LMFDB - Embedded newform 3825.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3825,2,Mod(1,3825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3825, 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("3825.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3825 = 3^{2} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3825.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:	$30.5427787731$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	3825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155 q^{2} +4.56155 q^{4} -6.56155 q^{8} +O(q^{10})$	$q-2.56155 q^{2} +4.56155 q^{4} -6.56155 q^{8} -1.56155 q^{11} -0.438447 q^{13} +7.68466 q^{16} +1.00000 q^{17} -4.68466 q^{19} +4.00000 q^{22} -2.43845 q^{23} +1.12311 q^{26} +8.24621 q^{29} +3.12311 q^{31} -6.56155 q^{32} -2.56155 q^{34} +5.12311 q^{37} +12.0000 q^{38} +3.56155 q^{41} -4.68466 q^{43} -7.12311 q^{44} +6.24621 q^{46} -11.1231 q^{47} -7.00000 q^{49} -2.00000 q^{52} +12.2462 q^{53} -21.1231 q^{58} -7.12311 q^{59} +9.12311 q^{61} -8.00000 q^{62} +1.43845 q^{64} -4.00000 q^{67} +4.56155 q^{68} +6.24621 q^{71} +12.2462 q^{73} -13.1231 q^{74} -21.3693 q^{76} -9.36932 q^{79} -9.12311 q^{82} -0.876894 q^{83} +12.0000 q^{86} +10.2462 q^{88} +1.12311 q^{89} -11.1231 q^{92} +28.4924 q^{94} +2.87689 q^{97} +17.9309 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 5 q^{4} - 9 q^{8}+O(q^{10}) $ 2 * q - q^2 + 5 * q^4 - 9 * q^8	$ 2 q - q^{2} + 5 q^{4} - 9 q^{8} + q^{11} - 5 q^{13} + 3 q^{16} + 2 q^{17} + 3 q^{19} + 8 q^{22} - 9 q^{23} - 6 q^{26} - 2 q^{31} - 9 q^{32} - q^{34} + 2 q^{37} + 24 q^{38} + 3 q^{41} + 3 q^{43} - 6 q^{44} - 4 q^{46} - 14 q^{47} - 14 q^{49} - 4 q^{52} + 8 q^{53} - 34 q^{58} - 6 q^{59} + 10 q^{61} - 16 q^{62} + 7 q^{64} - 8 q^{67} + 5 q^{68} - 4 q^{71} + 8 q^{73} - 18 q^{74} - 18 q^{76} + 6 q^{79} - 10 q^{82} - 10 q^{83} + 24 q^{86} + 4 q^{88} - 6 q^{89} - 14 q^{92} + 24 q^{94} + 14 q^{97} + 7 q^{98}+O(q^{100}) $ 2 * q - q^2 + 5 * q^4 - 9 * q^8 + q^11 - 5 * q^13 + 3 * q^16 + 2 * q^17 + 3 * q^19 + 8 * q^22 - 9 * q^23 - 6 * q^26 - 2 * q^31 - 9 * q^32 - q^34 + 2 * q^37 + 24 * q^38 + 3 * q^41 + 3 * q^43 - 6 * q^44 - 4 * q^46 - 14 * q^47 - 14 * q^49 - 4 * q^52 + 8 * q^53 - 34 * q^58 - 6 * q^59 + 10 * q^61 - 16 * q^62 + 7 * q^64 - 8 * q^67 + 5 * q^68 - 4 * q^71 + 8 * q^73 - 18 * q^74 - 18 * q^76 + 6 * q^79 - 10 * q^82 - 10 * q^83 + 24 * q^86 + 4 * q^88 - 6 * q^89 - 14 * q^92 + 24 * q^94 + 14 * q^97 + 7 * q^98

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 $

$2$	−2.56155	−1.81129	−0.905646	−	0.424035i	$-0.860613\pi$
$2$	−2.56155	−1.81129	−0.905646	+	0.424035i	$0.860613\pi$
$3$	0	0
$3$	0	0
$4$	4.56155	2.28078
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−6.56155	−2.31986
$9$	0	0
$10$	0	0
$11$	−1.56155	−0.470826	−0.235413	−	0.971895i	$-0.575644\pi$
$11$	−1.56155	−0.470826	−0.235413	+	0.971895i	$0.575644\pi$
$12$	0	0
$13$	−0.438447	−0.121603	−0.0608017	−	0.998150i	$-0.519366\pi$
$13$	−0.438447	−0.121603	−0.0608017	+	0.998150i	$0.519366\pi$
$14$	0	0
$15$	0	0
$16$	7.68466	1.92116
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−4.68466	−1.07473	−0.537367	−	0.843348i	$-0.680581\pi$
$19$	−4.68466	−1.07473	−0.537367	+	0.843348i	$0.680581\pi$
$20$	0	0
$21$	0	0
$22$	4.00000	0.852803
$23$	−2.43845	−0.508451	−0.254226	−	0.967145i	$-0.581821\pi$
$23$	−2.43845	−0.508451	−0.254226	+	0.967145i	$0.581821\pi$
$24$	0	0
$25$	0	0
$26$	1.12311	0.220259
$27$	0	0
$28$	0	0
$29$	8.24621	1.53128	0.765641	−	0.643268i	$-0.222422\pi$
$29$	8.24621	1.53128	0.765641	+	0.643268i	$0.222422\pi$
$30$	0	0
$31$	3.12311	0.560926	0.280463	−	0.959865i	$-0.409512\pi$
$31$	3.12311	0.560926	0.280463	+	0.959865i	$0.409512\pi$
$32$	−6.56155	−1.15993
$33$	0	0
$34$	−2.56155	−0.439303
$35$	0	0
$36$	0	0
$37$	5.12311	0.842233	0.421117	−	0.907006i	$-0.361638\pi$
$37$	5.12311	0.842233	0.421117	+	0.907006i	$0.361638\pi$
$38$	12.0000	1.94666
$39$	0	0
$40$	0	0
$41$	3.56155	0.556221	0.278111	−	0.960549i	$-0.410292\pi$
$41$	3.56155	0.556221	0.278111	+	0.960549i	$0.410292\pi$
$42$	0	0
$43$	−4.68466	−0.714404	−0.357202	−	0.934027i	$-0.616269\pi$
$43$	−4.68466	−0.714404	−0.357202	+	0.934027i	$0.616269\pi$
$44$	−7.12311	−1.07385
$45$	0	0
$46$	6.24621	0.920954
$47$	−11.1231	−1.62247	−0.811236	−	0.584719i	$-0.801205\pi$
$47$	−11.1231	−1.62247	−0.811236	+	0.584719i	$0.801205\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	−2.00000	−0.277350
$53$	12.2462	1.68215	0.841073	−	0.540921i	$-0.181924\pi$
$53$	12.2462	1.68215	0.841073	+	0.540921i	$0.181924\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−21.1231	−2.77360
$59$	−7.12311	−0.927349	−0.463675	−	0.886006i	$-0.653469\pi$
$59$	−7.12311	−0.927349	−0.463675	+	0.886006i	$0.653469\pi$
