LMFDB - Embedded newform 2475.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.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:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.193937 q^{2} -1.96239 q^{4} -3.35026 q^{7} +0.768452 q^{8} +O(q^{10})$	$q-0.193937 q^{2} -1.96239 q^{4} -3.35026 q^{7} +0.768452 q^{8} -1.00000 q^{11} -2.96239 q^{13} +0.649738 q^{14} +3.77575 q^{16} -4.57452 q^{17} -4.31265 q^{19} +0.193937 q^{22} -6.70052 q^{23} +0.574515 q^{26} +6.57452 q^{28} +3.61213 q^{29} +9.92478 q^{31} -2.26916 q^{32} +0.887166 q^{34} +2.00000 q^{37} +0.836381 q^{38} +4.38787 q^{41} +9.27504 q^{43} +1.96239 q^{44} +1.29948 q^{46} -9.92478 q^{47} +4.22425 q^{49} +5.81336 q^{52} +4.70052 q^{53} -2.57452 q^{56} -0.700523 q^{58} -10.7005 q^{59} -8.70052 q^{61} -1.92478 q^{62} -7.11142 q^{64} -5.92478 q^{67} +8.97698 q^{68} -9.92478 q^{71} +7.73813 q^{73} -0.387873 q^{74} +8.46310 q^{76} +3.35026 q^{77} +11.5369 q^{79} -0.850969 q^{82} +10.8872 q^{83} -1.79877 q^{86} -0.768452 q^{88} +2.77575 q^{89} +9.92478 q^{91} +13.1490 q^{92} +1.92478 q^{94} -0.0752228 q^{97} -0.819237 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 5 q^{4} - 9 q^{8}+O(q^{10}) $ 3 * q - q^2 + 5 * q^4 - 9 * q^8	$ 3 q - q^{2} + 5 q^{4} - 9 q^{8} - 3 q^{11} + 2 q^{13} + 12 q^{14} + 13 q^{16} - 2 q^{17} + 8 q^{19} + q^{22} - 10 q^{26} + 8 q^{28} + 10 q^{29} + 8 q^{31} - 29 q^{32} - 30 q^{34} + 6 q^{37} + 14 q^{41} - 4 q^{43} - 5 q^{44} + 24 q^{46} - 8 q^{47} + 11 q^{49} + 30 q^{52} - 6 q^{53} + 4 q^{56} + 18 q^{58} - 12 q^{59} - 6 q^{61} + 16 q^{62} + 13 q^{64} + 4 q^{67} + 42 q^{68} - 8 q^{71} + 14 q^{73} - 2 q^{74} + 48 q^{76} + 12 q^{79} - 26 q^{82} + 8 q^{86} + 9 q^{88} + 10 q^{89} + 8 q^{91} + 16 q^{92} - 16 q^{94} - 22 q^{97} + 39 q^{98}+O(q^{100}) $ 3 * q - q^2 + 5 * q^4 - 9 * q^8 - 3 * q^11 + 2 * q^13 + 12 * q^14 + 13 * q^16 - 2 * q^17 + 8 * q^19 + q^22 - 10 * q^26 + 8 * q^28 + 10 * q^29 + 8 * q^31 - 29 * q^32 - 30 * q^34 + 6 * q^37 + 14 * q^41 - 4 * q^43 - 5 * q^44 + 24 * q^46 - 8 * q^47 + 11 * q^49 + 30 * q^52 - 6 * q^53 + 4 * q^56 + 18 * q^58 - 12 * q^59 - 6 * q^61 + 16 * q^62 + 13 * q^64 + 4 * q^67 + 42 * q^68 - 8 * q^71 + 14 * q^73 - 2 * q^74 + 48 * q^76 + 12 * q^79 - 26 * q^82 + 8 * q^86 + 9 * q^88 + 10 * q^89 + 8 * q^91 + 16 * q^92 - 16 * q^94 - 22 * q^97 + 39 * 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 $

$2$	−0.193937	−0.137134	−0.0685669	−	0.997647i	$-0.521843\pi$
$2$	−0.193937	−0.137134	−0.0685669	+	0.997647i	$0.521843\pi$
$3$	0	0
$3$	0	0
$4$	−1.96239	−0.981194
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.35026	−1.26628	−0.633140	−	0.774037i	$-0.718234\pi$
$7$	−3.35026	−1.26628	−0.633140	+	0.774037i	$0.718234\pi$
$8$	0.768452	0.271689
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.96239	−0.821619	−0.410809	−	0.911721i	$-0.634754\pi$
$13$	−2.96239	−0.821619	−0.410809	+	0.911721i	$0.634754\pi$
$14$	0.649738	0.173650
$15$	0	0
$16$	3.77575	0.943937
$17$	−4.57452	−1.10948	−0.554741	−	0.832023i	$-0.687183\pi$
$17$	−4.57452	−1.10948	−0.554741	+	0.832023i	$0.687183\pi$
$18$	0	0
$19$	−4.31265	−0.989390	−0.494695	−	0.869067i	$-0.664720\pi$
$19$	−4.31265	−0.989390	−0.494695	+	0.869067i	$0.664720\pi$
$20$	0	0
$21$	0	0
$22$	0.193937	0.0413474
$23$	−6.70052	−1.39716	−0.698578	−	0.715534i	$-0.746183\pi$
$23$	−6.70052	−1.39716	−0.698578	+	0.715534i	$0.746183\pi$
$24$	0	0
$25$	0	0
$26$	0.574515	0.112672
$27$	0	0
$28$	6.57452	1.24247
$29$	3.61213	0.670755	0.335378	−	0.942084i	$-0.391136\pi$
$29$	3.61213	0.670755	0.335378	+	0.942084i	$0.391136\pi$
$30$	0	0
$31$	9.92478	1.78254	0.891271	−	0.453470i	$-0.149814\pi$
$31$	9.92478	1.78254	0.891271	+	0.453470i	$0.149814\pi$
$32$	−2.26916	−0.401134
$33$	0	0
$34$	0.887166	0.152148
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0.836381	0.135679
$39$	0	0
$40$	0	0
$41$	4.38787	0.685271	0.342635	−	0.939468i	$-0.388680\pi$
$41$	4.38787	0.685271	0.342635	+	0.939468i	$0.388680\pi$
$42$	0	0
$43$	9.27504	1.41443	0.707215	−	0.706998i	$-0.249951\pi$
$43$	9.27504	1.41443	0.707215	+	0.706998i	$0.249951\pi$
$44$	1.96239	0.295841
$45$	0	0
$46$	1.29948	0.191597
$47$	−9.92478	−1.44768	−0.723839	−	0.689969i	$-0.757624\pi$
$47$	−9.92478	−1.44768	−0.723839	+	0.689969i	$0.757624\pi$
$48$	0	0
$49$	4.22425	0.603465
$50$	0	0
$51$	0	0
$52$	5.81336	0.806168
$53$	4.70052	0.645667	0.322833	−	0.946456i	$-0.395365\pi$
$53$	4.70052	0.645667	0.322833	+	0.946456i	$0.395365\pi$
$54$	0	0
$55$	0	0
$56$	−2.57452	−0.344034
$57$	0	0
$58$	−0.700523	−0.0919832
$59$	−10.7005	−1.39309	−0.696545	−	0.717513i	$-0.745280\pi$
$59$	−10.7005	−1.39309	−0.696545	+	0.717513i	$0.745280\pi$
$60$	0	0
$61$	−8.70052	−1.11399	−0.556994	−	0.830517i	$-0.688045\pi$
$61$	−8.70052	−1.11399	−0.556994	+	0.830517i	$0.688045\pi$
$62$	−1.92478	−0.244447
$63$	0	0
$64$	−7.11142	−0.888927
$65$	0	0
$66$	0	0
