LMFDB - Embedded newform 4016.2.a.l.1.1

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			1.1
Root			$-3.12892$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.12892 q^{3} -3.70734 q^{5} -1.25809 q^{7} +6.79011 q^{9} +O(q^{10})$	$q-3.12892 q^{3} -3.70734 q^{5} -1.25809 q^{7} +6.79011 q^{9} +5.85699 q^{11} -5.70474 q^{13} +11.5999 q^{15} -0.899972 q^{17} +2.13431 q^{19} +3.93645 q^{21} -1.38725 q^{23} +8.74435 q^{25} -11.8589 q^{27} -1.64557 q^{29} +3.15019 q^{31} -18.3260 q^{33} +4.66416 q^{35} -9.50290 q^{37} +17.8496 q^{39} -7.18394 q^{41} -10.1356 q^{43} -25.1732 q^{45} -0.589261 q^{47} -5.41721 q^{49} +2.81594 q^{51} +3.68315 q^{53} -21.7138 q^{55} -6.67806 q^{57} -0.265882 q^{59} -12.0646 q^{61} -8.54256 q^{63} +21.1494 q^{65} +0.794876 q^{67} +4.34058 q^{69} +10.2272 q^{71} +8.75771 q^{73} -27.3603 q^{75} -7.36861 q^{77} -10.6715 q^{79} +16.7353 q^{81} +1.59118 q^{83} +3.33650 q^{85} +5.14886 q^{87} +0.0712382 q^{89} +7.17706 q^{91} -9.85667 q^{93} -7.91259 q^{95} +16.8255 q^{97} +39.7696 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.12892	−1.80648	−0.903240	−	0.429136i	$-0.858818\pi$
$3$	−3.12892	−1.80648	−0.903240	+	0.429136i	$0.858818\pi$
$4$	0	0
$5$	−3.70734	−1.65797	−0.828986	−	0.559270i	$-0.811082\pi$
$5$	−3.70734	−1.65797	−0.828986	+	0.559270i	$0.811082\pi$
$6$	0	0
$7$	−1.25809	−0.475513	−0.237756	−	0.971325i	$-0.576412\pi$
$7$	−1.25809	−0.475513	−0.237756	+	0.971325i	$0.576412\pi$
$8$	0	0
$9$	6.79011	2.26337
$10$	0	0
$11$	5.85699	1.76595	0.882975	−	0.469421i	$-0.155537\pi$
$11$	5.85699	1.76595	0.882975	+	0.469421i	$0.155537\pi$
$12$	0	0
$13$	−5.70474	−1.58221	−0.791105	−	0.611681i	$-0.790494\pi$
$13$	−5.70474	−1.58221	−0.791105	+	0.611681i	$0.790494\pi$
$14$	0	0
$15$	11.5999	2.99509
$16$	0	0
$17$	−0.899972	−0.218275	−0.109138	−	0.994027i	$-0.534809\pi$
$17$	−0.899972	−0.218275	−0.109138	+	0.994027i	$0.534809\pi$
$18$	0	0
$19$	2.13431	0.489643	0.244822	−	0.969568i	$-0.421271\pi$
$19$	2.13431	0.489643	0.244822	+	0.969568i	$0.421271\pi$
$20$	0	0
$21$	3.93645	0.859004
$22$	0	0
$23$	−1.38725	−0.289261	−0.144631	−	0.989486i	$-0.546199\pi$
$23$	−1.38725	−0.289261	−0.144631	+	0.989486i	$0.546199\pi$
$24$	0	0
$25$	8.74435	1.74887
$26$	0	0
$27$	−11.8589	−2.28225
$28$	0	0
$29$	−1.64557	−0.305575	−0.152788	−	0.988259i	$-0.548825\pi$
$29$	−1.64557	−0.305575	−0.152788	+	0.988259i	$0.548825\pi$
$30$	0	0
$31$	3.15019	0.565790	0.282895	−	0.959151i	$-0.408705\pi$
$31$	3.15019	0.565790	0.282895	+	0.959151i	$0.408705\pi$
$32$	0	0
$33$	−18.3260	−3.19015
$34$	0	0
$35$	4.66416	0.788386
$36$	0	0
$37$	−9.50290	−1.56227	−0.781134	−	0.624364i	$-0.785358\pi$
$37$	−9.50290	−1.56227	−0.781134	+	0.624364i	$0.785358\pi$
$38$	0	0
$39$	17.8496	2.85823
$40$	0	0
$41$	−7.18394	−1.12194	−0.560971	−	0.827835i	$-0.689572\pi$
$41$	−7.18394	−1.12194	−0.560971	+	0.827835i	$0.689572\pi$
$42$	0	0
$43$	−10.1356	−1.54566	−0.772831	−	0.634612i	$-0.781160\pi$
$43$	−10.1356	−1.54566	−0.772831	+	0.634612i	$0.781160\pi$
$44$	0	0
$45$	−25.1732	−3.75260
$46$	0	0
$47$	−0.589261	−0.0859525	−0.0429763	−	0.999076i	$-0.513684\pi$
$47$	−0.589261	−0.0859525	−0.0429763	+	0.999076i	$0.513684\pi$
$48$	0	0
$49$	−5.41721	−0.773888
$50$	0	0
$51$	2.81594	0.394310
$52$	0	0
$53$	3.68315	0.505919	0.252960	−	0.967477i	$-0.418596\pi$
$53$	3.68315	0.505919	0.252960	+	0.967477i	$0.418596\pi$
$54$	0	0
$55$	−21.7138	−2.92789
$56$	0	0
$57$	−6.67806	−0.884531
$58$	0	0
$59$	−0.265882	−0.0346149	−0.0173075	−	0.999850i	$-0.505509\pi$
$59$	−0.265882	−0.0346149	−0.0173075	+	0.999850i	$0.505509\pi$
$60$	0	0
$61$	−12.0646	−1.54471	−0.772357	−	0.635189i	$-0.780922\pi$
$61$	−12.0646	−1.54471	−0.772357	+	0.635189i	$0.780922\pi$
$62$	0	0
$63$	−8.54256	−1.07626
$64$	0	0
$65$	21.1494	2.62326
$66$	0	0
$67$	0.794876	0.0971095	0.0485548	−	0.998821i	$-0.484538\pi$
$67$	0.794876	0.0971095	0.0485548	+	0.998821i	$0.484538\pi$
$68$	0	0
$69$	4.34058	0.522545
$70$	0	0
$71$	10.2272	1.21375	0.606875	−	0.794797i	$-0.292423\pi$
$71$	10.2272	1.21375	0.606875	+	0.794797i	$0.292423\pi$
$72$	0	0
$73$	8.75771	1.02501	0.512506	−	0.858684i	$-0.328717\pi$
$73$	8.75771	1.02501	0.512506	+	0.858684i	$0.328717\pi$
$74$	0	0
$75$	−27.3603	−3.15930
$76$	0	0
$77$	−7.36861	−0.839731
$78$	0	0
$79$	−10.6715	−1.20064	−0.600318	−	0.799761i	$-0.704959\pi$
$79$	−10.6715	−1.20064	−0.600318	+	0.799761i	$0.704959\pi$
$80$	0	0
$81$	16.7353	1.85948
$82$	0	0
$83$	1.59118	0.174655	0.0873276	−	0.996180i	$-0.472167\pi$
