LMFDB - Embedded newform 8048.2.a.w.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, 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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$0$
Dimension:	$29$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.04161 q^{3} -4.02753 q^{5} -1.43352 q^{7} +1.16816 q^{9} +O(q^{10})$	$q-2.04161 q^{3} -4.02753 q^{5} -1.43352 q^{7} +1.16816 q^{9} +4.74425 q^{11} +2.52494 q^{13} +8.22263 q^{15} +2.51155 q^{17} -3.36749 q^{19} +2.92668 q^{21} -2.10695 q^{23} +11.2210 q^{25} +3.73990 q^{27} +3.73663 q^{29} +9.98999 q^{31} -9.68589 q^{33} +5.77353 q^{35} +11.0647 q^{37} -5.15493 q^{39} -11.3637 q^{41} +2.86718 q^{43} -4.70478 q^{45} -5.89374 q^{47} -4.94503 q^{49} -5.12759 q^{51} -10.3801 q^{53} -19.1076 q^{55} +6.87508 q^{57} -3.29972 q^{59} -13.7534 q^{61} -1.67457 q^{63} -10.1693 q^{65} -0.398643 q^{67} +4.30156 q^{69} +13.2376 q^{71} -7.28970 q^{73} -22.9089 q^{75} -6.80097 q^{77} +5.83784 q^{79} -11.1399 q^{81} +11.1641 q^{83} -10.1153 q^{85} -7.62873 q^{87} -5.60230 q^{89} -3.61954 q^{91} -20.3956 q^{93} +13.5627 q^{95} -3.96762 q^{97} +5.54203 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9	$ 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9} + 27 q^{11} + 16 q^{13} + 14 q^{15} - 15 q^{17} + 14 q^{19} + q^{21} + 25 q^{23} + 21 q^{25} + 25 q^{27} - 13 q^{29} + 27 q^{31} - 9 q^{33} + 29 q^{35} + 35 q^{37} + 38 q^{39} - 30 q^{41} + 38 q^{43} + q^{45} + 35 q^{47} + 14 q^{49} + 21 q^{51} + 2 q^{53} + 25 q^{55} - 25 q^{57} + 40 q^{59} + 10 q^{61} + 56 q^{63} - 50 q^{65} + 31 q^{67} + 11 q^{69} + 65 q^{71} - 23 q^{73} + 32 q^{75} + 13 q^{77} + 44 q^{79} - 7 q^{81} + 41 q^{83} + 26 q^{85} + 25 q^{87} - 48 q^{89} + 44 q^{91} + 25 q^{93} + 75 q^{95} - 18 q^{97} + 80 q^{99}+O(q^{100}) $ 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9 + 27 * q^11 + 16 * q^13 + 14 * q^15 - 15 * q^17 + 14 * q^19 + q^21 + 25 * q^23 + 21 * q^25 + 25 * q^27 - 13 * q^29 + 27 * q^31 - 9 * q^33 + 29 * q^35 + 35 * q^37 + 38 * q^39 - 30 * q^41 + 38 * q^43 + q^45 + 35 * q^47 + 14 * q^49 + 21 * q^51 + 2 * q^53 + 25 * q^55 - 25 * q^57 + 40 * q^59 + 10 * q^61 + 56 * q^63 - 50 * q^65 + 31 * q^67 + 11 * q^69 + 65 * q^71 - 23 * q^73 + 32 * q^75 + 13 * q^77 + 44 * q^79 - 7 * q^81 + 41 * q^83 + 26 * q^85 + 25 * q^87 - 48 * q^89 + 44 * q^91 + 25 * q^93 + 75 * q^95 - 18 * q^97 + 80 * 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
$3$	−2.04161	−1.17872	−0.589361	−	0.807870i	$-0.700620\pi$
$3$	−2.04161	−1.17872	−0.589361	+	0.807870i	$0.700620\pi$
$4$	0	0
$5$	−4.02753	−1.80117	−0.900583	−	0.434684i	$-0.856860\pi$
$5$	−4.02753	−1.80117	−0.900583	+	0.434684i	$0.856860\pi$
$6$	0	0
$7$	−1.43352	−0.541819	−0.270909	−	0.962605i	$-0.587324\pi$
$7$	−1.43352	−0.541819	−0.270909	+	0.962605i	$0.587324\pi$
$8$	0	0
$9$	1.16816	0.389385
$10$	0	0
$11$	4.74425	1.43045	0.715223	−	0.698896i	$-0.246325\pi$
$11$	4.74425	1.43045	0.715223	+	0.698896i	$0.246325\pi$
$12$	0	0
$13$	2.52494	0.700292	0.350146	−	0.936695i	$-0.386132\pi$
$13$	2.52494	0.700292	0.350146	+	0.936695i	$0.386132\pi$
$14$	0	0
$15$	8.22263	2.12307
$16$	0	0
$17$	2.51155	0.609140	0.304570	−	0.952490i	$-0.401487\pi$
$17$	2.51155	0.609140	0.304570	+	0.952490i	$0.401487\pi$
$18$	0	0
$19$	−3.36749	−0.772555	−0.386277	−	0.922383i	$-0.626239\pi$
$19$	−3.36749	−0.772555	−0.386277	+	0.922383i	$0.626239\pi$
$20$	0	0
$21$	2.92668	0.638654
$22$	0	0
$23$	−2.10695	−0.439329	−0.219664	−	0.975575i	$-0.570496\pi$
$23$	−2.10695	−0.439329	−0.219664	+	0.975575i	$0.570496\pi$
$24$	0	0
$25$	11.2210	2.24420
$26$	0	0
$27$	3.73990	0.719745
$28$	0	0
$29$	3.73663	0.693875	0.346937	−	0.937888i	$-0.387222\pi$
$29$	3.73663	0.693875	0.346937	+	0.937888i	$0.387222\pi$
$30$	0	0
$31$	9.98999	1.79425	0.897127	−	0.441772i	$-0.145650\pi$
$31$	9.98999	1.79425	0.897127	+	0.441772i	$0.145650\pi$
$32$	0	0
$33$	−9.68589	−1.68610
$34$	0	0
$35$	5.77353	0.975905
$36$	0	0
$37$	11.0647	1.81903	0.909516	−	0.415668i	$-0.136452\pi$
$37$	11.0647	1.81903	0.909516	+	0.415668i	$0.136452\pi$
$38$	0	0
$39$	−5.15493	−0.825449
$40$	0	0
$41$	−11.3637	−1.77471	−0.887354	−	0.461090i	$-0.847459\pi$
$41$	−11.3637	−1.77471	−0.887354	+	0.461090i	$0.847459\pi$
$42$	0	0
$43$	2.86718	0.437240	0.218620	−	0.975810i	$-0.429844\pi$
$43$	2.86718	0.437240	0.218620	+	0.975810i	$0.429844\pi$
$44$	0	0
$45$	−4.70478	−0.701348
$46$	0	0
$47$	−5.89374	−0.859690	−0.429845	−	0.902903i	$-0.641432\pi$
$47$	−5.89374	−0.859690	−0.429845	+	0.902903i	$0.641432\pi$
$48$	0	0
$49$	−4.94503	−0.706432
$50$	0	0
$51$	−5.12759	−0.718007
$52$	0	0
$53$	−10.3801	−1.42581	−0.712906	−	0.701259i	$-0.752622\pi$
$53$	−10.3801	−1.42581	−0.712906	+	0.701259i	$0.752622\pi$
$54$	0	0
$55$	−19.1076	−2.57647
