LMFDB - Embedded newform 4012.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.239123$ of defining polynomial
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.521753 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.521753 q^{5} +1.00000 q^{7} -2.00000 q^{9} -2.00000 q^{11} +0.478247 q^{13} -0.521753 q^{15} +1.00000 q^{17} -2.52175 q^{19} -1.00000 q^{21} +5.88564 q^{23} -4.72777 q^{25} +5.00000 q^{27} -1.47825 q^{29} +6.84213 q^{31} +2.00000 q^{33} +0.521753 q^{35} +7.32038 q^{37} -0.478247 q^{39} -2.52175 q^{41} +2.84213 q^{43} -1.04351 q^{45} -9.88564 q^{47} -6.00000 q^{49} -1.00000 q^{51} -5.72777 q^{53} -1.04351 q^{55} +2.52175 q^{57} -1.00000 q^{59} -2.47825 q^{61} -2.00000 q^{63} +0.249527 q^{65} -13.7713 q^{67} -5.88564 q^{69} -12.7278 q^{71} +2.00000 q^{73} +4.72777 q^{75} -2.00000 q^{77} -5.00000 q^{79} +1.00000 q^{81} +3.32038 q^{83} +0.521753 q^{85} +1.47825 q^{87} +8.84213 q^{89} +0.478247 q^{91} -6.84213 q^{93} -1.31573 q^{95} +7.20602 q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + q^{5} + 3 q^{7} - 6 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + q^5 + 3 * q^7 - 6 * q^9	$ 3 q - 3 q^{3} + q^{5} + 3 q^{7} - 6 q^{9} - 6 q^{11} + 2 q^{13} - q^{15} + 3 q^{17} - 7 q^{19} - 3 q^{21} + 20 q^{25} + 15 q^{27} - 5 q^{29} + 4 q^{31} + 6 q^{33} + q^{35} + 6 q^{37} - 2 q^{39} - 7 q^{41} - 8 q^{43} - 2 q^{45} - 12 q^{47} - 18 q^{49} - 3 q^{51} + 17 q^{53} - 2 q^{55} + 7 q^{57} - 3 q^{59} - 8 q^{61} - 6 q^{63} - 34 q^{65} - 6 q^{67} - 4 q^{71} + 6 q^{73} - 20 q^{75} - 6 q^{77} - 15 q^{79} + 3 q^{81} - 6 q^{83} + q^{85} + 5 q^{87} + 10 q^{89} + 2 q^{91} - 4 q^{93} - 37 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + q^5 + 3 * q^7 - 6 * q^9 - 6 * q^11 + 2 * q^13 - q^15 + 3 * q^17 - 7 * q^19 - 3 * q^21 + 20 * q^25 + 15 * q^27 - 5 * q^29 + 4 * q^31 + 6 * q^33 + q^35 + 6 * q^37 - 2 * q^39 - 7 * q^41 - 8 * q^43 - 2 * q^45 - 12 * q^47 - 18 * q^49 - 3 * q^51 + 17 * q^53 - 2 * q^55 + 7 * q^57 - 3 * q^59 - 8 * q^61 - 6 * q^63 - 34 * q^65 - 6 * q^67 - 4 * q^71 + 6 * q^73 - 20 * q^75 - 6 * q^77 - 15 * q^79 + 3 * q^81 - 6 * q^83 + q^85 + 5 * q^87 + 10 * q^89 + 2 * q^91 - 4 * q^93 - 37 * q^95 - 12 * q^97 + 12 * 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	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	0.521753	0.233335	0.116668	−	0.993171i	$-0.462779\pi$
$5$	0.521753	0.233335	0.116668	+	0.993171i	$0.462779\pi$
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	0.478247	0.132642	0.0663209	−	0.997798i	$-0.478874\pi$
$13$	0.478247	0.132642	0.0663209	+	0.997798i	$0.478874\pi$
$14$	0	0
$15$	−0.521753	−0.134716
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.52175	−0.578530	−0.289265	−	0.957249i	$-0.593411\pi$
$19$	−2.52175	−0.578530	−0.289265	+	0.957249i	$0.593411\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	5.88564	1.22724	0.613620	−	0.789601i	$-0.289713\pi$
$23$	5.88564	1.22724	0.613620	+	0.789601i	$0.289713\pi$
$24$	0	0
$25$	−4.72777	−0.945555
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	−1.47825	−0.274503	−0.137252	−	0.990536i	$-0.543827\pi$
$29$	−1.47825	−0.274503	−0.137252	+	0.990536i	$0.543827\pi$
$30$	0	0
$31$	6.84213	1.22888	0.614442	−	0.788962i	$-0.289381\pi$
$31$	6.84213	1.22888	0.614442	+	0.788962i	$0.289381\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	0.521753	0.0881924
$36$	0	0
$37$	7.32038	1.20346	0.601732	−	0.798698i	$-0.294478\pi$
$37$	7.32038	1.20346	0.601732	+	0.798698i	$0.294478\pi$
$38$	0	0
$39$	−0.478247	−0.0765807
$40$	0	0
$41$	−2.52175	−0.393832	−0.196916	−	0.980420i	$-0.563093\pi$
$41$	−2.52175	−0.393832	−0.196916	+	0.980420i	$0.563093\pi$
$42$	0	0
$43$	2.84213	0.433421	0.216711	−	0.976236i	$-0.430467\pi$
$43$	2.84213	0.433421	0.216711	+	0.976236i	$0.430467\pi$
$44$	0	0
$45$	−1.04351	−0.155557
$46$	0	0
$47$	−9.88564	−1.44197	−0.720984	−	0.692951i	$-0.756310\pi$
$47$	−9.88564	−1.44197	−0.720984	+	0.692951i	$0.756310\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	−5.72777	−0.786770	−0.393385	−	0.919374i	$-0.628696\pi$
$53$	−5.72777	−0.786770	−0.393385	+	0.919374i	$0.628696\pi$
$54$	0	0
$55$	−1.04351	−0.140706
$56$	0	0
$57$	2.52175	0.334014
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−2.47825	−0.317307	−0.158653	−	0.987334i	$-0.550715\pi$
$61$	−2.47825	−0.317307	−0.158653	+	0.987334i	$0.550715\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	0.249527	0.0309500
$66$	0	0
$67$	−13.7713	−1.68243	−0.841215	−	0.540701i	$-0.818159\pi$
$67$	−13.7713	−1.68243	−0.841215	+	0.540701i	$0.818159\pi$
$68$	0	0
$69$	−5.88564	−0.708548
$70$	0	0
$71$	−12.7278	−1.51051	−0.755254	−	0.655432i	$-0.772487\pi$
