LMFDB - Embedded newform 8024.2.a.bb.1.19

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:	$0$
Dimension:	$32$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q+0.792010 q^{3} -0.983575 q^{5} +1.99855 q^{7} -2.37272 q^{9} +O(q^{10})$	$q+0.792010 q^{3} -0.983575 q^{5} +1.99855 q^{7} -2.37272 q^{9} +1.73734 q^{11} +3.28098 q^{13} -0.779001 q^{15} +1.00000 q^{17} -2.40431 q^{19} +1.58287 q^{21} +7.43975 q^{23} -4.03258 q^{25} -4.25525 q^{27} +1.18279 q^{29} -2.58549 q^{31} +1.37599 q^{33} -1.96572 q^{35} +5.78997 q^{37} +2.59857 q^{39} +5.60174 q^{41} -5.69938 q^{43} +2.33375 q^{45} +8.65156 q^{47} -3.00580 q^{49} +0.792010 q^{51} -3.71785 q^{53} -1.70881 q^{55} -1.90423 q^{57} +1.00000 q^{59} +3.84667 q^{61} -4.74200 q^{63} -3.22709 q^{65} +11.5489 q^{67} +5.89236 q^{69} -3.48128 q^{71} -15.6536 q^{73} -3.19384 q^{75} +3.47216 q^{77} +17.2255 q^{79} +3.74796 q^{81} +10.4687 q^{83} -0.983575 q^{85} +0.936785 q^{87} -8.23391 q^{89} +6.55720 q^{91} -2.04773 q^{93} +2.36481 q^{95} +11.1786 q^{97} -4.12223 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9}+O(q^{10}) $ 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9	$ 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9} + 3 q^{11} + 13 q^{13} + 4 q^{15} + 32 q^{17} + 14 q^{19} - 7 q^{21} + 7 q^{23} + 38 q^{25} + 9 q^{27} + 17 q^{29} + 15 q^{31} + 18 q^{33} + 6 q^{35} + 21 q^{37} + 16 q^{39} + 49 q^{41} - 7 q^{43} + 14 q^{45} - 25 q^{47} + 37 q^{49} + 12 q^{53} + 15 q^{55} + 45 q^{57} + 32 q^{59} + 5 q^{61} - 12 q^{63} + 39 q^{65} + 12 q^{69} - 13 q^{71} + 70 q^{73} - 47 q^{75} - 10 q^{77} - q^{79} + 84 q^{81} - 17 q^{83} + 8 q^{85} + 20 q^{87} + 42 q^{89} + 36 q^{91} + 2 q^{93} - q^{95} + 58 q^{97} + 7 q^{99}+O(q^{100}) $ 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9 + 3 * q^11 + 13 * q^13 + 4 * q^15 + 32 * q^17 + 14 * q^19 - 7 * q^21 + 7 * q^23 + 38 * q^25 + 9 * q^27 + 17 * q^29 + 15 * q^31 + 18 * q^33 + 6 * q^35 + 21 * q^37 + 16 * q^39 + 49 * q^41 - 7 * q^43 + 14 * q^45 - 25 * q^47 + 37 * q^49 + 12 * q^53 + 15 * q^55 + 45 * q^57 + 32 * q^59 + 5 * q^61 - 12 * q^63 + 39 * q^65 + 12 * q^69 - 13 * q^71 + 70 * q^73 - 47 * q^75 - 10 * q^77 - q^79 + 84 * q^81 - 17 * q^83 + 8 * q^85 + 20 * q^87 + 42 * q^89 + 36 * q^91 + 2 * q^93 - q^95 + 58 * q^97 + 7 * 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
$3$	0.792010	0.457267	0.228634	−	0.973513i	$-0.426574\pi$
$3$	0.792010	0.457267	0.228634	+	0.973513i	$0.426574\pi$
$4$	0	0
$5$	−0.983575	−0.439868	−0.219934	−	0.975515i	$-0.570584\pi$
$5$	−0.983575	−0.439868	−0.219934	+	0.975515i	$0.570584\pi$
$6$	0	0
$7$	1.99855	0.755381	0.377690	−	0.925932i	$-0.376718\pi$
$7$	1.99855	0.755381	0.377690	+	0.925932i	$0.376718\pi$
$8$	0	0
$9$	−2.37272	−0.790907
$10$	0	0
$11$	1.73734	0.523828	0.261914	−	0.965091i	$-0.415646\pi$
$11$	1.73734	0.523828	0.261914	+	0.965091i	$0.415646\pi$
$12$	0	0
$13$	3.28098	0.909980	0.454990	−	0.890496i	$-0.349643\pi$
$13$	3.28098	0.909980	0.454990	+	0.890496i	$0.349643\pi$
$14$	0	0
$15$	−0.779001	−0.201137
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.40431	−0.551586	−0.275793	−	0.961217i	$-0.588940\pi$
$19$	−2.40431	−0.551586	−0.275793	+	0.961217i	$0.588940\pi$
$20$	0	0
$21$	1.58287	0.345411
$22$	0	0
$23$	7.43975	1.55130	0.775648	−	0.631166i	$-0.217423\pi$
$23$	7.43975	1.55130	0.775648	+	0.631166i	$0.217423\pi$
$24$	0	0
$25$	−4.03258	−0.806516
$26$	0	0
$27$	−4.25525	−0.818923
$28$	0	0
$29$	1.18279	0.219639	0.109820	−	0.993952i	$-0.464973\pi$
$29$	1.18279	0.219639	0.109820	+	0.993952i	$0.464973\pi$
$30$	0	0
$31$	−2.58549	−0.464368	−0.232184	−	0.972672i	$-0.574587\pi$
$31$	−2.58549	−0.464368	−0.232184	+	0.972672i	$0.574587\pi$
$32$	0	0
$33$	1.37599	0.239530
$34$	0	0
$35$	−1.96572	−0.332268
$36$	0	0
$37$	5.78997	0.951865	0.475932	−	0.879482i	$-0.342111\pi$
$37$	5.78997	0.951865	0.475932	+	0.879482i	$0.342111\pi$
$38$	0	0
$39$	2.59857	0.416104
$40$	0	0
$41$	5.60174	0.874844	0.437422	−	0.899256i	$-0.355892\pi$
$41$	5.60174	0.874844	0.437422	+	0.899256i	$0.355892\pi$
$42$	0	0
$43$	−5.69938	−0.869148	−0.434574	−	0.900636i	$-0.643101\pi$
$43$	−5.69938	−0.869148	−0.434574	+	0.900636i	$0.643101\pi$
$44$	0	0
$45$	2.33375	0.347895
$46$	0	0
$47$	8.65156	1.26196	0.630980	−	0.775799i	$-0.282653\pi$
$47$	8.65156	1.26196	0.630980	+	0.775799i	$0.282653\pi$
$48$	0	0
$49$	−3.00580	−0.429400
$50$	0	0
$51$	0.792010	0.110904
$52$	0	0
$53$	−3.71785	−0.510686	−0.255343	−	0.966851i	$-0.582188\pi$
$53$	−3.71785	−0.510686	−0.255343	+	0.966851i	$0.582188\pi$
$54$	0	0
$55$	−1.70881	−0.230415
$56$	0	0
$57$	−1.90423	−0.252222
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	3.84667	0.492515	0.246258	−	0.969204i	$-0.420799\pi$
$61$	3.84667	0.492515	0.246258	+	0.969204i	$0.420799\pi$
