LMFDB - Embedded newform 8044.2.a.a.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8044,2,Mod(1,8044)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8044, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8044 = 2^{2} \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8044.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:	$64.2316633859$
Analytic rank:	$1$
Dimension:	$80$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	8044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.57058 q^{3} +3.02835 q^{5} -3.19846 q^{7} +3.60789 q^{9} +O(q^{10})$	$q-2.57058 q^{3} +3.02835 q^{5} -3.19846 q^{7} +3.60789 q^{9} +4.21049 q^{11} +6.16044 q^{13} -7.78461 q^{15} -3.71804 q^{17} -7.54828 q^{19} +8.22190 q^{21} +0.211742 q^{23} +4.17088 q^{25} -1.56264 q^{27} -0.780170 q^{29} +5.05467 q^{31} -10.8234 q^{33} -9.68604 q^{35} +2.32745 q^{37} -15.8359 q^{39} -2.17463 q^{41} -7.13959 q^{43} +10.9260 q^{45} -12.7521 q^{47} +3.23014 q^{49} +9.55752 q^{51} +9.46724 q^{53} +12.7508 q^{55} +19.4035 q^{57} -8.74137 q^{59} -7.38093 q^{61} -11.5397 q^{63} +18.6560 q^{65} +4.57356 q^{67} -0.544300 q^{69} -4.65370 q^{71} +1.99217 q^{73} -10.7216 q^{75} -13.4671 q^{77} +4.78691 q^{79} -6.80678 q^{81} -12.2806 q^{83} -11.2595 q^{85} +2.00549 q^{87} -9.74472 q^{89} -19.7039 q^{91} -12.9935 q^{93} -22.8588 q^{95} -12.8065 q^{97} +15.1910 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) $ 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9	$ 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) $ 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9 - 34 * q^11 - q^13 - 24 * q^15 - 35 * q^17 - 31 * q^19 - 3 * q^21 - 43 * q^23 + 58 * q^25 - 49 * q^27 - 5 * q^29 - 56 * q^31 - 23 * q^33 - 72 * q^35 - 11 * q^37 - 74 * q^39 - 81 * q^41 - 34 * q^43 - 14 * q^45 - 64 * q^47 + 40 * q^49 - 59 * q^51 + 3 * q^53 - 53 * q^55 - 34 * q^57 - 116 * q^59 - 13 * q^61 - 61 * q^63 - 55 * q^65 - 22 * q^67 - 10 * q^69 - 86 * q^71 - 32 * q^73 - 85 * q^75 + 4 * q^77 - 88 * q^79 + 12 * q^81 - 83 * q^83 - 2 * q^85 - 87 * q^87 - 72 * q^89 - 49 * q^91 - 102 * q^95 - 34 * q^97 - 103 * 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
$3$	−2.57058	−1.48413	−0.742063	−	0.670330i	$-0.766153\pi$
$3$	−2.57058	−1.48413	−0.742063	+	0.670330i	$0.766153\pi$
$4$	0	0
$5$	3.02835	1.35432	0.677159	−	0.735837i	$-0.263211\pi$
$5$	3.02835	1.35432	0.677159	+	0.735837i	$0.263211\pi$
$6$	0	0
$7$	−3.19846	−1.20890	−0.604452	−	0.796642i	$-0.706608\pi$
$7$	−3.19846	−1.20890	−0.604452	+	0.796642i	$0.706608\pi$
$8$	0	0
$9$	3.60789	1.20263
$10$	0	0
$11$	4.21049	1.26951	0.634756	−	0.772713i	$-0.281101\pi$
$11$	4.21049	1.26951	0.634756	+	0.772713i	$0.281101\pi$
$12$	0	0
$13$	6.16044	1.70860	0.854300	−	0.519780i	$-0.173986\pi$
$13$	6.16044	1.70860	0.854300	+	0.519780i	$0.173986\pi$
$14$	0	0
$15$	−7.78461	−2.00998
$16$	0	0
$17$	−3.71804	−0.901756	−0.450878	−	0.892586i	$-0.648889\pi$
$17$	−3.71804	−0.901756	−0.450878	+	0.892586i	$0.648889\pi$
$18$	0	0
$19$	−7.54828	−1.73169	−0.865847	−	0.500309i	$-0.833220\pi$
$19$	−7.54828	−1.73169	−0.865847	+	0.500309i	$0.833220\pi$
$20$	0	0
$21$	8.22190	1.79417
$22$	0	0
$23$	0.211742	0.0441513	0.0220756	−	0.999756i	$-0.492973\pi$
$23$	0.211742	0.0441513	0.0220756	+	0.999756i	$0.492973\pi$
$24$	0	0
$25$	4.17088	0.834175
$26$	0	0
$27$	−1.56264	−0.300731
$28$	0	0
$29$	−0.780170	−0.144874	−0.0724369	−	0.997373i	$-0.523078\pi$
$29$	−0.780170	−0.144874	−0.0724369	+	0.997373i	$0.523078\pi$
$30$	0	0
$31$	5.05467	0.907846	0.453923	−	0.891041i	$-0.350024\pi$
$31$	5.05467	0.907846	0.453923	+	0.891041i	$0.350024\pi$
$32$	0	0
$33$	−10.8234	−1.88412
$34$	0	0
$35$	−9.68604	−1.63724
$36$	0	0
$37$	2.32745	0.382630	0.191315	−	0.981529i	$-0.438725\pi$
$37$	2.32745	0.382630	0.191315	+	0.981529i	$0.438725\pi$
$38$	0	0
$39$	−15.8359	−2.53578
$40$	0	0
$41$	−2.17463	−0.339620	−0.169810	−	0.985477i	$-0.554315\pi$
$41$	−2.17463	−0.339620	−0.169810	+	0.985477i	$0.554315\pi$
$42$	0	0
$43$	−7.13959	−1.08878	−0.544389	−	0.838833i	$-0.683238\pi$
$43$	−7.13959	−1.08878	−0.544389	+	0.838833i	$0.683238\pi$
$44$	0	0
$45$	10.9260	1.62874
$46$	0	0
$47$	−12.7521	−1.86008	−0.930042	−	0.367452i	$-0.880230\pi$
$47$	−12.7521	−1.86008	−0.930042	+	0.367452i	$0.880230\pi$
$48$	0	0
$49$	3.23014	0.461449
$50$	0	0
$51$	9.55752	1.33832
$52$	0	0
$53$	9.46724	1.30043	0.650213	−	0.759752i	$-0.274680\pi$
$53$	9.46724	1.30043	0.650213	+	0.759752i	$0.274680\pi$
$54$	0	0
$55$	12.7508	1.71932
$56$	0	0
$57$	19.4035	2.57005
$58$	0	0
$59$	−8.74137	−1.13803	−0.569014	−	0.822328i	$-0.692675\pi$
$59$	−8.74137	−1.13803	−0.569014	+	0.822328i	$0.692675\pi$
$60$	0	0
$61$	−7.38093	−0.945032	−0.472516	−	0.881322i	$-0.656654\pi$
$61$	−7.38093	−0.945032	−0.472516	+	0.881322i	$0.656654\pi$
