LMFDB - Embedded newform 8008.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 15x^{7} + 45x^{6} + 64x^{5} - 201x^{4} - 53x^{3} + 252x^{2} - 69x - 26 $ x^9 - 3x^8 - 15x^7 + 45x^6 + 64x^5 - 201x^4 - 53x^3 + 252x^2 - 69x - 26
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.78620$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.78620 q^{3} +0.423993 q^{5} +1.00000 q^{7} +4.76292 q^{9} +O(q^{10})$	$q-2.78620 q^{3} +0.423993 q^{5} +1.00000 q^{7} +4.76292 q^{9} +1.00000 q^{11} -1.00000 q^{13} -1.18133 q^{15} -4.54636 q^{17} -1.01363 q^{19} -2.78620 q^{21} -7.69364 q^{23} -4.82023 q^{25} -4.91185 q^{27} -6.92340 q^{29} +0.642629 q^{31} -2.78620 q^{33} +0.423993 q^{35} -0.164420 q^{37} +2.78620 q^{39} +3.76016 q^{41} +7.87421 q^{43} +2.01945 q^{45} +7.69364 q^{47} +1.00000 q^{49} +12.6671 q^{51} -0.377040 q^{53} +0.423993 q^{55} +2.82419 q^{57} +1.91902 q^{59} -5.47698 q^{61} +4.76292 q^{63} -0.423993 q^{65} +1.20021 q^{67} +21.4360 q^{69} +5.96520 q^{71} +6.32737 q^{73} +13.4301 q^{75} +1.00000 q^{77} -11.2111 q^{79} -0.603360 q^{81} +5.61774 q^{83} -1.92763 q^{85} +19.2900 q^{87} +1.37756 q^{89} -1.00000 q^{91} -1.79049 q^{93} -0.429774 q^{95} -14.9289 q^{97} +4.76292 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9	$ 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + q^{17} + 16 q^{19} + 3 q^{21} + 6 q^{23} + 11 q^{25} + 9 q^{27} - 2 q^{29} + 3 q^{31} + 3 q^{33} + 8 q^{35} - 4 q^{37} - 3 q^{39} + 20 q^{41} + 20 q^{43} + 8 q^{45} - 6 q^{47} + 9 q^{49} + 15 q^{51} + 15 q^{53} + 8 q^{55} + 16 q^{57} + 21 q^{59} + 12 q^{63} - 8 q^{65} + 18 q^{67} + 10 q^{69} + 12 q^{71} + 29 q^{73} + 12 q^{75} + 9 q^{77} - 6 q^{79} + 5 q^{81} + 25 q^{83} + 13 q^{85} + 17 q^{87} + 32 q^{89} - 9 q^{91} - 13 q^{93} + 38 q^{95} + 17 q^{97} + 12 q^{99}+O(q^{100}) $ 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 - 9 * q^15 + q^17 + 16 * q^19 + 3 * q^21 + 6 * q^23 + 11 * q^25 + 9 * q^27 - 2 * q^29 + 3 * q^31 + 3 * q^33 + 8 * q^35 - 4 * q^37 - 3 * q^39 + 20 * q^41 + 20 * q^43 + 8 * q^45 - 6 * q^47 + 9 * q^49 + 15 * q^51 + 15 * q^53 + 8 * q^55 + 16 * q^57 + 21 * q^59 + 12 * q^63 - 8 * q^65 + 18 * q^67 + 10 * q^69 + 12 * q^71 + 29 * q^73 + 12 * q^75 + 9 * q^77 - 6 * q^79 + 5 * q^81 + 25 * q^83 + 13 * q^85 + 17 * q^87 + 32 * q^89 - 9 * q^91 - 13 * q^93 + 38 * q^95 + 17 * q^97 + 12 * 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
$3$	−2.78620	−1.60861	−0.804307	−	0.594214i	$-0.797463\pi$
$3$	−2.78620	−1.60861	−0.804307	+	0.594214i	$0.797463\pi$
$4$	0	0
$5$	0.423993	0.189616	0.0948078	−	0.995496i	$-0.469776\pi$
$5$	0.423993	0.189616	0.0948078	+	0.995496i	$0.469776\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	4.76292	1.58764
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.18133	−0.305018
$16$	0	0
$17$	−4.54636	−1.10265	−0.551327	−	0.834289i	$-0.685878\pi$
$17$	−4.54636	−1.10265	−0.551327	+	0.834289i	$0.685878\pi$
$18$	0	0
$19$	−1.01363	−0.232544	−0.116272	−	0.993217i	$-0.537094\pi$
$19$	−1.01363	−0.232544	−0.116272	+	0.993217i	$0.537094\pi$
$20$	0	0
$21$	−2.78620	−0.607999
$22$	0	0
$23$	−7.69364	−1.60424	−0.802118	−	0.597166i	$-0.796293\pi$
$23$	−7.69364	−1.60424	−0.802118	+	0.597166i	$0.796293\pi$
$24$	0	0
$25$	−4.82023	−0.964046
$26$	0	0
$27$	−4.91185	−0.945285
$28$	0	0
$29$	−6.92340	−1.28564	−0.642822	−	0.766016i	$-0.722236\pi$
$29$	−6.92340	−1.28564	−0.642822	+	0.766016i	$0.722236\pi$
$30$	0	0
$31$	0.642629	0.115420	0.0577098	−	0.998333i	$-0.481620\pi$
$31$	0.642629	0.115420	0.0577098	+	0.998333i	$0.481620\pi$
$32$	0	0
$33$	−2.78620	−0.485015
$34$	0	0
$35$	0.423993	0.0716680
$36$	0	0
$37$	−0.164420	−0.0270304	−0.0135152	−	0.999909i	$-0.504302\pi$
$37$	−0.164420	−0.0270304	−0.0135152	+	0.999909i	$0.504302\pi$
$38$	0	0
$39$	2.78620	0.446149
$40$	0	0
$41$	3.76016	0.587238	0.293619	−	0.955922i	$-0.405140\pi$
$41$	3.76016	0.587238	0.293619	+	0.955922i	$0.405140\pi$
$42$	0	0
$43$	7.87421	1.20081	0.600403	−	0.799698i	$-0.295007\pi$
$43$	7.87421	1.20081	0.600403	+	0.799698i	$0.295007\pi$
$44$	0	0
$45$	2.01945	0.301041
$46$	0	0
$47$	7.69364	1.12223	0.561116	−	0.827737i	$-0.310372\pi$
$47$	7.69364	1.12223	0.561116	+	0.827737i	$0.310372\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	12.6671	1.77375
$52$	0	0
$53$	−0.377040	−0.0517904	−0.0258952	−	0.999665i	$-0.508244\pi$
$53$	−0.377040	−0.0517904	−0.0258952	+	0.999665i	$0.508244\pi$
$54$	0	0
$55$	0.423993	0.0571712
$56$	0	0
$57$	2.82419	0.374073
$58$	0	0
$59$	1.91902	0.249835	0.124917	−	0.992167i	$-0.460133\pi$
$59$	1.91902	0.249835	0.124917	+	0.992167i	$0.460133\pi$
$60$	0	0
$61$	−5.47698	−0.701255	−0.350628	−	0.936515i	$-0.614032\pi$
$61$	−5.47698	−0.701255	−0.350628	+	0.936515i	$0.614032\pi$