$56$	0	0
$57$	6.87508	0.910627
$58$	0	0
$59$	−3.29972	−0.429587	−0.214793	−	0.976660i	$-0.568908\pi$
$59$	−3.29972	−0.429587	−0.214793	+	0.976660i	$0.568908\pi$
$60$	0	0
$61$	−13.7534	−1.76095	−0.880473	−	0.474097i	$-0.842775\pi$
$61$	−13.7534	−1.76095	−0.880473	+	0.474097i	$0.842775\pi$
$62$	0	0
$63$	−1.67457	−0.210976
$64$	0	0
$65$	−10.1693	−1.26134
$66$	0	0
$67$	−0.398643	−0.0487020	−0.0243510	−	0.999703i	$-0.507752\pi$
$67$	−0.398643	−0.0487020	−0.0243510	+	0.999703i	$0.507752\pi$
$68$	0	0
$69$	4.30156	0.517847
$70$	0	0
$71$	13.2376	1.57101	0.785507	−	0.618853i	$-0.212402\pi$
$71$	13.2376	1.57101	0.785507	+	0.618853i	$0.212402\pi$
$72$	0	0
$73$	−7.28970	−0.853195	−0.426597	−	0.904442i	$-0.640288\pi$
$73$	−7.28970	−0.853195	−0.426597	+	0.904442i	$0.640288\pi$
$74$	0	0
$75$	−22.9089	−2.64529
$76$	0	0
$77$	−6.80097	−0.775042
$78$	0	0
$79$	5.83784	0.656809	0.328404	−	0.944537i	$-0.393489\pi$
$79$	5.83784	0.656809	0.328404	+	0.944537i	$0.393489\pi$
$80$	0	0
$81$	−11.1399	−1.23776
$82$	0	0
$83$	11.1641	1.22541	0.612707	−	0.790310i	$-0.290080\pi$
$83$	11.1641	1.22541	0.612707	+	0.790310i	$0.290080\pi$
$84$	0	0
$85$	−10.1153	−1.09716
$86$	0	0
$87$	−7.62873	−0.817886
$88$	0	0
$89$	−5.60230	−0.593843	−0.296921	−	0.954902i	$-0.595960\pi$
$89$	−5.60230	−0.593843	−0.296921	+	0.954902i	$0.595960\pi$
$90$	0	0
$91$	−3.61954	−0.379431
$92$	0	0
$93$	−20.3956	−2.11493
$94$	0	0
$95$	13.5627	1.39150
$96$	0	0
$97$	−3.96762	−0.402851	−0.201425	−	0.979504i	$-0.564557\pi$
$97$	−3.96762	−0.402851	−0.201425	+	0.979504i	$0.564557\pi$
$98$	0	0
$99$	5.54203	0.556995
$100$	0	0
$101$	14.6067	1.45342	0.726712	−	0.686942i	$-0.241047\pi$
$101$	14.6067	1.45342	0.726712	+	0.686942i	$0.241047\pi$
$102$	0	0
$103$	−6.39413	−0.630032	−0.315016	−	0.949086i	$-0.602010\pi$
$103$	−6.39413	−0.630032	−0.315016	+	0.949086i	$0.602010\pi$
$104$	0	0
$105$	−11.7873	−1.15032
$106$	0	0
$107$	15.0430	1.45426	0.727129	−	0.686501i	$-0.240854\pi$
$107$	15.0430	1.45426	0.727129	+	0.686501i	$0.240854\pi$
$108$	0	0
$109$	−2.68374	−0.257056	−0.128528	−	0.991706i	$-0.541025\pi$
$109$	−2.68374	−0.257056	−0.128528	+	0.991706i	$0.541025\pi$
$110$	0	0
$111$	−22.5899	−2.14413
$112$	0	0
$113$	−19.9114	−1.87311	−0.936554	−	0.350525i	$-0.886003\pi$
$113$	−19.9114	−1.87311	−0.936554	+	0.350525i	$0.886003\pi$
$114$	0	0
$115$	8.48579	0.791304
$116$	0	0
$117$	2.94952	0.272683
$118$	0	0
$119$	−3.60035	−0.330043
$120$	0	0
$121$	11.5079	1.04617
$122$	0	0
$123$	23.2001	2.09189
$124$	0	0
$125$	−25.0552	−2.24101
$126$	0	0
$127$	6.24712	0.554343	0.277171	−	0.960821i	$-0.410603\pi$
$127$	6.24712	0.554343	0.277171	+	0.960821i	$0.410603\pi$
$128$	0	0
$129$	−5.85364	−0.515385
$130$	0	0
$131$	−2.24915	−0.196509	−0.0982545	−	0.995161i	$-0.531326\pi$
$131$	−2.24915	−0.196509	−0.0982545	+	0.995161i	$0.531326\pi$
$132$	0	0
$133$	4.82735	0.418585
$134$	0	0
$135$	−15.0626	−1.29638
$136$	0	0
$137$	13.9870	1.19499	0.597494	−	0.801873i	$-0.296163\pi$
$137$	13.9870	1.19499	0.597494	+	0.801873i	$0.296163\pi$
$138$	0	0
$139$	−2.77964	−0.235766	−0.117883	−	0.993028i	$-0.537611\pi$
$139$	−2.77964	−0.235766	−0.117883	+	0.993028i	$0.537611\pi$
$140$	0	0
$141$	12.0327	1.01334
$142$	0	0
$143$	11.9789	1.00173
$144$	0	0
$145$	−15.0494	−1.24978
$146$	0	0
$147$	10.0958	0.832687
$148$	0	0
$149$	11.3469	0.929571	0.464785	−	0.885423i	$-0.346131\pi$
$149$	11.3469	0.929571	0.464785	+	0.885423i	$0.346131\pi$
$150$	0	0
$151$	11.6460	0.947736	0.473868	−	0.880596i	$-0.342857\pi$
$151$	11.6460	0.947736	0.473868	+	0.880596i	$0.342857\pi$
$152$	0	0
$153$	2.93388	0.237190
$154$	0	0
$155$	−40.2350	−3.23175
$156$	0	0
$157$	22.3343	1.78247	0.891235	−	0.453541i	$-0.149840\pi$
$157$	22.3343	1.78247	0.891235	+	0.453541i	$0.149840\pi$
$158$	0	0
$159$	21.1920	1.68064
$160$	0	0
$161$	3.02035	0.238037
$162$	0	0
$163$	4.79998	0.375963	0.187982	−	0.982173i	$-0.439805\pi$
$163$	4.79998	0.375963	0.187982	+	0.982173i	$0.439805\pi$
$164$	0	0
$165$	39.0102	3.03694
$166$	0	0
$167$	−3.51448	−0.271959	−0.135979	−	0.990712i	$-0.543418\pi$
$167$	−3.51448	−0.271959	−0.135979	+	0.990712i	$0.543418\pi$
$168$	0	0
$169$	−6.62469	−0.509591
$170$	0	0
$171$	−3.93375	−0.300821
$172$	0	0
$173$	−19.5122	−1.48348	−0.741742	−	0.670685i	$-0.766000\pi$
$173$	−19.5122	−1.48348	−0.741742	+	0.670685i	$0.766000\pi$
$174$	0	0
$175$	−16.0855	−1.21595
$176$	0	0
$177$	6.73673	0.506363
$178$	0	0
$179$	−10.3539	−0.773886	−0.386943	−	0.922104i	$-0.626469\pi$
$179$	−10.3539	−0.773886	−0.386943	+	0.922104i	$0.626469\pi$
$180$	0	0
$181$	18.7716	1.39528	0.697641	−	0.716447i	$-0.254233\pi$
$181$	18.7716	1.39528	0.697641	+	0.716447i	$0.254233\pi$
$182$	0	0
$183$	28.0791	2.07567
$184$	0	0
$185$	−44.5636	−3.27638
$186$	0	0
$187$	11.9154	0.871342
