LMFDB - Embedded newform 4016.2.a.k.1.7

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			1.7
Root			$-2.27410$ 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)

$f(q)$	$=$	$q-0.935470 q^{3} +3.41593 q^{5} -3.69332 q^{7} -2.12490 q^{9} +O(q^{10})$	$q-0.935470 q^{3} +3.41593 q^{5} -3.69332 q^{7} -2.12490 q^{9} +1.09538 q^{11} +0.0975293 q^{13} -3.19550 q^{15} -4.03572 q^{17} -5.04841 q^{19} +3.45499 q^{21} +0.592256 q^{23} +6.66859 q^{25} +4.79419 q^{27} -3.45917 q^{29} +0.372983 q^{31} -1.02469 q^{33} -12.6161 q^{35} +6.18783 q^{37} -0.0912357 q^{39} +11.3439 q^{41} +5.37250 q^{43} -7.25850 q^{45} +10.3815 q^{47} +6.64059 q^{49} +3.77530 q^{51} -11.2337 q^{53} +3.74174 q^{55} +4.72264 q^{57} +7.25509 q^{59} +0.601444 q^{61} +7.84791 q^{63} +0.333153 q^{65} -11.6371 q^{67} -0.554038 q^{69} +0.624713 q^{71} +9.71599 q^{73} -6.23827 q^{75} -4.04558 q^{77} +6.82487 q^{79} +1.88987 q^{81} +15.5486 q^{83} -13.7858 q^{85} +3.23595 q^{87} +15.9119 q^{89} -0.360206 q^{91} -0.348914 q^{93} -17.2450 q^{95} -7.70534 q^{97} -2.32757 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.935470	−0.540094	−0.270047	−	0.962847i	$-0.587039\pi$
$3$	−0.935470	−0.540094	−0.270047	+	0.962847i	$0.587039\pi$
$4$	0	0
$5$	3.41593	1.52765	0.763826	−	0.645423i	$-0.223319\pi$
$5$	3.41593	1.52765	0.763826	+	0.645423i	$0.223319\pi$
$6$	0	0
$7$	−3.69332	−1.39594	−0.697971	−	0.716126i	$-0.745914\pi$
$7$	−3.69332	−1.39594	−0.697971	+	0.716126i	$0.745914\pi$
$8$	0	0
$9$	−2.12490	−0.708299
$10$	0	0
$11$	1.09538	0.330269	0.165135	−	0.986271i	$-0.447194\pi$
$11$	1.09538	0.330269	0.165135	+	0.986271i	$0.447194\pi$
$12$	0	0
$13$	0.0975293	0.0270498	0.0135249	−	0.999909i	$-0.495695\pi$
$13$	0.0975293	0.0270498	0.0135249	+	0.999909i	$0.495695\pi$
$14$	0	0
$15$	−3.19550	−0.825075
$16$	0	0
$17$	−4.03572	−0.978807	−0.489403	−	0.872057i	$-0.662785\pi$
$17$	−4.03572	−0.978807	−0.489403	+	0.872057i	$0.662785\pi$
$18$	0	0
$19$	−5.04841	−1.15819	−0.579093	−	0.815262i	$-0.696593\pi$
$19$	−5.04841	−1.15819	−0.579093	+	0.815262i	$0.696593\pi$
$20$	0	0
$21$	3.45499	0.753940
$22$	0	0
$23$	0.592256	0.123494	0.0617469	−	0.998092i	$-0.480333\pi$
$23$	0.592256	0.123494	0.0617469	+	0.998092i	$0.480333\pi$
$24$	0	0
$25$	6.66859	1.33372
$26$	0	0
$27$	4.79419	0.922641
$28$	0	0
$29$	−3.45917	−0.642352	−0.321176	−	0.947020i	$-0.604078\pi$
$29$	−3.45917	−0.642352	−0.321176	+	0.947020i	$0.604078\pi$
$30$	0	0
$31$	0.372983	0.0669897	0.0334948	−	0.999439i	$-0.489336\pi$
$31$	0.372983	0.0669897	0.0334948	+	0.999439i	$0.489336\pi$
$32$	0	0
$33$	−1.02469	−0.178376
$34$	0	0
$35$	−12.6161	−2.13251
$36$	0	0
$37$	6.18783	1.01727	0.508636	−	0.860981i	$-0.330150\pi$
$37$	6.18783	1.01727	0.508636	+	0.860981i	$0.330150\pi$
$38$	0	0
$39$	−0.0912357	−0.0146094
$40$	0	0
$41$	11.3439	1.77161	0.885807	−	0.464054i	$-0.153605\pi$
$41$	11.3439	1.77161	0.885807	+	0.464054i	$0.153605\pi$
$42$	0	0
$43$	5.37250	0.819298	0.409649	−	0.912243i	$-0.365651\pi$
$43$	5.37250	0.819298	0.409649	+	0.912243i	$0.365651\pi$
$44$	0	0
$45$	−7.25850	−1.08203
$46$	0	0
$47$	10.3815	1.51430	0.757152	−	0.653239i	$-0.226590\pi$
$47$	10.3815	1.51430	0.757152	+	0.653239i	$0.226590\pi$
$48$	0	0
$49$	6.64059	0.948655
$50$	0	0
$51$	3.77530	0.528647
$52$	0	0
$53$	−11.2337	−1.54306	−0.771531	−	0.636192i	$-0.780508\pi$
$53$	−11.2337	−1.54306	−0.771531	+	0.636192i	$0.780508\pi$
$54$	0	0
$55$	3.74174	0.504537
$56$	0	0
$57$	4.72264	0.625529
$58$	0	0
$59$	7.25509	0.944533	0.472266	−	0.881456i	$-0.343436\pi$
$59$	7.25509	0.944533	0.472266	+	0.881456i	$0.343436\pi$
$60$	0	0
$61$	0.601444	0.0770070	0.0385035	−	0.999258i	$-0.487741\pi$
$61$	0.601444	0.0770070	0.0385035	+	0.999258i	$0.487741\pi$
$62$	0	0
$63$	7.84791	0.988744
$64$	0	0
$65$	0.333153	0.0413226
$66$	0	0
$67$	−11.6371	−1.42170	−0.710848	−	0.703346i	$-0.751689\pi$
$67$	−11.6371	−1.42170	−0.710848	+	0.703346i	$0.751689\pi$
$68$	0	0
$69$	−0.554038	−0.0666983
$70$	0	0
$71$	0.624713	0.0741397	0.0370699	−	0.999313i	$-0.488198\pi$
$71$	0.624713	0.0741397	0.0370699	+	0.999313i	$0.488198\pi$
$72$	0	0
$73$	9.71599	1.13717	0.568585	−	0.822624i	$-0.307491\pi$
$73$	9.71599	1.13717	0.568585	+	0.822624i	$0.307491\pi$
$74$	0	0
$75$	−6.23827	−0.720333
$76$	0	0
$77$	−4.04558	−0.461037
$78$	0	0
$79$	6.82487	0.767858	0.383929	−	0.923363i	$-0.374571\pi$
$79$	6.82487	0.767858	0.383929	+	0.923363i	$0.374571\pi$
$80$	0	0
$81$	1.88987	0.209986
$82$	0	0
$83$	15.5486	1.70668	0.853341	−	0.521353i	$-0.174572\pi$
