LMFDB - Embedded newform 6032.2.a.bd.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6032,2,Mod(1,6032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6032, 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("6032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6032 = 2^{4} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6032.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:	$48.1657624992$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 10 x^{10} + 98 x^{9} + 10 x^{8} - 585 x^{7} + 151 x^{6} + 1524 x^{5} - 445 x^{4} + \cdots + 68 $ x^12 - 6x^11 - 10x^10 + 98x^9 + 10x^8 - 585x^7 + 151x^6 + 1524x^5 - 445x^4 - 1567x^3 + 273x^2 + 424*x + 68
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.32893$ of defining polynomial
Character	$\chi$	$=$	6032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.328928 q^{3} -1.23337 q^{5} +3.96198 q^{7} -2.89181 q^{9} +O(q^{10})$	$q+0.328928 q^{3} -1.23337 q^{5} +3.96198 q^{7} -2.89181 q^{9} -0.303371 q^{11} +1.00000 q^{13} -0.405691 q^{15} -3.14782 q^{17} +2.50831 q^{19} +1.30321 q^{21} +3.71021 q^{23} -3.47879 q^{25} -1.93798 q^{27} +1.00000 q^{29} -9.43774 q^{31} -0.0997871 q^{33} -4.88660 q^{35} -7.40875 q^{37} +0.328928 q^{39} +1.65564 q^{41} +5.44682 q^{43} +3.56668 q^{45} -10.8886 q^{47} +8.69730 q^{49} -1.03540 q^{51} -3.56453 q^{53} +0.374169 q^{55} +0.825053 q^{57} +3.92484 q^{59} -0.998599 q^{61} -11.4573 q^{63} -1.23337 q^{65} -9.87167 q^{67} +1.22039 q^{69} +14.9053 q^{71} -3.64216 q^{73} -1.14427 q^{75} -1.20195 q^{77} -11.9586 q^{79} +8.03796 q^{81} -4.58822 q^{83} +3.88243 q^{85} +0.328928 q^{87} +12.3040 q^{89} +3.96198 q^{91} -3.10434 q^{93} -3.09368 q^{95} +7.54968 q^{97} +0.877289 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9}+O(q^{10}) $ 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9	$ 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9} - 14 q^{11} + 12 q^{13} - 8 q^{15} + 4 q^{17} - 11 q^{19} - 5 q^{21} - 15 q^{23} + 5 q^{25} - 24 q^{27} + 12 q^{29} - 13 q^{31} - q^{33} - 18 q^{35} - 23 q^{37} - 6 q^{39} - 2 q^{41} - 26 q^{43} - 9 q^{45} - 15 q^{47} + 16 q^{49} - 21 q^{51} + 31 q^{53} - 10 q^{55} - 10 q^{57} - 7 q^{59} + 2 q^{61} + 25 q^{63} + 3 q^{65} - 47 q^{67} - 8 q^{69} - 32 q^{71} - 25 q^{73} - 31 q^{75} - 4 q^{77} - 7 q^{79} + 64 q^{81} - 12 q^{83} + 7 q^{85} - 6 q^{87} + 6 q^{89} - 6 q^{91} + 17 q^{93} - q^{95} - 7 q^{97} - 44 q^{99}+O(q^{100}) $ 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9 - 14 * q^11 + 12 * q^13 - 8 * q^15 + 4 * q^17 - 11 * q^19 - 5 * q^21 - 15 * q^23 + 5 * q^25 - 24 * q^27 + 12 * q^29 - 13 * q^31 - q^33 - 18 * q^35 - 23 * q^37 - 6 * q^39 - 2 * q^41 - 26 * q^43 - 9 * q^45 - 15 * q^47 + 16 * q^49 - 21 * q^51 + 31 * q^53 - 10 * q^55 - 10 * q^57 - 7 * q^59 + 2 * q^61 + 25 * q^63 + 3 * q^65 - 47 * q^67 - 8 * q^69 - 32 * q^71 - 25 * q^73 - 31 * q^75 - 4 * q^77 - 7 * q^79 + 64 * q^81 - 12 * q^83 + 7 * q^85 - 6 * q^87 + 6 * q^89 - 6 * q^91 + 17 * q^93 - q^95 - 7 * q^97 - 44 * 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
$3$	0.328928	0.189907	0.0949533	−	0.995482i	$-0.469730\pi$
$3$	0.328928	0.189907	0.0949533	+	0.995482i	$0.469730\pi$
$4$	0	0
$5$	−1.23337	−0.551582	−0.275791	−	0.961218i	$-0.588940\pi$
$5$	−1.23337	−0.551582	−0.275791	+	0.961218i	$0.588940\pi$
$6$	0	0
$7$	3.96198	1.49749	0.748744	−	0.662859i	$-0.230657\pi$
$7$	3.96198	1.49749	0.748744	+	0.662859i	$0.230657\pi$
$8$	0	0
$9$	−2.89181	−0.963935
$10$	0	0
$11$	−0.303371	−0.0914697	−0.0457348	−	0.998954i	$-0.514563\pi$
$11$	−0.303371	−0.0914697	−0.0457348	+	0.998954i	$0.514563\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−0.405691	−0.104749
$16$	0	0
$17$	−3.14782	−0.763457	−0.381729	−	0.924274i	$-0.624671\pi$
$17$	−3.14782	−0.763457	−0.381729	+	0.924274i	$0.624671\pi$
$18$	0	0
$19$	2.50831	0.575446	0.287723	−	0.957714i	$-0.407102\pi$
$19$	2.50831	0.575446	0.287723	+	0.957714i	$0.407102\pi$
$20$	0	0
$21$	1.30321	0.284383
$22$	0	0
$23$	3.71021	0.773633	0.386817	−	0.922157i	$-0.373575\pi$
$23$	3.71021	0.773633	0.386817	+	0.922157i	$0.373575\pi$
$24$	0	0
$25$	−3.47879	−0.695758
$26$	0	0
$27$	−1.93798	−0.372964
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−9.43774	−1.69507	−0.847534	−	0.530742i	$-0.821913\pi$
$31$	−9.43774	−1.69507	−0.847534	+	0.530742i	$0.821913\pi$
$32$	0	0
$33$	−0.0997871	−0.0173707
$34$	0	0
$35$	−4.88660	−0.825987
$36$	0	0
$37$	−7.40875	−1.21799	−0.608996	−	0.793173i	$-0.708427\pi$
$37$	−7.40875	−1.21799	−0.608996	+	0.793173i	$0.708427\pi$
$38$	0	0
$39$	0.328928	0.0526706
$40$	0	0
$41$	1.65564	0.258568	0.129284	−	0.991608i	$-0.458732\pi$
$41$	1.65564	0.258568	0.129284	+	0.991608i	$0.458732\pi$
$42$	0	0
$43$	5.44682	0.830632	0.415316	−	0.909677i	$-0.363671\pi$
$43$	5.44682	0.830632	0.415316	+	0.909677i	$0.363671\pi$
$44$	0	0
$45$	3.56668	0.531689
$46$	0	0
