LMFDB - Embedded newform 1200.4.a.br.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(1,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1200.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.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:	$70.8022920069$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{109}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 27 $ x^2 - x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 600)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.72015$ of defining polynomial
Character	$\chi$	$=$	1200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +19.8806 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +19.8806 q^{7} +9.00000 q^{9} +70.6418 q^{11} -0.761226 q^{13} +108.403 q^{17} +125.881 q^{19} +59.6418 q^{21} -40.8806 q^{23} +27.0000 q^{27} -140.881 q^{29} -296.926 q^{31} +211.926 q^{33} -26.8061 q^{37} -2.28368 q^{39} -20.6418 q^{41} -32.5969 q^{43} -11.4326 q^{47} +52.2388 q^{49} +325.209 q^{51} +159.358 q^{53} +377.642 q^{57} +374.955 q^{59} +303.478 q^{61} +178.926 q^{63} -877.583 q^{67} -122.642 q^{69} -1159.54 q^{71} +120.716 q^{73} +1404.40 q^{77} +1248.66 q^{79} +81.0000 q^{81} -654.180 q^{83} -422.642 q^{87} +795.522 q^{89} -15.1336 q^{91} -890.777 q^{93} +850.910 q^{97} +635.777 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 - 2 * q^7 + 18 * q^9	$ 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9} + 16 q^{11} + 82 q^{13} + 8 q^{17} + 210 q^{19} - 6 q^{21} - 40 q^{23} + 54 q^{27} - 240 q^{29} - 218 q^{31} + 48 q^{33} + 364 q^{37} + 246 q^{39} + 84 q^{41} - 274 q^{43} - 524 q^{47} + 188 q^{49} + 24 q^{51} + 444 q^{53} + 630 q^{57} + 1084 q^{59} + 774 q^{61} - 18 q^{63} - 210 q^{67} - 120 q^{69} - 1108 q^{71} + 492 q^{73} + 2600 q^{77} + 1328 q^{79} + 162 q^{81} + 28 q^{83} - 720 q^{87} + 1424 q^{89} - 1826 q^{91} - 654 q^{93} + 2370 q^{97} + 144 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 2 * q^7 + 18 * q^9 + 16 * q^11 + 82 * q^13 + 8 * q^17 + 210 * q^19 - 6 * q^21 - 40 * q^23 + 54 * q^27 - 240 * q^29 - 218 * q^31 + 48 * q^33 + 364 * q^37 + 246 * q^39 + 84 * q^41 - 274 * q^43 - 524 * q^47 + 188 * q^49 + 24 * q^51 + 444 * q^53 + 630 * q^57 + 1084 * q^59 + 774 * q^61 - 18 * q^63 - 210 * q^67 - 120 * q^69 - 1108 * q^71 + 492 * q^73 + 2600 * q^77 + 1328 * q^79 + 162 * q^81 + 28 * q^83 - 720 * q^87 + 1424 * q^89 - 1826 * q^91 - 654 * q^93 + 2370 * q^97 + 144 * q^99

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	19.8806	1.07345	0.536726	−	0.843757i	$-0.319661\pi$
$7$	19.8806	1.07345	0.536726	+	0.843757i	$0.319661\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	70.6418	1.93630	0.968151	−	0.250368i	$-0.0805516\pi$
$11$	70.6418	1.93630	0.968151	+	0.250368i	$0.0805516\pi$
$12$	0	0
$13$	−0.761226	−0.0162405	−0.00812024	−	0.999967i	$-0.502585\pi$
$13$	−0.761226	−0.0162405	−0.00812024	+	0.999967i	$0.502585\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	108.403	1.54657	0.773283	−	0.634062i	$-0.218613\pi$
$17$	108.403	1.54657	0.773283	+	0.634062i	$0.218613\pi$
$18$	0	0
$19$	125.881	1.51995	0.759974	−	0.649954i	$-0.225212\pi$
$19$	125.881	1.51995	0.759974	+	0.649954i	$0.225212\pi$
$20$	0	0
$21$	59.6418	0.619758
$22$	0	0
$23$	−40.8806	−0.370617	−0.185309	−	0.982680i	$-0.559328\pi$
$23$	−40.8806	−0.370617	−0.185309	+	0.982680i	$0.559328\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−140.881	−0.902099	−0.451050	−	0.892499i	$-0.648950\pi$
$29$	−140.881	−0.902099	−0.451050	+	0.892499i	$0.648950\pi$
$30$	0	0
$31$	−296.926	−1.72030	−0.860152	−	0.510039i	$-0.829631\pi$
$31$	−296.926	−1.72030	−0.860152	+	0.510039i	$0.829631\pi$
$32$	0	0
$33$	211.926	1.11792
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−26.8061	−0.119105	−0.0595527	−	0.998225i	$-0.518967\pi$
$37$	−26.8061	−0.119105	−0.0595527	+	0.998225i	$0.518967\pi$
$38$	0	0
$39$	−2.28368	−0.00937644
$40$	0	0
$41$	−20.6418	−0.0786272	−0.0393136	−	0.999227i	$-0.512517\pi$
$41$	−20.6418	−0.0786272	−0.0393136	+	0.999227i	$0.512517\pi$
$42$	0	0
$43$	−32.5969	−0.115604	−0.0578022	−	0.998328i	$-0.518409\pi$
$43$	−32.5969	−0.115604	−0.0578022	+	0.998328i	$0.518409\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.4326	−0.0354813	−0.0177407	−	0.999843i	$-0.505647\pi$
$47$	−11.4326	−0.0354813	−0.0177407	+	0.999843i	$0.505647\pi$
$48$	0	0
$49$	52.2388	0.152300
$50$	0	0
$51$	325.209	0.892910
$52$	0	0
$53$	159.358	0.413010	0.206505	−	0.978446i	$-0.433791\pi$
$53$	159.358	0.413010	0.206505	+	0.978446i	$0.433791\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	377.642	0.877542
$58$	0	0
$59$	374.955	0.827373	0.413686	−	0.910419i	$-0.364241\pi$
$59$	374.955	0.827373	0.413686	+	0.910419i	$0.364241\pi$
$60$	0	0
$61$	303.478	0.636989	0.318494	−	0.947925i	$-0.396823\pi$
