LMFDB - Embedded newform 4016.2.a.l.1.18

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			1.18
Root			$3.03361$ 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)

$f(q)$	$=$	$q+3.03361 q^{3} -0.0964405 q^{5} +5.16490 q^{7} +6.20277 q^{9} +O(q^{10})$	$q+3.03361 q^{3} -0.0964405 q^{5} +5.16490 q^{7} +6.20277 q^{9} -2.15578 q^{11} -3.03319 q^{13} -0.292563 q^{15} +3.71166 q^{17} -6.01654 q^{19} +15.6683 q^{21} -2.42923 q^{23} -4.99070 q^{25} +9.71593 q^{27} +5.72633 q^{29} +5.57046 q^{31} -6.53978 q^{33} -0.498106 q^{35} +3.42745 q^{37} -9.20151 q^{39} +10.0269 q^{41} +9.75295 q^{43} -0.598198 q^{45} -2.69972 q^{47} +19.6762 q^{49} +11.2597 q^{51} -2.96594 q^{53} +0.207904 q^{55} -18.2518 q^{57} -1.35970 q^{59} -9.81032 q^{61} +32.0367 q^{63} +0.292523 q^{65} +8.93383 q^{67} -7.36933 q^{69} -3.08436 q^{71} +2.54726 q^{73} -15.1398 q^{75} -11.1344 q^{77} +17.3635 q^{79} +10.8660 q^{81} +5.00713 q^{83} -0.357954 q^{85} +17.3714 q^{87} +11.2899 q^{89} -15.6661 q^{91} +16.8986 q^{93} +0.580238 q^{95} +0.180641 q^{97} -13.3718 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	3.03361	1.75145	0.875727	−	0.482807i	$-0.160383\pi$
$3$	3.03361	1.75145	0.875727	+	0.482807i	$0.160383\pi$
$4$	0	0
$5$	−0.0964405	−0.0431295	−0.0215648	−	0.999767i	$-0.506865\pi$
$5$	−0.0964405	−0.0431295	−0.0215648	+	0.999767i	$0.506865\pi$
$6$	0	0
$7$	5.16490	1.95215	0.976074	−	0.217438i	$-0.0697701\pi$
$7$	5.16490	1.95215	0.976074	+	0.217438i	$0.0697701\pi$
$8$	0	0
$9$	6.20277	2.06759
$10$	0	0
$11$	−2.15578	−0.649991	−0.324996	−	0.945715i	$-0.605363\pi$
$11$	−2.15578	−0.649991	−0.324996	+	0.945715i	$0.605363\pi$
$12$	0	0
$13$	−3.03319	−0.841256	−0.420628	−	0.907233i	$-0.638190\pi$
$13$	−3.03319	−0.841256	−0.420628	+	0.907233i	$0.638190\pi$
$14$	0	0
$15$	−0.292563	−0.0755393
$16$	0	0
$17$	3.71166	0.900209	0.450105	−	0.892976i	$-0.351387\pi$
$17$	3.71166	0.900209	0.450105	+	0.892976i	$0.351387\pi$
$18$	0	0
$19$	−6.01654	−1.38029	−0.690144	−	0.723672i	$-0.742453\pi$
$19$	−6.01654	−1.38029	−0.690144	+	0.723672i	$0.742453\pi$
$20$	0	0
$21$	15.6683	3.41910
$22$	0	0
$23$	−2.42923	−0.506530	−0.253265	−	0.967397i	$-0.581504\pi$
$23$	−2.42923	−0.506530	−0.253265	+	0.967397i	$0.581504\pi$
$24$	0	0
$25$	−4.99070	−0.998140
$26$	0	0
$27$	9.71593	1.86983
$28$	0	0
$29$	5.72633	1.06335	0.531676	−	0.846948i	$-0.321562\pi$
$29$	5.72633	1.06335	0.531676	+	0.846948i	$0.321562\pi$
$30$	0	0
$31$	5.57046	1.00048	0.500242	−	0.865886i	$-0.333244\pi$
$31$	5.57046	1.00048	0.500242	+	0.865886i	$0.333244\pi$
$32$	0	0
$33$	−6.53978	−1.13843
$34$	0	0
$35$	−0.498106	−0.0841952
$36$	0	0
$37$	3.42745	0.563470	0.281735	−	0.959492i	$-0.409090\pi$
$37$	3.42745	0.563470	0.281735	+	0.959492i	$0.409090\pi$
$38$	0	0
$39$	−9.20151	−1.47342
$40$	0	0
$41$	10.0269	1.56593	0.782966	−	0.622065i	$-0.213706\pi$
$41$	10.0269	1.56593	0.782966	+	0.622065i	$0.213706\pi$
$42$	0	0
$43$	9.75295	1.48731	0.743655	−	0.668563i	$-0.233090\pi$
$43$	9.75295	1.48731	0.743655	+	0.668563i	$0.233090\pi$
$44$	0	0
$45$	−0.598198	−0.0891741
$46$	0	0
$47$	−2.69972	−0.393794	−0.196897	−	0.980424i	$-0.563087\pi$
$47$	−2.69972	−0.393794	−0.196897	+	0.980424i	$0.563087\pi$
$48$	0	0
$49$	19.6762	2.81088
$50$	0	0
$51$	11.2597	1.57667
$52$	0	0
$53$	−2.96594	−0.407404	−0.203702	−	0.979033i	$-0.565297\pi$
$53$	−2.96594	−0.407404	−0.203702	+	0.979033i	$0.565297\pi$
$54$	0	0
$55$	0.207904	0.0280338
$56$	0	0
$57$	−18.2518	−2.41751
$58$	0	0
$59$	−1.35970	−0.177018	−0.0885092	−	0.996075i	$-0.528210\pi$
$59$	−1.35970	−0.177018	−0.0885092	+	0.996075i	$0.528210\pi$
$60$	0	0
$61$	−9.81032	−1.25608	−0.628041	−	0.778180i	$-0.716143\pi$
$61$	−9.81032	−1.25608	−0.628041	+	0.778180i	$0.716143\pi$
$62$	0	0
$63$	32.0367	4.03624
$64$	0	0
$65$	0.292523	0.0362830
$66$	0	0
$67$	8.93383	1.09144	0.545720	−	0.837967i	$-0.316256\pi$
$67$	8.93383	1.09144	0.545720	+	0.837967i	$0.316256\pi$
$68$	0	0
$69$	−7.36933	−0.887163
$70$	0	0
$71$	−3.08436	−0.366046	−0.183023	−	0.983109i	$-0.558588\pi$
$71$	−3.08436	−0.366046	−0.183023	+	0.983109i	$0.558588\pi$
$72$	0	0
$73$	2.54726	0.298134	0.149067	−	0.988827i	$-0.452373\pi$
$73$	2.54726	0.298134	0.149067	+	0.988827i	$0.452373\pi$
$74$	0	0
$75$	−15.1398	−1.74820
$76$	0	0
$77$	−11.1344	−1.26888
$78$	0	0
$79$	17.3635	1.95355	0.976774	−	0.214272i	$-0.0687378\pi$
$79$	17.3635	1.95355	0.976774	+	0.214272i	$0.0687378\pi$
$80$	0	0
$81$	10.8660	1.20733
$82$	0	0
$83$	5.00713	0.549604	0.274802	−	0.961501i	$-0.411388\pi$
$83$	5.00713	0.549604	0.274802	+	0.961501i	$0.411388\pi$
$84$	0	0
$85$	−0.357954	−0.0388256
$86$	0	0
$87$	17.3714	1.86241
