LMFDB - Embedded newform 1452.4.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1452,4,Mod(1,1452)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1452, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1452.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1452 = 2^{2} \cdot 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1452.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:	$85.6707733283$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.20959101.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 90x^{2} + 91x + 2026 $ x^4 - 2x^3 - 90x^2 + 91*x + 2026
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.75283$ of defining polynomial
Character	$\chi$	$=$	1452.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -10.3041 q^{5} -18.4086 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -10.3041 q^{5} -18.4086 q^{7} +9.00000 q^{9} -93.4426 q^{13} -30.9124 q^{15} +75.0340 q^{17} -38.2168 q^{19} -55.2258 q^{21} -107.345 q^{23} -18.8248 q^{25} +27.0000 q^{27} -185.486 q^{29} +285.474 q^{31} +189.685 q^{35} +35.8248 q^{37} -280.328 q^{39} +446.005 q^{41} -295.937 q^{43} -92.7372 q^{45} -66.3868 q^{47} -4.12404 q^{49} +225.102 q^{51} +430.562 q^{53} -114.651 q^{57} -382.691 q^{59} -277.528 q^{61} -165.677 q^{63} +962.845 q^{65} -363.299 q^{67} -322.036 q^{69} +991.954 q^{71} +223.702 q^{73} -56.4744 q^{75} +310.147 q^{79} +81.0000 q^{81} +1073.30 q^{83} -773.161 q^{85} -556.457 q^{87} -519.917 q^{89} +1720.15 q^{91} +856.423 q^{93} +393.791 q^{95} +1091.12 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} + 12 q^{5} + 36 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 + 12 * q^5 + 36 * q^9	$ 4 q + 12 q^{3} + 12 q^{5} + 36 q^{9} + 36 q^{15} + 156 q^{23} + 244 q^{25} + 108 q^{27} + 184 q^{31} - 176 q^{37} + 108 q^{45} + 852 q^{47} + 1580 q^{49} + 924 q^{53} - 360 q^{59} - 176 q^{67} + 468 q^{69} + 3276 q^{71} + 732 q^{75} + 324 q^{81} - 3144 q^{89} - 144 q^{91} + 552 q^{93} + 2768 q^{97}+O(q^{100}) $ 4 * q + 12 * q^3 + 12 * q^5 + 36 * q^9 + 36 * q^15 + 156 * q^23 + 244 * q^25 + 108 * q^27 + 184 * q^31 - 176 * q^37 + 108 * q^45 + 852 * q^47 + 1580 * q^49 + 924 * q^53 - 360 * q^59 - 176 * q^67 + 468 * q^69 + 3276 * q^71 + 732 * q^75 + 324 * q^81 - 3144 * q^89 - 144 * q^91 + 552 * q^93 + 2768 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−10.3041	−0.921630	−0.460815	−	0.887496i	$-0.652443\pi$
$5$	−10.3041	−0.921630	−0.460815	+	0.887496i	$0.652443\pi$
$6$	0	0
$7$	−18.4086	−0.993970	−0.496985	−	0.867759i	$-0.665560\pi$
$7$	−18.4086	−0.993970	−0.496985	+	0.867759i	$0.665560\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−93.4426	−1.99356	−0.996781	−	0.0801697i	$-0.974454\pi$
$13$	−93.4426	−1.99356	−0.996781	+	0.0801697i	$0.974454\pi$
$14$	0	0
$15$	−30.9124	−0.532103
$16$	0	0
$17$	75.0340	1.07050	0.535248	−	0.844695i	$-0.320218\pi$
$17$	75.0340	1.07050	0.535248	+	0.844695i	$0.320218\pi$
$18$	0	0
$19$	−38.2168	−0.461450	−0.230725	−	0.973019i	$-0.574110\pi$
$19$	−38.2168	−0.461450	−0.230725	+	0.973019i	$0.574110\pi$
$20$	0	0
$21$	−55.2258	−0.573869
$22$	0	0
$23$	−107.345	−0.973177	−0.486589	−	0.873631i	$-0.661759\pi$
$23$	−107.345	−0.973177	−0.486589	+	0.873631i	$0.661759\pi$
$24$	0	0
$25$	−18.8248	−0.150598
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−185.486	−1.18772	−0.593859	−	0.804569i	$-0.702396\pi$
$29$	−185.486	−1.18772	−0.593859	+	0.804569i	$0.702396\pi$
$30$	0	0
$31$	285.474	1.65396	0.826979	−	0.562232i	$-0.190057\pi$
$31$	285.474	1.65396	0.826979	+	0.562232i	$0.190057\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	189.685	0.916072
$36$	0	0
$37$	35.8248	0.159177	0.0795887	−	0.996828i	$-0.474639\pi$
$37$	35.8248	0.159177	0.0795887	+	0.996828i	$0.474639\pi$
$38$	0	0
$39$	−280.328	−1.15098
$40$	0	0
$41$	446.005	1.69888	0.849442	−	0.527681i	$-0.176938\pi$
$41$	446.005	1.69888	0.849442	+	0.527681i	$0.176938\pi$
$42$	0	0
$43$	−295.937	−1.04953	−0.524767	−	0.851246i	$-0.675848\pi$
$43$	−295.937	−1.04953	−0.524767	+	0.851246i	$0.675848\pi$
$44$	0	0
$45$	−92.7372	−0.307210
$46$	0	0
$47$	−66.3868	−0.206032	−0.103016	−	0.994680i	$-0.532849\pi$
$47$	−66.3868	−0.206032	−0.103016	+	0.994680i	$0.532849\pi$
$48$	0	0
$49$	−4.12404	−0.0120234
$50$	0	0
$51$	225.102	0.618051
$52$	0	0
$53$	430.562	1.11589	0.557946	−	0.829877i	$-0.311590\pi$
$53$	430.562	1.11589	0.557946	+	0.829877i	$0.311590\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−114.651	−0.266418
$58$	0	0
$59$	−382.691	−0.844443	−0.422221	−	0.906493i	$-0.638750\pi$
$59$	−382.691	−0.844443	−0.422221	+	0.906493i	$0.638750\pi$
$60$	0	0
$61$	−277.528	−0.582523	−0.291261	−	0.956644i	$-0.594075\pi$
