LMFDB - Embedded newform 2300.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,2,Mod(1,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, 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("2300.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2300.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:	$18.3655924649$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.143376304.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 12x^{4} + 22x^{2} - 6x - 1 $ x^6 - 12x^4 + 22x^2 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 460)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.26443$ of defining polynomial
Character	$\chi$	$=$	2300.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.40050 q^{3} +4.41307 q^{7} +2.76241 q^{9} +O(q^{10})$	$q-2.40050 q^{3} +4.41307 q^{7} +2.76241 q^{9} +2.29289 q^{11} +6.92936 q^{13} +1.51387 q^{17} +2.89920 q^{19} -10.5936 q^{21} +1.00000 q^{23} +0.570328 q^{27} -7.68764 q^{29} +3.85746 q^{31} -5.50408 q^{33} +8.62830 q^{37} -16.6340 q^{39} -6.44324 q^{41} -3.48497 q^{43} -6.19747 q^{47} +12.4752 q^{49} -3.63405 q^{51} +2.17710 q^{53} -6.95953 q^{57} -11.7637 q^{59} -5.11443 q^{61} +12.1907 q^{63} +9.94597 q^{67} -2.40050 q^{69} +3.41407 q^{71} -8.95307 q^{73} +10.1187 q^{77} -1.92694 q^{79} -9.65631 q^{81} +8.04131 q^{83} +18.4542 q^{87} -1.09273 q^{89} +30.5798 q^{91} -9.25985 q^{93} -16.9208 q^{97} +6.33390 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} + 9 q^{7} + 10 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 + 9 * q^7 + 10 * q^9	$ 6 q + 4 q^{3} + 9 q^{7} + 10 q^{9} + 2 q^{11} + 8 q^{13} + 5 q^{17} + 4 q^{19} + 6 q^{23} + 22 q^{27} + 5 q^{29} + 9 q^{31} - 10 q^{33} + 21 q^{37} - 8 q^{39} - q^{41} + 16 q^{43} + 16 q^{47} + 19 q^{49} - 12 q^{51} - q^{53} + 12 q^{57} - 11 q^{59} - 4 q^{61} + 19 q^{63} + 25 q^{67} + 4 q^{69} - 17 q^{71} - 14 q^{73} + 20 q^{77} + 10 q^{79} + 14 q^{81} + 21 q^{83} + 64 q^{87} - 24 q^{89} - 4 q^{91} + 4 q^{97} - 16 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 + 9 * q^7 + 10 * q^9 + 2 * q^11 + 8 * q^13 + 5 * q^17 + 4 * q^19 + 6 * q^23 + 22 * q^27 + 5 * q^29 + 9 * q^31 - 10 * q^33 + 21 * q^37 - 8 * q^39 - q^41 + 16 * q^43 + 16 * q^47 + 19 * q^49 - 12 * q^51 - q^53 + 12 * q^57 - 11 * q^59 - 4 * q^61 + 19 * q^63 + 25 * q^67 + 4 * q^69 - 17 * q^71 - 14 * q^73 + 20 * q^77 + 10 * q^79 + 14 * q^81 + 21 * q^83 + 64 * q^87 - 24 * q^89 - 4 * q^91 + 4 * q^97 - 16 * 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
$3$	−2.40050	−1.38593	−0.692965	−	0.720971i	$-0.743696\pi$
$3$	−2.40050	−1.38593	−0.692965	+	0.720971i	$0.743696\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.41307	1.66798	0.833992	−	0.551777i	$-0.186050\pi$
$7$	4.41307	1.66798	0.833992	+	0.551777i	$0.186050\pi$
$8$	0	0
$9$	2.76241	0.920804
$10$	0	0
$11$	2.29289	0.691332	0.345666	−	0.938358i	$-0.387653\pi$
$11$	2.29289	0.691332	0.345666	+	0.938358i	$0.387653\pi$
$12$	0	0
$13$	6.92936	1.92186	0.960930	−	0.276792i	$-0.0892713\pi$
$13$	6.92936	1.92186	0.960930	+	0.276792i	$0.0892713\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.51387	0.367168	0.183584	−	0.983004i	$-0.441230\pi$
$17$	1.51387	0.367168	0.183584	+	0.983004i	$0.441230\pi$
$18$	0	0
$19$	2.89920	0.665121	0.332561	−	0.943082i	$-0.392087\pi$
$19$	2.89920	0.665121	0.332561	+	0.943082i	$0.392087\pi$
$20$	0	0
$21$	−10.5936	−2.31171
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0.570328	0.109760
$28$	0	0
$29$	−7.68764	−1.42756	−0.713779	−	0.700371i	$-0.753018\pi$
$29$	−7.68764	−1.42756	−0.713779	+	0.700371i	$0.753018\pi$
$30$	0	0
$31$	3.85746	0.692821	0.346410	−	0.938083i	$-0.387401\pi$
$31$	3.85746	0.692821	0.346410	+	0.938083i	$0.387401\pi$
$32$	0	0
$33$	−5.50408	−0.958138
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.62830	1.41848	0.709242	−	0.704965i	$-0.249037\pi$
$37$	8.62830	1.41848	0.709242	+	0.704965i	$0.249037\pi$
$38$	0	0
$39$	−16.6340	−2.66356
$40$	0	0
$41$	−6.44324	−1.00626	−0.503132	−	0.864209i	$-0.667819\pi$
$41$	−6.44324	−1.00626	−0.503132	+	0.864209i	$0.667819\pi$
$42$	0	0
$43$	−3.48497	−0.531453	−0.265727	−	0.964048i	$-0.585612\pi$
$43$	−3.48497	−0.531453	−0.265727	+	0.964048i	$0.585612\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.19747	−0.903994	−0.451997	−	0.892019i	$-0.649288\pi$
$47$	−6.19747	−0.903994	−0.451997	+	0.892019i	$0.649288\pi$
$48$	0	0
$49$	12.4752	1.78217
$50$	0	0
$51$	−3.63405	−0.508869
$52$	0	0
$53$	2.17710	0.299047	0.149524	−	0.988758i	$-0.452226\pi$
$53$	2.17710	0.299047	0.149524	+	0.988758i	$0.452226\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.95953	−0.921812
$58$	0	0
$59$	−11.7637	−1.53150	−0.765750	−	0.643138i	$-0.777632\pi$
$59$	−11.7637	−1.53150	−0.765750	+	0.643138i	$0.777632\pi$
