LMFDB - Embedded newform 4016.2.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, 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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 2 x^{16} - 28 x^{15} + 54 x^{14} + 317 x^{13} - 582 x^{12} - 1867 x^{11} + 3178 x^{10} + \cdots + 256 $ x^17 - 2x^16 - 28x^15 + 54x^14 + 317x^13 - 582x^12 - 1867x^11 + 3178x^10 + 6186x^9 - 9216x^8 - 11921x^7 + 13680x^6 + 13752x^5 - 9400x^4 - 8800x^3 + 1920x^2 + 2240x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 251)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.37907$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46273 q^{3} +1.31548 q^{5} +3.48956 q^{7} +3.06502 q^{9} +O(q^{10})$	$q-2.46273 q^{3} +1.31548 q^{5} +3.48956 q^{7} +3.06502 q^{9} -3.66893 q^{11} +4.41851 q^{13} -3.23967 q^{15} -7.90905 q^{17} +4.04690 q^{19} -8.59384 q^{21} -0.625539 q^{23} -3.26950 q^{25} -0.160129 q^{27} +10.4241 q^{29} +4.37191 q^{31} +9.03558 q^{33} +4.59046 q^{35} -3.01090 q^{37} -10.8816 q^{39} +12.0639 q^{41} -0.164454 q^{43} +4.03198 q^{45} +0.235969 q^{47} +5.17706 q^{49} +19.4778 q^{51} -11.4830 q^{53} -4.82642 q^{55} -9.96641 q^{57} -7.18778 q^{59} +7.92880 q^{61} +10.6956 q^{63} +5.81248 q^{65} -0.478919 q^{67} +1.54053 q^{69} -2.65624 q^{71} +9.29721 q^{73} +8.05189 q^{75} -12.8030 q^{77} -6.33824 q^{79} -8.80071 q^{81} +14.2551 q^{83} -10.4042 q^{85} -25.6717 q^{87} -9.55758 q^{89} +15.4187 q^{91} -10.7668 q^{93} +5.32363 q^{95} +3.65366 q^{97} -11.2454 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.46273	−1.42186	−0.710928	−	0.703265i	$-0.751725\pi$
$3$	−2.46273	−1.42186	−0.710928	+	0.703265i	$0.751725\pi$
$4$	0	0
$5$	1.31548	0.588302	0.294151	−	0.955759i	$-0.404963\pi$
$5$	1.31548	0.588302	0.294151	+	0.955759i	$0.404963\pi$
$6$	0	0
$7$	3.48956	1.31893	0.659466	−	0.751735i	$-0.270783\pi$
$7$	3.48956	1.31893	0.659466	+	0.751735i	$0.270783\pi$
$8$	0	0
$9$	3.06502	1.02167
$10$	0	0
$11$	−3.66893	−1.10623	−0.553113	−	0.833106i	$-0.686560\pi$
$11$	−3.66893	−1.10623	−0.553113	+	0.833106i	$0.686560\pi$
$12$	0	0
$13$	4.41851	1.22547	0.612737	−	0.790287i	$-0.290068\pi$
$13$	4.41851	1.22547	0.612737	+	0.790287i	$0.290068\pi$
$14$	0	0
$15$	−3.23967	−0.836480
$16$	0	0
$17$	−7.90905	−1.91823	−0.959113	−	0.283022i	$-0.908663\pi$
$17$	−7.90905	−1.91823	−0.959113	+	0.283022i	$0.908663\pi$
$18$	0	0
$19$	4.04690	0.928423	0.464211	−	0.885724i	$-0.346338\pi$
$19$	4.04690	0.928423	0.464211	+	0.885724i	$0.346338\pi$
$20$	0	0
$21$	−8.59384	−1.87533
$22$	0	0
$23$	−0.625539	−0.130434	−0.0652169	−	0.997871i	$-0.520774\pi$
$23$	−0.625539	−0.130434	−0.0652169	+	0.997871i	$0.520774\pi$
$24$	0	0
$25$	−3.26950	−0.653901
$26$	0	0
$27$	−0.160129	−0.0308168
$28$	0	0
$29$	10.4241	1.93571	0.967855	−	0.251509i	$-0.0809268\pi$
$29$	10.4241	1.93571	0.967855	+	0.251509i	$0.0809268\pi$
$30$	0	0
$31$	4.37191	0.785218	0.392609	−	0.919705i	$-0.371572\pi$
$31$	4.37191	0.785218	0.392609	+	0.919705i	$0.371572\pi$
$32$	0	0
$33$	9.03558	1.57289
$34$	0	0
$35$	4.59046	0.775930
$36$	0	0
$37$	−3.01090	−0.494989	−0.247495	−	0.968889i	$-0.579607\pi$
$37$	−3.01090	−0.494989	−0.247495	+	0.968889i	$0.579607\pi$
$38$	0	0
$39$	−10.8816	−1.74245
$40$	0	0
$41$	12.0639	1.88406	0.942032	−	0.335523i	$-0.108913\pi$
$41$	12.0639	1.88406	0.942032	+	0.335523i	$0.108913\pi$
$42$	0	0
$43$	−0.164454	−0.0250790	−0.0125395	−	0.999921i	$-0.503992\pi$
$43$	−0.164454	−0.0250790	−0.0125395	+	0.999921i	$0.503992\pi$
$44$	0	0
$45$	4.03198	0.601053
$46$	0	0
$47$	0.235969	0.0344196	0.0172098	−	0.999852i	$-0.494522\pi$
$47$	0.235969	0.0344196	0.0172098	+	0.999852i	$0.494522\pi$
$48$	0	0
$49$	5.17706	0.739579
$50$	0	0
$51$	19.4778	2.72744
$52$	0	0
$53$	−11.4830	−1.57731	−0.788656	−	0.614835i	$-0.789223\pi$
$53$	−11.4830	−1.57731	−0.788656	+	0.614835i	$0.789223\pi$
$54$	0	0
$55$	−4.82642	−0.650794
$56$	0	0
$57$	−9.96641	−1.32008
$58$	0	0
$59$	−7.18778	−0.935769	−0.467885	−	0.883790i	$-0.654984\pi$
$59$	−7.18778	−0.935769	−0.467885	+	0.883790i	$0.654984\pi$
$60$	0	0
$61$	7.92880	1.01518	0.507589	−	0.861599i	$-0.330537\pi$
$61$	7.92880	1.01518	0.507589	+	0.861599i	$0.330537\pi$
$62$	0	0
$63$	10.6956	1.34752
$64$	0	0
$65$	5.81248	0.720949
$66$	0	0
$67$	−0.478919	−0.0585093	−0.0292547	−	0.999572i	$-0.509313\pi$
$67$	−0.478919	−0.0585093	−0.0292547	+	0.999572i	$0.509313\pi$
$68$	0	0
$69$	1.54053	0.185458
$70$	0	0
$71$	−2.65624	−0.315238	−0.157619	−	0.987500i	$-0.550382\pi$
$71$	−2.65624	−0.315238	−0.157619	+	0.987500i	$0.550382\pi$
$72$	0	0
