LMFDB - Embedded newform 1232.4.a.t.1.4

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:	$1$
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} - 40x^{2} - 41x + 194 $ x^4 - 40x^2 - 41x + 194
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.80678$ 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)

$f(q)$	$=$	$q+8.36227 q^{3} -15.9758 q^{5} +7.00000 q^{7} +42.9275 q^{9} +O(q^{10})$	$q+8.36227 q^{3} -15.9758 q^{5} +7.00000 q^{7} +42.9275 q^{9} -11.0000 q^{11} -31.6092 q^{13} -133.594 q^{15} +78.5997 q^{17} -23.6753 q^{19} +58.5359 q^{21} -97.0623 q^{23} +130.227 q^{25} +133.190 q^{27} +25.6918 q^{29} -22.3475 q^{31} -91.9849 q^{33} -111.831 q^{35} -124.529 q^{37} -264.325 q^{39} -227.017 q^{41} -369.568 q^{43} -685.803 q^{45} -358.814 q^{47} +49.0000 q^{49} +657.272 q^{51} +95.6899 q^{53} +175.734 q^{55} -197.979 q^{57} +740.892 q^{59} -215.974 q^{61} +300.492 q^{63} +504.984 q^{65} +845.919 q^{67} -811.661 q^{69} -334.009 q^{71} -786.180 q^{73} +1088.99 q^{75} -77.0000 q^{77} -1087.95 q^{79} -45.2723 q^{81} -327.808 q^{83} -1255.70 q^{85} +214.842 q^{87} -818.881 q^{89} -221.265 q^{91} -186.876 q^{93} +378.233 q^{95} -297.061 q^{97} -472.202 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} - 19 q^{5} + 28 q^{7} + 37 q^{9}+O(q^{10}) $ 4 * q + 3 * q^3 - 19 * q^5 + 28 * q^7 + 37 * q^9	$ 4 q + 3 q^{3} - 19 q^{5} + 28 q^{7} + 37 q^{9} - 44 q^{11} + 2 q^{13} - 71 q^{15} + 8 q^{17} + 6 q^{19} + 21 q^{21} - 159 q^{23} - 119 q^{25} + 117 q^{27} + 144 q^{29} + 183 q^{31} - 33 q^{33} - 133 q^{35} - 475 q^{37} + 64 q^{39} - 768 q^{41} + 152 q^{43} - 1048 q^{45} + 228 q^{47} + 196 q^{49} + 882 q^{51} + 396 q^{53} + 209 q^{55} - 1286 q^{57} + 733 q^{59} - 1012 q^{61} + 259 q^{63} + 420 q^{65} + 171 q^{67} - 321 q^{69} + 1019 q^{71} - 1836 q^{73} + 1016 q^{75} - 308 q^{77} - 1374 q^{79} - 956 q^{81} + 1248 q^{83} + 46 q^{85} - 2238 q^{87} - 1401 q^{89} + 14 q^{91} - 1859 q^{93} - 238 q^{95} - 2559 q^{97} - 407 q^{99}+O(q^{100}) $ 4 * q + 3 * q^3 - 19 * q^5 + 28 * q^7 + 37 * q^9 - 44 * q^11 + 2 * q^13 - 71 * q^15 + 8 * q^17 + 6 * q^19 + 21 * q^21 - 159 * q^23 - 119 * q^25 + 117 * q^27 + 144 * q^29 + 183 * q^31 - 33 * q^33 - 133 * q^35 - 475 * q^37 + 64 * q^39 - 768 * q^41 + 152 * q^43 - 1048 * q^45 + 228 * q^47 + 196 * q^49 + 882 * q^51 + 396 * q^53 + 209 * q^55 - 1286 * q^57 + 733 * q^59 - 1012 * q^61 + 259 * q^63 + 420 * q^65 + 171 * q^67 - 321 * q^69 + 1019 * q^71 - 1836 * q^73 + 1016 * q^75 - 308 * q^77 - 1374 * q^79 - 956 * q^81 + 1248 * q^83 + 46 * q^85 - 2238 * q^87 - 1401 * q^89 + 14 * q^91 - 1859 * q^93 - 238 * q^95 - 2559 * q^97 - 407 * 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
$3$	8.36227	1.60932	0.804659	−	0.593737i	$-0.202348\pi$
$3$	8.36227	1.60932	0.804659	+	0.593737i	$0.202348\pi$
$4$	0	0
$5$	−15.9758	−1.42892	−0.714461	−	0.699675i	$-0.753328\pi$
$5$	−15.9758	−1.42892	−0.714461	+	0.699675i	$0.753328\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	42.9275	1.58991
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−31.6092	−0.674371	−0.337186	−	0.941438i	$-0.609475\pi$
$13$	−31.6092	−0.674371	−0.337186	+	0.941438i	$0.609475\pi$
$14$	0	0
$15$	−133.594	−2.29959
$16$	0	0
$17$	78.5997	1.12137	0.560683	−	0.828030i	$-0.310539\pi$
$17$	78.5997	1.12137	0.560683	+	0.828030i	$0.310539\pi$
$18$	0	0
$19$	−23.6753	−0.285868	−0.142934	−	0.989732i	$-0.545654\pi$
$19$	−23.6753	−0.285868	−0.142934	+	0.989732i	$0.545654\pi$
$20$	0	0
$21$	58.5359	0.608265
$22$	0	0
$23$	−97.0623	−0.879952	−0.439976	−	0.898010i	$-0.645013\pi$
$23$	−97.0623	−0.879952	−0.439976	+	0.898010i	$0.645013\pi$
$24$	0	0
$25$	130.227	1.04182
$26$	0	0
$27$	133.190	0.949349
$28$	0	0
$29$	25.6918	0.164512	0.0822560	−	0.996611i	$-0.473787\pi$
$29$	25.6918	0.164512	0.0822560	+	0.996611i	$0.473787\pi$
$30$	0	0
$31$	−22.3475	−0.129475	−0.0647377	−	0.997902i	$-0.520621\pi$
$31$	−22.3475	−0.129475	−0.0647377	+	0.997902i	$0.520621\pi$
$32$	0	0
$33$	−91.9849	−0.485228
$34$	0	0
$35$	−111.831	−0.540082
$36$	0	0
$37$	−124.529	−0.553307	−0.276654	−	0.960970i	$-0.589225\pi$
$37$	−124.529	−0.553307	−0.276654	+	0.960970i	$0.589225\pi$
$38$	0	0
$39$	−264.325	−1.08528
$40$	0	0
$41$	−227.017	−0.864733	−0.432367	−	0.901698i	$-0.642321\pi$
$41$	−227.017	−0.864733	−0.432367	+	0.901698i	$0.642321\pi$
$42$	0	0
$43$	−369.568	−1.31067	−0.655333	−	0.755340i	$-0.727472\pi$
$43$	−369.568	−1.31067	−0.655333	+	0.755340i	$0.727472\pi$
$44$	0	0
$45$	−685.803	−2.27185
$46$	0	0
$47$	−358.814	−1.11358	−0.556791	−	0.830652i	$-0.687968\pi$
$47$	−358.814	−1.11358	−0.556791	+	0.830652i	$0.687968\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	657.272	1.80464
$52$	0	0
$53$	95.6899	0.248000	0.124000	−	0.992282i	$-0.460428\pi$
$53$	95.6899	0.248000	0.124000	+	0.992282i	$0.460428\pi$