$60$	0	0
$61$	−5.11443	−0.654835	−0.327418	−	0.944880i	$-0.606178\pi$
$61$	−5.11443	−0.654835	−0.327418	+	0.944880i	$0.606178\pi$
$62$	0	0
$63$	12.1907	1.53589
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.94597	1.21509	0.607547	−	0.794284i	$-0.292154\pi$
$67$	9.94597	1.21509	0.607547	+	0.794284i	$0.292154\pi$
$68$	0	0
$69$	−2.40050	−0.288987
$70$	0	0
$71$	3.41407	0.405175	0.202588	−	0.979264i	$-0.435065\pi$
$71$	3.41407	0.405175	0.202588	+	0.979264i	$0.435065\pi$
$72$	0	0
$73$	−8.95307	−1.04788	−0.523939	−	0.851756i	$-0.675538\pi$
$73$	−8.95307	−1.04788	−0.523939	+	0.851756i	$0.675538\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.1187	1.15313
$78$	0	0
$79$	−1.92694	−0.216798	−0.108399	−	0.994107i	$-0.534572\pi$
$79$	−1.92694	−0.216798	−0.108399	+	0.994107i	$0.534572\pi$
$80$	0	0
$81$	−9.65631	−1.07292
$82$	0	0
$83$	8.04131	0.882648	0.441324	−	0.897348i	$-0.354509\pi$
$83$	8.04131	0.882648	0.441324	+	0.897348i	$0.354509\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	18.4542	1.97850
$88$	0	0
$89$	−1.09273	−0.115829	−0.0579147	−	0.998322i	$-0.518445\pi$
$89$	−1.09273	−0.115829	−0.0579147	+	0.998322i	$0.518445\pi$
$90$	0	0
$91$	30.5798	3.20563
$92$	0	0
$93$	−9.25985	−0.960201
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−16.9208	−1.71805	−0.859026	−	0.511933i	$-0.828930\pi$
$97$	−16.9208	−1.71805	−0.859026	+	0.511933i	$0.828930\pi$
$98$	0	0
$99$	6.33390	0.636581
$100$	0	0
$101$	12.9497	1.28855	0.644274	−	0.764795i	$-0.277160\pi$
$101$	12.9497	1.28855	0.644274	+	0.764795i	$0.277160\pi$
$102$	0	0
$103$	10.9754	1.08144	0.540721	−	0.841202i	$-0.318152\pi$
$103$	10.9754	1.08144	0.540721	+	0.841202i	$0.318152\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	19.8821	1.92208	0.961040	−	0.276411i	$-0.0891450\pi$
$107$	19.8821	1.92208	0.961040	+	0.276411i	$0.0891450\pi$
$108$	0	0
$109$	0.427249	0.0409231	0.0204615	−	0.999791i	$-0.493486\pi$
$109$	0.427249	0.0409231	0.0204615	+	0.999791i	$0.493486\pi$
$110$	0	0
$111$	−20.7123	−1.96592
$112$	0	0
$113$	−5.38533	−0.506609	−0.253304	−	0.967387i	$-0.581517\pi$
$113$	−5.38533	−0.506609	−0.253304	+	0.967387i	$0.581517\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	19.1418	1.76966
$118$	0	0
$119$	6.68082	0.612430
$120$	0	0
$121$	−5.74267	−0.522061
$122$	0	0
$123$	15.4670	1.39461
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.16014	0.546624	0.273312	−	0.961925i	$-0.411881\pi$
$127$	6.16014	0.546624	0.273312	+	0.961925i	$0.411881\pi$
$128$	0	0
$129$	8.36569	0.736558
$130$	0	0
$131$	14.6325	1.27845	0.639225	−	0.769020i	$-0.279255\pi$
$131$	14.6325	1.27845	0.639225	+	0.769020i	$0.279255\pi$
$132$	0	0
$133$	12.7944	1.10941
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.972256	0.0830654	0.0415327	−	0.999137i	$-0.486776\pi$
$137$	0.972256	0.0830654	0.0415327	+	0.999137i	$0.486776\pi$
$138$	0	0
$139$	7.74312	0.656763	0.328382	−	0.944545i	$-0.393497\pi$
$139$	7.74312	0.656763	0.328382	+	0.944545i	$0.393497\pi$
$140$	0	0
$141$	14.8770	1.25287
$142$	0	0
$143$	15.8883	1.32864
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−29.9467	−2.46996
$148$	0	0
$149$	−17.6823	−1.44859	−0.724293	−	0.689492i	$-0.757834\pi$
$149$	−17.6823	−1.44859	−0.724293	+	0.689492i	$0.757834\pi$
$150$	0	0
$151$	−2.01286	−0.163804	−0.0819022	−	0.996640i	$-0.526100\pi$
$151$	−2.01286	−0.163804	−0.0819022	+	0.996640i	$0.526100\pi$
$152$	0	0
$153$	4.18194	0.338090
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.82703	−0.465048	−0.232524	−	0.972591i	$-0.574698\pi$
$157$	−5.82703	−0.465048	−0.232524	+	0.972591i	$0.574698\pi$
$158$	0	0
$159$	−5.22612	−0.414459
$160$	0	0
$161$	4.41307	0.347799
$162$	0	0
$163$	−6.75147	−0.528816	−0.264408	−	0.964411i	$-0.585177\pi$
$163$	−6.75147	−0.528816	−0.264408	+	0.964411i	$0.585177\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−23.4541	−1.81493	−0.907466	−	0.420125i	$-0.861986\pi$
$167$	−23.4541	−1.81493	−0.907466	+	0.420125i	$0.861986\pi$
$168$	0	0
$169$	35.0161	2.69355
$170$	0	0
$171$	8.00878	0.612447
$172$	0	0
$173$	−8.28850	−0.630163	−0.315081	−	0.949065i	$-0.602032\pi$
$173$	−8.28850	−0.630163	−0.315081	+	0.949065i	$0.602032\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	28.2387	2.12255
$178$	0	0
$179$	1.04002	0.0777345	0.0388673	−	0.999244i	$-0.487625\pi$
$179$	1.04002	0.0777345	0.0388673	+	0.999244i	$0.487625\pi$
$180$	0	0
$181$	2.71850	0.202065	0.101032	−	0.994883i	$-0.467785\pi$
$181$	2.71850	0.202065	0.101032	+	0.994883i	$0.467785\pi$
$182$	0	0
$183$	12.2772	0.907557
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.47114	0.253835
$188$	0	0
$189$	2.51690	0.183077
$190$	0	0
$191$	8.04858	0.582375	0.291187	−	0.956666i	$-0.405950\pi$
$191$	8.04858	0.582375	0.291187	+	0.956666i	$0.405950\pi$
