LMFDB - Embedded newform 4016.2.a.l.1.3

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			1.3
Root			$-2.08912$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.08912 q^{3} -2.23992 q^{5} +1.92906 q^{7} +1.36442 q^{9} +O(q^{10})$	$q-2.08912 q^{3} -2.23992 q^{5} +1.92906 q^{7} +1.36442 q^{9} -4.71991 q^{11} -3.47433 q^{13} +4.67945 q^{15} -7.05174 q^{17} -7.10546 q^{19} -4.03004 q^{21} +4.99477 q^{23} +0.0172202 q^{25} +3.41692 q^{27} -1.87117 q^{29} +2.74748 q^{31} +9.86047 q^{33} -4.32093 q^{35} -0.747819 q^{37} +7.25828 q^{39} -11.3213 q^{41} +2.70221 q^{43} -3.05619 q^{45} -6.57542 q^{47} -3.27873 q^{49} +14.7319 q^{51} -8.29731 q^{53} +10.5722 q^{55} +14.8442 q^{57} -14.8388 q^{59} +12.1284 q^{61} +2.63205 q^{63} +7.78220 q^{65} -7.02061 q^{67} -10.4347 q^{69} +7.46670 q^{71} -2.65862 q^{73} -0.0359750 q^{75} -9.10500 q^{77} +12.5204 q^{79} -11.2316 q^{81} +6.62130 q^{83} +15.7953 q^{85} +3.90910 q^{87} +0.801340 q^{89} -6.70219 q^{91} -5.73981 q^{93} +15.9156 q^{95} -14.1208 q^{97} -6.43995 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) $ 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9	$ 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9} + 15 q^{11} - 8 q^{13} + 17 q^{15} - 4 q^{17} + 14 q^{19} - 9 q^{21} + 28 q^{23} + 25 q^{25} + 21 q^{27} - 13 q^{29} + 20 q^{31} - 6 q^{33} + 32 q^{35} - 16 q^{37} + 27 q^{39} + 2 q^{41} + 28 q^{43} - 29 q^{45} + 37 q^{47} + 36 q^{49} + 35 q^{51} - 37 q^{53} + 24 q^{55} - 11 q^{57} + 32 q^{59} - 7 q^{61} + 45 q^{63} + q^{65} + 45 q^{67} - 12 q^{69} + 49 q^{71} + 16 q^{73} + 35 q^{75} - 40 q^{77} + 33 q^{79} + 15 q^{81} + 43 q^{83} - 28 q^{85} + 48 q^{87} + 3 q^{89} + 56 q^{91} - 48 q^{93} + 43 q^{95} + 8 q^{97} + 74 q^{99}+O(q^{100}) $ 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9 + 15 * q^11 - 8 * q^13 + 17 * q^15 - 4 * q^17 + 14 * q^19 - 9 * q^21 + 28 * q^23 + 25 * q^25 + 21 * q^27 - 13 * q^29 + 20 * q^31 - 6 * q^33 + 32 * q^35 - 16 * q^37 + 27 * q^39 + 2 * q^41 + 28 * q^43 - 29 * q^45 + 37 * q^47 + 36 * q^49 + 35 * q^51 - 37 * q^53 + 24 * q^55 - 11 * q^57 + 32 * q^59 - 7 * q^61 + 45 * q^63 + q^65 + 45 * q^67 - 12 * q^69 + 49 * q^71 + 16 * q^73 + 35 * q^75 - 40 * q^77 + 33 * q^79 + 15 * q^81 + 43 * q^83 - 28 * q^85 + 48 * q^87 + 3 * q^89 + 56 * q^91 - 48 * q^93 + 43 * q^95 + 8 * q^97 + 74 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.08912	−1.20615	−0.603077	−	0.797683i	$-0.706059\pi$
$3$	−2.08912	−1.20615	−0.603077	+	0.797683i	$0.706059\pi$
$4$	0	0
$5$	−2.23992	−1.00172	−0.500860	−	0.865528i	$-0.666983\pi$
$5$	−2.23992	−1.00172	−0.500860	+	0.865528i	$0.666983\pi$
$6$	0	0
$7$	1.92906	0.729116	0.364558	−	0.931181i	$-0.381220\pi$
$7$	1.92906	0.729116	0.364558	+	0.931181i	$0.381220\pi$
$8$	0	0
$9$	1.36442	0.454807
$10$	0	0
$11$	−4.71991	−1.42311	−0.711554	−	0.702632i	$-0.752008\pi$
$11$	−4.71991	−1.42311	−0.711554	+	0.702632i	$0.752008\pi$
$12$	0	0
$13$	−3.47433	−0.963605	−0.481802	−	0.876280i	$-0.660018\pi$
$13$	−3.47433	−0.963605	−0.481802	+	0.876280i	$0.660018\pi$
$14$	0	0
$15$	4.67945	1.20823
$16$	0	0
$17$	−7.05174	−1.71030	−0.855149	−	0.518383i	$-0.826534\pi$
$17$	−7.05174	−1.71030	−0.855149	+	0.518383i	$0.826534\pi$
$18$	0	0
$19$	−7.10546	−1.63011	−0.815053	−	0.579387i	$-0.803292\pi$
$19$	−7.10546	−1.63011	−0.815053	+	0.579387i	$0.803292\pi$
$20$	0	0
$21$	−4.03004	−0.879427
$22$	0	0
$23$	4.99477	1.04148	0.520741	−	0.853715i	$-0.325656\pi$
$23$	4.99477	1.04148	0.520741	+	0.853715i	$0.325656\pi$
$24$	0	0
$25$	0.0172202	0.00344404
$26$	0	0
$27$	3.41692	0.657586
$28$	0	0
$29$	−1.87117	−0.347468	−0.173734	−	0.984793i	$-0.555583\pi$
$29$	−1.87117	−0.347468	−0.173734	+	0.984793i	$0.555583\pi$
$30$	0	0
$31$	2.74748	0.493461	0.246731	−	0.969084i	$-0.420644\pi$
$31$	2.74748	0.493461	0.246731	+	0.969084i	$0.420644\pi$
$32$	0	0
$33$	9.86047	1.71649
$34$	0	0
$35$	−4.32093	−0.730371
$36$	0	0
$37$	−0.747819	−0.122941	−0.0614703	−	0.998109i	$-0.519579\pi$
$37$	−0.747819	−0.122941	−0.0614703	+	0.998109i	$0.519579\pi$
$38$	0	0
$39$	7.25828	1.16226
$40$	0	0
$41$	−11.3213	−1.76809	−0.884044	−	0.467403i	$-0.845190\pi$
$41$	−11.3213	−1.76809	−0.884044	+	0.467403i	$0.845190\pi$
$42$	0	0
$43$	2.70221	0.412083	0.206041	−	0.978543i	$-0.433942\pi$
$43$	2.70221	0.412083	0.206041	+	0.978543i	$0.433942\pi$
$44$	0	0
$45$	−3.05619	−0.455590
$46$	0	0
$47$	−6.57542	−0.959124	−0.479562	−	0.877508i	$-0.659204\pi$
$47$	−6.57542	−0.959124	−0.479562	+	0.877508i	$0.659204\pi$
$48$	0	0
$49$	−3.27873	−0.468389
$50$	0	0
$51$	14.7319	2.06288
$52$	0	0
$53$	−8.29731	−1.13972	−0.569861	−	0.821741i	$-0.693003\pi$
