LMFDB - Embedded newform 4732.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4732,2,Mod(1,4732)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4732, 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("4732.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4732 = 2^{2} \cdot 7 \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4732.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:	$37.7852102365$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 3$ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 364)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	4732.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30278 q^{3} -0.302776 q^{5} -1.00000 q^{7} +2.30278 q^{9} +6.30278 q^{11} -0.697224 q^{15} -1.60555 q^{17} +3.69722 q^{19} -2.30278 q^{21} +2.00000 q^{23} -4.90833 q^{25} -1.60555 q^{27} -0.302776 q^{29} -0.394449 q^{31} +14.5139 q^{33} +0.302776 q^{35} +9.21110 q^{37} -1.39445 q^{41} -5.90833 q^{43} -0.697224 q^{45} +11.6056 q^{47} +1.00000 q^{49} -3.69722 q^{51} +10.8167 q^{53} -1.90833 q^{55} +8.51388 q^{57} -5.60555 q^{59} +7.21110 q^{61} -2.30278 q^{63} +13.0000 q^{67} +4.60555 q^{69} -9.81665 q^{71} +11.2111 q^{73} -11.3028 q^{75} -6.30278 q^{77} +8.00000 q^{79} -10.6056 q^{81} -4.39445 q^{83} +0.486122 q^{85} -0.697224 q^{87} +5.09167 q^{89} -0.908327 q^{93} -1.11943 q^{95} -6.09167 q^{97} +14.5139 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + q^{3} + 3 q^{5} - 2 q^{7} + q^{9} + 9 q^{11} - 5 q^{15} + 4 q^{17} + 11 q^{19} - q^{21} + 4 q^{23} + q^{25} + 4 q^{27} + 3 q^{29} - 8 q^{31} + 11 q^{33} - 3 q^{35} + 4 q^{37} - 10 q^{41} - q^{43}+ \cdots + 11 q^{99}+O(q^{100})$ 2 * q + q^3 + 3 * q^5 - 2 * q^7 + q^9 + 9 * q^11 - 5 * q^15 + 4 * q^17 + 11 * q^19 - q^21 + 4 * q^23 + q^25 + 4 * q^27 + 3 * q^29 - 8 * q^31 + 11 * q^33 - 3 * q^35 + 4 * q^37 - 10 * q^41 - q^43 - 5 * q^45 + 16 * q^47 + 2 * q^49 - 11 * q^51 + 7 * q^55 - q^57 - 4 * q^59 - q^63 + 26 * q^67 + 2 * q^69 + 2 * q^71 + 8 * q^73 - 19 * q^75 - 9 * q^77 + 16 * q^79 - 14 * q^81 - 16 * q^83 + 19 * q^85 - 5 * q^87 + 21 * q^89 + 9 * q^93 + 23 * q^95 - 23 * q^97 + 11 * q^99

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
$3$	2.30278	1.32951	0.664754	−	0.747062i	$-0.268536\pi$
$3$	2.30278	1.32951	0.664754	+	0.747062i	$0.268536\pi$
$4$	0	0
$5$	−0.302776	−0.135405	−0.0677027	−	0.997706i	$-0.521567\pi$
$5$	−0.302776	−0.135405	−0.0677027	+	0.997706i	$0.521567\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.30278	0.767592
$10$	0	0
$11$	6.30278	1.90036	0.950179	−	0.311704i	$-0.100900\pi$
$11$	6.30278	1.90036	0.950179	+	0.311704i	$0.100900\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−0.697224	−0.180023
$16$	0	0
$17$	−1.60555	−0.389403	−0.194702	−	0.980863i	$-0.562374\pi$
$17$	−1.60555	−0.389403	−0.194702	+	0.980863i	$0.562374\pi$
$18$	0	0
$19$	3.69722	0.848201	0.424101	−	0.905615i	$-0.360590\pi$
$19$	3.69722	0.848201	0.424101	+	0.905615i	$0.360590\pi$
$20$	0	0
$21$	−2.30278	−0.502507
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	−4.90833	−0.981665
$26$	0	0
$27$	−1.60555	−0.308988
$28$	0	0
$29$	−0.302776	−0.0562240	−0.0281120	−	0.999605i	$-0.508950\pi$
$29$	−0.302776	−0.0562240	−0.0281120	+	0.999605i	$0.508950\pi$
$30$	0	0
$31$	−0.394449	−0.0708451	−0.0354225	−	0.999372i	$-0.511278\pi$
$31$	−0.394449	−0.0708451	−0.0354225	+	0.999372i	$0.511278\pi$
$32$	0	0
$33$	14.5139	2.52654
$34$	0	0
$35$	0.302776	0.0511784
$36$	0	0
$37$	9.21110	1.51430	0.757148	−	0.653243i	$-0.226592\pi$
$37$	9.21110	1.51430	0.757148	+	0.653243i	$0.226592\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.39445	−0.217776	−0.108888	−	0.994054i	$-0.534729\pi$
$41$	−1.39445	−0.217776	−0.108888	+	0.994054i	$0.534729\pi$
$42$	0	0
$43$	−5.90833	−0.901011	−0.450506	−	0.892774i	$-0.648756\pi$
$43$	−5.90833	−0.901011	−0.450506	+	0.892774i	$0.648756\pi$
$44$	0	0
$45$	−0.697224	−0.103936
$46$	0	0
$47$	11.6056	1.69284	0.846422	−	0.532513i	$-0.178752\pi$
$47$	11.6056	1.69284	0.846422	+	0.532513i	$0.178752\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.69722	−0.517715
$52$	0	0
$53$	10.8167	1.48578	0.742891	−	0.669413i	$-0.233454\pi$
$53$	10.8167	1.48578	0.742891	+	0.669413i	$0.233454\pi$
$54$	0	0
$55$	−1.90833	−0.257319
$56$	0	0
$57$	8.51388	1.12769
$58$	0	0
$59$	−5.60555	−0.729781	−0.364890	−	0.931051i	$-0.618894\pi$
$59$	−5.60555	−0.729781	−0.364890	+	0.931051i	$0.618894\pi$
$60$	0	0
$61$	7.21110	0.923287	0.461644	−	0.887066i	$-0.347260\pi$
$61$	7.21110	0.923287	0.461644	+	0.887066i	$0.347260\pi$
$62$	0	0
$63$	−2.30278	−0.290122
