LMFDB - Embedded newform 567.4.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [567,4,Mod(1,567)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(567, 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("567.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 567 = 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	567.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:	$33.4540829733$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 49x^{6} + 138x^{5} + 708x^{4} - 1941x^{3} - 2506x^{2} + 8592x - 4616 $ x^8 - 3x^7 - 49x^6 + 138x^5 + 708x^4 - 1941x^3 - 2506x^2 + 8592*x - 4616
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.807372$ of defining polynomial
Character	$\chi$	$=$	567.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.807372 q^{2} -7.34815 q^{4} +18.2289 q^{5} -7.00000 q^{7} -12.3917 q^{8} +O(q^{10})$	\(q+0.807372 q^{2} -7.34815 q^{4} +18.2289 q^{5} -7.00000 q^{7} -12.3917 q^{8} +14.7175 q^{10} -49.4293 q^{11} +44.4860 q^{13} -5.65160 q^{14} +48.7805 q^{16} +47.2301 q^{17} +56.7545 q^{19} -133.949 q^{20} -39.9078 q^{22} -55.4232 q^{23} +207.292 q^{25} +35.9167 q^{26} +51.4371 q^{28} +150.281 q^{29} -167.549 q^{31} +138.517 q^{32} +38.1322 q^{34} -127.602 q^{35} +331.480 q^{37} +45.8220 q^{38} -225.886 q^{40} +295.432 q^{41} -317.788 q^{43} +363.214 q^{44} -44.7471 q^{46} -137.879 q^{47} +49.0000 q^{49} +167.362 q^{50} -326.890 q^{52} +411.775 q^{53} -901.041 q^{55} +86.7416 q^{56} +121.333 q^{58} +212.305 q^{59} -52.3569 q^{61} -135.275 q^{62} -278.409 q^{64} +810.929 q^{65} +706.772 q^{67} -347.054 q^{68} -103.022 q^{70} -78.7569 q^{71} +839.292 q^{73} +267.628 q^{74} -417.040 q^{76} +346.005 q^{77} +1015.33 q^{79} +889.214 q^{80} +238.523 q^{82} +1086.61 q^{83} +860.952 q^{85} -256.573 q^{86} +612.511 q^{88} -762.726 q^{89} -311.402 q^{91} +407.258 q^{92} -111.320 q^{94} +1034.57 q^{95} -1342.63 q^{97} +39.5612 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{2} + 43 q^{4} + 30 q^{5} - 56 q^{7} + 6 q^{8}+O(q^{10}) $ 8 * q + 3 * q^2 + 43 * q^4 + 30 * q^5 - 56 * q^7 + 6 * q^8	$ 8 q + 3 q^{2} + 43 q^{4} + 30 q^{5} - 56 q^{7} + 6 q^{8} - 14 q^{10} + 24 q^{11} + 68 q^{13} - 21 q^{14} + 103 q^{16} + 168 q^{17} + 176 q^{19} + 330 q^{20} + 151 q^{22} + 228 q^{23} + 244 q^{25} + 795 q^{26} - 301 q^{28} + 618 q^{29} + 72 q^{31} + 786 q^{32} - 261 q^{34} - 210 q^{35} + 210 q^{37} + 1032 q^{38} - 375 q^{40} + 420 q^{41} - 2 q^{43} + 387 q^{44} + 402 q^{46} + 570 q^{47} + 392 q^{49} + 1110 q^{50} - 431 q^{52} + 528 q^{53} - 838 q^{55} - 42 q^{56} + 37 q^{58} - 150 q^{59} + 578 q^{61} + 1170 q^{62} - 112 q^{64} - 366 q^{65} - 898 q^{67} + 2526 q^{68} + 98 q^{70} + 882 q^{71} + 972 q^{73} - 222 q^{74} + 1423 q^{76} - 168 q^{77} - 158 q^{79} + 2475 q^{80} - 211 q^{82} + 2958 q^{83} - 774 q^{85} - 114 q^{86} + 1317 q^{88} + 4380 q^{89} - 476 q^{91} + 4629 q^{92} - 3234 q^{94} + 930 q^{95} - 60 q^{97} + 147 q^{98}+O(q^{100}) $ 8 * q + 3 * q^2 + 43 * q^4 + 30 * q^5 - 56 * q^7 + 6 * q^8 - 14 * q^10 + 24 * q^11 + 68 * q^13 - 21 * q^14 + 103 * q^16 + 168 * q^17 + 176 * q^19 + 330 * q^20 + 151 * q^22 + 228 * q^23 + 244 * q^25 + 795 * q^26 - 301 * q^28 + 618 * q^29 + 72 * q^31 + 786 * q^32 - 261 * q^34 - 210 * q^35 + 210 * q^37 + 1032 * q^38 - 375 * q^40 + 420 * q^41 - 2 * q^43 + 387 * q^44 + 402 * q^46 + 570 * q^47 + 392 * q^49 + 1110 * q^50 - 431 * q^52 + 528 * q^53 - 838 * q^55 - 42 * q^56 + 37 * q^58 - 150 * q^59 + 578 * q^61 + 1170 * q^62 - 112 * q^64 - 366 * q^65 - 898 * q^67 + 2526 * q^68 + 98 * q^70 + 882 * q^71 + 972 * q^73 - 222 * q^74 + 1423 * q^76 - 168 * q^77 - 158 * q^79 + 2475 * q^80 - 211 * q^82 + 2958 * q^83 - 774 * q^85 - 114 * q^86 + 1317 * q^88 + 4380 * q^89 - 476 * q^91 + 4629 * q^92 - 3234 * q^94 + 930 * q^95 - 60 * q^97 + 147 * 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 $

$2$	0.807372	0.285449	0.142725	−	0.989762i	$-0.454414\pi$
$2$	0.807372	0.285449	0.142725	+	0.989762i	$0.454414\pi$
$3$	0	0
$3$	0	0
$4$	−7.34815	−0.918519
$5$	18.2289	1.63044	0.815220	−	0.579151i	$-0.196616\pi$
$5$	18.2289	1.63044	0.815220	+	0.579151i	$0.196616\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−12.3917	−0.547639
$9$	0	0
$10$	14.7175	0.465408
$11$	−49.4293	−1.35486	−0.677432	−	0.735586i	$-0.736907\pi$
$11$	−49.4293	−1.35486	−0.677432	+	0.735586i	$0.736907\pi$
$12$	0	0
$13$	44.4860	0.949091	0.474546	−	0.880231i	$-0.342612\pi$
$13$	44.4860	0.949091	0.474546	+	0.880231i	$0.342612\pi$
$14$	−5.65160	−0.107890
$15$	0	0
$16$	48.7805	0.762196
$17$	47.2301	0.673822	0.336911	−	0.941536i	$-0.390618\pi$
$17$	47.2301	0.673822	0.336911	+	0.941536i	$0.390618\pi$
$18$	0	0
$19$	56.7545	0.685283	0.342641	−	0.939466i	$-0.388678\pi$
$19$	56.7545	0.685283	0.342641	+	0.939466i	$0.388678\pi$
$20$	−133.949	−1.49759
$21$	0	0
$22$	−39.9078	−0.386744
$23$	−55.4232	−0.502458	−0.251229	−	0.967928i	$-0.580835\pi$
$23$	−55.4232	−0.502458	−0.251229	+	0.967928i	$0.580835\pi$
$24$	0	0
$25$	207.292	1.65834
$26$	35.9167	0.270917
$27$	0	0
$28$	51.4371	0.347167
$29$	150.281	0.962293	0.481146	−	0.876640i	$-0.340221\pi$
$29$	150.281	0.962293	0.481146	+	0.876640i	$0.340221\pi$
$30$	0	0
$31$	−167.549	−0.970733	−0.485367	−	0.874311i	$-0.661314\pi$
$31$	−167.549	−0.970733	−0.485367	+	0.874311i	$0.661314\pi$
$32$	138.517	0.765207
$33$	0	0
$34$	38.1322	0.192342
$35$	−127.602	−0.616248
$36$	0	0
