LMFDB - Embedded newform 8024.2.a.z.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.96328 q^{3} -2.74825 q^{5} -4.75780 q^{7} +0.854459 q^{9} +O(q^{10})$	$q-1.96328 q^{3} -2.74825 q^{5} -4.75780 q^{7} +0.854459 q^{9} -4.54979 q^{11} +2.89623 q^{13} +5.39558 q^{15} +1.00000 q^{17} -0.411253 q^{19} +9.34088 q^{21} -8.30979 q^{23} +2.55287 q^{25} +4.21229 q^{27} +9.19039 q^{29} -3.50170 q^{31} +8.93250 q^{33} +13.0756 q^{35} -8.19445 q^{37} -5.68611 q^{39} +0.0614700 q^{41} -5.65351 q^{43} -2.34827 q^{45} -5.64915 q^{47} +15.6367 q^{49} -1.96328 q^{51} +0.672233 q^{53} +12.5040 q^{55} +0.807403 q^{57} +1.00000 q^{59} +5.68228 q^{61} -4.06535 q^{63} -7.95957 q^{65} +15.6423 q^{67} +16.3144 q^{69} +10.1285 q^{71} +12.3224 q^{73} -5.01200 q^{75} +21.6470 q^{77} -9.01116 q^{79} -10.8333 q^{81} +13.2160 q^{83} -2.74825 q^{85} -18.0433 q^{87} -7.24833 q^{89} -13.7797 q^{91} +6.87481 q^{93} +1.13023 q^{95} +9.23654 q^{97} -3.88761 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9}+O(q^{10}) $ 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9	$ 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9} - 15 q^{11} - 17 q^{13} - 7 q^{15} + 24 q^{17} - q^{19} + 10 q^{21} - 21 q^{23} + 19 q^{25} - 26 q^{27} - 18 q^{29} - 23 q^{31} + 10 q^{33} - 5 q^{35} - 29 q^{37} - 44 q^{39} + 14 q^{41} - 23 q^{43} - 24 q^{45} - 17 q^{47} + 46 q^{49} - 5 q^{51} + 7 q^{53} - 33 q^{55} - 2 q^{57} + 24 q^{59} - 9 q^{61} - 52 q^{63} - 15 q^{65} + 2 q^{67} + 2 q^{69} - 51 q^{71} - 18 q^{73} + 7 q^{75} + 14 q^{77} - 36 q^{79} + 12 q^{81} - 39 q^{83} - 7 q^{85} - 5 q^{87} + 46 q^{89} - 52 q^{91} - 20 q^{93} - 52 q^{95} + 50 q^{97} - 53 q^{99}+O(q^{100}) $ 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9 - 15 * q^11 - 17 * q^13 - 7 * q^15 + 24 * q^17 - q^19 + 10 * q^21 - 21 * q^23 + 19 * q^25 - 26 * q^27 - 18 * q^29 - 23 * q^31 + 10 * q^33 - 5 * q^35 - 29 * q^37 - 44 * q^39 + 14 * q^41 - 23 * q^43 - 24 * q^45 - 17 * q^47 + 46 * q^49 - 5 * q^51 + 7 * q^53 - 33 * q^55 - 2 * q^57 + 24 * q^59 - 9 * q^61 - 52 * q^63 - 15 * q^65 + 2 * q^67 + 2 * q^69 - 51 * q^71 - 18 * q^73 + 7 * q^75 + 14 * q^77 - 36 * q^79 + 12 * q^81 - 39 * q^83 - 7 * q^85 - 5 * q^87 + 46 * q^89 - 52 * q^91 - 20 * q^93 - 52 * q^95 + 50 * q^97 - 53 * 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
$3$	−1.96328	−1.13350	−0.566749	−	0.823890i	$-0.691799\pi$
$3$	−1.96328	−1.13350	−0.566749	+	0.823890i	$0.691799\pi$
$4$	0	0
$5$	−2.74825	−1.22905	−0.614527	−	0.788896i	$-0.710653\pi$
$5$	−2.74825	−1.22905	−0.614527	+	0.788896i	$0.710653\pi$
$6$	0	0
$7$	−4.75780	−1.79828	−0.899140	−	0.437661i	$-0.855807\pi$
$7$	−4.75780	−1.79828	−0.899140	+	0.437661i	$0.855807\pi$
$8$	0	0
$9$	0.854459	0.284820
$10$	0	0
$11$	−4.54979	−1.37181	−0.685907	−	0.727690i	$-0.740594\pi$
$11$	−4.54979	−1.37181	−0.685907	+	0.727690i	$0.740594\pi$
$12$	0	0
$13$	2.89623	0.803270	0.401635	−	0.915800i	$-0.368442\pi$
$13$	2.89623	0.803270	0.401635	+	0.915800i	$0.368442\pi$
$14$	0	0
$15$	5.39558	1.39313
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−0.411253	−0.0943479	−0.0471739	−	0.998887i	$-0.515022\pi$
$19$	−0.411253	−0.0943479	−0.0471739	+	0.998887i	$0.515022\pi$
$20$	0	0
$21$	9.34088	2.03835
$22$	0	0
$23$	−8.30979	−1.73271	−0.866356	−	0.499428i	$-0.833544\pi$
$23$	−8.30979	−1.73271	−0.866356	+	0.499428i	$0.833544\pi$
$24$	0	0
$25$	2.55287	0.510574
$26$	0	0
$27$	4.21229	0.810656
$28$	0	0
$29$	9.19039	1.70661	0.853307	−	0.521409i	$-0.174594\pi$
$29$	9.19039	1.70661	0.853307	+	0.521409i	$0.174594\pi$
$30$	0	0
$31$	−3.50170	−0.628924	−0.314462	−	0.949270i	$-0.601824\pi$
$31$	−3.50170	−0.628924	−0.314462	+	0.949270i	$0.601824\pi$
$32$	0	0
$33$	8.93250	1.55495
$34$	0	0
$35$	13.0756	2.21018
$36$	0	0
$37$	−8.19445	−1.34716	−0.673580	−	0.739114i	$-0.735244\pi$
$37$	−8.19445	−1.34716	−0.673580	+	0.739114i	$0.735244\pi$
$38$	0	0
$39$	−5.68611	−0.910506
$40$	0	0
$41$	0.0614700	0.00960000	0.00480000	−	0.999988i	$-0.498472\pi$
$41$	0.0614700	0.00960000	0.00480000	+	0.999988i	$0.498472\pi$
$42$	0	0
$43$	−5.65351	−0.862152	−0.431076	−	0.902316i	$-0.641866\pi$
$43$	−5.65351	−0.862152	−0.431076	+	0.902316i	$0.641866\pi$
$44$	0	0
$45$	−2.34827	−0.350059
$46$	0	0
$47$	−5.64915	−0.824013	−0.412006	−	0.911181i	$-0.635172\pi$
$47$	−5.64915	−0.824013	−0.412006	+	0.911181i	$0.635172\pi$
$48$	0	0
$49$	15.6367	2.23381
$50$	0	0
$51$	−1.96328	−0.274914
$52$	0	0
$53$	0.672233	0.0923383	0.0461691	−	0.998934i	$-0.485299\pi$
$53$	0.672233	0.0923383	0.0461691	+	0.998934i	$0.485299\pi$
$54$	0	0
$55$	12.5040	1.68603
$56$	0	0
$57$	0.807403	0.106943
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	5.68228	0.727542	0.363771	−	0.931488i	$-0.381489\pi$
$61$	5.68228	0.727542	0.363771	+	0.931488i	$0.381489\pi$
