LMFDB - Embedded newform 8037.2.a.n.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 13 x^{14} + 65 x^{13} + 47 x^{12} - 390 x^{11} + 4 x^{10} + 1115 x^{9} - 320 x^{8} + \cdots - 25 $ x^16 - 4x^15 - 13x^14 + 65x^13 + 47x^12 - 390x^11 + 4x^10 + 1115x^9 - 320x^8 - 1639x^7 + 618x^6 + 1250x^5 - 487x^4 - 456x^3 + 179x^2 + 62*x - 25
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$2.59041$ of defining polynomial
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.59041 q^{2} +4.71020 q^{4} -1.03652 q^{5} -1.63930 q^{7} +7.02052 q^{8} +O(q^{10})$	\(q+2.59041 q^{2} +4.71020 q^{4} -1.03652 q^{5} -1.63930 q^{7} +7.02052 q^{8} -2.68500 q^{10} +1.66171 q^{11} -4.14978 q^{13} -4.24645 q^{14} +8.76560 q^{16} -2.70080 q^{17} -1.00000 q^{19} -4.88220 q^{20} +4.30451 q^{22} -3.28459 q^{23} -3.92563 q^{25} -10.7496 q^{26} -7.72144 q^{28} -6.51919 q^{29} -1.32698 q^{31} +8.66542 q^{32} -6.99618 q^{34} +1.69916 q^{35} +6.98134 q^{37} -2.59041 q^{38} -7.27689 q^{40} -11.4783 q^{41} +8.53675 q^{43} +7.82701 q^{44} -8.50843 q^{46} +1.00000 q^{47} -4.31269 q^{49} -10.1690 q^{50} -19.5463 q^{52} -2.69166 q^{53} -1.72239 q^{55} -11.5088 q^{56} -16.8873 q^{58} +9.54573 q^{59} -10.1324 q^{61} -3.43743 q^{62} +4.91575 q^{64} +4.30131 q^{65} -4.16063 q^{67} -12.7213 q^{68} +4.40152 q^{70} +8.09982 q^{71} -6.28581 q^{73} +18.0845 q^{74} -4.71020 q^{76} -2.72405 q^{77} +2.59249 q^{79} -9.08569 q^{80} -29.7335 q^{82} -8.70616 q^{83} +2.79943 q^{85} +22.1137 q^{86} +11.6661 q^{88} -11.2791 q^{89} +6.80273 q^{91} -15.4711 q^{92} +2.59041 q^{94} +1.03652 q^{95} -5.65342 q^{97} -11.1716 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8}+O(q^{10}) $ 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8	$ 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8} - 15 q^{10} - 19 q^{13} + 6 q^{14} + 10 q^{16} + 8 q^{17} - 16 q^{19} + 11 q^{20} - 12 q^{22} + 5 q^{23} - 3 q^{25} - 9 q^{26} - 17 q^{28} + 2 q^{29} - 18 q^{31} - 3 q^{32} - 14 q^{34} + 11 q^{35} - 24 q^{37} - 4 q^{38} - 50 q^{40} + 6 q^{41} - 34 q^{43} + 4 q^{44} - 3 q^{46} + 16 q^{47} + 5 q^{49} - 26 q^{50} - 44 q^{52} + 23 q^{53} - 48 q^{55} + 3 q^{56} - 26 q^{58} + 32 q^{59} - 16 q^{61} - 32 q^{62} + 7 q^{64} + 18 q^{65} - 67 q^{67} + 19 q^{68} + 24 q^{70} - 19 q^{71} - 2 q^{73} + 29 q^{74} - 10 q^{76} - 14 q^{77} - 27 q^{79} - 15 q^{80} - 56 q^{82} + 17 q^{83} + 15 q^{85} + q^{86} - 13 q^{88} - 20 q^{89} - 42 q^{91} - 45 q^{92} + 4 q^{94} - q^{95} - 50 q^{97} - 13 q^{98}+O(q^{100}) $ 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8 - 15 * q^10 - 19 * q^13 + 6 * q^14 + 10 * q^16 + 8 * q^17 - 16 * q^19 + 11 * q^20 - 12 * q^22 + 5 * q^23 - 3 * q^25 - 9 * q^26 - 17 * q^28 + 2 * q^29 - 18 * q^31 - 3 * q^32 - 14 * q^34 + 11 * q^35 - 24 * q^37 - 4 * q^38 - 50 * q^40 + 6 * q^41 - 34 * q^43 + 4 * q^44 - 3 * q^46 + 16 * q^47 + 5 * q^49 - 26 * q^50 - 44 * q^52 + 23 * q^53 - 48 * q^55 + 3 * q^56 - 26 * q^58 + 32 * q^59 - 16 * q^61 - 32 * q^62 + 7 * q^64 + 18 * q^65 - 67 * q^67 + 19 * q^68 + 24 * q^70 - 19 * q^71 - 2 * q^73 + 29 * q^74 - 10 * q^76 - 14 * q^77 - 27 * q^79 - 15 * q^80 - 56 * q^82 + 17 * q^83 + 15 * q^85 + q^86 - 13 * q^88 - 20 * q^89 - 42 * q^91 - 45 * q^92 + 4 * q^94 - q^95 - 50 * q^97 - 13 * q^98

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 $

$2$	2.59041	1.83169	0.915847	−	0.401528i	$-0.131521\pi$
$2$	2.59041	1.83169	0.915847	+	0.401528i	$0.131521\pi$
$3$	0	0
$3$	0	0
$4$	4.71020	2.35510
$5$	−1.03652	−0.463544	−0.231772	−	0.972770i	$-0.574452\pi$
$5$	−1.03652	−0.463544	−0.231772	+	0.972770i	$0.574452\pi$
$6$	0	0
$7$	−1.63930	−0.619597	−0.309799	−	0.950802i	$-0.600262\pi$
$7$	−1.63930	−0.619597	−0.309799	+	0.950802i	$0.600262\pi$
$8$	7.02052	2.48213
$9$	0	0
$10$	−2.68500	−0.849071
$11$	1.66171	0.501026	0.250513	−	0.968113i	$-0.419401\pi$
$11$	1.66171	0.501026	0.250513	+	0.968113i	$0.419401\pi$
$12$	0	0
$13$	−4.14978	−1.15094	−0.575471	−	0.817822i	$-0.695181\pi$
$13$	−4.14978	−1.15094	−0.575471	+	0.817822i	$0.695181\pi$
$14$	−4.24645	−1.13491
$15$	0	0
$16$	8.76560	2.19140
$17$	−2.70080	−0.655041	−0.327521	−	0.944844i	$-0.606213\pi$
$17$	−2.70080	−0.655041	−0.327521	+	0.944844i	$0.606213\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−4.88220	−1.09169
$21$	0	0
$22$	4.30451	0.917725
$23$	−3.28459	−0.684885	−0.342443	−	0.939539i	$-0.611254\pi$
$23$	−3.28459	−0.684885	−0.342443	+	0.939539i	$0.611254\pi$
$24$	0	0
$25$	−3.92563	−0.785127
$26$	−10.7496	−2.10817
$27$	0	0
$28$	−7.72144	−1.45921
$29$	−6.51919	−1.21058	−0.605292	−	0.796004i	$-0.706943\pi$
$29$	−6.51919	−1.21058	−0.605292	+	0.796004i	$0.706943\pi$
$30$	0	0
$31$	−1.32698	−0.238333	−0.119167	−	0.992874i	$-0.538022\pi$
$31$	−1.32698	−0.238333	−0.119167	+	0.992874i	$0.538022\pi$
$32$	8.66542	1.53184
$33$	0	0
$34$	−6.99618	−1.19983
$35$	1.69916	0.287211
$36$	0	0
$37$	6.98134	1.14773	0.573863	−	0.818952i	$-0.305444\pi$
$37$	6.98134	1.14773	0.573863	+	0.818952i	$0.305444\pi$
$38$	−2.59041	−0.420219
$39$	0	0
$40$	−7.27689	−1.15058
$41$	−11.4783	−1.79261	−0.896306	−	0.443436i	$-0.853759\pi$
$41$	−11.4783	−1.79261	−0.896306	+	0.443436i	$0.853759\pi$
$42$	0	0
$43$	8.53675	1.30184	0.650921	−	0.759145i	$-0.274383\pi$
