LMFDB - Embedded newform 4008.2.a.i.1.9

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			1.9
Root			$3.54323$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +2.54323 q^{5} -4.05015 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.54323 q^{5} -4.05015 q^{7} +1.00000 q^{9} +0.833896 q^{11} +2.46125 q^{13} +2.54323 q^{15} -1.83915 q^{17} -5.92708 q^{19} -4.05015 q^{21} -3.92292 q^{23} +1.46803 q^{25} +1.00000 q^{27} -7.78858 q^{29} -8.46269 q^{31} +0.833896 q^{33} -10.3005 q^{35} +3.96040 q^{37} +2.46125 q^{39} -11.0449 q^{41} -10.1144 q^{43} +2.54323 q^{45} +0.498289 q^{47} +9.40370 q^{49} -1.83915 q^{51} +2.45514 q^{53} +2.12079 q^{55} -5.92708 q^{57} +7.33541 q^{59} +2.76168 q^{61} -4.05015 q^{63} +6.25954 q^{65} +10.9183 q^{67} -3.92292 q^{69} -3.97515 q^{71} +6.83455 q^{73} +1.46803 q^{75} -3.37740 q^{77} +7.34900 q^{79} +1.00000 q^{81} -4.66163 q^{83} -4.67739 q^{85} -7.78858 q^{87} -4.56477 q^{89} -9.96844 q^{91} -8.46269 q^{93} -15.0739 q^{95} +10.8423 q^{97} +0.833896 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.54323	1.13737	0.568684	−	0.822556i	$-0.307453\pi$
$5$	2.54323	1.13737	0.568684	+	0.822556i	$0.307453\pi$
$6$	0	0
$7$	−4.05015	−1.53081	−0.765406	−	0.643548i	$-0.777462\pi$
$7$	−4.05015	−1.53081	−0.765406	+	0.643548i	$0.777462\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.833896	0.251429	0.125715	−	0.992066i	$-0.459878\pi$
$11$	0.833896	0.251429	0.125715	+	0.992066i	$0.459878\pi$
$12$	0	0
$13$	2.46125	0.682629	0.341314	−	0.939949i	$-0.389128\pi$
$13$	2.46125	0.682629	0.341314	+	0.939949i	$0.389128\pi$
$14$	0	0
$15$	2.54323	0.656660
$16$	0	0
$17$	−1.83915	−0.446060	−0.223030	−	0.974812i	$-0.571595\pi$
$17$	−1.83915	−0.446060	−0.223030	+	0.974812i	$0.571595\pi$
$18$	0	0
$19$	−5.92708	−1.35976	−0.679882	−	0.733321i	$-0.737969\pi$
$19$	−5.92708	−1.35976	−0.679882	+	0.733321i	$0.737969\pi$
$20$	0	0
$21$	−4.05015	−0.883815
$22$	0	0
$23$	−3.92292	−0.817986	−0.408993	−	0.912538i	$-0.634120\pi$
$23$	−3.92292	−0.817986	−0.408993	+	0.912538i	$0.634120\pi$
$24$	0	0
$25$	1.46803	0.293605
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.78858	−1.44630	−0.723152	−	0.690689i	$-0.757307\pi$
$29$	−7.78858	−1.44630	−0.723152	+	0.690689i	$0.757307\pi$
$30$	0	0
$31$	−8.46269	−1.51994	−0.759972	−	0.649956i	$-0.774787\pi$
$31$	−8.46269	−1.51994	−0.759972	+	0.649956i	$0.774787\pi$
$32$	0	0
$33$	0.833896	0.145163
$34$	0	0
$35$	−10.3005	−1.74110
$36$	0	0
$37$	3.96040	0.651086	0.325543	−	0.945527i	$-0.394453\pi$
$37$	3.96040	0.651086	0.325543	+	0.945527i	$0.394453\pi$
$38$	0	0
$39$	2.46125	0.394116
$40$	0	0
$41$	−11.0449	−1.72492	−0.862461	−	0.506124i	$-0.831078\pi$
$41$	−11.0449	−1.72492	−0.862461	+	0.506124i	$0.831078\pi$
$42$	0	0
$43$	−10.1144	−1.54243	−0.771213	−	0.636577i	$-0.780350\pi$
$43$	−10.1144	−1.54243	−0.771213	+	0.636577i	$0.780350\pi$
$44$	0	0
$45$	2.54323	0.379123
$46$	0	0
$47$	0.498289	0.0726830	0.0363415	−	0.999339i	$-0.488430\pi$
$47$	0.498289	0.0726830	0.0363415	+	0.999339i	$0.488430\pi$
$48$	0	0
$49$	9.40370	1.34339
$50$	0	0
$51$	−1.83915	−0.257533
$52$	0	0
$53$	2.45514	0.337240	0.168620	−	0.985681i	$-0.446069\pi$
$53$	2.45514	0.337240	0.168620	+	0.985681i	$0.446069\pi$
$54$	0	0
$55$	2.12079	0.285967
$56$	0	0
$57$	−5.92708	−0.785061
$58$	0	0
$59$	7.33541	0.954990	0.477495	−	0.878635i	$-0.341545\pi$
$59$	7.33541	0.954990	0.477495	+	0.878635i	$0.341545\pi$
$60$	0	0
$61$	2.76168	0.353597	0.176798	−	0.984247i	$-0.443426\pi$
$61$	2.76168	0.353597	0.176798	+	0.984247i	$0.443426\pi$
$62$	0	0
$63$	−4.05015	−0.510271
$64$	0	0
$65$	6.25954	0.776400
$66$	0	0
$67$	10.9183	1.33388	0.666942	−	0.745110i	$-0.267603\pi$
$67$	10.9183	1.33388	0.666942	+	0.745110i	$0.267603\pi$
$68$	0	0
$69$	−3.92292	−0.472264
$70$	0	0
$71$	−3.97515	−0.471763	−0.235882	−	0.971782i	$-0.575798\pi$
$71$	−3.97515	−0.471763	−0.235882	+	0.971782i	$0.575798\pi$
$72$	0	0
$73$	6.83455	0.799924	0.399962	−	0.916532i	$-0.369023\pi$
$73$	6.83455	0.799924	0.399962	+	0.916532i	$0.369023\pi$
$74$	0	0
$75$	1.46803	0.169513
$76$	0	0
$77$	−3.37740	−0.384891
$78$	0	0
$79$	7.34900	0.826827	0.413413	−	0.910543i	$-0.364336\pi$
$79$	7.34900	0.826827	0.413413	+	0.910543i	$0.364336\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.66163	−0.511680	−0.255840	−	0.966719i	$-0.582352\pi$
$83$	−4.66163	−0.511680	−0.255840	+	0.966719i	$0.582352\pi$
$84$	0	0
$85$	−4.67739	−0.507334
$86$	0	0