$188$	0	0
$189$	−5.36122	−0.389971
$190$	0	0
$191$	−10.1409	−0.733767	−0.366884	−	0.930267i	$-0.619575\pi$
$191$	−10.1409	−0.733767	−0.366884	+	0.930267i	$0.619575\pi$
$192$	0	0
$193$	3.24356	0.233476	0.116738	−	0.993163i	$-0.462756\pi$
$193$	3.24356	0.233476	0.116738	+	0.993163i	$0.462756\pi$
$194$	0	0
$195$	20.7616	1.48677
$196$	0	0
$197$	−13.1260	−0.935187	−0.467593	−	0.883944i	$-0.654879\pi$
$197$	−13.1260	−0.935187	−0.467593	+	0.883944i	$0.654879\pi$
$198$	0	0
$199$	1.97168	0.139768	0.0698842	−	0.997555i	$-0.477737\pi$
$199$	1.97168	0.139768	0.0698842	+	0.997555i	$0.477737\pi$
$200$	0	0
$201$	0.813872	0.0574061
$202$	0	0
$203$	−5.35653	−0.375954
$204$	0	0
$205$	45.7675	3.19654
$206$	0	0
$207$	−2.46124	−0.171068
$208$	0	0
$209$	−15.9762	−1.10510
$210$	0	0
$211$	−14.1879	−0.976732	−0.488366	−	0.872639i	$-0.662407\pi$
$211$	−14.1879	−0.976732	−0.488366	+	0.872639i	$0.662407\pi$
$212$	0	0
$213$	−27.0260	−1.85179
$214$	0	0
$215$	−11.5476	−0.787542
$216$	0	0
$217$	−14.3208	−0.972161
$218$	0	0
$219$	14.8827	1.00568
$220$	0	0
$221$	6.34150	0.426576
$222$	0	0
$223$	−8.99946	−0.602648	−0.301324	−	0.953522i	$-0.597429\pi$
$223$	−8.99946	−0.602648	−0.301324	+	0.953522i	$0.597429\pi$
$224$	0	0
$225$	13.1079	0.873858
$226$	0	0
$227$	9.36758	0.621748	0.310874	−	0.950451i	$-0.399378\pi$
$227$	9.36758	0.621748	0.310874	+	0.950451i	$0.399378\pi$
$228$	0	0
$229$	15.8433	1.04696	0.523478	−	0.852039i	$-0.324634\pi$
$229$	15.8433	1.04696	0.523478	+	0.852039i	$0.324634\pi$
$230$	0	0
$231$	13.8849	0.913559
$232$	0	0
$233$	−6.63659	−0.434778	−0.217389	−	0.976085i	$-0.569754\pi$
$233$	−6.63659	−0.434778	−0.217389	+	0.976085i	$0.569754\pi$
$234$	0	0
$235$	23.7372	1.54844
$236$	0	0
$237$	−11.9186	−0.774195
$238$	0	0
$239$	−13.6458	−0.882675	−0.441338	−	0.897341i	$-0.645496\pi$
$239$	−13.6458	−0.882675	−0.441338	+	0.897341i	$0.645496\pi$
$240$	0	0
$241$	−10.4290	−0.671794	−0.335897	−	0.941899i	$-0.609039\pi$
$241$	−10.4290	−0.671794	−0.335897	+	0.941899i	$0.609039\pi$
$242$	0	0
$243$	11.5235	0.739235
$244$	0	0
$245$	19.9162	1.27240
$246$	0	0
$247$	−8.50270	−0.541014
$248$	0	0
$249$	−22.7926	−1.44442
$250$	0	0
$251$	16.5095	1.04207	0.521035	−	0.853536i	$-0.325546\pi$
$251$	16.5095	1.04207	0.521035	+	0.853536i	$0.325546\pi$
$252$	0	0
$253$	−9.99589	−0.628436
$254$	0	0
$255$	20.6515	1.29325
$256$	0	0
$257$	−12.6399	−0.788456	−0.394228	−	0.919013i	$-0.628988\pi$
$257$	−12.6399	−0.788456	−0.394228	+	0.919013i	$0.628988\pi$
$258$	0	0
$259$	−15.8615	−0.985586
$260$	0	0
$261$	4.36497	0.270185
$262$	0	0
$263$	12.2907	0.757877	0.378938	−	0.925422i	$-0.376289\pi$
$263$	12.2907	0.757877	0.378938	+	0.925422i	$0.376289\pi$
$264$	0	0
$265$	41.8060	2.56813
$266$	0	0
$267$	11.4377	0.699975
$268$	0	0
$269$	−24.1380	−1.47172	−0.735859	−	0.677135i	$-0.763221\pi$
$269$	−24.1380	−1.47172	−0.735859	+	0.677135i	$0.763221\pi$
$270$	0	0
$271$	−8.09878	−0.491966	−0.245983	−	0.969274i	$-0.579111\pi$
$271$	−8.09878	−0.491966	−0.245983	+	0.969274i	$0.579111\pi$
$272$	0	0
$273$	7.38968	0.447244
$274$	0	0
$275$	53.2352	3.21020
$276$	0	0
$277$	10.5165	0.631878	0.315939	−	0.948780i	$-0.397681\pi$
$277$	10.5165	0.631878	0.315939	+	0.948780i	$0.397681\pi$
$278$	0	0
$279$	11.6699	0.698657
$280$	0	0
$281$	−9.71098	−0.579309	−0.289654	−	0.957131i	$-0.593540\pi$
$281$	−9.71098	−0.579309	−0.289654	+	0.957131i	$0.593540\pi$
$282$	0	0
$283$	−26.3036	−1.56359	−0.781794	−	0.623537i	$-0.785695\pi$
$283$	−26.3036	−1.56359	−0.781794	+	0.623537i	$0.785695\pi$
$284$	0	0
$285$	−27.6896	−1.64019
$286$	0	0
$287$	16.2900	0.961570
$288$	0	0
$289$	−10.6921	−0.628949
$290$	0	0
$291$	8.10032	0.474849
$292$	0	0
$293$	29.7985	1.74085	0.870424	−	0.492304i	$-0.163845\pi$
$293$	29.7985	1.74085	0.870424	+	0.492304i	$0.163845\pi$
$294$	0	0
$295$	13.2897	0.773757
$296$	0	0
$297$	17.7430	1.02956
$298$	0	0
$299$	−5.31991	−0.307658
$300$	0	0
$301$	−4.11015	−0.236905
$302$	0	0
$303$	−29.8212	−1.71318
$304$	0	0
$305$	55.3923	3.17175
$306$	0	0
$307$	16.3992	0.935950	0.467975	−	0.883742i	$-0.344984\pi$
$307$	16.3992	0.935950	0.467975	+	0.883742i	$0.344984\pi$
$308$	0	0
$309$	13.0543	0.742633
$310$	0	0
$311$	−14.7604	−0.836984	−0.418492	−	0.908221i	$-0.637441\pi$
$311$	−14.7604	−0.836984	−0.418492	+	0.908221i	$0.637441\pi$
$312$	0	0
$313$	−14.7459	−0.833490	−0.416745	−	0.909023i	$-0.636829\pi$
$313$	−14.7459	−0.833490	−0.416745	+	0.909023i	$0.636829\pi$
$314$	0	0
$315$	6.74439	0.380003
$316$	0	0
$317$	−4.88794	−0.274534	−0.137267	−	0.990534i	$-0.543832\pi$
$317$	−4.88794	−0.274534	−0.137267	+	0.990534i	$0.543832\pi$
$318$	0	0
$319$	17.7275	0.992550
$320$	0	0
$321$	−30.7118	−1.71416
$322$	0	0
$323$	−8.45761	−0.470594
$324$	0	0
$325$	28.3323	1.57159
$326$	0	0
$327$	5.47914	0.302997