$73$	9.29721	1.08816	0.544078	−	0.839035i	$-0.316879\pi$
$73$	9.29721	1.08816	0.544078	+	0.839035i	$0.316879\pi$
$74$	0	0
$75$	8.05189	0.929753
$76$	0	0
$77$	−12.8030	−1.45903
$78$	0	0
$79$	−6.33824	−0.713107	−0.356554	−	0.934275i	$-0.616048\pi$
$79$	−6.33824	−0.713107	−0.356554	+	0.934275i	$0.616048\pi$
$80$	0	0
$81$	−8.80071	−0.977857
$82$	0	0
$83$	14.2551	1.56470	0.782349	−	0.622840i	$-0.214021\pi$
$83$	14.2551	1.56470	0.782349	+	0.622840i	$0.214021\pi$
$84$	0	0
$85$	−10.4042	−1.12850
$86$	0	0
$87$	−25.6717	−2.75230
$88$	0	0
$89$	−9.55758	−1.01310	−0.506551	−	0.862210i	$-0.669080\pi$
$89$	−9.55758	−1.01310	−0.506551	+	0.862210i	$0.669080\pi$
$90$	0	0
$91$	15.4187	1.61632
$92$	0	0
$93$	−10.7668	−1.11647
$94$	0	0
$95$	5.32363	0.546193
$96$	0	0
$97$	3.65366	0.370973	0.185487	−	0.982647i	$-0.440614\pi$
$97$	3.65366	0.370973	0.185487	+	0.982647i	$0.440614\pi$
$98$	0	0
$99$	−11.2454	−1.13020
$100$	0	0
$101$	7.04820	0.701322	0.350661	−	0.936502i	$-0.385957\pi$
$101$	7.04820	0.701322	0.350661	+	0.936502i	$0.385957\pi$
$102$	0	0
$103$	−8.53341	−0.840822	−0.420411	−	0.907334i	$-0.638114\pi$
$103$	−8.53341	−0.840822	−0.420411	+	0.907334i	$0.638114\pi$
$104$	0	0
$105$	−11.3051	−1.10326
$106$	0	0
$107$	−7.02362	−0.678999	−0.339500	−	0.940606i	$-0.610258\pi$
$107$	−7.02362	−0.678999	−0.339500	+	0.940606i	$0.610258\pi$
$108$	0	0
$109$	10.9647	1.05023	0.525116	−	0.851031i	$-0.324022\pi$
$109$	10.9647	1.05023	0.525116	+	0.851031i	$0.324022\pi$
$110$	0	0
$111$	7.41503	0.703803
$112$	0	0
$113$	3.26571	0.307212	0.153606	−	0.988132i	$-0.450911\pi$
$113$	3.26571	0.307212	0.153606	+	0.988132i	$0.450911\pi$
$114$	0	0
$115$	−0.822885	−0.0767345
$116$	0	0
$117$	13.5428	1.25204
$118$	0	0
$119$	−27.5991	−2.53001
$120$	0	0
$121$	2.46108	0.223734
$122$	0	0
$123$	−29.7101	−2.67887
$124$	0	0
$125$	−10.8784	−0.972993
$126$	0	0
$127$	1.19962	0.106449	0.0532244	−	0.998583i	$-0.483050\pi$
$127$	1.19962	0.106449	0.0532244	+	0.998583i	$0.483050\pi$
$128$	0	0
$129$	0.405006	0.0356588
$130$	0	0
$131$	8.44030	0.737432	0.368716	−	0.929542i	$-0.379797\pi$
$131$	8.44030	0.737432	0.368716	+	0.929542i	$0.379797\pi$
$132$	0	0
$133$	14.1219	1.22453
$134$	0	0
$135$	−0.210647	−0.0181296
$136$	0	0
$137$	−13.0671	−1.11640	−0.558199	−	0.829707i	$-0.688507\pi$
$137$	−13.0671	−1.11640	−0.558199	+	0.829707i	$0.688507\pi$
$138$	0	0
$139$	5.12717	0.434881	0.217441	−	0.976074i	$-0.430229\pi$
$139$	5.12717	0.434881	0.217441	+	0.976074i	$0.430229\pi$
$140$	0	0
$141$	−0.581127	−0.0489397
$142$	0	0
$143$	−16.2112	−1.35565
$144$	0	0
$145$	13.7127	1.13878
$146$	0	0
$147$	−12.7497	−1.05158
$148$	0	0
$149$	−1.68186	−0.137784	−0.0688919	−	0.997624i	$-0.521946\pi$
$149$	−1.68186	−0.137784	−0.0688919	+	0.997624i	$0.521946\pi$
$150$	0	0
$151$	−4.85361	−0.394981	−0.197491	−	0.980305i	$-0.563279\pi$
$151$	−4.85361	−0.394981	−0.197491	+	0.980305i	$0.563279\pi$
$152$	0	0
$153$	−24.2414	−1.95980
$154$	0	0
$155$	5.75117	0.461945
$156$	0	0
$157$	14.9123	1.19013	0.595064	−	0.803678i	$-0.297127\pi$
$157$	14.9123	1.19013	0.595064	+	0.803678i	$0.297127\pi$
$158$	0	0
$159$	28.2795	2.24271
$160$	0	0
$161$	−2.18286	−0.172033
$162$	0	0
$163$	22.4041	1.75483	0.877413	−	0.479735i	$-0.159267\pi$
$163$	22.4041	1.75483	0.877413	+	0.479735i	$0.159267\pi$
$164$	0	0
$165$	11.8862	0.925336
$166$	0	0
$167$	2.57870	0.199546	0.0997729	−	0.995010i	$-0.468188\pi$
$167$	2.57870	0.199546	0.0997729	+	0.995010i	$0.468188\pi$
$168$	0	0
$169$	6.52325	0.501788
$170$	0	0
$171$	12.4038	0.948545
$172$	0	0
$173$	−2.69452	−0.204861	−0.102430	−	0.994740i	$-0.532662\pi$
$173$	−2.69452	−0.204861	−0.102430	+	0.994740i	$0.532662\pi$
$174$	0	0
$175$	−11.4091	−0.862450
$176$	0	0
$177$	17.7015	1.33053
$178$	0	0
$179$	24.9712	1.86644	0.933219	−	0.359309i	$-0.116988\pi$
$179$	24.9712	1.86644	0.933219	+	0.359309i	$0.116988\pi$
$180$	0	0
$181$	2.85397	0.212134	0.106067	−	0.994359i	$-0.466174\pi$
$181$	2.85397	0.212134	0.106067	+	0.994359i	$0.466174\pi$
$182$	0	0
$183$	−19.5265	−1.44344
$184$	0	0
$185$	−3.96079	−0.291203
$186$	0	0
$187$	29.0178	2.12199
$188$	0	0
$189$	−0.558780	−0.0406452
$190$	0	0
$191$	−3.43102	−0.248260	−0.124130	−	0.992266i	$-0.539614\pi$
$191$	−3.43102	−0.248260	−0.124130	+	0.992266i	$0.539614\pi$
$192$	0	0
$193$	10.7258	0.772060	0.386030	−	0.922486i	$-0.373846\pi$
$193$	10.7258	0.772060	0.386030	+	0.922486i	$0.373846\pi$
$194$	0	0
$195$	−14.3145	−1.02509
$196$	0	0
$197$	22.1350	1.57705	0.788527	−	0.615001i	$-0.210844\pi$
$197$	22.1350	1.57705	0.788527	+	0.615001i	$0.210844\pi$