$62$	0	0
$63$	−4.74200	−0.597436
$64$	0	0
$65$	−3.22709	−0.400271
$66$	0	0
$67$	11.5489	1.41092	0.705459	−	0.708750i	$-0.250741\pi$
$67$	11.5489	1.41092	0.705459	+	0.708750i	$0.250741\pi$
$68$	0	0
$69$	5.89236	0.709356
$70$	0	0
$71$	−3.48128	−0.413152	−0.206576	−	0.978431i	$-0.566232\pi$
$71$	−3.48128	−0.413152	−0.206576	+	0.978431i	$0.566232\pi$
$72$	0	0
$73$	−15.6536	−1.83212	−0.916059	−	0.401042i	$-0.868648\pi$
$73$	−15.6536	−1.83212	−0.916059	+	0.401042i	$0.868648\pi$
$74$	0	0
$75$	−3.19384	−0.368793
$76$	0	0
$77$	3.47216	0.395690
$78$	0	0
$79$	17.2255	1.93802	0.969009	−	0.247026i	$-0.0794532\pi$
$79$	17.2255	1.93802	0.969009	+	0.247026i	$0.0794532\pi$
$80$	0	0
$81$	3.74796	0.416440
$82$	0	0
$83$	10.4687	1.14909	0.574543	−	0.818474i	$-0.305180\pi$
$83$	10.4687	1.14909	0.574543	+	0.818474i	$0.305180\pi$
$84$	0	0
$85$	−0.983575	−0.106684
$86$	0	0
$87$	0.936785	0.100434
$88$	0	0
$89$	−8.23391	−0.872792	−0.436396	−	0.899755i	$-0.643745\pi$
$89$	−8.23391	−0.872792	−0.436396	+	0.899755i	$0.643745\pi$
$90$	0	0
$91$	6.55720	0.687382
$92$	0	0
$93$	−2.04773	−0.212340
$94$	0	0
$95$	2.36481	0.242625
$96$	0	0
$97$	11.1786	1.13501	0.567507	−	0.823368i	$-0.307908\pi$
$97$	11.1786	1.13501	0.567507	+	0.823368i	$0.307908\pi$
$98$	0	0
$99$	−4.12223	−0.414299
$100$	0	0
$101$	6.12668	0.609628	0.304814	−	0.952412i	$-0.401406\pi$
$101$	6.12668	0.609628	0.304814	+	0.952412i	$0.401406\pi$
$102$	0	0
$103$	3.58076	0.352822	0.176411	−	0.984317i	$-0.443551\pi$
$103$	3.58076	0.352822	0.176411	+	0.984317i	$0.443551\pi$
$104$	0	0
$105$	−1.55687	−0.151935
$106$	0	0
$107$	−10.2099	−0.987033	−0.493516	−	0.869736i	$-0.664289\pi$
$107$	−10.2099	−0.987033	−0.493516	+	0.869736i	$0.664289\pi$
$108$	0	0
$109$	−10.2108	−0.978014	−0.489007	−	0.872280i	$-0.662641\pi$
$109$	−10.2108	−0.978014	−0.489007	+	0.872280i	$0.662641\pi$
$110$	0	0
$111$	4.58571	0.435256
$112$	0	0
$113$	−11.7942	−1.10951	−0.554754	−	0.832014i	$-0.687188\pi$
$113$	−11.7942	−1.10951	−0.554754	+	0.832014i	$0.687188\pi$
$114$	0	0
$115$	−7.31755	−0.682365
$116$	0	0
$117$	−7.78485	−0.719709
$118$	0	0
$119$	1.99855	0.183207
$120$	0	0
$121$	−7.98164	−0.725604
$122$	0	0
$123$	4.43663	0.400038
$124$	0	0
$125$	8.88422	0.794629
$126$	0	0
$127$	−2.88205	−0.255740	−0.127870	−	0.991791i	$-0.540814\pi$
$127$	−2.88205	−0.255740	−0.127870	+	0.991791i	$0.540814\pi$
$128$	0	0
$129$	−4.51397	−0.397433
$130$	0	0
$131$	15.7099	1.37258	0.686291	−	0.727327i	$-0.259238\pi$
$131$	15.7099	1.37258	0.686291	+	0.727327i	$0.259238\pi$
$132$	0	0
$133$	−4.80513	−0.416657
$134$	0	0
$135$	4.18536	0.360218
$136$	0	0
$137$	21.2604	1.81640	0.908200	−	0.418537i	$-0.137457\pi$
$137$	21.2604	1.81640	0.908200	+	0.418537i	$0.137457\pi$
$138$	0	0
$139$	−15.7559	−1.33640	−0.668201	−	0.743981i	$-0.732935\pi$
$139$	−15.7559	−1.33640	−0.668201	+	0.743981i	$0.732935\pi$
$140$	0	0
$141$	6.85213	0.577053
$142$	0	0
$143$	5.70019	0.476674
$144$	0	0
$145$	−1.16337	−0.0966124
$146$	0	0
$147$	−2.38062	−0.196351
$148$	0	0
$149$	5.03026	0.412095	0.206048	−	0.978542i	$-0.433940\pi$
$149$	5.03026	0.412095	0.206048	+	0.978542i	$0.433940\pi$
$150$	0	0
$151$	22.0634	1.79549	0.897747	−	0.440512i	$-0.145203\pi$
$151$	22.0634	1.79549	0.897747	+	0.440512i	$0.145203\pi$
$152$	0	0
$153$	−2.37272	−0.191823
$154$	0	0
$155$	2.54302	0.204261
$156$	0	0
$157$	8.91626	0.711595	0.355798	−	0.934563i	$-0.384209\pi$
$157$	8.91626	0.711595	0.355798	+	0.934563i	$0.384209\pi$
$158$	0	0
$159$	−2.94458	−0.233520
$160$	0	0
$161$	14.8687	1.17182
$162$	0	0
$163$	18.6408	1.46006	0.730029	−	0.683416i	$-0.239506\pi$
$163$	18.6408	1.46006	0.730029	+	0.683416i	$0.239506\pi$
$164$	0	0
$165$	−1.35339	−0.105361
$166$	0	0
$167$	0.821415	0.0635630	0.0317815	−	0.999495i	$-0.489882\pi$
$167$	0.821415	0.0635630	0.0317815	+	0.999495i	$0.489882\pi$
$168$	0	0
$169$	−2.23517	−0.171936
$170$	0	0
$171$	5.70475	0.436253
$172$	0	0
$173$	−7.15910	−0.544296	−0.272148	−	0.962255i	$-0.587734\pi$
$173$	−7.15910	−0.544296	−0.272148	+	0.962255i	$0.587734\pi$
$174$	0	0
$175$	−8.05931	−0.609227
$176$	0	0
$177$	0.792010	0.0595311
$178$	0	0
$179$	−18.9328	−1.41511	−0.707553	−	0.706661i	$-0.750201\pi$
$179$	−18.9328	−1.41511	−0.707553	+	0.706661i	$0.750201\pi$
$180$	0	0
$181$	−18.8902	−1.40410	−0.702050	−	0.712128i	$-0.747732\pi$
$181$	−18.8902	−1.40410	−0.702050	+	0.712128i	$0.747732\pi$
$182$	0	0
$183$	3.04660	0.225211
$184$	0	0
$185$	−5.69486	−0.418695
$186$	0	0
$187$	1.73734	0.127047
$188$	0	0
$189$	−8.50433	−0.618599
$190$	0	0
$191$	−22.9765	−1.66252	−0.831259	−	0.555885i	$-0.812380\pi$
$191$	−22.9765	−1.66252	−0.831259	+	0.555885i	$0.812380\pi$
$192$	0	0
$193$	23.2384	1.67274	0.836368	−	0.548169i	$-0.184675\pi$
