LMFDB - Embedded newform 1936.4.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1936,4,Mod(1,1936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1936, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1936.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1936 = 2^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1936.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:	$114.227697771$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	1936.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.92820 q^{3} +14.8564 q^{5} +3.07180 q^{7} +35.8564 q^{9} +O(q^{10})$	$q+7.92820 q^{3} +14.8564 q^{5} +3.07180 q^{7} +35.8564 q^{9} -5.35898 q^{13} +117.785 q^{15} +41.2154 q^{17} +139.923 q^{19} +24.3538 q^{21} +111.354 q^{23} +95.7128 q^{25} +70.2154 q^{27} +24.9948 q^{29} -31.4974 q^{31} +45.6359 q^{35} +13.1436 q^{37} -42.4871 q^{39} -261.072 q^{41} -57.7128 q^{43} +532.697 q^{45} +343.846 q^{47} -333.564 q^{49} +326.764 q^{51} -342.995 q^{53} +1109.34 q^{57} -88.3693 q^{59} -738.697 q^{61} +110.144 q^{63} -79.6152 q^{65} -342.359 q^{67} +882.836 q^{69} +207.364 q^{71} +1010.60 q^{73} +758.831 q^{75} +1294.23 q^{79} -411.441 q^{81} +441.846 q^{83} +612.313 q^{85} +198.164 q^{87} -1489.11 q^{89} -16.4617 q^{91} -249.718 q^{93} +2078.75 q^{95} +1346.42 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 20 q^{7} + 44 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + 20 * q^7 + 44 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + 20 q^{7} + 44 q^{9} - 80 q^{13} + 194 q^{15} + 124 q^{17} + 72 q^{19} - 76 q^{21} + 98 q^{23} + 136 q^{25} + 182 q^{27} - 144 q^{29} + 34 q^{31} - 172 q^{35} + 54 q^{37} + 400 q^{39} - 536 q^{41} - 60 q^{43} + 428 q^{45} + 272 q^{47} - 390 q^{49} - 164 q^{51} - 492 q^{53} + 1512 q^{57} - 634 q^{59} - 840 q^{61} + 248 q^{63} + 880 q^{65} - 754 q^{67} + 962 q^{69} + 678 q^{71} + 400 q^{73} + 520 q^{75} + 316 q^{79} - 1294 q^{81} + 468 q^{83} - 452 q^{85} + 1200 q^{87} - 1842 q^{89} - 1280 q^{91} - 638 q^{93} + 2952 q^{95} + 2194 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 20 * q^7 + 44 * q^9 - 80 * q^13 + 194 * q^15 + 124 * q^17 + 72 * q^19 - 76 * q^21 + 98 * q^23 + 136 * q^25 + 182 * q^27 - 144 * q^29 + 34 * q^31 - 172 * q^35 + 54 * q^37 + 400 * q^39 - 536 * q^41 - 60 * q^43 + 428 * q^45 + 272 * q^47 - 390 * q^49 - 164 * q^51 - 492 * q^53 + 1512 * q^57 - 634 * q^59 - 840 * q^61 + 248 * q^63 + 880 * q^65 - 754 * q^67 + 962 * q^69 + 678 * q^71 + 400 * q^73 + 520 * q^75 + 316 * q^79 - 1294 * q^81 + 468 * q^83 - 452 * q^85 + 1200 * q^87 - 1842 * q^89 - 1280 * q^91 - 638 * q^93 + 2952 * q^95 + 2194 * 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	0
$2$	0	0
$3$	7.92820	1.52578	0.762892	−	0.646526i	$-0.223779\pi$
$3$	7.92820	1.52578	0.762892	+	0.646526i	$0.223779\pi$
$4$	0	0
$5$	14.8564	1.32880	0.664399	−	0.747378i	$-0.268688\pi$
$5$	14.8564	1.32880	0.664399	+	0.747378i	$0.268688\pi$
$6$	0	0
$7$	3.07180	0.165861	0.0829307	−	0.996555i	$-0.473572\pi$
$7$	3.07180	0.165861	0.0829307	+	0.996555i	$0.473572\pi$
$8$	0	0
$9$	35.8564	1.32802
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−5.35898	−0.114332	−0.0571659	−	0.998365i	$-0.518206\pi$
$13$	−5.35898	−0.114332	−0.0571659	+	0.998365i	$0.518206\pi$
$14$	0	0
$15$	117.785	2.02746
$16$	0	0
$17$	41.2154	0.588012	0.294006	−	0.955804i	$-0.405011\pi$
$17$	41.2154	0.588012	0.294006	+	0.955804i	$0.405011\pi$
$18$	0	0
$19$	139.923	1.68950	0.844751	−	0.535159i	$-0.179748\pi$
$19$	139.923	1.68950	0.844751	+	0.535159i	$0.179748\pi$
$20$	0	0
$21$	24.3538	0.253069
$22$	0	0
$23$	111.354	1.00952	0.504758	−	0.863261i	$-0.331582\pi$
$23$	111.354	1.00952	0.504758	+	0.863261i	$0.331582\pi$
$24$	0	0
$25$	95.7128	0.765703
$26$	0	0
$27$	70.2154	0.500480
$28$	0	0
$29$	24.9948	0.160049	0.0800246	−	0.996793i	$-0.474500\pi$
$29$	24.9948	0.160049	0.0800246	+	0.996793i	$0.474500\pi$
$30$	0	0
$31$	−31.4974	−0.182487	−0.0912436	−	0.995829i	$-0.529084\pi$
$31$	−31.4974	−0.182487	−0.0912436	+	0.995829i	$0.529084\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	45.6359	0.220396
$36$	0	0
$37$	13.1436	0.0583998	0.0291999	−	0.999574i	$-0.490704\pi$
$37$	13.1436	0.0583998	0.0291999	+	0.999574i	$0.490704\pi$
$38$	0	0
$39$	−42.4871	−0.174446
$40$	0	0
$41$	−261.072	−0.994453	−0.497226	−	0.867621i	$-0.665648\pi$
$41$	−261.072	−0.994453	−0.497226	+	0.867621i	$0.665648\pi$
$42$	0	0
$43$	−57.7128	−0.204677	−0.102339	−	0.994750i	$-0.532633\pi$
$43$	−57.7128	−0.204677	−0.102339	+	0.994750i	$0.532633\pi$
$44$	0	0
$45$	532.697	1.76466
$46$	0	0
$47$	343.846	1.06713	0.533565	−	0.845759i	$-0.320852\pi$
$47$	343.846	1.06713	0.533565	+	0.845759i	$0.320852\pi$
$48$	0	0
$49$	−333.564	−0.972490
$50$	0	0
$51$	326.764	0.897179
$52$	0	0
$53$	−342.995	−0.888943	−0.444471	−	0.895793i	$-0.646608\pi$
$53$	−342.995	−0.888943	−0.444471	+	0.895793i	$0.646608\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1109.34	2.57782
$58$	0	0
