LMFDB - Embedded newform 1150.4.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1150,4,Mod(1,1150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1150, 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("1150.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.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:	$67.8521965066$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 107x^{3} - 3x^{2} + 2151x - 2916 $ x^5 - 107x^3 - 3x^2 + 2151*x - 2916
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.21006$ of defining polynomial
Character	$\chi$	$=$	1150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} -7.21006 q^{3} +4.00000 q^{4} +14.4201 q^{6} -19.8879 q^{7} -8.00000 q^{8} +24.9850 q^{9} -31.0553 q^{11} -28.8402 q^{12} -7.15563 q^{13} +39.7757 q^{14} +16.0000 q^{16} -40.8326 q^{17} -49.9699 q^{18} +144.948 q^{19} +143.393 q^{21} +62.1106 q^{22} -23.0000 q^{23} +57.6805 q^{24} +14.3113 q^{26} +14.5286 q^{27} -79.5515 q^{28} -189.113 q^{29} +41.6222 q^{31} -32.0000 q^{32} +223.910 q^{33} +81.6652 q^{34} +99.9398 q^{36} -37.3828 q^{37} -289.896 q^{38} +51.5925 q^{39} +401.324 q^{41} -286.785 q^{42} -29.2550 q^{43} -124.221 q^{44} +46.0000 q^{46} -18.0210 q^{47} -115.361 q^{48} +52.5274 q^{49} +294.405 q^{51} -28.6225 q^{52} -214.024 q^{53} -29.0572 q^{54} +159.103 q^{56} -1045.08 q^{57} +378.226 q^{58} +313.495 q^{59} +452.293 q^{61} -83.2444 q^{62} -496.898 q^{63} +64.0000 q^{64} -447.821 q^{66} +858.562 q^{67} -163.330 q^{68} +165.831 q^{69} +451.196 q^{71} -199.880 q^{72} +742.705 q^{73} +74.7656 q^{74} +579.792 q^{76} +617.623 q^{77} -103.185 q^{78} +126.369 q^{79} -779.346 q^{81} -802.649 q^{82} +473.737 q^{83} +573.571 q^{84} +58.5100 q^{86} +1363.52 q^{87} +248.442 q^{88} -109.465 q^{89} +142.310 q^{91} -92.0000 q^{92} -300.099 q^{93} +36.0420 q^{94} +230.722 q^{96} +711.495 q^{97} -105.055 q^{98} -775.915 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 10 q^{2} - 5 q^{3} + 20 q^{4} + 10 q^{6} + 3 q^{7} - 40 q^{8} + 84 q^{9} - 26 q^{11} - 20 q^{12} + 61 q^{13} - 6 q^{14} + 80 q^{16} - 231 q^{17} - 168 q^{18} + 74 q^{19} - 88 q^{21} + 52 q^{22} - 115 q^{23}+ \cdots + 2397 q^{99}+O(q^{100}) $ 5 * q - 10 * q^2 - 5 * q^3 + 20 * q^4 + 10 * q^6 + 3 * q^7 - 40 * q^8 + 84 * q^9 - 26 * q^11 - 20 * q^12 + 61 * q^13 - 6 * q^14 + 80 * q^16 - 231 * q^17 - 168 * q^18 + 74 * q^19 - 88 * q^21 + 52 * q^22 - 115 * q^23 + 40 * q^24 - 122 * q^26 - 368 * q^27 + 12 * q^28 + 37 * q^29 + 220 * q^31 - 160 * q^32 - 391 * q^33 + 462 * q^34 + 336 * q^36 - 220 * q^37 - 148 * q^38 + 195 * q^39 + 112 * q^41 + 176 * q^42 - 97 * q^43 - 104 * q^44 + 230 * q^46 - 370 * q^47 - 80 * q^48 + 182 * q^49 + 563 * q^51 + 244 * q^52 - 1224 * q^53 + 736 * q^54 - 24 * q^56 + 1009 * q^57 - 74 * q^58 - 181 * q^59 + 1574 * q^61 - 440 * q^62 - 1829 * q^63 + 320 * q^64 + 782 * q^66 + 435 * q^67 - 924 * q^68 + 115 * q^69 - 30 * q^71 - 672 * q^72 - 2180 * q^73 + 440 * q^74 + 296 * q^76 + 1401 * q^77 - 390 * q^78 + 609 * q^79 + 1453 * q^81 - 224 * q^82 - 2018 * q^83 - 352 * q^84 + 194 * q^86 + 1485 * q^87 + 208 * q^88 - 1663 * q^89 + 361 * q^91 - 460 * q^92 - 2741 * q^93 + 740 * q^94 + 160 * q^96 + 1274 * q^97 - 364 * q^98 + 2397 * q^99

Coefficient data

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

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

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

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−7.21006	−1.38758	−0.693788	−	0.720179i	$-0.744060\pi$
$3$	−7.21006	−1.38758	−0.693788	+	0.720179i	$0.744060\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	14.4201	0.981165
$7$	−19.8879	−1.07384	−0.536922	−	0.843632i	$-0.680413\pi$
$7$	−19.8879	−1.07384	−0.536922	+	0.843632i	$0.680413\pi$
$8$	−8.00000	−0.353553
$9$	24.9850	0.925369
$10$	0	0
$11$	−31.0553	−0.851229	−0.425614	−	0.904905i	$-0.639942\pi$
$11$	−31.0553	−0.851229	−0.425614	+	0.904905i	$0.639942\pi$
$12$	−28.8402	−0.693788
$13$	−7.15563	−0.152663	−0.0763313	−	0.997083i	$-0.524321\pi$
$13$	−7.15563	−0.152663	−0.0763313	+	0.997083i	$0.524321\pi$
$14$	39.7757	0.759322
$15$	0	0
$16$	16.0000	0.250000
$17$	−40.8326	−0.582551	−0.291275	−	0.956639i	$-0.594080\pi$
$17$	−40.8326	−0.582551	−0.291275	+	0.956639i	$0.594080\pi$
$18$	−49.9699	−0.654335
$19$	144.948	1.75018	0.875089	−	0.483962i	$-0.160803\pi$
$19$	144.948	1.75018	0.875089	+	0.483962i	$0.160803\pi$
$20$	0	0
$21$	143.393	1.49004
$22$	62.1106	0.601910
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	57.6805	0.490582
$25$	0	0
$26$	14.3113	0.107949
$27$	14.5286	0.103557
$28$	−79.5515	−0.536922
$29$	−189.113	−1.21094	−0.605472	−	0.795866i	$-0.707016\pi$
$29$	−189.113	−1.21094	−0.605472	+	0.795866i	$0.707016\pi$
$30$	0	0
$31$	41.6222	0.241147	0.120574	−	0.992704i	$-0.461527\pi$
$31$	41.6222	0.241147	0.120574	+	0.992704i	$0.461527\pi$
$32$	−32.0000	−0.176777
$33$	223.910	1.18115
$34$	81.6652	0.411925
$35$	0	0
$36$	99.9398	0.462684
$37$	−37.3828	−0.166100	−0.0830500	−	0.996545i	$-0.526466\pi$
$37$	−37.3828	−0.166100	−0.0830500	+	0.996545i	$0.526466\pi$
$38$	−289.896	−1.23756
$39$	51.5925	0.211831
$40$	0	0
$41$	401.324	1.52869	0.764345	−	0.644807i	$-0.223062\pi$
$41$	401.324	1.52869	0.764345	+	0.644807i	$0.223062\pi$
$42$	−286.785	−1.05362
$43$	−29.2550	−0.103752	−0.0518761	−	0.998654i	$-0.516520\pi$
$43$	−29.2550	−0.103752	−0.0518761	+	0.998654i	$0.516520\pi$
$44$	−124.221	−0.425614
$45$	0	0
$46$	46.0000	0.147442
$47$	−18.0210	−0.0559284	−0.0279642	−	0.999609i	$-0.508902\pi$
