LMFDB - Embedded newform 153.4.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [153,4,Mod(1,153)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(153, 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("153.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 153 = 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	153.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:	$9.02729223088$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.5912.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 14x - 10 $ x^3 - x^2 - 14*x - 10
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.795427$ of defining polynomial
Character	$\chi$	$=$	153.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.79543 q^{2} -0.185590 q^{4} -7.18559 q^{5} +19.9241 q^{7} +22.8822 q^{8} +20.0868 q^{10} +20.2836 q^{11} -69.3212 q^{13} -55.6963 q^{14} -62.4808 q^{16} -17.0000 q^{17} -2.16352 q^{19} +1.33357 q^{20} -56.7013 q^{22} +13.0415 q^{23} -73.3673 q^{25} +193.782 q^{26} -3.69771 q^{28} -246.831 q^{29} -166.779 q^{31} -8.39714 q^{32} +47.5223 q^{34} -143.166 q^{35} -162.065 q^{37} +6.04795 q^{38} -164.422 q^{40} -253.572 q^{41} +556.702 q^{43} -3.76443 q^{44} -36.4565 q^{46} -198.218 q^{47} +53.9684 q^{49} +205.093 q^{50} +12.8653 q^{52} -496.904 q^{53} -145.750 q^{55} +455.907 q^{56} +689.997 q^{58} +343.050 q^{59} +332.776 q^{61} +466.218 q^{62} +523.320 q^{64} +498.114 q^{65} -169.821 q^{67} +3.15503 q^{68} +400.211 q^{70} +509.616 q^{71} -621.448 q^{73} +453.041 q^{74} +0.401527 q^{76} +404.132 q^{77} -1122.10 q^{79} +448.962 q^{80} +708.843 q^{82} +485.798 q^{83} +122.155 q^{85} -1556.22 q^{86} +464.134 q^{88} -1038.49 q^{89} -1381.16 q^{91} -2.42037 q^{92} +554.103 q^{94} +15.5462 q^{95} +1014.67 q^{97} -150.865 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 5 q^{2} + 13 q^{4} - 8 q^{5} - 8 q^{7} - 33 q^{8} - 38 q^{10} - 34 q^{11} + 36 q^{13} + 104 q^{14} - 79 q^{16} - 51 q^{17} - 142 q^{19} + 126 q^{20} - 248 q^{22} - 110 q^{23} - 193 q^{25} - 154 q^{26}+ \cdots - 4109 q^{98}+O(q^{100}) $ 3 * q - 5 * q^2 + 13 * q^4 - 8 * q^5 - 8 * q^7 - 33 * q^8 - 38 * q^10 - 34 * q^11 + 36 * q^13 + 104 * q^14 - 79 * q^16 - 51 * q^17 - 142 * q^19 + 126 * q^20 - 248 * q^22 - 110 * q^23 - 193 * q^25 - 154 * q^26 - 472 * q^28 - 90 * q^29 - 148 * q^31 + 151 * q^32 + 85 * q^34 - 416 * q^35 + 110 * q^37 - 80 * q^38 - 202 * q^40 - 720 * q^41 - 146 * q^43 + 192 * q^44 + 748 * q^46 - 500 * q^47 + 379 * q^49 + 385 * q^50 + 1218 * q^52 - 610 * q^53 + 430 * q^55 + 1368 * q^56 + 1006 * q^58 + 216 * q^59 - 18 * q^61 + 904 * q^62 + 377 * q^64 + 966 * q^65 - 1404 * q^67 - 221 * q^68 + 1472 * q^70 + 960 * q^71 - 794 * q^73 + 1874 * q^74 - 392 * q^76 - 48 * q^77 - 276 * q^79 + 1130 * q^80 + 382 * q^82 + 1552 * q^83 + 136 * q^85 - 16 * q^86 + 1724 * q^88 - 1394 * q^89 - 3968 * q^91 - 2244 * q^92 + 3960 * q^94 + 602 * q^95 + 402 * q^97 - 4109 * q^98

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 $

$2$	−2.79543	−0.988333	−0.494166	−	0.869367i	$-0.664527\pi$
$2$	−2.79543	−0.988333	−0.494166	+	0.869367i	$0.664527\pi$
$3$	0	0
$3$	0	0
$4$	−0.185590	−0.0231987
$5$	−7.18559	−0.642699	−0.321349	−	0.946961i	$-0.604136\pi$
$5$	−7.18559	−0.642699	−0.321349	+	0.946961i	$0.604136\pi$
$6$	0	0
$7$	19.9241	1.07580	0.537899	−	0.843009i	$-0.319218\pi$
$7$	19.9241	1.07580	0.537899	+	0.843009i	$0.319218\pi$
$8$	22.8822	1.01126
$9$	0	0
$10$	20.0868	0.635200
$11$	20.2836	0.555976	0.277988	−	0.960585i	$-0.410332\pi$
$11$	20.2836	0.555976	0.277988	+	0.960585i	$0.410332\pi$
$12$	0	0
$13$	−69.3212	−1.47894	−0.739471	−	0.673189i	$-0.764924\pi$
$13$	−69.3212	−1.47894	−0.739471	+	0.673189i	$0.764924\pi$
$14$	−55.6963	−1.06325
$15$	0	0
$16$	−62.4808	−0.976263
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	−2.16352	−0.0261234	−0.0130617	−	0.999915i	$-0.504158\pi$
$19$	−2.16352	−0.0261234	−0.0130617	+	0.999915i	$0.504158\pi$
$20$	1.33357	0.0149098
$21$	0	0
$22$	−56.7013	−0.549489
$23$	13.0415	0.118232	0.0591161	−	0.998251i	$-0.481172\pi$
$23$	13.0415	0.118232	0.0591161	+	0.998251i	$0.481172\pi$
$24$	0	0
$25$	−73.3673	−0.586938
$26$	193.782	1.46169
$27$	0	0
$28$	−3.69771	−0.0249572
$29$	−246.831	−1.58053	−0.790264	−	0.612767i	$-0.790057\pi$
$29$	−246.831	−1.58053	−0.790264	+	0.612767i	$0.790057\pi$
$30$	0	0
$31$	−166.779	−0.966269	−0.483135	−	0.875546i	$-0.660502\pi$
$31$	−166.779	−0.966269	−0.483135	+	0.875546i	$0.660502\pi$
$32$	−8.39714	−0.0463881
$33$	0	0
$34$	47.5223	0.239706
$35$	−143.166	−0.691414
$36$	0	0
$37$	−162.065	−0.720090	−0.360045	−	0.932935i	$-0.617239\pi$
$37$	−162.065	−0.720090	−0.360045	+	0.932935i	$0.617239\pi$
$38$	6.04795	0.0258186
$39$	0	0
$40$	−164.422	−0.649936
$41$	−253.572	−0.965886	−0.482943	−	0.875652i	$-0.660432\pi$
$41$	−253.572	−0.965886	−0.482943	+	0.875652i	$0.660432\pi$
$42$	0	0
$43$	556.702	1.97433	0.987166	−	0.159698i	$-0.0510520\pi$
$43$	556.702	1.97433	0.987166	+	0.159698i	$0.0510520\pi$
$44$	−3.76443	−0.0128979
$45$	0	0
$46$	−36.4565	−0.116853
$47$	−198.218	−0.615170	−0.307585	−	0.951521i	$-0.599521\pi$
$47$	−198.218	−0.615170	−0.307585	+	0.951521i	$0.599521\pi$
$48$	0	0
$49$	53.9684	0.157342
$50$	205.093	0.580090
$51$	0	0