$37$	331.480	1.47284	0.736419	−	0.676526i	$-0.236515\pi$
$37$	331.480	1.47284	0.736419	+	0.676526i	$0.236515\pi$
$38$	45.8220	0.195613
$39$	0	0
$40$	−225.886	−0.892893
$41$	295.432	1.12533	0.562667	−	0.826684i	$-0.309775\pi$
$41$	295.432	1.12533	0.562667	+	0.826684i	$0.309775\pi$
$42$	0	0
$43$	−317.788	−1.12703	−0.563514	−	0.826106i	$-0.690551\pi$
$43$	−317.788	−1.12703	−0.563514	+	0.826106i	$0.690551\pi$
$44$	363.214	1.24447
$45$	0	0
$46$	−44.7471	−0.143426
$47$	−137.879	−0.427909	−0.213955	−	0.976844i	$-0.568634\pi$
$47$	−137.879	−0.427909	−0.213955	+	0.976844i	$0.568634\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	167.362	0.473370
$51$	0	0
$52$	−326.890	−0.871758
$53$	411.775	1.06720	0.533600	−	0.845737i	$-0.320839\pi$
$53$	411.775	1.06720	0.533600	+	0.845737i	$0.320839\pi$
$54$	0	0
$55$	−901.041	−2.20902
$56$	86.7416	0.206988
$57$	0	0
$58$	121.333	0.274685
$59$	212.305	0.468470	0.234235	−	0.972180i	$-0.424741\pi$
$59$	212.305	0.468470	0.234235	+	0.972180i	$0.424741\pi$
$60$	0	0
$61$	−52.3569	−0.109895	−0.0549477	−	0.998489i	$-0.517499\pi$
$61$	−52.3569	−0.109895	−0.0549477	+	0.998489i	$0.517499\pi$
$62$	−135.275	−0.277095
$63$	0	0
$64$	−278.409	−0.543768
$65$	810.929	1.54744
$66$	0	0
$67$	706.772	1.28874	0.644372	−	0.764712i	$-0.277119\pi$
$67$	706.772	1.28874	0.644372	+	0.764712i	$0.277119\pi$
$68$	−347.054	−0.618919
$69$	0	0
$70$	−103.022	−0.175908
$71$	−78.7569	−0.131644	−0.0658220	−	0.997831i	$-0.520967\pi$
$71$	−78.7569	−0.131644	−0.0658220	+	0.997831i	$0.520967\pi$
$72$	0	0
$73$	839.292	1.34564	0.672820	−	0.739806i	$-0.265083\pi$
$73$	839.292	1.34564	0.672820	+	0.739806i	$0.265083\pi$
$74$	267.628	0.420420
$75$	0	0
$76$	−417.040	−0.629445
$77$	346.005	0.512090
$78$	0	0
$79$	1015.33	1.44599	0.722994	−	0.690854i	$-0.242765\pi$
$79$	1015.33	1.44599	0.722994	+	0.690854i	$0.242765\pi$
$80$	889.214	1.24271
$81$	0	0
$82$	238.523	0.321226
$83$	1086.61	1.43700	0.718500	−	0.695527i	$-0.244829\pi$
$83$	1086.61	1.43700	0.718500	+	0.695527i	$0.244829\pi$
$84$	0	0
$85$	860.952	1.09863
$86$	−256.573	−0.321709
$87$	0	0
$88$	612.511	0.741976
$89$	−762.726	−0.908414	−0.454207	−	0.890896i	$-0.650077\pi$
$89$	−762.726	−0.908414	−0.454207	+	0.890896i	$0.650077\pi$
$90$	0	0
$91$	−311.402	−0.358723
$92$	407.258	0.461517
$93$	0	0
$94$	−111.320	−0.122146
$95$	1034.57	1.11731
$96$	0	0
$97$	−1342.63	−1.40540	−0.702698	−	0.711488i	$-0.748021\pi$
$97$	−1342.63	−1.40540	−0.702698	+	0.711488i	$0.748021\pi$
$98$	39.5612	0.0407784
$99$	0	0
$100$	−1523.21	−1.52321
$101$	706.982	0.696509	0.348254	−	0.937400i	$-0.386775\pi$
$101$	706.982	0.696509	0.348254	+	0.937400i	$0.386775\pi$
$102$	0	0
$103$	514.976	0.492642	0.246321	−	0.969188i	$-0.420778\pi$
$103$	514.976	0.492642	0.246321	+	0.969188i	$0.420778\pi$
$104$	−551.255	−0.519760
$105$	0	0
$106$	332.455	0.304631
$107$	1186.50	1.07199	0.535997	−	0.844220i	$-0.319936\pi$
$107$	1186.50	1.07199	0.535997	+	0.844220i	$0.319936\pi$
$108$	0	0
$109$	1909.52	1.67797	0.838987	−	0.544151i	$-0.183148\pi$
$109$	1909.52	1.67797	0.838987	+	0.544151i	$0.183148\pi$
$110$	−727.475	−0.630564
$111$	0	0
$112$	−341.464	−0.288083
$113$	−941.939	−0.784161	−0.392080	−	0.919931i	$-0.628245\pi$
$113$	−941.939	−0.784161	−0.392080	+	0.919931i	$0.628245\pi$
$114$	0	0
$115$	−1010.30	−0.819228
$116$	−1104.29	−0.883884
$117$	0	0
$118$	171.409	0.133724
$119$	−330.611	−0.254681
$120$	0	0
$121$	1112.26	0.835655
$122$	−42.2715	−0.0313695
$123$	0	0
$124$	1231.18	0.891637
$125$	1500.09	1.07338
$126$	0	0
$127$	−1584.85	−1.10734	−0.553672	−	0.832735i	$-0.686774\pi$
$127$	−1584.85	−1.10734	−0.553672	+	0.832735i	$0.686774\pi$
$128$	−1332.92	−0.920425
$129$	0	0
$130$	654.721	0.441714
$131$	2173.85	1.44985	0.724926	−	0.688827i	$-0.241874\pi$
$131$	2173.85	1.44985	0.724926	+	0.688827i	$0.241874\pi$
$132$	0	0
$133$	−397.281	−0.259012
$134$	570.628	0.367871
$135$	0	0
$136$	−585.259	−0.369012
$137$	2094.51	1.30617	0.653087	−	0.757283i	$-0.273474\pi$
$137$	2094.51	1.30617	0.653087	+	0.757283i	$0.273474\pi$
$138$	0	0
$139$	−1313.32	−0.801400	−0.400700	−	0.916209i	$-0.631233\pi$
$139$	−1313.32	−0.801400	−0.400700	+	0.916209i	$0.631233\pi$
$140$	937.640	0.566036
$141$	0	0
$142$	−63.5861	−0.0375776
$143$	−2198.91	−1.28589
$144$	0	0
$145$	2739.45	1.56896
$146$	677.621	0.384112
$147$	0	0
$148$	−2435.77	−1.35283
$149$	−1487.45	−0.817833	−0.408916	−	0.912572i	$-0.634093\pi$
$149$	−1487.45	−0.817833	−0.408916	+	0.912572i	$0.634093\pi$
$150$	0	0
$151$	−2203.69	−1.18764	−0.593821	−	0.804597i	$-0.702381\pi$
$151$	−2203.69	−1.18764	−0.593821	+	0.804597i	$0.702381\pi$
$152$	−703.282	−0.375288
$153$	0	0
$154$	279.355	0.146176
$155$	−3054.23	−1.58272
$156$	0	0
$157$	−3445.34	−1.75139	−0.875696	−	0.482863i	$-0.839597\pi$
$157$	−3445.34	−1.75139	−0.875696	+	0.482863i	$0.839597\pi$
$158$	819.746	0.412756
$159$	0	0
$160$	2525.02	1.24762
$161$	387.963	0.189911
$162$	0	0
$163$	−2723.83	−1.30888	−0.654439	−	0.756115i	$-0.727095\pi$
$163$	−2723.83	−1.30888	−0.654439	+	0.756115i	$0.727095\pi$
$164$	−2170.88	−1.03364
$165$	0	0
$166$	877.298	0.410190
$167$	−1012.79	−0.469292	−0.234646	−	0.972081i	$-0.575393\pi$
$167$	−1012.79	−0.469292	−0.234646	+	0.972081i	$0.575393\pi$
$168$	0	0
$169$	−218.000	−0.0992261
$170$	695.108	0.313602
$171$	0	0
$172$	2335.16	1.03520
$173$	−1495.82	−0.657371	−0.328685	−	0.944440i	$-0.606606\pi$
$173$	−1495.82	−0.657371	−0.328685	+	0.944440i	$0.606606\pi$