$83$	1.59118	0.174655	0.0873276	+	0.996180i	$0.472167\pi$
$84$	0	0
$85$	3.33650	0.361894
$86$	0	0
$87$	5.14886	0.552016
$88$	0	0
$89$	0.0712382	0.00755123	0.00377562	−	0.999993i	$-0.498798\pi$
$89$	0.0712382	0.00755123	0.00377562	+	0.999993i	$0.498798\pi$
$90$	0	0
$91$	7.17706	0.752360
$92$	0	0
$93$	−9.85667	−1.02209
$94$	0	0
$95$	−7.91259	−0.811815
$96$	0	0
$97$	16.8255	1.70837	0.854187	−	0.519966i	$-0.174055\pi$
$97$	16.8255	1.70837	0.854187	+	0.519966i	$0.174055\pi$
$98$	0	0
$99$	39.7696	3.99700
$100$	0	0
$101$	−14.7752	−1.47019	−0.735095	−	0.677964i	$-0.762863\pi$
$101$	−14.7752	−1.47019	−0.735095	+	0.677964i	$0.762863\pi$
$102$	0	0
$103$	−16.8199	−1.65731	−0.828657	−	0.559757i	$-0.810895\pi$
$103$	−16.8199	−1.65731	−0.828657	+	0.559757i	$0.810895\pi$
$104$	0	0
$105$	−14.5937	−1.42420
$106$	0	0
$107$	−7.17174	−0.693318	−0.346659	−	0.937991i	$-0.612684\pi$
$107$	−7.17174	−0.693318	−0.346659	+	0.937991i	$0.612684\pi$
$108$	0	0
$109$	0.825823	0.0790995	0.0395497	−	0.999218i	$-0.487408\pi$
$109$	0.825823	0.0790995	0.0395497	+	0.999218i	$0.487408\pi$
$110$	0	0
$111$	29.7338	2.82221
$112$	0	0
$113$	−14.0602	−1.32268	−0.661338	−	0.750088i	$-0.730011\pi$
$113$	−14.0602	−1.32268	−0.661338	+	0.750088i	$0.730011\pi$
$114$	0	0
$115$	5.14300	0.479587
$116$	0	0
$117$	−38.7358	−3.58113
$118$	0	0
$119$	1.13224	0.103793
$120$	0	0
$121$	23.3043	2.11858
$122$	0	0
$123$	22.4779	2.02677
$124$	0	0
$125$	−13.8816	−1.24160
$126$	0	0
$127$	19.7719	1.75447	0.877235	−	0.480061i	$-0.159386\pi$
$127$	19.7719	1.75447	0.877235	+	0.480061i	$0.159386\pi$
$128$	0	0
$129$	31.7134	2.79221
$130$	0	0
$131$	−3.43587	−0.300194	−0.150097	−	0.988671i	$-0.547959\pi$
$131$	−3.43587	−0.300194	−0.150097	+	0.988671i	$0.547959\pi$
$132$	0	0
$133$	−2.68514	−0.232831
$134$	0	0
$135$	43.9651	3.78391
$136$	0	0
$137$	−20.8131	−1.77818	−0.889091	−	0.457730i	$-0.848663\pi$
$137$	−20.8131	−1.77818	−0.889091	+	0.457730i	$0.848663\pi$
$138$	0	0
$139$	−9.82967	−0.833742	−0.416871	−	0.908966i	$-0.636873\pi$
$139$	−9.82967	−0.833742	−0.416871	+	0.908966i	$0.636873\pi$
$140$	0	0
$141$	1.84375	0.155272
$142$	0	0
$143$	−33.4126	−2.79410
$144$	0	0
$145$	6.10070	0.506635
$146$	0	0
$147$	16.9500	1.39801
$148$	0	0
$149$	11.3956	0.933567	0.466784	−	0.884372i	$-0.345413\pi$
$149$	11.3956	0.933567	0.466784	+	0.884372i	$0.345413\pi$
$150$	0	0
$151$	7.33838	0.597189	0.298595	−	0.954380i	$-0.403482\pi$
$151$	7.33838	0.597189	0.298595	+	0.954380i	$0.403482\pi$
$152$	0	0
$153$	−6.11091	−0.494038
$154$	0	0
$155$	−11.6788	−0.938064
$156$	0	0
$157$	19.0538	1.52066	0.760329	−	0.649538i	$-0.225038\pi$
$157$	19.0538	1.52066	0.760329	+	0.649538i	$0.225038\pi$
$158$	0	0
$159$	−11.5243	−0.913933
$160$	0	0
$161$	1.74528	0.137547
$162$	0	0
$163$	5.21836	0.408734	0.204367	−	0.978894i	$-0.434486\pi$
$163$	5.21836	0.408734	0.204367	+	0.978894i	$0.434486\pi$
$164$	0	0
$165$	67.9408	5.28918
$166$	0	0
$167$	−13.4212	−1.03856	−0.519280	−	0.854604i	$-0.673800\pi$
$167$	−13.4212	−1.03856	−0.519280	+	0.854604i	$0.673800\pi$
$168$	0	0
$169$	19.5440	1.50339
$170$	0	0
$171$	14.4922	1.10824
$172$	0	0
$173$	−2.02422	−0.153899	−0.0769494	−	0.997035i	$-0.524518\pi$
$173$	−2.02422	−0.153899	−0.0769494	+	0.997035i	$0.524518\pi$
$174$	0	0
$175$	−11.0012	−0.831609
$176$	0	0
$177$	0.831923	0.0625312
$178$	0	0
$179$	22.9720	1.71701	0.858506	−	0.512804i	$-0.171393\pi$
$179$	22.9720	1.71701	0.858506	+	0.512804i	$0.171393\pi$
$180$	0	0
$181$	−19.1765	−1.42538	−0.712688	−	0.701481i	$-0.752523\pi$
$181$	−19.1765	−1.42538	−0.712688	+	0.701481i	$0.752523\pi$
$182$	0	0
$183$	37.7491	2.79050
$184$	0	0
$185$	35.2305	2.59020
$186$	0	0
$187$	−5.27113	−0.385463
$188$	0	0
$189$	14.9196	1.08524
$190$	0	0
$191$	−7.72631	−0.559056	−0.279528	−	0.960138i	$-0.590178\pi$
$191$	−7.72631	−0.559056	−0.279528	+	0.960138i	$0.590178\pi$
$192$	0	0
$193$	20.9494	1.50797	0.753986	−	0.656890i	$-0.228129\pi$
$193$	20.9494	1.50797	0.753986	+	0.656890i	$0.228129\pi$
$194$	0	0
$195$	−66.1746	−4.73886
$196$	0	0
$197$	−8.50552	−0.605993	−0.302997	−	0.952992i	$-0.597987\pi$
$197$	−8.50552	−0.605993	−0.302997	+	0.952992i	$0.597987\pi$
$198$	0	0
$199$	−2.70368	−0.191659	−0.0958293	−	0.995398i	$-0.530550\pi$
$199$	−2.70368	−0.191659	−0.0958293	+	0.995398i	$0.530550\pi$
$200$	0	0
$201$	−2.48710	−0.175426
$202$	0	0
$203$	2.07028	0.145305
$204$	0	0
$205$	26.6333	1.86015
$206$	0	0
$207$	−9.41957	−0.654705
$208$	0	0