$47$	−10.8886	−1.58826	−0.794130	−	0.607748i	$-0.792073\pi$
$47$	−10.8886	−1.58826	−0.794130	+	0.607748i	$0.792073\pi$
$48$	0	0
$49$	8.69730	1.24247
$50$	0	0
$51$	−1.03540	−0.144986
$52$	0	0
$53$	−3.56453	−0.489626	−0.244813	−	0.969570i	$-0.578727\pi$
$53$	−3.56453	−0.489626	−0.244813	+	0.969570i	$0.578727\pi$
$54$	0	0
$55$	0.374169	0.0504530
$56$	0	0
$57$	0.825053	0.109281
$58$	0	0
$59$	3.92484	0.510971	0.255485	−	0.966813i	$-0.417765\pi$
$59$	3.92484	0.510971	0.255485	+	0.966813i	$0.417765\pi$
$60$	0	0
$61$	−0.998599	−0.127858	−0.0639288	−	0.997954i	$-0.520363\pi$
$61$	−0.998599	−0.127858	−0.0639288	+	0.997954i	$0.520363\pi$
$62$	0	0
$63$	−11.4573	−1.44348
$64$	0	0
$65$	−1.23337	−0.152981
$66$	0	0
$67$	−9.87167	−1.20602	−0.603008	−	0.797735i	$-0.706031\pi$
$67$	−9.87167	−1.20602	−0.603008	+	0.797735i	$0.706031\pi$
$68$	0	0
$69$	1.22039	0.146918
$70$	0	0
$71$	14.9053	1.76893	0.884467	−	0.466602i	$-0.154522\pi$
$71$	14.9053	1.76893	0.884467	+	0.466602i	$0.154522\pi$
$72$	0	0
$73$	−3.64216	−0.426283	−0.213141	−	0.977021i	$-0.568370\pi$
$73$	−3.64216	−0.426283	−0.213141	+	0.977021i	$0.568370\pi$
$74$	0	0
$75$	−1.14427	−0.132129
$76$	0	0
$77$	−1.20195	−0.136975
$78$	0	0
$79$	−11.9586	−1.34545	−0.672726	−	0.739892i	$-0.734877\pi$
$79$	−11.9586	−1.34545	−0.672726	+	0.739892i	$0.734877\pi$
$80$	0	0
$81$	8.03796	0.893107
$82$	0	0
$83$	−4.58822	−0.503623	−0.251811	−	0.967776i	$-0.581026\pi$
$83$	−4.58822	−0.503623	−0.251811	+	0.967776i	$0.581026\pi$
$84$	0	0
$85$	3.88243	0.421109
$86$	0	0
$87$	0.328928	0.0352648
$88$	0	0
$89$	12.3040	1.30422	0.652112	−	0.758122i	$-0.273883\pi$
$89$	12.3040	1.30422	0.652112	+	0.758122i	$0.273883\pi$
$90$	0	0
$91$	3.96198	0.415329
$92$	0	0
$93$	−3.10434	−0.321905
$94$	0	0
$95$	−3.09368	−0.317405
$96$	0	0
$97$	7.54968	0.766553	0.383277	−	0.923634i	$-0.374796\pi$
$97$	7.54968	0.766553	0.383277	+	0.923634i	$0.374796\pi$
$98$	0	0
$99$	0.877289	0.0881709
$100$	0	0
$101$	−1.80586	−0.179690	−0.0898451	−	0.995956i	$-0.528637\pi$
$101$	−1.80586	−0.179690	−0.0898451	+	0.995956i	$0.528637\pi$
$102$	0	0
$103$	−17.0560	−1.68058	−0.840290	−	0.542137i	$-0.817615\pi$
$103$	−17.0560	−1.68058	−0.840290	+	0.542137i	$0.817615\pi$
$104$	0	0
$105$	−1.60734	−0.156860
$106$	0	0
$107$	3.73917	0.361479	0.180740	−	0.983531i	$-0.442151\pi$
$107$	3.73917	0.361479	0.180740	+	0.983531i	$0.442151\pi$
$108$	0	0
$109$	−5.71429	−0.547329	−0.273665	−	0.961825i	$-0.588236\pi$
$109$	−5.71429	−0.547329	−0.273665	+	0.961825i	$0.588236\pi$
$110$	0	0
$111$	−2.43695	−0.231305
$112$	0	0
$113$	15.9101	1.49670	0.748350	−	0.663304i	$-0.230846\pi$
$113$	15.9101	1.49670	0.748350	+	0.663304i	$0.230846\pi$
$114$	0	0
$115$	−4.57608	−0.426722
$116$	0	0
$117$	−2.89181	−0.267348
$118$	0	0
$119$	−12.4716	−1.14327
$120$	0	0
$121$	−10.9080	−0.991633
$122$	0	0
$123$	0.544588	0.0491038
$124$	0	0
$125$	10.4575	0.935349
$126$	0	0
$127$	3.84668	0.341338	0.170669	−	0.985328i	$-0.445407\pi$
$127$	3.84668	0.341338	0.170669	+	0.985328i	$0.445407\pi$
$128$	0	0
$129$	1.79161	0.157742
$130$	0	0
$131$	−17.5946	−1.53724	−0.768622	−	0.639703i	$-0.779057\pi$
$131$	−17.5946	−1.53724	−0.768622	+	0.639703i	$0.779057\pi$
$132$	0	0
$133$	9.93788	0.861723
$134$	0	0
$135$	2.39025	0.205720
$136$	0	0
$137$	−10.8104	−0.923597	−0.461798	−	0.886985i	$-0.652796\pi$
$137$	−10.8104	−0.923597	−0.461798	+	0.886985i	$0.652796\pi$
$138$	0	0
$139$	−11.2583	−0.954914	−0.477457	−	0.878655i	$-0.658441\pi$
$139$	−11.2583	−0.954914	−0.477457	+	0.878655i	$0.658441\pi$
$140$	0	0
$141$	−3.58155	−0.301621
$142$	0	0
$143$	−0.303371	−0.0253691
$144$	0	0
$145$	−1.23337	−0.102426
$146$	0	0
$147$	2.86079	0.235954
$148$	0	0
$149$	9.31314	0.762962	0.381481	−	0.924377i	$-0.375414\pi$
$149$	9.31314	0.762962	0.381481	+	0.924377i	$0.375414\pi$
$150$	0	0
$151$	3.45265	0.280973	0.140486	−	0.990083i	$-0.455133\pi$
$151$	3.45265	0.280973	0.140486	+	0.990083i	$0.455133\pi$
$152$	0	0
$153$	9.10287	0.735924
$154$	0	0
$155$	11.6403	0.934968
$156$	0	0
$157$	−14.0171	−1.11869	−0.559343	−	0.828936i	$-0.688947\pi$
$157$	−14.0171	−1.11869	−0.559343	+	0.828936i	$0.688947\pi$
$158$	0	0
$159$	−1.17247	−0.0929833
$160$	0	0
$161$	14.6998	1.15851
$162$	0	0
$163$	−18.6063	−1.45736	−0.728679	−	0.684856i	$-0.759865\pi$
$163$	−18.6063	−1.45736	−0.728679	+	0.684856i	$0.759865\pi$
$164$	0	0
$165$	0.123075	0.00958136
$166$	0	0
$167$	10.1579	0.786045	0.393022	−	0.919529i	$-0.371430\pi$
$167$	10.1579	0.786045	0.393022	+	0.919529i	$0.371430\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.25355	−0.554693
$172$	0	0
$173$	−18.8324	−1.43180	−0.715900	−	0.698203i	$-0.753983\pi$
$173$	−18.8324	−1.43180	−0.715900	+	0.698203i	$0.753983\pi$
$174$	0	0
$175$	−13.7829	−1.04189
$176$	0	0
$177$	1.29099	0.0970368
$178$	0	0
$179$	−22.0791	−1.65027	−0.825134	−	0.564937i	$-0.808901\pi$
$179$	−22.0791	−1.65027	−0.825134	+	0.564937i	$0.808901\pi$