$88$	0	0
$89$	11.2899	1.19673	0.598364	−	0.801224i	$-0.295818\pi$
$89$	11.2899	1.19673	0.598364	+	0.801224i	$0.295818\pi$
$90$	0	0
$91$	−15.6661	−1.64226
$92$	0	0
$93$	16.8986	1.75230
$94$	0	0
$95$	0.580238	0.0595312
$96$	0	0
$97$	0.180641	0.0183413	0.00917067	−	0.999958i	$-0.497081\pi$
$97$	0.180641	0.0183413	0.00917067	+	0.999958i	$0.497081\pi$
$98$	0	0
$99$	−13.3718	−1.34391
$100$	0	0
$101$	−7.73500	−0.769661	−0.384831	−	0.922987i	$-0.625740\pi$
$101$	−7.73500	−0.769661	−0.384831	+	0.922987i	$0.625740\pi$
$102$	0	0
$103$	−12.2262	−1.20468	−0.602340	−	0.798240i	$-0.705765\pi$
$103$	−12.2262	−1.20468	−0.602340	+	0.798240i	$0.705765\pi$
$104$	0	0
$105$	−1.51106	−0.147464
$106$	0	0
$107$	−7.26518	−0.702351	−0.351176	−	0.936310i	$-0.614218\pi$
$107$	−7.26518	−0.702351	−0.351176	+	0.936310i	$0.614218\pi$
$108$	0	0
$109$	−15.9292	−1.52574	−0.762869	−	0.646553i	$-0.776210\pi$
$109$	−15.9292	−1.52574	−0.762869	+	0.646553i	$0.776210\pi$
$110$	0	0
$111$	10.3975	0.986891
$112$	0	0
$113$	−4.24313	−0.399160	−0.199580	−	0.979882i	$-0.563958\pi$
$113$	−4.24313	−0.399160	−0.199580	+	0.979882i	$0.563958\pi$
$114$	0	0
$115$	0.234276	0.0218464
$116$	0	0
$117$	−18.8142	−1.73937
$118$	0	0
$119$	19.1703	1.75734
$120$	0	0
$121$	−6.35262	−0.577511
$122$	0	0
$123$	30.4175	2.74266
$124$	0	0
$125$	0.963508	0.0861788
$126$	0	0
$127$	−12.9139	−1.14592	−0.572960	−	0.819584i	$-0.694205\pi$
$127$	−12.9139	−1.14592	−0.572960	+	0.819584i	$0.694205\pi$
$128$	0	0
$129$	29.5866	2.60496
$130$	0	0
$131$	−7.44059	−0.650088	−0.325044	−	0.945699i	$-0.605379\pi$
$131$	−7.44059	−0.650088	−0.325044	+	0.945699i	$0.605379\pi$
$132$	0	0
$133$	−31.0748	−2.69453
$134$	0	0
$135$	−0.937010	−0.0806449
$136$	0	0
$137$	−11.4302	−0.976548	−0.488274	−	0.872690i	$-0.662373\pi$
$137$	−11.4302	−0.976548	−0.488274	+	0.872690i	$0.662373\pi$
$138$	0	0
$139$	−9.14378	−0.775565	−0.387783	−	0.921751i	$-0.626759\pi$
$139$	−9.14378	−0.775565	−0.387783	+	0.921751i	$0.626759\pi$
$140$	0	0
$141$	−8.18988	−0.689713
$142$	0	0
$143$	6.53889	0.546809
$144$	0	0
$145$	−0.552250	−0.0458619
$146$	0	0
$147$	59.6898	4.92313
$148$	0	0
$149$	−9.39271	−0.769481	−0.384740	−	0.923025i	$-0.625709\pi$
$149$	−9.39271	−0.769481	−0.384740	+	0.923025i	$0.625709\pi$
$150$	0	0
$151$	18.8614	1.53492	0.767459	−	0.641098i	$-0.221521\pi$
$151$	18.8614	1.53492	0.767459	+	0.641098i	$0.221521\pi$
$152$	0	0
$153$	23.0225	1.86126
$154$	0	0
$155$	−0.537218	−0.0431504
$156$	0	0
$157$	−15.4347	−1.23182	−0.615911	−	0.787816i	$-0.711212\pi$
$157$	−15.4347	−1.23182	−0.615911	+	0.787816i	$0.711212\pi$
$158$	0	0
$159$	−8.99750	−0.713548
$160$	0	0
$161$	−12.5467	−0.988821
$162$	0	0
$163$	3.77890	0.295986	0.147993	−	0.988988i	$-0.452719\pi$
$163$	3.77890	0.295986	0.147993	+	0.988988i	$0.452719\pi$
$164$	0	0
$165$	0.630700	0.0490999
$166$	0	0
$167$	−24.6580	−1.90810	−0.954048	−	0.299654i	$-0.903129\pi$
$167$	−24.6580	−1.90810	−0.954048	+	0.299654i	$0.903129\pi$
$168$	0	0
$169$	−3.79974	−0.292288
$170$	0	0
$171$	−37.3192	−2.85387
$172$	0	0
$173$	−23.3160	−1.77268	−0.886342	−	0.463031i	$-0.846762\pi$
$173$	−23.3160	−1.77268	−0.886342	+	0.463031i	$0.846762\pi$
$174$	0	0
$175$	−25.7765	−1.94852
$176$	0	0
$177$	−4.12481	−0.310039
$178$	0	0
$179$	25.9863	1.94231	0.971153	−	0.238458i	$-0.0766419\pi$
$179$	25.9863	1.94231	0.971153	+	0.238458i	$0.0766419\pi$
$180$	0	0
$181$	−5.07930	−0.377541	−0.188771	−	0.982021i	$-0.560450\pi$
$181$	−5.07930	−0.377541	−0.188771	+	0.982021i	$0.560450\pi$
$182$	0	0
$183$	−29.7606	−2.19997
$184$	0	0
$185$	−0.330545	−0.0243022
$186$	0	0
$187$	−8.00151	−0.585128
$188$	0	0
$189$	50.1818	3.65019
$190$	0	0
$191$	−15.5139	−1.12254	−0.561271	−	0.827632i	$-0.689688\pi$
$191$	−15.5139	−1.12254	−0.561271	+	0.827632i	$0.689688\pi$
$192$	0	0
$193$	−3.39023	−0.244034	−0.122017	−	0.992528i	$-0.538936\pi$
$193$	−3.39023	−0.244034	−0.122017	+	0.992528i	$0.538936\pi$
$194$	0	0
$195$	0.887399	0.0635479
$196$	0	0
$197$	−9.66801	−0.688817	−0.344408	−	0.938820i	$-0.611920\pi$
$197$	−9.66801	−0.688817	−0.344408	+	0.938820i	$0.611920\pi$
$198$	0	0
$199$	20.4995	1.45317	0.726585	−	0.687076i	$-0.241106\pi$
$199$	20.4995	1.45317	0.726585	+	0.687076i	$0.241106\pi$
$200$	0	0
$201$	27.1017	1.91161
$202$	0	0
$203$	29.5759	2.07582
$204$	0	0
$205$	−0.966995	−0.0675379
$206$	0	0
$207$	−15.0680	−1.04729
$208$	0	0
$209$	12.9703	0.897175
$210$	0	0
$211$	10.9754	0.755581	0.377790	−	0.925891i	$-0.376684\pi$
$211$	10.9754	0.755581	0.377790	+	0.925891i	$0.376684\pi$
$212$	0	0
$213$	−9.35674	−0.641113
$214$	0	0
$215$	−0.940580	−0.0641470
$216$	0	0
$217$	28.7709	1.95309