$83$	15.5486	1.70668	0.853341	+	0.521353i	$0.174572\pi$
$84$	0	0
$85$	−13.7858	−1.49528
$86$	0	0
$87$	3.23595	0.346930
$88$	0	0
$89$	15.9119	1.68665	0.843327	−	0.537401i	$-0.180594\pi$
$89$	15.9119	1.68665	0.843327	+	0.537401i	$0.180594\pi$
$90$	0	0
$91$	−0.360206	−0.0377599
$92$	0	0
$93$	−0.348914	−0.0361807
$94$	0	0
$95$	−17.2450	−1.76930
$96$	0	0
$97$	−7.70534	−0.782359	−0.391179	−	0.920314i	$-0.627933\pi$
$97$	−7.70534	−0.782359	−0.391179	+	0.920314i	$0.627933\pi$
$98$	0	0
$99$	−2.32757	−0.233929
$100$	0	0
$101$	18.0872	1.79975	0.899873	−	0.436153i	$-0.143659\pi$
$101$	18.0872	1.79975	0.899873	+	0.436153i	$0.143659\pi$
$102$	0	0
$103$	7.78184	0.766767	0.383384	−	0.923589i	$-0.374759\pi$
$103$	7.78184	0.766767	0.383384	+	0.923589i	$0.374759\pi$
$104$	0	0
$105$	11.8020	1.15176
$106$	0	0
$107$	2.56314	0.247788	0.123894	−	0.992295i	$-0.460462\pi$
$107$	2.56314	0.247788	0.123894	+	0.992295i	$0.460462\pi$
$108$	0	0
$109$	0.411462	0.0394109	0.0197054	−	0.999806i	$-0.493727\pi$
$109$	0.411462	0.0394109	0.0197054	+	0.999806i	$0.493727\pi$
$110$	0	0
$111$	−5.78853	−0.549423
$112$	0	0
$113$	−4.12662	−0.388200	−0.194100	−	0.980982i	$-0.562179\pi$
$113$	−4.12662	−0.388200	−0.194100	+	0.980982i	$0.562179\pi$
$114$	0	0
$115$	2.02311	0.188656
$116$	0	0
$117$	−0.207240	−0.0191593
$118$	0	0
$119$	14.9052	1.36636
$120$	0	0
$121$	−9.80014	−0.890922
$122$	0	0
$123$	−10.6118	−0.956838
$124$	0	0
$125$	5.69980	0.509805
$126$	0	0
$127$	11.8625	1.05263	0.526316	−	0.850289i	$-0.323573\pi$
$127$	11.8625	1.05263	0.526316	+	0.850289i	$0.323573\pi$
$128$	0	0
$129$	−5.02581	−0.442498
$130$	0	0
$131$	−22.4352	−1.96017	−0.980087	−	0.198570i	$-0.936370\pi$
$131$	−22.4352	−1.96017	−0.980087	+	0.198570i	$0.936370\pi$
$132$	0	0
$133$	18.6454	1.61676
$134$	0	0
$135$	16.3766	1.40947
$136$	0	0
$137$	−10.6248	−0.907740	−0.453870	−	0.891068i	$-0.649957\pi$
$137$	−10.6248	−0.907740	−0.453870	+	0.891068i	$0.649957\pi$
$138$	0	0
$139$	17.7360	1.50435	0.752175	−	0.658964i	$-0.229005\pi$
$139$	17.7360	1.50435	0.752175	+	0.658964i	$0.229005\pi$
$140$	0	0
$141$	−9.71162	−0.817866
$142$	0	0
$143$	0.106832	0.00893371
$144$	0	0
$145$	−11.8163	−0.981290
$146$	0	0
$147$	−6.21207	−0.512363
$148$	0	0
$149$	10.3371	0.846844	0.423422	−	0.905933i	$-0.360829\pi$
$149$	10.3371	0.846844	0.423422	+	0.905933i	$0.360829\pi$
$150$	0	0
$151$	−11.0755	−0.901312	−0.450656	−	0.892698i	$-0.648810\pi$
$151$	−11.0755	−0.901312	−0.450656	+	0.892698i	$0.648810\pi$
$152$	0	0
$153$	8.57550	0.693288
$154$	0	0
$155$	1.27408	0.102337
$156$	0	0
$157$	−24.3028	−1.93957	−0.969787	−	0.243955i	$-0.921555\pi$
$157$	−24.3028	−1.93957	−0.969787	+	0.243955i	$0.921555\pi$
$158$	0	0
$159$	10.5087	0.833398
$160$	0	0
$161$	−2.18739	−0.172390
$162$	0	0
$163$	12.6767	0.992913	0.496456	−	0.868062i	$-0.334634\pi$
$163$	12.6767	0.992913	0.496456	+	0.868062i	$0.334634\pi$
$164$	0	0
$165$	−3.50029	−0.272497
$166$	0	0
$167$	−8.35277	−0.646357	−0.323178	−	0.946338i	$-0.604751\pi$
$167$	−8.35277	−0.646357	−0.323178	+	0.946338i	$0.604751\pi$
$168$	0	0
$169$	−12.9905	−0.999268
$170$	0	0
$171$	10.7274	0.820342
$172$	0	0
$173$	12.8831	0.979487	0.489744	−	0.871866i	$-0.337090\pi$
$173$	12.8831	0.979487	0.489744	+	0.871866i	$0.337090\pi$
$174$	0	0
$175$	−24.6292	−1.86179
$176$	0	0
$177$	−6.78692	−0.510136
$178$	0	0
$179$	26.3530	1.96971	0.984856	−	0.173373i	$-0.0554666\pi$
$179$	26.3530	1.96971	0.984856	+	0.173373i	$0.0554666\pi$
$180$	0	0
$181$	−16.9891	−1.26279	−0.631396	−	0.775461i	$-0.717518\pi$
$181$	−16.9891	−1.26279	−0.631396	+	0.775461i	$0.717518\pi$
$182$	0	0
$183$	−0.562633	−0.0415910
$184$	0	0
$185$	21.1372	1.55404
$186$	0	0
$187$	−4.42065	−0.323270
$188$	0	0
$189$	−17.7064	−1.28795
$190$	0	0
$191$	−7.24390	−0.524151	−0.262075	−	0.965047i	$-0.584407\pi$
$191$	−7.24390	−0.524151	−0.262075	+	0.965047i	$0.584407\pi$
$192$	0	0
$193$	2.35410	0.169452	0.0847259	−	0.996404i	$-0.472999\pi$
$193$	2.35410	0.169452	0.0847259	+	0.996404i	$0.472999\pi$
$194$	0	0
$195$	−0.311655	−0.0223181
$196$	0	0
$197$	−12.2704	−0.874233	−0.437117	−	0.899405i	$-0.644000\pi$
$197$	−12.2704	−0.874233	−0.437117	+	0.899405i	$0.644000\pi$
$198$	0	0
$199$	−1.62720	−0.115349	−0.0576745	−	0.998335i	$-0.518369\pi$
$199$	−1.62720	−0.115349	−0.0576745	+	0.998335i	$0.518369\pi$
$200$	0	0
$201$	10.8861	0.767849
$202$	0	0
$203$	12.7758	0.896686
$204$	0	0
$205$	38.7499	2.70641
$206$	0	0
$207$	−1.25848	−0.0874706
$208$	0	0
$209$	−5.52993	−0.382513
$210$	0	0