$180$	0	0
$181$	9.69012	0.720261	0.360130	−	0.932902i	$-0.382732\pi$
$181$	9.69012	0.720261	0.360130	+	0.932902i	$0.382732\pi$
$182$	0	0
$183$	−0.328467	−0.0242810
$184$	0	0
$185$	9.13776	0.671822
$186$	0	0
$187$	0.954955	0.0698332
$188$	0	0
$189$	−7.67824	−0.558510
$190$	0	0
$191$	−4.11919	−0.298054	−0.149027	−	0.988833i	$-0.547614\pi$
$191$	−4.11919	−0.298054	−0.149027	+	0.988833i	$0.547614\pi$
$192$	0	0
$193$	−25.8152	−1.85822	−0.929109	−	0.369805i	$-0.879424\pi$
$193$	−25.8152	−1.85822	−0.929109	+	0.369805i	$0.879424\pi$
$194$	0	0
$195$	−0.405691	−0.0290521
$196$	0	0
$197$	−27.3699	−1.95002	−0.975011	−	0.222155i	$-0.928691\pi$
$197$	−27.3699	−1.95002	−0.975011	+	0.222155i	$0.928691\pi$
$198$	0	0
$199$	19.2292	1.36312	0.681560	−	0.731762i	$-0.261302\pi$
$199$	19.2292	1.36312	0.681560	+	0.731762i	$0.261302\pi$
$200$	0	0
$201$	−3.24707	−0.229031
$202$	0	0
$203$	3.96198	0.278077
$204$	0	0
$205$	−2.04203	−0.142621
$206$	0	0
$207$	−10.7292	−0.745732
$208$	0	0
$209$	−0.760947	−0.0526358
$210$	0	0
$211$	−9.90657	−0.681997	−0.340998	−	0.940064i	$-0.610765\pi$
$211$	−9.90657	−0.681997	−0.340998	+	0.940064i	$0.610765\pi$
$212$	0	0
$213$	4.90277	0.335932
$214$	0	0
$215$	−6.71796	−0.458161
$216$	0	0
$217$	−37.3921	−2.53834
$218$	0	0
$219$	−1.19801	−0.0809540
$220$	0	0
$221$	−3.14782	−0.211745
$222$	0	0
$223$	27.9930	1.87455	0.937274	−	0.348593i	$-0.113341\pi$
$223$	27.9930	1.87455	0.937274	+	0.348593i	$0.113341\pi$
$224$	0	0
$225$	10.0600	0.670666
$226$	0	0
$227$	16.3431	1.08473	0.542366	−	0.840142i	$-0.317528\pi$
$227$	16.3431	1.08473	0.542366	+	0.840142i	$0.317528\pi$
$228$	0	0
$229$	6.68265	0.441602	0.220801	−	0.975319i	$-0.429133\pi$
$229$	6.68265	0.441602	0.220801	+	0.975319i	$0.429133\pi$
$230$	0	0
$231$	−0.395355	−0.0260124
$232$	0	0
$233$	29.9546	1.96239	0.981197	−	0.193011i	$-0.0618254\pi$
$233$	29.9546	1.96239	0.981197	+	0.193011i	$0.0618254\pi$
$234$	0	0
$235$	13.4297	0.876055
$236$	0	0
$237$	−3.93353	−0.255510
$238$	0	0
$239$	9.66269	0.625027	0.312514	−	0.949913i	$-0.398829\pi$
$239$	9.66269	0.625027	0.312514	+	0.949913i	$0.398829\pi$
$240$	0	0
$241$	0.921718	0.0593730	0.0296865	−	0.999559i	$-0.490549\pi$
$241$	0.921718	0.0593730	0.0296865	+	0.999559i	$0.490549\pi$
$242$	0	0
$243$	8.45785	0.542571
$244$	0	0
$245$	−10.7270	−0.685324
$246$	0	0
$247$	2.50831	0.159600
$248$	0	0
$249$	−1.50919	−0.0956413
$250$	0	0
$251$	−6.61125	−0.417298	−0.208649	−	0.977991i	$-0.566907\pi$
$251$	−6.61125	−0.417298	−0.208649	+	0.977991i	$0.566907\pi$
$252$	0	0
$253$	−1.12557	−0.0707640
$254$	0	0
$255$	1.27704	0.0799714
$256$	0	0
$257$	5.22216	0.325750	0.162875	−	0.986647i	$-0.447923\pi$
$257$	5.22216	0.325750	0.162875	+	0.986647i	$0.447923\pi$
$258$	0	0
$259$	−29.3533	−1.82393
$260$	0	0
$261$	−2.89181	−0.178998
$262$	0	0
$263$	−1.67147	−0.103067	−0.0515335	−	0.998671i	$-0.516411\pi$
$263$	−1.67147	−0.103067	−0.0515335	+	0.998671i	$0.516411\pi$
$264$	0	0
$265$	4.39640	0.270069
$266$	0	0
$267$	4.04714	0.247681
$268$	0	0
$269$	22.7371	1.38631	0.693153	−	0.720791i	$-0.256221\pi$
$269$	22.7371	1.38631	0.693153	+	0.720791i	$0.256221\pi$
$270$	0	0
$271$	5.55353	0.337353	0.168676	−	0.985671i	$-0.446051\pi$
$271$	5.55353	0.337353	0.168676	+	0.985671i	$0.446051\pi$
$272$	0	0
$273$	1.30321	0.0788737
$274$	0	0
$275$	1.05536	0.0636407
$276$	0	0
$277$	−12.1926	−0.732584	−0.366292	−	0.930500i	$-0.619373\pi$
$277$	−12.1926	−0.732584	−0.366292	+	0.930500i	$0.619373\pi$
$278$	0	0
$279$	27.2921	1.63394
$280$	0	0
$281$	−9.39891	−0.560692	−0.280346	−	0.959899i	$-0.590449\pi$
$281$	−9.39891	−0.560692	−0.280346	+	0.959899i	$0.590449\pi$
$282$	0	0
$283$	−31.4725	−1.87085	−0.935423	−	0.353530i	$-0.884981\pi$
$283$	−31.4725	−1.87085	−0.935423	+	0.353530i	$0.884981\pi$
$284$	0	0
$285$	−1.01760	−0.0602774
$286$	0	0
$287$	6.55963	0.387203
$288$	0	0
$289$	−7.09126	−0.417133
$290$	0	0
$291$	2.48330	0.145574
$292$	0	0
$293$	17.3012	1.01074	0.505372	−	0.862901i	$-0.331355\pi$
$293$	17.3012	1.01074	0.505372	+	0.862901i	$0.331355\pi$
$294$	0	0
$295$	−4.84080	−0.281842
$296$	0	0
$297$	0.587926	0.0341149
$298$	0	0
$299$	3.71021	0.214567
$300$	0	0
$301$	21.5802	1.24386
$302$	0	0
$303$	−0.593999	−0.0341243
$304$	0	0
$305$	1.23165	0.0705238
$306$	0	0
$307$	−16.4554	−0.939161	−0.469580	−	0.882890i	$-0.655595\pi$
$307$	−16.4554	−0.939161	−0.469580	+	0.882890i	$0.655595\pi$
$308$	0	0
$309$	−5.61021	−0.319153
$310$	0	0
$311$	−24.4590	−1.38694	−0.693471	−	0.720484i	$-0.743920\pi$
$311$	−24.4590	−1.38694	−0.693471	+	0.720484i	$0.743920\pi$
$312$	0	0
$313$	30.2188	1.70807	0.854033	−	0.520219i	$-0.174150\pi$
$313$	30.2188	1.70807	0.854033	+	0.520219i	$0.174150\pi$
$314$	0	0
$315$	14.1311	0.796198
$316$	0	0
$317$	6.48628	0.364306	0.182153	−	0.983270i	$-0.441693\pi$
$317$	6.48628	0.364306	0.182153	+	0.983270i	$0.441693\pi$
$318$	0	0