$53$	−8.29731	−1.13972	−0.569861	+	0.821741i	$0.693003\pi$
$54$	0	0
$55$	10.5722	1.42556
$56$	0	0
$57$	14.8442	1.96616
$58$	0	0
$59$	−14.8388	−1.93184	−0.965921	−	0.258838i	$-0.916660\pi$
$59$	−14.8388	−1.93184	−0.965921	+	0.258838i	$0.916660\pi$
$60$	0	0
$61$	12.1284	1.55288	0.776440	−	0.630191i	$-0.217023\pi$
$61$	12.1284	1.55288	0.776440	+	0.630191i	$0.217023\pi$
$62$	0	0
$63$	2.63205	0.331607
$64$	0	0
$65$	7.78220	0.965263
$66$	0	0
$67$	−7.02061	−0.857704	−0.428852	−	0.903375i	$-0.641082\pi$
$67$	−7.02061	−0.857704	−0.428852	+	0.903375i	$0.641082\pi$
$68$	0	0
$69$	−10.4347	−1.25619
$70$	0	0
$71$	7.46670	0.886134	0.443067	−	0.896488i	$-0.353890\pi$
$71$	7.46670	0.886134	0.443067	+	0.896488i	$0.353890\pi$
$72$	0	0
$73$	−2.65862	−0.311168	−0.155584	−	0.987823i	$-0.549726\pi$
$73$	−2.65862	−0.311168	−0.155584	+	0.987823i	$0.549726\pi$
$74$	0	0
$75$	−0.0359750	−0.00415404
$76$	0	0
$77$	−9.10500	−1.03761
$78$	0	0
$79$	12.5204	1.40866	0.704329	−	0.709874i	$-0.251248\pi$
$79$	12.5204	1.40866	0.704329	+	0.709874i	$0.251248\pi$
$80$	0	0
$81$	−11.2316	−1.24796
$82$	0	0
$83$	6.62130	0.726782	0.363391	−	0.931637i	$-0.381619\pi$
$83$	6.62130	0.726782	0.363391	+	0.931637i	$0.381619\pi$
$84$	0	0
$85$	15.7953	1.71324
$86$	0	0
$87$	3.90910	0.419100
$88$	0	0
$89$	0.801340	0.0849419	0.0424709	−	0.999098i	$-0.486477\pi$
$89$	0.801340	0.0849419	0.0424709	+	0.999098i	$0.486477\pi$
$90$	0	0
$91$	−6.70219	−0.702580
$92$	0	0
$93$	−5.73981	−0.595190
$94$	0	0
$95$	15.9156	1.63291
$96$	0	0
$97$	−14.1208	−1.43375	−0.716874	−	0.697203i	$-0.754428\pi$
$97$	−14.1208	−1.43375	−0.716874	+	0.697203i	$0.754428\pi$
$98$	0	0
$99$	−6.43995	−0.647239
$100$	0	0
$101$	−16.3715	−1.62903	−0.814514	−	0.580144i	$-0.802996\pi$
$101$	−16.3715	−1.62903	−0.814514	+	0.580144i	$0.802996\pi$
$102$	0	0
$103$	14.2761	1.40666	0.703332	−	0.710861i	$-0.251695\pi$
$103$	14.2761	1.40666	0.703332	+	0.710861i	$0.251695\pi$
$104$	0	0
$105$	9.02694	0.880940
$106$	0	0
$107$	6.17704	0.597157	0.298578	−	0.954385i	$-0.403488\pi$
$107$	6.17704	0.597157	0.298578	+	0.954385i	$0.403488\pi$
$108$	0	0
$109$	−10.5512	−1.01063	−0.505313	−	0.862936i	$-0.668623\pi$
$109$	−10.5512	−1.01063	−0.505313	+	0.862936i	$0.668623\pi$
$110$	0	0
$111$	1.56228	0.148285
$112$	0	0
$113$	2.86033	0.269077	0.134539	−	0.990908i	$-0.457045\pi$
$113$	2.86033	0.269077	0.134539	+	0.990908i	$0.457045\pi$
$114$	0	0
$115$	−11.1879	−1.04327
$116$	0	0
$117$	−4.74044	−0.438254
$118$	0	0
$119$	−13.6032	−1.24701
$120$	0	0
$121$	11.2776	1.02524
$122$	0	0
$123$	23.6515	2.13259
$124$	0	0
$125$	11.1610	0.998271
$126$	0	0
$127$	−18.4069	−1.63335	−0.816675	−	0.577098i	$-0.804185\pi$
$127$	−18.4069	−1.63335	−0.816675	+	0.577098i	$0.804185\pi$
$128$	0	0
$129$	−5.64523	−0.497035
$130$	0	0
$131$	−21.3826	−1.86821	−0.934103	−	0.357004i	$-0.883798\pi$
$131$	−21.3826	−1.86821	−0.934103	+	0.357004i	$0.883798\pi$
$132$	0	0
$133$	−13.7069	−1.18854
$134$	0	0
$135$	−7.65361	−0.658718
$136$	0	0
$137$	10.6247	0.907729	0.453864	−	0.891071i	$-0.350045\pi$
$137$	10.6247	0.907729	0.453864	+	0.891071i	$0.350045\pi$
$138$	0	0
$139$	−8.17920	−0.693751	−0.346875	−	0.937911i	$-0.612757\pi$
$139$	−8.17920	−0.693751	−0.346875	+	0.937911i	$0.612757\pi$
$140$	0	0
$141$	13.7369	1.15685
$142$	0	0
$143$	16.3985	1.37131
$144$	0	0
$145$	4.19126	0.348066
$146$	0	0
$147$	6.84965	0.564950
$148$	0	0
$149$	10.8738	0.890813	0.445406	−	0.895328i	$-0.353059\pi$
$149$	10.8738	0.890813	0.445406	+	0.895328i	$0.353059\pi$
$150$	0	0
$151$	−13.0236	−1.05985	−0.529924	−	0.848045i	$-0.677779\pi$
$151$	−13.0236	−1.05985	−0.529924	+	0.848045i	$0.677779\pi$
$152$	0	0
$153$	−9.62154	−0.777855
$154$	0	0
$155$	−6.15411	−0.494310
$156$	0	0
$157$	−13.7750	−1.09937	−0.549683	−	0.835374i	$-0.685251\pi$
$157$	−13.7750	−1.09937	−0.549683	+	0.835374i	$0.685251\pi$
$158$	0	0
$159$	17.3341	1.37468
$160$	0	0
$161$	9.63521	0.759361
$162$	0	0
$163$	9.71944	0.761285	0.380642	−	0.924722i	$-0.375703\pi$
$163$	9.71944	0.761285	0.380642	+	0.924722i	$0.375703\pi$
$164$	0	0
$165$	−22.0866	−1.71944
$166$	0	0
$167$	3.97912	0.307913	0.153957	−	0.988078i	$-0.450798\pi$
$167$	3.97912	0.307913	0.153957	+	0.988078i	$0.450798\pi$
$168$	0	0
$169$	−0.929060	−0.0714661
$170$	0	0
$171$	−9.69485	−0.741384
$172$	0	0
$173$	5.12097	0.389340	0.194670	−	0.980869i	$-0.437636\pi$
$173$	5.12097	0.389340	0.194670	+	0.980869i	$0.437636\pi$
$174$	0	0
$175$	0.0332188	0.00251110
$176$	0	0
$177$	30.9999	2.33010
$178$	0	0
$179$	21.0214	1.57122	0.785608	−	0.618724i	$-0.212350\pi$
$179$	21.0214	1.57122	0.785608	+	0.618724i	$0.212350\pi$
$180$	0	0