$211$	−12.7353	−0.876736	−0.438368	−	0.898796i	$-0.644443\pi$
$211$	−12.7353	−0.876736	−0.438368	+	0.898796i	$0.644443\pi$
$212$	0	0
$213$	−0.584400	−0.0400424
$214$	0	0
$215$	18.3521	1.25160
$216$	0	0
$217$	−1.37754	−0.0935137
$218$	0	0
$219$	−9.08901	−0.614179
$220$	0	0
$221$	−0.393601	−0.0264765
$222$	0	0
$223$	−22.4100	−1.50069	−0.750343	−	0.661048i	$-0.770112\pi$
$223$	−22.4100	−1.50069	−0.750343	+	0.661048i	$0.770112\pi$
$224$	0	0
$225$	−14.1701	−0.944671
$226$	0	0
$227$	9.39073	0.623285	0.311642	−	0.950199i	$-0.399121\pi$
$227$	9.39073	0.623285	0.311642	+	0.950199i	$0.399121\pi$
$228$	0	0
$229$	14.8143	0.978957	0.489478	−	0.872015i	$-0.337187\pi$
$229$	14.8143	0.978957	0.489478	+	0.872015i	$0.337187\pi$
$230$	0	0
$231$	3.78452	0.249003
$232$	0	0
$233$	11.7123	0.767297	0.383648	−	0.923479i	$-0.374667\pi$
$233$	11.7123	0.767297	0.383648	+	0.923479i	$0.374667\pi$
$234$	0	0
$235$	35.4627	2.31333
$236$	0	0
$237$	−6.38446	−0.414715
$238$	0	0
$239$	25.8228	1.67034	0.835168	−	0.549995i	$-0.185371\pi$
$239$	25.8228	1.67034	0.835168	+	0.549995i	$0.185371\pi$
$240$	0	0
$241$	5.23762	0.337385	0.168692	−	0.985669i	$-0.446046\pi$
$241$	5.23762	0.337385	0.168692	+	0.985669i	$0.446046\pi$
$242$	0	0
$243$	−16.1505	−1.03605
$244$	0	0
$245$	22.6838	1.44921
$246$	0	0
$247$	−0.492368	−0.0313286
$248$	0	0
$249$	−14.5453	−0.921768
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	0.648745	0.0407863
$254$	0	0
$255$	12.8962	0.807589
$256$	0	0
$257$	25.3189	1.57935	0.789675	−	0.613525i	$-0.210249\pi$
$257$	25.3189	1.57935	0.789675	+	0.613525i	$0.210249\pi$
$258$	0	0
$259$	−22.8536	−1.42005
$260$	0	0
$261$	7.35038	0.454977
$262$	0	0
$263$	10.9977	0.678147	0.339074	−	0.940760i	$-0.389886\pi$
$263$	10.9977	0.678147	0.339074	+	0.940760i	$0.389886\pi$
$264$	0	0
$265$	−38.3734	−2.35726
$266$	0	0
$267$	−14.8851	−0.910951
$268$	0	0
$269$	24.8062	1.51246	0.756230	−	0.654305i	$-0.227039\pi$
$269$	24.8062	1.51246	0.756230	+	0.654305i	$0.227039\pi$
$270$	0	0
$271$	5.48722	0.333325	0.166662	−	0.986014i	$-0.446701\pi$
$271$	5.48722	0.333325	0.166662	+	0.986014i	$0.446701\pi$
$272$	0	0
$273$	0.336962	0.0203939
$274$	0	0
$275$	7.30464	0.440486
$276$	0	0
$277$	5.78730	0.347725	0.173862	−	0.984770i	$-0.444375\pi$
$277$	5.78730	0.347725	0.173862	+	0.984770i	$0.444375\pi$
$278$	0	0
$279$	−0.792550	−0.0474487
$280$	0	0
$281$	12.7109	0.758266	0.379133	−	0.925342i	$-0.376222\pi$
$281$	12.7109	0.758266	0.379133	+	0.925342i	$0.376222\pi$
$282$	0	0
$283$	−8.50578	−0.505616	−0.252808	−	0.967516i	$-0.581354\pi$
$283$	−8.50578	−0.505616	−0.252808	+	0.967516i	$0.581354\pi$
$284$	0	0
$285$	16.1322	0.955590
$286$	0	0
$287$	−41.8965	−2.47307
$288$	0	0
$289$	−0.712929	−0.0419370
$290$	0	0
$291$	7.20811	0.422547
$292$	0	0
$293$	19.2686	1.12569	0.562843	−	0.826564i	$-0.309708\pi$
$293$	19.2686	1.12569	0.562843	+	0.826564i	$0.309708\pi$
$294$	0	0
$295$	24.7829	1.44292
$296$	0	0
$297$	5.25145	0.304720
$298$	0	0
$299$	0.0577623	0.00334048
$300$	0	0
$301$	−19.8423	−1.14369
$302$	0	0
$303$	−16.9200	−0.972031
$304$	0	0
$305$	2.05449	0.117640
$306$	0	0
$307$	0.292654	0.0167026	0.00835132	−	0.999965i	$-0.497342\pi$
$307$	0.292654	0.0167026	0.00835132	+	0.999965i	$0.497342\pi$
$308$	0	0
$309$	−7.27967	−0.414126
$310$	0	0
$311$	−5.10705	−0.289594	−0.144797	−	0.989461i	$-0.546253\pi$
$311$	−5.10705	−0.289594	−0.144797	+	0.989461i	$0.546253\pi$
$312$	0	0
$313$	−7.20479	−0.407239	−0.203619	−	0.979050i	$-0.565271\pi$
$313$	−7.20479	−0.407239	−0.203619	+	0.979050i	$0.565271\pi$
$314$	0	0
$315$	26.8079	1.51046
$316$	0	0
$317$	−10.6486	−0.598086	−0.299043	−	0.954240i	$-0.596667\pi$
$317$	−10.6486	−0.598086	−0.299043	+	0.954240i	$0.596667\pi$
$318$	0	0
$319$	−3.78911	−0.212149
$320$	0	0
$321$	−2.39774	−0.133829
$322$	0	0
$323$	20.3740	1.13364
$324$	0	0
$325$	0.650383	0.0360768
$326$	0	0
$327$	−0.384910	−0.0212856
$328$	0	0
$329$	−38.3423	−2.11388
$330$	0	0
$331$	−16.4433	−0.903804	−0.451902	−	0.892068i	$-0.649254\pi$
$331$	−16.4433	−0.903804	−0.451902	+	0.892068i	$0.649254\pi$
$332$	0	0
$333$	−13.1485	−0.720533
$334$	0	0
$335$	−39.7515	−2.17186
$336$	0	0
$337$	14.0936	0.767728	0.383864	−	0.923390i	$-0.374593\pi$
$337$	14.0936	0.767728	0.383864	+	0.923390i	$0.374593\pi$
$338$	0	0
$339$	3.86033	0.209664
$340$	0	0
$341$	0.408558	0.0221246
$342$	0	0
$343$	1.32743	0.0716743
$344$	0	0
$345$	−1.89255	−0.101892
$346$	0	0
$347$	−10.7871	−0.579084	−0.289542	−	0.957165i	$-0.593503\pi$
