LMFDB - Embedded newform 1134.4.a.t.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1134,4,Mod(1,1134)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1134, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1134.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1134 = 2 \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1134.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:	$66.9081659465$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 151x^{4} + 79x^{3} + 5033x^{2} - 1640x - 9368 $ x^6 - 2x^5 - 151x^4 + 79x^3 + 5033x^2 - 1640*x - 9368
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 126)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.58228$ of defining polynomial
Character	$\chi$	$=$	1134.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} +2.90034 q^{5} +7.00000 q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+2.00000 q^{2} +4.00000 q^{4} +2.90034 q^{5} +7.00000 q^{7} +8.00000 q^{8} +5.80068 q^{10} +65.4249 q^{11} -39.3702 q^{13} +14.0000 q^{14} +16.0000 q^{16} +13.8868 q^{17} -11.6969 q^{19} +11.6014 q^{20} +130.850 q^{22} -8.25383 q^{23} -116.588 q^{25} -78.7403 q^{26} +28.0000 q^{28} +295.386 q^{29} +248.725 q^{31} +32.0000 q^{32} +27.7735 q^{34} +20.3024 q^{35} +289.301 q^{37} -23.3939 q^{38} +23.2027 q^{40} -221.242 q^{41} -129.899 q^{43} +261.700 q^{44} -16.5077 q^{46} -518.477 q^{47} +49.0000 q^{49} -233.176 q^{50} -157.481 q^{52} +264.820 q^{53} +189.754 q^{55} +56.0000 q^{56} +590.772 q^{58} -235.317 q^{59} -414.456 q^{61} +497.449 q^{62} +64.0000 q^{64} -114.187 q^{65} +139.770 q^{67} +55.5470 q^{68} +40.6047 q^{70} +145.858 q^{71} +771.401 q^{73} +578.601 q^{74} -46.7878 q^{76} +457.974 q^{77} -275.675 q^{79} +46.4054 q^{80} -442.483 q^{82} +1091.12 q^{83} +40.2763 q^{85} -259.797 q^{86} +523.399 q^{88} +852.941 q^{89} -275.591 q^{91} -33.0153 q^{92} -1036.95 q^{94} -33.9251 q^{95} +1516.75 q^{97} +98.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{2} + 24 q^{4} + 9 q^{5} + 42 q^{7} + 48 q^{8}+O(q^{10}) $ 6 * q + 12 * q^2 + 24 * q^4 + 9 * q^5 + 42 * q^7 + 48 * q^8	$ 6 q + 12 q^{2} + 24 q^{4} + 9 q^{5} + 42 q^{7} + 48 q^{8} + 18 q^{10} + 39 q^{11} + 21 q^{13} + 84 q^{14} + 96 q^{16} + 87 q^{17} + 96 q^{19} + 36 q^{20} + 78 q^{22} + 168 q^{23} + 309 q^{25} + 42 q^{26} + 168 q^{28} + 117 q^{29} + 129 q^{31} + 192 q^{32} + 174 q^{34} + 63 q^{35} + 471 q^{37} + 192 q^{38} + 72 q^{40} + 255 q^{41} + 942 q^{43} + 156 q^{44} + 336 q^{46} + 81 q^{47} + 294 q^{49} + 618 q^{50} + 84 q^{52} + 90 q^{53} + 696 q^{55} + 336 q^{56} + 234 q^{58} + 444 q^{59} + 978 q^{61} + 258 q^{62} + 384 q^{64} + 747 q^{65} + 663 q^{67} + 348 q^{68} + 126 q^{70} + 507 q^{71} + 144 q^{73} + 942 q^{74} + 384 q^{76} + 273 q^{77} + 609 q^{79} + 144 q^{80} + 510 q^{82} + 516 q^{83} + 1563 q^{85} + 1884 q^{86} + 312 q^{88} + 378 q^{89} + 147 q^{91} + 672 q^{92} + 162 q^{94} + 846 q^{95} + 2292 q^{97} + 588 q^{98}+O(q^{100}) $ 6 * q + 12 * q^2 + 24 * q^4 + 9 * q^5 + 42 * q^7 + 48 * q^8 + 18 * q^10 + 39 * q^11 + 21 * q^13 + 84 * q^14 + 96 * q^16 + 87 * q^17 + 96 * q^19 + 36 * q^20 + 78 * q^22 + 168 * q^23 + 309 * q^25 + 42 * q^26 + 168 * q^28 + 117 * q^29 + 129 * q^31 + 192 * q^32 + 174 * q^34 + 63 * q^35 + 471 * q^37 + 192 * q^38 + 72 * q^40 + 255 * q^41 + 942 * q^43 + 156 * q^44 + 336 * q^46 + 81 * q^47 + 294 * q^49 + 618 * q^50 + 84 * q^52 + 90 * q^53 + 696 * q^55 + 336 * q^56 + 234 * q^58 + 444 * q^59 + 978 * q^61 + 258 * q^62 + 384 * q^64 + 747 * q^65 + 663 * q^67 + 348 * q^68 + 126 * q^70 + 507 * q^71 + 144 * q^73 + 942 * q^74 + 384 * q^76 + 273 * q^77 + 609 * q^79 + 144 * q^80 + 510 * q^82 + 516 * q^83 + 1563 * q^85 + 1884 * q^86 + 312 * q^88 + 378 * q^89 + 147 * q^91 + 672 * q^92 + 162 * q^94 + 846 * q^95 + 2292 * q^97 + 588 * q^98

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	2.90034	0.259414	0.129707	−	0.991552i	$-0.458596\pi$
$5$	2.90034	0.259414	0.129707	+	0.991552i	$0.458596\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.00000	0.353553
$9$	0	0
$10$	5.80068	0.183434
$11$	65.4249	1.79330	0.896652	−	0.442735i	$-0.145992\pi$
$11$	65.4249	1.79330	0.896652	+	0.442735i	$0.145992\pi$
$12$	0	0
$13$	−39.3702	−0.839948	−0.419974	−	0.907536i	$-0.637961\pi$
$13$	−39.3702	−0.839948	−0.419974	+	0.907536i	$0.637961\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	13.8868	0.198120	0.0990598	−	0.995081i	$-0.468416\pi$
$17$	13.8868	0.198120	0.0990598	+	0.995081i	$0.468416\pi$
$18$	0	0
$19$	−11.6969	−0.141235	−0.0706174	−	0.997503i	$-0.522497\pi$
$19$	−11.6969	−0.141235	−0.0706174	+	0.997503i	$0.522497\pi$
$20$	11.6014	0.129707
$21$	0	0
$22$	130.850	1.26806
$23$	−8.25383	−0.0748279	−0.0374140	−	0.999300i	$-0.511912\pi$
$23$	−8.25383	−0.0748279	−0.0374140	+	0.999300i	$0.511912\pi$
$24$	0	0
$25$	−116.588	−0.932704
$26$	−78.7403	−0.593933
$27$	0	0
$28$	28.0000	0.188982
$29$	295.386	1.89144	0.945721	−	0.324979i	$-0.105357\pi$
$29$	295.386	1.89144	0.945721	+	0.324979i	$0.105357\pi$
$30$	0	0
$31$	248.725	1.44104	0.720520	−	0.693434i	$-0.243903\pi$
$31$	248.725	1.44104	0.720520	+	0.693434i	$0.243903\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	27.7735	0.140092
$35$	20.3024	0.0980494
$36$	0	0
$37$	289.301	1.28543	0.642713	−	0.766107i	$-0.277809\pi$
$37$	289.301	1.28543	0.642713	+	0.766107i	$0.277809\pi$
$38$	−23.3939	−0.0998681
$39$	0	0
$40$	23.2027	0.0917168