$218$	0	0
$219$	7.72738	0.522168
$220$	0	0
$221$	−11.2582	−0.757307
$222$	0	0
$223$	8.94794	0.599198	0.299599	−	0.954065i	$-0.403147\pi$
$223$	8.94794	0.599198	0.299599	+	0.954065i	$0.403147\pi$
$224$	0	0
$225$	−30.9561	−2.06374
$226$	0	0
$227$	−17.8618	−1.18553	−0.592764	−	0.805376i	$-0.701963\pi$
$227$	−17.8618	−1.18553	−0.592764	+	0.805376i	$0.701963\pi$
$228$	0	0
$229$	21.4150	1.41514	0.707571	−	0.706642i	$-0.249791\pi$
$229$	21.4150	1.41514	0.707571	+	0.706642i	$0.249791\pi$
$230$	0	0
$231$	−33.7773	−2.22238
$232$	0	0
$233$	27.6973	1.81451	0.907257	−	0.420578i	$-0.138173\pi$
$233$	27.6973	1.81451	0.907257	+	0.420578i	$0.138173\pi$
$234$	0	0
$235$	0.260362	0.0169842
$236$	0	0
$237$	52.6741	3.42155
$238$	0	0
$239$	5.11005	0.330542	0.165271	−	0.986248i	$-0.447150\pi$
$239$	5.11005	0.330542	0.165271	+	0.986248i	$0.447150\pi$
$240$	0	0
$241$	−3.40026	−0.219030	−0.109515	−	0.993985i	$-0.534930\pi$
$241$	−3.40026	−0.219030	−0.109515	+	0.993985i	$0.534930\pi$
$242$	0	0
$243$	3.81540	0.244758
$244$	0	0
$245$	−1.89758	−0.121232
$246$	0	0
$247$	18.2493	1.16118
$248$	0	0
$249$	15.1897	0.962606
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	5.23688	0.329240
$254$	0	0
$255$	−1.08589	−0.0680012
$256$	0	0
$257$	−21.1797	−1.32115	−0.660577	−	0.750759i	$-0.729688\pi$
$257$	−21.1797	−1.32115	−0.660577	+	0.750759i	$0.729688\pi$
$258$	0	0
$259$	17.7024	1.09998
$260$	0	0
$261$	35.5191	2.19858
$262$	0	0
$263$	−31.1440	−1.92042	−0.960211	−	0.279276i	$-0.909906\pi$
$263$	−31.1440	−1.92042	−0.960211	+	0.279276i	$0.909906\pi$
$264$	0	0
$265$	0.286037	0.0175711
$266$	0	0
$267$	34.2492	2.09601
$268$	0	0
$269$	−10.9223	−0.665941	−0.332971	−	0.942937i	$-0.608051\pi$
$269$	−10.9223	−0.665941	−0.332971	+	0.942937i	$0.608051\pi$
$270$	0	0
$271$	2.43771	0.148080	0.0740402	−	0.997255i	$-0.476411\pi$
$271$	2.43771	0.148080	0.0740402	+	0.997255i	$0.476411\pi$
$272$	0	0
$273$	−47.5249	−2.87634
$274$	0	0
$275$	10.7588	0.648782
$276$	0	0
$277$	5.75370	0.345707	0.172853	−	0.984948i	$-0.444701\pi$
$277$	5.75370	0.345707	0.172853	+	0.984948i	$0.444701\pi$
$278$	0	0
$279$	34.5523	2.06859
$280$	0	0
$281$	−12.3179	−0.734826	−0.367413	−	0.930058i	$-0.619756\pi$
$281$	−12.3179	−0.734826	−0.367413	+	0.930058i	$0.619756\pi$
$282$	0	0
$283$	3.45459	0.205354	0.102677	−	0.994715i	$-0.467259\pi$
$283$	3.45459	0.205354	0.102677	+	0.994715i	$0.467259\pi$
$284$	0	0
$285$	1.76021	0.104266
$286$	0	0
$287$	51.7877	3.05693
$288$	0	0
$289$	−3.22360	−0.189623
$290$	0	0
$291$	0.547994	0.0321240
$292$	0	0
$293$	−15.9215	−0.930146	−0.465073	−	0.885272i	$-0.653972\pi$
$293$	−15.9215	−0.930146	−0.465073	+	0.885272i	$0.653972\pi$
$294$	0	0
$295$	0.131131	0.00763472
$296$	0	0
$297$	−20.9454	−1.21537
$298$	0	0
$299$	7.36832	0.426121
$300$	0	0
$301$	50.3730	2.90345
$302$	0	0
$303$	−23.4649	−1.34803
$304$	0	0
$305$	0.946112	0.0541742
$306$	0	0
$307$	−2.15850	−0.123192	−0.0615961	−	0.998101i	$-0.519619\pi$
$307$	−2.15850	−0.123192	−0.0615961	+	0.998101i	$0.519619\pi$
$308$	0	0
$309$	−37.0894	−2.10994
$310$	0	0
$311$	−21.9622	−1.24536	−0.622680	−	0.782477i	$-0.713956\pi$
$311$	−21.9622	−1.24536	−0.622680	+	0.782477i	$0.713956\pi$
$312$	0	0
$313$	24.3832	1.37822	0.689109	−	0.724658i	$-0.258002\pi$
$313$	24.3832	1.37822	0.689109	+	0.724658i	$0.258002\pi$
$314$	0	0
$315$	−3.08963	−0.174081
$316$	0	0
$317$	−4.71013	−0.264547	−0.132274	−	0.991213i	$-0.542228\pi$
$317$	−4.71013	−0.264547	−0.132274	+	0.991213i	$0.542228\pi$
$318$	0	0
$319$	−12.3447	−0.691170
$320$	0	0
$321$	−22.0397	−1.23014
$322$	0	0
$323$	−22.3313	−1.24255
$324$	0	0
$325$	15.1378	0.839691
$326$	0	0
$327$	−48.3228	−2.67226
$328$	0	0
$329$	−13.9438	−0.768745
$330$	0	0
$331$	16.8534	0.926345	0.463172	−	0.886268i	$-0.346711\pi$
$331$	16.8534	0.926345	0.463172	+	0.886268i	$0.346711\pi$
$332$	0	0
$333$	21.2597	1.16502
$334$	0	0
$335$	−0.861583	−0.0470733
$336$	0	0
$337$	−7.64255	−0.416316	−0.208158	−	0.978095i	$-0.566747\pi$
$337$	−7.64255	−0.416316	−0.208158	+	0.978095i	$0.566747\pi$
$338$	0	0
$339$	−12.8720	−0.699110
$340$	0	0
$341$	−12.0087	−0.650306
$342$	0	0
$343$	65.4712	3.53511
$344$	0	0
$345$	0.710702	0.0382629
$346$	0	0
$347$	−15.2391	−0.818078	−0.409039	−	0.912517i	$-0.634136\pi$
$347$	−15.2391	−0.818078	−0.409039	+	0.912517i	$0.634136\pi$
$348$	0	0
$349$	−0.628090	−0.0336209	−0.0168104	−	0.999859i	$-0.505351\pi$
$349$	−0.628090	−0.0336209	−0.0168104	+	0.999859i	$0.505351\pi$
$350$	0	0
$351$	−29.4703	−1.57301
$352$	0	0
$353$	9.70340	0.516460	0.258230	−	0.966084i	$-0.416861\pi$
$353$	9.70340	0.516460	0.258230	+	0.966084i	$0.416861\pi$
