LMFDB - Embedded newform 6008.2.a.b.1.35

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			1.35
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)

$f(q)$	$=$	$q+1.65156 q^{3} +2.02555 q^{5} -1.52092 q^{7} -0.272343 q^{9} +O(q^{10})$	$q+1.65156 q^{3} +2.02555 q^{5} -1.52092 q^{7} -0.272343 q^{9} -3.27838 q^{11} +0.0809580 q^{13} +3.34532 q^{15} +5.20822 q^{17} -4.81722 q^{19} -2.51190 q^{21} -0.265548 q^{23} -0.897146 q^{25} -5.40448 q^{27} -7.75756 q^{29} -1.01023 q^{31} -5.41445 q^{33} -3.08070 q^{35} +8.64694 q^{37} +0.133707 q^{39} +6.42334 q^{41} -7.71123 q^{43} -0.551644 q^{45} -6.87795 q^{47} -4.68680 q^{49} +8.60170 q^{51} -4.57105 q^{53} -6.64052 q^{55} -7.95594 q^{57} +3.22742 q^{59} -3.60539 q^{61} +0.414212 q^{63} +0.163985 q^{65} +5.12386 q^{67} -0.438569 q^{69} -8.35316 q^{71} +1.23424 q^{73} -1.48169 q^{75} +4.98616 q^{77} -12.3509 q^{79} -8.10880 q^{81} -4.33943 q^{83} +10.5495 q^{85} -12.8121 q^{87} +6.77024 q^{89} -0.123131 q^{91} -1.66846 q^{93} -9.75753 q^{95} -5.92247 q^{97} +0.892843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.65156	0.953530	0.476765	−	0.879031i	$-0.341809\pi$
$3$	1.65156	0.953530	0.476765	+	0.879031i	$0.341809\pi$
$4$	0	0
$5$	2.02555	0.905854	0.452927	−	0.891548i	$-0.350380\pi$
$5$	2.02555	0.905854	0.452927	+	0.891548i	$0.350380\pi$
$6$	0	0
$7$	−1.52092	−0.574855	−0.287427	−	0.957802i	$-0.592800\pi$
$7$	−1.52092	−0.574855	−0.287427	+	0.957802i	$0.592800\pi$
$8$	0	0
$9$	−0.272343	−0.0907809
$10$	0	0
$11$	−3.27838	−0.988469	−0.494234	−	0.869329i	$-0.664552\pi$
$11$	−3.27838	−0.988469	−0.494234	+	0.869329i	$0.664552\pi$
$12$	0	0
$13$	0.0809580	0.0224537	0.0112269	−	0.999937i	$-0.496426\pi$
$13$	0.0809580	0.0224537	0.0112269	+	0.999937i	$0.496426\pi$
$14$	0	0
$15$	3.34532	0.863758
$16$	0	0
$17$	5.20822	1.26318	0.631590	−	0.775303i	$-0.282403\pi$
$17$	5.20822	1.26318	0.631590	+	0.775303i	$0.282403\pi$
$18$	0	0
$19$	−4.81722	−1.10515	−0.552573	−	0.833464i	$-0.686354\pi$
$19$	−4.81722	−1.10515	−0.552573	+	0.833464i	$0.686354\pi$
$20$	0	0
$21$	−2.51190	−0.548141
$22$	0	0
$23$	−0.265548	−0.0553706	−0.0276853	−	0.999617i	$-0.508814\pi$
$23$	−0.265548	−0.0553706	−0.0276853	+	0.999617i	$0.508814\pi$
$24$	0	0
$25$	−0.897146	−0.179429
$26$	0	0
$27$	−5.40448	−1.04009
$28$	0	0
$29$	−7.75756	−1.44054	−0.720272	−	0.693692i	$-0.755983\pi$
$29$	−7.75756	−1.44054	−0.720272	+	0.693692i	$0.755983\pi$
$30$	0	0
$31$	−1.01023	−0.181443	−0.0907215	−	0.995876i	$-0.528917\pi$
$31$	−1.01023	−0.181443	−0.0907215	+	0.995876i	$0.528917\pi$
$32$	0	0
$33$	−5.41445	−0.942535
$34$	0	0
$35$	−3.08070	−0.520734
$36$	0	0
$37$	8.64694	1.42155	0.710774	−	0.703420i	$-0.248345\pi$
$37$	8.64694	1.42155	0.710774	+	0.703420i	$0.248345\pi$
$38$	0	0
$39$	0.133707	0.0214103
$40$	0	0
$41$	6.42334	1.00316	0.501579	−	0.865112i	$-0.332753\pi$
$41$	6.42334	1.00316	0.501579	+	0.865112i	$0.332753\pi$
$42$	0	0
$43$	−7.71123	−1.17595	−0.587976	−	0.808878i	$-0.700075\pi$
$43$	−7.71123	−1.17595	−0.587976	+	0.808878i	$0.700075\pi$
$44$	0	0
$45$	−0.551644	−0.0822342
$46$	0	0
$47$	−6.87795	−1.00325	−0.501626	−	0.865085i	$-0.667265\pi$
$47$	−6.87795	−1.00325	−0.501626	+	0.865085i	$0.667265\pi$
$48$	0	0
$49$	−4.68680	−0.669542
$50$	0	0
$51$	8.60170	1.20448
$52$	0	0
$53$	−4.57105	−0.627883	−0.313941	−	0.949442i	$-0.601650\pi$
$53$	−4.57105	−0.627883	−0.313941	+	0.949442i	$0.601650\pi$
$54$	0	0
$55$	−6.64052	−0.895408
$56$	0	0
$57$	−7.95594	−1.05379
$58$	0	0
$59$	3.22742	0.420175	0.210087	−	0.977683i	$-0.432625\pi$
$59$	3.22742	0.420175	0.210087	+	0.977683i	$0.432625\pi$
$60$	0	0
$61$	−3.60539	−0.461623	−0.230812	−	0.972998i	$-0.574138\pi$
$61$	−3.60539	−0.461623	−0.230812	+	0.972998i	$0.574138\pi$
$62$	0	0
$63$	0.414212	0.0521858
$64$	0	0
$65$	0.163985	0.0203398
$66$	0	0
$67$	5.12386	0.625979	0.312989	−	0.949757i	$-0.398670\pi$
$67$	5.12386	0.625979	0.312989	+	0.949757i	$0.398670\pi$
$68$	0	0
$69$	−0.438569	−0.0527975
$70$	0	0
$71$	−8.35316	−0.991338	−0.495669	−	0.868511i	$-0.665077\pi$
$71$	−8.35316	−0.991338	−0.495669	+	0.868511i	$0.665077\pi$
$72$	0	0
$73$	1.23424	0.144457	0.0722287	−	0.997388i	$-0.476989\pi$
$73$	1.23424	0.144457	0.0722287	+	0.997388i	$0.476989\pi$
$74$	0	0
$75$	−1.48169	−0.171091
$76$	0	0
$77$	4.98616	0.568226
$78$	0	0
$79$	−12.3509	−1.38959	−0.694793	−	0.719209i	$-0.744504\pi$
$79$	−12.3509	−1.38959	−0.694793	+	0.719209i	$0.744504\pi$
$80$	0	0
$81$	−8.10880	−0.900978
$82$	0	0