$61$	303.478	0.636989	0.318494	+	0.947925i	$0.396823\pi$
$62$	0	0
$63$	178.926	0.357817
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−877.583	−1.60021	−0.800103	−	0.599863i	$-0.795222\pi$
$67$	−877.583	−1.60021	−0.800103	+	0.599863i	$0.795222\pi$
$68$	0	0
$69$	−122.642	−0.213976
$70$	0	0
$71$	−1159.54	−1.93819	−0.969097	−	0.246679i	$-0.920661\pi$
$71$	−1159.54	−1.93819	−0.969097	+	0.246679i	$0.920661\pi$
$72$	0	0
$73$	120.716	0.193545	0.0967724	−	0.995307i	$-0.469148\pi$
$73$	120.716	0.193545	0.0967724	+	0.995307i	$0.469148\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1404.40	2.07853
$78$	0	0
$79$	1248.66	1.77829	0.889145	−	0.457626i	$-0.151300\pi$
$79$	1248.66	1.77829	0.889145	+	0.457626i	$0.151300\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−654.180	−0.865127	−0.432564	−	0.901603i	$-0.642391\pi$
$83$	−654.180	−0.865127	−0.432564	+	0.901603i	$0.642391\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−422.642	−0.520827
$88$	0	0
$89$	795.522	0.947474	0.473737	−	0.880666i	$-0.342905\pi$
$89$	795.522	0.947474	0.473737	+	0.880666i	$0.342905\pi$
$90$	0	0
$91$	−15.1336	−0.0174334
$92$	0	0
$93$	−890.777	−0.993217
$94$	0	0
$95$	0	0
$96$	0	0
$97$	850.910	0.890689	0.445345	−	0.895359i	$-0.353081\pi$
$97$	850.910	0.890689	0.445345	+	0.895359i	$0.353081\pi$
$98$	0	0
$99$	635.777	0.645434
$100$	0	0
$101$	−149.777	−0.147558	−0.0737788	−	0.997275i	$-0.523506\pi$
$101$	−149.777	−0.147558	−0.0737788	+	0.997275i	$0.523506\pi$
$102$	0	0
$103$	−1176.81	−1.12577	−0.562884	−	0.826536i	$-0.690308\pi$
$103$	−1176.81	−1.12577	−0.562884	+	0.826536i	$0.690308\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−402.552	−0.363703	−0.181851	−	0.983326i	$-0.558209\pi$
$107$	−402.552	−0.363703	−0.181851	+	0.983326i	$0.558209\pi$
$108$	0	0
$109$	1650.31	1.45020	0.725098	−	0.688645i	$-0.241794\pi$
$109$	1650.31	1.45020	0.725098	+	0.688645i	$0.241794\pi$
$110$	0	0
$111$	−80.4184	−0.0687655
$112$	0	0
$113$	−1353.13	−1.12648	−0.563240	−	0.826293i	$-0.690445\pi$
$113$	−1353.13	−1.12648	−0.563240	+	0.826293i	$0.690445\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−6.85103	−0.00541349
$118$	0	0
$119$	2155.12	1.66016
$120$	0	0
$121$	3659.27	2.74926
$122$	0	0
$123$	−61.9255	−0.0453954
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1836.21	1.28297	0.641485	−	0.767135i	$-0.278318\pi$
$127$	1836.21	1.28297	0.641485	+	0.767135i	$0.278318\pi$
$128$	0	0
$129$	−97.7908	−0.0667442
$130$	0	0
$131$	1111.98	0.741638	0.370819	−	0.928705i	$-0.379077\pi$
$131$	1111.98	0.741638	0.370819	+	0.928705i	$0.379077\pi$
$132$	0	0
$133$	2502.58	1.63159
$134$	0	0
$135$	0	0
$136$	0	0
$137$	567.284	0.353769	0.176884	−	0.984232i	$-0.443398\pi$
$137$	567.284	0.353769	0.176884	+	0.984232i	$0.443398\pi$
$138$	0	0
$139$	−1968.87	−1.20142	−0.600709	−	0.799468i	$-0.705115\pi$
$139$	−1968.87	−1.20142	−0.600709	+	0.799468i	$0.705115\pi$
$140$	0	0
$141$	−34.2979	−0.0204852
$142$	0	0
$143$	−53.7744	−0.0314464
$144$	0	0
$145$	0	0
$146$	0	0
$147$	156.716	0.0879302
$148$	0	0
$149$	−1810.87	−0.995651	−0.497826	−	0.867277i	$-0.665868\pi$
$149$	−1810.87	−0.995651	−0.497826	+	0.867277i	$0.665868\pi$
$150$	0	0
$151$	−266.746	−0.143758	−0.0718791	−	0.997413i	$-0.522900\pi$
$151$	−266.746	−0.143758	−0.0718791	+	0.997413i	$0.522900\pi$
$152$	0	0
$153$	975.628	0.515522
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2993.33	1.52162	0.760808	−	0.648977i	$-0.224803\pi$
$157$	2993.33	1.52162	0.760808	+	0.648977i	$0.224803\pi$
$158$	0	0
$159$	478.074	0.238451
$160$	0	0
$161$	−812.732	−0.397840
$162$	0	0
$163$	−2356.36	−1.13230	−0.566148	−	0.824304i	$-0.691567\pi$
$163$	−2356.36	−1.13230	−0.566148	+	0.824304i	$0.691567\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3148.55	−1.45894	−0.729468	−	0.684015i	$-0.760232\pi$
$167$	−3148.55	−1.45894	−0.729468	+	0.684015i	$0.760232\pi$
$168$	0	0
$169$	−2196.42	−0.999736
$170$	0	0
$171$	1132.93	0.506649
$172$	0	0
$173$	−3747.12	−1.64675	−0.823376	−	0.567496i	$-0.807912\pi$
$173$	−3747.12	−1.64675	−0.823376	+	0.567496i	$0.807912\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1124.87	0.477684
$178$	0	0
$179$	2309.37	0.964305	0.482153	−	0.876087i	$-0.339855\pi$
$179$	2309.37	0.964305	0.482153	+	0.876087i	$0.339855\pi$
$180$	0	0
$181$	257.447	0.105723	0.0528615	−	0.998602i	$-0.483166\pi$
$181$	257.447	0.105723	0.0528615	+	0.998602i	$0.483166\pi$
$182$	0	0
$183$	910.433	0.367766
$184$	0	0
$185$	0	0
$186$	0	0
$187$	7657.79	2.99462
$188$	0	0
$189$	536.777	0.206586
$190$	0	0
$191$	4184.82	1.58536	0.792679	−	0.609640i	$-0.208686\pi$
$191$	4184.82	1.58536	0.792679	+	0.609640i	$0.208686\pi$
$192$	0	0
$193$	−3636.79	−1.35638	−0.678192	−	0.734885i	$-0.737236\pi$
