LMFDB - Embedded newform 4016.2.a.l.1.16

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:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 6 x^{18} - 21 x^{17} + 179 x^{16} + 90 x^{15} - 2109 x^{14} + 926 x^{13} + 12681 x^{12} + \cdots + 4 $ x^19 - 6x^18 - 21x^17 + 179x^16 + 90x^15 - 2109x^14 + 926x^13 + 12681x^12 - 10845x^11 - 41921x^10 + 43551x^9 + 76260x^8 - 80907x^7 - 72526x^6 + 64793x^5 + 33209x^4 - 13777x^3 - 7607x^2 - 747x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$2.50185$ 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)

$f(q)$	$=$	$q+2.50185 q^{3} +0.708265 q^{5} -3.63078 q^{7} +3.25924 q^{9} +O(q^{10})$	$q+2.50185 q^{3} +0.708265 q^{5} -3.63078 q^{7} +3.25924 q^{9} +4.92913 q^{11} -0.194151 q^{13} +1.77197 q^{15} +1.76138 q^{17} +2.48449 q^{19} -9.08365 q^{21} -1.92277 q^{23} -4.49836 q^{25} +0.648575 q^{27} -6.35253 q^{29} +9.91351 q^{31} +12.3319 q^{33} -2.57155 q^{35} +9.89186 q^{37} -0.485737 q^{39} +11.4933 q^{41} +4.40828 q^{43} +2.30840 q^{45} -7.19606 q^{47} +6.18253 q^{49} +4.40670 q^{51} +5.85281 q^{53} +3.49113 q^{55} +6.21583 q^{57} +15.0262 q^{59} -1.50997 q^{61} -11.8336 q^{63} -0.137511 q^{65} +2.37750 q^{67} -4.81047 q^{69} +12.2740 q^{71} -7.26662 q^{73} -11.2542 q^{75} -17.8966 q^{77} -8.51016 q^{79} -8.15508 q^{81} +11.9918 q^{83} +1.24752 q^{85} -15.8930 q^{87} -12.7961 q^{89} +0.704920 q^{91} +24.8021 q^{93} +1.75968 q^{95} -9.27317 q^{97} +16.0652 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) $ 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9	$ 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9} + 15 q^{11} - 8 q^{13} + 17 q^{15} - 4 q^{17} + 14 q^{19} - 9 q^{21} + 28 q^{23} + 25 q^{25} + 21 q^{27} - 13 q^{29} + 20 q^{31} - 6 q^{33} + 32 q^{35} - 16 q^{37} + 27 q^{39} + 2 q^{41} + 28 q^{43} - 29 q^{45} + 37 q^{47} + 36 q^{49} + 35 q^{51} - 37 q^{53} + 24 q^{55} - 11 q^{57} + 32 q^{59} - 7 q^{61} + 45 q^{63} + q^{65} + 45 q^{67} - 12 q^{69} + 49 q^{71} + 16 q^{73} + 35 q^{75} - 40 q^{77} + 33 q^{79} + 15 q^{81} + 43 q^{83} - 28 q^{85} + 48 q^{87} + 3 q^{89} + 56 q^{91} - 48 q^{93} + 43 q^{95} + 8 q^{97} + 74 q^{99}+O(q^{100}) $ 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9 + 15 * q^11 - 8 * q^13 + 17 * q^15 - 4 * q^17 + 14 * q^19 - 9 * q^21 + 28 * q^23 + 25 * q^25 + 21 * q^27 - 13 * q^29 + 20 * q^31 - 6 * q^33 + 32 * q^35 - 16 * q^37 + 27 * q^39 + 2 * q^41 + 28 * q^43 - 29 * q^45 + 37 * q^47 + 36 * q^49 + 35 * q^51 - 37 * q^53 + 24 * q^55 - 11 * q^57 + 32 * q^59 - 7 * q^61 + 45 * q^63 + q^65 + 45 * q^67 - 12 * q^69 + 49 * q^71 + 16 * q^73 + 35 * q^75 - 40 * q^77 + 33 * q^79 + 15 * q^81 + 43 * q^83 - 28 * q^85 + 48 * q^87 + 3 * q^89 + 56 * q^91 - 48 * q^93 + 43 * q^95 + 8 * q^97 + 74 * 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
$3$	2.50185	1.44444	0.722221	−	0.691662i	$-0.243121\pi$
$3$	2.50185	1.44444	0.722221	+	0.691662i	$0.243121\pi$
$4$	0	0
$5$	0.708265	0.316746	0.158373	−	0.987379i	$-0.449375\pi$
$5$	0.708265	0.316746	0.158373	+	0.987379i	$0.449375\pi$
$6$	0	0
$7$	−3.63078	−1.37230	−0.686152	−	0.727458i	$-0.740702\pi$
$7$	−3.63078	−1.37230	−0.686152	+	0.727458i	$0.740702\pi$
$8$	0	0
$9$	3.25924	1.08641
$10$	0	0
$11$	4.92913	1.48619	0.743095	−	0.669186i	$-0.233357\pi$
$11$	4.92913	1.48619	0.743095	+	0.669186i	$0.233357\pi$
$12$	0	0
$13$	−0.194151	−0.0538479	−0.0269240	−	0.999637i	$-0.508571\pi$
$13$	−0.194151	−0.0538479	−0.0269240	+	0.999637i	$0.508571\pi$
$14$	0	0
$15$	1.77197	0.457521
$16$	0	0
$17$	1.76138	0.427198	0.213599	−	0.976921i	$-0.431481\pi$
$17$	1.76138	0.427198	0.213599	+	0.976921i	$0.431481\pi$
$18$	0	0
$19$	2.48449	0.569982	0.284991	−	0.958530i	$-0.408009\pi$
$19$	2.48449	0.569982	0.284991	+	0.958530i	$0.408009\pi$
$20$	0	0
$21$	−9.08365	−1.98221
$22$	0	0
$23$	−1.92277	−0.400925	−0.200463	−	0.979701i	$-0.564244\pi$
$23$	−1.92277	−0.400925	−0.200463	+	0.979701i	$0.564244\pi$
$24$	0	0
$25$	−4.49836	−0.899672
$26$	0	0
$27$	0.648575	0.124818
$28$	0	0
$29$	−6.35253	−1.17963	−0.589817	−	0.807537i	$-0.700800\pi$
$29$	−6.35253	−1.17963	−0.589817	+	0.807537i	$0.700800\pi$
$30$	0	0
$31$	9.91351	1.78052	0.890259	−	0.455454i	$-0.150523\pi$
$31$	9.91351	1.78052	0.890259	+	0.455454i	$0.150523\pi$
$32$	0	0
$33$	12.3319	2.14671
$34$	0	0
$35$	−2.57155	−0.434671
$36$	0	0
$37$	9.89186	1.62621	0.813106	−	0.582115i	$-0.197775\pi$
$37$	9.89186	1.62621	0.813106	+	0.582115i	$0.197775\pi$
$38$	0	0
$39$	−0.485737	−0.0777802
$40$	0	0
$41$	11.4933	1.79495	0.897477	−	0.441061i	$-0.145398\pi$
$41$	11.4933	1.79495	0.897477	+	0.441061i	$0.145398\pi$
$42$	0	0
$43$	4.40828	0.672257	0.336129	−	0.941816i	$-0.390882\pi$
$43$	4.40828	0.672257	0.336129	+	0.941816i	$0.390882\pi$
$44$	0	0
$45$	2.30840	0.344117
$46$	0	0
$47$	−7.19606	−1.04965	−0.524827	−	0.851209i	$-0.675870\pi$