$52$	12.8653	0.0343096
$53$	−496.904	−1.28783	−0.643915	−	0.765097i	$-0.722691\pi$
$53$	−496.904	−1.28783	−0.643915	+	0.765097i	$0.722691\pi$
$54$	0	0
$55$	−145.750	−0.357325
$56$	455.907	1.08791
$57$	0	0
$58$	689.997	1.56209
$59$	343.050	0.756971	0.378486	−	0.925607i	$-0.376445\pi$
$59$	343.050	0.756971	0.378486	+	0.925607i	$0.376445\pi$
$60$	0	0
$61$	332.776	0.698484	0.349242	−	0.937033i	$-0.386439\pi$
$61$	332.776	0.698484	0.349242	+	0.937033i	$0.386439\pi$
$62$	466.218	0.954995
$63$	0	0
$64$	523.320	1.02211
$65$	498.114	0.950514
$66$	0	0
$67$	−169.821	−0.309656	−0.154828	−	0.987941i	$-0.549482\pi$
$67$	−169.821	−0.309656	−0.154828	+	0.987941i	$0.549482\pi$
$68$	3.15503	0.00562652
$69$	0	0
$70$	400.211	0.683347
$71$	509.616	0.851835	0.425918	−	0.904762i	$-0.359951\pi$
$71$	509.616	0.851835	0.425918	+	0.904762i	$0.359951\pi$
$72$	0	0
$73$	−621.448	−0.996370	−0.498185	−	0.867071i	$-0.666000\pi$
$73$	−621.448	−0.996370	−0.498185	+	0.867071i	$0.666000\pi$
$74$	453.041	0.711688
$75$	0	0
$76$	0.401527	0.000606031 0
$77$	404.132	0.598118
$78$	0	0
$79$	−1122.10	−1.59806	−0.799029	−	0.601292i	$-0.794653\pi$
$79$	−1122.10	−1.59806	−0.799029	+	0.601292i	$0.794653\pi$
$80$	448.962	0.627443
$81$	0	0
$82$	708.843	0.954617
$83$	485.798	0.642449	0.321224	−	0.947003i	$-0.395906\pi$
$83$	485.798	0.642449	0.321224	+	0.947003i	$0.395906\pi$
$84$	0	0
$85$	122.155	0.155877
$86$	−1556.22	−1.95130
$87$	0	0
$88$	464.134	0.562236
$89$	−1038.49	−1.23685	−0.618424	−	0.785844i	$-0.712229\pi$
$89$	−1038.49	−1.23685	−0.618424	+	0.785844i	$0.712229\pi$
$90$	0	0
$91$	−1381.16	−1.59104
$92$	−2.42037	−0.00274284
$93$	0	0
$94$	554.103	0.607993
$95$	15.5462	0.0167895
$96$	0	0
$97$	1014.67	1.06211	0.531054	−	0.847338i	$-0.321796\pi$
$97$	1014.67	1.06211	0.531054	+	0.847338i	$0.321796\pi$
$98$	−150.865	−0.155507
$99$	0	0
$100$	13.6162	0.0136162
$101$	653.602	0.643919	0.321959	−	0.946753i	$-0.395659\pi$
$101$	653.602	0.643919	0.321959	+	0.946753i	$0.395659\pi$
$102$	0	0
$103$	489.213	0.467996	0.233998	−	0.972237i	$-0.424819\pi$
$103$	489.213	0.467996	0.233998	+	0.972237i	$0.424819\pi$
$104$	−1586.22	−1.49560
$105$	0	0
$106$	1389.06	1.27281
$107$	−486.534	−0.439580	−0.219790	−	0.975547i	$-0.570537\pi$
$107$	−486.534	−0.439580	−0.219790	+	0.975547i	$0.570537\pi$
$108$	0	0
$109$	88.6813	0.0779278	0.0389639	−	0.999241i	$-0.487594\pi$
$109$	88.6813	0.0779278	0.0389639	+	0.999241i	$0.487594\pi$
$110$	407.432	0.353156
$111$	0	0
$112$	−1244.87	−1.05026
$113$	−1814.48	−1.51055	−0.755274	−	0.655409i	$-0.772496\pi$
$113$	−1814.48	−1.51055	−0.755274	+	0.655409i	$0.772496\pi$
$114$	0	0
$115$	−93.7108	−0.0759876
$116$	45.8093	0.0366663
$117$	0	0
$118$	−958.971	−0.748139
$119$	−338.709	−0.260919
$120$	0	0
$121$	−919.576	−0.690891
$122$	−930.250	−0.690335
$123$	0	0
$124$	30.9525	0.0224162
$125$	1425.39	1.01992
$126$	0	0
$127$	−2775.34	−1.93915	−0.969573	−	0.244803i	$-0.921277\pi$
$127$	−2775.34	−1.93915	−0.969573	+	0.244803i	$0.921277\pi$
$128$	−1395.73	−0.963796
$129$	0	0
$130$	−1392.44	−0.939424
$131$	1856.34	1.23808	0.619041	−	0.785359i	$-0.287521\pi$
$131$	1856.34	1.23808	0.619041	+	0.785359i	$0.287521\pi$
$132$	0	0
$133$	−43.1061	−0.0281035
$134$	474.722	0.306043
$135$	0	0
$136$	−388.998	−0.245267
$137$	1532.70	0.955823	0.477912	−	0.878408i	$-0.341394\pi$
$137$	1532.70	0.955823	0.477912	+	0.878408i	$0.341394\pi$
$138$	0	0
$139$	518.039	0.316111	0.158056	−	0.987430i	$-0.449477\pi$
$139$	518.039	0.316111	0.158056	+	0.987430i	$0.449477\pi$
$140$	26.5702	0.0160399
$141$	0	0
$142$	−1424.59	−0.841897
$143$	−1406.08	−0.822256
$144$	0	0
$145$	1773.62	1.01580
$146$	1737.21	0.984745
$147$	0	0
$148$	30.0776	0.0167052
$149$	−3401.05	−1.86997	−0.934983	−	0.354693i	$-0.884585\pi$
$149$	−3401.05	−1.86997	−0.934983	+	0.354693i	$0.884585\pi$
$150$	0	0
$151$	108.797	0.0586345	0.0293173	−	0.999570i	$-0.490667\pi$
$151$	108.797	0.0586345	0.0293173	+	0.999570i	$0.490667\pi$
$152$	−49.5061	−0.0264176
$153$	0	0
$154$	−1129.72	−0.591139
$155$	1198.40	0.621020
$156$	0	0
$157$	3553.34	1.80629	0.903145	−	0.429335i	$-0.141252\pi$
$157$	3553.34	1.80629	0.903145	+	0.429335i	$0.141252\pi$
$158$	3136.76	1.57941
$159$	0	0
$160$	60.3384	0.0298135
$161$	259.840	0.127194
$162$	0	0
$163$	−3875.94	−1.86250	−0.931249	−	0.364384i	$-0.881280\pi$
$163$	−3875.94	−1.86250	−0.931249	+	0.364384i	$0.881280\pi$
$164$	47.0605	0.0224074
$165$	0	0
$166$	−1358.01	−0.634953
$167$	−730.820	−0.338638	−0.169319	−	0.985561i	$-0.554157\pi$
$167$	−730.820	−0.338638	−0.169319	+	0.985561i	$0.554157\pi$
$168$	0	0
$169$	2608.43	1.18727
$170$	−341.475	−0.154059
$171$	0	0
$172$	−103.318	−0.0458020
$173$	−2979.89	−1.30958	−0.654789	−	0.755812i	$-0.727242\pi$
$173$	−2979.89	−1.30958	−0.654789	+	0.755812i	$0.727242\pi$
$174$	0	0
$175$	−1461.77	−0.631427
$176$	−1267.34	−0.542779
$177$	0	0
$178$	2903.02	1.22242
$179$	3112.68	1.29974	0.649868	−	0.760047i	$-0.274824\pi$
$179$	3112.68	1.29974	0.649868	+	0.760047i	$0.274824\pi$
$180$	0	0
$181$	2454.92	1.00814	0.504068	−	0.863664i	$-0.331836\pi$
$181$	2454.92	1.00814	0.504068	+	0.863664i	$0.331836\pi$
$182$	3860.93	1.57248
$183$	0	0
$184$	298.418	0.119563
$185$	1164.53	0.462801
$186$	0	0
$187$	−344.821	−0.134844