$198$	0	0
$199$	16.1670	1.14605	0.573024	−	0.819538i	$-0.305770\pi$
$199$	16.1670	1.14605	0.573024	+	0.819538i	$0.305770\pi$
$200$	0	0
$201$	1.17945	0.0831918
$202$	0	0
$203$	36.3756	2.55307
$204$	0	0
$205$	15.8699	1.10840
$206$	0	0
$207$	−1.91729	−0.133261
$208$	0	0
$209$	−14.8478	−1.02704
$210$	0	0
$211$	−7.10640	−0.489225	−0.244612	−	0.969621i	$-0.578661\pi$
$211$	−7.10640	−0.489225	−0.244612	+	0.969621i	$0.578661\pi$
$212$	0	0
$213$	6.54160	0.448223
$214$	0	0
$215$	−0.216337	−0.0147540
$216$	0	0
$217$	15.2561	1.03565
$218$	0	0
$219$	−22.8965	−1.54720
$220$	0	0
$221$	−34.9462	−2.35074
$222$	0	0
$223$	2.08114	0.139364	0.0696818	−	0.997569i	$-0.477802\pi$
$223$	2.08114	0.139364	0.0696818	+	0.997569i	$0.477802\pi$
$224$	0	0
$225$	−10.0211	−0.668073
$226$	0	0
$227$	18.1848	1.20697	0.603485	−	0.797374i	$-0.293778\pi$
$227$	18.1848	1.20697	0.603485	+	0.797374i	$0.293778\pi$
$228$	0	0
$229$	0.908662	0.0600461	0.0300230	−	0.999549i	$-0.490442\pi$
$229$	0.908662	0.0600461	0.0300230	+	0.999549i	$0.490442\pi$
$230$	0	0
$231$	31.5302	2.07454
$232$	0	0
$233$	12.2053	0.799594	0.399797	−	0.916604i	$-0.369081\pi$
$233$	12.2053	0.799594	0.399797	+	0.916604i	$0.369081\pi$
$234$	0	0
$235$	0.310413	0.0202491
$236$	0	0
$237$	15.6093	1.01394
$238$	0	0
$239$	−9.40536	−0.608382	−0.304191	−	0.952611i	$-0.598386\pi$
$239$	−9.40536	−0.608382	−0.304191	+	0.952611i	$0.598386\pi$
$240$	0	0
$241$	−7.45266	−0.480068	−0.240034	−	0.970764i	$-0.577159\pi$
$241$	−7.45266	−0.480068	−0.240034	+	0.970764i	$0.577159\pi$
$242$	0	0
$243$	22.1541	1.42119
$244$	0	0
$245$	6.81033	0.435096
$246$	0	0
$247$	17.8813	1.13776
$248$	0	0
$249$	−35.1064	−2.22477
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	2.29506	0.144289
$254$	0	0
$255$	25.6228	1.60456
$256$	0	0
$257$	−8.59388	−0.536071	−0.268036	−	0.963409i	$-0.586375\pi$
$257$	−8.59388	−0.536071	−0.268036	+	0.963409i	$0.586375\pi$
$258$	0	0
$259$	−10.5067	−0.652857
$260$	0	0
$261$	31.9501	1.97766
$262$	0	0
$263$	−5.00613	−0.308691	−0.154346	−	0.988017i	$-0.549327\pi$
$263$	−5.00613	−0.308691	−0.154346	+	0.988017i	$0.549327\pi$
$264$	0	0
$265$	−15.1057	−0.927935
$266$	0	0
$267$	23.5377	1.44048
$268$	0	0
$269$	−5.42980	−0.331061	−0.165530	−	0.986205i	$-0.552934\pi$
$269$	−5.42980	−0.331061	−0.165530	+	0.986205i	$0.552934\pi$
$270$	0	0
$271$	18.0972	1.09933	0.549664	−	0.835386i	$-0.314756\pi$
$271$	18.0972	1.09933	0.549664	+	0.835386i	$0.314756\pi$
$272$	0	0
$273$	−37.9720	−2.29817
$274$	0	0
$275$	11.9956	0.723362
$276$	0	0
$277$	16.4601	0.988994	0.494497	−	0.869179i	$-0.335352\pi$
$277$	16.4601	0.988994	0.494497	+	0.869179i	$0.335352\pi$
$278$	0	0
$279$	13.4000	0.802237
$280$	0	0
$281$	−3.66646	−0.218722	−0.109361	−	0.994002i	$-0.534881\pi$
$281$	−3.66646	−0.218722	−0.109361	+	0.994002i	$0.534881\pi$
$282$	0	0
$283$	−11.3048	−0.672000	−0.336000	−	0.941862i	$-0.609074\pi$
$283$	−11.3048	−0.672000	−0.336000	+	0.941862i	$0.609074\pi$
$284$	0	0
$285$	−13.1106	−0.776608
$286$	0	0
$287$	42.0977	2.48495
$288$	0	0
$289$	45.5531	2.67959
$290$	0	0
$291$	−8.99798	−0.527471
$292$	0	0
$293$	−27.8052	−1.62440	−0.812199	−	0.583380i	$-0.801730\pi$
$293$	−27.8052	−1.62440	−0.812199	+	0.583380i	$0.801730\pi$
$294$	0	0
$295$	−9.45540	−0.550515
$296$	0	0
$297$	0.587502	0.0340903
$298$	0	0
$299$	−2.76395	−0.159843
$300$	0	0
$301$	−0.573873	−0.0330775
$302$	0	0
$303$	−17.3578	−0.997178
$304$	0	0
$305$	10.4302	0.597231
$306$	0	0
$307$	18.2217	1.03997	0.519985	−	0.854176i	$-0.325938\pi$
$307$	18.2217	1.03997	0.519985	+	0.854176i	$0.325938\pi$
$308$	0	0
$309$	21.0155	1.19553
$310$	0	0
$311$	16.7179	0.947987	0.473994	−	0.880528i	$-0.342812\pi$
$311$	16.7179	0.947987	0.473994	+	0.880528i	$0.342812\pi$
$312$	0	0
$313$	22.1319	1.25097	0.625483	−	0.780238i	$-0.284902\pi$
$313$	22.1319	1.25097	0.625483	+	0.780238i	$0.284902\pi$
$314$	0	0
$315$	14.0699	0.792747
$316$	0	0
$317$	−2.71767	−0.152639	−0.0763197	−	0.997083i	$-0.524317\pi$
$317$	−2.71767	−0.152639	−0.0763197	+	0.997083i	$0.524317\pi$
$318$	0	0
$319$	−38.2454	−2.14133
$320$	0	0
$321$	17.2973	0.965439
$322$	0	0
$323$	−32.0072	−1.78093
$324$	0	0
$325$	−14.4463	−0.801339
$326$	0	0
$327$	−27.0032	−1.49328
$328$	0	0
$329$	0.823429	0.0453971
$330$	0	0
$331$	32.1394	1.76654	0.883270	−	0.468864i	$-0.155337\pi$
$331$	32.1394	1.76654	0.883270	+	0.468864i	$0.155337\pi$
$332$	0	0
$333$	−9.22848	−0.505717
$334$	0	0
$335$	−0.630010	−0.0344211
$336$	0	0
$337$	−26.4970	−1.44338	−0.721692	−	0.692215i	$-0.756635\pi$
