LMFDB - Embedded newform 4016.2.a.l.1.8

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			1.8
Root			$-0.312548$ 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)

$f(q)$	$=$	$q-0.312548 q^{3} +0.780222 q^{5} -4.05339 q^{7} -2.90231 q^{9} +O(q^{10})$	$q-0.312548 q^{3} +0.780222 q^{5} -4.05339 q^{7} -2.90231 q^{9} -0.0734478 q^{11} -6.70045 q^{13} -0.243857 q^{15} -6.03358 q^{17} +4.11909 q^{19} +1.26688 q^{21} +3.72623 q^{23} -4.39125 q^{25} +1.84476 q^{27} +2.88693 q^{29} +2.44904 q^{31} +0.0229560 q^{33} -3.16255 q^{35} -8.43214 q^{37} +2.09421 q^{39} -2.53725 q^{41} +1.58790 q^{43} -2.26445 q^{45} +4.24982 q^{47} +9.43000 q^{49} +1.88578 q^{51} +7.43071 q^{53} -0.0573056 q^{55} -1.28741 q^{57} -0.236686 q^{59} +12.7137 q^{61} +11.7642 q^{63} -5.22784 q^{65} +3.00209 q^{67} -1.16463 q^{69} -12.7361 q^{71} +13.2021 q^{73} +1.37248 q^{75} +0.297713 q^{77} -12.9823 q^{79} +8.13037 q^{81} +14.3903 q^{83} -4.70753 q^{85} -0.902305 q^{87} -4.02154 q^{89} +27.1596 q^{91} -0.765443 q^{93} +3.21380 q^{95} -0.211554 q^{97} +0.213168 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.312548	−0.180450	−0.0902249	−	0.995921i	$-0.528759\pi$
$3$	−0.312548	−0.180450	−0.0902249	+	0.995921i	$0.528759\pi$
$4$	0	0
$5$	0.780222	0.348926	0.174463	−	0.984664i	$-0.444181\pi$
$5$	0.780222	0.348926	0.174463	+	0.984664i	$0.444181\pi$
$6$	0	0
$7$	−4.05339	−1.53204	−0.766020	−	0.642817i	$-0.777765\pi$
$7$	−4.05339	−1.53204	−0.766020	+	0.642817i	$0.777765\pi$
$8$	0	0
$9$	−2.90231	−0.967438
$10$	0	0
$11$	−0.0734478	−0.0221453	−0.0110727	−	0.999939i	$-0.503525\pi$
$11$	−0.0734478	−0.0221453	−0.0110727	+	0.999939i	$0.503525\pi$
$12$	0	0
$13$	−6.70045	−1.85837	−0.929185	−	0.369614i	$-0.879490\pi$
$13$	−6.70045	−1.85837	−0.929185	+	0.369614i	$0.879490\pi$
$14$	0	0
$15$	−0.243857	−0.0629636
$16$	0	0
$17$	−6.03358	−1.46336	−0.731679	−	0.681649i	$-0.761263\pi$
$17$	−6.03358	−1.46336	−0.731679	+	0.681649i	$0.761263\pi$
$18$	0	0
$19$	4.11909	0.944983	0.472492	−	0.881335i	$-0.343355\pi$
$19$	4.11909	0.944983	0.472492	+	0.881335i	$0.343355\pi$
$20$	0	0
$21$	1.26688	0.276456
$22$	0	0
$23$	3.72623	0.776973	0.388486	−	0.921454i	$-0.372998\pi$
$23$	3.72623	0.776973	0.388486	+	0.921454i	$0.372998\pi$
$24$	0	0
$25$	−4.39125	−0.878251
$26$	0	0
$27$	1.84476	0.355024
$28$	0	0
$29$	2.88693	0.536089	0.268045	−	0.963406i	$-0.413623\pi$
$29$	2.88693	0.536089	0.268045	+	0.963406i	$0.413623\pi$
$30$	0	0
$31$	2.44904	0.439860	0.219930	−	0.975516i	$-0.429417\pi$
$31$	2.44904	0.439860	0.219930	+	0.975516i	$0.429417\pi$
$32$	0	0
$33$	0.0229560	0.00399612
$34$	0	0
$35$	−3.16255	−0.534568
$36$	0	0
$37$	−8.43214	−1.38623	−0.693117	−	0.720825i	$-0.743763\pi$
$37$	−8.43214	−1.38623	−0.693117	+	0.720825i	$0.743763\pi$
$38$	0	0
$39$	2.09421	0.335343
$40$	0	0
$41$	−2.53725	−0.396252	−0.198126	−	0.980177i	$-0.563485\pi$
$41$	−2.53725	−0.396252	−0.198126	+	0.980177i	$0.563485\pi$
$42$	0	0
$43$	1.58790	0.242152	0.121076	−	0.992643i	$-0.461365\pi$
$43$	1.58790	0.242152	0.121076	+	0.992643i	$0.461365\pi$
$44$	0	0
$45$	−2.26445	−0.337564
$46$	0	0
$47$	4.24982	0.619900	0.309950	−	0.950753i	$-0.399688\pi$
$47$	4.24982	0.619900	0.309950	+	0.950753i	$0.399688\pi$
$48$	0	0
$49$	9.43000	1.34714
$50$	0	0
$51$	1.88578	0.264063
$52$	0	0
$53$	7.43071	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$53$	7.43071	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$54$	0	0
$55$	−0.0573056	−0.00772708
$56$	0	0
$57$	−1.28741	−0.170522
$58$	0	0
$59$	−0.236686	−0.0308138	−0.0154069	−	0.999881i	$-0.504904\pi$
$59$	−0.236686	−0.0308138	−0.0154069	+	0.999881i	$0.504904\pi$
$60$	0	0
$61$	12.7137	1.62782	0.813912	−	0.580988i	$-0.197334\pi$
$61$	12.7137	1.62782	0.813912	+	0.580988i	$0.197334\pi$
$62$	0	0
$63$	11.7642	1.48215
$64$	0	0
$65$	−5.22784	−0.648434
$66$	0	0
$67$	3.00209	0.366764	0.183382	−	0.983042i	$-0.441295\pi$
$67$	3.00209	0.366764	0.183382	+	0.983042i	$0.441295\pi$
$68$	0	0
$69$	−1.16463	−0.140205
$70$	0	0
$71$	−12.7361	−1.51150	−0.755749	−	0.654862i	$-0.772727\pi$
$71$	−12.7361	−1.51150	−0.755749	+	0.654862i	$0.772727\pi$
$72$	0	0
$73$	13.2021	1.54519	0.772593	−	0.634902i	$-0.218960\pi$
$73$	13.2021	1.54519	0.772593	+	0.634902i	$0.218960\pi$
$74$	0	0
$75$	1.37248	0.158480
$76$	0	0
$77$	0.297713	0.0339275
$78$	0	0
$79$	−12.9823	−1.46062	−0.730311	−	0.683115i	$-0.760625\pi$
$79$	−12.9823	−1.46062	−0.730311	+	0.683115i	$0.760625\pi$
$80$	0	0
$81$	8.13037	0.903374
$82$	0	0