$174$	0	0
$175$	−1451.04	−0.626792
$176$	−2411.19	−1.03267
$177$	0	0
$178$	−615.804	−0.259306
$179$	1989.77	0.830851	0.415425	−	0.909627i	$-0.363633\pi$
$179$	1989.77	0.830851	0.415425	+	0.909627i	$0.363633\pi$
$180$	0	0
$181$	−2790.78	−1.14606	−0.573031	−	0.819534i	$-0.694232\pi$
$181$	−2790.78	−1.14606	−0.573031	+	0.819534i	$0.694232\pi$
$182$	−251.417	−0.102397
$183$	0	0
$184$	686.786	0.275166
$185$	6042.51	2.40137
$186$	0	0
$187$	−2334.55	−0.912937
$188$	1013.16	0.393043
$189$	0	0
$190$	835.283	0.318936
$191$	4335.69	1.64251	0.821256	−	0.570560i	$-0.193274\pi$
$191$	4335.69	1.64251	0.821256	+	0.570560i	$0.193274\pi$
$192$	0	0
$193$	−2429.90	−0.906260	−0.453130	−	0.891444i	$-0.649693\pi$
$193$	−2429.90	−0.906260	−0.453130	+	0.891444i	$0.649693\pi$
$194$	−1084.00	−0.401169
$195$	0	0
$196$	−360.059	−0.131217
$197$	273.113	0.0987742	0.0493871	−	0.998780i	$-0.484273\pi$
$197$	273.113	0.0987742	0.0493871	+	0.998780i	$0.484273\pi$
$198$	0	0
$199$	648.752	0.231100	0.115550	−	0.993302i	$-0.463137\pi$
$199$	648.752	0.231100	0.115550	+	0.993302i	$0.463137\pi$
$200$	−2568.69	−0.908170
$201$	0	0
$202$	570.797	0.198818
$203$	−1051.97	−0.363712
$204$	0	0
$205$	5385.39	1.83479
$206$	415.777	0.140624
$207$	0	0
$208$	2170.05	0.723393
$209$	−2805.33	−0.928464
$210$	0	0
$211$	243.086	0.0793114	0.0396557	−	0.999213i	$-0.487374\pi$
$211$	243.086	0.0793114	0.0396557	+	0.999213i	$0.487374\pi$
$212$	−3025.78	−0.980244
$213$	0	0
$214$	957.947	0.305999
$215$	−5792.92	−1.83755
$216$	0	0
$217$	1172.84	0.366903
$218$	1541.70	0.478976
$219$	0	0
$220$	6620.98	2.02903
$221$	2101.08	0.639519
$222$	0	0
$223$	1820.64	0.546721	0.273361	−	0.961912i	$-0.411865\pi$
$223$	1820.64	0.546721	0.273361	+	0.961912i	$0.411865\pi$
$224$	−969.621	−0.289221
$225$	0	0
$226$	−760.495	−0.223838
$227$	1700.20	0.497119	0.248560	−	0.968617i	$-0.420043\pi$
$227$	1700.20	0.497119	0.248560	+	0.968617i	$0.420043\pi$
$228$	0	0
$229$	188.977	0.0545326	0.0272663	−	0.999628i	$-0.491320\pi$
$229$	188.977	0.0545326	0.0272663	+	0.999628i	$0.491320\pi$
$230$	−815.690	−0.233848
$231$	0	0
$232$	−1862.23	−0.526989
$233$	−647.185	−0.181968	−0.0909840	−	0.995852i	$-0.529001\pi$
$233$	−647.185	−0.181968	−0.0909840	+	0.995852i	$0.529001\pi$
$234$	0	0
$235$	−2513.38	−0.697680
$236$	−1560.05	−0.430299
$237$	0	0
$238$	−266.926	−0.0726984
$239$	−3521.22	−0.953006	−0.476503	−	0.879173i	$-0.658096\pi$
$239$	−3521.22	−0.953006	−0.476503	+	0.879173i	$0.658096\pi$
$240$	0	0
$241$	324.501	0.0867341	0.0433671	−	0.999059i	$-0.486192\pi$
$241$	324.501	0.0867341	0.0433671	+	0.999059i	$0.486192\pi$
$242$	898.004	0.238537
$243$	0	0
$244$	384.727	0.100941
$245$	893.215	0.232920
$246$	0	0
$247$	2524.78	0.650396
$248$	2076.21	0.531612
$249$	0	0
$250$	1211.13	0.306394
$251$	3501.08	0.880424	0.440212	−	0.897894i	$-0.354903\pi$
$251$	3501.08	0.880424	0.440212	+	0.897894i	$0.354903\pi$
$252$	0	0
$253$	2739.53	0.680762
$254$	−1279.56	−0.316090
$255$	0	0
$256$	1151.11	0.281034
$257$	−1839.03	−0.446363	−0.223182	−	0.974777i	$-0.571644\pi$
$257$	−1839.03	−0.446363	−0.223182	+	0.974777i	$0.571644\pi$
$258$	0	0
$259$	−2320.36	−0.556680
$260$	−5958.83	−1.42135
$261$	0	0
$262$	1755.11	0.413859
$263$	111.346	0.0261060	0.0130530	−	0.999915i	$-0.495845\pi$
$263$	111.346	0.0261060	0.0130530	+	0.999915i	$0.495845\pi$
$264$	0	0
$265$	7506.19	1.74001
$266$	−320.754	−0.0739348
$267$	0	0
$268$	−5193.47	−1.18374
$269$	−2355.30	−0.533848	−0.266924	−	0.963718i	$-0.586007\pi$
$269$	−2355.30	−0.533848	−0.266924	+	0.963718i	$0.586007\pi$
$270$	0	0
$271$	−3281.29	−0.735514	−0.367757	−	0.929922i	$-0.619874\pi$
$271$	−3281.29	−0.735514	−0.367757	+	0.929922i	$0.619874\pi$
$272$	2303.91	0.513585
$273$	0	0
$274$	1691.04	0.372846
$275$	−10246.3	−2.24682
$276$	0	0
$277$	2220.83	0.481722	0.240861	−	0.970560i	$-0.422570\pi$
$277$	2220.83	0.481722	0.240861	+	0.970560i	$0.422570\pi$
$278$	−1060.34	−0.228759
$279$	0	0
$280$	1581.20	0.337482
$281$	−1189.50	−0.252526	−0.126263	−	0.991997i	$-0.540298\pi$
$281$	−1189.50	−0.252526	−0.126263	+	0.991997i	$0.540298\pi$
$282$	0	0
$283$	1159.61	0.243575	0.121788	−	0.992556i	$-0.461137\pi$
$283$	1159.61	0.243575	0.121788	+	0.992556i	$0.461137\pi$
$284$	578.717	0.120917
$285$	0	0
$286$	−1775.34	−0.367056
$287$	−2068.02	−0.425336
$288$	0	0
$289$	−2682.32	−0.545963
$290$	2211.76	0.447858
$291$	0	0
$292$	−6167.24	−1.23600
$293$	537.290	0.107129	0.0535645	−	0.998564i	$-0.482942\pi$
$293$	537.290	0.107129	0.0535645	+	0.998564i	$0.482942\pi$
$294$	0	0
$295$	3870.08	0.763813
$296$	−4107.59	−0.806584
$297$	0	0
$298$	−1200.93	−0.233450
$299$	−2465.55	−0.476879
$300$	0	0
$301$	2224.52	0.425977
$302$	−1779.20	−0.339011
$303$	0	0
$304$	2768.51	0.522319
$305$	−954.408	−0.179178
$306$	0	0
$307$	−1740.18	−0.323510	−0.161755	−	0.986831i	$-0.551715\pi$
$307$	−1740.18	−0.323510	−0.161755	+	0.986831i	$0.551715\pi$
$308$	−2542.50	−0.470365
$309$	0	0
$310$	−2465.90	−0.451787
$311$	−3917.90	−0.714353	−0.357177	−	0.934037i	$-0.616261\pi$
$311$	−3917.90	−0.714353	−0.357177	+	0.934037i	$0.616261\pi$
$312$	0	0
$313$	−9485.55	−1.71295	−0.856477	−	0.516185i	$-0.827352\pi$
$313$	−9485.55	−1.71295	−0.856477	+	0.516185i	$0.827352\pi$
$314$	−2781.67	−0.499933
$315$	0	0
$316$	−7460.77	−1.32817
$317$	−5674.37	−1.00538	−0.502688	−	0.864468i	$-0.667656\pi$
$317$	−5674.37	−1.00538	−0.502688	+	0.864468i	$0.667656\pi$
