$0.582595\pi$
$174$	0	0
$175$	10.9090	0.824642
$176$	0	0
$177$	3.16101	0.237596
$178$	0	0
$179$	−15.9034	−1.18868	−0.594339	−	0.804215i	$-0.702586\pi$
$179$	−15.9034	−1.18868	−0.594339	+	0.804215i	$0.702586\pi$
$180$	0	0
$181$	−20.1609	−1.49855	−0.749273	−	0.662262i	$-0.769597\pi$
$181$	−20.1609	−1.49855	−0.749273	+	0.662262i	$0.769597\pi$
$182$	0	0
$183$	−3.15259	−0.233046
$184$	0	0
$185$	−6.05155	−0.444919
$186$	0	0
$187$	30.8935	2.25916
$188$	0	0
$189$	−4.41944	−0.321467
$190$	0	0
$191$	−5.46129	−0.395165	−0.197583	−	0.980286i	$-0.563309\pi$
$191$	−5.46129	−0.395165	−0.197583	+	0.980286i	$0.563309\pi$
$192$	0	0
$193$	−13.9352	−1.00307	−0.501537	−	0.865136i	$-0.667232\pi$
$193$	−13.9352	−1.00307	−0.501537	+	0.865136i	$0.667232\pi$
$194$	0	0
$195$	−0.781789	−0.0559851
$196$	0	0
$197$	−20.1623	−1.43650	−0.718252	−	0.695783i	$-0.755058\pi$
$197$	−20.1623	−1.43650	−0.718252	+	0.695783i	$0.755058\pi$
$198$	0	0
$199$	13.4525	0.953626	0.476813	−	0.879005i	$-0.341792\pi$
$199$	13.4525	0.953626	0.476813	+	0.879005i	$0.341792\pi$
$200$	0	0
$201$	−0.204251	−0.0144068
$202$	0	0
$203$	25.8885	1.81702
$204$	0	0
$205$	8.14448	0.568835
$206$	0	0
$207$	−24.7785	−1.72222
$208$	0	0
$209$	−17.5611	−1.21472
$210$	0	0
$211$	−2.99575	−0.206236	−0.103118	−	0.994669i	$-0.532882\pi$
$211$	−2.99575	−0.206236	−0.103118	+	0.994669i	$0.532882\pi$
$212$	0	0
$213$	2.51969	0.172646
$214$	0	0
$215$	−11.6496	−0.794494
$216$	0	0
$217$	−15.5170	−1.05336
$218$	0	0
$219$	0.922544	0.0623397
$220$	0	0
$221$	−18.1960	−1.22399
$222$	0	0
$223$	−0.957053	−0.0640890	−0.0320445	−	0.999486i	$-0.510202\pi$
$223$	−0.957053	−0.0640890	−0.0320445	+	0.999486i	$0.510202\pi$
$224$	0	0
$225$	10.8445	0.722970
$226$	0	0
$227$	−23.6386	−1.56895	−0.784474	−	0.620162i	$-0.787067\pi$
$227$	−23.6386	−1.56895	−0.784474	+	0.620162i	$0.787067\pi$
$228$	0	0
$229$	−8.56225	−0.565809	−0.282905	−	0.959148i	$-0.591298\pi$
$229$	−8.56225	−0.565809	−0.282905	+	0.959148i	$0.591298\pi$
$230$	0	0
$231$	3.42966	0.225655
$232$	0	0
$233$	28.5277	1.86891	0.934456	−	0.356078i	$-0.115886\pi$
$233$	28.5277	1.86891	0.934456	+	0.356078i	$0.115886\pi$
$234$	0	0
$235$	5.55212	0.362180
$236$	0	0
$237$	1.45779	0.0946934
$238$	0	0
$239$	−13.7041	−0.886443	−0.443222	−	0.896412i	$-0.646165\pi$
$239$	−13.7041	−0.886443	−0.443222	+	0.896412i	$0.646165\pi$
$240$	0	0
$241$	−22.3406	−1.43908	−0.719541	−	0.694450i	$-0.755648\pi$
$241$	−22.3406	−1.43908	−0.719541	+	0.694450i	$0.755648\pi$
$242$	0	0
$243$	−6.61350	−0.424256
$244$	0	0
$245$	−1.97291	−0.126045
$246$	0	0
$247$	10.3433	0.658127
$248$	0	0
$249$	0.700777	0.0444099
$250$	0	0
$251$	−18.8213	−1.18799	−0.593994	−	0.804470i	$-0.702450\pi$
$251$	−18.8213	−1.18799	−0.593994	+	0.804470i	$0.702450\pi$
$252$	0	0
$253$	38.8741	2.44400
$254$	0	0
$255$	1.93207	0.120991
$256$	0	0
$257$	3.75055	0.233953	0.116976	−	0.993135i	$-0.462680\pi$
$257$	3.75055	0.233953	0.116976	+	0.993135i	$0.462680\pi$
$258$	0	0
$259$	−15.6364	−0.971597
$260$	0	0
$261$	25.7356	1.59299
$262$	0	0
$263$	−2.37751	−0.146604	−0.0733018	−	0.997310i	$-0.523354\pi$
$263$	−2.37751	−0.146604	−0.0733018	+	0.997310i	$0.523354\pi$
$264$	0	0
$265$	−10.7640	−0.661228
$266$	0	0
$267$	−1.30470	−0.0798463
$268$	0	0
$269$	14.5047	0.884369	0.442185	−	0.896924i	$-0.354204\pi$
$269$	14.5047	0.884369	0.442185	+	0.896924i	$0.354204\pi$
$270$	0	0
$271$	−8.82781	−0.536251	−0.268126	−	0.963384i	$-0.586404\pi$
$271$	−8.82781	−0.536251	−0.268126	+	0.963384i	$0.586404\pi$