$71$	−12.7278	−1.51051	−0.755254	+	0.655432i	$0.772487\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	4.72777	0.545916
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	−5.00000	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.32038	0.364459	0.182230	−	0.983256i	$-0.441669\pi$
$83$	3.32038	0.364459	0.182230	+	0.983256i	$0.441669\pi$
$84$	0	0
$85$	0.521753	0.0565921
$86$	0	0
$87$	1.47825	0.158485
$88$	0	0
$89$	8.84213	0.937264	0.468632	−	0.883393i	$-0.344747\pi$
$89$	8.84213	0.937264	0.468632	+	0.883393i	$0.344747\pi$
$90$	0	0
$91$	0.478247	0.0501339
$92$	0	0
$93$	−6.84213	−0.709496
$94$	0	0
$95$	−1.31573	−0.134991
$96$	0	0
$97$	7.20602	0.731660	0.365830	−	0.930682i	$-0.380785\pi$
$97$	7.20602	0.731660	0.365830	+	0.930682i	$0.380785\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	−0.592606	−0.0589665	−0.0294833	−	0.999565i	$-0.509386\pi$
$101$	−0.592606	−0.0589665	−0.0294833	+	0.999565i	$0.509386\pi$
$102$	0	0
$103$	−19.2930	−1.90100	−0.950500	−	0.310726i	$-0.899428\pi$
$103$	−19.2930	−1.90100	−0.950500	+	0.310726i	$0.899428\pi$
$104$	0	0
$105$	−0.521753	−0.0509179
$106$	0	0
$107$	2.04351	0.197553	0.0987766	−	0.995110i	$-0.468507\pi$
$107$	2.04351	0.197553	0.0987766	+	0.995110i	$0.468507\pi$
$108$	0	0
$109$	17.7713	1.70218	0.851090	−	0.525020i	$-0.175942\pi$
$109$	17.7713	1.70218	0.851090	+	0.525020i	$0.175942\pi$
$110$	0	0
$111$	−7.32038	−0.694820
$112$	0	0
$113$	11.2930	1.06236	0.531180	−	0.847259i	$-0.321749\pi$
$113$	11.2930	1.06236	0.531180	+	0.847259i	$0.321749\pi$
$114$	0	0
$115$	3.07085	0.286359
$116$	0	0
$117$	−0.956493	−0.0884278
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	2.52175	0.227379
$124$	0	0
$125$	−5.07550	−0.453966
$126$	0	0
$127$	7.56526	0.671308	0.335654	−	0.941985i	$-0.391043\pi$
$127$	7.56526	0.671308	0.335654	+	0.941985i	$0.391043\pi$
$128$	0	0
$129$	−2.84213	−0.250236
$130$	0	0
$131$	−21.6843	−1.89456	−0.947282	−	0.320402i	$-0.896182\pi$
$131$	−21.6843	−1.89456	−0.947282	+	0.320402i	$0.896182\pi$
$132$	0	0
$133$	−2.52175	−0.218664
$134$	0	0
$135$	2.60877	0.224527
$136$	0	0
$137$	6.04351	0.516332	0.258166	−	0.966101i	$-0.416882\pi$
$137$	6.04351	0.516332	0.258166	+	0.966101i	$0.416882\pi$
$138$	0	0
$139$	−12.7278	−1.07956	−0.539778	−	0.841808i	$-0.681492\pi$
$139$	−12.7278	−1.07956	−0.539778	+	0.841808i	$0.681492\pi$
$140$	0	0
$141$	9.88564	0.832521
$142$	0	0
$143$	−0.956493	−0.0799860
$144$	0	0
$145$	−0.771280	−0.0640513
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	−5.52175	−0.452360	−0.226180	−	0.974086i	$-0.572624\pi$
$149$	−5.52175	−0.452360	−0.226180	+	0.974086i	$0.572624\pi$
$150$	0	0
$151$	−16.3639	−1.33167	−0.665837	−	0.746097i	$-0.731925\pi$
$151$	−16.3639	−1.33167	−0.665837	+	0.746097i	$0.731925\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	3.56991	0.286742
$156$	0	0
$157$	9.20602	0.734720	0.367360	−	0.930079i	$-0.380262\pi$
$157$	9.20602	0.734720	0.367360	+	0.930079i	$0.380262\pi$
$158$	0	0
$159$	5.72777	0.454242
$160$	0	0
$161$	5.88564	0.463853
$162$	0	0
$163$	14.8148	1.16038	0.580192	−	0.814480i	$-0.302978\pi$
$163$	14.8148	1.16038	0.580192	+	0.814480i	$0.302978\pi$
$164$	0	0
$165$	1.04351	0.0812369
$166$	0	0
$167$	−14.7713	−1.14304	−0.571518	−	0.820590i	$-0.693645\pi$
$167$	−14.7713	−1.14304	−0.571518	+	0.820590i	$0.693645\pi$
$168$	0	0
$169$	−12.7713	−0.982406
$170$	0	0
$171$	5.04351	0.385687
$172$	0	0
$173$	−2.24953	−0.171028	−0.0855142	−	0.996337i	$-0.527253\pi$
$173$	−2.24953	−0.171028	−0.0855142	+	0.996337i	$0.527253\pi$
$174$	0	0
$175$	−4.72777	−0.357386
$176$	0	0
$177$	1.00000	0.0751646
$178$	0	0
$179$	−8.47825	−0.633694	−0.316847	−	0.948477i	$-0.602624\pi$
$179$	−8.47825	−0.633694	−0.316847	+	0.948477i	$0.602624\pi$
$180$	0	0
$181$	−22.9338	−1.70466	−0.852328	−	0.523008i	$-0.824810\pi$
$181$	−22.9338	−1.70466	−0.852328	+	0.523008i	$0.824810\pi$
$182$	0	0
$183$	2.47825	0.183197
$184$	0	0
$185$	3.81943	0.280810
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	5.00000	0.363696
$190$	0	0
$191$	−13.7986	−0.998434	−0.499217	−	0.866477i	$-0.666379\pi$
$191$	−13.7986	−0.998434	−0.499217	+	0.866477i	$0.666379\pi$
$192$	0	0
$193$	−0.434740	−0.0312932	−0.0156466	−	0.999878i	$-0.504981\pi$
$193$	−0.434740	−0.0312932	−0.0156466	+	0.999878i	$0.504981\pi$
$194$	0	0
$195$	−0.249527	−0.0178690
$196$	0	0
$197$	−22.4991	−1.60299	−0.801496	−	0.598001i	$-0.795962\pi$
$197$	−22.4991	−1.60299	−0.801496	+	0.598001i	$0.795962\pi$
$198$	0	0
$199$	1.72777	0.122479	0.0612393	−	0.998123i	$-0.480495\pi$
$199$	1.72777	0.122479	0.0612393	+	0.998123i	$0.480495\pi$
$200$	0	0
$201$	13.7713	0.971351
$202$	0	0
$203$	−1.47825	−0.103753