$61$	−277.528	−0.582523	−0.291261	+	0.956644i	$0.594075\pi$
$62$	0	0
$63$	−165.677	−0.331323
$64$	0	0
$65$	962.845	1.83733
$66$	0	0
$67$	−363.299	−0.662449	−0.331224	−	0.943552i	$-0.607462\pi$
$67$	−363.299	−0.662449	−0.331224	+	0.943552i	$0.607462\pi$
$68$	0	0
$69$	−322.036	−0.561864
$70$	0	0
$71$	991.954	1.65807	0.829037	−	0.559194i	$-0.188889\pi$
$71$	991.954	1.65807	0.829037	+	0.559194i	$0.188889\pi$
$72$	0	0
$73$	223.702	0.358663	0.179331	−	0.983789i	$-0.442607\pi$
$73$	223.702	0.358663	0.179331	+	0.983789i	$0.442607\pi$
$74$	0	0
$75$	−56.4744	−0.0869481
$76$	0	0
$77$	0	0
$78$	0	0
$79$	310.147	0.441699	0.220849	−	0.975308i	$-0.429117\pi$
$79$	310.147	0.441699	0.220849	+	0.975308i	$0.429117\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1073.30	1.41939	0.709696	−	0.704508i	$-0.248832\pi$
$83$	1073.30	1.41939	0.709696	+	0.704508i	$0.248832\pi$
$84$	0	0
$85$	−773.161	−0.986600
$86$	0	0
$87$	−556.457	−0.685729
$88$	0	0
$89$	−519.917	−0.619226	−0.309613	−	0.950863i	$-0.600200\pi$
$89$	−519.917	−0.619226	−0.309613	+	0.950863i	$0.600200\pi$
$90$	0	0
$91$	1720.15	1.98154
$92$	0	0
$93$	856.423	0.954914
$94$	0	0
$95$	393.791	0.425286
$96$	0	0
$97$	1091.12	1.14213	0.571066	−	0.820904i	$-0.306530\pi$
$97$	1091.12	1.14213	0.571066	+	0.820904i	$0.306530\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1187.95	−1.17035	−0.585174	−	0.810908i	$-0.698974\pi$
$101$	−1187.95	−1.17035	−0.585174	+	0.810908i	$0.698974\pi$
$102$	0	0
$103$	833.124	0.796992	0.398496	−	0.917170i	$-0.369532\pi$
$103$	833.124	0.796992	0.398496	+	0.917170i	$0.369532\pi$
$104$	0	0
$105$	569.054	0.528895
$106$	0	0
$107$	−829.573	−0.749513	−0.374756	−	0.927123i	$-0.622274\pi$
$107$	−829.573	−0.749513	−0.374756	+	0.927123i	$0.622274\pi$
$108$	0	0
$109$	201.095	0.176710	0.0883550	−	0.996089i	$-0.471839\pi$
$109$	201.095	0.176710	0.0883550	+	0.996089i	$0.471839\pi$
$110$	0	0
$111$	107.474	0.0919011
$112$	0	0
$113$	889.835	0.740784	0.370392	−	0.928876i	$-0.379223\pi$
$113$	889.835	0.740784	0.370392	+	0.928876i	$0.379223\pi$
$114$	0	0
$115$	1106.10	0.896909
$116$	0	0
$117$	−840.983	−0.664521
$118$	0	0
$119$	−1381.27	−1.06404
$120$	0	0
$121$	0	0
$122$	0	0
$123$	1338.02	0.980852
$124$	0	0
$125$	1481.99	1.06043
$126$	0	0
$127$	15.6092	0.0109063	0.00545313	−	0.999985i	$-0.498264\pi$
$127$	15.6092	0.0109063	0.00545313	+	0.999985i	$0.498264\pi$
$128$	0	0
$129$	−887.811	−0.605949
$130$	0	0
$131$	−418.986	−0.279442	−0.139721	−	0.990191i	$-0.544621\pi$
$131$	−418.986	−0.279442	−0.139721	+	0.990191i	$0.544621\pi$
$132$	0	0
$133$	703.518	0.458667
$134$	0	0
$135$	−278.212	−0.177368
$136$	0	0
$137$	2197.54	1.37043	0.685213	−	0.728343i	$-0.259709\pi$
$137$	2197.54	1.37043	0.685213	+	0.728343i	$0.259709\pi$
$138$	0	0
$139$	−32.6182	−0.0199039	−0.00995193	−	0.999950i	$-0.503168\pi$
$139$	−32.6182	−0.0199039	−0.00995193	+	0.999950i	$0.503168\pi$
$140$	0	0
$141$	−199.160	−0.118953
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1911.27	1.09464
$146$	0	0
$147$	−12.3721	−0.00694174
$148$	0	0
$149$	−3359.73	−1.84725	−0.923625	−	0.383298i	$-0.874788\pi$
$149$	−3359.73	−1.84725	−0.923625	+	0.383298i	$0.874788\pi$
$150$	0	0
$151$	3073.81	1.65658	0.828288	−	0.560303i	$-0.189315\pi$
$151$	3073.81	1.65658	0.828288	+	0.560303i	$0.189315\pi$
$152$	0	0
$153$	675.306	0.356832
$154$	0	0
$155$	−2941.57	−1.52434
$156$	0	0
$157$	2295.20	1.16673	0.583365	−	0.812210i	$-0.301736\pi$
$157$	2295.20	1.16673	0.583365	+	0.812210i	$0.301736\pi$
$158$	0	0
$159$	1291.69	0.644260
$160$	0	0
$161$	1976.08	0.967309
$162$	0	0
$163$	1173.55	0.563922	0.281961	−	0.959426i	$-0.409015\pi$
$163$	1173.55	0.563922	0.281961	+	0.959426i	$0.409015\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2874.11	−1.33177	−0.665885	−	0.746055i	$-0.731946\pi$
$167$	−2874.11	−1.33177	−0.665885	+	0.746055i	$0.731946\pi$
$168$	0	0
$169$	6534.52	2.97429
$170$	0	0
$171$	−343.952	−0.153817
$172$	0	0
$173$	2640.61	1.16047	0.580237	−	0.814448i	$-0.302960\pi$
$173$	2640.61	1.16047	0.580237	+	0.814448i	$0.302960\pi$
$174$	0	0
$175$	346.538	0.149690
$176$	0	0
$177$	−1148.07	−0.487539
$178$	0	0
$179$	3375.89	1.40964	0.704820	−	0.709386i	$-0.251028\pi$
$179$	3375.89	1.40964	0.704820	+	0.709386i	$0.251028\pi$
$180$	0	0
$181$	785.927	0.322749	0.161374	−	0.986893i	$-0.448407\pi$
$181$	785.927	0.322749	0.161374	+	0.986893i	$0.448407\pi$
$182$	0	0
$183$	−832.585	−0.336320
$184$	0	0
$185$	−369.144	−0.146703
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−497.032	−0.191290
$190$	0	0
$191$	−1039.91	−0.393956	−0.196978	−	0.980408i	$-0.563113\pi$
$191$	−1039.91	−0.393956	−0.196978	+	0.980408i	$0.563113\pi$