$188$	36.7872	0.0142712
$189$	0	0
$190$	−43.4581	−0.0165936
$191$	1806.74	0.684457	0.342228	−	0.939617i	$-0.388818\pi$
$191$	1806.74	0.684457	0.342228	+	0.939617i	$0.388818\pi$
$192$	0	0
$193$	−3252.39	−1.21302	−0.606508	−	0.795077i	$-0.707430\pi$
$193$	−3252.39	−1.21302	−0.606508	+	0.795077i	$0.707430\pi$
$194$	−2836.45	−1.04972
$195$	0	0
$196$	−10.0160	−0.00365015
$197$	2227.38	0.805554	0.402777	−	0.915298i	$-0.368045\pi$
$197$	2227.38	0.805554	0.402777	+	0.915298i	$0.368045\pi$
$198$	0	0
$199$	4309.13	1.53500	0.767502	−	0.641046i	$-0.221499\pi$
$199$	4309.13	1.53500	0.767502	+	0.641046i	$0.221499\pi$
$200$	−1678.81	−0.593548
$201$	0	0
$202$	−1827.10	−0.636406
$203$	−4917.87	−1.70033
$204$	0	0
$205$	1822.07	0.620774
$206$	−1367.56	−0.462536
$207$	0	0
$208$	4331.25	1.44384
$209$	−43.8839	−0.0145240
$210$	0	0
$211$	4219.39	1.37666	0.688330	−	0.725398i	$-0.258344\pi$
$211$	4219.39	1.37666	0.688330	+	0.725398i	$0.258344\pi$
$212$	92.2204	0.0298761
$213$	0	0
$214$	1360.07	0.434451
$215$	−4000.23	−1.26890
$216$	0	0
$217$	−3322.91	−1.03951
$218$	−247.902	−0.0770185
$219$	0	0
$220$	27.0497	0.00828949
$221$	1178.46	0.358696
$222$	0	0
$223$	−4825.85	−1.44916	−0.724580	−	0.689191i	$-0.757966\pi$
$223$	−4825.85	−1.44916	−0.724580	+	0.689191i	$0.757966\pi$
$224$	−167.305	−0.0499042
$225$	0	0
$226$	5072.25	1.49292
$227$	1665.18	0.486880	0.243440	−	0.969916i	$-0.421724\pi$
$227$	1665.18	0.486880	0.243440	+	0.969916i	$0.421724\pi$
$228$	0	0
$229$	2726.00	0.786633	0.393317	−	0.919403i	$-0.371328\pi$
$229$	2726.00	0.786633	0.393317	+	0.919403i	$0.371328\pi$
$230$	261.962	0.0751010
$231$	0	0
$232$	−5648.03	−1.59833
$233$	4836.51	1.35987	0.679937	−	0.733271i	$-0.262007\pi$
$233$	4836.51	1.35987	0.679937	+	0.733271i	$0.262007\pi$
$234$	0	0
$235$	1424.31	0.395369
$236$	−63.6666	−0.0175608
$237$	0	0
$238$	946.837	0.257875
$239$	−5200.13	−1.40740	−0.703700	−	0.710497i	$-0.748470\pi$
$239$	−5200.13	−1.40740	−0.703700	+	0.710497i	$0.748470\pi$
$240$	0	0
$241$	−372.453	−0.0995512	−0.0497756	−	0.998760i	$-0.515851\pi$
$241$	−372.453	−0.0995512	−0.0497756	+	0.998760i	$0.515851\pi$
$242$	2570.61	0.682830
$243$	0	0
$244$	−61.7598	−0.0162040
$245$	−387.795	−0.101124
$246$	0	0
$247$	149.978	0.0386350
$248$	−3816.27	−0.977150
$249$	0	0
$250$	−3984.56	−1.00802
$251$	902.694	0.227002	0.113501	−	0.993538i	$-0.463793\pi$
$251$	902.694	0.227002	0.113501	+	0.993538i	$0.463793\pi$
$252$	0	0
$253$	264.528	0.0657342
$254$	7758.26	1.91652
$255$	0	0
$256$	−284.912	−0.0695585
$257$	1361.43	0.330442	0.165221	−	0.986257i	$-0.447166\pi$
$257$	1361.43	0.330442	0.165221	+	0.986257i	$0.447166\pi$
$258$	0	0
$259$	−3228.99	−0.774671
$260$	−92.4449	−0.0220507
$261$	0	0
$262$	−5189.25	−1.22364
$263$	1065.57	0.249833	0.124916	−	0.992167i	$-0.460134\pi$
$263$	1065.57	0.249833	0.124916	+	0.992167i	$0.460134\pi$
$264$	0	0
$265$	3570.55	0.827687
$266$	120.500	0.0277756
$267$	0	0
$268$	31.5171	0.00718363
$269$	5516.38	1.25033	0.625166	−	0.780491i	$-0.285031\pi$
$269$	5516.38	1.25033	0.625166	+	0.780491i	$0.285031\pi$
$270$	0	0
$271$	6271.52	1.40578	0.702892	−	0.711296i	$-0.251892\pi$
$271$	6271.52	1.40578	0.702892	+	0.711296i	$0.251892\pi$
$272$	1062.17	0.236779
$273$	0	0
$274$	−4284.56	−0.944671
$275$	−1488.15	−0.326324
$276$	0	0
$277$	−612.909	−0.132946	−0.0664732	−	0.997788i	$-0.521175\pi$
$277$	−612.909	−0.132946	−0.0664732	+	0.997788i	$0.521175\pi$
$278$	−1448.14	−0.312423
$279$	0	0
$280$	−3275.96	−0.699200
$281$	8957.63	1.90166	0.950831	−	0.309709i	$-0.100232\pi$
$281$	8957.63	1.90166	0.950831	+	0.309709i	$0.100232\pi$
$282$	0	0
$283$	−366.649	−0.0770143	−0.0385072	−	0.999258i	$-0.512260\pi$
$283$	−366.649	−0.0770143	−0.0385072	+	0.999258i	$0.512260\pi$
$284$	−94.5797	−0.0197615
$285$	0	0
$286$	3930.60	0.812662
$287$	−5052.19	−1.03910
$288$	0	0
$289$	289.000	0.0588235
$290$	−4958.04	−1.00395
$291$	0	0
$292$	115.335	0.0231145
$293$	−544.260	−0.108519	−0.0542594	−	0.998527i	$-0.517280\pi$
$293$	−544.260	−0.108519	−0.0542594	+	0.998527i	$0.517280\pi$
$294$	0	0
$295$	−2465.02	−0.486504
$296$	−3708.41	−0.728198
$297$	0	0
$298$	9507.39	1.84815
$299$	−904.052	−0.174858
$300$	0	0
$301$	11091.8	2.12398
$302$	−304.135	−0.0579504
$303$	0	0
$304$	135.178	0.0255033
$305$	−2391.19	−0.448915
$306$	0	0
$307$	−6140.08	−1.14148	−0.570738	−	0.821132i	$-0.693343\pi$
$307$	−6140.08	−1.14148	−0.570738	+	0.821132i	$0.693343\pi$
$308$	−75.0028	−0.0138756
$309$	0	0
$310$	−3350.05	−0.613774
$311$	2449.54	0.446626	0.223313	−	0.974747i	$-0.428313\pi$
$311$	2449.54	0.446626	0.223313	+	0.974747i	$0.428313\pi$
$312$	0	0
$313$	−7062.20	−1.27533	−0.637666	−	0.770313i	$-0.720100\pi$
$313$	−7062.20	−1.27533	−0.637666	+	0.770313i	$0.720100\pi$
$314$	−9933.11	−1.78522
$315$	0	0
$316$	208.251	0.0370730
$317$	−4075.00	−0.722002	−0.361001	−	0.932565i	$-0.617565\pi$
$317$	−4075.00	−0.722002	−0.361001	+	0.932565i	$0.617565\pi$
$318$	0	0
$319$	−5006.61	−0.878735
$320$	−3760.36	−0.656909
$321$	0	0
$322$	−726.362	−0.125710
$323$	36.7798	0.00633586
$324$	0	0
$325$	5085.91	0.868048
$326$	10834.9	1.84077
$327$	0	0
$328$	−5802.30	−0.976763
$329$	−3949.30	−0.661799
$330$	0	0
$331$	3869.63	0.642580	0.321290	−	0.946981i	$-0.395883\pi$