$204$	0	0
$205$	−1.31573	−0.0918948
$206$	0	0
$207$	−11.7713	−0.818161
$208$	0	0
$209$	5.04351	0.348867
$210$	0	0
$211$	−6.92915	−0.477022	−0.238511	−	0.971140i	$-0.576659\pi$
$211$	−6.92915	−0.477022	−0.238511	+	0.971140i	$0.576659\pi$
$212$	0	0
$213$	12.7278	0.872093
$214$	0	0
$215$	1.48289	0.101132
$216$	0	0
$217$	6.84213	0.464474
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	0.478247	0.0321703
$222$	0	0
$223$	−25.6843	−1.71995	−0.859973	−	0.510340i	$-0.829520\pi$
$223$	−25.6843	−1.71995	−0.859973	+	0.510340i	$0.829520\pi$
$224$	0	0
$225$	9.45555	0.630370
$226$	0	0
$227$	23.4074	1.55360	0.776802	−	0.629745i	$-0.216841\pi$
$227$	23.4074	1.55360	0.776802	+	0.629745i	$0.216841\pi$
$228$	0	0
$229$	9.09166	0.600794	0.300397	−	0.953814i	$-0.402881\pi$
$229$	9.09166	0.600794	0.300397	+	0.953814i	$0.402881\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	−12.2769	−0.804285	−0.402142	−	0.915577i	$-0.631734\pi$
$233$	−12.2769	−0.804285	−0.402142	+	0.915577i	$0.631734\pi$
$234$	0	0
$235$	−5.15787	−0.336462
$236$	0	0
$237$	5.00000	0.324785
$238$	0	0
$239$	−4.52175	−0.292488	−0.146244	−	0.989249i	$-0.546718\pi$
$239$	−4.52175	−0.292488	−0.146244	+	0.989249i	$0.546718\pi$
$240$	0	0
$241$	−15.5653	−1.00265	−0.501323	−	0.865260i	$-0.667153\pi$
$241$	−15.5653	−1.00265	−0.501323	+	0.865260i	$0.667153\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	−3.13052	−0.200002
$246$	0	0
$247$	−1.20602	−0.0767372
$248$	0	0
$249$	−3.32038	−0.210421
$250$	0	0
$251$	13.1625	0.830810	0.415405	−	0.909636i	$-0.363640\pi$
$251$	13.1625	0.830810	0.415405	+	0.909636i	$0.363640\pi$
$252$	0	0
$253$	−11.7713	−0.740054
$254$	0	0
$255$	−0.521753	−0.0326735
$256$	0	0
$257$	18.6843	1.16549	0.582746	−	0.812654i	$-0.301978\pi$
$257$	18.6843	1.16549	0.582746	+	0.812654i	$0.301978\pi$
$258$	0	0
$259$	7.32038	0.454866
$260$	0	0
$261$	2.95649	0.183002
$262$	0	0
$263$	−0.521753	−0.0321727	−0.0160863	−	0.999871i	$-0.505121\pi$
$263$	−0.521753	−0.0321727	−0.0160863	+	0.999871i	$0.505121\pi$
$264$	0	0
$265$	−2.98849	−0.183581
$266$	0	0
$267$	−8.84213	−0.541130
$268$	0	0
$269$	−10.1352	−0.617952	−0.308976	−	0.951070i	$-0.599986\pi$
$269$	−10.1352	−0.617952	−0.308976	+	0.951070i	$0.599986\pi$
$270$	0	0
$271$	13.0208	0.790958	0.395479	−	0.918475i	$-0.370579\pi$
$271$	13.0208	0.790958	0.395479	+	0.918475i	$0.370579\pi$
$272$	0	0
$273$	−0.478247	−0.0289448
$274$	0	0
$275$	9.45555	0.570191
$276$	0	0
$277$	9.47825	0.569493	0.284746	−	0.958603i	$-0.408091\pi$
$277$	9.47825	0.569493	0.284746	+	0.958603i	$0.408091\pi$
$278$	0	0
$279$	−13.6843	−0.819256
$280$	0	0
$281$	−27.1833	−1.62162	−0.810810	−	0.585309i	$-0.800973\pi$
$281$	−27.1833	−1.62162	−0.810810	+	0.585309i	$0.800973\pi$
$282$	0	0
$283$	20.9773	1.24697	0.623486	−	0.781835i	$-0.285716\pi$
$283$	20.9773	1.24697	0.623486	+	0.781835i	$0.285716\pi$
$284$	0	0
$285$	1.31573	0.0779373
$286$	0	0
$287$	−2.52175	−0.148854
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−7.20602	−0.422424
$292$	0	0
$293$	−24.8583	−1.45224	−0.726119	−	0.687570i	$-0.758678\pi$
$293$	−24.8583	−1.45224	−0.726119	+	0.687570i	$0.758678\pi$
$294$	0	0
$295$	−0.521753	−0.0303777
$296$	0	0
$297$	−10.0000	−0.580259
$298$	0	0
$299$	2.81479	0.162783
$300$	0	0
$301$	2.84213	0.163818
$302$	0	0
$303$	0.592606	0.0340444
$304$	0	0
$305$	−1.29303	−0.0740389
$306$	0	0
$307$	28.2060	1.60980	0.804901	−	0.593409i	$-0.202218\pi$
$307$	28.2060	1.60980	0.804901	+	0.593409i	$0.202218\pi$
$308$	0	0
$309$	19.2930	1.09754
$310$	0	0
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	0	0
$313$	10.4782	0.592266	0.296133	−	0.955147i	$-0.404303\pi$
$313$	10.4782	0.592266	0.296133	+	0.955147i	$0.404303\pi$
$314$	0	0
$315$	−1.04351	−0.0587950
$316$	0	0
$317$	16.8148	0.944413	0.472206	−	0.881488i	$-0.343458\pi$
$317$	16.8148	0.944413	0.472206	+	0.881488i	$0.343458\pi$
$318$	0	0
$319$	2.95649	0.165532
$320$	0	0
$321$	−2.04351	−0.114057
$322$	0	0
$323$	−2.52175	−0.140314
$324$	0	0
$325$	−2.26104	−0.125420
$326$	0	0
$327$	−17.7713	−0.982754
$328$	0	0
$329$	−9.88564	−0.545013
$330$	0	0
$331$	−22.5218	−1.23791	−0.618954	−	0.785427i	$-0.712443\pi$
$331$	−22.5218	−1.23791	−0.618954	+	0.785427i	$0.712443\pi$
$332$	0	0
$333$	−14.6408	−0.802309
$334$	0	0
$335$	−7.18521	−0.392570
$336$	0	0
$337$	−10.6408	−0.579639	−0.289820	−	0.957081i	$-0.593595\pi$
$337$	−10.6408	−0.579639	−0.289820	+	0.957081i	$0.593595\pi$
$338$	0	0
$339$	−11.2930	−0.613353
$340$	0	0
$341$	−13.6843	−0.741045
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	−3.07085	−0.165329
$346$	0	0
$347$	−35.8194	−1.92289	−0.961444	−	0.275001i	$-0.911322\pi$