$83$	14.3903	1.57954	0.789771	−	0.613402i	$-0.210199\pi$
$83$	14.3903	1.57954	0.789771	+	0.613402i	$0.210199\pi$
$84$	0	0
$85$	−4.70753	−0.510604
$86$	0	0
$87$	−0.902305	−0.0967372
$88$	0	0
$89$	−4.02154	−0.426282	−0.213141	−	0.977021i	$-0.568369\pi$
$89$	−4.02154	−0.426282	−0.213141	+	0.977021i	$0.568369\pi$
$90$	0	0
$91$	27.1596	2.84710
$92$	0	0
$93$	−0.765443	−0.0793727
$94$	0	0
$95$	3.21380	0.329729
$96$	0	0
$97$	−0.211554	−0.0214800	−0.0107400	−	0.999942i	$-0.503419\pi$
$97$	−0.211554	−0.0214800	−0.0107400	+	0.999942i	$0.503419\pi$
$98$	0	0
$99$	0.213168	0.0214242
$100$	0	0
$101$	19.6498	1.95523	0.977613	−	0.210413i	$-0.0674808\pi$
$101$	19.6498	1.95523	0.977613	+	0.210413i	$0.0674808\pi$
$102$	0	0
$103$	−10.3523	−1.02004	−0.510022	−	0.860161i	$-0.670363\pi$
$103$	−10.3523	−1.02004	−0.510022	+	0.860161i	$0.670363\pi$
$104$	0	0
$105$	0.988448	0.0964627
$106$	0	0
$107$	17.5796	1.69948	0.849742	−	0.527198i	$-0.176757\pi$
$107$	17.5796	1.69948	0.849742	+	0.527198i	$0.176757\pi$
$108$	0	0
$109$	−10.4322	−0.999222	−0.499611	−	0.866250i	$-0.666524\pi$
$109$	−10.4322	−0.999222	−0.499611	+	0.866250i	$0.666524\pi$
$110$	0	0
$111$	2.63545	0.250146
$112$	0	0
$113$	13.5780	1.27731	0.638657	−	0.769492i	$-0.279490\pi$
$113$	13.5780	1.27731	0.638657	+	0.769492i	$0.279490\pi$
$114$	0	0
$115$	2.90729	0.271106
$116$	0	0
$117$	19.4468	1.79786
$118$	0	0
$119$	24.4565	2.24192
$120$	0	0
$121$	−10.9946	−0.999510
$122$	0	0
$123$	0.793012	0.0715035
$124$	0	0
$125$	−7.32726	−0.655370
$126$	0	0
$127$	−0.326628	−0.0289836	−0.0144918	−	0.999895i	$-0.504613\pi$
$127$	−0.326628	−0.0289836	−0.0144918	+	0.999895i	$0.504613\pi$
$128$	0	0
$129$	−0.496295	−0.0436964
$130$	0	0
$131$	9.06940	0.792397	0.396198	−	0.918165i	$-0.370329\pi$
$131$	9.06940	0.792397	0.396198	+	0.918165i	$0.370329\pi$
$132$	0	0
$133$	−16.6963	−1.44775
$134$	0	0
$135$	1.43932	0.123877
$136$	0	0
$137$	−13.0895	−1.11831	−0.559157	−	0.829062i	$-0.688875\pi$
$137$	−13.0895	−1.11831	−0.559157	+	0.829062i	$0.688875\pi$
$138$	0	0
$139$	−18.5764	−1.57563	−0.787813	−	0.615915i	$-0.788787\pi$
$139$	−18.5764	−1.57563	−0.787813	+	0.615915i	$0.788787\pi$
$140$	0	0
$141$	−1.32827	−0.111861
$142$	0	0
$143$	0.492133	0.0411542
$144$	0	0
$145$	2.25245	0.187055
$146$	0	0
$147$	−2.94733	−0.243092
$148$	0	0
$149$	−13.8779	−1.13692	−0.568459	−	0.822712i	$-0.692460\pi$
$149$	−13.8779	−1.13692	−0.568459	+	0.822712i	$0.692460\pi$
$150$	0	0
$151$	−5.01180	−0.407855	−0.203927	−	0.978986i	$-0.565371\pi$
$151$	−5.01180	−0.407855	−0.203927	+	0.978986i	$0.565371\pi$
$152$	0	0
$153$	17.5113	1.41571
$154$	0	0
$155$	1.91079	0.153479
$156$	0	0
$157$	−17.4311	−1.39115	−0.695575	−	0.718454i	$-0.744850\pi$
$157$	−17.4311	−1.39115	−0.695575	+	0.718454i	$0.744850\pi$
$158$	0	0
$159$	−2.32245	−0.184183
$160$	0	0
$161$	−15.1039	−1.19035
$162$	0	0
$163$	−22.7775	−1.78407	−0.892037	−	0.451962i	$-0.850724\pi$
$163$	−22.7775	−1.78407	−0.892037	+	0.451962i	$0.850724\pi$
$164$	0	0
$165$	0.0179108	0.00139435
$166$	0	0
$167$	13.6944	1.05970	0.529851	−	0.848091i	$-0.322248\pi$
$167$	13.6944	1.05970	0.529851	+	0.848091i	$0.322248\pi$
$168$	0	0
$169$	31.8960	2.45354
$170$	0	0
$171$	−11.9549	−0.914213
$172$	0	0
$173$	−6.62897	−0.503991	−0.251996	−	0.967728i	$-0.581087\pi$
$173$	−6.62897	−0.503991	−0.251996	+	0.967728i	$0.581087\pi$
$174$	0	0
$175$	17.7995	1.34551
$176$	0	0
$177$	0.0739757	0.00556035
$178$	0	0
$179$	18.2009	1.36040	0.680201	−	0.733025i	$-0.261892\pi$
$179$	18.2009	1.36040	0.680201	+	0.733025i	$0.261892\pi$
$180$	0	0
$181$	10.1700	0.755932	0.377966	−	0.925819i	$-0.376624\pi$
$181$	10.1700	0.755932	0.377966	+	0.925819i	$0.376624\pi$
$182$	0	0
$183$	−3.97365	−0.293741
$184$	0	0
$185$	−6.57894	−0.483693
$186$	0	0
$187$	0.443153	0.0324066
$188$	0	0
$189$	−7.47753	−0.543910
$190$	0	0
$191$	18.4226	1.33301	0.666505	−	0.745501i	$-0.267790\pi$
$191$	18.4226	1.33301	0.666505	+	0.745501i	$0.267790\pi$
$192$	0	0
$193$	−20.2356	−1.45659	−0.728297	−	0.685262i	$-0.759688\pi$
$193$	−20.2356	−1.45659	−0.728297	+	0.685262i	$0.759688\pi$
$194$	0	0
$195$	1.63395	0.117010
$196$	0	0
$197$	−14.0673	−1.00225	−0.501126	−	0.865374i	$-0.667081\pi$
$197$	−14.0673	−1.00225	−0.501126	+	0.865374i	$0.667081\pi$
$198$	0	0
$199$	1.13625	0.0805465	0.0402733	−	0.999189i	$-0.487177\pi$
$199$	1.13625	0.0805465	0.0402733	+	0.999189i	$0.487177\pi$
$200$	0	0
$201$	−0.938299	−0.0661825
$202$	0	0
$203$	−11.7019	−0.821310
$204$	0	0
$205$	−1.97962	−0.138262
$206$	0	0
$207$	−10.8147	−0.751673