$347$	−10.7871	−0.579084	−0.289542	+	0.957165i	$0.593503\pi$
$348$	0	0
$349$	7.97572	0.426930	0.213465	−	0.976951i	$-0.431525\pi$
$349$	7.97572	0.426930	0.213465	+	0.976951i	$0.431525\pi$
$350$	0	0
$351$	0.467573	0.0249572
$352$	0	0
$353$	11.2749	0.600104	0.300052	−	0.953923i	$-0.402996\pi$
$353$	11.2749	0.600104	0.300052	+	0.953923i	$0.402996\pi$
$354$	0	0
$355$	2.13398	0.113260
$356$	0	0
$357$	−13.9434	−0.737961
$358$	0	0
$359$	−13.1558	−0.694339	−0.347169	−	0.937802i	$-0.612857\pi$
$359$	−13.1558	−0.694339	−0.347169	+	0.937802i	$0.612857\pi$
$360$	0	0
$361$	6.48649	0.341394
$362$	0	0
$363$	9.16774	0.481181
$364$	0	0
$365$	33.1892	1.73720
$366$	0	0
$367$	−10.6315	−0.554961	−0.277481	−	0.960731i	$-0.589499\pi$
$367$	−10.6315	−0.554961	−0.277481	+	0.960731i	$0.589499\pi$
$368$	0	0
$369$	−24.1045	−1.25483
$370$	0	0
$371$	41.4894	2.15402
$372$	0	0
$373$	−12.1705	−0.630163	−0.315082	−	0.949065i	$-0.602032\pi$
$373$	−12.1705	−0.630163	−0.315082	+	0.949065i	$0.602032\pi$
$374$	0	0
$375$	−5.33199	−0.275343
$376$	0	0
$377$	−0.337370	−0.0173755
$378$	0	0
$379$	2.32877	0.119621	0.0598103	−	0.998210i	$-0.480950\pi$
$379$	2.32877	0.119621	0.0598103	+	0.998210i	$0.480950\pi$
$380$	0	0
$381$	−11.0971	−0.568519
$382$	0	0
$383$	−0.235563	−0.0120367	−0.00601835	−	0.999982i	$-0.501916\pi$
$383$	−0.235563	−0.0120367	−0.00601835	+	0.999982i	$0.501916\pi$
$384$	0	0
$385$	−13.8194	−0.704304
$386$	0	0
$387$	−11.4160	−0.580308
$388$	0	0
$389$	−22.8102	−1.15652	−0.578261	−	0.815852i	$-0.696268\pi$
$389$	−22.8102	−1.15652	−0.578261	+	0.815852i	$0.696268\pi$
$390$	0	0
$391$	−2.39018	−0.120877
$392$	0	0
$393$	20.9875	1.05868
$394$	0	0
$395$	23.3133	1.17302
$396$	0	0
$397$	23.5310	1.18099	0.590493	−	0.807043i	$-0.298933\pi$
$397$	23.5310	1.18099	0.590493	+	0.807043i	$0.298933\pi$
$398$	0	0
$399$	−17.4422	−0.873202
$400$	0	0
$401$	−0.765035	−0.0382040	−0.0191020	−	0.999818i	$-0.506081\pi$
$401$	−0.765035	−0.0382040	−0.0191020	+	0.999818i	$0.506081\pi$
$402$	0	0
$403$	0.0363767	0.00181205
$404$	0	0
$405$	6.45568	0.320785
$406$	0	0
$407$	6.77802	0.335974
$408$	0	0
$409$	−5.60286	−0.277044	−0.138522	−	0.990359i	$-0.544235\pi$
$409$	−5.60286	−0.277044	−0.138522	+	0.990359i	$0.544235\pi$
$410$	0	0
$411$	9.93920	0.490265
$412$	0	0
$413$	−26.7954	−1.31851
$414$	0	0
$415$	53.1130	2.60722
$416$	0	0
$417$	−16.5915	−0.812490
$418$	0	0
$419$	24.8299	1.21302	0.606509	−	0.795077i	$-0.292569\pi$
$419$	24.8299	1.21302	0.606509	+	0.795077i	$0.292569\pi$
$420$	0	0
$421$	12.2521	0.597133	0.298566	−	0.954389i	$-0.403492\pi$
$421$	12.2521	0.597133	0.298566	+	0.954389i	$0.403492\pi$
$422$	0	0
$423$	−22.0597	−1.07258
$424$	0	0
$425$	−26.9126	−1.30545
$426$	0	0
$427$	−2.22132	−0.107497
$428$	0	0
$429$	−0.0999377	−0.00482504
$430$	0	0
$431$	24.3396	1.17240	0.586199	−	0.810167i	$-0.300624\pi$
$431$	24.3396	1.17240	0.586199	+	0.810167i	$0.300624\pi$
$432$	0	0
$433$	8.35633	0.401580	0.200790	−	0.979634i	$-0.435649\pi$
$433$	8.35633	0.401580	0.200790	+	0.979634i	$0.435649\pi$
$434$	0	0
$435$	11.0538	0.529988
$436$	0	0
$437$	−2.98995	−0.143029
$438$	0	0
$439$	5.67395	0.270803	0.135402	−	0.990791i	$-0.456768\pi$
$439$	5.67395	0.270803	0.135402	+	0.990791i	$0.456768\pi$
$440$	0	0
$441$	−14.1106	−0.671931
$442$	0	0
$443$	−3.55735	−0.169015	−0.0845073	−	0.996423i	$-0.526932\pi$
$443$	−3.55735	−0.169015	−0.0845073	+	0.996423i	$0.526932\pi$
$444$	0	0
$445$	54.3538	2.57662
$446$	0	0
$447$	−9.67000	−0.457375
$448$	0	0
$449$	27.9540	1.31923	0.659616	−	0.751603i	$-0.270719\pi$
$449$	27.9540	1.31923	0.659616	+	0.751603i	$0.270719\pi$
$450$	0	0
$451$	12.4258	0.585110
$452$	0	0
$453$	10.3608	0.486793
$454$	0	0
$455$	−1.23044	−0.0576840
$456$	0	0
$457$	11.2752	0.527430	0.263715	−	0.964601i	$-0.415052\pi$
$457$	11.2752	0.527430	0.263715	+	0.964601i	$0.415052\pi$
$458$	0	0
$459$	−19.3480	−0.903088
$460$	0	0
$461$	20.2942	0.945193	0.472597	−	0.881279i	$-0.343317\pi$
$461$	20.2942	0.945193	0.472597	+	0.881279i	$0.343317\pi$
$462$	0	0
$463$	25.6264	1.19096	0.595481	−	0.803370i	$-0.296962\pi$
$463$	25.6264	1.19096	0.595481	+	0.803370i	$0.296962\pi$
$464$	0	0
$465$	−1.19187	−0.0552715
$466$	0	0
$467$	−19.0460	−0.881345	−0.440673	−	0.897668i	$-0.645260\pi$
$467$	−19.0460	−0.881345	−0.440673	+	0.897668i	$0.645260\pi$
$468$	0	0
$469$	42.9794	1.98461
$470$	0	0
$471$	22.7345	1.04755
$472$	0	0
$473$	5.88492	0.270589
$474$	0	0
$475$	−33.6658	−1.54469
$476$	0	0
$477$	23.8704	1.09295
$478$	0	0
$479$	−0.620088	−0.0283326	−0.0141663	−	0.999900i	$-0.504509\pi$
