LMFDB - Embedded newform 8024.2.a.x.1.15

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:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
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)

$f(q)$	$=$	$q+0.535004 q^{3} +2.05543 q^{5} +0.811137 q^{7} -2.71377 q^{9} +O(q^{10})$	$q+0.535004 q^{3} +2.05543 q^{5} +0.811137 q^{7} -2.71377 q^{9} -3.45488 q^{11} +1.72147 q^{13} +1.09966 q^{15} -1.00000 q^{17} -0.663347 q^{19} +0.433961 q^{21} +6.37055 q^{23} -0.775206 q^{25} -3.05689 q^{27} -9.09202 q^{29} +3.15275 q^{31} -1.84837 q^{33} +1.66724 q^{35} +2.00681 q^{37} +0.920994 q^{39} +4.70079 q^{41} -6.03154 q^{43} -5.57797 q^{45} -13.4182 q^{47} -6.34206 q^{49} -0.535004 q^{51} +7.07611 q^{53} -7.10127 q^{55} -0.354893 q^{57} +1.00000 q^{59} -4.24339 q^{61} -2.20124 q^{63} +3.53837 q^{65} -3.26237 q^{67} +3.40827 q^{69} -9.67438 q^{71} +10.4413 q^{73} -0.414738 q^{75} -2.80238 q^{77} +1.82818 q^{79} +6.50587 q^{81} +15.2156 q^{83} -2.05543 q^{85} -4.86427 q^{87} -0.405280 q^{89} +1.39635 q^{91} +1.68673 q^{93} -1.36346 q^{95} -16.2907 q^{97} +9.37576 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) $ 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9	$ 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9} + 2 q^{11} - 9 q^{13} - 7 q^{15} - 22 q^{17} - 10 q^{19} - 10 q^{21} + 14 q^{23} + 3 q^{25} + 6 q^{29} - 15 q^{31} - 52 q^{33} - 7 q^{35} + 9 q^{37} - 52 q^{41} - 7 q^{43} - 30 q^{45} - 7 q^{47} - 6 q^{49} + 3 q^{51} - 18 q^{53} - 39 q^{55} - 2 q^{57} + 22 q^{59} - 42 q^{61} - 35 q^{65} - 28 q^{67} - 10 q^{69} + 23 q^{71} - 33 q^{73} - 3 q^{75} - 28 q^{77} - 30 q^{79} - 2 q^{81} + 11 q^{83} - 3 q^{85} + 27 q^{87} - 34 q^{89} - 18 q^{91} - 28 q^{93} - 10 q^{95} - 3 q^{97} - 28 q^{99}+O(q^{100}) $ 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9 + 2 * q^11 - 9 * q^13 - 7 * q^15 - 22 * q^17 - 10 * q^19 - 10 * q^21 + 14 * q^23 + 3 * q^25 + 6 * q^29 - 15 * q^31 - 52 * q^33 - 7 * q^35 + 9 * q^37 - 52 * q^41 - 7 * q^43 - 30 * q^45 - 7 * q^47 - 6 * q^49 + 3 * q^51 - 18 * q^53 - 39 * q^55 - 2 * q^57 + 22 * q^59 - 42 * q^61 - 35 * q^65 - 28 * q^67 - 10 * q^69 + 23 * q^71 - 33 * q^73 - 3 * q^75 - 28 * q^77 - 30 * q^79 - 2 * q^81 + 11 * q^83 - 3 * q^85 + 27 * q^87 - 34 * q^89 - 18 * q^91 - 28 * q^93 - 10 * q^95 - 3 * q^97 - 28 * 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
$3$	0.535004	0.308885	0.154442	−	0.988002i	$-0.450642\pi$
$3$	0.535004	0.308885	0.154442	+	0.988002i	$0.450642\pi$
$4$	0	0
$5$	2.05543	0.919216	0.459608	−	0.888122i	$-0.347990\pi$
$5$	2.05543	0.919216	0.459608	+	0.888122i	$0.347990\pi$
$6$	0	0
$7$	0.811137	0.306581	0.153291	−	0.988181i	$-0.451013\pi$
$7$	0.811137	0.306581	0.153291	+	0.988181i	$0.451013\pi$
$8$	0	0
$9$	−2.71377	−0.904590
$10$	0	0
$11$	−3.45488	−1.04169	−0.520843	−	0.853652i	$-0.674382\pi$
$11$	−3.45488	−1.04169	−0.520843	+	0.853652i	$0.674382\pi$
$12$	0	0
$13$	1.72147	0.477450	0.238725	−	0.971087i	$-0.423270\pi$
$13$	1.72147	0.477450	0.238725	+	0.971087i	$0.423270\pi$
$14$	0	0
$15$	1.09966	0.283932
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−0.663347	−0.152182	−0.0760911	−	0.997101i	$-0.524244\pi$
$19$	−0.663347	−0.152182	−0.0760911	+	0.997101i	$0.524244\pi$
$20$	0	0
$21$	0.433961	0.0946981
$22$	0	0
$23$	6.37055	1.32835	0.664176	−	0.747576i	$-0.268783\pi$
$23$	6.37055	1.32835	0.664176	+	0.747576i	$0.268783\pi$
$24$	0	0
$25$	−0.775206	−0.155041
$26$	0	0
$27$	−3.05689	−0.588299
$28$	0	0
$29$	−9.09202	−1.68835	−0.844173	−	0.536071i	$-0.819908\pi$
$29$	−9.09202	−1.68835	−0.844173	+	0.536071i	$0.819908\pi$
$30$	0	0
$31$	3.15275	0.566250	0.283125	−	0.959083i	$-0.408629\pi$
$31$	3.15275	0.566250	0.283125	+	0.959083i	$0.408629\pi$
$32$	0	0
$33$	−1.84837	−0.321761
$34$	0	0
$35$	1.66724	0.281814
$36$	0	0
$37$	2.00681	0.329917	0.164959	−	0.986300i	$-0.447251\pi$
$37$	2.00681	0.329917	0.164959	+	0.986300i	$0.447251\pi$
$38$	0	0
$39$	0.920994	0.147477
$40$	0	0
$41$	4.70079	0.734141	0.367070	−	0.930193i	$-0.380361\pi$
$41$	4.70079	0.734141	0.367070	+	0.930193i	$0.380361\pi$
$42$	0	0
$43$	−6.03154	−0.919801	−0.459900	−	0.887970i	$-0.652115\pi$
$43$	−6.03154	−0.919801	−0.459900	+	0.887970i	$0.652115\pi$
$44$	0	0
$45$	−5.57797	−0.831514
$46$	0	0
$47$	−13.4182	−1.95724	−0.978620	−	0.205676i	$-0.934061\pi$
$47$	−13.4182	−1.95724	−0.978620	+	0.205676i	$0.934061\pi$
$48$	0	0
$49$	−6.34206	−0.906008
$50$	0	0
$51$	−0.535004	−0.0749155
$52$	0	0
$53$	7.07611	0.971979	0.485989	−	0.873965i	$-0.338459\pi$
$53$	7.07611	0.971979	0.485989	+	0.873965i	$0.338459\pi$
$54$	0	0
$55$	−7.10127	−0.957535
$56$	0	0
$57$	−0.354893	−0.0470067
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−4.24339	−0.543310	−0.271655	−	0.962395i	$-0.587571\pi$