$62$	0	0
$63$	−11.5397	−1.45387
$64$	0	0
$65$	18.6560	2.31399
$66$	0	0
$67$	4.57356	0.558750	0.279375	−	0.960182i	$-0.409873\pi$
$67$	4.57356	0.558750	0.279375	+	0.960182i	$0.409873\pi$
$68$	0	0
$69$	−0.544300	−0.0655261
$70$	0	0
$71$	−4.65370	−0.552292	−0.276146	−	0.961116i	$-0.589057\pi$
$71$	−4.65370	−0.552292	−0.276146	+	0.961116i	$0.589057\pi$
$72$	0	0
$73$	1.99217	0.233166	0.116583	−	0.993181i	$-0.462806\pi$
$73$	1.99217	0.233166	0.116583	+	0.993181i	$0.462806\pi$
$74$	0	0
$75$	−10.7216	−1.23802
$76$	0	0
$77$	−13.4671	−1.53472
$78$	0	0
$79$	4.78691	0.538569	0.269285	−	0.963061i	$-0.413213\pi$
$79$	4.78691	0.538569	0.269285	+	0.963061i	$0.413213\pi$
$80$	0	0
$81$	−6.80678	−0.756309
$82$	0	0
$83$	−12.2806	−1.34798	−0.673988	−	0.738742i	$-0.735420\pi$
$83$	−12.2806	−1.34798	−0.673988	+	0.738742i	$0.735420\pi$
$84$	0	0
$85$	−11.2595	−1.22126
$86$	0	0
$87$	2.00549	0.215011
$88$	0	0
$89$	−9.74472	−1.03294	−0.516469	−	0.856306i	$-0.672754\pi$
$89$	−9.74472	−1.03294	−0.516469	+	0.856306i	$0.672754\pi$
$90$	0	0
$91$	−19.7039	−2.06553
$92$	0	0
$93$	−12.9935	−1.34736
$94$	0	0
$95$	−22.8588	−2.34526
$96$	0	0
$97$	−12.8065	−1.30031	−0.650154	−	0.759802i	$-0.725296\pi$
$97$	−12.8065	−1.30031	−0.650154	+	0.759802i	$0.725296\pi$
$98$	0	0
$99$	15.1910	1.52675
$100$	0	0
$101$	−6.29609	−0.626485	−0.313242	−	0.949673i	$-0.601415\pi$
$101$	−6.29609	−0.626485	−0.313242	+	0.949673i	$0.601415\pi$
$102$	0	0
$103$	9.34414	0.920706	0.460353	−	0.887736i	$-0.347723\pi$
$103$	9.34414	0.920706	0.460353	+	0.887736i	$0.347723\pi$
$104$	0	0
$105$	24.8988	2.42987
$106$	0	0
$107$	9.04015	0.873944	0.436972	−	0.899475i	$-0.356051\pi$
$107$	9.04015	0.873944	0.436972	+	0.899475i	$0.356051\pi$
$108$	0	0
$109$	15.3702	1.47220	0.736099	−	0.676874i	$-0.236666\pi$
$109$	15.3702	1.47220	0.736099	+	0.676874i	$0.236666\pi$
$110$	0	0
$111$	−5.98289	−0.567871
$112$	0	0
$113$	6.36411	0.598685	0.299343	−	0.954146i	$-0.403233\pi$
$113$	6.36411	0.598685	0.299343	+	0.954146i	$0.403233\pi$
$114$	0	0
$115$	0.641228	0.0597948
$116$	0	0
$117$	22.2262	2.05482
$118$	0	0
$119$	11.8920	1.09014
$120$	0	0
$121$	6.72824	0.611659
$122$	0	0
$123$	5.59007	0.504040
$124$	0	0
$125$	−2.51087	−0.224579
$126$	0	0
$127$	4.74431	0.420990	0.210495	−	0.977595i	$-0.432492\pi$
$127$	4.74431	0.420990	0.210495	+	0.977595i	$0.432492\pi$
$128$	0	0
$129$	18.3529	1.61588
$130$	0	0
$131$	−0.447243	−0.0390758	−0.0195379	−	0.999809i	$-0.506220\pi$
$131$	−0.447243	−0.0390758	−0.0195379	+	0.999809i	$0.506220\pi$
$132$	0	0
$133$	24.1429	2.09345
$134$	0	0
$135$	−4.73222	−0.407285
$136$	0	0
$137$	13.9188	1.18917	0.594584	−	0.804034i	$-0.297317\pi$
$137$	13.9188	1.18917	0.594584	+	0.804034i	$0.297317\pi$
$138$	0	0
$139$	−5.54175	−0.470045	−0.235023	−	0.971990i	$-0.575516\pi$
$139$	−5.54175	−0.470045	−0.235023	+	0.971990i	$0.575516\pi$
$140$	0	0
$141$	32.7803	2.76060
$142$	0	0
$143$	25.9385	2.16909
$144$	0	0
$145$	−2.36262	−0.196205
$146$	0	0
$147$	−8.30335	−0.684848
$148$	0	0
$149$	6.09897	0.499647	0.249824	−	0.968291i	$-0.419627\pi$
$149$	6.09897	0.499647	0.249824	+	0.968291i	$0.419627\pi$
$150$	0	0
$151$	−3.26280	−0.265523	−0.132761	−	0.991148i	$-0.542384\pi$
$151$	−3.26280	−0.265523	−0.132761	+	0.991148i	$0.542384\pi$
$152$	0	0
$153$	−13.4143	−1.08448
$154$	0	0
$155$	15.3073	1.22951
$156$	0	0
$157$	−19.8229	−1.58204	−0.791022	−	0.611788i	$-0.790450\pi$
$157$	−19.8229	−1.58204	−0.791022	+	0.611788i	$0.790450\pi$
$158$	0	0
$159$	−24.3363	−1.93000
$160$	0	0
$161$	−0.677248	−0.0533747
$162$	0	0
$163$	6.10320	0.478040	0.239020	−	0.971015i	$-0.423174\pi$
$163$	6.10320	0.478040	0.239020	+	0.971015i	$0.423174\pi$
$164$	0	0
$165$	−32.7770	−2.55169
$166$	0	0
$167$	11.9698	0.926250	0.463125	−	0.886293i	$-0.346728\pi$
$167$	11.9698	0.926250	0.463125	+	0.886293i	$0.346728\pi$
$168$	0	0
$169$	24.9511	1.91931
$170$	0	0
$171$	−27.2334	−2.08259
$172$	0	0
$173$	2.79062	0.212167	0.106084	−	0.994357i	$-0.466169\pi$
$173$	2.79062	0.212167	0.106084	+	0.994357i	$0.466169\pi$
$174$	0	0
$175$	−13.3404	−1.00844
$176$	0	0
$177$	22.4704	1.68898
$178$	0	0
$179$	13.7433	1.02722	0.513611	−	0.858023i	$-0.328307\pi$
$179$	13.7433	1.02722	0.513611	+	0.858023i	$0.328307\pi$
$180$	0	0
$181$	−17.7090	−1.31630	−0.658148	−	0.752888i	$-0.728660\pi$
$181$	−17.7090	−1.31630	−0.658148	+	0.752888i	$0.728660\pi$
$182$	0	0
$183$	18.9733	1.40255
$184$	0	0
$185$	7.04831	0.518202
$186$	0	0
$187$	−15.6548	−1.14479
$188$	0	0
$189$	4.99805	0.363555
$190$	0	0
$191$	−23.9665	−1.73415	−0.867076	−	0.498176i	$-0.834003\pi$
$191$	−23.9665	−1.73415	−0.867076	+	0.498176i	$0.834003\pi$
$192$	0	0
$193$	−5.59733	−0.402905	−0.201452	−	0.979498i	$-0.564566\pi$
$193$	−5.59733	−0.402905	−0.201452	+	0.979498i	$0.564566\pi$
$194$	0	0