$354$	0	0
$355$	0.297457	0.0157874
$356$	0	0
$357$	58.1552	3.07790
$358$	0	0
$359$	−2.96845	−0.156669	−0.0783345	−	0.996927i	$-0.524960\pi$
$359$	−2.96845	−0.156669	−0.0783345	+	0.996927i	$0.524960\pi$
$360$	0	0
$361$	17.1987	0.905195
$362$	0	0
$363$	−19.2714	−1.01148
$364$	0	0
$365$	−0.245659	−0.0128584
$366$	0	0
$367$	2.48847	0.129897	0.0649487	−	0.997889i	$-0.479312\pi$
$367$	2.48847	0.129897	0.0649487	+	0.997889i	$0.479312\pi$
$368$	0	0
$369$	62.1942	3.23770
$370$	0	0
$371$	−15.3188	−0.795312
$372$	0	0
$373$	0.657165	0.0340267	0.0170134	−	0.999855i	$-0.494584\pi$
$373$	0.657165	0.0340267	0.0170134	+	0.999855i	$0.494584\pi$
$374$	0	0
$375$	2.92290	0.150938
$376$	0	0
$377$	−17.3691	−0.894552
$378$	0	0
$379$	−24.8342	−1.27565	−0.637824	−	0.770182i	$-0.720165\pi$
$379$	−24.8342	−1.27565	−0.637824	+	0.770182i	$0.720165\pi$
$380$	0	0
$381$	−39.1755	−2.00702
$382$	0	0
$383$	−36.7963	−1.88020	−0.940101	−	0.340897i	$-0.889269\pi$
$383$	−36.7963	−1.88020	−0.940101	+	0.340897i	$0.889269\pi$
$384$	0	0
$385$	1.07380	0.0547262
$386$	0	0
$387$	60.4953	3.07515
$388$	0	0
$389$	32.1920	1.63220	0.816101	−	0.577910i	$-0.196131\pi$
$389$	32.1920	1.63220	0.816101	+	0.577910i	$0.196131\pi$
$390$	0	0
$391$	−9.01647	−0.455983
$392$	0	0
$393$	−22.5718	−1.13860
$394$	0	0
$395$	−1.67455	−0.0842556
$396$	0	0
$397$	1.92852	0.0967895	0.0483948	−	0.998828i	$-0.484589\pi$
$397$	1.92852	0.0967895	0.0483948	+	0.998828i	$0.484589\pi$
$398$	0	0
$399$	−94.2687	−4.71934
$400$	0	0
$401$	−25.7193	−1.28436	−0.642181	−	0.766553i	$-0.721970\pi$
$401$	−25.7193	−1.28436	−0.642181	+	0.766553i	$0.721970\pi$
$402$	0	0
$403$	−16.8963	−0.841664
$404$	0	0
$405$	−1.04792	−0.0520717
$406$	0	0
$407$	−7.38882	−0.366250
$408$	0	0
$409$	9.57634	0.473520	0.236760	−	0.971568i	$-0.423915\pi$
$409$	9.57634	0.473520	0.236760	+	0.971568i	$0.423915\pi$
$410$	0	0
$411$	−34.6747	−1.71038
$412$	0	0
$413$	−7.02273	−0.345566
$414$	0	0
$415$	−0.482891	−0.0237042
$416$	0	0
$417$	−27.7386	−1.35837
$418$	0	0
$419$	9.62793	0.470355	0.235178	−	0.971952i	$-0.424433\pi$
$419$	9.62793	0.470355	0.235178	+	0.971952i	$0.424433\pi$
$420$	0	0
$421$	−9.10676	−0.443836	−0.221918	−	0.975065i	$-0.571232\pi$
$421$	−9.10676	−0.443836	−0.221918	+	0.975065i	$0.571232\pi$
$422$	0	0
$423$	−16.7457	−0.814205
$424$	0	0
$425$	−18.5238	−0.898535
$426$	0	0
$427$	−50.6693	−2.45206
$428$	0	0
$429$	19.8364	0.957711
$430$	0	0
$431$	11.9993	0.577987	0.288993	−	0.957331i	$-0.406679\pi$
$431$	11.9993	0.577987	0.288993	+	0.957331i	$0.406679\pi$
$432$	0	0
$433$	−6.60599	−0.317464	−0.158732	−	0.987322i	$-0.550741\pi$
$433$	−6.60599	−0.317464	−0.158732	+	0.987322i	$0.550741\pi$
$434$	0	0
$435$	−1.67531	−0.0803249
$436$	0	0
$437$	14.6156	0.699157
$438$	0	0
$439$	−5.17318	−0.246902	−0.123451	−	0.992351i	$-0.539396\pi$
$439$	−5.17318	−0.246902	−0.123451	+	0.992351i	$0.539396\pi$
$440$	0	0
$441$	122.047	5.81175
$442$	0	0
$443$	11.4620	0.544574	0.272287	−	0.962216i	$-0.412220\pi$
$443$	11.4620	0.544574	0.272287	+	0.962216i	$0.412220\pi$
$444$	0	0
$445$	−1.08881	−0.0516143
$446$	0	0
$447$	−28.4938	−1.34771
$448$	0	0
$449$	−15.1475	−0.714854	−0.357427	−	0.933941i	$-0.616346\pi$
$449$	−15.1475	−0.714854	−0.357427	+	0.933941i	$0.616346\pi$
$450$	0	0
$451$	−21.6157	−1.01784
$452$	0	0
$453$	57.2180	2.68834
$454$	0	0
$455$	1.51085	0.0708297
$456$	0	0
$457$	10.6276	0.497137	0.248568	−	0.968614i	$-0.420040\pi$
$457$	10.6276	0.497137	0.248568	+	0.968614i	$0.420040\pi$
$458$	0	0
$459$	36.0622	1.68324
$460$	0	0
$461$	−21.8974	−1.01986	−0.509931	−	0.860215i	$-0.670329\pi$
$461$	−21.8974	−1.01986	−0.509931	+	0.860215i	$0.670329\pi$
$462$	0	0
$463$	19.5520	0.908661	0.454330	−	0.890833i	$-0.349879\pi$
$463$	19.5520	0.908661	0.454330	+	0.890833i	$0.349879\pi$
$464$	0	0
$465$	−1.62971	−0.0755759
$466$	0	0
$467$	32.7002	1.51319	0.756593	−	0.653886i	$-0.226862\pi$
$467$	32.7002	1.51319	0.756593	+	0.653886i	$0.226862\pi$
$468$	0	0
$469$	46.1423	2.13065
$470$	0	0
$471$	−46.8228	−2.15748
$472$	0	0
$473$	−21.0252	−0.966739
$474$	0	0
$475$	30.0267	1.37772
$476$	0	0
$477$	−18.3970	−0.842343
$478$	0	0
$479$	12.3238	0.563090	0.281545	−	0.959548i	$-0.409153\pi$
$479$	12.3238	0.563090	0.281545	+	0.959548i	$0.409153\pi$
$480$	0	0
$481$	−10.3961	−0.474022
$482$	0	0
$483$	−38.0618	−1.73187
$484$	0	0
$485$	−0.0174211	−0.000791053 0
$486$	0	0
$487$	−0.885160	−0.0401104	−0.0200552	−	0.999799i	$-0.506384\pi$
$487$	−0.885160	−0.0401104	−0.0200552	+	0.999799i	$0.506384\pi$
$488$	0	0
$489$	11.4637	0.518405
$490$	0	0
$491$	24.5474	1.10781	0.553903	−	0.832581i	$-0.313138\pi$
