LMFDB - Embedded newform 735.4.a.bb.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,4,Mod(1,735)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("735.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 735 = 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	735.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:	$43.3664038542$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 55x^{6} + 80x^{5} + 969x^{4} - 866x^{3} - 5783x^{2} + 2328x + 9992 $ x^8 - 2x^7 - 55x^6 + 80x^5 + 969x^4 - 866x^3 - 5783x^2 + 2328*x + 9992
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.83170$ of defining polynomial
Character	$\chi$	$=$	735.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.83170 q^{2} -3.00000 q^{3} -4.64489 q^{4} -5.00000 q^{5} -5.49509 q^{6} -23.1616 q^{8} +9.00000 q^{9} -9.15848 q^{10} -13.3138 q^{11} +13.9347 q^{12} -11.2211 q^{13} +15.0000 q^{15} -5.26591 q^{16} -121.753 q^{17} +16.4853 q^{18} +7.26156 q^{19} +23.2244 q^{20} -24.3868 q^{22} -36.9105 q^{23} +69.4848 q^{24} +25.0000 q^{25} -20.5536 q^{26} -27.0000 q^{27} -53.5141 q^{29} +27.4754 q^{30} -47.4976 q^{31} +175.647 q^{32} +39.9413 q^{33} -223.014 q^{34} -41.8040 q^{36} +316.983 q^{37} +13.3010 q^{38} +33.6632 q^{39} +115.808 q^{40} -403.708 q^{41} +205.692 q^{43} +61.8410 q^{44} -45.0000 q^{45} -67.6088 q^{46} +166.899 q^{47} +15.7977 q^{48} +45.7924 q^{50} +365.258 q^{51} +52.1206 q^{52} +629.682 q^{53} -49.4558 q^{54} +66.5689 q^{55} -21.7847 q^{57} -98.0215 q^{58} +171.889 q^{59} -69.6733 q^{60} +430.858 q^{61} -87.0012 q^{62} +363.860 q^{64} +56.1054 q^{65} +73.1604 q^{66} -81.0331 q^{67} +565.527 q^{68} +110.731 q^{69} +97.0016 q^{71} -208.454 q^{72} -209.220 q^{73} +580.616 q^{74} -75.0000 q^{75} -33.7292 q^{76} +61.6608 q^{78} +997.659 q^{79} +26.3296 q^{80} +81.0000 q^{81} -739.470 q^{82} +330.407 q^{83} +608.763 q^{85} +376.765 q^{86} +160.542 q^{87} +308.368 q^{88} +62.9012 q^{89} -82.4263 q^{90} +171.445 q^{92} +142.493 q^{93} +305.708 q^{94} -36.3078 q^{95} -526.942 q^{96} -1713.93 q^{97} -119.824 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} - 24 q^{3} + 50 q^{4} - 40 q^{5} - 6 q^{6} + 66 q^{8} + 72 q^{9} - 10 q^{10} + 64 q^{11} - 150 q^{12} + 120 q^{15} + 206 q^{16} - 48 q^{17} + 18 q^{18} - 80 q^{19} - 250 q^{20} + 452 q^{22}+ \cdots + 576 q^{99}+O(q^{100}) $ 8 * q + 2 * q^2 - 24 * q^3 + 50 * q^4 - 40 * q^5 - 6 * q^6 + 66 * q^8 + 72 * q^9 - 10 * q^10 + 64 * q^11 - 150 * q^12 + 120 * q^15 + 206 * q^16 - 48 * q^17 + 18 * q^18 - 80 * q^19 - 250 * q^20 + 452 * q^22 + 120 * q^23 - 198 * q^24 + 200 * q^25 - 272 * q^26 - 216 * q^27 + 76 * q^29 + 30 * q^30 - 20 * q^31 + 770 * q^32 - 192 * q^33 - 320 * q^34 + 450 * q^36 + 348 * q^37 + 236 * q^38 - 330 * q^40 - 944 * q^41 + 1116 * q^43 + 172 * q^44 - 360 * q^45 + 496 * q^46 + 208 * q^47 - 618 * q^48 + 50 * q^50 + 144 * q^51 + 2272 * q^52 + 1144 * q^53 - 54 * q^54 - 320 * q^55 + 240 * q^57 + 560 * q^58 - 596 * q^59 + 750 * q^60 - 740 * q^61 + 1184 * q^62 + 1298 * q^64 - 1356 * q^66 + 1964 * q^67 - 96 * q^68 - 360 * q^69 - 4 * q^71 + 594 * q^72 + 1500 * q^73 + 3368 * q^74 - 600 * q^75 + 2912 * q^76 + 816 * q^78 - 460 * q^79 - 1030 * q^80 + 648 * q^81 + 5644 * q^82 - 700 * q^83 + 240 * q^85 - 1396 * q^86 - 228 * q^87 + 6892 * q^88 - 90 * q^90 + 644 * q^92 + 60 * q^93 + 6692 * q^94 + 400 * q^95 - 2310 * q^96 + 2052 * q^97 + 576 * 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 $

$2$	1.83170	0.647603	0.323801	−	0.946125i	$-0.395039\pi$
$2$	1.83170	0.647603	0.323801	+	0.946125i	$0.395039\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−4.64489	−0.580611
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−5.49509	−0.373893
$7$	0	0
$7$	0	0
$8$	−23.1616	−1.02361
$9$	9.00000	0.333333
$10$	−9.15848	−0.289617
$11$	−13.3138	−0.364932	−0.182466	−	0.983212i	$-0.558408\pi$
$11$	−13.3138	−0.364932	−0.182466	+	0.983212i	$0.558408\pi$
$12$	13.9347	0.335216
$13$	−11.2211	−0.239397	−0.119699	−	0.992810i	$-0.538193\pi$
$13$	−11.2211	−0.239397	−0.119699	+	0.992810i	$0.538193\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	−5.26591	−0.0822799
$17$	−121.753	−1.73702	−0.868511	−	0.495671i	$-0.834922\pi$
$17$	−121.753	−1.73702	−0.868511	+	0.495671i	$0.834922\pi$
$18$	16.4853	0.215868
$19$	7.26156	0.0876799	0.0438399	−	0.999039i	$-0.486041\pi$
$19$	7.26156	0.0876799	0.0438399	+	0.999039i	$0.486041\pi$
$20$	23.2244	0.259657
$21$	0	0
$22$	−24.3868	−0.236331
$23$	−36.9105	−0.334624	−0.167312	−	0.985904i	$-0.553509\pi$
$23$	−36.9105	−0.334624	−0.167312	+	0.985904i	$0.553509\pi$
$24$	69.4848	0.590980
$25$	25.0000	0.200000
$26$	−20.5536	−0.155034
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−53.5141	−0.342666	−0.171333	−	0.985213i	$-0.554807\pi$
$29$	−53.5141	−0.342666	−0.171333	+	0.985213i	$0.554807\pi$
$30$	27.4754	0.167210
$31$	−47.4976	−0.275188	−0.137594	−	0.990489i	$-0.543937\pi$
$31$	−47.4976	−0.275188	−0.137594	+	0.990489i	$0.543937\pi$
$32$	175.647	0.970323
$33$	39.9413	0.210694
$34$	−223.014	−1.12490
$35$	0	0
$36$	−41.8040	−0.193537
$37$	316.983	1.40842	0.704211	−	0.709990i	$-0.251301\pi$
$37$	316.983	1.40842	0.704211	+	0.709990i	$0.251301\pi$
$38$	13.3010	0.0567817
$39$	33.6632	0.138216
$40$	115.808	0.457771
$41$	−403.708	−1.53777	−0.768885	−	0.639387i	$-0.779188\pi$
$41$	−403.708	−1.53777	−0.768885	+	0.639387i	$0.779188\pi$
$42$	0	0
$43$	205.692	0.729481	0.364741	−	0.931109i	$-0.381158\pi$
$43$	205.692	0.729481	0.364741	+	0.931109i	$0.381158\pi$
$44$	61.8410	0.211884
$45$	−45.0000	−0.149071
$46$	−67.6088	−0.216704
$47$	166.899	0.517972	0.258986	−	0.965881i	$-0.416612\pi$