$193$	23.2384	1.67274	0.836368	+	0.548169i	$0.184675\pi$
$194$	0	0
$195$	−2.55589	−0.183031
$196$	0	0
$197$	11.9204	0.849297	0.424648	−	0.905358i	$-0.360398\pi$
$197$	11.9204	0.849297	0.424648	+	0.905358i	$0.360398\pi$
$198$	0	0
$199$	9.41635	0.667507	0.333754	−	0.942660i	$-0.391685\pi$
$199$	9.41635	0.667507	0.333754	+	0.942660i	$0.391685\pi$
$200$	0	0
$201$	9.14682	0.645167
$202$	0	0
$203$	2.36387	0.165911
$204$	0	0
$205$	−5.50973	−0.384816
$206$	0	0
$207$	−17.6524	−1.22693
$208$	0	0
$209$	−4.17710	−0.288936
$210$	0	0
$211$	4.68433	0.322483	0.161241	−	0.986915i	$-0.448450\pi$
$211$	4.68433	0.322483	0.161241	+	0.986915i	$0.448450\pi$
$212$	0	0
$213$	−2.75721	−0.188921
$214$	0	0
$215$	5.60577	0.382310
$216$	0	0
$217$	−5.16723	−0.350774
$218$	0	0
$219$	−12.3978	−0.837768
$220$	0	0
$221$	3.28098	0.220703
$222$	0	0
$223$	7.85434	0.525965	0.262983	−	0.964801i	$-0.415294\pi$
$223$	7.85434	0.525965	0.262983	+	0.964801i	$0.415294\pi$
$224$	0	0
$225$	9.56818	0.637879
$226$	0	0
$227$	−22.8943	−1.51954	−0.759772	−	0.650189i	$-0.774690\pi$
$227$	−22.8943	−1.51954	−0.759772	+	0.650189i	$0.774690\pi$
$228$	0	0
$229$	5.96029	0.393867	0.196933	−	0.980417i	$-0.436902\pi$
$229$	5.96029	0.393867	0.196933	+	0.980417i	$0.436902\pi$
$230$	0	0
$231$	2.74999	0.180936
$232$	0	0
$233$	−10.9409	−0.716763	−0.358381	−	0.933575i	$-0.616671\pi$
$233$	−10.9409	−0.716763	−0.358381	+	0.933575i	$0.616671\pi$
$234$	0	0
$235$	−8.50946	−0.555096
$236$	0	0
$237$	13.6428	0.886192
$238$	0	0
$239$	19.8609	1.28470	0.642349	−	0.766413i	$-0.277960\pi$
$239$	19.8609	1.28470	0.642349	+	0.766413i	$0.277960\pi$
$240$	0	0
$241$	5.64703	0.363757	0.181879	−	0.983321i	$-0.441782\pi$
$241$	5.64703	0.363757	0.181879	+	0.983321i	$0.441782\pi$
$242$	0	0
$243$	15.7342	1.00935
$244$	0	0
$245$	2.95643	0.188879
$246$	0	0
$247$	−7.88848	−0.501932
$248$	0	0
$249$	8.29129	0.525439
$250$	0	0
$251$	11.2511	0.710166	0.355083	−	0.934835i	$-0.384453\pi$
$251$	11.2511	0.710166	0.355083	+	0.934835i	$0.384453\pi$
$252$	0	0
$253$	12.9254	0.812612
$254$	0	0
$255$	−0.779001	−0.0487829
$256$	0	0
$257$	6.71003	0.418560	0.209280	−	0.977856i	$-0.432888\pi$
$257$	6.71003	0.418560	0.209280	+	0.977856i	$0.432888\pi$
$258$	0	0
$259$	11.5715	0.719020
$260$	0	0
$261$	−2.80644	−0.173714
$262$	0	0
$263$	15.2987	0.943359	0.471679	−	0.881770i	$-0.343648\pi$
$263$	15.2987	0.943359	0.471679	+	0.881770i	$0.343648\pi$
$264$	0	0
$265$	3.65679	0.224635
$266$	0	0
$267$	−6.52134	−0.399099
$268$	0	0
$269$	8.39004	0.511550	0.255775	−	0.966736i	$-0.417669\pi$
$269$	8.39004	0.511550	0.255775	+	0.966736i	$0.417669\pi$
$270$	0	0
$271$	8.40123	0.510338	0.255169	−	0.966896i	$-0.417869\pi$
$271$	8.40123	0.510338	0.255169	+	0.966896i	$0.417869\pi$
$272$	0	0
$273$	5.19337	0.314317
$274$	0	0
$275$	−7.00597	−0.422476
$276$	0	0
$277$	11.8863	0.714179	0.357090	−	0.934070i	$-0.383769\pi$
$277$	11.8863	0.714179	0.357090	+	0.934070i	$0.383769\pi$
$278$	0	0
$279$	6.13464	0.367272
$280$	0	0
$281$	12.1317	0.723719	0.361859	−	0.932233i	$-0.382142\pi$
$281$	12.1317	0.723719	0.361859	+	0.932233i	$0.382142\pi$
$282$	0	0
$283$	−4.75277	−0.282523	−0.141261	−	0.989972i	$-0.545116\pi$
$283$	−4.75277	−0.282523	−0.141261	+	0.989972i	$0.545116\pi$
$284$	0	0
$285$	1.87296	0.110944
$286$	0	0
$287$	11.1953	0.660840
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	8.85356	0.519005
$292$	0	0
$293$	15.4540	0.902830	0.451415	−	0.892314i	$-0.350919\pi$
$293$	15.4540	0.902830	0.451415	+	0.892314i	$0.350919\pi$
$294$	0	0
$295$	−0.983575	−0.0572659
$296$	0	0
$297$	−7.39282	−0.428975
$298$	0	0
$299$	24.4097	1.41165
$300$	0	0
$301$	−11.3905	−0.656537
$302$	0	0
$303$	4.85239	0.278763
$304$	0	0
$305$	−3.78349	−0.216642
$306$	0	0
$307$	27.0407	1.54330	0.771648	−	0.636050i	$-0.219433\pi$
$307$	27.0407	1.54330	0.771648	+	0.636050i	$0.219433\pi$
$308$	0	0
$309$	2.83599	0.161334
$310$	0	0
$311$	−30.1224	−1.70808	−0.854041	−	0.520206i	$-0.825855\pi$
$311$	−30.1224	−1.70808	−0.854041	+	0.520206i	$0.825855\pi$
$312$	0	0
$313$	12.6465	0.714824	0.357412	−	0.933947i	$-0.383659\pi$
$313$	12.6465	0.714824	0.357412	+	0.933947i	$0.383659\pi$
$314$	0	0
$315$	4.66411	0.262793
$316$	0	0
$317$	6.48794	0.364399	0.182200	−	0.983262i	$-0.441678\pi$
$317$	6.48794	0.364399	0.182200	+	0.983262i	$0.441678\pi$
$318$	0	0
$319$	2.05492	0.115053
$320$	0	0
$321$	−8.08638	−0.451338
$322$	0	0
$323$	−2.40431	−0.133779
$324$	0	0
$325$	−13.2308	−0.733914
$326$	0	0
$327$	−8.08703	−0.447214
$328$	0	0
$329$	17.2906	0.953261
$330$	0	0
$331$	8.07580	0.443886	0.221943	−	0.975060i	$-0.428760\pi$
$331$	8.07580	0.443886	0.221943	+	0.975060i	$0.428760\pi$
$332$	0	0
$333$	−13.7380	−0.752836
$334$	0	0
$335$	−11.3592	−0.620618
$336$	0	0
$337$	0.109820	0.00598227	0.00299114	−	0.999996i	$-0.499048\pi$