$67$	−5.92478	−0.723827	−0.361913	−	0.932212i	$-0.617876\pi$
$67$	−5.92478	−0.723827	−0.361913	+	0.932212i	$0.617876\pi$
$68$	8.97698	1.08862
$69$	0	0
$70$	0	0
$71$	−9.92478	−1.17785	−0.588927	−	0.808186i	$-0.700450\pi$
$71$	−9.92478	−1.17785	−0.588927	+	0.808186i	$0.700450\pi$
$72$	0	0
$73$	7.73813	0.905680	0.452840	−	0.891592i	$-0.350411\pi$
$73$	7.73813	0.905680	0.452840	+	0.891592i	$0.350411\pi$
$74$	−0.387873	−0.0450893
$75$	0	0
$76$	8.46310	0.970784
$77$	3.35026	0.381798
$78$	0	0
$79$	11.5369	1.29800	0.649002	−	0.760787i	$-0.275187\pi$
$79$	11.5369	1.29800	0.649002	+	0.760787i	$0.275187\pi$
$80$	0	0
$81$	0	0
$82$	−0.850969	−0.0939738
$83$	10.8872	1.19502	0.597511	−	0.801861i	$-0.296156\pi$
$83$	10.8872	1.19502	0.597511	+	0.801861i	$0.296156\pi$
$84$	0	0
$85$	0	0
$86$	−1.79877	−0.193966
$87$	0	0
$88$	−0.768452	−0.0819173
$89$	2.77575	0.294229	0.147114	−	0.989120i	$-0.453001\pi$
$89$	2.77575	0.294229	0.147114	+	0.989120i	$0.453001\pi$
$90$	0	0
$91$	9.92478	1.04040
$92$	13.1490	1.37088
$93$	0	0
$94$	1.92478	0.198526
$95$	0	0
$96$	0	0
$97$	−0.0752228	−0.00763772	−0.00381886	−	0.999993i	$-0.501216\pi$
$97$	−0.0752228	−0.00763772	−0.00381886	+	0.999993i	$0.501216\pi$
$98$	−0.819237	−0.0827555
$99$	0	0
$100$	0	0
$101$	15.0884	1.50135	0.750676	−	0.660671i	$-0.229728\pi$
$101$	15.0884	1.50135	0.750676	+	0.660671i	$0.229728\pi$
$102$	0	0
$103$	3.22425	0.317695	0.158848	−	0.987303i	$-0.449222\pi$
$103$	3.22425	0.317695	0.158848	+	0.987303i	$0.449222\pi$
$104$	−2.27645	−0.223225
$105$	0	0
$106$	−0.911603	−0.0885427
$107$	−0.962389	−0.0930376	−0.0465188	−	0.998917i	$-0.514813\pi$
$107$	−0.962389	−0.0930376	−0.0465188	+	0.998917i	$0.514813\pi$
$108$	0	0
$109$	11.4010	1.09202	0.546011	−	0.837778i	$-0.316146\pi$
$109$	11.4010	1.09202	0.546011	+	0.837778i	$0.316146\pi$
$110$	0	0
$111$	0	0
$112$	−12.6497	−1.19529
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	−7.08840	−0.658141
$117$	0	0
$118$	2.07522	0.191040
$119$	15.3258	1.40492
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.68735	0.152765
$123$	0	0
$124$	−19.4763	−1.74902
$125$	0	0
$126$	0	0
$127$	14.5745	1.29328	0.646640	−	0.762796i	$-0.276174\pi$
$127$	14.5745	1.29328	0.646640	+	0.762796i	$0.276174\pi$
$128$	5.91748	0.523037
$129$	0	0
$130$	0	0
$131$	5.92478	0.517650	0.258825	−	0.965924i	$-0.416665\pi$
$131$	5.92478	0.517650	0.258825	+	0.965924i	$0.416665\pi$
$132$	0	0
$133$	14.4485	1.25284
$134$	1.14903	0.0992612
$135$	0	0
$136$	−3.51530	−0.301434
$137$	13.8496	1.18325	0.591624	−	0.806214i	$-0.298487\pi$
$137$	13.8496	1.18325	0.591624	+	0.806214i	$0.298487\pi$
$138$	0	0
$139$	13.6121	1.15457	0.577283	−	0.816544i	$-0.304113\pi$
$139$	13.6121	1.15457	0.577283	+	0.816544i	$0.304113\pi$
$140$	0	0
$141$	0	0
$142$	1.92478	0.161524
$143$	2.96239	0.247727
$144$	0	0
$145$	0	0
$146$	−1.50071	−0.124199
$147$	0	0
$148$	−3.92478	−0.322615
$149$	−1.53690	−0.125908	−0.0629540	−	0.998016i	$-0.520052\pi$
$149$	−1.53690	−0.125908	−0.0629540	+	0.998016i	$0.520052\pi$
$150$	0	0
$151$	−6.76116	−0.550215	−0.275108	−	0.961413i	$-0.588713\pi$
$151$	−6.76116	−0.550215	−0.275108	+	0.961413i	$0.588713\pi$
$152$	−3.31406	−0.268806
$153$	0	0
$154$	−0.649738	−0.0523574
$155$	0	0
$156$	0	0
$157$	5.47627	0.437054	0.218527	−	0.975831i	$-0.429875\pi$
$157$	5.47627	0.437054	0.218527	+	0.975831i	$0.429875\pi$
$158$	−2.23743	−0.178000
$159$	0	0
$160$	0	0
$161$	22.4485	1.76919
$162$	0	0
$163$	−12.6253	−0.988890	−0.494445	−	0.869209i	$-0.664629\pi$
$163$	−12.6253	−0.988890	−0.494445	+	0.869209i	$0.664629\pi$
$164$	−8.61071	−0.672384
$165$	0	0
$166$	−2.11142	−0.163878
$167$	18.3634	1.42101	0.710503	−	0.703695i	$-0.248468\pi$
$167$	18.3634	1.42101	0.710503	+	0.703695i	$0.248468\pi$
$168$	0	0
$169$	−4.22425	−0.324943
$170$	0	0
$171$	0	0
$172$	−18.2012	−1.38783
$173$	−8.57452	−0.651908	−0.325954	−	0.945386i	$-0.605686\pi$
$173$	−8.57452	−0.651908	−0.325954	+	0.945386i	$0.605686\pi$
$174$	0	0
$175$	0	0
$176$	−3.77575	−0.284608
$177$	0	0
$178$	−0.538319	−0.0403487
$179$	−14.1768	−1.05962	−0.529812	−	0.848115i	$-0.677737\pi$
$179$	−14.1768	−1.05962	−0.529812	+	0.848115i	$0.677737\pi$
$180$	0	0
$181$	−5.22425	−0.388316	−0.194158	−	0.980970i	$-0.562197\pi$
$181$	−5.22425	−0.388316	−0.194158	+	0.980970i	$0.562197\pi$
$182$	−1.92478	−0.142674
$183$	0	0
$184$	−5.14903	−0.379592
$185$	0	0
$186$	0	0
$187$	4.57452	0.334522
$188$	19.4763	1.42045
$189$	0	0
$190$	0	0
$191$	16.6253	1.20296	0.601482	−	0.798886i	$-0.294577\pi$
$191$	16.6253	1.20296	0.601482	+	0.798886i	$0.294577\pi$
$192$	0	0
$193$	16.3634	1.17787	0.588933	−	0.808182i	$-0.299548\pi$
$193$	16.3634	1.17787	0.588933	+	0.808182i	$0.299548\pi$
$194$	0.0145884	0.00104739
$195$	0	0
$196$	−8.28963	−0.592116
$197$	−20.4241	−1.45515	−0.727577	−	0.686026i	$-0.759354\pi$
$197$	−20.4241	−1.45515	−0.727577	+	0.686026i	$0.759354\pi$
$198$	0	0