$43$	8.53675	1.30184	0.650921	+	0.759145i	$0.274383\pi$
$44$	7.82701	1.17997
$45$	0	0
$46$	−8.50843	−1.25450
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	−4.31269	−0.616099
$50$	−10.1690	−1.43811
$51$	0	0
$52$	−19.5463	−2.71058
$53$	−2.69166	−0.369728	−0.184864	−	0.982764i	$-0.559184\pi$
$53$	−2.69166	−0.369728	−0.184864	+	0.982764i	$0.559184\pi$
$54$	0	0
$55$	−1.72239	−0.232248
$56$	−11.5088	−1.53792
$57$	0	0
$58$	−16.8873	−2.21742
$59$	9.54573	1.24275	0.621374	−	0.783514i	$-0.286575\pi$
$59$	9.54573	1.24275	0.621374	+	0.783514i	$0.286575\pi$
$60$	0	0
$61$	−10.1324	−1.29733	−0.648663	−	0.761076i	$-0.724671\pi$
$61$	−10.1324	−1.29733	−0.648663	+	0.761076i	$0.724671\pi$
$62$	−3.43743	−0.436553
$63$	0	0
$64$	4.91575	0.614469
$65$	4.30131	0.533512
$66$	0	0
$67$	−4.16063	−0.508302	−0.254151	−	0.967165i	$-0.581796\pi$
$67$	−4.16063	−0.508302	−0.254151	+	0.967165i	$0.581796\pi$
$68$	−12.7213	−1.54269
$69$	0	0
$70$	4.40152	0.526082
$71$	8.09982	0.961272	0.480636	−	0.876920i	$-0.340406\pi$
$71$	8.09982	0.961272	0.480636	+	0.876920i	$0.340406\pi$
$72$	0	0
$73$	−6.28581	−0.735699	−0.367849	−	0.929885i	$-0.619906\pi$
$73$	−6.28581	−0.735699	−0.367849	+	0.929885i	$0.619906\pi$
$74$	18.0845	2.10228
$75$	0	0
$76$	−4.71020	−0.540297
$77$	−2.72405	−0.310434
$78$	0	0
$79$	2.59249	0.291678	0.145839	−	0.989308i	$-0.453412\pi$
$79$	2.59249	0.291678	0.145839	+	0.989308i	$0.453412\pi$
$80$	−9.08569	−1.01581
$81$	0	0
$82$	−29.7335	−3.28352
$83$	−8.70616	−0.955626	−0.477813	−	0.878462i	$-0.658570\pi$
$83$	−8.70616	−0.955626	−0.477813	+	0.878462i	$0.658570\pi$
$84$	0	0
$85$	2.79943	0.303641
$86$	22.1137	2.38458
$87$	0	0
$88$	11.6661	1.24361
$89$	−11.2791	−1.19559	−0.597794	−	0.801650i	$-0.703956\pi$
$89$	−11.2791	−1.19559	−0.597794	+	0.801650i	$0.703956\pi$
$90$	0	0
$91$	6.80273	0.713120
$92$	−15.4711	−1.61297
$93$	0	0
$94$	2.59041	0.267180
$95$	1.03652	0.106344
$96$	0	0
$97$	−5.65342	−0.574017	−0.287009	−	0.957928i	$-0.592661\pi$
$97$	−5.65342	−0.574017	−0.287009	+	0.957928i	$0.592661\pi$
$98$	−11.1716	−1.12850
$99$	0	0
$100$	−18.4905	−1.84905
$101$	18.6527	1.85601	0.928006	−	0.372565i	$-0.121522\pi$
$101$	18.6527	1.85601	0.928006	+	0.372565i	$0.121522\pi$
$102$	0	0
$103$	12.3031	1.21226	0.606129	−	0.795366i	$-0.292722\pi$
$103$	12.3031	1.21226	0.606129	+	0.795366i	$0.292722\pi$
$104$	−29.1336	−2.85679
$105$	0	0
$106$	−6.97249	−0.677229
$107$	13.5085	1.30592	0.652960	−	0.757393i	$-0.273527\pi$
$107$	13.5085	1.30592	0.652960	+	0.757393i	$0.273527\pi$
$108$	0	0
$109$	1.32498	0.126910	0.0634552	−	0.997985i	$-0.479788\pi$
$109$	1.32498	0.126910	0.0634552	+	0.997985i	$0.479788\pi$
$110$	−4.46170	−0.425406
$111$	0	0
$112$	−14.3695	−1.35779
$113$	5.38908	0.506962	0.253481	−	0.967340i	$-0.418424\pi$
$113$	5.38908	0.506962	0.253481	+	0.967340i	$0.418424\pi$
$114$	0	0
$115$	3.40454	0.317475
$116$	−30.7067	−2.85105
$117$	0	0
$118$	24.7273	2.27633
$119$	4.42743	0.405862
$120$	0	0
$121$	−8.23871	−0.748973
$122$	−26.2471	−2.37630
$123$	0	0
$124$	−6.25036	−0.561299
$125$	9.25157	0.827485
$126$	0	0
$127$	−8.43978	−0.748909	−0.374455	−	0.927245i	$-0.622170\pi$
$127$	−8.43978	−0.748909	−0.374455	+	0.927245i	$0.622170\pi$
$128$	−4.59705	−0.406326
$129$	0	0
$130$	11.1421	0.977231
$131$	−4.76110	−0.415980	−0.207990	−	0.978131i	$-0.566692\pi$
$131$	−4.76110	−0.415980	−0.207990	+	0.978131i	$0.566692\pi$
$132$	0	0
$133$	1.63930	0.142145
$134$	−10.7777	−0.931053
$135$	0	0
$136$	−18.9611	−1.62590
$137$	−17.1804	−1.46782	−0.733909	−	0.679248i	$-0.762306\pi$
$137$	−17.1804	−1.46782	−0.733909	+	0.679248i	$0.762306\pi$
$138$	0	0
$139$	−4.42331	−0.375180	−0.187590	−	0.982247i	$-0.560068\pi$
$139$	−4.42331	−0.375180	−0.187590	+	0.982247i	$0.560068\pi$
$140$	8.00340	0.676411
$141$	0	0
$142$	20.9818	1.76076
$143$	−6.89574	−0.576651
$144$	0	0
$145$	6.75725	0.561159
$146$	−16.2828	−1.34757
$147$	0	0
$148$	32.8835	2.70301
$149$	−16.3436	−1.33892	−0.669458	−	0.742850i	$-0.733474\pi$
$149$	−16.3436	−1.33892	−0.669458	+	0.742850i	$0.733474\pi$
$150$	0	0
$151$	14.1438	1.15101	0.575503	−	0.817800i	$-0.304806\pi$
$151$	14.1438	1.15101	0.575503	+	0.817800i	$0.304806\pi$
$152$	−7.02052	−0.569440
$153$	0	0
$154$	−7.05639	−0.568620
$155$	1.37544	0.110478
$156$	0	0
$157$	−11.2494	−0.897799	−0.448900	−	0.893582i	$-0.648184\pi$
$157$	−11.2494	−0.897799	−0.448900	+	0.893582i	$0.648184\pi$
$158$	6.71560	0.534264
$159$	0	0
$160$	−8.98185	−0.710078
$161$	5.38444	0.424353
$162$	0	0
$163$	−14.9086	−1.16773	−0.583865	−	0.811851i	$-0.698460\pi$
$163$	−14.9086	−1.16773	−0.583865	+	0.811851i	$0.698460\pi$
$164$	−54.0652	−4.22178
$165$	0	0
$166$	−22.5525	−1.75041
$167$	7.68176	0.594433	0.297216	−	0.954810i	$-0.403942\pi$
$167$	7.68176	0.594433	0.297216	+	0.954810i	$0.403942\pi$
$168$	0	0
$169$	4.22066	0.324666
$170$	7.25165	0.556176
$171$	0	0
$172$	40.2098	3.06597
$173$	17.5586	1.33495	0.667477	−	0.744631i	$-0.267374\pi$
$173$	17.5586	1.33495	0.667477	+	0.744631i	$0.267374\pi$
$174$	0	0
$175$	6.43529	0.486462
$176$	14.5659	1.09795
$177$	0	0
$178$	−29.2176	−2.18995
$179$	25.8151	1.92951	0.964757	−	0.263144i	$-0.0847594\pi$
$179$	25.8151	1.92951	0.964757	+	0.263144i	$0.0847594\pi$
$180$	0	0
$181$	−25.8827	−1.92384	−0.961922	−	0.273322i	$-0.911877\pi$