$87$	−7.78858	−0.835024
$88$	0	0
$89$	−4.56477	−0.483864	−0.241932	−	0.970293i	$-0.577781\pi$
$89$	−4.56477	−0.483864	−0.241932	+	0.970293i	$0.577781\pi$
$90$	0	0
$91$	−9.96844	−1.04498
$92$	0	0
$93$	−8.46269	−0.877540
$94$	0	0
$95$	−15.0739	−1.54655
$96$	0	0
$97$	10.8423	1.10087	0.550434	−	0.834879i	$-0.314462\pi$
$97$	10.8423	1.10087	0.550434	+	0.834879i	$0.314462\pi$
$98$	0	0
$99$	0.833896	0.0838097
$100$	0	0
$101$	1.34614	0.133946	0.0669731	−	0.997755i	$-0.478666\pi$
$101$	1.34614	0.133946	0.0669731	+	0.997755i	$0.478666\pi$
$102$	0	0
$103$	−17.7990	−1.75379	−0.876893	−	0.480686i	$-0.840388\pi$
$103$	−17.7990	−1.75379	−0.876893	+	0.480686i	$0.840388\pi$
$104$	0	0
$105$	−10.3005	−1.00522
$106$	0	0
$107$	−9.17441	−0.886923	−0.443462	−	0.896293i	$-0.646250\pi$
$107$	−9.17441	−0.886923	−0.443462	+	0.896293i	$0.646250\pi$
$108$	0	0
$109$	10.0390	0.961558	0.480779	−	0.876842i	$-0.340354\pi$
$109$	10.0390	0.961558	0.480779	+	0.876842i	$0.340354\pi$
$110$	0	0
$111$	3.96040	0.375905
$112$	0	0
$113$	18.2770	1.71936	0.859678	−	0.510837i	$-0.170664\pi$
$113$	18.2770	1.71936	0.859678	+	0.510837i	$0.170664\pi$
$114$	0	0
$115$	−9.97690	−0.930350
$116$	0	0
$117$	2.46125	0.227543
$118$	0	0
$119$	7.44884	0.682834
$120$	0	0
$121$	−10.3046	−0.936783
$122$	0	0
$123$	−11.0449	−0.995884
$124$	0	0
$125$	−8.98263	−0.803431
$126$	0	0
$127$	8.49866	0.754134	0.377067	−	0.926186i	$-0.376933\pi$
$127$	8.49866	0.754134	0.377067	+	0.926186i	$0.376933\pi$
$128$	0	0
$129$	−10.1144	−0.890520
$130$	0	0
$131$	−6.28622	−0.549230	−0.274615	−	0.961554i	$-0.588550\pi$
$131$	−6.28622	−0.549230	−0.274615	+	0.961554i	$0.588550\pi$
$132$	0	0
$133$	24.0055	2.08154
$134$	0	0
$135$	2.54323	0.218887
$136$	0	0
$137$	4.10350	0.350585	0.175293	−	0.984516i	$-0.443913\pi$
$137$	4.10350	0.350585	0.175293	+	0.984516i	$0.443913\pi$
$138$	0	0
$139$	4.54642	0.385622	0.192811	−	0.981236i	$-0.438240\pi$
$139$	4.54642	0.385622	0.192811	+	0.981236i	$0.438240\pi$
$140$	0	0
$141$	0.498289	0.0419635
$142$	0	0
$143$	2.05243	0.171633
$144$	0	0
$145$	−19.8082	−1.64498
$146$	0	0
$147$	9.40370	0.775604
$148$	0	0
$149$	−13.9805	−1.14533	−0.572663	−	0.819791i	$-0.694090\pi$
$149$	−13.9805	−1.14533	−0.572663	+	0.819791i	$0.694090\pi$
$150$	0	0
$151$	−2.44014	−0.198575	−0.0992877	−	0.995059i	$-0.531656\pi$
$151$	−2.44014	−0.198575	−0.0992877	+	0.995059i	$0.531656\pi$
$152$	0	0
$153$	−1.83915	−0.148687
$154$	0	0
$155$	−21.5226	−1.72874
$156$	0	0
$157$	−17.2013	−1.37281	−0.686407	−	0.727218i	$-0.740813\pi$
$157$	−17.2013	−1.37281	−0.686407	+	0.727218i	$0.740813\pi$
$158$	0	0
$159$	2.45514	0.194706
$160$	0	0
$161$	15.8884	1.25218
$162$	0	0
$163$	−0.421952	−0.0330498	−0.0165249	−	0.999863i	$-0.505260\pi$
$163$	−0.421952	−0.0330498	−0.0165249	+	0.999863i	$0.505260\pi$
$164$	0	0
$165$	2.12079	0.165103
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−6.94223	−0.534018
$170$	0	0
$171$	−5.92708	−0.453255
$172$	0	0
$173$	−16.6235	−1.26386	−0.631932	−	0.775024i	$-0.717738\pi$
$173$	−16.6235	−1.26386	−0.631932	+	0.775024i	$0.717738\pi$
$174$	0	0
$175$	−5.94572	−0.449454
$176$	0	0
$177$	7.33541	0.551364
$178$	0	0
$179$	1.25643	0.0939098	0.0469549	−	0.998897i	$-0.485048\pi$
$179$	1.25643	0.0939098	0.0469549	+	0.998897i	$0.485048\pi$
$180$	0	0
$181$	−4.10641	−0.305227	−0.152614	−	0.988286i	$-0.548769\pi$
$181$	−4.10641	−0.305227	−0.152614	+	0.988286i	$0.548769\pi$
$182$	0	0
$183$	2.76168	0.204149
$184$	0	0
$185$	10.0722	0.740524
$186$	0	0
$187$	−1.53366	−0.112152
$188$	0	0
$189$	−4.05015	−0.294605
$190$	0	0
$191$	14.8331	1.07328	0.536642	−	0.843810i	$-0.319693\pi$
$191$	14.8331	1.07328	0.536642	+	0.843810i	$0.319693\pi$
$192$	0	0
$193$	−10.2047	−0.734549	−0.367275	−	0.930113i	$-0.619709\pi$
$193$	−10.2047	−0.734549	−0.367275	+	0.930113i	$0.619709\pi$
$194$	0	0
$195$	6.25954	0.448255
$196$	0	0
$197$	−12.2783	−0.874796	−0.437398	−	0.899268i	$-0.644100\pi$
$197$	−12.2783	−0.874796	−0.437398	+	0.899268i	$0.644100\pi$
$198$	0	0
$199$	3.70118	0.262370	0.131185	−	0.991358i	$-0.458122\pi$
$199$	3.70118	0.262370	0.131185	+	0.991358i	$0.458122\pi$
$200$	0	0
$201$	10.9183	0.770118
$202$	0	0
$203$	31.5449	2.21402
$204$	0	0
$205$	−28.0897	−1.96187
$206$	0	0
$207$	−3.92292	−0.272662
$208$	0	0
$209$	−4.94257	−0.341885
$210$	0	0
$211$	−1.75230	−0.120633	−0.0603165	−	0.998179i	$-0.519211\pi$
$211$	−1.75230	−0.120633	−0.0603165	+	0.998179i	$0.519211\pi$
$212$	0	0
$213$	−3.97515	−0.272373