$209$	12.5006	0.864685
$210$	0	0
$211$	0.0169875	0.00116946	0.000584732	−	1.00000i	$-0.499814\pi$
$211$	0.0169875	0.00116946	0.000584732		1.00000i	$0.499814\pi$
$212$	0	0
$213$	−32.0002	−2.19262
$214$	0	0
$215$	37.5760	2.56266
$216$	0	0
$217$	−3.96321	−0.269040
$218$	0	0
$219$	−27.4021	−1.85166
$220$	0	0
$221$	5.13411	0.345357
$222$	0	0
$223$	3.64340	0.243980	0.121990	−	0.992531i	$-0.461072\pi$
$223$	3.64340	0.243980	0.121990	+	0.992531i	$0.461072\pi$
$224$	0	0
$225$	59.3751	3.95834
$226$	0	0
$227$	17.9854	1.19373	0.596865	−	0.802342i	$-0.296413\pi$
$227$	17.9854	1.19373	0.596865	+	0.802342i	$0.296413\pi$
$228$	0	0
$229$	−7.98418	−0.527609	−0.263805	−	0.964576i	$-0.584977\pi$
$229$	−7.98418	−0.527609	−0.263805	+	0.964576i	$0.584977\pi$
$230$	0	0
$231$	23.0558	1.51696
$232$	0	0
$233$	−26.2087	−1.71699	−0.858493	−	0.512825i	$-0.828599\pi$
$233$	−26.2087	−1.71699	−0.858493	+	0.512825i	$0.828599\pi$
$234$	0	0
$235$	2.18459	0.142507
$236$	0	0
$237$	33.3902	2.16892
$238$	0	0
$239$	11.5998	0.750328	0.375164	−	0.926959i	$-0.377586\pi$
$239$	11.5998	0.750328	0.375164	+	0.926959i	$0.377586\pi$
$240$	0	0
$241$	26.0545	1.67832	0.839159	−	0.543886i	$-0.183048\pi$
$241$	26.0545	1.67832	0.839159	+	0.543886i	$0.183048\pi$
$242$	0	0
$243$	−16.7865	−1.07685
$244$	0	0
$245$	20.0834	1.28308
$246$	0	0
$247$	−12.1756	−0.774718
$248$	0	0
$249$	−4.97868	−0.315511
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−8.12510	−0.510821
$254$	0	0
$255$	−10.4396	−0.653755
$256$	0	0
$257$	10.6358	0.663441	0.331720	−	0.943378i	$-0.392371\pi$
$257$	10.6358	0.663441	0.331720	+	0.943378i	$0.392371\pi$
$258$	0	0
$259$	11.9555	0.742878
$260$	0	0
$261$	−11.1736	−0.691630
$262$	0	0
$263$	1.86416	0.114949	0.0574744	−	0.998347i	$-0.481695\pi$
$263$	1.86416	0.114949	0.0574744	+	0.998347i	$0.481695\pi$
$264$	0	0
$265$	−13.6547	−0.838799
$266$	0	0
$267$	−0.222898	−0.0136411
$268$	0	0
$269$	−12.8454	−0.783196	−0.391598	−	0.920136i	$-0.628078\pi$
$269$	−12.8454	−0.783196	−0.391598	+	0.920136i	$0.628078\pi$
$270$	0	0
$271$	−6.52064	−0.396100	−0.198050	−	0.980192i	$-0.563461\pi$
$271$	−6.52064	−0.396100	−0.198050	+	0.980192i	$0.563461\pi$
$272$	0	0
$273$	−22.4564	−1.35912
$274$	0	0
$275$	51.2156	3.08841
$276$	0	0
$277$	21.0590	1.26531	0.632657	−	0.774432i	$-0.281964\pi$
$277$	21.0590	1.26531	0.632657	+	0.774432i	$0.281964\pi$
$278$	0	0
$279$	21.3901	1.28059
$280$	0	0
$281$	−23.2820	−1.38889	−0.694443	−	0.719548i	$-0.744349\pi$
$281$	−23.2820	−1.38889	−0.694443	+	0.719548i	$0.744349\pi$
$282$	0	0
$283$	−18.4689	−1.09786	−0.548932	−	0.835867i	$-0.684965\pi$
$283$	−18.4689	−1.09786	−0.548932	+	0.835867i	$0.684965\pi$
$284$	0	0
$285$	24.7578	1.46653
$286$	0	0
$287$	9.03802	0.533498
$288$	0	0
$289$	−16.1900	−0.952356
$290$	0	0
$291$	−52.6457	−3.08614
$292$	0	0
$293$	29.9591	1.75023	0.875115	−	0.483914i	$-0.160785\pi$
$293$	29.9591	1.75023	0.875115	+	0.483914i	$0.160785\pi$
$294$	0	0
$295$	0.985715	0.0573905
$296$	0	0
$297$	−69.4577	−4.03034
$298$	0	0
$299$	7.91388	0.457672
$300$	0	0
$301$	12.7514	0.734981
$302$	0	0
$303$	46.2304	2.65587
$304$	0	0
$305$	44.7275	2.56109
$306$	0	0
$307$	−9.77503	−0.557890	−0.278945	−	0.960307i	$-0.589985\pi$
$307$	−9.77503	−0.557890	−0.278945	+	0.960307i	$0.589985\pi$
$308$	0	0
$309$	52.6280	2.99390
$310$	0	0
$311$	23.1624	1.31342	0.656710	−	0.754143i	$-0.271948\pi$
$311$	23.1624	1.31342	0.656710	+	0.754143i	$0.271948\pi$
$312$	0	0
$313$	27.4787	1.55319	0.776595	−	0.630001i	$-0.216945\pi$
$313$	27.4787	1.55319	0.776595	+	0.630001i	$0.216945\pi$
$314$	0	0
$315$	31.6701	1.78441
$316$	0	0
$317$	−4.83590	−0.271611	−0.135806	−	0.990736i	$-0.543362\pi$
$317$	−4.83590	−0.271611	−0.135806	+	0.990736i	$0.543362\pi$
$318$	0	0
$319$	−9.63811	−0.539631
$320$	0	0
$321$	22.4398	1.25247
$322$	0	0
$323$	−1.92082	−0.106877
$324$	0	0
$325$	−49.8842	−2.76708
$326$	0	0
$327$	−2.58393	−0.142892
$328$	0	0
$329$	0.741342	0.0408715
$330$	0	0
$331$	24.8813	1.36760	0.683799	−	0.729670i	$-0.260326\pi$
$331$	24.8813	1.36760	0.683799	+	0.729670i	$0.260326\pi$
$332$	0	0
$333$	−64.5258	−3.53599
$334$	0	0
$335$	−2.94687	−0.161005
$336$	0	0
$337$	−8.17582	−0.445365	−0.222683	−	0.974891i	$-0.571481\pi$
$337$	−8.17582	−0.445365	−0.222683	+	0.974891i	$0.571481\pi$
$338$	0	0
$339$	43.9933	2.38939
$340$	0	0
$341$	18.4506	0.999157
$342$	0	0
$343$	15.6219	0.843506
$344$	0	0
$345$	−16.0920	−0.866364
$346$	0	0
$347$	−5.68704	−0.305296	−0.152648	−	0.988281i	$-0.548780\pi$
