LMFDB - Embedded newform 8880.2.a.cg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8880,2,Mod(1,8880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8880, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8880.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8880.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:	$70.9071569949$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.54764.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 3x + 2 $ x^4 - x^3 - 9x^2 + 3x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1110)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.339102$ of defining polynomial
Character	$\chi$	$=$	8880.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -1.84556 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -1.84556 q^{7} +1.00000 q^{9} -0.832640 q^{11} +3.21973 q^{13} -1.00000 q^{15} -7.95030 q^{17} -4.20681 q^{19} +1.84556 q^{21} +6.57614 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.03945 q^{29} +4.73057 q^{31} +0.832640 q^{33} -1.84556 q^{35} -1.00000 q^{37} -3.21973 q^{39} +2.73057 q^{41} +5.03945 q^{43} +1.00000 q^{45} -9.11766 q^{47} -3.59390 q^{49} +7.95030 q^{51} +3.95030 q^{53} -0.832640 q^{55} +4.20681 q^{57} +3.69113 q^{59} +5.40878 q^{61} -1.84556 q^{63} +3.21973 q^{65} -11.7959 q^{67} -6.57614 q^{69} +13.7700 q^{71} -3.52860 q^{73} -1.00000 q^{75} +1.53669 q^{77} -4.00000 q^{79} +1.00000 q^{81} +3.55445 q^{83} -7.95030 q^{85} -3.03945 q^{87} -6.55029 q^{89} -5.94222 q^{91} -4.73057 q^{93} -4.20681 q^{95} -10.4918 q^{97} -0.832640 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{11} - 3 q^{13} - 4 q^{15} + 6 q^{17} - 3 q^{19} + 4 q^{21} + q^{23} + 4 q^{25} - 4 q^{27} - 3 q^{29} - 3 q^{31} + 2 q^{33} - 4 q^{35} - 4 q^{37} + 3 q^{39} - 11 q^{41} + 5 q^{43} + 4 q^{45} + 14 q^{49} - 6 q^{51} - 22 q^{53} - 2 q^{55} + 3 q^{57} + 8 q^{59} - 5 q^{61} - 4 q^{63} - 3 q^{65} - 6 q^{67} - q^{69} + 18 q^{71} - 5 q^{73} - 4 q^{75} - 4 q^{77} - 16 q^{79} + 4 q^{81} + q^{83} + 6 q^{85} + 3 q^{87} - 5 q^{89} + 13 q^{91} + 3 q^{93} - 3 q^{95} + 7 q^{97} - 2 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9 - 2 * q^11 - 3 * q^13 - 4 * q^15 + 6 * q^17 - 3 * q^19 + 4 * q^21 + q^23 + 4 * q^25 - 4 * q^27 - 3 * q^29 - 3 * q^31 + 2 * q^33 - 4 * q^35 - 4 * q^37 + 3 * q^39 - 11 * q^41 + 5 * q^43 + 4 * q^45 + 14 * q^49 - 6 * q^51 - 22 * q^53 - 2 * q^55 + 3 * q^57 + 8 * q^59 - 5 * q^61 - 4 * q^63 - 3 * q^65 - 6 * q^67 - q^69 + 18 * q^71 - 5 * q^73 - 4 * q^75 - 4 * q^77 - 16 * q^79 + 4 * q^81 + q^83 + 6 * q^85 + 3 * q^87 - 5 * q^89 + 13 * q^91 + 3 * q^93 - 3 * q^95 + 7 * q^97 - 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.84556	−0.697557	−0.348779	−	0.937205i	$-0.613404\pi$
$7$	−1.84556	−0.697557	−0.348779	+	0.937205i	$0.613404\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.832640	−0.251050	−0.125525	−	0.992090i	$-0.540062\pi$
$11$	−0.832640	−0.251050	−0.125525	+	0.992090i	$0.540062\pi$
$12$	0	0
$13$	3.21973	0.892992	0.446496	−	0.894786i	$-0.352672\pi$
$13$	3.21973	0.892992	0.446496	+	0.894786i	$0.352672\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−7.95030	−1.92823	−0.964116	−	0.265482i	$-0.914469\pi$
$17$	−7.95030	−1.92823	−0.964116	+	0.265482i	$0.914469\pi$
$18$	0	0
$19$	−4.20681	−0.965108	−0.482554	−	0.875866i	$-0.660291\pi$
$19$	−4.20681	−0.965108	−0.482554	+	0.875866i	$0.660291\pi$
$20$	0	0
$21$	1.84556	0.402735
$22$	0	0
$23$	6.57614	1.37122	0.685610	−	0.727969i	$-0.259536\pi$
$23$	6.57614	1.37122	0.685610	+	0.727969i	$0.259536\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.03945	0.564411	0.282206	−	0.959354i	$-0.408934\pi$
$29$	3.03945	0.564411	0.282206	+	0.959354i	$0.408934\pi$
$30$	0	0
$31$	4.73057	0.849636	0.424818	−	0.905279i	$-0.360338\pi$
$31$	4.73057	0.849636	0.424818	+	0.905279i	$0.360338\pi$
$32$	0	0
$33$	0.832640	0.144944
$34$	0	0
$35$	−1.84556	−0.311957
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	−3.21973	−0.515569
$40$	0	0
$41$	2.73057	0.426444	0.213222	−	0.977004i	$-0.431604\pi$
$41$	2.73057	0.426444	0.213222	+	0.977004i	$0.431604\pi$
$42$	0	0
$43$	5.03945	0.768508	0.384254	−	0.923227i	$-0.374459\pi$
$43$	5.03945	0.768508	0.384254	+	0.923227i	$0.374459\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−9.11766	−1.32995	−0.664974	−	0.746867i	$-0.731557\pi$
$47$	−9.11766	−1.32995	−0.664974	+	0.746867i	$0.731557\pi$
$48$	0	0
$49$	−3.59390	−0.513414
$50$	0	0
$51$	7.95030	1.11327
$52$	0	0
$53$	3.95030	0.542616	0.271308	−	0.962493i	$-0.412544\pi$
$53$	3.95030	0.542616	0.271308	+	0.962493i	$0.412544\pi$
$54$	0	0
$55$	−0.832640	−0.112273
$56$	0	0
$57$	4.20681	0.557205
$58$	0	0
$59$	3.69113	0.480544	0.240272	−	0.970706i	$-0.422763\pi$
$59$	3.69113	0.480544	0.240272	+	0.970706i	$0.422763\pi$