$328$	0	0
$329$	8.44878	0.465796
$330$	0	0
$331$	18.2331	1.00218	0.501091	−	0.865395i	$-0.332932\pi$
$331$	18.2331	1.00218	0.501091	+	0.865395i	$0.332932\pi$
$332$	0	0
$333$	12.9254	0.708305
$334$	0	0
$335$	1.60555	0.0877204
$336$	0	0
$337$	14.9239	0.812956	0.406478	−	0.913660i	$-0.366757\pi$
$337$	14.9239	0.812956	0.406478	+	0.913660i	$0.366757\pi$
$338$	0	0
$339$	40.6512	2.20787
$340$	0	0
$341$	47.3950	2.56658
$342$	0	0
$343$	17.1234	0.924577
$344$	0	0
$345$	−17.3246	−0.932728
$346$	0	0
$347$	−4.75424	−0.255221	−0.127610	−	0.991824i	$-0.540731\pi$
$347$	−4.75424	−0.255221	−0.127610	+	0.991824i	$0.540731\pi$
$348$	0	0
$349$	26.8627	1.43793	0.718965	−	0.695047i	$-0.244616\pi$
$349$	26.8627	1.43793	0.718965	+	0.695047i	$0.244616\pi$
$350$	0	0
$351$	9.44302	0.504031
$352$	0	0
$353$	15.9662	0.849794	0.424897	−	0.905242i	$-0.360310\pi$
$353$	15.9662	0.849794	0.424897	+	0.905242i	$0.360310\pi$
$354$	0	0
$355$	−53.3148	−2.82966
$356$	0	0
$357$	7.35049	0.389029
$358$	0	0
$359$	22.0824	1.16546	0.582732	−	0.812664i	$-0.301984\pi$
$359$	22.0824	1.16546	0.582732	+	0.812664i	$0.301984\pi$
$360$	0	0
$361$	−7.66003	−0.403159
$362$	0	0
$363$	−23.4947	−1.23315
$364$	0	0
$365$	29.3595	1.53675
$366$	0	0
$367$	−11.4699	−0.598722	−0.299361	−	0.954140i	$-0.596773\pi$
$367$	−11.4699	−0.598722	−0.299361	+	0.954140i	$0.596773\pi$
$368$	0	0
$369$	−13.2745	−0.691045
$370$	0	0
$371$	14.8800	0.772532
$372$	0	0
$373$	26.8085	1.38809	0.694045	−	0.719932i	$-0.255827\pi$
$373$	26.8085	1.38809	0.694045	+	0.719932i	$0.255827\pi$
$374$	0	0
$375$	51.1529	2.64153
$376$	0	0
$377$	9.43476	0.485915
$378$	0	0
$379$	−16.7440	−0.860079	−0.430040	−	0.902810i	$-0.641500\pi$
$379$	−16.7440	−0.860079	−0.430040	+	0.902810i	$0.641500\pi$
$380$	0	0
$381$	−12.7542	−0.653416
$382$	0	0
$383$	24.4241	1.24801	0.624007	−	0.781419i	$-0.285504\pi$
$383$	24.4241	1.24801	0.624007	+	0.781419i	$0.285504\pi$
$384$	0	0
$385$	27.3911	1.39598
$386$	0	0
$387$	3.34931	0.170255
$388$	0	0
$389$	24.2151	1.22775	0.613877	−	0.789402i	$-0.289609\pi$
$389$	24.2151	1.22775	0.613877	+	0.789402i	$0.289609\pi$
$390$	0	0
$391$	−5.29170	−0.267613
$392$	0	0
$393$	4.59188	0.231630
$394$	0	0
$395$	−23.5121	−1.18302
$396$	0	0
$397$	6.34924	0.318659	0.159330	−	0.987225i	$-0.449067\pi$
$397$	6.34924	0.318659	0.159330	+	0.987225i	$0.449067\pi$
$398$	0	0
$399$	−9.85555	−0.493395
$400$	0	0
$401$	9.12056	0.455459	0.227729	−	0.973724i	$-0.426870\pi$
$401$	9.12056	0.455459	0.227729	+	0.973724i	$0.426870\pi$
$402$	0	0
$403$	25.2241	1.25650
$404$	0	0
$405$	44.8662	2.22942
$406$	0	0
$407$	52.4939	2.60203
$408$	0	0
$409$	38.8212	1.91958	0.959792	−	0.280714i	$-0.0905711\pi$
$409$	38.8212	1.91958	0.959792	+	0.280714i	$0.0905711\pi$
$410$	0	0
$411$	−28.5559	−1.40856
$412$	0	0
$413$	4.73021	0.232758
$414$	0	0
$415$	−44.9636	−2.20718
$416$	0	0
$417$	5.67492	0.277902
$418$	0	0
$419$	−38.4784	−1.87979	−0.939897	−	0.341459i	$-0.889079\pi$
$419$	−38.4784	−1.87979	−0.939897	+	0.341459i	$0.889079\pi$
$420$	0	0
$421$	−38.3070	−1.86697	−0.933485	−	0.358616i	$-0.883249\pi$
$421$	−38.3070	−1.86697	−0.933485	+	0.358616i	$0.883249\pi$
$422$	0	0
$423$	−6.88481	−0.334751
$424$	0	0
$425$	28.1821	1.36703
$426$	0	0
$427$	19.7158	0.954113
$428$	0	0
$429$	−24.4563	−1.18076
$430$	0	0
$431$	35.2293	1.69694	0.848469	−	0.529245i	$-0.177525\pi$
$431$	35.2293	1.69694	0.848469	+	0.529245i	$0.177525\pi$
$432$	0	0
$433$	0.163272	0.00784637	0.00392319	−	0.999992i	$-0.498751\pi$
$433$	0.163272	0.00784637	0.00392319	+	0.999992i	$0.498751\pi$
$434$	0	0
$435$	30.7249	1.47315
$436$	0	0
$437$	7.09512	0.339406
$438$	0	0
$439$	18.7407	0.894443	0.447221	−	0.894423i	$-0.352414\pi$
$439$	18.7407	0.894443	0.447221	+	0.894423i	$0.352414\pi$
$440$	0	0
$441$	−5.77656	−0.275075
$442$	0	0
$443$	−0.290170	−0.0137864	−0.00689319	−	0.999976i	$-0.502194\pi$
$443$	−0.290170	−0.0137864	−0.00689319	+	0.999976i	$0.502194\pi$
$444$	0	0
$445$	22.5634	1.06961
$446$	0	0
$447$	−23.1658	−1.09571
$448$	0	0
$449$	−5.35029	−0.252496	−0.126248	−	0.991999i	$-0.540293\pi$
$449$	−5.35029	−0.252496	−0.126248	+	0.991999i	$0.540293\pi$
$450$	0	0
$451$	−53.9121	−2.53862
$452$	0	0
$453$	−23.7765	−1.11712
$454$	0	0
$455$	14.5778	0.683419
$456$	0	0
$457$	−8.35570	−0.390863	−0.195432	−	0.980717i	$-0.562611\pi$
$457$	−8.35570	−0.390863	−0.195432	+	0.980717i	$0.562611\pi$
$458$	0	0
$459$	9.39295	0.438425
$460$	0	0
$461$	−10.6120	−0.494250	−0.247125	−	0.968984i	$-0.579486\pi$
$461$	−10.6120	−0.494250	−0.247125	+	0.968984i	$0.579486\pi$
$462$	0	0
$463$	−34.0167	−1.58089	−0.790446	−	0.612532i	$-0.790151\pi$
$463$	−34.0167	−1.58089	−0.790446	+	0.612532i	$0.790151\pi$
$464$	0	0
$465$	82.1440	3.80934
$466$	0	0
$467$	6.41597	0.296896	0.148448	−	0.988920i	$-0.452572\pi$