$83$	−4.33943	−0.476315	−0.238157	−	0.971227i	$-0.576543\pi$
$83$	−4.33943	−0.476315	−0.238157	+	0.971227i	$0.576543\pi$
$84$	0	0
$85$	10.5495	1.14426
$86$	0	0
$87$	−12.8121	−1.37360
$88$	0	0
$89$	6.77024	0.717644	0.358822	−	0.933406i	$-0.383178\pi$
$89$	6.77024	0.717644	0.358822	+	0.933406i	$0.383178\pi$
$90$	0	0
$91$	−0.123131	−0.0129076
$92$	0	0
$93$	−1.66846	−0.173011
$94$	0	0
$95$	−9.75753	−1.00110
$96$	0	0
$97$	−5.92247	−0.601336	−0.300668	−	0.953729i	$-0.597210\pi$
$97$	−5.92247	−0.601336	−0.300668	+	0.953729i	$0.597210\pi$
$98$	0	0
$99$	0.892843	0.0897341
$100$	0	0
$101$	−7.50709	−0.746984	−0.373492	−	0.927633i	$-0.621840\pi$
$101$	−7.50709	−0.746984	−0.373492	+	0.927633i	$0.621840\pi$
$102$	0	0
$103$	8.31877	0.819673	0.409836	−	0.912159i	$-0.365586\pi$
$103$	8.31877	0.819673	0.409836	+	0.912159i	$0.365586\pi$
$104$	0	0
$105$	−5.08797	−0.496535
$106$	0	0
$107$	−1.90024	−0.183703	−0.0918516	−	0.995773i	$-0.529279\pi$
$107$	−1.90024	−0.183703	−0.0918516	+	0.995773i	$0.529279\pi$
$108$	0	0
$109$	15.2695	1.46256	0.731279	−	0.682079i	$-0.238924\pi$
$109$	15.2695	1.46256	0.731279	+	0.682079i	$0.238924\pi$
$110$	0	0
$111$	14.2810	1.35549
$112$	0	0
$113$	2.06287	0.194058	0.0970292	−	0.995282i	$-0.469066\pi$
$113$	2.06287	0.194058	0.0970292	+	0.995282i	$0.469066\pi$
$114$	0	0
$115$	−0.537881	−0.0501576
$116$	0	0
$117$	−0.0220483	−0.00203837
$118$	0	0
$119$	−7.92130	−0.726145
$120$	0	0
$121$	−0.252220	−0.0229291
$122$	0	0
$123$	10.6085	0.956541
$124$	0	0
$125$	−11.9450	−1.06839
$126$	0	0
$127$	−2.46083	−0.218363	−0.109182	−	0.994022i	$-0.534823\pi$
$127$	−2.46083	−0.218363	−0.109182	+	0.994022i	$0.534823\pi$
$128$	0	0
$129$	−12.7356	−1.12131
$130$	0	0
$131$	−0.745435	−0.0651290	−0.0325645	−	0.999470i	$-0.510367\pi$
$131$	−0.745435	−0.0651290	−0.0325645	+	0.999470i	$0.510367\pi$
$132$	0	0
$133$	7.32662	0.635299
$134$	0	0
$135$	−10.9470	−0.942171
$136$	0	0
$137$	−5.48007	−0.468194	−0.234097	−	0.972213i	$-0.575213\pi$
$137$	−5.48007	−0.468194	−0.234097	+	0.972213i	$0.575213\pi$
$138$	0	0
$139$	19.6900	1.67009	0.835044	−	0.550183i	$-0.185442\pi$
$139$	19.6900	1.67009	0.835044	+	0.550183i	$0.185442\pi$
$140$	0	0
$141$	−11.3594	−0.956630
$142$	0	0
$143$	−0.265411	−0.0221948
$144$	0	0
$145$	−15.7133	−1.30492
$146$	0	0
$147$	−7.74053	−0.638428
$148$	0	0
$149$	−7.53611	−0.617382	−0.308691	−	0.951162i	$-0.599891\pi$
$149$	−7.53611	−0.617382	−0.308691	+	0.951162i	$0.599891\pi$
$150$	0	0
$151$	0.725528	0.0590426	0.0295213	−	0.999564i	$-0.490602\pi$
$151$	0.725528	0.0590426	0.0295213	+	0.999564i	$0.490602\pi$
$152$	0	0
$153$	−1.41842	−0.114673
$154$	0	0
$155$	−2.04628	−0.164361
$156$	0	0
$157$	15.4379	1.23208	0.616039	−	0.787716i	$-0.288736\pi$
$157$	15.4379	1.23208	0.616039	+	0.787716i	$0.288736\pi$
$158$	0	0
$159$	−7.54938	−0.598705
$160$	0	0
$161$	0.403878	0.0318300
$162$	0	0
$163$	−2.41843	−0.189426	−0.0947129	−	0.995505i	$-0.530193\pi$
$163$	−2.41843	−0.189426	−0.0947129	+	0.995505i	$0.530193\pi$
$164$	0	0
$165$	−10.9672	−0.853798
$166$	0	0
$167$	−4.10299	−0.317499	−0.158749	−	0.987319i	$-0.550746\pi$
$167$	−4.10299	−0.317499	−0.158749	+	0.987319i	$0.550746\pi$
$168$	0	0
$169$	−12.9934	−0.999496
$170$	0	0
$171$	1.31194	0.100326
$172$	0	0
$173$	11.4448	0.870134	0.435067	−	0.900398i	$-0.356725\pi$
$173$	11.4448	0.870134	0.435067	+	0.900398i	$0.356725\pi$
$174$	0	0
$175$	1.36449	0.103146
$176$	0	0
$177$	5.33029	0.400649
$178$	0	0
$179$	−20.5989	−1.53963	−0.769817	−	0.638265i	$-0.779653\pi$
$179$	−20.5989	−1.53963	−0.769817	+	0.638265i	$0.779653\pi$
$180$	0	0
$181$	−9.01334	−0.669956	−0.334978	−	0.942226i	$-0.608729\pi$
$181$	−9.01334	−0.669956	−0.334978	+	0.942226i	$0.608729\pi$
$182$	0	0
$183$	−5.95453	−0.440171
$184$	0	0
$185$	17.5148	1.28771
$186$	0	0
$187$	−17.0745	−1.24861
$188$	0	0
$189$	8.21979	0.597902
$190$	0	0
$191$	−7.13853	−0.516526	−0.258263	−	0.966075i	$-0.583150\pi$
$191$	−7.13853	−0.516526	−0.258263	+	0.966075i	$0.583150\pi$
$192$	0	0
$193$	−8.12956	−0.585178	−0.292589	−	0.956238i	$-0.594517\pi$
$193$	−8.12956	−0.585178	−0.292589	+	0.956238i	$0.594517\pi$
$194$	0	0
$195$	0.270831	0.0193946
$196$	0	0
$197$	−11.0465	−0.787031	−0.393516	−	0.919318i	$-0.628741\pi$
$197$	−11.0465	−0.787031	−0.393516	+	0.919318i	$0.628741\pi$
$198$	0	0
$199$	−27.8421	−1.97367	−0.986836	−	0.161723i	$-0.948295\pi$
$199$	−27.8421	−1.97367	−0.986836	+	0.161723i	$0.948295\pi$
$200$	0	0
$201$	8.46237	0.596889
$202$	0	0
$203$	11.7986	0.828103