$47$	−18.0210	−0.0559284	−0.0279642	+	0.999609i	$0.508902\pi$
$48$	−115.361	−0.346894
$49$	52.5274	0.153141
$50$	0	0
$51$	294.405	0.808333
$52$	−28.6225	−0.0763313
$53$	−214.024	−0.554689	−0.277344	−	0.960771i	$-0.589454\pi$
$53$	−214.024	−0.554689	−0.277344	+	0.960771i	$0.589454\pi$
$54$	−29.0572	−0.0732256
$55$	0	0
$56$	159.103	0.379661
$57$	−1045.08	−2.42851
$58$	378.226	0.856267
$59$	313.495	0.691754	0.345877	−	0.938280i	$-0.387581\pi$
$59$	313.495	0.691754	0.345877	+	0.938280i	$0.387581\pi$
$60$	0	0
$61$	452.293	0.949347	0.474674	−	0.880162i	$-0.342566\pi$
$61$	452.293	0.949347	0.474674	+	0.880162i	$0.342566\pi$
$62$	−83.2444	−0.170517
$63$	−496.898	−0.993702
$64$	64.0000	0.125000
$65$	0	0
$66$	−447.821	−0.835196
$67$	858.562	1.56552	0.782761	−	0.622322i	$-0.213811\pi$
$67$	858.562	1.56552	0.782761	+	0.622322i	$0.213811\pi$
$68$	−163.330	−0.291275
$69$	165.831	0.289330
$70$	0	0
$71$	451.196	0.754185	0.377093	−	0.926176i	$-0.376924\pi$
$71$	451.196	0.754185	0.377093	+	0.926176i	$0.376924\pi$
$72$	−199.880	−0.327167
$73$	742.705	1.19078	0.595390	−	0.803437i	$-0.296997\pi$
$73$	742.705	1.19078	0.595390	+	0.803437i	$0.296997\pi$
$74$	74.7656	0.117450
$75$	0	0
$76$	579.792	0.875089
$77$	617.623	0.914087
$78$	−103.185	−0.149787
$79$	126.369	0.179970	0.0899851	−	0.995943i	$-0.471318\pi$
$79$	126.369	0.179970	0.0899851	+	0.995943i	$0.471318\pi$
$80$	0	0
$81$	−779.346	−1.06906
$82$	−802.649	−1.08095
$83$	473.737	0.626499	0.313249	−	0.949671i	$-0.398582\pi$
$83$	473.737	0.626499	0.313249	+	0.949671i	$0.398582\pi$
$84$	573.571	0.745020
$85$	0	0
$86$	58.5100	0.0733639
$87$	1363.52	1.68028
$88$	248.442	0.300955
$89$	−109.465	−0.130374	−0.0651869	−	0.997873i	$-0.520764\pi$
$89$	−109.465	−0.130374	−0.0651869	+	0.997873i	$0.520764\pi$
$90$	0	0
$91$	142.310	0.163936
$92$	−92.0000	−0.104257
$93$	−300.099	−0.334610
$94$	36.0420	0.0395473
$95$	0	0
$96$	230.722	0.245291
$97$	711.495	0.744756	0.372378	−	0.928081i	$-0.378542\pi$
$97$	711.495	0.744756	0.372378	+	0.928081i	$0.378542\pi$
$98$	−105.055	−0.108287
$99$	−775.915	−0.787701
$100$	0	0
$101$	910.988	0.897492	0.448746	−	0.893659i	$-0.351871\pi$
$101$	910.988	0.897492	0.448746	+	0.893659i	$0.351871\pi$
$102$	−588.811	−0.571578
$103$	199.834	0.191167	0.0955834	−	0.995421i	$-0.469528\pi$
$103$	199.834	0.191167	0.0955834	+	0.995421i	$0.469528\pi$
$104$	57.2450	0.0539744
$105$	0	0
$106$	428.049	0.392224
$107$	−1574.87	−1.42288	−0.711440	−	0.702746i	$-0.751957\pi$
$107$	−1574.87	−1.42288	−0.711440	+	0.702746i	$0.751957\pi$
$108$	58.1144	0.0517783
$109$	−1530.65	−1.34504	−0.672522	−	0.740077i	$-0.734789\pi$
$109$	−1530.65	−1.34504	−0.672522	+	0.740077i	$0.734789\pi$
$110$	0	0
$111$	269.532	0.230476
$112$	−318.206	−0.268461
$113$	−1352.39	−1.12586	−0.562930	−	0.826505i	$-0.690326\pi$
$113$	−1352.39	−1.12586	−0.562930	+	0.826505i	$0.690326\pi$
$114$	2090.17	1.71721
$115$	0	0
$116$	−756.452	−0.605472
$117$	−178.783	−0.141269
$118$	−626.989	−0.489144
$119$	812.073	0.625568
$120$	0	0
$121$	−366.570	−0.275409
$122$	−904.586	−0.671290
$123$	−2893.57	−2.12118
$124$	166.489	0.120574
$125$	0	0
$126$	993.795	0.702653
$127$	1552.32	1.08461	0.542307	−	0.840180i	$-0.317551\pi$
$127$	1552.32	1.08461	0.542307	+	0.840180i	$0.317551\pi$
$128$	−128.000	−0.0883883
$129$	210.930	0.143964
$130$	0	0
$131$	416.739	0.277944	0.138972	−	0.990296i	$-0.455620\pi$
$131$	416.739	0.277944	0.138972	+	0.990296i	$0.455620\pi$
$132$	895.642	0.590573
$133$	−2882.71	−1.87942
$134$	−1717.12	−1.10699
$135$	0	0
$136$	326.661	0.205963
$137$	803.363	0.500992	0.250496	−	0.968118i	$-0.419406\pi$
$137$	803.363	0.500992	0.250496	+	0.968118i	$0.419406\pi$
$138$	−331.663	−0.204587
$139$	−1838.71	−1.12200	−0.560998	−	0.827817i	$-0.689582\pi$
$139$	−1838.71	−1.12200	−0.560998	+	0.827817i	$0.689582\pi$
$140$	0	0
$141$	129.932	0.0776049
$142$	−902.393	−0.533289
$143$	222.220	0.129951
$144$	399.759	0.231342
$145$	0	0
$146$	−1485.41	−0.842009
$147$	−378.726	−0.212495
$148$	−149.531	−0.0830500
$149$	−2083.27	−1.14542	−0.572711	−	0.819757i	$-0.694108\pi$
$149$	−2083.27	−1.14542	−0.572711	+	0.819757i	$0.694108\pi$
$150$	0	0
$151$	2370.61	1.27760	0.638799	−	0.769374i	$-0.279432\pi$
$151$	2370.61	1.27760	0.638799	+	0.769374i	$0.279432\pi$
$152$	−1159.58	−0.618781
$153$	−1020.20	−0.539074
$154$	−1235.25	−0.646357
$155$	0	0
$156$	206.370	0.105916
$157$	2855.47	1.45153	0.725767	−	0.687940i	$-0.241485\pi$
$157$	2855.47	1.45153	0.725767	+	0.687940i	$0.241485\pi$
$158$	−252.738	−0.127258
$159$	1543.13	0.769673
$160$	0	0
$161$	457.421	0.223912
$162$	1558.69	0.755941
$163$	−1719.64	−0.826335	−0.413167	−	0.910655i	$-0.635578\pi$
$163$	−1719.64	−0.826335	−0.413167	+	0.910655i	$0.635578\pi$
$164$	1605.30	0.764345
$165$	0	0
$166$	−947.474	−0.443001
$167$	2346.02	1.08707	0.543535	−	0.839387i	$-0.317086\pi$
$167$	2346.02	1.08707	0.543535	+	0.839387i	$0.317086\pi$
$168$	−1147.14	−0.526809
$169$	−2145.80	−0.976694
$170$	0	0
$171$	3621.52	1.61956
$172$	−117.020	−0.0518761
$173$	−3242.07	−1.42480	−0.712398	−	0.701776i	$-0.752391\pi$
$173$	−3242.07	−1.42480	−0.712398	+	0.701776i	$0.752391\pi$
$174$	−2727.03	−1.18814
$175$	0	0
$176$	−496.884	−0.212807
$177$	−2260.31	−0.959862
$178$	218.930	0.0921881
$179$	−1249.56	−0.521768	−0.260884	−	0.965370i	$-0.584014\pi$
$179$	−1249.56	−0.521768	−0.260884	+	0.965370i	$0.584014\pi$
$180$	0	0
$181$	−1369.10	−0.562236	−0.281118	−	0.959673i	$-0.590705\pi$
$181$	−1369.10	−0.562236	−0.281118	+	0.959673i	$0.590705\pi$
$182$	−284.620	−0.115920
