LMFDB - Embedded newform 4016.2.a.k.1.17

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			1.17
Root			$0.779516$ 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)

$f(q)$	$=$	$q+3.14584 q^{3} +1.66398 q^{5} -2.07256 q^{7} +6.89631 q^{9} +O(q^{10})$	$q+3.14584 q^{3} +1.66398 q^{5} -2.07256 q^{7} +6.89631 q^{9} -4.31307 q^{11} +0.691540 q^{13} +5.23461 q^{15} +4.59213 q^{17} -1.42603 q^{19} -6.51995 q^{21} +7.21258 q^{23} -2.23118 q^{25} +12.2572 q^{27} +6.96732 q^{29} -2.81029 q^{31} -13.5682 q^{33} -3.44870 q^{35} +5.24691 q^{37} +2.17548 q^{39} +4.77869 q^{41} +0.0696559 q^{43} +11.4753 q^{45} +11.4139 q^{47} -2.70449 q^{49} +14.4461 q^{51} +8.68993 q^{53} -7.17685 q^{55} -4.48607 q^{57} +7.99534 q^{59} -7.84861 q^{61} -14.2930 q^{63} +1.15071 q^{65} -14.4895 q^{67} +22.6896 q^{69} -10.8406 q^{71} -0.113363 q^{73} -7.01892 q^{75} +8.93909 q^{77} +6.17996 q^{79} +17.8702 q^{81} +11.1760 q^{83} +7.64120 q^{85} +21.9181 q^{87} -2.35477 q^{89} -1.43326 q^{91} -8.84072 q^{93} -2.37289 q^{95} -2.40260 q^{97} -29.7442 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.14584	1.81625	0.908126	−	0.418697i	$-0.137513\pi$
$3$	3.14584	1.81625	0.908126	+	0.418697i	$0.137513\pi$
$4$	0	0
$5$	1.66398	0.744154	0.372077	−	0.928202i	$-0.378646\pi$
$5$	1.66398	0.744154	0.372077	+	0.928202i	$0.378646\pi$
$6$	0	0
$7$	−2.07256	−0.783355	−0.391677	−	0.920103i	$-0.628105\pi$
$7$	−2.07256	−0.783355	−0.391677	+	0.920103i	$0.628105\pi$
$8$	0	0
$9$	6.89631	2.29877
$10$	0	0
$11$	−4.31307	−1.30044	−0.650219	−	0.759747i	$-0.725323\pi$
$11$	−4.31307	−1.30044	−0.650219	+	0.759747i	$0.725323\pi$
$12$	0	0
$13$	0.691540	0.191799	0.0958994	−	0.995391i	$-0.469427\pi$
$13$	0.691540	0.191799	0.0958994	+	0.995391i	$0.469427\pi$
$14$	0	0
$15$	5.23461	1.35157
$16$	0	0
$17$	4.59213	1.11375	0.556877	−	0.830595i	$-0.311999\pi$
$17$	4.59213	1.11375	0.556877	+	0.830595i	$0.311999\pi$
$18$	0	0
$19$	−1.42603	−0.327155	−0.163577	−	0.986531i	$-0.552303\pi$
$19$	−1.42603	−0.327155	−0.163577	+	0.986531i	$0.552303\pi$
$20$	0	0
$21$	−6.51995	−1.42277
$22$	0	0
$23$	7.21258	1.50393	0.751964	−	0.659204i	$-0.229107\pi$
$23$	7.21258	1.50393	0.751964	+	0.659204i	$0.229107\pi$
$24$	0	0
$25$	−2.23118	−0.446235
$26$	0	0
$27$	12.2572	2.35889
$28$	0	0
$29$	6.96732	1.29380	0.646900	−	0.762575i	$-0.276065\pi$
$29$	6.96732	1.29380	0.646900	+	0.762575i	$0.276065\pi$
$30$	0	0
$31$	−2.81029	−0.504743	−0.252371	−	0.967630i	$-0.581210\pi$
$31$	−2.81029	−0.504743	−0.252371	+	0.967630i	$0.581210\pi$
$32$	0	0
$33$	−13.5682	−2.36192
$34$	0	0
$35$	−3.44870	−0.582936
$36$	0	0
$37$	5.24691	0.862587	0.431294	−	0.902212i	$-0.358057\pi$
$37$	5.24691	0.862587	0.431294	+	0.902212i	$0.358057\pi$
$38$	0	0
$39$	2.17548	0.348355
$40$	0	0
$41$	4.77869	0.746306	0.373153	−	0.927770i	$-0.378277\pi$
$41$	4.77869	0.746306	0.373153	+	0.927770i	$0.378277\pi$
$42$	0	0
$43$	0.0696559	0.0106224	0.00531122	−	0.999986i	$-0.498309\pi$
$43$	0.0696559	0.0106224	0.00531122	+	0.999986i	$0.498309\pi$
$44$	0	0
$45$	11.4753	1.71064
$46$	0	0
$47$	11.4139	1.66489	0.832446	−	0.554106i	$-0.186940\pi$
$47$	11.4139	1.66489	0.832446	+	0.554106i	$0.186940\pi$
$48$	0	0
$49$	−2.70449	−0.386356
$50$	0	0
$51$	14.4461	2.02286
$52$	0	0
$53$	8.68993	1.19365	0.596827	−	0.802370i	$-0.296428\pi$
$53$	8.68993	1.19365	0.596827	+	0.802370i	$0.296428\pi$
$54$	0	0
$55$	−7.17685	−0.967726
$56$	0	0
$57$	−4.48607	−0.594195
$58$	0	0
$59$	7.99534	1.04090	0.520452	−	0.853891i	$-0.325763\pi$
$59$	7.99534	1.04090	0.520452	+	0.853891i	$0.325763\pi$
$60$	0	0
$61$	−7.84861	−1.00491	−0.502456	−	0.864603i	$-0.667570\pi$
$61$	−7.84861	−1.00491	−0.502456	+	0.864603i	$0.667570\pi$
$62$	0	0
$63$	−14.2930	−1.80075
$64$	0	0
$65$	1.15071	0.142728
$66$	0	0
$67$	−14.4895	−1.77017	−0.885087	−	0.465426i	$-0.845901\pi$
$67$	−14.4895	−1.77017	−0.885087	+	0.465426i	$0.845901\pi$
$68$	0	0
$69$	22.6896	2.73151
$70$	0	0
$71$	−10.8406	−1.28654	−0.643268	−	0.765641i	$-0.722422\pi$
$71$	−10.8406	−1.28654	−0.643268	+	0.765641i	$0.722422\pi$
$72$	0	0
$73$	−0.113363	−0.0132682	−0.00663409	−	0.999978i	$-0.502112\pi$
$73$	−0.113363	−0.0132682	−0.00663409	+	0.999978i	$0.502112\pi$
$74$	0	0
$75$	−7.01892	−0.810476
$76$	0	0
$77$	8.93909	1.01870
$78$	0	0
$79$	6.17996	0.695300	0.347650	−	0.937624i	$-0.386980\pi$
$79$	6.17996	0.695300	0.347650	+	0.937624i	$0.386980\pi$
$80$	0	0
$81$	17.8702	1.98557
$82$	0	0
$83$	11.1760	1.22673	0.613363	−	0.789801i	$-0.289816\pi$
