LMFDB - Embedded newform 6008.2.a.c.1.13

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			1.13
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)

$f(q)$	$=$	$q-1.63270 q^{3} +2.45098 q^{5} -0.216771 q^{7} -0.334292 q^{9} +O(q^{10})$	$q-1.63270 q^{3} +2.45098 q^{5} -0.216771 q^{7} -0.334292 q^{9} -0.722449 q^{11} +1.74800 q^{13} -4.00172 q^{15} -6.52699 q^{17} +3.06353 q^{19} +0.353923 q^{21} +2.33600 q^{23} +1.00731 q^{25} +5.44390 q^{27} -7.05907 q^{29} +4.04386 q^{31} +1.17954 q^{33} -0.531303 q^{35} +0.672407 q^{37} -2.85397 q^{39} +3.91174 q^{41} +8.40292 q^{43} -0.819343 q^{45} -12.0246 q^{47} -6.95301 q^{49} +10.6566 q^{51} +13.2471 q^{53} -1.77071 q^{55} -5.00182 q^{57} -8.35956 q^{59} -12.0828 q^{61} +0.0724649 q^{63} +4.28433 q^{65} +1.88237 q^{67} -3.81398 q^{69} +6.94202 q^{71} -7.10868 q^{73} -1.64464 q^{75} +0.156606 q^{77} -12.0345 q^{79} -7.88537 q^{81} +0.977452 q^{83} -15.9975 q^{85} +11.5253 q^{87} +0.649985 q^{89} -0.378917 q^{91} -6.60241 q^{93} +7.50865 q^{95} -12.7156 q^{97} +0.241509 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9	$ 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9} + 11 q^{11} - 36 q^{13} - 5 q^{15} - 10 q^{17} - 7 q^{19} - 42 q^{21} - 5 q^{23} + 29 q^{25} - 16 q^{27} - 57 q^{29} - 21 q^{31} - 32 q^{33} + 17 q^{35} - 52 q^{37} + 8 q^{39} - 16 q^{41} - 9 q^{43} - 84 q^{45} - q^{47} + 28 q^{49} - q^{51} - 52 q^{53} - 39 q^{55} - 15 q^{57} + 7 q^{59} - 85 q^{61} - 25 q^{63} - 9 q^{65} - 36 q^{67} - 72 q^{69} + 12 q^{71} - 60 q^{73} - 5 q^{75} - 81 q^{77} - 13 q^{79} + 20 q^{81} + 5 q^{83} - 72 q^{85} + 9 q^{87} - 37 q^{89} - 23 q^{91} - 60 q^{93} + 24 q^{95} - 79 q^{97} + 42 q^{99}+O(q^{100}) $ 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9 + 11 * q^11 - 36 * q^13 - 5 * q^15 - 10 * q^17 - 7 * q^19 - 42 * q^21 - 5 * q^23 + 29 * q^25 - 16 * q^27 - 57 * q^29 - 21 * q^31 - 32 * q^33 + 17 * q^35 - 52 * q^37 + 8 * q^39 - 16 * q^41 - 9 * q^43 - 84 * q^45 - q^47 + 28 * q^49 - q^51 - 52 * q^53 - 39 * q^55 - 15 * q^57 + 7 * q^59 - 85 * q^61 - 25 * q^63 - 9 * q^65 - 36 * q^67 - 72 * q^69 + 12 * q^71 - 60 * q^73 - 5 * q^75 - 81 * q^77 - 13 * q^79 + 20 * q^81 + 5 * q^83 - 72 * q^85 + 9 * q^87 - 37 * q^89 - 23 * q^91 - 60 * q^93 + 24 * q^95 - 79 * q^97 + 42 * 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
$3$	−1.63270	−0.942640	−0.471320	−	0.881962i	$-0.656222\pi$
$3$	−1.63270	−0.942640	−0.471320	+	0.881962i	$0.656222\pi$
$4$	0	0
$5$	2.45098	1.09611	0.548056	−	0.836442i	$-0.315368\pi$
$5$	2.45098	1.09611	0.548056	+	0.836442i	$0.315368\pi$
$6$	0	0
$7$	−0.216771	−0.0819319	−0.0409659	−	0.999161i	$-0.513044\pi$
$7$	−0.216771	−0.0819319	−0.0409659	+	0.999161i	$0.513044\pi$
$8$	0	0
$9$	−0.334292	−0.111431
$10$	0	0
$11$	−0.722449	−0.217826	−0.108913	−	0.994051i	$-0.534737\pi$
$11$	−0.722449	−0.217826	−0.108913	+	0.994051i	$0.534737\pi$
$12$	0	0
$13$	1.74800	0.484809	0.242405	−	0.970175i	$-0.422064\pi$
$13$	1.74800	0.484809	0.242405	+	0.970175i	$0.422064\pi$
$14$	0	0
$15$	−4.00172	−1.03324
$16$	0	0
$17$	−6.52699	−1.58303	−0.791513	−	0.611152i	$-0.790706\pi$
$17$	−6.52699	−1.58303	−0.791513	+	0.611152i	$0.790706\pi$
$18$	0	0
$19$	3.06353	0.702821	0.351411	−	0.936221i	$-0.385702\pi$
$19$	3.06353	0.702821	0.351411	+	0.936221i	$0.385702\pi$
$20$	0	0
$21$	0.353923	0.0772322
$22$	0	0
$23$	2.33600	0.487089	0.243544	−	0.969890i	$-0.421690\pi$
$23$	2.33600	0.487089	0.243544	+	0.969890i	$0.421690\pi$
$24$	0	0
$25$	1.00731	0.201462
$26$	0	0
$27$	5.44390	1.04768
$28$	0	0
$29$	−7.05907	−1.31084	−0.655418	−	0.755266i	$-0.727508\pi$
$29$	−7.05907	−1.31084	−0.655418	+	0.755266i	$0.727508\pi$
$30$	0	0
$31$	4.04386	0.726299	0.363149	−	0.931731i	$-0.381701\pi$
$31$	4.04386	0.726299	0.363149	+	0.931731i	$0.381701\pi$
$32$	0	0
$33$	1.17954	0.205332
$34$	0	0
$35$	−0.531303	−0.0898065
$36$	0	0
$37$	0.672407	0.110543	0.0552715	−	0.998471i	$-0.482398\pi$
$37$	0.672407	0.110543	0.0552715	+	0.998471i	$0.482398\pi$
$38$	0	0
$39$	−2.85397	−0.457000
$40$	0	0
$41$	3.91174	0.610911	0.305456	−	0.952206i	$-0.401191\pi$
$41$	3.91174	0.610911	0.305456	+	0.952206i	$0.401191\pi$
$42$	0	0
$43$	8.40292	1.28143	0.640716	−	0.767778i	$-0.278637\pi$
$43$	8.40292	1.28143	0.640716	+	0.767778i	$0.278637\pi$
$44$	0	0
$45$	−0.819343	−0.122140
$46$	0	0
$47$	−12.0246	−1.75397	−0.876986	−	0.480515i	$-0.840450\pi$
$47$	−12.0246	−1.75397	−0.876986	+	0.480515i	$0.840450\pi$
$48$	0	0
$49$	−6.95301	−0.993287
$50$	0	0
$51$	10.6566	1.49222
$52$	0	0
$53$	13.2471	1.81962	0.909811	−	0.415022i	$-0.136226\pi$
$53$	13.2471	1.81962	0.909811	+	0.415022i	$0.136226\pi$
$54$	0	0
$55$	−1.77071	−0.238762
$56$	0	0
$57$	−5.00182	−0.662507
$58$	0	0
$59$	−8.35956	−1.08832	−0.544161	−	0.838981i	$-0.683152\pi$