$347$	−5.68704	−0.305296	−0.152648	+	0.988281i	$0.548780\pi$
$348$	0	0
$349$	14.9708	0.801370	0.400685	−	0.916216i	$-0.368772\pi$
$349$	14.9708	0.801370	0.400685	+	0.916216i	$0.368772\pi$
$350$	0	0
$351$	67.6521	3.61100
$352$	0	0
$353$	−24.4545	−1.30158	−0.650790	−	0.759258i	$-0.725562\pi$
$353$	−24.4545	−1.30158	−0.650790	+	0.759258i	$0.725562\pi$
$354$	0	0
$355$	−37.9158	−2.01236
$356$	0	0
$357$	−3.54270	−0.187499
$358$	0	0
$359$	−24.6399	−1.30044	−0.650221	−	0.759745i	$-0.725324\pi$
$359$	−24.6399	−1.30044	−0.650221	+	0.759745i	$0.725324\pi$
$360$	0	0
$361$	−14.4447	−0.760250
$362$	0	0
$363$	−72.9173	−3.82717
$364$	0	0
$365$	−32.4678	−1.69944
$366$	0	0
$367$	−1.91192	−0.0998012	−0.0499006	−	0.998754i	$-0.515890\pi$
$367$	−1.91192	−0.0998012	−0.0499006	+	0.998754i	$0.515890\pi$
$368$	0	0
$369$	−48.7797	−2.53937
$370$	0	0
$371$	−4.63372	−0.240571
$372$	0	0
$373$	−22.7912	−1.18009	−0.590043	−	0.807372i	$-0.700889\pi$
$373$	−22.7912	−1.18009	−0.590043	+	0.807372i	$0.700889\pi$
$374$	0	0
$375$	43.4342	2.24293
$376$	0	0
$377$	9.38756	0.483484
$378$	0	0
$379$	13.2054	0.678315	0.339158	−	0.940730i	$-0.389858\pi$
$379$	13.2054	0.678315	0.339158	+	0.940730i	$0.389858\pi$
$380$	0	0
$381$	−61.8645	−3.16941
$382$	0	0
$383$	−2.23647	−0.114278	−0.0571391	−	0.998366i	$-0.518198\pi$
$383$	−2.23647	−0.114278	−0.0571391	+	0.998366i	$0.518198\pi$
$384$	0	0
$385$	27.3179	1.39225
$386$	0	0
$387$	−68.8217	−3.49840
$388$	0	0
$389$	−2.36649	−0.119986	−0.0599929	−	0.998199i	$-0.519108\pi$
$389$	−2.36649	−0.119986	−0.0599929	+	0.998199i	$0.519108\pi$
$390$	0	0
$391$	1.24849	0.0631386
$392$	0	0
$393$	10.7506	0.542294
$394$	0	0
$395$	39.5628	1.99062
$396$	0	0
$397$	18.4378	0.925368	0.462684	−	0.886523i	$-0.346887\pi$
$397$	18.4378	0.925368	0.462684	+	0.886523i	$0.346887\pi$
$398$	0	0
$399$	8.40159	0.420605
$400$	0	0
$401$	−14.3824	−0.718222	−0.359111	−	0.933295i	$-0.616920\pi$
$401$	−14.3824	−0.718222	−0.359111	+	0.933295i	$0.616920\pi$
$402$	0	0
$403$	−17.9710	−0.895199
$404$	0	0
$405$	−62.0433	−3.08296
$406$	0	0
$407$	−55.6584	−2.75889
$408$	0	0
$409$	−14.2711	−0.705659	−0.352829	−	0.935688i	$-0.614780\pi$
$409$	−14.2711	−0.705659	−0.352829	+	0.935688i	$0.614780\pi$
$410$	0	0
$411$	65.1224	3.21225
$412$	0	0
$413$	0.334503	0.0164598
$414$	0	0
$415$	−5.89906	−0.289573
$416$	0	0
$417$	30.7562	1.50614
$418$	0	0
$419$	−39.2197	−1.91601	−0.958004	−	0.286755i	$-0.907423\pi$
$419$	−39.2197	−1.91601	−0.958004	+	0.286755i	$0.907423\pi$
$420$	0	0
$421$	11.8231	0.576222	0.288111	−	0.957597i	$-0.406973\pi$
$421$	11.8231	0.576222	0.288111	+	0.957597i	$0.406973\pi$
$422$	0	0
$423$	−4.00115	−0.194542
$424$	0	0
$425$	−7.86967	−0.381735
$426$	0	0
$427$	15.1783	0.734531
$428$	0	0
$429$	104.545	5.04749
$430$	0	0
$431$	−36.6544	−1.76558	−0.882790	−	0.469768i	$-0.844337\pi$
$431$	−36.6544	−1.76558	−0.882790	+	0.469768i	$0.844337\pi$
$432$	0	0
$433$	−19.9902	−0.960670	−0.480335	−	0.877085i	$-0.659485\pi$
$433$	−19.9902	−0.960670	−0.480335	+	0.877085i	$0.659485\pi$
$434$	0	0
$435$	−19.0886	−0.915227
$436$	0	0
$437$	−2.96081	−0.141635
$438$	0	0
$439$	18.5431	0.885016	0.442508	−	0.896765i	$-0.354089\pi$
$439$	18.5431	0.885016	0.442508	+	0.896765i	$0.354089\pi$
$440$	0	0
$441$	−36.7835	−1.75160
$442$	0	0
$443$	19.5946	0.930969	0.465485	−	0.885056i	$-0.345880\pi$
$443$	19.5946	0.930969	0.465485	+	0.885056i	$0.345880\pi$
$444$	0	0
$445$	−0.264104	−0.0125197
$446$	0	0
$447$	−35.6560	−1.68647
$448$	0	0
$449$	34.3426	1.62073	0.810363	−	0.585928i	$-0.199270\pi$
$449$	34.3426	1.62073	0.810363	+	0.585928i	$0.199270\pi$
$450$	0	0
$451$	−42.0762	−1.98129
$452$	0	0
$453$	−22.9612	−1.07881
$454$	0	0
$455$	−26.6078	−1.24739
$456$	0	0
$457$	10.3250	0.482985	0.241493	−	0.970403i	$-0.422363\pi$
$457$	10.3250	0.482985	0.241493	+	0.970403i	$0.422363\pi$
$458$	0	0
$459$	10.6727	0.498160
$460$	0	0
$461$	38.4931	1.79280	0.896401	−	0.443245i	$-0.146173\pi$
$461$	38.4931	1.79280	0.896401	+	0.443245i	$0.146173\pi$
$462$	0	0
$463$	29.3827	1.36553	0.682765	−	0.730638i	$-0.260777\pi$
$463$	29.3827	1.36553	0.682765	+	0.730638i	$0.260777\pi$
$464$	0	0
$465$	36.5420	1.69459
$466$	0	0
$467$	−3.76540	−0.174242	−0.0871210	−	0.996198i	$-0.527767\pi$
$467$	−3.76540	−0.174242	−0.0871210	+	0.996198i	$0.527767\pi$
$468$	0	0
$469$	−1.00002	−0.0461768
$470$	0	0
$471$	−59.6177	−2.74704
$472$	0	0
$473$	−59.3640	−2.72956
$474$	0	0
$475$	18.6631	0.856322