$62$	0	0
$63$	−4.06535	−0.512186
$64$	0	0
$65$	−7.95957	−0.987263
$66$	0	0
$67$	15.6423	1.91101	0.955506	−	0.294971i	$-0.0953099\pi$
$67$	15.6423	1.91101	0.955506	+	0.294971i	$0.0953099\pi$
$68$	0	0
$69$	16.3144	1.96403
$70$	0	0
$71$	10.1285	1.20203	0.601017	−	0.799236i	$-0.294762\pi$
$71$	10.1285	1.20203	0.601017	+	0.799236i	$0.294762\pi$
$72$	0	0
$73$	12.3224	1.44223	0.721115	−	0.692815i	$-0.243630\pi$
$73$	12.3224	1.44223	0.721115	+	0.692815i	$0.243630\pi$
$74$	0	0
$75$	−5.01200	−0.578735
$76$	0	0
$77$	21.6470	2.46690
$78$	0	0
$79$	−9.01116	−1.01383	−0.506917	−	0.861995i	$-0.669215\pi$
$79$	−9.01116	−1.01383	−0.506917	+	0.861995i	$0.669215\pi$
$80$	0	0
$81$	−10.8333	−1.20370
$82$	0	0
$83$	13.2160	1.45064	0.725321	−	0.688411i	$-0.241691\pi$
$83$	13.2160	1.45064	0.725321	+	0.688411i	$0.241691\pi$
$84$	0	0
$85$	−2.74825	−0.298089
$86$	0	0
$87$	−18.0433	−1.93444
$88$	0	0
$89$	−7.24833	−0.768321	−0.384161	−	0.923266i	$-0.625509\pi$
$89$	−7.24833	−0.768321	−0.384161	+	0.923266i	$0.625509\pi$
$90$	0	0
$91$	−13.7797	−1.44450
$92$	0	0
$93$	6.87481	0.712884
$94$	0	0
$95$	1.13023	0.115959
$96$	0	0
$97$	9.23654	0.937828	0.468914	−	0.883244i	$-0.344645\pi$
$97$	9.23654	0.937828	0.468914	+	0.883244i	$0.344645\pi$
$98$	0	0
$99$	−3.88761	−0.390719
$100$	0	0
$101$	−2.87486	−0.286059	−0.143030	−	0.989718i	$-0.545684\pi$
$101$	−2.87486	−0.286059	−0.143030	+	0.989718i	$0.545684\pi$
$102$	0	0
$103$	11.2026	1.10383	0.551913	−	0.833902i	$-0.313898\pi$
$103$	11.2026	1.10383	0.551913	+	0.833902i	$0.313898\pi$
$104$	0	0
$105$	−25.6711	−2.50524
$106$	0	0
$107$	12.9939	1.25617	0.628084	−	0.778146i	$-0.283840\pi$
$107$	12.9939	1.25617	0.628084	+	0.778146i	$0.283840\pi$
$108$	0	0
$109$	−10.2495	−0.981722	−0.490861	−	0.871238i	$-0.663318\pi$
$109$	−10.2495	−0.981722	−0.490861	+	0.871238i	$0.663318\pi$
$110$	0	0
$111$	16.0880	1.52700
$112$	0	0
$113$	11.7232	1.10282	0.551411	−	0.834234i	$-0.314090\pi$
$113$	11.7232	1.10282	0.551411	+	0.834234i	$0.314090\pi$
$114$	0	0
$115$	22.8374	2.12960
$116$	0	0
$117$	2.47471	0.228787
$118$	0	0
$119$	−4.75780	−0.436147
$120$	0	0
$121$	9.70058	0.881871
$122$	0	0
$123$	−0.120683	−0.0108816
$124$	0	0
$125$	6.72532	0.601531
$126$	0	0
$127$	4.23859	0.376114	0.188057	−	0.982158i	$-0.439781\pi$
$127$	4.23859	0.376114	0.188057	+	0.982158i	$0.439781\pi$
$128$	0	0
$129$	11.0994	0.977248
$130$	0	0
$131$	−0.935003	−0.0816916	−0.0408458	−	0.999165i	$-0.513005\pi$
$131$	−0.935003	−0.0816916	−0.0408458	+	0.999165i	$0.513005\pi$
$132$	0	0
$133$	1.95666	0.169664
$134$	0	0
$135$	−11.5764	−0.996340
$136$	0	0
$137$	18.7236	1.59966	0.799831	−	0.600225i	$-0.204922\pi$
$137$	18.7236	1.59966	0.799831	+	0.600225i	$0.204922\pi$
$138$	0	0
$139$	−5.62515	−0.477119	−0.238559	−	0.971128i	$-0.576675\pi$
$139$	−5.62515	−0.477119	−0.238559	+	0.971128i	$0.576675\pi$
$140$	0	0
$141$	11.0908	0.934017
$142$	0	0
$143$	−13.1772	−1.10194
$144$	0	0
$145$	−25.2575	−2.09752
$146$	0	0
$147$	−30.6991	−2.53202
$148$	0	0
$149$	−7.61335	−0.623710	−0.311855	−	0.950130i	$-0.600950\pi$
$149$	−7.61335	−0.623710	−0.311855	+	0.950130i	$0.600950\pi$
$150$	0	0
$151$	−5.71065	−0.464726	−0.232363	−	0.972629i	$-0.574646\pi$
$151$	−5.71065	−0.464726	−0.232363	+	0.972629i	$0.574646\pi$
$152$	0	0
$153$	0.854459	0.0690789
$154$	0	0
$155$	9.62354	0.772981
$156$	0	0
$157$	−19.7706	−1.57787	−0.788935	−	0.614477i	$-0.789367\pi$
$157$	−19.7706	−1.57787	−0.788935	+	0.614477i	$0.789367\pi$
$158$	0	0
$159$	−1.31978	−0.104665
$160$	0	0
$161$	39.5363	3.11590
$162$	0	0
$163$	−12.1182	−0.949171	−0.474585	−	0.880209i	$-0.657402\pi$
$163$	−12.1182	−0.949171	−0.474585	+	0.880209i	$0.657402\pi$
$164$	0	0
$165$	−24.5487	−1.91112
$166$	0	0
$167$	15.3810	1.19022	0.595108	−	0.803646i	$-0.297109\pi$
$167$	15.3810	1.19022	0.595108	+	0.803646i	$0.297109\pi$
$168$	0	0
$169$	−4.61184	−0.354757
$170$	0	0
$171$	−0.351399	−0.0268721
$172$	0	0
$173$	−3.52643	−0.268109	−0.134055	−	0.990974i	$-0.542800\pi$
$173$	−3.52643	−0.268109	−0.134055	+	0.990974i	$0.542800\pi$
$174$	0	0
$175$	−12.1461	−0.918156
$176$	0	0
$177$	−1.96328	−0.147569
$178$	0	0
$179$	−25.5607	−1.91050	−0.955248	−	0.295807i	$-0.904412\pi$
$179$	−25.5607	−1.91050	−0.955248	+	0.295807i	$0.904412\pi$
$180$	0	0
$181$	−22.9055	−1.70255	−0.851275	−	0.524719i	$-0.824170\pi$
$181$	−22.9055	−1.70255	−0.851275	+	0.524719i	$0.824170\pi$
$182$	0	0
$183$	−11.1559	−0.824668
$184$	0	0
$185$	22.5204	1.65573
$186$	0	0
$187$	−4.54979	−0.332714
$188$	0	0
$189$	−20.0412	−1.45779
$190$	0	0
$191$	5.26077	0.380656	0.190328	−	0.981721i	$-0.439045\pi$
$191$	5.26077	0.380656	0.190328	+	0.981721i	$0.439045\pi$
$192$	0	0
$193$	−22.3365	−1.60782	−0.803908	−	0.594754i	$-0.797249\pi$