$59$	−8.35956	−1.08832	−0.544161	+	0.838981i	$0.683152\pi$
$60$	0	0
$61$	−12.0828	−1.54704	−0.773522	−	0.633769i	$-0.781507\pi$
$61$	−12.0828	−1.54704	−0.773522	+	0.633769i	$0.781507\pi$
$62$	0	0
$63$	0.0724649	0.00912972
$64$	0	0
$65$	4.28433	0.531405
$66$	0	0
$67$	1.88237	0.229968	0.114984	−	0.993367i	$-0.463318\pi$
$67$	1.88237	0.229968	0.114984	+	0.993367i	$0.463318\pi$
$68$	0	0
$69$	−3.81398	−0.459149
$70$	0	0
$71$	6.94202	0.823866	0.411933	−	0.911214i	$-0.364854\pi$
$71$	6.94202	0.823866	0.411933	+	0.911214i	$0.364854\pi$
$72$	0	0
$73$	−7.10868	−0.832008	−0.416004	−	0.909363i	$-0.636570\pi$
$73$	−7.10868	−0.832008	−0.416004	+	0.909363i	$0.636570\pi$
$74$	0	0
$75$	−1.64464	−0.189906
$76$	0	0
$77$	0.156606	0.0178469
$78$	0	0
$79$	−12.0345	−1.35398	−0.676992	−	0.735991i	$-0.736717\pi$
$79$	−12.0345	−1.35398	−0.676992	+	0.735991i	$0.736717\pi$
$80$	0	0
$81$	−7.88537	−0.876153
$82$	0	0
$83$	0.977452	0.107289	0.0536447	−	0.998560i	$-0.482916\pi$
$83$	0.977452	0.107289	0.0536447	+	0.998560i	$0.482916\pi$
$84$	0	0
$85$	−15.9975	−1.73518
$86$	0	0
$87$	11.5253	1.23565
$88$	0	0
$89$	0.649985	0.0688983	0.0344491	−	0.999406i	$-0.489032\pi$
$89$	0.649985	0.0688983	0.0344491	+	0.999406i	$0.489032\pi$
$90$	0	0
$91$	−0.378917	−0.0397213
$92$	0	0
$93$	−6.60241	−0.684638
$94$	0	0
$95$	7.50865	0.770371
$96$	0	0
$97$	−12.7156	−1.29107	−0.645535	−	0.763730i	$-0.723366\pi$
$97$	−12.7156	−1.29107	−0.645535	+	0.763730i	$0.723366\pi$
$98$	0	0
$99$	0.241509	0.0242725
$100$	0	0
$101$	9.99278	0.994318	0.497159	−	0.867659i	$-0.334376\pi$
$101$	9.99278	0.994318	0.497159	+	0.867659i	$0.334376\pi$
$102$	0	0
$103$	0.603426	0.0594573	0.0297287	−	0.999558i	$-0.490536\pi$
$103$	0.603426	0.0594573	0.0297287	+	0.999558i	$0.490536\pi$
$104$	0	0
$105$	0.867458	0.0846552
$106$	0	0
$107$	13.8596	1.33986	0.669929	−	0.742425i	$-0.266325\pi$
$107$	13.8596	1.33986	0.669929	+	0.742425i	$0.266325\pi$
$108$	0	0
$109$	−13.5180	−1.29479	−0.647397	−	0.762153i	$-0.724142\pi$
$109$	−13.5180	−1.29479	−0.647397	+	0.762153i	$0.724142\pi$
$110$	0	0
$111$	−1.09784	−0.104202
$112$	0	0
$113$	−5.06976	−0.476923	−0.238461	−	0.971152i	$-0.576643\pi$
$113$	−5.06976	−0.476923	−0.238461	+	0.971152i	$0.576643\pi$
$114$	0	0
$115$	5.72548	0.533904
$116$	0	0
$117$	−0.584344	−0.0540226
$118$	0	0
$119$	1.41486	0.129700
$120$	0	0
$121$	−10.4781	−0.952552
$122$	0	0
$123$	−6.38670	−0.575869
$124$	0	0
$125$	−9.78601	−0.875287
$126$	0	0
$127$	3.37240	0.299252	0.149626	−	0.988743i	$-0.452193\pi$
$127$	3.37240	0.299252	0.149626	+	0.988743i	$0.452193\pi$
$128$	0	0
$129$	−13.7194	−1.20793
$130$	0	0
$131$	10.7250	0.937049	0.468524	−	0.883451i	$-0.344786\pi$
$131$	10.7250	0.937049	0.468524	+	0.883451i	$0.344786\pi$
$132$	0	0
$133$	−0.664085	−0.0575835
$134$	0	0
$135$	13.3429	1.14837
$136$	0	0
$137$	5.64269	0.482087	0.241044	−	0.970514i	$-0.422510\pi$
$137$	5.64269	0.482087	0.241044	+	0.970514i	$0.422510\pi$
$138$	0	0
$139$	−12.6517	−1.07310	−0.536552	−	0.843867i	$-0.680273\pi$
$139$	−12.6517	−1.07310	−0.536552	+	0.843867i	$0.680273\pi$
$140$	0	0
$141$	19.6326	1.65336
$142$	0	0
$143$	−1.26284	−0.105604
$144$	0	0
$145$	−17.3016	−1.43682
$146$	0	0
$147$	11.3522	0.936312
$148$	0	0
$149$	0.442952	0.0362880	0.0181440	−	0.999835i	$-0.494224\pi$
$149$	0.442952	0.0362880	0.0181440	+	0.999835i	$0.494224\pi$
$150$	0	0
$151$	16.8401	1.37043	0.685214	−	0.728341i	$-0.259708\pi$
$151$	16.8401	1.37043	0.685214	+	0.728341i	$0.259708\pi$
$152$	0	0
$153$	2.18192	0.176398
$154$	0	0
$155$	9.91143	0.796105
$156$	0	0
$157$	0.746409	0.0595699	0.0297850	−	0.999556i	$-0.490518\pi$
$157$	0.746409	0.0595699	0.0297850	+	0.999556i	$0.490518\pi$
$158$	0	0
$159$	−21.6285	−1.71525
$160$	0	0
$161$	−0.506377	−0.0399081
$162$	0	0
$163$	−13.4720	−1.05521	−0.527603	−	0.849491i	$-0.676909\pi$
$163$	−13.4720	−1.05521	−0.527603	+	0.849491i	$0.676909\pi$
$164$	0	0
$165$	2.89104	0.225067
$166$	0	0
$167$	−4.95905	−0.383742	−0.191871	−	0.981420i	$-0.561456\pi$
$167$	−4.95905	−0.383742	−0.191871	+	0.981420i	$0.561456\pi$
$168$	0	0
$169$	−9.94448	−0.764960
$170$	0	0
$171$	−1.02411	−0.0783158
$172$	0	0
$173$	−11.6342	−0.884531	−0.442265	−	0.896884i	$-0.645825\pi$
$173$	−11.6342	−0.884531	−0.442265	+	0.896884i	$0.645825\pi$
$174$	0	0
$175$	−0.218356	−0.0165062
$176$	0	0
$177$	13.6486	1.02590
$178$	0	0
$179$	−1.16817	−0.0873130	−0.0436565	−	0.999047i	$-0.513901\pi$
$179$	−1.16817	−0.0873130	−0.0436565	+	0.999047i	$0.513901\pi$
$180$	0	0
$181$	1.63395	0.121450	0.0607251	−	0.998155i	$-0.480659\pi$
$181$	1.63395	0.121450	0.0607251	+	0.998155i	$0.480659\pi$
$182$	0	0
$183$	19.7276	1.45831
$184$	0	0
$185$	1.64806	0.121168
$186$	0	0
$187$	4.71541	0.344825
$188$	0	0
$189$	−1.18008	−0.0858383
$190$	0	0
$191$	−5.21157	−0.377096	−0.188548	−	0.982064i	$-0.560378\pi$