$193$	−3636.79	−1.35638	−0.678192	+	0.734885i	$0.737236\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3519.52	−1.27287	−0.636436	−	0.771330i	$-0.719592\pi$
$197$	−3519.52	−1.27287	−0.636436	+	0.771330i	$0.719592\pi$
$198$	0	0
$199$	1937.28	0.690103	0.345051	−	0.938584i	$-0.387861\pi$
$199$	1937.28	0.690103	0.345051	+	0.938584i	$0.387861\pi$
$200$	0	0
$201$	−2632.75	−0.923879
$202$	0	0
$203$	−2800.79	−0.968360
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−367.926	−0.123539
$208$	0	0
$209$	8892.44	2.94308
$210$	0	0
$211$	−2928.63	−0.955522	−0.477761	−	0.878490i	$-0.658552\pi$
$211$	−2928.63	−0.955522	−0.477761	+	0.878490i	$0.658552\pi$
$212$	0	0
$213$	−3478.61	−1.11902
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5903.06	−1.84666
$218$	0	0
$219$	362.149	0.111743
$220$	0	0
$221$	−82.5192	−0.0251169
$222$	0	0
$223$	−2637.07	−0.791890	−0.395945	−	0.918274i	$-0.629583\pi$
$223$	−2637.07	−0.791890	−0.395945	+	0.918274i	$0.629583\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3287.27	0.961162	0.480581	−	0.876950i	$-0.340426\pi$
$227$	3287.27	0.961162	0.480581	+	0.876950i	$0.340426\pi$
$228$	0	0
$229$	6382.02	1.84164	0.920820	−	0.389988i	$-0.127521\pi$
$229$	6382.02	1.84164	0.920820	+	0.389988i	$0.127521\pi$
$230$	0	0
$231$	4213.21	1.20004
$232$	0	0
$233$	2952.91	0.830265	0.415133	−	0.909761i	$-0.363735\pi$
$233$	2952.91	0.830265	0.415133	+	0.909761i	$0.363735\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	3745.97	1.02670
$238$	0	0
$239$	1384.34	0.374668	0.187334	−	0.982296i	$-0.440015\pi$
$239$	1384.34	0.374668	0.187334	+	0.982296i	$0.440015\pi$
$240$	0	0
$241$	4236.82	1.13244	0.566219	−	0.824255i	$-0.308406\pi$
$241$	4236.82	1.13244	0.566219	+	0.824255i	$0.308406\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−95.8236	−0.0246847
$248$	0	0
$249$	−1962.54	−0.499481
$250$	0	0
$251$	1861.22	0.468045	0.234023	−	0.972231i	$-0.424811\pi$
$251$	1861.22	0.468045	0.234023	+	0.972231i	$0.424811\pi$
$252$	0	0
$253$	−2887.88	−0.717627
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−347.251	−0.0842837	−0.0421419	−	0.999112i	$-0.513418\pi$
$257$	−347.251	−0.0842837	−0.0421419	+	0.999112i	$0.513418\pi$
$258$	0	0
$259$	−532.922	−0.127854
$260$	0	0
$261$	−1267.93	−0.300700
$262$	0	0
$263$	1489.39	0.349200	0.174600	−	0.984639i	$-0.444137\pi$
$263$	1489.39	0.349200	0.174600	+	0.984639i	$0.444137\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2386.57	0.547025
$268$	0	0
$269$	1570.88	0.356053	0.178026	−	0.984026i	$-0.443029\pi$
$269$	1570.88	0.356053	0.178026	+	0.984026i	$0.443029\pi$
$270$	0	0
$271$	−3014.42	−0.675693	−0.337847	−	0.941201i	$-0.609699\pi$
$271$	−3014.42	−0.675693	−0.337847	+	0.941201i	$0.609699\pi$
$272$	0	0
$273$	−45.4009	−0.0100652
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2792.41	−0.605702	−0.302851	−	0.953038i	$-0.597938\pi$
$277$	−2792.41	−0.605702	−0.302851	+	0.953038i	$0.597938\pi$
$278$	0	0
$279$	−2672.33	−0.573434
$280$	0	0
$281$	−1745.81	−0.370626	−0.185313	−	0.982680i	$-0.559330\pi$
$281$	−1745.81	−0.370626	−0.185313	+	0.982680i	$0.559330\pi$
$282$	0	0
$283$	2664.33	0.559639	0.279820	−	0.960053i	$-0.409725\pi$
$283$	2664.33	0.559639	0.279820	+	0.960053i	$0.409725\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−410.372	−0.0844025
$288$	0	0
$289$	6838.22	1.39186
$290$	0	0
$291$	2552.73	0.514240
$292$	0	0
$293$	−5927.79	−1.18193	−0.590965	−	0.806697i	$-0.701253\pi$
$293$	−5927.79	−1.18193	−0.590965	+	0.806697i	$0.701253\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1907.33	0.372641
$298$	0	0
$299$	31.1194	0.00601900
$300$	0	0
$301$	−648.047	−0.124096
$302$	0	0
$303$	−449.330	−0.0851925
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4634.48	0.861575	0.430788	−	0.902453i	$-0.358236\pi$
$307$	4634.48	0.861575	0.430788	+	0.902453i	$0.358236\pi$
$308$	0	0
$309$	−3530.42	−0.649963
$310$	0	0
$311$	−761.969	−0.138930	−0.0694651	−	0.997584i	$-0.522129\pi$
$311$	−761.969	−0.138930	−0.0694651	+	0.997584i	$0.522129\pi$
$312$	0	0
$313$	8734.32	1.57729	0.788647	−	0.614847i	$-0.210782\pi$
$313$	8734.32	1.57729	0.788647	+	0.614847i	$0.210782\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4428.21	−0.784584	−0.392292	−	0.919841i	$-0.628318\pi$
$317$	−4428.21	−0.784584	−0.392292	+	0.919841i	$0.628318\pi$
$318$	0	0
$319$	−9952.07	−1.74674
$320$	0	0
$321$	−1207.66	−0.209984
$322$	0	0
$323$	13645.8	2.35070
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4950.94	0.837271
$328$	0	0
$329$	−227.288	−0.0380875
$330$	0	0
$331$	2239.68	0.371915	0.185957	−	0.982558i	$-0.440461\pi$
$331$	2239.68	0.371915	0.185957	+	0.982558i	$0.440461\pi$
$332$	0	0
$333$	−241.255	−0.0397018
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−8459.21	−1.36737	−0.683683	−	0.729779i	$-0.739623\pi$
