LMFDB - Embedded newform 3025.2.a.bg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3025,2,Mod(1,3025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3025 = 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3025.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:	$24.1547466114$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.27433728.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 15x^{2} - 3 $ x^6 - 9x^4 + 15x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 605)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.480901$ of defining polynomial
Character	$\chi$	$=$	3025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.480901 q^{2} +1.60168 q^{3} -1.76873 q^{4} -0.770249 q^{6} +0.480901 q^{7} +1.81239 q^{8} -0.434624 q^{9} +O(q^{10})$	\(q-0.480901 q^{2} +1.60168 q^{3} -1.76873 q^{4} -0.770249 q^{6} +0.480901 q^{7} +1.81239 q^{8} -0.434624 q^{9} -2.83294 q^{12} -4.79559 q^{13} -0.231266 q^{14} +2.66589 q^{16} +2.50230 q^{17} +0.209011 q^{18} +5.75739 q^{19} +0.770249 q^{21} -4.43462 q^{23} +2.90286 q^{24} +2.30620 q^{26} -5.50117 q^{27} -0.850586 q^{28} -9.01248 q^{29} +7.97209 q^{31} -4.90680 q^{32} -1.20336 q^{34} +0.768734 q^{36} -5.20336 q^{37} -2.76873 q^{38} -7.68099 q^{39} +8.45124 q^{41} -0.370413 q^{42} +8.53158 q^{43} +2.13261 q^{46} -9.60168 q^{47} +4.26990 q^{48} -6.76873 q^{49} +4.00788 q^{51} +8.48212 q^{52} -6.10051 q^{53} +2.64552 q^{54} +0.871579 q^{56} +9.22149 q^{57} +4.33411 q^{58} +2.76873 q^{59} -4.02534 q^{61} -3.83379 q^{62} -0.209011 q^{63} -2.97209 q^{64} -7.60168 q^{67} -4.42590 q^{68} -7.10284 q^{69} -2.76873 q^{71} -0.787707 q^{72} -5.00460 q^{73} +2.50230 q^{74} -10.1833 q^{76} +3.69380 q^{78} +3.62478 q^{79} -7.50723 q^{81} -4.06421 q^{82} -1.71459 q^{83} -1.36237 q^{84} -4.10284 q^{86} -14.4351 q^{87} -15.7408 q^{89} -2.30620 q^{91} +7.84367 q^{92} +12.7687 q^{93} +4.61746 q^{94} -7.85913 q^{96} +3.63798 q^{97} +3.25509 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{4} + 12 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^4 + 12 * q^9	$ 6 q - 6 q^{3} + 6 q^{4} + 12 q^{9} - 18 q^{12} - 18 q^{14} + 18 q^{16} - 12 q^{23} - 36 q^{26} - 30 q^{27} + 24 q^{34} - 12 q^{36} + 30 q^{42} - 42 q^{47} + 6 q^{48} - 24 q^{49} - 24 q^{53} - 30 q^{56} + 24 q^{58} + 30 q^{64} - 30 q^{67} - 24 q^{69} + 72 q^{78} + 30 q^{81} - 42 q^{82} - 6 q^{86} - 30 q^{89} + 36 q^{91} - 36 q^{92} + 60 q^{93} - 24 q^{97}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^4 + 12 * q^9 - 18 * q^12 - 18 * q^14 + 18 * q^16 - 12 * q^23 - 36 * q^26 - 30 * q^27 + 24 * q^34 - 12 * q^36 + 30 * q^42 - 42 * q^47 + 6 * q^48 - 24 * q^49 - 24 * q^53 - 30 * q^56 + 24 * q^58 + 30 * q^64 - 30 * q^67 - 24 * q^69 + 72 * q^78 + 30 * q^81 - 42 * q^82 - 6 * q^86 - 30 * q^89 + 36 * q^91 - 36 * q^92 + 60 * q^93 - 24 * q^97

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 $

$2$	−0.480901	−0.340048	−0.170024	−	0.985440i	$-0.554385\pi$
$2$	−0.480901	−0.340048	−0.170024	+	0.985440i	$0.554385\pi$
$3$	1.60168	0.924730	0.462365	−	0.886690i	$-0.347001\pi$
$3$	1.60168	0.924730	0.462365	+	0.886690i	$0.347001\pi$
$4$	−1.76873	−0.884367
$5$	0	0
$5$	0	0
$6$	−0.770249	−0.314453
$7$	0.480901	0.181763	0.0908817	−	0.995862i	$-0.471031\pi$
$7$	0.480901	0.181763	0.0908817	+	0.995862i	$0.471031\pi$
$8$	1.81239	0.640776
$9$	−0.434624	−0.144875
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−2.83294	−0.817801
$13$	−4.79559	−1.33006	−0.665028	−	0.746818i	$-0.731581\pi$
$13$	−4.79559	−1.33006	−0.665028	+	0.746818i	$0.731581\pi$
$14$	−0.231266	−0.0618084
$15$	0	0
$16$	2.66589	0.666472
$17$	2.50230	0.606897	0.303448	−	0.952848i	$-0.401862\pi$
$17$	2.50230	0.606897	0.303448	+	0.952848i	$0.401862\pi$
$18$	0.209011	0.0492644
$19$	5.75739	1.32084	0.660418	−	0.750898i	$-0.270379\pi$
$19$	5.75739	1.32084	0.660418	+	0.750898i	$0.270379\pi$
$20$	0	0
$21$	0.770249	0.168082
$22$	0	0
$23$	−4.43462	−0.924683	−0.462342	−	0.886702i	$-0.652991\pi$
$23$	−4.43462	−0.924683	−0.462342	+	0.886702i	$0.652991\pi$
$24$	2.90286	0.592545
$25$	0	0
$26$	2.30620	0.452284
$27$	−5.50117	−1.05870
$28$	−0.850586	−0.160746
$29$	−9.01248	−1.67358	−0.836788	−	0.547527i	$-0.815569\pi$
$29$	−9.01248	−1.67358	−0.836788	+	0.547527i	$0.815569\pi$
$30$	0	0
$31$	7.97209	1.43183	0.715915	−	0.698187i	$-0.246010\pi$
$31$	7.97209	1.43183	0.715915	+	0.698187i	$0.246010\pi$
$32$	−4.90680	−0.867409
$33$	0	0
$34$	−1.20336	−0.206374
$35$	0	0
$36$	0.768734	0.128122
$37$	−5.20336	−0.855427	−0.427713	−	0.903914i	$-0.640681\pi$
$37$	−5.20336	−0.855427	−0.427713	+	0.903914i	$0.640681\pi$
$38$	−2.76873	−0.449148
$39$	−7.68099	−1.22994
$40$	0	0
$41$	8.45124	1.31986	0.659931	−	0.751326i	$-0.270585\pi$
$41$	8.45124	1.31986	0.659931	+	0.751326i	$0.270585\pi$
$42$	−0.370413	−0.0571560
$43$	8.53158	1.30105	0.650527	−	0.759483i	$-0.274548\pi$
$43$	8.53158	1.30105	0.650527	+	0.759483i	$0.274548\pi$
$44$	0	0
$45$	0	0
$46$	2.13261	0.314437
$47$	−9.60168	−1.40055	−0.700274	−	0.713874i	$-0.746939\pi$
$47$	−9.60168	−1.40055	−0.700274	+	0.713874i	$0.746939\pi$
$48$	4.26990	0.616307
$49$	−6.76873	−0.966962
$50$	0	0
$51$	4.00788	0.561216
$52$	8.48212	1.17626
$53$	−6.10051	−0.837970	−0.418985	−	0.907993i	$-0.637614\pi$
$53$	−6.10051	−0.837970	−0.418985	+	0.907993i	$0.637614\pi$
$54$	2.64552	0.360009
$55$	0	0
$56$	0.871579	0.116470
$57$	9.22149	1.22142
$58$	4.33411	0.569097
