LMFDB - Embedded newform 8032.2.a.j.1.20

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:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q+1.73507 q^{3} -3.61768 q^{5} -3.69185 q^{7} +0.0104735 q^{9} +O(q^{10})$	$q+1.73507 q^{3} -3.61768 q^{5} -3.69185 q^{7} +0.0104735 q^{9} -3.30209 q^{11} -0.935230 q^{13} -6.27693 q^{15} -7.33241 q^{17} +1.39616 q^{19} -6.40563 q^{21} -1.11286 q^{23} +8.08758 q^{25} -5.18704 q^{27} -0.862053 q^{29} -9.74616 q^{31} -5.72936 q^{33} +13.3559 q^{35} -7.53362 q^{37} -1.62269 q^{39} +5.35615 q^{41} +4.27366 q^{43} -0.0378896 q^{45} +1.64904 q^{47} +6.62976 q^{49} -12.7223 q^{51} -11.2127 q^{53} +11.9459 q^{55} +2.42244 q^{57} -2.76908 q^{59} -4.08098 q^{61} -0.0386665 q^{63} +3.38336 q^{65} -7.59365 q^{67} -1.93089 q^{69} +5.08595 q^{71} +2.99620 q^{73} +14.0325 q^{75} +12.1908 q^{77} -15.8032 q^{79} -9.03131 q^{81} +10.4361 q^{83} +26.5263 q^{85} -1.49572 q^{87} +4.02597 q^{89} +3.45273 q^{91} -16.9103 q^{93} -5.05087 q^{95} +9.60609 q^{97} -0.0345843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9	$ 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9} + 31 q^{11} + 5 q^{13} + 8 q^{15} - q^{17} + 29 q^{19} + 6 q^{21} + 27 q^{23} + 34 q^{25} + 36 q^{27} + q^{29} - 9 q^{31} - 8 q^{33} + 51 q^{35} + 5 q^{37} - 8 q^{39} - 3 q^{41} + 43 q^{43} + 42 q^{47} + 41 q^{49} + 54 q^{51} - 11 q^{53} - 10 q^{55} + 2 q^{57} + 49 q^{59} + 21 q^{61} + 15 q^{63} - 14 q^{65} + 68 q^{67} - 10 q^{69} + 44 q^{71} + 25 q^{73} + 51 q^{75} - 20 q^{77} - 49 q^{79} + 6 q^{81} + 88 q^{83} - 5 q^{85} + 16 q^{87} - 16 q^{89} + 46 q^{91} - 4 q^{93} + 28 q^{95} - 4 q^{97} + 71 q^{99}+O(q^{100}) $ 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9 + 31 * q^11 + 5 * q^13 + 8 * q^15 - q^17 + 29 * q^19 + 6 * q^21 + 27 * q^23 + 34 * q^25 + 36 * q^27 + q^29 - 9 * q^31 - 8 * q^33 + 51 * q^35 + 5 * q^37 - 8 * q^39 - 3 * q^41 + 43 * q^43 + 42 * q^47 + 41 * q^49 + 54 * q^51 - 11 * q^53 - 10 * q^55 + 2 * q^57 + 49 * q^59 + 21 * q^61 + 15 * q^63 - 14 * q^65 + 68 * q^67 - 10 * q^69 + 44 * q^71 + 25 * q^73 + 51 * q^75 - 20 * q^77 - 49 * q^79 + 6 * q^81 + 88 * q^83 - 5 * q^85 + 16 * q^87 - 16 * q^89 + 46 * q^91 - 4 * q^93 + 28 * q^95 - 4 * q^97 + 71 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.73507	1.00174	0.500872	−	0.865521i	$-0.333013\pi$
$3$	1.73507	1.00174	0.500872	+	0.865521i	$0.333013\pi$
$4$	0	0
$5$	−3.61768	−1.61787	−0.808937	−	0.587895i	$-0.799957\pi$
$5$	−3.61768	−1.61787	−0.808937	+	0.587895i	$0.799957\pi$
$6$	0	0
$7$	−3.69185	−1.39539	−0.697694	−	0.716396i	$-0.745791\pi$
$7$	−3.69185	−1.39539	−0.697694	+	0.716396i	$0.745791\pi$
$8$	0	0
$9$	0.0104735	0.00349115
$10$	0	0
$11$	−3.30209	−0.995616	−0.497808	−	0.867287i	$-0.665862\pi$
$11$	−3.30209	−0.995616	−0.497808	+	0.867287i	$0.665862\pi$
$12$	0	0
$13$	−0.935230	−0.259386	−0.129693	−	0.991554i	$-0.541399\pi$
$13$	−0.935230	−0.259386	−0.129693	+	0.991554i	$0.541399\pi$
$14$	0	0
$15$	−6.27693	−1.62070
$16$	0	0
$17$	−7.33241	−1.77837	−0.889185	−	0.457548i	$-0.848728\pi$
$17$	−7.33241	−1.77837	−0.889185	+	0.457548i	$0.848728\pi$
$18$	0	0
$19$	1.39616	0.320302	0.160151	−	0.987093i	$-0.448802\pi$
$19$	1.39616	0.320302	0.160151	+	0.987093i	$0.448802\pi$
$20$	0	0
$21$	−6.40563	−1.39782
$22$	0	0
$23$	−1.11286	−0.232047	−0.116023	−	0.993246i	$-0.537015\pi$
$23$	−1.11286	−0.232047	−0.116023	+	0.993246i	$0.537015\pi$
$24$	0	0
$25$	8.08758	1.61752
$26$	0	0
$27$	−5.18704	−0.998247
$28$	0	0
$29$	−0.862053	−0.160079	−0.0800396	−	0.996792i	$-0.525505\pi$
$29$	−0.862053	−0.160079	−0.0800396	+	0.996792i	$0.525505\pi$
$30$	0	0
$31$	−9.74616	−1.75046	−0.875231	−	0.483705i	$-0.839291\pi$
$31$	−9.74616	−1.75046	−0.875231	+	0.483705i	$0.839291\pi$
$32$	0	0
$33$	−5.72936	−0.997353
$34$	0	0
$35$	13.3559	2.25756
$36$	0	0
$37$	−7.53362	−1.23852	−0.619260	−	0.785186i	$-0.712567\pi$
$37$	−7.53362	−1.23852	−0.619260	+	0.785186i	$0.712567\pi$
$38$	0	0
$39$	−1.62269	−0.259838
$40$	0	0
$41$	5.35615	0.836491	0.418245	−	0.908334i	$-0.362645\pi$
$41$	5.35615	0.836491	0.418245	+	0.908334i	$0.362645\pi$
$42$	0	0
$43$	4.27366	0.651727	0.325864	−	0.945417i	$-0.394345\pi$
$43$	4.27366	0.651727	0.325864	+	0.945417i	$0.394345\pi$
$44$	0	0
$45$	−0.0378896	−0.00564825
$46$	0	0
$47$	1.64904	0.240538	0.120269	−	0.992741i	$-0.461624\pi$
$47$	1.64904	0.240538	0.120269	+	0.992741i	$0.461624\pi$
$48$	0	0
$49$	6.62976	0.947109
$50$	0	0
$51$	−12.7223	−1.78147
$52$	0	0
$53$	−11.2127	−1.54018	−0.770091	−	0.637934i	$-0.779789\pi$
$53$	−11.2127	−1.54018	−0.770091	+	0.637934i	$0.779789\pi$
$54$	0	0
$55$	11.9459	1.61078
$56$	0	0
$57$	2.42244	0.320860
$58$	0	0
$59$	−2.76908	−0.360504	−0.180252	−	0.983621i	$-0.557691\pi$
$59$	−2.76908	−0.360504	−0.180252	+	0.983621i	$0.557691\pi$