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.0000	1.58820	0.794101	−	0.607785i	$-0.207942\pi$
$67$	13.0000	1.58820	0.794101	+	0.607785i	$0.207942\pi$
$68$	0	0
$69$	4.60555	0.554443
$70$	0	0
$71$	−9.81665	−1.16502	−0.582511	−	0.812823i	$-0.697930\pi$
$71$	−9.81665	−1.16502	−0.582511	+	0.812823i	$0.697930\pi$
$72$	0	0
$73$	11.2111	1.31216	0.656080	−	0.754691i	$-0.272213\pi$
$73$	11.2111	1.31216	0.656080	+	0.754691i	$0.272213\pi$
$74$	0	0
$75$	−11.3028	−1.30513
$76$	0	0
$77$	−6.30278	−0.718268
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	−10.6056	−1.17839
$82$	0	0
$83$	−4.39445	−0.482353	−0.241177	−	0.970481i	$-0.577533\pi$
$83$	−4.39445	−0.482353	−0.241177	+	0.970481i	$0.577533\pi$
$84$	0	0
$85$	0.486122	0.0527273
$86$	0	0
$87$	−0.697224	−0.0747503
$88$	0	0
$89$	5.09167	0.539716	0.269858	−	0.962900i	$-0.413023\pi$
$89$	5.09167	0.539716	0.269858	+	0.962900i	$0.413023\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.908327	−0.0941891
$94$	0	0
$95$	−1.11943	−0.114851
$96$	0	0
$97$	−6.09167	−0.618516	−0.309258	−	0.950978i	$-0.600081\pi$
$97$	−6.09167	−0.618516	−0.309258	+	0.950978i	$0.600081\pi$
$98$	0	0
$99$	14.5139	1.45870
$100$	0	0
$101$	9.51388	0.946666	0.473333	−	0.880884i	$-0.343051\pi$
$101$	9.51388	0.946666	0.473333	+	0.880884i	$0.343051\pi$
$102$	0	0
$103$	−6.81665	−0.671665	−0.335832	−	0.941922i	$-0.609018\pi$
$103$	−6.81665	−0.671665	−0.335832	+	0.941922i	$0.609018\pi$
$104$	0	0
$105$	0.697224	0.0680421
$106$	0	0
$107$	−15.5139	−1.49978	−0.749892	−	0.661561i	$-0.769894\pi$
$107$	−15.5139	−1.49978	−0.749892	+	0.661561i	$0.769894\pi$
$108$	0	0
$109$	−3.60555	−0.345349	−0.172675	−	0.984979i	$-0.555241\pi$
$109$	−3.60555	−0.345349	−0.172675	+	0.984979i	$0.555241\pi$
$110$	0	0
$111$	21.2111	2.01327
$112$	0	0
$113$	−10.2111	−0.960580	−0.480290	−	0.877110i	$-0.659469\pi$
$113$	−10.2111	−0.960580	−0.480290	+	0.877110i	$0.659469\pi$
$114$	0	0
$115$	−0.605551	−0.0564679
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.60555	0.147181
$120$	0	0
$121$	28.7250	2.61136
$122$	0	0
$123$	−3.21110	−0.289535
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	−11.9083	−1.05669	−0.528347	−	0.849029i	$-0.677188\pi$
$127$	−11.9083	−1.05669	−0.528347	+	0.849029i	$0.677188\pi$
$128$	0	0
$129$	−13.6056	−1.19790
$130$	0	0
$131$	−12.5139	−1.09334	−0.546671	−	0.837347i	$-0.684105\pi$
$131$	−12.5139	−1.09334	−0.546671	+	0.837347i	$0.684105\pi$
$132$	0	0
$133$	−3.69722	−0.320590
$134$	0	0
$135$	0.486122	0.0418387
$136$	0	0
$137$	2.30278	0.196739	0.0983697	−	0.995150i	$-0.468637\pi$
$137$	2.30278	0.196739	0.0983697	+	0.995150i	$0.468637\pi$
$138$	0	0
$139$	7.90833	0.670776	0.335388	−	0.942080i	$-0.391133\pi$
$139$	7.90833	0.670776	0.335388	+	0.942080i	$0.391133\pi$
$140$	0	0
$141$	26.7250	2.25065
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0.0916731	0.00761304
$146$	0	0
$147$	2.30278	0.189930
$148$	0	0
$149$	−0.908327	−0.0744130	−0.0372065	−	0.999308i	$-0.511846\pi$
$149$	−0.908327	−0.0744130	−0.0372065	+	0.999308i	$0.511846\pi$
$150$	0	0
$151$	−12.2111	−0.993725	−0.496863	−	0.867829i	$-0.665515\pi$
$151$	−12.2111	−0.993725	−0.496863	+	0.867829i	$0.665515\pi$
$152$	0	0
$153$	−3.69722	−0.298903
$154$	0	0
$155$	0.119429	0.00959281
$156$	0	0
$157$	20.6972	1.65182	0.825909	−	0.563803i	$-0.190662\pi$
$157$	20.6972	1.65182	0.825909	+	0.563803i	$0.190662\pi$
$158$	0	0
$159$	24.9083	1.97536
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	−8.60555	−0.674039	−0.337019	−	0.941498i	$-0.609419\pi$
$163$	−8.60555	−0.674039	−0.337019	+	0.941498i	$0.609419\pi$
$164$	0	0
$165$	−4.39445	−0.342107
$166$	0	0
$167$	22.8167	1.76561	0.882803	−	0.469744i	$-0.155654\pi$
$167$	22.8167	1.76561	0.882803	+	0.469744i	$0.155654\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	8.51388	0.651073
$172$	0	0
$173$	−20.8167	−1.58266	−0.791330	−	0.611389i	$-0.790611\pi$
$173$	−20.8167	−1.58266	−0.791330	+	0.611389i	$0.790611\pi$
$174$	0	0
$175$	4.90833	0.371035
$176$	0	0
$177$	−12.9083	−0.970249
$178$	0	0
$179$	−1.18335	−0.0884474	−0.0442237	−	0.999022i	$-0.514081\pi$
$179$	−1.18335	−0.0884474	−0.0442237	+	0.999022i	$0.514081\pi$
$180$	0	0
$181$	1.39445	0.103649	0.0518243	−	0.998656i	$-0.483496\pi$
$181$	1.39445	0.103649	0.0518243	+	0.998656i	$0.483496\pi$
$182$	0	0
$183$	16.6056	1.22752
$184$	0	0
$185$	−2.78890	−0.205044
$186$	0	0
$187$	−10.1194	−0.740006
$188$	0	0
$189$	1.60555	0.116787
$190$	0	0
$191$	9.51388	0.688400	0.344200	−	0.938896i	$-0.388150\pi$
$191$	9.51388	0.688400	0.344200	+	0.938896i	$0.388150\pi$
$192$	0	0
$193$	−25.6333	−1.84513	−0.922563	−	0.385847i	$-0.873909\pi$