$61$	−4.24339	−0.543310	−0.271655	+	0.962395i	$0.587571\pi$
$62$	0	0
$63$	−2.20124	−0.277330
$64$	0	0
$65$	3.53837	0.438880
$66$	0	0
$67$	−3.26237	−0.398562	−0.199281	−	0.979942i	$-0.563861\pi$
$67$	−3.26237	−0.398562	−0.199281	+	0.979942i	$0.563861\pi$
$68$	0	0
$69$	3.40827	0.410307
$70$	0	0
$71$	−9.67438	−1.14814	−0.574069	−	0.818807i	$-0.694636\pi$
$71$	−9.67438	−1.14814	−0.574069	+	0.818807i	$0.694636\pi$
$72$	0	0
$73$	10.4413	1.22207	0.611033	−	0.791605i	$-0.290754\pi$
$73$	10.4413	1.22207	0.611033	+	0.791605i	$0.290754\pi$
$74$	0	0
$75$	−0.414738	−0.0478898
$76$	0	0
$77$	−2.80238	−0.319361
$78$	0	0
$79$	1.82818	0.205686	0.102843	−	0.994698i	$-0.467206\pi$
$79$	1.82818	0.205686	0.102843	+	0.994698i	$0.467206\pi$
$80$	0	0
$81$	6.50587	0.722874
$82$	0	0
$83$	15.2156	1.67013	0.835063	−	0.550155i	$-0.185431\pi$
$83$	15.2156	1.67013	0.835063	+	0.550155i	$0.185431\pi$
$84$	0	0
$85$	−2.05543	−0.222943
$86$	0	0
$87$	−4.86427	−0.521504
$88$	0	0
$89$	−0.405280	−0.0429596	−0.0214798	−	0.999769i	$-0.506838\pi$
$89$	−0.405280	−0.0429596	−0.0214798	+	0.999769i	$0.506838\pi$
$90$	0	0
$91$	1.39635	0.146377
$92$	0	0
$93$	1.68673	0.174906
$94$	0	0
$95$	−1.36346	−0.139888
$96$	0	0
$97$	−16.2907	−1.65407	−0.827033	−	0.562153i	$-0.809973\pi$
$97$	−16.2907	−1.65407	−0.827033	+	0.562153i	$0.809973\pi$
$98$	0	0
$99$	9.37576	0.942299
$100$	0	0
$101$	−2.43987	−0.242776	−0.121388	−	0.992605i	$-0.538735\pi$
$101$	−2.43987	−0.242776	−0.121388	+	0.992605i	$0.538735\pi$
$102$	0	0
$103$	0.782025	0.0770552	0.0385276	−	0.999258i	$-0.487733\pi$
$103$	0.782025	0.0770552	0.0385276	+	0.999258i	$0.487733\pi$
$104$	0	0
$105$	0.891977	0.0870481
$106$	0	0
$107$	7.69117	0.743533	0.371767	−	0.928326i	$-0.378752\pi$
$107$	7.69117	0.743533	0.371767	+	0.928326i	$0.378752\pi$
$108$	0	0
$109$	−6.09981	−0.584256	−0.292128	−	0.956379i	$-0.594363\pi$
$109$	−6.09981	−0.584256	−0.292128	+	0.956379i	$0.594363\pi$
$110$	0	0
$111$	1.07365	0.101906
$112$	0	0
$113$	−18.2292	−1.71486	−0.857430	−	0.514601i	$-0.827940\pi$
$113$	−18.2292	−1.71486	−0.857430	+	0.514601i	$0.827940\pi$
$114$	0	0
$115$	13.0942	1.22104
$116$	0	0
$117$	−4.67168	−0.431897
$118$	0	0
$119$	−0.811137	−0.0743568
$120$	0	0
$121$	0.936212	0.0851102
$122$	0	0
$123$	2.51494	0.226765
$124$	0	0
$125$	−11.8705	−1.06173
$126$	0	0
$127$	−2.69449	−0.239097	−0.119549	−	0.992828i	$-0.538145\pi$
$127$	−2.69449	−0.239097	−0.119549	+	0.992828i	$0.538145\pi$
$128$	0	0
$129$	−3.22690	−0.284112
$130$	0	0
$131$	−3.27305	−0.285968	−0.142984	−	0.989725i	$-0.545670\pi$
$131$	−3.27305	−0.285968	−0.142984	+	0.989725i	$0.545670\pi$
$132$	0	0
$133$	−0.538065	−0.0466562
$134$	0	0
$135$	−6.28322	−0.540774
$136$	0	0
$137$	−11.6998	−0.999580	−0.499790	−	0.866147i	$-0.666590\pi$
$137$	−11.6998	−0.999580	−0.499790	+	0.866147i	$0.666590\pi$
$138$	0	0
$139$	−19.8102	−1.68028	−0.840138	−	0.542373i	$-0.817526\pi$
$139$	−19.8102	−1.68028	−0.840138	+	0.542373i	$0.817526\pi$
$140$	0	0
$141$	−7.17877	−0.604561
$142$	0	0
$143$	−5.94748	−0.497353
$144$	0	0
$145$	−18.6880	−1.55196
$146$	0	0
$147$	−3.39302	−0.279852
$148$	0	0
$149$	−19.5826	−1.60427	−0.802136	−	0.597142i	$-0.796303\pi$
$149$	−19.5826	−1.60427	−0.802136	+	0.597142i	$0.796303\pi$
$150$	0	0
$151$	17.7854	1.44735	0.723676	−	0.690140i	$-0.242451\pi$
$151$	17.7854	1.44735	0.723676	+	0.690140i	$0.242451\pi$
$152$	0	0
$153$	2.71377	0.219395
$154$	0	0
$155$	6.48025	0.520506
$156$	0	0
$157$	−11.7298	−0.936137	−0.468068	−	0.883692i	$-0.655050\pi$
$157$	−11.7298	−0.936137	−0.468068	+	0.883692i	$0.655050\pi$
$158$	0	0
$159$	3.78575	0.300229
$160$	0	0
$161$	5.16739	0.407247
$162$	0	0
$163$	3.64161	0.285233	0.142616	−	0.989778i	$-0.454448\pi$
$163$	3.64161	0.285233	0.142616	+	0.989778i	$0.454448\pi$
$164$	0	0
$165$	−3.79921	−0.295768
$166$	0	0
$167$	−9.04055	−0.699579	−0.349789	−	0.936828i	$-0.613747\pi$
$167$	−9.04055	−0.699579	−0.349789	+	0.936828i	$0.613747\pi$
$168$	0	0
$169$	−10.0365	−0.772041
$170$	0	0
$171$	1.80017	0.137662
$172$	0	0
$173$	12.4771	0.948618	0.474309	−	0.880359i	$-0.342698\pi$
$173$	12.4771	0.948618	0.474309	+	0.880359i	$0.342698\pi$
$174$	0	0
$175$	−0.628798	−0.0475327
$176$	0	0
$177$	0.535004	0.0402133
$178$	0	0
$179$	5.36333	0.400874	0.200437	−	0.979707i	$-0.435764\pi$
$179$	5.36333	0.400874	0.200437	+	0.979707i	$0.435764\pi$
$180$	0	0
$181$	9.63798	0.716386	0.358193	−	0.933648i	$-0.383393\pi$
$181$	9.63798	0.716386	0.358193	+	0.933648i	$0.383393\pi$
$182$	0	0
$183$	−2.27023	−0.167820
$184$	0	0
$185$	4.12486	0.303265
$186$	0	0
$187$	3.45488	0.252646
$188$	0	0
$189$	−2.47956	−0.180361
$190$	0	0
$191$	3.84984	0.278565	0.139282	−	0.990253i	$-0.455520\pi$
$191$	3.84984	0.278565	0.139282	+	0.990253i	$0.455520\pi$
$192$	0	0
$193$	2.37091	0.170662	0.0853309	−	0.996353i	$-0.472805\pi$