$337$	−26.4970	−1.44338	−0.721692	+	0.692215i	$0.756635\pi$
$338$	0	0
$339$	−8.04255	−0.436811
$340$	0	0
$341$	−16.0403	−0.868629
$342$	0	0
$343$	−6.36128	−0.343477
$344$	0	0
$345$	2.02654	0.109105
$346$	0	0
$347$	22.2777	1.19593	0.597964	−	0.801523i	$-0.295977\pi$
$347$	22.2777	1.19593	0.597964	+	0.801523i	$0.295977\pi$
$348$	0	0
$349$	13.3137	0.712667	0.356333	−	0.934359i	$-0.384027\pi$
$349$	13.3137	0.712667	0.356333	+	0.934359i	$0.384027\pi$
$350$	0	0
$351$	−0.707531	−0.0377652
$352$	0	0
$353$	−20.7420	−1.10399	−0.551994	−	0.833848i	$-0.686133\pi$
$353$	−20.7420	−1.10399	−0.551994	+	0.833848i	$0.686133\pi$
$354$	0	0
$355$	−3.49424	−0.185455
$356$	0	0
$357$	67.9691	3.59731
$358$	0	0
$359$	8.64717	0.456380	0.228190	−	0.973617i	$-0.426719\pi$
$359$	8.64717	0.456380	0.228190	+	0.973617i	$0.426719\pi$
$360$	0	0
$361$	−2.62259	−0.138031
$362$	0	0
$363$	−6.06096	−0.318118
$364$	0	0
$365$	12.2303	0.640164
$366$	0	0
$367$	−25.1717	−1.31395	−0.656976	−	0.753912i	$-0.728165\pi$
$367$	−25.1717	−1.31395	−0.656976	+	0.753912i	$0.728165\pi$
$368$	0	0
$369$	36.9761	1.92490
$370$	0	0
$371$	−40.0707	−2.08037
$372$	0	0
$373$	−0.893363	−0.0462566	−0.0231283	−	0.999733i	$-0.507363\pi$
$373$	−0.893363	−0.0462566	−0.0231283	+	0.999733i	$0.507363\pi$
$374$	0	0
$375$	26.7905	1.38346
$376$	0	0
$377$	46.0591	2.37216
$378$	0	0
$379$	2.52089	0.129489	0.0647447	−	0.997902i	$-0.479377\pi$
$379$	2.52089	0.129489	0.0647447	+	0.997902i	$0.479377\pi$
$380$	0	0
$381$	−2.95433	−0.151355
$382$	0	0
$383$	−3.74890	−0.191560	−0.0957800	−	0.995403i	$-0.530535\pi$
$383$	−3.74890	−0.191560	−0.0957800	+	0.995403i	$0.530535\pi$
$384$	0	0
$385$	−16.8421	−0.858353
$386$	0	0
$387$	−0.504056	−0.0256226
$388$	0	0
$389$	0.193533	0.00981252	0.00490626	−	0.999988i	$-0.498438\pi$
$389$	0.193533	0.00981252	0.00490626	+	0.999988i	$0.498438\pi$
$390$	0	0
$391$	4.94742	0.250202
$392$	0	0
$393$	−20.7861	−1.04852
$394$	0	0
$395$	−8.33784	−0.419522
$396$	0	0
$397$	15.8009	0.793025	0.396513	−	0.918029i	$-0.370220\pi$
$397$	15.8009	0.793025	0.396513	+	0.918029i	$0.370220\pi$
$398$	0	0
$399$	−34.7784	−1.74110
$400$	0	0
$401$	15.2558	0.761839	0.380920	−	0.924608i	$-0.375607\pi$
$401$	15.2558	0.761839	0.380920	+	0.924608i	$0.375607\pi$
$402$	0	0
$403$	19.3173	0.962265
$404$	0	0
$405$	−11.5772	−0.575275
$406$	0	0
$407$	11.0468	0.547570
$408$	0	0
$409$	12.0065	0.593683	0.296842	−	0.954927i	$-0.404067\pi$
$409$	12.0065	0.593683	0.296842	+	0.954927i	$0.404067\pi$
$410$	0	0
$411$	32.1807	1.58736
$412$	0	0
$413$	−25.0822	−1.23421
$414$	0	0
$415$	18.7523	0.920515
$416$	0	0
$417$	−12.6268	−0.618338
$418$	0	0
$419$	15.9744	0.780402	0.390201	−	0.920730i	$-0.372405\pi$
$419$	15.9744	0.780402	0.390201	+	0.920730i	$0.372405\pi$
$420$	0	0
$421$	25.2569	1.23095	0.615474	−	0.788157i	$-0.288965\pi$
$421$	25.2569	1.23095	0.615474	+	0.788157i	$0.288965\pi$
$422$	0	0
$423$	0.723250	0.0351656
$424$	0	0
$425$	25.8587	1.25433
$426$	0	0
$427$	27.6680	1.33895
$428$	0	0
$429$	39.9238	1.92754
$430$	0	0
$431$	1.29490	0.0623733	0.0311867	−	0.999514i	$-0.490071\pi$
$431$	1.29490	0.0623733	0.0311867	+	0.999514i	$0.490071\pi$
$432$	0	0
$433$	23.8563	1.14646	0.573231	−	0.819394i	$-0.305690\pi$
$433$	23.8563	1.14646	0.573231	+	0.819394i	$0.305690\pi$
$434$	0	0
$435$	−33.7707	−1.61918
$436$	0	0
$437$	−2.53149	−0.121098
$438$	0	0
$439$	−13.5767	−0.647980	−0.323990	−	0.946060i	$-0.605024\pi$
$439$	−13.5767	−0.647980	−0.323990	+	0.946060i	$0.605024\pi$
$440$	0	0
$441$	15.8678	0.755609
$442$	0	0
$443$	−22.4586	−1.06704	−0.533520	−	0.845787i	$-0.679131\pi$
$443$	−22.4586	−1.06704	−0.533520	+	0.845787i	$0.679131\pi$
$444$	0	0
$445$	−12.5728	−0.596009
$446$	0	0
$447$	4.14197	0.195909
$448$	0	0
$449$	29.9262	1.41230	0.706152	−	0.708060i	$-0.250429\pi$
$449$	29.9262	1.41230	0.706152	+	0.708060i	$0.250429\pi$
$450$	0	0
$451$	−44.2616	−2.08420
$452$	0	0
$453$	11.9531	0.561606
$454$	0	0
$455$	20.2830	0.950882
$456$	0	0
$457$	−31.2410	−1.46139	−0.730696	−	0.682703i	$-0.760804\pi$
$457$	−31.2410	−1.46139	−0.730696	+	0.682703i	$0.760804\pi$
$458$	0	0
$459$	1.26647	0.0591136
$460$	0	0
$461$	−24.9115	−1.16025	−0.580123	−	0.814529i	$-0.696995\pi$
$461$	−24.9115	−1.16025	−0.580123	+	0.814529i	$0.696995\pi$
$462$	0	0
$463$	16.2898	0.757050	0.378525	−	0.925591i	$-0.376431\pi$
$463$	16.2898	0.757050	0.378525	+	0.925591i	$0.376431\pi$
$464$	0	0
$465$	−14.1636	−0.656820
$466$	0	0
$467$	27.7607	1.28461	0.642305	−	0.766449i	$-0.277978\pi$
$467$	27.7607	1.28461	0.642305	+	0.766449i	$0.277978\pi$