$59$	−88.3693	−0.194995	−0.0974975	−	0.995236i	$-0.531084\pi$
$59$	−88.3693	−0.194995	−0.0974975	+	0.995236i	$0.531084\pi$
$60$	0	0
$61$	−738.697	−1.55050	−0.775250	−	0.631654i	$-0.782376\pi$
$61$	−738.697	−1.55050	−0.775250	+	0.631654i	$0.782376\pi$
$62$	0	0
$63$	110.144	0.220266
$64$	0	0
$65$	−79.6152	−0.151924
$66$	0	0
$67$	−342.359	−0.624266	−0.312133	−	0.950038i	$-0.601043\pi$
$67$	−342.359	−0.624266	−0.312133	+	0.950038i	$0.601043\pi$
$68$	0	0
$69$	882.836	1.54030
$70$	0	0
$71$	207.364	0.346614	0.173307	−	0.984868i	$-0.444555\pi$
$71$	207.364	0.346614	0.173307	+	0.984868i	$0.444555\pi$
$72$	0	0
$73$	1010.60	1.62030	0.810149	−	0.586224i	$-0.199386\pi$
$73$	1010.60	1.62030	0.810149	+	0.586224i	$0.199386\pi$
$74$	0	0
$75$	758.831	1.16830
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1294.23	1.84319	0.921593	−	0.388157i	$-0.126888\pi$
$79$	1294.23	1.84319	0.921593	+	0.388157i	$0.126888\pi$
$80$	0	0
$81$	−411.441	−0.564391
$82$	0	0
$83$	441.846	0.584324	0.292162	−	0.956369i	$-0.405625\pi$
$83$	441.846	0.584324	0.292162	+	0.956369i	$0.405625\pi$
$84$	0	0
$85$	612.313	0.781349
$86$	0	0
$87$	198.164	0.244200
$88$	0	0
$89$	−1489.11	−1.77355	−0.886773	−	0.462205i	$-0.847058\pi$
$89$	−1489.11	−1.77355	−0.886773	+	0.462205i	$0.847058\pi$
$90$	0	0
$91$	−16.4617	−0.0189633
$92$	0	0
$93$	−249.718	−0.278436
$94$	0	0
$95$	2078.75	2.24501
$96$	0	0
$97$	1346.42	1.40936	0.704679	−	0.709526i	$-0.251091\pi$
$97$	1346.42	1.40936	0.704679	+	0.709526i	$0.251091\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	161.461	0.159069	0.0795347	−	0.996832i	$-0.474657\pi$
$101$	161.461	0.159069	0.0795347	+	0.996832i	$0.474657\pi$
$102$	0	0
$103$	34.7592	0.0332517	0.0166259	−	0.999862i	$-0.494708\pi$
$103$	34.7592	0.0332517	0.0166259	+	0.999862i	$0.494708\pi$
$104$	0	0
$105$	361.810	0.336277
$106$	0	0
$107$	832.179	0.751867	0.375934	−	0.926647i	$-0.377322\pi$
$107$	832.179	0.751867	0.375934	+	0.926647i	$0.377322\pi$
$108$	0	0
$109$	−1044.26	−0.917629	−0.458815	−	0.888532i	$-0.651726\pi$
$109$	−1044.26	−0.917629	−0.458815	+	0.888532i	$0.651726\pi$
$110$	0	0
$111$	104.205	0.0891055
$112$	0	0
$113$	295.082	0.245654	0.122827	−	0.992428i	$-0.460804\pi$
$113$	295.082	0.245654	0.122827	+	0.992428i	$0.460804\pi$
$114$	0	0
$115$	1654.32	1.34144
$116$	0	0
$117$	−192.154	−0.151834
$118$	0	0
$119$	126.605	0.0975285
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−2069.83	−1.51732
$124$	0	0
$125$	−435.102	−0.311334
$126$	0	0
$127$	−1317.60	−0.920618	−0.460309	−	0.887759i	$-0.652261\pi$
$127$	−1317.60	−0.920618	−0.460309	+	0.887759i	$0.652261\pi$
$128$	0	0
$129$	−457.559	−0.312293
$130$	0	0
$131$	−1600.71	−1.06759	−0.533797	−	0.845612i	$-0.679235\pi$
$131$	−1600.71	−1.06759	−0.533797	+	0.845612i	$0.679235\pi$
$132$	0	0
$133$	429.815	0.280223
$134$	0	0
$135$	1043.15	0.665036
$136$	0	0
$137$	1611.68	1.00507	0.502536	−	0.864556i	$-0.332400\pi$
$137$	1611.68	1.00507	0.502536	+	0.864556i	$0.332400\pi$
$138$	0	0
$139$	−31.8619	−0.0194424	−0.00972120	−	0.999953i	$-0.503094\pi$
$139$	−31.8619	−0.0194424	−0.00972120	+	0.999953i	$0.503094\pi$
$140$	0	0
$141$	2726.08	1.62821
$142$	0	0
$143$	0	0
$144$	0	0
$145$	371.334	0.212673
$146$	0	0
$147$	−2644.56	−1.48381
$148$	0	0
$149$	2428.34	1.33515	0.667576	−	0.744542i	$-0.267332\pi$
$149$	2428.34	1.33515	0.667576	+	0.744542i	$0.267332\pi$
$150$	0	0
$151$	−2576.68	−1.38866	−0.694328	−	0.719659i	$-0.744298\pi$
$151$	−2576.68	−1.38866	−0.694328	+	0.719659i	$0.744298\pi$
$152$	0	0
$153$	1477.84	0.780889
$154$	0	0
$155$	−467.939	−0.242489
$156$	0	0
$157$	2475.94	1.25861	0.629305	−	0.777158i	$-0.283340\pi$
$157$	2475.94	1.25861	0.629305	+	0.777158i	$0.283340\pi$
$158$	0	0
$159$	−2719.33	−1.35633
$160$	0	0
$161$	342.056	0.167440
$162$	0	0
$163$	2725.11	1.30949	0.654745	−	0.755850i	$-0.272776\pi$
$163$	2725.11	1.30949	0.654745	+	0.755850i	$0.272776\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2737.30	1.26837	0.634187	−	0.773180i	$-0.281335\pi$
$167$	2737.30	1.26837	0.634187	+	0.773180i	$0.281335\pi$
$168$	0	0
$169$	−2168.28	−0.986928
$170$	0	0
$171$	5017.14	2.24368
$172$	0	0
$173$	−2307.42	−1.01404	−0.507022	−	0.861933i	$-0.669254\pi$
$173$	−2307.42	−1.01404	−0.507022	+	0.861933i	$0.669254\pi$
$174$	0	0
$175$	294.010	0.127001
$176$	0	0
$177$	−700.610	−0.297520
$178$	0	0
$179$	1312.15	0.547905	0.273953	−	0.961743i	$-0.411669\pi$
$179$	1312.15	0.547905	0.273953	+	0.961743i	$0.411669\pi$
$180$	0	0
$181$	−803.174	−0.329831	−0.164916	−	0.986308i	$-0.552735\pi$
$181$	−803.174	−0.329831	−0.164916	+	0.986308i	$0.552735\pi$
$182$	0	0
$183$	−5856.54	−2.36573
$184$	0	0
$185$	195.267	0.0776015
$186$	0	0
$187$	0	0
$188$	0	0
$189$	215.687	0.0830103
$190$	0	0
$191$	−1718.25	−0.650932	−0.325466	−	0.945554i	$-0.605521\pi$
$191$	−1718.25	−0.650932	−0.325466	+	0.945554i	$0.605521\pi$
$192$	0	0
$193$	−1340.18	−0.499837	−0.249919	−	0.968267i	$-0.580404\pi$
$193$	−1340.18	−0.499837	−0.249919	+	0.968267i	$0.580404\pi$