$59$	2.76873	0.360459	0.180229	−	0.983625i	$-0.442316\pi$
$59$	2.76873	0.360459	0.180229	+	0.983625i	$0.442316\pi$
$60$	0	0
$61$	−4.02534	−0.515392	−0.257696	−	0.966226i	$-0.582963\pi$
$61$	−4.02534	−0.515392	−0.257696	+	0.966226i	$0.582963\pi$
$62$	−3.83379	−0.486891
$63$	−0.209011	−0.0263329
$64$	−2.97209	−0.371512
$65$	0	0
$66$	0	0
$67$	−7.60168	−0.928693	−0.464346	−	0.885654i	$-0.653711\pi$
$67$	−7.60168	−0.928693	−0.464346	+	0.885654i	$0.653711\pi$
$68$	−4.42590	−0.536720
$69$	−7.10284	−0.855082
$70$	0	0
$71$	−2.76873	−0.328588	−0.164294	−	0.986411i	$-0.552535\pi$
$71$	−2.76873	−0.328588	−0.164294	+	0.986411i	$0.552535\pi$
$72$	−0.787707	−0.0928322
$73$	−5.00460	−0.585744	−0.292872	−	0.956152i	$-0.594611\pi$
$73$	−5.00460	−0.585744	−0.292872	+	0.956152i	$0.594611\pi$
$74$	2.50230	0.290886
$75$	0	0
$76$	−10.1833	−1.16810
$77$	0	0
$78$	3.69380	0.418240
$79$	3.62478	0.407819	0.203910	−	0.978990i	$-0.434635\pi$
$79$	3.62478	0.407819	0.203910	+	0.978990i	$0.434635\pi$
$80$	0	0
$81$	−7.50723	−0.834137
$82$	−4.06421	−0.448817
$83$	−1.71459	−0.188201	−0.0941005	−	0.995563i	$-0.529998\pi$
$83$	−1.71459	−0.188201	−0.0941005	+	0.995563i	$0.529998\pi$
$84$	−1.36237	−0.148646
$85$	0	0
$86$	−4.10284	−0.442421
$87$	−14.4351	−1.54761
$88$	0	0
$89$	−15.7408	−1.66852	−0.834262	−	0.551368i	$-0.814106\pi$
$89$	−15.7408	−1.66852	−0.834262	+	0.551368i	$0.814106\pi$
$90$	0	0
$91$	−2.30620	−0.241756
$92$	7.84367	0.817759
$93$	12.7687	1.32406
$94$	4.61746	0.476254
$95$	0	0
$96$	−7.85913	−0.802119
$97$	3.63798	0.369381	0.184691	−	0.982797i	$-0.440872\pi$
$97$	3.63798	0.369381	0.184691	+	0.982797i	$0.440872\pi$
$98$	3.25509	0.328814
$99$	0	0
$100$	0	0
$101$	−7.69845	−0.766025	−0.383012	−	0.923743i	$-0.625113\pi$
$101$	−7.69845	−0.766025	−0.383012	+	0.923743i	$0.625113\pi$
$102$	−1.92739	−0.190840
$103$	−9.10284	−0.896930	−0.448465	−	0.893800i	$-0.648029\pi$
$103$	−9.10284	−0.896930	−0.448465	+	0.893800i	$0.648029\pi$
$104$	−8.69147	−0.852268
$105$	0	0
$106$	2.93374	0.284950
$107$	16.7913	1.62327	0.811637	−	0.584163i	$-0.198577\pi$
$107$	16.7913	1.62327	0.811637	+	0.584163i	$0.198577\pi$
$108$	9.73010	0.936279
$109$	−17.8817	−1.71276	−0.856380	−	0.516346i	$-0.827292\pi$
$109$	−17.8817	−1.71276	−0.856380	+	0.516346i	$0.827292\pi$
$110$	0	0
$111$	−8.33411	−0.791039
$112$	1.28203	0.121140
$113$	0.462531	0.0435113	0.0217556	−	0.999763i	$-0.493074\pi$
$113$	0.462531	0.0435113	0.0217556	+	0.999763i	$0.493074\pi$
$114$	−4.43462	−0.415341
$115$	0	0
$116$	15.9407	1.48006
$117$	2.08428	0.192692
$118$	−1.33149	−0.122573
$119$	1.20336	0.110312
$120$	0	0
$121$	0	0
$122$	1.93579	0.175258
$123$	13.5362	1.22052
$124$	−14.1005	−1.26626
$125$	0	0
$126$	0.100514	0.00895446
$127$	−15.8295	−1.40464	−0.702319	−	0.711862i	$-0.747852\pi$
$127$	−15.8295	−1.40464	−0.702319	+	0.711862i	$0.747852\pi$
$128$	11.2429	0.993741
$129$	13.6649	1.20312
$130$	0	0
$131$	9.60460	0.839158	0.419579	−	0.907719i	$-0.362178\pi$
$131$	9.60460	0.839158	0.419579	+	0.907719i	$0.362178\pi$
$132$	0	0
$133$	2.76873	0.240080
$134$	3.65565	0.315800
$135$	0	0
$136$	4.53514	0.388885
$137$	−3.97209	−0.339359	−0.169679	−	0.985499i	$-0.554273\pi$
$137$	−3.97209	−0.339359	−0.169679	+	0.985499i	$0.554273\pi$
$138$	3.41576	0.290769
$139$	9.22149	0.782157	0.391078	−	0.920357i	$-0.372102\pi$
$139$	9.22149	0.782157	0.391078	+	0.920357i	$0.372102\pi$
$140$	0	0
$141$	−15.3788	−1.29513
$142$	1.33149	0.111736
$143$	0	0
$144$	−1.15866	−0.0965550
$145$	0	0
$146$	2.40672	0.199181
$147$	−10.8413	−0.894179
$148$	9.20336	0.756511
$149$	7.65012	0.626722	0.313361	−	0.949634i	$-0.398545\pi$
$149$	7.65012	0.626722	0.313361	+	0.949634i	$0.398545\pi$
$150$	0	0
$151$	−2.87198	−0.233719	−0.116859	−	0.993148i	$-0.537283\pi$
$151$	−2.87198	−0.233719	−0.116859	+	0.993148i	$0.537283\pi$
$152$	10.4346	0.846360
$153$	−1.08756	−0.0879240
$154$	0	0
$155$	0	0
$156$	13.5856	1.08772
$157$	−17.4817	−1.39519	−0.697594	−	0.716493i	$-0.745746\pi$
$157$	−17.4817	−1.39519	−0.697594	+	0.716493i	$0.745746\pi$
$158$	−1.74316	−0.138678
$159$	−9.77107	−0.774896
$160$	0	0
$161$	−2.13261	−0.168074
$162$	3.61023	0.283647
$163$	−13.1671	−1.03132	−0.515662	−	0.856792i	$-0.672454\pi$
$163$	−13.1671	−1.03132	−0.515662	+	0.856792i	$0.672454\pi$
$164$	−14.9480	−1.16724
$165$	0	0
$166$	0.824549	0.0639974
$167$	15.8778	1.22866	0.614331	−	0.789049i	$-0.289426\pi$
$167$	15.8778	1.22866	0.614331	+	0.789049i	$0.289426\pi$
$168$	1.39599	0.107703
$169$	9.99767	0.769051
$170$	0	0
$171$	−2.50230	−0.191356
$172$	−15.0901	−1.15061
$173$	−22.8206	−1.73501	−0.867507	−	0.497425i	$-0.834279\pi$
$173$	−22.8206	−1.73501	−0.867507	+	0.497425i	$0.834279\pi$
$174$	6.94185	0.526261
$175$	0	0
$176$	0	0
$177$	4.43462	0.333327
$178$	7.56978	0.567379
$179$	−0.462531	−0.0345712	−0.0172856	−	0.999851i	$-0.505502\pi$
$179$	−0.462531	−0.0345712	−0.0172856	+	0.999851i	$0.505502\pi$
$180$	0	0
$181$	−3.12842	−0.232534	−0.116267	−	0.993218i	$-0.537093\pi$
$181$	−3.12842	−0.232534	−0.116267	+	0.993218i	$0.537093\pi$
$182$	1.10906	0.0822086
$183$	−6.44730	−0.476598
$184$	−8.03726	−0.592515
$185$	0	0
$186$	−6.14050	−0.450243
$187$	0	0
$188$	16.9828	1.23860
$189$	−2.64552	−0.192433
$190$	0	0
$191$	−5.43695	−0.393404	−0.196702	−	0.980463i	$-0.563023\pi$
$191$	−5.43695	−0.393404	−0.196702	+	0.980463i	$0.563023\pi$
$192$	−4.76034	−0.343548