$331$	3869.63	0.642580	0.321290	+	0.946981i	$0.395883\pi$
$332$	−90.1592	−0.0149040
$333$	0	0
$334$	2042.95	0.334687
$335$	1220.26	0.199015
$336$	0	0
$337$	−6253.84	−1.01089	−0.505443	−	0.862860i	$-0.668671\pi$
$337$	−6253.84	−1.01089	−0.505443	+	0.862860i	$0.668671\pi$
$338$	−7291.67	−1.17342
$339$	0	0
$340$	−22.6708	−0.00361616
$341$	−3382.87	−0.537222
$342$	0	0
$343$	−5758.68	−0.906530
$344$	12738.6	1.99656
$345$	0	0
$346$	8330.06	1.29430
$347$	6632.55	1.02609	0.513046	−	0.858361i	$-0.328517\pi$
$347$	6632.55	1.02609	0.513046	+	0.858361i	$0.328517\pi$
$348$	0	0
$349$	−3485.68	−0.534625	−0.267313	−	0.963610i	$-0.586136\pi$
$349$	−3485.68	−0.534625	−0.267313	+	0.963610i	$0.586136\pi$
$350$	4086.28	0.624060
$351$	0	0
$352$	−170.324	−0.0257906
$353$	−6439.69	−0.970963	−0.485481	−	0.874247i	$-0.661356\pi$
$353$	−6439.69	−0.970963	−0.485481	+	0.874247i	$0.661356\pi$
$354$	0	0
$355$	−3661.89	−0.547473
$356$	192.733	0.0286933
$357$	0	0
$358$	−8701.28	−1.28457
$359$	−7104.81	−1.04451	−0.522253	−	0.852791i	$-0.674908\pi$
$359$	−7104.81	−1.04451	−0.522253	+	0.852791i	$0.674908\pi$
$360$	0	0
$361$	−6854.32	−0.999318
$362$	−6862.54	−0.996374
$363$	0	0
$364$	256.330	0.0369102
$365$	4465.47	0.640366
$366$	0	0
$367$	7921.51	1.12670	0.563350	−	0.826218i	$-0.309512\pi$
$367$	7921.51	1.12670	0.563350	+	0.826218i	$0.309512\pi$
$368$	−814.843	−0.115426
$369$	0	0
$370$	−3255.37	−0.457401
$371$	−9900.35	−1.38545
$372$	0	0
$373$	−5560.92	−0.771940	−0.385970	−	0.922511i	$-0.626133\pi$
$373$	−5560.92	−0.771940	−0.385970	+	0.922511i	$0.626133\pi$
$374$	963.922	0.133271
$375$	0	0
$376$	−4535.66	−0.622098
$377$	17110.6	2.33751
$378$	0	0
$379$	3393.45	0.459920	0.229960	−	0.973200i	$-0.426140\pi$
$379$	3393.45	0.459920	0.229960	+	0.973200i	$0.426140\pi$
$380$	−2.88521	−0.000389495 0
$381$	0	0
$382$	−5050.61	−0.676471
$383$	2057.57	0.274509	0.137255	−	0.990536i	$-0.456172\pi$
$383$	2057.57	0.274509	0.137255	+	0.990536i	$0.456172\pi$
$384$	0	0
$385$	−2903.92	−0.384410
$386$	9091.82	1.19886
$387$	0	0
$388$	−188.313	−0.0246396
$389$	−5909.30	−0.770215	−0.385107	−	0.922872i	$-0.625836\pi$
$389$	−5909.30	−0.770215	−0.385107	+	0.922872i	$0.625836\pi$
$390$	0	0
$391$	−221.705	−0.0286755
$392$	1234.92	0.159114
$393$	0	0
$394$	−6226.47	−0.796155
$395$	8062.99	1.02707
$396$	0	0
$397$	8871.61	1.12154	0.560772	−	0.827970i	$-0.310504\pi$
$397$	8871.61	1.12154	0.560772	+	0.827970i	$0.310504\pi$
$398$	−12045.8	−1.51710
$399$	0	0
$400$	4584.05	0.573006
$401$	−5958.38	−0.742013	−0.371007	−	0.928630i	$-0.620987\pi$
$401$	−5958.38	−0.742013	−0.371007	+	0.928630i	$0.620987\pi$
$402$	0	0
$403$	11561.3	1.42906
$404$	−121.302	−0.0149381
$405$	0	0
$406$	13747.5	1.68049
$407$	−3287.26	−0.400352
$408$	0	0
$409$	7517.10	0.908793	0.454397	−	0.890799i	$-0.349855\pi$
$409$	7517.10	0.908793	0.454397	+	0.890799i	$0.349855\pi$
$410$	−5093.45	−0.613531
$411$	0	0
$412$	−90.7931	−0.0108569
$413$	6834.95	0.814348
$414$	0	0
$415$	−3490.74	−0.412901
$416$	582.100	0.0686052
$417$	0	0
$418$	122.674	0.0143545
$419$	−2102.77	−0.245172	−0.122586	−	0.992458i	$-0.539119\pi$
$419$	−2102.77	−0.245172	−0.122586	+	0.992458i	$0.539119\pi$
$420$	0	0
$421$	3480.77	0.402951	0.201476	−	0.979494i	$-0.435426\pi$
$421$	3480.77	0.402951	0.201476	+	0.979494i	$0.435426\pi$
$422$	−11795.0	−1.36060
$423$	0	0
$424$	−11370.3	−1.30233
$425$	1247.24	0.142353
$426$	0	0
$427$	6630.24	0.751428
$428$	90.2958	0.0101977
$429$	0	0
$430$	11182.4	1.25410
$431$	6815.12	0.761654	0.380827	−	0.924646i	$-0.375639\pi$
$431$	6815.12	0.761654	0.380827	+	0.924646i	$0.375639\pi$
$432$	0	0
$433$	2867.85	0.318292	0.159146	−	0.987255i	$-0.449126\pi$
$433$	2867.85	0.318292	0.159146	+	0.987255i	$0.449126\pi$
$434$	9288.95	1.02738
$435$	0	0
$436$	−16.4584	−0.00180783
$437$	−28.2155	−0.00308863
$438$	0	0
$439$	−3268.85	−0.355384	−0.177692	−	0.984086i	$-0.556863\pi$
$439$	−3268.85	−0.355384	−0.177692	+	0.984086i	$0.556863\pi$
$440$	−3335.07	−0.361349
$441$	0	0
$442$	−3294.30	−0.354511
$443$	−12387.8	−1.32858	−0.664289	−	0.747476i	$-0.731265\pi$
$443$	−12387.8	−1.32858	−0.664289	+	0.747476i	$0.731265\pi$
$444$	0	0
$445$	7462.15	0.794921
$446$	13490.3	1.43225
$447$	0	0
$448$	10426.7	1.09958
$449$	13922.0	1.46329	0.731646	−	0.681685i	$-0.238753\pi$
$449$	13922.0	1.46329	0.731646	+	0.681685i	$0.238753\pi$
$450$	0	0
$451$	−5143.36	−0.537009
$452$	336.749	0.0350428
$453$	0	0
$454$	−4654.88	−0.481199
$455$	9924.45	1.02256
$456$	0	0
$457$	7503.09	0.768008	0.384004	−	0.923331i	$-0.374545\pi$
$457$	7503.09	0.768008	0.384004	+	0.923331i	$0.374545\pi$
$458$	−7620.32	−0.777455
$459$	0	0
$460$	17.3918	0.00176282
$461$	1804.23	0.182280	0.0911402	−	0.995838i	$-0.470949\pi$
$461$	1804.23	0.182280	0.0911402	+	0.995838i	$0.470949\pi$
$462$	0	0
$463$	1843.06	0.184998	0.0924992	−	0.995713i	$-0.470514\pi$
$463$	1843.06	0.184998	0.0924992	+	0.995713i	$0.470514\pi$
$464$	15422.2	1.54301
$465$	0	0
$466$	−13520.1	−1.34401
$467$	−11884.3	−1.17760	−0.588799	−	0.808280i	$-0.700399\pi$
$467$	−11884.3	−1.17760	−0.588799	+	0.808280i	$0.700399\pi$
$468$	0	0
$469$	−3383.52	−0.333127
$470$	−3981.55	−0.390756
$471$	0	0
$472$	7849.74	0.765495
$473$	11291.9	1.09768
$474$	0	0