$337$	0.109820	0.00598227	0.00299114	+	0.999996i	$0.499048\pi$
$338$	0	0
$339$	−9.34116	−0.507342
$340$	0	0
$341$	−4.49188	−0.243249
$342$	0	0
$343$	−19.9971	−1.07974
$344$	0	0
$345$	−5.79557	−0.312023
$346$	0	0
$347$	−6.59413	−0.353991	−0.176996	−	0.984212i	$-0.556638\pi$
$347$	−6.59413	−0.353991	−0.176996	+	0.984212i	$0.556638\pi$
$348$	0	0
$349$	8.09237	0.433174	0.216587	−	0.976263i	$-0.430507\pi$
$349$	8.09237	0.433174	0.216587	+	0.976263i	$0.430507\pi$
$350$	0	0
$351$	−13.9614	−0.745204
$352$	0	0
$353$	18.2578	0.971766	0.485883	−	0.874024i	$-0.338498\pi$
$353$	18.2578	0.971766	0.485883	+	0.874024i	$0.338498\pi$
$354$	0	0
$355$	3.42410	0.181732
$356$	0	0
$357$	1.58287	0.0837744
$358$	0	0
$359$	−18.5135	−0.977108	−0.488554	−	0.872534i	$-0.662475\pi$
$359$	−18.5135	−0.977108	−0.488554	+	0.872534i	$0.662475\pi$
$360$	0	0
$361$	−13.2193	−0.695753
$362$	0	0
$363$	−6.32154	−0.331795
$364$	0	0
$365$	15.3965	0.805891
$366$	0	0
$367$	25.0736	1.30883	0.654417	−	0.756134i	$-0.272914\pi$
$367$	25.0736	1.30883	0.654417	+	0.756134i	$0.272914\pi$
$368$	0	0
$369$	−13.2913	−0.691920
$370$	0	0
$371$	−7.43031	−0.385763
$372$	0	0
$373$	−0.896182	−0.0464026	−0.0232013	−	0.999731i	$-0.507386\pi$
$373$	−0.896182	−0.0464026	−0.0232013	+	0.999731i	$0.507386\pi$
$374$	0	0
$375$	7.03639	0.363358
$376$	0	0
$377$	3.88073	0.199868
$378$	0	0
$379$	10.7235	0.550827	0.275414	−	0.961326i	$-0.411185\pi$
$379$	10.7235	0.550827	0.275414	+	0.961326i	$0.411185\pi$
$380$	0	0
$381$	−2.28261	−0.116942
$382$	0	0
$383$	8.58631	0.438740	0.219370	−	0.975642i	$-0.429600\pi$
$383$	8.58631	0.438740	0.219370	+	0.975642i	$0.429600\pi$
$384$	0	0
$385$	−3.41513	−0.174051
$386$	0	0
$387$	13.5230	0.687415
$388$	0	0
$389$	29.1038	1.47562	0.737810	−	0.675008i	$-0.235860\pi$
$389$	29.1038	1.47562	0.737810	+	0.675008i	$0.235860\pi$
$390$	0	0
$391$	7.43975	0.376244
$392$	0	0
$393$	12.4424	0.627636
$394$	0	0
$395$	−16.9425	−0.852472
$396$	0	0
$397$	11.8360	0.594030	0.297015	−	0.954873i	$-0.404009\pi$
$397$	11.8360	0.594030	0.297015	+	0.954873i	$0.404009\pi$
$398$	0	0
$399$	−3.80571	−0.190524
$400$	0	0
$401$	−17.1407	−0.855966	−0.427983	−	0.903787i	$-0.640776\pi$
$401$	−17.1407	−0.855966	−0.427983	+	0.903787i	$0.640776\pi$
$402$	0	0
$403$	−8.48294	−0.422565
$404$	0	0
$405$	−3.68640	−0.183179
$406$	0	0
$407$	10.0592	0.498614
$408$	0	0
$409$	6.71075	0.331825	0.165913	−	0.986140i	$-0.446943\pi$
$409$	6.71075	0.331825	0.165913	+	0.986140i	$0.446943\pi$
$410$	0	0
$411$	16.8385	0.830580
$412$	0	0
$413$	1.99855	0.0983422
$414$	0	0
$415$	−10.2967	−0.505446
$416$	0	0
$417$	−12.4789	−0.611093
$418$	0	0
$419$	−21.0300	−1.02738	−0.513690	−	0.857976i	$-0.671722\pi$
$419$	−21.0300	−1.02738	−0.513690	+	0.857976i	$0.671722\pi$
$420$	0	0
$421$	−2.60004	−0.126718	−0.0633590	−	0.997991i	$-0.520181\pi$
$421$	−2.60004	−0.126718	−0.0633590	+	0.997991i	$0.520181\pi$
$422$	0	0
$423$	−20.5277	−0.998093
$424$	0	0
$425$	−4.03258	−0.195609
$426$	0	0
$427$	7.68776	0.372037
$428$	0	0
$429$	4.51461	0.217967
$430$	0	0
$431$	−4.93896	−0.237901	−0.118951	−	0.992900i	$-0.537953\pi$
$431$	−4.93896	−0.237901	−0.118951	+	0.992900i	$0.537953\pi$
$432$	0	0
$433$	14.7975	0.711122	0.355561	−	0.934653i	$-0.384290\pi$
$433$	14.7975	0.711122	0.355561	+	0.934653i	$0.384290\pi$
$434$	0	0
$435$	−0.921398	−0.0441777
$436$	0	0
$437$	−17.8874	−0.855672
$438$	0	0
$439$	15.2775	0.729154	0.364577	−	0.931173i	$-0.381214\pi$
$439$	15.2775	0.729154	0.364577	+	0.931173i	$0.381214\pi$
$440$	0	0
$441$	7.13192	0.339615
$442$	0	0
$443$	−9.04734	−0.429852	−0.214926	−	0.976630i	$-0.568951\pi$
$443$	−9.04734	−0.429852	−0.214926	+	0.976630i	$0.568951\pi$
$444$	0	0
$445$	8.09866	0.383913
$446$	0	0
$447$	3.98402	0.188438
$448$	0	0
$449$	1.24712	0.0588555	0.0294277	−	0.999567i	$-0.490632\pi$
$449$	1.24712	0.0588555	0.0294277	+	0.999567i	$0.490632\pi$
$450$	0	0
$451$	9.73213	0.458268
$452$	0	0
$453$	17.4744	0.821021
$454$	0	0
$455$	−6.44950	−0.302357
$456$	0	0
$457$	35.4937	1.66033	0.830163	−	0.557520i	$-0.188247\pi$
$457$	35.4937	1.66033	0.830163	+	0.557520i	$0.188247\pi$
$458$	0	0
$459$	−4.25525	−0.198618
$460$	0	0
$461$	−32.5631	−1.51661	−0.758307	−	0.651898i	$-0.773973\pi$
$461$	−32.5631	−1.51661	−0.758307	+	0.651898i	$0.773973\pi$
$462$	0	0
$463$	−14.0147	−0.651319	−0.325659	−	0.945487i	$-0.605586\pi$
$463$	−14.0147	−0.651319	−0.325659	+	0.945487i	$0.605586\pi$
$464$	0	0
$465$	2.01410	0.0934016
$466$	0	0
$467$	−9.22763	−0.427004	−0.213502	−	0.976943i	$-0.568487\pi$
$467$	−9.22763	−0.427004	−0.213502	+	0.976943i	$0.568487\pi$
$468$	0	0
$469$	23.0810	1.06578
$470$	0	0
$471$	7.06177	0.325389
$472$	0	0
$473$	−9.90178	−0.455284
$474$	0	0
$475$	9.69556	0.444863
$476$	0	0
$477$	8.82142	0.403905
