LMFDB - Embedded newform 1150.4.a.t.1.3

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} - x^{4} - 38x^{3} + 38x^{2} + 202x + 101 $ x^5 - x^4 - 38x^3 + 38x^2 + 202*x + 101
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-5.76842$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} -1.80744 q^{3} +4.00000 q^{4} -3.61488 q^{6} -9.19471 q^{7} +8.00000 q^{8} -23.7332 q^{9} +43.1821 q^{11} -7.22976 q^{12} -25.6824 q^{13} -18.3894 q^{14} +16.0000 q^{16} -22.9669 q^{17} -47.4663 q^{18} +101.823 q^{19} +16.6189 q^{21} +86.3643 q^{22} +23.0000 q^{23} -14.4595 q^{24} -51.3647 q^{26} +91.6972 q^{27} -36.7788 q^{28} -146.682 q^{29} +23.9712 q^{31} +32.0000 q^{32} -78.0491 q^{33} -45.9339 q^{34} -94.9326 q^{36} -294.572 q^{37} +203.646 q^{38} +46.4193 q^{39} -127.823 q^{41} +33.2378 q^{42} +502.072 q^{43} +172.729 q^{44} +46.0000 q^{46} -607.879 q^{47} -28.9190 q^{48} -258.457 q^{49} +41.5114 q^{51} -102.729 q^{52} -674.254 q^{53} +183.394 q^{54} -73.5577 q^{56} -184.039 q^{57} -293.365 q^{58} +14.8056 q^{59} -478.549 q^{61} +47.9424 q^{62} +218.220 q^{63} +64.0000 q^{64} -156.098 q^{66} -458.422 q^{67} -91.8678 q^{68} -41.5711 q^{69} +1012.83 q^{71} -189.865 q^{72} +750.521 q^{73} -589.144 q^{74} +407.292 q^{76} -397.047 q^{77} +92.8387 q^{78} -641.648 q^{79} +475.058 q^{81} -255.645 q^{82} -900.187 q^{83} +66.4756 q^{84} +1004.14 q^{86} +265.120 q^{87} +345.457 q^{88} -923.770 q^{89} +236.142 q^{91} +92.0000 q^{92} -43.3265 q^{93} -1215.76 q^{94} -57.8381 q^{96} -168.230 q^{97} -516.915 q^{98} -1024.85 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 10 q^{2} - 12 q^{3} + 20 q^{4} - 24 q^{6} + 24 q^{7} + 40 q^{8} + 11 q^{9} - 54 q^{11} - 48 q^{12} - 36 q^{13} + 48 q^{14} + 80 q^{16} - 132 q^{17} + 22 q^{18} - 50 q^{19} - 158 q^{21} - 108 q^{22}+ \cdots - 1740 q^{99}+O(q^{100}) $ 5 * q + 10 * q^2 - 12 * q^3 + 20 * q^4 - 24 * q^6 + 24 * q^7 + 40 * q^8 + 11 * q^9 - 54 * q^11 - 48 * q^12 - 36 * q^13 + 48 * q^14 + 80 * q^16 - 132 * q^17 + 22 * q^18 - 50 * q^19 - 158 * q^21 - 108 * q^22 + 115 * q^23 - 96 * q^24 - 72 * q^26 - 489 * q^27 + 96 * q^28 - 104 * q^29 - 342 * q^31 + 160 * q^32 + 348 * q^33 - 264 * q^34 + 44 * q^36 - 380 * q^37 - 100 * q^38 + 193 * q^39 + 2 * q^41 - 316 * q^42 + 114 * q^43 - 216 * q^44 + 230 * q^46 - 966 * q^47 - 192 * q^48 - 441 * q^49 + 1200 * q^51 - 144 * q^52 - 308 * q^53 - 978 * q^54 + 192 * q^56 - 1410 * q^57 - 208 * q^58 - 277 * q^59 + 482 * q^61 - 684 * q^62 + 222 * q^63 + 320 * q^64 + 696 * q^66 - 2044 * q^67 - 528 * q^68 - 276 * q^69 + 208 * q^71 + 88 * q^72 - 646 * q^73 - 760 * q^74 - 200 * q^76 - 2124 * q^77 + 386 * q^78 - 3444 * q^79 + 2933 * q^81 + 4 * q^82 - 1218 * q^83 - 632 * q^84 + 228 * q^86 - 103 * q^87 - 432 * q^88 + 140 * q^89 + 1976 * q^91 + 460 * q^92 - 473 * q^93 - 1932 * q^94 - 384 * q^96 - 2816 * q^97 - 882 * q^98 - 1740 * 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