$272$	0	0
$273$	−2.02004	−0.122258
$274$	0	0
$275$	−17.0136	−1.02596
$276$	0	0
$277$	24.3478	1.46292	0.731458	−	0.681887i	$-0.238840\pi$
$277$	24.3478	1.46292	0.731458	+	0.681887i	$0.238840\pi$
$278$	0	0
$279$	−15.4253	−0.923490
$280$	0	0
$281$	−16.4674	−0.982365	−0.491182	−	0.871057i	$-0.663435\pi$
$281$	−16.4674	−0.982365	−0.491182	+	0.871057i	$0.663435\pi$
$282$	0	0
$283$	1.07647	0.0639897	0.0319949	−	0.999488i	$-0.489814\pi$
$283$	1.07647	0.0639897	0.0319949	+	0.999488i	$0.489814\pi$
$284$	0	0
$285$	−1.09826	−0.0650553
$286$	0	0
$287$	21.0442	1.24220
$288$	0	0
$289$	27.9685	1.64520
$290$	0	0
$291$	−0.172154	−0.0100919
$292$	0	0
$293$	8.83987	0.516430	0.258215	−	0.966087i	$-0.416866\pi$
$293$	8.83987	0.516430	0.258215	+	0.966087i	$0.416866\pi$
$294$	0	0
$295$	14.3391	0.834854
$296$	0	0
$297$	6.89254	0.399946
$298$	0	0
$299$	−22.8965	−1.32414
$300$	0	0
$301$	−30.1009	−1.73499
$302$	0	0
$303$	−4.47273	−0.256952
$304$	0	0
$305$	−14.3009	−0.818867
$306$	0	0
$307$	−1.14588	−0.0653987	−0.0326993	−	0.999465i	$-0.510410\pi$
$307$	−1.14588	−0.0653987	−0.0326993	+	0.999465i	$0.510410\pi$
$308$	0	0
$309$	−2.81655	−0.160228
$310$	0	0
$311$	−0.980364	−0.0555914	−0.0277957	−	0.999614i	$-0.508849\pi$
$311$	−0.980364	−0.0555914	−0.0277957	+	0.999614i	$0.508849\pi$
$312$	0	0
$313$	−26.6505	−1.50637	−0.753186	−	0.657807i	$-0.771484\pi$
$313$	−26.6505	−1.50637	−0.753186	+	0.657807i	$0.771484\pi$
$314$	0	0
$315$	−9.91656	−0.558735
$316$	0	0
$317$	4.87754	0.273950	0.136975	−	0.990575i	$-0.456262\pi$
$317$	4.87754	0.273950	0.136975	+	0.990575i	$0.456262\pi$
$318$	0	0
$319$	−40.3757	−2.26061
$320$	0	0
$321$	1.23136	0.0687280
$322$	0	0
$323$	−25.5618	−1.42229
$324$	0	0
$325$	10.0209	0.555857
$326$	0	0
$327$	2.04865	0.113291
$328$	0	0
$329$	14.3459	0.790915
$330$	0	0
$331$	0.797406	0.0438294	0.0219147	−	0.999760i	$-0.493024\pi$
$331$	0.797406	0.0438294	0.0219147	+	0.999760i	$0.493024\pi$
$332$	0	0
$333$	−15.5440	−0.851806
$334$	0	0
$335$	−0.926531	−0.0506218
$336$	0	0
$337$	28.4113	1.54766	0.773832	−	0.633391i	$-0.218338\pi$
$337$	28.4113	1.54766	0.773832	+	0.633391i	$0.218338\pi$
$338$	0	0
$339$	4.47853	0.243240
$340$	0	0
$341$	24.2003	1.31052
$342$	0	0
$343$	15.5798	0.841231
$344$	0	0
$345$	2.43117	0.130890
$346$	0	0
$347$	−13.6649	−0.733569	−0.366784	−	0.930306i	$-0.619541\pi$
$347$	−13.6649	−0.733569	−0.366784	+	0.930306i	$0.619541\pi$
$348$	0	0
$349$	27.6224	1.47859	0.739296	−	0.673380i	$-0.235158\pi$
$349$	27.6224	1.47859	0.739296	+	0.673380i	$0.235158\pi$
$350$	0	0
$351$	−4.05964	−0.216687
$352$	0	0
$353$	−9.72895	−0.517820	−0.258910	−	0.965902i	$-0.583363\pi$
$353$	−9.72895	−0.517820	−0.258910	+	0.965902i	$0.583363\pi$
$354$	0	0
$355$	11.4299	0.606635
$356$	0	0
$357$	4.99220	0.264215
$358$	0	0
$359$	−32.0359	−1.69079	−0.845396	−	0.534140i	$-0.820636\pi$
$359$	−32.0359	−1.69079	−0.845396	+	0.534140i	$0.820636\pi$
$360$	0	0
$361$	−4.46973	−0.235249
$362$	0	0
$363$	−2.57666	−0.135240
$364$	0	0
$365$	4.18488	0.219046
$366$	0	0
$367$	5.86530	0.306166	0.153083	−	0.988213i	$-0.451080\pi$
$367$	5.86530	0.306166	0.153083	+	0.988213i	$0.451080\pi$
$368$	0	0
$369$	20.9199	1.08905
$370$	0	0
$371$	−27.8127	−1.44396
$372$	0	0
$373$	15.7819	0.817154	0.408577	−	0.912724i	$-0.366025\pi$
$373$	15.7819	0.817154	0.408577	+	0.912724i	$0.366025\pi$
$374$	0	0
$375$	−2.50461	−0.129337
$376$	0	0
$377$	23.7809	1.22478
$378$	0	0
$379$	16.2902	0.836771	0.418385	−	0.908270i	$-0.362596\pi$