$194$	0	0
$195$	−631.206	−0.231803
$196$	0	0
$197$	3518.33	1.27244	0.636220	−	0.771508i	$-0.280497\pi$
$197$	3518.33	1.27244	0.636220	+	0.771508i	$0.280497\pi$
$198$	0	0
$199$	−823.692	−0.293417	−0.146709	−	0.989180i	$-0.546868\pi$
$199$	−823.692	−0.293417	−0.146709	+	0.989180i	$0.546868\pi$
$200$	0	0
$201$	−2714.29	−0.952494
$202$	0	0
$203$	76.7791	0.0265460
$204$	0	0
$205$	−3878.59	−1.32143
$206$	0	0
$207$	3992.75	1.34065
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−107.343	−0.0350228	−0.0175114	−	0.999847i	$-0.505574\pi$
$211$	−107.343	−0.0350228	−0.0175114	+	0.999847i	$0.505574\pi$
$212$	0	0
$213$	1644.03	0.528858
$214$	0	0
$215$	−857.405	−0.271975
$216$	0	0
$217$	−96.7537	−0.0302676
$218$	0	0
$219$	8012.24	2.47222
$220$	0	0
$221$	−220.873	−0.0672285
$222$	0	0
$223$	3933.68	1.18125	0.590625	−	0.806946i	$-0.298881\pi$
$223$	3933.68	1.18125	0.590625	+	0.806946i	$0.298881\pi$
$224$	0	0
$225$	3431.92	1.01686
$226$	0	0
$227$	−1771.90	−0.518085	−0.259042	−	0.965866i	$-0.583407\pi$
$227$	−1771.90	−0.518085	−0.259042	+	0.965866i	$0.583407\pi$
$228$	0	0
$229$	1915.37	0.552713	0.276356	−	0.961055i	$-0.410873\pi$
$229$	1915.37	0.552713	0.276356	+	0.961055i	$0.410873\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4396.32	−1.23610	−0.618052	−	0.786137i	$-0.712078\pi$
$233$	−4396.32	−1.23610	−0.618052	+	0.786137i	$0.712078\pi$
$234$	0	0
$235$	5108.32	1.41800
$236$	0	0
$237$	10260.9	2.81230
$238$	0	0
$239$	−4084.49	−1.10546	−0.552728	−	0.833362i	$-0.686413\pi$
$239$	−4084.49	−1.10546	−0.552728	+	0.833362i	$0.686413\pi$
$240$	0	0
$241$	−3908.58	−1.04471	−0.522353	−	0.852730i	$-0.674946\pi$
$241$	−3908.58	−1.04471	−0.522353	+	0.852730i	$0.674946\pi$
$242$	0	0
$243$	−5157.80	−1.36162
$244$	0	0
$245$	−4955.56	−1.29224
$246$	0	0
$247$	−749.845	−0.193164
$248$	0	0
$249$	3503.05	0.891552
$250$	0	0
$251$	−1094.89	−0.275335	−0.137667	−	0.990479i	$-0.543960\pi$
$251$	−1094.89	−0.275335	−0.137667	+	0.990479i	$0.543960\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	4854.54	1.19217
$256$	0	0
$257$	783.179	0.190091	0.0950454	−	0.995473i	$-0.469700\pi$
$257$	783.179	0.190091	0.0950454	+	0.995473i	$0.469700\pi$
$258$	0	0
$259$	40.3744	0.00968628
$260$	0	0
$261$	896.225	0.212548
$262$	0	0
$263$	6180.06	1.44897	0.724484	−	0.689292i	$-0.242078\pi$
$263$	6180.06	1.44897	0.724484	+	0.689292i	$0.242078\pi$
$264$	0	0
$265$	−5095.67	−1.18122
$266$	0	0
$267$	−11806.0	−2.70605
$268$	0	0
$269$	986.965	0.223704	0.111852	−	0.993725i	$-0.464322\pi$
$269$	986.965	0.223704	0.111852	+	0.993725i	$0.464322\pi$
$270$	0	0
$271$	4576.99	1.02595	0.512975	−	0.858404i	$-0.328543\pi$
$271$	4576.99	1.02595	0.512975	+	0.858404i	$0.328543\pi$
$272$	0	0
$273$	−130.512	−0.0289338
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−567.836	−0.123169	−0.0615847	−	0.998102i	$-0.519615\pi$
$277$	−567.836	−0.123169	−0.0615847	+	0.998102i	$0.519615\pi$
$278$	0	0
$279$	−1129.38	−0.242346
$280$	0	0
$281$	−5311.01	−1.12750	−0.563752	−	0.825944i	$-0.690643\pi$
$281$	−5311.01	−1.12750	−0.563752	+	0.825944i	$0.690643\pi$
$282$	0	0
$283$	−4728.44	−0.993204	−0.496602	−	0.867978i	$-0.665419\pi$
$283$	−4728.44	−0.993204	−0.496602	+	0.867978i	$0.665419\pi$
$284$	0	0
$285$	16480.8	3.42539
$286$	0	0
$287$	−801.960	−0.164941
$288$	0	0
$289$	−3214.29	−0.654242
$290$	0	0
$291$	10674.7	2.15038
$292$	0	0
$293$	−2328.92	−0.464358	−0.232179	−	0.972673i	$-0.574585\pi$
$293$	−2328.92	−0.464358	−0.232179	+	0.972673i	$0.574585\pi$
$294$	0	0
$295$	−1312.85	−0.259109
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−596.743	−0.115420
$300$	0	0
$301$	−177.282	−0.0339481
$302$	0	0
$303$	1280.10	0.242705
$304$	0	0
$305$	−10974.4	−2.06030
$306$	0	0
$307$	−1678.07	−0.311962	−0.155981	−	0.987760i	$-0.549854\pi$
$307$	−1678.07	−0.311962	−0.155981	+	0.987760i	$0.549854\pi$
$308$	0	0
$309$	275.578	0.0507349
$310$	0	0
$311$	−3572.71	−0.651413	−0.325707	−	0.945471i	$-0.605602\pi$
$311$	−3572.71	−0.651413	−0.325707	+	0.945471i	$0.605602\pi$
$312$	0	0
$313$	7184.36	1.29739	0.648697	−	0.761047i	$-0.275314\pi$
$313$	7184.36	1.29739	0.648697	+	0.761047i	$0.275314\pi$
$314$	0	0
$315$	1636.34	0.292690
$316$	0	0
$317$	−15.7077	−0.00278306	−0.00139153	−	0.999999i	$-0.500443\pi$
$317$	−15.7077	−0.00278306	−0.00139153	+	0.999999i	$0.500443\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	6597.69	1.14719
$322$	0	0
$323$	5766.98	0.993447
$324$	0	0
$325$	−512.923	−0.0875442
$326$	0	0
$327$	−8279.08	−1.40010
$328$	0	0
$329$	1056.23	0.176996
$330$	0	0
$331$	1318.95	0.219022	0.109511	−	0.993986i	$-0.465072\pi$
$331$	1318.95	0.219022	0.109511	+	0.993986i	$0.465072\pi$
$332$	0	0
$333$	471.282	0.0775558
$334$	0	0
$335$	−5086.22	−0.829523
$336$	0	0
$337$	239.183	0.0386621	0.0193310	−	0.999813i	$-0.493846\pi$
$337$	239.183	0.0386621	0.0193310	+	0.999813i	$0.493846\pi$
$338$	0	0
$339$	2339.47	0.374816
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−2078.27	−0.327160
