LMFDB - Embedded newform 153.4.a.i.1.1

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:	$0$
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			1.1
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)

$f(q)$	$=$	\(q-3.56787 q^{2} +4.72971 q^{4} +20.6288 q^{5} -5.24213 q^{7} +11.6680 q^{8} -73.6008 q^{10} +5.21023 q^{11} -14.8007 q^{13} +18.7033 q^{14} -79.4675 q^{16} -17.0000 q^{17} +26.2601 q^{19} +97.5682 q^{20} -18.5894 q^{22} +165.443 q^{23} +300.546 q^{25} +52.8069 q^{26} -24.7938 q^{28} -42.3983 q^{29} +263.956 q^{31} +190.186 q^{32} +60.6538 q^{34} -108.139 q^{35} -322.217 q^{37} -93.6927 q^{38} +240.696 q^{40} +321.529 q^{41} +385.799 q^{43} +24.6429 q^{44} -590.280 q^{46} +309.308 q^{47} -315.520 q^{49} -1072.31 q^{50} -70.0029 q^{52} +192.701 q^{53} +107.481 q^{55} -61.1650 q^{56} +151.272 q^{58} -587.082 q^{59} -241.163 q^{61} -941.763 q^{62} -42.8203 q^{64} -305.320 q^{65} +205.396 q^{67} -80.4052 q^{68} +385.825 q^{70} +933.035 q^{71} -869.875 q^{73} +1149.63 q^{74} +124.203 q^{76} -27.3127 q^{77} +102.161 q^{79} -1639.32 q^{80} -1147.17 q^{82} +298.886 q^{83} -350.689 q^{85} -1376.48 q^{86} +60.7928 q^{88} -666.125 q^{89} +77.5871 q^{91} +782.499 q^{92} -1103.57 q^{94} +541.714 q^{95} +1351.60 q^{97} +1125.74 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 26 q^{4} + 22 q^{5} - 24 q^{7} + 102 q^{8} - 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}+ \cdots - 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

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	$-0.717239\pi$
$2$	−3.56787	−1.26143	−0.630717	+	0.776013i	$0.717239\pi$
$3$	0	0
$3$	0	0
$4$	4.72971	0.591214
$5$	20.6288	1.84509	0.922547	−	0.385885i	$-0.126104\pi$
$5$	20.6288	1.84509	0.922547	+	0.385885i	$0.126104\pi$
$6$	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	$-0.477251\pi$
$11$	5.21023	0.142813	0.0714065	+	0.997447i	$0.477251\pi$
$12$	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	$-0.230082\pi$
$23$	165.443	1.49988	0.749941	+	0.661504i	$0.230082\pi$
$24$	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	$-0.543343\pi$
$29$	−42.3983	−0.271488	−0.135744	+	0.990744i	$0.543343\pi$
$30$	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	$-0.290216\pi$
$41$	321.529	1.22474	0.612371	+	0.790571i	$0.290216\pi$
$42$	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	$-0.340647\pi$
$47$	309.308	0.959942	0.479971	+	0.877284i	$0.340647\pi$
$48$	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	$-0.419664\pi$
$53$	192.701	0.499426	0.249713	+	0.968320i	$0.419664\pi$
$54$	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	$-0.724279\pi$
$59$	−587.082	−1.29545	−0.647725	+	0.761874i	$0.724279\pi$
$60$	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	$-0.215323\pi$
$71$	933.035	1.55959	0.779795	+	0.626035i	$0.215323\pi$
$72$	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	$-0.436675\pi$
$83$	298.886	0.395265	0.197633	+	0.980276i	$0.436675\pi$
$84$	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	$-0.629838\pi$
$89$	−666.125	−0.793361	−0.396680	+	0.917957i	$0.629838\pi$
$90$	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	$-0.796257\pi$
$101$	−1628.22	−1.60410	−0.802050	+	0.597256i	$0.796257\pi$
$102$	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	$-0.264268\pi$
$107$	1493.56	1.34942	0.674711	+	0.738082i	$0.264268\pi$
$108$	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	$-0.317660\pi$
$113$	1302.15	1.08404	0.542018	+	0.840367i	$0.317660\pi$
$114$	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	$-0.604094\pi$
$131$	−963.258	−0.642444	−0.321222	+	0.947004i	$0.604094\pi$
$132$	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	$-0.469007\pi$