$54$	0	0
$55$	175.734	0.430836
$56$	0	0
$57$	−197.979	−0.460053
$58$	0	0
$59$	740.892	1.63485	0.817423	−	0.576038i	$-0.195402\pi$
$59$	740.892	1.63485	0.817423	+	0.576038i	$0.195402\pi$
$60$	0	0
$61$	−215.974	−0.453323	−0.226661	−	0.973974i	$-0.572781\pi$
$61$	−215.974	−0.453323	−0.226661	+	0.973974i	$0.572781\pi$
$62$	0	0
$63$	300.492	0.600928
$64$	0	0
$65$	504.984	0.963624
$66$	0	0
$67$	845.919	1.54247	0.771234	−	0.636551i	$-0.219640\pi$
$67$	845.919	1.54247	0.771234	+	0.636551i	$0.219640\pi$
$68$	0	0
$69$	−811.661	−1.41612
$70$	0	0
$71$	−334.009	−0.558304	−0.279152	−	0.960247i	$-0.590053\pi$
$71$	−334.009	−0.558304	−0.279152	+	0.960247i	$0.590053\pi$
$72$	0	0
$73$	−786.180	−1.26049	−0.630243	−	0.776398i	$-0.717045\pi$
$73$	−786.180	−1.26049	−0.630243	+	0.776398i	$0.717045\pi$
$74$	0	0
$75$	1088.99	1.67662
$76$	0	0
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−1087.95	−1.54942	−0.774712	−	0.632315i	$-0.782105\pi$
$79$	−1087.95	−1.54942	−0.774712	+	0.632315i	$0.782105\pi$
$80$	0	0
$81$	−45.2723	−0.0621020
$82$	0	0
$83$	−327.808	−0.433514	−0.216757	−	0.976226i	$-0.569548\pi$
$83$	−327.808	−0.433514	−0.216757	+	0.976226i	$0.569548\pi$
$84$	0	0
$85$	−1255.70	−1.60235
$86$	0	0
$87$	214.842	0.264752
$88$	0	0
$89$	−818.881	−0.975295	−0.487648	−	0.873041i	$-0.662145\pi$
$89$	−818.881	−0.975295	−0.487648	+	0.873041i	$0.662145\pi$
$90$	0	0
$91$	−221.265	−0.254888
$92$	0	0
$93$	−186.876	−0.208367
$94$	0	0
$95$	378.233	0.408483
$96$	0	0
$97$	−297.061	−0.310948	−0.155474	−	0.987840i	$-0.549691\pi$
$97$	−297.061	−0.310948	−0.155474	+	0.987840i	$0.549691\pi$
$98$	0	0
$99$	−472.202	−0.479375
$100$	0	0
$101$	−906.772	−0.893339	−0.446669	−	0.894699i	$-0.647390\pi$
$101$	−906.772	−0.893339	−0.446669	+	0.894699i	$0.647390\pi$
$102$	0	0
$103$	−1515.86	−1.45011	−0.725057	−	0.688688i	$-0.758187\pi$
$103$	−1515.86	−1.45011	−0.725057	+	0.688688i	$0.758187\pi$
$104$	0	0
$105$	−935.159	−0.869164
$106$	0	0
$107$	−413.671	−0.373749	−0.186874	−	0.982384i	$-0.559836\pi$
$107$	−413.671	−0.373749	−0.186874	+	0.982384i	$0.559836\pi$
$108$	0	0
$109$	2164.77	1.90227	0.951136	−	0.308772i	$-0.0999180\pi$
$109$	2164.77	1.90227	0.951136	+	0.308772i	$0.0999180\pi$
$110$	0	0
$111$	−1041.34	−0.890448
$112$	0	0
$113$	−1540.59	−1.28254	−0.641269	−	0.767317i	$-0.721592\pi$
$113$	−1540.59	−1.28254	−0.641269	+	0.767317i	$0.721592\pi$
$114$	0	0
$115$	1550.65	1.25738
$116$	0	0
$117$	−1356.91	−1.07219
$118$	0	0
$119$	550.198	0.423837
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−1898.37	−1.39163
$124$	0	0
$125$	−83.5094	−0.0597545
$126$	0	0
$127$	−2375.64	−1.65987	−0.829936	−	0.557859i	$-0.811623\pi$
$127$	−2375.64	−1.65987	−0.829936	+	0.557859i	$0.811623\pi$
$128$	0	0
$129$	−3090.43	−2.10928
$130$	0	0
$131$	773.841	0.516113	0.258056	−	0.966130i	$-0.416918\pi$
$131$	773.841	0.516113	0.258056	+	0.966130i	$0.416918\pi$
$132$	0	0
$133$	−165.727	−0.108048
$134$	0	0
$135$	−2127.82	−1.35655
$136$	0	0
$137$	794.520	0.495477	0.247739	−	0.968827i	$-0.420313\pi$
$137$	794.520	0.495477	0.247739	+	0.968827i	$0.420313\pi$
$138$	0	0
$139$	1661.91	1.01411	0.507056	−	0.861913i	$-0.330734\pi$
$139$	1661.91	1.01411	0.507056	+	0.861913i	$0.330734\pi$
$140$	0	0
$141$	−3000.50	−1.79211
$142$	0	0
$143$	347.702	0.203331
$144$	0	0
$145$	−410.448	−0.235075
$146$	0	0
$147$	409.751	0.229903
$148$	0	0
$149$	−3078.53	−1.69264	−0.846319	−	0.532676i	$-0.821186\pi$
$149$	−3078.53	−1.69264	−0.846319	+	0.532676i	$0.821186\pi$
$150$	0	0
$151$	2691.57	1.45058	0.725288	−	0.688446i	$-0.241707\pi$
$151$	2691.57	1.45058	0.725288	+	0.688446i	$0.241707\pi$
$152$	0	0
$153$	3374.09	1.78287
$154$	0	0
$155$	357.021	0.185010
$156$	0	0
$157$	−727.129	−0.369626	−0.184813	−	0.982774i	$-0.559168\pi$
$157$	−727.129	−0.369626	−0.184813	+	0.982774i	$0.559168\pi$
$158$	0	0
$159$	800.184	0.399112
$160$	0	0
$161$	−679.436	−0.332590
$162$	0	0
$163$	−2917.24	−1.40181	−0.700907	−	0.713252i	$-0.747221\pi$
$163$	−2917.24	−1.40181	−0.700907	+	0.713252i	$0.747221\pi$
$164$	0	0
$165$	1469.54	0.693353
$166$	0	0
$167$	1080.10	0.500481	0.250241	−	0.968184i	$-0.419490\pi$
$167$	1080.10	0.500481	0.250241	+	0.968184i	$0.419490\pi$
$168$	0	0
$169$	−1197.86	−0.545223
$170$	0	0
$171$	−1016.32	−0.454504
$172$	0	0
$173$	254.908	0.112025	0.0560124	−	0.998430i	$-0.482161\pi$
$173$	254.908	0.112025	0.0560124	+	0.998430i	$0.482161\pi$
$174$	0	0
$175$	911.591	0.393770
$176$	0	0
$177$	6195.53	2.63099
$178$	0	0
$179$	2748.88	1.14783	0.573913	−	0.818916i	$-0.305425\pi$
$179$	2748.88	1.14783	0.573913	+	0.818916i	$0.305425\pi$
$180$	0	0
$181$	4268.67	1.75297	0.876486	−	0.481428i	$-0.159882\pi$
$181$	4268.67	1.75297	0.876486	+	0.481428i	$0.159882\pi$
$182$	0	0
$183$	−1806.04	−0.729541
$184$	0	0
$185$	1989.45	0.790633
$186$	0	0
$187$	−864.597	−0.338105
