LMFDB - Embedded newform 8032.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8032,2,Mod(1,8032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8032 = 2^{5} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8032.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:	$64.1358429035$
Analytic rank:	$1$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.07132 q^{3} +2.45715 q^{5} -4.52565 q^{7} +6.43300 q^{9} +O(q^{10})$	$q-3.07132 q^{3} +2.45715 q^{5} -4.52565 q^{7} +6.43300 q^{9} +0.159132 q^{11} -0.488460 q^{13} -7.54668 q^{15} +0.619933 q^{17} -2.21667 q^{19} +13.8997 q^{21} +1.05816 q^{23} +1.03757 q^{25} -10.5438 q^{27} +7.64709 q^{29} +5.50602 q^{31} -0.488745 q^{33} -11.1202 q^{35} -10.5821 q^{37} +1.50022 q^{39} -9.02540 q^{41} -1.71361 q^{43} +15.8068 q^{45} +0.546604 q^{47} +13.4816 q^{49} -1.90401 q^{51} -3.96050 q^{53} +0.391011 q^{55} +6.80812 q^{57} -1.79925 q^{59} +6.54703 q^{61} -29.1135 q^{63} -1.20022 q^{65} +13.5783 q^{67} -3.24996 q^{69} +12.8337 q^{71} +3.82490 q^{73} -3.18672 q^{75} -0.720176 q^{77} -6.87798 q^{79} +13.0845 q^{81} +6.43055 q^{83} +1.52327 q^{85} -23.4866 q^{87} +15.9538 q^{89} +2.21060 q^{91} -16.9108 q^{93} -5.44670 q^{95} -18.6648 q^{97} +1.02370 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 3 q^{3} - 13 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q - 3 * q^3 - 13 * q^7 + 35 * q^9	$ 30 q - 3 q^{3} - 13 q^{7} + 35 q^{9} - 13 q^{11} - 7 q^{13} - 28 q^{15} - 9 q^{17} + 17 q^{19} - 6 q^{21} - 43 q^{23} + 34 q^{25} - 12 q^{27} - q^{29} - 39 q^{31} - 17 q^{35} - q^{37} - 48 q^{39} - 3 q^{41} + 19 q^{43} - 66 q^{47} + 25 q^{49} + 14 q^{51} + 3 q^{53} - 50 q^{55} - 14 q^{57} - 27 q^{59} + 15 q^{61} - 75 q^{63} - 6 q^{65} - 8 q^{67} + 18 q^{69} - 64 q^{71} - 15 q^{73} - 9 q^{75} - 71 q^{79} + 6 q^{81} - 60 q^{83} + 15 q^{85} - 64 q^{87} - 32 q^{89} + 26 q^{91} - 4 q^{93} - 72 q^{95} - 4 q^{97} - 13 q^{99}+O(q^{100}) $ 30 * q - 3 * q^3 - 13 * q^7 + 35 * q^9 - 13 * q^11 - 7 * q^13 - 28 * q^15 - 9 * q^17 + 17 * q^19 - 6 * q^21 - 43 * q^23 + 34 * q^25 - 12 * q^27 - q^29 - 39 * q^31 - 17 * q^35 - q^37 - 48 * q^39 - 3 * q^41 + 19 * q^43 - 66 * q^47 + 25 * q^49 + 14 * q^51 + 3 * q^53 - 50 * q^55 - 14 * q^57 - 27 * q^59 + 15 * q^61 - 75 * q^63 - 6 * q^65 - 8 * q^67 + 18 * q^69 - 64 * q^71 - 15 * q^73 - 9 * q^75 - 71 * q^79 + 6 * q^81 - 60 * q^83 + 15 * q^85 - 64 * q^87 - 32 * q^89 + 26 * q^91 - 4 * q^93 - 72 * q^95 - 4 * q^97 - 13 * 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
$3$	−3.07132	−1.77323	−0.886613	−	0.462511i	$-0.846949\pi$
$3$	−3.07132	−1.77323	−0.886613	+	0.462511i	$0.846949\pi$
$4$	0	0
$5$	2.45715	1.09887	0.549435	−	0.835537i	$-0.314843\pi$
$5$	2.45715	1.09887	0.549435	+	0.835537i	$0.314843\pi$
$6$	0	0
$7$	−4.52565	−1.71054	−0.855268	−	0.518185i	$-0.826608\pi$
$7$	−4.52565	−1.71054	−0.855268	+	0.518185i	$0.826608\pi$
$8$	0	0
$9$	6.43300	2.14433
$10$	0	0
$11$	0.159132	0.0479801	0.0239900	−	0.999712i	$-0.492363\pi$
$11$	0.159132	0.0479801	0.0239900	+	0.999712i	$0.492363\pi$
$12$	0	0
$13$	−0.488460	−0.135474	−0.0677372	−	0.997703i	$-0.521578\pi$
$13$	−0.488460	−0.135474	−0.0677372	+	0.997703i	$0.521578\pi$
$14$	0	0
$15$	−7.54668	−1.94855
$16$	0	0
$17$	0.619933	0.150356	0.0751779	−	0.997170i	$-0.476048\pi$
$17$	0.619933	0.150356	0.0751779	+	0.997170i	$0.476048\pi$
$18$	0	0
$19$	−2.21667	−0.508540	−0.254270	−	0.967133i	$-0.581835\pi$
$19$	−2.21667	−0.508540	−0.254270	+	0.967133i	$0.581835\pi$
$20$	0	0
$21$	13.8997	3.03317
$22$	0	0
$23$	1.05816	0.220642	0.110321	−	0.993896i	$-0.464812\pi$
$23$	1.05816	0.220642	0.110321	+	0.993896i	$0.464812\pi$
$24$	0	0
$25$	1.03757	0.207515
$26$	0	0
$27$	−10.5438	−2.02916
$28$	0	0
$29$	7.64709	1.42003	0.710014	−	0.704187i	$-0.248689\pi$
$29$	7.64709	1.42003	0.710014	+	0.704187i	$0.248689\pi$
$30$	0	0
$31$	5.50602	0.988911	0.494455	−	0.869203i	$-0.335368\pi$
$31$	5.50602	0.988911	0.494455	+	0.869203i	$0.335368\pi$
$32$	0	0
$33$	−0.488745	−0.0850796
$34$	0	0
$35$	−11.1202	−1.87966
$36$	0	0
$37$	−10.5821	−1.73969	−0.869846	−	0.493323i	$-0.835782\pi$
$37$	−10.5821	−1.73969	−0.869846	+	0.493323i	$0.835782\pi$
$38$	0	0
$39$	1.50022	0.240227
$40$	0	0
$41$	−9.02540	−1.40953	−0.704765	−	0.709441i	$-0.748948\pi$
$41$	−9.02540	−1.40953	−0.704765	+	0.709441i	$0.748948\pi$
$42$	0	0
$43$	−1.71361	−0.261323	−0.130662	−	0.991427i	$-0.541710\pi$
$43$	−1.71361	−0.261323	−0.130662	+	0.991427i	$0.541710\pi$
$44$	0	0
$45$	15.8068	2.35634
$46$	0	0
$47$	0.546604	0.0797303	0.0398652	−	0.999205i	$-0.487307\pi$
$47$	0.546604	0.0797303	0.0398652	+	0.999205i	$0.487307\pi$
$48$	0	0
$49$	13.4816	1.92594
$50$	0	0
$51$	−1.90401	−0.266615
$52$	0	0
$53$	−3.96050	−0.544016	−0.272008	−	0.962295i	$-0.587688\pi$
$53$	−3.96050	−0.544016	−0.272008	+	0.962295i	$0.587688\pi$