$476$	0	0
$477$	25.0090	1.14508
$478$	0	0
$479$	41.9253	1.91562	0.957808	−	0.287408i	$-0.0927937\pi$
$479$	41.9253	1.91562	0.957808	+	0.287408i	$0.0927937\pi$
$480$	0	0
$481$	54.2116	2.47183
$482$	0	0
$483$	−5.46083	−0.248477
$484$	0	0
$485$	−62.3779	−2.83244
$486$	0	0
$487$	10.2675	0.465267	0.232633	−	0.972564i	$-0.425266\pi$
$487$	10.2675	0.465267	0.232633	+	0.972564i	$0.425266\pi$
$488$	0	0
$489$	−16.3278	−0.738369
$490$	0	0
$491$	−17.3266	−0.781939	−0.390970	−	0.920404i	$-0.627860\pi$
$491$	−17.3266	−0.781939	−0.390970	+	0.920404i	$0.627860\pi$
$492$	0	0
$493$	1.48097	0.0666996
$494$	0	0
$495$	−147.439	−6.62691
$496$	0	0
$497$	−12.8668	−0.577154
$498$	0	0
$499$	34.6014	1.54897	0.774485	−	0.632592i	$-0.218009\pi$
$499$	34.6014	1.54897	0.774485	+	0.632592i	$0.218009\pi$
$500$	0	0
$501$	41.9937	1.87614
$502$	0	0
$503$	−27.5024	−1.22627	−0.613135	−	0.789978i	$-0.710092\pi$
$503$	−27.5024	−1.22627	−0.613135	+	0.789978i	$0.710092\pi$
$504$	0	0
$505$	54.7768	2.43753
$506$	0	0
$507$	−61.1516	−2.71584
$508$	0	0
$509$	2.33282	0.103401	0.0517003	−	0.998663i	$-0.483536\pi$
$509$	2.33282	0.103401	0.0517003	+	0.998663i	$0.483536\pi$
$510$	0	0
$511$	−11.0180	−0.487406
$512$	0	0
$513$	−25.3106	−1.11749
$514$	0	0
$515$	62.3570	2.74778
$516$	0	0
$517$	−3.45130	−0.151788
$518$	0	0
$519$	6.33362	0.278015
$520$	0	0
$521$	−10.1548	−0.444889	−0.222444	−	0.974945i	$-0.571404\pi$
$521$	−10.1548	−0.444889	−0.222444	+	0.974945i	$0.571404\pi$
$522$	0	0
$523$	15.6757	0.685450	0.342725	−	0.939436i	$-0.388650\pi$
$523$	15.6757	0.685450	0.342725	+	0.939436i	$0.388650\pi$
$524$	0	0
$525$	34.4217	1.50229
$526$	0	0
$527$	−2.83508	−0.123498
$528$	0	0
$529$	−21.0755	−0.916328
$530$	0	0
$531$	−1.80537	−0.0783464
$532$	0	0
$533$	40.9825	1.77515
$534$	0	0
$535$	26.5880	1.14950
$536$	0	0
$537$	−71.8776	−3.10175
$538$	0	0
$539$	−31.7286	−1.36665
$540$	0	0
$541$	−15.1509	−0.651387	−0.325694	−	0.945475i	$-0.605598\pi$
$541$	−15.1509	−0.651387	−0.325694	+	0.945475i	$0.605598\pi$
$542$	0	0
$543$	60.0016	2.57491
$544$	0	0
$545$	−3.06160	−0.131145
$546$	0	0
$547$	7.39742	0.316291	0.158145	−	0.987416i	$-0.449449\pi$
$547$	7.39742	0.316291	0.158145	+	0.987416i	$0.449449\pi$
$548$	0	0
$549$	−81.9200	−3.49626
$550$	0	0
$551$	−3.51216	−0.149623
$552$	0	0
$553$	13.4257	0.570917
$554$	0	0
$555$	−110.233	−4.67914
$556$	0	0
$557$	25.1951	1.06755	0.533775	−	0.845626i	$-0.320773\pi$
$557$	25.1951	1.06755	0.533775	+	0.845626i	$0.320773\pi$
$558$	0	0
$559$	57.8208	2.44556
$560$	0	0
$561$	16.4929	0.696332
$562$	0	0
$563$	6.36348	0.268189	0.134094	−	0.990969i	$-0.457187\pi$
$563$	6.36348	0.268189	0.134094	+	0.990969i	$0.457187\pi$
$564$	0	0
$565$	52.1261	2.19296
$566$	0	0
$567$	−21.0545	−0.884204
$568$	0	0
$569$	32.5668	1.36527	0.682636	−	0.730759i	$-0.260833\pi$
$569$	32.5668	1.36527	0.682636	+	0.730759i	$0.260833\pi$
$570$	0	0
$571$	4.99409	0.208996	0.104498	−	0.994525i	$-0.466676\pi$
$571$	4.99409	0.208996	0.104498	+	0.994525i	$0.466676\pi$
$572$	0	0
$573$	24.1750	1.00992
$574$	0	0
$575$	−12.1306	−0.505880
$576$	0	0
$577$	4.85143	0.201968	0.100984	−	0.994888i	$-0.467801\pi$
$577$	4.85143	0.201968	0.100984	+	0.994888i	$0.467801\pi$
$578$	0	0
$579$	−65.5490	−2.72412
$580$	0	0
$581$	−2.00185	−0.0830507
$582$	0	0
$583$	21.5722	0.893427
$584$	0	0
$585$	143.607	5.93740
$586$	0	0
$587$	−6.79625	−0.280511	−0.140256	−	0.990115i	$-0.544792\pi$
$587$	−6.79625	−0.280511	−0.140256	+	0.990115i	$0.544792\pi$
$588$	0	0
$589$	6.72346	0.277035
$590$	0	0
$591$	26.6131	1.09472
$592$	0	0
$593$	1.54630	0.0634991	0.0317496	−	0.999496i	$-0.489892\pi$
$593$	1.54630	0.0634991	0.0317496	+	0.999496i	$0.489892\pi$
$594$	0	0
$595$	−4.19761	−0.172085
$596$	0	0
$597$	8.45958	0.346227
$598$	0	0
$599$	27.0661	1.10589	0.552945	−	0.833218i	$-0.313504\pi$
$599$	27.0661	1.10589	0.552945	+	0.833218i	$0.313504\pi$
$600$	0	0
$601$	−26.2702	−1.07158	−0.535792	−	0.844350i	$-0.679987\pi$
$601$	−26.2702	−1.07158	−0.535792	+	0.844350i	$0.679987\pi$
$602$	0	0
$603$	5.39730	0.219795
$604$	0	0
$605$	−86.3971	−3.51254
$606$	0	0
$607$	33.5584	1.36209	0.681047	−	0.732239i	$-0.261525\pi$
$607$	33.5584	1.36209	0.681047	+	0.732239i	$0.261525\pi$
$608$	0	0
$609$	−6.47772	−0.262490
$610$	0	0
$611$	3.36158	0.135995
$612$	0	0
$613$	−11.9867	−0.484139	−0.242070	−	0.970259i	$-0.577826\pi$
$613$	−11.9867	−0.484139	−0.242070	+	0.970259i	$0.577826\pi$
$614$	0	0
$615$	−83.3332	−3.36032
$616$	0	0