$188$	0	0
$189$	932.330	0.358820
$190$	0	0
$191$	−1444.07	−0.547063	−0.273531	−	0.961863i	$-0.588192\pi$
$191$	−1444.07	−0.547063	−0.273531	+	0.961863i	$0.588192\pi$
$192$	0	0
$193$	−1783.70	−0.665252	−0.332626	−	0.943059i	$-0.607935\pi$
$193$	−1783.70	−0.665252	−0.332626	+	0.943059i	$0.607935\pi$
$194$	0	0
$195$	4222.81	1.55078
$196$	0	0
$197$	1432.47	0.518069	0.259034	−	0.965868i	$-0.416596\pi$
$197$	1432.47	0.518069	0.259034	+	0.965868i	$0.416596\pi$
$198$	0	0
$199$	−3325.30	−1.18454	−0.592272	−	0.805738i	$-0.701769\pi$
$199$	−3325.30	−1.18454	−0.592272	+	0.805738i	$0.701769\pi$
$200$	0	0
$201$	7073.80	2.48232
$202$	0	0
$203$	179.843	0.0621797
$204$	0	0
$205$	3626.78	1.23564
$206$	0	0
$207$	−4166.64	−1.39904
$208$	0	0
$209$	260.429	0.0861924
$210$	0	0
$211$	−1638.52	−0.534600	−0.267300	−	0.963613i	$-0.586132\pi$
$211$	−1638.52	−0.534600	−0.267300	+	0.963613i	$0.586132\pi$
$212$	0	0
$213$	−2793.07	−0.898488
$214$	0	0
$215$	5904.16	1.87284
$216$	0	0
$217$	−156.433	−0.0489371
$218$	0	0
$219$	−6574.25	−2.02852
$220$	0	0
$221$	−2484.48	−0.756217
$222$	0	0
$223$	−513.181	−0.154104	−0.0770518	−	0.997027i	$-0.524551\pi$
$223$	−513.181	−0.154104	−0.0770518	+	0.997027i	$0.524551\pi$
$224$	0	0
$225$	5590.33	1.65639
$226$	0	0
$227$	5977.46	1.74774	0.873872	−	0.486156i	$-0.161601\pi$
$227$	5977.46	1.74774	0.873872	+	0.486156i	$0.161601\pi$
$228$	0	0
$229$	1199.39	0.346103	0.173052	−	0.984913i	$-0.444637\pi$
$229$	1199.39	0.346103	0.173052	+	0.984913i	$0.444637\pi$
$230$	0	0
$231$	−643.895	−0.183399
$232$	0	0
$233$	4211.03	1.18401	0.592004	−	0.805935i	$-0.298337\pi$
$233$	4211.03	1.18401	0.592004	+	0.805935i	$0.298337\pi$
$234$	0	0
$235$	5732.35	1.59122
$236$	0	0
$237$	−9097.76	−2.49352
$238$	0	0
$239$	4224.72	1.14341	0.571704	−	0.820460i	$-0.306283\pi$
$239$	4224.72	1.14341	0.571704	+	0.820460i	$0.306283\pi$
$240$	0	0
$241$	−611.081	−0.163333	−0.0816663	−	0.996660i	$-0.526024\pi$
$241$	−611.081	−0.163333	−0.0816663	+	0.996660i	$0.526024\pi$
$242$	0	0
$243$	−3974.71	−1.04929
$244$	0	0
$245$	−782.816	−0.204132
$246$	0	0
$247$	748.359	0.192781
$248$	0	0
$249$	−2741.22	−0.697662
$250$	0	0
$251$	1370.15	0.344553	0.172276	−	0.985049i	$-0.444888\pi$
$251$	1370.15	0.344553	0.172276	+	0.985049i	$0.444888\pi$
$252$	0	0
$253$	1067.69	0.265315
$254$	0	0
$255$	−10500.5	−2.57868
$256$	0	0
$257$	2938.32	0.713180	0.356590	−	0.934261i	$-0.383939\pi$
$257$	2938.32	0.713180	0.356590	+	0.934261i	$0.383939\pi$
$258$	0	0
$259$	−871.700	−0.209130
$260$	0	0
$261$	1102.88	0.261559
$262$	0	0
$263$	421.989	0.0989390	0.0494695	−	0.998776i	$-0.484247\pi$
$263$	421.989	0.0989390	0.0494695	+	0.998776i	$0.484247\pi$
$264$	0	0
$265$	−1528.73	−0.354373
$266$	0	0
$267$	−6847.70	−1.56956
$268$	0	0
$269$	−4570.09	−1.03585	−0.517924	−	0.855426i	$-0.673295\pi$
$269$	−4570.09	−1.03585	−0.517924	+	0.855426i	$0.673295\pi$
$270$	0	0
$271$	2731.47	0.612269	0.306134	−	0.951988i	$-0.400964\pi$
$271$	2731.47	0.612269	0.306134	+	0.951988i	$0.400964\pi$
$272$	0	0
$273$	−1850.27	−0.410197
$274$	0	0
$275$	−1432.50	−0.314120
$276$	0	0
$277$	−7552.38	−1.63819	−0.819095	−	0.573658i	$-0.805524\pi$
$277$	−7552.38	−1.63819	−0.819095	+	0.573658i	$0.805524\pi$
$278$	0	0
$279$	−959.324	−0.205854
$280$	0	0
$281$	−1083.80	−0.230085	−0.115043	−	0.993361i	$-0.536700\pi$
$281$	−1083.80	−0.230085	−0.115043	+	0.993361i	$0.536700\pi$
$282$	0	0
$283$	−5198.90	−1.09202	−0.546012	−	0.837777i	$-0.683855\pi$
$283$	−5198.90	−1.09202	−0.546012	+	0.837777i	$0.683855\pi$
$284$	0	0
$285$	3162.89	0.657380
$286$	0	0
$287$	−1589.12	−0.326838
$288$	0	0
$289$	1264.91	0.257463
$290$	0	0
$291$	−2484.10	−0.500415
$292$	0	0
$293$	1246.35	0.248507	0.124253	−	0.992251i	$-0.460346\pi$
$293$	1246.35	0.248507	0.124253	+	0.992251i	$0.460346\pi$
$294$	0	0
$295$	−11836.4	−2.33607
$296$	0	0
$297$	−1465.09	−0.286240
$298$	0	0
$299$	3068.07	0.593414
$300$	0	0
$301$	−2586.98	−0.495385
$302$	0	0
$303$	−7582.67	−1.43767
$304$	0	0
$305$	3450.37	0.647763
$306$	0	0
$307$	3667.39	0.681789	0.340894	−	0.940102i	$-0.389270\pi$
$307$	3667.39	0.681789	0.340894	+	0.940102i	$0.389270\pi$
$308$	0	0
$309$	−12676.0	−2.33370
$310$	0	0
$311$	6596.96	1.20283	0.601413	−	0.798938i	$-0.294605\pi$
$311$	6596.96	1.20283	0.601413	+	0.798938i	$0.294605\pi$
$312$	0	0
$313$	6345.94	1.14599	0.572993	−	0.819560i	$-0.305782\pi$
$313$	6345.94	1.14599	0.572993	+	0.819560i	$0.305782\pi$
$314$	0	0
$315$	−4800.62	−0.858680
$316$	0	0
$317$	6078.31	1.07695	0.538473	−	0.842643i	$-0.319001\pi$
$317$	6078.31	1.07695	0.538473	+	0.842643i	$0.319001\pi$
$318$	0	0
$319$	−282.610	−0.0496022
$320$	0	0
$321$	−3459.23	−0.601481
$322$	0	0
$323$	−1860.87	−0.320563
$324$	0	0
$325$	−4116.38	−0.702572
$326$	0	0
$327$	18102.4	3.06136
$328$	0	0
$329$	−2511.70	−0.420895
$330$	0	0