$214$	0	0
$215$	−25.7232	−1.75431
$216$	0	0
$217$	34.2752	2.32675
$218$	0	0
$219$	6.83455	0.461836
$220$	0	0
$221$	−4.52662	−0.304493
$222$	0	0
$223$	−6.79814	−0.455237	−0.227619	−	0.973750i	$-0.573094\pi$
$223$	−6.79814	−0.455237	−0.227619	+	0.973750i	$0.573094\pi$
$224$	0	0
$225$	1.46803	0.0978684
$226$	0	0
$227$	2.28779	0.151846	0.0759231	−	0.997114i	$-0.475810\pi$
$227$	2.28779	0.151846	0.0759231	+	0.997114i	$0.475810\pi$
$228$	0	0
$229$	−21.2604	−1.40493	−0.702463	−	0.711720i	$-0.747916\pi$
$229$	−21.2604	−1.40493	−0.702463	+	0.711720i	$0.747916\pi$
$230$	0	0
$231$	−3.37740	−0.222217
$232$	0	0
$233$	17.9559	1.17633	0.588164	−	0.808741i	$-0.299851\pi$
$233$	17.9559	1.17633	0.588164	+	0.808741i	$0.299851\pi$
$234$	0	0
$235$	1.26727	0.0826673
$236$	0	0
$237$	7.34900	0.477369
$238$	0	0
$239$	−9.32061	−0.602900	−0.301450	−	0.953482i	$-0.597471\pi$
$239$	−9.32061	−0.602900	−0.301450	+	0.953482i	$0.597471\pi$
$240$	0	0
$241$	10.7267	0.690969	0.345485	−	0.938424i	$-0.387715\pi$
$241$	10.7267	0.690969	0.345485	+	0.938424i	$0.387715\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	23.9158	1.52792
$246$	0	0
$247$	−14.5880	−0.928215
$248$	0	0
$249$	−4.66163	−0.295419
$250$	0	0
$251$	14.6533	0.924910	0.462455	−	0.886643i	$-0.346969\pi$
$251$	14.6533	0.924910	0.462455	+	0.886643i	$0.346969\pi$
$252$	0	0
$253$	−3.27131	−0.205665
$254$	0	0
$255$	−4.67739	−0.292909
$256$	0	0
$257$	4.98675	0.311065	0.155532	−	0.987831i	$-0.450291\pi$
$257$	4.98675	0.311065	0.155532	+	0.987831i	$0.450291\pi$
$258$	0	0
$259$	−16.0402	−0.996690
$260$	0	0
$261$	−7.78858	−0.482101
$262$	0	0
$263$	14.5146	0.895008	0.447504	−	0.894282i	$-0.352313\pi$
$263$	14.5146	0.895008	0.447504	+	0.894282i	$0.352313\pi$
$264$	0	0
$265$	6.24400	0.383566
$266$	0	0
$267$	−4.56477	−0.279359
$268$	0	0
$269$	17.8787	1.09008	0.545042	−	0.838409i	$-0.316514\pi$
$269$	17.8787	1.09008	0.545042	+	0.838409i	$0.316514\pi$
$270$	0	0
$271$	26.2459	1.59433	0.797163	−	0.603764i	$-0.206333\pi$
$271$	26.2459	1.59433	0.797163	+	0.603764i	$0.206333\pi$
$272$	0	0
$273$	−9.96844	−0.603318
$274$	0	0
$275$	1.22418	0.0738209
$276$	0	0
$277$	−10.7923	−0.648445	−0.324222	−	0.945981i	$-0.605103\pi$
$277$	−10.7923	−0.648445	−0.324222	+	0.945981i	$0.605103\pi$
$278$	0	0
$279$	−8.46269	−0.506648
$280$	0	0
$281$	3.45938	0.206369	0.103185	−	0.994662i	$-0.467097\pi$
$281$	3.45938	0.206369	0.103185	+	0.994662i	$0.467097\pi$
$282$	0	0
$283$	−7.01935	−0.417257	−0.208628	−	0.977995i	$-0.566900\pi$
$283$	−7.01935	−0.417257	−0.208628	+	0.977995i	$0.566900\pi$
$284$	0	0
$285$	−15.0739	−0.892902
$286$	0	0
$287$	44.7334	2.64053
$288$	0	0
$289$	−13.6175	−0.801031
$290$	0	0
$291$	10.8423	0.635586
$292$	0	0
$293$	−21.2689	−1.24254	−0.621270	−	0.783596i	$-0.713383\pi$
$293$	−21.2689	−1.24254	−0.621270	+	0.783596i	$0.713383\pi$
$294$	0	0
$295$	18.6557	1.08617
$296$	0	0
$297$	0.833896	0.0483876
$298$	0	0
$299$	−9.65531	−0.558381
$300$	0	0
$301$	40.9647	2.36116
$302$	0	0
$303$	1.34614	0.0773338
$304$	0	0
$305$	7.02359	0.402169
$306$	0	0
$307$	6.59119	0.376179	0.188090	−	0.982152i	$-0.439770\pi$
$307$	6.59119	0.376179	0.188090	+	0.982152i	$0.439770\pi$
$308$	0	0
$309$	−17.7990	−1.01255
$310$	0	0
$311$	−9.88758	−0.560673	−0.280337	−	0.959902i	$-0.590446\pi$
$311$	−9.88758	−0.560673	−0.280337	+	0.959902i	$0.590446\pi$
$312$	0	0
$313$	15.8236	0.894405	0.447202	−	0.894433i	$-0.352420\pi$
$313$	15.8236	0.894405	0.447202	+	0.894433i	$0.352420\pi$
$314$	0	0
$315$	−10.3005	−0.580365
$316$	0	0
$317$	19.7065	1.10683	0.553415	−	0.832906i	$-0.313324\pi$
$317$	19.7065	1.10683	0.553415	+	0.832906i	$0.313324\pi$
$318$	0	0
$319$	−6.49487	−0.363643
$320$	0	0
$321$	−9.17441	−0.512065
$322$	0	0
$323$	10.9008	0.606536
$324$	0	0
$325$	3.61318	0.200423
$326$	0	0
$327$	10.0390	0.555156
$328$	0	0
$329$	−2.01815	−0.111264
$330$	0	0
$331$	−10.2190	−0.561687	−0.280844	−	0.959754i	$-0.590614\pi$
$331$	−10.2190	−0.561687	−0.280844	+	0.959754i	$0.590614\pi$
$332$	0	0
$333$	3.96040	0.217029
$334$	0	0
$335$	27.7678	1.51712
$336$	0	0
$337$	11.2996	0.615529	0.307764	−	0.951463i	$-0.400419\pi$
$337$	11.2996	0.615529	0.307764	+	0.951463i	$0.400419\pi$
$338$	0	0
$339$	18.2770	0.992670
$340$	0	0
$341$	−7.05700	−0.382158
$342$	0	0
$343$	−9.73536	−0.525660
$344$	0	0
$345$	−9.97690	−0.537138
$346$	0	0
$347$	−5.76457	−0.309459	−0.154729	−	0.987957i	$-0.549451\pi$
$347$	−5.76457	−0.309459	−0.154729	+	0.987957i	$0.549451\pi$