$479$	−0.620088	−0.0283326	−0.0141663	+	0.999900i	$0.504509\pi$
$480$	0	0
$481$	0.603495	0.0275170
$482$	0	0
$483$	2.04624	0.0931070
$484$	0	0
$485$	−26.3209	−1.19517
$486$	0	0
$487$	−3.10719	−0.140800	−0.0704000	−	0.997519i	$-0.522428\pi$
$487$	−3.10719	−0.140800	−0.0704000	+	0.997519i	$0.522428\pi$
$488$	0	0
$489$	−11.8586	−0.536266
$490$	0	0
$491$	1.67757	0.0757079	0.0378539	−	0.999283i	$-0.487948\pi$
$491$	1.67757	0.0757079	0.0378539	+	0.999283i	$0.487948\pi$
$492$	0	0
$493$	13.9603	0.628739
$494$	0	0
$495$	−7.95082	−0.357363
$496$	0	0
$497$	−2.30726	−0.103495
$498$	0	0
$499$	18.6142	0.833284	0.416642	−	0.909071i	$-0.363207\pi$
$499$	18.6142	0.833284	0.416642	+	0.909071i	$0.363207\pi$
$500$	0	0
$501$	7.81376	0.349093
$502$	0	0
$503$	29.5856	1.31915	0.659577	−	0.751637i	$-0.270735\pi$
$503$	29.5856	1.31915	0.659577	+	0.751637i	$0.270735\pi$
$504$	0	0
$505$	61.7847	2.74938
$506$	0	0
$507$	12.1522	0.539699
$508$	0	0
$509$	25.2807	1.12055	0.560274	−	0.828307i	$-0.310696\pi$
$509$	25.2807	1.12055	0.560274	+	0.828307i	$0.310696\pi$
$510$	0	0
$511$	−35.8842	−1.58742
$512$	0	0
$513$	−24.2030	−1.06859
$514$	0	0
$515$	26.5822	1.17135
$516$	0	0
$517$	11.3717	0.500128
$518$	0	0
$519$	−12.0518	−0.529015
$520$	0	0
$521$	31.4315	1.37704	0.688520	−	0.725217i	$-0.258261\pi$
$521$	31.4315	1.37704	0.688520	+	0.725217i	$0.258261\pi$
$522$	0	0
$523$	−2.72046	−0.118957	−0.0594786	−	0.998230i	$-0.518944\pi$
$523$	−2.72046	−0.118957	−0.0594786	+	0.998230i	$0.518944\pi$
$524$	0	0
$525$	23.0399	1.00554
$526$	0	0
$527$	−1.50526	−0.0655699
$528$	0	0
$529$	−22.6492	−0.984749
$530$	0	0
$531$	−15.4163	−0.669011
$532$	0	0
$533$	1.10636	0.0479217
$534$	0	0
$535$	8.75551	0.378534
$536$	0	0
$537$	−24.6524	−1.06383
$538$	0	0
$539$	7.27396	0.313312
$540$	0	0
$541$	33.7433	1.45074	0.725369	−	0.688360i	$-0.241669\pi$
$541$	33.7433	1.45074	0.725369	+	0.688360i	$0.241669\pi$
$542$	0	0
$543$	15.8928	0.682026
$544$	0	0
$545$	1.40553	0.0602061
$546$	0	0
$547$	−1.51530	−0.0647894	−0.0323947	−	0.999475i	$-0.510313\pi$
$547$	−1.51530	−0.0647894	−0.0323947	+	0.999475i	$0.510313\pi$
$548$	0	0
$549$	−1.27801	−0.0545440
$550$	0	0
$551$	17.4633	0.743963
$552$	0	0
$553$	−25.2064	−1.07189
$554$	0	0
$555$	−19.7732	−0.839326
$556$	0	0
$557$	−4.08594	−0.173127	−0.0865633	−	0.996246i	$-0.527588\pi$
$557$	−4.08594	−0.173127	−0.0865633	+	0.996246i	$0.527588\pi$
$558$	0	0
$559$	0.523976	0.0221618
$560$	0	0
$561$	4.13539	0.174596
$562$	0	0
$563$	−4.30244	−0.181326	−0.0906631	−	0.995882i	$-0.528899\pi$
$563$	−4.30244	−0.181326	−0.0906631	+	0.995882i	$0.528899\pi$
$564$	0	0
$565$	−14.0962	−0.593034
$566$	0	0
$567$	−6.97990	−0.293128
$568$	0	0
$569$	−25.8210	−1.08247	−0.541236	−	0.840871i	$-0.682043\pi$
$569$	−25.8210	−1.08247	−0.541236	+	0.840871i	$0.682043\pi$
$570$	0	0
$571$	−26.3752	−1.10377	−0.551884	−	0.833921i	$-0.686091\pi$
$571$	−26.3752	−1.10377	−0.551884	+	0.833921i	$0.686091\pi$
$572$	0	0
$573$	6.77645	0.283090
$574$	0	0
$575$	3.94951	0.164706
$576$	0	0
$577$	−28.3402	−1.17982	−0.589909	−	0.807470i	$-0.700836\pi$
$577$	−28.3402	−1.17982	−0.589909	+	0.807470i	$0.700836\pi$
$578$	0	0
$579$	−2.20219	−0.0915199
$580$	0	0
$581$	−57.4260	−2.38243
$582$	0	0
$583$	−12.3051	−0.509626
$584$	0	0
$585$	−0.707916	−0.0292687
$586$	0	0
$587$	−21.1726	−0.873885	−0.436942	−	0.899489i	$-0.643939\pi$
$587$	−21.1726	−0.873885	−0.436942	+	0.899489i	$0.643939\pi$
$588$	0	0
$589$	−1.88297	−0.0775865
$590$	0	0
$591$	11.4786	0.472168
$592$	0	0
$593$	−34.9627	−1.43575	−0.717873	−	0.696174i	$-0.754884\pi$
$593$	−34.9627	−1.43575	−0.717873	+	0.696174i	$0.754884\pi$
$594$	0	0
$595$	50.9152	2.08732
$596$	0	0
$597$	1.52219	0.0622993
$598$	0	0
$599$	−36.2409	−1.48076	−0.740381	−	0.672188i	$-0.765355\pi$
$599$	−36.2409	−1.48076	−0.740381	+	0.672188i	$0.765355\pi$
$600$	0	0
$601$	−26.6251	−1.08606	−0.543030	−	0.839713i	$-0.682723\pi$
$601$	−26.6251	−1.08606	−0.543030	+	0.839713i	$0.682723\pi$
$602$	0	0
$603$	24.7276	1.00699
$604$	0	0
$605$	−33.4766	−1.36102
$606$	0	0
$607$	−37.0123	−1.50228	−0.751142	−	0.660140i	$-0.770497\pi$
$607$	−37.0123	−1.50228	−0.751142	+	0.660140i	$0.770497\pi$
$608$	0	0
$609$	−11.9514	−0.484295
$610$	0	0
$611$	1.01250	0.0409616
$612$	0	0
$613$	13.4713	0.544100	0.272050	−	0.962283i	$-0.412298\pi$
$613$	13.4713	0.544100	0.272050	+	0.962283i	$0.412298\pi$
$614$	0	0
$615$	−36.2493	−1.46171
$616$	0	0
$617$	−34.0747	−1.37180	−0.685899	−	0.727697i	$-0.740591\pi$