$208$	0	0
$209$	−0.302538	−0.0209270
$210$	0	0
$211$	26.7118	1.83892	0.919458	−	0.393188i	$-0.128628\pi$
$211$	26.7118	1.83892	0.919458	+	0.393188i	$0.128628\pi$
$212$	0	0
$213$	3.98065	0.272749
$214$	0	0
$215$	1.23891	0.0844932
$216$	0	0
$217$	−9.92692	−0.673883
$218$	0	0
$219$	−4.12628	−0.278828
$220$	0	0
$221$	40.4277	2.71946
$222$	0	0
$223$	−9.47532	−0.634515	−0.317257	−	0.948340i	$-0.602762\pi$
$223$	−9.47532	−0.634515	−0.317257	+	0.948340i	$0.602762\pi$
$224$	0	0
$225$	12.7448	0.849653
$226$	0	0
$227$	−18.8249	−1.24946	−0.624728	−	0.780843i	$-0.714790\pi$
$227$	−18.8249	−1.24946	−0.624728	+	0.780843i	$0.714790\pi$
$228$	0	0
$229$	2.95530	0.195292	0.0976459	−	0.995221i	$-0.468869\pi$
$229$	2.95530	0.195292	0.0976459	+	0.995221i	$0.468869\pi$
$230$	0	0
$231$	−0.0930496	−0.00612221
$232$	0	0
$233$	17.1275	1.12206	0.561031	−	0.827795i	$-0.310405\pi$
$233$	17.1275	1.12206	0.561031	+	0.827795i	$0.310405\pi$
$234$	0	0
$235$	3.31580	0.216299
$236$	0	0
$237$	4.05759	0.263569
$238$	0	0
$239$	24.9176	1.61178	0.805892	−	0.592063i	$-0.201686\pi$
$239$	24.9176	1.61178	0.805892	+	0.592063i	$0.201686\pi$
$240$	0	0
$241$	28.2012	1.81660	0.908300	−	0.418319i	$-0.137380\pi$
$241$	28.2012	1.81660	0.908300	+	0.418319i	$0.137380\pi$
$242$	0	0
$243$	−8.07540	−0.518037
$244$	0	0
$245$	7.35750	0.470053
$246$	0	0
$247$	−27.5997	−1.75613
$248$	0	0
$249$	−4.49767	−0.285028
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−0.273683	−0.0172063
$254$	0	0
$255$	1.47133	0.0921383
$256$	0	0
$257$	7.69037	0.479712	0.239856	−	0.970808i	$-0.422900\pi$
$257$	7.69037	0.479712	0.239856	+	0.970808i	$0.422900\pi$
$258$	0	0
$259$	34.1788	2.12377
$260$	0	0
$261$	−8.37877	−0.518633
$262$	0	0
$263$	−26.2295	−1.61738	−0.808691	−	0.588234i	$-0.799823\pi$
$263$	−26.2295	−1.61738	−0.808691	+	0.588234i	$0.799823\pi$
$264$	0	0
$265$	5.79760	0.356144
$266$	0	0
$267$	1.25693	0.0769226
$268$	0	0
$269$	−3.54811	−0.216332	−0.108166	−	0.994133i	$-0.534498\pi$
$269$	−3.54811	−0.216332	−0.108166	+	0.994133i	$0.534498\pi$
$270$	0	0
$271$	19.4471	1.18133	0.590664	−	0.806918i	$-0.298866\pi$
$271$	19.4471	1.18133	0.590664	+	0.806918i	$0.298866\pi$
$272$	0	0
$273$	−8.48867	−0.513758
$274$	0	0
$275$	0.322528	0.0194492
$276$	0	0
$277$	−17.5952	−1.05719	−0.528597	−	0.848873i	$-0.677282\pi$
$277$	−17.5952	−1.05719	−0.528597	+	0.848873i	$0.677282\pi$
$278$	0	0
$279$	−7.10788	−0.425537
$280$	0	0
$281$	−5.92586	−0.353507	−0.176754	−	0.984255i	$-0.556560\pi$
$281$	−5.92586	−0.353507	−0.176754	+	0.984255i	$0.556560\pi$
$282$	0	0
$283$	−28.0937	−1.67000	−0.834999	−	0.550252i	$-0.814532\pi$
$283$	−28.0937	−1.67000	−0.834999	+	0.550252i	$0.814532\pi$
$284$	0	0
$285$	−1.00447	−0.0594995
$286$	0	0
$287$	10.2845	0.607073
$288$	0	0
$289$	19.4041	1.14142
$290$	0	0
$291$	0.0661207	0.00387607
$292$	0	0
$293$	−5.93353	−0.346641	−0.173320	−	0.984866i	$-0.555450\pi$
$293$	−5.93353	−0.346641	−0.173320	+	0.984866i	$0.555450\pi$
$294$	0	0
$295$	−0.184667	−0.0107517
$296$	0	0
$297$	−0.135493	−0.00786212
$298$	0	0
$299$	−24.9674	−1.44390
$300$	0	0
$301$	−6.43638	−0.370987
$302$	0	0
$303$	−6.14150	−0.352820
$304$	0	0
$305$	9.91952	0.567990
$306$	0	0
$307$	23.3817	1.33446	0.667231	−	0.744851i	$-0.267479\pi$
$307$	23.3817	1.33446	0.667231	+	0.744851i	$0.267479\pi$
$308$	0	0
$309$	3.23560	0.184067
$310$	0	0
$311$	−13.8484	−0.785271	−0.392635	−	0.919694i	$-0.628437\pi$
$311$	−13.8484	−0.785271	−0.392635	+	0.919694i	$0.628437\pi$
$312$	0	0
$313$	31.1120	1.75856	0.879278	−	0.476310i	$-0.158026\pi$
$313$	31.1120	1.75856	0.879278	+	0.476310i	$0.158026\pi$
$314$	0	0
$315$	9.17870	0.517161
$316$	0	0
$317$	−24.3516	−1.36772	−0.683861	−	0.729613i	$-0.739700\pi$
$317$	−24.3516	−1.36772	−0.683861	+	0.729613i	$0.739700\pi$
$318$	0	0
$319$	−0.212039	−0.0118719
$320$	0	0
$321$	−5.49447	−0.306672
$322$	0	0
$323$	−24.8528	−1.38285
$324$	0	0
$325$	29.4234	1.63212
$326$	0	0
$327$	3.26056	0.180309
$328$	0	0
$329$	−17.2262	−0.949712
$330$	0	0
$331$	−0.136312	−0.00749239	−0.00374619	−	0.999993i	$-0.501192\pi$
$331$	−0.136312	−0.00749239	−0.00374619	+	0.999993i	$0.501192\pi$
$332$	0	0
$333$	24.4727	1.34110
$334$	0	0
$335$	2.34230	0.127973
$336$	0	0
$337$	18.9540	1.03249	0.516246	−	0.856440i	$-0.327329\pi$
$337$	18.9540	1.03249	0.516246	+	0.856440i	$0.327329\pi$
$338$	0	0
$339$	−4.24379	−0.230491
$340$	0	0
$341$	−0.179876	−0.00974085
$342$	0	0
$343$	−9.84977	−0.531837
$344$	0	0
$345$	−0.908667	−0.0489210
$346$	0	0
