LMFDB - Embedded newform 153.4.a.h.1.4

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:	$4$
Coefficient field:	4.4.1506848.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 17x^{2} + 18 $ x^4 - 17*x^2 + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.06515$ 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)

$f(q)$	$=$	$q+3.56787 q^{2} +4.72971 q^{4} -20.6288 q^{5} -5.24213 q^{7} -11.6680 q^{8} +O(q^{10})$	\(q+3.56787 q^{2} +4.72971 q^{4} -20.6288 q^{5} -5.24213 q^{7} -11.6680 q^{8} -73.6008 q^{10} -5.21023 q^{11} -14.8007 q^{13} -18.7033 q^{14} -79.4675 q^{16} +17.0000 q^{17} +26.2601 q^{19} -97.5682 q^{20} -18.5894 q^{22} -165.443 q^{23} +300.546 q^{25} -52.8069 q^{26} -24.7938 q^{28} +42.3983 q^{29} +263.956 q^{31} -190.186 q^{32} +60.6538 q^{34} +108.139 q^{35} -322.217 q^{37} +93.6927 q^{38} +240.696 q^{40} -321.529 q^{41} +385.799 q^{43} -24.6429 q^{44} -590.280 q^{46} -309.308 q^{47} -315.520 q^{49} +1072.31 q^{50} -70.0029 q^{52} -192.701 q^{53} +107.481 q^{55} +61.1650 q^{56} +151.272 q^{58} +587.082 q^{59} -241.163 q^{61} +941.763 q^{62} -42.8203 q^{64} +305.320 q^{65} +205.396 q^{67} +80.4052 q^{68} +385.825 q^{70} -933.035 q^{71} -869.875 q^{73} -1149.63 q^{74} +124.203 q^{76} +27.3127 q^{77} +102.161 q^{79} +1639.32 q^{80} -1147.17 q^{82} -298.886 q^{83} -350.689 q^{85} +1376.48 q^{86} +60.7928 q^{88} +666.125 q^{89} +77.5871 q^{91} -782.499 q^{92} -1103.57 q^{94} -541.714 q^{95} +1351.60 q^{97} -1125.74 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 24 q^{7} - 102 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 24 * q^7 - 102 * q^8	$ 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 24 q^{7} - 102 q^{8} - 2 q^{10} - 50 q^{11} + 26 q^{13} - 80 q^{14} + 138 q^{16} + 68 q^{17} + 34 q^{19} - 312 q^{20} - 254 q^{22} - 382 q^{23} + 138 q^{25} + 22 q^{26} + 52 q^{28} - 540 q^{29} - 356 q^{31} - 730 q^{32} - 68 q^{34} - 304 q^{35} - 404 q^{37} - 298 q^{38} + 332 q^{40} + 114 q^{41} + 570 q^{43} + 1368 q^{44} - 290 q^{46} - 496 q^{47} - 224 q^{49} + 1862 q^{50} - 1012 q^{52} - 92 q^{53} - 482 q^{55} + 1428 q^{56} + 1324 q^{58} + 48 q^{59} - 1036 q^{61} + 2564 q^{62} + 2898 q^{64} + 342 q^{65} + 812 q^{67} + 442 q^{68} + 152 q^{70} - 1044 q^{71} - 1212 q^{73} + 1444 q^{74} + 2268 q^{76} + 564 q^{77} + 488 q^{79} + 1000 q^{80} - 938 q^{82} - 1708 q^{83} - 374 q^{85} + 2446 q^{86} - 3868 q^{88} + 8 q^{89} + 716 q^{91} - 1356 q^{92} - 1224 q^{94} - 1010 q^{95} - 76 q^{97} + 1472 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 24 * q^7 - 102 * q^8 - 2 * q^10 - 50 * q^11 + 26 * q^13 - 80 * q^14 + 138 * q^16 + 68 * q^17 + 34 * q^19 - 312 * q^20 - 254 * q^22 - 382 * q^23 + 138 * q^25 + 22 * q^26 + 52 * q^28 - 540 * q^29 - 356 * q^31 - 730 * q^32 - 68 * q^34 - 304 * q^35 - 404 * q^37 - 298 * q^38 + 332 * q^40 + 114 * q^41 + 570 * q^43 + 1368 * q^44 - 290 * q^46 - 496 * q^47 - 224 * q^49 + 1862 * q^50 - 1012 * q^52 - 92 * q^53 - 482 * q^55 + 1428 * q^56 + 1324 * q^58 + 48 * q^59 - 1036 * q^61 + 2564 * q^62 + 2898 * q^64 + 342 * q^65 + 812 * q^67 + 442 * q^68 + 152 * q^70 - 1044 * q^71 - 1212 * q^73 + 1444 * q^74 + 2268 * q^76 + 564 * q^77 + 488 * q^79 + 1000 * q^80 - 938 * q^82 - 1708 * q^83 - 374 * q^85 + 2446 * q^86 - 3868 * q^88 + 8 * q^89 + 716 * q^91 - 1356 * q^92 - 1224 * q^94 - 1010 * q^95 - 76 * q^97 + 1472 * 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 $

$2$	3.56787	1.26143	0.630717	−	0.776013i	$-0.282761\pi$
$2$	3.56787	1.26143	0.630717	+	0.776013i	$0.282761\pi$
$3$	0	0
$3$	0	0
$4$	4.72971	0.591214
$5$	−20.6288	−1.84509	−0.922547	−	0.385885i	$-0.873896\pi$
$5$	−20.6288	−1.84509	−0.922547	+	0.385885i	$0.873896\pi$
$6$	0	0
$7$	−5.24213	−0.283049	−0.141524	−	0.989935i	$-0.545200\pi$
$7$	−5.24213	−0.283049	−0.141524	+	0.989935i	$0.545200\pi$
$8$	−11.6680	−0.515656
$9$	0	0
$10$	−73.6008	−2.32746
$11$	−5.21023	−0.142813	−0.0714065	−	0.997447i	$-0.522749\pi$
$11$	−5.21023	−0.142813	−0.0714065	+	0.997447i	$0.522749\pi$
$12$	0	0
$13$	−14.8007	−0.315767	−0.157883	−	0.987458i	$-0.550467\pi$
$13$	−14.8007	−0.315767	−0.157883	+	0.987458i	$0.550467\pi$
$14$	−18.7033	−0.357047
$15$	0	0
$16$	−79.4675	−1.24168
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	26.2601	0.317078	0.158539	−	0.987353i	$-0.449322\pi$
$19$	26.2601	0.317078	0.158539	+	0.987353i	$0.449322\pi$
$20$	−97.5682	−1.09085
$21$	0	0
$22$	−18.5894	−0.180149
$23$	−165.443	−1.49988	−0.749941	−	0.661504i	$-0.769918\pi$
$23$	−165.443	−1.49988	−0.749941	+	0.661504i	$0.769918\pi$
$24$	0	0
$25$	300.546	2.40437
$26$	−52.8069	−0.398319
$27$	0	0
$28$	−24.7938	−0.167342
$29$	42.3983	0.271488	0.135744	−	0.990744i	$-0.456657\pi$
$29$	42.3983	0.271488	0.135744	+	0.990744i	$0.456657\pi$
$30$	0	0
$31$	263.956	1.52929	0.764644	−	0.644452i	$-0.222915\pi$
$31$	263.956	1.52929	0.764644	+	0.644452i	$0.222915\pi$
$32$	−190.186	−1.05064
$33$	0	0
$34$	60.6538	0.305943
$35$	108.139	0.522251
$36$	0	0
$37$	−322.217	−1.43168	−0.715840	−	0.698264i	$-0.753956\pi$
$37$	−322.217	−1.43168	−0.715840	+	0.698264i	$0.753956\pi$
$38$	93.6927	0.399973
$39$	0	0
$40$	240.696	0.951434
$41$	−321.529	−1.22474	−0.612371	−	0.790571i	$-0.709784\pi$
$41$	−321.529	−1.22474	−0.612371	+	0.790571i	$0.709784\pi$
$42$	0	0
$43$	385.799	1.36823	0.684114	−	0.729375i	$-0.260189\pi$