$379$	16.2902	0.836771	0.418385	+	0.908270i	$0.362596\pi$
$380$	0	0
$381$	0.536243	0.0274725
$382$	0	0
$383$	7.61829	0.389277	0.194638	−	0.980875i	$-0.437647\pi$
$383$	7.61829	0.389277	0.194638	+	0.980875i	$0.437647\pi$
$384$	0	0
$385$	15.5578	0.792896
$386$	0	0
$387$	−29.9230	−1.52107
$388$	0	0
$389$	6.91050	0.350376	0.175188	−	0.984535i	$-0.443947\pi$
$389$	6.91050	0.350376	0.175188	+	0.984535i	$0.443947\pi$
$390$	0	0
$391$	56.5849	2.86162
$392$	0	0
$393$	0.560193	0.0282580
$394$	0	0
$395$	6.61286	0.332729
$396$	0	0
$397$	−18.3598	−0.921453	−0.460726	−	0.887542i	$-0.652411\pi$
$397$	−18.3598	−0.921453	−0.460726	+	0.887542i	$0.652411\pi$
$398$	0	0
$399$	−2.83775	−0.142065
$400$	0	0
$401$	4.42851	0.221149	0.110575	−	0.993868i	$-0.464731\pi$
$401$	4.42851	0.221149	0.110575	+	0.993868i	$0.464731\pi$
$402$	0	0
$403$	−14.2537	−0.710027
$404$	0	0
$405$	−9.64014	−0.479022
$406$	0	0
$407$	24.3864	1.20879
$408$	0	0
$409$	−36.0535	−1.78273	−0.891367	−	0.453283i	$-0.850253\pi$
$409$	−36.0535	−1.78273	−0.891367	+	0.453283i	$0.850253\pi$
$410$	0	0
$411$	−1.01023	−0.0498312
$412$	0	0
$413$	37.0502	1.82312
$414$	0	0
$415$	3.17889	0.156045
$416$	0	0
$417$	4.55261	0.222942
$418$	0	0
$419$	14.1124	0.689434	0.344717	−	0.938707i	$-0.387975\pi$
$419$	14.1124	0.689434	0.344717	+	0.938707i	$0.387975\pi$
$420$	0	0
$421$	8.08360	0.393970	0.196985	−	0.980406i	$-0.436885\pi$
$421$	8.08360	0.393970	0.196985	+	0.980406i	$0.436885\pi$
$422$	0	0
$423$	14.2611	0.693401
$424$	0	0
$425$	−24.7649	−1.20128
$426$	0	0
$427$	−36.9516	−1.78821
$428$	0	0
$429$	3.15044	0.152105
$430$	0	0
$431$	31.8067	1.53208	0.766038	−	0.642796i	$-0.222226\pi$
$431$	31.8067	1.53208	0.766038	+	0.642796i	$0.222226\pi$
$432$	0	0
$433$	−31.0657	−1.49292	−0.746461	−	0.665430i	$-0.768248\pi$
$433$	−31.0657	−1.49292	−0.746461	+	0.665430i	$0.768248\pi$
$434$	0	0
$435$	−2.52508	−0.121068
$436$	0	0
$437$	−32.1650	−1.53866
$438$	0	0
$439$	−29.0275	−1.38541	−0.692704	−	0.721222i	$-0.743581\pi$
$439$	−29.0275	−1.38541	−0.692704	+	0.721222i	$0.743581\pi$
$440$	0	0
$441$	−5.06762	−0.241315
$442$	0	0
$443$	19.3683	0.920217	0.460109	−	0.887863i	$-0.347810\pi$
$443$	19.3683	0.920217	0.460109	+	0.887863i	$0.347810\pi$
$444$	0	0
$445$	−5.91842	−0.280560
$446$	0	0
$447$	0.0713158	0.00337312
$448$	0	0
$449$	18.0937	0.853895	0.426947	−	0.904276i	$-0.359589\pi$
$449$	18.0937	0.853895	0.426947	+	0.904276i	$0.359589\pi$
$450$	0	0
$451$	−32.8205	−1.54546
$452$	0	0
$453$	−4.43390	−0.208323
$454$	0	0
$455$	−9.16335	−0.429585
$456$	0	0
$457$	4.21494	0.197167	0.0985834	−	0.995129i	$-0.468569\pi$
$457$	4.21494	0.197167	0.0985834	+	0.995129i	$0.468569\pi$
$458$	0	0
$459$	10.0327	0.468289
$460$	0	0
$461$	20.0523	0.933927	0.466963	−	0.884277i	$-0.345348\pi$
$461$	20.0523	0.933927	0.466963	+	0.884277i	$0.345348\pi$
$462$	0	0
$463$	−40.6145	−1.88752	−0.943759	−	0.330634i	$-0.892737\pi$
$463$	−40.6145	−1.88752	−0.943759	+	0.330634i	$0.892737\pi$
$464$	0	0
$465$	1.51347	0.0701856
$466$	0	0
$467$	−33.2401	−1.53817	−0.769084	−	0.639148i	$-0.779287\pi$
$467$	−33.2401	−1.53817	−0.769084	+	0.639148i	$0.779287\pi$
$468$	0	0
$469$	−2.39403	−0.110546
$470$	0	0
$471$	0.771260	0.0355378
$472$	0	0
$473$	46.9452	2.15854
$474$	0	0
$475$	14.0773	0.645912
$476$	0	0
$477$	−27.6484	−1.26593
$478$	0	0
$479$	−17.2081	−0.786260	−0.393130	−	0.919483i	$-0.628608\pi$
$479$	−17.2081	−0.786260	−0.393130	+	0.919483i	$0.628608\pi$
$480$	0	0
$481$	−14.3634	−0.654913
$482$	0	0
$483$	6.28181	0.285832
$484$	0	0