$195$	−47.9567	−3.43425
$196$	0	0
$197$	−21.6987	−1.54597	−0.772983	−	0.634427i	$-0.781236\pi$
$197$	−21.6987	−1.54597	−0.772983	+	0.634427i	$0.781236\pi$
$198$	0	0
$199$	14.7068	1.04253	0.521267	−	0.853394i	$-0.325460\pi$
$199$	14.7068	1.04253	0.521267	+	0.853394i	$0.325460\pi$
$200$	0	0
$201$	−11.7567	−0.829255
$202$	0	0
$203$	2.49534	0.175139
$204$	0	0
$205$	−6.58554	−0.459954
$206$	0	0
$207$	0.763943	0.0530977
$208$	0	0
$209$	−31.7820	−2.19840
$210$	0	0
$211$	−0.0959008	−0.00660208	−0.00330104	−	0.999995i	$-0.501051\pi$
$211$	−0.0959008	−0.00660208	−0.00330104	+	0.999995i	$0.501051\pi$
$212$	0	0
$213$	11.9627	0.819672
$214$	0	0
$215$	−21.6211	−1.47455
$216$	0	0
$217$	−16.1672	−1.09750
$218$	0	0
$219$	−5.12103	−0.346047
$220$	0	0
$221$	−22.9047	−1.54074
$222$	0	0
$223$	8.29701	0.555609	0.277804	−	0.960638i	$-0.410393\pi$
$223$	8.29701	0.555609	0.277804	+	0.960638i	$0.410393\pi$
$224$	0	0
$225$	15.0481	1.00321
$226$	0	0
$227$	6.50808	0.431956	0.215978	−	0.976398i	$-0.430706\pi$
$227$	6.50808	0.431956	0.215978	+	0.976398i	$0.430706\pi$
$228$	0	0
$229$	−26.1523	−1.72819	−0.864097	−	0.503325i	$-0.832110\pi$
$229$	−26.1523	−1.72819	−0.864097	+	0.503325i	$0.832110\pi$
$230$	0	0
$231$	34.6183	2.27771
$232$	0	0
$233$	24.3316	1.59402	0.797008	−	0.603969i	$-0.206415\pi$
$233$	24.3316	1.59402	0.797008	+	0.603969i	$0.206415\pi$
$234$	0	0
$235$	−38.6178	−2.51914
$236$	0	0
$237$	−12.3051	−0.799305
$238$	0	0
$239$	19.6446	1.27071	0.635353	−	0.772222i	$-0.280854\pi$
$239$	19.6446	1.27071	0.635353	+	0.772222i	$0.280854\pi$
$240$	0	0
$241$	−3.47465	−0.223822	−0.111911	−	0.993718i	$-0.535697\pi$
$241$	−3.47465	−0.223822	−0.111911	+	0.993718i	$0.535697\pi$
$242$	0	0
$243$	22.1853	1.42319
$244$	0	0
$245$	9.78198	0.624948
$246$	0	0
$247$	−46.5008	−2.95877
$248$	0	0
$249$	31.5684	2.00057
$250$	0	0
$251$	−6.52917	−0.412117	−0.206059	−	0.978540i	$-0.566064\pi$
$251$	−6.52917	−0.412117	−0.206059	+	0.978540i	$0.566064\pi$
$252$	0	0
$253$	0.891538	0.0560505
$254$	0	0
$255$	28.9435	1.81251
$256$	0	0
$257$	26.0153	1.62279	0.811396	−	0.584496i	$-0.198708\pi$
$257$	26.0153	1.62279	0.811396	+	0.584496i	$0.198708\pi$
$258$	0	0
$259$	−7.44424	−0.462563
$260$	0	0
$261$	−2.81477	−0.174230
$262$	0	0
$263$	−27.3843	−1.68859	−0.844293	−	0.535882i	$-0.819979\pi$
$263$	−27.3843	−1.68859	−0.844293	+	0.535882i	$0.819979\pi$
$264$	0	0
$265$	28.6701	1.76119
$266$	0	0
$267$	25.0496	1.53301
$268$	0	0
$269$	0.176304	0.0107495	0.00537473	−	0.999986i	$-0.498289\pi$
$269$	0.176304	0.0107495	0.00537473	+	0.999986i	$0.498289\pi$
$270$	0	0
$271$	−16.4894	−1.00166	−0.500831	−	0.865545i	$-0.666972\pi$
$271$	−16.4894	−1.00166	−0.500831	+	0.865545i	$0.666972\pi$
$272$	0	0
$273$	50.6506	3.06551
$274$	0	0
$275$	17.5614	1.05899
$276$	0	0
$277$	8.53113	0.512586	0.256293	−	0.966599i	$-0.417499\pi$
$277$	8.53113	0.512586	0.256293	+	0.966599i	$0.417499\pi$
$278$	0	0
$279$	18.2367	1.09180
$280$	0	0
$281$	−25.7439	−1.53575	−0.767876	−	0.640598i	$-0.778686\pi$
$281$	−25.7439	−1.53575	−0.767876	+	0.640598i	$0.778686\pi$
$282$	0	0
$283$	6.65169	0.395402	0.197701	−	0.980262i	$-0.436652\pi$
$283$	6.65169	0.395402	0.197701	+	0.980262i	$0.436652\pi$
$284$	0	0
$285$	58.7604	3.48067
$286$	0	0
$287$	6.95547	0.410568
$288$	0	0
$289$	−3.17622	−0.186836
$290$	0	0
$291$	32.9203	1.92982
$292$	0	0
$293$	−8.01358	−0.468158	−0.234079	−	0.972218i	$-0.575207\pi$
$293$	−8.01358	−0.468158	−0.234079	+	0.972218i	$0.575207\pi$
$294$	0	0
$295$	−26.4719	−1.54125
$296$	0	0
$297$	−6.57950	−0.381781
$298$	0	0
$299$	1.30443	0.0754369
$300$	0	0
$301$	22.8357	1.31623
$302$	0	0
$303$	16.1846	0.929782
$304$	0	0
$305$	−22.3520	−1.27987
$306$	0	0
$307$	−20.5837	−1.17477	−0.587386	−	0.809307i	$-0.699843\pi$
$307$	−20.5837	−1.17477	−0.587386	+	0.809307i	$0.699843\pi$
$308$	0	0
$309$	−24.0199	−1.36644
$310$	0	0
$311$	−29.6873	−1.68341	−0.841707	−	0.539935i	$-0.818449\pi$
$311$	−29.6873	−1.68341	−0.841707	+	0.539935i	$0.818449\pi$
$312$	0	0
$313$	11.4776	0.648750	0.324375	−	0.945929i	$-0.394846\pi$
$313$	11.4776	0.648750	0.324375	+	0.945929i	$0.394846\pi$
$314$	0	0
$315$	−34.9462	−1.96900
$316$	0	0
$317$	4.39015	0.246576	0.123288	−	0.992371i	$-0.460656\pi$
$317$	4.39015	0.246576	0.123288	+	0.992371i	$0.460656\pi$
$318$	0	0
$319$	−3.28490	−0.183919
$320$	0	0
$321$	−23.2384	−1.29704
$322$	0	0
$323$	28.0648	1.56157
$324$	0	0
$325$	25.6945	1.42527
$326$	0	0
$327$	−39.5104	−2.18493
$328$	0	0
$329$	40.7871	2.24866
$330$	0	0
$331$	−23.8892	−1.31307	−0.656533	−	0.754297i	$-0.727978\pi$
$331$	−23.8892	−1.31307	−0.656533	+	0.754297i	$0.727978\pi$
$332$	0	0
$333$	8.39718	0.460163
$334$	0	0
$335$	13.8503	0.756725
$336$	0	0
$337$	−7.49097	−0.408059	−0.204029	−	0.978965i	$-0.565404\pi$