$47$	−7.19606	−1.04965	−0.524827	+	0.851209i	$0.675870\pi$
$48$	0	0
$49$	6.18253	0.883219
$50$	0	0
$51$	4.40670	0.617062
$52$	0	0
$53$	5.85281	0.803946	0.401973	−	0.915652i	$-0.368325\pi$
$53$	5.85281	0.803946	0.401973	+	0.915652i	$0.368325\pi$
$54$	0	0
$55$	3.49113	0.470744
$56$	0	0
$57$	6.21583	0.823306
$58$	0	0
$59$	15.0262	1.95624	0.978119	−	0.208045i	$-0.0667101\pi$
$59$	15.0262	1.95624	0.978119	+	0.208045i	$0.0667101\pi$
$60$	0	0
$61$	−1.50997	−0.193332	−0.0966658	−	0.995317i	$-0.530818\pi$
$61$	−1.50997	−0.193332	−0.0966658	+	0.995317i	$0.530818\pi$
$62$	0	0
$63$	−11.8336	−1.49089
$64$	0	0
$65$	−0.137511	−0.0170561
$66$	0	0
$67$	2.37750	0.290457	0.145229	−	0.989398i	$-0.453608\pi$
$67$	2.37750	0.290457	0.145229	+	0.989398i	$0.453608\pi$
$68$	0	0
$69$	−4.81047	−0.579113
$70$	0	0
$71$	12.2740	1.45666	0.728328	−	0.685229i	$-0.240298\pi$
$71$	12.2740	1.45666	0.728328	+	0.685229i	$0.240298\pi$
$72$	0	0
$73$	−7.26662	−0.850494	−0.425247	−	0.905077i	$-0.639813\pi$
$73$	−7.26662	−0.850494	−0.425247	+	0.905077i	$0.639813\pi$
$74$	0	0
$75$	−11.2542	−1.29952
$76$	0	0
$77$	−17.8966	−2.03950
$78$	0	0
$79$	−8.51016	−0.957468	−0.478734	−	0.877960i	$-0.658904\pi$
$79$	−8.51016	−0.957468	−0.478734	+	0.877960i	$0.658904\pi$
$80$	0	0
$81$	−8.15508	−0.906120
$82$	0	0
$83$	11.9918	1.31627	0.658134	−	0.752901i	$-0.271346\pi$
$83$	11.9918	1.31627	0.658134	+	0.752901i	$0.271346\pi$
$84$	0	0
$85$	1.24752	0.135313
$86$	0	0
$87$	−15.8930	−1.70391
$88$	0	0
$89$	−12.7961	−1.35639	−0.678193	−	0.734884i	$-0.737237\pi$
$89$	−12.7961	−1.35639	−0.678193	+	0.734884i	$0.737237\pi$
$90$	0	0
$91$	0.704920	0.0738957
$92$	0	0
$93$	24.8021	2.57186
$94$	0	0
$95$	1.75968	0.180539
$96$	0	0
$97$	−9.27317	−0.941547	−0.470774	−	0.882254i	$-0.656025\pi$
$97$	−9.27317	−0.941547	−0.470774	+	0.882254i	$0.656025\pi$
$98$	0	0
$99$	16.0652	1.61462
$100$	0	0
$101$	−6.86753	−0.683345	−0.341673	−	0.939819i	$-0.610993\pi$
$101$	−6.86753	−0.683345	−0.341673	+	0.939819i	$0.610993\pi$
$102$	0	0
$103$	−16.1492	−1.59123	−0.795614	−	0.605804i	$-0.792851\pi$
$103$	−16.1492	−1.59123	−0.795614	+	0.605804i	$0.792851\pi$
$104$	0	0
$105$	−6.43363	−0.627858
$106$	0	0
$107$	17.5806	1.69958	0.849791	−	0.527120i	$-0.176728\pi$
$107$	17.5806	1.69958	0.849791	+	0.527120i	$0.176728\pi$
$108$	0	0
$109$	16.5817	1.58824	0.794122	−	0.607759i	$-0.207931\pi$
$109$	16.5817	1.58824	0.794122	+	0.607759i	$0.207931\pi$
$110$	0	0
$111$	24.7479	2.34897
$112$	0	0
$113$	1.22854	0.115571	0.0577856	−	0.998329i	$-0.481596\pi$
$113$	1.22854	0.115571	0.0577856	+	0.998329i	$0.481596\pi$
$114$	0	0
$115$	−1.36183	−0.126991
$116$	0	0
$117$	−0.632786	−0.0585011
$118$	0	0
$119$	−6.39518	−0.586245
$120$	0	0
$121$	13.2963	1.20876
$122$	0	0
$123$	28.7545	2.59271
$124$	0	0
$125$	−6.72736	−0.601713
$126$	0	0
$127$	−6.49059	−0.575946	−0.287973	−	0.957638i	$-0.592981\pi$
$127$	−6.49059	−0.575946	−0.287973	+	0.957638i	$0.592981\pi$
$128$	0	0
$129$	11.0289	0.971036
$130$	0	0
$131$	9.36573	0.818288	0.409144	−	0.912470i	$-0.365827\pi$
$131$	9.36573	0.818288	0.409144	+	0.912470i	$0.365827\pi$
$132$	0	0
$133$	−9.02064	−0.782189
$134$	0	0
$135$	0.459363	0.0395356
$136$	0	0
$137$	−13.6423	−1.16554	−0.582772	−	0.812636i	$-0.698032\pi$
$137$	−13.6423	−1.16554	−0.582772	+	0.812636i	$0.698032\pi$
$138$	0	0
$139$	−7.38047	−0.626004	−0.313002	−	0.949753i	$-0.601335\pi$
$139$	−7.38047	−0.626004	−0.313002	+	0.949753i	$0.601335\pi$
$140$	0	0
$141$	−18.0034	−1.51616
$142$	0	0
$143$	−0.956998	−0.0800282
$144$	0	0
$145$	−4.49927	−0.373644
$146$	0	0
$147$	15.4677	1.27576
$148$	0	0
$149$	5.67957	0.465288	0.232644	−	0.972562i	$-0.425262\pi$
$149$	5.67957	0.465288	0.232644	+	0.972562i	$0.425262\pi$
$150$	0	0
$151$	15.9604	1.29884	0.649419	−	0.760431i	$-0.275012\pi$
$151$	15.9604	1.29884	0.649419	+	0.760431i	$0.275012\pi$
$152$	0	0
$153$	5.74076	0.464113
$154$	0	0
$155$	7.02139	0.563972
$156$	0	0
$157$	−8.50032	−0.678399	−0.339200	−	0.940714i	$-0.610156\pi$
$157$	−8.50032	−0.678399	−0.339200	+	0.940714i	$0.610156\pi$
$158$	0	0
$159$	14.6428	1.16125
$160$	0	0
$161$	6.98114	0.550191
$162$	0	0
$163$	13.2949	1.04133	0.520667	−	0.853760i	$-0.325683\pi$
$163$	13.2949	1.04133	0.520667	+	0.853760i	$0.325683\pi$
$164$	0	0
$165$	8.73428	0.679963
$166$	0	0
$167$	22.0874	1.70918	0.854588	−	0.519307i	$-0.173810\pi$
$167$	22.0874	1.70918	0.854588	+	0.519307i	$0.173810\pi$
$168$	0	0
$169$	−12.9623	−0.997100
$170$	0	0
$171$	8.09756	0.619236
$172$	0	0
$173$	−18.5313	−1.40891	−0.704454	−	0.709750i	$-0.748808\pi$
$173$	−18.5313	−1.40891	−0.704454	+	0.709750i	$0.748808\pi$
$174$	0	0
$175$	16.3325	1.23462
$176$	0	0
$177$	37.5931	2.82567
$178$	0	0
$179$	−7.91203	−0.591373	−0.295687	−	0.955285i	$-0.595548\pi$
$179$	−7.91203	−0.591373	−0.295687	+	0.955285i	$0.595548\pi$