$193$	−22.3365	−1.60782	−0.803908	+	0.594754i	$0.797249\pi$
$194$	0	0
$195$	15.6268	1.11906
$196$	0	0
$197$	−14.6153	−1.04130	−0.520648	−	0.853771i	$-0.674310\pi$
$197$	−14.6153	−1.04130	−0.520648	+	0.853771i	$0.674310\pi$
$198$	0	0
$199$	19.0397	1.34969	0.674844	−	0.737961i	$-0.264211\pi$
$199$	19.0397	1.34969	0.674844	+	0.737961i	$0.264211\pi$
$200$	0	0
$201$	−30.7102	−2.16613
$202$	0	0
$203$	−43.7261	−3.06897
$204$	0	0
$205$	−0.168935	−0.0117989
$206$	0	0
$207$	−7.10038	−0.493510
$208$	0	0
$209$	1.87111	0.129428
$210$	0	0
$211$	0.926998	0.0638171	0.0319086	−	0.999491i	$-0.489841\pi$
$211$	0.926998	0.0638171	0.0319086	+	0.999491i	$0.489841\pi$
$212$	0	0
$213$	−19.8851	−1.36250
$214$	0	0
$215$	15.5372	1.05963
$216$	0	0
$217$	16.6604	1.13098
$218$	0	0
$219$	−24.1923	−1.63477
$220$	0	0
$221$	2.89623	0.194822
$222$	0	0
$223$	13.4816	0.902792	0.451396	−	0.892324i	$-0.350926\pi$
$223$	13.4816	0.902792	0.451396	+	0.892324i	$0.350926\pi$
$224$	0	0
$225$	2.18132	0.145422
$226$	0	0
$227$	−1.35655	−0.0900373	−0.0450186	−	0.998986i	$-0.514335\pi$
$227$	−1.35655	−0.0900373	−0.0450186	+	0.998986i	$0.514335\pi$
$228$	0	0
$229$	18.4179	1.21709	0.608545	−	0.793520i	$-0.291754\pi$
$229$	18.4179	1.21709	0.608545	+	0.793520i	$0.291754\pi$
$230$	0	0
$231$	−42.4991	−2.79623
$232$	0	0
$233$	−15.4383	−1.01140	−0.505699	−	0.862710i	$-0.668765\pi$
$233$	−15.4383	−1.01140	−0.505699	+	0.862710i	$0.668765\pi$
$234$	0	0
$235$	15.5253	1.01276
$236$	0	0
$237$	17.6914	1.14918
$238$	0	0
$239$	8.88794	0.574913	0.287457	−	0.957794i	$-0.407190\pi$
$239$	8.88794	0.574913	0.287457	+	0.957794i	$0.407190\pi$
$240$	0	0
$241$	7.93978	0.511446	0.255723	−	0.966750i	$-0.417686\pi$
$241$	7.93978	0.511446	0.255723	+	0.966750i	$0.417686\pi$
$242$	0	0
$243$	8.63185	0.553734
$244$	0	0
$245$	−42.9735	−2.74547
$246$	0	0
$247$	−1.19108	−0.0757868
$248$	0	0
$249$	−25.9466	−1.64430
$250$	0	0
$251$	−11.6654	−0.736315	−0.368158	−	0.929763i	$-0.620011\pi$
$251$	−11.6654	−0.736315	−0.368158	+	0.929763i	$0.620011\pi$
$252$	0	0
$253$	37.8078	2.37696
$254$	0	0
$255$	5.39558	0.337884
$256$	0	0
$257$	0.879250	0.0548461	0.0274230	−	0.999624i	$-0.491270\pi$
$257$	0.879250	0.0548461	0.0274230	+	0.999624i	$0.491270\pi$
$258$	0	0
$259$	38.9876	2.42257
$260$	0	0
$261$	7.85282	0.486077
$262$	0	0
$263$	−3.81079	−0.234984	−0.117492	−	0.993074i	$-0.537485\pi$
$263$	−3.81079	−0.234984	−0.117492	+	0.993074i	$0.537485\pi$
$264$	0	0
$265$	−1.84746	−0.113489
$266$	0	0
$267$	14.2305	0.870891
$268$	0	0
$269$	−24.2269	−1.47714	−0.738571	−	0.674176i	$-0.764499\pi$
$269$	−24.2269	−1.47714	−0.738571	+	0.674176i	$0.764499\pi$
$270$	0	0
$271$	5.05251	0.306918	0.153459	−	0.988155i	$-0.450959\pi$
$271$	5.05251	0.306918	0.153459	+	0.988155i	$0.450959\pi$
$272$	0	0
$273$	27.0534	1.63734
$274$	0	0
$275$	−11.6150	−0.700413
$276$	0	0
$277$	3.26328	0.196071	0.0980357	−	0.995183i	$-0.468744\pi$
$277$	3.26328	0.196071	0.0980357	+	0.995183i	$0.468744\pi$
$278$	0	0
$279$	−2.99206	−0.179130
$280$	0	0
$281$	17.1739	1.02451	0.512253	−	0.858834i	$-0.328811\pi$
$281$	17.1739	1.02451	0.512253	+	0.858834i	$0.328811\pi$
$282$	0	0
$283$	−9.30702	−0.553245	−0.276622	−	0.960979i	$-0.589215\pi$
$283$	−9.30702	−0.553245	−0.276622	+	0.960979i	$0.589215\pi$
$284$	0	0
$285$	−2.21895	−0.131439
$286$	0	0
$287$	−0.292462	−0.0172635
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−18.1339	−1.06303
$292$	0	0
$293$	29.1452	1.70268	0.851340	−	0.524614i	$-0.175790\pi$
$293$	29.1452	1.70268	0.851340	+	0.524614i	$0.175790\pi$
$294$	0	0
$295$	−2.74825	−0.160009
$296$	0	0
$297$	−19.1650	−1.11207
$298$	0	0
$299$	−24.0671	−1.39184
$300$	0	0
$301$	26.8983	1.55039
$302$	0	0
$303$	5.64415	0.324248
$304$	0	0
$305$	−15.6163	−0.894188
$306$	0	0
$307$	−32.0156	−1.82723	−0.913613	−	0.406584i	$-0.866720\pi$
$307$	−32.0156	−1.82723	−0.913613	+	0.406584i	$0.866720\pi$
$308$	0	0
$309$	−21.9938	−1.25119
$310$	0	0
$311$	−20.9753	−1.18940	−0.594701	−	0.803947i	$-0.702730\pi$
$311$	−20.9753	−1.18940	−0.594701	+	0.803947i	$0.702730\pi$
$312$	0	0
$313$	−1.22538	−0.0692625	−0.0346312	−	0.999400i	$-0.511026\pi$
$313$	−1.22538	−0.0692625	−0.0346312	+	0.999400i	$0.511026\pi$
$314$	0	0
$315$	11.1726	0.629504
$316$	0	0
$317$	−5.10903	−0.286952	−0.143476	−	0.989654i	$-0.545828\pi$
$317$	−5.10903	−0.286952	−0.143476	+	0.989654i	$0.545828\pi$
$318$	0	0
$319$	−41.8144	−2.34115
$320$	0	0
$321$	−25.5106	−1.42386
$322$	0	0
$323$	−0.411253	−0.0228827
$324$	0	0
$325$	7.39371	0.410129
$326$	0	0
$327$	20.1226	1.11278
$328$	0	0
$329$	26.8775	1.48180
$330$	0	0
$331$	−10.2706	−0.564523	−0.282262	−	0.959337i	$-0.591085\pi$
$331$	−10.2706	−0.564523	−0.282262	+	0.959337i	$0.591085\pi$
$332$	0	0
$333$	−7.00183	−0.383698
$334$	0	0
$335$	−42.9890	−2.34874
$336$	0	0