$60$	0	0
$61$	5.40878	0.692523	0.346261	−	0.938138i	$-0.387451\pi$
$61$	5.40878	0.692523	0.346261	+	0.938138i	$0.387451\pi$
$62$	0	0
$63$	−1.84556	−0.232519
$64$	0	0
$65$	3.21973	0.399358
$66$	0	0
$67$	−11.7959	−1.44109	−0.720547	−	0.693406i	$-0.756109\pi$
$67$	−11.7959	−1.44109	−0.720547	+	0.693406i	$0.756109\pi$
$68$	0	0
$69$	−6.57614	−0.791674
$70$	0	0
$71$	13.7700	1.63420	0.817100	−	0.576495i	$-0.195580\pi$
$71$	13.7700	1.63420	0.817100	+	0.576495i	$0.195580\pi$
$72$	0	0
$73$	−3.52860	−0.412992	−0.206496	−	0.978447i	$-0.566206\pi$
$73$	−3.52860	−0.412992	−0.206496	+	0.978447i	$0.566206\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	1.53669	0.175122
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.55445	0.390151	0.195076	−	0.980788i	$-0.437505\pi$
$83$	3.55445	0.390151	0.195076	+	0.980788i	$0.437505\pi$
$84$	0	0
$85$	−7.95030	−0.862331
$86$	0	0
$87$	−3.03945	−0.325863
$88$	0	0
$89$	−6.55029	−0.694329	−0.347165	−	0.937804i	$-0.612856\pi$
$89$	−6.55029	−0.694329	−0.347165	+	0.937804i	$0.612856\pi$
$90$	0	0
$91$	−5.94222	−0.622913
$92$	0	0
$93$	−4.73057	−0.490538
$94$	0	0
$95$	−4.20681	−0.431609
$96$	0	0
$97$	−10.4918	−1.06528	−0.532642	−	0.846341i	$-0.678801\pi$
$97$	−10.4918	−1.06528	−0.532642	+	0.846341i	$0.678801\pi$
$98$	0	0
$99$	−0.832640	−0.0836835
$100$	0	0
$101$	−3.63067	−0.361265	−0.180633	−	0.983551i	$-0.557814\pi$
$101$	−3.63067	−0.361265	−0.180633	+	0.983551i	$0.557814\pi$
$102$	0	0
$103$	9.11766	0.898390	0.449195	−	0.893434i	$-0.351711\pi$
$103$	9.11766	0.898390	0.449195	+	0.893434i	$0.351711\pi$
$104$	0	0
$105$	1.84556	0.180109
$106$	0	0
$107$	−8.91086	−0.861445	−0.430723	−	0.902484i	$-0.641741\pi$
$107$	−8.91086	−0.861445	−0.430723	+	0.902484i	$0.641741\pi$
$108$	0	0
$109$	11.2979	1.08215	0.541073	−	0.840975i	$-0.318018\pi$
$109$	11.2979	1.08215	0.541073	+	0.840975i	$0.318018\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	0	0
$113$	−11.0394	−1.03850	−0.519252	−	0.854621i	$-0.673789\pi$
$113$	−11.0394	−1.03850	−0.519252	+	0.854621i	$0.673789\pi$
$114$	0	0
$115$	6.57614	0.613228
$116$	0	0
$117$	3.21973	0.297664
$118$	0	0
$119$	14.6728	1.34505
$120$	0	0
$121$	−10.3067	−0.936974
$122$	0	0
$123$	−2.73057	−0.246208
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.136678	0.0121282	0.00606410	−	0.999982i	$-0.498070\pi$
$127$	0.136678	0.0121282	0.00606410	+	0.999982i	$0.498070\pi$
$128$	0	0
$129$	−5.03945	−0.443699
$130$	0	0
$131$	−2.07889	−0.181634	−0.0908169	−	0.995868i	$-0.528948\pi$
$131$	−2.07889	−0.181634	−0.0908169	+	0.995868i	$0.528948\pi$
$132$	0	0
$133$	7.76393	0.673218
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	3.06597	0.261943	0.130972	−	0.991386i	$-0.458190\pi$
$137$	3.06597	0.261943	0.130972	+	0.991386i	$0.458190\pi$
$138$	0	0
$139$	−22.1917	−1.88228	−0.941139	−	0.338021i	$-0.890243\pi$
$139$	−22.1917	−1.88228	−0.941139	+	0.338021i	$0.890243\pi$
$140$	0	0
$141$	9.11766	0.767846
$142$	0	0
$143$	−2.68088	−0.224186
$144$	0	0
$145$	3.03945	0.252412
$146$	0	0
$147$	3.59390	0.296420
$148$	0	0
$149$	−14.4999	−1.18788	−0.593940	−	0.804510i	$-0.702428\pi$
$149$	−14.4999	−1.18788	−0.593940	+	0.804510i	$0.702428\pi$
$150$	0	0
$151$	−8.24142	−0.670677	−0.335339	−	0.942098i	$-0.608851\pi$
$151$	−8.24142	−0.670677	−0.335339	+	0.942098i	$0.608851\pi$
$152$	0	0
$153$	−7.95030	−0.642744
$154$	0	0
$155$	4.73057	0.379969
$156$	0	0
$157$	−6.11283	−0.487857	−0.243928	−	0.969793i	$-0.578436\pi$
$157$	−6.11283	−0.487857	−0.243928	+	0.969793i	$0.578436\pi$
$158$	0	0
$159$	−3.95030	−0.313279
$160$	0	0
$161$	−12.1367	−0.956504
$162$	0	0
$163$	21.4373	1.67910	0.839549	−	0.543283i	$-0.182819\pi$
$163$	21.4373	1.67910	0.839549	+	0.543283i	$0.182819\pi$
$164$	0	0
$165$	0.832640	0.0648210
$166$	0	0
$167$	22.5815	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$167$	22.5815	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$168$	0	0
$169$	−2.63334	−0.202565
$170$	0	0
$171$	−4.20681	−0.321703
$172$	0	0
$173$	−6.48916	−0.493361	−0.246681	−	0.969097i	$-0.579340\pi$
$173$	−6.48916	−0.493361	−0.246681	+	0.969097i	$0.579340\pi$
$174$	0	0
$175$	−1.84556	−0.139511
$176$	0	0
$177$	−3.69113	−0.277442
$178$	0	0
$179$	−16.2095	−1.21155	−0.605777	−	0.795635i	$-0.707138\pi$
$179$	−16.2095	−1.21155	−0.605777	+	0.795635i	$0.707138\pi$
$180$	0	0
$181$	5.38225	0.400060	0.200030	−	0.979790i	$-0.435896\pi$
$181$	5.38225	0.400060	0.200030	+	0.979790i	$0.435896\pi$
$182$	0	0
$183$	−5.40878	−0.399828
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	6.61974	0.484083
$188$	0	0
$189$	1.84556	0.134245
$190$	0	0
$191$	10.5680	0.764677	0.382339	−	0.924022i	$-0.375119\pi$
$191$	10.5680	0.764677	0.382339	+	0.924022i	$0.375119\pi$
$192$	0	0