$199$	−8.62530	−0.611431	−0.305716	−	0.952123i	$-0.598896\pi$
$199$	−8.62530	−0.611431	−0.305716	+	0.952123i	$0.598896\pi$
$200$	0	0
$201$	0	0
$202$	−2.92619	−0.205886
$203$	−12.1016	−0.849364
$204$	0	0
$205$	0	0
$206$	−0.625301	−0.0435668
$207$	0	0
$208$	−11.1852	−0.775556
$209$	4.31265	0.298312
$210$	0	0
$211$	9.08840	0.625671	0.312836	−	0.949807i	$-0.398721\pi$
$211$	9.08840	0.625671	0.312836	+	0.949807i	$0.398721\pi$
$212$	−9.22425	−0.633524
$213$	0	0
$214$	0.186642	0.0127586
$215$	0	0
$216$	0	0
$217$	−33.2506	−2.25720
$218$	−2.21108	−0.149753
$219$	0	0
$220$	0	0
$221$	13.5515	0.911572
$222$	0	0
$223$	6.70052	0.448700	0.224350	−	0.974509i	$-0.427974\pi$
$223$	6.70052	0.448700	0.224350	+	0.974509i	$0.427974\pi$
$224$	7.60228	0.507949
$225$	0	0
$226$	1.16362	0.0774028
$227$	16.9624	1.12583	0.562917	−	0.826514i	$-0.309679\pi$
$227$	16.9624	1.12583	0.562917	+	0.826514i	$0.309679\pi$
$228$	0	0
$229$	25.8496	1.70819	0.854093	−	0.520120i	$-0.174113\pi$
$229$	25.8496	1.70819	0.854093	+	0.520120i	$0.174113\pi$
$230$	0	0
$231$	0	0
$232$	2.77575	0.182237
$233$	−19.2750	−1.26275	−0.631375	−	0.775478i	$-0.717509\pi$
$233$	−19.2750	−1.26275	−0.631375	+	0.775478i	$0.717509\pi$
$234$	0	0
$235$	0	0
$236$	20.9986	1.36689
$237$	0	0
$238$	−2.97224	−0.192662
$239$	−26.5501	−1.71738	−0.858691	−	0.512494i	$-0.828722\pi$
$239$	−26.5501	−1.71738	−0.858691	+	0.512494i	$0.828722\pi$
$240$	0	0
$241$	28.5501	1.83907	0.919536	−	0.393006i	$-0.128565\pi$
$241$	28.5501	1.83907	0.919536	+	0.393006i	$0.128565\pi$
$242$	−0.193937	−0.0124667
$243$	0	0
$244$	17.0738	1.09304
$245$	0	0
$246$	0	0
$247$	12.7757	0.812901
$248$	7.62672	0.484297
$249$	0	0
$250$	0	0
$251$	−29.9248	−1.88884	−0.944418	−	0.328748i	$-0.893373\pi$
$251$	−29.9248	−1.88884	−0.944418	+	0.328748i	$0.893373\pi$
$252$	0	0
$253$	6.70052	0.421258
$254$	−2.82653	−0.177352
$255$	0	0
$256$	13.0752	0.817201
$257$	8.70052	0.542724	0.271362	−	0.962477i	$-0.412526\pi$
$257$	8.70052	0.542724	0.271362	+	0.962477i	$0.412526\pi$
$258$	0	0
$259$	−6.70052	−0.416350
$260$	0	0
$261$	0	0
$262$	−1.14903	−0.0709874
$263$	12.2882	0.757724	0.378862	−	0.925453i	$-0.376316\pi$
$263$	12.2882	0.757724	0.378862	+	0.925453i	$0.376316\pi$
$264$	0	0
$265$	0	0
$266$	−2.80209	−0.171807
$267$	0	0
$268$	11.6267	0.710215
$269$	5.84955	0.356654	0.178327	−	0.983971i	$-0.442932\pi$
$269$	5.84955	0.356654	0.178327	+	0.983971i	$0.442932\pi$
$270$	0	0
$271$	−5.08840	−0.309098	−0.154549	−	0.987985i	$-0.549392\pi$
$271$	−5.08840	−0.309098	−0.154549	+	0.987985i	$0.549392\pi$
$272$	−17.2722	−1.04728
$273$	0	0
$274$	−2.68594	−0.162263
$275$	0	0
$276$	0	0
$277$	−1.41090	−0.0847725	−0.0423863	−	0.999101i	$-0.513496\pi$
$277$	−1.41090	−0.0847725	−0.0423863	+	0.999101i	$0.513496\pi$
$278$	−2.63989	−0.158330
$279$	0	0
$280$	0	0
$281$	4.38787	0.261759	0.130879	−	0.991398i	$-0.458220\pi$
$281$	4.38787	0.261759	0.130879	+	0.991398i	$0.458220\pi$
$282$	0	0
$283$	−26.5745	−1.57969	−0.789845	−	0.613306i	$-0.789839\pi$
$283$	−26.5745	−1.57969	−0.789845	+	0.613306i	$0.789839\pi$
$284$	19.4763	1.15570
$285$	0	0
$286$	−0.574515	−0.0339718
$287$	−14.7005	−0.867744
$288$	0	0
$289$	3.92619	0.230952
$290$	0	0
$291$	0	0
$292$	−15.1852	−0.888648
$293$	−3.42548	−0.200119	−0.100059	−	0.994981i	$-0.531903\pi$
$293$	−3.42548	−0.200119	−0.100059	+	0.994981i	$0.531903\pi$
$294$	0	0
$295$	0	0
$296$	1.53690	0.0893307
$297$	0	0
$298$	0.298062	0.0172663
$299$	19.8496	1.14793
$300$	0	0
$301$	−31.0738	−1.79106
$302$	1.31124	0.0754531
$303$	0	0
$304$	−16.2835	−0.933921
$305$	0	0
$306$	0	0
$307$	16.6497	0.950251	0.475125	−	0.879918i	$-0.342403\pi$
$307$	16.6497	0.950251	0.475125	+	0.879918i	$0.342403\pi$
$308$	−6.57452	−0.374618
$309$	0	0
$310$	0	0
$311$	−32.9986	−1.87118	−0.935589	−	0.353091i	$-0.885131\pi$
$311$	−32.9986	−1.87118	−0.935589	+	0.353091i	$0.885131\pi$
$312$	0	0
$313$	−15.4010	−0.870519	−0.435259	−	0.900305i	$-0.643343\pi$
$313$	−15.4010	−0.870519	−0.435259	+	0.900305i	$0.643343\pi$
$314$	−1.06205	−0.0599349
$315$	0	0
$316$	−22.6399	−1.27359
$317$	2.15045	0.120781	0.0603905	−	0.998175i	$-0.480765\pi$
$317$	2.15045	0.120781	0.0603905	+	0.998175i	$0.480765\pi$
$318$	0	0
$319$	−3.61213	−0.202240
$320$	0	0
$321$	0	0
$322$	−4.35359	−0.242616
$323$	19.7283	1.09771
$324$	0	0
$325$	0	0
$326$	2.44851	0.135610
$327$	0	0
$328$	3.37187	0.186180
$329$	33.2506	1.83316
$330$	0	0
$331$	−14.5501	−0.799745	−0.399872	−	0.916571i	$-0.630946\pi$
$331$	−14.5501	−0.799745	−0.399872	+	0.916571i	$0.630946\pi$
$332$	−21.3649	−1.17255
$333$	0	0
$334$	−3.56134	−0.194868
$335$	0	0
$336$	0	0
$337$	−16.2619	−0.885840	−0.442920	−	0.896561i	$-0.646057\pi$
$337$	−16.2619	−0.885840	−0.442920	+	0.896561i	$0.646057\pi$
$338$	0.819237	0.0445606
$339$	0	0
$340$	0	0
$341$	−9.92478	−0.537457
$342$	0	0
$343$	9.29948	0.502125
$344$	7.12742	0.384285
$345$	0	0