$337$	−8.12236	−0.442453	−0.221227	−	0.975222i	$-0.571006\pi$
$337$	−8.12236	−0.442453	−0.221227	+	0.975222i	$0.571006\pi$
$338$	0	0
$339$	−23.0158	−1.25005
$340$	0	0
$341$	15.9320	0.862766
$342$	0	0
$343$	−41.0916	−2.21874
$344$	0	0
$345$	−44.8361	−2.41389
$346$	0	0
$347$	−10.3183	−0.553914	−0.276957	−	0.960882i	$-0.589326\pi$
$347$	−10.3183	−0.553914	−0.276957	+	0.960882i	$0.589326\pi$
$348$	0	0
$349$	−1.23807	−0.0662725	−0.0331363	−	0.999451i	$-0.510550\pi$
$349$	−1.23807	−0.0662725	−0.0331363	+	0.999451i	$0.510550\pi$
$350$	0	0
$351$	12.1998	0.651176
$352$	0	0
$353$	24.3270	1.29480	0.647399	−	0.762151i	$-0.275857\pi$
$353$	24.3270	1.29480	0.647399	+	0.762151i	$0.275857\pi$
$354$	0	0
$355$	−27.8357	−1.47736
$356$	0	0
$357$	9.34088	0.494372
$358$	0	0
$359$	18.2291	0.962095	0.481047	−	0.876695i	$-0.340256\pi$
$359$	18.2291	0.962095	0.481047	+	0.876695i	$0.340256\pi$
$360$	0	0
$361$	−18.8309	−0.991098
$362$	0	0
$363$	−19.0449	−0.999600
$364$	0	0
$365$	−33.8651	−1.77258
$366$	0	0
$367$	19.6543	1.02595	0.512974	−	0.858404i	$-0.328544\pi$
$367$	19.6543	1.02595	0.512974	+	0.858404i	$0.328544\pi$
$368$	0	0
$369$	0.0525236	0.00273427
$370$	0	0
$371$	−3.19835	−0.166050
$372$	0	0
$373$	−0.456215	−0.0236219	−0.0118109	−	0.999930i	$-0.503760\pi$
$373$	−0.456215	−0.0236219	−0.0118109	+	0.999930i	$0.503760\pi$
$374$	0	0
$375$	−13.2037	−0.681834
$376$	0	0
$377$	26.6175	1.37087
$378$	0	0
$379$	14.8753	0.764094	0.382047	−	0.924143i	$-0.375219\pi$
$379$	14.8753	0.764094	0.382047	+	0.924143i	$0.375219\pi$
$380$	0	0
$381$	−8.32152	−0.426324
$382$	0	0
$383$	0.788654	0.0402983	0.0201492	−	0.999797i	$-0.493586\pi$
$383$	0.788654	0.0402983	0.0201492	+	0.999797i	$0.493586\pi$
$384$	0	0
$385$	−59.4913	−3.03196
$386$	0	0
$387$	−4.83069	−0.245558
$388$	0	0
$389$	−8.36644	−0.424195	−0.212098	−	0.977248i	$-0.568029\pi$
$389$	−8.36644	−0.424195	−0.212098	+	0.977248i	$0.568029\pi$
$390$	0	0
$391$	−8.30979	−0.420244
$392$	0	0
$393$	1.83567	0.0925973
$394$	0	0
$395$	24.7649	1.24606
$396$	0	0
$397$	11.6745	0.585927	0.292963	−	0.956124i	$-0.405359\pi$
$397$	11.6745	0.585927	0.292963	+	0.956124i	$0.405359\pi$
$398$	0	0
$399$	−3.84146	−0.192314
$400$	0	0
$401$	20.4020	1.01883	0.509413	−	0.860522i	$-0.329863\pi$
$401$	20.4020	1.01883	0.509413	+	0.860522i	$0.329863\pi$
$402$	0	0
$403$	−10.1417	−0.505196
$404$	0	0
$405$	29.7725	1.47941
$406$	0	0
$407$	37.2830	1.84805
$408$	0	0
$409$	−25.9866	−1.28496	−0.642478	−	0.766304i	$-0.722094\pi$
$409$	−25.9866	−1.28496	−0.642478	+	0.766304i	$0.722094\pi$
$410$	0	0
$411$	−36.7596	−1.81322
$412$	0	0
$413$	−4.75780	−0.234116
$414$	0	0
$415$	−36.3208	−1.78292
$416$	0	0
$417$	11.0437	0.540814
$418$	0	0
$419$	1.35547	0.0662188	0.0331094	−	0.999452i	$-0.489459\pi$
$419$	1.35547	0.0662188	0.0331094	+	0.999452i	$0.489459\pi$
$420$	0	0
$421$	−12.5643	−0.612345	−0.306172	−	0.951976i	$-0.599048\pi$
$421$	−12.5643	−0.612345	−0.306172	+	0.951976i	$0.599048\pi$
$422$	0	0
$423$	−4.82696	−0.234695
$424$	0	0
$425$	2.55287	0.123832
$426$	0	0
$427$	−27.0352	−1.30832
$428$	0	0
$429$	25.8706	1.24904
$430$	0	0
$431$	37.0280	1.78358	0.891789	−	0.452451i	$-0.149450\pi$
$431$	37.0280	1.78358	0.891789	+	0.452451i	$0.149450\pi$
$432$	0	0
$433$	−4.68449	−0.225122	−0.112561	−	0.993645i	$-0.535905\pi$
$433$	−4.68449	−0.225122	−0.112561	+	0.993645i	$0.535905\pi$
$434$	0	0
$435$	49.5875	2.37754
$436$	0	0
$437$	3.41742	0.163478
$438$	0	0
$439$	−18.4130	−0.878805	−0.439403	−	0.898290i	$-0.644810\pi$
$439$	−18.4130	−0.878805	−0.439403	+	0.898290i	$0.644810\pi$
$440$	0	0
$441$	13.3609	0.636233
$442$	0	0
$443$	3.86601	0.183680	0.0918399	−	0.995774i	$-0.470725\pi$
$443$	3.86601	0.183680	0.0918399	+	0.995774i	$0.470725\pi$
$444$	0	0
$445$	19.9202	0.944308
$446$	0	0
$447$	14.9471	0.706974
$448$	0	0
$449$	−27.4598	−1.29591	−0.647954	−	0.761679i	$-0.724375\pi$
$449$	−27.4598	−1.29591	−0.647954	+	0.761679i	$0.724375\pi$
$450$	0	0
$451$	−0.279676	−0.0131694
$452$	0	0
$453$	11.2116	0.526766
$454$	0	0
$455$	37.8700	1.77537
$456$	0	0
$457$	−11.1768	−0.522827	−0.261413	−	0.965227i	$-0.584189\pi$
$457$	−11.1768	−0.522827	−0.261413	+	0.965227i	$0.584189\pi$
$458$	0	0
$459$	4.21229	0.196613
$460$	0	0
$461$	3.67820	0.171311	0.0856554	−	0.996325i	$-0.472702\pi$
$461$	3.67820	0.171311	0.0856554	+	0.996325i	$0.472702\pi$
$462$	0	0
$463$	23.6611	1.09963	0.549813	−	0.835288i	$-0.314699\pi$
$463$	23.6611	1.09963	0.549813	+	0.835288i	$0.314699\pi$
$464$	0	0
$465$	−18.8937	−0.876173
$466$	0	0
$467$	35.6696	1.65059	0.825297	−	0.564699i	$-0.191008\pi$
$467$	35.6696	1.65059	0.825297	+	0.564699i	$0.191008\pi$
$468$	0	0
$469$	−74.4230	−3.43654
$470$	0	0
$471$	38.8153	1.78851
$472$	0	0
$473$	25.7223	1.18271
$474$	0	0
$475$	−1.04988	−0.0481716