$485$	−0.780932	−0.0354603
$486$	0	0
$487$	−25.4301	−1.15235	−0.576175	−	0.817327i	$-0.695455\pi$
$487$	−25.4301	−1.15235	−0.576175	+	0.817327i	$0.695455\pi$
$488$	0	0
$489$	5.61366	0.253859
$490$	0	0
$491$	−2.16525	−0.0977164	−0.0488582	−	0.998806i	$-0.515558\pi$
$491$	−2.16525	−0.0977164	−0.0488582	+	0.998806i	$0.515558\pi$
$492$	0	0
$493$	−58.7706	−2.64690
$494$	0	0
$495$	15.4658	0.695138
$496$	0	0
$497$	29.5332	1.32475
$498$	0	0
$499$	22.0046	0.985063	0.492532	−	0.870295i	$-0.336072\pi$
$499$	22.0046	0.985063	0.492532	+	0.870295i	$0.336072\pi$
$500$	0	0
$501$	3.38025	0.151019
$502$	0	0
$503$	33.9725	1.51476	0.757379	−	0.652976i	$-0.226480\pi$
$503$	33.9725	1.51476	0.757379	+	0.652976i	$0.226480\pi$
$504$	0	0
$505$	−20.2894	−0.902864
$506$	0	0
$507$	1.42069	0.0630952
$508$	0	0
$509$	−28.1600	−1.24817	−0.624085	−	0.781356i	$-0.714528\pi$
$509$	−28.1600	−1.24817	−0.624085	+	0.781356i	$0.714528\pi$
$510$	0	0
$511$	10.8131	0.478345
$512$	0	0
$513$	−5.70299	−0.251793
$514$	0	0
$515$	−12.7765	−0.563001
$516$	0	0
$517$	−22.3738	−0.984000
$518$	0	0
$519$	1.70102	0.0746667
$520$	0	0
$521$	−5.99942	−0.262839	−0.131420	−	0.991327i	$-0.541954\pi$
$521$	−5.99942	−0.262839	−0.131420	+	0.991327i	$0.541954\pi$
$522$	0	0
$523$	2.86485	0.125271	0.0626355	−	0.998036i	$-0.480049\pi$
$523$	2.86485	0.125271	0.0626355	+	0.998036i	$0.480049\pi$
$524$	0	0
$525$	−2.74929	−0.119989
$526$	0	0
$527$	35.2257	1.53446
$528$	0	0
$529$	48.2023	2.09575
$530$	0	0
$531$	36.8313	1.59834
$532$	0	0
$533$	19.3309	0.837316
$534$	0	0
$535$	5.58575	0.241493
$536$	0	0
$537$	4.00800	0.172958
$538$	0	0
$539$	7.95041	0.342449
$540$	0	0
$541$	20.9419	0.900362	0.450181	−	0.892937i	$-0.351360\pi$
$541$	20.9419	0.900362	0.450181	+	0.892937i	$0.351360\pi$
$542$	0	0
$543$	5.08096	0.218045
$544$	0	0
$545$	9.29315	0.398075
$546$	0	0
$547$	−17.3985	−0.743908	−0.371954	−	0.928251i	$-0.621312\pi$
$547$	−17.3985	−0.743908	−0.371954	+	0.928251i	$0.621312\pi$
$548$	0	0
$549$	−36.7333	−1.56774
$550$	0	0
$551$	33.4075	1.42321
$552$	0	0
$553$	17.0867	0.726601
$554$	0	0
$555$	1.52512	0.0647376
$556$	0	0
$557$	−10.5066	−0.445180	−0.222590	−	0.974912i	$-0.571451\pi$
$557$	−10.5066	−0.445180	−0.222590	+	0.974912i	$0.571451\pi$
$558$	0	0
$559$	−27.6503	−1.16948
$560$	0	0
$561$	−7.78582	−0.328717
$562$	0	0
$563$	22.3769	0.943074	0.471537	−	0.881846i	$-0.343699\pi$
$563$	22.3769	0.943074	0.471537	+	0.881846i	$0.343699\pi$
$564$	0	0
$565$	20.3156	0.854685
$566$	0	0
$567$	−24.9088	−1.04607
$568$	0	0
$569$	−10.6913	−0.448201	−0.224100	−	0.974566i	$-0.571944\pi$
$569$	−10.6913	−0.448201	−0.224100	+	0.974566i	$0.571944\pi$
$570$	0	0
$571$	3.53095	0.147766	0.0738829	−	0.997267i	$-0.476461\pi$
$571$	3.53095	0.147766	0.0738829	+	0.997267i	$0.476461\pi$
$572$	0	0
$573$	1.37636	0.0574982
$574$	0	0
$575$	−31.1624	−1.29956
$576$	0	0
$577$	−1.96274	−0.0817098	−0.0408549	−	0.999165i	$-0.513008\pi$
$577$	−1.96274	−0.0817098	−0.0408549	+	0.999165i	$0.513008\pi$
$578$	0	0
$579$	3.51195	0.145952
$580$	0	0
$581$	8.21381	0.340766
$582$	0	0
$583$	43.3766	1.79648
$584$	0	0
$585$	−9.10922	−0.376620
$586$	0	0
$587$	−40.0967	−1.65497	−0.827484	−	0.561489i	$-0.810229\pi$
$587$	−40.0967	−1.65497	−0.827484	+	0.561489i	$0.810229\pi$
$588$	0	0
$589$	−20.0236	−0.825060
$590$	0	0
$591$	5.08132	0.209018
$592$	0	0
$593$	38.0326	1.56181	0.780905	−	0.624650i	$-0.214758\pi$
$593$	38.0326	1.56181	0.780905	+	0.624650i	$0.214758\pi$
$594$	0	0
$595$	22.6458	0.928386
$596$	0	0