$191$	−5.21157	−0.377096	−0.188548	+	0.982064i	$0.560378\pi$
$192$	0	0
$193$	−2.72307	−0.196011	−0.0980054	−	0.995186i	$-0.531246\pi$
$193$	−2.72307	−0.196011	−0.0980054	+	0.995186i	$0.531246\pi$
$194$	0	0
$195$	−6.99502	−0.500924
$196$	0	0
$197$	11.4955	0.819021	0.409511	−	0.912305i	$-0.365699\pi$
$197$	11.4955	0.819021	0.409511	+	0.912305i	$0.365699\pi$
$198$	0	0
$199$	−5.79455	−0.410765	−0.205382	−	0.978682i	$-0.565844\pi$
$199$	−5.79455	−0.410765	−0.205382	+	0.978682i	$0.565844\pi$
$200$	0	0
$201$	−3.07335	−0.216777
$202$	0	0
$203$	1.53020	0.107399
$204$	0	0
$205$	9.58760	0.669627
$206$	0	0
$207$	−0.780904	−0.0542766
$208$	0	0
$209$	−2.21324	−0.153093
$210$	0	0
$211$	1.89344	0.130350	0.0651749	−	0.997874i	$-0.479239\pi$
$211$	1.89344	0.130350	0.0651749	+	0.997874i	$0.479239\pi$
$212$	0	0
$213$	−11.3342	−0.776609
$214$	0	0
$215$	20.5954	1.40459
$216$	0	0
$217$	−0.876593	−0.0595070
$218$	0	0
$219$	11.6063	0.784283
$220$	0	0
$221$	−11.4092	−0.767466
$222$	0	0
$223$	0.110645	0.00740936	0.00370468	−	0.999993i	$-0.498821\pi$
$223$	0.110645	0.00740936	0.00370468	+	0.999993i	$0.498821\pi$
$224$	0	0
$225$	−0.336736	−0.0224491
$226$	0	0
$227$	−8.36255	−0.555042	−0.277521	−	0.960720i	$-0.589513\pi$
$227$	−8.36255	−0.555042	−0.277521	+	0.960720i	$0.589513\pi$
$228$	0	0
$229$	21.1121	1.39513	0.697563	−	0.716523i	$-0.254268\pi$
$229$	21.1121	1.39513	0.697563	+	0.716523i	$0.254268\pi$
$230$	0	0
$231$	−0.255691	−0.0168232
$232$	0	0
$233$	−24.0549	−1.57589	−0.787946	−	0.615745i	$-0.788855\pi$
$233$	−24.0549	−1.57589	−0.787946	+	0.615745i	$0.788855\pi$
$234$	0	0
$235$	−29.4721	−1.92255
$236$	0	0
$237$	19.6487	1.27632
$238$	0	0
$239$	10.8969	0.704862	0.352431	−	0.935838i	$-0.385355\pi$
$239$	10.8969	0.704862	0.352431	+	0.935838i	$0.385355\pi$
$240$	0	0
$241$	7.52561	0.484767	0.242384	−	0.970180i	$-0.422071\pi$
$241$	7.52561	0.484767	0.242384	+	0.970180i	$0.422071\pi$
$242$	0	0
$243$	−3.45725	−0.221782
$244$	0	0
$245$	−17.0417	−1.08875
$246$	0	0
$247$	5.35506	0.340734
$248$	0	0
$249$	−1.59589	−0.101135
$250$	0	0
$251$	−7.08174	−0.446996	−0.223498	−	0.974704i	$-0.571748\pi$
$251$	−7.08174	−0.446996	−0.223498	+	0.974704i	$0.571748\pi$
$252$	0	0
$253$	−1.68764	−0.106101
$254$	0	0
$255$	26.1192	1.63564
$256$	0	0
$257$	−11.9031	−0.742496	−0.371248	−	0.928534i	$-0.621070\pi$
$257$	−11.9031	−0.742496	−0.371248	+	0.928534i	$0.621070\pi$
$258$	0	0
$259$	−0.145759	−0.00905700
$260$	0	0
$261$	2.35979	0.146067
$262$	0	0
$263$	−10.2708	−0.633326	−0.316663	−	0.948538i	$-0.602562\pi$
$263$	−10.2708	−0.633326	−0.316663	+	0.948538i	$0.602562\pi$
$264$	0	0
$265$	32.4683	1.99451
$266$	0	0
$267$	−1.06123	−0.0649463
$268$	0	0
$269$	16.6077	1.01259	0.506295	−	0.862360i	$-0.331015\pi$
$269$	16.6077	1.01259	0.506295	+	0.862360i	$0.331015\pi$
$270$	0	0
$271$	−3.09259	−0.187862	−0.0939308	−	0.995579i	$-0.529943\pi$
$271$	−3.09259	−0.187862	−0.0939308	+	0.995579i	$0.529943\pi$
$272$	0	0
$273$	0.618658	0.0374429
$274$	0	0
$275$	−0.727731	−0.0438838
$276$	0	0
$277$	−3.74912	−0.225263	−0.112632	−	0.993637i	$-0.535928\pi$
$277$	−3.74912	−0.225263	−0.112632	+	0.993637i	$0.535928\pi$
$278$	0	0
$279$	−1.35183	−0.0809319
$280$	0	0
$281$	1.14713	0.0684322	0.0342161	−	0.999414i	$-0.489107\pi$
$281$	1.14713	0.0684322	0.0342161	+	0.999414i	$0.489107\pi$
$282$	0	0
$283$	−9.29719	−0.552660	−0.276330	−	0.961063i	$-0.589118\pi$
$283$	−9.29719	−0.552660	−0.276330	+	0.961063i	$0.589118\pi$
$284$	0	0
$285$	−12.2594	−0.726182
$286$	0	0
$287$	−0.847953	−0.0500531
$288$	0	0
$289$	25.6015	1.50597
$290$	0	0
$291$	20.7607	1.21701
$292$	0	0
$293$	−27.6804	−1.61711	−0.808553	−	0.588424i	$-0.799749\pi$
$293$	−27.6804	−1.61711	−0.808553	+	0.588424i	$0.799749\pi$
$294$	0	0
$295$	−20.4891	−1.19292
$296$	0	0
$297$	−3.93294	−0.228212
$298$	0	0
$299$	4.08333	0.236145
$300$	0	0
$301$	−1.82151	−0.104990
$302$	0	0
$303$	−16.3152	−0.937284
$304$	0	0
$305$	−29.6147	−1.69574
$306$	0	0
$307$	−31.0143	−1.77008	−0.885039	−	0.465517i	$-0.845868\pi$
$307$	−31.0143	−1.77008	−0.885039	+	0.465517i	$0.845868\pi$
$308$	0	0
$309$	−0.985214	−0.0560468
$310$	0	0
$311$	−3.46185	−0.196303	−0.0981516	−	0.995171i	$-0.531293\pi$
$311$	−3.46185	−0.196303	−0.0981516	+	0.995171i	$0.531293\pi$
$312$	0	0
$313$	−26.4342	−1.49415	−0.747074	−	0.664741i	$-0.768542\pi$
$313$	−26.4342	−1.49415	−0.747074	+	0.664741i	$0.768542\pi$
$314$	0	0
$315$	0.177610	0.0100072
$316$	0	0
$317$	−0.931881	−0.0523396	−0.0261698	−	0.999658i	$-0.508331\pi$
$317$	−0.931881	−0.0523396	−0.0261698	+	0.999658i	$0.508331\pi$
$318$	0	0
$319$	5.09981	0.285535
$320$	0	0
$321$	−22.6286	−1.26300
$322$	0	0
$323$	−19.9956	−1.11258
$324$	0	0
$325$	1.76079	0.0976708
$326$	0	0
$327$	22.0709	1.22052
$328$	0	0
$329$	2.60660	0.143706