$204$	0	0
$205$	13.0108	0.908714
$206$	0	0
$207$	0.0723201	0.00502659
$208$	0	0
$209$	15.7927	1.09240
$210$	0	0
$211$	3.89496	0.268140	0.134070	−	0.990972i	$-0.457195\pi$
$211$	3.89496	0.268140	0.134070	+	0.990972i	$0.457195\pi$
$212$	0	0
$213$	−13.7958	−0.945270
$214$	0	0
$215$	−15.6195	−1.06524
$216$	0	0
$217$	1.53648	0.104303
$218$	0	0
$219$	2.03843	0.137744
$220$	0	0
$221$	0.421647	0.0283631
$222$	0	0
$223$	−2.27199	−0.152144	−0.0760718	−	0.997102i	$-0.524238\pi$
$223$	−2.27199	−0.152144	−0.0760718	+	0.997102i	$0.524238\pi$
$224$	0	0
$225$	0.244331	0.0162888
$226$	0	0
$227$	−4.95870	−0.329120	−0.164560	−	0.986367i	$-0.552620\pi$
$227$	−4.95870	−0.329120	−0.164560	+	0.986367i	$0.552620\pi$
$228$	0	0
$229$	−9.60292	−0.634579	−0.317289	−	0.948329i	$-0.602773\pi$
$229$	−9.60292	−0.634579	−0.317289	+	0.948329i	$0.602773\pi$
$230$	0	0
$231$	8.23496	0.541820
$232$	0	0
$233$	10.5066	0.688309	0.344155	−	0.938913i	$-0.388166\pi$
$233$	10.5066	0.688309	0.344155	+	0.938913i	$0.388166\pi$
$234$	0	0
$235$	−13.9316	−0.908799
$236$	0	0
$237$	−20.3983	−1.32501
$238$	0	0
$239$	20.8737	1.35021	0.675105	−	0.737722i	$-0.264098\pi$
$239$	20.8737	1.35021	0.675105	+	0.737722i	$0.264098\pi$
$240$	0	0
$241$	−19.8148	−1.27638	−0.638192	−	0.769877i	$-0.720317\pi$
$241$	−19.8148	−1.27638	−0.638192	+	0.769877i	$0.720317\pi$
$242$	0	0
$243$	2.82124	0.180983
$244$	0	0
$245$	−9.49334	−0.606507
$246$	0	0
$247$	−0.389993	−0.0248146
$248$	0	0
$249$	−7.16684	−0.454180
$250$	0	0
$251$	3.33357	0.210413	0.105206	−	0.994450i	$-0.466450\pi$
$251$	3.33357	0.210413	0.105206	+	0.994450i	$0.466450\pi$
$252$	0	0
$253$	0.870567	0.0547321
$254$	0	0
$255$	17.4232	1.09108
$256$	0	0
$257$	−11.7012	−0.729902	−0.364951	−	0.931027i	$-0.618914\pi$
$257$	−11.7012	−0.729902	−0.364951	+	0.931027i	$0.618914\pi$
$258$	0	0
$259$	−13.1513	−0.817183
$260$	0	0
$261$	2.11272	0.130774
$262$	0	0
$263$	9.19127	0.566758	0.283379	−	0.959008i	$-0.408545\pi$
$263$	9.19127	0.566758	0.283379	+	0.959008i	$0.408545\pi$
$264$	0	0
$265$	−9.25890	−0.568770
$266$	0	0
$267$	11.1815	0.684295
$268$	0	0
$269$	−3.63385	−0.221560	−0.110780	−	0.993845i	$-0.535335\pi$
$269$	−3.63385	−0.221560	−0.110780	+	0.993845i	$0.535335\pi$
$270$	0	0
$271$	−0.180685	−0.0109758	−0.00548792	−	0.999985i	$-0.501747\pi$
$271$	−0.180685	−0.0109758	−0.00548792	+	0.999985i	$0.501747\pi$
$272$	0	0
$273$	−0.203358	−0.0123078
$274$	0	0
$275$	2.94119	0.177360
$276$	0	0
$277$	11.5231	0.692358	0.346179	−	0.938168i	$-0.387479\pi$
$277$	11.5231	0.692358	0.346179	+	0.938168i	$0.387479\pi$
$278$	0	0
$279$	0.275130	0.0164716
$280$	0	0
$281$	−12.7358	−0.759757	−0.379878	−	0.925036i	$-0.624034\pi$
$281$	−12.7358	−0.759757	−0.379878	+	0.925036i	$0.624034\pi$
$282$	0	0
$283$	14.5097	0.862513	0.431257	−	0.902229i	$-0.358070\pi$
$283$	14.5097	0.862513	0.431257	+	0.902229i	$0.358070\pi$
$284$	0	0
$285$	−16.1152	−0.954580
$286$	0	0
$287$	−9.76940	−0.576670
$288$	0	0
$289$	10.1256	0.595623
$290$	0	0
$291$	−9.78133	−0.573392
$292$	0	0
$293$	5.15572	0.301200	0.150600	−	0.988595i	$-0.451879\pi$
$293$	5.15572	0.301200	0.150600	+	0.988595i	$0.451879\pi$
$294$	0	0
$295$	6.53731	0.380617
$296$	0	0
$297$	17.7179	1.02810
$298$	0	0
$299$	−0.0214982	−0.00124328
$300$	0	0
$301$	11.7282	0.676001
$302$	0	0
$303$	−12.3984	−0.712271
$304$	0	0
$305$	−7.30290	−0.418163
$306$	0	0
$307$	−6.52933	−0.372648	−0.186324	−	0.982488i	$-0.559657\pi$
$307$	−6.52933	−0.372648	−0.186324	+	0.982488i	$0.559657\pi$
$308$	0	0
$309$	13.7390	0.781583
$310$	0	0
$311$	28.4650	1.61410	0.807052	−	0.590480i	$-0.201062\pi$
$311$	28.4650	1.61410	0.807052	+	0.590480i	$0.201062\pi$
$312$	0	0
$313$	25.3691	1.43395	0.716974	−	0.697100i	$-0.245527\pi$
$313$	25.3691	1.43395	0.716974	+	0.697100i	$0.245527\pi$
$314$	0	0
$315$	0.839008	0.0472727
$316$	0	0
$317$	11.6306	0.653240	0.326620	−	0.945156i	$-0.394090\pi$
$317$	11.6306	0.653240	0.326620	+	0.945156i	$0.394090\pi$
$318$	0	0
$319$	25.4322	1.42393
$320$	0	0
$321$	−3.13836	−0.175166
$322$	0	0
$323$	−25.0892	−1.39600
$324$	0	0
$325$	−0.0726312	−0.00402885
$326$	0	0
$327$	25.2186	1.39459
$328$	0	0
$329$	10.4608	0.576724
$330$	0	0
$331$	24.2703	1.33402	0.667008	−	0.745050i	$-0.267574\pi$
$331$	24.2703	1.33402	0.667008	+	0.745050i	$0.267574\pi$
$332$	0	0
$333$	−2.35493	−0.129049
$334$	0	0
$335$	10.3786	0.567045
$336$	0	0
$337$	−18.5276	−1.00926	−0.504632	−	0.863335i	$-0.668372\pi$
$337$	−18.5276	−1.00926	−0.504632	+	0.863335i	$0.668372\pi$