$478$	0	0
$479$	−28.6477	−1.30895	−0.654475	−	0.756084i	$-0.727110\pi$
$479$	−28.6477	−1.30895	−0.654475	+	0.756084i	$0.727110\pi$
$480$	0	0
$481$	18.9968	0.866178
$482$	0	0
$483$	11.7762	0.535834
$484$	0	0
$485$	−10.9950	−0.499257
$486$	0	0
$487$	8.48457	0.384473	0.192236	−	0.981349i	$-0.438426\pi$
$487$	8.48457	0.384473	0.192236	+	0.981349i	$0.438426\pi$
$488$	0	0
$489$	14.7637	0.667637
$490$	0	0
$491$	−17.2688	−0.779329	−0.389664	−	0.920957i	$-0.627409\pi$
$491$	−17.2688	−0.779329	−0.389664	+	0.920957i	$0.627409\pi$
$492$	0	0
$493$	1.18279	0.0532704
$494$	0	0
$495$	4.05452	0.182237
$496$	0	0
$497$	−6.95752	−0.312087
$498$	0	0
$499$	17.1996	0.769958	0.384979	−	0.922925i	$-0.374209\pi$
$499$	17.1996	0.769958	0.384979	+	0.922925i	$0.374209\pi$
$500$	0	0
$501$	0.650569	0.0290653
$502$	0	0
$503$	−28.1919	−1.25701	−0.628506	−	0.777804i	$-0.716333\pi$
$503$	−28.1919	−1.25701	−0.628506	+	0.777804i	$0.716333\pi$
$504$	0	0
$505$	−6.02605	−0.268156
$506$	0	0
$507$	−1.77027	−0.0786206
$508$	0	0
$509$	−18.8756	−0.836645	−0.418322	−	0.908299i	$-0.637382\pi$
$509$	−18.8756	−0.836645	−0.418322	+	0.908299i	$0.637382\pi$
$510$	0	0
$511$	−31.2846	−1.38395
$512$	0	0
$513$	10.2309	0.451706
$514$	0	0
$515$	−3.52194	−0.155195
$516$	0	0
$517$	15.0307	0.661051
$518$	0	0
$519$	−5.67008	−0.248889
$520$	0	0
$521$	−17.3344	−0.759434	−0.379717	−	0.925103i	$-0.623979\pi$
$521$	−17.3344	−0.759434	−0.379717	+	0.925103i	$0.623979\pi$
$522$	0	0
$523$	2.25099	0.0984291	0.0492145	−	0.998788i	$-0.484328\pi$
$523$	2.25099	0.0984291	0.0492145	+	0.998788i	$0.484328\pi$
$524$	0	0
$525$	−6.38306	−0.278579
$526$	0	0
$527$	−2.58549	−0.112626
$528$	0	0
$529$	32.3499	1.40652
$530$	0	0
$531$	−2.37272	−0.102967
$532$	0	0
$533$	18.3792	0.796091
$534$	0	0
$535$	10.0422	0.434164
$536$	0	0
$537$	−14.9950	−0.647081
$538$	0	0
$539$	−5.22210	−0.224932
$540$	0	0
$541$	−7.99838	−0.343877	−0.171939	−	0.985108i	$-0.555003\pi$
$541$	−7.99838	−0.343877	−0.171939	+	0.985108i	$0.555003\pi$
$542$	0	0
$543$	−14.9612	−0.642049
$544$	0	0
$545$	10.0431	0.430197
$546$	0	0
$547$	38.3416	1.63937	0.819684	−	0.572815i	$-0.194149\pi$
$547$	38.3416	1.63937	0.819684	+	0.572815i	$0.194149\pi$
$548$	0	0
$549$	−9.12707	−0.389534
$550$	0	0
$551$	−2.84380	−0.121150
$552$	0	0
$553$	34.4260	1.46394
$554$	0	0
$555$	−4.51039	−0.191455
$556$	0	0
$557$	35.0063	1.48326	0.741631	−	0.670808i	$-0.234052\pi$
$557$	35.0063	1.48326	0.741631	+	0.670808i	$0.234052\pi$
$558$	0	0
$559$	−18.6996	−0.790907
$560$	0	0
$561$	1.37599	0.0580945
$562$	0	0
$563$	13.7343	0.578832	0.289416	−	0.957203i	$-0.406539\pi$
$563$	13.7343	0.578832	0.289416	+	0.957203i	$0.406539\pi$
$564$	0	0
$565$	11.6005	0.488037
$566$	0	0
$567$	7.49048	0.314571
$568$	0	0
$569$	−7.79955	−0.326974	−0.163487	−	0.986545i	$-0.552274\pi$
$569$	−7.79955	−0.326974	−0.163487	+	0.986545i	$0.552274\pi$
$570$	0	0
$571$	−13.4180	−0.561524	−0.280762	−	0.959777i	$-0.590587\pi$
$571$	−13.4180	−0.561524	−0.280762	+	0.959777i	$0.590587\pi$
$572$	0	0
$573$	−18.1976	−0.760215
$574$	0	0
$575$	−30.0014	−1.25114
$576$	0	0
$577$	29.3180	1.22052	0.610262	−	0.792200i	$-0.291064\pi$
$577$	29.3180	1.22052	0.610262	+	0.792200i	$0.291064\pi$
$578$	0	0
$579$	18.4050	0.764887
$580$	0	0
$581$	20.9222	0.867997
$582$	0	0
$583$	−6.45918	−0.267512
$584$	0	0
$585$	7.65698	0.316577
$586$	0	0
$587$	24.2390	1.00045	0.500226	−	0.865895i	$-0.333250\pi$
$587$	24.2390	1.00045	0.500226	+	0.865895i	$0.333250\pi$
$588$	0	0
$589$	6.21631	0.256139
$590$	0	0
$591$	9.44111	0.388356
$592$	0	0
$593$	−11.8677	−0.487347	−0.243674	−	0.969857i	$-0.578353\pi$
$593$	−11.8677	−0.487347	−0.243674	+	0.969857i	$0.578353\pi$
$594$	0	0
$595$	−1.96572	−0.0805868
$596$	0	0
$597$	7.45784	0.305229
$598$	0	0
$599$	−30.1882	−1.23346	−0.616728	−	0.787176i	$-0.711542\pi$
$599$	−30.1882	−1.23346	−0.616728	+	0.787176i	$0.711542\pi$
$600$	0	0
$601$	−10.6800	−0.435646	−0.217823	−	0.975988i	$-0.569895\pi$
$601$	−10.6800	−0.435646	−0.217823	+	0.975988i	$0.569895\pi$
$602$	0	0
$603$	−27.4022	−1.11591
$604$	0	0
$605$	7.85054	0.319170
$606$	0	0
$607$	16.6265	0.674850	0.337425	−	0.941352i	$-0.390444\pi$
$607$	16.6265	0.674850	0.337425	+	0.941352i	$0.390444\pi$
$608$	0	0
$609$	1.87221	0.0758659
$610$	0	0
$611$	28.3856	1.14836
$612$	0	0
$613$	−5.38670	−0.217567	−0.108783	−	0.994065i	$-0.534695\pi$
$613$	−5.38670	−0.217567	−0.108783	+	0.994065i	$0.534695\pi$
$614$	0	0
$615$	−4.36376	−0.175964
$616$	0	0
$617$	25.8387	1.04023	0.520114	−	0.854097i	$-0.325890\pi$
$617$	25.8387	1.04023	0.520114	+	0.854097i	$0.325890\pi$
$618$	0	0
$619$	−5.23636	−0.210467	−0.105234	−	0.994448i	$-0.533559\pi$
$619$	−5.23636	−0.210467	−0.105234	+	0.994448i	$0.533559\pi$
$620$	0	0
$621$	−31.6580	−1.27039
$622$	0	0
