LMFDB - Embedded newform 2254.4.a.bb.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2254,4,Mod(1,2254)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2254.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2254, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2254 = 2 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2254.a (trivial)

Newform invariants

comment:select newform

sage:traces = [18,-36,0,72,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$132.990305153$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 344 x^{16} + 46921 x^{14} - 3221102 x^{12} + 116812804 x^{10} - 2177115434 x^{8} + \cdots - 512512128 $ x^18 - 344x^16 + 46921x^14 - 3221102x^12 + 116812804x^10 - 2177115434x^8 + 19893950452x^6 - 77101162400x^4 + 74670805696x^2 - 512512128
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{6}\cdot 7^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-8.06273$ of defining polynomial
Character	$\chi$	$=$	2254.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} -8.06273 q^{3} +4.00000 q^{4} +5.73327 q^{5} +16.1255 q^{6} -8.00000 q^{8} +38.0077 q^{9} -11.4665 q^{10} +8.05436 q^{11} -32.2509 q^{12} -82.2442 q^{13} -46.2259 q^{15} +16.0000 q^{16} -80.5834 q^{17} -76.0154 q^{18} +78.2292 q^{19} +22.9331 q^{20} -16.1087 q^{22} -23.0000 q^{23} +64.5019 q^{24} -92.1296 q^{25} +164.488 q^{26} -88.7521 q^{27} -3.23669 q^{29} +92.4517 q^{30} -302.949 q^{31} -32.0000 q^{32} -64.9402 q^{33} +161.167 q^{34} +152.031 q^{36} -230.847 q^{37} -156.458 q^{38} +663.113 q^{39} -45.8662 q^{40} -179.690 q^{41} +62.8847 q^{43} +32.2174 q^{44} +217.908 q^{45} +46.0000 q^{46} -209.075 q^{47} -129.004 q^{48} +184.259 q^{50} +649.723 q^{51} -328.977 q^{52} +528.755 q^{53} +177.504 q^{54} +46.1778 q^{55} -630.741 q^{57} +6.47338 q^{58} -14.3437 q^{59} -184.903 q^{60} +110.750 q^{61} +605.898 q^{62} +64.0000 q^{64} -471.528 q^{65} +129.880 q^{66} +875.144 q^{67} -322.334 q^{68} +185.443 q^{69} -951.845 q^{71} -304.062 q^{72} +749.523 q^{73} +461.694 q^{74} +742.816 q^{75} +312.917 q^{76} -1326.23 q^{78} +551.767 q^{79} +91.7324 q^{80} -310.623 q^{81} +359.379 q^{82} -233.135 q^{83} -462.007 q^{85} -125.769 q^{86} +26.0966 q^{87} -64.4349 q^{88} -189.121 q^{89} -435.817 q^{90} -92.0000 q^{92} +2442.60 q^{93} +418.150 q^{94} +448.509 q^{95} +258.008 q^{96} -620.958 q^{97} +306.128 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 36 q^{2} + 72 q^{4} - 144 q^{8} + 202 q^{9} + 76 q^{11} + 348 q^{15} + 288 q^{16} - 404 q^{18} - 152 q^{22} - 414 q^{23} + 662 q^{25} + 1164 q^{29} - 696 q^{30} - 576 q^{32} + 808 q^{36} + 188 q^{37}+ \cdots - 3864 q^{99}+O(q^{100}) $ 18 * q - 36 * q^2 + 72 * q^4 - 144 * q^8 + 202 * q^9 + 76 * q^11 + 348 * q^15 + 288 * q^16 - 404 * q^18 - 152 * q^22 - 414 * q^23 + 662 * q^25 + 1164 * q^29 - 696 * q^30 - 576 * q^32 + 808 * q^36 + 188 * q^37 + 1780 * q^39 - 248 * q^43 + 304 * q^44 + 828 * q^46 - 1324 * q^50 + 1596 * q^53 + 108 * q^57 - 2328 * q^58 + 1392 * q^60 + 1152 * q^64 + 1748 * q^65 + 172 * q^67 + 500 * q^71 - 1616 * q^72 - 376 * q^74 - 3560 * q^78 - 1772 * q^79 + 6382 * q^81 - 4344 * q^85 + 496 * q^86 - 608 * q^88 - 1656 * q^92 - 1192 * q^93 + 2456 * q^95 - 3864 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−8.06273	−1.55167	−0.775837	−	0.630933i	$-0.782672\pi$
$3$	−8.06273	−1.55167	−0.775837	+	0.630933i	$0.782672\pi$
$4$	4.00000	0.500000
$5$	5.73327	0.512799	0.256400	−	0.966571i	$-0.417464\pi$
$5$	5.73327	0.512799	0.256400	+	0.966571i	$0.417464\pi$
$6$	16.1255	1.09720
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	38.0077	1.40769
$10$	−11.4665	−0.362604
$11$	8.05436	0.220771	0.110386	−	0.993889i	$-0.464791\pi$
$11$	8.05436	0.220771	0.110386	+	0.993889i	$0.464791\pi$
$12$	−32.2509	−0.775837
$13$	−82.2442	−1.75465	−0.877324	−	0.479898i	$-0.840674\pi$
$13$	−82.2442	−1.75465	−0.877324	+	0.479898i	$0.840674\pi$
$14$	0	0
$15$	−46.2259	−0.795698
$16$	16.0000	0.250000
$17$	−80.5834	−1.14967	−0.574834	−	0.818270i	$-0.694933\pi$
$17$	−80.5834	−1.14967	−0.574834	+	0.818270i	$0.694933\pi$
$18$	−76.0154	−0.995389
$19$	78.2292	0.944580	0.472290	−	0.881443i	$-0.343428\pi$
$19$	78.2292	0.944580	0.472290	+	0.881443i	$0.343428\pi$
$20$	22.9331	0.256400
$21$	0	0
$22$	−16.1087	−0.156109
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	64.5019	0.548600
$25$	−92.1296	−0.737037
$26$	164.488	1.24072
$27$	−88.7521	−0.632606
$28$	0	0
$29$	−3.23669	−0.0207254	−0.0103627	−	0.999946i	$-0.503299\pi$
$29$	−3.23669	−0.0207254	−0.0103627	+	0.999946i	$0.503299\pi$
$30$	92.4517	0.562643
$31$	−302.949	−1.75520	−0.877600	−	0.479393i	$-0.840857\pi$
$31$	−302.949	−1.75520	−0.877600	+	0.479393i	$0.840857\pi$
$32$	−32.0000	−0.176777
$33$	−64.9402	−0.342565
$34$	161.167	0.812938
$35$	0	0
$36$	152.031	0.703846
$37$	−230.847	−1.02570	−0.512852	−	0.858477i	$-0.671411\pi$
$37$	−230.847	−1.02570	−0.512852	+	0.858477i	$0.671411\pi$
$38$	−156.458	−0.667919
$39$	663.113	2.72264
$40$	−45.8662	−0.181302
$41$	−179.690	−0.684459	−0.342229	−	0.939616i	$-0.611182\pi$
$41$	−179.690	−0.684459	−0.342229	+	0.939616i	$0.611182\pi$
$42$	0	0
$43$	62.8847	0.223019	0.111510	−	0.993763i	$-0.464431\pi$
$43$	62.8847	0.223019	0.111510	+	0.993763i	$0.464431\pi$
$44$	32.2174	0.110386
$45$	217.908	0.721864
$46$	46.0000	0.147442
$47$	−209.075	−0.648867	−0.324433	−	0.945909i	$-0.605174\pi$
$47$	−209.075	−0.648867	−0.324433	+	0.945909i	$0.605174\pi$
$48$	−129.004	−0.387919
$49$	0	0
$50$	184.259	0.521164
$51$	649.723	1.78391