$347$	14.1804	0.761243	0.380621	−	0.924731i	$-0.375710\pi$
$347$	14.1804	0.761243	0.380621	+	0.924731i	$0.375710\pi$
$348$	0	0
$349$	0.0336059	0.00179889	0.000899443	−	1.00000i	$-0.499714\pi$
$349$	0.0336059	0.00179889	0.000899443		1.00000i	$0.499714\pi$
$350$	0	0
$351$	−12.3607	−0.659766
$352$	0	0
$353$	−18.0676	−0.961641	−0.480821	−	0.876819i	$-0.659661\pi$
$353$	−18.0676	−0.961641	−0.480821	+	0.876819i	$0.659661\pi$
$354$	0	0
$355$	−9.93699	−0.527401
$356$	0	0
$357$	−7.64383	−0.404554
$358$	0	0
$359$	20.8423	1.10002	0.550008	−	0.835159i	$-0.314625\pi$
$359$	20.8423	1.10002	0.550008	+	0.835159i	$0.314625\pi$
$360$	0	0
$361$	−2.03313	−0.107007
$362$	0	0
$363$	3.43634	0.180361
$364$	0	0
$365$	10.3005	0.539155
$366$	0	0
$367$	17.3065	0.903391	0.451696	−	0.892172i	$-0.350819\pi$
$367$	17.3065	0.903391	0.451696	+	0.892172i	$0.350819\pi$
$368$	0	0
$369$	7.36389	0.383349
$370$	0	0
$371$	−30.1196	−1.56373
$372$	0	0
$373$	6.52984	0.338102	0.169051	−	0.985607i	$-0.445930\pi$
$373$	6.52984	0.338102	0.169051	+	0.985607i	$0.445930\pi$
$374$	0	0
$375$	2.29012	0.118261
$376$	0	0
$377$	−19.3437	−0.996253
$378$	0	0
$379$	8.65156	0.444401	0.222200	−	0.975001i	$-0.428676\pi$
$379$	8.65156	0.444401	0.222200	+	0.975001i	$0.428676\pi$
$380$	0	0
$381$	0.102087	0.00523008
$382$	0	0
$383$	20.3623	1.04046	0.520232	−	0.854025i	$-0.325845\pi$
$383$	20.3623	1.04046	0.520232	+	0.854025i	$0.325845\pi$
$384$	0	0
$385$	0.232282	0.0118382
$386$	0	0
$387$	−4.60858	−0.234267
$388$	0	0
$389$	24.8422	1.25955	0.629776	−	0.776777i	$-0.283147\pi$
$389$	24.8422	1.25955	0.629776	+	0.776777i	$0.283147\pi$
$390$	0	0
$391$	−22.4825	−1.13699
$392$	0	0
$393$	−2.83462	−0.142988
$394$	0	0
$395$	−10.1291	−0.509649
$396$	0	0
$397$	−14.6677	−0.736152	−0.368076	−	0.929796i	$-0.619983\pi$
$397$	−14.6677	−0.736152	−0.368076	+	0.929796i	$0.619983\pi$
$398$	0	0
$399$	5.21839	0.261246
$400$	0	0
$401$	11.0097	0.549799	0.274899	−	0.961473i	$-0.411355\pi$
$401$	11.0097	0.549799	0.274899	+	0.961473i	$0.411355\pi$
$402$	0	0
$403$	−16.4097	−0.817423
$404$	0	0
$405$	6.34349	0.315211
$406$	0	0
$407$	0.619322	0.0306986
$408$	0	0
$409$	−2.25333	−0.111420	−0.0557100	−	0.998447i	$-0.517742\pi$
$409$	−2.25333	−0.111420	−0.0557100	+	0.998447i	$0.517742\pi$
$410$	0	0
$411$	4.09111	0.201800
$412$	0	0
$413$	0.959380	0.0472080
$414$	0	0
$415$	11.2276	0.551143
$416$	0	0
$417$	5.80600	0.284321
$418$	0	0
$419$	13.7628	0.672356	0.336178	−	0.941799i	$-0.390866\pi$
$419$	13.7628	0.672356	0.336178	+	0.941799i	$0.390866\pi$
$420$	0	0
$421$	12.0144	0.585548	0.292774	−	0.956182i	$-0.405422\pi$
$421$	12.0144	0.585548	0.292774	+	0.956182i	$0.405422\pi$
$422$	0	0
$423$	−12.3343	−0.599715
$424$	0	0
$425$	26.4950	1.28520
$426$	0	0
$427$	−51.5337	−2.49389
$428$	0	0
$429$	−0.153815	−0.00742627
$430$	0	0
$431$	13.9884	0.673795	0.336898	−	0.941541i	$-0.390622\pi$
$431$	13.9884	0.673795	0.336898	+	0.941541i	$0.390622\pi$
$432$	0	0
$433$	−31.2697	−1.50273	−0.751364	−	0.659888i	$-0.770604\pi$
$433$	−31.2697	−1.50273	−0.751364	+	0.659888i	$0.770604\pi$
$434$	0	0
$435$	−0.703998	−0.0337541
$436$	0	0
$437$	15.3487	0.734226
$438$	0	0
$439$	−16.3536	−0.780517	−0.390259	−	0.920705i	$-0.627614\pi$
$439$	−16.3536	−0.780517	−0.390259	+	0.920705i	$0.627614\pi$
$440$	0	0
$441$	−27.3688	−1.30328
$442$	0	0
$443$	40.9943	1.94770	0.973849	−	0.227195i	$-0.0729553\pi$
$443$	40.9943	1.94770	0.973849	+	0.227195i	$0.0729553\pi$
$444$	0	0
$445$	−3.13769	−0.148741
$446$	0	0
$447$	4.33750	0.205157
$448$	0	0
$449$	16.3120	0.769813	0.384907	−	0.922956i	$-0.374234\pi$
$449$	16.3120	0.769813	0.384907	+	0.922956i	$0.374234\pi$
$450$	0	0
$451$	0.186355	0.00877512
$452$	0	0
$453$	1.56643	0.0735973
$454$	0	0
$455$	21.1905	0.993426
$456$	0	0
$457$	−15.6367	−0.731455	−0.365728	−	0.930722i	$-0.619180\pi$
$457$	−15.6367	−0.731455	−0.365728	+	0.930722i	$0.619180\pi$
$458$	0	0
$459$	−11.1305	−0.519527
$460$	0	0
$461$	−24.4146	−1.13710	−0.568550	−	0.822648i	$-0.692496\pi$
$461$	−24.4146	−1.13710	−0.568550	+	0.822648i	$0.692496\pi$
$462$	0	0
$463$	16.1423	0.750198	0.375099	−	0.926985i	$-0.377609\pi$
$463$	16.1423	0.750198	0.375099	+	0.926985i	$0.377609\pi$
$464$	0	0
$465$	−0.597215	−0.0276952
$466$	0	0
$467$	3.11320	0.144062	0.0720309	−	0.997402i	$-0.477052\pi$
$467$	3.11320	0.144062	0.0720309	+	0.997402i	$0.477052\pi$
$468$	0	0
$469$	−12.1687	−0.561897
$470$	0	0
$471$	5.44804	0.251033
$472$	0	0
$473$	−0.116628	−0.00536255
$474$	0	0
$475$	−18.0880	−0.829932
$476$	0	0
