LMFDB - Embedded newform 4016.2.a.l.1.12

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			1.12
Root			$1.27063$ 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)

$f(q)$	$=$	$q+1.27063 q^{3} -4.07798 q^{5} +2.09947 q^{7} -1.38551 q^{9} +O(q^{10})$	$q+1.27063 q^{3} -4.07798 q^{5} +2.09947 q^{7} -1.38551 q^{9} -5.74597 q^{11} +4.45538 q^{13} -5.18159 q^{15} +1.29179 q^{17} -1.28292 q^{19} +2.66764 q^{21} +1.98443 q^{23} +11.6299 q^{25} -5.57234 q^{27} -0.687538 q^{29} +7.43482 q^{31} -7.30099 q^{33} -8.56158 q^{35} -9.14574 q^{37} +5.66113 q^{39} +6.98332 q^{41} -5.76564 q^{43} +5.65006 q^{45} -9.07427 q^{47} -2.59223 q^{49} +1.64139 q^{51} -11.0517 q^{53} +23.4319 q^{55} -1.63011 q^{57} +13.3282 q^{59} +0.329869 q^{61} -2.90882 q^{63} -18.1689 q^{65} +8.50653 q^{67} +2.52148 q^{69} +8.72459 q^{71} +15.8781 q^{73} +14.7773 q^{75} -12.0635 q^{77} -6.44366 q^{79} -2.92386 q^{81} +13.9067 q^{83} -5.26790 q^{85} -0.873605 q^{87} -3.38571 q^{89} +9.35392 q^{91} +9.44689 q^{93} +5.23171 q^{95} +5.52372 q^{97} +7.96107 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.27063	0.733597	0.366799	−	0.930300i	$-0.380454\pi$
$3$	1.27063	0.733597	0.366799	+	0.930300i	$0.380454\pi$
$4$	0	0
$5$	−4.07798	−1.82373	−0.911864	−	0.410493i	$-0.865356\pi$
$5$	−4.07798	−1.82373	−0.911864	+	0.410493i	$0.865356\pi$
$6$	0	0
$7$	2.09947	0.793524	0.396762	−	0.917921i	$-0.370134\pi$
$7$	2.09947	0.793524	0.396762	+	0.917921i	$0.370134\pi$
$8$	0	0
$9$	−1.38551	−0.461835
$10$	0	0
$11$	−5.74597	−1.73248	−0.866238	−	0.499632i	$-0.833468\pi$
$11$	−5.74597	−1.73248	−0.866238	+	0.499632i	$0.833468\pi$
$12$	0	0
$13$	4.45538	1.23570	0.617850	−	0.786296i	$-0.288004\pi$
$13$	4.45538	1.23570	0.617850	+	0.786296i	$0.288004\pi$
$14$	0	0
$15$	−5.18159	−1.33788
$16$	0	0
$17$	1.29179	0.313305	0.156653	−	0.987654i	$-0.449930\pi$
$17$	1.29179	0.313305	0.156653	+	0.987654i	$0.449930\pi$
$18$	0	0
$19$	−1.28292	−0.294322	−0.147161	−	0.989113i	$-0.547013\pi$
$19$	−1.28292	−0.294322	−0.147161	+	0.989113i	$0.547013\pi$
$20$	0	0
$21$	2.66764	0.582127
$22$	0	0
$23$	1.98443	0.413783	0.206891	−	0.978364i	$-0.433665\pi$
$23$	1.98443	0.413783	0.206891	+	0.978364i	$0.433665\pi$
$24$	0	0
$25$	11.6299	2.32598
$26$	0	0
$27$	−5.57234	−1.07240
$28$	0	0
$29$	−0.687538	−0.127673	−0.0638363	−	0.997960i	$-0.520334\pi$
$29$	−0.687538	−0.127673	−0.0638363	+	0.997960i	$0.520334\pi$
$30$	0	0
$31$	7.43482	1.33533	0.667667	−	0.744460i	$-0.267293\pi$
$31$	7.43482	1.33533	0.667667	+	0.744460i	$0.267293\pi$
$32$	0	0
$33$	−7.30099	−1.27094
$34$	0	0
$35$	−8.56158	−1.44717
$36$	0	0
$37$	−9.14574	−1.50355	−0.751775	−	0.659420i	$-0.770802\pi$
$37$	−9.14574	−1.50355	−0.751775	+	0.659420i	$0.770802\pi$
$38$	0	0
$39$	5.66113	0.906506
$40$	0	0
$41$	6.98332	1.09061	0.545306	−	0.838237i	$-0.316414\pi$
$41$	6.98332	1.09061	0.545306	+	0.838237i	$0.316414\pi$
$42$	0	0
$43$	−5.76564	−0.879251	−0.439626	−	0.898181i	$-0.644889\pi$
$43$	−5.76564	−0.879251	−0.439626	+	0.898181i	$0.644889\pi$
$44$	0	0
$45$	5.65006	0.842261
$46$	0	0
$47$	−9.07427	−1.32362	−0.661809	−	0.749673i	$-0.730211\pi$
$47$	−9.07427	−1.32362	−0.661809	+	0.749673i	$0.730211\pi$
$48$	0	0
$49$	−2.59223	−0.370319
$50$	0	0
$51$	1.64139	0.229840
$52$	0	0
$53$	−11.0517	−1.51807	−0.759034	−	0.651051i	$-0.774328\pi$
$53$	−11.0517	−1.51807	−0.759034	+	0.651051i	$0.774328\pi$
$54$	0	0
$55$	23.4319	3.15956
$56$	0	0
$57$	−1.63011	−0.215914
$58$	0	0
$59$	13.3282	1.73519	0.867595	−	0.497272i	$-0.165665\pi$
$59$	13.3282	1.73519	0.867595	+	0.497272i	$0.165665\pi$
$60$	0	0
$61$	0.329869	0.0422354	0.0211177	−	0.999777i	$-0.493278\pi$
$61$	0.329869	0.0422354	0.0211177	+	0.999777i	$0.493278\pi$
$62$	0	0
$63$	−2.90882	−0.366477
$64$	0	0
$65$	−18.1689	−2.25358
$66$	0	0
$67$	8.50653	1.03924	0.519619	−	0.854398i	$-0.326074\pi$
$67$	8.50653	1.03924	0.519619	+	0.854398i	$0.326074\pi$
$68$	0	0
$69$	2.52148	0.303550
$70$	0	0
$71$	8.72459	1.03542	0.517709	−	0.855557i	$-0.326785\pi$
$71$	8.72459	1.03542	0.517709	+	0.855557i	$0.326785\pi$
$72$	0	0
$73$	15.8781	1.85839	0.929197	−	0.369584i	$-0.120500\pi$
$73$	15.8781	1.85839	0.929197	+	0.369584i	$0.120500\pi$
$74$	0	0
$75$	14.7773	1.70633
$76$	0	0
$77$	−12.0635	−1.37476
$78$	0	0
$79$	−6.44366	−0.724969	−0.362484	−	0.931990i	$-0.618071\pi$
$79$	−6.44366	−0.724969	−0.362484	+	0.931990i	$0.618071\pi$
$80$	0	0
$81$	−2.92386	−0.324873
$82$	0	0