$43$	385.799	1.36823	0.684114	+	0.729375i	$0.260189\pi$
$44$	−24.6429	−0.0844331
$45$	0	0
$46$	−590.280	−1.89200
$47$	−309.308	−0.959942	−0.479971	−	0.877284i	$-0.659353\pi$
$47$	−309.308	−0.959942	−0.479971	+	0.877284i	$0.659353\pi$
$48$	0	0
$49$	−315.520	−0.919884
$50$	1072.31	3.03295
$51$	0	0
$52$	−70.0029	−0.186686
$53$	−192.701	−0.499426	−0.249713	−	0.968320i	$-0.580336\pi$
$53$	−192.701	−0.499426	−0.249713	+	0.968320i	$0.580336\pi$
$54$	0	0
$55$	107.481	0.263504
$56$	61.1650	0.145956
$57$	0	0
$58$	151.272	0.342464
$59$	587.082	1.29545	0.647725	−	0.761874i	$-0.275721\pi$
$59$	587.082	1.29545	0.647725	+	0.761874i	$0.275721\pi$
$60$	0	0
$61$	−241.163	−0.506192	−0.253096	−	0.967441i	$-0.581449\pi$
$61$	−241.163	−0.506192	−0.253096	+	0.967441i	$0.581449\pi$
$62$	941.763	1.92910
$63$	0	0
$64$	−42.8203	−0.0836334
$65$	305.320	0.582619
$66$	0	0
$67$	205.396	0.374525	0.187262	−	0.982310i	$-0.440039\pi$
$67$	205.396	0.374525	0.187262	+	0.982310i	$0.440039\pi$
$68$	80.4052	0.143391
$69$	0	0
$70$	385.825	0.658785
$71$	−933.035	−1.55959	−0.779795	−	0.626035i	$-0.784677\pi$
$71$	−933.035	−1.55959	−0.779795	+	0.626035i	$0.784677\pi$
$72$	0	0
$73$	−869.875	−1.39467	−0.697337	−	0.716744i	$-0.745632\pi$
$73$	−869.875	−1.39467	−0.697337	+	0.716744i	$0.745632\pi$
$74$	−1149.63	−1.80597
$75$	0	0
$76$	124.203	0.187461
$77$	27.3127	0.0404230
$78$	0	0
$79$	102.161	0.145494	0.0727470	−	0.997350i	$-0.476823\pi$
$79$	102.161	0.145494	0.0727470	+	0.997350i	$0.476823\pi$
$80$	1639.32	2.29102
$81$	0	0
$82$	−1147.17	−1.54493
$83$	−298.886	−0.395265	−0.197633	−	0.980276i	$-0.563325\pi$
$83$	−298.886	−0.395265	−0.197633	+	0.980276i	$0.563325\pi$
$84$	0	0
$85$	−350.689	−0.447501
$86$	1376.48	1.72593
$87$	0	0
$88$	60.7928	0.0736424
$89$	666.125	0.793361	0.396680	−	0.917957i	$-0.370162\pi$
$89$	666.125	0.793361	0.396680	+	0.917957i	$0.370162\pi$
$90$	0	0
$91$	77.5871	0.0893773
$92$	−782.499	−0.886752
$93$	0	0
$94$	−1103.57	−1.21090
$95$	−541.714	−0.585038
$96$	0	0
$97$	1351.60	1.41478	0.707391	−	0.706822i	$-0.249872\pi$
$97$	1351.60	1.41478	0.707391	+	0.706822i	$0.249872\pi$
$98$	−1125.74	−1.16037
$99$	0	0
$100$	1421.50	1.42150
$101$	1628.22	1.60410	0.802050	−	0.597256i	$-0.203743\pi$
$101$	1628.22	1.60410	0.802050	+	0.597256i	$0.203743\pi$
$102$	0	0
$103$	48.4374	0.0463367	0.0231684	−	0.999732i	$-0.492625\pi$
$103$	48.4374	0.0463367	0.0231684	+	0.999732i	$0.492625\pi$
$104$	172.694	0.162827
$105$	0	0
$106$	−687.534	−0.629993
$107$	−1493.56	−1.34942	−0.674711	−	0.738082i	$-0.735732\pi$
$107$	−1493.56	−1.34942	−0.674711	+	0.738082i	$0.735732\pi$
$108$	0	0
$109$	694.642	0.610410	0.305205	−	0.952287i	$-0.401275\pi$
$109$	694.642	0.610410	0.305205	+	0.952287i	$0.401275\pi$
$110$	383.477	0.332392
$111$	0	0
$112$	416.579	0.351456
$113$	−1302.15	−1.08404	−0.542018	−	0.840367i	$-0.682340\pi$
$113$	−1302.15	−1.08404	−0.542018	+	0.840367i	$0.682340\pi$
$114$	0	0
$115$	3412.89	2.76742
$116$	200.532	0.160508
$117$	0	0
$118$	2094.63	1.63412
$119$	−89.1163	−0.0686494
$120$	0	0
$121$	−1303.85	−0.979604
$122$	−860.438	−0.638528
$123$	0	0
$124$	1248.44	0.904138
$125$	−3621.31	−2.59120
$126$	0	0
$127$	−2100.00	−1.46728	−0.733642	−	0.679536i	$-0.762181\pi$
$127$	−2100.00	−1.46728	−0.733642	+	0.679536i	$0.762181\pi$
$128$	1368.71	0.945143
$129$	0	0
$130$	1089.34	0.734935
$131$	963.258	0.642444	0.321222	−	0.947004i	$-0.395906\pi$
$131$	963.258	0.642444	0.321222	+	0.947004i	$0.395906\pi$
$132$	0	0
$133$	−137.659	−0.0897484
$134$	732.828	0.472438
$135$	0	0
$136$	−198.355	−0.125065
$137$	−311.771	−0.194426	−0.0972130	−	0.995264i	$-0.530993\pi$
$137$	−311.771	−0.194426	−0.0972130	+	0.995264i	$0.530993\pi$
$138$	0	0
$139$	−1039.05	−0.634035	−0.317018	−	0.948420i	$-0.602681\pi$
$139$	−1039.05	−0.634035	−0.317018	+	0.948420i	$0.602681\pi$
$140$	511.466	0.308762
$141$	0	0
$142$	−3328.95	−1.96732
$143$	77.1149	0.0450956
$144$	0	0
$145$	−874.624	−0.500921
$146$	−3103.60	−1.75929
$147$	0	0
$148$	−1524.00	−0.846430
$149$	−375.216	−0.206301	−0.103151	−	0.994666i	$-0.532892\pi$
$149$	−375.216	−0.206301	−0.103151	+	0.994666i	$0.532892\pi$
$150$	0	0
$151$	−2602.60	−1.40262	−0.701312	−	0.712854i	$-0.747402\pi$
$151$	−2602.60	−1.40262	−0.701312	+	0.712854i	$0.747402\pi$
$152$	−306.402	−0.163503
$153$	0	0
$154$	97.4483	0.0509910
$155$	−5445.10	−2.82168
$156$	0	0
$157$	−3134.77	−1.59351	−0.796756	−	0.604301i	$-0.793453\pi$
$157$	−3134.77	−1.59351	−0.796756	+	0.604301i	$0.793453\pi$
$158$	364.498	0.183531
$159$	0	0
$160$	3923.31	1.93853
$161$	867.276	0.424540
$162$	0	0
$163$	−52.9329	−0.0254357	−0.0127179	−	0.999919i	$-0.504048\pi$
$163$	−52.9329	−0.0254357	−0.0127179	+	0.999919i	$0.504048\pi$
$164$	−1520.74	−0.724085
$165$	0	0
$166$	−1066.39	−0.498601
$167$	3554.79	1.64717	0.823586	−	0.567192i	$-0.191970\pi$
$167$	3554.79	1.64717	0.823586	+	0.567192i	$0.191970\pi$
$168$	0	0
$169$	−1977.94	−0.900291
$170$	−1251.21	−0.564493
$171$	0	0
$172$	1824.72	0.808916
$173$	−134.253	−0.0590006	−0.0295003	−	0.999565i	$-0.509392\pi$
$173$	−134.253	−0.0590006	−0.0295003	+	0.999565i	$0.509392\pi$
$174$	0	0
$175$	−1575.50	−0.680554
$176$	414.044	0.177328
$177$	0	0
$178$	2376.65	1.00077
$179$	2368.80	0.989119	0.494559	−	0.869144i	$-0.335329\pi$
$179$	2368.80	0.989119	0.494559	+	0.869144i	$0.335329\pi$
$180$	0	0