$41$	−221.242	−0.842735	−0.421367	−	0.906890i	$-0.638450\pi$
$41$	−221.242	−0.842735	−0.421367	+	0.906890i	$0.638450\pi$
$42$	0	0
$43$	−129.899	−0.460683	−0.230341	−	0.973110i	$-0.573984\pi$
$43$	−129.899	−0.460683	−0.230341	+	0.973110i	$0.573984\pi$
$44$	261.700	0.896652
$45$	0	0
$46$	−16.5077	−0.0529113
$47$	−518.477	−1.60910	−0.804550	−	0.593885i	$-0.797594\pi$
$47$	−518.477	−1.60910	−0.804550	+	0.593885i	$0.797594\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−233.176	−0.659522
$51$	0	0
$52$	−157.481	−0.419974
$53$	264.820	0.686337	0.343168	−	0.939274i	$-0.388500\pi$
$53$	264.820	0.686337	0.343168	+	0.939274i	$0.388500\pi$
$54$	0	0
$55$	189.754	0.465209
$56$	56.0000	0.133631
$57$	0	0
$58$	590.772	1.33745
$59$	−235.317	−0.519248	−0.259624	−	0.965710i	$-0.583599\pi$
$59$	−235.317	−0.519248	−0.259624	+	0.965710i	$0.583599\pi$
$60$	0	0
$61$	−414.456	−0.869929	−0.434964	−	0.900448i	$-0.643239\pi$
$61$	−414.456	−0.869929	−0.434964	+	0.900448i	$0.643239\pi$
$62$	497.449	1.01897
$63$	0	0
$64$	64.0000	0.125000
$65$	−114.187	−0.217894
$66$	0	0
$67$	139.770	0.254859	0.127430	−	0.991848i	$-0.459327\pi$
$67$	139.770	0.254859	0.127430	+	0.991848i	$0.459327\pi$
$68$	55.5470	0.0990598
$69$	0	0
$70$	40.6047	0.0693314
$71$	145.858	0.243805	0.121902	−	0.992542i	$-0.461101\pi$
$71$	145.858	0.243805	0.121902	+	0.992542i	$0.461101\pi$
$72$	0	0
$73$	771.401	1.23679	0.618395	−	0.785867i	$-0.287783\pi$
$73$	771.401	1.23679	0.618395	+	0.785867i	$0.287783\pi$
$74$	578.601	0.908933
$75$	0	0
$76$	−46.7878	−0.0706174
$77$	457.974	0.677805
$78$	0	0
$79$	−275.675	−0.392605	−0.196303	−	0.980543i	$-0.562894\pi$
$79$	−275.675	−0.392605	−0.196303	+	0.980543i	$0.562894\pi$
$80$	46.4054	0.0648536
$81$	0	0
$82$	−442.483	−0.595903
$83$	1091.12	1.44296	0.721479	−	0.692436i	$-0.243463\pi$
$83$	1091.12	1.44296	0.721479	+	0.692436i	$0.243463\pi$
$84$	0	0
$85$	40.2763	0.0513950
$86$	−259.797	−0.325752
$87$	0	0
$88$	523.399	0.634029
$89$	852.941	1.01586	0.507930	−	0.861398i	$-0.330411\pi$
$89$	852.941	1.01586	0.507930	+	0.861398i	$0.330411\pi$
$90$	0	0
$91$	−275.591	−0.317470
$92$	−33.0153	−0.0374140
$93$	0	0
$94$	−1036.95	−1.13781
$95$	−33.9251	−0.0366383
$96$	0	0
$97$	1516.75	1.58766	0.793830	−	0.608140i	$-0.208084\pi$
$97$	1516.75	1.58766	0.793830	+	0.608140i	$0.208084\pi$
$98$	98.0000	0.101015
$99$	0	0
$100$	−466.352	−0.466352
$101$	−728.143	−0.717356	−0.358678	−	0.933461i	$-0.616772\pi$
$101$	−728.143	−0.717356	−0.358678	+	0.933461i	$0.616772\pi$
$102$	0	0
$103$	−37.1768	−0.0355644	−0.0177822	−	0.999842i	$-0.505661\pi$
$103$	−37.1768	−0.0355644	−0.0177822	+	0.999842i	$0.505661\pi$
$104$	−314.961	−0.296966
$105$	0	0
$106$	529.640	0.485313
$107$	−1213.30	−1.09621	−0.548103	−	0.836411i	$-0.684650\pi$
$107$	−1213.30	−1.09621	−0.548103	+	0.836411i	$0.684650\pi$
$108$	0	0
$109$	727.706	0.639464	0.319732	−	0.947508i	$-0.396407\pi$
$109$	727.706	0.639464	0.319732	+	0.947508i	$0.396407\pi$
$110$	379.509	0.328952
$111$	0	0
$112$	112.000	0.0944911
$113$	1762.53	1.46730	0.733649	−	0.679528i	$-0.237816\pi$
$113$	1762.53	1.46730	0.733649	+	0.679528i	$0.237816\pi$
$114$	0	0
$115$	−23.9389	−0.0194114
$116$	1181.54	0.945721
$117$	0	0
$118$	−470.633	−0.367164
$119$	97.2073	0.0748822
$120$	0	0
$121$	2949.42	2.21594
$122$	−828.912	−0.615132
$123$	0	0
$124$	994.898	0.720520
$125$	−700.687	−0.501371
$126$	0	0
$127$	1643.70	1.14846	0.574232	−	0.818692i	$-0.305301\pi$
$127$	1643.70	1.14846	0.574232	+	0.818692i	$0.305301\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−228.374	−0.154075
$131$	443.911	0.296066	0.148033	−	0.988982i	$-0.452706\pi$
$131$	443.911	0.296066	0.148033	+	0.988982i	$0.452706\pi$
$132$	0	0
$133$	−81.8786	−0.0533818
$134$	279.539	0.180213
$135$	0	0
$136$	111.094	0.0700458
$137$	−1183.69	−0.738171	−0.369085	−	0.929395i	$-0.620329\pi$
$137$	−1183.69	−0.738171	−0.369085	+	0.929395i	$0.620329\pi$
$138$	0	0
$139$	−536.955	−0.327654	−0.163827	−	0.986489i	$-0.552384\pi$
$139$	−536.955	−0.327654	−0.163827	+	0.986489i	$0.552384\pi$
$140$	81.2095	0.0490247
$141$	0	0
$142$	291.716	0.172396
$143$	−2575.79	−1.50628
$144$	0	0
$145$	856.720	0.490667
$146$	1542.80	0.874543
$147$	0	0
$148$	1157.20	0.642713
$149$	1333.95	0.733432	0.366716	−	0.930333i	$-0.380482\pi$
$149$	1333.95	0.733432	0.366716	+	0.930333i	$0.380482\pi$
$150$	0	0
$151$	1518.12	0.818164	0.409082	−	0.912498i	$-0.365849\pi$
$151$	1518.12	0.818164	0.409082	+	0.912498i	$0.365849\pi$
$152$	−93.5755	−0.0499341
$153$	0	0
$154$	915.949	0.479281
$155$	721.386	0.373826
$156$	0	0
$157$	−3217.94	−1.63580	−0.817898	−	0.575364i	$-0.804860\pi$
$157$	−3217.94	−1.63580	−0.817898	+	0.575364i	$0.804860\pi$
$158$	−551.349	−0.277614
$159$	0	0
$160$	92.8109	0.0458584
$161$	−57.7768	−0.0282823
$162$	0	0
$163$	−3272.13	−1.57235	−0.786176	−	0.618003i	$-0.787942\pi$
$163$	−3272.13	−1.57235	−0.786176	+	0.618003i	$0.787942\pi$
$164$	−884.966	−0.421367
$165$	0	0
$166$	2182.23	1.02033
$167$	−592.797	−0.274683	−0.137341	−	0.990524i	$-0.543856\pi$
$167$	−592.797	−0.274683	−0.137341	+	0.990524i	$0.543856\pi$
$168$	0	0
$169$	−646.990	−0.294488
$170$	80.5526	0.0363418
$171$	0	0
$172$	−519.595	−0.230341
$173$	−3373.65	−1.48262	−0.741311	−	0.671162i	$-0.765796\pi$
$173$	−3373.65	−1.48262	−0.741311	+	0.671162i	$0.765796\pi$
$174$	0	0
$175$	−816.116	−0.352529
$176$	1046.80	0.448326
$177$	0	0