$331$	5880.62	0.976520	0.488260	−	0.872698i	$-0.337632\pi$
$331$	5880.62	0.976520	0.488260	+	0.872698i	$0.337632\pi$
$332$	0	0
$333$	−5345.70	−0.879707
$334$	0	0
$335$	−13514.3	−2.20407
$336$	0	0
$337$	1068.26	0.172676	0.0863378	−	0.996266i	$-0.472484\pi$
$337$	1068.26	0.172676	0.0863378	+	0.996266i	$0.472484\pi$
$338$	0	0
$339$	−12882.8	−2.06401
$340$	0	0
$341$	245.823	0.0390383
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	12967.0	2.02353
$346$	0	0
$347$	8599.94	1.33046	0.665229	−	0.746639i	$-0.268334\pi$
$347$	8599.94	1.33046	0.665229	+	0.746639i	$0.268334\pi$
$348$	0	0
$349$	−2036.07	−0.312287	−0.156144	−	0.987734i	$-0.549906\pi$
$349$	−2036.07	−0.312287	−0.156144	+	0.987734i	$0.549906\pi$
$350$	0	0
$351$	−4210.03	−0.640214
$352$	0	0
$353$	−4278.13	−0.645047	−0.322524	−	0.946561i	$-0.604531\pi$
$353$	−4278.13	−0.645047	−0.322524	+	0.946561i	$0.604531\pi$
$354$	0	0
$355$	5336.07	0.797772
$356$	0	0
$357$	4600.90	0.682088
$358$	0	0
$359$	12106.0	1.77975	0.889873	−	0.456209i	$-0.150793\pi$
$359$	12106.0	1.77975	0.889873	+	0.456209i	$0.150793\pi$
$360$	0	0
$361$	−6298.48	−0.918279
$362$	0	0
$363$	1011.83	0.146302
$364$	0	0
$365$	12559.9	1.80113
$366$	0	0
$367$	−4321.55	−0.614668	−0.307334	−	0.951602i	$-0.599437\pi$
$367$	−4321.55	−0.614668	−0.307334	+	0.951602i	$0.599437\pi$
$368$	0	0
$369$	−9745.26	−1.37485
$370$	0	0
$371$	669.829	0.0937353
$372$	0	0
$373$	−5331.49	−0.740091	−0.370045	−	0.929014i	$-0.620658\pi$
$373$	−5331.49	−0.740091	−0.370045	+	0.929014i	$0.620658\pi$
$374$	0	0
$375$	−698.328	−0.0961640
$376$	0	0
$377$	−812.098	−0.110942
$378$	0	0
$379$	−13879.0	−1.88105	−0.940526	−	0.339723i	$-0.889667\pi$
$379$	−13879.0	−1.88105	−0.940526	+	0.339723i	$0.889667\pi$
$380$	0	0
$381$	−19865.7	−2.67126
$382$	0	0
$383$	−10311.5	−1.37570	−0.687848	−	0.725855i	$-0.741444\pi$
$383$	−10311.5	−1.37570	−0.687848	+	0.725855i	$0.741444\pi$
$384$	0	0
$385$	1230.14	0.162841
$386$	0	0
$387$	−15864.6	−2.08384
$388$	0	0
$389$	−3705.46	−0.482967	−0.241484	−	0.970405i	$-0.577634\pi$
$389$	−3705.46	−0.482967	−0.241484	+	0.970405i	$0.577634\pi$
$390$	0	0
$391$	−7629.07	−0.986748
$392$	0	0
$393$	6471.06	0.830590
$394$	0	0
$395$	17381.0	2.21400
$396$	0	0
$397$	−14332.6	−1.81192	−0.905962	−	0.423359i	$-0.860851\pi$
$397$	−14332.6	−1.81192	−0.905962	+	0.423359i	$0.860851\pi$
$398$	0	0
$399$	−1385.86	−0.173884
$400$	0	0
$401$	−1996.80	−0.248666	−0.124333	−	0.992241i	$-0.539679\pi$
$401$	−1996.80	−0.248666	−0.124333	+	0.992241i	$0.539679\pi$
$402$	0	0
$403$	706.389	0.0873145
$404$	0	0
$405$	723.263	0.0887389
$406$	0	0
$407$	1369.81	0.166828
$408$	0	0
$409$	5984.20	0.723471	0.361735	−	0.932281i	$-0.382184\pi$
$409$	5984.20	0.723471	0.361735	+	0.932281i	$0.382184\pi$
$410$	0	0
$411$	6643.98	0.797381
$412$	0	0
$413$	5186.24	0.617913
$414$	0	0
$415$	5237.01	0.619457
$416$	0	0
$417$	13897.3	1.63203
$418$	0	0
$419$	610.976	0.0712366	0.0356183	−	0.999365i	$-0.488660\pi$
$419$	610.976	0.0712366	0.0356183	+	0.999365i	$0.488660\pi$
$420$	0	0
$421$	1738.27	0.201231	0.100616	−	0.994925i	$-0.467919\pi$
$421$	1738.27	0.201231	0.100616	+	0.994925i	$0.467919\pi$
$422$	0	0
$423$	−15403.0	−1.77049
$424$	0	0
$425$	10235.8	1.16826
$426$	0	0
$427$	−1511.82	−0.171340
$428$	0	0
$429$	2907.57	0.327224
$430$	0	0
$431$	5680.70	0.634871	0.317436	−	0.948280i	$-0.397178\pi$
$431$	5680.70	0.634871	0.317436	+	0.948280i	$0.397178\pi$
$432$	0	0
$433$	−5841.83	−0.648361	−0.324180	−	0.945995i	$-0.605088\pi$
$433$	−5841.83	−0.648361	−0.324180	+	0.945995i	$0.605088\pi$
$434$	0	0
$435$	−3432.27	−0.378310
$436$	0	0
$437$	2297.98	0.251550
$438$	0	0
$439$	5538.70	0.602159	0.301080	−	0.953599i	$-0.402653\pi$
$439$	5538.70	0.602159	0.301080	+	0.953599i	$0.402653\pi$
$440$	0	0
$441$	2103.45	0.227130
$442$	0	0
$443$	13206.6	1.41640	0.708198	−	0.706014i	$-0.249508\pi$
$443$	13206.6	1.41640	0.708198	+	0.706014i	$0.249508\pi$
$444$	0	0
$445$	13082.3	1.39362
$446$	0	0
$447$	−25743.5	−2.72400
$448$	0	0
$449$	−15907.6	−1.67199	−0.835996	−	0.548736i	$-0.815109\pi$
$449$	−15907.6	−1.67199	−0.835996	+	0.548736i	$0.815109\pi$
$450$	0	0
$451$	2497.18	0.260727
$452$	0	0
$453$	22507.6	2.33444
$454$	0	0
$455$	3534.89	0.364216
$456$	0	0
$457$	−11519.7	−1.17914	−0.589572	−	0.807716i	$-0.700703\pi$
$457$	−11519.7	−1.17914	−0.589572	+	0.807716i	$0.700703\pi$
$458$	0	0
$459$	10468.7	1.06457
$460$	0	0
$461$	4410.30	0.445571	0.222785	−	0.974867i	$-0.428485\pi$
$461$	4410.30	0.445571	0.222785	+	0.974867i	$0.428485\pi$
$462$	0	0
$463$	18523.9	1.85935	0.929673	−	0.368385i	$-0.120089\pi$
$463$	18523.9	1.85935	0.929673	+	0.368385i	$0.120089\pi$
$464$	0	0
$465$	2985.50	0.297740
$466$	0	0
$467$	−6399.09	−0.634078	−0.317039	−	0.948413i	$-0.602689\pi$
$467$	−6399.09	−0.634078	−0.317039	+	0.948413i	$0.602689\pi$
$468$	0	0