$467$	6.41597	0.296896	0.148448	+	0.988920i	$0.452572\pi$
$468$	0	0
$469$	0.571462	0.0263876
$470$	0	0
$471$	−45.5978	−2.10104
$472$	0	0
$473$	13.6026	0.625448
$474$	0	0
$475$	−37.7866	−1.73377
$476$	0	0
$477$	−12.1255	−0.555191
$478$	0	0
$479$	39.7618	1.81676	0.908380	−	0.418145i	$-0.137320\pi$
$479$	39.7618	1.81676	0.908380	+	0.418145i	$0.137320\pi$
$480$	0	0
$481$	27.9378	1.27385
$482$	0	0
$483$	−6.16636	−0.280579
$484$	0	0
$485$	15.9797	0.725601
$486$	0	0
$487$	25.3925	1.15064	0.575321	−	0.817928i	$-0.304877\pi$
$487$	25.3925	1.15064	0.575321	+	0.817928i	$0.304877\pi$
$488$	0	0
$489$	−9.79966	−0.443156
$490$	0	0
$491$	−33.8772	−1.52886	−0.764429	−	0.644707i	$-0.776979\pi$
$491$	−33.8772	−1.52886	−0.764429	+	0.644707i	$0.776979\pi$
$492$	0	0
$493$	9.38473	0.422667
$494$	0	0
$495$	−22.3207	−1.00324
$496$	0	0
$497$	−18.9763	−0.851205
$498$	0	0
$499$	20.1826	0.903497	0.451748	−	0.892145i	$-0.350801\pi$
$499$	20.1826	0.903497	0.451748	+	0.892145i	$0.350801\pi$
$500$	0	0
$501$	7.17519	0.320564
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−58.8291	−2.61786
$506$	0	0
$507$	13.5250	0.600667
$508$	0	0
$509$	−29.1665	−1.29278	−0.646391	−	0.763006i	$-0.723722\pi$
$509$	−29.1665	−1.29278	−0.646391	+	0.763006i	$0.723722\pi$
$510$	0	0
$511$	10.4499	0.462277
$512$	0	0
$513$	−12.5941	−0.556042
$514$	0	0
$515$	25.7525	1.13479
$516$	0	0
$517$	−27.9614	−1.22974
$518$	0	0
$519$	39.8362	1.74862
$520$	0	0
$521$	−22.5951	−0.989908	−0.494954	−	0.868919i	$-0.664815\pi$
$521$	−22.5951	−0.989908	−0.494954	+	0.868919i	$0.664815\pi$
$522$	0	0
$523$	28.7676	1.25792	0.628960	−	0.777438i	$-0.283481\pi$
$523$	28.7676	1.25792	0.628960	+	0.777438i	$0.283481\pi$
$524$	0	0
$525$	32.8402	1.43327
$526$	0	0
$527$	25.0903	1.09295
$528$	0	0
$529$	−18.5608	−0.806990
$530$	0	0
$531$	−3.85459	−0.167275
$532$	0	0
$533$	−28.6926	−1.24281
$534$	0	0
$535$	−60.5859	−2.61936
$536$	0	0
$537$	21.1386	0.912197
$538$	0	0
$539$	−23.4605	−1.01051
$540$	0	0
$541$	−8.82436	−0.379389	−0.189694	−	0.981843i	$-0.560750\pi$
$541$	−8.82436	−0.379389	−0.189694	+	0.981843i	$0.560750\pi$
$542$	0	0
$543$	−38.3242	−1.64465
$544$	0	0
$545$	10.8088	0.463000
$546$	0	0
$547$	32.7352	1.39965	0.699827	−	0.714312i	$-0.253260\pi$
$547$	32.7352	1.39965	0.699827	+	0.714312i	$0.253260\pi$
$548$	0	0
$549$	−16.0661	−0.685686
$550$	0	0
$551$	−12.5831	−0.536056
$552$	0	0
$553$	−8.36865	−0.355871
$554$	0	0
$555$	90.9813	3.86194
$556$	0	0
$557$	15.8358	0.670984	0.335492	−	0.942043i	$-0.391097\pi$
$557$	15.8358	0.670984	0.335492	+	0.942043i	$0.391097\pi$
$558$	0	0
$559$	7.23944	0.306196
$560$	0	0
$561$	−24.3266	−1.02707
$562$	0	0
$563$	11.6467	0.490850	0.245425	−	0.969416i	$-0.421072\pi$
$563$	11.6467	0.490850	0.245425	+	0.969416i	$0.421072\pi$
$564$	0	0
$565$	80.1938	3.37378
$566$	0	0
$567$	15.9692	0.670644
$568$	0	0
$569$	30.2305	1.26733	0.633664	−	0.773608i	$-0.281550\pi$
$569$	30.2305	1.26733	0.633664	+	0.773608i	$0.281550\pi$
$570$	0	0
$571$	18.6965	0.782423	0.391211	−	0.920301i	$-0.372056\pi$
$571$	18.6965	0.782423	0.391211	+	0.920301i	$0.372056\pi$
$572$	0	0
$573$	20.7037	0.864908
$574$	0	0
$575$	−23.6420	−0.985941
$576$	0	0
$577$	−29.5326	−1.22946	−0.614728	−	0.788739i	$-0.710734\pi$
$577$	−29.5326	−1.22946	−0.614728	+	0.788739i	$0.710734\pi$
$578$	0	0
$579$	−6.62206	−0.275204
$580$	0	0
$581$	−16.0039	−0.663953
$582$	0	0
$583$	−49.2457	−2.03955
$584$	0	0
$585$	−11.8793	−0.491148
$586$	0	0
$587$	−24.4316	−1.00840	−0.504200	−	0.863587i	$-0.668213\pi$
$587$	−24.4316	−1.00840	−0.504200	+	0.863587i	$0.668213\pi$
$588$	0	0
$589$	−33.6412	−1.38616
$590$	0	0
$591$	26.7981	1.10233
$592$	0	0
$593$	−46.6888	−1.91728	−0.958639	−	0.284624i	$-0.908131\pi$
$593$	−46.6888	−1.91728	−0.958639	+	0.284624i	$0.908131\pi$
$594$	0	0
$595$	14.5005	0.594463
$596$	0	0
$597$	−4.02539	−0.164748
$598$	0	0
$599$	−41.9010	−1.71203	−0.856015	−	0.516952i	$-0.827067\pi$
$599$	−41.9010	−1.71203	−0.856015	+	0.516952i	$0.827067\pi$
$600$	0	0
$601$	43.2750	1.76522	0.882612	−	0.470101i	$-0.155783\pi$
$601$	43.2750	1.76522	0.882612	+	0.470101i	$0.155783\pi$
$602$	0	0
$603$	−0.465677	−0.0189638
$604$	0	0
$605$	−46.3485	−1.88433
$606$	0	0
$607$	−17.8186	−0.723236	−0.361618	−	0.932326i	$-0.617776\pi$
$607$	−17.8186	−0.723236	−0.361618	+	0.932326i	$0.617776\pi$
$608$	0	0
$609$	10.9359	0.443146
$610$	0	0
$611$	−14.8813	−0.602034
$612$	0	0
$613$	32.8059	1.32502	0.662508	−	0.749055i	$-0.269492\pi$
$613$	32.8059	1.32502	0.662508	+	0.749055i	$0.269492\pi$
$614$	0	0
$615$	−93.4393	−3.76783
$616$	0	0
$617$	−24.5072	−0.986624	−0.493312	−	0.869852i	$-0.664214\pi$
$617$	−24.5072	−0.986624	−0.493312	+	0.869852i	$0.664214\pi$
$618$	0	0
$619$	37.3665	1.50188	0.750942	−	0.660368i	$-0.229600\pi$