$318$	0	0
$319$	−7428.29	−1.30378
$320$	−5075.09	−0.886581
$321$	0	0
$322$	313.230	0.0542100
$323$	2680.52	0.461759
$324$	0	0
$325$	9221.58	1.57391
$326$	−2199.15	−0.373618
$327$	0	0
$328$	−3660.89	−0.616277
$329$	965.153	0.161734
$330$	0	0
$331$	4732.59	0.785881	0.392940	−	0.919564i	$-0.371458\pi$
$331$	4732.59	0.785881	0.392940	+	0.919564i	$0.371458\pi$
$332$	−7984.57	−1.31991
$333$	0	0
$334$	−817.695	−0.133959
$335$	12883.7	2.10122
$336$	0	0
$337$	−4043.29	−0.653567	−0.326783	−	0.945099i	$-0.605965\pi$
$337$	−4043.29	−0.653567	−0.326783	+	0.945099i	$0.605965\pi$
$338$	−176.007	−0.0283240
$339$	0	0
$340$	−6326.40	−1.00911
$341$	8281.84	1.31521
$342$	0	0
$343$	−343.000	−0.0539949
$344$	3937.92	0.617205
$345$	0	0
$346$	−1207.68	−0.187646
$347$	−3893.86	−0.602402	−0.301201	−	0.953561i	$-0.597388\pi$
$347$	−3893.86	−0.602402	−0.301201	+	0.953561i	$0.597388\pi$
$348$	0	0
$349$	−944.608	−0.144882	−0.0724408	−	0.997373i	$-0.523079\pi$
$349$	−944.608	−0.144882	−0.0724408	+	0.997373i	$0.523079\pi$
$350$	−1171.53	−0.178917
$351$	0	0
$352$	−6846.82	−1.03675
$353$	−9398.52	−1.41709	−0.708545	−	0.705666i	$-0.750648\pi$
$353$	−9398.52	−1.41709	−0.708545	+	0.705666i	$0.750648\pi$
$354$	0	0
$355$	−1435.65	−0.214638
$356$	5604.63	0.834395
$357$	0	0
$358$	1606.48	0.237165
$359$	6522.25	0.958861	0.479430	−	0.877580i	$-0.340843\pi$
$359$	6522.25	0.958861	0.479430	+	0.877580i	$0.340843\pi$
$360$	0	0
$361$	−3637.93	−0.530388
$362$	−2253.20	−0.327142
$363$	0	0
$364$	2288.23	0.329494
$365$	15299.3	2.19398
$366$	0	0
$367$	4736.35	0.673666	0.336833	−	0.941564i	$-0.390644\pi$
$367$	4736.35	0.673666	0.336833	+	0.941564i	$0.390644\pi$
$368$	−2703.57	−0.382972
$369$	0	0
$370$	4878.55	0.685470
$371$	−2882.42	−0.403364
$372$	0	0
$373$	5726.01	0.794857	0.397429	−	0.917633i	$-0.369903\pi$
$373$	5726.01	0.794857	0.397429	+	0.917633i	$0.369903\pi$
$374$	−1884.85	−0.260597
$375$	0	0
$376$	1708.55	0.234340
$377$	6685.39	0.913303
$378$	0	0
$379$	−2131.56	−0.288894	−0.144447	−	0.989513i	$-0.546140\pi$
$379$	−2131.56	−0.288894	−0.144447	+	0.989513i	$0.546140\pi$
$380$	−7602.18	−1.02627
$381$	0	0
$382$	3500.52	0.468853
$383$	10190.6	1.35957	0.679785	−	0.733411i	$-0.262073\pi$
$383$	10190.6	1.35957	0.679785	+	0.733411i	$0.262073\pi$
$384$	0	0
$385$	6307.28	0.834932
$386$	−1961.83	−0.258691
$387$	0	0
$388$	9865.85	1.29088
$389$	4749.46	0.619042	0.309521	−	0.950893i	$-0.399831\pi$
$389$	4749.46	0.619042	0.309521	+	0.950893i	$0.399831\pi$
$390$	0	0
$391$	−2617.64	−0.338568
$392$	−607.191	−0.0782342
$393$	0	0
$394$	220.504	0.0281950
$395$	18508.3	2.35760
$396$	0	0
$397$	−4178.89	−0.528294	−0.264147	−	0.964482i	$-0.585090\pi$
$397$	−4178.89	−0.528294	−0.264147	+	0.964482i	$0.585090\pi$
$398$	523.784	0.0659672
$399$	0	0
$400$	10111.8	1.26398
$401$	3539.08	0.440732	0.220366	−	0.975417i	$-0.429275\pi$
$401$	3539.08	0.440732	0.220366	+	0.975417i	$0.429275\pi$
$402$	0	0
$403$	−7453.59	−0.921314
$404$	−5195.01	−0.639756
$405$	0	0
$406$	−849.329	−0.103821
$407$	−16384.8	−1.99549
$408$	0	0
$409$	−5969.43	−0.721686	−0.360843	−	0.932627i	$-0.617511\pi$
$409$	−5969.43	−0.721686	−0.360843	+	0.932627i	$0.617511\pi$
$410$	4348.01	0.523739
$411$	0	0
$412$	−3784.12	−0.452501
$413$	−1486.13	−0.177065
$414$	0	0
$415$	19807.7	2.34294
$416$	6162.08	0.726251
$417$	0	0
$418$	−2264.95	−0.265029
$419$	11549.0	1.34656	0.673279	−	0.739389i	$-0.264885\pi$
$419$	11549.0	1.34656	0.673279	+	0.739389i	$0.264885\pi$
$420$	0	0
$421$	2694.98	0.311985	0.155992	−	0.987758i	$-0.450143\pi$
$421$	2694.98	0.311985	0.155992	+	0.987758i	$0.450143\pi$
$422$	196.260	0.0226394
$423$	0	0
$424$	−5102.57	−0.584441
$425$	9790.42	1.11742
$426$	0	0
$427$	366.499	0.0415366
$428$	−8718.58	−0.984646
$429$	0	0
$430$	−4677.04	−0.524528
$431$	2267.88	0.253457	0.126729	−	0.991937i	$-0.459552\pi$
$431$	2267.88	0.253457	0.126729	+	0.991937i	$0.459552\pi$
$432$	0	0
$433$	11979.8	1.32958	0.664792	−	0.747028i	$-0.268520\pi$
$433$	11979.8	1.32958	0.664792	+	0.747028i	$0.268520\pi$
$434$	946.922	0.104732
$435$	0	0
$436$	−14031.5	−1.54125
$437$	−3145.52	−0.344326
$438$	0	0
$439$	10900.6	1.18509	0.592547	−	0.805536i	$-0.298123\pi$
$439$	10900.6	1.18509	0.592547	+	0.805536i	$0.298123\pi$
$440$	11165.4	1.20975
$441$	0	0
$442$	1696.35	0.182550
$443$	−368.489	−0.0395202	−0.0197601	−	0.999805i	$-0.506290\pi$
$443$	−368.489	−0.0395202	−0.0197601	+	0.999805i	$0.506290\pi$
$444$	0	0
$445$	−13903.6	−1.48111
$446$	1469.93	0.156061
$447$	0	0
$448$	1948.86	0.205525
$449$	−10077.9	−1.05925	−0.529625	−	0.848232i	$-0.677667\pi$
$449$	−10077.9	−1.05925	−0.529625	+	0.848232i	$0.677667\pi$
$450$	0	0
$451$	−14603.0	−1.52467
$452$	6921.51	0.720267
$453$	0	0
$454$	1372.69	0.141902
$455$	−5676.50	−0.584876
$456$	0	0
$457$	−2184.73	−0.223627	−0.111813	−	0.993729i	$-0.535666\pi$
$457$	−2184.73	−0.223627	−0.111813	+	0.993729i	$0.535666\pi$
$458$	152.575	0.0155663
$459$	0	0
$460$	7423.86	0.752476
$461$	16602.4	1.67734	0.838669	−	0.544641i	$-0.183334\pi$
$461$	16602.4	1.67734	0.838669	+	0.544641i	$0.183334\pi$
$462$	0	0
$463$	−3836.72	−0.385113	−0.192557	−	0.981286i	$-0.561678\pi$
$463$	−3836.72	−0.385113	−0.192557	+	0.981286i	$0.561678\pi$
$464$	7330.79	0.733455
$465$	0	0
$466$	−522.519	−0.0519426
$467$	9751.89	0.966304	0.483152	−	0.875537i	$-0.339492\pi$
$467$	9751.89	0.966304	0.483152	+	0.875537i	$0.339492\pi$
$468$	0	0