$178$	1705.88	0.718322
$179$	597.839	0.249634	0.124817	−	0.992180i	$-0.460166\pi$
$179$	597.839	0.249634	0.124817	+	0.992180i	$0.460166\pi$
$180$	0	0
$181$	4639.73	1.90535	0.952675	−	0.303991i	$-0.0983192\pi$
$181$	4639.73	1.90535	0.952675	+	0.303991i	$0.0983192\pi$
$182$	−551.182	−0.224485
$183$	0	0
$184$	−66.0306	−0.0264557
$185$	839.070	0.333458
$186$	0	0
$187$	908.539	0.355289
$188$	−2073.91	−0.804550
$189$	0	0
$190$	−67.8502	−0.0259072
$191$	1031.31	0.390695	0.195347	−	0.980734i	$-0.437417\pi$
$191$	1031.31	0.390695	0.195347	+	0.980734i	$0.437417\pi$
$192$	0	0
$193$	2807.68	1.04716	0.523578	−	0.851978i	$-0.324597\pi$
$193$	2807.68	1.04716	0.523578	+	0.851978i	$0.324597\pi$
$194$	3033.51	1.12265
$195$	0	0
$196$	196.000	0.0714286
$197$	−3913.63	−1.41540	−0.707702	−	0.706511i	$-0.750268\pi$
$197$	−3913.63	−1.41540	−0.707702	+	0.706511i	$0.750268\pi$
$198$	0	0
$199$	822.179	0.292878	0.146439	−	0.989220i	$-0.453219\pi$
$199$	822.179	0.292878	0.146439	+	0.989220i	$0.453219\pi$
$200$	−932.704	−0.329761
$201$	0	0
$202$	−1456.29	−0.507247
$203$	2067.70	0.714898
$204$	0	0
$205$	−641.675	−0.218617
$206$	−74.3536	−0.0251479
$207$	0	0
$208$	−629.923	−0.209987
$209$	−765.271	−0.253277
$210$	0	0
$211$	2893.58	0.944087	0.472043	−	0.881575i	$-0.343517\pi$
$211$	2893.58	0.944087	0.472043	+	0.881575i	$0.343517\pi$
$212$	1059.28	0.343168
$213$	0	0
$214$	−2426.60	−0.775135
$215$	−376.750	−0.119508
$216$	0	0
$217$	1741.07	0.544662
$218$	1455.41	0.452169
$219$	0	0
$220$	759.018	0.232604
$221$	−546.724	−0.166410
$222$	0	0
$223$	4582.74	1.37616	0.688078	−	0.725637i	$-0.258455\pi$
$223$	4582.74	1.37616	0.688078	+	0.725637i	$0.258455\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	3525.06	1.03754
$227$	2593.00	0.758165	0.379082	−	0.925363i	$-0.376240\pi$
$227$	2593.00	0.758165	0.379082	+	0.925363i	$0.376240\pi$
$228$	0	0
$229$	−3079.57	−0.888664	−0.444332	−	0.895862i	$-0.646559\pi$
$229$	−3079.57	−0.888664	−0.444332	+	0.895862i	$0.646559\pi$
$230$	−47.8778	−0.0137260
$231$	0	0
$232$	2363.09	0.668726
$233$	2940.24	0.826702	0.413351	−	0.910572i	$-0.364358\pi$
$233$	2940.24	0.826702	0.413351	+	0.910572i	$0.364358\pi$
$234$	0	0
$235$	−1503.76	−0.417424
$236$	−941.266	−0.259624
$237$	0	0
$238$	194.415	0.0529497
$239$	−5898.63	−1.59645	−0.798223	−	0.602362i	$-0.794226\pi$
$239$	−5898.63	−1.59645	−0.798223	+	0.602362i	$0.794226\pi$
$240$	0	0
$241$	4574.05	1.22257	0.611287	−	0.791409i	$-0.290652\pi$
$241$	4574.05	1.22257	0.611287	+	0.791409i	$0.290652\pi$
$242$	5898.84	1.56691
$243$	0	0
$244$	−1657.82	−0.434964
$245$	142.117	0.0370592
$246$	0	0
$247$	460.511	0.118630
$248$	1989.80	0.509485
$249$	0	0
$250$	−1401.37	−0.354523
$251$	−1514.83	−0.380938	−0.190469	−	0.981693i	$-0.561001\pi$
$251$	−1514.83	−0.380938	−0.190469	+	0.981693i	$0.561001\pi$
$252$	0	0
$253$	−540.006	−0.134189
$254$	3287.40	0.812087
$255$	0	0
$256$	256.000	0.0625000
$257$	−3921.32	−0.951772	−0.475886	−	0.879507i	$-0.657873\pi$
$257$	−3921.32	−0.951772	−0.475886	+	0.879507i	$0.657873\pi$
$258$	0	0
$259$	2025.10	0.485845
$260$	−456.747	−0.108947
$261$	0	0
$262$	887.822	0.209351
$263$	−6500.42	−1.52408	−0.762040	−	0.647530i	$-0.775802\pi$
$263$	−6500.42	−1.52408	−0.762040	+	0.647530i	$0.775802\pi$
$264$	0	0
$265$	768.068	0.178045
$266$	−163.757	−0.0377466
$267$	0	0
$268$	559.079	0.127430
$269$	−2072.23	−0.469689	−0.234844	−	0.972033i	$-0.575458\pi$
$269$	−2072.23	−0.469689	−0.234844	+	0.972033i	$0.575458\pi$
$270$	0	0
$271$	19.3588	0.00433934	0.00216967	−	0.999998i	$-0.499309\pi$
$271$	19.3588	0.00433934	0.00216967	+	0.999998i	$0.499309\pi$
$272$	222.188	0.0495299
$273$	0	0
$274$	−2367.38	−0.521966
$275$	−7627.76	−1.67262
$276$	0	0
$277$	4334.03	0.940097	0.470048	−	0.882641i	$-0.344236\pi$
$277$	4334.03	0.940097	0.470048	+	0.882641i	$0.344236\pi$
$278$	−1073.91	−0.231686
$279$	0	0
$280$	162.419	0.0346657
$281$	−5360.76	−1.13806	−0.569032	−	0.822315i	$-0.692682\pi$
$281$	−5360.76	−1.13806	−0.569032	+	0.822315i	$0.692682\pi$
$282$	0	0
$283$	1381.15	0.290109	0.145055	−	0.989424i	$-0.453664\pi$
$283$	1381.15	0.290109	0.145055	+	0.989424i	$0.453664\pi$
$284$	583.431	0.121902
$285$	0	0
$286$	−5151.58	−1.06510
$287$	−1548.69	−0.318524
$288$	0	0
$289$	−4720.16	−0.960749
$290$	1713.44	0.346954
$291$	0	0
$292$	3085.60	0.618395
$293$	−7616.56	−1.51865	−0.759324	−	0.650712i	$-0.774470\pi$
$293$	−7616.56	−1.51865	−0.759324	+	0.650712i	$0.774470\pi$
$294$	0	0
$295$	−682.498	−0.134700
$296$	2314.41	0.454467
$297$	0	0
$298$	2667.90	0.518614
$299$	324.955	0.0628515
$300$	0	0
$301$	−909.291	−0.174122
$302$	3036.24	0.578529
$303$	0	0
$304$	−187.151	−0.0353087
$305$	−1202.06	−0.225672
$306$	0	0
$307$	1394.89	0.259318	0.129659	−	0.991559i	$-0.458612\pi$
$307$	1394.89	0.259318	0.129659	+	0.991559i	$0.458612\pi$
$308$	1831.90	0.338903
$309$	0	0
$310$	1442.77	0.264335
$311$	3513.58	0.640634	0.320317	−	0.947310i	$-0.396211\pi$
$311$	3513.58	0.640634	0.320317	+	0.947310i	$0.396211\pi$
$312$	0	0
$313$	−6394.20	−1.15470	−0.577351	−	0.816496i	$-0.695914\pi$
$313$	−6394.20	−1.15470	−0.577351	+	0.816496i	$0.695914\pi$
$314$	−6435.89	−1.15668
$315$	0	0
$316$	−1102.70	−0.196303
$317$	3301.56	0.584966	0.292483	−	0.956271i	$-0.405519\pi$
$317$	3301.56	0.584966	0.292483	+	0.956271i	$0.405519\pi$
$318$	0	0
$319$	19325.6	3.39193
$320$	185.622	0.0324268
$321$	0	0
$322$	−115.554	−0.0199986