$337$	−7.49097	−0.408059	−0.204029	+	0.978965i	$0.565404\pi$
$338$	0	0
$339$	−16.3595	−0.888525
$340$	0	0
$341$	21.2827	1.15252
$342$	0	0
$343$	12.0577	0.651057
$344$	0	0
$345$	−1.64833	−0.0887431
$346$	0	0
$347$	−13.9258	−0.747578	−0.373789	−	0.927514i	$-0.621942\pi$
$347$	−13.9258	−0.747578	−0.373789	+	0.927514i	$0.621942\pi$
$348$	0	0
$349$	−11.0178	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$349$	−11.0178	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$350$	0	0
$351$	−9.62658	−0.513829
$352$	0	0
$353$	−0.568947	−0.0302820	−0.0151410	−	0.999885i	$-0.504820\pi$
$353$	−0.568947	−0.0302820	−0.0151410	+	0.999885i	$0.504820\pi$
$354$	0	0
$355$	−14.0930	−0.747979
$356$	0	0
$357$	−30.5693	−1.61790
$358$	0	0
$359$	−6.11293	−0.322628	−0.161314	−	0.986903i	$-0.551573\pi$
$359$	−6.11293	−0.322628	−0.161314	+	0.986903i	$0.551573\pi$
$360$	0	0
$361$	37.9765	1.99876
$362$	0	0
$363$	−17.2955	−0.907779
$364$	0	0
$365$	6.03297	0.315780
$366$	0	0
$367$	4.57649	0.238891	0.119445	−	0.992841i	$-0.461888\pi$
$367$	4.57649	0.238891	0.119445	+	0.992841i	$0.461888\pi$
$368$	0	0
$369$	−7.84584	−0.408438
$370$	0	0
$371$	−30.2806	−1.57209
$372$	0	0
$373$	−30.1925	−1.56331	−0.781655	−	0.623711i	$-0.785624\pi$
$373$	−30.1925	−1.56331	−0.781655	+	0.623711i	$0.785624\pi$
$374$	0	0
$375$	6.45440	0.333304
$376$	0	0
$377$	−4.80619	−0.247531
$378$	0	0
$379$	−9.10912	−0.467904	−0.233952	−	0.972248i	$-0.575166\pi$
$379$	−9.10912	−0.467904	−0.233952	+	0.972248i	$0.575166\pi$
$380$	0	0
$381$	−12.1956	−0.624802
$382$	0	0
$383$	18.9140	0.966463	0.483231	−	0.875493i	$-0.339463\pi$
$383$	18.9140	0.966463	0.483231	+	0.875493i	$0.339463\pi$
$384$	0	0
$385$	−40.7830	−2.07849
$386$	0	0
$387$	−25.7589	−1.30940
$388$	0	0
$389$	10.1862	0.516460	0.258230	−	0.966083i	$-0.416861\pi$
$389$	10.1862	0.516460	0.258230	+	0.966083i	$0.416861\pi$
$390$	0	0
$391$	−0.787264	−0.0398137
$392$	0	0
$393$	1.14968	0.0579935
$394$	0	0
$395$	14.4964	0.729394
$396$	0	0
$397$	4.65076	0.233415	0.116707	−	0.993166i	$-0.462766\pi$
$397$	4.65076	0.233415	0.116707	+	0.993166i	$0.462766\pi$
$398$	0	0
$399$	−62.0612	−3.10695
$400$	0	0
$401$	27.9538	1.39595	0.697973	−	0.716124i	$-0.254085\pi$
$401$	27.9538	1.39595	0.697973	+	0.716124i	$0.254085\pi$
$402$	0	0
$403$	31.1390	1.55115
$404$	0	0
$405$	−20.6133	−1.02428
$406$	0	0
$407$	9.79969	0.485753
$408$	0	0
$409$	−5.37710	−0.265880	−0.132940	−	0.991124i	$-0.542442\pi$
$409$	−5.37710	−0.265880	−0.132940	+	0.991124i	$0.542442\pi$
$410$	0	0
$411$	−35.7795	−1.76487
$412$	0	0
$413$	27.9589	1.37577
$414$	0	0
$415$	−37.1900	−1.82559
$416$	0	0
$417$	14.2455	0.697607
$418$	0	0
$419$	11.4709	0.560388	0.280194	−	0.959943i	$-0.409601\pi$
$419$	11.4709	0.560388	0.280194	+	0.959943i	$0.409601\pi$
$420$	0	0
$421$	−18.9432	−0.923236	−0.461618	−	0.887079i	$-0.652731\pi$
$421$	−18.9432	−0.923236	−0.461618	+	0.887079i	$0.652731\pi$
$422$	0	0
$423$	−46.0082	−2.23700
$424$	0	0
$425$	−15.5075	−0.752223
$426$	0	0
$427$	23.6076	1.14245
$428$	0	0
$429$	−66.6771	−3.21920
$430$	0	0
$431$	−20.0499	−0.965771	−0.482885	−	0.875684i	$-0.660411\pi$
$431$	−20.0499	−0.965771	−0.482885	+	0.875684i	$0.660411\pi$
$432$	0	0
$433$	4.95646	0.238193	0.119096	−	0.992883i	$-0.462000\pi$
$433$	4.95646	0.238193	0.119096	+	0.992883i	$0.462000\pi$
$434$	0	0
$435$	6.07332	0.291193
$436$	0	0
$437$	−1.59829	−0.0764565
$438$	0	0
$439$	31.0736	1.48306	0.741531	−	0.670919i	$-0.234100\pi$
$439$	31.0736	1.48306	0.741531	+	0.670919i	$0.234100\pi$
$440$	0	0
$441$	11.6540	0.554953
$442$	0	0
$443$	−31.9614	−1.51853	−0.759265	−	0.650781i	$-0.774442\pi$
$443$	−31.9614	−1.51853	−0.759265	+	0.650781i	$0.774442\pi$
$444$	0	0
$445$	−29.5104	−1.39893
$446$	0	0
$447$	−15.6779	−0.741540
$448$	0	0
$449$	8.83102	0.416762	0.208381	−	0.978048i	$-0.433181\pi$
$449$	8.83102	0.416762	0.208381	+	0.978048i	$0.433181\pi$
$450$	0	0
$451$	−9.15627	−0.431152
$452$	0	0
$453$	8.38730	0.394070
$454$	0	0
$455$	−59.6703	−2.79739
$456$	0	0
$457$	5.89538	0.275774	0.137887	−	0.990448i	$-0.455969\pi$
$457$	5.89538	0.275774	0.137887	+	0.990448i	$0.455969\pi$
$458$	0	0
$459$	5.80996	0.271186
$460$	0	0
$461$	−35.7938	−1.66708	−0.833541	−	0.552457i	$-0.813690\pi$
$461$	−35.7938	−1.66708	−0.833541	+	0.552457i	$0.813690\pi$
$462$	0	0
$463$	26.3204	1.22321	0.611607	−	0.791161i	$-0.290523\pi$
$463$	26.3204	1.22321	0.611607	+	0.791161i	$0.290523\pi$
$464$	0	0
$465$	−39.3487	−1.82475
$466$	0	0
$467$	−7.84453	−0.363001	−0.181501	−	0.983391i	$-0.558095\pi$
$467$	−7.84453	−0.363001	−0.181501	+	0.983391i	$0.558095\pi$
$468$	0	0
$469$	−14.6284	−0.675475
$470$	0	0
$471$	50.9565	2.34795
$472$	0	0
$473$	−30.0612	−1.38221
$474$	0	0
$475$	−31.4829	−1.44454
$476$	0	0
$477$	34.1568	1.56393
$478$	0	0
$479$	−18.4423	−0.842652	−0.421326	−	0.906909i	$-0.638435\pi$