$52$	−328.977	−0.877324
$53$	528.755	1.37038	0.685189	−	0.728365i	$-0.259719\pi$
$53$	528.755	1.37038	0.685189	+	0.728365i	$0.259719\pi$
$54$	177.504	0.447320
$55$	46.1778	0.113211
$56$	0	0
$57$	−630.741	−1.46568
$58$	6.47338	0.0146551
$59$	−14.3437	−0.0316508	−0.0158254	−	0.999875i	$-0.505038\pi$
$59$	−14.3437	−0.0316508	−0.0158254	+	0.999875i	$0.505038\pi$
$60$	−184.903	−0.397849
$61$	110.750	0.232461	0.116230	−	0.993222i	$-0.462919\pi$
$61$	110.750	0.232461	0.116230	+	0.993222i	$0.462919\pi$
$62$	605.898	1.24111
$63$	0	0
$64$	64.0000	0.125000
$65$	−471.528	−0.899783
$66$	129.880	0.242230
$67$	875.144	1.59576	0.797880	−	0.602817i	$-0.205955\pi$
$67$	875.144	1.59576	0.797880	+	0.602817i	$0.205955\pi$
$68$	−322.334	−0.574834
$69$	185.443	0.323546
$70$	0	0
$71$	−951.845	−1.59103	−0.795516	−	0.605933i	$-0.792800\pi$
$71$	−951.845	−1.59103	−0.795516	+	0.605933i	$0.792800\pi$
$72$	−304.062	−0.497694
$73$	749.523	1.20171	0.600856	−	0.799357i	$-0.294826\pi$
$73$	749.523	1.20171	0.600856	+	0.799357i	$0.294826\pi$
$74$	461.694	0.725282
$75$	742.816	1.14364
$76$	312.917	0.472290
$77$	0	0
$78$	−1326.23	−1.92520
$79$	551.767	0.785806	0.392903	−	0.919580i	$-0.371471\pi$
$79$	551.767	0.785806	0.392903	+	0.919580i	$0.371471\pi$
$80$	91.7324	0.128200
$81$	−310.623	−0.426095
$82$	359.379	0.483985
$83$	−233.135	−0.308312	−0.154156	−	0.988047i	$-0.549266\pi$
$83$	−233.135	−0.308312	−0.154156	+	0.988047i	$0.549266\pi$
$84$	0	0
$85$	−462.007	−0.589549
$86$	−125.769	−0.157698
$87$	26.0966	0.0321591
$88$	−64.4349	−0.0780543
$89$	−189.121	−0.225245	−0.112622	−	0.993638i	$-0.535925\pi$
$89$	−189.121	−0.225245	−0.112622	+	0.993638i	$0.535925\pi$
$90$	−435.817	−0.510435
$91$	0	0
$92$	−92.0000	−0.104257
$93$	2442.60	2.72350
$94$	418.150	0.458818
$95$	448.509	0.484380
$96$	258.008	0.274300
$97$	−620.958	−0.649987	−0.324993	−	0.945716i	$-0.605362\pi$
$97$	−620.958	−0.649987	−0.324993	+	0.945716i	$0.605362\pi$
$98$	0	0
$99$	306.128	0.310778
$100$	−368.518	−0.368518
$101$	−159.250	−0.156891	−0.0784453	−	0.996918i	$-0.524996\pi$
$101$	−159.250	−0.156891	−0.0784453	+	0.996918i	$0.524996\pi$
$102$	−1299.45	−1.26141
$103$	−827.993	−0.792083	−0.396042	−	0.918233i	$-0.629616\pi$
$103$	−827.993	−0.792083	−0.396042	+	0.918233i	$0.629616\pi$
$104$	657.953	0.620362
$105$	0	0
$106$	−1057.51	−0.969004
$107$	−1320.80	−1.19333	−0.596667	−	0.802489i	$-0.703509\pi$
$107$	−1320.80	−1.19333	−0.596667	+	0.802489i	$0.703509\pi$
$108$	−355.008	−0.316303
$109$	1072.77	0.942689	0.471344	−	0.881949i	$-0.343769\pi$
$109$	1072.77	0.942689	0.471344	+	0.881949i	$0.343769\pi$
$110$	−92.3557	−0.0800524
$111$	1861.26	1.59156
$112$	0	0
$113$	−1290.46	−1.07430	−0.537150	−	0.843487i	$-0.680499\pi$
$113$	−1290.46	−1.07430	−0.537150	+	0.843487i	$0.680499\pi$
$114$	1261.48	1.03639
$115$	−131.865	−0.106926
$116$	−12.9468	−0.0103627
$117$	−3125.91	−2.47000
$118$	28.6875	0.0223805
$119$	0	0
$120$	369.807	0.281322
$121$	−1266.13	−0.951260
$122$	−221.500	−0.164375
$123$	1448.79	1.06206
$124$	−1211.80	−0.877600
$125$	−1244.86	−0.890751
$126$	0	0
$127$	−2432.34	−1.69949	−0.849744	−	0.527195i	$-0.823244\pi$
$127$	−2432.34	−1.69949	−0.849744	+	0.527195i	$0.823244\pi$
$128$	−128.000	−0.0883883
$129$	−507.023	−0.346053
$130$	943.056	0.636242
$131$	−2351.44	−1.56829	−0.784146	−	0.620576i	$-0.786899\pi$
$131$	−2351.44	−1.56829	−0.784146	+	0.620576i	$0.786899\pi$
$132$	−259.761	−0.171282
$133$	0	0
$134$	−1750.29	−1.12837
$135$	−508.840	−0.324400
$136$	644.667	0.406469
$137$	286.763	0.178831	0.0894154	−	0.995994i	$-0.471500\pi$
$137$	286.763	0.178831	0.0894154	+	0.995994i	$0.471500\pi$
$138$	−370.886	−0.228782
$139$	261.920	0.159826	0.0799128	−	0.996802i	$-0.474536\pi$
$139$	261.920	0.159826	0.0799128	+	0.996802i	$0.474536\pi$
$140$	0	0
$141$	1685.72	1.00683
$142$	1903.69	1.12503
$143$	−662.424	−0.387375
$144$	608.123	0.351923
$145$	−18.5568	−0.0106280
$146$	−1499.05	−0.849739
$147$	0	0
$148$	−923.388	−0.512852
$149$	1403.36	0.771593	0.385797	−	0.922584i	$-0.373927\pi$
$149$	1403.36	0.771593	0.385797	+	0.922584i	$0.373927\pi$
$150$	−1485.63	−0.808676
$151$	−1335.61	−0.719804	−0.359902	−	0.932990i	$-0.617190\pi$
$151$	−1335.61	−0.719804	−0.359902	+	0.932990i	$0.617190\pi$
$152$	−625.834	−0.333959
$153$	−3062.79	−1.61838
$154$	0	0
$155$	−1736.89	−0.900066
$156$	2652.45	1.36132
$157$	−2904.00	−1.47621	−0.738103	−	0.674688i	$-0.764278\pi$
$157$	−2904.00	−1.47621	−0.738103	+	0.674688i	$0.764278\pi$
$158$	−1103.53	−0.555649
$159$	−4263.21	−2.12638
$160$	−183.465	−0.0906510
$161$	0	0
$162$	621.246	0.301294
$163$	−2370.66	−1.13917	−0.569583	−	0.821934i	$-0.692895\pi$
$163$	−2370.66	−1.13917	−0.569583	+	0.821934i	$0.692895\pi$
$164$	−718.759	−0.342229
$165$	−372.320	−0.175667
$166$	466.270	0.218010
$167$	−1529.11	−0.708541	−0.354271	−	0.935143i	$-0.615271\pi$
$167$	−1529.11	−0.708541	−0.354271	+	0.935143i	$0.615271\pi$
$168$	0	0
$169$	4567.10	2.07879
$170$	924.013	0.416874
$171$	2973.31	1.32968
$172$	251.539	0.111510
$173$	−560.674	−0.246400	−0.123200	−	0.992382i	$-0.539316\pi$
$173$	−560.674	−0.246400	−0.123200	+	0.992382i	$0.539316\pi$
$174$	−52.1931	−0.0227399
$175$	0	0
$176$	128.870	0.0551928
$177$	115.650	0.0491117
$178$	378.242	0.159272
$179$	1243.65	0.519298	0.259649	−	0.965703i	$-0.416393\pi$
$179$	1243.65	0.519298	0.259649	+	0.965703i	$0.416393\pi$
$180$	871.634	0.360932
$181$	−1779.50	−0.730769	−0.365385	−	0.930857i	$-0.619062\pi$
$181$	−1779.50	−0.730769	−0.365385	+	0.930857i	$0.619062\pi$
$182$	0	0