$193$	−25.6333	−1.84513	−0.922563	+	0.385847i	$0.873909\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.90833	0.492198	0.246099	−	0.969245i	$-0.420851\pi$
$197$	6.90833	0.492198	0.246099	+	0.969245i	$0.420851\pi$
$198$	0	0
$199$	17.4222	1.23503	0.617514	−	0.786560i	$-0.288140\pi$
$199$	17.4222	1.23503	0.617514	+	0.786560i	$0.288140\pi$
$200$	0	0
$201$	29.9361	2.11153
$202$	0	0
$203$	0.302776	0.0212507
$204$	0	0
$205$	0.422205	0.0294881
$206$	0	0
$207$	4.60555	0.320108
$208$	0	0
$209$	23.3028	1.61189
$210$	0	0
$211$	17.4222	1.19939	0.599697	−	0.800227i	$-0.295288\pi$
$211$	17.4222	1.19939	0.599697	+	0.800227i	$0.295288\pi$
$212$	0	0
$213$	−22.6056	−1.54891
$214$	0	0
$215$	1.78890	0.122002
$216$	0	0
$217$	0.394449	0.0267769
$218$	0	0
$219$	25.8167	1.74453
$220$	0	0
$221$	0	0
$222$	0	0
$223$	14.0917	0.943648	0.471824	−	0.881693i	$-0.343596\pi$
$223$	14.0917	0.943648	0.471824	+	0.881693i	$0.343596\pi$
$224$	0	0
$225$	−11.3028	−0.753518
$226$	0	0
$227$	12.2111	0.810479	0.405240	−	0.914210i	$-0.367188\pi$
$227$	12.2111	0.810479	0.405240	+	0.914210i	$0.367188\pi$
$228$	0	0
$229$	1.39445	0.0921478	0.0460739	−	0.998938i	$-0.485329\pi$
$229$	1.39445	0.0921478	0.0460739	+	0.998938i	$0.485329\pi$
$230$	0	0
$231$	−14.5139	−0.954943
$232$	0	0
$233$	8.11943	0.531922	0.265961	−	0.963984i	$-0.414311\pi$
$233$	8.11943	0.531922	0.265961	+	0.963984i	$0.414311\pi$
$234$	0	0
$235$	−3.51388	−0.229220
$236$	0	0
$237$	18.4222	1.19665
$238$	0	0
$239$	9.00000	0.582162	0.291081	−	0.956698i	$-0.405985\pi$
$239$	9.00000	0.582162	0.291081	+	0.956698i	$0.405985\pi$
$240$	0	0
$241$	4.88057	0.314385	0.157193	−	0.987568i	$-0.449756\pi$
$241$	4.88057	0.314385	0.157193	+	0.987568i	$0.449756\pi$
$242$	0	0
$243$	−19.6056	−1.25770
$244$	0	0
$245$	−0.302776	−0.0193436
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−10.1194	−0.641293
$250$	0	0
$251$	26.2389	1.65618	0.828091	−	0.560594i	$-0.189427\pi$
$251$	26.2389	1.65618	0.828091	+	0.560594i	$0.189427\pi$
$252$	0	0
$253$	12.6056	0.792504
$254$	0	0
$255$	1.11943	0.0701014
$256$	0	0
$257$	−2.09167	−0.130475	−0.0652375	−	0.997870i	$-0.520780\pi$
$257$	−2.09167	−0.130475	−0.0652375	+	0.997870i	$0.520780\pi$
$258$	0	0
$259$	−9.21110	−0.572350
$260$	0	0
$261$	−0.697224	−0.0431571
$262$	0	0
$263$	10.3944	0.640949	0.320475	−	0.947257i	$-0.396158\pi$
$263$	10.3944	0.640949	0.320475	+	0.947257i	$0.396158\pi$
$264$	0	0
$265$	−3.27502	−0.201183
$266$	0	0
$267$	11.7250	0.717557
$268$	0	0
$269$	−25.5416	−1.55730	−0.778650	−	0.627458i	$-0.784095\pi$
$269$	−25.5416	−1.55730	−0.778650	+	0.627458i	$0.784095\pi$
$270$	0	0
$271$	−2.21110	−0.134315	−0.0671575	−	0.997742i	$-0.521393\pi$
$271$	−2.21110	−0.134315	−0.0671575	+	0.997742i	$0.521393\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−30.9361	−1.86552
$276$	0	0
$277$	−3.42221	−0.205620	−0.102810	−	0.994701i	$-0.532783\pi$
$277$	−3.42221	−0.205620	−0.102810	+	0.994701i	$0.532783\pi$
$278$	0	0
$279$	−0.908327	−0.0543801
$280$	0	0
$281$	−15.8167	−0.943542	−0.471771	−	0.881721i	$-0.656385\pi$
$281$	−15.8167	−0.943542	−0.471771	+	0.881721i	$0.656385\pi$
$282$	0	0
$283$	−20.4222	−1.21397	−0.606987	−	0.794712i	$-0.707622\pi$
$283$	−20.4222	−1.21397	−0.606987	+	0.794712i	$0.707622\pi$
$284$	0	0
$285$	−2.57779	−0.152695
$286$	0	0
$287$	1.39445	0.0823117
$288$	0	0
$289$	−14.4222	−0.848365
$290$	0	0
$291$	−14.0278	−0.822322
$292$	0	0
$293$	23.3944	1.36672	0.683359	−	0.730082i	$-0.260518\pi$
$293$	23.3944	1.36672	0.683359	+	0.730082i	$0.260518\pi$
$294$	0	0
$295$	1.69722	0.0988162
$296$	0	0
$297$	−10.1194	−0.587189
$298$	0	0
$299$	0	0
$300$	0	0
$301$	5.90833	0.340550
$302$	0	0
$303$	21.9083	1.25860
$304$	0	0
$305$	−2.18335	−0.125018
$306$	0	0
$307$	−12.5139	−0.714205	−0.357102	−	0.934065i	$-0.616235\pi$
$307$	−12.5139	−0.714205	−0.357102	+	0.934065i	$0.616235\pi$
$308$	0	0
$309$	−15.6972	−0.892984
$310$	0	0
$311$	2.09167	0.118608	0.0593039	−	0.998240i	$-0.481112\pi$
$311$	2.09167	0.118608	0.0593039	+	0.998240i	$0.481112\pi$
$312$	0	0
$313$	25.0278	1.41465	0.707326	−	0.706887i	$-0.249901\pi$
$313$	25.0278	1.41465	0.707326	+	0.706887i	$0.249901\pi$
$314$	0	0
$315$	0.697224	0.0392841
$316$	0	0
$317$	29.2389	1.64222	0.821109	−	0.570771i	$-0.193356\pi$
$317$	29.2389	1.64222	0.821109	+	0.570771i	$0.193356\pi$
$318$	0	0
$319$	−1.90833	−0.106846
$320$	0	0
$321$	−35.7250	−1.99397
$322$	0	0
$323$	−5.93608	−0.330293
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−8.30278	−0.459145
$328$	0	0
$329$	−11.6056	−0.639835
$330$	0	0
$331$	−29.3305	−1.61215	−0.806076	−	0.591812i	$-0.798413\pi$
$331$	−29.3305	−1.61215	−0.806076	+	0.591812i	$0.798413\pi$
