LMFDB - Embedded newform 4008.2.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.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:	$32.0040411301$
Analytic rank:	$1$
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} - 16x^{7} + 45x^{6} + 67x^{5} - 166x^{4} - 83x^{3} + 152x^{2} + 51x - 10 $ x^9 - 3x^8 - 16x^7 + 45x^6 + 67x^5 - 166x^4 - 83x^3 + 152x^2 + 51x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.482381$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.48238 q^{5} -1.03908 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.48238 q^{5} -1.03908 q^{7} +1.00000 q^{9} +5.64105 q^{11} -4.47983 q^{13} -1.48238 q^{15} -6.13922 q^{17} +5.87154 q^{19} -1.03908 q^{21} -1.22710 q^{23} -2.80255 q^{25} +1.00000 q^{27} -6.91991 q^{29} +3.70123 q^{31} +5.64105 q^{33} +1.54031 q^{35} -6.62294 q^{37} -4.47983 q^{39} +1.85846 q^{41} -2.29781 q^{43} -1.48238 q^{45} +3.59229 q^{47} -5.92031 q^{49} -6.13922 q^{51} -0.299721 q^{53} -8.36218 q^{55} +5.87154 q^{57} +0.102376 q^{59} +3.64463 q^{61} -1.03908 q^{63} +6.64082 q^{65} -5.27810 q^{67} -1.22710 q^{69} +2.00339 q^{71} +9.34023 q^{73} -2.80255 q^{75} -5.86150 q^{77} -3.92469 q^{79} +1.00000 q^{81} -11.4441 q^{83} +9.10067 q^{85} -6.91991 q^{87} -10.4583 q^{89} +4.65490 q^{91} +3.70123 q^{93} -8.70385 q^{95} -3.40524 q^{97} +5.64105 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} - 6 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^3 - 6 * q^5 - 11 * q^7 + 9 * q^9	$ 9 q + 9 q^{3} - 6 q^{5} - 11 q^{7} + 9 q^{9} + q^{11} - 4 q^{13} - 6 q^{15} - 9 q^{17} - 8 q^{19} - 11 q^{21} - 7 q^{23} - q^{25} + 9 q^{27} - 9 q^{29} - 25 q^{31} + q^{33} + 5 q^{35} - 6 q^{37} - 4 q^{39} - 4 q^{41} - 24 q^{43} - 6 q^{45} - 16 q^{47} + 4 q^{49} - 9 q^{51} - 26 q^{53} - 29 q^{55} - 8 q^{57} - 4 q^{59} - 20 q^{61} - 11 q^{63} - 8 q^{65} - 25 q^{67} - 7 q^{69} - 15 q^{71} - 10 q^{73} - q^{75} - 20 q^{77} - 34 q^{79} + 9 q^{81} - 4 q^{83} - 13 q^{85} - 9 q^{87} + 13 q^{89} - 21 q^{91} - 25 q^{93} - 7 q^{95} - 4 q^{97} + q^{99}+O(q^{100}) $ 9 * q + 9 * q^3 - 6 * q^5 - 11 * q^7 + 9 * q^9 + q^11 - 4 * q^13 - 6 * q^15 - 9 * q^17 - 8 * q^19 - 11 * q^21 - 7 * q^23 - q^25 + 9 * q^27 - 9 * q^29 - 25 * q^31 + q^33 + 5 * q^35 - 6 * q^37 - 4 * q^39 - 4 * q^41 - 24 * q^43 - 6 * q^45 - 16 * q^47 + 4 * q^49 - 9 * q^51 - 26 * q^53 - 29 * q^55 - 8 * q^57 - 4 * q^59 - 20 * q^61 - 11 * q^63 - 8 * q^65 - 25 * q^67 - 7 * q^69 - 15 * q^71 - 10 * q^73 - q^75 - 20 * q^77 - 34 * q^79 + 9 * q^81 - 4 * q^83 - 13 * q^85 - 9 * q^87 + 13 * q^89 - 21 * q^91 - 25 * q^93 - 7 * q^95 - 4 * q^97 + 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.48238	−0.662941	−0.331470	−	0.943466i	$-0.607545\pi$
$5$	−1.48238	−0.662941	−0.331470	+	0.943466i	$0.607545\pi$
$6$	0	0
$7$	−1.03908	−0.392735	−0.196368	−	0.980530i	$-0.562915\pi$
$7$	−1.03908	−0.392735	−0.196368	+	0.980530i	$0.562915\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.64105	1.70084	0.850420	−	0.526105i	$-0.176348\pi$
$11$	5.64105	1.70084	0.850420	+	0.526105i	$0.176348\pi$
$12$	0	0
$13$	−4.47983	−1.24248	−0.621241	−	0.783620i	$-0.713371\pi$
$13$	−4.47983	−1.24248	−0.621241	+	0.783620i	$0.713371\pi$
$14$	0	0
$15$	−1.48238	−0.382749
$16$	0	0
$17$	−6.13922	−1.48898	−0.744490	−	0.667634i	$-0.767307\pi$
$17$	−6.13922	−1.48898	−0.744490	+	0.667634i	$0.767307\pi$
$18$	0	0
$19$	5.87154	1.34702	0.673511	−	0.739177i	$-0.264785\pi$
$19$	5.87154	1.34702	0.673511	+	0.739177i	$0.264785\pi$
$20$	0	0
$21$	−1.03908	−0.226746
$22$	0	0
$23$	−1.22710	−0.255869	−0.127934	−	0.991783i	$-0.540835\pi$
$23$	−1.22710	−0.255869	−0.127934	+	0.991783i	$0.540835\pi$
$24$	0	0
$25$	−2.80255	−0.560509
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.91991	−1.28500	−0.642498	−	0.766288i	$-0.722102\pi$
$29$	−6.91991	−1.28500	−0.642498	+	0.766288i	$0.722102\pi$
$30$	0	0
$31$	3.70123	0.664760	0.332380	−	0.943146i	$-0.392148\pi$
$31$	3.70123	0.664760	0.332380	+	0.943146i	$0.392148\pi$
$32$	0	0
$33$	5.64105	0.981980
$34$	0	0
$35$	1.54031	0.260360
$36$	0	0
$37$	−6.62294	−1.08880	−0.544402	−	0.838824i	$-0.683244\pi$
$37$	−6.62294	−1.08880	−0.544402	+	0.838824i	$0.683244\pi$
$38$	0	0
$39$	−4.47983	−0.717347
$40$	0	0
$41$	1.85846	0.290242	0.145121	−	0.989414i	$-0.453643\pi$
$41$	1.85846	0.290242	0.145121	+	0.989414i	$0.453643\pi$
$42$	0	0
$43$	−2.29781	−0.350413	−0.175207	−	0.984532i	$-0.556059\pi$
$43$	−2.29781	−0.350413	−0.175207	+	0.984532i	$0.556059\pi$
$44$	0	0
$45$	−1.48238	−0.220980
$46$	0	0
$47$	3.59229	0.523990	0.261995	−	0.965069i	$-0.415620\pi$
$47$	3.59229	0.523990	0.261995	+	0.965069i	$0.415620\pi$
$48$	0	0
$49$	−5.92031	−0.845759
$50$	0	0
$51$	−6.13922	−0.859663
$52$	0	0
$53$	−0.299721	−0.0411699	−0.0205849	−	0.999788i	$-0.506553\pi$