$83$	11.1760	1.22673	0.613363	+	0.789801i	$0.289816\pi$
$84$	0	0
$85$	7.64120	0.828805
$86$	0	0
$87$	21.9181	2.34986
$88$	0	0
$89$	−2.35477	−0.249605	−0.124803	−	0.992182i	$-0.539830\pi$
$89$	−2.35477	−0.249605	−0.124803	+	0.992182i	$0.539830\pi$
$90$	0	0
$91$	−1.43326	−0.150246
$92$	0	0
$93$	−8.84072	−0.916740
$94$	0	0
$95$	−2.37289	−0.243453
$96$	0	0
$97$	−2.40260	−0.243947	−0.121973	−	0.992533i	$-0.538922\pi$
$97$	−2.40260	−0.243947	−0.121973	+	0.992533i	$0.538922\pi$
$98$	0	0
$99$	−29.7442	−2.98941
$100$	0	0
$101$	−0.196664	−0.0195688	−0.00978438	−	0.999952i	$-0.503115\pi$
$101$	−0.196664	−0.0195688	−0.00978438	+	0.999952i	$0.503115\pi$
$102$	0	0
$103$	−8.71179	−0.858398	−0.429199	−	0.903210i	$-0.641204\pi$
$103$	−8.71179	−0.858398	−0.429199	+	0.903210i	$0.641204\pi$
$104$	0	0
$105$	−10.8490	−1.05876
$106$	0	0
$107$	4.84459	0.468344	0.234172	−	0.972195i	$-0.424762\pi$
$107$	4.84459	0.468344	0.234172	+	0.972195i	$0.424762\pi$
$108$	0	0
$109$	20.3490	1.94908	0.974541	−	0.224208i	$-0.0719796\pi$
$109$	20.3490	1.94908	0.974541	+	0.224208i	$0.0719796\pi$
$110$	0	0
$111$	16.5060	1.56668
$112$	0	0
$113$	−16.1506	−1.51932	−0.759659	−	0.650322i	$-0.774634\pi$
$113$	−16.1506	−1.51932	−0.759659	+	0.650322i	$0.774634\pi$
$114$	0	0
$115$	12.0016	1.11915
$116$	0	0
$117$	4.76908	0.440901
$118$	0	0
$119$	−9.51747	−0.872465
$120$	0	0
$121$	7.60253	0.691139
$122$	0	0
$123$	15.0330	1.35548
$124$	0	0
$125$	−12.0325	−1.07622
$126$	0	0
$127$	21.5000	1.90782	0.953909	−	0.300097i	$-0.0970191\pi$
$127$	21.5000	1.90782	0.953909	+	0.300097i	$0.0970191\pi$
$128$	0	0
$129$	0.219126	0.0192930
$130$	0	0
$131$	10.4234	0.910697	0.455349	−	0.890313i	$-0.349515\pi$
$131$	10.4234	0.910697	0.455349	+	0.890313i	$0.349515\pi$
$132$	0	0
$133$	2.95554	0.256278
$134$	0	0
$135$	20.3957	1.75538
$136$	0	0
$137$	−17.0007	−1.45246	−0.726232	−	0.687449i	$-0.758730\pi$
$137$	−17.0007	−1.45246	−0.726232	+	0.687449i	$0.758730\pi$
$138$	0	0
$139$	−15.9865	−1.35596	−0.677980	−	0.735081i	$-0.737144\pi$
$139$	−15.9865	−1.35596	−0.677980	+	0.735081i	$0.737144\pi$
$140$	0	0
$141$	35.9064	3.02386
$142$	0	0
$143$	−2.98266	−0.249422
$144$	0	0
$145$	11.5935	0.962785
$146$	0	0
$147$	−8.50789	−0.701719
$148$	0	0
$149$	−14.2480	−1.16725	−0.583623	−	0.812025i	$-0.698365\pi$
$149$	−14.2480	−1.16725	−0.583623	+	0.812025i	$0.698365\pi$
$150$	0	0
$151$	−13.4756	−1.09663	−0.548316	−	0.836271i	$-0.684731\pi$
$151$	−13.4756	−1.09663	−0.548316	+	0.836271i	$0.684731\pi$
$152$	0	0
$153$	31.6687	2.56027
$154$	0	0
$155$	−4.67626	−0.375606
$156$	0	0
$157$	10.2729	0.819870	0.409935	−	0.912115i	$-0.365551\pi$
$157$	10.2729	0.819870	0.409935	+	0.912115i	$0.365551\pi$
$158$	0	0
$159$	27.3371	2.16798
$160$	0	0
$161$	−14.9485	−1.17811
$162$	0	0
$163$	−12.4576	−0.975753	−0.487877	−	0.872913i	$-0.662228\pi$
$163$	−12.4576	−0.975753	−0.487877	+	0.872913i	$0.662228\pi$
$164$	0	0
$165$	−22.5772	−1.75763
$166$	0	0
$167$	17.4629	1.35132	0.675659	−	0.737214i	$-0.263859\pi$
$167$	17.4629	1.35132	0.675659	+	0.737214i	$0.263859\pi$
$168$	0	0
$169$	−12.5218	−0.963213
$170$	0	0
$171$	−9.83437	−0.752053
$172$	0	0
$173$	−13.2820	−1.00982	−0.504908	−	0.863173i	$-0.668473\pi$
$173$	−13.2820	−1.00982	−0.504908	+	0.863173i	$0.668473\pi$
$174$	0	0
$175$	4.62425	0.349560
$176$	0	0
$177$	25.1521	1.89054
$178$	0	0
$179$	−16.9053	−1.26357	−0.631783	−	0.775146i	$-0.717676\pi$
$179$	−16.9053	−1.26357	−0.631783	+	0.775146i	$0.717676\pi$
$180$	0	0
$181$	−8.95143	−0.665354	−0.332677	−	0.943041i	$-0.607952\pi$
$181$	−8.95143	−0.665354	−0.332677	+	0.943041i	$0.607952\pi$
$182$	0	0
$183$	−24.6905	−1.82517
$184$	0	0
$185$	8.73075	0.641898
$186$	0	0
$187$	−19.8062	−1.44837
$188$	0	0
$189$	−25.4037	−1.84785
$190$	0	0
$191$	−6.87340	−0.497342	−0.248671	−	0.968588i	$-0.579994\pi$
$191$	−6.87340	−0.497342	−0.248671	+	0.968588i	$0.579994\pi$
$192$	0	0
$193$	10.7691	0.775174	0.387587	−	0.921833i	$-0.373309\pi$
$193$	10.7691	0.775174	0.387587	+	0.921833i	$0.373309\pi$
$194$	0	0
$195$	3.61994	0.259230
$196$	0	0
$197$	−15.3371	−1.09273	−0.546363	−	0.837548i	$-0.683988\pi$
$197$	−15.3371	−1.09273	−0.546363	+	0.837548i	$0.683988\pi$
$198$	0	0
$199$	5.96752	0.423026	0.211513	−	0.977375i	$-0.432161\pi$
$199$	5.96752	0.423026	0.211513	+	0.977375i	$0.432161\pi$
$200$	0	0
$201$	−45.5816	−3.21508
$202$	0	0
$203$	−14.4402	−1.01350
$204$	0	0
$205$	7.95163	0.555366
$206$	0	0
$207$	49.7402	3.45718
$208$	0	0
$209$	6.15058	0.425444
$210$	0	0