$619$	37.3665	1.50188	0.750942	+	0.660368i	$0.229600\pi$
$620$	0	0
$621$	−7.87978	−0.316205
$622$	0	0
$623$	8.03099	0.321755
$624$	0	0
$625$	44.8057	1.79223
$626$	0	0
$627$	32.6171	1.30260
$628$	0	0
$629$	27.7896	1.10805
$630$	0	0
$631$	−14.9567	−0.595418	−0.297709	−	0.954657i	$-0.596223\pi$
$631$	−14.9567	−0.595418	−0.297709	+	0.954657i	$0.596223\pi$
$632$	0	0
$633$	28.9660	1.15130
$634$	0	0
$635$	−25.1605	−0.998463
$636$	0	0
$637$	−12.4859	−0.494709
$638$	0	0
$639$	15.4636	0.611730
$640$	0	0
$641$	−16.0305	−0.633166	−0.316583	−	0.948565i	$-0.602536\pi$
$641$	−16.0305	−0.633166	−0.316583	+	0.948565i	$0.602536\pi$
$642$	0	0
$643$	19.1138	0.753773	0.376887	−	0.926259i	$-0.376995\pi$
$643$	19.1138	0.753773	0.376887	+	0.926259i	$0.376995\pi$
$644$	0	0
$645$	23.5757	0.928293
$646$	0	0
$647$	6.95218	0.273318	0.136659	−	0.990618i	$-0.456363\pi$
$647$	6.95218	0.273318	0.136659	+	0.990618i	$0.456363\pi$
$648$	0	0
$649$	−15.6547	−0.614501
$650$	0	0
$651$	29.2375	1.14591
$652$	0	0
$653$	10.6819	0.418016	0.209008	−	0.977914i	$-0.432977\pi$
$653$	10.6819	0.418016	0.209008	+	0.977914i	$0.432977\pi$
$654$	0	0
$655$	9.05852	0.353945
$656$	0	0
$657$	−8.51551	−0.332222
$658$	0	0
$659$	−27.8496	−1.08487	−0.542434	−	0.840098i	$-0.682497\pi$
$659$	−27.8496	−1.08487	−0.542434	+	0.840098i	$0.682497\pi$
$660$	0	0
$661$	−30.5988	−1.19016	−0.595078	−	0.803668i	$-0.702879\pi$
$661$	−30.5988	−1.19016	−0.595078	+	0.803668i	$0.702879\pi$
$662$	0	0
$663$	−12.9469	−0.502814
$664$	0	0
$665$	−19.4423	−0.753940
$666$	0	0
$667$	−7.87288	−0.304839
$668$	0	0
$669$	18.3733	0.710355
$670$	0	0
$671$	−65.2497	−2.51894
$672$	0	0
$673$	45.5188	1.75462	0.877311	−	0.479922i	$-0.159335\pi$
$673$	45.5188	1.75462	0.877311	+	0.479922i	$0.159335\pi$
$674$	0	0
$675$	41.9654	1.61525
$676$	0	0
$677$	7.62580	0.293083	0.146542	−	0.989204i	$-0.453186\pi$
$677$	7.62580	0.293083	0.146542	+	0.989204i	$0.453186\pi$
$678$	0	0
$679$	5.68766	0.218272
$680$	0	0
$681$	−19.1249	−0.732868
$682$	0	0
$683$	38.0889	1.45743	0.728716	−	0.684816i	$-0.240117\pi$
$683$	38.0889	1.45743	0.728716	+	0.684816i	$0.240117\pi$
$684$	0	0
$685$	−56.3330	−2.15237
$686$	0	0
$687$	−32.3458	−1.23407
$688$	0	0
$689$	−26.2090	−0.998485
$690$	0	0
$691$	4.60275	0.175097	0.0875484	−	0.996160i	$-0.472097\pi$
$691$	4.60275	0.175097	0.0875484	+	0.996160i	$0.472097\pi$
$692$	0	0
$693$	−7.94459	−0.301790
$694$	0	0
$695$	11.1951	0.424653
$696$	0	0
$697$	−28.5404	−1.08105
$698$	0	0
$699$	13.5493	0.512482
$700$	0	0
$701$	16.3413	0.617202	0.308601	−	0.951192i	$-0.400139\pi$
$701$	16.3413	0.617202	0.308601	+	0.951192i	$0.400139\pi$
$702$	0	0
$703$	−37.2604	−1.40530
$704$	0	0
$705$	−48.4620	−1.82519
$706$	0	0
$707$	−20.9390	−0.787493
$708$	0	0
$709$	23.9152	0.898155	0.449078	−	0.893493i	$-0.351753\pi$
$709$	23.9152	0.898155	0.449078	+	0.893493i	$0.351753\pi$
$710$	0	0
$711$	6.81951	0.255752
$712$	0	0
$713$	−21.0484	−0.788268
$714$	0	0
$715$	−48.2455	−1.80428
$716$	0	0
$717$	27.8594	1.04043
$718$	0	0
$719$	29.9814	1.11812	0.559060	−	0.829127i	$-0.311162\pi$
$719$	29.9814	1.11812	0.559060	+	0.829127i	$0.311162\pi$
$720$	0	0
$721$	9.16610	0.341363
$722$	0	0
$723$	21.2920	0.791858
$724$	0	0
$725$	41.9287	1.55719
$726$	0	0
$727$	5.33674	0.197929	0.0989644	−	0.995091i	$-0.468447\pi$
$727$	5.33674	0.197929	0.0989644	+	0.995091i	$0.468447\pi$
$728$	0	0
$729$	9.89311	0.366412
$730$	0	0
$731$	7.20105	0.266340
$732$	0	0
$733$	−40.7523	−1.50522	−0.752610	−	0.658467i	$-0.771205\pi$
$733$	−40.7523	−1.50522	−0.752610	+	0.658467i	$0.771205\pi$
$734$	0	0
$735$	−40.6611	−1.49981
$736$	0	0
$737$	−1.89126	−0.0696655
$738$	0	0
$739$	14.7824	0.543780	0.271890	−	0.962328i	$-0.412351\pi$
$739$	14.7824	0.543780	0.271890	+	0.962328i	$0.412351\pi$
$740$	0	0
$741$	17.3592	0.637705
$742$	0	0
$743$	9.93503	0.364481	0.182240	−	0.983254i	$-0.441665\pi$
$743$	9.93503	0.364481	0.182240	+	0.983254i	$0.441665\pi$
$744$	0	0
$745$	−45.6998	−1.67431
$746$	0	0
$747$	13.0414	0.477159
$748$	0	0
$749$	−21.5643	−0.787944
$750$	0	0
$751$	14.8421	0.541598	0.270799	−	0.962636i	$-0.412712\pi$
$751$	14.8421	0.541598	0.270799	+	0.962636i	$0.412712\pi$
$752$	0	0
$753$	−33.7059	−1.22831
$754$	0	0
$755$	−46.9045	−1.70703
$756$	0	0
$757$	−20.8979	−0.759547	−0.379773	−	0.925080i	$-0.623998\pi$
$757$	−20.8979	−0.759547	−0.379773	+	0.925080i	$0.623998\pi$
$758$	0	0
$759$	20.4077	0.740752
$760$	0	0
$761$	10.4952	0.380452	0.190226	−	0.981740i	$-0.439078\pi$
$761$	10.4952	0.380452	0.190226	+	0.981740i	$0.439078\pi$
$762$	0	0
$763$	3.84719	0.139278
$764$	0	0
$765$	−11.8163	−0.427219
$766$	0	0
$767$	−8.33159	−0.300836
$768$	0	0
$769$	−8.98967	−0.324176	−0.162088	−	0.986776i	$-0.551823\pi$
$769$	−8.98967	−0.324176	−0.162088	+	0.986776i	$0.551823\pi$