$53$	−0.299721	−0.0411699	−0.0205849	+	0.999788i	$0.506553\pi$
$54$	0	0
$55$	−8.36218	−1.12756
$56$	0	0
$57$	5.87154	0.777704
$58$	0	0
$59$	0.102376	0.0133282	0.00666409	−	0.999978i	$-0.497879\pi$
$59$	0.102376	0.0133282	0.00666409	+	0.999978i	$0.497879\pi$
$60$	0	0
$61$	3.64463	0.466647	0.233323	−	0.972399i	$-0.425040\pi$
$61$	3.64463	0.466647	0.233323	+	0.972399i	$0.425040\pi$
$62$	0	0
$63$	−1.03908	−0.130912
$64$	0	0
$65$	6.64082	0.823692
$66$	0	0
$67$	−5.27810	−0.644823	−0.322412	−	0.946600i	$-0.604493\pi$
$67$	−5.27810	−0.644823	−0.322412	+	0.946600i	$0.604493\pi$
$68$	0	0
$69$	−1.22710	−0.147726
$70$	0	0
$71$	2.00339	0.237759	0.118879	−	0.992909i	$-0.462070\pi$
$71$	2.00339	0.237759	0.118879	+	0.992909i	$0.462070\pi$
$72$	0	0
$73$	9.34023	1.09319	0.546596	−	0.837397i	$-0.315923\pi$
$73$	9.34023	1.09319	0.546596	+	0.837397i	$0.315923\pi$
$74$	0	0
$75$	−2.80255	−0.323610
$76$	0	0
$77$	−5.86150	−0.667979
$78$	0	0
$79$	−3.92469	−0.441562	−0.220781	−	0.975323i	$-0.570861\pi$
$79$	−3.92469	−0.441562	−0.220781	+	0.975323i	$0.570861\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.4441	−1.25615	−0.628075	−	0.778153i	$-0.716157\pi$
$83$	−11.4441	−1.25615	−0.628075	+	0.778153i	$0.716157\pi$
$84$	0	0
$85$	9.10067	0.987106
$86$	0	0
$87$	−6.91991	−0.741893
$88$	0	0
$89$	−10.4583	−1.10858	−0.554290	−	0.832324i	$-0.687010\pi$
$89$	−10.4583	−1.10858	−0.554290	+	0.832324i	$0.687010\pi$
$90$	0	0
$91$	4.65490	0.487966
$92$	0	0
$93$	3.70123	0.383799
$94$	0	0
$95$	−8.70385	−0.892997
$96$	0	0
$97$	−3.40524	−0.345749	−0.172875	−	0.984944i	$-0.555306\pi$
$97$	−3.40524	−0.345749	−0.172875	+	0.984944i	$0.555306\pi$
$98$	0	0
$99$	5.64105	0.566947
$100$	0	0
$101$	−12.6052	−1.25426	−0.627132	−	0.778913i	$-0.715771\pi$
$101$	−12.6052	−1.25426	−0.627132	+	0.778913i	$0.715771\pi$
$102$	0	0
$103$	−15.3704	−1.51449	−0.757245	−	0.653131i	$-0.773455\pi$
$103$	−15.3704	−1.51449	−0.757245	+	0.653131i	$0.773455\pi$
$104$	0	0
$105$	1.54031	0.150319
$106$	0	0
$107$	6.62482	0.640446	0.320223	−	0.947342i	$-0.396242\pi$
$107$	6.62482	0.640446	0.320223	+	0.947342i	$0.396242\pi$
$108$	0	0
$109$	−18.9911	−1.81901	−0.909507	−	0.415690i	$-0.863540\pi$
$109$	−18.9911	−1.81901	−0.909507	+	0.415690i	$0.863540\pi$
$110$	0	0
$111$	−6.62294	−0.628622
$112$	0	0
$113$	16.3656	1.53955	0.769774	−	0.638316i	$-0.220369\pi$
$113$	16.3656	1.53955	0.769774	+	0.638316i	$0.220369\pi$
$114$	0	0
$115$	1.81903	0.169626
$116$	0	0
$117$	−4.47983	−0.414160
$118$	0	0
$119$	6.37914	0.584775
$120$	0	0
$121$	20.8214	1.89286
$122$	0	0
$123$	1.85846	0.167571
$124$	0	0
$125$	11.5663	1.03453
$126$	0	0
$127$	17.6141	1.56300	0.781499	−	0.623906i	$-0.214455\pi$
$127$	17.6141	1.56300	0.781499	+	0.623906i	$0.214455\pi$
$128$	0	0
$129$	−2.29781	−0.202311
$130$	0	0
$131$	9.79697	0.855965	0.427982	−	0.903787i	$-0.359224\pi$
$131$	9.79697	0.855965	0.427982	+	0.903787i	$0.359224\pi$
$132$	0	0
$133$	−6.10099	−0.529023
$134$	0	0
$135$	−1.48238	−0.127583
$136$	0	0
$137$	−6.81856	−0.582549	−0.291274	−	0.956640i	$-0.594079\pi$
$137$	−6.81856	−0.582549	−0.291274	+	0.956640i	$0.594079\pi$
$138$	0	0
$139$	−5.34794	−0.453606	−0.226803	−	0.973941i	$-0.572827\pi$
$139$	−5.34794	−0.453606	−0.226803	+	0.973941i	$0.572827\pi$
$140$	0	0
$141$	3.59229	0.302526
$142$	0	0
$143$	−25.2709	−2.11326
$144$	0	0
$145$	10.2579	0.851876
$146$	0	0
$147$	−5.92031	−0.488299
$148$	0	0
$149$	−20.0944	−1.64620	−0.823100	−	0.567896i	$-0.807757\pi$
$149$	−20.0944	−1.64620	−0.823100	+	0.567896i	$0.807757\pi$
$150$	0	0
$151$	−15.4596	−1.25808	−0.629041	−	0.777372i	$-0.716552\pi$
$151$	−15.4596	−1.25808	−0.629041	+	0.777372i	$0.716552\pi$
$152$	0	0
$153$	−6.13922	−0.496327
$154$	0	0
$155$	−5.48663	−0.440696
$156$	0	0
$157$	10.6576	0.850568	0.425284	−	0.905060i	$-0.360174\pi$
$157$	10.6576	0.850568	0.425284	+	0.905060i	$0.360174\pi$
$158$	0	0
$159$	−0.299721	−0.0237694
$160$	0	0
$161$	1.27506	0.100489
$162$	0	0
$163$	−21.2802	−1.66680	−0.833399	−	0.552672i	$-0.813608\pi$
$163$	−21.2802	−1.66680	−0.833399	+	0.552672i	$0.813608\pi$
$164$	0	0
$165$	−8.36218	−0.650995
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	7.06888	0.543760
$170$	0	0
$171$	5.87154	0.449008
$172$	0	0
$173$	−16.4057	−1.24730	−0.623650	−	0.781704i	$-0.714351\pi$
$173$	−16.4057	−1.24730	−0.623650	+	0.781704i	$0.714351\pi$
$174$	0	0
$175$	2.91207	0.220132
$176$	0	0
$177$	0.102376	0.00769503
$178$	0	0
$179$	1.21604	0.0908913	0.0454456	−	0.998967i	$-0.485529\pi$
$179$	1.21604	0.0908913	0.0454456	+	0.998967i	$0.485529\pi$
$180$	0	0
$181$	−6.24694	−0.464331	−0.232166	−	0.972676i	$-0.574581\pi$
$181$	−6.24694	−0.464331	−0.232166	+	0.972676i	$0.574581\pi$
$182$	0	0
$183$	3.64463	0.269419
$184$	0	0
$185$	9.81772	0.721813
$186$	0	0
$187$	−34.6316	−2.53252
$188$	0	0
$189$	−1.03908	−0.0755819
$190$	0	0