$60$	0	0
$61$	−4.08098	−0.522516	−0.261258	−	0.965269i	$-0.584137\pi$
$61$	−4.08098	−0.522516	−0.261258	+	0.965269i	$0.584137\pi$
$62$	0	0
$63$	−0.0386665	−0.00487152
$64$	0	0
$65$	3.38336	0.419654
$66$	0	0
$67$	−7.59365	−0.927712	−0.463856	−	0.885911i	$-0.653534\pi$
$67$	−7.59365	−0.927712	−0.463856	+	0.885911i	$0.653534\pi$
$68$	0	0
$69$	−1.93089	−0.232451
$70$	0	0
$71$	5.08595	0.603591	0.301796	−	0.953373i	$-0.402414\pi$
$71$	5.08595	0.603591	0.301796	+	0.953373i	$0.402414\pi$
$72$	0	0
$73$	2.99620	0.350678	0.175339	−	0.984508i	$-0.443898\pi$
$73$	2.99620	0.350678	0.175339	+	0.984508i	$0.443898\pi$
$74$	0	0
$75$	14.0325	1.62034
$76$	0	0
$77$	12.1908	1.38927
$78$	0	0
$79$	−15.8032	−1.77800	−0.888999	−	0.457910i	$-0.848598\pi$
$79$	−15.8032	−1.77800	−0.888999	+	0.457910i	$0.848598\pi$
$80$	0	0
$81$	−9.03131	−1.00348
$82$	0	0
$83$	10.4361	1.14551	0.572756	−	0.819726i	$-0.305874\pi$
$83$	10.4361	1.14551	0.572756	+	0.819726i	$0.305874\pi$
$84$	0	0
$85$	26.5263	2.87718
$86$	0	0
$87$	−1.49572	−0.160358
$88$	0	0
$89$	4.02597	0.426752	0.213376	−	0.976970i	$-0.431554\pi$
$89$	4.02597	0.426752	0.213376	+	0.976970i	$0.431554\pi$
$90$	0	0
$91$	3.45273	0.361944
$92$	0	0
$93$	−16.9103	−1.75352
$94$	0	0
$95$	−5.05087	−0.518208
$96$	0	0
$97$	9.60609	0.975351	0.487675	−	0.873025i	$-0.337845\pi$
$97$	9.60609	0.975351	0.487675	+	0.873025i	$0.337845\pi$
$98$	0	0
$99$	−0.0345843	−0.00347585
$100$	0	0
$101$	16.1227	1.60427	0.802135	−	0.597142i	$-0.203697\pi$
$101$	16.1227	1.60427	0.802135	+	0.597142i	$0.203697\pi$
$102$	0	0
$103$	−0.536966	−0.0529089	−0.0264544	−	0.999650i	$-0.508422\pi$
$103$	−0.536966	−0.0529089	−0.0264544	+	0.999650i	$0.508422\pi$
$104$	0	0
$105$	23.1735	2.26150
$106$	0	0
$107$	−7.16816	−0.692972	−0.346486	−	0.938055i	$-0.612625\pi$
$107$	−7.16816	−0.692972	−0.346486	+	0.938055i	$0.612625\pi$
$108$	0	0
$109$	−10.8259	−1.03693	−0.518464	−	0.855099i	$-0.673496\pi$
$109$	−10.8259	−1.03693	−0.518464	+	0.855099i	$0.673496\pi$
$110$	0	0
$111$	−13.0714	−1.24068
$112$	0	0
$113$	−3.37488	−0.317482	−0.158741	−	0.987320i	$-0.550743\pi$
$113$	−3.37488	−0.317482	−0.158741	+	0.987320i	$0.550743\pi$
$114$	0	0
$115$	4.02596	0.375422
$116$	0	0
$117$	−0.00979509	−0.000905557 0
$118$	0	0
$119$	27.0702	2.48152
$120$	0	0
$121$	−0.0962295	−0.00874814
$122$	0	0
$123$	9.29331	0.837950
$124$	0	0
$125$	−11.1699	−0.999064
$126$	0	0
$127$	5.59941	0.496867	0.248434	−	0.968649i	$-0.420084\pi$
$127$	5.59941	0.496867	0.248434	+	0.968649i	$0.420084\pi$
$128$	0	0
$129$	7.41511	0.652864
$130$	0	0
$131$	−7.52950	−0.657856	−0.328928	−	0.944355i	$-0.606687\pi$
$131$	−7.52950	−0.657856	−0.328928	+	0.944355i	$0.606687\pi$
$132$	0	0
$133$	−5.15443	−0.446946
$134$	0	0
$135$	18.7650	1.61504
$136$	0	0
$137$	−0.522133	−0.0446088	−0.0223044	−	0.999751i	$-0.507100\pi$
$137$	−0.522133	−0.0446088	−0.0223044	+	0.999751i	$0.507100\pi$
$138$	0	0
$139$	−5.60292	−0.475233	−0.237617	−	0.971359i	$-0.576366\pi$
$139$	−5.60292	−0.475233	−0.237617	+	0.971359i	$0.576366\pi$
$140$	0	0
$141$	2.86121	0.240957
$142$	0	0
$143$	3.08821	0.258249
$144$	0	0
$145$	3.11863	0.258988
$146$	0	0
$147$	11.5031	0.948761
$148$	0	0
$149$	16.7364	1.37110	0.685549	−	0.728027i	$-0.259562\pi$
$149$	16.7364	1.37110	0.685549	+	0.728027i	$0.259562\pi$
$150$	0	0
$151$	−1.96976	−0.160297	−0.0801485	−	0.996783i	$-0.525539\pi$
$151$	−1.96976	−0.160297	−0.0801485	+	0.996783i	$0.525539\pi$
$152$	0	0
$153$	−0.0767957	−0.00620856
$154$	0	0
$155$	35.2585	2.83203
$156$	0	0
$157$	16.2318	1.29544	0.647719	−	0.761880i	$-0.275723\pi$
$157$	16.2318	1.29544	0.647719	+	0.761880i	$0.275723\pi$
$158$	0	0
$159$	−19.4548	−1.54287
$160$	0	0
$161$	4.10850	0.323795
$162$	0	0
$163$	−3.94125	−0.308703	−0.154351	−	0.988016i	$-0.549329\pi$
$163$	−3.94125	−0.308703	−0.154351	+	0.988016i	$0.549329\pi$
$164$	0	0
$165$	20.7270	1.61359
$166$	0	0
$167$	4.03044	0.311885	0.155943	−	0.987766i	$-0.450159\pi$
$167$	4.03044	0.311885	0.155943	+	0.987766i	$0.450159\pi$
$168$	0	0
$169$	−12.1253	−0.932719
$170$	0	0
$171$	0.0146227	0.00111822
$172$	0	0
$173$	−11.0233	−0.838085	−0.419042	−	0.907967i	$-0.637634\pi$
$173$	−11.0233	−0.838085	−0.419042	+	0.907967i	$0.637634\pi$
$174$	0	0
$175$	−29.8582	−2.25706
$176$	0	0
$177$	−4.80455	−0.361132
$178$	0	0
$179$	10.3370	0.772626	0.386313	−	0.922368i	$-0.373748\pi$
$179$	10.3370	0.772626	0.386313	+	0.922368i	$0.373748\pi$
$180$	0	0
$181$	10.1300	0.752954	0.376477	−	0.926426i	$-0.377135\pi$
$181$	10.1300	0.752954	0.376477	+	0.926426i	$0.377135\pi$
$182$	0	0
$183$	−7.08079	−0.523427
$184$	0	0
$185$	27.2542	2.00377
$186$	0	0
$187$	24.2122	1.77057
$188$	0	0
$189$	19.1498	1.39294
$190$	0	0
$191$	0.519429	0.0375845	0.0187923	−	0.999823i	$-0.494018\pi$
$191$	0.519429	0.0375845	0.0187923	+	0.999823i	$0.494018\pi$