$337$	−8459.21	−1.36737	−0.683683	+	0.729779i	$0.739623\pi$
$338$	0	0
$339$	−4059.40	−0.650373
$340$	0	0
$341$	−20975.4	−3.33103
$342$	0	0
$343$	−5780.51	−0.909966
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9253.53	−1.43157	−0.715786	−	0.698320i	$-0.753931\pi$
$347$	−9253.53	−1.43157	−0.715786	+	0.698320i	$0.753931\pi$
$348$	0	0
$349$	−1630.53	−0.250087	−0.125044	−	0.992151i	$-0.539907\pi$
$349$	−1630.53	−0.250087	−0.125044	+	0.992151i	$0.539907\pi$
$350$	0	0
$351$	−20.5531	−0.00312548
$352$	0	0
$353$	−8148.98	−1.22869	−0.614343	−	0.789039i	$-0.710579\pi$
$353$	−8148.98	−1.22869	−0.614343	+	0.789039i	$0.710579\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6465.36	0.958496
$358$	0	0
$359$	5850.67	0.860130	0.430065	−	0.902798i	$-0.358491\pi$
$359$	5850.67	0.860130	0.430065	+	0.902798i	$0.358491\pi$
$360$	0	0
$361$	8986.93	1.31024
$362$	0	0
$363$	10977.8	1.58729
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7205.40	1.02485	0.512424	−	0.858733i	$-0.328748\pi$
$367$	7205.40	1.02485	0.512424	+	0.858733i	$0.328748\pi$
$368$	0	0
$369$	−185.777	−0.0262091
$370$	0	0
$371$	3168.14	0.443346
$372$	0	0
$373$	8708.52	1.20887	0.604437	−	0.796653i	$-0.293398\pi$
$373$	8708.52	1.20887	0.604437	+	0.796653i	$0.293398\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	107.242	0.0146505
$378$	0	0
$379$	4153.83	0.562975	0.281488	−	0.959565i	$-0.409172\pi$
$379$	4153.83	0.562975	0.281488	+	0.959565i	$0.409172\pi$
$380$	0	0
$381$	5508.63	0.740724
$382$	0	0
$383$	−10362.0	−1.38244	−0.691219	−	0.722645i	$-0.742926\pi$
$383$	−10362.0	−1.38244	−0.691219	+	0.722645i	$0.742926\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−293.372	−0.0385348
$388$	0	0
$389$	−3056.19	−0.398343	−0.199171	−	0.979965i	$-0.563825\pi$
$389$	−3056.19	−0.398343	−0.199171	+	0.979965i	$0.563825\pi$
$390$	0	0
$391$	−4431.58	−0.573184
$392$	0	0
$393$	3335.95	0.428185
$394$	0	0
$395$	0	0
$396$	0	0
$397$	14305.4	1.80849	0.904243	−	0.427019i	$-0.140436\pi$
$397$	14305.4	1.80849	0.904243	+	0.427019i	$0.140436\pi$
$398$	0	0
$399$	7507.75	0.941999
$400$	0	0
$401$	11391.2	1.41858	0.709289	−	0.704917i	$-0.249016\pi$
$401$	11391.2	1.41858	0.709289	+	0.704917i	$0.249016\pi$
$402$	0	0
$403$	226.027	0.0279385
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1893.63	−0.230624
$408$	0	0
$409$	−5689.18	−0.687804	−0.343902	−	0.939006i	$-0.611749\pi$
$409$	−5689.18	−0.687804	−0.343902	+	0.939006i	$0.611749\pi$
$410$	0	0
$411$	1701.85	0.204248
$412$	0	0
$413$	7454.34	0.888145
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−5906.60	−0.693639
$418$	0	0
$419$	−2086.17	−0.243236	−0.121618	−	0.992577i	$-0.538808\pi$
$419$	−2086.17	−0.243236	−0.121618	+	0.992577i	$0.538808\pi$
$420$	0	0
$421$	661.557	0.0765851	0.0382926	−	0.999267i	$-0.487808\pi$
$421$	661.557	0.0765851	0.0382926	+	0.999267i	$0.487808\pi$
$422$	0	0
$423$	−102.894	−0.0118271
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6033.32	0.683777
$428$	0	0
$429$	−161.323	−0.0181556
$430$	0	0
$431$	−14508.1	−1.62142	−0.810709	−	0.585449i	$-0.800918\pi$
$431$	−14508.1	−1.62142	−0.810709	+	0.585449i	$0.800918\pi$
$432$	0	0
$433$	11584.2	1.28568	0.642840	−	0.766000i	$-0.277756\pi$
$433$	11584.2	1.28568	0.642840	+	0.766000i	$0.277756\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5146.08	−0.563319
$438$	0	0
$439$	−10554.6	−1.14748	−0.573740	−	0.819038i	$-0.694508\pi$
$439$	−10554.6	−1.14748	−0.573740	+	0.819038i	$0.694508\pi$
$440$	0	0
$441$	470.149	0.0507665
$442$	0	0
$443$	−14791.1	−1.58634	−0.793169	−	0.609001i	$-0.791571\pi$
$443$	−14791.1	−1.58634	−0.793169	+	0.609001i	$0.791571\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−5432.60	−0.574840
$448$	0	0
$449$	540.276	0.0567866	0.0283933	−	0.999597i	$-0.490961\pi$
$449$	540.276	0.0567866	0.0283933	+	0.999597i	$0.490961\pi$
$450$	0	0
$451$	−1458.18	−0.152246
$452$	0	0
$453$	−800.238	−0.0829988
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8022.12	0.821136	0.410568	−	0.911830i	$-0.365330\pi$
$457$	8022.12	0.821136	0.410568	+	0.911830i	$0.365330\pi$
$458$	0	0
$459$	2926.88	0.297637
$460$	0	0
$461$	−11830.8	−1.19526	−0.597632	−	0.801771i	$-0.703891\pi$
$461$	−11830.8	−1.19526	−0.597632	+	0.801771i	$0.703891\pi$
$462$	0	0
$463$	12268.5	1.23145	0.615727	−	0.787959i	$-0.288862\pi$
$463$	12268.5	1.23145	0.615727	+	0.787959i	$0.288862\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12567.2	−1.24526	−0.622632	−	0.782515i	$-0.713937\pi$
$467$	−12567.2	−1.24526	−0.622632	+	0.782515i	$0.713937\pi$
$468$	0	0
$469$	−17446.9	−1.71774
$470$	0	0
$471$	8979.99	0.878506
$472$	0	0
$473$	−2302.71	−0.223845
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1434.22	0.137670
$478$	0	0
$479$	5686.36	0.542414	0.271207	−	0.962521i	$-0.412577\pi$