$54$	0	0
$55$	0.391011	0.0527239
$56$	0	0
$57$	6.80812	0.901757
$58$	0	0
$59$	−1.79925	−0.234243	−0.117121	−	0.993118i	$-0.537367\pi$
$59$	−1.79925	−0.234243	−0.117121	+	0.993118i	$0.537367\pi$
$60$	0	0
$61$	6.54703	0.838261	0.419131	−	0.907926i	$-0.362335\pi$
$61$	6.54703	0.838261	0.419131	+	0.907926i	$0.362335\pi$
$62$	0	0
$63$	−29.1135	−3.66796
$64$	0	0
$65$	−1.20022	−0.148869
$66$	0	0
$67$	13.5783	1.65886	0.829428	−	0.558614i	$-0.188667\pi$
$67$	13.5783	1.65886	0.829428	+	0.558614i	$0.188667\pi$
$68$	0	0
$69$	−3.24996	−0.391249
$70$	0	0
$71$	12.8337	1.52308	0.761539	−	0.648119i	$-0.224444\pi$
$71$	12.8337	1.52308	0.761539	+	0.648119i	$0.224444\pi$
$72$	0	0
$73$	3.82490	0.447671	0.223835	−	0.974627i	$-0.428142\pi$
$73$	3.82490	0.447671	0.223835	+	0.974627i	$0.428142\pi$
$74$	0	0
$75$	−3.18672	−0.367970
$76$	0	0
$77$	−0.720176	−0.0820717
$78$	0	0
$79$	−6.87798	−0.773833	−0.386916	−	0.922115i	$-0.626460\pi$
$79$	−6.87798	−0.773833	−0.386916	+	0.922115i	$0.626460\pi$
$80$	0	0
$81$	13.0845	1.45383
$82$	0	0
$83$	6.43055	0.705845	0.352922	−	0.935653i	$-0.385188\pi$
$83$	6.43055	0.705845	0.352922	+	0.935653i	$0.385188\pi$
$84$	0	0
$85$	1.52327	0.165222
$86$	0	0
$87$	−23.4866	−2.51803
$88$	0	0
$89$	15.9538	1.69110	0.845550	−	0.533897i	$-0.179273\pi$
$89$	15.9538	1.69110	0.845550	+	0.533897i	$0.179273\pi$
$90$	0	0
$91$	2.21060	0.231734
$92$	0	0
$93$	−16.9108	−1.75356
$94$	0	0
$95$	−5.44670	−0.558819
$96$	0	0
$97$	−18.6648	−1.89512	−0.947560	−	0.319577i	$-0.896459\pi$
$97$	−18.6648	−1.89512	−0.947560	+	0.319577i	$0.896459\pi$
$98$	0	0
$99$	1.02370	0.102885
$100$	0	0
$101$	−4.22543	−0.420446	−0.210223	−	0.977653i	$-0.567419\pi$
$101$	−4.22543	−0.420446	−0.210223	+	0.977653i	$0.567419\pi$
$102$	0	0
$103$	3.71518	0.366068	0.183034	−	0.983107i	$-0.441408\pi$
$103$	3.71518	0.366068	0.183034	+	0.983107i	$0.441408\pi$
$104$	0	0
$105$	34.1537	3.33306
$106$	0	0
$107$	2.61949	0.253235	0.126618	−	0.991952i	$-0.459588\pi$
$107$	2.61949	0.253235	0.126618	+	0.991952i	$0.459588\pi$
$108$	0	0
$109$	−0.0806750	−0.00772727	−0.00386363	−	0.999993i	$-0.501230\pi$
$109$	−0.0806750	−0.00772727	−0.00386363	+	0.999993i	$0.501230\pi$
$110$	0	0
$111$	32.5011	3.08487
$112$	0	0
$113$	−10.1601	−0.955784	−0.477892	−	0.878419i	$-0.658599\pi$
$113$	−10.1601	−0.955784	−0.477892	+	0.878419i	$0.658599\pi$
$114$	0	0
$115$	2.60006	0.242457
$116$	0	0
$117$	−3.14227	−0.290503
$118$	0	0
$119$	−2.80560	−0.257189
$120$	0	0
$121$	−10.9747	−0.997698
$122$	0	0
$123$	27.7199	2.49942
$124$	0	0
$125$	−9.73627	−0.870838
$126$	0	0
$127$	−14.8943	−1.32166	−0.660828	−	0.750538i	$-0.729795\pi$
$127$	−14.8943	−1.32166	−0.660828	+	0.750538i	$0.729795\pi$
$128$	0	0
$129$	5.26305	0.463386
$130$	0	0
$131$	5.21400	0.455550	0.227775	−	0.973714i	$-0.426855\pi$
$131$	5.21400	0.455550	0.227775	+	0.973714i	$0.426855\pi$
$132$	0	0
$133$	10.0319	0.869876
$134$	0	0
$135$	−25.9078	−2.22979
$136$	0	0
$137$	6.35824	0.543221	0.271611	−	0.962407i	$-0.412444\pi$
$137$	6.35824	0.543221	0.271611	+	0.962407i	$0.412444\pi$
$138$	0	0
$139$	−3.46635	−0.294012	−0.147006	−	0.989136i	$-0.546964\pi$
$139$	−3.46635	−0.294012	−0.147006	+	0.989136i	$0.546964\pi$
$140$	0	0
$141$	−1.67879	−0.141380
$142$	0	0
$143$	−0.0777296	−0.00650008
$144$	0	0
$145$	18.7900	1.56043
$146$	0	0
$147$	−41.4062	−3.41512
$148$	0	0
$149$	7.47589	0.612448	0.306224	−	0.951959i	$-0.400934\pi$
$149$	7.47589	0.612448	0.306224	+	0.951959i	$0.400934\pi$
$150$	0	0
$151$	−11.3307	−0.922079	−0.461039	−	0.887380i	$-0.652523\pi$
$151$	−11.3307	−0.922079	−0.461039	+	0.887380i	$0.652523\pi$
$152$	0	0
$153$	3.98803	0.322413
$154$	0	0
$155$	13.5291	1.08668
$156$	0	0
$157$	−0.846230	−0.0675365	−0.0337682	−	0.999430i	$-0.510751\pi$
$157$	−0.846230	−0.0675365	−0.0337682	+	0.999430i	$0.510751\pi$
$158$	0	0
$159$	12.1640	0.964665
$160$	0	0
$161$	−4.78888	−0.377417
$162$	0	0
$163$	8.45438	0.662198	0.331099	−	0.943596i	$-0.392581\pi$
$163$	8.45438	0.662198	0.331099	+	0.943596i	$0.392581\pi$
$164$	0	0
$165$	−1.20092	−0.0934914
$166$	0	0
$167$	1.83101	0.141688	0.0708439	−	0.997487i	$-0.477431\pi$
$167$	1.83101	0.141688	0.0708439	+	0.997487i	$0.477431\pi$
$168$	0	0
$169$	−12.7614	−0.981647
$170$	0	0
$171$	−14.2599	−1.09048
$172$	0	0
$173$	18.3178	1.39268	0.696340	−	0.717713i	$-0.254811\pi$
$173$	18.3178	1.39268	0.696340	+	0.717713i	$0.254811\pi$
$174$	0	0
$175$	−4.69570	−0.354961
$176$	0	0
$177$	5.52608	0.415366
$178$	0	0
$179$	19.3196	1.44401	0.722007	−	0.691885i	$-0.243220\pi$
$179$	19.3196	1.44401	0.722007	+	0.691885i	$0.243220\pi$
$180$	0	0
$181$	2.21696	0.164785	0.0823927	−	0.996600i	$-0.473744\pi$
$181$	2.21696	0.164785	0.0823927	+	0.996600i	$0.473744\pi$
$182$	0	0
$183$	−20.1080	−1.48643
$184$	0	0
$185$	−26.0019	−1.91169
$186$	0	0
$187$	0.0986512	0.00721409