$181$	−25.8827	−1.92384	−0.961922	+	0.273322i	$0.911877\pi$
$182$	17.6218	1.30622
$183$	0	0
$184$	−23.0596	−1.69997
$185$	−7.23628	−0.532022
$186$	0	0
$187$	−4.48796	−0.328192
$188$	4.71020	0.343527
$189$	0	0
$190$	2.68500	0.194790
$191$	0.152788	0.0110554	0.00552768	−	0.999985i	$-0.498240\pi$
$191$	0.152788	0.0110554	0.00552768	+	0.999985i	$0.498240\pi$
$192$	0	0
$193$	−0.627275	−0.0451523	−0.0225761	−	0.999745i	$-0.507187\pi$
$193$	−0.627275	−0.0451523	−0.0225761	+	0.999745i	$0.507187\pi$
$194$	−14.6446	−1.05142
$195$	0	0
$196$	−20.3137	−1.45098
$197$	11.8138	0.841699	0.420849	−	0.907131i	$-0.361732\pi$
$197$	11.8138	0.841699	0.420849	+	0.907131i	$0.361732\pi$
$198$	0	0
$199$	−13.0552	−0.925460	−0.462730	−	0.886499i	$-0.653130\pi$
$199$	−13.0552	−0.925460	−0.462730	+	0.886499i	$0.653130\pi$
$200$	−27.5600	−1.94879
$201$	0	0
$202$	48.3180	3.39965
$203$	10.6869	0.750074
$204$	0	0
$205$	11.8975	0.830955
$206$	31.8700	2.22048
$207$	0	0
$208$	−36.3753	−2.52217
$209$	−1.66171	−0.114943
$210$	0	0
$211$	10.7714	0.741531	0.370766	−	0.928726i	$-0.379095\pi$
$211$	10.7714	0.741531	0.370766	+	0.928726i	$0.379095\pi$
$212$	−12.6783	−0.870747
$213$	0	0
$214$	34.9926	2.39204
$215$	−8.84849	−0.603462
$216$	0	0
$217$	2.17532	0.147671
$218$	3.43225	0.232461
$219$	0	0
$220$	−8.11282	−0.546967
$221$	11.2077	0.753914
$222$	0	0
$223$	−13.0014	−0.870641	−0.435320	−	0.900276i	$-0.643365\pi$
$223$	−13.0014	−0.870641	−0.435320	+	0.900276i	$0.643365\pi$
$224$	−14.2052	−0.949127
$225$	0	0
$226$	13.9599	0.928599
$227$	10.9054	0.723818	0.361909	−	0.932213i	$-0.382125\pi$
$227$	10.9054	0.723818	0.361909	+	0.932213i	$0.382125\pi$
$228$	0	0
$229$	−3.89713	−0.257529	−0.128765	−	0.991675i	$-0.541101\pi$
$229$	−3.89713	−0.257529	−0.128765	+	0.991675i	$0.541101\pi$
$230$	8.81913	0.581516
$231$	0	0
$232$	−45.7681	−3.00483
$233$	5.71607	0.374472	0.187236	−	0.982315i	$-0.440047\pi$
$233$	5.71607	0.374472	0.187236	+	0.982315i	$0.440047\pi$
$234$	0	0
$235$	−1.03652	−0.0676149
$236$	44.9623	2.92680
$237$	0	0
$238$	11.4688	0.743414
$239$	24.0369	1.55482	0.777408	−	0.628996i	$-0.216534\pi$
$239$	24.0369	1.55482	0.777408	+	0.628996i	$0.216534\pi$
$240$	0	0
$241$	26.0615	1.67877	0.839384	−	0.543539i	$-0.182916\pi$
$241$	26.0615	1.67877	0.839384	+	0.543539i	$0.182916\pi$
$242$	−21.3416	−1.37189
$243$	0	0
$244$	−47.7258	−3.05533
$245$	4.47018	0.285589
$246$	0	0
$247$	4.14978	0.264044
$248$	−9.31612	−0.591574
$249$	0	0
$250$	23.9653	1.51570
$251$	20.2194	1.27624	0.638118	−	0.769938i	$-0.279713\pi$
$251$	20.2194	1.27624	0.638118	+	0.769938i	$0.279713\pi$
$252$	0	0
$253$	−5.45806	−0.343145
$254$	−21.8624	−1.37177
$255$	0	0
$256$	−21.7397	−1.35873
$257$	−28.0281	−1.74834	−0.874171	−	0.485618i	$-0.838594\pi$
$257$	−28.0281	−1.74834	−0.874171	+	0.485618i	$0.838594\pi$
$258$	0	0
$259$	−11.4445	−0.711128
$260$	20.2601	1.25648
$261$	0	0
$262$	−12.3332	−0.761947
$263$	−22.3411	−1.37761	−0.688807	−	0.724945i	$-0.741865\pi$
$263$	−22.3411	−1.37761	−0.688807	+	0.724945i	$0.741865\pi$
$264$	0	0
$265$	2.78995	0.171385
$266$	4.24645	0.260367
$267$	0	0
$268$	−19.5974	−1.19710
$269$	7.22303	0.440396	0.220198	−	0.975455i	$-0.429330\pi$
$269$	7.22303	0.440396	0.220198	+	0.975455i	$0.429330\pi$
$270$	0	0
$271$	−3.06905	−0.186432	−0.0932159	−	0.995646i	$-0.529715\pi$
$271$	−3.06905	−0.186432	−0.0932159	+	0.995646i	$0.529715\pi$
$272$	−23.6742	−1.43546
$273$	0	0
$274$	−44.5041	−2.68859
$275$	−6.52328	−0.393368
$276$	0	0
$277$	2.89697	0.174062	0.0870309	−	0.996206i	$-0.472262\pi$
$277$	2.89697	0.174062	0.0870309	+	0.996206i	$0.472262\pi$
$278$	−11.4582	−0.687215
$279$	0	0
$280$	11.9290	0.712895
$281$	29.8820	1.78261	0.891306	−	0.453403i	$-0.149790\pi$
$281$	29.8820	1.78261	0.891306	+	0.453403i	$0.149790\pi$
$282$	0	0
$283$	16.3862	0.974057	0.487029	−	0.873386i	$-0.338081\pi$
$283$	16.3862	0.974057	0.487029	+	0.873386i	$0.338081\pi$
$284$	38.1518	2.26389
$285$	0	0
$286$	−17.8628	−1.05625
$287$	18.8164	1.11070
$288$	0	0
$289$	−9.70566	−0.570921
$290$	17.5040	1.02787
$291$	0	0
$292$	−29.6074	−1.73264
$293$	−22.4154	−1.30952	−0.654761	−	0.755836i	$-0.727231\pi$
$293$	−22.4154	−1.30952	−0.654761	+	0.755836i	$0.727231\pi$
$294$	0	0
$295$	−9.89431	−0.576069
$296$	49.0127	2.84880
$297$	0	0
$298$	−42.3365	−2.45249
$299$	13.6303	0.788263
$300$	0	0
$301$	−13.9943	−0.806618
$302$	36.6381	2.10829
$303$	0	0
$304$	−8.76560	−0.502742
$305$	10.5024	0.601368
$306$	0	0
$307$	−22.6770	−1.29425	−0.647124	−	0.762385i	$-0.724028\pi$
$307$	−22.6770	−1.29425	−0.647124	+	0.762385i	$0.724028\pi$
$308$	−12.8308	−0.731104
$309$	0	0
$310$	3.56295	0.202362
$311$	−16.0197	−0.908395	−0.454198	−	0.890901i	$-0.650074\pi$
$311$	−16.0197	−0.908395	−0.454198	+	0.890901i	$0.650074\pi$
$312$	0	0
$313$	−12.4293	−0.702548	−0.351274	−	0.936273i	$-0.614251\pi$
$313$	−12.4293	−0.702548	−0.351274	+	0.936273i	$0.614251\pi$
$314$	−29.1405	−1.64449
$315$	0	0
$316$	12.2111	0.686931
$317$	−9.41108	−0.528579	−0.264289	−	0.964443i	$-0.585137\pi$
$317$	−9.41108	−0.528579	−0.264289	+	0.964443i	$0.585137\pi$
$318$	0	0
$319$	−10.8330	−0.606533
$320$	−5.09526	−0.284834
$321$	0	0
$322$	13.9479	0.777285
$323$	2.70080	0.150277
$324$	0	0
$325$	16.2905	0.903635