$348$	0	0
$349$	29.3610	1.57166	0.785828	−	0.618445i	$-0.212237\pi$
$349$	29.3610	1.57166	0.785828	+	0.618445i	$0.212237\pi$
$350$	0	0
$351$	2.46125	0.131372
$352$	0	0
$353$	3.57956	0.190521	0.0952603	−	0.995452i	$-0.469632\pi$
$353$	3.57956	0.190521	0.0952603	+	0.995452i	$0.469632\pi$
$354$	0	0
$355$	−10.1097	−0.536568
$356$	0	0
$357$	7.44884	0.394234
$358$	0	0
$359$	−5.20874	−0.274907	−0.137453	−	0.990508i	$-0.543892\pi$
$359$	−5.20874	−0.274907	−0.137453	+	0.990508i	$0.543892\pi$
$360$	0	0
$361$	16.1302	0.848960
$362$	0	0
$363$	−10.3046	−0.540852
$364$	0	0
$365$	17.3818	0.909808
$366$	0	0
$367$	−7.05201	−0.368112	−0.184056	−	0.982916i	$-0.558923\pi$
$367$	−7.05201	−0.368112	−0.184056	+	0.982916i	$0.558923\pi$
$368$	0	0
$369$	−11.0449	−0.574974
$370$	0	0
$371$	−9.94370	−0.516251
$372$	0	0
$373$	−28.5516	−1.47835	−0.739173	−	0.673515i	$-0.764784\pi$
$373$	−28.5516	−1.47835	−0.739173	+	0.673515i	$0.764784\pi$
$374$	0	0
$375$	−8.98263	−0.463861
$376$	0	0
$377$	−19.1697	−0.987289
$378$	0	0
$379$	−37.0375	−1.90249	−0.951244	−	0.308438i	$-0.900194\pi$
$379$	−37.0375	−1.90249	−0.951244	+	0.308438i	$0.900194\pi$
$380$	0	0
$381$	8.49866	0.435400
$382$	0	0
$383$	−14.3367	−0.732571	−0.366285	−	0.930503i	$-0.619371\pi$
$383$	−14.3367	−0.732571	−0.366285	+	0.930503i	$0.619371\pi$
$384$	0	0
$385$	−8.58952	−0.437762
$386$	0	0
$387$	−10.1144	−0.514142
$388$	0	0
$389$	27.7940	1.40921	0.704606	−	0.709599i	$-0.251124\pi$
$389$	27.7940	1.40921	0.704606	+	0.709599i	$0.251124\pi$
$390$	0	0
$391$	7.21485	0.364871
$392$	0	0
$393$	−6.28622	−0.317098
$394$	0	0
$395$	18.6902	0.940406
$396$	0	0
$397$	−18.2225	−0.914562	−0.457281	−	0.889322i	$-0.651177\pi$
$397$	−18.2225	−0.914562	−0.457281	+	0.889322i	$0.651177\pi$
$398$	0	0
$399$	24.0055	1.20178
$400$	0	0
$401$	15.4736	0.772715	0.386357	−	0.922349i	$-0.373733\pi$
$401$	15.4736	0.772715	0.386357	+	0.922349i	$0.373733\pi$
$402$	0	0
$403$	−20.8288	−1.03756
$404$	0	0
$405$	2.54323	0.126374
$406$	0	0
$407$	3.30256	0.163702
$408$	0	0
$409$	−28.8220	−1.42515	−0.712577	−	0.701594i	$-0.752472\pi$
$409$	−28.8220	−1.42515	−0.712577	+	0.701594i	$0.752472\pi$
$410$	0	0
$411$	4.10350	0.202411
$412$	0	0
$413$	−29.7095	−1.46191
$414$	0	0
$415$	−11.8556	−0.581968
$416$	0	0
$417$	4.54642	0.222639
$418$	0	0
$419$	−5.39623	−0.263623	−0.131812	−	0.991275i	$-0.542079\pi$
$419$	−5.39623	−0.263623	−0.131812	+	0.991275i	$0.542079\pi$
$420$	0	0
$421$	−39.5566	−1.92787	−0.963935	−	0.266137i	$-0.914253\pi$
$421$	−39.5566	−1.92787	−0.963935	+	0.266137i	$0.914253\pi$
$422$	0	0
$423$	0.498289	0.0242277
$424$	0	0
$425$	−2.69992	−0.130965
$426$	0	0
$427$	−11.1852	−0.541290
$428$	0	0
$429$	2.05243	0.0990923
$430$	0	0
$431$	13.6111	0.655625	0.327813	−	0.944743i	$-0.393689\pi$
$431$	13.6111	0.655625	0.327813	+	0.944743i	$0.393689\pi$
$432$	0	0
$433$	8.00025	0.384468	0.192234	−	0.981349i	$-0.438427\pi$
$433$	8.00025	0.384468	0.192234	+	0.981349i	$0.438427\pi$
$434$	0	0
$435$	−19.8082	−0.949729
$436$	0	0
$437$	23.2515	1.11227
$438$	0	0
$439$	−2.97754	−0.142110	−0.0710551	−	0.997472i	$-0.522637\pi$
$439$	−2.97754	−0.142110	−0.0710551	+	0.997472i	$0.522637\pi$
$440$	0	0
$441$	9.40370	0.447795
$442$	0	0
$443$	21.5745	1.02504	0.512518	−	0.858677i	$-0.328713\pi$
$443$	21.5745	1.02504	0.512518	+	0.858677i	$0.328713\pi$
$444$	0	0
$445$	−11.6093	−0.550331
$446$	0	0
$447$	−13.9805	−0.661254
$448$	0	0
$449$	30.7847	1.45282	0.726409	−	0.687263i	$-0.241188\pi$
$449$	30.7847	1.45282	0.726409	+	0.687263i	$0.241188\pi$
$450$	0	0
$451$	−9.21029	−0.433696
$452$	0	0
$453$	−2.44014	−0.114648
$454$	0	0
$455$	−25.3521	−1.18852
$456$	0	0
$457$	8.42006	0.393874	0.196937	−	0.980416i	$-0.436901\pi$
$457$	8.42006	0.393874	0.196937	+	0.980416i	$0.436901\pi$
$458$	0	0
$459$	−1.83915	−0.0858442
$460$	0	0
$461$	23.3100	1.08565	0.542827	−	0.839844i	$-0.317354\pi$
$461$	23.3100	1.08565	0.542827	+	0.839844i	$0.317354\pi$
$462$	0	0
$463$	1.50270	0.0698364	0.0349182	−	0.999390i	$-0.488883\pi$
$463$	1.50270	0.0698364	0.0349182	+	0.999390i	$0.488883\pi$
$464$	0	0
$465$	−21.5226	−0.998086
$466$	0	0
$467$	−8.68794	−0.402030	−0.201015	−	0.979588i	$-0.564424\pi$
$467$	−8.68794	−0.402030	−0.201015	+	0.979588i	$0.564424\pi$
$468$	0	0
$469$	−44.2208	−2.04193
$470$	0	0
$471$	−17.2013	−0.792594
$472$	0	0
$473$	−8.43433	−0.387811
$474$	0	0
$475$	−8.70110	−0.399234
$476$	0	0
$477$	2.45514	0.112413
$478$	0	0
$479$	−33.2228	−1.51799	−0.758994	−	0.651098i	$-0.774309\pi$