$477$	−21.5662	−0.987450
$478$	0	0
$479$	−20.6491	−0.943481	−0.471740	−	0.881737i	$-0.656374\pi$
$479$	−20.6491	−0.943481	−0.471740	+	0.881737i	$0.656374\pi$
$480$	0	0
$481$	56.4991	2.57614
$482$	0	0
$483$	4.72069	0.214799
$484$	0	0
$485$	−0.165059	−0.00749494
$486$	0	0
$487$	33.0244	1.49648	0.748239	−	0.663429i	$-0.230900\pi$
$487$	33.0244	1.49648	0.748239	+	0.663429i	$0.230900\pi$
$488$	0	0
$489$	7.11908	0.321936
$490$	0	0
$491$	14.7947	0.667674	0.333837	−	0.942631i	$-0.391657\pi$
$491$	14.7947	0.667674	0.333837	+	0.942631i	$0.391657\pi$
$492$	0	0
$493$	−17.4185	−0.784491
$494$	0	0
$495$	0.166319	0.00747547
$496$	0	0
$497$	51.6245	2.31567
$498$	0	0
$499$	16.7660	0.750551	0.375276	−	0.926913i	$-0.377548\pi$
$499$	16.7660	0.750551	0.375276	+	0.926913i	$0.377548\pi$
$500$	0	0
$501$	−4.28015	−0.191223
$502$	0	0
$503$	−7.73246	−0.344773	−0.172387	−	0.985029i	$-0.555148\pi$
$503$	−7.73246	−0.344773	−0.172387	+	0.985029i	$0.555148\pi$
$504$	0	0
$505$	15.3312	0.682229
$506$	0	0
$507$	−9.96905	−0.442741
$508$	0	0
$509$	1.83461	0.0813176	0.0406588	−	0.999173i	$-0.487054\pi$
$509$	1.83461	0.0813176	0.0406588	+	0.999173i	$0.487054\pi$
$510$	0	0
$511$	−53.5132	−2.36728
$512$	0	0
$513$	7.59872	0.335491
$514$	0	0
$515$	−8.07710	−0.355920
$516$	0	0
$517$	−0.312140	−0.0137279
$518$	0	0
$519$	2.07187	0.0909451
$520$	0	0
$521$	−43.2455	−1.89462	−0.947309	−	0.320322i	$-0.896209\pi$
$521$	−43.2455	−1.89462	−0.947309	+	0.320322i	$0.896209\pi$
$522$	0	0
$523$	6.99595	0.305912	0.152956	−	0.988233i	$-0.451121\pi$
$523$	6.99595	0.305912	0.152956	+	0.988233i	$0.451121\pi$
$524$	0	0
$525$	−5.56320	−0.242798
$526$	0	0
$527$	−14.7765	−0.643673
$528$	0	0
$529$	−9.11521	−0.396313
$530$	0	0
$531$	0.686936	0.0298105
$532$	0	0
$533$	17.0007	0.736382
$534$	0	0
$535$	13.7160	0.592994
$536$	0	0
$537$	−5.68867	−0.245484
$538$	0	0
$539$	−0.692613	−0.0298329
$540$	0	0
$541$	−7.49538	−0.322252	−0.161126	−	0.986934i	$-0.551513\pi$
$541$	−7.49538	−0.322252	−0.161126	+	0.986934i	$0.551513\pi$
$542$	0	0
$543$	−3.17862	−0.136408
$544$	0	0
$545$	−8.13942	−0.348655
$546$	0	0
$547$	−35.2273	−1.50621	−0.753106	−	0.657899i	$-0.771445\pi$
$547$	−35.2273	−1.50621	−0.753106	+	0.657899i	$0.771445\pi$
$548$	0	0
$549$	−36.8992	−1.57482
$550$	0	0
$551$	11.8915	0.506595
$552$	0	0
$553$	52.6223	2.23773
$554$	0	0
$555$	2.05624	0.0872823
$556$	0	0
$557$	−15.2377	−0.645641	−0.322821	−	0.946460i	$-0.604631\pi$
$557$	−15.2377	−0.645641	−0.322821	+	0.946460i	$0.604631\pi$
$558$	0	0
$559$	−10.6396	−0.450009
$560$	0	0
$561$	−0.138507	−0.00584776
$562$	0	0
$563$	17.9240	0.755408	0.377704	−	0.925926i	$-0.376714\pi$
$563$	17.9240	0.755408	0.377704	+	0.925926i	$0.376714\pi$
$564$	0	0
$565$	10.5939	0.445688
$566$	0	0
$567$	−32.9556	−1.38400
$568$	0	0
$569$	29.0549	1.21805	0.609023	−	0.793153i	$-0.291562\pi$
$569$	29.0549	1.21805	0.609023	+	0.793153i	$0.291562\pi$
$570$	0	0
$571$	0.812908	0.0340192	0.0170096	−	0.999855i	$-0.494585\pi$
$571$	0.812908	0.0340192	0.0170096	+	0.999855i	$0.494585\pi$
$572$	0	0
$573$	−5.75794	−0.240541
$574$	0	0
$575$	−16.3628	−0.682377
$576$	0	0
$577$	12.4189	0.517007	0.258503	−	0.966010i	$-0.416771\pi$
$577$	12.4189	0.517007	0.258503	+	0.966010i	$0.416771\pi$
$578$	0	0
$579$	6.32461	0.262842
$580$	0	0
$581$	−58.3296	−2.41992
$582$	0	0
$583$	−0.545769	−0.0226034
$584$	0	0
$585$	15.1728	0.627319
$586$	0	0
$587$	−26.4700	−1.09253	−0.546267	−	0.837611i	$-0.683951\pi$
$587$	−26.4700	−1.09253	−0.546267	+	0.837611i	$0.683951\pi$
$588$	0	0
$589$	10.0878	0.415661
$590$	0	0
$591$	4.39670	0.180856
$592$	0	0
$593$	39.7573	1.63264	0.816319	−	0.577601i	$-0.196011\pi$
$593$	39.7573	1.63264	0.816319	+	0.577601i	$0.196011\pi$
$594$	0	0
$595$	19.0815	0.782265
$596$	0	0
$597$	−0.355132	−0.0145346
$598$	0	0
$599$	21.8468	0.892636	0.446318	−	0.894875i	$-0.352735\pi$
$599$	21.8468	0.892636	0.446318	+	0.894875i	$0.352735\pi$
$600$	0	0
$601$	36.0099	1.46887	0.734437	−	0.678677i	$-0.237446\pi$
$601$	36.0099	1.46887	0.734437	+	0.678677i	$0.237446\pi$
$602$	0	0
$603$	−8.71302	−0.354822
$604$	0	0
$605$	−8.57823	−0.348755
$606$	0	0
$607$	9.35489	0.379703	0.189852	−	0.981813i	$-0.439199\pi$
$607$	9.35489	0.379703	0.189852	+	0.981813i	$0.439199\pi$
$608$	0	0
$609$	3.65740	0.148205
$610$	0	0
$611$	−28.4757	−1.15200
$612$	0	0
$613$	4.02976	0.162760	0.0813802	−	0.996683i	$-0.474067\pi$
$613$	4.02976	0.162760	0.0813802	+	0.996683i	$0.474067\pi$
$614$	0	0
$615$	0.618726	0.0249494
$616$	0	0