$83$	13.9067	1.52646	0.763229	−	0.646129i	$-0.223613\pi$
$83$	13.9067	1.52646	0.763229	+	0.646129i	$0.223613\pi$
$84$	0	0
$85$	−5.26790	−0.571383
$86$	0	0
$87$	−0.873605	−0.0936603
$88$	0	0
$89$	−3.38571	−0.358885	−0.179443	−	0.983768i	$-0.557429\pi$
$89$	−3.38571	−0.358885	−0.179443	+	0.983768i	$0.557429\pi$
$90$	0	0
$91$	9.35392	0.980558
$92$	0	0
$93$	9.44689	0.979597
$94$	0	0
$95$	5.23171	0.536762
$96$	0	0
$97$	5.52372	0.560848	0.280424	−	0.959876i	$-0.409525\pi$
$97$	5.52372	0.560848	0.280424	+	0.959876i	$0.409525\pi$
$98$	0	0
$99$	7.96107	0.800118
$100$	0	0
$101$	9.82506	0.977630	0.488815	−	0.872388i	$-0.337429\pi$
$101$	9.82506	0.977630	0.488815	+	0.872388i	$0.337429\pi$
$102$	0	0
$103$	7.71382	0.760066	0.380033	−	0.924973i	$-0.375913\pi$
$103$	7.71382	0.760066	0.380033	+	0.924973i	$0.375913\pi$
$104$	0	0
$105$	−10.8786	−1.06164
$106$	0	0
$107$	17.3121	1.67363	0.836813	−	0.547489i	$-0.184416\pi$
$107$	17.3121	1.67363	0.836813	+	0.547489i	$0.184416\pi$
$108$	0	0
$109$	11.0896	1.06219	0.531097	−	0.847311i	$-0.321780\pi$
$109$	11.0896	1.06219	0.531097	+	0.847311i	$0.321780\pi$
$110$	0	0
$111$	−11.6208	−1.10300
$112$	0	0
$113$	16.2016	1.52412	0.762059	−	0.647508i	$-0.224189\pi$
$113$	16.2016	1.52412	0.762059	+	0.647508i	$0.224189\pi$
$114$	0	0
$115$	−8.09247	−0.754627
$116$	0	0
$117$	−6.17295	−0.570689
$118$	0	0
$119$	2.71207	0.248615
$120$	0	0
$121$	22.0162	2.00147
$122$	0	0
$123$	8.87320	0.800070
$124$	0	0
$125$	−27.0366	−2.41823
$126$	0	0
$127$	14.8561	1.31826	0.659132	−	0.752028i	$-0.270924\pi$
$127$	14.8561	1.31826	0.659132	+	0.752028i	$0.270924\pi$
$128$	0	0
$129$	−7.32598	−0.645016
$130$	0	0
$131$	9.40880	0.822051	0.411025	−	0.911624i	$-0.365171\pi$
$131$	9.40880	0.822051	0.411025	+	0.911624i	$0.365171\pi$
$132$	0	0
$133$	−2.69345	−0.233551
$134$	0	0
$135$	22.7239	1.95576
$136$	0	0
$137$	−15.7005	−1.34138	−0.670691	−	0.741737i	$-0.734002\pi$
$137$	−15.7005	−1.34138	−0.670691	+	0.741737i	$0.734002\pi$
$138$	0	0
$139$	1.96051	0.166288	0.0831439	−	0.996538i	$-0.473504\pi$
$139$	1.96051	0.166288	0.0831439	+	0.996538i	$0.473504\pi$
$140$	0	0
$141$	−11.5300	−0.971002
$142$	0	0
$143$	−25.6005	−2.14082
$144$	0	0
$145$	2.80376	0.232840
$146$	0	0
$147$	−3.29376	−0.271665
$148$	0	0
$149$	1.15853	0.0949102	0.0474551	−	0.998873i	$-0.484889\pi$
$149$	1.15853	0.0949102	0.0474551	+	0.998873i	$0.484889\pi$
$150$	0	0
$151$	−14.7477	−1.20015	−0.600076	−	0.799943i	$-0.704863\pi$
$151$	−14.7477	−1.20015	−0.600076	+	0.799943i	$0.704863\pi$
$152$	0	0
$153$	−1.78978	−0.144695
$154$	0	0
$155$	−30.3190	−2.43528
$156$	0	0
$157$	17.6512	1.40872	0.704360	−	0.709843i	$-0.251234\pi$
$157$	17.6512	1.40872	0.704360	+	0.709843i	$0.251234\pi$
$158$	0	0
$159$	−14.0426	−1.11365
$160$	0	0
$161$	4.16625	0.328347
$162$	0	0
$163$	1.81765	0.142369	0.0711847	−	0.997463i	$-0.477322\pi$
$163$	1.81765	0.142369	0.0711847	+	0.997463i	$0.477322\pi$
$164$	0	0
$165$	29.7733	2.31785
$166$	0	0
$167$	−4.91374	−0.380236	−0.190118	−	0.981761i	$-0.560887\pi$
$167$	−4.91374	−0.380236	−0.190118	+	0.981761i	$0.560887\pi$
$168$	0	0
$169$	6.85039	0.526953
$170$	0	0
$171$	1.77749	0.135928
$172$	0	0
$173$	−11.8817	−0.903348	−0.451674	−	0.892183i	$-0.649173\pi$
$173$	−11.8817	−0.903348	−0.451674	+	0.892183i	$0.649173\pi$
$174$	0	0
$175$	24.4166	1.84572
$176$	0	0
$177$	16.9352	1.27293
$178$	0	0
$179$	1.93193	0.144399	0.0721996	−	0.997390i	$-0.476998\pi$
$179$	1.93193	0.144399	0.0721996	+	0.997390i	$0.476998\pi$
$180$	0	0
$181$	9.70885	0.721653	0.360827	−	0.932633i	$-0.382495\pi$
$181$	9.70885	0.721653	0.360827	+	0.932633i	$0.382495\pi$
$182$	0	0
$183$	0.419140	0.0309838
$184$	0	0
$185$	37.2961	2.74206
$186$	0	0
$187$	−7.42259	−0.542794
$188$	0	0
$189$	−11.6990	−0.850974
$190$	0	0
$191$	0.472941	0.0342208	0.0171104	−	0.999854i	$-0.494553\pi$
$191$	0.472941	0.0342208	0.0171104	+	0.999854i	$0.494553\pi$
$192$	0	0
$193$	−13.0184	−0.937088	−0.468544	−	0.883440i	$-0.655221\pi$
$193$	−13.0184	−0.937088	−0.468544	+	0.883440i	$0.655221\pi$
$194$	0	0
$195$	−23.0859	−1.65322
$196$	0	0
$197$	−24.1877	−1.72330	−0.861651	−	0.507502i	$-0.830569\pi$
$197$	−24.1877	−1.72330	−0.861651	+	0.507502i	$0.830569\pi$
$198$	0	0
$199$	−0.689063	−0.0488464	−0.0244232	−	0.999702i	$-0.507775\pi$
$199$	−0.689063	−0.0488464	−0.0244232	+	0.999702i	$0.507775\pi$
$200$	0	0
$201$	10.8086	0.762382
$202$	0	0
$203$	−1.44346	−0.101311
$204$	0	0
$205$	−28.4778	−1.98898
$206$	0	0
