LMFDB - Embedded newform 1936.4.a.br.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1936,4,Mod(1,1936)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [6,0,-3,0,-12,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$114.227697771$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 82x^{4} + 161x^{3} + 1730x^{2} - 2271x - 5931 $ x^6 - 3x^5 - 82x^4 + 161x^3 + 1730x^2 - 2271*x - 5931
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 11 $
Twist minimal:	no (minimal twist has level 44)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$5.21558$ 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)

$f(q)$	$=$	$q+4.21558 q^{3} +16.9944 q^{5} -3.71712 q^{7} -9.22891 q^{9} -62.6552 q^{13} +71.6414 q^{15} -83.4146 q^{17} -49.7405 q^{19} -15.6698 q^{21} +42.7247 q^{23} +163.811 q^{25} -152.726 q^{27} +154.890 q^{29} +202.202 q^{31} -63.1704 q^{35} +95.9126 q^{37} -264.128 q^{39} -450.262 q^{41} +343.078 q^{43} -156.840 q^{45} -392.379 q^{47} -329.183 q^{49} -351.641 q^{51} -192.684 q^{53} -209.685 q^{57} +529.767 q^{59} -337.496 q^{61} +34.3050 q^{63} -1064.79 q^{65} -831.356 q^{67} +180.109 q^{69} -676.018 q^{71} -1215.12 q^{73} +690.559 q^{75} +1072.29 q^{79} -394.646 q^{81} +80.8887 q^{83} -1417.58 q^{85} +652.951 q^{87} -30.9863 q^{89} +232.897 q^{91} +852.400 q^{93} -845.313 q^{95} +91.1400 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} - 12 q^{5} - 8 q^{7} + 11 q^{9} - 80 q^{13} + 98 q^{15} - 113 q^{17} + 53 q^{19} - 152 q^{21} + 194 q^{23} + 476 q^{25} - 72 q^{27} - 374 q^{29} - 16 q^{31} + 1044 q^{35} - 456 q^{37} + 592 q^{39}+ \cdots + 683 q^{97}+O(q^{100}) $ 6 * q - 3 * q^3 - 12 * q^5 - 8 * q^7 + 11 * q^9 - 80 * q^13 + 98 * q^15 - 113 * q^17 + 53 * q^19 - 152 * q^21 + 194 * q^23 + 476 * q^25 - 72 * q^27 - 374 * q^29 - 16 * q^31 + 1044 * q^35 - 456 * q^37 + 592 * q^39 - 825 * q^41 + 1165 * q^43 - 670 * q^45 - 1248 * q^47 - 4 * q^49 + 988 * q^51 - 122 * q^53 - 2282 * q^57 + 767 * q^59 - 1010 * q^61 + 254 * q^63 - 1946 * q^65 + 2197 * q^67 + 2648 * q^69 - 598 * q^71 - 1523 * q^73 - 609 * q^75 + 2116 * q^79 - 2106 * q^81 + 1213 * q^83 - 1066 * q^85 + 1518 * q^87 - 477 * q^89 - 2420 * q^91 - 714 * q^93 + 404 * q^95 + 683 * q^97

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
$3$	4.21558	0.811288	0.405644	−	0.914031i	$-0.367047\pi$
$3$	4.21558	0.811288	0.405644	+	0.914031i	$0.367047\pi$
$4$	0	0
$5$	16.9944	1.52003	0.760015	−	0.649906i	$-0.225192\pi$
$5$	16.9944	1.52003	0.760015	+	0.649906i	$0.225192\pi$
$6$	0	0
$7$	−3.71712	−0.200706	−0.100353	−	0.994952i	$-0.531997\pi$
$7$	−3.71712	−0.200706	−0.100353	+	0.994952i	$0.531997\pi$
$8$	0	0
$9$	−9.22891	−0.341812
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−62.6552	−1.33673	−0.668363	−	0.743836i	$-0.733004\pi$
$13$	−62.6552	−1.33673	−0.668363	+	0.743836i	$0.733004\pi$
$14$	0	0
$15$	71.6414	1.23318
$16$	0	0
$17$	−83.4146	−1.19006	−0.595030	−	0.803704i	$-0.702860\pi$
$17$	−83.4146	−1.19006	−0.595030	+	0.803704i	$0.702860\pi$
$18$	0	0
$19$	−49.7405	−0.600592	−0.300296	−	0.953846i	$-0.597086\pi$
$19$	−49.7405	−0.600592	−0.300296	+	0.953846i	$0.597086\pi$
$20$	0	0
$21$	−15.6698	−0.162830
$22$	0	0
$23$	42.7247	0.387335	0.193668	−	0.981067i	$-0.437962\pi$
$23$	42.7247	0.387335	0.193668	+	0.981067i	$0.437962\pi$
$24$	0	0
$25$	163.811	1.31049
$26$	0	0
$27$	−152.726	−1.08860
$28$	0	0
$29$	154.890	0.991805	0.495903	−	0.868378i	$-0.334837\pi$
$29$	154.890	0.991805	0.495903	+	0.868378i	$0.334837\pi$
$30$	0	0
$31$	202.202	1.17150	0.585752	−	0.810490i	$-0.300799\pi$
$31$	202.202	1.17150	0.585752	+	0.810490i	$0.300799\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−63.1704	−0.305079
$36$	0	0
$37$	95.9126	0.426160	0.213080	−	0.977035i	$-0.431650\pi$
$37$	95.9126	0.426160	0.213080	+	0.977035i	$0.431650\pi$
$38$	0	0
$39$	−264.128	−1.08447
$40$	0	0
$41$	−450.262	−1.71510	−0.857550	−	0.514400i	$-0.828015\pi$
$41$	−450.262	−1.71510	−0.857550	+	0.514400i	$0.828015\pi$
$42$	0	0
$43$	343.078	1.21672	0.608360	−	0.793661i	$-0.291828\pi$
$43$	343.078	1.21672	0.608360	+	0.793661i	$0.291828\pi$
$44$	0	0
$45$	−156.840	−0.519564
$46$	0	0
$47$	−392.379	−1.21775	−0.608876	−	0.793265i	$-0.708379\pi$
$47$	−392.379	−1.21775	−0.608876	+	0.793265i	$0.708379\pi$
$48$	0	0
$49$	−329.183	−0.959717
$50$	0	0
$51$	−351.641	−0.965481
$52$	0	0
$53$	−192.684	−0.499380	−0.249690	−	0.968326i	$-0.580329\pi$
$53$	−192.684	−0.499380	−0.249690	+	0.968326i	$0.580329\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−209.685	−0.487254
$58$	0	0
$59$	529.767	1.16898	0.584490	−	0.811401i	$-0.301295\pi$
$59$	529.767	1.16898	0.584490	+	0.811401i	$0.301295\pi$
$60$	0	0
$61$	−337.496	−0.708393	−0.354197	−	0.935171i	$-0.615246\pi$