$479$	−18.4423	−0.842652	−0.421326	+	0.906909i	$0.638435\pi$
$480$	0	0
$481$	14.3381	0.653761
$482$	0	0
$483$	1.74092	0.0792147
$484$	0	0
$485$	−38.7827	−1.76103
$486$	0	0
$487$	−28.3886	−1.28641	−0.643204	−	0.765694i	$-0.722395\pi$
$487$	−28.3886	−1.28641	−0.643204	+	0.765694i	$0.722395\pi$
$488$	0	0
$489$	−15.6888	−0.709471
$490$	0	0
$491$	24.5526	1.10804	0.554021	−	0.832503i	$-0.313093\pi$
$491$	24.5526	1.10804	0.554021	+	0.832503i	$0.313093\pi$
$492$	0	0
$493$	2.90070	0.130641
$494$	0	0
$495$	46.0036	2.06771
$496$	0	0
$497$	14.8847	0.667668
$498$	0	0
$499$	11.4277	0.511573	0.255786	−	0.966733i	$-0.417666\pi$
$499$	11.4277	0.511573	0.255786	+	0.966733i	$0.417666\pi$
$500$	0	0
$501$	−30.7693	−1.37467
$502$	0	0
$503$	13.8608	0.618024	0.309012	−	0.951058i	$-0.400002\pi$
$503$	13.8608	0.618024	0.309012	+	0.951058i	$0.400002\pi$
$504$	0	0
$505$	−19.0667	−0.848459
$506$	0	0
$507$	−64.1388	−2.84850
$508$	0	0
$509$	18.8205	0.834204	0.417102	−	0.908860i	$-0.363046\pi$
$509$	18.8205	0.834204	0.417102	+	0.908860i	$0.363046\pi$
$510$	0	0
$511$	−6.37187	−0.281875
$512$	0	0
$513$	11.7953	0.520774
$514$	0	0
$515$	28.2973	1.24693
$516$	0	0
$517$	−53.6926	−2.36140
$518$	0	0
$519$	−7.17353	−0.314883
$520$	0	0
$521$	−13.7351	−0.601747	−0.300874	−	0.953664i	$-0.597278\pi$
$521$	−13.7351	−0.601747	−0.300874	+	0.953664i	$0.597278\pi$
$522$	0	0
$523$	−1.87386	−0.0819383	−0.0409692	−	0.999160i	$-0.513045\pi$
$523$	−1.87386	−0.0819383	−0.0409692	+	0.999160i	$0.513045\pi$
$524$	0	0
$525$	34.2925	1.49665
$526$	0	0
$527$	−18.7935	−0.818656
$528$	0	0
$529$	−22.9552	−0.998051
$530$	0	0
$531$	−31.5379	−1.36863
$532$	0	0
$533$	−13.3967	−0.580275
$534$	0	0
$535$	27.3767	1.18360
$536$	0	0
$537$	−35.3283	−1.52453
$538$	0	0
$539$	13.6005	0.585814
$540$	0	0
$541$	−26.4084	−1.13539	−0.567693	−	0.823240i	$-0.692164\pi$
$541$	−26.4084	−1.13539	−0.567693	+	0.823240i	$0.692164\pi$
$542$	0	0
$543$	45.5223	1.95355
$544$	0	0
$545$	46.5463	1.99382
$546$	0	0
$547$	−8.44677	−0.361158	−0.180579	−	0.983560i	$-0.557797\pi$
$547$	−8.44677	−0.361158	−0.180579	+	0.983560i	$0.557797\pi$
$548$	0	0
$549$	−26.6296	−1.13652
$550$	0	0
$551$	5.88894	0.250877
$552$	0	0
$553$	−15.3107	−0.651078
$554$	0	0
$555$	−18.1183	−0.769078
$556$	0	0
$557$	−29.6652	−1.25695	−0.628477	−	0.777829i	$-0.716321\pi$
$557$	−29.6652	−1.25695	−0.628477	+	0.777829i	$0.716321\pi$
$558$	0	0
$559$	−43.9830	−1.86028
$560$	0	0
$561$	40.2418	1.69901
$562$	0	0
$563$	1.58399	0.0667571	0.0333785	−	0.999443i	$-0.489373\pi$
$563$	1.58399	0.0667571	0.0333785	+	0.999443i	$0.489373\pi$
$564$	0	0
$565$	19.2727	0.810810
$566$	0	0
$567$	21.7712	0.914305
$568$	0	0
$569$	−2.17545	−0.0911997	−0.0455998	−	0.998960i	$-0.514520\pi$
$569$	−2.17545	−0.0911997	−0.0455998	+	0.998960i	$0.514520\pi$
$570$	0	0
$571$	−2.19667	−0.0919278	−0.0459639	−	0.998943i	$-0.514636\pi$
$571$	−2.19667	−0.0919278	−0.0459639	+	0.998943i	$0.514636\pi$
$572$	0	0
$573$	61.6077	2.57370
$574$	0	0
$575$	0.883150	0.0368299
$576$	0	0
$577$	−15.4938	−0.645015	−0.322507	−	0.946567i	$-0.604526\pi$
$577$	−15.4938	−0.645015	−0.322507	+	0.946567i	$0.604526\pi$
$578$	0	0
$579$	14.3884	0.597962
$580$	0	0
$581$	39.2791	1.62957
$582$	0	0
$583$	39.8617	1.65090
$584$	0	0
$585$	67.3087	2.78287
$586$	0	0
$587$	−31.7461	−1.31030	−0.655151	−	0.755498i	$-0.727395\pi$
$587$	−31.7461	−1.31030	−0.655151	+	0.755498i	$0.727395\pi$
$588$	0	0
$589$	−38.1541	−1.57211
$590$	0	0
$591$	55.7782	2.29441
$592$	0	0
$593$	4.80759	0.197424	0.0987120	−	0.995116i	$-0.468528\pi$
$593$	4.80759	0.197424	0.0987120	+	0.995116i	$0.468528\pi$
$594$	0	0
$595$	36.0130	1.47639
$596$	0	0
$597$	−37.8049	−1.54725
$598$	0	0
$599$	−40.7234	−1.66391	−0.831957	−	0.554839i	$-0.812780\pi$
$599$	−40.7234	−1.66391	−0.831957	+	0.554839i	$0.812780\pi$
$600$	0	0
$601$	−4.22140	−0.172195	−0.0860973	−	0.996287i	$-0.527440\pi$
$601$	−4.22140	−0.172195	−0.0860973	+	0.996287i	$0.527440\pi$
$602$	0	0
$603$	16.5009	0.671970
$604$	0	0
$605$	20.3754	0.828380
$606$	0	0
$607$	−42.3613	−1.71939	−0.859696	−	0.510807i	$-0.829347\pi$
$607$	−42.3613	−1.71939	−0.859696	+	0.510807i	$0.829347\pi$
$608$	0	0
$609$	−6.41448	−0.259928
$610$	0	0
$611$	−78.5586	−3.17814
$612$	0	0
$613$	3.90527	0.157732	0.0788661	−	0.996885i	$-0.474870\pi$
$613$	3.90527	0.157732	0.0788661	+	0.996885i	$0.474870\pi$
$614$	0	0
$615$	16.9287	0.682630
$616$	0	0
$617$	12.1073	0.487423	0.243712	−	0.969848i	$-0.421635\pi$
$617$	12.1073	0.487423	0.243712	+	0.969848i	$0.421635\pi$
$618$	0	0
$619$	6.73347	0.270641	0.135321	−	0.990802i	$-0.456794\pi$
$619$	6.73347	0.270641	0.135321	+	0.990802i	$0.456794\pi$
$620$	0	0
$621$	−0.330877	−0.0132777
$622$	0	0
$623$	31.1681	1.24872
$624$	0	0
$625$	−28.4582	−1.13833
$626$	0	0