$597$	−3.39032	−0.138757
$598$	0	0
$599$	39.7839	1.62552	0.812762	−	0.582595i	$-0.197963\pi$
$599$	39.7839	1.62552	0.812762	+	0.582595i	$0.197963\pi$
$600$	0	0
$601$	3.28308	0.133920	0.0669598	−	0.997756i	$-0.478670\pi$
$601$	3.28308	0.133920	0.0669598	+	0.997756i	$0.478670\pi$
$602$	0	0
$603$	−2.37988	−0.0969163
$604$	0	0
$605$	−11.6883	−0.475199
$606$	0	0
$607$	−24.1249	−0.979198	−0.489599	−	0.871948i	$-0.662857\pi$
$607$	−24.1249	−0.979198	−0.489599	+	0.871948i	$0.662857\pi$
$608$	0	0
$609$	−6.52445	−0.264384
$610$	0	0
$611$	13.1780	0.533123
$612$	0	0
$613$	−11.7593	−0.474955	−0.237478	−	0.971393i	$-0.576321\pi$
$613$	−11.7593	−0.474955	−0.237478	+	0.971393i	$0.576321\pi$
$614$	0	0
$615$	−2.05258	−0.0827680
$616$	0	0
$617$	10.0660	0.405243	0.202621	−	0.979257i	$-0.435054\pi$
$617$	10.0660	0.405243	0.202621	+	0.979257i	$0.435054\pi$
$618$	0	0
$619$	32.5890	1.30986	0.654931	−	0.755689i	$-0.272698\pi$
$619$	32.5890	1.30986	0.654931	+	0.755689i	$0.272698\pi$
$620$	0	0
$621$	12.6245	0.506602
$622$	0	0
$623$	−15.2924	−0.612677
$624$	0	0
$625$	7.10368	0.284147
$626$	0	0
$627$	4.42575	0.176748
$628$	0	0
$629$	35.4968	1.41535
$630$	0	0
$631$	−13.6390	−0.542959	−0.271479	−	0.962444i	$-0.587513\pi$
$631$	−13.6390	−0.542959	−0.271479	+	0.962444i	$0.587513\pi$
$632$	0	0
$633$	0.754993	0.0300083
$634$	0	0
$635$	2.43252	0.0965317
$636$	0	0
$637$	−4.68271	−0.185536
$638$	0	0
$639$	29.3588	1.16141
$640$	0	0
$641$	48.3226	1.90863	0.954314	−	0.298805i	$-0.0965881\pi$
$641$	48.3226	1.90863	0.954314	+	0.298805i	$0.0965881\pi$
$642$	0	0
$643$	−8.93728	−0.352452	−0.176226	−	0.984350i	$-0.556389\pi$
$643$	−8.93728	−0.352452	−0.176226	+	0.984350i	$0.556389\pi$
$644$	0	0
$645$	2.93593	0.115602
$646$	0	0
$647$	−39.7198	−1.56155	−0.780773	−	0.624814i	$-0.785175\pi$
$647$	−39.7198	−1.56155	−0.780773	+	0.624814i	$0.785175\pi$
$648$	0	0
$649$	−57.7834	−2.26820
$650$	0	0
$651$	3.91061	0.153269
$652$	0	0
$653$	−17.4501	−0.682874	−0.341437	−	0.939905i	$-0.610914\pi$
$653$	−17.4501	−0.682874	−0.341437	+	0.939905i	$0.610914\pi$
$654$	0	0
$655$	2.54117	0.0992916
$656$	0	0
$657$	10.7493	0.419368
$658$	0	0
$659$	34.8645	1.35813	0.679063	−	0.734080i	$-0.262386\pi$
$659$	34.8645	1.35813	0.679063	+	0.734080i	$0.262386\pi$
$660$	0	0
$661$	−35.2295	−1.37027	−0.685134	−	0.728417i	$-0.740256\pi$
$661$	−35.2295	−1.37027	−0.685134	+	0.728417i	$0.740256\pi$
$662$	0	0
$663$	4.58576	0.178096
$664$	0	0
$665$	−12.8727	−0.499182
$666$	0	0
$667$	−73.9526	−2.86346
$668$	0	0
$669$	0.241197	0.00932523
$670$	0	0
$671$	57.6295	2.22476
$672$	0	0
$673$	−7.35195	−0.283397	−0.141698	−	0.989910i	$-0.545256\pi$
$673$	−7.35195	−0.283397	−0.141698	+	0.989910i	$0.545256\pi$
$674$	0	0
$675$	−5.52522	−0.212666
$676$	0	0
$677$	−9.50512	−0.365311	−0.182656	−	0.983177i	$-0.558469\pi$
$677$	−9.50512	−0.365311	−0.182656	+	0.983177i	$0.558469\pi$
$678$	0	0
$679$	−2.01782	−0.0774368
$680$	0	0
$681$	5.95742	0.228289
$682$	0	0
$683$	−23.3338	−0.892842	−0.446421	−	0.894823i	$-0.647302\pi$
$683$	−23.3338	−0.892842	−0.446421	+	0.894823i	$0.647302\pi$
$684$	0	0
$685$	−4.58266	−0.175094
$686$	0	0
$687$	2.15787	0.0823277
$688$	0	0
$689$	−25.5484	−0.973315
$690$	0	0
$691$	−26.0906	−0.992534	−0.496267	−	0.868170i	$-0.665296\pi$
$691$	−26.0906	−0.992534	−0.496267	+	0.868170i	$0.665296\pi$
$692$	0	0
$693$	39.9616	1.51802
$694$	0	0
$695$	20.6517	0.783364
$696$	0	0
$697$	−47.7733	−1.80954
$698$	0	0
$699$	−7.18958	−0.271935
$700$	0	0