$347$	−35.8194	−1.92289	−0.961444	+	0.275001i	$0.911322\pi$
$348$	0	0
$349$	−5.40739	−0.289451	−0.144726	−	0.989472i	$-0.546230\pi$
$349$	−5.40739	−0.289451	−0.144726	+	0.989472i	$0.546230\pi$
$350$	0	0
$351$	2.39123	0.127635
$352$	0	0
$353$	8.27687	0.440533	0.220267	−	0.975440i	$-0.429307\pi$
$353$	8.27687	0.440533	0.220267	+	0.975440i	$0.429307\pi$
$354$	0	0
$355$	−6.64076	−0.352455
$356$	0	0
$357$	−1.00000	−0.0529256
$358$	0	0
$359$	−15.2495	−0.804839	−0.402420	−	0.915455i	$-0.631831\pi$
$359$	−15.2495	−0.804839	−0.402420	+	0.915455i	$0.631831\pi$
$360$	0	0
$361$	−12.6408	−0.665303
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	1.04351	0.0546196
$366$	0	0
$367$	15.8194	0.825768	0.412884	−	0.910784i	$-0.364522\pi$
$367$	15.8194	0.825768	0.412884	+	0.910784i	$0.364522\pi$
$368$	0	0
$369$	5.04351	0.262554
$370$	0	0
$371$	−5.72777	−0.297371
$372$	0	0
$373$	6.72777	0.348351	0.174175	−	0.984715i	$-0.444274\pi$
$373$	6.72777	0.348351	0.174175	+	0.984715i	$0.444274\pi$
$374$	0	0
$375$	5.07550	0.262098
$376$	0	0
$377$	−0.706966	−0.0364106
$378$	0	0
$379$	−13.0870	−0.672235	−0.336117	−	0.941820i	$-0.609114\pi$
$379$	−13.0870	−0.672235	−0.336117	+	0.941820i	$0.609114\pi$
$380$	0	0
$381$	−7.56526	−0.387580
$382$	0	0
$383$	16.4991	0.843062	0.421531	−	0.906814i	$-0.361493\pi$
$383$	16.4991	0.843062	0.421531	+	0.906814i	$0.361493\pi$
$384$	0	0
$385$	−1.04351	−0.0531820
$386$	0	0
$387$	−5.68427	−0.288948
$388$	0	0
$389$	−15.6843	−0.795224	−0.397612	−	0.917554i	$-0.630161\pi$
$389$	−15.6843	−0.795224	−0.397612	+	0.917554i	$0.630161\pi$
$390$	0	0
$391$	5.88564	0.297650
$392$	0	0
$393$	21.6843	1.09383
$394$	0	0
$395$	−2.60877	−0.131261
$396$	0	0
$397$	34.9773	1.75546	0.877730	−	0.479155i	$-0.159057\pi$
$397$	34.9773	1.75546	0.877730	+	0.479155i	$0.159057\pi$
$398$	0	0
$399$	2.52175	0.126246
$400$	0	0
$401$	−20.2495	−1.01121	−0.505607	−	0.862764i	$-0.668731\pi$
$401$	−20.2495	−1.01121	−0.505607	+	0.862764i	$0.668731\pi$
$402$	0	0
$403$	3.27223	0.163001
$404$	0	0
$405$	0.521753	0.0259261
$406$	0	0
$407$	−14.6408	−0.725716
$408$	0	0
$409$	24.2495	1.19906	0.599531	−	0.800352i	$-0.295354\pi$
$409$	24.2495	1.19906	0.599531	+	0.800352i	$0.295354\pi$
$410$	0	0
$411$	−6.04351	−0.298104
$412$	0	0
$413$	−1.00000	−0.0492068
$414$	0	0
$415$	1.73242	0.0850411
$416$	0	0
$417$	12.7278	0.623282
$418$	0	0
$419$	−33.2060	−1.62222	−0.811110	−	0.584893i	$-0.801136\pi$
$419$	−33.2060	−1.62222	−0.811110	+	0.584893i	$0.801136\pi$
$420$	0	0
$421$	31.3412	1.52748	0.763738	−	0.645526i	$-0.223362\pi$
$421$	31.3412	1.52748	0.763738	+	0.645526i	$0.223362\pi$
$422$	0	0
$423$	19.7713	0.961313
$424$	0	0
$425$	−4.72777	−0.229331
$426$	0	0
$427$	−2.47825	−0.119931
$428$	0	0
$429$	0.956493	0.0461799
$430$	0	0
$431$	12.2287	0.589037	0.294518	−	0.955646i	$-0.404841\pi$
$431$	12.2287	0.589037	0.294518	+	0.955646i	$0.404841\pi$
$432$	0	0
$433$	−26.6843	−1.28236	−0.641182	−	0.767389i	$-0.721556\pi$
$433$	−26.6843	−1.28236	−0.641182	+	0.767389i	$0.721556\pi$
$434$	0	0
$435$	0.771280	0.0369801
$436$	0	0
$437$	−14.8421	−0.709996
$438$	0	0
$439$	4.72777	0.225644	0.112822	−	0.993615i	$-0.464011\pi$
$439$	4.72777	0.225644	0.112822	+	0.993615i	$0.464011\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	20.7551	0.986105	0.493053	−	0.869999i	$-0.335881\pi$
$443$	20.7551	0.986105	0.493053	+	0.869999i	$0.335881\pi$
$444$	0	0
$445$	4.61341	0.218697
$446$	0	0
$447$	5.52175	0.261170
$448$	0	0
$449$	−13.9773	−0.659630	−0.329815	−	0.944046i	$-0.606986\pi$
$449$	−13.9773	−0.659630	−0.329815	+	0.944046i	$0.606986\pi$
$450$	0	0
$451$	5.04351	0.237489
$452$	0	0
$453$	16.3639	0.768842
$454$	0	0
$455$	0.249527	0.0116980
$456$	0	0
$457$	1.81014	0.0846748	0.0423374	−	0.999103i	$-0.486520\pi$
$457$	1.81014	0.0846748	0.0423374	+	0.999103i	$0.486520\pi$
$458$	0	0
$459$	5.00000	0.233380
$460$	0	0
$461$	35.6843	1.66198	0.830991	−	0.556286i	$-0.187774\pi$
$461$	35.6843	1.66198	0.830991	+	0.556286i	$0.187774\pi$
$462$	0	0
$463$	−10.0597	−0.467512	−0.233756	−	0.972295i	$-0.575102\pi$
$463$	−10.0597	−0.467512	−0.233756	+	0.972295i	$0.575102\pi$
$464$	0	0
$465$	−3.56991	−0.165550
$466$	0	0
$467$	9.04351	0.418484	0.209242	−	0.977864i	$-0.432900\pi$
$467$	9.04351	0.418484	0.209242	+	0.977864i	$0.432900\pi$
$468$	0	0
$469$	−13.7713	−0.635899
$470$	0	0
$471$	−9.20602	−0.424191
$472$	0	0
$473$	−5.68427	−0.261363
$474$	0	0
$475$	11.9223	0.547032
$476$	0	0
$477$	11.4555	0.524513
$478$	0	0
$479$	−23.5973	−1.07819	−0.539093	−	0.842246i	$-0.681233\pi$
$479$	−23.5973	−1.07819	−0.539093	+	0.842246i	$0.681233\pi$
$480$	0	0
$481$	3.50095	0.159629
$482$	0	0
$483$	−5.88564	−0.267806
$484$	0	0