$207$	−2.74944	−0.191099
$208$	0	0
$209$	7.37161	0.509905
$210$	0	0
$211$	−6.68495	−0.460211	−0.230105	−	0.973166i	$-0.573907\pi$
$211$	−6.68495	−0.460211	−0.230105	+	0.973166i	$0.573907\pi$
$212$	0	0
$213$	11.0857	0.759580
$214$	0	0
$215$	23.5121	1.60351
$216$	0	0
$217$	15.6092	1.05962
$218$	0	0
$219$	20.1752	1.36331
$220$	0	0
$221$	5.75542	0.387151
$222$	0	0
$223$	22.0985	1.47982	0.739911	−	0.672704i	$-0.234867\pi$
$223$	22.0985	1.47982	0.739911	+	0.672704i	$0.234867\pi$
$224$	0	0
$225$	−16.1133	−1.07422
$226$	0	0
$227$	9.03384	0.599597	0.299799	−	0.954003i	$-0.403081\pi$
$227$	9.03384	0.599597	0.299799	+	0.954003i	$0.403081\pi$
$228$	0	0
$229$	−20.2993	−1.34142	−0.670708	−	0.741721i	$-0.734010\pi$
$229$	−20.2993	−1.34142	−0.670708	+	0.741721i	$0.734010\pi$
$230$	0	0
$231$	−15.3282	−1.00852
$232$	0	0
$233$	−9.17237	−0.600902	−0.300451	−	0.953797i	$-0.597137\pi$
$233$	−9.17237	−0.600902	−0.300451	+	0.953797i	$0.597137\pi$
$234$	0	0
$235$	37.0047	2.41392
$236$	0	0
$237$	−8.18750	−0.531835
$238$	0	0
$239$	20.6737	1.33727	0.668636	−	0.743590i	$-0.266878\pi$
$239$	20.6737	1.33727	0.668636	+	0.743590i	$0.266878\pi$
$240$	0	0
$241$	20.6847	1.33242	0.666208	−	0.745766i	$-0.267916\pi$
$241$	20.6847	1.33242	0.666208	+	0.745766i	$0.267916\pi$
$242$	0	0
$243$	13.0019	0.834072
$244$	0	0
$245$	10.5711	0.675361
$246$	0	0
$247$	−5.71588	−0.363693
$248$	0	0
$249$	17.6702	1.11980
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−11.4025	−0.716869
$254$	0	0
$255$	−6.69353	−0.419165
$256$	0	0
$257$	−17.3220	−1.08052	−0.540260	−	0.841498i	$-0.681674\pi$
$257$	−17.3220	−1.08052	−0.540260	+	0.841498i	$0.681674\pi$
$258$	0	0
$259$	−19.2012	−1.19310
$260$	0	0
$261$	0.952587	0.0589637
$262$	0	0
$263$	−0.290383	−0.0179058	−0.00895290	−	0.999960i	$-0.502850\pi$
$263$	−0.290383	−0.0179058	−0.00895290	+	0.999960i	$0.502850\pi$
$264$	0	0
$265$	45.0686	2.76854
$266$	0	0
$267$	−4.30198	−0.263277
$268$	0	0
$269$	17.8551	1.08864	0.544322	−	0.838876i	$-0.316787\pi$
$269$	17.8551	1.08864	0.544322	+	0.838876i	$0.316787\pi$
$270$	0	0
$271$	9.63591	0.585340	0.292670	−	0.956214i	$-0.405456\pi$
$271$	9.63591	0.585340	0.292670	+	0.956214i	$0.405456\pi$
$272$	0	0
$273$	11.8854	0.719334
$274$	0	0
$275$	−66.8251	−4.02970
$276$	0	0
$277$	−2.95462	−0.177526	−0.0887631	−	0.996053i	$-0.528291\pi$
$277$	−2.95462	−0.177526	−0.0887631	+	0.996053i	$0.528291\pi$
$278$	0	0
$279$	−10.3010	−0.616704
$280$	0	0
$281$	−2.53220	−0.151058	−0.0755292	−	0.997144i	$-0.524065\pi$
$281$	−2.53220	−0.151058	−0.0755292	+	0.997144i	$0.524065\pi$
$282$	0	0
$283$	29.5538	1.75679	0.878396	−	0.477933i	$-0.158614\pi$
$283$	29.5538	1.75679	0.878396	+	0.477933i	$0.158614\pi$
$284$	0	0
$285$	6.64756	0.393767
$286$	0	0
$287$	14.6613	0.865427
$288$	0	0
$289$	−15.3313	−0.901840
$290$	0	0
$291$	7.01859	0.411437
$292$	0	0
$293$	−18.7684	−1.09646	−0.548230	−	0.836328i	$-0.684698\pi$
$293$	−18.7684	−1.09646	−0.548230	+	0.836328i	$0.684698\pi$
$294$	0	0
$295$	−54.3523	−3.16451
$296$	0	0
$297$	32.0185	1.85790
$298$	0	0
$299$	8.84140	0.511311
$300$	0	0
$301$	−12.1048	−0.697707
$302$	0	0
$303$	12.4840	0.717186
$304$	0	0
$305$	−1.34520	−0.0770258
$306$	0	0
$307$	15.4950	0.884348	0.442174	−	0.896929i	$-0.354207\pi$
$307$	15.4950	0.884348	0.442174	+	0.896929i	$0.354207\pi$
$308$	0	0
$309$	9.80140	0.557582
$310$	0	0
$311$	−12.1236	−0.687466	−0.343733	−	0.939067i	$-0.611692\pi$
$311$	−12.1236	−0.687466	−0.343733	+	0.939067i	$0.611692\pi$
$312$	0	0
$313$	−9.67230	−0.546711	−0.273355	−	0.961913i	$-0.588133\pi$
$313$	−9.67230	−0.546711	−0.273355	+	0.961913i	$0.588133\pi$
$314$	0	0
$315$	11.8621	0.668355
$316$	0	0
$317$	−20.5449	−1.15392	−0.576959	−	0.816773i	$-0.695761\pi$
$317$	−20.5449	−1.15392	−0.576959	+	0.816773i	$0.695761\pi$
$318$	0	0
$319$	3.95057	0.221190
$320$	0	0
$321$	21.9973	1.22777
$322$	0	0
$323$	−1.65726	−0.0922125
$324$	0	0
$325$	51.8156	2.87421
$326$	0	0
$327$	14.0908	0.779223
$328$	0	0
$329$	−19.0511	−1.05032
$330$	0	0
$331$	−7.83056	−0.430406	−0.215203	−	0.976569i	$-0.569041\pi$
$331$	−7.83056	−0.430406	−0.215203	+	0.976569i	$0.569041\pi$
$332$	0	0
$333$	12.6715	0.694392
$334$	0	0
$335$	−34.6894	−1.89529
$336$	0	0
$337$	17.4251	0.949205	0.474602	−	0.880200i	$-0.342592\pi$
$337$	17.4251	0.949205	0.474602	+	0.880200i	$0.342592\pi$
$338$	0	0
$339$	20.5862	1.11809
$340$	0	0
$341$	−42.7203	−2.31343
$342$	0	0
$343$	−20.1386	−1.08738
$344$	0	0
$345$	−10.2825	−0.553592