$181$	15.1736	1.12784	0.563922	−	0.825828i	$-0.309292\pi$
$181$	15.1736	1.12784	0.563922	+	0.825828i	$0.309292\pi$
$182$	0	0
$183$	−25.3377	−1.87301
$184$	0	0
$185$	1.67505	0.123152
$186$	0	0
$187$	33.2836	2.43394
$188$	0	0
$189$	6.59144	0.479457
$190$	0	0
$191$	−16.7552	−1.21237	−0.606183	−	0.795325i	$-0.707300\pi$
$191$	−16.7552	−1.21237	−0.606183	+	0.795325i	$0.707300\pi$
$192$	0	0
$193$	19.3271	1.39120	0.695598	−	0.718431i	$-0.255139\pi$
$193$	19.3271	1.39120	0.695598	+	0.718431i	$0.255139\pi$
$194$	0	0
$195$	−16.2579	−1.16426
$196$	0	0
$197$	6.20110	0.441810	0.220905	−	0.975295i	$-0.429099\pi$
$197$	6.20110	0.441810	0.220905	+	0.975295i	$0.429099\pi$
$198$	0	0
$199$	12.3616	0.876289	0.438144	−	0.898905i	$-0.355636\pi$
$199$	12.3616	0.876289	0.438144	+	0.898905i	$0.355636\pi$
$200$	0	0
$201$	14.6669	1.03452
$202$	0	0
$203$	−3.60960	−0.253344
$204$	0	0
$205$	25.3587	1.77113
$206$	0	0
$207$	6.81497	0.473673
$208$	0	0
$209$	33.5372	2.31982
$210$	0	0
$211$	0.232196	0.0159850	0.00799251	−	0.999968i	$-0.497456\pi$
$211$	0.232196	0.0159850	0.00799251	+	0.999968i	$0.497456\pi$
$212$	0	0
$213$	−15.5988	−1.06881
$214$	0	0
$215$	−6.05271	−0.412792
$216$	0	0
$217$	5.30005	0.359791
$218$	0	0
$219$	5.55418	0.375317
$220$	0	0
$221$	24.5000	1.64805
$222$	0	0
$223$	−5.21681	−0.349344	−0.174672	−	0.984627i	$-0.555886\pi$
$223$	−5.21681	−0.349344	−0.174672	+	0.984627i	$0.555886\pi$
$224$	0	0
$225$	0.0234956	0.00156637
$226$	0	0
$227$	9.28962	0.616574	0.308287	−	0.951293i	$-0.400244\pi$
$227$	9.28962	0.616574	0.308287	+	0.951293i	$0.400244\pi$
$228$	0	0
$229$	27.3407	1.80672	0.903362	−	0.428879i	$-0.141091\pi$
$229$	27.3407	1.80672	0.903362	+	0.428879i	$0.141091\pi$
$230$	0	0
$231$	19.0214	1.25152
$232$	0	0
$233$	−21.2416	−1.39158	−0.695792	−	0.718244i	$-0.744946\pi$
$233$	−21.2416	−1.39158	−0.695792	+	0.718244i	$0.744946\pi$
$234$	0	0
$235$	14.7284	0.960774
$236$	0	0
$237$	−26.1567	−1.69906
$238$	0	0
$239$	−6.06753	−0.392476	−0.196238	−	0.980556i	$-0.562872\pi$
$239$	−6.06753	−0.392476	−0.196238	+	0.980556i	$0.562872\pi$
$240$	0	0
$241$	29.3153	1.88836	0.944182	−	0.329425i	$-0.106855\pi$
$241$	29.3153	1.88836	0.944182	+	0.329425i	$0.106855\pi$
$242$	0	0
$243$	13.2134	0.847642
$244$	0	0
$245$	7.34407	0.469195
$246$	0	0
$247$	24.6867	1.57078
$248$	0	0
$249$	−13.8327	−0.876611
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−23.5749	−1.48214
$254$	0	0
$255$	−32.9983	−2.06643
$256$	0	0
$257$	−14.9538	−0.932794	−0.466397	−	0.884575i	$-0.654448\pi$
$257$	−14.9538	−0.932794	−0.466397	+	0.884575i	$0.654448\pi$
$258$	0	0
$259$	−1.44259	−0.0896381
$260$	0	0
$261$	−2.55307	−0.158031
$262$	0	0
$263$	23.2943	1.43639	0.718193	−	0.695844i	$-0.244970\pi$
$263$	23.2943	1.43639	0.718193	+	0.695844i	$0.244970\pi$
$264$	0	0
$265$	18.5853	1.14168
$266$	0	0
$267$	−1.67410	−0.102453
$268$	0	0
$269$	0.318179	0.0193997	0.00969986	−	0.999953i	$-0.496912\pi$
$269$	0.318179	0.0193997	0.00969986	+	0.999953i	$0.496912\pi$
$270$	0	0
$271$	7.18569	0.436500	0.218250	−	0.975893i	$-0.429965\pi$
$271$	7.18569	0.436500	0.218250	+	0.975893i	$0.429965\pi$
$272$	0	0
$273$	14.0017	0.847419
$274$	0	0
$275$	−0.0812778	−0.00490124
$276$	0	0
$277$	19.7159	1.18461	0.592306	−	0.805713i	$-0.298218\pi$
$277$	19.7159	1.18461	0.592306	+	0.805713i	$0.298218\pi$
$278$	0	0
$279$	3.74872	0.224430
$280$	0	0
$281$	−24.1133	−1.43848	−0.719240	−	0.694762i	$-0.755510\pi$
$281$	−24.1133	−1.43848	−0.719240	+	0.694762i	$0.755510\pi$
$282$	0	0
$283$	−29.4029	−1.74782	−0.873911	−	0.486087i	$-0.838424\pi$
$283$	−29.4029	−1.74782	−0.873911	+	0.486087i	$0.838424\pi$
$284$	0	0
$285$	−33.2497	−1.96954
$286$	0	0
$287$	−21.8395	−1.28914
$288$	0	0
$289$	32.7270	1.92512
$290$	0	0
$291$	29.5000	1.72932
$292$	0	0
$293$	−14.8488	−0.867475	−0.433738	−	0.901039i	$-0.642806\pi$
$293$	−14.8488	−0.867475	−0.433738	+	0.901039i	$0.642806\pi$
$294$	0	0
$295$	33.2376	1.93517
$296$	0	0
$297$	−16.1276	−0.935816
$298$	0	0
$299$	−17.3535	−1.00358
$300$	0	0
$301$	5.21272	0.300456
$302$	0	0
$303$	34.2021	1.96486
$304$	0	0
$305$	−27.1666	−1.55555
$306$	0	0
$307$	0.165341	0.00943652	0.00471826	−	0.999989i	$-0.498498\pi$
$307$	0.165341	0.00943652	0.00471826	+	0.999989i	$0.498498\pi$
$308$	0	0
$309$	−29.8245	−1.69665
$310$	0	0
$311$	−6.72604	−0.381399	−0.190699	−	0.981649i	$-0.561076\pi$
$311$	−6.72604	−0.381399	−0.190699	+	0.981649i	$0.561076\pi$
$312$	0	0
$313$	−7.71428	−0.436037	−0.218019	−	0.975945i	$-0.569959\pi$
$313$	−7.71428	−0.436037	−0.218019	+	0.975945i	$0.569959\pi$
$314$	0	0
$315$	−5.89557	−0.332178
$316$	0	0
$317$	24.5341	1.37797	0.688987	−	0.724774i	$-0.258056\pi$