$192$	0	0
$193$	−17.2055	−1.23848	−0.619240	−	0.785202i	$-0.712559\pi$
$193$	−17.2055	−1.23848	−0.619240	+	0.785202i	$0.712559\pi$
$194$	0	0
$195$	5.87037	0.420386
$196$	0	0
$197$	−10.5292	−0.750175	−0.375087	−	0.926989i	$-0.622387\pi$
$197$	−10.5292	−0.750175	−0.375087	+	0.926989i	$0.622387\pi$
$198$	0	0
$199$	−25.2100	−1.78709	−0.893543	−	0.448977i	$-0.851789\pi$
$199$	−25.2100	−1.78709	−0.893543	+	0.448977i	$0.851789\pi$
$200$	0	0
$201$	−13.1755	−0.929330
$202$	0	0
$203$	3.18257	0.223373
$204$	0	0
$205$	−19.3768	−1.35334
$206$	0	0
$207$	−0.0116555	−0.000810111 0
$208$	0	0
$209$	−4.61025	−0.318898
$210$	0	0
$211$	8.01788	0.551974	0.275987	−	0.961161i	$-0.410995\pi$
$211$	8.01788	0.551974	0.275987	+	0.961161i	$0.410995\pi$
$212$	0	0
$213$	8.82449	0.604644
$214$	0	0
$215$	−15.4607	−1.05441
$216$	0	0
$217$	35.9814	2.44258
$218$	0	0
$219$	5.19862	0.351290
$220$	0	0
$221$	6.85749	0.461284
$222$	0	0
$223$	−11.3712	−0.761473	−0.380737	−	0.924684i	$-0.624330\pi$
$223$	−11.3712	−0.761473	−0.380737	+	0.924684i	$0.624330\pi$
$224$	0	0
$225$	0.0847050	0.00564700
$226$	0	0
$227$	16.0529	1.06547	0.532735	−	0.846282i	$-0.321164\pi$
$227$	16.0529	1.06547	0.532735	+	0.846282i	$0.321164\pi$
$228$	0	0
$229$	−1.83383	−0.121183	−0.0605913	−	0.998163i	$-0.519299\pi$
$229$	−1.83383	−0.121183	−0.0605913	+	0.998163i	$0.519299\pi$
$230$	0	0
$231$	21.1519	1.39169
$232$	0	0
$233$	−14.6774	−0.961552	−0.480776	−	0.876844i	$-0.659645\pi$
$233$	−14.6774	−0.961552	−0.480776	+	0.876844i	$0.659645\pi$
$234$	0	0
$235$	−5.96570	−0.389160
$236$	0	0
$237$	−27.4197	−1.78110
$238$	0	0
$239$	12.3649	0.799818	0.399909	−	0.916555i	$-0.369042\pi$
$239$	12.3649	0.799818	0.399909	+	0.916555i	$0.369042\pi$
$240$	0	0
$241$	8.31250	0.535455	0.267728	−	0.963495i	$-0.413727\pi$
$241$	8.31250	0.535455	0.267728	+	0.963495i	$0.413727\pi$
$242$	0	0
$243$	−0.108843	−0.00698228
$244$	0	0
$245$	−23.9843	−1.53230
$246$	0	0
$247$	−1.30573	−0.0830818
$248$	0	0
$249$	18.1074	1.14751
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	3.67475	0.231029
$254$	0	0
$255$	46.0250	2.88220
$256$	0	0
$257$	18.3297	1.14337	0.571687	−	0.820472i	$-0.306289\pi$
$257$	18.3297	1.14337	0.571687	+	0.820472i	$0.306289\pi$
$258$	0	0
$259$	27.8130	1.72822
$260$	0	0
$261$	−0.00902868	−0.000558861 0
$262$	0	0
$263$	4.03375	0.248732	0.124366	−	0.992236i	$-0.460310\pi$
$263$	4.03375	0.248732	0.124366	+	0.992236i	$0.460310\pi$
$264$	0	0
$265$	40.5639	2.49182
$266$	0	0
$267$	6.98535	0.427496
$268$	0	0
$269$	−19.2925	−1.17628	−0.588142	−	0.808758i	$-0.700140\pi$
$269$	−19.2925	−1.17628	−0.588142	+	0.808758i	$0.700140\pi$
$270$	0	0
$271$	−22.9907	−1.39659	−0.698294	−	0.715812i	$-0.746057\pi$
$271$	−22.9907	−1.39659	−0.698294	+	0.715812i	$0.746057\pi$
$272$	0	0
$273$	5.99073	0.362576
$274$	0	0
$275$	−26.7059	−1.61043
$276$	0	0
$277$	−8.36040	−0.502328	−0.251164	−	0.967945i	$-0.580813\pi$
$277$	−8.36040	−0.502328	−0.251164	+	0.967945i	$0.580813\pi$
$278$	0	0
$279$	−0.102076	−0.00611113
$280$	0	0
$281$	−16.3140	−0.973210	−0.486605	−	0.873622i	$-0.661765\pi$
$281$	−16.3140	−0.973210	−0.486605	+	0.873622i	$0.661765\pi$
$282$	0	0
$283$	−1.98774	−0.118159	−0.0590793	−	0.998253i	$-0.518816\pi$
$283$	−1.98774	−0.118159	−0.0590793	+	0.998253i	$0.518816\pi$
$284$	0	0
$285$	−8.76362	−0.519112
$286$	0	0
$287$	−19.7741	−1.16723
$288$	0	0
$289$	36.7642	2.16260
$290$	0	0
$291$	16.6673	0.977052
$292$	0	0
$293$	−6.58820	−0.384887	−0.192443	−	0.981308i	$-0.561641\pi$
$293$	−6.58820	−0.384887	−0.192443	+	0.981308i	$0.561641\pi$
$294$	0	0
$295$	10.0176	0.583249
$296$	0	0
$297$	17.1281	0.993871
$298$	0	0
$299$	1.04078	0.0601897
$300$	0	0
$301$	−15.7777	−0.909413
$302$	0	0
$303$	27.9741	1.60707
$304$	0	0
$305$	14.7637	0.845365
$306$	0	0
$307$	26.7347	1.52583	0.762914	−	0.646500i	$-0.223768\pi$
$307$	26.7347	1.52583	0.762914	+	0.646500i	$0.223768\pi$
$308$	0	0
$309$	−0.931675	−0.0530011
$310$	0	0
$311$	8.42648	0.477822	0.238911	−	0.971041i	$-0.423210\pi$
$311$	8.42648	0.477822	0.238911	+	0.971041i	$0.423210\pi$
$312$	0	0
$313$	−12.4174	−0.701876	−0.350938	−	0.936399i	$-0.614137\pi$
$313$	−12.4174	−0.701876	−0.350938	+	0.936399i	$0.614137\pi$
$314$	0	0
$315$	0.139883	0.00788150
$316$	0	0
$317$	34.5711	1.94170	0.970852	−	0.239678i	$-0.0770420\pi$
$317$	34.5711	1.94170	0.970852	+	0.239678i	$0.0770420\pi$
$318$	0	0
$319$	2.84657	0.159377
$320$	0	0
$321$	−12.4373	−0.694180
$322$	0	0
$323$	−10.2372	−0.569615
$324$	0	0
$325$	−7.56375	−0.419561
$326$	0	0
$327$	−18.7836	−1.03874
$328$	0	0
$329$	−6.08802	−0.335643
$330$	0	0
$331$	−21.1859	−1.16449	−0.582243	−	0.813015i	$-0.697825\pi$
$331$	−21.1859	−1.16449	−0.582243	+	0.813015i	$0.697825\pi$
$332$	0	0
$333$	−0.0789031	−0.00432386
$334$	0	0
$335$	27.4714	1.50092
