LMFDB - Embedded newform 952.2.a.c.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [952,2,Mod(1,952)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(952, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("952.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 952 = 2^{3} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	952.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$7.60175827243$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	952.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.745898 q^{3} +0.935432 q^{5} +1.00000 q^{7} -2.44364 q^{9} -0.508203 q^{11} -5.36266 q^{13} -0.697737 q^{15} -1.00000 q^{17} -5.87086 q^{19} -0.745898 q^{21} +3.36266 q^{23} -4.12497 q^{25} +4.06040 q^{27} +1.87086 q^{29} +3.91903 q^{31} +0.379068 q^{33} +0.935432 q^{35} -2.12914 q^{37} +4.00000 q^{39} -9.12497 q^{41} -2.10856 q^{43} -2.28586 q^{45} -2.98359 q^{47} +1.00000 q^{49} +0.745898 q^{51} -11.3145 q^{53} -0.475390 q^{55} +4.37907 q^{57} +6.72532 q^{59} -9.25410 q^{61} -2.44364 q^{63} -5.01641 q^{65} -2.80630 q^{67} -2.50820 q^{69} -3.83805 q^{71} +6.14137 q^{73} +3.07681 q^{75} -0.508203 q^{77} +4.00000 q^{79} +4.30226 q^{81} -9.87086 q^{83} -0.935432 q^{85} -1.39547 q^{87} -17.1044 q^{89} -5.36266 q^{91} -2.92319 q^{93} -5.49180 q^{95} +9.28169 q^{97} +1.24186 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 5 q^{5} + 3 q^{7} + 2 q^{9} - 2 q^{13} + 8 q^{15} - 3 q^{17} - 2 q^{19} - 3 q^{21} - 4 q^{23} + 4 q^{25} - 12 q^{27} - 10 q^{29} + 7 q^{31} - 16 q^{33} - 5 q^{35} - 22 q^{37} + 12 q^{39}+ \cdots + 38 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 5 * q^5 + 3 * q^7 + 2 * q^9 - 2 * q^13 + 8 * q^15 - 3 * q^17 - 2 * q^19 - 3 * q^21 - 4 * q^23 + 4 * q^25 - 12 * q^27 - 10 * q^29 + 7 * q^31 - 16 * q^33 - 5 * q^35 - 22 * q^37 + 12 * q^39 - 11 * q^41 + 7 * q^43 - 20 * q^45 - 12 * q^47 + 3 * q^49 + 3 * q^51 - 9 * q^53 - 6 * q^55 - 4 * q^57 - 8 * q^59 - 27 * q^61 + 2 * q^63 - 12 * q^65 + 15 * q^67 - 6 * q^69 - 2 * q^71 - q^73 - 18 * q^75 + 12 * q^79 + 23 * q^81 - 14 * q^83 + 5 * q^85 + 16 * q^87 - 6 * q^89 - 2 * q^91 - 36 * q^93 - 18 * q^95 + 9 * q^97 + 38 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.745898	−0.430645	−0.215322	−	0.976543i	$-0.569080\pi$
$3$	−0.745898	−0.430645	−0.215322	+	0.976543i	$0.569080\pi$
$4$	0	0
$5$	0.935432	0.418338	0.209169	−	0.977880i	$-0.432924\pi$
$5$	0.935432	0.418338	0.209169	+	0.977880i	$0.432924\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.44364	−0.814545
$10$	0	0
$11$	−0.508203	−0.153229	−0.0766145	−	0.997061i	$-0.524411\pi$
$11$	−0.508203	−0.153229	−0.0766145	+	0.997061i	$0.524411\pi$
$12$	0	0
$13$	−5.36266	−1.48733	−0.743667	−	0.668550i	$-0.766915\pi$
$13$	−5.36266	−1.48733	−0.743667	+	0.668550i	$0.766915\pi$
$14$	0	0
$15$	−0.697737	−0.180155
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−5.87086	−1.34687	−0.673434	−	0.739247i	$-0.735182\pi$
$19$	−5.87086	−1.34687	−0.673434	+	0.739247i	$0.735182\pi$
$20$	0	0
$21$	−0.745898	−0.162768
$22$	0	0
$23$	3.36266	0.701163	0.350582	−	0.936532i	$-0.385984\pi$
$23$	3.36266	0.701163	0.350582	+	0.936532i	$0.385984\pi$
$24$	0	0
$25$	−4.12497	−0.824993
$26$	0	0
$27$	4.06040	0.781424
$28$	0	0
$29$	1.87086	0.347411	0.173705	−	0.984798i	$-0.444426\pi$
$29$	1.87086	0.347411	0.173705	+	0.984798i	$0.444426\pi$
$30$	0	0
$31$	3.91903	0.703878	0.351939	−	0.936023i	$-0.385522\pi$
$31$	3.91903	0.703878	0.351939	+	0.936023i	$0.385522\pi$
$32$	0	0
$33$	0.379068	0.0659873
$34$	0	0
$35$	0.935432	0.158117
$36$	0	0
$37$	−2.12914	−0.350028	−0.175014	−	0.984566i	$-0.555997\pi$
$37$	−2.12914	−0.350028	−0.175014	+	0.984566i	$0.555997\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−9.12497	−1.42508	−0.712540	−	0.701631i	$-0.752455\pi$
$41$	−9.12497	−1.42508	−0.712540	+	0.701631i	$0.752455\pi$
$42$	0	0
$43$	−2.10856	−0.321552	−0.160776	−	0.986991i	$-0.551400\pi$
$43$	−2.10856	−0.321552	−0.160776	+	0.986991i	$0.551400\pi$
$44$	0	0
$45$	−2.28586	−0.340755
$46$	0	0
$47$	−2.98359	−0.435202	−0.217601	−	0.976038i	$-0.569823\pi$
$47$	−2.98359	−0.435202	−0.217601	+	0.976038i	$0.569823\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.745898	0.104447
$52$	0	0
$53$	−11.3145	−1.55417	−0.777083	−	0.629398i	$-0.783302\pi$
$53$	−11.3145	−1.55417	−0.777083	+	0.629398i	$0.783302\pi$
$54$	0	0
$55$	−0.475390	−0.0641016
$56$	0	0
$57$	4.37907	0.580022
$58$	0	0
$59$	6.72532	0.875562	0.437781	−	0.899082i	$-0.355764\pi$
$59$	6.72532	0.875562	0.437781	+	0.899082i	$0.355764\pi$
$60$	0	0
$61$	−9.25410	−1.18487	−0.592433	−	0.805620i	$-0.701833\pi$
$61$	−9.25410	−1.18487	−0.592433	+	0.805620i	$0.701833\pi$
$62$	0	0
$63$	−2.44364	−0.307869
$64$	0	0
$65$	−5.01641	−0.622209