$326$	−38.6192	−2.13892
$327$	0	0
$328$	−80.5838	−4.44950
$329$	−1.63930	−0.0903776
$330$	0	0
$331$	7.77619	0.427418	0.213709	−	0.976897i	$-0.431446\pi$
$331$	7.77619	0.427418	0.213709	+	0.976897i	$0.431446\pi$
$332$	−41.0078	−2.25060
$333$	0	0
$334$	19.8989	1.08882
$335$	4.31256	0.235620
$336$	0	0
$337$	−17.9771	−0.979277	−0.489639	−	0.871925i	$-0.662871\pi$
$337$	−17.9771	−0.979277	−0.489639	+	0.871925i	$0.662871\pi$
$338$	10.9332	0.594689
$339$	0	0
$340$	13.1859	0.715104
$341$	−2.20507	−0.119411
$342$	0	0
$343$	18.5449	1.00133
$344$	59.9325	3.23134
$345$	0	0
$346$	45.4838	2.44523
$347$	5.40684	0.290254	0.145127	−	0.989413i	$-0.453641\pi$
$347$	5.40684	0.290254	0.145127	+	0.989413i	$0.453641\pi$
$348$	0	0
$349$	−2.33624	−0.125056	−0.0625279	−	0.998043i	$-0.519916\pi$
$349$	−2.33624	−0.125056	−0.0625279	+	0.998043i	$0.519916\pi$
$350$	16.6700	0.891050
$351$	0	0
$352$	14.3994	0.767493
$353$	32.0625	1.70652	0.853258	−	0.521489i	$-0.174623\pi$
$353$	32.0625	1.70652	0.853258	+	0.521489i	$0.174623\pi$
$354$	0	0
$355$	−8.39560	−0.445592
$356$	−53.1271	−2.81573
$357$	0	0
$358$	66.8717	3.53428
$359$	−17.9690	−0.948369	−0.474184	−	0.880426i	$-0.657257\pi$
$359$	−17.9690	−0.948369	−0.474184	+	0.880426i	$0.657257\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−67.0467	−3.52389
$363$	0	0
$364$	32.0423	1.67947
$365$	6.51535	0.341029
$366$	0	0
$367$	−22.5498	−1.17709	−0.588545	−	0.808464i	$-0.700299\pi$
$367$	−22.5498	−1.17709	−0.588545	+	0.808464i	$0.700299\pi$
$368$	−28.7915	−1.50086
$369$	0	0
$370$	−18.7449	−0.974500
$371$	4.41244	0.229083
$372$	0	0
$373$	−30.6586	−1.58744	−0.793721	−	0.608282i	$-0.791859\pi$
$373$	−30.6586	−1.58744	−0.793721	+	0.608282i	$0.791859\pi$
$374$	−11.6256	−0.601148
$375$	0	0
$376$	7.02052	0.362056
$377$	27.0532	1.39331
$378$	0	0
$379$	−6.19393	−0.318161	−0.159080	−	0.987266i	$-0.550853\pi$
$379$	−6.19393	−0.318161	−0.159080	+	0.987266i	$0.550853\pi$
$380$	4.88220	0.250452
$381$	0	0
$382$	0.395783	0.0202500
$383$	−15.0624	−0.769651	−0.384825	−	0.922989i	$-0.625738\pi$
$383$	−15.0624	−0.769651	−0.384825	+	0.922989i	$0.625738\pi$
$384$	0	0
$385$	2.82352	0.143900
$386$	−1.62490	−0.0827051
$387$	0	0
$388$	−26.6287	−1.35187
$389$	3.74433	0.189845	0.0949225	−	0.995485i	$-0.469740\pi$
$389$	3.74433	0.189845	0.0949225	+	0.995485i	$0.469740\pi$
$390$	0	0
$391$	8.87104	0.448628
$392$	−30.2774	−1.52924
$393$	0	0
$394$	30.6026	1.54173
$395$	−2.68716	−0.135206
$396$	0	0
$397$	26.0913	1.30949	0.654743	−	0.755852i	$-0.272777\pi$
$397$	26.0913	1.30949	0.654743	+	0.755852i	$0.272777\pi$
$398$	−33.8183	−1.69516
$399$	0	0
$400$	−34.4105	−1.72053
$401$	−11.6915	−0.583843	−0.291922	−	0.956442i	$-0.594295\pi$
$401$	−11.6915	−0.583843	−0.291922	+	0.956442i	$0.594295\pi$
$402$	0	0
$403$	5.50669	0.274308
$404$	87.8580	4.37110
$405$	0	0
$406$	27.6834	1.37391
$407$	11.6010	0.575040
$408$	0	0
$409$	1.13721	0.0562316	0.0281158	−	0.999605i	$-0.491049\pi$
$409$	1.13721	0.0562316	0.0281158	+	0.999605i	$0.491049\pi$
$410$	30.8193	1.52206
$411$	0	0
$412$	57.9500	2.85499
$413$	−15.6483	−0.770004
$414$	0	0
$415$	9.02408	0.442975
$416$	−35.9596	−1.76306
$417$	0	0
$418$	−4.30451	−0.210541
$419$	1.75915	0.0859399	0.0429699	−	0.999076i	$-0.486318\pi$
$419$	1.75915	0.0859399	0.0429699	+	0.999076i	$0.486318\pi$
$420$	0	0
$421$	6.26556	0.305365	0.152682	−	0.988275i	$-0.451209\pi$
$421$	6.26556	0.305365	0.152682	+	0.988275i	$0.451209\pi$
$422$	27.9022	1.35826
$423$	0	0
$424$	−18.8969	−0.917713
$425$	10.6024	0.514290
$426$	0	0
$427$	16.6101	0.803819
$428$	63.6279	3.07557
$429$	0	0
$430$	−22.9212	−1.10536
$431$	−13.8753	−0.668348	−0.334174	−	0.942511i	$-0.608457\pi$
$431$	−13.8753	−0.668348	−0.334174	+	0.942511i	$0.608457\pi$
$432$	0	0
$433$	−29.7756	−1.43093	−0.715463	−	0.698651i	$-0.753784\pi$
$433$	−29.7756	−1.43093	−0.715463	+	0.698651i	$0.753784\pi$
$434$	5.63497	0.270487
$435$	0	0
$436$	6.24094	0.298887
$437$	3.28459	0.157123
$438$	0	0
$439$	14.9414	0.713112	0.356556	−	0.934274i	$-0.383951\pi$
$439$	14.9414	0.713112	0.356556	+	0.934274i	$0.383951\pi$
$440$	−12.0921	−0.576469
$441$	0	0
$442$	29.0326	1.38094
$443$	22.9943	1.09249	0.546247	−	0.837624i	$-0.316056\pi$
$443$	22.9943	1.09249	0.546247	+	0.837624i	$0.316056\pi$
$444$	0	0
$445$	11.6910	0.554208
$446$	−33.6790	−1.59475
$447$	0	0
$448$	−8.05839	−0.380723
$449$	11.4576	0.540718	0.270359	−	0.962760i	$-0.412858\pi$
$449$	11.4576	0.540718	0.270359	+	0.962760i	$0.412858\pi$
$450$	0	0
$451$	−19.0737	−0.898145
$452$	25.3837	1.19395
$453$	0	0
$454$	28.2495	1.32581
$455$	−7.05115	−0.330563
$456$	0	0
$457$	12.0531	0.563821	0.281910	−	0.959441i	$-0.409032\pi$
$457$	12.0531	0.563821	0.281910	+	0.959441i	$0.409032\pi$
$458$	−10.0951	−0.471715
$459$	0	0
$460$	16.0361	0.747685
$461$	−19.3112	−0.899414	−0.449707	−	0.893176i	$-0.648472\pi$
$461$	−19.3112	−0.899414	−0.449707	+	0.893176i	$0.648472\pi$
$462$	0	0
$463$	1.83390	0.0852285	0.0426143	−	0.999092i	$-0.486431\pi$
$463$	1.83390	0.0852285	0.0426143	+	0.999092i	$0.486431\pi$
$464$	−57.1446	−2.65287
$465$	0	0
$466$	14.8069	0.685919
$467$	15.9242	0.736886	0.368443	−	0.929650i	$-0.379891\pi$
$467$	15.9242	0.736886	0.368443	+	0.929650i	$0.379891\pi$
$468$	0	0
$469$	6.82052	0.314942