$323$	−162.433	−0.0279814
$324$	0	0
$325$	4590.09	0.783423
$326$	−6544.27	−1.11182
$327$	0	0
$328$	−1769.93	−0.297952
$329$	−3629.34	−0.608183
$330$	0	0
$331$	−1332.01	−0.221190	−0.110595	−	0.993866i	$-0.535276\pi$
$331$	−1332.01	−0.221190	−0.110595	+	0.993866i	$0.535276\pi$
$332$	4364.46	0.721479
$333$	0	0
$334$	−1185.59	−0.194230
$335$	405.379	0.0661142
$336$	0	0
$337$	−11276.5	−1.82276	−0.911380	−	0.411567i	$-0.864982\pi$
$337$	−11276.5	−1.82276	−0.911380	+	0.411567i	$0.864982\pi$
$338$	−1293.98	−0.208235
$339$	0	0
$340$	161.105	0.0256975
$341$	16272.8	2.58422
$342$	0	0
$343$	343.000	0.0539949
$344$	−1039.19	−0.162876
$345$	0	0
$346$	−6747.29	−1.04837
$347$	−8458.91	−1.30864	−0.654320	−	0.756218i	$-0.727045\pi$
$347$	−8458.91	−1.30864	−0.654320	+	0.756218i	$0.727045\pi$
$348$	0	0
$349$	−6721.93	−1.03099	−0.515497	−	0.856892i	$-0.672393\pi$
$349$	−6721.93	−1.03099	−0.515497	+	0.856892i	$0.672393\pi$
$350$	−1632.23	−0.249276
$351$	0	0
$352$	2093.60	0.317014
$353$	−1709.45	−0.257747	−0.128874	−	0.991661i	$-0.541136\pi$
$353$	−1709.45	−0.257747	−0.128874	+	0.991661i	$0.541136\pi$
$354$	0	0
$355$	423.037	0.0632464
$356$	3411.76	0.507930
$357$	0	0
$358$	1195.68	0.176518
$359$	2066.56	0.303814	0.151907	−	0.988395i	$-0.451459\pi$
$359$	2066.56	0.303814	0.151907	+	0.988395i	$0.451459\pi$
$360$	0	0
$361$	−6722.18	−0.980053
$362$	9279.46	1.34729
$363$	0	0
$364$	−1102.36	−0.158735
$365$	2237.32	0.320841
$366$	0	0
$367$	−11876.1	−1.68917	−0.844587	−	0.535419i	$-0.820154\pi$
$367$	−11876.1	−1.68917	−0.844587	+	0.535419i	$0.820154\pi$
$368$	−132.061	−0.0187070
$369$	0	0
$370$	1678.14	0.235790
$371$	1853.74	0.259411
$372$	0	0
$373$	2799.74	0.388646	0.194323	−	0.980938i	$-0.437749\pi$
$373$	2799.74	0.388646	0.194323	+	0.980938i	$0.437749\pi$
$374$	1817.08	0.251227
$375$	0	0
$376$	−4147.82	−0.568903
$377$	−11629.4	−1.58871
$378$	0	0
$379$	3922.24	0.531588	0.265794	−	0.964030i	$-0.414366\pi$
$379$	3922.24	0.531588	0.265794	+	0.964030i	$0.414366\pi$
$380$	−135.700	−0.0183192
$381$	0	0
$382$	2062.61	0.276263
$383$	−4990.59	−0.665816	−0.332908	−	0.942959i	$-0.608030\pi$
$383$	−4990.59	−0.665816	−0.332908	+	0.942959i	$0.608030\pi$
$384$	0	0
$385$	1328.28	0.175832
$386$	5615.35	0.740451
$387$	0	0
$388$	6067.01	0.793830
$389$	−11866.4	−1.54666	−0.773331	−	0.634002i	$-0.781411\pi$
$389$	−11866.4	−1.54666	−0.773331	+	0.634002i	$0.781411\pi$
$390$	0	0
$391$	−114.619	−0.0148249
$392$	392.000	0.0505076
$393$	0	0
$394$	−7827.26	−1.00084
$395$	−799.550	−0.101847
$396$	0	0
$397$	−10596.6	−1.33962	−0.669812	−	0.742531i	$-0.733625\pi$
$397$	−10596.6	−1.33962	−0.669812	+	0.742531i	$0.733625\pi$
$398$	1644.36	0.207096
$399$	0	0
$400$	−1865.41	−0.233176
$401$	7857.35	0.978497	0.489248	−	0.872144i	$-0.337271\pi$
$401$	7857.35	0.978497	0.489248	+	0.872144i	$0.337271\pi$
$402$	0	0
$403$	−9792.33	−1.21040
$404$	−2912.57	−0.358678
$405$	0	0
$406$	4135.41	0.505509
$407$	18927.5	2.30516
$408$	0	0
$409$	−4647.94	−0.561921	−0.280961	−	0.959719i	$-0.590653\pi$
$409$	−4647.94	−0.561921	−0.280961	+	0.959719i	$0.590653\pi$
$410$	−1283.35	−0.154586
$411$	0	0
$412$	−148.707	−0.0177822
$413$	−1647.22	−0.196257
$414$	0	0
$415$	3164.61	0.374324
$416$	−1259.85	−0.148483
$417$	0	0
$418$	−1530.54	−0.179094
$419$	7737.32	0.902131	0.451065	−	0.892491i	$-0.351044\pi$
$419$	7737.32	0.902131	0.451065	+	0.892491i	$0.351044\pi$
$420$	0	0
$421$	−2171.16	−0.251344	−0.125672	−	0.992072i	$-0.540109\pi$
$421$	−2171.16	−0.251344	−0.125672	+	0.992072i	$0.540109\pi$
$422$	5787.16	0.667570
$423$	0	0
$424$	2118.56	0.242657
$425$	−1619.03	−0.184787
$426$	0	0
$427$	−2901.19	−0.328802
$428$	−4853.20	−0.548103
$429$	0	0
$430$	−753.501	−0.0845047
$431$	4714.95	0.526940	0.263470	−	0.964668i	$-0.415133\pi$
$431$	4714.95	0.526940	0.263470	+	0.964668i	$0.415133\pi$
$432$	0	0
$433$	−6862.68	−0.761661	−0.380830	−	0.924645i	$-0.624362\pi$
$433$	−6862.68	−0.761661	−0.380830	+	0.924645i	$0.624362\pi$
$434$	3482.14	0.385134
$435$	0	0
$436$	2910.82	0.319732
$437$	96.5446	0.0105683
$438$	0	0
$439$	500.020	0.0543614	0.0271807	−	0.999631i	$-0.491347\pi$
$439$	500.020	0.0543614	0.0271807	+	0.999631i	$0.491347\pi$
$440$	1518.04	0.164476
$441$	0	0
$442$	−1093.45	−0.117670
$443$	−1752.34	−0.187938	−0.0939688	−	0.995575i	$-0.529955\pi$
$443$	−1752.34	−0.187938	−0.0939688	+	0.995575i	$0.529955\pi$
$444$	0	0
$445$	2473.82	0.263529
$446$	9165.47	0.973089
$447$	0	0
$448$	448.000	0.0472456
$449$	−11846.4	−1.24514	−0.622568	−	0.782565i	$-0.713911\pi$
$449$	−11846.4	−1.24514	−0.622568	+	0.782565i	$0.713911\pi$
$450$	0	0
$451$	−14474.7	−1.51128
$452$	7050.11	0.733649
$453$	0	0
$454$	5186.00	0.536103
$455$	−799.308	−0.0823563
$456$	0	0
$457$	12807.2	1.31093	0.655464	−	0.755226i	$-0.272473\pi$
$457$	12807.2	1.31093	0.655464	+	0.755226i	$0.272473\pi$
$458$	−6159.15	−0.628380
$459$	0	0
$460$	−95.7556	−0.00970572
$461$	−10740.2	−1.08508	−0.542539	−	0.840030i	$-0.682537\pi$
$461$	−10740.2	−1.08508	−0.542539	+	0.840030i	$0.682537\pi$
$462$	0	0
$463$	3237.31	0.324947	0.162473	−	0.986713i	$-0.448053\pi$
$463$	3237.31	0.324947	0.162473	+	0.986713i	$0.448053\pi$
$464$	4726.18	0.472861
$465$	0	0
$466$	5880.48	0.584566
$467$	13705.1	1.35802	0.679009	−	0.734130i	$-0.262410\pi$
$467$	13705.1	1.35802	0.679009	+	0.734130i	$0.262410\pi$
$468$	0	0
$469$	978.388	0.0963278
$470$	−3007.52	−0.295163
$471$	0	0
$472$	−1882.53	−0.183582