$627$	81.6982	3.26271
$628$	0	0
$629$	−8.65353	−0.345039
$630$	0	0
$631$	−10.5640	−0.420544	−0.210272	−	0.977643i	$-0.567435\pi$
$631$	−10.5640	−0.420544	−0.210272	+	0.977643i	$0.567435\pi$
$632$	0	0
$633$	0.246521	0.00979833
$634$	0	0
$635$	14.3674	0.570154
$636$	0	0
$637$	19.8991	0.788431
$638$	0	0
$639$	−16.7900	−0.664204
$640$	0	0
$641$	−43.0557	−1.70060	−0.850298	−	0.526301i	$-0.823579\pi$
$641$	−43.0557	−1.70060	−0.850298	+	0.526301i	$0.823579\pi$
$642$	0	0
$643$	−46.8444	−1.84736	−0.923680	−	0.383164i	$-0.874834\pi$
$643$	−46.8444	−1.84736	−0.923680	+	0.383164i	$0.874834\pi$
$644$	0	0
$645$	55.5789	2.18842
$646$	0	0
$647$	−19.5962	−0.770406	−0.385203	−	0.922832i	$-0.625869\pi$
$647$	−19.5962	−0.770406	−0.385203	+	0.922832i	$0.625869\pi$
$648$	0	0
$649$	−36.8055	−1.44474
$650$	0	0
$651$	41.5590	1.62883
$652$	0	0
$653$	−11.9594	−0.468008	−0.234004	−	0.972236i	$-0.575183\pi$
$653$	−11.9594	−0.468008	−0.234004	+	0.972236i	$0.575183\pi$
$654$	0	0
$655$	−1.35441	−0.0529211
$656$	0	0
$657$	7.18753	0.280412
$658$	0	0
$659$	25.2792	0.984739	0.492369	−	0.870386i	$-0.336131\pi$
$659$	25.2792	0.984739	0.492369	+	0.870386i	$0.336131\pi$
$660$	0	0
$661$	−8.94775	−0.348027	−0.174014	−	0.984743i	$-0.555674\pi$
$661$	−8.94775	−0.348027	−0.174014	+	0.984743i	$0.555674\pi$
$662$	0	0
$663$	58.8785	2.28665
$664$	0	0
$665$	73.1129	2.83520
$666$	0	0
$667$	−0.165195	−0.00639637
$668$	0	0
$669$	−21.3281	−0.824594
$670$	0	0
$671$	−31.0774	−1.19973
$672$	0	0
$673$	−18.6715	−0.719732	−0.359866	−	0.933004i	$-0.617178\pi$
$673$	−18.6715	−0.719732	−0.359866	+	0.933004i	$0.617178\pi$
$674$	0	0
$675$	−6.51759	−0.250862
$676$	0	0
$677$	44.7673	1.72055	0.860274	−	0.509832i	$-0.170293\pi$
$677$	44.7673	1.72055	0.860274	+	0.509832i	$0.170293\pi$
$678$	0	0
$679$	40.9612	1.57195
$680$	0	0
$681$	−16.7296	−0.641078
$682$	0	0
$683$	2.16125	0.0826981	0.0413490	−	0.999145i	$-0.486834\pi$
$683$	2.16125	0.0826981	0.0413490	+	0.999145i	$0.486834\pi$
$684$	0	0
$685$	42.1511	1.61051
$686$	0	0
$687$	67.2267	2.56486
$688$	0	0
$689$	58.3224	2.22191
$690$	0	0
$691$	23.0105	0.875359	0.437680	−	0.899131i	$-0.355800\pi$
$691$	23.0105	0.875359	0.437680	+	0.899131i	$0.355800\pi$
$692$	0	0
$693$	−48.5878	−1.84570
$694$	0	0
$695$	−16.7823	−0.636590
$696$	0	0
$697$	8.08536	0.306255
$698$	0	0
$699$	−62.5464	−2.36572
$700$	0	0
$701$	15.6671	0.591740	0.295870	−	0.955228i	$-0.404391\pi$
$701$	15.6671	0.591740	0.295870	+	0.955228i	$0.404391\pi$
$702$	0	0
$703$	−17.5682	−0.662598
$704$	0	0
$705$	99.2701	3.73873
$706$	0	0
$707$	20.1378	0.757360
$708$	0	0
$709$	24.7988	0.931339	0.465670	−	0.884959i	$-0.345814\pi$
$709$	24.7988	0.931339	0.465670	+	0.884959i	$0.345814\pi$
$710$	0	0
$711$	17.2707	0.647700
$712$	0	0
$713$	1.07029	0.0400826
$714$	0	0
$715$	78.5507	2.93763
$716$	0	0
$717$	−50.4982	−1.88589
$718$	0	0
$719$	−26.6347	−0.993308	−0.496654	−	0.867949i	$-0.665438\pi$
$719$	−26.6347	−0.993308	−0.496654	+	0.867949i	$0.665438\pi$
$720$	0	0
$721$	−29.8869	−1.11305
$722$	0	0
$723$	8.93187	0.332180
$724$	0	0
$725$	−3.25399	−0.120850
$726$	0	0
$727$	46.6716	1.73095	0.865476	−	0.500950i	$-0.167016\pi$
$727$	46.6716	1.73095	0.865476	+	0.500950i	$0.167016\pi$
$728$	0	0
$729$	−36.6089	−1.35588
$730$	0	0
$731$	26.5452	0.981811
$732$	0	0
$733$	−44.0549	−1.62721	−0.813603	−	0.581421i	$-0.802497\pi$
$733$	−44.0549	−1.62721	−0.813603	+	0.581421i	$0.802497\pi$
$734$	0	0
$735$	−25.1454	−0.927502
$736$	0	0
$737$	19.2570	0.709339
$738$	0	0
$739$	−35.3170	−1.29916	−0.649578	−	0.760295i	$-0.725054\pi$
$739$	−35.3170	−1.29916	−0.649578	+	0.760295i	$0.725054\pi$
$740$	0	0
$741$	119.534	4.39119
$742$	0	0
$743$	−0.518242	−0.0190125	−0.00950623	−	0.999955i	$-0.503026\pi$
$743$	−0.518242	−0.0190125	−0.00950623	+	0.999955i	$0.503026\pi$
$744$	0	0
$745$	18.4698	0.676681
$746$	0	0
$747$	−44.3073	−1.62112
$748$	0	0
$749$	−28.9145	−1.05651
$750$	0	0
$751$	−51.8855	−1.89333	−0.946664	−	0.322223i	$-0.895570\pi$
$751$	−51.8855	−1.89333	−0.946664	+	0.322223i	$0.895570\pi$
$752$	0	0
$753$	16.7838	0.611634
$754$	0	0
$755$	−9.88089	−0.359602
$756$	0	0
$757$	−1.69435	−0.0615822	−0.0307911	−	0.999526i	$-0.509803\pi$
$757$	−1.69435	−0.0615822	−0.0307911	+	0.999526i	$0.509803\pi$
$758$	0	0
$759$	−2.29177	−0.0831861
$760$	0	0
$761$	14.0743	0.510192	0.255096	−	0.966916i	$-0.417893\pi$
$761$	14.0743	0.510192	0.255096	+	0.966916i	$0.417893\pi$
$762$	0	0
$763$	−49.1610	−1.77975
$764$	0	0
$765$	−40.6231	−1.46873
$766$	0	0
$767$	−53.8507	−1.94444
$768$	0	0
$769$	−10.6198	−0.382959	−0.191480	−	0.981497i	$-0.561329\pi$
$769$	−10.6198	−0.382959	−0.191480	+	0.981497i	$0.561329\pi$
$770$	0	0
$771$	−66.8746	−2.40843
$772$	0	0
$773$	−4.71058	−0.169428	−0.0847139	−	0.996405i	$-0.526998\pi$