$61$	−337.496	−0.708393	−0.354197	+	0.935171i	$0.615246\pi$
$62$	0	0
$63$	34.3050	0.0686035
$64$	0	0
$65$	−1064.79	−2.03186
$66$	0	0
$67$	−831.356	−1.51591	−0.757957	−	0.652304i	$-0.773803\pi$
$67$	−831.356	−1.51591	−0.757957	+	0.652304i	$0.773803\pi$
$68$	0	0
$69$	180.109	0.314240
$70$	0	0
$71$	−676.018	−1.12998	−0.564990	−	0.825098i	$-0.691120\pi$
$71$	−676.018	−1.12998	−0.564990	+	0.825098i	$0.691120\pi$
$72$	0	0
$73$	−1215.12	−1.94821	−0.974106	−	0.226094i	$-0.927404\pi$
$73$	−1215.12	−1.94821	−0.974106	+	0.226094i	$0.927404\pi$
$74$	0	0
$75$	690.559	1.06319
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1072.29	1.52712	0.763558	−	0.645739i	$-0.223451\pi$
$79$	1072.29	1.52712	0.763558	+	0.645739i	$0.223451\pi$
$80$	0	0
$81$	−394.646	−0.541353
$82$	0	0
$83$	80.8887	0.106972	0.0534861	−	0.998569i	$-0.482967\pi$
$83$	80.8887	0.106972	0.0534861	+	0.998569i	$0.482967\pi$
$84$	0	0
$85$	−1417.58	−1.80893
$86$	0	0
$87$	652.951	0.804640
$88$	0	0
$89$	−30.9863	−0.0369049	−0.0184525	−	0.999830i	$-0.505874\pi$
$89$	−30.9863	−0.0369049	−0.0184525	+	0.999830i	$0.505874\pi$
$90$	0	0
$91$	232.897	0.268288
$92$	0	0
$93$	852.400	0.950427
$94$	0	0
$95$	−845.313	−0.912918
$96$	0	0
$97$	91.1400	0.0954007	0.0477004	−	0.998862i	$-0.484811\pi$
$97$	91.1400	0.0954007	0.0477004	+	0.998862i	$0.484811\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1203.84	−1.18600	−0.593002	−	0.805201i	$-0.702057\pi$
$101$	−1203.84	−1.18600	−0.593002	+	0.805201i	$0.702057\pi$
$102$	0	0
$103$	834.238	0.798058	0.399029	−	0.916938i	$-0.369347\pi$
$103$	834.238	0.798058	0.399029	+	0.916938i	$0.369347\pi$
$104$	0	0
$105$	−266.300	−0.247507
$106$	0	0
$107$	−401.263	−0.362538	−0.181269	−	0.983434i	$-0.558020\pi$
$107$	−401.263	−0.362538	−0.181269	+	0.983434i	$0.558020\pi$
$108$	0	0
$109$	−1195.49	−1.05052	−0.525262	−	0.850941i	$-0.676033\pi$
$109$	−1195.49	−1.05052	−0.525262	+	0.850941i	$0.676033\pi$
$110$	0	0
$111$	404.327	0.345739
$112$	0	0
$113$	871.176	0.725251	0.362625	−	0.931935i	$-0.381880\pi$
$113$	871.176	0.725251	0.362625	+	0.931935i	$0.381880\pi$
$114$	0	0
$115$	726.082	0.588761
$116$	0	0
$117$	578.240	0.456908
$118$	0	0
$119$	310.062	0.238852
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−1898.11	−1.39144
$124$	0	0
$125$	659.577	0.471955
$126$	0	0
$127$	−865.182	−0.604508	−0.302254	−	0.953227i	$-0.597739\pi$
$127$	−865.182	−0.604508	−0.302254	+	0.953227i	$0.597739\pi$
$128$	0	0
$129$	1446.27	0.987110
$130$	0	0
$131$	−2598.55	−1.73310	−0.866551	−	0.499088i	$-0.833668\pi$
$131$	−2598.55	−1.73310	−0.866551	+	0.499088i	$0.833668\pi$
$132$	0	0
$133$	184.892	0.120542
$134$	0	0
$135$	−2595.49	−1.65470
$136$	0	0
$137$	−844.622	−0.526722	−0.263361	−	0.964697i	$-0.584831\pi$
$137$	−844.622	−0.526722	−0.263361	+	0.964697i	$0.584831\pi$
$138$	0	0
$139$	−1247.71	−0.761364	−0.380682	−	0.924706i	$-0.624311\pi$
$139$	−1247.71	−0.761364	−0.380682	+	0.924706i	$0.624311\pi$
$140$	0	0
$141$	−1654.10	−0.987948
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2632.27	1.50757
$146$	0	0
$147$	−1387.70	−0.778607
$148$	0	0
$149$	−324.398	−0.178361	−0.0891804	−	0.996015i	$-0.528425\pi$
$149$	−324.398	−0.178361	−0.0891804	+	0.996015i	$0.528425\pi$
$150$	0	0
$151$	1840.05	0.991662	0.495831	−	0.868419i	$-0.334864\pi$
$151$	1840.05	0.991662	0.495831	+	0.868419i	$0.334864\pi$
$152$	0	0
$153$	769.826	0.406776
$154$	0	0
$155$	3436.32	1.78072
$156$	0	0
$157$	−2590.85	−1.31702	−0.658510	−	0.752572i	$-0.728813\pi$
$157$	−2590.85	−1.31702	−0.658510	+	0.752572i	$0.728813\pi$
$158$	0	0
$159$	−812.272	−0.405141
$160$	0	0
$161$	−158.813	−0.0777404
$162$	0	0
$163$	−2942.42	−1.41392	−0.706958	−	0.707255i	$-0.749933\pi$
$163$	−2942.42	−1.41392	−0.706958	+	0.707255i	$0.749933\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1895.29	0.878213	0.439107	−	0.898435i	$-0.355295\pi$
$167$	1895.29	0.878213	0.439107	+	0.898435i	$0.355295\pi$
$168$	0	0
$169$	1728.67	0.786834
$170$	0	0
$171$	459.051	0.205290
$172$	0	0
$173$	2032.61	0.893273	0.446636	−	0.894716i	$-0.352622\pi$
$173$	2032.61	0.893273	0.446636	+	0.894716i	$0.352622\pi$
$174$	0	0
$175$	−608.906	−0.263023
$176$	0	0
$177$	2233.27	0.948379
$178$	0	0
$179$	1974.79	0.824598	0.412299	−	0.911049i	$-0.364726\pi$
$179$	1974.79	0.824598	0.412299	+	0.911049i	$0.364726\pi$
$180$	0	0
$181$	−907.762	−0.372781	−0.186391	−	0.982476i	$-0.559679\pi$
$181$	−907.762	−0.372781	−0.186391	+	0.982476i	$0.559679\pi$
$182$	0	0
$183$	−1422.74	−0.574711
$184$	0	0
$185$	1629.98	0.647777
$186$	0	0
$187$	0	0
$188$	0	0
$189$	567.700	0.218487
$190$	0	0
$191$	1670.92	0.633002	0.316501	−	0.948592i	$-0.397492\pi$
$191$	1670.92	0.633002	0.316501	+	0.948592i	$0.397492\pi$
$192$	0	0
$193$	2410.93	0.899184	0.449592	−	0.893234i	$-0.351569\pi$