$485$	3.75977	0.170722
$486$	0	0
$487$	−36.2268	−1.64159	−0.820797	−	0.571220i	$-0.806470\pi$
$487$	−36.2268	−1.64159	−0.820797	+	0.571220i	$0.806470\pi$
$488$	0	0
$489$	−14.8148	−0.669948
$490$	0	0
$491$	−5.07550	−0.229054	−0.114527	−	0.993420i	$-0.536535\pi$
$491$	−5.07550	−0.229054	−0.114527	+	0.993420i	$0.536535\pi$
$492$	0	0
$493$	−1.47825	−0.0665769
$494$	0	0
$495$	2.08701	0.0938043
$496$	0	0
$497$	−12.7278	−0.570919
$498$	0	0
$499$	14.7713	0.661253	0.330627	−	0.943762i	$-0.392740\pi$
$499$	14.7713	0.661253	0.330627	+	0.943762i	$0.392740\pi$
$500$	0	0
$501$	14.7713	0.659932
$502$	0	0
$503$	−43.0046	−1.91748	−0.958741	−	0.284280i	$-0.908245\pi$
$503$	−43.0046	−1.91748	−0.958741	+	0.284280i	$0.908245\pi$
$504$	0	0
$505$	−0.309194	−0.0137590
$506$	0	0
$507$	12.7713	0.567192
$508$	0	0
$509$	5.60877	0.248604	0.124302	−	0.992244i	$-0.460331\pi$
$509$	5.60877	0.248604	0.124302	+	0.992244i	$0.460331\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	−12.6088	−0.556691
$514$	0	0
$515$	−10.0662	−0.443570
$516$	0	0
$517$	19.7713	0.869540
$518$	0	0
$519$	2.24953	0.0987433
$520$	0	0
$521$	19.6843	0.862383	0.431192	−	0.902260i	$-0.358093\pi$
$521$	19.6843	0.862383	0.431192	+	0.902260i	$0.358093\pi$
$522$	0	0
$523$	−40.0643	−1.75189	−0.875945	−	0.482411i	$-0.839761\pi$
$523$	−40.0643	−1.75189	−0.875945	+	0.482411i	$0.839761\pi$
$524$	0	0
$525$	4.72777	0.206337
$526$	0	0
$527$	6.84213	0.298048
$528$	0	0
$529$	11.6408	0.506120
$530$	0	0
$531$	2.00000	0.0867926
$532$	0	0
$533$	−1.20602	−0.0522385
$534$	0	0
$535$	1.06621	0.0460961
$536$	0	0
$537$	8.47825	0.365863
$538$	0	0
$539$	12.0000	0.516877
$540$	0	0
$541$	41.0255	1.76382	0.881911	−	0.471416i	$-0.156257\pi$
$541$	41.0255	1.76382	0.881911	+	0.471416i	$0.156257\pi$
$542$	0	0
$543$	22.9338	0.984183
$544$	0	0
$545$	9.27223	0.397179
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	0	0
$549$	4.95649	0.211538
$550$	0	0
$551$	3.72777	0.158808
$552$	0	0
$553$	−5.00000	−0.212622
$554$	0	0
$555$	−3.81943	−0.162126
$556$	0	0
$557$	13.8695	0.587669	0.293834	−	0.955856i	$-0.405069\pi$
$557$	13.8695	0.587669	0.293834	+	0.955856i	$0.405069\pi$
$558$	0	0
$559$	1.35924	0.0574897
$560$	0	0
$561$	2.00000	0.0844401
$562$	0	0
$563$	−13.6843	−0.576723	−0.288362	−	0.957522i	$-0.593110\pi$
$563$	−13.6843	−0.576723	−0.288362	+	0.957522i	$0.593110\pi$
$564$	0	0
$565$	5.89218	0.247886
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	35.8194	1.50163	0.750814	−	0.660513i	$-0.229661\pi$
$569$	35.8194	1.50163	0.750814	+	0.660513i	$0.229661\pi$
$570$	0	0
$571$	−34.3185	−1.43618	−0.718092	−	0.695948i	$-0.754984\pi$
$571$	−34.3185	−1.43618	−0.718092	+	0.695948i	$0.754984\pi$
$572$	0	0
$573$	13.7986	0.576446
$574$	0	0
$575$	−27.8260	−1.16042
$576$	0	0
$577$	25.0000	1.04076	0.520382	−	0.853934i	$-0.325790\pi$
$577$	25.0000	1.04076	0.520382	+	0.853934i	$0.325790\pi$
$578$	0	0
$579$	0.434740	0.0180672
$580$	0	0
$581$	3.32038	0.137753
$582$	0	0
$583$	11.4555	0.474440
$584$	0	0
$585$	−0.499054	−0.0206333
$586$	0	0
$587$	30.7004	1.26714	0.633571	−	0.773684i	$-0.281588\pi$
$587$	30.7004	1.26714	0.633571	+	0.773684i	$0.281588\pi$
$588$	0	0
$589$	−17.2542	−0.710946
$590$	0	0
$591$	22.4991	0.925487
$592$	0	0
$593$	−35.2703	−1.44838	−0.724190	−	0.689601i	$-0.757786\pi$
$593$	−35.2703	−1.44838	−0.724190	+	0.689601i	$0.757786\pi$
$594$	0	0
$595$	0.521753	0.0213898
$596$	0	0
$597$	−1.72777	−0.0707131
$598$	0	0
$599$	−23.9773	−0.979686	−0.489843	−	0.871811i	$-0.662946\pi$
$599$	−23.9773	−0.979686	−0.489843	+	0.871811i	$0.662946\pi$
$600$	0	0
$601$	32.6731	1.33276	0.666381	−	0.745611i	$-0.267842\pi$
$601$	32.6731	1.33276	0.666381	+	0.745611i	$0.267842\pi$
$602$	0	0
$603$	27.5426	1.12162
$604$	0	0
$605$	−3.65227	−0.148486
$606$	0	0
$607$	38.6296	1.56793	0.783963	−	0.620808i	$-0.213195\pi$
$607$	38.6296	1.56793	0.783963	+	0.620808i	$0.213195\pi$
$608$	0	0
$609$	1.47825	0.0599016
$610$	0	0
$611$	−4.72777	−0.191265
$612$	0	0
$613$	−9.24488	−0.373397	−0.186699	−	0.982417i	$-0.559779\pi$
$613$	−9.24488	−0.373397	−0.186699	+	0.982417i	$0.559779\pi$
$614$	0	0
$615$	1.31573	0.0530555
$616$	0	0
$617$	7.97730	0.321154	0.160577	−	0.987023i	$-0.448665\pi$
$617$	7.97730	0.321154	0.160577	+	0.987023i	$0.448665\pi$
$618$	0	0
$619$	23.8148	0.957197	0.478599	−	0.878034i	$-0.341145\pi$
$619$	23.8148	0.957197	0.478599	+	0.878034i	$0.341145\pi$
$620$	0	0
$621$	29.4282	1.18091
$622$	0	0
$623$	8.84213	0.354253
$624$	0	0
$625$	20.9907	0.839628
$626$	0	0
$627$	−5.04351	−0.201418
$628$	0	0
$629$	7.32038	0.291883
$630$	0	0
$631$	25.2268	1.00426	0.502132	−	0.864791i	$-0.332549\pi$
$631$	25.2268	1.00426	0.502132	+	0.864791i	$0.332549\pi$