$346$	0	0
$347$	−4.62509	−0.248288	−0.124144	−	0.992264i	$-0.539618\pi$
$347$	−4.62509	−0.248288	−0.124144	+	0.992264i	$0.539618\pi$
$348$	0	0
$349$	2.64243	0.141446	0.0707230	−	0.997496i	$-0.477469\pi$
$349$	2.64243	0.141446	0.0707230	+	0.997496i	$0.477469\pi$
$350$	0	0
$351$	−24.8269	−1.32516
$352$	0	0
$353$	19.7926	1.05345	0.526727	−	0.850035i	$-0.323419\pi$
$353$	19.7926	1.05345	0.526727	+	0.850035i	$0.323419\pi$
$354$	0	0
$355$	−35.5787	−1.88832
$356$	0	0
$357$	3.44604	0.182384
$358$	0	0
$359$	9.70209	0.512057	0.256028	−	0.966669i	$-0.417586\pi$
$359$	9.70209	0.512057	0.256028	+	0.966669i	$0.417586\pi$
$360$	0	0
$361$	−17.3541	−0.913375
$362$	0	0
$363$	27.9744	1.46827
$364$	0	0
$365$	−64.7507	−3.38921
$366$	0	0
$367$	7.64672	0.399155	0.199578	−	0.979882i	$-0.436043\pi$
$367$	7.64672	0.399155	0.199578	+	0.979882i	$0.436043\pi$
$368$	0	0
$369$	−9.67543	−0.503683
$370$	0	0
$371$	−23.2027	−1.20462
$372$	0	0
$373$	−9.63477	−0.498869	−0.249435	−	0.968392i	$-0.580245\pi$
$373$	−9.63477	−0.498869	−0.249435	+	0.968392i	$0.580245\pi$
$374$	0	0
$375$	−34.3535	−1.77400
$376$	0	0
$377$	−3.06324	−0.157765
$378$	0	0
$379$	−23.3789	−1.20090	−0.600448	−	0.799664i	$-0.705011\pi$
$379$	−23.3789	−1.20090	−0.600448	+	0.799664i	$0.705011\pi$
$380$	0	0
$381$	18.8765	0.967074
$382$	0	0
$383$	−20.4517	−1.04503	−0.522515	−	0.852630i	$-0.675006\pi$
$383$	−20.4517	−1.04503	−0.522515	+	0.852630i	$0.675006\pi$
$384$	0	0
$385$	49.1946	2.50719
$386$	0	0
$387$	7.98832	0.406069
$388$	0	0
$389$	−39.2010	−1.98757	−0.993784	−	0.111328i	$-0.964490\pi$
$389$	−39.2010	−1.98757	−0.993784	+	0.111328i	$0.964490\pi$
$390$	0	0
$391$	2.56347	0.129640
$392$	0	0
$393$	11.9551	0.603054
$394$	0	0
$395$	26.2771	1.32214
$396$	0	0
$397$	12.7704	0.640928	0.320464	−	0.947261i	$-0.396161\pi$
$397$	12.7704	0.640928	0.320464	+	0.947261i	$0.396161\pi$
$398$	0	0
$399$	−3.42237	−0.171333
$400$	0	0
$401$	−25.2221	−1.25953	−0.629765	−	0.776786i	$-0.716849\pi$
$401$	−25.2221	−1.25953	−0.629765	+	0.776786i	$0.716849\pi$
$402$	0	0
$403$	33.1249	1.65007
$404$	0	0
$405$	11.9234	0.592480
$406$	0	0
$407$	52.5511	2.60486
$408$	0	0
$409$	7.62798	0.377179	0.188590	−	0.982056i	$-0.439608\pi$
$409$	7.62798	0.377179	0.188590	+	0.982056i	$0.439608\pi$
$410$	0	0
$411$	−19.9495	−0.984035
$412$	0	0
$413$	27.9822	1.37692
$414$	0	0
$415$	−56.7112	−2.78384
$416$	0	0
$417$	2.49107	0.121988
$418$	0	0
$419$	−24.9826	−1.22048	−0.610240	−	0.792216i	$-0.708927\pi$
$419$	−24.9826	−1.22048	−0.610240	+	0.792216i	$0.708927\pi$
$420$	0	0
$421$	12.5445	0.611382	0.305691	−	0.952131i	$-0.401113\pi$
$421$	12.5445	0.611382	0.305691	+	0.952131i	$0.401113\pi$
$422$	0	0
$423$	12.5724	0.611293
$424$	0	0
$425$	15.0234	0.728742
$426$	0	0
$427$	0.692549	0.0335148
$428$	0	0
$429$	−32.5287	−1.57050
$430$	0	0
$431$	8.86619	0.427069	0.213535	−	0.976935i	$-0.431502\pi$
$431$	8.86619	0.427069	0.213535	+	0.976935i	$0.431502\pi$
$432$	0	0
$433$	13.1928	0.634005	0.317003	−	0.948425i	$-0.397324\pi$
$433$	13.1928	0.634005	0.317003	+	0.948425i	$0.397324\pi$
$434$	0	0
$435$	3.56254	0.170811
$436$	0	0
$437$	−2.54587	−0.121785
$438$	0	0
$439$	−2.12443	−0.101394	−0.0506968	−	0.998714i	$-0.516144\pi$
$439$	−2.12443	−0.101394	−0.0506968	+	0.998714i	$0.516144\pi$
$440$	0	0
$441$	3.59155	0.171026
$442$	0	0
$443$	29.9876	1.42475	0.712377	−	0.701797i	$-0.247619\pi$
$443$	29.9876	1.42475	0.712377	+	0.701797i	$0.247619\pi$
$444$	0	0
$445$	13.8069	0.654508
$446$	0	0
$447$	1.47206	0.0696259
$448$	0	0
$449$	−40.3281	−1.90320	−0.951601	−	0.307337i	$-0.900562\pi$
$449$	−40.3281	−1.90320	−0.951601	+	0.307337i	$0.900562\pi$
$450$	0	0
$451$	−40.1260	−1.88946
$452$	0	0
$453$	−18.7388	−0.880428
$454$	0	0
$455$	−38.1451	−1.78827
$456$	0	0
$457$	10.3046	0.482028	0.241014	−	0.970522i	$-0.422520\pi$
$457$	10.3046	0.482028	0.241014	+	0.970522i	$0.422520\pi$
$458$	0	0
$459$	−7.19830	−0.335988
$460$	0	0
$461$	7.68761	0.358048	0.179024	−	0.983845i	$-0.442706\pi$
$461$	7.68761	0.358048	0.179024	+	0.983845i	$0.442706\pi$
$462$	0	0
$463$	−0.198326	−0.00921699	−0.00460850	−	0.999989i	$-0.501467\pi$
$463$	−0.198326	−0.00921699	−0.00460850	+	0.999989i	$0.501467\pi$
$464$	0	0
$465$	−38.5242	−1.78652
$466$	0	0
$467$	23.5806	1.09118	0.545590	−	0.838052i	$-0.316306\pi$
$467$	23.5806	1.09118	0.545590	+	0.838052i	$0.316306\pi$
$468$	0	0
$469$	17.8592	0.824660
$470$	0	0
$471$	22.4281	1.03343
$472$	0	0