$479$	−33.2228	−1.51799	−0.758994	+	0.651098i	$0.774309\pi$
$480$	0	0
$481$	9.74755	0.444450
$482$	0	0
$483$	15.8884	0.722948
$484$	0	0
$485$	27.5744	1.25209
$486$	0	0
$487$	−19.4634	−0.881971	−0.440985	−	0.897514i	$-0.645371\pi$
$487$	−19.4634	−0.881971	−0.440985	+	0.897514i	$0.645371\pi$
$488$	0	0
$489$	−0.421952	−0.0190813
$490$	0	0
$491$	42.1354	1.90154	0.950772	−	0.309892i	$-0.100293\pi$
$491$	42.1354	1.90154	0.950772	+	0.309892i	$0.100293\pi$
$492$	0	0
$493$	14.3244	0.645138
$494$	0	0
$495$	2.12079	0.0953225
$496$	0	0
$497$	16.0999	0.722181
$498$	0	0
$499$	21.5365	0.964107	0.482053	−	0.876142i	$-0.339891\pi$
$499$	21.5365	0.964107	0.482053	+	0.876142i	$0.339891\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−36.5611	−1.63018	−0.815090	−	0.579334i	$-0.803313\pi$
$503$	−36.5611	−1.63018	−0.815090	+	0.579334i	$0.803313\pi$
$504$	0	0
$505$	3.42355	0.152346
$506$	0	0
$507$	−6.94223	−0.308315
$508$	0	0
$509$	−20.4301	−0.905547	−0.452773	−	0.891626i	$-0.649565\pi$
$509$	−20.4301	−0.905547	−0.452773	+	0.891626i	$0.649565\pi$
$510$	0	0
$511$	−27.6810	−1.22453
$512$	0	0
$513$	−5.92708	−0.261687
$514$	0	0
$515$	−45.2669	−1.99470
$516$	0	0
$517$	0.415522	0.0182746
$518$	0	0
$519$	−16.6235	−0.729692
$520$	0	0
$521$	16.7009	0.731678	0.365839	−	0.930678i	$-0.380782\pi$
$521$	16.7009	0.731678	0.365839	+	0.930678i	$0.380782\pi$
$522$	0	0
$523$	23.3167	1.01957	0.509784	−	0.860302i	$-0.329725\pi$
$523$	23.3167	1.01957	0.509784	+	0.860302i	$0.329725\pi$
$524$	0	0
$525$	−5.94572	−0.259493
$526$	0	0
$527$	15.5642	0.677986
$528$	0	0
$529$	−7.61069	−0.330899
$530$	0	0
$531$	7.33541	0.318330
$532$	0	0
$533$	−27.1843	−1.17748
$534$	0	0
$535$	−23.3326	−1.00876
$536$	0	0
$537$	1.25643	0.0542188
$538$	0	0
$539$	7.84171	0.337766
$540$	0	0
$541$	−25.6590	−1.10317	−0.551584	−	0.834119i	$-0.685976\pi$
$541$	−25.6590	−1.10317	−0.551584	+	0.834119i	$0.685976\pi$
$542$	0	0
$543$	−4.10641	−0.176223
$544$	0	0
$545$	25.5314	1.09364
$546$	0	0
$547$	−36.7011	−1.56923	−0.784613	−	0.619985i	$-0.787138\pi$
$547$	−36.7011	−1.56923	−0.784613	+	0.619985i	$0.787138\pi$
$548$	0	0
$549$	2.76168	0.117866
$550$	0	0
$551$	46.1635	1.96663
$552$	0	0
$553$	−29.7645	−1.26572
$554$	0	0
$555$	10.0722	0.427542
$556$	0	0
$557$	9.41547	0.398946	0.199473	−	0.979903i	$-0.436077\pi$
$557$	9.41547	0.398946	0.199473	+	0.979903i	$0.436077\pi$
$558$	0	0
$559$	−24.8940	−1.05290
$560$	0	0
$561$	−1.53366	−0.0647512
$562$	0	0
$563$	−2.64266	−0.111375	−0.0556874	−	0.998448i	$-0.517735\pi$
$563$	−2.64266	−0.111375	−0.0556874	+	0.998448i	$0.517735\pi$
$564$	0	0
$565$	46.4826	1.95554
$566$	0	0
$567$	−4.05015	−0.170090
$568$	0	0
$569$	33.1152	1.38826	0.694130	−	0.719850i	$-0.255789\pi$
$569$	33.1152	1.38826	0.694130	+	0.719850i	$0.255789\pi$
$570$	0	0
$571$	11.1000	0.464521	0.232261	−	0.972654i	$-0.425388\pi$
$571$	11.1000	0.464521	0.232261	+	0.972654i	$0.425388\pi$
$572$	0	0
$573$	14.8331	0.619661
$574$	0	0
$575$	−5.75895	−0.240165
$576$	0	0
$577$	−13.0188	−0.541981	−0.270991	−	0.962582i	$-0.587351\pi$
$577$	−13.0188	−0.541981	−0.270991	+	0.962582i	$0.587351\pi$
$578$	0	0
$579$	−10.2047	−0.424092
$580$	0	0
$581$	18.8803	0.783286
$582$	0	0
$583$	2.04734	0.0847920
$584$	0	0
$585$	6.25954	0.258800
$586$	0	0
$587$	23.7173	0.978917	0.489458	−	0.872027i	$-0.337194\pi$
$587$	23.7173	0.978917	0.489458	+	0.872027i	$0.337194\pi$
$588$	0	0
$589$	50.1590	2.06677
$590$	0	0
$591$	−12.2783	−0.505064
$592$	0	0
$593$	−24.6798	−1.01348	−0.506739	−	0.862100i	$-0.669149\pi$
$593$	−24.6798	−1.01348	−0.506739	+	0.862100i	$0.669149\pi$
$594$	0	0
$595$	18.9441	0.776633
$596$	0	0
$597$	3.70118	0.151479
$598$	0	0
$599$	−41.1844	−1.68275	−0.841374	−	0.540454i	$-0.818252\pi$
$599$	−41.1844	−1.68275	−0.841374	+	0.540454i	$0.818252\pi$
$600$	0	0
$601$	−3.01949	−0.123168	−0.0615838	−	0.998102i	$-0.519615\pi$
$601$	−3.01949	−0.123168	−0.0615838	+	0.998102i	$0.519615\pi$
$602$	0	0
$603$	10.9183	0.444628
$604$	0	0
$605$	−26.2070	−1.06547
$606$	0	0
$607$	−21.3274	−0.865653	−0.432826	−	0.901477i	$-0.642484\pi$
$607$	−21.3274	−0.865653	−0.432826	+	0.901477i	$0.642484\pi$
$608$	0	0
$609$	31.5449	1.27826
$610$	0	0
$611$	1.22642	0.0496155
$612$	0	0
$613$	38.6813	1.56232	0.781161	−	0.624330i	$-0.214628\pi$
$613$	38.6813	1.56232	0.781161	+	0.624330i	$0.214628\pi$
$614$	0	0
$615$	−28.0897	−1.13269
$616$	0	0
$617$	43.0717	1.73400	0.867001	−	0.498307i	$-0.166045\pi$