$192$	0	0
$193$	1245.76	0.464620	0.232310	−	0.972642i	$-0.425372\pi$
$193$	1245.76	0.464620	0.232310	+	0.972642i	$0.425372\pi$
$194$	0	0
$195$	2888.54	1.06078
$196$	0	0
$197$	−2892.73	−1.04619	−0.523093	−	0.852275i	$-0.675222\pi$
$197$	−2892.73	−1.04619	−0.523093	+	0.852275i	$0.675222\pi$
$198$	0	0
$199$	−5130.39	−1.82756	−0.913779	−	0.406212i	$-0.866849\pi$
$199$	−5130.39	−1.82756	−0.913779	+	0.406212i	$0.866849\pi$
$200$	0	0
$201$	−1089.90	−0.382465
$202$	0	0
$203$	3414.53	1.18056
$204$	0	0
$205$	−4595.70	−1.56574
$206$	0	0
$207$	−966.109	−0.324392
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−2977.57	−0.971489	−0.485744	−	0.874101i	$-0.661451\pi$
$211$	−2977.57	−0.971489	−0.485744	+	0.874101i	$0.661451\pi$
$212$	0	0
$213$	2975.86	0.957289
$214$	0	0
$215$	3049.37	0.967282
$216$	0	0
$217$	−5255.18	−1.64399
$218$	0	0
$219$	671.107	0.207074
$220$	0	0
$221$	−7011.37	−2.13410
$222$	0	0
$223$	−1865.55	−0.560208	−0.280104	−	0.959970i	$-0.590369\pi$
$223$	−1865.55	−0.560208	−0.280104	+	0.959970i	$0.590369\pi$
$224$	0	0
$225$	−169.423	−0.0501995
$226$	0	0
$227$	−1950.88	−0.570417	−0.285209	−	0.958465i	$-0.592063\pi$
$227$	−1950.88	−0.570417	−0.285209	+	0.958465i	$0.592063\pi$
$228$	0	0
$229$	443.547	0.127993	0.0639966	−	0.997950i	$-0.479615\pi$
$229$	443.547	0.127993	0.0639966	+	0.997950i	$0.479615\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5814.86	1.63495	0.817477	−	0.575961i	$-0.195372\pi$
$233$	5814.86	1.63495	0.817477	+	0.575961i	$0.195372\pi$
$234$	0	0
$235$	684.059	0.189885
$236$	0	0
$237$	930.440	0.255015
$238$	0	0
$239$	−5208.02	−1.40953	−0.704767	−	0.709439i	$-0.748948\pi$
$239$	−5208.02	−1.40953	−0.704767	+	0.709439i	$0.748948\pi$
$240$	0	0
$241$	−1871.23	−0.500150	−0.250075	−	0.968226i	$-0.580455\pi$
$241$	−1871.23	−0.500150	−0.250075	+	0.968226i	$0.580455\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	42.4947	0.0110812
$246$	0	0
$247$	3571.08	0.919929
$248$	0	0
$249$	3219.89	0.819487
$250$	0	0
$251$	501.085	0.126009	0.0630044	−	0.998013i	$-0.479932\pi$
$251$	501.085	0.126009	0.0630044	+	0.998013i	$0.479932\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−2319.48	−0.569614
$256$	0	0
$257$	3578.85	0.868647	0.434324	−	0.900757i	$-0.356987\pi$
$257$	3578.85	0.868647	0.434324	+	0.900757i	$0.356987\pi$
$258$	0	0
$259$	−659.484	−0.158218
$260$	0	0
$261$	−1669.37	−0.395906
$262$	0	0
$263$	1769.60	0.414898	0.207449	−	0.978246i	$-0.433484\pi$
$263$	1769.60	0.414898	0.207449	+	0.978246i	$0.433484\pi$
$264$	0	0
$265$	−4436.57	−1.02844
$266$	0	0
$267$	−1559.75	−0.357510
$268$	0	0
$269$	−3356.68	−0.760819	−0.380409	−	0.924818i	$-0.624217\pi$
$269$	−3356.68	−0.760819	−0.380409	+	0.924818i	$0.624217\pi$
$270$	0	0
$271$	7222.10	1.61886	0.809430	−	0.587216i	$-0.199776\pi$
$271$	7222.10	1.61886	0.809430	+	0.587216i	$0.199776\pi$
$272$	0	0
$273$	5160.44	1.14404
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5161.74	−1.11963	−0.559817	−	0.828616i	$-0.689129\pi$
$277$	−5161.74	−1.11963	−0.559817	+	0.828616i	$0.689129\pi$
$278$	0	0
$279$	2569.27	0.551320
$280$	0	0
$281$	−958.646	−0.203516	−0.101758	−	0.994809i	$-0.532447\pi$
$281$	−958.646	−0.203516	−0.101758	+	0.994809i	$0.532447\pi$
$282$	0	0
$283$	1283.98	0.269698	0.134849	−	0.990866i	$-0.456945\pi$
$283$	1283.98	0.269698	0.134849	+	0.990866i	$0.456945\pi$
$284$	0	0
$285$	1181.37	0.245539
$286$	0	0
$287$	−8210.32	−1.68864
$288$	0	0
$289$	717.102	0.145960
$290$	0	0
$291$	3273.37	0.659411
$292$	0	0
$293$	−6400.71	−1.27622	−0.638112	−	0.769944i	$-0.720284\pi$
$293$	−6400.71	−1.27622	−0.638112	+	0.769944i	$0.720284\pi$
$294$	0	0
$295$	3943.30	0.778264
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10030.6	1.94009
$300$	0	0
$301$	5447.78	1.04321
$302$	0	0
$303$	−3563.84	−0.675701
$304$	0	0
$305$	2859.69	0.536870
$306$	0	0
$307$	5637.23	1.04799	0.523996	−	0.851721i	$-0.324441\pi$
$307$	5637.23	1.04799	0.523996	+	0.851721i	$0.324441\pi$
$308$	0	0
$309$	2499.37	0.460143
$310$	0	0
$311$	8785.63	1.60189	0.800944	−	0.598739i	$-0.204331\pi$
$311$	8785.63	1.60189	0.800944	+	0.598739i	$0.204331\pi$
$312$	0	0
$313$	5970.23	1.07814	0.539069	−	0.842261i	$-0.318776\pi$
$313$	5970.23	1.07814	0.539069	+	0.842261i	$0.318776\pi$
$314$	0	0
$315$	1707.16	0.305357
$316$	0	0
$317$	−2490.40	−0.441245	−0.220622	−	0.975359i	$-0.570809\pi$
$317$	−2490.40	−0.441245	−0.220622	+	0.975359i	$0.570809\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−2488.72	−0.432731
$322$	0	0
$323$	−2867.56	−0.493980
$324$	0	0
$325$	1759.04	0.300227
$326$	0	0
$327$	603.284	0.102024
$328$	0	0
$329$	1222.09	0.204790
$330$	0	0
$331$	297.606	0.0494197	0.0247098	−	0.999695i	$-0.492134\pi$
