LMFDB - Embedded newform 275.4.a.k.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,4,Mod(1,275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, 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("275.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	275.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:	$16.2255252516$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 72x^{8} + 1771x^{6} - 17056x^{4} + 52892x^{2} - 3136 $ x^10 - 72x^8 + 1771x^6 - 17056x^4 + 52892x^2 - 3136
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.245890$ of defining polynomial
Character	$\chi$	$=$	275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.245890 q^{2} +2.52576 q^{3} -7.93954 q^{4} +0.621060 q^{6} -10.5016 q^{7} -3.91938 q^{8} -20.6205 q^{9} +11.0000 q^{11} -20.0534 q^{12} +63.0258 q^{13} -2.58224 q^{14} +62.5526 q^{16} +133.900 q^{17} -5.07039 q^{18} +76.1851 q^{19} -26.5245 q^{21} +2.70479 q^{22} -169.684 q^{23} -9.89940 q^{24} +15.4974 q^{26} -120.278 q^{27} +83.3777 q^{28} +202.349 q^{29} +191.639 q^{31} +46.7361 q^{32} +27.7833 q^{33} +32.9246 q^{34} +163.718 q^{36} -21.5915 q^{37} +18.7332 q^{38} +159.188 q^{39} +305.966 q^{41} -6.52211 q^{42} -285.731 q^{43} -87.3349 q^{44} -41.7236 q^{46} -123.908 q^{47} +157.993 q^{48} -232.717 q^{49} +338.198 q^{51} -500.396 q^{52} +480.068 q^{53} -29.5752 q^{54} +41.1597 q^{56} +192.425 q^{57} +49.7556 q^{58} +364.714 q^{59} +9.11965 q^{61} +47.1222 q^{62} +216.548 q^{63} -488.929 q^{64} +6.83166 q^{66} +568.827 q^{67} -1063.10 q^{68} -428.580 q^{69} -157.805 q^{71} +80.8197 q^{72} +212.831 q^{73} -5.30915 q^{74} -604.874 q^{76} -115.517 q^{77} +39.1428 q^{78} +792.027 q^{79} +252.962 q^{81} +75.2340 q^{82} -587.824 q^{83} +210.592 q^{84} -70.2586 q^{86} +511.084 q^{87} -43.1132 q^{88} +698.243 q^{89} -661.871 q^{91} +1347.21 q^{92} +484.033 q^{93} -30.4677 q^{94} +118.044 q^{96} -1837.23 q^{97} -57.2228 q^{98} -226.826 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 64 q^{4} + 26 q^{6} + 152 q^{9} + 110 q^{11} + 34 q^{14} + 468 q^{16} + 90 q^{19} + 302 q^{21} + 206 q^{24} + 392 q^{26} - 58 q^{29} + 1242 q^{31} - 66 q^{34} + 1786 q^{36} - 384 q^{39} + 416 q^{41}+ \cdots + 1672 q^{99}+O(q^{100}) $ 10 * q + 64 * q^4 + 26 * q^6 + 152 * q^9 + 110 * q^11 + 34 * q^14 + 468 * q^16 + 90 * q^19 + 302 * q^21 + 206 * q^24 + 392 * q^26 - 58 * q^29 + 1242 * q^31 - 66 * q^34 + 1786 * q^36 - 384 * q^39 + 416 * q^41 + 704 * q^44 + 1816 * q^46 + 1980 * q^49 + 510 * q^51 + 1882 * q^54 + 2626 * q^56 + 476 * q^59 + 1650 * q^61 + 2576 * q^64 + 286 * q^66 + 4492 * q^69 - 498 * q^71 - 5374 * q^74 + 1410 * q^76 - 416 * q^79 - 710 * q^81 - 9018 * q^84 - 6872 * q^86 + 1918 * q^89 + 1384 * q^91 - 2860 * q^94 - 10114 * q^96 + 1672 * 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 $

$2$	0.245890	0.0869354	0.0434677	−	0.999055i	$-0.486159\pi$
$2$	0.245890	0.0869354	0.0434677	+	0.999055i	$0.486159\pi$
$3$	2.52576	0.486082	0.243041	−	0.970016i	$-0.421855\pi$
$3$	2.52576	0.486082	0.243041	+	0.970016i	$0.421855\pi$
$4$	−7.93954	−0.992442
$5$	0	0
$5$	0	0
$6$	0.621060	0.0422578
$7$	−10.5016	−0.567032	−0.283516	−	0.958967i	$-0.591501\pi$
$7$	−10.5016	−0.567032	−0.283516	+	0.958967i	$0.591501\pi$
$8$	−3.91938	−0.173214
$9$	−20.6205	−0.763724
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−20.0534	−0.482409
$13$	63.0258	1.34463	0.672316	−	0.740265i	$-0.265300\pi$
$13$	63.0258	1.34463	0.672316	+	0.740265i	$0.265300\pi$
$14$	−2.58224	−0.0492952
$15$	0	0
$16$	62.5526	0.977384
$17$	133.900	1.91032	0.955160	−	0.296091i	$-0.0956832\pi$
$17$	133.900	1.91032	0.955160	+	0.296091i	$0.0956832\pi$
$18$	−5.07039	−0.0663946
$19$	76.1851	0.919898	0.459949	−	0.887945i	$-0.347868\pi$
$19$	76.1851	0.919898	0.459949	+	0.887945i	$0.347868\pi$
$20$	0	0
$21$	−26.5245	−0.275624
$22$	2.70479	0.0262120
$23$	−169.684	−1.53833	−0.769163	−	0.639053i	$-0.779326\pi$
$23$	−169.684	−1.53833	−0.769163	+	0.639053i	$0.779326\pi$
$24$	−9.89940	−0.0841961
$25$	0	0
$26$	15.4974	0.116896
$27$	−120.278	−0.857315
$28$	83.3777	0.562747
$29$	202.349	1.29570	0.647849	−	0.761769i	$-0.275669\pi$
$29$	202.349	1.29570	0.647849	+	0.761769i	$0.275669\pi$
$30$	0	0
$31$	191.639	1.11030	0.555151	−	0.831750i	$-0.312660\pi$
$31$	191.639	1.11030	0.555151	+	0.831750i	$0.312660\pi$
$32$	46.7361	0.258183
$33$	27.7833	0.146559
$34$	32.9246	0.166074
$35$	0	0
$36$	163.718	0.757952
$37$	−21.5915	−0.0959359	−0.0479679	−	0.998849i	$-0.515275\pi$
$37$	−21.5915	−0.0959359	−0.0479679	+	0.998849i	$0.515275\pi$
$38$	18.7332	0.0799717
$39$	159.188	0.653602
$40$	0	0
$41$	305.966	1.16546	0.582729	−	0.812666i	$-0.301985\pi$
$41$	305.966	1.16546	0.582729	+	0.812666i	$0.301985\pi$
$42$	−6.52211	−0.0239615
$43$	−285.731	−1.01334	−0.506670	−	0.862140i	$-0.669124\pi$
$43$	−285.731	−1.01334	−0.506670	+	0.862140i	$0.669124\pi$
$44$	−87.3349	−0.299233
$45$	0	0
$46$	−41.7236	−0.133735
$47$	−123.908	−0.384548	−0.192274	−	0.981341i	$-0.561586\pi$
$47$	−123.908	−0.384548	−0.192274	+	0.981341i	$0.561586\pi$
$48$	157.993	0.475089
$49$	−232.717	−0.678474
$50$	0	0
$51$	338.198	0.928573
$52$	−500.396	−1.33447
$53$	480.068	1.24420	0.622098	−	0.782939i	$-0.286280\pi$
$53$	480.068	1.24420	0.622098	+	0.782939i	$0.286280\pi$
$54$	−29.5752	−0.0745310
$55$	0	0
$56$	41.1597	0.0982178
$57$	192.425	0.447146
$58$	49.7556	0.112642
$59$	364.714	0.804776	0.402388	−	0.915469i	$-0.368180\pi$
$59$	364.714	0.804776	0.402388	+	0.915469i	$0.368180\pi$