$193$	2410.93	0.899184	0.449592	+	0.893234i	$0.351569\pi$
$194$	0	0
$195$	−4488.71	−1.64843
$196$	0	0
$197$	1966.58	0.711234	0.355617	−	0.934632i	$-0.384271\pi$
$197$	1966.58	0.711234	0.355617	+	0.934632i	$0.384271\pi$
$198$	0	0
$199$	−136.896	−0.0487653	−0.0243827	−	0.999703i	$-0.507762\pi$
$199$	−136.896	−0.0487653	−0.0243827	+	0.999703i	$0.507762\pi$
$200$	0	0
$201$	−3504.64	−1.22984
$202$	0	0
$203$	−575.745	−0.199061
$204$	0	0
$205$	−7651.95	−2.60700
$206$	0	0
$207$	−394.302	−0.132396
$208$	0	0
$209$	0	0
$210$	0	0
$211$	905.311	0.295375	0.147688	−	0.989034i	$-0.452817\pi$
$211$	905.311	0.295375	0.147688	+	0.989034i	$0.452817\pi$
$212$	0	0
$213$	−2849.81	−0.916740
$214$	0	0
$215$	5830.43	1.84945
$216$	0	0
$217$	−751.611	−0.235128
$218$	0	0
$219$	−5122.44	−1.58056
$220$	0	0
$221$	5226.36	1.59078
$222$	0	0
$223$	1096.94	0.329402	0.164701	−	0.986344i	$-0.447334\pi$
$223$	1096.94	0.329402	0.164701	+	0.986344i	$0.447334\pi$
$224$	0	0
$225$	−1511.80	−0.447941
$226$	0	0
$227$	3409.73	0.996968	0.498484	−	0.866899i	$-0.333890\pi$
$227$	3409.73	0.996968	0.498484	+	0.866899i	$0.333890\pi$
$228$	0	0
$229$	5535.14	1.59726	0.798630	−	0.601822i	$-0.205558\pi$
$229$	5535.14	1.59726	0.798630	+	0.601822i	$0.205558\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	238.649	0.0671005	0.0335502	−	0.999437i	$-0.489319\pi$
$233$	238.649	0.0671005	0.0335502	+	0.999437i	$0.489319\pi$
$234$	0	0
$235$	−6668.26	−1.85102
$236$	0	0
$237$	4520.32	1.23893
$238$	0	0
$239$	−26.7935	−0.00725158	−0.00362579	−	0.999993i	$-0.501154\pi$
$239$	−26.7935	−0.00725158	−0.00362579	+	0.999993i	$0.501154\pi$
$240$	0	0
$241$	−440.617	−0.117770	−0.0588851	−	0.998265i	$-0.518755\pi$
$241$	−440.617	−0.117770	−0.0588851	+	0.998265i	$0.518755\pi$
$242$	0	0
$243$	2459.93	0.649402
$244$	0	0
$245$	−5594.28	−1.45880
$246$	0	0
$247$	3116.50	0.802827
$248$	0	0
$249$	340.992	0.0867852
$250$	0	0
$251$	−1061.78	−0.267008	−0.133504	−	0.991048i	$-0.542623\pi$
$251$	−1061.78	−0.267008	−0.133504	+	0.991048i	$0.542623\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−5975.94	−1.46756
$256$	0	0
$257$	2735.93	0.664057	0.332029	−	0.943269i	$-0.392267\pi$
$257$	2735.93	0.664057	0.332029	+	0.943269i	$0.392267\pi$
$258$	0	0
$259$	−356.519	−0.0855328
$260$	0	0
$261$	−1429.47	−0.339011
$262$	0	0
$263$	−165.974	−0.0389140	−0.0194570	−	0.999811i	$-0.506194\pi$
$263$	−165.974	−0.0389140	−0.0194570	+	0.999811i	$0.506194\pi$
$264$	0	0
$265$	−3274.55	−0.759072
$266$	0	0
$267$	−130.625	−0.0299405
$268$	0	0
$269$	−2076.50	−0.470657	−0.235329	−	0.971916i	$-0.575617\pi$
$269$	−2076.50	−0.470657	−0.235329	+	0.971916i	$0.575617\pi$
$270$	0	0
$271$	−8425.15	−1.88853	−0.944265	−	0.329186i	$-0.893226\pi$
$271$	−8425.15	−1.88853	−0.944265	+	0.329186i	$0.893226\pi$
$272$	0	0
$273$	981.795	0.217659
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2406.96	0.522095	0.261047	−	0.965326i	$-0.415932\pi$
$277$	2406.96	0.522095	0.261047	+	0.965326i	$0.415932\pi$
$278$	0	0
$279$	−1866.11	−0.400434
$280$	0	0
$281$	3346.88	0.710527	0.355264	−	0.934766i	$-0.384391\pi$
$281$	3346.88	0.710527	0.355264	+	0.934766i	$0.384391\pi$
$282$	0	0
$283$	−4789.70	−1.00607	−0.503035	−	0.864266i	$-0.667783\pi$
$283$	−4789.70	−1.00607	−0.503035	+	0.864266i	$0.667783\pi$
$284$	0	0
$285$	−3563.48	−0.740640
$286$	0	0
$287$	1673.68	0.344230
$288$	0	0
$289$	2044.99	0.416241
$290$	0	0
$291$	384.208	0.0773975
$292$	0	0
$293$	−3070.70	−0.612259	−0.306130	−	0.951990i	$-0.599034\pi$
$293$	−3070.70	−0.612259	−0.306130	+	0.951990i	$0.599034\pi$
$294$	0	0
$295$	9003.10	1.77688
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2676.92	−0.517761
$300$	0	0
$301$	−1275.26	−0.244203
$302$	0	0
$303$	−5074.87	−0.962191
$304$	0	0
$305$	−5735.56	−1.07678
$306$	0	0
$307$	−9278.28	−1.72488	−0.862442	−	0.506156i	$-0.831066\pi$
$307$	−9278.28	−1.72488	−0.862442	+	0.506156i	$0.831066\pi$
$308$	0	0
$309$	3516.80	0.647455
$310$	0	0
$311$	−6967.22	−1.27034	−0.635169	−	0.772373i	$-0.719069\pi$
$311$	−6967.22	−1.27034	−0.635169	+	0.772373i	$0.719069\pi$
$312$	0	0
$313$	−5346.77	−0.965550	−0.482775	−	0.875744i	$-0.660371\pi$
$313$	−5346.77	−0.965550	−0.482775	+	0.875744i	$0.660371\pi$
$314$	0	0
$315$	582.995	0.104279
$316$	0	0
$317$	8018.39	1.42069	0.710343	−	0.703856i	$-0.248540\pi$
$317$	8018.39	1.42069	0.710343	+	0.703856i	$0.248540\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−1691.55	−0.294122
$322$	0	0
$323$	4149.08	0.714741
$324$	0	0
$325$	−10263.6	−1.75177
$326$	0	0
$327$	−5039.68	−0.852277
$328$	0	0
$329$	1458.52	0.244410
$330$	0	0
$331$	6738.30	1.11894	0.559472	−	0.828849i	$-0.311004\pi$
$331$	6738.30	1.11894	0.559472	+	0.828849i	$0.311004\pi$
$332$	0	0
$333$	−885.169	−0.145667