$180$	0	0
$181$	−9.90996	−0.736602	−0.368301	−	0.929707i	$-0.620060\pi$
$181$	−9.90996	−0.736602	−0.368301	+	0.929707i	$0.620060\pi$
$182$	0	0
$183$	−3.77771	−0.279256
$184$	0	0
$185$	7.00606	0.515096
$186$	0	0
$187$	8.68208	0.634896
$188$	0	0
$189$	−2.35483	−0.171289
$190$	0	0
$191$	1.98957	0.143960	0.0719800	−	0.997406i	$-0.477068\pi$
$191$	1.98957	0.143960	0.0719800	+	0.997406i	$0.477068\pi$
$192$	0	0
$193$	−3.81309	−0.274472	−0.137236	−	0.990538i	$-0.543822\pi$
$193$	−3.81309	−0.274472	−0.137236	+	0.990538i	$0.543822\pi$
$194$	0	0
$195$	−0.344031	−0.0246365
$196$	0	0
$197$	16.8381	1.19967	0.599833	−	0.800126i	$-0.295234\pi$
$197$	16.8381	1.19967	0.599833	+	0.800126i	$0.295234\pi$
$198$	0	0
$199$	−19.4881	−1.38148	−0.690738	−	0.723105i	$-0.742714\pi$
$199$	−19.4881	−1.38148	−0.690738	+	0.723105i	$0.742714\pi$
$200$	0	0
$201$	5.94813	0.419549
$202$	0	0
$203$	23.0646	1.61882
$204$	0	0
$205$	8.14031	0.568544
$206$	0	0
$207$	−6.26676	−0.435570
$208$	0	0
$209$	12.2464	0.847102
$210$	0	0
$211$	3.35737	0.231131	0.115565	−	0.993300i	$-0.463132\pi$
$211$	3.35737	0.231131	0.115565	+	0.993300i	$0.463132\pi$
$212$	0	0
$213$	30.7077	2.10406
$214$	0	0
$215$	3.12223	0.212935
$216$	0	0
$217$	−35.9937	−2.44341
$218$	0	0
$219$	−18.1800	−1.22849
$220$	0	0
$221$	−0.341975	−0.0230037
$222$	0	0
$223$	−29.2345	−1.95768	−0.978842	−	0.204617i	$-0.934405\pi$
$223$	−29.2345	−1.95768	−0.978842	+	0.204617i	$0.934405\pi$
$224$	0	0
$225$	−14.6612	−0.977415
$226$	0	0
$227$	24.0650	1.59725	0.798626	−	0.601828i	$-0.205561\pi$
$227$	24.0650	1.59725	0.798626	+	0.601828i	$0.205561\pi$
$228$	0	0
$229$	13.0539	0.862625	0.431312	−	0.902203i	$-0.358051\pi$
$229$	13.0539	0.862625	0.431312	+	0.902203i	$0.358051\pi$
$230$	0	0
$231$	−44.7745	−2.94595
$232$	0	0
$233$	−23.3572	−1.53018	−0.765091	−	0.643922i	$-0.777306\pi$
$233$	−23.3572	−1.53018	−0.765091	+	0.643922i	$0.777306\pi$
$234$	0	0
$235$	−5.09672	−0.332473
$236$	0	0
$237$	−21.2911	−1.38301
$238$	0	0
$239$	4.22725	0.273438	0.136719	−	0.990610i	$-0.456344\pi$
$239$	4.22725	0.273438	0.136719	+	0.990610i	$0.456344\pi$
$240$	0	0
$241$	−7.04645	−0.453902	−0.226951	−	0.973906i	$-0.572876\pi$
$241$	−7.04645	−0.453902	−0.226951	+	0.973906i	$0.572876\pi$
$242$	0	0
$243$	−22.3485	−1.43366
$244$	0	0
$245$	4.37887	0.279756
$246$	0	0
$247$	−0.482368	−0.0306924
$248$	0	0
$249$	30.0016	1.90127
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−9.47759	−0.595851
$254$	0	0
$255$	3.12111	0.195452
$256$	0	0
$257$	19.5354	1.21859	0.609293	−	0.792945i	$-0.291453\pi$
$257$	19.5354	1.21859	0.609293	+	0.792945i	$0.291453\pi$
$258$	0	0
$259$	−35.9151	−2.23166
$260$	0	0
$261$	−20.7044	−1.28157
$262$	0	0
$263$	2.50465	0.154444	0.0772218	−	0.997014i	$-0.475395\pi$
$263$	2.50465	0.154444	0.0772218	+	0.997014i	$0.475395\pi$
$264$	0	0
$265$	4.14534	0.254646
$266$	0	0
$267$	−32.0139	−1.95922
$268$	0	0
$269$	7.60409	0.463629	0.231815	−	0.972760i	$-0.425534\pi$
$269$	7.60409	0.463629	0.231815	+	0.972760i	$0.425534\pi$
$270$	0	0
$271$	12.7713	0.775802	0.387901	−	0.921701i	$-0.373200\pi$
$271$	12.7713	0.775802	0.387901	+	0.921701i	$0.373200\pi$
$272$	0	0
$273$	1.76360	0.106738
$274$	0	0
$275$	−22.1730	−1.33708
$276$	0	0
$277$	12.4965	0.750840	0.375420	−	0.926855i	$-0.377498\pi$
$277$	12.4965	0.750840	0.375420	+	0.926855i	$0.377498\pi$
$278$	0	0
$279$	32.3105	1.93438
$280$	0	0
$281$	−6.78176	−0.404566	−0.202283	−	0.979327i	$-0.564836\pi$
$281$	−6.78176	−0.404566	−0.202283	+	0.979327i	$0.564836\pi$
$282$	0	0
$283$	−5.08324	−0.302167	−0.151084	−	0.988521i	$-0.548276\pi$
$283$	−5.08324	−0.302167	−0.151084	+	0.988521i	$0.548276\pi$
$284$	0	0
$285$	4.40245	0.260779
$286$	0	0
$287$	−41.7296	−2.46322
$288$	0	0
$289$	−13.8975	−0.817502
$290$	0	0
$291$	−23.2000	−1.36001
$292$	0	0
$293$	14.1749	0.828108	0.414054	−	0.910252i	$-0.364112\pi$
$293$	14.1749	0.828108	0.414054	+	0.910252i	$0.364112\pi$
$294$	0	0
$295$	10.6425	0.619630
$296$	0	0
$297$	3.19691	0.185504
$298$	0	0
$299$	0.373308	0.0215890
$300$	0	0
$301$	−16.0055	−0.922541
$302$	0	0
$303$	−17.1815	−0.987052
$304$	0	0
$305$	−1.06946	−0.0612370
$306$	0	0
$307$	−22.1877	−1.26632	−0.633160	−	0.774021i	$-0.718243\pi$
$307$	−22.1877	−1.26632	−0.633160	+	0.774021i	$0.718243\pi$
$308$	0	0
$309$	−40.4028	−2.29844
$310$	0	0
$311$	11.0097	0.624305	0.312153	−	0.950032i	$-0.398950\pi$
$311$	11.0097	0.624305	0.312153	+	0.950032i	$0.398950\pi$
$312$	0	0
$313$	7.06437	0.399302	0.199651	−	0.979867i	$-0.436019\pi$
$313$	7.06437	0.399302	0.199651	+	0.979867i	$0.436019\pi$
$314$	0	0
$315$	−8.38130	−0.472233
$316$	0	0
$317$	−28.2945	−1.58918	−0.794588	−	0.607149i	$-0.792313\pi$
$317$	−28.2945	−1.58918	−0.794588	+	0.607149i	$0.792313\pi$
$318$	0	0
$319$	−31.3124	−1.75316
$320$	0	0
$321$	43.9840	2.45495
$322$	0	0
$323$	4.37614	0.243495