$60$	0	0
$61$	9.11965	0.0191418	0.00957091	−	0.999954i	$-0.496953\pi$
$61$	9.11965	0.0191418	0.00957091	+	0.999954i	$0.496953\pi$
$62$	47.1222	0.0965245
$63$	216.548	0.433056
$64$	−488.929	−0.954939
$65$	0	0
$66$	6.83166	0.0127412
$67$	568.827	1.03721	0.518607	−	0.855013i	$-0.326451\pi$
$67$	568.827	1.03721	0.518607	+	0.855013i	$0.326451\pi$
$68$	−1063.10	−1.89588
$69$	−428.580	−0.747753
$70$	0	0
$71$	−157.805	−0.263775	−0.131887	−	0.991265i	$-0.542104\pi$
$71$	−157.805	−0.263775	−0.131887	+	0.991265i	$0.542104\pi$
$72$	80.8197	0.132287
$73$	212.831	0.341232	0.170616	−	0.985338i	$-0.445424\pi$
$73$	212.831	0.341232	0.170616	+	0.985338i	$0.445424\pi$
$74$	−5.30915	−0.00834022
$75$	0	0
$76$	−604.874	−0.912945
$77$	−115.517	−0.170967
$78$	39.1428	0.0568211
$79$	792.027	1.12798	0.563988	−	0.825783i	$-0.309267\pi$
$79$	792.027	1.12798	0.563988	+	0.825783i	$0.309267\pi$
$80$	0	0
$81$	252.962	0.346998
$82$	75.2340	0.101320
$83$	−587.824	−0.777375	−0.388687	−	0.921370i	$-0.627071\pi$
$83$	−587.824	−0.777375	−0.388687	+	0.921370i	$0.627071\pi$
$84$	210.592	0.273541
$85$	0	0
$86$	−70.2586	−0.0880951
$87$	511.084	0.629816
$88$	−43.1132	−0.0522259
$89$	698.243	0.831613	0.415807	−	0.909453i	$-0.363499\pi$
$89$	698.243	0.831613	0.415807	+	0.909453i	$0.363499\pi$
$90$	0	0
$91$	−661.871	−0.762449
$92$	1347.21	1.52670
$93$	484.033	0.539698
$94$	−30.4677	−0.0334309
$95$	0	0
$96$	118.044	0.125498
$97$	−1837.23	−1.92312	−0.961558	−	0.274601i	$-0.911454\pi$
$97$	−1837.23	−1.92312	−0.961558	+	0.274601i	$0.911454\pi$
$98$	−57.2228	−0.0589834
$99$	−226.826	−0.230271
$100$	0	0
$101$	81.1351	0.0799331	0.0399666	−	0.999201i	$-0.487275\pi$
$101$	81.1351	0.0799331	0.0399666	+	0.999201i	$0.487275\pi$
$102$	83.1597	0.0807258
$103$	17.0321	0.0162935	0.00814673	−	0.999967i	$-0.497407\pi$
$103$	17.0321	0.0162935	0.00814673	+	0.999967i	$0.497407\pi$
$104$	−247.022	−0.232909
$105$	0	0
$106$	118.044	0.108165
$107$	128.189	0.115817	0.0579087	−	0.998322i	$-0.481557\pi$
$107$	128.189	0.115817	0.0579087	+	0.998322i	$0.481557\pi$
$108$	954.952	0.850836
$109$	−206.739	−0.181670	−0.0908351	−	0.995866i	$-0.528954\pi$
$109$	−206.739	−0.181670	−0.0908351	+	0.995866i	$0.528954\pi$
$110$	0	0
$111$	−54.5350	−0.0466327
$112$	−656.901	−0.554208
$113$	291.669	0.242814	0.121407	−	0.992603i	$-0.461259\pi$
$113$	291.669	0.242814	0.121407	+	0.992603i	$0.461259\pi$
$114$	47.3155	0.0388728
$115$	0	0
$116$	−1606.56	−1.28590
$117$	−1299.63	−1.02693
$118$	89.6797	0.0699635
$119$	−1406.16	−1.08321
$120$	0	0
$121$	121.000	0.0909091
$122$	2.24243	0.00166410
$123$	772.795	0.566509
$124$	−1521.52	−1.10191
$125$	0	0
$126$	53.2472	0.0376479
$127$	1241.35	0.867336	0.433668	−	0.901073i	$-0.357219\pi$
$127$	1241.35	0.867336	0.433668	+	0.901073i	$0.357219\pi$
$128$	−494.112	−0.341201
$129$	−721.689	−0.492567
$130$	0	0
$131$	−2013.62	−1.34298	−0.671491	−	0.741013i	$-0.734346\pi$
$131$	−2013.62	−1.34298	−0.671491	+	0.741013i	$0.734346\pi$
$132$	−220.587	−0.145452
$133$	−800.064	−0.521612
$134$	139.869	0.0901706
$135$	0	0
$136$	−524.803	−0.330894
$137$	−214.973	−0.134061	−0.0670307	−	0.997751i	$-0.521353\pi$
$137$	−214.973	−0.134061	−0.0670307	+	0.997751i	$0.521353\pi$
$138$	−105.384	−0.0650062
$139$	2073.76	1.26543	0.632714	−	0.774386i	$-0.281941\pi$
$139$	2073.76	1.26543	0.632714	+	0.774386i	$0.281941\pi$
$140$	0	0
$141$	−312.960	−0.186922
$142$	−38.8028	−0.0229314
$143$	693.284	0.405422
$144$	−1289.87	−0.746451
$145$	0	0
$146$	52.3330	0.0296652
$147$	−587.786	−0.329794
$148$	171.427	0.0952108
$149$	−357.210	−0.196401	−0.0982007	−	0.995167i	$-0.531309\pi$
$149$	−357.210	−0.196401	−0.0982007	+	0.995167i	$0.531309\pi$
$150$	0	0
$151$	−1097.04	−0.591229	−0.295615	−	0.955307i	$-0.595524\pi$
$151$	−1097.04	−0.591229	−0.295615	+	0.955307i	$0.595524\pi$
$152$	−298.598	−0.159339
$153$	−2761.08	−1.45896
$154$	−28.4046	−0.0148631
$155$	0	0
$156$	−1263.88	−0.648662
$157$	804.745	0.409081	0.204540	−	0.978858i	$-0.434430\pi$
$157$	804.745	0.409081	0.204540	+	0.978858i	$0.434430\pi$
$158$	194.752	0.0980610
$159$	1212.54	0.604782
$160$	0	0
$161$	1781.95	0.872280
$162$	62.2008	0.0301664
$163$	−2112.00	−1.01487	−0.507437	−	0.861689i	$-0.669407\pi$
$163$	−2112.00	−1.01487	−0.507437	+	0.861689i	$0.669407\pi$
$164$	−2429.23	−1.15665
$165$	0	0
$166$	−144.540	−0.0675814
$167$	−45.0510	−0.0208752	−0.0104376	−	0.999946i	$-0.503322\pi$
$167$	−45.0510	−0.0208752	−0.0104376	+	0.999946i	$0.503322\pi$
$168$	103.959	0.0477419
$169$	1775.25	0.808034
$170$	0	0
$171$	−1570.98	−0.702548
$172$	2268.58	1.00568
$173$	1381.34	0.607058	0.303529	−	0.952822i	$-0.401835\pi$
$173$	1381.34	0.607058	0.303529	+	0.952822i	$0.401835\pi$
$174$	125.671	0.0547533
$175$	0	0
$176$	688.078	0.294692
$177$	921.180	0.391187
$178$	171.691	0.0722966
$179$	−92.1182	−0.0384650	−0.0192325	−	0.999815i	$-0.506122\pi$
$179$	−92.1182	−0.0384650	−0.0192325	+	0.999815i	$0.506122\pi$
$180$	0	0
$181$	−759.559	−0.311920	−0.155960	−	0.987763i	$-0.549847\pi$
$181$	−759.559	−0.311920	−0.155960	+	0.987763i	$0.549847\pi$
$182$	−162.748	−0.0662838
$183$	23.0340	0.00930450
$184$	665.054	0.266459
$185$	0	0
$186$	119.019	0.0469189
$187$	1472.90	0.575983
$188$	983.769	0.381642
$189$	1263.11	0.486125
$190$	0	0
$191$	1496.04	0.566752	0.283376	−	0.959009i	$-0.408546\pi$
$191$	1496.04	0.566752	0.283376	+	0.959009i	$0.408546\pi$
$192$	−1234.92	−0.464179
$193$	3483.32	1.29914	0.649572	−	0.760300i	$-0.274948\pi$
