LMFDB - Embedded newform 1232.4.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,4,Mod(1,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, 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("1232.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1232.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:	$72.6903531271$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 29x^{2} + 3x + 114 $ x^4 - x^3 - 29x^2 + 3x + 114
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.49248$ of defining polynomial
Character	$\chi$	$=$	1232.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.49248 q^{3} -8.18240 q^{5} -7.00000 q^{7} -6.81760 q^{9} -11.0000 q^{11} +6.07157 q^{13} +36.7593 q^{15} -108.555 q^{17} -40.0288 q^{19} +31.4474 q^{21} -49.1578 q^{23} -58.0484 q^{25} +151.925 q^{27} -89.1584 q^{29} -111.551 q^{31} +49.4173 q^{33} +57.2768 q^{35} -195.258 q^{37} -27.2764 q^{39} -226.927 q^{41} +17.0565 q^{43} +55.7843 q^{45} -61.5015 q^{47} +49.0000 q^{49} +487.680 q^{51} -691.929 q^{53} +90.0064 q^{55} +179.829 q^{57} +240.751 q^{59} -694.558 q^{61} +47.7232 q^{63} -49.6800 q^{65} -396.960 q^{67} +220.840 q^{69} +232.999 q^{71} +43.3791 q^{73} +260.781 q^{75} +77.0000 q^{77} +230.462 q^{79} -498.445 q^{81} -210.747 q^{83} +888.237 q^{85} +400.543 q^{87} +328.953 q^{89} -42.5010 q^{91} +501.140 q^{93} +327.532 q^{95} +1765.90 q^{97} +74.9936 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} - 13 q^{5} - 28 q^{7} - 47 q^{9} - 44 q^{11} - 34 q^{13} - 63 q^{15} - 58 q^{17} - 60 q^{19} - 21 q^{21} + 93 q^{23} - 19 q^{25} - 63 q^{27} - 144 q^{29} + 129 q^{31} - 33 q^{33} + 91 q^{35}+ \cdots + 517 q^{99}+O(q^{100}) $ 4 * q + 3 * q^3 - 13 * q^5 - 28 * q^7 - 47 * q^9 - 44 * q^11 - 34 * q^13 - 63 * q^15 - 58 * q^17 - 60 * q^19 - 21 * q^21 + 93 * q^23 - 19 * q^25 - 63 * q^27 - 144 * q^29 + 129 * q^31 - 33 * q^33 + 91 * q^35 - 187 * q^37 + 244 * q^39 + 110 * q^41 + 360 * q^43 - 286 * q^45 + 438 * q^47 + 196 * q^49 + 902 * q^51 + 56 * q^53 + 143 * q^55 - 94 * q^57 + 1209 * q^59 + 104 * q^61 + 329 * q^63 - 1256 * q^65 + 1075 * q^67 + 451 * q^69 + 963 * q^71 - 646 * q^73 + 804 * q^75 + 308 * q^77 + 1838 * q^79 - 656 * q^81 + 1238 * q^83 - 278 * q^85 + 914 * q^87 - 1453 * q^89 + 238 * q^91 + 521 * q^93 + 2730 * q^95 + 573 * q^97 + 517 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−4.49248	−0.864579	−0.432289	−	0.901735i	$-0.642294\pi$
$3$	−4.49248	−0.864579	−0.432289	+	0.901735i	$0.642294\pi$
$4$	0	0
$5$	−8.18240	−0.731856	−0.365928	−	0.930643i	$-0.619248\pi$
$5$	−8.18240	−0.731856	−0.365928	+	0.930643i	$0.619248\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−6.81760	−0.252504
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	6.07157	0.129535	0.0647673	−	0.997900i	$-0.479369\pi$
$13$	6.07157	0.129535	0.0647673	+	0.997900i	$0.479369\pi$
$14$	0	0
$15$	36.7593	0.632747
$16$	0	0
$17$	−108.555	−1.54873	−0.774363	−	0.632741i	$-0.781930\pi$
$17$	−108.555	−1.54873	−0.774363	+	0.632741i	$0.781930\pi$
$18$	0	0
$19$	−40.0288	−0.483328	−0.241664	−	0.970360i	$-0.577693\pi$
$19$	−40.0288	−0.483328	−0.241664	+	0.970360i	$0.577693\pi$
$20$	0	0
$21$	31.4474	0.326780
$22$	0	0
$23$	−49.1578	−0.445657	−0.222828	−	0.974858i	$-0.571529\pi$
$23$	−49.1578	−0.445657	−0.222828	+	0.974858i	$0.571529\pi$
$24$	0	0
$25$	−58.0484	−0.464387
$26$	0	0
$27$	151.925	1.08289
$28$	0	0
$29$	−89.1584	−0.570907	−0.285454	−	0.958393i	$-0.592144\pi$
$29$	−89.1584	−0.570907	−0.285454	+	0.958393i	$0.592144\pi$
$30$	0	0
$31$	−111.551	−0.646294	−0.323147	−	0.946349i	$-0.604741\pi$
$31$	−111.551	−0.646294	−0.323147	+	0.946349i	$0.604741\pi$
$32$	0	0
$33$	49.4173	0.260680
$34$	0	0
$35$	57.2768	0.276616
$36$	0	0
$37$	−195.258	−0.867572	−0.433786	−	0.901016i	$-0.642823\pi$
$37$	−195.258	−0.867572	−0.433786	+	0.901016i	$0.642823\pi$
$38$	0	0
$39$	−27.2764	−0.111993
$40$	0	0
$41$	−226.927	−0.864393	−0.432197	−	0.901779i	$-0.642261\pi$
$41$	−226.927	−0.864393	−0.432197	+	0.901779i	$0.642261\pi$
$42$	0	0
$43$	17.0565	0.0604904	0.0302452	−	0.999543i	$-0.490371\pi$
$43$	17.0565	0.0604904	0.0302452	+	0.999543i	$0.490371\pi$
$44$	0	0
$45$	55.7843	0.184796
$46$	0	0
$47$	−61.5015	−0.190870	−0.0954352	−	0.995436i	$-0.530424\pi$
$47$	−61.5015	−0.190870	−0.0954352	+	0.995436i	$0.530424\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	487.680	1.33900
$52$	0	0
$53$	−691.929	−1.79328	−0.896639	−	0.442762i	$-0.853999\pi$
$53$	−691.929	−1.79328	−0.896639	+	0.442762i	$0.853999\pi$
$54$	0	0
$55$	90.0064	0.220663
$56$	0	0
$57$	179.829	0.417875
$58$	0	0
$59$	240.751	0.531238	0.265619	−	0.964078i	$-0.414424\pi$
$59$	240.751	0.531238	0.265619	+	0.964078i	$0.414424\pi$
$60$	0	0