$192$	0	0
$193$	−16.0276	−1.15370	−0.576848	−	0.816852i	$-0.695717\pi$
$193$	−16.0276	−1.15370	−0.576848	+	0.816852i	$0.695717\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.06316	0.289488	0.144744	−	0.989469i	$-0.453764\pi$
$197$	4.06316	0.289488	0.144744	+	0.989469i	$0.453764\pi$
$198$	0	0
$199$	18.7042	1.32590	0.662952	−	0.748662i	$-0.269303\pi$
$199$	18.7042	1.32590	0.662952	+	0.748662i	$0.269303\pi$
$200$	0	0
$201$	−23.8753	−1.68404
$202$	0	0
$203$	−33.9261	−2.38114
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.76241	0.192001
$208$	0	0
$209$	6.64753	0.459819
$210$	0	0
$211$	7.87839	0.542371	0.271185	−	0.962527i	$-0.412584\pi$
$211$	7.87839	0.542371	0.271185	+	0.962527i	$0.412584\pi$
$212$	0	0
$213$	−8.19548	−0.561545
$214$	0	0
$215$	0	0
$216$	0	0
$217$	17.0232	1.15561
$218$	0	0
$219$	21.4919	1.45229
$220$	0	0
$221$	10.4902	0.705645
$222$	0	0
$223$	−2.29263	−0.153526	−0.0767628	−	0.997049i	$-0.524458\pi$
$223$	−2.29263	−0.153526	−0.0767628	+	0.997049i	$0.524458\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.8325	0.851725	0.425862	−	0.904788i	$-0.359971\pi$
$227$	12.8325	0.851725	0.425862	+	0.904788i	$0.359971\pi$
$228$	0	0
$229$	−16.2528	−1.07401	−0.537007	−	0.843578i	$-0.680445\pi$
$229$	−16.2528	−1.07401	−0.537007	+	0.843578i	$0.680445\pi$
$230$	0	0
$231$	−24.2899	−1.59816
$232$	0	0
$233$	21.5410	1.41120	0.705598	−	0.708613i	$-0.250679\pi$
$233$	21.5410	1.41120	0.705598	+	0.708613i	$0.250679\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.62563	0.300467
$238$	0	0
$239$	−11.0622	−0.715555	−0.357778	−	0.933807i	$-0.616465\pi$
$239$	−11.0622	−0.715555	−0.357778	+	0.933807i	$0.616465\pi$
$240$	0	0
$241$	18.2164	1.17342	0.586710	−	0.809797i	$-0.300423\pi$
$241$	18.2164	1.17342	0.586710	+	0.809797i	$0.300423\pi$
$242$	0	0
$243$	21.4690	1.37724
$244$	0	0
$245$	0	0
$246$	0	0
$247$	20.0896	1.27827
$248$	0	0
$249$	−19.3032	−1.22329
$250$	0	0
$251$	−10.0182	−0.632345	−0.316172	−	0.948702i	$-0.602398\pi$
$251$	−10.0182	−0.632345	−0.316172	+	0.948702i	$0.602398\pi$
$252$	0	0
$253$	2.29289	0.144153
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.3021	−0.892141	−0.446070	−	0.894998i	$-0.647177\pi$
$257$	−14.3021	−0.892141	−0.446070	+	0.894998i	$0.647177\pi$
$258$	0	0
$259$	38.0773	2.36601
$260$	0	0
$261$	−21.2364	−1.31450
$262$	0	0
$263$	−9.44361	−0.582318	−0.291159	−	0.956675i	$-0.594041\pi$
$263$	−9.44361	−0.582318	−0.291159	+	0.956675i	$0.594041\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.62311	0.160531
$268$	0	0
$269$	23.7630	1.44886	0.724428	−	0.689350i	$-0.242104\pi$
$269$	23.7630	1.44886	0.724428	+	0.689350i	$0.242104\pi$
$270$	0	0
$271$	−17.5939	−1.06875	−0.534375	−	0.845247i	$-0.679453\pi$
$271$	−17.5939	−1.06875	−0.534375	+	0.845247i	$0.679453\pi$
$272$	0	0
$273$	−73.4068	−4.44278
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−29.6582	−1.78199	−0.890995	−	0.454014i	$-0.849992\pi$
$277$	−29.6582	−1.78199	−0.890995	+	0.454014i	$0.849992\pi$
$278$	0	0
$279$	10.6559	0.637952
$280$	0	0
$281$	−8.25484	−0.492442	−0.246221	−	0.969214i	$-0.579189\pi$
$281$	−8.25484	−0.492442	−0.246221	+	0.969214i	$0.579189\pi$
$282$	0	0
$283$	9.21317	0.547666	0.273833	−	0.961777i	$-0.411708\pi$
$283$	9.21317	0.547666	0.273833	+	0.961777i	$0.411708\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−28.4344	−1.67843
$288$	0	0
$289$	−14.7082	−0.865188
$290$	0	0
$291$	40.6185	2.38110
$292$	0	0
$293$	11.1138	0.649277	0.324639	−	0.945838i	$-0.394757\pi$
$293$	11.1138	0.649277	0.324639	+	0.945838i	$0.394757\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.30770	0.0758804
$298$	0	0
$299$	6.92936	0.400735
$300$	0	0
$301$	−15.3794	−0.886455
$302$	0	0
$303$	−31.0859	−1.78584
$304$	0	0
$305$	0	0
$306$	0	0
$307$	18.7800	1.07183	0.535916	−	0.844272i	$-0.319967\pi$
$307$	18.7800	1.07183	0.535916	+	0.844272i	$0.319967\pi$
$308$	0	0
$309$	−26.3465	−1.49880
$310$	0	0
$311$	2.51258	0.142475	0.0712377	−	0.997459i	$-0.477305\pi$
$311$	2.51258	0.142475	0.0712377	+	0.997459i	$0.477305\pi$
$312$	0	0
$313$	19.9278	1.12638	0.563192	−	0.826326i	$-0.309573\pi$
$313$	19.9278	1.12638	0.563192	+	0.826326i	$0.309573\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.8849	−1.56617	−0.783087	−	0.621912i	$-0.786356\pi$
$317$	−27.8849	−1.56617	−0.783087	+	0.621912i	$0.786356\pi$
$318$	0	0
$319$	−17.6269	−0.986916
$320$	0	0
$321$	−47.7271	−2.66387
$322$	0	0
$323$	4.38901	0.244211
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.02561	−0.0567165
$328$	0	0
$329$	−27.3499	−1.50785
$330$	0	0
$331$	−28.5409	−1.56875	−0.784375	−	0.620287i	$-0.787016\pi$
$331$	−28.5409	−1.56875	−0.784375	+	0.620287i	$0.787016\pi$
$332$	0	0
$333$	23.8349	1.30615