$181$	1320.79	0.542396	0.271198	−	0.962524i	$-0.412580\pi$
$181$	1320.79	0.542396	0.271198	+	0.962524i	$0.412580\pi$
$182$	276.821	0.112744
$183$	0	0
$184$	1930.39	0.773423
$185$	6646.95	2.64158
$186$	0	0
$187$	−88.5739	−0.0346373
$188$	−1462.94	−0.567531
$189$	0	0
$190$	−1932.77	−0.737987
$191$	549.534	0.208183	0.104091	−	0.994568i	$-0.466807\pi$
$191$	549.534	0.208183	0.104091	+	0.994568i	$0.466807\pi$
$192$	0	0
$193$	4191.18	1.56315	0.781575	−	0.623811i	$-0.214417\pi$
$193$	4191.18	1.56315	0.781575	+	0.623811i	$0.214417\pi$
$194$	4822.33	1.78465
$195$	0	0
$196$	−1492.32	−0.543848
$197$	3963.36	1.43339	0.716695	−	0.697386i	$-0.245654\pi$
$197$	3963.36	1.43339	0.716695	+	0.697386i	$0.245654\pi$
$198$	0	0
$199$	−4603.54	−1.63988	−0.819940	−	0.572450i	$-0.805993\pi$
$199$	−4603.54	−1.63988	−0.819940	+	0.572450i	$0.805993\pi$
$200$	−3506.76	−1.23983
$201$	0	0
$202$	5809.29	2.02347
$203$	−222.257	−0.0768444
$204$	0	0
$205$	6632.75	2.25976
$206$	172.819	0.0584507
$207$	0	0
$208$	1176.17	0.392081
$209$	−136.821	−0.0452829
$210$	0	0
$211$	−3972.74	−1.29618	−0.648091	−	0.761563i	$-0.724432\pi$
$211$	−3972.74	−1.29618	−0.648091	+	0.761563i	$0.724432\pi$
$212$	−911.423	−0.295268
$213$	0	0
$214$	−5328.84	−1.70221
$215$	−7958.56	−2.52451
$216$	0	0
$217$	−1383.69	−0.432863
$218$	2478.40	0.769992
$219$	0	0
$220$	508.353	0.155787
$221$	−251.611	−0.0765847
$222$	0	0
$223$	3347.58	1.00525	0.502624	−	0.864505i	$-0.332368\pi$
$223$	3347.58	1.00525	0.502624	+	0.864505i	$0.332368\pi$
$224$	996.982	0.297382
$225$	0	0
$226$	−4645.91	−1.36744
$227$	−2809.86	−0.821572	−0.410786	−	0.911732i	$-0.634746\pi$
$227$	−2809.86	−0.821572	−0.410786	+	0.911732i	$0.634746\pi$
$228$	0	0
$229$	5978.12	1.72509	0.862544	−	0.505981i	$-0.168870\pi$
$229$	5978.12	1.72509	0.862544	+	0.505981i	$0.168870\pi$
$230$	12176.8	3.49092
$231$	0	0
$232$	−494.701	−0.139995
$233$	2877.85	0.809160	0.404580	−	0.914503i	$-0.367418\pi$
$233$	2877.85	0.809160	0.404580	+	0.914503i	$0.367418\pi$
$234$	0	0
$235$	6380.65	1.77118
$236$	2776.73	0.765888
$237$	0	0
$238$	−317.955	−0.0865966
$239$	836.275	0.226335	0.113168	−	0.993576i	$-0.463900\pi$
$239$	836.275	0.226335	0.113168	+	0.993576i	$0.463900\pi$
$240$	0	0
$241$	1336.21	0.357148	0.178574	−	0.983926i	$-0.442852\pi$
$241$	1336.21	0.357148	0.178574	+	0.983926i	$0.442852\pi$
$242$	−4651.98	−1.23571
$243$	0	0
$244$	−1140.63	−0.299268
$245$	6508.79	1.69727
$246$	0	0
$247$	−388.667	−0.100123
$248$	−3079.83	−0.788587
$249$	0	0
$250$	−12920.4	−3.26862
$251$	5582.36	1.40381	0.701903	−	0.712273i	$-0.252334\pi$
$251$	5582.36	1.40381	0.701903	+	0.712273i	$0.252334\pi$
$252$	0	0
$253$	861.998	0.214203
$254$	−7492.55	−1.85088
$255$	0	0
$256$	5225.96	1.27587
$257$	−4627.76	−1.12324	−0.561618	−	0.827397i	$-0.689821\pi$
$257$	−4627.76	−1.12324	−0.561618	+	0.827397i	$0.689821\pi$
$258$	0	0
$259$	1689.11	0.405235
$260$	1444.07	0.344453
$261$	0	0
$262$	3436.78	0.810401
$263$	−399.519	−0.0936708	−0.0468354	−	0.998903i	$-0.514914\pi$
$263$	−399.519	−0.0936708	−0.0468354	+	0.998903i	$0.514914\pi$
$264$	0	0
$265$	3975.20	0.921488
$266$	−491.149	−0.113212
$267$	0	0
$268$	971.466	0.221424
$269$	2204.10	0.499577	0.249789	−	0.968300i	$-0.419639\pi$
$269$	2204.10	0.499577	0.249789	+	0.968300i	$0.419639\pi$
$270$	0	0
$271$	−2363.70	−0.529833	−0.264916	−	0.964271i	$-0.585344\pi$
$271$	−2363.70	−0.529833	−0.264916	+	0.964271i	$0.585344\pi$
$272$	−1350.95	−0.301152
$273$	0	0
$274$	−1112.36	−0.245255
$275$	−1565.92	−0.343376
$276$	0	0
$277$	−3988.65	−0.865179	−0.432589	−	0.901591i	$-0.642400\pi$
$277$	−3988.65	−0.865179	−0.432589	+	0.901591i	$0.642400\pi$
$278$	−3707.19	−0.799793
$279$	0	0
$280$	−1261.76	−0.269302
$281$	1494.96	0.317373	0.158686	−	0.987329i	$-0.449274\pi$
$281$	1494.96	0.317373	0.158686	+	0.987329i	$0.449274\pi$
$282$	0	0
$283$	5973.29	1.25468	0.627341	−	0.778744i	$-0.284143\pi$
$283$	5973.29	1.25468	0.627341	+	0.778744i	$0.284143\pi$
$284$	−4412.99	−0.922052
$285$	0	0
$286$	275.136	0.0568851
$287$	1685.50	0.346661
$288$	0	0
$289$	289.000	0.0588235
$290$	−3120.55	−0.631879
$291$	0	0
$292$	−4114.26	−0.824551
$293$	5706.31	1.13777	0.568884	−	0.822417i	$-0.307375\pi$
$293$	5706.31	1.13777	0.568884	+	0.822417i	$0.307375\pi$
$294$	0	0
$295$	−12110.8	−2.39023
$296$	3759.62	0.738254
$297$	0	0
$298$	−1338.72	−0.260235
$299$	2448.67	0.473613
$300$	0	0
$301$	−2022.41	−0.387275
$302$	−9285.73	−1.76932
$303$	0	0
$304$	−2086.82	−0.393709
$305$	4974.89	0.933973
$306$	0	0
$307$	4684.64	0.870900	0.435450	−	0.900213i	$-0.356589\pi$
$307$	4684.64	0.870900	0.435450	+	0.900213i	$0.356589\pi$
$308$	129.181	0.0238987
$309$	0	0
$310$	−19427.4	−3.55936
$311$	−4086.74	−0.745138	−0.372569	−	0.928005i	$-0.621523\pi$
$311$	−4086.74	−0.745138	−0.372569	+	0.928005i	$0.621523\pi$
$312$	0	0
$313$	6893.79	1.24492	0.622460	−	0.782651i	$-0.286133\pi$
$313$	6893.79	1.24492	0.622460	+	0.782651i	$0.286133\pi$
$314$	−11184.4	−2.01011
$315$	0	0
$316$	483.193	0.0860182
$317$	−4907.72	−0.869542	−0.434771	−	0.900541i	$-0.643171\pi$
$317$	−4907.72	−0.869542	−0.434771	+	0.900541i	$0.643171\pi$
$318$	0	0
$319$	−220.905	−0.0387721
$320$	883.330	0.154311
$321$	0	0
$322$	3094.33	0.535529
$323$	446.422	0.0769027
$324$	0	0
$325$	−4448.29	−0.759220
$326$	−188.858	−0.0320855
$327$	0	0
$328$	3751.59	0.631545