$183$	−3261.06	−1.31729
$184$	184.000	0.0737210
$185$	0	0
$186$	600.197	0.236605
$187$	1268.07	0.495884
$188$	−72.0840	−0.0279642
$189$	−288.943	−0.111204
$190$	0	0
$191$	394.212	0.149341	0.0746706	−	0.997208i	$-0.476209\pi$
$191$	394.212	0.149341	0.0746706	+	0.997208i	$0.476209\pi$
$192$	−461.444	−0.173447
$193$	−3700.44	−1.38012	−0.690060	−	0.723752i	$-0.742416\pi$
$193$	−3700.44	−1.38012	−0.690060	+	0.723752i	$0.742416\pi$
$194$	−1422.99	−0.526622
$195$	0	0
$196$	210.110	0.0765706
$197$	−1211.64	−0.438200	−0.219100	−	0.975702i	$-0.570312\pi$
$197$	−1211.64	−0.438200	−0.219100	+	0.975702i	$0.570312\pi$
$198$	1551.83	0.556988
$199$	−5141.99	−1.83169	−0.915845	−	0.401532i	$-0.868478\pi$
$199$	−5141.99	−1.83169	−0.915845	+	0.401532i	$0.868478\pi$
$200$	0	0
$201$	−6190.28	−2.17228
$202$	−1821.98	−0.634622
$203$	3761.05	1.30037
$204$	1177.62	0.404167
$205$	0	0
$206$	−399.667	−0.135175
$207$	−574.654	−0.192953
$208$	−114.490	−0.0381657
$209$	−4501.40	−1.48980
$210$	0	0
$211$	−2390.46	−0.779932	−0.389966	−	0.920829i	$-0.627513\pi$
$211$	−2390.46	−0.779932	−0.389966	+	0.920829i	$0.627513\pi$
$212$	−856.097	−0.277344
$213$	−3253.15	−1.04649
$214$	3149.74	1.00613
$215$	0	0
$216$	−116.229	−0.0366128
$217$	−827.777	−0.258955
$218$	3061.30	0.951090
$219$	−5354.94	−1.65230
$220$	0	0
$221$	292.183	0.0889337
$222$	−539.065	−0.162971
$223$	313.417	0.0941164	0.0470582	−	0.998892i	$-0.485015\pi$
$223$	313.417	0.0941164	0.0470582	+	0.998892i	$0.485015\pi$
$224$	636.412	0.189831
$225$	0	0
$226$	2704.78	0.796103
$227$	778.016	0.227483	0.113742	−	0.993510i	$-0.463716\pi$
$227$	778.016	0.227483	0.113742	+	0.993510i	$0.463716\pi$
$228$	−4180.34	−1.21425
$229$	280.476	0.0809360	0.0404680	−	0.999181i	$-0.487115\pi$
$229$	280.476	0.0809360	0.0404680	+	0.999181i	$0.487115\pi$
$230$	0	0
$231$	−4453.10	−1.26837
$232$	1512.90	0.428134
$233$	−1284.93	−0.361283	−0.180641	−	0.983549i	$-0.557817\pi$
$233$	−1284.93	−0.361283	−0.180641	+	0.983549i	$0.557817\pi$
$234$	357.566	0.0998924
$235$	0	0
$236$	1253.98	0.345877
$237$	−911.130	−0.249722
$238$	−1624.15	−0.442344
$239$	3726.71	1.00862	0.504311	−	0.863522i	$-0.331747\pi$
$239$	3726.71	1.00862	0.504311	+	0.863522i	$0.331747\pi$
$240$	0	0
$241$	−3089.79	−0.825854	−0.412927	−	0.910764i	$-0.635493\pi$
$241$	−3089.79	−0.825854	−0.412927	+	0.910764i	$0.635493\pi$
$242$	733.140	0.194744
$243$	5226.86	1.37985
$244$	1809.17	0.474674
$245$	0	0
$246$	5787.14	1.49990
$247$	−1037.19	−0.267187
$248$	−332.978	−0.0852585
$249$	−3415.67	−0.869315
$250$	0	0
$251$	1139.61	0.286580	0.143290	−	0.989681i	$-0.454232\pi$
$251$	1139.61	0.286580	0.143290	+	0.989681i	$0.454232\pi$
$252$	−1987.59	−0.496851
$253$	714.271	0.177493
$254$	−3104.64	−0.766939
$255$	0	0
$256$	256.000	0.0625000
$257$	−8186.05	−1.98689	−0.993447	−	0.114289i	$-0.963541\pi$
$257$	−8186.05	−1.98689	−0.993447	+	0.114289i	$0.963541\pi$
$258$	−421.860	−0.101798
$259$	743.465	0.178365
$260$	0	0
$261$	−4724.98	−1.12057
$262$	−833.478	−0.196536
$263$	5077.03	1.19035	0.595177	−	0.803595i	$-0.297082\pi$
$263$	5077.03	1.19035	0.595177	+	0.803595i	$0.297082\pi$
$264$	−1791.28	−0.417598
$265$	0	0
$266$	5765.42	1.32895
$267$	789.249	0.180904
$268$	3434.25	0.782761
$269$	4729.95	1.07208	0.536041	−	0.844192i	$-0.319919\pi$
$269$	4729.95	1.07208	0.536041	+	0.844192i	$0.319919\pi$
$270$	0	0
$271$	−4388.84	−0.983774	−0.491887	−	0.870659i	$-0.663693\pi$
$271$	−4388.84	−0.983774	−0.491887	+	0.870659i	$0.663693\pi$
$272$	−653.322	−0.145638
$273$	−1026.07	−0.227474
$274$	−1606.73	−0.354255
$275$	0	0
$276$	663.325	0.144665
$277$	−3382.55	−0.733709	−0.366855	−	0.930278i	$-0.619565\pi$
$277$	−3382.55	−0.733709	−0.366855	+	0.930278i	$0.619565\pi$
$278$	3677.42	0.793370
$279$	1039.93	0.223150
$280$	0	0
$281$	3328.66	0.706660	0.353330	−	0.935499i	$-0.385049\pi$
$281$	3328.66	0.706660	0.353330	+	0.935499i	$0.385049\pi$
$282$	−259.865	−0.0548749
$283$	8202.56	1.72294	0.861469	−	0.507810i	$-0.169545\pi$
$283$	8202.56	1.72294	0.861469	+	0.507810i	$0.169545\pi$
$284$	1804.79	0.377093
$285$	0	0
$286$	−444.440	−0.0918891
$287$	−7981.49	−1.64158
$288$	−799.519	−0.163584
$289$	−3245.70	−0.660635
$290$	0	0
$291$	−5129.92	−1.03341
$292$	2970.82	0.595390
$293$	−5893.87	−1.17517	−0.587583	−	0.809164i	$-0.699921\pi$
$293$	−5893.87	−1.17517	−0.587583	+	0.809164i	$0.699921\pi$
$294$	757.451	0.150257
$295$	0	0
$296$	299.063	0.0587252
$297$	−451.189	−0.0881504
$298$	4166.54	0.809936
$299$	164.579	0.0318324
$300$	0	0
$301$	581.819	0.111414
$302$	−4741.21	−0.903398
$303$	−6568.27	−1.24534
$304$	2319.17	0.437544
$305$	0	0
$306$	2040.40	0.381183
$307$	−378.111	−0.0702930	−0.0351465	−	0.999382i	$-0.511190\pi$
$307$	−378.111	−0.0702930	−0.0351465	+	0.999382i	$0.511190\pi$
$308$	2470.49	0.457044
$309$	−1440.81	−0.265259
$310$	0	0
$311$	−5368.01	−0.978752	−0.489376	−	0.872073i	$-0.662775\pi$
$311$	−5368.01	−0.978752	−0.489376	+	0.872073i	$0.662775\pi$
$312$	−412.740	−0.0748936
$313$	−331.040	−0.0597810	−0.0298905	−	0.999553i	$-0.509516\pi$
$313$	−331.040	−0.0597810	−0.0298905	+	0.999553i	$0.509516\pi$
$314$	−5710.93	−1.02639
$315$	0	0
$316$	505.477	0.0899851
$317$	7028.20	1.24525	0.622623	−	0.782522i	$-0.286067\pi$
$317$	7028.20	1.24525	0.622623	+	0.782522i	$0.286067\pi$
$318$	−3086.26	−0.544241
$319$	5872.95	1.03079
$320$	0	0
$321$	11354.9	1.97436
$322$	−914.842	−0.158330
$323$	−5918.61	−1.01957
$324$	−3117.38	−0.534531
$325$	0	0
$326$	3439.28	0.584307
$327$	11036.1	1.86635
$328$	−3210.59	−0.540474
$329$	358.399	0.0600583