$770$	0	0
$771$	25.8057	0.929370
$772$	0	0
$773$	−0.626074	−0.0225183	−0.0112592	−	0.999937i	$-0.503584\pi$
$773$	−0.626074	−0.0225183	−0.0112592	+	0.999937i	$0.503584\pi$
$774$	0	0
$775$	112.098	4.02666
$776$	0	0
$777$	32.3830	1.16173
$778$	0	0
$779$	38.2670	1.37106
$780$	0	0
$781$	62.8025	2.24725
$782$	0	0
$783$	13.9746	0.499413
$784$	0	0
$785$	−89.9520	−3.21052
$786$	0	0
$787$	46.1734	1.64590	0.822952	−	0.568111i	$-0.192326\pi$
$787$	46.1734	1.64590	0.822952	+	0.568111i	$0.192326\pi$
$788$	0	0
$789$	−25.0928	−0.893326
$790$	0	0
$791$	28.5433	1.01488
$792$	0	0
$793$	−34.7265	−1.23318
$794$	0	0
$795$	−85.3515	−3.02711
$796$	0	0
$797$	30.2424	1.07124	0.535620	−	0.844459i	$-0.320078\pi$
$797$	30.2424	1.07124	0.535620	+	0.844459i	$0.320078\pi$
$798$	0	0
$799$	−14.8024	−0.523672
$800$	0	0
$801$	−6.54436	−0.231234
$802$	0	0
$803$	−34.5842	−1.22045
$804$	0	0
$805$	−12.1645	−0.428743
$806$	0	0
$807$	49.2802	1.73475
$808$	0	0
$809$	6.78649	0.238600	0.119300	−	0.992858i	$-0.461935\pi$
$809$	6.78649	0.238600	0.119300	+	0.992858i	$0.461935\pi$
$810$	0	0
$811$	−7.35125	−0.258137	−0.129069	−	0.991636i	$-0.541199\pi$
$811$	−7.35125	−0.258137	−0.129069	+	0.991636i	$0.541199\pi$
$812$	0	0
$813$	16.5345	0.579891
$814$	0	0
$815$	−19.3320	−0.677172
$816$	0	0
$817$	−9.65518	−0.337792
$818$	0	0
$819$	−4.22819	−0.147745
$820$	0	0
$821$	−3.55064	−0.123918	−0.0619591	−	0.998079i	$-0.519735\pi$
$821$	−3.55064	−0.123918	−0.0619591	+	0.998079i	$0.519735\pi$
$822$	0	0
$823$	28.6601	0.999028	0.499514	−	0.866306i	$-0.333512\pi$
$823$	28.6601	0.999028	0.499514	+	0.866306i	$0.333512\pi$
$824$	0	0
$825$	−108.685	−3.78394
$826$	0	0
$827$	29.4927	1.02556	0.512782	−	0.858519i	$-0.328615\pi$
$827$	29.4927	1.02556	0.512782	+	0.858519i	$0.328615\pi$
$828$	0	0
$829$	34.8454	1.21023	0.605115	−	0.796138i	$-0.293127\pi$
$829$	34.8454	1.21023	0.605115	+	0.796138i	$0.293127\pi$
$830$	0	0
$831$	−21.4706	−0.744808
$832$	0	0
$833$	−12.4197	−0.430316
$834$	0	0
$835$	14.1547	0.489843
$836$	0	0
$837$	37.3616	1.29141
$838$	0	0
$839$	−53.8614	−1.85950	−0.929751	−	0.368188i	$-0.879978\pi$
$839$	−53.8614	−1.85950	−0.929751	+	0.368188i	$0.879978\pi$
$840$	0	0
$841$	−15.0376	−0.518538
$842$	0	0
$843$	19.8260	0.682844
$844$	0	0
$845$	26.6811	0.917859
$846$	0	0
$847$	−16.4968	−0.566837
$848$	0	0
$849$	53.7016	1.84304
$850$	0	0
$851$	−23.3128	−0.799154
$852$	0	0
$853$	0.446115	0.0152747	0.00763734	−	0.999971i	$-0.497569\pi$
$853$	0.446115	0.0152747	0.00763734	+	0.999971i	$0.497569\pi$
$854$	0	0
$855$	15.8433	0.541829
$856$	0	0
$857$	34.2839	1.17112	0.585558	−	0.810630i	$-0.300875\pi$
$857$	34.2839	1.17112	0.585558	+	0.810630i	$0.300875\pi$
$858$	0	0
$859$	−42.5154	−1.45061	−0.725304	−	0.688429i	$-0.758301\pi$
$859$	−42.5154	−1.45061	−0.725304	+	0.688429i	$0.758301\pi$
$860$	0	0
$861$	−33.2578	−1.13342
$862$	0	0
$863$	27.0050	0.919260	0.459630	−	0.888111i	$-0.347982\pi$
$863$	27.0050	0.919260	0.459630	+	0.888111i	$0.347982\pi$
$864$	0	0
$865$	78.5859	2.67200
$866$	0	0
$867$	21.8291	0.741355
$868$	0	0
$869$	27.6962	0.939529
$870$	0	0
$871$	−1.00655	−0.0341056
$872$	0	0
$873$	−4.63480	−0.156864
$874$	0	0
$875$	35.9171	1.21422
$876$	0	0
$877$	7.58347	0.256076	0.128038	−	0.991769i	$-0.459132\pi$
$877$	7.58347	0.256076	0.128038	+	0.991769i	$0.459132\pi$
$878$	0	0
$879$	−60.8368	−2.05197
$880$	0	0
$881$	40.8783	1.37722	0.688612	−	0.725130i	$-0.258220\pi$
$881$	40.8783	1.37722	0.688612	+	0.725130i	$0.258220\pi$
$882$	0	0
$883$	37.2883	1.25485	0.627425	−	0.778677i	$-0.284109\pi$
$883$	37.2883	1.25485	0.627425	+	0.778677i	$0.284109\pi$
$884$	0	0
$885$	−27.1324	−0.912045
$886$	0	0
$887$	8.46297	0.284159	0.142079	−	0.989855i	$-0.454621\pi$
$887$	8.46297	0.284159	0.142079	+	0.989855i	$0.454621\pi$
$888$	0	0
$889$	−8.95536	−0.300353
$890$	0	0
$891$	−52.8504	−1.77055
$892$	0	0
$893$	19.8471	0.664158
$894$	0	0
$895$	41.7006	1.39390
$896$	0	0
$897$	10.8612	0.362644
$898$	0	0
$899$	37.3289	1.24499
$900$	0	0
$901$	−26.0701	−0.868519
$902$	0	0
$903$	8.39130	0.279245
$904$	0	0
$905$	−75.6032	−2.51314
$906$	0	0
$907$	−13.7716	−0.457277	−0.228638	−	0.973511i	$-0.573427\pi$
$907$	−13.7716	−0.457277	−0.228638	+	0.973511i	$0.573427\pi$
$908$	0	0
$909$	17.0630	0.565942
$910$	0	0
$911$	−50.9286	−1.68734	−0.843670	−	0.536863i	$-0.819609\pi$
$911$	−50.9286	−1.68734	−0.843670	+	0.536863i	$0.819609\pi$
$912$	0	0
$913$	52.9651	1.75289
$914$	0	0
$915$	−113.089	−3.73862
$916$	0	0
$917$	3.22420	0.106472
$918$	0	0
$919$	7.76364	0.256099	0.128049	−	0.991768i	$-0.459128\pi$
$919$	7.76364	0.256099	0.128049	+	0.991768i	$0.459128\pi$
$920$	0	0
$921$	−33.4806	−1.10322
$922$	0	0
$923$	33.4241	1.10017
$924$	0	0
$925$	124.157	4.08227
$926$	0	0
$927$	−7.46934	−0.245325