$617$	15.6956	0.631879	0.315940	−	0.948779i	$-0.397680\pi$
$617$	15.6956	0.631879	0.315940	+	0.948779i	$0.397680\pi$
$618$	0	0
$619$	38.2430	1.53712	0.768558	−	0.639780i	$-0.220975\pi$
$619$	38.2430	1.53712	0.768558	+	0.639780i	$0.220975\pi$
$620$	0	0
$621$	6.87399	0.275844
$622$	0	0
$623$	16.3009	0.653081
$624$	0	0
$625$	16.2394	0.649575
$626$	0	0
$627$	0.0945576	0.00377627
$628$	0	0
$629$	50.8760	2.02856
$630$	0	0
$631$	−5.66509	−0.225524	−0.112762	−	0.993622i	$-0.535970\pi$
$631$	−5.66509	−0.225524	−0.112762	+	0.993622i	$0.535970\pi$
$632$	0	0
$633$	−8.34873	−0.331832
$634$	0	0
$635$	−0.254843	−0.0101131
$636$	0	0
$637$	−63.1853	−2.50349
$638$	0	0
$639$	36.9642	1.46228
$640$	0	0
$641$	14.8971	0.588402	0.294201	−	0.955744i	$-0.404947\pi$
$641$	14.8971	0.588402	0.294201	+	0.955744i	$0.404947\pi$
$642$	0	0
$643$	−13.0615	−0.515094	−0.257547	−	0.966266i	$-0.582914\pi$
$643$	−13.0615	−0.515094	−0.257547	+	0.966266i	$0.582914\pi$
$644$	0	0
$645$	−0.387220	−0.0152468
$646$	0	0
$647$	−44.1799	−1.73689	−0.868446	−	0.495784i	$-0.834881\pi$
$647$	−44.1799	−1.73689	−0.868446	+	0.495784i	$0.834881\pi$
$648$	0	0
$649$	0.0173840	0.000682383 0
$650$	0	0
$651$	3.10264	0.121602
$652$	0	0
$653$	4.52495	0.177075	0.0885375	−	0.996073i	$-0.471781\pi$
$653$	4.52495	0.177075	0.0885375	+	0.996073i	$0.471781\pi$
$654$	0	0
$655$	7.07614	0.276488
$656$	0	0
$657$	−38.3165	−1.49487
$658$	0	0
$659$	−27.6961	−1.07889	−0.539443	−	0.842022i	$-0.681365\pi$
$659$	−27.6961	−1.07889	−0.539443	+	0.842022i	$0.681365\pi$
$660$	0	0
$661$	−13.3594	−0.519620	−0.259810	−	0.965660i	$-0.583660\pi$
$661$	−13.3594	−0.519620	−0.259810	+	0.965660i	$0.583660\pi$
$662$	0	0
$663$	−12.6356	−0.490726
$664$	0	0
$665$	−13.0268	−0.505158
$666$	0	0
$667$	10.7574	0.416527
$668$	0	0
$669$	2.96149	0.114498
$670$	0	0
$671$	−0.933794	−0.0360487
$672$	0	0
$673$	42.3277	1.63161	0.815807	−	0.578324i	$-0.196293\pi$
$673$	42.3277	1.63161	0.815807	+	0.578324i	$0.196293\pi$
$674$	0	0
$675$	−8.10080	−0.311800
$676$	0	0
$677$	−24.0185	−0.923104	−0.461552	−	0.887113i	$-0.652707\pi$
$677$	−24.0185	−0.923104	−0.461552	+	0.887113i	$0.652707\pi$
$678$	0	0
$679$	0.857511	0.0329082
$680$	0	0
$681$	5.88370	0.225464
$682$	0	0
$683$	16.1293	0.617172	0.308586	−	0.951197i	$-0.400144\pi$
$683$	16.1293	0.617172	0.308586	+	0.951197i	$0.400144\pi$
$684$	0	0
$685$	−10.2127	−0.390209
$686$	0	0
$687$	−0.923674	−0.0352404
$688$	0	0
$689$	−49.7891	−1.89681
$690$	0	0
$691$	7.32037	0.278480	0.139240	−	0.990259i	$-0.455534\pi$
$691$	7.32037	0.278480	0.139240	+	0.990259i	$0.455534\pi$
$692$	0	0
$693$	−0.864056	−0.0328228
$694$	0	0
$695$	−14.4937	−0.549777
$696$	0	0
$697$	15.3087	0.579858
$698$	0	0
$699$	−5.35318	−0.202476
$700$	0	0
$701$	−26.3497	−0.995214	−0.497607	−	0.867403i	$-0.665788\pi$
$701$	−26.3497	−0.995214	−0.497607	+	0.867403i	$0.665788\pi$
$702$	0	0
$703$	−34.7327	−1.30997
$704$	0	0
$705$	−1.03635	−0.0390312
$706$	0	0
$707$	−79.6483	−2.99548
$708$	0	0
$709$	−3.17194	−0.119125	−0.0595623	−	0.998225i	$-0.518970\pi$
$709$	−3.17194	−0.119125	−0.0595623	+	0.998225i	$0.518970\pi$
$710$	0	0
$711$	37.6787	1.41306
$712$	0	0
$713$	9.12568	0.341759
$714$	0	0
$715$	0.383973	0.0143598
$716$	0	0
$717$	−7.78794	−0.290846
$718$	0	0
$719$	36.5622	1.36354	0.681771	−	0.731566i	$-0.261210\pi$
$719$	36.5622	1.36354	0.681771	+	0.731566i	$0.261210\pi$
$720$	0	0
$721$	41.9620	1.56275
$722$	0	0
$723$	−8.81424	−0.327805
$724$	0	0
$725$	−12.6772	−0.470821
$726$	0	0
$727$	−29.7363	−1.10286	−0.551428	−	0.834222i	$-0.685917\pi$
$727$	−29.7363	−1.10286	−0.551428	+	0.834222i	$0.685917\pi$
$728$	0	0
$729$	−21.8671	−0.809894
$730$	0	0
$731$	−9.58072	−0.354356
$732$	0	0
$733$	−3.74485	−0.138319	−0.0691595	−	0.997606i	$-0.522032\pi$
$733$	−3.74485	−0.138319	−0.0691595	+	0.997606i	$0.522032\pi$
$734$	0	0
$735$	−2.29957	−0.0848210
$736$	0	0
$737$	−0.220497	−0.00812212
$738$	0	0
$739$	−13.5250	−0.497526	−0.248763	−	0.968564i	$-0.580024\pi$
$739$	−13.5250	−0.497526	−0.248763	+	0.968564i	$0.580024\pi$
$740$	0	0
$741$	8.62625	0.316893
$742$	0	0
$743$	17.6397	0.647136	0.323568	−	0.946205i	$-0.395118\pi$
$743$	17.6397	0.647136	0.323568	+	0.946205i	$0.395118\pi$
$744$	0	0
$745$	−10.8278	−0.396700
$746$	0	0
$747$	−41.7652	−1.52811
$748$	0	0
$749$	−71.2571	−2.60368
$750$	0	0
$751$	36.7457	1.34087	0.670435	−	0.741969i	$-0.266108\pi$
$751$	36.7457	1.34087	0.670435	+	0.741969i	$0.266108\pi$
$752$	0	0
$753$	−0.312548	−0.0113899
$754$	0	0
$755$	−3.91032	−0.142311
$756$	0	0