$188$	0	0
$189$	47.7178	3.47096
$190$	0	0
$191$	5.13756	0.371741	0.185870	−	0.982574i	$-0.440490\pi$
$191$	5.13756	0.371741	0.185870	+	0.982574i	$0.440490\pi$
$192$	0	0
$193$	21.6528	1.55861	0.779303	−	0.626648i	$-0.215574\pi$
$193$	21.6528	1.55861	0.779303	+	0.626648i	$0.215574\pi$
$194$	0	0
$195$	3.68625	0.263978
$196$	0	0
$197$	17.7080	1.26164	0.630820	−	0.775929i	$-0.282719\pi$
$197$	17.7080	1.26164	0.630820	+	0.775929i	$0.282719\pi$
$198$	0	0
$199$	1.86356	0.132104	0.0660520	−	0.997816i	$-0.478960\pi$
$199$	1.86356	0.132104	0.0660520	+	0.997816i	$0.478960\pi$
$200$	0	0
$201$	−41.7033	−2.94153
$202$	0	0
$203$	−34.6081	−2.42901
$204$	0	0
$205$	−22.1767	−1.54889
$206$	0	0
$207$	6.80716	0.473131
$208$	0	0
$209$	−0.352744	−0.0243998
$210$	0	0
$211$	3.47462	0.239203	0.119601	−	0.992822i	$-0.461838\pi$
$211$	3.47462	0.239203	0.119601	+	0.992822i	$0.461838\pi$
$212$	0	0
$213$	−39.4163	−2.70076
$214$	0	0
$215$	−4.21060	−0.287160
$216$	0	0
$217$	−24.9184	−1.69157
$218$	0	0
$219$	−11.7475	−0.793822
$220$	0	0
$221$	−0.302813	−0.0203694
$222$	0	0
$223$	−23.0097	−1.54085	−0.770423	−	0.637533i	$-0.779955\pi$
$223$	−23.0097	−1.54085	−0.770423	+	0.637533i	$0.779955\pi$
$224$	0	0
$225$	6.67471	0.444980
$226$	0	0
$227$	4.01890	0.266744	0.133372	−	0.991066i	$-0.457420\pi$
$227$	4.01890	0.266744	0.133372	+	0.991066i	$0.457420\pi$
$228$	0	0
$229$	7.69655	0.508602	0.254301	−	0.967125i	$-0.418155\pi$
$229$	7.69655	0.508602	0.254301	+	0.967125i	$0.418155\pi$
$230$	0	0
$231$	2.21189	0.145532
$232$	0	0
$233$	−17.9132	−1.17353	−0.586766	−	0.809756i	$-0.699599\pi$
$233$	−17.9132	−1.17353	−0.586766	+	0.809756i	$0.699599\pi$
$234$	0	0
$235$	1.34309	0.0876132
$236$	0	0
$237$	21.1245	1.37218
$238$	0	0
$239$	3.03155	0.196094	0.0980472	−	0.995182i	$-0.468740\pi$
$239$	3.03155	0.196094	0.0980472	+	0.995182i	$0.468740\pi$
$240$	0	0
$241$	−12.8954	−0.830665	−0.415332	−	0.909670i	$-0.636335\pi$
$241$	−12.8954	−0.830665	−0.415332	+	0.909670i	$0.636335\pi$
$242$	0	0
$243$	−8.55516	−0.548814
$244$	0	0
$245$	33.1262	2.11635
$246$	0	0
$247$	1.08276	0.0688942
$248$	0	0
$249$	−19.7503	−1.25162
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	0.168388	0.0105864
$254$	0	0
$255$	−4.67844	−0.292975
$256$	0	0
$257$	10.5542	0.658353	0.329176	−	0.944268i	$-0.393229\pi$
$257$	10.5542	0.658353	0.329176	+	0.944268i	$0.393229\pi$
$258$	0	0
$259$	47.8911	2.97581
$260$	0	0
$261$	49.1937	3.04502
$262$	0	0
$263$	−4.71679	−0.290850	−0.145425	−	0.989369i	$-0.546455\pi$
$263$	−4.71679	−0.290850	−0.145425	+	0.989369i	$0.546455\pi$
$264$	0	0
$265$	−9.73153	−0.597803
$266$	0	0
$267$	−48.9992	−2.99870
$268$	0	0
$269$	20.9187	1.27544	0.637718	−	0.770270i	$-0.279879\pi$
$269$	20.9187	1.27544	0.637718	+	0.770270i	$0.279879\pi$
$270$	0	0
$271$	−26.6317	−1.61776	−0.808879	−	0.587975i	$-0.799925\pi$
$271$	−26.6317	−1.61776	−0.808879	+	0.587975i	$0.799925\pi$
$272$	0	0
$273$	−6.78947	−0.410917
$274$	0	0
$275$	0.165111	0.00995657
$276$	0	0
$277$	−17.2671	−1.03748	−0.518739	−	0.854933i	$-0.673598\pi$
$277$	−17.2671	−1.03748	−0.518739	+	0.854933i	$0.673598\pi$
$278$	0	0
$279$	35.4203	2.12055
$280$	0	0
$281$	−19.8239	−1.18259	−0.591297	−	0.806454i	$-0.701384\pi$
$281$	−19.8239	−1.18259	−0.591297	+	0.806454i	$0.701384\pi$
$282$	0	0
$283$	−8.96729	−0.533050	−0.266525	−	0.963828i	$-0.585875\pi$
$283$	−8.96729	−0.533050	−0.266525	+	0.963828i	$0.585875\pi$
$284$	0	0
$285$	16.7285	0.990913
$286$	0	0
$287$	40.8458	2.41105
$288$	0	0
$289$	−16.6157	−0.977393
$290$	0	0
$291$	57.3255	3.36048
$292$	0	0
$293$	12.1731	0.711161	0.355581	−	0.934646i	$-0.384283\pi$
$293$	12.1731	0.711161	0.355581	+	0.934646i	$0.384283\pi$
$294$	0	0
$295$	−4.42103	−0.257402
$296$	0	0
$297$	−1.67786	−0.0973595
$298$	0	0
$299$	−0.516870	−0.0298914
$300$	0	0
$301$	7.75522	0.447003
$302$	0	0
$303$	12.9776	0.745546
$304$	0	0
$305$	16.0870	0.921140
$306$	0	0
$307$	6.33479	0.361545	0.180773	−	0.983525i	$-0.442140\pi$
$307$	6.33479	0.361545	0.180773	+	0.983525i	$0.442140\pi$
$308$	0	0
$309$	−11.4105	−0.649121
$310$	0	0
$311$	12.4901	0.708250	0.354125	−	0.935198i	$-0.384779\pi$
$311$	12.4901	0.708250	0.354125	+	0.935198i	$0.384779\pi$
$312$	0	0
$313$	−0.364589	−0.0206078	−0.0103039	−	0.999947i	$-0.503280\pi$
$313$	−0.364589	−0.0206078	−0.0103039	+	0.999947i	$0.503280\pi$
$314$	0	0
$315$	−71.5363	−4.03061
$316$	0	0
$317$	−23.8232	−1.33804	−0.669022	−	0.743243i	$-0.733287\pi$
$317$	−23.8232	−1.33804	−0.669022	+	0.743243i	$0.733287\pi$
$318$	0	0
$319$	1.21690	0.0681331
$320$	0	0
$321$	−8.04528	−0.449044
$322$	0	0
$323$	−1.37419	−0.0764620
$324$	0	0
$325$	−0.506813	−0.0281129
$326$	0	0
$327$	0.247779	0.0137022
$328$	0	0
$329$	−2.47374	−0.136382
$330$	0	0
$331$	−18.7999	−1.03334	−0.516668	−	0.856186i	$-0.672828\pi$