$66$	0	0
$67$	−2.80630	−0.342844	−0.171422	−	0.985198i	$-0.554836\pi$
$67$	−2.80630	−0.342844	−0.171422	+	0.985198i	$0.554836\pi$
$68$	0	0
$69$	−2.50820	−0.301952
$70$	0	0
$71$	−3.83805	−0.455493	−0.227746	−	0.973720i	$-0.573136\pi$
$71$	−3.83805	−0.455493	−0.227746	+	0.973720i	$0.573136\pi$
$72$	0	0
$73$	6.14137	0.718793	0.359397	−	0.933185i	$-0.382983\pi$
$73$	6.14137	0.718793	0.359397	+	0.933185i	$0.382983\pi$
$74$	0	0
$75$	3.07681	0.355279
$76$	0	0
$77$	−0.508203	−0.0579151
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	4.30226	0.478029
$82$	0	0
$83$	−9.87086	−1.08347	−0.541734	−	0.840550i	$-0.682232\pi$
$83$	−9.87086	−1.08347	−0.541734	+	0.840550i	$0.682232\pi$
$84$	0	0
$85$	−0.935432	−0.101462
$86$	0	0
$87$	−1.39547	−0.149611
$88$	0	0
$89$	−17.1044	−1.81306	−0.906531	−	0.422139i	$-0.861279\pi$
$89$	−17.1044	−1.81306	−0.906531	+	0.422139i	$0.861279\pi$
$90$	0	0
$91$	−5.36266	−0.562160
$92$	0	0
$93$	−2.92319	−0.303121
$94$	0	0
$95$	−5.49180	−0.563446
$96$	0	0
$97$	9.28169	0.942413	0.471206	−	0.882023i	$-0.343819\pi$
$97$	9.28169	0.942413	0.471206	+	0.882023i	$0.343819\pi$
$98$	0	0
$99$	1.24186	0.124812
$100$	0	0
$101$	6.21712	0.618626	0.309313	−	0.950960i	$-0.399901\pi$
$101$	6.21712	0.618626	0.309313	+	0.950960i	$0.399901\pi$
$102$	0	0
$103$	−1.87086	−0.184342	−0.0921709	−	0.995743i	$-0.529381\pi$
$103$	−1.87086	−0.184342	−0.0921709	+	0.995743i	$0.529381\pi$
$104$	0	0
$105$	−0.697737	−0.0680922
$106$	0	0
$107$	3.65375	0.353221	0.176610	−	0.984281i	$-0.443487\pi$
$107$	3.65375	0.353221	0.176610	+	0.984281i	$0.443487\pi$
$108$	0	0
$109$	11.1044	1.06361	0.531804	−	0.846868i	$-0.321514\pi$
$109$	11.1044	1.06361	0.531804	+	0.846868i	$0.321514\pi$
$110$	0	0
$111$	1.58812	0.150738
$112$	0	0
$113$	−3.32985	−0.313246	−0.156623	−	0.987658i	$-0.550061\pi$
$113$	−3.32985	−0.313246	−0.156623	+	0.987658i	$0.550061\pi$
$114$	0	0
$115$	3.14554	0.293323
$116$	0	0
$117$	13.1044	1.21150
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−10.7417	−0.976521
$122$	0	0
$123$	6.80630	0.613703
$124$	0	0
$125$	−8.53579	−0.763464
$126$	0	0
$127$	13.7899	1.22365	0.611827	−	0.790991i	$-0.290435\pi$
$127$	13.7899	1.22365	0.611827	+	0.790991i	$0.290435\pi$
$128$	0	0
$129$	1.57277	0.138475
$130$	0	0
$131$	−14.3791	−1.25631	−0.628153	−	0.778090i	$-0.716189\pi$
$131$	−14.3791	−1.25631	−0.628153	+	0.778090i	$0.716189\pi$
$132$	0	0
$133$	−5.87086	−0.509069
$134$	0	0
$135$	3.79823	0.326899
$136$	0	0
$137$	15.6332	1.33563	0.667816	−	0.744326i	$-0.267229\pi$
$137$	15.6332	1.33563	0.667816	+	0.744326i	$0.267229\pi$
$138$	0	0
$139$	15.0562	1.27705	0.638526	−	0.769600i	$-0.279544\pi$
$139$	15.0562	1.27705	0.638526	+	0.769600i	$0.279544\pi$
$140$	0	0
$141$	2.22546	0.187417
$142$	0	0
$143$	2.72532	0.227903
$144$	0	0
$145$	1.75007	0.145335
$146$	0	0
$147$	−0.745898	−0.0615207
$148$	0	0
$149$	−0.366830	−0.0300519	−0.0150260	−	0.999887i	$-0.504783\pi$
$149$	−0.366830	−0.0300519	−0.0150260	+	0.999887i	$0.504783\pi$
$150$	0	0
$151$	16.3585	1.33124	0.665618	−	0.746293i	$-0.268168\pi$
$151$	16.3585	1.33124	0.665618	+	0.746293i	$0.268168\pi$
$152$	0	0
$153$	2.44364	0.197556
$154$	0	0
$155$	3.66598	0.294459
$156$	0	0
$157$	−0.887271	−0.0708120	−0.0354060	−	0.999373i	$-0.511272\pi$
$157$	−0.887271	−0.0708120	−0.0354060	+	0.999373i	$0.511272\pi$
$158$	0	0
$159$	8.43947	0.669293
$160$	0	0
$161$	3.36266	0.265015
$162$	0	0
$163$	−13.7417	−1.07634	−0.538168	−	0.842838i	$-0.680883\pi$
$163$	−13.7417	−1.07634	−0.538168	+	0.842838i	$0.680883\pi$
$164$	0	0
$165$	0.354593	0.0276050
$166$	0	0
$167$	−5.09215	−0.394043	−0.197021	−	0.980399i	$-0.563127\pi$
$167$	−5.09215	−0.394043	−0.197021	+	0.980399i	$0.563127\pi$
$168$	0	0
$169$	15.7581	1.21216
$170$	0	0
$171$	14.3463	1.09709
$172$	0	0
$173$	−8.23769	−0.626300	−0.313150	−	0.949704i	$-0.601384\pi$
$173$	−8.23769	−0.626300	−0.313150	+	0.949704i	$0.601384\pi$
$174$	0	0
$175$	−4.12497	−0.311818
$176$	0	0
$177$	−5.01641	−0.377056
$178$	0	0
$179$	11.8831	0.888185	0.444092	−	0.895981i	$-0.353526\pi$
$179$	11.8831	0.888185	0.444092	+	0.895981i	$0.353526\pi$
$180$	0	0
$181$	−13.3955	−0.995678	−0.497839	−	0.867270i	$-0.665873\pi$
$181$	−13.3955	−0.995678	−0.497839	+	0.867270i	$0.665873\pi$
$182$	0	0
$183$	6.90262	0.510256
$184$	0	0
$185$	−1.99166	−0.146430
$186$	0	0
$187$	0.508203	0.0371635
$188$	0	0
$189$	4.06040	0.295351
$190$	0	0
$191$	16.4189	1.18803	0.594015	−	0.804454i	$-0.297542\pi$
$191$	16.4189	1.18803	0.594015	+	0.804454i	$0.297542\pi$
$192$	0	0
$193$	22.1208	1.59229	0.796145	−	0.605106i	$-0.206869\pi$
$193$	22.1208	1.59229	0.796145	+	0.605106i	$0.206869\pi$
$194$	0	0
$195$	3.74173	0.267951
$196$	0	0
$197$	1.32985	0.0947477	0.0473739	−	0.998877i	$-0.484915\pi$