$329$	1621.44	0.271710
$330$	0	0
$331$	−8884.92	−1.47541	−0.737703	−	0.675126i	$-0.764089\pi$
$331$	−8884.92	−1.47541	−0.737703	+	0.675126i	$0.764089\pi$
$332$	−1413.65	−0.233687
$333$	0	0
$334$	12683.0	2.07780
$335$	−4237.07	−0.691033
$336$	0	0
$337$	−5398.17	−0.872572	−0.436286	−	0.899808i	$-0.643706\pi$
$337$	−5398.17	−0.872572	−0.436286	+	0.899808i	$0.643706\pi$
$338$	−7057.04	−1.13566
$339$	0	0
$340$	−1658.66	−0.264569
$341$	−1375.27	−0.218402
$342$	0	0
$343$	3452.05	0.543420
$344$	−4501.49	−0.705534
$345$	0	0
$346$	−478.999	−0.0744253
$347$	−6120.24	−0.946835	−0.473417	−	0.880838i	$-0.656980\pi$
$347$	−6120.24	−0.946835	−0.473417	+	0.880838i	$0.656980\pi$
$348$	0	0
$349$	−4211.65	−0.645972	−0.322986	−	0.946404i	$-0.604687\pi$
$349$	−4211.65	−0.645972	−0.322986	+	0.946404i	$0.604687\pi$
$350$	−5621.20	−0.858473
$351$	0	0
$352$	990.915	0.150045
$353$	982.908	0.148201	0.0741005	−	0.997251i	$-0.476391\pi$
$353$	982.908	0.148201	0.0741005	+	0.997251i	$0.476391\pi$
$354$	0	0
$355$	19247.4	2.87759
$356$	3150.58	0.469046
$357$	0	0
$358$	8451.57	1.24771
$359$	−12216.9	−1.79605	−0.898027	−	0.439940i	$-0.855000\pi$
$359$	−12216.9	−1.79605	−0.898027	+	0.439940i	$0.855000\pi$
$360$	0	0
$361$	−6169.41	−0.899462
$362$	4712.42	0.684197
$363$	0	0
$364$	366.965	0.0528411
$365$	17944.5	2.57330
$366$	0	0
$367$	−3054.02	−0.434383	−0.217192	−	0.976129i	$-0.569690\pi$
$367$	−3054.02	−0.434383	−0.217192	+	0.976129i	$0.569690\pi$
$368$	13147.4	1.86237
$369$	0	0
$370$	23715.5	3.33218
$371$	1010.17	0.141362
$372$	0	0
$373$	5547.11	0.770023	0.385012	−	0.922912i	$-0.374197\pi$
$373$	5547.11	0.770023	0.385012	+	0.922912i	$0.374197\pi$
$374$	−316.021	−0.0436926
$375$	0	0
$376$	3609.00	0.495000
$377$	−627.523	−0.0857269
$378$	0	0
$379$	3793.70	0.514168	0.257084	−	0.966389i	$-0.417238\pi$
$379$	3793.70	0.514168	0.257084	+	0.966389i	$0.417238\pi$
$380$	−2562.15	−0.345883
$381$	0	0
$382$	1960.67	0.262609
$383$	−10645.4	−1.42024	−0.710122	−	0.704079i	$-0.751360\pi$
$383$	−10645.4	−1.42024	−0.710122	+	0.704079i	$0.751360\pi$
$384$	0	0
$385$	−563.428	−0.0745843
$386$	14953.6	1.97181
$387$	0	0
$388$	6392.67	0.836440
$389$	−551.008	−0.0718180	−0.0359090	−	0.999355i	$-0.511433\pi$
$389$	−551.008	−0.0718180	−0.0359090	+	0.999355i	$0.511433\pi$
$390$	0	0
$391$	−2812.54	−0.363775
$392$	3681.48	0.474343
$393$	0	0
$394$	14140.8	1.80813
$395$	−2107.46	−0.268450
$396$	0	0
$397$	−12508.7	−1.58135	−0.790674	−	0.612238i	$-0.790269\pi$
$397$	−12508.7	−1.58135	−0.790674	+	0.612238i	$0.790269\pi$
$398$	−16424.8	−2.06860
$399$	0	0
$400$	−23883.7	−2.98546
$401$	9724.10	1.21097	0.605484	−	0.795857i	$-0.292979\pi$
$401$	9724.10	1.21097	0.605484	+	0.795857i	$0.292979\pi$
$402$	0	0
$403$	−3906.73	−0.482898
$404$	7701.03	0.948367
$405$	0	0
$406$	−792.986	−0.0969341
$407$	1678.83	0.204463
$408$	0	0
$409$	−1698.84	−0.205385	−0.102693	−	0.994713i	$-0.532746\pi$
$409$	−1698.84	−0.205385	−0.102693	+	0.994713i	$0.532746\pi$
$410$	23664.8	2.85054
$411$	0	0
$412$	229.095	0.0273949
$413$	−3077.56	−0.366675
$414$	0	0
$415$	6165.66	0.729302
$416$	2814.88	0.331757
$417$	0	0
$418$	−488.161	−0.0571213
$419$	−2663.86	−0.310592	−0.155296	−	0.987868i	$-0.549633\pi$
$419$	−2663.86	−0.310592	−0.155296	+	0.987868i	$0.549633\pi$
$420$	0	0
$421$	776.756	0.0899211	0.0449606	−	0.998989i	$-0.485684\pi$
$421$	776.756	0.0899211	0.0449606	+	0.998989i	$0.485684\pi$
$422$	−14174.2	−1.63505
$423$	0	0
$424$	2248.43	0.257532
$425$	5109.29	0.583146
$426$	0	0
$427$	1264.21	0.143277
$428$	−7064.12	−0.797798
$429$	0	0
$430$	−28395.1	−3.18450
$431$	5951.70	0.665158	0.332579	−	0.943075i	$-0.392081\pi$
$431$	5951.70	0.665158	0.332579	+	0.943075i	$0.392081\pi$
$432$	0	0
$433$	154.417	0.0171382	0.00856908	−	0.999963i	$-0.497272\pi$
$433$	154.417	0.0171382	0.00856908	+	0.999963i	$0.497272\pi$
$434$	−4936.84	−0.546028
$435$	0	0
$436$	3285.46	0.360883
$437$	−4344.56	−0.475580
$438$	0	0
$439$	−12859.5	−1.39806	−0.699031	−	0.715091i	$-0.746385\pi$
$439$	−12859.5	−1.39806	−0.699031	+	0.715091i	$0.746385\pi$
$440$	−1254.08	−0.135877
$441$	0	0
$442$	−897.717	−0.0966065
$443$	16428.4	1.76193	0.880967	−	0.473177i	$-0.156893\pi$
$443$	16428.4	1.76193	0.880967	+	0.473177i	$0.156893\pi$
$444$	0	0
$445$	−13741.3	−1.46383
$446$	11943.7	1.26805
$447$	0	0
$448$	224.470	0.0236723
$449$	700.731	0.0736516	0.0368258	−	0.999322i	$-0.488275\pi$
$449$	700.731	0.0736516	0.0368258	+	0.999322i	$0.488275\pi$
$450$	0	0
$451$	1675.24	0.174909
$452$	−6158.80	−0.640898
$453$	0	0
$454$	−10025.2	−1.03636
$455$	−1600.53	−0.164910
$456$	0	0
$457$	3127.81	0.320159	0.160080	−	0.987104i	$-0.448825\pi$
$457$	3127.81	0.320159	0.160080	+	0.987104i	$0.448825\pi$
$458$	21329.2	2.17608
$459$	0	0
$460$	16142.0	1.63614
$461$	12198.4	1.23240	0.616200	−	0.787590i	$-0.288671\pi$
$461$	12198.4	1.23240	0.616200	+	0.787590i	$0.288671\pi$
$462$	0	0
$463$	−5054.14	−0.507313	−0.253657	−	0.967294i	$-0.581633\pi$
$463$	−5054.14	−0.507313	−0.253657	+	0.967294i	$0.581633\pi$
$464$	−3369.28	−0.337102
$465$	0	0
$466$	10267.8	1.02070
$467$	−8366.25	−0.829002	−0.414501	−	0.910049i	$-0.636044\pi$
$467$	−8366.25	−0.829002	−0.414501	+	0.910049i	$0.636044\pi$
$468$	0	0
$469$	−1076.71	−0.106009
$470$	22765.4	2.23423
$471$	0	0
$472$	−6850.05	−0.668006
$473$	−2010.10	−0.195401
$474$	0	0
$475$	7892.38	0.762373