$62$	0	0
$63$	4.76292	0.600071
$64$	0	0
$65$	−0.423993	−0.0525899
$66$	0	0
$67$	1.20021	0.146628	0.0733142	−	0.997309i	$-0.476642\pi$
$67$	1.20021	0.146628	0.0733142	+	0.997309i	$0.476642\pi$
$68$	0	0
$69$	21.4360	2.58060
$70$	0	0
$71$	5.96520	0.707939	0.353970	−	0.935257i	$-0.384832\pi$
$71$	5.96520	0.707939	0.353970	+	0.935257i	$0.384832\pi$
$72$	0	0
$73$	6.32737	0.740563	0.370281	−	0.928920i	$-0.379261\pi$
$73$	6.32737	0.740563	0.370281	+	0.928920i	$0.379261\pi$
$74$	0	0
$75$	13.4301	1.55078
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−11.2111	−1.26135	−0.630676	−	0.776046i	$-0.717222\pi$
$79$	−11.2111	−1.26135	−0.630676	+	0.776046i	$0.717222\pi$
$80$	0	0
$81$	−0.603360	−0.0670400
$82$	0	0
$83$	5.61774	0.616627	0.308313	−	0.951285i	$-0.400235\pi$
$83$	5.61774	0.616627	0.308313	+	0.951285i	$0.400235\pi$
$84$	0	0
$85$	−1.92763	−0.209081
$86$	0	0
$87$	19.2900	2.06810
$88$	0	0
$89$	1.37756	0.146021	0.0730105	−	0.997331i	$-0.476739\pi$
$89$	1.37756	0.146021	0.0730105	+	0.997331i	$0.476739\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−1.79049	−0.185666
$94$	0	0
$95$	−0.429774	−0.0440939
$96$	0	0
$97$	−14.9289	−1.51580	−0.757901	−	0.652370i	$-0.773775\pi$
$97$	−14.9289	−1.51580	−0.757901	+	0.652370i	$0.773775\pi$
$98$	0	0
$99$	4.76292	0.478691
$100$	0	0
$101$	−2.85631	−0.284213	−0.142107	−	0.989851i	$-0.545388\pi$
$101$	−2.85631	−0.284213	−0.142107	+	0.989851i	$0.545388\pi$
$102$	0	0
$103$	−4.92831	−0.485600	−0.242800	−	0.970076i	$-0.578066\pi$
$103$	−4.92831	−0.485600	−0.242800	+	0.970076i	$0.578066\pi$
$104$	0	0
$105$	−1.18133	−0.115286
$106$	0	0
$107$	1.66939	0.161386	0.0806932	−	0.996739i	$-0.474287\pi$
$107$	1.66939	0.161386	0.0806932	+	0.996739i	$0.474287\pi$
$108$	0	0
$109$	−10.5114	−1.00681	−0.503403	−	0.864052i	$-0.667919\pi$
$109$	−10.5114	−1.00681	−0.503403	+	0.864052i	$0.667919\pi$
$110$	0	0
$111$	0.458106	0.0434815
$112$	0	0
$113$	−1.44509	−0.135943	−0.0679714	−	0.997687i	$-0.521653\pi$
$113$	−1.44509	−0.135943	−0.0679714	+	0.997687i	$0.521653\pi$
$114$	0	0
$115$	−3.26205	−0.304188
$116$	0	0
$117$	−4.76292	−0.440332
$118$	0	0
$119$	−4.54636	−0.416764
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−10.4766	−0.944640
$124$	0	0
$125$	−4.16371	−0.372414
$126$	0	0
$127$	20.3014	1.80146	0.900728	−	0.434384i	$-0.143034\pi$
$127$	20.3014	1.80146	0.900728	+	0.434384i	$0.143034\pi$
$128$	0	0
$129$	−21.9391	−1.93163
$130$	0	0
$131$	5.33912	0.466481	0.233240	−	0.972419i	$-0.425067\pi$
$131$	5.33912	0.466481	0.233240	+	0.972419i	$0.425067\pi$
$132$	0	0
$133$	−1.01363	−0.0878932
$134$	0	0
$135$	−2.08259	−0.179241
$136$	0	0
$137$	19.6188	1.67615	0.838075	−	0.545555i	$-0.183681\pi$
$137$	19.6188	1.67615	0.838075	+	0.545555i	$0.183681\pi$
$138$	0	0
$139$	13.4487	1.14070	0.570352	−	0.821400i	$-0.306807\pi$
$139$	13.4487	1.14070	0.570352	+	0.821400i	$0.306807\pi$
$140$	0	0
$141$	−21.4360	−1.80524
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−2.93548	−0.243778
$146$	0	0
$147$	−2.78620	−0.229802
$148$	0	0
$149$	−21.2029	−1.73701	−0.868506	−	0.495678i	$-0.834920\pi$
$149$	−21.2029	−1.73701	−0.868506	+	0.495678i	$0.834920\pi$
$150$	0	0
$151$	−21.1298	−1.71952	−0.859759	−	0.510701i	$-0.829386\pi$
$151$	−21.1298	−1.71952	−0.859759	+	0.510701i	$0.829386\pi$
$152$	0	0
$153$	−21.6540	−1.75062
$154$	0	0
$155$	0.272470	0.0218853
$156$	0	0
$157$	−3.30525	−0.263788	−0.131894	−	0.991264i	$-0.542106\pi$
$157$	−3.30525	−0.263788	−0.131894	+	0.991264i	$0.542106\pi$
$158$	0	0
$159$	1.05051	0.0833109
$160$	0	0
$161$	−7.69364	−0.606344
$162$	0	0
$163$	−9.31290	−0.729443	−0.364721	−	0.931117i	$-0.618836\pi$
$163$	−9.31290	−0.729443	−0.364721	+	0.931117i	$0.618836\pi$
$164$	0	0
$165$	−1.18133	−0.0919665
$166$	0	0
$167$	−2.68898	−0.208079	−0.104040	−	0.994573i	$-0.533177\pi$
$167$	−2.68898	−0.208079	−0.104040	+	0.994573i	$0.533177\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.82786	−0.369196
$172$	0	0
$173$	21.5856	1.64112	0.820561	−	0.571559i	$-0.193661\pi$
$173$	21.5856	1.64112	0.820561	+	0.571559i	$0.193661\pi$
$174$	0	0
$175$	−4.82023	−0.364375
$176$	0	0
$177$	−5.34677	−0.401888
$178$	0	0
$179$	1.74761	0.130622	0.0653112	−	0.997865i	$-0.479196\pi$
$179$	1.74761	0.130622	0.0653112	+	0.997865i	$0.479196\pi$
$180$	0	0
$181$	11.1853	0.831400	0.415700	−	0.909502i	$-0.363537\pi$
$181$	11.1853	0.831400	0.415700	+	0.909502i	$0.363537\pi$
$182$	0	0
$183$	15.2600	1.12805
$184$	0	0
$185$	−0.0697129	−0.00512539
$186$	0	0
$187$	−4.54636	−0.332463
$188$	0	0
$189$	−4.91185	−0.357284
$190$	0	0
$191$	−7.26088	−0.525379	−0.262689	−	0.964880i	$-0.584609\pi$
$191$	−7.26088	−0.525379	−0.262689	+	0.964880i	$0.584609\pi$
$192$	0	0
$193$	−6.86891	−0.494435	−0.247218	−	0.968960i	$-0.579516\pi$