$183$	−892.949	−0.360703
$184$	184.000	0.0737210
$185$	−1323.51	−0.525980
$186$	−4885.19	−1.92580
$187$	−649.048	−0.253813
$188$	−836.300	−0.324433
$189$	0	0
$190$	−897.019	−0.342508
$191$	2793.42	1.05824	0.529122	−	0.848546i	$-0.322521\pi$
$191$	2793.42	1.05824	0.529122	+	0.848546i	$0.322521\pi$
$192$	−516.015	−0.193959
$193$	31.4223	0.0117193	0.00585965	−	0.999983i	$-0.498135\pi$
$193$	31.4223	0.0117193	0.00585965	+	0.999983i	$0.498135\pi$
$194$	1241.92	0.459610
$195$	3801.81	1.39617
$196$	0	0
$197$	2452.19	0.886859	0.443429	−	0.896309i	$-0.353762\pi$
$197$	2452.19	0.886859	0.443429	+	0.896309i	$0.353762\pi$
$198$	−612.255	−0.219753
$199$	4590.38	1.63519	0.817596	−	0.575792i	$-0.195306\pi$
$199$	4590.38	1.63519	0.817596	+	0.575792i	$0.195306\pi$
$200$	737.037	0.260582
$201$	−7056.05	−2.47610
$202$	318.500	0.110938
$203$	0	0
$204$	2598.89	0.891955
$205$	−1030.21	−0.350990
$206$	1655.99	0.560088
$207$	−874.177	−0.293524
$208$	−1315.91	−0.438662
$209$	630.086	0.208536
$210$	0	0
$211$	5461.36	1.78188	0.890938	−	0.454125i	$-0.150048\pi$
$211$	5461.36	1.78188	0.890938	+	0.454125i	$0.150048\pi$
$212$	2115.02	0.685189
$213$	7674.47	2.46876
$214$	2641.60	0.843815
$215$	360.535	0.114364
$216$	710.017	0.223660
$217$	0	0
$218$	−2145.55	−0.666582
$219$	−6043.20	−1.86467
$220$	184.711	0.0566056
$221$	6627.51	2.01726
$222$	−3722.52	−1.12540
$223$	−2170.97	−0.651922	−0.325961	−	0.945383i	$-0.605688\pi$
$223$	−2170.97	−0.651922	−0.325961	+	0.945383i	$0.605688\pi$
$224$	0	0
$225$	−3501.63	−1.03752
$226$	2580.91	0.759645
$227$	−4725.13	−1.38158	−0.690789	−	0.723056i	$-0.742737\pi$
$227$	−4725.13	−1.38158	−0.690789	+	0.723056i	$0.742737\pi$
$228$	−2522.97	−0.732840
$229$	5289.14	1.52627	0.763135	−	0.646239i	$-0.223659\pi$
$229$	5289.14	1.52627	0.763135	+	0.646239i	$0.223659\pi$
$230$	263.731	0.0756082
$231$	0	0
$232$	25.8935	0.00732755
$233$	5198.96	1.46178	0.730890	−	0.682495i	$-0.239105\pi$
$233$	5198.96	1.46178	0.730890	+	0.682495i	$0.239105\pi$
$234$	6251.82	1.74656
$235$	−1198.68	−0.332738
$236$	−57.3750	−0.0158254
$237$	−4448.75	−1.21931
$238$	0	0
$239$	−3438.27	−0.930558	−0.465279	−	0.885164i	$-0.654046\pi$
$239$	−3438.27	−0.930558	−0.465279	+	0.885164i	$0.654046\pi$
$240$	−739.614	−0.198924
$241$	−4241.45	−1.13368	−0.566838	−	0.823830i	$-0.691833\pi$
$241$	−4241.45	−1.13368	−0.566838	+	0.823830i	$0.691833\pi$
$242$	2532.25	0.672643
$243$	4900.78	1.29377
$244$	443.001	0.116230
$245$	0	0
$246$	−2897.58	−0.750988
$247$	−6433.90	−1.65740
$248$	2423.59	0.620557
$249$	1879.71	0.478400
$250$	2489.73	0.629856
$251$	1639.03	0.412171	0.206085	−	0.978534i	$-0.433928\pi$
$251$	1639.03	0.412171	0.206085	+	0.978534i	$0.433928\pi$
$252$	0	0
$253$	−185.250	−0.0460339
$254$	4864.68	1.20172
$255$	3725.04	0.914788
$256$	256.000	0.0625000
$257$	4308.80	1.04582	0.522910	−	0.852388i	$-0.324846\pi$
$257$	4308.80	1.04582	0.522910	+	0.852388i	$0.324846\pi$
$258$	1014.05	0.244697
$259$	0	0
$260$	−1886.11	−0.449891
$261$	−123.019	−0.0291751
$262$	4702.88	1.10895
$263$	5837.74	1.36871	0.684354	−	0.729150i	$-0.260084\pi$
$263$	5837.74	1.36871	0.684354	+	0.729150i	$0.260084\pi$
$264$	519.521	0.121115
$265$	3031.50	0.702730
$266$	0	0
$267$	1524.83	0.349507
$268$	3500.58	0.797880
$269$	2831.26	0.641730	0.320865	−	0.947125i	$-0.396026\pi$
$269$	2831.26	0.641730	0.320865	+	0.947125i	$0.396026\pi$
$270$	1017.68	0.229385
$271$	276.373	0.0619500	0.0309750	−	0.999520i	$-0.490139\pi$
$271$	276.373	0.0619500	0.0309750	+	0.999520i	$0.490139\pi$
$272$	−1289.33	−0.287417
$273$	0	0
$274$	−573.526	−0.126452
$275$	−742.045	−0.162716
$276$	741.772	0.161773
$277$	−4568.54	−0.990963	−0.495482	−	0.868618i	$-0.665008\pi$
$277$	−4568.54	−0.990963	−0.495482	+	0.868618i	$0.665008\pi$
$278$	−523.840	−0.113014
$279$	−11514.4	−2.47078
$280$	0	0
$281$	3963.88	0.841513	0.420756	−	0.907174i	$-0.361765\pi$
$281$	3963.88	0.841513	0.420756	+	0.907174i	$0.361765\pi$
$282$	−3371.43	−0.711936
$283$	828.631	0.174053	0.0870265	−	0.996206i	$-0.472264\pi$
$283$	828.631	0.174053	0.0870265	+	0.996206i	$0.472264\pi$
$284$	−3807.38	−0.795516
$285$	−3616.21	−0.751600
$286$	1324.85	0.273916
$287$	0	0
$288$	−1216.25	−0.248847
$289$	1580.69	0.321735
$290$	37.1136	0.00751513
$291$	5006.62	1.00857
$292$	2998.09	0.600856
$293$	−5247.43	−1.04627	−0.523137	−	0.852249i	$-0.675238\pi$
$293$	−5247.43	−1.04627	−0.523137	+	0.852249i	$0.675238\pi$
$294$	0	0
$295$	−82.2365	−0.0162305
$296$	1846.78	0.362641
$297$	−714.842	−0.139661
$298$	−2806.71	−0.545599
$299$	1891.62	0.365869
$300$	2971.27	0.571820
$301$	0	0
$302$	2671.22	0.508978
$303$	1283.99	0.243443
$304$	1251.67	0.236145
$305$	634.961	0.119206
$306$	6125.58	1.14437
$307$	−5792.69	−1.07689	−0.538447	−	0.842659i	$-0.680989\pi$
$307$	−5792.69	−1.07689	−0.538447	+	0.842659i	$0.680989\pi$
$308$	0	0
$309$	6675.89	1.22906
$310$	3473.78	0.636443
$311$	6877.97	1.25406	0.627032	−	0.778994i	$-0.284270\pi$
$311$	6877.97	1.25406	0.627032	+	0.778994i	$0.284270\pi$
$312$	−5304.90	−0.962599
$313$	−8107.52	−1.46410	−0.732051	−	0.681250i	$-0.761437\pi$
$313$	−8107.52	−1.46410	−0.732051	+	0.681250i	$0.761437\pi$
$314$	5808.00	1.04384
$315$	0	0
$316$	2207.07	0.392903
$317$	−664.259	−0.117692	−0.0588462	−	0.998267i	$-0.518742\pi$
$317$	−664.259	−0.117692	−0.0588462	+	0.998267i	$0.518742\pi$
$318$	8526.42	1.50358
$319$	−26.0695	−0.00457558
$320$	366.929	0.0640999
$321$	10649.3	1.85167
$322$	0	0
$323$	−6303.98	−1.08595
$324$	−1242.49	−0.213047
$325$	7577.12	1.29324
$326$	4741.31	0.805512
