LMFDB - Embedded newform 525.4.a.u.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [525,4,Mod(1,525)] 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("525.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	525.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,12,16,0,0,28,9,36,0,21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$30.9760027530$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 24x^{2} - 3x + 46 $ x^4 - 24x^2 - 3x + 46
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.37627$ of defining polynomial
Character	$\chi$	$=$	525.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.37627 q^{2} +3.00000 q^{3} -6.10588 q^{4} +4.12881 q^{6} +7.00000 q^{7} -19.4135 q^{8} +9.00000 q^{9} +15.2804 q^{11} -18.3176 q^{12} +76.7787 q^{13} +9.63389 q^{14} +22.1288 q^{16} -96.7073 q^{17} +12.3864 q^{18} -14.1659 q^{19} +21.0000 q^{21} +21.0300 q^{22} -75.7097 q^{23} -58.2405 q^{24} +105.668 q^{26} +27.0000 q^{27} -42.7412 q^{28} +89.9625 q^{29} +289.563 q^{31} +185.763 q^{32} +45.8412 q^{33} -133.095 q^{34} -54.9529 q^{36} +14.2345 q^{37} -19.4961 q^{38} +230.336 q^{39} +318.710 q^{41} +28.9017 q^{42} +389.707 q^{43} -93.3003 q^{44} -104.197 q^{46} +228.849 q^{47} +66.3864 q^{48} +49.0000 q^{49} -290.122 q^{51} -468.801 q^{52} +679.493 q^{53} +37.1593 q^{54} -135.895 q^{56} -42.4977 q^{57} +123.813 q^{58} -398.628 q^{59} -146.361 q^{61} +398.517 q^{62} +63.0000 q^{63} +78.6300 q^{64} +63.0899 q^{66} +291.493 q^{67} +590.483 q^{68} -227.129 q^{69} +333.233 q^{71} -174.722 q^{72} +891.133 q^{73} +19.5905 q^{74} +86.4953 q^{76} +106.963 q^{77} +317.005 q^{78} -416.731 q^{79} +81.0000 q^{81} +438.631 q^{82} -814.323 q^{83} -128.223 q^{84} +536.342 q^{86} +269.887 q^{87} -296.646 q^{88} -650.076 q^{89} +537.451 q^{91} +462.275 q^{92} +868.690 q^{93} +314.958 q^{94} +557.290 q^{96} -1585.25 q^{97} +67.4372 q^{98} +137.524 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} + 16 q^{4} + 28 q^{7} + 9 q^{8} + 36 q^{9} + 21 q^{11} + 48 q^{12} + 5 q^{13} + 72 q^{16} + 99 q^{17} + 72 q^{19} + 84 q^{21} + 221 q^{22} + 102 q^{23} + 27 q^{24} + 129 q^{26} + 108 q^{27}+ \cdots + 189 q^{99}+O(q^{100}) $ 4 * q + 12 * q^3 + 16 * q^4 + 28 * q^7 + 9 * q^8 + 36 * q^9 + 21 * q^11 + 48 * q^12 + 5 * q^13 + 72 * q^16 + 99 * q^17 + 72 * q^19 + 84 * q^21 + 221 * q^22 + 102 * q^23 + 27 * q^24 + 129 * q^26 + 108 * q^27 + 112 * q^28 - 240 * q^29 + 351 * q^31 + 72 * q^32 + 63 * q^33 - 285 * q^34 + 144 * q^36 + 399 * q^37 + 324 * q^38 + 15 * q^39 + 381 * q^41 + 460 * q^43 - 975 * q^44 + 550 * q^46 + 60 * q^47 + 216 * q^48 + 196 * q^49 + 297 * q^51 + 223 * q^52 + 873 * q^53 + 63 * q^56 + 216 * q^57 + 1408 * q^58 - 855 * q^59 + 687 * q^61 - 477 * q^62 + 252 * q^63 - 1285 * q^64 + 663 * q^66 - 503 * q^67 + 1725 * q^68 + 306 * q^69 - 681 * q^71 + 81 * q^72 + 1228 * q^73 - 3369 * q^74 + 1910 * q^76 + 147 * q^77 + 387 * q^78 + 345 * q^79 + 324 * q^81 - 495 * q^82 + 1509 * q^83 + 336 * q^84 - 540 * q^86 - 720 * q^87 + 2266 * q^88 - 198 * q^89 + 35 * q^91 + 2994 * q^92 + 1053 * q^93 - 1732 * q^94 + 216 * q^96 - 372 * q^97 + 189 * 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 $

$2$	1.37627	0.486585	0.243293	−	0.969953i	$-0.421773\pi$
$2$	1.37627	0.486585	0.243293	+	0.969953i	$0.421773\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−6.10588	−0.763235
$5$	0	0
$5$	0	0
$6$	4.12881	0.280930
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−19.4135	−0.857964
$9$	9.00000	0.333333
$10$	0	0
$11$	15.2804	0.418838	0.209419	−	0.977826i	$-0.432843\pi$
$11$	15.2804	0.418838	0.209419	+	0.977826i	$0.432843\pi$
$12$	−18.3176	−0.440654
$13$	76.7787	1.63804	0.819022	−	0.573762i	$-0.194517\pi$
$13$	76.7787	1.63804	0.819022	+	0.573762i	$0.194517\pi$
$14$	9.63389	0.183912
$15$	0	0
$16$	22.1288	0.345763
$17$	−96.7073	−1.37970	−0.689852	−	0.723951i	$-0.742324\pi$
$17$	−96.7073	−1.37970	−0.689852	+	0.723951i	$0.742324\pi$
$18$	12.3864	0.162195
$19$	−14.1659	−0.171046	−0.0855232	−	0.996336i	$-0.527256\pi$
$19$	−14.1659	−0.171046	−0.0855232	+	0.996336i	$0.527256\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	21.0300	0.203800
$23$	−75.7097	−0.686373	−0.343186	−	0.939267i	$-0.611506\pi$
$23$	−75.7097	−0.686373	−0.343186	+	0.939267i	$0.611506\pi$
$24$	−58.2405	−0.495346
$25$	0	0
$26$	105.668	0.797048
$27$	27.0000	0.192450
$28$	−42.7412	−0.288476
$29$	89.9625	0.576055	0.288028	−	0.957622i	$-0.407000\pi$
$29$	89.9625	0.576055	0.288028	+	0.957622i	$0.407000\pi$
$30$	0	0
$31$	289.563	1.67765	0.838824	−	0.544403i	$-0.183244\pi$
$31$	289.563	1.67765	0.838824	+	0.544403i	$0.183244\pi$
$32$	185.763	1.02621
$33$	45.8412	0.241816
$34$	−133.095	−0.671343
$35$	0	0
$36$	−54.9529	−0.254412
$37$	14.2345	0.0632470	0.0316235	−	0.999500i	$-0.489932\pi$
$37$	14.2345	0.0632470	0.0316235	+	0.999500i	$0.489932\pi$
$38$	−19.4961	−0.0832286
$39$	230.336	0.945725
$40$	0	0
$41$	318.710	1.21400	0.607002	−	0.794700i	$-0.292372\pi$
$41$	318.710	1.21400	0.607002	+	0.794700i	$0.292372\pi$
$42$	28.9017	0.106182
$43$	389.707	1.38209	0.691044	−	0.722813i	$-0.257151\pi$
$43$	389.707	1.38209	0.691044	+	0.722813i	$0.257151\pi$
$44$	−93.3003	−0.319672
$45$	0	0
$46$	−104.197	−0.333979
$47$	228.849	0.710236	0.355118	−	0.934821i	$-0.384441\pi$
$47$	228.849	0.710236	0.355118	+	0.934821i	$0.384441\pi$
$48$	66.3864	0.199626
$49$	49.0000	0.142857
$50$	0	0
$51$	−290.122	−0.796572
$52$	−468.801	−1.25021
$53$	679.493	1.76105	0.880524	−	0.474002i	$-0.157191\pi$