$475$	158.731	0.0153328
$476$	62.8610	0.00605300
$477$	0	0
$478$	14536.6	1.39098
$479$	−4591.44	−0.437971	−0.218986	−	0.975728i	$-0.570275\pi$
$479$	−4591.44	−0.437971	−0.218986	+	0.975728i	$0.570275\pi$
$480$	0	0
$481$	11234.5	1.06497
$482$	1041.17	0.0983897
$483$	0	0
$484$	170.664	0.0160278
$485$	−7291.03	−0.682616
$486$	0	0
$487$	3075.97	0.286213	0.143106	−	0.989707i	$-0.454291\pi$
$487$	3075.97	0.286213	0.143106	+	0.989707i	$0.454291\pi$
$488$	7614.64	0.706350
$489$	0	0
$490$	1084.05	0.0999439
$491$	2556.76	0.235000	0.117500	−	0.993073i	$-0.462512\pi$
$491$	2556.76	0.235000	0.117500	+	0.993073i	$0.462512\pi$
$492$	0	0
$493$	4196.12	0.383334
$494$	−419.251	−0.0381842
$495$	0	0
$496$	10420.5	0.943333
$497$	10153.6	0.916403
$498$	0	0
$499$	3551.49	0.318610	0.159305	−	0.987229i	$-0.449075\pi$
$499$	3551.49	0.318610	0.159305	+	0.987229i	$0.449075\pi$
$500$	−264.537	−0.0236609
$501$	0	0
$502$	−2523.42	−0.224354
$503$	96.3146	0.00853768	0.00426884	−	0.999991i	$-0.498641\pi$
$503$	96.3146	0.00853768	0.00426884	+	0.999991i	$0.498641\pi$
$504$	0	0
$505$	−4696.51	−0.413846
$506$	−739.469	−0.0649672
$507$	0	0
$508$	515.075	0.0449858
$509$	−19081.4	−1.66163	−0.830815	−	0.556549i	$-0.812125\pi$
$509$	−19081.4	−1.66163	−0.830815	+	0.556549i	$0.812125\pi$
$510$	0	0
$511$	−12381.8	−1.07189
$512$	11962.3	1.03254
$513$	0	0
$514$	−3805.77	−0.326587
$515$	−3515.29	−0.300781
$516$	0	0
$517$	−4020.57	−0.342020
$518$	9026.41	0.765633
$519$	0	0
$520$	11397.9	0.961217
$521$	−9570.95	−0.804820	−0.402410	−	0.915460i	$-0.631827\pi$
$521$	−9570.95	−0.804820	−0.402410	+	0.915460i	$0.631827\pi$
$522$	0	0
$523$	−16979.2	−1.41959	−0.709797	−	0.704406i	$-0.751214\pi$
$523$	−16979.2	−1.41959	−0.709797	+	0.704406i	$0.751214\pi$
$524$	−344.517	−0.0287220
$525$	0	0
$526$	−2978.73	−0.246918
$527$	2835.24	0.234355
$528$	0	0
$529$	−11996.9	−0.986021
$530$	−9981.21	−0.818030
$531$	0	0
$532$	8.00005	0.000651967 0
$533$	17577.9	1.42849
$534$	0	0
$535$	3496.03	0.282517
$536$	−3885.88	−0.313143
$537$	0	0
$538$	−15420.6	−1.23574
$539$	1094.67	0.0874785
$540$	0	0
$541$	−9339.39	−0.742203	−0.371101	−	0.928592i	$-0.621020\pi$
$541$	−9339.39	−0.742203	−0.371101	+	0.928592i	$0.621020\pi$
$542$	−17531.6	−1.38938
$543$	0	0
$544$	142.751	0.0112508
$545$	−637.227	−0.0500841
$546$	0	0
$547$	3347.48	0.261660	0.130830	−	0.991405i	$-0.458236\pi$
$547$	3347.48	0.261660	0.130830	+	0.991405i	$0.458236\pi$
$548$	−284.455	−0.0221739
$549$	0	0
$550$	4160.02	0.322516
$551$	534.022	0.0412888
$552$	0	0
$553$	−22356.9	−1.71919
$554$	1713.34	0.131395
$555$	0	0
$556$	−96.1428	−0.00733339
$557$	−20271.9	−1.54209	−0.771047	−	0.636778i	$-0.780267\pi$
$557$	−20271.9	−1.54209	−0.771047	+	0.636778i	$0.780267\pi$
$558$	0	0
$559$	−38591.3	−2.91992
$560$	8945.14	0.675002
$561$	0	0
$562$	−25040.4	−1.87947
$563$	6144.58	0.459970	0.229985	−	0.973194i	$-0.426132\pi$
$563$	6144.58	0.459970	0.229985	+	0.973194i	$0.426132\pi$
$564$	0	0
$565$	13038.1	0.970828
$566$	1024.94	0.0761158
$567$	0	0
$568$	11661.1	0.861428
$569$	−19198.3	−1.41447	−0.707237	−	0.706977i	$-0.750058\pi$
$569$	−19198.3	−1.41447	−0.707237	+	0.706977i	$0.750058\pi$
$570$	0	0
$571$	−2944.07	−0.215772	−0.107886	−	0.994163i	$-0.534408\pi$
$571$	−2944.07	−0.215772	−0.107886	+	0.994163i	$0.534408\pi$
$572$	260.955	0.0190753
$573$	0	0
$574$	14123.0	1.02698
$575$	−956.819	−0.0693950
$576$	0	0
$577$	−1802.77	−0.130070	−0.0650351	−	0.997883i	$-0.520716\pi$
$577$	−1802.77	−0.130070	−0.0650351	+	0.997883i	$0.520716\pi$
$578$	−807.878	−0.0581372
$579$	0	0
$580$	−329.167	−0.0235654
$581$	9679.07	0.691145
$582$	0	0
$583$	−10079.0	−0.716003
$584$	−14220.1	−1.00759
$585$	0	0
$586$	1521.44	0.107253
$587$	−24517.6	−1.72394	−0.861969	−	0.506961i	$-0.830769\pi$
$587$	−24517.6	−1.72394	−0.861969	+	0.506961i	$0.830769\pi$
$588$	0	0
$589$	360.829	0.0252423
$590$	6890.77	0.480828
$591$	0	0
$592$	10126.0	0.702997
$593$	14269.5	0.988161	0.494081	−	0.869416i	$-0.335505\pi$
$593$	14269.5	0.988161	0.494081	+	0.869416i	$0.335505\pi$
$594$	0	0
$595$	2433.82	0.167693
$596$	631.201	0.0433809
$597$	0	0
$598$	2527.21	0.172818
$599$	−15362.8	−1.04792	−0.523961	−	0.851742i	$-0.675546\pi$
$599$	−15362.8	−1.04792	−0.523961	+	0.851742i	$0.675546\pi$
$600$	0	0
$601$	19852.3	1.34741	0.673704	−	0.739001i	$-0.264702\pi$
$601$	19852.3	1.34741	0.673704	+	0.739001i	$0.264702\pi$
$602$	−31006.2	−2.09920
$603$	0	0
$604$	−20.1917	−0.00136025
$605$	6607.69	0.444035
$606$	0	0
$607$	−19495.1	−1.30359	−0.651796	−	0.758394i	$-0.725984\pi$
$607$	−19495.1	−1.30359	−0.651796	+	0.758394i	$0.725984\pi$
$608$	18.1674	0.00121181
$609$	0	0
$610$	6684.39	0.443677
$611$	13740.7	0.909801
$612$	0	0
$613$	16796.4	1.10669	0.553343	−	0.832954i	$-0.313352\pi$
$613$	16796.4	1.10669	0.553343	+	0.832954i	$0.313352\pi$
$614$	17164.2	1.12816
$615$	0	0
$616$	9247.43	0.604853
$617$	−1185.68	−0.0773642	−0.0386821	−	0.999252i	$-0.512316\pi$
$617$	−1185.68	−0.0773642	−0.0386821	+	0.999252i	$0.512316\pi$
$618$	0	0
$619$	11747.1	0.762771	0.381385	−	0.924416i	$-0.375447\pi$
$619$	11747.1	0.762771	0.381385	+	0.924416i	$0.375447\pi$
$620$	−222.412	−0.0144069
$621$	0	0
$622$	−6847.51	−0.441415
$623$	−20690.9	−1.33060
$624$	0	0