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.71371	0.202298	0.101149	−	0.994871i	$-0.467748\pi$
$337$	3.71371	0.202298	0.101149	+	0.994871i	$0.467748\pi$
$338$	0	0
$339$	12.9275	0.702125
$340$	0	0
$341$	8.84472	0.478969
$342$	0	0
$343$	24.1624	1.30464
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.1459	0.759393	0.379696	−	0.925111i	$-0.376028\pi$
$347$	14.1459	0.759393	0.379696	+	0.925111i	$0.376028\pi$
$348$	0	0
$349$	−21.2802	−1.13910	−0.569550	−	0.821957i	$-0.692883\pi$
$349$	−21.2802	−1.13910	−0.569550	+	0.821957i	$0.692883\pi$
$350$	0	0
$351$	3.95201	0.210943
$352$	0	0
$353$	24.4641	1.30209	0.651046	−	0.759038i	$-0.274331\pi$
$353$	24.4641	1.30209	0.651046	+	0.759038i	$0.274331\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−16.0373	−0.848786
$358$	0	0
$359$	−23.0252	−1.21522	−0.607612	−	0.794234i	$-0.707872\pi$
$359$	−23.0252	−1.21522	−0.607612	+	0.794234i	$0.707872\pi$
$360$	0	0
$361$	−10.5947	−0.557613
$362$	0	0
$363$	13.7853	0.723540
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0.478057	0.0249544	0.0124772	−	0.999922i	$-0.496028\pi$
$367$	0.478057	0.0249544	0.0124772	+	0.999922i	$0.496028\pi$
$368$	0	0
$369$	−17.7989	−0.926573
$370$	0	0
$371$	9.60767	0.498806
$372$	0	0
$373$	27.3508	1.41617	0.708085	−	0.706127i	$-0.249559\pi$
$373$	27.3508	1.41617	0.708085	+	0.706127i	$0.249559\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−53.2704	−2.74357
$378$	0	0
$379$	−7.35358	−0.377728	−0.188864	−	0.982003i	$-0.560481\pi$
$379$	−7.35358	−0.377728	−0.188864	+	0.982003i	$0.560481\pi$
$380$	0	0
$381$	−14.7874	−0.757583
$382$	0	0
$383$	−3.34108	−0.170721	−0.0853606	−	0.996350i	$-0.527204\pi$
$383$	−3.34108	−0.170721	−0.0853606	+	0.996350i	$0.527204\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.62693	−0.489364
$388$	0	0
$389$	0.366568	0.0185858	0.00929288	−	0.999957i	$-0.497042\pi$
$389$	0.366568	0.0185858	0.00929288	+	0.999957i	$0.497042\pi$
$390$	0	0
$391$	1.51387	0.0765598
$392$	0	0
$393$	−35.1254	−1.77184
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−11.3222	−0.568245	−0.284122	−	0.958788i	$-0.591702\pi$
$397$	−11.3222	−0.568245	−0.284122	+	0.958788i	$0.591702\pi$
$398$	0	0
$399$	−30.7129	−1.53757
$400$	0	0
$401$	37.1673	1.85605	0.928024	−	0.372520i	$-0.121506\pi$
$401$	37.1673	1.85605	0.928024	+	0.372520i	$0.121506\pi$
$402$	0	0
$403$	26.7298	1.33150
$404$	0	0
$405$	0	0
$406$	0	0
$407$	19.7837	0.980643
$408$	0	0
$409$	−13.7745	−0.681107	−0.340553	−	0.940225i	$-0.610614\pi$
$409$	−13.7745	−0.681107	−0.340553	+	0.940225i	$0.610614\pi$
$410$	0	0
$411$	−2.33390	−0.115123
$412$	0	0
$413$	−51.9139	−2.55452
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−18.5874	−0.910228
$418$	0	0
$419$	7.50726	0.366753	0.183377	−	0.983043i	$-0.441297\pi$
$419$	7.50726	0.366753	0.183377	+	0.983043i	$0.441297\pi$
$420$	0	0
$421$	−7.65685	−0.373172	−0.186586	−	0.982439i	$-0.559742\pi$
$421$	−7.65685	−0.373172	−0.186586	+	0.982439i	$0.559742\pi$
$422$	0	0
$423$	−17.1200	−0.832402
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−22.5703	−1.09225
$428$	0	0
$429$	−38.1398	−1.84141
$430$	0	0
$431$	6.22764	0.299975	0.149987	−	0.988688i	$-0.452077\pi$
$431$	6.22764	0.299975	0.149987	+	0.988688i	$0.452077\pi$
$432$	0	0
$433$	0.704087	0.0338362	0.0169181	−	0.999857i	$-0.494615\pi$
$433$	0.704087	0.0338362	0.0169181	+	0.999857i	$0.494615\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.89920	0.138687
$438$	0	0
$439$	−34.8570	−1.66363	−0.831816	−	0.555052i	$-0.812699\pi$
$439$	−34.8570	−1.66363	−0.831816	+	0.555052i	$0.812699\pi$
$440$	0	0
$441$	34.4616	1.64103
$442$	0	0
$443$	−24.8165	−1.17907	−0.589533	−	0.807745i	$-0.700688\pi$
$443$	−24.8165	−1.17907	−0.589533	+	0.807745i	$0.700688\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	42.4463	2.00764
$448$	0	0
$449$	6.77738	0.319844	0.159922	−	0.987130i	$-0.448876\pi$
$449$	6.77738	0.319844	0.159922	+	0.987130i	$0.448876\pi$
$450$	0	0
$451$	−14.7736	−0.695662
$452$	0	0
$453$	4.83188	0.227021
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.47169	0.0688429	0.0344214	−	0.999407i	$-0.489041\pi$
$457$	1.47169	0.0688429	0.0344214	+	0.999407i	$0.489041\pi$
$458$	0	0
$459$	0.863404	0.0403003
$460$	0	0
$461$	25.8037	1.20180	0.600899	−	0.799325i	$-0.294809\pi$
$461$	25.8037	1.20180	0.600899	+	0.799325i	$0.294809\pi$
$462$	0	0
$463$	−15.3469	−0.713231	−0.356616	−	0.934251i	$-0.616069\pi$
$463$	−15.3469	−0.713231	−0.356616	+	0.934251i	$0.616069\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.2933	0.476315	0.238157	−	0.971227i	$-0.423457\pi$
$467$	10.2933	0.476315	0.238157	+	0.971227i	$0.423457\pi$
$468$	0	0
$469$	43.8922	2.02676
$470$	0	0
$471$	13.9878	0.644524
$472$	0	0
$473$	−7.99065	−0.367410