$623$	−16.4559	−0.659290
$624$	0	0
$625$	11.4246	0.456984
$626$	0	0
$627$	−3.30831	−0.132121
$628$	0	0
$629$	5.78997	0.230861
$630$	0	0
$631$	−21.2604	−0.846362	−0.423181	−	0.906045i	$-0.639087\pi$
$631$	−21.2604	−0.846362	−0.423181	+	0.906045i	$0.639087\pi$
$632$	0	0
$633$	3.71004	0.147461
$634$	0	0
$635$	2.83471	0.112492
$636$	0	0
$637$	−9.86197	−0.390745
$638$	0	0
$639$	8.26011	0.326765
$640$	0	0
$641$	19.9331	0.787311	0.393655	−	0.919258i	$-0.371210\pi$
$641$	19.9331	0.787311	0.393655	+	0.919258i	$0.371210\pi$
$642$	0	0
$643$	−27.2667	−1.07529	−0.537647	−	0.843170i	$-0.680687\pi$
$643$	−27.2667	−1.07529	−0.537647	+	0.843170i	$0.680687\pi$
$644$	0	0
$645$	4.43983	0.174818
$646$	0	0
$647$	−15.6926	−0.616940	−0.308470	−	0.951234i	$-0.599817\pi$
$647$	−15.6926	−0.616940	−0.308470	+	0.951234i	$0.599817\pi$
$648$	0	0
$649$	1.73734	0.0681966
$650$	0	0
$651$	−4.09250	−0.160398
$652$	0	0
$653$	−44.6718	−1.74814	−0.874071	−	0.485798i	$-0.838529\pi$
$653$	−44.6718	−1.74814	−0.874071	+	0.485798i	$0.838529\pi$
$654$	0	0
$655$	−15.4519	−0.603755
$656$	0	0
$657$	37.1417	1.44904
$658$	0	0
$659$	−38.5271	−1.50080	−0.750400	−	0.660983i	$-0.770139\pi$
$659$	−38.5271	−1.50080	−0.750400	+	0.660983i	$0.770139\pi$
$660$	0	0
$661$	9.31270	0.362222	0.181111	−	0.983463i	$-0.442031\pi$
$661$	9.31270	0.362222	0.181111	+	0.983463i	$0.442031\pi$
$662$	0	0
$663$	2.59857	0.100920
$664$	0	0
$665$	4.72620	0.183274
$666$	0	0
$667$	8.79970	0.340726
$668$	0	0
$669$	6.22071	0.240507
$670$	0	0
$671$	6.68298	0.257994
$672$	0	0
$673$	38.2049	1.47269	0.736346	−	0.676605i	$-0.236549\pi$
$673$	38.2049	1.47269	0.736346	+	0.676605i	$0.236549\pi$
$674$	0	0
$675$	17.1596	0.660475
$676$	0	0
$677$	47.9887	1.84435	0.922177	−	0.386769i	$-0.126409\pi$
$677$	47.9887	1.84435	0.922177	+	0.386769i	$0.126409\pi$
$678$	0	0
$679$	22.3410	0.857368
$680$	0	0
$681$	−18.1325	−0.694838
$682$	0	0
$683$	−9.57340	−0.366316	−0.183158	−	0.983083i	$-0.558632\pi$
$683$	−9.57340	−0.366316	−0.183158	+	0.983083i	$0.558632\pi$
$684$	0	0
$685$	−20.9112	−0.798976
$686$	0	0
$687$	4.72061	0.180102
$688$	0	0
$689$	−12.1982	−0.464714
$690$	0	0
$691$	17.1678	0.653092	0.326546	−	0.945181i	$-0.394115\pi$
$691$	17.1678	0.653092	0.326546	+	0.945181i	$0.394115\pi$
$692$	0	0
$693$	−8.23847	−0.312954
$694$	0	0
$695$	15.4971	0.587840
$696$	0	0
$697$	5.60174	0.212181
$698$	0	0
$699$	−8.66531	−0.327752
$700$	0	0
$701$	−14.0081	−0.529078	−0.264539	−	0.964375i	$-0.585220\pi$
$701$	−14.0081	−0.529078	−0.264539	+	0.964375i	$0.585220\pi$
$702$	0	0
$703$	−13.9209	−0.525035
$704$	0	0
$705$	−6.73958	−0.253827
$706$	0	0
$707$	12.2445	0.460501
$708$	0	0
$709$	28.3432	1.06445	0.532226	−	0.846602i	$-0.321356\pi$
$709$	28.3432	1.06445	0.532226	+	0.846602i	$0.321356\pi$
$710$	0	0
$711$	−40.8712	−1.53279
$712$	0	0
$713$	−19.2354	−0.720371
$714$	0	0
$715$	−5.60656	−0.209673
$716$	0	0
$717$	15.7301	0.587450
$718$	0	0
$719$	11.5402	0.430378	0.215189	−	0.976572i	$-0.430963\pi$
$719$	11.5402	0.430378	0.215189	+	0.976572i	$0.430963\pi$
$720$	0	0
$721$	7.15632	0.266515
$722$	0	0
$723$	4.47251	0.166334
$724$	0	0
$725$	−4.76972	−0.177143
$726$	0	0
$727$	45.5113	1.68792	0.843960	−	0.536406i	$-0.180219\pi$
$727$	45.5113	1.68792	0.843960	+	0.536406i	$0.180219\pi$
$728$	0	0
$729$	1.21774	0.0451015
$730$	0	0
$731$	−5.69938	−0.210799
$732$	0	0
$733$	−2.32368	−0.0858272	−0.0429136	−	0.999079i	$-0.513664\pi$
$733$	−2.32368	−0.0858272	−0.0429136	+	0.999079i	$0.513664\pi$
$734$	0	0
$735$	2.34152	0.0863683
$736$	0	0
$737$	20.0643	0.739079
$738$	0	0
$739$	−46.4485	−1.70864	−0.854318	−	0.519751i	$-0.826025\pi$
$739$	−46.4485	−1.70864	−0.854318	+	0.519751i	$0.826025\pi$
$740$	0	0
$741$	−6.24776	−0.229517
$742$	0	0
$743$	−6.66269	−0.244430	−0.122215	−	0.992504i	$-0.539000\pi$
$743$	−6.66269	−0.244430	−0.122215	+	0.992504i	$0.539000\pi$
$744$	0	0
$745$	−4.94764	−0.181267
$746$	0	0
$747$	−24.8392	−0.908820
$748$	0	0
$749$	−20.4051	−0.745586
$750$	0	0
$751$	2.21986	0.0810037	0.0405019	−	0.999179i	$-0.487104\pi$
$751$	2.21986	0.0810037	0.0405019	+	0.999179i	$0.487104\pi$
$752$	0	0
$753$	8.91102	0.324736
$754$	0	0
$755$	−21.7010	−0.789780
$756$	0	0
$757$	−31.3385	−1.13902	−0.569508	−	0.821986i	$-0.692866\pi$
$757$	−31.3385	−1.13902	−0.569508	+	0.821986i	$0.692866\pi$
$758$	0	0
$759$	10.2370	0.371581
$760$	0	0
$761$	−45.4496	−1.64755	−0.823774	−	0.566919i	$-0.808135\pi$
$761$	−45.4496	−1.64755	−0.823774	+	0.566919i	$0.808135\pi$
$762$	0	0
$763$	−20.4067	−0.738773
$764$	0	0
$765$	2.33375	0.0843768
$766$	0	0
$767$	3.28098	0.118469
$768$	0	0
$769$	9.69182	0.349496	0.174748	−	0.984613i	$-0.444089\pi$
$769$	9.69182	0.349496	0.174748	+	0.984613i	$0.444089\pi$
$770$	0	0
$771$	5.31441	0.191394
$772$	0	0