$330$	0	0
$331$	446.633	0.0741667	0.0370833	−	0.999312i	$-0.488193\pi$
$331$	446.633	0.0741667	0.0370833	+	0.999312i	$0.488193\pi$
$332$	1894.95	0.313249
$333$	−934.008	−0.153704
$334$	−4692.04	−0.768674
$335$	0	0
$336$	2294.28	0.372510
$337$	−9601.38	−1.55199	−0.775995	−	0.630739i	$-0.782752\pi$
$337$	−9601.38	−1.55199	−0.775995	+	0.630739i	$0.782752\pi$
$338$	4291.59	0.690627
$339$	9750.82	1.56222
$340$	0	0
$341$	−1292.59	−0.205272
$342$	−7243.04	−1.14520
$343$	5776.88	0.909394
$344$	234.040	0.0366819
$345$	0	0
$346$	6484.13	1.00748
$347$	−1757.75	−0.271934	−0.135967	−	0.990713i	$-0.543414\pi$
$347$	−1757.75	−0.271934	−0.135967	+	0.990713i	$0.543414\pi$
$348$	5454.06	0.840139
$349$	2119.93	0.325150	0.162575	−	0.986696i	$-0.448020\pi$
$349$	2119.93	0.325150	0.162575	+	0.986696i	$0.448020\pi$
$350$	0	0
$351$	−103.961	−0.0158092
$352$	993.769	0.150477
$353$	6770.14	1.02079	0.510394	−	0.859940i	$-0.329499\pi$
$353$	6770.14	1.02079	0.510394	+	0.859940i	$0.329499\pi$
$354$	4520.63	0.678725
$355$	0	0
$356$	−437.860	−0.0651869
$357$	−5855.10	−0.868024
$358$	2499.12	0.368946
$359$	−5360.80	−0.788112	−0.394056	−	0.919086i	$-0.628928\pi$
$359$	−5360.80	−0.788112	−0.394056	+	0.919086i	$0.628928\pi$
$360$	0	0
$361$	14151.0	2.06312
$362$	2738.21	0.397561
$363$	2642.99	0.382152
$364$	569.241	0.0819679
$365$	0	0
$366$	6522.12	0.931466
$367$	9655.69	1.37336	0.686680	−	0.726960i	$-0.259068\pi$
$367$	9655.69	1.37336	0.686680	+	0.726960i	$0.259068\pi$
$368$	−368.000	−0.0521286
$369$	10027.1	1.41460
$370$	0	0
$371$	4256.49	0.595649
$372$	−1200.39	−0.167305
$373$	3035.77	0.421411	0.210705	−	0.977550i	$-0.432424\pi$
$373$	3035.77	0.421411	0.210705	+	0.977550i	$0.432424\pi$
$374$	−2536.13	−0.350643
$375$	0	0
$376$	144.168	0.0197737
$377$	1353.22	0.184866
$378$	577.885	0.0786328
$379$	395.166	0.0535576	0.0267788	−	0.999641i	$-0.491475\pi$
$379$	395.166	0.0535576	0.0267788	+	0.999641i	$0.491475\pi$
$380$	0	0
$381$	−11192.3	−1.50499
$382$	−788.423	−0.105600
$383$	−10699.6	−1.42748	−0.713740	−	0.700410i	$-0.753000\pi$
$383$	−10699.6	−1.42748	−0.713740	+	0.700410i	$0.753000\pi$
$384$	922.888	0.122646
$385$	0	0
$386$	7400.87	0.975892
$387$	−730.935	−0.0960090
$388$	2845.98	0.372378
$389$	−1598.78	−0.208384	−0.104192	−	0.994557i	$-0.533226\pi$
$389$	−1598.78	−0.208384	−0.104192	+	0.994557i	$0.533226\pi$
$390$	0	0
$391$	939.150	0.121470
$392$	−420.219	−0.0541436
$393$	−3004.71	−0.385669
$394$	2423.27	0.309855
$395$	0	0
$396$	−3103.66	−0.393850
$397$	7841.03	0.991260	0.495630	−	0.868534i	$-0.334937\pi$
$397$	7841.03	0.991260	0.495630	+	0.868534i	$0.334937\pi$
$398$	10284.0	1.29520
$399$	20784.5	2.60784
$400$	0	0
$401$	−11218.9	−1.39712	−0.698562	−	0.715550i	$-0.746176\pi$
$401$	−11218.9	−1.39712	−0.698562	+	0.715550i	$0.746176\pi$
$402$	12380.6	1.53604
$403$	−297.833	−0.0368142
$404$	3643.95	0.448746
$405$	0	0
$406$	−7522.11	−0.919497
$407$	1160.93	0.141389
$408$	−2355.24	−0.285789
$409$	−3978.47	−0.480985	−0.240492	−	0.970651i	$-0.577309\pi$
$409$	−3978.47	−0.480985	−0.240492	+	0.970651i	$0.577309\pi$
$410$	0	0
$411$	−5792.29	−0.695165
$412$	799.334	0.0955834
$413$	−6234.74	−0.742836
$414$	1149.31	0.136438
$415$	0	0
$416$	228.980	0.0269872
$417$	13257.2	1.55685
$418$	9002.81	1.05345
$419$	−10561.2	−1.23138	−0.615692	−	0.787987i	$-0.711124\pi$
$419$	−10561.2	−1.23138	−0.615692	+	0.787987i	$0.711124\pi$
$420$	0	0
$421$	9325.67	1.07958	0.539792	−	0.841798i	$-0.318503\pi$
$421$	9325.67	1.07958	0.539792	+	0.841798i	$0.318503\pi$
$422$	4780.91	0.551495
$423$	−450.254	−0.0517543
$424$	1712.19	0.196112
$425$	0	0
$426$	6506.30	0.739980
$427$	−8995.15	−1.01945
$428$	−6299.47	−0.711440
$429$	−1602.22	−0.180317
$430$	0	0
$431$	−4826.62	−0.539420	−0.269710	−	0.962942i	$-0.586928\pi$
$431$	−4826.62	−0.539420	−0.269710	+	0.962942i	$0.586928\pi$
$432$	232.457	0.0258891
$433$	13581.4	1.50735	0.753674	−	0.657248i	$-0.228280\pi$
$433$	13581.4	1.50735	0.753674	+	0.657248i	$0.228280\pi$
$434$	1655.55	0.183109
$435$	0	0
$436$	−6122.61	−0.672522
$437$	−3333.81	−0.364937
$438$	10709.9	1.16835
$439$	−3991.11	−0.433907	−0.216953	−	0.976182i	$-0.569612\pi$
$439$	−3991.11	−0.433907	−0.216953	+	0.976182i	$0.569612\pi$
$440$	0	0
$441$	1312.39	0.141712
$442$	−584.366	−0.0628856
$443$	749.024	0.0803322	0.0401661	−	0.999193i	$-0.487211\pi$
$443$	749.024	0.0803322	0.0401661	+	0.999193i	$0.487211\pi$
$444$	1078.13	0.115238
$445$	0	0
$446$	−626.834	−0.0665504
$447$	15020.5	1.58936
$448$	−1272.82	−0.134231
$449$	−7842.42	−0.824291	−0.412146	−	0.911118i	$-0.635221\pi$
$449$	−7842.42	−0.824291	−0.412146	+	0.911118i	$0.635221\pi$
$450$	0	0
$451$	−12463.2	−1.30127
$452$	−5409.56	−0.562930
$453$	−17092.2	−1.77276
$454$	−1556.03	−0.160855
$455$	0	0
$456$	8360.68	0.858606
$457$	−11131.4	−1.13939	−0.569697	−	0.821855i	$-0.692940\pi$
$457$	−11131.4	−1.13939	−0.569697	+	0.821855i	$0.692940\pi$
$458$	−560.951	−0.0572304
$459$	−593.240	−0.0603270
$460$	0	0
$461$	11089.8	1.12039	0.560197	−	0.828359i	$-0.310725\pi$
$461$	11089.8	1.12039	0.560197	+	0.828359i	$0.310725\pi$
$462$	8906.20	0.896870
$463$	11853.1	1.18977	0.594883	−	0.803812i	$-0.297198\pi$
$463$	11853.1	1.18977	0.594883	+	0.803812i	$0.297198\pi$
$464$	−3025.81	−0.302736
$465$	0	0
$466$	2569.87	0.255465
$467$	13.9804	0.00138530	0.000692650	−	1.00000i	$-0.499780\pi$
$467$	13.9804	0.00138530	0.000692650		1.00000i	$0.499780\pi$
$468$	−715.132	−0.0706346
$469$	−17075.0	−1.68113
$470$	0	0
$471$	−20588.1	−2.01412
$472$	−2507.96	−0.244572