$470$	−2.68500	−0.123850
$471$	0	0
$472$	67.0160	3.08466
$473$	14.1856	0.652256
$474$	0	0
$475$	3.92563	0.180120
$476$	20.8541	0.955845
$477$	0	0
$478$	62.2653	2.84795
$479$	−36.5556	−1.67027	−0.835133	−	0.550048i	$-0.814609\pi$
$479$	−36.5556	−1.67027	−0.835133	+	0.550048i	$0.814609\pi$
$480$	0	0
$481$	−28.9710	−1.32096
$482$	67.5098	3.07499
$483$	0	0
$484$	−38.8060	−1.76391
$485$	5.85986	0.266083
$486$	0	0
$487$	18.0024	0.815766	0.407883	−	0.913034i	$-0.366267\pi$
$487$	18.0024	0.815766	0.407883	+	0.913034i	$0.366267\pi$
$488$	−71.1350	−3.22013
$489$	0	0
$490$	11.5796	0.523112
$491$	−37.5165	−1.69310	−0.846548	−	0.532313i	$-0.821323\pi$
$491$	−37.5165	−1.69310	−0.846548	+	0.532313i	$0.821323\pi$
$492$	0	0
$493$	17.6071	0.792982
$494$	10.7496	0.483648
$495$	0	0
$496$	−11.6318	−0.522284
$497$	−13.2780	−0.595601
$498$	0	0
$499$	23.8158	1.06614	0.533071	−	0.846071i	$-0.321038\pi$
$499$	23.8158	1.06614	0.533071	+	0.846071i	$0.321038\pi$
$500$	43.5768	1.94881
$501$	0	0
$502$	52.3764	2.33767
$503$	29.8568	1.33125	0.665624	−	0.746287i	$-0.268165\pi$
$503$	29.8568	1.33125	0.665624	+	0.746287i	$0.268165\pi$
$504$	0	0
$505$	−19.3338	−0.860344
$506$	−14.1386	−0.628536
$507$	0	0
$508$	−39.7531	−1.76376
$509$	27.3239	1.21111	0.605556	−	0.795803i	$-0.292951\pi$
$509$	27.3239	1.21111	0.605556	+	0.795803i	$0.292951\pi$
$510$	0	0
$511$	10.3043	0.455837
$512$	−47.1206	−2.08246
$513$	0	0
$514$	−72.6040	−3.20243
$515$	−12.7523	−0.561935
$516$	0	0
$517$	1.66171	0.0730821
$518$	−29.6459	−1.30257
$519$	0	0
$520$	30.1975	1.32425
$521$	33.3640	1.46170	0.730851	−	0.682537i	$-0.239124\pi$
$521$	33.3640	1.46170	0.730851	+	0.682537i	$0.239124\pi$
$522$	0	0
$523$	2.06026	0.0900887	0.0450443	−	0.998985i	$-0.485657\pi$
$523$	2.06026	0.0900887	0.0450443	+	0.998985i	$0.485657\pi$
$524$	−22.4258	−0.979674
$525$	0	0
$526$	−57.8726	−2.52337
$527$	3.58392	0.156118
$528$	0	0
$529$	−12.2114	−0.530932
$530$	7.22711	0.313925
$531$	0	0
$532$	7.72144	0.334767
$533$	47.6325	2.06319
$534$	0	0
$535$	−14.0018	−0.605351
$536$	−29.2098	−1.26167
$537$	0	0
$538$	18.7106	0.806670
$539$	−7.16646	−0.308681
$540$	0	0
$541$	32.7315	1.40724	0.703619	−	0.710577i	$-0.251566\pi$
$541$	32.7315	1.40724	0.703619	+	0.710577i	$0.251566\pi$
$542$	−7.95010	−0.341486
$543$	0	0
$544$	−23.4036	−1.00342
$545$	−1.37337	−0.0588286
$546$	0	0
$547$	27.2935	1.16698	0.583492	−	0.812119i	$-0.301686\pi$
$547$	27.2935	1.16698	0.583492	+	0.812119i	$0.301686\pi$
$548$	−80.9230	−3.45686
$549$	0	0
$550$	−16.8979	−0.720531
$551$	6.51919	0.277727
$552$	0	0
$553$	−4.24987	−0.180723
$554$	7.50432	0.318828
$555$	0	0
$556$	−20.8347	−0.883587
$557$	2.81265	0.119176	0.0595880	−	0.998223i	$-0.481021\pi$
$557$	2.81265	0.119176	0.0595880	+	0.998223i	$0.481021\pi$
$558$	0	0
$559$	−35.4256	−1.49834
$560$	14.8942	0.629394
$561$	0	0
$562$	77.4066	3.26520
$563$	4.65226	0.196069	0.0980346	−	0.995183i	$-0.468744\pi$
$563$	4.65226	0.196069	0.0980346	+	0.995183i	$0.468744\pi$
$564$	0	0
$565$	−5.58587	−0.234999
$566$	42.4469	1.78417
$567$	0	0
$568$	56.8650	2.38600
$569$	6.79476	0.284851	0.142426	−	0.989806i	$-0.454510\pi$
$569$	6.79476	0.284851	0.142426	+	0.989806i	$0.454510\pi$
$570$	0	0
$571$	1.91697	0.0802228	0.0401114	−	0.999195i	$-0.487229\pi$
$571$	1.91697	0.0802228	0.0401114	+	0.999195i	$0.487229\pi$
$572$	−32.4803	−1.35807
$573$	0	0
$574$	48.7422	2.03446
$575$	12.8941	0.537722
$576$	0	0
$577$	−11.7830	−0.490532	−0.245266	−	0.969456i	$-0.578875\pi$
$577$	−11.7830	−0.490532	−0.245266	+	0.969456i	$0.578875\pi$
$578$	−25.1416	−1.04575
$579$	0	0
$580$	31.8280	1.32159
$581$	14.2720	0.592103
$582$	0	0
$583$	−4.47277	−0.185243
$584$	−44.1297	−1.82610
$585$	0	0
$586$	−58.0650	−2.39864
$587$	−20.9616	−0.865176	−0.432588	−	0.901592i	$-0.642400\pi$
$587$	−20.9616	−0.865176	−0.432588	+	0.901592i	$0.642400\pi$
$588$	0	0
$589$	1.32698	0.0546774
$590$	−25.6303	−1.05518
$591$	0	0
$592$	61.1957	2.51513
$593$	−34.1723	−1.40329	−0.701644	−	0.712528i	$-0.747550\pi$
$593$	−34.1723	−1.40329	−0.701644	+	0.712528i	$0.747550\pi$
$594$	0	0
$595$	−4.58910	−0.188135
$596$	−76.9815	−3.15328
$597$	0	0
$598$	35.3081	1.44386
$599$	−12.2765	−0.501604	−0.250802	−	0.968038i	$-0.580694\pi$
$599$	−12.2765	−0.501604	−0.250802	+	0.968038i	$0.580694\pi$
$600$	0	0
$601$	−8.78289	−0.358261	−0.179131	−	0.983825i	$-0.557329\pi$
$601$	−8.78289	−0.358261	−0.179131	+	0.983825i	$0.557329\pi$
$602$	−36.2509	−1.47748
$603$	0	0
$604$	66.6201	2.71073
$605$	8.53956	0.347182
$606$	0	0
$607$	−32.0309	−1.30009	−0.650047	−	0.759894i	$-0.725251\pi$
$607$	−32.0309	−1.30009	−0.650047	+	0.759894i	$0.725251\pi$
$608$	−8.66542	−0.351429
$609$	0	0
$610$	27.2056	1.10152
$611$	−4.14978	−0.167882
$612$	0	0
$613$	−41.7084	−1.68459	−0.842294	−	0.539018i	$-0.818795\pi$
$613$	−41.7084	−1.68459	−0.842294	+	0.539018i	$0.818795\pi$
$614$	−58.7428	−2.37066
$615$	0	0
$616$	−19.1242	−0.770538
$617$	15.1961	0.611771	0.305885	−	0.952068i	$-0.401048\pi$
$617$	15.1961	0.611771	0.305885	+	0.952068i	$0.401048\pi$
$618$	0	0
$619$	18.3339	0.736902	0.368451	−	0.929647i	$-0.379888\pi$
$619$	18.3339	0.736902	0.368451	+	0.929647i	$0.379888\pi$
$620$	6.47860	0.260187
$621$	0	0
$622$	−41.4976	−1.66390
$623$	18.4899	0.740783