$61$	−694.558	−1.45785	−0.728927	−	0.684591i	$-0.759981\pi$
$61$	−694.558	−1.45785	−0.728927	+	0.684591i	$0.759981\pi$
$62$	0	0
$63$	47.7232	0.0954375
$64$	0	0
$65$	−49.6800	−0.0948006
$66$	0	0
$67$	−396.960	−0.723826	−0.361913	−	0.932212i	$-0.617876\pi$
$67$	−396.960	−0.723826	−0.361913	+	0.932212i	$0.617876\pi$
$68$	0	0
$69$	220.840	0.385305
$70$	0	0
$71$	232.999	0.389463	0.194732	−	0.980857i	$-0.437616\pi$
$71$	232.999	0.389463	0.194732	+	0.980857i	$0.437616\pi$
$72$	0	0
$73$	43.3791	0.0695499	0.0347749	−	0.999395i	$-0.488929\pi$
$73$	43.3791	0.0695499	0.0347749	+	0.999395i	$0.488929\pi$
$74$	0	0
$75$	260.781	0.401499
$76$	0	0
$77$	77.0000	0.113961
$78$	0	0
$79$	230.462	0.328215	0.164108	−	0.986442i	$-0.447526\pi$
$79$	230.462	0.328215	0.164108	+	0.986442i	$0.447526\pi$
$80$	0	0
$81$	−498.445	−0.683738
$82$	0	0
$83$	−210.747	−0.278705	−0.139353	−	0.990243i	$-0.544502\pi$
$83$	−210.747	−0.278705	−0.139353	+	0.990243i	$0.544502\pi$
$84$	0	0
$85$	888.237	1.13344
$86$	0	0
$87$	400.543	0.493594
$88$	0	0
$89$	328.953	0.391786	0.195893	−	0.980625i	$-0.437239\pi$
$89$	328.953	0.391786	0.195893	+	0.980625i	$0.437239\pi$
$90$	0	0
$91$	−42.5010	−0.0489595
$92$	0	0
$93$	501.140	0.558772
$94$	0	0
$95$	327.532	0.353727
$96$	0	0
$97$	1765.90	1.84846	0.924228	−	0.381841i	$-0.124710\pi$
$97$	1765.90	1.84846	0.924228	+	0.381841i	$0.124710\pi$
$98$	0	0
$99$	74.9936	0.0761328
$100$	0	0
$101$	−1038.52	−1.02314	−0.511569	−	0.859242i	$-0.670935\pi$
$101$	−1038.52	−1.02314	−0.511569	+	0.859242i	$0.670935\pi$
$102$	0	0
$103$	1756.90	1.68070	0.840352	−	0.542041i	$-0.182348\pi$
$103$	1756.90	1.68070	0.840352	+	0.542041i	$0.182348\pi$
$104$	0	0
$105$	−257.315	−0.239156
$106$	0	0
$107$	−708.981	−0.640559	−0.320279	−	0.947323i	$-0.603777\pi$
$107$	−708.981	−0.640559	−0.320279	+	0.947323i	$0.603777\pi$
$108$	0	0
$109$	−631.308	−0.554755	−0.277377	−	0.960761i	$-0.589465\pi$
$109$	−631.308	−0.554755	−0.277377	+	0.960761i	$0.589465\pi$
$110$	0	0
$111$	877.191	0.750084
$112$	0	0
$113$	−694.277	−0.577983	−0.288991	−	0.957332i	$-0.593320\pi$
$113$	−694.277	−0.577983	−0.288991	+	0.957332i	$0.593320\pi$
$114$	0	0
$115$	402.229	0.326157
$116$	0	0
$117$	−41.3935	−0.0327080
$118$	0	0
$119$	759.882	0.585364
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	1019.47	0.747336
$124$	0	0
$125$	1497.77	1.07172
$126$	0	0
$127$	−566.147	−0.395571	−0.197785	−	0.980245i	$-0.563375\pi$
$127$	−566.147	−0.395571	−0.197785	+	0.980245i	$0.563375\pi$
$128$	0	0
$129$	−76.6259	−0.0522987
$130$	0	0
$131$	646.992	0.431511	0.215756	−	0.976447i	$-0.430779\pi$
$131$	646.992	0.431511	0.215756	+	0.976447i	$0.430779\pi$
$132$	0	0
$133$	280.202	0.182681
$134$	0	0
$135$	−1243.11	−0.792518
$136$	0	0
$137$	−415.390	−0.259045	−0.129522	−	0.991576i	$-0.541344\pi$
$137$	−415.390	−0.259045	−0.129522	+	0.991576i	$0.541344\pi$
$138$	0	0
$139$	−2005.54	−1.22379	−0.611897	−	0.790937i	$-0.709593\pi$
$139$	−2005.54	−1.22379	−0.611897	+	0.790937i	$0.709593\pi$
$140$	0	0
$141$	276.294	0.165022
$142$	0	0
$143$	−66.7872	−0.0390561
$144$	0	0
$145$	729.530	0.417822
$146$	0	0
$147$	−220.132	−0.123511
$148$	0	0
$149$	1271.61	0.699156	0.349578	−	0.936907i	$-0.386325\pi$
$149$	1271.61	0.699156	0.349578	+	0.936907i	$0.386325\pi$
$150$	0	0
$151$	2653.44	1.43003	0.715014	−	0.699110i	$-0.246420\pi$
$151$	2653.44	1.43003	0.715014	+	0.699110i	$0.246420\pi$
$152$	0	0
$153$	740.082	0.391059
$154$	0	0
$155$	912.754	0.472994
$156$	0	0
$157$	986.712	0.501581	0.250790	−	0.968041i	$-0.419309\pi$
$157$	986.712	0.501581	0.250790	+	0.968041i	$0.419309\pi$
$158$	0	0
$159$	3108.48	1.55043
$160$	0	0
$161$	344.104	0.168442
$162$	0	0
$163$	807.865	0.388202	0.194101	−	0.980982i	$-0.437821\pi$
$163$	807.865	0.388202	0.194101	+	0.980982i	$0.437821\pi$
$164$	0	0
$165$	−404.352	−0.190780
$166$	0	0
$167$	2269.50	1.05161	0.525806	−	0.850604i	$-0.323764\pi$
$167$	2269.50	1.05161	0.525806	+	0.850604i	$0.323764\pi$
$168$	0	0
$169$	−2160.14	−0.983221
$170$	0	0
$171$	272.900	0.122042
$172$	0	0
$173$	−3029.98	−1.33159	−0.665794	−	0.746135i	$-0.731907\pi$
$173$	−3029.98	−1.33159	−0.665794	+	0.746135i	$0.731907\pi$
$174$	0	0
$175$	406.339	0.175522
$176$	0	0
$177$	−1081.57	−0.459297
$178$	0	0
$179$	−2013.28	−0.840667	−0.420333	−	0.907370i	$-0.638087\pi$
$179$	−2013.28	−0.840667	−0.420333	+	0.907370i	$0.638087\pi$
$180$	0	0
$181$	897.906	0.368734	0.184367	−	0.982857i	$-0.440977\pi$
$181$	897.906	0.368734	0.184367	+	0.982857i	$0.440977\pi$
$182$	0	0
$183$	3120.29	1.26043
$184$	0	0
$185$	1597.68	0.634938
$186$	0	0
$187$	1194.10	0.466959
$188$	0	0
$189$	−1063.47	−0.409293
$190$	0	0
$191$	1725.88	0.653823	0.326911	−	0.945055i	$-0.393992\pi$
$191$	1725.88	0.653823	0.326911	+	0.945055i	$0.393992\pi$