$336$	0	0
$337$	−26.5134	−1.44428	−0.722138	−	0.691749i	$-0.756840\pi$
$337$	−26.5134	−1.44428	−0.722138	+	0.691749i	$0.756840\pi$
$338$	0	0
$339$	−5.85566	−0.318036
$340$	0	0
$341$	32.1827	1.74279
$342$	0	0
$343$	1.36685	0.0738032
$344$	0	0
$345$	6.98532	0.376077
$346$	0	0
$347$	32.1482	1.72581	0.862903	−	0.505370i	$-0.168644\pi$
$347$	32.1482	1.72581	0.862903	+	0.505370i	$0.168644\pi$
$348$	0	0
$349$	−36.5811	−1.95814	−0.979070	−	0.203522i	$-0.934761\pi$
$349$	−36.5811	−1.95814	−0.979070	+	0.203522i	$0.934761\pi$
$350$	0	0
$351$	4.85108	0.258931
$352$	0	0
$353$	−18.4276	−0.980801	−0.490400	−	0.871497i	$-0.663149\pi$
$353$	−18.4276	−0.980801	−0.490400	+	0.871497i	$0.663149\pi$
$354$	0	0
$355$	−18.3993	−0.976535
$356$	0	0
$357$	46.9687	2.48585
$358$	0	0
$359$	−16.8839	−0.891099	−0.445549	−	0.895257i	$-0.646992\pi$
$359$	−16.8839	−0.891099	−0.445549	+	0.895257i	$0.646992\pi$
$360$	0	0
$361$	−17.0507	−0.897407
$362$	0	0
$363$	−0.166965	−0.00876340
$364$	0	0
$365$	−10.8393	−0.567353
$366$	0	0
$367$	9.97587	0.520736	0.260368	−	0.965509i	$-0.416156\pi$
$367$	9.97587	0.520736	0.260368	+	0.965509i	$0.416156\pi$
$368$	0	0
$369$	0.0560975	0.00292032
$370$	0	0
$371$	41.3956	2.14915
$372$	0	0
$373$	−6.03861	−0.312667	−0.156334	−	0.987704i	$-0.549968\pi$
$373$	−6.03861	−0.312667	−0.156334	+	0.987704i	$0.549968\pi$
$374$	0	0
$375$	−19.3805	−1.00081
$376$	0	0
$377$	0.806218	0.0415223
$378$	0	0
$379$	23.9364	1.22953	0.614765	−	0.788710i	$-0.289251\pi$
$379$	23.9364	1.22953	0.614765	+	0.788710i	$0.289251\pi$
$380$	0	0
$381$	9.71538	0.497734
$382$	0	0
$383$	29.6841	1.51679	0.758394	−	0.651797i	$-0.225984\pi$
$383$	29.6841	1.51679	0.758394	+	0.651797i	$0.225984\pi$
$384$	0	0
$385$	−44.1024	−2.24767
$386$	0	0
$387$	0.0447600	0.00227528
$388$	0	0
$389$	−27.4608	−1.39232	−0.696159	−	0.717888i	$-0.745109\pi$
$389$	−27.4608	−1.39232	−0.696159	+	0.717888i	$0.745109\pi$
$390$	0	0
$391$	8.15992	0.412665
$392$	0	0
$393$	−13.0642	−0.659003
$394$	0	0
$395$	57.1708	2.87658
$396$	0	0
$397$	−13.4684	−0.675958	−0.337979	−	0.941154i	$-0.609743\pi$
$397$	−13.4684	−0.675958	−0.337979	+	0.941154i	$0.609743\pi$
$398$	0	0
$399$	−8.94330	−0.447725
$400$	0	0
$401$	−16.6174	−0.829835	−0.414917	−	0.909859i	$-0.636190\pi$
$401$	−16.6174	−0.829835	−0.414917	+	0.909859i	$0.636190\pi$
$402$	0	0
$403$	9.11490	0.454046
$404$	0	0
$405$	32.6724	1.62350
$406$	0	0
$407$	24.8767	1.23309
$408$	0	0
$409$	−0.695346	−0.0343827	−0.0171913	−	0.999852i	$-0.505472\pi$
$409$	−0.695346	−0.0343827	−0.0171913	+	0.999852i	$0.505472\pi$
$410$	0	0
$411$	−0.905937	−0.0446866
$412$	0	0
$413$	10.2230	0.503043
$414$	0	0
$415$	−37.7545	−1.85330
$416$	0	0
$417$	−9.72146	−0.476062
$418$	0	0
$419$	17.6020	0.859912	0.429956	−	0.902850i	$-0.358529\pi$
$419$	17.6020	0.859912	0.429956	+	0.902850i	$0.358529\pi$
$420$	0	0
$421$	−21.0428	−1.02556	−0.512782	−	0.858519i	$-0.671385\pi$
$421$	−21.0428	−1.02556	−0.512782	+	0.858519i	$0.671385\pi$
$422$	0	0
$423$	0.0172712	0.000839754 0
$424$	0	0
$425$	−59.3015	−2.87654
$426$	0	0
$427$	15.0664	0.729112
$428$	0	0
$429$	5.35826	0.258699
$430$	0	0
$431$	−6.33808	−0.305295	−0.152647	−	0.988281i	$-0.548780\pi$
$431$	−6.33808	−0.305295	−0.152647	+	0.988281i	$0.548780\pi$
$432$	0	0
$433$	−15.9345	−0.765765	−0.382882	−	0.923797i	$-0.625069\pi$
$433$	−15.9345	−0.765765	−0.382882	+	0.923797i	$0.625069\pi$
$434$	0	0
$435$	5.41104	0.259440
$436$	0	0
$437$	−1.55373	−0.0743250
$438$	0	0
$439$	−2.94654	−0.140631	−0.0703155	−	0.997525i	$-0.522401\pi$
$439$	−2.94654	−0.140631	−0.0703155	+	0.997525i	$0.522401\pi$
$440$	0	0
$441$	0.0694366	0.00330650
$442$	0	0
$443$	−1.01321	−0.0481388	−0.0240694	−	0.999710i	$-0.507662\pi$
$443$	−1.01321	−0.0481388	−0.0240694	+	0.999710i	$0.507662\pi$
$444$	0	0
$445$	−14.5647	−0.690431
$446$	0	0
$447$	29.0388	1.37349
$448$	0	0
$449$	14.0402	0.662598	0.331299	−	0.943526i	$-0.392513\pi$
$449$	14.0402	0.662598	0.331299	+	0.943526i	$0.392513\pi$
$450$	0	0
$451$	−17.6865	−0.832824
$452$	0	0
$453$	−3.41768	−0.160577
$454$	0	0
$455$	−12.4909	−0.585580
$456$	0	0
$457$	−6.32825	−0.296023	−0.148012	−	0.988986i	$-0.547287\pi$
$457$	−6.32825	−0.296023	−0.148012	+	0.988986i	$0.547287\pi$
$458$	0	0
$459$	38.0335	1.77525
$460$	0	0
$461$	8.18026	0.380993	0.190496	−	0.981688i	$-0.438990\pi$
$461$	8.18026	0.380993	0.190496	+	0.981688i	$0.438990\pi$
$462$	0	0
$463$	−7.27005	−0.337868	−0.168934	−	0.985627i	$-0.554032\pi$
$463$	−7.27005	−0.337868	−0.168934	+	0.985627i	$0.554032\pi$
$464$	0	0
$465$	61.1760	2.83697
$466$	0	0
$467$	13.5717	0.628021	0.314011	−	0.949419i	$-0.398327\pi$
$467$	13.5717	0.628021	0.314011	+	0.949419i	$0.398327\pi$
$468$	0	0
$469$	28.0346	1.29452
$470$	0	0
$471$	28.1633	1.29770
$472$	0	0
$473$	−14.1120	−0.648870