$473$	908.522	0.0883168
$474$	1822.26	0.176580
$475$	0	0
$476$	3248.29	0.312784
$477$	−5347.39	−0.513292
$478$	−7453.41	−0.713203
$479$	3710.42	0.353932	0.176966	−	0.984217i	$-0.443372\pi$
$479$	3710.42	0.353932	0.176966	+	0.984217i	$0.443372\pi$
$480$	0	0
$481$	267.498	0.0253572
$482$	6179.58	0.583967
$483$	−3298.03	−0.310695
$484$	−1466.28	−0.137705
$485$	0	0
$486$	−10453.7	−0.975700
$487$	6095.23	0.567148	0.283574	−	0.958950i	$-0.408480\pi$
$487$	6095.23	0.567148	0.283574	+	0.958950i	$0.408480\pi$
$488$	−3618.34	−0.335645
$489$	12398.7	1.14660
$490$	0	0
$491$	1517.73	0.139500	0.0697499	−	0.997565i	$-0.477780\pi$
$491$	1517.73	0.139500	0.0697499	+	0.997565i	$0.477780\pi$
$492$	−11574.3	−1.06059
$493$	7721.97	0.705436
$494$	2074.39	0.188930
$495$	0	0
$496$	665.955	0.0602868
$497$	−8973.33	−0.809877
$498$	6831.34	0.614698
$499$	7433.49	0.666871	0.333435	−	0.942773i	$-0.391792\pi$
$499$	7433.49	0.666871	0.333435	+	0.942773i	$0.391792\pi$
$500$	0	0
$501$	−16915.0	−1.50839
$502$	−2279.22	−0.202643
$503$	3281.58	0.290891	0.145446	−	0.989366i	$-0.453538\pi$
$503$	3281.58	0.290891	0.145446	+	0.989366i	$0.453538\pi$
$504$	3975.18	0.351327
$505$	0	0
$506$	−1428.54	−0.125507
$507$	15471.3	1.35524
$508$	6209.28	0.542307
$509$	−16218.6	−1.41233	−0.706165	−	0.708048i	$-0.749576\pi$
$509$	−16218.6	−1.41233	−0.706165	+	0.708048i	$0.749576\pi$
$510$	0	0
$511$	−14770.8	−1.27871
$512$	−512.000	−0.0441942
$513$	2105.89	0.181242
$514$	16372.1	1.40495
$515$	0	0
$516$	843.721	0.0719820
$517$	559.647	0.0476078
$518$	−1486.93	−0.126123
$519$	23375.5	1.97701
$520$	0	0
$521$	−18519.3	−1.55729	−0.778644	−	0.627466i	$-0.784092\pi$
$521$	−18519.3	−1.55729	−0.778644	+	0.627466i	$0.784092\pi$
$522$	9449.96	0.792363
$523$	−1761.43	−0.147270	−0.0736348	−	0.997285i	$-0.523460\pi$
$523$	−1761.43	−0.147270	−0.0736348	+	0.997285i	$0.523460\pi$
$524$	1666.96	0.138972
$525$	0	0
$526$	−10154.1	−0.841708
$527$	−1699.54	−0.140480
$528$	3582.57	0.295286
$529$	529.000	0.0434783
$530$	0	0
$531$	7832.65	0.640128
$532$	−11530.8	−0.939709
$533$	−2871.73	−0.233374
$534$	−1578.50	−0.127918
$535$	0	0
$536$	−6868.49	−0.553496
$537$	9009.40	0.723993
$538$	−9459.89	−0.758076
$539$	−1631.25	−0.130358
$540$	0	0
$541$	23184.0	1.84244	0.921220	−	0.389043i	$-0.127194\pi$
$541$	23184.0	1.84244	0.921220	+	0.389043i	$0.127194\pi$
$542$	8777.67	0.695633
$543$	9871.32	0.780146
$544$	1306.64	0.102981
$545$	0	0
$546$	2052.13	0.160848
$547$	−5111.78	−0.399569	−0.199784	−	0.979840i	$-0.564024\pi$
$547$	−5111.78	−0.399569	−0.199784	+	0.979840i	$0.564024\pi$
$548$	3213.45	0.250496
$549$	11300.5	0.878496
$550$	0	0
$551$	−27411.6	−2.11937
$552$	−1326.65	−0.102294
$553$	−2513.21	−0.193260
$554$	6765.09	0.518811
$555$	0	0
$556$	−7354.84	−0.560998
$557$	−11154.6	−0.848540	−0.424270	−	0.905536i	$-0.639469\pi$
$557$	−11154.6	−0.848540	−0.424270	+	0.905536i	$0.639469\pi$
$558$	−2079.86	−0.157791
$559$	209.338	0.0158391
$560$	0	0
$561$	−9142.84	−0.688077
$562$	−6657.33	−0.499684
$563$	−15250.7	−1.14163	−0.570817	−	0.821078i	$-0.693373\pi$
$563$	−15250.7	−1.14163	−0.570817	+	0.821078i	$0.693373\pi$
$564$	519.730	0.0388024
$565$	0	0
$566$	−16405.1	−1.21830
$567$	15499.5	1.14801
$568$	−3609.57	−0.266645
$569$	−758.697	−0.0558985	−0.0279492	−	0.999609i	$-0.508898\pi$
$569$	−758.697	−0.0558985	−0.0279492	+	0.999609i	$0.508898\pi$
$570$	0	0
$571$	−7021.46	−0.514604	−0.257302	−	0.966331i	$-0.582834\pi$
$571$	−7021.46	−0.514604	−0.257302	+	0.966331i	$0.582834\pi$
$572$	888.880	0.0649754
$573$	−2842.29	−0.207222
$574$	15963.0	1.16077
$575$	0	0
$576$	1599.04	0.115671
$577$	23370.4	1.68618	0.843088	−	0.537775i	$-0.180735\pi$
$577$	23370.4	1.68618	0.843088	+	0.537775i	$0.180735\pi$
$578$	6491.40	0.467139
$579$	26680.4	1.91502
$580$	0	0
$581$	−9421.62	−0.672762
$582$	10259.8	0.730729
$583$	6646.59	0.472167
$584$	−5941.64	−0.421005
$585$	0	0
$586$	11787.7	0.830968
$587$	8041.44	0.565427	0.282713	−	0.959204i	$-0.408765\pi$
$587$	8041.44	0.565427	0.282713	+	0.959204i	$0.408765\pi$
$588$	−1514.90	−0.106248
$589$	6033.06	0.422051
$590$	0	0
$591$	8735.97	0.608037
$592$	−598.125	−0.0415250
$593$	5040.16	0.349030	0.174515	−	0.984655i	$-0.444164\pi$
$593$	5040.16	0.349030	0.174515	+	0.984655i	$0.444164\pi$
$594$	902.379	0.0623317
$595$	0	0
$596$	−8333.07	−0.572711
$597$	37074.1	2.54161
$598$	−329.159	−0.0225089
$599$	25123.5	1.71372	0.856862	−	0.515546i	$-0.172411\pi$
$599$	25123.5	1.71372	0.856862	+	0.515546i	$0.172411\pi$
$600$	0	0
$601$	23406.4	1.58863	0.794316	−	0.607504i	$-0.207829\pi$
$601$	23406.4	1.58863	0.794316	+	0.607504i	$0.207829\pi$
$602$	−1163.64	−0.0787814
$603$	21451.1	1.44869
$604$	9482.42	0.638799
$605$	0	0
$606$	13136.5	0.880587
$607$	−16932.3	−1.13222	−0.566112	−	0.824328i	$-0.691553\pi$
$607$	−16932.3	−1.13222	−0.566112	+	0.824328i	$0.691553\pi$
$608$	−4638.34	−0.309391
$609$	−27117.4	−1.80436
$610$	0	0
$611$	128.952	0.00853817
$612$	−4080.80	−0.269537
$613$	18473.4	1.21719	0.608593	−	0.793482i	$-0.291734\pi$
$613$	18473.4	1.21719	0.608593	+	0.793482i	$0.291734\pi$
$614$	756.222	0.0497046
$615$	0	0
$616$	−4940.99	−0.323179
$617$	18550.8	1.21042	0.605208	−	0.796067i	$-0.293090\pi$
$617$	18550.8	1.21042	0.605208	+	0.796067i	$0.293090\pi$
$618$	2881.62	0.187566
$619$	−5080.21	−0.329872	−0.164936	−	0.986304i	$-0.552742\pi$
$619$	−5080.21	−0.329872	−0.164936	+	0.986304i	$0.552742\pi$
$620$	0	0
$621$	−334.158	−0.0215930
$622$	10736.0	0.692082
$623$	2177.02	0.140001
$624$	825.480	0.0529578
$625$	0	0
$626$	662.079	0.0422716