$617$	18.9117	0.761355	0.380678	−	0.924708i	$-0.375691\pi$
$617$	18.9117	0.761355	0.380678	+	0.924708i	$0.375691\pi$
$618$	0	0
$619$	−23.4045	−0.940705	−0.470352	−	0.882479i	$-0.655873\pi$
$619$	−23.4045	−0.940705	−0.470352	+	0.882479i	$0.655873\pi$
$620$	0	0
$621$	16.4513	0.660168
$622$	0	0
$623$	−0.0896239	−0.00359070
$624$	0	0
$625$	7.74187	0.309675
$626$	0	0
$627$	−39.1133	−1.56204
$628$	0	0
$629$	8.55235	0.341005
$630$	0	0
$631$	−4.22524	−0.168204	−0.0841022	−	0.996457i	$-0.526802\pi$
$631$	−4.22524	−0.168204	−0.0841022	+	0.996457i	$0.526802\pi$
$632$	0	0
$633$	−0.0531523	−0.00211261
$634$	0	0
$635$	−73.3010	−2.90886
$636$	0	0
$637$	30.9038	1.22445
$638$	0	0
$639$	69.4441	2.74717
$640$	0	0
$641$	−27.0129	−1.06695	−0.533473	−	0.845817i	$-0.679113\pi$
$641$	−27.0129	−1.06695	−0.533473	+	0.845817i	$0.679113\pi$
$642$	0	0
$643$	−21.2789	−0.839156	−0.419578	−	0.907719i	$-0.637822\pi$
$643$	−21.2789	−0.839156	−0.419578	+	0.907719i	$0.637822\pi$
$644$	0	0
$645$	−117.572	−4.62940
$646$	0	0
$647$	31.5132	1.23891	0.619456	−	0.785032i	$-0.287353\pi$
$647$	31.5132	1.23891	0.619456	+	0.785032i	$0.287353\pi$
$648$	0	0
$649$	−1.55727	−0.0611282
$650$	0	0
$651$	12.4006	0.486016
$652$	0	0
$653$	32.7855	1.28300	0.641499	−	0.767124i	$-0.278313\pi$
$653$	32.7855	1.28300	0.641499	+	0.767124i	$0.278313\pi$
$654$	0	0
$655$	12.7379	0.497712
$656$	0	0
$657$	59.4658	2.31998
$658$	0	0
$659$	36.9878	1.44084	0.720419	−	0.693539i	$-0.243949\pi$
$659$	36.9878	1.44084	0.720419	+	0.693539i	$0.243949\pi$
$660$	0	0
$661$	17.1004	0.665128	0.332564	−	0.943081i	$-0.392086\pi$
$661$	17.1004	0.665128	0.332564	+	0.943081i	$0.392086\pi$
$662$	0	0
$663$	−16.0642	−0.623881
$664$	0	0
$665$	9.95473	0.386028
$666$	0	0
$667$	2.28282	0.0883911
$668$	0	0
$669$	−11.3999	−0.440745
$670$	0	0
$671$	−70.6623	−2.72789
$672$	0	0
$673$	36.8403	1.42009	0.710045	−	0.704156i	$-0.248675\pi$
$673$	36.8403	1.42009	0.710045	+	0.704156i	$0.248675\pi$
$674$	0	0
$675$	−103.699	−3.99136
$676$	0	0
$677$	−13.7957	−0.530210	−0.265105	−	0.964220i	$-0.585407\pi$
$677$	−13.7957	−0.530210	−0.265105	+	0.964220i	$0.585407\pi$
$678$	0	0
$679$	−21.1680	−0.812353
$680$	0	0
$681$	−56.2747	−2.15645
$682$	0	0
$683$	−5.82871	−0.223029	−0.111515	−	0.993763i	$-0.535570\pi$
$683$	−5.82871	−0.223029	−0.111515	+	0.993763i	$0.535570\pi$
$684$	0	0
$685$	77.1612	2.94818
$686$	0	0
$687$	24.9818	0.953116
$688$	0	0
$689$	−21.0114	−0.800470
$690$	0	0
$691$	36.0008	1.36953	0.684767	−	0.728762i	$-0.259904\pi$
$691$	36.0008	1.36953	0.684767	+	0.728762i	$0.259904\pi$
$692$	0	0
$693$	−50.0337	−1.90062
$694$	0	0
$695$	36.4419	1.38232
$696$	0	0
$697$	6.46534	0.244892
$698$	0	0
$699$	82.0047	3.10170
$700$	0	0
$701$	−28.7503	−1.08588	−0.542942	−	0.839770i	$-0.682690\pi$
$701$	−28.7503	−1.08588	−0.542942	+	0.839770i	$0.682690\pi$
$702$	0	0
$703$	−20.2821	−0.764954
$704$	0	0
$705$	−6.83539	−0.257436
$706$	0	0
$707$	18.5885	0.699094
$708$	0	0
$709$	−13.6871	−0.514030	−0.257015	−	0.966407i	$-0.582739\pi$
$709$	−13.6871	−0.514030	−0.257015	+	0.966407i	$0.582739\pi$
$710$	0	0
$711$	−72.4606	−2.71748
$712$	0	0
$713$	−4.37009	−0.163661
$714$	0	0
$715$	123.872	4.63254
$716$	0	0
$717$	−36.2947	−1.35545
$718$	0	0
$719$	−22.5128	−0.839586	−0.419793	−	0.907620i	$-0.637897\pi$
$719$	−22.5128	−0.839586	−0.419793	+	0.907620i	$0.637897\pi$
$720$	0	0
$721$	21.1609	0.788073
$722$	0	0
$723$	−81.5223	−3.03185
$724$	0	0
$725$	−14.3895	−0.534411
$726$	0	0
$727$	15.8328	0.587206	0.293603	−	0.955927i	$-0.405146\pi$
$727$	15.8328	0.587206	0.293603	+	0.955927i	$0.405146\pi$
$728$	0	0
$729$	2.31760	0.0858369
$730$	0	0
$731$	9.12174	0.337380
$732$	0	0
$733$	−31.7810	−1.17386	−0.586929	−	0.809638i	$-0.699663\pi$
$733$	−31.7810	−1.17386	−0.586929	+	0.809638i	$0.699663\pi$
$734$	0	0
$735$	−62.8394	−2.31787
$736$	0	0
$737$	4.65558	0.171490
$738$	0	0
$739$	44.2736	1.62863	0.814316	−	0.580422i	$-0.197113\pi$
$739$	44.2736	1.62863	0.814316	+	0.580422i	$0.197113\pi$
$740$	0	0
$741$	38.0966	1.39951
$742$	0	0
$743$	−28.6041	−1.04938	−0.524691	−	0.851293i	$-0.675819\pi$
$743$	−28.6041	−1.04938	−0.524691	+	0.851293i	$0.675819\pi$
$744$	0	0
$745$	−42.2475	−1.54783
$746$	0	0
$747$	10.8043	0.395309
$748$	0	0
$749$	9.02268	0.329681
$750$	0	0
$751$	−11.4568	−0.418065	−0.209032	−	0.977909i	$-0.567031\pi$
$751$	−11.4568	−0.418065	−0.209032	+	0.977909i	$0.567031\pi$
$752$	0	0
$753$	−3.12892	−0.114024
$754$	0	0
$755$	−27.2059	−0.990122
$756$	0	0