$197$	1.32985	0.0947477	0.0473739	+	0.998877i	$0.484915\pi$
$198$	0	0
$199$	−9.85029	−0.698268	−0.349134	−	0.937073i	$-0.613524\pi$
$199$	−9.85029	−0.698268	−0.349134	+	0.937073i	$0.613524\pi$
$200$	0	0
$201$	2.09321	0.147644
$202$	0	0
$203$	1.87086	0.131309
$204$	0	0
$205$	−8.53579	−0.596165
$206$	0	0
$207$	−8.21712	−0.571129
$208$	0	0
$209$	2.98359	0.206379
$210$	0	0
$211$	16.8873	1.16257	0.581284	−	0.813701i	$-0.302551\pi$
$211$	16.8873	1.16257	0.581284	+	0.813701i	$0.302551\pi$
$212$	0	0
$213$	2.86280	0.196156
$214$	0	0
$215$	−1.97241	−0.134518
$216$	0	0
$217$	3.91903	0.266041
$218$	0	0
$219$	−4.58084	−0.309544
$220$	0	0
$221$	5.36266	0.360732
$222$	0	0
$223$	−20.8545	−1.39652	−0.698259	−	0.715845i	$-0.746042\pi$
$223$	−20.8545	−1.39652	−0.698259	+	0.715845i	$0.746042\pi$
$224$	0	0
$225$	10.0799	0.671994
$226$	0	0
$227$	−15.9107	−1.05603	−0.528015	−	0.849235i	$-0.677063\pi$
$227$	−15.9107	−1.05603	−0.528015	+	0.849235i	$0.677063\pi$
$228$	0	0
$229$	−7.49180	−0.495072	−0.247536	−	0.968879i	$-0.579621\pi$
$229$	−7.49180	−0.495072	−0.247536	+	0.968879i	$0.579621\pi$
$230$	0	0
$231$	0.379068	0.0249408
$232$	0	0
$233$	22.3379	1.46341	0.731703	−	0.681624i	$-0.238726\pi$
$233$	22.3379	1.46341	0.731703	+	0.681624i	$0.238726\pi$
$234$	0	0
$235$	−2.79095	−0.182061
$236$	0	0
$237$	−2.98359	−0.193805
$238$	0	0
$239$	15.6004	1.00910	0.504552	−	0.863382i	$-0.331658\pi$
$239$	15.6004	1.00910	0.504552	+	0.863382i	$0.331658\pi$
$240$	0	0
$241$	12.2653	0.790076	0.395038	−	0.918665i	$-0.370731\pi$
$241$	12.2653	0.790076	0.395038	+	0.918665i	$0.370731\pi$
$242$	0	0
$243$	−15.3902	−0.987285
$244$	0	0
$245$	0.935432	0.0597626
$246$	0	0
$247$	31.4835	2.00324
$248$	0	0
$249$	7.36266	0.466590
$250$	0	0
$251$	−1.87086	−0.118088	−0.0590440	−	0.998255i	$-0.518805\pi$
$251$	−1.87086	−0.118088	−0.0590440	+	0.998255i	$0.518805\pi$
$252$	0	0
$253$	−1.70892	−0.107439
$254$	0	0
$255$	0.697737	0.0436940
$256$	0	0
$257$	−2.31344	−0.144308	−0.0721542	−	0.997393i	$-0.522987\pi$
$257$	−2.31344	−0.144308	−0.0721542	+	0.997393i	$0.522987\pi$
$258$	0	0
$259$	−2.12914	−0.132298
$260$	0	0
$261$	−4.57171	−0.282982
$262$	0	0
$263$	−29.4506	−1.81600	−0.908002	−	0.418965i	$-0.862393\pi$
$263$	−29.4506	−1.81600	−0.908002	+	0.418965i	$0.862393\pi$
$264$	0	0
$265$	−10.5839	−0.650167
$266$	0	0
$267$	12.7581	0.780785
$268$	0	0
$269$	−6.08798	−0.371191	−0.185595	−	0.982626i	$-0.559421\pi$
$269$	−6.08798	−0.371191	−0.185595	+	0.982626i	$0.559421\pi$
$270$	0	0
$271$	18.3463	1.11446	0.557228	−	0.830360i	$-0.311865\pi$
$271$	18.3463	1.11446	0.557228	+	0.830360i	$0.311865\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	2.09632	0.126413
$276$	0	0
$277$	−22.8461	−1.37269	−0.686345	−	0.727276i	$-0.740786\pi$
$277$	−22.8461	−1.37269	−0.686345	+	0.727276i	$0.740786\pi$
$278$	0	0
$279$	−9.57667	−0.573340
$280$	0	0
$281$	17.2817	1.03094	0.515470	−	0.856908i	$-0.327618\pi$
$281$	17.2817	1.03094	0.515470	+	0.856908i	$0.327618\pi$
$282$	0	0
$283$	7.47645	0.444429	0.222214	−	0.974998i	$-0.428672\pi$
$283$	7.47645	0.444429	0.222214	+	0.974998i	$0.428672\pi$
$284$	0	0
$285$	4.09632	0.242645
$286$	0	0
$287$	−9.12497	−0.538630
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−6.92319	−0.405845
$292$	0	0
$293$	−14.3791	−0.840034	−0.420017	−	0.907516i	$-0.637976\pi$
$293$	−14.3791	−0.840034	−0.420017	+	0.907516i	$0.637976\pi$
$294$	0	0
$295$	6.29108	0.366281
$296$	0	0
$297$	−2.06351	−0.119737
$298$	0	0
$299$	−18.0328	−1.04286
$300$	0	0
$301$	−2.10856	−0.121535
$302$	0	0
$303$	−4.63734	−0.266408
$304$	0	0
$305$	−8.65659	−0.495675
$306$	0	0
$307$	25.7334	1.46868	0.734341	−	0.678781i	$-0.237491\pi$
$307$	25.7334	1.46868	0.734341	+	0.678781i	$0.237491\pi$
$308$	0	0
$309$	1.39547	0.0793858
$310$	0	0
$311$	14.5480	0.824943	0.412471	−	0.910971i	$-0.364666\pi$
$311$	14.5480	0.824943	0.412471	+	0.910971i	$0.364666\pi$
$312$	0	0
$313$	7.05623	0.398842	0.199421	−	0.979914i	$-0.436094\pi$
$313$	7.05623	0.398842	0.199421	+	0.979914i	$0.436094\pi$
$314$	0	0
$315$	−2.28586	−0.128793
$316$	0	0
$317$	−26.7253	−1.50104	−0.750522	−	0.660846i	$-0.770198\pi$
$317$	−26.7253	−1.50104	−0.750522	+	0.660846i	$0.770198\pi$
$318$	0	0
$319$	−0.950780	−0.0532334
$320$	0	0
$321$	−2.72532	−0.152113
$322$	0	0
$323$	5.87086	0.326664
$324$	0	0
$325$	22.1208	1.22704
$326$	0	0
$327$	−8.28275	−0.458037
$328$	0	0
$329$	−2.98359	−0.164491
$330$	0	0
$331$	−27.4025	−1.50618	−0.753088	−	0.657919i	$-0.771437\pi$
$331$	−27.4025	−1.50618	−0.753088	+	0.657919i	$0.771437\pi$
$332$	0	0
$333$	5.20283	0.285113
$334$	0	0
$335$	−2.62510	−0.143425
$336$	0	0
$337$	14.8133	0.806932	0.403466	−	0.914995i	$-0.367805\pi$
$337$	14.8133	0.806932	0.403466	+	0.914995i	$0.367805\pi$