$211$	0.312277	0.0214981	0.0107490	−	0.999942i	$-0.496578\pi$
$211$	0.312277	0.0214981	0.0107490	+	0.999942i	$0.496578\pi$
$212$	0	0
$213$	−34.1026	−2.33667
$214$	0	0
$215$	0.115906	0.00790472
$216$	0	0
$217$	5.82449	0.395392
$218$	0	0
$219$	−0.356623	−0.0240984
$220$	0	0
$221$	3.17564	0.213617
$222$	0	0
$223$	9.92618	0.664706	0.332353	−	0.943155i	$-0.392157\pi$
$223$	9.92618	0.664706	0.332353	+	0.943155i	$0.392157\pi$
$224$	0	0
$225$	−15.3869	−1.02579
$226$	0	0
$227$	21.8928	1.45307	0.726537	−	0.687128i	$-0.241129\pi$
$227$	21.8928	1.45307	0.726537	+	0.687128i	$0.241129\pi$
$228$	0	0
$229$	−3.74494	−0.247473	−0.123736	−	0.992315i	$-0.539488\pi$
$229$	−3.74494	−0.247473	−0.123736	+	0.992315i	$0.539488\pi$
$230$	0	0
$231$	28.1210	1.85022
$232$	0	0
$233$	22.6997	1.48710	0.743552	−	0.668678i	$-0.233140\pi$
$233$	22.6997	1.48710	0.743552	+	0.668678i	$0.233140\pi$
$234$	0	0
$235$	18.9925	1.23894
$236$	0	0
$237$	19.4412	1.26284
$238$	0	0
$239$	−4.32556	−0.279797	−0.139899	−	0.990166i	$-0.544678\pi$
$239$	−4.32556	−0.279797	−0.139899	+	0.990166i	$0.544678\pi$
$240$	0	0
$241$	−12.9477	−0.834034	−0.417017	−	0.908899i	$-0.636924\pi$
$241$	−12.9477	−0.834034	−0.417017	+	0.908899i	$0.636924\pi$
$242$	0	0
$243$	19.4452	1.24741
$244$	0	0
$245$	−4.50021	−0.287508
$246$	0	0
$247$	−0.986160	−0.0627478
$248$	0	0
$249$	35.1580	2.22804
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−31.1083	−1.95576
$254$	0	0
$255$	24.0380	1.50532
$256$	0	0
$257$	−0.394093	−0.0245828	−0.0122914	−	0.999924i	$-0.503913\pi$
$257$	−0.394093	−0.0245828	−0.0122914	+	0.999924i	$0.503913\pi$
$258$	0	0
$259$	−10.8746	−0.675712
$260$	0	0
$261$	48.0488	2.97415
$262$	0	0
$263$	−1.53116	−0.0944155	−0.0472077	−	0.998885i	$-0.515032\pi$
$263$	−1.53116	−0.0944155	−0.0472077	+	0.998885i	$0.515032\pi$
$264$	0	0
$265$	14.4599	0.888262
$266$	0	0
$267$	−7.40773	−0.453346
$268$	0	0
$269$	−3.68392	−0.224612	−0.112306	−	0.993674i	$-0.535824\pi$
$269$	−3.68392	−0.224612	−0.112306	+	0.993674i	$0.535824\pi$
$270$	0	0
$271$	−10.8632	−0.659892	−0.329946	−	0.944000i	$-0.607031\pi$
$271$	−10.8632	−0.659892	−0.329946	+	0.944000i	$0.607031\pi$
$272$	0	0
$273$	−4.50881	−0.272885
$274$	0	0
$275$	9.62321	0.580301
$276$	0	0
$277$	−3.92556	−0.235864	−0.117932	−	0.993022i	$-0.537627\pi$
$277$	−3.92556	−0.235864	−0.117932	+	0.993022i	$0.537627\pi$
$278$	0	0
$279$	−19.3806	−1.16029
$280$	0	0
$281$	−1.81456	−0.108248	−0.0541239	−	0.998534i	$-0.517237\pi$
$281$	−1.81456	−0.108248	−0.0541239	+	0.998534i	$0.517237\pi$
$282$	0	0
$283$	−20.4093	−1.21321	−0.606604	−	0.795004i	$-0.707469\pi$
$283$	−20.4093	−1.21321	−0.606604	+	0.795004i	$0.707469\pi$
$284$	0	0
$285$	−7.46473	−0.442172
$286$	0	0
$287$	−9.90412	−0.584622
$288$	0	0
$289$	4.08765	0.240450
$290$	0	0
$291$	−7.55819	−0.443069
$292$	0	0
$293$	−19.2694	−1.12573	−0.562865	−	0.826549i	$-0.690301\pi$
$293$	−19.2694	−1.12573	−0.562865	+	0.826549i	$0.690301\pi$
$294$	0	0
$295$	13.3041	0.774593
$296$	0	0
$297$	−52.8660	−3.06759
$298$	0	0
$299$	4.98779	0.288452
$300$	0	0
$301$	−0.144366	−0.00832113
$302$	0	0
$303$	−0.618672	−0.0355418
$304$	0	0
$305$	−13.0599	−0.747808
$306$	0	0
$307$	−19.9296	−1.13744	−0.568720	−	0.822531i	$-0.692561\pi$
$307$	−19.9296	−1.13744	−0.568720	+	0.822531i	$0.692561\pi$
$308$	0	0
$309$	−27.4059	−1.55907
$310$	0	0
$311$	−29.0820	−1.64909	−0.824545	−	0.565797i	$-0.808569\pi$
$311$	−29.0820	−1.64909	−0.824545	+	0.565797i	$0.808569\pi$
$312$	0	0
$313$	−25.4848	−1.44049	−0.720243	−	0.693722i	$-0.755970\pi$
$313$	−25.4848	−1.44049	−0.720243	+	0.693722i	$0.755970\pi$
$314$	0	0
$315$	−23.7833	−1.34004
$316$	0	0
$317$	22.9102	1.28677	0.643383	−	0.765544i	$-0.277530\pi$
$317$	22.9102	1.28677	0.643383	+	0.765544i	$0.277530\pi$
$318$	0	0
$319$	−30.0505	−1.68251
$320$	0	0
$321$	15.2403	0.850631
$322$	0	0
$323$	−6.54853	−0.364370
$324$	0	0
$325$	−1.54295	−0.0855874
$326$	0	0
$327$	64.0148	3.54002
$328$	0	0
$329$	−23.6561	−1.30420
$330$	0	0
$331$	−24.7108	−1.35823	−0.679113	−	0.734034i	$-0.737635\pi$
$331$	−24.7108	−1.35823	−0.679113	+	0.734034i	$0.737635\pi$
$332$	0	0
$333$	36.1843	1.98289
$334$	0	0
$335$	−24.1102	−1.31728
$336$	0	0
$337$	17.8271	0.971102	0.485551	−	0.874208i	$-0.338619\pi$
$337$	17.8271	0.971102	0.485551	+	0.874208i	$0.338619\pi$
$338$	0	0
$339$	−50.8071	−2.75946
$340$	0	0
$341$	12.1210	0.656387
$342$	0	0
$343$	20.1131	1.08601
$344$	0	0
$345$	37.7551	2.03266
$346$	0	0
$347$	26.3885	1.41661	0.708305	−	0.705907i	$-0.249460\pi$