$624$	0	0
$625$	10.0388	0.401550
$626$	−32.1971	−1.28685
$627$	0	0
$628$	−52.9869	−2.11441
$629$	−18.8552	−0.751807
$630$	0	0
$631$	−29.2078	−1.16274	−0.581372	−	0.813638i	$-0.697484\pi$
$631$	−29.2078	−1.16274	−0.581372	+	0.813638i	$0.697484\pi$
$632$	18.2006	0.723982
$633$	0	0
$634$	−24.3785	−0.968195
$635$	8.74797	0.347153
$636$	0	0
$637$	17.8967	0.709094
$638$	−28.0619	−1.11098
$639$	0	0
$640$	4.76492	0.188350
$641$	−23.5885	−0.931689	−0.465845	−	0.884867i	$-0.654249\pi$
$641$	−23.5885	−0.931689	−0.465845	+	0.884867i	$0.654249\pi$
$642$	0	0
$643$	28.6384	1.12939	0.564694	−	0.825300i	$-0.308994\pi$
$643$	28.6384	1.12939	0.564694	+	0.825300i	$0.308994\pi$
$644$	25.3618	0.999395
$645$	0	0
$646$	6.99618	0.275261
$647$	25.6634	1.00893	0.504465	−	0.863432i	$-0.331690\pi$
$647$	25.6634	1.00893	0.504465	+	0.863432i	$0.331690\pi$
$648$	0	0
$649$	15.8623	0.622649
$650$	42.1990	1.65518
$651$	0	0
$652$	−70.2224	−2.75012
$653$	23.0338	0.901381	0.450690	−	0.892680i	$-0.351178\pi$
$653$	23.0338	0.901381	0.450690	+	0.892680i	$0.351178\pi$
$654$	0	0
$655$	4.93496	0.192825
$656$	−100.614	−3.92833
$657$	0	0
$658$	−4.24645	−0.165544
$659$	−4.23045	−0.164795	−0.0823975	−	0.996600i	$-0.526258\pi$
$659$	−4.23045	−0.164795	−0.0823975	+	0.996600i	$0.526258\pi$
$660$	0	0
$661$	9.35821	0.363992	0.181996	−	0.983299i	$-0.441744\pi$
$661$	9.35821	0.363992	0.181996	+	0.983299i	$0.441744\pi$
$662$	20.1435	0.782898
$663$	0	0
$664$	−61.1218	−2.37199
$665$	−1.69916	−0.0658907
$666$	0	0
$667$	21.4129	0.829111
$668$	36.1827	1.39995
$669$	0	0
$670$	11.1713	0.431584
$671$	−16.8372	−0.649993
$672$	0	0
$673$	11.2093	0.432085	0.216042	−	0.976384i	$-0.430685\pi$
$673$	11.2093	0.432085	0.216042	+	0.976384i	$0.430685\pi$
$674$	−46.5681	−1.79374
$675$	0	0
$676$	19.8802	0.764622
$677$	1.92961	0.0741611	0.0370806	−	0.999312i	$-0.488194\pi$
$677$	1.92961	0.0741611	0.0370806	+	0.999312i	$0.488194\pi$
$678$	0	0
$679$	9.26765	0.355660
$680$	19.6535	0.753675
$681$	0	0
$682$	−5.71202	−0.218724
$683$	−1.87681	−0.0718140	−0.0359070	−	0.999355i	$-0.511432\pi$
$683$	−1.87681	−0.0718140	−0.0359070	+	0.999355i	$0.511432\pi$
$684$	0	0
$685$	17.8077	0.680399
$686$	48.0388	1.83413
$687$	0	0
$688$	74.8298	2.85286
$689$	11.1698	0.425535
$690$	0	0
$691$	19.8063	0.753469	0.376734	−	0.926321i	$-0.377047\pi$
$691$	19.8063	0.753469	0.376734	+	0.926321i	$0.377047\pi$
$692$	82.7044	3.14395
$693$	0	0
$694$	14.0059	0.531657
$695$	4.58483	0.173912
$696$	0	0
$697$	31.0007	1.17423
$698$	−6.05180	−0.229064
$699$	0	0
$700$	30.3115	1.14567
$701$	37.0276	1.39851	0.699257	−	0.714870i	$-0.253514\pi$
$701$	37.0276	1.39851	0.699257	+	0.714870i	$0.253514\pi$
$702$	0	0
$703$	−6.98134	−0.263306
$704$	8.16857	0.307865
$705$	0	0
$706$	83.0550	3.12582
$707$	−30.5774	−1.14998
$708$	0	0
$709$	−44.9806	−1.68928	−0.844641	−	0.535333i	$-0.820186\pi$
$709$	−44.9806	−1.68928	−0.844641	+	0.535333i	$0.820186\pi$
$710$	−21.7480	−0.816188
$711$	0	0
$712$	−79.1855	−2.96760
$713$	4.35860	0.163231
$714$	0	0
$715$	7.14755	0.267303
$716$	121.594	4.54420
$717$	0	0
$718$	−46.5471	−1.73712
$719$	29.6174	1.10454	0.552271	−	0.833665i	$-0.313761\pi$
$719$	29.6174	1.10454	0.552271	+	0.833665i	$0.313761\pi$
$720$	0	0
$721$	−20.1684	−0.751112
$722$	2.59041	0.0964049
$723$	0	0
$724$	−121.913	−4.53085
$725$	25.5919	0.950461
$726$	0	0
$727$	35.5861	1.31981	0.659907	−	0.751347i	$-0.270595\pi$
$727$	35.5861	1.31981	0.659907	+	0.751347i	$0.270595\pi$
$728$	47.7588	1.77006
$729$	0	0
$730$	16.8774	0.624661
$731$	−23.0561	−0.852760
$732$	0	0
$733$	28.5956	1.05620	0.528102	−	0.849181i	$-0.322904\pi$
$733$	28.5956	1.05620	0.528102	+	0.849181i	$0.322904\pi$
$734$	−58.4132	−2.15607
$735$	0	0
$736$	−28.4624	−1.04914
$737$	−6.91377	−0.254672
$738$	0	0
$739$	−15.8126	−0.581677	−0.290839	−	0.956772i	$-0.593934\pi$
$739$	−15.8126	−0.581677	−0.290839	+	0.956772i	$0.593934\pi$
$740$	−34.0843	−1.25296
$741$	0	0
$742$	11.4300	0.419609
$743$	35.7904	1.31302	0.656511	−	0.754317i	$-0.272032\pi$
$743$	35.7904	1.31302	0.656511	+	0.754317i	$0.272032\pi$
$744$	0	0
$745$	16.9404	0.620647
$746$	−79.4182	−2.90771
$747$	0	0
$748$	−21.1392	−0.772926
$749$	−22.1445	−0.809144
$750$	0	0
$751$	8.14956	0.297382	0.148691	−	0.988884i	$-0.452494\pi$
$751$	8.14956	0.297382	0.148691	+	0.988884i	$0.452494\pi$
$752$	8.76560	0.319649
$753$	0	0
$754$	70.0788	2.55212
$755$	−14.6603	−0.533542
$756$	0	0
$757$	42.5047	1.54486	0.772430	−	0.635100i	$-0.219041\pi$
$757$	42.5047	1.54486	0.772430	+	0.635100i	$0.219041\pi$
$758$	−16.0448	−0.582773
$759$	0	0
$760$	7.27689	0.263961
$761$	−3.22865	−0.117038	−0.0585192	−	0.998286i	$-0.518638\pi$
$761$	−3.22865	−0.117038	−0.0585192	+	0.998286i	$0.518638\pi$
$762$	0	0
$763$	−2.17205	−0.0786334
$764$	0.719663	0.0260365
$765$	0	0
$766$	−39.0176	−1.40976
$767$	−39.6127	−1.43033
$768$	0	0
$769$	2.61873	0.0944337	0.0472169	−	0.998885i	$-0.484965\pi$
$769$	2.61873	0.0944337	0.0472169	+	0.998885i	$0.484965\pi$
$770$	7.31407	0.263581
$771$	0	0
$772$	−2.95459	−0.106338
$773$	−14.4679	−0.520375	−0.260188	−	0.965558i	$-0.583784\pi$
$773$	−14.4679	−0.520375	−0.260188	+	0.965558i	$0.583784\pi$
$774$	0	0
$775$	5.20925	0.187122
$776$	−39.6899	−1.42479
$777$	0	0
$778$	9.69933	0.347738