$491$	24.5474	1.10781	0.553903	+	0.832581i	$0.313138\pi$
$492$	0	0
$493$	21.2542	0.957240
$494$	0	0
$495$	1.28958	0.0579624
$496$	0	0
$497$	−15.9304	−0.714577
$498$	0	0
$499$	−35.8211	−1.60357	−0.801787	−	0.597610i	$-0.796117\pi$
$499$	−35.8211	−1.60357	−0.801787	+	0.597610i	$0.796117\pi$
$500$	0	0
$501$	−74.8028	−3.34194
$502$	0	0
$503$	4.38016	0.195302	0.0976508	−	0.995221i	$-0.468867\pi$
$503$	4.38016	0.195302	0.0976508	+	0.995221i	$0.468867\pi$
$504$	0	0
$505$	0.745968	0.0331951
$506$	0	0
$507$	−11.5269	−0.511929
$508$	0	0
$509$	−18.9696	−0.840814	−0.420407	−	0.907336i	$-0.638113\pi$
$509$	−18.9696	−0.840814	−0.420407	+	0.907336i	$0.638113\pi$
$510$	0	0
$511$	13.1563	0.582002
$512$	0	0
$513$	−58.4563	−2.58091
$514$	0	0
$515$	1.17910	0.0519572
$516$	0	0
$517$	5.81999	0.255963
$518$	0	0
$519$	−70.7316	−3.10477
$520$	0	0
$521$	7.41948	0.325053	0.162527	−	0.986704i	$-0.448036\pi$
$521$	7.41948	0.325053	0.162527	+	0.986704i	$0.448036\pi$
$522$	0	0
$523$	16.2675	0.711328	0.355664	−	0.934614i	$-0.384255\pi$
$523$	16.2675	0.711328	0.355664	+	0.934614i	$0.384255\pi$
$524$	0	0
$525$	−78.1956	−3.41274
$526$	0	0
$527$	20.6756	0.900645
$528$	0	0
$529$	−17.0988	−0.743428
$530$	0	0
$531$	−8.43393	−0.366001
$532$	0	0
$533$	−30.4134	−1.31735
$534$	0	0
$535$	0.700658	0.0302921
$536$	0	0
$537$	78.8321	3.40186
$538$	0	0
$539$	−42.4175	−1.82705
$540$	0	0
$541$	−20.0661	−0.862709	−0.431354	−	0.902183i	$-0.641964\pi$
$541$	−20.0661	−0.862709	−0.431354	+	0.902183i	$0.641964\pi$
$542$	0	0
$543$	−15.4086	−0.661246
$544$	0	0
$545$	1.53622	0.0658043
$546$	0	0
$547$	−7.30927	−0.312522	−0.156261	−	0.987716i	$-0.549944\pi$
$547$	−7.30927	−0.312522	−0.156261	+	0.987716i	$0.549944\pi$
$548$	0	0
$549$	−60.8511	−2.59706
$550$	0	0
$551$	−34.4527	−1.46773
$552$	0	0
$553$	89.6808	3.81362
$554$	0	0
$555$	−1.00274	−0.0425641
$556$	0	0
$557$	−19.1133	−0.809857	−0.404929	−	0.914348i	$-0.632704\pi$
$557$	−19.1133	−0.809857	−0.404929	+	0.914348i	$0.632704\pi$
$558$	0	0
$559$	−29.5826	−1.25121
$560$	0	0
$561$	−24.2734	−1.02482
$562$	0	0
$563$	37.4255	1.57729	0.788647	−	0.614846i	$-0.210782\pi$
$563$	37.4255	1.57729	0.788647	+	0.614846i	$0.210782\pi$
$564$	0	0
$565$	0.409209	0.0172156
$566$	0	0
$567$	56.1218	2.35690
$568$	0	0
$569$	0.373874	0.0156736	0.00783681	−	0.999969i	$-0.497505\pi$
$569$	0.373874	0.0156736	0.00783681	+	0.999969i	$0.497505\pi$
$570$	0	0
$571$	24.7804	1.03703	0.518514	−	0.855069i	$-0.326486\pi$
$571$	24.7804	1.03703	0.518514	+	0.855069i	$0.326486\pi$
$572$	0	0
$573$	−47.0629	−1.96608
$574$	0	0
$575$	12.1236	0.505587
$576$	0	0
$577$	−36.1038	−1.50302	−0.751510	−	0.659722i	$-0.770674\pi$
$577$	−36.1038	−1.50302	−0.751510	+	0.659722i	$0.770674\pi$
$578$	0	0
$579$	−10.2846	−0.427414
$580$	0	0
$581$	25.8613	1.07291
$582$	0	0
$583$	6.39391	0.264809
$584$	0	0
$585$	1.81445	0.0750183
$586$	0	0
$587$	25.1401	1.03764	0.518822	−	0.854882i	$-0.326371\pi$
$587$	25.1401	1.03764	0.518822	+	0.854882i	$0.326371\pi$
$588$	0	0
$589$	−33.5149	−1.38096
$590$	0	0
$591$	−29.3289	−1.20643
$592$	0	0
$593$	−8.15748	−0.334987	−0.167494	−	0.985873i	$-0.553567\pi$
$593$	−8.15748	−0.334987	−0.167494	+	0.985873i	$0.553567\pi$
$594$	0	0
$595$	−1.84880	−0.0757933
$596$	0	0
$597$	62.1874	2.54516
$598$	0	0
$599$	42.5313	1.73778	0.868890	−	0.495005i	$-0.164834\pi$
$599$	42.5313	1.73778	0.868890	+	0.495005i	$0.164834\pi$
$600$	0	0
$601$	17.2395	0.703213	0.351606	−	0.936148i	$-0.385636\pi$
$601$	17.2395	0.703213	0.351606	+	0.936148i	$0.385636\pi$
$602$	0	0
$603$	55.4144	2.25665
$604$	0	0
$605$	0.612650	0.0249078
$606$	0	0
$607$	18.2098	0.739113	0.369556	−	0.929208i	$-0.379510\pi$
$607$	18.2098	0.739113	0.369556	+	0.929208i	$0.379510\pi$
$608$	0	0
$609$	89.7216	3.63570
$610$	0	0
$611$	8.18877	0.331282
$612$	0	0
$613$	11.5918	0.468187	0.234094	−	0.972214i	$-0.424788\pi$
$613$	11.5918	0.468187	0.234094	+	0.972214i	$0.424788\pi$
$614$	0	0
$615$	−2.93348	−0.118289
$616$	0	0
$617$	−29.7234	−1.19662	−0.598309	−	0.801265i	$-0.704161\pi$
$617$	−29.7234	−1.19662	−0.598309	+	0.801265i	$0.704161\pi$
$618$	0	0
$619$	−26.4020	−1.06119	−0.530594	−	0.847626i	$-0.678031\pi$
$619$	−26.4020	−1.06119	−0.530594	+	0.847626i	$0.678031\pi$
$620$	0	0
$621$	−23.6022	−0.947125
$622$	0	0
$623$	58.3113	2.33619
$624$	0	0
$625$	24.8606	0.994423
$626$	0	0
$627$	39.3468	1.57136
$628$	0	0
$629$	12.7215	0.507241
$630$	0	0
$631$	33.6085	1.33793	0.668967	−	0.743292i	$-0.266737\pi$
$631$	33.6085	1.33793	0.668967	+	0.743292i	$0.266737\pi$
$632$	0	0
$633$	33.2952	1.32336
$634$	0	0
$635$	1.24542	0.0494229
$636$	0	0
$637$	−59.6816	−2.36467
$638$	0	0