$469$	5921.43	0.582998
$470$	0	0
$471$	−6080.45	−0.594846
$472$	0	0
$473$	4065.25	0.395181
$474$	0	0
$475$	−3083.17	−0.297822
$476$	0	0
$477$	4107.73	0.394298
$478$	0	0
$479$	9520.12	0.908111	0.454056	−	0.890973i	$-0.349977\pi$
$479$	9520.12	0.908111	0.454056	+	0.890973i	$0.349977\pi$
$480$	0	0
$481$	3936.25	0.373134
$482$	0	0
$483$	−5681.63	−0.535244
$484$	0	0
$485$	4745.80	0.444321
$486$	0	0
$487$	−1238.63	−0.115252	−0.0576260	−	0.998338i	$-0.518353\pi$
$487$	−1238.63	−0.115252	−0.0576260	+	0.998338i	$0.518353\pi$
$488$	0	0
$489$	−24394.7	−2.25597
$490$	0	0
$491$	3474.64	0.319365	0.159682	−	0.987168i	$-0.448953\pi$
$491$	3474.64	0.319365	0.159682	+	0.987168i	$0.448953\pi$
$492$	0	0
$493$	2019.37	0.184478
$494$	0	0
$495$	7543.83	0.684990
$496$	0	0
$497$	−2338.06	−0.211019
$498$	0	0
$499$	21026.7	1.88634	0.943169	−	0.332315i	$-0.107830\pi$
$499$	21026.7	1.88634	0.943169	+	0.332315i	$0.107830\pi$
$500$	0	0
$501$	9032.05	0.805434
$502$	0	0
$503$	7902.24	0.700484	0.350242	−	0.936659i	$-0.386099\pi$
$503$	7902.24	0.700484	0.350242	+	0.936659i	$0.386099\pi$
$504$	0	0
$505$	14486.4	1.27651
$506$	0	0
$507$	−10016.8	−0.877438
$508$	0	0
$509$	9434.00	0.821522	0.410761	−	0.911743i	$-0.365263\pi$
$509$	9434.00	0.821522	0.410761	+	0.911743i	$0.365263\pi$
$510$	0	0
$511$	−5503.26	−0.476419
$512$	0	0
$513$	−3153.32	−0.271389
$514$	0	0
$515$	24217.1	2.07210
$516$	0	0
$517$	3946.95	0.335758
$518$	0	0
$519$	2131.61	0.180283
$520$	0	0
$521$	13173.1	1.10772	0.553861	−	0.832609i	$-0.313154\pi$
$521$	13173.1	1.10772	0.553861	+	0.832609i	$0.313154\pi$
$522$	0	0
$523$	−3020.73	−0.252557	−0.126279	−	0.991995i	$-0.540303\pi$
$523$	−3020.73	−0.252557	−0.126279	+	0.991995i	$0.540303\pi$
$524$	0	0
$525$	7622.96	0.633702
$526$	0	0
$527$	−1756.51	−0.145189
$528$	0	0
$529$	−2745.91	−0.225685
$530$	0	0
$531$	31804.6	2.59925
$532$	0	0
$533$	7175.83	0.583151
$534$	0	0
$535$	6608.75	0.534058
$536$	0	0
$537$	22986.9	1.84722
$538$	0	0
$539$	−539.000	−0.0430730
$540$	0	0
$541$	3415.08	0.271397	0.135699	−	0.990750i	$-0.456672\pi$
$541$	3415.08	0.271397	0.135699	+	0.990750i	$0.456672\pi$
$542$	0	0
$543$	35695.8	2.82109
$544$	0	0
$545$	−34584.1	−2.71820
$546$	0	0
$547$	−11520.6	−0.900518	−0.450259	−	0.892898i	$-0.648668\pi$
$547$	−11520.6	−0.900518	−0.450259	+	0.892898i	$0.648668\pi$
$548$	0	0
$549$	−9271.24	−0.720741
$550$	0	0
$551$	−608.262	−0.0470287
$552$	0	0
$553$	−7615.68	−0.585627
$554$	0	0
$555$	16636.3	1.27238
$556$	0	0
$557$	−12422.2	−0.944962	−0.472481	−	0.881341i	$-0.656641\pi$
$557$	−12422.2	−0.944962	−0.472481	+	0.881341i	$0.656641\pi$
$558$	0	0
$559$	11681.8	0.883876
$560$	0	0
$561$	−7229.99	−0.544118
$562$	0	0
$563$	−18530.6	−1.38716	−0.693579	−	0.720381i	$-0.743967\pi$
$563$	−18530.6	−1.38716	−0.693579	+	0.720381i	$0.743967\pi$
$564$	0	0
$565$	24612.2	1.83265
$566$	0	0
$567$	−316.906	−0.0234723
$568$	0	0
$569$	22037.4	1.62365	0.811823	−	0.583904i	$-0.198476\pi$
$569$	22037.4	1.62365	0.811823	+	0.583904i	$0.198476\pi$
$570$	0	0
$571$	10650.5	0.780580	0.390290	−	0.920692i	$-0.372375\pi$
$571$	10650.5	0.780580	0.390290	+	0.920692i	$0.372375\pi$
$572$	0	0
$573$	−12075.7	−0.880399
$574$	0	0
$575$	−12640.2	−0.916749
$576$	0	0
$577$	−14465.6	−1.04369	−0.521846	−	0.853040i	$-0.674756\pi$
$577$	−14465.6	−1.04369	−0.521846	+	0.853040i	$0.674756\pi$
$578$	0	0
$579$	−14915.8	−1.07060
$580$	0	0
$581$	−2294.66	−0.163853
$582$	0	0
$583$	−1052.59	−0.0747749
$584$	0	0
$585$	21677.7	1.53207
$586$	0	0
$587$	18990.5	1.33530	0.667652	−	0.744474i	$-0.267299\pi$
$587$	18990.5	1.33530	0.667652	+	0.744474i	$0.267299\pi$
$588$	0	0
$589$	529.085	0.0370129
$590$	0	0
$591$	11978.7	0.833738
$592$	0	0
$593$	−10033.9	−0.694845	−0.347422	−	0.937709i	$-0.612943\pi$
$593$	−10033.9	−0.694845	−0.347422	+	0.937709i	$0.612943\pi$
$594$	0	0
$595$	−8789.87	−0.605630
$596$	0	0
$597$	−27807.0	−1.90631
$598$	0	0
$599$	11775.1	0.803201	0.401601	−	0.915815i	$-0.368454\pi$
$599$	11775.1	0.803201	0.401601	+	0.915815i	$0.368454\pi$
$600$	0	0
$601$	−10328.8	−0.701030	−0.350515	−	0.936557i	$-0.613993\pi$
$601$	−10328.8	−0.701030	−0.350515	+	0.936557i	$0.613993\pi$
$602$	0	0
$603$	36313.2	2.45238
$604$	0	0
$605$	−1933.08	−0.129902
$606$	0	0
$607$	9829.63	0.657285	0.328643	−	0.944454i	$-0.393409\pi$
$607$	9829.63	0.657285	0.328643	+	0.944454i	$0.393409\pi$
$608$	0	0
$609$	1503.89	0.100067
$610$	0	0
$611$	11341.8	0.750968
$612$	0	0
$613$	−3352.44	−0.220887	−0.110443	−	0.993882i	$-0.535227\pi$
$613$	−3352.44	−0.220887	−0.110443	+	0.993882i	$0.535227\pi$
$614$	0	0
$615$	30328.1	1.98853
$616$	0	0
$617$	−15579.2	−1.01652	−0.508260	−	0.861203i	$-0.669711\pi$
$617$	−15579.2	−1.01652	−0.508260	+	0.861203i	$0.669711\pi$
$618$	0	0
$619$	15669.5	1.01747	0.508733	−	0.860924i	$-0.330114\pi$
$619$	15669.5	1.01747	0.508733	+	0.860924i	$0.330114\pi$