$3$	−1.80744	−0.347842	−0.173921	−	0.984760i	$-0.555644\pi$
$3$	−1.80744	−0.347842	−0.173921	+	0.984760i	$0.555644\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−3.61488	−0.245961
$7$	−9.19471	−0.496468	−0.248234	−	0.968700i	$-0.579850\pi$
$7$	−9.19471	−0.496468	−0.248234	+	0.968700i	$0.579850\pi$
$8$	8.00000	0.353553
$9$	−23.7332	−0.879006
$10$	0	0
$11$	43.1821	1.18363	0.591814	−	0.806075i	$-0.298412\pi$
$11$	43.1821	1.18363	0.591814	+	0.806075i	$0.298412\pi$
$12$	−7.22976	−0.173921
$13$	−25.6824	−0.547924	−0.273962	−	0.961741i	$-0.588334\pi$
$13$	−25.6824	−0.547924	−0.273962	+	0.961741i	$0.588334\pi$
$14$	−18.3894	−0.351056
$15$	0	0
$16$	16.0000	0.250000
$17$	−22.9669	−0.327665	−0.163832	−	0.986488i	$-0.552386\pi$
$17$	−22.9669	−0.327665	−0.163832	+	0.986488i	$0.552386\pi$
$18$	−47.4663	−0.621551
$19$	101.823	1.22946	0.614732	−	0.788736i	$-0.289264\pi$
$19$	101.823	1.22946	0.614732	+	0.788736i	$0.289264\pi$
$20$	0	0
$21$	16.6189	0.172692
$22$	86.3643	0.836951
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−14.4595	−0.122981
$25$	0	0
$26$	−51.3647	−0.387440
$27$	91.6972	0.653597
$28$	−36.7788	−0.248234
$29$	−146.682	−0.939250	−0.469625	−	0.882866i	$-0.655611\pi$
$29$	−146.682	−0.939250	−0.469625	+	0.882866i	$0.655611\pi$
$30$	0	0
$31$	23.9712	0.138882	0.0694412	−	0.997586i	$-0.477878\pi$
$31$	23.9712	0.138882	0.0694412	+	0.997586i	$0.477878\pi$
$32$	32.0000	0.176777
$33$	−78.0491	−0.411715
$34$	−45.9339	−0.231694
$35$	0	0
$36$	−94.9326	−0.439503
$37$	−294.572	−1.30885	−0.654423	−	0.756129i	$-0.727088\pi$
$37$	−294.572	−1.30885	−0.654423	+	0.756129i	$0.727088\pi$
$38$	203.646	0.869362
$39$	46.4193	0.190591
$40$	0	0
$41$	−127.823	−0.486891	−0.243445	−	0.969915i	$-0.578278\pi$
$41$	−127.823	−0.486891	−0.243445	+	0.969915i	$0.578278\pi$
$42$	33.2378	0.122112
$43$	502.072	1.78059	0.890293	−	0.455388i	$-0.150499\pi$
$43$	502.072	1.78059	0.890293	+	0.455388i	$0.150499\pi$
$44$	172.729	0.591814
$45$	0	0
$46$	46.0000	0.147442
$47$	−607.879	−1.88656	−0.943279	−	0.332002i	$-0.892276\pi$
$47$	−607.879	−1.88656	−0.943279	+	0.332002i	$0.892276\pi$
$48$	−28.9190	−0.0869605
$49$	−258.457	−0.753520
$50$	0	0
$51$	41.5114	0.113976
$52$	−102.729	−0.273962
$53$	−674.254	−1.74747	−0.873735	−	0.486401i	$-0.838309\pi$
$53$	−674.254	−1.74747	−0.873735	+	0.486401i	$0.838309\pi$
$54$	183.394	0.462163
$55$	0	0
$56$	−73.5577	−0.175528
$57$	−184.039	−0.427659
$58$	−293.365	−0.664150
$59$	14.8056	0.0326700	0.0163350	−	0.999867i	$-0.494800\pi$
$59$	14.8056	0.0326700	0.0163350	+	0.999867i	$0.494800\pi$