$334$	0	0
$335$	−14128.4	−2.30424
$336$	0	0
$337$	6001.47	0.970092	0.485046	−	0.874489i	$-0.338803\pi$
$337$	6001.47	0.970092	0.485046	+	0.874489i	$0.338803\pi$
$338$	0	0
$339$	3672.51	0.588387
$340$	0	0
$341$	0	0
$342$	0	0
$343$	2498.59	0.393326
$344$	0	0
$345$	3060.85	0.477655
$346$	0	0
$347$	6961.34	1.07696	0.538479	−	0.842639i	$-0.318999\pi$
$347$	6961.34	1.07696	0.538479	+	0.842639i	$0.318999\pi$
$348$	0	0
$349$	230.792	0.0353983	0.0176992	−	0.999843i	$-0.494366\pi$
$349$	230.792	0.0353983	0.0176992	+	0.999843i	$0.494366\pi$
$350$	0	0
$351$	9569.06	1.45515
$352$	0	0
$353$	6730.46	1.01481	0.507403	−	0.861709i	$-0.330606\pi$
$353$	6730.46	1.01481	0.507403	+	0.861709i	$0.330606\pi$
$354$	0	0
$355$	−11488.6	−1.71760
$356$	0	0
$357$	1307.09	0.193778
$358$	0	0
$359$	−797.152	−0.117192	−0.0585962	−	0.998282i	$-0.518662\pi$
$359$	−797.152	−0.117192	−0.0585962	+	0.998282i	$0.518662\pi$
$360$	0	0
$361$	−4384.88	−0.639289
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−20650.3	−2.96134
$366$	0	0
$367$	10698.7	1.52171	0.760853	−	0.648925i	$-0.224781\pi$
$367$	10698.7	1.52171	0.760853	+	0.648925i	$0.224781\pi$
$368$	0	0
$369$	4155.43	0.586241
$370$	0	0
$371$	716.228	0.100228
$372$	0	0
$373$	−748.042	−0.103839	−0.0519197	−	0.998651i	$-0.516534\pi$
$373$	−748.042	−0.103839	−0.0519197	+	0.998651i	$0.516534\pi$
$374$	0	0
$375$	2780.50	0.382891
$376$	0	0
$377$	−9704.67	−1.32577
$378$	0	0
$379$	−9005.92	−1.22059	−0.610294	−	0.792175i	$-0.708949\pi$
$379$	−9005.92	−1.22059	−0.610294	+	0.792175i	$0.708949\pi$
$380$	0	0
$381$	−3647.24	−0.490430
$382$	0	0
$383$	6749.89	0.900531	0.450265	−	0.892895i	$-0.351329\pi$
$383$	6749.89	0.900531	0.450265	+	0.892895i	$0.351329\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3166.24	−0.415889
$388$	0	0
$389$	−10884.9	−1.41873	−0.709367	−	0.704839i	$-0.751019\pi$
$389$	−10884.9	−1.41873	−0.709367	+	0.704839i	$0.751019\pi$
$390$	0	0
$391$	−3563.86	−0.460952
$392$	0	0
$393$	−10954.4	−1.40605
$394$	0	0
$395$	18223.0	2.32126
$396$	0	0
$397$	−2715.71	−0.343318	−0.171659	−	0.985156i	$-0.554913\pi$
$397$	−2715.71	−0.343318	−0.171659	+	0.985156i	$0.554913\pi$
$398$	0	0
$399$	779.424	0.0977946
$400$	0	0
$401$	346.044	0.0430938	0.0215469	−	0.999768i	$-0.493141\pi$
$401$	346.044	0.0430938	0.0215469	+	0.999768i	$0.493141\pi$
$402$	0	0
$403$	−12669.0	−1.56598
$404$	0	0
$405$	−6706.80	−0.822873
$406$	0	0
$407$	0	0
$408$	0	0
$409$	4639.27	0.560874	0.280437	−	0.959872i	$-0.409521\pi$
$409$	4639.27	0.560874	0.280437	+	0.959872i	$0.409521\pi$
$410$	0	0
$411$	−3560.57	−0.427323
$412$	0	0
$413$	−1969.21	−0.234621
$414$	0	0
$415$	1374.66	0.162601
$416$	0	0
$417$	−5259.83	−0.617686
$418$	0	0
$419$	−4439.38	−0.517609	−0.258804	−	0.965930i	$-0.583328\pi$
$419$	−4439.38	−0.517609	−0.258804	+	0.965930i	$0.583328\pi$
$420$	0	0
$421$	14103.2	1.63266	0.816328	−	0.577589i	$-0.196006\pi$
$421$	14103.2	1.63266	0.816328	+	0.577589i	$0.196006\pi$
$422$	0	0
$423$	3621.23	0.416242
$424$	0	0
$425$	−13664.3	−1.55956
$426$	0	0
$427$	1254.51	0.142179
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12381.2	1.38372	0.691860	−	0.722032i	$-0.256791\pi$
$431$	12381.2	1.38372	0.691860	+	0.722032i	$0.256791\pi$
$432$	0	0
$433$	6176.00	0.685449	0.342725	−	0.939436i	$-0.388650\pi$
$433$	6176.00	0.685449	0.342725	+	0.939436i	$0.388650\pi$
$434$	0	0
$435$	11096.5	1.22308
$436$	0	0
$437$	−2125.15	−0.232631
$438$	0	0
$439$	−6768.55	−0.735867	−0.367933	−	0.929852i	$-0.619935\pi$
$439$	−6768.55	−0.735867	−0.367933	+	0.929852i	$0.619935\pi$
$440$	0	0
$441$	3038.00	0.328043
$442$	0	0
$443$	1375.61	0.147534	0.0737668	−	0.997276i	$-0.476498\pi$
$443$	1375.61	0.147534	0.0737668	+	0.997276i	$0.476498\pi$
$444$	0	0
$445$	−526.594	−0.0560966
$446$	0	0
$447$	−1367.53	−0.144702
$448$	0	0
$449$	−880.495	−0.0925459	−0.0462730	−	0.998929i	$-0.514734\pi$
$449$	−880.495	−0.0925459	−0.0462730	+	0.998929i	$0.514734\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	7756.86	0.804523
$454$	0	0
$455$	3957.96	0.407806
$456$	0	0
$457$	9385.58	0.960697	0.480349	−	0.877078i	$-0.340510\pi$
$457$	9385.58	0.960697	0.480349	+	0.877078i	$0.340510\pi$
$458$	0	0
$459$	12739.6	1.29549
$460$	0	0
$461$	−14837.4	−1.49902	−0.749508	−	0.661996i	$-0.769710\pi$
$461$	−14837.4	−1.49902	−0.749508	+	0.661996i	$0.769710\pi$
$462$	0	0
$463$	−8010.57	−0.804067	−0.402033	−	0.915625i	$-0.631696\pi$
$463$	−8010.57	−0.804067	−0.402033	+	0.915625i	$0.631696\pi$
$464$	0	0
$465$	14486.1	1.44468
$466$	0	0
$467$	5729.52	0.567732	0.283866	−	0.958864i	$-0.408383\pi$
$467$	5729.52	0.567732	0.283866	+	0.958864i	$0.408383\pi$
$468$	0	0
$469$	3090.25	0.304253
$470$	0	0
$471$	−10921.9	−1.06848
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−8148.06	−0.787071