$479$	5686.36	0.542414	0.271207	+	0.962521i	$0.412577\pi$
$480$	0	0
$481$	20.4055	0.00193433
$482$	0	0
$483$	−2438.19	−0.229693
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−9429.19	−0.877367	−0.438683	−	0.898642i	$-0.644555\pi$
$487$	−9429.19	−0.877367	−0.438683	+	0.898642i	$0.644555\pi$
$488$	0	0
$489$	−7069.07	−0.653731
$490$	0	0
$491$	−19869.3	−1.82625	−0.913126	−	0.407677i	$-0.866339\pi$
$491$	−19869.3	−1.82625	−0.913126	+	0.407677i	$0.866339\pi$
$492$	0	0
$493$	−15271.9	−1.39515
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−23052.3	−2.08056
$498$	0	0
$499$	−4160.66	−0.373260	−0.186630	−	0.982430i	$-0.559757\pi$
$499$	−4160.66	−0.373260	−0.186630	+	0.982430i	$0.559757\pi$
$500$	0	0
$501$	−9445.66	−0.842318
$502$	0	0
$503$	9817.15	0.870229	0.435114	−	0.900375i	$-0.356708\pi$
$503$	9817.15	0.870229	0.435114	+	0.900375i	$0.356708\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6589.26	−0.577198
$508$	0	0
$509$	−8709.02	−0.758390	−0.379195	−	0.925317i	$-0.623799\pi$
$509$	−8709.02	−0.758390	−0.379195	+	0.925317i	$0.623799\pi$
$510$	0	0
$511$	2399.91	0.207761
$512$	0	0
$513$	3398.78	0.292514
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−807.623	−0.0687025
$518$	0	0
$519$	−11241.4	−0.950753
$520$	0	0
$521$	−1711.84	−0.143948	−0.0719742	−	0.997406i	$-0.522930\pi$
$521$	−1711.84	−0.143948	−0.0719742	+	0.997406i	$0.522930\pi$
$522$	0	0
$523$	1901.43	0.158975	0.0794875	−	0.996836i	$-0.474672\pi$
$523$	1901.43	0.158975	0.0794875	+	0.996836i	$0.474672\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−32187.6	−2.66056
$528$	0	0
$529$	−10495.8	−0.862643
$530$	0	0
$531$	3374.60	0.275791
$532$	0	0
$533$	15.7131	0.00127694
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6928.12	0.556742
$538$	0	0
$539$	3690.24	0.294898
$540$	0	0
$541$	10469.9	0.832043	0.416021	−	0.909355i	$-0.363424\pi$
$541$	10469.9	0.832043	0.416021	+	0.909355i	$0.363424\pi$
$542$	0	0
$543$	772.341	0.0610392
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−23357.6	−1.82578	−0.912889	−	0.408207i	$-0.866154\pi$
$547$	−23357.6	−1.82578	−0.912889	+	0.408207i	$0.866154\pi$
$548$	0	0
$549$	2731.30	0.212330
$550$	0	0
$551$	−17734.1	−1.37114
$552$	0	0
$553$	24824.1	1.90891
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7925.47	0.602896	0.301448	−	0.953483i	$-0.402530\pi$
$557$	7925.47	0.602896	0.301448	+	0.953483i	$0.402530\pi$
$558$	0	0
$559$	24.8136	0.00187747
$560$	0	0
$561$	22973.4	1.72894
$562$	0	0
$563$	−11846.1	−0.886770	−0.443385	−	0.896331i	$-0.646223\pi$
$563$	−11846.1	−0.886770	−0.443385	+	0.896331i	$0.646223\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1610.33	0.119272
$568$	0	0
$569$	5807.33	0.427866	0.213933	−	0.976848i	$-0.431373\pi$
$569$	5807.33	0.427866	0.213933	+	0.976848i	$0.431373\pi$
$570$	0	0
$571$	2520.62	0.184737	0.0923685	−	0.995725i	$-0.470556\pi$
$571$	2520.62	0.184737	0.0923685	+	0.995725i	$0.470556\pi$
$572$	0	0
$573$	12554.5	0.915306
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8634.70	0.622994	0.311497	−	0.950247i	$-0.399170\pi$
$577$	8634.70	0.622994	0.311497	+	0.950247i	$0.399170\pi$
$578$	0	0
$579$	−10910.4	−0.783109
$580$	0	0
$581$	−13005.5	−0.928672
$582$	0	0
$583$	11257.4	0.799712
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22165.7	−1.55856	−0.779282	−	0.626674i	$-0.784416\pi$
$587$	−22165.7	−1.55856	−0.779282	+	0.626674i	$0.784416\pi$
$588$	0	0
$589$	−37377.2	−2.61477
$590$	0	0
$591$	−10558.6	−0.734893
$592$	0	0
$593$	17874.0	1.23777	0.618883	−	0.785483i	$-0.287585\pi$
$593$	17874.0	1.23777	0.618883	+	0.785483i	$0.287585\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5811.85	0.398431
$598$	0	0
$599$	−11051.4	−0.753839	−0.376919	−	0.926246i	$-0.623017\pi$
$599$	−11051.4	−0.753839	−0.376919	+	0.926246i	$0.623017\pi$
$600$	0	0
$601$	17452.3	1.18452	0.592258	−	0.805749i	$-0.298237\pi$
$601$	17452.3	1.18452	0.592258	+	0.805749i	$0.298237\pi$
$602$	0	0
$603$	−7898.24	−0.533402
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4557.98	0.304782	0.152391	−	0.988320i	$-0.451303\pi$
$607$	4557.98	0.304782	0.152391	+	0.988320i	$0.451303\pi$
$608$	0	0
$609$	−8402.38	−0.559083
$610$	0	0
$611$	8.70283	0.000576233 0
$612$	0	0
$613$	−26741.1	−1.76193	−0.880963	−	0.473185i	$-0.843104\pi$
$613$	−26741.1	−1.76193	−0.880963	+	0.473185i	$0.843104\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6569.62	−0.428659	−0.214330	−	0.976761i	$-0.568757\pi$
$617$	−6569.62	−0.428659	−0.214330	+	0.976761i	$0.568757\pi$
$618$	0	0
$619$	−12947.7	−0.840729	−0.420365	−	0.907355i	$-0.638098\pi$
$619$	−12947.7	−0.840729	−0.420365	+	0.907355i	$0.638098\pi$
$620$	0	0
$621$	−1103.78	−0.0713253
$622$	0	0
$623$	15815.5	1.01707
$624$	0	0
$625$	0	0
$626$	0	0
$627$	26677.3	1.69919
$628$	0	0