$330$	0	0
$331$	7.82835	0.430285	0.215142	−	0.976583i	$-0.430978\pi$
$331$	7.82835	0.430285	0.215142	+	0.976583i	$0.430978\pi$
$332$	0	0
$333$	−0.224780	−0.0123179
$334$	0	0
$335$	4.61366	0.252071
$336$	0	0
$337$	−6.23331	−0.339550	−0.169775	−	0.985483i	$-0.554304\pi$
$337$	−6.23331	−0.339550	−0.169775	+	0.985483i	$0.554304\pi$
$338$	0	0
$339$	8.27740	0.449566
$340$	0	0
$341$	−2.92148	−0.158207
$342$	0	0
$343$	3.02461	0.163314
$344$	0	0
$345$	−9.34799	−0.503279
$346$	0	0
$347$	−33.0875	−1.77623	−0.888115	−	0.459622i	$-0.847985\pi$
$347$	−33.0875	−1.77623	−0.888115	+	0.459622i	$0.847985\pi$
$348$	0	0
$349$	13.2394	0.708690	0.354345	−	0.935115i	$-0.384704\pi$
$349$	13.2394	0.708690	0.354345	+	0.935115i	$0.384704\pi$
$350$	0	0
$351$	9.51595	0.507924
$352$	0	0
$353$	24.7848	1.31916	0.659582	−	0.751633i	$-0.270733\pi$
$353$	24.7848	1.31916	0.659582	+	0.751633i	$0.270733\pi$
$354$	0	0
$355$	17.0148	0.903050
$356$	0	0
$357$	−2.31005	−0.122261
$358$	0	0
$359$	−1.80295	−0.0951560	−0.0475780	−	0.998868i	$-0.515150\pi$
$359$	−1.80295	−0.0951560	−0.0475780	+	0.998868i	$0.515150\pi$
$360$	0	0
$361$	−9.61481	−0.506042
$362$	0	0
$363$	17.1075	0.897913
$364$	0	0
$365$	−17.4232	−0.911974
$366$	0	0
$367$	18.8016	0.981434	0.490717	−	0.871319i	$-0.336735\pi$
$367$	18.8016	0.981434	0.490717	+	0.871319i	$0.336735\pi$
$368$	0	0
$369$	−1.30766	−0.0680742
$370$	0	0
$371$	−2.87158	−0.149085
$372$	0	0
$373$	20.1253	1.04205	0.521024	−	0.853542i	$-0.325550\pi$
$373$	20.1253	1.04205	0.521024	+	0.853542i	$0.325550\pi$
$374$	0	0
$375$	15.9776	0.825080
$376$	0	0
$377$	−12.3393	−0.635505
$378$	0	0
$379$	−22.7595	−1.16908	−0.584539	−	0.811365i	$-0.698725\pi$
$379$	−22.7595	−1.16908	−0.584539	+	0.811365i	$0.698725\pi$
$380$	0	0
$381$	−5.50611	−0.282087
$382$	0	0
$383$	−14.4688	−0.739319	−0.369659	−	0.929167i	$-0.620526\pi$
$383$	−14.4688	−0.739319	−0.369659	+	0.929167i	$0.620526\pi$
$384$	0	0
$385$	0.383839	0.0195622
$386$	0	0
$387$	−2.80903	−0.142791
$388$	0	0
$389$	−30.8828	−1.56582	−0.782910	−	0.622136i	$-0.786265\pi$
$389$	−30.8828	−1.56582	−0.782910	+	0.622136i	$0.786265\pi$
$390$	0	0
$391$	−15.2470	−0.771074
$392$	0	0
$393$	−17.5107	−0.883299
$394$	0	0
$395$	−29.4963	−1.48412
$396$	0	0
$397$	−20.3597	−1.02183	−0.510913	−	0.859632i	$-0.670693\pi$
$397$	−20.3597	−1.02183	−0.510913	+	0.859632i	$0.670693\pi$
$398$	0	0
$399$	1.08425	0.0542804
$400$	0	0
$401$	−8.89366	−0.444128	−0.222064	−	0.975032i	$-0.571279\pi$
$401$	−8.89366	−0.444128	−0.222064	+	0.975032i	$0.571279\pi$
$402$	0	0
$403$	7.06868	0.352116
$404$	0	0
$405$	−19.3269	−0.960362
$406$	0	0
$407$	−0.485780	−0.0240792
$408$	0	0
$409$	14.2994	0.707062	0.353531	−	0.935423i	$-0.384981\pi$
$409$	14.2994	0.707062	0.353531	+	0.935423i	$0.384981\pi$
$410$	0	0
$411$	−9.21281	−0.454434
$412$	0	0
$413$	1.81211	0.0891682
$414$	0	0
$415$	2.39572	0.117601
$416$	0	0
$417$	20.6564	1.01155
$418$	0	0
$419$	−5.36618	−0.262155	−0.131078	−	0.991372i	$-0.541844\pi$
$419$	−5.36618	−0.262155	−0.131078	+	0.991372i	$0.541844\pi$
$420$	0	0
$421$	5.40879	0.263608	0.131804	−	0.991276i	$-0.457923\pi$
$421$	5.40879	0.263608	0.131804	+	0.991276i	$0.457923\pi$
$422$	0	0
$423$	4.01974	0.195446
$424$	0	0
$425$	−6.57471	−0.318920
$426$	0	0
$427$	2.61921	0.126752
$428$	0	0
$429$	2.06184	0.0995468
$430$	0	0
$431$	−14.8058	−0.713171	−0.356586	−	0.934263i	$-0.616059\pi$
$431$	−14.8058	−0.713171	−0.356586	+	0.934263i	$0.616059\pi$
$432$	0	0
$433$	−16.4306	−0.789602	−0.394801	−	0.918767i	$-0.629187\pi$
$433$	−16.4306	−0.789602	−0.394801	+	0.918767i	$0.629187\pi$
$434$	0	0
$435$	28.2484	1.35441
$436$	0	0
$437$	7.15638	0.342336
$438$	0	0
$439$	−19.2426	−0.918401	−0.459201	−	0.888333i	$-0.651864\pi$
$439$	−19.2426	−0.918401	−0.459201	+	0.888333i	$0.651864\pi$
$440$	0	0
$441$	2.32433	0.110683
$442$	0	0
$443$	33.9368	1.61238	0.806192	−	0.591654i	$-0.201525\pi$
$443$	33.9368	1.61238	0.806192	+	0.591654i	$0.201525\pi$
$444$	0	0
$445$	1.59310	0.0755203
$446$	0	0
$447$	−0.723208	−0.0342065
$448$	0	0
$449$	−26.7481	−1.26232	−0.631161	−	0.775652i	$-0.717421\pi$
$449$	−26.7481	−1.26232	−0.631161	+	0.775652i	$0.717421\pi$
$450$	0	0
$451$	−2.82603	−0.133073
$452$	0	0
$453$	−27.4948	−1.29182
$454$	0	0
$455$	−0.928719	−0.0435390
$456$	0	0
$457$	−36.0791	−1.68771	−0.843855	−	0.536571i	$-0.819719\pi$
$457$	−36.0791	−1.68771	−0.843855	+	0.536571i	$0.819719\pi$
$458$	0	0
$459$	−35.5322	−1.65850
$460$	0	0
$461$	19.3001	0.898893	0.449447	−	0.893307i	$-0.351621\pi$
$461$	19.3001	0.898893	0.449447	+	0.893307i	$0.351621\pi$
$462$	0	0
$463$	−21.9353	−1.01942	−0.509710	−	0.860346i	$-0.670247\pi$
$463$	−21.9353	−1.01942	−0.509710	+	0.860346i	$0.670247\pi$
$464$	0	0
$465$	−16.1824	−0.750440
$466$	0	0
$467$	−12.1837	−0.563792	−0.281896	−	0.959445i	$-0.590963\pi$