$331$	297.606	0.0494197	0.0247098	+	0.999695i	$0.492134\pi$
$332$	0	0
$333$	322.423	0.0530591
$334$	0	0
$335$	3743.48	0.610533
$336$	0	0
$337$	11726.6	1.89552	0.947758	−	0.318991i	$-0.103344\pi$
$337$	11726.6	1.89552	0.947758	+	0.318991i	$0.103344\pi$
$338$	0	0
$339$	2669.50	0.427692
$340$	0	0
$341$	0	0
$342$	0	0
$343$	6390.06	1.00592
$344$	0	0
$345$	3318.31	0.517831
$346$	0	0
$347$	−1627.93	−0.251849	−0.125925	−	0.992040i	$-0.540190\pi$
$347$	−1627.93	−0.251849	−0.125925	+	0.992040i	$0.540190\pi$
$348$	0	0
$349$	11665.9	1.78929	0.894644	−	0.446780i	$-0.147429\pi$
$349$	11665.9	1.78929	0.894644	+	0.446780i	$0.147429\pi$
$350$	0	0
$351$	−2522.95	−0.383661
$352$	0	0
$353$	−2269.34	−0.342166	−0.171083	−	0.985257i	$-0.554727\pi$
$353$	−2269.34	−0.342166	−0.171083	+	0.985257i	$0.554727\pi$
$354$	0	0
$355$	−10221.2	−1.52813
$356$	0	0
$357$	−4143.81	−0.614324
$358$	0	0
$359$	2809.30	0.413006	0.206503	−	0.978446i	$-0.433792\pi$
$359$	2809.30	0.413006	0.206503	+	0.978446i	$0.433792\pi$
$360$	0	0
$361$	−5398.47	−0.787064
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2305.06	−0.330554
$366$	0	0
$367$	−3366.60	−0.478842	−0.239421	−	0.970916i	$-0.576958\pi$
$367$	−3366.60	−0.478842	−0.239421	+	0.970916i	$0.576958\pi$
$368$	0	0
$369$	4014.05	0.566295
$370$	0	0
$371$	−7926.04	−1.10916
$372$	0	0
$373$	2071.89	0.287610	0.143805	−	0.989606i	$-0.454066\pi$
$373$	2071.89	0.287610	0.143805	+	0.989606i	$0.454066\pi$
$374$	0	0
$375$	4445.97	0.612237
$376$	0	0
$377$	17332.2	2.36779
$378$	0	0
$379$	9563.59	1.29617	0.648085	−	0.761568i	$-0.275570\pi$
$379$	9563.59	1.29617	0.648085	+	0.761568i	$0.275570\pi$
$380$	0	0
$381$	46.8277	0.00629674
$382$	0	0
$383$	956.509	0.127612	0.0638059	−	0.997962i	$-0.479676\pi$
$383$	956.509	0.127612	0.0638059	+	0.997962i	$0.479676\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2663.43	−0.349845
$388$	0	0
$389$	−14004.3	−1.82531	−0.912655	−	0.408731i	$-0.865972\pi$
$389$	−14004.3	−1.82531	−0.912655	+	0.408731i	$0.865972\pi$
$390$	0	0
$391$	−8054.56	−1.04178
$392$	0	0
$393$	−1256.96	−0.161336
$394$	0	0
$395$	−3195.79	−0.407083
$396$	0	0
$397$	11361.4	1.43630	0.718152	−	0.695887i	$-0.244988\pi$
$397$	11361.4	1.43630	0.718152	+	0.695887i	$0.244988\pi$
$398$	0	0
$399$	2110.55	0.264812
$400$	0	0
$401$	−6752.88	−0.840954	−0.420477	−	0.907303i	$-0.638137\pi$
$401$	−6752.88	−0.840954	−0.420477	+	0.907303i	$0.638137\pi$
$402$	0	0
$403$	−26675.5	−3.29727
$404$	0	0
$405$	−834.635	−0.102403
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−1362.24	−0.164690	−0.0823450	−	0.996604i	$-0.526241\pi$
$409$	−1362.24	−0.164690	−0.0823450	+	0.996604i	$0.526241\pi$
$410$	0	0
$411$	6592.61	0.791216
$412$	0	0
$413$	7044.80	0.839351
$414$	0	0
$415$	−11059.4	−1.30815
$416$	0	0
$417$	−97.8545	−0.0114915
$418$	0	0
$419$	6496.80	0.757493	0.378746	−	0.925500i	$-0.376355\pi$
$419$	6496.80	0.757493	0.378746	+	0.925500i	$0.376355\pi$
$420$	0	0
$421$	−2898.48	−0.335542	−0.167771	−	0.985826i	$-0.553657\pi$
$421$	−2898.48	−0.335542	−0.167771	+	0.985826i	$0.553657\pi$
$422$	0	0
$423$	−597.481	−0.0686774
$424$	0	0
$425$	−1412.50	−0.161215
$426$	0	0
$427$	5108.91	0.579010
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16041.8	−1.79282	−0.896409	−	0.443228i	$-0.853833\pi$
$431$	−16041.8	−1.79282	−0.896409	+	0.443228i	$0.853833\pi$
$432$	0	0
$433$	8413.94	0.933830	0.466915	−	0.884302i	$-0.345365\pi$
$433$	8413.94	0.933830	0.466915	+	0.884302i	$0.345365\pi$
$434$	0	0
$435$	5733.80	0.631988
$436$	0	0
$437$	4102.41	0.449072
$438$	0	0
$439$	−15686.4	−1.70540	−0.852700	−	0.522400i	$-0.825037\pi$
$439$	−15686.4	−1.70540	−0.852700	+	0.522400i	$0.825037\pi$
$440$	0	0
$441$	−37.1164	−0.00400781
$442$	0	0
$443$	8628.26	0.925374	0.462687	−	0.886522i	$-0.346885\pi$
$443$	8628.26	0.925374	0.462687	+	0.886522i	$0.346885\pi$
$444$	0	0
$445$	5357.30	0.570697
$446$	0	0
$447$	−10079.2	−1.06651
$448$	0	0
$449$	−5832.05	−0.612988	−0.306494	−	0.951873i	$-0.599156\pi$
$449$	−5832.05	−0.612988	−0.306494	+	0.951873i	$0.599156\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	9221.42	0.956424
$454$	0	0
$455$	−17724.6	−1.82625
$456$	0	0
$457$	4795.06	0.490817	0.245408	−	0.969420i	$-0.421078\pi$
$457$	4795.06	0.490817	0.245408	+	0.969420i	$0.421078\pi$
$458$	0	0
$459$	2025.92	0.206017
$460$	0	0
$461$	−11452.6	−1.15706	−0.578528	−	0.815663i	$-0.696372\pi$
$461$	−11452.6	−1.15706	−0.578528	+	0.815663i	$0.696372\pi$
$462$	0	0
$463$	13765.5	1.38172	0.690859	−	0.722989i	$-0.257232\pi$
$463$	13765.5	1.38172	0.690859	+	0.722989i	$0.257232\pi$
$464$	0	0
$465$	−8824.70	−0.880077
$466$	0	0
$467$	8670.67	0.859166	0.429583	−	0.903027i	$-0.358661\pi$