$324$	0	0
$325$	0.873363	0.0484455
$326$	0	0
$327$	41.4850	2.29413
$328$	0	0
$329$	26.1273	1.44044
$330$	0	0
$331$	−9.62814	−0.529211	−0.264605	−	0.964357i	$-0.585242\pi$
$331$	−9.62814	−0.529211	−0.264605	+	0.964357i	$0.585242\pi$
$332$	0	0
$333$	32.2399	1.76674
$334$	0	0
$335$	1.68390	0.0920011
$336$	0	0
$337$	12.1512	0.661916	0.330958	−	0.943645i	$-0.392628\pi$
$337$	12.1512	0.661916	0.330958	+	0.943645i	$0.392628\pi$
$338$	0	0
$339$	3.07361	0.166936
$340$	0	0
$341$	48.8650	2.64619
$342$	0	0
$343$	2.96804	0.160259
$344$	0	0
$345$	−3.40709	−0.183432
$346$	0	0
$347$	1.98182	0.106390	0.0531949	−	0.998584i	$-0.483060\pi$
$347$	1.98182	0.106390	0.0531949	+	0.998584i	$0.483060\pi$
$348$	0	0
$349$	11.6374	0.622934	0.311467	−	0.950257i	$-0.399180\pi$
$349$	11.6374	0.622934	0.311467	+	0.950257i	$0.399180\pi$
$350$	0	0
$351$	−0.125922	−0.00672120
$352$	0	0
$353$	−28.0985	−1.49553	−0.747765	−	0.663964i	$-0.768873\pi$
$353$	−28.0985	−1.49553	−0.747765	+	0.663964i	$0.768873\pi$
$354$	0	0
$355$	8.69325	0.461390
$356$	0	0
$357$	−15.9998	−0.846797
$358$	0	0
$359$	−36.4000	−1.92112	−0.960558	−	0.278078i	$-0.910303\pi$
$359$	−36.4000	−1.92112	−0.960558	+	0.278078i	$0.910303\pi$
$360$	0	0
$361$	−12.8273	−0.675120
$362$	0	0
$363$	33.2654	1.74598
$364$	0	0
$365$	−5.14669	−0.269390
$366$	0	0
$367$	4.94302	0.258023	0.129012	−	0.991643i	$-0.458820\pi$
$367$	4.94302	0.258023	0.129012	+	0.991643i	$0.458820\pi$
$368$	0	0
$369$	37.4595	1.95006
$370$	0	0
$371$	−21.2502	−1.10326
$372$	0	0
$373$	3.56829	0.184759	0.0923794	−	0.995724i	$-0.470553\pi$
$373$	3.56829	0.184759	0.0923794	+	0.995724i	$0.470553\pi$
$374$	0	0
$375$	−16.8308	−0.869139
$376$	0	0
$377$	1.23335	0.0635209
$378$	0	0
$379$	15.5585	0.799188	0.399594	−	0.916692i	$-0.369151\pi$
$379$	15.5585	0.799188	0.399594	+	0.916692i	$0.369151\pi$
$380$	0	0
$381$	−16.2385	−0.831921
$382$	0	0
$383$	−9.89360	−0.505539	−0.252770	−	0.967526i	$-0.581341\pi$
$383$	−9.89360	−0.505539	−0.252770	+	0.967526i	$0.581341\pi$
$384$	0	0
$385$	−12.6755	−0.646004
$386$	0	0
$387$	14.3677	0.730349
$388$	0	0
$389$	−14.0737	−0.713565	−0.356782	−	0.934188i	$-0.616126\pi$
$389$	−14.0737	−0.713565	−0.356782	+	0.934188i	$0.616126\pi$
$390$	0	0
$391$	−3.38673	−0.171274
$392$	0	0
$393$	23.4316	1.18197
$394$	0	0
$395$	−6.02745	−0.303274
$396$	0	0
$397$	−14.7019	−0.737866	−0.368933	−	0.929456i	$-0.620277\pi$
$397$	−14.7019	−0.737866	−0.368933	+	0.929456i	$0.620277\pi$
$398$	0	0
$399$	−22.5683	−1.12983
$400$	0	0
$401$	−14.4470	−0.721448	−0.360724	−	0.932673i	$-0.617470\pi$
$401$	−14.4470	−0.721448	−0.360724	+	0.932673i	$0.617470\pi$
$402$	0	0
$403$	−1.92472	−0.0958772
$404$	0	0
$405$	−5.77596	−0.287010
$406$	0	0
$407$	48.7583	2.41686
$408$	0	0
$409$	12.4541	0.615816	0.307908	−	0.951416i	$-0.400371\pi$
$409$	12.4541	0.615816	0.307908	+	0.951416i	$0.400371\pi$
$410$	0	0
$411$	−34.1311	−1.68356
$412$	0	0
$413$	−54.5566	−2.68455
$414$	0	0
$415$	8.49335	0.416922
$416$	0	0
$417$	−18.4648	−0.904226
$418$	0	0
$419$	14.1621	0.691866	0.345933	−	0.938259i	$-0.387562\pi$
$419$	14.1621	0.691866	0.345933	+	0.938259i	$0.387562\pi$
$420$	0	0
$421$	15.3416	0.747703	0.373852	−	0.927489i	$-0.378037\pi$
$421$	15.3416	0.747703	0.373852	+	0.927489i	$0.378037\pi$
$422$	0	0
$423$	−23.4537	−1.14036
$424$	0	0
$425$	−7.92333	−0.384338
$426$	0	0
$427$	5.48236	0.265310
$428$	0	0
$429$	−2.39426	−0.115596
$430$	0	0
$431$	4.14844	0.199824	0.0999118	−	0.994996i	$-0.468144\pi$
$431$	4.14844	0.199824	0.0999118	+	0.994996i	$0.468144\pi$
$432$	0	0
$433$	30.8966	1.48479	0.742397	−	0.669960i	$-0.233689\pi$
$433$	30.8966	1.48479	0.742397	+	0.669960i	$0.233689\pi$
$434$	0	0
$435$	−11.2565	−0.539707
$436$	0	0
$437$	−4.77711	−0.228520
$438$	0	0
$439$	−7.13840	−0.340697	−0.170349	−	0.985384i	$-0.554489\pi$
$439$	−7.13840	−0.340697	−0.170349	+	0.985384i	$0.554489\pi$
$440$	0	0
$441$	20.1503	0.959540
$442$	0	0
$443$	−27.9735	−1.32906	−0.664531	−	0.747261i	$-0.731369\pi$
$443$	−27.9735	−1.32906	−0.664531	+	0.747261i	$0.731369\pi$
$444$	0	0
$445$	−9.06304	−0.429629
$446$	0	0
$447$	14.2094	0.672082
$448$	0	0
$449$	−18.5781	−0.876756	−0.438378	−	0.898791i	$-0.644447\pi$
$449$	−18.5781	−0.876756	−0.438378	+	0.898791i	$0.644447\pi$
$450$	0	0
$451$	56.6521	2.66764
$452$	0	0
$453$	39.9305	1.87610
$454$	0	0
$455$	0.499270	0.0234061
$456$	0	0
$457$	−15.0942	−0.706078	−0.353039	−	0.935609i	$-0.614852\pi$
$457$	−15.0942	−0.706078	−0.353039	+	0.935609i	$0.614852\pi$
$458$	0	0
$459$	1.14239	0.0533221
$460$	0	0
$461$	22.4990	1.04788	0.523941	−	0.851755i	$-0.324461\pi$
$461$	22.4990	1.04788	0.523941	+	0.851755i	$0.324461\pi$
$462$	0	0
$463$	−10.7596	−0.500043	−0.250021	−	0.968240i	$-0.580438\pi$
$463$	−10.7596	−0.500043	−0.250021	+	0.968240i	$0.580438\pi$
$464$	0	0
$465$	17.5664	0.814624
$466$	0	0