$627$	32455.4	2.06721
$628$	11421.9	0.725767
$629$	1526.44	0.0967616
$630$	0	0
$631$	−15351.4	−0.968508	−0.484254	−	0.874928i	$-0.660909\pi$
$631$	−15351.4	−0.968508	−0.484254	+	0.874928i	$0.660909\pi$
$632$	−1010.95	−0.0636291
$633$	17235.3	1.08222
$634$	−14056.4	−0.880523
$635$	0	0
$636$	6172.51	0.384837
$637$	−375.867	−0.0233789
$638$	−11745.9	−0.728879
$639$	11273.1	0.697899
$640$	0	0
$641$	−19024.4	−1.17226	−0.586130	−	0.810217i	$-0.699349\pi$
$641$	−19024.4	−1.17226	−0.586130	+	0.810217i	$0.699349\pi$
$642$	−22709.8	−1.39608
$643$	3926.85	0.240840	0.120420	−	0.992723i	$-0.461576\pi$
$643$	3926.85	0.240840	0.120420	+	0.992723i	$0.461576\pi$
$644$	1829.68	0.111956
$645$	0	0
$646$	11837.2	0.720943
$647$	−9550.19	−0.580304	−0.290152	−	0.956981i	$-0.593706\pi$
$647$	−9550.19	−0.580304	−0.290152	+	0.956981i	$0.593706\pi$
$648$	6234.77	0.377970
$649$	−9735.66	−0.588841
$650$	0	0
$651$	5968.32	0.359319
$652$	−6878.56	−0.413167
$653$	−23074.9	−1.38283	−0.691416	−	0.722457i	$-0.743013\pi$
$653$	−23074.9	−1.38283	−0.691416	+	0.722457i	$0.743013\pi$
$654$	−22072.2	−1.31971
$655$	0	0
$656$	6421.19	0.382173
$657$	18556.4	1.10191
$658$	−716.799	−0.0424677
$659$	−20087.4	−1.18740	−0.593698	−	0.804688i	$-0.702333\pi$
$659$	−20087.4	−1.18740	−0.593698	+	0.804688i	$0.702333\pi$
$660$	0	0
$661$	27061.7	1.59240	0.796201	−	0.605032i	$-0.206840\pi$
$661$	27061.7	1.59240	0.796201	+	0.605032i	$0.206840\pi$
$662$	−893.266	−0.0524438
$663$	−2106.66	−0.123402
$664$	−3789.90	−0.221501
$665$	0	0
$666$	1868.02	0.108685
$667$	4349.60	0.252499
$668$	9384.09	0.543535
$669$	−2259.76	−0.130594
$670$	0	0
$671$	−14046.1	−0.808112
$672$	−4588.57	−0.263405
$673$	−24014.8	−1.37549	−0.687743	−	0.725954i	$-0.741399\pi$
$673$	−24014.8	−1.37549	−0.687743	+	0.725954i	$0.741399\pi$
$674$	19202.8	1.09742
$675$	0	0
$676$	−8583.19	−0.488347
$677$	9473.17	0.537790	0.268895	−	0.963170i	$-0.413342\pi$
$677$	9473.17	0.537790	0.268895	+	0.963170i	$0.413342\pi$
$678$	−19501.6	−1.10465
$679$	−14150.1	−0.799752
$680$	0	0
$681$	−5609.54	−0.315651
$682$	2585.18	0.145149
$683$	−26177.4	−1.46655	−0.733274	−	0.679934i	$-0.762009\pi$
$683$	−26177.4	−1.46655	−0.733274	+	0.679934i	$0.762009\pi$
$684$	14486.1	0.809780
$685$	0	0
$686$	−11553.8	−0.643039
$687$	−2022.25	−0.112305
$688$	−468.080	−0.0259380
$689$	1531.48	0.0846802
$690$	0	0
$691$	5912.51	0.325503	0.162751	−	0.986667i	$-0.447963\pi$
$691$	5912.51	0.325503	0.162751	+	0.986667i	$0.447963\pi$
$692$	−12968.3	−0.712398
$693$	15431.3	0.845868
$694$	3515.51	0.192287
$695$	0	0
$696$	−10908.1	−0.594068
$697$	−16387.1	−0.890540
$698$	−4239.87	−0.229916
$699$	9264.45	0.501307
$700$	0	0
$701$	−7228.60	−0.389473	−0.194737	−	0.980856i	$-0.562385\pi$
$701$	−7228.60	−0.389473	−0.194737	+	0.980856i	$0.562385\pi$
$702$	207.922	0.0111788
$703$	−5418.57	−0.290704
$704$	−1987.54	−0.106404
$705$	0	0
$706$	−13540.3	−0.721807
$707$	−18117.6	−0.963766
$708$	−9041.26	−0.479931
$709$	9704.32	0.514038	0.257019	−	0.966406i	$-0.417260\pi$
$709$	9704.32	0.514038	0.257019	+	0.966406i	$0.417260\pi$
$710$	0	0
$711$	3157.33	0.166539
$712$	875.720	0.0460941
$713$	−957.311	−0.0502827
$714$	11710.2	0.613786
$715$	0	0
$716$	−4998.24	−0.260884
$717$	−26869.8	−1.39954
$718$	10721.6	0.557279
$719$	1257.53	0.0652265	0.0326132	−	0.999468i	$-0.489617\pi$
$719$	1257.53	0.0652265	0.0326132	+	0.999468i	$0.489617\pi$
$720$	0	0
$721$	−3974.26	−0.205283
$722$	−28301.9	−1.45885
$723$	22277.6	1.14594
$724$	−5476.42	−0.281118
$725$	0	0
$726$	−5285.98	−0.270222
$727$	−5083.78	−0.259349	−0.129675	−	0.991557i	$-0.541393\pi$
$727$	−5083.78	−0.259349	−0.129675	+	0.991557i	$0.541393\pi$
$728$	−1138.48	−0.0579601
$729$	−16643.6	−0.845583
$730$	0	0
$731$	1194.56	0.0604409
$732$	−13044.2	−0.658646
$733$	2278.52	0.114815	0.0574074	−	0.998351i	$-0.481717\pi$
$733$	2278.52	0.114815	0.0574074	+	0.998351i	$0.481717\pi$
$734$	−19311.4	−0.971112
$735$	0	0
$736$	736.000	0.0368605
$737$	−26662.9	−1.33262
$738$	−20054.1	−1.00028
$739$	−34632.0	−1.72389	−0.861947	−	0.506998i	$-0.830755\pi$
$739$	−34632.0	−1.72389	−0.861947	+	0.506998i	$0.830755\pi$
$740$	0	0
$741$	7478.24	0.370742
$742$	−8512.98	−0.421188
$743$	27831.5	1.37421	0.687105	−	0.726558i	$-0.258881\pi$
$743$	27831.5	1.37421	0.687105	+	0.726558i	$0.258881\pi$
$744$	2400.79	0.118303
$745$	0	0
$746$	−6071.54	−0.297982
$747$	11836.3	0.579742
$748$	5072.27	0.247942
$749$	31320.8	1.52795
$750$	0	0
$751$	31443.6	1.52782	0.763910	−	0.645323i	$-0.223277\pi$
$751$	31443.6	1.52782	0.763910	+	0.645323i	$0.223277\pi$
$752$	−288.336	−0.0139821
$753$	−8216.66	−0.397652
$754$	−2706.44	−0.130720
$755$	0	0
$756$	−1155.77	−0.0556018
$757$	−22166.6	−1.06428	−0.532138	−	0.846657i	$-0.678611\pi$
$757$	−22166.6	−1.06428	−0.532138	+	0.846657i	$0.678611\pi$
$758$	−790.332	−0.0378709
$759$	−5149.94	−0.246286
$760$	0	0
$761$	−1609.00	−0.0766443	−0.0383221	−	0.999265i	$-0.512201\pi$
$761$	−1609.00	−0.0766443	−0.0383221	+	0.999265i	$0.512201\pi$
$762$	22384.6	1.06419
$763$	30441.4	1.44437
$764$	1576.85	0.0746706
$765$	0	0
$766$	21399.2	1.00938
$767$	−2243.25	−0.105605
$768$	−1845.78	−0.0867235
$769$	−16630.8	−0.779872	−0.389936	−	0.920842i	$-0.627503\pi$
$769$	−16630.8	−0.779872	−0.389936	+	0.920842i	$0.627503\pi$
$770$	0	0
$771$	59021.9	2.75697
$772$	−14801.7	−0.690060
$773$	42083.0	1.95811	0.979057	−	0.203587i	$-0.0652601\pi$
$773$	42083.0	1.95811	0.979057	+	0.203587i	$0.0652601\pi$
$774$	1461.87	0.0678886
$775$	0	0
$776$	−5691.96	−0.263311
$777$	−5360.42	−0.247496