$319$	−0.303371	−0.0169855
$320$	0	0
$321$	1.22992	0.0686473
$322$	0	0
$323$	−7.89570	−0.439328
$324$	0	0
$325$	−3.47879	−0.192968
$326$	0	0
$327$	−1.87959	−0.103942
$328$	0	0
$329$	−43.1403	−2.37840
$330$	0	0
$331$	−19.8488	−1.09099	−0.545495	−	0.838114i	$-0.683658\pi$
$331$	−19.8488	−1.09099	−0.545495	+	0.838114i	$0.683658\pi$
$332$	0	0
$333$	21.4247	1.17407
$334$	0	0
$335$	12.1755	0.665216
$336$	0	0
$337$	−0.971447	−0.0529181	−0.0264590	−	0.999650i	$-0.508423\pi$
$337$	−0.971447	−0.0529181	−0.0264590	+	0.999650i	$0.508423\pi$
$338$	0	0
$339$	5.23329	0.284233
$340$	0	0
$341$	2.86313	0.155047
$342$	0	0
$343$	6.72468	0.363099
$344$	0	0
$345$	−1.50520	−0.0810373
$346$	0	0
$347$	−8.18191	−0.439228	−0.219614	−	0.975587i	$-0.570480\pi$
$347$	−8.18191	−0.439228	−0.219614	+	0.975587i	$0.570480\pi$
$348$	0	0
$349$	−13.1713	−0.705045	−0.352523	−	0.935803i	$-0.614676\pi$
$349$	−13.1713	−0.705045	−0.352523	+	0.935803i	$0.614676\pi$
$350$	0	0
$351$	−1.93798	−0.103442
$352$	0	0
$353$	−27.8572	−1.48269	−0.741344	−	0.671125i	$-0.765811\pi$
$353$	−27.8572	−1.48269	−0.741344	+	0.671125i	$0.765811\pi$
$354$	0	0
$355$	−18.3838	−0.975712
$356$	0	0
$357$	−4.10225	−0.217114
$358$	0	0
$359$	−2.16245	−0.114130	−0.0570649	−	0.998370i	$-0.518174\pi$
$359$	−2.16245	−0.114130	−0.0570649	+	0.998370i	$0.518174\pi$
$360$	0	0
$361$	−12.7084	−0.668862
$362$	0	0
$363$	−3.58794	−0.188318
$364$	0	0
$365$	4.49215	0.235130
$366$	0	0
$367$	8.78188	0.458411	0.229205	−	0.973378i	$-0.426387\pi$
$367$	8.78188	0.458411	0.229205	+	0.973378i	$0.426387\pi$
$368$	0	0
$369$	−4.78780	−0.249243
$370$	0	0
$371$	−14.1226	−0.733210
$372$	0	0
$373$	17.8276	0.923079	0.461540	−	0.887120i	$-0.347297\pi$
$373$	17.8276	0.923079	0.461540	+	0.887120i	$0.347297\pi$
$374$	0	0
$375$	3.43977	0.177629
$376$	0	0
$377$	1.00000	0.0515026
$378$	0	0
$379$	−22.5689	−1.15928	−0.579642	−	0.814871i	$-0.696808\pi$
$379$	−22.5689	−1.15928	−0.579642	+	0.814871i	$0.696808\pi$
$380$	0	0
$381$	1.26528	0.0648223
$382$	0	0
$383$	6.06223	0.309765	0.154883	−	0.987933i	$-0.450500\pi$
$383$	6.06223	0.309765	0.154883	+	0.987933i	$0.450500\pi$
$384$	0	0
$385$	1.48245	0.0755528
$386$	0	0
$387$	−15.7511	−0.800675
$388$	0	0
$389$	−29.9625	−1.51916	−0.759580	−	0.650414i	$-0.774595\pi$
$389$	−29.9625	−1.51916	−0.759580	+	0.650414i	$0.774595\pi$
$390$	0	0
$391$	−11.6791	−0.590636
$392$	0	0
$393$	−5.78735	−0.291933
$394$	0	0
$395$	14.7495	0.742126
$396$	0	0
$397$	−37.8829	−1.90129	−0.950645	−	0.310280i	$-0.899577\pi$
$397$	−37.8829	−1.90129	−0.950645	+	0.310280i	$0.899577\pi$
$398$	0	0
$399$	3.26885	0.163647
$400$	0	0
$401$	3.69123	0.184331	0.0921657	−	0.995744i	$-0.470621\pi$
$401$	3.69123	0.184331	0.0921657	+	0.995744i	$0.470621\pi$
$402$	0	0
$403$	−9.43774	−0.470127
$404$	0	0
$405$	−9.91381	−0.492621
$406$	0	0
$407$	2.24760	0.111409
$408$	0	0
$409$	−1.67872	−0.0830072	−0.0415036	−	0.999138i	$-0.513215\pi$
$409$	−1.67872	−0.0830072	−0.0415036	+	0.999138i	$0.513215\pi$
$410$	0	0
$411$	−3.55585	−0.175397
$412$	0	0
$413$	15.5502	0.765173
$414$	0	0
$415$	5.65899	0.277789
$416$	0	0
$417$	−3.70316	−0.181344
$418$	0	0
$419$	16.6122	0.811560	0.405780	−	0.913971i	$-0.367000\pi$
$419$	16.6122	0.811560	0.405780	+	0.913971i	$0.367000\pi$
$420$	0	0
$421$	−12.5060	−0.609505	−0.304752	−	0.952432i	$-0.598574\pi$
$421$	−12.5060	−0.609505	−0.304752	+	0.952432i	$0.598574\pi$
$422$	0	0
$423$	31.4876	1.53098
$424$	0	0
$425$	10.9506	0.531181
$426$	0	0
$427$	−3.95643	−0.191465
$428$	0	0
$429$	−0.0997871	−0.00481777
$430$	0	0
$431$	23.5550	1.13460	0.567302	−	0.823510i	$-0.307987\pi$
$431$	23.5550	1.13460	0.567302	+	0.823510i	$0.307987\pi$
$432$	0	0
$433$	−23.3296	−1.12115	−0.560575	−	0.828104i	$-0.689420\pi$
$433$	−23.3296	−1.12115	−0.560575	+	0.828104i	$0.689420\pi$
$434$	0	0
$435$	−0.405691	−0.0194514
$436$	0	0
$437$	9.30636	0.445184
$438$	0	0
$439$	−28.8531	−1.37709	−0.688543	−	0.725196i	$-0.741749\pi$
$439$	−28.8531	−1.37709	−0.688543	+	0.725196i	$0.741749\pi$
$440$	0	0
$441$	−25.1509	−1.19766
$442$	0	0
$443$	−36.1847	−1.71919	−0.859594	−	0.510978i	$-0.829283\pi$
$443$	−36.1847	−1.71919	−0.859594	+	0.510978i	$0.829283\pi$
$444$	0	0
$445$	−15.1755	−0.719386
$446$	0	0
$447$	3.06335	0.144892
$448$	0	0
$449$	27.1491	1.28125	0.640623	−	0.767855i	$-0.278676\pi$
$449$	27.1491	1.28125	0.640623	+	0.767855i	$0.278676\pi$
$450$	0	0
$451$	−0.502274	−0.0236511
$452$	0	0
$453$	1.13567	0.0533586
$454$	0	0
$455$	−4.88660	−0.229088
$456$	0	0
$457$	17.5765	0.822196	0.411098	−	0.911591i	$-0.365145\pi$
$457$	17.5765	0.822196	0.411098	+	0.911591i	$0.365145\pi$
$458$	0	0
$459$	6.10040	0.284742
$460$	0	0
$461$	17.9206	0.834645	0.417323	−	0.908758i	$-0.362968\pi$
$461$	17.9206	0.834645	0.417323	+	0.908758i	$0.362968\pi$
$462$	0	0
$463$	−10.0361	−0.466415	−0.233208	−	0.972427i	$-0.574922\pi$
$463$	−10.0361	−0.466415	−0.233208	+	0.972427i	$0.574922\pi$