$617$	43.0717	1.73400	0.867001	+	0.498307i	$0.166045\pi$
$618$	0	0
$619$	−9.34350	−0.375547	−0.187773	−	0.982212i	$-0.560127\pi$
$619$	−9.34350	−0.375547	−0.187773	+	0.982212i	$0.560127\pi$
$620$	0	0
$621$	−3.92292	−0.157421
$622$	0	0
$623$	18.4880	0.740705
$624$	0	0
$625$	−30.1850	−1.20740
$626$	0	0
$627$	−4.94257	−0.197387
$628$	0	0
$629$	−7.28378	−0.290423
$630$	0	0
$631$	34.3258	1.36649	0.683245	−	0.730190i	$-0.260568\pi$
$631$	34.3258	1.36649	0.683245	+	0.730190i	$0.260568\pi$
$632$	0	0
$633$	−1.75230	−0.0696475
$634$	0	0
$635$	21.6141	0.857728
$636$	0	0
$637$	23.1449	0.917034
$638$	0	0
$639$	−3.97515	−0.157254
$640$	0	0
$641$	29.5973	1.16902	0.584511	−	0.811386i	$-0.301286\pi$
$641$	29.5973	1.16902	0.584511	+	0.811386i	$0.301286\pi$
$642$	0	0
$643$	−24.8427	−0.979701	−0.489850	−	0.871807i	$-0.662949\pi$
$643$	−24.8427	−0.979701	−0.489850	+	0.871807i	$0.662949\pi$
$644$	0	0
$645$	−25.7232	−1.01285
$646$	0	0
$647$	21.8126	0.857542	0.428771	−	0.903413i	$-0.358947\pi$
$647$	21.8126	0.857542	0.428771	+	0.903413i	$0.358947\pi$
$648$	0	0
$649$	6.11697	0.240112
$650$	0	0
$651$	34.2752	1.34335
$652$	0	0
$653$	−31.0957	−1.21687	−0.608435	−	0.793604i	$-0.708202\pi$
$653$	−31.0957	−1.21687	−0.608435	+	0.793604i	$0.708202\pi$
$654$	0	0
$655$	−15.9873	−0.624676
$656$	0	0
$657$	6.83455	0.266641
$658$	0	0
$659$	8.55446	0.333234	0.166617	−	0.986022i	$-0.446716\pi$
$659$	8.55446	0.333234	0.166617	+	0.986022i	$0.446716\pi$
$660$	0	0
$661$	24.3444	0.946887	0.473444	−	0.880824i	$-0.343011\pi$
$661$	24.3444	0.946887	0.473444	+	0.880824i	$0.343011\pi$
$662$	0	0
$663$	−4.52662	−0.175799
$664$	0	0
$665$	61.0517	2.36748
$666$	0	0
$667$	30.5540	1.18306
$668$	0	0
$669$	−6.79814	−0.262831
$670$	0	0
$671$	2.30295	0.0889045
$672$	0	0
$673$	−6.31282	−0.243341	−0.121671	−	0.992571i	$-0.538825\pi$
$673$	−6.31282	−0.243341	−0.121671	+	0.992571i	$0.538825\pi$
$674$	0	0
$675$	1.46803	0.0565043
$676$	0	0
$677$	−20.7946	−0.799200	−0.399600	−	0.916690i	$-0.630851\pi$
$677$	−20.7946	−0.799200	−0.399600	+	0.916690i	$0.630851\pi$
$678$	0	0
$679$	−43.9129	−1.68522
$680$	0	0
$681$	2.28779	0.0876685
$682$	0	0
$683$	−36.3346	−1.39030	−0.695152	−	0.718862i	$-0.744663\pi$
$683$	−36.3346	−1.39030	−0.695152	+	0.718862i	$0.744663\pi$
$684$	0	0
$685$	10.4361	0.398744
$686$	0	0
$687$	−21.2604	−0.811134
$688$	0	0
$689$	6.04273	0.230210
$690$	0	0
$691$	20.2826	0.771587	0.385794	−	0.922585i	$-0.373928\pi$
$691$	20.2826	0.771587	0.385794	+	0.922585i	$0.373928\pi$
$692$	0	0
$693$	−3.37740	−0.128297
$694$	0	0
$695$	11.5626	0.438594
$696$	0	0
$697$	20.3132	0.769418
$698$	0	0
$699$	17.9559	0.679154
$700$	0	0
$701$	−1.45086	−0.0547982	−0.0273991	−	0.999625i	$-0.508723\pi$
$701$	−1.45086	−0.0547982	−0.0273991	+	0.999625i	$0.508723\pi$
$702$	0	0
$703$	−23.4736	−0.885324
$704$	0	0
$705$	1.26727	0.0477280
$706$	0	0
$707$	−5.45208	−0.205046
$708$	0	0
$709$	21.8610	0.821006	0.410503	−	0.911859i	$-0.365353\pi$
$709$	21.8610	0.821006	0.410503	+	0.911859i	$0.365353\pi$
$710$	0	0
$711$	7.34900	0.275609
$712$	0	0
$713$	33.1985	1.24329
$714$	0	0
$715$	5.21980	0.195210
$716$	0	0
$717$	−9.32061	−0.348085
$718$	0	0
$719$	48.2100	1.79793	0.898965	−	0.438020i	$-0.144320\pi$
$719$	48.2100	1.79793	0.898965	+	0.438020i	$0.144320\pi$
$720$	0	0
$721$	72.0885	2.68472
$722$	0	0
$723$	10.7267	0.398931
$724$	0	0
$725$	−11.4338	−0.424642
$726$	0	0
$727$	5.64488	0.209357	0.104679	−	0.994506i	$-0.466619\pi$
$727$	5.64488	0.209357	0.104679	+	0.994506i	$0.466619\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	18.6018	0.688014
$732$	0	0
$733$	9.56879	0.353431	0.176716	−	0.984262i	$-0.443453\pi$
$733$	9.56879	0.353431	0.176716	+	0.984262i	$0.443453\pi$
$734$	0	0
$735$	23.9158	0.882147
$736$	0	0
$737$	9.10473	0.335377
$738$	0	0
$739$	29.7475	1.09428	0.547139	−	0.837041i	$-0.315717\pi$
$739$	29.7475	1.09428	0.547139	+	0.837041i	$0.315717\pi$
$740$	0	0
$741$	−14.5880	−0.535905
$742$	0	0
$743$	−36.8506	−1.35192	−0.675959	−	0.736939i	$-0.736271\pi$
$743$	−36.8506	−1.35192	−0.675959	+	0.736939i	$0.736271\pi$
$744$	0	0
$745$	−35.5556	−1.30266
$746$	0	0
$747$	−4.66163	−0.170560
$748$	0	0
$749$	37.1577	1.35771
$750$	0	0
$751$	24.0585	0.877908	0.438954	−	0.898509i	$-0.355349\pi$
$751$	24.0585	0.877908	0.438954	+	0.898509i	$0.355349\pi$
$752$	0	0
$753$	14.6533	0.533997
$754$	0	0
$755$	−6.20583	−0.225853
$756$	0	0
$757$	−7.73664	−0.281193	−0.140596	−	0.990067i	$-0.544902\pi$