$474$	0	0
$475$	11.2916	0.518094
$476$	0	0
$477$	−0.117436	−0.00537701
$478$	0	0
$479$	16.1117	0.736161	0.368081	−	0.929794i	$-0.380015\pi$
$479$	16.1117	0.736161	0.368081	+	0.929794i	$0.380015\pi$
$480$	0	0
$481$	7.04567	0.321255
$482$	0	0
$483$	7.12854	0.324360
$484$	0	0
$485$	−34.7517	−1.57800
$486$	0	0
$487$	21.6147	0.979453	0.489727	−	0.871876i	$-0.337097\pi$
$487$	21.6147	0.979453	0.489727	+	0.871876i	$0.337097\pi$
$488$	0	0
$489$	−6.83836	−0.309241
$490$	0	0
$491$	21.7084	0.979686	0.489843	−	0.871811i	$-0.337054\pi$
$491$	21.7084	0.979686	0.489843	+	0.871811i	$0.337054\pi$
$492$	0	0
$493$	6.32092	0.284680
$494$	0	0
$495$	0.125115	0.00562349
$496$	0	0
$497$	−18.7766	−0.842245
$498$	0	0
$499$	−3.90013	−0.174594	−0.0872969	−	0.996182i	$-0.527823\pi$
$499$	−3.90013	−0.174594	−0.0872969	+	0.996182i	$0.527823\pi$
$500$	0	0
$501$	6.99311	0.312429
$502$	0	0
$503$	−7.00677	−0.312416	−0.156208	−	0.987724i	$-0.549927\pi$
$503$	−7.00677	−0.312416	−0.156208	+	0.987724i	$0.549927\pi$
$504$	0	0
$505$	−58.3268	−2.59551
$506$	0	0
$507$	−21.0383	−0.934346
$508$	0	0
$509$	28.5799	1.26678	0.633391	−	0.773832i	$-0.281663\pi$
$509$	28.5799	1.26678	0.633391	+	0.773832i	$0.281663\pi$
$510$	0	0
$511$	−11.0615	−0.489332
$512$	0	0
$513$	−7.24196	−0.319740
$514$	0	0
$515$	1.94257	0.0855999
$516$	0	0
$517$	−5.44528	−0.239483
$518$	0	0
$519$	−19.1262	−0.839547
$520$	0	0
$521$	−37.2914	−1.63376	−0.816882	−	0.576805i	$-0.804299\pi$
$521$	−37.2914	−1.63376	−0.816882	+	0.576805i	$0.804299\pi$
$522$	0	0
$523$	−25.3214	−1.10723	−0.553614	−	0.832774i	$-0.686752\pi$
$523$	−25.3214	−1.10723	−0.553614	+	0.832774i	$0.686752\pi$
$524$	0	0
$525$	−51.8060	−2.26100
$526$	0	0
$527$	71.4628	3.11297
$528$	0	0
$529$	−21.7615	−0.946154
$530$	0	0
$531$	−0.0290019	−0.00125857
$532$	0	0
$533$	−5.00923	−0.216974
$534$	0	0
$535$	25.9321	1.12114
$536$	0	0
$537$	17.9355	0.773974
$538$	0	0
$539$	−21.8921	−0.942957
$540$	0	0
$541$	−24.2805	−1.04390	−0.521951	−	0.852976i	$-0.674796\pi$
$541$	−24.2805	−1.04390	−0.521951	+	0.852976i	$0.674796\pi$
$542$	0	0
$543$	17.5762	0.754268
$544$	0	0
$545$	39.1644	1.67762
$546$	0	0
$547$	27.2666	1.16583	0.582917	−	0.812532i	$-0.301911\pi$
$547$	27.2666	1.16583	0.582917	+	0.812532i	$0.301911\pi$
$548$	0	0
$549$	−0.0427420	−0.00182418
$550$	0	0
$551$	−1.20357	−0.0512737
$552$	0	0
$553$	58.3430	2.48100
$554$	0	0
$555$	47.2880	2.00726
$556$	0	0
$557$	29.1224	1.23396	0.616978	−	0.786980i	$-0.288357\pi$
$557$	29.1224	1.23396	0.616978	+	0.786980i	$0.288357\pi$
$558$	0	0
$559$	−3.99686	−0.169049
$560$	0	0
$561$	42.0100	1.77366
$562$	0	0
$563$	−37.4886	−1.57996	−0.789978	−	0.613135i	$-0.789908\pi$
$563$	−37.4886	−1.57996	−0.789978	+	0.613135i	$0.789908\pi$
$564$	0	0
$565$	12.2092	0.513646
$566$	0	0
$567$	33.3423	1.40024
$568$	0	0
$569$	−15.4515	−0.647760	−0.323880	−	0.946098i	$-0.604987\pi$
$569$	−15.4515	−0.647760	−0.323880	+	0.946098i	$0.604987\pi$
$570$	0	0
$571$	22.3576	0.935637	0.467818	−	0.883825i	$-0.345040\pi$
$571$	22.3576	0.935637	0.467818	+	0.883825i	$0.345040\pi$
$572$	0	0
$573$	0.901246	0.0376501
$574$	0	0
$575$	−9.00032	−0.375339
$576$	0	0
$577$	36.8589	1.53445	0.767227	−	0.641375i	$-0.221636\pi$
$577$	36.8589	1.53445	0.767227	+	0.641375i	$0.221636\pi$
$578$	0	0
$579$	−29.8528	−1.24064
$580$	0	0
$581$	−38.5286	−1.59844
$582$	0	0
$583$	37.0253	1.53343
$584$	0	0
$585$	0.0354355	0.00146508
$586$	0	0
$587$	−6.68998	−0.276125	−0.138063	−	0.990424i	$-0.544088\pi$
$587$	−6.68998	−0.276125	−0.138063	+	0.990424i	$0.544088\pi$
$588$	0	0
$589$	−13.6072	−0.560676
$590$	0	0
$591$	−18.2689	−0.751483
$592$	0	0
$593$	−9.13582	−0.375163	−0.187582	−	0.982249i	$-0.560065\pi$
$593$	−9.13582	−0.375163	−0.187582	+	0.982249i	$0.560065\pi$
$594$	0	0
$595$	−97.9311	−4.01478
$596$	0	0
$597$	−43.7411	−1.79020
$598$	0	0
$599$	−11.0380	−0.451000	−0.225500	−	0.974243i	$-0.572402\pi$
$599$	−11.0380	−0.451000	−0.225500	+	0.974243i	$0.572402\pi$
$600$	0	0
$601$	9.40706	0.383722	0.191861	−	0.981422i	$-0.438548\pi$
$601$	9.40706	0.383722	0.191861	+	0.981422i	$0.438548\pi$
$602$	0	0
$603$	−0.0795318	−0.00323878
$604$	0	0
$605$	0.348127	0.0141534
$606$	0	0
$607$	14.6319	0.593889	0.296944	−	0.954895i	$-0.404032\pi$
$607$	14.6319	0.593889	0.296944	+	0.954895i	$0.404032\pi$
$608$	0	0
$609$	5.52199	0.223762
$610$	0	0
$611$	−1.54223	−0.0623921
$612$	0	0
$613$	17.3985	0.702721	0.351360	−	0.936240i	$-0.385719\pi$
$613$	17.3985	0.702721	0.351360	+	0.936240i	$0.385719\pi$
$614$	0	0
$615$	−33.6202	−1.35570
$616$	0	0
$617$	−36.8898	−1.48513	−0.742564	−	0.669775i	$-0.766390\pi$
$617$	−36.8898	−1.48513	−0.742564	+	0.669775i	$0.766390\pi$
$618$	0	0
$619$	36.5100	1.46746	0.733731	−	0.679441i	$-0.237777\pi$
$619$	36.5100	1.46746	0.733731	+	0.679441i	$0.237777\pi$