$464$	0	0
$465$	3.82881	0.177557
$466$	0	0
$467$	−27.7951	−1.28620	−0.643101	−	0.765781i	$-0.722353\pi$
$467$	−27.7951	−1.28620	−0.643101	+	0.765781i	$0.722353\pi$
$468$	0	0
$469$	−39.1114	−1.80600
$470$	0	0
$471$	−4.61062	−0.212446
$472$	0	0
$473$	−1.65240	−0.0759776
$474$	0	0
$475$	−8.72588	−0.400371
$476$	0	0
$477$	10.3079	0.471968
$478$	0	0
$479$	12.4135	0.567188	0.283594	−	0.958944i	$-0.408473\pi$
$479$	12.4135	0.567188	0.283594	+	0.958944i	$0.408473\pi$
$480$	0	0
$481$	−7.40875	−0.337810
$482$	0	0
$483$	4.83518	0.220008
$484$	0	0
$485$	−9.31157	−0.422817
$486$	0	0
$487$	14.2492	0.645695	0.322848	−	0.946451i	$-0.395360\pi$
$487$	14.2492	0.645695	0.322848	+	0.946451i	$0.395360\pi$
$488$	0	0
$489$	−6.12013	−0.276762
$490$	0	0
$491$	−4.35385	−0.196486	−0.0982432	−	0.995162i	$-0.531322\pi$
$491$	−4.35385	−0.196486	−0.0982432	+	0.995162i	$0.531322\pi$
$492$	0	0
$493$	−3.14782	−0.141770
$494$	0	0
$495$	−1.08203	−0.0486334
$496$	0	0
$497$	59.0546	2.64896
$498$	0	0
$499$	31.8350	1.42513	0.712565	−	0.701606i	$-0.247533\pi$
$499$	31.8350	1.42513	0.712565	+	0.701606i	$0.247533\pi$
$500$	0	0
$501$	3.34123	0.149275
$502$	0	0
$503$	−25.9380	−1.15652	−0.578260	−	0.815853i	$-0.696268\pi$
$503$	−25.9380	−1.15652	−0.578260	+	0.815853i	$0.696268\pi$
$504$	0	0
$505$	2.22730	0.0991137
$506$	0	0
$507$	0.328928	0.0146082
$508$	0	0
$509$	1.34548	0.0596375	0.0298188	−	0.999555i	$-0.490507\pi$
$509$	1.34548	0.0596375	0.0298188	+	0.999555i	$0.490507\pi$
$510$	0	0
$511$	−14.4302	−0.638354
$512$	0	0
$513$	−4.86105	−0.214621
$514$	0	0
$515$	21.0365	0.926977
$516$	0	0
$517$	3.30327	0.145278
$518$	0	0
$519$	−6.19450	−0.271908
$520$	0	0
$521$	−20.7235	−0.907912	−0.453956	−	0.891024i	$-0.649988\pi$
$521$	−20.7235	−0.907912	−0.453956	+	0.891024i	$0.649988\pi$
$522$	0	0
$523$	26.6997	1.16750	0.583748	−	0.811935i	$-0.301585\pi$
$523$	26.6997	1.16750	0.583748	+	0.811935i	$0.301585\pi$
$524$	0	0
$525$	−4.53358	−0.197862
$526$	0	0
$527$	29.7082	1.29411
$528$	0	0
$529$	−9.23432	−0.401492
$530$	0	0
$531$	−11.3499	−0.492543
$532$	0	0
$533$	1.65564	0.0717139
$534$	0	0
$535$	−4.61179	−0.199385
$536$	0	0
$537$	−7.26243	−0.313397
$538$	0	0
$539$	−2.63851	−0.113648
$540$	0	0
$541$	−22.6584	−0.974162	−0.487081	−	0.873357i	$-0.661938\pi$
$541$	−22.6584	−0.974162	−0.487081	+	0.873357i	$0.661938\pi$
$542$	0	0
$543$	3.18735	0.136782
$544$	0	0
$545$	7.04785	0.301897
$546$	0	0
$547$	12.0092	0.513478	0.256739	−	0.966481i	$-0.417352\pi$
$547$	12.0092	0.513478	0.256739	+	0.966481i	$0.417352\pi$
$548$	0	0
$549$	2.88776	0.123246
$550$	0	0
$551$	2.50831	0.106858
$552$	0	0
$553$	−47.3799	−2.01480
$554$	0	0
$555$	3.00567	0.127583
$556$	0	0
$557$	7.89146	0.334372	0.167186	−	0.985925i	$-0.446532\pi$
$557$	7.89146	0.334372	0.167186	+	0.985925i	$0.446532\pi$
$558$	0	0
$559$	5.44682	0.230376
$560$	0	0
$561$	0.314111	0.0132618
$562$	0	0
$563$	28.3383	1.19432	0.597159	−	0.802123i	$-0.296296\pi$
$563$	28.3383	1.19432	0.597159	+	0.802123i	$0.296296\pi$
$564$	0	0
$565$	−19.6232	−0.825552
$566$	0	0
$567$	31.8463	1.33742
$568$	0	0
$569$	12.1480	0.509273	0.254636	−	0.967037i	$-0.418044\pi$
$569$	12.1480	0.509273	0.254636	+	0.967037i	$0.418044\pi$
$570$	0	0
$571$	40.0123	1.67446	0.837231	−	0.546850i	$-0.184173\pi$
$571$	40.0123	1.67446	0.837231	+	0.546850i	$0.184173\pi$
$572$	0	0
$573$	−1.35492	−0.0566024
$574$	0	0
$575$	−12.9071	−0.538261
$576$	0	0
$577$	−34.8150	−1.44937	−0.724683	−	0.689083i	$-0.758014\pi$
$577$	−34.8150	−1.44937	−0.724683	+	0.689083i	$0.758014\pi$
$578$	0	0
$579$	−8.49134	−0.352888
$580$	0	0
$581$	−18.1785	−0.754169
$582$	0	0
$583$	1.08137	0.0447860
$584$	0	0
$585$	3.56668	0.147464
$586$	0	0
$587$	21.0586	0.869180	0.434590	−	0.900628i	$-0.356893\pi$
$587$	21.0586	0.869180	0.434590	+	0.900628i	$0.356893\pi$
$588$	0	0
$589$	−23.6728	−0.975419
$590$	0	0
$591$	−9.00272	−0.370322
$592$	0	0
$593$	−33.0806	−1.35846	−0.679229	−	0.733927i	$-0.737686\pi$
$593$	−33.0806	−1.35846	−0.679229	+	0.733927i	$0.737686\pi$
$594$	0	0
$595$	15.3821	0.630606
$596$	0	0
$597$	6.32502	0.258866
$598$	0	0
$599$	−6.10448	−0.249422	−0.124711	−	0.992193i	$-0.539800\pi$
$599$	−6.10448	−0.249422	−0.124711	+	0.992193i	$0.539800\pi$
$600$	0	0
$601$	8.36370	0.341163	0.170581	−	0.985344i	$-0.445435\pi$
$601$	8.36370	0.341163	0.170581	+	0.985344i	$0.445435\pi$
$602$	0	0
$603$	28.5470	1.16252
$604$	0	0
$605$	13.4536	0.546967
$606$	0	0
$607$	30.5321	1.23926	0.619629	−	0.784895i	$-0.287283\pi$
$607$	30.5321	1.23926	0.619629	+	0.784895i	$0.287283\pi$
$608$	0	0
$609$	1.30321	0.0528086
$610$	0	0
$611$	−10.8886	−0.440504
$612$	0	0
$613$	40.7026	1.64396	0.821982	−	0.569514i	$-0.192869\pi$
$613$	40.7026	1.64396	0.821982	+	0.569514i	$0.192869\pi$
$614$	0	0
$615$	−0.671680	−0.0270848
$616$	0	0
$617$	−15.1770	−0.611004	−0.305502	−	0.952191i	$-0.598824\pi$