$344$	0	0
$345$	13115.8	2.04675
$346$	0	0
$347$	−5862.79	−0.907006	−0.453503	−	0.891255i	$-0.649826\pi$
$347$	−5862.79	−0.907006	−0.453503	+	0.891255i	$0.649826\pi$
$348$	0	0
$349$	−3491.73	−0.535553	−0.267776	−	0.963481i	$-0.586289\pi$
$349$	−3491.73	−0.535553	−0.267776	+	0.963481i	$0.586289\pi$
$350$	0	0
$351$	−376.283	−0.0572208
$352$	0	0
$353$	−10916.7	−1.64600	−0.822999	−	0.568043i	$-0.807701\pi$
$353$	−10916.7	−1.64600	−0.822999	+	0.568043i	$0.807701\pi$
$354$	0	0
$355$	3080.69	0.460580
$356$	0	0
$357$	1003.75	0.148807
$358$	0	0
$359$	−11500.7	−1.69077	−0.845384	−	0.534160i	$-0.820628\pi$
$359$	−11500.7	−1.69077	−0.845384	+	0.534160i	$0.820628\pi$
$360$	0	0
$361$	12719.5	1.85442
$362$	0	0
$363$	0	0
$364$	0	0
$365$	15013.9	2.15305
$366$	0	0
$367$	−6767.01	−0.962493	−0.481246	−	0.876585i	$-0.659816\pi$
$367$	−6767.01	−0.962493	−0.481246	+	0.876585i	$0.659816\pi$
$368$	0	0
$369$	−9361.10	−1.32065
$370$	0	0
$371$	−1053.61	−0.147441
$372$	0	0
$373$	5310.22	0.737139	0.368569	−	0.929600i	$-0.379848\pi$
$373$	5310.22	0.737139	0.368569	+	0.929600i	$0.379848\pi$
$374$	0	0
$375$	−3449.58	−0.475028
$376$	0	0
$377$	−133.947	−0.0182987
$378$	0	0
$379$	838.267	0.113612	0.0568059	−	0.998385i	$-0.481908\pi$
$379$	838.267	0.113612	0.0568059	+	0.998385i	$0.481908\pi$
$380$	0	0
$381$	−10446.2	−1.40466
$382$	0	0
$383$	2832.16	0.377851	0.188925	−	0.981991i	$-0.439500\pi$
$383$	2832.16	0.377851	0.188925	+	0.981991i	$0.439500\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2069.37	−0.271814
$388$	0	0
$389$	3111.25	0.405519	0.202759	−	0.979229i	$-0.435009\pi$
$389$	3111.25	0.405519	0.202759	+	0.979229i	$0.435009\pi$
$390$	0	0
$391$	4589.49	0.593608
$392$	0	0
$393$	−12690.8	−1.62892
$394$	0	0
$395$	19227.5	2.44922
$396$	0	0
$397$	14208.7	1.79626	0.898131	−	0.439728i	$-0.144925\pi$
$397$	14208.7	1.79626	0.898131	+	0.439728i	$0.144925\pi$
$398$	0	0
$399$	3407.66	0.427560
$400$	0	0
$401$	−6261.68	−0.779784	−0.389892	−	0.920861i	$-0.627488\pi$
$401$	−6261.68	−0.779784	−0.389892	+	0.920861i	$0.627488\pi$
$402$	0	0
$403$	168.794	0.0208641
$404$	0	0
$405$	−6112.54	−0.749961
$406$	0	0
$407$	0	0
$408$	0	0
$409$	4192.50	0.506860	0.253430	−	0.967354i	$-0.418441\pi$
$409$	4192.50	0.506860	0.253430	+	0.967354i	$0.418441\pi$
$410$	0	0
$411$	12777.7	1.53352
$412$	0	0
$413$	−271.453	−0.0323421
$414$	0	0
$415$	6564.25	0.776448
$416$	0	0
$417$	−252.608	−0.0296649
$418$	0	0
$419$	9287.15	1.08283	0.541416	−	0.840755i	$-0.317888\pi$
$419$	9287.15	1.08283	0.541416	+	0.840755i	$0.317888\pi$
$420$	0	0
$421$	13146.0	1.52185	0.760923	−	0.648842i	$-0.224746\pi$
$421$	13146.0	1.52185	0.760923	+	0.648842i	$0.224746\pi$
$422$	0	0
$423$	12329.1	1.41716
$424$	0	0
$425$	3944.84	0.450242
$426$	0	0
$427$	−2269.13	−0.257168
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4909.67	0.548701	0.274351	−	0.961630i	$-0.411537\pi$
$431$	4909.67	0.548701	0.274351	+	0.961630i	$0.411537\pi$
$432$	0	0
$433$	−11743.3	−1.30334	−0.651671	−	0.758502i	$-0.725932\pi$
$433$	−11743.3	−1.30334	−0.651671	+	0.758502i	$0.725932\pi$
$434$	0	0
$435$	2944.01	0.324493
$436$	0	0
$437$	15581.0	1.70558
$438$	0	0
$439$	−11824.2	−1.28551	−0.642754	−	0.766073i	$-0.722208\pi$
$439$	−11824.2	−1.28551	−0.642754	+	0.766073i	$0.722208\pi$
$440$	0	0
$441$	−11960.4	−1.29148
$442$	0	0
$443$	−10102.1	−1.08344	−0.541722	−	0.840558i	$-0.682228\pi$
$443$	−10102.1	−1.08344	−0.541722	+	0.840558i	$0.682228\pi$
$444$	0	0
$445$	−22122.9	−2.35668
$446$	0	0
$447$	19252.4	2.03715
$448$	0	0
$449$	−345.254	−0.0362885	−0.0181443	−	0.999835i	$-0.505776\pi$
$449$	−345.254	−0.0362885	−0.0181443	+	0.999835i	$0.505776\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−20428.4	−2.11879
$454$	0	0
$455$	−244.562	−0.0251983
$456$	0	0
$457$	10567.1	1.08164	0.540821	−	0.841138i	$-0.318114\pi$
$457$	10567.1	1.08164	0.540821	+	0.841138i	$0.318114\pi$
$458$	0	0
$459$	2893.95	0.294288
$460$	0	0
$461$	−4733.96	−0.478270	−0.239135	−	0.970986i	$-0.576864\pi$
$461$	−4733.96	−0.478270	−0.239135	+	0.970986i	$0.576864\pi$
$462$	0	0
$463$	−3431.20	−0.344409	−0.172204	−	0.985061i	$-0.555089\pi$
$463$	−3431.20	−0.344409	−0.172204	+	0.985061i	$0.555089\pi$
$464$	0	0
$465$	−3709.91	−0.369985
$466$	0	0
$467$	−5116.96	−0.507034	−0.253517	−	0.967331i	$-0.581587\pi$
$467$	−5116.96	−0.507034	−0.253517	+	0.967331i	$0.581587\pi$
$468$	0	0
$469$	−1051.66	−0.103542
$470$	0	0
$471$	19629.8	1.92037
$472$	0	0
$473$	0	0
$474$	0	0
$475$	13392.4	1.29366
$476$	0	0
$477$	−12298.6	−1.18053
$478$	0	0
$479$	11566.9	1.10335	0.551675	−	0.834059i	$-0.313989\pi$
$479$	11566.9	1.10335	0.551675	+	0.834059i	$0.313989\pi$
$480$	0	0
$481$	−70.4363	−0.00667696
$482$	0	0
$483$	2711.89	0.255477
$484$	0	0
$485$	20002.9	1.87275
$486$	0	0
$487$	18326.5	1.70525	0.852623	−	0.522527i	$-0.175010\pi$
$487$	18326.5	1.70525	0.852623	+	0.522527i	$0.175010\pi$
$488$	0	0
$489$	21605.2	1.99800
$490$	0	0