$338$	0	0
$339$	3.40696	0.185040
$340$	0	0
$341$	3.31193	0.179351
$342$	0	0
$343$	17.7747	0.959744
$344$	0	0
$345$	−0.888343	−0.0478268
$346$	0	0
$347$	12.2744	0.658923	0.329462	−	0.944169i	$-0.393133\pi$
$347$	12.2744	0.658923	0.329462	+	0.944169i	$0.393133\pi$
$348$	0	0
$349$	15.5611	0.832966	0.416483	−	0.909144i	$-0.363263\pi$
$349$	15.5611	0.832966	0.416483	+	0.909144i	$0.363263\pi$
$350$	0	0
$351$	−0.437536	−0.0233539
$352$	0	0
$353$	20.7158	1.10259	0.551296	−	0.834310i	$-0.314133\pi$
$353$	20.7158	1.10259	0.551296	+	0.834310i	$0.314133\pi$
$354$	0	0
$355$	−16.9198	−0.898007
$356$	0	0
$357$	−13.0825	−0.692400
$358$	0	0
$359$	9.42963	0.497677	0.248838	−	0.968545i	$-0.419951\pi$
$359$	9.42963	0.497677	0.248838	+	0.968545i	$0.419951\pi$
$360$	0	0
$361$	4.20564	0.221349
$362$	0	0
$363$	−0.416558	−0.0218636
$364$	0	0
$365$	2.50002	0.130857
$366$	0	0
$367$	23.5343	1.22848	0.614240	−	0.789119i	$-0.289463\pi$
$367$	23.5343	1.22848	0.614240	+	0.789119i	$0.289463\pi$
$368$	0	0
$369$	−1.74935	−0.0910676
$370$	0	0
$371$	6.95222	0.360941
$372$	0	0
$373$	17.7774	0.920478	0.460239	−	0.887795i	$-0.347764\pi$
$373$	17.7774	0.920478	0.460239	+	0.887795i	$0.347764\pi$
$374$	0	0
$375$	−19.7279	−1.01874
$376$	0	0
$377$	−0.628037	−0.0323455
$378$	0	0
$379$	−28.6316	−1.47071	−0.735353	−	0.677685i	$-0.762983\pi$
$379$	−28.6316	−1.47071	−0.735353	+	0.677685i	$0.762983\pi$
$380$	0	0
$381$	−4.06421	−0.208216
$382$	0	0
$383$	14.2635	0.728832	0.364416	−	0.931236i	$-0.381269\pi$
$383$	14.2635	0.728832	0.364416	+	0.931236i	$0.381269\pi$
$384$	0	0
$385$	10.0997	0.514729
$386$	0	0
$387$	2.10010	0.106754
$388$	0	0
$389$	6.81876	0.345725	0.172862	−	0.984946i	$-0.444698\pi$
$389$	6.81876	0.345725	0.172862	+	0.984946i	$0.444698\pi$
$390$	0	0
$391$	−1.38303	−0.0699430
$392$	0	0
$393$	−1.23113	−0.0621024
$394$	0	0
$395$	−25.0174	−1.25876
$396$	0	0
$397$	8.87183	0.445264	0.222632	−	0.974903i	$-0.428535\pi$
$397$	8.87183	0.445264	0.222632	+	0.974903i	$0.428535\pi$
$398$	0	0
$399$	12.1004	0.605776
$400$	0	0
$401$	−10.2486	−0.511792	−0.255896	−	0.966704i	$-0.582370\pi$
$401$	−10.2486	−0.511792	−0.255896	+	0.966704i	$0.582370\pi$
$402$	0	0
$403$	−0.0817864	−0.00407407
$404$	0	0
$405$	−16.4248	−0.816154
$406$	0	0
$407$	−28.3480	−1.40516
$408$	0	0
$409$	8.48827	0.419718	0.209859	−	0.977732i	$-0.432700\pi$
$409$	8.48827	0.419718	0.209859	+	0.977732i	$0.432700\pi$
$410$	0	0
$411$	−9.05067	−0.446437
$412$	0	0
$413$	−4.90866	−0.241539
$414$	0	0
$415$	−8.78974	−0.431471
$416$	0	0
$417$	32.5193	1.59248
$418$	0	0
$419$	−31.8502	−1.55598	−0.777992	−	0.628274i	$-0.783762\pi$
$419$	−31.8502	−1.55598	−0.777992	+	0.628274i	$0.783762\pi$
$420$	0	0
$421$	6.65712	0.324448	0.162224	−	0.986754i	$-0.448133\pi$
$421$	6.65712	0.324448	0.162224	+	0.986754i	$0.448133\pi$
$422$	0	0
$423$	1.87316	0.0910761
$424$	0	0
$425$	−4.67254	−0.226651
$426$	0	0
$427$	5.48352	0.265366
$428$	0	0
$429$	−0.438343	−0.0211634
$430$	0	0
$431$	6.25257	0.301176	0.150588	−	0.988597i	$-0.451883\pi$
$431$	6.25257	0.301176	0.150588	+	0.988597i	$0.451883\pi$
$432$	0	0
$433$	14.5819	0.700761	0.350381	−	0.936607i	$-0.386052\pi$
$433$	14.5819	0.700761	0.350381	+	0.936607i	$0.386052\pi$
$434$	0	0
$435$	−25.9515	−1.24428
$436$	0	0
$437$	1.27920	0.0611926
$438$	0	0
$439$	−13.8347	−0.660293	−0.330147	−	0.943930i	$-0.607098\pi$
$439$	−13.8347	−0.660293	−0.330147	+	0.943930i	$0.607098\pi$
$440$	0	0
$441$	1.27642	0.0607817
$442$	0	0
$443$	−6.68785	−0.317749	−0.158875	−	0.987299i	$-0.550787\pi$
$443$	−6.68785	−0.317749	−0.158875	+	0.987299i	$0.550787\pi$
$444$	0	0
$445$	13.7135	0.650081
$446$	0	0
$447$	−12.4464	−0.588692
$448$	0	0
$449$	−31.8338	−1.50233	−0.751164	−	0.660116i	$-0.770507\pi$
$449$	−31.8338	−1.50233	−0.751164	+	0.660116i	$0.770507\pi$
$450$	0	0
$451$	−21.0582	−0.991590
$452$	0	0
$453$	1.19825	0.0562989
$454$	0	0
$455$	−0.249408	−0.0116924
$456$	0	0
$457$	36.5502	1.70975	0.854874	−	0.518836i	$-0.173634\pi$
$457$	36.5502	1.70975	0.854874	+	0.518836i	$0.173634\pi$
$458$	0	0
$459$	−28.1477	−1.31382
$460$	0	0
$461$	10.6315	0.495159	0.247580	−	0.968868i	$-0.420365\pi$
$461$	10.6315	0.495159	0.247580	+	0.968868i	$0.420365\pi$
$462$	0	0
$463$	13.0731	0.607560	0.303780	−	0.952742i	$-0.401751\pi$
$463$	13.0731	0.607560	0.303780	+	0.952742i	$0.401751\pi$
$464$	0	0
$465$	−3.37955	−0.156723
$466$	0	0
$467$	23.0024	1.06442	0.532212	−	0.846611i	$-0.321361\pi$