$476$	0	0
$477$	1778.26	0.170694
$478$	0	0
$479$	−8428.60	−0.803993	−0.401996	−	0.915641i	$-0.631684\pi$
$479$	−8428.60	−0.803993	−0.401996	+	0.915641i	$0.631684\pi$
$480$	0	0
$481$	−6009.42	−0.569659
$482$	0	0
$483$	−669.487	−0.0630698
$484$	0	0
$485$	1548.87	0.145012
$486$	0	0
$487$	−1764.43	−0.164176	−0.0820880	−	0.996625i	$-0.526159\pi$
$487$	−1764.43	−0.164176	−0.0820880	+	0.996625i	$0.526159\pi$
$488$	0	0
$489$	−12404.0	−1.14709
$490$	0	0
$491$	5191.70	0.477185	0.238593	−	0.971120i	$-0.423314\pi$
$491$	5191.70	0.477185	0.238593	+	0.971120i	$0.423314\pi$
$492$	0	0
$493$	−12920.1	−1.18031
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2512.84	0.226793
$498$	0	0
$499$	−4552.88	−0.408447	−0.204224	−	0.978924i	$-0.565467\pi$
$499$	−4552.88	−0.408447	−0.204224	+	0.978924i	$0.565467\pi$
$500$	0	0
$501$	7989.72	0.712484
$502$	0	0
$503$	−5982.19	−0.530284	−0.265142	−	0.964209i	$-0.585419\pi$
$503$	−5982.19	−0.530284	−0.265142	+	0.964209i	$0.585419\pi$
$504$	0	0
$505$	−20458.6	−1.80276
$506$	0	0
$507$	7287.36	0.638349
$508$	0	0
$509$	−14809.0	−1.28959	−0.644793	−	0.764357i	$-0.723056\pi$
$509$	−14809.0	−1.28959	−0.644793	+	0.764357i	$0.723056\pi$
$510$	0	0
$511$	4516.76	0.391017
$512$	0	0
$513$	7596.66	0.653802
$514$	0	0
$515$	14177.4	1.21307
$516$	0	0
$517$	0	0
$518$	0	0
$519$	8568.61	0.724702
$520$	0	0
$521$	−14071.4	−1.18326	−0.591630	−	0.806210i	$-0.701515\pi$
$521$	−14071.4	−1.18326	−0.591630	+	0.806210i	$0.701515\pi$
$522$	0	0
$523$	−8061.03	−0.673966	−0.336983	−	0.941511i	$-0.609406\pi$
$523$	−8061.03	−0.673966	−0.336983	+	0.941511i	$0.609406\pi$
$524$	0	0
$525$	−2566.89	−0.213387
$526$	0	0
$527$	−16866.6	−1.39416
$528$	0	0
$529$	−10341.6	−0.849972
$530$	0	0
$531$	−4889.17	−0.399571
$532$	0	0
$533$	28211.3	2.29262
$534$	0	0
$535$	−6819.24	−0.551068
$536$	0	0
$537$	8324.90	0.668987
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2315.31	0.183998	0.0919989	−	0.995759i	$-0.470674\pi$
$541$	2315.31	0.183998	0.0919989	+	0.995759i	$0.470674\pi$
$542$	0	0
$543$	−3826.74	−0.302433
$544$	0	0
$545$	−20316.7	−1.59683
$546$	0	0
$547$	9940.92	0.777044	0.388522	−	0.921439i	$-0.372986\pi$
$547$	9940.92	0.777044	0.388522	+	0.921439i	$0.372986\pi$
$548$	0	0
$549$	3114.73	0.242137
$550$	0	0
$551$	−7704.31	−0.595671
$552$	0	0
$553$	−3985.84	−0.306501
$554$	0	0
$555$	6871.31	0.525533
$556$	0	0
$557$	−6849.75	−0.521065	−0.260533	−	0.965465i	$-0.583898\pi$
$557$	−6849.75	−0.521065	−0.260533	+	0.965465i	$0.583898\pi$
$558$	0	0
$559$	−21495.6	−1.62642
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23630.2	1.76891	0.884453	−	0.466629i	$-0.154532\pi$
$563$	23630.2	1.76891	0.884453	+	0.466629i	$0.154532\pi$
$564$	0	0
$565$	14805.2	1.10240
$566$	0	0
$567$	1466.95	0.108653
$568$	0	0
$569$	26066.9	1.92053	0.960266	−	0.279085i	$-0.0900311\pi$
$569$	26066.9	1.92053	0.960266	+	0.279085i	$0.0900311\pi$
$570$	0	0
$571$	10583.8	0.775688	0.387844	−	0.921725i	$-0.373220\pi$
$571$	10583.8	0.775688	0.387844	+	0.921725i	$0.373220\pi$
$572$	0	0
$573$	7043.88	0.513547
$574$	0	0
$575$	6998.78	0.507599
$576$	0	0
$577$	10687.3	0.771086	0.385543	−	0.922690i	$-0.374014\pi$
$577$	10687.3	0.771086	0.385543	+	0.922690i	$0.374014\pi$
$578$	0	0
$579$	10163.5	0.729497
$580$	0	0
$581$	−300.673	−0.0214699
$582$	0	0
$583$	0	0
$584$	0	0
$585$	9826.86	0.694514
$586$	0	0
$587$	9410.72	0.661707	0.330853	−	0.943682i	$-0.392663\pi$
$587$	9410.72	0.661707	0.330853	+	0.943682i	$0.392663\pi$
$588$	0	0
$589$	−10057.7	−0.703596
$590$	0	0
$591$	8290.27	0.577015
$592$	0	0
$593$	21740.9	1.50555	0.752776	−	0.658277i	$-0.228714\pi$
$593$	21740.9	1.50555	0.752776	+	0.658277i	$0.228714\pi$
$594$	0	0
$595$	5269.34	0.363062
$596$	0	0
$597$	−577.095	−0.0395627
$598$	0	0
$599$	−981.913	−0.0669781	−0.0334891	−	0.999439i	$-0.510662\pi$
$599$	−981.913	−0.0669781	−0.0334891	+	0.999439i	$0.510662\pi$
$600$	0	0
$601$	−7190.74	−0.488047	−0.244024	−	0.969769i	$-0.578467\pi$
$601$	−7190.74	−0.488047	−0.244024	+	0.969769i	$0.578467\pi$
$602$	0	0
$603$	7672.51	0.518157
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−13920.4	−0.930829	−0.465415	−	0.885093i	$-0.654095\pi$
$607$	−13920.4	−0.930829	−0.465415	+	0.885093i	$0.654095\pi$
$608$	0	0
$609$	−2427.10	−0.161496
$610$	0	0
$611$	24584.6	1.62780
$612$	0	0
$613$	26512.0	1.74684	0.873418	−	0.486971i	$-0.161898\pi$
$613$	26512.0	1.74684	0.873418	+	0.486971i	$0.161898\pi$
$614$	0	0
$615$	−32257.4	−2.11503
$616$	0	0
$617$	808.541	0.0527563	0.0263782	−	0.999652i	$-0.491603\pi$
$617$	808.541	0.0527563	0.0263782	+	0.999652i	$0.491603\pi$
$618$	0	0
$619$	4953.14	0.321621	0.160811	−	0.986985i	$-0.448589\pi$
$619$	4953.14	0.321621	0.160811	+	0.986985i	$0.448589\pi$
$620$	0	0