$468$	0	0
$469$	−1.67122	−0.0771698
$470$	0	0
$471$	−36.7248	−1.69219
$472$	0	0
$473$	0.603372	0.0277431
$474$	0	0
$475$	−13.2314	−0.607097
$476$	0	0
$477$	−35.1956	−1.61150
$478$	0	0
$479$	33.4396	1.52789	0.763947	−	0.645279i	$-0.223259\pi$
$479$	33.4396	1.52789	0.763947	+	0.645279i	$0.223259\pi$
$480$	0	0
$481$	−13.3037	−0.606597
$482$	0	0
$483$	5.37578	0.244606
$484$	0	0
$485$	4.80633	0.218244
$486$	0	0
$487$	5.85497	0.265314	0.132657	−	0.991162i	$-0.457649\pi$
$487$	5.85497	0.265314	0.132657	+	0.991162i	$0.457649\pi$
$488$	0	0
$489$	−55.1752	−2.49511
$490$	0	0
$491$	−27.9694	−1.26224	−0.631121	−	0.775684i	$-0.717405\pi$
$491$	−27.9694	−1.26224	−0.631121	+	0.775684i	$0.717405\pi$
$492$	0	0
$493$	−82.4449	−3.71313
$494$	0	0
$495$	−14.7931	−0.664900
$496$	0	0
$497$	−9.26912	−0.415777
$498$	0	0
$499$	−28.6178	−1.28111	−0.640553	−	0.767914i	$-0.721295\pi$
$499$	−28.6178	−1.28111	−0.640553	+	0.767914i	$0.721295\pi$
$500$	0	0
$501$	−6.35063	−0.283725
$502$	0	0
$503$	−4.06882	−0.181420	−0.0907098	−	0.995877i	$-0.528914\pi$
$503$	−4.06882	−0.181420	−0.0907098	+	0.995877i	$0.528914\pi$
$504$	0	0
$505$	9.27178	0.412589
$506$	0	0
$507$	−16.0650	−0.713471
$508$	0	0
$509$	−3.27751	−0.145273	−0.0726364	−	0.997358i	$-0.523141\pi$
$509$	−3.27751	−0.145273	−0.0726364	+	0.997358i	$0.523141\pi$
$510$	0	0
$511$	32.4432	1.43520
$512$	0	0
$513$	−0.648026	−0.0286110
$514$	0	0
$515$	−11.2256	−0.494657
$516$	0	0
$517$	−0.865754	−0.0380758
$518$	0	0
$519$	6.63587	0.291282
$520$	0	0
$521$	19.6775	0.862085	0.431043	−	0.902332i	$-0.358146\pi$
$521$	19.6775	0.862085	0.431043	+	0.902332i	$0.358146\pi$
$522$	0	0
$523$	−35.9586	−1.57236	−0.786181	−	0.617996i	$-0.787945\pi$
$523$	−35.9586	−1.57236	−0.786181	+	0.617996i	$0.787945\pi$
$524$	0	0
$525$	28.0976	1.22628
$526$	0	0
$527$	−34.5777	−1.50623
$528$	0	0
$529$	−22.6087	−0.982987
$530$	0	0
$531$	−22.0307	−0.956051
$532$	0	0
$533$	53.3045	2.30887
$534$	0	0
$535$	−9.23946	−0.399457
$536$	0	0
$537$	−61.4973	−2.65380
$538$	0	0
$539$	−18.9943	−0.818141
$540$	0	0
$541$	−14.0585	−0.604421	−0.302211	−	0.953241i	$-0.597725\pi$
$541$	−14.0585	−0.604421	−0.302211	+	0.953241i	$0.597725\pi$
$542$	0	0
$543$	−7.02854	−0.301623
$544$	0	0
$545$	14.4239	0.617854
$546$	0	0
$547$	−18.0040	−0.769795	−0.384898	−	0.922959i	$-0.625763\pi$
$547$	−18.0040	−0.769795	−0.384898	+	0.922959i	$0.625763\pi$
$548$	0	0
$549$	24.3019	1.03718
$550$	0	0
$551$	42.1854	1.79716
$552$	0	0
$553$	−22.1177	−0.940540
$554$	0	0
$555$	9.75434	0.414049
$556$	0	0
$557$	7.94515	0.336647	0.168323	−	0.985732i	$-0.446165\pi$
$557$	7.94515	0.336647	0.168323	+	0.985732i	$0.446165\pi$
$558$	0	0
$559$	−0.726643	−0.0307337
$560$	0	0
$561$	−71.4629	−3.01716
$562$	0	0
$563$	−17.1327	−0.722059	−0.361029	−	0.932554i	$-0.617575\pi$
$563$	−17.1327	−0.722059	−0.361029	+	0.932554i	$0.617575\pi$
$564$	0	0
$565$	4.29598	0.180733
$566$	0	0
$567$	−30.7106	−1.28973
$568$	0	0
$569$	22.1683	0.929343	0.464672	−	0.885483i	$-0.346172\pi$
$569$	22.1683	0.929343	0.464672	+	0.885483i	$0.346172\pi$
$570$	0	0
$571$	−6.04758	−0.253083	−0.126542	−	0.991961i	$-0.540388\pi$
$571$	−6.04758	−0.253083	−0.126542	+	0.991961i	$0.540388\pi$
$572$	0	0
$573$	8.44966	0.352990
$574$	0	0
$575$	2.04520	0.0852908
$576$	0	0
$577$	23.3358	0.971483	0.485742	−	0.874102i	$-0.338550\pi$
$577$	23.3358	0.971483	0.485742	+	0.874102i	$0.338550\pi$
$578$	0	0
$579$	−26.4147	−1.09776
$580$	0	0
$581$	49.7440	2.06373
$582$	0	0
$583$	42.1304	1.74486
$584$	0	0
$585$	17.8154	0.736575
$586$	0	0
$587$	−5.01691	−0.207070	−0.103535	−	0.994626i	$-0.533015\pi$
$587$	−5.01691	−0.207070	−0.103535	+	0.994626i	$0.533015\pi$
$588$	0	0
$589$	17.6927	0.729015
$590$	0	0
$591$	−54.5124	−2.24234
$592$	0	0
$593$	14.9574	0.614228	0.307114	−	0.951673i	$-0.400637\pi$
$593$	14.9574	0.614228	0.307114	+	0.951673i	$0.400637\pi$
$594$	0	0
$595$	−36.3062	−1.48841
$596$	0	0
$597$	−39.8149	−1.62952
$598$	0	0
$599$	37.6200	1.53711	0.768555	−	0.639784i	$-0.220976\pi$
$599$	37.6200	1.53711	0.768555	+	0.639784i	$0.220976\pi$
$600$	0	0
$601$	−25.2201	−1.02875	−0.514374	−	0.857566i	$-0.671976\pi$
$601$	−25.2201	−1.02875	−0.514374	+	0.857566i	$0.671976\pi$
$602$	0	0
$603$	−1.46790	−0.0597774
$604$	0	0
$605$	3.23751	0.131623
$606$	0	0
$607$	27.6339	1.12162	0.560812	−	0.827943i	$-0.310489\pi$
$607$	27.6339	1.12162	0.560812	+	0.827943i	$0.310489\pi$
$608$	0	0
$609$	−89.5832	−3.63009
$610$	0	0
$611$	1.04263	0.0421804
$612$	0	0
$613$	−33.9869	−1.37272	−0.686359	−	0.727263i	$-0.740792\pi$