$347$	26.3885	1.41661	0.708305	+	0.705907i	$0.249460\pi$
$348$	0	0
$349$	−16.0414	−0.858677	−0.429339	−	0.903144i	$-0.641253\pi$
$349$	−16.0414	−0.858677	−0.429339	+	0.903144i	$0.641253\pi$
$350$	0	0
$351$	8.47633	0.452433
$352$	0	0
$353$	9.77533	0.520289	0.260144	−	0.965570i	$-0.416230\pi$
$353$	9.77533	0.520289	0.260144	+	0.965570i	$0.416230\pi$
$354$	0	0
$355$	−18.0384	−0.957381
$356$	0	0
$357$	−29.9404	−1.58462
$358$	0	0
$359$	−9.93170	−0.524175	−0.262088	−	0.965044i	$-0.584411\pi$
$359$	−9.93170	−0.524175	−0.262088	+	0.965044i	$0.584411\pi$
$360$	0	0
$361$	−16.9664	−0.892970
$362$	0	0
$363$	23.9163	1.25528
$364$	0	0
$365$	−0.188634	−0.00987357
$366$	0	0
$367$	−7.88573	−0.411632	−0.205816	−	0.978591i	$-0.565985\pi$
$367$	−7.88573	−0.411632	−0.205816	+	0.978591i	$0.565985\pi$
$368$	0	0
$369$	32.9553	1.71559
$370$	0	0
$371$	−18.0104	−0.935054
$372$	0	0
$373$	−9.68592	−0.501518	−0.250759	−	0.968050i	$-0.580680\pi$
$373$	−9.68592	−0.501518	−0.250759	+	0.968050i	$0.580680\pi$
$374$	0	0
$375$	−37.8524	−1.95469
$376$	0	0
$377$	4.81818	0.248149
$378$	0	0
$379$	−6.40384	−0.328943	−0.164471	−	0.986382i	$-0.552592\pi$
$379$	−6.40384	−0.328943	−0.164471	+	0.986382i	$0.552592\pi$
$380$	0	0
$381$	67.6356	3.46508
$382$	0	0
$383$	3.01090	0.153850	0.0769248	−	0.997037i	$-0.475490\pi$
$383$	3.01090	0.153850	0.0769248	+	0.997037i	$0.475490\pi$
$384$	0	0
$385$	14.8745	0.758072
$386$	0	0
$387$	0.480369	0.0244185
$388$	0	0
$389$	−3.05939	−0.155117	−0.0775586	−	0.996988i	$-0.524713\pi$
$389$	−3.05939	−0.155117	−0.0775586	+	0.996988i	$0.524713\pi$
$390$	0	0
$391$	33.1211	1.67501
$392$	0	0
$393$	32.7904	1.65406
$394$	0	0
$395$	10.2833	0.517410
$396$	0	0
$397$	11.8686	0.595668	0.297834	−	0.954618i	$-0.403736\pi$
$397$	11.8686	0.595668	0.297834	+	0.954618i	$0.403736\pi$
$398$	0	0
$399$	9.29766	0.465465
$400$	0	0
$401$	−3.21244	−0.160422	−0.0802109	−	0.996778i	$-0.525559\pi$
$401$	−3.21244	−0.160422	−0.0802109	+	0.996778i	$0.525559\pi$
$402$	0	0
$403$	−1.94343	−0.0968090
$404$	0	0
$405$	29.7355	1.47757
$406$	0	0
$407$	−22.6303	−1.12174
$408$	0	0
$409$	−39.8967	−1.97277	−0.986383	−	0.164462i	$-0.947411\pi$
$409$	−39.8967	−1.97277	−0.986383	+	0.164462i	$0.947411\pi$
$410$	0	0
$411$	−53.4814	−2.63804
$412$	0	0
$413$	−16.5708	−0.815397
$414$	0	0
$415$	18.5966	0.912873
$416$	0	0
$417$	−50.2911	−2.46276
$418$	0	0
$419$	−21.7394	−1.06204	−0.531020	−	0.847359i	$-0.678191\pi$
$419$	−21.7394	−1.06204	−0.531020	+	0.847359i	$0.678191\pi$
$420$	0	0
$421$	4.07730	0.198715	0.0993577	−	0.995052i	$-0.468321\pi$
$421$	4.07730	0.198715	0.0993577	+	0.995052i	$0.468321\pi$
$422$	0	0
$423$	78.7140	3.82721
$424$	0	0
$425$	−10.2459	−0.496997
$426$	0	0
$427$	16.2667	0.787202
$428$	0	0
$429$	−9.38297	−0.453014
$430$	0	0
$431$	−12.9221	−0.622436	−0.311218	−	0.950339i	$-0.600737\pi$
$431$	−12.9221	−0.622436	−0.311218	+	0.950339i	$0.600737\pi$
$432$	0	0
$433$	24.6228	1.18330	0.591649	−	0.806196i	$-0.298477\pi$
$433$	24.6228	1.18330	0.591649	+	0.806196i	$0.298477\pi$
$434$	0	0
$435$	36.4712	1.74866
$436$	0	0
$437$	−10.2854	−0.492017
$438$	0	0
$439$	21.7282	1.03703	0.518515	−	0.855068i	$-0.326485\pi$
$439$	21.7282	1.03703	0.518515	+	0.855068i	$0.326485\pi$
$440$	0	0
$441$	−18.6510	−0.888143
$442$	0	0
$443$	35.5501	1.68904	0.844518	−	0.535527i	$-0.179887\pi$
$443$	35.5501	1.68904	0.844518	+	0.535527i	$0.179887\pi$
$444$	0	0
$445$	−3.91829	−0.185745
$446$	0	0
$447$	−44.8221	−2.12001
$448$	0	0
$449$	−30.4969	−1.43924	−0.719619	−	0.694370i	$-0.755683\pi$
$449$	−30.4969	−1.43924	−0.719619	+	0.694370i	$0.755683\pi$
$450$	0	0
$451$	−20.6108	−0.970524
$452$	0	0
$453$	−42.3922	−1.99176
$454$	0	0
$455$	−2.38491	−0.111806
$456$	0	0
$457$	23.1694	1.08382	0.541908	−	0.840437i	$-0.317702\pi$
$457$	23.1694	1.08382	0.541908	+	0.840437i	$0.317702\pi$
$458$	0	0
$459$	56.2865	2.62723
$460$	0	0
$461$	24.0832	1.12167	0.560833	−	0.827929i	$-0.310481\pi$
$461$	24.0832	1.12167	0.560833	+	0.827929i	$0.310481\pi$
$462$	0	0
$463$	−5.04661	−0.234536	−0.117268	−	0.993100i	$-0.537414\pi$
$463$	−5.04661	−0.234536	−0.117268	+	0.993100i	$0.537414\pi$
$464$	0	0
$465$	−14.7108	−0.682195
$466$	0	0
$467$	15.4324	0.714128	0.357064	−	0.934080i	$-0.383778\pi$
$467$	15.4324	0.714128	0.357064	+	0.934080i	$0.383778\pi$
$468$	0	0
$469$	30.0304	1.38667
$470$	0	0
$471$	32.3170	1.48909
$472$	0	0
$473$	−0.300431	−0.0138138
$474$	0	0
$475$	3.18173	0.145988
$476$	0	0
$477$	59.9285	2.74394
$478$	0	0
$479$	−3.51010	−0.160381	−0.0801903	−	0.996780i	$-0.525553\pi$