$53$	679.493	1.76105	0.880524	+	0.474002i	$0.157191\pi$
$54$	37.1593	0.0936433
$55$	0	0
$56$	−135.895	−0.324280
$57$	−42.4977	−0.0987536
$58$	123.813	0.280300
$59$	−398.628	−0.879610	−0.439805	−	0.898093i	$-0.644952\pi$
$59$	−398.628	−0.879610	−0.439805	+	0.898093i	$0.644952\pi$
$60$	0	0
$61$	−146.361	−0.307206	−0.153603	−	0.988133i	$-0.549088\pi$
$61$	−146.361	−0.307206	−0.153603	+	0.988133i	$0.549088\pi$
$62$	398.517	0.816318
$63$	63.0000	0.125988
$64$	78.6300	0.153574
$65$	0	0
$66$	63.0899	0.117664
$67$	291.493	0.531516	0.265758	−	0.964040i	$-0.414378\pi$
$67$	291.493	0.531516	0.265758	+	0.964040i	$0.414378\pi$
$68$	590.483	1.05304
$69$	−227.129	−0.396278
$70$	0	0
$71$	333.233	0.557007	0.278503	−	0.960435i	$-0.410162\pi$
$71$	333.233	0.557007	0.278503	+	0.960435i	$0.410162\pi$
$72$	−174.722	−0.285988
$73$	891.133	1.42876	0.714378	−	0.699760i	$-0.246710\pi$
$73$	891.133	1.42876	0.714378	+	0.699760i	$0.246710\pi$
$74$	19.5905	0.0307751
$75$	0	0
$76$	86.4953	0.130549
$77$	106.963	0.158306
$78$	317.005	0.460176
$79$	−416.731	−0.593493	−0.296746	−	0.954956i	$-0.595902\pi$
$79$	−416.731	−0.593493	−0.296746	+	0.954956i	$0.595902\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	438.631	0.590716
$83$	−814.323	−1.07691	−0.538455	−	0.842654i	$-0.680992\pi$
$83$	−814.323	−1.07691	−0.538455	+	0.842654i	$0.680992\pi$
$84$	−128.223	−0.166552
$85$	0	0
$86$	536.342	0.672503
$87$	269.887	0.332586
$88$	−296.646	−0.359348
$89$	−650.076	−0.774247	−0.387123	−	0.922028i	$-0.626531\pi$
$89$	−650.076	−0.774247	−0.387123	+	0.922028i	$0.626531\pi$
$90$	0	0
$91$	537.451	0.619123
$92$	462.275	0.523864
$93$	868.690	0.968590
$94$	314.958	0.345590
$95$	0	0
$96$	557.290	0.592481
$97$	−1585.25	−1.65936	−0.829678	−	0.558243i	$-0.811476\pi$
$97$	−1585.25	−1.65936	−0.829678	+	0.558243i	$0.811476\pi$
$98$	67.4372	0.0695121
$99$	137.524	0.139613
$100$	0	0
$101$	−496.139	−0.488789	−0.244395	−	0.969676i	$-0.578589\pi$
$101$	−496.139	−0.488789	−0.244395	+	0.969676i	$0.578589\pi$
$102$	−399.286	−0.387600
$103$	468.440	0.448124	0.224062	−	0.974575i	$-0.428068\pi$
$103$	468.440	0.448124	0.224062	+	0.974575i	$0.428068\pi$
$104$	−1490.54	−1.40538
$105$	0	0
$106$	935.166	0.856900
$107$	1978.79	1.78782	0.893909	−	0.448249i	$-0.147952\pi$
$107$	1978.79	1.78782	0.893909	+	0.448249i	$0.147952\pi$
$108$	−164.859	−0.146885
$109$	−1009.70	−0.887260	−0.443630	−	0.896210i	$-0.646309\pi$
$109$	−1009.70	−0.887260	−0.443630	+	0.896210i	$0.646309\pi$
$110$	0	0
$111$	42.7035	0.0365157
$112$	154.902	0.130686
$113$	120.953	0.100693	0.0503466	−	0.998732i	$-0.483967\pi$
$113$	120.953	0.100693	0.0503466	+	0.998732i	$0.483967\pi$
$114$	−58.4883	−0.0480520
$115$	0	0
$116$	−549.300	−0.439666
$117$	691.008	0.546015
$118$	−548.620	−0.428005
$119$	−676.951	−0.521479
$120$	0	0
$121$	−1097.51	−0.824575
$122$	−201.432	−0.149482
$123$	956.130	0.700905
$124$	−1768.04	−1.28044
$125$	0	0
$126$	86.7050	0.0613040
$127$	−1545.52	−1.07986	−0.539932	−	0.841709i	$-0.681550\pi$
$127$	−1545.52	−1.07986	−0.539932	+	0.841709i	$0.681550\pi$
$128$	−1377.89	−0.951480
$129$	1169.12	0.797949
$130$	0	0
$131$	−2258.15	−1.50607	−0.753035	−	0.657980i	$-0.771411\pi$
$131$	−2258.15	−1.50607	−0.753035	+	0.657980i	$0.771411\pi$
$132$	−279.901	−0.184563
$133$	−99.1613	−0.0646494
$134$	401.174	0.258628
$135$	0	0
$136$	1877.43	1.18374
$137$	−1242.72	−0.774982	−0.387491	−	0.921873i	$-0.626658\pi$
$137$	−1242.72	−0.774982	−0.387491	+	0.921873i	$0.626658\pi$
$138$	−312.591	−0.192823
$139$	405.672	0.247544	0.123772	−	0.992311i	$-0.460501\pi$
$139$	405.672	0.247544	0.123772	+	0.992311i	$0.460501\pi$
$140$	0	0
$141$	686.548	0.410055
$142$	458.619	0.271031
$143$	1173.21	0.686075
$144$	199.159	0.115254
$145$	0	0
$146$	1226.44	0.695211
$147$	147.000	0.0824786
$148$	−86.9142	−0.0482723
$149$	882.453	0.485191	0.242595	−	0.970128i	$-0.422001\pi$
$149$	882.453	0.485191	0.242595	+	0.970128i	$0.422001\pi$
$150$	0	0
$151$	2246.66	1.21080	0.605400	−	0.795921i	$-0.293013\pi$
$151$	2246.66	1.21080	0.605400	+	0.795921i	$0.293013\pi$
$152$	275.010	0.146752
$153$	−870.366	−0.459901
$154$	147.210	0.0770292
$155$	0	0
$156$	−1406.40	−0.721811
$157$	2953.43	1.50133	0.750667	−	0.660681i	$-0.229732\pi$
$157$	2953.43	1.50133	0.750667	+	0.660681i	$0.229732\pi$
$158$	−573.535	−0.288785
$159$	2038.48	1.01674
$160$	0	0
$161$	−529.968	−0.259425
$162$	111.478	0.0540650
$163$	−104.301	−0.0501197	−0.0250599	−	0.999686i	$-0.507978\pi$
$163$	−104.301	−0.0501197	−0.0250599	+	0.999686i	$0.507978\pi$
$164$	−1946.01	−0.926570
$165$	0	0
$166$	−1120.73	−0.524008
$167$	2220.68	1.02899	0.514495	−	0.857493i	$-0.327979\pi$
$167$	2220.68	1.02899	0.514495	+	0.857493i	$0.327979\pi$
$168$	−407.684	−0.187223
$169$	3697.97	1.68319
$170$	0	0
$171$	−127.493	−0.0570154
$172$	−2379.50	−1.05486
$173$	−1727.64	−0.759247	−0.379623	−	0.925141i	$-0.623946\pi$
$173$	−1727.64	−0.759247	−0.379623	+	0.925141i	$0.623946\pi$
$174$	371.438	0.161831
$175$	0	0
$176$	338.137	0.144818
$177$	−1195.89	−0.507843
$178$	−894.681	−0.376737
$179$	−3332.33	−1.39145	−0.695727	−	0.718306i	$-0.744918\pi$
$179$	−3332.33	−1.39145	−0.695727	+	0.718306i	$0.744918\pi$
$180$	0	0
$181$	1308.68	0.537422	0.268711	−	0.963221i	$-0.413402\pi$
$181$	1308.68	0.537422	0.268711	+	0.963221i	$0.413402\pi$
$182$	739.678	0.301256
$183$	−439.083	−0.177366
$184$	1469.79	0.588883
$185$	0	0
$186$	1195.55	0.471302
$187$	−1477.73	−0.577872
$188$	−1397.33	−0.542077
$189$	189.000	0.0727393