$613$	−33.9869	−1.37272	−0.686359	+	0.727263i	$0.740792\pi$
$614$	0	0
$615$	−39.0831	−1.57598
$616$	0	0
$617$	33.4920	1.34834	0.674169	−	0.738577i	$-0.264502\pi$
$617$	33.4920	1.34834	0.674169	+	0.738577i	$0.264502\pi$
$618$	0	0
$619$	−26.5574	−1.06743	−0.533717	−	0.845663i	$-0.679205\pi$
$619$	−26.5574	−1.06743	−0.533717	+	0.845663i	$0.679205\pi$
$620$	0	0
$621$	0.100167	0.00401955
$622$	0	0
$623$	−33.3518	−1.33621
$624$	0	0
$625$	2.03718	0.0814873
$626$	0	0
$627$	36.5661	1.46031
$628$	0	0
$629$	23.8134	0.949501
$630$	0	0
$631$	0.456653	0.0181791	0.00908953	−	0.999959i	$-0.497107\pi$
$631$	0.456653	0.0181791	0.00908953	+	0.999959i	$0.497107\pi$
$632$	0	0
$633$	17.5011	0.695607
$634$	0	0
$635$	1.57808	0.0626240
$636$	0	0
$637$	22.8749	0.906336
$638$	0	0
$639$	−8.14143	−0.322070
$640$	0	0
$641$	33.0081	1.30374	0.651871	−	0.758330i	$-0.273984\pi$
$641$	33.0081	1.30374	0.651871	+	0.758330i	$0.273984\pi$
$642$	0	0
$643$	28.9957	1.14348	0.571739	−	0.820436i	$-0.306269\pi$
$643$	28.9957	1.14348	0.571739	+	0.820436i	$0.306269\pi$
$644$	0	0
$645$	0.532778	0.0209781
$646$	0	0
$647$	−5.00136	−0.196624	−0.0983119	−	0.995156i	$-0.531344\pi$
$647$	−5.00136	−0.196624	−0.0983119	+	0.995156i	$0.531344\pi$
$648$	0	0
$649$	26.3715	1.03517
$650$	0	0
$651$	−37.5715	−1.47254
$652$	0	0
$653$	5.31388	0.207948	0.103974	−	0.994580i	$-0.466844\pi$
$653$	5.31388	0.207948	0.103974	+	0.994580i	$0.466844\pi$
$654$	0	0
$655$	11.1031	0.433833
$656$	0	0
$657$	28.4961	1.11174
$658$	0	0
$659$	19.1561	0.746214	0.373107	−	0.927788i	$-0.378292\pi$
$659$	19.1561	0.746214	0.373107	+	0.927788i	$0.378292\pi$
$660$	0	0
$661$	22.9498	0.892643	0.446322	−	0.894873i	$-0.352734\pi$
$661$	22.9498	0.892643	0.446322	+	0.894873i	$0.352734\pi$
$662$	0	0
$663$	86.0630	3.34241
$664$	0	0
$665$	18.5771	0.720391
$666$	0	0
$667$	−6.52069	−0.252482
$668$	0	0
$669$	−5.12529	−0.198155
$670$	0	0
$671$	−29.0902	−1.12302
$672$	0	0
$673$	19.6759	0.758450	0.379225	−	0.925305i	$-0.376191\pi$
$673$	19.6759	0.758450	0.379225	+	0.925305i	$0.376191\pi$
$674$	0	0
$675$	0.523542	0.0201511
$676$	0	0
$677$	−32.7409	−1.25834	−0.629168	−	0.777269i	$-0.716604\pi$
$677$	−32.7409	−1.25834	−0.629168	+	0.777269i	$0.716604\pi$
$678$	0	0
$679$	12.7497	0.489288
$680$	0	0
$681$	−44.7843	−1.71614
$682$	0	0
$683$	−4.74368	−0.181512	−0.0907560	−	0.995873i	$-0.528928\pi$
$683$	−4.74368	−0.181512	−0.0907560	+	0.995873i	$0.528928\pi$
$684$	0	0
$685$	−17.1896	−0.656779
$686$	0	0
$687$	−2.23779	−0.0853769
$688$	0	0
$689$	−50.7378	−1.93296
$690$	0	0
$691$	17.4515	0.663886	0.331943	−	0.943300i	$-0.392296\pi$
$691$	17.4515	0.663886	0.331943	+	0.943300i	$0.392296\pi$
$692$	0	0
$693$	−39.2414	−1.49066
$694$	0	0
$695$	6.74471	0.255841
$696$	0	0
$697$	−95.4140	−3.61406
$698$	0	0
$699$	−30.0583	−1.13691
$700$	0	0
$701$	45.8717	1.73255	0.866274	−	0.499569i	$-0.166508\pi$
$701$	45.8717	1.73255	0.866274	+	0.499569i	$0.166508\pi$
$702$	0	0
$703$	−12.1848	−0.459559
$704$	0	0
$705$	−0.764462	−0.0287913
$706$	0	0
$707$	24.5951	0.924995
$708$	0	0
$709$	−20.8242	−0.782069	−0.391035	−	0.920376i	$-0.627883\pi$
$709$	−20.8242	−0.782069	−0.391035	+	0.920376i	$0.627883\pi$
$710$	0	0
$711$	−19.4268	−0.728563
$712$	0	0
$713$	−2.73480	−0.102419
$714$	0	0
$715$	−21.3256	−0.797532
$716$	0	0
$717$	23.1628	0.865031
$718$	0	0
$719$	−38.1880	−1.42417	−0.712086	−	0.702092i	$-0.752249\pi$
$719$	−38.1880	−1.42417	−0.712086	+	0.702092i	$0.752249\pi$
$720$	0	0
$721$	−29.7779	−1.10899
$722$	0	0
$723$	18.3539	0.682588
$724$	0	0
$725$	−34.0817	−1.26576
$726$	0	0
$727$	−37.1384	−1.37739	−0.688693	−	0.725053i	$-0.741815\pi$
$727$	−37.1384	−1.37739	−0.688693	+	0.725053i	$0.741815\pi$
$728$	0	0
$729$	−28.1574	−1.04287
$730$	0	0
$731$	1.30068	0.0481073
$732$	0	0
$733$	27.7780	1.02601	0.513003	−	0.858387i	$-0.328533\pi$
$733$	27.7780	1.02601	0.513003	+	0.858387i	$0.328533\pi$
$734$	0	0
$735$	−16.7720	−0.618644
$736$	0	0
$737$	1.75712	0.0647245
$738$	0	0
$739$	−24.3250	−0.894809	−0.447405	−	0.894332i	$-0.647652\pi$
$739$	−24.3250	−0.894809	−0.447405	+	0.894332i	$0.647652\pi$
$740$	0	0
$741$	−44.0367	−1.61773
$742$	0	0
$743$	−22.6948	−0.832590	−0.416295	−	0.909230i	$-0.636672\pi$
$743$	−22.6948	−0.832590	−0.416295	+	0.909230i	$0.636672\pi$
$744$	0	0
$745$	−2.21246	−0.0810584
$746$	0	0
$747$	43.6921	1.59861
$748$	0	0
$749$	−24.5094	−0.895553
$750$	0	0
$751$	−53.8859	−1.96632	−0.983162	−	0.182738i	$-0.941504\pi$
$751$	−53.8859	−1.96632	−0.983162	+	0.182738i	$0.941504\pi$
$752$	0	0
$753$	2.46273	0.0897467
$754$	0	0