$773$	20.7587	0.746639	0.373319	−	0.927703i	$-0.378220\pi$
$773$	20.7587	0.746639	0.373319	+	0.927703i	$0.378220\pi$
$774$	0	0
$775$	10.4262	0.374520
$776$	0	0
$777$	9.16477	0.328784
$778$	0	0
$779$	−13.4683	−0.482551
$780$	0	0
$781$	−6.04818	−0.216421
$782$	0	0
$783$	−5.03309	−0.179868
$784$	0	0
$785$	−8.76981	−0.313008
$786$	0	0
$787$	−26.7522	−0.953613	−0.476806	−	0.879008i	$-0.658206\pi$
$787$	−26.7522	−0.953613	−0.476806	+	0.879008i	$0.658206\pi$
$788$	0	0
$789$	12.1167	0.431367
$790$	0	0
$791$	−23.5714	−0.838101
$792$	0	0
$793$	12.6208	0.448179
$794$	0	0
$795$	2.89621	0.102718
$796$	0	0
$797$	−27.5061	−0.974318	−0.487159	−	0.873313i	$-0.661967\pi$
$797$	−27.5061	−0.974318	−0.487159	+	0.873313i	$0.661967\pi$
$798$	0	0
$799$	8.65156	0.306070
$800$	0	0
$801$	19.5368	0.690297
$802$	0	0
$803$	−27.1957	−0.959716
$804$	0	0
$805$	−14.6245	−0.515445
$806$	0	0
$807$	6.64500	0.233915
$808$	0	0
$809$	31.1064	1.09364	0.546822	−	0.837249i	$-0.315837\pi$
$809$	31.1064	1.09364	0.546822	+	0.837249i	$0.315837\pi$
$810$	0	0
$811$	−36.9478	−1.29741	−0.648706	−	0.761040i	$-0.724689\pi$
$811$	−36.9478	−1.29741	−0.648706	+	0.761040i	$0.724689\pi$
$812$	0	0
$813$	6.65386	0.233361
$814$	0	0
$815$	−18.3346	−0.642233
$816$	0	0
$817$	13.7031	0.479409
$818$	0	0
$819$	−15.5584	−0.543655
$820$	0	0
$821$	38.2016	1.33324	0.666622	−	0.745396i	$-0.267739\pi$
$821$	38.2016	1.33324	0.666622	+	0.745396i	$0.267739\pi$
$822$	0	0
$823$	−22.2370	−0.775132	−0.387566	−	0.921842i	$-0.626684\pi$
$823$	−22.2370	−0.775132	−0.387566	+	0.921842i	$0.626684\pi$
$824$	0	0
$825$	−5.54880	−0.193184
$826$	0	0
$827$	2.67314	0.0929541	0.0464771	−	0.998919i	$-0.485201\pi$
$827$	2.67314	0.0929541	0.0464771	+	0.998919i	$0.485201\pi$
$828$	0	0
$829$	−21.2728	−0.738834	−0.369417	−	0.929264i	$-0.620443\pi$
$829$	−21.2728	−0.738834	−0.369417	+	0.929264i	$0.620443\pi$
$830$	0	0
$831$	9.41408	0.326571
$832$	0	0
$833$	−3.00580	−0.104145
$834$	0	0
$835$	−0.807923	−0.0279593
$836$	0	0
$837$	11.0019	0.380281
$838$	0	0
$839$	−17.3968	−0.600604	−0.300302	−	0.953844i	$-0.597087\pi$
$839$	−17.3968	−0.600604	−0.300302	+	0.953844i	$0.597087\pi$
$840$	0	0
$841$	−27.6010	−0.951758
$842$	0	0
$843$	9.60846	0.330933
$844$	0	0
$845$	2.19845	0.0756291
$846$	0	0
$847$	−15.9517	−0.548107
$848$	0	0
$849$	−3.76424	−0.129188
$850$	0	0
$851$	43.0759	1.47662
$852$	0	0
$853$	−15.8175	−0.541582	−0.270791	−	0.962638i	$-0.587285\pi$
$853$	−15.8175	−0.541582	−0.270791	+	0.962638i	$0.587285\pi$
$854$	0	0
$855$	−5.61104	−0.191894
$856$	0	0
$857$	−13.0454	−0.445624	−0.222812	−	0.974861i	$-0.571524\pi$
$857$	−13.0454	−0.445624	−0.222812	+	0.974861i	$0.571524\pi$
$858$	0	0
$859$	−17.4244	−0.594512	−0.297256	−	0.954798i	$-0.596072\pi$
$859$	−17.4244	−0.594512	−0.297256	+	0.954798i	$0.596072\pi$
$860$	0	0
$861$	8.86683	0.302181
$862$	0	0
$863$	36.0778	1.22810	0.614051	−	0.789267i	$-0.289539\pi$
$863$	36.0778	1.22810	0.614051	+	0.789267i	$0.289539\pi$
$864$	0	0
$865$	7.04151	0.239418
$866$	0	0
$867$	0.792010	0.0268981
$868$	0	0
$869$	29.9266	1.01519
$870$	0	0
$871$	37.8916	1.28391
$872$	0	0
$873$	−26.5237	−0.897691
$874$	0	0
$875$	17.7556	0.600247
$876$	0	0
$877$	−36.6145	−1.23638	−0.618191	−	0.786028i	$-0.712134\pi$
$877$	−36.6145	−1.23638	−0.618191	+	0.786028i	$0.712134\pi$
$878$	0	0
$879$	12.2397	0.412834
$880$	0	0
$881$	50.3424	1.69608	0.848040	−	0.529933i	$-0.177783\pi$
$881$	50.3424	1.69608	0.848040	+	0.529933i	$0.177783\pi$
$882$	0	0
$883$	40.0319	1.34718	0.673591	−	0.739104i	$-0.264751\pi$
$883$	40.0319	1.34718	0.673591	+	0.739104i	$0.264751\pi$
$884$	0	0
$885$	−0.779001	−0.0261858
$886$	0	0
$887$	20.4343	0.686116	0.343058	−	0.939314i	$-0.388537\pi$
$887$	20.4343	0.686116	0.343058	+	0.939314i	$0.388537\pi$
$888$	0	0
$889$	−5.75991	−0.193181
$890$	0	0
$891$	6.51149	0.218143
$892$	0	0
$893$	−20.8010	−0.696079
$894$	0	0
$895$	18.6218	0.622460
$896$	0	0
$897$	19.3327	0.645500
$898$	0	0
$899$	−3.05810	−0.101993
$900$	0	0
$901$	−3.71785	−0.123860
$902$	0	0
$903$	−9.02139	−0.300213
$904$	0	0
$905$	18.5799	0.617618
$906$	0	0
$907$	−27.1507	−0.901524	−0.450762	−	0.892644i	$-0.648848\pi$
$907$	−27.1507	−0.901524	−0.450762	+	0.892644i	$0.648848\pi$
$908$	0	0
$909$	−14.5369	−0.482159
$910$	0	0
$911$	−22.9287	−0.759662	−0.379831	−	0.925056i	$-0.624018\pi$
$911$	−22.9287	−0.759662	−0.379831	+	0.925056i	$0.624018\pi$
$912$	0	0
$913$	18.1877	0.601924
$914$	0	0
$915$	−2.99656	−0.0990632
$916$	0	0
$917$	31.3970	1.03682
$918$	0	0
$919$	−3.92308	−0.129410	−0.0647052	−	0.997904i	$-0.520611\pi$
$919$	−3.92308	−0.129410	−0.0647052	+	0.997904i	$0.520611\pi$
$920$	0	0
$921$	21.4165	0.705699
$922$	0	0
$923$	−11.4220	−0.375960
$924$	0	0
$925$	−23.3485	−0.767694
$926$	0	0
$927$	−8.49613	−0.279050
$928$	0	0