$193$	3483.32	1.29914	0.649572	+	0.760300i	$0.274948\pi$
$194$	−451.757	−0.167187
$195$	0	0
$196$	1847.66	0.673347
$197$	−1648.00	−0.596017	−0.298009	−	0.954563i	$-0.596322\pi$
$197$	−1648.00	−0.596017	−0.298009	+	0.954563i	$0.596322\pi$
$198$	−55.7743	−0.0200187
$199$	2761.91	0.983851	0.491925	−	0.870637i	$-0.336293\pi$
$199$	2761.91	0.983851	0.491925	+	0.870637i	$0.336293\pi$
$200$	0	0
$201$	1436.72	0.504171
$202$	19.9503	0.00694902
$203$	−2124.98	−0.734702
$204$	−2685.14	−0.921555
$205$	0	0
$206$	4.18804	0.00141648
$207$	3498.97	1.17486
$208$	3942.42	1.31422
$209$	838.036	0.277360
$210$	0	0
$211$	3345.09	1.09140	0.545701	−	0.837980i	$-0.316264\pi$
$211$	3345.09	1.09140	0.545701	+	0.837980i	$0.316264\pi$
$212$	−3811.52	−1.23479
$213$	−398.578	−0.128216
$214$	31.5204	0.0100686
$215$	0	0
$216$	471.415	0.148499
$217$	−2012.51	−0.629577
$218$	−50.8352	−0.0157936
$219$	537.559	0.165867
$220$	0	0
$221$	8439.13	2.56868
$222$	−13.4096	−0.00405403
$223$	−3935.15	−1.18169	−0.590846	−	0.806784i	$-0.701206\pi$
$223$	−3935.15	−1.18169	−0.590846	+	0.806784i	$0.701206\pi$
$224$	−490.803	−0.146398
$225$	0	0
$226$	71.7187	0.0211091
$227$	−3157.35	−0.923175	−0.461588	−	0.887095i	$-0.652720\pi$
$227$	−3157.35	−0.923175	−0.461588	+	0.887095i	$0.652720\pi$
$228$	−1527.77	−0.443767
$229$	3945.40	1.13851	0.569257	−	0.822160i	$-0.307231\pi$
$229$	3945.40	1.13851	0.569257	+	0.822160i	$0.307231\pi$
$230$	0	0
$231$	−291.769	−0.0831039
$232$	−793.082	−0.224433
$233$	144.268	0.0405636	0.0202818	−	0.999794i	$-0.493544\pi$
$233$	144.268	0.0405636	0.0202818	+	0.999794i	$0.493544\pi$
$234$	−319.566	−0.0892763
$235$	0	0
$236$	−2895.66	−0.798693
$237$	2000.47	0.548289
$238$	−345.761	−0.0941695
$239$	655.848	0.177503	0.0887517	−	0.996054i	$-0.471712\pi$
$239$	655.848	0.177503	0.0887517	+	0.996054i	$0.471712\pi$
$240$	0	0
$241$	1949.94	0.521191	0.260595	−	0.965448i	$-0.416081\pi$
$241$	1949.94	0.521191	0.260595	+	0.965448i	$0.416081\pi$
$242$	29.7527	0.00790322
$243$	3886.43	1.02598
$244$	−72.4058	−0.0189972
$245$	0	0
$246$	190.023	0.0492497
$247$	4801.62	1.23692
$248$	−751.105	−0.192320
$249$	−1484.70	−0.377868
$250$	0	0
$251$	−6016.61	−1.51301	−0.756504	−	0.653989i	$-0.773094\pi$
$251$	−6016.61	−1.51301	−0.756504	+	0.653989i	$0.773094\pi$
$252$	−1719.29	−0.429783
$253$	−1866.52	−0.463823
$254$	305.235	0.0754022
$255$	0	0
$256$	3789.93	0.925276
$257$	649.498	0.157644	0.0788221	−	0.996889i	$-0.474884\pi$
$257$	649.498	0.157644	0.0788221	+	0.996889i	$0.474884\pi$
$258$	−177.456	−0.0428215
$259$	226.745	0.0543987
$260$	0	0
$261$	−4172.54	−0.989555
$262$	−495.130	−0.116753
$263$	4534.73	1.06321	0.531604	−	0.846993i	$-0.321589\pi$
$263$	4534.73	1.06321	0.531604	+	0.846993i	$0.321589\pi$
$264$	−108.893	−0.0253861
$265$	0	0
$266$	−196.728	−0.0453465
$267$	1763.59	0.404233
$268$	−4516.23	−1.02937
$269$	−3873.76	−0.878019	−0.439009	−	0.898482i	$-0.644671\pi$
$269$	−3873.76	−0.878019	−0.439009	+	0.898482i	$0.644671\pi$
$270$	0	0
$271$	8052.53	1.80501	0.902503	−	0.430684i	$-0.141728\pi$
$271$	8052.53	1.80501	0.902503	+	0.430684i	$0.141728\pi$
$272$	8375.77	1.86712
$273$	−1671.73	−0.370613
$274$	−52.8599	−0.0116547
$275$	0	0
$276$	3402.73	0.742101
$277$	−6923.67	−1.50182	−0.750908	−	0.660407i	$-0.770384\pi$
$277$	−6923.67	−1.50182	−0.750908	+	0.660407i	$0.770384\pi$
$278$	509.919	0.110010
$279$	−3951.70	−0.847964
$280$	0	0
$281$	−1994.96	−0.423520	−0.211760	−	0.977322i	$-0.567920\pi$
$281$	−1994.96	−0.423520	−0.211760	+	0.977322i	$0.567920\pi$
$282$	−76.9540	−0.0162501
$283$	−4636.33	−0.973855	−0.486928	−	0.873442i	$-0.661882\pi$
$283$	−4636.33	−0.973855	−0.486928	+	0.873442i	$0.661882\pi$
$284$	1252.90	0.261781
$285$	0	0
$286$	170.472	0.0352455
$287$	−3213.12	−0.660853
$288$	−963.724	−0.197181
$289$	13016.1	2.64932
$290$	0	0
$291$	−4640.40	−0.934793
$292$	−1689.78	−0.338653
$293$	326.024	0.0650052	0.0325026	−	0.999472i	$-0.489652\pi$
$293$	326.024	0.0650052	0.0325026	+	0.999472i	$0.489652\pi$
$294$	−144.531	−0.0286708
$295$	0	0
$296$	84.6254	0.0166174
$297$	−1323.06	−0.258490
$298$	−87.8346	−0.0170742
$299$	−10694.4	−2.06848
$300$	0	0
$301$	3000.63	0.574597
$302$	−269.751	−0.0513987
$303$	204.928	0.0388541
$304$	4765.57	0.899093
$305$	0	0
$306$	−678.924	−0.126835
$307$	−2137.32	−0.397339	−0.198669	−	0.980067i	$-0.563662\pi$
$307$	−2137.32	−0.397339	−0.198669	+	0.980067i	$0.563662\pi$
$308$	917.155	0.169675
$309$	43.0191	0.00791996
$310$	0	0
$311$	2422.78	0.441747	0.220874	−	0.975302i	$-0.429109\pi$
$311$	2422.78	0.441747	0.220874	+	0.975302i	$0.429109\pi$
$312$	−623.918	−0.113213
$313$	4607.87	0.832115	0.416058	−	0.909338i	$-0.363411\pi$
$313$	4607.87	0.832115	0.416058	+	0.909338i	$0.363411\pi$
$314$	197.879	0.0355636
$315$	0	0
$316$	−6288.33	−1.11945
$317$	5773.18	1.02288	0.511442	−	0.859318i	$-0.329111\pi$
$317$	5773.18	1.02288	0.511442	+	0.859318i	$0.329111\pi$
$318$	298.151	0.0525770
$319$	2225.84	0.390668
$320$	0	0
$321$	323.774	0.0562968
$322$	438.164	0.0758320
$323$	10201.2	1.75730
$324$	−2008.40	−0.344376
$325$	0	0
$326$	−519.320	−0.0882285
$327$	−522.174	−0.0883066
$328$	−1199.20	−0.201873
$329$	1301.23	0.218051
$330$	0	0
$331$	8976.09	1.49054	0.745272	−	0.666760i	$-0.232319\pi$
$331$	8976.09	1.49054	0.745272	+	0.666760i	$0.232319\pi$
$332$	4667.05	0.771499
$333$	445.229	0.0732685
$334$	−11.0776	−0.00181479
$335$	0	0
$336$	−1659.17	−0.269391
$337$	3868.16	0.625258	0.312629	−	0.949875i	$-0.398790\pi$