$192$	0	0
$193$	2822.38	1.05264	0.526319	−	0.850287i	$-0.323572\pi$
$193$	2822.38	1.05264	0.526319	+	0.850287i	$0.323572\pi$
$194$	0	0
$195$	223.186	0.0819626
$196$	0	0
$197$	2324.71	0.840755	0.420378	−	0.907349i	$-0.361898\pi$
$197$	2324.71	0.840755	0.420378	+	0.907349i	$0.361898\pi$
$198$	0	0
$199$	1221.56	0.435145	0.217573	−	0.976044i	$-0.430186\pi$
$199$	1221.56	0.435145	0.217573	+	0.976044i	$0.430186\pi$
$200$	0	0
$201$	1783.33	0.625804
$202$	0	0
$203$	624.109	0.215783
$204$	0	0
$205$	1856.81	0.632611
$206$	0	0
$207$	335.138	0.112530
$208$	0	0
$209$	440.317	0.145729
$210$	0	0
$211$	395.480	0.129033	0.0645165	−	0.997917i	$-0.479449\pi$
$211$	395.480	0.129033	0.0645165	+	0.997917i	$0.479449\pi$
$212$	0	0
$213$	−1046.74	−0.336722
$214$	0	0
$215$	−139.563	−0.0442703
$216$	0	0
$217$	780.856	0.244276
$218$	0	0
$219$	−194.880	−0.0601314
$220$	0	0
$221$	−659.096	−0.200614
$222$	0	0
$223$	−4280.19	−1.28530	−0.642652	−	0.766158i	$-0.722166\pi$
$223$	−4280.19	−1.28530	−0.642652	+	0.766158i	$0.722166\pi$
$224$	0	0
$225$	395.751	0.117259
$226$	0	0
$227$	2834.12	0.828667	0.414333	−	0.910125i	$-0.364015\pi$
$227$	2834.12	0.828667	0.414333	+	0.910125i	$0.364015\pi$
$228$	0	0
$229$	3886.81	1.12161	0.560803	−	0.827949i	$-0.310493\pi$
$229$	3886.81	1.12161	0.560803	+	0.827949i	$0.310493\pi$
$230$	0	0
$231$	−345.921	−0.0985279
$232$	0	0
$233$	568.968	0.159976	0.0799878	−	0.996796i	$-0.474512\pi$
$233$	568.968	0.159976	0.0799878	+	0.996796i	$0.474512\pi$
$234$	0	0
$235$	503.229	0.139690
$236$	0	0
$237$	−1035.35	−0.283768
$238$	0	0
$239$	−4685.38	−1.26808	−0.634041	−	0.773299i	$-0.718605\pi$
$239$	−4685.38	−1.26808	−0.634041	+	0.773299i	$0.718605\pi$
$240$	0	0
$241$	6788.02	1.81433	0.907167	−	0.420770i	$-0.138240\pi$
$241$	6788.02	1.81433	0.907167	+	0.420770i	$0.138240\pi$
$242$	0	0
$243$	−1862.72	−0.491743
$244$	0	0
$245$	−400.938	−0.104551
$246$	0	0
$247$	−243.037	−0.0626077
$248$	0	0
$249$	946.779	0.240962
$250$	0	0
$251$	−888.807	−0.223510	−0.111755	−	0.993736i	$-0.535647\pi$
$251$	−888.807	−0.223510	−0.111755	+	0.993736i	$0.535647\pi$
$252$	0	0
$253$	540.736	0.134371
$254$	0	0
$255$	−3990.39	−0.979952
$256$	0	0
$257$	513.349	0.124599	0.0622993	−	0.998058i	$-0.480157\pi$
$257$	513.349	0.124599	0.0622993	+	0.998058i	$0.480157\pi$
$258$	0	0
$259$	1366.80	0.327911
$260$	0	0
$261$	607.847	0.144156
$262$	0	0
$263$	−2569.03	−0.602331	−0.301166	−	0.953572i	$-0.597376\pi$
$263$	−2569.03	−0.602331	−0.301166	+	0.953572i	$0.597376\pi$
$264$	0	0
$265$	5661.64	1.31242
$266$	0	0
$267$	−1477.82	−0.338730
$268$	0	0
$269$	−5056.44	−1.14608	−0.573042	−	0.819526i	$-0.694237\pi$
$269$	−5056.44	−1.14608	−0.573042	+	0.819526i	$0.694237\pi$
$270$	0	0
$271$	−702.927	−0.157564	−0.0787818	−	0.996892i	$-0.525103\pi$
$271$	−702.927	−0.157564	−0.0787818	+	0.996892i	$0.525103\pi$
$272$	0	0
$273$	190.935	0.0423293
$274$	0	0
$275$	638.532	0.140018
$276$	0	0
$277$	−3032.51	−0.657783	−0.328891	−	0.944368i	$-0.606675\pi$
$277$	−3032.51	−0.657783	−0.328891	+	0.944368i	$0.606675\pi$
$278$	0	0
$279$	760.509	0.163192
$280$	0	0
$281$	5593.01	1.18737	0.593686	−	0.804697i	$-0.297672\pi$
$281$	5593.01	1.18737	0.593686	+	0.804697i	$0.297672\pi$
$282$	0	0
$283$	−6042.44	−1.26921	−0.634604	−	0.772837i	$-0.718837\pi$
$283$	−6042.44	−1.26921	−0.634604	+	0.772837i	$0.718837\pi$
$284$	0	0
$285$	−1471.43	−0.305824
$286$	0	0
$287$	1588.49	0.326710
$288$	0	0
$289$	6871.10	1.39856
$290$	0	0
$291$	−7933.29	−1.59814
$292$	0	0
$293$	5871.31	1.17067	0.585334	−	0.810793i	$-0.300964\pi$
$293$	5871.31	1.17067	0.585334	+	0.810793i	$0.300964\pi$
$294$	0	0
$295$	−1969.92	−0.388790
$296$	0	0
$297$	−1671.17	−0.326503
$298$	0	0
$299$	−298.465	−0.0577280
$300$	0	0
$301$	−119.395	−0.0228632
$302$	0	0
$303$	4665.54	0.884582
$304$	0	0
$305$	5683.15	1.06694
$306$	0	0
$307$	−3040.06	−0.565165	−0.282582	−	0.959243i	$-0.591191\pi$
$307$	−3040.06	−0.565165	−0.282582	+	0.959243i	$0.591191\pi$
$308$	0	0
$309$	−7892.84	−1.45310
$310$	0	0
$311$	3692.41	0.673240	0.336620	−	0.941641i	$-0.390716\pi$
$311$	3692.41	0.673240	0.336620	+	0.941641i	$0.390716\pi$
$312$	0	0
$313$	−5510.85	−0.995180	−0.497590	−	0.867412i	$-0.665782\pi$
$313$	−5510.85	−0.995180	−0.497590	+	0.867412i	$0.665782\pi$
$314$	0	0
$315$	−390.490	−0.0698465
$316$	0	0
$317$	5290.26	0.937320	0.468660	−	0.883379i	$-0.344737\pi$
$317$	5290.26	0.937320	0.468660	+	0.883379i	$0.344737\pi$
$318$	0	0
$319$	980.743	0.172135
$320$	0	0
$321$	3185.09	0.553813
$322$	0	0
$323$	4345.31	0.748543
$324$	0	0
$325$	−352.444	−0.0601542
$326$	0	0
$327$	2836.14	0.479629
$328$	0	0
$329$	430.510	0.0721422
$330$	0	0
$331$	−9949.58	−1.65220	−0.826100	−	0.563523i	$-0.809446\pi$