$467$	23.0024	1.06442	0.532212	+	0.846611i	$0.321361\pi$
$468$	0	0
$469$	−7.79299	−0.359847
$470$	0	0
$471$	25.4966	1.17482
$472$	0	0
$473$	25.2804	1.16239
$474$	0	0
$475$	4.32175	0.198296
$476$	0	0
$477$	1.24489	0.0569998
$478$	0	0
$479$	−36.5372	−1.66943	−0.834713	−	0.550685i	$-0.814367\pi$
$479$	−36.5372	−1.66943	−0.834713	+	0.550685i	$0.814367\pi$
$480$	0	0
$481$	0.700039	0.0319190
$482$	0	0
$483$	0.667029	0.0303509
$484$	0	0
$485$	−11.9963	−0.544722
$486$	0	0
$487$	6.14639	0.278519	0.139260	−	0.990256i	$-0.455528\pi$
$487$	6.14639	0.278519	0.139260	+	0.990256i	$0.455528\pi$
$488$	0	0
$489$	−3.99418	−0.180623
$490$	0	0
$491$	−21.1480	−0.954396	−0.477198	−	0.878796i	$-0.658348\pi$
$491$	−21.1480	−0.954396	−0.477198	+	0.878796i	$0.658348\pi$
$492$	0	0
$493$	−40.4031	−1.81966
$494$	0	0
$495$	1.80850	0.0812860
$496$	0	0
$497$	12.7045	0.569875
$498$	0	0
$499$	−12.2945	−0.550376	−0.275188	−	0.961390i	$-0.588740\pi$
$499$	−12.2945	−0.550376	−0.275188	+	0.961390i	$0.588740\pi$
$500$	0	0
$501$	−6.77634	−0.302745
$502$	0	0
$503$	2.97430	0.132617	0.0663086	−	0.997799i	$-0.478878\pi$
$503$	2.97430	0.132617	0.0663086	+	0.997799i	$0.478878\pi$
$504$	0	0
$505$	−15.2060	−0.676658
$506$	0	0
$507$	−21.4595	−0.953049
$508$	0	0
$509$	−29.8990	−1.32525	−0.662625	−	0.748952i	$-0.730557\pi$
$509$	−29.8990	−1.32525	−0.662625	+	0.748952i	$0.730557\pi$
$510$	0	0
$511$	−1.87719	−0.0830420
$512$	0	0
$513$	26.0346	1.14945
$514$	0	0
$515$	16.8501	0.742504
$516$	0	0
$517$	22.5485	0.991683
$518$	0	0
$519$	18.9018	0.829699
$520$	0	0
$521$	−10.8845	−0.476861	−0.238430	−	0.971160i	$-0.576633\pi$
$521$	−10.8845	−0.476861	−0.238430	+	0.971160i	$0.576633\pi$
$522$	0	0
$523$	−28.3864	−1.24125	−0.620624	−	0.784108i	$-0.713121\pi$
$523$	−28.3864	−1.24125	−0.620624	+	0.784108i	$0.713121\pi$
$524$	0	0
$525$	2.25354	0.0983525
$526$	0	0
$527$	−5.26151	−0.229195
$528$	0	0
$529$	−22.9295	−0.996934
$530$	0	0
$531$	−0.878966	−0.0381439
$532$	0	0
$533$	0.520021	0.0225246
$534$	0	0
$535$	−3.84903	−0.166408
$536$	0	0
$537$	−34.0204	−1.46809
$538$	0	0
$539$	15.3651	0.661822
$540$	0	0
$541$	4.36222	0.187547	0.0937733	−	0.995594i	$-0.470107\pi$
$541$	4.36222	0.187547	0.0937733	+	0.995594i	$0.470107\pi$
$542$	0	0
$543$	−14.8861	−0.638823
$544$	0	0
$545$	30.9292	1.32486
$546$	0	0
$547$	−33.1092	−1.41565	−0.707823	−	0.706390i	$-0.750323\pi$
$547$	−33.1092	−1.41565	−0.707823	+	0.706390i	$0.750323\pi$
$548$	0	0
$549$	0.981903	0.0419066
$550$	0	0
$551$	37.3699	1.59201
$552$	0	0
$553$	18.7848	0.798810
$554$	0	0
$555$	28.9268	1.22787
$556$	0	0
$557$	2.09450	0.0887466	0.0443733	−	0.999015i	$-0.485871\pi$
$557$	2.09450	0.0887466	0.0443733	+	0.999015i	$0.485871\pi$
$558$	0	0
$559$	−0.624286	−0.0264045
$560$	0	0
$561$	−28.1997	−1.19059
$562$	0	0
$563$	−5.42252	−0.228532	−0.114266	−	0.993450i	$-0.536452\pi$
$563$	−5.42252	−0.228532	−0.114266	+	0.993450i	$0.536452\pi$
$564$	0	0
$565$	4.17844	0.175788
$566$	0	0
$567$	12.3329	0.517931
$568$	0	0
$569$	9.82849	0.412032	0.206016	−	0.978549i	$-0.433950\pi$
$569$	9.82849	0.412032	0.206016	+	0.978549i	$0.433950\pi$
$570$	0	0
$571$	−6.01362	−0.251662	−0.125831	−	0.992052i	$-0.540160\pi$
$571$	−6.01362	−0.251662	−0.125831	+	0.992052i	$0.540160\pi$
$572$	0	0
$573$	−11.7897	−0.492523
$574$	0	0
$575$	0.238235	0.00993510
$576$	0	0
$577$	−17.7537	−0.739098	−0.369549	−	0.929211i	$-0.620488\pi$
$577$	−17.7537	−0.739098	−0.369549	+	0.929211i	$0.620488\pi$
$578$	0	0
$579$	−13.4265	−0.557985
$580$	0	0
$581$	6.59994	0.273812
$582$	0	0
$583$	14.9857	0.620643
$584$	0	0
$585$	−0.0446600	−0.00184646
$586$	0	0
$587$	27.2355	1.12413	0.562065	−	0.827093i	$-0.310007\pi$
$587$	27.2355	1.12413	0.562065	+	0.827093i	$0.310007\pi$
$588$	0	0
$589$	4.86651	0.200521
$590$	0	0
$591$	−18.2440	−0.750458
$592$	0	0
$593$	−18.7228	−0.768851	−0.384426	−	0.923156i	$-0.625601\pi$
$593$	−18.7228	−0.768851	−0.384426	+	0.923156i	$0.625601\pi$
$594$	0	0
$595$	−16.0450	−0.657781
$596$	0	0
$597$	−45.9829	−1.88196
$598$	0	0
$599$	−14.5647	−0.595099	−0.297549	−	0.954706i	$-0.596169\pi$
$599$	−14.5647	−0.595099	−0.297549	+	0.954706i	$0.596169\pi$
$600$	0	0
$601$	−4.82199	−0.196693	−0.0983465	−	0.995152i	$-0.531355\pi$
$601$	−4.82199	−0.196693	−0.0983465	+	0.995152i	$0.531355\pi$
$602$	0	0
$603$	−1.39545	−0.0568269
$604$	0	0
$605$	−0.510885	−0.0207704
$606$	0	0
$607$	38.8119	1.57532	0.787662	−	0.616107i	$-0.211291\pi$
$607$	38.8119	1.57532	0.787662	+	0.616107i	$0.211291\pi$
$608$	0	0