$193$	−6.86891	−0.494435	−0.247218	+	0.968960i	$0.579516\pi$
$194$	0	0
$195$	1.18133	0.0845969
$196$	0	0
$197$	−6.49895	−0.463031	−0.231516	−	0.972831i	$-0.574368\pi$
$197$	−6.49895	−0.463031	−0.231516	+	0.972831i	$0.574368\pi$
$198$	0	0
$199$	−14.4754	−1.02613	−0.513067	−	0.858349i	$-0.671491\pi$
$199$	−14.4754	−1.02613	−0.513067	+	0.858349i	$0.671491\pi$
$200$	0	0
$201$	−3.34401	−0.235869
$202$	0	0
$203$	−6.92340	−0.485928
$204$	0	0
$205$	1.59428	0.111350
$206$	0	0
$207$	−36.6442	−2.54695
$208$	0	0
$209$	−1.01363	−0.0701146
$210$	0	0
$211$	−1.06530	−0.0733382	−0.0366691	−	0.999327i	$-0.511675\pi$
$211$	−1.06530	−0.0733382	−0.0366691	+	0.999327i	$0.511675\pi$
$212$	0	0
$213$	−16.6203	−1.13880
$214$	0	0
$215$	3.33861	0.227691
$216$	0	0
$217$	0.642629	0.0436245
$218$	0	0
$219$	−17.6293	−1.19128
$220$	0	0
$221$	4.54636	0.305821
$222$	0	0
$223$	19.0197	1.27365	0.636827	−	0.771007i	$-0.280246\pi$
$223$	19.0197	1.27365	0.636827	+	0.771007i	$0.280246\pi$
$224$	0	0
$225$	−22.9584	−1.53056
$226$	0	0
$227$	29.8788	1.98312	0.991561	−	0.129640i	$-0.0413821\pi$
$227$	29.8788	1.98312	0.991561	+	0.129640i	$0.0413821\pi$
$228$	0	0
$229$	7.13941	0.471785	0.235893	−	0.971779i	$-0.424199\pi$
$229$	7.13941	0.471785	0.235893	+	0.971779i	$0.424199\pi$
$230$	0	0
$231$	−2.78620	−0.183319
$232$	0	0
$233$	−0.956539	−0.0626649	−0.0313325	−	0.999509i	$-0.509975\pi$
$233$	−0.956539	−0.0626649	−0.0313325	+	0.999509i	$0.509975\pi$
$234$	0	0
$235$	3.26205	0.212793
$236$	0	0
$237$	31.2365	2.02903
$238$	0	0
$239$	14.9866	0.969403	0.484701	−	0.874680i	$-0.338928\pi$
$239$	14.9866	0.969403	0.484701	+	0.874680i	$0.338928\pi$
$240$	0	0
$241$	−8.56823	−0.551928	−0.275964	−	0.961168i	$-0.588997\pi$
$241$	−8.56823	−0.551928	−0.275964	+	0.961168i	$0.588997\pi$
$242$	0	0
$243$	16.4166	1.05313
$244$	0	0
$245$	0.423993	0.0270879
$246$	0	0
$247$	1.01363	0.0644960
$248$	0	0
$249$	−15.6522	−0.991915
$250$	0	0
$251$	24.9997	1.57797	0.788984	−	0.614413i	$-0.210607\pi$
$251$	24.9997	1.57797	0.788984	+	0.614413i	$0.210607\pi$
$252$	0	0
$253$	−7.69364	−0.483695
$254$	0	0
$255$	5.37076	0.336330
$256$	0	0
$257$	−7.55991	−0.471575	−0.235787	−	0.971805i	$-0.575767\pi$
$257$	−7.55991	−0.471575	−0.235787	+	0.971805i	$0.575767\pi$
$258$	0	0
$259$	−0.164420	−0.0102165
$260$	0	0
$261$	−32.9756	−2.04114
$262$	0	0
$263$	1.52156	0.0938232	0.0469116	−	0.998899i	$-0.485062\pi$
$263$	1.52156	0.0938232	0.0469116	+	0.998899i	$0.485062\pi$
$264$	0	0
$265$	−0.159863	−0.00982028
$266$	0	0
$267$	−3.83816	−0.234891
$268$	0	0
$269$	21.8612	1.33290	0.666452	−	0.745548i	$-0.267812\pi$
$269$	21.8612	1.33290	0.666452	+	0.745548i	$0.267812\pi$
$270$	0	0
$271$	−3.61750	−0.219748	−0.109874	−	0.993946i	$-0.535045\pi$
$271$	−3.61750	−0.219748	−0.109874	+	0.993946i	$0.535045\pi$
$272$	0	0
$273$	2.78620	0.168629
$274$	0	0
$275$	−4.82023	−0.290671
$276$	0	0
$277$	−11.4181	−0.686044	−0.343022	−	0.939327i	$-0.611451\pi$
$277$	−11.4181	−0.686044	−0.343022	+	0.939327i	$0.611451\pi$
$278$	0	0
$279$	3.06079	0.183245
$280$	0	0
$281$	7.69281	0.458914	0.229457	−	0.973319i	$-0.426305\pi$
$281$	7.69281	0.458914	0.229457	+	0.973319i	$0.426305\pi$
$282$	0	0
$283$	25.3184	1.50502	0.752512	−	0.658579i	$-0.228842\pi$
$283$	25.3184	1.50502	0.752512	+	0.658579i	$0.228842\pi$
$284$	0	0
$285$	1.19744	0.0709301
$286$	0	0
$287$	3.76016	0.221955
$288$	0	0
$289$	3.66941	0.215848
$290$	0	0
$291$	41.5950	2.43834
$292$	0	0
$293$	17.1986	1.00475	0.502376	−	0.864649i	$-0.332459\pi$
$293$	17.1986	1.00475	0.502376	+	0.864649i	$0.332459\pi$
$294$	0	0
$295$	0.813651	0.0473726
$296$	0	0
$297$	−4.91185	−0.285014
$298$	0	0
$299$	7.69364	0.444935
$300$	0	0
$301$	7.87421	0.453862
$302$	0	0
$303$	7.95825	0.457189
$304$	0	0
$305$	−2.32220	−0.132969
$306$	0	0
$307$	−26.6157	−1.51904	−0.759518	−	0.650486i	$-0.774565\pi$
$307$	−26.6157	−1.51904	−0.759518	+	0.650486i	$0.774565\pi$
$308$	0	0
$309$	13.7313	0.781144
$310$	0	0
$311$	−2.46468	−0.139759	−0.0698795	−	0.997555i	$-0.522261\pi$
$311$	−2.46468	−0.139759	−0.0698795	+	0.997555i	$0.522261\pi$
$312$	0	0
$313$	15.0770	0.852200	0.426100	−	0.904676i	$-0.359887\pi$
$313$	15.0770	0.852200	0.426100	+	0.904676i	$0.359887\pi$
$314$	0	0
$315$	2.01945	0.113783
$316$	0	0
$317$	26.4955	1.48814	0.744068	−	0.668104i	$-0.232894\pi$
$317$	26.4955	1.48814	0.744068	+	0.668104i	$0.232894\pi$
$318$	0	0
$319$	−6.92340	−0.387636
$320$	0	0
$321$	−4.65127	−0.259609
$322$	0	0
$323$	4.60835	0.256415
$324$	0	0
$325$	4.82023	0.267378
$326$	0	0
$327$	29.2868	1.61956
$328$	0	0
$329$	7.69364	0.424164
$330$	0	0
$331$	28.0038	1.53923	0.769614	−	0.638509i	$-0.220449\pi$
$331$	28.0038	1.53923	0.769614	+	0.638509i	$0.220449\pi$
$332$	0	0
$333$	−0.783118	−0.0429146