$473$	33.1292	1.52328
$474$	0	0
$475$	−14.9202	−0.684586
$476$	0	0
$477$	15.3122	0.701097
$478$	0	0
$479$	−16.9678	−0.775279	−0.387640	−	0.921811i	$-0.626709\pi$
$479$	−16.9678	−0.775279	−0.387640	+	0.921811i	$0.626709\pi$
$480$	0	0
$481$	−40.7477	−1.85794
$482$	0	0
$483$	5.29376	0.240874
$484$	0	0
$485$	−22.5256	−1.02283
$486$	0	0
$487$	−16.4337	−0.744681	−0.372341	−	0.928096i	$-0.621445\pi$
$487$	−16.4337	−0.744681	−0.372341	+	0.928096i	$0.621445\pi$
$488$	0	0
$489$	2.30956	0.104442
$490$	0	0
$491$	40.5450	1.82977	0.914884	−	0.403716i	$-0.132282\pi$
$491$	40.5450	1.82977	0.914884	+	0.403716i	$0.132282\pi$
$492$	0	0
$493$	−0.888155	−0.0400005
$494$	0	0
$495$	−32.4651	−1.45920
$496$	0	0
$497$	18.3170	0.821630
$498$	0	0
$499$	8.49458	0.380270	0.190135	−	0.981758i	$-0.439108\pi$
$499$	8.49458	0.380270	0.190135	+	0.981758i	$0.439108\pi$
$500$	0	0
$501$	−6.24353	−0.278940
$502$	0	0
$503$	−31.9330	−1.42382	−0.711910	−	0.702270i	$-0.752170\pi$
$503$	−31.9330	−1.42382	−0.711910	+	0.702270i	$0.752170\pi$
$504$	0	0
$505$	−40.0664	−1.78293
$506$	0	0
$507$	8.70429	0.386571
$508$	0	0
$509$	29.0855	1.28919	0.644596	−	0.764523i	$-0.277025\pi$
$509$	29.0855	1.28919	0.644596	+	0.764523i	$0.277025\pi$
$510$	0	0
$511$	33.3356	1.47468
$512$	0	0
$513$	7.14886	0.315630
$514$	0	0
$515$	−31.4568	−1.38615
$516$	0	0
$517$	52.1405	2.29314
$518$	0	0
$519$	−15.0972	−0.662693
$520$	0	0
$521$	36.7946	1.61200	0.806001	−	0.591914i	$-0.201627\pi$
$521$	36.7946	1.61200	0.806001	+	0.591914i	$0.201627\pi$
$522$	0	0
$523$	−3.46392	−0.151467	−0.0757333	−	0.997128i	$-0.524130\pi$
$523$	−3.46392	−0.151467	−0.0757333	+	0.997128i	$0.524130\pi$
$524$	0	0
$525$	31.0244	1.35402
$526$	0	0
$527$	9.60423	0.418367
$528$	0	0
$529$	−19.0620	−0.828784
$530$	0	0
$531$	−18.4663	−0.801371
$532$	0	0
$533$	31.1133	1.34767
$534$	0	0
$535$	−70.5984	−3.05224
$536$	0	0
$537$	2.45476	0.105931
$538$	0	0
$539$	14.8949	0.641569
$540$	0	0
$541$	20.1924	0.868140	0.434070	−	0.900879i	$-0.357077\pi$
$541$	20.1924	0.868140	0.434070	+	0.900879i	$0.357077\pi$
$542$	0	0
$543$	12.3363	0.529403
$544$	0	0
$545$	−45.2233	−1.93715
$546$	0	0
$547$	26.4747	1.13198	0.565988	−	0.824413i	$-0.308495\pi$
$547$	26.4747	1.13198	0.565988	+	0.824413i	$0.308495\pi$
$548$	0	0
$549$	−0.457035	−0.0195058
$550$	0	0
$551$	0.882055	0.0375768
$552$	0	0
$553$	−13.5283	−0.575280
$554$	0	0
$555$	47.3895	2.01157
$556$	0	0
$557$	−37.0277	−1.56892	−0.784458	−	0.620182i	$-0.787059\pi$
$557$	−37.0277	−1.56892	−0.784458	+	0.620182i	$0.787059\pi$
$558$	0	0
$559$	−25.6881	−1.08649
$560$	0	0
$561$	−9.43135	−0.398192
$562$	0	0
$563$	7.57305	0.319166	0.159583	−	0.987184i	$-0.448985\pi$
$563$	7.57305	0.319166	0.159583	+	0.987184i	$0.448985\pi$
$564$	0	0
$565$	−66.0698	−2.77958
$566$	0	0
$567$	−6.13855	−0.257795
$568$	0	0
$569$	22.0963	0.926326	0.463163	−	0.886273i	$-0.346714\pi$
$569$	22.0963	0.926326	0.463163	+	0.886273i	$0.346714\pi$
$570$	0	0
$571$	−24.7923	−1.03753	−0.518764	−	0.854918i	$-0.673607\pi$
$571$	−24.7923	−1.03753	−0.518764	+	0.854918i	$0.673607\pi$
$572$	0	0
$573$	0.600932	0.0251043
$574$	0	0
$575$	23.0788	0.962451
$576$	0	0
$577$	−46.4720	−1.93465	−0.967327	−	0.253533i	$-0.918407\pi$
$577$	−46.4720	−1.93465	−0.967327	+	0.253533i	$0.918407\pi$
$578$	0	0
$579$	−16.5416	−0.687445
$580$	0	0
$581$	29.1966	1.21128
$582$	0	0
$583$	63.5027	2.63001
$584$	0	0
$585$	25.1731	1.04078
$586$	0	0
$587$	5.69596	0.235097	0.117549	−	0.993067i	$-0.462496\pi$
$587$	5.69596	0.235097	0.117549	+	0.993067i	$0.462496\pi$
$588$	0	0
$589$	−9.53827	−0.393017
$590$	0	0
$591$	−30.7335	−1.26421
$592$	0	0
$593$	28.4158	1.16690	0.583449	−	0.812150i	$-0.301703\pi$
$593$	28.4158	1.16690	0.583449	+	0.812150i	$0.301703\pi$
$594$	0	0
$595$	−11.0598	−0.453407
$596$	0	0
$597$	−0.875543	−0.0358336
$598$	0	0
$599$	18.6140	0.760546	0.380273	−	0.924874i	$-0.375830\pi$
$599$	18.6140	0.760546	0.380273	+	0.924874i	$0.375830\pi$
$600$	0	0
$601$	21.1822	0.864039	0.432020	−	0.901864i	$-0.357801\pi$
$601$	21.1822	0.864039	0.432020	+	0.901864i	$0.357801\pi$
$602$	0	0
$603$	−11.7858	−0.479956
$604$	0	0
$605$	−89.7815	−3.65014
$606$	0	0
$607$	18.6485	0.756920	0.378460	−	0.925618i	$-0.376454\pi$
$607$	18.6485	0.756920	0.378460	+	0.925618i	$0.376454\pi$
$608$	0	0
$609$	−1.83411	−0.0743217
$610$	0	0
$611$	−40.4293	−1.63559
$612$	0	0
$613$	−29.8764	−1.20670	−0.603349	−	0.797477i	$-0.706167\pi$
$613$	−29.8764	−1.20670	−0.603349	+	0.797477i	$0.706167\pi$