$755$	−6.38484	−0.232368
$756$	0	0
$757$	38.8162	1.41080	0.705400	−	0.708810i	$-0.250768\pi$
$757$	38.8162	1.41080	0.705400	+	0.708810i	$0.250768\pi$
$758$	0	0
$759$	−5.65211	−0.205158
$760$	0	0
$761$	−28.5923	−1.03647	−0.518235	−	0.855238i	$-0.673411\pi$
$761$	−28.5923	−1.03647	−0.518235	+	0.855238i	$0.673411\pi$
$762$	0	0
$763$	38.2622	1.38518
$764$	0	0
$765$	−31.8892	−1.15295
$766$	0	0
$767$	−31.7593	−1.14676
$768$	0	0
$769$	−36.2609	−1.30760	−0.653800	−	0.756667i	$-0.726826\pi$
$769$	−36.2609	−1.30760	−0.653800	+	0.756667i	$0.726826\pi$
$770$	0	0
$771$	21.1644	0.762216
$772$	0	0
$773$	−41.4690	−1.49154	−0.745768	−	0.666205i	$-0.767917\pi$
$773$	−41.4690	−1.49154	−0.745768	+	0.666205i	$0.767917\pi$
$774$	0	0
$775$	−14.2940	−0.513455
$776$	0	0
$777$	25.8752	0.928268
$778$	0	0
$779$	48.8214	1.74921
$780$	0	0
$781$	9.74557	0.348724
$782$	0	0
$783$	−1.66920	−0.0596524
$784$	0	0
$785$	19.6168	0.700155
$786$	0	0
$787$	46.6878	1.66424	0.832119	−	0.554597i	$-0.187127\pi$
$787$	46.6878	1.66424	0.832119	+	0.554597i	$0.187127\pi$
$788$	0	0
$789$	12.3287	0.438914
$790$	0	0
$791$	11.3959	0.405191
$792$	0	0
$793$	35.0335	1.24408
$794$	0	0
$795$	37.2012	1.31939
$796$	0	0
$797$	−45.0698	−1.59646	−0.798228	−	0.602356i	$-0.794229\pi$
$797$	−45.0698	−1.59646	−0.798228	+	0.602356i	$0.794229\pi$
$798$	0	0
$799$	−1.86629	−0.0660246
$800$	0	0
$801$	−29.2942	−1.03506
$802$	0	0
$803$	−34.1109	−1.20375
$804$	0	0
$805$	−2.87151	−0.101207
$806$	0	0
$807$	13.3721	0.470720
$808$	0	0
$809$	0.551761	0.0193989	0.00969944	−	0.999953i	$-0.496913\pi$
$809$	0.551761	0.0193989	0.00969944	+	0.999953i	$0.496913\pi$
$810$	0	0
$811$	−4.33715	−0.152298	−0.0761490	−	0.997096i	$-0.524262\pi$
$811$	−4.33715	−0.152298	−0.0761490	+	0.997096i	$0.524262\pi$
$812$	0	0
$813$	−44.5685	−1.56309
$814$	0	0
$815$	29.4722	1.03237
$816$	0	0
$817$	−0.665530	−0.0232840
$818$	0	0
$819$	47.2586	1.65135
$820$	0	0
$821$	7.79694	0.272115	0.136058	−	0.990701i	$-0.456557\pi$
$821$	7.79694	0.272115	0.136058	+	0.990701i	$0.456557\pi$
$822$	0	0
$823$	−23.2012	−0.808741	−0.404371	−	0.914595i	$-0.632509\pi$
$823$	−23.2012	−0.808741	−0.404371	+	0.914595i	$0.632509\pi$
$824$	0	0
$825$	−29.5419	−1.02852
$826$	0	0
$827$	−40.8965	−1.42211	−0.711055	−	0.703136i	$-0.751783\pi$
$827$	−40.8965	−1.42211	−0.711055	+	0.703136i	$0.751783\pi$
$828$	0	0
$829$	13.7145	0.476324	0.238162	−	0.971225i	$-0.423455\pi$
$829$	13.7145	0.476324	0.238162	+	0.971225i	$0.423455\pi$
$830$	0	0
$831$	−40.5368	−1.40621
$832$	0	0
$833$	−40.9456	−1.41868
$834$	0	0
$835$	3.39224	0.117393
$836$	0	0
$837$	−0.700069	−0.0241979
$838$	0	0
$839$	32.4698	1.12098	0.560491	−	0.828161i	$-0.310613\pi$
$839$	32.4698	1.12098	0.560491	+	0.828161i	$0.310613\pi$
$840$	0	0
$841$	79.6622	2.74697
$842$	0	0
$843$	9.02948	0.310992
$844$	0	0
$845$	8.58122	0.295203
$846$	0	0
$847$	8.58809	0.295090
$848$	0	0
$849$	27.8406	0.955487
$850$	0	0
$851$	1.88344	0.0645633
$852$	0	0
$853$	−0.229410	−0.00785485	−0.00392743	−	0.999992i	$-0.501250\pi$
$853$	−0.229410	−0.00785485	−0.00392743	+	0.999992i	$0.501250\pi$
$854$	0	0
$855$	16.3170	0.558031
$856$	0	0
$857$	−33.4957	−1.14419	−0.572096	−	0.820186i	$-0.693870\pi$
$857$	−33.4957	−1.14419	−0.572096	+	0.820186i	$0.693870\pi$
$858$	0	0
$859$	18.0814	0.616931	0.308465	−	0.951236i	$-0.400185\pi$
$859$	18.0814	0.616931	0.308465	+	0.951236i	$0.400185\pi$
$860$	0	0
$861$	−103.675	−3.53324
$862$	0	0
$863$	−10.3333	−0.351751	−0.175875	−	0.984412i	$-0.556276\pi$
$863$	−10.3333	−0.351751	−0.175875	+	0.984412i	$0.556276\pi$
$864$	0	0
$865$	−3.54460	−0.120520
$866$	0	0
$867$	−112.185	−3.80999
$868$	0	0
$869$	23.2546	0.788857
$870$	0	0
$871$	−2.11611	−0.0717017
$872$	0	0
$873$	11.1986	0.379014
$874$	0	0
$875$	−37.9608	−1.28331
$876$	0	0
$877$	32.9398	1.11230	0.556149	−	0.831083i	$-0.312278\pi$
$877$	32.9398	1.11230	0.556149	+	0.831083i	$0.312278\pi$
$878$	0	0
$879$	68.4767	2.30966
$880$	0	0
$881$	14.5017	0.488573	0.244287	−	0.969703i	$-0.421446\pi$
$881$	14.5017	0.488573	0.244287	+	0.969703i	$0.421446\pi$
$882$	0	0
$883$	−28.1025	−0.945725	−0.472862	−	0.881136i	$-0.656779\pi$
$883$	−28.1025	−0.945725	−0.472862	+	0.881136i	$0.656779\pi$
$884$	0	0
$885$	23.2861	0.782752
$886$	0	0
$887$	−10.2546	−0.344314	−0.172157	−	0.985070i	$-0.555074\pi$
$887$	−10.2546	−0.344314	−0.172157	+	0.985070i	$0.555074\pi$
$888$	0	0
$889$	4.18614	0.140399
$890$	0	0
$891$	32.2892	1.08173
$892$	0	0
$893$	0.954943	0.0319559
$894$	0	0
$895$	32.8492	1.09803
$896$	0	0
$897$	6.80685	0.227274