$469$	−4947.40	−0.487100
$470$	−2029.23	−0.199152
$471$	0	0
$472$	−2630.81	−0.256553
$473$	15708.1	1.52697
$474$	0	0
$475$	11764.7	1.13643
$476$	2429.38	0.233929
$477$	0	0
$478$	−2842.93	−0.272035
$479$	7039.89	0.671526	0.335763	−	0.941947i	$-0.391006\pi$
$479$	7039.89	0.671526	0.335763	+	0.941947i	$0.391006\pi$
$480$	0	0
$481$	14746.2	1.39786
$482$	261.993	0.0247582
$483$	0	0
$484$	−8173.03	−0.767565
$485$	−24474.6	−2.29141
$486$	0	0
$487$	−8005.18	−0.744865	−0.372433	−	0.928059i	$-0.621476\pi$
$487$	−8005.18	−0.744865	−0.372433	+	0.928059i	$0.621476\pi$
$488$	648.790	0.0601830
$489$	0	0
$490$	721.156	0.0664868
$491$	−7893.96	−0.725559	−0.362779	−	0.931875i	$-0.618172\pi$
$491$	−7893.96	−0.725559	−0.362779	+	0.931875i	$0.618172\pi$
$492$	0	0
$493$	7097.79	0.648414
$494$	2038.43	0.185655
$495$	0	0
$496$	−8173.14	−0.739889
$497$	551.298	0.0497567
$498$	0	0
$499$	2740.47	0.245852	0.122926	−	0.992416i	$-0.460772\pi$
$499$	2740.47	0.245852	0.122926	+	0.992416i	$0.460772\pi$
$500$	−11022.9	−0.985916
$501$	0	0
$502$	2826.68	0.251316
$503$	5275.06	0.467601	0.233801	−	0.972285i	$-0.424884\pi$
$503$	5275.06	0.467601	0.233801	+	0.972285i	$0.424884\pi$
$504$	0	0
$505$	12887.5	1.13562
$506$	2211.82	0.194323
$507$	0	0
$508$	11645.7	1.01712
$509$	−15702.1	−1.36735	−0.683676	−	0.729786i	$-0.739620\pi$
$509$	−15702.1	−1.36735	−0.683676	+	0.729786i	$0.739620\pi$
$510$	0	0
$511$	−5875.04	−0.508604
$512$	11592.7	1.00065
$513$	0	0
$514$	−1484.78	−0.127414
$515$	9387.44	0.803223
$516$	0	0
$517$	6815.27	0.579758
$518$	−1873.39	−0.158904
$519$	0	0
$520$	−10048.8	−0.847437
$521$	17171.3	1.44393	0.721966	−	0.691928i	$-0.243238\pi$
$521$	17171.3	1.44393	0.721966	+	0.691928i	$0.243238\pi$
$522$	0	0
$523$	−8658.40	−0.723911	−0.361955	−	0.932195i	$-0.617891\pi$
$523$	−8658.40	−0.723911	−0.361955	+	0.932195i	$0.617891\pi$
$524$	−15973.8	−1.33172
$525$	0	0
$526$	89.8975	0.00745193
$527$	−7913.37	−0.654102
$528$	0	0
$529$	−9095.27	−0.747536
$530$	6060.29	0.496683
$531$	0	0
$532$	2919.28	0.237908
$533$	13142.6	1.06804
$534$	0	0
$535$	21628.6	1.74782
$536$	−8758.08	−0.705767
$537$	0	0
$538$	−1901.60	−0.152386
$539$	−2422.04	−0.193552
$540$	0	0
$541$	11733.7	0.932480	0.466240	−	0.884658i	$-0.345608\pi$
$541$	11733.7	0.932480	0.466240	+	0.884658i	$0.345608\pi$
$542$	−2649.22	−0.209952
$543$	0	0
$544$	6542.19	0.515614
$545$	34808.5	2.73584
$546$	0	0
$547$	2534.11	0.198082	0.0990408	−	0.995083i	$-0.468423\pi$
$547$	2534.11	0.198082	0.0990408	+	0.995083i	$0.468423\pi$
$548$	−15390.7	−1.19974
$549$	0	0
$550$	−8272.57	−0.641352
$551$	8529.12	0.659442
$552$	0	0
$553$	−7107.28	−0.546532
$554$	1793.04	0.137507
$555$	0	0
$556$	9650.49	0.736101
$557$	2697.62	0.205210	0.102605	−	0.994722i	$-0.467282\pi$
$557$	2697.62	0.205210	0.102605	+	0.994722i	$0.467282\pi$
$558$	0	0
$559$	−14137.1	−1.06965
$560$	−6224.50	−0.469702
$561$	0	0
$562$	−960.370	−0.0720832
$563$	17399.6	1.30250	0.651249	−	0.758864i	$-0.274245\pi$
$563$	17399.6	1.30250	0.651249	+	0.758864i	$0.274245\pi$
$564$	0	0
$565$	−17170.5	−1.27853
$566$	936.238	0.0695283
$567$	0	0
$568$	975.929	0.0720934
$569$	−10356.7	−0.763052	−0.381526	−	0.924358i	$-0.624601\pi$
$569$	−10356.7	−0.763052	−0.381526	+	0.924358i	$0.624601\pi$
$570$	0	0
$571$	15384.1	1.12750	0.563751	−	0.825945i	$-0.309358\pi$
$571$	15384.1	1.12750	0.563751	+	0.825945i	$0.309358\pi$
$572$	16157.9	1.18111
$573$	0	0
$574$	−1669.66	−0.121412
$575$	−11488.8	−0.833244
$576$	0	0
$577$	−25346.5	−1.82875	−0.914375	−	0.404868i	$-0.867318\pi$
$577$	−25346.5	−1.82875	−0.914375	+	0.404868i	$0.867318\pi$
$578$	−2165.63	−0.155845
$579$	0	0
$580$	−20129.9	−1.44112
$581$	−7606.27	−0.543135
$582$	0	0
$583$	−20353.7	−1.44591
$584$	−10400.2	−0.736925
$585$	0	0
$586$	433.793	0.0305799
$587$	7759.32	0.545590	0.272795	−	0.962072i	$-0.412052\pi$
$587$	7759.32	0.545590	0.272795	+	0.962072i	$0.412052\pi$
$588$	0	0
$589$	−9509.17	−0.665227
$590$	3124.59	0.218030
$591$	0	0
$592$	16169.8	1.12259
$593$	−16418.8	−1.13700	−0.568500	−	0.822684i	$-0.692476\pi$
$593$	−16418.8	−1.13700	−0.568500	+	0.822684i	$0.692476\pi$
$594$	0	0
$595$	−6026.66	−0.415242
$596$	10930.0	0.751195
$597$	0	0
$598$	−1990.62	−0.136125
$599$	−25648.9	−1.74956	−0.874779	−	0.484523i	$-0.838993\pi$
$599$	−25648.9	−1.74956	−0.874779	+	0.484523i	$0.838993\pi$
$600$	0	0
$601$	−6219.23	−0.422110	−0.211055	−	0.977474i	$-0.567690\pi$
$601$	−6219.23	−0.422110	−0.211055	+	0.977474i	$0.567690\pi$
$602$	1796.01	0.121595
$603$	0	0
$604$	16193.1	1.09087
$605$	20275.2	1.36249
$606$	0	0
$607$	−9558.23	−0.639138	−0.319569	−	0.947563i	$-0.603538\pi$
$607$	−9558.23	−0.639138	−0.319569	+	0.947563i	$0.603538\pi$
$608$	7861.48	0.524383
$609$	0	0
$610$	−770.562	−0.0511461
$611$	−6133.68	−0.406125
$612$	0	0
$613$	23697.2	1.56137	0.780685	−	0.624925i	$-0.214870\pi$
$613$	23697.2	1.56137	0.780685	+	0.624925i	$0.214870\pi$
$614$	−1404.97	−0.0923455
$615$	0	0
$616$	−4287.58	−0.280441
$617$	−10364.7	−0.676284	−0.338142	−	0.941095i	$-0.609798\pi$
$617$	−10364.7	−0.676284	−0.338142	+	0.941095i	$0.609798\pi$
$618$	0	0
$619$	−11818.5	−0.767408	−0.383704	−	0.923456i	$-0.625352\pi$
$619$	−11818.5	−0.767408	−0.383704	+	0.923456i	$0.625352\pi$
$620$	22443.0	1.45376
$621$	0	0
$622$	−3163.20	−0.203911
$623$	5339.08	0.343348
$624$	0	0
$625$	1433.44	0.0917402
$626$	−7658.36	−0.488961
$627$	0	0
$628$	25316.9	1.60869