$317$	24.5341	1.37797	0.688987	+	0.724774i	$0.258056\pi$
$318$	0	0
$319$	8.83177	0.494484
$320$	0	0
$321$	−12.9046	−0.720263
$322$	0	0
$323$	50.1059	2.78796
$324$	0	0
$325$	−0.0598285	−0.00331869
$326$	0	0
$327$	22.0428	1.21897
$328$	0	0
$329$	−12.6844	−0.699313
$330$	0	0
$331$	−12.9905	−0.714022	−0.357011	−	0.934100i	$-0.616204\pi$
$331$	−12.9905	−0.714022	−0.357011	+	0.934100i	$0.616204\pi$
$332$	0	0
$333$	−1.02034	−0.0559143
$334$	0	0
$335$	15.7256	0.859180
$336$	0	0
$337$	16.2295	0.884075	0.442038	−	0.896997i	$-0.354256\pi$
$337$	16.2295	0.884075	0.442038	+	0.896997i	$0.354256\pi$
$338$	0	0
$339$	−5.97557	−0.324548
$340$	0	0
$341$	−12.9678	−0.702248
$342$	0	0
$343$	−19.8283	−1.07063
$344$	0	0
$345$	23.3728	1.25835
$346$	0	0
$347$	22.7521	1.22140	0.610698	−	0.791863i	$-0.290889\pi$
$347$	22.7521	1.22140	0.610698	+	0.791863i	$0.290889\pi$
$348$	0	0
$349$	−1.08120	−0.0578751	−0.0289376	−	0.999581i	$-0.509212\pi$
$349$	−1.08120	−0.0578751	−0.0289376	+	0.999581i	$0.509212\pi$
$350$	0	0
$351$	−11.8715	−0.633653
$352$	0	0
$353$	−3.20396	−0.170529	−0.0852647	−	0.996358i	$-0.527174\pi$
$353$	−3.20396	−0.170529	−0.0852647	+	0.996358i	$0.527174\pi$
$354$	0	0
$355$	−16.7248	−0.887659
$356$	0	0
$357$	28.4188	1.50408
$358$	0	0
$359$	18.1902	0.960042	0.480021	−	0.877257i	$-0.340629\pi$
$359$	18.1902	0.960042	0.480021	+	0.877257i	$0.340629\pi$
$360$	0	0
$361$	31.4876	1.65724
$362$	0	0
$363$	−23.5602	−1.23659
$364$	0	0
$365$	5.95509	0.311704
$366$	0	0
$367$	5.13829	0.268216	0.134108	−	0.990967i	$-0.457183\pi$
$367$	5.13829	0.268216	0.134108	+	0.990967i	$0.457183\pi$
$368$	0	0
$369$	−15.4470	−0.804139
$370$	0	0
$371$	−16.0060	−0.830990
$372$	0	0
$373$	−2.38096	−0.123281	−0.0616406	−	0.998098i	$-0.519633\pi$
$373$	−2.38096	−0.123281	−0.0616406	+	0.998098i	$0.519633\pi$
$374$	0	0
$375$	−23.3167	−1.20407
$376$	0	0
$377$	6.50106	0.334822
$378$	0	0
$379$	38.0153	1.95272	0.976358	−	0.216160i	$-0.0693531\pi$
$379$	38.0153	1.95272	0.976358	+	0.216160i	$0.0693531\pi$
$380$	0	0
$381$	38.4543	1.97007
$382$	0	0
$383$	−8.26406	−0.422274	−0.211137	−	0.977456i	$-0.567717\pi$
$383$	−8.26406	−0.422274	−0.211137	+	0.977456i	$0.567717\pi$
$384$	0	0
$385$	20.3944	1.03940
$386$	0	0
$387$	3.68695	0.187418
$388$	0	0
$389$	−22.1736	−1.12425	−0.562124	−	0.827053i	$-0.690016\pi$
$389$	−22.1736	−1.12425	−0.562124	+	0.827053i	$0.690016\pi$
$390$	0	0
$391$	−35.2218	−1.78124
$392$	0	0
$393$	44.6708	2.25334
$394$	0	0
$395$	−28.0447	−1.41108
$396$	0	0
$397$	9.92472	0.498107	0.249054	−	0.968490i	$-0.419880\pi$
$397$	9.92472	0.498107	0.249054	+	0.968490i	$0.419880\pi$
$398$	0	0
$399$	28.6353	1.43356
$400$	0	0
$401$	−28.2702	−1.41175	−0.705874	−	0.708338i	$-0.749445\pi$
$401$	−28.2702	−1.41175	−0.705874	+	0.708338i	$0.749445\pi$
$402$	0	0
$403$	−9.54563	−0.475502
$404$	0	0
$405$	25.1579	1.25010
$406$	0	0
$407$	3.52964	0.174958
$408$	0	0
$409$	19.8966	0.983824	0.491912	−	0.870645i	$-0.336298\pi$
$409$	19.8966	0.983824	0.491912	+	0.870645i	$0.336298\pi$
$410$	0	0
$411$	−22.1963	−1.09486
$412$	0	0
$413$	−28.6249	−1.40854
$414$	0	0
$415$	−14.8312	−0.728033
$416$	0	0
$417$	17.0873	0.836770
$418$	0	0
$419$	−24.0361	−1.17424	−0.587120	−	0.809500i	$-0.699738\pi$
$419$	−24.0361	−1.17424	−0.587120	+	0.809500i	$0.699738\pi$
$420$	0	0
$421$	−17.9078	−0.872773	−0.436386	−	0.899759i	$-0.643742\pi$
$421$	−17.9078	−0.872773	−0.436386	+	0.899759i	$0.643742\pi$
$422$	0	0
$423$	−8.97165	−0.436217
$424$	0	0
$425$	−0.121432	−0.00589033
$426$	0	0
$427$	23.3964	1.13223
$428$	0	0
$429$	−34.2585	−1.65401
$430$	0	0
$431$	−12.7088	−0.612164	−0.306082	−	0.952005i	$-0.599018\pi$
$431$	−12.7088	−0.612164	−0.306082	+	0.952005i	$0.599018\pi$
$432$	0	0
$433$	−1.48718	−0.0714692	−0.0357346	−	0.999361i	$-0.511377\pi$
$433$	−1.48718	−0.0714692	−0.0357346	+	0.999361i	$0.511377\pi$
$434$	0	0
$435$	−8.75605	−0.419821
$436$	0	0
$437$	−35.4901	−1.69772
$438$	0	0
$439$	−11.7894	−0.562679	−0.281340	−	0.959608i	$-0.590779\pi$
$439$	−11.7894	−0.562679	−0.281340	+	0.959608i	$0.590779\pi$
$440$	0	0
$441$	−4.47356	−0.213027
$442$	0	0
$443$	−15.6087	−0.741590	−0.370795	−	0.928715i	$-0.620915\pi$
$443$	−15.6087	−0.741590	−0.370795	+	0.928715i	$0.620915\pi$
$444$	0	0
$445$	−1.79493	−0.0850880
$446$	0	0
$447$	−22.7166	−1.07446
$448$	0	0
$449$	−25.6521	−1.21060	−0.605298	−	0.795999i	$-0.706946\pi$
$449$	−25.6521	−1.21060	−0.605298	+	0.795999i	$0.706946\pi$
$450$	0	0
$451$	53.4355	2.51618
$452$	0	0
$453$	27.2079	1.27834
$454$	0	0
$455$	15.0123	0.703789
$456$	0	0
$457$	−5.93653	−0.277699	−0.138850	−	0.990313i	$-0.544340\pi$