$614$	0	0
$615$	−36.1847	−1.45911
$616$	0	0
$617$	1.98925	0.0800843	0.0400422	−	0.999198i	$-0.487251\pi$
$617$	1.98925	0.0800843	0.0400422	+	0.999198i	$0.487251\pi$
$618$	0	0
$619$	5.42262	0.217953	0.108977	−	0.994044i	$-0.465243\pi$
$619$	5.42262	0.217953	0.108977	+	0.994044i	$0.465243\pi$
$620$	0	0
$621$	−11.0579	−0.443740
$622$	0	0
$623$	−7.10820	−0.284784
$624$	0	0
$625$	52.1051	2.08421
$626$	0	0
$627$	9.36657	0.374065
$628$	0	0
$629$	−11.8144	−0.471070
$630$	0	0
$631$	−1.46384	−0.0582744	−0.0291372	−	0.999575i	$-0.509276\pi$
$631$	−1.46384	−0.0582744	−0.0291372	+	0.999575i	$0.509276\pi$
$632$	0	0
$633$	−8.49408	−0.337609
$634$	0	0
$635$	−60.5827	−2.40415
$636$	0	0
$637$	−11.5494	−0.457603
$638$	0	0
$639$	−12.0880	−0.478193
$640$	0	0
$641$	35.0555	1.38461	0.692304	−	0.721606i	$-0.256596\pi$
$641$	35.0555	1.38461	0.692304	+	0.721606i	$0.256596\pi$
$642$	0	0
$643$	18.2894	0.721265	0.360632	−	0.932708i	$-0.382561\pi$
$643$	18.2894	0.721265	0.360632	+	0.932708i	$0.382561\pi$
$644$	0	0
$645$	29.8752	1.17633
$646$	0	0
$647$	41.3769	1.62669	0.813346	−	0.581780i	$-0.197643\pi$
$647$	41.3769	1.62669	0.813346	+	0.581780i	$0.197643\pi$
$648$	0	0
$649$	−76.5837	−3.00617
$650$	0	0
$651$	19.8334	0.777334
$652$	0	0
$653$	24.9378	0.975893	0.487946	−	0.872874i	$-0.337746\pi$
$653$	24.9378	0.975893	0.487946	+	0.872874i	$0.337746\pi$
$654$	0	0
$655$	−38.3689	−1.49920
$656$	0	0
$657$	−21.9992	−0.858272
$658$	0	0
$659$	−29.4443	−1.14699	−0.573493	−	0.819210i	$-0.694412\pi$
$659$	−29.4443	−1.14699	−0.573493	+	0.819210i	$0.694412\pi$
$660$	0	0
$661$	−2.94304	−0.114471	−0.0572356	−	0.998361i	$-0.518229\pi$
$661$	−2.94304	−0.114471	−0.0572356	+	0.998361i	$0.518229\pi$
$662$	0	0
$663$	7.31299	0.284013
$664$	0	0
$665$	10.9838	0.425934
$666$	0	0
$667$	−1.36437	−0.0528287
$668$	0	0
$669$	28.0789	1.08559
$670$	0	0
$671$	−1.89542	−0.0731717
$672$	0	0
$673$	−9.81948	−0.378513	−0.189257	−	0.981928i	$-0.560608\pi$
$673$	−9.81948	−0.378513	−0.189257	+	0.981928i	$0.560608\pi$
$674$	0	0
$675$	−64.8058	−2.49438
$676$	0	0
$677$	4.68327	0.179993	0.0899964	−	0.995942i	$-0.471314\pi$
$677$	4.68327	0.179993	0.0899964	+	0.995942i	$0.471314\pi$
$678$	0	0
$679$	11.5969	0.445047
$680$	0	0
$681$	11.4786	0.439863
$682$	0	0
$683$	−8.71397	−0.333431	−0.166715	−	0.986005i	$-0.553316\pi$
$683$	−8.71397	−0.333431	−0.166715	+	0.986005i	$0.553316\pi$
$684$	0	0
$685$	64.0262	2.44632
$686$	0	0
$687$	−25.7929	−0.984059
$688$	0	0
$689$	−49.2395	−1.87587
$690$	0	0
$691$	−12.1931	−0.463848	−0.231924	−	0.972734i	$-0.574502\pi$
$691$	−12.1931	−0.463848	−0.231924	+	0.972734i	$0.574502\pi$
$692$	0	0
$693$	16.7140	0.634913
$694$	0	0
$695$	−7.99490	−0.303264
$696$	0	0
$697$	9.02099	0.341694
$698$	0	0
$699$	−11.6547	−0.440820
$700$	0	0
$701$	−33.7007	−1.27286	−0.636429	−	0.771335i	$-0.719589\pi$
$701$	−33.7007	−1.27286	−0.636429	+	0.771335i	$0.719589\pi$
$702$	0	0
$703$	11.7332	0.442527
$704$	0	0
$705$	47.0191	1.77084
$706$	0	0
$707$	20.6274	0.775773
$708$	0	0
$709$	−7.98925	−0.300043	−0.150021	−	0.988683i	$-0.547934\pi$
$709$	−7.98925	−0.300043	−0.150021	+	0.988683i	$0.547934\pi$
$710$	0	0
$711$	8.92773	0.334816
$712$	0	0
$713$	14.7539	0.552538
$714$	0	0
$715$	104.398	3.90427
$716$	0	0
$717$	26.2686	0.981019
$718$	0	0
$719$	21.8001	0.813006	0.406503	−	0.913649i	$-0.366748\pi$
$719$	21.8001	0.813006	0.406503	+	0.913649i	$0.366748\pi$
$720$	0	0
$721$	16.1949	0.603131
$722$	0	0
$723$	26.2825	0.977457
$724$	0	0
$725$	−7.99600	−0.296964
$726$	0	0
$727$	20.2534	0.751155	0.375578	−	0.926791i	$-0.377444\pi$
$727$	20.2534	0.751155	0.375578	+	0.926791i	$0.377444\pi$
$728$	0	0
$729$	25.2921	0.936746
$730$	0	0
$731$	−7.44800	−0.275474
$732$	0	0
$733$	33.8328	1.24964	0.624821	−	0.780768i	$-0.285172\pi$
$733$	33.8328	1.24964	0.624821	+	0.780768i	$0.285172\pi$
$734$	0	0
$735$	13.4319	0.495443
$736$	0	0
$737$	−48.8783	−1.80045
$738$	0	0
$739$	−28.5878	−1.05162	−0.525810	−	0.850602i	$-0.676238\pi$
$739$	−28.5878	−1.05162	−0.525810	+	0.850602i	$0.676238\pi$
$740$	0	0
$741$	−7.26276	−0.266804
$742$	0	0
$743$	−13.7425	−0.504162	−0.252081	−	0.967706i	$-0.581115\pi$
$743$	−13.7425	−0.504162	−0.252081	+	0.967706i	$0.581115\pi$
$744$	0	0
$745$	−4.72445	−0.173090
$746$	0	0
$747$	−19.2678	−0.704971
$748$	0	0
$749$	36.3462	1.32806
$750$	0	0
$751$	13.3612	0.487557	0.243779	−	0.969831i	$-0.421613\pi$
$751$	13.3612	0.487557	0.243779	+	0.969831i	$0.421613\pi$