$327$	−8649.49	−1.46275
$328$	1437.52	0.241993
$329$	0	0
$330$	744.639	0.124215
$331$	−1866.80	−0.309996	−0.154998	−	0.987915i	$-0.549537\pi$
$331$	−1866.80	−0.309996	−0.154998	+	0.987915i	$0.549537\pi$
$332$	−932.541	−0.154156
$333$	−8773.96	−1.44387
$334$	3058.23	0.501014
$335$	5017.44	0.818304
$336$	0	0
$337$	1019.88	0.164855	0.0824275	−	0.996597i	$-0.473733\pi$
$337$	1019.88	0.164855	0.0824275	+	0.996597i	$0.473733\pi$
$338$	−9134.20	−1.46993
$339$	10404.6	1.66696
$340$	−1848.03	−0.294774
$341$	−2440.06	−0.387497
$342$	−5946.62	−0.940224
$343$	0	0
$344$	−503.078	−0.0788492
$345$	1063.19	0.165914
$346$	1121.35	0.174231
$347$	−4495.83	−0.695530	−0.347765	−	0.937582i	$-0.613059\pi$
$347$	−4495.83	−0.695530	−0.347765	+	0.937582i	$0.613059\pi$
$348$	104.386	0.0160796
$349$	8373.24	1.28427	0.642133	−	0.766593i	$-0.278050\pi$
$349$	8373.24	1.28427	0.642133	+	0.766593i	$0.278050\pi$
$350$	0	0
$351$	7299.34	1.11000
$352$	−257.740	−0.0390272
$353$	−8401.21	−1.26672	−0.633359	−	0.773858i	$-0.718324\pi$
$353$	−8401.21	−1.26672	−0.633359	+	0.773858i	$0.718324\pi$
$354$	−231.300	−0.0347272
$355$	−5457.19	−0.815880
$356$	−756.484	−0.112622
$357$	0	0
$358$	−2487.29	−0.367199
$359$	8053.38	1.18396	0.591979	−	0.805953i	$-0.298347\pi$
$359$	8053.38	1.18396	0.591979	+	0.805953i	$0.298347\pi$
$360$	−1743.27	−0.255217
$361$	−739.189	−0.107769
$362$	3559.00	0.516732
$363$	10208.4	1.47605
$364$	0	0
$365$	4297.22	0.616237
$366$	1785.90	0.255056
$367$	399.862	0.0568737	0.0284369	−	0.999596i	$-0.490947\pi$
$367$	399.862	0.0568737	0.0284369	+	0.999596i	$0.490947\pi$
$368$	−368.000	−0.0521286
$369$	−6829.59	−0.963507
$370$	2647.02	0.371924
$371$	0	0
$372$	9770.39	1.36175
$373$	−821.834	−0.114083	−0.0570415	−	0.998372i	$-0.518167\pi$
$373$	−821.834	−0.114083	−0.0570415	+	0.998372i	$0.518167\pi$
$374$	1298.10	0.179473
$375$	10037.0	1.38216
$376$	1672.60	0.229409
$377$	266.199	0.0363659
$378$	0	0
$379$	6222.74	0.843379	0.421689	−	0.906740i	$-0.361437\pi$
$379$	6222.74	0.843379	0.421689	+	0.906740i	$0.361437\pi$
$380$	1794.04	0.242190
$381$	19611.3	2.63705
$382$	−5586.83	−0.748291
$383$	−10627.8	−1.41790	−0.708950	−	0.705259i	$-0.750831\pi$
$383$	−10627.8	−1.41790	−0.708950	+	0.705259i	$0.750831\pi$
$384$	1032.03	0.137150
$385$	0	0
$386$	−62.8445	−0.00828679
$387$	2390.10	0.313943
$388$	−2483.83	−0.324993
$389$	3512.15	0.457771	0.228885	−	0.973453i	$-0.426492\pi$
$389$	3512.15	0.457771	0.228885	+	0.973453i	$0.426492\pi$
$390$	−7603.61	−0.987241
$391$	1853.42	0.239722
$392$	0	0
$393$	18959.0	2.43348
$394$	−4904.38	−0.627104
$395$	3163.43	0.402961
$396$	1224.51	0.155389
$397$	9383.03	1.18620	0.593099	−	0.805129i	$-0.297904\pi$
$397$	9383.03	1.18620	0.593099	+	0.805129i	$0.297904\pi$
$398$	−9180.76	−1.15626
$399$	0	0
$400$	−1474.07	−0.184259
$401$	14633.9	1.82239	0.911197	−	0.411970i	$-0.135159\pi$
$401$	14633.9	1.82239	0.911197	+	0.411970i	$0.135159\pi$
$402$	14112.1	1.75087
$403$	24915.8	3.07976
$404$	−636.999	−0.0784453
$405$	−1780.89	−0.218501
$406$	0	0
$407$	−1859.33	−0.226446
$408$	−5197.78	−0.630707
$409$	7726.66	0.934129	0.467064	−	0.884223i	$-0.345312\pi$
$409$	7726.66	0.934129	0.467064	+	0.884223i	$0.345312\pi$
$410$	2060.42	0.248187
$411$	−2312.09	−0.277487
$412$	−3311.97	−0.396042
$413$	0	0
$414$	1748.35	0.207553
$415$	−1336.63	−0.158102
$416$	2631.81	0.310181
$417$	−2111.79	−0.247997
$418$	−1260.17	−0.147457
$419$	2664.86	0.310708	0.155354	−	0.987859i	$-0.450348\pi$
$419$	2664.86	0.310708	0.155354	+	0.987859i	$0.450348\pi$
$420$	0	0
$421$	−10914.3	−1.26349	−0.631746	−	0.775176i	$-0.717661\pi$
$421$	−10914.3	−1.26349	−0.631746	+	0.775176i	$0.717661\pi$
$422$	−10922.7	−1.25998
$423$	−7946.46	−0.913405
$424$	−4230.04	−0.484502
$425$	7424.12	0.847347
$426$	−15348.9	−1.74568
$427$	0	0
$428$	−5283.21	−0.596667
$429$	5340.95	0.601080
$430$	−721.070	−0.0808677
$431$	−15213.0	−1.70019	−0.850097	−	0.526626i	$-0.823457\pi$
$431$	−15213.0	−1.70019	−0.850097	+	0.526626i	$0.823457\pi$
$432$	−1420.03	−0.158151
$433$	1713.28	0.190150	0.0950749	−	0.995470i	$-0.469691\pi$
$433$	1713.28	0.190150	0.0950749	+	0.995470i	$0.469691\pi$
$434$	0	0
$435$	149.619	0.0164912
$436$	4291.09	0.471344
$437$	−1799.27	−0.196958
$438$	12086.4	1.31852
$439$	−6591.04	−0.716567	−0.358284	−	0.933613i	$-0.616638\pi$
$439$	−6591.04	−0.716567	−0.358284	+	0.933613i	$0.616638\pi$
$440$	−369.423	−0.0400262
$441$	0	0
$442$	−13255.0	−1.42642
$443$	−7061.06	−0.757294	−0.378647	−	0.925541i	$-0.623610\pi$
$443$	−7061.06	−0.757294	−0.378647	+	0.925541i	$0.623610\pi$
$444$	7445.03	0.795779
$445$	−1084.28	−0.115505
$446$	4341.93	0.460979
$447$	−11314.9	−1.19726
$448$	0	0
$449$	13203.1	1.38773	0.693866	−	0.720104i	$-0.255906\pi$
$449$	13203.1	1.38773	0.693866	+	0.720104i	$0.255906\pi$
$450$	7003.27	0.733638
$451$	−1447.29	−0.151109
$452$	−5161.83	−0.537150
$453$	10768.7	1.11690
$454$	9450.27	0.976923
$455$	0	0
$456$	5045.93	0.518196
$457$	17520.6	1.79339	0.896697	−	0.442646i	$-0.145960\pi$
$457$	17520.6	1.79339	0.896697	+	0.442646i	$0.145960\pi$
$458$	−10578.3	−1.07924
$459$	7151.95	0.727286
$460$	−527.461	−0.0534630
$461$	−2297.57	−0.232123	−0.116061	−	0.993242i	$-0.537027\pi$
$461$	−2297.57	−0.232123	−0.116061	+	0.993242i	$0.537027\pi$
$462$	0	0
$463$	−754.837	−0.0757673	−0.0378837	−	0.999282i	$-0.512062\pi$
$463$	−754.837	−0.0757673	−0.0378837	+	0.999282i	$0.512062\pi$
$464$	−51.7870	−0.00518136
$465$	14004.1	1.39661
$466$	−10397.9	−1.03363
$467$	−4396.25	−0.435619	−0.217810	−	0.975991i	$-0.569891\pi$