$457$	−5.93653	−0.277699	−0.138850	+	0.990313i	$0.544340\pi$
$458$	0	0
$459$	−24.0952	−1.12467
$460$	0	0
$461$	13.9658	0.650452	0.325226	−	0.945636i	$-0.394560\pi$
$461$	13.9658	0.650452	0.325226	+	0.945636i	$0.394560\pi$
$462$	0	0
$463$	−33.2620	−1.54582	−0.772908	−	0.634518i	$-0.781199\pi$
$463$	−33.2620	−1.54582	−0.772908	+	0.634518i	$0.781199\pi$
$464$	0	0
$465$	12.8567	0.596214
$466$	0	0
$467$	3.42155	0.158330	0.0791652	−	0.996862i	$-0.474775\pi$
$467$	3.42155	0.158330	0.0791652	+	0.996862i	$0.474775\pi$
$468$	0	0
$469$	−13.5432	−0.625366
$470$	0	0
$471$	28.7776	1.32600
$472$	0	0
$473$	−12.7542	−0.586438
$474$	0	0
$475$	−0.122357	−0.00561414
$476$	0	0
$477$	−11.3210	−0.518354
$478$	0	0
$479$	22.0707	1.00843	0.504217	−	0.863577i	$-0.331781\pi$
$479$	22.0707	1.00843	0.504217	+	0.863577i	$0.331781\pi$
$480$	0	0
$481$	2.59817	0.118466
$482$	0	0
$483$	−20.1291	−0.915906
$484$	0	0
$485$	31.6293	1.43621
$486$	0	0
$487$	−14.3950	−0.652301	−0.326150	−	0.945318i	$-0.605752\pi$
$487$	−14.3950	−0.652301	−0.326150	+	0.945318i	$0.605752\pi$
$488$	0	0
$489$	−20.3051	−0.918227
$490$	0	0
$491$	13.3831	0.603970	0.301985	−	0.953313i	$-0.402351\pi$
$491$	13.3831	0.603970	0.301985	+	0.953313i	$0.402351\pi$
$492$	0	0
$493$	13.1950	0.594273
$494$	0	0
$495$	14.4249	0.648353
$496$	0	0
$497$	14.4037	0.646095
$498$	0	0
$499$	−25.1196	−1.12451	−0.562253	−	0.826965i	$-0.690065\pi$
$499$	−25.1196	−1.12451	−0.562253	+	0.826965i	$0.690065\pi$
$500$	0	0
$501$	−8.31286	−0.371391
$502$	0	0
$503$	13.5485	0.604098	0.302049	−	0.953292i	$-0.402329\pi$
$503$	13.5485	0.604098	0.302049	+	0.953292i	$0.402329\pi$
$504$	0	0
$505$	36.6708	1.63183
$506$	0	0
$507$	1.94092	0.0861991
$508$	0	0
$509$	−13.4794	−0.597463	−0.298731	−	0.954337i	$-0.596563\pi$
$509$	−13.4794	−0.597463	−0.298731	+	0.954337i	$0.596563\pi$
$510$	0	0
$511$	−5.12864	−0.226878
$512$	0	0
$513$	−24.2788	−1.07194
$514$	0	0
$515$	−31.9772	−1.40908
$516$	0	0
$517$	31.0354	1.36494
$518$	0	0
$519$	−10.6983	−0.469604
$520$	0	0
$521$	−28.6806	−1.25652	−0.628259	−	0.778004i	$-0.716232\pi$
$521$	−28.6806	−1.25652	−0.628259	+	0.778004i	$0.716232\pi$
$522$	0	0
$523$	4.44537	0.194382	0.0971912	−	0.995266i	$-0.469014\pi$
$523$	4.44537	0.194382	0.0971912	+	0.995266i	$0.469014\pi$
$524$	0	0
$525$	−0.0693980	−0.00302878
$526$	0	0
$527$	−19.3745	−0.843966
$528$	0	0
$529$	1.94771	0.0846832
$530$	0	0
$531$	−20.2463	−0.878615
$532$	0	0
$533$	39.3339	1.70374
$534$	0	0
$535$	−13.8360	−0.598184
$536$	0	0
$537$	−43.9163	−1.89513
$538$	0	0
$539$	15.4753	0.666568
$540$	0	0
$541$	7.67999	0.330188	0.165094	−	0.986278i	$-0.447207\pi$
$541$	7.67999	0.330188	0.165094	+	0.986278i	$0.447207\pi$
$542$	0	0
$543$	−31.6995	−1.36035
$544$	0	0
$545$	23.6339	1.01236
$546$	0	0
$547$	−20.3148	−0.868600	−0.434300	−	0.900768i	$-0.643004\pi$
$547$	−20.3148	−0.868600	−0.434300	+	0.900768i	$0.643004\pi$
$548$	0	0
$549$	16.5482	0.706261
$550$	0	0
$551$	13.2955	0.566409
$552$	0	0
$553$	24.1526	1.02708
$554$	0	0
$555$	−3.49938	−0.148541
$556$	0	0
$557$	−42.1821	−1.78731	−0.893656	−	0.448753i	$-0.851868\pi$
$557$	−42.1821	−1.78731	−0.893656	+	0.448753i	$0.851868\pi$
$558$	0	0
$559$	−9.38835	−0.397085
$560$	0	0
$561$	−69.5334	−2.93570
$562$	0	0
$563$	−31.6167	−1.33248	−0.666242	−	0.745736i	$-0.732098\pi$
$563$	−31.6167	−1.33248	−0.666242	+	0.745736i	$0.732098\pi$
$564$	0	0
$565$	−6.40689	−0.269540
$566$	0	0
$567$	−21.6665	−0.909906
$568$	0	0
$569$	−5.08040	−0.212982	−0.106491	−	0.994314i	$-0.533961\pi$
$569$	−5.08040	−0.212982	−0.106491	+	0.994314i	$0.533961\pi$
$570$	0	0
$571$	−20.6733	−0.865152	−0.432576	−	0.901597i	$-0.642395\pi$
$571$	−20.6733	−0.865152	−0.432576	+	0.901597i	$0.642395\pi$
$572$	0	0
$573$	35.0037	1.46230
$574$	0	0
$575$	0.0860108	0.00358690
$576$	0	0
$577$	−14.5901	−0.607394	−0.303697	−	0.952769i	$-0.598221\pi$
$577$	−14.5901	−0.607394	−0.303697	+	0.952769i	$0.598221\pi$
$578$	0	0
$579$	−40.3767	−1.67800
$580$	0	0
$581$	12.7729	0.529909
$582$	0	0
$583$	39.1626	1.62195
$584$	0	0
$585$	10.6182	0.439008
$586$	0	0
$587$	23.9218	0.987360	0.493680	−	0.869644i	$-0.335651\pi$
$587$	23.9218	0.987360	0.493680	+	0.869644i	$0.335651\pi$
$588$	0	0
$589$	−19.5221	−0.804394
$590$	0	0
$591$	−12.9548	−0.532891
$592$	0	0
$593$	−38.4877	−1.58050	−0.790251	−	0.612783i	$-0.790050\pi$
$593$	−38.4877	−1.58050	−0.790251	+	0.612783i	$0.790050\pi$
$594$	0	0
$595$	30.4701	1.24915
$596$	0	0
$597$	−25.8248	−1.05694
$598$	0	0
$599$	25.3411	1.03541	0.517704	−	0.855560i	$-0.326787\pi$
$599$	25.3411	1.03541	0.517704	+	0.855560i	$0.326787\pi$
$600$	0	0
$601$	−5.98959	−0.244321	−0.122160	−	0.992510i	$-0.538982\pi$