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.01404	0.275364
$478$	0	0
$479$	4.50453	0.205817	0.102909	−	0.994691i	$-0.467185\pi$
$479$	4.50453	0.205817	0.102909	+	0.994691i	$0.467185\pi$
$480$	0	0
$481$	59.7886	2.72613
$482$	0	0
$483$	−10.5936	−0.482025
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−27.2669	−1.23558	−0.617790	−	0.786343i	$-0.711972\pi$
$487$	−27.2669	−1.23558	−0.617790	+	0.786343i	$0.711972\pi$
$488$	0	0
$489$	16.2069	0.732902
$490$	0	0
$491$	24.4866	1.10507	0.552533	−	0.833491i	$-0.313661\pi$
$491$	24.4866	1.10507	0.552533	+	0.833491i	$0.313661\pi$
$492$	0	0
$493$	−11.6381	−0.524154
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.0665	0.675826
$498$	0	0
$499$	−24.1892	−1.08286	−0.541429	−	0.840746i	$-0.682117\pi$
$499$	−24.1892	−1.08286	−0.541429	+	0.840746i	$0.682117\pi$
$500$	0	0
$501$	56.3016	2.51537
$502$	0	0
$503$	30.6230	1.36541	0.682706	−	0.730693i	$-0.260803\pi$
$503$	30.6230	1.36541	0.682706	+	0.730693i	$0.260803\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−84.0562	−3.73307
$508$	0	0
$509$	−2.56826	−0.113836	−0.0569181	−	0.998379i	$-0.518127\pi$
$509$	−2.56826	−0.113836	−0.0569181	+	0.998379i	$0.518127\pi$
$510$	0	0
$511$	−39.5105	−1.74784
$512$	0	0
$513$	1.65349	0.0730035
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−14.2101	−0.624960
$518$	0	0
$519$	19.8966	0.873362
$520$	0	0
$521$	−44.1355	−1.93361	−0.966807	−	0.255509i	$-0.917757\pi$
$521$	−44.1355	−1.93361	−0.966807	+	0.255509i	$0.917757\pi$
$522$	0	0
$523$	42.2884	1.84914	0.924572	−	0.381008i	$-0.124423\pi$
$523$	42.2884	1.84914	0.924572	+	0.381008i	$0.124423\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.83970	0.254381
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−32.4961	−1.41021
$532$	0	0
$533$	−44.6475	−1.93390
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−2.49656	−0.107735
$538$	0	0
$539$	28.6042	1.23207
$540$	0	0
$541$	10.2776	0.441871	0.220935	−	0.975288i	$-0.429089\pi$
$541$	10.2776	0.441871	0.220935	+	0.975288i	$0.429089\pi$
$542$	0	0
$543$	−6.52577	−0.280048
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−3.87305	−0.165600	−0.0827998	−	0.996566i	$-0.526386\pi$
$547$	−3.87305	−0.165600	−0.0827998	+	0.996566i	$0.526386\pi$
$548$	0	0
$549$	−14.1282	−0.602975
$550$	0	0
$551$	−22.2880	−0.949500
$552$	0	0
$553$	−8.50373	−0.361615
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.62364	−0.407767	−0.203883	−	0.978995i	$-0.565356\pi$
$557$	−9.62364	−0.407767	−0.203883	+	0.978995i	$0.565356\pi$
$558$	0	0
$559$	−24.1486	−1.02138
$560$	0	0
$561$	−8.33248	−0.351797
$562$	0	0
$563$	3.35865	0.141550	0.0707751	−	0.997492i	$-0.477453\pi$
$563$	3.35865	0.141550	0.0707751	+	0.997492i	$0.477453\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−42.6140	−1.78962
$568$	0	0
$569$	−4.26939	−0.178982	−0.0894910	−	0.995988i	$-0.528524\pi$
$569$	−4.26939	−0.178982	−0.0894910	+	0.995988i	$0.528524\pi$
$570$	0	0
$571$	16.4567	0.688689	0.344345	−	0.938843i	$-0.388101\pi$
$571$	16.4567	0.688689	0.344345	+	0.938843i	$0.388101\pi$
$572$	0	0
$573$	−19.3206	−0.807131
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.98780	0.0827530	0.0413765	−	0.999144i	$-0.486826\pi$
$577$	1.98780	0.0827530	0.0413765	+	0.999144i	$0.486826\pi$
$578$	0	0
$579$	38.4744	1.59894
$580$	0	0
$581$	35.4869	1.47224
$582$	0	0
$583$	4.99184	0.206741
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.9720	−1.19580	−0.597900	−	0.801570i	$-0.703998\pi$
$587$	−28.9720	−1.19580	−0.597900	+	0.801570i	$0.703998\pi$
$588$	0	0
$589$	11.1835	0.460810
$590$	0	0
$591$	−9.75363	−0.401211
$592$	0	0
$593$	−19.4799	−0.799944	−0.399972	−	0.916527i	$-0.630980\pi$
$593$	−19.4799	−0.799944	−0.399972	+	0.916527i	$0.630980\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−44.8994	−1.83761
$598$	0	0
$599$	37.8442	1.54627	0.773136	−	0.634240i	$-0.218687\pi$
$599$	37.8442	1.54627	0.773136	+	0.634240i	$0.218687\pi$
$600$	0	0
$601$	−20.2347	−0.825389	−0.412695	−	0.910869i	$-0.635412\pi$
$601$	−20.2347	−0.825389	−0.412695	+	0.910869i	$0.635412\pi$
$602$	0	0
$603$	27.4749	1.11886
$604$	0	0
$605$	0	0
$606$	0	0
$607$	24.4075	0.990670	0.495335	−	0.868702i	$-0.335045\pi$
$607$	24.4075	0.990670	0.495335	+	0.868702i	$0.335045\pi$
$608$	0	0
$609$	81.4396	3.30010
$610$	0	0
$611$	−42.9445	−1.73735
$612$	0	0
$613$	10.6964	0.432025	0.216012	−	0.976391i	$-0.430695\pi$
$613$	10.6964	0.432025	0.216012	+	0.976391i	$0.430695\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−27.4738	−1.10605	−0.553026	−	0.833164i	$-0.686527\pi$
$617$	−27.4738	−1.10605	−0.553026	+	0.833164i	$0.686527\pi$
$618$	0	0
$619$	−0.389330	−0.0156485	−0.00782425	−	0.999969i	$-0.502491\pi$
$619$	−0.389330	−0.0156485	−0.00782425	+	0.999969i	$0.502491\pi$
$620$	0	0