$191$	16.2291	1.17429	0.587146	−	0.809481i	$-0.300251\pi$
$191$	16.2291	1.17429	0.587146	+	0.809481i	$0.300251\pi$
$192$	0	0
$193$	−8.97087	−0.645737	−0.322869	−	0.946444i	$-0.604647\pi$
$193$	−8.97087	−0.645737	−0.322869	+	0.946444i	$0.604647\pi$
$194$	0	0
$195$	6.64082	0.475559
$196$	0	0
$197$	−23.1926	−1.65240	−0.826201	−	0.563376i	$-0.809502\pi$
$197$	−23.1926	−1.65240	−0.826201	+	0.563376i	$0.809502\pi$
$198$	0	0
$199$	−13.8934	−0.984876	−0.492438	−	0.870348i	$-0.663894\pi$
$199$	−13.8934	−0.984876	−0.492438	+	0.870348i	$0.663894\pi$
$200$	0	0
$201$	−5.27810	−0.372289
$202$	0	0
$203$	7.19034	0.504663
$204$	0	0
$205$	−2.75494	−0.192413
$206$	0	0
$207$	−1.22710	−0.0852895
$208$	0	0
$209$	33.1216	2.29107
$210$	0	0
$211$	−11.9560	−0.823082	−0.411541	−	0.911391i	$-0.635009\pi$
$211$	−11.9560	−0.823082	−0.411541	+	0.911391i	$0.635009\pi$
$212$	0	0
$213$	2.00339	0.137270
$214$	0	0
$215$	3.40623	0.232303
$216$	0	0
$217$	−3.84587	−0.261074
$218$	0	0
$219$	9.34023	0.631155
$220$	0	0
$221$	27.5027	1.85003
$222$	0	0
$223$	−25.7496	−1.72432	−0.862159	−	0.506638i	$-0.830888\pi$
$223$	−25.7496	−1.72432	−0.862159	+	0.506638i	$0.830888\pi$
$224$	0	0
$225$	−2.80255	−0.186836
$226$	0	0
$227$	23.2899	1.54580	0.772901	−	0.634527i	$-0.218805\pi$
$227$	23.2899	1.54580	0.772901	+	0.634527i	$0.218805\pi$
$228$	0	0
$229$	−10.4690	−0.691813	−0.345906	−	0.938269i	$-0.612429\pi$
$229$	−10.4690	−0.691813	−0.345906	+	0.938269i	$0.612429\pi$
$230$	0	0
$231$	−5.86150	−0.385658
$232$	0	0
$233$	−19.6195	−1.28532	−0.642658	−	0.766153i	$-0.722168\pi$
$233$	−19.6195	−1.28532	−0.642658	+	0.766153i	$0.722168\pi$
$234$	0	0
$235$	−5.32515	−0.347374
$236$	0	0
$237$	−3.92469	−0.254936
$238$	0	0
$239$	15.5821	1.00793	0.503963	−	0.863725i	$-0.331875\pi$
$239$	15.5821	1.00793	0.503963	+	0.863725i	$0.331875\pi$
$240$	0	0
$241$	−30.8313	−1.98602	−0.993011	−	0.118025i	$-0.962344\pi$
$241$	−30.8313	−1.98602	−0.993011	+	0.118025i	$0.962344\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	8.77616	0.560688
$246$	0	0
$247$	−26.3035	−1.67365
$248$	0	0
$249$	−11.4441	−0.725238
$250$	0	0
$251$	−5.66860	−0.357799	−0.178899	−	0.983867i	$-0.557254\pi$
$251$	−5.66860	−0.357799	−0.178899	+	0.983867i	$0.557254\pi$
$252$	0	0
$253$	−6.92214	−0.435191
$254$	0	0
$255$	9.10067	0.569906
$256$	0	0
$257$	−6.81618	−0.425182	−0.212591	−	0.977141i	$-0.568190\pi$
$257$	−6.81618	−0.425182	−0.212591	+	0.977141i	$0.568190\pi$
$258$	0	0
$259$	6.88176	0.427612
$260$	0	0
$261$	−6.91991	−0.428332
$262$	0	0
$263$	4.22632	0.260606	0.130303	−	0.991474i	$-0.458405\pi$
$263$	4.22632	0.260606	0.130303	+	0.991474i	$0.458405\pi$
$264$	0	0
$265$	0.444301	0.0272932
$266$	0	0
$267$	−10.4583	−0.640039
$268$	0	0
$269$	20.5783	1.25468	0.627340	−	0.778745i	$-0.284144\pi$
$269$	20.5783	1.25468	0.627340	+	0.778745i	$0.284144\pi$
$270$	0	0
$271$	−6.93172	−0.421072	−0.210536	−	0.977586i	$-0.567521\pi$
$271$	−6.93172	−0.421072	−0.210536	+	0.977586i	$0.567521\pi$
$272$	0	0
$273$	4.65490	0.281727
$274$	0	0
$275$	−15.8093	−0.953336
$276$	0	0
$277$	22.5380	1.35418	0.677088	−	0.735902i	$-0.263242\pi$
$277$	22.5380	1.35418	0.677088	+	0.735902i	$0.263242\pi$
$278$	0	0
$279$	3.70123	0.221587
$280$	0	0
$281$	20.5684	1.22701	0.613504	−	0.789691i	$-0.289759\pi$
$281$	20.5684	1.22701	0.613504	+	0.789691i	$0.289759\pi$
$282$	0	0
$283$	29.4021	1.74777	0.873886	−	0.486131i	$-0.161592\pi$
$283$	29.4021	1.74777	0.873886	+	0.486131i	$0.161592\pi$
$284$	0	0
$285$	−8.70385	−0.515572
$286$	0	0
$287$	−1.93108	−0.113988
$288$	0	0
$289$	20.6900	1.21706
$290$	0	0
$291$	−3.40524	−0.199618
$292$	0	0
$293$	20.2839	1.18500	0.592500	−	0.805571i	$-0.298141\pi$
$293$	20.2839	1.18500	0.592500	+	0.805571i	$0.298141\pi$
$294$	0	0
$295$	−0.151760	−0.00883580
$296$	0	0
$297$	5.64105	0.327327
$298$	0	0
$299$	5.49721	0.317912
$300$	0	0
$301$	2.38761	0.137619
$302$	0	0
$303$	−12.6052	−0.724149
$304$	0	0
$305$	−5.40273	−0.309359
$306$	0	0
$307$	10.9899	0.627224	0.313612	−	0.949551i	$-0.398461\pi$
$307$	10.9899	0.627224	0.313612	+	0.949551i	$0.398461\pi$
$308$	0	0
$309$	−15.3704	−0.874391
$310$	0	0
$311$	26.9703	1.52935	0.764673	−	0.644418i	$-0.222900\pi$
$311$	26.9703	1.52935	0.764673	+	0.644418i	$0.222900\pi$
$312$	0	0
$313$	−32.1109	−1.81501	−0.907507	−	0.420036i	$-0.862017\pi$
$313$	−32.1109	−1.81501	−0.907507	+	0.420036i	$0.862017\pi$
$314$	0	0
$315$	1.54031	0.0867867
$316$	0	0
$317$	−21.3657	−1.20002	−0.600008	−	0.799994i	$-0.704836\pi$
$317$	−21.3657	−1.20002	−0.600008	+	0.799994i	$0.704836\pi$
$318$	0	0
$319$	−39.0356	−2.18557
$320$	0	0
$321$	6.62482	0.369761
$322$	0	0
$323$	−36.0467	−2.00569
$324$	0	0
$325$	12.5549	0.696422
$326$	0	0
$327$	−18.9911	−1.05021
$328$	0	0
$329$	−3.73268	−0.205789
$330$	0	0
$331$	23.6701	1.30102	0.650512	−	0.759496i	$-0.274554\pi$