$190$	0	0
$191$	−3545.30	−1.34308	−0.671541	−	0.740968i	$-0.734367\pi$
$191$	−3545.30	−1.34308	−0.671541	+	0.740968i	$0.734367\pi$
$192$	235.890	0.0886661
$193$	1077.32	0.401798	0.200899	−	0.979612i	$-0.435614\pi$
$193$	1077.32	0.401798	0.200899	+	0.979612i	$0.435614\pi$
$194$	−2181.73	−0.807417
$195$	0	0
$196$	−299.188	−0.109034
$197$	3800.24	1.37439	0.687197	−	0.726471i	$-0.258841\pi$
$197$	3800.24	1.37439	0.687197	+	0.726471i	$0.258841\pi$
$198$	189.270	0.0679334
$199$	2494.27	0.888513	0.444256	−	0.895900i	$-0.353468\pi$
$199$	2494.27	0.888513	0.444256	+	0.895900i	$0.353468\pi$
$200$	0	0
$201$	874.480	0.306871
$202$	−682.822	−0.237838
$203$	629.737	0.217728
$204$	1771.45	0.607972
$205$	0	0
$206$	644.700	0.218050
$207$	−681.388	−0.228791
$208$	1699.02	0.566375
$209$	−216.461	−0.0716407
$210$	0	0
$211$	−4235.73	−1.38199	−0.690995	−	0.722859i	$-0.742827\pi$
$211$	−4235.73	−1.38199	−0.690995	+	0.722859i	$0.742827\pi$
$212$	−4148.90	−1.34409
$213$	999.699	0.321588
$214$	2723.34	0.869925
$215$	0	0
$216$	−524.165	−0.165115
$217$	2026.94	0.634091
$218$	−1389.61	−0.431727
$219$	2673.40	0.824893
$220$	0	0
$221$	−7425.06	−2.26002
$222$	58.7716	0.0177680
$223$	−1274.70	−0.382781	−0.191391	−	0.981514i	$-0.561300\pi$
$223$	−1274.70	−0.382781	−0.191391	+	0.981514i	$0.561300\pi$
$224$	1300.34	0.387870
$225$	0	0
$226$	166.465	0.0489958
$227$	−2710.17	−0.792425	−0.396213	−	0.918159i	$-0.629676\pi$
$227$	−2710.17	−0.792425	−0.396213	+	0.918159i	$0.629676\pi$
$228$	259.486	0.0753722
$229$	1396.94	0.403111	0.201555	−	0.979477i	$-0.435400\pi$
$229$	1396.94	0.403111	0.201555	+	0.979477i	$0.435400\pi$
$230$	0	0
$231$	320.889	0.0913979
$232$	−1746.49	−0.494235
$233$	−5628.80	−1.58264	−0.791319	−	0.611403i	$-0.790605\pi$
$233$	−5628.80	−1.58264	−0.791319	+	0.611403i	$0.790605\pi$
$234$	951.014	0.265683
$235$	0	0
$236$	2433.98	0.671349
$237$	−1250.19	−0.342653
$238$	−931.668	−0.253744
$239$	3490.10	0.944585	0.472293	−	0.881442i	$-0.343427\pi$
$239$	3490.10	0.944585	0.472293	+	0.881442i	$0.343427\pi$
$240$	0	0
$241$	3077.81	0.822653	0.411326	−	0.911488i	$-0.365066\pi$
$241$	3077.81	0.822653	0.411326	+	0.911488i	$0.365066\pi$
$242$	−1510.47	−0.401226
$243$	243.000	0.0641500
$244$	893.662	0.234471
$245$	0	0
$246$	1315.89	0.341050
$247$	−1087.64	−0.280181
$248$	−5621.44	−1.43936
$249$	−2442.97	−0.621754
$250$	0	0
$251$	−3094.27	−0.778123	−0.389061	−	0.921212i	$-0.627201\pi$
$251$	−3094.27	−0.778123	−0.389061	+	0.921212i	$0.627201\pi$
$252$	−384.670	−0.0961586
$253$	−1156.88	−0.287479
$254$	−2127.05	−0.525445
$255$	0	0
$256$	−2525.39	−0.616550
$257$	1966.11	0.477208	0.238604	−	0.971117i	$-0.423310\pi$
$257$	1966.11	0.477208	0.238604	+	0.971117i	$0.423310\pi$
$258$	1609.03	0.388270
$259$	99.6416	0.0239051
$260$	0	0
$261$	809.662	0.192018
$262$	−3107.82	−0.732831
$263$	2828.34	0.663129	0.331565	−	0.943432i	$-0.392423\pi$
$263$	2828.34	0.663129	0.331565	+	0.943432i	$0.392423\pi$
$264$	−889.939	−0.207469
$265$	0	0
$266$	−136.473	−0.0314574
$267$	−1950.23	−0.447011
$268$	−1779.82	−0.405672
$269$	−1653.14	−0.374698	−0.187349	−	0.982293i	$-0.559989\pi$
$269$	−1653.14	−0.374698	−0.187349	+	0.982293i	$0.559989\pi$
$270$	0	0
$271$	−5326.26	−1.19390	−0.596950	−	0.802278i	$-0.703621\pi$
$271$	−5326.26	−1.19390	−0.596950	+	0.802278i	$0.703621\pi$
$272$	−2140.02	−0.477050
$273$	1612.35	0.357451
$274$	−1710.32	−0.377095
$275$	0	0
$276$	1386.82	0.302453
$277$	−5317.24	−1.15336	−0.576682	−	0.816968i	$-0.695653\pi$
$277$	−5317.24	−1.15336	−0.576682	+	0.816968i	$0.695653\pi$
$278$	558.314	0.120451
$279$	2606.07	0.559216
$280$	0	0
$281$	−3584.56	−0.760985	−0.380493	−	0.924784i	$-0.624246\pi$
$281$	−3584.56	−0.760985	−0.380493	+	0.924784i	$0.624246\pi$
$282$	944.875	0.199527
$283$	−1230.41	−0.258446	−0.129223	−	0.991616i	$-0.541248\pi$
$283$	−1230.41	−0.258446	−0.129223	+	0.991616i	$0.541248\pi$
$284$	−2034.68	−0.425127
$285$	0	0
$286$	1614.65	0.333834
$287$	2230.97	0.458850
$288$	1671.87	0.342069
$289$	4439.30	0.903582
$290$	0	0
$291$	−4755.74	−0.958029
$292$	−5441.15	−1.09048
$293$	−8544.91	−1.70375	−0.851875	−	0.523745i	$-0.824535\pi$
$293$	−8544.91	−1.70375	−0.851875	+	0.523745i	$0.824535\pi$
$294$	202.312	0.0401329
$295$	0	0
$296$	−276.342	−0.0542636
$297$	412.571	0.0806054
$298$	1214.49	0.236086
$299$	−5812.89	−1.12431
$300$	0	0
$301$	2727.95	0.522380
$302$	3092.02	0.589158
$303$	−1488.42	−0.282203
$304$	−313.474	−0.0591414
$305$	0	0
$306$	−1197.86	−0.223781
$307$	2066.36	0.384148	0.192074	−	0.981380i	$-0.438479\pi$
$307$	2066.36	0.384148	0.192074	+	0.981380i	$0.438479\pi$
$308$	−653.102	−0.120825
$309$	1405.32	0.258725
$310$	0	0
$311$	7346.18	1.33943	0.669717	−	0.742617i	$-0.266416\pi$
$311$	7346.18	1.33943	0.669717	+	0.742617i	$0.266416\pi$
$312$	−4471.63	−0.811398
$313$	6421.62	1.15965	0.579826	−	0.814740i	$-0.303120\pi$
$313$	6421.62	1.15965	0.579826	+	0.814740i	$0.303120\pi$
$314$	4064.72	0.730526
$315$	0	0
$316$	2544.51	0.452974
$317$	−799.972	−0.141738	−0.0708689	−	0.997486i	$-0.522577\pi$
$317$	−799.972	−0.141738	−0.0708689	+	0.997486i	$0.522577\pi$
$318$	2805.50	0.494731
$319$	1374.66	0.241274
$320$	0	0
$321$	5936.36	1.03220
$322$	−729.380	−0.126232
$323$	1369.95	0.235993
$324$	−494.576	−0.0848039
$325$	0	0
$326$	−143.547	−0.0243875
$327$	−3029.09	−0.512260
$328$	−6187.28	−1.04157
$329$	1601.94	0.268444
$330$	0	0
$331$	−4453.67	−0.739565	−0.369782	−	0.929118i	$-0.620568\pi$
$331$	−4453.67	−0.739565	−0.369782	+	0.929118i	$0.620568\pi$