$346$	1.66291	0.0893987
$347$	−0.962389	−0.0516637	−0.0258319	−	0.999666i	$-0.508223\pi$
$347$	−0.962389	−0.0516637	−0.0258319	+	0.999666i	$0.508223\pi$
$348$	0	0
$349$	20.7005	1.10807	0.554037	−	0.832492i	$-0.313087\pi$
$349$	20.7005	1.10807	0.554037	+	0.832492i	$0.313087\pi$
$350$	0	0
$351$	0	0
$352$	2.26916	0.120947
$353$	20.5501	1.09377	0.546885	−	0.837208i	$-0.315813\pi$
$353$	20.5501	1.09377	0.546885	+	0.837208i	$0.315813\pi$
$354$	0	0
$355$	0	0
$356$	−5.44709	−0.288695
$357$	0	0
$358$	2.74940	0.145310
$359$	−17.9248	−0.946034	−0.473017	−	0.881053i	$-0.656835\pi$
$359$	−17.9248	−0.946034	−0.473017	+	0.881053i	$0.656835\pi$
$360$	0	0
$361$	−0.401047	−0.0211077
$362$	1.01317	0.0532512
$363$	0	0
$364$	−19.4763	−1.02083
$365$	0	0
$366$	0	0
$367$	29.6531	1.54788	0.773939	−	0.633261i	$-0.218284\pi$
$367$	29.6531	1.54788	0.773939	+	0.633261i	$0.218284\pi$
$368$	−25.2995	−1.31883
$369$	0	0
$370$	0	0
$371$	−15.7480	−0.817595
$372$	0	0
$373$	9.13918	0.473209	0.236604	−	0.971606i	$-0.423965\pi$
$373$	9.13918	0.473209	0.236604	+	0.971606i	$0.423965\pi$
$374$	−0.887166	−0.0458743
$375$	0	0
$376$	−7.62672	−0.393318
$377$	−10.7005	−0.551105
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	0	0
$382$	−3.22425	−0.164967
$383$	−34.9234	−1.78450	−0.892250	−	0.451541i	$-0.850874\pi$
$383$	−34.9234	−1.78450	−0.892250	+	0.451541i	$0.850874\pi$
$384$	0	0
$385$	0	0
$386$	−3.17347	−0.161525
$387$	0	0
$388$	0.147616	0.00749408
$389$	−2.77575	−0.140736	−0.0703680	−	0.997521i	$-0.522417\pi$
$389$	−2.77575	−0.140736	−0.0703680	+	0.997521i	$0.522417\pi$
$390$	0	0
$391$	30.6516	1.55012
$392$	3.24614	0.163955
$393$	0	0
$394$	3.96097	0.199551
$395$	0	0
$396$	0	0
$397$	19.9248	0.999996	0.499998	−	0.866027i	$-0.333334\pi$
$397$	19.9248	0.999996	0.499998	+	0.866027i	$0.333334\pi$
$398$	1.67276	0.0838479
$399$	0	0
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	−29.4010	−1.46457
$404$	−29.6093	−1.47312
$405$	0	0
$406$	2.34694	0.116477
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	−13.0738	−0.646458	−0.323229	−	0.946321i	$-0.604768\pi$
$409$	−13.0738	−0.646458	−0.323229	+	0.946321i	$0.604768\pi$
$410$	0	0
$411$	0	0
$412$	−6.32724	−0.311721
$413$	35.8496	1.76404
$414$	0	0
$415$	0	0
$416$	6.72213	0.329580
$417$	0	0
$418$	−0.836381	−0.0409087
$419$	−7.22425	−0.352928	−0.176464	−	0.984307i	$-0.556466\pi$
$419$	−7.22425	−0.352928	−0.176464	+	0.984307i	$0.556466\pi$
$420$	0	0
$421$	30.6253	1.49259	0.746293	−	0.665618i	$-0.231832\pi$
$421$	30.6253	1.49259	0.746293	+	0.665618i	$0.231832\pi$
$422$	−1.76257	−0.0858007
$423$	0	0
$424$	3.61213	0.175420
$425$	0	0
$426$	0	0
$427$	29.1490	1.41062
$428$	1.88858	0.0912880
$429$	0	0
$430$	0	0
$431$	33.8759	1.63174	0.815872	−	0.578232i	$-0.196257\pi$
$431$	33.8759	1.63174	0.815872	+	0.578232i	$0.196257\pi$
$432$	0	0
$433$	9.47627	0.455400	0.227700	−	0.973731i	$-0.426879\pi$
$433$	9.47627	0.455400	0.227700	+	0.973731i	$0.426879\pi$
$434$	6.44851	0.309538
$435$	0	0
$436$	−22.3733	−1.07149
$437$	28.8970	1.38233
$438$	0	0
$439$	−29.4617	−1.40613	−0.703065	−	0.711126i	$-0.748186\pi$
$439$	−29.4617	−1.40613	−0.703065	+	0.711126i	$0.748186\pi$
$440$	0	0
$441$	0	0
$442$	−2.62813	−0.125007
$443$	−19.0738	−0.906224	−0.453112	−	0.891454i	$-0.649686\pi$
$443$	−19.0738	−0.906224	−0.453112	+	0.891454i	$0.649686\pi$
$444$	0	0
$445$	0	0
$446$	−1.29948	−0.0615320
$447$	0	0
$448$	23.8251	1.12563
$449$	−35.8759	−1.69309	−0.846544	−	0.532318i	$-0.821321\pi$
$449$	−35.8759	−1.69309	−0.846544	+	0.532318i	$0.821321\pi$
$450$	0	0
$451$	−4.38787	−0.206617
$452$	11.7743	0.553818
$453$	0	0
$454$	−3.28963	−0.154390
$455$	0	0
$456$	0	0
$457$	−5.28963	−0.247438	−0.123719	−	0.992317i	$-0.539482\pi$
$457$	−5.28963	−0.247438	−0.123719	+	0.992317i	$0.539482\pi$
$458$	−5.01317	−0.234250
$459$	0	0
$460$	0	0
$461$	−36.3390	−1.69248	−0.846238	−	0.532805i	$-0.821138\pi$
$461$	−36.3390	−1.69248	−0.846238	+	0.532805i	$0.821138\pi$
$462$	0	0
$463$	−10.5501	−0.490304	−0.245152	−	0.969485i	$-0.578838\pi$
$463$	−10.5501	−0.490304	−0.245152	+	0.969485i	$0.578838\pi$
$464$	13.6385	0.633150
$465$	0	0
$466$	3.73813	0.173166
$467$	18.7005	0.865357	0.432679	−	0.901548i	$-0.357569\pi$
$467$	18.7005	0.865357	0.432679	+	0.901548i	$0.357569\pi$
$468$	0	0
$469$	19.8496	0.916567
$470$	0	0
$471$	0	0
$472$	−8.22284	−0.378487
$473$	−9.27504	−0.426467
$474$	0	0
$475$	0	0
$476$	−30.0752	−1.37850
$477$	0	0
$478$	5.14903	0.235511
$479$	9.29948	0.424904	0.212452	−	0.977172i	$-0.431855\pi$
$479$	9.29948	0.424904	0.212452	+	0.977172i	$0.431855\pi$
$480$	0	0
$481$	−5.92478	−0.270147
$482$	−5.53690	−0.252199
$483$	0	0
$484$	−1.96239	−0.0891995
$485$	0	0
$486$	0	0
$487$	35.4763	1.60758	0.803792	−	0.594911i	$-0.202813\pi$
$487$	35.4763	1.60758	0.803792	+	0.594911i	$0.202813\pi$
$488$	−6.68594	−0.302658
$489$	0	0
$490$	0	0
$491$	−24.7757	−1.11811	−0.559057	−	0.829129i	$-0.688837\pi$