$467$	−4396.25	−0.435619	−0.217810	+	0.975991i	$0.569891\pi$
$468$	−12503.6	−1.23500
$469$	0	0
$470$	2397.37	0.235282
$471$	23414.2	2.29059
$472$	114.750	0.0111902
$473$	506.496	0.0492362
$474$	8897.51	0.862186
$475$	−7207.23	−0.696190
$476$	0	0
$477$	20096.8	1.92907
$478$	6876.54	0.658004
$479$	−7707.47	−0.735205	−0.367603	−	0.929983i	$-0.619821\pi$
$479$	−7707.47	−0.735205	−0.367603	+	0.929983i	$0.619821\pi$
$480$	1479.23	0.140661
$481$	18985.8	1.79975
$482$	8482.90	0.801629
$483$	0	0
$484$	−5064.51	−0.475630
$485$	−3560.12	−0.333313
$486$	−9801.56	−0.914830
$487$	14060.4	1.30829	0.654145	−	0.756369i	$-0.273029\pi$
$487$	14060.4	1.30829	0.654145	+	0.756369i	$0.273029\pi$
$488$	−886.001	−0.0821873
$489$	19114.0	1.76761
$490$	0	0
$491$	−2111.81	−0.194103	−0.0970517	−	0.995279i	$-0.530941\pi$
$491$	−2111.81	−0.194103	−0.0970517	+	0.995279i	$0.530941\pi$
$492$	5795.16	0.531028
$493$	260.823	0.0238274
$494$	12867.8	1.17196
$495$	1755.11	0.159367
$496$	−4847.18	−0.438800
$497$	0	0
$498$	−3759.41	−0.338280
$499$	6857.92	0.615236	0.307618	−	0.951510i	$-0.400468\pi$
$499$	6857.92	0.615236	0.307618	+	0.951510i	$0.400468\pi$
$500$	−4979.45	−0.445376
$501$	12328.8	1.09943
$502$	−3278.07	−0.291449
$503$	12971.2	1.14981	0.574907	−	0.818219i	$-0.305038\pi$
$503$	12971.2	1.14981	0.574907	+	0.818219i	$0.305038\pi$
$504$	0	0
$505$	−913.022	−0.0804534
$506$	370.501	0.0325509
$507$	−36823.3	−3.22560
$508$	−9729.35	−0.849744
$509$	−1009.22	−0.0878839	−0.0439420	−	0.999034i	$-0.513992\pi$
$509$	−1009.22	−0.0878839	−0.0439420	+	0.999034i	$0.513992\pi$
$510$	−7450.07	−0.646853
$511$	0	0
$512$	−512.000	−0.0441942
$513$	−6943.01	−0.597546
$514$	−8617.61	−0.739507
$515$	−4747.11	−0.406180
$516$	−2028.09	−0.173027
$517$	−1683.97	−0.143251
$518$	0	0
$519$	4520.57	0.382333
$520$	3772.23	0.318121
$521$	20277.5	1.70513	0.852567	−	0.522618i	$-0.175044\pi$
$521$	20277.5	1.70513	0.852567	+	0.522618i	$0.175044\pi$
$522$	246.038	0.0206299
$523$	−1109.66	−0.0927766	−0.0463883	−	0.998923i	$-0.514771\pi$
$523$	−1109.66	−0.0927766	−0.0463883	+	0.998923i	$0.514771\pi$
$524$	−9405.76	−0.784146
$525$	0	0
$526$	−11675.5	−0.967823
$527$	24412.7	2.01790
$528$	−1039.04	−0.0856412
$529$	529.000	0.0434783
$530$	−6062.99	−0.496905
$531$	−545.172	−0.0445546
$532$	0	0
$533$	14778.4	1.20098
$534$	−3049.67	−0.247139
$535$	−7572.52	−0.611941
$536$	−7001.15	−0.564186
$537$	−10027.2	−0.805782
$538$	−5662.53	−0.453771
$539$	0	0
$540$	−2035.36	−0.162200
$541$	2354.97	0.187150	0.0935748	−	0.995612i	$-0.470171\pi$
$541$	2354.97	0.187150	0.0935748	+	0.995612i	$0.470171\pi$
$542$	−552.746	−0.0438053
$543$	14347.6	1.13392
$544$	2578.67	0.203234
$545$	6150.50	0.483410
$546$	0	0
$547$	−14261.1	−1.11474	−0.557368	−	0.830265i	$-0.688189\pi$
$547$	−14261.1	−1.11474	−0.557368	+	0.830265i	$0.688189\pi$
$548$	1147.05	0.0894154
$549$	4209.36	0.327233
$550$	1484.09	0.115058
$551$	−253.204	−0.0195768
$552$	−1483.54	−0.114391
$553$	0	0
$554$	9137.08	0.700717
$555$	10671.1	0.816150
$556$	1047.68	0.0799128
$557$	5411.60	0.411664	0.205832	−	0.978587i	$-0.434010\pi$
$557$	5411.60	0.411664	0.205832	+	0.978587i	$0.434010\pi$
$558$	23028.8	1.74711
$559$	−5171.90	−0.391320
$560$	0	0
$561$	5233.10	0.393835
$562$	−7927.75	−0.595039
$563$	−618.052	−0.0462661	−0.0231330	−	0.999732i	$-0.507364\pi$
$563$	−618.052	−0.0462661	−0.0231330	+	0.999732i	$0.507364\pi$
$564$	6742.87	0.503415
$565$	−7398.54	−0.550901
$566$	−1657.26	−0.123074
$567$	0	0
$568$	7614.76	0.562514
$569$	−1488.24	−0.109649	−0.0548244	−	0.998496i	$-0.517460\pi$
$569$	−1488.24	−0.109649	−0.0548244	+	0.998496i	$0.517460\pi$
$570$	7232.42	0.531461
$571$	−3213.38	−0.235510	−0.117755	−	0.993043i	$-0.537570\pi$
$571$	−3213.38	−0.235510	−0.117755	+	0.993043i	$0.537570\pi$
$572$	−2649.70	−0.193688
$573$	−22522.6	−1.64205
$574$	0	0
$575$	2118.98	0.153683
$576$	2432.49	0.175962
$577$	18951.0	1.36732	0.683658	−	0.729802i	$-0.260388\pi$
$577$	18951.0	1.36732	0.683658	+	0.729802i	$0.260388\pi$
$578$	−3161.37	−0.227501
$579$	−253.349	−0.0181845
$580$	−74.2273	−0.00531400
$581$	0	0
$582$	−10013.2	−0.713165
$583$	4258.78	0.302540
$584$	−5996.18	−0.424869
$585$	−17921.7	−1.26662
$586$	10494.9	0.739827
$587$	−25108.2	−1.76546	−0.882731	−	0.469878i	$-0.844298\pi$
$587$	−25108.2	−1.76546	−0.882731	+	0.469878i	$0.844298\pi$
$588$	0	0
$589$	−23699.5	−1.65793
$590$	164.473	0.0114767
$591$	−19771.3	−1.37612
$592$	−3693.55	−0.256426
$593$	8509.71	0.589295	0.294648	−	0.955606i	$-0.404798\pi$
$593$	8509.71	0.589295	0.294648	+	0.955606i	$0.404798\pi$
$594$	1429.68	0.0987552
$595$	0	0
$596$	5613.43	0.385797
$597$	−37011.0	−2.53729
$598$	−3783.23	−0.258709
$599$	−4981.05	−0.339767	−0.169883	−	0.985464i	$-0.554339\pi$
$599$	−4981.05	−0.339767	−0.169883	+	0.985464i	$0.554339\pi$
$600$	−5942.53	−0.404338
$601$	10639.1	0.722096	0.361048	−	0.932547i	$-0.382419\pi$
$601$	10639.1	0.722096	0.361048	+	0.932547i	$0.382419\pi$
$602$	0	0
$603$	33262.2	2.24634
$604$	−5342.44	−0.359902
$605$	−7259.05	−0.487806
$606$	−2567.98	−0.172140
$607$	18008.1	1.20416	0.602080	−	0.798436i	$-0.294339\pi$
$607$	18008.1	1.20416	0.602080	+	0.798436i	$0.294339\pi$
$608$	−2503.34	−0.166980
$609$	0	0
$610$	−1269.92	−0.0842912
$611$	17195.2	1.13853
$612$	−12251.2	−0.809189
$613$	9879.09	0.650918	0.325459	−	0.945556i	$-0.394481\pi$
$613$	9879.09	0.650918	0.325459	+	0.945556i	$0.394481\pi$
$614$	11585.4	0.761479
$615$	8306.31	0.544622
$616$	0	0
$617$	3364.06	0.219501	0.109750	−	0.993959i	$-0.464995\pi$