$337$	3868.16	0.625258	0.312629	+	0.949875i	$0.398790\pi$
$338$	436.517	0.0702467
$339$	736.686	0.118027
$340$	0	0
$341$	2108.03	0.334769
$342$	−386.288	−0.0610763
$343$	6045.94	0.951749
$344$	1119.89	0.175524
$345$	0	0
$346$	339.657	0.0527749
$347$	−11307.6	−1.74935	−0.874677	−	0.484706i	$-0.838926\pi$
$347$	−11307.6	−1.74935	−0.874677	+	0.484706i	$0.838926\pi$
$348$	−4057.77	−0.625056
$349$	−3296.03	−0.505537	−0.252768	−	0.967527i	$-0.581341\pi$
$349$	−3296.03	−0.505537	−0.252768	+	0.967527i	$0.581341\pi$
$350$	0	0
$351$	−7580.61	−1.15277
$352$	514.097	0.0778451
$353$	−3675.46	−0.554179	−0.277089	−	0.960844i	$-0.589370\pi$
$353$	−3675.46	−0.554179	−0.277089	+	0.960844i	$0.589370\pi$
$354$	226.509	0.0340080
$355$	0	0
$356$	−5543.72	−0.825328
$357$	−3551.62	−0.526531
$358$	−22.6510	−0.00334397
$359$	−5319.46	−0.782034	−0.391017	−	0.920383i	$-0.627877\pi$
$359$	−5319.46	−0.782034	−0.391017	+	0.920383i	$0.627877\pi$
$360$	0	0
$361$	−1054.83	−0.153788
$362$	−186.768	−0.0271169
$363$	305.617	0.0441893
$364$	5254.95	0.756687
$365$	0	0
$366$	5.66385	0.000808891 0
$367$	−1408.85	−0.200386	−0.100193	−	0.994968i	$-0.531946\pi$
$367$	−1408.85	−0.200386	−0.100193	+	0.994968i	$0.531946\pi$
$368$	−10614.1	−1.50353
$369$	−6309.18	−0.890089
$370$	0	0
$371$	−5041.47	−0.705500
$372$	−3843.00	−0.535619
$373$	−7785.80	−1.08079	−0.540393	−	0.841413i	$-0.681725\pi$
$373$	−7785.80	−1.08079	−0.540393	+	0.841413i	$0.681725\pi$
$374$	362.171	0.0500733
$375$	0	0
$376$	485.641	0.0666091
$377$	12753.2	1.74224
$378$	310.586	0.0422615
$379$	355.192	0.0481398	0.0240699	−	0.999710i	$-0.492338\pi$
$379$	355.192	0.0481398	0.0240699	+	0.999710i	$0.492338\pi$
$380$	0	0
$381$	3135.34	0.421597
$382$	367.861	0.0492708
$383$	−2611.83	−0.348454	−0.174227	−	0.984705i	$-0.555743\pi$
$383$	−2611.83	−0.348454	−0.174227	+	0.984705i	$0.555743\pi$
$384$	−1248.01	−0.165852
$385$	0	0
$386$	856.515	0.112942
$387$	5891.94	0.773912
$388$	14586.7	1.90858
$389$	−5258.80	−0.685428	−0.342714	−	0.939440i	$-0.611346\pi$
$389$	−5258.80	−0.685428	−0.342714	+	0.939440i	$0.611346\pi$
$390$	0	0
$391$	−22720.6	−2.93869
$392$	912.105	0.117521
$393$	−5085.91	−0.652800
$394$	−405.228	−0.0518150
$395$	0	0
$396$	1800.89	0.228531
$397$	967.463	0.122306	0.0611531	−	0.998128i	$-0.480522\pi$
$397$	967.463	0.122306	0.0611531	+	0.998128i	$0.480522\pi$
$398$	679.126	0.0855315
$399$	−2020.77	−0.253546
$400$	0	0
$401$	−12713.0	−1.58318	−0.791592	−	0.611050i	$-0.790747\pi$
$401$	−12713.0	−1.58318	−0.791592	+	0.611050i	$0.790747\pi$
$402$	353.276	0.0438303
$403$	12078.2	1.49295
$404$	−644.175	−0.0793290
$405$	0	0
$406$	−522.513	−0.0638716
$407$	−237.507	−0.0289258
$408$	−1325.53	−0.160842
$409$	874.540	0.105729	0.0528646	−	0.998602i	$-0.483165\pi$
$409$	874.540	0.105729	0.0528646	+	0.998602i	$0.483165\pi$
$410$	0	0
$411$	−542.971	−0.0651649
$412$	−135.227	−0.0161703
$413$	−3830.08	−0.456334
$414$	860.363	0.102137
$415$	0	0
$416$	2945.58	0.347161
$417$	5237.83	0.615102
$418$	206.065	0.0241124
$419$	2682.23	0.312734	0.156367	−	0.987699i	$-0.450022\pi$
$419$	2682.23	0.312734	0.156367	+	0.987699i	$0.450022\pi$
$420$	0	0
$421$	6853.07	0.793345	0.396672	−	0.917960i	$-0.370165\pi$
$421$	6853.07	0.793345	0.396672	+	0.917960i	$0.370165\pi$
$422$	822.526	0.0948814
$423$	2555.04	0.293689
$424$	−1881.57	−0.215512
$425$	0	0
$426$	−98.0064	−0.0111465
$427$	−95.7708	−0.0108540
$428$	−1017.76	−0.114942
$429$	1751.07	0.197068
$430$	0	0
$431$	−12404.3	−1.38630	−0.693151	−	0.720793i	$-0.743778\pi$
$431$	−12404.3	−1.38630	−0.693151	+	0.720793i	$0.743778\pi$
$432$	−7523.70	−0.837926
$433$	−10034.7	−1.11372	−0.556858	−	0.830608i	$-0.687993\pi$
$433$	−10034.7	−1.11372	−0.556858	+	0.830608i	$0.687993\pi$
$434$	−494.857	−0.0547325
$435$	0	0
$436$	1641.42	0.180297
$437$	−12927.4	−1.41510
$438$	132.181	0.0144197
$439$	−1016.19	−0.110479	−0.0552393	−	0.998473i	$-0.517592\pi$
$439$	−1016.19	−0.110479	−0.0552393	+	0.998473i	$0.517592\pi$
$440$	0	0
$441$	4798.75	0.518167
$442$	2075.10	0.223309
$443$	−13425.8	−1.43991	−0.719955	−	0.694020i	$-0.755838\pi$
$443$	−13425.8	−1.43991	−0.719955	+	0.694020i	$0.755838\pi$
$444$	432.983	0.0462803
$445$	0	0
$446$	−967.616	−0.102731
$447$	−902.227	−0.0954672
$448$	5134.52	0.541481
$449$	−1006.56	−0.105796	−0.0528982	−	0.998600i	$-0.516846\pi$
$449$	−1006.56	−0.105796	−0.0528982	+	0.998600i	$0.516846\pi$
$450$	0	0
$451$	3365.62	0.351399
$452$	−2315.72	−0.240979
$453$	−2770.85	−0.287386
$454$	−776.363	−0.0802566
$455$	0	0
$456$	−754.187	−0.0774518
$457$	11568.6	1.18415	0.592077	−	0.805882i	$-0.298308\pi$
$457$	11568.6	1.18415	0.592077	+	0.805882i	$0.298308\pi$
$458$	970.137	0.0989771
$459$	−16105.2	−1.63775
$460$	0	0
$461$	−17359.4	−1.75381	−0.876906	−	0.480663i	$-0.840396\pi$
$461$	−17359.4	−1.75381	−0.876906	+	0.480663i	$0.840396\pi$
$462$	−71.7432	−0.00722467
$463$	−4837.20	−0.485537	−0.242768	−	0.970084i	$-0.578056\pi$
$463$	−4837.20	−0.485537	−0.242768	+	0.970084i	$0.578056\pi$
$464$	12657.4	1.26639
$465$	0	0
$466$	35.4742	0.00352641
$467$	−666.327	−0.0660256	−0.0330128	−	0.999455i	$-0.510510\pi$
$467$	−666.327	−0.0660256	−0.0330128	+	0.999455i	$0.510510\pi$
$468$	10318.4	1.01917
$469$	−5973.59	−0.588134
$470$	0	0
$471$	2032.59	0.198847
$472$	−1429.45	−0.139398
$473$	−3143.05	−0.305534
$474$	491.896	0.0476657
$475$	0	0
$476$	11164.2	1.07503
$477$	−9899.26	−0.950223
$478$	161.267	0.0154313
$479$	−6115.67	−0.583366	−0.291683	−	0.956515i	$-0.594215\pi$