$467$	−12.1837	−0.563792	−0.281896	+	0.959445i	$0.590963\pi$
$468$	0	0
$469$	−0.408044	−0.0188417
$470$	0	0
$471$	−1.21866	−0.0561530
$472$	0	0
$473$	−6.07068	−0.279130
$474$	0	0
$475$	3.08593	0.141592
$476$	0	0
$477$	−4.42838	−0.202762
$478$	0	0
$479$	13.5541	0.619304	0.309652	−	0.950850i	$-0.399787\pi$
$479$	13.5541	0.619304	0.309652	+	0.950850i	$0.399787\pi$
$480$	0	0
$481$	1.17537	0.0535923
$482$	0	0
$483$	0.826762	0.0376190
$484$	0	0
$485$	−31.1656	−1.41516
$486$	0	0
$487$	16.7308	0.758144	0.379072	−	0.925367i	$-0.376243\pi$
$487$	16.7308	0.758144	0.379072	+	0.925367i	$0.376243\pi$
$488$	0	0
$489$	21.9957	0.994678
$490$	0	0
$491$	42.8942	1.93579	0.967894	−	0.251358i	$-0.0808773\pi$
$491$	42.8942	1.93579	0.967894	+	0.251358i	$0.0808773\pi$
$492$	0	0
$493$	46.0744	2.07509
$494$	0	0
$495$	0.591934	0.0266054
$496$	0	0
$497$	−1.50483	−0.0675009
$498$	0	0
$499$	−27.2406	−1.21946	−0.609729	−	0.792610i	$-0.708722\pi$
$499$	−27.2406	−1.21946	−0.609729	+	0.792610i	$0.708722\pi$
$500$	0	0
$501$	8.09663	0.361731
$502$	0	0
$503$	−11.8683	−0.529182	−0.264591	−	0.964361i	$-0.585237\pi$
$503$	−11.8683	−0.529182	−0.264591	+	0.964361i	$0.585237\pi$
$504$	0	0
$505$	24.4921	1.08988
$506$	0	0
$507$	16.2364	0.721082
$508$	0	0
$509$	−12.2059	−0.541017	−0.270509	−	0.962718i	$-0.587192\pi$
$509$	−12.2059	−0.541017	−0.270509	+	0.962718i	$0.587192\pi$
$510$	0	0
$511$	1.54096	0.0681679
$512$	0	0
$513$	16.6775	0.736331
$514$	0	0
$515$	1.47899	0.0651719
$516$	0	0
$517$	8.68718	0.382062
$518$	0	0
$519$	18.9951	0.833794
$520$	0	0
$521$	23.8313	1.04407	0.522034	−	0.852925i	$-0.325174\pi$
$521$	23.8313	1.04407	0.522034	+	0.852925i	$0.325174\pi$
$522$	0	0
$523$	29.2004	1.27685	0.638423	−	0.769686i	$-0.279587\pi$
$523$	29.2004	1.27685	0.638423	+	0.769686i	$0.279587\pi$
$524$	0	0
$525$	0.356510	0.0155594
$526$	0	0
$527$	−26.3942	−1.14975
$528$	0	0
$529$	−17.5431	−0.762745
$530$	0	0
$531$	2.79453	0.121272
$532$	0	0
$533$	6.83774	0.296175
$534$	0	0
$535$	33.9696	1.46863
$536$	0	0
$537$	1.90727	0.0823047
$538$	0	0
$539$	5.02319	0.216364
$540$	0	0
$541$	−41.1650	−1.76982	−0.884910	−	0.465761i	$-0.845781\pi$
$541$	−41.1650	−1.76982	−0.884910	+	0.465761i	$0.845781\pi$
$542$	0	0
$543$	−2.66774	−0.114484
$544$	0	0
$545$	−33.1325	−1.41924
$546$	0	0
$547$	−45.2808	−1.93607	−0.968034	−	0.250819i	$-0.919300\pi$
$547$	−45.2808	−1.93607	−0.968034	+	0.250819i	$0.919300\pi$
$548$	0	0
$549$	4.03918	0.172388
$550$	0	0
$551$	−21.6256	−0.921283
$552$	0	0
$553$	2.60873	0.110934
$554$	0	0
$555$	−2.69078	−0.114217
$556$	0	0
$557$	−7.07523	−0.299787	−0.149893	−	0.988702i	$-0.547893\pi$
$557$	−7.07523	−0.299787	−0.149893	+	0.988702i	$0.547893\pi$
$558$	0	0
$559$	14.6883	0.621250
$560$	0	0
$561$	−7.69885	−0.325046
$562$	0	0
$563$	30.6808	1.29304	0.646521	−	0.762896i	$-0.276223\pi$
$563$	30.6808	1.29304	0.646521	+	0.762896i	$0.276223\pi$
$564$	0	0
$565$	−12.4259	−0.522761
$566$	0	0
$567$	1.70932	0.0717848
$568$	0	0
$569$	30.2766	1.26926	0.634632	−	0.772815i	$-0.281152\pi$
$569$	30.2766	1.26926	0.634632	+	0.772815i	$0.281152\pi$
$570$	0	0
$571$	2.63372	0.110218	0.0551088	−	0.998480i	$-0.482449\pi$
$571$	2.63372	0.110218	0.0551088	+	0.998480i	$0.482449\pi$
$572$	0	0
$573$	8.50893	0.355465
$574$	0	0
$575$	2.35308	0.0981301
$576$	0	0
$577$	19.3538	0.805710	0.402855	−	0.915264i	$-0.368018\pi$
$577$	19.3538	0.805710	0.402855	+	0.915264i	$0.368018\pi$
$578$	0	0
$579$	4.44596	0.184768
$580$	0	0
$581$	−0.211884	−0.00879042
$582$	0	0
$583$	−9.57032	−0.396362
$584$	0	0
$585$	−1.43222	−0.0592148
$586$	0	0
$587$	−27.0409	−1.11610	−0.558049	−	0.829808i	$-0.688450\pi$
$587$	−27.0409	−1.11610	−0.558049	+	0.829808i	$0.688450\pi$
$588$	0	0
$589$	12.3885	0.510458
$590$	0	0
$591$	−18.7687	−0.772042
$592$	0	0
$593$	−43.3177	−1.77885	−0.889423	−	0.457084i	$-0.848894\pi$
$593$	−43.3177	−1.77885	−0.889423	+	0.457084i	$0.848894\pi$
$594$	0	0
$595$	3.46781	0.142166
$596$	0	0
$597$	9.46076	0.387203
$598$	0	0
$599$	9.61337	0.392792	0.196396	−	0.980525i	$-0.437076\pi$
$599$	9.61337	0.392792	0.196396	+	0.980525i	$0.437076\pi$
$600$	0	0
$601$	17.7859	0.725503	0.362752	−	0.931886i	$-0.381837\pi$
$601$	17.7859	0.725503	0.362752	+	0.931886i	$0.381837\pi$
$602$	0	0
$603$	−0.629261	−0.0256255
$604$	0	0
$605$	−25.6816	−1.04410
$606$	0	0
$607$	5.73656	0.232840	0.116420	−	0.993200i	$-0.462858\pi$
$607$	5.73656	0.232840	0.116420	+	0.993200i	$0.462858\pi$
$608$	0	0
$609$	−2.49836	−0.101239
$610$	0	0
$611$	−21.0191	−0.850342
$612$	0	0
$613$	6.95489	0.280905	0.140453	−	0.990087i	$-0.455144\pi$
$613$	6.95489	0.280905	0.140453	+	0.990087i	$0.455144\pi$
$614$	0	0
$615$	−15.6537	−0.631217
$616$	0	0
$617$	14.5321	0.585041	0.292521	−	0.956259i	$-0.405506\pi$
$617$	14.5321	0.585041	0.292521	+	0.956259i	$0.405506\pi$