$331$	23.6701	1.30102	0.650512	+	0.759496i	$0.274554\pi$
$332$	0	0
$333$	−6.62294	−0.362935
$334$	0	0
$335$	7.82416	0.427480
$336$	0	0
$337$	−23.6003	−1.28559	−0.642796	−	0.766037i	$-0.722226\pi$
$337$	−23.6003	−1.28559	−0.642796	+	0.766037i	$0.722226\pi$
$338$	0	0
$339$	16.3656	0.888859
$340$	0	0
$341$	20.8788	1.13065
$342$	0	0
$343$	13.4252	0.724894
$344$	0	0
$345$	1.81903	0.0979335
$346$	0	0
$347$	17.2073	0.923736	0.461868	−	0.886949i	$-0.347179\pi$
$347$	17.2073	0.923736	0.461868	+	0.886949i	$0.347179\pi$
$348$	0	0
$349$	20.7148	1.10884	0.554419	−	0.832238i	$-0.312941\pi$
$349$	20.7148	1.10884	0.554419	+	0.832238i	$0.312941\pi$
$350$	0	0
$351$	−4.47983	−0.239116
$352$	0	0
$353$	1.81530	0.0966187	0.0483094	−	0.998832i	$-0.484617\pi$
$353$	1.81530	0.0966187	0.0483094	+	0.998832i	$0.484617\pi$
$354$	0	0
$355$	−2.96979	−0.157620
$356$	0	0
$357$	6.37914	0.337620
$358$	0	0
$359$	−7.55958	−0.398979	−0.199490	−	0.979900i	$-0.563928\pi$
$359$	−7.55958	−0.398979	−0.199490	+	0.979900i	$0.563928\pi$
$360$	0	0
$361$	15.4749	0.814471
$362$	0	0
$363$	20.8214	1.09284
$364$	0	0
$365$	−13.8458	−0.724722
$366$	0	0
$367$	−24.2242	−1.26449	−0.632247	−	0.774767i	$-0.717867\pi$
$367$	−24.2242	−1.26449	−0.632247	+	0.774767i	$0.717867\pi$
$368$	0	0
$369$	1.85846	0.0967473
$370$	0	0
$371$	0.311434	0.0161689
$372$	0	0
$373$	8.50121	0.440176	0.220088	−	0.975480i	$-0.429366\pi$
$373$	8.50121	0.440176	0.220088	+	0.975480i	$0.429366\pi$
$374$	0	0
$375$	11.5663	0.597284
$376$	0	0
$377$	31.0000	1.59658
$378$	0	0
$379$	−4.72626	−0.242772	−0.121386	−	0.992605i	$-0.538734\pi$
$379$	−4.72626	−0.242772	−0.121386	+	0.992605i	$0.538734\pi$
$380$	0	0
$381$	17.6141	0.902398
$382$	0	0
$383$	22.7605	1.16301	0.581505	−	0.813543i	$-0.302464\pi$
$383$	22.7605	1.16301	0.581505	+	0.813543i	$0.302464\pi$
$384$	0	0
$385$	8.68897	0.442831
$386$	0	0
$387$	−2.29781	−0.116804
$388$	0	0
$389$	26.2141	1.32911	0.664554	−	0.747241i	$-0.268622\pi$
$389$	26.2141	1.32911	0.664554	+	0.747241i	$0.268622\pi$
$390$	0	0
$391$	7.53346	0.380983
$392$	0	0
$393$	9.79697	0.494192
$394$	0	0
$395$	5.81789	0.292730
$396$	0	0
$397$	8.99098	0.451244	0.225622	−	0.974215i	$-0.427559\pi$
$397$	8.99098	0.451244	0.225622	+	0.974215i	$0.427559\pi$
$398$	0	0
$399$	−6.10099	−0.305432
$400$	0	0
$401$	4.85669	0.242531	0.121266	−	0.992620i	$-0.461305\pi$
$401$	4.85669	0.242531	0.121266	+	0.992620i	$0.461305\pi$
$402$	0	0
$403$	−16.5809	−0.825952
$404$	0	0
$405$	−1.48238	−0.0736601
$406$	0	0
$407$	−37.3603	−1.85188
$408$	0	0
$409$	28.4764	1.40807	0.704034	−	0.710166i	$-0.251380\pi$
$409$	28.4764	1.40807	0.704034	+	0.710166i	$0.251380\pi$
$410$	0	0
$411$	−6.81856	−0.336335
$412$	0	0
$413$	−0.106376	−0.00523444
$414$	0	0
$415$	16.9645	0.832753
$416$	0	0
$417$	−5.34794	−0.261890
$418$	0	0
$419$	22.7505	1.11143	0.555717	−	0.831371i	$-0.312444\pi$
$419$	22.7505	1.11143	0.555717	+	0.831371i	$0.312444\pi$
$420$	0	0
$421$	9.05605	0.441365	0.220682	−	0.975346i	$-0.429172\pi$
$421$	9.05605	0.441365	0.220682	+	0.975346i	$0.429172\pi$
$422$	0	0
$423$	3.59229	0.174663
$424$	0	0
$425$	17.2055	0.834587
$426$	0	0
$427$	−3.78706	−0.183269
$428$	0	0
$429$	−25.2709	−1.22009
$430$	0	0
$431$	7.21053	0.347319	0.173660	−	0.984806i	$-0.444441\pi$
$431$	7.21053	0.347319	0.173660	+	0.984806i	$0.444441\pi$
$432$	0	0
$433$	−26.9366	−1.29449	−0.647246	−	0.762281i	$-0.724079\pi$
$433$	−26.9366	−1.29449	−0.647246	+	0.762281i	$0.724079\pi$
$434$	0	0
$435$	10.2579	0.491831
$436$	0	0
$437$	−7.20498	−0.344661
$438$	0	0
$439$	−24.4736	−1.16806	−0.584031	−	0.811731i	$-0.698525\pi$
$439$	−24.4736	−1.16806	−0.584031	+	0.811731i	$0.698525\pi$
$440$	0	0
$441$	−5.92031	−0.281920
$442$	0	0
$443$	32.1730	1.52858	0.764292	−	0.644870i	$-0.223088\pi$
$443$	32.1730	1.52858	0.764292	+	0.644870i	$0.223088\pi$
$444$	0	0
$445$	15.5032	0.734923
$446$	0	0
$447$	−20.0944	−0.950434
$448$	0	0
$449$	37.2697	1.75887	0.879434	−	0.476021i	$-0.157921\pi$
$449$	37.2697	1.75887	0.879434	+	0.476021i	$0.157921\pi$
$450$	0	0
$451$	10.4836	0.493655
$452$	0	0
$453$	−15.4596	−0.726354
$454$	0	0
$455$	−6.90033	−0.323493
$456$	0	0
$457$	−3.52936	−0.165096	−0.0825482	−	0.996587i	$-0.526306\pi$
$457$	−3.52936	−0.165096	−0.0825482	+	0.996587i	$0.526306\pi$
$458$	0	0
$459$	−6.13922	−0.286554
$460$	0	0
$461$	−5.01289	−0.233474	−0.116737	−	0.993163i	$-0.537243\pi$
$461$	−5.01289	−0.233474	−0.116737	+	0.993163i	$0.537243\pi$
$462$	0	0
$463$	−11.8928	−0.552707	−0.276353	−	0.961056i	$-0.589126\pi$
$463$	−11.8928	−0.552707	−0.276353	+	0.961056i	$0.589126\pi$
$464$	0	0
$465$	−5.48663	−0.254436
$466$	0	0
$467$	−8.68147	−0.401730	−0.200865	−	0.979619i	$-0.564375\pi$
$467$	−8.68147	−0.401730	−0.200865	+	0.979619i	$0.564375\pi$
$468$	0	0
$469$	5.48437	0.253245
$470$	0	0