$467$	−28.6240	−1.32456	−0.662281	−	0.749256i	$-0.730411\pi$
$467$	−28.6240	−1.32456	−0.662281	+	0.749256i	$0.730411\pi$
$468$	0	0
$469$	−8.63216	−0.398596
$470$	0	0
$471$	−21.2665	−0.979908
$472$	0	0
$473$	21.7290	0.999101
$474$	0	0
$475$	−11.1762	−0.512797
$476$	0	0
$477$	19.0757	0.873417
$478$	0	0
$479$	−19.0211	−0.869094	−0.434547	−	0.900649i	$-0.643092\pi$
$479$	−19.0211	−0.869094	−0.434547	+	0.900649i	$0.643092\pi$
$480$	0	0
$481$	−1.92052	−0.0875681
$482$	0	0
$483$	17.4658	0.794719
$484$	0	0
$485$	−6.56786	−0.298231
$486$	0	0
$487$	−5.28526	−0.239498	−0.119749	−	0.992804i	$-0.538209\pi$
$487$	−5.28526	−0.239498	−0.119749	+	0.992804i	$0.538209\pi$
$488$	0	0
$489$	33.2617	1.50415
$490$	0	0
$491$	−5.50545	−0.248458	−0.124229	−	0.992254i	$-0.539646\pi$
$491$	−5.50545	−0.248458	−0.124229	+	0.992254i	$0.539646\pi$
$492$	0	0
$493$	−11.1892	−0.503937
$494$	0	0
$495$	11.3784	0.511422
$496$	0	0
$497$	−44.5642	−1.99898
$498$	0	0
$499$	−14.8675	−0.665560	−0.332780	−	0.943004i	$-0.607987\pi$
$499$	−14.8675	−0.665560	−0.332780	+	0.943004i	$0.607987\pi$
$500$	0	0
$501$	55.2593	2.46880
$502$	0	0
$503$	−21.1764	−0.944210	−0.472105	−	0.881542i	$-0.656506\pi$
$503$	−21.1764	−0.944210	−0.472105	+	0.881542i	$0.656506\pi$
$504$	0	0
$505$	−4.86403	−0.216447
$506$	0	0
$507$	−32.4297	−1.44025
$508$	0	0
$509$	−13.1779	−0.584100	−0.292050	−	0.956403i	$-0.594337\pi$
$509$	−13.1779	−0.584100	−0.292050	+	0.956403i	$0.594337\pi$
$510$	0	0
$511$	26.3835	1.16714
$512$	0	0
$513$	1.61138	0.0711442
$514$	0	0
$515$	−11.4379	−0.504014
$516$	0	0
$517$	−35.4703	−1.55998
$518$	0	0
$519$	−46.3624	−2.03509
$520$	0	0
$521$	43.5466	1.90781	0.953905	−	0.300110i	$-0.0970234\pi$
$521$	43.5466	1.90781	0.953905	+	0.300110i	$0.0970234\pi$
$522$	0	0
$523$	−17.6565	−0.772066	−0.386033	−	0.922485i	$-0.626155\pi$
$523$	−17.6565	−0.772066	−0.386033	+	0.922485i	$0.626155\pi$
$524$	0	0
$525$	40.8615	1.78334
$526$	0	0
$527$	17.4615	0.760633
$528$	0	0
$529$	−19.3030	−0.839259
$530$	0	0
$531$	48.9738	2.12528
$532$	0	0
$533$	−2.23144	−0.0966545
$534$	0	0
$535$	12.4517	0.538335
$536$	0	0
$537$	−19.7947	−0.854204
$538$	0	0
$539$	30.4745	1.31263
$540$	0	0
$541$	36.3343	1.56213	0.781067	−	0.624448i	$-0.214676\pi$
$541$	36.3343	1.56213	0.781067	+	0.624448i	$0.214676\pi$
$542$	0	0
$543$	−24.7932	−1.06398
$544$	0	0
$545$	11.7443	0.503069
$546$	0	0
$547$	23.8290	1.01885	0.509427	−	0.860514i	$-0.329857\pi$
$547$	23.8290	1.01885	0.509427	+	0.860514i	$0.329857\pi$
$548$	0	0
$549$	−4.92135	−0.210038
$550$	0	0
$551$	−15.7828	−0.672371
$552$	0	0
$553$	30.8985	1.31394
$554$	0	0
$555$	17.5281	0.744026
$556$	0	0
$557$	4.72772	0.200320	0.100160	−	0.994971i	$-0.468065\pi$
$557$	4.72772	0.200320	0.100160	+	0.994971i	$0.468065\pi$
$558$	0	0
$559$	−0.855875	−0.0361996
$560$	0	0
$561$	21.7212	0.917071
$562$	0	0
$563$	−32.2407	−1.35878	−0.679392	−	0.733776i	$-0.737756\pi$
$563$	−32.2407	−1.35878	−0.679392	+	0.733776i	$0.737756\pi$
$564$	0	0
$565$	0.870130	0.0366067
$566$	0	0
$567$	29.6093	1.24347
$568$	0	0
$569$	−42.9715	−1.80146	−0.900729	−	0.434382i	$-0.856967\pi$
$569$	−42.9715	−1.80146	−0.900729	+	0.434382i	$0.856967\pi$
$570$	0	0
$571$	37.9600	1.58858	0.794289	−	0.607540i	$-0.207844\pi$
$571$	37.9600	1.58858	0.794289	+	0.607540i	$0.207844\pi$
$572$	0	0
$573$	4.97760	0.207942
$574$	0	0
$575$	8.64931	0.360701
$576$	0	0
$577$	−31.6282	−1.31670	−0.658349	−	0.752713i	$-0.728745\pi$
$577$	−31.6282	−1.31670	−0.658349	+	0.752713i	$0.728745\pi$
$578$	0	0
$579$	−9.53977	−0.396459
$580$	0	0
$581$	−43.5394	−1.80632
$582$	0	0
$583$	28.8493	1.19482
$584$	0	0
$585$	−0.448180	−0.0185300
$586$	0	0
$587$	4.17170	0.172185	0.0860923	−	0.996287i	$-0.472562\pi$
$587$	4.17170	0.172185	0.0860923	+	0.996287i	$0.472562\pi$
$588$	0	0
$589$	24.6301	1.01486
$590$	0	0
$591$	42.1264	1.73285
$592$	0	0
$593$	−31.4228	−1.29038	−0.645190	−	0.764023i	$-0.723222\pi$
$593$	−31.4228	−1.29038	−0.645190	+	0.764023i	$0.723222\pi$
$594$	0	0
$595$	−4.52948	−0.185691
$596$	0	0
$597$	−48.7563	−1.99546
$598$	0	0
$599$	2.85254	0.116552	0.0582758	−	0.998301i	$-0.481440\pi$
$599$	2.85254	0.116552	0.0582758	+	0.998301i	$0.481440\pi$
$600$	0	0
$601$	43.2661	1.76486	0.882430	−	0.470443i	$-0.155906\pi$
$601$	43.2661	1.76486	0.882430	+	0.470443i	$0.155906\pi$
$602$	0	0
$603$	7.74883	0.315557
$604$	0	0
$605$	9.41734	0.382869
$606$	0	0
$607$	−36.7948	−1.49346	−0.746728	−	0.665130i	$-0.768376\pi$
$607$	−36.7948	−1.49346	−0.746728	+	0.665130i	$0.768376\pi$
$608$	0	0
$609$	57.7041	2.33829
$610$	0	0
$611$	1.39712	0.0565216
$612$	0	0
$613$	−26.5668	−1.07302	−0.536511	−	0.843893i	$-0.680258\pi$
$613$	−26.5668	−1.07302	−0.536511	+	0.843893i	$0.680258\pi$
$614$	0	0
$615$	20.3658	0.821229
$616$	0	0
$617$	1.11508	0.0448916	0.0224458	−	0.999748i	$-0.492855\pi$