$620$	0	0
$621$	5.77244	0.231640
$622$	0	0
$623$	−14.8633	−0.595485
$624$	0	0
$625$	−0.0289089	−0.00115636
$626$	0	0
$627$	−7.99912	−0.319454
$628$	0	0
$629$	55.2396	2.20255
$630$	0	0
$631$	−40.2313	−1.60158	−0.800791	−	0.598944i	$-0.795587\pi$
$631$	−40.2313	−1.60158	−0.800791	+	0.598944i	$0.795587\pi$
$632$	0	0
$633$	13.9116	0.552936
$634$	0	0
$635$	−20.2569	−0.803869
$636$	0	0
$637$	−6.20035	−0.245667
$638$	0	0
$639$	0.0532675	0.00210723
$640$	0	0
$641$	−46.7827	−1.84781	−0.923903	−	0.382628i	$-0.875019\pi$
$641$	−46.7827	−1.84781	−0.923903	+	0.382628i	$0.875019\pi$
$642$	0	0
$643$	−43.7073	−1.72365	−0.861823	−	0.507209i	$-0.830677\pi$
$643$	−43.7073	−1.72365	−0.861823	+	0.507209i	$0.830677\pi$
$644$	0	0
$645$	−26.8255	−1.05625
$646$	0	0
$647$	−44.2043	−1.73785	−0.868924	−	0.494945i	$-0.835188\pi$
$647$	−44.2043	−1.73785	−0.868924	+	0.494945i	$0.835188\pi$
$648$	0	0
$649$	9.14374	0.358923
$650$	0	0
$651$	62.4303	2.44684
$652$	0	0
$653$	−23.0004	−0.900075	−0.450037	−	0.893010i	$-0.648589\pi$
$653$	−23.0004	−0.900075	−0.450037	+	0.893010i	$0.648589\pi$
$654$	0	0
$655$	27.2393	1.06433
$656$	0	0
$657$	0.0313805	0.00122427
$658$	0	0
$659$	−39.3107	−1.53133	−0.765664	−	0.643240i	$-0.777590\pi$
$659$	−39.3107	−1.53133	−0.765664	+	0.643240i	$0.777590\pi$
$660$	0	0
$661$	17.9700	0.698951	0.349476	−	0.936945i	$-0.386360\pi$
$661$	17.9700	0.698951	0.349476	+	0.936945i	$0.386360\pi$
$662$	0	0
$663$	11.8982	0.462089
$664$	0	0
$665$	18.6471	0.723102
$666$	0	0
$667$	0.959342	0.0371459
$668$	0	0
$669$	−19.7299	−0.762801
$670$	0	0
$671$	13.4757	0.520225
$672$	0	0
$673$	15.1263	0.583077	0.291538	−	0.956559i	$-0.405833\pi$
$673$	15.1263	0.583077	0.291538	+	0.956559i	$0.405833\pi$
$674$	0	0
$675$	−41.9506	−1.61468
$676$	0	0
$677$	−0.0882823	−0.00339296	−0.00169648	−	0.999999i	$-0.500540\pi$
$677$	−0.0882823	−0.00339296	−0.00169648	+	0.999999i	$0.500540\pi$
$678$	0	0
$679$	−35.4643	−1.36099
$680$	0	0
$681$	27.8530	1.06733
$682$	0	0
$683$	6.97917	0.267050	0.133525	−	0.991045i	$-0.457370\pi$
$683$	6.97917	0.267050	0.133525	+	0.991045i	$0.457370\pi$
$684$	0	0
$685$	1.88891	0.0721714
$686$	0	0
$687$	−3.18182	−0.121394
$688$	0	0
$689$	10.4864	0.399502
$690$	0	0
$691$	−50.2362	−1.91108	−0.955538	−	0.294868i	$-0.904724\pi$
$691$	−50.2362	−1.91108	−0.955538	+	0.294868i	$0.904724\pi$
$692$	0	0
$693$	0.127680	0.00485016
$694$	0	0
$695$	20.2695	0.768867
$696$	0	0
$697$	−39.2735	−1.48759
$698$	0	0
$699$	−25.4664	−0.963229
$700$	0	0
$701$	28.5850	1.07964	0.539820	−	0.841780i	$-0.318492\pi$
$701$	28.5850	1.07964	0.539820	+	0.841780i	$0.318492\pi$
$702$	0	0
$703$	−10.5182	−0.396700
$704$	0	0
$705$	−10.3509	−0.389838
$706$	0	0
$707$	−59.5227	−2.23858
$708$	0	0
$709$	−3.68252	−0.138300	−0.0691499	−	0.997606i	$-0.522029\pi$
$709$	−3.68252	−0.138300	−0.0691499	+	0.997606i	$0.522029\pi$
$710$	0	0
$711$	−0.165514	−0.00620726
$712$	0	0
$713$	10.8461	0.406189
$714$	0	0
$715$	−11.1721	−0.417814
$716$	0	0
$717$	21.4540	0.801213
$718$	0	0
$719$	−27.2440	−1.01603	−0.508015	−	0.861348i	$-0.669620\pi$
$719$	−27.2440	−1.01603	−0.508015	+	0.861348i	$0.669620\pi$
$720$	0	0
$721$	1.98240	0.0738284
$722$	0	0
$723$	14.4228	0.536389
$724$	0	0
$725$	−6.97193	−0.258931
$726$	0	0
$727$	5.84556	0.216800	0.108400	−	0.994107i	$-0.465427\pi$
$727$	5.84556	0.216800	0.108400	+	0.994107i	$0.465427\pi$
$728$	0	0
$729$	26.9051	0.996485
$730$	0	0
$731$	−31.3362	−1.15901
$732$	0	0
$733$	−34.6177	−1.27863	−0.639317	−	0.768944i	$-0.720783\pi$
$733$	−34.6177	−1.27863	−0.639317	+	0.768944i	$0.720783\pi$
$734$	0	0
$735$	−41.6146	−1.53498
$736$	0	0
$737$	25.0749	0.923645
$738$	0	0
$739$	30.4826	1.12132	0.560660	−	0.828046i	$-0.310548\pi$
$739$	30.4826	1.12132	0.560660	+	0.828046i	$0.310548\pi$
$740$	0	0
$741$	−2.26554	−0.0832267
$742$	0	0
$743$	3.78708	0.138935	0.0694673	−	0.997584i	$-0.477870\pi$
$743$	3.78708	0.138935	0.0694673	+	0.997584i	$0.477870\pi$
$744$	0	0
$745$	−60.5468	−2.21826
$746$	0	0
$747$	0.109302	0.00399916
$748$	0	0
$749$	26.4638	0.966965
$750$	0	0
$751$	−5.42952	−0.198126	−0.0990630	−	0.995081i	$-0.531585\pi$
$751$	−5.42952	−0.198126	−0.0990630	+	0.995081i	$0.531585\pi$
$752$	0	0
$753$	−1.73507	−0.0632295
$754$	0	0
$755$	7.12597	0.259340
$756$	0	0
$757$	24.2095	0.879909	0.439955	−	0.898020i	$-0.354995\pi$
$757$	24.2095	0.879909	0.439955	+	0.898020i	$0.354995\pi$
$758$	0	0
$759$	6.37595	0.231432
$760$	0	0
$761$	−21.9066	−0.794112	−0.397056	−	0.917794i	$-0.629968\pi$
$761$	−21.9066	−0.794112	−0.397056	+	0.917794i	$0.629968\pi$
$762$	0	0
$763$	39.9674	1.44692
$764$	0	0
$765$	0.277822	0.0100447
$766$	0	0
$767$	2.58973	0.0935096
$768$	0	0
$769$	−37.7527	−1.36140	−0.680698	−	0.732564i	$-0.738323\pi$
$769$	−37.7527	−1.36140	−0.680698	+	0.732564i	$0.738323\pi$