$473$	−8498.61	−0.826145
$474$	0	0
$475$	1363.72	0.131730
$476$	388.829	0.0374411
$477$	0	0
$478$	−11797.3	−1.12886
$479$	10660.4	1.01688	0.508441	−	0.861097i	$-0.330222\pi$
$479$	10660.4	1.01688	0.508441	+	0.861097i	$0.330222\pi$
$480$	0	0
$481$	−11389.8	−1.07969
$482$	9148.10	0.864491
$483$	0	0
$484$	11797.7	1.10797
$485$	4399.10	0.411862
$486$	0	0
$487$	−14014.9	−1.30406	−0.652030	−	0.758193i	$-0.726082\pi$
$487$	−14014.9	−1.30406	−0.652030	+	0.758193i	$0.726082\pi$
$488$	−3315.65	−0.307566
$489$	0	0
$490$	284.233	0.0262048
$491$	1022.51	0.0939819	0.0469910	−	0.998895i	$-0.485037\pi$
$491$	1022.51	0.0939819	0.0469910	+	0.998895i	$0.485037\pi$
$492$	0	0
$493$	4101.96	0.374732
$494$	921.021	0.0838840
$495$	0	0
$496$	3979.59	0.360260
$497$	1021.00	0.0921495
$498$	0	0
$499$	−15241.9	−1.36738	−0.683691	−	0.729772i	$-0.739626\pi$
$499$	−15241.9	−1.36738	−0.683691	+	0.729772i	$0.739626\pi$
$500$	−2802.75	−0.250685
$501$	0	0
$502$	−3029.67	−0.269364
$503$	−11798.5	−1.04586	−0.522930	−	0.852375i	$-0.675161\pi$
$503$	−11798.5	−1.04586	−0.522930	+	0.852375i	$0.675161\pi$
$504$	0	0
$505$	−2111.86	−0.186092
$506$	−1080.01	−0.0948862
$507$	0	0
$508$	6574.81	0.574232
$509$	3568.49	0.310747	0.155374	−	0.987856i	$-0.450342\pi$
$509$	3568.49	0.310747	0.155374	+	0.987856i	$0.450342\pi$
$510$	0	0
$511$	5399.81	0.467463
$512$	512.000	0.0441942
$513$	0	0
$514$	−7842.65	−0.673005
$515$	−107.825	−0.00922592
$516$	0	0
$517$	−33921.3	−2.88561
$518$	4050.21	0.343544
$519$	0	0
$520$	−913.495	−0.0770373
$521$	−20499.0	−1.72376	−0.861878	−	0.507116i	$-0.830712\pi$
$521$	−20499.0	−1.72376	−0.861878	+	0.507116i	$0.830712\pi$
$522$	0	0
$523$	−14517.6	−1.21379	−0.606893	−	0.794783i	$-0.707584\pi$
$523$	−14517.6	−1.21379	−0.606893	+	0.794783i	$0.707584\pi$
$524$	1775.64	0.148033
$525$	0	0
$526$	−13000.8	−1.07769
$527$	3453.98	0.285498
$528$	0	0
$529$	−12098.9	−0.994401
$530$	1536.14	0.125897
$531$	0	0
$532$	−327.514	−0.0266909
$533$	8710.31	0.707853
$534$	0	0
$535$	−3518.98	−0.284372
$536$	1118.16	0.0901064
$537$	0	0
$538$	−4144.47	−0.332120
$539$	3205.82	0.256186
$540$	0	0
$541$	18273.3	1.45218	0.726090	−	0.687600i	$-0.241336\pi$
$541$	18273.3	1.45218	0.726090	+	0.687600i	$0.241336\pi$
$542$	38.7176	0.00306838
$543$	0	0
$544$	444.376	0.0350229
$545$	2110.59	0.165886
$546$	0	0
$547$	−9031.76	−0.705979	−0.352989	−	0.935627i	$-0.614835\pi$
$547$	−9031.76	−0.705979	−0.352989	+	0.935627i	$0.614835\pi$
$548$	−4734.76	−0.369085
$549$	0	0
$550$	−15255.5	−1.18272
$551$	−3455.12	−0.267138
$552$	0	0
$553$	−1929.72	−0.148391
$554$	8668.07	0.664749
$555$	0	0
$556$	−2147.82	−0.163827
$557$	18384.7	1.39854	0.699268	−	0.714859i	$-0.253509\pi$
$557$	18384.7	1.39854	0.699268	+	0.714859i	$0.253509\pi$
$558$	0	0
$559$	5114.13	0.386949
$560$	324.838	0.0245123
$561$	0	0
$562$	−10721.5	−0.804733
$563$	−1085.56	−0.0812624	−0.0406312	−	0.999174i	$-0.512937\pi$
$563$	−1085.56	−0.0812624	−0.0406312	+	0.999174i	$0.512937\pi$
$564$	0	0
$565$	5111.93	0.380638
$566$	2762.30	0.205138
$567$	0	0
$568$	1166.86	0.0861980
$569$	−13845.2	−1.02007	−0.510035	−	0.860154i	$-0.670368\pi$
$569$	−13845.2	−1.02007	−0.510035	+	0.860154i	$0.670368\pi$
$570$	0	0
$571$	6746.87	0.494480	0.247240	−	0.968954i	$-0.420476\pi$
$571$	6746.87	0.494480	0.247240	+	0.968954i	$0.420476\pi$
$572$	−10303.2	−0.753141
$573$	0	0
$574$	−3097.38	−0.225230
$575$	962.298	0.0697923
$576$	0	0
$577$	19670.9	1.41926	0.709629	−	0.704575i	$-0.248863\pi$
$577$	19670.9	1.41926	0.709629	+	0.704575i	$0.248863\pi$
$578$	−9440.32	−0.679352
$579$	0	0
$580$	3426.88	0.245334
$581$	7637.81	0.545387
$582$	0	0
$583$	17325.8	1.23081
$584$	6171.21	0.437271
$585$	0	0
$586$	−15233.1	−1.07385
$587$	−18881.4	−1.32763	−0.663816	−	0.747896i	$-0.731064\pi$
$587$	−18881.4	−1.32763	−0.663816	+	0.747896i	$0.731064\pi$
$588$	0	0
$589$	−2909.32	−0.203525
$590$	−1365.00	−0.0952474
$591$	0	0
$592$	4628.81	0.321356
$593$	−14887.6	−1.03096	−0.515480	−	0.856902i	$-0.672386\pi$
$593$	−14887.6	−1.03096	−0.515480	+	0.856902i	$0.672386\pi$
$594$	0	0
$595$	281.934	0.0194255
$596$	5335.79	0.366716
$597$	0	0
$598$	649.909	0.0444428
$599$	25645.0	1.74929	0.874647	−	0.484760i	$-0.161093\pi$
$599$	25645.0	1.74929	0.874647	+	0.484760i	$0.161093\pi$
$600$	0	0
$601$	22644.6	1.53693	0.768464	−	0.639893i	$-0.221022\pi$
$601$	22644.6	1.53693	0.768464	+	0.639893i	$0.221022\pi$
$602$	−1818.58	−0.123123
$603$	0	0
$604$	6072.48	0.409082
$605$	8554.31	0.574847
$606$	0	0
$607$	−1710.22	−0.114359	−0.0571794	−	0.998364i	$-0.518211\pi$
$607$	−1710.22	−0.114359	−0.0571794	+	0.998364i	$0.518211\pi$
$608$	−374.302	−0.0249670
$609$	0	0
$610$	−2404.13	−0.159574
$611$	20412.5	1.35156
$612$	0	0
$613$	8644.91	0.569600	0.284800	−	0.958587i	$-0.408073\pi$
$613$	8644.91	0.569600	0.284800	+	0.958587i	$0.408073\pi$
$614$	2789.79	0.183366
$615$	0	0
$616$	3663.79	0.239640
$617$	21078.6	1.37535	0.687677	−	0.726016i	$-0.258630\pi$
$617$	21078.6	1.37535	0.687677	+	0.726016i	$0.258630\pi$
$618$	0	0
$619$	−2631.57	−0.170875	−0.0854376	−	0.996344i	$-0.527229\pi$
$619$	−2631.57	−0.170875	−0.0854376	+	0.996344i	$0.527229\pi$
$620$	2885.54	0.186913
$621$	0	0
$622$	7027.17	0.452996
$623$	5970.59	0.383959
$624$	0	0
$625$	12541.3	0.802641
$626$	−12788.4	−0.816498
$627$	0	0
$628$	−12871.8	−0.817898
$629$	4017.45	0.254668
$630$	0	0
$631$	11022.1	0.695378	0.347689	−	0.937610i	$-0.386966\pi$