$621$	0.570328	0.0228865
$622$	0	0
$623$	−4.82230	−0.193201
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−15.9574	−0.637278
$628$	0	0
$629$	13.0621	0.520822
$630$	0	0
$631$	−11.9330	−0.475047	−0.237523	−	0.971382i	$-0.576336\pi$
$631$	−11.9330	−0.475047	−0.237523	+	0.971382i	$0.576336\pi$
$632$	0	0
$633$	−18.9121	−0.751689
$634$	0	0
$635$	0	0
$636$	0	0
$637$	86.4451	3.42508
$638$	0	0
$639$	9.43106	0.373087
$640$	0	0
$641$	22.8525	0.902621	0.451311	−	0.892367i	$-0.350957\pi$
$641$	22.8525	0.902621	0.451311	+	0.892367i	$0.350957\pi$
$642$	0	0
$643$	34.9279	1.37742	0.688711	−	0.725036i	$-0.258177\pi$
$643$	34.9279	1.37742	0.688711	+	0.725036i	$0.258177\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.3738	1.03686	0.518430	−	0.855120i	$-0.326517\pi$
$647$	26.3738	1.03686	0.518430	+	0.855120i	$0.326517\pi$
$648$	0	0
$649$	−26.9728	−1.05877
$650$	0	0
$651$	−40.8643	−1.60160
$652$	0	0
$653$	−2.17022	−0.0849274	−0.0424637	−	0.999098i	$-0.513521\pi$
$653$	−2.17022	−0.0849274	−0.0424637	+	0.999098i	$0.513521\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−24.7321	−0.964891
$658$	0	0
$659$	19.7820	0.770596	0.385298	−	0.922792i	$-0.374099\pi$
$659$	19.7820	0.770596	0.385298	+	0.922792i	$0.374099\pi$
$660$	0	0
$661$	45.7505	1.77949	0.889743	−	0.456461i	$-0.150883\pi$
$661$	45.7505	1.77949	0.889743	+	0.456461i	$0.150883\pi$
$662$	0	0
$663$	−25.1817	−0.977976
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.68764	−0.297666
$668$	0	0
$669$	5.50346	0.212776
$670$	0	0
$671$	−11.7268	−0.452708
$672$	0	0
$673$	6.99940	0.269807	0.134904	−	0.990859i	$-0.456928\pi$
$673$	6.99940	0.269807	0.134904	+	0.990859i	$0.456928\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	26.7178	1.02685	0.513425	−	0.858134i	$-0.328376\pi$
$677$	26.7178	1.02685	0.513425	+	0.858134i	$0.328376\pi$
$678$	0	0
$679$	−74.6728	−2.86568
$680$	0	0
$681$	−30.8045	−1.18043
$682$	0	0
$683$	−19.8947	−0.761248	−0.380624	−	0.924730i	$-0.624291\pi$
$683$	−19.8947	−0.761248	−0.380624	+	0.924730i	$0.624291\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	39.0148	1.48851
$688$	0	0
$689$	15.0859	0.574727
$690$	0	0
$691$	1.24438	0.0473385	0.0236693	−	0.999720i	$-0.492465\pi$
$691$	1.24438	0.0473385	0.0236693	+	0.999720i	$0.492465\pi$
$692$	0	0
$693$	27.9520	1.06181
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−9.75424	−0.369468
$698$	0	0
$699$	−51.7091	−1.95582
$700$	0	0
$701$	−29.8454	−1.12724	−0.563622	−	0.826033i	$-0.690592\pi$
$701$	−29.8454	−1.12724	−0.563622	+	0.826033i	$0.690592\pi$
$702$	0	0
$703$	25.0151	0.943464
$704$	0	0
$705$	0	0
$706$	0	0
$707$	57.1481	2.14928
$708$	0	0
$709$	25.2255	0.947362	0.473681	−	0.880697i	$-0.342925\pi$
$709$	25.2255	0.947362	0.473681	+	0.880697i	$0.342925\pi$
$710$	0	0
$711$	−5.32301	−0.199628
$712$	0	0
$713$	3.85746	0.144463
$714$	0	0
$715$	0	0
$716$	0	0
$717$	26.5549	0.991710
$718$	0	0
$719$	−23.7787	−0.886797	−0.443398	−	0.896325i	$-0.646227\pi$
$719$	−23.7787	−0.886797	−0.443398	+	0.896325i	$0.646227\pi$
$720$	0	0
$721$	48.4353	1.80383
$722$	0	0
$723$	−43.7285	−1.62628
$724$	0	0
$725$	0	0
$726$	0	0
$727$	36.5849	1.35686	0.678429	−	0.734666i	$-0.262661\pi$
$727$	36.5849	1.35686	0.678429	+	0.734666i	$0.262661\pi$
$728$	0	0
$729$	−22.5675	−0.835833
$730$	0	0
$731$	−5.27580	−0.195133
$732$	0	0
$733$	−21.7593	−0.803698	−0.401849	−	0.915706i	$-0.631632\pi$
$733$	−21.7593	−0.803698	−0.401849	+	0.915706i	$0.631632\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22.8050	0.840032
$738$	0	0
$739$	2.72303	0.100168	0.0500842	−	0.998745i	$-0.484051\pi$
$739$	2.72303	0.100168	0.0500842	+	0.998745i	$0.484051\pi$
$740$	0	0
$741$	−48.2251	−1.77159
$742$	0	0
$743$	−32.0161	−1.17456	−0.587278	−	0.809385i	$-0.699800\pi$
$743$	−32.0161	−1.17456	−0.587278	+	0.809385i	$0.699800\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	22.2134	0.812746
$748$	0	0
$749$	87.7413	3.20600
$750$	0	0
$751$	−36.0949	−1.31712	−0.658561	−	0.752527i	$-0.728835\pi$
$751$	−36.0949	−1.31712	−0.658561	+	0.752527i	$0.728835\pi$
$752$	0	0
$753$	24.0488	0.876386
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−46.8188	−1.70166	−0.850830	−	0.525442i	$-0.823900\pi$
$757$	−46.8188	−1.70166	−0.850830	+	0.525442i	$0.823900\pi$
$758$	0	0
$759$	−5.50408	−0.199786
$760$	0	0
$761$	−28.8182	−1.04466	−0.522330	−	0.852744i	$-0.674937\pi$
$761$	−28.8182	−1.04466	−0.522330	+	0.852744i	$0.674937\pi$
$762$	0	0
$763$	1.88548	0.0682590
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−81.5148	−2.94333
$768$	0	0
$769$	51.6243	1.86162	0.930811	−	0.365501i	$-0.119102\pi$
$769$	51.6243	1.86162	0.930811	+	0.365501i	$0.119102\pi$
$770$	0	0
$771$	34.3322	1.23645
$772$	0	0
$773$	25.9610	0.933753	0.466877	−	0.884322i	$-0.345379\pi$