$617$	−34.0747	−1.37180	−0.685899	+	0.727697i	$0.740591\pi$
$618$	0	0
$619$	−14.1852	−0.570150	−0.285075	−	0.958505i	$-0.592019\pi$
$619$	−14.1852	−0.570150	−0.285075	+	0.958505i	$0.592019\pi$
$620$	0	0
$621$	2.83938	0.113941
$622$	0	0
$623$	−58.7675	−2.35447
$624$	0	0
$625$	−13.8728	−0.554914
$626$	0	0
$627$	5.17308	0.206593
$628$	0	0
$629$	−24.9724	−0.995714
$630$	0	0
$631$	27.9613	1.11312	0.556560	−	0.830807i	$-0.312121\pi$
$631$	27.9613	1.11312	0.556560	+	0.830807i	$0.312121\pi$
$632$	0	0
$633$	11.9135	0.473520
$634$	0	0
$635$	40.5217	1.60805
$636$	0	0
$637$	0.647652	0.0256609
$638$	0	0
$639$	−1.32745	−0.0525131
$640$	0	0
$641$	32.4718	1.28256	0.641279	−	0.767308i	$-0.278404\pi$
$641$	32.4718	1.28256	0.641279	+	0.767308i	$0.278404\pi$
$642$	0	0
$643$	−28.4307	−1.12120	−0.560599	−	0.828087i	$-0.689429\pi$
$643$	−28.4307	−1.12120	−0.560599	+	0.828087i	$0.689429\pi$
$644$	0	0
$645$	−17.1678	−0.675982
$646$	0	0
$647$	12.0028	0.471877	0.235938	−	0.971768i	$-0.424184\pi$
$647$	12.0028	0.471877	0.235938	+	0.971768i	$0.424184\pi$
$648$	0	0
$649$	7.94708	0.311950
$650$	0	0
$651$	1.28865	0.0505062
$652$	0	0
$653$	−10.9544	−0.428677	−0.214338	−	0.976759i	$-0.568760\pi$
$653$	−10.9544	−0.428677	−0.214338	+	0.976759i	$0.568760\pi$
$654$	0	0
$655$	−76.6372	−2.99446
$656$	0	0
$657$	−20.6455	−0.805456
$658$	0	0
$659$	−7.12266	−0.277460	−0.138730	−	0.990330i	$-0.544302\pi$
$659$	−7.12266	−0.277460	−0.138730	+	0.990330i	$0.544302\pi$
$660$	0	0
$661$	30.5505	1.18828	0.594138	−	0.804363i	$-0.297493\pi$
$661$	30.5505	1.18828	0.594138	+	0.804363i	$0.297493\pi$
$662$	0	0
$663$	0.368202	0.0142998
$664$	0	0
$665$	63.6914	2.46985
$666$	0	0
$667$	−2.04871	−0.0793266
$668$	0	0
$669$	20.9639	0.810511
$670$	0	0
$671$	0.658810	0.0254331
$672$	0	0
$673$	−17.8945	−0.689782	−0.344891	−	0.938643i	$-0.612084\pi$
$673$	−17.8945	−0.689782	−0.344891	+	0.938643i	$0.612084\pi$
$674$	0	0
$675$	31.9705	1.23054
$676$	0	0
$677$	18.2603	0.701802	0.350901	−	0.936413i	$-0.385875\pi$
$677$	18.2603	0.701802	0.350901	+	0.936413i	$0.385875\pi$
$678$	0	0
$679$	28.4583	1.09213
$680$	0	0
$681$	−8.78474	−0.336632
$682$	0	0
$683$	−36.9531	−1.41397	−0.706985	−	0.707229i	$-0.749945\pi$
$683$	−36.9531	−1.41397	−0.706985	+	0.707229i	$0.749945\pi$
$684$	0	0
$685$	−36.2937	−1.38671
$686$	0	0
$687$	−13.8583	−0.528728
$688$	0	0
$689$	−1.09561	−0.0417394
$690$	0	0
$691$	−30.4630	−1.15887	−0.579434	−	0.815019i	$-0.696726\pi$
$691$	−30.4630	−1.15887	−0.579434	+	0.815019i	$0.696726\pi$
$692$	0	0
$693$	8.59645	0.326552
$694$	0	0
$695$	60.5850	2.29812
$696$	0	0
$697$	−45.7807	−1.73407
$698$	0	0
$699$	−10.9565	−0.414412
$700$	0	0
$701$	23.1239	0.873376	0.436688	−	0.899613i	$-0.356151\pi$
$701$	23.1239	0.873376	0.436688	+	0.899613i	$0.356151\pi$
$702$	0	0
$703$	−31.2387	−1.17819
$704$	0	0
$705$	−33.1742	−1.24941
$706$	0	0
$707$	−66.8018	−2.51234
$708$	0	0
$709$	33.4421	1.25595	0.627973	−	0.778235i	$-0.283885\pi$
$709$	33.4421	1.25595	0.627973	+	0.778235i	$0.283885\pi$
$710$	0	0
$711$	−14.5021	−0.543873
$712$	0	0
$713$	0.220901	0.00827281
$714$	0	0
$715$	0.364929	0.0136476
$716$	0	0
$717$	−24.1564	−0.902138
$718$	0	0
$719$	−32.2243	−1.20176	−0.600882	−	0.799338i	$-0.705184\pi$
$719$	−32.2243	−1.20176	−0.600882	+	0.799338i	$0.705184\pi$
$720$	0	0
$721$	−28.7408	−1.07036
$722$	0	0
$723$	−4.89964	−0.182219
$724$	0	0
$725$	−23.0678	−0.856717
$726$	0	0
$727$	15.5274	0.575881	0.287940	−	0.957648i	$-0.407030\pi$
$727$	15.5274	0.575881	0.287940	+	0.957648i	$0.407030\pi$
$728$	0	0
$729$	9.43866	0.349580
$730$	0	0
$731$	−21.6819	−0.801935
$732$	0	0
$733$	11.3519	0.419293	0.209646	−	0.977777i	$-0.432769\pi$
$733$	11.3519	0.419293	0.209646	+	0.977777i	$0.432769\pi$
$734$	0	0
$735$	−21.2200	−0.782712
$736$	0	0
$737$	−12.7470	−0.469543
$738$	0	0
$739$	34.6988	1.27641	0.638207	−	0.769864i	$-0.279676\pi$
$739$	34.6988	1.27641	0.638207	+	0.769864i	$0.279676\pi$
$740$	0	0
$741$	0.460596	0.0169204
$742$	0	0
$743$	17.1702	0.629912	0.314956	−	0.949106i	$-0.398010\pi$
$743$	17.1702	0.629912	0.314956	+	0.949106i	$0.398010\pi$
$744$	0	0
$745$	35.3107	1.29368
$746$	0	0
$747$	−33.0392	−1.20884
$748$	0	0
$749$	−9.46649	−0.345898
$750$	0	0
$751$	−18.5379	−0.676457	−0.338229	−	0.941064i	$-0.609828\pi$
$751$	−18.5379	−0.676457	−0.338229	+	0.941064i	$0.609828\pi$
$752$	0	0
$753$	0.935470	0.0340904
$754$	0	0
$755$	−37.8332	−1.37689
$756$	0	0
$757$	19.1921	0.697550	0.348775	−	0.937207i	$-0.386598\pi$
$757$	19.1921	0.697550	0.348775	+	0.937207i	$0.386598\pi$