$701$	−35.5202	−1.34158	−0.670791	−	0.741647i	$-0.734045\pi$
$701$	−35.5202	−1.34158	−0.670791	+	0.741647i	$0.734045\pi$
$702$	0	0
$703$	−20.1777	−0.761017
$704$	0	0
$705$	−1.39925	−0.0526988
$706$	0	0
$707$	−52.4249	−1.97164
$708$	0	0
$709$	47.0545	1.76717	0.883584	−	0.468273i	$-0.155124\pi$
$709$	47.0545	1.76717	0.883584	+	0.468273i	$0.155124\pi$
$710$	0	0
$711$	16.9858	0.637016
$712$	0	0
$713$	44.3255	1.66000
$714$	0	0
$715$	14.2911	0.534458
$716$	0	0
$717$	3.45371	0.128981
$718$	0	0
$719$	−6.33724	−0.236339	−0.118170	−	0.992993i	$-0.537703\pi$
$719$	−6.33724	−0.236339	−0.118170	+	0.992993i	$0.537703\pi$
$720$	0	0
$721$	−33.0128	−1.22946
$722$	0	0
$723$	5.63029	0.209393
$724$	0	0
$725$	32.3661	1.20205
$726$	0	0
$727$	−10.6592	−0.395328	−0.197664	−	0.980270i	$-0.563335\pi$
$727$	−10.6592	−0.395328	−0.197664	+	0.980270i	$0.563335\pi$
$728$	0	0
$729$	−23.6305	−0.875203
$730$	0	0
$731$	68.3332	2.52739
$732$	0	0
$733$	40.8304	1.50810	0.754052	−	0.656814i	$-0.228096\pi$
$733$	40.8304	1.50810	0.754052	+	0.656814i	$0.228096\pi$
$734$	0	0
$735$	0.497215	0.0183401
$736$	0	0
$737$	3.73372	0.137533
$738$	0	0
$739$	−13.1822	−0.484917	−0.242458	−	0.970162i	$-0.577954\pi$
$739$	−13.1822	−0.484917	−0.242458	+	0.970162i	$0.577954\pi$
$740$	0	0
$741$	−2.60672	−0.0957603
$742$	0	0
$743$	−30.1895	−1.10754	−0.553772	−	0.832668i	$-0.686812\pi$
$743$	−30.1895	−1.10754	−0.553772	+	0.832668i	$0.686812\pi$
$744$	0	0
$745$	0.323505	0.0118523
$746$	0	0
$747$	8.16528	0.298752
$748$	0	0
$749$	14.4328	0.527363
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	4.74335	0.172857
$754$	0	0
$755$	−20.1132	−0.731995
$756$	0	0
$757$	7.94987	0.288943	0.144471	−	0.989509i	$-0.453852\pi$
$757$	7.94987	0.288943	0.144471	+	0.989509i	$0.453852\pi$
$758$	0	0
$759$	−9.79709	−0.355612
$760$	0	0
$761$	−22.0572	−0.799571	−0.399786	−	0.916609i	$-0.630915\pi$
$761$	−22.0572	−0.799571	−0.399786	+	0.916609i	$0.630915\pi$
$762$	0	0
$763$	24.0122	0.869300
$764$	0	0
$765$	22.5120	0.813923
$766$	0	0
$767$	34.0338	1.22889
$768$	0	0
$769$	−40.1229	−1.44687	−0.723435	−	0.690393i	$-0.757438\pi$
$769$	−40.1229	−1.44687	−0.723435	+	0.690393i	$0.757438\pi$
$770$	0	0
$771$	−0.945217	−0.0340412
$772$	0	0
$773$	−28.5766	−1.02783	−0.513915	−	0.857841i	$-0.671805\pi$
$773$	−28.5766	−1.02783	−0.513915	+	0.857841i	$0.671805\pi$
$774$	0	0
$775$	−19.3995	−0.696849
$776$	0	0
$777$	3.94069	0.141372
$778$	0	0
$779$	27.1562	0.972970
$780$	0	0
$781$	−46.0599	−1.64815
$782$	0	0
$783$	−13.1121	−0.468588
$784$	0	0
$785$	3.49862	0.124871
$786$	0	0
$787$	−3.05876	−0.109033	−0.0545164	−	0.998513i	$-0.517362\pi$
$787$	−3.05876	−0.109033	−0.0545164	+	0.998513i	$0.517362\pi$
$788$	0	0
$789$	0.599182	0.0213314
$790$	0	0
$791$	52.4928	1.86643
$792$	0	0
$793$	−33.9432	−1.20536
$794$	0	0
$795$	2.71275	0.0962115
$796$	0	0
$797$	29.5429	1.04646	0.523232	−	0.852191i	$-0.324726\pi$
$797$	29.5429	1.04646	0.523232	+	0.852191i	$0.324726\pi$
$798$	0	0
$799$	−32.5672	−1.15215
$800$	0	0
$801$	−15.2020	−0.537138
$802$	0	0
$803$	−16.8641	−0.595123
$804$	0	0
$805$	28.4958	1.00434
$806$	0	0
$807$	−3.65550	−0.128680
$808$	0	0
$809$	18.4865	0.649950	0.324975	−	0.945723i	$-0.394644\pi$
$809$	18.4865	0.649950	0.324975	+	0.945723i	$0.394644\pi$
$810$	0	0
$811$	−14.3391	−0.503516	−0.251758	−	0.967790i	$-0.581009\pi$
$811$	−14.3391	−0.503516	−0.251758	+	0.967790i	$0.581009\pi$
$812$	0	0
$813$	2.22479	0.0780269
$814$	0	0
$815$	25.4649	0.891996
$816$	0	0
$817$	−38.8432	−1.35895
$818$	0	0