$47$	166.899	0.517972	0.258986	+	0.965881i	$0.416612\pi$
$48$	15.7977	0.0475043
$49$	0	0
$50$	45.7924	0.129521
$51$	365.258	1.00287
$52$	52.1206	0.138997
$53$	629.682	1.63195	0.815976	−	0.578085i	$-0.196200\pi$
$53$	629.682	1.63195	0.815976	+	0.578085i	$0.196200\pi$
$54$	−49.4558	−0.124631
$55$	66.5689	0.163203
$56$	0	0
$57$	−21.7847	−0.0506220
$58$	−98.0215	−0.221911
$59$	171.889	0.379289	0.189644	−	0.981853i	$-0.439267\pi$
$59$	171.889	0.379289	0.189644	+	0.981853i	$0.439267\pi$
$60$	−69.6733	−0.149913
$61$	430.858	0.904356	0.452178	−	0.891928i	$-0.350647\pi$
$61$	430.858	0.904356	0.452178	+	0.891928i	$0.350647\pi$
$62$	−87.0012	−0.178212
$63$	0	0
$64$	363.860	0.710664
$65$	56.1054	0.107062
$66$	73.1604	0.136446
$67$	−81.0331	−0.147758	−0.0738788	−	0.997267i	$-0.523538\pi$
$67$	−81.0331	−0.147758	−0.0738788	+	0.997267i	$0.523538\pi$
$68$	565.527	1.00853
$69$	110.731	0.193196
$70$	0	0
$71$	97.0016	0.162140	0.0810702	−	0.996708i	$-0.474166\pi$
$71$	97.0016	0.162140	0.0810702	+	0.996708i	$0.474166\pi$
$72$	−208.454	−0.341203
$73$	−209.220	−0.335443	−0.167721	−	0.985834i	$-0.553641\pi$
$73$	−209.220	−0.335443	−0.167721	+	0.985834i	$0.553641\pi$
$74$	580.616	0.912098
$75$	−75.0000	−0.115470
$76$	−33.7292	−0.0509079
$77$	0	0
$78$	61.6608	0.0895091
$79$	997.659	1.42083	0.710414	−	0.703784i	$-0.248508\pi$
$79$	997.659	1.42083	0.710414	+	0.703784i	$0.248508\pi$
$80$	26.3296	0.0367967
$81$	81.0000	0.111111
$82$	−739.470	−0.995863
$83$	330.407	0.436950	0.218475	−	0.975843i	$-0.429892\pi$
$83$	330.407	0.436950	0.218475	+	0.975843i	$0.429892\pi$
$84$	0	0
$85$	608.763	0.776819
$86$	376.765	0.472414
$87$	160.542	0.197838
$88$	308.368	0.373547
$89$	62.9012	0.0749159	0.0374580	−	0.999298i	$-0.488074\pi$
$89$	62.9012	0.0749159	0.0374580	+	0.999298i	$0.488074\pi$
$90$	−82.4263	−0.0965389
$91$	0	0
$92$	171.445	0.194287
$93$	142.493	0.158880
$94$	305.708	0.335440
$95$	−36.3078	−0.0392116
$96$	−526.942	−0.560216
$97$	−1713.93	−1.79405	−0.897027	−	0.441977i	$-0.854277\pi$
$97$	−1713.93	−1.79405	−0.897027	+	0.441977i	$0.854277\pi$
$98$	0	0
$99$	−119.824	−0.121644
$100$	−116.122	−0.116122
$101$	590.869	0.582116	0.291058	−	0.956705i	$-0.405993\pi$
$101$	590.869	0.582116	0.291058	+	0.956705i	$0.405993\pi$
$102$	669.042	0.649461
$103$	1309.69	1.25289	0.626443	−	0.779467i	$-0.284510\pi$
$103$	1309.69	1.25289	0.626443	+	0.779467i	$0.284510\pi$
$104$	259.898	0.245049
$105$	0	0
$106$	1153.39	1.05686
$107$	−1567.21	−1.41596	−0.707980	−	0.706232i	$-0.750394\pi$
$107$	−1567.21	−1.41596	−0.707980	+	0.706232i	$0.750394\pi$
$108$	125.412	0.111739
$109$	1077.16	0.946540	0.473270	−	0.880917i	$-0.343073\pi$
$109$	1077.16	0.946540	0.473270	+	0.880917i	$0.343073\pi$
$110$	121.934	0.105690
$111$	−950.948	−0.813153
$112$	0	0
$113$	1273.29	1.06001	0.530005	−	0.847994i	$-0.322190\pi$
$113$	1273.29	1.06001	0.530005	+	0.847994i	$0.322190\pi$
$114$	−39.9029	−0.0327829
$115$	184.552	0.149649
$116$	248.567	0.198956
$117$	−100.990	−0.0797991
$118$	314.848	0.245628
$119$	0	0
$120$	−347.424	−0.264294
$121$	−1153.74	−0.866825
$122$	789.201	0.585663
$123$	1211.12	0.887832
$124$	220.621	0.159777
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1417.11	0.990143	0.495072	−	0.868852i	$-0.335142\pi$
$127$	1417.11	0.990143	0.495072	+	0.868852i	$0.335142\pi$
$128$	−738.697	−0.510095
$129$	−617.075	−0.421166
$130$	102.768	0.0693335
$131$	1641.98	1.09512	0.547558	−	0.836768i	$-0.315558\pi$
$131$	1641.98	1.09512	0.547558	+	0.836768i	$0.315558\pi$
$132$	−185.523	−0.122331
$133$	0	0
$134$	−148.428	−0.0956882
$135$	135.000	0.0860663
$136$	2819.99	1.77803
$137$	1170.21	0.729763	0.364881	−	0.931054i	$-0.381110\pi$
$137$	1170.21	0.729763	0.364881	+	0.931054i	$0.381110\pi$
$138$	202.826	0.125114
$139$	−2779.78	−1.69624	−0.848122	−	0.529802i	$-0.822266\pi$
$139$	−2779.78	−1.69624	−0.848122	+	0.529802i	$0.822266\pi$
$140$	0	0
$141$	−500.696	−0.299051
$142$	177.678	0.105003
$143$	149.395	0.0873638
$144$	−47.3932	−0.0274266
$145$	267.570	0.153245
$146$	−383.227	−0.217233
$147$	0	0
$148$	−1472.35	−0.817746
$149$	−596.391	−0.327908	−0.163954	−	0.986468i	$-0.552425\pi$
$149$	−596.391	−0.327908	−0.163954	+	0.986468i	$0.552425\pi$
$150$	−137.377	−0.0747787
$151$	−1714.83	−0.924177	−0.462088	−	0.886834i	$-0.652900\pi$
$151$	−1714.83	−0.924177	−0.462088	+	0.886834i	$0.652900\pi$
$152$	−168.189	−0.0897498
$153$	−1095.77	−0.579007
$154$	0	0
$155$	237.488	0.123068
$156$	−156.362	−0.0802498
$157$	−1009.54	−0.513186	−0.256593	−	0.966520i	$-0.582600\pi$
$157$	−1009.54	−0.513186	−0.256593	+	0.966520i	$0.582600\pi$
$158$	1827.41	0.920132
$159$	−1889.05	−0.942208
$160$	−878.236	−0.433942
$161$	0	0
$162$	148.367	0.0719558
$163$	1578.01	0.758277	0.379138	−	0.925340i	$-0.376220\pi$
$163$	1578.01	0.758277	0.379138	+	0.925340i	$0.376220\pi$
$164$	1875.18	0.892846
$165$	−199.707	−0.0942251
$166$	605.205	0.282970
$167$	−1136.75	−0.526733	−0.263367	−	0.964696i	$-0.584833\pi$
$167$	−1136.75	−0.526733	−0.263367	+	0.964696i	$0.584833\pi$
$168$	0	0
$169$	−2071.09	−0.942689
$170$	1115.07	0.503070
$171$	65.3541	0.0292266
$172$	−955.415	−0.423545
$173$	159.521	0.0701048	0.0350524	−	0.999385i	$-0.488840\pi$
$173$	159.521	0.0701048	0.0350524	+	0.999385i	$0.488840\pi$
$174$	294.065	0.128121
$175$	0	0
$176$	70.1092	0.0300266
$177$	−515.667	−0.218982
$178$	115.216	0.0485158
$179$	702.548	0.293357	0.146679	−	0.989184i	$-0.453142\pi$
$179$	702.548	0.293357	0.146679	+	0.989184i	$0.453142\pi$
$180$	209.020	0.0865524
$181$	1407.17	0.577867	0.288933	−	0.957349i	$-0.406699\pi$
$181$	1407.17	0.577867	0.288933	+	0.957349i	$0.406699\pi$
$182$	0	0
$183$	−1292.57	−0.522130
$184$	854.905	0.342524