$625$	−1071.33	−0.0685650
$626$	19741.9	1.26045
$627$	0	0
$628$	−659.465	−0.0419037
$629$	2755.10	0.174647
$630$	0	0
$631$	−11744.3	−0.740943	−0.370471	−	0.928844i	$-0.620804\pi$
$631$	−11744.3	−0.740943	−0.370471	+	0.928844i	$0.620804\pi$
$632$	−25676.2	−1.61605
$633$	0	0
$634$	11391.4	0.713578
$635$	19942.4	1.24629
$636$	0	0
$637$	−3741.16	−0.232700
$638$	13995.6	0.868483
$639$	0	0
$640$	10029.1	0.619431
$641$	−4318.07	−0.266074	−0.133037	−	0.991111i	$-0.542473\pi$
$641$	−4318.07	−0.266074	−0.133037	+	0.991111i	$0.542473\pi$
$642$	0	0
$643$	−16920.9	−1.03779	−0.518893	−	0.854839i	$-0.673656\pi$
$643$	−16920.9	−1.03779	−0.518893	+	0.854839i	$0.673656\pi$
$644$	−48.2236	−0.00295074
$645$	0	0
$646$	−102.815	−0.00626194
$647$	−12459.5	−0.757084	−0.378542	−	0.925584i	$-0.623574\pi$
$647$	−12459.5	−0.757084	−0.378542	+	0.925584i	$0.623574\pi$
$648$	0	0
$649$	6958.29	0.420858
$650$	−14217.3	−0.857920
$651$	0	0
$652$	719.336	0.0432076
$653$	−12781.8	−0.765990	−0.382995	−	0.923750i	$-0.625107\pi$
$653$	−12781.8	−0.765990	−0.382995	+	0.923750i	$0.625107\pi$
$654$	0	0
$655$	−13338.9	−0.795714
$656$	15843.4	0.942959
$657$	0	0
$658$	11040.0	0.654078
$659$	7400.62	0.437461	0.218731	−	0.975785i	$-0.429808\pi$
$659$	7400.62	0.437461	0.218731	+	0.975785i	$0.429808\pi$
$660$	0	0
$661$	−21906.0	−1.28902	−0.644512	−	0.764594i	$-0.722939\pi$
$661$	−21906.0	−1.28902	−0.644512	+	0.764594i	$0.722939\pi$
$662$	−10817.3	−0.635083
$663$	0	0
$664$	11116.1	0.649683
$665$	309.743	0.0180621
$666$	0	0
$667$	−3219.04	−0.186869
$668$	135.633	0.00785598
$669$	0	0
$670$	−3411.16	−0.196693
$671$	6749.89	0.388340
$672$	0	0
$673$	−440.817	−0.0252485	−0.0126243	−	0.999920i	$-0.504019\pi$
$673$	−440.817	−0.0252485	−0.0126243	+	0.999920i	$0.504019\pi$
$674$	17482.2	0.999091
$675$	0	0
$676$	−484.098	−0.0275431
$677$	−17950.8	−1.01906	−0.509532	−	0.860452i	$-0.670181\pi$
$677$	−17950.8	−1.01906	−0.509532	+	0.860452i	$0.670181\pi$
$678$	0	0
$679$	20216.4	1.14261
$680$	2795.18	0.157633
$681$	0	0
$682$	9456.57	0.530954
$683$	32522.6	1.82203	0.911013	−	0.412377i	$-0.135301\pi$
$683$	32522.6	1.82203	0.911013	+	0.412377i	$0.135301\pi$
$684$	0	0
$685$	−11013.4	−0.614306
$686$	16098.0	0.895953
$687$	0	0
$688$	−34783.2	−1.92747
$689$	34446.0	1.90463
$690$	0	0
$691$	29316.7	1.61398	0.806991	−	0.590564i	$-0.201095\pi$
$691$	29316.7	1.61398	0.806991	+	0.590564i	$0.201095\pi$
$692$	553.038	0.0303805
$693$	0	0
$694$	−18540.8	−1.01412
$695$	−3722.42	−0.203164
$696$	0	0
$697$	4310.73	0.234262
$698$	9743.96	0.528387
$699$	0	0
$700$	271.291	0.0146483
$701$	−25948.5	−1.39809	−0.699045	−	0.715077i	$-0.746392\pi$
$701$	−25948.5	−1.39809	−0.699045	+	0.715077i	$0.746392\pi$
$702$	0	0
$703$	350.630	0.0188112
$704$	10614.8	0.568268
$705$	0	0
$706$	18001.7	0.959634
$707$	13022.4	0.692727
$708$	0	0
$709$	−22102.6	−1.17077	−0.585387	−	0.810754i	$-0.699057\pi$
$709$	−22102.6	−1.17077	−0.585387	+	0.810754i	$0.699057\pi$
$710$	10236.6	0.541086
$711$	0	0
$712$	−23762.9	−1.25078
$713$	−2175.04	−0.114244
$714$	0	0
$715$	10103.5	0.528463
$716$	−577.683	−0.0301523
$717$	0	0
$718$	19861.0	1.03232
$719$	9876.18	0.512266	0.256133	−	0.966642i	$-0.417551\pi$
$719$	9876.18	0.512266	0.256133	+	0.966642i	$0.417551\pi$
$720$	0	0
$721$	9747.12	0.503470
$722$	19160.7	0.987658
$723$	0	0
$724$	−455.608	−0.0233875
$725$	18109.3	0.927672
$726$	0	0
$727$	7187.21	0.366656	0.183328	−	0.983052i	$-0.441313\pi$
$727$	7187.21	0.366656	0.183328	+	0.983052i	$0.441313\pi$
$728$	−31604.0	−1.60896
$729$	0	0
$730$	−12482.9	−0.632894
$731$	−9463.94	−0.478846
$732$	0	0
$733$	28840.8	1.45328	0.726642	−	0.687016i	$-0.241080\pi$
$733$	28840.8	1.45328	0.726642	+	0.687016i	$0.241080\pi$
$734$	−22144.0	−1.11356
$735$	0	0
$736$	−109.511	−0.00548456
$737$	−3444.58	−0.172161
$738$	0	0
$739$	13008.1	0.647514	0.323757	−	0.946140i	$-0.395054\pi$
$739$	13008.1	0.647514	0.323757	+	0.946140i	$0.395054\pi$
$740$	−216.126	−0.0107364
$741$	0	0
$742$	27675.7	1.36928
$743$	29485.3	1.45587	0.727934	−	0.685647i	$-0.240481\pi$
$743$	29485.3	1.45587	0.727934	+	0.685647i	$0.240481\pi$
$744$	0	0
$745$	24438.6	1.20182
$746$	15545.2	0.762934
$747$	0	0
$748$	63.9954	0.00312821
$749$	−9693.73	−0.472899
$750$	0	0
$751$	17194.3	0.835458	0.417729	−	0.908572i	$-0.362826\pi$
$751$	17194.3	0.835458	0.417729	+	0.908572i	$0.362826\pi$
$752$	12384.8	0.600568
$753$	0	0
$754$	−47831.4	−2.31024
$755$	−781.774	−0.0376843
$756$	0	0
$757$	7578.25	0.363852	0.181926	−	0.983312i	$-0.441767\pi$
$757$	7578.25	0.363852	0.181926	+	0.983312i	$0.441767\pi$
$758$	−9486.14	−0.454554
$759$	0	0
$760$	355.730	0.0169785
$761$	−12964.9	−0.617577	−0.308789	−	0.951131i	$-0.599924\pi$
$761$	−12964.9	−0.617577	−0.308789	+	0.951131i	$0.599924\pi$
$762$	0	0
$763$	1766.89	0.0838346
$764$	−335.313	−0.0158785
$765$	0	0
$766$	−5751.79	−0.271306
$767$	−23780.6	−1.11952
$768$	0	0
$769$	−5997.37	−0.281236	−0.140618	−	0.990064i	$-0.544909\pi$
$769$	−5997.37	−0.281236	−0.140618	+	0.990064i	$0.544909\pi$
$770$	8117.71	0.379925
$771$	0	0
$772$	603.611	0.0281405
$773$	457.468	0.0212859	0.0106429	−	0.999943i	$-0.496612\pi$
$773$	457.468	0.0212859	0.0106429	+	0.999943i	$0.496612\pi$
$774$	0	0
$775$	12236.1	0.567140