$137$	311.771	0.194426	0.0972130	+	0.995264i	$0.469007\pi$
$138$	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	$-0.467108\pi$
$149$	375.216	0.206301	0.103151	+	0.994666i	$0.467108\pi$
$150$	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	$-0.808030\pi$
$167$	−3554.79	−1.64717	−0.823586	+	0.567192i	$0.808030\pi$
$168$	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	$-0.490608\pi$
$173$	134.253	0.0590006	0.0295003	+	0.999565i	$0.490608\pi$
$174$	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	$-0.664671\pi$
$179$	−2368.80	−0.989119	−0.494559	+	0.869144i	$0.664671\pi$
$180$	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	$-0.533193\pi$
$191$	−549.534	−0.208183	−0.104091	+	0.994568i	$0.533193\pi$
$192$	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	$-0.754346\pi$
$197$	−3963.36	−1.43339	−0.716695	+	0.697386i	$0.754346\pi$
$198$	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	$-0.365254\pi$
$227$	2809.86	0.821572	0.410786	+	0.911732i	$0.365254\pi$
$228$	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	$-0.632582\pi$
$233$	−2877.85	−0.809160	−0.404580	+	0.914503i	$0.632582\pi$
$234$	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	$-0.536100\pi$
$239$	−836.275	−0.226335	−0.113168	+	0.993576i	$0.536100\pi$
$240$	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	$-0.747666\pi$
$251$	−5582.36	−1.40381	−0.701903	+	0.712273i	$0.747666\pi$
$252$	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	$-0.310179\pi$
$257$	4627.76	1.12324	0.561618	+	0.827397i	$0.310179\pi$
$258$	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	$-0.485086\pi$
$263$	399.519	0.0936708	0.0468354	+	0.998903i	$0.485086\pi$
$264$	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	$-0.580361\pi$
$269$	−2204.10	−0.499577	−0.249789	+	0.968300i	$0.580361\pi$
$270$	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	$-0.550726\pi$
$281$	−1494.96	−0.317373	−0.158686	+	0.987329i	$0.550726\pi$
$282$	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	$-0.692625\pi$
$293$	−5706.31	−1.13777	−0.568884	+	0.822417i	$0.692625\pi$
$294$	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	$-0.378477\pi$
$311$	4086.74	0.745138	0.372569	+	0.928005i	$0.378477\pi$
$312$	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	$-0.356829\pi$
$317$	4907.72	0.869542	0.434771	+	0.900541i	$0.356829\pi$
$318$	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	$-0.343020\pi$
$347$	6120.24	0.946835	0.473417	+	0.880838i	$0.343020\pi$
$348$	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	$-0.523609\pi$
$353$	−982.908	−0.148201	−0.0741005	+	0.997251i	$0.523609\pi$
$354$	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	$-0.145000\pi$
$359$	12216.9	1.79605	0.898027	+	0.439940i	$0.145000\pi$
$360$	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	$-0.248640\pi$
$383$	10645.4	1.42024	0.710122	+	0.704079i	$0.248640\pi$
$384$	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	$-0.488567\pi$
$389$	551.008	0.0718180	0.0359090	+	0.999355i	$0.488567\pi$
$390$	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	$-0.707021\pi$
$401$	−9724.10	−1.21097	−0.605484	+	0.795857i	$0.707021\pi$
$402$	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	$-0.450367\pi$
$419$	2663.86	0.310592	0.155296	+	0.987868i	$0.450367\pi$
$420$	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	$-0.607919\pi$
$431$	−5951.70	−0.665158	−0.332579	+	0.943075i	$0.607919\pi$
$432$	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	$-0.843107\pi$
$443$	−16428.4	−1.76193	−0.880967	+	0.473177i	$0.843107\pi$
$444$	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	$-0.511725\pi$