$770$	0	0
$771$	31.8033	1.14537
$772$	0	0
$773$	−44.7764	−1.61050	−0.805248	−	0.592938i	$-0.797968\pi$
$773$	−44.7764	−1.61050	−0.805248	+	0.592938i	$0.797968\pi$
$774$	0	0
$775$	−78.8229	−2.83140
$776$	0	0
$777$	48.2576	1.73123
$778$	0	0
$779$	7.47807	0.267930
$780$	0	0
$781$	−16.7942	−0.600945
$782$	0	0
$783$	4.47151	0.159799
$784$	0	0
$785$	−58.7213	−2.09585
$786$	0	0
$787$	−15.5163	−0.553098	−0.276549	−	0.961000i	$-0.589191\pi$
$787$	−15.5163	−0.553098	−0.276549	+	0.961000i	$0.589191\pi$
$788$	0	0
$789$	6.99885	0.249166
$790$	0	0
$791$	12.4596	0.443011
$792$	0	0
$793$	3.81665	0.135533
$794$	0	0
$795$	70.3813	2.49617
$796$	0	0
$797$	−8.07191	−0.285922	−0.142961	−	0.989728i	$-0.545662\pi$
$797$	−8.07191	−0.285922	−0.142961	+	0.989728i	$0.545662\pi$
$798$	0	0
$799$	−12.0915	−0.427765
$800$	0	0
$801$	0.0421658	0.00148986
$802$	0	0
$803$	−9.89370	−0.349141
$804$	0	0
$805$	−14.8632	−0.523860
$806$	0	0
$807$	−33.4738	−1.17833
$808$	0	0
$809$	−23.3950	−0.822525	−0.411263	−	0.911517i	$-0.634912\pi$
$809$	−23.3950	−0.822525	−0.411263	+	0.911517i	$0.634912\pi$
$810$	0	0
$811$	−1.03314	−0.0362783	−0.0181392	−	0.999835i	$-0.505774\pi$
$811$	−1.03314	−0.0362783	−0.0181392	+	0.999835i	$0.505774\pi$
$812$	0	0
$813$	−39.8906	−1.39902
$814$	0	0
$815$	14.2582	0.499442
$816$	0	0
$817$	5.96673	0.208749
$818$	0	0
$819$	0.0361620	0.00126360
$820$	0	0
$821$	3.33498	0.116392	0.0581959	−	0.998305i	$-0.481465\pi$
$821$	3.33498	0.116392	0.0581959	+	0.998305i	$0.481465\pi$
$822$	0	0
$823$	−3.34900	−0.116739	−0.0583695	−	0.998295i	$-0.518590\pi$
$823$	−3.34900	−0.116739	−0.0583695	+	0.998295i	$0.518590\pi$
$824$	0	0
$825$	−46.3366	−1.61323
$826$	0	0
$827$	−8.52979	−0.296610	−0.148305	−	0.988942i	$-0.547382\pi$
$827$	−8.52979	−0.296610	−0.148305	+	0.988942i	$0.547382\pi$
$828$	0	0
$829$	−54.3076	−1.88618	−0.943090	−	0.332537i	$-0.892096\pi$
$829$	−54.3076	−1.88618	−0.943090	+	0.332537i	$0.892096\pi$
$830$	0	0
$831$	−14.5059	−0.503204
$832$	0	0
$833$	−48.6121	−1.68431
$834$	0	0
$835$	−14.5808	−0.504591
$836$	0	0
$837$	50.5538	1.74739
$838$	0	0
$839$	50.3357	1.73778	0.868891	−	0.495004i	$-0.164833\pi$
$839$	50.3357	1.73778	0.868891	+	0.495004i	$0.164833\pi$
$840$	0	0
$841$	−28.2569	−0.974375
$842$	0	0
$843$	−28.3059	−0.974907
$844$	0	0
$845$	43.8656	1.50902
$846$	0	0
$847$	0.355265	0.0122071
$848$	0	0
$849$	−3.44887	−0.118365
$850$	0	0
$851$	8.38384	0.287394
$852$	0	0
$853$	−5.99146	−0.205144	−0.102572	−	0.994726i	$-0.532707\pi$
$853$	−5.99146	−0.205144	−0.102572	+	0.994726i	$0.532707\pi$
$854$	0	0
$855$	−0.0529001	−0.00180914
$856$	0	0
$857$	−33.9607	−1.16008	−0.580038	−	0.814590i	$-0.696962\pi$
$857$	−33.9607	−1.16008	−0.580038	+	0.814590i	$0.696962\pi$
$858$	0	0
$859$	−39.7472	−1.35616	−0.678079	−	0.734989i	$-0.737187\pi$
$859$	−39.7472	−1.35616	−0.678079	+	0.734989i	$0.737187\pi$
$860$	0	0
$861$	−34.3095	−1.16927
$862$	0	0
$863$	53.8262	1.83226	0.916132	−	0.400876i	$-0.131294\pi$
$863$	53.8262	1.83226	0.916132	+	0.400876i	$0.131294\pi$
$864$	0	0
$865$	39.8787	1.35592
$866$	0	0
$867$	63.7885	2.16637
$868$	0	0
$869$	52.1835	1.77020
$870$	0	0
$871$	7.10181	0.240635
$872$	0	0
$873$	0.100609	0.00340510
$874$	0	0
$875$	41.2375	1.39408
$876$	0	0
$877$	55.9421	1.88903	0.944516	−	0.328465i	$-0.106531\pi$
$877$	55.9421	1.88903	0.944516	+	0.328465i	$0.106531\pi$
$878$	0	0
$879$	−11.4310	−0.385558
$880$	0	0
$881$	−42.9217	−1.44607	−0.723035	−	0.690811i	$-0.757254\pi$
$881$	−42.9217	−1.44607	−0.723035	+	0.690811i	$0.757254\pi$
$882$	0	0
$883$	−9.92776	−0.334096	−0.167048	−	0.985949i	$-0.553423\pi$
$883$	−9.92776	−0.334096	−0.167048	+	0.985949i	$0.553423\pi$
$884$	0	0
$885$	17.3813	0.584267
$886$	0	0
$887$	12.3747	0.415502	0.207751	−	0.978182i	$-0.433386\pi$
$887$	12.3747	0.415502	0.207751	+	0.978182i	$0.433386\pi$
$888$	0	0
$889$	−20.6722	−0.693323
$890$	0	0
$891$	29.8222	0.999080
$892$	0	0
$893$	2.30233	0.0770446
$894$	0	0
$895$	−37.3961	−1.25001
$896$	0	0
$897$	1.80582	0.0602946
$898$	0	0
$899$	8.40171	0.280213
$900$	0	0
$901$	82.2160	2.73901
$902$	0	0
$903$	−27.3755	−0.910999
$904$	0	0
$905$	−36.6469	−1.21819
$906$	0	0
$907$	−49.1545	−1.63215	−0.816074	−	0.577948i	$-0.803854\pi$
$907$	−49.1545	−1.63215	−0.816074	+	0.577948i	$0.803854\pi$
$908$	0	0
$909$	0.168861	0.00560076
$910$	0	0
$911$	−54.7477	−1.81387	−0.906936	−	0.421269i	$-0.861585\pi$
$911$	−54.7477	−1.81387	−0.906936	+	0.421269i	$0.861585\pi$
$912$	0	0
$913$	−34.4610	−1.14049
$914$	0	0
$915$	25.6160	0.846839
$916$	0	0
$917$	27.7978	0.917964
$918$	0	0
$919$	9.88350	0.326026	0.163013	−	0.986624i	$-0.447879\pi$
$919$	9.88350	0.326026	0.163013	+	0.986624i	$0.447879\pi$
$920$	0	0
$921$	46.3866	1.52849
$922$	0	0
$923$	−4.75653	−0.156563
$924$	0	0
$925$	−60.9288	−2.00333
$926$	0	0