$332$	0	0
$333$	21.2111	1.16236
$334$	0	0
$335$	−3.93608	−0.215051
$336$	0	0
$337$	10.3028	0.561228	0.280614	−	0.959821i	$-0.409462\pi$
$337$	10.3028	0.561228	0.280614	+	0.959821i	$0.409462\pi$
$338$	0	0
$339$	−23.5139	−1.27710
$340$	0	0
$341$	−2.48612	−0.134631
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.39445	−0.0750746
$346$	0	0
$347$	−16.6056	−0.891433	−0.445716	−	0.895174i	$-0.647051\pi$
$347$	−16.6056	−0.891433	−0.445716	+	0.895174i	$0.647051\pi$
$348$	0	0
$349$	−7.78890	−0.416930	−0.208465	−	0.978030i	$-0.566847\pi$
$349$	−7.78890	−0.416930	−0.208465	+	0.978030i	$0.566847\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.9083	0.580592	0.290296	−	0.956937i	$-0.406246\pi$
$353$	10.9083	0.580592	0.290296	+	0.956937i	$0.406246\pi$
$354$	0	0
$355$	2.97224	0.157750
$356$	0	0
$357$	3.69722	0.195678
$358$	0	0
$359$	−5.72498	−0.302153	−0.151076	−	0.988522i	$-0.548274\pi$
$359$	−5.72498	−0.302153	−0.151076	+	0.988522i	$0.548274\pi$
$360$	0	0
$361$	−5.33053	−0.280554
$362$	0	0
$363$	66.1472	3.47183
$364$	0	0
$365$	−3.39445	−0.177674
$366$	0	0
$367$	11.2111	0.585215	0.292607	−	0.956233i	$-0.405477\pi$
$367$	11.2111	0.585215	0.292607	+	0.956233i	$0.405477\pi$
$368$	0	0
$369$	−3.21110	−0.167163
$370$	0	0
$371$	−10.8167	−0.561573
$372$	0	0
$373$	−1.48612	−0.0769485	−0.0384742	−	0.999260i	$-0.512250\pi$
$373$	−1.48612	−0.0769485	−0.0384742	+	0.999260i	$0.512250\pi$
$374$	0	0
$375$	6.90833	0.356744
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−14.9083	−0.765789	−0.382895	−	0.923792i	$-0.625073\pi$
$379$	−14.9083	−0.765789	−0.382895	+	0.923792i	$0.625073\pi$
$380$	0	0
$381$	−27.4222	−1.40488
$382$	0	0
$383$	−9.90833	−0.506292	−0.253146	−	0.967428i	$-0.581465\pi$
$383$	−9.90833	−0.506292	−0.253146	+	0.967428i	$0.581465\pi$
$384$	0	0
$385$	1.90833	0.0972573
$386$	0	0
$387$	−13.6056	−0.691609
$388$	0	0
$389$	−27.2111	−1.37966	−0.689829	−	0.723973i	$-0.742314\pi$
$389$	−27.2111	−1.37966	−0.689829	+	0.723973i	$0.742314\pi$
$390$	0	0
$391$	−3.21110	−0.162392
$392$	0	0
$393$	−28.8167	−1.45361
$394$	0	0
$395$	−2.42221	−0.121874
$396$	0	0
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	0	0
$399$	−8.51388	−0.426227
$400$	0	0
$401$	15.6972	0.783882	0.391941	−	0.919990i	$-0.371804\pi$
$401$	15.6972	0.783882	0.391941	+	0.919990i	$0.371804\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	3.21110	0.159561
$406$	0	0
$407$	58.0555	2.87770
$408$	0	0
$409$	21.9361	1.08467	0.542335	−	0.840162i	$-0.317540\pi$
$409$	21.9361	1.08467	0.542335	+	0.840162i	$0.317540\pi$
$410$	0	0
$411$	5.30278	0.261567
$412$	0	0
$413$	5.60555	0.275831
$414$	0	0
$415$	1.33053	0.0653132
$416$	0	0
$417$	18.2111	0.891802
$418$	0	0
$419$	−36.2111	−1.76903	−0.884514	−	0.466514i	$-0.845510\pi$
$419$	−36.2111	−1.76903	−0.884514	+	0.466514i	$0.845510\pi$
$420$	0	0
$421$	−14.0000	−0.682318	−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000	−0.682318	−0.341159	+	0.940006i	$0.610819\pi$
$422$	0	0
$423$	26.7250	1.29941
$424$	0	0
$425$	7.88057	0.382264
$426$	0	0
$427$	−7.21110	−0.348970
$428$	0	0
$429$	0	0
$430$	0	0
$431$	40.3305	1.94265	0.971327	−	0.237749i	$-0.0764095\pi$
$431$	40.3305	1.94265	0.971327	+	0.237749i	$0.0764095\pi$
$432$	0	0
$433$	11.6056	0.557727	0.278864	−	0.960331i	$-0.410042\pi$
$433$	11.6056	0.557727	0.278864	+	0.960331i	$0.410042\pi$
$434$	0	0
$435$	0.211103	0.0101216
$436$	0	0
$437$	7.39445	0.353724
$438$	0	0
$439$	−29.1194	−1.38979	−0.694897	−	0.719109i	$-0.744550\pi$
$439$	−29.1194	−1.38979	−0.694897	+	0.719109i	$0.744550\pi$
$440$	0	0
$441$	2.30278	0.109656
$442$	0	0
$443$	1.81665	0.0863118	0.0431559	−	0.999068i	$-0.486259\pi$
$443$	1.81665	0.0863118	0.0431559	+	0.999068i	$0.486259\pi$
$444$	0	0
$445$	−1.54163	−0.0730805
$446$	0	0
$447$	−2.09167	−0.0989327
$448$	0	0
$449$	−23.2111	−1.09540	−0.547700	−	0.836675i	$-0.684496\pi$
$449$	−23.2111	−1.09540	−0.547700	+	0.836675i	$0.684496\pi$
$450$	0	0
$451$	−8.78890	−0.413853
$452$	0	0
$453$	−28.1194	−1.32117
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.57779	−0.0738061	−0.0369031	−	0.999319i	$-0.511749\pi$
$457$	−1.57779	−0.0738061	−0.0369031	+	0.999319i	$0.511749\pi$
$458$	0	0
$459$	2.57779	0.120321
$460$	0	0
$461$	−33.1472	−1.54382	−0.771909	−	0.635733i	$-0.780698\pi$
$461$	−33.1472	−1.54382	−0.771909	+	0.635733i	$0.780698\pi$
$462$	0	0
$463$	−35.8444	−1.66583	−0.832916	−	0.553400i	$-0.813330\pi$
$463$	−35.8444	−1.66583	−0.832916	+	0.553400i	$0.813330\pi$
$464$	0	0
$465$	0.275019	0.0127537
$466$	0	0
$467$	−36.0278	−1.66717	−0.833583	−	0.552394i	$-0.813714\pi$