$331$	−18.7999	−1.03334	−0.516668	+	0.856186i	$0.672828\pi$
$332$	0	0
$333$	−68.0749	−3.73048
$334$	0	0
$335$	33.3639	1.82287
$336$	0	0
$337$	−2.91875	−0.158994	−0.0794971	−	0.996835i	$-0.525331\pi$
$337$	−2.91875	−0.158994	−0.0794971	+	0.996835i	$0.525331\pi$
$338$	0	0
$339$	31.2050	1.69482
$340$	0	0
$341$	0.876184	0.0474480
$342$	0	0
$343$	−29.3333	−1.58385
$344$	0	0
$345$	−7.98562	−0.429931
$346$	0	0
$347$	−6.77301	−0.363594	−0.181797	−	0.983336i	$-0.558191\pi$
$347$	−6.77301	−0.363594	−0.181797	+	0.983336i	$0.558191\pi$
$348$	0	0
$349$	24.6427	1.31909	0.659547	−	0.751663i	$-0.270748\pi$
$349$	24.6427	1.31909	0.659547	+	0.751663i	$0.270748\pi$
$350$	0	0
$351$	5.15025	0.274900
$352$	0	0
$353$	11.8159	0.628894	0.314447	−	0.949275i	$-0.398181\pi$
$353$	11.8159	0.628894	0.314447	+	0.949275i	$0.398181\pi$
$354$	0	0
$355$	31.5342	1.67366
$356$	0	0
$357$	8.61691	0.456055
$358$	0	0
$359$	0.180014	0.00950076	0.00475038	−	0.999989i	$-0.498488\pi$
$359$	0.180014	0.00950076	0.00475038	+	0.999989i	$0.498488\pi$
$360$	0	0
$361$	−14.0864	−0.741387
$362$	0	0
$363$	33.7067	1.76914
$364$	0	0
$365$	9.39835	0.491932
$366$	0	0
$367$	8.00138	0.417669	0.208834	−	0.977951i	$-0.433033\pi$
$367$	8.00138	0.417669	0.208834	+	0.977951i	$0.433033\pi$
$368$	0	0
$369$	−58.0604	−3.02250
$370$	0	0
$371$	17.9239	0.930560
$372$	0	0
$373$	−35.9965	−1.86383	−0.931914	−	0.362679i	$-0.881862\pi$
$373$	−35.9965	−1.86383	−0.931914	+	0.362679i	$0.881862\pi$
$374$	0	0
$375$	29.9032	1.54419
$376$	0	0
$377$	−3.73530	−0.192378
$378$	0	0
$379$	8.39465	0.431204	0.215602	−	0.976481i	$-0.430829\pi$
$379$	8.39465	0.431204	0.215602	+	0.976481i	$0.430829\pi$
$380$	0	0
$381$	45.7452	2.34360
$382$	0	0
$383$	−25.4699	−1.30145	−0.650725	−	0.759314i	$-0.725535\pi$
$383$	−25.4699	−1.30145	−0.650725	+	0.759314i	$0.725535\pi$
$384$	0	0
$385$	−1.76958	−0.0901861
$386$	0	0
$387$	−11.0237	−0.560365
$388$	0	0
$389$	−2.88170	−0.146108	−0.0730540	−	0.997328i	$-0.523275\pi$
$389$	−2.88170	−0.146108	−0.0730540	+	0.997328i	$0.523275\pi$
$390$	0	0
$391$	0.655990	0.0331749
$392$	0	0
$393$	−16.0139	−0.807793
$394$	0	0
$395$	−16.9002	−0.850342
$396$	0	0
$397$	−25.1263	−1.26105	−0.630526	−	0.776168i	$-0.717161\pi$
$397$	−25.1263	−1.26105	−0.630526	+	0.776168i	$0.717161\pi$
$398$	0	0
$399$	−30.8112	−1.54249
$400$	0	0
$401$	30.4086	1.51854	0.759268	−	0.650779i	$-0.225557\pi$
$401$	30.4086	1.51854	0.759268	+	0.650779i	$0.225557\pi$
$402$	0	0
$403$	−2.68947	−0.133972
$404$	0	0
$405$	32.1506	1.59757
$406$	0	0
$407$	−1.68396	−0.0834706
$408$	0	0
$409$	9.02755	0.446384	0.223192	−	0.974775i	$-0.428352\pi$
$409$	9.02755	0.446384	0.223192	+	0.974775i	$0.428352\pi$
$410$	0	0
$411$	−19.5282	−0.963255
$412$	0	0
$413$	8.14280	0.400681
$414$	0	0
$415$	15.8008	0.775631
$416$	0	0
$417$	10.6463	0.521350
$418$	0	0
$419$	−26.3727	−1.28839	−0.644196	−	0.764860i	$-0.722808\pi$
$419$	−26.3727	−1.28839	−0.644196	+	0.764860i	$0.722808\pi$
$420$	0	0
$421$	−20.5813	−1.00307	−0.501535	−	0.865137i	$-0.667231\pi$
$421$	−20.5813	−1.00307	−0.501535	+	0.865137i	$0.667231\pi$
$422$	0	0
$423$	3.51630	0.170968
$424$	0	0
$425$	0.643226	0.0312010
$426$	0	0
$427$	−29.6296	−1.43388
$428$	0	0
$429$	0.238733	0.0115261
$430$	0	0
$431$	28.8710	1.39067	0.695333	−	0.718688i	$-0.255257\pi$
$431$	28.8710	1.39067	0.695333	+	0.718688i	$0.255257\pi$
$432$	0	0
$433$	−34.8162	−1.67316	−0.836581	−	0.547843i	$-0.815449\pi$
$433$	−34.8162	−1.67316	−0.836581	+	0.547843i	$0.815449\pi$
$434$	0	0
$435$	−57.7101	−2.76699
$436$	0	0
$437$	−2.34560	−0.112205
$438$	0	0
$439$	−14.8469	−0.708605	−0.354303	−	0.935131i	$-0.615282\pi$
$439$	−14.8469	−0.708605	−0.354303	+	0.935131i	$0.615282\pi$
$440$	0	0
$441$	86.7269	4.12985
$442$	0	0
$443$	−21.0475	−0.999997	−0.499998	−	0.866026i	$-0.666666\pi$
$443$	−21.0475	−0.999997	−0.499998	+	0.866026i	$0.666666\pi$
$444$	0	0
$445$	39.2008	1.85830
$446$	0	0
$447$	−22.9608	−1.08601
$448$	0	0
$449$	23.4801	1.10809	0.554047	−	0.832485i	$-0.313083\pi$
$449$	23.4801	1.10809	0.554047	+	0.832485i	$0.313083\pi$
$450$	0	0
$451$	−1.43623	−0.0676294
$452$	0	0
$453$	34.8002	1.63505
$454$	0	0
$455$	5.43178	0.254646
$456$	0	0
$457$	−11.1230	−0.520311	−0.260156	−	0.965567i	$-0.583774\pi$
$457$	−11.1230	−0.520311	−0.260156	+	0.965567i	$0.583774\pi$
$458$	0	0
$459$	−6.53648	−0.305097
$460$	0	0
$461$	−7.44962	−0.346963	−0.173482	−	0.984837i	$-0.555502\pi$
$461$	−7.44962	−0.346963	−0.173482	+	0.984837i	$0.555502\pi$
$462$	0	0
$463$	−42.3394	−1.96768	−0.983840	−	0.179049i	$-0.942698\pi$
$463$	−42.3394	−1.96768	−0.983840	+	0.179049i	$0.942698\pi$
$464$	0	0
$465$	−41.5522	−1.92694
$466$	0	0
$467$	−30.5710	−1.41466	−0.707329	−	0.706885i	$-0.750100\pi$
$467$	−30.5710	−1.41466	−0.707329	+	0.706885i	$0.750100\pi$
$468$	0	0
$469$	−61.4508	−2.83753
$470$	0	0