$757$	−7.73664	−0.281193	−0.140596	+	0.990067i	$0.544902\pi$
$758$	0	0
$759$	−3.27131	−0.118741
$760$	0	0
$761$	−17.0837	−0.619285	−0.309643	−	0.950853i	$-0.600209\pi$
$761$	−17.0837	−0.619285	−0.309643	+	0.950853i	$0.600209\pi$
$762$	0	0
$763$	−40.6593	−1.47196
$764$	0	0
$765$	−4.67739	−0.169111
$766$	0	0
$767$	18.0543	0.651904
$768$	0	0
$769$	−39.0339	−1.40760	−0.703800	−	0.710398i	$-0.748515\pi$
$769$	−39.0339	−1.40760	−0.703800	+	0.710398i	$0.748515\pi$
$770$	0	0
$771$	4.98675	0.179593
$772$	0	0
$773$	18.5873	0.668540	0.334270	−	0.942477i	$-0.391510\pi$
$773$	18.5873	0.668540	0.334270	+	0.942477i	$0.391510\pi$
$774$	0	0
$775$	−12.4234	−0.446263
$776$	0	0
$777$	−16.0402	−0.575439
$778$	0	0
$779$	65.4639	2.34549
$780$	0	0
$781$	−3.31486	−0.118615
$782$	0	0
$783$	−7.78858	−0.278341
$784$	0	0
$785$	−43.7469	−1.56139
$786$	0	0
$787$	−49.1552	−1.75219	−0.876097	−	0.482134i	$-0.839862\pi$
$787$	−49.1552	−1.75219	−0.876097	+	0.482134i	$0.839862\pi$
$788$	0	0
$789$	14.5146	0.516733
$790$	0	0
$791$	−74.0246	−2.63201
$792$	0	0
$793$	6.79719	0.241375
$794$	0	0
$795$	6.24400	0.221452
$796$	0	0
$797$	−45.5162	−1.61227	−0.806133	−	0.591734i	$-0.798443\pi$
$797$	−45.5162	−1.61227	−0.806133	+	0.591734i	$0.798443\pi$
$798$	0	0
$799$	−0.916430	−0.0324210
$800$	0	0
$801$	−4.56477	−0.161288
$802$	0	0
$803$	5.69931	0.201124
$804$	0	0
$805$	40.4079	1.42419
$806$	0	0
$807$	17.8787	0.629360
$808$	0	0
$809$	−17.0876	−0.600768	−0.300384	−	0.953818i	$-0.597115\pi$
$809$	−17.0876	−0.600768	−0.300384	+	0.953818i	$0.597115\pi$
$810$	0	0
$811$	37.2942	1.30958	0.654789	−	0.755812i	$-0.272757\pi$
$811$	37.2942	1.30958	0.654789	+	0.755812i	$0.272757\pi$
$812$	0	0
$813$	26.2459	0.920484
$814$	0	0
$815$	−1.07312	−0.0375898
$816$	0	0
$817$	59.9486	2.09734
$818$	0	0
$819$	−9.96844	−0.348326
$820$	0	0
$821$	35.4980	1.23889	0.619444	−	0.785041i	$-0.287358\pi$
$821$	35.4980	1.23889	0.619444	+	0.785041i	$0.287358\pi$
$822$	0	0
$823$	−52.0090	−1.81292	−0.906460	−	0.422291i	$-0.861226\pi$
$823$	−52.0090	−1.81292	−0.906460	+	0.422291i	$0.861226\pi$
$824$	0	0
$825$	1.22418	0.0426205
$826$	0	0
$827$	−17.5986	−0.611962	−0.305981	−	0.952038i	$-0.598984\pi$
$827$	−17.5986	−0.611962	−0.305981	+	0.952038i	$0.598984\pi$
$828$	0	0
$829$	31.8869	1.10748	0.553738	−	0.832691i	$-0.313201\pi$
$829$	31.8869	1.10748	0.553738	+	0.832691i	$0.313201\pi$
$830$	0	0
$831$	−10.7923	−0.374380
$832$	0	0
$833$	−17.2948	−0.599231
$834$	0	0
$835$	−2.54323	−0.0880121
$836$	0	0
$837$	−8.46269	−0.292513
$838$	0	0
$839$	−22.6755	−0.782846	−0.391423	−	0.920211i	$-0.628017\pi$
$839$	−22.6755	−0.782846	−0.391423	+	0.920211i	$0.628017\pi$
$840$	0	0
$841$	31.6620	1.09179
$842$	0	0
$843$	3.45938	0.119147
$844$	0	0
$845$	−17.6557	−0.607374
$846$	0	0
$847$	41.7352	1.43404
$848$	0	0
$849$	−7.01935	−0.240903
$850$	0	0
$851$	−15.5363	−0.532579
$852$	0	0
$853$	−40.0438	−1.37107	−0.685537	−	0.728038i	$-0.740433\pi$
$853$	−40.0438	−1.37107	−0.685537	+	0.728038i	$0.740433\pi$
$854$	0	0
$855$	−15.0739	−0.515517
$856$	0	0
$857$	−15.5167	−0.530039	−0.265019	−	0.964243i	$-0.585378\pi$
$857$	−15.5167	−0.530039	−0.265019	+	0.964243i	$0.585378\pi$
$858$	0	0
$859$	9.92115	0.338505	0.169253	−	0.985573i	$-0.445865\pi$
$859$	9.92115	0.338505	0.169253	+	0.985573i	$0.445865\pi$
$860$	0	0
$861$	44.7334	1.52451
$862$	0	0
$863$	−5.06567	−0.172437	−0.0862187	−	0.996276i	$-0.527478\pi$
$863$	−5.06567	−0.172437	−0.0862187	+	0.996276i	$0.527478\pi$
$864$	0	0
$865$	−42.2775	−1.43748
$866$	0	0
$867$	−13.6175	−0.462475
$868$	0	0
$869$	6.12830	0.207888
$870$	0	0
$871$	26.8727	0.910547
$872$	0	0
$873$	10.8423	0.366956
$874$	0	0
$875$	36.3810	1.22990
$876$	0	0
$877$	42.0614	1.42031	0.710157	−	0.704043i	$-0.248624\pi$
$877$	42.0614	1.42031	0.710157	+	0.704043i	$0.248624\pi$
$878$	0	0
$879$	−21.2689	−0.717381
$880$	0	0
$881$	2.70081	0.0909927	0.0454964	−	0.998965i	$-0.485513\pi$
$881$	2.70081	0.0909927	0.0454964	+	0.998965i	$0.485513\pi$
$882$	0	0
$883$	30.7249	1.03397	0.516987	−	0.855993i	$-0.327054\pi$
$883$	30.7249	1.03397	0.516987	+	0.855993i	$0.327054\pi$
$884$	0	0
$885$	18.6557	0.627103
$886$	0	0
$887$	52.0588	1.74796	0.873981	−	0.485960i	$-0.161530\pi$
$887$	52.0588	1.74796	0.873981	+	0.485960i	$0.161530\pi$
$888$	0	0
$889$	−34.4208	−1.15444
$890$	0	0
$891$	0.833896	0.0279366
$892$	0	0
$893$	−2.95340	−0.0988318
$894$	0	0
$895$	3.19538	0.106810
$896$	0	0
$897$	−9.65531	−0.322381
$898$	0	0