$331$	−9949.58	−1.65220	−0.826100	+	0.563523i	$0.809446\pi$
$332$	0	0
$333$	1331.19	0.219065
$334$	0	0
$335$	3248.08	0.529736
$336$	0	0
$337$	−626.780	−0.101314	−0.0506571	−	0.998716i	$-0.516132\pi$
$337$	−626.780	−0.101314	−0.0506571	+	0.998716i	$0.516132\pi$
$338$	0	0
$339$	3119.03	0.499712
$340$	0	0
$341$	1227.06	0.194865
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−1807.00	−0.281988
$346$	0	0
$347$	−2666.53	−0.412527	−0.206263	−	0.978496i	$-0.566130\pi$
$347$	−2666.53	−0.412527	−0.206263	+	0.978496i	$0.566130\pi$
$348$	0	0
$349$	11653.7	1.78742	0.893711	−	0.448643i	$-0.148093\pi$
$349$	11653.7	1.78742	0.893711	+	0.448643i	$0.148093\pi$
$350$	0	0
$351$	922.423	0.140271
$352$	0	0
$353$	−3044.06	−0.458977	−0.229488	−	0.973311i	$-0.573705\pi$
$353$	−3044.06	−0.458977	−0.229488	+	0.973311i	$0.573705\pi$
$354$	0	0
$355$	−1906.49	−0.285031
$356$	0	0
$357$	−3413.76	−0.506093
$358$	0	0
$359$	−4702.31	−0.691305	−0.345652	−	0.938363i	$-0.612342\pi$
$359$	−4702.31	−0.691305	−0.345652	+	0.938363i	$0.612342\pi$
$360$	0	0
$361$	−5256.70	−0.766394
$362$	0	0
$363$	−543.590	−0.0785981
$364$	0	0
$365$	−354.945	−0.0509005
$366$	0	0
$367$	2435.50	0.346410	0.173205	−	0.984886i	$-0.444588\pi$
$367$	2435.50	0.346410	0.173205	+	0.984886i	$0.444588\pi$
$368$	0	0
$369$	1547.10	0.218263
$370$	0	0
$371$	4843.50	0.677795
$372$	0	0
$373$	−8577.64	−1.19071	−0.595353	−	0.803464i	$-0.702988\pi$
$373$	−8577.64	−1.19071	−0.595353	+	0.803464i	$0.702988\pi$
$374$	0	0
$375$	−6728.73	−0.926586
$376$	0	0
$377$	−541.331	−0.0739522
$378$	0	0
$379$	1258.81	0.170609	0.0853045	−	0.996355i	$-0.472814\pi$
$379$	1258.81	0.170609	0.0853045	+	0.996355i	$0.472814\pi$
$380$	0	0
$381$	2543.41	0.342002
$382$	0	0
$383$	5723.93	0.763653	0.381826	−	0.924234i	$-0.375295\pi$
$383$	5723.93	0.763653	0.381826	+	0.924234i	$0.375295\pi$
$384$	0	0
$385$	−630.045	−0.0834027
$386$	0	0
$387$	−116.284	−0.0152741
$388$	0	0
$389$	−14276.0	−1.86073	−0.930363	−	0.366640i	$-0.880508\pi$
$389$	−14276.0	−1.86073	−0.930363	+	0.366640i	$0.880508\pi$
$390$	0	0
$391$	5336.30	0.690201
$392$	0	0
$393$	−2906.60	−0.373075
$394$	0	0
$395$	−1885.73	−0.240206
$396$	0	0
$397$	1715.60	0.216886	0.108443	−	0.994103i	$-0.465414\pi$
$397$	1715.60	0.216886	0.108443	+	0.994103i	$0.465414\pi$
$398$	0	0
$399$	−1258.80	−0.157942
$400$	0	0
$401$	−14622.1	−1.82093	−0.910466	−	0.413583i	$-0.864277\pi$
$401$	−14622.1	−1.82093	−0.910466	+	0.413583i	$0.864277\pi$
$402$	0	0
$403$	−677.288	−0.0837175
$404$	0	0
$405$	4078.48	0.500398
$406$	0	0
$407$	2147.83	0.261583
$408$	0	0
$409$	910.825	0.110116	0.0550580	−	0.998483i	$-0.482466\pi$
$409$	910.825	0.110116	0.0550580	+	0.998483i	$0.482466\pi$
$410$	0	0
$411$	1866.13	0.223965
$412$	0	0
$413$	−1685.25	−0.200789
$414$	0	0
$415$	1724.42	0.203972
$416$	0	0
$417$	9009.84	1.05807
$418$	0	0
$419$	−13824.4	−1.61185	−0.805927	−	0.592015i	$-0.798332\pi$
$419$	−13824.4	−1.61185	−0.805927	+	0.592015i	$0.798332\pi$
$420$	0	0
$421$	−6206.97	−0.718550	−0.359275	−	0.933232i	$-0.616976\pi$
$421$	−6206.97	−0.718550	−0.359275	+	0.933232i	$0.616976\pi$
$422$	0	0
$423$	419.292	0.0481955
$424$	0	0
$425$	6301.42	0.719208
$426$	0	0
$427$	4861.91	0.551017
$428$	0	0
$429$	300.040	0.0337671
$430$	0	0
$431$	−13193.6	−1.47450	−0.737252	−	0.675618i	$-0.763877\pi$
$431$	−13193.6	−1.47450	−0.737252	+	0.675618i	$0.763877\pi$
$432$	0	0
$433$	1764.74	0.195861	0.0979306	−	0.995193i	$-0.468778\pi$
$433$	1764.74	0.195861	0.0979306	+	0.995193i	$0.468778\pi$
$434$	0	0
$435$	−3277.40	−0.361240
$436$	0	0
$437$	1967.73	0.215398
$438$	0	0
$439$	403.375	0.0438543	0.0219271	−	0.999760i	$-0.493020\pi$
$439$	403.375	0.0438543	0.0219271	+	0.999760i	$0.493020\pi$
$440$	0	0
$441$	−334.062	−0.0360720
$442$	0	0
$443$	4424.25	0.474498	0.237249	−	0.971449i	$-0.423754\pi$
$443$	4424.25	0.474498	0.237249	+	0.971449i	$0.423754\pi$
$444$	0	0
$445$	−2691.63	−0.286731
$446$	0	0
$447$	−5712.68	−0.604475
$448$	0	0
$449$	−12791.4	−1.34446	−0.672230	−	0.740343i	$-0.734663\pi$
$449$	−12791.4	−1.34446	−0.672230	+	0.740343i	$0.734663\pi$
$450$	0	0
$451$	2496.20	0.260624
$452$	0	0
$453$	−11920.6	−1.23637
$454$	0	0
$455$	347.760	0.0358313
$456$	0	0
$457$	−14733.4	−1.50810	−0.754048	−	0.656819i	$-0.771902\pi$
$457$	−14733.4	−1.50810	−0.754048	+	0.656819i	$0.771902\pi$
$458$	0	0
$459$	−16492.2	−1.67710
$460$	0	0
$461$	383.371	0.0387319	0.0193659	−	0.999812i	$-0.493835\pi$
$461$	383.371	0.0387319	0.0193659	+	0.999812i	$0.493835\pi$
$462$	0	0
$463$	17603.9	1.76700	0.883502	−	0.468428i	$-0.155179\pi$
$463$	17603.9	1.76700	0.883502	+	0.468428i	$0.155179\pi$
$464$	0	0
$465$	−4100.53	−0.408941
$466$	0	0
$467$	2841.15	0.281526	0.140763	−	0.990043i	$-0.455044\pi$