$185$	−1584.91	−0.629866
$186$	261.004	0.102891
$187$	1620.99	0.633895
$188$	−775.226	−0.300740
$189$	0	0
$190$	−66.5049	−0.0253935
$191$	−3630.53	−1.37537	−0.687686	−	0.726008i	$-0.741373\pi$
$191$	−3630.53	−1.37537	−0.687686	+	0.726008i	$0.741373\pi$
$192$	−1091.58	−0.410302
$193$	−3155.74	−1.17697	−0.588486	−	0.808508i	$-0.700276\pi$
$193$	−3155.74	−1.17697	−0.588486	+	0.808508i	$0.700276\pi$
$194$	−3139.40	−1.16183
$195$	−168.316	−0.0618121
$196$	0	0
$197$	2368.58	0.856622	0.428311	−	0.903631i	$-0.359109\pi$
$197$	2368.58	0.856622	0.428311	+	0.903631i	$0.359109\pi$
$198$	−219.481	−0.0787770
$199$	3803.91	1.35504	0.677518	−	0.735506i	$-0.263056\pi$
$199$	3803.91	1.35504	0.677518	+	0.735506i	$0.263056\pi$
$200$	−579.040	−0.204722
$201$	243.099	0.0853079
$202$	1082.29	0.376980
$203$	0	0
$204$	−1696.58	−0.582277
$205$	2018.54	0.687711
$206$	2398.95	0.811372
$207$	−332.194	−0.111541
$208$	59.0892	0.0196976
$209$	−96.6788	−0.0319972
$210$	0	0
$211$	−326.870	−0.106648	−0.0533239	−	0.998577i	$-0.516982\pi$
$211$	−326.870	−0.106648	−0.0533239	+	0.998577i	$0.516982\pi$
$212$	−2924.80	−0.947530
$213$	−291.005	−0.0936118
$214$	−2870.65	−0.916980
$215$	−1028.46	−0.326234
$216$	625.363	0.196993
$217$	0	0
$218$	1973.02	0.612982
$219$	627.659	0.193668
$220$	−309.205	−0.0947572
$221$	1366.20	0.415838
$222$	−1741.85	−0.526600
$223$	4109.05	1.23391	0.616956	−	0.786998i	$-0.288366\pi$
$223$	4109.05	1.23391	0.616956	+	0.786998i	$0.288366\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	2332.28	0.686466
$227$	−5449.66	−1.59342	−0.796710	−	0.604361i	$-0.793428\pi$
$227$	−5449.66	−1.59342	−0.796710	+	0.604361i	$0.793428\pi$
$228$	101.187	0.0293917
$229$	3689.67	1.06472	0.532359	−	0.846519i	$-0.321306\pi$
$229$	3689.67	1.06472	0.532359	+	0.846519i	$0.321306\pi$
$230$	338.044	0.0969128
$231$	0	0
$232$	1239.47	0.350755
$233$	1884.41	0.529835	0.264918	−	0.964271i	$-0.414655\pi$
$233$	1884.41	0.529835	0.264918	+	0.964271i	$0.414655\pi$
$234$	−184.982	−0.0516781
$235$	−834.494	−0.231644
$236$	−798.405	−0.220219
$237$	−2992.98	−0.820315
$238$	0	0
$239$	−5885.01	−1.59276	−0.796381	−	0.604796i	$-0.793255\pi$
$239$	−5885.01	−1.59276	−0.796381	+	0.604796i	$0.793255\pi$
$240$	−78.9887	−0.0212446
$241$	1643.66	0.439324	0.219662	−	0.975576i	$-0.429504\pi$
$241$	1643.66	0.439324	0.219662	+	0.975576i	$0.429504\pi$
$242$	−2113.31	−0.561358
$243$	−243.000	−0.0641500
$244$	−2001.29	−0.525079
$245$	0	0
$246$	2218.41	0.574962
$247$	−81.4826	−0.0209903
$248$	1100.12	0.281684
$249$	−991.221	−0.252273
$250$	−228.962	−0.0579233
$251$	3803.30	0.956422	0.478211	−	0.878245i	$-0.341285\pi$
$251$	3803.30	0.956422	0.478211	+	0.878245i	$0.341285\pi$
$252$	0	0
$253$	491.417	0.122115
$254$	2595.72	0.641219
$255$	−1826.29	−0.448497
$256$	−4263.95	−1.04100
$257$	−3752.95	−0.910906	−0.455453	−	0.890260i	$-0.650523\pi$
$257$	−3752.95	−0.910906	−0.455453	+	0.890260i	$0.650523\pi$
$258$	−1130.29	−0.272748
$259$	0	0
$260$	−260.603	−0.0621612
$261$	−481.627	−0.114222
$262$	3007.60	0.709200
$263$	5889.59	1.38087	0.690433	−	0.723397i	$-0.257420\pi$
$263$	5889.59	1.38087	0.690433	+	0.723397i	$0.257420\pi$
$264$	−925.105	−0.215668
$265$	−3148.41	−0.729831
$266$	0	0
$267$	−188.704	−0.0432527
$268$	376.389	0.0857897
$269$	−5530.04	−1.25343	−0.626715	−	0.779249i	$-0.715601\pi$
$269$	−5530.04	−1.25343	−0.626715	+	0.779249i	$0.715601\pi$
$270$	247.279	0.0557368
$271$	−44.6615	−0.0100110	−0.00500552	−	0.999987i	$-0.501593\pi$
$271$	−44.6615	−0.0100110	−0.00500552	+	0.999987i	$0.501593\pi$
$272$	641.139	0.142922
$273$	0	0
$274$	2143.46	0.472596
$275$	−332.844	−0.0729864
$276$	−514.335	−0.112171
$277$	−3276.62	−0.710734	−0.355367	−	0.934727i	$-0.615644\pi$
$277$	−3276.62	−0.710734	−0.355367	+	0.934727i	$0.615644\pi$
$278$	−5091.71	−1.09849
$279$	−427.478	−0.0917293
$280$	0	0
$281$	731.877	0.155374	0.0776871	−	0.996978i	$-0.475246\pi$
$281$	731.877	0.155374	0.0776871	+	0.996978i	$0.475246\pi$
$282$	−917.124	−0.193666
$283$	−2793.06	−0.586678	−0.293339	−	0.956008i	$-0.594766\pi$
$283$	−2793.06	−0.586678	−0.293339	+	0.956008i	$0.594766\pi$
$284$	−450.562	−0.0941405
$285$	108.923	0.0226388
$286$	273.646	0.0565770
$287$	0	0
$288$	1580.82	0.323441
$289$	9910.71	2.01724
$290$	490.108	0.0992418
$291$	5141.79	1.03580
$292$	971.802	0.194762
$293$	9681.56	1.93039	0.965193	−	0.261539i	$-0.0842301\pi$
$293$	9681.56	1.93039	0.965193	+	0.261539i	$0.0842301\pi$
$294$	0	0
$295$	−859.445	−0.169623
$296$	−7341.83	−1.44167
$297$	359.472	0.0702312
$298$	−1092.41	−0.212354
$299$	414.175	0.0801082
$300$	348.367	0.0670432
$301$	0	0
$302$	−3141.04	−0.598499
$303$	−1772.61	−0.336085
$304$	−38.2388	−0.00721429
$305$	−2154.29	−0.404440
$306$	−2007.13	−0.374966
$307$	8369.52	1.55594	0.777971	−	0.628301i	$-0.216249\pi$
$307$	8369.52	1.55594	0.777971	+	0.628301i	$0.216249\pi$
$308$	0	0
$309$	−3929.06	−0.723354
$310$	435.006	0.0796990
$311$	−8821.56	−1.60844	−0.804219	−	0.594332i	$-0.797416\pi$
$311$	−8821.56	−1.60844	−0.804219	+	0.594332i	$0.797416\pi$
$312$	−779.694	−0.141479
$313$	10237.7	1.84878	0.924391	−	0.381447i	$-0.124574\pi$
$313$	10237.7	1.84878	0.924391	+	0.381447i	$0.124574\pi$
$314$	−1849.17	−0.332341
$315$	0	0
$316$	−4634.01	−0.824948
$317$	−2085.93	−0.369582	−0.184791	−	0.982778i	$-0.559161\pi$
$317$	−2085.93	−0.369582	−0.184791	+	0.982778i	$0.559161\pi$
$318$	−3460.16	−0.610176
$319$	712.474	0.125050
$320$	−1819.30	−0.317818
$321$	4701.63	0.817505
$322$	0	0
$323$	−884.115	−0.152302
$324$	−376.236	−0.0645123
$325$	−280.527	−0.0478795
$326$	2890.43	0.491062
$327$	−3231.47	−0.546485
$328$	9350.52	1.57407
$329$	0	0
$330$	−365.802	−0.0610204