$757$	−9.12761	−0.331749	−0.165874	−	0.986147i	$-0.553045\pi$
$757$	−9.12761	−0.331749	−0.165874	+	0.986147i	$0.553045\pi$
$758$	0	0
$759$	25.4227	0.922787
$760$	0	0
$761$	14.8993	0.540099	0.270049	−	0.962846i	$-0.412960\pi$
$761$	14.8993	0.540099	0.270049	+	0.962846i	$0.412960\pi$
$762$	0	0
$763$	−1.03896	−0.0376128
$764$	0	0
$765$	22.6552	0.819101
$766$	0	0
$767$	1.51679	0.0547680
$768$	0	0
$769$	−4.62541	−0.166797	−0.0833983	−	0.996516i	$-0.526577\pi$
$769$	−4.62541	−0.166797	−0.0833983	+	0.996516i	$0.526577\pi$
$770$	0	0
$771$	−33.2784	−1.19849
$772$	0	0
$773$	30.3423	1.09134	0.545668	−	0.838001i	$-0.316276\pi$
$773$	30.3423	1.09134	0.545668	+	0.838001i	$0.316276\pi$
$774$	0	0
$775$	27.5463	0.989493
$776$	0	0
$777$	−37.4077	−1.34199
$778$	0	0
$779$	−15.3327	−0.549351
$780$	0	0
$781$	59.9009	2.14342
$782$	0	0
$783$	19.5148	0.697401
$784$	0	0
$785$	−70.6388	−2.52121
$786$	0	0
$787$	3.74681	0.133560	0.0667798	−	0.997768i	$-0.478728\pi$
$787$	3.74681	0.133560	0.0667798	+	0.997768i	$0.478728\pi$
$788$	0	0
$789$	−5.83279	−0.207653
$790$	0	0
$791$	17.6890	0.628949
$792$	0	0
$793$	68.8254	2.44406
$794$	0	0
$795$	42.7243	1.51527
$796$	0	0
$797$	29.2769	1.03704	0.518521	−	0.855065i	$-0.326483\pi$
$797$	29.2769	1.03704	0.518521	+	0.855065i	$0.326483\pi$
$798$	0	0
$799$	0.530319	0.0187613
$800$	0	0
$801$	0.483715	0.0170912
$802$	0	0
$803$	51.2938	1.81012
$804$	0	0
$805$	−6.47034	−0.228050
$806$	0	0
$807$	40.1921	1.41483
$808$	0	0
$809$	−8.47722	−0.298043	−0.149022	−	0.988834i	$-0.547612\pi$
$809$	−8.47722	−0.298043	−0.149022	+	0.988834i	$0.547612\pi$
$810$	0	0
$811$	30.0176	1.05406	0.527031	−	0.849846i	$-0.323305\pi$
$811$	30.0176	1.05406	0.527031	+	0.849846i	$0.323305\pi$
$812$	0	0
$813$	20.4025	0.715548
$814$	0	0
$815$	−19.3462	−0.677669
$816$	0	0
$817$	−21.6324	−0.756823
$818$	0	0
$819$	48.7330	1.70287
$820$	0	0
$821$	−2.75828	−0.0962646	−0.0481323	−	0.998841i	$-0.515327\pi$
$821$	−2.75828	−0.0962646	−0.0481323	+	0.998841i	$0.515327\pi$
$822$	0	0
$823$	−15.7953	−0.550591	−0.275296	−	0.961360i	$-0.588776\pi$
$823$	−15.7953	−0.550591	−0.275296	+	0.961360i	$0.588776\pi$
$824$	0	0
$825$	−160.249	−5.57916
$826$	0	0
$827$	12.4615	0.433327	0.216664	−	0.976246i	$-0.430482\pi$
$827$	12.4615	0.433327	0.216664	+	0.976246i	$0.430482\pi$
$828$	0	0
$829$	−32.3305	−1.12288	−0.561442	−	0.827516i	$-0.689753\pi$
$829$	−32.3305	−1.12288	−0.561442	+	0.827516i	$0.689753\pi$
$830$	0	0
$831$	−65.8919	−2.28576
$832$	0	0
$833$	4.87534	0.168921
$834$	0	0
$835$	49.7567	1.72190
$836$	0	0
$837$	−37.3579	−1.29128
$838$	0	0
$839$	−7.01321	−0.242123	−0.121062	−	0.992645i	$-0.538630\pi$
$839$	−7.01321	−0.242123	−0.121062	+	0.992645i	$0.538630\pi$
$840$	0	0
$841$	−26.2921	−0.906624
$842$	0	0
$843$	72.8473	2.50899
$844$	0	0
$845$	−72.4562	−2.49257
$846$	0	0
$847$	−29.3189	−1.00741
$848$	0	0
$849$	57.7877	1.98327
$850$	0	0
$851$	13.1829	0.451903
$852$	0	0
$853$	44.4453	1.52178	0.760889	−	0.648882i	$-0.224763\pi$
$853$	44.4453	1.52178	0.760889	+	0.648882i	$0.224763\pi$
$854$	0	0
$855$	−53.7274	−1.83744
$856$	0	0
$857$	46.7681	1.59757	0.798784	−	0.601618i	$-0.205477\pi$
$857$	46.7681	1.59757	0.798784	+	0.601618i	$0.205477\pi$
$858$	0	0
$859$	33.3770	1.13881	0.569405	−	0.822057i	$-0.307174\pi$
$859$	33.3770	1.13881	0.569405	+	0.822057i	$0.307174\pi$
$860$	0	0
$861$	−28.2792	−0.963753
$862$	0	0
$863$	49.4660	1.68384	0.841922	−	0.539600i	$-0.181425\pi$
$863$	49.4660	1.68384	0.841922	+	0.539600i	$0.181425\pi$
$864$	0	0
$865$	7.50448	0.255160
$866$	0	0
$867$	50.6573	1.72041
$868$	0	0
$869$	−62.5028	−2.12026
$870$	0	0
$871$	−4.53456	−0.153648
$872$	0	0
$873$	114.247	3.86668
$874$	0	0
$875$	17.4642	0.590398
$876$	0	0
$877$	−16.6639	−0.562702	−0.281351	−	0.959605i	$-0.590782\pi$
$877$	−16.6639	−0.562702	−0.281351	+	0.959605i	$0.590782\pi$
$878$	0	0
$879$	−93.7396	−3.16176
$880$	0	0
$881$	−2.92902	−0.0986811	−0.0493406	−	0.998782i	$-0.515712\pi$
$881$	−2.92902	−0.0986811	−0.0493406	+	0.998782i	$0.515712\pi$
$882$	0	0
$883$	0.692218	0.0232950	0.0116475	−	0.999932i	$-0.496292\pi$
$883$	0.692218	0.0232950	0.0116475	+	0.999932i	$0.496292\pi$
$884$	0	0
$885$	−3.08422	−0.103675
$886$	0	0
$887$	21.7438	0.730085	0.365042	−	0.930991i	$-0.381054\pi$
$887$	21.7438	0.730085	0.365042	+	0.930991i	$0.381054\pi$
$888$	0	0
$889$	−24.8747	−0.834272
$890$	0	0
$891$	98.0184	3.28374
$892$	0	0
$893$	−1.25766	−0.0420861
$894$	0	0
$895$	−85.1651	−2.84676
$896$	0	0