$601$	−5.98959	−0.244321	−0.122160	+	0.992510i	$0.538982\pi$
$602$	0	0
$603$	−9.57907	−0.390090
$604$	0	0
$605$	−25.2608	−1.02700
$606$	0	0
$607$	2.07377	0.0841716	0.0420858	−	0.999114i	$-0.486600\pi$
$607$	2.07377	0.0841716	0.0420858	+	0.999114i	$0.486600\pi$
$608$	0	0
$609$	7.54089	0.305572
$610$	0	0
$611$	22.8452	0.924217
$612$	0	0
$613$	−27.2104	−1.09902	−0.549508	−	0.835488i	$-0.685185\pi$
$613$	−27.2104	−1.09902	−0.549508	+	0.835488i	$0.685185\pi$
$614$	0	0
$615$	−52.9774	−2.13626
$616$	0	0
$617$	−14.9398	−0.601453	−0.300727	−	0.953710i	$-0.597229\pi$
$617$	−14.9398	−0.601453	−0.300727	+	0.953710i	$0.597229\pi$
$618$	0	0
$619$	32.9887	1.32593	0.662964	−	0.748652i	$-0.269298\pi$
$619$	32.9887	1.32593	0.662964	+	0.748652i	$0.269298\pi$
$620$	0	0
$621$	17.0667	0.684864
$622$	0	0
$623$	1.54583	0.0619325
$624$	0	0
$625$	−25.0858	−1.00343
$626$	0	0
$627$	−70.0632	−2.79805
$628$	0	0
$629$	5.27342	0.210265
$630$	0	0
$631$	−32.1989	−1.28182	−0.640908	−	0.767618i	$-0.721442\pi$
$631$	−32.1989	−1.28182	−0.640908	+	0.767618i	$0.721442\pi$
$632$	0	0
$633$	−0.485085	−0.0192804
$634$	0	0
$635$	41.2299	1.63616
$636$	0	0
$637$	11.3914	0.451342
$638$	0	0
$639$	10.1877	0.403020
$640$	0	0
$641$	27.0801	1.06960	0.534799	−	0.844979i	$-0.320387\pi$
$641$	27.0801	1.06960	0.534799	+	0.844979i	$0.320387\pi$
$642$	0	0
$643$	−28.5959	−1.12771	−0.563857	−	0.825872i	$-0.690683\pi$
$643$	−28.5959	−1.12771	−0.563857	+	0.825872i	$0.690683\pi$
$644$	0	0
$645$	12.6448	0.497890
$646$	0	0
$647$	−44.6366	−1.75484	−0.877422	−	0.479719i	$-0.840738\pi$
$647$	−44.6366	−1.75484	−0.877422	+	0.479719i	$0.840738\pi$
$648$	0	0
$649$	70.0377	2.74922
$650$	0	0
$651$	−11.0724	−0.433963
$652$	0	0
$653$	−32.7268	−1.28070	−0.640350	−	0.768083i	$-0.721211\pi$
$653$	−32.7268	−1.28070	−0.640350	+	0.768083i	$0.721211\pi$
$654$	0	0
$655$	47.8952	1.87142
$656$	0	0
$657$	−3.62748	−0.141522
$658$	0	0
$659$	19.8446	0.773036	0.386518	−	0.922282i	$-0.373678\pi$
$659$	19.8446	0.773036	0.386518	+	0.922282i	$0.373678\pi$
$660$	0	0
$661$	5.30415	0.206308	0.103154	−	0.994665i	$-0.467107\pi$
$661$	5.30415	0.206308	0.103154	+	0.994665i	$0.467107\pi$
$662$	0	0
$663$	−51.1835	−1.98780
$664$	0	0
$665$	30.7022	1.19058
$666$	0	0
$667$	−9.34607	−0.361881
$668$	0	0
$669$	10.8985	0.421362
$670$	0	0
$671$	−57.2449	−2.20992
$672$	0	0
$673$	−26.9322	−1.03816	−0.519080	−	0.854726i	$-0.673725\pi$
$673$	−26.9322	−1.03816	−0.519080	+	0.854726i	$0.673725\pi$
$674$	0	0
$675$	0.0588400	0.00226475
$676$	0	0
$677$	9.50137	0.365167	0.182584	−	0.983190i	$-0.441554\pi$
$677$	9.50137	0.365167	0.182584	+	0.983190i	$0.441554\pi$
$678$	0	0
$679$	−27.2398	−1.04537
$680$	0	0
$681$	−19.4071	−0.743683
$682$	0	0
$683$	−10.0354	−0.383993	−0.191997	−	0.981396i	$-0.561496\pi$
$683$	−10.0354	−0.383993	−0.191997	+	0.981396i	$0.561496\pi$
$684$	0	0
$685$	−23.7984	−0.909291
$686$	0	0
$687$	−57.1180	−2.17919
$688$	0	0
$689$	28.8275	1.09824
$690$	0	0
$691$	−32.1385	−1.22261	−0.611303	−	0.791396i	$-0.709354\pi$
$691$	−32.1385	−1.22261	−0.611303	+	0.791396i	$0.709354\pi$
$692$	0	0
$693$	−12.4231	−0.471913
$694$	0	0
$695$	18.3207	0.694944
$696$	0	0
$697$	79.8348	3.02396
$698$	0	0
$699$	44.3763	1.67846
$700$	0	0
$701$	29.0189	1.09603	0.548014	−	0.836469i	$-0.315384\pi$
$701$	29.0189	1.09603	0.548014	+	0.836469i	$0.315384\pi$
$702$	0	0
$703$	5.31360	0.200406
$704$	0	0
$705$	−30.7694	−1.15884
$706$	0	0
$707$	−31.5817	−1.18775
$708$	0	0
$709$	−39.4571	−1.48184	−0.740921	−	0.671592i	$-0.765611\pi$
$709$	−39.4571	−1.48184	−0.740921	+	0.671592i	$0.765611\pi$
$710$	0	0
$711$	17.0831	0.640667
$712$	0	0
$713$	13.7230	0.513931
$714$	0	0
$715$	−36.7313	−1.37367
$716$	0	0
$717$	12.6758	0.473386
$718$	0	0
$719$	50.6254	1.88801	0.944005	−	0.329932i	$-0.107026\pi$
$719$	50.6254	1.88801	0.944005	+	0.329932i	$0.107026\pi$
$720$	0	0
$721$	27.5394	1.02562
$722$	0	0
$723$	−61.2431	−2.27766
$724$	0	0
$725$	−0.0322219	−0.00119669
$726$	0	0
$727$	26.1217	0.968800	0.484400	−	0.874847i	$-0.339038\pi$
$727$	26.1217	0.968800	0.484400	+	0.874847i	$0.339038\pi$
$728$	0	0
$729$	6.09040	0.225570
$730$	0	0
$731$	−19.0552	−0.704784
$732$	0	0
$733$	−28.2198	−1.04232	−0.521162	−	0.853458i	$-0.674501\pi$
$733$	−28.2198	−1.04232	−0.521162	+	0.853458i	$0.674501\pi$
$734$	0	0
$735$	−15.3426	−0.565922
$736$	0	0
$737$	33.1367	1.22060
$738$	0	0
$739$	34.6524	1.27471	0.637355	−	0.770570i	$-0.280028\pi$
$739$	34.6524	1.27471	0.637355	+	0.770570i	$0.280028\pi$
$740$	0	0
$741$	−51.5735	−1.89460
$742$	0	0
$743$	2.33064	0.0855030	0.0427515	−	0.999086i	$-0.486388\pi$
$743$	2.33064	0.0855030	0.0427515	+	0.999086i	$0.486388\pi$