$491$	−7617.58	−0.700156	−0.350078	−	0.936721i	$-0.613845\pi$
$491$	−7617.58	−0.700156	−0.350078	+	0.936721i	$0.613845\pi$
$492$	0	0
$493$	1030.17	0.0941108
$494$	0	0
$495$	0	0
$496$	0	0
$497$	636.980	0.0574899
$498$	0	0
$499$	−12909.1	−1.15810	−0.579050	−	0.815292i	$-0.696576\pi$
$499$	−12909.1	−1.15810	−0.579050	+	0.815292i	$0.696576\pi$
$500$	0	0
$501$	21701.8	1.93526
$502$	0	0
$503$	10165.7	0.901121	0.450561	−	0.892746i	$-0.351224\pi$
$503$	10165.7	0.901121	0.450561	+	0.892746i	$0.351224\pi$
$504$	0	0
$505$	2398.74	0.211371
$506$	0	0
$507$	−17190.6	−1.50584
$508$	0	0
$509$	6449.93	0.561666	0.280833	−	0.959757i	$-0.409389\pi$
$509$	6449.93	0.561666	0.280833	+	0.959757i	$0.409389\pi$
$510$	0	0
$511$	3104.36	0.268745
$512$	0	0
$513$	9824.75	0.845562
$514$	0	0
$515$	516.397	0.0441848
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−18293.7	−1.54721
$520$	0	0
$521$	−19327.4	−1.62524	−0.812620	−	0.582794i	$-0.801959\pi$
$521$	−19327.4	−1.62524	−0.812620	+	0.582794i	$0.801959\pi$
$522$	0	0
$523$	6259.09	0.523310	0.261655	−	0.965161i	$-0.415732\pi$
$523$	6259.09	0.523310	0.261655	+	0.965161i	$0.415732\pi$
$524$	0	0
$525$	2330.97	0.193775
$526$	0	0
$527$	−1298.18	−0.107305
$528$	0	0
$529$	232.675	0.0191235
$530$	0	0
$531$	−3168.61	−0.258956
$532$	0	0
$533$	1399.08	0.113698
$534$	0	0
$535$	12363.2	0.999079
$536$	0	0
$537$	10403.0	0.835985
$538$	0	0
$539$	0	0
$540$	0	0
$541$	14008.2	1.11323	0.556616	−	0.830770i	$-0.312100\pi$
$541$	14008.2	1.11323	0.556616	+	0.830770i	$0.312100\pi$
$542$	0	0
$543$	−6367.72	−0.503251
$544$	0	0
$545$	−15513.9	−1.21934
$546$	0	0
$547$	−4949.45	−0.386879	−0.193440	−	0.981112i	$-0.561964\pi$
$547$	−4949.45	−0.386879	−0.193440	+	0.981112i	$0.561964\pi$
$548$	0	0
$549$	−26487.0	−2.05909
$550$	0	0
$551$	3497.35	0.270404
$552$	0	0
$553$	3975.60	0.305714
$554$	0	0
$555$	1548.11	0.118403
$556$	0	0
$557$	3801.58	0.289188	0.144594	−	0.989491i	$-0.453812\pi$
$557$	3801.58	0.289188	0.144594	+	0.989491i	$0.453812\pi$
$558$	0	0
$559$	309.282	0.0234011
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9900.11	−0.741101	−0.370551	−	0.928812i	$-0.620831\pi$
$563$	−9900.11	−0.741101	−0.370551	+	0.928812i	$0.620831\pi$
$564$	0	0
$565$	4383.85	0.326425
$566$	0	0
$567$	−1263.86	−0.0936107
$568$	0	0
$569$	−5329.16	−0.392636	−0.196318	−	0.980540i	$-0.562898\pi$
$569$	−5329.16	−0.392636	−0.196318	+	0.980540i	$0.562898\pi$
$570$	0	0
$571$	−16962.6	−1.24319	−0.621597	−	0.783337i	$-0.713516\pi$
$571$	−16962.6	−1.24319	−0.621597	+	0.783337i	$0.713516\pi$
$572$	0	0
$573$	−13622.6	−0.993181
$574$	0	0
$575$	10658.0	0.772989
$576$	0	0
$577$	−15487.0	−1.11738	−0.558692	−	0.829375i	$-0.688697\pi$
$577$	−15487.0	−1.11738	−0.558692	+	0.829375i	$0.688697\pi$
$578$	0	0
$579$	−10625.3	−0.762643
$580$	0	0
$581$	1357.26	0.0969169
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−2854.72	−0.201757
$586$	0	0
$587$	−11084.2	−0.779373	−0.389686	−	0.920948i	$-0.627417\pi$
$587$	−11084.2	−0.779373	−0.389686	+	0.920948i	$0.627417\pi$
$588$	0	0
$589$	−4407.22	−0.308313
$590$	0	0
$591$	27894.0	1.94147
$592$	0	0
$593$	−4349.68	−0.301214	−0.150607	−	0.988594i	$-0.548123\pi$
$593$	−4349.68	−0.301214	−0.150607	+	0.988594i	$0.548123\pi$
$594$	0	0
$595$	1880.90	0.129596
$596$	0	0
$597$	−6530.40	−0.447691
$598$	0	0
$599$	−13183.9	−0.899299	−0.449650	−	0.893205i	$-0.648451\pi$
$599$	−13183.9	−0.899299	−0.449650	+	0.893205i	$0.648451\pi$
$600$	0	0
$601$	18765.0	1.27361	0.636806	−	0.771024i	$-0.280255\pi$
$601$	18765.0	1.27361	0.636806	+	0.771024i	$0.280255\pi$
$602$	0	0
$603$	−12275.8	−0.829034
$604$	0	0
$605$	0	0
$606$	0	0
$607$	21871.4	1.46249	0.731244	−	0.682116i	$-0.238940\pi$
$607$	21871.4	1.46249	0.731244	+	0.682116i	$0.238940\pi$
$608$	0	0
$609$	608.720	0.0405034
$610$	0	0
$611$	−1842.67	−0.122007
$612$	0	0
$613$	3527.85	0.232445	0.116222	−	0.993223i	$-0.462921\pi$
$613$	3527.85	0.232445	0.116222	+	0.993223i	$0.462921\pi$
$614$	0	0
$615$	−30750.2	−2.01621
$616$	0	0
$617$	−22728.1	−1.48298	−0.741490	−	0.670963i	$-0.765881\pi$
$617$	−22728.1	−1.48298	−0.741490	+	0.670963i	$0.765881\pi$
$618$	0	0
$619$	21443.3	1.39237	0.696187	−	0.717861i	$-0.254879\pi$
$619$	21443.3	1.39237	0.696187	+	0.717861i	$0.254879\pi$
$620$	0	0
$621$	7818.75	0.505243
$622$	0	0
$623$	−4574.25	−0.294163
$624$	0	0
$625$	−18428.2	−1.17940
$626$	0	0
$627$	0	0
$628$	0	0
$629$	541.718	0.0343398
$630$	0	0
$631$	−21532.0	−1.35844	−0.679219	−	0.733936i	$-0.737681\pi$
$631$	−21532.0	−1.35844	−0.679219	+	0.733936i	$0.737681\pi$
$632$	0	0
$633$	−851.038	−0.0534372
$634$	0	0
$635$	−19574.9	−1.22332
$636$	0	0
$637$	1787.56	0.111187
$638$	0	0
$639$	7435.33	0.460309
$640$	0	0
$641$	20148.3	1.24151	0.620756	−	0.784004i	$-0.286826\pi$
$641$	20148.3	1.24151	0.620756	+	0.784004i	$0.286826\pi$
$642$	0	0
$643$	−28869.7	−1.77062	−0.885310	−	0.465000i	$-0.846054\pi$
$643$	−28869.7	−1.77062	−0.885310	+	0.465000i	$0.846054\pi$
$644$	0	0
$645$	−6797.68	−0.414974
$646$	0	0