$60$	0	0
$61$	9.12311	1.16809	0.584047	−	0.811720i	$-0.301468\pi$
$61$	9.12311	1.16809	0.584047	+	0.811720i	$0.301468\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	1.43845	0.179806
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	4.56155	0.553170
$69$	0	0
$70$	0	0
$71$	6.24621	0.741289	0.370644	−	0.928775i	$-0.379137\pi$
$71$	6.24621	0.741289	0.370644	+	0.928775i	$0.379137\pi$
$72$	0	0
$73$	12.2462	1.43331	0.716655	−	0.697428i	$-0.245672\pi$
$73$	12.2462	1.43331	0.716655	+	0.697428i	$0.245672\pi$
$74$	−13.1231	−1.52553
$75$	0	0
$76$	−21.3693	−2.45123
$77$	0	0
$78$	0	0
$79$	−9.36932	−1.05413	−0.527065	−	0.849825i	$-0.676708\pi$
$79$	−9.36932	−1.05413	−0.527065	+	0.849825i	$0.676708\pi$
$80$	0	0
$81$	0	0
$82$	−9.12311	−1.00748
$83$	−0.876894	−0.0962517	−0.0481258	−	0.998841i	$-0.515325\pi$
$83$	−0.876894	−0.0962517	−0.0481258	+	0.998841i	$0.515325\pi$
$84$	0	0
$85$	0	0
$86$	12.0000	1.29399
$87$	0	0
$88$	10.2462	1.09225
$89$	1.12311	0.119049	0.0595245	−	0.998227i	$-0.481042\pi$
$89$	1.12311	0.119049	0.0595245	+	0.998227i	$0.481042\pi$
$90$	0	0
$91$	0	0
$92$	−11.1231	−1.15966
$93$	0	0
$94$	28.4924	2.93877
$95$	0	0
$96$	0	0
$97$	2.87689	0.292104	0.146052	−	0.989277i	$-0.453343\pi$
$97$	2.87689	0.292104	0.146052	+	0.989277i	$0.453343\pi$
$98$	17.9309	1.81129
$99$	0	0
$100$	0	0
$101$	−10.8769	−1.08229	−0.541146	−	0.840929i	$-0.682009\pi$
$101$	−10.8769	−1.08229	−0.541146	+	0.840929i	$0.682009\pi$
$102$	0	0
$103$	−16.6847	−1.64399	−0.821994	−	0.569496i	$-0.807138\pi$
$103$	−16.6847	−1.64399	−0.821994	+	0.569496i	$0.807138\pi$
$104$	2.87689	0.282103
$105$	0	0
$106$	−31.3693	−3.04686
$107$	−4.68466	−0.452883	−0.226442	−	0.974025i	$-0.572709\pi$
$107$	−4.68466	−0.452883	−0.226442	+	0.974025i	$0.572709\pi$
$108$	0	0
$109$	−6.87689	−0.658687	−0.329344	−	0.944210i	$-0.606827\pi$
$109$	−6.87689	−0.658687	−0.329344	+	0.944210i	$0.606827\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.438447	−0.0412456	−0.0206228	−	0.999787i	$-0.506565\pi$
$113$	−0.438447	−0.0412456	−0.0206228	+	0.999787i	$0.506565\pi$
$114$	0	0
$115$	0	0
$116$	37.6155	3.49251
$117$	0	0
$118$	18.2462	1.67970
$119$	0	0
$120$	0	0
$121$	−8.56155	−0.778323
$122$	−23.3693	−2.11576
$123$	0	0
$124$	14.2462	1.27935
$125$	0	0
$126$	0	0
$127$	−19.8078	−1.75765	−0.878827	−	0.477140i	$-0.841674\pi$
$127$	−19.8078	−1.75765	−0.878827	+	0.477140i	$0.841674\pi$
$128$	9.43845	0.834249
$129$	0	0
$130$	0	0
$131$	−14.4384	−1.26149	−0.630746	−	0.775989i	$-0.717251\pi$
$131$	−14.4384	−1.26149	−0.630746	+	0.775989i	$0.717251\pi$
$132$	0	0
$133$	0	0
$134$	10.2462	0.885138
$135$	0	0
$136$	−6.56155	−0.562649
$137$	0.246211	0.0210352	0.0105176	−	0.999945i	$-0.496652\pi$
$137$	0.246211	0.0210352	0.0105176	+	0.999945i	$0.496652\pi$
$138$	0	0
$139$	−0.876894	−0.0743772	−0.0371886	−	0.999308i	$-0.511840\pi$
$139$	−0.876894	−0.0743772	−0.0371886	+	0.999308i	$0.511840\pi$
$140$	0	0
$141$	0	0
$142$	−16.0000	−1.34269
$143$	0.684658	0.0572540
$144$	0	0
$145$	0	0
$146$	−31.3693	−2.59614
$147$	0	0
$148$	23.3693	1.92095
$149$	−12.2462	−1.00325	−0.501624	−	0.865086i	$-0.667264\pi$
$149$	−12.2462	−1.00325	−0.501624	+	0.865086i	$0.667264\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	30.7386	2.49323
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−6.68466	−0.533494	−0.266747	−	0.963767i	$-0.585949\pi$
$157$	−6.68466	−0.533494	−0.266747	+	0.963767i	$0.585949\pi$
$158$	24.0000	1.90934
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	15.1231	1.18453	0.592267	−	0.805742i	$-0.298233\pi$
$163$	15.1231	1.18453	0.592267	+	0.805742i	$0.298233\pi$
$164$	16.2462	1.26862
$165$	0	0
$166$	2.24621	0.174340
$167$	19.8078	1.53277	0.766385	−	0.642381i	$-0.222053\pi$
$167$	19.8078	1.53277	0.766385	+	0.642381i	$0.222053\pi$
$168$	0	0
$169$	−12.8078	−0.985213
$170$	0	0
$171$	0	0
$172$	−21.3693	−1.62940
$173$	1.80776	0.137442	0.0687209	−	0.997636i	$-0.478108\pi$
$173$	1.80776	0.137442	0.0687209	+	0.997636i	$0.478108\pi$
$174$	0	0
$175$	0	0
$176$	−12.0000	−0.904534
$177$	0	0
$178$	−2.87689	−0.215632
$179$	0.876894	0.0655422	0.0327711	−	0.999463i	$-0.489567\pi$
$179$	0.876894	0.0655422	0.0327711	+	0.999463i	$0.489567\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	0	0
$184$	16.0000	1.17954
$185$	0	0
$186$	0	0
$187$	−1.56155	−0.114192
$188$	−50.7386	−3.70050
$189$	0	0
$190$	0	0
$191$	4.87689	0.352880	0.176440	−	0.984311i	$-0.443542\pi$
$191$	4.87689	0.352880	0.176440	+	0.984311i	$0.443542\pi$
$192$	0	0
$193$	7.75379	0.558130	0.279065	−	0.960272i	$-0.409976\pi$
$193$	7.75379	0.558130	0.279065	+	0.960272i	$0.409976\pi$
$194$	−7.36932	−0.529086
$195$	0	0
$196$	−31.9309	−2.28078
$197$	−8.93087	−0.636298	−0.318149	−	0.948041i	$-0.603061\pi$