$758$	0	0
$759$	−0.606881	−0.0220284
$760$	0	0
$761$	44.6608	1.61895	0.809477	−	0.587152i	$-0.199751\pi$
$761$	44.6608	1.61895	0.809477	+	0.587152i	$0.199751\pi$
$762$	0	0
$763$	−1.51966	−0.0550153
$764$	0	0
$765$	29.2933	1.05910
$766$	0	0
$767$	0.707584	0.0255494
$768$	0	0
$769$	11.1082	0.400572	0.200286	−	0.979737i	$-0.435813\pi$
$769$	11.1082	0.400572	0.200286	+	0.979737i	$0.435813\pi$
$770$	0	0
$771$	−23.6851	−0.852997
$772$	0	0
$773$	−54.3686	−1.95550	−0.977751	−	0.209769i	$-0.932729\pi$
$773$	−54.3686	−1.95550	−0.977751	+	0.209769i	$0.932729\pi$
$774$	0	0
$775$	2.48727	0.0893454
$776$	0	0
$777$	21.3789	0.766963
$778$	0	0
$779$	−57.2685	−2.05186
$780$	0	0
$781$	0.684298	0.0244861
$782$	0	0
$783$	−16.5839	−0.592661
$784$	0	0
$785$	−83.0167	−2.96299
$786$	0	0
$787$	−0.0428803	−0.00152852	−0.000764259	−	1.00000i	$-0.500243\pi$
$787$	−0.0428803	−0.00152852	−0.000764259		1.00000i	$0.500243\pi$
$788$	0	0
$789$	−10.2880	−0.366263
$790$	0	0
$791$	15.2409	0.541904
$792$	0	0
$793$	0.0586584	0.00208302
$794$	0	0
$795$	35.8972	1.27314
$796$	0	0
$797$	−42.0336	−1.48891	−0.744454	−	0.667674i	$-0.767290\pi$
$797$	−42.0336	−1.48891	−0.744454	+	0.667674i	$0.767290\pi$
$798$	0	0
$799$	−41.8971	−1.48221
$800$	0	0
$801$	−33.8110	−1.19465
$802$	0	0
$803$	10.6427	0.375573
$804$	0	0
$805$	−7.47197	−0.263352
$806$	0	0
$807$	−23.2055	−0.816871
$808$	0	0
$809$	−35.3621	−1.24327	−0.621633	−	0.783309i	$-0.713530\pi$
$809$	−35.3621	−1.24327	−0.621633	+	0.783309i	$0.713530\pi$
$810$	0	0
$811$	44.0189	1.54571	0.772856	−	0.634582i	$-0.218828\pi$
$811$	44.0189	1.54571	0.772856	+	0.634582i	$0.218828\pi$
$812$	0	0
$813$	−5.13313	−0.180027
$814$	0	0
$815$	43.3026	1.51682
$816$	0	0
$817$	−27.1226	−0.948899
$818$	0	0
$819$	0.765401	0.0267453
$820$	0	0
$821$	3.78216	0.131998	0.0659991	−	0.997820i	$-0.478977\pi$
$821$	3.78216	0.131998	0.0659991	+	0.997820i	$0.478977\pi$
$822$	0	0
$823$	41.0108	1.42955	0.714774	−	0.699356i	$-0.246530\pi$
$823$	41.0108	1.42955	0.714774	+	0.699356i	$0.246530\pi$
$824$	0	0
$825$	−6.83327	−0.237904
$826$	0	0
$827$	−43.0262	−1.49617	−0.748083	−	0.663605i	$-0.769026\pi$
$827$	−43.0262	−1.49617	−0.748083	+	0.663605i	$0.769026\pi$
$828$	0	0
$829$	22.6915	0.788108	0.394054	−	0.919087i	$-0.371072\pi$
$829$	22.6915	0.788108	0.394054	+	0.919087i	$0.371072\pi$
$830$	0	0
$831$	−5.41384	−0.187804
$832$	0	0
$833$	−26.7996	−0.928550
$834$	0	0
$835$	−28.5325	−0.987407
$836$	0	0
$837$	1.78815	0.0618074
$838$	0	0
$839$	−49.4834	−1.70835	−0.854177	−	0.519982i	$-0.825939\pi$
$839$	−49.4834	−1.70835	−0.854177	+	0.519982i	$0.825939\pi$
$840$	0	0
$841$	−17.0341	−0.587384
$842$	0	0
$843$	−11.8906	−0.409535
$844$	0	0
$845$	−44.3746	−1.52653
$846$	0	0
$847$	36.1950	1.24368
$848$	0	0
$849$	7.95690	0.273080
$850$	0	0
$851$	3.66478	0.125627
$852$	0	0
$853$	23.5427	0.806087	0.403043	−	0.915181i	$-0.367952\pi$
$853$	23.5427	0.806087	0.403043	+	0.915181i	$0.367952\pi$
$854$	0	0
$855$	36.6439	1.25320
$856$	0	0
$857$	−28.5719	−0.975996	−0.487998	−	0.872845i	$-0.662273\pi$
$857$	−28.5719	−0.975996	−0.487998	+	0.872845i	$0.662273\pi$
$858$	0	0
$859$	30.5642	1.04284	0.521418	−	0.853301i	$-0.325403\pi$
$859$	30.5642	1.04284	0.521418	+	0.853301i	$0.325403\pi$
$860$	0	0
$861$	39.1929	1.33569
$862$	0	0
$863$	20.3242	0.691845	0.345922	−	0.938263i	$-0.387566\pi$
$863$	20.3242	0.691845	0.345922	+	0.938263i	$0.387566\pi$
$864$	0	0
$865$	44.0079	1.49632
$866$	0	0
$867$	0.666923	0.0226499
$868$	0	0
$869$	7.47583	0.253600
$870$	0	0
$871$	−1.13496	−0.0384565
$872$	0	0
$873$	16.3730	0.554144
$874$	0	0
$875$	−21.0512	−0.711659
$876$	0	0
$877$	15.6408	0.528152	0.264076	−	0.964502i	$-0.414933\pi$
$877$	15.6408	0.528152	0.264076	+	0.964502i	$0.414933\pi$
$878$	0	0
$879$	−18.0252	−0.607975
$880$	0	0
$881$	−46.2975	−1.55980	−0.779901	−	0.625902i	$-0.784731\pi$
$881$	−46.2975	−1.55980	−0.779901	+	0.625902i	$0.784731\pi$
$882$	0	0
$883$	−23.7796	−0.800246	−0.400123	−	0.916461i	$-0.631033\pi$
$883$	−23.7796	−0.800246	−0.400123	+	0.916461i	$0.631033\pi$
$884$	0	0
$885$	−23.1837	−0.779310
$886$	0	0
$887$	−4.44067	−0.149103	−0.0745515	−	0.997217i	$-0.523753\pi$
$887$	−4.44067	−0.149103	−0.0745515	+	0.997217i	$0.523753\pi$
$888$	0	0
$889$	−43.8121	−1.46941
$890$	0	0
$891$	2.07013	0.0693519
$892$	0	0
$893$	−52.4104	−1.75385
$894$	0	0
$895$	90.0199	3.00903
$896$	0	0
$897$	−0.0540349	−0.00180417
$898$	0	0
$899$	−1.29021	−0.0430309
$900$	0	0
$901$	45.3359	1.51036
$902$	0	0
$903$	18.5619	0.617701