$334$	0	0
$335$	0.508879	0.0278030
$336$	0	0
$337$	20.8199	1.13413	0.567066	−	0.823672i	$-0.308078\pi$
$337$	20.8199	1.13413	0.567066	+	0.823672i	$0.308078\pi$
$338$	0	0
$339$	4.02632	0.218680
$340$	0	0
$341$	0.642629	0.0348003
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	9.08874	0.489321
$346$	0	0
$347$	28.5884	1.53470	0.767352	−	0.641226i	$-0.221574\pi$
$347$	28.5884	1.53470	0.767352	+	0.641226i	$0.221574\pi$
$348$	0	0
$349$	−30.0107	−1.60644	−0.803219	−	0.595684i	$-0.796881\pi$
$349$	−30.0107	−1.60644	−0.803219	+	0.595684i	$0.796881\pi$
$350$	0	0
$351$	4.91185	0.262175
$352$	0	0
$353$	−15.2403	−0.811158	−0.405579	−	0.914060i	$-0.632930\pi$
$353$	−15.2403	−0.811158	−0.405579	+	0.914060i	$0.632930\pi$
$354$	0	0
$355$	2.52921	0.134236
$356$	0	0
$357$	12.6671	0.670413
$358$	0	0
$359$	−14.1605	−0.747363	−0.373681	−	0.927557i	$-0.621905\pi$
$359$	−14.1605	−0.747363	−0.373681	+	0.927557i	$0.621905\pi$
$360$	0	0
$361$	−17.9725	−0.945923
$362$	0	0
$363$	−2.78620	−0.146238
$364$	0	0
$365$	2.68276	0.140422
$366$	0	0
$367$	1.87408	0.0978261	0.0489131	−	0.998803i	$-0.484424\pi$
$367$	1.87408	0.0978261	0.0489131	+	0.998803i	$0.484424\pi$
$368$	0	0
$369$	17.9093	0.932323
$370$	0	0
$371$	−0.377040	−0.0195750
$372$	0	0
$373$	32.9865	1.70798	0.853988	−	0.520293i	$-0.174177\pi$
$373$	32.9865	1.70798	0.853988	+	0.520293i	$0.174177\pi$
$374$	0	0
$375$	11.6009	0.599070
$376$	0	0
$377$	6.92340	0.356573
$378$	0	0
$379$	10.9443	0.562172	0.281086	−	0.959683i	$-0.409305\pi$
$379$	10.9443	0.562172	0.281086	+	0.959683i	$0.409305\pi$
$380$	0	0
$381$	−56.5637	−2.89785
$382$	0	0
$383$	−20.8328	−1.06451	−0.532253	−	0.846585i	$-0.678655\pi$
$383$	−20.8328	−1.06451	−0.532253	+	0.846585i	$0.678655\pi$
$384$	0	0
$385$	0.423993	0.0216087
$386$	0	0
$387$	37.5042	1.90645
$388$	0	0
$389$	−34.4691	−1.74765	−0.873825	−	0.486240i	$-0.838368\pi$
$389$	−34.4691	−1.74765	−0.873825	+	0.486240i	$0.838368\pi$
$390$	0	0
$391$	34.9781	1.76892
$392$	0	0
$393$	−14.8759	−0.750388
$394$	0	0
$395$	−4.75345	−0.239172
$396$	0	0
$397$	9.55479	0.479541	0.239771	−	0.970830i	$-0.422928\pi$
$397$	9.55479	0.479541	0.239771	+	0.970830i	$0.422928\pi$
$398$	0	0
$399$	2.82419	0.141386
$400$	0	0
$401$	−28.0996	−1.40323	−0.701614	−	0.712557i	$-0.747537\pi$
$401$	−28.0996	−1.40323	−0.701614	+	0.712557i	$0.747537\pi$
$402$	0	0
$403$	−0.642629	−0.0320116
$404$	0	0
$405$	−0.255821	−0.0127118
$406$	0	0
$407$	−0.164420	−0.00814998
$408$	0	0
$409$	17.3248	0.856658	0.428329	−	0.903623i	$-0.359102\pi$
$409$	17.3248	0.856658	0.428329	+	0.903623i	$0.359102\pi$
$410$	0	0
$411$	−54.6620	−2.69628
$412$	0	0
$413$	1.91902	0.0944287
$414$	0	0
$415$	2.38188	0.116922
$416$	0	0
$417$	−37.4708	−1.83495
$418$	0	0
$419$	−34.7997	−1.70008	−0.850039	−	0.526719i	$-0.823422\pi$
$419$	−34.7997	−1.70008	−0.850039	+	0.526719i	$0.823422\pi$
$420$	0	0
$421$	17.0965	0.833231	0.416616	−	0.909083i	$-0.363216\pi$
$421$	17.0965	0.833231	0.416616	+	0.909083i	$0.363216\pi$
$422$	0	0
$423$	36.6442	1.78170
$424$	0	0
$425$	21.9145	1.06301
$426$	0	0
$427$	−5.47698	−0.265050
$428$	0	0
$429$	2.78620	0.134519
$430$	0	0
$431$	−14.8926	−0.717354	−0.358677	−	0.933462i	$-0.616772\pi$
$431$	−14.8926	−0.717354	−0.358677	+	0.933462i	$0.616772\pi$
$432$	0	0
$433$	−4.43730	−0.213243	−0.106622	−	0.994300i	$-0.534003\pi$
$433$	−4.43730	−0.213243	−0.106622	+	0.994300i	$0.534003\pi$
$434$	0	0
$435$	8.17883	0.392145
$436$	0	0
$437$	7.79854	0.373055
$438$	0	0
$439$	13.9173	0.664235	0.332117	−	0.943238i	$-0.392237\pi$
$439$	13.9173	0.664235	0.332117	+	0.943238i	$0.392237\pi$
$440$	0	0
$441$	4.76292	0.226806
$442$	0	0
$443$	−18.5354	−0.880643	−0.440321	−	0.897840i	$-0.645135\pi$
$443$	−18.5354	−0.880643	−0.440321	+	0.897840i	$0.645135\pi$
$444$	0	0
$445$	0.584076	0.0276878
$446$	0	0
$447$	59.0757	2.79418
$448$	0	0
$449$	4.42137	0.208658	0.104329	−	0.994543i	$-0.466731\pi$
$449$	4.42137	0.208658	0.104329	+	0.994543i	$0.466731\pi$
$450$	0	0
$451$	3.76016	0.177059
$452$	0	0
$453$	58.8718	2.76604
$454$	0	0
$455$	−0.423993	−0.0198771
$456$	0	0
$457$	−10.9535	−0.512382	−0.256191	−	0.966626i	$-0.582468\pi$
$457$	−10.9535	−0.512382	−0.256191	+	0.966626i	$0.582468\pi$
$458$	0	0
$459$	22.3310	1.04232
$460$	0	0
$461$	29.3391	1.36646	0.683228	−	0.730205i	$-0.260576\pi$
$461$	29.3391	1.36646	0.683228	+	0.730205i	$0.260576\pi$
$462$	0	0
$463$	12.6837	0.589461	0.294730	−	0.955580i	$-0.404770\pi$
$463$	12.6837	0.589461	0.294730	+	0.955580i	$0.404770\pi$
$464$	0	0
$465$	−0.759157	−0.0352051
$466$	0	0
$467$	28.0911	1.29990	0.649951	−	0.759976i	$-0.274789\pi$
$467$	28.0911	1.29990	0.649951	+	0.759976i	$0.274789\pi$
$468$	0	0
$469$	1.20021	0.0554203
$470$	0	0
$471$	9.20909	0.424333
$472$	0	0