$193$	3.11766	0.224414	0.112207	−	0.993685i	$-0.464208\pi$
$193$	3.11766	0.224414	0.112207	+	0.993685i	$0.464208\pi$
$194$	0	0
$195$	−3.21973	−0.230570
$196$	0	0
$197$	0.576137	0.0410481	0.0205240	−	0.999789i	$-0.493467\pi$
$197$	0.576137	0.0410481	0.0205240	+	0.999789i	$0.493467\pi$
$198$	0	0
$199$	−6.64359	−0.470952	−0.235476	−	0.971880i	$-0.575665\pi$
$199$	−6.64359	−0.470952	−0.235476	+	0.971880i	$0.575665\pi$
$200$	0	0
$201$	11.7959	0.832016
$202$	0	0
$203$	−5.60949	−0.393709
$204$	0	0
$205$	2.73057	0.190712
$206$	0	0
$207$	6.57614	0.457073
$208$	0	0
$209$	3.50276	0.242291
$210$	0	0
$211$	8.83531	0.608248	0.304124	−	0.952632i	$-0.401636\pi$
$211$	8.83531	0.608248	0.304124	+	0.952632i	$0.401636\pi$
$212$	0	0
$213$	−13.7700	−0.943506
$214$	0	0
$215$	5.03945	0.343687
$216$	0	0
$217$	−8.73057	−0.592670
$218$	0	0
$219$	3.52860	0.238441
$220$	0	0
$221$	−25.5978	−1.72190
$222$	0	0
$223$	−17.8828	−1.19752	−0.598762	−	0.800927i	$-0.704340\pi$
$223$	−17.8828	−1.19752	−0.598762	+	0.800927i	$0.704340\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−25.1959	−1.67231	−0.836155	−	0.548494i	$-0.815201\pi$
$227$	−25.1959	−1.67231	−0.836155	+	0.548494i	$0.815201\pi$
$228$	0	0
$229$	−5.48699	−0.362591	−0.181295	−	0.983429i	$-0.558029\pi$
$229$	−5.48699	−0.362591	−0.181295	+	0.983429i	$0.558029\pi$
$230$	0	0
$231$	−1.53669	−0.101107
$232$	0	0
$233$	−28.5788	−1.87226	−0.936130	−	0.351654i	$-0.885619\pi$
$233$	−28.5788	−1.87226	−0.936130	+	0.351654i	$0.885619\pi$
$234$	0	0
$235$	−9.11766	−0.594771
$236$	0	0
$237$	4.00000	0.259828
$238$	0	0
$239$	−1.77811	−0.115016	−0.0575081	−	0.998345i	$-0.518316\pi$
$239$	−1.77811	−0.115016	−0.0575081	+	0.998345i	$0.518316\pi$
$240$	0	0
$241$	22.2612	1.43397	0.716984	−	0.697090i	$-0.245522\pi$
$241$	22.2612	1.43397	0.716984	+	0.697090i	$0.245522\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.59390	−0.229606
$246$	0	0
$247$	−13.5448	−0.861834
$248$	0	0
$249$	−3.55445	−0.225254
$250$	0	0
$251$	28.6748	1.80994	0.904968	−	0.425479i	$-0.139894\pi$
$251$	28.6748	1.80994	0.904968	+	0.425479i	$0.139894\pi$
$252$	0	0
$253$	−5.47556	−0.344245
$254$	0	0
$255$	7.95030	0.497867
$256$	0	0
$257$	9.19388	0.573499	0.286749	−	0.958006i	$-0.407425\pi$
$257$	9.19388	0.573499	0.286749	+	0.958006i	$0.407425\pi$
$258$	0	0
$259$	1.84556	0.114678
$260$	0	0
$261$	3.03945	0.188137
$262$	0	0
$263$	8.02652	0.494937	0.247468	−	0.968896i	$-0.420401\pi$
$263$	8.02652	0.494937	0.247468	+	0.968896i	$0.420401\pi$
$264$	0	0
$265$	3.95030	0.242665
$266$	0	0
$267$	6.55029	0.400871
$268$	0	0
$269$	21.3849	1.30386	0.651931	−	0.758278i	$-0.273959\pi$
$269$	21.3849	1.30386	0.651931	+	0.758278i	$0.273959\pi$
$270$	0	0
$271$	−2.64359	−0.160587	−0.0802934	−	0.996771i	$-0.525586\pi$
$271$	−2.64359	−0.160587	−0.0802934	+	0.996771i	$0.525586\pi$
$272$	0	0
$273$	5.94222	0.359639
$274$	0	0
$275$	−0.832640	−0.0502101
$276$	0	0
$277$	−14.4456	−0.867949	−0.433975	−	0.900925i	$-0.642889\pi$
$277$	−14.4456	−0.867949	−0.433975	+	0.900925i	$0.642889\pi$
$278$	0	0
$279$	4.73057	0.283212
$280$	0	0
$281$	−11.2197	−0.669313	−0.334656	−	0.942340i	$-0.608620\pi$
$281$	−11.2197	−0.669313	−0.334656	+	0.942340i	$0.608620\pi$
$282$	0	0
$283$	−3.86332	−0.229651	−0.114825	−	0.993386i	$-0.536631\pi$
$283$	−3.86332	−0.229651	−0.114825	+	0.993386i	$0.536631\pi$
$284$	0	0
$285$	4.20681	0.249190
$286$	0	0
$287$	−5.03945	−0.297469
$288$	0	0
$289$	46.2073	2.71808
$290$	0	0
$291$	10.4918	0.615042
$292$	0	0
$293$	−16.1544	−0.943752	−0.471876	−	0.881665i	$-0.656423\pi$
$293$	−16.1544	−0.943752	−0.471876	+	0.881665i	$0.656423\pi$
$294$	0	0
$295$	3.69113	0.214906
$296$	0	0
$297$	0.832640	0.0483147
$298$	0	0
$299$	21.1734	1.22449
$300$	0	0
$301$	−9.30062	−0.536079
$302$	0	0
$303$	3.63067	0.208577
$304$	0	0
$305$	5.40878	0.309706
$306$	0	0
$307$	−6.28303	−0.358591	−0.179296	−	0.983795i	$-0.557382\pi$
$307$	−6.28303	−0.358591	−0.179296	+	0.983795i	$0.557382\pi$
$308$	0	0
$309$	−9.11766	−0.518686
$310$	0	0
$311$	−20.6312	−1.16989	−0.584943	−	0.811074i	$-0.698883\pi$
$311$	−20.6312	−1.16989	−0.584943	+	0.811074i	$0.698883\pi$
$312$	0	0
$313$	−7.32180	−0.413852	−0.206926	−	0.978357i	$-0.566346\pi$
$313$	−7.32180	−0.413852	−0.206926	+	0.978357i	$0.566346\pi$
$314$	0	0
$315$	−1.84556	−0.103986
$316$	0	0
$317$	19.1442	1.07524	0.537622	−	0.843186i	$-0.319323\pi$
$317$	19.1442	1.07524	0.537622	+	0.843186i	$0.319323\pi$
$318$	0	0
$319$	−2.53077	−0.141696
$320$	0	0
$321$	8.91086	0.497356
$322$	0	0
$323$	33.4454	1.86095
$324$	0	0
$325$	3.21973	0.178598
$326$	0	0
$327$	−11.2979	−0.624778
$328$	0	0
$329$	16.8272	0.927715
$330$	0	0
$331$	−0.908184	−0.0499183	−0.0249591	−	0.999688i	$-0.507946\pi$
$331$	−0.908184	−0.0499183	−0.0249591	+	0.999688i	$0.507946\pi$
$332$	0	0