$331$	160.030	0.0265742	0.0132871	−	0.999912i	$-0.495770\pi$
$331$	160.030	0.0265742	0.0132871	+	0.999912i	$0.495770\pi$
$332$	−1534.70	−0.253698
$333$	2852.84	0.469474
$334$	−2082.18	−0.341114
$335$	405.165	0.0660792
$336$	0	0
$337$	9314.75	1.50566	0.752829	−	0.658217i	$-0.228689\pi$
$337$	9314.75	1.50566	0.752829	+	0.658217i	$0.228689\pi$
$338$	−3793.60	−0.610488
$339$	−3819.87	−0.611997
$340$	−2827.64	−0.451030
$341$	632.372	0.100425
$342$	119.709	0.0189272
$343$	0	0
$344$	−4764.15	−0.746703
$345$	−553.657	−0.0863997
$346$	292.193	0.0454000
$347$	−5906.18	−0.913719	−0.456860	−	0.889539i	$-0.651026\pi$
$347$	−5906.18	−0.913719	−0.456860	+	0.889539i	$0.651026\pi$
$348$	−745.700	−0.114867
$349$	9121.45	1.39903	0.699513	−	0.714620i	$-0.253400\pi$
$349$	9121.45	1.39903	0.699513	+	0.714620i	$0.253400\pi$
$350$	0	0
$351$	302.969	0.0460720
$352$	−2338.53	−0.354102
$353$	−330.631	−0.0498519	−0.0249260	−	0.999689i	$-0.507935\pi$
$353$	−330.631	−0.0498519	−0.0249260	+	0.999689i	$0.507935\pi$
$354$	−944.545	−0.141814
$355$	−485.008	−0.0725114
$356$	−292.169	−0.0434970
$357$	0	0
$358$	1286.85	0.189979
$359$	9546.43	1.40346	0.701729	−	0.712444i	$-0.252412\pi$
$359$	9546.43	1.40346	0.701729	+	0.712444i	$0.252412\pi$
$360$	1042.27	0.152590
$361$	−6806.27	−0.992312
$362$	2577.50	0.374228
$363$	3461.23	0.500461
$364$	0	0
$365$	1046.10	0.150014
$366$	−2367.60	−0.338133
$367$	−7546.11	−1.07331	−0.536653	−	0.843803i	$-0.680312\pi$
$367$	−7546.11	−1.07331	−0.536653	+	0.843803i	$0.680312\pi$
$368$	194.367	0.0275329
$369$	−3633.37	−0.512590
$370$	−2903.08	−0.407903
$371$	0	0
$372$	−661.863	−0.0922473
$373$	−9406.12	−1.30571	−0.652856	−	0.757482i	$-0.726429\pi$
$373$	−9406.12	−1.30571	−0.652856	+	0.757482i	$0.726429\pi$
$374$	2969.16	0.410512
$375$	375.000	0.0516398
$376$	−3865.64	−0.530200
$377$	600.485	0.0820333
$378$	0	0
$379$	−13028.9	−1.76584	−0.882918	−	0.469526i	$-0.844425\pi$
$379$	−13028.9	−1.76584	−0.882918	+	0.469526i	$0.844425\pi$
$380$	168.646	0.0227667
$381$	−4251.33	−0.571659
$382$	−6650.03	−0.890695
$383$	8038.42	1.07244	0.536219	−	0.844079i	$-0.319852\pi$
$383$	8038.42	1.07244	0.536219	+	0.844079i	$0.319852\pi$
$384$	2216.09	0.294504
$385$	0	0
$386$	−5780.37	−0.762210
$387$	1851.23	0.243160
$388$	7961.01	1.04165
$389$	−5950.65	−0.775604	−0.387802	−	0.921743i	$-0.626766\pi$
$389$	−5950.65	−0.775604	−0.387802	+	0.921743i	$0.626766\pi$
$390$	−308.304	−0.0400297
$391$	4493.95	0.581250
$392$	0	0
$393$	−4925.93	−0.632266
$394$	4338.53	0.554751
$395$	−4988.30	−0.635414
$396$	556.569	0.0706279
$397$	11519.9	1.45634	0.728172	−	0.685395i	$-0.240370\pi$
$397$	11519.9	1.45634	0.728172	+	0.685395i	$0.240370\pi$
$398$	6967.61	0.877524
$399$	0	0
$400$	−131.648	−0.0164560
$401$	8174.35	1.01797	0.508987	−	0.860774i	$-0.330020\pi$
$401$	8174.35	1.01797	0.508987	+	0.860774i	$0.330020\pi$
$402$	445.284	0.0552456
$403$	532.974	0.0658792
$404$	−2744.52	−0.337983
$405$	−405.000	−0.0496904
$406$	0	0
$407$	−4220.24	−0.513979
$408$	−8459.96	−1.02654
$409$	125.495	0.0151720	0.00758598	−	0.999971i	$-0.497585\pi$
$409$	125.495	0.0151720	0.00758598	+	0.999971i	$0.497585\pi$
$410$	3697.35	0.445364
$411$	−3510.62	−0.421329
$412$	−6083.35	−0.727439
$413$	0	0
$414$	−608.479	−0.0722345
$415$	−1652.04	−0.195410
$416$	−1970.95	−0.232293
$417$	8339.34	0.979326
$418$	−177.086	−0.0207215
$419$	−16735.8	−1.95131	−0.975654	−	0.219314i	$-0.929618\pi$
$419$	−16735.8	−1.95131	−0.975654	+	0.219314i	$0.929618\pi$
$420$	0	0
$421$	9780.23	1.13221	0.566104	−	0.824334i	$-0.308450\pi$
$421$	9780.23	1.13221	0.566104	+	0.824334i	$0.308450\pi$
$422$	−598.727	−0.0690654
$423$	1502.09	0.172657
$424$	−14584.4	−1.67048
$425$	−3043.82	−0.347404
$426$	−533.033	−0.0606233
$427$	0	0
$428$	7279.51	0.822122
$429$	−448.185	−0.0504395
$430$	−1883.82	−0.211270
$431$	5238.31	0.585430	0.292715	−	0.956200i	$-0.405441\pi$
$431$	5238.31	0.585430	0.292715	+	0.956200i	$0.405441\pi$
$432$	142.180	0.0158348
$433$	7563.10	0.839398	0.419699	−	0.907663i	$-0.362136\pi$
$433$	7563.10	0.839398	0.419699	+	0.907663i	$0.362136\pi$
$434$	0	0
$435$	−802.711	−0.0884760
$436$	−5003.27	−0.549572
$437$	−268.028	−0.0293398
$438$	1149.68	0.125420
$439$	2039.02	0.221679	0.110839	−	0.993838i	$-0.464646\pi$
$439$	2039.02	0.221679	0.110839	+	0.993838i	$0.464646\pi$
$440$	−1541.84	−0.167055
$441$	0	0
$442$	2502.46	0.269298
$443$	−12956.2	−1.38954	−0.694769	−	0.719233i	$-0.744493\pi$
$443$	−12956.2	−1.38954	−0.694769	+	0.719233i	$0.744493\pi$
$444$	4417.05	0.472126
$445$	−314.506	−0.0335034
$446$	7526.53	0.799085
$447$	1789.17	0.189318
$448$	0	0
$449$	−4257.39	−0.447480	−0.223740	−	0.974649i	$-0.571827\pi$
$449$	−4257.39	−0.447480	−0.223740	+	0.974649i	$0.571827\pi$
$450$	412.132	0.0431735
$451$	5374.87	0.561182
$452$	−5914.30	−0.615454
$453$	5144.48	0.533574
$454$	−9982.12	−1.03190
$455$	0	0
$456$	504.568	0.0518171
$457$	15880.9	1.62555	0.812777	−	0.582575i	$-0.197955\pi$
$457$	15880.9	1.62555	0.812777	+	0.582575i	$0.197955\pi$
$458$	6758.36	0.689514
$459$	3287.32	0.334290
$460$	−857.225	−0.0868876
$461$	−187.474	−0.0189404	−0.00947022	−	0.999955i	$-0.503015\pi$
$461$	−187.474	−0.0189404	−0.00947022	+	0.999955i	$0.503015\pi$
$462$	0	0
$463$	10377.8	1.04168	0.520838	−	0.853656i	$-0.325620\pi$
$463$	10377.8	1.04168	0.520838	+	0.853656i	$0.325620\pi$
$464$	281.801	0.0281945
$465$	−712.464	−0.0710532
$466$	3451.66	0.343123
$467$	−7514.27	−0.744581	−0.372290	−	0.928116i	$-0.621427\pi$
$467$	−7514.27	−0.744581	−0.372290	+	0.928116i	$0.621427\pi$
$468$	469.086	0.0463323
$469$	0	0
$470$	−1528.54	−0.150013
$471$	3028.63	0.296288
$472$	−3981.22	−0.388243