$479$	−6115.67	−0.583366	−0.291683	+	0.956515i	$0.594215\pi$
$480$	0	0
$481$	−1360.82	−0.128998
$482$	479.473	0.0453099
$483$	4500.77	0.424000
$484$	−960.684	−0.0902220
$485$	0	0
$486$	955.635	0.0891944
$487$	−17161.4	−1.59683	−0.798417	−	0.602105i	$-0.794329\pi$
$487$	−17161.4	−1.59683	−0.798417	+	0.602105i	$0.794329\pi$
$488$	−35.7434	−0.00331563
$489$	−5334.40	−0.493313
$490$	0	0
$491$	16482.4	1.51495	0.757477	−	0.652862i	$-0.226432\pi$
$491$	16482.4	1.51495	0.757477	+	0.652862i	$0.226432\pi$
$492$	−6135.64	−0.562227
$493$	27094.4	2.47520
$494$	1180.67	0.107532
$495$	0	0
$496$	11987.5	1.08519
$497$	1657.20	0.149569
$498$	−365.074	−0.0328501
$499$	17008.9	1.52590	0.762951	−	0.646456i	$-0.223750\pi$
$499$	17008.9	1.52590	0.762951	+	0.646456i	$0.223750\pi$
$500$	0	0
$501$	−113.788	−0.0101470
$502$	−1479.43	−0.131534
$503$	12515.0	1.10938	0.554689	−	0.832058i	$-0.312837\pi$
$503$	12515.0	1.10938	0.554689	+	0.832058i	$0.312837\pi$
$504$	−848.735	−0.0750113
$505$	0	0
$506$	−458.959	−0.0403226
$507$	4483.85	0.392771
$508$	−9855.71	−0.860781
$509$	1833.43	0.159657	0.0798284	−	0.996809i	$-0.474563\pi$
$509$	1833.43	0.159657	0.0798284	+	0.996809i	$0.474563\pi$
$510$	0	0
$511$	−2235.06	−0.193490
$512$	4884.80	0.421640
$513$	−9163.39	−0.788642
$514$	159.705	0.0137049
$515$	0	0
$516$	5729.87	0.488844
$517$	−1362.98	−0.115946
$518$	55.7545	0.00472917
$519$	3488.92	0.295080
$520$	0	0
$521$	9607.94	0.807930	0.403965	−	0.914774i	$-0.367632\pi$
$521$	9607.94	0.807930	0.403965	+	0.914774i	$0.367632\pi$
$522$	−1025.99	−0.0860274
$523$	−22737.4	−1.90103	−0.950515	−	0.310680i	$-0.899443\pi$
$523$	−22737.4	−1.90103	−0.950515	+	0.310680i	$0.899443\pi$
$524$	15987.2	1.33283
$525$	0	0
$526$	1115.05	0.0924304
$527$	25660.4	2.12103
$528$	1737.92	0.143245
$529$	16625.5	1.36644
$530$	0	0
$531$	−7520.61	−0.614626
$532$	6352.14	0.517669
$533$	19283.7	1.56711
$534$	433.650	0.0351421
$535$	0	0
$536$	−2229.45	−0.179660
$537$	−232.668	−0.0186972
$538$	−952.519	−0.0763309
$539$	−2559.88	−0.204568
$540$	0	0
$541$	17169.2	1.36444	0.682218	−	0.731149i	$-0.261015\pi$
$541$	17169.2	1.36444	0.682218	+	0.731149i	$0.261015\pi$
$542$	1980.04	0.156919
$543$	−1918.46	−0.151619
$544$	6257.95	0.493212
$545$	0	0
$546$	−411.061	−0.0322194
$547$	8825.65	0.689868	0.344934	−	0.938627i	$-0.387901\pi$
$547$	8825.65	0.689868	0.344934	+	0.938627i	$0.387901\pi$
$548$	1706.79	0.133048
$549$	−188.052	−0.0146191
$550$	0	0
$551$	15416.0	1.19191
$552$	1679.77	0.129521
$553$	−8317.54	−0.639598
$554$	−1702.46	−0.130561
$555$	0	0
$556$	−16464.7	−1.25586
$557$	5127.25	0.390034	0.195017	−	0.980800i	$-0.437524\pi$
$557$	5127.25	0.390034	0.195017	+	0.980800i	$0.437524\pi$
$558$	−971.685	−0.0737181
$559$	−18008.5	−1.36257
$560$	0	0
$561$	3720.18	0.279975
$562$	−490.541	−0.0368189
$563$	−7376.40	−0.552182	−0.276091	−	0.961132i	$-0.589039\pi$
$563$	−7376.40	−0.552182	−0.276091	+	0.961132i	$0.589039\pi$
$564$	2484.76	0.185509
$565$	0	0
$566$	−1140.03	−0.0846625
$567$	−2656.50	−0.196759
$568$	618.498	0.0456894
$569$	−4994.96	−0.368013	−0.184007	−	0.982925i	$-0.558907\pi$
$569$	−4994.96	−0.368013	−0.184007	+	0.982925i	$0.558907\pi$
$570$	0	0
$571$	−17437.0	−1.27796	−0.638981	−	0.769223i	$-0.720644\pi$
$571$	−17437.0	−1.27796	−0.638981	+	0.769223i	$0.720644\pi$
$572$	−5504.35	−0.402358
$573$	3778.63	0.275488
$574$	−790.076	−0.0574515
$575$	0	0
$576$	10082.0	0.729309
$577$	−6214.31	−0.448362	−0.224181	−	0.974548i	$-0.571971\pi$
$577$	−6214.31	−0.448362	−0.224181	+	0.974548i	$0.571971\pi$
$578$	3200.54	0.230320
$579$	8798.02	0.631491
$580$	0	0
$581$	6173.09	0.440796
$582$	−1141.03	−0.0812666
$583$	5280.75	0.375139
$584$	−834.165	−0.0591061
$585$	0	0
$586$	80.1661	0.00565125
$587$	−17894.4	−1.25823	−0.629115	−	0.777312i	$-0.716583\pi$
$587$	−17894.4	−1.25823	−0.629115	+	0.777312i	$0.716583\pi$
$588$	4666.75	0.327302
$589$	14600.0	1.02136
$590$	0	0
$591$	−4162.46	−0.289714
$592$	−1350.61	−0.0937662
$593$	−9135.21	−0.632611	−0.316305	−	0.948657i	$-0.602442\pi$
$593$	−9135.21	−0.632611	−0.316305	+	0.948657i	$0.602442\pi$
$594$	−325.327	−0.0224719
$595$	0	0
$596$	2836.08	0.194917
$597$	6975.91	0.478233
$598$	−2629.66	−0.179824
$599$	−10733.3	−0.732138	−0.366069	−	0.930588i	$-0.619297\pi$
$599$	−10733.3	−0.732138	−0.366069	+	0.930588i	$0.619297\pi$
$600$	0	0
$601$	22991.7	1.56048	0.780241	−	0.625479i	$-0.215096\pi$
$601$	22991.7	1.56048	0.780241	+	0.625479i	$0.215096\pi$
$602$	737.827	0.0499528
$603$	−11729.5	−0.792145
$604$	8709.97	0.586761
$605$	0	0
$606$	50.3897	0.00337779
$607$	7737.80	0.517410	0.258705	−	0.965956i	$-0.416704\pi$
$607$	7737.80	0.517410	0.258705	+	0.965956i	$0.416704\pi$
$608$	3560.59	0.237502
$609$	−5367.19	−0.357126
$610$	0	0
$611$	−7809.37	−0.517076
$612$	21921.7	1.44793
$613$	7184.13	0.473351	0.236676	−	0.971589i	$-0.423942\pi$
$613$	7184.13	0.473351	0.236676	+	0.971589i	$0.423942\pi$
$614$	−525.545	−0.0345428
$615$	0	0
$616$	452.757	0.0296138
$617$	8472.83	0.552842	0.276421	−	0.961037i	$-0.410852\pi$
$617$	8472.83	0.552842	0.276421	+	0.961037i	$0.410852\pi$
$618$	10.5780	0.000688525 0
$619$	−19905.0	−1.29249	−0.646243	−	0.763132i	$-0.723661\pi$
$619$	−19905.0	−1.29249	−0.646243	+	0.763132i	$0.723661\pi$
$620$	0	0
$621$	20409.2	1.31883
$622$	595.739	0.0384035
$623$	−7332.66	−0.471552
$624$	9957.61	0.638820
$625$	0	0
$626$	1133.03	0.0723403
$627$	2116.68	0.134820
$628$	−6389.30	−0.405989
$629$	−2891.10	−0.183268