$449$	−700.731	−0.0736516	−0.0368258	+	0.999322i	$0.511725\pi$
$450$	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	$-0.711329\pi$
$461$	−12198.4	−1.23240	−0.616200	+	0.787590i	$0.711329\pi$
$462$	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	$-0.363956\pi$
$467$	8366.25	0.829002	0.414501	+	0.910049i	$0.363956\pi$
$468$	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	$-0.560207\pi$
$479$	−3942.17	−0.376038	−0.188019	+	0.982165i	$0.560207\pi$
$480$	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	$-0.205496\pi$
$491$	17380.5	1.59750	0.798748	+	0.601666i	$0.205496\pi$
$492$	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	$-0.466541\pi$
$503$	2367.22	0.209839	0.104920	+	0.994481i	$0.466541\pi$
$504$	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	$-0.291631\pi$
$509$	13983.5	1.21770	0.608850	+	0.793286i	$0.291631\pi$
$510$	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	$-0.700533\pi$
$521$	−14012.1	−1.17828	−0.589139	+	0.808032i	$0.700533\pi$
$522$	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	$-0.630100\pi$
$557$	−10449.2	−0.794874	−0.397437	+	0.917629i	$0.630100\pi$
$558$	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	$-0.420466\pi$
$563$	6606.44	0.494544	0.247272	+	0.968946i	$0.420466\pi$
$564$	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	$-0.571146\pi$
$569$	−6016.94	−0.443310	−0.221655	+	0.975125i	$0.571146\pi$
$570$	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	$-0.299672\pi$
$587$	16742.5	1.17724	0.588618	+	0.808411i	$0.299672\pi$
$588$	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	$-0.302115\pi$
$593$	16820.2	1.16480	0.582398	+	0.812904i	$0.302115\pi$
$594$	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	$-0.309434\pi$
$599$	16523.6	1.12711	0.563553	+	0.826080i	$0.309434\pi$
$600$	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	$-0.276699\pi$
$617$	19782.2	1.29076	0.645381	+	0.763861i	$0.276699\pi$
$618$	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	$-0.529644\pi$
$641$	−3018.43	−0.185992	−0.0929961	+	0.995666i	$0.529644\pi$
$642$	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	$-0.460018\pi$
$647$	4123.41	0.250553	0.125277	+	0.992122i	$0.460018\pi$
$648$	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	$-0.681484\pi$
$653$	−18013.5	−1.07951	−0.539757	+	0.841821i	$0.681484\pi$
$654$	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	$-0.444336\pi$
$659$	5886.59	0.347965	0.173982	+	0.984749i	$0.444336\pi$
$660$	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	$-0.540694\pi$
$677$	−4491.71	−0.254993	−0.127497	+	0.991839i	$0.540694\pi$
$678$	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	$-0.604713\pi$
$683$	−11533.2	−0.646131	−0.323065	+	0.946377i	$0.604713\pi$
$684$	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	$-0.353429\pi$
$701$	16494.8	0.888730	0.444365	+	0.895846i	$0.353429\pi$
$702$	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	$-0.630877\pi$
$719$	−15410.9	−0.799348	−0.399674	+	0.916657i	$0.630877\pi$
$720$	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	$-0.363059\pi$
$743$	16893.3	0.834126	0.417063	+	0.908878i	$0.363059\pi$
$744$	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	$-0.325531\pi$
$761$	21878.0	1.04215	0.521076	+	0.853510i	$0.325531\pi$
$762$	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	$-0.474174\pi$
$773$	3483.57	0.162090	0.0810448	+	0.996710i	$0.474174\pi$
$774$	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	$-0.249467\pi$
$797$	31873.4	1.41658	0.708289	+	0.705922i	$0.249467\pi$
$798$	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	$-0.637323\pi$
$809$	−19243.8	−0.836312	−0.418156	+	0.908375i	$0.637323\pi$