$338$	0	0
$339$	2.48373	0.134898
$340$	0	0
$341$	−1.99166	−0.107855
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.34625	−0.126318
$346$	0	0
$347$	5.67610	0.304709	0.152355	−	0.988326i	$-0.451314\pi$
$347$	5.67610	0.304709	0.152355	+	0.988326i	$0.451314\pi$
$348$	0	0
$349$	7.61259	0.407493	0.203746	−	0.979024i	$-0.434688\pi$
$349$	7.61259	0.407493	0.203746	+	0.979024i	$0.434688\pi$
$350$	0	0
$351$	−21.7745	−1.16224
$352$	0	0
$353$	19.2992	1.02719	0.513595	−	0.858033i	$-0.328313\pi$
$353$	19.2992	1.02719	0.513595	+	0.858033i	$0.328313\pi$
$354$	0	0
$355$	−3.59024	−0.190550
$356$	0	0
$357$	0.745898	0.0394771
$358$	0	0
$359$	−19.4025	−1.02402	−0.512012	−	0.858978i	$-0.671100\pi$
$359$	−19.4025	−1.02402	−0.512012	+	0.858978i	$0.671100\pi$
$360$	0	0
$361$	15.4671	0.814055
$362$	0	0
$363$	8.01224	0.420533
$364$	0	0
$365$	5.74484	0.300699
$366$	0	0
$367$	−24.1414	−1.26017	−0.630085	−	0.776526i	$-0.716980\pi$
$367$	−24.1414	−1.26017	−0.630085	+	0.776526i	$0.716980\pi$
$368$	0	0
$369$	22.2981	1.16079
$370$	0	0
$371$	−11.3145	−0.587420
$372$	0	0
$373$	−26.4241	−1.36819	−0.684095	−	0.729393i	$-0.739802\pi$
$373$	−26.4241	−1.36819	−0.684095	+	0.729393i	$0.739802\pi$
$374$	0	0
$375$	6.36683	0.328782
$376$	0	0
$377$	−10.0328	−0.516716
$378$	0	0
$379$	33.9588	1.74435	0.872174	−	0.489195i	$-0.162709\pi$
$379$	33.9588	1.74435	0.872174	+	0.489195i	$0.162709\pi$
$380$	0	0
$381$	−10.2859	−0.526960
$382$	0	0
$383$	−27.9037	−1.42581	−0.712906	−	0.701260i	$-0.752621\pi$
$383$	−27.9037	−1.42581	−0.712906	+	0.701260i	$0.752621\pi$
$384$	0	0
$385$	−0.475390	−0.0242281
$386$	0	0
$387$	5.15255	0.261919
$388$	0	0
$389$	31.6524	1.60484	0.802421	−	0.596759i	$-0.203545\pi$
$389$	31.6524	1.60484	0.802421	+	0.596759i	$0.203545\pi$
$390$	0	0
$391$	−3.36266	−0.170057
$392$	0	0
$393$	10.7253	0.541021
$394$	0	0
$395$	3.74173	0.188267
$396$	0	0
$397$	1.19370	0.0599102	0.0299551	−	0.999551i	$-0.490464\pi$
$397$	1.19370	0.0599102	0.0299551	+	0.999551i	$0.490464\pi$
$398$	0	0
$399$	4.37907	0.219228
$400$	0	0
$401$	0.821644	0.0410310	0.0205155	−	0.999790i	$-0.493469\pi$
$401$	0.821644	0.0410310	0.0205155	+	0.999790i	$0.493469\pi$
$402$	0	0
$403$	−21.0164	−1.04690
$404$	0	0
$405$	4.02448	0.199978
$406$	0	0
$407$	1.08203	0.0536344
$408$	0	0
$409$	32.6290	1.61340	0.806700	−	0.590961i	$-0.201251\pi$
$409$	32.6290	1.61340	0.806700	+	0.590961i	$0.201251\pi$
$410$	0	0
$411$	−11.6608	−0.575183
$412$	0	0
$413$	6.72532	0.330931
$414$	0	0
$415$	−9.23353	−0.453256
$416$	0	0
$417$	−11.2304	−0.549956
$418$	0	0
$419$	−35.8144	−1.74965	−0.874823	−	0.484442i	$-0.839023\pi$
$419$	−35.8144	−1.74965	−0.874823	+	0.484442i	$0.839023\pi$
$420$	0	0
$421$	−30.0346	−1.46380	−0.731898	−	0.681414i	$-0.761365\pi$
$421$	−30.0346	−1.46380	−0.731898	+	0.681414i	$0.761365\pi$
$422$	0	0
$423$	7.29081	0.354492
$424$	0	0
$425$	4.12497	0.200090
$426$	0	0
$427$	−9.25410	−0.447837
$428$	0	0
$429$	−2.03281	−0.0981452
$430$	0	0
$431$	−19.9588	−0.961384	−0.480692	−	0.876890i	$-0.659614\pi$
$431$	−19.9588	−0.961384	−0.480692	+	0.876890i	$0.659614\pi$
$432$	0	0
$433$	−13.0492	−0.627106	−0.313553	−	0.949571i	$-0.601519\pi$
$433$	−13.0492	−0.627106	−0.313553	+	0.949571i	$0.601519\pi$
$434$	0	0
$435$	−1.30537	−0.0625878
$436$	0	0
$437$	−19.7417	−0.944375
$438$	0	0
$439$	10.1742	0.485587	0.242794	−	0.970078i	$-0.421936\pi$
$439$	10.1742	0.485587	0.242794	+	0.970078i	$0.421936\pi$
$440$	0	0
$441$	−2.44364	−0.116364
$442$	0	0
$443$	6.72532	0.319530	0.159765	−	0.987155i	$-0.448926\pi$
$443$	6.72532	0.319530	0.159765	+	0.987155i	$0.448926\pi$
$444$	0	0
$445$	−16.0000	−0.758473
$446$	0	0
$447$	0.273618	0.0129417
$448$	0	0
$449$	−19.1924	−0.905744	−0.452872	−	0.891576i	$-0.649601\pi$
$449$	−19.1924	−0.905744	−0.452872	+	0.891576i	$0.649601\pi$
$450$	0	0
$451$	4.63734	0.218364
$452$	0	0
$453$	−12.2018	−0.573289
$454$	0	0
$455$	−5.01641	−0.235173
$456$	0	0
$457$	−17.8779	−0.836292	−0.418146	−	0.908380i	$-0.637320\pi$
$457$	−17.8779	−0.836292	−0.418146	+	0.908380i	$0.637320\pi$
$458$	0	0
$459$	−4.06040	−0.189523
$460$	0	0
$461$	19.1924	0.893878	0.446939	−	0.894564i	$-0.352514\pi$
$461$	19.1924	0.893878	0.446939	+	0.894564i	$0.352514\pi$
$462$	0	0
$463$	−24.8339	−1.15413	−0.577064	−	0.816699i	$-0.695802\pi$
$463$	−24.8339	−1.15413	−0.577064	+	0.816699i	$0.695802\pi$
$464$	0	0
$465$	−2.73445	−0.126807
$466$	0	0
$467$	−41.1372	−1.90360	−0.951801	−	0.306715i	$-0.900770\pi$
$467$	−41.1372	−1.90360	−0.951801	+	0.306715i	$0.900770\pi$
$468$	0	0
$469$	−2.80630	−0.129583
$470$	0	0
$471$	0.661814	0.0304948
$472$	0	0
$473$	1.07158	0.0492712
$474$	0	0
$475$	24.2171	1.11116
$476$	0	0
$477$	27.6485	1.26594
$478$	0	0