$476$	0	0
$477$	0.574395	0.0262998
$478$	0	0
$479$	−31.9331	−1.45906	−0.729531	−	0.683947i	$-0.760262\pi$
$479$	−31.9331	−1.45906	−0.729531	+	0.683947i	$0.760262\pi$
$480$	0	0
$481$	−23.7330	−1.08213
$482$	0	0
$483$	−77.6208	−3.53187
$484$	0	0
$485$	−25.3843	−1.15264
$486$	0	0
$487$	20.5059	0.929210	0.464605	−	0.885518i	$-0.346196\pi$
$487$	20.5059	0.929210	0.464605	+	0.885518i	$0.346196\pi$
$488$	0	0
$489$	23.7914	1.07588
$490$	0	0
$491$	31.3833	1.41631	0.708154	−	0.706058i	$-0.249528\pi$
$491$	31.3833	1.41631	0.708154	+	0.706058i	$0.249528\pi$
$492$	0	0
$493$	9.19039	0.413915
$494$	0	0
$495$	10.6841	0.480215
$496$	0	0
$497$	−48.1895	−2.16159
$498$	0	0
$499$	−11.1194	−0.497774	−0.248887	−	0.968532i	$-0.580065\pi$
$499$	−11.1194	−0.497774	−0.248887	+	0.968532i	$0.580065\pi$
$500$	0	0
$501$	−30.1971	−1.34911
$502$	0	0
$503$	−7.85372	−0.350180	−0.175090	−	0.984552i	$-0.556022\pi$
$503$	−7.85372	−0.350180	−0.175090	+	0.984552i	$0.556022\pi$
$504$	0	0
$505$	7.90084	0.351583
$506$	0	0
$507$	9.05432	0.402116
$508$	0	0
$509$	0.553126	0.0245169	0.0122584	−	0.999925i	$-0.496098\pi$
$509$	0.553126	0.0245169	0.0122584	+	0.999925i	$0.496098\pi$
$510$	0	0
$511$	−58.6276	−2.59353
$512$	0	0
$513$	−1.73232	−0.0764837
$514$	0	0
$515$	−30.7876	−1.35666
$516$	0	0
$517$	25.7024	1.13039
$518$	0	0
$519$	6.92336	0.303902
$520$	0	0
$521$	30.9321	1.35516	0.677580	−	0.735449i	$-0.263029\pi$
$521$	30.9321	1.35516	0.677580	+	0.735449i	$0.263029\pi$
$522$	0	0
$523$	−8.89845	−0.389102	−0.194551	−	0.980892i	$-0.562325\pi$
$523$	−8.89845	−0.389102	−0.194551	+	0.980892i	$0.562325\pi$
$524$	0	0
$525$	23.8461	1.04073
$526$	0	0
$527$	−3.50170	−0.152536
$528$	0	0
$529$	46.0526	2.00229
$530$	0	0
$531$	0.854459	0.0370804
$532$	0	0
$533$	0.178031	0.00771140
$534$	0	0
$535$	−35.7105	−1.54390
$536$	0	0
$537$	50.1827	2.16554
$538$	0	0
$539$	−71.1436	−3.06437
$540$	0	0
$541$	31.0858	1.33648	0.668242	−	0.743944i	$-0.267047\pi$
$541$	31.0858	1.33648	0.668242	+	0.743944i	$0.267047\pi$
$542$	0	0
$543$	44.9698	1.92984
$544$	0	0
$545$	28.1681	1.20659
$546$	0	0
$547$	−11.0063	−0.470595	−0.235297	−	0.971923i	$-0.575606\pi$
$547$	−11.0063	−0.470595	−0.235297	+	0.971923i	$0.575606\pi$
$548$	0	0
$549$	4.85528	0.207218
$550$	0	0
$551$	−3.77958	−0.161015
$552$	0	0
$553$	42.8733	1.82316
$554$	0	0
$555$	−44.2138	−1.87677
$556$	0	0
$557$	4.42234	0.187381	0.0936904	−	0.995601i	$-0.470134\pi$
$557$	4.42234	0.187381	0.0936904	+	0.995601i	$0.470134\pi$
$558$	0	0
$559$	−16.3739	−0.692541
$560$	0	0
$561$	8.93250	0.377130
$562$	0	0
$563$	19.3614	0.815986	0.407993	−	0.912985i	$-0.366229\pi$
$563$	19.3614	0.815986	0.407993	+	0.912985i	$0.366229\pi$
$564$	0	0
$565$	−32.2181	−1.35543
$566$	0	0
$567$	51.5426	2.16458
$568$	0	0
$569$	16.8987	0.708431	0.354216	−	0.935164i	$-0.384748\pi$
$569$	16.8987	0.708431	0.354216	+	0.935164i	$0.384748\pi$
$570$	0	0
$571$	9.98914	0.418033	0.209016	−	0.977912i	$-0.432974\pi$
$571$	9.98914	0.418033	0.209016	+	0.977912i	$0.432974\pi$
$572$	0	0
$573$	−10.3284	−0.431473
$574$	0	0
$575$	−21.2138	−0.884678
$576$	0	0
$577$	3.48963	0.145275	0.0726375	−	0.997358i	$-0.476858\pi$
$577$	3.48963	0.145275	0.0726375	+	0.997358i	$0.476858\pi$
$578$	0	0
$579$	43.8527	1.82246
$580$	0	0
$581$	−62.8790	−2.60866
$582$	0	0
$583$	−3.05852	−0.126671
$584$	0	0
$585$	−6.80113	−0.281192
$586$	0	0
$587$	−30.2067	−1.24676	−0.623381	−	0.781918i	$-0.714242\pi$
$587$	−30.2067	−1.24676	−0.623381	+	0.781918i	$0.714242\pi$
$588$	0	0
$589$	1.44008	0.0593376
$590$	0	0
$591$	28.6939	1.18031
$592$	0	0
$593$	33.3834	1.37089	0.685445	−	0.728124i	$-0.259608\pi$
$593$	33.3834	1.37089	0.685445	+	0.728124i	$0.259608\pi$
$594$	0	0
$595$	13.0756	0.536048
$596$	0	0
$597$	−37.3802	−1.52987
$598$	0	0
$599$	0.414480	0.0169352	0.00846760	−	0.999964i	$-0.497305\pi$
$599$	0.414480	0.0169352	0.00846760	+	0.999964i	$0.497305\pi$
$600$	0	0
$601$	−22.1375	−0.903005	−0.451503	−	0.892270i	$-0.649112\pi$
$601$	−22.1375	−0.903005	−0.451503	+	0.892270i	$0.649112\pi$
$602$	0	0
$603$	13.3657	0.544294
$604$	0	0
$605$	−26.6596	−1.08387
$606$	0	0
$607$	30.3970	1.23378	0.616888	−	0.787051i	$-0.288393\pi$
$607$	30.3970	1.23378	0.616888	+	0.787051i	$0.288393\pi$
$608$	0	0
$609$	85.8464	3.47867
$610$	0	0
$611$	−16.3612	−0.661905
$612$	0	0
$613$	9.00277	0.363618	0.181809	−	0.983334i	$-0.441805\pi$
$613$	9.00277	0.363618	0.181809	+	0.983334i	$0.441805\pi$
$614$	0	0
$615$	0.331666	0.0133741
$616$	0	0
$617$	43.1484	1.73709	0.868545	−	0.495610i	$-0.165056\pi$
$617$	43.1484	1.73709	0.868545	+	0.495610i	$0.165056\pi$
$618$	0	0
$619$	40.5818	1.63112	0.815561	−	0.578672i	$-0.196429\pi$
$619$	40.5818	1.63112	0.815561	+	0.578672i	$0.196429\pi$
$620$	0	0
$621$	−35.0033	−1.40463
$622$	0	0