$332$	4972.16	0.821935
$333$	128.111	0.0210823
$334$	3056.25	0.500691
$335$	0	0
$336$	464.705	0.0754516
$337$	10777.5	1.74210	0.871051	−	0.491193i	$-0.163439\pi$
$337$	10777.5	1.74210	0.871051	+	0.491193i	$0.163439\pi$
$338$	5089.40	0.819015
$339$	362.860	0.0581353
$340$	0	0
$341$	4424.64	0.702662
$342$	−175.465	−0.0277429
$343$	343.000	0.0539949
$344$	−7565.58	−1.18578
$345$	0	0
$346$	−2377.69	−0.369438
$347$	7096.10	1.09781	0.548903	−	0.835886i	$-0.315046\pi$
$347$	7096.10	1.09781	0.548903	+	0.835886i	$0.315046\pi$
$348$	−1647.90	−0.253841
$349$	8300.35	1.27309	0.636543	−	0.771241i	$-0.280364\pi$
$349$	8300.35	1.27309	0.636543	+	0.771241i	$0.280364\pi$
$350$	0	0
$351$	2073.02	0.315242
$352$	2838.54	0.429814
$353$	−9084.35	−1.36972	−0.684860	−	0.728674i	$-0.740137\pi$
$353$	−9084.35	−1.36972	−0.684860	+	0.728674i	$0.740137\pi$
$354$	−1645.86	−0.247109
$355$	0	0
$356$	3969.29	0.590932
$357$	−2030.85	−0.301076
$358$	−4586.19	−0.677061
$359$	−5129.81	−0.754153	−0.377077	−	0.926182i	$-0.623071\pi$
$359$	−5129.81	−0.754153	−0.377077	+	0.926182i	$0.623071\pi$
$360$	0	0
$361$	−6658.33	−0.970743
$362$	1801.10	0.261501
$363$	−3292.53	−0.476069
$364$	−3281.61	−0.472536
$365$	0	0
$366$	−604.296	−0.0863035
$367$	13074.0	1.85956	0.929780	−	0.368116i	$-0.119997\pi$
$367$	13074.0	1.85956	0.929780	+	0.368116i	$0.119997\pi$
$368$	−1675.37	−0.237322
$369$	2868.39	0.404668
$370$	0	0
$371$	4756.45	0.665614
$372$	−5304.11	−0.739262
$373$	−4056.31	−0.563076	−0.281538	−	0.959550i	$-0.590845\pi$
$373$	−4056.31	−0.563076	−0.281538	+	0.959550i	$0.590845\pi$
$374$	−2033.75	−0.281184
$375$	0	0
$376$	−4442.77	−0.609357
$377$	6907.20	0.943604
$378$	260.115	0.0353939
$379$	−11594.6	−1.57143	−0.785715	−	0.618588i	$-0.787705\pi$
$379$	−11594.6	−1.57143	−0.785715	+	0.618588i	$0.787705\pi$
$380$	0	0
$381$	−4636.56	−0.623460
$382$	−4879.29	−0.653523
$383$	10950.0	1.46089	0.730444	−	0.682972i	$-0.239313\pi$
$383$	10950.0	1.46089	0.730444	+	0.682972i	$0.239313\pi$
$384$	−4133.67	−0.549337
$385$	0	0
$386$	1482.68	0.195509
$387$	3507.36	0.460696
$388$	9679.33	1.26648
$389$	−2180.03	−0.284143	−0.142072	−	0.989856i	$-0.545376\pi$
$389$	−2180.03	−0.284143	−0.142072	+	0.989856i	$0.545376\pi$
$390$	0	0
$391$	7321.68	0.946991
$392$	−951.262	−0.122566
$393$	−6774.44	−0.869530
$394$	5230.16	0.668760
$395$	0	0
$396$	−839.703	−0.106557
$397$	6139.23	0.776119	0.388059	−	0.921634i	$-0.373146\pi$
$397$	6139.23	0.776119	0.388059	+	0.921634i	$0.373146\pi$
$398$	3432.79	0.432337
$399$	−297.484	−0.0373254
$400$	0	0
$401$	−11689.1	−1.45567	−0.727835	−	0.685752i	$-0.759473\pi$
$401$	−11689.1	−1.45567	−0.727835	+	0.685752i	$0.759473\pi$
$402$	1203.52	0.149319
$403$	22232.3	2.74806
$404$	3029.37	0.373061
$405$	0	0
$406$	866.689	0.105943
$407$	217.509	0.0264902
$408$	5632.28	0.683430
$409$	662.397	0.0800818	0.0400409	−	0.999198i	$-0.487251\pi$
$409$	662.397	0.0800818	0.0400409	+	0.999198i	$0.487251\pi$
$410$	0	0
$411$	−3728.15	−0.447436
$412$	−2860.24	−0.342024
$413$	−2790.40	−0.332461
$414$	−937.774	−0.111326
$415$	0	0
$416$	14262.7	1.68097
$417$	1217.01	0.142920
$418$	−297.908	−0.0348593
$419$	−7414.29	−0.864467	−0.432234	−	0.901762i	$-0.642274\pi$
$419$	−7414.29	−0.864467	−0.432234	+	0.901762i	$0.642274\pi$
$420$	0	0
$421$	9767.88	1.13078	0.565389	−	0.824825i	$-0.308726\pi$
$421$	9767.88	1.13078	0.565389	+	0.824825i	$0.308726\pi$
$422$	−5829.52	−0.672456
$423$	2059.64	0.236745
$424$	−13191.3	−1.51092
$425$	0	0
$426$	1375.86	0.156480
$427$	−1024.53	−0.116113
$428$	−12082.2	−1.36452
$429$	3519.63	0.396106
$430$	0	0
$431$	−16905.3	−1.88933	−0.944664	−	0.328041i	$-0.893612\pi$
$431$	−16905.3	−1.88933	−0.944664	+	0.328041i	$0.893612\pi$
$432$	597.478	0.0665421
$433$	11118.3	1.23397	0.616985	−	0.786975i	$-0.288354\pi$
$433$	11118.3	1.23397	0.616985	+	0.786975i	$0.288354\pi$
$434$	2789.62	0.308539
$435$	0	0
$436$	6165.08	0.677188
$437$	1072.50	0.117402
$438$	3679.32	0.401380
$439$	2480.01	0.269623	0.134811	−	0.990871i	$-0.456957\pi$
$439$	2480.01	0.269623	0.134811	+	0.990871i	$0.456957\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	−10218.9	−1.09969
$443$	1734.62	0.186036	0.0930182	−	0.995664i	$-0.470349\pi$
$443$	1734.62	0.186036	0.0930182	+	0.995664i	$0.470349\pi$
$444$	−260.743	−0.0278700
$445$	0	0
$446$	−1754.33	−0.186256
$447$	2647.36	0.280125
$448$	550.410	0.0580456
$449$	3488.08	0.366621	0.183311	−	0.983055i	$-0.441319\pi$
$449$	3488.08	0.366621	0.183311	+	0.983055i	$0.441319\pi$
$450$	0	0
$451$	4870.02	0.508471
$452$	−738.527	−0.0768526
$453$	6739.99	0.699056
$454$	−3729.93	−0.385582
$455$	0	0
$456$	825.029	0.0847270
$457$	3634.10	0.371983	0.185991	−	0.982551i	$-0.440450\pi$
$457$	3634.10	0.371983	0.185991	+	0.982551i	$0.440450\pi$
$458$	1922.57	0.196148
$459$	−2611.10	−0.265524
$460$	0	0
$461$	7589.30	0.766744	0.383372	−	0.923594i	$-0.374763\pi$
$461$	7589.30	0.766744	0.383372	+	0.923594i	$0.374763\pi$
$462$	441.629	0.0444729
$463$	−6275.14	−0.629871	−0.314936	−	0.949113i	$-0.601983\pi$
$463$	−6275.14	−0.629871	−0.314936	+	0.949113i	$0.601983\pi$
$464$	1990.76	0.199178
$465$	0	0
$466$	−7746.75	−0.770088
$467$	−11705.9	−1.15992	−0.579962	−	0.814644i	$-0.696933\pi$
$467$	−11705.9	−1.15992	−0.579962	+	0.814644i	$0.696933\pi$
$468$	−4219.21	−0.416738
$469$	2040.45	0.200894
$470$	0	0
$471$	8860.29	0.866795
$472$	7738.77	0.754674
$473$	5954.88	0.578871
$474$	−1720.60	−0.166730
$475$	0	0
$476$	4133.38	0.398011