$479$	−28.3585	−1.29573	−0.647866	−	0.761754i	$-0.724338\pi$
$479$	−28.3585	−1.29573	−0.647866	+	0.761754i	$0.724338\pi$
$480$	0	0
$481$	11.4178	0.520608
$482$	0	0
$483$	−2.50820	−0.114127
$484$	0	0
$485$	8.68239	0.394247
$486$	0	0
$487$	41.4095	1.87644	0.938222	−	0.346035i	$-0.112472\pi$
$487$	41.4095	1.87644	0.938222	+	0.346035i	$0.112472\pi$
$488$	0	0
$489$	10.2499	0.463518
$490$	0	0
$491$	−0.0809744	−0.00365432	−0.00182716	−	0.999998i	$-0.500582\pi$
$491$	−0.0809744	−0.00365432	−0.00182716	+	0.999998i	$0.500582\pi$
$492$	0	0
$493$	−1.87086	−0.0842595
$494$	0	0
$495$	1.16168	0.0522136
$496$	0	0
$497$	−3.83805	−0.172160
$498$	0	0
$499$	34.5962	1.54874	0.774369	−	0.632734i	$-0.218067\pi$
$499$	34.5962	1.54874	0.774369	+	0.632734i	$0.218067\pi$
$500$	0	0
$501$	3.79823	0.169692
$502$	0	0
$503$	6.64435	0.296257	0.148128	−	0.988968i	$-0.452675\pi$
$503$	6.64435	0.296257	0.148128	+	0.988968i	$0.452675\pi$
$504$	0	0
$505$	5.81569	0.258795
$506$	0	0
$507$	−11.7540	−0.522012
$508$	0	0
$509$	9.74173	0.431795	0.215897	−	0.976416i	$-0.430732\pi$
$509$	9.74173	0.431795	0.215897	+	0.976416i	$0.430732\pi$
$510$	0	0
$511$	6.14137	0.271678
$512$	0	0
$513$	−23.8381	−1.05248
$514$	0	0
$515$	−1.75007	−0.0771172
$516$	0	0
$517$	1.51627	0.0666856
$518$	0	0
$519$	6.14448	0.269713
$520$	0	0
$521$	−21.6660	−0.949204	−0.474602	−	0.880201i	$-0.657408\pi$
$521$	−21.6660	−0.949204	−0.474602	+	0.880201i	$0.657408\pi$
$522$	0	0
$523$	−3.42829	−0.149909	−0.0749543	−	0.997187i	$-0.523881\pi$
$523$	−3.42829	−0.149909	−0.0749543	+	0.997187i	$0.523881\pi$
$524$	0	0
$525$	3.07681	0.134283
$526$	0	0
$527$	−3.91903	−0.170715
$528$	0	0
$529$	−11.6925	−0.508370
$530$	0	0
$531$	−16.4342	−0.713185
$532$	0	0
$533$	48.9341	2.11957
$534$	0	0
$535$	3.41783	0.147766
$536$	0	0
$537$	−8.86359	−0.382492
$538$	0	0
$539$	−0.508203	−0.0218899
$540$	0	0
$541$	−29.3955	−1.26381	−0.631905	−	0.775046i	$-0.717727\pi$
$541$	−29.3955	−1.26381	−0.631905	+	0.775046i	$0.717727\pi$
$542$	0	0
$543$	9.99166	0.428783
$544$	0	0
$545$	10.3874	0.444948
$546$	0	0
$547$	31.5470	1.34885	0.674425	−	0.738343i	$-0.264391\pi$
$547$	31.5470	1.34885	0.674425	+	0.738343i	$0.264391\pi$
$548$	0	0
$549$	22.6137	0.965127
$550$	0	0
$551$	−10.9836	−0.467917
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	1.48558	0.0630592
$556$	0	0
$557$	−21.6761	−0.918446	−0.459223	−	0.888321i	$-0.651872\pi$
$557$	−21.6761	−0.918446	−0.459223	+	0.888321i	$0.651872\pi$
$558$	0	0
$559$	11.3075	0.478256
$560$	0	0
$561$	−0.379068	−0.0160043
$562$	0	0
$563$	−16.9201	−0.713097	−0.356548	−	0.934277i	$-0.616046\pi$
$563$	−16.9201	−0.713097	−0.356548	+	0.934277i	$0.616046\pi$
$564$	0	0
$565$	−3.11485	−0.131043
$566$	0	0
$567$	4.30226	0.180678
$568$	0	0
$569$	−21.8831	−0.917387	−0.458694	−	0.888594i	$-0.651682\pi$
$569$	−21.8831	−0.917387	−0.458694	+	0.888594i	$0.651682\pi$
$570$	0	0
$571$	−24.6842	−1.03300	−0.516500	−	0.856287i	$-0.672765\pi$
$571$	−24.6842	−1.03300	−0.516500	+	0.856287i	$0.672765\pi$
$572$	0	0
$573$	−12.2468	−0.511618
$574$	0	0
$575$	−13.8709	−0.578455
$576$	0	0
$577$	22.7581	0.947434	0.473717	−	0.880677i	$-0.342912\pi$
$577$	22.7581	0.947434	0.473717	+	0.880677i	$0.342912\pi$
$578$	0	0
$579$	−16.4999	−0.685711
$580$	0	0
$581$	−9.87086	−0.409512
$582$	0	0
$583$	5.75007	0.238143
$584$	0	0
$585$	12.2583	0.506817
$586$	0	0
$587$	−20.1208	−0.830474	−0.415237	−	0.909713i	$-0.636301\pi$
$587$	−20.1208	−0.830474	−0.415237	+	0.909713i	$0.636301\pi$
$588$	0	0
$589$	−23.0081	−0.948031
$590$	0	0
$591$	−0.991931	−0.0408026
$592$	0	0
$593$	14.9753	0.614960	0.307480	−	0.951555i	$-0.400514\pi$
$593$	14.9753	0.614960	0.307480	+	0.951555i	$0.400514\pi$
$594$	0	0
$595$	−0.935432	−0.0383490
$596$	0	0
$597$	7.34731	0.300706
$598$	0	0
$599$	−2.58395	−0.105577	−0.0527887	−	0.998606i	$-0.516811\pi$
$599$	−2.58395	−0.105577	−0.0527887	+	0.998606i	$0.516811\pi$
$600$	0	0
$601$	−3.01641	−0.123042	−0.0615209	−	0.998106i	$-0.519595\pi$
$601$	−3.01641	−0.123042	−0.0615209	+	0.998106i	$0.519595\pi$
$602$	0	0
$603$	6.85757	0.279262
$604$	0	0
$605$	−10.0482	−0.408516
$606$	0	0
$607$	−5.62794	−0.228431	−0.114216	−	0.993456i	$-0.536435\pi$
$607$	−5.62794	−0.228431	−0.114216	+	0.993456i	$0.536435\pi$
$608$	0	0
$609$	−1.39547	−0.0565475
$610$	0	0
$611$	16.0000	0.647291
$612$	0	0
$613$	−6.99060	−0.282348	−0.141174	−	0.989985i	$-0.545088\pi$
$613$	−6.99060	−0.282348	−0.141174	+	0.989985i	$0.545088\pi$
$614$	0	0
$615$	6.36683	0.256735
$616$	0	0
$617$	−41.7006	−1.67880	−0.839401	−	0.543513i	$-0.817094\pi$
$617$	−41.7006	−1.67880	−0.839401	+	0.543513i	$0.817094\pi$
$618$	0	0
$619$	−21.1044	−0.848257	−0.424129	−	0.905602i	$-0.639420\pi$
$619$	−21.1044	−0.848257	−0.424129	+	0.905602i	$0.639420\pi$