$647$	1590.02	0.0966155	0.0483077	−	0.998833i	$-0.484617\pi$
$647$	1590.02	0.0966155	0.0483077	+	0.998833i	$0.484617\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−767.083	−0.0461818
$652$	0	0
$653$	20028.1	1.20024	0.600122	−	0.799909i	$-0.295119\pi$
$653$	20028.1	1.20024	0.600122	+	0.799909i	$0.295119\pi$
$654$	0	0
$655$	−23780.8	−1.41862
$656$	0	0
$657$	36236.5	2.15178
$658$	0	0
$659$	−10520.7	−0.621897	−0.310948	−	0.950427i	$-0.600647\pi$
$659$	−10520.7	−0.621897	−0.310948	+	0.950427i	$0.600647\pi$
$660$	0	0
$661$	3295.83	0.193938	0.0969690	−	0.995287i	$-0.469085\pi$
$661$	3295.83	0.193938	0.0969690	+	0.995287i	$0.469085\pi$
$662$	0	0
$663$	−1751.12	−0.102576
$664$	0	0
$665$	6385.51	0.372360
$666$	0	0
$667$	2783.27	0.161572
$668$	0	0
$669$	31187.0	1.80233
$670$	0	0
$671$	0	0
$672$	0	0
$673$	1187.64	0.0680239	0.0340119	−	0.999421i	$-0.489172\pi$
$673$	1187.64	0.0680239	0.0340119	+	0.999421i	$0.489172\pi$
$674$	0	0
$675$	6720.51	0.383219
$676$	0	0
$677$	−13221.4	−0.750574	−0.375287	−	0.926909i	$-0.622456\pi$
$677$	−13221.4	−0.750574	−0.375287	+	0.926909i	$0.622456\pi$
$678$	0	0
$679$	4135.91	0.233758
$680$	0	0
$681$	−14048.0	−0.790485
$682$	0	0
$683$	13831.4	0.774882	0.387441	−	0.921894i	$-0.373359\pi$
$683$	13831.4	0.774882	0.387441	+	0.921894i	$0.373359\pi$
$684$	0	0
$685$	23943.7	1.33554
$686$	0	0
$687$	15185.4	0.843320
$688$	0	0
$689$	1838.10	0.101635
$690$	0	0
$691$	9817.07	0.540462	0.270231	−	0.962796i	$-0.412900\pi$
$691$	9817.07	0.540462	0.270231	+	0.962796i	$0.412900\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−473.354	−0.0258350
$696$	0	0
$697$	−10760.2	−0.584750
$698$	0	0
$699$	−34854.9	−1.88603
$700$	0	0
$701$	−29949.8	−1.61368	−0.806838	−	0.590773i	$-0.798823\pi$
$701$	−29949.8	−1.61368	−0.806838	+	0.590773i	$0.798823\pi$
$702$	0	0
$703$	1839.09	0.0986667
$704$	0	0
$705$	40499.8	2.16356
$706$	0	0
$707$	495.976	0.0263835
$708$	0	0
$709$	11307.5	0.598959	0.299479	−	0.954103i	$-0.403187\pi$
$709$	11307.5	0.598959	0.299479	+	0.954103i	$0.403187\pi$
$710$	0	0
$711$	46406.3	2.44778
$712$	0	0
$713$	−3507.36	−0.184224
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−32382.7	−1.68669
$718$	0	0
$719$	32623.4	1.69214	0.846070	−	0.533071i	$-0.178962\pi$
$719$	32623.4	1.69214	0.846070	+	0.533071i	$0.178962\pi$
$720$	0	0
$721$	106.773	0.00551518
$722$	0	0
$723$	−30988.0	−1.59399
$724$	0	0
$725$	2392.33	0.122550
$726$	0	0
$727$	502.545	0.0256373	0.0128187	−	0.999918i	$-0.495920\pi$
$727$	502.545	0.0256373	0.0128187	+	0.999918i	$0.495920\pi$
$728$	0	0
$729$	−29783.2	−1.51314
$730$	0	0
$731$	−2378.66	−0.120353
$732$	0	0
$733$	−8631.37	−0.434935	−0.217467	−	0.976068i	$-0.569780\pi$
$733$	−8631.37	−0.434935	−0.217467	+	0.976068i	$0.569780\pi$
$734$	0	0
$735$	−39288.7	−1.97168
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−18357.5	−0.913792	−0.456896	−	0.889520i	$-0.651039\pi$
$739$	−18357.5	−0.913792	−0.456896	+	0.889520i	$0.651039\pi$
$740$	0	0
$741$	−5944.93	−0.294726
$742$	0	0
$743$	11182.6	0.552155	0.276078	−	0.961135i	$-0.410965\pi$
$743$	11182.6	0.552155	0.276078	+	0.961135i	$0.410965\pi$
$744$	0	0
$745$	36076.4	1.77415
$746$	0	0
$747$	15843.0	0.775991
$748$	0	0
$749$	2556.29	0.124706
$750$	0	0
$751$	−16733.4	−0.813063	−0.406531	−	0.913637i	$-0.633262\pi$
$751$	−16733.4	−0.813063	−0.406531	+	0.913637i	$0.633262\pi$
$752$	0	0
$753$	−8680.53	−0.420101
$754$	0	0
$755$	−38280.2	−1.84524
$756$	0	0
$757$	−24402.4	−1.17163	−0.585813	−	0.810446i	$-0.699225\pi$
$757$	−24402.4	−1.17163	−0.585813	+	0.810446i	$0.699225\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−8469.33	−0.403434	−0.201717	−	0.979444i	$-0.564652\pi$
$761$	−8469.33	−0.403434	−0.201717	+	0.979444i	$0.564652\pi$
$762$	0	0
$763$	−3207.74	−0.152199
$764$	0	0
$765$	21955.3	1.03764
$766$	0	0
$767$	473.570	0.0222941
$768$	0	0
$769$	−32834.7	−1.53973	−0.769864	−	0.638208i	$-0.779676\pi$
$769$	−32834.7	−1.53973	−0.769864	+	0.638208i	$0.779676\pi$
$770$	0	0
$771$	6209.20	0.290038
$772$	0	0
$773$	−35571.4	−1.65513	−0.827564	−	0.561371i	$-0.810274\pi$
$773$	−35571.4	−1.65513	−0.827564	+	0.561371i	$0.810274\pi$
$774$	0	0
$775$	−3014.71	−0.139731
$776$	0	0
$777$	320.097	0.0147792
$778$	0	0
$779$	−36530.0	−1.68013
$780$	0	0
$781$	0	0
$782$	0	0
$783$	1755.02	0.0801014
$784$	0	0
$785$	36783.6	1.67244
$786$	0	0
$787$	15729.6	0.712452	0.356226	−	0.934400i	$-0.384063\pi$
$787$	15729.6	0.712452	0.356226	+	0.934400i	$0.384063\pi$
$788$	0	0
$789$	48996.7	2.21081
$790$	0	0
$791$	906.431	0.0407446
$792$	0	0
$793$	3958.67	0.177272
$794$	0	0
$795$	−40399.5	−1.80229
$796$	0	0
$797$	7888.07	0.350577	0.175288	−	0.984517i	$-0.443914\pi$
$797$	7888.07	0.350577	0.175288	+	0.984517i	$0.443914\pi$
$798$	0	0
$799$	14171.8	0.627485
$800$	0	0
$801$	−53394.2	−2.35530
$802$	0	0
$803$	0	0
$804$	0	0
$805$	5081.73	0.222494
$806$	0	0
$807$	7824.86	0.341323
$808$	0	0
$809$	−5896.97	−0.256275	−0.128138	−	0.991756i	$-0.540900\pi$