$467$	8670.67	0.859166	0.429583	+	0.903027i	$0.358661\pi$
$468$	0	0
$469$	6687.82	0.658454
$470$	0	0
$471$	6885.59	0.673612
$472$	0	0
$473$	0	0
$474$	0	0
$475$	719.425	0.0694936
$476$	0	0
$477$	3875.06	0.371964
$478$	0	0
$479$	−12507.3	−1.19306	−0.596528	−	0.802592i	$-0.703453\pi$
$479$	−12507.3	−1.19306	−0.596528	+	0.802592i	$0.703453\pi$
$480$	0	0
$481$	−3347.56	−0.317330
$482$	0	0
$483$	5928.23	0.558476
$484$	0	0
$485$	−11243.1	−1.05262
$486$	0	0
$487$	−17148.5	−1.59563	−0.797817	−	0.602900i	$-0.794012\pi$
$487$	−17148.5	−1.59563	−0.797817	+	0.602900i	$0.794012\pi$
$488$	0	0
$489$	3520.64	0.325581
$490$	0	0
$491$	15140.1	1.39157	0.695786	−	0.718249i	$-0.255056\pi$
$491$	15140.1	1.39157	0.695786	+	0.718249i	$0.255056\pi$
$492$	0	0
$493$	−13917.7	−1.27145
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−18260.5	−1.64808
$498$	0	0
$499$	−11101.7	−0.995952	−0.497976	−	0.867191i	$-0.665923\pi$
$499$	−11101.7	−0.995952	−0.497976	+	0.867191i	$0.665923\pi$
$500$	0	0
$501$	−8622.34	−0.768898
$502$	0	0
$503$	18622.9	1.65080	0.825400	−	0.564549i	$-0.190950\pi$
$503$	18622.9	1.65080	0.825400	+	0.564549i	$0.190950\pi$
$504$	0	0
$505$	12240.8	1.07863
$506$	0	0
$507$	19603.6	1.71721
$508$	0	0
$509$	2403.28	0.209280	0.104640	−	0.994510i	$-0.466631\pi$
$509$	2403.28	0.209280	0.104640	+	0.994510i	$0.466631\pi$
$510$	0	0
$511$	−4118.04	−0.356500
$512$	0	0
$513$	−1031.85	−0.0888060
$514$	0	0
$515$	−8584.62	−0.734531
$516$	0	0
$517$	0	0
$518$	0	0
$519$	7921.84	0.670000
$520$	0	0
$521$	−7624.25	−0.641122	−0.320561	−	0.947228i	$-0.603871\pi$
$521$	−7624.25	−0.641122	−0.320561	+	0.947228i	$0.603871\pi$
$522$	0	0
$523$	12095.3	1.01126	0.505632	−	0.862749i	$-0.331259\pi$
$523$	12095.3	1.01126	0.505632	+	0.862749i	$0.331259\pi$
$524$	0	0
$525$	1039.61	0.0864238
$526$	0	0
$527$	21420.3	1.77056
$528$	0	0
$529$	−643.948	−0.0529257
$530$	0	0
$531$	−3444.22	−0.281481
$532$	0	0
$533$	−41675.9	−3.38683
$534$	0	0
$535$	8548.03	0.690773
$536$	0	0
$537$	10127.7	0.813857
$538$	0	0
$539$	0	0
$540$	0	0
$541$	16614.3	1.32034	0.660169	−	0.751117i	$-0.270485\pi$
$541$	16614.3	1.32034	0.660169	+	0.751117i	$0.270485\pi$
$542$	0	0
$543$	2357.78	0.186339
$544$	0	0
$545$	−2072.11	−0.162861
$546$	0	0
$547$	12871.3	1.00610	0.503050	−	0.864258i	$-0.332211\pi$
$547$	12871.3	1.00610	0.503050	+	0.864258i	$0.332211\pi$
$548$	0	0
$549$	−2497.76	−0.194174
$550$	0	0
$551$	7088.67	0.548072
$552$	0	0
$553$	−5709.36	−0.439036
$554$	0	0
$555$	−1107.43	−0.0846988
$556$	0	0
$557$	5691.26	0.432938	0.216469	−	0.976289i	$-0.430546\pi$
$557$	5691.26	0.432938	0.216469	+	0.976289i	$0.430546\pi$
$558$	0	0
$559$	27653.1	2.09231
$560$	0	0
$561$	0	0
$562$	0	0
$563$	17890.6	1.33925	0.669626	−	0.742699i	$-0.266455\pi$
$563$	17890.6	1.33925	0.669626	+	0.742699i	$0.266455\pi$
$564$	0	0
$565$	−9168.98	−0.682729
$566$	0	0
$567$	−1491.10	−0.110441
$568$	0	0
$569$	−9099.56	−0.670428	−0.335214	−	0.942142i	$-0.608809\pi$
$569$	−9099.56	−0.670428	−0.335214	+	0.942142i	$0.608809\pi$
$570$	0	0
$571$	−3320.97	−0.243394	−0.121697	−	0.992567i	$-0.538834\pi$
$571$	−3320.97	−0.243394	−0.121697	+	0.992567i	$0.538834\pi$
$572$	0	0
$573$	−3119.74	−0.227451
$574$	0	0
$575$	2020.76	0.146559
$576$	0	0
$577$	−24181.1	−1.74467	−0.872333	−	0.488912i	$-0.837394\pi$
$577$	−24181.1	−1.74467	−0.872333	+	0.488912i	$0.837394\pi$
$578$	0	0
$579$	3737.28	0.268249
$580$	0	0
$581$	−19757.9	−1.41083
$582$	0	0
$583$	0	0
$584$	0	0
$585$	8665.61	0.612442
$586$	0	0
$587$	−26367.9	−1.85403	−0.927017	−	0.375018i	$-0.877636\pi$
$587$	−26367.9	−1.85403	−0.927017	+	0.375018i	$0.877636\pi$
$588$	0	0
$589$	−10909.9	−0.763219
$590$	0	0
$591$	−8678.20	−0.604016
$592$	0	0
$593$	3262.61	0.225935	0.112967	−	0.993599i	$-0.463964\pi$
$593$	3262.61	0.225935	0.112967	+	0.993599i	$0.463964\pi$
$594$	0	0
$595$	14232.8	0.980651
$596$	0	0
$597$	−15391.2	−1.05514
$598$	0	0
$599$	−10205.7	−0.696148	−0.348074	−	0.937467i	$-0.613164\pi$
$599$	−10205.7	−0.696148	−0.348074	+	0.937467i	$0.613164\pi$
$600$	0	0
$601$	−19629.0	−1.33225	−0.666125	−	0.745840i	$-0.732048\pi$
$601$	−19629.0	−1.33225	−0.666125	+	0.745840i	$0.732048\pi$
$602$	0	0
$603$	−3269.69	−0.220816
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−18047.2	−1.20678	−0.603388	−	0.797448i	$-0.706183\pi$
$607$	−18047.2	−1.20678	−0.603388	+	0.797448i	$0.706183\pi$
$608$	0	0
$609$	10243.6	0.681594
$610$	0	0
$611$	6203.36	0.410738
$612$	0	0
$613$	−21199.2	−1.39678	−0.698390	−	0.715717i	$-0.746100\pi$
$613$	−21199.2	−1.39678	−0.698390	+	0.715717i	$0.746100\pi$
$614$	0	0
$615$	−13787.1	−0.903982
$616$	0	0
$617$	−24424.5	−1.59367	−0.796835	−	0.604196i	$-0.793494\pi$