$477$	6115.44	0.587016
$478$	4803.32	0.459621
$479$	−842.832	−0.0803966	−0.0401983	−	0.999192i	$-0.512799\pi$
$479$	−842.832	−0.0803966	−0.0401983	+	0.999192i	$0.512799\pi$
$480$	0	0
$481$	1092.91	0.103601
$482$	4235.90	0.400290
$483$	−1589.90	−0.149779
$484$	6701.26	0.629344
$485$	0	0
$486$	334.434	0.0312144
$487$	1570.99	0.146177	0.0730885	−	0.997325i	$-0.476714\pi$
$487$	1570.99	0.146177	0.0730885	+	0.997325i	$0.476714\pi$
$488$	2841.38	0.263572
$489$	−312.904	−0.0289366
$490$	0	0
$491$	12555.4	1.15400	0.577001	−	0.816743i	$-0.304223\pi$
$491$	12555.4	1.15400	0.577001	+	0.816743i	$0.304223\pi$
$492$	−5838.02	−0.534955
$493$	−8700.02	−0.794786
$494$	−1496.89	−0.136332
$495$	0	0
$496$	6407.69	0.580068
$497$	2332.63	0.210529
$498$	−3362.18	−0.302536
$499$	−19319.0	−1.73314	−0.866570	−	0.499055i	$-0.833680\pi$
$499$	−19319.0	−1.73314	−0.866570	+	0.499055i	$0.833680\pi$
$500$	0	0
$501$	6662.04	0.594088
$502$	−4258.56	−0.378623
$503$	12824.8	1.13684	0.568421	−	0.822738i	$-0.307554\pi$
$503$	12824.8	1.13684	0.568421	+	0.822738i	$0.307554\pi$
$504$	−1223.05	−0.108093
$505$	0	0
$506$	−1592.17	−0.139883
$507$	11093.9	0.971790
$508$	9436.76	0.824190
$509$	−5978.67	−0.520628	−0.260314	−	0.965524i	$-0.583826\pi$
$509$	−5978.67	−0.520628	−0.260314	+	0.965524i	$0.583826\pi$
$510$	0	0
$511$	6237.93	0.540019
$512$	7547.50	0.651476
$513$	−382.479	−0.0329179
$514$	2705.90	0.232202
$515$	0	0
$516$	−7138.51	−0.609022
$517$	3496.91	0.297474
$518$	137.134	0.0116319
$519$	−5182.91	−0.438351
$520$	0	0
$521$	4380.56	0.368361	0.184180	−	0.982892i	$-0.441037\pi$
$521$	4380.56	0.368361	0.184180	+	0.982892i	$0.441037\pi$
$522$	1114.31	0.0934333
$523$	−3438.46	−0.287482	−0.143741	−	0.989615i	$-0.545913\pi$
$523$	−3438.46	−0.287482	−0.143741	+	0.989615i	$0.545913\pi$
$524$	13788.0	1.14949
$525$	0	0
$526$	3892.56	0.322669
$527$	−28002.9	−2.31466
$528$	1014.41	0.0836110
$529$	−6435.03	−0.528892
$530$	0	0
$531$	−3587.66	−0.293203
$532$	605.467	0.0493427
$533$	24470.1	1.98859
$534$	−2684.04	−0.217509
$535$	0	0
$536$	−5658.91	−0.456022
$537$	−9997.00	−0.803357
$538$	−2275.17	−0.182322
$539$	748.740	0.0598340
$540$	0	0
$541$	−13321.2	−1.05864	−0.529319	−	0.848423i	$-0.677552\pi$
$541$	−13321.2	−1.05864	−0.529319	+	0.848423i	$0.677552\pi$
$542$	−7330.37	−0.580934
$543$	3926.04	0.310281
$544$	−17964.7	−1.41586
$545$	0	0
$546$	2219.03	0.173930
$547$	14729.0	1.15131	0.575655	−	0.817693i	$-0.304747\pi$
$547$	14729.0	1.15131	0.575655	+	0.817693i	$0.304747\pi$
$548$	7587.88	0.591493
$549$	−1317.25	−0.102402
$550$	0	0
$551$	−1274.40	−0.0985322
$552$	4409.37	0.339992
$553$	−2917.12	−0.224319
$554$	−7317.96	−0.561210
$555$	0	0
$556$	−2476.98	−0.188934
$557$	−18675.8	−1.42068	−0.710341	−	0.703858i	$-0.751459\pi$
$557$	−18675.8	−1.42068	−0.710341	+	0.703858i	$0.751459\pi$
$558$	3586.66	0.272106
$559$	29921.2	2.26392
$560$	0	0
$561$	−4433.18	−0.333635
$562$	−4933.32	−0.370284
$563$	−952.378	−0.0712929	−0.0356465	−	0.999364i	$-0.511349\pi$
$563$	−952.378	−0.0712929	−0.0356465	+	0.999364i	$0.511349\pi$
$564$	−4191.98	−0.312968
$565$	0	0
$566$	−1693.38	−0.125756
$567$	567.000	0.0419961
$568$	−6469.22	−0.477892
$569$	9623.89	0.709059	0.354530	−	0.935045i	$-0.384641\pi$
$569$	9623.89	0.709059	0.354530	+	0.935045i	$0.384641\pi$
$570$	0	0
$571$	−13532.6	−0.991804	−0.495902	−	0.868378i	$-0.665163\pi$
$571$	−13532.6	−0.991804	−0.495902	+	0.868378i	$0.665163\pi$
$572$	−7163.48	−0.523636
$573$	−10635.9	−0.775429
$574$	3070.42	0.223270
$575$	0	0
$576$	707.670	0.0511914
$577$	−14504.0	−1.04646	−0.523230	−	0.852191i	$-0.675273\pi$
$577$	−14504.0	−1.04646	−0.523230	+	0.852191i	$0.675273\pi$
$578$	6109.68	0.439670
$579$	3231.95	0.231978
$580$	0	0
$581$	−5700.26	−0.407034
$582$	−6545.19	−0.466163
$583$	10382.9	0.737593
$584$	−17300.0	−1.22582
$585$	0	0
$586$	−11760.1	−0.829019
$587$	25601.0	1.80011	0.900056	−	0.435775i	$-0.143525\pi$
$587$	25601.0	1.80011	0.900056	+	0.435775i	$0.143525\pi$
$588$	−897.564	−0.0629506
$589$	−4101.92	−0.286955
$590$	0	0
$591$	11400.7	0.793507
$592$	314.993	0.0218685
$593$	27481.9	1.90311	0.951556	−	0.307475i	$-0.0994840\pi$
$593$	27481.9	1.90311	0.951556	+	0.307475i	$0.0994840\pi$
$594$	567.809	0.0392214
$595$	0	0
$596$	−5388.15	−0.370314
$597$	7482.81	0.512983
$598$	−8000.11	−0.547072
$599$	23862.6	1.62771	0.813855	−	0.581067i	$-0.197365\pi$
$599$	23862.6	1.62771	0.813855	+	0.581067i	$0.197365\pi$
$600$	0	0
$601$	22709.4	1.54132	0.770661	−	0.637245i	$-0.219926\pi$
$601$	22709.4	1.54132	0.770661	+	0.637245i	$0.219926\pi$
$602$	3754.40	0.254182
$603$	2623.44	0.177172
$604$	−13717.9	−0.924126
$605$	0	0
$606$	−2048.47	−0.137316
$607$	−7717.69	−0.516065	−0.258032	−	0.966136i	$-0.583074\pi$
$607$	−7717.69	−0.516065	−0.258032	+	0.966136i	$0.583074\pi$
$608$	−2631.50	−0.175529
$609$	1889.21	0.125706
$610$	0	0
$611$	17570.7	1.16340
$612$	5314.35	0.351013
$613$	−12662.3	−0.834296	−0.417148	−	0.908839i	$-0.636970\pi$
$613$	−12662.3	−0.834296	−0.417148	+	0.908839i	$0.636970\pi$
$614$	2843.87	0.186921
$615$	0	0
$616$	−2076.52	−0.135821
$617$	7990.11	0.521345	0.260672	−	0.965427i	$-0.416056\pi$
$617$	7990.11	0.521345	0.260672	+	0.965427i	$0.416056\pi$
$618$	1934.10	0.125891
$619$	16702.4	1.08453	0.542266	−	0.840207i	$-0.317567\pi$
$619$	16702.4	1.08453	0.542266	+	0.840207i	$0.317567\pi$
$620$	0	0
$621$	−2044.16	−0.132093
$622$	10110.3	0.651748
$623$	−4550.53	−0.292638
$624$	5097.06	0.326997
$625$	0	0
$626$	8837.88	0.564270
$627$	−649.382	−0.0413618