$467$	2841.15	0.281526	0.140763	+	0.990043i	$0.455044\pi$
$468$	0	0
$469$	2778.72	0.273580
$470$	0	0
$471$	−4432.79	−0.433656
$472$	0	0
$473$	−187.621	−0.0182385
$474$	0	0
$475$	2323.61	0.224451
$476$	0	0
$477$	4717.30	0.452810
$478$	0	0
$479$	−3302.93	−0.315062	−0.157531	−	0.987514i	$-0.550353\pi$
$479$	−3302.93	−0.315062	−0.157531	+	0.987514i	$0.550353\pi$
$480$	0	0
$481$	−1185.52	−0.112381
$482$	0	0
$483$	−1545.88	−0.145632
$484$	0	0
$485$	−14449.3	−1.35280
$486$	0	0
$487$	−11185.4	−1.04078	−0.520388	−	0.853930i	$-0.674213\pi$
$487$	−11185.4	−1.04078	−0.520388	+	0.853930i	$0.674213\pi$
$488$	0	0
$489$	−3629.32	−0.335631
$490$	0	0
$491$	−2930.79	−0.269379	−0.134689	−	0.990888i	$-0.543004\pi$
$491$	−2930.79	−0.269379	−0.134689	+	0.990888i	$0.543004\pi$
$492$	0	0
$493$	9678.56	0.884179
$494$	0	0
$495$	−613.628	−0.0557182
$496$	0	0
$497$	−1630.99	−0.147203
$498$	0	0
$499$	11809.6	1.05946	0.529732	−	0.848165i	$-0.322293\pi$
$499$	11809.6	1.05946	0.529732	+	0.848165i	$0.322293\pi$
$500$	0	0
$501$	−10195.7	−0.909202
$502$	0	0
$503$	10205.3	0.904636	0.452318	−	0.891857i	$-0.350597\pi$
$503$	10205.3	0.904636	0.452318	+	0.891857i	$0.350597\pi$
$504$	0	0
$505$	8497.60	0.748789
$506$	0	0
$507$	9704.37	0.850072
$508$	0	0
$509$	−6050.14	−0.526852	−0.263426	−	0.964680i	$-0.584852\pi$
$509$	−6050.14	−0.526852	−0.263426	+	0.964680i	$0.584852\pi$
$510$	0	0
$511$	−303.654	−0.0262874
$512$	0	0
$513$	−6081.37	−0.523390
$514$	0	0
$515$	−14375.7	−1.23003
$516$	0	0
$517$	676.516	0.0575496
$518$	0	0
$519$	13612.1	1.15126
$520$	0	0
$521$	2309.38	0.194195	0.0970975	−	0.995275i	$-0.469044\pi$
$521$	2309.38	0.194195	0.0970975	+	0.995275i	$0.469044\pi$
$522$	0	0
$523$	11491.2	0.960753	0.480376	−	0.877063i	$-0.340500\pi$
$523$	11491.2	0.960753	0.480376	+	0.877063i	$0.340500\pi$
$524$	0	0
$525$	−1825.47	−0.151752
$526$	0	0
$527$	12109.4	1.00093
$528$	0	0
$529$	−9750.51	−0.801390
$530$	0	0
$531$	−1641.34	−0.134140
$532$	0	0
$533$	−1377.81	−0.111969
$534$	0	0
$535$	5801.17	0.468797
$536$	0	0
$537$	9044.61	0.726823
$538$	0	0
$539$	−539.000	−0.0430730
$540$	0	0
$541$	4.36215	0.000346661 0	0.000173330	−	1.00000i	$-0.499945\pi$
$541$	4.36215	0.000346661 0	0.000173330		1.00000i	$0.499945\pi$
$542$	0	0
$543$	−4033.83	−0.318799
$544$	0	0
$545$	5165.61	0.406001
$546$	0	0
$547$	2129.15	0.166428	0.0832139	−	0.996532i	$-0.473482\pi$
$547$	2129.15	0.166428	0.0832139	+	0.996532i	$0.473482\pi$
$548$	0	0
$549$	4735.22	0.368114
$550$	0	0
$551$	3568.90	0.275935
$552$	0	0
$553$	−1613.23	−0.124054
$554$	0	0
$555$	−7177.53	−0.548953
$556$	0	0
$557$	9651.66	0.734208	0.367104	−	0.930180i	$-0.380349\pi$
$557$	9651.66	0.734208	0.367104	+	0.930180i	$0.380349\pi$
$558$	0	0
$559$	103.560	0.00783560
$560$	0	0
$561$	−5364.48	−0.403723
$562$	0	0
$563$	−15382.2	−1.15148	−0.575740	−	0.817633i	$-0.695286\pi$
$563$	−15382.2	−1.15148	−0.575740	+	0.817633i	$0.695286\pi$
$564$	0	0
$565$	5680.85	0.423000
$566$	0	0
$567$	3489.12	0.258429
$568$	0	0
$569$	−17553.4	−1.29328	−0.646640	−	0.762795i	$-0.723826\pi$
$569$	−17553.4	−1.29328	−0.646640	+	0.762795i	$0.723826\pi$
$570$	0	0
$571$	−2243.19	−0.164404	−0.0822020	−	0.996616i	$-0.526195\pi$
$571$	−2243.19	−0.164404	−0.0822020	+	0.996616i	$0.526195\pi$
$572$	0	0
$573$	−7753.47	−0.565281
$574$	0	0
$575$	2853.53	0.206957
$576$	0	0
$577$	−10345.0	−0.746393	−0.373197	−	0.927752i	$-0.621738\pi$
$577$	−10345.0	−0.746393	−0.373197	+	0.927752i	$0.621738\pi$
$578$	0	0
$579$	−12679.5	−0.910088
$580$	0	0
$581$	1475.23	0.105341
$582$	0	0
$583$	7611.22	0.540694
$584$	0	0
$585$	338.698	0.0239375
$586$	0	0
$587$	14295.9	1.00520	0.502601	−	0.864518i	$-0.332376\pi$
$587$	14295.9	1.00520	0.502601	+	0.864518i	$0.332376\pi$
$588$	0	0
$589$	4465.25	0.312372
$590$	0	0
$591$	−10443.7	−0.726899
$592$	0	0
$593$	1637.20	0.113376	0.0566878	−	0.998392i	$-0.481946\pi$
$593$	1637.20	0.113376	0.0566878	+	0.998392i	$0.481946\pi$
$594$	0	0
$595$	−6217.66	−0.428402
$596$	0	0
$597$	−5487.82	−0.376217
$598$	0	0
$599$	12291.3	0.838415	0.419208	−	0.907890i	$-0.362308\pi$
$599$	12291.3	0.838415	0.419208	+	0.907890i	$0.362308\pi$
$600$	0	0
$601$	16840.4	1.14298	0.571492	−	0.820608i	$-0.306365\pi$
$601$	16840.4	1.14298	0.571492	+	0.820608i	$0.306365\pi$
$602$	0	0
$603$	2706.31	0.182769
$604$	0	0
$605$	−990.070	−0.0665324
$606$	0	0
$607$	−14037.1	−0.938627	−0.469313	−	0.883032i	$-0.655499\pi$
$607$	−14037.1	−0.938627	−0.469313	+	0.883032i	$0.655499\pi$
$608$	0	0
$609$	−2803.80	−0.186561
$610$	0	0
$611$	−373.410	−0.0247243
$612$	0	0
$613$	−8255.31	−0.543929	−0.271965	−	0.962307i	$-0.587673\pi$
$613$	−8255.31	−0.543929	−0.271965	+	0.962307i	$0.587673\pi$
$614$	0	0
$615$	−8341.69	−0.546942
$616$	0	0