$471$	2.59904	0.119758
$472$	0	0
$473$	−0.272691	−0.0125383
$474$	0	0
$475$	−2.29996	−0.105529
$476$	0	0
$477$	−25.4779	−1.16655
$478$	0	0
$479$	−34.5171	−1.57713	−0.788563	−	0.614954i	$-0.789174\pi$
$479$	−34.5171	−1.57713	−0.788563	+	0.614954i	$0.789174\pi$
$480$	0	0
$481$	5.16895	0.235684
$482$	0	0
$483$	14.7082	0.669245
$484$	0	0
$485$	−45.8621	−2.08249
$486$	0	0
$487$	−36.9794	−1.67570	−0.837848	−	0.545903i	$-0.816187\pi$
$487$	−36.9794	−1.67570	−0.837848	+	0.545903i	$0.816187\pi$
$488$	0	0
$489$	−25.9661	−1.17423
$490$	0	0
$491$	8.67750	0.391610	0.195805	−	0.980643i	$-0.437268\pi$
$491$	8.67750	0.391610	0.195805	+	0.980643i	$0.437268\pi$
$492$	0	0
$493$	4.74068	0.213510
$494$	0	0
$495$	2.51537	0.113058
$496$	0	0
$497$	−58.0808	−2.60528
$498$	0	0
$499$	−21.5352	−0.964049	−0.482025	−	0.876158i	$-0.660098\pi$
$499$	−21.5352	−0.964049	−0.482025	+	0.876158i	$0.660098\pi$
$500$	0	0
$501$	−5.62361	−0.251245
$502$	0	0
$503$	−17.9046	−0.798327	−0.399164	−	0.916880i	$-0.630699\pi$
$503$	−17.9046	−0.798327	−0.399164	+	0.916880i	$0.630699\pi$
$504$	0	0
$505$	−10.3825	−0.462015
$506$	0	0
$507$	39.1944	1.74068
$508$	0	0
$509$	−23.5326	−1.04306	−0.521532	−	0.853232i	$-0.674639\pi$
$509$	−23.5326	−1.04306	−0.521532	+	0.853232i	$0.674639\pi$
$510$	0	0
$511$	−17.3102	−0.765757
$512$	0	0
$513$	23.3723	1.03191
$514$	0	0
$515$	9.12875	0.402261
$516$	0	0
$517$	0.0869821	0.00382547
$518$	0	0
$519$	−56.2599	−2.46954
$520$	0	0
$521$	−20.0013	−0.876272	−0.438136	−	0.898909i	$-0.644361\pi$
$521$	−20.0013	−0.876272	−0.438136	+	0.898909i	$0.644361\pi$
$522$	0	0
$523$	16.5436	0.723403	0.361701	−	0.932294i	$-0.382196\pi$
$523$	16.5436	0.723403	0.361701	+	0.932294i	$0.382196\pi$
$524$	0	0
$525$	14.4220	0.629427
$526$	0	0
$527$	3.41337	0.148689
$528$	0	0
$529$	−21.8803	−0.951317
$530$	0	0
$531$	−11.5746	−0.502295
$532$	0	0
$533$	4.40855	0.190955
$534$	0	0
$535$	6.43647	0.278273
$536$	0	0
$537$	−59.3367	−2.56057
$538$	0	0
$539$	2.14535	0.0924066
$540$	0	0
$541$	−15.0980	−0.649114	−0.324557	−	0.945866i	$-0.605215\pi$
$541$	−15.0980	−0.649114	−0.324557	+	0.945866i	$0.605215\pi$
$542$	0	0
$543$	−6.80899	−0.292202
$544$	0	0
$545$	−0.198230	−0.00849126
$546$	0	0
$547$	−14.4937	−0.619705	−0.309853	−	0.950785i	$-0.600280\pi$
$547$	−14.4937	−0.619705	−0.309853	+	0.950785i	$0.600280\pi$
$548$	0	0
$549$	42.1170	1.79751
$550$	0	0
$551$	−16.9511	−0.722141
$552$	0	0
$553$	31.1274	1.32367
$554$	0	0
$555$	79.8600	3.38987
$556$	0	0
$557$	13.4419	0.569552	0.284776	−	0.958594i	$-0.408081\pi$
$557$	13.4419	0.569552	0.284776	+	0.958594i	$0.408081\pi$
$558$	0	0
$559$	0.837031	0.0354027
$560$	0	0
$561$	−0.302989	−0.0127922
$562$	0	0
$563$	1.75794	0.0740883	0.0370441	−	0.999314i	$-0.488206\pi$
$563$	1.75794	0.0740883	0.0370441	+	0.999314i	$0.488206\pi$
$564$	0	0
$565$	−24.9649	−1.05028
$566$	0	0
$567$	−59.2160	−2.48684
$568$	0	0
$569$	2.42550	0.101682	0.0508410	−	0.998707i	$-0.483810\pi$
$569$	2.42550	0.101682	0.0508410	+	0.998707i	$0.483810\pi$
$570$	0	0
$571$	−28.7021	−1.20114	−0.600572	−	0.799571i	$-0.705060\pi$
$571$	−28.7021	−1.20114	−0.600572	+	0.799571i	$0.705060\pi$
$572$	0	0
$573$	−15.7791	−0.659181
$574$	0	0
$575$	1.09792	0.0457865
$576$	0	0
$577$	32.0454	1.33407	0.667033	−	0.745028i	$-0.267564\pi$
$577$	32.0454	1.33407	0.667033	+	0.745028i	$0.267564\pi$
$578$	0	0
$579$	−66.5028	−2.76376
$580$	0	0
$581$	−29.1025	−1.20737
$582$	0	0
$583$	−0.630242	−0.0261020
$584$	0	0
$585$	−7.72101	−0.319224
$586$	0	0
$587$	−10.6013	−0.437562	−0.218781	−	0.975774i	$-0.570208\pi$
$587$	−10.6013	−0.437562	−0.218781	+	0.975774i	$0.570208\pi$
$588$	0	0
$589$	−12.2051	−0.502901
$590$	0	0
$591$	−54.3868	−2.23717
$592$	0	0
$593$	17.5967	0.722609	0.361305	−	0.932448i	$-0.382331\pi$
$593$	17.5967	0.722609	0.361305	+	0.932448i	$0.382331\pi$
$594$	0	0
$595$	−6.89378	−0.282617
$596$	0	0
$597$	−5.72358	−0.234250
$598$	0	0
$599$	−44.7814	−1.82972	−0.914858	−	0.403775i	$-0.867698\pi$
$599$	−44.7814	−1.82972	−0.914858	+	0.403775i	$0.867698\pi$
$600$	0	0
$601$	−44.7935	−1.82716	−0.913582	−	0.406655i	$-0.866695\pi$
$601$	−44.7935	−1.82716	−0.913582	+	0.406655i	$0.866695\pi$
$602$	0	0
$603$	87.3493	3.55714
$604$	0	0
$605$	−26.9664	−1.09634
$606$	0	0
$607$	38.8941	1.57866	0.789332	−	0.613967i	$-0.210427\pi$
$607$	38.8941	1.57866	0.789332	+	0.613967i	$0.210427\pi$
$608$	0	0
$609$	106.292	4.30719
$610$	0	0
$611$	−0.266994	−0.0108014
$612$	0	0
$613$	1.50170	0.0606533	0.0303266	−	0.999540i	$-0.490345\pi$
$613$	1.50170	0.0606533	0.0303266	+	0.999540i	$0.490345\pi$
$614$	0	0
$615$	68.1118	2.74653
$616$	0	0
$617$	26.6482	1.07281	0.536407	−	0.843959i	$-0.319781\pi$
$617$	26.6482	1.07281	0.536407	+	0.843959i	$0.319781\pi$
$618$	0	0
$619$	−16.9980	−0.683207	−0.341604	−	0.939844i	$-0.610970\pi$