$628$	−18033.3	−1.14587
$629$	−1376.58	−0.0872621
$630$	0	0
$631$	12762.2	0.805162	0.402581	−	0.915384i	$-0.368113\pi$
$631$	12762.2	0.805162	0.402581	+	0.915384i	$0.368113\pi$
$632$	8090.21	0.509195
$633$	−12707.2	−0.797893
$634$	−1100.98	−0.0689675
$635$	0	0
$636$	−12446.7	−0.776013
$637$	3762.16	0.234006
$638$	1891.91	0.117400
$639$	2999.10	0.185669
$640$	0	0
$641$	−29847.0	−1.83914	−0.919568	−	0.392931i	$-0.871461\pi$
$641$	−29847.0	−1.83914	−0.919568	+	0.392931i	$0.871461\pi$
$642$	8170.03	0.502251
$643$	−11676.6	−0.716141	−0.358070	−	0.933695i	$-0.616565\pi$
$643$	−11676.6	−0.716141	−0.358070	+	0.933695i	$0.616565\pi$
$644$	3235.92	0.198002
$645$	0	0
$646$	1885.42	0.114831
$647$	−7200.41	−0.437523	−0.218762	−	0.975778i	$-0.570202\pi$
$647$	−7200.41	−0.437523	−0.218762	+	0.975778i	$0.570202\pi$
$648$	−1572.49	−0.0953293
$649$	−6091.20	−0.368414
$650$	0	0
$651$	6080.83	0.366093
$652$	636.852	0.0382531
$653$	−18481.6	−1.10756	−0.553782	−	0.832662i	$-0.686816\pi$
$653$	−18481.6	−1.10756	−0.553782	+	0.832662i	$0.686816\pi$
$654$	−4168.84	−0.249258
$655$	0	0
$656$	7052.67	0.419757
$657$	8020.19	0.476252
$658$	2204.71	0.130621
$659$	26203.7	1.54894	0.774470	−	0.632611i	$-0.218017\pi$
$659$	26203.7	1.54894	0.774470	+	0.632611i	$0.218017\pi$
$660$	0	0
$661$	4432.97	0.260851	0.130426	−	0.991458i	$-0.458366\pi$
$661$	4432.97	0.260851	0.130426	+	0.991458i	$0.458366\pi$
$662$	−6129.46	−0.359861
$663$	−22275.2	−1.30482
$664$	15808.9	0.923950
$665$	0	0
$666$	176.315	0.0102584
$667$	−6811.03	−0.395389
$668$	−13559.2	−0.785361
$669$	−3824.10	−0.220999
$670$	0	0
$671$	−2236.45	−0.128670
$672$	3901.03	0.223937
$673$	−14535.9	−0.832570	−0.416285	−	0.909234i	$-0.636668\pi$
$673$	−14535.9	−0.832570	−0.416285	+	0.909234i	$0.636668\pi$
$674$	14832.8	0.847681
$675$	0	0
$676$	−22579.3	−1.28467
$677$	−14611.2	−0.829475	−0.414737	−	0.909941i	$-0.636127\pi$
$677$	−14611.2	−0.829475	−0.414737	+	0.909941i	$0.636127\pi$
$678$	499.394	0.0282878
$679$	−11096.7	−0.627177
$680$	0	0
$681$	−8130.52	−0.457507
$682$	6089.51	0.341905
$683$	15735.8	0.881570	0.440785	−	0.897613i	$-0.354700\pi$
$683$	15735.8	0.881570	0.440785	+	0.897613i	$0.354700\pi$
$684$	778.457	0.0435162
$685$	0	0
$686$	472.061	0.0262731
$687$	4190.82	0.232736
$688$	8623.75	0.477874
$689$	52170.6	2.88467
$690$	0	0
$691$	18704.7	1.02975	0.514877	−	0.857264i	$-0.327838\pi$
$691$	18704.7	1.02975	0.514877	+	0.857264i	$0.327838\pi$
$692$	10548.7	0.579484
$693$	962.666	0.0527686
$694$	9766.15	0.534176
$695$	0	0
$696$	−5239.46	−0.285347
$697$	−30821.6	−1.67497
$698$	11423.5	0.619465
$699$	−16886.4	−0.913737
$700$	0	0
$701$	14341.3	0.772699	0.386349	−	0.922353i	$-0.373736\pi$
$701$	14341.3	0.772699	0.386349	+	0.922353i	$0.373736\pi$
$702$	2853.04	0.153392
$703$	−201.645	−0.0108182
$704$	1201.50	0.0643227
$705$	0	0
$706$	−12502.5	−0.666486
$707$	−3472.98	−0.184745
$708$	7301.93	0.387604
$709$	−21041.9	−1.11459	−0.557295	−	0.830315i	$-0.688161\pi$
$709$	−21041.9	−1.11459	−0.557295	+	0.830315i	$0.688161\pi$
$710$	0	0
$711$	−3750.58	−0.197831
$712$	12620.3	0.664276
$713$	−21922.8	−1.15149
$714$	−2795.00	−0.146499
$715$	0	0
$716$	20346.8	1.06201
$717$	10470.3	0.545356
$718$	−7060.01	−0.366960
$719$	−18439.2	−0.956422	−0.478211	−	0.878245i	$-0.658715\pi$
$719$	−18439.2	−0.956422	−0.478211	+	0.878245i	$0.658715\pi$
$720$	0	0
$721$	3279.08	0.169375
$722$	−9163.66	−0.472349
$723$	9233.43	0.474959
$724$	−7990.64	−0.410179
$725$	0	0
$726$	−4531.41	−0.231648
$727$	5826.21	0.297224	0.148612	−	0.988896i	$-0.452519\pi$
$727$	5826.21	0.297224	0.148612	+	0.988896i	$0.452519\pi$
$728$	−10433.8	−0.531185
$729$	729.000	0.0370370
$730$	0	0
$731$	−37687.5	−1.90687
$732$	2680.99	0.135372
$733$	−27711.8	−1.39640	−0.698198	−	0.715905i	$-0.746014\pi$
$733$	−27711.8	−1.39640	−0.698198	+	0.715905i	$0.746014\pi$
$734$	17993.4	0.904834
$735$	0	0
$736$	−14064.1	−0.704360
$737$	4454.14	0.222619
$738$	3947.68	0.196905
$739$	−32108.2	−1.59827	−0.799133	−	0.601154i	$-0.794708\pi$
$739$	−32108.2	−1.59827	−0.799133	+	0.601154i	$0.794708\pi$
$740$	0	0
$741$	−3262.92	−0.161763
$742$	6546.16	0.323878
$743$	−3004.06	−0.148329	−0.0741643	−	0.997246i	$-0.523629\pi$
$743$	−3004.06	−0.148329	−0.0741643	+	0.997246i	$0.523629\pi$
$744$	−16864.3	−0.831016
$745$	0	0
$746$	−5582.57	−0.273985
$747$	−7328.91	−0.358970
$748$	9022.82	0.441052
$749$	13851.5	0.675731
$750$	0	0
$751$	−12573.6	−0.610942	−0.305471	−	0.952201i	$-0.598814\pi$
$751$	−12573.6	−0.610942	−0.305471	+	0.952201i	$0.598814\pi$
$752$	5064.16	0.245573
$753$	−9282.82	−0.449249
$754$	9506.17	0.459144
$755$	0	0
$756$	−1154.01	−0.0555172
$757$	−7518.76	−0.360996	−0.180498	−	0.983575i	$-0.557771\pi$
$757$	−7518.76	−0.360996	−0.180498	+	0.983575i	$0.557771\pi$
$758$	−15957.2	−0.764635
$759$	−3470.63	−0.165976
$760$	0	0
$761$	−41187.7	−1.96196	−0.980981	−	0.194105i	$-0.937820\pi$
$761$	−41187.7	−1.96196	−0.980981	+	0.194105i	$0.937820\pi$
$762$	−6381.16	−0.303366
$763$	−7067.87	−0.335353
$764$	21647.1	1.02509
$765$	0	0
$766$	15070.2	0.710847
$767$	−30606.2	−1.44084
$768$	−7576.17	−0.355965
$769$	−28909.5	−1.35566	−0.677832	−	0.735217i	$-0.737080\pi$
$769$	−28909.5	−1.35566	−0.677832	+	0.735217i	$0.737080\pi$
$770$	0	0
$771$	5898.33	0.275516
$772$	−6577.97	−0.306666
$773$	−17242.4	−0.802285	−0.401142	−	0.916016i	$-0.631387\pi$
$773$	−17242.4	−0.802285	−0.401142	+	0.916016i	$0.631387\pi$
$774$	4827.08	0.224168
$775$	0	0
$776$	30775.2	1.42367