$618$	0	0
$619$	−35.2203	−1.41562	−0.707812	−	0.706400i	$-0.750318\pi$
$619$	−35.2203	−1.41562	−0.707812	+	0.706400i	$0.750318\pi$
$620$	0	0
$621$	12.7169	0.510312
$622$	0	0
$623$	−0.140898	−0.00564497
$624$	0	0
$625$	−29.0219	−1.16088
$626$	0	0
$627$	3.61356	0.144312
$628$	0	0
$629$	−4.38879	−0.174993
$630$	0	0
$631$	0.384100	0.0152908	0.00764540	−	0.999971i	$-0.497566\pi$
$631$	0.384100	0.0152908	0.00764540	+	0.999971i	$0.497566\pi$
$632$	0	0
$633$	−3.09142	−0.122873
$634$	0	0
$635$	8.26568	0.328014
$636$	0	0
$637$	−12.1539	−0.481555
$638$	0	0
$639$	−2.32066	−0.0918039
$640$	0	0
$641$	16.6923	0.659307	0.329654	−	0.944102i	$-0.393068\pi$
$641$	16.6923	0.659307	0.329654	+	0.944102i	$0.393068\pi$
$642$	0	0
$643$	−25.2636	−0.996301	−0.498150	−	0.867091i	$-0.665987\pi$
$643$	−25.2636	−0.996301	−0.498150	+	0.867091i	$0.665987\pi$
$644$	0	0
$645$	−33.6261	−1.32403
$646$	0	0
$647$	27.2406	1.07094	0.535470	−	0.844554i	$-0.320135\pi$
$647$	27.2406	1.07094	0.535470	+	0.844554i	$0.320135\pi$
$648$	0	0
$649$	6.03935	0.237065
$650$	0	0
$651$	1.43121	0.0560937
$652$	0	0
$653$	−5.11824	−0.200292	−0.100146	−	0.994973i	$-0.531931\pi$
$653$	−5.11824	−0.200292	−0.100146	+	0.994973i	$0.531931\pi$
$654$	0	0
$655$	26.2868	1.02711
$656$	0	0
$657$	2.37637	0.0927111
$658$	0	0
$659$	21.2093	0.826195	0.413098	−	0.910687i	$-0.364447\pi$
$659$	21.2093	0.826195	0.413098	+	0.910687i	$0.364447\pi$
$660$	0	0
$661$	30.6518	1.19222	0.596109	−	0.802903i	$-0.296712\pi$
$661$	30.6518	1.19222	0.596109	+	0.802903i	$0.296712\pi$
$662$	0	0
$663$	18.6278	0.723444
$664$	0	0
$665$	−1.62766	−0.0631179
$666$	0	0
$667$	−16.4900	−0.638494
$668$	0	0
$669$	−0.180651	−0.00698436
$670$	0	0
$671$	8.72921	0.336987
$672$	0	0
$673$	10.6681	0.411226	0.205613	−	0.978633i	$-0.434081\pi$
$673$	10.6681	0.411226	0.205613	+	0.978633i	$0.434081\pi$
$674$	0	0
$675$	5.48370	0.211068
$676$	0	0
$677$	−8.32641	−0.320010	−0.160005	−	0.987116i	$-0.551151\pi$
$677$	−8.32641	−0.320010	−0.160005	+	0.987116i	$0.551151\pi$
$678$	0	0
$679$	2.75637	0.105780
$680$	0	0
$681$	13.6535	0.523205
$682$	0	0
$683$	8.91915	0.341282	0.170641	−	0.985333i	$-0.445416\pi$
$683$	8.91915	0.341282	0.170641	+	0.985333i	$0.445416\pi$
$684$	0	0
$685$	13.8301	0.528422
$686$	0	0
$687$	−34.4697	−1.31510
$688$	0	0
$689$	23.1559	0.882170
$690$	0	0
$691$	−39.8769	−1.51699	−0.758495	−	0.651679i	$-0.774065\pi$
$691$	−39.8769	−1.51699	−0.758495	+	0.651679i	$0.774065\pi$
$692$	0	0
$693$	−0.0523522	−0.00198869
$694$	0	0
$695$	−31.0091	−1.17624
$696$	0	0
$697$	−25.5319	−0.967089
$698$	0	0
$699$	39.2745	1.48550
$700$	0	0
$701$	16.2094	0.612220	0.306110	−	0.951996i	$-0.400972\pi$
$701$	16.2094	0.612220	0.306110	+	0.951996i	$0.400972\pi$
$702$	0	0
$703$	2.05994	0.0776920
$704$	0	0
$705$	48.1192	1.81227
$706$	0	0
$707$	−2.16615	−0.0814664
$708$	0	0
$709$	−21.9447	−0.824149	−0.412074	−	0.911150i	$-0.635196\pi$
$709$	−21.9447	−0.824149	−0.412074	+	0.911150i	$0.635196\pi$
$710$	0	0
$711$	4.02302	0.150875
$712$	0	0
$713$	9.44644	0.353772
$714$	0	0
$715$	−3.09521	−0.115754
$716$	0	0
$717$	−17.7914	−0.664430
$718$	0	0
$719$	14.3711	0.535952	0.267976	−	0.963426i	$-0.413645\pi$
$719$	14.3711	0.535952	0.267976	+	0.963426i	$0.413645\pi$
$720$	0	0
$721$	−0.130805	−0.00487145
$722$	0	0
$723$	−12.2871	−0.456961
$724$	0	0
$725$	−7.11068	−0.264084
$726$	0	0
$727$	−47.9070	−1.77677	−0.888387	−	0.459095i	$-0.848174\pi$
$727$	−47.9070	−1.77677	−0.888387	+	0.459095i	$0.848174\pi$
$728$	0	0
$729$	29.3008	1.08521
$730$	0	0
$731$	−54.8457	−2.02854
$732$	0	0
$733$	−51.9621	−1.91926	−0.959631	−	0.281261i	$-0.909247\pi$
$733$	−51.9621	−1.91926	−0.959631	+	0.281261i	$0.909247\pi$
$734$	0	0
$735$	27.8240	1.02630
$736$	0	0
$737$	−1.35992	−0.0500932
$738$	0	0
$739$	−39.5752	−1.45580	−0.727898	−	0.685685i	$-0.759503\pi$
$739$	−39.5752	−1.45580	−0.727898	+	0.685685i	$0.759503\pi$
$740$	0	0
$741$	−8.74320	−0.321189
$742$	0	0
$743$	18.9308	0.694504	0.347252	−	0.937772i	$-0.387115\pi$
$743$	18.9308	0.694504	0.347252	+	0.937772i	$0.387115\pi$
$744$	0	0
$745$	1.08567	0.0397758
$746$	0	0
$747$	−0.326754	−0.0119553
$748$	0	0
$749$	−3.00436	−0.109777
$750$	0	0
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	0	0
$753$	11.5624	0.421356
$754$	0	0
$755$	41.2748	1.50214
$756$	0	0
$757$	4.40445	0.160082	0.0800412	−	0.996792i	$-0.474495\pi$
$757$	4.40445	0.160082	0.0800412	+	0.996792i	$0.474495\pi$
$758$	0	0
$759$	2.75540	0.100015
$760$	0	0
$761$	−28.9490	−1.04940	−0.524701	−	0.851287i	$-0.675823\pi$
$761$	−28.9490	−1.04940	−0.524701	+	0.851287i	$0.675823\pi$
$762$	0	0
$763$	2.93032	0.106085
$764$	0	0
$765$	5.34784	0.193352
$766$	0	0
$767$	−14.6125	−0.527628
$768$	0	0
$769$	8.11891	0.292776	0.146388	−	0.989227i	$-0.453235\pi$