$630$	0	0
$631$	−10106.1	−0.637585	−0.318792	−	0.947825i	$-0.603277\pi$
$631$	−10106.1	−0.637585	−0.318792	+	0.947825i	$0.603277\pi$
$632$	−3104.26	−0.195381
$633$	8448.90	0.530511
$634$	1419.57	0.0889248
$635$	0	0
$636$	−9626.97	−0.600211
$637$	−14667.2	−0.912298
$638$	547.312	0.0339628
$639$	3254.03	0.201451
$640$	0	0
$641$	9871.80	0.608288	0.304144	−	0.952626i	$-0.401630\pi$
$641$	9871.80	0.608288	0.304144	+	0.952626i	$0.401630\pi$
$642$	79.6128	0.00489419
$643$	−17121.0	−1.05006	−0.525029	−	0.851085i	$-0.675945\pi$
$643$	−17121.0	−1.05006	−0.525029	+	0.851085i	$0.675945\pi$
$644$	−14147.8	−0.865688
$645$	0	0
$646$	2508.37	0.152771
$647$	−31134.9	−1.89187	−0.945934	−	0.324360i	$-0.894851\pi$
$647$	−31134.9	−1.89187	−0.945934	+	0.324360i	$0.894851\pi$
$648$	−991.453	−0.0601048
$649$	4011.86	0.242649
$650$	0	0
$651$	−5083.12	−0.306026
$652$	16768.3	1.00720
$653$	6956.15	0.416868	0.208434	−	0.978036i	$-0.433163\pi$
$653$	6956.15	0.416868	0.208434	+	0.978036i	$0.433163\pi$
$654$	−128.397	−0.00767697
$655$	0	0
$656$	19138.9	1.13910
$657$	−4388.69	−0.260607
$658$	319.959	0.0189564
$659$	−24539.2	−1.45055	−0.725273	−	0.688461i	$-0.758287\pi$
$659$	−24539.2	−1.45055	−0.725273	+	0.688461i	$0.758287\pi$
$660$	0	0
$661$	8010.03	0.471338	0.235669	−	0.971833i	$-0.424272\pi$
$661$	8010.03	0.471338	0.235669	+	0.971833i	$0.424272\pi$
$662$	2207.13	0.129581
$663$	21315.2	1.24859
$664$	2303.91	0.134652
$665$	0	0
$666$	109.478	0.00636963
$667$	−34335.3	−1.99320
$668$	357.684	0.0207174
$669$	−9939.24	−0.574400
$670$	0	0
$671$	100.316	0.00577148
$672$	−1239.65	−0.0711615
$673$	32455.9	1.85896	0.929481	−	0.368870i	$-0.120255\pi$
$673$	32455.9	1.85896	0.929481	+	0.368870i	$0.120255\pi$
$674$	951.143	0.0543571
$675$	0	0
$676$	−14094.7	−0.801927
$677$	−29149.4	−1.65480	−0.827402	−	0.561610i	$-0.810182\pi$
$677$	−29149.4	−1.65480	−0.827402	+	0.561610i	$0.810182\pi$
$678$	181.144	0.0102608
$679$	19293.8	1.09047
$680$	0	0
$681$	−7974.71	−0.448739
$682$	518.344	0.0291032
$683$	4649.07	0.260457	0.130228	−	0.991484i	$-0.458429\pi$
$683$	4649.07	0.260457	0.130228	+	0.991484i	$0.458429\pi$
$684$	12472.8	0.697238
$685$	0	0
$686$	1486.64	0.0827407
$687$	9965.14	0.553411
$688$	−17873.2	−0.990423
$689$	30256.7	1.67299
$690$	0	0
$691$	4286.03	0.235960	0.117980	−	0.993016i	$-0.462358\pi$
$691$	4286.03	0.235960	0.117980	+	0.993016i	$0.462358\pi$
$692$	−10967.2	−0.602470
$693$	2382.03	0.130571
$694$	−2780.44	−0.152081
$695$	0	0
$696$	−2003.13	−0.109093
$697$	40968.7	2.22640
$698$	−810.462	−0.0439490
$699$	364.387	0.0197173
$700$	0	0
$701$	22751.0	1.22581	0.612905	−	0.790157i	$-0.290001\pi$
$701$	22751.0	1.22581	0.612905	+	0.790157i	$0.290001\pi$
$702$	−1864.00	−0.100217
$703$	−1644.95	−0.0882512
$704$	−5378.21	−0.287925
$705$	0	0
$706$	−903.761	−0.0481777
$707$	−852.047	−0.0453247
$708$	−7313.74	−0.388231
$709$	−541.209	−0.0286679	−0.0143339	−	0.999897i	$-0.504563\pi$
$709$	−541.209	−0.0286679	−0.0143339	+	0.999897i	$0.504563\pi$
$710$	0	0
$711$	−16332.0	−0.861462
$712$	−2736.68	−0.144047
$713$	−32518.0	−1.70801
$714$	−873.308	−0.0457741
$715$	0	0
$716$	731.376	0.0381743
$717$	1656.51	0.0862813
$718$	−1308.00	−0.0679865
$719$	11819.9	0.613085	0.306543	−	0.951857i	$-0.400828\pi$
$719$	11819.9	0.613085	0.306543	+	0.951857i	$0.400828\pi$
$720$	0	0
$721$	−178.864	−0.00923892
$722$	−259.374	−0.0133696
$723$	4925.09	0.253342
$724$	6030.55	0.309563
$725$	0	0
$726$	75.1482	0.00384161
$727$	28.3305	0.00144528	0.000722642	−	1.00000i	$-0.499770\pi$
$727$	28.3305	0.00144528	0.000722642		1.00000i	$0.499770\pi$
$728$	2594.12	0.132067
$729$	2986.21	0.151715
$730$	0	0
$731$	−38259.3	−1.93580
$732$	−182.880	−0.00923418
$733$	3239.61	0.163244	0.0816220	−	0.996663i	$-0.473990\pi$
$733$	3239.61	0.163244	0.0816220	+	0.996663i	$0.473990\pi$
$734$	−346.423	−0.0174206
$735$	0	0
$736$	−7930.35	−0.397169
$737$	6257.10	0.312732
$738$	−1551.37	−0.0773802
$739$	−10380.6	−0.516720	−0.258360	−	0.966049i	$-0.583182\pi$
$739$	−10380.6	−0.516720	−0.258360	+	0.966049i	$0.583182\pi$
$740$	0	0
$741$	12127.7	0.601247
$742$	−1239.65	−0.0613329
$743$	12344.3	0.609515	0.304758	−	0.952430i	$-0.401425\pi$
$743$	12344.3	0.609515	0.304758	+	0.952430i	$0.401425\pi$
$744$	−1897.11	−0.0934831
$745$	0	0
$746$	−1914.45	−0.0939586
$747$	12121.3	0.593700
$748$	−11694.1	−0.571630
$749$	−1346.18	−0.0656722
$750$	0	0
$751$	−5340.50	−0.259491	−0.129745	−	0.991547i	$-0.541416\pi$
$751$	−5340.50	−0.259491	−0.129745	+	0.991547i	$0.541416\pi$
$752$	−7750.73	−0.375851
$753$	−15196.5	−0.735447
$754$	3135.89	0.151462
$755$	0	0
$756$	−10028.5	−0.482451
$757$	22616.5	1.08588	0.542939	−	0.839772i	$-0.317312\pi$
$757$	22616.5	1.08588	0.542939	+	0.839772i	$0.317312\pi$
$758$	87.3383	0.00418505
$759$	−4714.38	−0.225456
$760$	0	0
$761$	−36847.6	−1.75522	−0.877611	−	0.479373i	$-0.840864\pi$
$761$	−36847.6	−1.75522	−0.877611	+	0.479373i	$0.840864\pi$
$762$	770.950	0.0366517
$763$	2171.09	0.103013
$764$	−11877.9	−0.562468
$765$	0	0
$766$	−642.223	−0.0302930
$767$	22986.4	1.08213
$768$	9572.45	0.449760
$769$	−30448.8	−1.42785	−0.713923	−	0.700225i	$-0.753083\pi$
$769$	−30448.8	−1.42785	−0.713923	+	0.700225i	$0.753083\pi$
$770$	0	0
$771$	1640.47	0.0766281
$772$	−27656.0	−1.28933
$773$	8119.13	0.377781	0.188890	−	0.981998i	$-0.439511\pi$
$773$	8119.13	0.377781	0.188890	+	0.981998i	$0.439511\pi$
$774$	1448.77	0.0672804
$775$	0	0
$776$	7200.80	0.333110
$777$	572.704	0.0264423