$471$	10.6576	0.491076
$472$	0	0
$473$	−12.9621	−0.595996
$474$	0	0
$475$	−16.4553	−0.755019
$476$	0	0
$477$	−0.299721	−0.0137233
$478$	0	0
$479$	−11.9188	−0.544585	−0.272293	−	0.962215i	$-0.587782\pi$
$479$	−11.9188	−0.544585	−0.272293	+	0.962215i	$0.587782\pi$
$480$	0	0
$481$	29.6697	1.35282
$482$	0	0
$483$	1.27506	0.0580171
$484$	0	0
$485$	5.04786	0.229211
$486$	0	0
$487$	34.7303	1.57378	0.786889	−	0.617095i	$-0.211690\pi$
$487$	34.7303	1.57378	0.786889	+	0.617095i	$0.211690\pi$
$488$	0	0
$489$	−21.2802	−0.962326
$490$	0	0
$491$	−8.62022	−0.389025	−0.194513	−	0.980900i	$-0.562313\pi$
$491$	−8.62022	−0.389025	−0.194513	+	0.980900i	$0.562313\pi$
$492$	0	0
$493$	42.4829	1.91333
$494$	0	0
$495$	−8.36218	−0.375852
$496$	0	0
$497$	−2.08168	−0.0933762
$498$	0	0
$499$	19.1457	0.857080	0.428540	−	0.903523i	$-0.359028\pi$
$499$	19.1457	0.857080	0.428540	+	0.903523i	$0.359028\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−24.8619	−1.10854	−0.554269	−	0.832337i	$-0.687002\pi$
$503$	−24.8619	−1.10854	−0.554269	+	0.832337i	$0.687002\pi$
$504$	0	0
$505$	18.6857	0.831502
$506$	0	0
$507$	7.06888	0.313940
$508$	0	0
$509$	2.12427	0.0941567	0.0470783	−	0.998891i	$-0.485009\pi$
$509$	2.12427	0.0941567	0.0470783	+	0.998891i	$0.485009\pi$
$510$	0	0
$511$	−9.70525	−0.429335
$512$	0	0
$513$	5.87154	0.259235
$514$	0	0
$515$	22.7848	1.00402
$516$	0	0
$517$	20.2643	0.891222
$518$	0	0
$519$	−16.4057	−0.720129
$520$	0	0
$521$	−2.08693	−0.0914302	−0.0457151	−	0.998955i	$-0.514557\pi$
$521$	−2.08693	−0.0914302	−0.0457151	+	0.998955i	$0.514557\pi$
$522$	0	0
$523$	−21.6413	−0.946306	−0.473153	−	0.880980i	$-0.656884\pi$
$523$	−21.6413	−0.946306	−0.473153	+	0.880980i	$0.656884\pi$
$524$	0	0
$525$	2.91207	0.127093
$526$	0	0
$527$	−22.7226	−0.989814
$528$	0	0
$529$	−21.4942	−0.934531
$530$	0	0
$531$	0.102376	0.00444273
$532$	0	0
$533$	−8.32556	−0.360620
$534$	0	0
$535$	−9.82051	−0.424578
$536$	0	0
$537$	1.21604	0.0524761
$538$	0	0
$539$	−33.3968	−1.43850
$540$	0	0
$541$	24.7817	1.06545	0.532725	−	0.846289i	$-0.321168\pi$
$541$	24.7817	1.06545	0.532725	+	0.846289i	$0.321168\pi$
$542$	0	0
$543$	−6.24694	−0.268082
$544$	0	0
$545$	28.1520	1.20590
$546$	0	0
$547$	2.13808	0.0914179	0.0457089	−	0.998955i	$-0.485445\pi$
$547$	2.13808	0.0914179	0.0457089	+	0.998955i	$0.485445\pi$
$548$	0	0
$549$	3.64463	0.155549
$550$	0	0
$551$	−40.6305	−1.73092
$552$	0	0
$553$	4.07807	0.173417
$554$	0	0
$555$	9.81772	0.416739
$556$	0	0
$557$	−11.7687	−0.498656	−0.249328	−	0.968419i	$-0.580210\pi$
$557$	−11.7687	−0.498656	−0.249328	+	0.968419i	$0.580210\pi$
$558$	0	0
$559$	10.2938	0.435382
$560$	0	0
$561$	−34.6316	−1.46215
$562$	0	0
$563$	14.6893	0.619079	0.309540	−	0.950887i	$-0.399825\pi$
$563$	14.6893	0.619079	0.309540	+	0.950887i	$0.399825\pi$
$564$	0	0
$565$	−24.2601	−1.02063
$566$	0	0
$567$	−1.03908	−0.0436372
$568$	0	0
$569$	−16.2827	−0.682608	−0.341304	−	0.939953i	$-0.610869\pi$
$569$	−16.2827	−0.682608	−0.341304	+	0.939953i	$0.610869\pi$
$570$	0	0
$571$	14.0677	0.588714	0.294357	−	0.955696i	$-0.404895\pi$
$571$	14.0677	0.588714	0.294357	+	0.955696i	$0.404895\pi$
$572$	0	0
$573$	16.2291	0.677978
$574$	0	0
$575$	3.43901	0.143417
$576$	0	0
$577$	15.5023	0.645369	0.322684	−	0.946507i	$-0.395415\pi$
$577$	15.5023	0.645369	0.322684	+	0.946507i	$0.395415\pi$
$578$	0	0
$579$	−8.97087	−0.372816
$580$	0	0
$581$	11.8913	0.493334
$582$	0	0
$583$	−1.69074	−0.0700234
$584$	0	0
$585$	6.64082	0.274564
$586$	0	0
$587$	−22.7257	−0.937989	−0.468995	−	0.883201i	$-0.655384\pi$
$587$	−22.7257	−0.937989	−0.468995	+	0.883201i	$0.655384\pi$
$588$	0	0
$589$	21.7319	0.895447
$590$	0	0
$591$	−23.1926	−0.954015
$592$	0	0
$593$	−34.1852	−1.40382	−0.701909	−	0.712267i	$-0.747669\pi$
$593$	−34.1852	−1.40382	−0.701909	+	0.712267i	$0.747669\pi$
$594$	0	0
$595$	−9.45631	−0.387671
$596$	0	0
$597$	−13.8934	−0.568618
$598$	0	0
$599$	5.88943	0.240636	0.120318	−	0.992735i	$-0.461609\pi$
$599$	5.88943	0.240636	0.120318	+	0.992735i	$0.461609\pi$
$600$	0	0
$601$	−39.0125	−1.59135	−0.795676	−	0.605723i	$-0.792884\pi$
$601$	−39.0125	−1.59135	−0.795676	+	0.605723i	$0.792884\pi$
$602$	0	0
$603$	−5.27810	−0.214941
$604$	0	0
$605$	−30.8653	−1.25485
$606$	0	0
$607$	−5.36469	−0.217746	−0.108873	−	0.994056i	$-0.534724\pi$
$607$	−5.36469	−0.217746	−0.108873	+	0.994056i	$0.534724\pi$
$608$	0	0
$609$	7.19034	0.291367
$610$	0	0
$611$	−16.0929	−0.651047
$612$	0	0
$613$	−44.6516	−1.80346	−0.901731	−	0.432297i	$-0.857703\pi$
$613$	−44.6516	−1.80346	−0.901731	+	0.432297i	$0.857703\pi$
$614$	0	0
$615$	−2.75494	−0.111090
$616$	0	0
$617$	−15.1280	−0.609030	−0.304515	−	0.952508i	$-0.598494\pi$
$617$	−15.1280	−0.609030	−0.304515	+	0.952508i	$0.598494\pi$
$618$	0	0
$619$	−38.8437	−1.56126	−0.780631	−	0.624992i	$-0.785102\pi$