$778$	3197.57	0.147350
$779$	58171.2	2.67548
$780$	0	0
$781$	−14012.0	−0.641984
$782$	−1878.30	−0.0858924
$783$	−2747.54	−0.125401
$784$	840.438	0.0382853
$785$	0	0
$786$	6009.43	0.272709
$787$	−5370.59	−0.243254	−0.121627	−	0.992576i	$-0.538811\pi$
$787$	−5370.59	−0.243254	−0.121627	+	0.992576i	$0.538811\pi$
$788$	−4846.54	−0.219100
$789$	−36605.7	−1.65171
$790$	0	0
$791$	26896.2	1.20900
$792$	6207.32	0.278494
$793$	−3236.44	−0.144930
$794$	−15682.1	−0.700926
$795$	0	0
$796$	−20568.0	−0.915845
$797$	−39819.9	−1.76975	−0.884876	−	0.465827i	$-0.845757\pi$
$797$	−39819.9	−1.76975	−0.884876	+	0.465827i	$0.845757\pi$
$798$	−41569.0	−1.84402
$799$	735.844	0.0325811
$800$	0	0
$801$	−2734.98	−0.120644
$802$	22437.8	0.987915
$803$	−23064.9	−1.01363
$804$	−24761.1	−1.08614
$805$	0	0
$806$	595.666	0.0260316
$807$	−34103.2	−1.48759
$808$	−7287.90	−0.317311
$809$	−16747.3	−0.727815	−0.363908	−	0.931435i	$-0.618558\pi$
$809$	−16747.3	−0.727815	−0.363908	+	0.931435i	$0.618558\pi$
$810$	0	0
$811$	41283.4	1.78749	0.893746	−	0.448573i	$-0.148068\pi$
$811$	41283.4	1.78749	0.893746	+	0.448573i	$0.148068\pi$
$812$	15044.2	0.650183
$813$	31643.8	1.36506
$814$	−2321.87	−0.0999772
$815$	0	0
$816$	4710.49	0.202083
$817$	−4240.45	−0.181585
$818$	7956.94	0.340107
$819$	3555.61	0.151701
$820$	0	0
$821$	996.361	0.0423547	0.0211774	−	0.999776i	$-0.493259\pi$
$821$	996.361	0.0423547	0.0211774	+	0.999776i	$0.493259\pi$
$822$	11584.6	0.491556
$823$	25984.5	1.10056	0.550282	−	0.834979i	$-0.314520\pi$
$823$	25984.5	1.10056	0.550282	+	0.834979i	$0.314520\pi$
$824$	−1598.67	−0.0675877
$825$	0	0
$826$	12469.5	0.525265
$827$	21559.7	0.906533	0.453267	−	0.891375i	$-0.350259\pi$
$827$	21559.7	0.906533	0.453267	+	0.891375i	$0.350259\pi$
$828$	−2298.62	−0.0964764
$829$	−23510.9	−0.985005	−0.492502	−	0.870311i	$-0.663918\pi$
$829$	−23510.9	−0.985005	−0.492502	+	0.870311i	$0.663918\pi$
$830$	0	0
$831$	24388.4	1.01808
$832$	−457.960	−0.0190828
$833$	−2144.83	−0.0892124
$834$	−26514.4	−1.10086
$835$	0	0
$836$	−18005.6	−0.744901
$837$	604.712	0.0249724
$838$	21122.5	0.870720
$839$	−7341.21	−0.302082	−0.151041	−	0.988528i	$-0.548263\pi$
$839$	−7341.21	−0.302082	−0.151041	+	0.988528i	$0.548263\pi$
$840$	0	0
$841$	11374.7	0.466386
$842$	−18651.3	−0.763382
$843$	−23999.9	−0.980545
$844$	−9561.82	−0.389966
$845$	0	0
$846$	900.508	0.0365959
$847$	7290.29	0.295747
$848$	−3424.39	−0.138672
$849$	−59140.9	−2.39071
$850$	0	0
$851$	859.805	0.0346342
$852$	−13012.6	−0.523245
$853$	−8666.53	−0.347874	−0.173937	−	0.984757i	$-0.555649\pi$
$853$	−8666.53	−0.347874	−0.173937	+	0.984757i	$0.555649\pi$
$854$	17990.3	0.720861
$855$	0	0
$856$	12598.9	0.503064
$857$	−1719.95	−0.0685557	−0.0342778	−	0.999412i	$-0.510913\pi$
$857$	−1719.95	−0.0685557	−0.0342778	+	0.999412i	$0.510913\pi$
$858$	3204.44	0.127503
$859$	32940.3	1.30839	0.654196	−	0.756325i	$-0.273007\pi$
$859$	32940.3	1.30839	0.654196	+	0.756325i	$0.273007\pi$
$860$	0	0
$861$	57547.0	2.27781
$862$	9653.24	0.381428
$863$	−15387.2	−0.606936	−0.303468	−	0.952842i	$-0.598145\pi$
$863$	−15387.2	−0.606936	−0.303468	+	0.952842i	$0.598145\pi$
$864$	−464.915	−0.0183064
$865$	0	0
$866$	−27162.9	−1.06586
$867$	23401.7	0.916681
$868$	−3311.11	−0.129477
$869$	−3924.43	−0.153196
$870$	0	0
$871$	−6143.55	−0.238997
$872$	12245.2	0.475545
$873$	17776.7	0.689174
$874$	6667.61	0.258050
$875$	0	0
$876$	−21419.8	−0.826150
$877$	13393.3	0.515691	0.257845	−	0.966186i	$-0.416987\pi$
$877$	13393.3	0.515691	0.257845	+	0.966186i	$0.416987\pi$
$878$	7982.21	0.306818
$879$	42495.2	1.63063
$880$	0	0
$881$	−43474.3	−1.66253	−0.831263	−	0.555879i	$-0.812382\pi$
$881$	−43474.3	−1.66253	−0.831263	+	0.555879i	$0.812382\pi$
$882$	−2624.79	−0.100206
$883$	−10494.8	−0.399976	−0.199988	−	0.979798i	$-0.564090\pi$
$883$	−10494.8	−0.399976	−0.199988	+	0.979798i	$0.564090\pi$
$884$	1168.73	0.0444668
$885$	0	0
$886$	−1498.05	−0.0568035
$887$	−36634.1	−1.38676	−0.693378	−	0.720574i	$-0.743878\pi$
$887$	−36634.1	−1.38676	−0.693378	+	0.720574i	$0.743878\pi$
$888$	−2156.26	−0.0814857
$889$	−30872.3	−1.16471
$890$	0	0
$891$	24202.8	0.910016
$892$	1253.67	0.0470582
$893$	−2612.11	−0.0978846
$894$	−30041.0	−1.12385
$895$	0	0
$896$	2545.65	0.0949153
$897$	−1186.63	−0.0441698
$898$	15684.8	0.582862
$899$	−7871.30	−0.292016
$900$	0	0
$901$	8739.17	0.323134
$902$	24926.5	0.920134
$903$	−4194.95	−0.154595
$904$	10819.1	0.398052
$905$	0	0
$906$	34184.4	1.25353
$907$	−3552.27	−0.130045	−0.0650227	−	0.997884i	$-0.520712\pi$
$907$	−3552.27	−0.130045	−0.0650227	+	0.997884i	$0.520712\pi$
$908$	3112.06	0.113742
$909$	22761.0	0.830511
$910$	0	0
$911$	7216.97	0.262469	0.131234	−	0.991351i	$-0.458106\pi$
$911$	7216.97	0.262469	0.131234	+	0.991351i	$0.458106\pi$
$912$	−16721.4	−0.607126
$913$	−14712.0	−0.533294
$914$	22262.7	0.805674
$915$	0	0
$916$	1121.90	0.0404680
$917$	−8288.05	−0.298469
$918$	1186.48	0.0426576
$919$	12618.4	0.452932	0.226466	−	0.974019i	$-0.427283\pi$
$919$	12618.4	0.452932	0.226466	+	0.974019i	$0.427283\pi$
$920$	0	0
$921$	2726.20	0.0975369
$922$	−22179.5	−0.792239
$923$	−3228.59	−0.115136
$924$	−17812.4	−0.634183
$925$	0	0
$926$	−23706.3	−0.841292
$927$	4992.83	0.176900
$928$	6051.61	0.214067
$929$	−39009.3	−1.37767	−0.688834	−	0.724919i	$-0.741877\pi$
$929$	−39009.3	−1.37767	−0.688834	+	0.724919i	$0.741877\pi$
$930$	0	0
$931$	7613.75	0.268024
$932$	−5139.74	−0.180641
$933$	38703.6	1.35809
$934$	−27.9608	−0.000979555 0
$935$	0	0
$936$	1430.26	0.0499462