$193$	3.20675	0.230827	0.115414	−	0.993318i	$-0.463181\pi$
$193$	3.20675	0.230827	0.115414	+	0.993318i	$0.463181\pi$
$194$	−1.74951	−0.125607
$195$	0	0
$196$	11.9721	0.855149
$197$	−19.4048	−1.38253	−0.691267	−	0.722600i	$-0.742947\pi$
$197$	−19.4048	−1.38253	−0.691267	+	0.722600i	$0.742947\pi$
$198$	0	0
$199$	20.3509	1.44264	0.721319	−	0.692603i	$-0.243536\pi$
$199$	20.3509	1.44264	0.721319	+	0.692603i	$0.243536\pi$
$200$	0	0
$201$	−12.1755	−0.858790
$202$	3.70219	0.260485
$203$	−4.33411	−0.304195
$204$	−7.08888	−0.496321
$205$	0	0
$206$	4.37757	0.305000
$207$	1.92739	0.133963
$208$	−12.7845	−0.886446
$209$	0	0
$210$	0	0
$211$	−6.87987	−0.473630	−0.236815	−	0.971555i	$-0.576103\pi$
$211$	−6.87987	−0.473630	−0.236815	+	0.971555i	$0.576103\pi$
$212$	10.7902	0.741073
$213$	−4.43462	−0.303855
$214$	−8.07494	−0.551991
$215$	0	0
$216$	−9.97025	−0.678389
$217$	3.83379	0.260254
$218$	8.59935	0.582421
$219$	−8.01576	−0.541655
$220$	0	0
$221$	−12.0000	−0.807207
$222$	4.00788	0.268991
$223$	1.03863	0.0695521	0.0347760	−	0.999395i	$-0.488928\pi$
$223$	1.03863	0.0695521	0.0347760	+	0.999395i	$0.488928\pi$
$224$	−2.35969	−0.157663
$225$	0	0
$226$	−0.222432	−0.0147959
$227$	0.480901	0.0319185	0.0159593	−	0.999873i	$-0.494920\pi$
$227$	0.480901	0.0319185	0.0159593	+	0.999873i	$0.494920\pi$
$228$	−16.3104	−1.08018
$229$	13.5398	0.894735	0.447368	−	0.894350i	$-0.352362\pi$
$229$	13.5398	0.894735	0.447368	+	0.894350i	$0.352362\pi$
$230$	0	0
$231$	0	0
$232$	−16.3341	−1.07239
$233$	−21.1543	−1.38586	−0.692932	−	0.721003i	$-0.743681\pi$
$233$	−21.1543	−1.38586	−0.692932	+	0.721003i	$0.743681\pi$
$234$	−1.00233	−0.0655244
$235$	0	0
$236$	−4.89716	−0.318778
$237$	5.80573	0.377123
$238$	−0.578696	−0.0375113
$239$	−2.67639	−0.173122	−0.0865608	−	0.996247i	$-0.527588\pi$
$239$	−2.67639	−0.173122	−0.0865608	+	0.996247i	$0.527588\pi$
$240$	0	0
$241$	4.56912	0.294323	0.147161	−	0.989112i	$-0.452986\pi$
$241$	4.56912	0.294323	0.147161	+	0.989112i	$0.452986\pi$
$242$	0	0
$243$	4.47932	0.287349
$244$	7.11976	0.455796
$245$	0	0
$246$	−6.50956	−0.415034
$247$	−27.6101	−1.75679
$248$	14.4485	0.917482
$249$	−2.74623	−0.174035
$250$	0	0
$251$	−11.4370	−0.721894	−0.360947	−	0.932586i	$-0.617547\pi$
$251$	−11.4370	−0.721894	−0.360947	+	0.932586i	$0.617547\pi$
$252$	0.369685	0.0232880
$253$	0	0
$254$	7.61241	0.477645
$255$	0	0
$256$	0.537469	0.0335918
$257$	20.0447	1.25035	0.625177	−	0.780483i	$-0.285027\pi$
$257$	20.0447	1.25035	0.625177	+	0.780483i	$0.285027\pi$
$258$	−6.57144	−0.409120
$259$	−2.50230	−0.155485
$260$	0	0
$261$	3.91704	0.242459
$262$	−4.61886	−0.285354
$263$	−6.55852	−0.404416	−0.202208	−	0.979343i	$-0.564812\pi$
$263$	−6.55852	−0.404416	−0.202208	+	0.979343i	$0.564812\pi$
$264$	0	0
$265$	0	0
$266$	−1.33149	−0.0816387
$267$	−25.2118	−1.54293
$268$	13.4454	0.821306
$269$	1.89716	0.115672	0.0578358	−	0.998326i	$-0.481580\pi$
$269$	1.89716	0.115672	0.0578358	+	0.998326i	$0.481580\pi$
$270$	0	0
$271$	−11.3541	−0.689713	−0.344856	−	0.938655i	$-0.612072\pi$
$271$	−11.3541	−0.689713	−0.344856	+	0.938655i	$0.612072\pi$
$272$	6.67086	0.404480
$273$	−3.69380	−0.223559
$274$	1.91018	0.115398
$275$	0	0
$276$	12.5630	0.756206
$277$	−22.8340	−1.37196	−0.685980	−	0.727620i	$-0.740626\pi$
$277$	−22.8340	−1.37196	−0.685980	+	0.727620i	$0.740626\pi$
$278$	−4.43462	−0.265971
$279$	−3.46486	−0.207436
$280$	0	0
$281$	−22.6115	−1.34889	−0.674446	−	0.738324i	$-0.735617\pi$
$281$	−22.6115	−1.34889	−0.674446	+	0.738324i	$0.735617\pi$
$282$	7.39568	0.440407
$283$	27.5667	1.63867	0.819335	−	0.573316i	$-0.194343\pi$
$283$	27.5667	1.63867	0.819335	+	0.573316i	$0.194343\pi$
$284$	4.89716	0.290593
$285$	0	0
$286$	0	0
$287$	4.06421	0.239903
$288$	2.13261	0.125666
$289$	−10.7385	−0.631676
$290$	0	0
$291$	5.82688	0.341578
$292$	8.85181	0.518013
$293$	21.8587	1.27700	0.638501	−	0.769621i	$-0.279555\pi$
$293$	21.8587	1.27700	0.638501	+	0.769621i	$0.279555\pi$
$294$	5.21361	0.304064
$295$	0	0
$296$	−9.43050	−0.548137
$297$	0	0
$298$	−3.67895	−0.213116
$299$	21.2666	1.22988
$300$	0	0
$301$	4.10284	0.236484
$302$	1.38114	0.0794757
$303$	−12.3305	−0.708366
$304$	15.3486	0.880301
$305$	0	0
$306$	0.523008	0.0298984
$307$	9.80019	0.559326	0.279663	−	0.960098i	$-0.409777\pi$
$307$	9.80019	0.559326	0.279663	+	0.960098i	$0.409777\pi$
$308$	0	0
$309$	−14.5798	−0.829418
$310$	0	0
$311$	9.77107	0.554066	0.277033	−	0.960860i	$-0.410649\pi$
$311$	9.77107	0.554066	0.277033	+	0.960860i	$0.410649\pi$
$312$	−13.9209	−0.788118
$313$	0.897155	0.0507102	0.0253551	−	0.999679i	$-0.491928\pi$
$313$	0.897155	0.0507102	0.0253551	+	0.999679i	$0.491928\pi$
$314$	8.40694	0.474431
$315$	0	0
$316$	−6.41126	−0.360662
$317$	2.66822	0.149862	0.0749311	−	0.997189i	$-0.476126\pi$
$317$	2.66822	0.149862	0.0749311	+	0.997189i	$0.476126\pi$
$318$	4.69891	0.263502
$319$	0	0
$320$	0	0
$321$	26.8942	1.50109
$322$	1.02558	0.0571531
$323$	14.4067	0.801611
$324$	13.2783	0.737683
$325$	0	0
$326$	6.33205	0.350700
$327$	−28.6408	−1.58384
$328$	15.3169	0.845736
$329$	−4.61746	−0.254569
$330$	0	0
$331$	18.9744	1.04293	0.521464	−	0.853273i	$-0.325386\pi$
$331$	18.9744	1.04293	0.521464	+	0.853273i	$0.325386\pi$
$332$	3.03266	0.166439
$333$	2.26150	0.123930
$334$	−7.63565	−0.417804
$335$	0	0
$336$	2.05340	0.112022
$337$	4.79559	0.261232	0.130616	−	0.991433i	$-0.458304\pi$
$337$	4.79559	0.261232	0.130616	+	0.991433i	$0.458304\pi$