$193$	2.37091	0.170662	0.0853309	+	0.996353i	$0.472805\pi$
$194$	0	0
$195$	1.89304	0.135563
$196$	0	0
$197$	−13.2614	−0.944838	−0.472419	−	0.881374i	$-0.656619\pi$
$197$	−13.2614	−0.944838	−0.472419	+	0.881374i	$0.656619\pi$
$198$	0	0
$199$	−24.4714	−1.73473	−0.867367	−	0.497670i	$-0.834189\pi$
$199$	−24.4714	−1.73473	−0.867367	+	0.497670i	$0.834189\pi$
$200$	0	0
$201$	−1.74538	−0.123110
$202$	0	0
$203$	−7.37488	−0.517615
$204$	0	0
$205$	9.66215	0.674834
$206$	0	0
$207$	−17.2882	−1.20161
$208$	0	0
$209$	2.29178	0.158526
$210$	0	0
$211$	18.4842	1.27251	0.636253	−	0.771481i	$-0.280484\pi$
$211$	18.4842	1.27251	0.636253	+	0.771481i	$0.280484\pi$
$212$	0	0
$213$	−5.17583	−0.354642
$214$	0	0
$215$	−12.3974	−0.845496
$216$	0	0
$217$	2.55731	0.173601
$218$	0	0
$219$	5.58615	0.377477
$220$	0	0
$221$	−1.72147	−0.115799
$222$	0	0
$223$	−24.1876	−1.61972	−0.809862	−	0.586621i	$-0.800458\pi$
$223$	−24.1876	−1.61972	−0.809862	+	0.586621i	$0.800458\pi$
$224$	0	0
$225$	2.10373	0.140249
$226$	0	0
$227$	16.2615	1.07932	0.539658	−	0.841884i	$-0.318554\pi$
$227$	16.2615	1.07932	0.539658	+	0.841884i	$0.318554\pi$
$228$	0	0
$229$	−21.5433	−1.42362	−0.711811	−	0.702371i	$-0.752125\pi$
$229$	−21.5433	−1.42362	−0.711811	+	0.702371i	$0.752125\pi$
$230$	0	0
$231$	−1.49929	−0.0986457
$232$	0	0
$233$	8.29511	0.543430	0.271715	−	0.962378i	$-0.412409\pi$
$233$	8.29511	0.543430	0.271715	+	0.962378i	$0.412409\pi$
$234$	0	0
$235$	−27.5801	−1.79913
$236$	0	0
$237$	0.978081	0.0635332
$238$	0	0
$239$	19.4255	1.25653	0.628266	−	0.777998i	$-0.283765\pi$
$239$	19.4255	1.25653	0.628266	+	0.777998i	$0.283765\pi$
$240$	0	0
$241$	−0.0959857	−0.00618298	−0.00309149	−	0.999995i	$-0.500984\pi$
$241$	−0.0959857	−0.00618298	−0.00309149	+	0.999995i	$0.500984\pi$
$242$	0	0
$243$	12.6513	0.811583
$244$	0	0
$245$	−13.0357	−0.832818
$246$	0	0
$247$	−1.14193	−0.0726594
$248$	0	0
$249$	8.14039	0.515876
$250$	0	0
$251$	8.82037	0.556737	0.278368	−	0.960474i	$-0.410206\pi$
$251$	8.82037	0.556737	0.278368	+	0.960474i	$0.410206\pi$
$252$	0	0
$253$	−22.0095	−1.38373
$254$	0	0
$255$	−1.09966	−0.0688636
$256$	0	0
$257$	17.2640	1.07690	0.538451	−	0.842657i	$-0.319010\pi$
$257$	17.2640	1.07690	0.538451	+	0.842657i	$0.319010\pi$
$258$	0	0
$259$	1.62780	0.101146
$260$	0	0
$261$	24.6737	1.52726
$262$	0	0
$263$	−28.6534	−1.76684	−0.883422	−	0.468579i	$-0.844766\pi$
$263$	−28.6534	−1.76684	−0.883422	+	0.468579i	$0.844766\pi$
$264$	0	0
$265$	14.5445	0.893459
$266$	0	0
$267$	−0.216826	−0.0132695
$268$	0	0
$269$	14.2456	0.868571	0.434285	−	0.900775i	$-0.357001\pi$
$269$	14.2456	0.868571	0.434285	+	0.900775i	$0.357001\pi$
$270$	0	0
$271$	5.02808	0.305434	0.152717	−	0.988270i	$-0.451198\pi$
$271$	5.02808	0.305434	0.152717	+	0.988270i	$0.451198\pi$
$272$	0	0
$273$	0.747052	0.0452137
$274$	0	0
$275$	2.67824	0.161504
$276$	0	0
$277$	−23.4335	−1.40798	−0.703992	−	0.710208i	$-0.748601\pi$
$277$	−23.4335	−1.40798	−0.703992	+	0.710208i	$0.748601\pi$
$278$	0	0
$279$	−8.55583	−0.512224
$280$	0	0
$281$	13.9407	0.831634	0.415817	−	0.909448i	$-0.363496\pi$
$281$	13.9407	0.831634	0.415817	+	0.909448i	$0.363496\pi$
$282$	0	0
$283$	−11.6397	−0.691907	−0.345954	−	0.938252i	$-0.612445\pi$
$283$	−11.6397	−0.691907	−0.345954	+	0.938252i	$0.612445\pi$
$284$	0	0
$285$	−0.729458	−0.0432093
$286$	0	0
$287$	3.81299	0.225074
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−8.71557	−0.510916
$292$	0	0
$293$	−3.64041	−0.212675	−0.106338	−	0.994330i	$-0.533912\pi$
$293$	−3.64041	−0.212675	−0.106338	+	0.994330i	$0.533912\pi$
$294$	0	0
$295$	2.05543	0.119672
$296$	0	0
$297$	10.5612	0.612822
$298$	0	0
$299$	10.9667	0.634222
$300$	0	0
$301$	−4.89240	−0.281993
$302$	0	0
$303$	−1.30534	−0.0749898
$304$	0	0
$305$	−8.72199	−0.499420
$306$	0	0
$307$	19.0822	1.08908	0.544538	−	0.838736i	$-0.316705\pi$
$307$	19.0822	1.08908	0.544538	+	0.838736i	$0.316705\pi$
$308$	0	0
$309$	0.418386	0.0238012
$310$	0	0
$311$	−6.08661	−0.345140	−0.172570	−	0.984997i	$-0.555207\pi$
$311$	−6.08661	−0.345140	−0.172570	+	0.984997i	$0.555207\pi$
$312$	0	0
$313$	−3.18360	−0.179948	−0.0899740	−	0.995944i	$-0.528678\pi$
$313$	−3.18360	−0.179948	−0.0899740	+	0.995944i	$0.528678\pi$
$314$	0	0
$315$	−4.52450	−0.254927
$316$	0	0
$317$	−11.6266	−0.653018	−0.326509	−	0.945194i	$-0.605872\pi$
$317$	−11.6266	−0.653018	−0.326509	+	0.945194i	$0.605872\pi$
$318$	0	0
$319$	31.4119	1.75873
$320$	0	0
$321$	4.11480	0.229666
$322$	0	0
$323$	0.663347	0.0369096
$324$	0	0
$325$	−1.33449	−0.0740244
$326$	0	0
$327$	−3.26342	−0.180468
$328$	0	0
$329$	−10.8840	−0.600053
$330$	0	0
$331$	−10.3831	−0.570709	−0.285354	−	0.958422i	$-0.592111\pi$
$331$	−10.3831	−0.570709	−0.285354	+	0.958422i	$0.592111\pi$
$332$	0	0
$333$	−5.44602	−0.298440
$334$	0	0