$769$	8.11891	0.292776	0.146388	+	0.989227i	$0.453235\pi$
$770$	0	0
$771$	19.4342	0.699906
$772$	0	0
$773$	20.6159	0.741502	0.370751	−	0.928732i	$-0.379100\pi$
$773$	20.6159	0.741502	0.370751	+	0.928732i	$0.379100\pi$
$774$	0	0
$775$	4.07343	0.146322
$776$	0	0
$777$	0.237980	0.00853748
$778$	0	0
$779$	11.9837	0.429361
$780$	0	0
$781$	−5.01525	−0.179460
$782$	0	0
$783$	−38.4288	−1.37333
$784$	0	0
$785$	1.82944	0.0652953
$786$	0	0
$787$	52.0811	1.85649	0.928245	−	0.371970i	$-0.121318\pi$
$787$	52.0811	1.85649	0.928245	+	0.371970i	$0.121318\pi$
$788$	0	0
$789$	16.7692	0.596998
$790$	0	0
$791$	1.09898	0.0390752
$792$	0	0
$793$	−21.1208	−0.750021
$794$	0	0
$795$	−53.0110	−1.88010
$796$	0	0
$797$	−35.5260	−1.25839	−0.629197	−	0.777246i	$-0.716616\pi$
$797$	−35.5260	−1.25839	−0.629197	+	0.777246i	$0.716616\pi$
$798$	0	0
$799$	78.4846	2.77659
$800$	0	0
$801$	−0.217285	−0.00767738
$802$	0	0
$803$	5.13565	0.181233
$804$	0	0
$805$	−1.24112	−0.0437438
$806$	0	0
$807$	−27.1154	−0.954507
$808$	0	0
$809$	17.7961	0.625677	0.312838	−	0.949806i	$-0.398720\pi$
$809$	17.7961	0.625677	0.312838	+	0.949806i	$0.398720\pi$
$810$	0	0
$811$	31.9819	1.12304	0.561519	−	0.827464i	$-0.310217\pi$
$811$	31.9819	1.12304	0.561519	+	0.827464i	$0.310217\pi$
$812$	0	0
$813$	5.04928	0.177086
$814$	0	0
$815$	−33.0195	−1.15662
$816$	0	0
$817$	25.7426	0.900618
$818$	0	0
$819$	0.126669	0.00442617
$820$	0	0
$821$	−35.6748	−1.24506	−0.622529	−	0.782597i	$-0.713895\pi$
$821$	−35.6748	−1.24506	−0.622529	+	0.782597i	$0.713895\pi$
$822$	0	0
$823$	10.0246	0.349437	0.174718	−	0.984618i	$-0.444099\pi$
$823$	10.0246	0.349437	0.174718	+	0.984618i	$0.444099\pi$
$824$	0	0
$825$	1.18817	0.0413666
$826$	0	0
$827$	−47.9793	−1.66840	−0.834201	−	0.551460i	$-0.814071\pi$
$827$	−47.9793	−1.66840	−0.834201	+	0.551460i	$0.814071\pi$
$828$	0	0
$829$	−15.1850	−0.527396	−0.263698	−	0.964605i	$-0.584942\pi$
$829$	−15.1850	−0.527396	−0.263698	+	0.964605i	$0.584942\pi$
$830$	0	0
$831$	6.12119	0.212342
$832$	0	0
$833$	45.3822	1.57240
$834$	0	0
$835$	−12.1545	−0.420625
$836$	0	0
$837$	22.0144	0.760928
$838$	0	0
$839$	29.6370	1.02318	0.511591	−	0.859229i	$-0.329056\pi$
$839$	29.6370	1.02318	0.511591	+	0.859229i	$0.329056\pi$
$840$	0	0
$841$	20.8304	0.718291
$842$	0	0
$843$	−1.87292	−0.0645069
$844$	0	0
$845$	−24.3737	−0.838482
$846$	0	0
$847$	2.27135	0.0780443
$848$	0	0
$849$	15.1795	0.520960
$850$	0	0
$851$	1.57074	0.0538443
$852$	0	0
$853$	−39.3001	−1.34561	−0.672805	−	0.739820i	$-0.734911\pi$
$853$	−39.3001	−1.34561	−0.672805	+	0.739820i	$0.734911\pi$
$854$	0	0
$855$	−2.51008	−0.0858429
$856$	0	0
$857$	10.6417	0.363514	0.181757	−	0.983343i	$-0.441822\pi$
$857$	10.6417	0.363514	0.181757	+	0.983343i	$0.441822\pi$
$858$	0	0
$859$	15.7384	0.536989	0.268494	−	0.963281i	$-0.413474\pi$
$859$	15.7384	0.536989	0.268494	+	0.963281i	$0.413474\pi$
$860$	0	0
$861$	1.38445	0.0471820
$862$	0	0
$863$	−23.3684	−0.795470	−0.397735	−	0.917500i	$-0.630204\pi$
$863$	−23.3684	−0.795470	−0.397735	+	0.917500i	$0.630204\pi$
$864$	0	0
$865$	−28.5152	−0.969545
$866$	0	0
$867$	−41.7996	−1.41959
$868$	0	0
$869$	8.69429	0.294933
$870$	0	0
$871$	3.29039	0.111491
$872$	0	0
$873$	4.25071	0.143865
$874$	0	0
$875$	2.12133	0.0717139
$876$	0	0
$877$	31.8808	1.07654	0.538270	−	0.842773i	$-0.319078\pi$
$877$	31.8808	1.07654	0.538270	+	0.842773i	$0.319078\pi$
$878$	0	0
$879$	45.1938	1.52435
$880$	0	0
$881$	30.3238	1.02163	0.510817	−	0.859689i	$-0.329343\pi$
$881$	30.3238	1.02163	0.510817	+	0.859689i	$0.329343\pi$
$882$	0	0
$883$	−4.01819	−0.135223	−0.0676114	−	0.997712i	$-0.521538\pi$
$883$	−4.01819	−0.135223	−0.0676114	+	0.997712i	$0.521538\pi$
$884$	0	0
$885$	33.4526	1.12450
$886$	0	0
$887$	8.36563	0.280890	0.140445	−	0.990088i	$-0.455147\pi$
$887$	8.36563	0.280890	0.140445	+	0.990088i	$0.455147\pi$
$888$	0	0
$889$	−0.731039	−0.0245183
$890$	0	0
$891$	5.69678	0.190849
$892$	0	0
$893$	−36.8378	−1.23273
$894$	0	0
$895$	−2.86316	−0.0957049
$896$	0	0
$897$	−6.66685	−0.222600
$898$	0	0
$899$	−28.5459	−0.952059
$900$	0	0
$901$	−86.4633	−2.88051
$902$	0	0
$903$	2.97398	0.0989679
$904$	0	0
$905$	4.00477	0.133123
$906$	0	0
$907$	11.1342	0.369706	0.184853	−	0.982766i	$-0.440819\pi$
$907$	11.1342	0.369706	0.184853	+	0.982766i	$0.440819\pi$
$908$	0	0
$909$	−3.34050	−0.110798
$910$	0	0
$911$	−9.36090	−0.310141	−0.155070	−	0.987903i	$-0.549560\pi$
$911$	−9.36090	−0.310141	−0.155070	+	0.987903i	$0.549560\pi$
$912$	0	0
$913$	−0.706159	−0.0233705
$914$	0	0
$915$	48.3520	1.59847
$916$	0	0
$917$	−2.32488	−0.0767742
$918$	0	0
$919$	26.0683	0.859915	0.429958	−	0.902849i	$-0.358528\pi$
$919$	26.0683	0.859915	0.429958	+	0.902849i	$0.358528\pi$
$920$	0	0
$921$	50.6370	1.66855
$922$	0	0
$923$	12.1347	0.399418
$924$	0	0
$925$	0.677323	0.0222703