$197$	−8.93087	−0.636298	−0.318149	+	0.948041i	$0.603061\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	27.8617	1.96035
$203$	0	0
$204$	0	0
$205$	0	0
$206$	42.7386	2.97774
$207$	0	0
$208$	−3.36932	−0.233620
$209$	7.31534	0.506013
$210$	0	0
$211$	13.3693	0.920382	0.460191	−	0.887820i	$-0.347781\pi$
$211$	13.3693	0.920382	0.460191	+	0.887820i	$0.347781\pi$
$212$	55.8617	3.83660
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	0	0
$217$	0	0
$218$	17.6155	1.19307
$219$	0	0
$220$	0	0
$221$	−0.438447	−0.0294931
$222$	0	0
$223$	14.9309	0.999845	0.499922	−	0.866070i	$-0.333362\pi$
$223$	14.9309	0.999845	0.499922	+	0.866070i	$0.333362\pi$
$224$	0	0
$225$	0	0
$226$	1.12311	0.0747079
$227$	−14.0540	−0.932795	−0.466398	−	0.884575i	$-0.654448\pi$
$227$	−14.0540	−0.932795	−0.466398	+	0.884575i	$0.654448\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	−54.1080	−3.55236
$233$	−3.56155	−0.233325	−0.116663	−	0.993172i	$-0.537220\pi$
$233$	−3.56155	−0.233325	−0.116663	+	0.993172i	$0.537220\pi$
$234$	0	0
$235$	0	0
$236$	−32.4924	−2.11508
$237$	0	0
$238$	0	0
$239$	6.24621	0.404034	0.202017	−	0.979382i	$-0.435250\pi$
$239$	6.24621	0.404034	0.202017	+	0.979382i	$0.435250\pi$
$240$	0	0
$241$	3.36932	0.217037	0.108518	−	0.994094i	$-0.465389\pi$
$241$	3.36932	0.217037	0.108518	+	0.994094i	$0.465389\pi$
$242$	21.9309	1.40977
$243$	0	0
$244$	41.6155	2.66416
$245$	0	0
$246$	0	0
$247$	2.05398	0.130691
$248$	−20.4924	−1.30127
$249$	0	0
$250$	0	0
$251$	−8.49242	−0.536037	−0.268018	−	0.963414i	$-0.586369\pi$
$251$	−8.49242	−0.536037	−0.268018	+	0.963414i	$0.586369\pi$
$252$	0	0
$253$	3.80776	0.239392
$254$	50.7386	3.18363
$255$	0	0
$256$	−27.0540	−1.69087
$257$	−15.3693	−0.958712	−0.479356	−	0.877621i	$-0.659130\pi$
$257$	−15.3693	−0.958712	−0.479356	+	0.877621i	$0.659130\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	36.9848	2.28493
$263$	−20.4924	−1.26362	−0.631808	−	0.775125i	$-0.717687\pi$
$263$	−20.4924	−1.26362	−0.631808	+	0.775125i	$0.717687\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−18.2462	−1.11456
$269$	−16.4384	−1.00227	−0.501135	−	0.865369i	$-0.667084\pi$
$269$	−16.4384	−1.00227	−0.501135	+	0.865369i	$0.667084\pi$
$270$	0	0
$271$	19.8078	1.20324	0.601618	−	0.798784i	$-0.294523\pi$
$271$	19.8078	1.20324	0.601618	+	0.798784i	$0.294523\pi$
$272$	7.68466	0.465951
$273$	0	0
$274$	−0.630683	−0.0381010
$275$	0	0
$276$	0	0
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	2.24621	0.134719
$279$	0	0
$280$	0	0
$281$	10.8769	0.648861	0.324431	−	0.945910i	$-0.394827\pi$
$281$	10.8769	0.648861	0.324431	+	0.945910i	$0.394827\pi$
$282$	0	0
$283$	−21.3693	−1.27027	−0.635137	−	0.772399i	$-0.719056\pi$
$283$	−21.3693	−1.27027	−0.635137	+	0.772399i	$0.719056\pi$
$284$	28.4924	1.69071
$285$	0	0
$286$	−1.75379	−0.103704
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	55.8617	3.26906
$293$	1.12311	0.0656125	0.0328063	−	0.999462i	$-0.489556\pi$
$293$	1.12311	0.0656125	0.0328063	+	0.999462i	$0.489556\pi$
$294$	0	0
$295$	0	0
$296$	−33.6155	−1.95386
$297$	0	0
$298$	31.3693	1.81718
$299$	1.06913	0.0618294
$300$	0	0
$301$	0	0
$302$	−20.4924	−1.17921
$303$	0	0
$304$	−36.0000	−2.06474
$305$	0	0
$306$	0	0
$307$	−32.4924	−1.85444	−0.927220	−	0.374516i	$-0.877809\pi$
$307$	−32.4924	−1.85444	−0.927220	+	0.374516i	$0.877809\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−33.6155	−1.90006	−0.950031	−	0.312156i	$-0.898949\pi$
$313$	−33.6155	−1.90006	−0.950031	+	0.312156i	$0.898949\pi$
$314$	17.1231	0.966313
$315$	0	0
$316$	−42.7386	−2.40424
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−12.8769	−0.720968
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.68466	−0.260661
$324$	0	0
$325$	0	0
$326$	−38.7386	−2.14553
$327$	0	0
$328$	−23.3693	−1.29035
$329$	0	0
$330$	0	0
$331$	−34.9309	−1.91997	−0.959987	−	0.280044i	$-0.909651\pi$
$331$	−34.9309	−1.91997	−0.959987	+	0.280044i	$0.909651\pi$
$332$	−4.00000	−0.219529
$333$	0	0
$334$	−50.7386	−2.77629
$335$	0	0
$336$	0	0
$337$	16.7386	0.911811	0.455906	−	0.890028i	$-0.349315\pi$
$337$	16.7386	0.911811	0.455906	+	0.890028i	$0.349315\pi$
$338$	32.8078	1.78451
$339$	0	0
$340$	0	0
$341$	−4.87689	−0.264099
$342$	0	0
$343$	0	0
$344$	30.7386	1.65732
$345$	0	0
$346$	−4.63068	−0.248947
$347$	−8.49242	−0.455897	−0.227949	−	0.973673i	$-0.573202\pi$
$347$	−8.49242	−0.455897	−0.227949	+	0.973673i	$0.573202\pi$
$348$	0	0
$349$	11.5616	0.618876	0.309438	−	0.950920i	$-0.399859\pi$
$349$	11.5616	0.618876	0.309438	+	0.950920i	$0.399859\pi$
$350$	0	0
$351$	0	0
$352$	10.2462	0.546125
$353$	−10.4924	−0.558455	−0.279228	−	0.960225i	$-0.590078\pi$
$353$	−10.4924	−0.558455	−0.279228	+	0.960225i	$0.590078\pi$
$354$	0	0
$355$	0	0