$335$	−6.70558	−0.366365
$336$	0	0
$337$	−8.57722	−0.467231	−0.233616	−	0.972329i	$-0.575056\pi$
$337$	−8.57722	−0.467231	−0.233616	+	0.972329i	$0.575056\pi$
$338$	0	0
$339$	−9.75269	−0.529694
$340$	0	0
$341$	−10.8924	−0.589855
$342$	0	0
$343$	−10.8222	−0.584346
$344$	0	0
$345$	7.00546	0.377161
$346$	0	0
$347$	−17.0394	−0.914723	−0.457361	−	0.889281i	$-0.651205\pi$
$347$	−17.0394	−0.914723	−0.457361	+	0.889281i	$0.651205\pi$
$348$	0	0
$349$	−1.35418	−0.0724876	−0.0362438	−	0.999343i	$-0.511539\pi$
$349$	−1.35418	−0.0724876	−0.0362438	+	0.999343i	$0.511539\pi$
$350$	0	0
$351$	−5.26235	−0.280883
$352$	0	0
$353$	7.68875	0.409231	0.204615	−	0.978842i	$-0.434406\pi$
$353$	7.68875	0.409231	0.204615	+	0.978842i	$0.434406\pi$
$354$	0	0
$355$	−19.8850	−1.05539
$356$	0	0
$357$	−0.433961	−0.0229677
$358$	0	0
$359$	24.7027	1.30376	0.651879	−	0.758323i	$-0.273981\pi$
$359$	24.7027	1.30376	0.651879	+	0.758323i	$0.273981\pi$
$360$	0	0
$361$	−18.5600	−0.976841
$362$	0	0
$363$	0.500877	0.0262892
$364$	0	0
$365$	21.4614	1.12334
$366$	0	0
$367$	9.42529	0.491996	0.245998	−	0.969270i	$-0.420884\pi$
$367$	9.42529	0.491996	0.245998	+	0.969270i	$0.420884\pi$
$368$	0	0
$369$	−12.7569	−0.664096
$370$	0	0
$371$	5.73970	0.297990
$372$	0	0
$373$	30.3875	1.57341	0.786703	−	0.617332i	$-0.211786\pi$
$373$	30.3875	1.57341	0.786703	+	0.617332i	$0.211786\pi$
$374$	0	0
$375$	−6.35078	−0.327953
$376$	0	0
$377$	−15.6517	−0.806101
$378$	0	0
$379$	−0.0182744	−0.000938693 0	−0.000469346	−	1.00000i	$-0.500149\pi$
$379$	−0.0182744	−0.000938693 0	−0.000469346		1.00000i	$0.500149\pi$
$380$	0	0
$381$	−1.44156	−0.0738535
$382$	0	0
$383$	31.0406	1.58610	0.793052	−	0.609155i	$-0.208491\pi$
$383$	31.0406	1.58610	0.793052	+	0.609155i	$0.208491\pi$
$384$	0	0
$385$	−5.76010	−0.293562
$386$	0	0
$387$	16.3682	0.832043
$388$	0	0
$389$	7.39790	0.375088	0.187544	−	0.982256i	$-0.439947\pi$
$389$	7.39790	0.375088	0.187544	+	0.982256i	$0.439947\pi$
$390$	0	0
$391$	−6.37055	−0.322173
$392$	0	0
$393$	−1.75110	−0.0883311
$394$	0	0
$395$	3.75769	0.189070
$396$	0	0
$397$	1.29282	0.0648848	0.0324424	−	0.999474i	$-0.489671\pi$
$397$	1.29282	0.0648848	0.0324424	+	0.999474i	$0.489671\pi$
$398$	0	0
$399$	−0.287867	−0.0144114
$400$	0	0
$401$	2.87476	0.143558	0.0717792	−	0.997421i	$-0.477132\pi$
$401$	2.87476	0.143558	0.0717792	+	0.997421i	$0.477132\pi$
$402$	0	0
$403$	5.42736	0.270356
$404$	0	0
$405$	13.3724	0.664478
$406$	0	0
$407$	−6.93329	−0.343670
$408$	0	0
$409$	14.4562	0.714811	0.357405	−	0.933949i	$-0.383661\pi$
$409$	14.4562	0.714811	0.357405	+	0.933949i	$0.383661\pi$
$410$	0	0
$411$	−6.25943	−0.308755
$412$	0	0
$413$	0.811137	0.0399134
$414$	0	0
$415$	31.2745	1.53521
$416$	0	0
$417$	−10.5985	−0.519011
$418$	0	0
$419$	−3.11640	−0.152246	−0.0761230	−	0.997098i	$-0.524254\pi$
$419$	−3.11640	−0.152246	−0.0761230	+	0.997098i	$0.524254\pi$
$420$	0	0
$421$	−15.8531	−0.772633	−0.386316	−	0.922366i	$-0.626253\pi$
$421$	−15.8531	−0.772633	−0.386316	+	0.922366i	$0.626253\pi$
$422$	0	0
$423$	36.4138	1.77050
$424$	0	0
$425$	0.775206	0.0376030
$426$	0	0
$427$	−3.44197	−0.166569
$428$	0	0
$429$	−3.18192	−0.153625
$430$	0	0
$431$	17.0313	0.820369	0.410185	−	0.912003i	$-0.365464\pi$
$431$	17.0313	0.820369	0.410185	+	0.912003i	$0.365464\pi$
$432$	0	0
$433$	−0.282550	−0.0135785	−0.00678925	−	0.999977i	$-0.502161\pi$
$433$	−0.282550	−0.0135785	−0.00678925	+	0.999977i	$0.502161\pi$
$434$	0	0
$435$	−9.99816	−0.479375
$436$	0	0
$437$	−4.22588	−0.202151
$438$	0	0
$439$	18.3905	0.877730	0.438865	−	0.898553i	$-0.355381\pi$
$439$	18.3905	0.877730	0.438865	+	0.898553i	$0.355381\pi$
$440$	0	0
$441$	17.2109	0.819566
$442$	0	0
$443$	18.2877	0.868876	0.434438	−	0.900702i	$-0.356947\pi$
$443$	18.2877	0.868876	0.434438	+	0.900702i	$0.356947\pi$
$444$	0	0
$445$	−0.833024	−0.0394891
$446$	0	0
$447$	−10.4768	−0.495535
$448$	0	0
$449$	4.09225	0.193125	0.0965627	−	0.995327i	$-0.469215\pi$
$449$	4.09225	0.193125	0.0965627	+	0.995327i	$0.469215\pi$
$450$	0	0
$451$	−16.2407	−0.764744
$452$	0	0
$453$	9.51523	0.447065
$454$	0	0
$455$	2.87010	0.134552
$456$	0	0
$457$	5.04616	0.236049	0.118025	−	0.993011i	$-0.462344\pi$
$457$	5.04616	0.236049	0.118025	+	0.993011i	$0.462344\pi$
$458$	0	0
$459$	3.05689	0.142683
$460$	0	0
$461$	−18.2077	−0.848015	−0.424007	−	0.905659i	$-0.639377\pi$
$461$	−18.2077	−0.848015	−0.424007	+	0.905659i	$0.639377\pi$
$462$	0	0
$463$	−12.7079	−0.590587	−0.295294	−	0.955407i	$-0.595417\pi$
$463$	−12.7079	−0.590587	−0.295294	+	0.955407i	$0.595417\pi$
$464$	0	0
$465$	3.46696	0.160776
$466$	0	0
$467$	3.89516	0.180247	0.0901234	−	0.995931i	$-0.471274\pi$
$467$	3.89516	0.180247	0.0901234	+	0.995931i	$0.471274\pi$
$468$	0	0
$469$	−2.64623	−0.122192
$470$	0	0
$471$	−6.27547	−0.289158