$338$	−4.80789	−0.261515
$339$	0.740827	0.0402362
$340$	0	0
$341$	0	0
$342$	1.20336	0.0650702
$343$	−6.62140	−0.357522
$344$	15.4625	0.833684
$345$	0	0
$346$	10.9744	0.589989
$347$	0.258469	0.0138754	0.00693768	−	0.999976i	$-0.497792\pi$
$347$	0.258469	0.0138754	0.00693768	+	0.999976i	$0.497792\pi$
$348$	25.5319	1.36865
$349$	1.34491	0.0719913	0.0359956	−	0.999352i	$-0.488540\pi$
$349$	1.34491	0.0719913	0.0359956	+	0.999352i	$0.488540\pi$
$350$	0	0
$351$	26.3813	1.40813
$352$	0	0
$353$	−35.4537	−1.88701	−0.943506	−	0.331355i	$-0.892494\pi$
$353$	−35.4537	−1.88701	−0.943506	+	0.331355i	$0.892494\pi$
$354$	−2.13261	−0.113347
$355$	0	0
$356$	27.8413	1.47559
$357$	1.92739	0.102008
$358$	0.222432	0.0117559
$359$	−7.84167	−0.413867	−0.206934	−	0.978355i	$-0.566348\pi$
$359$	−7.84167	−0.413867	−0.206934	+	0.978355i	$0.566348\pi$
$360$	0	0
$361$	14.1475	0.744608
$362$	1.50446	0.0790727
$363$	0	0
$364$	4.07906	0.213801
$365$	0	0
$366$	3.10051	0.162066
$367$	−1.06654	−0.0556730	−0.0278365	−	0.999612i	$-0.508862\pi$
$367$	−1.06654	−0.0556730	−0.0278365	+	0.999612i	$0.508862\pi$
$368$	−11.8222	−0.616276
$369$	−3.67311	−0.191215
$370$	0	0
$371$	−2.93374	−0.152312
$372$	−22.5845	−1.17095
$373$	37.6388	1.94886	0.974431	−	0.224689i	$-0.0721366\pi$
$373$	37.6388	1.94886	0.974431	+	0.224689i	$0.0721366\pi$
$374$	0	0
$375$	0	0
$376$	−17.4020	−0.897438
$377$	43.2201	2.22595
$378$	1.27223	0.0654365
$379$	35.1475	1.80541	0.902704	−	0.430262i	$-0.141579\pi$
$379$	35.1475	1.80541	0.902704	+	0.430262i	$0.141579\pi$
$380$	0	0
$381$	−25.3537	−1.29891
$382$	2.61464	0.133776
$383$	−0.824549	−0.0421325	−0.0210662	−	0.999778i	$-0.506706\pi$
$383$	−0.824549	−0.0421325	−0.0210662	+	0.999778i	$0.506706\pi$
$384$	18.0075	0.918942
$385$	0	0
$386$	−1.54213	−0.0784924
$387$	−3.70803	−0.188490
$388$	−6.43462	−0.326669
$389$	4.84367	0.245584	0.122792	−	0.992432i	$-0.460815\pi$
$389$	4.84367	0.245584	0.122792	+	0.992432i	$0.460815\pi$
$390$	0	0
$391$	−11.0968	−0.561187
$392$	−12.2676	−0.619606
$393$	15.3835	0.775994
$394$	9.33178	0.470128
$395$	0	0
$396$	0	0
$397$	7.33178	0.367971	0.183986	−	0.982929i	$-0.441100\pi$
$397$	7.33178	0.367971	0.183986	+	0.982929i	$0.441100\pi$
$398$	−9.78677	−0.490566
$399$	4.43462	0.222009
$400$	0	0
$401$	−31.1499	−1.55555	−0.777775	−	0.628542i	$-0.783652\pi$
$401$	−31.1499	−1.55555	−0.777775	+	0.628542i	$0.783652\pi$
$402$	5.85519	0.292030
$403$	−38.2309	−1.90442
$404$	13.6165	0.677447
$405$	0	0
$406$	2.08428	0.103441
$407$	0	0
$408$	7.26384	0.359613
$409$	−7.23209	−0.357604	−0.178802	−	0.983885i	$-0.557222\pi$
$409$	−7.23209	−0.357604	−0.178802	+	0.983885i	$0.557222\pi$
$410$	0	0
$411$	−6.36202	−0.313815
$412$	16.1005	0.793215
$413$	1.33149	0.0655182
$414$	−0.926885	−0.0455539
$415$	0	0
$416$	23.5310	1.15370
$417$	14.7699	0.723284
$418$	0	0
$419$	−2.30620	−0.112665	−0.0563327	−	0.998412i	$-0.517941\pi$
$419$	−2.30620	−0.112665	−0.0563327	+	0.998412i	$0.517941\pi$
$420$	0	0
$421$	6.63798	0.323515	0.161758	−	0.986831i	$-0.448284\pi$
$421$	6.63798	0.323515	0.161758	+	0.986831i	$0.448284\pi$
$422$	3.30853	0.161057
$423$	4.17312	0.202904
$424$	−11.0565	−0.536951
$425$	0	0
$426$	2.13261	0.103326
$427$	−1.93579	−0.0936794
$428$	−29.6993	−1.43557
$429$	0	0
$430$	0	0
$431$	−32.8432	−1.58200	−0.791000	−	0.611816i	$-0.790439\pi$
$431$	−32.8432	−1.58200	−0.791000	+	0.611816i	$0.790439\pi$
$432$	−14.6655	−0.705594
$433$	−2.36202	−0.113511	−0.0567557	−	0.998388i	$-0.518076\pi$
$433$	−2.36202	−0.113511	−0.0567557	+	0.998388i	$0.518076\pi$
$434$	−1.84367	−0.0884991
$435$	0	0
$436$	31.6281	1.51471
$437$	−25.5319	−1.22135
$438$	3.85479	0.184189
$439$	30.8847	1.47404	0.737022	−	0.675869i	$-0.236231\pi$
$439$	30.8847	1.47404	0.737022	+	0.675869i	$0.236231\pi$
$440$	0	0
$441$	2.94185	0.140088
$442$	5.77081	0.274489
$443$	13.7748	0.654460	0.327230	−	0.944945i	$-0.393885\pi$
$443$	13.7748	0.654460	0.327230	+	0.944945i	$0.393885\pi$
$444$	14.7408	0.699569
$445$	0	0
$446$	−0.499480	−0.0236511
$447$	12.2530	0.579548
$448$	−1.42928	−0.0675272
$449$	−38.9163	−1.83657	−0.918286	−	0.395917i	$-0.870427\pi$
$449$	−38.9163	−1.83657	−0.918286	+	0.395917i	$0.870427\pi$
$450$	0	0
$451$	0	0
$452$	−0.818095	−0.0384800
$453$	−4.60000	−0.216127
$454$	−0.231266	−0.0108538
$455$	0	0
$456$	16.7129	0.782654
$457$	1.33149	0.0622843	0.0311422	−	0.999515i	$-0.490086\pi$
$457$	1.33149	0.0622843	0.0311422	+	0.999515i	$0.490086\pi$
$458$	−6.51130	−0.304253
$459$	−13.7656	−0.642522
$460$	0	0
$461$	−5.73993	−0.267335	−0.133668	−	0.991026i	$-0.542675\pi$
$461$	−5.73993	−0.267335	−0.133668	+	0.991026i	$0.542675\pi$
$462$	0	0
$463$	3.44535	0.160119	0.0800595	−	0.996790i	$-0.474489\pi$
$463$	3.44535	0.160119	0.0800595	+	0.996790i	$0.474489\pi$
$464$	−24.0263	−1.11539
$465$	0	0
$466$	10.1731	0.471261
$467$	27.2164	1.25943	0.629713	−	0.776828i	$-0.283173\pi$
$467$	27.2164	1.25943	0.629713	+	0.776828i	$0.283173\pi$
$468$	−3.68653	−0.170410
$469$	−3.65565	−0.168802
$470$	0	0
$471$	−28.0000	−1.29017
$472$	5.01802	0.230973
$473$	0	0
$474$	−2.79198	−0.128240
$475$	0	0
$476$	−2.12842	−0.0975560
$477$	2.65143	0.121401
$478$	1.28708	0.0588697
$479$	−13.9822	−0.638861	−0.319431	−	0.947610i	$-0.603492\pi$
$479$	−13.9822	−0.638861	−0.319431	+	0.947610i	$0.603492\pi$
$480$	0	0
$481$	24.9532	1.13777
$482$	−2.19729	−0.100084
$483$	−3.41576	−0.155423
$484$	0	0
$485$	0	0