$617$	−15.1770	−0.611004	−0.305502	+	0.952191i	$0.598824\pi$
$618$	0	0
$619$	6.46598	0.259889	0.129945	−	0.991521i	$-0.458520\pi$
$619$	6.46598	0.259889	0.129945	+	0.991521i	$0.458520\pi$
$620$	0	0
$621$	−7.19032	−0.288538
$622$	0	0
$623$	48.7483	1.95306
$624$	0	0
$625$	4.49592	0.179837
$626$	0	0
$627$	−0.250297	−0.00999590
$628$	0	0
$629$	23.3214	0.929885
$630$	0	0
$631$	−12.6649	−0.504183	−0.252091	−	0.967703i	$-0.581118\pi$
$631$	−12.6649	−0.504183	−0.252091	+	0.967703i	$0.581118\pi$
$632$	0	0
$633$	−3.25855	−0.129516
$634$	0	0
$635$	−4.74439	−0.188275
$636$	0	0
$637$	8.69730	0.344600
$638$	0	0
$639$	−43.1033	−1.70514
$640$	0	0
$641$	20.0714	0.792772	0.396386	−	0.918084i	$-0.370264\pi$
$641$	20.0714	0.792772	0.396386	+	0.918084i	$0.370264\pi$
$642$	0	0
$643$	−26.9914	−1.06444	−0.532218	−	0.846607i	$-0.678641\pi$
$643$	−26.9914	−1.06444	−0.532218	+	0.846607i	$0.678641\pi$
$644$	0	0
$645$	−2.20972	−0.0870078
$646$	0	0
$647$	18.7826	0.738422	0.369211	−	0.929346i	$-0.379628\pi$
$647$	18.7826	0.738422	0.369211	+	0.929346i	$0.379628\pi$
$648$	0	0
$649$	−1.19068	−0.0467383
$650$	0	0
$651$	−12.2993	−0.482048
$652$	0	0
$653$	25.0496	0.980267	0.490133	−	0.871647i	$-0.336948\pi$
$653$	25.0496	0.980267	0.490133	+	0.871647i	$0.336948\pi$
$654$	0	0
$655$	21.7007	0.847916
$656$	0	0
$657$	10.5324	0.410909
$658$	0	0
$659$	−45.4099	−1.76892	−0.884460	−	0.466616i	$-0.845473\pi$
$659$	−45.4099	−1.76892	−0.884460	+	0.466616i	$0.845473\pi$
$660$	0	0
$661$	−47.5719	−1.85033	−0.925167	−	0.379561i	$-0.876075\pi$
$661$	−47.5719	−1.85033	−0.925167	+	0.379561i	$0.876075\pi$
$662$	0	0
$663$	−1.03540	−0.0402118
$664$	0	0
$665$	−12.2571	−0.475311
$666$	0	0
$667$	3.71021	0.143660
$668$	0	0
$669$	9.20767	0.355989
$670$	0	0
$671$	0.302946	0.0116951
$672$	0	0
$673$	9.09795	0.350700	0.175350	−	0.984506i	$-0.443894\pi$
$673$	9.09795	0.350700	0.175350	+	0.984506i	$0.443894\pi$
$674$	0	0
$675$	6.74182	0.259493
$676$	0	0
$677$	31.5583	1.21289	0.606443	−	0.795127i	$-0.292596\pi$
$677$	31.5583	1.21289	0.606443	+	0.795127i	$0.292596\pi$
$678$	0	0
$679$	29.9117	1.14790
$680$	0	0
$681$	5.37572	0.205998
$682$	0	0
$683$	−30.2567	−1.15774	−0.578871	−	0.815419i	$-0.696506\pi$
$683$	−30.2567	−1.15774	−0.578871	+	0.815419i	$0.696506\pi$
$684$	0	0
$685$	13.3333	0.509439
$686$	0	0
$687$	2.19811	0.0838632
$688$	0	0
$689$	−3.56453	−0.135798
$690$	0	0
$691$	−43.0346	−1.63712	−0.818558	−	0.574424i	$-0.805226\pi$
$691$	−43.0346	−1.63712	−0.818558	+	0.574424i	$0.805226\pi$
$692$	0	0
$693$	3.47580	0.132035
$694$	0	0
$695$	13.8857	0.526713
$696$	0	0
$697$	−5.21166	−0.197406
$698$	0	0
$699$	9.85292	0.372671
$700$	0	0
$701$	−4.63055	−0.174894	−0.0874468	−	0.996169i	$-0.527871\pi$
$701$	−4.63055	−0.174894	−0.0874468	+	0.996169i	$0.527871\pi$
$702$	0	0
$703$	−18.5834	−0.700888
$704$	0	0
$705$	4.41739	0.166369
$706$	0	0
$707$	−7.15480	−0.269084
$708$	0	0
$709$	−18.3541	−0.689301	−0.344651	−	0.938731i	$-0.612003\pi$
$709$	−18.3541	−0.689301	−0.344651	+	0.938731i	$0.612003\pi$
$710$	0	0
$711$	34.5821	1.29693
$712$	0	0
$713$	−35.0160	−1.31136
$714$	0	0
$715$	0.374169	0.0139931
$716$	0	0
$717$	3.17833	0.118697
$718$	0	0
$719$	−16.3013	−0.607937	−0.303968	−	0.952682i	$-0.598312\pi$
$719$	−16.3013	−0.607937	−0.303968	+	0.952682i	$0.598312\pi$
$720$	0	0
$721$	−67.5757	−2.51665
$722$	0	0
$723$	0.303179	0.0112753
$724$	0	0
$725$	−3.47879	−0.129199
$726$	0	0
$727$	23.6899	0.878610	0.439305	−	0.898338i	$-0.355225\pi$
$727$	23.6899	0.878610	0.439305	+	0.898338i	$0.355225\pi$
$728$	0	0
$729$	−21.3319	−0.790069
$730$	0	0
$731$	−17.1456	−0.634152
$732$	0	0
$733$	24.8363	0.917349	0.458674	−	0.888604i	$-0.348324\pi$
$733$	24.8363	0.917349	0.458674	+	0.888604i	$0.348324\pi$
$734$	0	0
$735$	−3.52842	−0.130148
$736$	0	0
$737$	2.99477	0.110314
$738$	0	0
$739$	22.1444	0.814597	0.407298	−	0.913295i	$-0.366471\pi$
$739$	22.1444	0.814597	0.407298	+	0.913295i	$0.366471\pi$
$740$	0	0
$741$	0.825053	0.0303091
$742$	0	0
$743$	18.7999	0.689702	0.344851	−	0.938657i	$-0.387929\pi$
$743$	18.7999	0.689702	0.344851	+	0.938657i	$0.387929\pi$
$744$	0	0
$745$	−11.4866	−0.420836
$746$	0	0
$747$	13.2682	0.485460
$748$	0	0
$749$	14.8145	0.541311
$750$	0	0
$751$	−22.0112	−0.803200	−0.401600	−	0.915815i	$-0.631546\pi$
$751$	−22.0112	−0.803200	−0.401600	+	0.915815i	$0.631546\pi$
$752$	0	0
$753$	−2.17462	−0.0792477
$754$	0	0
$755$	−4.25841	−0.154979
$756$	0	0
$757$	39.4206	1.43277	0.716383	−	0.697708i	$-0.245796\pi$
$757$	39.4206	1.43277	0.716383	+	0.697708i	$0.245796\pi$
$758$	0	0
$759$	−0.370231	−0.0134385
$760$	0	0
$761$	46.8466	1.69819	0.849093	−	0.528243i	$-0.177149\pi$
$761$	46.8466	1.69819	0.849093	+	0.528243i	$0.177149\pi$
$762$	0	0
$763$	−22.6399	−0.819620
$764$	0	0
$765$	−11.2272	−0.405922
$766$	0	0
$767$	3.92484	0.141718
$768$	0	0
$769$	−18.9194	−0.682252	−0.341126	−	0.940018i	$-0.610808\pi$