$609$	19.4862	0.789621
$610$	0	0
$611$	−0.556825	−0.0225267
$612$	0	0
$613$	−1.16083	−0.0468855	−0.0234427	−	0.999725i	$-0.507463\pi$
$613$	−1.16083	−0.0468855	−0.0234427	+	0.999725i	$0.507463\pi$
$614$	0	0
$615$	21.4881	0.866486
$616$	0	0
$617$	23.7024	0.954222	0.477111	−	0.878843i	$-0.341684\pi$
$617$	23.7024	0.954222	0.477111	+	0.878843i	$0.341684\pi$
$618$	0	0
$619$	15.1857	0.610364	0.305182	−	0.952294i	$-0.401283\pi$
$619$	15.1857	0.610364	0.305182	+	0.952294i	$0.401283\pi$
$620$	0	0
$621$	1.43515	0.0575905
$622$	0	0
$623$	−10.2970	−0.412541
$624$	0	0
$625$	−19.7094	−0.788376
$626$	0	0
$627$	26.0826	1.04164
$628$	0	0
$629$	45.0352	1.79567
$630$	0	0
$631$	−32.9269	−1.31080	−0.655400	−	0.755282i	$-0.727500\pi$
$631$	−32.9269	−1.31080	−0.655400	+	0.755282i	$0.727500\pi$
$632$	0	0
$633$	6.43277	0.255680
$634$	0	0
$635$	−4.98453	−0.197805
$636$	0	0
$637$	−0.379434	−0.0150337
$638$	0	0
$639$	2.27492	0.0899946
$640$	0	0
$641$	−26.8032	−1.05866	−0.529332	−	0.848415i	$-0.677557\pi$
$641$	−26.8032	−1.05866	−0.529332	+	0.848415i	$0.677557\pi$
$642$	0	0
$643$	5.89363	0.232422	0.116211	−	0.993225i	$-0.462925\pi$
$643$	5.89363	0.232422	0.116211	+	0.993225i	$0.462925\pi$
$644$	0	0
$645$	−25.7966	−1.01574
$646$	0	0
$647$	13.4327	0.528096	0.264048	−	0.964510i	$-0.414942\pi$
$647$	13.4327	0.528096	0.264048	+	0.964510i	$0.414942\pi$
$648$	0	0
$649$	−10.5807	−0.415330
$650$	0	0
$651$	2.53760	0.0994564
$652$	0	0
$653$	−25.3103	−0.990470	−0.495235	−	0.868759i	$-0.664918\pi$
$653$	−25.3103	−0.990470	−0.495235	+	0.868759i	$0.664918\pi$
$654$	0	0
$655$	−1.50992	−0.0589973
$656$	0	0
$657$	−0.336138	−0.0131140
$658$	0	0
$659$	−41.0118	−1.59759	−0.798796	−	0.601603i	$-0.794529\pi$
$659$	−41.0118	−1.59759	−0.798796	+	0.601603i	$0.794529\pi$
$660$	0	0
$661$	−23.8401	−0.927271	−0.463636	−	0.886026i	$-0.653455\pi$
$661$	−23.8401	−0.927271	−0.463636	+	0.886026i	$0.653455\pi$
$662$	0	0
$663$	0.696377	0.0270450
$664$	0	0
$665$	14.8404	0.575488
$666$	0	0
$667$	2.06000	0.0797637
$668$	0	0
$669$	−3.75233	−0.145073
$670$	0	0
$671$	11.8198	0.456300
$672$	0	0
$673$	11.2906	0.435220	0.217610	−	0.976036i	$-0.430174\pi$
$673$	11.2906	0.435220	0.217610	+	0.976036i	$0.430174\pi$
$674$	0	0
$675$	4.84860	0.186623
$676$	0	0
$677$	31.2844	1.20236	0.601178	−	0.799115i	$-0.294698\pi$
$677$	31.2844	1.20236	0.601178	+	0.799115i	$0.294698\pi$
$678$	0	0
$679$	9.00762	0.345681
$680$	0	0
$681$	−8.18960	−0.313826
$682$	0	0
$683$	13.3600	0.511204	0.255602	−	0.966782i	$-0.417726\pi$
$683$	13.3600	0.511204	0.255602	+	0.966782i	$0.417726\pi$
$684$	0	0
$685$	−11.1001	−0.424115
$686$	0	0
$687$	−15.8598	−0.605090
$688$	0	0
$689$	−0.370064	−0.0140983
$690$	0	0
$691$	−19.4304	−0.739166	−0.369583	−	0.929198i	$-0.620499\pi$
$691$	−19.4304	−0.739166	−0.369583	+	0.929198i	$0.620499\pi$
$692$	0	0
$693$	−1.35795	−0.0515841
$694$	0	0
$695$	39.8832	1.51285
$696$	0	0
$697$	33.4542	1.26717
$698$	0	0
$699$	17.3523	0.656323
$700$	0	0
$701$	20.2429	0.764563	0.382281	−	0.924046i	$-0.375139\pi$
$701$	20.2429	0.764563	0.382281	+	0.924046i	$0.375139\pi$
$702$	0	0
$703$	−41.6542	−1.57102
$704$	0	0
$705$	−23.0089	−0.866567
$706$	0	0
$707$	11.4177	0.429407
$708$	0	0
$709$	28.9129	1.08585	0.542924	−	0.839782i	$-0.317317\pi$
$709$	28.9129	1.08585	0.542924	+	0.839782i	$0.317317\pi$
$710$	0	0
$711$	3.36368	0.126148
$712$	0	0
$713$	0.268265	0.0100466
$714$	0	0
$715$	−0.537604	−0.0201052
$716$	0	0
$717$	34.4743	1.28747
$718$	0	0
$719$	−26.6620	−0.994325	−0.497162	−	0.867657i	$-0.665625\pi$
$719$	−26.6620	−0.994325	−0.497162	+	0.867657i	$0.665625\pi$
$720$	0	0
$721$	−12.6522	−0.471193
$722$	0	0
$723$	−32.7254	−1.21707
$724$	0	0
$725$	6.95967	0.258475
$726$	0	0
$727$	−29.2675	−1.08547	−0.542735	−	0.839904i	$-0.682611\pi$
$727$	−29.2675	−1.08547	−0.542735	+	0.839904i	$0.682611\pi$
$728$	0	0
$729$	28.9859	1.07355
$730$	0	0
$731$	−40.1618	−1.48544
$732$	0	0
$733$	−31.8329	−1.17578	−0.587888	−	0.808943i	$-0.700040\pi$
$733$	−31.8329	−1.17578	−0.587888	+	0.808943i	$0.700040\pi$
$734$	0	0
$735$	−15.6788	−0.578323
$736$	0	0
$737$	−16.7980	−0.618761
$738$	0	0
$739$	1.67685	0.0616838	0.0308419	−	0.999524i	$-0.490181\pi$
$739$	1.67685	0.0616838	0.0308419	+	0.999524i	$0.490181\pi$
$740$	0	0
$741$	−0.644097	−0.0236615
$742$	0	0
$743$	−20.3345	−0.746002	−0.373001	−	0.927831i	$-0.621671\pi$
$743$	−20.3345	−0.746002	−0.373001	+	0.927831i	$0.621671\pi$
$744$	0	0
$745$	−15.2648	−0.559258
$746$	0	0