$777$	298.925	0.0138016
$778$	−3000.31	−0.138260
$779$	−4514.81	−0.207651
$780$	0	0
$781$	5091.93	0.233295
$782$	10076.6	0.460792
$783$	2428.99	0.110862
$784$	1084.31	0.0493947
$785$	0	0
$786$	−9323.46	−0.423100
$787$	19396.4	0.878535	0.439267	−	0.898356i	$-0.355238\pi$
$787$	19396.4	0.878535	0.439267	+	0.898356i	$0.355238\pi$
$788$	−23203.8	−1.04899
$789$	8485.03	0.382858
$790$	0	0
$791$	846.674	0.0380585
$792$	−2669.82	−0.119783
$793$	−11237.4	−0.503218
$794$	8449.24	0.377648
$795$	0	0
$796$	−15229.7	−0.678144
$797$	−14898.8	−0.662161	−0.331080	−	0.943603i	$-0.607413\pi$
$797$	−14898.8	−0.662161	−0.331080	+	0.943603i	$0.607413\pi$
$798$	−409.418	−0.0181620
$799$	−22131.4	−0.979915
$800$	0	0
$801$	−5850.69	−0.258082
$802$	−16087.3	−0.708307
$803$	13616.9	0.598417
$804$	−5339.47	−0.234215
$805$	0	0
$806$	30597.6	1.33717
$807$	−4959.42	−0.216332
$808$	9631.80	0.419363
$809$	−12088.9	−0.525367	−0.262684	−	0.964882i	$-0.584608\pi$
$809$	−12088.9	−0.525367	−0.262684	+	0.964882i	$0.584608\pi$
$810$	0	0
$811$	9177.34	0.397361	0.198681	−	0.980064i	$-0.436334\pi$
$811$	9177.34	0.397361	0.198681	+	0.980064i	$0.436334\pi$
$812$	−3845.10	−0.166178
$813$	−15978.8	−0.689299
$814$	299.351	0.0128898
$815$	0	0
$816$	−6420.05	−0.275425
$817$	−5520.55	−0.236401
$818$	911.638	0.0389666
$819$	4837.06	0.206374
$820$	0	0
$821$	−5604.88	−0.238260	−0.119130	−	0.992879i	$-0.538011\pi$
$821$	−5604.88	−0.238260	−0.119130	+	0.992879i	$0.538011\pi$
$822$	−5130.95	−0.217716
$823$	39788.8	1.68524	0.842619	−	0.538511i	$-0.181013\pi$
$823$	39788.8	1.68524	0.842619	+	0.538511i	$0.181013\pi$
$824$	−9094.06	−0.384474
$825$	0	0
$826$	−3840.34	−0.161771
$827$	−36668.7	−1.54183	−0.770916	−	0.636937i	$-0.780201\pi$
$827$	−36668.7	−1.54183	−0.770916	+	0.636937i	$0.780201\pi$
$828$	4160.47	0.174621
$829$	16027.0	0.671460	0.335730	−	0.941958i	$-0.391017\pi$
$829$	16027.0	0.671460	0.335730	+	0.941958i	$0.391017\pi$
$830$	0	0
$831$	−15951.7	−0.665895
$832$	6037.11	0.251561
$833$	−4738.66	−0.197101
$834$	1674.94	0.0695425
$835$	0	0
$836$	1321.68	0.0546787
$837$	7818.21	0.322863
$838$	−10204.1	−0.420637
$839$	−26562.9	−1.09303	−0.546515	−	0.837449i	$-0.684046\pi$
$839$	−26562.9	−1.09303	−0.546515	+	0.837449i	$0.684046\pi$
$840$	0	0
$841$	−16295.8	−0.668160
$842$	13443.2	0.550219
$843$	−10753.7	−0.439355
$844$	25862.9	1.05478
$845$	0	0
$846$	2834.63	0.115197
$847$	−7682.56	−0.311660
$848$	15036.4	0.608905
$849$	−3691.23	−0.149214
$850$	0	0
$851$	−1077.69	−0.0434110
$852$	−6104.04	−0.245447
$853$	−26089.3	−1.04722	−0.523611	−	0.851957i	$-0.675416\pi$
$853$	−26089.3	−1.04722	−0.523611	+	0.851957i	$0.675416\pi$
$854$	−1410.03	−0.0564989
$855$	0	0
$856$	−38415.2	−1.53388
$857$	−9551.99	−0.380735	−0.190367	−	0.981713i	$-0.560968\pi$
$857$	−9551.99	−0.380735	−0.190367	+	0.981713i	$0.560968\pi$
$858$	4843.96	0.192739
$859$	29646.7	1.17757	0.588784	−	0.808290i	$-0.299607\pi$
$859$	29646.7	1.17757	0.588784	+	0.808290i	$0.299607\pi$
$860$	0	0
$861$	6692.91	0.264917
$862$	−23266.3	−0.919318
$863$	5981.74	0.235945	0.117973	−	0.993017i	$-0.462360\pi$
$863$	5981.74	0.235945	0.117973	+	0.993017i	$0.462360\pi$
$864$	5015.61	0.197494
$865$	0	0
$866$	15301.7	0.600431
$867$	13317.9	0.521683
$868$	−12376.3	−0.483961
$869$	−6367.82	−0.248577
$870$	0	0
$871$	22380.5	0.870647
$872$	19601.7	0.761237
$873$	−14267.2	−0.553118
$874$	1476.05	0.0571258
$875$	0	0
$876$	−16323.4	−0.629587
$877$	27503.4	1.05898	0.529488	−	0.848317i	$-0.322384\pi$
$877$	27503.4	1.05898	0.529488	+	0.848317i	$0.322384\pi$
$878$	3413.16	0.131194
$879$	−25634.7	−0.983661
$880$	0	0
$881$	27005.9	1.03275	0.516374	−	0.856363i	$-0.327281\pi$
$881$	27005.9	1.03275	0.516374	+	0.856363i	$0.327281\pi$
$882$	606.935	0.0231707
$883$	−11434.3	−0.435782	−0.217891	−	0.975973i	$-0.569918\pi$
$883$	−11434.3	−0.435782	−0.217891	+	0.975973i	$0.569918\pi$
$884$	45336.5	1.72492
$885$	0	0
$886$	2387.30	0.0905225
$887$	−13464.3	−0.509683	−0.254841	−	0.966983i	$-0.582023\pi$
$887$	−13464.3	−0.509683	−0.254841	+	0.966983i	$0.582023\pi$
$888$	−829.025	−0.0313291
$889$	−10818.6	−0.408150
$890$	0	0
$891$	1237.71	0.0465375
$892$	7783.16	0.292152
$893$	−3241.85	−0.121483
$894$	3643.48	0.136305
$895$	0	0
$896$	−9645.23	−0.359626
$897$	−17438.7	−0.649120
$898$	4800.55	0.178392
$899$	26049.8	0.966418
$900$	0	0
$901$	−65711.9	−2.42972
$902$	6702.46	0.247414
$903$	8183.85	0.301596
$904$	−2348.13	−0.0863912
$905$	0	0
$906$	9276.05	0.340150
$907$	1220.35	0.0446760	0.0223380	−	0.999750i	$-0.492889\pi$
$907$	1220.35	0.0446760	0.0223380	+	0.999750i	$0.492889\pi$
$908$	16548.0	0.604807
$909$	−4465.25	−0.162930
$910$	0	0
$911$	−2651.77	−0.0964403	−0.0482201	−	0.998837i	$-0.515355\pi$
$911$	−2651.77	−0.0964403	−0.0482201	+	0.998837i	$0.515355\pi$
$912$	−940.423	−0.0341453
$913$	−12443.2	−0.451051
$914$	5001.51	0.181001
$915$	0	0
$916$	−8529.54	−0.307668
$917$	−15807.0	−0.569241
$918$	−3593.57	−0.129200
$919$	−47600.3	−1.70858	−0.854291	−	0.519795i	$-0.826008\pi$
$919$	−47600.3	−1.70858	−0.854291	+	0.519795i	$0.826008\pi$
$920$	0	0
$921$	6199.08	0.221788
$922$	10444.9	0.373086
$923$	25585.2	0.912402
$924$	−1959.31	−0.0697581
$925$	0	0
$926$	−8636.28	−0.306486
$927$	4215.96	0.149375
$928$	16711.7	0.591152
$929$	1545.26	0.0545729	0.0272865	−	0.999628i	$-0.491313\pi$
$929$	1545.26	0.0545729	0.0272865	+	0.999628i	$0.491313\pi$
$930$	0	0
$931$	−694.129	−0.0244352
$932$	34368.8	1.20793
$933$	22038.5	0.773322
$934$	−16110.5	−0.564402
$935$	0	0