$333$	−1.00000	−0.0547997
$334$	0	0
$335$	−11.7959	−0.644477
$336$	0	0
$337$	−8.11083	−0.441825	−0.220913	−	0.975294i	$-0.570904\pi$
$337$	−8.11083	−0.441825	−0.220913	+	0.975294i	$0.570904\pi$
$338$	0	0
$339$	11.0394	0.599580
$340$	0	0
$341$	−3.93887	−0.213302
$342$	0	0
$343$	19.5517	1.05569
$344$	0	0
$345$	−6.57614	−0.354047
$346$	0	0
$347$	−9.35641	−0.502278	−0.251139	−	0.967951i	$-0.580805\pi$
$347$	−9.35641	−0.502278	−0.251139	+	0.967951i	$0.580805\pi$
$348$	0	0
$349$	−35.4665	−1.89848	−0.949239	−	0.314556i	$-0.898144\pi$
$349$	−35.4665	−1.89848	−0.949239	+	0.314556i	$0.898144\pi$
$350$	0	0
$351$	−3.21973	−0.171856
$352$	0	0
$353$	−36.0923	−1.92100	−0.960500	−	0.278279i	$-0.910236\pi$
$353$	−36.0923	−1.92100	−0.960500	+	0.278279i	$0.910236\pi$
$354$	0	0
$355$	13.7700	0.730837
$356$	0	0
$357$	−14.6728	−0.776566
$358$	0	0
$359$	−2.07889	−0.109720	−0.0548599	−	0.998494i	$-0.517471\pi$
$359$	−2.07889	−0.109720	−0.0548599	+	0.998494i	$0.517471\pi$
$360$	0	0
$361$	−1.30278	−0.0685674
$362$	0	0
$363$	10.3067	0.540962
$364$	0	0
$365$	−3.52860	−0.184696
$366$	0	0
$367$	−34.2645	−1.78859	−0.894297	−	0.447474i	$-0.852324\pi$
$367$	−34.2645	−1.78859	−0.894297	+	0.447474i	$0.852324\pi$
$368$	0	0
$369$	2.73057	0.142148
$370$	0	0
$371$	−7.29053	−0.378506
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	9.78620	0.504015
$378$	0	0
$379$	−29.8217	−1.53184	−0.765919	−	0.642937i	$-0.777716\pi$
$379$	−29.8217	−1.53184	−0.765919	+	0.642937i	$0.777716\pi$
$380$	0	0
$381$	−0.136678	−0.00700222
$382$	0	0
$383$	6.78027	0.346456	0.173228	−	0.984882i	$-0.444580\pi$
$383$	6.78027	0.346456	0.173228	+	0.984882i	$0.444580\pi$
$384$	0	0
$385$	1.53669	0.0783170
$386$	0	0
$387$	5.03945	0.256169
$388$	0	0
$389$	35.1959	1.78450	0.892251	−	0.451540i	$-0.149125\pi$
$389$	35.1959	1.78450	0.892251	+	0.451540i	$0.149125\pi$
$390$	0	0
$391$	−52.2823	−2.64403
$392$	0	0
$393$	2.07889	0.104866
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	−3.56054	−0.178698	−0.0893492	−	0.996000i	$-0.528479\pi$
$397$	−3.56054	−0.178698	−0.0893492	+	0.996000i	$0.528479\pi$
$398$	0	0
$399$	−7.76393	−0.388683
$400$	0	0
$401$	1.44971	0.0723950	0.0361975	−	0.999345i	$-0.488475\pi$
$401$	1.44971	0.0723950	0.0361975	+	0.999345i	$0.488475\pi$
$402$	0	0
$403$	15.2312	0.758718
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	0.832640	0.0412724
$408$	0	0
$409$	25.1781	1.24498	0.622489	−	0.782629i	$-0.286122\pi$
$409$	25.1781	1.24498	0.622489	+	0.782629i	$0.286122\pi$
$410$	0	0
$411$	−3.06597	−0.151233
$412$	0	0
$413$	−6.81221	−0.335207
$414$	0	0
$415$	3.55445	0.174481
$416$	0	0
$417$	22.1917	1.08673
$418$	0	0
$419$	−31.6938	−1.54834	−0.774172	−	0.632976i	$-0.781833\pi$
$419$	−31.6938	−1.54834	−0.774172	+	0.632976i	$0.781833\pi$
$420$	0	0
$421$	−5.14351	−0.250679	−0.125340	−	0.992114i	$-0.540002\pi$
$421$	−5.14351	−0.250679	−0.125340	+	0.992114i	$0.540002\pi$
$422$	0	0
$423$	−9.11766	−0.443316
$424$	0	0
$425$	−7.95030	−0.385646
$426$	0	0
$427$	−9.98224	−0.483075
$428$	0	0
$429$	2.68088	0.129434
$430$	0	0
$431$	−2.72448	−0.131234	−0.0656168	−	0.997845i	$-0.520902\pi$
$431$	−2.72448	−0.131234	−0.0656168	+	0.997845i	$0.520902\pi$
$432$	0	0
$433$	−8.68088	−0.417176	−0.208588	−	0.978004i	$-0.566887\pi$
$433$	−8.68088	−0.417176	−0.208588	+	0.978004i	$0.566887\pi$
$434$	0	0
$435$	−3.03945	−0.145730
$436$	0	0
$437$	−27.6645	−1.32337
$438$	0	0
$439$	−7.64752	−0.364996	−0.182498	−	0.983206i	$-0.558418\pi$
$439$	−7.64752	−0.364996	−0.182498	+	0.983206i	$0.558418\pi$
$440$	0	0
$441$	−3.59390	−0.171138
$442$	0	0
$443$	−16.7917	−0.797798	−0.398899	−	0.916995i	$-0.630608\pi$
$443$	−16.7917	−0.797798	−0.398899	+	0.916995i	$0.630608\pi$
$444$	0	0
$445$	−6.55029	−0.310514
$446$	0	0
$447$	14.4999	0.685823
$448$	0	0
$449$	26.1047	1.23196	0.615979	−	0.787762i	$-0.288760\pi$
$449$	26.1047	1.23196	0.615979	+	0.787762i	$0.288760\pi$
$450$	0	0
$451$	−2.27359	−0.107059
$452$	0	0
$453$	8.24142	0.387216
$454$	0	0
$455$	−5.94222	−0.278575
$456$	0	0
$457$	−17.6646	−0.826315	−0.413158	−	0.910660i	$-0.635574\pi$
$457$	−17.6646	−0.826315	−0.413158	+	0.910660i	$0.635574\pi$
$458$	0	0
$459$	7.95030	0.371088
$460$	0	0
$461$	−21.2231	−0.988457	−0.494229	−	0.869332i	$-0.664549\pi$
$461$	−21.2231	−0.988457	−0.494229	+	0.869332i	$0.664549\pi$
$462$	0	0
$463$	−32.5530	−1.51286	−0.756432	−	0.654072i	$-0.773059\pi$
$463$	−32.5530	−1.51286	−0.756432	+	0.654072i	$0.773059\pi$
$464$	0	0
$465$	−4.73057	−0.219375
$466$	0	0
$467$	−15.1170	−0.699531	−0.349765	−	0.936837i	$-0.613739\pi$
$467$	−15.1170	−0.699531	−0.349765	+	0.936837i	$0.613739\pi$
$468$	0	0
$469$	21.7700	1.00525
$470$	0	0
$471$	6.11283	0.281664
$472$	0	0
$473$	−4.19605	−0.192934
$474$	0	0
$475$	−4.20681	−0.193022
$476$	0	0