$479$	−3.51010	−0.160381	−0.0801903	+	0.996780i	$0.525553\pi$
$480$	0	0
$481$	3.62845	0.165443
$482$	0	0
$483$	−47.0257	−2.13974
$484$	0	0
$485$	−3.99787	−0.181534
$486$	0	0
$487$	−25.8708	−1.17232	−0.586158	−	0.810196i	$-0.699360\pi$
$487$	−25.8708	−1.17232	−0.586158	+	0.810196i	$0.699360\pi$
$488$	0	0
$489$	−39.1896	−1.77221
$490$	0	0
$491$	20.4715	0.923868	0.461934	−	0.886914i	$-0.347156\pi$
$491$	20.4715	0.923868	0.461934	+	0.886914i	$0.347156\pi$
$492$	0	0
$493$	31.9948	1.44098
$494$	0	0
$495$	−49.4938	−2.22458
$496$	0	0
$497$	22.4677	1.00781
$498$	0	0
$499$	31.6343	1.41615	0.708073	−	0.706139i	$-0.249565\pi$
$499$	31.6343	1.41615	0.708073	+	0.706139i	$0.249565\pi$
$500$	0	0
$501$	54.9354	2.45433
$502$	0	0
$503$	7.48882	0.333910	0.166955	−	0.985965i	$-0.446607\pi$
$503$	7.48882	0.333910	0.166955	+	0.985965i	$0.446607\pi$
$504$	0	0
$505$	−0.327244	−0.0145622
$506$	0	0
$507$	−39.3915	−1.74944
$508$	0	0
$509$	3.71394	0.164618	0.0823088	−	0.996607i	$-0.473771\pi$
$509$	3.71394	0.164618	0.0823088	+	0.996607i	$0.473771\pi$
$510$	0	0
$511$	0.234953	0.0103937
$512$	0	0
$513$	−17.4791	−0.771722
$514$	0	0
$515$	−14.4962	−0.638780
$516$	0	0
$517$	−49.2290	−2.16509
$518$	0	0
$519$	−41.7832	−1.83408
$520$	0	0
$521$	−17.6405	−0.772846	−0.386423	−	0.922322i	$-0.626289\pi$
$521$	−17.6405	−0.772846	−0.386423	+	0.922322i	$0.626289\pi$
$522$	0	0
$523$	11.7757	0.514915	0.257458	−	0.966290i	$-0.417115\pi$
$523$	11.7757	0.514915	0.257458	+	0.966290i	$0.417115\pi$
$524$	0	0
$525$	14.5472	0.634890
$526$	0	0
$527$	−12.9052	−0.562160
$528$	0	0
$529$	29.0214	1.26180
$530$	0	0
$531$	55.1383	2.39280
$532$	0	0
$533$	3.30466	0.143141
$534$	0	0
$535$	8.06130	0.348520
$536$	0	0
$537$	−53.1815	−2.29495
$538$	0	0
$539$	11.6646	0.502432
$540$	0	0
$541$	17.0425	0.732715	0.366357	−	0.930474i	$-0.380605\pi$
$541$	17.0425	0.732715	0.366357	+	0.930474i	$0.380605\pi$
$542$	0	0
$543$	−28.1598	−1.20845
$544$	0	0
$545$	33.8603	1.45042
$546$	0	0
$547$	20.9090	0.894004	0.447002	−	0.894533i	$-0.352492\pi$
$547$	20.9090	0.894004	0.447002	+	0.894533i	$0.352492\pi$
$548$	0	0
$549$	−54.1264	−2.31006
$550$	0	0
$551$	−9.93563	−0.423272
$552$	0	0
$553$	−12.8084	−0.544666
$554$	0	0
$555$	27.4655	1.16585
$556$	0	0
$557$	−14.3266	−0.607037	−0.303518	−	0.952826i	$-0.598161\pi$
$557$	−14.3266	−0.607037	−0.303518	+	0.952826i	$0.598161\pi$
$558$	0	0
$559$	0.0481699	0.00203737
$560$	0	0
$561$	−62.3070	−2.63060
$562$	0	0
$563$	2.48884	0.104892	0.0524460	−	0.998624i	$-0.483298\pi$
$563$	2.48884	0.104892	0.0524460	+	0.998624i	$0.483298\pi$
$564$	0	0
$565$	−26.8742	−1.13061
$566$	0	0
$567$	−37.0370	−1.55541
$568$	0	0
$569$	−5.07011	−0.212550	−0.106275	−	0.994337i	$-0.533892\pi$
$569$	−5.07011	−0.212550	−0.106275	+	0.994337i	$0.533892\pi$
$570$	0	0
$571$	35.7515	1.49615	0.748077	−	0.663612i	$-0.230977\pi$
$571$	35.7515	1.49615	0.748077	+	0.663612i	$0.230977\pi$
$572$	0	0
$573$	−21.6226	−0.903298
$574$	0	0
$575$	−16.0925	−0.671106
$576$	0	0
$577$	12.8559	0.535197	0.267599	−	0.963530i	$-0.413770\pi$
$577$	12.8559	0.535197	0.267599	+	0.963530i	$0.413770\pi$
$578$	0	0
$579$	33.8777	1.40791
$580$	0	0
$581$	−23.1630	−0.960962
$582$	0	0
$583$	−37.4803	−1.55227
$584$	0	0
$585$	7.93564	0.328098
$586$	0	0
$587$	−30.4149	−1.25536	−0.627679	−	0.778472i	$-0.715995\pi$
$587$	−30.4149	−1.25536	−0.627679	+	0.778472i	$0.715995\pi$
$588$	0	0
$589$	4.00757	0.165129
$590$	0	0
$591$	−48.2482	−1.98467
$592$	0	0
$593$	28.6314	1.17575	0.587876	−	0.808951i	$-0.299964\pi$
$593$	28.6314	1.17575	0.587876	+	0.808951i	$0.299964\pi$
$594$	0	0
$595$	−15.8369	−0.649248
$596$	0	0
$597$	18.7729	0.768322
$598$	0	0
$599$	11.0037	0.449600	0.224800	−	0.974405i	$-0.427827\pi$
$599$	11.0037	0.449600	0.224800	+	0.974405i	$0.427827\pi$
$600$	0	0
$601$	0.423259	0.0172651	0.00863255	−	0.999963i	$-0.497252\pi$
$601$	0.423259	0.0172651	0.00863255	+	0.999963i	$0.497252\pi$
$602$	0	0
$603$	−99.9240	−4.06922
$604$	0	0
$605$	12.6504	0.514314
$606$	0	0
$607$	40.7175	1.65267	0.826336	−	0.563177i	$-0.190421\pi$
$607$	40.7175	1.65267	0.826336	+	0.563177i	$0.190421\pi$
$608$	0	0
$609$	−45.4266	−1.84078
$610$	0	0
$611$	7.89319	0.319324
$612$	0	0
$613$	4.87460	0.196883	0.0984417	−	0.995143i	$-0.468614\pi$
$613$	4.87460	0.196883	0.0984417	+	0.995143i	$0.468614\pi$
$614$	0	0
$615$	25.0146	1.00868
$616$	0	0
$617$	−7.06448	−0.284405	−0.142203	−	0.989838i	$-0.545418\pi$
$617$	−7.06448	−0.284405	−0.142203	+	0.989838i	$0.545418\pi$