$476$	−421.494	−0.0405865
$477$	0	0
$478$	2983.72	0.285507
$479$	3942.17	0.376038	0.188019	−	0.982165i	$-0.439793\pi$
$479$	3942.17	0.376038	0.188019	+	0.982165i	$0.439793\pi$
$480$	0	0
$481$	4769.03	0.452077
$482$	4767.42	0.450519
$483$	0	0
$484$	−6166.86	−0.579156
$485$	−27881.8	−2.61041
$486$	0	0
$487$	160.598	0.0149433	0.00747165	−	0.999972i	$-0.497622\pi$
$487$	160.598	0.0149433	0.00747165	+	0.999972i	$0.497622\pi$
$488$	2813.88	0.261021
$489$	0	0
$490$	23222.5	2.14099
$491$	−17380.5	−1.59750	−0.798748	−	0.601666i	$-0.794504\pi$
$491$	−17380.5	−1.59750	−0.798748	+	0.601666i	$0.794504\pi$
$492$	0	0
$493$	720.770	0.0658456
$494$	−1386.71	−0.126298
$495$	0	0
$496$	−20976.0	−1.89889
$497$	4891.09	0.441440
$498$	0	0
$499$	17648.5	1.58328	0.791640	−	0.610988i	$-0.209228\pi$
$499$	17648.5	1.58328	0.791640	+	0.610988i	$0.209228\pi$
$500$	−17127.8	−1.53195
$501$	0	0
$502$	19917.1	1.77081
$503$	−2367.22	−0.209839	−0.104920	−	0.994481i	$-0.533459\pi$
$503$	−2367.22	−0.209839	−0.104920	+	0.994481i	$0.533459\pi$
$504$	0	0
$505$	−33588.2	−2.95972
$506$	3075.50	0.270203
$507$	0	0
$508$	−9932.42	−0.867480
$509$	−13983.5	−1.21770	−0.608850	−	0.793286i	$-0.708369\pi$
$509$	−13983.5	−1.21770	−0.608850	+	0.793286i	$0.708369\pi$
$510$	0	0
$511$	4560.00	0.394760
$512$	7695.84	0.664280
$513$	0	0
$514$	−16511.2	−1.41689
$515$	−999.205	−0.0854956
$516$	0	0
$517$	1611.57	0.137092
$518$	6026.51	0.511177
$519$	0	0
$520$	−3562.46	−0.300431
$521$	14012.1	1.17828	0.589139	−	0.808032i	$-0.299467\pi$
$521$	14012.1	1.17828	0.589139	+	0.808032i	$0.299467\pi$
$522$	0	0
$523$	−6665.75	−0.557310	−0.278655	−	0.960391i	$-0.589889\pi$
$523$	−6665.75	−0.557310	−0.278655	+	0.960391i	$0.589889\pi$
$524$	4555.93	0.379822
$525$	0	0
$526$	−1425.43	−0.118159
$527$	4487.26	0.370907
$528$	0	0
$529$	15204.5	1.24965
$530$	14183.0	1.16240
$531$	0	0
$532$	−651.087	−0.0530606
$533$	4758.85	0.386733
$534$	0	0
$535$	30810.4	2.48981
$536$	−2396.56	−0.193126
$537$	0	0
$538$	7863.94	0.630183
$539$	1643.93	0.131371
$540$	0	0
$541$	−2658.37	−0.211261	−0.105631	−	0.994405i	$-0.533686\pi$
$541$	−2658.37	−0.211261	−0.105631	+	0.994405i	$0.533686\pi$
$542$	−8433.39	−0.668349
$543$	0	0
$544$	−3233.17	−0.254818
$545$	−14329.6	−1.12626
$546$	0	0
$547$	−20140.6	−1.57431	−0.787156	−	0.616755i	$-0.788447\pi$
$547$	−20140.6	−1.57431	−0.787156	+	0.616755i	$0.788447\pi$
$548$	−1474.59	−0.114947
$549$	0	0
$550$	−5586.99	−0.433146
$551$	1113.38	0.0860829
$552$	0	0
$553$	−535.543	−0.0411819
$554$	−14231.0	−1.09137
$555$	0	0
$556$	−4914.40	−0.374851
$557$	10449.2	0.794874	0.397437	−	0.917629i	$-0.369900\pi$
$557$	10449.2	0.794874	0.397437	+	0.917629i	$0.369900\pi$
$558$	0	0
$559$	−5710.08	−0.432041
$560$	−8593.52	−0.648469
$561$	0	0
$562$	5333.82	0.400345
$563$	−6606.44	−0.494544	−0.247272	−	0.968946i	$-0.579534\pi$
$563$	−6606.44	−0.494544	−0.247272	+	0.968946i	$0.579534\pi$
$564$	0	0
$565$	26861.8	2.00015
$566$	21311.9	1.58270
$567$	0	0
$568$	10886.6	0.804212
$569$	6016.94	0.443310	0.221655	−	0.975125i	$-0.428854\pi$
$569$	6016.94	0.443310	0.221655	+	0.975125i	$0.428854\pi$
$570$	0	0
$571$	−10560.4	−0.773974	−0.386987	−	0.922085i	$-0.626484\pi$
$571$	−10560.4	−0.773974	−0.386987	+	0.922085i	$0.626484\pi$
$572$	364.731	0.0266612
$573$	0	0
$574$	6013.64	0.437290
$575$	−49723.4	−3.60627
$576$	0	0
$577$	−9374.56	−0.676374	−0.338187	−	0.941079i	$-0.609814\pi$
$577$	−9374.56	−0.676374	−0.338187	+	0.941079i	$0.609814\pi$
$578$	1031.12	0.0742020
$579$	0	0
$580$	−4136.72	−0.296152
$581$	1566.80	0.111879
$582$	0	0
$583$	1004.02	0.0713246
$584$	10149.7	0.719172
$585$	0	0
$586$	20359.4	1.43522
$587$	−16742.5	−1.17724	−0.588618	−	0.808411i	$-0.700328\pi$
$587$	−16742.5	−1.17724	−0.588618	+	0.808411i	$0.700328\pi$
$588$	0	0
$589$	6931.52	0.484904
$590$	−43209.7	−3.01511
$591$	0	0
$592$	25605.8	1.77769
$593$	−16820.2	−1.16480	−0.582398	−	0.812904i	$-0.697885\pi$
$593$	−16820.2	−1.16480	−0.582398	+	0.812904i	$0.697885\pi$
$594$	0	0
$595$	1838.36	0.126665
$596$	−1774.66	−0.121968
$597$	0	0
$598$	8736.54	0.597431
$599$	−16523.6	−1.12711	−0.563553	−	0.826080i	$-0.690566\pi$
$599$	−16523.6	−1.12711	−0.563553	+	0.826080i	$0.690566\pi$
$600$	0	0
$601$	1133.13	0.0769075	0.0384537	−	0.999260i	$-0.487757\pi$
$601$	1133.13	0.0769075	0.0384537	+	0.999260i	$0.487757\pi$
$602$	−7215.70	−0.488521
$603$	0	0
$604$	−12309.5	−0.829252
$605$	26896.9	1.80746
$606$	0	0
$607$	210.365	0.0140667	0.00703333	−	0.999975i	$-0.497761\pi$
$607$	210.365	0.0140667	0.00703333	+	0.999975i	$0.497761\pi$
$608$	−4994.31	−0.333135
$609$	0	0
$610$	17749.8	1.17814
$611$	4577.97	0.303118
$612$	0	0
$613$	−4813.64	−0.317163	−0.158582	−	0.987346i	$-0.550692\pi$
$613$	−4813.64	−0.317163	−0.158582	+	0.987346i	$0.550692\pi$
$614$	16714.2	1.09858
$615$	0	0
$616$	−318.684	−0.0208444
$617$	−19782.2	−1.29076	−0.645381	−	0.763861i	$-0.723301\pi$
$617$	−19782.2	−1.29076	−0.645381	+	0.763861i	$0.723301\pi$
$618$	0	0
$619$	6340.53	0.411709	0.205854	−	0.978583i	$-0.434003\pi$
$619$	6340.53	0.411709	0.205854	+	0.978583i	$0.434003\pi$
$620$	−25753.8	−1.66822
$621$	0	0
$622$	−14581.0	−0.939942
$623$	−3491.92	−0.224560
$624$	0	0
$625$	37134.8	2.37663
$626$	24596.2	1.57038
$627$	0	0
$628$	−14826.5	−0.942108
$629$	−5477.69	−0.347233
$630$	0	0
$631$	−4591.83	−0.289696	−0.144848	−	0.989454i	$-0.546269\pi$