$467$	−36.0278	−1.66717	−0.833583	+	0.552394i	$0.813714\pi$
$468$	0	0
$469$	−13.0000	−0.600284
$470$	0	0
$471$	47.6611	2.19611
$472$	0	0
$473$	−37.2389	−1.71224
$474$	0	0
$475$	−18.1472	−0.832650
$476$	0	0
$477$	24.9083	1.14047
$478$	0	0
$479$	−26.7250	−1.22110	−0.610548	−	0.791979i	$-0.709051\pi$
$479$	−26.7250	−1.22110	−0.610548	+	0.791979i	$0.709051\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−4.60555	−0.209560
$484$	0	0
$485$	1.84441	0.0837504
$486$	0	0
$487$	4.09167	0.185411	0.0927057	−	0.995694i	$-0.470448\pi$
$487$	4.09167	0.185411	0.0927057	+	0.995694i	$0.470448\pi$
$488$	0	0
$489$	−19.8167	−0.896140
$490$	0	0
$491$	−10.0917	−0.455431	−0.227715	−	0.973728i	$-0.573126\pi$
$491$	−10.0917	−0.455431	−0.227715	+	0.973728i	$0.573126\pi$
$492$	0	0
$493$	0.486122	0.0218938
$494$	0	0
$495$	−4.39445	−0.197516
$496$	0	0
$497$	9.81665	0.440337
$498$	0	0
$499$	29.5416	1.32247	0.661233	−	0.750181i	$-0.270034\pi$
$499$	29.5416	1.32247	0.661233	+	0.750181i	$0.270034\pi$
$500$	0	0
$501$	52.5416	2.34739
$502$	0	0
$503$	20.1194	0.897081	0.448541	−	0.893763i	$-0.351944\pi$
$503$	20.1194	0.897081	0.448541	+	0.893763i	$0.351944\pi$
$504$	0	0
$505$	−2.88057	−0.128184
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7.42221	0.328983	0.164492	−	0.986378i	$-0.447402\pi$
$509$	7.42221	0.328983	0.164492	+	0.986378i	$0.447402\pi$
$510$	0	0
$511$	−11.2111	−0.495950
$512$	0	0
$513$	−5.93608	−0.262084
$514$	0	0
$515$	2.06392	0.0909470
$516$	0	0
$517$	73.1472	3.21701
$518$	0	0
$519$	−47.9361	−2.10416
$520$	0	0
$521$	−23.4500	−1.02736	−0.513681	−	0.857981i	$-0.671718\pi$
$521$	−23.4500	−1.02736	−0.513681	+	0.857981i	$0.671718\pi$
$522$	0	0
$523$	−35.4500	−1.55012	−0.775059	−	0.631889i	$-0.782280\pi$
$523$	−35.4500	−1.55012	−0.775059	+	0.631889i	$0.782280\pi$
$524$	0	0
$525$	11.3028	0.493294
$526$	0	0
$527$	0.633308	0.0275873
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−12.9083	−0.560174
$532$	0	0
$533$	0	0
$534$	0	0
$535$	4.69722	0.203079
$536$	0	0
$537$	−2.72498	−0.117592
$538$	0	0
$539$	6.30278	0.271480
$540$	0	0
$541$	−8.11943	−0.349082	−0.174541	−	0.984650i	$-0.555844\pi$
$541$	−8.11943	−0.349082	−0.174541	+	0.984650i	$0.555844\pi$
$542$	0	0
$543$	3.21110	0.137802
$544$	0	0
$545$	1.09167	0.0467621
$546$	0	0
$547$	−37.2389	−1.59222	−0.796109	−	0.605153i	$-0.793112\pi$
$547$	−37.2389	−1.59222	−0.796109	+	0.605153i	$0.793112\pi$
$548$	0	0
$549$	16.6056	0.708708
$550$	0	0
$551$	−1.11943	−0.0476893
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	−6.42221	−0.272607
$556$	0	0
$557$	5.30278	0.224686	0.112343	−	0.993669i	$-0.464164\pi$
$557$	5.30278	0.224686	0.112343	+	0.993669i	$0.464164\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−23.3028	−0.983844
$562$	0	0
$563$	−37.6333	−1.58605	−0.793027	−	0.609186i	$-0.791496\pi$
$563$	−37.6333	−1.58605	−0.793027	+	0.609186i	$0.791496\pi$
$564$	0	0
$565$	3.09167	0.130068
$566$	0	0
$567$	10.6056	0.445391
$568$	0	0
$569$	−4.78890	−0.200761	−0.100381	−	0.994949i	$-0.532006\pi$
$569$	−4.78890	−0.200761	−0.100381	+	0.994949i	$0.532006\pi$
$570$	0	0
$571$	19.8806	0.831976	0.415988	−	0.909370i	$-0.363436\pi$
$571$	19.8806	0.831976	0.415988	+	0.909370i	$0.363436\pi$
$572$	0	0
$573$	21.9083	0.915233
$574$	0	0
$575$	−9.81665	−0.409383
$576$	0	0
$577$	−28.8167	−1.19965	−0.599826	−	0.800130i	$-0.704764\pi$
$577$	−28.8167	−1.19965	−0.599826	+	0.800130i	$0.704764\pi$
$578$	0	0
$579$	−59.0278	−2.45311
$580$	0	0
$581$	4.39445	0.182312
$582$	0	0
$583$	68.1749	2.82352
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.4861	0.969376	0.484688	−	0.874687i	$-0.338933\pi$
$587$	23.4861	0.969376	0.484688	+	0.874687i	$0.338933\pi$
$588$	0	0
$589$	−1.45837	−0.0600909
$590$	0	0
$591$	15.9083	0.654381
$592$	0	0
$593$	−18.0917	−0.742936	−0.371468	−	0.928446i	$-0.621145\pi$
$593$	−18.0917	−0.742936	−0.371468	+	0.928446i	$0.621145\pi$
$594$	0	0
$595$	−0.486122	−0.0199291
$596$	0	0
$597$	40.1194	1.64198
$598$	0	0
$599$	16.1472	0.659756	0.329878	−	0.944024i	$-0.392992\pi$
$599$	16.1472	0.659756	0.329878	+	0.944024i	$0.392992\pi$
$600$	0	0
$601$	20.1194	0.820689	0.410344	−	0.911931i	$-0.365409\pi$
$601$	20.1194	0.820689	0.410344	+	0.911931i	$0.365409\pi$
$602$	0	0
$603$	29.9361	1.21909
$604$	0	0
$605$	−8.69722	−0.353592
$606$	0	0
$607$	−39.8444	−1.61723	−0.808617	−	0.588335i	$-0.799784\pi$
$607$	−39.8444	−1.61723	−0.808617	+	0.588335i	$0.799784\pi$
$608$	0	0
$609$	0.697224	0.0282530
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−33.7250	−1.36214	−0.681070	−	0.732219i	$-0.738485\pi$
$613$	−33.7250	−1.36214	−0.681070	+	0.732219i	$0.738485\pi$
$614$	0	0