$356$	5.12311	0.271524
$357$	0	0
$358$	−2.24621	−0.118716
$359$	−14.2462	−0.751886	−0.375943	−	0.926643i	$-0.622681\pi$
$359$	−14.2462	−0.751886	−0.375943	+	0.926643i	$0.622681\pi$
$360$	0	0
$361$	2.94602	0.155054
$362$	−15.3693	−0.807793
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.75379	0.0915470	0.0457735	−	0.998952i	$-0.485425\pi$
$367$	1.75379	0.0915470	0.0457735	+	0.998952i	$0.485425\pi$
$368$	−18.7386	−0.976819
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0.246211	0.0127483	0.00637417	−	0.999980i	$-0.497971\pi$
$373$	0.246211	0.0127483	0.00637417	+	0.999980i	$0.497971\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	72.9848	3.76391
$377$	−3.61553	−0.186209
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	0	0
$382$	−12.4924	−0.639168
$383$	6.24621	0.319166	0.159583	−	0.987184i	$-0.448985\pi$
$383$	6.24621	0.319166	0.159583	+	0.987184i	$0.448985\pi$
$384$	0	0
$385$	0	0
$386$	−19.8617	−1.01094
$387$	0	0
$388$	13.1231	0.666225
$389$	−35.8617	−1.81826	−0.909131	−	0.416510i	$-0.863253\pi$
$389$	−35.8617	−1.81826	−0.909131	+	0.416510i	$0.863253\pi$
$390$	0	0
$391$	−2.43845	−0.123318
$392$	45.9309	2.31986
$393$	0	0
$394$	22.8769	1.15252
$395$	0	0
$396$	0	0
$397$	19.3693	0.972118	0.486059	−	0.873926i	$-0.338434\pi$
$397$	19.3693	0.972118	0.486059	+	0.873926i	$0.338434\pi$
$398$	−40.9848	−2.05438
$399$	0	0
$400$	0	0
$401$	−39.1771	−1.95641	−0.978205	−	0.207641i	$-0.933421\pi$
$401$	−39.1771	−1.95641	−0.978205	+	0.207641i	$0.933421\pi$
$402$	0	0
$403$	−1.36932	−0.0682105
$404$	−49.6155	−2.46846
$405$	0	0
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	−14.6847	−0.726110	−0.363055	−	0.931768i	$-0.618266\pi$
$409$	−14.6847	−0.726110	−0.363055	+	0.931768i	$0.618266\pi$
$410$	0	0
$411$	0	0
$412$	−76.1080	−3.74957
$413$	0	0
$414$	0	0
$415$	0	0
$416$	2.87689	0.141051
$417$	0	0
$418$	−18.7386	−0.916537
$419$	0.492423	0.0240564	0.0120282	−	0.999928i	$-0.496171\pi$
$419$	0.492423	0.0240564	0.0120282	+	0.999928i	$0.496171\pi$
$420$	0	0
$421$	24.4384	1.19106	0.595529	−	0.803334i	$-0.296943\pi$
$421$	24.4384	1.19106	0.595529	+	0.803334i	$0.296943\pi$
$422$	−34.2462	−1.66708
$423$	0	0
$424$	−80.3542	−3.90234
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−21.3693	−1.03292
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	−26.6847	−1.28238	−0.641191	−	0.767381i	$-0.721560\pi$
$433$	−26.6847	−1.28238	−0.641191	+	0.767381i	$0.721560\pi$
$434$	0	0
$435$	0	0
$436$	−31.3693	−1.50232
$437$	11.4233	0.546450
$438$	0	0
$439$	−22.2462	−1.06175	−0.530877	−	0.847449i	$-0.678137\pi$
$439$	−22.2462	−1.06175	−0.530877	+	0.847449i	$0.678137\pi$
$440$	0	0
$441$	0	0
$442$	1.12311	0.0534207
$443$	−31.1231	−1.47870	−0.739352	−	0.673319i	$-0.764868\pi$
$443$	−31.1231	−1.47870	−0.739352	+	0.673319i	$0.764868\pi$
$444$	0	0
$445$	0	0
$446$	−38.2462	−1.81101
$447$	0	0
$448$	0	0
$449$	−36.7386	−1.73380	−0.866902	−	0.498479i	$-0.833892\pi$
$449$	−36.7386	−1.73380	−0.866902	+	0.498479i	$0.833892\pi$
$450$	0	0
$451$	−5.56155	−0.261883
$452$	−2.00000	−0.0940721
$453$	0	0
$454$	36.0000	1.68956
$455$	0	0
$456$	0	0
$457$	−13.8078	−0.645900	−0.322950	−	0.946416i	$-0.604675\pi$
$457$	−13.8078	−0.645900	−0.322950	+	0.946416i	$0.604675\pi$
$458$	−15.3693	−0.718161
$459$	0	0
$460$	0	0
$461$	8.24621	0.384064	0.192032	−	0.981389i	$-0.438492\pi$
$461$	8.24621	0.384064	0.192032	+	0.981389i	$0.438492\pi$
$462$	0	0
$463$	40.9848	1.90473	0.952364	−	0.304965i	$-0.0986447\pi$
$463$	40.9848	1.90473	0.952364	+	0.304965i	$0.0986447\pi$
$464$	63.3693	2.94185
$465$	0	0
$466$	9.12311	0.422620
$467$	21.3693	0.988854	0.494427	−	0.869219i	$-0.335378\pi$
$467$	21.3693	0.988854	0.494427	+	0.869219i	$0.335378\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	46.7386	2.15132
$473$	7.31534	0.336360
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−16.0000	−0.731823
$479$	−24.3002	−1.11030	−0.555152	−	0.831749i	$-0.687340\pi$
$479$	−24.3002	−1.11030	−0.555152	+	0.831749i	$0.687340\pi$
$480$	0	0
$481$	−2.24621	−0.102418
$482$	−8.63068	−0.393117
$483$	0	0
$484$	−39.0540	−1.77518
$485$	0	0
$486$	0	0
$487$	−17.3693	−0.787079	−0.393539	−	0.919308i	$-0.628750\pi$
$487$	−17.3693	−0.787079	−0.393539	+	0.919308i	$0.628750\pi$
$488$	−59.8617	−2.70981
$489$	0	0
$490$	0	0
$491$	21.3693	0.964384	0.482192	−	0.876066i	$-0.339841\pi$
$491$	21.3693	0.964384	0.482192	+	0.876066i	$0.339841\pi$
$492$	0	0
$493$	8.24621	0.371391
$494$	−5.26137	−0.236720
$495$	0	0
$496$	24.0000	1.07763
$497$	0	0
$498$	0	0
$499$	13.3693	0.598493	0.299246	−	0.954176i	$-0.403265\pi$
$499$	13.3693	0.598493	0.299246	+	0.954176i	$0.403265\pi$
$500$	0	0
$501$	0	0
$502$	21.7538	0.970919
$503$	−29.5616	−1.31808	−0.659042	−	0.752106i	$-0.729038\pi$