$619$	−16.9980	−0.683207	−0.341604	+	0.939844i	$0.610970\pi$
$620$	0	0
$621$	−11.1571	−0.447719
$622$	0	0
$623$	−72.2014	−2.89269
$624$	0	0
$625$	−29.1113	−1.16445
$626$	0	0
$627$	1.08339	0.0432664
$628$	0	0
$629$	−6.56022	−0.261573
$630$	0	0
$631$	5.32667	0.212051	0.106026	−	0.994363i	$-0.466187\pi$
$631$	5.32667	0.212051	0.106026	+	0.994363i	$0.466187\pi$
$632$	0	0
$633$	−10.6717	−0.424160
$634$	0	0
$635$	−36.5975	−1.45233
$636$	0	0
$637$	−6.58520	−0.260915
$638$	0	0
$639$	82.5591	3.26599
$640$	0	0
$641$	15.3700	0.607078	0.303539	−	0.952819i	$-0.401832\pi$
$641$	15.3700	0.607078	0.303539	+	0.952819i	$0.401832\pi$
$642$	0	0
$643$	26.9311	1.06206	0.531030	−	0.847353i	$-0.321805\pi$
$643$	26.9311	1.06206	0.531030	+	0.847353i	$0.321805\pi$
$644$	0	0
$645$	12.9321	0.509201
$646$	0	0
$647$	−16.9953	−0.668156	−0.334078	−	0.942546i	$-0.608425\pi$
$647$	−16.9953	−0.668156	−0.334078	+	0.942546i	$0.608425\pi$
$648$	0	0
$649$	−0.286319	−0.0112390
$650$	0	0
$651$	76.5322	2.99953
$652$	0	0
$653$	29.8061	1.16640	0.583202	−	0.812328i	$-0.301800\pi$
$653$	29.8061	1.16640	0.583202	+	0.812328i	$0.301800\pi$
$654$	0	0
$655$	12.8116	0.500590
$656$	0	0
$657$	24.6056	0.959956
$658$	0	0
$659$	44.7863	1.74463	0.872314	−	0.488947i	$-0.162619\pi$
$659$	44.7863	1.74463	0.872314	+	0.488947i	$0.162619\pi$
$660$	0	0
$661$	−4.55523	−0.177178	−0.0885890	−	0.996068i	$-0.528236\pi$
$661$	−4.55523	−0.177178	−0.0885890	+	0.996068i	$0.528236\pi$
$662$	0	0
$663$	0.930034	0.0361195
$664$	0	0
$665$	24.6499	0.955881
$666$	0	0
$667$	8.09186	0.313318
$668$	0	0
$669$	70.6703	2.73227
$670$	0	0
$671$	1.04184	0.0402198
$672$	0	0
$673$	−10.2243	−0.394118	−0.197059	−	0.980392i	$-0.563139\pi$
$673$	−10.2243	−0.394118	−0.197059	+	0.980392i	$0.563139\pi$
$674$	0	0
$675$	−10.9400	−0.421081
$676$	0	0
$677$	13.2303	0.508480	0.254240	−	0.967141i	$-0.418175\pi$
$677$	13.2303	0.508480	0.254240	+	0.967141i	$0.418175\pi$
$678$	0	0
$679$	84.4703	3.24167
$680$	0	0
$681$	−12.3433	−0.472997
$682$	0	0
$683$	−33.3959	−1.27786	−0.638929	−	0.769265i	$-0.720622\pi$
$683$	−33.3959	−1.27786	−0.638929	+	0.769265i	$0.720622\pi$
$684$	0	0
$685$	15.6231	0.596929
$686$	0	0
$687$	−23.6386	−0.901868
$688$	0	0
$689$	1.93455	0.0737003
$690$	0	0
$691$	−8.11332	−0.308645	−0.154323	−	0.988021i	$-0.549320\pi$
$691$	−8.11332	−0.308645	−0.154323	+	0.988021i	$0.549320\pi$
$692$	0	0
$693$	−4.63290	−0.175989
$694$	0	0
$695$	−8.51734	−0.323081
$696$	0	0
$697$	−5.59514	−0.211931
$698$	0	0
$699$	55.0172	2.08094
$700$	0	0
$701$	−4.99690	−0.188730	−0.0943651	−	0.995538i	$-0.530082\pi$
$701$	−4.99690	−0.188730	−0.0943651	+	0.995538i	$0.530082\pi$
$702$	0	0
$703$	23.4571	0.884703
$704$	0	0
$705$	−4.12504	−0.155358
$706$	0	0
$707$	19.1228	0.719188
$708$	0	0
$709$	−38.8883	−1.46048	−0.730241	−	0.683190i	$-0.760592\pi$
$709$	−38.8883	−1.46048	−0.730241	+	0.683190i	$0.760592\pi$
$710$	0	0
$711$	−44.2460	−1.65936
$712$	0	0
$713$	5.82627	0.218195
$714$	0	0
$715$	−0.190993	−0.00714274
$716$	0	0
$717$	−9.31085	−0.347720
$718$	0	0
$719$	−46.1101	−1.71962	−0.859809	−	0.510616i	$-0.829417\pi$
$719$	−46.1101	−1.71962	−0.859809	+	0.510616i	$0.829417\pi$
$720$	0	0
$721$	−16.8136	−0.626172
$722$	0	0
$723$	39.6059	1.47296
$724$	0	0
$725$	7.93441	0.294677
$726$	0	0
$727$	−48.2741	−1.79039	−0.895194	−	0.445676i	$-0.852963\pi$
$727$	−48.2741	−1.79039	−0.895194	+	0.445676i	$0.852963\pi$
$728$	0	0
$729$	−12.9779	−0.480662
$730$	0	0
$731$	−1.06233	−0.0392915
$732$	0	0
$733$	−4.44992	−0.164362	−0.0821808	−	0.996617i	$-0.526188\pi$
$733$	−4.44992	−0.164362	−0.0821808	+	0.996617i	$0.526188\pi$
$734$	0	0
$735$	−101.741	−3.75277
$736$	0	0
$737$	2.16074	0.0795920
$738$	0	0
$739$	28.2954	1.04086	0.520431	−	0.853904i	$-0.325771\pi$
$739$	28.2954	1.04086	0.520431	+	0.853904i	$0.325771\pi$
$740$	0	0
$741$	−3.32549	−0.122165
$742$	0	0
$743$	−27.1447	−0.995843	−0.497922	−	0.867222i	$-0.665903\pi$
$743$	−27.1447	−0.995843	−0.497922	+	0.867222i	$0.665903\pi$
$744$	0	0
$745$	18.3694	0.673001
$746$	0	0
$747$	41.3677	1.51357
$748$	0	0
$749$	−11.8549	−0.433169
$750$	0	0
$751$	19.1982	0.700553	0.350276	−	0.936646i	$-0.386088\pi$
$751$	19.1982	0.700553	0.350276	+	0.936646i	$0.386088\pi$
$752$	0	0
$753$	3.07132	0.111925
$754$	0	0
$755$	−27.8412	−1.01324
$756$	0	0
$757$	−48.4212	−1.75990	−0.879949	−	0.475069i	$-0.842423\pi$
$757$	−48.4212	−1.75990	−0.879949	+	0.475069i	$0.842423\pi$
$758$	0	0
$759$	−0.517172	−0.0187721
$760$	0	0
$761$	−16.6940	−0.605158	−0.302579	−	0.953124i	$-0.597848\pi$
$761$	−16.6940	−0.605158	−0.302579	+	0.953124i	$0.597848\pi$
$762$	0	0
$763$	0.365107	0.0132178
$764$	0	0
$765$	9.79918	0.354290
$766$	0	0
$767$	0.878864	0.0317339
$768$	0	0
$769$	40.1783	1.44887	0.724433	−	0.689345i	$-0.242102\pi$
$769$	40.1783	1.44887	0.724433	+	0.689345i	$0.242102\pi$
$770$	0	0