$623$	34.4861	1.38166
$624$	0	0
$625$	−31.2472	−1.24989
$626$	0	0
$627$	−3.67352	−0.146706
$628$	0	0
$629$	−8.19445	−0.326734
$630$	0	0
$631$	−2.73306	−0.108802	−0.0544008	−	0.998519i	$-0.517325\pi$
$631$	−2.73306	−0.108802	−0.0544008	+	0.998519i	$0.517325\pi$
$632$	0	0
$633$	−1.81995	−0.0723367
$634$	0	0
$635$	−11.6487	−0.462264
$636$	0	0
$637$	45.2874	1.79435
$638$	0	0
$639$	8.65440	0.342363
$640$	0	0
$641$	−8.62511	−0.340671	−0.170336	−	0.985386i	$-0.554485\pi$
$641$	−8.62511	−0.340671	−0.170336	+	0.985386i	$0.554485\pi$
$642$	0	0
$643$	22.0870	0.871025	0.435512	−	0.900183i	$-0.356567\pi$
$643$	22.0870	0.871025	0.435512	+	0.900183i	$0.356567\pi$
$644$	0	0
$645$	−30.5039	−1.20109
$646$	0	0
$647$	−18.7439	−0.736898	−0.368449	−	0.929648i	$-0.620111\pi$
$647$	−18.7439	−0.736898	−0.368449	+	0.929648i	$0.620111\pi$
$648$	0	0
$649$	−4.54979	−0.178595
$650$	0	0
$651$	−32.7090	−1.28197
$652$	0	0
$653$	48.8623	1.91213	0.956065	−	0.293154i	$-0.0947049\pi$
$653$	48.8623	1.91213	0.956065	+	0.293154i	$0.0947049\pi$
$654$	0	0
$655$	2.56962	0.100403
$656$	0	0
$657$	10.5290	0.410776
$658$	0	0
$659$	32.3326	1.25950	0.629750	−	0.776798i	$-0.283157\pi$
$659$	32.3326	1.25950	0.629750	+	0.776798i	$0.283157\pi$
$660$	0	0
$661$	−24.3698	−0.947875	−0.473938	−	0.880558i	$-0.657168\pi$
$661$	−24.3698	−0.947875	−0.473938	+	0.880558i	$0.657168\pi$
$662$	0	0
$663$	−5.68611	−0.220830
$664$	0	0
$665$	−5.37739	−0.208526
$666$	0	0
$667$	−76.3703	−2.95707
$668$	0	0
$669$	−26.4680	−1.02331
$670$	0	0
$671$	−25.8532	−0.998051
$672$	0	0
$673$	6.77324	0.261089	0.130545	−	0.991442i	$-0.458327\pi$
$673$	6.77324	0.261089	0.130545	+	0.991442i	$0.458327\pi$
$674$	0	0
$675$	10.7534	0.413900
$676$	0	0
$677$	−15.5443	−0.597415	−0.298707	−	0.954345i	$-0.596555\pi$
$677$	−15.5443	−0.597415	−0.298707	+	0.954345i	$0.596555\pi$
$678$	0	0
$679$	−43.9456	−1.68648
$680$	0	0
$681$	2.66328	0.102057
$682$	0	0
$683$	−5.29223	−0.202502	−0.101251	−	0.994861i	$-0.532284\pi$
$683$	−5.29223	−0.202502	−0.101251	+	0.994861i	$0.532284\pi$
$684$	0	0
$685$	−51.4570	−1.96607
$686$	0	0
$687$	−36.1595	−1.37957
$688$	0	0
$689$	1.94694	0.0741726
$690$	0	0
$691$	30.0627	1.14364	0.571819	−	0.820380i	$-0.306238\pi$
$691$	30.0627	1.14364	0.571819	+	0.820380i	$0.306238\pi$
$692$	0	0
$693$	18.4965	0.702623
$694$	0	0
$695$	15.4593	0.586405
$696$	0	0
$697$	0.0614700	0.00232834
$698$	0	0
$699$	30.3097	1.14642
$700$	0	0
$701$	−14.3150	−0.540672	−0.270336	−	0.962766i	$-0.587135\pi$
$701$	−14.3150	−0.540672	−0.270336	+	0.962766i	$0.587135\pi$
$702$	0	0
$703$	3.36999	0.127102
$704$	0	0
$705$	−30.4804	−1.14796
$706$	0	0
$707$	13.6780	0.514415
$708$	0	0
$709$	23.9663	0.900072	0.450036	−	0.893010i	$-0.351411\pi$
$709$	23.9663	0.900072	0.450036	+	0.893010i	$0.351411\pi$
$710$	0	0
$711$	−7.69967	−0.288760
$712$	0	0
$713$	29.0984	1.08974
$714$	0	0
$715$	36.2144	1.35434
$716$	0	0
$717$	−17.4495	−0.651664
$718$	0	0
$719$	−8.29409	−0.309317	−0.154659	−	0.987968i	$-0.549428\pi$
$719$	−8.29409	−0.309317	−0.154659	+	0.987968i	$0.549428\pi$
$720$	0	0
$721$	−53.2998	−1.98499
$722$	0	0
$723$	−15.5880	−0.579724
$724$	0	0
$725$	23.4619	0.871353
$726$	0	0
$727$	34.1590	1.26689	0.633443	−	0.773789i	$-0.281641\pi$
$727$	34.1590	1.26689	0.633443	+	0.773789i	$0.281641\pi$
$728$	0	0
$729$	15.5531	0.576041
$730$	0	0
$731$	−5.65351	−0.209102
$732$	0	0
$733$	−11.6201	−0.429196	−0.214598	−	0.976702i	$-0.568844\pi$
$733$	−11.6201	−0.429196	−0.214598	+	0.976702i	$0.568844\pi$
$734$	0	0
$735$	84.3688	3.11199
$736$	0	0
$737$	−71.1692	−2.62155
$738$	0	0
$739$	−4.78725	−0.176102	−0.0880509	−	0.996116i	$-0.528064\pi$
$739$	−4.78725	−0.176102	−0.0880509	+	0.996116i	$0.528064\pi$
$740$	0	0
$741$	2.33843	0.0859043
$742$	0	0
$743$	−41.5233	−1.52334	−0.761671	−	0.647964i	$-0.775621\pi$
$743$	−41.5233	−1.52334	−0.761671	+	0.647964i	$0.775621\pi$
$744$	0	0
$745$	20.9234	0.766573
$746$	0	0
$747$	11.2925	0.413171
$748$	0	0
$749$	−61.8224	−2.25894
$750$	0	0
$751$	−38.2938	−1.39736	−0.698680	−	0.715435i	$-0.746229\pi$
$751$	−38.2938	−1.39736	−0.698680	+	0.715435i	$0.746229\pi$
$752$	0	0
$753$	22.9025	0.834613
$754$	0	0
$755$	15.6943	0.571173
$756$	0	0
$757$	10.9679	0.398633	0.199317	−	0.979935i	$-0.436128\pi$
$757$	10.9679	0.398633	0.199317	+	0.979935i	$0.436128\pi$
$758$	0	0
$759$	−74.2272	−2.69428
$760$	0	0
$761$	46.2867	1.67789	0.838945	−	0.544216i	$-0.183173\pi$
$761$	46.2867	1.67789	0.838945	+	0.544216i	$0.183173\pi$
$762$	0	0
$763$	48.7650	1.76541
$764$	0	0
$765$	−2.34827	−0.0849017
$766$	0	0
$767$	2.89623	0.104577
$768$	0	0
$769$	6.80182	0.245280	0.122640	−	0.992451i	$-0.460864\pi$
$769$	6.80182	0.245280	0.122640	+	0.992451i	$0.460864\pi$
$770$	0	0
$771$	−1.72621	−0.0621680
$772$	0	0