$617$	−24424.5	−1.59367	−0.796835	+	0.604196i	$0.793494\pi$
$618$	0	0
$619$	−6954.04	−0.451545	−0.225773	−	0.974180i	$-0.572491\pi$
$619$	−6954.04	−0.451545	−0.225773	+	0.974180i	$0.572491\pi$
$620$	0	0
$621$	−2898.33	−0.187288
$622$	0	0
$623$	9570.94	0.615492
$624$	0	0
$625$	−12917.5	−0.826722
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2688.08	0.170399
$630$	0	0
$631$	10767.5	0.679314	0.339657	−	0.940549i	$-0.389689\pi$
$631$	10767.5	0.679314	0.339657	+	0.940549i	$0.389689\pi$
$632$	0	0
$633$	−8932.70	−0.560889
$634$	0	0
$635$	−160.840	−0.0100515
$636$	0	0
$637$	385.361	0.0239695
$638$	0	0
$639$	8927.58	0.552691
$640$	0	0
$641$	−4627.73	−0.285155	−0.142577	−	0.989784i	$-0.545539\pi$
$641$	−4627.73	−0.285155	−0.142577	+	0.989784i	$0.545539\pi$
$642$	0	0
$643$	−21050.2	−1.29104	−0.645521	−	0.763743i	$-0.723360\pi$
$643$	−21050.2	−1.29104	−0.645521	+	0.763743i	$0.723360\pi$
$644$	0	0
$645$	9148.12	0.558461
$646$	0	0
$647$	4869.46	0.295886	0.147943	−	0.988996i	$-0.452735\pi$
$647$	4869.46	0.295886	0.147943	+	0.988996i	$0.452735\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−15765.5	−0.949155
$652$	0	0
$653$	−30026.5	−1.79943	−0.899715	−	0.436477i	$-0.856226\pi$
$653$	−30026.5	−1.79943	−0.899715	+	0.436477i	$0.856226\pi$
$654$	0	0
$655$	4317.28	0.257542
$656$	0	0
$657$	2013.32	0.119554
$658$	0	0
$659$	10255.0	0.606189	0.303095	−	0.952960i	$-0.401980\pi$
$659$	10255.0	0.606189	0.303095	+	0.952960i	$0.401980\pi$
$660$	0	0
$661$	15754.2	0.927031	0.463515	−	0.886089i	$-0.346588\pi$
$661$	15754.2	0.927031	0.463515	+	0.886089i	$0.346588\pi$
$662$	0	0
$663$	−21034.1	−1.23212
$664$	0	0
$665$	−7249.14	−0.422721
$666$	0	0
$667$	19911.0	1.15586
$668$	0	0
$669$	−5596.64	−0.323436
$670$	0	0
$671$	0	0
$672$	0	0
$673$	6016.90	0.344628	0.172314	−	0.985042i	$-0.444876\pi$
$673$	6016.90	0.344628	0.172314	+	0.985042i	$0.444876\pi$
$674$	0	0
$675$	−508.270	−0.0289827
$676$	0	0
$677$	25373.0	1.44042	0.720209	−	0.693757i	$-0.244046\pi$
$677$	25373.0	1.44042	0.720209	+	0.693757i	$0.244046\pi$
$678$	0	0
$679$	−20086.0	−1.13525
$680$	0	0
$681$	−5852.65	−0.329331
$682$	0	0
$683$	−3803.49	−0.213084	−0.106542	−	0.994308i	$-0.533978\pi$
$683$	−3803.49	−0.213084	−0.106542	+	0.994308i	$0.533978\pi$
$684$	0	0
$685$	−22643.7	−1.26303
$686$	0	0
$687$	1330.64	0.0738969
$688$	0	0
$689$	−40232.8	−2.22460
$690$	0	0
$691$	15703.7	0.864540	0.432270	−	0.901744i	$-0.357713\pi$
$691$	15703.7	0.864540	0.432270	+	0.901744i	$0.357713\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	336.102	0.0183440
$696$	0	0
$697$	33465.5	1.81865
$698$	0	0
$699$	17444.6	0.943941
$700$	0	0
$701$	−16367.6	−0.881879	−0.440939	−	0.897537i	$-0.645355\pi$
$701$	−16367.6	−0.881879	−0.440939	+	0.897537i	$0.645355\pi$
$702$	0	0
$703$	−1369.11	−0.0734523
$704$	0	0
$705$	2052.18	0.109630
$706$	0	0
$707$	21868.4	1.16329
$708$	0	0
$709$	−31262.6	−1.65598	−0.827992	−	0.560739i	$-0.810517\pi$
$709$	−31262.6	−1.65598	−0.827992	+	0.560739i	$0.810517\pi$
$710$	0	0
$711$	2791.32	0.147233
$712$	0	0
$713$	−30644.4	−1.60960
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−15624.1	−0.813795
$718$	0	0
$719$	−29448.9	−1.52748	−0.763741	−	0.645523i	$-0.776640\pi$
$719$	−29448.9	−1.52748	−0.763741	+	0.645523i	$0.776640\pi$
$720$	0	0
$721$	−15336.6	−0.792186
$722$	0	0
$723$	−5613.68	−0.288762
$724$	0	0
$725$	3491.73	0.178868
$726$	0	0
$727$	23185.3	1.18280	0.591399	−	0.806379i	$-0.298576\pi$
$727$	23185.3	1.18280	0.591399	+	0.806379i	$0.298576\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−22205.3	−1.12352
$732$	0	0
$733$	19688.4	0.992097	0.496049	−	0.868295i	$-0.334784\pi$
$733$	19688.4	0.992097	0.496049	+	0.868295i	$0.334784\pi$
$734$	0	0
$735$	127.484	0.00639771
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−3261.15	−0.162332	−0.0811660	−	0.996701i	$-0.525864\pi$
$739$	−3261.15	−0.162332	−0.0811660	+	0.996701i	$0.525864\pi$
$740$	0	0
$741$	10713.2	0.531121
$742$	0	0
$743$	20001.0	0.987573	0.493787	−	0.869583i	$-0.335612\pi$
$743$	20001.0	0.987573	0.493787	+	0.869583i	$0.335612\pi$
$744$	0	0
$745$	34619.2	1.70248
$746$	0	0
$747$	9659.67	0.473131
$748$	0	0
$749$	15271.3	0.744993
$750$	0	0
$751$	−10624.2	−0.516224	−0.258112	−	0.966115i	$-0.583100\pi$
$751$	−10624.2	−0.516224	−0.258112	+	0.966115i	$0.583100\pi$
$752$	0	0
$753$	1503.25	0.0727512
$754$	0	0
$755$	−31672.9	−1.52675
$756$	0	0
$757$	−14839.5	−0.712486	−0.356243	−	0.934393i	$-0.615942\pi$
$757$	−14839.5	−0.712486	−0.356243	+	0.934393i	$0.615942\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4401.81	0.209679	0.104839	−	0.994489i	$-0.466567\pi$