$60$	0	0
$61$	−478.549	−1.00446	−0.502229	−	0.864735i	$-0.667487\pi$
$61$	−478.549	−1.00446	−0.502229	+	0.864735i	$0.667487\pi$
$62$	47.9424	0.0982047
$63$	218.220	0.436398
$64$	64.0000	0.125000
$65$	0	0
$66$	−156.098	−0.291127
$67$	−458.422	−0.835898	−0.417949	−	0.908471i	$-0.637251\pi$
$67$	−458.422	−0.835898	−0.417949	+	0.908471i	$0.637251\pi$
$68$	−91.8678	−0.163832
$69$	−41.5711	−0.0725301
$70$	0	0
$71$	1012.83	1.69296	0.846481	−	0.532419i	$-0.178717\pi$
$71$	1012.83	1.69296	0.846481	+	0.532419i	$0.178717\pi$
$72$	−189.865	−0.310776
$73$	750.521	1.20331	0.601657	−	0.798755i	$-0.294508\pi$
$73$	750.521	1.20331	0.601657	+	0.798755i	$0.294508\pi$
$74$	−589.144	−0.925494
$75$	0	0
$76$	407.292	0.614732
$77$	−397.047	−0.587633
$78$	92.8387	0.134768
$79$	−641.648	−0.913811	−0.456905	−	0.889515i	$-0.651042\pi$
$79$	−641.648	−0.913811	−0.456905	+	0.889515i	$0.651042\pi$
$80$	0	0
$81$	475.058	0.651657
$82$	−255.645	−0.344284
$83$	−900.187	−1.19046	−0.595231	−	0.803554i	$-0.702940\pi$
$83$	−900.187	−1.19046	−0.595231	+	0.803554i	$0.702940\pi$
$84$	66.4756	0.0863462
$85$	0	0
$86$	1004.14	1.25906
$87$	265.120	0.326711
$88$	345.457	0.418476
$89$	−923.770	−1.10022	−0.550109	−	0.835093i	$-0.685414\pi$
$89$	−923.770	−1.10022	−0.550109	+	0.835093i	$0.685414\pi$
$90$	0	0
$91$	236.142	0.272026
$92$	92.0000	0.104257
$93$	−43.3265	−0.0483091
$94$	−1215.76	−1.33400
$95$	0	0
$96$	−57.8381	−0.0614904
$97$	−168.230	−0.176095	−0.0880473	−	0.996116i	$-0.528063\pi$
$97$	−168.230	−0.176095	−0.0880473	+	0.996116i	$0.528063\pi$
$98$	−516.915	−0.532819
$99$	−1024.85	−1.04042
$100$	0	0
$101$	−633.410	−0.624026	−0.312013	−	0.950078i	$-0.601003\pi$
$101$	−633.410	−0.624026	−0.312013	+	0.950078i	$0.601003\pi$
$102$	83.0228	0.0805929
$103$	−1300.63	−1.24422	−0.622111	−	0.782929i	$-0.713725\pi$
$103$	−1300.63	−1.24422	−0.622111	+	0.782929i	$0.713725\pi$
$104$	−205.459	−0.193720
$105$	0	0
$106$	−1348.51	−1.23565
$107$	495.620	0.447789	0.223894	−	0.974613i	$-0.428123\pi$
$107$	495.620	0.447789	0.223894	+	0.974613i	$0.428123\pi$
$108$	366.789	0.326799
$109$	−198.210	−0.174175	−0.0870874	−	0.996201i	$-0.527756\pi$
$109$	−198.210	−0.174175	−0.0870874	+	0.996201i	$0.527756\pi$
$110$	0	0
$111$	532.421	0.455272
$112$	−147.115	−0.124117
$113$	−868.555	−0.723069	−0.361534	−	0.932359i	$-0.617747\pi$
$113$	−868.555	−0.723069	−0.361534	+	0.932359i	$0.617747\pi$
$114$	−368.078	−0.302401
$115$	0	0
$116$	−586.730	−0.469625
$117$	609.524	0.481628
$118$	29.6113	0.0231012
$119$	211.174	0.162675
$120$	0	0
$121$	533.696	0.400974
$122$	−957.098	−0.710259
$123$	231.032	0.169361
$124$	95.8848	0.0694412
$125$	0	0
$126$	436.439	0.308580
$127$	−2152.32	−1.50384	−0.751918	−	0.659257i	$-0.770871\pi$