$928$	0	0
$929$	−25.1700	−0.825799	−0.412900	−	0.910777i	$-0.635484\pi$
$929$	−25.1700	−0.825799	−0.412900	+	0.910777i	$0.635484\pi$
$930$	0	0
$931$	16.6523	0.545758
$932$	0	0
$933$	30.1349	0.986571
$934$	0	0
$935$	−47.9897	−1.56943
$936$	0	0
$937$	8.66015	0.282915	0.141457	−	0.989944i	$-0.454821\pi$
$937$	8.66015	0.282915	0.141457	+	0.989944i	$0.454821\pi$
$938$	0	0
$939$	30.1054	0.982453
$940$	0	0
$941$	−26.7089	−0.870685	−0.435342	−	0.900265i	$-0.643373\pi$
$941$	−26.7089	−0.870685	−0.435342	+	0.900265i	$0.643373\pi$
$942$	0	0
$943$	23.9427	0.779680
$944$	0	0
$945$	21.5925	0.702403
$946$	0	0
$947$	35.2619	1.14586	0.572929	−	0.819605i	$-0.305807\pi$
$947$	35.2619	1.14586	0.572929	+	0.819605i	$0.305807\pi$
$948$	0	0
$949$	−18.4060	−0.597485
$950$	0	0
$951$	9.97925	0.323599
$952$	0	0
$953$	20.2501	0.655965	0.327983	−	0.944684i	$-0.393631\pi$
$953$	20.2501	0.655965	0.327983	+	0.944684i	$0.393631\pi$
$954$	0	0
$955$	40.8426	1.32164
$956$	0	0
$957$	−36.1926	−1.16994
$958$	0	0
$959$	−20.0506	−0.647467
$960$	0	0
$961$	68.7999	2.21935
$962$	0	0
$963$	17.5725	0.566267
$964$	0	0
$965$	−13.0635	−0.420530
$966$	0	0
$967$	−15.0414	−0.483699	−0.241850	−	0.970314i	$-0.577754\pi$
$967$	−15.0414	−0.483699	−0.241850	+	0.970314i	$0.577754\pi$
$968$	0	0
$969$	17.2671	0.554699
$970$	0	0
$971$	−34.6488	−1.11193	−0.555966	−	0.831205i	$-0.687652\pi$
$971$	−34.6488	−1.11193	−0.555966	+	0.831205i	$0.687652\pi$
$972$	0	0
$973$	3.98466	0.127742
$974$	0	0
$975$	−57.8434	−1.85247
$976$	0	0
$977$	−11.5906	−0.370815	−0.185408	−	0.982662i	$-0.559361\pi$
$977$	−11.5906	−0.370815	−0.185408	+	0.982662i	$0.559361\pi$
$978$	0	0
$979$	−26.5787	−0.849460
$980$	0	0
$981$	−3.13503	−0.100094
$982$	0	0
$983$	25.3940	0.809943	0.404972	−	0.914329i	$-0.367281\pi$
$983$	25.3940	0.809943	0.404972	+	0.914329i	$0.367281\pi$
$984$	0	0
$985$	52.8652	1.68443
$986$	0	0
$987$	−17.2491	−0.549044
$988$	0	0
$989$	−6.04099	−0.192092
$990$	0	0
$991$	−40.3841	−1.28284	−0.641422	−	0.767188i	$-0.721655\pi$
$991$	−40.3841	−1.28284	−0.641422	+	0.767188i	$0.721655\pi$
$992$	0	0
$993$	−37.2248	−1.18129
$994$	0	0
$995$	−7.94098	−0.251746
$996$	0	0
$997$	−33.1846	−1.05097	−0.525484	−	0.850804i	$-0.676116\pi$
$997$	−33.1846	−1.05097	−0.525484	+	0.850804i	$0.676116\pi$
$998$	0	0
$999$	41.3811	1.30924

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.w.1.5		29
4.3	odd	2		4024.2.a.e.1.25	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.25	✓	29	4.3	odd	2
8048.2.a.w.1.5		29	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-2.04161 q^{3} -4.02753 q^{5} -1.43352 q^{7} +1.16816 q^{9} +O(q^{10})\)	\(q-2.04161 q^{3} -4.02753 q^{5} -1.43352 q^{7} +1.16816 q^{9} +4.74425 q^{11} +2.52494 q^{13} +8.22263 q^{15} +2.51155 q^{17} -3.36749 q^{19} +2.92668 q^{21} -2.10695 q^{23} +11.2210 q^{25} +3.73990 q^{27} +3.73663 q^{29} +9.98999 q^{31} -9.68589 q^{33} +5.77353 q^{35} +11.0647 q^{37} -5.15493 q^{39} -11.3637 q^{41} +2.86718 q^{43} -4.70478 q^{45} -5.89374 q^{47} -4.94503 q^{49} -5.12759 q^{51} -10.3801 q^{53} -19.1076 q^{55} +6.87508 q^{57} -3.29972 q^{59} -13.7534 q^{61} -1.67457 q^{63} -10.1693 q^{65} -0.398643 q^{67} +4.30156 q^{69} +13.2376 q^{71} -7.28970 q^{73} -22.9089 q^{75} -6.80097 q^{77} +5.83784 q^{79} -11.1399 q^{81} +11.1641 q^{83} -10.1153 q^{85} -7.62873 q^{87} -5.60230 q^{89} -3.61954 q^{91} -20.3956 q^{93} +13.5627 q^{95} -3.96762 q^{97} +5.54203 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9	\( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9} + 27 q^{11} + 16 q^{13} + 14 q^{15} - 15 q^{17} + 14 q^{19} + q^{21} + 25 q^{23} + 21 q^{25} + 25 q^{27} - 13 q^{29} + 27 q^{31} - 9 q^{33} + 29 q^{35} + 35 q^{37} + 38 q^{39} - 30 q^{41} + 38 q^{43} + q^{45} + 35 q^{47} + 14 q^{49} + 21 q^{51} + 2 q^{53} + 25 q^{55} - 25 q^{57} + 40 q^{59} + 10 q^{61} + 56 q^{63} - 50 q^{65} + 31 q^{67} + 11 q^{69} + 65 q^{71} - 23 q^{73} + 32 q^{75} + 13 q^{77} + 44 q^{79} - 7 q^{81} + 41 q^{83} + 26 q^{85} + 25 q^{87} - 48 q^{89} + 44 q^{91} + 25 q^{93} + 75 q^{95} - 18 q^{97} + 80 q^{99}+O(q^{100}) \) 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9 + 27 * q^11 + 16 * q^13 + 14 * q^15 - 15 * q^17 + 14 * q^19 + q^21 + 25 * q^23 + 21 * q^25 + 25 * q^27 - 13 * q^29 + 27 * q^31 - 9 * q^33 + 29 * q^35 + 35 * q^37 + 38 * q^39 - 30 * q^41 + 38 * q^43 + q^45 + 35 * q^47 + 14 * q^49 + 21 * q^51 + 2 * q^53 + 25 * q^55 - 25 * q^57 + 40 * q^59 + 10 * q^61 + 56 * q^63 - 50 * q^65 + 31 * q^67 + 11 * q^69 + 65 * q^71 - 23 * q^73 + 32 * q^75 + 13 * q^77 + 44 * q^79 - 7 * q^81 + 41 * q^83 + 26 * q^85 + 25 * q^87 - 48 * q^89 + 44 * q^91 + 25 * q^93 + 75 * q^95 - 18 * q^97 + 80 * q^99

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