$632$	0	0
$633$	6.92915	0.275409
$634$	0	0
$635$	3.94720	0.156640
$636$	0	0
$637$	−2.86948	−0.113693
$638$	0	0
$639$	25.4555	1.00701
$640$	0	0
$641$	40.4120	1.59618	0.798090	−	0.602539i	$-0.205844\pi$
$641$	40.4120	1.59618	0.798090	+	0.602539i	$0.205844\pi$
$642$	0	0
$643$	−46.0528	−1.81615	−0.908073	−	0.418813i	$-0.862446\pi$
$643$	−46.0528	−1.81615	−0.908073	+	0.418813i	$0.862446\pi$
$644$	0	0
$645$	−1.48289	−0.0583888
$646$	0	0
$647$	−30.3800	−1.19436	−0.597181	−	0.802106i	$-0.703713\pi$
$647$	−30.3800	−1.19436	−0.597181	+	0.802106i	$0.703713\pi$
$648$	0	0
$649$	2.00000	0.0785069
$650$	0	0
$651$	−6.84213	−0.268164
$652$	0	0
$653$	−6.69578	−0.262026	−0.131013	−	0.991381i	$-0.541823\pi$
$653$	−6.69578	−0.262026	−0.131013	+	0.991381i	$0.541823\pi$
$654$	0	0
$655$	−11.3138	−0.442068
$656$	0	0
$657$	−4.00000	−0.156055
$658$	0	0
$659$	29.2657	1.14003	0.570015	−	0.821635i	$-0.306938\pi$
$659$	29.2657	1.14003	0.570015	+	0.821635i	$0.306938\pi$
$660$	0	0
$661$	10.5973	0.412185	0.206093	−	0.978532i	$-0.433925\pi$
$661$	10.5973	0.412185	0.206093	+	0.978532i	$0.433925\pi$
$662$	0	0
$663$	−0.478247	−0.0185736
$664$	0	0
$665$	−1.31573	−0.0510220
$666$	0	0
$667$	−8.70043	−0.336882
$668$	0	0
$669$	25.6843	0.993011
$670$	0	0
$671$	4.95649	0.191343
$672$	0	0
$673$	40.1560	1.54790	0.773950	−	0.633247i	$-0.218278\pi$
$673$	40.1560	1.54790	0.773950	+	0.633247i	$0.218278\pi$
$674$	0	0
$675$	−23.6389	−0.909860
$676$	0	0
$677$	−38.4991	−1.47964	−0.739819	−	0.672805i	$-0.765089\pi$
$677$	−38.4991	−1.47964	−0.739819	+	0.672805i	$0.765089\pi$
$678$	0	0
$679$	7.20602	0.276542
$680$	0	0
$681$	−23.4074	−0.896973
$682$	0	0
$683$	20.6616	0.790593	0.395296	−	0.918554i	$-0.370642\pi$
$683$	20.6616	0.790593	0.395296	+	0.918554i	$0.370642\pi$
$684$	0	0
$685$	3.15322	0.120478
$686$	0	0
$687$	−9.09166	−0.346868
$688$	0	0
$689$	−2.73929	−0.104359
$690$	0	0
$691$	−25.2542	−0.960714	−0.480357	−	0.877073i	$-0.659493\pi$
$691$	−25.2542	−0.960714	−0.480357	+	0.877073i	$0.659493\pi$
$692$	0	0
$693$	4.00000	0.151947
$694$	0	0
$695$	−6.64076	−0.251898
$696$	0	0
$697$	−2.52175	−0.0955182
$698$	0	0
$699$	12.2769	0.464354
$700$	0	0
$701$	36.2977	1.37094	0.685472	−	0.728099i	$-0.259596\pi$
$701$	36.2977	1.37094	0.685472	+	0.728099i	$0.259596\pi$
$702$	0	0
$703$	−18.4602	−0.696239
$704$	0	0
$705$	5.15787	0.194257
$706$	0	0
$707$	−0.592606	−0.0222873
$708$	0	0
$709$	14.7505	0.553966	0.276983	−	0.960875i	$-0.410665\pi$
$709$	14.7505	0.553966	0.276983	+	0.960875i	$0.410665\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	0	0
$713$	40.2703	1.50814
$714$	0	0
$715$	−0.499054	−0.0186635
$716$	0	0
$717$	4.52175	0.168868
$718$	0	0
$719$	4.73929	0.176746	0.0883728	−	0.996087i	$-0.471833\pi$
$719$	4.73929	0.176746	0.0883728	+	0.996087i	$0.471833\pi$
$720$	0	0
$721$	−19.2930	−0.718510
$722$	0	0
$723$	15.5653	0.578878
$724$	0	0
$725$	6.98881	0.259558
$726$	0	0
$727$	20.4991	0.760268	0.380134	−	0.924931i	$-0.375878\pi$
$727$	20.4991	0.760268	0.380134	+	0.924931i	$0.375878\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	2.84213	0.105120
$732$	0	0
$733$	−40.1833	−1.48420	−0.742102	−	0.670287i	$-0.766171\pi$
$733$	−40.1833	−1.48420	−0.742102	+	0.670287i	$0.766171\pi$
$734$	0	0
$735$	3.13052	0.115471
$736$	0	0
$737$	27.5426	1.01454
$738$	0	0
$739$	33.8791	1.24626	0.623131	−	0.782117i	$-0.285860\pi$
$739$	33.8791	1.24626	0.623131	+	0.782117i	$0.285860\pi$
$740$	0	0
$741$	1.20602	0.0443042
$742$	0	0
$743$	3.36853	0.123579	0.0617897	−	0.998089i	$-0.480319\pi$
$743$	3.36853	0.123579	0.0617897	+	0.998089i	$0.480319\pi$
$744$	0	0
$745$	−2.88099	−0.105551
$746$	0	0
$747$	−6.64076	−0.242973
$748$	0	0
$749$	2.04351	0.0746681
$750$	0	0
$751$	−39.6843	−1.44810	−0.724050	−	0.689748i	$-0.757721\pi$
$751$	−39.6843	−1.44810	−0.724050	+	0.689748i	$0.757721\pi$
$752$	0	0
$753$	−13.1625	−0.479669
$754$	0	0
$755$	−8.53791	−0.310727
$756$	0	0
$757$	−45.8148	−1.66517	−0.832583	−	0.553900i	$-0.813139\pi$
$757$	−45.8148	−1.66517	−0.832583	+	0.553900i	$0.813139\pi$
$758$	0	0
$759$	11.7713	0.427270
$760$	0	0
$761$	26.2268	0.950722	0.475361	−	0.879791i	$-0.342318\pi$
$761$	26.2268	0.950722	0.475361	+	0.879791i	$0.342318\pi$
$762$	0	0
$763$	17.7713	0.643364
$764$	0	0
$765$	−1.04351	−0.0377281
$766$	0	0
$767$	−0.478247	−0.0172685
$768$	0	0
$769$	9.52175	0.343363	0.171682	−	0.985152i	$-0.445080\pi$
$769$	9.52175	0.343363	0.171682	+	0.985152i	$0.445080\pi$
$770$	0	0
$771$	−18.6843	−0.672897
$772$	0	0
$773$	25.5907	0.920434	0.460217	−	0.887806i	$-0.347772\pi$
$773$	25.5907	0.920434	0.460217	+	0.887806i	$0.347772\pi$
$774$	0	0
$775$	−32.3481	−1.16198
$776$	0	0