$473$	−2738.53	−0.266211
$474$	−5482.23	−0.531238
$475$	181.539	0.0175360
$476$	0	0
$477$	5667.14	0.543984
$478$	−10779.6	−1.03148
$479$	4624.70	0.441144	0.220572	−	0.975371i	$-0.429208\pi$
$479$	4624.70	0.441144	0.220572	+	0.975371i	$0.429208\pi$
$480$	2634.71	0.250536
$481$	−3556.89	−0.337173
$482$	3010.68	0.284508
$483$	0	0
$484$	5359.01	0.503288
$485$	8569.65	0.802325
$486$	−445.102	−0.0415437
$487$	7475.38	0.695568	0.347784	−	0.937575i	$-0.386934\pi$
$487$	7475.38	0.695568	0.347784	+	0.937575i	$0.386934\pi$
$488$	−9979.36	−0.925706
$489$	−4734.02	−0.437791
$490$	0	0
$491$	−10657.8	−0.979596	−0.489798	−	0.871836i	$-0.662929\pi$
$491$	−10657.8	−0.979596	−0.489798	+	0.871836i	$0.662929\pi$
$492$	−5625.53	−0.515485
$493$	6515.48	0.595218
$494$	−149.251	−0.0135934
$495$	599.120	0.0544009
$496$	250.118	0.0226424
$497$	0	0
$498$	−1815.62	−0.163373
$499$	18206.5	1.63334	0.816670	−	0.577106i	$-0.195818\pi$
$499$	18206.5	1.63334	0.816670	+	0.577106i	$0.195818\pi$
$500$	580.611	0.0519314
$501$	3410.25	0.304110
$502$	6966.49	0.619381
$503$	−6417.85	−0.568902	−0.284451	−	0.958691i	$-0.591811\pi$
$503$	−6417.85	−0.568902	−0.284451	+	0.958691i	$0.591811\pi$
$504$	0	0
$505$	−2954.35	−0.260330
$506$	900.128	0.0790821
$507$	6213.26	0.544262
$508$	−6582.32	−0.574888
$509$	−19812.9	−1.72532	−0.862662	−	0.505781i	$-0.831204\pi$
$509$	−19812.9	−1.72532	−0.862662	+	0.505781i	$0.831204\pi$
$510$	−3345.21	−0.290448
$511$	0	0
$512$	−1900.68	−0.164060
$513$	−196.062	−0.0168740
$514$	−6874.27	−0.589905
$515$	−6548.43	−0.560308
$516$	2866.25	0.244534
$517$	−2222.05	−0.189025
$518$	0	0
$519$	−478.562	−0.0404750
$520$	−1299.49	−0.109589
$521$	−12726.1	−1.07013	−0.535067	−	0.844810i	$-0.679714\pi$
$521$	−12726.1	−1.07013	−0.535067	+	0.844810i	$0.679714\pi$
$522$	−882.194	−0.0739704
$523$	15027.0	1.25638	0.628189	−	0.778060i	$-0.283796\pi$
$523$	15027.0	1.25638	0.628189	+	0.778060i	$0.283796\pi$
$524$	−7626.80	−0.635836
$525$	0	0
$526$	10787.9	0.894252
$527$	5782.96	0.478007
$528$	−210.328	−0.0173359
$529$	−10804.6	−0.888026
$530$	−5766.93	−0.472641
$531$	1547.00	0.126430
$532$	0	0
$533$	4530.03	0.368138
$534$	−345.648	−0.0280106
$535$	7836.05	0.633237
$536$	1876.86	0.151246
$537$	−2107.64	−0.169370
$538$	−10129.4	−0.811724
$539$	0	0
$540$	−627.060	−0.0499710
$541$	−21168.2	−1.68224	−0.841122	−	0.540846i	$-0.818104\pi$
$541$	−21168.2	−1.68224	−0.841122	+	0.540846i	$0.818104\pi$
$542$	−81.8063	−0.00648317
$543$	−4221.50	−0.333632
$544$	−21385.5	−1.68547
$545$	−5385.78	−0.423306
$546$	0	0
$547$	420.240	0.0328486	0.0164243	−	0.999865i	$-0.494772\pi$
$547$	420.240	0.0328486	0.0164243	+	0.999865i	$0.494772\pi$
$548$	−5435.48	−0.423708
$549$	3877.72	0.301452
$550$	−609.670	−0.0472662
$551$	−388.596	−0.0300449
$552$	−2564.72	−0.197756
$553$	0	0
$554$	−6001.78	−0.460273
$555$	4754.74	0.363653
$556$	12911.8	0.984857
$557$	1441.25	0.109637	0.0548184	−	0.998496i	$-0.482542\pi$
$557$	1441.25	0.109637	0.0548184	+	0.998496i	$0.482542\pi$
$558$	−783.011	−0.0594041
$559$	−2308.08	−0.174636
$560$	0	0
$561$	−4862.96	−0.365979
$562$	1340.58	0.100621
$563$	10789.1	0.807646	0.403823	−	0.914837i	$-0.367681\pi$
$563$	10789.1	0.807646	0.403823	+	0.914837i	$0.367681\pi$
$564$	2325.68	0.173633
$565$	−6366.46	−0.474051
$566$	−5116.03	−0.379934
$567$	0	0
$568$	−2246.71	−0.165968
$569$	18119.7	1.33500	0.667501	−	0.744609i	$-0.267364\pi$
$569$	18119.7	1.33500	0.667501	+	0.744609i	$0.267364\pi$
$570$	199.515	0.0146610
$571$	12036.4	0.882151	0.441076	−	0.897470i	$-0.354597\pi$
$571$	12036.4	0.882151	0.441076	+	0.897470i	$0.354597\pi$
$572$	−693.922	−0.0507244
$573$	10891.6	0.794072
$574$	0	0
$575$	−922.761	−0.0669249
$576$	3274.74	0.236888
$577$	10430.6	0.752567	0.376284	−	0.926504i	$-0.377202\pi$
$577$	10430.6	0.752567	0.376284	+	0.926504i	$0.377202\pi$
$578$	18153.4	1.30637
$579$	9467.23	0.679525
$580$	−1242.83	−0.0889757
$581$	0	0
$582$	9418.19	0.670785
$583$	−8383.45	−0.595552
$584$	4845.86	0.343362
$585$	504.948	0.0356873
$586$	17733.7	1.25012
$587$	8903.66	0.626054	0.313027	−	0.949744i	$-0.398657\pi$
$587$	8903.66	0.626054	0.313027	+	0.949744i	$0.398657\pi$
$588$	0	0
$589$	−344.907	−0.0241284
$590$	−1574.24	−0.109848
$591$	−7105.75	−0.494571
$592$	−1669.20	−0.115885
$593$	−5276.59	−0.365402	−0.182701	−	0.983168i	$-0.558484\pi$
$593$	−5276.59	−0.365402	−0.182701	+	0.983168i	$0.558484\pi$
$594$	658.443	0.0454819
$595$	0	0
$596$	2770.17	0.190387
$597$	−11411.7	−0.782330
$598$	758.643	0.0518783
$599$	23429.2	1.59815	0.799073	−	0.601233i	$-0.205324\pi$
$599$	23429.2	1.59815	0.799073	+	0.601233i	$0.205324\pi$
$600$	1737.12	0.118196
$601$	8166.03	0.554241	0.277121	−	0.960835i	$-0.410620\pi$
$601$	8166.03	0.554241	0.277121	+	0.960835i	$0.410620\pi$
$602$	0	0
$603$	−729.298	−0.0492526
$604$	7965.18	0.536587
$605$	5768.72	0.387656
$606$	−3246.88	−0.217649
$607$	2034.40	0.136036	0.0680178	−	0.997684i	$-0.478333\pi$
$607$	2034.40	0.136036	0.0680178	+	0.997684i	$0.478333\pi$
$608$	1275.47	0.0850778
$609$	0	0
$610$	−3946.00	−0.261917
$611$	−1872.78	−0.124001
$612$	5089.75	0.336178
$613$	13091.6	0.862586	0.431293	−	0.902212i	$-0.358057\pi$
$613$	13091.6	0.862586	0.431293	+	0.902212i	$0.358057\pi$
$614$	15330.4	1.00763
$615$	−6055.62	−0.397050
$616$	0	0
$617$	1800.49	0.117479	0.0587397	−	0.998273i	$-0.481292\pi$
$617$	1800.49	0.117479	0.0587397	+	0.998273i	$0.481292\pi$
$618$	−7196.85	−0.468446
$619$	−15939.3	−1.03498	−0.517490	−	0.855689i	$-0.673134\pi$
$619$	−15939.3	−1.03498	−0.517490	+	0.855689i	$0.673134\pi$
$620$	−1103.11	−0.0714545
$621$	996.582	0.0643985
$622$	−16158.4	−1.04163
$623$	0	0
$624$	−177.268	−0.0113724