$491$	−24.7757	−1.11811	−0.559057	+	0.829129i	$0.688837\pi$
$492$	0	0
$493$	−16.5237	−0.744191
$494$	−2.47768	−0.111476
$495$	0	0
$496$	37.4734	1.68261
$497$	33.2506	1.49149
$498$	0	0
$499$	14.1768	0.634640	0.317320	−	0.948318i	$-0.397217\pi$
$499$	14.1768	0.634640	0.317320	+	0.948318i	$0.397217\pi$
$500$	0	0
$501$	0	0
$502$	5.80351	0.259023
$503$	−8.43866	−0.376261	−0.188131	−	0.982144i	$-0.560243\pi$
$503$	−8.43866	−0.376261	−0.188131	+	0.982144i	$0.560243\pi$
$504$	0	0
$505$	0	0
$506$	−1.29948	−0.0577688
$507$	0	0
$508$	−28.6009	−1.26896
$509$	−1.10299	−0.0488890	−0.0244445	−	0.999701i	$-0.507782\pi$
$509$	−1.10299	−0.0488890	−0.0244445	+	0.999701i	$0.507782\pi$
$510$	0	0
$511$	−25.9248	−1.14684
$512$	−14.3707	−0.635103
$513$	0	0
$514$	−1.68735	−0.0744258
$515$	0	0
$516$	0	0
$517$	9.92478	0.436491
$518$	1.29948	0.0570957
$519$	0	0
$520$	0	0
$521$	12.4485	0.545379	0.272690	−	0.962102i	$-0.412087\pi$
$521$	12.4485	0.545379	0.272690	+	0.962102i	$0.412087\pi$
$522$	0	0
$523$	−30.0508	−1.31403	−0.657015	−	0.753878i	$-0.728181\pi$
$523$	−30.0508	−1.31403	−0.657015	+	0.753878i	$0.728181\pi$
$524$	−11.6267	−0.507915
$525$	0	0
$526$	−2.38313	−0.103910
$527$	−45.4010	−1.97770
$528$	0	0
$529$	21.8970	0.952044
$530$	0	0
$531$	0	0
$532$	−28.3536	−1.22928
$533$	−12.9986	−0.563031
$534$	0	0
$535$	0	0
$536$	−4.55291	−0.196656
$537$	0	0
$538$	−1.13444	−0.0489093
$539$	−4.22425	−0.181951
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0.986826	0.0423878
$543$	0	0
$544$	10.3803	0.445052
$545$	0	0
$546$	0	0
$547$	14.3028	0.611544	0.305772	−	0.952105i	$-0.401086\pi$
$547$	14.3028	0.611544	0.305772	+	0.952105i	$0.401086\pi$
$548$	−27.1782	−1.16100
$549$	0	0
$550$	0	0
$551$	−15.5778	−0.663638
$552$	0	0
$553$	−38.6516	−1.64364
$554$	0.273624	0.0116252
$555$	0	0
$556$	−26.7123	−1.13285
$557$	−11.7988	−0.499930	−0.249965	−	0.968255i	$-0.580419\pi$
$557$	−11.7988	−0.499930	−0.249965	+	0.968255i	$0.580419\pi$
$558$	0	0
$559$	−27.4763	−1.16212
$560$	0	0
$561$	0	0
$562$	−0.850969	−0.0358960
$563$	30.4847	1.28478	0.642389	−	0.766379i	$-0.277944\pi$
$563$	30.4847	1.28478	0.642389	+	0.766379i	$0.277944\pi$
$564$	0	0
$565$	0	0
$566$	5.15377	0.216629
$567$	0	0
$568$	−7.62672	−0.320010
$569$	27.0884	1.13560	0.567802	−	0.823165i	$-0.307794\pi$
$569$	27.0884	1.13560	0.567802	+	0.823165i	$0.307794\pi$
$570$	0	0
$571$	7.28489	0.304863	0.152432	−	0.988314i	$-0.451290\pi$
$571$	7.28489	0.304863	0.152432	+	0.988314i	$0.451290\pi$
$572$	−5.81336	−0.243069
$573$	0	0
$574$	2.85097	0.118997
$575$	0	0
$576$	0	0
$577$	31.6239	1.31652	0.658260	−	0.752791i	$-0.271293\pi$
$577$	31.6239	1.31652	0.658260	+	0.752791i	$0.271293\pi$
$578$	−0.761432	−0.0316714
$579$	0	0
$580$	0	0
$581$	−36.4749	−1.51323
$582$	0	0
$583$	−4.70052	−0.194676
$584$	5.94639	0.246063
$585$	0	0
$586$	0.664327	0.0274431
$587$	33.1490	1.36821	0.684103	−	0.729385i	$-0.260194\pi$
$587$	33.1490	1.36821	0.684103	+	0.729385i	$0.260194\pi$
$588$	0	0
$589$	−42.8021	−1.76363
$590$	0	0
$591$	0	0
$592$	7.55149	0.310364
$593$	34.4993	1.41672	0.708358	−	0.705853i	$-0.249436\pi$
$593$	34.4993	1.41672	0.708358	+	0.705853i	$0.249436\pi$
$594$	0	0
$595$	0	0
$596$	3.01600	0.123540
$597$	0	0
$598$	−3.84955	−0.157420
$599$	14.4485	0.590350	0.295175	−	0.955443i	$-0.404622\pi$
$599$	14.4485	0.590350	0.295175	+	0.955443i	$0.404622\pi$
$600$	0	0
$601$	−15.9248	−0.649585	−0.324793	−	0.945785i	$-0.605295\pi$
$601$	−15.9248	−0.649585	−0.324793	+	0.945785i	$0.605295\pi$
$602$	6.02635	0.245616
$603$	0	0
$604$	13.2680	0.539868
$605$	0	0
$606$	0	0
$607$	14.5745	0.591561	0.295781	−	0.955256i	$-0.404420\pi$
$607$	14.5745	0.591561	0.295781	+	0.955256i	$0.404420\pi$
$608$	9.78609	0.396878
$609$	0	0
$610$	0	0
$611$	29.4010	1.18944
$612$	0	0
$613$	−16.4123	−0.662887	−0.331443	−	0.943475i	$-0.607536\pi$
$613$	−16.4123	−0.662887	−0.331443	+	0.943475i	$0.607536\pi$
$614$	−3.22899	−0.130312
$615$	0	0
$616$	2.57452	0.103730
$617$	−17.8496	−0.718596	−0.359298	−	0.933223i	$-0.616984\pi$
$617$	−17.8496	−0.718596	−0.359298	+	0.933223i	$0.616984\pi$
$618$	0	0
$619$	−0.402462	−0.0161763	−0.00808815	−	0.999967i	$-0.502575\pi$
$619$	−0.402462	−0.0161763	−0.00808815	+	0.999967i	$0.502575\pi$
$620$	0	0
$621$	0	0
$622$	6.39963	0.256602
$623$	−9.29948	−0.372576
$624$	0	0
$625$	0	0
$626$	2.98683	0.119378
$627$	0	0
$628$	−10.7466	−0.428835
$629$	−9.14903	−0.364796
$630$	0	0
$631$	−38.0263	−1.51380	−0.756902	−	0.653528i	$-0.773288\pi$
$631$	−38.0263	−1.51380	−0.756902	+	0.653528i	$0.773288\pi$
$632$	8.86556	0.352653
$633$	0	0
$634$	−0.417050	−0.0165632
$635$	0	0
$636$	0	0
$637$	−12.5139	−0.495818
$638$	0.700523	0.0277340
$639$	0	0
$640$	0	0
$641$	28.0263	1.10697	0.553487	−	0.832858i	$-0.313297\pi$
$641$	28.0263	1.10697	0.553487	+	0.832858i	$0.313297\pi$
$642$	0	0
$643$	4.62530	0.182404	0.0912020	−	0.995832i	$-0.470929\pi$