$810$	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	$-0.475321\pi$
$821$	3644.00	0.154904	0.0774522	+	0.996996i	$0.475321\pi$
$822$	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	$-0.464077\pi$
$827$	5356.59	0.225232	0.112616	+	0.993639i	$0.464077\pi$
$828$	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	$-0.585592\pi$
$839$	−12912.5	−0.531333	−0.265667	+	0.964065i	$0.585592\pi$
$840$	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	$-0.626581\pi$
$857$	−19431.8	−0.774536	−0.387268	+	0.921967i	$0.626581\pi$
$858$	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	$-0.417033\pi$
$863$	13066.9	0.515412	0.257706	+	0.966223i	$0.417033\pi$
$864$	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	$-0.398229\pi$
$881$	16437.8	0.628608	0.314304	+	0.949322i	$0.398229\pi$
$882$	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	$-0.268344\pi$
$887$	35145.6	1.33041	0.665205	+	0.746661i	$0.268344\pi$
$888$	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	$-0.198050\pi$
$911$	44687.5	1.62521	0.812603	+	0.582818i	$0.198050\pi$
$912$	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	$-0.684620\pi$
$929$	−31035.2	−1.09605	−0.548025	+	0.836462i	$0.684620\pi$
$930$	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	$-0.433635\pi$
$941$	11949.5	0.413966	0.206983	+	0.978344i	$0.433635\pi$
$942$	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	$-0.612324\pi$
$947$	−20143.1	−0.691197	−0.345599	+	0.938382i	$0.612324\pi$
$948$	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	$-0.286690\pi$
$953$	36544.6	1.24218	0.621090	+	0.783739i	$0.286690\pi$
$954$	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	$-0.496259\pi$
$971$	711.265	0.0235073	0.0117536	+	0.999931i	$0.496259\pi$
$972$	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	$-0.896324\pi$
$977$	−57865.1	−1.89485	−0.947425	+	0.319979i	$0.896324\pi$
$978$	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	$-0.474235\pi$
$983$	4983.77	0.161707	0.0808534	+	0.996726i	$0.474235\pi$
$984$	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.i.1.1	yes	4
3.2	odd	2		153.4.a.h.1.4	✓	4
4.3	odd	2		2448.4.a.bs.1.4		4
12.11	even	2		2448.4.a.bo.1.1		4

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

\(f(q)\)	\(=\)	\(q-3.56787 q^{2} +4.72971 q^{4} +20.6288 q^{5} -5.24213 q^{7} +11.6680 q^{8} -73.6008 q^{10} +5.21023 q^{11} -14.8007 q^{13} +18.7033 q^{14} -79.4675 q^{16} -17.0000 q^{17} +26.2601 q^{19} +97.5682 q^{20} -18.5894 q^{22} +165.443 q^{23} +300.546 q^{25} +52.8069 q^{26} -24.7938 q^{28} -42.3983 q^{29} +263.956 q^{31} +190.186 q^{32} +60.6538 q^{34} -108.139 q^{35} -322.217 q^{37} -93.6927 q^{38} +240.696 q^{40} +321.529 q^{41} +385.799 q^{43} +24.6429 q^{44} -590.280 q^{46} +309.308 q^{47} -315.520 q^{49} -1072.31 q^{50} -70.0029 q^{52} +192.701 q^{53} +107.481 q^{55} -61.1650 q^{56} +151.272 q^{58} -587.082 q^{59} -241.163 q^{61} -941.763 q^{62} -42.8203 q^{64} -305.320 q^{65} +205.396 q^{67} -80.4052 q^{68} +385.825 q^{70} +933.035 q^{71} -869.875 q^{73} +1149.63 q^{74} +124.203 q^{76} -27.3127 q^{77} +102.161 q^{79} -1639.32 q^{80} -1147.17 q^{82} +298.886 q^{83} -350.689 q^{85} -1376.48 q^{86} +60.7928 q^{88} -666.125 q^{89} +77.5871 q^{91} +782.499 q^{92} -1103.57 q^{94} +541.714 q^{95} +1351.60 q^{97} +1125.74 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 26 q^{4} + 22 q^{5} - 24 q^{7} + 102 q^{8} - 2 q^{10} + 50 q^{11} + 26 q^{13} + 80 q^{14} + 138 q^{16} - 68 q^{17} + 34 q^{19} + 312 q^{20} - 254 q^{22} + 382 q^{23} + 138 q^{25} - 22 q^{26}+ \cdots - 1472 q^{98}+O(q^{100}) \) 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

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