$625$	625.000	0.0400000
$626$	18752.3	1.19728
$627$	290.036	0.0184736
$628$	4689.21	0.297961
$629$	−38593.5	−2.44646
$630$	0	0
$631$	9594.47	0.605309	0.302654	−	0.953100i	$-0.402127\pi$
$631$	9594.47	0.605309	0.302654	+	0.953100i	$0.402127\pi$
$632$	−23107.4	−1.45437
$633$	980.611	0.0615731
$634$	−3820.79	−0.239342
$635$	−7085.55	−0.442806
$636$	8774.41	0.547056
$637$	0	0
$638$	1305.04	0.0809826
$639$	873.014	0.0540468
$640$	3693.49	0.228122
$641$	17390.0	1.07155	0.535774	−	0.844362i	$-0.320020\pi$
$641$	17390.0	1.07155	0.535774	+	0.844362i	$0.320020\pi$
$642$	8611.95	0.529419
$643$	−8460.25	−0.518880	−0.259440	−	0.965759i	$-0.583538\pi$
$643$	−8460.25	−0.518880	−0.259440	+	0.965759i	$0.583538\pi$
$644$	0	0
$645$	3085.38	0.188351
$646$	−1619.43	−0.0986310
$647$	661.813	0.0402141	0.0201071	−	0.999798i	$-0.493599\pi$
$647$	661.813	0.0402141	0.0201071	+	0.999798i	$0.493599\pi$
$648$	−1876.09	−0.113734
$649$	−2288.49	−0.138415
$650$	−513.840	−0.0310069
$651$	0	0
$652$	−7329.67	−0.440264
$653$	10775.2	0.645739	0.322869	−	0.946444i	$-0.395353\pi$
$653$	10775.2	0.645739	0.322869	+	0.946444i	$0.395353\pi$
$654$	−5919.07	−0.353905
$655$	−8209.89	−0.489751
$656$	2125.89	0.126528
$657$	−1882.98	−0.111814
$658$	0	0
$659$	23424.7	1.38467	0.692335	−	0.721576i	$-0.256582\pi$
$659$	23424.7	1.38467	0.692335	+	0.721576i	$0.256582\pi$
$660$	927.615	0.0547081
$661$	4881.40	0.287238	0.143619	−	0.989633i	$-0.454126\pi$
$661$	4881.40	0.287238	0.143619	+	0.989633i	$0.454126\pi$
$662$	293.127	0.0172095
$663$	−4098.59	−0.240084
$664$	−7652.75	−0.447266
$665$	0	0
$666$	5225.55	0.304033
$667$	1975.23	0.114664
$668$	5280.08	0.305827
$669$	−12327.2	−0.712400
$670$	742.140	0.0427931
$671$	−5736.35	−0.330029
$672$	0	0
$673$	18795.2	1.07653	0.538264	−	0.842776i	$-0.319080\pi$
$673$	18795.2	1.07653	0.538264	+	0.842776i	$0.319080\pi$
$674$	17061.8	0.975067
$675$	−675.000	−0.0384900
$676$	9619.97	0.547336
$677$	3149.57	0.178800	0.0894002	−	0.995996i	$-0.471505\pi$
$677$	3149.57	0.178800	0.0894002	+	0.995996i	$0.471505\pi$
$678$	−6996.85	−0.396331
$679$	0	0
$680$	−14099.9	−0.795158
$681$	16349.0	0.919962
$682$	1158.31	0.0650354
$683$	−2156.91	−0.120837	−0.0604187	−	0.998173i	$-0.519244\pi$
$683$	−2156.91	−0.120837	−0.0604187	+	0.998173i	$0.519244\pi$
$684$	−303.562	−0.0169693
$685$	−5851.03	−0.326360
$686$	0	0
$687$	−11069.0	−0.614715
$688$	−1083.16	−0.0600217
$689$	−7065.71	−0.390685
$690$	−1014.13	−0.0559526
$691$	19743.6	1.08695	0.543475	−	0.839425i	$-0.317108\pi$
$691$	19743.6	1.08695	0.543475	+	0.839425i	$0.317108\pi$
$692$	−740.955	−0.0407036
$693$	0	0
$694$	−10818.3	−0.591727
$695$	13898.9	0.758583
$696$	−3718.41	−0.202509
$697$	49152.5	2.67114
$698$	16707.7	0.906013
$699$	−5653.22	−0.305901
$700$	0	0
$701$	33027.9	1.77952	0.889761	−	0.456427i	$-0.150871\pi$
$701$	33027.9	1.77952	0.889761	+	0.456427i	$0.150871\pi$
$702$	554.947	0.0298364
$703$	2301.79	0.123490
$704$	−4844.35	−0.259344
$705$	2503.48	0.133740
$706$	−605.616	−0.0322842
$707$	0	0
$708$	2395.21	0.127144
$709$	7162.52	0.379399	0.189700	−	0.981842i	$-0.439249\pi$
$709$	7162.52	0.379399	0.189700	+	0.981842i	$0.439249\pi$
$710$	−888.388	−0.0469586
$711$	8978.93	0.473609
$712$	−1456.89	−0.0766845
$713$	1753.16	0.0920846
$714$	0	0
$715$	−746.974	−0.0390703
$716$	−3263.26	−0.170326
$717$	17655.0	0.919581
$718$	17486.2	0.908883
$719$	15221.7	0.789534	0.394767	−	0.918781i	$-0.370825\pi$
$719$	15221.7	0.789534	0.394767	+	0.918781i	$0.370825\pi$
$720$	236.966	0.0122656
$721$	0	0
$722$	−12467.0	−0.642624
$723$	−4930.97	−0.253644
$724$	−6536.13	−0.335516
$725$	−1337.85	−0.0685332
$726$	6339.92	0.324100
$727$	21110.6	1.07696	0.538479	−	0.842639i	$-0.318999\pi$
$727$	21110.6	1.07696	0.538479	+	0.842639i	$0.318999\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	1916.13	0.0971498
$731$	−25043.5	−1.26712
$732$	6003.86	0.303154
$733$	3192.56	0.160873	0.0804365	−	0.996760i	$-0.474369\pi$
$733$	3192.56	0.160873	0.0804365	+	0.996760i	$0.474369\pi$
$734$	−13822.2	−0.695076
$735$	0	0
$736$	−6483.22	−0.324694
$737$	1078.86	0.0539215
$738$	−6655.23	−0.331954
$739$	−2086.23	−0.103847	−0.0519237	−	0.998651i	$-0.516535\pi$
$739$	−2086.23	−0.103847	−0.0519237	+	0.998651i	$0.516535\pi$
$740$	7361.75	0.365707
$741$	244.448	0.0121188
$742$	0	0
$743$	24803.5	1.22470	0.612351	−	0.790586i	$-0.290224\pi$
$743$	24803.5	1.22470	0.612351	+	0.790586i	$0.290224\pi$
$744$	−3300.36	−0.162631
$745$	2981.96	0.146645
$746$	−17229.2	−0.845582
$747$	2973.66	0.145650
$748$	−7529.30	−0.368046
$749$	0	0
$750$	686.886	0.0334421
$751$	−31030.6	−1.50776	−0.753878	−	0.657015i	$-0.771819\pi$
$751$	−31030.6	−1.50776	−0.753878	+	0.657015i	$0.771819\pi$
$752$	−878.875	−0.0426187
$753$	−11409.9	−0.552191
$754$	1099.91	0.0531250
$755$	8574.14	0.413304
$756$	0	0
$757$	8976.32	0.430977	0.215489	−	0.976506i	$-0.430866\pi$
$757$	8976.32	0.430977	0.215489	+	0.976506i	$0.430866\pi$
$758$	−23865.1	−1.14356
$759$	−1474.25	−0.0705033
$760$	840.947	0.0401373
$761$	32657.8	1.55564	0.777822	−	0.628484i	$-0.216324\pi$
$761$	32657.8	1.55564	0.777822	+	0.628484i	$0.216324\pi$
$762$	−7787.15	−0.370208
$763$	0	0
$764$	16863.4	0.798556
$765$	5478.87	0.258940
$766$	14723.9	0.694514
$767$	−1928.78	−0.0908007
$768$	12791.8	0.601023
$769$	6824.24	0.320011	0.160005	−	0.987116i	$-0.448849\pi$
$769$	6824.24	0.320011	0.160005	+	0.987116i	$0.448849\pi$
$770$	0	0
$771$	11258.9	0.525912
$772$	14658.1	0.683363
$773$	−8628.75	−0.401494	−0.200747	−	0.979643i	$-0.564337\pi$
$773$	−8628.75	−0.401494	−0.200747	+	0.979643i	$0.564337\pi$
$774$	3390.88	0.157471
$775$	−1187.44	−0.0550376
$776$	39697.3	1.83641
$777$	0	0