$639$	−19.1316	−0.756833
$640$	0	0
$641$	−1.80600	−0.0713329	−0.0356664	−	0.999364i	$-0.511355\pi$
$641$	−1.80600	−0.0713329	−0.0356664	+	0.999364i	$0.511355\pi$
$642$	0	0
$643$	−16.6525	−0.656711	−0.328356	−	0.944554i	$-0.606494\pi$
$643$	−16.6525	−0.656711	−0.328356	+	0.944554i	$0.606494\pi$
$644$	0	0
$645$	−2.85335	−0.112350
$646$	0	0
$647$	31.7178	1.24695	0.623477	−	0.781842i	$-0.285719\pi$
$647$	31.7178	1.24695	0.623477	+	0.781842i	$0.285719\pi$
$648$	0	0
$649$	2.93122	0.115060
$650$	0	0
$651$	87.2795	3.42075
$652$	0	0
$653$	−15.7418	−0.616023	−0.308012	−	0.951383i	$-0.599664\pi$
$653$	−15.7418	−0.616023	−0.308012	+	0.951383i	$0.599664\pi$
$654$	0	0
$655$	0.717575	0.0280380
$656$	0	0
$657$	15.8001	0.616419
$658$	0	0
$659$	−10.6040	−0.413074	−0.206537	−	0.978439i	$-0.566219\pi$
$659$	−10.6040	−0.413074	−0.206537	+	0.978439i	$0.566219\pi$
$660$	0	0
$661$	15.0013	0.583483	0.291742	−	0.956497i	$-0.405765\pi$
$661$	15.0013	0.583483	0.291742	+	0.956497i	$0.405765\pi$
$662$	0	0
$663$	−34.1529	−1.32639
$664$	0	0
$665$	2.99687	0.116214
$666$	0	0
$667$	−13.9106	−0.538619
$668$	0	0
$669$	27.1445	1.04947
$670$	0	0
$671$	21.1489	0.816443
$672$	0	0
$673$	31.9220	1.23050	0.615252	−	0.788330i	$-0.289054\pi$
$673$	31.9220	1.23050	0.615252	+	0.788330i	$0.289054\pi$
$674$	0	0
$675$	−48.4893	−1.86635
$676$	0	0
$677$	−35.3400	−1.35823	−0.679114	−	0.734033i	$-0.737636\pi$
$677$	−35.3400	−1.35823	−0.679114	+	0.734033i	$0.737636\pi$
$678$	0	0
$679$	0.932993	0.0358050
$680$	0	0
$681$	−54.1856	−2.07640
$682$	0	0
$683$	−31.8650	−1.21928	−0.609640	−	0.792678i	$-0.708686\pi$
$683$	−31.8650	−1.21928	−0.609640	+	0.792678i	$0.708686\pi$
$684$	0	0
$685$	1.10233	0.0421180
$686$	0	0
$687$	64.9647	2.47856
$688$	0	0
$689$	8.99627	0.342731
$690$	0	0
$691$	47.3986	1.80313	0.901563	−	0.432647i	$-0.142420\pi$
$691$	47.3986	1.80313	0.901563	+	0.432647i	$0.142420\pi$
$692$	0	0
$693$	−69.0639	−2.62352
$694$	0	0
$695$	0.881831	0.0334497
$696$	0	0
$697$	37.2163	1.40967
$698$	0	0
$699$	84.0228	3.17803
$700$	0	0
$701$	48.8273	1.84418	0.922091	−	0.386972i	$-0.126479\pi$
$701$	48.8273	1.84418	0.922091	+	0.386972i	$0.126479\pi$
$702$	0	0
$703$	−20.6214	−0.777750
$704$	0	0
$705$	0.789837	0.0297470
$706$	0	0
$707$	−39.9505	−1.50249
$708$	0	0
$709$	−33.7424	−1.26722	−0.633612	−	0.773651i	$-0.718428\pi$
$709$	−33.7424	−1.26722	−0.633612	+	0.773651i	$0.718428\pi$
$710$	0	0
$711$	107.702	4.03913
$712$	0	0
$713$	−13.5319	−0.506775
$714$	0	0
$715$	−0.630614	−0.0235836
$716$	0	0
$717$	15.5019	0.578929
$718$	0	0
$719$	−18.7893	−0.700721	−0.350361	−	0.936615i	$-0.613941\pi$
$719$	−18.7893	−0.700721	−0.350361	+	0.936615i	$0.613941\pi$
$720$	0	0
$721$	−63.1469	−2.35171
$722$	0	0
$723$	−10.3151	−0.383621
$724$	0	0
$725$	−28.5784	−1.06137
$726$	0	0
$727$	36.3738	1.34903	0.674514	−	0.738262i	$-0.264353\pi$
$727$	36.3738	1.34903	0.674514	+	0.738262i	$0.264353\pi$
$728$	0	0
$729$	−21.0236	−0.778652
$730$	0	0
$731$	36.1996	1.33889
$732$	0	0
$733$	−10.7511	−0.397100	−0.198550	−	0.980091i	$-0.563623\pi$
$733$	−10.7511	−0.397100	−0.198550	+	0.980091i	$0.563623\pi$
$734$	0	0
$735$	−5.75651	−0.212332
$736$	0	0
$737$	−19.2593	−0.709427
$738$	0	0
$739$	30.5561	1.12402	0.562011	−	0.827130i	$-0.310028\pi$
$739$	30.5561	1.12402	0.562011	+	0.827130i	$0.310028\pi$
$740$	0	0
$741$	55.3612	2.03375
$742$	0	0
$743$	−39.2948	−1.44159	−0.720793	−	0.693151i	$-0.756222\pi$
$743$	−39.2948	−1.44159	−0.720793	+	0.693151i	$0.756222\pi$
$744$	0	0
$745$	0.905838	0.0331873
$746$	0	0
$747$	31.0581	1.13636
$748$	0	0
$749$	−37.5239	−1.37109
$750$	0	0
$751$	6.49150	0.236878	0.118439	−	0.992961i	$-0.462211\pi$
$751$	6.49150	0.236878	0.118439	+	0.992961i	$0.462211\pi$
$752$	0	0
$753$	3.03361	0.110551
$754$	0	0
$755$	−1.81900	−0.0662003
$756$	0	0
$757$	35.0192	1.27280	0.636398	−	0.771361i	$-0.280424\pi$
$757$	35.0192	1.27280	0.636398	+	0.771361i	$0.280424\pi$
$758$	0	0
$759$	15.8866	0.576648
$760$	0	0
$761$	47.1671	1.70981	0.854904	−	0.518787i	$-0.173616\pi$
$761$	47.1671	1.70981	0.854904	+	0.518787i	$0.173616\pi$
$762$	0	0
$763$	−82.2726	−2.97847
$764$	0	0
$765$	−2.22031	−0.0802753
$766$	0	0
$767$	4.12424	0.148918
$768$	0	0
$769$	−26.0705	−0.940127	−0.470063	−	0.882633i	$-0.655769\pi$
$769$	−26.0705	−0.940127	−0.470063	+	0.882633i	$0.655769\pi$
$770$	0	0
$771$	−64.2509	−2.31394
$772$	0	0
$773$	4.18846	0.150649	0.0753243	−	0.997159i	$-0.476001\pi$
$773$	4.18846	0.150649	0.0753243	+	0.997159i	$0.476001\pi$
$774$	0	0
$775$	−27.8005	−0.998623
$776$	0	0
$777$	53.7022	1.92656
$778$	0	0
$779$	−60.3269	−2.16144
$780$	0	0
$781$	6.64920	0.237927
$782$	0	0