$897$	−24.7619	−0.826775
$898$	0	0
$899$	−5.18387	−0.172892
$900$	0	0
$901$	−3.31473	−0.110430
$902$	0	0
$903$	−39.8982	−1.32773
$904$	0	0
$905$	71.0937	2.36323
$906$	0	0
$907$	1.10254	0.0366092	0.0183046	−	0.999832i	$-0.494173\pi$
$907$	1.10254	0.0366092	0.0183046	+	0.999832i	$0.494173\pi$
$908$	0	0
$909$	−100.325	−3.32759
$910$	0	0
$911$	−58.1279	−1.92586	−0.962931	−	0.269747i	$-0.913060\pi$
$911$	−58.1279	−1.92586	−0.962931	+	0.269747i	$0.913060\pi$
$912$	0	0
$913$	9.31955	0.308432
$914$	0	0
$915$	−139.949	−4.62656
$916$	0	0
$917$	4.32263	0.142746
$918$	0	0
$919$	34.3829	1.13419	0.567093	−	0.823654i	$-0.308068\pi$
$919$	34.3829	1.13419	0.567093	+	0.823654i	$0.308068\pi$
$920$	0	0
$921$	30.5852	1.00782
$922$	0	0
$923$	−58.3437	−1.92041
$924$	0	0
$925$	−83.0967	−2.73220
$926$	0	0
$927$	−114.209	−3.75112
$928$	0	0
$929$	−16.9394	−0.555764	−0.277882	−	0.960615i	$-0.589632\pi$
$929$	−16.9394	−0.555764	−0.277882	+	0.960615i	$0.589632\pi$
$930$	0	0
$931$	−11.5620	−0.378929
$932$	0	0
$933$	−72.4732	−2.37267
$934$	0	0
$935$	19.5419	0.639087
$936$	0	0
$937$	−48.3763	−1.58039	−0.790193	−	0.612859i	$-0.790020\pi$
$937$	−48.3763	−1.58039	−0.790193	+	0.612859i	$0.790020\pi$
$938$	0	0
$939$	−85.9786	−2.80581
$940$	0	0
$941$	−16.2881	−0.530978	−0.265489	−	0.964114i	$-0.585533\pi$
$941$	−16.2881	−0.530978	−0.265489	+	0.964114i	$0.585533\pi$
$942$	0	0
$943$	9.96590	0.324534
$944$	0	0
$945$	−55.3119	−1.79930
$946$	0	0
$947$	−0.165839	−0.00538904	−0.00269452	−	0.999996i	$-0.500858\pi$
$947$	−0.165839	−0.00538904	−0.00269452	+	0.999996i	$0.500858\pi$
$948$	0	0
$949$	−49.9604	−1.62178
$950$	0	0
$951$	15.1311	0.490660
$952$	0	0
$953$	−28.2554	−0.915283	−0.457641	−	0.889137i	$-0.651306\pi$
$953$	−28.2554	−0.915283	−0.457641	+	0.889137i	$0.651306\pi$
$954$	0	0
$955$	28.6440	0.926899
$956$	0	0
$957$	30.1568	0.974832
$958$	0	0
$959$	26.1847	0.845548
$960$	0	0
$961$	−21.0763	−0.679881
$962$	0	0
$963$	−48.6969	−1.56924
$964$	0	0
$965$	−77.6666	−2.50018
$966$	0	0
$967$	−27.7950	−0.893828	−0.446914	−	0.894577i	$-0.647477\pi$
$967$	−27.7950	−0.893828	−0.446914	+	0.894577i	$0.647477\pi$
$968$	0	0
$969$	6.01007	0.193071
$970$	0	0
$971$	7.39433	0.237295	0.118648	−	0.992936i	$-0.462144\pi$
$971$	7.39433	0.237295	0.118648	+	0.992936i	$0.462144\pi$
$972$	0	0
$973$	12.3666	0.396455
$974$	0	0
$975$	156.083	4.99867
$976$	0	0
$977$	12.4734	0.399058	0.199529	−	0.979892i	$-0.436059\pi$
$977$	12.4734	0.399058	0.199529	+	0.979892i	$0.436059\pi$
$978$	0	0
$979$	0.417241	0.0133351
$980$	0	0
$981$	5.60743	0.179031
$982$	0	0
$983$	−29.7533	−0.948983	−0.474492	−	0.880260i	$-0.657368\pi$
$983$	−29.7533	−0.948983	−0.474492	+	0.880260i	$0.657368\pi$
$984$	0	0
$985$	31.5328	1.00472
$986$	0	0
$987$	−2.31960	−0.0738336
$988$	0	0
$989$	14.0606	0.447100
$990$	0	0
$991$	−41.0767	−1.30484	−0.652422	−	0.757856i	$-0.726247\pi$
$991$	−41.0767	−1.30484	−0.652422	+	0.757856i	$0.726247\pi$
$992$	0	0
$993$	−77.8514	−2.47054
$994$	0	0
$995$	10.0234	0.317764
$996$	0	0
$997$	52.3269	1.65721	0.828604	−	0.559835i	$-0.189135\pi$
$997$	52.3269	1.65721	0.828604	+	0.559835i	$0.189135\pi$
$998$	0	0
$999$	112.694	3.56549

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.l.1.1		19
4.3	odd	2		2008.2.a.c.1.19	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.c.1.19	✓	19	4.3	odd	2
4016.2.a.l.1.1		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)

\(f(q)\)	\(=\)	\(q-3.12892 q^{3} -3.70734 q^{5} -1.25809 q^{7} +6.79011 q^{9} +O(q^{10})\)	\(q-3.12892 q^{3} -3.70734 q^{5} -1.25809 q^{7} +6.79011 q^{9} +5.85699 q^{11} -5.70474 q^{13} +11.5999 q^{15} -0.899972 q^{17} +2.13431 q^{19} +3.93645 q^{21} -1.38725 q^{23} +8.74435 q^{25} -11.8589 q^{27} -1.64557 q^{29} +3.15019 q^{31} -18.3260 q^{33} +4.66416 q^{35} -9.50290 q^{37} +17.8496 q^{39} -7.18394 q^{41} -10.1356 q^{43} -25.1732 q^{45} -0.589261 q^{47} -5.41721 q^{49} +2.81594 q^{51} +3.68315 q^{53} -21.7138 q^{55} -6.67806 q^{57} -0.265882 q^{59} -12.0646 q^{61} -8.54256 q^{63} +21.1494 q^{65} +0.794876 q^{67} +4.34058 q^{69} +10.2272 q^{71} +8.75771 q^{73} -27.3603 q^{75} -7.36861 q^{77} -10.6715 q^{79} +16.7353 q^{81} +1.59118 q^{83} +3.33650 q^{85} +5.14886 q^{87} +0.0712382 q^{89} +7.17706 q^{91} -9.85667 q^{93} -7.91259 q^{95} +16.8255 q^{97} +39.7696 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) \) 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

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