$747$	1.18181	0.0432403
$748$	0	0
$749$	2.89012	0.105603
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	5.50559	0.200635
$754$	0	0
$755$	1.46959	0.0534840
$756$	0	0
$757$	24.1189	0.876615	0.438308	−	0.898825i	$-0.355578\pi$
$757$	24.1189	0.876615	0.438308	+	0.898825i	$0.355578\pi$
$758$	0	0
$759$	1.43780	0.0521887
$760$	0	0
$761$	0.454644	0.0164808	0.00824042	−	0.999966i	$-0.497377\pi$
$761$	0.454644	0.0164808	0.00824042	+	0.999966i	$0.497377\pi$
$762$	0	0
$763$	−23.2238	−0.840758
$764$	0	0
$765$	−2.87309	−0.103877
$766$	0	0
$767$	0.261286	0.00943448
$768$	0	0
$769$	12.2703	0.442479	0.221240	−	0.975219i	$-0.428990\pi$
$769$	12.2703	0.442479	0.221240	+	0.975219i	$0.428990\pi$
$770$	0	0
$771$	−19.3253	−0.695983
$772$	0	0
$773$	18.1483	0.652751	0.326375	−	0.945240i	$-0.394173\pi$
$773$	18.1483	0.652751	0.326375	+	0.945240i	$0.394173\pi$
$774$	0	0
$775$	0.906326	0.0325562
$776$	0	0
$777$	−21.7202	−0.779209
$778$	0	0
$779$	−30.9427	−1.10864
$780$	0	0
$781$	27.3848	0.979907
$782$	0	0
$783$	41.9256	1.49830
$784$	0	0
$785$	31.2702	1.11608
$786$	0	0
$787$	32.1092	1.14457	0.572285	−	0.820055i	$-0.306057\pi$
$787$	32.1092	1.14457	0.572285	+	0.820055i	$0.306057\pi$
$788$	0	0
$789$	15.1800	0.540421
$790$	0	0
$791$	−3.13746	−0.111555
$792$	0	0
$793$	−0.291885	−0.0103652
$794$	0	0
$795$	−15.2917	−0.542339
$796$	0	0
$797$	−17.8073	−0.630768	−0.315384	−	0.948964i	$-0.602133\pi$
$797$	−17.8073	−0.630768	−0.315384	+	0.948964i	$0.602133\pi$
$798$	0	0
$799$	−35.8219	−1.26729
$800$	0	0
$801$	−1.84383	−0.0651484
$802$	0	0
$803$	−4.04632	−0.142792
$804$	0	0
$805$	0.818075	0.0288333
$806$	0	0
$807$	−6.00153	−0.211264
$808$	0	0
$809$	−25.3856	−0.892511	−0.446255	−	0.894906i	$-0.647243\pi$
$809$	−25.3856	−0.892511	−0.446255	+	0.894906i	$0.647243\pi$
$810$	0	0
$811$	35.7235	1.25442	0.627211	−	0.778849i	$-0.284196\pi$
$811$	35.7235	1.25442	0.627211	+	0.778849i	$0.284196\pi$
$812$	0	0
$813$	−0.298413	−0.0104658
$814$	0	0
$815$	−4.89865	−0.171592
$816$	0	0
$817$	37.1467	1.29960
$818$	0	0
$819$	0.0335338	0.00117177
$820$	0	0
$821$	12.8050	0.446897	0.223448	−	0.974716i	$-0.428269\pi$
$821$	12.8050	0.446897	0.223448	+	0.974716i	$0.428269\pi$
$822$	0	0
$823$	−40.6550	−1.41715	−0.708573	−	0.705638i	$-0.750661\pi$
$823$	−40.6550	−1.41715	−0.708573	+	0.705638i	$0.750661\pi$
$824$	0	0
$825$	4.85755	0.169118
$826$	0	0
$827$	17.3910	0.604745	0.302372	−	0.953190i	$-0.402221\pi$
$827$	17.3910	0.604745	0.302372	+	0.953190i	$0.402221\pi$
$828$	0	0
$829$	34.1291	1.18535	0.592676	−	0.805441i	$-0.298071\pi$
$829$	34.1291	1.18535	0.592676	+	0.805441i	$0.298071\pi$
$830$	0	0
$831$	19.0312	0.660184
$832$	0	0
$833$	−24.4099	−0.845752
$834$	0	0
$835$	−8.31081	−0.287608
$836$	0	0
$837$	5.45978	0.188718
$838$	0	0
$839$	40.5944	1.40147	0.700737	−	0.713420i	$-0.252855\pi$
$839$	40.5944	1.40147	0.700737	+	0.713420i	$0.252855\pi$
$840$	0	0
$841$	31.1798	1.07516
$842$	0	0
$843$	−21.0340	−0.724451
$844$	0	0
$845$	−26.3189	−0.905397
$846$	0	0
$847$	0.383608	0.0131809
$848$	0	0
$849$	23.9637	0.822432
$850$	0	0
$851$	−2.29618	−0.0787119
$852$	0	0
$853$	−29.2136	−1.00026	−0.500128	−	0.865952i	$-0.666714\pi$
$853$	−29.2136	−1.00026	−0.500128	+	0.865952i	$0.666714\pi$
$854$	0	0
$855$	2.65739	0.0908809
$856$	0	0
$857$	53.3210	1.82141	0.910706	−	0.413055i	$-0.135538\pi$
$857$	53.3210	1.82141	0.910706	+	0.413055i	$0.135538\pi$
$858$	0	0
$859$	−44.1761	−1.50727	−0.753635	−	0.657294i	$-0.771701\pi$
$859$	−44.1761	−1.50727	−0.753635	+	0.657294i	$0.771701\pi$
$860$	0	0
$861$	−16.1348	−0.549872
$862$	0	0
$863$	12.5556	0.427398	0.213699	−	0.976900i	$-0.431449\pi$
$863$	12.5556	0.427398	0.213699	+	0.976900i	$0.431449\pi$
$864$	0	0
$865$	23.1821	0.788214
$866$	0	0
$867$	16.7230	0.567944
$868$	0	0
$869$	40.4910	1.37356
$870$	0	0
$871$	0.414817	0.0140555
$872$	0	0
$873$	1.61294	0.0545898
$874$	0	0
$875$	18.1674	0.614169
$876$	0	0
$877$	25.1403	0.848926	0.424463	−	0.905445i	$-0.360463\pi$
$877$	25.1403	0.848926	0.424463	+	0.905445i	$0.360463\pi$
$878$	0	0
$879$	8.51499	0.287204
$880$	0	0
$881$	−0.517665	−0.0174406	−0.00872029	−	0.999962i	$-0.502776\pi$
$881$	−0.517665	−0.0174406	−0.00872029	+	0.999962i	$0.502776\pi$
$882$	0	0
$883$	−41.9585	−1.41202	−0.706009	−	0.708203i	$-0.749506\pi$
$883$	−41.9585	−1.41202	−0.706009	+	0.708203i	$0.749506\pi$
$884$	0	0
$885$	10.7968	0.362929
$886$	0	0
$887$	−14.4419	−0.484911	−0.242455	−	0.970163i	$-0.577953\pi$
$887$	−14.4419	−0.484911	−0.242455	+	0.970163i	$0.577953\pi$