$771$	−32.4153	−1.16741
$772$	0	0
$773$	44.8521	1.61322	0.806608	−	0.591086i	$-0.201301\pi$
$773$	44.8521	1.61322	0.806608	+	0.591086i	$0.201301\pi$
$774$	0	0
$775$	5.71290	0.205213
$776$	0	0
$777$	−147.089	−5.27678
$778$	0	0
$779$	20.0064	0.716803
$780$	0	0
$781$	2.04225	0.0730774
$782$	0	0
$783$	−80.6297	−2.88147
$784$	0	0
$785$	−2.07931	−0.0742138
$786$	0	0
$787$	−38.6992	−1.37948	−0.689739	−	0.724058i	$-0.742275\pi$
$787$	−38.6992	−1.37948	−0.689739	+	0.724058i	$0.742275\pi$
$788$	0	0
$789$	14.4868	0.515743
$790$	0	0
$791$	45.9812	1.63490
$792$	0	0
$793$	−3.19796	−0.113563
$794$	0	0
$795$	29.8886	1.06004
$796$	0	0
$797$	6.16688	0.218442	0.109221	−	0.994017i	$-0.465164\pi$
$797$	6.16688	0.218442	0.109221	+	0.994017i	$0.465164\pi$
$798$	0	0
$799$	0.338858	0.0119879
$800$	0	0
$801$	102.631	3.62628
$802$	0	0
$803$	0.608664	0.0214793
$804$	0	0
$805$	−11.7670	−0.414732
$806$	0	0
$807$	−64.2480	−2.26164
$808$	0	0
$809$	−21.3139	−0.749355	−0.374678	−	0.927155i	$-0.622247\pi$
$809$	−21.3139	−0.749355	−0.374678	+	0.927155i	$0.622247\pi$
$810$	0	0
$811$	7.45869	0.261910	0.130955	−	0.991388i	$-0.458196\pi$
$811$	7.45869	0.261910	0.130955	+	0.991388i	$0.458196\pi$
$812$	0	0
$813$	81.7944	2.86865
$814$	0	0
$815$	20.7737	0.727670
$816$	0	0
$817$	3.79852	0.132893
$818$	0	0
$819$	14.2208	0.496915
$820$	0	0
$821$	−4.16187	−0.145250	−0.0726251	−	0.997359i	$-0.523138\pi$
$821$	−4.16187	−0.145250	−0.0726251	+	0.997359i	$0.523138\pi$
$822$	0	0
$823$	−47.9890	−1.67279	−0.836396	−	0.548126i	$-0.815341\pi$
$823$	−47.9890	−1.67279	−0.836396	+	0.548126i	$0.815341\pi$
$824$	0	0
$825$	−0.507109	−0.0176553
$826$	0	0
$827$	35.9438	1.24989	0.624945	−	0.780669i	$-0.285121\pi$
$827$	35.9438	1.24989	0.624945	+	0.780669i	$0.285121\pi$
$828$	0	0
$829$	25.2396	0.876606	0.438303	−	0.898827i	$-0.355580\pi$
$829$	25.2396	0.876606	0.438303	+	0.898827i	$0.355580\pi$
$830$	0	0
$831$	53.0327	1.83968
$832$	0	0
$833$	8.35766	0.289576
$834$	0	0
$835$	4.49906	0.155696
$836$	0	0
$837$	−58.0546	−2.00666
$838$	0	0
$839$	−13.9880	−0.482918	−0.241459	−	0.970411i	$-0.577626\pi$
$839$	−13.9880	−0.482918	−0.241459	+	0.970411i	$0.577626\pi$
$840$	0	0
$841$	29.4779	1.01648
$842$	0	0
$843$	60.8855	2.09701
$844$	0	0
$845$	−31.3567	−1.07870
$846$	0	0
$847$	49.6676	1.70660
$848$	0	0
$849$	27.5414	0.945219
$850$	0	0
$851$	−11.1976	−0.383849
$852$	0	0
$853$	24.7325	0.846824	0.423412	−	0.905937i	$-0.360832\pi$
$853$	24.7325	0.846824	0.423412	+	0.905937i	$0.360832\pi$
$854$	0	0
$855$	−35.0386	−1.19829
$856$	0	0
$857$	−23.1446	−0.790603	−0.395302	−	0.918551i	$-0.629360\pi$
$857$	−23.1446	−0.790603	−0.395302	+	0.918551i	$0.629360\pi$
$858$	0	0
$859$	−2.74572	−0.0936828	−0.0468414	−	0.998902i	$-0.514916\pi$
$859$	−2.74572	−0.0936828	−0.0468414	+	0.998902i	$0.514916\pi$
$860$	0	0
$861$	−125.451	−4.27535
$862$	0	0
$863$	−36.6589	−1.24788	−0.623942	−	0.781471i	$-0.714470\pi$
$863$	−36.6589	−1.24788	−0.623942	+	0.781471i	$0.714470\pi$
$864$	0	0
$865$	45.0096	1.53037
$866$	0	0
$867$	51.0321	1.73314
$868$	0	0
$869$	−1.09451	−0.0371286
$870$	0	0
$871$	−6.63247	−0.224733
$872$	0	0
$873$	−120.071	−4.06377
$874$	0	0
$875$	44.0630	1.48960
$876$	0	0
$877$	13.0546	0.440822	0.220411	−	0.975407i	$-0.429260\pi$
$877$	13.0546	0.440822	0.220411	+	0.975407i	$0.429260\pi$
$878$	0	0
$879$	−37.3875	−1.26105
$880$	0	0
$881$	8.02669	0.270426	0.135213	−	0.990817i	$-0.456828\pi$
$881$	8.02669	0.270426	0.135213	+	0.990817i	$0.456828\pi$
$882$	0	0
$883$	48.5117	1.63255	0.816275	−	0.577664i	$-0.196035\pi$
$883$	48.5117	1.63255	0.816275	+	0.577664i	$0.196035\pi$
$884$	0	0
$885$	13.5784	0.456433
$886$	0	0
$887$	53.2507	1.78798	0.893992	−	0.448083i	$-0.147893\pi$
$887$	53.2507	1.78798	0.893992	+	0.448083i	$0.147893\pi$
$888$	0	0
$889$	67.4065	2.26074
$890$	0	0
$891$	2.08216	0.0697551
$892$	0	0
$893$	−1.21164	−0.0405461
$894$	0	0
$895$	47.4711	1.58678
$896$	0	0
$897$	1.58747	0.0530042
$898$	0	0
$899$	42.1050	1.40428
$900$	0	0
$901$	−2.45525	−0.0817961
$902$	0	0
$903$	−23.8188	−0.792638
$904$	0	0
$905$	5.44740	0.181078
$906$	0	0
$907$	−11.7536	−0.390271	−0.195135	−	0.980776i	$-0.562515\pi$
$907$	−11.7536	−0.390271	−0.195135	+	0.980776i	$0.562515\pi$
$908$	0	0
$909$	−27.1822	−0.901576
$910$	0	0
$911$	−14.2886	−0.473404	−0.236702	−	0.971582i	$-0.576066\pi$
$911$	−14.2886	−0.473404	−0.236702	+	0.971582i	$0.576066\pi$
$912$	0	0
$913$	1.02331	0.0338665
$914$	0	0
$915$	−49.4084	−1.63339
$916$	0	0
$917$	−23.5968	−0.779234
$918$	0	0
$919$	5.10388	0.168361	0.0841807	−	0.996451i	$-0.473173\pi$
$919$	5.10388	0.168361	0.0841807	+	0.996451i	$0.473173\pi$
$920$	0	0
$921$	−19.4561	−0.641102
$922$	0	0
$923$	−6.26874	−0.206338
$924$	0	0
$925$	−10.9797	−0.361011
$926$	0	0
$927$	23.8998	0.784971
$928$	0	0
$929$	−37.6777	−1.23616	−0.618082	−	0.786114i	$-0.712090\pi$