$503$	−29.5616	−1.31808	−0.659042	+	0.752106i	$0.729038\pi$
$504$	0	0
$505$	0	0
$506$	−9.75379	−0.433609
$507$	0	0
$508$	−90.3542	−4.00882
$509$	−25.1231	−1.11356	−0.556781	−	0.830659i	$-0.687964\pi$
$509$	−25.1231	−1.11356	−0.556781	+	0.830659i	$0.687964\pi$
$510$	0	0
$511$	0	0
$512$	50.4233	2.22842
$513$	0	0
$514$	39.3693	1.73651
$515$	0	0
$516$	0	0
$517$	17.3693	0.763902
$518$	0	0
$519$	0	0
$520$	0	0
$521$	35.5616	1.55798	0.778990	−	0.627036i	$-0.215732\pi$
$521$	35.5616	1.55798	0.778990	+	0.627036i	$0.215732\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	−65.8617	−2.87718
$525$	0	0
$526$	52.4924	2.28878
$527$	3.12311	0.136045
$528$	0	0
$529$	−17.0540	−0.741477
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.56155	−0.0676384
$534$	0	0
$535$	0	0
$536$	26.2462	1.13366
$537$	0	0
$538$	42.1080	1.81540
$539$	10.9309	0.470826
$540$	0	0
$541$	34.1080	1.46642	0.733208	−	0.680005i	$-0.238022\pi$
$541$	34.1080	1.46642	0.733208	+	0.680005i	$0.238022\pi$
$542$	−50.7386	−2.17941
$543$	0	0
$544$	−6.56155	−0.281324
$545$	0	0
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	1.12311	0.0479767
$549$	0	0
$550$	0	0
$551$	−38.6307	−1.64572
$552$	0	0
$553$	0	0
$554$	15.3693	0.652980
$555$	0	0
$556$	−4.00000	−0.169638
$557$	26.4924	1.12252	0.561260	−	0.827640i	$-0.310317\pi$
$557$	26.4924	1.12252	0.561260	+	0.827640i	$0.310317\pi$
$558$	0	0
$559$	2.05398	0.0868739
$560$	0	0
$561$	0	0
$562$	−27.8617	−1.17528
$563$	31.1231	1.31168	0.655841	−	0.754899i	$-0.272314\pi$
$563$	31.1231	1.31168	0.655841	+	0.754899i	$0.272314\pi$
$564$	0	0
$565$	0	0
$566$	54.7386	2.30084
$567$	0	0
$568$	−40.9848	−1.71969
$569$	−21.1231	−0.885527	−0.442763	−	0.896639i	$-0.646002\pi$
$569$	−21.1231	−0.885527	−0.442763	+	0.896639i	$0.646002\pi$
$570$	0	0
$571$	−30.7386	−1.28637	−0.643186	−	0.765710i	$-0.722388\pi$
$571$	−30.7386	−1.28637	−0.643186	+	0.765710i	$0.722388\pi$
$572$	3.12311	0.130584
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3.94602	0.164275	0.0821376	−	0.996621i	$-0.473825\pi$
$577$	3.94602	0.164275	0.0821376	+	0.996621i	$0.473825\pi$
$578$	−2.56155	−0.106547
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−19.1231	−0.791998
$584$	−80.3542	−3.32508
$585$	0	0
$586$	−2.87689	−0.118843
$587$	−28.9848	−1.19633	−0.598166	−	0.801372i	$-0.704104\pi$
$587$	−28.9848	−1.19633	−0.598166	+	0.801372i	$0.704104\pi$
$588$	0	0
$589$	−14.6307	−0.602847
$590$	0	0
$591$	0	0
$592$	39.3693	1.61807
$593$	27.7538	1.13971	0.569856	−	0.821745i	$-0.306999\pi$
$593$	27.7538	1.13971	0.569856	+	0.821745i	$0.306999\pi$
$594$	0	0
$595$	0	0
$596$	−55.8617	−2.28819
$597$	0	0
$598$	−2.73863	−0.111991
$599$	0.384472	0.0157091	0.00785455	−	0.999969i	$-0.497500\pi$
$599$	0.384472	0.0157091	0.00785455	+	0.999969i	$0.497500\pi$
$600$	0	0
$601$	−30.9848	−1.26390	−0.631949	−	0.775010i	$-0.717745\pi$
$601$	−30.9848	−1.26390	−0.631949	+	0.775010i	$0.717745\pi$
$602$	0	0
$603$	0	0
$604$	36.4924	1.48486
$605$	0	0
$606$	0	0
$607$	9.36932	0.380289	0.190144	−	0.981756i	$-0.439104\pi$
$607$	9.36932	0.380289	0.190144	+	0.981756i	$0.439104\pi$
$608$	30.7386	1.24662
$609$	0	0
$610$	0	0
$611$	4.87689	0.197298
$612$	0	0
$613$	−14.6847	−0.593108	−0.296554	−	0.955016i	$-0.595837\pi$
$613$	−14.6847	−0.593108	−0.296554	+	0.955016i	$0.595837\pi$
$614$	83.2311	3.35893
$615$	0	0
$616$	0	0
$617$	−44.2462	−1.78129	−0.890643	−	0.454704i	$-0.849745\pi$
$617$	−44.2462	−1.78129	−0.890643	+	0.454704i	$0.849745\pi$
$618$	0	0
$619$	5.36932	0.215811	0.107906	−	0.994161i	$-0.465586\pi$
$619$	5.36932	0.215811	0.107906	+	0.994161i	$0.465586\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	86.1080	3.44157
$627$	0	0
$628$	−30.4924	−1.21678
$629$	5.12311	0.204272
$630$	0	0
$631$	−0.684658	−0.0272558	−0.0136279	−	0.999907i	$-0.504338\pi$
$631$	−0.684658	−0.0272558	−0.0136279	+	0.999907i	$0.504338\pi$
$632$	61.4773	2.44543
$633$	0	0
$634$	46.1080	1.83118
$635$	0	0
$636$	0	0
$637$	3.06913	0.121603
$638$	32.9848	1.30588
$639$	0	0
$640$	0	0
$641$	28.9309	1.14270	0.571350	−	0.820706i	$-0.306420\pi$
$641$	28.9309	1.14270	0.571350	+	0.820706i	$0.306420\pi$
$642$	0	0
$643$	13.7538	0.542396	0.271198	−	0.962524i	$-0.412580\pi$
$643$	13.7538	0.542396	0.271198	+	0.962524i	$0.412580\pi$
$644$	0	0
$645$	0	0
$646$	12.0000	0.472134
$647$	−9.36932	−0.368346	−0.184173	−	0.982894i	$-0.558961\pi$
$647$	−9.36932	−0.368346	−0.184173	+	0.982894i	$0.558961\pi$
$648$	0	0
$649$	11.1231	0.436620
$650$	0	0
$651$	0	0
$652$	68.9848	2.70166
$653$	−32.9309	−1.28868	−0.644342	−	0.764737i	$-0.722869\pi$
$653$	−32.9309	−1.28868	−0.644342	+	0.764737i	$0.722869\pi$
$654$	0	0
$655$	0	0
$656$	27.3693	1.06859
$657$	0	0
$658$	0	0
$659$	9.86174	0.384159	0.192079	−	0.981379i	$-0.438477\pi$