$778$	−1293.09	−0.0595880
$779$	23310.0	1.07210
$780$	0	0
$781$	−1735.86	−0.0795311
$782$	−5586.77	−0.255476
$783$	−24338.1	−1.11082
$784$	−14557.0	−0.663130
$785$	0	0
$786$	−1250.58	−0.0567514
$787$	39807.6	1.80303	0.901517	−	0.432744i	$-0.142455\pi$
$787$	39807.6	1.80303	0.901517	+	0.432744i	$0.142455\pi$
$788$	13084.4	0.591513
$789$	11453.6	0.516807
$790$	0	0
$791$	−3062.99	−0.137683
$792$	889.017	0.0398862
$793$	574.773	0.0257387
$794$	237.890	0.0106327
$795$	0	0
$796$	−21928.3	−0.976415
$797$	26083.3	1.15924	0.579621	−	0.814886i	$-0.303201\pi$
$797$	26083.3	1.15924	0.579621	+	0.814886i	$0.303201\pi$
$798$	−496.887	−0.0220421
$799$	−16591.2	−0.734610
$800$	0	0
$801$	−14398.1	−0.635123
$802$	−3126.00	−0.137635
$803$	2341.14	0.102885
$804$	−11406.9	−0.500361
$805$	0	0
$806$	2969.91	0.129790
$807$	−9784.17	−0.426789
$808$	−317.999	−0.0138455
$809$	−24719.7	−1.07429	−0.537143	−	0.843491i	$-0.680496\pi$
$809$	−24719.7	−1.07429	−0.537143	+	0.843491i	$0.680496\pi$
$810$	0	0
$811$	−39107.6	−1.69328	−0.846642	−	0.532163i	$-0.821379\pi$
$811$	−39107.6	−1.69328	−0.846642	+	0.532163i	$0.821379\pi$
$812$	16871.4	0.729150
$813$	20338.7	0.877381
$814$	−58.4007	−0.00251467
$815$	0	0
$816$	21155.2	0.907572
$817$	−21768.5	−0.932170
$818$	215.041	0.00919161
$819$	13648.1	0.582301
$820$	0	0
$821$	−8149.66	−0.346438	−0.173219	−	0.984883i	$-0.555417\pi$
$821$	−8149.66	−0.346438	−0.173219	+	0.984883i	$0.555417\pi$
$822$	−133.511	−0.00566514
$823$	1052.83	0.0445923	0.0222962	−	0.999751i	$-0.492902\pi$
$823$	1052.83	0.0445923	0.0222962	+	0.999751i	$0.492902\pi$
$824$	−66.7554	−0.00282225
$825$	0	0
$826$	−941.779	−0.0396715
$827$	−14414.1	−0.606078	−0.303039	−	0.952978i	$-0.598001\pi$
$827$	−14414.1	−0.606078	−0.303039	+	0.952978i	$0.598001\pi$
$828$	−27780.2	−1.16598
$829$	25958.2	1.08753	0.543767	−	0.839236i	$-0.316997\pi$
$829$	25958.2	1.08753	0.543767	+	0.839236i	$0.316997\pi$
$830$	0	0
$831$	−17487.5	−0.730006
$832$	−30815.1	−1.28404
$833$	−31160.7	−1.29610
$834$	1287.93	0.0534741
$835$	0	0
$836$	−6653.62	−0.275263
$837$	−23049.9	−0.951879
$838$	659.535	0.0271877
$839$	−39.9909	−0.00164558	−0.000822789	−	1.00000i	$-0.500262\pi$
$839$	−39.9909	−0.00164558	−0.000822789		1.00000i	$0.500262\pi$
$840$	0	0
$841$	16556.0	0.678832
$842$	1685.10	0.0689697
$843$	−5038.78	−0.205866
$844$	−26558.5	−1.08315
$845$	0	0
$846$	628.260	0.0255319
$847$	−1270.69	−0.0515484
$848$	30029.5	1.21606
$849$	−11710.2	−0.473374
$850$	0	0
$851$	3663.73	0.147581
$852$	3164.52	0.127247
$853$	10756.0	0.431743	0.215871	−	0.976422i	$-0.430741\pi$
$853$	10756.0	0.431743	0.215871	+	0.976422i	$0.430741\pi$
$854$	−23.5491	−0.000943599 0
$855$	0	0
$856$	−502.420	−0.0200612
$857$	−37977.4	−1.51375	−0.756875	−	0.653560i	$-0.773275\pi$
$857$	−37977.4	−1.51375	−0.756875	+	0.653560i	$0.773275\pi$
$858$	430.570	0.0171322
$859$	−12471.3	−0.495361	−0.247681	−	0.968842i	$-0.579668\pi$
$859$	−12471.3	−0.495361	−0.247681	+	0.968842i	$0.579668\pi$
$860$	0	0
$861$	−8115.57	−0.321229
$862$	−3050.11	−0.120519
$863$	30920.2	1.21963	0.609813	−	0.792545i	$-0.291245\pi$
$863$	30920.2	1.21963	0.609813	+	0.792545i	$0.291245\pi$
$864$	−5621.32	−0.221344
$865$	0	0
$866$	−2467.45	−0.0968212
$867$	32875.5	1.28779
$868$	15978.4	0.624819
$869$	8712.30	0.340097
$870$	0	0
$871$	35850.8	1.39467
$872$	810.290	0.0314678
$873$	37884.7	1.46873
$874$	−3178.71	−0.123022
$875$	0	0
$876$	−4267.97	−0.164613
$877$	1896.09	0.0730061	0.0365030	−	0.999334i	$-0.488378\pi$
$877$	1896.09	0.0730061	0.0365030	+	0.999334i	$0.488378\pi$
$878$	−249.871	−0.00960450
$879$	823.457	0.0315979
$880$	0	0
$881$	−9451.40	−0.361437	−0.180718	−	0.983535i	$-0.557842\pi$
$881$	−9451.40	−0.361437	−0.180718	+	0.983535i	$0.557842\pi$
$882$	1179.97	0.0450471
$883$	−17623.7	−0.671669	−0.335834	−	0.941921i	$-0.609018\pi$
$883$	−17623.7	−0.671669	−0.335834	+	0.941921i	$0.609018\pi$
$884$	−67002.8	−2.54926
$885$	0	0
$886$	−3301.28	−0.125179
$887$	43411.7	1.64332	0.821658	−	0.569980i	$-0.193049\pi$
$887$	43411.7	1.64332	0.821658	+	0.569980i	$0.193049\pi$
$888$	213.743	0.00807743
$889$	−13036.1	−0.491807
$890$	0	0
$891$	2782.58	0.104624
$892$	31243.3	1.17276
$893$	−9439.91	−0.353745
$894$	−221.849	−0.00829948
$895$	0	0
$896$	5188.96	0.193472
$897$	−27011.6	−1.00545
$898$	−247.504	−0.00919745
$899$	38777.9	1.43862
$900$	0	0
$901$	64280.9	2.37681
$902$	827.574	0.0305490
$903$	7578.87	0.279301
$904$	−1143.16	−0.0420587
$905$	0	0
$906$	−681.325	−0.0249840
$907$	−37888.6	−1.38707	−0.693534	−	0.720423i	$-0.743947\pi$
$907$	−37888.6	−1.38707	−0.693534	+	0.720423i	$0.743947\pi$
$908$	25067.9	0.916198
$909$	−1673.05	−0.0610468
$910$	0	0
$911$	−42276.8	−1.53753	−0.768766	−	0.639530i	$-0.779129\pi$
$911$	−42276.8	−1.53753	−0.768766	+	0.639530i	$0.779129\pi$
$912$	12036.7	0.437033
$913$	−6466.07	−0.234387
$914$	2844.62	0.102945
$915$	0	0
$916$	−31324.7	−1.12991
$917$	21146.2	0.761514
$918$	−3960.11	−0.142378
$919$	32790.8	1.17701	0.588504	−	0.808494i	$-0.299717\pi$
$919$	32790.8	1.17701	0.588504	+	0.808494i	$0.299717\pi$
$920$	0	0
$921$	−5398.34	−0.193139
$922$	−4268.51	−0.152468
$923$	−9945.79	−0.354680
$924$	2316.51	0.0824758
$925$	0	0
$926$	−1189.42	−0.0422103
$927$	−351.212	−0.0124437
$928$	9457.00	0.334527
$929$	25997.1	0.918122	0.459061	−	0.888405i	$-0.348186\pi$
$929$	25997.1	0.918122	0.459061	+	0.888405i	$0.348186\pi$
$930$	0	0
$931$	−17729.5	−0.624127
$932$	−1145.42	−0.0402571
$933$	6119.36	0.214726