$629$	−2905.87	−0.184204
$630$	0	0
$631$	15799.6	0.996785	0.498393	−	0.866951i	$-0.333924\pi$
$631$	15799.6	0.996785	0.498393	+	0.866951i	$0.333924\pi$
$632$	0	0
$633$	−8785.89	−0.551671
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−39.7655	−0.00247342
$638$	0	0
$639$	−10435.8	−0.646065
$640$	0	0
$641$	7372.39	0.454278	0.227139	−	0.973862i	$-0.427063\pi$
$641$	7372.39	0.454278	0.227139	+	0.973862i	$0.427063\pi$
$642$	0	0
$643$	−3238.37	−0.198614	−0.0993070	−	0.995057i	$-0.531663\pi$
$643$	−3238.37	−0.198614	−0.0993070	+	0.995057i	$0.531663\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	23260.8	1.41341	0.706707	−	0.707507i	$-0.250180\pi$
$647$	23260.8	1.41341	0.706707	+	0.707507i	$0.250180\pi$
$648$	0	0
$649$	26487.5	1.60204
$650$	0	0
$651$	−17709.2	−1.06617
$652$	0	0
$653$	−20129.6	−1.20633	−0.603163	−	0.797618i	$-0.706093\pi$
$653$	−20129.6	−1.20633	−0.603163	+	0.797618i	$0.706093\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1086.45	0.0645150
$658$	0	0
$659$	−12381.4	−0.731882	−0.365941	−	0.930638i	$-0.619253\pi$
$659$	−12381.4	−0.731882	−0.365941	+	0.930638i	$0.619253\pi$
$660$	0	0
$661$	−24627.1	−1.44914	−0.724572	−	0.689199i	$-0.757963\pi$
$661$	−24627.1	−1.44914	−0.724572	+	0.689199i	$0.757963\pi$
$662$	0	0
$663$	−247.558	−0.0145013
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5759.29	0.334333
$668$	0	0
$669$	−7911.22	−0.457198
$670$	0	0
$671$	21438.2	1.23340
$672$	0	0
$673$	5211.65	0.298506	0.149253	−	0.988799i	$-0.452313\pi$
$673$	5211.65	0.298506	0.149253	+	0.988799i	$0.452313\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18402.6	1.04471	0.522354	−	0.852728i	$-0.325054\pi$
$677$	18402.6	1.04471	0.522354	+	0.852728i	$0.325054\pi$
$678$	0	0
$679$	16916.6	0.956112
$680$	0	0
$681$	9861.81	0.554927
$682$	0	0
$683$	−30347.2	−1.70015	−0.850076	−	0.526659i	$-0.823444\pi$
$683$	−30347.2	−1.70015	−0.850076	+	0.526659i	$0.823444\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	19146.0	1.06327
$688$	0	0
$689$	−121.308	−0.00670748
$690$	0	0
$691$	−18853.1	−1.03792	−0.518961	−	0.854798i	$-0.673681\pi$
$691$	−18853.1	−1.03792	−0.518961	+	0.854798i	$0.673681\pi$
$692$	0	0
$693$	12639.6	0.692842
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−2237.64	−0.121602
$698$	0	0
$699$	8858.74	0.479354
$700$	0	0
$701$	11008.1	0.593109	0.296555	−	0.955016i	$-0.404162\pi$
$701$	11008.1	0.593109	0.296555	+	0.955016i	$0.404162\pi$
$702$	0	0
$703$	−3374.37	−0.181034
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2977.65	−0.158396
$708$	0	0
$709$	18670.6	0.988981	0.494491	−	0.869183i	$-0.335355\pi$
$709$	18670.6	0.988981	0.494491	+	0.869183i	$0.335355\pi$
$710$	0	0
$711$	11237.9	0.592763
$712$	0	0
$713$	12138.5	0.637574
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4153.03	0.216315
$718$	0	0
$719$	26208.7	1.35941	0.679707	−	0.733484i	$-0.262107\pi$
$719$	26208.7	1.35941	0.679707	+	0.733484i	$0.262107\pi$
$720$	0	0
$721$	−23395.6	−1.20846
$722$	0	0
$723$	12710.5	0.653813
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−24977.6	−1.27424	−0.637118	−	0.770767i	$-0.719873\pi$
$727$	−24977.6	−1.27424	−0.637118	+	0.770767i	$0.719873\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−3533.61	−0.178790
$732$	0	0
$733$	−1491.96	−0.0751800	−0.0375900	−	0.999293i	$-0.511968\pi$
$733$	−1491.96	−0.0751800	−0.0375900	+	0.999293i	$0.511968\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−61994.1	−3.09848
$738$	0	0
$739$	−5276.36	−0.262644	−0.131322	−	0.991340i	$-0.541922\pi$
$739$	−5276.36	−0.262644	−0.131322	+	0.991340i	$0.541922\pi$
$740$	0	0
$741$	−287.471	−0.0142517
$742$	0	0
$743$	23611.8	1.16586	0.582930	−	0.812522i	$-0.301906\pi$
$743$	23611.8	1.16586	0.582930	+	0.812522i	$0.301906\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−5887.62	−0.288376
$748$	0	0
$749$	−8002.98	−0.390417
$750$	0	0
$751$	31848.5	1.54749	0.773747	−	0.633494i	$-0.218380\pi$
$751$	31848.5	1.54749	0.773747	+	0.633494i	$0.218380\pi$
$752$	0	0
$753$	5583.67	0.270226
$754$	0	0
$755$	0	0
$756$	0	0
$757$	19974.7	0.959041	0.479521	−	0.877531i	$-0.340811\pi$
$757$	19974.7	0.959041	0.479521	+	0.877531i	$0.340811\pi$
$758$	0	0
$759$	−8663.65	−0.414322
$760$	0	0
$761$	27224.0	1.29680	0.648402	−	0.761298i	$-0.275438\pi$
$761$	27224.0	1.29680	0.648402	+	0.761298i	$0.275438\pi$
$762$	0	0
$763$	32809.3	1.55672
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−285.426	−0.0134369
$768$	0	0
$769$	−29987.8	−1.40623	−0.703113	−	0.711078i	$-0.748207\pi$
$769$	−29987.8	−1.40623	−0.703113	+	0.711078i	$0.748207\pi$
$770$	0	0
$771$	−1041.75	−0.0486612
$772$	0	0
$773$	2856.40	0.132908	0.0664538	−	0.997790i	$-0.478831\pi$
$773$	2856.40	0.132908	0.0664538	+	0.997790i	$0.478831\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1598.77	−0.0738165