$809$	−5896.97	−0.256275	−0.128138	+	0.991756i	$0.540900\pi$
$810$	0	0
$811$	14197.9	0.614744	0.307372	−	0.951589i	$-0.400550\pi$
$811$	14197.9	0.614744	0.307372	+	0.951589i	$0.400550\pi$
$812$	0	0
$813$	36287.3	1.56538
$814$	0	0
$815$	40485.3	1.74005
$816$	0	0
$817$	−8075.35	−0.345803
$818$	0	0
$819$	−590.258	−0.0251835
$820$	0	0
$821$	19841.7	0.843459	0.421729	−	0.906722i	$-0.361423\pi$
$821$	19841.7	0.843459	0.421729	+	0.906722i	$0.361423\pi$
$822$	0	0
$823$	28202.2	1.19449	0.597246	−	0.802058i	$-0.296262\pi$
$823$	28202.2	1.19449	0.597246	+	0.802058i	$0.296262\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34031.0	1.43092	0.715462	−	0.698651i	$-0.246216\pi$
$827$	34031.0	1.43092	0.715462	+	0.698651i	$0.246216\pi$
$828$	0	0
$829$	4931.55	0.206610	0.103305	−	0.994650i	$-0.467058\pi$
$829$	4931.55	0.206610	0.103305	+	0.994650i	$0.467058\pi$
$830$	0	0
$831$	−4501.92	−0.187930
$832$	0	0
$833$	−13748.0	−0.571836
$834$	0	0
$835$	40666.4	1.68541
$836$	0	0
$837$	−2211.60	−0.0913312
$838$	0	0
$839$	38189.8	1.57146	0.785731	−	0.618568i	$-0.212287\pi$
$839$	38189.8	1.57146	0.785731	+	0.618568i	$0.212287\pi$
$840$	0	0
$841$	−23764.3	−0.974384
$842$	0	0
$843$	−42106.8	−1.72033
$844$	0	0
$845$	−32212.9	−1.31143
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−37488.0	−1.51541
$850$	0	0
$851$	1463.59	0.0589556
$852$	0	0
$853$	−42966.8	−1.72469	−0.862343	−	0.506325i	$-0.831003\pi$
$853$	−42966.8	−1.72469	−0.862343	+	0.506325i	$0.831003\pi$
$854$	0	0
$855$	74536.6	2.98140
$856$	0	0
$857$	17281.5	0.688828	0.344414	−	0.938818i	$-0.388078\pi$
$857$	17281.5	0.688828	0.344414	+	0.938818i	$0.388078\pi$
$858$	0	0
$859$	−9316.75	−0.370062	−0.185031	−	0.982733i	$-0.559239\pi$
$859$	−9316.75	−0.370062	−0.185031	+	0.982733i	$0.559239\pi$
$860$	0	0
$861$	−6358.10	−0.251665
$862$	0	0
$863$	9647.65	0.380544	0.190272	−	0.981731i	$-0.439063\pi$
$863$	9647.65	0.380544	0.190272	+	0.981731i	$0.439063\pi$
$864$	0	0
$865$	−34279.9	−1.34746
$866$	0	0
$867$	−25483.6	−0.998232
$868$	0	0
$869$	0	0
$870$	0	0
$871$	1834.70	0.0713735
$872$	0	0
$873$	48277.6	1.87165
$874$	0	0
$875$	−1336.55	−0.0516383
$876$	0	0
$877$	−19728.7	−0.759624	−0.379812	−	0.925064i	$-0.624011\pi$
$877$	−19728.7	−0.759624	−0.379812	+	0.925064i	$0.624011\pi$
$878$	0	0
$879$	−18464.1	−0.708509
$880$	0	0
$881$	19473.9	0.744712	0.372356	−	0.928090i	$-0.378550\pi$
$881$	19473.9	0.744712	0.372356	+	0.928090i	$0.378550\pi$
$882$	0	0
$883$	−49092.4	−1.87100	−0.935499	−	0.353329i	$-0.885050\pi$
$883$	−49092.4	−1.87100	−0.935499	+	0.353329i	$0.885050\pi$
$884$	0	0
$885$	−10408.5	−0.395344
$886$	0	0
$887$	9292.86	0.351774	0.175887	−	0.984410i	$-0.443721\pi$
$887$	9292.86	0.351774	0.175887	+	0.984410i	$0.443721\pi$
$888$	0	0
$889$	−4047.41	−0.152695
$890$	0	0
$891$	0	0
$892$	0	0
$893$	48112.0	1.80292
$894$	0	0
$895$	19493.9	0.728055
$896$	0	0
$897$	−4731.10	−0.176106
$898$	0	0
$899$	−787.273	−0.0292069
$900$	0	0
$901$	−14136.7	−0.522709
$902$	0	0
$903$	−1405.53	−0.0517974
$904$	0	0
$905$	−11932.3	−0.438279
$906$	0	0
$907$	−37688.7	−1.37975	−0.689875	−	0.723928i	$-0.742335\pi$
$907$	−37688.7	−1.37975	−0.689875	+	0.723928i	$0.742335\pi$
$908$	0	0
$909$	5789.42	0.211246
$910$	0	0
$911$	−33049.6	−1.20196	−0.600979	−	0.799265i	$-0.705222\pi$
$911$	−33049.6	−1.20196	−0.600979	+	0.799265i	$0.705222\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−87007.2	−3.14357
$916$	0	0
$917$	−4917.06	−0.177073
$918$	0	0
$919$	−23148.0	−0.830883	−0.415442	−	0.909620i	$-0.636373\pi$
$919$	−23148.0	−0.830883	−0.415442	+	0.909620i	$0.636373\pi$
$920$	0	0
$921$	−13304.1	−0.475986
$922$	0	0
$923$	−1111.26	−0.0396290
$924$	0	0
$925$	1258.01	0.0447169
$926$	0	0
$927$	1246.34	0.0441588
$928$	0	0
$929$	−23177.9	−0.818561	−0.409280	−	0.912409i	$-0.634220\pi$
$929$	−23177.9	−0.818561	−0.409280	+	0.912409i	$0.634220\pi$
$930$	0	0
$931$	−46673.3	−1.64302
$932$	0	0
$933$	−28325.1	−0.993916
$934$	0	0
$935$	0	0
$936$	0	0
$937$	34574.7	1.20545	0.602724	−	0.797950i	$-0.294082\pi$
$937$	34574.7	1.20545	0.602724	+	0.797950i	$0.294082\pi$
$938$	0	0
$939$	56959.1	1.97954
$940$	0	0
$941$	−41831.2	−1.44916	−0.724578	−	0.689192i	$-0.757966\pi$
$941$	−41831.2	−1.44916	−0.724578	+	0.689192i	$0.757966\pi$
$942$	0	0
$943$	−29071.3	−1.00392
$944$	0	0
$945$	3204.34	0.110304
$946$	0	0
$947$	−27231.2	−0.934419	−0.467209	−	0.884147i	$-0.654741\pi$
$947$	−27231.2	−0.934419	−0.467209	+	0.884147i	$0.654741\pi$
$948$	0	0
$949$	−5415.79	−0.185252
$950$	0	0
$951$	−124.534	−0.00424635
$952$	0	0
$953$	−40939.4	−1.39156	−0.695781	−	0.718254i	$-0.744942\pi$
$953$	−40939.4	−1.39156	−0.695781	+	0.718254i	$0.744942\pi$
$954$	0	0
$955$	−25527.0	−0.864956
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4950.74	0.166703
$960$	0	0
$961$	−28798.9	−0.966698
$962$	0	0
$963$	29839.0	0.998491
$964$	0	0
$965$	−19910.3	−0.664182
$966$	0	0
$967$	−46173.1	−1.53550	−0.767750	−	0.640750i	$-0.778624\pi$