$617$	3364.06	0.219501	0.109750	+	0.993959i	$0.464995\pi$
$618$	−13351.8	−0.869073
$619$	−19316.3	−1.25426	−0.627132	−	0.778913i	$-0.715771\pi$
$619$	−19316.3	−1.25426	−0.627132	+	0.778913i	$0.715771\pi$
$620$	−6947.55	−0.450033
$621$	2041.30	0.131907
$622$	−13755.9	−0.886757
$623$	0	0
$624$	10609.8	0.680660
$625$	4379.06	0.280260
$626$	16215.0	1.03528
$627$	−5080.22	−0.323580
$628$	−11616.0	−0.738103
$629$	18602.4	1.17922
$630$	0	0
$631$	−25499.9	−1.60877	−0.804385	−	0.594108i	$-0.797505\pi$
$631$	−25499.9	−1.60877	−0.804385	+	0.594108i	$0.797505\pi$
$632$	−4414.14	−0.277824
$633$	−44033.5	−2.76489
$634$	1328.52	0.0832211
$635$	−13945.3	−0.871497
$636$	−17052.8	−1.06319
$637$	0	0
$638$	52.1389	0.00323542
$639$	−36177.4	−2.23968
$640$	−733.859	−0.0453255
$641$	7348.27	0.452791	0.226396	−	0.974035i	$-0.427306\pi$
$641$	7348.27	0.452791	0.226396	+	0.974035i	$0.427306\pi$
$642$	−21298.6	−1.30933
$643$	5608.49	0.343977	0.171989	−	0.985099i	$-0.444981\pi$
$643$	5608.49	0.343977	0.171989	+	0.985099i	$0.444981\pi$
$644$	0	0
$645$	−2906.90	−0.177456
$646$	12608.0	0.767884
$647$	−8396.22	−0.510185	−0.255092	−	0.966917i	$-0.582106\pi$
$647$	−8396.22	−0.510185	−0.255092	+	0.966917i	$0.582106\pi$
$648$	2484.98	0.150647
$649$	−115.530	−0.00698757
$650$	−15154.2	−0.914459
$651$	0	0
$652$	−9482.62	−0.569583
$653$	22116.6	1.32540	0.662702	−	0.748884i	$-0.269410\pi$
$653$	22116.6	1.32540	0.662702	+	0.748884i	$0.269410\pi$
$654$	17299.0	1.03432
$655$	−13481.4	−0.804219
$656$	−2875.03	−0.171115
$657$	28487.6	1.69164
$658$	0	0
$659$	−9873.26	−0.583623	−0.291811	−	0.956476i	$-0.594258\pi$
$659$	−9873.26	−0.583623	−0.291811	+	0.956476i	$0.594258\pi$
$660$	−1489.28	−0.0878335
$661$	−13624.6	−0.801719	−0.400859	−	0.916140i	$-0.631288\pi$
$661$	−13624.6	−0.801719	−0.400859	+	0.916140i	$0.631288\pi$
$662$	3733.61	0.219201
$663$	−53435.9	−3.13013
$664$	1865.08	0.109005
$665$	0	0
$666$	17547.9	1.02097
$667$	74.4438	0.00432156
$668$	−6116.46	−0.354271
$669$	17503.9	1.01157
$670$	−10034.9	−0.578629
$671$	892.022	0.0513206
$672$	0	0
$673$	137.840	0.00789503	0.00394752	−	0.999992i	$-0.498743\pi$
$673$	137.840	0.00789503	0.00394752	+	0.999992i	$0.498743\pi$
$674$	−2039.75	−0.116570
$675$	8176.70	0.466254
$676$	18268.4	1.03940
$677$	120.348	0.00683210	0.00341605	−	0.999994i	$-0.498913\pi$
$677$	120.348	0.00683210	0.00341605	+	0.999994i	$0.498913\pi$
$678$	−20809.2	−1.17872
$679$	0	0
$680$	3696.05	0.208437
$681$	38097.5	2.14376
$682$	4880.12	0.274002
$683$	−7488.16	−0.419512	−0.209756	−	0.977754i	$-0.567267\pi$
$683$	−7488.16	−0.419512	−0.209756	+	0.977754i	$0.567267\pi$
$684$	11893.2	0.664839
$685$	1644.09	0.0917043
$686$	0	0
$687$	−42644.9	−2.36827
$688$	1006.16	0.0557548
$689$	−43487.0	−2.40453
$690$	−2126.39	−0.117319
$691$	15292.7	0.841914	0.420957	−	0.907081i	$-0.361694\pi$
$691$	15292.7	0.841914	0.420957	+	0.907081i	$0.361694\pi$
$692$	−2242.70	−0.123200
$693$	0	0
$694$	8991.66	0.491814
$695$	1501.66	0.0819585
$696$	−208.773	−0.0113700
$697$	14480.0	0.786900
$698$	−16746.5	−0.908114
$699$	−41917.8	−2.26821
$700$	0	0
$701$	22848.7	1.23108	0.615538	−	0.788107i	$-0.288939\pi$
$701$	22848.7	1.23108	0.615538	+	0.788107i	$0.288939\pi$
$702$	−14598.7	−0.784889
$703$	−18059.0	−0.968858
$704$	515.479	0.0275964
$705$	9664.67	0.516302
$706$	16802.4	0.895705
$707$	0	0
$708$	462.599	0.0245558
$709$	−23568.8	−1.24844	−0.624220	−	0.781249i	$-0.714583\pi$
$709$	−23568.8	−1.24844	−0.624220	+	0.781249i	$0.714583\pi$
$710$	10914.4	0.576914
$711$	20971.4	1.10617
$712$	1512.97	0.0796361
$713$	6967.82	0.365985
$714$	0	0
$715$	−3797.86	−0.198646
$716$	4974.58	0.259649
$717$	27721.9	1.44392
$718$	−16106.8	−0.837185
$719$	5245.39	0.272073	0.136036	−	0.990704i	$-0.456564\pi$
$719$	5245.39	0.272073	0.136036	+	0.990704i	$0.456564\pi$
$720$	3486.54	0.180466
$721$	0	0
$722$	1478.38	0.0762044
$723$	34197.7	1.75909
$724$	−7118.00	−0.365385
$725$	298.195	0.0152754
$726$	−20416.9	−1.04372
$727$	2106.51	0.107464	0.0537319	−	0.998555i	$-0.482888\pi$
$727$	2106.51	0.107464	0.0537319	+	0.998555i	$0.482888\pi$
$728$	0	0
$729$	−31126.9	−1.58141
$730$	−8594.43	−0.435746
$731$	−5067.46	−0.256398
$732$	−3571.80	−0.180352
$733$	27902.9	1.40602	0.703012	−	0.711178i	$-0.251838\pi$
$733$	27902.9	1.40602	0.703012	+	0.711178i	$0.251838\pi$
$734$	−799.725	−0.0402158
$735$	0	0
$736$	736.000	0.0368605
$737$	7048.73	0.352297
$738$	13659.2	0.681303
$739$	−8956.27	−0.445821	−0.222911	−	0.974839i	$-0.571556\pi$
$739$	−8956.27	−0.445821	−0.222911	+	0.974839i	$0.571556\pi$
$740$	−5294.04	−0.262990
$741$	51874.8	2.57175
$742$	0	0
$743$	−19289.5	−0.952438	−0.476219	−	0.879327i	$-0.657993\pi$
$743$	−19289.5	−0.952438	−0.476219	+	0.879327i	$0.657993\pi$
$744$	−19540.8	−0.962902
$745$	8045.82	0.395673
$746$	1643.67	0.0806688
$747$	−8860.93	−0.434009
$748$	−2596.19	−0.126907
$749$	0	0
$750$	−20074.0	−0.977332
$751$	5439.62	0.264307	0.132154	−	0.991229i	$-0.457811\pi$
$751$	5439.62	0.264307	0.132154	+	0.991229i	$0.457811\pi$
$752$	−3345.20	−0.162217
$753$	−13215.1	−0.639555
$754$	−532.398	−0.0257146
$755$	−7657.42	−0.369115
$756$	0	0
$757$	−24706.4	−1.18622	−0.593110	−	0.805122i	$-0.702100\pi$
$757$	−24706.4	−1.18622	−0.593110	+	0.805122i	$0.702100\pi$
$758$	−12445.5	−0.596359
$759$	1493.62	0.0714297
$760$	−3588.08	−0.171254
$761$	16060.5	0.765034	0.382517	−	0.923948i	$-0.375057\pi$
$761$	16060.5	0.765034	0.382517	+	0.923948i	$0.375057\pi$
$762$	−39222.6	−1.86468
$763$	0	0
$764$	11173.7	0.529122
$765$	−17559.8	−0.829903