$631$	−4591.83	−0.289696	−0.144848	+	0.989454i	$0.546269\pi$
$632$	−1192.01	−0.0750249
$633$	0	0
$634$	−17510.1	−1.09687
$635$	43320.5	2.70728
$636$	0	0
$637$	4669.91	0.290469
$638$	−788.160	−0.0489084
$639$	0	0
$640$	−28234.9	−1.74388
$641$	3018.43	0.185992	0.0929961	−	0.995666i	$-0.470356\pi$
$641$	3018.43	0.185992	0.0929961	+	0.995666i	$0.470356\pi$
$642$	0	0
$643$	−25834.1	−1.58444	−0.792222	−	0.610233i	$-0.791076\pi$
$643$	−25834.1	−1.58444	−0.792222	+	0.610233i	$0.791076\pi$
$644$	4101.97	0.250994
$645$	0	0
$646$	1592.78	0.0970076
$647$	−4123.41	−0.250553	−0.125277	−	0.992122i	$-0.539982\pi$
$647$	−4123.41	−0.250553	−0.125277	+	0.992122i	$0.539982\pi$
$648$	0	0
$649$	−3058.83	−0.185007
$650$	−15870.9	−0.957706
$651$	0	0
$652$	−250.358	−0.0150380
$653$	18013.5	1.07951	0.539757	−	0.841821i	$-0.318516\pi$
$653$	18013.5	1.07951	0.539757	+	0.841821i	$0.318516\pi$
$654$	0	0
$655$	−19870.8	−1.18537
$656$	25551.1	1.52074
$657$	0	0
$658$	5785.08	0.342744
$659$	−5886.59	−0.347965	−0.173982	−	0.984749i	$-0.555664\pi$
$659$	−5886.59	−0.347965	−0.173982	+	0.984749i	$0.555664\pi$
$660$	0	0
$661$	−1099.06	−0.0646722	−0.0323361	−	0.999477i	$-0.510295\pi$
$661$	−1099.06	−0.0646722	−0.0323361	+	0.999477i	$0.510295\pi$
$662$	−31700.2	−1.86113
$663$	0	0
$664$	3487.39	0.203821
$665$	2839.74	0.165594
$666$	0	0
$667$	−7014.51	−0.407201
$668$	16813.1	0.973832
$669$	0	0
$670$	−15117.3	−0.871692
$671$	1256.51	0.0722909
$672$	0	0
$673$	−5682.62	−0.325481	−0.162741	−	0.986669i	$-0.552033\pi$
$673$	−5682.62	−0.325481	−0.162741	+	0.986669i	$0.552033\pi$
$674$	−19260.0	−1.10069
$675$	0	0
$676$	−9355.09	−0.532265
$677$	4491.71	0.254993	0.127497	−	0.991839i	$-0.459306\pi$
$677$	4491.71	0.254993	0.127497	+	0.991839i	$0.459306\pi$
$678$	0	0
$679$	−7085.25	−0.400452
$680$	4091.83	0.230757
$681$	0	0
$682$	−4906.80	−0.275500
$683$	11533.2	0.646131	0.323065	−	0.946377i	$-0.395287\pi$
$683$	11533.2	0.646131	0.323065	+	0.946377i	$0.395287\pi$
$684$	0	0
$685$	6431.45	0.358734
$686$	12316.5	0.685489
$687$	0	0
$688$	−30658.5	−1.69890
$689$	2852.11	0.157702
$690$	0	0
$691$	28818.2	1.58654	0.793268	−	0.608873i	$-0.208378\pi$
$691$	28818.2	1.58654	0.793268	+	0.608873i	$0.208378\pi$
$692$	−634.980	−0.0348820
$693$	0	0
$694$	−21836.2	−1.19437
$695$	21434.3	1.16985
$696$	0	0
$697$	−5466.00	−0.297044
$698$	−15026.6	−0.814851
$699$	0	0
$700$	−7451.69	−0.402353
$701$	−16494.8	−0.888730	−0.444365	−	0.895846i	$-0.646571\pi$
$701$	−16494.8	−0.888730	−0.444365	+	0.895846i	$0.646571\pi$
$702$	0	0
$703$	−8461.45	−0.453954
$704$	223.104	0.0119439
$705$	0	0
$706$	3506.89	0.186946
$707$	−8535.36	−0.454038
$708$	0	0
$709$	26363.9	1.39650	0.698248	−	0.715856i	$-0.253963\pi$
$709$	26363.9	1.39650	0.698248	+	0.715856i	$0.253963\pi$
$710$	68672.2	3.62989
$711$	0	0
$712$	−7772.32	−0.409101
$713$	−43669.8	−2.29375
$714$	0	0
$715$	−1590.79	−0.0832056
$716$	11203.7	0.584781
$717$	0	0
$718$	−43588.3	−2.26560
$719$	15410.9	0.799348	0.399674	−	0.916657i	$-0.369123\pi$
$719$	15410.9	0.799348	0.399674	+	0.916657i	$0.369123\pi$
$720$	0	0
$721$	−253.915	−0.0131155
$722$	−22011.7	−1.13461
$723$	0	0
$724$	6246.97	0.320672
$725$	12742.6	0.652759
$726$	0	0
$727$	−3399.02	−0.173401	−0.0867005	−	0.996234i	$-0.527632\pi$
$727$	−3399.02	−0.173401	−0.0867005	+	0.996234i	$0.527632\pi$
$728$	−905.283	−0.0460879
$729$	0	0
$730$	64023.5	3.24605
$731$	6558.58	0.331844
$732$	0	0
$733$	−1731.67	−0.0872585	−0.0436293	−	0.999048i	$-0.513892\pi$
$733$	−1731.67	−0.0872585	−0.0436293	+	0.999048i	$0.513892\pi$
$734$	−10896.4	−0.547945
$735$	0	0
$736$	31465.0	1.57584
$737$	−1070.16	−0.0534870
$738$	0	0
$739$	−34116.1	−1.69821	−0.849107	−	0.528221i	$-0.822859\pi$
$739$	−34116.1	−1.69821	−0.849107	+	0.528221i	$0.822859\pi$
$740$	31438.2	1.56174
$741$	0	0
$742$	3604.15	0.178319
$743$	−16893.3	−0.834126	−0.417063	−	0.908878i	$-0.636941\pi$
$743$	−16893.3	−0.834126	−0.417063	+	0.908878i	$0.636941\pi$
$744$	0	0
$745$	7740.24	0.380645
$746$	19791.4	0.971333
$747$	0	0
$748$	−418.929	−0.0204780
$749$	7829.45	0.381952
$750$	0	0
$751$	32492.2	1.57877	0.789385	−	0.613898i	$-0.210399\pi$
$751$	32492.2	1.57877	0.789385	+	0.613898i	$0.210399\pi$
$752$	24580.0	1.19194
$753$	0	0
$754$	−2238.92	−0.108139
$755$	53688.4	2.58797
$756$	0	0
$757$	31568.5	1.51569	0.757845	−	0.652434i	$-0.226252\pi$
$757$	31568.5	1.51569	0.757845	+	0.652434i	$0.226252\pi$
$758$	13535.5	0.648588
$759$	0	0
$760$	6320.69	0.301679
$761$	−21878.0	−1.04215	−0.521076	−	0.853510i	$-0.674469\pi$
$761$	−21878.0	−1.04215	−0.521076	+	0.853510i	$0.674469\pi$
$762$	0	0
$763$	−3641.41	−0.172776
$764$	2599.14	0.123081
$765$	0	0
$766$	−37981.4	−1.79154
$767$	−8689.20	−0.409060
$768$	0	0
$769$	12510.4	0.586653	0.293327	−	0.956012i	$-0.405238\pi$
$769$	12510.4	0.586653	0.293327	+	0.956012i	$0.405238\pi$
$770$	−2010.24	−0.0940831
$771$	0	0
$772$	19823.1	0.924157
$773$	−3483.57	−0.162090	−0.0810448	−	0.996710i	$-0.525826\pi$
$773$	−3483.57	−0.162090	−0.0810448	+	0.996710i	$0.525826\pi$
$774$	0	0
$775$	79331.1	3.67698
$776$	−15770.4	−0.729541
$777$	0	0
$778$	−1965.93	−0.0905936
$779$	−8443.39	−0.388338
$780$	0	0
$781$	4861.33	0.222730
$782$	−10034.8	−0.458878
$783$	0	0
$784$	25073.6	1.14220
$785$	64666.4	2.94018
$786$	0	0
$787$	−21827.4	−0.988643	−0.494322	−	0.869279i	$-0.664584\pi$