$617$	1.11508	0.0448916	0.0224458	+	0.999748i	$0.492855\pi$
$618$	0	0
$619$	−13.2792	−0.533736	−0.266868	−	0.963733i	$-0.585989\pi$
$619$	−13.2792	−0.533736	−0.266868	+	0.963733i	$0.585989\pi$
$620$	0	0
$621$	−1.24706	−0.0500428
$622$	0	0
$623$	46.4598	1.86137
$624$	0	0
$625$	17.7271	0.709082
$626$	0	0
$627$	30.6386	1.22359
$628$	0	0
$629$	17.4233	0.694714
$630$	0	0
$631$	26.4250	1.05196	0.525982	−	0.850496i	$-0.323698\pi$
$631$	26.4250	1.05196	0.525982	+	0.850496i	$0.323698\pi$
$632$	0	0
$633$	8.39963	0.333855
$634$	0	0
$635$	−4.59705	−0.182429
$636$	0	0
$637$	−1.20035	−0.0475595
$638$	0	0
$639$	40.0039	1.58253
$640$	0	0
$641$	−29.9336	−1.18230	−0.591152	−	0.806560i	$-0.701327\pi$
$641$	−29.9336	−1.18230	−0.591152	+	0.806560i	$0.701327\pi$
$642$	0	0
$643$	31.7782	1.25321	0.626606	−	0.779336i	$-0.284444\pi$
$643$	31.7782	1.25321	0.626606	+	0.779336i	$0.284444\pi$
$644$	0	0
$645$	7.81135	0.307572
$646$	0	0
$647$	−40.1262	−1.57753	−0.788763	−	0.614698i	$-0.789278\pi$
$647$	−40.1262	−1.57753	−0.788763	+	0.614698i	$0.789278\pi$
$648$	0	0
$649$	74.0659	2.90734
$650$	0	0
$651$	−90.0508	−3.52937
$652$	0	0
$653$	−12.8925	−0.504521	−0.252260	−	0.967659i	$-0.581174\pi$
$653$	−12.8925	−0.504521	−0.252260	+	0.967659i	$0.581174\pi$
$654$	0	0
$655$	6.63342	0.259189
$656$	0	0
$657$	−23.6837	−0.923988
$658$	0	0
$659$	−7.67257	−0.298881	−0.149440	−	0.988771i	$-0.547747\pi$
$659$	−7.67257	−0.298881	−0.149440	+	0.988771i	$0.547747\pi$
$660$	0	0
$661$	−28.7887	−1.11975	−0.559875	−	0.828577i	$-0.689151\pi$
$661$	−28.7887	−1.11975	−0.559875	+	0.828577i	$0.689151\pi$
$662$	0	0
$663$	−0.855568	−0.0332275
$664$	0	0
$665$	−6.38900	−0.247755
$666$	0	0
$667$	12.2144	0.472945
$668$	0	0
$669$	−73.1401	−2.82776
$670$	0	0
$671$	−7.44284	−0.287327
$672$	0	0
$673$	−5.35674	−0.206487	−0.103244	−	0.994656i	$-0.532922\pi$
$673$	−5.35674	−0.206487	−0.103244	+	0.994656i	$0.532922\pi$
$674$	0	0
$675$	−2.91752	−0.112296
$676$	0	0
$677$	8.14007	0.312848	0.156424	−	0.987690i	$-0.450003\pi$
$677$	8.14007	0.312848	0.156424	+	0.987690i	$0.450003\pi$
$678$	0	0
$679$	33.6688	1.29209
$680$	0	0
$681$	60.2070	2.30714
$682$	0	0
$683$	−8.42132	−0.322233	−0.161116	−	0.986935i	$-0.551509\pi$
$683$	−8.42132	−0.322233	−0.161116	+	0.986935i	$0.551509\pi$
$684$	0	0
$685$	−9.66240	−0.369181
$686$	0	0
$687$	32.6588	1.24601
$688$	0	0
$689$	−1.13633	−0.0432908
$690$	0	0
$691$	−12.3286	−0.469002	−0.234501	−	0.972116i	$-0.575346\pi$
$691$	−12.3286	−0.469002	−0.234501	+	0.972116i	$0.575346\pi$
$692$	0	0
$693$	−58.3292	−2.21574
$694$	0	0
$695$	−5.22733	−0.198284
$696$	0	0
$697$	20.2441	0.766800
$698$	0	0
$699$	−58.4362	−2.21026
$700$	0	0
$701$	−4.22882	−0.159721	−0.0798603	−	0.996806i	$-0.525447\pi$
$701$	−4.22882	−0.159721	−0.0798603	+	0.996806i	$0.525447\pi$
$702$	0	0
$703$	24.5763	0.926912
$704$	0	0
$705$	−12.7512	−0.480238
$706$	0	0
$707$	24.9345	0.937757
$708$	0	0
$709$	−3.29968	−0.123922	−0.0619610	−	0.998079i	$-0.519735\pi$
$709$	−3.29968	−0.123922	−0.0619610	+	0.998079i	$0.519735\pi$
$710$	0	0
$711$	−27.7366	−1.04021
$712$	0	0
$713$	−19.0614	−0.713855
$714$	0	0
$715$	−0.677808	−0.0253486
$716$	0	0
$717$	10.5759	0.394965
$718$	0	0
$719$	12.9747	0.483874	0.241937	−	0.970292i	$-0.422217\pi$
$719$	12.9747	0.483874	0.241937	+	0.970292i	$0.422217\pi$
$720$	0	0
$721$	58.6341	2.18365
$722$	0	0
$723$	−17.6291	−0.655635
$724$	0	0
$725$	28.5760	1.06128
$726$	0	0
$727$	44.2490	1.64110	0.820551	−	0.571573i	$-0.193667\pi$
$727$	44.2490	1.64110	0.820551	+	0.571573i	$0.193667\pi$
$728$	0	0
$729$	−31.4473	−1.16471
$730$	0	0
$731$	7.76467	0.287187
$732$	0	0
$733$	−5.72315	−0.211389	−0.105695	−	0.994399i	$-0.533707\pi$
$733$	−5.72315	−0.211389	−0.105695	+	0.994399i	$0.533707\pi$
$734$	0	0
$735$	10.9553	0.404091
$736$	0	0
$737$	11.7190	0.431675
$738$	0	0
$739$	15.7959	0.581060	0.290530	−	0.956866i	$-0.406168\pi$
$739$	15.7959	0.581060	0.290530	+	0.956866i	$0.406168\pi$
$740$	0	0
$741$	−1.20681	−0.0443333
$742$	0	0
$743$	25.8582	0.948644	0.474322	−	0.880352i	$-0.342693\pi$
$743$	25.8582	0.948644	0.474322	+	0.880352i	$0.342693\pi$
$744$	0	0
$745$	4.02264	0.147378
$746$	0	0
$747$	39.0840	1.43001
$748$	0	0
$749$	−63.8312	−2.33234
$750$	0	0
$751$	−35.6589	−1.30121	−0.650606	−	0.759415i	$-0.725485\pi$
$751$	−35.6589	−1.30121	−0.650606	+	0.759415i	$0.725485\pi$
$752$	0	0
$753$	2.50185	0.0911724
$754$	0	0
$755$	11.3042	0.411401
$756$	0	0
$757$	11.2988	0.410661	0.205331	−	0.978693i	$-0.434173\pi$
$757$	11.2988	0.410661	0.205331	+	0.978693i	$0.434173\pi$
$758$	0	0
$759$	−23.7115	−0.860672
$760$	0	0
$761$	−16.4750	−0.597217	−0.298608	−	0.954376i	$-0.596522\pi$
$761$	−16.4750	−0.597217	−0.298608	+	0.954376i	$0.596522\pi$
$762$	0	0
$763$	−60.2046	−2.17955
$764$	0	0
$765$	4.06598	0.147006
$766$	0	0
$767$	−2.91735	−0.105339
$768$	0	0