$127$	−2152.32	−1.50384	−0.751918	+	0.659257i	$0.770871\pi$
$128$	128.000	0.0883883
$129$	−907.464	−0.619363
$130$	0	0
$131$	1726.29	1.15135	0.575673	−	0.817680i	$-0.304740\pi$
$131$	1726.29	1.15135	0.575673	+	0.817680i	$0.304740\pi$
$132$	−312.196	−0.205858
$133$	−936.233	−0.610389
$134$	−916.844	−0.591069
$135$	0	0
$136$	−183.736	−0.115847
$137$	1771.09	1.10448	0.552242	−	0.833684i	$-0.313773\pi$
$137$	1771.09	1.10448	0.552242	+	0.833684i	$0.313773\pi$
$138$	−83.1423	−0.0512865
$139$	2789.08	1.70192	0.850959	−	0.525232i	$-0.176021\pi$
$139$	2789.08	1.70192	0.850959	+	0.525232i	$0.176021\pi$
$140$	0	0
$141$	1098.70	0.656224
$142$	2025.65	1.19710
$143$	−1109.02	−0.648537
$144$	−379.731	−0.219751
$145$	0	0
$146$	1501.04	0.850871
$147$	467.146	0.262106
$148$	−1178.29	−0.654423
$149$	−2190.05	−1.20413	−0.602067	−	0.798445i	$-0.705656\pi$
$149$	−2190.05	−1.20413	−0.602067	+	0.798445i	$0.705656\pi$
$150$	0	0
$151$	−1569.59	−0.845902	−0.422951	−	0.906153i	$-0.639006\pi$
$151$	−1569.59	−0.845902	−0.422951	+	0.906153i	$0.639006\pi$
$152$	814.584	0.434681
$153$	545.078	0.288019
$154$	−794.094	−0.415519
$155$	0	0
$156$	185.677	0.0952954
$157$	−2633.86	−1.33888	−0.669442	−	0.742864i	$-0.733467\pi$
$157$	−2633.86	−1.33888	−0.669442	+	0.742864i	$0.733467\pi$
$158$	−1283.30	−0.646162
$159$	1218.67	0.607844
$160$	0	0
$161$	−211.478	−0.103521
$162$	950.116	0.460791
$163$	287.657	0.138227	0.0691135	−	0.997609i	$-0.477983\pi$
$163$	287.657	0.138227	0.0691135	+	0.997609i	$0.477983\pi$
$164$	−511.290	−0.243445
$165$	0	0
$166$	−1800.37	−0.841784
$167$	−1295.54	−0.600310	−0.300155	−	0.953890i	$-0.597038\pi$
$167$	−1295.54	−0.600310	−0.300155	+	0.953890i	$0.597038\pi$
$168$	132.951	0.0610560
$169$	−1537.42	−0.699780
$170$	0	0
$171$	−2416.58	−1.08071
$172$	2008.29	0.890293
$173$	1313.46	0.577230	0.288615	−	0.957445i	$-0.406805\pi$
$173$	1313.46	0.577230	0.288615	+	0.957445i	$0.406805\pi$
$174$	530.240	0.231019
$175$	0	0
$176$	690.914	0.295907
$177$	−26.7603	−0.0113640
$178$	−1847.54	−0.777972
$179$	1916.37	0.800204	0.400102	−	0.916471i	$-0.368975\pi$
$179$	1916.37	0.800204	0.400102	+	0.916471i	$0.368975\pi$
$180$	0	0
$181$	4443.32	1.82469	0.912345	−	0.409421i	$-0.134269\pi$
$181$	4443.32	1.82469	0.912345	+	0.409421i	$0.134269\pi$
$182$	472.284	0.192352
$183$	864.949	0.349393
$184$	184.000	0.0737210
$185$	0	0
$186$	−86.6530	−0.0341597
$187$	−991.762	−0.387833
$188$	−2431.51	−0.943279
$189$	−843.129	−0.324490
$190$	0	0
$191$	1903.54	0.721129	0.360564	−	0.932734i	$-0.382584\pi$
$191$	1903.54	0.721129	0.360564	+	0.932734i	$0.382584\pi$
$192$	−115.676	−0.0434803
$193$	−3417.60	−1.27463	−0.637317	−	0.770602i	$-0.719956\pi$
$193$	−3417.60	−1.27463	−0.637317	+	0.770602i	$0.719956\pi$