$927$	−0.00562390	−0.000184713 0
$928$	0	0
$929$	−33.4341	−1.09694	−0.548469	−	0.836171i	$-0.684789\pi$
$929$	−33.4341	−1.09694	−0.548469	+	0.836171i	$0.684789\pi$
$930$	0	0
$931$	9.25623	0.303361
$932$	0	0
$933$	14.6205	0.478655
$934$	0	0
$935$	−87.5921	−2.86457
$936$	0	0
$937$	34.8936	1.13992	0.569962	−	0.821671i	$-0.306958\pi$
$937$	34.8936	1.13992	0.569962	+	0.821671i	$0.306958\pi$
$938$	0	0
$939$	−21.5452	−0.703100
$940$	0	0
$941$	−27.3194	−0.890588	−0.445294	−	0.895385i	$-0.646901\pi$
$941$	−27.3194	−0.890588	−0.445294	+	0.895385i	$0.646901\pi$
$942$	0	0
$943$	−5.96063	−0.194105
$944$	0	0
$945$	−69.2777	−2.25361
$946$	0	0
$947$	37.3935	1.21513	0.607563	−	0.794272i	$-0.292147\pi$
$947$	37.3935	1.21513	0.607563	+	0.794272i	$0.292147\pi$
$948$	0	0
$949$	−2.80213	−0.0909611
$950$	0	0
$951$	59.9833	1.94509
$952$	0	0
$953$	34.0873	1.10420	0.552098	−	0.833779i	$-0.313827\pi$
$953$	34.0873	1.10420	0.552098	+	0.833779i	$0.313827\pi$
$954$	0	0
$955$	−1.87913	−0.0608071
$956$	0	0
$957$	4.93901	0.159655
$958$	0	0
$959$	1.92764	0.0622466
$960$	0	0
$961$	63.9877	2.06412
$962$	0	0
$963$	−0.0750754	−0.00241927
$964$	0	0
$965$	62.2439	2.00370
$966$	0	0
$967$	42.7935	1.37615	0.688073	−	0.725641i	$-0.258457\pi$
$967$	42.7935	1.37615	0.688073	+	0.725641i	$0.258457\pi$
$968$	0	0
$969$	−17.7623	−0.570609
$970$	0	0
$971$	46.3893	1.48870	0.744351	−	0.667789i	$-0.232759\pi$
$971$	46.3893	1.48870	0.744351	+	0.667789i	$0.232759\pi$
$972$	0	0
$973$	20.6851	0.663135
$974$	0	0
$975$	−13.1236	−0.420293
$976$	0	0
$977$	9.36047	0.299468	0.149734	−	0.988726i	$-0.452158\pi$
$977$	9.36047	0.299468	0.149734	+	0.988726i	$0.452158\pi$
$978$	0	0
$979$	−13.2941	−0.424881
$980$	0	0
$981$	−0.113384	−0.00362008
$982$	0	0
$983$	−53.4619	−1.70517	−0.852586	−	0.522588i	$-0.824967\pi$
$983$	−53.4619	−1.70517	−0.852586	+	0.522588i	$0.824967\pi$
$984$	0	0
$985$	38.0913	1.21369
$986$	0	0
$987$	−10.5631	−0.336229
$988$	0	0
$989$	−4.75597	−0.151231
$990$	0	0
$991$	54.1399	1.71981	0.859905	−	0.510455i	$-0.170523\pi$
$991$	54.1399	1.71981	0.859905	+	0.510455i	$0.170523\pi$
$992$	0	0
$993$	−36.7591	−1.16652
$994$	0	0
$995$	91.2015	2.89128
$996$	0	0
$997$	−23.6940	−0.750396	−0.375198	−	0.926945i	$-0.622425\pi$
$997$	−23.6940	−0.750396	−0.375198	+	0.926945i	$0.622425\pi$
$998$	0	0
$999$	39.0772	1.23635

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.j.1.20	yes	30
4.3	odd	2		8032.2.a.g.1.11	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.g.1.11	✓	30	4.3	odd	2
8032.2.a.j.1.20	yes	30	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.73507 q^{3} -3.61768 q^{5} -3.69185 q^{7} +0.0104735 q^{9} +O(q^{10})\)	\(q+1.73507 q^{3} -3.61768 q^{5} -3.69185 q^{7} +0.0104735 q^{9} -3.30209 q^{11} -0.935230 q^{13} -6.27693 q^{15} -7.33241 q^{17} +1.39616 q^{19} -6.40563 q^{21} -1.11286 q^{23} +8.08758 q^{25} -5.18704 q^{27} -0.862053 q^{29} -9.74616 q^{31} -5.72936 q^{33} +13.3559 q^{35} -7.53362 q^{37} -1.62269 q^{39} +5.35615 q^{41} +4.27366 q^{43} -0.0378896 q^{45} +1.64904 q^{47} +6.62976 q^{49} -12.7223 q^{51} -11.2127 q^{53} +11.9459 q^{55} +2.42244 q^{57} -2.76908 q^{59} -4.08098 q^{61} -0.0386665 q^{63} +3.38336 q^{65} -7.59365 q^{67} -1.93089 q^{69} +5.08595 q^{71} +2.99620 q^{73} +14.0325 q^{75} +12.1908 q^{77} -15.8032 q^{79} -9.03131 q^{81} +10.4361 q^{83} +26.5263 q^{85} -1.49572 q^{87} +4.02597 q^{89} +3.45273 q^{91} -16.9103 q^{93} -5.05087 q^{95} +9.60609 q^{97} -0.0345843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9	\( 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9} + 31 q^{11} + 5 q^{13} + 8 q^{15} - q^{17} + 29 q^{19} + 6 q^{21} + 27 q^{23} + 34 q^{25} + 36 q^{27} + q^{29} - 9 q^{31} - 8 q^{33} + 51 q^{35} + 5 q^{37} - 8 q^{39} - 3 q^{41} + 43 q^{43} + 42 q^{47} + 41 q^{49} + 54 q^{51} - 11 q^{53} - 10 q^{55} + 2 q^{57} + 49 q^{59} + 21 q^{61} + 15 q^{63} - 14 q^{65} + 68 q^{67} - 10 q^{69} + 44 q^{71} + 25 q^{73} + 51 q^{75} - 20 q^{77} - 49 q^{79} + 6 q^{81} + 88 q^{83} - 5 q^{85} + 16 q^{87} - 16 q^{89} + 46 q^{91} - 4 q^{93} + 28 q^{95} - 4 q^{97} + 71 q^{99}+O(q^{100}) \) 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9 + 31 * q^11 + 5 * q^13 + 8 * q^15 - q^17 + 29 * q^19 + 6 * q^21 + 27 * q^23 + 34 * q^25 + 36 * q^27 + q^29 - 9 * q^31 - 8 * q^33 + 51 * q^35 + 5 * q^37 - 8 * q^39 - 3 * q^41 + 43 * q^43 + 42 * q^47 + 41 * q^49 + 54 * q^51 - 11 * q^53 - 10 * q^55 + 2 * q^57 + 49 * q^59 + 21 * q^61 + 15 * q^63 - 14 * q^65 + 68 * q^67 - 10 * q^69 + 44 * q^71 + 25 * q^73 + 51 * q^75 - 20 * q^77 - 49 * q^79 + 6 * q^81 + 88 * q^83 - 5 * q^85 + 16 * q^87 - 16 * q^89 + 46 * q^91 - 4 * q^93 + 28 * q^95 - 4 * q^97 + 71 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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