$617$	−25428.4	−1.65917	−0.829587	−	0.558377i	$-0.811424\pi$
$617$	−25428.4	−1.65917	−0.829587	+	0.558377i	$0.811424\pi$
$618$	0	0
$619$	−15165.2	−0.984716	−0.492358	−	0.870393i	$-0.663865\pi$
$619$	−15165.2	−0.984716	−0.492358	+	0.870393i	$0.663865\pi$
$620$	0	0
$621$	−7468.30	−0.482596
$622$	0	0
$623$	−2302.67	−0.148081
$624$	0	0
$625$	−4999.34	−0.319958
$626$	0	0
$627$	−1978.12	−0.125994
$628$	0	0
$629$	21196.1	1.34363
$630$	0	0
$631$	30014.5	1.89359	0.946797	−	0.321832i	$-0.104299\pi$
$631$	30014.5	1.89359	0.946797	+	0.321832i	$0.104299\pi$
$632$	0	0
$633$	−1776.69	−0.111559
$634$	0	0
$635$	4632.44	0.289501
$636$	0	0
$637$	297.507	0.0185049
$638$	0	0
$639$	−1588.49	−0.0983410
$640$	0	0
$641$	−24408.0	−1.50399	−0.751996	−	0.659168i	$-0.770909\pi$
$641$	−24408.0	−1.50399	−0.751996	+	0.659168i	$0.770909\pi$
$642$	0	0
$643$	−27986.9	−1.71648	−0.858240	−	0.513248i	$-0.828442\pi$
$643$	−27986.9	−1.71648	−0.858240	+	0.513248i	$0.828442\pi$
$644$	0	0
$645$	626.984	0.0382751
$646$	0	0
$647$	−13488.5	−0.819610	−0.409805	−	0.912173i	$-0.634403\pi$
$647$	−13488.5	−0.819610	−0.409805	+	0.912173i	$0.634403\pi$
$648$	0	0
$649$	−2648.26	−0.160174
$650$	0	0
$651$	−3507.98	−0.211196
$652$	0	0
$653$	−19495.0	−1.16830	−0.584148	−	0.811647i	$-0.698571\pi$
$653$	−19495.0	−1.16830	−0.584148	+	0.811647i	$0.698571\pi$
$654$	0	0
$655$	−5293.95	−0.315804
$656$	0	0
$657$	−295.742	−0.0175616
$658$	0	0
$659$	13211.9	0.780978	0.390489	−	0.920608i	$-0.372306\pi$
$659$	13211.9	0.780978	0.390489	+	0.920608i	$0.372306\pi$
$660$	0	0
$661$	16876.6	0.993077	0.496539	−	0.868015i	$-0.334604\pi$
$661$	16876.6	0.993077	0.496539	+	0.868015i	$0.334604\pi$
$662$	0	0
$663$	2960.98	0.173446
$664$	0	0
$665$	−2292.72	−0.133696
$666$	0	0
$667$	4382.83	0.254429
$668$	0	0
$669$	19228.7	1.11125
$670$	0	0
$671$	7640.14	0.439560
$672$	0	0
$673$	20616.1	1.18082	0.590411	−	0.807103i	$-0.298966\pi$
$673$	20616.1	1.18082	0.590411	+	0.807103i	$0.298966\pi$
$674$	0	0
$675$	−8819.00	−0.502879
$676$	0	0
$677$	−26690.2	−1.51519	−0.757597	−	0.652723i	$-0.773627\pi$
$677$	−26690.2	−1.51519	−0.757597	+	0.652723i	$0.773627\pi$
$678$	0	0
$679$	−12361.3	−0.698651
$680$	0	0
$681$	−12732.2	−0.716448
$682$	0	0
$683$	18297.1	1.02506	0.512532	−	0.858668i	$-0.328708\pi$
$683$	18297.1	1.02506	0.512532	+	0.858668i	$0.328708\pi$
$684$	0	0
$685$	3398.89	0.189584
$686$	0	0
$687$	−17461.4	−0.969716
$688$	0	0
$689$	−4201.09	−0.232292
$690$	0	0
$691$	11807.2	0.650022	0.325011	−	0.945710i	$-0.394632\pi$
$691$	11807.2	0.650022	0.325011	+	0.945710i	$0.394632\pi$
$692$	0	0
$693$	−524.955	−0.0287755
$694$	0	0
$695$	16410.1	0.895641
$696$	0	0
$697$	24634.0	1.33871
$698$	0	0
$699$	−2556.08	−0.138311
$700$	0	0
$701$	−3279.56	−0.176701	−0.0883503	−	0.996089i	$-0.528159\pi$
$701$	−3279.56	−0.176701	−0.0883503	+	0.996089i	$0.528159\pi$
$702$	0	0
$703$	7815.93	0.419322
$704$	0	0
$705$	−2260.75	−0.120773
$706$	0	0
$707$	7269.66	0.386709
$708$	0	0
$709$	5192.80	0.275063	0.137531	−	0.990497i	$-0.456083\pi$
$709$	5192.80	0.275063	0.137531	+	0.990497i	$0.456083\pi$
$710$	0	0
$711$	−1571.20	−0.0828756
$712$	0	0
$713$	5483.59	0.288025
$714$	0	0
$715$	546.480	0.0285835
$716$	0	0
$717$	21049.0	1.09636
$718$	0	0
$719$	18770.1	0.973581	0.486791	−	0.873519i	$-0.338167\pi$
$719$	18770.1	0.973581	0.486791	+	0.873519i	$0.338167\pi$
$720$	0	0
$721$	−12298.3	−0.635246
$722$	0	0
$723$	−30495.1	−1.56864
$724$	0	0
$725$	5175.50	0.265122
$726$	0	0
$727$	−10399.6	−0.530537	−0.265269	−	0.964175i	$-0.585461\pi$
$727$	−10399.6	−0.530537	−0.265269	+	0.964175i	$0.585461\pi$
$728$	0	0
$729$	21826.2	1.10889
$730$	0	0
$731$	−1851.56	−0.0936831
$732$	0	0
$733$	22751.6	1.14645	0.573226	−	0.819397i	$-0.305692\pi$
$733$	22751.6	1.14645	0.573226	+	0.819397i	$0.305692\pi$
$734$	0	0
$735$	1801.20	0.0903924
$736$	0	0
$737$	4366.56	0.218242
$738$	0	0
$739$	−15447.5	−0.768936	−0.384468	−	0.923138i	$-0.625615\pi$
$739$	−15447.5	−0.768936	−0.384468	+	0.923138i	$0.625615\pi$
$740$	0	0
$741$	1091.84	0.0541293
$742$	0	0
$743$	36291.9	1.79195	0.895976	−	0.444103i	$-0.146478\pi$
$743$	36291.9	1.79195	0.895976	+	0.444103i	$0.146478\pi$
$744$	0	0
$745$	−10404.8	−0.511681
$746$	0	0
$747$	1436.79	0.0703741
$748$	0	0
$749$	4962.87	0.242108
$750$	0	0
$751$	21750.9	1.05686	0.528429	−	0.848978i	$-0.322781\pi$
$751$	21750.9	1.05686	0.528429	+	0.848978i	$0.322781\pi$
$752$	0	0
$753$	3992.95	0.193242
$754$	0	0
$755$	−21711.5	−1.04657
$756$	0	0
$757$	13059.9	0.627040	0.313520	−	0.949582i	$-0.398492\pi$
$757$	13059.9	0.627040	0.313520	+	0.949582i	$0.398492\pi$
$758$	0	0
$759$	−2429.25	−0.116174
$760$	0	0
$761$	−6072.64	−0.289268	−0.144634	−	0.989485i	$-0.546200\pi$
$761$	−6072.64	−0.289268	−0.144634	+	0.989485i	$0.546200\pi$