$486$	−2.15411	−0.0977124
$487$	7.45375	0.337761	0.168881	−	0.985636i	$-0.445985\pi$
$487$	7.45375	0.337761	0.168881	+	0.985636i	$0.445985\pi$
$488$	−7.29548	−0.330251
$489$	−21.0894	−0.953696
$490$	0	0
$491$	15.9756	0.720969	0.360484	−	0.932765i	$-0.382611\pi$
$491$	15.9756	0.720969	0.360484	+	0.932765i	$0.382611\pi$
$492$	−23.9419	−1.07938
$493$	−22.5519	−1.01569
$494$	13.2777	0.597392
$495$	0	0
$496$	21.2527	0.954275
$497$	−1.33149	−0.0597253
$498$	1.32066	0.0591803
$499$	−24.3509	−1.09010	−0.545048	−	0.838405i	$-0.683489\pi$
$499$	−24.3509	−1.09010	−0.545048	+	0.838405i	$0.683489\pi$
$500$	0	0
$501$	25.4311	1.13618
$502$	5.50004	0.245479
$503$	27.5667	1.22914	0.614569	−	0.788863i	$-0.289330\pi$
$503$	27.5667	1.22914	0.614569	+	0.788863i	$0.289330\pi$
$504$	−0.378809	−0.0168735
$505$	0	0
$506$	0	0
$507$	16.0131	0.711165
$508$	27.9981	1.24222
$509$	17.6850	0.783874	0.391937	−	0.919992i	$-0.371805\pi$
$509$	17.6850	0.783874	0.391937	+	0.919992i	$0.371805\pi$
$510$	0	0
$511$	−2.40672	−0.106467
$512$	−22.7443	−1.00516
$513$	−31.6724	−1.39837
$514$	−9.63951	−0.425181
$515$	0	0
$516$	−24.1695	−1.06400
$517$	0	0
$518$	1.20336	0.0528725
$519$	−36.5512	−1.60442
$520$	0	0
$521$	−5.38993	−0.236137	−0.118068	−	0.993005i	$-0.537670\pi$
$521$	−5.38993	−0.236137	−0.118068	+	0.993005i	$0.537670\pi$
$522$	−1.88371	−0.0824477
$523$	−37.6737	−1.64735	−0.823677	−	0.567059i	$-0.808081\pi$
$523$	−37.6737	−1.64735	−0.823677	+	0.567059i	$0.808081\pi$
$524$	−16.9880	−0.742123
$525$	0	0
$526$	3.15400	0.137521
$527$	19.9486	0.868973
$528$	0	0
$529$	−3.33411	−0.144961
$530$	0	0
$531$	−1.20336	−0.0522213
$532$	−4.89716	−0.212319
$533$	−40.5287	−1.75549
$534$	12.1244	0.524672
$535$	0	0
$536$	−13.7772	−0.595084
$537$	−0.740827	−0.0319690
$538$	−0.912344	−0.0393339
$539$	0	0
$540$	0	0
$541$	28.9651	1.24531	0.622653	−	0.782498i	$-0.286055\pi$
$541$	28.9651	1.24531	0.622653	+	0.782498i	$0.286055\pi$
$542$	5.46020	0.234536
$543$	−5.01073	−0.215031
$544$	−12.2783	−0.526428
$545$	0	0
$546$	1.77635	0.0760208
$547$	19.7396	0.844002	0.422001	−	0.906595i	$-0.361328\pi$
$547$	19.7396	0.844002	0.422001	+	0.906595i	$0.361328\pi$
$548$	7.02558	0.300118
$549$	1.74951	0.0746672
$550$	0	0
$551$	−51.8884	−2.21052
$552$	−12.8731	−0.547916
$553$	1.74316	0.0741266
$554$	10.9809	0.466533
$555$	0	0
$556$	−16.3104	−0.691714
$557$	−16.6801	−0.706757	−0.353378	−	0.935481i	$-0.614967\pi$
$557$	−16.6801	−0.706757	−0.353378	+	0.935481i	$0.614967\pi$
$558$	1.66626	0.0705382
$559$	−40.9139	−1.73048
$560$	0	0
$561$	0	0
$562$	10.8739	0.458688
$563$	4.90680	0.206797	0.103399	−	0.994640i	$-0.467028\pi$
$563$	4.90680	0.206797	0.103399	+	0.994640i	$0.467028\pi$
$564$	27.2010	1.14537
$565$	0	0
$566$	−13.2568	−0.557227
$567$	−3.61023	−0.151616
$568$	−5.01802	−0.210551
$569$	−22.4334	−0.940457	−0.470229	−	0.882545i	$-0.655829\pi$
$569$	−22.4334	−0.940457	−0.470229	+	0.882545i	$0.655829\pi$
$570$	0	0
$571$	0.195590	0.00818520	0.00409260	−	0.999992i	$-0.498697\pi$
$571$	0.195590	0.00818520	0.00409260	+	0.999992i	$0.498697\pi$
$572$	0	0
$573$	−8.70826	−0.363793
$574$	−1.95448	−0.0815785
$575$	0	0
$576$	1.29174	0.0538226
$577$	41.2481	1.71718	0.858590	−	0.512664i	$-0.171341\pi$
$577$	41.2481	1.71718	0.858590	+	0.512664i	$0.171341\pi$
$578$	5.16415	0.214800
$579$	5.13619	0.213453
$580$	0	0
$581$	−0.824549	−0.0342081
$582$	−2.80215	−0.116153
$583$	0	0
$584$	−9.07028	−0.375331
$585$	0	0
$586$	−10.5119	−0.434242
$587$	1.77480	0.0732538	0.0366269	−	0.999329i	$-0.488339\pi$
$587$	1.77480	0.0732538	0.0366269	+	0.999329i	$0.488339\pi$
$588$	19.1755	0.790782
$589$	45.8984	1.89121
$590$	0	0
$591$	−31.0802	−1.27847
$592$	−13.8716	−0.570118
$593$	29.0094	1.19127	0.595636	−	0.803254i	$-0.296900\pi$
$593$	29.0094	1.19127	0.595636	+	0.803254i	$0.296900\pi$
$594$	0	0
$595$	0	0
$596$	−13.5310	−0.554252
$597$	32.5956	1.33405
$598$	−10.2271	−0.418219
$599$	40.2672	1.64527	0.822636	−	0.568568i	$-0.192502\pi$
$599$	40.2672	1.64527	0.822636	+	0.568568i	$0.192502\pi$
$600$	0	0
$601$	−13.6957	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$601$	−13.6957	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$602$	−1.97306	−0.0804160
$603$	3.30387	0.134544
$604$	5.07978	0.206693
$605$	0	0
$606$	5.92972	0.240879
$607$	−21.1543	−0.858626	−0.429313	−	0.903156i	$-0.641244\pi$
$607$	−21.1543	−0.858626	−0.429313	+	0.903156i	$0.641244\pi$
$608$	−28.2504	−1.14570
$609$	−6.94185	−0.281298
$610$	0	0
$611$	46.0457	1.86281
$612$	1.92360	0.0777571
$613$	23.2520	0.939139	0.469570	−	0.882895i	$-0.344409\pi$
$613$	23.2520	0.939139	0.469570	+	0.882895i	$0.344409\pi$
$614$	−4.71292	−0.190198
$615$	0	0
$616$	0	0
$617$	13.1028	0.527501	0.263750	−	0.964591i	$-0.415040\pi$
$617$	13.1028	0.527501	0.263750	+	0.964591i	$0.415040\pi$
$618$	7.01146	0.282042
$619$	16.6682	0.669952	0.334976	−	0.942227i	$-0.391272\pi$
$619$	16.6682	0.669952	0.334976	+	0.942227i	$0.391272\pi$
$620$	0	0
$621$	24.3956	0.978962
$622$	−4.69891	−0.188409
$623$	−7.56978	−0.303277
$624$	−20.4767	−0.819723
$625$	0	0
$626$	−0.431443	−0.0172439
$627$	0	0
$628$	30.9204	1.23386
$629$	−13.0204	−0.519156
$630$	0	0
$631$	−2.66356	−0.106035	−0.0530173	−	0.998594i	$-0.516884\pi$
$631$	−2.66356	−0.106035	−0.0530173	+	0.998594i	$0.516884\pi$
$632$	6.56950	0.261321
$633$	−11.0193	−0.437979
$634$	−1.28315	−0.0509604
$635$	0	0
$636$	17.2824	0.685292
$637$	32.4601	1.28611