$615$	0.972244	0.0392046
$616$	0	0
$617$	28.6333	1.15273	0.576367	−	0.817191i	$-0.304470\pi$
$617$	28.6333	1.15273	0.576367	+	0.817191i	$0.304470\pi$
$618$	0	0
$619$	−8.42221	−0.338517	−0.169259	−	0.985572i	$-0.554137\pi$
$619$	−8.42221	−0.338517	−0.169259	+	0.985572i	$0.554137\pi$
$620$	0	0
$621$	−3.21110	−0.128857
$622$	0	0
$623$	−5.09167	−0.203994
$624$	0	0
$625$	23.6333	0.945332
$626$	0	0
$627$	53.6611	2.14302
$628$	0	0
$629$	−14.7889	−0.589672
$630$	0	0
$631$	−31.0917	−1.23774	−0.618870	−	0.785493i	$-0.712409\pi$
$631$	−31.0917	−1.23774	−0.618870	+	0.785493i	$0.712409\pi$
$632$	0	0
$633$	40.1194	1.59460
$634$	0	0
$635$	3.60555	0.143082
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−22.6056	−0.894262
$640$	0	0
$641$	−8.69722	−0.343520	−0.171760	−	0.985139i	$-0.554945\pi$
$641$	−8.69722	−0.343520	−0.171760	+	0.985139i	$0.554945\pi$
$642$	0	0
$643$	−19.0000	−0.749287	−0.374643	−	0.927169i	$-0.622235\pi$
$643$	−19.0000	−0.749287	−0.374643	+	0.927169i	$0.622235\pi$
$644$	0	0
$645$	4.11943	0.162202
$646$	0	0
$647$	14.6056	0.574203	0.287102	−	0.957900i	$-0.407308\pi$
$647$	14.6056	0.574203	0.287102	+	0.957900i	$0.407308\pi$
$648$	0	0
$649$	−35.3305	−1.38684
$650$	0	0
$651$	0.908327	0.0356001
$652$	0	0
$653$	−39.3583	−1.54021	−0.770104	−	0.637918i	$-0.779796\pi$
$653$	−39.3583	−1.54021	−0.770104	+	0.637918i	$0.779796\pi$
$654$	0	0
$655$	3.78890	0.148044
$656$	0	0
$657$	25.8167	1.00720
$658$	0	0
$659$	49.2666	1.91915	0.959577	−	0.281445i	$-0.0908136\pi$
$659$	49.2666	1.91915	0.959577	+	0.281445i	$0.0908136\pi$
$660$	0	0
$661$	−5.39445	−0.209820	−0.104910	−	0.994482i	$-0.533455\pi$
$661$	−5.39445	−0.209820	−0.104910	+	0.994482i	$0.533455\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.11943	0.0434096
$666$	0	0
$667$	−0.605551	−0.0234470
$668$	0	0
$669$	32.4500	1.25459
$670$	0	0
$671$	45.4500	1.75458
$672$	0	0
$673$	33.7889	1.30247	0.651233	−	0.758878i	$-0.274252\pi$
$673$	33.7889	1.30247	0.651233	+	0.758878i	$0.274252\pi$
$674$	0	0
$675$	7.88057	0.303323
$676$	0	0
$677$	14.3028	0.549700	0.274850	−	0.961487i	$-0.411372\pi$
$677$	14.3028	0.549700	0.274850	+	0.961487i	$0.411372\pi$
$678$	0	0
$679$	6.09167	0.233777
$680$	0	0
$681$	28.1194	1.07754
$682$	0	0
$683$	11.1833	0.427919	0.213959	−	0.976843i	$-0.431364\pi$
$683$	11.1833	0.427919	0.213959	+	0.976843i	$0.431364\pi$
$684$	0	0
$685$	−0.697224	−0.0266396
$686$	0	0
$687$	3.21110	0.122511
$688$	0	0
$689$	0	0
$690$	0	0
$691$	44.5416	1.69444	0.847222	−	0.531239i	$-0.178273\pi$
$691$	44.5416	1.69444	0.847222	+	0.531239i	$0.178273\pi$
$692$	0	0
$693$	−14.5139	−0.551337
$694$	0	0
$695$	−2.39445	−0.0908266
$696$	0	0
$697$	2.23886	0.0848028
$698$	0	0
$699$	18.6972	0.707194
$700$	0	0
$701$	−19.8167	−0.748465	−0.374232	−	0.927335i	$-0.622094\pi$
$701$	−19.8167	−0.748465	−0.374232	+	0.927335i	$0.622094\pi$
$702$	0	0
$703$	34.0555	1.28443
$704$	0	0
$705$	−8.09167	−0.304750
$706$	0	0
$707$	−9.51388	−0.357806
$708$	0	0
$709$	−3.51388	−0.131966	−0.0659832	−	0.997821i	$-0.521018\pi$
$709$	−3.51388	−0.131966	−0.0659832	+	0.997821i	$0.521018\pi$
$710$	0	0
$711$	18.4222	0.690887
$712$	0	0
$713$	−0.788897	−0.0295444
$714$	0	0
$715$	0	0
$716$	0	0
$717$	20.7250	0.773989
$718$	0	0
$719$	−20.8167	−0.776330	−0.388165	−	0.921590i	$-0.626891\pi$
$719$	−20.8167	−0.776330	−0.388165	+	0.921590i	$0.626891\pi$
$720$	0	0
$721$	6.81665	0.253865
$722$	0	0
$723$	11.2389	0.417978
$724$	0	0
$725$	1.48612	0.0551932
$726$	0	0
$727$	41.6056	1.54306	0.771532	−	0.636190i	$-0.219491\pi$
$727$	41.6056	1.54306	0.771532	+	0.636190i	$0.219491\pi$
$728$	0	0
$729$	−13.3305	−0.493723
$730$	0	0
$731$	9.48612	0.350857
$732$	0	0
$733$	16.9361	0.625549	0.312774	−	0.949827i	$-0.398742\pi$
$733$	16.9361	0.625549	0.312774	+	0.949827i	$0.398742\pi$
$734$	0	0
$735$	−0.697224	−0.0257175
$736$	0	0
$737$	81.9361	3.01815
$738$	0	0
$739$	−34.6056	−1.27299	−0.636493	−	0.771283i	$-0.719616\pi$
$739$	−34.6056	−1.27299	−0.636493	+	0.771283i	$0.719616\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4.27502	0.156835	0.0784176	−	0.996921i	$-0.475013\pi$
$743$	4.27502	0.156835	0.0784176	+	0.996921i	$0.475013\pi$
$744$	0	0
$745$	0.275019	0.0100759
$746$	0	0
$747$	−10.1194	−0.370251
$748$	0	0
$749$	15.5139	0.566865
$750$	0	0
$751$	−17.7889	−0.649126	−0.324563	−	0.945864i	$-0.605217\pi$
$751$	−17.7889	−0.649126	−0.324563	+	0.945864i	$0.605217\pi$
$752$	0	0
$753$	60.4222	2.20191
$754$	0	0
$755$	3.69722	0.134556
$756$	0	0
$757$	−42.4222	−1.54186	−0.770931	−	0.636919i	$-0.780209\pi$
$757$	−42.4222	−1.54186	−0.770931	+	0.636919i	$0.780209\pi$
$758$	0	0
$759$	29.0278	1.05364