$773$	25.9610	0.933753	0.466877	+	0.884322i	$0.345379\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−91.4046	−3.27912
$778$	0	0
$779$	−18.6802	−0.669288
$780$	0	0
$781$	7.82807	0.280110
$782$	0	0
$783$	−4.38448	−0.156688
$784$	0	0
$785$	0	0
$786$	0	0
$787$	53.2258	1.89730	0.948648	−	0.316333i	$-0.102452\pi$
$787$	53.2258	1.89730	0.948648	+	0.316333i	$0.102452\pi$
$788$	0	0
$789$	22.6694	0.807053
$790$	0	0
$791$	−23.7658	−0.845015
$792$	0	0
$793$	−35.4397	−1.25850
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31.1874	1.10471	0.552357	−	0.833608i	$-0.313729\pi$
$797$	31.1874	1.10471	0.552357	+	0.833608i	$0.313729\pi$
$798$	0	0
$799$	−9.38218	−0.331918
$800$	0	0
$801$	−3.01858	−0.106656
$802$	0	0
$803$	−20.5284	−0.724431
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−57.0432	−2.00802
$808$	0	0
$809$	−1.60251	−0.0563414	−0.0281707	−	0.999603i	$-0.508968\pi$
$809$	−1.60251	−0.0563414	−0.0281707	+	0.999603i	$0.508968\pi$
$810$	0	0
$811$	−36.9545	−1.29765	−0.648824	−	0.760938i	$-0.724739\pi$
$811$	−36.9545	−1.29765	−0.648824	+	0.760938i	$0.724739\pi$
$812$	0	0
$813$	42.2341	1.48121
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−10.1036	−0.353481
$818$	0	0
$819$	84.4739	2.95176
$820$	0	0
$821$	−16.5583	−0.577890	−0.288945	−	0.957346i	$-0.593304\pi$
$821$	−16.5583	−0.577890	−0.288945	+	0.957346i	$0.593304\pi$
$822$	0	0
$823$	−7.63201	−0.266035	−0.133018	−	0.991114i	$-0.542467\pi$
$823$	−7.63201	−0.266035	−0.133018	+	0.991114i	$0.542467\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−19.1969	−0.667543	−0.333772	−	0.942654i	$-0.608321\pi$
$827$	−19.1969	−0.667543	−0.333772	+	0.942654i	$0.608321\pi$
$828$	0	0
$829$	−6.32413	−0.219646	−0.109823	−	0.993951i	$-0.535028\pi$
$829$	−6.32413	−0.219646	−0.109823	+	0.993951i	$0.535028\pi$
$830$	0	0
$831$	71.1946	2.46971
$832$	0	0
$833$	18.8858	0.654355
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.20002	0.0760438
$838$	0	0
$839$	23.1158	0.798046	0.399023	−	0.916941i	$-0.369349\pi$
$839$	23.1158	0.798046	0.399023	+	0.916941i	$0.369349\pi$
$840$	0	0
$841$	30.0997	1.03792
$842$	0	0
$843$	19.8158	0.682491
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−25.3428	−0.870789
$848$	0	0
$849$	−22.1163	−0.759028
$850$	0	0
$851$	8.62830	0.295774
$852$	0	0
$853$	−37.7049	−1.29099	−0.645496	−	0.763764i	$-0.723349\pi$
$853$	−37.7049	−1.29099	−0.645496	+	0.763764i	$0.723349\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.7133	0.673395	0.336697	−	0.941613i	$-0.390690\pi$
$857$	19.7133	0.673395	0.336697	+	0.941613i	$0.390690\pi$
$858$	0	0
$859$	2.00609	0.0684471	0.0342235	−	0.999414i	$-0.489104\pi$
$859$	2.00609	0.0684471	0.0342235	+	0.999414i	$0.489104\pi$
$860$	0	0
$861$	68.2570	2.32619
$862$	0	0
$863$	16.8127	0.572312	0.286156	−	0.958183i	$-0.407623\pi$
$863$	16.8127	0.572312	0.286156	+	0.958183i	$0.407623\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	35.3071	1.19909
$868$	0	0
$869$	−4.41826	−0.149879
$870$	0	0
$871$	68.9192	2.33524
$872$	0	0
$873$	−46.7424	−1.58199
$874$	0	0
$875$	0	0
$876$	0	0
$877$	14.8922	0.502874	0.251437	−	0.967874i	$-0.419097\pi$
$877$	14.8922	0.502874	0.251437	+	0.967874i	$0.419097\pi$
$878$	0	0
$879$	−26.6788	−0.899853
$880$	0	0
$881$	−16.5218	−0.556632	−0.278316	−	0.960490i	$-0.589776\pi$
$881$	−16.5218	−0.556632	−0.278316	+	0.960490i	$0.589776\pi$
$882$	0	0
$883$	38.7254	1.30321	0.651607	−	0.758557i	$-0.274095\pi$
$883$	38.7254	1.30321	0.651607	+	0.758557i	$0.274095\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.16364	0.0390711	0.0195355	−	0.999809i	$-0.493781\pi$
$887$	1.16364	0.0390711	0.0195355	+	0.999809i	$0.493781\pi$
$888$	0	0
$889$	27.1851	0.911760
$890$	0	0
$891$	−22.1408	−0.741746
$892$	0	0
$893$	−17.9677	−0.601266
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−16.6340	−0.555392
$898$	0	0
$899$	−29.6548	−0.989042
$900$	0	0
$901$	3.29584	0.109800
$902$	0	0
$903$	36.9183	1.22857
$904$	0	0
$905$	0	0
$906$	0	0
$907$	30.2796	1.00542	0.502709	−	0.864456i	$-0.332337\pi$
$907$	30.2796	1.00542	0.502709	+	0.864456i	$0.332337\pi$
$908$	0	0
$909$	35.7725	1.18650
$910$	0	0
$911$	−28.2296	−0.935290	−0.467645	−	0.883916i	$-0.654897\pi$
$911$	−28.2296	−0.935290	−0.467645	+	0.883916i	$0.654897\pi$
$912$	0	0
$913$	18.4378	0.610203
$914$	0	0
$915$	0	0
$916$	0	0
$917$	64.5744	2.13243
$918$	0	0
$919$	3.28150	0.108247	0.0541233	−	0.998534i	$-0.482764\pi$
$919$	3.28150	0.108247	0.0541233	+	0.998534i	$0.482764\pi$
$920$	0	0
$921$	−45.0814	−1.48548
$922$	0	0
$923$	23.6573	0.778690
$924$	0	0
$925$	0	0
$926$	0	0
$927$	30.3187	0.995796
$928$	0	0
$929$	7.64745	0.250905	0.125452	−	0.992100i	$-0.459962\pi$
$929$	7.64745	0.250905	0.125452	+	0.992100i	$0.459962\pi$
$930$	0	0
$931$	36.1680	1.18536