$638$	0	0
$639$	1.20336	0.0476041
$640$	0	0
$641$	−0.462531	−0.0182689	−0.00913445	−	0.999958i	$-0.502908\pi$
$641$	−0.462531	−0.0182689	−0.00913445	+	0.999958i	$0.502908\pi$
$642$	−12.9335	−0.510443
$643$	33.4621	1.31962	0.659809	−	0.751433i	$-0.270637\pi$
$643$	33.4621	1.31962	0.659809	+	0.751433i	$0.270637\pi$
$644$	3.77203	0.148639
$645$	0	0
$646$	−6.92820	−0.272587
$647$	40.2588	1.58274	0.791368	−	0.611340i	$-0.209369\pi$
$647$	40.2588	1.58274	0.791368	+	0.611340i	$0.209369\pi$
$648$	−13.6060	−0.534495
$649$	0	0
$650$	0	0
$651$	6.14050	0.240665
$652$	23.2890	0.912068
$653$	−15.3486	−0.600636	−0.300318	−	0.953839i	$-0.597093\pi$
$653$	−15.3486	−0.600636	−0.300318	+	0.953839i	$0.597093\pi$
$654$	13.7734	0.538582
$655$	0	0
$656$	22.5301	0.879652
$657$	2.17512	0.0848595
$658$	2.22054	0.0865656
$659$	36.6635	1.42821	0.714104	−	0.700039i	$-0.246834\pi$
$659$	36.6635	1.42821	0.714104	+	0.700039i	$0.246834\pi$
$660$	0	0
$661$	−8.03024	−0.312340	−0.156170	−	0.987730i	$-0.549915\pi$
$661$	−8.03024	−0.312340	−0.156170	+	0.987730i	$0.549915\pi$
$662$	−9.12482	−0.354646
$663$	−19.2201	−0.746449
$664$	−3.10751	−0.120595
$665$	0	0
$666$	−1.08756	−0.0421421
$667$	39.9670	1.54753
$668$	−28.0836	−1.08659
$669$	1.66356	0.0643169
$670$	0	0
$671$	0	0
$672$	−3.77946	−0.145796
$673$	11.7587	0.453265	0.226632	−	0.973980i	$-0.427228\pi$
$673$	11.7587	0.453265	0.226632	+	0.973980i	$0.427228\pi$
$674$	−2.30620	−0.0888316
$675$	0	0
$676$	−17.6832	−0.680124
$677$	27.2948	1.04902	0.524512	−	0.851403i	$-0.324248\pi$
$677$	27.2948	1.04902	0.524512	+	0.851403i	$0.324248\pi$
$678$	−0.356264	−0.0136822
$679$	1.74951	0.0671400
$680$	0	0
$681$	0.770249	0.0295160
$682$	0	0
$683$	−49.5738	−1.89689	−0.948444	−	0.316945i	$-0.897343\pi$
$683$	−49.5738	−1.89689	−0.948444	+	0.316945i	$0.897343\pi$
$684$	4.42590	0.169229
$685$	0	0
$686$	3.18424	0.121575
$687$	21.6864	0.827388
$688$	22.7443	0.867116
$689$	29.2556	1.11455
$690$	0	0
$691$	−25.3253	−0.963421	−0.481710	−	0.876330i	$-0.659984\pi$
$691$	−25.3253	−0.963421	−0.481710	+	0.876330i	$0.659984\pi$
$692$	40.3635	1.53439
$693$	0	0
$694$	−0.124298	−0.00471829
$695$	0	0
$696$	−26.1620	−0.991668
$697$	21.1475	0.801020
$698$	−0.646767	−0.0244805
$699$	−33.8824	−1.28155
$700$	0	0
$701$	−18.2555	−0.689500	−0.344750	−	0.938695i	$-0.612036\pi$
$701$	−18.2555	−0.689500	−0.344750	+	0.938695i	$0.612036\pi$
$702$	−12.6868	−0.478833
$703$	−29.9578	−1.12988
$704$	0	0
$705$	0	0
$706$	17.0497	0.641675
$707$	−3.70219	−0.139235
$708$	−7.84367	−0.294783
$709$	4.33178	0.162683	0.0813417	−	0.996686i	$-0.474079\pi$
$709$	4.33178	0.162683	0.0813417	+	0.996686i	$0.474079\pi$
$710$	0	0
$711$	−1.57541	−0.0590827
$712$	−28.5285	−1.06915
$713$	−35.3532	−1.32399
$714$	−0.926885	−0.0346878
$715$	0	0
$716$	0.818095	0.0305737
$717$	−4.28673	−0.160091
$718$	3.77107	0.140735
$719$	−24.1005	−0.898797	−0.449399	−	0.893331i	$-0.648362\pi$
$719$	−24.1005	−0.898797	−0.449399	+	0.893331i	$0.648362\pi$
$720$	0	0
$721$	−4.37757	−0.163029
$722$	−6.80357	−0.253203
$723$	7.31826	0.272169
$724$	5.53335	0.205645
$725$	0	0
$726$	0	0
$727$	25.1066	0.931151	0.465576	−	0.885008i	$-0.345847\pi$
$727$	25.1066	0.931151	0.465576	+	0.885008i	$0.345847\pi$
$728$	−4.17973	−0.154911
$729$	29.6961	1.09986
$730$	0	0
$731$	21.3486	0.789605
$732$	11.4036	0.421488
$733$	17.4463	0.644393	0.322196	−	0.946673i	$-0.395579\pi$
$733$	17.4463	0.644393	0.322196	+	0.946673i	$0.395579\pi$
$734$	0.512901	0.0189315
$735$	0	0
$736$	21.7598	0.802078
$737$	0	0
$738$	1.76640	0.0650222
$739$	−5.20019	−0.191292	−0.0956460	−	0.995415i	$-0.530492\pi$
$739$	−5.20019	−0.191292	−0.0956460	+	0.995415i	$0.530492\pi$
$740$	0	0
$741$	−44.2225	−1.62455
$742$	1.41084	0.0517935
$743$	37.1928	1.36447	0.682235	−	0.731133i	$-0.261008\pi$
$743$	37.1928	1.36447	0.682235	+	0.731133i	$0.261008\pi$
$744$	23.1419	0.848423
$745$	0	0
$746$	−18.1005	−0.662707
$747$	0.745203	0.0272656
$748$	0	0
$749$	8.07494	0.295052
$750$	0	0
$751$	41.4212	1.51148	0.755740	−	0.654872i	$-0.227277\pi$
$751$	41.4212	1.51148	0.755740	+	0.654872i	$0.227277\pi$
$752$	−25.5970	−0.933427
$753$	−18.3183	−0.667557
$754$	−20.7846	−0.756931
$755$	0	0
$756$	4.67921	0.170181
$757$	−3.91628	−0.142340	−0.0711698	−	0.997464i	$-0.522673\pi$
$757$	−3.91628	−0.142340	−0.0711698	+	0.997464i	$0.522673\pi$
$758$	−16.9025	−0.613926
$759$	0	0
$760$	0	0
$761$	−39.1309	−1.41849	−0.709247	−	0.704960i	$-0.750965\pi$
$761$	−39.1309	−1.41849	−0.709247	+	0.704960i	$0.750965\pi$
$762$	12.1926	0.441692
$763$	−8.59935	−0.311317
$764$	9.61653	0.347914
$765$	0	0
$766$	0.396526	0.0143271
$767$	−13.2777	−0.479430
$768$	0.860852	0.0310633
$769$	11.6755	0.421028	0.210514	−	0.977591i	$-0.432486\pi$
$769$	11.6755	0.421028	0.210514	+	0.977591i	$0.432486\pi$
$770$	0	0
$771$	32.1052	1.15624
$772$	−5.67189	−0.204136
$773$	7.76640	0.279338	0.139669	−	0.990198i	$-0.455396\pi$
$773$	7.76640	0.279338	0.139669	+	0.990198i	$0.455396\pi$
$774$	1.78319	0.0640956
$775$	0	0
$776$	6.59343	0.236691
$777$	−4.00788	−0.143782
$778$	−2.32933	−0.0835104
$779$	48.6571	1.74332
$780$	0	0
$781$	0	0
$782$	5.33644	0.190831
$783$	49.5791	1.77181
$784$	−18.0447	−0.644454
$785$	0	0
$786$	−7.39793	−0.263875
$787$	1.42928	0.0509484	0.0254742	−	0.999675i	$-0.491890\pi$
$787$	1.42928	0.0509484	0.0254742	+	0.999675i	$0.491890\pi$
$788$	34.3219	1.22267
$789$	−10.5046	−0.373975
$790$	0	0