$777$	−7.32038	−0.262617
$778$	0	0
$779$	6.35924	0.227843
$780$	0	0
$781$	25.4555	0.910871
$782$	0	0
$783$	−7.39123	−0.264141
$784$	0	0
$785$	4.80327	0.171436
$786$	0	0
$787$	4.49905	0.160374	0.0801870	−	0.996780i	$-0.474448\pi$
$787$	4.49905	0.160374	0.0801870	+	0.996780i	$0.474448\pi$
$788$	0	0
$789$	0.521753	0.0185749
$790$	0	0
$791$	11.2930	0.401534
$792$	0	0
$793$	−1.18521	−0.0420881
$794$	0	0
$795$	2.98849	0.105991
$796$	0	0
$797$	−10.0755	−0.356892	−0.178446	−	0.983950i	$-0.557107\pi$
$797$	−10.0755	−0.356892	−0.178446	+	0.983950i	$0.557107\pi$
$798$	0	0
$799$	−9.88564	−0.349729
$800$	0	0
$801$	−17.6843	−0.624843
$802$	0	0
$803$	−4.00000	−0.141157
$804$	0	0
$805$	3.07085	0.108233
$806$	0	0
$807$	10.1352	0.356775
$808$	0	0
$809$	34.8629	1.22572	0.612858	−	0.790193i	$-0.290020\pi$
$809$	34.8629	1.22572	0.612858	+	0.790193i	$0.290020\pi$
$810$	0	0
$811$	26.6889	0.937174	0.468587	−	0.883417i	$-0.344763\pi$
$811$	26.6889	0.937174	0.468587	+	0.883417i	$0.344763\pi$
$812$	0	0
$813$	−13.0208	−0.456660
$814$	0	0
$815$	7.72967	0.270758
$816$	0	0
$817$	−7.16716	−0.250747
$818$	0	0
$819$	−0.956493	−0.0334226
$820$	0	0
$821$	−10.0482	−0.350683	−0.175341	−	0.984508i	$-0.556103\pi$
$821$	−10.0482	−0.350683	−0.175341	+	0.984508i	$0.556103\pi$
$822$	0	0
$823$	39.2060	1.36664	0.683318	−	0.730121i	$-0.260536\pi$
$823$	39.2060	1.36664	0.683318	+	0.730121i	$0.260536\pi$
$824$	0	0
$825$	−9.45555	−0.329200
$826$	0	0
$827$	−22.8148	−0.793348	−0.396674	−	0.917960i	$-0.629836\pi$
$827$	−22.8148	−0.793348	−0.396674	+	0.917960i	$0.629836\pi$
$828$	0	0
$829$	−23.0963	−0.802168	−0.401084	−	0.916041i	$-0.631366\pi$
$829$	−23.0963	−0.802168	−0.401084	+	0.916041i	$0.631366\pi$
$830$	0	0
$831$	−9.47825	−0.328797
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	−7.70697	−0.266711
$836$	0	0
$837$	34.2107	1.18249
$838$	0	0
$839$	−13.5491	−0.467767	−0.233883	−	0.972265i	$-0.575143\pi$
$839$	−13.5491	−0.467767	−0.233883	+	0.972265i	$0.575143\pi$
$840$	0	0
$841$	−26.8148	−0.924648
$842$	0	0
$843$	27.1833	0.936243
$844$	0	0
$845$	−6.66346	−0.229230
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	−20.9773	−0.719939
$850$	0	0
$851$	43.0851	1.47694
$852$	0	0
$853$	24.0320	0.822840	0.411420	−	0.911446i	$-0.365033\pi$
$853$	24.0320	0.822840	0.411420	+	0.911446i	$0.365033\pi$
$854$	0	0
$855$	2.63147	0.0899943
$856$	0	0
$857$	33.5699	1.14673	0.573363	−	0.819301i	$-0.305638\pi$
$857$	33.5699	1.14673	0.573363	+	0.819301i	$0.305638\pi$
$858$	0	0
$859$	32.4717	1.10792	0.553960	−	0.832543i	$-0.313116\pi$
$859$	32.4717	1.10792	0.553960	+	0.832543i	$0.313116\pi$
$860$	0	0
$861$	2.52175	0.0859411
$862$	0	0
$863$	1.50095	0.0510928	0.0255464	−	0.999674i	$-0.491867\pi$
$863$	1.50095	0.0510928	0.0255464	+	0.999674i	$0.491867\pi$
$864$	0	0
$865$	−1.17370	−0.0399069
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	10.0000	0.339227
$870$	0	0
$871$	−6.58607	−0.223160
$872$	0	0
$873$	−14.4120	−0.487774
$874$	0	0
$875$	−5.07550	−0.171583
$876$	0	0
$877$	1.61995	0.0547019	0.0273510	−	0.999626i	$-0.491293\pi$
$877$	1.61995	0.0547019	0.0273510	+	0.999626i	$0.491293\pi$
$878$	0	0
$879$	24.8583	0.838449
$880$	0	0
$881$	−31.2657	−1.05337	−0.526684	−	0.850061i	$-0.676565\pi$
$881$	−31.2657	−1.05337	−0.526684	+	0.850061i	$0.676565\pi$
$882$	0	0
$883$	13.6616	0.459748	0.229874	−	0.973220i	$-0.426169\pi$
$883$	13.6616	0.459748	0.229874	+	0.973220i	$0.426169\pi$
$884$	0	0
$885$	0.521753	0.0175386
$886$	0	0
$887$	−6.90180	−0.231740	−0.115870	−	0.993264i	$-0.536966\pi$
$887$	−6.90180	−0.231740	−0.115870	+	0.993264i	$0.536966\pi$
$888$	0	0
$889$	7.56526	0.253731
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	24.9291	0.834222
$894$	0	0
$895$	−4.42355	−0.147863
$896$	0	0
$897$	−2.81479	−0.0939830
$898$	0	0
$899$	−10.1144	−0.337333
$900$	0	0
$901$	−5.72777	−0.190820
$902$	0	0
$903$	−2.84213	−0.0945803
$904$	0	0
$905$	−11.9658	−0.397756
$906$	0	0
$907$	24.9130	0.827222	0.413611	−	0.910454i	$-0.364267\pi$
$907$	24.9130	0.827222	0.413611	+	0.910454i	$0.364267\pi$
$908$	0	0
$909$	1.18521	0.0393110
$910$	0	0
$911$	41.4444	1.37311	0.686556	−	0.727076i	$-0.259121\pi$
$911$	41.4444	1.37311	0.686556	+	0.727076i	$0.259121\pi$
$912$	0	0
$913$	−6.64076	−0.219777
$914$	0	0
$915$	1.29303	0.0427464
$916$	0	0
$917$	−21.6843	−0.716078
$918$	0	0
$919$	0.180567	0.00595634	0.00297817	−	0.999996i	$-0.499052\pi$
$919$	0.180567	0.00595634	0.00297817	+	0.999996i	$0.499052\pi$
$920$	0	0
$921$	−28.2060	−0.929420
$922$	0	0
$923$	−6.08701	−0.200356
$924$	0	0
$925$	−34.6091	−1.13794
$926$	0	0
$927$	38.5861	1.26733
$928$	0	0
$929$	−15.3592	−0.503920	−0.251960	−	0.967738i	$-0.581075\pi$