$819$	−23.5370	−0.822448
$820$	0	0
$821$	−7.24580	−0.252880	−0.126440	−	0.991974i	$-0.540355\pi$
$821$	−7.24580	−0.252880	−0.126440	+	0.991974i	$0.540355\pi$
$822$	0	0
$823$	−25.0662	−0.873751	−0.436876	−	0.899522i	$-0.643915\pi$
$823$	−25.0662	−0.873751	−0.436876	+	0.899522i	$0.643915\pi$
$824$	0	0
$825$	4.28779	0.149282
$826$	0	0
$827$	23.7084	0.824423	0.412211	−	0.911088i	$-0.364757\pi$
$827$	23.7084	0.824423	0.412211	+	0.911088i	$0.364757\pi$
$828$	0	0
$829$	4.17291	0.144931	0.0724656	−	0.997371i	$-0.476913\pi$
$829$	4.17291	0.144931	0.0724656	+	0.997371i	$0.476913\pi$
$830$	0	0
$831$	−6.13615	−0.212861
$832$	0	0
$833$	11.5726	0.400966
$834$	0	0
$835$	15.3336	0.530642
$836$	0	0
$837$	7.85909	0.271650
$838$	0	0
$839$	−36.3469	−1.25483	−0.627417	−	0.778684i	$-0.715888\pi$
$839$	−36.3469	−1.25483	−0.627417	+	0.778684i	$0.715888\pi$
$840$	0	0
$841$	47.8091	1.64859
$842$	0	0
$843$	4.15014	0.142938
$844$	0	0
$845$	6.44460	0.221701
$846$	0	0
$847$	−30.2011	−1.03772
$848$	0	0
$849$	−0.271294	−0.00931078
$850$	0	0
$851$	44.6665	1.53115
$852$	0	0
$853$	−2.60157	−0.0890759	−0.0445380	−	0.999008i	$-0.514182\pi$
$853$	−2.60157	−0.0890759	−0.0445380	+	0.999008i	$0.514182\pi$
$854$	0	0
$855$	−12.7967	−0.437637
$856$	0	0
$857$	2.25317	0.0769670	0.0384835	−	0.999259i	$-0.487747\pi$
$857$	2.25317	0.0769670	0.0384835	+	0.999259i	$0.487747\pi$
$858$	0	0
$859$	17.7809	0.606678	0.303339	−	0.952883i	$-0.401899\pi$
$859$	17.7809	0.606678	0.303339	+	0.952883i	$0.401899\pi$
$860$	0	0
$861$	−5.30358	−0.180746
$862$	0	0
$863$	−49.5856	−1.68791	−0.843957	−	0.536411i	$-0.819780\pi$
$863$	−49.5856	−1.68791	−0.843957	+	0.536411i	$0.819780\pi$
$864$	0	0
$865$	7.71624	0.262360
$866$	0	0
$867$	−7.04863	−0.239384
$868$	0	0
$869$	−26.6484	−0.903985
$870$	0	0
$871$	−2.19912	−0.0745144
$872$	0	0
$873$	−2.00590	−0.0678894
$874$	0	0
$875$	−29.3565	−0.992432
$876$	0	0
$877$	36.8074	1.24290	0.621448	−	0.783455i	$-0.286545\pi$
$877$	36.8074	1.24290	0.621448	+	0.783455i	$0.286545\pi$
$878$	0	0
$879$	−2.22783	−0.0751429
$880$	0	0
$881$	−20.5873	−0.693604	−0.346802	−	0.937938i	$-0.612732\pi$
$881$	−20.5873	−0.693604	−0.346802	+	0.937938i	$0.612732\pi$
$882$	0	0
$883$	42.5771	1.43283	0.716417	−	0.697673i	$-0.245781\pi$
$883$	42.5771	1.43283	0.716417	+	0.697673i	$0.245781\pi$
$884$	0	0
$885$	−3.61375	−0.121475
$886$	0	0
$887$	3.91496	0.131452	0.0657258	−	0.997838i	$-0.479064\pi$
$887$	3.91496	0.131452	0.0657258	+	0.997838i	$0.479064\pi$
$888$	0	0
$889$	6.28530	0.210802
$890$	0	0
$891$	38.8477	1.30145
$892$	0	0
$893$	18.5124	0.619495
$894$	0	0
$895$	18.1812	0.607731
$896$	0	0
$897$	5.77038	0.192668
$898$	0	0
$899$	−46.0376	−1.53544
$900$	0	0
$901$	63.1387	2.10346
$902$	0	0
$903$	7.58605	0.252448
$904$	0	0
$905$	23.0484	0.766155
$906$	0	0
$907$	−9.46207	−0.314183	−0.157091	−	0.987584i	$-0.550212\pi$
$907$	−9.46207	−0.314183	−0.157091	+	0.987584i	$0.550212\pi$
$908$	0	0
$909$	−52.1152	−1.72855
$910$	0	0
$911$	−9.48825	−0.314360	−0.157180	−	0.987570i	$-0.550240\pi$
$911$	−9.48825	−0.314360	−0.157180	+	0.987570i	$0.550240\pi$
$912$	0	0
$913$	−12.8102	−0.423957
$914$	0	0
$915$	3.60413	0.119149
$916$	0	0
$917$	6.56602	0.216829
$918$	0	0
$919$	−30.8507	−1.01767	−0.508835	−	0.860864i	$-0.669924\pi$
$919$	−30.8507	−1.01767	−0.508835	+	0.860864i	$0.669924\pi$
$920$	0	0
$921$	0.288785	0.00951579
$922$	0	0
$923$	27.1288	0.892956
$924$	0	0
$925$	−19.5487	−0.642758
$926$	0	0
$927$	−32.8177	−1.07788
$928$	0	0