$791$	0.222432	0.00790876
$792$	0	0
$793$	19.3039	0.685501
$794$	−3.52586	−0.125128
$795$	0	0
$796$	−35.9953	−1.27582
$797$	36.3788	1.28860	0.644302	−	0.764771i	$-0.277148\pi$
$797$	36.3788	1.28860	0.644302	+	0.764771i	$0.277148\pi$
$798$	−2.13261	−0.0754937
$799$	−24.0263	−0.849989
$800$	0	0
$801$	6.84134	0.241727
$802$	14.9800	0.528962
$803$	0	0
$804$	21.5351	0.759486
$805$	0	0
$806$	18.3853	0.647593
$807$	3.03863	0.106965
$808$	−13.9526	−0.490850
$809$	27.8735	0.979980	0.489990	−	0.871728i	$-0.337000\pi$
$809$	27.8735	0.979980	0.489990	+	0.871728i	$0.337000\pi$
$810$	0	0
$811$	39.1793	1.37577	0.687885	−	0.725820i	$-0.258539\pi$
$811$	39.1793	1.37577	0.687885	+	0.725820i	$0.258539\pi$
$812$	7.66589	0.269020
$813$	−18.1856	−0.637798
$814$	0	0
$815$	0	0
$816$	10.6846	0.374035
$817$	49.1196	1.71848
$818$	3.47792	0.121603
$819$	1.00233	0.0350243
$820$	0	0
$821$	−13.7772	−0.480827	−0.240414	−	0.970671i	$-0.577283\pi$
$821$	−13.7772	−0.480827	−0.240414	+	0.970671i	$0.577283\pi$
$822$	3.05950	0.106712
$823$	19.8800	0.692972	0.346486	−	0.938055i	$-0.387375\pi$
$823$	19.8800	0.692972	0.346486	+	0.938055i	$0.387375\pi$
$824$	−16.4979	−0.574731
$825$	0	0
$826$	−0.640313	−0.0222793
$827$	−13.3272	−0.463431	−0.231716	−	0.972784i	$-0.574434\pi$
$827$	−13.3272	−0.463431	−0.231716	+	0.972784i	$0.574434\pi$
$828$	−3.40905	−0.118473
$829$	51.1899	1.77790	0.888950	−	0.458005i	$-0.151436\pi$
$829$	51.1899	1.77790	0.888950	+	0.458005i	$0.151436\pi$
$830$	0	0
$831$	−36.5727	−1.26869
$832$	14.2529	0.494132
$833$	−16.9374	−0.586846
$834$	−7.10284	−0.245951
$835$	0	0
$836$	0	0
$837$	−43.8558	−1.51588
$838$	1.10906	0.0383117
$839$	−16.8739	−0.582552	−0.291276	−	0.956639i	$-0.594080\pi$
$839$	−16.8739	−0.582552	−0.291276	+	0.956639i	$0.594080\pi$
$840$	0	0
$841$	52.2248	1.80086
$842$	−3.19221	−0.110011
$843$	−36.2164	−1.24736
$844$	12.1687	0.418862
$845$	0	0
$846$	−2.00686	−0.0689972
$847$	0	0
$848$	−16.2633	−0.558484
$849$	44.1530	1.51533
$850$	0	0
$851$	23.0749	0.790999
$852$	7.84367	0.268720
$853$	−5.42262	−0.185667	−0.0928335	−	0.995682i	$-0.529592\pi$
$853$	−5.42262	−0.185667	−0.0928335	+	0.995682i	$0.529592\pi$
$854$	0.930923	0.0318555
$855$	0	0
$856$	30.4323	1.04015
$857$	−27.8386	−0.950947	−0.475474	−	0.879730i	$-0.657723\pi$
$857$	−27.8386	−0.950947	−0.475474	+	0.879730i	$0.657723\pi$
$858$	0	0
$859$	48.1899	1.64422	0.822109	−	0.569330i	$-0.192797\pi$
$859$	48.1899	1.64422	0.822109	+	0.569330i	$0.192797\pi$
$860$	0	0
$861$	6.50956	0.221845
$862$	15.7943	0.537956
$863$	−26.8609	−0.914354	−0.457177	−	0.889376i	$-0.651139\pi$
$863$	−26.8609	−0.914354	−0.457177	+	0.889376i	$0.651139\pi$
$864$	26.9931	0.918325
$865$	0	0
$866$	1.13590	0.0385993
$867$	−17.1996	−0.584130
$868$	−6.78095	−0.230160
$869$	0	0
$870$	0	0
$871$	36.4545	1.23521
$872$	−32.4087	−1.09750
$873$	−1.58115	−0.0535140
$874$	12.2783	0.415320
$875$	0	0
$876$	14.1778	0.479022
$877$	−39.8837	−1.34678	−0.673389	−	0.739289i	$-0.735162\pi$
$877$	−39.8837	−1.34678	−0.673389	+	0.739289i	$0.735162\pi$
$878$	−14.8525	−0.501246
$879$	35.0107	1.18088
$880$	0	0
$881$	−2.82222	−0.0950829	−0.0475415	−	0.998869i	$-0.515139\pi$
$881$	−2.82222	−0.0950829	−0.0475415	+	0.998869i	$0.515139\pi$
$882$	−1.41474	−0.0476368
$883$	−50.5287	−1.70043	−0.850213	−	0.526439i	$-0.823527\pi$
$883$	−50.5287	−1.70043	−0.850213	+	0.526439i	$0.823527\pi$
$884$	21.2248	0.713868
$885$	0	0
$886$	−6.62431	−0.222548
$887$	28.3195	0.950875	0.475437	−	0.879750i	$-0.342290\pi$
$887$	28.3195	0.950875	0.475437	+	0.879750i	$0.342290\pi$
$888$	−15.1046	−0.506879
$889$	−7.61241	−0.255312
$890$	0	0
$891$	0	0
$892$	−1.83707	−0.0615096
$893$	−55.2806	−1.84990
$894$	−5.89249	−0.197074
$895$	0	0
$896$	5.40672	0.180626
$897$	34.0623	1.13731
$898$	18.7149	0.624523
$899$	−71.8483	−2.39628
$900$	0	0
$901$	−15.2653	−0.508561
$902$	0	0
$903$	6.57144	0.218684
$904$	0.838286	0.0278810
$905$	0	0
$906$	2.21214	0.0734935
$907$	−51.0508	−1.69511	−0.847556	−	0.530705i	$-0.821927\pi$
$907$	−51.0508	−1.69511	−0.847556	+	0.530705i	$0.821927\pi$
$908$	−0.850586	−0.0282277
$909$	3.34593	0.110978
$910$	0	0
$911$	−25.1429	−0.833021	−0.416510	−	0.909131i	$-0.636747\pi$
$911$	−25.1429	−0.833021	−0.416510	+	0.909131i	$0.636747\pi$
$912$	24.5835	0.814040
$913$	0	0
$914$	−0.640313	−0.0211797
$915$	0	0
$916$	−23.9483	−0.791274
$917$	4.61886	0.152528
$918$	6.61987	0.218488
$919$	−25.6151	−0.844965	−0.422482	−	0.906371i	$-0.638841\pi$
$919$	−25.6151	−0.844965	−0.422482	+	0.906371i	$0.638841\pi$
$920$	0	0
$921$	15.6968	0.517226
$922$	2.76034	0.0909069
$923$	13.2777	0.437041
$924$	0	0
$925$	0	0
$926$	−1.65687	−0.0544482
$927$	3.95631	0.129942
$928$	44.2225	1.45167
$929$	−8.60774	−0.282411	−0.141205	−	0.989980i	$-0.545098\pi$
$929$	−8.60774	−0.282411	−0.141205	+	0.989980i	$0.545098\pi$
$930$	0	0
$931$	−38.9702	−1.27720
$932$	37.4163	1.22561
$933$	15.6501	0.512362
$934$	−13.0884	−0.428266
$935$	0	0
$936$	3.77752	0.123472
$937$	−5.96640	−0.194914	−0.0974569	−	0.995240i	$-0.531071\pi$
$937$	−5.96640	−0.194914	−0.0974569	+	0.995240i	$0.531071\pi$
$938$	1.75801	0.0574010
$939$	1.43695	0.0468933
$940$	0	0
$941$	22.8380	0.744498	0.372249	−	0.928133i	$-0.378587\pi$
$941$	22.8380	0.744498	0.372249	+	0.928133i	$0.378587\pi$
$942$	13.4652	0.438721
$943$	−37.4781	−1.22045
$944$	7.38114	0.240236
$945$	0	0
$946$	0	0
$947$	2.30620	0.0749415	0.0374708	−	0.999298i	$-0.488070\pi$