$477$	3.95030	0.180872
$478$	0	0
$479$	−3.44971	−0.157621	−0.0788106	−	0.996890i	$-0.525112\pi$
$479$	−3.44971	−0.157621	−0.0788106	+	0.996890i	$0.525112\pi$
$480$	0	0
$481$	−3.21973	−0.146807
$482$	0	0
$483$	12.1367	0.552238
$484$	0	0
$485$	−10.4918	−0.476409
$486$	0	0
$487$	15.6619	0.709707	0.354853	−	0.934922i	$-0.384531\pi$
$487$	15.6619	0.709707	0.354853	+	0.934922i	$0.384531\pi$
$488$	0	0
$489$	−21.4373	−0.969428
$490$	0	0
$491$	−7.89793	−0.356429	−0.178214	−	0.983992i	$-0.557032\pi$
$491$	−7.89793	−0.356429	−0.178214	+	0.983992i	$0.557032\pi$
$492$	0	0
$493$	−24.1645	−1.08832
$494$	0	0
$495$	−0.832640	−0.0374244
$496$	0	0
$497$	−25.4134	−1.13995
$498$	0	0
$499$	−5.87209	−0.262871	−0.131435	−	0.991325i	$-0.541959\pi$
$499$	−5.87209	−0.262871	−0.131435	+	0.991325i	$0.541959\pi$
$500$	0	0
$501$	−22.5815	−1.00887
$502$	0	0
$503$	−3.33056	−0.148502	−0.0742512	−	0.997240i	$-0.523657\pi$
$503$	−3.33056	−0.148502	−0.0742512	+	0.997240i	$0.523657\pi$
$504$	0	0
$505$	−3.63067	−0.161563
$506$	0	0
$507$	2.63334	0.116951
$508$	0	0
$509$	−30.9508	−1.37187	−0.685935	−	0.727663i	$-0.740607\pi$
$509$	−30.9508	−1.37187	−0.685935	+	0.727663i	$0.740607\pi$
$510$	0	0
$511$	6.51226	0.288085
$512$	0	0
$513$	4.20681	0.185735
$514$	0	0
$515$	9.11766	0.401772
$516$	0	0
$517$	7.59173	0.333884
$518$	0	0
$519$	6.48916	0.284842
$520$	0	0
$521$	7.45306	0.326524	0.163262	−	0.986583i	$-0.447798\pi$
$521$	7.45306	0.326524	0.163262	+	0.986583i	$0.447798\pi$
$522$	0	0
$523$	−31.5917	−1.38141	−0.690705	−	0.723137i	$-0.742700\pi$
$523$	−31.5917	−1.38141	−0.690705	+	0.723137i	$0.742700\pi$
$524$	0	0
$525$	1.84556	0.0805470
$526$	0	0
$527$	−37.6095	−1.63830
$528$	0	0
$529$	20.2456	0.880242
$530$	0	0
$531$	3.69113	0.160181
$532$	0	0
$533$	8.79171	0.380811
$534$	0	0
$535$	−8.91086	−0.385250
$536$	0	0
$537$	16.2095	0.699491
$538$	0	0
$539$	2.99242	0.128893
$540$	0	0
$541$	36.0816	1.55127	0.775634	−	0.631183i	$-0.217430\pi$
$541$	36.0816	1.55127	0.775634	+	0.631183i	$0.217430\pi$
$542$	0	0
$543$	−5.38225	−0.230975
$544$	0	0
$545$	11.2979	0.483951
$546$	0	0
$547$	38.7420	1.65649	0.828244	−	0.560367i	$-0.189340\pi$
$547$	38.7420	1.65649	0.828244	+	0.560367i	$0.189340\pi$
$548$	0	0
$549$	5.40878	0.230841
$550$	0	0
$551$	−12.7864	−0.544717
$552$	0	0
$553$	7.38225	0.313925
$554$	0	0
$555$	1.00000	0.0424476
$556$	0	0
$557$	−23.3618	−0.989869	−0.494935	−	0.868930i	$-0.664808\pi$
$557$	−23.3618	−0.989869	−0.494935	+	0.868930i	$0.664808\pi$
$558$	0	0
$559$	16.2257	0.686272
$560$	0	0
$561$	−6.61974	−0.279486
$562$	0	0
$563$	−39.0351	−1.64513	−0.822567	−	0.568668i	$-0.807459\pi$
$563$	−39.0351	−1.64513	−0.822567	+	0.568668i	$0.807459\pi$
$564$	0	0
$565$	−11.0394	−0.464433
$566$	0	0
$567$	−1.84556	−0.0775064
$568$	0	0
$569$	35.4809	1.48744	0.743718	−	0.668493i	$-0.233060\pi$
$569$	35.4809	1.48744	0.743718	+	0.668493i	$0.233060\pi$
$570$	0	0
$571$	−37.0093	−1.54879	−0.774395	−	0.632702i	$-0.781946\pi$
$571$	−37.0093	−1.54879	−0.774395	+	0.632702i	$0.781946\pi$
$572$	0	0
$573$	−10.5680	−0.441487
$574$	0	0
$575$	6.57614	0.274244
$576$	0	0
$577$	−15.7354	−0.655074	−0.327537	−	0.944838i	$-0.606219\pi$
$577$	−15.7354	−0.655074	−0.327537	+	0.944838i	$0.606219\pi$
$578$	0	0
$579$	−3.11766	−0.129566
$580$	0	0
$581$	−6.55996	−0.272153
$582$	0	0
$583$	−3.28918	−0.136224
$584$	0	0
$585$	3.21973	0.133119
$586$	0	0
$587$	−37.3006	−1.53956	−0.769781	−	0.638309i	$-0.779634\pi$
$587$	−37.3006	−1.53956	−0.769781	+	0.638309i	$0.779634\pi$
$588$	0	0
$589$	−19.9006	−0.819990
$590$	0	0
$591$	−0.576137	−0.0236991
$592$	0	0
$593$	−17.9829	−0.738470	−0.369235	−	0.929336i	$-0.620380\pi$
$593$	−17.9829	−0.738470	−0.369235	+	0.929336i	$0.620380\pi$
$594$	0	0
$595$	14.6728	0.601526
$596$	0	0
$597$	6.64359	0.271904
$598$	0	0
$599$	32.9998	1.34834	0.674168	−	0.738578i	$-0.264502\pi$
$599$	32.9998	1.34834	0.674168	+	0.738578i	$0.264502\pi$
$600$	0	0
$601$	33.0334	1.34746	0.673729	−	0.738978i	$-0.264691\pi$
$601$	33.0334	1.34746	0.673729	+	0.738978i	$0.264691\pi$
$602$	0	0
$603$	−11.7959	−0.480365
$604$	0	0
$605$	−10.3067	−0.419027
$606$	0	0
$607$	−27.6102	−1.12066	−0.560331	−	0.828269i	$-0.689326\pi$
$607$	−27.6102	−1.12066	−0.560331	+	0.828269i	$0.689326\pi$
$608$	0	0
$609$	5.60949	0.227308
$610$	0	0
$611$	−29.3564	−1.18763
$612$	0	0
$613$	1.62999	0.0658348	0.0329174	−	0.999458i	$-0.489520\pi$
$613$	1.62999	0.0658348	0.0329174	+	0.999458i	$0.489520\pi$
$614$	0	0
$615$	−2.73057	−0.110107
$616$	0	0
$617$	28.6143	1.15197	0.575985	−	0.817460i	$-0.304619\pi$
$617$	28.6143	1.15197	0.575985	+	0.817460i	$0.304619\pi$
$618$	0	0
$619$	−19.6830	−0.791128	−0.395564	−	0.918438i	$-0.629451\pi$
$619$	−19.6830	−0.791128	−0.395564	+	0.918438i	$0.629451\pi$
$620$	0	0