$659$	9.86174	0.384159	0.192079	+	0.981379i	$0.438477\pi$
$660$	0	0
$661$	13.3153	0.517907	0.258953	−	0.965890i	$-0.416622\pi$
$661$	13.3153	0.517907	0.258953	+	0.965890i	$0.416622\pi$
$662$	89.4773	3.47763
$663$	0	0
$664$	5.75379	0.223290
$665$	0	0
$666$	0	0
$667$	−20.1080	−0.778583
$668$	90.3542	3.49591
$669$	0	0
$670$	0	0
$671$	−14.2462	−0.549969
$672$	0	0
$673$	0.738634	0.0284722	0.0142361	−	0.999899i	$-0.495468\pi$
$673$	0.738634	0.0284722	0.0142361	+	0.999899i	$0.495468\pi$
$674$	−42.8769	−1.65156
$675$	0	0
$676$	−58.4233	−2.24705
$677$	−1.31534	−0.0505527	−0.0252763	−	0.999681i	$-0.508047\pi$
$677$	−1.31534	−0.0505527	−0.0252763	+	0.999681i	$0.508047\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	12.4924	0.478360
$683$	−9.56155	−0.365863	−0.182931	−	0.983126i	$-0.558559\pi$
$683$	−9.56155	−0.365863	−0.182931	+	0.983126i	$0.558559\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−36.0000	−1.37249
$689$	−5.36932	−0.204555
$690$	0	0
$691$	−28.9848	−1.10264	−0.551318	−	0.834295i	$-0.685875\pi$
$691$	−28.9848	−1.10264	−0.551318	+	0.834295i	$0.685875\pi$
$692$	8.24621	0.313474
$693$	0	0
$694$	21.7538	0.825763
$695$	0	0
$696$	0	0
$697$	3.56155	0.134903
$698$	−29.6155	−1.12096
$699$	0	0
$700$	0	0
$701$	−15.3693	−0.580491	−0.290246	−	0.956952i	$-0.593737\pi$
$701$	−15.3693	−0.580491	−0.290246	+	0.956952i	$0.593737\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	−2.24621	−0.0846573
$705$	0	0
$706$	26.8769	1.01153
$707$	0	0
$708$	0	0
$709$	−44.7386	−1.68019	−0.840097	−	0.542436i	$-0.817502\pi$
$709$	−44.7386	−1.68019	−0.840097	+	0.542436i	$0.817502\pi$
$710$	0	0
$711$	0	0
$712$	−7.36932	−0.276177
$713$	−7.61553	−0.285204
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	0	0
$718$	36.4924	1.36189
$719$	11.8078	0.440355	0.220178	−	0.975460i	$-0.429336\pi$
$719$	11.8078	0.440355	0.220178	+	0.975460i	$0.429336\pi$
$720$	0	0
$721$	0	0
$722$	−7.54640	−0.280848
$723$	0	0
$724$	27.3693	1.01717
$725$	0	0
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.68466	−0.173268
$732$	0	0
$733$	11.7538	0.434136	0.217068	−	0.976156i	$-0.430351\pi$
$733$	11.7538	0.434136	0.217068	+	0.976156i	$0.430351\pi$
$734$	−4.49242	−0.165818
$735$	0	0
$736$	16.0000	0.589768
$737$	6.24621	0.230082
$738$	0	0
$739$	−20.6847	−0.760897	−0.380449	−	0.924802i	$-0.624230\pi$
$739$	−20.6847	−0.760897	−0.380449	+	0.924802i	$0.624230\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	28.4924	1.04529	0.522643	−	0.852552i	$-0.324946\pi$
$743$	28.4924	1.04529	0.522643	+	0.852552i	$0.324946\pi$
$744$	0	0
$745$	0	0
$746$	−0.630683	−0.0230909
$747$	0	0
$748$	−7.12311	−0.260447
$749$	0	0
$750$	0	0
$751$	−25.3693	−0.925740	−0.462870	−	0.886426i	$-0.653180\pi$
$751$	−25.3693	−0.925740	−0.462870	+	0.886426i	$0.653180\pi$
$752$	−85.4773	−3.11704
$753$	0	0
$754$	9.26137	0.337279
$755$	0	0
$756$	0	0
$757$	−16.0540	−0.583492	−0.291746	−	0.956496i	$-0.594236\pi$
$757$	−16.0540	−0.583492	−0.291746	+	0.956496i	$0.594236\pi$
$758$	30.7386	1.11648
$759$	0	0
$760$	0	0
$761$	15.7538	0.571074	0.285537	−	0.958368i	$-0.407828\pi$
$761$	15.7538	0.571074	0.285537	+	0.958368i	$0.407828\pi$
$762$	0	0
$763$	0	0
$764$	22.2462	0.804840
$765$	0	0
$766$	−16.0000	−0.578103
$767$	3.12311	0.112769
$768$	0	0
$769$	40.5464	1.46214	0.731070	−	0.682302i	$-0.239021\pi$
$769$	40.5464	1.46214	0.731070	+	0.682302i	$0.239021\pi$
$770$	0	0
$771$	0	0
$772$	35.3693	1.27297
$773$	−8.63068	−0.310424	−0.155212	−	0.987881i	$-0.549606\pi$
$773$	−8.63068	−0.310424	−0.155212	+	0.987881i	$0.549606\pi$
$774$	0	0
$775$	0	0
$776$	−18.8769	−0.677641
$777$	0	0
$778$	91.8617	3.29340
$779$	−16.6847	−0.597790
$780$	0	0
$781$	−9.75379	−0.349018
$782$	6.24621	0.223364
$783$	0	0
$784$	−53.7926	−1.92116
$785$	0	0
$786$	0	0
$787$	10.2462	0.365238	0.182619	−	0.983184i	$-0.441543\pi$
$787$	10.2462	0.365238	0.182619	+	0.983184i	$0.441543\pi$
$788$	−40.7386	−1.45125
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4.00000	−0.142044
$794$	−49.6155	−1.76079
$795$	0	0
$796$	72.9848	2.58688
$797$	−9.61553	−0.340599	−0.170300	−	0.985392i	$-0.554474\pi$
$797$	−9.61553	−0.340599	−0.170300	+	0.985392i	$0.554474\pi$
$798$	0	0
$799$	−11.1231	−0.393507
$800$	0	0
$801$	0	0
$802$	100.354	3.54363
$803$	−19.1231	−0.674840
$804$	0	0
$805$	0	0
$806$	3.50758	0.123549
$807$	0	0
$808$	71.3693	2.51076
$809$	−15.9460	−0.560632	−0.280316	−	0.959908i	$-0.590439\pi$
$809$	−15.9460	−0.560632	−0.280316	+	0.959908i	$0.590439\pi$
$810$	0	0
$811$	−45.3693	−1.59313	−0.796566	−	0.604551i	$-0.793352\pi$
$811$	−45.3693	−1.59313	−0.796566	+	0.604551i	$0.793352\pi$
$812$	0	0
$813$	0	0
$814$	20.4924	0.718259
$815$	0	0
$816$	0	0
$817$	21.9460	0.767794
$818$	37.6155	1.31520