$619$	−38.8437	−1.56126	−0.780631	+	0.624992i	$0.785102\pi$
$620$	0	0
$621$	−1.22710	−0.0492419
$622$	0	0
$623$	10.8670	0.435378
$624$	0	0
$625$	−3.13300	−0.125320
$626$	0	0
$627$	33.1216	1.32275
$628$	0	0
$629$	40.6597	1.62121
$630$	0	0
$631$	23.0968	0.919470	0.459735	−	0.888056i	$-0.347944\pi$
$631$	23.0968	0.919470	0.459735	+	0.888056i	$0.347944\pi$
$632$	0	0
$633$	−11.9560	−0.475207
$634$	0	0
$635$	−26.1108	−1.03618
$636$	0	0
$637$	26.5220	1.05084
$638$	0	0
$639$	2.00339	0.0792529
$640$	0	0
$641$	28.3497	1.11974	0.559872	−	0.828579i	$-0.310850\pi$
$641$	28.3497	1.11974	0.559872	+	0.828579i	$0.310850\pi$
$642$	0	0
$643$	−40.8208	−1.60981	−0.804907	−	0.593401i	$-0.797785\pi$
$643$	−40.8208	−1.60981	−0.804907	+	0.593401i	$0.797785\pi$
$644$	0	0
$645$	3.40623	0.134120
$646$	0	0
$647$	−31.6874	−1.24576	−0.622881	−	0.782317i	$-0.714038\pi$
$647$	−31.6874	−1.24576	−0.622881	+	0.782317i	$0.714038\pi$
$648$	0	0
$649$	0.577506	0.0226691
$650$	0	0
$651$	−3.84587	−0.150731
$652$	0	0
$653$	17.1808	0.672337	0.336169	−	0.941802i	$-0.390869\pi$
$653$	17.1808	0.672337	0.336169	+	0.941802i	$0.390869\pi$
$654$	0	0
$655$	−14.5228	−0.567454
$656$	0	0
$657$	9.34023	0.364397
$658$	0	0
$659$	29.0908	1.13322	0.566608	−	0.823988i	$-0.308255\pi$
$659$	29.0908	1.13322	0.566608	+	0.823988i	$0.308255\pi$
$660$	0	0
$661$	2.34936	0.0913794	0.0456897	−	0.998956i	$-0.485451\pi$
$661$	2.34936	0.0913794	0.0456897	+	0.998956i	$0.485451\pi$
$662$	0	0
$663$	27.5027	1.06812
$664$	0	0
$665$	9.04400	0.350711
$666$	0	0
$667$	8.49144	0.328790
$668$	0	0
$669$	−25.7496	−0.995535
$670$	0	0
$671$	20.5595	0.793691
$672$	0	0
$673$	19.7107	0.759792	0.379896	−	0.925029i	$-0.375960\pi$
$673$	19.7107	0.759792	0.379896	+	0.925029i	$0.375960\pi$
$674$	0	0
$675$	−2.80255	−0.107870
$676$	0	0
$677$	13.2631	0.509744	0.254872	−	0.966975i	$-0.417967\pi$
$677$	13.2631	0.509744	0.254872	+	0.966975i	$0.417967\pi$
$678$	0	0
$679$	3.53831	0.135788
$680$	0	0
$681$	23.2899	0.892469
$682$	0	0
$683$	−32.2080	−1.23241	−0.616203	−	0.787587i	$-0.711330\pi$
$683$	−32.2080	−1.23241	−0.616203	+	0.787587i	$0.711330\pi$
$684$	0	0
$685$	10.1077	0.386195
$686$	0	0
$687$	−10.4690	−0.399418
$688$	0	0
$689$	1.34270	0.0511528
$690$	0	0
$691$	−18.3513	−0.698115	−0.349058	−	0.937101i	$-0.613498\pi$
$691$	−18.3513	−0.698115	−0.349058	+	0.937101i	$0.613498\pi$
$692$	0	0
$693$	−5.86150	−0.222660
$694$	0	0
$695$	7.92768	0.300714
$696$	0	0
$697$	−11.4095	−0.432164
$698$	0	0
$699$	−19.6195	−0.742078
$700$	0	0
$701$	−15.0603	−0.568821	−0.284411	−	0.958703i	$-0.591798\pi$
$701$	−15.0603	−0.568821	−0.284411	+	0.958703i	$0.591798\pi$
$702$	0	0
$703$	−38.8868	−1.46665
$704$	0	0
$705$	−5.32515	−0.200557
$706$	0	0
$707$	13.0978	0.492593
$708$	0	0
$709$	22.7311	0.853684	0.426842	−	0.904326i	$-0.359626\pi$
$709$	22.7311	0.853684	0.426842	+	0.904326i	$0.359626\pi$
$710$	0	0
$711$	−3.92469	−0.147187
$712$	0	0
$713$	−4.54178	−0.170091
$714$	0	0
$715$	37.4611	1.40097
$716$	0	0
$717$	15.5821	0.581926
$718$	0	0
$719$	22.0804	0.823460	0.411730	−	0.911306i	$-0.364925\pi$
$719$	22.0804	0.823460	0.411730	+	0.911306i	$0.364925\pi$
$720$	0	0
$721$	15.9711	0.594793
$722$	0	0
$723$	−30.8313	−1.14663
$724$	0	0
$725$	19.3934	0.720252
$726$	0	0
$727$	−1.19378	−0.0442749	−0.0221374	−	0.999755i	$-0.507047\pi$
$727$	−1.19378	−0.0442749	−0.0221374	+	0.999755i	$0.507047\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	14.1068	0.521758
$732$	0	0
$733$	−40.8588	−1.50915	−0.754577	−	0.656212i	$-0.772158\pi$
$733$	−40.8588	−1.50915	−0.754577	+	0.656212i	$0.772158\pi$
$734$	0	0
$735$	8.77616	0.323714
$736$	0	0
$737$	−29.7740	−1.09674
$738$	0	0
$739$	43.2993	1.59279	0.796396	−	0.604776i	$-0.206737\pi$
$739$	43.2993	1.59279	0.796396	+	0.604776i	$0.206737\pi$
$740$	0	0
$741$	−26.3035	−0.966283
$742$	0	0
$743$	30.6987	1.12622	0.563112	−	0.826380i	$-0.309604\pi$
$743$	30.6987	1.12622	0.563112	+	0.826380i	$0.309604\pi$
$744$	0	0
$745$	29.7876	1.09133
$746$	0	0
$747$	−11.4441	−0.418717
$748$	0	0
$749$	−6.88372	−0.251525
$750$	0	0
$751$	38.3723	1.40022	0.700112	−	0.714033i	$-0.253133\pi$
$751$	38.3723	1.40022	0.700112	+	0.714033i	$0.253133\pi$
$752$	0	0
$753$	−5.66860	−0.206575
$754$	0	0
$755$	22.9170	0.834034
$756$	0	0
$757$	15.5003	0.563367	0.281684	−	0.959507i	$-0.409107\pi$
$757$	15.5003	0.563367	0.281684	+	0.959507i	$0.409107\pi$
$758$	0	0
$759$	−6.92214	−0.251258
$760$	0	0
$761$	41.5796	1.50726	0.753629	−	0.657300i	$-0.228301\pi$
$761$	41.5796	1.50726	0.753629	+	0.657300i	$0.228301\pi$
$762$	0	0
$763$	19.7332	0.714390
$764$	0	0
$765$	9.10067	0.329035
$766$	0	0
$767$	−0.458626	−0.0165600
$768$	0	0
$769$	42.7687	1.54228	0.771140	−	0.636666i	$-0.219687\pi$
$769$	42.7687	1.54228	0.771140	+	0.636666i	$0.219687\pi$
$770$	0	0
$771$	−6.81618	−0.245479