$621$	−6.57614	−0.263891
$622$	0	0
$623$	12.0890	0.484335
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−3.50276	−0.139887
$628$	0	0
$629$	7.95030	0.316999
$630$	0	0
$631$	5.76193	0.229379	0.114689	−	0.993401i	$-0.463413\pi$
$631$	5.76193	0.229379	0.114689	+	0.993401i	$0.463413\pi$
$632$	0	0
$633$	−8.83531	−0.351172
$634$	0	0
$635$	0.136678	0.00542390
$636$	0	0
$637$	−11.5714	−0.458474
$638$	0	0
$639$	13.7700	0.544734
$640$	0	0
$641$	20.8967	0.825369	0.412685	−	0.910874i	$-0.364591\pi$
$641$	20.8967	0.825369	0.412685	+	0.910874i	$0.364591\pi$
$642$	0	0
$643$	−30.6197	−1.20752	−0.603762	−	0.797164i	$-0.706332\pi$
$643$	−30.6197	−1.20752	−0.603762	+	0.797164i	$0.706332\pi$
$644$	0	0
$645$	−5.03945	−0.198428
$646$	0	0
$647$	10.6278	0.417823	0.208912	−	0.977935i	$-0.433008\pi$
$647$	10.6278	0.417823	0.208912	+	0.977935i	$0.433008\pi$
$648$	0	0
$649$	−3.07338	−0.120641
$650$	0	0
$651$	8.73057	0.342178
$652$	0	0
$653$	−5.90061	−0.230909	−0.115454	−	0.993313i	$-0.536832\pi$
$653$	−5.90061	−0.230909	−0.115454	+	0.993313i	$0.536832\pi$
$654$	0	0
$655$	−2.07889	−0.0812291
$656$	0	0
$657$	−3.52860	−0.137664
$658$	0	0
$659$	−19.8305	−0.772486	−0.386243	−	0.922397i	$-0.626227\pi$
$659$	−19.8305	−0.772486	−0.386243	+	0.922397i	$0.626227\pi$
$660$	0	0
$661$	−9.32379	−0.362653	−0.181327	−	0.983423i	$-0.558039\pi$
$661$	−9.32379	−0.362653	−0.181327	+	0.983423i	$0.558039\pi$
$662$	0	0
$663$	25.5978	0.994137
$664$	0	0
$665$	7.76393	0.301072
$666$	0	0
$667$	19.9878	0.773931
$668$	0	0
$669$	17.8828	0.691391
$670$	0	0
$671$	−4.50357	−0.173858
$672$	0	0
$673$	33.0113	1.27249	0.636245	−	0.771487i	$-0.280487\pi$
$673$	33.0113	1.27249	0.636245	+	0.771487i	$0.280487\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	26.1245	1.00405	0.502023	−	0.864854i	$-0.332589\pi$
$677$	26.1245	1.00405	0.502023	+	0.864854i	$0.332589\pi$
$678$	0	0
$679$	19.3633	0.743097
$680$	0	0
$681$	25.1959	0.965508
$682$	0	0
$683$	39.3957	1.50743	0.753717	−	0.657199i	$-0.228259\pi$
$683$	39.3957	1.50743	0.753717	+	0.657199i	$0.228259\pi$
$684$	0	0
$685$	3.06597	0.117145
$686$	0	0
$687$	5.48699	0.209342
$688$	0	0
$689$	12.7189	0.484552
$690$	0	0
$691$	9.45306	0.359611	0.179806	−	0.983702i	$-0.442453\pi$
$691$	9.45306	0.359611	0.179806	+	0.983702i	$0.442453\pi$
$692$	0	0
$693$	1.53669	0.0583740
$694$	0	0
$695$	−22.1917	−0.841780
$696$	0	0
$697$	−21.7089	−0.822283
$698$	0	0
$699$	28.5788	1.08095
$700$	0	0
$701$	37.2842	1.40821	0.704103	−	0.710098i	$-0.251349\pi$
$701$	37.2842	1.40821	0.704103	+	0.710098i	$0.251349\pi$
$702$	0	0
$703$	4.20681	0.158663
$704$	0	0
$705$	9.11766	0.343391
$706$	0	0
$707$	6.70063	0.252003
$708$	0	0
$709$	−19.8110	−0.744016	−0.372008	−	0.928230i	$-0.621331\pi$
$709$	−19.8110	−0.744016	−0.372008	+	0.928230i	$0.621331\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	31.1089	1.16504
$714$	0	0
$715$	−2.68088	−0.100259
$716$	0	0
$717$	1.77811	0.0664046
$718$	0	0
$719$	19.2136	0.716548	0.358274	−	0.933617i	$-0.383365\pi$
$719$	19.2136	0.716548	0.358274	+	0.933617i	$0.383365\pi$
$720$	0	0
$721$	−16.8272	−0.626679
$722$	0	0
$723$	−22.2612	−0.827902
$724$	0	0
$725$	3.03945	0.112882
$726$	0	0
$727$	45.0183	1.66964	0.834818	−	0.550527i	$-0.185573\pi$
$727$	45.0183	1.66964	0.834818	+	0.550527i	$0.185573\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−40.0651	−1.48186
$732$	0	0
$733$	−20.7262	−0.765541	−0.382771	−	0.923843i	$-0.625030\pi$
$733$	−20.7262	−0.765541	−0.382771	+	0.923843i	$0.625030\pi$
$734$	0	0
$735$	3.59390	0.132563
$736$	0	0
$737$	9.82171	0.361787
$738$	0	0
$739$	−24.2176	−0.890858	−0.445429	−	0.895317i	$-0.646949\pi$
$739$	−24.2176	−0.890858	−0.445429	+	0.895317i	$0.646949\pi$
$740$	0	0
$741$	13.5448	0.497580
$742$	0	0
$743$	−13.9530	−0.511885	−0.255943	−	0.966692i	$-0.582386\pi$
$743$	−13.9530	−0.511885	−0.255943	+	0.966692i	$0.582386\pi$
$744$	0	0
$745$	−14.4999	−0.531236
$746$	0	0
$747$	3.55445	0.130050
$748$	0	0
$749$	16.4456	0.600907
$750$	0	0
$751$	−41.6176	−1.51865	−0.759324	−	0.650713i	$-0.774470\pi$
$751$	−41.6176	−1.51865	−0.759324	+	0.650713i	$0.774470\pi$
$752$	0	0
$753$	−28.6748	−1.04497
$754$	0	0
$755$	−8.24142	−0.299936
$756$	0	0
$757$	−45.4034	−1.65021	−0.825107	−	0.564977i	$-0.808885\pi$
$757$	−45.4034	−1.65021	−0.825107	+	0.564977i	$0.808885\pi$
$758$	0	0
$759$	5.47556	0.198750
$760$	0	0
$761$	18.9347	0.686383	0.343191	−	0.939266i	$-0.388492\pi$
$761$	18.9347	0.686383	0.343191	+	0.939266i	$0.388492\pi$
$762$	0	0
$763$	−20.8511	−0.754860
$764$	0	0
$765$	−7.95030	−0.287444
$766$	0	0
$767$	11.8844	0.429122
$768$	0	0
$769$	31.0884	1.12108	0.560538	−	0.828129i	$-0.310594\pi$
$769$	31.0884	1.12108	0.560538	+	0.828129i	$0.310594\pi$
$770$	0	0
$771$	−9.19388	−0.331110
$772$	0	0