$773$	−34.6053	−1.24467	−0.622334	−	0.782752i	$-0.713815\pi$
$773$	−34.6053	−1.24467	−0.622334	+	0.782752i	$0.713815\pi$
$774$	0	0
$775$	−8.93939	−0.321112
$776$	0	0
$777$	−76.5435	−2.74598
$778$	0	0
$779$	−0.0252797	−0.000905740 0
$780$	0	0
$781$	−46.0826	−1.64897
$782$	0	0
$783$	38.7126	1.38348
$784$	0	0
$785$	54.3346	1.93929
$786$	0	0
$787$	−12.2263	−0.435822	−0.217911	−	0.975969i	$-0.569924\pi$
$787$	−12.2263	−0.435822	−0.217911	+	0.975969i	$0.569924\pi$
$788$	0	0
$789$	7.48164	0.266354
$790$	0	0
$791$	−55.7764	−1.98318
$792$	0	0
$793$	16.4572	0.584413
$794$	0	0
$795$	3.62708	0.128639
$796$	0	0
$797$	17.6272	0.624389	0.312195	−	0.950018i	$-0.398936\pi$
$797$	17.6272	0.624389	0.312195	+	0.950018i	$0.398936\pi$
$798$	0	0
$799$	−5.64915	−0.199852
$800$	0	0
$801$	−6.19340	−0.218833
$802$	0	0
$803$	−56.0644	−1.97847
$804$	0	0
$805$	−108.656	−3.82961
$806$	0	0
$807$	47.5642	1.67434
$808$	0	0
$809$	39.2330	1.37936	0.689679	−	0.724115i	$-0.257752\pi$
$809$	39.2330	1.37936	0.689679	+	0.724115i	$0.257752\pi$
$810$	0	0
$811$	−14.1716	−0.497631	−0.248815	−	0.968551i	$-0.580041\pi$
$811$	−14.1716	−0.497631	−0.248815	+	0.968551i	$0.580041\pi$
$812$	0	0
$813$	−9.91948	−0.347891
$814$	0	0
$815$	33.3038	1.16658
$816$	0	0
$817$	2.32502	0.0813422
$818$	0	0
$819$	−11.7742	−0.411423
$820$	0	0
$821$	−16.0977	−0.561812	−0.280906	−	0.959735i	$-0.590635\pi$
$821$	−16.0977	−0.561812	−0.280906	+	0.959735i	$0.590635\pi$
$822$	0	0
$823$	−32.5048	−1.13305	−0.566524	−	0.824046i	$-0.691712\pi$
$823$	−32.5048	−1.13305	−0.566524	+	0.824046i	$0.691712\pi$
$824$	0	0
$825$	22.8035	0.793917
$826$	0	0
$827$	−53.9474	−1.87594	−0.937968	−	0.346721i	$-0.887295\pi$
$827$	−53.9474	−1.87594	−0.937968	+	0.346721i	$0.887295\pi$
$828$	0	0
$829$	12.3528	0.429029	0.214514	−	0.976721i	$-0.431183\pi$
$829$	12.3528	0.429029	0.214514	+	0.976721i	$0.431183\pi$
$830$	0	0
$831$	−6.40672	−0.222247
$832$	0	0
$833$	15.6367	0.541779
$834$	0	0
$835$	−42.2707	−1.46284
$836$	0	0
$837$	−14.7502	−0.509841
$838$	0	0
$839$	−22.1804	−0.765751	−0.382875	−	0.923800i	$-0.625066\pi$
$839$	−22.1804	−0.765751	−0.382875	+	0.923800i	$0.625066\pi$
$840$	0	0
$841$	55.4633	1.91253
$842$	0	0
$843$	−33.7171	−1.16128
$844$	0	0
$845$	12.6745	0.436015
$846$	0	0
$847$	−46.1534	−1.58585
$848$	0	0
$849$	18.2723	0.627102
$850$	0	0
$851$	68.0942	2.33424
$852$	0	0
$853$	20.0557	0.686694	0.343347	−	0.939209i	$-0.388439\pi$
$853$	20.0557	0.686694	0.343347	+	0.939209i	$0.388439\pi$
$854$	0	0
$855$	0.965731	0.0330273
$856$	0	0
$857$	50.1527	1.71318	0.856592	−	0.515994i	$-0.172577\pi$
$857$	50.1527	1.71318	0.856592	+	0.515994i	$0.172577\pi$
$858$	0	0
$859$	−7.48987	−0.255551	−0.127776	−	0.991803i	$-0.540784\pi$
$859$	−7.48987	−0.255551	−0.127776	+	0.991803i	$0.540784\pi$
$860$	0	0
$861$	0.574184	0.0195681
$862$	0	0
$863$	−34.9509	−1.18974	−0.594871	−	0.803821i	$-0.702797\pi$
$863$	−34.9509	−1.18974	−0.594871	+	0.803821i	$0.702797\pi$
$864$	0	0
$865$	9.69150	0.329521
$866$	0	0
$867$	−1.96328	−0.0666764
$868$	0	0
$869$	40.9989	1.39079
$870$	0	0
$871$	45.3038	1.53506
$872$	0	0
$873$	7.89224	0.267112
$874$	0	0
$875$	−31.9977	−1.08172
$876$	0	0
$877$	−43.2470	−1.46035	−0.730174	−	0.683261i	$-0.760561\pi$
$877$	−43.2470	−1.46035	−0.730174	+	0.683261i	$0.760561\pi$
$878$	0	0
$879$	−57.2201	−1.92999
$880$	0	0
$881$	−32.3120	−1.08862	−0.544309	−	0.838885i	$-0.683208\pi$
$881$	−32.3120	−1.08862	−0.544309	+	0.838885i	$0.683208\pi$
$882$	0	0
$883$	−15.2718	−0.513937	−0.256968	−	0.966420i	$-0.582724\pi$
$883$	−15.2718	−0.513937	−0.256968	+	0.966420i	$0.582724\pi$
$884$	0	0
$885$	5.39558	0.181370
$886$	0	0
$887$	−44.6641	−1.49967	−0.749837	−	0.661622i	$-0.769868\pi$
$887$	−44.6641	−1.49967	−0.749837	+	0.661622i	$0.769868\pi$
$888$	0	0
$889$	−20.1664	−0.676358
$890$	0	0
$891$	49.2891	1.65125
$892$	0	0
$893$	2.32323	0.0777438
$894$	0	0
$895$	70.2471	2.34810
$896$	0	0
$897$	47.2504	1.57764
$898$	0	0
$899$	−32.1820	−1.07333
$900$	0	0
$901$	0.672233	0.0223953
$902$	0	0
$903$	−52.8087	−1.75736
$904$	0	0
$905$	62.9499	2.09253
$906$	0	0
$907$	54.0521	1.79477	0.897386	−	0.441246i	$-0.145463\pi$
$907$	54.0521	1.79477	0.897386	+	0.441246i	$0.145463\pi$
$908$	0	0
$909$	−2.45645	−0.0814754
$910$	0	0
$911$	27.2083	0.901450	0.450725	−	0.892663i	$-0.351166\pi$
$911$	27.2083	0.901450	0.450725	+	0.892663i	$0.351166\pi$
$912$	0	0
$913$	−60.1299	−1.99001
$914$	0	0
$915$	30.6592	1.01356
$916$	0	0
$917$	4.44856	0.146904
$918$	0	0
$919$	−2.16087	−0.0712805	−0.0356403	−	0.999365i	$-0.511347\pi$
$919$	−2.16087	−0.0712805	−0.0356403	+	0.999365i	$0.511347\pi$
$920$	0	0
$921$	62.8555	2.07116
$922$	0	0
$923$	29.3345	0.965558
$924$	0	0
$925$	−20.9194	−0.687825
$926$	0	0
$927$	9.57217	0.314391
$928$	0	0