$761$	4401.81	0.209679	0.104839	+	0.994489i	$0.466567\pi$
$762$	0	0
$763$	−3701.87	−0.175644
$764$	0	0
$765$	−6958.44	−0.328867
$766$	0	0
$767$	35759.6	1.68345
$768$	0	0
$769$	−34100.9	−1.59910	−0.799552	−	0.600597i	$-0.794929\pi$
$769$	−34100.9	−1.59910	−0.799552	+	0.600597i	$0.794929\pi$
$770$	0	0
$771$	10736.5	0.501514
$772$	0	0
$773$	−15004.5	−0.698157	−0.349079	−	0.937093i	$-0.613505\pi$
$773$	−15004.5	−0.698157	−0.349079	+	0.937093i	$0.613505\pi$
$774$	0	0
$775$	−5374.00	−0.249084
$776$	0	0
$777$	−1978.45	−0.0913469
$778$	0	0
$779$	−17044.9	−0.783950
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−5008.11	−0.228576
$784$	0	0
$785$	−23650.0	−1.07529
$786$	0	0
$787$	−15766.9	−0.714143	−0.357072	−	0.934077i	$-0.616225\pi$
$787$	−15766.9	−0.714143	−0.357072	+	0.934077i	$0.616225\pi$
$788$	0	0
$789$	5308.79	0.239541
$790$	0	0
$791$	−16380.6	−0.736317
$792$	0	0
$793$	25933.0	1.16130
$794$	0	0
$795$	−13309.7	−0.593769
$796$	0	0
$797$	−8490.24	−0.377340	−0.188670	−	0.982041i	$-0.560418\pi$
$797$	−8490.24	−0.377340	−0.188670	+	0.982041i	$0.560418\pi$
$798$	0	0
$799$	−4981.27	−0.220557
$800$	0	0
$801$	−4679.26	−0.206409
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−20361.8	−0.891501
$806$	0	0
$807$	−10070.0	−0.439259
$808$	0	0
$809$	−41661.0	−1.81054	−0.905268	−	0.424841i	$-0.860330\pi$
$809$	−41661.0	−1.81054	−0.905268	+	0.424841i	$0.860330\pi$
$810$	0	0
$811$	38286.1	1.65772	0.828858	−	0.559458i	$-0.188991\pi$
$811$	38286.1	1.65772	0.828858	+	0.559458i	$0.188991\pi$
$812$	0	0
$813$	21666.3	0.934650
$814$	0	0
$815$	−12092.4	−0.519728
$816$	0	0
$817$	11309.8	0.484307
$818$	0	0
$819$	15481.3	0.660514
$820$	0	0
$821$	18942.7	0.805244	0.402622	−	0.915366i	$-0.368099\pi$
$821$	18942.7	0.805244	0.402622	+	0.915366i	$0.368099\pi$
$822$	0	0
$823$	22445.5	0.950668	0.475334	−	0.879805i	$-0.342327\pi$
$823$	22445.5	0.950668	0.475334	+	0.879805i	$0.342327\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31648.8	1.33076	0.665380	−	0.746505i	$-0.268270\pi$
$827$	31648.8	1.33076	0.665380	+	0.746505i	$0.268270\pi$
$828$	0	0
$829$	13695.0	0.573760	0.286880	−	0.957967i	$-0.407382\pi$
$829$	13695.0	0.573760	0.286880	+	0.957967i	$0.407382\pi$
$830$	0	0
$831$	−15485.2	−0.646421
$832$	0	0
$833$	−309.443	−0.0128710
$834$	0	0
$835$	29615.2	1.22740
$836$	0	0
$837$	7707.81	0.318305
$838$	0	0
$839$	34249.7	1.40934	0.704668	−	0.709537i	$-0.251096\pi$
$839$	34249.7	1.40934	0.704668	+	0.709537i	$0.251096\pi$
$840$	0	0
$841$	10015.9	0.410672
$842$	0	0
$843$	−2875.94	−0.117500
$844$	0	0
$845$	−67332.6	−2.74120
$846$	0	0
$847$	0	0
$848$	0	0
$849$	3851.93	0.155710
$850$	0	0
$851$	−3845.63	−0.154908
$852$	0	0
$853$	2046.03	0.0821275	0.0410637	−	0.999157i	$-0.486925\pi$
$853$	2046.03	0.0821275	0.0410637	+	0.999157i	$0.486925\pi$
$854$	0	0
$855$	3544.12	0.141762
$856$	0	0
$857$	281.695	0.0112281	0.00561407	−	0.999984i	$-0.498213\pi$
$857$	281.695	0.0112281	0.00561407	+	0.999984i	$0.498213\pi$
$858$	0	0
$859$	19534.2	0.775901	0.387951	−	0.921680i	$-0.373183\pi$
$859$	19534.2	0.775901	0.387951	+	0.921680i	$0.373183\pi$
$860$	0	0
$861$	−24631.0	−0.974937
$862$	0	0
$863$	25836.9	1.01912	0.509559	−	0.860435i	$-0.329808\pi$
$863$	25836.9	1.01912	0.509559	+	0.860435i	$0.329808\pi$
$864$	0	0
$865$	−27209.2	−1.06953
$866$	0	0
$867$	2151.31	0.0842701
$868$	0	0
$869$	0	0
$870$	0	0
$871$	33947.6	1.32063
$872$	0	0
$873$	9820.12	0.380711
$874$	0	0
$875$	−27281.3	−1.05403
$876$	0	0
$877$	−7433.88	−0.286230	−0.143115	−	0.989706i	$-0.545712\pi$
$877$	−7433.88	−0.286230	−0.143115	+	0.989706i	$0.545712\pi$
$878$	0	0
$879$	−19202.1	−0.736828
$880$	0	0
$881$	−26580.5	−1.01648	−0.508240	−	0.861215i	$-0.669704\pi$
$881$	−26580.5	−1.01648	−0.508240	+	0.861215i	$0.669704\pi$
$882$	0	0
$883$	47009.3	1.79161	0.895804	−	0.444449i	$-0.146600\pi$
$883$	47009.3	1.79161	0.895804	+	0.444449i	$0.146600\pi$
$884$	0	0
$885$	11829.9	0.449331
$886$	0	0
$887$	23948.4	0.906550	0.453275	−	0.891371i	$-0.350256\pi$
$887$	23948.4	0.906550	0.453275	+	0.891371i	$0.350256\pi$
$888$	0	0
$889$	−287.344	−0.0108405
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2537.09	0.0950735
$894$	0	0
$895$	−34785.6	−1.29917
$896$	0	0
$897$	30091.9	1.12011
$898$	0	0
$899$	−52951.4	−1.96443
$900$	0	0
$901$	32306.8	1.19456
$902$	0	0
$903$	16343.3	0.602295
$904$	0	0
$905$	−8098.30	−0.297455
$906$	0	0
$907$	8876.24	0.324951	0.162475	−	0.986713i	$-0.448052\pi$
$907$	8876.24	0.324951	0.162475	+	0.986713i	$0.448052\pi$
$908$	0	0
$909$	−10691.5	−0.390116
$910$	0	0
$911$	29594.4	1.07630	0.538148	−	0.842851i	$-0.319124\pi$
$911$	29594.4	1.07630	0.538148	+	0.842851i	$0.319124\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	8579.07	0.309962