$621$	−6525.16	−0.421651
$622$	0	0
$623$	115.180	0.00740702
$624$	0	0
$625$	−9267.27	−0.593105
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−8000.51	−0.507156
$630$	0	0
$631$	12261.5	0.773567	0.386784	−	0.922170i	$-0.373586\pi$
$631$	12261.5	0.773567	0.386784	+	0.922170i	$0.373586\pi$
$632$	0	0
$633$	3816.41	0.239634
$634$	0	0
$635$	−14703.3	−0.918870
$636$	0	0
$637$	20625.0	1.28288
$638$	0	0
$639$	6238.91	0.386240
$640$	0	0
$641$	9977.35	0.614792	0.307396	−	0.951582i	$-0.400542\pi$
$641$	9977.35	0.614792	0.307396	+	0.951582i	$0.400542\pi$
$642$	0	0
$643$	3262.86	0.200116	0.100058	−	0.994982i	$-0.468097\pi$
$643$	3262.86	0.200116	0.100058	+	0.994982i	$0.468097\pi$
$644$	0	0
$645$	24578.6	1.50044
$646$	0	0
$647$	−25310.8	−1.53797	−0.768987	−	0.639264i	$-0.779239\pi$
$647$	−25310.8	−1.53797	−0.768987	+	0.639264i	$0.779239\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−3168.47	−0.190756
$652$	0	0
$653$	−2567.01	−0.153836	−0.0769180	−	0.997037i	$-0.524508\pi$
$653$	−2567.01	−0.153836	−0.0769180	+	0.997037i	$0.524508\pi$
$654$	0	0
$655$	−44160.9	−2.63437
$656$	0	0
$657$	11214.3	0.665921
$658$	0	0
$659$	21712.9	1.28348	0.641742	−	0.766920i	$-0.278212\pi$
$659$	21712.9	1.28348	0.641742	+	0.766920i	$0.278212\pi$
$660$	0	0
$661$	−9544.46	−0.561629	−0.280814	−	0.959762i	$-0.590605\pi$
$661$	−9544.46	−0.561629	−0.280814	+	0.959762i	$0.590605\pi$
$662$	0	0
$663$	22032.1	1.29058
$664$	0	0
$665$	3142.13	0.183228
$666$	0	0
$667$	6617.62	0.384161
$668$	0	0
$669$	4624.24	0.267240
$670$	0	0
$671$	0	0
$672$	0	0
$673$	23359.0	1.33792	0.668961	−	0.743297i	$-0.266739\pi$
$673$	23359.0	1.33792	0.668961	+	0.743297i	$0.266739\pi$
$674$	0	0
$675$	−25018.2	−1.42659
$676$	0	0
$677$	−343.591	−0.0195056	−0.00975278	−	0.999952i	$-0.503104\pi$
$677$	−343.591	−0.0195056	−0.00975278	+	0.999952i	$0.503104\pi$
$678$	0	0
$679$	−338.779	−0.0191475
$680$	0	0
$681$	14374.0	0.808828
$682$	0	0
$683$	−1268.81	−0.0710831	−0.0355415	−	0.999368i	$-0.511316\pi$
$683$	−1268.81	−0.0710831	−0.0355415	+	0.999368i	$0.511316\pi$
$684$	0	0
$685$	−14353.9	−0.800634
$686$	0	0
$687$	23333.8	1.29584
$688$	0	0
$689$	12072.6	0.667533
$690$	0	0
$691$	−6528.70	−0.359426	−0.179713	−	0.983719i	$-0.557517\pi$
$691$	−6528.70	−0.359426	−0.179713	+	0.983719i	$0.557517\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−21204.2	−1.15730
$696$	0	0
$697$	37558.4	2.04107
$698$	0	0
$699$	1006.04	0.0544378
$700$	0	0
$701$	−25058.5	−1.35014	−0.675070	−	0.737754i	$-0.735887\pi$
$701$	−25058.5	−1.35014	−0.675070	+	0.737754i	$0.735887\pi$
$702$	0	0
$703$	−4770.74	−0.255949
$704$	0	0
$705$	−28110.6	−1.50171
$706$	0	0
$707$	4474.81	0.238038
$708$	0	0
$709$	−29270.4	−1.55045	−0.775227	−	0.631683i	$-0.782364\pi$
$709$	−29270.4	−1.55045	−0.775227	+	0.631683i	$0.782364\pi$
$710$	0	0
$711$	−9896.08	−0.521986
$712$	0	0
$713$	8639.03	0.453765
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−112.950	−0.00588312
$718$	0	0
$719$	−14554.8	−0.754940	−0.377470	−	0.926022i	$-0.623206\pi$
$719$	−14554.8	−0.754940	−0.377470	+	0.926022i	$0.623206\pi$
$720$	0	0
$721$	−3100.97	−0.160175
$722$	0	0
$723$	−1857.45	−0.0955456
$724$	0	0
$725$	25372.7	1.29975
$726$	0	0
$727$	−25556.6	−1.30377	−0.651887	−	0.758316i	$-0.726022\pi$
$727$	−25556.6	−1.30377	−0.651887	+	0.758316i	$0.726022\pi$
$728$	0	0
$729$	21025.5	1.06821
$730$	0	0
$731$	−28617.7	−1.44797
$732$	0	0
$733$	−22890.8	−1.15347	−0.576733	−	0.816933i	$-0.695673\pi$
$733$	−22890.8	−1.15347	−0.576733	+	0.816933i	$0.695673\pi$
$734$	0	0
$735$	−23583.1	−1.18351
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−23671.7	−1.17832	−0.589161	−	0.808016i	$-0.700541\pi$
$739$	−23671.7	−1.17832	−0.589161	+	0.808016i	$0.700541\pi$
$740$	0	0
$741$	13137.9	0.651324
$742$	0	0
$743$	−8042.68	−0.397116	−0.198558	−	0.980089i	$-0.563626\pi$
$743$	−8042.68	−0.397116	−0.198558	+	0.980089i	$0.563626\pi$
$744$	0	0
$745$	−5512.97	−0.271114
$746$	0	0
$747$	−746.515	−0.0365643
$748$	0	0
$749$	1491.54	0.0727634
$750$	0	0
$751$	31048.1	1.50860	0.754302	−	0.656527i	$-0.227975\pi$
$751$	31048.1	1.50860	0.754302	+	0.656527i	$0.227975\pi$
$752$	0	0
$753$	−4476.02	−0.216620
$754$	0	0
$755$	31270.6	1.50736
$756$	0	0
$757$	2488.39	0.119474	0.0597372	−	0.998214i	$-0.480974\pi$
$757$	2488.39	0.119474	0.0597372	+	0.998214i	$0.480974\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−15606.5	−0.743408	−0.371704	−	0.928351i	$-0.621226\pi$
$761$	−15606.5	−0.743408	−0.371704	+	0.928351i	$0.621226\pi$
$762$	0	0
$763$	4443.78	0.210846
$764$	0	0
$765$	13082.8	0.618312
$766$	0	0
$767$	−33192.7	−1.56260
$768$	0	0
$769$	17212.1	0.807129	0.403565	−	0.914951i	$-0.367771\pi$
$769$	17212.1	0.807129	0.403565	+	0.914951i	$0.367771\pi$
$770$	0	0