$762$	0	0
$763$	4419.15	0.209678
$764$	0	0
$765$	−6055.65	−0.286199
$766$	0	0
$767$	1461.73	0.0688137
$768$	0	0
$769$	−7464.45	−0.350033	−0.175016	−	0.984566i	$-0.555998\pi$
$769$	−7464.45	−0.350033	−0.175016	+	0.984566i	$0.555998\pi$
$770$	0	0
$771$	−2306.21	−0.107725
$772$	0	0
$773$	−15112.2	−0.703165	−0.351583	−	0.936157i	$-0.614356\pi$
$773$	−15112.2	−0.703165	−0.351583	+	0.936157i	$0.614356\pi$
$774$	0	0
$775$	6475.34	0.300131
$776$	0	0
$777$	−6140.34	−0.283505
$778$	0	0
$779$	9083.63	0.417786
$780$	0	0
$781$	−2562.99	−0.117428
$782$	0	0
$783$	−13545.4	−0.618228
$784$	0	0
$785$	−8073.67	−0.367085
$786$	0	0
$787$	−16109.1	−0.729642	−0.364821	−	0.931078i	$-0.618870\pi$
$787$	−16109.1	−0.729642	−0.364821	+	0.931078i	$0.618870\pi$
$788$	0	0
$789$	11541.3	0.520763
$790$	0	0
$791$	4859.94	0.218457
$792$	0	0
$793$	−4217.06	−0.188842
$794$	0	0
$795$	−25434.8	−1.13469
$796$	0	0
$797$	17206.3	0.764717	0.382358	−	0.924014i	$-0.375112\pi$
$797$	17206.3	0.764717	0.382358	+	0.924014i	$0.375112\pi$
$798$	0	0
$799$	6676.27	0.295606
$800$	0	0
$801$	−2242.67	−0.0989275
$802$	0	0
$803$	−477.170	−0.0209701
$804$	0	0
$805$	−2815.60	−0.123276
$806$	0	0
$807$	22716.0	0.990880
$808$	0	0
$809$	−23746.9	−1.03201	−0.516006	−	0.856585i	$-0.672582\pi$
$809$	−23746.9	−1.03201	−0.516006	+	0.856585i	$0.672582\pi$
$810$	0	0
$811$	−23541.5	−1.01930	−0.509651	−	0.860381i	$-0.670225\pi$
$811$	−23541.5	−1.01930	−0.509651	+	0.860381i	$0.670225\pi$
$812$	0	0
$813$	3157.89	0.136226
$814$	0	0
$815$	−6610.27	−0.284108
$816$	0	0
$817$	−682.750	−0.0292367
$818$	0	0
$819$	289.755	0.0123625
$820$	0	0
$821$	−28342.3	−1.20482	−0.602408	−	0.798188i	$-0.705792\pi$
$821$	−28342.3	−1.20482	−0.602408	+	0.798188i	$0.705792\pi$
$822$	0	0
$823$	21565.3	0.913388	0.456694	−	0.889624i	$-0.349033\pi$
$823$	21565.3	0.913388	0.456694	+	0.889624i	$0.349033\pi$
$824$	0	0
$825$	−2868.59	−0.121056
$826$	0	0
$827$	−46430.8	−1.95230	−0.976152	−	0.217087i	$-0.930344\pi$
$827$	−46430.8	−1.95230	−0.976152	+	0.217087i	$0.930344\pi$
$828$	0	0
$829$	11019.8	0.461679	0.230839	−	0.972992i	$-0.425853\pi$
$829$	11019.8	0.461679	0.230839	+	0.972992i	$0.425853\pi$
$830$	0	0
$831$	13623.5	0.568705
$832$	0	0
$833$	−5319.18	−0.221247
$834$	0	0
$835$	−18570.0	−0.769629
$836$	0	0
$837$	−16947.4	−0.699864
$838$	0	0
$839$	−19541.4	−0.804107	−0.402053	−	0.915616i	$-0.631703\pi$
$839$	−19541.4	−0.804107	−0.402053	+	0.915616i	$0.631703\pi$
$840$	0	0
$841$	−16439.8	−0.674065
$842$	0	0
$843$	−25126.5	−1.02658
$844$	0	0
$845$	17675.1	0.719576
$846$	0	0
$847$	−847.000	−0.0343604
$848$	0	0
$849$	27145.6	1.09733
$850$	0	0
$851$	9598.43	0.386639
$852$	0	0
$853$	5718.29	0.229532	0.114766	−	0.993393i	$-0.463388\pi$
$853$	5718.29	0.229532	0.114766	+	0.993393i	$0.463388\pi$
$854$	0	0
$855$	−2232.98	−0.0893173
$856$	0	0
$857$	15047.3	0.599774	0.299887	−	0.953975i	$-0.403051\pi$
$857$	15047.3	0.599774	0.299887	+	0.953975i	$0.403051\pi$
$858$	0	0
$859$	5866.22	0.233007	0.116503	−	0.993190i	$-0.462831\pi$
$859$	5866.22	0.233007	0.116503	+	0.993190i	$0.462831\pi$
$860$	0	0
$861$	−7136.27	−0.282466
$862$	0	0
$863$	662.351	0.0261260	0.0130630	−	0.999915i	$-0.495842\pi$
$863$	662.351	0.0261260	0.0130630	+	0.999915i	$0.495842\pi$
$864$	0	0
$865$	24792.5	0.974531
$866$	0	0
$867$	−30868.3	−1.20916
$868$	0	0
$869$	−2535.08	−0.0989606
$870$	0	0
$871$	−2410.17	−0.0937605
$872$	0	0
$873$	−12039.2	−0.466742
$874$	0	0
$875$	−10484.4	−0.405072
$876$	0	0
$877$	−24887.2	−0.958247	−0.479124	−	0.877747i	$-0.659045\pi$
$877$	−24887.2	−0.958247	−0.479124	+	0.877747i	$0.659045\pi$
$878$	0	0
$879$	−26376.7	−1.01213
$880$	0	0
$881$	−11782.4	−0.450577	−0.225289	−	0.974292i	$-0.572333\pi$
$881$	−11782.4	−0.450577	−0.225289	+	0.974292i	$0.572333\pi$
$882$	0	0
$883$	26577.4	1.01291	0.506456	−	0.862266i	$-0.330955\pi$
$883$	26577.4	1.01291	0.506456	+	0.862266i	$0.330955\pi$
$884$	0	0
$885$	8849.82	0.336140
$886$	0	0
$887$	40493.4	1.53285	0.766424	−	0.642335i	$-0.222034\pi$
$887$	40493.4	1.53285	0.766424	+	0.642335i	$0.222034\pi$
$888$	0	0
$889$	3963.03	0.149512
$890$	0	0
$891$	5482.90	0.206155
$892$	0	0
$893$	2461.83	0.0922530
$894$	0	0
$895$	16473.4	0.615247
$896$	0	0
$897$	1340.85	0.0499104
$898$	0	0
$899$	9945.70	0.368974
$900$	0	0
$901$	75112.1	2.77730
$902$	0	0
$903$	536.381	0.0197671
$904$	0	0
$905$	−7347.02	−0.269860
$906$	0	0
$907$	160.206	0.00586498	0.00293249	−	0.999996i	$-0.499067\pi$
$907$	160.206	0.00586498	0.00293249	+	0.999996i	$0.499067\pi$
$908$	0	0
$909$	7080.23	0.258346
$910$	0	0
$911$	2990.04	0.108743	0.0543713	−	0.998521i	$-0.482685\pi$
$911$	2990.04	0.108743	0.0543713	+	0.998521i	$0.482685\pi$
$912$	0	0
$913$	2318.22	0.0840328
$914$	0	0