$629$	15655.8	0.992431
$630$	0	0
$631$	16150.5	1.01892	0.509462	−	0.860493i	$-0.329845\pi$
$631$	16150.5	1.01892	0.509462	+	0.860493i	$0.329845\pi$
$632$	−12581.6	−0.791880
$633$	0	0
$634$	−4581.33	−0.286984
$635$	−28890.0	−1.80546
$636$	0	0
$637$	2179.81	0.135584
$638$	−5997.39	−0.372161
$639$	0	0
$640$	−24297.6	−1.50070
$641$	−11812.0	−0.727838	−0.363919	−	0.931431i	$-0.618561\pi$
$641$	−11812.0	−0.727838	−0.363919	+	0.931431i	$0.618561\pi$
$642$	0	0
$643$	6197.93	0.380128	0.190064	−	0.981772i	$-0.439130\pi$
$643$	6197.93	0.380128	0.190064	+	0.981772i	$0.439130\pi$
$644$	−2850.81	−0.174437
$645$	0	0
$646$	2164.18	0.131809
$647$	−4568.63	−0.277606	−0.138803	−	0.990320i	$-0.544326\pi$
$647$	−4568.63	−0.277606	−0.138803	+	0.990320i	$0.544326\pi$
$648$	0	0
$649$	−10494.1	−0.634713
$650$	7445.24	0.449271
$651$	0	0
$652$	20015.1	1.20223
$653$	27075.6	1.62259	0.811294	−	0.584639i	$-0.198764\pi$
$653$	27075.6	1.62259	0.811294	+	0.584639i	$0.198764\pi$
$654$	0	0
$655$	39626.9	2.36390
$656$	14411.3	0.857725
$657$	0	0
$658$	779.238	0.0461669
$659$	−18043.1	−1.06656	−0.533278	−	0.845940i	$-0.679040\pi$
$659$	−18043.1	−1.06656	−0.533278	+	0.845940i	$0.679040\pi$
$660$	0	0
$661$	−8182.05	−0.481460	−0.240730	−	0.970592i	$-0.577387\pi$
$661$	−8182.05	−0.481460	−0.240730	+	0.970592i	$0.577387\pi$
$662$	3820.96	0.224329
$663$	0	0
$664$	−13464.9	−0.786957
$665$	−7241.99	−0.422304
$666$	0	0
$667$	−8329.06	−0.483512
$668$	7442.10	0.431053
$669$	0	0
$670$	10401.9	0.599792
$671$	2587.97	0.148893
$672$	0	0
$673$	−19102.9	−1.09415	−0.547075	−	0.837084i	$-0.684259\pi$
$673$	−19102.9	−1.09415	−0.547075	+	0.837084i	$0.684259\pi$
$674$	−3264.44	−0.186560
$675$	0	0
$676$	1601.90	0.0911411
$677$	8162.77	0.463399	0.231699	−	0.972787i	$-0.425571\pi$
$677$	8162.77	0.463399	0.231699	+	0.972787i	$0.425571\pi$
$678$	0	0
$679$	9398.41	0.531190
$680$	−10668.6	−0.601651
$681$	0	0
$682$	6686.53	0.375426
$683$	15581.2	0.872909	0.436454	−	0.899726i	$-0.356234\pi$
$683$	15581.2	0.872909	0.436454	+	0.899726i	$0.356234\pi$
$684$	0	0
$685$	38180.5	2.12964
$686$	−276.929	−0.0154128
$687$	0	0
$688$	−15501.9	−0.859017
$689$	18318.2	1.01287
$690$	0	0
$691$	23733.8	1.30662	0.653310	−	0.757090i	$-0.273380\pi$
$691$	23733.8	1.30662	0.653310	+	0.757090i	$0.273380\pi$
$692$	10991.5	0.603807
$693$	0	0
$694$	−3143.80	−0.171955
$695$	−23940.4	−1.30663
$696$	0	0
$697$	13953.3	0.758275
$698$	−762.649	−0.0413563
$699$	0	0
$700$	10662.5	0.575720
$701$	32518.0	1.75205	0.876025	−	0.482266i	$-0.160186\pi$
$701$	32518.0	1.75205	0.876025	+	0.482266i	$0.160186\pi$
$702$	0	0
$703$	18813.0	1.00931
$704$	13761.6	0.736731
$705$	0	0
$706$	−7588.10	−0.404507
$707$	−4948.88	−0.263255
$708$	0	0
$709$	13915.9	0.737128	0.368564	−	0.929602i	$-0.379850\pi$
$709$	13915.9	0.737128	0.368564	+	0.929602i	$0.379850\pi$
$710$	−1159.10	−0.0612681
$711$	0	0
$712$	9451.45	0.497483
$713$	9286.12	0.487753
$714$	0	0
$715$	−40083.7	−2.09656
$716$	−14621.1	−0.763152
$717$	0	0
$718$	5265.88	0.273706
$719$	−20884.8	−1.08327	−0.541635	−	0.840614i	$-0.682194\pi$
$719$	−20884.8	−1.08327	−0.541635	+	0.840614i	$0.682194\pi$
$720$	0	0
$721$	−3604.83	−0.186201
$722$	−2937.16	−0.151399
$723$	0	0
$724$	20507.1	1.05268
$725$	31152.0	1.59580
$726$	0	0
$727$	−20890.9	−1.06575	−0.532875	−	0.846194i	$-0.678888\pi$
$727$	−20890.9	−1.06575	−0.532875	+	0.846194i	$0.678888\pi$
$728$	3858.78	0.196451
$729$	0	0
$730$	12352.3	0.626271
$731$	−15009.2	−0.759417
$732$	0	0
$733$	−8252.87	−0.415862	−0.207931	−	0.978144i	$-0.566673\pi$
$733$	−8252.87	−0.415862	−0.207931	+	0.978144i	$0.566673\pi$
$734$	3823.99	0.192297
$735$	0	0
$736$	−7677.08	−0.384485
$737$	−34935.2	−1.74607
$738$	0	0
$739$	29252.3	1.45611	0.728054	−	0.685520i	$-0.240425\pi$
$739$	29252.3	1.45611	0.728054	+	0.685520i	$0.240425\pi$
$740$	−44401.3	−2.20571
$741$	0	0
$742$	−2327.19	−0.115140
$743$	27403.2	1.35306	0.676532	−	0.736413i	$-0.263482\pi$
$743$	27403.2	1.35306	0.676532	+	0.736413i	$0.263482\pi$
$744$	0	0
$745$	−27114.6	−1.33343
$746$	4623.02	0.226891
$747$	0	0
$748$	17154.6	0.838550
$749$	−8305.50	−0.405175
$750$	0	0
$751$	19031.6	0.924731	0.462366	−	0.886689i	$-0.347001\pi$
$751$	19031.6	0.924731	0.462366	+	0.886689i	$0.347001\pi$
$752$	−6725.81	−0.326150
$753$	0	0
$754$	5397.60	0.260702
$755$	−40170.8	−1.93638
$756$	0	0
$757$	−15036.1	−0.721926	−0.360963	−	0.932580i	$-0.617552\pi$
$757$	−15036.1	−0.721926	−0.360963	+	0.932580i	$0.617552\pi$
$758$	−1720.96	−0.0824646
$759$	0	0
$760$	−12820.0	−0.611884
$761$	−20216.9	−0.963025	−0.481513	−	0.876439i	$-0.659912\pi$
$761$	−20216.9	−0.963025	−0.481513	+	0.876439i	$0.659912\pi$
$762$	0	0
$763$	−13366.7	−0.634215
$764$	−31859.3	−1.50868
$765$	0	0
$766$	8227.60	0.388088
$767$	9444.59	0.444621
$768$	0	0
$769$	−12156.3	−0.570049	−0.285025	−	0.958520i	$-0.592002\pi$
$769$	−12156.3	−0.570049	−0.285025	+	0.958520i	$0.592002\pi$
$770$	5092.32	0.238331
$771$	0	0
$772$	17855.3	0.832417
$773$	2842.00	0.132237	0.0661187	−	0.997812i	$-0.478938\pi$
$773$	2842.00	0.132237	0.0661187	+	0.997812i	$0.478938\pi$
$774$	0	0
$775$	−34731.6	−1.60980
$776$	16637.4	0.769650
$777$	0	0
$778$	3834.58	0.176705
$779$	16767.1	0.771172
$780$	0	0
$781$	3892.90	0.178360
$782$	−2113.41	−0.0966438
$783$	0	0
$784$	2390.25	0.108885
$785$	−62804.8	−2.85554
$786$	0	0
$787$	−28170.4	−1.27594	−0.637970	−	0.770061i	$-0.720226\pi$