$769$	−18.9194	−0.682252	−0.341126	+	0.940018i	$0.610808\pi$
$770$	0	0
$771$	1.71772	0.0618620
$772$	0	0
$773$	27.0667	0.973520	0.486760	−	0.873536i	$-0.338179\pi$
$773$	27.0667	0.973520	0.486760	+	0.873536i	$0.338179\pi$
$774$	0	0
$775$	32.8319	1.17936
$776$	0	0
$777$	−9.65514	−0.346376
$778$	0	0
$779$	4.15287	0.148792
$780$	0	0
$781$	−4.52183	−0.161804
$782$	0	0
$783$	−1.93798	−0.0692578
$784$	0	0
$785$	17.2883	0.617047
$786$	0	0
$787$	38.4915	1.37208	0.686038	−	0.727566i	$-0.259348\pi$
$787$	38.4915	1.37208	0.686038	+	0.727566i	$0.259348\pi$
$788$	0	0
$789$	−0.549792	−0.0195731
$790$	0	0
$791$	63.0357	2.24129
$792$	0	0
$793$	−0.998599	−0.0354613
$794$	0	0
$795$	1.44610	0.0512879
$796$	0	0
$797$	22.6501	0.802307	0.401154	−	0.916011i	$-0.368609\pi$
$797$	22.6501	0.802307	0.401154	+	0.916011i	$0.368609\pi$
$798$	0	0
$799$	34.2752	1.21257
$800$	0	0
$801$	−35.5809	−1.25719
$802$	0	0
$803$	1.10493	0.0389920
$804$	0	0
$805$	−18.1303	−0.639011
$806$	0	0
$807$	7.47887	0.263269
$808$	0	0
$809$	15.9731	0.561584	0.280792	−	0.959769i	$-0.409403\pi$
$809$	15.9731	0.561584	0.280792	+	0.959769i	$0.409403\pi$
$810$	0	0
$811$	−19.6124	−0.688684	−0.344342	−	0.938844i	$-0.611898\pi$
$811$	−19.6124	−0.688684	−0.344342	+	0.938844i	$0.611898\pi$
$812$	0	0
$813$	1.82671	0.0640655
$814$	0	0
$815$	22.9485	0.803851
$816$	0	0
$817$	13.6623	0.477983
$818$	0	0
$819$	−11.4573	−0.400350
$820$	0	0
$821$	21.3536	0.745247	0.372624	−	0.927983i	$-0.378458\pi$
$821$	21.3536	0.745247	0.372624	+	0.927983i	$0.378458\pi$
$822$	0	0
$823$	−1.87454	−0.0653423	−0.0326711	−	0.999466i	$-0.510401\pi$
$823$	−1.87454	−0.0653423	−0.0326711	+	0.999466i	$0.510401\pi$
$824$	0	0
$825$	0.347138	0.0120858
$826$	0	0
$827$	18.0390	0.627276	0.313638	−	0.949543i	$-0.398452\pi$
$827$	18.0390	0.627276	0.313638	+	0.949543i	$0.398452\pi$
$828$	0	0
$829$	13.6326	0.473481	0.236741	−	0.971573i	$-0.423921\pi$
$829$	13.6326	0.473481	0.236741	+	0.971573i	$0.423921\pi$
$830$	0	0
$831$	−4.01050	−0.139123
$832$	0	0
$833$	−27.3775	−0.948574
$834$	0	0
$835$	−12.5285	−0.433568
$836$	0	0
$837$	18.2901	0.632200
$838$	0	0
$839$	−32.9070	−1.13607	−0.568037	−	0.823003i	$-0.692297\pi$
$839$	−32.9070	−1.13607	−0.568037	+	0.823003i	$0.692297\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−3.09157	−0.106479
$844$	0	0
$845$	−1.23337	−0.0424293
$846$	0	0
$847$	−43.2172	−1.48496
$848$	0	0
$849$	−10.3522	−0.355286
$850$	0	0
$851$	−27.4881	−0.942278
$852$	0	0
$853$	−28.7142	−0.983155	−0.491577	−	0.870834i	$-0.663579\pi$
$853$	−28.7142	−0.983155	−0.491577	+	0.870834i	$0.663579\pi$
$854$	0	0
$855$	8.94633	0.305958
$856$	0	0
$857$	2.30405	0.0787048	0.0393524	−	0.999225i	$-0.487471\pi$
$857$	2.30405	0.0787048	0.0393524	+	0.999225i	$0.487471\pi$
$858$	0	0
$859$	13.2619	0.452491	0.226245	−	0.974070i	$-0.427355\pi$
$859$	13.2619	0.452491	0.226245	+	0.974070i	$0.427355\pi$
$860$	0	0
$861$	2.15765	0.0735324
$862$	0	0
$863$	−51.4654	−1.75190	−0.875952	−	0.482399i	$-0.839766\pi$
$863$	−51.4654	−1.75190	−0.875952	+	0.482399i	$0.839766\pi$
$864$	0	0
$865$	23.2274	0.789754
$866$	0	0
$867$	−2.33251	−0.0792163
$868$	0	0
$869$	3.62790	0.123068
$870$	0	0
$871$	−9.87167	−0.334489
$872$	0	0
$873$	−21.8322	−0.738908
$874$	0	0
$875$	41.4325	1.40067
$876$	0	0
$877$	17.3910	0.587252	0.293626	−	0.955920i	$-0.405138\pi$
$877$	17.3910	0.587252	0.293626	+	0.955920i	$0.405138\pi$
$878$	0	0
$879$	5.69084	0.191947
$880$	0	0
$881$	48.8804	1.64682	0.823412	−	0.567444i	$-0.192068\pi$
$881$	48.8804	1.64682	0.823412	+	0.567444i	$0.192068\pi$
$882$	0	0
$883$	19.5812	0.658959	0.329479	−	0.944163i	$-0.393127\pi$
$883$	19.5812	0.658959	0.329479	+	0.944163i	$0.393127\pi$
$884$	0	0
$885$	−1.59227	−0.0535237
$886$	0	0
$887$	14.9575	0.502223	0.251112	−	0.967958i	$-0.419204\pi$
$887$	14.9575	0.502223	0.251112	+	0.967958i	$0.419204\pi$
$888$	0	0
$889$	15.2405	0.511149
$890$	0	0
$891$	−2.43848	−0.0816922
$892$	0	0
$893$	−27.3119	−0.913957
$894$	0	0
$895$	27.2318	0.910258
$896$	0	0
$897$	1.22039	0.0407477
$898$	0	0
$899$	−9.43774	−0.314766
$900$	0	0
$901$	11.2205	0.373809
$902$	0	0
$903$	7.09833	0.236218
$904$	0	0
$905$	−11.9515	−0.397282
$906$	0	0
$907$	13.6643	0.453714	0.226857	−	0.973928i	$-0.427155\pi$
$907$	13.6643	0.453714	0.226857	+	0.973928i	$0.427155\pi$
$908$	0	0
$909$	5.22221	0.173210
$910$	0	0
$911$	−32.2887	−1.06977	−0.534886	−	0.844924i	$-0.679645\pi$
$911$	−32.2887	−1.06977	−0.534886	+	0.844924i	$0.679645\pi$
$912$	0	0
$913$	1.39193	0.0460662
$914$	0	0
$915$	0.405123	0.0133929
$916$	0	0
$917$	−69.7094	−2.30201
$918$	0	0
$919$	45.8033	1.51091	0.755456	−	0.655199i	$-0.227415\pi$
$919$	45.8033	1.51091	0.755456	+	0.655199i	$0.227415\pi$
$920$	0	0
$921$	−5.41265	−0.178353
$922$	0	0
$923$	14.9053	0.490614
$924$	0	0
$925$	25.7735	0.847427
$926$	0	0
$927$	49.3227	1.61997
$928$	0	0
$929$	−9.75510	−0.320054	−0.160027	−	0.987113i	$-0.551158\pi$