$776$	23218.0	1.07407
$777$	0	0
$778$	16519.0	0.761228
$779$	548.608	0.0252323
$780$	0	0
$781$	10336.8	0.473600
$782$	619.761	0.0283409
$783$	0	0
$784$	−3371.99	−0.153608
$785$	−25532.9	−1.16090
$786$	0	0
$787$	−837.185	−0.0379192	−0.0189596	−	0.999820i	$-0.506035\pi$
$787$	−837.185	−0.0379192	−0.0189596	+	0.999820i	$0.506035\pi$
$788$	−413.379	−0.0186878
$789$	0	0
$790$	−22539.5	−1.01509
$791$	−36151.8	−1.62505
$792$	0	0
$793$	−23068.4	−1.03302
$794$	−24799.9	−1.10846
$795$	0	0
$796$	−799.731	−0.0356102
$797$	−21061.7	−0.936063	−0.468032	−	0.883712i	$-0.655037\pi$
$797$	−21061.7	−0.936063	−0.468032	+	0.883712i	$0.655037\pi$
$798$	0	0
$799$	3369.70	0.149201
$800$	616.075	0.0272269
$801$	0	0
$802$	16656.2	0.733356
$803$	−12605.2	−0.553957
$804$	0	0
$805$	−1867.10	−0.0817474
$806$	−32318.8	−1.41238
$807$	0	0
$808$	14955.9	0.651170
$809$	13593.4	0.590752	0.295376	−	0.955381i	$-0.404555\pi$
$809$	13593.4	0.590752	0.295376	+	0.955381i	$0.404555\pi$
$810$	0	0
$811$	21346.1	0.924243	0.462122	−	0.886817i	$-0.347088\pi$
$811$	21346.1	0.924243	0.462122	+	0.886817i	$0.347088\pi$
$812$	912.707	0.0394455
$813$	0	0
$814$	9189.30	0.395681
$815$	27850.9	1.19702
$816$	0	0
$817$	−1204.43	−0.0515763
$818$	−21013.5	−0.898190
$819$	0	0
$820$	−338.157	−0.0144012
$821$	−10609.9	−0.451019	−0.225510	−	0.974241i	$-0.572405\pi$
$821$	−10609.9	−0.451019	−0.225510	+	0.974241i	$0.572405\pi$
$822$	0	0
$823$	−16152.8	−0.684145	−0.342072	−	0.939674i	$-0.611129\pi$
$823$	−16152.8	−0.684145	−0.342072	+	0.939674i	$0.611129\pi$
$824$	11194.3	0.473266
$825$	0	0
$826$	−19106.6	−0.804847
$827$	17755.3	0.746570	0.373285	−	0.927717i	$-0.378231\pi$
$827$	17755.3	0.746570	0.373285	+	0.927717i	$0.378231\pi$
$828$	0	0
$829$	16425.3	0.688149	0.344075	−	0.938942i	$-0.388193\pi$
$829$	16425.3	0.688149	0.344075	+	0.938942i	$0.388193\pi$
$830$	9758.12	0.408084
$831$	0	0
$832$	−36277.2	−1.51164
$833$	−917.463	−0.0381611
$834$	0	0
$835$	5251.37	0.217642
$836$	8.14442	0.000336938 0
$837$	0	0
$838$	5878.15	0.242312
$839$	1680.00	0.0691298	0.0345649	−	0.999402i	$-0.488995\pi$
$839$	1680.00	0.0691298	0.0345649	+	0.999402i	$0.488995\pi$
$840$	0	0
$841$	36536.4	1.49807
$842$	−9730.24	−0.398250
$843$	0	0
$844$	−783.077	−0.0319368
$845$	−18743.1	−0.763056
$846$	0	0
$847$	−18321.7	−0.743259
$848$	31047.0	1.25726
$849$	0	0
$850$	−3486.58	−0.140693
$851$	−2113.57	−0.0851377
$852$	0	0
$853$	−14637.8	−0.587560	−0.293780	−	0.955873i	$-0.594913\pi$
$853$	−14637.8	−0.587560	−0.293780	+	0.955873i	$0.594913\pi$
$854$	−18534.4	−0.742661
$855$	0	0
$856$	−11133.0	−0.444529
$857$	28175.6	1.12306	0.561528	−	0.827458i	$-0.310214\pi$
$857$	28175.6	1.12306	0.561528	+	0.827458i	$0.310214\pi$
$858$	0	0
$859$	−941.593	−0.0374002	−0.0187001	−	0.999825i	$-0.505953\pi$
$859$	−941.593	−0.0374002	−0.0187001	+	0.999825i	$0.505953\pi$
$860$	742.403	0.0294369
$861$	0	0
$862$	−19051.2	−0.752768
$863$	17106.4	0.674750	0.337375	−	0.941370i	$-0.390461\pi$
$863$	17106.4	0.674750	0.337375	+	0.941370i	$0.390461\pi$
$864$	0	0
$865$	21412.3	0.841663
$866$	−8016.87	−0.314578
$867$	0	0
$868$	616.699	0.0241154
$869$	−22760.3	−0.888482
$870$	0	0
$871$	11772.2	0.457963
$872$	2029.22	0.0788053
$873$	0	0
$874$	78.8743	0.00305259
$875$	28399.5	1.09723
$876$	0	0
$877$	−16734.4	−0.644332	−0.322166	−	0.946683i	$-0.604411\pi$
$877$	−16734.4	−0.644332	−0.322166	+	0.946683i	$0.604411\pi$
$878$	9137.82	0.351237
$879$	0	0
$880$	9106.56	0.348843
$881$	19999.2	0.764802	0.382401	−	0.923996i	$-0.375097\pi$
$881$	19999.2	0.764802	0.382401	+	0.923996i	$0.375097\pi$
$882$	0	0
$883$	10029.6	0.382248	0.191124	−	0.981566i	$-0.438787\pi$
$883$	10029.6	0.382248	0.191124	+	0.981566i	$0.438787\pi$
$884$	−218.710	−0.00832130
$885$	0	0
$886$	34629.1	1.31308
$887$	40481.7	1.53240	0.766202	−	0.642600i	$-0.222144\pi$
$887$	40481.7	1.53240	0.766202	+	0.642600i	$0.222144\pi$
$888$	0	0
$889$	−55296.0	−2.08613
$890$	−20859.9	−0.785646
$891$	0	0
$892$	895.629	0.0336187
$893$	428.847	0.0160704
$894$	0	0
$895$	−22366.5	−0.835339
$896$	−27808.5	−1.03685
$897$	0	0
$898$	−38917.8	−1.44622
$899$	41166.1	1.52722
$900$	0	0
$901$	8447.37	0.312345
$902$	14377.9	0.530744
$903$	0	0
$904$	−41519.3	−1.52756
$905$	−17640.0	−0.647928
$906$	0	0
$907$	−22918.7	−0.839034	−0.419517	−	0.907747i	$-0.637801\pi$
$907$	−22918.7	−0.839034	−0.419517	+	0.907747i	$0.637801\pi$
$908$	−309.040	−0.0112950
$909$	0	0
$910$	−27743.1	−1.01063
$911$	−20326.2	−0.739230	−0.369615	−	0.929185i	$-0.620510\pi$
$911$	−20326.2	−0.739230	−0.369615	+	0.929185i	$0.620510\pi$
$912$	0	0
$913$	9853.73	0.357186
$914$	−20974.3	−0.759048
$915$	0	0
$916$	−505.918	−0.0182489
$917$	36985.7	1.33193
$918$	0	0
$919$	29975.7	1.07596	0.537980	−	0.842958i	$-0.319188\pi$
$919$	29975.7	1.07596	0.537980	+	0.842958i	$0.319188\pi$
$920$	−2144.31	−0.0768433
$921$	0	0
$922$	−5043.59	−0.180154
$923$	−35327.2	−1.25981
$924$	0	0
$925$	11890.3	0.422648
$926$	−5152.14	−0.182840
$927$	0	0
$928$	2072.67	0.0733176
$929$	−18708.2	−0.660706	−0.330353	−	0.943857i	$-0.607168\pi$
$929$	−18708.2	−0.660706	−0.330353	+	0.943857i	$0.607168\pi$
$930$	0	0
$931$	−116.762	−0.00411032
$932$	−897.608	−0.0315474
$933$	0	0
$934$	33221.6	1.16386
$935$	2477.74	0.0866640