$760$	0	0
$761$	−42.6972	−1.54777	−0.773887	−	0.633324i	$-0.781690\pi$
$761$	−42.6972	−1.54777	−0.773887	+	0.633324i	$0.781690\pi$
$762$	0	0
$763$	3.60555	0.130530
$764$	0	0
$765$	1.11943	0.0404731
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−25.6056	−0.923360	−0.461680	−	0.887047i	$-0.652753\pi$
$769$	−25.6056	−0.923360	−0.461680	+	0.887047i	$0.652753\pi$
$770$	0	0
$771$	−4.81665	−0.173468
$772$	0	0
$773$	−26.8167	−0.964528	−0.482264	−	0.876026i	$-0.660185\pi$
$773$	−26.8167	−0.964528	−0.482264	+	0.876026i	$0.660185\pi$
$774$	0	0
$775$	1.93608	0.0695462
$776$	0	0
$777$	−21.2111	−0.760944
$778$	0	0
$779$	−5.15559	−0.184718
$780$	0	0
$781$	−61.8722	−2.21396
$782$	0	0
$783$	0.486122	0.0173726
$784$	0	0
$785$	−6.26662	−0.223665
$786$	0	0
$787$	38.0000	1.35455	0.677277	−	0.735728i	$-0.263160\pi$
$787$	38.0000	1.35455	0.677277	+	0.735728i	$0.263160\pi$
$788$	0	0
$789$	23.9361	0.852147
$790$	0	0
$791$	10.2111	0.363065
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−7.54163	−0.267474
$796$	0	0
$797$	11.4500	0.405578	0.202789	−	0.979222i	$-0.434999\pi$
$797$	11.4500	0.405578	0.202789	+	0.979222i	$0.434999\pi$
$798$	0	0
$799$	−18.6333	−0.659199
$800$	0	0
$801$	11.7250	0.414282
$802$	0	0
$803$	70.6611	2.49357
$804$	0	0
$805$	0.605551	0.0213429
$806$	0	0
$807$	−58.8167	−2.07044
$808$	0	0
$809$	−5.36669	−0.188683	−0.0943414	−	0.995540i	$-0.530075\pi$
$809$	−5.36669	−0.188683	−0.0943414	+	0.995540i	$0.530075\pi$
$810$	0	0
$811$	16.7250	0.587294	0.293647	−	0.955914i	$-0.405131\pi$
$811$	16.7250	0.587294	0.293647	+	0.955914i	$0.405131\pi$
$812$	0	0
$813$	−5.09167	−0.178573
$814$	0	0
$815$	2.60555	0.0912685
$816$	0	0
$817$	−21.8444	−0.764239
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−43.8444	−1.53018	−0.765090	−	0.643923i	$-0.777306\pi$
$821$	−43.8444	−1.53018	−0.765090	+	0.643923i	$0.777306\pi$
$822$	0	0
$823$	−37.6056	−1.31085	−0.655424	−	0.755262i	$-0.727510\pi$
$823$	−37.6056	−1.31085	−0.655424	+	0.755262i	$0.727510\pi$
$824$	0	0
$825$	−71.2389	−2.48022
$826$	0	0
$827$	38.9361	1.35394	0.676970	−	0.736010i	$-0.263293\pi$
$827$	38.9361	1.35394	0.676970	+	0.736010i	$0.263293\pi$
$828$	0	0
$829$	−43.9361	−1.52596	−0.762982	−	0.646420i	$-0.776265\pi$
$829$	−43.9361	−1.52596	−0.762982	+	0.646420i	$0.776265\pi$
$830$	0	0
$831$	−7.88057	−0.273374
$832$	0	0
$833$	−1.60555	−0.0556291
$834$	0	0
$835$	−6.90833	−0.239073
$836$	0	0
$837$	0.633308	0.0218903
$838$	0	0
$839$	37.9361	1.30970	0.654850	−	0.755759i	$-0.272732\pi$
$839$	37.9361	1.30970	0.654850	+	0.755759i	$0.272732\pi$
$840$	0	0
$841$	−28.9083	−0.996839
$842$	0	0
$843$	−36.4222	−1.25445
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−28.7250	−0.987002
$848$	0	0
$849$	−47.0278	−1.61399
$850$	0	0
$851$	18.4222	0.631505
$852$	0	0
$853$	−11.0000	−0.376633	−0.188316	−	0.982108i	$-0.560303\pi$
$853$	−11.0000	−0.376633	−0.188316	+	0.982108i	$0.560303\pi$
$854$	0	0
$855$	−2.57779	−0.0881587
$856$	0	0
$857$	−46.9361	−1.60331	−0.801653	−	0.597790i	$-0.796046\pi$
$857$	−46.9361	−1.60331	−0.801653	+	0.597790i	$0.796046\pi$
$858$	0	0
$859$	−43.2111	−1.47434	−0.737172	−	0.675705i	$-0.763839\pi$
$859$	−43.2111	−1.47434	−0.737172	+	0.675705i	$0.763839\pi$
$860$	0	0
$861$	3.21110	0.109434
$862$	0	0
$863$	36.7889	1.25231	0.626154	−	0.779699i	$-0.284628\pi$
$863$	36.7889	1.25231	0.626154	+	0.779699i	$0.284628\pi$
$864$	0	0
$865$	6.30278	0.214301
$866$	0	0
$867$	−33.2111	−1.12791
$868$	0	0
$869$	50.4222	1.71046
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−14.0278	−0.474768
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	−2.02776	−0.0684725	−0.0342362	−	0.999414i	$-0.510900\pi$
$877$	−2.02776	−0.0684725	−0.0342362	+	0.999414i	$0.510900\pi$
$878$	0	0
$879$	53.8722	1.81706
$880$	0	0
$881$	48.2111	1.62427	0.812137	−	0.583467i	$-0.198304\pi$
$881$	48.2111	1.62427	0.812137	+	0.583467i	$0.198304\pi$
$882$	0	0
$883$	−5.97224	−0.200982	−0.100491	−	0.994938i	$-0.532041\pi$
$883$	−5.97224	−0.200982	−0.100491	+	0.994938i	$0.532041\pi$
$884$	0	0
$885$	3.90833	0.131377
$886$	0	0
$887$	−42.6056	−1.43055	−0.715277	−	0.698841i	$-0.753700\pi$
$887$	−42.6056	−1.43055	−0.715277	+	0.698841i	$0.753700\pi$
$888$	0	0
$889$	11.9083	0.399392
$890$	0	0
$891$	−66.8444	−2.23937
$892$	0	0
$893$	42.9083	1.43587
$894$	0	0
$895$	0.358288	0.0119763
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.119429	0.00398320
$900$	0	0
$901$	−17.3667	−0.578568
$902$	0	0
$903$	13.6056	0.452764
$904$	0	0
$905$	−0.422205	−0.0140346
$906$	0	0
$907$	51.6333	1.71446	0.857228	−	0.514937i	$-0.172185\pi$
$907$	51.6333	1.71446	0.857228	+	0.514937i	$0.172185\pi$