$631$	11022.1	0.695378	0.347689	+	0.937610i	$0.386966\pi$
$632$	−2205.40	−0.138807
$633$	0	0
$634$	6603.12	0.413633
$635$	4767.29	0.297928
$636$	0	0
$637$	−1929.14	−0.119993
$638$	38651.2	2.39846
$639$	0	0
$640$	371.243	0.0229292
$641$	3620.49	0.223090	0.111545	−	0.993759i	$-0.464420\pi$
$641$	3620.49	0.223090	0.111545	+	0.993759i	$0.464420\pi$
$642$	0	0
$643$	5427.59	0.332882	0.166441	−	0.986051i	$-0.446772\pi$
$643$	5427.59	0.332882	0.166441	+	0.986051i	$0.446772\pi$
$644$	−231.107	−0.0141412
$645$	0	0
$646$	−324.865	−0.0197858
$647$	−25639.9	−1.55797	−0.778986	−	0.627041i	$-0.784266\pi$
$647$	−25639.9	−1.55797	−0.778986	+	0.627041i	$0.784266\pi$
$648$	0	0
$649$	−15395.6	−0.931169
$650$	9180.18	0.553963
$651$	0	0
$652$	−13088.5	−0.786176
$653$	10437.8	0.625518	0.312759	−	0.949833i	$-0.398747\pi$
$653$	10437.8	0.625518	0.312759	+	0.949833i	$0.398747\pi$
$654$	0	0
$655$	1287.49	0.0768038
$656$	−3539.86	−0.210684
$657$	0	0
$658$	−7258.68	−0.430050
$659$	6718.99	0.397170	0.198585	−	0.980084i	$-0.436365\pi$
$659$	6718.99	0.397170	0.198585	+	0.980084i	$0.436365\pi$
$660$	0	0
$661$	11617.4	0.683605	0.341803	−	0.939772i	$-0.388963\pi$
$661$	11617.4	0.683605	0.341803	+	0.939772i	$0.388963\pi$
$662$	−2664.01	−0.156405
$663$	0	0
$664$	8728.93	0.510163
$665$	−237.476	−0.0138480
$666$	0	0
$667$	−2438.07	−0.141533
$668$	−2371.19	−0.137341
$669$	0	0
$670$	810.759	0.0467498
$671$	−27115.7	−1.56005
$672$	0	0
$673$	−33260.4	−1.90504	−0.952522	−	0.304470i	$-0.901521\pi$
$673$	−33260.4	−1.90504	−0.952522	+	0.304470i	$0.901521\pi$
$674$	−22553.0	−1.28889
$675$	0	0
$676$	−2587.96	−0.147244
$677$	−12726.9	−0.722504	−0.361252	−	0.932468i	$-0.617651\pi$
$677$	−12726.9	−0.722504	−0.361252	+	0.932468i	$0.617651\pi$
$678$	0	0
$679$	10617.3	0.600079
$680$	322.210	0.0181709
$681$	0	0
$682$	32545.6	1.82732
$683$	−12156.0	−0.681020	−0.340510	−	0.940241i	$-0.610600\pi$
$683$	−12156.0	−0.681020	−0.340510	+	0.940241i	$0.610600\pi$
$684$	0	0
$685$	−3433.10	−0.191492
$686$	686.000	0.0381802
$687$	0	0
$688$	−2078.38	−0.115171
$689$	−10426.0	−0.576487
$690$	0	0
$691$	−27996.8	−1.54131	−0.770657	−	0.637251i	$-0.780072\pi$
$691$	−27996.8	−1.54131	−0.770657	+	0.637251i	$0.780072\pi$
$692$	−13494.6	−0.741311
$693$	0	0
$694$	−16917.8	−0.925348
$695$	−1557.35	−0.0849981
$696$	0	0
$697$	−3072.33	−0.166962
$698$	−13443.9	−0.729022
$699$	0	0
$700$	−3264.46	−0.176265
$701$	−11244.9	−0.605869	−0.302935	−	0.953011i	$-0.597966\pi$
$701$	−11244.9	−0.605869	−0.302935	+	0.953011i	$0.597966\pi$
$702$	0	0
$703$	−3383.93	−0.181547
$704$	4187.19	0.224163
$705$	0	0
$706$	−3418.90	−0.182255
$707$	−5097.00	−0.271135
$708$	0	0
$709$	−8312.18	−0.440297	−0.220148	−	0.975466i	$-0.570654\pi$
$709$	−8312.18	−0.440297	−0.220148	+	0.975466i	$0.570654\pi$
$710$	846.074	0.0447220
$711$	0	0
$712$	6823.53	0.359161
$713$	−2052.93	−0.107830
$714$	0	0
$715$	−7470.66	−0.390751
$716$	2391.35	0.124817
$717$	0	0
$718$	4133.13	0.214829
$719$	−2599.23	−0.134819	−0.0674096	−	0.997725i	$-0.521473\pi$
$719$	−2599.23	−0.134819	−0.0674096	+	0.997725i	$0.521473\pi$
$720$	0	0
$721$	−260.237	−0.0134421
$722$	−13444.4	−0.693002
$723$	0	0
$724$	18558.9	0.952675
$725$	−34438.5	−1.76416
$726$	0	0
$727$	−35759.8	−1.82429	−0.912143	−	0.409871i	$-0.865574\pi$
$727$	−35759.8	−1.82429	−0.912143	+	0.409871i	$0.865574\pi$
$728$	−2204.73	−0.112243
$729$	0	0
$730$	4474.65	0.226869
$731$	−1803.87	−0.0912703
$732$	0	0
$733$	16403.7	0.826580	0.413290	−	0.910599i	$-0.364380\pi$
$733$	16403.7	0.826580	0.413290	+	0.910599i	$0.364380\pi$
$734$	−23752.2	−1.19443
$735$	0	0
$736$	−264.123	−0.0132278
$737$	9144.42	0.457041
$738$	0	0
$739$	37710.3	1.87713	0.938563	−	0.345107i	$-0.112157\pi$
$739$	37710.3	1.87713	0.938563	+	0.345107i	$0.112157\pi$
$740$	3356.28	0.166729
$741$	0	0
$742$	3707.48	0.183431
$743$	23577.0	1.16414	0.582070	−	0.813139i	$-0.302243\pi$
$743$	23577.0	1.16414	0.582070	+	0.813139i	$0.302243\pi$
$744$	0	0
$745$	3868.90	0.190263
$746$	5599.48	0.274814
$747$	0	0
$748$	3634.16	0.177644
$749$	−8493.10	−0.414327
$750$	0	0
$751$	20928.4	1.01690	0.508448	−	0.861093i	$-0.330219\pi$
$751$	20928.4	1.01690	0.508448	+	0.861093i	$0.330219\pi$
$752$	−8295.64	−0.402275
$753$	0	0
$754$	−23258.8	−1.12339
$755$	4403.06	0.212243
$756$	0	0
$757$	−726.989	−0.0349047	−0.0174523	−	0.999848i	$-0.505556\pi$
$757$	−726.989	−0.0349047	−0.0174523	+	0.999848i	$0.505556\pi$
$758$	7844.47	0.375889
$759$	0	0
$760$	−271.401	−0.0129536
$761$	8942.09	0.425954	0.212977	−	0.977057i	$-0.431684\pi$
$761$	8942.09	0.425954	0.212977	+	0.977057i	$0.431684\pi$
$762$	0	0
$763$	5093.94	0.241695
$764$	4125.22	0.195347
$765$	0	0
$766$	−9981.19	−0.470803
$767$	9264.45	0.436141
$768$	0	0
$769$	−34900.2	−1.63659	−0.818293	−	0.574801i	$-0.805079\pi$
$769$	−34900.2	−1.63659	−0.818293	+	0.574801i	$0.805079\pi$
$770$	2656.56	0.124332
$771$	0	0
$772$	11230.7	0.523578
$773$	9031.51	0.420234	0.210117	−	0.977676i	$-0.432616\pi$
$773$	9031.51	0.420234	0.210117	+	0.977676i	$0.432616\pi$
$774$	0	0
$775$	−28998.3	−1.34406
$776$	12134.0	0.561323
$777$	0	0
$778$	−23732.9	−1.09366
$779$	2587.85	0.119024
$780$	0	0
$781$	9542.73	0.437216
$782$	−229.238	−0.0104828
$783$	0	0
$784$	784.000	0.0357143
$785$	−9333.13	−0.424349
$786$	0	0
$787$	−10089.6	−0.456995	−0.228497	−	0.973545i	$-0.573381\pi$
$787$	−10089.6	−0.456995	−0.228497	+	0.973545i	$0.573381\pi$