$787$	−21827.4	−0.988643	−0.494322	+	0.869279i	$0.664584\pi$
$788$	18745.6	0.847441
$789$	0	0
$790$	−7519.15	−0.338632
$791$	6826.05	0.306835
$792$	0	0
$793$	3569.37	0.159839
$794$	−44629.5	−1.99476
$795$	0	0
$796$	−21773.4	−0.969520
$797$	−31873.4	−1.41658	−0.708289	−	0.705922i	$-0.750533\pi$
$797$	−31873.4	−1.41658	−0.708289	+	0.705922i	$0.750533\pi$
$798$	0	0
$799$	−5258.24	−0.232820
$800$	−57159.8	−2.52613
$801$	0	0
$802$	34694.3	1.52756
$803$	4532.25	0.199178
$804$	0	0
$805$	−17890.8	−0.783316
$806$	−13938.7	−0.609144
$807$	0	0
$808$	−18998.0	−0.827164
$809$	19243.8	0.836312	0.418156	−	0.908375i	$-0.362677\pi$
$809$	19243.8	0.836312	0.418156	+	0.908375i	$0.362677\pi$
$810$	0	0
$811$	8969.81	0.388376	0.194188	−	0.980964i	$-0.437793\pi$
$811$	8969.81	0.388376	0.194188	+	0.980964i	$0.437793\pi$
$812$	−1051.21	−0.0454315
$813$	0	0
$814$	5989.84	0.257916
$815$	1091.94	0.0469313
$816$	0	0
$817$	10131.1	0.433835
$818$	−6061.26	−0.259080
$819$	0	0
$820$	31371.0	1.33600
$821$	−3644.00	−0.154904	−0.0774522	−	0.996996i	$-0.524679\pi$
$821$	−3644.00	−0.154904	−0.0774522	+	0.996996i	$0.524679\pi$
$822$	0	0
$823$	−7219.82	−0.305793	−0.152896	−	0.988242i	$-0.548860\pi$
$823$	−7219.82	−0.305793	−0.152896	+	0.988242i	$0.548860\pi$
$824$	−565.166	−0.0238938
$825$	0	0
$826$	−10980.3	−0.462536
$827$	−5356.59	−0.225232	−0.112616	−	0.993639i	$-0.535923\pi$
$827$	−5356.59	−0.225232	−0.112616	+	0.993639i	$0.535923\pi$
$828$	0	0
$829$	−14161.8	−0.593319	−0.296659	−	0.954983i	$-0.595873\pi$
$829$	−14161.8	−0.593319	−0.296659	+	0.954983i	$0.595873\pi$
$830$	21998.3	0.919966
$831$	0	0
$832$	633.769	0.0264086
$833$	−5363.84	−0.223105
$834$	0	0
$835$	−73330.9	−3.03919
$836$	−647.125	−0.0267719
$837$	0	0
$838$	−9504.30	−0.391791
$839$	12912.5	0.531333	0.265667	−	0.964065i	$-0.414408\pi$
$839$	12912.5	0.531333	0.265667	+	0.964065i	$0.414408\pi$
$840$	0	0
$841$	−22591.4	−0.926294
$842$	2771.37	0.113430
$843$	0	0
$844$	−18789.9	−0.766321
$845$	40802.5	1.66112
$846$	0	0
$847$	6834.97	0.277276
$848$	15313.5	0.620127
$849$	0	0
$850$	18229.3	0.735599
$851$	53308.7	2.14735
$852$	0	0
$853$	−6526.03	−0.261954	−0.130977	−	0.991385i	$-0.541811\pi$
$853$	−6526.03	−0.261954	−0.130977	+	0.991385i	$0.541811\pi$
$854$	4510.53	0.180734
$855$	0	0
$856$	17426.8	0.695837
$857$	19431.8	0.774536	0.387268	−	0.921967i	$-0.373419\pi$
$857$	19431.8	0.774536	0.387268	+	0.921967i	$0.373419\pi$
$858$	0	0
$859$	8373.86	0.332610	0.166305	−	0.986074i	$-0.446816\pi$
$859$	8373.86	0.332610	0.166305	+	0.986074i	$0.446816\pi$
$860$	−37641.7	−1.49253
$861$	0	0
$862$	21234.9	0.839053
$863$	−13066.9	−0.515412	−0.257706	−	0.966223i	$-0.582967\pi$
$863$	−13066.9	−0.515412	−0.257706	+	0.966223i	$0.582967\pi$
$864$	0	0
$865$	2769.48	0.108862
$866$	550.941	0.0216186
$867$	0	0
$868$	−6544.48	−0.255915
$869$	−532.283	−0.0207785
$870$	0	0
$871$	−3040.00	−0.118262
$872$	−8105.06	−0.314761
$873$	0	0
$874$	−15500.8	−0.599912
$875$	18983.4	0.733435
$876$	0	0
$877$	−40889.1	−1.57438	−0.787188	−	0.616713i	$-0.788464\pi$
$877$	−40889.1	−1.57438	−0.787188	+	0.616713i	$0.788464\pi$
$878$	−45881.0	−1.76356
$879$	0	0
$880$	−8541.22	−0.327187
$881$	−16437.8	−0.628608	−0.314304	−	0.949322i	$-0.601771\pi$
$881$	−16437.8	−0.628608	−0.314304	+	0.949322i	$0.601771\pi$
$882$	0	0
$883$	−37873.1	−1.44341	−0.721705	−	0.692201i	$-0.756641\pi$
$883$	−37873.1	−1.44341	−0.721705	+	0.692201i	$0.756641\pi$
$884$	−1190.05	−0.0452780
$885$	0	0
$886$	58614.5	2.22256
$887$	−35145.6	−1.33041	−0.665205	−	0.746661i	$-0.731656\pi$
$887$	−35145.6	−1.33041	−0.665205	+	0.746661i	$0.731656\pi$
$888$	0	0
$889$	11008.5	0.415313
$890$	−49027.4	−1.84652
$891$	0	0
$892$	15833.1	0.594317
$893$	−8122.47	−0.304376
$894$	0	0
$895$	−48865.4	−1.82502
$896$	−7174.98	−0.267521
$897$	0	0
$898$	2500.12	0.0929066
$899$	11191.3	0.415184
$900$	0	0
$901$	−3275.93	−0.121129
$902$	5977.05	0.220636
$903$	0	0
$904$	15193.4	0.558990
$905$	−27246.3	−1.00077
$906$	0	0
$907$	4884.66	0.178823	0.0894116	−	0.995995i	$-0.471501\pi$
$907$	4884.66	0.178823	0.0894116	+	0.995995i	$0.471501\pi$
$908$	−13289.8	−0.485725
$909$	0	0
$910$	−5710.47	−0.208022
$911$	−44687.5	−1.62521	−0.812603	−	0.582818i	$-0.801950\pi$
$911$	−44687.5	−1.62521	−0.812603	+	0.582818i	$0.801950\pi$
$912$	0	0
$913$	1557.27	0.0564491
$914$	11159.6	0.403860
$915$	0	0
$916$	28274.8	1.01990
$917$	−5049.52	−0.181843
$918$	0	0
$919$	43322.8	1.55505	0.777524	−	0.628853i	$-0.216475\pi$
$919$	43322.8	1.55505	0.777524	+	0.628853i	$0.216475\pi$
$920$	−39821.5	−1.42704
$921$	0	0
$922$	43522.3	1.55459
$923$	13809.5	0.492467
$924$	0	0
$925$	−96841.2	−3.44229
$926$	−18032.5	−0.639942
$927$	0	0
$928$	−8063.57	−0.285237
$929$	31035.2	1.09605	0.548025	−	0.836462i	$-0.315380\pi$
$929$	31035.2	1.09605	0.548025	+	0.836462i	$0.315380\pi$
$930$	0	0
$931$	−8285.59	−0.291675
$932$	13611.4	0.478387
$933$	0	0
$934$	−29849.7	−1.04573
$935$	1827.17	0.0639090
$936$	0	0
$937$	−28794.9	−1.00394	−0.501968	−	0.864886i	$-0.667390\pi$
$937$	−28794.9	−1.00394	−0.501968	+	0.864886i	$0.667390\pi$
$938$	−3841.58	−0.133723
$939$	0	0
$940$	30178.7	1.04715
$941$	−11949.5	−0.413966	−0.206983	−	0.978344i	$-0.566365\pi$
$941$	−11949.5	−0.413966	−0.206983	+	0.978344i	$0.566365\pi$
$942$	0	0
$943$	53194.8	1.83697
$944$	−46653.9	−1.60853