$752$	0	0
$753$	1.27063	0.0463042
$754$	0	0
$755$	60.1408	2.18875
$756$	0	0
$757$	50.7678	1.84519	0.922593	−	0.385775i	$-0.126066\pi$
$757$	50.7678	1.84519	0.922593	+	0.385775i	$0.126066\pi$
$758$	0	0
$759$	−14.4883	−0.525893
$760$	0	0
$761$	−10.2687	−0.372239	−0.186119	−	0.982527i	$-0.559591\pi$
$761$	−10.2687	−0.372239	−0.186119	+	0.982527i	$0.559591\pi$
$762$	0	0
$763$	23.2823	0.842877
$764$	0	0
$765$	7.29870	0.263885
$766$	0	0
$767$	59.3824	2.14417
$768$	0	0
$769$	−22.8470	−0.823883	−0.411942	−	0.911210i	$-0.635149\pi$
$769$	−22.8470	−0.823883	−0.411942	+	0.911210i	$0.635149\pi$
$770$	0	0
$771$	−22.0099	−0.792666
$772$	0	0
$773$	18.2658	0.656974	0.328487	−	0.944508i	$-0.393461\pi$
$773$	18.2658	0.656974	0.328487	+	0.944508i	$0.393461\pi$
$774$	0	0
$775$	86.4662	3.10596
$776$	0	0
$777$	−24.3976	−0.875257
$778$	0	0
$779$	−8.95903	−0.320991
$780$	0	0
$781$	−50.1313	−1.79384
$782$	0	0
$783$	3.83120	0.136916
$784$	0	0
$785$	−71.9812	−2.56912
$786$	0	0
$787$	−8.57089	−0.305519	−0.152760	−	0.988263i	$-0.548816\pi$
$787$	−8.57089	−0.305519	−0.152760	+	0.988263i	$0.548816\pi$
$788$	0	0
$789$	−0.368969	−0.0131356
$790$	0	0
$791$	34.0147	1.20942
$792$	0	0
$793$	1.46969	0.0521902
$794$	0	0
$795$	57.2654	2.03099
$796$	0	0
$797$	28.4791	1.00878	0.504390	−	0.863476i	$-0.331717\pi$
$797$	28.4791	1.00878	0.504390	+	0.863476i	$0.331717\pi$
$798$	0	0
$799$	−11.7221	−0.414697
$800$	0	0
$801$	4.69093	0.165746
$802$	0	0
$803$	−91.2353	−3.21962
$804$	0	0
$805$	−16.9899	−0.598815
$806$	0	0
$807$	22.6872	0.798627
$808$	0	0
$809$	18.3119	0.643813	0.321907	−	0.946771i	$-0.395676\pi$
$809$	18.3119	0.643813	0.321907	+	0.946771i	$0.395676\pi$
$810$	0	0
$811$	−0.351002	−0.0123254	−0.00616268	−	0.999981i	$-0.501962\pi$
$811$	−0.351002	−0.0123254	−0.00616268	+	0.999981i	$0.501962\pi$
$812$	0	0
$813$	12.2437	0.429404
$814$	0	0
$815$	−7.41234	−0.259643
$816$	0	0
$817$	7.39684	0.258783
$818$	0	0
$819$	−12.9599	−0.452856
$820$	0	0
$821$	5.32180	0.185732	0.0928660	−	0.995679i	$-0.470397\pi$
$821$	5.32180	0.185732	0.0928660	+	0.995679i	$0.470397\pi$
$822$	0	0
$823$	−14.2386	−0.496326	−0.248163	−	0.968718i	$-0.579827\pi$
$823$	−14.2386	−0.496326	−0.248163	+	0.968718i	$0.579827\pi$
$824$	0	0
$825$	−84.9098	−2.95618
$826$	0	0
$827$	1.00907	0.0350888	0.0175444	−	0.999846i	$-0.494415\pi$
$827$	1.00907	0.0350888	0.0175444	+	0.999846i	$0.494415\pi$
$828$	0	0
$829$	30.4099	1.05618	0.528091	−	0.849188i	$-0.322908\pi$
$829$	30.4099	1.05618	0.528091	+	0.849188i	$0.322908\pi$
$830$	0	0
$831$	−3.75423	−0.130233
$832$	0	0
$833$	−3.34862	−0.116023
$834$	0	0
$835$	20.0381	0.693447
$836$	0	0
$837$	−41.4294	−1.43201
$838$	0	0
$839$	41.1515	1.42071	0.710354	−	0.703845i	$-0.248535\pi$
$839$	41.1515	1.42071	0.710354	+	0.703845i	$0.248535\pi$
$840$	0	0
$841$	−28.5273	−0.983700
$842$	0	0
$843$	−3.21749	−0.110816
$844$	0	0
$845$	−27.9357	−0.961018
$846$	0	0
$847$	46.2223	1.58822
$848$	0	0
$849$	37.5519	1.28878
$850$	0	0
$851$	−18.1491	−0.622143
$852$	0	0
$853$	−31.7317	−1.08647	−0.543236	−	0.839580i	$-0.682801\pi$
$853$	−31.7317	−1.08647	−0.543236	+	0.839580i	$0.682801\pi$
$854$	0	0
$855$	−7.24856	−0.247896
$856$	0	0
$857$	−7.56093	−0.258277	−0.129138	−	0.991627i	$-0.541221\pi$
$857$	−7.56093	−0.258277	−0.129138	+	0.991627i	$0.541221\pi$
$858$	0	0
$859$	1.41905	0.0484173	0.0242086	−	0.999707i	$-0.492293\pi$
$859$	1.41905	0.0484173	0.0242086	+	0.999707i	$0.492293\pi$
$860$	0	0
$861$	18.6290	0.634875
$862$	0	0
$863$	49.7544	1.69366	0.846830	−	0.531864i	$-0.178508\pi$
$863$	49.7544	1.69366	0.846830	+	0.531864i	$0.178508\pi$
$864$	0	0
$865$	48.4532	1.64746
$866$	0	0
$867$	−19.4803	−0.661587
$868$	0	0
$869$	37.0251	1.25599
$870$	0	0
$871$	37.8998	1.28419
$872$	0	0
$873$	−7.65314	−0.259019
$874$	0	0
$875$	−56.7625	−1.91892
$876$	0	0
$877$	−22.8695	−0.772248	−0.386124	−	0.922447i	$-0.626186\pi$
$877$	−22.8695	−0.772248	−0.386124	+	0.922447i	$0.626186\pi$
$878$	0	0
$879$	−23.8476	−0.804359
$880$	0	0
$881$	26.3281	0.887016	0.443508	−	0.896270i	$-0.353734\pi$
$881$	26.3281	0.887016	0.443508	+	0.896270i	$0.353734\pi$
$882$	0	0
$883$	−42.6713	−1.43601	−0.718003	−	0.696040i	$-0.754943\pi$
$883$	−42.6713	−1.43601	−0.718003	+	0.696040i	$0.754943\pi$
$884$	0	0
$885$	−69.0615	−2.32148
$886$	0	0
$887$	−59.0291	−1.98200	−0.991001	−	0.133856i	$-0.957264\pi$
$887$	−59.0291	−1.98200	−0.991001	+	0.133856i	$0.957264\pi$
$888$	0	0