$620$	0	0
$621$	−12927.7	−0.835381
$622$	0	0
$623$	−5732.17	−0.368627
$624$	0	0
$625$	−14944.3	−0.956433
$626$	0	0
$627$	2177.77	0.138711
$628$	0	0
$629$	−9787.91	−0.620460
$630$	0	0
$631$	−235.955	−0.0148863	−0.00744313	−	0.999972i	$-0.502369\pi$
$631$	−235.955	−0.0148863	−0.00744313	+	0.999972i	$0.502369\pi$
$632$	0	0
$633$	−13701.8	−0.860342
$634$	0	0
$635$	37952.8	2.37183
$636$	0	0
$637$	−1548.85	−0.0963387
$638$	0	0
$639$	−14338.2	−0.887651
$640$	0	0
$641$	6720.50	0.414109	0.207055	−	0.978329i	$-0.433612\pi$
$641$	6720.50	0.414109	0.207055	+	0.978329i	$0.433612\pi$
$642$	0	0
$643$	25302.1	1.55181	0.775906	−	0.630848i	$-0.217293\pi$
$643$	25302.1	1.55181	0.775906	+	0.630848i	$0.217293\pi$
$644$	0	0
$645$	49372.2	3.01400
$646$	0	0
$647$	−17494.7	−1.06304	−0.531521	−	0.847045i	$-0.678379\pi$
$647$	−17494.7	−1.06304	−0.531521	+	0.847045i	$0.678379\pi$
$648$	0	0
$649$	−8149.81	−0.492924
$650$	0	0
$651$	−1308.13	−0.0787554
$652$	0	0
$653$	−2045.20	−0.122565	−0.0612824	−	0.998120i	$-0.519519\pi$
$653$	−2045.20	−0.122565	−0.0612824	+	0.998120i	$0.519519\pi$
$654$	0	0
$655$	−12362.8	−0.737485
$656$	0	0
$657$	−33748.7	−2.00405
$658$	0	0
$659$	17535.3	1.03654	0.518269	−	0.855218i	$-0.326577\pi$
$659$	17535.3	1.03654	0.518269	+	0.855218i	$0.326577\pi$
$660$	0	0
$661$	32935.5	1.93803	0.969017	−	0.246993i	$-0.0794426\pi$
$661$	32935.5	1.93803	0.969017	+	0.246993i	$0.0794426\pi$
$662$	0	0
$663$	−20775.9	−1.21699
$664$	0	0
$665$	2647.63	0.154392
$666$	0	0
$667$	−2493.70	−0.144763
$668$	0	0
$669$	−4291.35	−0.248002
$670$	0	0
$671$	2375.72	0.136682
$672$	0	0
$673$	13574.1	0.777477	0.388739	−	0.921348i	$-0.372911\pi$
$673$	13574.1	0.777477	0.388739	+	0.921348i	$0.372911\pi$
$674$	0	0
$675$	17345.0	0.989049
$676$	0	0
$677$	3889.26	0.220792	0.110396	−	0.993888i	$-0.464788\pi$
$677$	3889.26	0.220792	0.110396	+	0.993888i	$0.464788\pi$
$678$	0	0
$679$	−2079.43	−0.117527
$680$	0	0
$681$	49985.1	2.81268
$682$	0	0
$683$	−10029.9	−0.561908	−0.280954	−	0.959721i	$-0.590651\pi$
$683$	−10029.9	−0.561908	−0.280954	+	0.959721i	$0.590651\pi$
$684$	0	0
$685$	−12693.1	−0.707998
$686$	0	0
$687$	10029.6	0.556990
$688$	0	0
$689$	−3024.68	−0.167244
$690$	0	0
$691$	−21305.0	−1.17291	−0.586456	−	0.809981i	$-0.699477\pi$
$691$	−21305.0	−1.17291	−0.586456	+	0.809981i	$0.699477\pi$
$692$	0	0
$693$	−3305.42	−0.181187
$694$	0	0
$695$	−26550.4	−1.44909
$696$	0	0
$697$	−17843.4	−0.969683
$698$	0	0
$699$	35213.8	1.90545
$700$	0	0
$701$	−5093.90	−0.274457	−0.137228	−	0.990539i	$-0.543819\pi$
$701$	−5093.90	−0.274457	−0.137228	+	0.990539i	$0.543819\pi$
$702$	0	0
$703$	2948.25	0.158173
$704$	0	0
$705$	47935.4	2.56078
$706$	0	0
$707$	−6347.41	−0.337650
$708$	0	0
$709$	−17744.6	−0.939933	−0.469966	−	0.882684i	$-0.655734\pi$
$709$	−17744.6	−0.939933	−0.469966	+	0.882684i	$0.655734\pi$
$710$	0	0
$711$	−46703.2	−2.46344
$712$	0	0
$713$	2169.10	0.113932
$714$	0	0
$715$	−5554.82	−0.290544
$716$	0	0
$717$	35328.2	1.84011
$718$	0	0
$719$	−14695.1	−0.762220	−0.381110	−	0.924530i	$-0.624458\pi$
$719$	−14695.1	−0.762220	−0.381110	+	0.924530i	$0.624458\pi$
$720$	0	0
$721$	−10611.0	−0.548092
$722$	0	0
$723$	−5110.02	−0.262854
$724$	0	0
$725$	3345.77	0.171392
$726$	0	0
$727$	22609.9	1.15344	0.576722	−	0.816941i	$-0.304332\pi$
$727$	22609.9	1.15344	0.576722	+	0.816941i	$0.304332\pi$
$728$	0	0
$729$	−32015.2	−1.62654
$730$	0	0
$731$	−29048.0	−1.46974
$732$	0	0
$733$	25799.9	1.30006	0.650028	−	0.759911i	$-0.274757\pi$
$733$	25799.9	1.30006	0.650028	+	0.759911i	$0.274757\pi$
$734$	0	0
$735$	−6546.11	−0.328513
$736$	0	0
$737$	−9305.10	−0.465072
$738$	0	0
$739$	−29131.3	−1.45008	−0.725042	−	0.688705i	$-0.758180\pi$
$739$	−29131.3	−1.45008	−0.725042	+	0.688705i	$0.758180\pi$
$740$	0	0
$741$	6257.98	0.310246
$742$	0	0
$743$	1435.32	0.0708704	0.0354352	−	0.999372i	$-0.488718\pi$
$743$	1435.32	0.0708704	0.0354352	+	0.999372i	$0.488718\pi$
$744$	0	0
$745$	49182.1	2.41865
$746$	0	0
$747$	−14072.0	−0.689247
$748$	0	0
$749$	−2895.70	−0.141264
$750$	0	0
$751$	−35081.8	−1.70460	−0.852299	−	0.523054i	$-0.824792\pi$
$751$	−35081.8	−1.70460	−0.852299	+	0.523054i	$0.824792\pi$
$752$	0	0
$753$	11457.5	0.554496
$754$	0	0
$755$	−43000.1	−2.07276
$756$	0	0
$757$	35527.7	1.70578	0.852891	−	0.522090i	$-0.174847\pi$
$757$	35527.7	1.70578	0.852891	+	0.522090i	$0.174847\pi$
$758$	0	0
$759$	8928.27	0.426977
$760$	0	0
$761$	−7608.21	−0.362414	−0.181207	−	0.983445i	$-0.558000\pi$
$761$	−7608.21	−0.362414	−0.181207	+	0.983445i	$0.558000\pi$
$762$	0	0
$763$	15153.4	0.718991
$764$	0	0
$765$	−53903.9	−2.54758
$766$	0	0
$767$	−23419.0	−1.10249
$768$	0	0
$769$	37184.7	1.74371	0.871856	−	0.489763i	$-0.162917\pi$
$769$	37184.7	1.74371	0.871856	+	0.489763i	$0.162917\pi$
$770$	0	0
$771$	24571.0	1.14773
$772$	0	0