$473$	7.87421	0.362056
$474$	0	0
$475$	4.88595	0.224183
$476$	0	0
$477$	−1.79581	−0.0822246
$478$	0	0
$479$	−17.3210	−0.791418	−0.395709	−	0.918376i	$-0.629501\pi$
$479$	−17.3210	−0.791418	−0.395709	+	0.918376i	$0.629501\pi$
$480$	0	0
$481$	0.164420	0.00749689
$482$	0	0
$483$	21.4360	0.975373
$484$	0	0
$485$	−6.32976	−0.287420
$486$	0	0
$487$	−40.3872	−1.83012	−0.915058	−	0.403321i	$-0.867856\pi$
$487$	−40.3872	−1.83012	−0.915058	+	0.403321i	$0.867856\pi$
$488$	0	0
$489$	25.9476	1.17339
$490$	0	0
$491$	36.7943	1.66050	0.830252	−	0.557388i	$-0.188196\pi$
$491$	36.7943	1.66050	0.830252	+	0.557388i	$0.188196\pi$
$492$	0	0
$493$	31.4763	1.41762
$494$	0	0
$495$	2.01945	0.0907673
$496$	0	0
$497$	5.96520	0.267576
$498$	0	0
$499$	−4.17013	−0.186681	−0.0933404	−	0.995634i	$-0.529754\pi$
$499$	−4.17013	−0.186681	−0.0933404	+	0.995634i	$0.529754\pi$
$500$	0	0
$501$	7.49203	0.334719
$502$	0	0
$503$	21.3039	0.949895	0.474948	−	0.880014i	$-0.342467\pi$
$503$	21.3039	0.949895	0.474948	+	0.880014i	$0.342467\pi$
$504$	0	0
$505$	−1.21105	−0.0538912
$506$	0	0
$507$	−2.78620	−0.123740
$508$	0	0
$509$	39.0643	1.73149	0.865746	−	0.500483i	$-0.166844\pi$
$509$	39.0643	1.73149	0.865746	+	0.500483i	$0.166844\pi$
$510$	0	0
$511$	6.32737	0.279906
$512$	0	0
$513$	4.97882	0.219820
$514$	0	0
$515$	−2.08957	−0.0920774
$516$	0	0
$517$	7.69364	0.338366
$518$	0	0
$519$	−60.1418	−2.63993
$520$	0	0
$521$	−23.0094	−1.00806	−0.504031	−	0.863686i	$-0.668150\pi$
$521$	−23.0094	−1.00806	−0.504031	+	0.863686i	$0.668150\pi$
$522$	0	0
$523$	0.337323	0.0147501	0.00737505	−	0.999973i	$-0.497652\pi$
$523$	0.337323	0.0147501	0.00737505	+	0.999973i	$0.497652\pi$
$524$	0	0
$525$	13.4301	0.586139
$526$	0	0
$527$	−2.92162	−0.127268
$528$	0	0
$529$	36.1921	1.57357
$530$	0	0
$531$	9.14013	0.396648
$532$	0	0
$533$	−3.76016	−0.162871
$534$	0	0
$535$	0.707812	0.0306014
$536$	0	0
$537$	−4.86919	−0.210121
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−27.1501	−1.16727	−0.583636	−	0.812015i	$-0.698371\pi$
$541$	−27.1501	−1.16727	−0.583636	+	0.812015i	$0.698371\pi$
$542$	0	0
$543$	−31.1646	−1.33740
$544$	0	0
$545$	−4.45675	−0.190906
$546$	0	0
$547$	18.9516	0.810313	0.405156	−	0.914247i	$-0.367217\pi$
$547$	18.9516	0.810313	0.405156	+	0.914247i	$0.367217\pi$
$548$	0	0
$549$	−26.0864	−1.11334
$550$	0	0
$551$	7.01780	0.298968
$552$	0	0
$553$	−11.2111	−0.476746
$554$	0	0
$555$	0.194234	0.00824478
$556$	0	0
$557$	−23.1364	−0.980322	−0.490161	−	0.871632i	$-0.663062\pi$
$557$	−23.1364	−0.980322	−0.490161	+	0.871632i	$0.663062\pi$
$558$	0	0
$559$	−7.87421	−0.333043
$560$	0	0
$561$	12.6671	0.534805
$562$	0	0
$563$	19.1179	0.805723	0.402862	−	0.915261i	$-0.368016\pi$
$563$	19.1179	0.805723	0.402862	+	0.915261i	$0.368016\pi$
$564$	0	0
$565$	−0.612709	−0.0257769
$566$	0	0
$567$	−0.603360	−0.0253388
$568$	0	0
$569$	−24.5764	−1.03030	−0.515149	−	0.857101i	$-0.672263\pi$
$569$	−24.5764	−1.03030	−0.515149	+	0.857101i	$0.672263\pi$
$570$	0	0
$571$	11.1388	0.466143	0.233071	−	0.972460i	$-0.425122\pi$
$571$	11.1388	0.466143	0.233071	+	0.972460i	$0.425122\pi$
$572$	0	0
$573$	20.2303	0.845132
$574$	0	0
$575$	37.0851	1.54656
$576$	0	0
$577$	26.8760	1.11886	0.559430	−	0.828877i	$-0.311020\pi$
$577$	26.8760	1.11886	0.559430	+	0.828877i	$0.311020\pi$
$578$	0	0
$579$	19.1382	0.795356
$580$	0	0
$581$	5.61774	0.233063
$582$	0	0
$583$	−0.377040	−0.0156154
$584$	0	0
$585$	−2.01945	−0.0834938
$586$	0	0
$587$	20.9858	0.866178	0.433089	−	0.901351i	$-0.357424\pi$
$587$	20.9858	0.866178	0.433089	+	0.901351i	$0.357424\pi$
$588$	0	0
$589$	−0.651391	−0.0268401
$590$	0	0
$591$	18.1074	0.744839
$592$	0	0
$593$	27.5858	1.13281	0.566407	−	0.824126i	$-0.308333\pi$
$593$	27.5858	1.13281	0.566407	+	0.824126i	$0.308333\pi$
$594$	0	0
$595$	−1.92763	−0.0790250
$596$	0	0
$597$	40.3313	1.65065
$598$	0	0
$599$	17.0157	0.695244	0.347622	−	0.937635i	$-0.386989\pi$
$599$	17.0157	0.695244	0.347622	+	0.937635i	$0.386989\pi$
$600$	0	0
$601$	−6.29993	−0.256980	−0.128490	−	0.991711i	$-0.541013\pi$
$601$	−6.29993	−0.256980	−0.128490	+	0.991711i	$0.541013\pi$
$602$	0	0
$603$	5.71648	0.232793
$604$	0	0
$605$	0.423993	0.0172378
$606$	0	0
$607$	1.05606	0.0428643	0.0214322	−	0.999770i	$-0.493177\pi$
$607$	1.05606	0.0428643	0.0214322	+	0.999770i	$0.493177\pi$
$608$	0	0
$609$	19.2900	0.781670
$610$	0	0
$611$	−7.69364	−0.311251
$612$	0	0
$613$	7.49123	0.302568	0.151284	−	0.988490i	$-0.451659\pi$
$613$	7.49123	0.302568	0.151284	+	0.988490i	$0.451659\pi$
$614$	0	0
$615$	−4.44199	−0.179118
$616$	0	0
$617$	−16.6317	−0.669568	−0.334784	−	0.942295i	$-0.608663\pi$
$617$	−16.6317	−0.669568	−0.334784	+	0.942295i	$0.608663\pi$
$618$	0	0
$619$	−28.3733	−1.14042	−0.570210	−	0.821499i	$-0.693138\pi$