$819$	0	0
$820$	0	0
$821$	12.4384	0.434105	0.217052	−	0.976160i	$-0.430356\pi$
$821$	12.4384	0.434105	0.217052	+	0.976160i	$0.430356\pi$
$822$	0	0
$823$	−3.50758	−0.122266	−0.0611332	−	0.998130i	$-0.519471\pi$
$823$	−3.50758	−0.122266	−0.0611332	+	0.998130i	$0.519471\pi$
$824$	109.477	3.81382
$825$	0	0
$826$	0	0
$827$	47.4233	1.64907	0.824535	−	0.565811i	$-0.191437\pi$
$827$	47.4233	1.64907	0.824535	+	0.565811i	$0.191437\pi$
$828$	0	0
$829$	17.5076	0.608063	0.304032	−	0.952662i	$-0.401667\pi$
$829$	17.5076	0.608063	0.304032	+	0.952662i	$0.401667\pi$
$830$	0	0
$831$	0	0
$832$	−0.630683	−0.0218650
$833$	−7.00000	−0.242536
$834$	0	0
$835$	0	0
$836$	33.3693	1.15410
$837$	0	0
$838$	−1.26137	−0.0435732
$839$	26.0540	0.899483	0.449742	−	0.893159i	$-0.351516\pi$
$839$	26.0540	0.899483	0.449742	+	0.893159i	$0.351516\pi$
$840$	0	0
$841$	39.0000	1.34483
$842$	−62.6004	−2.15735
$843$	0	0
$844$	60.9848	2.09918
$845$	0	0
$846$	0	0
$847$	0	0
$848$	94.1080	3.23168
$849$	0	0
$850$	0	0
$851$	−12.4924	−0.428235
$852$	0	0
$853$	28.7386	0.983992	0.491996	−	0.870597i	$-0.336267\pi$
$853$	28.7386	0.983992	0.491996	+	0.870597i	$0.336267\pi$
$854$	0	0
$855$	0	0
$856$	30.7386	1.05062
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	12.0000	0.409435	0.204717	−	0.978821i	$-0.434372\pi$
$859$	12.0000	0.409435	0.204717	+	0.978821i	$0.434372\pi$
$860$	0	0
$861$	0	0
$862$	−61.4773	−2.09392
$863$	−9.75379	−0.332023	−0.166011	−	0.986124i	$-0.553089\pi$
$863$	−9.75379	−0.332023	−0.166011	+	0.986124i	$0.553089\pi$
$864$	0	0
$865$	0	0
$866$	68.3542	2.32277
$867$	0	0
$868$	0	0
$869$	14.6307	0.496312
$870$	0	0
$871$	1.75379	0.0594249
$872$	45.1231	1.52806
$873$	0	0
$874$	−29.2614	−0.989780
$875$	0	0
$876$	0	0
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	56.9848	1.92315
$879$	0	0
$880$	0	0
$881$	−40.2462	−1.35593	−0.677965	−	0.735095i	$-0.737138\pi$
$881$	−40.2462	−1.35593	−0.677965	+	0.735095i	$0.737138\pi$
$882$	0	0
$883$	23.4233	0.788257	0.394128	−	0.919055i	$-0.371047\pi$
$883$	23.4233	0.788257	0.394128	+	0.919055i	$0.371047\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	79.7235	2.67836
$887$	18.4384	0.619102	0.309551	−	0.950883i	$-0.399821\pi$
$887$	18.4384	0.619102	0.309551	+	0.950883i	$0.399821\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	68.1080	2.28042
$893$	52.1080	1.74373
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	94.1080	3.14042
$899$	25.7538	0.858937
$900$	0	0
$901$	12.2462	0.407980
$902$	14.2462	0.474347
$903$	0	0
$904$	2.87689	0.0956841
$905$	0	0
$906$	0	0
$907$	9.86174	0.327454	0.163727	−	0.986506i	$-0.447648\pi$
$907$	9.86174	0.327454	0.163727	+	0.986506i	$0.447648\pi$
$908$	−64.1080	−2.12750
$909$	0	0
$910$	0	0
$911$	−24.3002	−0.805101	−0.402551	−	0.915398i	$-0.631876\pi$
$911$	−24.3002	−0.805101	−0.402551	+	0.915398i	$0.631876\pi$
$912$	0	0
$913$	1.36932	0.0453178
$914$	35.3693	1.16991
$915$	0	0
$916$	27.3693	0.904308
$917$	0	0
$918$	0	0
$919$	−16.6847	−0.550376	−0.275188	−	0.961390i	$-0.588740\pi$
$919$	−16.6847	−0.550376	−0.275188	+	0.961390i	$0.588740\pi$
$920$	0	0
$921$	0	0
$922$	−21.1231	−0.695652
$923$	−2.73863	−0.0901432
$924$	0	0
$925$	0	0
$926$	−104.985	−3.45002
$927$	0	0
$928$	−54.1080	−1.77618
$929$	−3.06913	−0.100695	−0.0503474	−	0.998732i	$-0.516033\pi$
$929$	−3.06913	−0.100695	−0.0503474	+	0.998732i	$0.516033\pi$
$930$	0	0
$931$	32.7926	1.07473
$932$	−16.2462	−0.532162
$933$	0	0
$934$	−54.7386	−1.79110
$935$	0	0
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	−8.68466	−0.282811
$944$	−54.7386	−1.78159
$945$	0	0
$946$	−18.7386	−0.609246
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	−5.36932	−0.174295
$950$	0	0
$951$	0	0
$952$	0	0
$953$	36.3542	1.17763	0.588813	−	0.808269i	$-0.299595\pi$
$953$	36.3542	1.17763	0.588813	+	0.808269i	$0.299595\pi$
$954$	0	0
$955$	0	0
$956$	28.4924	0.921511
$957$	0	0
$958$	62.2462	2.01108
$959$	0	0
$960$	0	0
$961$	−21.2462	−0.685362
$962$	5.75379	0.185510
$963$	0	0
$964$	15.3693	0.495012
$965$	0	0
$966$	0	0
$967$	−42.4384	−1.36473	−0.682364	−	0.731012i	$-0.739048\pi$
$967$	−42.4384	−1.36473	−0.682364	+	0.731012i	$0.739048\pi$
$968$	56.1771	1.80560
$969$	0	0
$970$	0	0
$971$	43.6155	1.39969	0.699844	−	0.714295i	$-0.253253\pi$
$971$	43.6155	1.39969	0.699844	+	0.714295i	$0.253253\pi$
$972$	0	0
$973$	0	0
$974$	44.4924	1.42563
$975$	0	0
$976$	70.1080	2.24410
$977$	8.24621	0.263820	0.131910	−	0.991262i	$-0.457889\pi$
$977$	8.24621	0.263820	0.131910	+	0.991262i	$0.457889\pi$
$978$	0	0
$979$	−1.75379	−0.0560513
$980$	0	0
$981$	0	0
$982$	−54.7386	−1.74678
$983$	30.9309	0.986542	0.493271	−	0.869876i	$-0.335801\pi$
$983$	30.9309	0.986542	0.493271	+	0.869876i	$0.335801\pi$
$984$	0	0
$985$	0	0
$986$	−21.1231	−0.672697