$783$	55.6366	1.98829
$784$	0	0
$785$	1.48853	0.0531279
$786$	0	0
$787$	24.5656	0.875670	0.437835	−	0.899055i	$-0.355745\pi$
$787$	24.5656	0.875670	0.437835	+	0.899055i	$0.355745\pi$
$788$	0	0
$789$	−94.4786	−3.36353
$790$	0	0
$791$	−21.9153	−0.779219
$792$	0	0
$793$	29.7566	1.05669
$794$	0	0
$795$	0.867724	0.0307750
$796$	0	0
$797$	−0.725326	−0.0256923	−0.0128462	−	0.999917i	$-0.504089\pi$
$797$	−0.725326	−0.0256923	−0.0128462	+	0.999917i	$0.504089\pi$
$798$	0	0
$799$	−10.0204	−0.354497
$800$	0	0
$801$	70.0287	2.47434
$802$	0	0
$803$	−5.49132	−0.193785
$804$	0	0
$805$	1.21001	0.0426474
$806$	0	0
$807$	−33.1338	−1.16637
$808$	0	0
$809$	−44.2386	−1.55535	−0.777674	−	0.628668i	$-0.783600\pi$
$809$	−44.2386	−1.55535	−0.777674	+	0.628668i	$0.783600\pi$
$810$	0	0
$811$	−44.0261	−1.54597	−0.772983	−	0.634427i	$-0.781236\pi$
$811$	−44.0261	−1.54597	−0.772983	+	0.634427i	$0.781236\pi$
$812$	0	0
$813$	7.39505	0.259356
$814$	0	0
$815$	−0.364439	−0.0127657
$816$	0	0
$817$	−58.6790	−2.05292
$818$	0	0
$819$	−97.1733	−3.39551
$820$	0	0
$821$	35.4513	1.23726	0.618630	−	0.785683i	$-0.287688\pi$
$821$	35.4513	1.23726	0.618630	+	0.785683i	$0.287688\pi$
$822$	0	0
$823$	5.69709	0.198588	0.0992941	−	0.995058i	$-0.468342\pi$
$823$	5.69709	0.198588	0.0992941	+	0.995058i	$0.468342\pi$
$824$	0	0
$825$	32.6381	1.13631
$826$	0	0
$827$	33.7475	1.17352	0.586758	−	0.809763i	$-0.300404\pi$
$827$	33.7475	1.17352	0.586758	+	0.809763i	$0.300404\pi$
$828$	0	0
$829$	−10.7319	−0.372735	−0.186368	−	0.982480i	$-0.559672\pi$
$829$	−10.7319	−0.372735	−0.186368	+	0.982480i	$0.559672\pi$
$830$	0	0
$831$	17.4545	0.605489
$832$	0	0
$833$	73.0312	2.53038
$834$	0	0
$835$	2.37803	0.0822953
$836$	0	0
$837$	54.1222	1.87074
$838$	0	0
$839$	12.0332	0.415431	0.207716	−	0.978189i	$-0.433397\pi$
$839$	12.0332	0.415431	0.207716	+	0.978189i	$0.433397\pi$
$840$	0	0
$841$	3.79082	0.130718
$842$	0	0
$843$	−37.3677	−1.28701
$844$	0	0
$845$	0.366449	0.0126062
$846$	0	0
$847$	−32.8107	−1.12739
$848$	0	0
$849$	10.4799	0.359668
$850$	0	0
$851$	−8.32607	−0.285414
$852$	0	0
$853$	23.6309	0.809106	0.404553	−	0.914514i	$-0.367427\pi$
$853$	23.6309	0.809106	0.404553	+	0.914514i	$0.367427\pi$
$854$	0	0
$855$	3.59908	0.123086
$856$	0	0
$857$	30.3708	1.03745	0.518724	−	0.854942i	$-0.326407\pi$
$857$	30.3708	1.03745	0.518724	+	0.854942i	$0.326407\pi$
$858$	0	0
$859$	18.1338	0.618717	0.309359	−	0.950945i	$-0.399886\pi$
$859$	18.1338	0.618717	0.309359	+	0.950945i	$0.399886\pi$
$860$	0	0
$861$	157.103	5.35407
$862$	0	0
$863$	22.6191	0.769962	0.384981	−	0.922925i	$-0.374208\pi$
$863$	22.6191	0.769962	0.384981	+	0.922925i	$0.374208\pi$
$864$	0	0
$865$	2.24861	0.0764550
$866$	0	0
$867$	−9.77912	−0.332116
$868$	0	0
$869$	−37.4319	−1.26979
$870$	0	0
$871$	−27.0980	−0.918181
$872$	0	0
$873$	1.12048	0.0379223
$874$	0	0
$875$	4.97642	0.168234
$876$	0	0
$877$	10.3353	0.349000	0.174500	−	0.984657i	$-0.444169\pi$
$877$	10.3353	0.349000	0.174500	+	0.984657i	$0.444169\pi$
$878$	0	0
$879$	−48.2997	−1.62911
$880$	0	0
$881$	−23.8773	−0.804448	−0.402224	−	0.915541i	$-0.631763\pi$
$881$	−23.8773	−0.804448	−0.402224	+	0.915541i	$0.631763\pi$
$882$	0	0
$883$	55.0193	1.85155	0.925773	−	0.378080i	$-0.123416\pi$
$883$	55.0193	1.85155	0.925773	+	0.378080i	$0.123416\pi$
$884$	0	0
$885$	0.397799	0.0133719
$886$	0	0
$887$	−4.40585	−0.147934	−0.0739670	−	0.997261i	$-0.523566\pi$
$887$	−4.40585	−0.147934	−0.0739670	+	0.997261i	$0.523566\pi$
$888$	0	0
$889$	−66.6987	−2.23700
$890$	0	0
$891$	−23.4247	−0.784757
$892$	0	0
$893$	16.2430	0.543550
$894$	0	0
$895$	−2.50613	−0.0837707
$896$	0	0
$897$	22.3526	0.746331
$898$	0	0
$899$	31.8983	1.06387
$900$	0	0
$901$	−11.0086	−0.366748
$902$	0	0
$903$	152.812	5.08526
$904$	0	0
$905$	0.489850	0.0162832
$906$	0	0
$907$	17.9845	0.597167	0.298584	−	0.954384i	$-0.403486\pi$
$907$	17.9845	0.597167	0.298584	+	0.954384i	$0.403486\pi$
$908$	0	0
$909$	−47.9784	−1.59134
$910$	0	0
$911$	−23.8121	−0.788930	−0.394465	−	0.918911i	$-0.629070\pi$
$911$	−23.8121	−0.788930	−0.394465	+	0.918911i	$0.629070\pi$
$912$	0	0
$913$	−10.7943	−0.357238
$914$	0	0
$915$	2.87013	0.0948836
$916$	0	0
$917$	−38.4299	−1.26907
$918$	0	0
$919$	−6.72074	−0.221697	−0.110848	−	0.993837i	$-0.535357\pi$
$919$	−6.72074	−0.221697	−0.110848	+	0.993837i	$0.535357\pi$
$920$	0	0
$921$	−6.54804	−0.215765
$922$	0	0
$923$	9.35546	0.307939
$924$	0	0
$925$	−17.1054	−0.562422
$926$	0	0
$927$	−75.8360	−2.49078
$928$	0	0
$929$	−14.4334	−0.473544	−0.236772	−	0.971565i	$-0.576089\pi$
$929$	−14.4334	−0.473544	−0.236772	+	0.971565i	$0.576089\pi$
$930$	0	0
$931$	−118.382	−3.87983