$945$	0	0
$946$	−7171.79	−0.246485
$947$	20143.1	0.691197	0.345599	−	0.938382i	$-0.387676\pi$
$947$	20143.1	0.691197	0.345599	+	0.938382i	$0.387676\pi$
$948$	0	0
$949$	12874.7	0.440391
$950$	28159.0	0.961683
$951$	0	0
$952$	1039.81	0.0353994
$953$	−36544.6	−1.24218	−0.621090	−	0.783739i	$-0.713310\pi$
$953$	−36544.6	−1.24218	−0.621090	+	0.783739i	$0.713310\pi$
$954$	0	0
$955$	−11336.2	−0.384117
$956$	3955.34	0.133813
$957$	0	0
$958$	14065.2	0.474347
$959$	1634.34	0.0550320
$960$	0	0
$961$	39881.9	1.33872
$962$	17015.3	0.570265
$963$	0	0
$964$	6319.88	0.211151
$965$	−86459.0	−2.88416
$966$	0	0
$967$	−40645.1	−1.35166	−0.675831	−	0.737056i	$-0.736215\pi$
$967$	−40645.1	−1.35166	−0.675831	+	0.737056i	$0.736215\pi$
$968$	15213.3	0.505139
$969$	0	0
$970$	−99478.7	−3.29285
$971$	−711.265	−0.0235073	−0.0117536	−	0.999931i	$-0.503741\pi$
$971$	−711.265	−0.0235073	−0.0117536	+	0.999931i	$0.503741\pi$
$972$	0	0
$973$	5446.83	0.179463
$974$	572.993	0.0188500
$975$	0	0
$976$	19164.6	0.628529
$977$	57865.1	1.89485	0.947425	−	0.319979i	$-0.103676\pi$
$977$	57865.1	1.89485	0.947425	+	0.319979i	$0.103676\pi$
$978$	0	0
$979$	−3470.67	−0.113302
$980$	30784.7	1.00345
$981$	0	0
$982$	−62011.4	−2.01514
$983$	−4983.77	−0.161707	−0.0808534	−	0.996726i	$-0.525765\pi$
$983$	−4983.77	−0.161707	−0.0808534	+	0.996726i	$0.525765\pi$
$984$	0	0
$985$	−81759.4	−2.64474
$986$	2571.62	0.0830598
$987$	0	0
$988$	−1838.28	−0.0591939
$989$	−63827.8	−2.05218
$990$	0	0
$991$	20463.6	0.655952	0.327976	−	0.944686i	$-0.393633\pi$
$991$	20463.6	0.655952	0.327976	+	0.944686i	$0.393633\pi$
$992$	−50200.9	−1.60673
$993$	0	0
$994$	17450.8	0.556847
$995$	94965.3	3.02573
$996$	0	0
$997$	37140.8	1.17980	0.589900	−	0.807476i	$-0.299167\pi$
$997$	37140.8	1.17980	0.589900	+	0.807476i	$0.299167\pi$
$998$	62967.7	1.99720
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.a.h.1.4	✓	4
3.2	odd	2		153.4.a.i.1.1	yes	4
4.3	odd	2		2448.4.a.bo.1.1		4
12.11	even	2		2448.4.a.bs.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.4.a.h.1.4	✓	4	1.1	even	1	trivial
153.4.a.i.1.1	yes	4	3.2	odd	2
2448.4.a.bo.1.1		4	4.3	odd	2
2448.4.a.bs.1.4		4	12.11	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)

\(f(q)\)	\(=\)	\(q+3.56787 q^{2} +4.72971 q^{4} -20.6288 q^{5} -5.24213 q^{7} -11.6680 q^{8} +O(q^{10})\)	\(q+3.56787 q^{2} +4.72971 q^{4} -20.6288 q^{5} -5.24213 q^{7} -11.6680 q^{8} -73.6008 q^{10} -5.21023 q^{11} -14.8007 q^{13} -18.7033 q^{14} -79.4675 q^{16} +17.0000 q^{17} +26.2601 q^{19} -97.5682 q^{20} -18.5894 q^{22} -165.443 q^{23} +300.546 q^{25} -52.8069 q^{26} -24.7938 q^{28} +42.3983 q^{29} +263.956 q^{31} -190.186 q^{32} +60.6538 q^{34} +108.139 q^{35} -322.217 q^{37} +93.6927 q^{38} +240.696 q^{40} -321.529 q^{41} +385.799 q^{43} -24.6429 q^{44} -590.280 q^{46} -309.308 q^{47} -315.520 q^{49} +1072.31 q^{50} -70.0029 q^{52} -192.701 q^{53} +107.481 q^{55} +61.1650 q^{56} +151.272 q^{58} +587.082 q^{59} -241.163 q^{61} +941.763 q^{62} -42.8203 q^{64} +305.320 q^{65} +205.396 q^{67} +80.4052 q^{68} +385.825 q^{70} -933.035 q^{71} -869.875 q^{73} -1149.63 q^{74} +124.203 q^{76} +27.3127 q^{77} +102.161 q^{79} +1639.32 q^{80} -1147.17 q^{82} -298.886 q^{83} -350.689 q^{85} +1376.48 q^{86} +60.7928 q^{88} +666.125 q^{89} +77.5871 q^{91} -782.499 q^{92} -1103.57 q^{94} -541.714 q^{95} +1351.60 q^{97} -1125.74 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 24 q^{7} - 102 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 24 * q^7 - 102 * q^8	\( 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 24 q^{7} - 102 q^{8} - 2 q^{10} - 50 q^{11} + 26 q^{13} - 80 q^{14} + 138 q^{16} + 68 q^{17} + 34 q^{19} - 312 q^{20} - 254 q^{22} - 382 q^{23} + 138 q^{25} + 22 q^{26} + 52 q^{28} - 540 q^{29} - 356 q^{31} - 730 q^{32} - 68 q^{34} - 304 q^{35} - 404 q^{37} - 298 q^{38} + 332 q^{40} + 114 q^{41} + 570 q^{43} + 1368 q^{44} - 290 q^{46} - 496 q^{47} - 224 q^{49} + 1862 q^{50} - 1012 q^{52} - 92 q^{53} - 482 q^{55} + 1428 q^{56} + 1324 q^{58} + 48 q^{59} - 1036 q^{61} + 2564 q^{62} + 2898 q^{64} + 342 q^{65} + 812 q^{67} + 442 q^{68} + 152 q^{70} - 1044 q^{71} - 1212 q^{73} + 1444 q^{74} + 2268 q^{76} + 564 q^{77} + 488 q^{79} + 1000 q^{80} - 938 q^{82} - 1708 q^{83} - 374 q^{85} + 2446 q^{86} - 3868 q^{88} + 8 q^{89} + 716 q^{91} - 1356 q^{92} - 1224 q^{94} - 1010 q^{95} - 76 q^{97} + 1472 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 24 * q^7 - 102 * q^8 - 2 * q^10 - 50 * q^11 + 26 * q^13 - 80 * q^14 + 138 * q^16 + 68 * q^17 + 34 * q^19 - 312 * q^20 - 254 * q^22 - 382 * q^23 + 138 * q^25 + 22 * q^26 + 52 * q^28 - 540 * q^29 - 356 * q^31 - 730 * q^32 - 68 * q^34 - 304 * q^35 - 404 * q^37 - 298 * q^38 + 332 * q^40 + 114 * q^41 + 570 * q^43 + 1368 * q^44 - 290 * q^46 - 496 * q^47 - 224 * q^49 + 1862 * q^50 - 1012 * q^52 - 92 * q^53 - 482 * q^55 + 1428 * q^56 + 1324 * q^58 + 48 * q^59 - 1036 * q^61 + 2564 * q^62 + 2898 * q^64 + 342 * q^65 + 812 * q^67 + 442 * q^68 + 152 * q^70 - 1044 * q^71 - 1212 * q^73 + 1444 * q^74 + 2268 * q^76 + 564 * q^77 + 488 * q^79 + 1000 * q^80 - 938 * q^82 - 1708 * q^83 - 374 * q^85 + 2446 * q^86 - 3868 * q^88 + 8 * q^89 + 716 * q^91 - 1356 * q^92 - 1224 * q^94 - 1010 * q^95 - 76 * q^97 + 1472 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more