$898$	0	0
$899$	45.5733	1.51996
$900$	0	0
$901$	90.8196	3.02564
$902$	0	0
$903$	1.41329	0.0470315
$904$	0	0
$905$	3.75434	0.124799
$906$	0	0
$907$	−34.6173	−1.14945	−0.574725	−	0.818347i	$-0.694891\pi$
$907$	−34.6173	−1.14945	−0.574725	+	0.818347i	$0.694891\pi$
$908$	0	0
$909$	21.6029	0.716522
$910$	0	0
$911$	−19.5034	−0.646178	−0.323089	−	0.946369i	$-0.604721\pi$
$911$	−19.5034	−0.646178	−0.323089	+	0.946369i	$0.604721\pi$
$912$	0	0
$913$	−52.3009	−1.73091
$914$	0	0
$915$	−25.6867	−0.849177
$916$	0	0
$917$	29.4530	0.972622
$918$	0	0
$919$	26.3765	0.870081	0.435041	−	0.900411i	$-0.356734\pi$
$919$	26.3765	0.870081	0.435041	+	0.900411i	$0.356734\pi$
$920$	0	0
$921$	−44.8752	−1.47869
$922$	0	0
$923$	−11.7366	−0.386316
$924$	0	0
$925$	9.84416	0.323674
$926$	0	0
$927$	−26.1551	−0.859046
$928$	0	0
$929$	−28.6371	−0.939553	−0.469776	−	0.882785i	$-0.655665\pi$
$929$	−28.6371	−0.939553	−0.469776	+	0.882785i	$0.655665\pi$
$930$	0	0
$931$	20.9510	0.686642
$932$	0	0
$933$	−41.1717	−1.34790
$934$	0	0
$935$	38.1724	1.24837
$936$	0	0
$937$	29.0180	0.947977	0.473989	−	0.880531i	$-0.342814\pi$
$937$	29.0180	0.947977	0.473989	+	0.880531i	$0.342814\pi$
$938$	0	0
$939$	−54.5047	−1.77869
$940$	0	0
$941$	28.8763	0.941339	0.470670	−	0.882310i	$-0.344012\pi$
$941$	28.8763	0.941339	0.470670	+	0.882310i	$0.344012\pi$
$942$	0	0
$943$	−7.54643	−0.245746
$944$	0	0
$945$	−0.735065	−0.0239117
$946$	0	0
$947$	−20.2389	−0.657676	−0.328838	−	0.944386i	$-0.606657\pi$
$947$	−20.2389	−0.657676	−0.328838	+	0.944386i	$0.606657\pi$
$948$	0	0
$949$	41.0798	1.33351
$950$	0	0
$951$	6.69287	0.217031
$952$	0	0
$953$	−12.2731	−0.397564	−0.198782	−	0.980044i	$-0.563699\pi$
$953$	−12.2731	−0.397564	−0.198782	+	0.980044i	$0.563699\pi$
$954$	0	0
$955$	−4.51345	−0.146052
$956$	0	0
$957$	94.1879	3.04466
$958$	0	0
$959$	−45.5985	−1.47245
$960$	0	0
$961$	−11.8864	−0.383432
$962$	0	0
$963$	−21.5276	−0.693716
$964$	0	0
$965$	14.1096	0.454204
$966$	0	0
$967$	7.18637	0.231098	0.115549	−	0.993302i	$-0.463137\pi$
$967$	7.18637	0.231098	0.115549	+	0.993302i	$0.463137\pi$
$968$	0	0
$969$	78.8249	2.53222
$970$	0	0
$971$	−27.1179	−0.870256	−0.435128	−	0.900369i	$-0.643297\pi$
$971$	−27.1179	−0.870256	−0.435128	+	0.900369i	$0.643297\pi$
$972$	0	0
$973$	17.8916	0.573578
$974$	0	0
$975$	35.5774	1.13939
$976$	0	0
$977$	−29.1247	−0.931782	−0.465891	−	0.884842i	$-0.654266\pi$
$977$	−29.1247	−0.931782	−0.465891	+	0.884842i	$0.654266\pi$
$978$	0	0
$979$	35.0661	1.12072
$980$	0	0
$981$	33.6072	1.07299
$982$	0	0
$983$	47.7824	1.52402	0.762010	−	0.647565i	$-0.224212\pi$
$983$	47.7824	1.52402	0.762010	+	0.647565i	$0.224212\pi$
$984$	0	0
$985$	29.1182	0.927783
$986$	0	0
$987$	−2.02788	−0.0645481
$988$	0	0
$989$	0.102872	0.00327115
$990$	0	0
$991$	−42.7324	−1.35744	−0.678720	−	0.734397i	$-0.737465\pi$
$991$	−42.7324	−1.35744	−0.678720	+	0.734397i	$0.737465\pi$
$992$	0	0
$993$	−79.1505	−2.51177
$994$	0	0
$995$	21.2674	0.674223
$996$	0	0
$997$	−20.8160	−0.659250	−0.329625	−	0.944112i	$-0.606922\pi$
$997$	−20.8160	−0.659250	−0.329625	+	0.944112i	$0.606922\pi$
$998$	0	0
$999$	0.482132	0.0152540

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.k.1.3		17
4.3	odd	2		251.2.a.b.1.12	✓	17
12.11	even	2		2259.2.a.k.1.6		17
20.19	odd	2		6275.2.a.e.1.6		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
251.2.a.b.1.12	✓	17	4.3	odd	2
2259.2.a.k.1.6		17	12.11	even	2
4016.2.a.k.1.3		17	1.1	even	1	trivial
6275.2.a.e.1.6		17	20.19	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q-2.46273 q^{3} +1.31548 q^{5} +3.48956 q^{7} +3.06502 q^{9} +O(q^{10})\)	\(q-2.46273 q^{3} +1.31548 q^{5} +3.48956 q^{7} +3.06502 q^{9} -3.66893 q^{11} +4.41851 q^{13} -3.23967 q^{15} -7.90905 q^{17} +4.04690 q^{19} -8.59384 q^{21} -0.625539 q^{23} -3.26950 q^{25} -0.160129 q^{27} +10.4241 q^{29} +4.37191 q^{31} +9.03558 q^{33} +4.59046 q^{35} -3.01090 q^{37} -10.8816 q^{39} +12.0639 q^{41} -0.164454 q^{43} +4.03198 q^{45} +0.235969 q^{47} +5.17706 q^{49} +19.4778 q^{51} -11.4830 q^{53} -4.82642 q^{55} -9.96641 q^{57} -7.18778 q^{59} +7.92880 q^{61} +10.6956 q^{63} +5.81248 q^{65} -0.478919 q^{67} +1.54053 q^{69} -2.65624 q^{71} +9.29721 q^{73} +8.05189 q^{75} -12.8030 q^{77} -6.33824 q^{79} -8.80071 q^{81} +14.2551 q^{83} -10.4042 q^{85} -25.6717 q^{87} -9.55758 q^{89} +15.4187 q^{91} -10.7668 q^{93} +5.32363 q^{95} +3.65366 q^{97} -11.2454 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * q^99

Introduction

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