$778$	0	0
$779$	−2598.41	−0.119509
$780$	0	0
$781$	−81911.9	−3.75293
$782$	0	0
$783$	−3803.78	−0.173609
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−16704.1	−0.756590	−0.378295	−	0.925685i	$-0.623489\pi$
$787$	−16704.1	−0.756590	−0.378295	+	0.925685i	$0.623489\pi$
$788$	0	0
$789$	4468.16	0.201611
$790$	0	0
$791$	−26901.1	−1.20922
$792$	0	0
$793$	−231.015	−0.0103450
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−33709.4	−1.49818	−0.749090	−	0.662468i	$-0.769509\pi$
$797$	−33709.4	−1.49818	−0.749090	+	0.662468i	$0.769509\pi$
$798$	0	0
$799$	−1239.33	−0.0548742
$800$	0	0
$801$	7159.70	0.315825
$802$	0	0
$803$	8527.62	0.374761
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4712.64	0.205567
$808$	0	0
$809$	21582.4	0.937944	0.468972	−	0.883213i	$-0.344625\pi$
$809$	21582.4	0.937944	0.468972	+	0.883213i	$0.344625\pi$
$810$	0	0
$811$	40451.0	1.75145	0.875726	−	0.482809i	$-0.160384\pi$
$811$	40451.0	1.75145	0.875726	+	0.482809i	$0.160384\pi$
$812$	0	0
$813$	−9043.26	−0.390112
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−4103.32	−0.175712
$818$	0	0
$819$	−136.203	−0.00581112
$820$	0	0
$821$	−2857.45	−0.121468	−0.0607342	−	0.998154i	$-0.519344\pi$
$821$	−2857.45	−0.121468	−0.0607342	+	0.998154i	$0.519344\pi$
$822$	0	0
$823$	−15150.8	−0.641707	−0.320853	−	0.947129i	$-0.603970\pi$
$823$	−15150.8	−0.641707	−0.320853	+	0.947129i	$0.603970\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12171.5	0.511784	0.255892	−	0.966705i	$-0.417631\pi$
$827$	12171.5	0.511784	0.255892	+	0.966705i	$0.417631\pi$
$828$	0	0
$829$	−13543.0	−0.567392	−0.283696	−	0.958914i	$-0.591561\pi$
$829$	−13543.0	−0.567392	−0.283696	+	0.958914i	$0.591561\pi$
$830$	0	0
$831$	−8377.22	−0.349702
$832$	0	0
$833$	5662.84	0.235541
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8016.99	−0.331072
$838$	0	0
$839$	−25376.2	−1.04420	−0.522101	−	0.852884i	$-0.674851\pi$
$839$	−25376.2	−1.04420	−0.522101	+	0.852884i	$0.674851\pi$
$840$	0	0
$841$	−4541.65	−0.186217
$842$	0	0
$843$	−5237.42	−0.213981
$844$	0	0
$845$	0	0
$846$	0	0
$847$	72748.5	2.95120
$848$	0	0
$849$	7992.99	0.323108
$850$	0	0
$851$	1095.85	0.0441425
$852$	0	0
$853$	−10225.0	−0.410431	−0.205215	−	0.978717i	$-0.565789\pi$
$853$	−10225.0	−0.410431	−0.205215	+	0.978717i	$0.565789\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−35735.9	−1.42440	−0.712202	−	0.701975i	$-0.752302\pi$
$857$	−35735.9	−1.42440	−0.712202	+	0.701975i	$0.752302\pi$
$858$	0	0
$859$	3972.20	0.157776	0.0788881	−	0.996883i	$-0.474863\pi$
$859$	3972.20	0.157776	0.0788881	+	0.996883i	$0.474863\pi$
$860$	0	0
$861$	−1231.12	−0.0487298
$862$	0	0
$863$	−22682.0	−0.894673	−0.447337	−	0.894366i	$-0.647627\pi$
$863$	−22682.0	−0.894673	−0.447337	+	0.894366i	$0.647627\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	20514.7	0.803593
$868$	0	0
$869$	88207.4	3.44331
$870$	0	0
$871$	668.039	0.0259881
$872$	0	0
$873$	7658.19	0.296896
$874$	0	0
$875$	0	0
$876$	0	0
$877$	12622.0	0.485993	0.242997	−	0.970027i	$-0.421870\pi$
$877$	12622.0	0.485993	0.242997	+	0.970027i	$0.421870\pi$
$878$	0	0
$879$	−17783.4	−0.682387
$880$	0	0
$881$	−5343.08	−0.204328	−0.102164	−	0.994768i	$-0.532577\pi$
$881$	−5343.08	−0.204328	−0.102164	+	0.994768i	$0.532577\pi$
$882$	0	0
$883$	21649.7	0.825110	0.412555	−	0.910933i	$-0.364637\pi$
$883$	21649.7	0.825110	0.412555	+	0.910933i	$0.364637\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30808.6	1.16624	0.583118	−	0.812388i	$-0.301833\pi$
$887$	30808.6	1.16624	0.583118	+	0.812388i	$0.301833\pi$
$888$	0	0
$889$	36505.0	1.37721
$890$	0	0
$891$	5721.99	0.215145
$892$	0	0
$893$	−1439.15	−0.0539297
$894$	0	0
$895$	0	0
$896$	0	0
$897$	93.3582	0.00347507
$898$	0	0
$899$	41831.0	1.55188
$900$	0	0
$901$	17274.9	0.638747
$902$	0	0
$903$	−1944.14	−0.0716467
$904$	0	0
$905$	0	0
$906$	0	0
$907$	26499.7	0.970131	0.485065	−	0.874478i	$-0.338796\pi$
$907$	26499.7	0.970131	0.485065	+	0.874478i	$0.338796\pi$
$908$	0	0
$909$	−1347.99	−0.0491859
$910$	0	0
$911$	−25861.3	−0.940532	−0.470266	−	0.882525i	$-0.655842\pi$
$911$	−25861.3	−0.940532	−0.470266	+	0.882525i	$0.655842\pi$
$912$	0	0
$913$	−46212.5	−1.67515
$914$	0	0
$915$	0	0
$916$	0	0
$917$	22106.9	0.796113
$918$	0	0
$919$	−10296.7	−0.369595	−0.184798	−	0.982777i	$-0.559163\pi$
$919$	−10296.7	−0.369595	−0.184798	+	0.982777i	$0.559163\pi$
$920$	0	0
$921$	13903.4	0.497431
$922$	0	0
$923$	882.670	0.0314772
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−10591.3	−0.375256
$928$	0	0
$929$	271.470	0.00958734	0.00479367	−	0.999989i	$-0.498474\pi$
$929$	271.470	0.00958734	0.00479367	+	0.999989i	$0.498474\pi$
$930$	0	0
$931$	6575.85	0.231487
$932$	0	0
$933$	−2285.91	−0.0802114
$934$	0	0
$935$	0	0
$936$	0	0