$618$	0	0
$619$	10.4543	0.420195	0.210098	−	0.977680i	$-0.432622\pi$
$619$	10.4543	0.420195	0.210098	+	0.977680i	$0.432622\pi$
$620$	0	0
$621$	88.4058	3.54760
$622$	0	0
$623$	4.88041	0.195529
$624$	0	0
$625$	−8.86597	−0.354639
$626$	0	0
$627$	19.3487	0.772714
$628$	0	0
$629$	24.0945	0.960711
$630$	0	0
$631$	8.50745	0.338676	0.169338	−	0.985558i	$-0.445837\pi$
$631$	8.50745	0.338676	0.169338	+	0.985558i	$0.445837\pi$
$632$	0	0
$633$	0.982375	0.0390459
$634$	0	0
$635$	35.7755	1.41971
$636$	0	0
$637$	−1.87026	−0.0741026
$638$	0	0
$639$	−74.7598	−2.95745
$640$	0	0
$641$	17.1989	0.679315	0.339657	−	0.940549i	$-0.389689\pi$
$641$	17.1989	0.679315	0.339657	+	0.940549i	$0.389689\pi$
$642$	0	0
$643$	30.4905	1.20243	0.601215	−	0.799088i	$-0.294684\pi$
$643$	30.4905	1.20243	0.601215	+	0.799088i	$0.294684\pi$
$644$	0	0
$645$	0.364622	0.0143570
$646$	0	0
$647$	23.5857	0.927250	0.463625	−	0.886031i	$-0.346548\pi$
$647$	23.5857	0.927250	0.463625	+	0.886031i	$0.346548\pi$
$648$	0	0
$649$	−34.4844	−1.35363
$650$	0	0
$651$	18.3229	0.718132
$652$	0	0
$653$	11.0925	0.434081	0.217041	−	0.976163i	$-0.430360\pi$
$653$	11.0925	0.434081	0.217041	+	0.976163i	$0.430360\pi$
$654$	0	0
$655$	17.3443	0.677699
$656$	0	0
$657$	−0.781789	−0.0305005
$658$	0	0
$659$	−31.0542	−1.20970	−0.604850	−	0.796339i	$-0.706767\pi$
$659$	−31.0542	−1.20970	−0.604850	+	0.796339i	$0.706767\pi$
$660$	0	0
$661$	7.18698	0.279541	0.139771	−	0.990184i	$-0.455364\pi$
$661$	7.18698	0.279541	0.139771	+	0.990184i	$0.455364\pi$
$662$	0	0
$663$	9.99006	0.387982
$664$	0	0
$665$	4.91796	0.190710
$666$	0	0
$667$	50.2524	1.94578
$668$	0	0
$669$	31.2262	1.20727
$670$	0	0
$671$	33.8515	1.30682
$672$	0	0
$673$	−29.4721	−1.13607	−0.568034	−	0.823005i	$-0.692296\pi$
$673$	−29.4721	−1.13607	−0.568034	+	0.823005i	$0.692296\pi$
$674$	0	0
$675$	−27.3479	−1.05262
$676$	0	0
$677$	−46.4622	−1.78569	−0.892843	−	0.450368i	$-0.851293\pi$
$677$	−46.4622	−1.78569	−0.892843	+	0.450368i	$0.851293\pi$
$678$	0	0
$679$	4.97953	0.191097
$680$	0	0
$681$	68.8711	2.63915
$682$	0	0
$683$	9.60289	0.367444	0.183722	−	0.982978i	$-0.441185\pi$
$683$	9.60289	0.367444	0.183722	+	0.982978i	$0.441185\pi$
$684$	0	0
$685$	−28.2887	−1.08086
$686$	0	0
$687$	−11.7810	−0.449473
$688$	0	0
$689$	6.00944	0.228941
$690$	0	0
$691$	−8.92238	−0.339423	−0.169712	−	0.985494i	$-0.554284\pi$
$691$	−8.92238	−0.339423	−0.169712	+	0.985494i	$0.554284\pi$
$692$	0	0
$693$	61.6467	2.34177
$694$	0	0
$695$	−26.6012	−1.00904
$696$	0	0
$697$	21.9444	0.831202
$698$	0	0
$699$	71.4095	2.70095
$700$	0	0
$701$	−3.79254	−0.143242	−0.0716212	−	0.997432i	$-0.522817\pi$
$701$	−3.79254	−0.143242	−0.0716212	+	0.997432i	$0.522817\pi$
$702$	0	0
$703$	−7.48228	−0.282199
$704$	0	0
$705$	59.7475	2.25022
$706$	0	0
$707$	0.407597	0.0153293
$708$	0	0
$709$	−13.0451	−0.489920	−0.244960	−	0.969533i	$-0.578775\pi$
$709$	−13.0451	−0.489920	−0.244960	+	0.969533i	$0.578775\pi$
$710$	0	0
$711$	42.6189	1.59833
$712$	0	0
$713$	−20.2694	−0.759097
$714$	0	0
$715$	−4.96308	−0.185609
$716$	0	0
$717$	−13.6075	−0.508182
$718$	0	0
$719$	49.9742	1.86372	0.931861	−	0.362814i	$-0.118184\pi$
$719$	49.9742	1.86372	0.931861	+	0.362814i	$0.118184\pi$
$720$	0	0
$721$	18.0557	0.672430
$722$	0	0
$723$	−40.7313	−1.51482
$724$	0	0
$725$	−15.5453	−0.577339
$726$	0	0
$727$	−44.0322	−1.63306	−0.816532	−	0.577301i	$-0.804106\pi$
$727$	−44.0322	−1.63306	−0.816532	+	0.577301i	$0.804106\pi$
$728$	0	0
$729$	7.56087	0.280032
$730$	0	0
$731$	0.319869	0.0118308
$732$	0	0
$733$	26.0442	0.961963	0.480981	−	0.876731i	$-0.340280\pi$
$733$	26.0442	0.961963	0.480981	+	0.876731i	$0.340280\pi$
$734$	0	0
$735$	−14.1569	−0.522187
$736$	0	0
$737$	62.4941	2.30200
$738$	0	0
$739$	−48.8930	−1.79856	−0.899278	−	0.437377i	$-0.855908\pi$
$739$	−48.8930	−1.79856	−0.899278	+	0.437377i	$0.855908\pi$
$740$	0	0
$741$	−3.10230	−0.113966
$742$	0	0
$743$	−12.1636	−0.446241	−0.223120	−	0.974791i	$-0.571624\pi$
$743$	−12.1636	−0.446241	−0.223120	+	0.974791i	$0.571624\pi$
$744$	0	0
$745$	−23.7084	−0.868610
$746$	0	0
$747$	77.0733	2.81996
$748$	0	0
$749$	−10.0407	−0.366880
$750$	0	0
$751$	24.7050	0.901500	0.450750	−	0.892650i	$-0.351157\pi$
$751$	24.7050	0.901500	0.450750	+	0.892650i	$0.351157\pi$
$752$	0	0
$753$	−3.14584	−0.114641
$754$	0	0
$755$	−22.4232	−0.816063
$756$	0	0
$757$	−11.9146	−0.433042	−0.216521	−	0.976278i	$-0.569471\pi$
$757$	−11.9146	−0.433042	−0.216521	+	0.976278i	$0.569471\pi$
$758$	0	0