$899$	65.9124	2.19830
$900$	0	0
$901$	−4.51538	−0.150429
$902$	0	0
$903$	40.9647	1.36322
$904$	0	0
$905$	−10.4436	−0.347155
$906$	0	0
$907$	−14.7456	−0.489621	−0.244811	−	0.969571i	$-0.578726\pi$
$907$	−14.7456	−0.489621	−0.244811	+	0.969571i	$0.578726\pi$
$908$	0	0
$909$	1.34614	0.0446487
$910$	0	0
$911$	19.6554	0.651212	0.325606	−	0.945506i	$-0.394432\pi$
$911$	19.6554	0.651212	0.325606	+	0.945506i	$0.394432\pi$
$912$	0	0
$913$	−3.88731	−0.128651
$914$	0	0
$915$	7.02359	0.232193
$916$	0	0
$917$	25.4601	0.840768
$918$	0	0
$919$	27.8835	0.919792	0.459896	−	0.887973i	$-0.347887\pi$
$919$	27.8835	0.919792	0.459896	+	0.887973i	$0.347887\pi$
$920$	0	0
$921$	6.59119	0.217187
$922$	0	0
$923$	−9.78384	−0.322039
$924$	0	0
$925$	5.81397	0.191162
$926$	0	0
$927$	−17.7990	−0.584595
$928$	0	0
$929$	13.7960	0.452631	0.226316	−	0.974054i	$-0.427332\pi$
$929$	13.7960	0.452631	0.226316	+	0.974054i	$0.427332\pi$
$930$	0	0
$931$	−55.7365	−1.82669
$932$	0	0
$933$	−9.88758	−0.323705
$934$	0	0
$935$	−3.90046	−0.127559
$936$	0	0
$937$	−5.96881	−0.194993	−0.0974963	−	0.995236i	$-0.531083\pi$
$937$	−5.96881	−0.194993	−0.0974963	+	0.995236i	$0.531083\pi$
$938$	0	0
$939$	15.8236	0.516385
$940$	0	0
$941$	−57.6012	−1.87775	−0.938873	−	0.344264i	$-0.888128\pi$
$941$	−57.6012	−1.87775	−0.938873	+	0.344264i	$0.888128\pi$
$942$	0	0
$943$	43.3282	1.41096
$944$	0	0
$945$	−10.3005	−0.335074
$946$	0	0
$947$	50.1673	1.63022	0.815109	−	0.579308i	$-0.196677\pi$
$947$	50.1673	1.63022	0.815109	+	0.579308i	$0.196677\pi$
$948$	0	0
$949$	16.8216	0.546051
$950$	0	0
$951$	19.7065	0.639028
$952$	0	0
$953$	−19.9024	−0.644703	−0.322352	−	0.946620i	$-0.604473\pi$
$953$	−19.9024	−0.644703	−0.322352	+	0.946620i	$0.604473\pi$
$954$	0	0
$955$	37.7240	1.22072
$956$	0	0
$957$	−6.49487	−0.209949
$958$	0	0
$959$	−16.6198	−0.536680
$960$	0	0
$961$	40.6171	1.31023
$962$	0	0
$963$	−9.17441	−0.295641
$964$	0	0
$965$	−25.9529	−0.835453
$966$	0	0
$967$	−44.9296	−1.44484	−0.722419	−	0.691456i	$-0.756970\pi$
$967$	−44.9296	−1.44484	−0.722419	+	0.691456i	$0.756970\pi$
$968$	0	0
$969$	10.9008	0.350184
$970$	0	0
$971$	35.3274	1.13371	0.566855	−	0.823817i	$-0.308160\pi$
$971$	35.3274	1.13371	0.566855	+	0.823817i	$0.308160\pi$
$972$	0	0
$973$	−18.4137	−0.590315
$974$	0	0
$975$	3.61318	0.115714
$976$	0	0
$977$	−0.366186	−0.0117153	−0.00585766	−	0.999983i	$-0.501865\pi$
$977$	−0.366186	−0.0117153	−0.00585766	+	0.999983i	$0.501865\pi$
$978$	0	0
$979$	−3.80654	−0.121658
$980$	0	0
$981$	10.0390	0.320519
$982$	0	0
$983$	26.9995	0.861149	0.430575	−	0.902555i	$-0.358311\pi$
$983$	26.9995	0.861149	0.430575	+	0.902555i	$0.358311\pi$
$984$	0	0
$985$	−31.2267	−0.994965
$986$	0	0
$987$	−2.01815	−0.0642383
$988$	0	0
$989$	39.6779	1.26168
$990$	0	0
$991$	53.3689	1.69532	0.847659	−	0.530542i	$-0.178011\pi$
$991$	53.3689	1.69532	0.847659	+	0.530542i	$0.178011\pi$
$992$	0	0
$993$	−10.2190	−0.324290
$994$	0	0
$995$	9.41296	0.298411
$996$	0	0
$997$	55.2046	1.74835	0.874174	−	0.485613i	$-0.161404\pi$
$997$	55.2046	1.74835	0.874174	+	0.485613i	$0.161404\pi$
$998$	0	0
$999$	3.96040	0.125302

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.i.1.9	✓	9
4.3	odd	2		8016.2.a.ba.1.9		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.i.1.9	✓	9	1.1	even	1	trivial
8016.2.a.ba.1.9		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.54323 q^{5} -4.05015 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.54323 q^{5} -4.05015 q^{7} +1.00000 q^{9} +0.833896 q^{11} +2.46125 q^{13} +2.54323 q^{15} -1.83915 q^{17} -5.92708 q^{19} -4.05015 q^{21} -3.92292 q^{23} +1.46803 q^{25} +1.00000 q^{27} -7.78858 q^{29} -8.46269 q^{31} +0.833896 q^{33} -10.3005 q^{35} +3.96040 q^{37} +2.46125 q^{39} -11.0449 q^{41} -10.1144 q^{43} +2.54323 q^{45} +0.498289 q^{47} +9.40370 q^{49} -1.83915 q^{51} +2.45514 q^{53} +2.12079 q^{55} -5.92708 q^{57} +7.33541 q^{59} +2.76168 q^{61} -4.05015 q^{63} +6.25954 q^{65} +10.9183 q^{67} -3.92292 q^{69} -3.97515 q^{71} +6.83455 q^{73} +1.46803 q^{75} -3.37740 q^{77} +7.34900 q^{79} +1.00000 q^{81} -4.66163 q^{83} -4.67739 q^{85} -7.78858 q^{87} -4.56477 q^{89} -9.96844 q^{91} -8.46269 q^{93} -15.0739 q^{95} +10.8423 q^{97} +0.833896 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} - 6 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) \) 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

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