$773$	34.3542	1.23564	0.617818	−	0.786321i	$-0.288017\pi$
$773$	34.3542	1.23564	0.617818	+	0.786321i	$0.288017\pi$
$774$	0	0
$775$	4.73057	0.169927
$776$	0	0
$777$	−1.84556	−0.0662092
$778$	0	0
$779$	−11.4870	−0.411564
$780$	0	0
$781$	−11.4655	−0.410267
$782$	0	0
$783$	−3.03945	−0.108621
$784$	0	0
$785$	−6.11283	−0.218176
$786$	0	0
$787$	−36.0053	−1.28345	−0.641726	−	0.766934i	$-0.721781\pi$
$787$	−36.0053	−1.28345	−0.641726	+	0.766934i	$0.721781\pi$
$788$	0	0
$789$	−8.02652	−0.285752
$790$	0	0
$791$	20.3740	0.724416
$792$	0	0
$793$	17.4148	0.618418
$794$	0	0
$795$	−3.95030	−0.140103
$796$	0	0
$797$	43.2107	1.53060	0.765300	−	0.643674i	$-0.222591\pi$
$797$	43.2107	1.53060	0.765300	+	0.643674i	$0.222591\pi$
$798$	0	0
$799$	72.4882	2.56445
$800$	0	0
$801$	−6.55029	−0.231443
$802$	0	0
$803$	2.93806	0.103682
$804$	0	0
$805$	−12.1367	−0.427762
$806$	0	0
$807$	−21.3849	−0.752785
$808$	0	0
$809$	−32.2209	−1.13283	−0.566414	−	0.824121i	$-0.691669\pi$
$809$	−32.2209	−1.13283	−0.566414	+	0.824121i	$0.691669\pi$
$810$	0	0
$811$	−8.87892	−0.311781	−0.155890	−	0.987774i	$-0.549825\pi$
$811$	−8.87892	−0.311781	−0.155890	+	0.987774i	$0.549825\pi$
$812$	0	0
$813$	2.64359	0.0927148
$814$	0	0
$815$	21.4373	0.750916
$816$	0	0
$817$	−21.2000	−0.741693
$818$	0	0
$819$	−5.94222	−0.207638
$820$	0	0
$821$	2.31155	0.0806735	0.0403368	−	0.999186i	$-0.487157\pi$
$821$	2.31155	0.0806735	0.0403368	+	0.999186i	$0.487157\pi$
$822$	0	0
$823$	−13.6280	−0.475042	−0.237521	−	0.971382i	$-0.576335\pi$
$823$	−13.6280	−0.475042	−0.237521	+	0.971382i	$0.576335\pi$
$824$	0	0
$825$	0.832640	0.0289888
$826$	0	0
$827$	17.8298	0.620003	0.310001	−	0.950736i	$-0.399670\pi$
$827$	17.8298	0.620003	0.310001	+	0.950736i	$0.399670\pi$
$828$	0	0
$829$	−29.7388	−1.03287	−0.516435	−	0.856326i	$-0.672741\pi$
$829$	−29.7388	−1.03287	−0.516435	+	0.856326i	$0.672741\pi$
$830$	0	0
$831$	14.4456	0.501111
$832$	0	0
$833$	28.5726	0.989980
$834$	0	0
$835$	22.5815	0.781464
$836$	0	0
$837$	−4.73057	−0.163513
$838$	0	0
$839$	10.4556	0.360969	0.180484	−	0.983578i	$-0.442234\pi$
$839$	10.4556	0.360969	0.180484	+	0.983578i	$0.442234\pi$
$840$	0	0
$841$	−19.7618	−0.681440
$842$	0	0
$843$	11.2197	0.386428
$844$	0	0
$845$	−2.63334	−0.0905897
$846$	0	0
$847$	19.0217	0.653593
$848$	0	0
$849$	3.86332	0.132589
$850$	0	0
$851$	−6.57614	−0.225427
$852$	0	0
$853$	−15.6333	−0.535275	−0.267638	−	0.963520i	$-0.586243\pi$
$853$	−15.6333	−0.535275	−0.267638	+	0.963520i	$0.586243\pi$
$854$	0	0
$855$	−4.20681	−0.143870
$856$	0	0
$857$	−14.4945	−0.495123	−0.247561	−	0.968872i	$-0.579629\pi$
$857$	−14.4945	−0.495123	−0.247561	+	0.968872i	$0.579629\pi$
$858$	0	0
$859$	−3.38493	−0.115492	−0.0577461	−	0.998331i	$-0.518391\pi$
$859$	−3.38493	−0.115492	−0.0577461	+	0.998331i	$0.518391\pi$
$860$	0	0
$861$	5.03945	0.171744
$862$	0	0
$863$	56.3924	1.91962	0.959810	−	0.280649i	$-0.0905498\pi$
$863$	56.3924	1.91962	0.959810	+	0.280649i	$0.0905498\pi$
$864$	0	0
$865$	−6.48916	−0.220638
$866$	0	0
$867$	−46.2073	−1.56928
$868$	0	0
$869$	3.33056	0.112982
$870$	0	0
$871$	−37.9795	−1.28689
$872$	0	0
$873$	−10.4918	−0.355095
$874$	0	0
$875$	−1.84556	−0.0623914
$876$	0	0
$877$	−32.1755	−1.08649	−0.543245	−	0.839574i	$-0.682805\pi$
$877$	−32.1755	−1.08649	−0.543245	+	0.839574i	$0.682805\pi$
$878$	0	0
$879$	16.1544	0.544876
$880$	0	0
$881$	14.6097	0.492212	0.246106	−	0.969243i	$-0.420849\pi$
$881$	14.6097	0.492212	0.246106	+	0.969243i	$0.420849\pi$
$882$	0	0
$883$	−18.5203	−0.623259	−0.311630	−	0.950204i	$-0.600875\pi$
$883$	−18.5203	−0.623259	−0.311630	+	0.950204i	$0.600875\pi$
$884$	0	0
$885$	−3.69113	−0.124076
$886$	0	0
$887$	28.5611	0.958986	0.479493	−	0.877546i	$-0.340821\pi$
$887$	28.5611	0.958986	0.479493	+	0.877546i	$0.340821\pi$
$888$	0	0
$889$	−0.252248	−0.00846012
$890$	0	0
$891$	−0.832640	−0.0278945
$892$	0	0
$893$	38.3562	1.28354
$894$	0	0
$895$	−16.2095	−0.541823
$896$	0	0
$897$	−21.1734	−0.706959
$898$	0	0
$899$	14.3783	0.479544
$900$	0	0
$901$	−31.4061	−1.04629
$902$	0	0
$903$	9.30062	0.309505
$904$	0	0
$905$	5.38225	0.178912
$906$	0	0
$907$	25.8428	0.858097	0.429048	−	0.903282i	$-0.358849\pi$
$907$	25.8428	0.858097	0.429048	+	0.903282i	$0.358849\pi$
$908$	0	0
$909$	−3.63067	−0.120422
$910$	0	0
$911$	−24.9998	−0.828281	−0.414141	−	0.910213i	$-0.635918\pi$
$911$	−24.9998	−0.828281	−0.414141	+	0.910213i	$0.635918\pi$
$912$	0	0
$913$	−2.95958	−0.0979477
$914$	0	0
$915$	−5.40878	−0.178809
$916$	0	0
$917$	3.83673	0.126700
$918$	0	0
$919$	34.3931	1.13452	0.567262	−	0.823537i	$-0.308003\pi$
$919$	34.3931	1.13452	0.567262	+	0.823537i	$0.308003\pi$
$920$	0	0
$921$	6.28303	0.207033
$922$	0	0
$923$	44.3357	1.45933
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	9.11766	0.299463
$928$	0	0