$773$	−4.71058	−0.169428	−0.0847139	+	0.996405i	$0.526998\pi$
$774$	0	0
$775$	21.0824	0.757303
$776$	0	0
$777$	19.1360	0.686501
$778$	0	0
$779$	16.4147	0.588119
$780$	0	0
$781$	−19.5944	−0.701141
$782$	0	0
$783$	1.21913	0.0435680
$784$	0	0
$785$	−60.0307	−2.14259
$786$	0	0
$787$	−18.7269	−0.667543	−0.333772	−	0.942654i	$-0.608321\pi$
$787$	−18.7269	−0.667543	−0.333772	+	0.942654i	$0.608321\pi$
$788$	0	0
$789$	70.3935	2.50607
$790$	0	0
$791$	−20.3554	−0.723753
$792$	0	0
$793$	−45.4698	−1.61468
$794$	0	0
$795$	−73.6988	−2.61383
$796$	0	0
$797$	−2.52134	−0.0893106	−0.0446553	−	0.999002i	$-0.514219\pi$
$797$	−2.52134	−0.0893106	−0.0446553	+	0.999002i	$0.514219\pi$
$798$	0	0
$799$	47.4127	1.67734
$800$	0	0
$801$	−35.1579	−1.24224
$802$	0	0
$803$	8.38801	0.296006
$804$	0	0
$805$	−2.05094	−0.0722862
$806$	0	0
$807$	−0.453205	−0.0159536
$808$	0	0
$809$	−33.4115	−1.17468	−0.587342	−	0.809339i	$-0.699826\pi$
$809$	−33.4115	−1.17468	−0.587342	+	0.809339i	$0.699826\pi$
$810$	0	0
$811$	6.33016	0.222282	0.111141	−	0.993805i	$-0.464549\pi$
$811$	6.33016	0.222282	0.111141	+	0.993805i	$0.464549\pi$
$812$	0	0
$813$	42.3874	1.48659
$814$	0	0
$815$	18.4826	0.647417
$816$	0	0
$817$	53.8916	1.88543
$818$	0	0
$819$	−71.0897	−2.48408
$820$	0	0
$821$	−21.1083	−0.736684	−0.368342	−	0.929690i	$-0.620074\pi$
$821$	−21.1083	−0.736684	−0.368342	+	0.929690i	$0.620074\pi$
$822$	0	0
$823$	−54.6368	−1.90452	−0.952260	−	0.305290i	$-0.901247\pi$
$823$	−54.6368	−1.90452	−0.952260	+	0.305290i	$0.901247\pi$
$824$	0	0
$825$	−45.1431	−1.57168
$826$	0	0
$827$	−22.4266	−0.779848	−0.389924	−	0.920847i	$-0.627499\pi$
$827$	−22.4266	−0.779848	−0.389924	+	0.920847i	$0.627499\pi$
$828$	0	0
$829$	12.0758	0.419411	0.209706	−	0.977765i	$-0.432749\pi$
$829$	12.0758	0.419411	0.209706	+	0.977765i	$0.432749\pi$
$830$	0	0
$831$	−21.9300	−0.760742
$832$	0	0
$833$	−12.0098	−0.416114
$834$	0	0
$835$	36.2487	1.25444
$836$	0	0
$837$	−7.89865	−0.273017
$838$	0	0
$839$	8.15417	0.281513	0.140757	−	0.990044i	$-0.455046\pi$
$839$	8.15417	0.281513	0.140757	+	0.990044i	$0.455046\pi$
$840$	0	0
$841$	−28.3913	−0.979012
$842$	0	0
$843$	66.1768	2.27925
$844$	0	0
$845$	75.5605	2.59936
$846$	0	0
$847$	−21.5200	−0.739436
$848$	0	0
$849$	−17.0987	−0.586827
$850$	0	0
$851$	0.492818	0.0168936
$852$	0	0
$853$	18.2910	0.626270	0.313135	−	0.949709i	$-0.398621\pi$
$853$	18.2910	0.626270	0.313135	+	0.949709i	$0.398621\pi$
$854$	0	0
$855$	−82.4721	−2.82049
$856$	0	0
$857$	32.5636	1.11235	0.556176	−	0.831065i	$-0.312268\pi$
$857$	32.5636	1.11235	0.556176	+	0.831065i	$0.312268\pi$
$858$	0	0
$859$	16.7210	0.570513	0.285256	−	0.958451i	$-0.407921\pi$
$859$	16.7210	0.570513	0.285256	+	0.958451i	$0.407921\pi$
$860$	0	0
$861$	−17.8796	−0.609335
$862$	0	0
$863$	34.9477	1.18963	0.594817	−	0.803861i	$-0.297225\pi$
$863$	34.9477	1.18963	0.594817	+	0.803861i	$0.297225\pi$
$864$	0	0
$865$	8.45097	0.287342
$866$	0	0
$867$	8.16472	0.277289
$868$	0	0
$869$	20.1552	0.683720
$870$	0	0
$871$	28.1752	0.954680
$872$	0	0
$873$	−46.2047	−1.56379
$874$	0	0
$875$	8.03092	0.271495
$876$	0	0
$877$	40.3848	1.36370	0.681848	−	0.731493i	$-0.261176\pi$
$877$	40.3848	1.36370	0.681848	+	0.731493i	$0.261176\pi$
$878$	0	0
$879$	20.5996	0.694806
$880$	0	0
$881$	24.4793	0.824730	0.412365	−	0.911019i	$-0.364703\pi$
$881$	24.4793	0.824730	0.412365	+	0.911019i	$0.364703\pi$
$882$	0	0
$883$	22.4219	0.754558	0.377279	−	0.926100i	$-0.376860\pi$
$883$	22.4219	0.754558	0.377279	+	0.926100i	$0.376860\pi$
$884$	0	0
$885$	68.0481	2.28741
$886$	0	0
$887$	−27.5267	−0.924258	−0.462129	−	0.886813i	$-0.652914\pi$
$887$	−27.5267	−0.924258	−0.462129	+	0.886813i	$0.652914\pi$
$888$	0	0
$889$	−15.1745	−0.508936
$890$	0	0
$891$	−28.6599	−0.960143
$892$	0	0
$893$	96.2564	3.22110
$894$	0	0
$895$	41.6194	1.39118
$896$	0	0
$897$	−3.35313	−0.111958
$898$	0	0
$899$	−3.94350	−0.131523
$900$	0	0
$901$	−35.1995	−1.17267
$902$	0	0
$903$	−58.7010	−1.95345
$904$	0	0
$905$	−53.6288	−1.78268
$906$	0	0
$907$	23.3424	0.775070	0.387535	−	0.921855i	$-0.373327\pi$
$907$	23.3424	0.775070	0.387535	+	0.921855i	$0.373327\pi$
$908$	0	0
$909$	−22.7156	−0.753430
$910$	0	0
$911$	−30.3675	−1.00612	−0.503060	−	0.864251i	$-0.667793\pi$
$911$	−30.3675	−1.00612	−0.503060	+	0.864251i	$0.667793\pi$
$912$	0	0
$913$	−51.7075	−1.71127
$914$	0	0
$915$	57.4577	1.89949
$916$	0	0
$917$	1.43049	0.0472389
$918$	0	0
$919$	45.9998	1.51739	0.758697	−	0.651444i	$-0.225837\pi$
$919$	45.9998	1.51739	0.758697	+	0.651444i	$0.225837\pi$
$920$	0	0
$921$	52.9120	1.74351
$922$	0	0
$923$	−28.6688	−0.943646
$924$	0	0
$925$	9.70749	0.319180
$926$	0	0
$927$	33.7127	1.10727
$928$	0	0
$929$	−9.09647	−0.298446	−0.149223	−	0.988804i	$-0.547677\pi$
$929$	−9.09647	−0.298446	−0.149223	+	0.988804i	$0.547677\pi$