$472$	0	0
$473$	20.8383	0.958144
$474$	0	0
$475$	0.514230	0.0235945
$476$	0	0
$477$	−19.2030	−0.879243
$478$	0	0
$479$	14.8162	0.676968	0.338484	−	0.940972i	$-0.390086\pi$
$479$	14.8162	0.676968	0.338484	+	0.940972i	$0.390086\pi$
$480$	0	0
$481$	3.45466	0.157519
$482$	0	0
$483$	2.76457	0.125792
$484$	0	0
$485$	−33.4843	−1.52044
$486$	0	0
$487$	11.8415	0.536588	0.268294	−	0.963337i	$-0.413540\pi$
$487$	11.8415	0.536588	0.268294	+	0.963337i	$0.413540\pi$
$488$	0	0
$489$	1.94828	0.0881041
$490$	0	0
$491$	−23.7355	−1.07117	−0.535584	−	0.844482i	$-0.679909\pi$
$491$	−23.7355	−1.07117	−0.535584	+	0.844482i	$0.679909\pi$
$492$	0	0
$493$	9.09202	0.409484
$494$	0	0
$495$	19.2712	0.866177
$496$	0	0
$497$	−7.84725	−0.351997
$498$	0	0
$499$	−19.5274	−0.874164	−0.437082	−	0.899422i	$-0.643988\pi$
$499$	−19.5274	−0.874164	−0.437082	+	0.899422i	$0.643988\pi$
$500$	0	0
$501$	−4.83673	−0.216089
$502$	0	0
$503$	−26.9581	−1.20200	−0.601001	−	0.799248i	$-0.705231\pi$
$503$	−26.9581	−1.20200	−0.601001	+	0.799248i	$0.705231\pi$
$504$	0	0
$505$	−5.01498	−0.223164
$506$	0	0
$507$	−5.36958	−0.238472
$508$	0	0
$509$	−27.2506	−1.20786	−0.603930	−	0.797037i	$-0.706399\pi$
$509$	−27.2506	−1.20786	−0.603930	+	0.797037i	$0.706399\pi$
$510$	0	0
$511$	8.46935	0.374662
$512$	0	0
$513$	2.02778	0.0895285
$514$	0	0
$515$	1.60740	0.0708304
$516$	0	0
$517$	46.3582	2.03883
$518$	0	0
$519$	6.67530	0.293013
$520$	0	0
$521$	−14.4301	−0.632195	−0.316098	−	0.948727i	$-0.602373\pi$
$521$	−14.4301	−0.632195	−0.316098	+	0.948727i	$0.602373\pi$
$522$	0	0
$523$	4.86193	0.212597	0.106299	−	0.994334i	$-0.466100\pi$
$523$	4.86193	0.212597	0.106299	+	0.994334i	$0.466100\pi$
$524$	0	0
$525$	−0.336409	−0.0146821
$526$	0	0
$527$	−3.15275	−0.137336
$528$	0	0
$529$	17.5839	0.764519
$530$	0	0
$531$	−2.71377	−0.117768
$532$	0	0
$533$	8.09228	0.350516
$534$	0	0
$535$	15.8087	0.683468
$536$	0	0
$537$	2.86940	0.123824
$538$	0	0
$539$	21.9111	0.943776
$540$	0	0
$541$	−7.08620	−0.304659	−0.152330	−	0.988330i	$-0.548678\pi$
$541$	−7.08620	−0.304659	−0.152330	+	0.988330i	$0.548678\pi$
$542$	0	0
$543$	5.15636	0.221280
$544$	0	0
$545$	−12.5377	−0.537058
$546$	0	0
$547$	−0.312421	−0.0133582	−0.00667908	−	0.999978i	$-0.502126\pi$
$547$	−0.312421	−0.0133582	−0.00667908	+	0.999978i	$0.502126\pi$
$548$	0	0
$549$	11.5156	0.491473
$550$	0	0
$551$	6.03116	0.256936
$552$	0	0
$553$	1.48290	0.0630594
$554$	0	0
$555$	2.20681	0.0936740
$556$	0	0
$557$	−36.7081	−1.55537	−0.777686	−	0.628652i	$-0.783607\pi$
$557$	−36.7081	−1.55537	−0.777686	+	0.628652i	$0.783607\pi$
$558$	0	0
$559$	−10.3831	−0.439159
$560$	0	0
$561$	1.84837	0.0780384
$562$	0	0
$563$	−43.6133	−1.83808	−0.919042	−	0.394161i	$-0.871035\pi$
$563$	−43.6133	−1.83808	−0.919042	+	0.394161i	$0.871035\pi$
$564$	0	0
$565$	−37.4689	−1.57633
$566$	0	0
$567$	5.27715	0.221619
$568$	0	0
$569$	7.28373	0.305350	0.152675	−	0.988276i	$-0.451211\pi$
$569$	7.28373	0.305350	0.152675	+	0.988276i	$0.451211\pi$
$570$	0	0
$571$	23.9611	1.00274	0.501371	−	0.865232i	$-0.332829\pi$
$571$	23.9611	1.00274	0.501371	+	0.865232i	$0.332829\pi$
$572$	0	0
$573$	2.05968	0.0860443
$574$	0	0
$575$	−4.93849	−0.205949
$576$	0	0
$577$	−34.8836	−1.45223	−0.726113	−	0.687576i	$-0.758675\pi$
$577$	−34.8836	−1.45223	−0.726113	+	0.687576i	$0.758675\pi$
$578$	0	0
$579$	1.26845	0.0527148
$580$	0	0
$581$	12.3419	0.512029
$582$	0	0
$583$	−24.4471	−1.01250
$584$	0	0
$585$	−9.60231	−0.397007
$586$	0	0
$587$	38.5894	1.59275	0.796377	−	0.604800i	$-0.206747\pi$
$587$	38.5894	1.59275	0.796377	+	0.604800i	$0.206747\pi$
$588$	0	0
$589$	−2.09136	−0.0861731
$590$	0	0
$591$	−7.09491	−0.291846
$592$	0	0
$593$	−11.2064	−0.460192	−0.230096	−	0.973168i	$-0.573904\pi$
$593$	−11.2064	−0.460192	−0.230096	+	0.973168i	$0.573904\pi$
$594$	0	0
$595$	−1.66724	−0.0683500
$596$	0	0
$597$	−13.0923	−0.535832
$598$	0	0
$599$	34.1483	1.39526	0.697631	−	0.716457i	$-0.254237\pi$
$599$	34.1483	1.39526	0.697631	+	0.716457i	$0.254237\pi$
$600$	0	0
$601$	4.59503	0.187435	0.0937175	−	0.995599i	$-0.470125\pi$
$601$	4.59503	0.187435	0.0937175	+	0.995599i	$0.470125\pi$
$602$	0	0
$603$	8.85333	0.360536
$604$	0	0
$605$	1.92432	0.0782347
$606$	0	0
$607$	−2.68115	−0.108825	−0.0544124	−	0.998519i	$-0.517329\pi$
$607$	−2.68115	−0.108825	−0.0544124	+	0.998519i	$0.517329\pi$
$608$	0	0
$609$	−3.94559	−0.159883
$610$	0	0
$611$	−23.0990	−0.934485
$612$	0	0
$613$	−32.5189	−1.31343	−0.656713	−	0.754140i	$-0.728054\pi$
$613$	−32.5189	−1.31343	−0.656713	+	0.754140i	$0.728054\pi$
$614$	0	0
$615$	5.16929	0.208446
$616$	0	0
$617$	−20.8890	−0.840961	−0.420480	−	0.907302i	$-0.638138\pi$
$617$	−20.8890	−0.840961	−0.420480	+	0.907302i	$0.638138\pi$
$618$	0	0
$619$	45.4907	1.82843	0.914213	−	0.405235i	$-0.132810\pi$