$915$	−25531.5	−0.922453
$916$	0	0
$917$	−4528.94	−0.163096
$918$	0	0
$919$	1276.07	0.0458036	0.0229018	−	0.999738i	$-0.492709\pi$
$919$	1276.07	0.0458036	0.0229018	+	0.999738i	$0.492709\pi$
$920$	0	0
$921$	13657.4	0.488630
$922$	0	0
$923$	1414.67	0.0504490
$924$	0	0
$925$	11334.4	0.402889
$926$	0	0
$927$	−11977.8	−0.424384
$928$	0	0
$929$	33935.0	1.19846	0.599231	−	0.800576i	$-0.295473\pi$
$929$	33935.0	1.19846	0.599231	+	0.800576i	$0.295473\pi$
$930$	0	0
$931$	−1961.41	−0.0690469
$932$	0	0
$933$	−16588.1	−0.582069
$934$	0	0
$935$	−9770.61	−0.341747
$936$	0	0
$937$	22344.1	0.779029	0.389515	−	0.921020i	$-0.372643\pi$
$937$	22344.1	0.779029	0.389515	+	0.921020i	$0.372643\pi$
$938$	0	0
$939$	24757.4	0.860412
$940$	0	0
$941$	−23429.8	−0.811677	−0.405839	−	0.913945i	$-0.633020\pi$
$941$	−23429.8	−0.811677	−0.405839	+	0.913945i	$0.633020\pi$
$942$	0	0
$943$	11155.3	0.385223
$944$	0	0
$945$	8701.77	0.299544
$946$	0	0
$947$	−55034.4	−1.88847	−0.944234	−	0.329276i	$-0.893195\pi$
$947$	−55034.4	−1.88847	−0.944234	+	0.329276i	$0.893195\pi$
$948$	0	0
$949$	263.379	0.00900912
$950$	0	0
$951$	−23766.4	−0.810387
$952$	0	0
$953$	8403.70	0.285648	0.142824	−	0.989748i	$-0.454382\pi$
$953$	8403.70	0.285648	0.142824	+	0.989748i	$0.454382\pi$
$954$	0	0
$955$	−14121.8	−0.478504
$956$	0	0
$957$	−4405.97	−0.148824
$958$	0	0
$959$	2907.73	0.0979098
$960$	0	0
$961$	−17347.4	−0.582304
$962$	0	0
$963$	4833.55	0.161744
$964$	0	0
$965$	−23093.8	−0.770379
$966$	0	0
$967$	42446.0	1.41155	0.705776	−	0.708435i	$-0.250598\pi$
$967$	42446.0	1.41155	0.705776	+	0.708435i	$0.250598\pi$
$968$	0	0
$969$	−19521.2	−0.647175
$970$	0	0
$971$	23798.6	0.786544	0.393272	−	0.919422i	$-0.371343\pi$
$971$	23798.6	0.786544	0.393272	+	0.919422i	$0.371343\pi$
$972$	0	0
$973$	14038.8	0.462551
$974$	0	0
$975$	1583.35	0.0520080
$976$	0	0
$977$	8379.23	0.274386	0.137193	−	0.990544i	$-0.456192\pi$
$977$	8379.23	0.274386	0.137193	+	0.990544i	$0.456192\pi$
$978$	0	0
$979$	−3618.48	−0.118128
$980$	0	0
$981$	4304.00	0.140078
$982$	0	0
$983$	36510.6	1.18464	0.592322	−	0.805701i	$-0.298211\pi$
$983$	36510.6	1.18464	0.592322	+	0.805701i	$0.298211\pi$
$984$	0	0
$985$	−19021.7	−0.615312
$986$	0	0
$987$	−1934.06	−0.0623726
$988$	0	0
$989$	−838.459	−0.0269580
$990$	0	0
$991$	−41796.2	−1.33976	−0.669880	−	0.742470i	$-0.733654\pi$
$991$	−41796.2	−1.33976	−0.669880	+	0.742470i	$0.733654\pi$
$992$	0	0
$993$	44698.3	1.42846
$994$	0	0
$995$	−9995.27	−0.318464
$996$	0	0
$997$	31523.2	1.00136	0.500678	−	0.865634i	$-0.333084\pi$
$997$	31523.2	1.00136	0.500678	+	0.865634i	$0.333084\pi$
$998$	0	0
$999$	−29664.5	−0.939483

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.u.1.1		4
4.3	odd	2		616.4.a.f.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.4.a.f.1.4	✓	4	4.3	odd	2
1232.4.a.u.1.1		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1232.a (trivial)

\(f(q)\)	\(=\)	\(q-4.49248 q^{3} -8.18240 q^{5} -7.00000 q^{7} -6.81760 q^{9} -11.0000 q^{11} +6.07157 q^{13} +36.7593 q^{15} -108.555 q^{17} -40.0288 q^{19} +31.4474 q^{21} -49.1578 q^{23} -58.0484 q^{25} +151.925 q^{27} -89.1584 q^{29} -111.551 q^{31} +49.4173 q^{33} +57.2768 q^{35} -195.258 q^{37} -27.2764 q^{39} -226.927 q^{41} +17.0565 q^{43} +55.7843 q^{45} -61.5015 q^{47} +49.0000 q^{49} +487.680 q^{51} -691.929 q^{53} +90.0064 q^{55} +179.829 q^{57} +240.751 q^{59} -694.558 q^{61} +47.7232 q^{63} -49.6800 q^{65} -396.960 q^{67} +220.840 q^{69} +232.999 q^{71} +43.3791 q^{73} +260.781 q^{75} +77.0000 q^{77} +230.462 q^{79} -498.445 q^{81} -210.747 q^{83} +888.237 q^{85} +400.543 q^{87} +328.953 q^{89} -42.5010 q^{91} +501.140 q^{93} +327.532 q^{95} +1765.90 q^{97} +74.9936 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} - 13 q^{5} - 28 q^{7} - 47 q^{9} - 44 q^{11} - 34 q^{13} - 63 q^{15} - 58 q^{17} - 60 q^{19} - 21 q^{21} + 93 q^{23} - 19 q^{25} - 63 q^{27} - 144 q^{29} + 129 q^{31} - 33 q^{33} + 91 q^{35}+ \cdots + 517 q^{99}+O(q^{100}) \) 4 * q + 3 * q^3 - 13 * q^5 - 28 * q^7 - 47 * q^9 - 44 * q^11 - 34 * q^13 - 63 * q^15 - 58 * q^17 - 60 * q^19 - 21 * q^21 + 93 * q^23 - 19 * q^25 - 63 * q^27 - 144 * q^29 + 129 * q^31 - 33 * q^33 + 91 * q^35 - 187 * q^37 + 244 * q^39 + 110 * q^41 + 360 * q^43 - 286 * q^45 + 438 * q^47 + 196 * q^49 + 902 * q^51 + 56 * q^53 + 143 * q^55 - 94 * q^57 + 1209 * q^59 + 104 * q^61 + 329 * q^63 - 1256 * q^65 + 1075 * q^67 + 451 * q^69 + 963 * q^71 - 646 * q^73 + 804 * q^75 + 308 * q^77 + 1838 * q^79 - 656 * q^81 + 1238 * q^83 - 278 * q^85 + 914 * q^87 - 1453 * q^89 + 238 * q^91 + 521 * q^93 + 2730 * q^95 + 573 * q^97 + 517 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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