$619$	−28.3733	−1.14042	−0.570210	+	0.821499i	$0.693138\pi$
$620$	0	0
$621$	37.7900	1.51646
$622$	0	0
$623$	1.37756	0.0551907
$624$	0	0
$625$	22.3358	0.893430
$626$	0	0
$627$	2.82419	0.112787
$628$	0	0
$629$	0.747512	0.0298052
$630$	0	0
$631$	17.2962	0.688551	0.344276	−	0.938869i	$-0.388125\pi$
$631$	17.2962	0.688551	0.344276	+	0.938869i	$0.388125\pi$
$632$	0	0
$633$	2.96814	0.117973
$634$	0	0
$635$	8.60765	0.341584
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	28.4118	1.12395
$640$	0	0
$641$	10.4164	0.411422	0.205711	−	0.978613i	$-0.434049\pi$
$641$	10.4164	0.411422	0.205711	+	0.978613i	$0.434049\pi$
$642$	0	0
$643$	−19.4159	−0.765690	−0.382845	−	0.923813i	$-0.625056\pi$
$643$	−19.4159	−0.765690	−0.382845	+	0.923813i	$0.625056\pi$
$644$	0	0
$645$	−9.30204	−0.366268
$646$	0	0
$647$	9.03166	0.355071	0.177536	−	0.984114i	$-0.443188\pi$
$647$	9.03166	0.355071	0.177536	+	0.984114i	$0.443188\pi$
$648$	0	0
$649$	1.91902	0.0753281
$650$	0	0
$651$	−1.79049	−0.0701750
$652$	0	0
$653$	−25.4490	−0.995896	−0.497948	−	0.867207i	$-0.665913\pi$
$653$	−25.4490	−0.995896	−0.497948	+	0.867207i	$0.665913\pi$
$654$	0	0
$655$	2.26375	0.0884520
$656$	0	0
$657$	30.1368	1.17575
$658$	0	0
$659$	16.6034	0.646775	0.323387	−	0.946267i	$-0.395178\pi$
$659$	16.6034	0.646775	0.323387	+	0.946267i	$0.395178\pi$
$660$	0	0
$661$	5.09967	0.198354	0.0991771	−	0.995070i	$-0.468379\pi$
$661$	5.09967	0.198354	0.0991771	+	0.995070i	$0.468379\pi$
$662$	0	0
$663$	−12.6671	−0.491949
$664$	0	0
$665$	−0.429774	−0.0166659
$666$	0	0
$667$	53.2662	2.06247
$668$	0	0
$669$	−52.9928	−2.04882
$670$	0	0
$671$	−5.47698	−0.211436
$672$	0	0
$673$	−30.0358	−1.15779	−0.578897	−	0.815401i	$-0.696517\pi$
$673$	−30.0358	−1.15779	−0.578897	+	0.815401i	$0.696517\pi$
$674$	0	0
$675$	23.6762	0.911299
$676$	0	0
$677$	−27.0939	−1.04130	−0.520652	−	0.853769i	$-0.674311\pi$
$677$	−27.0939	−1.04130	−0.520652	+	0.853769i	$0.674311\pi$
$678$	0	0
$679$	−14.9289	−0.572919
$680$	0	0
$681$	−83.2482	−3.19008
$682$	0	0
$683$	−31.2117	−1.19428	−0.597141	−	0.802137i	$-0.703697\pi$
$683$	−31.2117	−1.19428	−0.597141	+	0.802137i	$0.703697\pi$
$684$	0	0
$685$	8.31825	0.317824
$686$	0	0
$687$	−19.8918	−0.758921
$688$	0	0
$689$	0.377040	0.0143641
$690$	0	0
$691$	21.4608	0.816409	0.408205	−	0.912891i	$-0.366155\pi$
$691$	21.4608	0.816409	0.408205	+	0.912891i	$0.366155\pi$
$692$	0	0
$693$	4.76292	0.180928
$694$	0	0
$695$	5.70216	0.216295
$696$	0	0
$697$	−17.0951	−0.647521
$698$	0	0
$699$	2.66511	0.100804
$700$	0	0
$701$	15.3187	0.578579	0.289289	−	0.957242i	$-0.406581\pi$
$701$	15.3187	0.578579	0.289289	+	0.957242i	$0.406581\pi$
$702$	0	0
$703$	0.166661	0.00628576
$704$	0	0
$705$	−9.08874	−0.342302
$706$	0	0
$707$	−2.85631	−0.107422
$708$	0	0
$709$	21.7511	0.816881	0.408441	−	0.912785i	$-0.366073\pi$
$709$	21.7511	0.816881	0.408441	+	0.912785i	$0.366073\pi$
$710$	0	0
$711$	−53.3977	−2.00257
$712$	0	0
$713$	−4.94416	−0.185160
$714$	0	0
$715$	−0.423993	−0.0158565
$716$	0	0
$717$	−41.7557	−1.55939
$718$	0	0
$719$	−13.6688	−0.509759	−0.254879	−	0.966973i	$-0.582036\pi$
$719$	−13.6688	−0.509759	−0.254879	+	0.966973i	$0.582036\pi$
$720$	0	0
$721$	−4.92831	−0.183540
$722$	0	0
$723$	23.8728	0.887840
$724$	0	0
$725$	33.3724	1.23942
$726$	0	0
$727$	30.8012	1.14235	0.571177	−	0.820827i	$-0.306487\pi$
$727$	30.8012	1.14235	0.571177	+	0.820827i	$0.306487\pi$
$728$	0	0
$729$	−43.9299	−1.62704
$730$	0	0
$731$	−35.7990	−1.32407
$732$	0	0
$733$	25.6470	0.947295	0.473648	−	0.880714i	$-0.342937\pi$
$733$	25.6470	0.947295	0.473648	+	0.880714i	$0.342937\pi$
$734$	0	0
$735$	−1.18133	−0.0435740
$736$	0	0
$737$	1.20021	0.0442101
$738$	0	0
$739$	16.3243	0.600498	0.300249	−	0.953861i	$-0.402930\pi$
$739$	16.3243	0.600498	0.300249	+	0.953861i	$0.402930\pi$
$740$	0	0
$741$	−2.82419	−0.103749
$742$	0	0
$743$	−21.2844	−0.780848	−0.390424	−	0.920635i	$-0.627672\pi$
$743$	−21.2844	−0.780848	−0.390424	+	0.920635i	$0.627672\pi$
$744$	0	0
$745$	−8.98991	−0.329365
$746$	0	0
$747$	26.7568	0.978981
$748$	0	0
$749$	1.66939	0.0609983
$750$	0	0
$751$	−14.5471	−0.530832	−0.265416	−	0.964134i	$-0.585509\pi$
$751$	−14.5471	−0.530832	−0.265416	+	0.964134i	$0.585509\pi$
$752$	0	0
$753$	−69.6543	−2.53834
$754$	0	0
$755$	−8.95889	−0.326047
$756$	0	0
$757$	41.2567	1.49950	0.749750	−	0.661721i	$-0.230174\pi$
$757$	41.2567	1.49950	0.749750	+	0.661721i	$0.230174\pi$
$758$	0	0
$759$	21.4360	0.778079
$760$	0	0
$761$	47.5539	1.72383	0.861913	−	0.507056i	$-0.169266\pi$
$761$	47.5539	1.72383	0.861913	+	0.507056i	$0.169266\pi$
$762$	0	0
$763$	−10.5114	−0.380537
$764$	0	0
$765$	−9.18113	−0.331945
$766$	0	0
$767$	−1.91902	−0.0692917
$768$	0	0
$769$	−35.5060	−1.28038	−0.640189	−	0.768217i	$-0.721144\pi$