$937$	−30772.4	−1.07288	−0.536441	−	0.843938i	$-0.680232\pi$
$937$	−30772.4	−1.07288	−0.536441	+	0.843938i	$0.680232\pi$
$938$	34149.9	1.18874
$939$	2386.81	0.0829507
$940$	0	0
$941$	29910.9	1.03620	0.518101	−	0.855319i	$-0.326639\pi$
$941$	29910.9	1.03620	0.518101	+	0.855319i	$0.326639\pi$
$942$	41176.2	1.42419
$943$	−9230.46	−0.318754
$944$	5015.91	0.172939
$945$	0	0
$946$	−1817.04	−0.0624494
$947$	36389.7	1.24869	0.624344	−	0.781149i	$-0.285366\pi$
$947$	36389.7	1.24869	0.624344	+	0.781149i	$0.285366\pi$
$948$	−3644.52	−0.124861
$949$	−5314.52	−0.181788
$950$	0	0
$951$	−50673.8	−1.72788
$952$	−6496.59	−0.221172
$953$	1937.31	0.0658506	0.0329253	−	0.999458i	$-0.489518\pi$
$953$	1937.31	0.0658506	0.0329253	+	0.999458i	$0.489518\pi$
$954$	10694.8	0.362952
$955$	0	0
$956$	14906.8	0.504311
$957$	−42344.3	−1.43030
$958$	−7420.85	−0.250268
$959$	−15977.2	−0.537987
$960$	0	0
$961$	−28058.6	−0.941848
$962$	−534.995	−0.0179303
$963$	−39348.0	−1.31669
$964$	−12359.2	−0.412927
$965$	0	0
$966$	6596.07	0.219695
$967$	2138.21	0.0711066	0.0355533	−	0.999368i	$-0.488681\pi$
$967$	2138.21	0.0711066	0.0355533	+	0.999368i	$0.488681\pi$
$968$	2932.56	0.0973719
$969$	42673.5	1.41473
$970$	0	0
$971$	−7943.13	−0.262520	−0.131260	−	0.991348i	$-0.541902\pi$
$971$	−7943.13	−0.262520	−0.131260	+	0.991348i	$0.541902\pi$
$972$	20907.4	0.689924
$973$	36568.0	1.20485
$974$	−12190.5	−0.401034
$975$	0	0
$976$	7236.69	0.237337
$977$	−10563.3	−0.345904	−0.172952	−	0.984930i	$-0.555331\pi$
$977$	−10563.3	−0.345904	−0.172952	+	0.984930i	$0.555331\pi$
$978$	−24797.4	−0.810770
$979$	3399.46	0.110978
$980$	0	0
$981$	−38243.3	−1.24466
$982$	−3035.47	−0.0986413
$983$	−59851.9	−1.94199	−0.970997	−	0.239091i	$-0.923151\pi$
$983$	−59851.9	−1.94199	−0.970997	+	0.239091i	$0.923151\pi$
$984$	23148.6	0.749949
$985$	0	0
$986$	−15443.9	−0.498819
$987$	−2584.08	−0.0833355
$988$	−4148.78	−0.133593
$989$	672.865	0.0216338
$990$	0	0
$991$	2534.12	0.0812300	0.0406150	−	0.999175i	$-0.487068\pi$
$991$	2534.12	0.0812300	0.0406150	+	0.999175i	$0.487068\pi$
$992$	−1331.91	−0.0426292
$993$	−3220.25	−0.102912
$994$	17946.7	0.572670
$995$	0	0
$996$	−13662.7	−0.434657
$997$	42439.2	1.34811	0.674054	−	0.738682i	$-0.264551\pi$
$997$	42439.2	1.34811	0.674054	+	0.738682i	$0.264551\pi$
$998$	−14867.0	−0.471549
$999$	−543.120	−0.0172007

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.a.q.1.2	✓	5
5.2	odd	4		1150.4.b.r.599.4		10
5.3	odd	4		1150.4.b.r.599.7		10
5.4	even	2		1150.4.a.v.1.4	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1150.4.a.q.1.2	✓	5	1.1	even	1	trivial
1150.4.a.v.1.4	yes	5	5.4	even	2
1150.4.b.r.599.4		10	5.2	odd	4
1150.4.b.r.599.7		10	5.3	odd	4

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:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1150.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -7.21006 q^{3} +4.00000 q^{4} +14.4201 q^{6} -19.8879 q^{7} -8.00000 q^{8} +24.9850 q^{9} -31.0553 q^{11} -28.8402 q^{12} -7.15563 q^{13} +39.7757 q^{14} +16.0000 q^{16} -40.8326 q^{17} -49.9699 q^{18} +144.948 q^{19} +143.393 q^{21} +62.1106 q^{22} -23.0000 q^{23} +57.6805 q^{24} +14.3113 q^{26} +14.5286 q^{27} -79.5515 q^{28} -189.113 q^{29} +41.6222 q^{31} -32.0000 q^{32} +223.910 q^{33} +81.6652 q^{34} +99.9398 q^{36} -37.3828 q^{37} -289.896 q^{38} +51.5925 q^{39} +401.324 q^{41} -286.785 q^{42} -29.2550 q^{43} -124.221 q^{44} +46.0000 q^{46} -18.0210 q^{47} -115.361 q^{48} +52.5274 q^{49} +294.405 q^{51} -28.6225 q^{52} -214.024 q^{53} -29.0572 q^{54} +159.103 q^{56} -1045.08 q^{57} +378.226 q^{58} +313.495 q^{59} +452.293 q^{61} -83.2444 q^{62} -496.898 q^{63} +64.0000 q^{64} -447.821 q^{66} +858.562 q^{67} -163.330 q^{68} +165.831 q^{69} +451.196 q^{71} -199.880 q^{72} +742.705 q^{73} +74.7656 q^{74} +579.792 q^{76} +617.623 q^{77} -103.185 q^{78} +126.369 q^{79} -779.346 q^{81} -802.649 q^{82} +473.737 q^{83} +573.571 q^{84} +58.5100 q^{86} +1363.52 q^{87} +248.442 q^{88} -109.465 q^{89} +142.310 q^{91} -92.0000 q^{92} -300.099 q^{93} +36.0420 q^{94} +230.722 q^{96} +711.495 q^{97} -105.055 q^{98} -775.915 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{2} - 5 q^{3} + 20 q^{4} + 10 q^{6} + 3 q^{7} - 40 q^{8} + 84 q^{9} - 26 q^{11} - 20 q^{12} + 61 q^{13} - 6 q^{14} + 80 q^{16} - 231 q^{17} - 168 q^{18} + 74 q^{19} - 88 q^{21} + 52 q^{22} - 115 q^{23}+ \cdots + 2397 q^{99}+O(q^{100}) \) 5 * q - 10 * q^2 - 5 * q^3 + 20 * q^4 + 10 * q^6 + 3 * q^7 - 40 * q^8 + 84 * q^9 - 26 * q^11 - 20 * q^12 + 61 * q^13 - 6 * q^14 + 80 * q^16 - 231 * q^17 - 168 * q^18 + 74 * q^19 - 88 * q^21 + 52 * q^22 - 115 * q^23 + 40 * q^24 - 122 * q^26 - 368 * q^27 + 12 * q^28 + 37 * q^29 + 220 * q^31 - 160 * q^32 - 391 * q^33 + 462 * q^34 + 336 * q^36 - 220 * q^37 - 148 * q^38 + 195 * q^39 + 112 * q^41 + 176 * q^42 - 97 * q^43 - 104 * q^44 + 230 * q^46 - 370 * q^47 - 80 * q^48 + 182 * q^49 + 563 * q^51 + 244 * q^52 - 1224 * q^53 + 736 * q^54 - 24 * q^56 + 1009 * q^57 - 74 * q^58 - 181 * q^59 + 1574 * q^61 - 440 * q^62 - 1829 * q^63 + 320 * q^64 + 782 * q^66 + 435 * q^67 - 924 * q^68 + 115 * q^69 - 30 * q^71 - 672 * q^72 - 2180 * q^73 + 440 * q^74 + 296 * q^76 + 1401 * q^77 - 390 * q^78 + 609 * q^79 + 1453 * q^81 - 224 * q^82 - 2018 * q^83 - 352 * q^84 + 194 * q^86 + 1485 * q^87 + 208 * q^88 - 1663 * q^89 + 361 * q^91 - 460 * q^92 - 2741 * q^93 + 740 * q^94 + 160 * q^96 + 1274 * q^97 - 364 * q^98 + 2397 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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