$620$	0	0
$621$	13.6537	0.547906
$622$	0	0
$623$	−17.1044	−0.685273
$624$	0	0
$625$	12.6402	0.505607
$626$	0	0
$627$	−2.22546	−0.0888762
$628$	0	0
$629$	2.12914	0.0848942
$630$	0	0
$631$	−28.8391	−1.14807	−0.574033	−	0.818832i	$-0.694622\pi$
$631$	−28.8391	−1.14807	−0.574033	+	0.818832i	$0.694622\pi$
$632$	0	0
$633$	−12.5962	−0.500653
$634$	0	0
$635$	12.8995	0.511901
$636$	0	0
$637$	−5.36266	−0.212476
$638$	0	0
$639$	9.37880	0.371020
$640$	0	0
$641$	15.8709	0.626861	0.313431	−	0.949611i	$-0.398522\pi$
$641$	15.8709	0.626861	0.313431	+	0.949611i	$0.398522\pi$
$642$	0	0
$643$	−38.2377	−1.50795	−0.753974	−	0.656905i	$-0.771865\pi$
$643$	−38.2377	−1.50795	−0.753974	+	0.656905i	$0.771865\pi$
$644$	0	0
$645$	1.47122	0.0579293
$646$	0	0
$647$	−7.64541	−0.300572	−0.150286	−	0.988643i	$-0.548019\pi$
$647$	−7.64541	−0.300572	−0.150286	+	0.988643i	$0.548019\pi$
$648$	0	0
$649$	−3.41783	−0.134162
$650$	0	0
$651$	−2.92319	−0.114569
$652$	0	0
$653$	−43.5386	−1.70380	−0.851899	−	0.523706i	$-0.824549\pi$
$653$	−43.5386	−1.70380	−0.851899	+	0.523706i	$0.824549\pi$
$654$	0	0
$655$	−13.4506	−0.525560
$656$	0	0
$657$	−15.0073	−0.585490
$658$	0	0
$659$	18.3861	0.716220	0.358110	−	0.933679i	$-0.383421\pi$
$659$	18.3861	0.716220	0.358110	+	0.933679i	$0.383421\pi$
$660$	0	0
$661$	30.0140	1.16741	0.583705	−	0.811966i	$-0.301602\pi$
$661$	30.0140	1.16741	0.583705	+	0.811966i	$0.301602\pi$
$662$	0	0
$663$	−4.00000	−0.155347
$664$	0	0
$665$	−5.49180	−0.212963
$666$	0	0
$667$	6.29108	0.243592
$668$	0	0
$669$	15.5553	0.601403
$670$	0	0
$671$	4.70297	0.181556
$672$	0	0
$673$	−16.2088	−0.624803	−0.312401	−	0.949950i	$-0.601133\pi$
$673$	−16.2088	−0.624803	−0.312401	+	0.949950i	$0.601133\pi$
$674$	0	0
$675$	−16.7490	−0.644670
$676$	0	0
$677$	8.05517	0.309585	0.154793	−	0.987947i	$-0.450529\pi$
$677$	8.05517	0.309585	0.154793	+	0.987947i	$0.450529\pi$
$678$	0	0
$679$	9.28169	0.356198
$680$	0	0
$681$	11.8678	0.454773
$682$	0	0
$683$	−19.4506	−0.744258	−0.372129	−	0.928181i	$-0.621372\pi$
$683$	−19.4506	−0.744258	−0.372129	+	0.928181i	$0.621372\pi$
$684$	0	0
$685$	14.6238	0.558746
$686$	0	0
$687$	5.58812	0.213200
$688$	0	0
$689$	60.6758	2.31157
$690$	0	0
$691$	22.7376	0.864978	0.432489	−	0.901639i	$-0.357636\pi$
$691$	22.7376	0.864978	0.432489	+	0.901639i	$0.357636\pi$
$692$	0	0
$693$	1.24186	0.0471745
$694$	0	0
$695$	14.0841	0.534240
$696$	0	0
$697$	9.12497	0.345633
$698$	0	0
$699$	−16.6618	−0.630208
$700$	0	0
$701$	−8.22546	−0.310671	−0.155336	−	0.987862i	$-0.549646\pi$
$701$	−8.22546	−0.310671	−0.155336	+	0.987862i	$0.549646\pi$
$702$	0	0
$703$	12.4999	0.471441
$704$	0	0
$705$	2.08176	0.0784038
$706$	0	0
$707$	6.21712	0.233819
$708$	0	0
$709$	−18.7909	−0.705709	−0.352854	−	0.935678i	$-0.614789\pi$
$709$	−18.7909	−0.705709	−0.352854	+	0.935678i	$0.614789\pi$
$710$	0	0
$711$	−9.77454	−0.366574
$712$	0	0
$713$	13.1784	0.493533
$714$	0	0
$715$	2.54935	0.0953405
$716$	0	0
$717$	−11.6363	−0.434565
$718$	0	0
$719$	18.6496	0.695512	0.347756	−	0.937585i	$-0.386944\pi$
$719$	18.6496	0.695512	0.347756	+	0.937585i	$0.386944\pi$
$720$	0	0
$721$	−1.87086	−0.0696746
$722$	0	0
$723$	−9.14865	−0.340242
$724$	0	0
$725$	−7.71725	−0.286612
$726$	0	0
$727$	17.9917	0.667274	0.333637	−	0.942702i	$-0.391724\pi$
$727$	17.9917	0.667274	0.333637	+	0.942702i	$0.391724\pi$
$728$	0	0
$729$	−1.42723	−0.0528603
$730$	0	0
$731$	2.10856	0.0779879
$732$	0	0
$733$	−8.98359	−0.331817	−0.165908	−	0.986141i	$-0.553056\pi$
$733$	−8.98359	−0.331817	−0.165908	+	0.986141i	$0.553056\pi$
$734$	0	0
$735$	−0.697737	−0.0257364
$736$	0	0
$737$	1.42617	0.0525336
$738$	0	0
$739$	49.6576	1.82669	0.913343	−	0.407191i	$-0.133492\pi$
$739$	49.6576	1.82669	0.913343	+	0.407191i	$0.133492\pi$
$740$	0	0
$741$	−23.4835	−0.862686
$742$	0	0
$743$	−17.8709	−0.655618	−0.327809	−	0.944744i	$-0.606310\pi$
$743$	−17.8709	−0.655618	−0.327809	+	0.944744i	$0.606310\pi$
$744$	0	0
$745$	−0.343145	−0.0125719
$746$	0	0
$747$	24.1208	0.882534
$748$	0	0
$749$	3.65375	0.133505
$750$	0	0
$751$	21.1127	0.770414	0.385207	−	0.922830i	$-0.374130\pi$
$751$	21.1127	0.770414	0.385207	+	0.922830i	$0.374130\pi$
$752$	0	0
$753$	1.39547	0.0508539
$754$	0	0
$755$	15.3023	0.556906
$756$	0	0
$757$	22.8339	0.829912	0.414956	−	0.909842i	$-0.363797\pi$
$757$	22.8339	0.829912	0.414956	+	0.909842i	$0.363797\pi$
$758$	0	0
$759$	1.27468	0.0462679
$760$	0	0
$761$	1.78288	0.0646294	0.0323147	−	0.999478i	$-0.489712\pi$
$761$	1.78288	0.0646294	0.0323147	+	0.999478i	$0.489712\pi$
$762$	0	0
$763$	11.1044	0.402006
$764$	0	0
$765$	2.28586	0.0826453
$766$	0	0
$767$	−36.0656	−1.30225
$768$	0	0
$769$	−34.5962	−1.24757	−0.623785	−	0.781596i	$-0.714406\pi$