$769$	−35.5060	−1.28038	−0.640189	+	0.768217i	$0.721144\pi$
$770$	0	0
$771$	21.0634	0.758581
$772$	0	0
$773$	18.3161	0.658785	0.329393	−	0.944193i	$-0.393156\pi$
$773$	18.3161	0.658785	0.329393	+	0.944193i	$0.393156\pi$
$774$	0	0
$775$	−3.09762	−0.111270
$776$	0	0
$777$	0.458106	0.0164345
$778$	0	0
$779$	−3.81143	−0.136559
$780$	0	0
$781$	5.96520	0.213452
$782$	0	0
$783$	34.0067	1.21530
$784$	0	0
$785$	−1.40140	−0.0500182
$786$	0	0
$787$	19.6687	0.701115	0.350557	−	0.936541i	$-0.385992\pi$
$787$	19.6687	0.701115	0.350557	+	0.936541i	$0.385992\pi$
$788$	0	0
$789$	−4.23936	−0.150925
$790$	0	0
$791$	−1.44509	−0.0513816
$792$	0	0
$793$	5.47698	0.194493
$794$	0	0
$795$	0.445409	0.0157970
$796$	0	0
$797$	38.3160	1.35722	0.678611	−	0.734498i	$-0.262582\pi$
$797$	38.3160	1.35722	0.678611	+	0.734498i	$0.262582\pi$
$798$	0	0
$799$	−34.9781	−1.23744
$800$	0	0
$801$	6.56120	0.231829
$802$	0	0
$803$	6.32737	0.223288
$804$	0	0
$805$	−3.26205	−0.114972
$806$	0	0
$807$	−60.9098	−2.14413
$808$	0	0
$809$	54.6342	1.92084	0.960418	−	0.278561i	$-0.0898576\pi$
$809$	54.6342	1.92084	0.960418	+	0.278561i	$0.0898576\pi$
$810$	0	0
$811$	32.4200	1.13842	0.569211	−	0.822192i	$-0.307249\pi$
$811$	32.4200	1.13842	0.569211	+	0.822192i	$0.307249\pi$
$812$	0	0
$813$	10.0791	0.353489
$814$	0	0
$815$	−3.94861	−0.138314
$816$	0	0
$817$	−7.98157	−0.279240
$818$	0	0
$819$	−4.76292	−0.166430
$820$	0	0
$821$	52.7974	1.84264	0.921320	−	0.388805i	$-0.127112\pi$
$821$	52.7974	1.84264	0.921320	+	0.388805i	$0.127112\pi$
$822$	0	0
$823$	−5.94715	−0.207304	−0.103652	−	0.994614i	$-0.533053\pi$
$823$	−5.94715	−0.207304	−0.103652	+	0.994614i	$0.533053\pi$
$824$	0	0
$825$	13.4301	0.467577
$826$	0	0
$827$	21.0842	0.733170	0.366585	−	0.930385i	$-0.380527\pi$
$827$	21.0842	0.733170	0.366585	+	0.930385i	$0.380527\pi$
$828$	0	0
$829$	−25.8003	−0.896082	−0.448041	−	0.894013i	$-0.647878\pi$
$829$	−25.8003	−0.896082	−0.448041	+	0.894013i	$0.647878\pi$
$830$	0	0
$831$	31.8130	1.10358
$832$	0	0
$833$	−4.54636	−0.157522
$834$	0	0
$835$	−1.14011	−0.0394551
$836$	0	0
$837$	−3.15650	−0.109104
$838$	0	0
$839$	29.6067	1.02214	0.511069	−	0.859540i	$-0.329250\pi$
$839$	29.6067	1.02214	0.511069	+	0.859540i	$0.329250\pi$
$840$	0	0
$841$	18.9335	0.652879
$842$	0	0
$843$	−21.4337	−0.738216
$844$	0	0
$845$	0.423993	0.0145858
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−70.5422	−2.42100
$850$	0	0
$851$	1.26499	0.0433632
$852$	0	0
$853$	0.650155	0.0222609	0.0111304	−	0.999938i	$-0.496457\pi$
$853$	0.650155	0.0222609	0.0111304	+	0.999938i	$0.496457\pi$
$854$	0	0
$855$	−2.04698	−0.0700052
$856$	0	0
$857$	−36.7576	−1.25562	−0.627808	−	0.778368i	$-0.716048\pi$
$857$	−36.7576	−1.25562	−0.627808	+	0.778368i	$0.716048\pi$
$858$	0	0
$859$	−38.5084	−1.31389	−0.656944	−	0.753939i	$-0.728151\pi$
$859$	−38.5084	−1.31389	−0.656944	+	0.753939i	$0.728151\pi$
$860$	0	0
$861$	−10.4766	−0.357040
$862$	0	0
$863$	12.9093	0.439439	0.219720	−	0.975563i	$-0.429486\pi$
$863$	12.9093	0.439439	0.219720	+	0.975563i	$0.429486\pi$
$864$	0	0
$865$	9.15214	0.311182
$866$	0	0
$867$	−10.2237	−0.347216
$868$	0	0
$869$	−11.2111	−0.380312
$870$	0	0
$871$	−1.20021	−0.0406674
$872$	0	0
$873$	−71.1052	−2.40655
$874$	0	0
$875$	−4.16371	−0.140759
$876$	0	0
$877$	7.43402	0.251029	0.125514	−	0.992092i	$-0.459942\pi$
$877$	7.43402	0.251029	0.125514	+	0.992092i	$0.459942\pi$
$878$	0	0
$879$	−47.9187	−1.61626
$880$	0	0
$881$	16.0776	0.541669	0.270835	−	0.962626i	$-0.412700\pi$
$881$	16.0776	0.541669	0.270835	+	0.962626i	$0.412700\pi$
$882$	0	0
$883$	−22.4488	−0.755462	−0.377731	−	0.925915i	$-0.623296\pi$
$883$	−22.4488	−0.755462	−0.377731	+	0.925915i	$0.623296\pi$
$884$	0	0
$885$	−2.26700	−0.0762042
$886$	0	0
$887$	20.0596	0.673536	0.336768	−	0.941588i	$-0.390666\pi$
$887$	20.0596	0.673536	0.336768	+	0.941588i	$0.390666\pi$
$888$	0	0
$889$	20.3014	0.680886
$890$	0	0
$891$	−0.603360	−0.0202133
$892$	0	0
$893$	−7.79854	−0.260968
$894$	0	0
$895$	0.740975	0.0247681
$896$	0	0
$897$	−21.4360	−0.715728
$898$	0	0
$899$	−4.44918	−0.148388
$900$	0	0
$901$	1.71416	0.0571070
$902$	0	0
$903$	−21.9391	−0.730088
$904$	0	0
$905$	4.74251	0.157646
$906$	0	0
$907$	28.8714	0.958659	0.479330	−	0.877635i	$-0.340880\pi$
$907$	28.8714	0.958659	0.479330	+	0.877635i	$0.340880\pi$
$908$	0	0
$909$	−13.6044	−0.451228
$910$	0	0
$911$	−33.0632	−1.09543	−0.547716	−	0.836664i	$-0.684503\pi$
$911$	−33.0632	−1.09543	−0.547716	+	0.836664i	$0.684503\pi$
$912$	0	0
$913$	5.61774	0.185920
$914$	0	0
$915$	6.47012	0.213896
$916$	0	0
$917$	5.33912	0.176313
$918$	0	0
$919$	35.1250	1.15867	0.579334	−	0.815090i	$-0.303313\pi$
$919$	35.1250	1.15867	0.579334	+	0.815090i	$0.303313\pi$
$920$	0	0
$921$	74.1566	2.44354