$619$	45.4907	1.82843	0.914213	+	0.405235i	$0.132810\pi$
$620$	0	0
$621$	−19.4741	−0.781467
$622$	0	0
$623$	−0.328737	−0.0131706
$624$	0	0
$625$	−20.5230	−0.820921
$626$	0	0
$627$	1.22611	0.0489662
$628$	0	0
$629$	−2.00681	−0.0800167
$630$	0	0
$631$	−0.193160	−0.00768957	−0.00384478	−	0.999993i	$-0.501224\pi$
$631$	−0.193160	−0.00768957	−0.00384478	+	0.999993i	$0.501224\pi$
$632$	0	0
$633$	9.88912	0.393057
$634$	0	0
$635$	−5.53834	−0.219782
$636$	0	0
$637$	−10.9177	−0.432574
$638$	0	0
$639$	26.2541	1.03859
$640$	0	0
$641$	−15.6499	−0.618135	−0.309068	−	0.951040i	$-0.600017\pi$
$641$	−15.6499	−0.618135	−0.309068	+	0.951040i	$0.600017\pi$
$642$	0	0
$643$	18.8110	0.741833	0.370917	−	0.928666i	$-0.379044\pi$
$643$	18.8110	0.741833	0.370917	+	0.928666i	$0.379044\pi$
$644$	0	0
$645$	−6.63266	−0.261161
$646$	0	0
$647$	−4.93916	−0.194178	−0.0970892	−	0.995276i	$-0.530953\pi$
$647$	−4.93916	−0.194178	−0.0970892	+	0.995276i	$0.530953\pi$
$648$	0	0
$649$	−3.45488	−0.135616
$650$	0	0
$651$	1.36817	0.0536228
$652$	0	0
$653$	35.0251	1.37064	0.685319	−	0.728243i	$-0.259663\pi$
$653$	35.0251	1.37064	0.685319	+	0.728243i	$0.259663\pi$
$654$	0	0
$655$	−6.72754	−0.262867
$656$	0	0
$657$	−28.3354	−1.10547
$658$	0	0
$659$	−7.37858	−0.287429	−0.143714	−	0.989619i	$-0.545905\pi$
$659$	−7.37858	−0.287429	−0.143714	+	0.989619i	$0.545905\pi$
$660$	0	0
$661$	21.2592	0.826886	0.413443	−	0.910530i	$-0.364326\pi$
$661$	21.2592	0.826886	0.413443	+	0.910530i	$0.364326\pi$
$662$	0	0
$663$	−0.920994	−0.0357684
$664$	0	0
$665$	−1.10596	−0.0428871
$666$	0	0
$667$	−57.9212	−2.24272
$668$	0	0
$669$	−12.9405	−0.500308
$670$	0	0
$671$	14.6604	0.565959
$672$	0	0
$673$	−35.2030	−1.35698	−0.678488	−	0.734611i	$-0.737365\pi$
$673$	−35.2030	−1.35698	−0.678488	+	0.734611i	$0.737365\pi$
$674$	0	0
$675$	2.36972	0.0912105
$676$	0	0
$677$	−32.6909	−1.25641	−0.628206	−	0.778047i	$-0.716210\pi$
$677$	−32.6909	−1.25641	−0.628206	+	0.778047i	$0.716210\pi$
$678$	0	0
$679$	−13.2140	−0.507105
$680$	0	0
$681$	8.69998	0.333384
$682$	0	0
$683$	−22.5233	−0.861830	−0.430915	−	0.902393i	$-0.641809\pi$
$683$	−22.5233	−0.861830	−0.430915	+	0.902393i	$0.641809\pi$
$684$	0	0
$685$	−24.0481	−0.918830
$686$	0	0
$687$	−11.5257	−0.439735
$688$	0	0
$689$	12.1813	0.464072
$690$	0	0
$691$	−6.31457	−0.240217	−0.120109	−	0.992761i	$-0.538324\pi$
$691$	−6.31457	−0.240217	−0.120109	+	0.992761i	$0.538324\pi$
$692$	0	0
$693$	7.60503	0.288891
$694$	0	0
$695$	−40.7184	−1.54454
$696$	0	0
$697$	−4.70079	−0.178055
$698$	0	0
$699$	4.43791	0.167857
$700$	0	0
$701$	30.7292	1.16063	0.580313	−	0.814394i	$-0.302930\pi$
$701$	30.7292	1.16063	0.580313	+	0.814394i	$0.302930\pi$
$702$	0	0
$703$	−1.33121	−0.0502075
$704$	0	0
$705$	−14.7555	−0.555723
$706$	0	0
$707$	−1.97907	−0.0744305
$708$	0	0
$709$	−30.1587	−1.13264	−0.566318	−	0.824187i	$-0.691632\pi$
$709$	−30.1587	−1.13264	−0.566318	+	0.824187i	$0.691632\pi$
$710$	0	0
$711$	−4.96125	−0.186062
$712$	0	0
$713$	20.0847	0.752179
$714$	0	0
$715$	−12.2246	−0.457175
$716$	0	0
$717$	10.3927	0.388124
$718$	0	0
$719$	−33.3510	−1.24378	−0.621891	−	0.783104i	$-0.713635\pi$
$719$	−33.3510	−1.24378	−0.621891	+	0.783104i	$0.713635\pi$
$720$	0	0
$721$	0.634330	0.0236237
$722$	0	0
$723$	−0.0513527	−0.00190983
$724$	0	0
$725$	7.04819	0.261763
$726$	0	0
$727$	−19.0210	−0.705451	−0.352725	−	0.935727i	$-0.614745\pi$
$727$	−19.0210	−0.705451	−0.352725	+	0.935727i	$0.614745\pi$
$728$	0	0
$729$	−12.7491	−0.472189
$730$	0	0
$731$	6.03154	0.223084
$732$	0	0
$733$	24.0526	0.888404	0.444202	−	0.895927i	$-0.353487\pi$
$733$	24.0526	0.888404	0.444202	+	0.895927i	$0.353487\pi$
$734$	0	0
$735$	−6.97412	−0.257244
$736$	0	0
$737$	11.2711	0.415177
$738$	0	0
$739$	1.55816	0.0573177	0.0286589	−	0.999589i	$-0.490876\pi$
$739$	1.55816	0.0573177	0.0286589	+	0.999589i	$0.490876\pi$
$740$	0	0
$741$	−0.610938	−0.0224434
$742$	0	0
$743$	−7.07918	−0.259710	−0.129855	−	0.991533i	$-0.541451\pi$
$743$	−7.07918	−0.259710	−0.129855	+	0.991533i	$0.541451\pi$
$744$	0	0
$745$	−40.2507	−1.47467
$746$	0	0
$747$	−41.2916	−1.51078
$748$	0	0
$749$	6.23859	0.227953
$750$	0	0
$751$	44.6214	1.62826	0.814128	−	0.580685i	$-0.197215\pi$
$751$	44.6214	1.62826	0.814128	+	0.580685i	$0.197215\pi$
$752$	0	0
$753$	4.71893	0.171967
$754$	0	0
$755$	36.5566	1.33043
$756$	0	0
$757$	12.5013	0.454368	0.227184	−	0.973852i	$-0.427048\pi$
$757$	12.5013	0.454368	0.227184	+	0.973852i	$0.427048\pi$
$758$	0	0
$759$	−11.7752	−0.427412
$760$	0	0
$761$	−4.31323	−0.156355	−0.0781773	−	0.996939i	$-0.524910\pi$
$761$	−4.31323	−0.156355	−0.0781773	+	0.996939i	$0.524910\pi$
$762$	0	0
$763$	−4.94778	−0.179122
$764$	0	0
$765$	5.57797	0.201672
$766$	0	0
$767$	1.72147	0.0621587
$768$	0	0
$769$	38.0144	1.37083	0.685417	−	0.728151i	$-0.259620\pi$