$757$	−14.2476	−0.517838	−0.258919	−	0.965899i	$-0.583366\pi$
$757$	−14.2476	−0.517838	−0.258919	+	0.965899i	$0.583366\pi$
$758$	0	0
$759$	0.0855392	0.00310488
$760$	0	0
$761$	−8.60149	−0.311804	−0.155902	−	0.987773i	$-0.549828\pi$
$761$	−8.60149	−0.311804	−0.155902	+	0.987773i	$0.549828\pi$
$762$	0	0
$763$	42.2858	1.53085
$764$	0	0
$765$	13.6627	0.493977
$766$	0	0
$767$	1.58590	0.0572636
$768$	0	0
$769$	10.7431	0.387406	0.193703	−	0.981060i	$-0.437950\pi$
$769$	10.7431	0.387406	0.193703	+	0.981060i	$0.437950\pi$
$770$	0	0
$771$	−2.40361	−0.0865640
$772$	0	0
$773$	−18.1392	−0.652422	−0.326211	−	0.945297i	$-0.605772\pi$
$773$	−18.1392	−0.652422	−0.326211	+	0.945297i	$0.605772\pi$
$774$	0	0
$775$	−10.7543	−0.386308
$776$	0	0
$777$	−10.6825	−0.383233
$778$	0	0
$779$	−10.4511	−0.374451
$780$	0	0
$781$	0.935439	0.0334726
$782$	0	0
$783$	5.32568	0.190324
$784$	0	0
$785$	−13.6001	−0.485408
$786$	0	0
$787$	−17.0308	−0.607084	−0.303542	−	0.952818i	$-0.598169\pi$
$787$	−17.0308	−0.607084	−0.303542	+	0.952818i	$0.598169\pi$
$788$	0	0
$789$	8.19799	0.291856
$790$	0	0
$791$	−55.0371	−1.95689
$792$	0	0
$793$	−85.1876	−3.02510
$794$	0	0
$795$	−1.81203	−0.0642661
$796$	0	0
$797$	4.93046	0.174646	0.0873229	−	0.996180i	$-0.472169\pi$
$797$	4.93046	0.174646	0.0873229	+	0.996180i	$0.472169\pi$
$798$	0	0
$799$	−25.6416	−0.907136
$800$	0	0
$801$	11.6718	0.412402
$802$	0	0
$803$	−0.969663	−0.0342186
$804$	0	0
$805$	−11.7844	−0.415345
$806$	0	0
$807$	1.10896	0.0390371
$808$	0	0
$809$	−33.5710	−1.18029	−0.590146	−	0.807296i	$-0.700930\pi$
$809$	−33.5710	−1.18029	−0.590146	+	0.807296i	$0.700930\pi$
$810$	0	0
$811$	32.9415	1.15673	0.578366	−	0.815777i	$-0.303691\pi$
$811$	32.9415	1.15673	0.578366	+	0.815777i	$0.303691\pi$
$812$	0	0
$813$	−6.07816	−0.213170
$814$	0	0
$815$	−17.7715	−0.622510
$816$	0	0
$817$	6.54070	0.228830
$818$	0	0
$819$	−78.8256	−2.75439
$820$	0	0
$821$	−16.1872	−0.564938	−0.282469	−	0.959276i	$-0.591153\pi$
$821$	−16.1872	−0.564938	−0.282469	+	0.959276i	$0.591153\pi$
$822$	0	0
$823$	−15.3791	−0.536080	−0.268040	−	0.963408i	$-0.586376\pi$
$823$	−15.3791	−0.536080	−0.268040	+	0.963408i	$0.586376\pi$
$824$	0	0
$825$	−0.100805	−0.00350960
$826$	0	0
$827$	39.8319	1.38509	0.692545	−	0.721374i	$-0.256489\pi$
$827$	39.8319	1.38509	0.692545	+	0.721374i	$0.256489\pi$
$828$	0	0
$829$	43.4836	1.51025	0.755124	−	0.655582i	$-0.227577\pi$
$829$	43.4836	1.51025	0.755124	+	0.655582i	$0.227577\pi$
$830$	0	0
$831$	5.49935	0.190770
$832$	0	0
$833$	−56.8967	−1.97135
$834$	0	0
$835$	10.6846	0.369757
$836$	0	0
$837$	4.51788	0.156161
$838$	0	0
$839$	−37.6838	−1.30099	−0.650495	−	0.759511i	$-0.725438\pi$
$839$	−37.6838	−1.30099	−0.650495	+	0.759511i	$0.725438\pi$
$840$	0	0
$841$	−20.6656	−0.712608
$842$	0	0
$843$	1.85212	0.0637903
$844$	0	0
$845$	24.8860	0.856104
$846$	0	0
$847$	44.5655	1.53129
$848$	0	0
$849$	8.78064	0.301351
$850$	0	0
$851$	−31.4201	−1.07707
$852$	0	0
$853$	−44.4785	−1.52292	−0.761458	−	0.648215i	$-0.775516\pi$
$853$	−44.4785	−1.52292	−0.761458	+	0.648215i	$0.775516\pi$
$854$	0	0
$855$	−9.32746	−0.318992
$856$	0	0
$857$	10.0187	0.342233	0.171117	−	0.985251i	$-0.445263\pi$
$857$	10.0187	0.342233	0.171117	+	0.985251i	$0.445263\pi$
$858$	0	0
$859$	−7.41120	−0.252867	−0.126433	−	0.991975i	$-0.540353\pi$
$859$	−7.41120	−0.252867	−0.126433	+	0.991975i	$0.540353\pi$
$860$	0	0
$861$	−3.21439	−0.109546
$862$	0	0
$863$	−14.9090	−0.507509	−0.253755	−	0.967269i	$-0.581666\pi$
$863$	−14.9090	−0.507509	−0.253755	+	0.967269i	$0.581666\pi$
$864$	0	0
$865$	−5.17207	−0.175856
$866$	0	0
$867$	−6.06472	−0.205969
$868$	0	0
$869$	0.953520	0.0323460
$870$	0	0
$871$	−20.1154	−0.681584
$872$	0	0
$873$	0.613995	0.0207806
$874$	0	0
$875$	29.7003	1.00405
$876$	0	0
$877$	0.438135	0.0147948	0.00739739	−	0.999973i	$-0.497645\pi$
$877$	0.438135	0.0147948	0.00739739	+	0.999973i	$0.497645\pi$
$878$	0	0
$879$	1.85451	0.0625512
$880$	0	0
$881$	−23.7272	−0.799388	−0.399694	−	0.916649i	$-0.630884\pi$
$881$	−23.7272	−0.799388	−0.399694	+	0.916649i	$0.630884\pi$
$882$	0	0
$883$	54.2105	1.82433	0.912164	−	0.409825i	$-0.134410\pi$
$883$	54.2105	1.82433	0.912164	+	0.409825i	$0.134410\pi$
$884$	0	0
$885$	0.0577174	0.00194015
$886$	0	0
$887$	10.0052	0.335943	0.167972	−	0.985792i	$-0.446278\pi$
$887$	10.0052	0.335943	0.167972	+	0.985792i	$0.446278\pi$
$888$	0	0
$889$	1.32395	0.0444040
$890$	0	0
$891$	−0.597157	−0.0200055
$892$	0	0
$893$	17.5054	0.585795
$894$	0	0
$895$	14.2008	0.474680
$896$	0	0