$926$	0	0
$927$	−0.201720	−0.00662537
$928$	0	0
$929$	39.6291	1.30019	0.650095	−	0.759853i	$-0.274729\pi$
$929$	39.6291	1.30019	0.650095	+	0.759853i	$0.274729\pi$
$930$	0	0
$931$	−21.3007	−0.698103
$932$	0	0
$933$	5.65215	0.185043
$934$	0	0
$935$	11.5574	0.377967
$936$	0	0
$937$	4.64247	0.151663	0.0758315	−	0.997121i	$-0.475839\pi$
$937$	4.64247	0.151663	0.0758315	+	0.997121i	$0.475839\pi$
$938$	0	0
$939$	43.1591	1.40844
$940$	0	0
$941$	12.4373	0.405444	0.202722	−	0.979236i	$-0.435021\pi$
$941$	12.4373	0.405444	0.202722	+	0.979236i	$0.435021\pi$
$942$	0	0
$943$	9.13781	0.297568
$944$	0	0
$945$	−2.89236	−0.0940884
$946$	0	0
$947$	−42.6020	−1.38438	−0.692190	−	0.721716i	$-0.743354\pi$
$947$	−42.6020	−1.38438	−0.692190	+	0.721716i	$0.743354\pi$
$948$	0	0
$949$	−12.4260	−0.403365
$950$	0	0
$951$	1.52148	0.0493374
$952$	0	0
$953$	19.7829	0.640832	0.320416	−	0.947277i	$-0.396177\pi$
$953$	19.7829	0.640832	0.320416	+	0.947277i	$0.396177\pi$
$954$	0	0
$955$	−12.7735	−0.413339
$956$	0	0
$957$	−8.32647	−0.269156
$958$	0	0
$959$	−1.22317	−0.0394983
$960$	0	0
$961$	−14.6472	−0.472490
$962$	0	0
$963$	−4.63315	−0.149301
$964$	0	0
$965$	−6.67420	−0.214850
$966$	0	0
$967$	55.8191	1.79502	0.897510	−	0.440993i	$-0.145374\pi$
$967$	55.8191	1.79502	0.897510	+	0.440993i	$0.145374\pi$
$968$	0	0
$969$	32.6468	1.04877
$970$	0	0
$971$	−37.5676	−1.20560	−0.602801	−	0.797892i	$-0.705949\pi$
$971$	−37.5676	−1.20560	−0.602801	+	0.797892i	$0.705949\pi$
$972$	0	0
$973$	2.74253	0.0879214
$974$	0	0
$975$	−2.87483	−0.0920684
$976$	0	0
$977$	40.4503	1.29412	0.647060	−	0.762439i	$-0.275998\pi$
$977$	40.4503	1.29412	0.647060	+	0.762439i	$0.275998\pi$
$978$	0	0
$979$	−0.469581	−0.0150079
$980$	0	0
$981$	4.51897	0.144280
$982$	0	0
$983$	42.9552	1.37006	0.685029	−	0.728516i	$-0.259790\pi$
$983$	42.9552	1.37006	0.685029	+	0.728516i	$0.259790\pi$
$984$	0	0
$985$	28.1753	0.897739
$986$	0	0
$987$	−4.25579	−0.135463
$988$	0	0
$989$	19.6292	0.624172
$990$	0	0
$991$	53.4814	1.69889	0.849447	−	0.527674i	$-0.176936\pi$
$991$	53.4814	1.69889	0.849447	+	0.527674i	$0.176936\pi$
$992$	0	0
$993$	−12.7813	−0.405603
$994$	0	0
$995$	−14.2023	−0.450244
$996$	0	0
$997$	8.16721	0.258658	0.129329	−	0.991602i	$-0.458718\pi$
$997$	8.16721	0.258658	0.129329	+	0.991602i	$0.458718\pi$
$998$	0	0
$999$	3.66051	0.115814

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.c.1.13	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.c.1.13	✓	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)

\(f(q)\)	\(=\)	\(q-1.63270 q^{3} +2.45098 q^{5} -0.216771 q^{7} -0.334292 q^{9} +O(q^{10})\)	\(q-1.63270 q^{3} +2.45098 q^{5} -0.216771 q^{7} -0.334292 q^{9} -0.722449 q^{11} +1.74800 q^{13} -4.00172 q^{15} -6.52699 q^{17} +3.06353 q^{19} +0.353923 q^{21} +2.33600 q^{23} +1.00731 q^{25} +5.44390 q^{27} -7.05907 q^{29} +4.04386 q^{31} +1.17954 q^{33} -0.531303 q^{35} +0.672407 q^{37} -2.85397 q^{39} +3.91174 q^{41} +8.40292 q^{43} -0.819343 q^{45} -12.0246 q^{47} -6.95301 q^{49} +10.6566 q^{51} +13.2471 q^{53} -1.77071 q^{55} -5.00182 q^{57} -8.35956 q^{59} -12.0828 q^{61} +0.0724649 q^{63} +4.28433 q^{65} +1.88237 q^{67} -3.81398 q^{69} +6.94202 q^{71} -7.10868 q^{73} -1.64464 q^{75} +0.156606 q^{77} -12.0345 q^{79} -7.88537 q^{81} +0.977452 q^{83} -15.9975 q^{85} +11.5253 q^{87} +0.649985 q^{89} -0.378917 q^{91} -6.60241 q^{93} +7.50865 q^{95} -12.7156 q^{97} +0.241509 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9	\( 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9} + 11 q^{11} - 36 q^{13} - 5 q^{15} - 10 q^{17} - 7 q^{19} - 42 q^{21} - 5 q^{23} + 29 q^{25} - 16 q^{27} - 57 q^{29} - 21 q^{31} - 32 q^{33} + 17 q^{35} - 52 q^{37} + 8 q^{39} - 16 q^{41} - 9 q^{43} - 84 q^{45} - q^{47} + 28 q^{49} - q^{51} - 52 q^{53} - 39 q^{55} - 15 q^{57} + 7 q^{59} - 85 q^{61} - 25 q^{63} - 9 q^{65} - 36 q^{67} - 72 q^{69} + 12 q^{71} - 60 q^{73} - 5 q^{75} - 81 q^{77} - 13 q^{79} + 20 q^{81} + 5 q^{83} - 72 q^{85} + 9 q^{87} - 37 q^{89} - 23 q^{91} - 60 q^{93} + 24 q^{95} - 79 q^{97} + 42 q^{99}+O(q^{100}) \) 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9 + 11 * q^11 - 36 * q^13 - 5 * q^15 - 10 * q^17 - 7 * q^19 - 42 * q^21 - 5 * q^23 + 29 * q^25 - 16 * q^27 - 57 * q^29 - 21 * q^31 - 32 * q^33 + 17 * q^35 - 52 * q^37 + 8 * q^39 - 16 * q^41 - 9 * q^43 - 84 * q^45 - q^47 + 28 * q^49 - q^51 - 52 * q^53 - 39 * q^55 - 15 * q^57 + 7 * q^59 - 85 * q^61 - 25 * q^63 - 9 * q^65 - 36 * q^67 - 72 * q^69 + 12 * q^71 - 60 * q^73 - 5 * q^75 - 81 * q^77 - 13 * q^79 + 20 * q^81 + 5 * q^83 - 72 * q^85 + 9 * q^87 - 37 * q^89 - 23 * q^91 - 60 * q^93 + 24 * q^95 - 79 * q^97 + 42 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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