$922$	0	0
$923$	−5.96520	−0.196347
$924$	0	0
$925$	0.792541	0.0260586
$926$	0	0
$927$	−23.4731	−0.770958
$928$	0	0
$929$	33.7942	1.10875	0.554375	−	0.832267i	$-0.312957\pi$
$929$	33.7942	1.10875	0.554375	+	0.832267i	$0.312957\pi$
$930$	0	0
$931$	−1.01363	−0.0332205
$932$	0	0
$933$	6.86708	0.224818
$934$	0	0
$935$	−1.92763	−0.0630402
$936$	0	0
$937$	25.7477	0.841139	0.420570	−	0.907260i	$-0.361830\pi$
$937$	25.7477	0.841139	0.420570	+	0.907260i	$0.361830\pi$
$938$	0	0
$939$	−42.0075	−1.37086
$940$	0	0
$941$	−13.1645	−0.429149	−0.214575	−	0.976708i	$-0.568837\pi$
$941$	−13.1645	−0.429149	−0.214575	+	0.976708i	$0.568837\pi$
$942$	0	0
$943$	−28.9293	−0.942069
$944$	0	0
$945$	−2.08259	−0.0677467
$946$	0	0
$947$	53.1051	1.72568	0.862842	−	0.505474i	$-0.168682\pi$
$947$	53.1051	1.72568	0.862842	+	0.505474i	$0.168682\pi$
$948$	0	0
$949$	−6.32737	−0.205395
$950$	0	0
$951$	−73.8218	−2.39384
$952$	0	0
$953$	2.77677	0.0899484	0.0449742	−	0.998988i	$-0.485679\pi$
$953$	2.77677	0.0899484	0.0449742	+	0.998988i	$0.485679\pi$
$954$	0	0
$955$	−3.07856	−0.0996200
$956$	0	0
$957$	19.2900	0.623557
$958$	0	0
$959$	19.6188	0.633525
$960$	0	0
$961$	−30.5870	−0.986678
$962$	0	0
$963$	7.95119	0.256224
$964$	0	0
$965$	−2.91237	−0.0937526
$966$	0	0
$967$	−30.6959	−0.987115	−0.493557	−	0.869713i	$-0.664304\pi$
$967$	−30.6959	−0.987115	−0.493557	+	0.869713i	$0.664304\pi$
$968$	0	0
$969$	−12.8398	−0.412473
$970$	0	0
$971$	24.2344	0.777719	0.388860	−	0.921297i	$-0.372869\pi$
$971$	24.2344	0.777719	0.388860	+	0.921297i	$0.372869\pi$
$972$	0	0
$973$	13.4487	0.431146
$974$	0	0
$975$	−13.4301	−0.430108
$976$	0	0
$977$	21.5958	0.690910	0.345455	−	0.938435i	$-0.387725\pi$
$977$	21.5958	0.690910	0.345455	+	0.938435i	$0.387725\pi$
$978$	0	0
$979$	1.37756	0.0440270
$980$	0	0
$981$	−50.0648	−1.59844
$982$	0	0
$983$	21.5394	0.687001	0.343501	−	0.939152i	$-0.388387\pi$
$983$	21.5394	0.687001	0.343501	+	0.939152i	$0.388387\pi$
$984$	0	0
$985$	−2.75551	−0.0877979
$986$	0	0
$987$	−21.4360	−0.682316
$988$	0	0
$989$	−60.5813	−1.92637
$990$	0	0
$991$	48.8557	1.55195	0.775976	−	0.630762i	$-0.217258\pi$
$991$	48.8557	1.55195	0.775976	+	0.630762i	$0.217258\pi$
$992$	0	0
$993$	−78.0242	−2.47602
$994$	0	0
$995$	−6.13747	−0.194571
$996$	0	0
$997$	−26.8771	−0.851205	−0.425603	−	0.904910i	$-0.639938\pi$
$997$	−26.8771	−0.851205	−0.425603	+	0.904910i	$0.639938\pi$
$998$	0	0
$999$	0.807605	0.0255515

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.q.1.1	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.q.1.1	✓	9	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q-2.78620 q^{3} +0.423993 q^{5} +1.00000 q^{7} +4.76292 q^{9} +O(q^{10})\)	\(q-2.78620 q^{3} +0.423993 q^{5} +1.00000 q^{7} +4.76292 q^{9} +1.00000 q^{11} -1.00000 q^{13} -1.18133 q^{15} -4.54636 q^{17} -1.01363 q^{19} -2.78620 q^{21} -7.69364 q^{23} -4.82023 q^{25} -4.91185 q^{27} -6.92340 q^{29} +0.642629 q^{31} -2.78620 q^{33} +0.423993 q^{35} -0.164420 q^{37} +2.78620 q^{39} +3.76016 q^{41} +7.87421 q^{43} +2.01945 q^{45} +7.69364 q^{47} +1.00000 q^{49} +12.6671 q^{51} -0.377040 q^{53} +0.423993 q^{55} +2.82419 q^{57} +1.91902 q^{59} -5.47698 q^{61} +4.76292 q^{63} -0.423993 q^{65} +1.20021 q^{67} +21.4360 q^{69} +5.96520 q^{71} +6.32737 q^{73} +13.4301 q^{75} +1.00000 q^{77} -11.2111 q^{79} -0.603360 q^{81} +5.61774 q^{83} -1.92763 q^{85} +19.2900 q^{87} +1.37756 q^{89} -1.00000 q^{91} -1.79049 q^{93} -0.429774 q^{95} -14.9289 q^{97} +4.76292 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9	\( 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + q^{17} + 16 q^{19} + 3 q^{21} + 6 q^{23} + 11 q^{25} + 9 q^{27} - 2 q^{29} + 3 q^{31} + 3 q^{33} + 8 q^{35} - 4 q^{37} - 3 q^{39} + 20 q^{41} + 20 q^{43} + 8 q^{45} - 6 q^{47} + 9 q^{49} + 15 q^{51} + 15 q^{53} + 8 q^{55} + 16 q^{57} + 21 q^{59} + 12 q^{63} - 8 q^{65} + 18 q^{67} + 10 q^{69} + 12 q^{71} + 29 q^{73} + 12 q^{75} + 9 q^{77} - 6 q^{79} + 5 q^{81} + 25 q^{83} + 13 q^{85} + 17 q^{87} + 32 q^{89} - 9 q^{91} - 13 q^{93} + 38 q^{95} + 17 q^{97} + 12 q^{99}+O(q^{100}) \) 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 - 9 * q^15 + q^17 + 16 * q^19 + 3 * q^21 + 6 * q^23 + 11 * q^25 + 9 * q^27 - 2 * q^29 + 3 * q^31 + 3 * q^33 + 8 * q^35 - 4 * q^37 - 3 * q^39 + 20 * q^41 + 20 * q^43 + 8 * q^45 - 6 * q^47 + 9 * q^49 + 15 * q^51 + 15 * q^53 + 8 * q^55 + 16 * q^57 + 21 * q^59 + 12 * q^63 - 8 * q^65 + 18 * q^67 + 10 * q^69 + 12 * q^71 + 29 * q^73 + 12 * q^75 + 9 * q^77 - 6 * q^79 + 5 * q^81 + 25 * q^83 + 13 * q^85 + 17 * q^87 + 32 * q^89 - 9 * q^91 - 13 * q^93 + 38 * q^95 + 17 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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