$929$	−2.08105	−0.0682769	−0.0341384	−	0.999417i	$-0.510869\pi$
$929$	−2.08105	−0.0682769	−0.0341384	+	0.999417i	$0.510869\pi$
$930$	0	0
$931$	7.22686	0.236851
$932$	0	0
$933$	−23.8572	−0.781050
$934$	0	0
$935$	−1.70881	−0.0558839
$936$	0	0
$937$	2.69947	0.0881879	0.0440939	−	0.999027i	$-0.485960\pi$
$937$	2.69947	0.0881879	0.0440939	+	0.999027i	$0.485960\pi$
$938$	0	0
$939$	10.0162	0.326866
$940$	0	0
$941$	−13.8315	−0.450894	−0.225447	−	0.974255i	$-0.572384\pi$
$941$	−13.8315	−0.450894	−0.225447	+	0.974255i	$0.572384\pi$
$942$	0	0
$943$	41.6755	1.35714
$944$	0	0
$945$	8.36464	0.272102
$946$	0	0
$947$	4.86549	0.158107	0.0790535	−	0.996870i	$-0.474810\pi$
$947$	4.86549	0.158107	0.0790535	+	0.996870i	$0.474810\pi$
$948$	0	0
$949$	−51.3593	−1.66719
$950$	0	0
$951$	5.13852	0.166628
$952$	0	0
$953$	−15.5778	−0.504614	−0.252307	−	0.967647i	$-0.581189\pi$
$953$	−15.5778	−0.504614	−0.252307	+	0.967647i	$0.581189\pi$
$954$	0	0
$955$	22.5991	0.731289
$956$	0	0
$957$	1.62752	0.0526101
$958$	0	0
$959$	42.4900	1.37207
$960$	0	0
$961$	−24.3152	−0.784363
$962$	0	0
$963$	24.2253	0.780651
$964$	0	0
$965$	−22.8567	−0.735783
$966$	0	0
$967$	50.2549	1.61609	0.808044	−	0.589122i	$-0.200526\pi$
$967$	50.2549	1.61609	0.808044	+	0.589122i	$0.200526\pi$
$968$	0	0
$969$	−1.90423	−0.0611728
$970$	0	0
$971$	5.29935	0.170064	0.0850321	−	0.996378i	$-0.472901\pi$
$971$	5.29935	0.170064	0.0850321	+	0.996378i	$0.472901\pi$
$972$	0	0
$973$	−31.4890	−1.00949
$974$	0	0
$975$	−10.4789	−0.335595
$976$	0	0
$977$	−41.7597	−1.33601	−0.668006	−	0.744156i	$-0.732852\pi$
$977$	−41.7597	−1.33601	−0.668006	+	0.744156i	$0.732852\pi$
$978$	0	0
$979$	−14.3051	−0.457193
$980$	0	0
$981$	24.2273	0.773518
$982$	0	0
$983$	11.0906	0.353736	0.176868	−	0.984235i	$-0.443403\pi$
$983$	11.0906	0.353736	0.176868	+	0.984235i	$0.443403\pi$
$984$	0	0
$985$	−11.7247	−0.373578
$986$	0	0
$987$	13.6943	0.435895
$988$	0	0
$989$	−42.4020	−1.34830
$990$	0	0
$991$	−35.2782	−1.12065	−0.560324	−	0.828273i	$-0.689323\pi$
$991$	−35.2782	−1.12065	−0.560324	+	0.828273i	$0.689323\pi$
$992$	0	0
$993$	6.39612	0.202975
$994$	0	0
$995$	−9.26168	−0.293615
$996$	0	0
$997$	32.5793	1.03180	0.515899	−	0.856650i	$-0.327458\pi$
$997$	32.5793	1.03180	0.515899	+	0.856650i	$0.327458\pi$
$998$	0	0
$999$	−24.6377	−0.779504

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.bb.1.19	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.bb.1.19	✓	32	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)

\(f(q)\)	\(=\)	\(q+0.792010 q^{3} -0.983575 q^{5} +1.99855 q^{7} -2.37272 q^{9} +O(q^{10})\)	\(q+0.792010 q^{3} -0.983575 q^{5} +1.99855 q^{7} -2.37272 q^{9} +1.73734 q^{11} +3.28098 q^{13} -0.779001 q^{15} +1.00000 q^{17} -2.40431 q^{19} +1.58287 q^{21} +7.43975 q^{23} -4.03258 q^{25} -4.25525 q^{27} +1.18279 q^{29} -2.58549 q^{31} +1.37599 q^{33} -1.96572 q^{35} +5.78997 q^{37} +2.59857 q^{39} +5.60174 q^{41} -5.69938 q^{43} +2.33375 q^{45} +8.65156 q^{47} -3.00580 q^{49} +0.792010 q^{51} -3.71785 q^{53} -1.70881 q^{55} -1.90423 q^{57} +1.00000 q^{59} +3.84667 q^{61} -4.74200 q^{63} -3.22709 q^{65} +11.5489 q^{67} +5.89236 q^{69} -3.48128 q^{71} -15.6536 q^{73} -3.19384 q^{75} +3.47216 q^{77} +17.2255 q^{79} +3.74796 q^{81} +10.4687 q^{83} -0.983575 q^{85} +0.936785 q^{87} -8.23391 q^{89} +6.55720 q^{91} -2.04773 q^{93} +2.36481 q^{95} +11.1786 q^{97} -4.12223 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9}+O(q^{10}) \) 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9	\( 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9} + 3 q^{11} + 13 q^{13} + 4 q^{15} + 32 q^{17} + 14 q^{19} - 7 q^{21} + 7 q^{23} + 38 q^{25} + 9 q^{27} + 17 q^{29} + 15 q^{31} + 18 q^{33} + 6 q^{35} + 21 q^{37} + 16 q^{39} + 49 q^{41} - 7 q^{43} + 14 q^{45} - 25 q^{47} + 37 q^{49} + 12 q^{53} + 15 q^{55} + 45 q^{57} + 32 q^{59} + 5 q^{61} - 12 q^{63} + 39 q^{65} + 12 q^{69} - 13 q^{71} + 70 q^{73} - 47 q^{75} - 10 q^{77} - q^{79} + 84 q^{81} - 17 q^{83} + 8 q^{85} + 20 q^{87} + 42 q^{89} + 36 q^{91} + 2 q^{93} - q^{95} + 58 q^{97} + 7 q^{99}+O(q^{100}) \) 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9 + 3 * q^11 + 13 * q^13 + 4 * q^15 + 32 * q^17 + 14 * q^19 - 7 * q^21 + 7 * q^23 + 38 * q^25 + 9 * q^27 + 17 * q^29 + 15 * q^31 + 18 * q^33 + 6 * q^35 + 21 * q^37 + 16 * q^39 + 49 * q^41 - 7 * q^43 + 14 * q^45 - 25 * q^47 + 37 * q^49 + 12 * q^53 + 15 * q^55 + 45 * q^57 + 32 * q^59 + 5 * q^61 - 12 * q^63 + 39 * q^65 + 12 * q^69 - 13 * q^71 + 70 * q^73 - 47 * q^75 - 10 * q^77 - q^79 + 84 * q^81 - 17 * q^83 + 8 * q^85 + 20 * q^87 + 42 * q^89 + 36 * q^91 + 2 * q^93 - q^95 + 58 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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