$779$	11.4783	0.411254
$780$	0	0
$781$	13.4596	0.481622
$782$	22.9796	0.821749
$783$	0	0
$784$	−37.8034	−1.35012
$785$	11.6602	0.416170
$786$	0	0
$787$	−27.2183	−0.970227	−0.485113	−	0.874451i	$-0.661222\pi$
$787$	−27.2183	−0.970227	−0.485113	+	0.874451i	$0.661222\pi$
$788$	55.6454	1.98229
$789$	0	0
$790$	−6.96083	−0.247655
$791$	−8.83432	−0.314112
$792$	0	0
$793$	42.0473	1.49315
$794$	67.5871	2.39858
$795$	0	0
$796$	−61.4928	−2.17955
$797$	24.4884	0.867422	0.433711	−	0.901052i	$-0.357204\pi$
$797$	24.4884	0.867422	0.433711	+	0.901052i	$0.357204\pi$
$798$	0	0
$799$	−2.70080	−0.0955476
$800$	−34.0173	−1.20269
$801$	0	0
$802$	−30.2856	−1.06942
$803$	−10.4452	−0.368604
$804$	0	0
$805$	−5.58106	−0.196706
$806$	14.2646	0.502447
$807$	0	0
$808$	130.952	4.60686
$809$	−27.8391	−0.978772	−0.489386	−	0.872067i	$-0.662779\pi$
$809$	−27.8391	−0.978772	−0.489386	+	0.872067i	$0.662779\pi$
$810$	0	0
$811$	−13.9073	−0.488352	−0.244176	−	0.969731i	$-0.578517\pi$
$811$	−13.9073	−0.488352	−0.244176	+	0.969731i	$0.578517\pi$
$812$	50.3375	1.76650
$813$	0	0
$814$	30.0513	1.05330
$815$	15.4530	0.541294
$816$	0	0
$817$	−8.53675	−0.298663
$818$	2.94584	0.102999
$819$	0	0
$820$	56.0395	1.95698
$821$	−6.85117	−0.239107	−0.119554	−	0.992828i	$-0.538146\pi$
$821$	−6.85117	−0.239107	−0.119554	+	0.992828i	$0.538146\pi$
$822$	0	0
$823$	17.3004	0.603056	0.301528	−	0.953457i	$-0.402503\pi$
$823$	17.3004	0.603056	0.301528	+	0.953457i	$0.402503\pi$
$824$	86.3740	3.00898
$825$	0	0
$826$	−40.5355	−1.41041
$827$	55.7504	1.93863	0.969315	−	0.245822i	$-0.0790578\pi$
$827$	55.7504	1.93863	0.969315	+	0.245822i	$0.0790578\pi$
$828$	0	0
$829$	−47.0166	−1.63295	−0.816477	−	0.577378i	$-0.804076\pi$
$829$	−47.0166	−1.63295	−0.816477	+	0.577378i	$0.804076\pi$
$830$	23.3760	0.811394
$831$	0	0
$832$	−20.3993	−0.707218
$833$	11.6477	0.403570
$834$	0	0
$835$	−7.96228	−0.275546
$836$	−7.82701	−0.270703
$837$	0	0
$838$	4.55690	0.157416
$839$	27.1567	0.937555	0.468778	−	0.883316i	$-0.344695\pi$
$839$	27.1567	0.937555	0.468778	+	0.883316i	$0.344695\pi$
$840$	0	0
$841$	13.4998	0.465512
$842$	16.2304	0.559335
$843$	0	0
$844$	50.7353	1.74638
$845$	−4.37478	−0.150497
$846$	0	0
$847$	13.5057	0.464062
$848$	−23.5940	−0.810222
$849$	0	0
$850$	27.4644	0.942022
$851$	−22.9309	−0.786060
$852$	0	0
$853$	−16.7316	−0.572880	−0.286440	−	0.958098i	$-0.592472\pi$
$853$	−16.7316	−0.572880	−0.286440	+	0.958098i	$0.592472\pi$
$854$	43.0269	1.47235
$855$	0	0
$856$	94.8370	3.24146
$857$	−54.7756	−1.87110	−0.935550	−	0.353194i	$-0.885096\pi$
$857$	−54.7756	−1.87110	−0.935550	+	0.353194i	$0.885096\pi$
$858$	0	0
$859$	−34.0285	−1.16104	−0.580518	−	0.814247i	$-0.697150\pi$
$859$	−34.0285	−1.16104	−0.580518	+	0.814247i	$0.697150\pi$
$860$	−41.6782	−1.42121
$861$	0	0
$862$	−35.9426	−1.22421
$863$	13.7842	0.469219	0.234609	−	0.972090i	$-0.424619\pi$
$863$	13.7842	0.469219	0.234609	+	0.972090i	$0.424619\pi$
$864$	0	0
$865$	−18.1998	−0.618810
$866$	−77.1310	−2.62102
$867$	0	0
$868$	10.2462	0.347779
$869$	4.30797	0.146138
$870$	0	0
$871$	17.2657	0.585025
$872$	9.30208	0.315008
$873$	0	0
$874$	8.50843	0.287802
$875$	−15.1661	−0.512708
$876$	0	0
$877$	−25.5582	−0.863040	−0.431520	−	0.902103i	$-0.642023\pi$
$877$	−25.5582	−0.863040	−0.431520	+	0.902103i	$0.642023\pi$
$878$	38.7042	1.30620
$879$	0	0
$880$	−15.0978	−0.508947
$881$	−35.6865	−1.20231	−0.601154	−	0.799133i	$-0.705292\pi$
$881$	−35.6865	−1.20231	−0.601154	+	0.799133i	$0.705292\pi$
$882$	0	0
$883$	−44.7144	−1.50476	−0.752380	−	0.658729i	$-0.771094\pi$
$883$	−44.7144	−1.50476	−0.752380	+	0.658729i	$0.771094\pi$
$884$	52.7907	1.77554
$885$	0	0
$886$	59.5647	2.00111
$887$	−44.2937	−1.48724	−0.743618	−	0.668605i	$-0.766892\pi$
$887$	−44.2937	−1.48724	−0.743618	+	0.668605i	$0.766892\pi$
$888$	0	0
$889$	13.8353	0.464022
$890$	30.2845	1.01514
$891$	0	0
$892$	−61.2394	−2.05045
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	−26.7578	−0.894415
$896$	7.53595	0.251758
$897$	0	0
$898$	29.6798	0.990429
$899$	8.65086	0.288522
$900$	0	0
$901$	7.26965	0.242187
$902$	−49.4086	−1.64513
$903$	0	0
$904$	37.8342	1.25835
$905$	26.8278	0.891787
$906$	0	0
$907$	18.6288	0.618559	0.309280	−	0.950971i	$-0.399912\pi$
$907$	18.6288	0.618559	0.309280	+	0.950971i	$0.399912\pi$
$908$	51.3667	1.70467
$909$	0	0
$910$	−18.2653	−0.605490
$911$	−35.1415	−1.16429	−0.582144	−	0.813086i	$-0.697786\pi$
$911$	−35.1415	−1.16429	−0.582144	+	0.813086i	$0.697786\pi$
$912$	0	0
$913$	−14.4672	−0.478793
$914$	31.2225	1.03275
$915$	0	0
$916$	−18.3563	−0.606508
$917$	7.80488	0.257740
$918$	0	0
$919$	−27.0044	−0.890792	−0.445396	−	0.895334i	$-0.646937\pi$
$919$	−27.0044	−0.890792	−0.445396	+	0.895334i	$0.646937\pi$
$920$	23.9016	0.788014
$921$	0	0
$922$	−50.0240	−1.64745
$923$	−33.6125	−1.10637
$924$	0	0
$925$	−27.4062	−0.901110
$926$	4.75054	0.156113
$927$	0	0
$928$	−56.4915	−1.85443
$929$	27.4384	0.900226	0.450113	−	0.892972i	$-0.351384\pi$
$929$	27.4384	0.900226	0.450113	+	0.892972i	$0.351384\pi$
$930$	0	0
$931$	4.31269	0.141343
$932$	26.9239	0.881920
$933$	0	0
$934$	41.2502	1.34975
$935$	4.65185	0.152132
$936$	0	0
$937$	15.2600	0.498522	0.249261	−	0.968436i	$-0.419812\pi$
$937$	15.2600	0.498522	0.249261	+	0.968436i	$0.419812\pi$