$643$	4.62530	0.182404	0.0912020	+	0.995832i	$0.470929\pi$
$644$	−44.0527	−1.73592
$645$	0	0
$646$	−3.82604	−0.150533
$647$	23.5778	0.926941	0.463470	−	0.886112i	$-0.346604\pi$
$647$	23.5778	0.926941	0.463470	+	0.886112i	$0.346604\pi$
$648$	0	0
$649$	10.7005	0.420032
$650$	0	0
$651$	0	0
$652$	24.7757	0.970293
$653$	−2.25202	−0.0881282	−0.0440641	−	0.999029i	$-0.514031\pi$
$653$	−2.25202	−0.0881282	−0.0440641	+	0.999029i	$0.514031\pi$
$654$	0	0
$655$	0	0
$656$	16.5675	0.646852
$657$	0	0
$658$	−6.44851	−0.251389
$659$	41.4010	1.61276	0.806378	−	0.591401i	$-0.201425\pi$
$659$	41.4010	1.61276	0.806378	+	0.591401i	$0.201425\pi$
$660$	0	0
$661$	3.40105	0.132285	0.0661427	−	0.997810i	$-0.478931\pi$
$661$	3.40105	0.132285	0.0661427	+	0.997810i	$0.478931\pi$
$662$	2.82179	0.109672
$663$	0	0
$664$	8.36626	0.324674
$665$	0	0
$666$	0	0
$667$	−24.2031	−0.937149
$668$	−36.0362	−1.39428
$669$	0	0
$670$	0	0
$671$	8.70052	0.335880
$672$	0	0
$673$	−0.887166	−0.0341977	−0.0170989	−	0.999854i	$-0.505443\pi$
$673$	−0.887166	−0.0341977	−0.0170989	+	0.999854i	$0.505443\pi$
$674$	3.15377	0.121479
$675$	0	0
$676$	8.28963	0.318832
$677$	18.9018	0.726453	0.363227	−	0.931701i	$-0.381675\pi$
$677$	18.9018	0.726453	0.363227	+	0.931701i	$0.381675\pi$
$678$	0	0
$679$	0.252016	0.00967149
$680$	0	0
$681$	0	0
$682$	1.92478	0.0737035
$683$	−20.8773	−0.798848	−0.399424	−	0.916766i	$-0.630790\pi$
$683$	−20.8773	−0.798848	−0.399424	+	0.916766i	$0.630790\pi$
$684$	0	0
$685$	0	0
$686$	−1.80351	−0.0688583
$687$	0	0
$688$	35.0202	1.33513
$689$	−13.9248	−0.530492
$690$	0	0
$691$	−2.44851	−0.0931456	−0.0465728	−	0.998915i	$-0.514830\pi$
$691$	−2.44851	−0.0931456	−0.0465728	+	0.998915i	$0.514830\pi$
$692$	16.8265	0.639649
$693$	0	0
$694$	0.186642	0.00708485
$695$	0	0
$696$	0	0
$697$	−20.0724	−0.760296
$698$	−4.01459	−0.151954
$699$	0	0
$700$	0	0
$701$	2.98683	0.112811	0.0564054	−	0.998408i	$-0.482036\pi$
$701$	2.98683	0.112811	0.0564054	+	0.998408i	$0.482036\pi$
$702$	0	0
$703$	−8.62530	−0.325309
$704$	7.11142	0.268022
$705$	0	0
$706$	−3.98541	−0.149993
$707$	−50.5501	−1.90113
$708$	0	0
$709$	24.1768	0.907979	0.453989	−	0.891007i	$-0.350000\pi$
$709$	24.1768	0.907979	0.453989	+	0.891007i	$0.350000\pi$
$710$	0	0
$711$	0	0
$712$	2.13303	0.0799386
$713$	−66.5012	−2.49049
$714$	0	0
$715$	0	0
$716$	27.8204	1.03970
$717$	0	0
$718$	3.47627	0.129733
$719$	30.0263	1.11979	0.559897	−	0.828562i	$-0.310841\pi$
$719$	30.0263	1.11979	0.559897	+	0.828562i	$0.310841\pi$
$720$	0	0
$721$	−10.8021	−0.402291
$722$	0.0777777	0.00289459
$723$	0	0
$724$	10.2520	0.381013
$725$	0	0
$726$	0	0
$727$	−14.9525	−0.554559	−0.277279	−	0.960789i	$-0.589433\pi$
$727$	−14.9525	−0.554559	−0.277279	+	0.960789i	$0.589433\pi$
$728$	7.62672	0.282665
$729$	0	0
$730$	0	0
$731$	−42.4288	−1.56929
$732$	0	0
$733$	19.1128	0.705949	0.352974	−	0.935633i	$-0.385170\pi$
$733$	19.1128	0.705949	0.352974	+	0.935633i	$0.385170\pi$
$734$	−5.75081	−0.212266
$735$	0	0
$736$	15.2046	0.560447
$737$	5.92478	0.218242
$738$	0	0
$739$	−3.31406	−0.121910	−0.0609549	−	0.998141i	$-0.519415\pi$
$739$	−3.31406	−0.121910	−0.0609549	+	0.998141i	$0.519415\pi$
$740$	0	0
$741$	0	0
$742$	3.05411	0.112120
$743$	−34.9887	−1.28361	−0.641806	−	0.766867i	$-0.721815\pi$
$743$	−34.9887	−1.28361	−0.641806	+	0.766867i	$0.721815\pi$
$744$	0	0
$745$	0	0
$746$	−1.77242	−0.0648930
$747$	0	0
$748$	−8.97698	−0.328231
$749$	3.22425	0.117812
$750$	0	0
$751$	−26.9234	−0.982447	−0.491224	−	0.871033i	$-0.663450\pi$
$751$	−26.9234	−0.982447	−0.491224	+	0.871033i	$0.663450\pi$
$752$	−37.4734	−1.36652
$753$	0	0
$754$	2.07522	0.0755752
$755$	0	0
$756$	0	0
$757$	−15.9248	−0.578796	−0.289398	−	0.957209i	$-0.593455\pi$
$757$	−15.9248	−0.578796	−0.289398	+	0.957209i	$0.593455\pi$
$758$	3.87873	0.140882
$759$	0	0
$760$	0	0
$761$	30.9380	1.12150	0.560750	−	0.827985i	$-0.310513\pi$
$761$	30.9380	1.12150	0.560750	+	0.827985i	$0.310513\pi$
$762$	0	0
$763$	−38.1965	−1.38281
$764$	−32.6253	−1.18034
$765$	0	0
$766$	6.77292	0.244715
$767$	31.6991	1.14459
$768$	0	0
$769$	9.32582	0.336298	0.168149	−	0.985762i	$-0.446221\pi$
$769$	9.32582	0.336298	0.168149	+	0.985762i	$0.446221\pi$
$770$	0	0
$771$	0	0
$772$	−32.1114	−1.15572
$773$	44.7005	1.60777	0.803883	−	0.594787i	$-0.202764\pi$
$773$	44.7005	1.60777	0.803883	+	0.594787i	$0.202764\pi$
$774$	0	0
$775$	0	0
$776$	−0.0578051	−0.00207508
$777$	0	0
$778$	0.538319	0.0192997
$779$	−18.9234	−0.678000
$780$	0	0
$781$	9.92478	0.355136
$782$	−5.94448	−0.212574
$783$	0	0
$784$	15.9497	0.569633
$785$	0	0
$786$	0	0
$787$	−21.6775	−0.772719	−0.386360	−	0.922348i	$-0.626268\pi$
$787$	−21.6775	−0.772719	−0.386360	+	0.922348i	$0.626268\pi$
$788$	40.0800	1.42779
$789$	0	0
$790$	0	0
$791$	20.1016	0.714730
$792$	0	0
$793$	25.7743	0.915273
$794$	−3.86414	−0.137133
$795$	0	0
$796$	16.9262	0.599933