$759$	−97.8619	−3.55216
$760$	0	0
$761$	−7.04012	−0.255204	−0.127602	−	0.991825i	$-0.540728\pi$
$761$	−7.04012	−0.255204	−0.127602	+	0.991825i	$0.540728\pi$
$762$	0	0
$763$	−42.1746	−1.52682
$764$	0	0
$765$	52.6961	1.90523
$766$	0	0
$767$	5.52910	0.199644
$768$	0	0
$769$	12.7885	0.461164	0.230582	−	0.973053i	$-0.425937\pi$
$769$	12.7885	0.461164	0.230582	+	0.973053i	$0.425937\pi$
$770$	0	0
$771$	−1.23975	−0.0446486
$772$	0	0
$773$	43.1745	1.55288	0.776439	−	0.630192i	$-0.217024\pi$
$773$	43.1745	1.55288	0.776439	+	0.630192i	$0.217024\pi$
$774$	0	0
$775$	6.27025	0.225234
$776$	0	0
$777$	−34.2096	−1.22726
$778$	0	0
$779$	−6.81457	−0.244157
$780$	0	0
$781$	46.7560	1.67306
$782$	0	0
$783$	85.3996	3.05193
$784$	0	0
$785$	17.0939	0.610109
$786$	0	0
$787$	−22.3365	−0.796212	−0.398106	−	0.917339i	$-0.630332\pi$
$787$	−22.3365	−0.796212	−0.398106	+	0.917339i	$0.630332\pi$
$788$	0	0
$789$	−4.81679	−0.171482
$790$	0	0
$791$	33.4731	1.19016
$792$	0	0
$793$	−5.42763	−0.192741
$794$	0	0
$795$	45.4884	1.61331
$796$	0	0
$797$	39.2565	1.39054	0.695269	−	0.718750i	$-0.255285\pi$
$797$	39.2565	1.39054	0.695269	+	0.718750i	$0.255285\pi$
$798$	0	0
$799$	52.4142	1.85428
$800$	0	0
$801$	−16.2392	−0.573785
$802$	0	0
$803$	0.488944	0.0172545
$804$	0	0
$805$	−24.8740	−0.876694
$806$	0	0
$807$	−11.5890	−0.407952
$808$	0	0
$809$	−4.53751	−0.159530	−0.0797651	−	0.996814i	$-0.525417\pi$
$809$	−4.53751	−0.159530	−0.0797651	+	0.996814i	$0.525417\pi$
$810$	0	0
$811$	−12.1635	−0.427118	−0.213559	−	0.976930i	$-0.568506\pi$
$811$	−12.1635	−0.427118	−0.213559	+	0.976930i	$0.568506\pi$
$812$	0	0
$813$	−34.1739	−1.19853
$814$	0	0
$815$	−20.7291	−0.726110
$816$	0	0
$817$	−0.0993317	−0.00347518
$818$	0	0
$819$	−9.88420	−0.345382
$820$	0	0
$821$	−12.4541	−0.434650	−0.217325	−	0.976099i	$-0.569733\pi$
$821$	−12.4541	−0.434650	−0.217325	+	0.976099i	$0.569733\pi$
$822$	0	0
$823$	21.5162	0.750009	0.375005	−	0.927023i	$-0.377641\pi$
$823$	21.5162	0.750009	0.375005	+	0.927023i	$0.377641\pi$
$824$	0	0
$825$	30.2731	1.05397
$826$	0	0
$827$	−21.6544	−0.752997	−0.376499	−	0.926417i	$-0.622872\pi$
$827$	−21.6544	−0.752997	−0.376499	+	0.926417i	$0.622872\pi$
$828$	0	0
$829$	23.6207	0.820382	0.410191	−	0.912000i	$-0.365462\pi$
$829$	23.6207	0.820382	0.410191	+	0.912000i	$0.365462\pi$
$830$	0	0
$831$	−12.3492	−0.428389
$832$	0	0
$833$	−12.4194	−0.430306
$834$	0	0
$835$	29.0578	1.00559
$836$	0	0
$837$	−34.4462	−1.19063
$838$	0	0
$839$	−7.82348	−0.270096	−0.135048	−	0.990839i	$-0.543119\pi$
$839$	−7.82348	−0.270096	−0.135048	+	0.990839i	$0.543119\pi$
$840$	0	0
$841$	19.5436	0.673916
$842$	0	0
$843$	−5.70833	−0.196605
$844$	0	0
$845$	−20.8360	−0.716779
$846$	0	0
$847$	−15.7567	−0.541407
$848$	0	0
$849$	−64.2045	−2.20349
$850$	0	0
$851$	37.8438	1.29727
$852$	0	0
$853$	−20.1784	−0.690896	−0.345448	−	0.938438i	$-0.612273\pi$
$853$	−20.1784	−0.690896	−0.345448	+	0.938438i	$0.612273\pi$
$854$	0	0
$855$	−16.3642	−0.559643
$856$	0	0
$857$	−7.59263	−0.259359	−0.129680	−	0.991556i	$-0.541395\pi$
$857$	−7.59263	−0.259359	−0.129680	+	0.991556i	$0.541395\pi$
$858$	0	0
$859$	26.8210	0.915122	0.457561	−	0.889178i	$-0.348723\pi$
$859$	26.8210	0.915122	0.457561	+	0.889178i	$0.348723\pi$
$860$	0	0
$861$	−31.1568	−1.06182
$862$	0	0
$863$	2.72517	0.0927660	0.0463830	−	0.998924i	$-0.485231\pi$
$863$	2.72517	0.0927660	0.0463830	+	0.998924i	$0.485231\pi$
$864$	0	0
$865$	−22.1010	−0.751458
$866$	0	0
$867$	12.8591	0.436718
$868$	0	0
$869$	−26.6546	−0.904195
$870$	0	0
$871$	−10.0201	−0.339517
$872$	0	0
$873$	−16.5691	−0.560778
$874$	0	0
$875$	24.9381	0.843063
$876$	0	0
$877$	44.9386	1.51747	0.758735	−	0.651400i	$-0.225818\pi$
$877$	44.9386	1.51747	0.758735	+	0.651400i	$0.225818\pi$
$878$	0	0
$879$	−60.6185	−2.04461
$880$	0	0
$881$	−56.3147	−1.89729	−0.948645	−	0.316344i	$-0.897545\pi$
$881$	−56.3147	−1.89729	−0.948645	+	0.316344i	$0.897545\pi$
$882$	0	0
$883$	46.7714	1.57398	0.786992	−	0.616963i	$-0.211637\pi$
$883$	46.7714	1.57398	0.786992	+	0.616963i	$0.211637\pi$
$884$	0	0
$885$	41.8525	1.40686
$886$	0	0
$887$	43.3251	1.45472	0.727358	−	0.686259i	$-0.240748\pi$
$887$	43.3251	1.45472	0.727358	+	0.686259i	$0.240748\pi$
$888$	0	0
$889$	−44.5601	−1.49450
$890$	0	0
$891$	−77.0751	−2.58211
$892$	0	0
$893$	−16.2766	−0.544677
$894$	0	0
$895$	−28.1301	−0.940287
$896$	0	0
$897$	15.6908	0.523901
$898$	0	0
$899$	−19.5802	−0.653036
$900$	0	0
$901$	39.9053	1.32944
$902$	0	0
$903$	−0.454153	−0.0151133
$904$	0	0