$888$	0	0
$889$	3.74273	0.125527
$890$	0	0
$891$	26.5837	0.890589
$892$	0	0
$893$	33.1326	1.10874
$894$	0	0
$895$	−41.7241	−1.39468
$896$	0	0
$897$	−0.0355057	−0.00118550
$898$	0	0
$899$	7.83694	0.261377
$900$	0	0
$901$	−23.8071	−0.793129
$902$	0	0
$903$	19.3698	0.644587
$904$	0	0
$905$	−18.2570	−0.606882
$906$	0	0
$907$	−16.0472	−0.532837	−0.266418	−	0.963857i	$-0.585840\pi$
$907$	−16.0472	−0.532837	−0.266418	+	0.963857i	$0.585840\pi$
$908$	0	0
$909$	2.04450	0.0678119
$910$	0	0
$911$	40.2080	1.33215	0.666075	−	0.745885i	$-0.267973\pi$
$911$	40.2080	1.33215	0.666075	+	0.745885i	$0.267973\pi$
$912$	0	0
$913$	14.2263	0.470822
$914$	0	0
$915$	−12.0612	−0.398731
$916$	0	0
$917$	1.13375	0.0374397
$918$	0	0
$919$	−39.3644	−1.29851	−0.649256	−	0.760570i	$-0.724920\pi$
$919$	−39.3644	−1.29851	−0.649256	+	0.760570i	$0.724920\pi$
$920$	0	0
$921$	−10.7836	−0.355331
$922$	0	0
$923$	−0.676255	−0.0222592
$924$	0	0
$925$	−7.75757	−0.255067
$926$	0	0
$927$	−2.26556	−0.0744107
$928$	0	0
$929$	42.8018	1.40428	0.702141	−	0.712038i	$-0.252228\pi$
$929$	42.8018	1.40428	0.702141	+	0.712038i	$0.252228\pi$
$930$	0	0
$931$	22.5773	0.739942
$932$	0	0
$933$	47.0118	1.53910
$934$	0	0
$935$	−34.5853	−1.13106
$936$	0	0
$937$	−13.8128	−0.451243	−0.225622	−	0.974215i	$-0.572441\pi$
$937$	−13.8128	−0.451243	−0.225622	+	0.974215i	$0.572441\pi$
$938$	0	0
$939$	41.8987	1.36731
$940$	0	0
$941$	−7.82865	−0.255207	−0.127603	−	0.991825i	$-0.540728\pi$
$941$	−7.82865	−0.255207	−0.127603	+	0.991825i	$0.540728\pi$
$942$	0	0
$943$	−1.70571	−0.0555454
$944$	0	0
$945$	16.6496	0.541611
$946$	0	0
$947$	−31.0874	−1.01021	−0.505103	−	0.863059i	$-0.668545\pi$
$947$	−31.0874	−1.01021	−0.505103	+	0.863059i	$0.668545\pi$
$948$	0	0
$949$	0.0999220	0.00324360
$950$	0	0
$951$	19.2087	0.622884
$952$	0	0
$953$	16.6169	0.538276	0.269138	−	0.963102i	$-0.413261\pi$
$953$	16.6169	0.538276	0.269138	+	0.963102i	$0.413261\pi$
$954$	0	0
$955$	−14.4595	−0.467897
$956$	0	0
$957$	42.0029	1.35776
$958$	0	0
$959$	8.33475	0.269143
$960$	0	0
$961$	−29.9794	−0.967078
$962$	0	0
$963$	0.517517	0.0166767
$964$	0	0
$965$	−16.4668	−0.530086
$966$	0	0
$967$	37.5255	1.20674	0.603369	−	0.797462i	$-0.293825\pi$
$967$	37.5255	1.20674	0.603369	+	0.797462i	$0.293825\pi$
$968$	0	0
$969$	−41.4363	−1.33113
$970$	0	0
$971$	21.7506	0.698010	0.349005	−	0.937121i	$-0.386520\pi$
$971$	21.7506	0.698010	0.349005	+	0.937121i	$0.386520\pi$
$972$	0	0
$973$	−29.9470	−0.960057
$974$	0	0
$975$	−0.119955	−0.00384163
$976$	0	0
$977$	8.46868	0.270937	0.135469	−	0.990782i	$-0.456746\pi$
$977$	8.46868	0.270937	0.135469	+	0.990782i	$0.456746\pi$
$978$	0	0
$979$	−22.1954	−0.709369
$980$	0	0
$981$	−4.15855	−0.132772
$982$	0	0
$983$	−3.60789	−0.115074	−0.0575369	−	0.998343i	$-0.518325\pi$
$983$	−3.60789	−0.115074	−0.0575369	+	0.998343i	$0.518325\pi$
$984$	0	0
$985$	−22.3753	−0.712935
$986$	0	0
$987$	17.2767	0.549923
$988$	0	0
$989$	2.04770	0.0651131
$990$	0	0
$991$	10.3797	0.329721	0.164860	−	0.986317i	$-0.447283\pi$
$991$	10.3797	0.329721	0.164860	+	0.986317i	$0.447283\pi$
$992$	0	0
$993$	40.0839	1.27202
$994$	0	0
$995$	−56.3955	−1.78786
$996$	0	0
$997$	8.81127	0.279056	0.139528	−	0.990218i	$-0.455442\pi$
$997$	8.81127	0.279056	0.139528	+	0.990218i	$0.455442\pi$
$998$	0	0
$999$	−46.7322	−1.47854

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.35	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.35	✓	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)

\(f(q)\)	\(=\)	\(q+1.65156 q^{3} +2.02555 q^{5} -1.52092 q^{7} -0.272343 q^{9} +O(q^{10})\)	\(q+1.65156 q^{3} +2.02555 q^{5} -1.52092 q^{7} -0.272343 q^{9} -3.27838 q^{11} +0.0809580 q^{13} +3.34532 q^{15} +5.20822 q^{17} -4.81722 q^{19} -2.51190 q^{21} -0.265548 q^{23} -0.897146 q^{25} -5.40448 q^{27} -7.75756 q^{29} -1.01023 q^{31} -5.41445 q^{33} -3.08070 q^{35} +8.64694 q^{37} +0.133707 q^{39} +6.42334 q^{41} -7.71123 q^{43} -0.551644 q^{45} -6.87795 q^{47} -4.68680 q^{49} +8.60170 q^{51} -4.57105 q^{53} -6.64052 q^{55} -7.95594 q^{57} +3.22742 q^{59} -3.60539 q^{61} +0.414212 q^{63} +0.163985 q^{65} +5.12386 q^{67} -0.438569 q^{69} -8.35316 q^{71} +1.23424 q^{73} -1.48169 q^{75} +4.98616 q^{77} -12.3509 q^{79} -8.10880 q^{81} -4.33943 q^{83} +10.5495 q^{85} -12.8121 q^{87} +6.77024 q^{89} -0.123131 q^{91} -1.66846 q^{93} -9.75753 q^{95} -5.92247 q^{97} +0.892843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more