$938$	17.6679	0.576878
$939$	0	0
$940$	−4.88220	−0.159240
$941$	20.1737	0.657645	0.328822	−	0.944392i	$-0.393348\pi$
$941$	20.1737	0.657645	0.328822	+	0.944392i	$0.393348\pi$
$942$	0	0
$943$	37.7016	1.22773
$944$	83.6741	2.72336
$945$	0	0
$946$	36.7466	1.19473
$947$	−56.9520	−1.85069	−0.925345	−	0.379126i	$-0.876225\pi$
$947$	−56.9520	−1.85069	−0.925345	+	0.379126i	$0.876225\pi$
$948$	0	0
$949$	26.0847	0.846746
$950$	10.1690	0.329925
$951$	0	0
$952$	31.0829	1.00740
$953$	28.4406	0.921280	0.460640	−	0.887587i	$-0.347620\pi$
$953$	28.4406	0.921280	0.460640	+	0.887587i	$0.347620\pi$
$954$	0	0
$955$	−0.158367	−0.00512465
$956$	113.219	3.66175
$957$	0	0
$958$	−94.6937	−3.05942
$959$	28.1638	0.909456
$960$	0	0
$961$	−29.2391	−0.943197
$962$	−75.0467	−2.41960
$963$	0	0
$964$	122.755	3.95367
$965$	0.650181	0.0209301
$966$	0	0
$967$	15.3574	0.493862	0.246931	−	0.969033i	$-0.420578\pi$
$967$	15.3574	0.493862	0.246931	+	0.969033i	$0.420578\pi$
$968$	−57.8401	−1.85905
$969$	0	0
$970$	15.1794	0.487382
$971$	34.5064	1.10736	0.553681	−	0.832729i	$-0.313223\pi$
$971$	34.5064	1.10736	0.553681	+	0.832729i	$0.313223\pi$
$972$	0	0
$973$	7.25113	0.232460
$974$	46.6335	1.49423
$975$	0	0
$976$	−88.8169	−2.84296
$977$	−41.0151	−1.31219	−0.656095	−	0.754679i	$-0.727793\pi$
$977$	−41.0151	−1.31219	−0.656095	+	0.754679i	$0.727793\pi$
$978$	0	0
$979$	−18.7427	−0.599020
$980$	21.0554	0.672592
$981$	0	0
$982$	−97.1829	−3.10123
$983$	22.1295	0.705822	0.352911	−	0.935657i	$-0.385192\pi$
$983$	22.1295	0.705822	0.352911	+	0.935657i	$0.385192\pi$
$984$	0	0
$985$	−12.2452	−0.390165
$986$	45.6094	1.45250
$987$	0	0
$988$	19.5463	0.621851
$989$	−28.0398	−0.891613
$990$	0	0
$991$	40.3075	1.28041	0.640206	−	0.768204i	$-0.278849\pi$
$991$	40.3075	1.28041	0.640206	+	0.768204i	$0.278849\pi$
$992$	−11.4989	−0.365089
$993$	0	0
$994$	−34.3955	−1.09096
$995$	13.5320	0.428992
$996$	0	0
$997$	−9.32424	−0.295302	−0.147651	−	0.989040i	$-0.547171\pi$
$997$	−9.32424	−0.295302	−0.147651	+	0.989040i	$0.547171\pi$
$998$	61.6926	1.95285
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.n.1.16		16
3.2	odd	2		893.2.a.b.1.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.b.1.1	✓	16	3.2	odd	2
8037.2.a.n.1.16		16	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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q+2.59041 q^{2} +4.71020 q^{4} -1.03652 q^{5} -1.63930 q^{7} +7.02052 q^{8} +O(q^{10})\)	\(q+2.59041 q^{2} +4.71020 q^{4} -1.03652 q^{5} -1.63930 q^{7} +7.02052 q^{8} -2.68500 q^{10} +1.66171 q^{11} -4.14978 q^{13} -4.24645 q^{14} +8.76560 q^{16} -2.70080 q^{17} -1.00000 q^{19} -4.88220 q^{20} +4.30451 q^{22} -3.28459 q^{23} -3.92563 q^{25} -10.7496 q^{26} -7.72144 q^{28} -6.51919 q^{29} -1.32698 q^{31} +8.66542 q^{32} -6.99618 q^{34} +1.69916 q^{35} +6.98134 q^{37} -2.59041 q^{38} -7.27689 q^{40} -11.4783 q^{41} +8.53675 q^{43} +7.82701 q^{44} -8.50843 q^{46} +1.00000 q^{47} -4.31269 q^{49} -10.1690 q^{50} -19.5463 q^{52} -2.69166 q^{53} -1.72239 q^{55} -11.5088 q^{56} -16.8873 q^{58} +9.54573 q^{59} -10.1324 q^{61} -3.43743 q^{62} +4.91575 q^{64} +4.30131 q^{65} -4.16063 q^{67} -12.7213 q^{68} +4.40152 q^{70} +8.09982 q^{71} -6.28581 q^{73} +18.0845 q^{74} -4.71020 q^{76} -2.72405 q^{77} +2.59249 q^{79} -9.08569 q^{80} -29.7335 q^{82} -8.70616 q^{83} +2.79943 q^{85} +22.1137 q^{86} +11.6661 q^{88} -11.2791 q^{89} +6.80273 q^{91} -15.4711 q^{92} +2.59041 q^{94} +1.03652 q^{95} -5.65342 q^{97} -11.1716 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8}+O(q^{10}) \) 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8	\( 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8} - 15 q^{10} - 19 q^{13} + 6 q^{14} + 10 q^{16} + 8 q^{17} - 16 q^{19} + 11 q^{20} - 12 q^{22} + 5 q^{23} - 3 q^{25} - 9 q^{26} - 17 q^{28} + 2 q^{29} - 18 q^{31} - 3 q^{32} - 14 q^{34} + 11 q^{35} - 24 q^{37} - 4 q^{38} - 50 q^{40} + 6 q^{41} - 34 q^{43} + 4 q^{44} - 3 q^{46} + 16 q^{47} + 5 q^{49} - 26 q^{50} - 44 q^{52} + 23 q^{53} - 48 q^{55} + 3 q^{56} - 26 q^{58} + 32 q^{59} - 16 q^{61} - 32 q^{62} + 7 q^{64} + 18 q^{65} - 67 q^{67} + 19 q^{68} + 24 q^{70} - 19 q^{71} - 2 q^{73} + 29 q^{74} - 10 q^{76} - 14 q^{77} - 27 q^{79} - 15 q^{80} - 56 q^{82} + 17 q^{83} + 15 q^{85} + q^{86} - 13 q^{88} - 20 q^{89} - 42 q^{91} - 45 q^{92} + 4 q^{94} - q^{95} - 50 q^{97} - 13 q^{98}+O(q^{100}) \) 16 * q + 4 * q^2 + 10 * q^4 + q^5 - 9 * q^7 + 9 * q^8 - 15 * q^10 - 19 * q^13 + 6 * q^14 + 10 * q^16 + 8 * q^17 - 16 * q^19 + 11 * q^20 - 12 * q^22 + 5 * q^23 - 3 * q^25 - 9 * q^26 - 17 * q^28 + 2 * q^29 - 18 * q^31 - 3 * q^32 - 14 * q^34 + 11 * q^35 - 24 * q^37 - 4 * q^38 - 50 * q^40 + 6 * q^41 - 34 * q^43 + 4 * q^44 - 3 * q^46 + 16 * q^47 + 5 * q^49 - 26 * q^50 - 44 * q^52 + 23 * q^53 - 48 * q^55 + 3 * q^56 - 26 * q^58 + 32 * q^59 - 16 * q^61 - 32 * q^62 + 7 * q^64 + 18 * q^65 - 67 * q^67 + 19 * q^68 + 24 * q^70 - 19 * q^71 - 2 * q^73 + 29 * q^74 - 10 * q^76 - 14 * q^77 - 27 * q^79 - 15 * q^80 - 56 * q^82 + 17 * q^83 + 15 * q^85 + q^86 - 13 * q^88 - 20 * q^89 - 42 * q^91 - 45 * q^92 + 4 * q^94 - q^95 - 50 * q^97 - 13 * q^98

Introduction

L-functions

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