$772$	0	0
$773$	32.3512	1.16359	0.581795	−	0.813335i	$-0.302350\pi$
$773$	32.3512	1.16359	0.581795	+	0.813335i	$0.302350\pi$
$774$	0	0
$775$	−10.3729	−0.372604
$776$	0	0
$777$	6.88176	0.246882
$778$	0	0
$779$	10.9120	0.390963
$780$	0	0
$781$	11.3012	0.404389
$782$	0	0
$783$	−6.91991	−0.247298
$784$	0	0
$785$	−15.7986	−0.563877
$786$	0	0
$787$	0.830709	0.0296116	0.0148058	−	0.999890i	$-0.495287\pi$
$787$	0.830709	0.0296116	0.0148058	+	0.999890i	$0.495287\pi$
$788$	0	0
$789$	4.22632	0.150461
$790$	0	0
$791$	−17.0052	−0.604635
$792$	0	0
$793$	−16.3273	−0.579800
$794$	0	0
$795$	0.444301	0.0157577
$796$	0	0
$797$	44.9143	1.59095	0.795473	−	0.605988i	$-0.207222\pi$
$797$	44.9143	1.59095	0.795473	+	0.605988i	$0.207222\pi$
$798$	0	0
$799$	−22.0539	−0.780210
$800$	0	0
$801$	−10.4583	−0.369527
$802$	0	0
$803$	52.6887	1.85934
$804$	0	0
$805$	−1.89012	−0.0666180
$806$	0	0
$807$	20.5783	0.724390
$808$	0	0
$809$	−25.0553	−0.880897	−0.440449	−	0.897778i	$-0.645181\pi$
$809$	−25.0553	−0.880897	−0.440449	+	0.897778i	$0.645181\pi$
$810$	0	0
$811$	30.9063	1.08527	0.542634	−	0.839969i	$-0.317427\pi$
$811$	30.9063	1.08527	0.542634	+	0.839969i	$0.317427\pi$
$812$	0	0
$813$	−6.93172	−0.243106
$814$	0	0
$815$	31.5454	1.10499
$816$	0	0
$817$	−13.4917	−0.472014
$818$	0	0
$819$	4.65490	0.162655
$820$	0	0
$821$	4.78573	0.167023	0.0835115	−	0.996507i	$-0.473386\pi$
$821$	4.78573	0.167023	0.0835115	+	0.996507i	$0.473386\pi$
$822$	0	0
$823$	21.7901	0.759555	0.379778	−	0.925078i	$-0.376001\pi$
$823$	21.7901	0.759555	0.379778	+	0.925078i	$0.376001\pi$
$824$	0	0
$825$	−15.8093	−0.550409
$826$	0	0
$827$	37.7470	1.31259	0.656295	−	0.754504i	$-0.272123\pi$
$827$	37.7470	1.31259	0.656295	+	0.754504i	$0.272123\pi$
$828$	0	0
$829$	−13.6115	−0.472748	−0.236374	−	0.971662i	$-0.575959\pi$
$829$	−13.6115	−0.472748	−0.236374	+	0.971662i	$0.575959\pi$
$830$	0	0
$831$	22.5380	0.781833
$832$	0	0
$833$	36.3461	1.25932
$834$	0	0
$835$	1.48238	0.0512999
$836$	0	0
$837$	3.70123	0.127933
$838$	0	0
$839$	−20.7228	−0.715431	−0.357715	−	0.933831i	$-0.616444\pi$
$839$	−20.7228	−0.715431	−0.357715	+	0.933831i	$0.616444\pi$
$840$	0	0
$841$	18.8852	0.651214
$842$	0	0
$843$	20.5684	0.708414
$844$	0	0
$845$	−10.4788	−0.360481
$846$	0	0
$847$	−21.6351	−0.743391
$848$	0	0
$849$	29.4021	1.00908
$850$	0	0
$851$	8.12703	0.278591
$852$	0	0
$853$	−18.7796	−0.643001	−0.321501	−	0.946909i	$-0.604187\pi$
$853$	−18.7796	−0.643001	−0.321501	+	0.946909i	$0.604187\pi$
$854$	0	0
$855$	−8.70385	−0.297666
$856$	0	0
$857$	−47.7602	−1.63146	−0.815729	−	0.578434i	$-0.803664\pi$
$857$	−47.7602	−1.63146	−0.815729	+	0.578434i	$0.803664\pi$
$858$	0	0
$859$	41.5608	1.41804	0.709018	−	0.705190i	$-0.249138\pi$
$859$	41.5608	1.41804	0.709018	+	0.705190i	$0.249138\pi$
$860$	0	0
$861$	−1.93108	−0.0658111
$862$	0	0
$863$	0.525151	0.0178764	0.00893818	−	0.999960i	$-0.497155\pi$
$863$	0.525151	0.0178764	0.00893818	+	0.999960i	$0.497155\pi$
$864$	0	0
$865$	24.3194	0.826886
$866$	0	0
$867$	20.6900	0.702671
$868$	0	0
$869$	−22.1394	−0.751027
$870$	0	0
$871$	23.6450	0.801181
$872$	0	0
$873$	−3.40524	−0.115250
$874$	0	0
$875$	−12.0184	−0.406294
$876$	0	0
$877$	−3.76515	−0.127140	−0.0635701	−	0.997977i	$-0.520249\pi$
$877$	−3.76515	−0.127140	−0.0635701	+	0.997977i	$0.520249\pi$
$878$	0	0
$879$	20.2839	0.684160
$880$	0	0
$881$	−30.4381	−1.02549	−0.512743	−	0.858542i	$-0.671371\pi$
$881$	−30.4381	−1.02549	−0.512743	+	0.858542i	$0.671371\pi$
$882$	0	0
$883$	−3.63578	−0.122354	−0.0611768	−	0.998127i	$-0.519485\pi$
$883$	−3.63578	−0.122354	−0.0611768	+	0.998127i	$0.519485\pi$
$884$	0	0
$885$	−0.151760	−0.00510135
$886$	0	0
$887$	−19.9846	−0.671018	−0.335509	−	0.942037i	$-0.608908\pi$
$887$	−19.9846	−0.671018	−0.335509	+	0.942037i	$0.608908\pi$
$888$	0	0
$889$	−18.3024	−0.613844
$890$	0	0
$891$	5.64105	0.188982
$892$	0	0
$893$	21.0923	0.705826
$894$	0	0
$895$	−1.80264	−0.0602555
$896$	0	0
$897$	5.49721	0.183547
$898$	0	0
$899$	−25.6122	−0.854213
$900$	0	0
$901$	1.84006	0.0613011
$902$	0	0
$903$	2.38761	0.0794547
$904$	0	0
$905$	9.26035	0.307824
$906$	0	0
$907$	−20.7963	−0.690529	−0.345264	−	0.938505i	$-0.612211\pi$
$907$	−20.7963	−0.690529	−0.345264	+	0.938505i	$0.612211\pi$
$908$	0	0
$909$	−12.6052	−0.418088
$910$	0	0
$911$	54.5814	1.80836	0.904181	−	0.427150i	$-0.140482\pi$
$911$	54.5814	1.80836	0.904181	+	0.427150i	$0.140482\pi$
$912$	0	0
$913$	−64.5565	−2.13651
$914$	0	0
$915$	−5.40273	−0.178609
$916$	0	0
$917$	−10.1798	−0.336168
$918$	0	0
$919$	8.31975	0.274443	0.137222	−	0.990540i	$-0.456183\pi$
$919$	8.31975	0.274443	0.137222	+	0.990540i	$0.456183\pi$
$920$	0	0
$921$	10.9899	0.362128
$922$	0	0
$923$	−8.97485	−0.295411
$924$	0	0
$925$	18.5611	0.610285
$926$	0	0
$927$	−15.3704	−0.504830
$928$	0	0