$771$	11533.5	0.538742
$772$	0	0
$773$	−1251.01	−0.0582090	−0.0291045	−	0.999576i	$-0.509266\pi$
$773$	−1251.01	−0.0582090	−0.0291045	+	0.999576i	$0.509266\pi$
$774$	0	0
$775$	33123.0	1.53524
$776$	0	0
$777$	−1502.93	−0.0693918
$778$	0	0
$779$	22396.3	1.03008
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−23655.7	−1.07968
$784$	0	0
$785$	−44030.0	−2.00191
$786$	0	0
$787$	14423.8	0.653306	0.326653	−	0.945144i	$-0.394079\pi$
$787$	14423.8	0.653306	0.326653	+	0.945144i	$0.394079\pi$
$788$	0	0
$789$	−699.675	−0.0315704
$790$	0	0
$791$	−3238.27	−0.145562
$792$	0	0
$793$	21145.9	0.946927
$794$	0	0
$795$	−13804.1	−0.615826
$796$	0	0
$797$	−1350.83	−0.0600362	−0.0300181	−	0.999549i	$-0.509556\pi$
$797$	−1350.83	−0.0600362	−0.0300181	+	0.999549i	$0.509556\pi$
$798$	0	0
$799$	32730.1	1.44920
$800$	0	0
$801$	285.970	0.0126145
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−2698.94	−0.118168
$806$	0	0
$807$	−8753.66	−0.381838
$808$	0	0
$809$	−6954.69	−0.302242	−0.151121	−	0.988515i	$-0.548288\pi$
$809$	−6954.69	−0.302242	−0.151121	+	0.988515i	$0.548288\pi$
$810$	0	0
$811$	17155.8	0.742812	0.371406	−	0.928471i	$-0.378876\pi$
$811$	17155.8	0.742812	0.371406	+	0.928471i	$0.378876\pi$
$812$	0	0
$813$	−35516.9	−1.53214
$814$	0	0
$815$	−50004.9	−2.14920
$816$	0	0
$817$	−17064.9	−0.730753
$818$	0	0
$819$	−2149.39	−0.0917041
$820$	0	0
$821$	21672.1	0.921268	0.460634	−	0.887590i	$-0.347622\pi$
$821$	21672.1	0.921268	0.460634	+	0.887590i	$0.347622\pi$
$822$	0	0
$823$	−7925.66	−0.335688	−0.167844	−	0.985814i	$-0.553681\pi$
$823$	−7925.66	−0.335688	−0.167844	+	0.985814i	$0.553681\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38551.8	1.62101	0.810506	−	0.585731i	$-0.199192\pi$
$827$	38551.8	1.62101	0.810506	+	0.585731i	$0.199192\pi$
$828$	0	0
$829$	−6636.34	−0.278033	−0.139017	−	0.990290i	$-0.544394\pi$
$829$	−6636.34	−0.278033	−0.139017	+	0.990290i	$0.544394\pi$
$830$	0	0
$831$	10146.7	0.423569
$832$	0	0
$833$	27458.7	1.14212
$834$	0	0
$835$	32209.3	1.33491
$836$	0	0
$837$	−30881.5	−1.27529
$838$	0	0
$839$	46877.6	1.92896	0.964479	−	0.264159i	$-0.0850943\pi$
$839$	46877.6	1.92896	0.964479	+	0.264159i	$0.0850943\pi$
$840$	0	0
$841$	−398.075	−0.0163219
$842$	0	0
$843$	14109.0	0.576442
$844$	0	0
$845$	29377.9	1.19601
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−20191.3	−0.816213
$850$	0	0
$851$	4097.83	0.165067
$852$	0	0
$853$	−1699.59	−0.0682215	−0.0341108	−	0.999418i	$-0.510860\pi$
$853$	−1699.59	−0.0682215	−0.0341108	+	0.999418i	$0.510860\pi$
$854$	0	0
$855$	7801.32	0.312046
$856$	0	0
$857$	1506.84	0.0600615	0.0300308	−	0.999549i	$-0.490439\pi$
$857$	1506.84	0.0600615	0.0300308	+	0.999549i	$0.490439\pi$
$858$	0	0
$859$	−11830.1	−0.469892	−0.234946	−	0.972008i	$-0.575491\pi$
$859$	−11830.1	−0.469892	−0.234946	+	0.972008i	$0.575491\pi$
$860$	0	0
$861$	7055.52	0.279270
$862$	0	0
$863$	−40299.3	−1.58958	−0.794788	−	0.606887i	$-0.792418\pi$
$863$	−40299.3	−1.58958	−0.794788	+	0.606887i	$0.792418\pi$
$864$	0	0
$865$	34543.0	1.35780
$866$	0	0
$867$	8620.83	0.337692
$868$	0	0
$869$	0	0
$870$	0	0
$871$	52088.8	2.02636
$872$	0	0
$873$	−841.124	−0.0326091
$874$	0	0
$875$	−2451.73	−0.0947240
$876$	0	0
$877$	−12867.6	−0.495449	−0.247724	−	0.968831i	$-0.579683\pi$
$877$	−12867.6	−0.495449	−0.247724	+	0.968831i	$0.579683\pi$
$878$	0	0
$879$	−12944.8	−0.496719
$880$	0	0
$881$	−31803.0	−1.21620	−0.608098	−	0.793862i	$-0.708067\pi$
$881$	−31803.0	−1.21620	−0.608098	+	0.793862i	$0.708067\pi$
$882$	0	0
$883$	−33129.6	−1.26263	−0.631314	−	0.775527i	$-0.717484\pi$
$883$	−33129.6	−1.26263	−0.631314	+	0.775527i	$0.717484\pi$
$884$	0	0
$885$	37953.2	1.44156
$886$	0	0
$887$	11134.0	0.421471	0.210735	−	0.977543i	$-0.432414\pi$
$887$	11134.0	0.421471	0.210735	+	0.977543i	$0.432414\pi$
$888$	0	0
$889$	3215.99	0.121328
$890$	0	0
$891$	0	0
$892$	0	0
$893$	19517.1	0.731373
$894$	0	0
$895$	33560.5	1.25341
$896$	0	0
$897$	−11284.8	−0.420053
$898$	0	0
$899$	31319.1	1.16190
$900$	0	0
$901$	16072.6	0.594291
$902$	0	0
$903$	−5375.97	−0.198119
$904$	0	0
$905$	−15426.9	−0.566639
$906$	0	0
$907$	15892.9	0.581825	0.290913	−	0.956750i	$-0.406041\pi$
$907$	15892.9	0.581825	0.290913	+	0.956750i	$0.406041\pi$
$908$	0	0
$909$	11110.1	0.405390
$910$	0	0
$911$	44565.1	1.62075	0.810377	−	0.585909i	$-0.199262\pi$
$911$	44565.1	1.62075	0.810377	+	0.585909i	$0.199262\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−24178.7	−0.873578
$916$	0	0
$917$	9659.13	0.347843
$918$	0	0
$919$	−17058.6	−0.612310	−0.306155	−	0.951982i	$-0.599043\pi$
$919$	−17058.6	−0.612310	−0.306155	+	0.951982i	$0.599043\pi$
$920$	0	0
$921$	−39113.3	−1.39938
$922$	0	0
$923$	42356.1	1.51047
$924$	0	0
$925$	15711.6	0.558479
$926$	0	0
$927$	−7699.11	−0.272785