$934$	−163.843	−0.00573996
$935$	0	0
$936$	5093.73	0.177878
$937$	−23318.7	−0.813008	−0.406504	−	0.913649i	$-0.633252\pi$
$937$	−23318.7	−0.813008	−0.406504	+	0.913649i	$0.633252\pi$
$938$	−1468.85	−0.0511296
$939$	11638.4	0.404477
$940$	0	0
$941$	47178.4	1.63440	0.817201	−	0.576353i	$-0.195525\pi$
$941$	47178.4	1.63440	0.817201	+	0.576353i	$0.195525\pi$
$942$	499.795	0.0172868
$943$	−51917.4	−1.79285
$944$	22813.8	0.786575
$945$	0	0
$946$	−772.845	−0.0265617
$947$	−18988.7	−0.651584	−0.325792	−	0.945441i	$-0.605631\pi$
$947$	−18988.7	−0.651584	−0.325792	+	0.945441i	$0.605631\pi$
$948$	−15882.8	−0.544145
$949$	13413.8	0.458832
$950$	0	0
$951$	14581.7	0.497206
$952$	5511.27	0.187627
$953$	−33650.1	−1.14379	−0.571896	−	0.820326i	$-0.693792\pi$
$953$	−33650.1	−1.14379	−0.571896	+	0.820326i	$0.693792\pi$
$954$	−2434.13	−0.0826080
$955$	0	0
$956$	−5207.13	−0.176162
$957$	5621.93	0.189897
$958$	−1503.79	−0.0507151
$959$	2257.56	0.0760172
$960$	0	0
$961$	6934.46	0.232770
$962$	−334.613	−0.0112145
$963$	−2643.32	−0.0884526
$964$	−15481.7	−0.517252
$965$	0	0
$966$	1106.70	0.0368606
$967$	−25555.9	−0.849869	−0.424934	−	0.905224i	$-0.639703\pi$
$967$	−25555.9	−0.849869	−0.424934	+	0.905224i	$0.639703\pi$
$968$	−474.245	−0.0157467
$969$	25765.6	0.854192
$970$	0	0
$971$	23379.4	0.772690	0.386345	−	0.922354i	$-0.373737\pi$
$971$	23379.4	0.772690	0.386345	+	0.922354i	$0.373737\pi$
$972$	−30856.4	−1.01823
$973$	−21777.8	−0.717538
$974$	−4219.83	−0.138821
$975$	0	0
$976$	570.457	0.0187089
$977$	−5631.85	−0.184421	−0.0922103	−	0.995740i	$-0.529393\pi$
$977$	−5631.85	−0.184421	−0.0922103	+	0.995740i	$0.529393\pi$
$978$	−1311.68	−0.0428863
$979$	7680.67	0.250741
$980$	0	0
$981$	4263.08	0.138746
$982$	4052.88	0.131703
$983$	−24438.5	−0.792946	−0.396473	−	0.918046i	$-0.629766\pi$
$983$	−24438.5	−0.792946	−0.396473	+	0.918046i	$0.629766\pi$
$984$	−3028.88	−0.0981271
$985$	0	0
$986$	6662.26	0.215182
$987$	3286.58	0.105991
$988$	−38122.7	−1.22757
$989$	48484.0	1.55885
$990$	0	0
$991$	−13590.4	−0.435635	−0.217817	−	0.975990i	$-0.569894\pi$
$991$	−13590.4	−0.435635	−0.217817	+	0.975990i	$0.569894\pi$
$992$	8956.46	0.286661
$993$	22671.4	0.724528
$994$	407.491	0.0130028
$995$	0	0
$996$	11787.8	0.375012
$997$	24783.1	0.787249	0.393624	−	0.919271i	$-0.371221\pi$
$997$	24783.1	0.787249	0.393624	+	0.919271i	$0.371221\pi$
$998$	4182.34	0.132655
$999$	2596.99	0.0822473

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.4.a.k.1.6		10
3.2	odd	2		2475.4.a.bw.1.5		10
5.2	odd	4		55.4.b.b.34.6	yes	10
5.3	odd	4		55.4.b.b.34.5	✓	10
5.4	even	2	inner	275.4.a.k.1.5		10
15.2	even	4		495.4.c.b.199.5		10
15.8	even	4		495.4.c.b.199.6		10
15.14	odd	2		2475.4.a.bw.1.6		10
20.3	even	4		880.4.b.i.529.5		10
20.7	even	4		880.4.b.i.529.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.4.b.b.34.5	✓	10	5.3	odd	4
55.4.b.b.34.6	yes	10	5.2	odd	4
275.4.a.k.1.5		10	5.4	even	2	inner
275.4.a.k.1.6		10	1.1	even	1	trivial
495.4.c.b.199.5		10	15.2	even	4
495.4.c.b.199.6		10	15.8	even	4
880.4.b.i.529.5		10	20.3	even	4
880.4.b.i.529.6		10	20.7	even	4
2475.4.a.bw.1.5		10	3.2	odd	2
2475.4.a.bw.1.6		10	15.14	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:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	275.a (trivial)

\(f(q)\)	\(=\)	\(q+0.245890 q^{2} +2.52576 q^{3} -7.93954 q^{4} +0.621060 q^{6} -10.5016 q^{7} -3.91938 q^{8} -20.6205 q^{9} +11.0000 q^{11} -20.0534 q^{12} +63.0258 q^{13} -2.58224 q^{14} +62.5526 q^{16} +133.900 q^{17} -5.07039 q^{18} +76.1851 q^{19} -26.5245 q^{21} +2.70479 q^{22} -169.684 q^{23} -9.89940 q^{24} +15.4974 q^{26} -120.278 q^{27} +83.3777 q^{28} +202.349 q^{29} +191.639 q^{31} +46.7361 q^{32} +27.7833 q^{33} +32.9246 q^{34} +163.718 q^{36} -21.5915 q^{37} +18.7332 q^{38} +159.188 q^{39} +305.966 q^{41} -6.52211 q^{42} -285.731 q^{43} -87.3349 q^{44} -41.7236 q^{46} -123.908 q^{47} +157.993 q^{48} -232.717 q^{49} +338.198 q^{51} -500.396 q^{52} +480.068 q^{53} -29.5752 q^{54} +41.1597 q^{56} +192.425 q^{57} +49.7556 q^{58} +364.714 q^{59} +9.11965 q^{61} +47.1222 q^{62} +216.548 q^{63} -488.929 q^{64} +6.83166 q^{66} +568.827 q^{67} -1063.10 q^{68} -428.580 q^{69} -157.805 q^{71} +80.8197 q^{72} +212.831 q^{73} -5.30915 q^{74} -604.874 q^{76} -115.517 q^{77} +39.1428 q^{78} +792.027 q^{79} +252.962 q^{81} +75.2340 q^{82} -587.824 q^{83} +210.592 q^{84} -70.2586 q^{86} +511.084 q^{87} -43.1132 q^{88} +698.243 q^{89} -661.871 q^{91} +1347.21 q^{92} +484.033 q^{93} -30.4677 q^{94} +118.044 q^{96} -1837.23 q^{97} -57.2228 q^{98} -226.826 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 64 q^{4} + 26 q^{6} + 152 q^{9} + 110 q^{11} + 34 q^{14} + 468 q^{16} + 90 q^{19} + 302 q^{21} + 206 q^{24} + 392 q^{26} - 58 q^{29} + 1242 q^{31} - 66 q^{34} + 1786 q^{36} - 384 q^{39} + 416 q^{41}+ \cdots + 1672 q^{99}+O(q^{100}) \) 10 * q + 64 * q^4 + 26 * q^6 + 152 * q^9 + 110 * q^11 + 34 * q^14 + 468 * q^16 + 90 * q^19 + 302 * q^21 + 206 * q^24 + 392 * q^26 - 58 * q^29 + 1242 * q^31 - 66 * q^34 + 1786 * q^36 - 384 * q^39 + 416 * q^41 + 704 * q^44 + 1816 * q^46 + 1980 * q^49 + 510 * q^51 + 1882 * q^54 + 2626 * q^56 + 476 * q^59 + 1650 * q^61 + 2576 * q^64 + 286 * q^66 + 4492 * q^69 - 498 * q^71 - 5374 * q^74 + 1410 * q^76 - 416 * q^79 - 710 * q^81 - 9018 * q^84 - 6872 * q^86 + 1918 * q^89 + 1384 * q^91 - 2860 * q^94 - 10114 * q^96 + 1672 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more