$967$	−46173.1	−1.53550	−0.767750	+	0.640750i	$0.778624\pi$
$968$	0	0
$969$	45721.8	1.51579
$970$	0	0
$971$	5153.91	0.170337	0.0851683	−	0.996367i	$-0.472857\pi$
$971$	5153.91	0.170337	0.0851683	+	0.996367i	$0.472857\pi$
$972$	0	0
$973$	−97.8734	−0.00322474
$974$	0	0
$975$	−4066.56	−0.133574
$976$	0	0
$977$	9692.13	0.317378	0.158689	−	0.987329i	$-0.449273\pi$
$977$	9692.13	0.317378	0.158689	+	0.987329i	$0.449273\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−37443.3	−1.21863
$982$	0	0
$983$	−32915.7	−1.06800	−0.534002	−	0.845483i	$-0.679313\pi$
$983$	−32915.7	−1.06800	−0.534002	+	0.845483i	$0.679313\pi$
$984$	0	0
$985$	52269.7	1.69081
$986$	0	0
$987$	8373.97	0.270057
$988$	0	0
$989$	−6426.54	−0.206625
$990$	0	0
$991$	−29477.9	−0.944901	−0.472451	−	0.881357i	$-0.656630\pi$
$991$	−29477.9	−0.944901	−0.472451	+	0.881357i	$0.656630\pi$
$992$	0	0
$993$	10456.9	0.334180
$994$	0	0
$995$	−12237.1	−0.389892
$996$	0	0
$997$	31944.4	1.01473	0.507366	−	0.861731i	$-0.330619\pi$
$997$	31944.4	1.01473	0.507366	+	0.861731i	$0.330619\pi$
$998$	0	0
$999$	922.883	0.0292279

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1936.4.a.w.1.2		2
4.3	odd	2		121.4.a.c.1.1		2
11.10	odd	2		176.4.a.i.1.2		2
12.11	even	2		1089.4.a.v.1.2		2
33.32	even	2		1584.4.a.bc.1.1		2
44.3	odd	10		121.4.c.f.9.1		8
44.7	even	10		121.4.c.c.27.2		8
44.15	odd	10		121.4.c.f.27.1		8
44.19	even	10		121.4.c.c.9.2		8
44.27	odd	10		121.4.c.f.3.2		8
44.31	odd	10		121.4.c.f.81.2		8
44.35	even	10		121.4.c.c.81.1		8
44.39	even	10		121.4.c.c.3.1		8
44.43	even	2		11.4.a.a.1.2	✓	2
88.21	odd	2		704.4.a.n.1.1		2
88.43	even	2		704.4.a.p.1.2		2
132.131	odd	2		99.4.a.c.1.1		2
220.43	odd	4		275.4.b.c.199.1		4
220.87	odd	4		275.4.b.c.199.4		4
220.219	even	2		275.4.a.b.1.1		2
308.307	odd	2		539.4.a.e.1.2		2
572.571	even	2		1859.4.a.a.1.1		2
660.659	odd	2		2475.4.a.q.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.4.a.a.1.2	✓	2	44.43	even	2
99.4.a.c.1.1		2	132.131	odd	2
121.4.a.c.1.1		2	4.3	odd	2
121.4.c.c.3.1		8	44.39	even	10
121.4.c.c.9.2		8	44.19	even	10
121.4.c.c.27.2		8	44.7	even	10
121.4.c.c.81.1		8	44.35	even	10
121.4.c.f.3.2		8	44.27	odd	10
121.4.c.f.9.1		8	44.3	odd	10
121.4.c.f.27.1		8	44.15	odd	10
121.4.c.f.81.2		8	44.31	odd	10
176.4.a.i.1.2		2	11.10	odd	2
275.4.a.b.1.1		2	220.219	even	2
275.4.b.c.199.1		4	220.43	odd	4
275.4.b.c.199.4		4	220.87	odd	4
539.4.a.e.1.2		2	308.307	odd	2
704.4.a.n.1.1		2	88.21	odd	2
704.4.a.p.1.2		2	88.43	even	2
1089.4.a.v.1.2		2	12.11	even	2
1584.4.a.bc.1.1		2	33.32	even	2
1859.4.a.a.1.1		2	572.571	even	2
1936.4.a.w.1.2		2	1.1	even	1	trivial
2475.4.a.q.1.2		2	660.659	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1936 = 2^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1936.a (trivial)

\(f(q)\)	\(=\)	\(q+7.92820 q^{3} +14.8564 q^{5} +3.07180 q^{7} +35.8564 q^{9} +O(q^{10})\)	\(q+7.92820 q^{3} +14.8564 q^{5} +3.07180 q^{7} +35.8564 q^{9} -5.35898 q^{13} +117.785 q^{15} +41.2154 q^{17} +139.923 q^{19} +24.3538 q^{21} +111.354 q^{23} +95.7128 q^{25} +70.2154 q^{27} +24.9948 q^{29} -31.4974 q^{31} +45.6359 q^{35} +13.1436 q^{37} -42.4871 q^{39} -261.072 q^{41} -57.7128 q^{43} +532.697 q^{45} +343.846 q^{47} -333.564 q^{49} +326.764 q^{51} -342.995 q^{53} +1109.34 q^{57} -88.3693 q^{59} -738.697 q^{61} +110.144 q^{63} -79.6152 q^{65} -342.359 q^{67} +882.836 q^{69} +207.364 q^{71} +1010.60 q^{73} +758.831 q^{75} +1294.23 q^{79} -411.441 q^{81} +441.846 q^{83} +612.313 q^{85} +198.164 q^{87} -1489.11 q^{89} -16.4617 q^{91} -249.718 q^{93} +2078.75 q^{95} +1346.42 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 20 q^{7} + 44 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + 20 * q^7 + 44 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + 20 q^{7} + 44 q^{9} - 80 q^{13} + 194 q^{15} + 124 q^{17} + 72 q^{19} - 76 q^{21} + 98 q^{23} + 136 q^{25} + 182 q^{27} - 144 q^{29} + 34 q^{31} - 172 q^{35} + 54 q^{37} + 400 q^{39} - 536 q^{41} - 60 q^{43} + 428 q^{45} + 272 q^{47} - 390 q^{49} - 164 q^{51} - 492 q^{53} + 1512 q^{57} - 634 q^{59} - 840 q^{61} + 248 q^{63} + 880 q^{65} - 754 q^{67} + 962 q^{69} + 678 q^{71} + 400 q^{73} + 520 q^{75} + 316 q^{79} - 1294 q^{81} + 468 q^{83} - 452 q^{85} + 1200 q^{87} - 1842 q^{89} - 1280 q^{91} - 638 q^{93} + 2952 q^{95} + 2194 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 20 * q^7 + 44 * q^9 - 80 * q^13 + 194 * q^15 + 124 * q^17 + 72 * q^19 - 76 * q^21 + 98 * q^23 + 136 * q^25 + 182 * q^27 - 144 * q^29 + 34 * q^31 - 172 * q^35 + 54 * q^37 + 400 * q^39 - 536 * q^41 - 60 * q^43 + 428 * q^45 + 272 * q^47 - 390 * q^49 - 164 * q^51 - 492 * q^53 + 1512 * q^57 - 634 * q^59 - 840 * q^61 + 248 * q^63 + 880 * q^65 - 754 * q^67 + 962 * q^69 + 678 * q^71 + 400 * q^73 + 520 * q^75 + 316 * q^79 - 1294 * q^81 + 468 * q^83 - 452 * q^85 + 1200 * q^87 - 1842 * q^89 - 1280 * q^91 - 638 * q^93 + 2952 * q^95 + 2194 * q^97

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