$905$	−14.8950	−0.495126
$906$	0	0
$907$	−34.0973	−1.13218	−0.566091	−	0.824343i	$-0.691545\pi$
$907$	−34.0973	−1.13218	−0.566091	+	0.824343i	$0.691545\pi$
$908$	0	0
$909$	−1.35625	−0.0449841
$910$	0	0
$911$	9.81857	0.325304	0.162652	−	0.986684i	$-0.447995\pi$
$911$	9.81857	0.325304	0.162652	+	0.986684i	$0.447995\pi$
$912$	0	0
$913$	−48.2029	−1.59528
$914$	0	0
$915$	−41.0844	−1.35821
$916$	0	0
$917$	−21.6032	−0.713399
$918$	0	0
$919$	−38.2375	−1.26134	−0.630670	−	0.776051i	$-0.717220\pi$
$919$	−38.2375	−1.26134	−0.630670	+	0.776051i	$0.717220\pi$
$920$	0	0
$921$	−62.6952	−2.06588
$922$	0	0
$923$	−7.49668	−0.246756
$924$	0	0
$925$	−11.7068	−0.384917
$926$	0	0
$927$	−60.0792	−1.97326
$928$	0	0
$929$	−8.67986	−0.284777	−0.142388	−	0.989811i	$-0.545478\pi$
$929$	−8.67986	−0.284777	−0.142388	+	0.989811i	$0.545478\pi$
$930$	0	0
$931$	3.85669	0.126398
$932$	0	0
$933$	−91.4873	−2.99516
$934$	0	0
$935$	−32.9570	−1.07781
$936$	0	0
$937$	25.5923	0.836063	0.418032	−	0.908432i	$-0.362720\pi$
$937$	25.5923	0.836063	0.418032	+	0.908432i	$0.362720\pi$
$938$	0	0
$939$	−80.1711	−2.61628
$940$	0	0
$941$	−12.9227	−0.421267	−0.210633	−	0.977565i	$-0.567553\pi$
$941$	−12.9227	−0.421267	−0.210633	+	0.977565i	$0.567553\pi$
$942$	0	0
$943$	34.4667	1.12239
$944$	0	0
$945$	−42.2713	−1.37508
$946$	0	0
$947$	−51.5358	−1.67469	−0.837345	−	0.546675i	$-0.815893\pi$
$947$	−51.5358	−1.67469	−0.837345	+	0.546675i	$0.815893\pi$
$948$	0	0
$949$	−0.0783954	−0.00254482
$950$	0	0
$951$	72.0719	2.33709
$952$	0	0
$953$	−20.2468	−0.655859	−0.327929	−	0.944702i	$-0.606351\pi$
$953$	−20.2468	−0.655859	−0.327929	+	0.944702i	$0.606351\pi$
$954$	0	0
$955$	−11.4372	−0.370099
$956$	0	0
$957$	−94.5341	−3.05585
$958$	0	0
$959$	35.2349	1.13779
$960$	0	0
$961$	−23.1023	−0.745235
$962$	0	0
$963$	33.4098	1.07662
$964$	0	0
$965$	17.9195	0.576848
$966$	0	0
$967$	−16.6869	−0.536616	−0.268308	−	0.963333i	$-0.586465\pi$
$967$	−16.6869	−0.536616	−0.268308	+	0.963333i	$0.586465\pi$
$968$	0	0
$969$	−20.6006	−0.661787
$970$	0	0
$971$	−37.5064	−1.20364	−0.601819	−	0.798633i	$-0.705557\pi$
$971$	−37.5064	−1.20364	−0.601819	+	0.798633i	$0.705557\pi$
$972$	0	0
$973$	33.1331	1.06220
$974$	0	0
$975$	−4.85387	−0.155448
$976$	0	0
$977$	−8.01221	−0.256333	−0.128167	−	0.991753i	$-0.540909\pi$
$977$	−8.01221	−0.256333	−0.128167	+	0.991753i	$0.540909\pi$
$978$	0	0
$979$	10.1563	0.324596
$980$	0	0
$981$	140.333	4.48049
$982$	0	0
$983$	−27.5705	−0.879362	−0.439681	−	0.898154i	$-0.644909\pi$
$983$	−27.5705	−0.879362	−0.439681	+	0.898154i	$0.644909\pi$
$984$	0	0
$985$	−25.5207	−0.813156
$986$	0	0
$987$	−74.4182	−2.36876
$988$	0	0
$989$	0.502399	0.0159754
$990$	0	0
$991$	−11.0011	−0.349463	−0.174731	−	0.984616i	$-0.555906\pi$
$991$	−11.0011	−0.349463	−0.174731	+	0.984616i	$0.555906\pi$
$992$	0	0
$993$	−77.7361	−2.46688
$994$	0	0
$995$	9.92982	0.314796
$996$	0	0
$997$	−29.1077	−0.921849	−0.460925	−	0.887439i	$-0.652482\pi$
$997$	−29.1077	−0.921849	−0.460925	+	0.887439i	$0.652482\pi$
$998$	0	0
$999$	64.3123	2.03475

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.k.1.17		17
4.3	odd	2		251.2.a.b.1.10	✓	17
12.11	even	2		2259.2.a.k.1.8		17
20.19	odd	2		6275.2.a.e.1.8		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
251.2.a.b.1.10	✓	17	4.3	odd	2
2259.2.a.k.1.8		17	12.11	even	2
4016.2.a.k.1.17		17	1.1	even	1	trivial
6275.2.a.e.1.8		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)

\(f(q)\)	\(=\)	\(q+3.14584 q^{3} +1.66398 q^{5} -2.07256 q^{7} +6.89631 q^{9} +O(q^{10})\)	\(q+3.14584 q^{3} +1.66398 q^{5} -2.07256 q^{7} +6.89631 q^{9} -4.31307 q^{11} +0.691540 q^{13} +5.23461 q^{15} +4.59213 q^{17} -1.42603 q^{19} -6.51995 q^{21} +7.21258 q^{23} -2.23118 q^{25} +12.2572 q^{27} +6.96732 q^{29} -2.81029 q^{31} -13.5682 q^{33} -3.44870 q^{35} +5.24691 q^{37} +2.17548 q^{39} +4.77869 q^{41} +0.0696559 q^{43} +11.4753 q^{45} +11.4139 q^{47} -2.70449 q^{49} +14.4461 q^{51} +8.68993 q^{53} -7.17685 q^{55} -4.48607 q^{57} +7.99534 q^{59} -7.84861 q^{61} -14.2930 q^{63} +1.15071 q^{65} -14.4895 q^{67} +22.6896 q^{69} -10.8406 q^{71} -0.113363 q^{73} -7.01892 q^{75} +8.93909 q^{77} +6.17996 q^{79} +17.8702 q^{81} +11.1760 q^{83} +7.64120 q^{85} +21.9181 q^{87} -2.35477 q^{89} -1.43326 q^{91} -8.84072 q^{93} -2.37289 q^{95} -2.40260 q^{97} -29.7442 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) \) 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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