$769$	−34.5962	−1.24757	−0.623785	+	0.781596i	$0.714406\pi$
$770$	0	0
$771$	1.72559	0.0621457
$772$	0	0
$773$	−25.1700	−0.905303	−0.452651	−	0.891688i	$-0.649522\pi$
$773$	−25.1700	−0.905303	−0.452651	+	0.891688i	$0.649522\pi$
$774$	0	0
$775$	−16.1658	−0.580694
$776$	0	0
$777$	1.58812	0.0569734
$778$	0	0
$779$	53.5714	1.91940
$780$	0	0
$781$	1.95051	0.0697948
$782$	0	0
$783$	7.59646	0.271475
$784$	0	0
$785$	−0.829982	−0.0296233
$786$	0	0
$787$	13.3627	0.476327	0.238164	−	0.971225i	$-0.423455\pi$
$787$	13.3627	0.476327	0.238164	+	0.971225i	$0.423455\pi$
$788$	0	0
$789$	21.9672	0.782053
$790$	0	0
$791$	−3.32985	−0.118396
$792$	0	0
$793$	49.6266	1.76229
$794$	0	0
$795$	7.89455	0.279991
$796$	0	0
$797$	−23.4751	−0.831531	−0.415766	−	0.909472i	$-0.636486\pi$
$797$	−23.4751	−0.831531	−0.415766	+	0.909472i	$0.636486\pi$
$798$	0	0
$799$	2.98359	0.105552
$800$	0	0
$801$	41.7969	1.47682
$802$	0	0
$803$	−3.12107	−0.110140
$804$	0	0
$805$	3.14554	0.110866
$806$	0	0
$807$	4.54102	0.159851
$808$	0	0
$809$	21.4835	0.755318	0.377659	−	0.925945i	$-0.376729\pi$
$809$	21.4835	0.755318	0.377659	+	0.925945i	$0.376729\pi$
$810$	0	0
$811$	−15.7540	−0.553197	−0.276598	−	0.960986i	$-0.589207\pi$
$811$	−15.7540	−0.553197	−0.276598	+	0.960986i	$0.589207\pi$
$812$	0	0
$813$	−13.6844	−0.479934
$814$	0	0
$815$	−12.8545	−0.450272
$816$	0	0
$817$	12.3791	0.433089
$818$	0	0
$819$	13.1044	0.457904
$820$	0	0
$821$	30.6290	1.06896	0.534480	−	0.845181i	$-0.320508\pi$
$821$	30.6290	1.06896	0.534480	+	0.845181i	$0.320508\pi$
$822$	0	0
$823$	−28.0796	−0.978795	−0.489397	−	0.872061i	$-0.662783\pi$
$823$	−28.0796	−0.978795	−0.489397	+	0.872061i	$0.662783\pi$
$824$	0	0
$825$	−1.56364	−0.0544391
$826$	0	0
$827$	24.1291	0.839052	0.419526	−	0.907743i	$-0.362196\pi$
$827$	24.1291	0.839052	0.419526	+	0.907743i	$0.362196\pi$
$828$	0	0
$829$	12.9284	0.449023	0.224511	−	0.974471i	$-0.427921\pi$
$829$	12.9284	0.449023	0.224511	+	0.974471i	$0.427921\pi$
$830$	0	0
$831$	17.0409	0.591142
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−4.76336	−0.164843
$836$	0	0
$837$	15.9128	0.550027
$838$	0	0
$839$	41.5163	1.43330	0.716650	−	0.697433i	$-0.245674\pi$
$839$	41.5163	1.43330	0.716650	+	0.697433i	$0.245674\pi$
$840$	0	0
$841$	−25.4999	−0.879306
$842$	0	0
$843$	−12.8904	−0.443968
$844$	0	0
$845$	14.7407	0.507094
$846$	0	0
$847$	−10.7417	−0.369090
$848$	0	0
$849$	−5.57667	−0.191391
$850$	0	0
$851$	−7.15956	−0.245427
$852$	0	0
$853$	−1.39547	−0.0477801	−0.0238901	−	0.999715i	$-0.507605\pi$
$853$	−1.39547	−0.0477801	−0.0238901	+	0.999715i	$0.507605\pi$
$854$	0	0
$855$	13.4200	0.458953
$856$	0	0
$857$	−47.1219	−1.60965	−0.804826	−	0.593511i	$-0.797741\pi$
$857$	−47.1219	−1.60965	−0.804826	+	0.593511i	$0.797741\pi$
$858$	0	0
$859$	12.0656	0.411674	0.205837	−	0.978586i	$-0.434008\pi$
$859$	12.0656	0.411674	0.205837	+	0.978586i	$0.434008\pi$
$860$	0	0
$861$	6.80630	0.231958
$862$	0	0
$863$	14.3064	0.486997	0.243498	−	0.969901i	$-0.421705\pi$
$863$	14.3064	0.486997	0.243498	+	0.969901i	$0.421705\pi$
$864$	0	0
$865$	−7.70581	−0.262005
$866$	0	0
$867$	−0.745898	−0.0253320
$868$	0	0
$869$	−2.03281	−0.0689585
$870$	0	0
$871$	15.0492	0.509923
$872$	0	0
$873$	−22.6811	−0.767638
$874$	0	0
$875$	−8.53579	−0.288562
$876$	0	0
$877$	−15.8792	−0.536203	−0.268101	−	0.963391i	$-0.586396\pi$
$877$	−15.8792	−0.536203	−0.268101	+	0.963391i	$0.586396\pi$
$878$	0	0
$879$	10.7253	0.361756
$880$	0	0
$881$	−10.5564	−0.355653	−0.177826	−	0.984062i	$-0.556907\pi$
$881$	−10.5564	−0.355653	−0.177826	+	0.984062i	$0.556907\pi$
$882$	0	0
$883$	17.0974	0.575373	0.287686	−	0.957725i	$-0.407114\pi$
$883$	17.0974	0.575373	0.287686	+	0.957725i	$0.407114\pi$
$884$	0	0
$885$	−4.69251	−0.157737
$886$	0	0
$887$	47.8367	1.60620	0.803100	−	0.595844i	$-0.203182\pi$
$887$	47.8367	1.60620	0.803100	+	0.595844i	$0.203182\pi$
$888$	0	0
$889$	13.7899	0.462498
$890$	0	0
$891$	−2.18642	−0.0732480
$892$	0	0
$893$	17.5163	0.586160
$894$	0	0
$895$	11.1158	0.371561
$896$	0	0
$897$	13.4506	0.449104
$898$	0	0
$899$	7.33197	0.244535
$900$	0	0
$901$	11.3145	0.376941
$902$	0	0
$903$	1.57277	0.0523385
$904$	0	0
$905$	−12.5306	−0.416530
$906$	0	0
$907$	50.4342	1.67464	0.837321	−	0.546712i	$-0.184121\pi$
$907$	50.4342	1.67464	0.837321	+	0.546712i	$0.184121\pi$
$908$	0	0
$909$	−15.1924	−0.503899
$910$	0	0
$911$	−47.0081	−1.55745	−0.778723	−	0.627367i	$-0.784132\pi$
$911$	−47.0081	−1.55745	−0.778723	+	0.627367i	$0.784132\pi$
$912$	0	0
$913$	5.01641	0.166019
$914$	0	0
$915$	6.45693	0.213460
$916$	0	0
$917$	−14.3791	−0.474839
$918$	0	0
$919$	30.8011	1.01603	0.508017	−	0.861347i	$-0.330379\pi$
$919$	30.8011	1.01603	0.508017	+	0.861347i	$0.330379\pi$
$920$	0	0
$921$	−19.1945	−0.632480
$922$	0	0