$778$	−10899.8	−0.502283
$779$	−2931.55	−0.134831
$780$	781.809	0.0358888
$781$	−1291.46	−0.0591703
$782$	8231.55	0.376419
$783$	1444.88	0.0659461
$784$	0	0
$785$	5047.71	0.229504
$786$	−9022.81	−0.409457
$787$	−11886.5	−0.538385	−0.269193	−	0.963086i	$-0.586757\pi$
$787$	−11886.5	−0.538385	−0.269193	+	0.963086i	$0.586757\pi$
$788$	−11001.8	−0.497364
$789$	−17668.8	−0.797243
$790$	−9137.04	−0.411495
$791$	0	0
$792$	2775.31	0.124516
$793$	−4834.69	−0.216500
$794$	21101.0	0.943132
$795$	9445.23	0.421368
$796$	−17668.7	−0.786748
$797$	40174.0	1.78549	0.892745	−	0.450563i	$-0.148777\pi$
$797$	40174.0	1.78549	0.892745	+	0.450563i	$0.148777\pi$
$798$	0	0
$799$	−20320.4	−0.899729
$800$	4391.18	0.194065
$801$	566.111	0.0249720
$802$	14972.9	0.659243
$803$	2785.50	0.122414
$804$	−1129.17	−0.0495307
$805$	0	0
$806$	976.247	0.0426636
$807$	16590.1	0.723668
$808$	−13685.5	−0.595858
$809$	−15041.0	−0.653661	−0.326831	−	0.945083i	$-0.605981\pi$
$809$	−15041.0	−0.653661	−0.326831	+	0.945083i	$0.605981\pi$
$810$	−741.837	−0.0321796
$811$	−19457.6	−0.842478	−0.421239	−	0.906950i	$-0.638405\pi$
$811$	−19457.6	−0.842478	−0.421239	+	0.906950i	$0.638405\pi$
$812$	0	0
$813$	133.984	0.00577988
$814$	−7730.19	−0.332854
$815$	−7890.04	−0.339112
$816$	−1923.42	−0.0825160
$817$	1493.64	0.0639608
$818$	229.869	0.00982540
$819$	0	0
$820$	−9375.89	−0.399293
$821$	12604.3	0.535802	0.267901	−	0.963446i	$-0.413670\pi$
$821$	12604.3	0.535802	0.267901	+	0.963446i	$0.413670\pi$
$822$	−6430.39	−0.272853
$823$	31260.2	1.32401	0.662005	−	0.749499i	$-0.269706\pi$
$823$	31260.2	1.32401	0.662005	+	0.749499i	$0.269706\pi$
$824$	−30334.4	−1.28246
$825$	998.533	0.0421387
$826$	0	0
$827$	−26591.8	−1.11812	−0.559062	−	0.829126i	$-0.688839\pi$
$827$	−26591.8	−1.11812	−0.559062	+	0.829126i	$0.688839\pi$
$828$	1543.00	0.0647622
$829$	−29416.3	−1.23241	−0.616206	−	0.787585i	$-0.711331\pi$
$829$	−29416.3	−1.23241	−0.616206	+	0.787585i	$0.711331\pi$
$830$	−3026.03	−0.126548
$831$	9829.87	0.410342
$832$	−4082.90	−0.170131
$833$	0	0
$834$	15275.1	0.634214
$835$	5683.76	0.235562
$836$	449.062	0.0185779
$837$	1282.44	0.0529599
$838$	−30654.9	−1.26367
$839$	−30432.1	−1.25224	−0.626122	−	0.779725i	$-0.715359\pi$
$839$	−30432.1	−1.25224	−0.626122	+	0.779725i	$0.715359\pi$
$840$	0	0
$841$	−21525.2	−0.882580
$842$	17914.4	0.733220
$843$	−2195.63	−0.0897053
$844$	1518.28	0.0619209
$845$	10355.4	0.421583
$846$	2751.37	0.111813
$847$	0	0
$848$	−3315.85	−0.134277
$849$	8379.17	0.338719
$850$	−5575.35	−0.224980
$851$	−11700.0	−0.471293
$852$	1351.68	0.0543521
$853$	42852.3	1.72009	0.860045	−	0.510219i	$-0.170436\pi$
$853$	42852.3	1.72009	0.860045	+	0.510219i	$0.170436\pi$
$854$	0	0
$855$	−326.770	−0.0130705
$856$	36299.1	1.44939
$857$	13976.6	0.557098	0.278549	−	0.960422i	$-0.410147\pi$
$857$	13976.6	0.557098	0.278549	+	0.960422i	$0.410147\pi$
$858$	−820.938	−0.0326648
$859$	−24712.9	−0.981599	−0.490799	−	0.871273i	$-0.663295\pi$
$859$	−24712.9	−0.981599	−0.490799	+	0.871273i	$0.663295\pi$
$860$	4777.08	0.189415
$861$	0	0
$862$	9594.99	0.379126
$863$	−18395.3	−0.725591	−0.362795	−	0.931869i	$-0.618178\pi$
$863$	−18395.3	−0.725591	−0.362795	+	0.931869i	$0.618178\pi$
$864$	−4742.47	−0.186739
$865$	−797.603	−0.0313518
$866$	13853.3	0.543596
$867$	−29732.1	−1.16466
$868$	0	0
$869$	−13282.6	−0.518506
$870$	−1470.32	−0.0572973
$871$	909.278	0.0353728
$872$	−24948.7	−0.968886
$873$	−15425.4	−0.598018
$874$	−490.945	−0.0190005
$875$	0	0
$876$	−2915.41	−0.112446
$877$	15590.6	0.600294	0.300147	−	0.953893i	$-0.402964\pi$
$877$	15590.6	0.600294	0.300147	+	0.953893i	$0.402964\pi$
$878$	3734.86	0.143560
$879$	−29044.7	−1.11451
$880$	−350.546	−0.0134283
$881$	−10854.5	−0.415095	−0.207547	−	0.978225i	$-0.566548\pi$
$881$	−10854.5	−0.415095	−0.207547	+	0.978225i	$0.566548\pi$
$882$	0	0
$883$	537.994	0.0205039	0.0102519	−	0.999947i	$-0.496737\pi$
$883$	537.994	0.0205039	0.0102519	+	0.999947i	$0.496737\pi$
$884$	−6345.83	−0.241440
$885$	2578.33	0.0979319
$886$	−23731.7	−0.899868
$887$	−28177.3	−1.06663	−0.533316	−	0.845916i	$-0.679054\pi$
$887$	−28177.3	−1.06663	−0.533316	+	0.845916i	$0.679054\pi$
$888$	22025.5	0.832350
$889$	0	0
$890$	−576.080	−0.0216969
$891$	−1078.42	−0.0405480
$892$	−19086.1	−0.716423
$893$	1211.95	0.0454157
$894$	3277.22	0.122603
$895$	−3512.74	−0.131193
$896$	0	0
$897$	−1242.52	−0.0462505
$898$	−7798.24	−0.289789
$899$	2541.79	0.0942975
$900$	−1045.10	−0.0387074
$901$	−76665.5	−2.83474
$902$	9845.14	0.363423
$903$	0	0
$904$	−29491.5	−1.08503
$905$	−7035.84	−0.258430
$906$	9423.13	0.345544
$907$	48750.3	1.78470	0.892352	−	0.451339i	$-0.149054\pi$
$907$	48750.3	1.78470	0.892352	+	0.451339i	$0.149054\pi$
$908$	25313.0	0.925157
$909$	5317.82	0.194039
$910$	0	0
$911$	29158.5	1.06044	0.530222	−	0.847859i	$-0.322109\pi$
$911$	29158.5	1.06044	0.530222	+	0.847859i	$0.322109\pi$
$912$	114.716	0.00416517
$913$	−4398.96	−0.159457
$914$	29089.0	1.05271
$915$	6462.87	0.233504
$916$	−17138.1	−0.618187
$917$	0	0
$918$	6021.38	0.216487
$919$	−22696.7	−0.814684	−0.407342	−	0.913276i	$-0.633544\pi$
$919$	−22696.7	−0.814684	−0.407342	+	0.913276i	$0.633544\pi$
$920$	−4274.53	−0.153181
$921$	−25108.6	−0.898323
$922$	−343.396	−0.0122659
$923$	−1088.46	−0.0388160
$924$	0	0
$925$	7924.57	0.281685
$926$	19008.9	0.674591
$927$	11787.2	0.417629
$928$	−9399.60	−0.332497
$929$	31842.1	1.12455	0.562274	−	0.826951i	$-0.309927\pi$
$929$	31842.1	1.12455	0.562274	+	0.826951i	$0.309927\pi$
$930$	−1305.02	−0.0460142
$931$	0	0
$932$	−8752.86	−0.307628
$933$	26464.7	0.928633
$934$	−13763.9	−0.482192
$935$	−8104.94	−0.283486
$936$	2339.08	0.0816830