$744$	0	0
$745$	−24.3563	−0.892346
$746$	0	0
$747$	9.03424	0.330546
$748$	0	0
$749$	11.9159	0.435397
$750$	0	0
$751$	44.2505	1.61472	0.807362	−	0.590056i	$-0.200894\pi$
$751$	44.2505	1.61472	0.807362	+	0.590056i	$0.200894\pi$
$752$	0	0
$753$	−2.08912	−0.0761318
$754$	0	0
$755$	29.1718	1.06167
$756$	0	0
$757$	−8.89721	−0.323374	−0.161687	−	0.986842i	$-0.551694\pi$
$757$	−8.89721	−0.323374	−0.161687	+	0.986842i	$0.551694\pi$
$758$	0	0
$759$	49.2507	1.78769
$760$	0	0
$761$	−17.8081	−0.645544	−0.322772	−	0.946477i	$-0.604615\pi$
$761$	−17.8081	−0.645544	−0.322772	+	0.946477i	$0.604615\pi$
$762$	0	0
$763$	−20.3540	−0.736864
$764$	0	0
$765$	21.5514	0.779194
$766$	0	0
$767$	51.5547	1.86153
$768$	0	0
$769$	4.05987	0.146403	0.0732013	−	0.997317i	$-0.476678\pi$
$769$	4.05987	0.146403	0.0732013	+	0.997317i	$0.476678\pi$
$770$	0	0
$771$	31.2403	1.12509
$772$	0	0
$773$	−23.2116	−0.834864	−0.417432	−	0.908708i	$-0.637070\pi$
$773$	−23.2116	−0.834864	−0.417432	+	0.908708i	$0.637070\pi$
$774$	0	0
$775$	0.0473120	0.00169950
$776$	0	0
$777$	3.01374	0.108117
$778$	0	0
$779$	80.4430	2.88217
$780$	0	0
$781$	−35.2422	−1.26106
$782$	0	0
$783$	−6.39364	−0.228490
$784$	0	0
$785$	30.8549	1.10126
$786$	0	0
$787$	−11.1949	−0.399054	−0.199527	−	0.979892i	$-0.563941\pi$
$787$	−11.1949	−0.399054	−0.199527	+	0.979892i	$0.563941\pi$
$788$	0	0
$789$	−48.6645	−1.73250
$790$	0	0
$791$	5.51775	0.196189
$792$	0	0
$793$	−42.1380	−1.49636
$794$	0	0
$795$	−38.8268	−1.37705
$796$	0	0
$797$	46.1950	1.63631	0.818155	−	0.574997i	$-0.194997\pi$
$797$	46.1950	1.63631	0.818155	+	0.574997i	$0.194997\pi$
$798$	0	0
$799$	46.3682	1.64039
$800$	0	0
$801$	1.09337	0.0386322
$802$	0	0
$803$	12.5485	0.442826
$804$	0	0
$805$	−21.5821	−0.760668
$806$	0	0
$807$	−0.664714	−0.0233991
$808$	0	0
$809$	44.6151	1.56858	0.784291	−	0.620393i	$-0.213027\pi$
$809$	44.6151	1.56858	0.784291	+	0.620393i	$0.213027\pi$
$810$	0	0
$811$	−7.58514	−0.266350	−0.133175	−	0.991093i	$-0.542517\pi$
$811$	−7.58514	−0.266350	−0.133175	+	0.991093i	$0.542517\pi$
$812$	0	0
$813$	−15.0118	−0.526486
$814$	0	0
$815$	−21.7707	−0.762595
$816$	0	0
$817$	−19.2004	−0.671738
$818$	0	0
$819$	−9.14461	−0.319538
$820$	0	0
$821$	−52.3945	−1.82858	−0.914290	−	0.405060i	$-0.867251\pi$
$821$	−52.3945	−1.82858	−0.914290	+	0.405060i	$0.867251\pi$
$822$	0	0
$823$	23.7784	0.828863	0.414431	−	0.910081i	$-0.363981\pi$
$823$	23.7784	0.828863	0.414431	+	0.910081i	$0.363981\pi$
$824$	0	0
$825$	0.169799	0.00591164
$826$	0	0
$827$	−2.88587	−0.100352	−0.0501758	−	0.998740i	$-0.515978\pi$
$827$	−2.88587	−0.100352	−0.0501758	+	0.998740i	$0.515978\pi$
$828$	0	0
$829$	17.9426	0.623172	0.311586	−	0.950218i	$-0.399140\pi$
$829$	17.9426	0.623172	0.311586	+	0.950218i	$0.399140\pi$
$830$	0	0
$831$	−41.1888	−1.42882
$832$	0	0
$833$	23.1207	0.801085
$834$	0	0
$835$	−8.91289	−0.308443
$836$	0	0
$837$	9.38790	0.324493
$838$	0	0
$839$	−28.5081	−0.984209	−0.492104	−	0.870536i	$-0.663772\pi$
$839$	−28.5081	−0.984209	−0.492104	+	0.870536i	$0.663772\pi$
$840$	0	0
$841$	−25.4987	−0.879266
$842$	0	0
$843$	50.3756	1.73503
$844$	0	0
$845$	2.08101	0.0715891
$846$	0	0
$847$	21.7551	0.747516
$848$	0	0
$849$	61.4262	2.10814
$850$	0	0
$851$	−3.73518	−0.128040
$852$	0	0
$853$	−13.8304	−0.473545	−0.236773	−	0.971565i	$-0.576090\pi$
$853$	−13.8304	−0.473545	−0.236773	+	0.971565i	$0.576090\pi$
$854$	0	0
$855$	21.7156	0.742659
$856$	0	0
$857$	33.3614	1.13960	0.569802	−	0.821782i	$-0.307020\pi$
$857$	33.3614	1.13960	0.569802	+	0.821782i	$0.307020\pi$
$858$	0	0
$859$	44.3297	1.51251	0.756254	−	0.654278i	$-0.227027\pi$
$859$	44.3297	1.51251	0.756254	+	0.654278i	$0.227027\pi$
$860$	0	0
$861$	45.6252	1.55490
$862$	0	0
$863$	23.8335	0.811303	0.405651	−	0.914028i	$-0.367045\pi$
$863$	23.8335	0.811303	0.405651	+	0.914028i	$0.367045\pi$
$864$	0	0
$865$	−11.4705	−0.390010
$866$	0	0
$867$	−68.3706	−2.32199
$868$	0	0
$869$	−59.0953	−2.00467
$870$	0	0
$871$	24.3919	0.826487
$872$	0	0
$873$	−19.2667	−0.652079
$874$	0	0
$875$	21.5303	0.727855
$876$	0	0
$877$	23.8260	0.804548	0.402274	−	0.915519i	$-0.368220\pi$
$877$	23.8260	0.804548	0.402274	+	0.915519i	$0.368220\pi$
$878$	0	0
$879$	31.0209	1.04631
$880$	0	0
$881$	3.47940	0.117224	0.0586119	−	0.998281i	$-0.481333\pi$
$881$	3.47940	0.117224	0.0586119	+	0.998281i	$0.481333\pi$
$882$	0	0
$883$	48.6555	1.63739	0.818694	−	0.574231i	$-0.194699\pi$
$883$	48.6555	1.63739	0.818694	+	0.574231i	$0.194699\pi$
$884$	0	0
$885$	−69.4372	−2.33411
$886$	0	0
$887$	−54.5160	−1.83047	−0.915233	−	0.402924i	$-0.867994\pi$
$887$	−54.5160	−1.83047	−0.915233	+	0.402924i	$0.867994\pi$