$897$	7.80352	0.260552
$898$	0	0
$899$	7.07020	0.235804
$900$	0	0
$901$	−44.8338	−1.49363
$902$	0	0
$903$	2.01168	0.0669445
$904$	0	0
$905$	7.93488	0.263764
$906$	0	0
$907$	31.7955	1.05575	0.527875	−	0.849322i	$-0.322989\pi$
$907$	31.7955	1.05575	0.527875	+	0.849322i	$0.322989\pi$
$908$	0	0
$909$	−57.0298	−1.89156
$910$	0	0
$911$	−7.59951	−0.251783	−0.125892	−	0.992044i	$-0.540179\pi$
$911$	−7.59951	−0.251783	−0.125892	+	0.992044i	$0.540179\pi$
$912$	0	0
$913$	−1.05694	−0.0349795
$914$	0	0
$915$	−3.10033	−0.102494
$916$	0	0
$917$	−36.7618	−1.21398
$918$	0	0
$919$	14.4606	0.477011	0.238505	−	0.971141i	$-0.423342\pi$
$919$	14.4606	0.477011	0.238505	+	0.971141i	$0.423342\pi$
$920$	0	0
$921$	−7.30790	−0.240803
$922$	0	0
$923$	85.3376	2.80892
$924$	0	0
$925$	37.0277	1.21746
$926$	0	0
$927$	30.0457	0.986829
$928$	0	0
$929$	−18.6426	−0.611643	−0.305821	−	0.952089i	$-0.598931\pi$
$929$	−18.6426	−0.611643	−0.305821	+	0.952089i	$0.598931\pi$
$930$	0	0
$931$	38.8430	1.27303
$932$	0	0
$933$	4.32829	0.141702
$934$	0	0
$935$	0.345758	0.0113075
$936$	0	0
$937$	31.9634	1.04420	0.522099	−	0.852885i	$-0.325149\pi$
$937$	31.9634	1.04420	0.522099	+	0.852885i	$0.325149\pi$
$938$	0	0
$939$	−9.72400	−0.317331
$940$	0	0
$941$	−4.79135	−0.156194	−0.0780968	−	0.996946i	$-0.524884\pi$
$941$	−4.79135	−0.156194	−0.0780968	+	0.996946i	$0.524884\pi$
$942$	0	0
$943$	−9.45437	−0.307877
$944$	0	0
$945$	−5.83413	−0.189784
$946$	0	0
$947$	−50.7682	−1.64975	−0.824873	−	0.565319i	$-0.808753\pi$
$947$	−50.7682	−1.64975	−0.824873	+	0.565319i	$0.808753\pi$
$948$	0	0
$949$	−88.4598	−2.87153
$950$	0	0
$951$	7.61104	0.246805
$952$	0	0
$953$	−31.9128	−1.03376	−0.516878	−	0.856059i	$-0.672906\pi$
$953$	−31.9128	−1.03376	−0.516878	+	0.856059i	$0.672906\pi$
$954$	0	0
$955$	14.3737	0.465122
$956$	0	0
$957$	0.0662723	0.00214228
$958$	0	0
$959$	53.0570	1.71330
$960$	0	0
$961$	−25.0022	−0.806523
$962$	0	0
$963$	−51.0215	−1.64415
$964$	0	0
$965$	−15.7883	−0.508243
$966$	0	0
$967$	18.8812	0.607177	0.303589	−	0.952803i	$-0.401815\pi$
$967$	18.8812	0.607177	0.303589	+	0.952803i	$0.401815\pi$
$968$	0	0
$969$	7.76771	0.249535
$970$	0	0
$971$	−37.5618	−1.20542	−0.602708	−	0.797962i	$-0.705912\pi$
$971$	−37.5618	−1.20542	−0.602708	+	0.797962i	$0.705912\pi$
$972$	0	0
$973$	75.2973	2.41392
$974$	0	0
$975$	−9.19622	−0.294515
$976$	0	0
$977$	26.5967	0.850905	0.425453	−	0.904981i	$-0.360115\pi$
$977$	26.5967	0.850905	0.425453	+	0.904981i	$0.360115\pi$
$978$	0	0
$979$	0.295373	0.00944017
$980$	0	0
$981$	30.2775	0.966686
$982$	0	0
$983$	−19.4290	−0.619687	−0.309844	−	0.950787i	$-0.600277\pi$
$983$	−19.4290	−0.619687	−0.309844	+	0.950787i	$0.600277\pi$
$984$	0	0
$985$	−10.9756	−0.349712
$986$	0	0
$987$	5.38402	0.171375
$988$	0	0
$989$	5.91688	0.188146
$990$	0	0
$991$	−36.8447	−1.17041	−0.585205	−	0.810885i	$-0.698986\pi$
$991$	−36.8447	−1.17041	−0.585205	+	0.810885i	$0.698986\pi$
$992$	0	0
$993$	0.0426041	0.00135200
$994$	0	0
$995$	0.886526	0.0281048
$996$	0	0
$997$	−0.253257	−0.00802074	−0.00401037	−	0.999992i	$-0.501277\pi$
$997$	−0.253257	−0.00802074	−0.00401037	+	0.999992i	$0.501277\pi$
$998$	0	0
$999$	−15.5552	−0.492146

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.l.1.8		19
4.3	odd	2		2008.2.a.c.1.12	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.c.1.12	✓	19	4.3	odd	2
4016.2.a.l.1.8		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)

\(f(q)\)	\(=\)	\(q-0.312548 q^{3} +0.780222 q^{5} -4.05339 q^{7} -2.90231 q^{9} +O(q^{10})\)	\(q-0.312548 q^{3} +0.780222 q^{5} -4.05339 q^{7} -2.90231 q^{9} -0.0734478 q^{11} -6.70045 q^{13} -0.243857 q^{15} -6.03358 q^{17} +4.11909 q^{19} +1.26688 q^{21} +3.72623 q^{23} -4.39125 q^{25} +1.84476 q^{27} +2.88693 q^{29} +2.44904 q^{31} +0.0229560 q^{33} -3.16255 q^{35} -8.43214 q^{37} +2.09421 q^{39} -2.53725 q^{41} +1.58790 q^{43} -2.26445 q^{45} +4.24982 q^{47} +9.43000 q^{49} +1.88578 q^{51} +7.43071 q^{53} -0.0573056 q^{55} -1.28741 q^{57} -0.236686 q^{59} +12.7137 q^{61} +11.7642 q^{63} -5.22784 q^{65} +3.00209 q^{67} -1.16463 q^{69} -12.7361 q^{71} +13.2021 q^{73} +1.37248 q^{75} +0.297713 q^{77} -12.9823 q^{79} +8.13037 q^{81} +14.3903 q^{83} -4.70753 q^{85} -0.902305 q^{87} -4.02154 q^{89} +27.1596 q^{91} -0.765443 q^{93} +3.21380 q^{95} -0.211554 q^{97} +0.213168 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) \) 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

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