$923$	20.5822	0.677470
$924$	0	0
$925$	8.78261	0.288770
$926$	0	0
$927$	4.57171	0.150155
$928$	0	0
$929$	−25.7816	−0.845865	−0.422933	−	0.906161i	$-0.638999\pi$
$929$	−25.7816	−0.845865	−0.422933	+	0.906161i	$0.638999\pi$
$930$	0	0
$931$	−5.87086	−0.192410
$932$	0	0
$933$	−10.8513	−0.355257
$934$	0	0
$935$	0.475390	0.0155469
$936$	0	0
$937$	−34.5550	−1.12886	−0.564432	−	0.825480i	$-0.690905\pi$
$937$	−34.5550	−1.12886	−0.564432	+	0.825480i	$0.690905\pi$
$938$	0	0
$939$	−5.26323	−0.171759
$940$	0	0
$941$	33.9711	1.10743	0.553713	−	0.832708i	$-0.313211\pi$
$941$	33.9711	1.10743	0.553713	+	0.832708i	$0.313211\pi$
$942$	0	0
$943$	−30.6842	−0.999214
$944$	0	0
$945$	3.79823	0.123556
$946$	0	0
$947$	−6.54102	−0.212554	−0.106277	−	0.994337i	$-0.533893\pi$
$947$	−6.54102	−0.212554	−0.106277	+	0.994337i	$0.533893\pi$
$948$	0	0
$949$	−32.9341	−1.06909
$950$	0	0
$951$	19.9344	0.646416
$952$	0	0
$953$	16.4272	0.532130	0.266065	−	0.963955i	$-0.414276\pi$
$953$	16.4272	0.532130	0.266065	+	0.963955i	$0.414276\pi$
$954$	0	0
$955$	15.3588	0.496998
$956$	0	0
$957$	0.709185	0.0229247
$958$	0	0
$959$	15.6332	0.504821
$960$	0	0
$961$	−15.6412	−0.504556
$962$	0	0
$963$	−8.92842	−0.287714
$964$	0	0
$965$	20.6925	0.666115
$966$	0	0
$967$	−43.1083	−1.38627	−0.693135	−	0.720808i	$-0.743771\pi$
$967$	−43.1083	−1.38627	−0.693135	+	0.720808i	$0.743771\pi$
$968$	0	0
$969$	−4.37907	−0.140676
$970$	0	0
$971$	36.0140	1.15574	0.577872	−	0.816127i	$-0.303883\pi$
$971$	36.0140	1.15574	0.577872	+	0.816127i	$0.303883\pi$
$972$	0	0
$973$	15.0562	0.482681
$974$	0	0
$975$	−16.4999	−0.528419
$976$	0	0
$977$	−17.9602	−0.574597	−0.287298	−	0.957841i	$-0.592757\pi$
$977$	−17.9602	−0.574597	−0.287298	+	0.957841i	$0.592757\pi$
$978$	0	0
$979$	8.69251	0.277814
$980$	0	0
$981$	−27.1351	−0.866357
$982$	0	0
$983$	1.00106	0.0319288	0.0159644	−	0.999873i	$-0.494918\pi$
$983$	1.00106	0.0319288	0.0159644	+	0.999873i	$0.494918\pi$
$984$	0	0
$985$	1.24398	0.0396366
$986$	0	0
$987$	2.22546	0.0708371
$988$	0	0
$989$	−7.09037	−0.225461
$990$	0	0
$991$	33.5714	1.06643	0.533216	−	0.845979i	$-0.320983\pi$
$991$	33.5714	1.06643	0.533216	+	0.845979i	$0.320983\pi$
$992$	0	0
$993$	20.4395	0.648627
$994$	0	0
$995$	−9.21428	−0.292112
$996$	0	0
$997$	−32.2210	−1.02045	−0.510225	−	0.860041i	$-0.670438\pi$
$997$	−32.2210	−1.02045	−0.510225	+	0.860041i	$0.670438\pi$
$998$	0	0
$999$	−8.64514	−0.273520

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	952.2.a.c.1.2	✓	3
3.2	odd	2		8568.2.a.be.1.1		3
4.3	odd	2		1904.2.a.o.1.2		3
7.6	odd	2		6664.2.a.m.1.2		3
8.3	odd	2		7616.2.a.bb.1.2		3
8.5	even	2		7616.2.a.bh.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.a.c.1.2	✓	3	1.1	even	1	trivial
1904.2.a.o.1.2		3	4.3	odd	2
6664.2.a.m.1.2		3	7.6	odd	2
7616.2.a.bb.1.2		3	8.3	odd	2
7616.2.a.bh.1.2		3	8.5	even	2
8568.2.a.be.1.1		3	3.2	odd	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 952 = 2^{3} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	952.a (trivial)

\(f(q)\)	\(=\)	\(q-0.745898 q^{3} +0.935432 q^{5} +1.00000 q^{7} -2.44364 q^{9} -0.508203 q^{11} -5.36266 q^{13} -0.697737 q^{15} -1.00000 q^{17} -5.87086 q^{19} -0.745898 q^{21} +3.36266 q^{23} -4.12497 q^{25} +4.06040 q^{27} +1.87086 q^{29} +3.91903 q^{31} +0.379068 q^{33} +0.935432 q^{35} -2.12914 q^{37} +4.00000 q^{39} -9.12497 q^{41} -2.10856 q^{43} -2.28586 q^{45} -2.98359 q^{47} +1.00000 q^{49} +0.745898 q^{51} -11.3145 q^{53} -0.475390 q^{55} +4.37907 q^{57} +6.72532 q^{59} -9.25410 q^{61} -2.44364 q^{63} -5.01641 q^{65} -2.80630 q^{67} -2.50820 q^{69} -3.83805 q^{71} +6.14137 q^{73} +3.07681 q^{75} -0.508203 q^{77} +4.00000 q^{79} +4.30226 q^{81} -9.87086 q^{83} -0.935432 q^{85} -1.39547 q^{87} -17.1044 q^{89} -5.36266 q^{91} -2.92319 q^{93} -5.49180 q^{95} +9.28169 q^{97} +1.24186 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 5 q^{5} + 3 q^{7} + 2 q^{9} - 2 q^{13} + 8 q^{15} - 3 q^{17} - 2 q^{19} - 3 q^{21} - 4 q^{23} + 4 q^{25} - 12 q^{27} - 10 q^{29} + 7 q^{31} - 16 q^{33} - 5 q^{35} - 22 q^{37} + 12 q^{39}+ \cdots + 38 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 5 * q^5 + 3 * q^7 + 2 * q^9 - 2 * q^13 + 8 * q^15 - 3 * q^17 - 2 * q^19 - 3 * q^21 - 4 * q^23 + 4 * q^25 - 12 * q^27 - 10 * q^29 + 7 * q^31 - 16 * q^33 - 5 * q^35 - 22 * q^37 + 12 * q^39 - 11 * q^41 + 7 * q^43 - 20 * q^45 - 12 * q^47 + 3 * q^49 + 3 * q^51 - 9 * q^53 - 6 * q^55 - 4 * q^57 - 8 * q^59 - 27 * q^61 + 2 * q^63 - 12 * q^65 + 15 * q^67 - 6 * q^69 - 2 * q^71 - q^73 - 18 * q^75 + 12 * q^79 + 23 * q^81 - 14 * q^83 + 5 * q^85 + 16 * q^87 - 6 * q^89 - 2 * q^91 - 36 * q^93 - 18 * q^95 + 9 * q^97 + 38 * q^99

Introduction

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