$929$	−37.6777	−1.23616	−0.618082	+	0.786114i	$0.712090\pi$
$930$	0	0
$931$	−29.8842	−0.979416
$932$	0	0
$933$	−38.3612	−1.25589
$934$	0	0
$935$	0.242401	0.00792734
$936$	0	0
$937$	24.9815	0.816110	0.408055	−	0.912957i	$-0.366207\pi$
$937$	24.9815	0.816110	0.408055	+	0.912957i	$0.366207\pi$
$938$	0	0
$939$	1.11977	0.0365423
$940$	0	0
$941$	−3.43231	−0.111890	−0.0559450	−	0.998434i	$-0.517817\pi$
$941$	−3.43231	−0.111890	−0.0559450	+	0.998434i	$0.517817\pi$
$942$	0	0
$943$	−9.55034	−0.311002
$944$	0	0
$945$	117.250	3.81413
$946$	0	0
$947$	−34.5973	−1.12426	−0.562130	−	0.827049i	$-0.690018\pi$
$947$	−34.5973	−1.12426	−0.562130	+	0.827049i	$0.690018\pi$
$948$	0	0
$949$	−1.86831	−0.0606480
$950$	0	0
$951$	73.1686	2.37266
$952$	0	0
$953$	37.7562	1.22304	0.611521	−	0.791228i	$-0.290558\pi$
$953$	37.7562	1.22304	0.611521	+	0.791228i	$0.290558\pi$
$954$	0	0
$955$	12.6237	0.408495
$956$	0	0
$957$	−3.73748	−0.120815
$958$	0	0
$959$	−28.7752	−0.929200
$960$	0	0
$961$	−0.683720	−0.0220555
$962$	0	0
$963$	16.8512	0.543021
$964$	0	0
$965$	53.2042	1.71270
$966$	0	0
$967$	−11.8035	−0.379574	−0.189787	−	0.981825i	$-0.560780\pi$
$967$	−11.8035	−0.379574	−0.189787	+	0.981825i	$0.560780\pi$
$968$	0	0
$969$	4.22058	0.135584
$970$	0	0
$971$	18.4652	0.592578	0.296289	−	0.955098i	$-0.404251\pi$
$971$	18.4652	0.592578	0.296289	+	0.955098i	$0.404251\pi$
$972$	0	0
$973$	15.6875	0.502919
$974$	0	0
$975$	1.55658	0.0498506
$976$	0	0
$977$	−10.3947	−0.332557	−0.166279	−	0.986079i	$-0.553175\pi$
$977$	−10.3947	−0.332557	−0.166279	+	0.986079i	$0.553175\pi$
$978$	0	0
$979$	2.53876	0.0811391
$980$	0	0
$981$	−0.518983	−0.0165698
$982$	0	0
$983$	−12.3760	−0.394732	−0.197366	−	0.980330i	$-0.563239\pi$
$983$	−12.3760	−0.394732	−0.197366	+	0.980330i	$0.563239\pi$
$984$	0	0
$985$	43.5111	1.38638
$986$	0	0
$987$	7.59764	0.241836
$988$	0	0
$989$	−1.81328	−0.0576590
$990$	0	0
$991$	−32.4045	−1.02936	−0.514682	−	0.857381i	$-0.672090\pi$
$991$	−32.4045	−1.02936	−0.514682	+	0.857381i	$0.672090\pi$
$992$	0	0
$993$	57.7405	1.83234
$994$	0	0
$995$	4.57903	0.145165
$996$	0	0
$997$	12.4094	0.393011	0.196505	−	0.980503i	$-0.437041\pi$
$997$	12.4094	0.393011	0.196505	+	0.980503i	$0.437041\pi$
$998$	0	0
$999$	111.576	3.53012

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.h.1.2	✓	30
4.3	odd	2		8032.2.a.i.1.29	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.h.1.2	✓	30	1.1	even	1	trivial
8032.2.a.i.1.29	yes	30	4.3	odd	2

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:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q-3.07132 q^{3} +2.45715 q^{5} -4.52565 q^{7} +6.43300 q^{9} +O(q^{10})\)	\(q-3.07132 q^{3} +2.45715 q^{5} -4.52565 q^{7} +6.43300 q^{9} +0.159132 q^{11} -0.488460 q^{13} -7.54668 q^{15} +0.619933 q^{17} -2.21667 q^{19} +13.8997 q^{21} +1.05816 q^{23} +1.03757 q^{25} -10.5438 q^{27} +7.64709 q^{29} +5.50602 q^{31} -0.488745 q^{33} -11.1202 q^{35} -10.5821 q^{37} +1.50022 q^{39} -9.02540 q^{41} -1.71361 q^{43} +15.8068 q^{45} +0.546604 q^{47} +13.4816 q^{49} -1.90401 q^{51} -3.96050 q^{53} +0.391011 q^{55} +6.80812 q^{57} -1.79925 q^{59} +6.54703 q^{61} -29.1135 q^{63} -1.20022 q^{65} +13.5783 q^{67} -3.24996 q^{69} +12.8337 q^{71} +3.82490 q^{73} -3.18672 q^{75} -0.720176 q^{77} -6.87798 q^{79} +13.0845 q^{81} +6.43055 q^{83} +1.52327 q^{85} -23.4866 q^{87} +15.9538 q^{89} +2.21060 q^{91} -16.9108 q^{93} -5.44670 q^{95} -18.6648 q^{97} +1.02370 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 3 q^{3} - 13 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q - 3 * q^3 - 13 * q^7 + 35 * q^9	\( 30 q - 3 q^{3} - 13 q^{7} + 35 q^{9} - 13 q^{11} - 7 q^{13} - 28 q^{15} - 9 q^{17} + 17 q^{19} - 6 q^{21} - 43 q^{23} + 34 q^{25} - 12 q^{27} - q^{29} - 39 q^{31} - 17 q^{35} - q^{37} - 48 q^{39} - 3 q^{41} + 19 q^{43} - 66 q^{47} + 25 q^{49} + 14 q^{51} + 3 q^{53} - 50 q^{55} - 14 q^{57} - 27 q^{59} + 15 q^{61} - 75 q^{63} - 6 q^{65} - 8 q^{67} + 18 q^{69} - 64 q^{71} - 15 q^{73} - 9 q^{75} - 71 q^{79} + 6 q^{81} - 60 q^{83} + 15 q^{85} - 64 q^{87} - 32 q^{89} + 26 q^{91} - 4 q^{93} - 72 q^{95} - 4 q^{97} - 13 q^{99}+O(q^{100}) \) 30 * q - 3 * q^3 - 13 * q^7 + 35 * q^9 - 13 * q^11 - 7 * q^13 - 28 * q^15 - 9 * q^17 + 17 * q^19 - 6 * q^21 - 43 * q^23 + 34 * q^25 - 12 * q^27 - q^29 - 39 * q^31 - 17 * q^35 - q^37 - 48 * q^39 - 3 * q^41 + 19 * q^43 - 66 * q^47 + 25 * q^49 + 14 * q^51 + 3 * q^53 - 50 * q^55 - 14 * q^57 - 27 * q^59 + 15 * q^61 - 75 * q^63 - 6 * q^65 - 8 * q^67 + 18 * q^69 - 64 * q^71 - 15 * q^73 - 9 * q^75 - 71 * q^79 + 6 * q^81 - 60 * q^83 + 15 * q^85 - 64 * q^87 - 32 * q^89 + 26 * q^91 - 4 * q^93 - 72 * q^95 - 4 * q^97 - 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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