$937$	8739.10	0.304689	0.152345	−	0.988327i	$-0.451318\pi$
$937$	8739.10	0.304689	0.152345	+	0.988327i	$0.451318\pi$
$938$	0	0
$939$	26202.9	0.910651
$940$	0	0
$941$	−55104.2	−1.90897	−0.954487	−	0.298252i	$-0.903597\pi$
$941$	−55104.2	−1.90897	−0.954487	+	0.298252i	$0.903597\pi$
$942$	0	0
$943$	843.851	0.0291406
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20552.8	−0.705256	−0.352628	−	0.935764i	$-0.614712\pi$
$947$	−20552.8	−0.705256	−0.352628	+	0.935764i	$0.614712\pi$
$948$	0	0
$949$	−91.8924	−0.00314326
$950$	0	0
$951$	−13284.6	−0.452980
$952$	0	0
$953$	5642.08	0.191778	0.0958892	−	0.995392i	$-0.469431\pi$
$953$	5642.08	0.191778	0.0958892	+	0.995392i	$0.469431\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−29856.2	−1.00848
$958$	0	0
$959$	11277.9	0.379754
$960$	0	0
$961$	58373.8	1.95944
$962$	0	0
$963$	−3622.97	−0.121234
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−21996.2	−0.731489	−0.365745	−	0.930715i	$-0.619186\pi$
$967$	−21996.2	−0.731489	−0.365745	+	0.930715i	$0.619186\pi$
$968$	0	0
$969$	40937.5	1.35718
$970$	0	0
$971$	43179.9	1.42710	0.713548	−	0.700606i	$-0.247087\pi$
$971$	43179.9	1.42710	0.713548	+	0.700606i	$0.247087\pi$
$972$	0	0
$973$	−39142.3	−1.28967
$974$	0	0
$975$	0	0
$976$	0	0
$977$	50887.5	1.66636	0.833181	−	0.553001i	$-0.186517\pi$
$977$	50887.5	1.66636	0.833181	+	0.553001i	$0.186517\pi$
$978$	0	0
$979$	56197.2	1.83460
$980$	0	0
$981$	14852.8	0.483399
$982$	0	0
$983$	−13150.8	−0.426700	−0.213350	−	0.976976i	$-0.568438\pi$
$983$	−13150.8	−0.426700	−0.213350	+	0.976976i	$0.568438\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−681.864	−0.0219898
$988$	0	0
$989$	1332.58	0.0428450
$990$	0	0
$991$	49196.4	1.57697	0.788483	−	0.615056i	$-0.210867\pi$
$991$	49196.4	1.57697	0.788483	+	0.615056i	$0.210867\pi$
$992$	0	0
$993$	6719.03	0.214725
$994$	0	0
$995$	0	0
$996$	0	0
$997$	39419.0	1.25217	0.626085	−	0.779755i	$-0.284656\pi$
$997$	39419.0	1.25217	0.626085	+	0.779755i	$0.284656\pi$
$998$	0	0
$999$	−723.766	−0.0229218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.a.br.1.2		2
4.3	odd	2		600.4.a.s.1.1	✓	2
5.2	odd	4		1200.4.f.x.49.2		4
5.3	odd	4		1200.4.f.x.49.3		4
5.4	even	2		1200.4.a.bp.1.1		2
12.11	even	2		1800.4.a.bo.1.1		2
20.3	even	4		600.4.f.j.49.2		4
20.7	even	4		600.4.f.j.49.3		4
20.19	odd	2		600.4.a.u.1.2	yes	2
60.23	odd	4		1800.4.f.z.649.3		4
60.47	odd	4		1800.4.f.z.649.2		4
60.59	even	2		1800.4.a.bm.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.4.a.s.1.1	✓	2	4.3	odd	2
600.4.a.u.1.2	yes	2	20.19	odd	2
600.4.f.j.49.2		4	20.3	even	4
600.4.f.j.49.3		4	20.7	even	4
1200.4.a.bp.1.1		2	5.4	even	2
1200.4.a.br.1.2		2	1.1	even	1	trivial
1200.4.f.x.49.2		4	5.2	odd	4
1200.4.f.x.49.3		4	5.3	odd	4
1800.4.a.bm.1.2		2	60.59	even	2
1800.4.a.bo.1.1		2	12.11	even	2
1800.4.f.z.649.2		4	60.47	odd	4
1800.4.f.z.649.3		4	60.23	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +19.8806 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +19.8806 q^{7} +9.00000 q^{9} +70.6418 q^{11} -0.761226 q^{13} +108.403 q^{17} +125.881 q^{19} +59.6418 q^{21} -40.8806 q^{23} +27.0000 q^{27} -140.881 q^{29} -296.926 q^{31} +211.926 q^{33} -26.8061 q^{37} -2.28368 q^{39} -20.6418 q^{41} -32.5969 q^{43} -11.4326 q^{47} +52.2388 q^{49} +325.209 q^{51} +159.358 q^{53} +377.642 q^{57} +374.955 q^{59} +303.478 q^{61} +178.926 q^{63} -877.583 q^{67} -122.642 q^{69} -1159.54 q^{71} +120.716 q^{73} +1404.40 q^{77} +1248.66 q^{79} +81.0000 q^{81} -654.180 q^{83} -422.642 q^{87} +795.522 q^{89} -15.1336 q^{91} -890.777 q^{93} +850.910 q^{97} +635.777 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 - 2 * q^7 + 18 * q^9	\( 2 q + 6 q^{3} - 2 q^{7} + 18 q^{9} + 16 q^{11} + 82 q^{13} + 8 q^{17} + 210 q^{19} - 6 q^{21} - 40 q^{23} + 54 q^{27} - 240 q^{29} - 218 q^{31} + 48 q^{33} + 364 q^{37} + 246 q^{39} + 84 q^{41} - 274 q^{43} - 524 q^{47} + 188 q^{49} + 24 q^{51} + 444 q^{53} + 630 q^{57} + 1084 q^{59} + 774 q^{61} - 18 q^{63} - 210 q^{67} - 120 q^{69} - 1108 q^{71} + 492 q^{73} + 2600 q^{77} + 1328 q^{79} + 162 q^{81} + 28 q^{83} - 720 q^{87} + 1424 q^{89} - 1826 q^{91} - 654 q^{93} + 2370 q^{97} + 144 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 2 * q^7 + 18 * q^9 + 16 * q^11 + 82 * q^13 + 8 * q^17 + 210 * q^19 - 6 * q^21 - 40 * q^23 + 54 * q^27 - 240 * q^29 - 218 * q^31 + 48 * q^33 + 364 * q^37 + 246 * q^39 + 84 * q^41 - 274 * q^43 - 524 * q^47 + 188 * q^49 + 24 * q^51 + 444 * q^53 + 630 * q^57 + 1084 * q^59 + 774 * q^61 - 18 * q^63 - 210 * q^67 - 120 * q^69 - 1108 * q^71 + 492 * q^73 + 2600 * q^77 + 1328 * q^79 + 162 * q^81 + 28 * q^83 - 720 * q^87 + 1424 * q^89 - 1826 * q^91 - 654 * q^93 + 2370 * q^97 + 144 * q^99

Introduction

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