$947$	2.30620	0.0749415	0.0374708	+	0.999298i	$0.488070\pi$
$948$	−10.2688	−0.333515
$949$	24.0000	0.779073
$950$	0	0
$951$	4.27363	0.138582
$952$	2.18095	0.0706851
$953$	28.6129	0.926861	0.463431	−	0.886133i	$-0.346618\pi$
$953$	28.6129	0.926861	0.463431	+	0.886133i	$0.346618\pi$
$954$	−1.27507	−0.0412821
$955$	0	0
$956$	4.73383	0.153103
$957$	0	0
$958$	6.72404	0.217244
$959$	−1.91018	−0.0616830
$960$	0	0
$961$	32.5543	1.05014
$962$	−12.0000	−0.386896
$963$	−7.29789	−0.235171
$964$	−8.08156	−0.260289
$965$	0	0
$966$	1.64264	0.0528512
$967$	−31.7422	−1.02076	−0.510380	−	0.859949i	$-0.670495\pi$
$967$	−31.7422	−1.02076	−0.510380	+	0.859949i	$0.670495\pi$
$968$	0	0
$969$	23.0749	0.741274
$970$	0	0
$971$	−37.0023	−1.18746	−0.593731	−	0.804664i	$-0.702346\pi$
$971$	−37.0023	−1.18746	−0.593731	+	0.804664i	$0.702346\pi$
$972$	−7.92273	−0.254122
$973$	4.43462	0.142168
$974$	−3.58451	−0.114855
$975$	0	0
$976$	−10.7311	−0.343494
$977$	−33.8716	−1.08365	−0.541824	−	0.840492i	$-0.682266\pi$
$977$	−33.8716	−1.08365	−0.541824	+	0.840492i	$0.682266\pi$
$978$	10.1419	0.324303
$979$	0	0
$980$	0	0
$981$	7.77184	0.248136
$982$	−7.68268	−0.245164
$983$	37.7516	1.20409	0.602044	−	0.798463i	$-0.294353\pi$
$983$	37.7516	1.20409	0.602044	+	0.798463i	$0.294353\pi$
$984$	24.5328	0.782077
$985$	0	0
$986$	10.8452	0.345383
$987$	−7.39568	−0.235407
$988$	48.8349	1.55364
$989$	−37.8343	−1.20306
$990$	0	0
$991$	−44.9354	−1.42742	−0.713710	−	0.700441i	$-0.752987\pi$
$991$	−44.9354	−1.42742	−0.713710	+	0.700441i	$0.752987\pi$
$992$	−39.1175	−1.24198
$993$	30.3909	0.964427
$994$	0.640313	0.0203095
$995$	0	0
$996$	4.85735	0.153911
$997$	−0.396526	−0.0125581	−0.00627906	−	0.999980i	$-0.501999\pi$
$997$	−0.396526	−0.0125581	−0.00627906	+	0.999980i	$0.501999\pi$
$998$	11.7104	0.370685
$999$	28.6245	0.905640

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.bg.1.3		6
5.4	even	2		605.2.a.m.1.4	yes	6
11.10	odd	2	inner	3025.2.a.bg.1.4		6
15.14	odd	2		5445.2.a.bx.1.3		6
20.19	odd	2		9680.2.a.cw.1.6		6
55.4	even	10		605.2.g.q.511.4		24
55.9	even	10		605.2.g.q.81.3		24
55.14	even	10		605.2.g.q.251.4		24
55.19	odd	10		605.2.g.q.251.3		24
55.24	odd	10		605.2.g.q.81.4		24
55.29	odd	10		605.2.g.q.511.3		24
55.39	odd	10		605.2.g.q.366.4		24
55.49	even	10		605.2.g.q.366.3		24
55.54	odd	2		605.2.a.m.1.3	✓	6
165.164	even	2		5445.2.a.bx.1.4		6
220.219	even	2		9680.2.a.cw.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.a.m.1.3	✓	6	55.54	odd	2
605.2.a.m.1.4	yes	6	5.4	even	2
605.2.g.q.81.3		24	55.9	even	10
605.2.g.q.81.4		24	55.24	odd	10
605.2.g.q.251.3		24	55.19	odd	10
605.2.g.q.251.4		24	55.14	even	10
605.2.g.q.366.3		24	55.49	even	10
605.2.g.q.366.4		24	55.39	odd	10
605.2.g.q.511.3		24	55.29	odd	10
605.2.g.q.511.4		24	55.4	even	10
3025.2.a.bg.1.3		6	1.1	even	1	trivial
3025.2.a.bg.1.4		6	11.10	odd	2	inner
5445.2.a.bx.1.3		6	15.14	odd	2
5445.2.a.bx.1.4		6	165.164	even	2
9680.2.a.cw.1.5		6	220.219	even	2
9680.2.a.cw.1.6		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:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3025.a (trivial)

\(f(q)\)	\(=\)	\(q-0.480901 q^{2} +1.60168 q^{3} -1.76873 q^{4} -0.770249 q^{6} +0.480901 q^{7} +1.81239 q^{8} -0.434624 q^{9} +O(q^{10})\)	\(q-0.480901 q^{2} +1.60168 q^{3} -1.76873 q^{4} -0.770249 q^{6} +0.480901 q^{7} +1.81239 q^{8} -0.434624 q^{9} -2.83294 q^{12} -4.79559 q^{13} -0.231266 q^{14} +2.66589 q^{16} +2.50230 q^{17} +0.209011 q^{18} +5.75739 q^{19} +0.770249 q^{21} -4.43462 q^{23} +2.90286 q^{24} +2.30620 q^{26} -5.50117 q^{27} -0.850586 q^{28} -9.01248 q^{29} +7.97209 q^{31} -4.90680 q^{32} -1.20336 q^{34} +0.768734 q^{36} -5.20336 q^{37} -2.76873 q^{38} -7.68099 q^{39} +8.45124 q^{41} -0.370413 q^{42} +8.53158 q^{43} +2.13261 q^{46} -9.60168 q^{47} +4.26990 q^{48} -6.76873 q^{49} +4.00788 q^{51} +8.48212 q^{52} -6.10051 q^{53} +2.64552 q^{54} +0.871579 q^{56} +9.22149 q^{57} +4.33411 q^{58} +2.76873 q^{59} -4.02534 q^{61} -3.83379 q^{62} -0.209011 q^{63} -2.97209 q^{64} -7.60168 q^{67} -4.42590 q^{68} -7.10284 q^{69} -2.76873 q^{71} -0.787707 q^{72} -5.00460 q^{73} +2.50230 q^{74} -10.1833 q^{76} +3.69380 q^{78} +3.62478 q^{79} -7.50723 q^{81} -4.06421 q^{82} -1.71459 q^{83} -1.36237 q^{84} -4.10284 q^{86} -14.4351 q^{87} -15.7408 q^{89} -2.30620 q^{91} +7.84367 q^{92} +12.7687 q^{93} +4.61746 q^{94} -7.85913 q^{96} +3.63798 q^{97} +3.25509 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{4} + 12 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^4 + 12 * q^9	\( 6 q - 6 q^{3} + 6 q^{4} + 12 q^{9} - 18 q^{12} - 18 q^{14} + 18 q^{16} - 12 q^{23} - 36 q^{26} - 30 q^{27} + 24 q^{34} - 12 q^{36} + 30 q^{42} - 42 q^{47} + 6 q^{48} - 24 q^{49} - 24 q^{53} - 30 q^{56} + 24 q^{58} + 30 q^{64} - 30 q^{67} - 24 q^{69} + 72 q^{78} + 30 q^{81} - 42 q^{82} - 6 q^{86} - 30 q^{89} + 36 q^{91} - 36 q^{92} + 60 q^{93} - 24 q^{97}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^4 + 12 * q^9 - 18 * q^12 - 18 * q^14 + 18 * q^16 - 12 * q^23 - 36 * q^26 - 30 * q^27 + 24 * q^34 - 12 * q^36 + 30 * q^42 - 42 * q^47 + 6 * q^48 - 24 * q^49 - 24 * q^53 - 30 * q^56 + 24 * q^58 + 30 * q^64 - 30 * q^67 - 24 * q^69 + 72 * q^78 + 30 * q^81 - 42 * q^82 - 6 * q^86 - 30 * q^89 + 36 * q^91 - 36 * q^92 + 60 * q^93 - 24 * q^97

Introduction

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