$788$	−15654.5	−0.707702
$789$	0	0
$790$	−1599.10	−0.0720170
$791$	12337.7	0.554587
$792$	0	0
$793$	16317.2	0.730694
$794$	−21193.3	−0.947257
$795$	0	0
$796$	3288.72	0.146439
$797$	−10278.4	−0.456813	−0.228407	−	0.973566i	$-0.573352\pi$
$797$	−10278.4	−0.456813	−0.228407	+	0.973566i	$0.573352\pi$
$798$	0	0
$799$	−7199.97	−0.318794
$800$	−3730.82	−0.164880
$801$	0	0
$802$	15714.7	0.691902
$803$	50468.8	2.21794
$804$	0	0
$805$	−167.572	−0.00733683
$806$	−19584.7	−0.855881
$807$	0	0
$808$	−5825.14	−0.253624
$809$	−20199.7	−0.877855	−0.438927	−	0.898523i	$-0.644641\pi$
$809$	−20199.7	−0.877855	−0.438927	+	0.898523i	$0.644641\pi$
$810$	0	0
$811$	−34950.3	−1.51328	−0.756640	−	0.653832i	$-0.773160\pi$
$811$	−34950.3	−1.51328	−0.756640	+	0.653832i	$0.773160\pi$
$812$	8270.81	0.357449
$813$	0	0
$814$	37854.9	1.62999
$815$	−9490.30	−0.407890
$816$	0	0
$817$	1519.42	0.0650645
$818$	−9295.88	−0.397339
$819$	0	0
$820$	−2566.70	−0.109309
$821$	−26672.3	−1.13382	−0.566911	−	0.823779i	$-0.691862\pi$
$821$	−26672.3	−1.13382	−0.566911	+	0.823779i	$0.691862\pi$
$822$	0	0
$823$	7626.51	0.323018	0.161509	−	0.986871i	$-0.448364\pi$
$823$	7626.51	0.323018	0.161509	+	0.986871i	$0.448364\pi$
$824$	−297.414	−0.0125739
$825$	0	0
$826$	−3294.43	−0.138775
$827$	−26226.5	−1.10276	−0.551381	−	0.834253i	$-0.685899\pi$
$827$	−26226.5	−1.10276	−0.551381	+	0.834253i	$0.685899\pi$
$828$	0	0
$829$	36002.3	1.50834	0.754169	−	0.656680i	$-0.228040\pi$
$829$	36002.3	1.50834	0.754169	+	0.656680i	$0.228040\pi$
$830$	6329.21	0.264687
$831$	0	0
$832$	−2519.69	−0.104993
$833$	680.451	0.0283028
$834$	0	0
$835$	−1719.31	−0.0712567
$836$	−3061.09	−0.126639
$837$	0	0
$838$	15474.6	0.637903
$839$	8594.00	0.353633	0.176816	−	0.984244i	$-0.443420\pi$
$839$	8594.00	0.353633	0.176816	+	0.984244i	$0.443420\pi$
$840$	0	0
$841$	62864.0	2.57756
$842$	−4342.31	−0.177727
$843$	0	0
$844$	11574.3	0.472043
$845$	−1876.49	−0.0763944
$846$	0	0
$847$	20645.9	0.837547
$848$	4237.12	0.171584
$849$	0	0
$850$	−3238.06	−0.130664
$851$	−2387.84	−0.0961857
$852$	0	0
$853$	3443.01	0.138202	0.0691011	−	0.997610i	$-0.477987\pi$
$853$	3443.01	0.138202	0.0691011	+	0.997610i	$0.477987\pi$
$854$	−5802.38	−0.232498
$855$	0	0
$856$	−9706.40	−0.387568
$857$	869.959	0.0346759	0.0173379	−	0.999850i	$-0.494481\pi$
$857$	869.959	0.0346759	0.0173379	+	0.999850i	$0.494481\pi$
$858$	0	0
$859$	−48875.7	−1.94135	−0.970673	−	0.240403i	$-0.922720\pi$
$859$	−48875.7	−1.94135	−0.970673	+	0.240403i	$0.922720\pi$
$860$	−1507.00	−0.0597538
$861$	0	0
$862$	9429.89	0.372603
$863$	2645.88	0.104365	0.0521823	−	0.998638i	$-0.483382\pi$
$863$	2645.88	0.104365	0.0521823	+	0.998638i	$0.483382\pi$
$864$	0	0
$865$	−9784.72	−0.384613
$866$	−13725.4	−0.538576
$867$	0	0
$868$	6964.29	0.272331
$869$	−18036.0	−0.704061
$870$	0	0
$871$	−5502.75	−0.214069
$872$	5821.65	0.226085
$873$	0	0
$874$	193.089	0.00747293
$875$	−4904.81	−0.189500
$876$	0	0
$877$	−7592.31	−0.292331	−0.146165	−	0.989260i	$-0.546693\pi$
$877$	−7592.31	−0.292331	−0.146165	+	0.989260i	$0.546693\pi$
$878$	1000.04	0.0384393
$879$	0	0
$880$	3036.07	0.116302
$881$	23336.2	0.892413	0.446206	−	0.894930i	$-0.352775\pi$
$881$	23336.2	0.892413	0.446206	+	0.894930i	$0.352775\pi$
$882$	0	0
$883$	−36313.8	−1.38398	−0.691992	−	0.721905i	$-0.743267\pi$
$883$	−36313.8	−1.38398	−0.691992	+	0.721905i	$0.743267\pi$
$884$	−2186.89	−0.0832050
$885$	0	0
$886$	−3504.69	−0.132892
$887$	11034.9	0.417719	0.208859	−	0.977946i	$-0.433025\pi$
$887$	11034.9	0.417719	0.208859	+	0.977946i	$0.433025\pi$
$888$	0	0
$889$	11505.9	0.434079
$890$	4947.64	0.186343
$891$	0	0
$892$	18330.9	0.688078
$893$	6064.60	0.227261
$894$	0	0
$895$	1733.93	0.0647587
$896$	896.000	0.0334077
$897$	0	0
$898$	−23692.8	−0.880445
$899$	73469.8	2.72564
$900$	0	0
$901$	3677.49	0.135977
$902$	−28949.4	−1.06864
$903$	0	0
$904$	14100.2	0.518768
$905$	13456.8	0.494275
$906$	0	0
$907$	34256.2	1.25409	0.627045	−	0.778983i	$-0.284264\pi$
$907$	34256.2	1.25409	0.627045	+	0.778983i	$0.284264\pi$
$908$	10372.0	0.379082
$909$	0	0
$910$	−1598.62	−0.0582347
$911$	14989.0	0.545124	0.272562	−	0.962138i	$-0.412129\pi$
$911$	14989.0	0.545124	0.272562	+	0.962138i	$0.412129\pi$
$912$	0	0
$913$	71386.2	2.58766
$914$	25614.3	0.926966
$915$	0	0
$916$	−12318.3	−0.444332
$917$	3107.38	0.111903
$918$	0	0
$919$	25897.1	0.929561	0.464780	−	0.885426i	$-0.346133\pi$
$919$	25897.1	0.929561	0.464780	+	0.885426i	$0.346133\pi$
$920$	−191.511	−0.00686298
$921$	0	0
$922$	−21480.4	−0.767267
$923$	−5742.45	−0.204783
$924$	0	0
$925$	−33729.0	−1.19892
$926$	6474.62	0.229772
$927$	0	0
$928$	9452.36	0.334363
$929$	−48768.7	−1.72234	−0.861168	−	0.508320i	$-0.830267\pi$
$929$	−48768.7	−1.72234	−0.861168	+	0.508320i	$0.830267\pi$
$930$	0	0
$931$	−573.150	−0.0201764
$932$	11761.0	0.413351
$933$	0	0
$934$	27410.1	0.960263
$935$	2635.07	0.0921669
$936$	0	0
$937$	30690.4	1.07002	0.535011	−	0.844845i	$-0.320307\pi$
$937$	30690.4	1.07002	0.535011	+	0.844845i	$0.320307\pi$
$938$	1956.78	0.0681140
$939$	0	0
$940$	−6015.04	−0.208712
$941$	8085.91	0.280120	0.140060	−	0.990143i	$-0.455270\pi$
$941$	8085.91	0.280120	0.140060	+	0.990143i	$0.455270\pi$
$942$	0	0
$943$	1826.09	0.0630601
$944$	−3765.07	−0.129812
$945$	0	0
$946$	−16997.2	−0.584173
$947$	30723.5	1.05425	0.527127	−	0.849787i	$-0.323269\pi$
$947$	30723.5	1.05425	0.527127	+	0.849787i	$0.323269\pi$