$930$	0	0
$931$	−24.3820	−0.799088
$932$	0	0
$933$	76.3137	2.49840
$934$	0	0
$935$	−47.4080	−1.55041
$936$	0	0
$937$	−56.6548	−1.85083	−0.925416	−	0.378953i	$-0.876284\pi$
$937$	−56.6548	−1.85083	−0.925416	+	0.378953i	$0.876284\pi$
$938$	0	0
$939$	−29.5040	−0.962827
$940$	0	0
$941$	33.5716	1.09440	0.547202	−	0.837001i	$-0.315693\pi$
$941$	33.5716	1.09440	0.547202	+	0.837001i	$0.315693\pi$
$942$	0	0
$943$	−0.460461	−0.0149947
$944$	0	0
$945$	15.1358	0.492368
$946$	0	0
$947$	12.3691	0.401941	0.200970	−	0.979597i	$-0.435590\pi$
$947$	12.3691	0.401941	0.200970	+	0.979597i	$0.435590\pi$
$948$	0	0
$949$	12.2726	0.398387
$950$	0	0
$951$	−11.2853	−0.365949
$952$	0	0
$953$	−17.2518	−0.558840	−0.279420	−	0.960169i	$-0.590142\pi$
$953$	−17.2518	−0.558840	−0.279420	+	0.960169i	$0.590142\pi$
$954$	0	0
$955$	−72.5787	−2.34859
$956$	0	0
$957$	8.44410	0.272959
$958$	0	0
$959$	−44.5189	−1.43759
$960$	0	0
$961$	−5.45026	−0.175815
$962$	0	0
$963$	32.6159	1.05103
$964$	0	0
$965$	−16.9507	−0.545661
$966$	0	0
$967$	30.1824	0.970602	0.485301	−	0.874347i	$-0.338710\pi$
$967$	30.1824	0.970602	0.485301	+	0.874347i	$0.338710\pi$
$968$	0	0
$969$	−72.1428	−2.31756
$970$	0	0
$971$	−0.613211	−0.0196789	−0.00983944	−	0.999952i	$-0.503132\pi$
$971$	−0.613211	−0.0196789	−0.00983944	+	0.999952i	$0.503132\pi$
$972$	0	0
$973$	17.7251	0.568240
$974$	0	0
$975$	−66.0497	−2.11528
$976$	0	0
$977$	40.3744	1.29169	0.645846	−	0.763468i	$-0.276505\pi$
$977$	40.3744	1.29169	0.645846	+	0.763468i	$0.276505\pi$
$978$	0	0
$979$	−41.0301	−1.31133
$980$	0	0
$981$	55.4541	1.77051
$982$	0	0
$983$	39.3110	1.25383	0.626913	−	0.779089i	$-0.284318\pi$
$983$	39.3110	1.25383	0.626913	+	0.779089i	$0.284318\pi$
$984$	0	0
$985$	−65.7110	−2.09373
$986$	0	0
$987$	−104.847	−3.33730
$988$	0	0
$989$	−1.51175	−0.0480709
$990$	0	0
$991$	57.8530	1.83776	0.918881	−	0.394534i	$-0.129094\pi$
$991$	57.8530	1.83776	0.918881	+	0.394534i	$0.129094\pi$
$992$	0	0
$993$	61.4090	1.94876
$994$	0	0
$995$	44.5371	1.41192
$996$	0	0
$997$	38.1616	1.20859	0.604295	−	0.796760i	$-0.293455\pi$
$997$	38.1616	1.20859	0.604295	+	0.796760i	$0.293455\pi$
$998$	0	0
$999$	−3.63697	−0.115069

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8044.2.a.a.1.13	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8044.2.a.a.1.13	✓	80	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:	\( N \)	\(=\)	\( 8044 = 2^{2} \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8044.a (trivial)

\(f(q)\)	\(=\)	\(q-2.57058 q^{3} +3.02835 q^{5} -3.19846 q^{7} +3.60789 q^{9} +O(q^{10})\)	\(q-2.57058 q^{3} +3.02835 q^{5} -3.19846 q^{7} +3.60789 q^{9} +4.21049 q^{11} +6.16044 q^{13} -7.78461 q^{15} -3.71804 q^{17} -7.54828 q^{19} +8.22190 q^{21} +0.211742 q^{23} +4.17088 q^{25} -1.56264 q^{27} -0.780170 q^{29} +5.05467 q^{31} -10.8234 q^{33} -9.68604 q^{35} +2.32745 q^{37} -15.8359 q^{39} -2.17463 q^{41} -7.13959 q^{43} +10.9260 q^{45} -12.7521 q^{47} +3.23014 q^{49} +9.55752 q^{51} +9.46724 q^{53} +12.7508 q^{55} +19.4035 q^{57} -8.74137 q^{59} -7.38093 q^{61} -11.5397 q^{63} +18.6560 q^{65} +4.57356 q^{67} -0.544300 q^{69} -4.65370 q^{71} +1.99217 q^{73} -10.7216 q^{75} -13.4671 q^{77} +4.78691 q^{79} -6.80678 q^{81} -12.2806 q^{83} -11.2595 q^{85} +2.00549 q^{87} -9.74472 q^{89} -19.7039 q^{91} -12.9935 q^{93} -22.8588 q^{95} -12.8065 q^{97} +15.1910 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9}+O(q^{10}) \) 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9	\( 80 q - 13 q^{3} - 2 q^{5} - 12 q^{7} + 63 q^{9} - 34 q^{11} - q^{13} - 24 q^{15} - 35 q^{17} - 31 q^{19} - 3 q^{21} - 43 q^{23} + 58 q^{25} - 49 q^{27} - 5 q^{29} - 56 q^{31} - 23 q^{33} - 72 q^{35} - 11 q^{37} - 74 q^{39} - 81 q^{41} - 34 q^{43} - 14 q^{45} - 64 q^{47} + 40 q^{49} - 59 q^{51} + 3 q^{53} - 53 q^{55} - 34 q^{57} - 116 q^{59} - 13 q^{61} - 61 q^{63} - 55 q^{65} - 22 q^{67} - 10 q^{69} - 86 q^{71} - 32 q^{73} - 85 q^{75} + 4 q^{77} - 88 q^{79} + 12 q^{81} - 83 q^{83} - 2 q^{85} - 87 q^{87} - 72 q^{89} - 49 q^{91} - 102 q^{95} - 34 q^{97} - 103 q^{99}+O(q^{100}) \) 80 * q - 13 * q^3 - 2 * q^5 - 12 * q^7 + 63 * q^9 - 34 * q^11 - q^13 - 24 * q^15 - 35 * q^17 - 31 * q^19 - 3 * q^21 - 43 * q^23 + 58 * q^25 - 49 * q^27 - 5 * q^29 - 56 * q^31 - 23 * q^33 - 72 * q^35 - 11 * q^37 - 74 * q^39 - 81 * q^41 - 34 * q^43 - 14 * q^45 - 64 * q^47 + 40 * q^49 - 59 * q^51 + 3 * q^53 - 53 * q^55 - 34 * q^57 - 116 * q^59 - 13 * q^61 - 61 * q^63 - 55 * q^65 - 22 * q^67 - 10 * q^69 - 86 * q^71 - 32 * q^73 - 85 * q^75 + 4 * q^77 - 88 * q^79 + 12 * q^81 - 83 * q^83 - 2 * q^85 - 87 * q^87 - 72 * q^89 - 49 * q^91 - 102 * q^95 - 34 * q^97 - 103 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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