$932$	0	0
$933$	−66.6245	−2.18119
$934$	0	0
$935$	0.771670	0.0252363
$936$	0	0
$937$	−39.3358	−1.28504	−0.642522	−	0.766267i	$-0.722112\pi$
$937$	−39.3358	−1.28504	−0.642522	+	0.766267i	$0.722112\pi$
$938$	0	0
$939$	73.9689	2.41388
$940$	0	0
$941$	40.4272	1.31789	0.658944	−	0.752192i	$-0.271003\pi$
$941$	40.4272	1.31789	0.658944	+	0.752192i	$0.271003\pi$
$942$	0	0
$943$	−24.3575	−0.793191
$944$	0	0
$945$	−4.83956	−0.157431
$946$	0	0
$947$	42.1858	1.37085	0.685427	−	0.728142i	$-0.259616\pi$
$947$	42.1858	1.37085	0.685427	+	0.728142i	$0.259616\pi$
$948$	0	0
$949$	−7.72633	−0.250807
$950$	0	0
$951$	−14.2887	−0.463342
$952$	0	0
$953$	19.9525	0.646325	0.323162	−	0.946344i	$-0.395254\pi$
$953$	19.9525	0.646325	0.323162	+	0.946344i	$0.395254\pi$
$954$	0	0
$955$	1.49616	0.0484147
$956$	0	0
$957$	−37.4489	−1.21055
$958$	0	0
$959$	−59.0358	−1.90637
$960$	0	0
$961$	0.0300371	0.000968940 0
$962$	0	0
$963$	−45.0642	−1.45217
$964$	0	0
$965$	0.326956	0.0105251
$966$	0	0
$967$	27.8133	0.894414	0.447207	−	0.894430i	$-0.352419\pi$
$967$	27.8133	0.894414	0.447207	+	0.894430i	$0.352419\pi$
$968$	0	0
$969$	−67.7444	−2.17626
$970$	0	0
$971$	−21.6785	−0.695695	−0.347848	−	0.937551i	$-0.613087\pi$
$971$	−21.6785	−0.695695	−0.347848	+	0.937551i	$0.613087\pi$
$972$	0	0
$973$	−47.2267	−1.51402
$974$	0	0
$975$	45.9220	1.47068
$976$	0	0
$977$	14.2725	0.456617	0.228309	−	0.973589i	$-0.426681\pi$
$977$	14.2725	0.456617	0.228309	+	0.973589i	$0.426681\pi$
$978$	0	0
$979$	−24.3385	−0.777863
$980$	0	0
$981$	−98.8049	−3.15460
$982$	0	0
$983$	−8.41810	−0.268496	−0.134248	−	0.990948i	$-0.542862\pi$
$983$	−8.41810	−0.268496	−0.134248	+	0.990948i	$0.542862\pi$
$984$	0	0
$985$	0.932388	0.0297083
$986$	0	0
$987$	−42.2999	−1.34642
$988$	0	0
$989$	−23.6922	−0.753367
$990$	0	0
$991$	46.7987	1.48661	0.743305	−	0.668953i	$-0.233257\pi$
$991$	46.7987	1.48661	0.743305	+	0.668953i	$0.233257\pi$
$992$	0	0
$993$	51.1265	1.62245
$994$	0	0
$995$	−1.97698	−0.0626745
$996$	0	0
$997$	−59.3196	−1.87867	−0.939336	−	0.342999i	$-0.888557\pi$
$997$	−59.3196	−1.87867	−0.939336	+	0.342999i	$0.888557\pi$
$998$	0	0
$999$	33.3009	1.05359

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.l.1.18		19
4.3	odd	2		2008.2.a.c.1.2	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.c.1.2	✓	19	4.3	odd	2
4016.2.a.l.1.18		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)

\(f(q)\)	\(=\)	\(q+3.03361 q^{3} -0.0964405 q^{5} +5.16490 q^{7} +6.20277 q^{9} +O(q^{10})\)	\(q+3.03361 q^{3} -0.0964405 q^{5} +5.16490 q^{7} +6.20277 q^{9} -2.15578 q^{11} -3.03319 q^{13} -0.292563 q^{15} +3.71166 q^{17} -6.01654 q^{19} +15.6683 q^{21} -2.42923 q^{23} -4.99070 q^{25} +9.71593 q^{27} +5.72633 q^{29} +5.57046 q^{31} -6.53978 q^{33} -0.498106 q^{35} +3.42745 q^{37} -9.20151 q^{39} +10.0269 q^{41} +9.75295 q^{43} -0.598198 q^{45} -2.69972 q^{47} +19.6762 q^{49} +11.2597 q^{51} -2.96594 q^{53} +0.207904 q^{55} -18.2518 q^{57} -1.35970 q^{59} -9.81032 q^{61} +32.0367 q^{63} +0.292523 q^{65} +8.93383 q^{67} -7.36933 q^{69} -3.08436 q^{71} +2.54726 q^{73} -15.1398 q^{75} -11.1344 q^{77} +17.3635 q^{79} +10.8660 q^{81} +5.00713 q^{83} -0.357954 q^{85} +17.3714 q^{87} +11.2899 q^{89} -15.6661 q^{91} +16.8986 q^{93} +0.580238 q^{95} +0.180641 q^{97} -13.3718 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) \) 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9} + 15 q^{11} - 8 q^{13} + 17 q^{15} - 4 q^{17} + 14 q^{19} - 9 q^{21} + 28 q^{23} + 25 q^{25} + 21 q^{27} - 13 q^{29} + 20 q^{31} - 6 q^{33} + 32 q^{35} - 16 q^{37} + 27 q^{39} + 2 q^{41} + 28 q^{43} - 29 q^{45} + 37 q^{47} + 36 q^{49} + 35 q^{51} - 37 q^{53} + 24 q^{55} - 11 q^{57} + 32 q^{59} - 7 q^{61} + 45 q^{63} + q^{65} + 45 q^{67} - 12 q^{69} + 49 q^{71} + 16 q^{73} + 35 q^{75} - 40 q^{77} + 33 q^{79} + 15 q^{81} + 43 q^{83} - 28 q^{85} + 48 q^{87} + 3 q^{89} + 56 q^{91} - 48 q^{93} + 43 q^{95} + 8 q^{97} + 74 q^{99}+O(q^{100}) \) 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9 + 15 * q^11 - 8 * q^13 + 17 * q^15 - 4 * q^17 + 14 * q^19 - 9 * q^21 + 28 * q^23 + 25 * q^25 + 21 * q^27 - 13 * q^29 + 20 * q^31 - 6 * q^33 + 32 * q^35 - 16 * q^37 + 27 * q^39 + 2 * q^41 + 28 * q^43 - 29 * q^45 + 37 * q^47 + 36 * q^49 + 35 * q^51 - 37 * q^53 + 24 * q^55 - 11 * q^57 + 32 * q^59 - 7 * q^61 + 45 * q^63 + q^65 + 45 * q^67 - 12 * q^69 + 49 * q^71 + 16 * q^73 + 35 * q^75 - 40 * q^77 + 33 * q^79 + 15 * q^81 + 43 * q^83 - 28 * q^85 + 48 * q^87 + 3 * q^89 + 56 * q^91 - 48 * q^93 + 43 * q^95 + 8 * q^97 + 74 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more