$916$	0	0
$917$	7712.93	0.277757
$918$	0	0
$919$	37066.9	1.33049	0.665247	−	0.746623i	$-0.268326\pi$
$919$	37066.9	1.33049	0.665247	+	0.746623i	$0.268326\pi$
$920$	0	0
$921$	16911.7	0.605058
$922$	0	0
$923$	−92690.7	−3.30547
$924$	0	0
$925$	−674.395	−0.0239719
$926$	0	0
$927$	7498.12	0.265664
$928$	0	0
$929$	−39465.4	−1.39378	−0.696888	−	0.717180i	$-0.745432\pi$
$929$	−39465.4	−1.39378	−0.696888	+	0.717180i	$0.745432\pi$
$930$	0	0
$931$	157.608	0.00554821
$932$	0	0
$933$	26356.9	0.924851
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−29964.1	−1.04470	−0.522350	−	0.852731i	$-0.674944\pi$
$937$	−29964.1	−1.04470	−0.522350	+	0.852731i	$0.674944\pi$
$938$	0	0
$939$	17910.7	0.622464
$940$	0	0
$941$	13533.3	0.468835	0.234417	−	0.972136i	$-0.424682\pi$
$941$	13533.3	0.468835	0.234417	+	0.972136i	$0.424682\pi$
$942$	0	0
$943$	−47876.6	−1.65332
$944$	0	0
$945$	5121.48	0.176298
$946$	0	0
$947$	23764.6	0.815466	0.407733	−	0.913101i	$-0.366319\pi$
$947$	23764.6	0.815466	0.407733	+	0.913101i	$0.366319\pi$
$948$	0	0
$949$	−20903.3	−0.715017
$950$	0	0
$951$	−7471.19	−0.254753
$952$	0	0
$953$	44164.0	1.50117	0.750583	−	0.660776i	$-0.229773\pi$
$953$	44164.0	1.50117	0.750583	+	0.660776i	$0.229773\pi$
$954$	0	0
$955$	10715.4	0.363081
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−40453.6	−1.36216
$960$	0	0
$961$	51704.6	1.73558
$962$	0	0
$963$	−7466.16	−0.249838
$964$	0	0
$965$	−12836.5	−0.428208
$966$	0	0
$967$	12183.8	0.405174	0.202587	−	0.979264i	$-0.435065\pi$
$967$	12183.8	0.405174	0.202587	+	0.979264i	$0.435065\pi$
$968$	0	0
$969$	−8602.69	−0.285199
$970$	0	0
$971$	53903.0	1.78149	0.890747	−	0.454500i	$-0.150182\pi$
$971$	53903.0	1.78149	0.890747	+	0.454500i	$0.150182\pi$
$972$	0	0
$973$	600.454	0.0197838
$974$	0	0
$975$	5277.12	0.173336
$976$	0	0
$977$	23888.9	0.782264	0.391132	−	0.920334i	$-0.372084\pi$
$977$	23888.9	0.782264	0.391132	+	0.920334i	$0.372084\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	1809.85	0.0589033
$982$	0	0
$983$	−54001.2	−1.75216	−0.876079	−	0.482167i	$-0.839850\pi$
$983$	−54001.2	−1.75216	−0.876079	+	0.482167i	$0.839850\pi$
$984$	0	0
$985$	29807.1	0.964197
$986$	0	0
$987$	3666.26	0.118235
$988$	0	0
$989$	31767.5	1.02138
$990$	0	0
$991$	−8358.61	−0.267931	−0.133966	−	0.990986i	$-0.542771\pi$
$991$	−8358.61	−0.267931	−0.133966	+	0.990986i	$0.542771\pi$
$992$	0	0
$993$	892.818	0.0285325
$994$	0	0
$995$	52864.3	1.68433
$996$	0	0
$997$	−11623.2	−0.369219	−0.184610	−	0.982812i	$-0.559102\pi$
$997$	−11623.2	−0.369219	−0.184610	+	0.982812i	$0.559102\pi$
$998$	0	0
$999$	967.270	0.0306337

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1452.4.a.q.1.1	✓	4
11.10	odd	2	inner	1452.4.a.q.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1452.4.a.q.1.1	✓	4	1.1	even	1	trivial
1452.4.a.q.1.2	yes	4	11.10	odd	2	inner

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:	\( N \)	\(=\)	\( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1452.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -10.3041 q^{5} -18.4086 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -10.3041 q^{5} -18.4086 q^{7} +9.00000 q^{9} -93.4426 q^{13} -30.9124 q^{15} +75.0340 q^{17} -38.2168 q^{19} -55.2258 q^{21} -107.345 q^{23} -18.8248 q^{25} +27.0000 q^{27} -185.486 q^{29} +285.474 q^{31} +189.685 q^{35} +35.8248 q^{37} -280.328 q^{39} +446.005 q^{41} -295.937 q^{43} -92.7372 q^{45} -66.3868 q^{47} -4.12404 q^{49} +225.102 q^{51} +430.562 q^{53} -114.651 q^{57} -382.691 q^{59} -277.528 q^{61} -165.677 q^{63} +962.845 q^{65} -363.299 q^{67} -322.036 q^{69} +991.954 q^{71} +223.702 q^{73} -56.4744 q^{75} +310.147 q^{79} +81.0000 q^{81} +1073.30 q^{83} -773.161 q^{85} -556.457 q^{87} -519.917 q^{89} +1720.15 q^{91} +856.423 q^{93} +393.791 q^{95} +1091.12 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} + 12 q^{5} + 36 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 + 12 * q^5 + 36 * q^9	\( 4 q + 12 q^{3} + 12 q^{5} + 36 q^{9} + 36 q^{15} + 156 q^{23} + 244 q^{25} + 108 q^{27} + 184 q^{31} - 176 q^{37} + 108 q^{45} + 852 q^{47} + 1580 q^{49} + 924 q^{53} - 360 q^{59} - 176 q^{67} + 468 q^{69} + 3276 q^{71} + 732 q^{75} + 324 q^{81} - 3144 q^{89} - 144 q^{91} + 552 q^{93} + 2768 q^{97}+O(q^{100}) \) 4 * q + 12 * q^3 + 12 * q^5 + 36 * q^9 + 36 * q^15 + 156 * q^23 + 244 * q^25 + 108 * q^27 + 184 * q^31 - 176 * q^37 + 108 * q^45 + 852 * q^47 + 1580 * q^49 + 924 * q^53 - 360 * q^59 - 176 * q^67 + 468 * q^69 + 3276 * q^71 + 732 * q^75 + 324 * q^81 - 3144 * q^89 - 144 * q^91 + 552 * q^93 + 2768 * q^97

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