$937$	24904.2	0.868287	0.434143	−	0.900844i	$-0.357051\pi$
$937$	24904.2	0.868287	0.434143	+	0.900844i	$0.357051\pi$
$938$	0	0
$939$	−30713.1	−1.06739
$940$	3876.13	0.134495
$941$	16348.0	0.566346	0.283173	−	0.959069i	$-0.408613\pi$
$941$	16348.0	0.566346	0.283173	+	0.959069i	$0.408613\pi$
$942$	5547.52	0.191877
$943$	14901.0	0.514575
$944$	−905.152	−0.0312078
$945$	0	0
$946$	−5016.16	−0.172399
$947$	−17252.5	−0.592009	−0.296004	−	0.955187i	$-0.595654\pi$
$947$	−17252.5	−0.592009	−0.296004	+	0.955187i	$0.595654\pi$
$948$	13902.0	0.476284
$949$	2347.67	0.0803041
$950$	332.525	0.0113563
$951$	6257.78	0.213378
$952$	0	0
$953$	45959.4	1.56219	0.781096	−	0.624411i	$-0.214661\pi$
$953$	45959.4	1.56219	0.781096	+	0.624411i	$0.214661\pi$
$954$	10380.5	0.352286
$955$	18152.7	0.615085
$956$	27335.2	0.924775
$957$	−2137.42	−0.0721976
$958$	8471.05	0.285686
$959$	0	0
$960$	5457.90	0.183493
$961$	−27535.0	−0.924272
$962$	−6515.14	−0.218354
$963$	−14104.9	−0.471987
$964$	−7634.60	−0.255077
$965$	15778.7	0.526358
$966$	0	0
$967$	−46208.8	−1.53668	−0.768342	−	0.640039i	$-0.778918\pi$
$967$	−46208.8	−1.53668	−0.768342	+	0.640039i	$0.778918\pi$
$968$	26722.5	0.887288
$969$	2652.34	0.0879315
$970$	15697.0	0.519588
$971$	−23927.9	−0.790818	−0.395409	−	0.918505i	$-0.629397\pi$
$971$	−23927.9	−0.790818	−0.395409	+	0.918505i	$0.629397\pi$
$972$	1128.71	0.0372462
$973$	0	0
$974$	13692.6	0.450452
$975$	841.581	0.0276432
$976$	−2268.86	−0.0744103
$977$	−34491.2	−1.12945	−0.564724	−	0.825279i	$-0.691018\pi$
$977$	−34491.2	−1.12945	−0.564724	+	0.825279i	$0.691018\pi$
$978$	−8671.29	−0.283515
$979$	−837.453	−0.0273392
$980$	0	0
$981$	9694.41	0.315513
$982$	−19521.9	−0.634389
$983$	853.837	0.0277041	0.0138521	−	0.999904i	$-0.495591\pi$
$983$	853.837	0.0277041	0.0138521	+	0.999904i	$0.495591\pi$
$984$	−28051.5	−0.908791
$985$	−11842.9	−0.383093
$986$	11934.4	0.385465
$987$	0	0
$988$	378.477	0.0121872
$989$	−7592.18	−0.244102
$990$	1097.41	0.0352301
$991$	−39350.8	−1.26137	−0.630686	−	0.776038i	$-0.717227\pi$
$991$	−39350.8	−1.26137	−0.630686	+	0.776038i	$0.717227\pi$
$992$	−8342.82	−0.267021
$993$	−480.090	−0.0153426
$994$	0	0
$995$	−19019.6	−0.605990
$996$	4604.11	0.146473
$997$	31911.5	1.01369	0.506844	−	0.862038i	$-0.330812\pi$
$997$	31911.5	1.01369	0.506844	+	0.862038i	$0.330812\pi$
$998$	33348.8	1.05775
$999$	−8558.53	−0.271051

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.4.a.bb.1.5	✓	8
3.2	odd	2		2205.4.a.ce.1.4		8
7.6	odd	2		735.4.a.bc.1.5	yes	8
21.20	even	2		2205.4.a.cd.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.4.a.bb.1.5	✓	8	1.1	even	1	trivial
735.4.a.bc.1.5	yes	8	7.6	odd	2
2205.4.a.cd.1.4		8	21.20	even	2
2205.4.a.ce.1.4		8	3.2	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:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	735.a (trivial)

\(f(q)\)	\(=\)	\(q+1.83170 q^{2} -3.00000 q^{3} -4.64489 q^{4} -5.00000 q^{5} -5.49509 q^{6} -23.1616 q^{8} +9.00000 q^{9} -9.15848 q^{10} -13.3138 q^{11} +13.9347 q^{12} -11.2211 q^{13} +15.0000 q^{15} -5.26591 q^{16} -121.753 q^{17} +16.4853 q^{18} +7.26156 q^{19} +23.2244 q^{20} -24.3868 q^{22} -36.9105 q^{23} +69.4848 q^{24} +25.0000 q^{25} -20.5536 q^{26} -27.0000 q^{27} -53.5141 q^{29} +27.4754 q^{30} -47.4976 q^{31} +175.647 q^{32} +39.9413 q^{33} -223.014 q^{34} -41.8040 q^{36} +316.983 q^{37} +13.3010 q^{38} +33.6632 q^{39} +115.808 q^{40} -403.708 q^{41} +205.692 q^{43} +61.8410 q^{44} -45.0000 q^{45} -67.6088 q^{46} +166.899 q^{47} +15.7977 q^{48} +45.7924 q^{50} +365.258 q^{51} +52.1206 q^{52} +629.682 q^{53} -49.4558 q^{54} +66.5689 q^{55} -21.7847 q^{57} -98.0215 q^{58} +171.889 q^{59} -69.6733 q^{60} +430.858 q^{61} -87.0012 q^{62} +363.860 q^{64} +56.1054 q^{65} +73.1604 q^{66} -81.0331 q^{67} +565.527 q^{68} +110.731 q^{69} +97.0016 q^{71} -208.454 q^{72} -209.220 q^{73} +580.616 q^{74} -75.0000 q^{75} -33.7292 q^{76} +61.6608 q^{78} +997.659 q^{79} +26.3296 q^{80} +81.0000 q^{81} -739.470 q^{82} +330.407 q^{83} +608.763 q^{85} +376.765 q^{86} +160.542 q^{87} +308.368 q^{88} +62.9012 q^{89} -82.4263 q^{90} +171.445 q^{92} +142.493 q^{93} +305.708 q^{94} -36.3078 q^{95} -526.942 q^{96} -1713.93 q^{97} -119.824 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - 24 q^{3} + 50 q^{4} - 40 q^{5} - 6 q^{6} + 66 q^{8} + 72 q^{9} - 10 q^{10} + 64 q^{11} - 150 q^{12} + 120 q^{15} + 206 q^{16} - 48 q^{17} + 18 q^{18} - 80 q^{19} - 250 q^{20} + 452 q^{22}+ \cdots + 576 q^{99}+O(q^{100}) \) 8 * q + 2 * q^2 - 24 * q^3 + 50 * q^4 - 40 * q^5 - 6 * q^6 + 66 * q^8 + 72 * q^9 - 10 * q^10 + 64 * q^11 - 150 * q^12 + 120 * q^15 + 206 * q^16 - 48 * q^17 + 18 * q^18 - 80 * q^19 - 250 * q^20 + 452 * q^22 + 120 * q^23 - 198 * q^24 + 200 * q^25 - 272 * q^26 - 216 * q^27 + 76 * q^29 + 30 * q^30 - 20 * q^31 + 770 * q^32 - 192 * q^33 - 320 * q^34 + 450 * q^36 + 348 * q^37 + 236 * q^38 - 330 * q^40 - 944 * q^41 + 1116 * q^43 + 172 * q^44 - 360 * q^45 + 496 * q^46 + 208 * q^47 - 618 * q^48 + 50 * q^50 + 144 * q^51 + 2272 * q^52 + 1144 * q^53 - 54 * q^54 - 320 * q^55 + 240 * q^57 + 560 * q^58 - 596 * q^59 + 750 * q^60 - 740 * q^61 + 1184 * q^62 + 1298 * q^64 - 1356 * q^66 + 1964 * q^67 - 96 * q^68 - 360 * q^69 - 4 * q^71 + 594 * q^72 + 1500 * q^73 + 3368 * q^74 - 600 * q^75 + 2912 * q^76 + 816 * q^78 - 460 * q^79 - 1030 * q^80 + 648 * q^81 + 5644 * q^82 - 700 * q^83 + 240 * q^85 - 1396 * q^86 - 228 * q^87 + 6892 * q^88 - 90 * q^90 + 644 * q^92 + 60 * q^93 + 6692 * q^94 + 400 * q^95 - 2310 * q^96 + 2052 * q^97 + 576 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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