$787$	−28170.4	−1.27594	−0.637970	+	0.770061i	$0.720226\pi$
$788$	−2006.88	−0.0907259
$789$	0	0
$790$	14943.0	0.672974
$791$	6593.57	0.296385
$792$	0	0
$793$	−2329.15	−0.104301
$794$	−3373.92	−0.150801
$795$	0	0
$796$	−4767.13	−0.212269
$797$	−32377.2	−1.43897	−0.719486	−	0.694507i	$-0.755623\pi$
$797$	−32377.2	−1.43897	−0.719486	+	0.694507i	$0.755623\pi$
$798$	0	0
$799$	−6512.04	−0.288335
$800$	28713.5	1.26897
$801$	0	0
$802$	2857.36	0.125806
$803$	−41485.6	−1.82316
$804$	0	0
$805$	7072.12	0.309639
$806$	−6017.82	−0.262988
$807$	0	0
$808$	−8760.69	−0.381435
$809$	11154.6	0.484764	0.242382	−	0.970181i	$-0.422071\pi$
$809$	11154.6	0.484764	0.242382	+	0.970181i	$0.422071\pi$
$810$	0	0
$811$	−21528.0	−0.932122	−0.466061	−	0.884753i	$-0.654327\pi$
$811$	−21528.0	−0.932122	−0.466061	+	0.884753i	$0.654327\pi$
$812$	7730.01	0.334077
$813$	0	0
$814$	−13228.6	−0.569612
$815$	−49652.4	−2.13405
$816$	0	0
$817$	−18035.9	−0.772333
$818$	−4819.55	−0.206005
$819$	0	0
$820$	−39572.7	−1.68529
$821$	6631.22	0.281889	0.140945	−	0.990017i	$-0.454986\pi$
$821$	6631.22	0.281889	0.140945	+	0.990017i	$0.454986\pi$
$822$	0	0
$823$	21583.7	0.914169	0.457084	−	0.889423i	$-0.348894\pi$
$823$	21583.7	0.914169	0.457084	+	0.889423i	$0.348894\pi$
$824$	−6381.41	−0.269790
$825$	0	0
$826$	−1199.86	−0.0505431
$827$	−39987.3	−1.68137	−0.840686	−	0.541523i	$-0.817848\pi$
$827$	−39987.3	−1.68137	−0.840686	+	0.541523i	$0.817848\pi$
$828$	0	0
$829$	−28422.3	−1.19077	−0.595384	−	0.803442i	$-0.703000\pi$
$829$	−28422.3	−1.19077	−0.595384	+	0.803442i	$0.703000\pi$
$830$	15992.2	0.668790
$831$	0	0
$832$	−12385.3	−0.516085
$833$	2314.27	0.0962603
$834$	0	0
$835$	−18461.9	−0.765152
$836$	20614.0	0.852812
$837$	0	0
$838$	9324.37	0.384374
$839$	22461.8	0.924275	0.462138	−	0.886808i	$-0.347083\pi$
$839$	22461.8	0.924275	0.462138	+	0.886808i	$0.347083\pi$
$840$	0	0
$841$	−1804.61	−0.0739929
$842$	2175.85	0.0890557
$843$	0	0
$844$	−1786.23	−0.0728490
$845$	−3973.89	−0.161782
$846$	0	0
$847$	−7785.80	−0.315848
$848$	20086.6	0.813415
$849$	0	0
$850$	7904.50	0.318967
$851$	−18371.7	−0.740039
$852$	0	0
$853$	−11163.9	−0.448119	−0.224060	−	0.974575i	$-0.571931\pi$
$853$	−11163.9	−0.448119	−0.224060	+	0.974575i	$0.571931\pi$
$854$	295.901	0.0118566
$855$	0	0
$856$	−14702.7	−0.587066
$857$	3033.25	0.120903	0.0604515	−	0.998171i	$-0.480746\pi$
$857$	3033.25	0.120903	0.0604515	+	0.998171i	$0.480746\pi$
$858$	0	0
$859$	3447.28	0.136926	0.0684632	−	0.997654i	$-0.478190\pi$
$859$	3447.28	0.136926	0.0684632	+	0.997654i	$0.478190\pi$
$860$	42567.3	1.68783
$861$	0	0
$862$	1831.03	0.0723492
$863$	15012.0	0.592135	0.296068	−	0.955167i	$-0.404325\pi$
$863$	15012.0	0.592135	0.296068	+	0.955167i	$0.404325\pi$
$864$	0	0
$865$	−27267.1	−1.07180
$866$	9672.11	0.379529
$867$	0	0
$868$	−8618.24	−0.337007
$869$	−50186.9	−1.95912
$870$	0	0
$871$	31441.4	1.22314
$872$	−23662.2	−0.918925
$873$	0	0
$874$	−2539.60	−0.0982875
$875$	−10500.6	−0.405698
$876$	0	0
$877$	−5658.60	−0.217876	−0.108938	−	0.994049i	$-0.534745\pi$
$877$	−5658.60	−0.217876	−0.108938	+	0.994049i	$0.534745\pi$
$878$	8800.81	0.338284
$879$	0	0
$880$	−43953.2	−1.68371
$881$	38874.9	1.48664	0.743320	−	0.668936i	$-0.233250\pi$
$881$	38874.9	1.48664	0.743320	+	0.668936i	$0.233250\pi$
$882$	0	0
$883$	13946.0	0.531507	0.265753	−	0.964041i	$-0.414379\pi$
$883$	13946.0	0.531507	0.265753	+	0.964041i	$0.414379\pi$
$884$	−15439.0	−0.587410
$885$	0	0
$886$	−297.508	−0.0112810
$887$	−20128.5	−0.761951	−0.380975	−	0.924585i	$-0.624412\pi$
$887$	−20128.5	−0.761951	−0.380975	+	0.924585i	$0.624412\pi$
$888$	0	0
$889$	11094.0	0.418537
$890$	−11225.4	−0.422783
$891$	0	0
$892$	−13378.3	−0.502174
$893$	−7825.25	−0.293239
$894$	0	0
$895$	36271.2	1.35465
$896$	9330.43	0.347888
$897$	0	0
$898$	−8136.58	−0.302362
$899$	−25179.5	−0.934130
$900$	0	0
$901$	19448.2	0.719103
$902$	−11790.0	−0.435217
$903$	0	0
$904$	11672.2	0.429437
$905$	−50872.8	−1.86858
$906$	0	0
$907$	−16771.0	−0.613971	−0.306986	−	0.951714i	$-0.599320\pi$
$907$	−16771.0	−0.613971	−0.306986	+	0.951714i	$0.599320\pi$
$908$	−12493.3	−0.456613
$909$	0	0
$910$	−4583.05	−0.166952
$911$	6626.66	0.241000	0.120500	−	0.992713i	$-0.461550\pi$
$911$	6626.66	0.241000	0.120500	+	0.992713i	$0.461550\pi$
$912$	0	0
$913$	−53710.4	−1.94694
$914$	−1763.89	−0.0638340
$915$	0	0
$916$	−1388.63	−0.0500892
$917$	−15217.0	−0.547992
$918$	0	0
$919$	23808.0	0.854573	0.427287	−	0.904116i	$-0.359469\pi$
$919$	23808.0	0.854573	0.427287	+	0.904116i	$0.359469\pi$
$920$	12519.3	0.448642
$921$	0	0
$922$	13404.3	0.478795
$923$	−3503.57	−0.124942
$924$	0	0
$925$	68713.1	2.44246
$926$	−3097.66	−0.109930
$927$	0	0
$928$	20816.5	0.736353
$929$	52057.6	1.83849	0.919243	−	0.393690i	$-0.128802\pi$
$929$	52057.6	1.83849	0.919243	+	0.393690i	$0.128802\pi$
$930$	0	0
$931$	2780.97	0.0978975
$932$	4755.62	0.167141
$933$	0	0
$934$	7873.40	0.275830
$935$	−42556.2	−1.48849
$936$	0	0
$937$	−23455.8	−0.817789	−0.408895	−	0.912582i	$-0.634086\pi$
$937$	−23455.8	−0.817789	−0.408895	+	0.912582i	$0.634086\pi$
$938$	−3994.39	−0.139042
$939$	0	0
$940$	18468.7	0.640832
$941$	40102.5	1.38927	0.694636	−	0.719362i	$-0.255566\pi$
$941$	40102.5	1.38927	0.694636	+	0.719362i	$0.255566\pi$
$942$	0	0
$943$	−16373.8	−0.565433
$944$	10356.3	0.357066
$945$	0	0
$946$	12682.2	0.435872
$947$	42213.2	1.44852	0.724259	−	0.689529i	$-0.242182\pi$