$987$	0	0
$988$	9.36932	0.298078
$989$	11.4233	0.363240
$990$	0	0
$991$	42.7386	1.35764	0.678819	−	0.734306i	$-0.262492\pi$
$991$	42.7386	1.35764	0.678819	+	0.734306i	$0.262492\pi$
$992$	−20.4924	−0.650635
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	−34.2462	−1.08404
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3825.2.a.s.1.1		2
3.2	odd	2		1275.2.a.n.1.2		2
5.4	even	2		153.2.a.e.1.2		2
15.2	even	4		1275.2.b.d.1174.4		4
15.8	even	4		1275.2.b.d.1174.1		4
15.14	odd	2		51.2.a.b.1.1	✓	2
20.19	odd	2		2448.2.a.v.1.1		2
35.34	odd	2		7497.2.a.v.1.2		2
40.19	odd	2		9792.2.a.cz.1.2		2
40.29	even	2		9792.2.a.cy.1.2		2
60.59	even	2		816.2.a.m.1.2		2
85.84	even	2		2601.2.a.t.1.2		2
105.104	even	2		2499.2.a.o.1.1		2
120.29	odd	2		3264.2.a.bl.1.1		2
120.59	even	2		3264.2.a.bg.1.1		2
165.164	even	2		6171.2.a.p.1.2		2
195.194	odd	2		8619.2.a.q.1.2		2
255.14	even	16		867.2.h.j.757.2		16
255.29	even	16		867.2.h.j.688.3		16
255.44	even	16		867.2.h.j.712.3		16
255.59	odd	8		867.2.e.f.829.4		8
255.74	even	16		867.2.h.j.733.1		16
255.89	odd	4		867.2.d.c.577.4		4
255.104	odd	8		867.2.e.f.616.1		8
255.134	odd	8		867.2.e.f.616.2		8
255.149	odd	4		867.2.d.c.577.3		4
255.164	even	16		867.2.h.j.733.2		16
255.179	odd	8		867.2.e.f.829.3		8
255.194	even	16		867.2.h.j.712.4		16
255.209	even	16		867.2.h.j.688.4		16
255.224	even	16		867.2.h.j.757.1		16
255.254	odd	2		867.2.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.a.b.1.1	✓	2	15.14	odd	2
153.2.a.e.1.2		2	5.4	even	2
816.2.a.m.1.2		2	60.59	even	2
867.2.a.f.1.1		2	255.254	odd	2
867.2.d.c.577.3		4	255.149	odd	4
867.2.d.c.577.4		4	255.89	odd	4
867.2.e.f.616.1		8	255.104	odd	8
867.2.e.f.616.2		8	255.134	odd	8
867.2.e.f.829.3		8	255.179	odd	8
867.2.e.f.829.4		8	255.59	odd	8
867.2.h.j.688.3		16	255.29	even	16
867.2.h.j.688.4		16	255.209	even	16
867.2.h.j.712.3		16	255.44	even	16
867.2.h.j.712.4		16	255.194	even	16
867.2.h.j.733.1		16	255.74	even	16
867.2.h.j.733.2		16	255.164	even	16
867.2.h.j.757.1		16	255.224	even	16
867.2.h.j.757.2		16	255.14	even	16
1275.2.a.n.1.2		2	3.2	odd	2
1275.2.b.d.1174.1		4	15.8	even	4
1275.2.b.d.1174.4		4	15.2	even	4
2448.2.a.v.1.1		2	20.19	odd	2
2499.2.a.o.1.1		2	105.104	even	2
2601.2.a.t.1.2		2	85.84	even	2
3264.2.a.bg.1.1		2	120.59	even	2
3264.2.a.bl.1.1		2	120.29	odd	2
3825.2.a.s.1.1		2	1.1	even	1	trivial
6171.2.a.p.1.2		2	165.164	even	2
7497.2.a.v.1.2		2	35.34	odd	2
8619.2.a.q.1.2		2	195.194	odd	2
9792.2.a.cy.1.2		2	40.29	even	2
9792.2.a.cz.1.2		2	40.19	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:	\( N \)	\(=\)	\( 3825 = 3^{2} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3825.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56155 q^{2} +4.56155 q^{4} -6.56155 q^{8} +O(q^{10})\)	\(q-2.56155 q^{2} +4.56155 q^{4} -6.56155 q^{8} -1.56155 q^{11} -0.438447 q^{13} +7.68466 q^{16} +1.00000 q^{17} -4.68466 q^{19} +4.00000 q^{22} -2.43845 q^{23} +1.12311 q^{26} +8.24621 q^{29} +3.12311 q^{31} -6.56155 q^{32} -2.56155 q^{34} +5.12311 q^{37} +12.0000 q^{38} +3.56155 q^{41} -4.68466 q^{43} -7.12311 q^{44} +6.24621 q^{46} -11.1231 q^{47} -7.00000 q^{49} -2.00000 q^{52} +12.2462 q^{53} -21.1231 q^{58} -7.12311 q^{59} +9.12311 q^{61} -8.00000 q^{62} +1.43845 q^{64} -4.00000 q^{67} +4.56155 q^{68} +6.24621 q^{71} +12.2462 q^{73} -13.1231 q^{74} -21.3693 q^{76} -9.36932 q^{79} -9.12311 q^{82} -0.876894 q^{83} +12.0000 q^{86} +10.2462 q^{88} +1.12311 q^{89} -11.1231 q^{92} +28.4924 q^{94} +2.87689 q^{97} +17.9309 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 5 q^{4} - 9 q^{8}+O(q^{10}) \) 2 * q - q^2 + 5 * q^4 - 9 * q^8	\( 2 q - q^{2} + 5 q^{4} - 9 q^{8} + q^{11} - 5 q^{13} + 3 q^{16} + 2 q^{17} + 3 q^{19} + 8 q^{22} - 9 q^{23} - 6 q^{26} - 2 q^{31} - 9 q^{32} - q^{34} + 2 q^{37} + 24 q^{38} + 3 q^{41} + 3 q^{43} - 6 q^{44} - 4 q^{46} - 14 q^{47} - 14 q^{49} - 4 q^{52} + 8 q^{53} - 34 q^{58} - 6 q^{59} + 10 q^{61} - 16 q^{62} + 7 q^{64} - 8 q^{67} + 5 q^{68} - 4 q^{71} + 8 q^{73} - 18 q^{74} - 18 q^{76} + 6 q^{79} - 10 q^{82} - 10 q^{83} + 24 q^{86} + 4 q^{88} - 6 q^{89} - 14 q^{92} + 24 q^{94} + 14 q^{97} + 7 q^{98}+O(q^{100}) \) 2 * q - q^2 + 5 * q^4 - 9 * q^8 + q^11 - 5 * q^13 + 3 * q^16 + 2 * q^17 + 3 * q^19 + 8 * q^22 - 9 * q^23 - 6 * q^26 - 2 * q^31 - 9 * q^32 - q^34 + 2 * q^37 + 24 * q^38 + 3 * q^41 + 3 * q^43 - 6 * q^44 - 4 * q^46 - 14 * q^47 - 14 * q^49 - 4 * q^52 + 8 * q^53 - 34 * q^58 - 6 * q^59 + 10 * q^61 - 16 * q^62 + 7 * q^64 - 8 * q^67 + 5 * q^68 - 4 * q^71 + 8 * q^73 - 18 * q^74 - 18 * q^76 + 6 * q^79 - 10 * q^82 - 10 * q^83 + 24 * q^86 + 4 * q^88 - 6 * q^89 - 14 * q^92 + 24 * q^94 + 14 * q^97 + 7 * q^98

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