$797$	22.7466	0.805725	0.402862	−	0.915261i	$-0.368015\pi$
$797$	22.7466	0.805725	0.402862	+	0.915261i	$0.368015\pi$
$798$	0	0
$799$	45.4010	1.60617
$800$	0	0
$801$	0	0
$802$	0.387873	0.0136963
$803$	−7.73813	−0.273073
$804$	0	0
$805$	0	0
$806$	5.70194	0.200842
$807$	0	0
$808$	11.5947	0.407900
$809$	23.6121	0.830158	0.415079	−	0.909785i	$-0.363754\pi$
$809$	23.6121	0.830158	0.415079	+	0.909785i	$0.363754\pi$
$810$	0	0
$811$	−26.0870	−0.916038	−0.458019	−	0.888942i	$-0.651441\pi$
$811$	−26.0870	−0.916038	−0.458019	+	0.888942i	$0.651441\pi$
$812$	23.7480	0.833391
$813$	0	0
$814$	0.387873	0.0135949
$815$	0	0
$816$	0	0
$817$	−40.0000	−1.39942
$818$	2.53549	0.0886513
$819$	0	0
$820$	0	0
$821$	54.4142	1.89907	0.949535	−	0.313662i	$-0.101556\pi$
$821$	54.4142	1.89907	0.949535	+	0.313662i	$0.101556\pi$
$822$	0	0
$823$	−0.121269	−0.00422716	−0.00211358	−	0.999998i	$-0.500673\pi$
$823$	−0.121269	−0.00422716	−0.00211358	+	0.999998i	$0.500673\pi$
$824$	2.47768	0.0863142
$825$	0	0
$826$	−6.95254	−0.241910
$827$	−18.2130	−0.633328	−0.316664	−	0.948538i	$-0.602563\pi$
$827$	−18.2130	−0.633328	−0.316664	+	0.948538i	$0.602563\pi$
$828$	0	0
$829$	13.0738	0.454072	0.227036	−	0.973886i	$-0.427096\pi$
$829$	13.0738	0.454072	0.227036	+	0.973886i	$0.427096\pi$
$830$	0	0
$831$	0	0
$832$	21.0668	0.730359
$833$	−19.3239	−0.669534
$834$	0	0
$835$	0	0
$836$	−8.46310	−0.292702
$837$	0	0
$838$	1.40105	0.0483984
$839$	26.5501	0.916610	0.458305	−	0.888795i	$-0.348457\pi$
$839$	26.5501	0.916610	0.458305	+	0.888795i	$0.348457\pi$
$840$	0	0
$841$	−15.9525	−0.550088
$842$	−5.93937	−0.204684
$843$	0	0
$844$	−17.8350	−0.613905
$845$	0	0
$846$	0	0
$847$	−3.35026	−0.115116
$848$	17.7480	0.609468
$849$	0	0
$850$	0	0
$851$	−13.4010	−0.459382
$852$	0	0
$853$	−40.6155	−1.39065	−0.695323	−	0.718697i	$-0.744739\pi$
$853$	−40.6155	−1.39065	−0.695323	+	0.718697i	$0.744739\pi$
$854$	−5.65306	−0.193444
$855$	0	0
$856$	−0.739549	−0.0252773
$857$	20.1721	0.689064	0.344532	−	0.938775i	$-0.388038\pi$
$857$	20.1721	0.689064	0.344532	+	0.938775i	$0.388038\pi$
$858$	0	0
$859$	21.8035	0.743926	0.371963	−	0.928248i	$-0.378685\pi$
$859$	21.8035	0.743926	0.371963	+	0.928248i	$0.378685\pi$
$860$	0	0
$861$	0	0
$862$	−6.56978	−0.223767
$863$	35.4274	1.20596	0.602981	−	0.797755i	$-0.293979\pi$
$863$	35.4274	1.20596	0.602981	+	0.797755i	$0.293979\pi$
$864$	0	0
$865$	0	0
$866$	−1.83780	−0.0624508
$867$	0	0
$868$	65.2506	2.21475
$869$	−11.5369	−0.391363
$870$	0	0
$871$	17.5515	0.594710
$872$	8.76116	0.296690
$873$	0	0
$874$	−5.60419	−0.189564
$875$	0	0
$876$	0	0
$877$	14.0362	0.473969	0.236984	−	0.971513i	$-0.423841\pi$
$877$	14.0362	0.473969	0.236984	+	0.971513i	$0.423841\pi$
$878$	5.71370	0.192828
$879$	0	0
$880$	0	0
$881$	21.0738	0.709995	0.354997	−	0.934867i	$-0.384482\pi$
$881$	21.0738	0.709995	0.354997	+	0.934867i	$0.384482\pi$
$882$	0	0
$883$	42.1476	1.41838	0.709190	−	0.705017i	$-0.249061\pi$
$883$	42.1476	1.41838	0.709190	+	0.705017i	$0.249061\pi$
$884$	−26.5933	−0.894429
$885$	0	0
$886$	3.69911	0.124274
$887$	6.93604	0.232889	0.116445	−	0.993197i	$-0.462850\pi$
$887$	6.93604	0.232889	0.116445	+	0.993197i	$0.462850\pi$
$888$	0	0
$889$	−48.8284	−1.63765
$890$	0	0
$891$	0	0
$892$	−13.1490	−0.440262
$893$	42.8021	1.43232
$894$	0	0
$895$	0	0
$896$	−19.8251	−0.662311
$897$	0	0
$898$	6.95765	0.232180
$899$	35.8496	1.19565
$900$	0	0
$901$	−21.5026	−0.716356
$902$	0.850969	0.0283342
$903$	0	0
$904$	−4.61071	−0.153350
$905$	0	0
$906$	0	0
$907$	−53.2017	−1.76653	−0.883267	−	0.468870i	$-0.844661\pi$
$907$	−53.2017	−1.76653	−0.883267	+	0.468870i	$0.844661\pi$
$908$	−33.2868	−1.10466
$909$	0	0
$910$	0	0
$911$	−36.4749	−1.20847	−0.604233	−	0.796808i	$-0.706520\pi$
$911$	−36.4749	−1.20847	−0.604233	+	0.796808i	$0.706520\pi$
$912$	0	0
$913$	−10.8872	−0.360313
$914$	1.02585	0.0339322
$915$	0	0
$916$	−50.7269	−1.67606
$917$	−19.8496	−0.655490
$918$	0	0
$919$	9.73340	0.321075	0.160538	−	0.987030i	$-0.448677\pi$
$919$	9.73340	0.321075	0.160538	+	0.987030i	$0.448677\pi$
$920$	0	0
$921$	0	0
$922$	7.04746	0.232096
$923$	29.4010	0.967747
$924$	0	0
$925$	0	0
$926$	2.04605	0.0672372
$927$	0	0
$928$	−8.19649	−0.269063
$929$	24.1768	0.793215	0.396607	−	0.917988i	$-0.370187\pi$
$929$	24.1768	0.793215	0.396607	+	0.917988i	$0.370187\pi$
$930$	0	0
$931$	−18.2177	−0.597062
$932$	37.8251	1.23900
$933$	0	0
$934$	−3.62672	−0.118670
$935$	0	0
$936$	0	0
$937$	7.48612	0.244561	0.122280	−	0.992496i	$-0.460979\pi$
$937$	7.48612	0.244561	0.122280	+	0.992496i	$0.460979\pi$
$938$	−3.84955	−0.125692
$939$	0	0
$940$	0	0
$941$	21.2360	0.692274	0.346137	−	0.938184i	$-0.387493\pi$
$941$	21.2360	0.692274	0.346137	+	0.938184i	$0.387493\pi$
$942$	0	0
$943$	−29.4010	−0.957430
$944$	−40.4025	−1.31499
$945$	0	0
$946$	1.79877	0.0584830
$947$	−15.4763	−0.502911	−0.251456	−	0.967869i	$-0.580909\pi$