$929$	−26.2065	−0.859808	−0.429904	−	0.902875i	$-0.641453\pi$
$929$	−26.2065	−0.859808	−0.429904	+	0.902875i	$0.641453\pi$
$930$	0	0
$931$	−6.43062	−0.210755
$932$	0	0
$933$	41.1804	1.34819
$934$	0	0
$935$	12.5040	0.408923
$936$	0	0
$937$	−27.2240	−0.889370	−0.444685	−	0.895687i	$-0.646684\pi$
$937$	−27.2240	−0.889370	−0.444685	+	0.895687i	$0.646684\pi$
$938$	0	0
$939$	2.40576	0.0785089
$940$	0	0
$941$	35.3760	1.15322	0.576612	−	0.817018i	$-0.304375\pi$
$941$	35.3760	1.15322	0.576612	+	0.817018i	$0.304375\pi$
$942$	0	0
$943$	−0.510803	−0.0166340
$944$	0	0
$945$	55.0783	1.79170
$946$	0	0
$947$	−17.8520	−0.580111	−0.290056	−	0.957010i	$-0.593674\pi$
$947$	−17.8520	−0.580111	−0.290056	+	0.957010i	$0.593674\pi$
$948$	0	0
$949$	35.6886	1.15850
$950$	0	0
$951$	10.0305	0.325260
$952$	0	0
$953$	43.0650	1.39501	0.697507	−	0.716578i	$-0.254293\pi$
$953$	43.0650	1.39501	0.697507	+	0.716578i	$0.254293\pi$
$954$	0	0
$955$	−14.4579	−0.467847
$956$	0	0
$957$	82.0932	2.65370
$958$	0	0
$959$	−89.0830	−2.87664
$960$	0	0
$961$	−18.7381	−0.604455
$962$	0	0
$963$	11.1028	0.357781
$964$	0	0
$965$	61.3862	1.97609
$966$	0	0
$967$	−55.4728	−1.78388	−0.891942	−	0.452150i	$-0.850657\pi$
$967$	−55.4728	−1.78388	−0.891942	+	0.452150i	$0.850657\pi$
$968$	0	0
$969$	0.807403	0.0259375
$970$	0	0
$971$	15.5712	0.499704	0.249852	−	0.968284i	$-0.419618\pi$
$971$	15.5712	0.499704	0.249852	+	0.968284i	$0.419618\pi$
$972$	0	0
$973$	26.7633	0.857993
$974$	0	0
$975$	−14.5159	−0.464881
$976$	0	0
$977$	3.12352	0.0999302	0.0499651	−	0.998751i	$-0.484089\pi$
$977$	3.12352	0.0999302	0.0499651	+	0.998751i	$0.484089\pi$
$978$	0	0
$979$	32.9784	1.05399
$980$	0	0
$981$	−8.75776	−0.279614
$982$	0	0
$983$	43.6675	1.39278	0.696388	−	0.717665i	$-0.254789\pi$
$983$	43.6675	1.39278	0.696388	+	0.717665i	$0.254789\pi$
$984$	0	0
$985$	40.1665	1.27981
$986$	0	0
$987$	−52.7680	−1.67962
$988$	0	0
$989$	46.9795	1.49386
$990$	0	0
$991$	5.26848	0.167359	0.0836795	−	0.996493i	$-0.473333\pi$
$991$	5.26848	0.167359	0.0836795	+	0.996493i	$0.473333\pi$
$992$	0	0
$993$	20.1640	0.639886
$994$	0	0
$995$	−52.3258	−1.65884
$996$	0	0
$997$	−49.9106	−1.58068	−0.790342	−	0.612666i	$-0.790097\pi$
$997$	−49.9106	−1.58068	−0.790342	+	0.612666i	$0.790097\pi$
$998$	0	0
$999$	−34.5174	−1.09208

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.z.1.8	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.z.1.8	✓	24	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.96328 q^{3} -2.74825 q^{5} -4.75780 q^{7} +0.854459 q^{9} +O(q^{10})\)	\(q-1.96328 q^{3} -2.74825 q^{5} -4.75780 q^{7} +0.854459 q^{9} -4.54979 q^{11} +2.89623 q^{13} +5.39558 q^{15} +1.00000 q^{17} -0.411253 q^{19} +9.34088 q^{21} -8.30979 q^{23} +2.55287 q^{25} +4.21229 q^{27} +9.19039 q^{29} -3.50170 q^{31} +8.93250 q^{33} +13.0756 q^{35} -8.19445 q^{37} -5.68611 q^{39} +0.0614700 q^{41} -5.65351 q^{43} -2.34827 q^{45} -5.64915 q^{47} +15.6367 q^{49} -1.96328 q^{51} +0.672233 q^{53} +12.5040 q^{55} +0.807403 q^{57} +1.00000 q^{59} +5.68228 q^{61} -4.06535 q^{63} -7.95957 q^{65} +15.6423 q^{67} +16.3144 q^{69} +10.1285 q^{71} +12.3224 q^{73} -5.01200 q^{75} +21.6470 q^{77} -9.01116 q^{79} -10.8333 q^{81} +13.2160 q^{83} -2.74825 q^{85} -18.0433 q^{87} -7.24833 q^{89} -13.7797 q^{91} +6.87481 q^{93} +1.13023 q^{95} +9.23654 q^{97} -3.88761 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9}+O(q^{10}) \) 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9	\( 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9} - 15 q^{11} - 17 q^{13} - 7 q^{15} + 24 q^{17} - q^{19} + 10 q^{21} - 21 q^{23} + 19 q^{25} - 26 q^{27} - 18 q^{29} - 23 q^{31} + 10 q^{33} - 5 q^{35} - 29 q^{37} - 44 q^{39} + 14 q^{41} - 23 q^{43} - 24 q^{45} - 17 q^{47} + 46 q^{49} - 5 q^{51} + 7 q^{53} - 33 q^{55} - 2 q^{57} + 24 q^{59} - 9 q^{61} - 52 q^{63} - 15 q^{65} + 2 q^{67} + 2 q^{69} - 51 q^{71} - 18 q^{73} + 7 q^{75} + 14 q^{77} - 36 q^{79} + 12 q^{81} - 39 q^{83} - 7 q^{85} - 5 q^{87} + 46 q^{89} - 52 q^{91} - 20 q^{93} - 52 q^{95} + 50 q^{97} - 53 q^{99}+O(q^{100}) \) 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9 - 15 * q^11 - 17 * q^13 - 7 * q^15 + 24 * q^17 - q^19 + 10 * q^21 - 21 * q^23 + 19 * q^25 - 26 * q^27 - 18 * q^29 - 23 * q^31 + 10 * q^33 - 5 * q^35 - 29 * q^37 - 44 * q^39 + 14 * q^41 - 23 * q^43 - 24 * q^45 - 17 * q^47 + 46 * q^49 - 5 * q^51 + 7 * q^53 - 33 * q^55 - 2 * q^57 + 24 * q^59 - 9 * q^61 - 52 * q^63 - 15 * q^65 + 2 * q^67 + 2 * q^69 - 51 * q^71 - 18 * q^73 + 7 * q^75 + 14 * q^77 - 36 * q^79 + 12 * q^81 - 39 * q^83 - 7 * q^85 - 5 * q^87 + 46 * q^89 - 52 * q^91 - 20 * q^93 - 52 * q^95 + 50 * q^97 - 53 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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