$769$	−17.3686	−0.626327	−0.313164	−	0.949699i	$-0.601389\pi$
$769$	−17.3686	−0.626327	−0.313164	+	0.949699i	$0.601389\pi$
$770$	0	0
$771$	48.8746	1.76018
$772$	0	0
$773$	36.6737	1.31906	0.659531	−	0.751678i	$-0.270755\pi$
$773$	36.6737	1.31906	0.659531	+	0.751678i	$0.270755\pi$
$774$	0	0
$775$	−44.5945	−1.60188
$776$	0	0
$777$	−89.8542	−3.22350
$778$	0	0
$779$	28.5551	1.02309
$780$	0	0
$781$	60.5002	2.16487
$782$	0	0
$783$	−4.12009	−0.147240
$784$	0	0
$785$	−6.02048	−0.214880
$786$	0	0
$787$	−44.8445	−1.59854	−0.799268	−	0.600975i	$-0.794779\pi$
$787$	−44.8445	−1.59854	−0.799268	+	0.600975i	$0.794779\pi$
$788$	0	0
$789$	6.26626	0.223085
$790$	0	0
$791$	−4.46055	−0.158599
$792$	0	0
$793$	0.293163	0.0104105
$794$	0	0
$795$	10.3710	0.367822
$796$	0	0
$797$	−23.1703	−0.820736	−0.410368	−	0.911920i	$-0.634600\pi$
$797$	−23.1703	−0.820736	−0.410368	+	0.911920i	$0.634600\pi$
$798$	0	0
$799$	−12.6750	−0.448409
$800$	0	0
$801$	−41.7056	−1.47359
$802$	0	0
$803$	−35.8182	−1.26400
$804$	0	0
$805$	4.94450	0.174271
$806$	0	0
$807$	19.0243	0.669686
$808$	0	0
$809$	−15.2524	−0.536244	−0.268122	−	0.963385i	$-0.586403\pi$
$809$	−15.2524	−0.536244	−0.268122	+	0.963385i	$0.586403\pi$
$810$	0	0
$811$	1.85778	0.0652356	0.0326178	−	0.999468i	$-0.489616\pi$
$811$	1.85778	0.0652356	0.0326178	+	0.999468i	$0.489616\pi$
$812$	0	0
$813$	31.9519	1.12060
$814$	0	0
$815$	9.41628	0.329838
$816$	0	0
$817$	10.9524	0.383175
$818$	0	0
$819$	2.29750	0.0802813
$820$	0	0
$821$	−26.9542	−0.940709	−0.470355	−	0.882477i	$-0.655874\pi$
$821$	−26.9542	−0.940709	−0.470355	+	0.882477i	$0.655874\pi$
$822$	0	0
$823$	−29.1031	−1.01447	−0.507235	−	0.861808i	$-0.669332\pi$
$823$	−29.1031	−1.01447	−0.507235	+	0.861808i	$0.669332\pi$
$824$	0	0
$825$	−55.4735	−1.93134
$826$	0	0
$827$	13.5785	0.472172	0.236086	−	0.971732i	$-0.424135\pi$
$827$	13.5785	0.472172	0.236086	+	0.971732i	$0.424135\pi$
$828$	0	0
$829$	−16.7623	−0.582178	−0.291089	−	0.956696i	$-0.594018\pi$
$829$	−16.7623	−0.582178	−0.291089	+	0.956696i	$0.594018\pi$
$830$	0	0
$831$	31.2642	1.08454
$832$	0	0
$833$	10.8898	0.377309
$834$	0	0
$835$	15.6437	0.541374
$836$	0	0
$837$	6.42965	0.222241
$838$	0	0
$839$	5.96928	0.206082	0.103041	−	0.994677i	$-0.467143\pi$
$839$	5.96928	0.206082	0.103041	+	0.994677i	$0.467143\pi$
$840$	0	0
$841$	11.3546	0.391538
$842$	0	0
$843$	−16.9669	−0.584372
$844$	0	0
$845$	−9.18075	−0.315827
$846$	0	0
$847$	−48.2761	−1.65879
$848$	0	0
$849$	−12.7175	−0.436463
$850$	0	0
$851$	−19.0198	−0.651989
$852$	0	0
$853$	2.43052	0.0832195	0.0416098	−	0.999134i	$-0.486751\pi$
$853$	2.43052	0.0832195	0.0416098	+	0.999134i	$0.486751\pi$
$854$	0	0
$855$	5.73522	0.196140
$856$	0	0
$857$	34.5403	1.17988	0.589938	−	0.807449i	$-0.299152\pi$
$857$	34.5403	1.17988	0.589938	+	0.807449i	$0.299152\pi$
$858$	0	0
$859$	44.8663	1.53082	0.765410	−	0.643543i	$-0.222536\pi$
$859$	44.8663	1.53082	0.765410	+	0.643543i	$0.222536\pi$
$860$	0	0
$861$	−104.401	−3.55798
$862$	0	0
$863$	9.43890	0.321304	0.160652	−	0.987011i	$-0.448640\pi$
$863$	9.43890	0.321304	0.160652	+	0.987011i	$0.448640\pi$
$864$	0	0
$865$	−13.1251	−0.446265
$866$	0	0
$867$	−34.7695	−1.18083
$868$	0	0
$869$	−41.9477	−1.42298
$870$	0	0
$871$	−0.461594	−0.0156405
$872$	0	0
$873$	−30.2235	−1.02291
$874$	0	0
$875$	24.4255	0.825733
$876$	0	0
$877$	−44.0156	−1.48630	−0.743150	−	0.669125i	$-0.766669\pi$
$877$	−44.0156	−1.48630	−0.743150	+	0.669125i	$0.766669\pi$
$878$	0	0
$879$	35.4635	1.19615
$880$	0	0
$881$	43.2200	1.45612	0.728059	−	0.685515i	$-0.240423\pi$
$881$	43.2200	1.45612	0.728059	+	0.685515i	$0.240423\pi$
$882$	0	0
$883$	8.18810	0.275551	0.137776	−	0.990463i	$-0.456005\pi$
$883$	8.18810	0.275551	0.137776	+	0.990463i	$0.456005\pi$
$884$	0	0
$885$	26.6259	0.895020
$886$	0	0
$887$	−34.7143	−1.16559	−0.582796	−	0.812618i	$-0.698041\pi$
$887$	−34.7143	−1.16559	−0.582796	+	0.812618i	$0.698041\pi$
$888$	0	0
$889$	23.5659	0.790374
$890$	0	0
$891$	−40.1975	−1.34667
$892$	0	0
$893$	−17.8786	−0.598284
$894$	0	0
$895$	−5.60382	−0.187315
$896$	0	0
$897$	0.933960	0.0311840
$898$	0	0
$899$	−62.9758	−2.10036
$900$	0	0
$901$	10.3090	0.343444
$902$	0	0
$903$	−40.0433	−1.33256
$904$	0	0
$905$	−7.01888	−0.233315
$906$	0	0
$907$	−36.9689	−1.22753	−0.613765	−	0.789488i	$-0.710346\pi$
$907$	−36.9689	−1.22753	−0.613765	+	0.789488i	$0.710346\pi$
$908$	0	0
$909$	−22.3829	−0.742395
$910$	0	0
$911$	−43.2185	−1.43189	−0.715946	−	0.698156i	$-0.754004\pi$
$911$	−43.2185	−1.43189	−0.715946	+	0.698156i	$0.754004\pi$
$912$	0	0
$913$	59.1090	1.95622
$914$	0	0
$915$	−2.67562	−0.0884532
$916$	0	0
$917$	−34.0049	−1.12294
$918$	0	0
$919$	28.7728	0.949127	0.474564	−	0.880221i	$-0.342606\pi$
$919$	28.7728	0.949127	0.474564	+	0.880221i	$0.342606\pi$
$920$	0	0
$921$	−55.5103	−1.82913
$922$	0	0
$923$	−2.38302	−0.0784379