$947$	−15.4763	−0.502911	−0.251456	+	0.967869i	$0.580909\pi$
$948$	0	0
$949$	−22.9234	−0.744124
$950$	0	0
$951$	0	0
$952$	11.7772	0.381700
$953$	−32.0508	−1.03823	−0.519113	−	0.854705i	$-0.673738\pi$
$953$	−32.0508	−1.03823	−0.519113	+	0.854705i	$0.673738\pi$
$954$	0	0
$955$	0	0
$956$	52.1016	1.68509
$957$	0	0
$958$	−1.80351	−0.0582687
$959$	−46.3996	−1.49832
$960$	0	0
$961$	67.5012	2.17746
$962$	1.14903	0.0370462
$963$	0	0
$964$	−56.0263	−1.80449
$965$	0	0
$966$	0	0
$967$	17.3766	0.558794	0.279397	−	0.960176i	$-0.409865\pi$
$967$	17.3766	0.558794	0.279397	+	0.960176i	$0.409865\pi$
$968$	0.768452	0.0246990
$969$	0	0
$970$	0	0
$971$	−36.2031	−1.16181	−0.580907	−	0.813970i	$-0.697302\pi$
$971$	−36.2031	−1.16181	−0.580907	+	0.813970i	$0.697302\pi$
$972$	0	0
$973$	−45.6042	−1.46200
$974$	−6.88015	−0.220454
$975$	0	0
$976$	−32.8510	−1.05153
$977$	−28.1476	−0.900522	−0.450261	−	0.892897i	$-0.648669\pi$
$977$	−28.1476	−0.900522	−0.450261	+	0.892897i	$0.648669\pi$
$978$	0	0
$979$	−2.77575	−0.0887132
$980$	0	0
$981$	0	0
$982$	4.80492	0.153331
$983$	7.07381	0.225619	0.112810	−	0.993617i	$-0.464015\pi$
$983$	7.07381	0.225619	0.112810	+	0.993617i	$0.464015\pi$
$984$	0	0
$985$	0	0
$986$	3.20456	0.102054
$987$	0	0
$988$	−25.0710	−0.797614
$989$	−62.1476	−1.97618
$990$	0	0
$991$	44.4260	1.41124	0.705619	−	0.708592i	$-0.250669\pi$
$991$	44.4260	1.41124	0.705619	+	0.708592i	$0.250669\pi$
$992$	−22.5209	−0.715039
$993$	0	0
$994$	−6.44851	−0.204534
$995$	0	0
$996$	0	0
$997$	28.4847	0.902120	0.451060	−	0.892494i	$-0.351046\pi$
$997$	28.4847	0.902120	0.451060	+	0.892494i	$0.351046\pi$
$998$	−2.74940	−0.0870307
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.bb.1.2		3
3.2	odd	2		825.2.a.k.1.2		3
5.2	odd	4		2475.2.c.r.199.3		6
5.3	odd	4		2475.2.c.r.199.4		6
5.4	even	2		495.2.a.e.1.2		3
15.2	even	4		825.2.c.g.199.4		6
15.8	even	4		825.2.c.g.199.3		6
15.14	odd	2		165.2.a.c.1.2	✓	3
20.19	odd	2		7920.2.a.cj.1.1		3
33.32	even	2		9075.2.a.cf.1.2		3
55.54	odd	2		5445.2.a.z.1.2		3
60.59	even	2		2640.2.a.be.1.1		3
105.104	even	2		8085.2.a.bk.1.2		3
165.164	even	2		1815.2.a.m.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.c.1.2	✓	3	15.14	odd	2
495.2.a.e.1.2		3	5.4	even	2
825.2.a.k.1.2		3	3.2	odd	2
825.2.c.g.199.3		6	15.8	even	4
825.2.c.g.199.4		6	15.2	even	4
1815.2.a.m.1.2		3	165.164	even	2
2475.2.a.bb.1.2		3	1.1	even	1	trivial
2475.2.c.r.199.3		6	5.2	odd	4
2475.2.c.r.199.4		6	5.3	odd	4
2640.2.a.be.1.1		3	60.59	even	2
5445.2.a.z.1.2		3	55.54	odd	2
7920.2.a.cj.1.1		3	20.19	odd	2
8085.2.a.bk.1.2		3	105.104	even	2
9075.2.a.cf.1.2		3	33.32	even	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q-0.193937 q^{2} -1.96239 q^{4} -3.35026 q^{7} +0.768452 q^{8} +O(q^{10})\)	\(q-0.193937 q^{2} -1.96239 q^{4} -3.35026 q^{7} +0.768452 q^{8} -1.00000 q^{11} -2.96239 q^{13} +0.649738 q^{14} +3.77575 q^{16} -4.57452 q^{17} -4.31265 q^{19} +0.193937 q^{22} -6.70052 q^{23} +0.574515 q^{26} +6.57452 q^{28} +3.61213 q^{29} +9.92478 q^{31} -2.26916 q^{32} +0.887166 q^{34} +2.00000 q^{37} +0.836381 q^{38} +4.38787 q^{41} +9.27504 q^{43} +1.96239 q^{44} +1.29948 q^{46} -9.92478 q^{47} +4.22425 q^{49} +5.81336 q^{52} +4.70052 q^{53} -2.57452 q^{56} -0.700523 q^{58} -10.7005 q^{59} -8.70052 q^{61} -1.92478 q^{62} -7.11142 q^{64} -5.92478 q^{67} +8.97698 q^{68} -9.92478 q^{71} +7.73813 q^{73} -0.387873 q^{74} +8.46310 q^{76} +3.35026 q^{77} +11.5369 q^{79} -0.850969 q^{82} +10.8872 q^{83} -1.79877 q^{86} -0.768452 q^{88} +2.77575 q^{89} +9.92478 q^{91} +13.1490 q^{92} +1.92478 q^{94} -0.0752228 q^{97} -0.819237 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 5 q^{4} - 9 q^{8}+O(q^{10}) \) 3 * q - q^2 + 5 * q^4 - 9 * q^8	\( 3 q - q^{2} + 5 q^{4} - 9 q^{8} - 3 q^{11} + 2 q^{13} + 12 q^{14} + 13 q^{16} - 2 q^{17} + 8 q^{19} + q^{22} - 10 q^{26} + 8 q^{28} + 10 q^{29} + 8 q^{31} - 29 q^{32} - 30 q^{34} + 6 q^{37} + 14 q^{41} - 4 q^{43} - 5 q^{44} + 24 q^{46} - 8 q^{47} + 11 q^{49} + 30 q^{52} - 6 q^{53} + 4 q^{56} + 18 q^{58} - 12 q^{59} - 6 q^{61} + 16 q^{62} + 13 q^{64} + 4 q^{67} + 42 q^{68} - 8 q^{71} + 14 q^{73} - 2 q^{74} + 48 q^{76} + 12 q^{79} - 26 q^{82} + 8 q^{86} + 9 q^{88} + 10 q^{89} + 8 q^{91} + 16 q^{92} - 16 q^{94} - 22 q^{97} + 39 q^{98}+O(q^{100}) \) 3 * q - q^2 + 5 * q^4 - 9 * q^8 - 3 * q^11 + 2 * q^13 + 12 * q^14 + 13 * q^16 - 2 * q^17 + 8 * q^19 + q^22 - 10 * q^26 + 8 * q^28 + 10 * q^29 + 8 * q^31 - 29 * q^32 - 30 * q^34 + 6 * q^37 + 14 * q^41 - 4 * q^43 - 5 * q^44 + 24 * q^46 - 8 * q^47 + 11 * q^49 + 30 * q^52 - 6 * q^53 + 4 * q^56 + 18 * q^58 - 12 * q^59 - 6 * q^61 + 16 * q^62 + 13 * q^64 + 4 * q^67 + 42 * q^68 - 8 * q^71 + 14 * q^73 - 2 * q^74 + 48 * q^76 + 12 * q^79 - 26 * q^82 + 8 * q^86 + 9 * q^88 + 10 * q^89 + 8 * q^91 + 16 * q^92 - 16 * q^94 - 22 * q^97 + 39 * q^98

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