$769$	38.0144	1.37083	0.685417	+	0.728151i	$0.259620\pi$
$770$	0	0
$771$	9.23633	0.332638
$772$	0	0
$773$	25.5511	0.919010	0.459505	−	0.888175i	$-0.348027\pi$
$773$	25.5511	0.919010	0.459505	+	0.888175i	$0.348027\pi$
$774$	0	0
$775$	−2.44403	−0.0877920
$776$	0	0
$777$	0.870878	0.0312426
$778$	0	0
$779$	−3.11826	−0.111723
$780$	0	0
$781$	33.4239	1.19600
$782$	0	0
$783$	27.7933	0.993252
$784$	0	0
$785$	−24.1097	−0.860512
$786$	0	0
$787$	−13.3799	−0.476943	−0.238471	−	0.971150i	$-0.576646\pi$
$787$	−13.3799	−0.476943	−0.238471	+	0.971150i	$0.576646\pi$
$788$	0	0
$789$	−15.3297	−0.545751
$790$	0	0
$791$	−14.7864	−0.525743
$792$	0	0
$793$	−7.30487	−0.259404
$794$	0	0
$795$	7.78134	0.275976
$796$	0	0
$797$	−8.55424	−0.303007	−0.151503	−	0.988457i	$-0.548411\pi$
$797$	−8.55424	−0.303007	−0.151503	+	0.988457i	$0.548411\pi$
$798$	0	0
$799$	13.4182	0.474700
$800$	0	0
$801$	1.09984	0.0388608
$802$	0	0
$803$	−36.0736	−1.27301
$804$	0	0
$805$	10.6212	0.374349
$806$	0	0
$807$	7.62146	0.268288
$808$	0	0
$809$	−21.8777	−0.769180	−0.384590	−	0.923087i	$-0.625657\pi$
$809$	−21.8777	−0.769180	−0.384590	+	0.923087i	$0.625657\pi$
$810$	0	0
$811$	6.96136	0.244447	0.122223	−	0.992503i	$-0.460998\pi$
$811$	6.96136	0.244447	0.122223	+	0.992503i	$0.460998\pi$
$812$	0	0
$813$	2.69004	0.0943439
$814$	0	0
$815$	7.48508	0.262191
$816$	0	0
$817$	4.00100	0.139977
$818$	0	0
$819$	−3.78937	−0.132411
$820$	0	0
$821$	−10.1922	−0.355709	−0.177854	−	0.984057i	$-0.556916\pi$
$821$	−10.1922	−0.355709	−0.177854	+	0.984057i	$0.556916\pi$
$822$	0	0
$823$	16.5328	0.576298	0.288149	−	0.957586i	$-0.406960\pi$
$823$	16.5328	0.576298	0.288149	+	0.957586i	$0.406960\pi$
$824$	0	0
$825$	1.43287	0.0498862
$826$	0	0
$827$	28.3995	0.987547	0.493773	−	0.869591i	$-0.335617\pi$
$827$	28.3995	0.987547	0.493773	+	0.869591i	$0.335617\pi$
$828$	0	0
$829$	16.1424	0.560649	0.280325	−	0.959905i	$-0.409558\pi$
$829$	16.1424	0.560649	0.280325	+	0.959905i	$0.409558\pi$
$830$	0	0
$831$	−12.5370	−0.434904
$832$	0	0
$833$	6.34206	0.219739
$834$	0	0
$835$	−18.5822	−0.643064
$836$	0	0
$837$	−9.63759	−0.333124
$838$	0	0
$839$	48.1991	1.66402	0.832009	−	0.554762i	$-0.187191\pi$
$839$	48.1991	1.66402	0.832009	+	0.554762i	$0.187191\pi$
$840$	0	0
$841$	53.6649	1.85051
$842$	0	0
$843$	7.45834	0.256879
$844$	0	0
$845$	−20.6294	−0.709673
$846$	0	0
$847$	0.759396	0.0260932
$848$	0	0
$849$	−6.22727	−0.213719
$850$	0	0
$851$	12.7845	0.438246
$852$	0	0
$853$	−47.0722	−1.61172	−0.805860	−	0.592106i	$-0.798297\pi$
$853$	−47.0722	−1.61172	−0.805860	+	0.592106i	$0.798297\pi$
$854$	0	0
$855$	3.70013	0.126542
$856$	0	0
$857$	−22.5819	−0.771381	−0.385691	−	0.922628i	$-0.626037\pi$
$857$	−22.5819	−0.771381	−0.385691	+	0.922628i	$0.626037\pi$
$858$	0	0
$859$	−0.962769	−0.0328493	−0.0164246	−	0.999865i	$-0.505228\pi$
$859$	−0.962769	−0.0328493	−0.0164246	+	0.999865i	$0.505228\pi$
$860$	0	0
$861$	2.03996	0.0695217
$862$	0	0
$863$	−0.996503	−0.0339213	−0.0169607	−	0.999856i	$-0.505399\pi$
$863$	−0.996503	−0.0339213	−0.0169607	+	0.999856i	$0.505399\pi$
$864$	0	0
$865$	25.6458	0.871985
$866$	0	0
$867$	0.535004	0.0181697
$868$	0	0
$869$	−6.31614	−0.214260
$870$	0	0
$871$	−5.61608	−0.190294
$872$	0	0
$873$	44.2091	1.49625
$874$	0	0
$875$	−9.62863	−0.325507
$876$	0	0
$877$	9.21554	0.311187	0.155593	−	0.987821i	$-0.450271\pi$
$877$	9.21554	0.311187	0.155593	+	0.987821i	$0.450271\pi$
$878$	0	0
$879$	−1.94763	−0.0656921
$880$	0	0
$881$	18.7606	0.632060	0.316030	−	0.948749i	$-0.397650\pi$
$881$	18.7606	0.632060	0.316030	+	0.948749i	$0.397650\pi$
$882$	0	0
$883$	−16.0294	−0.539433	−0.269716	−	0.962940i	$-0.586930\pi$
$883$	−16.0294	−0.539433	−0.269716	+	0.962940i	$0.586930\pi$
$884$	0	0
$885$	1.09966	0.0369648
$886$	0	0
$887$	14.6881	0.493179	0.246590	−	0.969120i	$-0.420690\pi$
$887$	14.6881	0.493179	0.246590	+	0.969120i	$0.420690\pi$
$888$	0	0
$889$	−2.18560	−0.0733027
$890$	0	0
$891$	−22.4770	−0.753008
$892$	0	0
$893$	8.90089	0.297857
$894$	0	0
$895$	11.0240	0.368490
$896$	0	0
$897$	5.86724	0.195901
$898$	0	0
$899$	−28.6648	−0.956026
$900$	0	0
$901$	−7.07611	−0.235739
$902$	0	0
$903$	−2.61745	−0.0871034
$904$	0	0
$905$	19.8102	0.658513
$906$	0	0
$907$	42.5564	1.41306	0.706531	−	0.707682i	$-0.250259\pi$
$907$	42.5564	1.41306	0.706531	+	0.707682i	$0.250259\pi$
$908$	0	0
$909$	6.62125	0.219613
$910$	0	0
$911$	27.0779	0.897131	0.448565	−	0.893750i	$-0.351935\pi$
$911$	27.0779	0.897131	0.448565	+	0.893750i	$0.351935\pi$
$912$	0	0
$913$	−52.5680	−1.73975
$914$	0	0
$915$	−4.66630	−0.154263
$916$	0	0
$917$	−2.65490	−0.0876724
$918$	0	0
$919$	−20.4281	−0.673860	−0.336930	−	0.941530i	$-0.609389\pi$
$919$	−20.4281	−0.673860	−0.336930	+	0.941530i	$0.609389\pi$
$920$	0	0
$921$	10.2090	0.336399
$922$	0	0