$908$	0	0
$909$	21.9083	0.726653
$910$	0	0
$911$	4.54163	0.150471	0.0752355	−	0.997166i	$-0.476029\pi$
$911$	4.54163	0.150471	0.0752355	+	0.997166i	$0.476029\pi$
$912$	0	0
$913$	−27.6972	−0.916644
$914$	0	0
$915$	−5.02776	−0.166212
$916$	0	0
$917$	12.5139	0.413245
$918$	0	0
$919$	−27.2111	−0.897611	−0.448806	−	0.893629i	$-0.648150\pi$
$919$	−27.2111	−0.897611	−0.448806	+	0.893629i	$0.648150\pi$
$920$	0	0
$921$	−28.8167	−0.949541
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−45.2111	−1.48653
$926$	0	0
$927$	−15.6972	−0.515564
$928$	0	0
$929$	−42.6333	−1.39875	−0.699377	−	0.714753i	$-0.746539\pi$
$929$	−42.6333	−1.39875	−0.699377	+	0.714753i	$0.746539\pi$
$930$	0	0
$931$	3.69722	0.121172
$932$	0	0
$933$	4.81665	0.157690
$934$	0	0
$935$	3.06392	0.100201
$936$	0	0
$937$	−5.36669	−0.175322	−0.0876611	−	0.996150i	$-0.527939\pi$
$937$	−5.36669	−0.175322	−0.0876611	+	0.996150i	$0.527939\pi$
$938$	0	0
$939$	57.6333	1.88079
$940$	0	0
$941$	−7.78890	−0.253911	−0.126955	−	0.991908i	$-0.540521\pi$
$941$	−7.78890	−0.253911	−0.126955	+	0.991908i	$0.540521\pi$
$942$	0	0
$943$	−2.78890	−0.0908190
$944$	0	0
$945$	−0.486122	−0.0158135
$946$	0	0
$947$	−27.6972	−0.900039	−0.450019	−	0.893019i	$-0.648583\pi$
$947$	−27.6972	−0.900039	−0.450019	+	0.893019i	$0.648583\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	67.3305	2.18334
$952$	0	0
$953$	−16.6333	−0.538806	−0.269403	−	0.963028i	$-0.586826\pi$
$953$	−16.6333	−0.538806	−0.269403	+	0.963028i	$0.586826\pi$
$954$	0	0
$955$	−2.88057	−0.0932131
$956$	0	0
$957$	−4.39445	−0.142052
$958$	0	0
$959$	−2.30278	−0.0743605
$960$	0	0
$961$	−30.8444	−0.994981
$962$	0	0
$963$	−35.7250	−1.15122
$964$	0	0
$965$	7.76114	0.249840
$966$	0	0
$967$	30.3944	0.977420	0.488710	−	0.872446i	$-0.337468\pi$
$967$	30.3944	0.977420	0.488710	+	0.872446i	$0.337468\pi$
$968$	0	0
$969$	−13.6695	−0.439127
$970$	0	0
$971$	39.0000	1.25157	0.625785	−	0.779996i	$-0.284779\pi$
$971$	39.0000	1.25157	0.625785	+	0.779996i	$0.284779\pi$
$972$	0	0
$973$	−7.90833	−0.253529
$974$	0	0
$975$	0	0
$976$	0	0
$977$	30.6333	0.980046	0.490023	−	0.871709i	$-0.336988\pi$
$977$	30.6333	0.980046	0.490023	+	0.871709i	$0.336988\pi$
$978$	0	0
$979$	32.0917	1.02565
$980$	0	0
$981$	−8.30278	−0.265087
$982$	0	0
$983$	−5.54163	−0.176751	−0.0883753	−	0.996087i	$-0.528167\pi$
$983$	−5.54163	−0.176751	−0.0883753	+	0.996087i	$0.528167\pi$
$984$	0	0
$985$	−2.09167	−0.0666462
$986$	0	0
$987$	−26.7250	−0.850666
$988$	0	0
$989$	−11.8167	−0.375748
$990$	0	0
$991$	−24.9083	−0.791239	−0.395620	−	0.918414i	$-0.629470\pi$
$991$	−24.9083	−0.791239	−0.395620	+	0.918414i	$0.629470\pi$
$992$	0	0
$993$	−67.5416	−2.14337
$994$	0	0
$995$	−5.27502	−0.167229
$996$	0	0
$997$	−32.3944	−1.02594	−0.512971	−	0.858406i	$-0.671455\pi$
$997$	−32.3944	−1.02594	−0.512971	+	0.858406i	$0.671455\pi$
$998$	0	0
$999$	−14.7889	−0.467900

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4732.2.a.k.1.2		2
13.3	even	3		364.2.k.c.113.1	yes	4
13.5	odd	4		4732.2.g.g.337.4		4
13.8	odd	4		4732.2.g.g.337.3		4
13.9	even	3		364.2.k.c.29.1	✓	4
13.12	even	2		4732.2.a.j.1.2		2
39.29	odd	6		3276.2.z.d.3025.2		4
39.35	odd	6		3276.2.z.d.757.2		4
52.3	odd	6		1456.2.s.m.113.2		4
52.35	odd	6		1456.2.s.m.1121.2		4
91.3	odd	6		2548.2.l.i.373.1		4
91.9	even	3		2548.2.l.k.1537.2		4
91.16	even	3		2548.2.i.j.165.1		4
91.48	odd	6		2548.2.k.f.393.2		4
91.55	odd	6		2548.2.k.f.1569.2		4
91.61	odd	6		2548.2.l.i.1537.1		4
91.68	odd	6		2548.2.i.l.165.2		4
91.74	even	3		2548.2.i.j.1745.1		4
91.81	even	3		2548.2.l.k.373.2		4
91.87	odd	6		2548.2.i.l.1745.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.k.c.29.1	✓	4	13.9	even	3
364.2.k.c.113.1	yes	4	13.3	even	3
1456.2.s.m.113.2		4	52.3	odd	6
1456.2.s.m.1121.2		4	52.35	odd	6
2548.2.i.j.165.1		4	91.16	even	3
2548.2.i.j.1745.1		4	91.74	even	3
2548.2.i.l.165.2		4	91.68	odd	6
2548.2.i.l.1745.2		4	91.87	odd	6
2548.2.k.f.393.2		4	91.48	odd	6
2548.2.k.f.1569.2		4	91.55	odd	6
2548.2.l.i.373.1		4	91.3	odd	6
2548.2.l.i.1537.1		4	91.61	odd	6
2548.2.l.k.373.2		4	91.81	even	3
2548.2.l.k.1537.2		4	91.9	even	3
3276.2.z.d.757.2		4	39.35	odd	6
3276.2.z.d.3025.2		4	39.29	odd	6
4732.2.a.j.1.2		2	13.12	even	2
4732.2.a.k.1.2		2	1.1	even	1	trivial
4732.2.g.g.337.3		4	13.8	odd	4
4732.2.g.g.337.4		4	13.5	odd	4

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Classical	Maass
Hilbert	Bianchi

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4732.2.a.k.1.2

Level	$4732$
Weight	$2$
Character	4732.1
Self dual	yes
Analytic conductor	$37.785$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$