$929$	−15.3592	−0.503920	−0.251960	+	0.967738i	$0.581075\pi$
$930$	0	0
$931$	15.1305	0.495883
$932$	0	0
$933$	15.0000	0.491078
$934$	0	0
$935$	−1.04351	−0.0341263
$936$	0	0
$937$	2.26104	0.0738650	0.0369325	−	0.999318i	$-0.488241\pi$
$937$	2.26104	0.0738650	0.0369325	+	0.999318i	$0.488241\pi$
$938$	0	0
$939$	−10.4782	−0.341945
$940$	0	0
$941$	−25.5310	−0.832288	−0.416144	−	0.909299i	$-0.636619\pi$
$941$	−25.5310	−0.832288	−0.416144	+	0.909299i	$0.636619\pi$
$942$	0	0
$943$	−14.8421	−0.483326
$944$	0	0
$945$	2.60877	0.0848632
$946$	0	0
$947$	−10.1305	−0.329198	−0.164599	−	0.986361i	$-0.552633\pi$
$947$	−10.1305	−0.329198	−0.164599	+	0.986361i	$0.552633\pi$
$948$	0	0
$949$	0.956493	0.0310491
$950$	0	0
$951$	−16.8148	−0.545257
$952$	0	0
$953$	−55.2268	−1.78897	−0.894486	−	0.447096i	$-0.852458\pi$
$953$	−55.2268	−1.78897	−0.894486	+	0.447096i	$0.852458\pi$
$954$	0	0
$955$	−7.19948	−0.232970
$956$	0	0
$957$	−2.95649	−0.0955698
$958$	0	0
$959$	6.04351	0.195155
$960$	0	0
$961$	15.8148	0.510154
$962$	0	0
$963$	−4.08701	−0.131702
$964$	0	0
$965$	−0.226827	−0.00730182
$966$	0	0
$967$	33.1787	1.06695	0.533477	−	0.845815i	$-0.320885\pi$
$967$	33.1787	1.06695	0.533477	+	0.845815i	$0.320885\pi$
$968$	0	0
$969$	2.52175	0.0810104
$970$	0	0
$971$	−5.97730	−0.191821	−0.0959103	−	0.995390i	$-0.530576\pi$
$971$	−5.97730	−0.191821	−0.0959103	+	0.995390i	$0.530576\pi$
$972$	0	0
$973$	−12.7278	−0.408034
$974$	0	0
$975$	2.26104	0.0724113
$976$	0	0
$977$	−44.7874	−1.43288	−0.716439	−	0.697650i	$-0.754229\pi$
$977$	−44.7874	−1.43288	−0.716439	+	0.697650i	$0.754229\pi$
$978$	0	0
$979$	−17.6843	−0.565192
$980$	0	0
$981$	−35.5426	−1.13479
$982$	0	0
$983$	18.9018	0.602874	0.301437	−	0.953486i	$-0.402534\pi$
$983$	18.9018	0.602874	0.301437	+	0.953486i	$0.402534\pi$
$984$	0	0
$985$	−11.7390	−0.374034
$986$	0	0
$987$	9.88564	0.314663
$988$	0	0
$989$	16.7278	0.531912
$990$	0	0
$991$	26.4509	0.840241	0.420120	−	0.907468i	$-0.361988\pi$
$991$	26.4509	0.840241	0.420120	+	0.907468i	$0.361988\pi$
$992$	0	0
$993$	22.5218	0.714706
$994$	0	0
$995$	0.901472	0.0285786
$996$	0	0
$997$	−3.21721	−0.101890	−0.0509450	−	0.998701i	$-0.516223\pi$
$997$	−3.21721	−0.101890	−0.0509450	+	0.998701i	$0.516223\pi$
$998$	0	0
$999$	36.6019	1.15803

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.f.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.f.1.2	✓	3	1.1	even	1	trivial

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Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.521753 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.521753 q^{5} +1.00000 q^{7} -2.00000 q^{9} -2.00000 q^{11} +0.478247 q^{13} -0.521753 q^{15} +1.00000 q^{17} -2.52175 q^{19} -1.00000 q^{21} +5.88564 q^{23} -4.72777 q^{25} +5.00000 q^{27} -1.47825 q^{29} +6.84213 q^{31} +2.00000 q^{33} +0.521753 q^{35} +7.32038 q^{37} -0.478247 q^{39} -2.52175 q^{41} +2.84213 q^{43} -1.04351 q^{45} -9.88564 q^{47} -6.00000 q^{49} -1.00000 q^{51} -5.72777 q^{53} -1.04351 q^{55} +2.52175 q^{57} -1.00000 q^{59} -2.47825 q^{61} -2.00000 q^{63} +0.249527 q^{65} -13.7713 q^{67} -5.88564 q^{69} -12.7278 q^{71} +2.00000 q^{73} +4.72777 q^{75} -2.00000 q^{77} -5.00000 q^{79} +1.00000 q^{81} +3.32038 q^{83} +0.521753 q^{85} +1.47825 q^{87} +8.84213 q^{89} +0.478247 q^{91} -6.84213 q^{93} -1.31573 q^{95} +7.20602 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + q^{5} + 3 q^{7} - 6 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + q^5 + 3 * q^7 - 6 * q^9	\( 3 q - 3 q^{3} + q^{5} + 3 q^{7} - 6 q^{9} - 6 q^{11} + 2 q^{13} - q^{15} + 3 q^{17} - 7 q^{19} - 3 q^{21} + 20 q^{25} + 15 q^{27} - 5 q^{29} + 4 q^{31} + 6 q^{33} + q^{35} + 6 q^{37} - 2 q^{39} - 7 q^{41} - 8 q^{43} - 2 q^{45} - 12 q^{47} - 18 q^{49} - 3 q^{51} + 17 q^{53} - 2 q^{55} + 7 q^{57} - 3 q^{59} - 8 q^{61} - 6 q^{63} - 34 q^{65} - 6 q^{67} - 4 q^{71} + 6 q^{73} - 20 q^{75} - 6 q^{77} - 15 q^{79} + 3 q^{81} - 6 q^{83} + q^{85} + 5 q^{87} + 10 q^{89} + 2 q^{91} - 4 q^{93} - 37 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + q^5 + 3 * q^7 - 6 * q^9 - 6 * q^11 + 2 * q^13 - q^15 + 3 * q^17 - 7 * q^19 - 3 * q^21 + 20 * q^25 + 15 * q^27 - 5 * q^29 + 4 * q^31 + 6 * q^33 + q^35 + 6 * q^37 - 2 * q^39 - 7 * q^41 - 8 * q^43 - 2 * q^45 - 12 * q^47 - 18 * q^49 - 3 * q^51 + 17 * q^53 - 2 * q^55 + 7 * q^57 - 3 * q^59 - 8 * q^61 - 6 * q^63 - 34 * q^65 - 6 * q^67 - 4 * q^71 + 6 * q^73 - 20 * q^75 - 6 * q^77 - 15 * q^79 + 3 * q^81 - 6 * q^83 + q^85 + 5 * q^87 + 10 * q^89 + 2 * q^91 - 4 * q^93 - 37 * q^95 - 12 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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