$904$	0	0
$905$	−58.0337	−1.92911
$906$	0	0
$907$	−14.8394	−0.492735	−0.246368	−	0.969176i	$-0.579237\pi$
$907$	−14.8394	−0.492735	−0.246368	+	0.969176i	$0.579237\pi$
$908$	0	0
$909$	−38.4335	−1.27476
$910$	0	0
$911$	−36.2128	−1.19978	−0.599892	−	0.800081i	$-0.704790\pi$
$911$	−36.2128	−1.19978	−0.599892	+	0.800081i	$0.704790\pi$
$912$	0	0
$913$	17.0316	0.563665
$914$	0	0
$915$	−1.92192	−0.0635366
$916$	0	0
$917$	82.8603	2.73629
$918$	0	0
$919$	−45.1422	−1.48910	−0.744551	−	0.667565i	$-0.767337\pi$
$919$	−45.1422	−1.48910	−0.744551	+	0.667565i	$0.767337\pi$
$920$	0	0
$921$	−0.273769	−0.00902099
$922$	0	0
$923$	0.0609278	0.00200546
$924$	0	0
$925$	41.2641	1.35676
$926$	0	0
$927$	−16.5356	−0.543100
$928$	0	0
$929$	47.1799	1.54792	0.773961	−	0.633233i	$-0.218273\pi$
$929$	47.1799	1.54792	0.773961	+	0.633233i	$0.218273\pi$
$930$	0	0
$931$	−33.5244	−1.09872
$932$	0	0
$933$	4.77749	0.156408
$934$	0	0
$935$	−15.1006	−0.493844
$936$	0	0
$937$	−44.0230	−1.43817	−0.719084	−	0.694923i	$-0.755439\pi$
$937$	−44.0230	−1.43817	−0.719084	+	0.694923i	$0.755439\pi$
$938$	0	0
$939$	6.73987	0.219947
$940$	0	0
$941$	35.1353	1.14538	0.572688	−	0.819773i	$-0.305900\pi$
$941$	35.1353	1.14538	0.572688	+	0.819773i	$0.305900\pi$
$942$	0	0
$943$	6.71847	0.218784
$944$	0	0
$945$	−60.4840	−1.96755
$946$	0	0
$947$	−18.6892	−0.607316	−0.303658	−	0.952781i	$-0.598208\pi$
$947$	−18.6892	−0.607316	−0.303658	+	0.952781i	$0.598208\pi$
$948$	0	0
$949$	0.947593	0.0307602
$950$	0	0
$951$	9.96146	0.323022
$952$	0	0
$953$	−9.80725	−0.317688	−0.158844	−	0.987304i	$-0.550777\pi$
$953$	−9.80725	−0.317688	−0.158844	+	0.987304i	$0.550777\pi$
$954$	0	0
$955$	−24.7447	−0.800719
$956$	0	0
$957$	3.54459	0.114580
$958$	0	0
$959$	39.2408	1.26715
$960$	0	0
$961$	−30.8609	−0.995512
$962$	0	0
$963$	−5.44641	−0.175508
$964$	0	0
$965$	8.04145	0.258863
$966$	0	0
$967$	10.4028	0.334533	0.167266	−	0.985912i	$-0.446506\pi$
$967$	10.4028	0.334533	0.167266	+	0.985912i	$0.446506\pi$
$968$	0	0
$969$	−19.0593	−0.612272
$970$	0	0
$971$	36.5079	1.17159	0.585797	−	0.810458i	$-0.300782\pi$
$971$	36.5079	1.17159	0.585797	+	0.810458i	$0.300782\pi$
$972$	0	0
$973$	−65.5047	−2.09999
$974$	0	0
$975$	−0.608414	−0.0194848
$976$	0	0
$977$	−2.60773	−0.0834288	−0.0417144	−	0.999130i	$-0.513282\pi$
$977$	−2.60773	−0.0834288	−0.0417144	+	0.999130i	$0.513282\pi$
$978$	0	0
$979$	17.4295	0.557050
$980$	0	0
$981$	−0.874314	−0.0279147
$982$	0	0
$983$	14.7408	0.470160	0.235080	−	0.971976i	$-0.424465\pi$
$983$	14.7408	0.470160	0.235080	+	0.971976i	$0.424465\pi$
$984$	0	0
$985$	−41.9150	−1.33552
$986$	0	0
$987$	35.8681	1.14169
$988$	0	0
$989$	3.18189	0.101178
$990$	0	0
$991$	22.1779	0.704505	0.352252	−	0.935905i	$-0.385416\pi$
$991$	22.1779	0.704505	0.352252	+	0.935905i	$0.385416\pi$
$992$	0	0
$993$	15.3822	0.488139
$994$	0	0
$995$	−5.55840	−0.176213
$996$	0	0
$997$	−34.6820	−1.09839	−0.549194	−	0.835695i	$-0.685065\pi$
$997$	−34.6820	−1.09839	−0.549194	+	0.835695i	$0.685065\pi$
$998$	0	0
$999$	29.6656	0.938578

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.k.1.7		17
4.3	odd	2		251.2.a.b.1.4	✓	17
12.11	even	2		2259.2.a.k.1.14		17
20.19	odd	2		6275.2.a.e.1.14		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
251.2.a.b.1.4	✓	17	4.3	odd	2
2259.2.a.k.1.14		17	12.11	even	2
4016.2.a.k.1.7		17	1.1	even	1	trivial
6275.2.a.e.1.14		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)

\(f(q)\)	\(=\)	\(q-0.935470 q^{3} +3.41593 q^{5} -3.69332 q^{7} -2.12490 q^{9} +O(q^{10})\)	\(q-0.935470 q^{3} +3.41593 q^{5} -3.69332 q^{7} -2.12490 q^{9} +1.09538 q^{11} +0.0975293 q^{13} -3.19550 q^{15} -4.03572 q^{17} -5.04841 q^{19} +3.45499 q^{21} +0.592256 q^{23} +6.66859 q^{25} +4.79419 q^{27} -3.45917 q^{29} +0.372983 q^{31} -1.02469 q^{33} -12.6161 q^{35} +6.18783 q^{37} -0.0912357 q^{39} +11.3439 q^{41} +5.37250 q^{43} -7.25850 q^{45} +10.3815 q^{47} +6.64059 q^{49} +3.77530 q^{51} -11.2337 q^{53} +3.74174 q^{55} +4.72264 q^{57} +7.25509 q^{59} +0.601444 q^{61} +7.84791 q^{63} +0.333153 q^{65} -11.6371 q^{67} -0.554038 q^{69} +0.624713 q^{71} +9.71599 q^{73} -6.23827 q^{75} -4.04558 q^{77} +6.82487 q^{79} +1.88987 q^{81} +15.5486 q^{83} -13.7858 q^{85} +3.23595 q^{87} +15.9119 q^{89} -0.360206 q^{91} -0.348914 q^{93} -17.2450 q^{95} -7.70534 q^{97} -2.32757 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) \) 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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