$947$	42213.2	1.44852	0.724259	+	0.689529i	$0.242182\pi$
$948$	0	0
$949$	37336.7	1.27713
$950$	9498.52	0.324392
$951$	0	0
$952$	4096.82	0.139473
$953$	−26903.6	−0.914475	−0.457237	−	0.889345i	$-0.651161\pi$
$953$	−26903.6	−0.914475	−0.457237	+	0.889345i	$0.651161\pi$
$954$	0	0
$955$	79034.8	2.67802
$956$	25874.4	0.875354
$957$	0	0
$958$	5683.81	0.191686
$959$	−14661.5	−0.493687
$960$	0	0
$961$	−1718.24	−0.0576766
$962$	11905.7	0.399017
$963$	0	0
$964$	−2384.48	−0.0796669
$965$	−44294.4	−1.47760
$966$	0	0
$967$	47783.4	1.58905	0.794525	−	0.607231i	$-0.207720\pi$
$967$	47783.4	1.58905	0.794525	+	0.607231i	$0.207720\pi$
$968$	−13782.7	−0.457637
$969$	0	0
$970$	−19760.1	−0.654082
$971$	−35958.5	−1.18843	−0.594214	−	0.804307i	$-0.702537\pi$
$971$	−35958.5	−1.18843	−0.594214	+	0.804307i	$0.702537\pi$
$972$	0	0
$973$	9193.26	0.302901
$974$	−6463.16	−0.212621
$975$	0	0
$976$	−2554.00	−0.0837618
$977$	33204.6	1.08732	0.543659	−	0.839307i	$-0.317039\pi$
$977$	33204.6	1.08732	0.543659	+	0.839307i	$0.317039\pi$
$978$	0	0
$979$	37701.0	1.23078
$980$	−6563.48	−0.213941
$981$	0	0
$982$	−6373.36	−0.207110
$983$	−16216.8	−0.526182	−0.263091	−	0.964771i	$-0.584742\pi$
$983$	−16216.8	−0.526182	−0.263091	+	0.964771i	$0.584742\pi$
$984$	0	0
$985$	4978.55	0.161045
$986$	5730.55	0.185089
$987$	0	0
$988$	−18552.4	−0.597401
$989$	17612.8	0.566285
$990$	0	0
$991$	−45661.6	−1.46366	−0.731830	−	0.681487i	$-0.761334\pi$
$991$	−45661.6	−1.46366	−0.731830	+	0.681487i	$0.761334\pi$
$992$	−23208.5	−0.742812
$993$	0	0
$994$	445.102	0.0142030
$995$	11826.0	0.376794
$996$	0	0
$997$	−16483.6	−0.523610	−0.261805	−	0.965121i	$-0.584318\pi$
$997$	−16483.6	−0.523610	−0.261805	+	0.965121i	$0.584318\pi$
$998$	2212.58	0.0701782
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.4.a.i.1.4		8
3.2	odd	2		567.4.a.g.1.5		8
9.2	odd	6		189.4.f.b.64.4		16
9.4	even	3		63.4.f.b.43.5	yes	16
9.5	odd	6		189.4.f.b.127.4		16
9.7	even	3		63.4.f.b.22.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.f.b.22.5	✓	16	9.7	even	3
63.4.f.b.43.5	yes	16	9.4	even	3
189.4.f.b.64.4		16	9.2	odd	6
189.4.f.b.127.4		16	9.5	odd	6
567.4.a.g.1.5		8	3.2	odd	2
567.4.a.i.1.4		8	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:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	567.a (trivial)

\(f(q)\)	\(=\)	\(q+0.807372 q^{2} -7.34815 q^{4} +18.2289 q^{5} -7.00000 q^{7} -12.3917 q^{8} +O(q^{10})\)	\(q+0.807372 q^{2} -7.34815 q^{4} +18.2289 q^{5} -7.00000 q^{7} -12.3917 q^{8} +14.7175 q^{10} -49.4293 q^{11} +44.4860 q^{13} -5.65160 q^{14} +48.7805 q^{16} +47.2301 q^{17} +56.7545 q^{19} -133.949 q^{20} -39.9078 q^{22} -55.4232 q^{23} +207.292 q^{25} +35.9167 q^{26} +51.4371 q^{28} +150.281 q^{29} -167.549 q^{31} +138.517 q^{32} +38.1322 q^{34} -127.602 q^{35} +331.480 q^{37} +45.8220 q^{38} -225.886 q^{40} +295.432 q^{41} -317.788 q^{43} +363.214 q^{44} -44.7471 q^{46} -137.879 q^{47} +49.0000 q^{49} +167.362 q^{50} -326.890 q^{52} +411.775 q^{53} -901.041 q^{55} +86.7416 q^{56} +121.333 q^{58} +212.305 q^{59} -52.3569 q^{61} -135.275 q^{62} -278.409 q^{64} +810.929 q^{65} +706.772 q^{67} -347.054 q^{68} -103.022 q^{70} -78.7569 q^{71} +839.292 q^{73} +267.628 q^{74} -417.040 q^{76} +346.005 q^{77} +1015.33 q^{79} +889.214 q^{80} +238.523 q^{82} +1086.61 q^{83} +860.952 q^{85} -256.573 q^{86} +612.511 q^{88} -762.726 q^{89} -311.402 q^{91} +407.258 q^{92} -111.320 q^{94} +1034.57 q^{95} -1342.63 q^{97} +39.5612 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{2} + 43 q^{4} + 30 q^{5} - 56 q^{7} + 6 q^{8}+O(q^{10}) \) 8 * q + 3 * q^2 + 43 * q^4 + 30 * q^5 - 56 * q^7 + 6 * q^8	\( 8 q + 3 q^{2} + 43 q^{4} + 30 q^{5} - 56 q^{7} + 6 q^{8} - 14 q^{10} + 24 q^{11} + 68 q^{13} - 21 q^{14} + 103 q^{16} + 168 q^{17} + 176 q^{19} + 330 q^{20} + 151 q^{22} + 228 q^{23} + 244 q^{25} + 795 q^{26} - 301 q^{28} + 618 q^{29} + 72 q^{31} + 786 q^{32} - 261 q^{34} - 210 q^{35} + 210 q^{37} + 1032 q^{38} - 375 q^{40} + 420 q^{41} - 2 q^{43} + 387 q^{44} + 402 q^{46} + 570 q^{47} + 392 q^{49} + 1110 q^{50} - 431 q^{52} + 528 q^{53} - 838 q^{55} - 42 q^{56} + 37 q^{58} - 150 q^{59} + 578 q^{61} + 1170 q^{62} - 112 q^{64} - 366 q^{65} - 898 q^{67} + 2526 q^{68} + 98 q^{70} + 882 q^{71} + 972 q^{73} - 222 q^{74} + 1423 q^{76} - 168 q^{77} - 158 q^{79} + 2475 q^{80} - 211 q^{82} + 2958 q^{83} - 774 q^{85} - 114 q^{86} + 1317 q^{88} + 4380 q^{89} - 476 q^{91} + 4629 q^{92} - 3234 q^{94} + 930 q^{95} - 60 q^{97} + 147 q^{98}+O(q^{100}) \) 8 * q + 3 * q^2 + 43 * q^4 + 30 * q^5 - 56 * q^7 + 6 * q^8 - 14 * q^10 + 24 * q^11 + 68 * q^13 - 21 * q^14 + 103 * q^16 + 168 * q^17 + 176 * q^19 + 330 * q^20 + 151 * q^22 + 228 * q^23 + 244 * q^25 + 795 * q^26 - 301 * q^28 + 618 * q^29 + 72 * q^31 + 786 * q^32 - 261 * q^34 - 210 * q^35 + 210 * q^37 + 1032 * q^38 - 375 * q^40 + 420 * q^41 - 2 * q^43 + 387 * q^44 + 402 * q^46 + 570 * q^47 + 392 * q^49 + 1110 * q^50 - 431 * q^52 + 528 * q^53 - 838 * q^55 - 42 * q^56 + 37 * q^58 - 150 * q^59 + 578 * q^61 + 1170 * q^62 - 112 * q^64 - 366 * q^65 - 898 * q^67 + 2526 * q^68 + 98 * q^70 + 882 * q^71 + 972 * q^73 - 222 * q^74 + 1423 * q^76 - 168 * q^77 - 158 * q^79 + 2475 * q^80 - 211 * q^82 + 2958 * q^83 - 774 * q^85 - 114 * q^86 + 1317 * q^88 + 4380 * q^89 - 476 * q^91 + 4629 * q^92 - 3234 * q^94 + 930 * q^95 - 60 * q^97 + 147 * q^98

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