$766$	21255.6	1.00261
$767$	1179.69	0.0555360
$768$	−2064.06	−0.0969796
$769$	5911.30	0.277200	0.138600	−	0.990348i	$-0.455740\pi$
$769$	5911.30	0.277200	0.138600	+	0.990348i	$0.455740\pi$
$770$	0	0
$771$	−34740.8	−1.62277
$772$	125.689	0.00585965
$773$	36469.3	1.69691	0.848454	−	0.529269i	$-0.177534\pi$
$773$	36469.3	1.69691	0.848454	+	0.529269i	$0.177534\pi$
$774$	−4780.21	−0.221991
$775$	27910.6	1.29365
$776$	4967.66	0.229805
$777$	0	0
$778$	−7024.29	−0.323693
$779$	−14057.0	−0.646526
$780$	15207.2	0.698085
$781$	−7666.50	−0.351254
$782$	−3706.84	−0.169509
$783$	287.263	0.0131110
$784$	0	0
$785$	−16649.4	−0.756998
$786$	−37918.1	−1.72073
$787$	29488.7	1.33565	0.667826	−	0.744318i	$-0.267225\pi$
$787$	29488.7	1.33565	0.667826	+	0.744318i	$0.267225\pi$
$788$	9808.75	0.443429
$789$	−47068.1	−2.12379
$790$	−6326.86	−0.284936
$791$	0	0
$792$	−2449.02	−0.109876
$793$	−9108.55	−0.407887
$794$	−18766.1	−0.838769
$795$	−24442.1	−1.09041
$796$	18361.5	0.817596
$797$	−15888.3	−0.706140	−0.353070	−	0.935597i	$-0.614862\pi$
$797$	−15888.3	−0.706140	−0.353070	+	0.935597i	$0.614862\pi$
$798$	0	0
$799$	16848.0	0.745981
$800$	2948.15	0.130291
$801$	−7188.06	−0.317076
$802$	−29267.7	−1.28863
$803$	6036.93	0.265303
$804$	−28224.2	−1.23805
$805$	0	0
$806$	−49831.5	−2.17772
$807$	−22827.7	−0.995755
$808$	1274.00	0.0554692
$809$	17338.2	0.753496	0.376748	−	0.926316i	$-0.377042\pi$
$809$	17338.2	0.753496	0.376748	+	0.926316i	$0.377042\pi$
$810$	3561.77	0.154504
$811$	18183.7	0.787318	0.393659	−	0.919257i	$-0.371209\pi$
$811$	18183.7	0.787318	0.393659	+	0.919257i	$0.371209\pi$
$812$	0	0
$813$	−2228.32	−0.0961262
$814$	3718.65	0.160121
$815$	−13591.6	−0.584164
$816$	10395.6	0.445977
$817$	4919.42	0.210659
$818$	−15453.3	−0.660529
$819$	0	0
$820$	−4120.84	−0.175495
$821$	−23071.0	−0.980734	−0.490367	−	0.871516i	$-0.663137\pi$
$821$	−23071.0	−0.980734	−0.490367	+	0.871516i	$0.663137\pi$
$822$	4624.19	0.196213
$823$	16533.5	0.700271	0.350135	−	0.936699i	$-0.386136\pi$
$823$	16533.5	0.700271	0.350135	+	0.936699i	$0.386136\pi$
$824$	6623.95	0.280044
$825$	5982.91	0.252483
$826$	0	0
$827$	11548.1	0.485571	0.242786	−	0.970080i	$-0.421939\pi$
$827$	11548.1	0.485571	0.242786	+	0.970080i	$0.421939\pi$
$828$	−3496.71	−0.146762
$829$	−37170.8	−1.55729	−0.778646	−	0.627463i	$-0.784093\pi$
$829$	−37170.8	−1.55729	−0.778646	+	0.627463i	$0.784093\pi$
$830$	2673.25	0.111795
$831$	36834.9	1.53765
$832$	−5263.63	−0.219331
$833$	0	0
$834$	4223.58	0.175361
$835$	−8766.83	−0.363340
$836$	2520.35	0.104268
$837$	26887.3	1.11035
$838$	−5329.72	−0.219704
$839$	46912.7	1.93040	0.965200	−	0.261511i	$-0.0842209\pi$
$839$	46912.7	1.93040	0.965200	+	0.261511i	$0.0842209\pi$
$840$	0	0
$841$	−24378.5	−0.999570
$842$	21828.6	0.893423
$843$	−31959.7	−1.30575
$844$	21845.5	0.890938
$845$	26184.4	1.06600
$846$	15892.9	0.645875
$847$	0	0
$848$	8460.08	0.342595
$849$	−6681.03	−0.270074
$850$	−14848.2	−0.599165
$851$	5309.48	0.213874
$852$	30697.9	1.23438
$853$	−691.252	−0.0277468	−0.0138734	−	0.999904i	$-0.504416\pi$
$853$	−691.252	−0.0277468	−0.0138734	+	0.999904i	$0.504416\pi$
$854$	0	0
$855$	17046.8	0.681858
$856$	10566.4	0.421907
$857$	47287.8	1.88485	0.942427	−	0.334411i	$-0.108537\pi$
$857$	47287.8	1.88485	0.942427	+	0.334411i	$0.108537\pi$
$858$	−10681.9	−0.425028
$859$	23061.3	0.915999	0.457999	−	0.888952i	$-0.348566\pi$
$859$	23061.3	0.915999	0.457999	+	0.888952i	$0.348566\pi$
$860$	1442.14	0.0571821
$861$	0	0
$862$	30426.0	1.20222
$863$	8754.48	0.345314	0.172657	−	0.984982i	$-0.444765\pi$
$863$	8754.48	0.345314	0.172657	+	0.984982i	$0.444765\pi$
$864$	2840.07	0.111830
$865$	−3214.50	−0.126354
$866$	−3426.56	−0.134456
$867$	−12744.7	−0.499228
$868$	0	0
$869$	4444.13	0.173483
$870$	−299.237	−0.0116610
$871$	−71975.5	−2.80000
$872$	−8582.19	−0.333291
$873$	−23601.2	−0.914981
$874$	3598.54	0.139271
$875$	0	0
$876$	−24172.8	−0.932333
$877$	8203.34	0.315858	0.157929	−	0.987450i	$-0.449518\pi$
$877$	8203.34	0.315858	0.157929	+	0.987450i	$0.449518\pi$
$878$	13182.1	0.506689
$879$	42308.6	1.62348
$880$	738.845	0.0283028
$881$	10816.5	0.413641	0.206820	−	0.978379i	$-0.433688\pi$
$881$	10816.5	0.413641	0.206820	+	0.978379i	$0.433688\pi$
$882$	0	0
$883$	−3591.96	−0.136896	−0.0684479	−	0.997655i	$-0.521805\pi$
$883$	−3591.96	−0.136896	−0.0684479	+	0.997655i	$0.521805\pi$
$884$	26510.1	1.00863
$885$	663.051	0.0251844
$886$	14122.1	0.535487
$887$	36493.6	1.38144	0.690718	−	0.723124i	$-0.257295\pi$
$887$	36493.6	1.38144	0.690718	+	0.723124i	$0.257295\pi$
$888$	−14890.1	−0.562700
$889$	0	0
$890$	2168.57	0.0816747
$891$	−2501.87	−0.0940694
$892$	−8683.86	−0.325961
$893$	−16355.8	−0.612906
$894$	22629.8	0.846592
$895$	7130.16	0.266296
$896$	0	0
$897$	−15251.6	−0.567710
$898$	−26406.1	−0.981274
$899$	980.551	0.0363773
$900$	−14006.5	−0.518760
$901$	−42608.9	−1.57548
$902$	2894.57	0.106850
$903$	0	0
$904$	10323.7	0.379823
$905$	−10202.4	−0.374738
$906$	−21537.3	−0.789769
$907$	30703.1	1.12401	0.562006	−	0.827133i	$-0.310030\pi$
$907$	30703.1	1.12401	0.562006	+	0.827133i	$0.310030\pi$
$908$	−18900.5	−0.690789
$909$	−6052.72	−0.220854
$910$	0	0
$911$	−34556.4	−1.25676	−0.628378	−	0.777908i	$-0.716281\pi$
$911$	−34556.4	−1.25676	−0.628378	+	0.777908i	$0.716281\pi$
$912$	−10091.9	−0.366420
$913$	−1877.75	−0.0680664
$914$	−35041.3	−1.26812
$915$	−5119.52	−0.184968
$916$	21156.5	0.763135
$917$	0	0
$918$	−14303.9	−0.514269
$919$	−45661.9	−1.63901	−0.819503	−	0.573075i	$-0.805750\pi$