$888$	0	0
$889$	−35.5081	−1.19090
$890$	0	0
$891$	53.0123	1.77598
$892$	0	0
$893$	46.7214	1.56347
$894$	0	0
$895$	−47.0863	−1.57392
$896$	0	0
$897$	36.2534	1.21047
$898$	0	0
$899$	−5.14100	−0.171462
$900$	0	0
$901$	58.5104	1.94926
$902$	0	0
$903$	−10.8900	−0.362396
$904$	0	0
$905$	−33.9876	−1.12979
$906$	0	0
$907$	5.24593	0.174188	0.0870942	−	0.996200i	$-0.472242\pi$
$907$	5.24593	0.174188	0.0870942	+	0.996200i	$0.472242\pi$
$908$	0	0
$909$	−22.3377	−0.740894
$910$	0	0
$911$	−12.3934	−0.410613	−0.205307	−	0.978698i	$-0.565819\pi$
$911$	−12.3934	−0.410613	−0.205307	+	0.978698i	$0.565819\pi$
$912$	0	0
$913$	−31.2520	−1.03429
$914$	0	0
$915$	56.7542	1.87624
$916$	0	0
$917$	−41.2483	−1.36214
$918$	0	0
$919$	38.2402	1.26143	0.630715	−	0.776015i	$-0.282762\pi$
$919$	38.2402	1.26143	0.630715	+	0.776015i	$0.282762\pi$
$920$	0	0
$921$	−0.345418	−0.0113819
$922$	0	0
$923$	−25.9417	−0.853883
$924$	0	0
$925$	−0.0128776	−0.000423412 0
$926$	0	0
$927$	19.4786	0.639761
$928$	0	0
$929$	17.4054	0.571052	0.285526	−	0.958371i	$-0.407832\pi$
$929$	17.4054	0.571052	0.285526	+	0.958371i	$0.407832\pi$
$930$	0	0
$931$	23.2969	0.763524
$932$	0	0
$933$	14.0515	0.460025
$934$	0	0
$935$	−74.5524	−2.43812
$936$	0	0
$937$	23.0316	0.752409	0.376205	−	0.926537i	$-0.377229\pi$
$937$	23.0316	0.752409	0.376205	+	0.926537i	$0.377229\pi$
$938$	0	0
$939$	16.1161	0.525928
$940$	0	0
$941$	−7.18215	−0.234131	−0.117066	−	0.993124i	$-0.537349\pi$
$941$	−7.18215	−0.234131	−0.117066	+	0.993124i	$0.537349\pi$
$942$	0	0
$943$	−56.5472	−1.84143
$944$	0	0
$945$	−14.7643	−0.480282
$946$	0	0
$947$	18.0753	0.587369	0.293685	−	0.955902i	$-0.405118\pi$
$947$	18.0753	0.587369	0.293685	+	0.955902i	$0.405118\pi$
$948$	0	0
$949$	9.23692	0.299843
$950$	0	0
$951$	−51.2547	−1.66205
$952$	0	0
$953$	−56.4467	−1.82849	−0.914244	−	0.405164i	$-0.867214\pi$
$953$	−56.4467	−1.82849	−0.914244	+	0.405164i	$0.867214\pi$
$954$	0	0
$955$	37.5303	1.21445
$956$	0	0
$957$	−18.4506	−0.596424
$958$	0	0
$959$	20.4957	0.661840
$960$	0	0
$961$	−23.4514	−0.756496
$962$	0	0
$963$	8.42808	0.271591
$964$	0	0
$965$	−43.2911	−1.39359
$966$	0	0
$967$	−38.9829	−1.25361	−0.626803	−	0.779178i	$-0.715637\pi$
$967$	−38.9829	−1.25361	−0.626803	+	0.779178i	$0.715637\pi$
$968$	0	0
$969$	−104.677	−3.36271
$970$	0	0
$971$	−36.5699	−1.17358	−0.586792	−	0.809737i	$-0.699610\pi$
$971$	−36.5699	−1.17358	−0.586792	+	0.809737i	$0.699610\pi$
$972$	0	0
$973$	−15.7782	−0.505825
$974$	0	0
$975$	0.124989	0.00400285
$976$	0	0
$977$	−9.76866	−0.312527	−0.156264	−	0.987715i	$-0.549945\pi$
$977$	−9.76866	−0.312527	−0.156264	+	0.987715i	$0.549945\pi$
$978$	0	0
$979$	−3.78226	−0.120881
$980$	0	0
$981$	−14.3963	−0.459640
$982$	0	0
$983$	−54.0083	−1.72260	−0.861298	−	0.508100i	$-0.830348\pi$
$983$	−54.0083	−1.72260	−0.861298	+	0.508100i	$0.830348\pi$
$984$	0	0
$985$	−13.8899	−0.442570
$986$	0	0
$987$	26.4992	0.843479
$988$	0	0
$989$	13.4969	0.429176
$990$	0	0
$991$	−48.3638	−1.53633	−0.768163	−	0.640254i	$-0.778829\pi$
$991$	−48.3638	−1.53633	−0.768163	+	0.640254i	$0.778829\pi$
$992$	0	0
$993$	27.1387	0.861220
$994$	0	0
$995$	−27.6889	−0.877796
$996$	0	0
$997$	−11.1542	−0.353257	−0.176628	−	0.984278i	$-0.556519\pi$
$997$	−11.1542	−0.353257	−0.176628	+	0.984278i	$0.556519\pi$
$998$	0	0
$999$	−2.55524	−0.0808441

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.l.1.3		19
4.3	odd	2		2008.2.a.c.1.17	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.c.1.17	✓	19	4.3	odd	2
4016.2.a.l.1.3		19	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.08912 q^{3} -2.23992 q^{5} +1.92906 q^{7} +1.36442 q^{9} +O(q^{10})\)	\(q-2.08912 q^{3} -2.23992 q^{5} +1.92906 q^{7} +1.36442 q^{9} -4.71991 q^{11} -3.47433 q^{13} +4.67945 q^{15} -7.05174 q^{17} -7.10546 q^{19} -4.03004 q^{21} +4.99477 q^{23} +0.0172202 q^{25} +3.41692 q^{27} -1.87117 q^{29} +2.74748 q^{31} +9.86047 q^{33} -4.32093 q^{35} -0.747819 q^{37} +7.25828 q^{39} -11.3213 q^{41} +2.70221 q^{43} -3.05619 q^{45} -6.57542 q^{47} -3.27873 q^{49} +14.7319 q^{51} -8.29731 q^{53} +10.5722 q^{55} +14.8442 q^{57} -14.8388 q^{59} +12.1284 q^{61} +2.63205 q^{63} +7.78220 q^{65} -7.02061 q^{67} -10.4347 q^{69} +7.46670 q^{71} -2.65862 q^{73} -0.0359750 q^{75} -9.10500 q^{77} +12.5204 q^{79} -11.2316 q^{81} +6.62130 q^{83} +15.7953 q^{85} +3.90910 q^{87} +0.801340 q^{89} -6.70219 q^{91} -5.73981 q^{93} +15.9156 q^{95} -14.1208 q^{97} -6.43995 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) \) 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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