$889$	31.1898	1.04607
$890$	0	0
$891$	16.8004	0.562835
$892$	0	0
$893$	11.6415	0.389569
$894$	0	0
$895$	−7.87837	−0.263345
$896$	0	0
$897$	11.2341	0.375097
$898$	0	0
$899$	−5.11172	−0.170485
$900$	0	0
$901$	−14.2765	−0.475619
$902$	0	0
$903$	−15.3807	−0.511836
$904$	0	0
$905$	−39.5925	−1.31610
$906$	0	0
$907$	9.49031	0.315121	0.157560	−	0.987509i	$-0.449637\pi$
$907$	9.49031	0.315121	0.157560	+	0.987509i	$0.449637\pi$
$908$	0	0
$909$	−13.6127	−0.451504
$910$	0	0
$911$	44.6095	1.47798	0.738989	−	0.673717i	$-0.235303\pi$
$911$	44.6095	1.47798	0.738989	+	0.673717i	$0.235303\pi$
$912$	0	0
$913$	−79.9074	−2.64455
$914$	0	0
$915$	−1.70925	−0.0565059
$916$	0	0
$917$	19.7535	0.652317
$918$	0	0
$919$	−0.909500	−0.0300016	−0.0150008	−	0.999887i	$-0.504775\pi$
$919$	−0.909500	−0.0300016	−0.0150008	+	0.999887i	$0.504775\pi$
$920$	0	0
$921$	19.6884	0.648755
$922$	0	0
$923$	38.8714	1.27947
$924$	0	0
$925$	−106.364	−3.49723
$926$	0	0
$927$	−10.6875	−0.351025
$928$	0	0
$929$	36.0888	1.18403	0.592017	−	0.805925i	$-0.298332\pi$
$929$	36.0888	1.18403	0.592017	+	0.805925i	$0.298332\pi$
$930$	0	0
$931$	3.32562	0.108993
$932$	0	0
$933$	−15.4046	−0.504324
$934$	0	0
$935$	30.2692	0.989908
$936$	0	0
$937$	−17.0356	−0.556530	−0.278265	−	0.960504i	$-0.589759\pi$
$937$	−17.0356	−0.556530	−0.278265	+	0.960504i	$0.589759\pi$
$938$	0	0
$939$	−12.2899	−0.401065
$940$	0	0
$941$	−45.1460	−1.47172	−0.735859	−	0.677135i	$-0.763221\pi$
$941$	−45.1460	−1.47172	−0.735859	+	0.677135i	$0.763221\pi$
$942$	0	0
$943$	13.8579	0.451276
$944$	0	0
$945$	47.7081	1.55194
$946$	0	0
$947$	53.8474	1.74981	0.874903	−	0.484297i	$-0.160925\pi$
$947$	53.8474	1.74981	0.874903	+	0.484297i	$0.160925\pi$
$948$	0	0
$949$	70.7431	2.29642
$950$	0	0
$951$	−26.1049	−0.846510
$952$	0	0
$953$	29.4499	0.953976	0.476988	−	0.878910i	$-0.341728\pi$
$953$	29.4499	0.953976	0.476988	+	0.878910i	$0.341728\pi$
$954$	0	0
$955$	−1.92864	−0.0624094
$956$	0	0
$957$	5.01971	0.162264
$958$	0	0
$959$	−32.9627	−1.06442
$960$	0	0
$961$	24.2766	0.783115
$962$	0	0
$963$	−23.9860	−0.772939
$964$	0	0
$965$	53.0889	1.70899
$966$	0	0
$967$	38.6706	1.24356	0.621781	−	0.783191i	$-0.286409\pi$
$967$	38.6706	1.24356	0.621781	+	0.783191i	$0.286409\pi$
$968$	0	0
$969$	−2.10576	−0.0676469
$970$	0	0
$971$	19.7886	0.635045	0.317522	−	0.948251i	$-0.397149\pi$
$971$	19.7886	0.635045	0.317522	+	0.948251i	$0.397149\pi$
$972$	0	0
$973$	4.11602	0.131953
$974$	0	0
$975$	65.8383	2.10851
$976$	0	0
$977$	26.7402	0.855496	0.427748	−	0.903898i	$-0.359307\pi$
$977$	26.7402	0.855496	0.427748	+	0.903898i	$0.359307\pi$
$978$	0	0
$979$	19.4542	0.621759
$980$	0	0
$981$	−15.3648	−0.490559
$982$	0	0
$983$	−42.6381	−1.35995	−0.679973	−	0.733237i	$-0.738008\pi$
$983$	−42.6381	−1.35995	−0.679973	+	0.733237i	$0.738008\pi$
$984$	0	0
$985$	98.6368	3.14283
$986$	0	0
$987$	−24.2069	−0.770514
$988$	0	0
$989$	−11.4415	−0.363819
$990$	0	0
$991$	−6.21324	−0.197370	−0.0986850	−	0.995119i	$-0.531464\pi$
$991$	−6.21324	−0.197370	−0.0986850	+	0.995119i	$0.531464\pi$
$992$	0	0
$993$	−9.94972	−0.315745
$994$	0	0
$995$	2.80998	0.0890825
$996$	0	0
$997$	14.9268	0.472736	0.236368	−	0.971664i	$-0.424043\pi$
$997$	14.9268	0.472736	0.236368	+	0.971664i	$0.424043\pi$
$998$	0	0
$999$	50.9632	1.61240

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.27063 q^{3} -4.07798 q^{5} +2.09947 q^{7} -1.38551 q^{9} +O(q^{10})\)	\(q+1.27063 q^{3} -4.07798 q^{5} +2.09947 q^{7} -1.38551 q^{9} -5.74597 q^{11} +4.45538 q^{13} -5.18159 q^{15} +1.29179 q^{17} -1.28292 q^{19} +2.66764 q^{21} +1.98443 q^{23} +11.6299 q^{25} -5.57234 q^{27} -0.687538 q^{29} +7.43482 q^{31} -7.30099 q^{33} -8.56158 q^{35} -9.14574 q^{37} +5.66113 q^{39} +6.98332 q^{41} -5.76564 q^{43} +5.65006 q^{45} -9.07427 q^{47} -2.59223 q^{49} +1.64139 q^{51} -11.0517 q^{53} +23.4319 q^{55} -1.63011 q^{57} +13.3282 q^{59} +0.329869 q^{61} -2.90882 q^{63} -18.1689 q^{65} +8.50653 q^{67} +2.52148 q^{69} +8.72459 q^{71} +15.8781 q^{73} +14.7773 q^{75} -12.0635 q^{77} -6.44366 q^{79} -2.92386 q^{81} +13.9067 q^{83} -5.26790 q^{85} -0.873605 q^{87} -3.38571 q^{89} +9.35392 q^{91} +9.44689 q^{93} +5.23171 q^{95} +5.52372 q^{97} +7.96107 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) \) 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

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Database

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