$919$	−45661.9	−1.63901	−0.819503	+	0.573075i	$0.805750\pi$
$920$	1054.92	0.0378041
$921$	46705.0	1.67099
$922$	4595.14	0.164135
$923$	78283.7	2.79170
$924$	0	0
$925$	21267.8	0.755981
$926$	1509.67	0.0535756
$927$	−31470.1	−1.11501
$928$	103.574	0.00366378
$929$	−16229.2	−0.573158	−0.286579	−	0.958057i	$-0.592518\pi$
$929$	−16229.2	−0.573158	−0.286579	+	0.958057i	$0.592518\pi$
$930$	−28008.1	−0.987552
$931$	0	0
$932$	20795.8	0.730890
$933$	−55455.2	−1.94590
$934$	8792.50	0.308029
$935$	−3721.17	−0.130155
$936$	25007.3	0.873279
$937$	10960.3	0.382132	0.191066	−	0.981577i	$-0.438806\pi$
$937$	10960.3	0.382132	0.191066	+	0.981577i	$0.438806\pi$
$938$	0	0
$939$	65368.7	2.27181
$940$	−4794.74	−0.166369
$941$	50970.1	1.76576	0.882879	−	0.469600i	$-0.155602\pi$
$941$	50970.1	1.76576	0.882879	+	0.469600i	$0.155602\pi$
$942$	−46828.4	−1.61969
$943$	4132.86	0.142720
$944$	−229.500	−0.00791269
$945$	0	0
$946$	−1012.99	−0.0348152
$947$	47089.4	1.61584	0.807919	−	0.589294i	$-0.200594\pi$
$947$	47089.4	1.61584	0.807919	+	0.589294i	$0.200594\pi$
$948$	−17795.0	−0.609657
$949$	−61643.9	−2.10858
$950$	14414.5	0.492281
$951$	5355.75	0.182620
$952$	0	0
$953$	26950.6	0.916069	0.458035	−	0.888934i	$-0.348554\pi$
$953$	26950.6	0.916069	0.458035	+	0.888934i	$0.348554\pi$
$954$	−40193.5	−1.36406
$955$	16015.4	0.542667
$956$	−13753.1	−0.465279
$957$	210.191	0.00709981
$958$	15414.9	0.519869
$959$	0	0
$960$	−2958.45	−0.0994622
$961$	61987.0	2.08073
$962$	−37971.6	−1.27261
$963$	−50200.7	−1.67985
$964$	−16965.8	−0.566838
$965$	180.152	0.00600965
$966$	0	0
$967$	−19117.1	−0.635744	−0.317872	−	0.948134i	$-0.602968\pi$
$967$	−19117.1	−0.635744	−0.317872	+	0.948134i	$0.602968\pi$
$968$	10129.0	0.336321
$969$	50827.3	1.68504
$970$	7120.24	0.235688
$971$	42540.0	1.40595	0.702973	−	0.711217i	$-0.251856\pi$
$971$	42540.0	1.40595	0.702973	+	0.711217i	$0.251856\pi$
$972$	19603.1	0.646883
$973$	0	0
$974$	−28120.8	−0.925101
$975$	−61092.3	−2.00669
$976$	1772.00	0.0581152
$977$	−22248.0	−0.728534	−0.364267	−	0.931295i	$-0.618680\pi$
$977$	−22248.0	−0.728534	−0.364267	+	0.931295i	$0.618680\pi$
$978$	−38227.9	−1.24989
$979$	−1523.25	−0.0497275
$980$	0	0
$981$	40773.7	1.32702
$982$	4223.62	0.137252
$983$	−24224.8	−0.786014	−0.393007	−	0.919535i	$-0.628565\pi$
$983$	−24224.8	−0.786014	−0.393007	+	0.919535i	$0.628565\pi$
$984$	−11590.3	−0.375494
$985$	14059.1	0.454781
$986$	−521.647	−0.0168485
$987$	0	0
$988$	−25735.6	−0.828702
$989$	−1446.35	−0.0465027
$990$	−3510.23	−0.112689
$991$	−39354.7	−1.26150	−0.630748	−	0.775988i	$-0.717252\pi$
$991$	−39354.7	−1.26150	−0.630748	+	0.775988i	$0.717252\pi$
$992$	9694.36	0.310279
$993$	15051.5	0.481013
$994$	0	0
$995$	26317.9	0.838526
$996$	7518.83	0.239200
$997$	−36219.1	−1.15052	−0.575261	−	0.817970i	$-0.695100\pi$
$997$	−36219.1	−1.15052	−0.575261	+	0.817970i	$0.695100\pi$
$998$	−13715.8	−0.435037
$999$	20488.2	0.648866

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.bb.1.3	✓	18
7.6	odd	2	inner	2254.4.a.bb.1.16	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2254.4.a.bb.1.3	✓	18	1.1	even	1	trivial
2254.4.a.bb.1.16	yes	18	7.6	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2254.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -8.06273 q^{3} +4.00000 q^{4} +5.73327 q^{5} +16.1255 q^{6} -8.00000 q^{8} +38.0077 q^{9} -11.4665 q^{10} +8.05436 q^{11} -32.2509 q^{12} -82.2442 q^{13} -46.2259 q^{15} +16.0000 q^{16} -80.5834 q^{17} -76.0154 q^{18} +78.2292 q^{19} +22.9331 q^{20} -16.1087 q^{22} -23.0000 q^{23} +64.5019 q^{24} -92.1296 q^{25} +164.488 q^{26} -88.7521 q^{27} -3.23669 q^{29} +92.4517 q^{30} -302.949 q^{31} -32.0000 q^{32} -64.9402 q^{33} +161.167 q^{34} +152.031 q^{36} -230.847 q^{37} -156.458 q^{38} +663.113 q^{39} -45.8662 q^{40} -179.690 q^{41} +62.8847 q^{43} +32.2174 q^{44} +217.908 q^{45} +46.0000 q^{46} -209.075 q^{47} -129.004 q^{48} +184.259 q^{50} +649.723 q^{51} -328.977 q^{52} +528.755 q^{53} +177.504 q^{54} +46.1778 q^{55} -630.741 q^{57} +6.47338 q^{58} -14.3437 q^{59} -184.903 q^{60} +110.750 q^{61} +605.898 q^{62} +64.0000 q^{64} -471.528 q^{65} +129.880 q^{66} +875.144 q^{67} -322.334 q^{68} +185.443 q^{69} -951.845 q^{71} -304.062 q^{72} +749.523 q^{73} +461.694 q^{74} +742.816 q^{75} +312.917 q^{76} -1326.23 q^{78} +551.767 q^{79} +91.7324 q^{80} -310.623 q^{81} +359.379 q^{82} -233.135 q^{83} -462.007 q^{85} -125.769 q^{86} +26.0966 q^{87} -64.4349 q^{88} -189.121 q^{89} -435.817 q^{90} -92.0000 q^{92} +2442.60 q^{93} +418.150 q^{94} +448.509 q^{95} +258.008 q^{96} -620.958 q^{97} +306.128 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 36 q^{2} + 72 q^{4} - 144 q^{8} + 202 q^{9} + 76 q^{11} + 348 q^{15} + 288 q^{16} - 404 q^{18} - 152 q^{22} - 414 q^{23} + 662 q^{25} + 1164 q^{29} - 696 q^{30} - 576 q^{32} + 808 q^{36} + 188 q^{37}+ \cdots - 3864 q^{99}+O(q^{100}) \) 18 * q - 36 * q^2 + 72 * q^4 - 144 * q^8 + 202 * q^9 + 76 * q^11 + 348 * q^15 + 288 * q^16 - 404 * q^18 - 152 * q^22 - 414 * q^23 + 662 * q^25 + 1164 * q^29 - 696 * q^30 - 576 * q^32 + 808 * q^36 + 188 * q^37 + 1780 * q^39 - 248 * q^43 + 304 * q^44 + 828 * q^46 - 1324 * q^50 + 1596 * q^53 + 108 * q^57 - 2328 * q^58 + 1392 * q^60 + 1152 * q^64 + 1748 * q^65 + 172 * q^67 + 500 * q^71 - 1616 * q^72 - 376 * q^74 - 3560 * q^78 - 1772 * q^79 + 6382 * q^81 - 4344 * q^85 + 496 * q^86 - 608 * q^88 - 1656 * q^92 - 1192 * q^93 + 2456 * q^95 - 3864 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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