$928$	0	0
$929$	27486.2	0.970715	0.485358	−	0.874316i	$-0.338689\pi$
$929$	27486.2	0.970715	0.485358	+	0.874316i	$0.338689\pi$
$930$	0	0
$931$	16373.7	0.576399
$932$	0	0
$933$	−29370.9	−1.03061
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−2159.77	−0.0753007	−0.0376503	−	0.999291i	$-0.511987\pi$
$937$	−2159.77	−0.0753007	−0.0376503	+	0.999291i	$0.511987\pi$
$938$	0	0
$939$	−22539.7	−0.783340
$940$	0	0
$941$	−19886.4	−0.688925	−0.344462	−	0.938800i	$-0.611939\pi$
$941$	−19886.4	−0.688925	−0.344462	+	0.938800i	$0.611939\pi$
$942$	0	0
$943$	−19237.3	−0.664319
$944$	0	0
$945$	9647.75	0.332107
$946$	0	0
$947$	22957.8	0.787780	0.393890	−	0.919158i	$-0.371129\pi$
$947$	22957.8	0.787780	0.393890	+	0.919158i	$0.371129\pi$
$948$	0	0
$949$	76133.8	2.60422
$950$	0	0
$951$	33802.1	1.15259
$952$	0	0
$953$	2899.62	0.0985604	0.0492802	−	0.998785i	$-0.484307\pi$
$953$	2899.62	0.0985604	0.0492802	+	0.998785i	$0.484307\pi$
$954$	0	0
$955$	28396.3	0.962182
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3139.56	0.105716
$960$	0	0
$961$	11094.8	0.372421
$962$	0	0
$963$	3703.22	0.123920
$964$	0	0
$965$	40972.4	1.36679
$966$	0	0
$967$	33626.4	1.11825	0.559127	−	0.829082i	$-0.311136\pi$
$967$	33626.4	1.11825	0.559127	+	0.829082i	$0.311136\pi$
$968$	0	0
$969$	17490.8	0.579861
$970$	0	0
$971$	−47179.6	−1.55929	−0.779643	−	0.626225i	$-0.784599\pi$
$971$	−47179.6	−1.55929	−0.779643	+	0.626225i	$0.784599\pi$
$972$	0	0
$973$	4637.90	0.152810
$974$	0	0
$975$	−43267.1	−1.42119
$976$	0	0
$977$	−10162.1	−0.332769	−0.166384	−	0.986061i	$-0.553209\pi$
$977$	−10162.1	−0.332769	−0.166384	+	0.986061i	$0.553209\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	11033.1	0.359081
$982$	0	0
$983$	41639.0	1.35105	0.675523	−	0.737339i	$-0.263918\pi$
$983$	41639.0	1.35105	0.675523	+	0.737339i	$0.263918\pi$
$984$	0	0
$985$	33420.9	1.08110
$986$	0	0
$987$	6148.50	0.198287
$988$	0	0
$989$	14657.9	0.471278
$990$	0	0
$991$	23893.3	0.765888	0.382944	−	0.923772i	$-0.374910\pi$
$991$	23893.3	0.765888	0.382944	+	0.923772i	$0.374910\pi$
$992$	0	0
$993$	28405.8	0.907786
$994$	0	0
$995$	−2326.47	−0.0741247
$996$	0	0
$997$	59471.8	1.88916	0.944580	−	0.328282i	$-0.106470\pi$
$997$	59471.8	1.88916	0.944580	+	0.328282i	$0.106470\pi$
$998$	0	0
$999$	−14648.3	−0.463916

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1936.4.a.br.1.5		6
4.3	odd	2		484.4.a.i.1.2		6
11.5	even	5		176.4.m.d.113.3		12
11.9	even	5		176.4.m.d.81.3		12
11.10	odd	2		1936.4.a.bs.1.5		6
44.27	odd	10		44.4.e.a.25.1	✓	12
44.31	odd	10		44.4.e.a.37.1	yes	12
44.43	even	2		484.4.a.h.1.2		6
132.71	even	10		396.4.j.d.289.3		12
132.119	even	10		396.4.j.d.37.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.4.e.a.25.1	✓	12	44.27	odd	10
44.4.e.a.37.1	yes	12	44.31	odd	10
176.4.m.d.81.3		12	11.9	even	5
176.4.m.d.113.3		12	11.5	even	5
396.4.j.d.37.3		12	132.119	even	10
396.4.j.d.289.3		12	132.71	even	10
484.4.a.h.1.2		6	44.43	even	2
484.4.a.i.1.2		6	4.3	odd	2
1936.4.a.br.1.5		6	1.1	even	1	trivial
1936.4.a.bs.1.5		6	11.10	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+4.21558 q^{3} +16.9944 q^{5} -3.71712 q^{7} -9.22891 q^{9} -62.6552 q^{13} +71.6414 q^{15} -83.4146 q^{17} -49.7405 q^{19} -15.6698 q^{21} +42.7247 q^{23} +163.811 q^{25} -152.726 q^{27} +154.890 q^{29} +202.202 q^{31} -63.1704 q^{35} +95.9126 q^{37} -264.128 q^{39} -450.262 q^{41} +343.078 q^{43} -156.840 q^{45} -392.379 q^{47} -329.183 q^{49} -351.641 q^{51} -192.684 q^{53} -209.685 q^{57} +529.767 q^{59} -337.496 q^{61} +34.3050 q^{63} -1064.79 q^{65} -831.356 q^{67} +180.109 q^{69} -676.018 q^{71} -1215.12 q^{73} +690.559 q^{75} +1072.29 q^{79} -394.646 q^{81} +80.8887 q^{83} -1417.58 q^{85} +652.951 q^{87} -30.9863 q^{89} +232.897 q^{91} +852.400 q^{93} -845.313 q^{95} +91.1400 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - 12 q^{5} - 8 q^{7} + 11 q^{9} - 80 q^{13} + 98 q^{15} - 113 q^{17} + 53 q^{19} - 152 q^{21} + 194 q^{23} + 476 q^{25} - 72 q^{27} - 374 q^{29} - 16 q^{31} + 1044 q^{35} - 456 q^{37} + 592 q^{39}+ \cdots + 683 q^{97}+O(q^{100}) \) 6 * q - 3 * q^3 - 12 * q^5 - 8 * q^7 + 11 * q^9 - 80 * q^13 + 98 * q^15 - 113 * q^17 + 53 * q^19 - 152 * q^21 + 194 * q^23 + 476 * q^25 - 72 * q^27 - 374 * q^29 - 16 * q^31 + 1044 * q^35 - 456 * q^37 + 592 * q^39 - 825 * q^41 + 1165 * q^43 - 670 * q^45 - 1248 * q^47 - 4 * q^49 + 988 * q^51 - 122 * q^53 - 2282 * q^57 + 767 * q^59 - 1010 * q^61 + 254 * q^63 - 1946 * q^65 + 2197 * q^67 + 2648 * q^69 - 598 * q^71 - 1523 * q^73 - 609 * q^75 + 2116 * q^79 - 2106 * q^81 + 1213 * q^83 - 1066 * q^85 + 1518 * q^87 - 477 * q^89 - 2420 * q^91 - 714 * q^93 + 404 * q^95 + 683 * q^97

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