$923$	−16.6542	−0.548179
$924$	0	0
$925$	−1.55569	−0.0511508
$926$	0	0
$927$	−2.12224	−0.0697034
$928$	0	0
$929$	14.4978	0.475656	0.237828	−	0.971307i	$-0.423565\pi$
$929$	14.4978	0.475656	0.237828	+	0.971307i	$0.423565\pi$
$930$	0	0
$931$	4.20698	0.137878
$932$	0	0
$933$	−3.25636	−0.106608
$934$	0	0
$935$	7.10127	0.232236
$936$	0	0
$937$	17.7924	0.581251	0.290625	−	0.956837i	$-0.406137\pi$
$937$	17.7924	0.581251	0.290625	+	0.956837i	$0.406137\pi$
$938$	0	0
$939$	−1.70324	−0.0555831
$940$	0	0
$941$	7.45827	0.243133	0.121566	−	0.992583i	$-0.461208\pi$
$941$	7.45827	0.243133	0.121566	+	0.992583i	$0.461208\pi$
$942$	0	0
$943$	29.9466	0.975197
$944$	0	0
$945$	−5.09655	−0.165791
$946$	0	0
$947$	−32.2780	−1.04889	−0.524446	−	0.851443i	$-0.675728\pi$
$947$	−32.2780	−1.04889	−0.524446	+	0.851443i	$0.675728\pi$
$948$	0	0
$949$	17.9745	0.583476
$950$	0	0
$951$	−6.22030	−0.201707
$952$	0	0
$953$	−42.5324	−1.37776	−0.688880	−	0.724875i	$-0.741897\pi$
$953$	−42.5324	−1.37776	−0.688880	+	0.724875i	$0.741897\pi$
$954$	0	0
$955$	7.91308	0.256061
$956$	0	0
$957$	16.8055	0.543244
$958$	0	0
$959$	−9.49013	−0.306452
$960$	0	0
$961$	−21.0602	−0.679361
$962$	0	0
$963$	−20.8721	−0.672593
$964$	0	0
$965$	4.87324	0.156875
$966$	0	0
$967$	52.0820	1.67484	0.837422	−	0.546557i	$-0.184062\pi$
$967$	52.0820	1.67484	0.837422	+	0.546557i	$0.184062\pi$
$968$	0	0
$969$	0.354893	0.0114008
$970$	0	0
$971$	33.4756	1.07428	0.537141	−	0.843492i	$-0.319504\pi$
$971$	33.4756	1.07428	0.537141	+	0.843492i	$0.319504\pi$
$972$	0	0
$973$	−16.0688	−0.515141
$974$	0	0
$975$	−0.713960	−0.0228650
$976$	0	0
$977$	−37.9639	−1.21457	−0.607286	−	0.794483i	$-0.707742\pi$
$977$	−37.9639	−1.21457	−0.607286	+	0.794483i	$0.707742\pi$
$978$	0	0
$979$	1.40019	0.0447504
$980$	0	0
$981$	16.5535	0.528512
$982$	0	0
$983$	−9.21261	−0.293837	−0.146918	−	0.989149i	$-0.546935\pi$
$983$	−9.21261	−0.293837	−0.146918	+	0.989149i	$0.546935\pi$
$984$	0	0
$985$	−27.2579	−0.868510
$986$	0	0
$987$	−5.82296	−0.185347
$988$	0	0
$989$	−38.4242	−1.22182
$990$	0	0
$991$	−58.3543	−1.85368	−0.926842	−	0.375451i	$-0.877488\pi$
$991$	−58.3543	−1.85368	−0.926842	+	0.375451i	$0.877488\pi$
$992$	0	0
$993$	−5.55501	−0.176283
$994$	0	0
$995$	−50.2993	−1.59460
$996$	0	0
$997$	45.7383	1.44855	0.724273	−	0.689513i	$-0.242176\pi$
$997$	45.7383	1.44855	0.724273	+	0.689513i	$0.242176\pi$
$998$	0	0
$999$	−6.13459	−0.194090

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.x.1.15	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.x.1.15	✓	22	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+0.535004 q^{3} +2.05543 q^{5} +0.811137 q^{7} -2.71377 q^{9} +O(q^{10})\)	\(q+0.535004 q^{3} +2.05543 q^{5} +0.811137 q^{7} -2.71377 q^{9} -3.45488 q^{11} +1.72147 q^{13} +1.09966 q^{15} -1.00000 q^{17} -0.663347 q^{19} +0.433961 q^{21} +6.37055 q^{23} -0.775206 q^{25} -3.05689 q^{27} -9.09202 q^{29} +3.15275 q^{31} -1.84837 q^{33} +1.66724 q^{35} +2.00681 q^{37} +0.920994 q^{39} +4.70079 q^{41} -6.03154 q^{43} -5.57797 q^{45} -13.4182 q^{47} -6.34206 q^{49} -0.535004 q^{51} +7.07611 q^{53} -7.10127 q^{55} -0.354893 q^{57} +1.00000 q^{59} -4.24339 q^{61} -2.20124 q^{63} +3.53837 q^{65} -3.26237 q^{67} +3.40827 q^{69} -9.67438 q^{71} +10.4413 q^{73} -0.414738 q^{75} -2.80238 q^{77} +1.82818 q^{79} +6.50587 q^{81} +15.2156 q^{83} -2.05543 q^{85} -4.86427 q^{87} -0.405280 q^{89} +1.39635 q^{91} +1.68673 q^{93} -1.36346 q^{95} -16.2907 q^{97} +9.37576 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9}+O(q^{10}) \) 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9	\( 22 q - 3 q^{3} + 3 q^{5} + 13 q^{9} + 2 q^{11} - 9 q^{13} - 7 q^{15} - 22 q^{17} - 10 q^{19} - 10 q^{21} + 14 q^{23} + 3 q^{25} + 6 q^{29} - 15 q^{31} - 52 q^{33} - 7 q^{35} + 9 q^{37} - 52 q^{41} - 7 q^{43} - 30 q^{45} - 7 q^{47} - 6 q^{49} + 3 q^{51} - 18 q^{53} - 39 q^{55} - 2 q^{57} + 22 q^{59} - 42 q^{61} - 35 q^{65} - 28 q^{67} - 10 q^{69} + 23 q^{71} - 33 q^{73} - 3 q^{75} - 28 q^{77} - 30 q^{79} - 2 q^{81} + 11 q^{83} - 3 q^{85} + 27 q^{87} - 34 q^{89} - 18 q^{91} - 28 q^{93} - 10 q^{95} - 3 q^{97} - 28 q^{99}+O(q^{100}) \) 22 * q - 3 * q^3 + 3 * q^5 + 13 * q^9 + 2 * q^11 - 9 * q^13 - 7 * q^15 - 22 * q^17 - 10 * q^19 - 10 * q^21 + 14 * q^23 + 3 * q^25 + 6 * q^29 - 15 * q^31 - 52 * q^33 - 7 * q^35 + 9 * q^37 - 52 * q^41 - 7 * q^43 - 30 * q^45 - 7 * q^47 - 6 * q^49 + 3 * q^51 - 18 * q^53 - 39 * q^55 - 2 * q^57 + 22 * q^59 - 42 * q^61 - 35 * q^65 - 28 * q^67 - 10 * q^69 + 23 * q^71 - 33 * q^73 - 3 * q^75 - 28 * q^77 - 30 * q^79 - 2 * q^81 + 11 * q^83 - 3 * q^85 + 27 * q^87 - 34 * q^89 - 18 * q^91 - 28 * q^93 - 10 * q^95 - 3 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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