$929$	14.6371	0.480229	0.240114	−	0.970745i	$-0.422815\pi$
$929$	14.6371	0.480229	0.240114	+	0.970745i	$0.422815\pi$
$930$	0	0
$931$	−34.7613	−1.13926
$932$	0	0
$933$	26.9703	0.882969
$934$	0	0
$935$	51.3373	1.67891
$936$	0	0
$937$	10.0788	0.329260	0.164630	−	0.986355i	$-0.447357\pi$
$937$	10.0788	0.329260	0.164630	+	0.986355i	$0.447357\pi$
$938$	0	0
$939$	−32.1109	−1.04790
$940$	0	0
$941$	12.5806	0.410116	0.205058	−	0.978750i	$-0.434262\pi$
$941$	12.5806	0.410116	0.205058	+	0.978750i	$0.434262\pi$
$942$	0	0
$943$	−2.28052	−0.0742638
$944$	0	0
$945$	1.54031	0.0501063
$946$	0	0
$947$	15.0438	0.488858	0.244429	−	0.969667i	$-0.421399\pi$
$947$	15.0438	0.488858	0.244429	+	0.969667i	$0.421399\pi$
$948$	0	0
$949$	−41.8427	−1.35827
$950$	0	0
$951$	−21.3657	−0.692830
$952$	0	0
$953$	61.0466	1.97749	0.988746	−	0.149603i	$-0.0477996\pi$
$953$	61.0466	1.97749	0.988746	+	0.149603i	$0.0477996\pi$
$954$	0	0
$955$	−24.0576	−0.778487
$956$	0	0
$957$	−39.0356	−1.26184
$958$	0	0
$959$	7.08502	0.228787
$960$	0	0
$961$	−17.3009	−0.558095
$962$	0	0
$963$	6.62482	0.213482
$964$	0	0
$965$	13.2982	0.428085
$966$	0	0
$967$	6.42784	0.206705	0.103353	−	0.994645i	$-0.467043\pi$
$967$	6.42784	0.206705	0.103353	+	0.994645i	$0.467043\pi$
$968$	0	0
$969$	−36.0467	−1.15799
$970$	0	0
$971$	28.9135	0.927879	0.463940	−	0.885867i	$-0.346435\pi$
$971$	28.9135	0.927879	0.463940	+	0.885867i	$0.346435\pi$
$972$	0	0
$973$	5.55693	0.178147
$974$	0	0
$975$	12.5549	0.402080
$976$	0	0
$977$	−5.12031	−0.163813	−0.0819066	−	0.996640i	$-0.526101\pi$
$977$	−5.12031	−0.163813	−0.0819066	+	0.996640i	$0.526101\pi$
$978$	0	0
$979$	−58.9959	−1.88552
$980$	0	0
$981$	−18.9911	−0.606338
$982$	0	0
$983$	−15.0970	−0.481518	−0.240759	−	0.970585i	$-0.577396\pi$
$983$	−15.0970	−0.481518	−0.240759	+	0.970585i	$0.577396\pi$
$984$	0	0
$985$	34.3802	1.09544
$986$	0	0
$987$	−3.73268	−0.118812
$988$	0	0
$989$	2.81965	0.0896597
$990$	0	0
$991$	−24.5344	−0.779360	−0.389680	−	0.920950i	$-0.627414\pi$
$991$	−24.5344	−0.779360	−0.389680	+	0.920950i	$0.627414\pi$
$992$	0	0
$993$	23.6701	0.751147
$994$	0	0
$995$	20.5953	0.652915
$996$	0	0
$997$	−2.08691	−0.0660930	−0.0330465	−	0.999454i	$-0.510521\pi$
$997$	−2.08691	−0.0660930	−0.0330465	+	0.999454i	$0.510521\pi$
$998$	0	0
$999$	−6.62294	−0.209541

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.i.1.4	✓	9
4.3	odd	2		8016.2.a.ba.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.i.1.4	✓	9	1.1	even	1	trivial
8016.2.a.ba.1.4		9	4.3	odd	2

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:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.48238 q^{5} -1.03908 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.48238 q^{5} -1.03908 q^{7} +1.00000 q^{9} +5.64105 q^{11} -4.47983 q^{13} -1.48238 q^{15} -6.13922 q^{17} +5.87154 q^{19} -1.03908 q^{21} -1.22710 q^{23} -2.80255 q^{25} +1.00000 q^{27} -6.91991 q^{29} +3.70123 q^{31} +5.64105 q^{33} +1.54031 q^{35} -6.62294 q^{37} -4.47983 q^{39} +1.85846 q^{41} -2.29781 q^{43} -1.48238 q^{45} +3.59229 q^{47} -5.92031 q^{49} -6.13922 q^{51} -0.299721 q^{53} -8.36218 q^{55} +5.87154 q^{57} +0.102376 q^{59} +3.64463 q^{61} -1.03908 q^{63} +6.64082 q^{65} -5.27810 q^{67} -1.22710 q^{69} +2.00339 q^{71} +9.34023 q^{73} -2.80255 q^{75} -5.86150 q^{77} -3.92469 q^{79} +1.00000 q^{81} -11.4441 q^{83} +9.10067 q^{85} -6.91991 q^{87} -10.4583 q^{89} +4.65490 q^{91} +3.70123 q^{93} -8.70385 q^{95} -3.40524 q^{97} +5.64105 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} - 6 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^3 - 6 * q^5 - 11 * q^7 + 9 * q^9	\( 9 q + 9 q^{3} - 6 q^{5} - 11 q^{7} + 9 q^{9} + q^{11} - 4 q^{13} - 6 q^{15} - 9 q^{17} - 8 q^{19} - 11 q^{21} - 7 q^{23} - q^{25} + 9 q^{27} - 9 q^{29} - 25 q^{31} + q^{33} + 5 q^{35} - 6 q^{37} - 4 q^{39} - 4 q^{41} - 24 q^{43} - 6 q^{45} - 16 q^{47} + 4 q^{49} - 9 q^{51} - 26 q^{53} - 29 q^{55} - 8 q^{57} - 4 q^{59} - 20 q^{61} - 11 q^{63} - 8 q^{65} - 25 q^{67} - 7 q^{69} - 15 q^{71} - 10 q^{73} - q^{75} - 20 q^{77} - 34 q^{79} + 9 q^{81} - 4 q^{83} - 13 q^{85} - 9 q^{87} + 13 q^{89} - 21 q^{91} - 25 q^{93} - 7 q^{95} - 4 q^{97} + q^{99}+O(q^{100}) \) 9 * q + 9 * q^3 - 6 * q^5 - 11 * q^7 + 9 * q^9 + q^11 - 4 * q^13 - 6 * q^15 - 9 * q^17 - 8 * q^19 - 11 * q^21 - 7 * q^23 - q^25 + 9 * q^27 - 9 * q^29 - 25 * q^31 + q^33 + 5 * q^35 - 6 * q^37 - 4 * q^39 - 4 * q^41 - 24 * q^43 - 6 * q^45 - 16 * q^47 + 4 * q^49 - 9 * q^51 - 26 * q^53 - 29 * q^55 - 8 * q^57 - 4 * q^59 - 20 * q^61 - 11 * q^63 - 8 * q^65 - 25 * q^67 - 7 * q^69 - 15 * q^71 - 10 * q^73 - q^75 - 20 * q^77 - 34 * q^79 + 9 * q^81 - 4 * q^83 - 13 * q^85 - 9 * q^87 + 13 * q^89 - 21 * q^91 - 25 * q^93 - 7 * q^95 - 4 * q^97 + q^99

Introduction

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