$936$	−13414.9	−0.468461
$937$	−53013.1	−1.84830	−0.924152	−	0.382025i	$-0.875227\pi$
$937$	−53013.1	−1.84830	−0.924152	+	0.382025i	$0.875227\pi$
$938$	2808.22	0.0977522
$939$	19264.9	0.669526
$940$	0	0
$941$	19832.7	0.687066	0.343533	−	0.939141i	$-0.388376\pi$
$941$	19832.7	0.687066	0.343533	+	0.939141i	$0.388376\pi$
$942$	12194.2	0.421770
$943$	−24129.5	−0.833259
$944$	−8821.17	−0.304136
$945$	0	0
$946$	8195.53	0.281670
$947$	14700.2	0.504428	0.252214	−	0.967671i	$-0.418841\pi$
$947$	14700.2	0.504428	0.252214	+	0.967671i	$0.418841\pi$
$948$	7633.53	0.261525
$949$	68420.0	2.34037
$950$	0	0
$951$	−2399.92	−0.0818324
$952$	13142.0	0.447410
$953$	39820.9	1.35354	0.676770	−	0.736194i	$-0.263379\pi$
$953$	39820.9	1.35354	0.676770	+	0.736194i	$0.263379\pi$
$954$	8416.49	0.285633
$955$	0	0
$956$	−21310.1	−0.720940
$957$	4123.99	0.139299
$958$	−1159.96	−0.0391198
$959$	−8699.02	−0.292916
$960$	0	0
$961$	54055.8	1.81450
$962$	1504.14	0.0504109
$963$	17809.1	0.595939
$964$	−18792.7	−0.627877
$965$	0	0
$966$	−2188.14	−0.0728801
$967$	50708.3	1.68632	0.843160	−	0.537663i	$-0.180693\pi$
$967$	50708.3	1.68632	0.843160	+	0.537663i	$0.180693\pi$
$968$	21306.5	0.707455
$969$	4109.84	0.136251
$970$	0	0
$971$	58395.2	1.92996	0.964980	−	0.262325i	$-0.0844892\pi$
$971$	58395.2	1.92996	0.964980	+	0.262325i	$0.0844892\pi$
$972$	−1483.73	−0.0489615
$973$	2839.70	0.0935628
$974$	2162.10	0.0711275
$975$	0	0
$976$	−3238.79	−0.106221
$977$	−31028.9	−1.01607	−0.508036	−	0.861336i	$-0.669628\pi$
$977$	−31028.9	−1.01607	−0.508036	+	0.861336i	$0.669628\pi$
$978$	−430.641	−0.0140801
$979$	−9933.43	−0.324284
$980$	0	0
$981$	−9087.26	−0.295753
$982$	17279.6	0.561520
$983$	−4769.34	−0.154749	−0.0773746	−	0.997002i	$-0.524654\pi$
$983$	−4769.34	−0.154749	−0.0773746	+	0.997002i	$0.524654\pi$
$984$	−18561.8	−0.601351
$985$	0	0
$986$	−11973.6	−0.386731
$987$	4805.83	0.154986
$988$	6640.99	0.213844
$989$	−29504.6	−0.948627
$990$	0	0
$991$	6849.75	0.219566	0.109783	−	0.993956i	$-0.464984\pi$
$991$	6849.75	0.219566	0.109783	+	0.993956i	$0.464984\pi$
$992$	53790.2	1.72161
$993$	−13361.0	−0.426988
$994$	3210.33	0.102440
$995$	0	0
$996$	14916.5	0.474545
$997$	−37344.1	−1.18626	−0.593130	−	0.805107i	$-0.702108\pi$
$997$	−37344.1	−1.18626	−0.593130	+	0.805107i	$0.702108\pi$
$998$	−26588.2	−0.843320
$999$	384.332	0.0121719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.4.a.u.1.3	yes	4
3.2	odd	2		1575.4.a.bk.1.2		4
5.2	odd	4		525.4.d.n.274.5		8
5.3	odd	4		525.4.d.n.274.4		8
5.4	even	2		525.4.a.t.1.2	✓	4
15.14	odd	2		1575.4.a.bj.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.4.a.t.1.2	✓	4	5.4	even	2
525.4.a.u.1.3	yes	4	1.1	even	1	trivial
525.4.d.n.274.4		8	5.3	odd	4
525.4.d.n.274.5		8	5.2	odd	4
1575.4.a.bj.1.3		4	15.14	odd	2
1575.4.a.bk.1.2		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)

\(f(q)\)	\(=\)	\(q+1.37627 q^{2} +3.00000 q^{3} -6.10588 q^{4} +4.12881 q^{6} +7.00000 q^{7} -19.4135 q^{8} +9.00000 q^{9} +15.2804 q^{11} -18.3176 q^{12} +76.7787 q^{13} +9.63389 q^{14} +22.1288 q^{16} -96.7073 q^{17} +12.3864 q^{18} -14.1659 q^{19} +21.0000 q^{21} +21.0300 q^{22} -75.7097 q^{23} -58.2405 q^{24} +105.668 q^{26} +27.0000 q^{27} -42.7412 q^{28} +89.9625 q^{29} +289.563 q^{31} +185.763 q^{32} +45.8412 q^{33} -133.095 q^{34} -54.9529 q^{36} +14.2345 q^{37} -19.4961 q^{38} +230.336 q^{39} +318.710 q^{41} +28.9017 q^{42} +389.707 q^{43} -93.3003 q^{44} -104.197 q^{46} +228.849 q^{47} +66.3864 q^{48} +49.0000 q^{49} -290.122 q^{51} -468.801 q^{52} +679.493 q^{53} +37.1593 q^{54} -135.895 q^{56} -42.4977 q^{57} +123.813 q^{58} -398.628 q^{59} -146.361 q^{61} +398.517 q^{62} +63.0000 q^{63} +78.6300 q^{64} +63.0899 q^{66} +291.493 q^{67} +590.483 q^{68} -227.129 q^{69} +333.233 q^{71} -174.722 q^{72} +891.133 q^{73} +19.5905 q^{74} +86.4953 q^{76} +106.963 q^{77} +317.005 q^{78} -416.731 q^{79} +81.0000 q^{81} +438.631 q^{82} -814.323 q^{83} -128.223 q^{84} +536.342 q^{86} +269.887 q^{87} -296.646 q^{88} -650.076 q^{89} +537.451 q^{91} +462.275 q^{92} +868.690 q^{93} +314.958 q^{94} +557.290 q^{96} -1585.25 q^{97} +67.4372 q^{98} +137.524 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} + 16 q^{4} + 28 q^{7} + 9 q^{8} + 36 q^{9} + 21 q^{11} + 48 q^{12} + 5 q^{13} + 72 q^{16} + 99 q^{17} + 72 q^{19} + 84 q^{21} + 221 q^{22} + 102 q^{23} + 27 q^{24} + 129 q^{26} + 108 q^{27}+ \cdots + 189 q^{99}+O(q^{100}) \) 4 * q + 12 * q^3 + 16 * q^4 + 28 * q^7 + 9 * q^8 + 36 * q^9 + 21 * q^11 + 48 * q^12 + 5 * q^13 + 72 * q^16 + 99 * q^17 + 72 * q^19 + 84 * q^21 + 221 * q^22 + 102 * q^23 + 27 * q^24 + 129 * q^26 + 108 * q^27 + 112 * q^28 - 240 * q^29 + 351 * q^31 + 72 * q^32 + 63 * q^33 - 285 * q^34 + 144 * q^36 + 399 * q^37 + 324 * q^38 + 15 * q^39 + 381 * q^41 + 460 * q^43 - 975 * q^44 + 550 * q^46 + 60 * q^47 + 216 * q^48 + 196 * q^49 + 297 * q^51 + 223 * q^52 + 873 * q^53 + 63 * q^56 + 216 * q^57 + 1408 * q^58 - 855 * q^59 + 687 * q^61 - 477 * q^62 + 252 * q^63 - 1285 * q^64 + 663 * q^66 - 503 * q^67 + 1725 * q^68 + 306 * q^69 - 681 * q^71 + 81 * q^72 + 1228 * q^73 - 3369 * q^74 + 1910 * q^76 + 147 * q^77 + 387 * q^78 + 345 * q^79 + 324 * q^81 - 495 * q^82 + 1509 * q^83 + 336 * q^84 - 540 * q^86 - 720 * q^87 + 2266 * q^88 - 198 * q^89 + 35 * q^91 + 2994 * q^92 + 1053 * q^93 - 1732 * q^94 + 216 * q^96 - 372 * q^97 + 189 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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