$924$	0	0
$925$	−44.4972	−1.46306
$926$	0	0
$927$	−52.6341	−1.72873
$928$	0	0
$929$	51.1374	1.67776	0.838881	−	0.544314i	$-0.183210\pi$
$929$	51.1374	1.67776	0.838881	+	0.544314i	$0.183210\pi$
$930$	0	0
$931$	15.3605	0.503419
$932$	0	0
$933$	27.5447	0.901773
$934$	0	0
$935$	6.14921	0.201101
$936$	0	0
$937$	23.7919	0.777246	0.388623	−	0.921397i	$-0.372951\pi$
$937$	23.7919	0.777246	0.388623	+	0.921397i	$0.372951\pi$
$938$	0	0
$939$	17.6740	0.576769
$940$	0	0
$941$	17.4595	0.569164	0.284582	−	0.958652i	$-0.408145\pi$
$941$	17.4595	0.569164	0.284582	+	0.958652i	$0.408145\pi$
$942$	0	0
$943$	−22.0990	−0.719642
$944$	0	0
$945$	−1.66784	−0.0542549
$946$	0	0
$947$	−15.9339	−0.517784	−0.258892	−	0.965906i	$-0.583357\pi$
$947$	−15.9339	−0.517784	−0.258892	+	0.965906i	$0.583357\pi$
$948$	0	0
$949$	1.41083	0.0457973
$950$	0	0
$951$	−70.7884	−2.29547
$952$	0	0
$953$	−19.9351	−0.645761	−0.322881	−	0.946440i	$-0.604651\pi$
$953$	−19.9351	−0.645761	−0.322881	+	0.946440i	$0.604651\pi$
$954$	0	0
$955$	1.40914	0.0455987
$956$	0	0
$957$	−78.3389	−2.53234
$958$	0	0
$959$	49.5323	1.59948
$960$	0	0
$961$	67.2777	2.17025
$962$	0	0
$963$	57.2994	1.84645
$964$	0	0
$965$	−2.70068	−0.0869379
$966$	0	0
$967$	−56.5192	−1.81753	−0.908767	−	0.417303i	$-0.862975\pi$
$967$	−56.5192	−1.81753	−0.908767	+	0.417303i	$0.862975\pi$
$968$	0	0
$969$	10.9484	0.351714
$970$	0	0
$971$	26.5268	0.851286	0.425643	−	0.904891i	$-0.360048\pi$
$971$	26.5268	0.851286	0.425643	+	0.904891i	$0.360048\pi$
$972$	0	0
$973$	26.7968	0.859067
$974$	0	0
$975$	2.18502	0.0699767
$976$	0	0
$977$	−49.3417	−1.57858	−0.789291	−	0.614019i	$-0.789552\pi$
$977$	−49.3417	−1.57858	−0.789291	+	0.614019i	$0.789552\pi$
$978$	0	0
$979$	−63.0738	−2.01585
$980$	0	0
$981$	54.0439	1.72549
$982$	0	0
$983$	−49.4224	−1.57633	−0.788166	−	0.615463i	$-0.788969\pi$
$983$	−49.4224	−1.57633	−0.788166	+	0.615463i	$0.788969\pi$
$984$	0	0
$985$	11.9258	0.379989
$986$	0	0
$987$	65.3664	2.08064
$988$	0	0
$989$	−8.47611	−0.269525
$990$	0	0
$991$	−36.6203	−1.16328	−0.581641	−	0.813446i	$-0.697589\pi$
$991$	−36.6203	−1.16328	−0.581641	+	0.813446i	$0.697589\pi$
$992$	0	0
$993$	−24.0881	−0.764414
$994$	0	0
$995$	−13.8028	−0.437577
$996$	0	0
$997$	46.0130	1.45725	0.728623	−	0.684915i	$-0.240161\pi$
$997$	46.0130	1.45725	0.728623	+	0.684915i	$0.240161\pi$
$998$	0	0
$999$	6.41561	0.202981

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.l.1.16		19
4.3	odd	2		2008.2.a.c.1.4	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.c.1.4	✓	19	4.3	odd	2
4016.2.a.l.1.16		19	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+2.50185 q^{3} +0.708265 q^{5} -3.63078 q^{7} +3.25924 q^{9} +O(q^{10})\)	\(q+2.50185 q^{3} +0.708265 q^{5} -3.63078 q^{7} +3.25924 q^{9} +4.92913 q^{11} -0.194151 q^{13} +1.77197 q^{15} +1.76138 q^{17} +2.48449 q^{19} -9.08365 q^{21} -1.92277 q^{23} -4.49836 q^{25} +0.648575 q^{27} -6.35253 q^{29} +9.91351 q^{31} +12.3319 q^{33} -2.57155 q^{35} +9.89186 q^{37} -0.485737 q^{39} +11.4933 q^{41} +4.40828 q^{43} +2.30840 q^{45} -7.19606 q^{47} +6.18253 q^{49} +4.40670 q^{51} +5.85281 q^{53} +3.49113 q^{55} +6.21583 q^{57} +15.0262 q^{59} -1.50997 q^{61} -11.8336 q^{63} -0.137511 q^{65} +2.37750 q^{67} -4.81047 q^{69} +12.2740 q^{71} -7.26662 q^{73} -11.2542 q^{75} -17.8966 q^{77} -8.51016 q^{79} -8.15508 q^{81} +11.9918 q^{83} +1.24752 q^{85} -15.8930 q^{87} -12.7961 q^{89} +0.704920 q^{91} +24.8021 q^{93} +1.75968 q^{95} -9.27317 q^{97} +16.0652 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) \) 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9} + 15 q^{11} - 8 q^{13} + 17 q^{15} - 4 q^{17} + 14 q^{19} - 9 q^{21} + 28 q^{23} + 25 q^{25} + 21 q^{27} - 13 q^{29} + 20 q^{31} - 6 q^{33} + 32 q^{35} - 16 q^{37} + 27 q^{39} + 2 q^{41} + 28 q^{43} - 29 q^{45} + 37 q^{47} + 36 q^{49} + 35 q^{51} - 37 q^{53} + 24 q^{55} - 11 q^{57} + 32 q^{59} - 7 q^{61} + 45 q^{63} + q^{65} + 45 q^{67} - 12 q^{69} + 49 q^{71} + 16 q^{73} + 35 q^{75} - 40 q^{77} + 33 q^{79} + 15 q^{81} + 43 q^{83} - 28 q^{85} + 48 q^{87} + 3 q^{89} + 56 q^{91} - 48 q^{93} + 43 q^{95} + 8 q^{97} + 74 q^{99}+O(q^{100}) \) 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9 + 15 * q^11 - 8 * q^13 + 17 * q^15 - 4 * q^17 + 14 * q^19 - 9 * q^21 + 28 * q^23 + 25 * q^25 + 21 * q^27 - 13 * q^29 + 20 * q^31 - 6 * q^33 + 32 * q^35 - 16 * q^37 + 27 * q^39 + 2 * q^41 + 28 * q^43 - 29 * q^45 + 37 * q^47 + 36 * q^49 + 35 * q^51 - 37 * q^53 + 24 * q^55 - 11 * q^57 + 32 * q^59 - 7 * q^61 + 45 * q^63 + q^65 + 45 * q^67 - 12 * q^69 + 49 * q^71 + 16 * q^73 + 35 * q^75 - 40 * q^77 + 33 * q^79 + 15 * q^81 + 43 * q^83 - 28 * q^85 + 48 * q^87 + 3 * q^89 + 56 * q^91 - 48 * q^93 + 43 * q^95 + 8 * q^97 + 74 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more