$936$	0	0
$937$	−23835.6	−0.831032	−0.415516	−	0.909586i	$-0.636399\pi$
$937$	−23835.6	−0.831032	−0.415516	+	0.909586i	$0.636399\pi$
$938$	9458.39	0.329240
$939$	0	0
$940$	−264.338	−0.00917207
$941$	29086.9	1.00766	0.503829	−	0.863803i	$-0.331924\pi$
$941$	29086.9	1.00766	0.503829	+	0.863803i	$0.331924\pi$
$942$	0	0
$943$	−3306.96	−0.114199
$944$	−21434.0	−0.739003
$945$	0	0
$946$	−31565.7	−1.08487
$947$	−4232.84	−0.145247	−0.0726234	−	0.997359i	$-0.523137\pi$
$947$	−4232.84	−0.145247	−0.0726234	+	0.997359i	$0.523137\pi$
$948$	0	0
$949$	43079.5	1.47357
$950$	−443.722	−0.0151539
$951$	0	0
$952$	−7750.42	−0.263858
$953$	34662.0	1.17819	0.589093	−	0.808065i	$-0.299485\pi$
$953$	34662.0	1.17819	0.589093	+	0.808065i	$0.299485\pi$
$954$	0	0
$955$	−12982.5	−0.439899
$956$	965.093	0.0326499
$957$	0	0
$958$	12835.0	0.432861
$959$	30537.7	1.02827
$960$	0	0
$961$	−1975.85	−0.0663238
$962$	−31405.3	−1.05255
$963$	0	0
$964$	69.1236	0.00230946
$965$	23370.3	0.779604
$966$	0	0
$967$	21658.2	0.720248	0.360124	−	0.932904i	$-0.382734\pi$
$967$	21658.2	0.720248	0.360124	+	0.932904i	$0.382734\pi$
$968$	−21041.9	−0.698671
$969$	0	0
$970$	20381.5	0.674651
$971$	−4143.58	−0.136945	−0.0684727	−	0.997653i	$-0.521813\pi$
$971$	−4143.58	−0.136945	−0.0684727	+	0.997653i	$0.521813\pi$
$972$	0	0
$973$	10321.4	0.340072
$974$	−8598.65	−0.282873
$975$	0	0
$976$	−20792.1	−0.681904
$977$	9390.27	0.307494	0.153747	−	0.988110i	$-0.450866\pi$
$977$	9390.27	0.307494	0.153747	+	0.988110i	$0.450866\pi$
$978$	0	0
$979$	−21064.3	−0.687658
$980$	71.9709	0.00234594
$981$	0	0
$982$	−7147.22	−0.232258
$983$	42182.9	1.36869	0.684346	−	0.729157i	$-0.260088\pi$
$983$	42182.9	1.36869	0.684346	+	0.729157i	$0.260088\pi$
$984$	0	0
$985$	−16005.0	−0.517728
$986$	−11729.9	−0.378862
$987$	0	0
$988$	−27.8343	−0.000896284 0
$989$	7260.22	0.233429
$990$	0	0
$991$	−9493.78	−0.304319	−0.152159	−	0.988356i	$-0.548623\pi$
$991$	−9493.78	−0.304319	−0.152159	+	0.988356i	$0.548623\pi$
$992$	1400.46	0.0448234
$993$	0	0
$994$	−28383.7	−0.905711
$995$	−30963.6	−0.986546
$996$	0	0
$997$	−53688.9	−1.70546	−0.852731	−	0.522350i	$-0.825055\pi$
$997$	−53688.9	−1.70546	−0.852731	+	0.522350i	$0.825055\pi$
$998$	−9927.93	−0.314893
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.a.f.1.2		3
3.2	odd	2		51.4.a.e.1.2	✓	3
4.3	odd	2		2448.4.a.bd.1.2		3
12.11	even	2		816.4.a.s.1.2		3
15.14	odd	2		1275.4.a.q.1.2		3
21.20	even	2		2499.4.a.n.1.2		3
51.50	odd	2		867.4.a.k.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.a.e.1.2	✓	3	3.2	odd	2
153.4.a.f.1.2		3	1.1	even	1	trivial
816.4.a.s.1.2		3	12.11	even	2
867.4.a.k.1.2		3	51.50	odd	2
1275.4.a.q.1.2		3	15.14	odd	2
2448.4.a.bd.1.2		3	4.3	odd	2
2499.4.a.n.1.2		3	21.20	even	2

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:	\( N \)	\(=\)	\( 153 = 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	153.a (trivial)

\(f(q)\)	\(=\)	\(q-2.79543 q^{2} -0.185590 q^{4} -7.18559 q^{5} +19.9241 q^{7} +22.8822 q^{8} +20.0868 q^{10} +20.2836 q^{11} -69.3212 q^{13} -55.6963 q^{14} -62.4808 q^{16} -17.0000 q^{17} -2.16352 q^{19} +1.33357 q^{20} -56.7013 q^{22} +13.0415 q^{23} -73.3673 q^{25} +193.782 q^{26} -3.69771 q^{28} -246.831 q^{29} -166.779 q^{31} -8.39714 q^{32} +47.5223 q^{34} -143.166 q^{35} -162.065 q^{37} +6.04795 q^{38} -164.422 q^{40} -253.572 q^{41} +556.702 q^{43} -3.76443 q^{44} -36.4565 q^{46} -198.218 q^{47} +53.9684 q^{49} +205.093 q^{50} +12.8653 q^{52} -496.904 q^{53} -145.750 q^{55} +455.907 q^{56} +689.997 q^{58} +343.050 q^{59} +332.776 q^{61} +466.218 q^{62} +523.320 q^{64} +498.114 q^{65} -169.821 q^{67} +3.15503 q^{68} +400.211 q^{70} +509.616 q^{71} -621.448 q^{73} +453.041 q^{74} +0.401527 q^{76} +404.132 q^{77} -1122.10 q^{79} +448.962 q^{80} +708.843 q^{82} +485.798 q^{83} +122.155 q^{85} -1556.22 q^{86} +464.134 q^{88} -1038.49 q^{89} -1381.16 q^{91} -2.42037 q^{92} +554.103 q^{94} +15.5462 q^{95} +1014.67 q^{97} -150.865 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 5 q^{2} + 13 q^{4} - 8 q^{5} - 8 q^{7} - 33 q^{8} - 38 q^{10} - 34 q^{11} + 36 q^{13} + 104 q^{14} - 79 q^{16} - 51 q^{17} - 142 q^{19} + 126 q^{20} - 248 q^{22} - 110 q^{23} - 193 q^{25} - 154 q^{26}+ \cdots - 4109 q^{98}+O(q^{100}) \) 3 * q - 5 * q^2 + 13 * q^4 - 8 * q^5 - 8 * q^7 - 33 * q^8 - 38 * q^10 - 34 * q^11 + 36 * q^13 + 104 * q^14 - 79 * q^16 - 51 * q^17 - 142 * q^19 + 126 * q^20 - 248 * q^22 - 110 * q^23 - 193 * q^25 - 154 * q^26 - 472 * q^28 - 90 * q^29 - 148 * q^31 + 151 * q^32 + 85 * q^34 - 416 * q^35 + 110 * q^37 - 80 * q^38 - 202 * q^40 - 720 * q^41 - 146 * q^43 + 192 * q^44 + 748 * q^46 - 500 * q^47 + 379 * q^49 + 385 * q^50 + 1218 * q^52 - 610 * q^53 + 430 * q^55 + 1368 * q^56 + 1006 * q^58 + 216 * q^59 - 18 * q^61 + 904 * q^62 + 377 * q^64 + 966 * q^65 - 1404 * q^67 - 221 * q^68 + 1472 * q^70 + 960 * q^71 - 794 * q^73 + 1874 * q^74 - 392 * q^76 - 48 * q^77 - 276 * q^79 + 1130 * q^80 + 382 * q^82 + 1552 * q^83 + 136 * q^85 - 16 * q^86 + 1724 * q^88 - 1394 * q^89 - 3968 * q^91 - 2244 * q^92 + 3960 * q^94 + 602 * q^95 + 402 * q^97 - 4109 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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