$948$	0	0
$949$	−30370.2	−1.03884
$950$	2727.45	0.0931474
$951$	0	0
$952$	777.658	0.0264748
$953$	17357.3	0.589988	0.294994	−	0.955499i	$-0.404682\pi$
$953$	17357.3	0.589988	0.294994	+	0.955499i	$0.404682\pi$
$954$	0	0
$955$	2991.14	0.101352
$956$	−23594.5	−0.798223
$957$	0	0
$958$	21320.8	0.719044
$959$	−8285.83	−0.279002
$960$	0	0
$961$	32072.9	1.07660
$962$	−22779.6	−0.763456
$963$	0	0
$964$	18296.2	0.611287
$965$	8143.21	0.271647
$966$	0	0
$967$	3138.10	0.104358	0.0521792	−	0.998638i	$-0.483383\pi$
$967$	3138.10	0.104358	0.0521792	+	0.998638i	$0.483383\pi$
$968$	23595.3	0.783454
$969$	0	0
$970$	8798.20	0.291230
$971$	31486.5	1.04063	0.520313	−	0.853975i	$-0.325815\pi$
$971$	31486.5	1.04063	0.520313	+	0.853975i	$0.325815\pi$
$972$	0	0
$973$	−3758.68	−0.123842
$974$	−28029.9	−0.922109
$975$	0	0
$976$	−6631.29	−0.217482
$977$	−25913.7	−0.848570	−0.424285	−	0.905529i	$-0.639475\pi$
$977$	−25913.7	−0.848570	−0.424285	+	0.905529i	$0.639475\pi$
$978$	0	0
$979$	55803.6	1.82175
$980$	568.466	0.0185296
$981$	0	0
$982$	2045.01	0.0664552
$983$	16370.5	0.531169	0.265584	−	0.964088i	$-0.414435\pi$
$983$	16370.5	0.531169	0.265584	+	0.964088i	$0.414435\pi$
$984$	0	0
$985$	−11350.9	−0.367176
$986$	8203.91	0.264975
$987$	0	0
$988$	1842.04	0.0593149
$989$	1072.16	0.0344720
$990$	0	0
$991$	37103.8	1.18935	0.594673	−	0.803967i	$-0.297281\pi$
$991$	37103.8	1.18935	0.594673	+	0.803967i	$0.297281\pi$
$992$	7959.19	0.254742
$993$	0	0
$994$	2042.01	0.0651596
$995$	2384.60	0.0759767
$996$	0	0
$997$	−7834.90	−0.248881	−0.124440	−	0.992227i	$-0.539714\pi$
$997$	−7834.90	−0.248881	−0.124440	+	0.992227i	$0.539714\pi$
$998$	−30483.9	−0.966885
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1134.4.a.t.1.3		6
3.2	odd	2		1134.4.a.s.1.4		6
9.2	odd	6		378.4.f.d.253.3		12
9.4	even	3		126.4.f.d.43.6	✓	12
9.5	odd	6		378.4.f.d.127.3		12
9.7	even	3		126.4.f.d.85.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.4.f.d.43.6	✓	12	9.4	even	3
126.4.f.d.85.6	yes	12	9.7	even	3
378.4.f.d.127.3		12	9.5	odd	6
378.4.f.d.253.3		12	9.2	odd	6
1134.4.a.s.1.4		6	3.2	odd	2
1134.4.a.t.1.3		6	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:	\( N \)	\(=\)	\( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1134.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +2.90034 q^{5} +7.00000 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} +2.90034 q^{5} +7.00000 q^{7} +8.00000 q^{8} +5.80068 q^{10} +65.4249 q^{11} -39.3702 q^{13} +14.0000 q^{14} +16.0000 q^{16} +13.8868 q^{17} -11.6969 q^{19} +11.6014 q^{20} +130.850 q^{22} -8.25383 q^{23} -116.588 q^{25} -78.7403 q^{26} +28.0000 q^{28} +295.386 q^{29} +248.725 q^{31} +32.0000 q^{32} +27.7735 q^{34} +20.3024 q^{35} +289.301 q^{37} -23.3939 q^{38} +23.2027 q^{40} -221.242 q^{41} -129.899 q^{43} +261.700 q^{44} -16.5077 q^{46} -518.477 q^{47} +49.0000 q^{49} -233.176 q^{50} -157.481 q^{52} +264.820 q^{53} +189.754 q^{55} +56.0000 q^{56} +590.772 q^{58} -235.317 q^{59} -414.456 q^{61} +497.449 q^{62} +64.0000 q^{64} -114.187 q^{65} +139.770 q^{67} +55.5470 q^{68} +40.6047 q^{70} +145.858 q^{71} +771.401 q^{73} +578.601 q^{74} -46.7878 q^{76} +457.974 q^{77} -275.675 q^{79} +46.4054 q^{80} -442.483 q^{82} +1091.12 q^{83} +40.2763 q^{85} -259.797 q^{86} +523.399 q^{88} +852.941 q^{89} -275.591 q^{91} -33.0153 q^{92} -1036.95 q^{94} -33.9251 q^{95} +1516.75 q^{97} +98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{2} + 24 q^{4} + 9 q^{5} + 42 q^{7} + 48 q^{8}+O(q^{10}) \) 6 * q + 12 * q^2 + 24 * q^4 + 9 * q^5 + 42 * q^7 + 48 * q^8	\( 6 q + 12 q^{2} + 24 q^{4} + 9 q^{5} + 42 q^{7} + 48 q^{8} + 18 q^{10} + 39 q^{11} + 21 q^{13} + 84 q^{14} + 96 q^{16} + 87 q^{17} + 96 q^{19} + 36 q^{20} + 78 q^{22} + 168 q^{23} + 309 q^{25} + 42 q^{26} + 168 q^{28} + 117 q^{29} + 129 q^{31} + 192 q^{32} + 174 q^{34} + 63 q^{35} + 471 q^{37} + 192 q^{38} + 72 q^{40} + 255 q^{41} + 942 q^{43} + 156 q^{44} + 336 q^{46} + 81 q^{47} + 294 q^{49} + 618 q^{50} + 84 q^{52} + 90 q^{53} + 696 q^{55} + 336 q^{56} + 234 q^{58} + 444 q^{59} + 978 q^{61} + 258 q^{62} + 384 q^{64} + 747 q^{65} + 663 q^{67} + 348 q^{68} + 126 q^{70} + 507 q^{71} + 144 q^{73} + 942 q^{74} + 384 q^{76} + 273 q^{77} + 609 q^{79} + 144 q^{80} + 510 q^{82} + 516 q^{83} + 1563 q^{85} + 1884 q^{86} + 312 q^{88} + 378 q^{89} + 147 q^{91} + 672 q^{92} + 162 q^{94} + 846 q^{95} + 2292 q^{97} + 588 q^{98}+O(q^{100}) \) 6 * q + 12 * q^2 + 24 * q^4 + 9 * q^5 + 42 * q^7 + 48 * q^8 + 18 * q^10 + 39 * q^11 + 21 * q^13 + 84 * q^14 + 96 * q^16 + 87 * q^17 + 96 * q^19 + 36 * q^20 + 78 * q^22 + 168 * q^23 + 309 * q^25 + 42 * q^26 + 168 * q^28 + 117 * q^29 + 129 * q^31 + 192 * q^32 + 174 * q^34 + 63 * q^35 + 471 * q^37 + 192 * q^38 + 72 * q^40 + 255 * q^41 + 942 * q^43 + 156 * q^44 + 336 * q^46 + 81 * q^47 + 294 * q^49 + 618 * q^50 + 84 * q^52 + 90 * q^53 + 696 * q^55 + 336 * q^56 + 234 * q^58 + 444 * q^59 + 978 * q^61 + 258 * q^62 + 384 * q^64 + 747 * q^65 + 663 * q^67 + 348 * q^68 + 126 * q^70 + 507 * q^71 + 144 * q^73 + 942 * q^74 + 384 * q^76 + 273 * q^77 + 609 * q^79 + 144 * q^80 + 510 * q^82 + 516 * q^83 + 1563 * q^85 + 1884 * q^86 + 312 * q^88 + 378 * q^89 + 147 * q^91 + 672 * q^92 + 162 * q^94 + 846 * q^95 + 2292 * q^97 + 588 * q^98

Introduction

L-functions

Modular forms

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