$929$	55.6310	1.82519	0.912597	−	0.408860i	$-0.134074\pi$
$929$	55.6310	1.82519	0.912597	+	0.408860i	$0.134074\pi$
$930$	0	0
$931$	15.1188	0.495499
$932$	0	0
$933$	20.6312	0.675434
$934$	0	0
$935$	6.61974	0.216489
$936$	0	0
$937$	−53.9576	−1.76272	−0.881360	−	0.472446i	$-0.843371\pi$
$937$	−53.9576	−1.76272	−0.881360	+	0.472446i	$0.843371\pi$
$938$	0	0
$939$	7.32180	0.238938
$940$	0	0
$941$	−50.1092	−1.63351	−0.816756	−	0.576983i	$-0.804230\pi$
$941$	−50.1092	−1.63351	−0.816756	+	0.576983i	$0.804230\pi$
$942$	0	0
$943$	17.9566	0.584748
$944$	0	0
$945$	1.84556	0.0600362
$946$	0	0
$947$	−39.3795	−1.27966	−0.639831	−	0.768516i	$-0.720996\pi$
$947$	−39.3795	−1.27966	−0.639831	+	0.768516i	$0.720996\pi$
$948$	0	0
$949$	−11.3611	−0.368798
$950$	0	0
$951$	−19.1442	−0.620793
$952$	0	0
$953$	−20.5096	−0.664371	−0.332185	−	0.943214i	$-0.607786\pi$
$953$	−20.5096	−0.664371	−0.332185	+	0.943214i	$0.607786\pi$
$954$	0	0
$955$	10.5680	0.341974
$956$	0	0
$957$	2.53077	0.0818080
$958$	0	0
$959$	−5.65844	−0.182721
$960$	0	0
$961$	−8.62168	−0.278119
$962$	0	0
$963$	−8.91086	−0.287148
$964$	0	0
$965$	3.11766	0.100361
$966$	0	0
$967$	−18.6265	−0.598988	−0.299494	−	0.954098i	$-0.596818\pi$
$967$	−18.6265	−0.598988	−0.299494	+	0.954098i	$0.596818\pi$
$968$	0	0
$969$	−33.4454	−1.07442
$970$	0	0
$971$	3.56656	0.114456	0.0572282	−	0.998361i	$-0.481774\pi$
$971$	3.56656	0.114456	0.0572282	+	0.998361i	$0.481774\pi$
$972$	0	0
$973$	40.9562	1.31300
$974$	0	0
$975$	−3.21973	−0.103114
$976$	0	0
$977$	−43.6998	−1.39808	−0.699041	−	0.715082i	$-0.746389\pi$
$977$	−43.6998	−1.39808	−0.699041	+	0.715082i	$0.746389\pi$
$978$	0	0
$979$	5.45404	0.174312
$980$	0	0
$981$	11.2979	0.360716
$982$	0	0
$983$	−53.1096	−1.69393	−0.846966	−	0.531647i	$-0.821573\pi$
$983$	−53.1096	−1.69393	−0.846966	+	0.531647i	$0.821573\pi$
$984$	0	0
$985$	0.576137	0.0183572
$986$	0	0
$987$	−16.8272	−0.535616
$988$	0	0
$989$	33.1401	1.05379
$990$	0	0
$991$	9.14283	0.290432	0.145216	−	0.989400i	$-0.453612\pi$
$991$	9.14283	0.290432	0.145216	+	0.989400i	$0.453612\pi$
$992$	0	0
$993$	0.908184	0.0288203
$994$	0	0
$995$	−6.64359	−0.210616
$996$	0	0
$997$	−15.6280	−0.494944	−0.247472	−	0.968895i	$-0.579600\pi$
$997$	−15.6280	−0.494944	−0.247472	+	0.968895i	$0.579600\pi$
$998$	0	0
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8880.2.a.cg.1.2		4
4.3	odd	2		1110.2.a.s.1.3	✓	4
12.11	even	2		3330.2.a.bj.1.3		4
20.19	odd	2		5550.2.a.cj.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.s.1.3	✓	4	4.3	odd	2
3330.2.a.bj.1.3		4	12.11	even	2
5550.2.a.cj.1.2		4	20.19	odd	2
8880.2.a.cg.1.2		4	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:	\( N \)	\(=\)	\( 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8880.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -1.84556 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -1.84556 q^{7} +1.00000 q^{9} -0.832640 q^{11} +3.21973 q^{13} -1.00000 q^{15} -7.95030 q^{17} -4.20681 q^{19} +1.84556 q^{21} +6.57614 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.03945 q^{29} +4.73057 q^{31} +0.832640 q^{33} -1.84556 q^{35} -1.00000 q^{37} -3.21973 q^{39} +2.73057 q^{41} +5.03945 q^{43} +1.00000 q^{45} -9.11766 q^{47} -3.59390 q^{49} +7.95030 q^{51} +3.95030 q^{53} -0.832640 q^{55} +4.20681 q^{57} +3.69113 q^{59} +5.40878 q^{61} -1.84556 q^{63} +3.21973 q^{65} -11.7959 q^{67} -6.57614 q^{69} +13.7700 q^{71} -3.52860 q^{73} -1.00000 q^{75} +1.53669 q^{77} -4.00000 q^{79} +1.00000 q^{81} +3.55445 q^{83} -7.95030 q^{85} -3.03945 q^{87} -6.55029 q^{89} -5.94222 q^{91} -4.73057 q^{93} -4.20681 q^{95} -10.4918 q^{97} -0.832640 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{11} - 3 q^{13} - 4 q^{15} + 6 q^{17} - 3 q^{19} + 4 q^{21} + q^{23} + 4 q^{25} - 4 q^{27} - 3 q^{29} - 3 q^{31} + 2 q^{33} - 4 q^{35} - 4 q^{37} + 3 q^{39} - 11 q^{41} + 5 q^{43} + 4 q^{45} + 14 q^{49} - 6 q^{51} - 22 q^{53} - 2 q^{55} + 3 q^{57} + 8 q^{59} - 5 q^{61} - 4 q^{63} - 3 q^{65} - 6 q^{67} - q^{69} + 18 q^{71} - 5 q^{73} - 4 q^{75} - 4 q^{77} - 16 q^{79} + 4 q^{81} + q^{83} + 6 q^{85} + 3 q^{87} - 5 q^{89} + 13 q^{91} + 3 q^{93} - 3 q^{95} + 7 q^{97} - 2 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9 - 2 * q^11 - 3 * q^13 - 4 * q^15 + 6 * q^17 - 3 * q^19 + 4 * q^21 + q^23 + 4 * q^25 - 4 * q^27 - 3 * q^29 - 3 * q^31 + 2 * q^33 - 4 * q^35 - 4 * q^37 + 3 * q^39 - 11 * q^41 + 5 * q^43 + 4 * q^45 + 14 * q^49 - 6 * q^51 - 22 * q^53 - 2 * q^55 + 3 * q^57 + 8 * q^59 - 5 * q^61 - 4 * q^63 - 3 * q^65 - 6 * q^67 - q^69 + 18 * q^71 - 5 * q^73 - 4 * q^75 - 4 * q^77 - 16 * q^79 + 4 * q^81 + q^83 + 6 * q^85 + 3 * q^87 - 5 * q^89 + 13 * q^91 + 3 * q^93 - 3 * q^95 + 7 * q^97 - 2 * q^99

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