$932$	0	0
$933$	−6.03146	−0.197461
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−4.95226	−0.161783	−0.0808917	−	0.996723i	$-0.525777\pi$
$937$	−4.95226	−0.161783	−0.0808917	+	0.996723i	$0.525777\pi$
$938$	0	0
$939$	−47.8366	−1.56109
$940$	0	0
$941$	−8.62677	−0.281225	−0.140612	−	0.990065i	$-0.544907\pi$
$941$	−8.62677	−0.281225	−0.140612	+	0.990065i	$0.544907\pi$
$942$	0	0
$943$	−6.44324	−0.209821
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−46.6807	−1.51692	−0.758460	−	0.651720i	$-0.774048\pi$
$947$	−46.6807	−1.51692	−0.758460	+	0.651720i	$0.774048\pi$
$948$	0	0
$949$	−62.0391	−2.01387
$950$	0	0
$951$	66.9378	2.17061
$952$	0	0
$953$	−12.2343	−0.396307	−0.198154	−	0.980171i	$-0.563495\pi$
$953$	−12.2343	−0.396307	−0.198154	+	0.980171i	$0.563495\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	42.3134	1.36780
$958$	0	0
$959$	4.29063	0.138552
$960$	0	0
$961$	−16.1200	−0.520000
$962$	0	0
$963$	54.9227	1.76986
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−54.8318	−1.76327	−0.881635	−	0.471932i	$-0.843557\pi$
$967$	−54.8318	−1.76327	−0.881635	+	0.471932i	$0.843557\pi$
$968$	0	0
$969$	−10.5358	−0.338460
$970$	0	0
$971$	40.3804	1.29587	0.647935	−	0.761696i	$-0.275633\pi$
$971$	40.3804	1.29587	0.647935	+	0.761696i	$0.275633\pi$
$972$	0	0
$973$	34.1709	1.09547
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.6700	−0.341363	−0.170681	−	0.985326i	$-0.554597\pi$
$977$	−10.6700	−0.341363	−0.170681	+	0.985326i	$0.554597\pi$
$978$	0	0
$979$	−2.50551	−0.0800765
$980$	0	0
$981$	1.18024	0.0376821
$982$	0	0
$983$	6.27812	0.200241	0.100120	−	0.994975i	$-0.468077\pi$
$983$	6.27812	0.200241	0.100120	+	0.994975i	$0.468077\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	65.6535	2.08977
$988$	0	0
$989$	−3.48497	−0.110816
$990$	0	0
$991$	61.2864	1.94683	0.973413	−	0.229057i	$-0.0735643\pi$
$991$	61.2864	1.94683	0.973413	+	0.229057i	$0.0735643\pi$
$992$	0	0
$993$	68.5125	2.17418
$994$	0	0
$995$	0	0
$996$	0	0
$997$	39.9799	1.26618	0.633088	−	0.774080i	$-0.281787\pi$
$997$	39.9799	1.26618	0.633088	+	0.774080i	$0.281787\pi$
$998$	0	0
$999$	4.92096	0.155692

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.2.a.o.1.1		6
4.3	odd	2		9200.2.a.cx.1.6		6
5.2	odd	4		460.2.c.a.369.10	yes	12
5.3	odd	4		460.2.c.a.369.3	✓	12
5.4	even	2		2300.2.a.n.1.6		6
15.2	even	4		4140.2.f.b.829.7		12
15.8	even	4		4140.2.f.b.829.8		12
20.3	even	4		1840.2.e.f.369.10		12
20.7	even	4		1840.2.e.f.369.3		12
20.19	odd	2		9200.2.a.cy.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.c.a.369.3	✓	12	5.3	odd	4
460.2.c.a.369.10	yes	12	5.2	odd	4
1840.2.e.f.369.3		12	20.7	even	4
1840.2.e.f.369.10		12	20.3	even	4
2300.2.a.n.1.6		6	5.4	even	2
2300.2.a.o.1.1		6	1.1	even	1	trivial
4140.2.f.b.829.7		12	15.2	even	4
4140.2.f.b.829.8		12	15.8	even	4
9200.2.a.cx.1.6		6	4.3	odd	2
9200.2.a.cy.1.1		6	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2300.a (trivial)

\(f(q)\)	\(=\)	\(q-2.40050 q^{3} +4.41307 q^{7} +2.76241 q^{9} +O(q^{10})\)	\(q-2.40050 q^{3} +4.41307 q^{7} +2.76241 q^{9} +2.29289 q^{11} +6.92936 q^{13} +1.51387 q^{17} +2.89920 q^{19} -10.5936 q^{21} +1.00000 q^{23} +0.570328 q^{27} -7.68764 q^{29} +3.85746 q^{31} -5.50408 q^{33} +8.62830 q^{37} -16.6340 q^{39} -6.44324 q^{41} -3.48497 q^{43} -6.19747 q^{47} +12.4752 q^{49} -3.63405 q^{51} +2.17710 q^{53} -6.95953 q^{57} -11.7637 q^{59} -5.11443 q^{61} +12.1907 q^{63} +9.94597 q^{67} -2.40050 q^{69} +3.41407 q^{71} -8.95307 q^{73} +10.1187 q^{77} -1.92694 q^{79} -9.65631 q^{81} +8.04131 q^{83} +18.4542 q^{87} -1.09273 q^{89} +30.5798 q^{91} -9.25985 q^{93} -16.9208 q^{97} +6.33390 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} + 9 q^{7} + 10 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 + 9 * q^7 + 10 * q^9	\( 6 q + 4 q^{3} + 9 q^{7} + 10 q^{9} + 2 q^{11} + 8 q^{13} + 5 q^{17} + 4 q^{19} + 6 q^{23} + 22 q^{27} + 5 q^{29} + 9 q^{31} - 10 q^{33} + 21 q^{37} - 8 q^{39} - q^{41} + 16 q^{43} + 16 q^{47} + 19 q^{49} - 12 q^{51} - q^{53} + 12 q^{57} - 11 q^{59} - 4 q^{61} + 19 q^{63} + 25 q^{67} + 4 q^{69} - 17 q^{71} - 14 q^{73} + 20 q^{77} + 10 q^{79} + 14 q^{81} + 21 q^{83} + 64 q^{87} - 24 q^{89} - 4 q^{91} + 4 q^{97} - 16 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 + 9 * q^7 + 10 * q^9 + 2 * q^11 + 8 * q^13 + 5 * q^17 + 4 * q^19 + 6 * q^23 + 22 * q^27 + 5 * q^29 + 9 * q^31 - 10 * q^33 + 21 * q^37 - 8 * q^39 - q^41 + 16 * q^43 + 16 * q^47 + 19 * q^49 - 12 * q^51 - q^53 + 12 * q^57 - 11 * q^59 - 4 * q^61 + 19 * q^63 + 25 * q^67 + 4 * q^69 - 17 * q^71 - 14 * q^73 + 20 * q^77 + 10 * q^79 + 14 * q^81 + 21 * q^83 + 64 * q^87 - 24 * q^89 - 4 * q^91 + 4 * q^97 - 16 * q^99

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