$929$	−9.75510	−0.320054	−0.160027	+	0.987113i	$0.551158\pi$
$930$	0	0
$931$	21.8155	0.714975
$932$	0	0
$933$	−8.04525	−0.263390
$934$	0	0
$935$	−1.17782	−0.0385187
$936$	0	0
$937$	1.36411	0.0445634	0.0222817	−	0.999752i	$-0.492907\pi$
$937$	1.36411	0.0445634	0.0222817	+	0.999752i	$0.492907\pi$
$938$	0	0
$939$	9.93980	0.324373
$940$	0	0
$941$	−1.60288	−0.0522524	−0.0261262	−	0.999659i	$-0.508317\pi$
$941$	−1.60288	−0.0522524	−0.0261262	+	0.999659i	$0.508317\pi$
$942$	0	0
$943$	6.14279	0.200037
$944$	0	0
$945$	9.47014	0.308064
$946$	0	0
$947$	29.9110	0.971976	0.485988	−	0.873966i	$-0.338460\pi$
$947$	29.9110	0.971976	0.485988	+	0.873966i	$0.338460\pi$
$948$	0	0
$949$	−3.64216	−0.118230
$950$	0	0
$951$	2.13352	0.0691840
$952$	0	0
$953$	−12.5192	−0.405537	−0.202769	−	0.979227i	$-0.564994\pi$
$953$	−12.5192	−0.405537	−0.202769	+	0.979227i	$0.564994\pi$
$954$	0	0
$955$	5.08050	0.164401
$956$	0	0
$957$	−0.0997871	−0.00322566
$958$	0	0
$959$	−42.8307	−1.38308
$960$	0	0
$961$	58.0709	1.87325
$962$	0	0
$963$	−10.8129	−0.348442
$964$	0	0
$965$	31.8398	1.02496
$966$	0	0
$967$	34.3886	1.10586	0.552932	−	0.833226i	$-0.313509\pi$
$967$	34.3886	1.10586	0.552932	+	0.833226i	$0.313509\pi$
$968$	0	0
$969$	−2.59712	−0.0834314
$970$	0	0
$971$	50.0724	1.60690	0.803450	−	0.595373i	$-0.202996\pi$
$971$	50.0724	1.60690	0.803450	+	0.595373i	$0.202996\pi$
$972$	0	0
$973$	−44.6051	−1.42997
$974$	0	0
$975$	−1.14427	−0.0366460
$976$	0	0
$977$	23.5717	0.754126	0.377063	−	0.926188i	$-0.376934\pi$
$977$	23.5717	0.754126	0.377063	+	0.926188i	$0.376934\pi$
$978$	0	0
$979$	−3.73268	−0.119297
$980$	0	0
$981$	16.5246	0.527590
$982$	0	0
$983$	−35.8749	−1.14423	−0.572116	−	0.820172i	$-0.693877\pi$
$983$	−35.8749	−1.14423	−0.572116	+	0.820172i	$0.693877\pi$
$984$	0	0
$985$	33.7573	1.07560
$986$	0	0
$987$	−14.1900	−0.451674
$988$	0	0
$989$	20.2088	0.642604
$990$	0	0
$991$	−8.46300	−0.268836	−0.134418	−	0.990925i	$-0.542916\pi$
$991$	−8.46300	−0.268836	−0.134418	+	0.990925i	$0.542916\pi$
$992$	0	0
$993$	−6.52883	−0.207186
$994$	0	0
$995$	−23.7168	−0.751872
$996$	0	0
$997$	35.3585	1.11981	0.559907	−	0.828556i	$-0.310837\pi$
$997$	35.3585	1.11981	0.559907	+	0.828556i	$0.310837\pi$
$998$	0	0
$999$	14.3580	0.454267

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.bd.1.8		12
4.3	odd	2		3016.2.a.j.1.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.j.1.5	✓	12	4.3	odd	2
6032.2.a.bd.1.8		12	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:	\( N \)	\(=\)	\( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6032.a (trivial)

\(f(q)\)	\(=\)	\(q+0.328928 q^{3} -1.23337 q^{5} +3.96198 q^{7} -2.89181 q^{9} +O(q^{10})\)	\(q+0.328928 q^{3} -1.23337 q^{5} +3.96198 q^{7} -2.89181 q^{9} -0.303371 q^{11} +1.00000 q^{13} -0.405691 q^{15} -3.14782 q^{17} +2.50831 q^{19} +1.30321 q^{21} +3.71021 q^{23} -3.47879 q^{25} -1.93798 q^{27} +1.00000 q^{29} -9.43774 q^{31} -0.0997871 q^{33} -4.88660 q^{35} -7.40875 q^{37} +0.328928 q^{39} +1.65564 q^{41} +5.44682 q^{43} +3.56668 q^{45} -10.8886 q^{47} +8.69730 q^{49} -1.03540 q^{51} -3.56453 q^{53} +0.374169 q^{55} +0.825053 q^{57} +3.92484 q^{59} -0.998599 q^{61} -11.4573 q^{63} -1.23337 q^{65} -9.87167 q^{67} +1.22039 q^{69} +14.9053 q^{71} -3.64216 q^{73} -1.14427 q^{75} -1.20195 q^{77} -11.9586 q^{79} +8.03796 q^{81} -4.58822 q^{83} +3.88243 q^{85} +0.328928 q^{87} +12.3040 q^{89} +3.96198 q^{91} -3.10434 q^{93} -3.09368 q^{95} +7.54968 q^{97} +0.877289 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9}+O(q^{10}) \) 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9	\( 12 q - 6 q^{3} + 3 q^{5} - 6 q^{7} + 20 q^{9} - 14 q^{11} + 12 q^{13} - 8 q^{15} + 4 q^{17} - 11 q^{19} - 5 q^{21} - 15 q^{23} + 5 q^{25} - 24 q^{27} + 12 q^{29} - 13 q^{31} - q^{33} - 18 q^{35} - 23 q^{37} - 6 q^{39} - 2 q^{41} - 26 q^{43} - 9 q^{45} - 15 q^{47} + 16 q^{49} - 21 q^{51} + 31 q^{53} - 10 q^{55} - 10 q^{57} - 7 q^{59} + 2 q^{61} + 25 q^{63} + 3 q^{65} - 47 q^{67} - 8 q^{69} - 32 q^{71} - 25 q^{73} - 31 q^{75} - 4 q^{77} - 7 q^{79} + 64 q^{81} - 12 q^{83} + 7 q^{85} - 6 q^{87} + 6 q^{89} - 6 q^{91} + 17 q^{93} - q^{95} - 7 q^{97} - 44 q^{99}+O(q^{100}) \) 12 * q - 6 * q^3 + 3 * q^5 - 6 * q^7 + 20 * q^9 - 14 * q^11 + 12 * q^13 - 8 * q^15 + 4 * q^17 - 11 * q^19 - 5 * q^21 - 15 * q^23 + 5 * q^25 - 24 * q^27 + 12 * q^29 - 13 * q^31 - q^33 - 18 * q^35 - 23 * q^37 - 6 * q^39 - 2 * q^41 - 26 * q^43 - 9 * q^45 - 15 * q^47 + 16 * q^49 - 21 * q^51 + 31 * q^53 - 10 * q^55 - 10 * q^57 - 7 * q^59 + 2 * q^61 + 25 * q^63 + 3 * q^65 - 47 * q^67 - 8 * q^69 - 32 * q^71 - 25 * q^73 - 31 * q^75 - 4 * q^77 - 7 * q^79 + 64 * q^81 - 12 * q^83 + 7 * q^85 - 6 * q^87 + 6 * q^89 - 6 * q^91 + 17 * q^93 - q^95 - 7 * q^97 - 44 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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