$773$	3878.73	0.180476	0.0902382	−	0.995920i	$-0.471237\pi$
$773$	3878.73	0.180476	0.0902382	+	0.995920i	$0.471237\pi$
$774$	0	0
$775$	−2910.26	−0.134890
$776$	0	0
$777$	−7289.38	−0.336558
$778$	0	0
$779$	5374.69	0.247200
$780$	0	0
$781$	3674.10	0.168335
$782$	0	0
$783$	3421.89	0.156179
$784$	0	0
$785$	11616.5	0.528166
$786$	0	0
$787$	−30639.9	−1.38779	−0.693897	−	0.720074i	$-0.744108\pi$
$787$	−30639.9	−1.38779	−0.693897	+	0.720074i	$0.744108\pi$
$788$	0	0
$789$	3528.78	0.159224
$790$	0	0
$791$	−10784.1	−0.484753
$792$	0	0
$793$	6826.79	0.305708
$794$	0	0
$795$	−12783.6	−0.570299
$796$	0	0
$797$	−2413.82	−0.107280	−0.0536399	−	0.998560i	$-0.517082\pi$
$797$	−2413.82	−0.107280	−0.0536399	+	0.998560i	$0.517082\pi$
$798$	0	0
$799$	−28202.7	−1.24873
$800$	0	0
$801$	−35152.5	−1.55063
$802$	0	0
$803$	8647.98	0.380051
$804$	0	0
$805$	10854.6	0.475246
$806$	0	0
$807$	−38216.3	−1.66701
$808$	0	0
$809$	−31164.1	−1.35435	−0.677177	−	0.735820i	$-0.736797\pi$
$809$	−31164.1	−1.35435	−0.677177	+	0.735820i	$0.736797\pi$
$810$	0	0
$811$	−19483.9	−0.843615	−0.421808	−	0.906685i	$-0.638604\pi$
$811$	−19483.9	−0.843615	−0.421808	+	0.906685i	$0.638604\pi$
$812$	0	0
$813$	22841.3	0.985336
$814$	0	0
$815$	46605.3	2.00308
$816$	0	0
$817$	8749.65	0.374678
$818$	0	0
$819$	−9498.34	−0.405249
$820$	0	0
$821$	−5127.86	−0.217983	−0.108991	−	0.994043i	$-0.534762\pi$
$821$	−5127.86	−0.217983	−0.108991	+	0.994043i	$0.534762\pi$
$822$	0	0
$823$	−3802.51	−0.161054	−0.0805269	−	0.996752i	$-0.525660\pi$
$823$	−3802.51	−0.161054	−0.0805269	+	0.996752i	$0.525660\pi$
$824$	0	0
$825$	−11978.9	−0.505519
$826$	0	0
$827$	42059.1	1.76848	0.884242	−	0.467028i	$-0.154675\pi$
$827$	42059.1	1.76848	0.884242	+	0.467028i	$0.154675\pi$
$828$	0	0
$829$	−37314.0	−1.56329	−0.781647	−	0.623721i	$-0.785620\pi$
$829$	−37314.0	−1.56329	−0.781647	+	0.623721i	$0.785620\pi$
$830$	0	0
$831$	−63155.0	−2.63637
$832$	0	0
$833$	3851.39	0.160195
$834$	0	0
$835$	−17255.4	−0.715149
$836$	0	0
$837$	−2976.47	−0.122917
$838$	0	0
$839$	13141.6	0.540762	0.270381	−	0.962753i	$-0.412850\pi$
$839$	13141.6	0.540762	0.270381	+	0.962753i	$0.412850\pi$
$840$	0	0
$841$	−23728.9	−0.972936
$842$	0	0
$843$	−9063.01	−0.370281
$844$	0	0
$845$	19136.7	0.779082
$846$	0	0
$847$	847.000	0.0343604
$848$	0	0
$849$	−43474.6	−1.75741
$850$	0	0
$851$	12087.0	0.486883
$852$	0	0
$853$	26338.3	1.05722	0.528609	−	0.848865i	$-0.322714\pi$
$853$	26338.3	1.05722	0.528609	+	0.848865i	$0.322714\pi$
$854$	0	0
$855$	16236.6	0.649450
$856$	0	0
$857$	9167.45	0.365407	0.182704	−	0.983168i	$-0.441515\pi$
$857$	9167.45	0.365407	0.182704	+	0.983168i	$0.441515\pi$
$858$	0	0
$859$	19375.7	0.769606	0.384803	−	0.922999i	$-0.374269\pi$
$859$	19375.7	0.769606	0.384803	+	0.922999i	$0.374269\pi$
$860$	0	0
$861$	−13288.6	−0.525987
$862$	0	0
$863$	−24320.2	−0.959292	−0.479646	−	0.877462i	$-0.659235\pi$
$863$	−24320.2	−0.959292	−0.479646	+	0.877462i	$0.659235\pi$
$864$	0	0
$865$	−4072.36	−0.160075
$866$	0	0
$867$	10577.6	0.414340
$868$	0	0
$869$	11967.5	0.467169
$870$	0	0
$871$	−26738.8	−1.04020
$872$	0	0
$873$	−12752.1	−0.494379
$874$	0	0
$875$	−584.566	−0.0225851
$876$	0	0
$877$	46273.7	1.78170	0.890851	−	0.454295i	$-0.150109\pi$
$877$	46273.7	1.78170	0.890851	+	0.454295i	$0.150109\pi$
$878$	0	0
$879$	10422.3	0.399926
$880$	0	0
$881$	−14312.2	−0.547322	−0.273661	−	0.961826i	$-0.588235\pi$
$881$	−14312.2	−0.547322	−0.273661	+	0.961826i	$0.588235\pi$
$882$	0	0
$883$	−42398.5	−1.61588	−0.807940	−	0.589265i	$-0.799417\pi$
$883$	−42398.5	−1.61588	−0.807940	+	0.589265i	$0.799417\pi$
$884$	0	0
$885$	−98978.8	−3.75948
$886$	0	0
$887$	41762.8	1.58090	0.790449	−	0.612528i	$-0.209847\pi$
$887$	41762.8	1.58090	0.790449	+	0.612528i	$0.209847\pi$
$888$	0	0
$889$	−16629.5	−0.627372
$890$	0	0
$891$	497.996	0.0187245
$892$	0	0
$893$	8495.04	0.318338
$894$	0	0
$895$	−43915.6	−1.64015
$896$	0	0
$897$	25656.0	0.954992
$898$	0	0
$899$	−574.148	−0.0213002
$900$	0	0
$901$	7521.20	0.278099
$902$	0	0
$903$	−21633.0	−0.797233
$904$	0	0
$905$	−68195.6	−2.50486
$906$	0	0
$907$	−43484.7	−1.59194	−0.795968	−	0.605339i	$-0.793038\pi$
$907$	−43484.7	−1.59194	−0.795968	+	0.605339i	$0.793038\pi$
$908$	0	0
$909$	−38925.5	−1.42033
$910$	0	0
$911$	−42461.2	−1.54424	−0.772120	−	0.635477i	$-0.780803\pi$
$911$	−42461.2	−1.54424	−0.772120	+	0.635477i	$0.780803\pi$
$912$	0	0
$913$	3605.89	0.130709
$914$	0	0
$915$	28852.9	1.04246
$916$	0	0
$917$	5416.89	0.195072
$918$	0	0
$919$	8812.11	0.316305	0.158153	−	0.987415i	$-0.449446\pi$
$919$	8812.11	0.316305	0.158153	+	0.987415i	$0.449446\pi$
$920$	0	0
$921$	30667.7	1.09722
$922$	0	0
$923$	10557.8	0.376504
$924$	0	0
$925$	−16217.0	−0.576445
$926$	0	0
$927$	−65071.9	−2.30555
$928$	0	0
$929$	49257.0	1.73958	0.869790	−	0.493422i	$-0.164254\pi$