$929$	38.1553	1.25184	0.625918	−	0.779889i	$-0.284725\pi$
$929$	38.1553	1.25184	0.625918	+	0.779889i	$0.284725\pi$
$930$	0	0
$931$	−6.57829	−0.215595
$932$	0	0
$933$	0.247072	0.00808878
$934$	0	0
$935$	−35.3183	−1.15503
$936$	0	0
$937$	−32.8566	−1.07338	−0.536689	−	0.843780i	$-0.680325\pi$
$937$	−32.8566	−1.07338	−0.536689	+	0.843780i	$0.680325\pi$
$938$	0	0
$939$	6.71647	0.219184
$940$	0	0
$941$	31.3849	1.02312	0.511559	−	0.859248i	$-0.329068\pi$
$941$	31.3849	1.02312	0.511559	+	0.859248i	$0.329068\pi$
$942$	0	0
$943$	−60.1144	−1.95759
$944$	0	0
$945$	5.05242	0.164355
$946$	0	0
$947$	2.54658	0.0827527	0.0413763	−	0.999144i	$-0.486826\pi$
$947$	2.54658	0.0827527	0.0413763	+	0.999144i	$0.486826\pi$
$948$	0	0
$949$	9.93280	0.322432
$950$	0	0
$951$	−1.22924	−0.0398609
$952$	0	0
$953$	−37.8956	−1.22756	−0.613780	−	0.789477i	$-0.710352\pi$
$953$	−37.8956	−1.22756	−0.613780	+	0.789477i	$0.710352\pi$
$954$	0	0
$955$	6.24349	0.202035
$956$	0	0
$957$	10.1755	0.328928
$958$	0	0
$959$	−11.8410	−0.382364
$960$	0	0
$961$	−3.40614	−0.109875
$962$	0	0
$963$	14.3475	0.462343
$964$	0	0
$965$	15.9310	0.512838
$966$	0	0
$967$	7.21263	0.231943	0.115971	−	0.993253i	$-0.463002\pi$
$967$	7.21263	0.231943	0.115971	+	0.993253i	$0.463002\pi$
$968$	0	0
$969$	6.44210	0.206950
$970$	0	0
$971$	34.8698	1.11903	0.559513	−	0.828821i	$-0.310988\pi$
$971$	34.8698	1.11903	0.559513	+	0.828821i	$0.310988\pi$
$972$	0	0
$973$	53.3611	1.71068
$974$	0	0
$975$	−2.52546	−0.0808796
$976$	0	0
$977$	8.00216	0.256012	0.128006	−	0.991773i	$-0.459142\pi$
$977$	8.00216	0.256012	0.128006	+	0.991773i	$0.459142\pi$
$978$	0	0
$979$	23.8500	0.762248
$980$	0	0
$981$	23.8703	0.762122
$982$	0	0
$983$	−26.1736	−0.834808	−0.417404	−	0.908721i	$-0.637060\pi$
$983$	−26.1736	−0.834808	−0.417404	+	0.908721i	$0.637060\pi$
$984$	0	0
$985$	23.0501	0.734436
$986$	0	0
$987$	−3.61547	−0.115082
$988$	0	0
$989$	85.9854	2.73418
$990$	0	0
$991$	−39.1644	−1.24410	−0.622050	−	0.782978i	$-0.713700\pi$
$991$	−39.1644	−1.24410	−0.622050	+	0.782978i	$0.713700\pi$
$992$	0	0
$993$	−0.200963	−0.00637736
$994$	0	0
$995$	−15.3793	−0.487557
$996$	0	0
$997$	55.9094	1.77067	0.885334	−	0.464955i	$-0.153929\pi$
$997$	55.9094	1.77067	0.885334	+	0.464955i	$0.153929\pi$
$998$	0	0
$999$	7.91956	0.250564

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.24	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.24	✓	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)

\(f(q)\)	\(=\)	\(q-0.252021 q^{3} -1.14323 q^{5} -2.95394 q^{7} -2.93649 q^{9} +O(q^{10})\)	\(q-0.252021 q^{3} -1.14323 q^{5} -2.95394 q^{7} -2.93649 q^{9} +4.60695 q^{11} -2.71345 q^{13} +0.288117 q^{15} +6.70585 q^{17} -3.81186 q^{19} +0.744454 q^{21} +8.43814 q^{23} -3.69304 q^{25} +1.49612 q^{27} -8.76408 q^{29} +5.25299 q^{31} -1.16105 q^{33} +3.37702 q^{35} +5.29340 q^{37} +0.683845 q^{39} -7.12412 q^{41} +10.1901 q^{43} +3.35707 q^{45} -4.85654 q^{47} +1.72574 q^{49} -1.69001 q^{51} +9.41547 q^{53} -5.26679 q^{55} +0.960668 q^{57} -12.5427 q^{59} +12.5093 q^{61} +8.67419 q^{63} +3.10208 q^{65} +0.810453 q^{67} -2.12659 q^{69} -9.99792 q^{71} -3.66059 q^{73} +0.930722 q^{75} -13.6086 q^{77} -5.78439 q^{79} +8.43240 q^{81} -2.78063 q^{83} -7.66630 q^{85} +2.20873 q^{87} +5.17695 q^{89} +8.01535 q^{91} -1.32386 q^{93} +4.35782 q^{95} +0.683095 q^{97} -13.5282 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 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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