$929$	49257.0	1.73958	0.869790	+	0.493422i	$0.164254\pi$
$930$	0	0
$931$	−1160.09	−0.0408383
$932$	0	0
$933$	55165.5	1.93573
$934$	0	0
$935$	13812.7	0.483125
$936$	0	0
$937$	−40006.7	−1.39484	−0.697418	−	0.716664i	$-0.745668\pi$
$937$	−40006.7	−1.39484	−0.697418	+	0.716664i	$0.745668\pi$
$938$	0	0
$939$	53066.5	1.84426
$940$	0	0
$941$	41911.4	1.45194	0.725969	−	0.687727i	$-0.241392\pi$
$941$	41911.4	1.45194	0.725969	+	0.687727i	$0.241392\pi$
$942$	0	0
$943$	22034.8	0.760923
$944$	0	0
$945$	−14894.7	−0.512726
$946$	0	0
$947$	−1397.25	−0.0479457	−0.0239728	−	0.999713i	$-0.507632\pi$
$947$	−1397.25	−0.0479457	−0.0239728	+	0.999713i	$0.507632\pi$
$948$	0	0
$949$	24850.6	0.850035
$950$	0	0
$951$	50828.5	1.73315
$952$	0	0
$953$	−14897.7	−0.506382	−0.253191	−	0.967416i	$-0.581480\pi$
$953$	−14897.7	−0.506382	−0.253191	+	0.967416i	$0.581480\pi$
$954$	0	0
$955$	23070.2	0.781710
$956$	0	0
$957$	−2363.26	−0.0798258
$958$	0	0
$959$	5561.64	0.187273
$960$	0	0
$961$	−29291.6	−0.983236
$962$	0	0
$963$	−17757.9	−0.594226
$964$	0	0
$965$	28496.1	0.950593
$966$	0	0
$967$	−11386.6	−0.378664	−0.189332	−	0.981913i	$-0.560632\pi$
$967$	−11386.6	−0.378664	−0.189332	+	0.981913i	$0.560632\pi$
$968$	0	0
$969$	−15561.1	−0.515888
$970$	0	0
$971$	−38787.0	−1.28191	−0.640955	−	0.767578i	$-0.721462\pi$
$971$	−38787.0	−1.28191	−0.640955	+	0.767578i	$0.721462\pi$
$972$	0	0
$973$	11633.4	0.383298
$974$	0	0
$975$	−34422.3	−1.13066
$976$	0	0
$977$	31977.0	1.04712	0.523559	−	0.851989i	$-0.324604\pi$
$977$	31977.0	1.04712	0.523559	+	0.851989i	$0.324604\pi$
$978$	0	0
$979$	9007.70	0.294063
$980$	0	0
$981$	92928.3	3.02444
$982$	0	0
$983$	37231.4	1.20803	0.604017	−	0.796972i	$-0.293566\pi$
$983$	37231.4	1.20803	0.604017	+	0.796972i	$0.293566\pi$
$984$	0	0
$985$	−22885.0	−0.740280
$986$	0	0
$987$	−21003.5	−0.677354
$988$	0	0
$989$	35871.2	1.15332
$990$	0	0
$991$	−17636.1	−0.565318	−0.282659	−	0.959220i	$-0.591217\pi$
$991$	−17636.1	−0.565318	−0.282659	+	0.959220i	$0.591217\pi$
$992$	0	0
$993$	49175.3	1.57153
$994$	0	0
$995$	53124.4	1.69262
$996$	0	0
$997$	44423.7	1.41115	0.705573	−	0.708637i	$-0.250690\pi$
$997$	44423.7	1.41115	0.705573	+	0.708637i	$0.250690\pi$
$998$	0	0
$999$	−16586.0	−0.525282

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.4.a.e.1.1	✓	4	4.3	odd	2
1232.4.a.t.1.4		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)

\(f(q)\)	\(=\)	\(q+8.36227 q^{3} -15.9758 q^{5} +7.00000 q^{7} +42.9275 q^{9} +O(q^{10})\)	\(q+8.36227 q^{3} -15.9758 q^{5} +7.00000 q^{7} +42.9275 q^{9} -11.0000 q^{11} -31.6092 q^{13} -133.594 q^{15} +78.5997 q^{17} -23.6753 q^{19} +58.5359 q^{21} -97.0623 q^{23} +130.227 q^{25} +133.190 q^{27} +25.6918 q^{29} -22.3475 q^{31} -91.9849 q^{33} -111.831 q^{35} -124.529 q^{37} -264.325 q^{39} -227.017 q^{41} -369.568 q^{43} -685.803 q^{45} -358.814 q^{47} +49.0000 q^{49} +657.272 q^{51} +95.6899 q^{53} +175.734 q^{55} -197.979 q^{57} +740.892 q^{59} -215.974 q^{61} +300.492 q^{63} +504.984 q^{65} +845.919 q^{67} -811.661 q^{69} -334.009 q^{71} -786.180 q^{73} +1088.99 q^{75} -77.0000 q^{77} -1087.95 q^{79} -45.2723 q^{81} -327.808 q^{83} -1255.70 q^{85} +214.842 q^{87} -818.881 q^{89} -221.265 q^{91} -186.876 q^{93} +378.233 q^{95} -297.061 q^{97} -472.202 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} - 19 q^{5} + 28 q^{7} + 37 q^{9}+O(q^{10}) \) 4 * q + 3 * q^3 - 19 * q^5 + 28 * q^7 + 37 * q^9	\( 4 q + 3 q^{3} - 19 q^{5} + 28 q^{7} + 37 q^{9} - 44 q^{11} + 2 q^{13} - 71 q^{15} + 8 q^{17} + 6 q^{19} + 21 q^{21} - 159 q^{23} - 119 q^{25} + 117 q^{27} + 144 q^{29} + 183 q^{31} - 33 q^{33} - 133 q^{35} - 475 q^{37} + 64 q^{39} - 768 q^{41} + 152 q^{43} - 1048 q^{45} + 228 q^{47} + 196 q^{49} + 882 q^{51} + 396 q^{53} + 209 q^{55} - 1286 q^{57} + 733 q^{59} - 1012 q^{61} + 259 q^{63} + 420 q^{65} + 171 q^{67} - 321 q^{69} + 1019 q^{71} - 1836 q^{73} + 1016 q^{75} - 308 q^{77} - 1374 q^{79} - 956 q^{81} + 1248 q^{83} + 46 q^{85} - 2238 q^{87} - 1401 q^{89} + 14 q^{91} - 1859 q^{93} - 238 q^{95} - 2559 q^{97} - 407 q^{99}+O(q^{100}) \) 4 * q + 3 * q^3 - 19 * q^5 + 28 * q^7 + 37 * q^9 - 44 * q^11 + 2 * q^13 - 71 * q^15 + 8 * q^17 + 6 * q^19 + 21 * q^21 - 159 * q^23 - 119 * q^25 + 117 * q^27 + 144 * q^29 + 183 * q^31 - 33 * q^33 - 133 * q^35 - 475 * q^37 + 64 * q^39 - 768 * q^41 + 152 * q^43 - 1048 * q^45 + 228 * q^47 + 196 * q^49 + 882 * q^51 + 396 * q^53 + 209 * q^55 - 1286 * q^57 + 733 * q^59 - 1012 * q^61 + 259 * q^63 + 420 * q^65 + 171 * q^67 - 321 * q^69 + 1019 * q^71 - 1836 * q^73 + 1016 * q^75 - 308 * q^77 - 1374 * q^79 - 956 * q^81 + 1248 * q^83 + 46 * q^85 - 2238 * q^87 - 1401 * q^89 + 14 * q^91 - 1859 * q^93 - 238 * q^95 - 2559 * q^97 - 407 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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