$194$	−336.460	−0.124518
$195$	0	0
$196$	−1033.83	−0.376760
$197$	−1609.57	−0.582117	−0.291059	−	0.956705i	$-0.594007\pi$
$197$	−1609.57	−0.582117	−0.291059	+	0.956705i	$0.594007\pi$
$198$	−2049.70	−0.735685
$199$	−4518.91	−1.60973	−0.804866	−	0.593456i	$-0.797763\pi$
$199$	−4518.91	−1.60973	−0.804866	+	0.593456i	$0.797763\pi$
$200$	0	0
$201$	828.570	0.290760
$202$	−1266.82	−0.441253
$203$	1348.70	0.466307
$204$	166.046	0.0569878
$205$	0	0
$206$	−2601.26	−0.879798
$207$	−545.863	−0.183285
$208$	−410.918	−0.136981
$209$	4396.94	1.45523
$210$	0	0
$211$	5408.64	1.76467	0.882337	−	0.470618i	$-0.155969\pi$
$211$	5408.64	1.76467	0.882337	+	0.470618i	$0.155969\pi$
$212$	−2697.02	−0.873735
$213$	−1830.62	−0.588883
$214$	991.240	0.316634
$215$	0	0
$216$	733.577	0.231082
$217$	−220.408	−0.0689506
$218$	−396.420	−0.123160
$219$	−1356.52	−0.418563
$220$	0	0
$221$	589.846	0.179535
$222$	1064.84	0.321926
$223$	−5812.74	−1.74552	−0.872758	−	0.488154i	$-0.837671\pi$
$223$	−5812.74	−1.74552	−0.872758	+	0.488154i	$0.837671\pi$
$224$	−294.231	−0.0877639
$225$	0	0
$226$	−1737.11	−0.511287
$227$	−2033.30	−0.594514	−0.297257	−	0.954798i	$-0.596072\pi$
$227$	−2033.30	−0.594514	−0.297257	+	0.954798i	$0.596072\pi$
$228$	−736.156	−0.213830
$229$	−435.903	−0.125787	−0.0628936	−	0.998020i	$-0.520033\pi$
$229$	−435.903	−0.125787	−0.0628936	+	0.998020i	$0.520033\pi$
$230$	0	0
$231$	717.639	0.204403
$232$	−1173.46	−0.332075
$233$	1582.05	0.444822	0.222411	−	0.974953i	$-0.428607\pi$
$233$	1582.05	0.444822	0.222411	+	0.974953i	$0.428607\pi$
$234$	1219.05	0.340562
$235$	0	0
$236$	59.2226	0.0163350
$237$	1159.74	0.317862
$238$	422.349	0.115029
$239$	−1732.60	−0.468923	−0.234461	−	0.972125i	$-0.575333\pi$
$239$	−1732.60	−0.468923	−0.234461	+	0.972125i	$0.575333\pi$
$240$	0	0
$241$	−4441.73	−1.18721	−0.593604	−	0.804757i	$-0.702296\pi$
$241$	−4441.73	−1.18721	−0.593604	+	0.804757i	$0.702296\pi$
$242$	1067.39	0.283531
$243$	−3334.46	−0.880271
$244$	−1914.20	−0.502229
$245$	0	0
$246$	462.063	0.119756
$247$	−2615.06	−0.673652
$248$	191.770	0.0491023
$249$	1627.03	0.414093
$250$	0	0
$251$	4714.54	1.18557	0.592787	−	0.805359i	$-0.298028\pi$
$251$	4714.54	1.18557	0.592787	+	0.805359i	$0.298028\pi$
$252$	872.878	0.218199
$253$	993.189	0.246803
$254$	−4304.63	−1.06337
$255$	0	0
$256$	256.000	0.0625000
$257$	−4229.41	−1.02655	−0.513275	−	0.858224i	$-0.671568\pi$
$257$	−4229.41	−1.02655	−0.513275	+	0.858224i	$0.671568\pi$
$258$	−1814.93	−0.437956
$259$	2708.50	0.649800
$260$	0	0
$261$	3481.24	0.825606
$262$	3452.57	0.814125
$263$	−4848.80	−1.13684	−0.568422	−	0.822737i	$-0.692446\pi$
$263$	−4848.80	−1.13684	−0.568422	+	0.822737i	$0.692446\pi$
$264$	−624.393	−0.145563
$265$	0	0
$266$	−1872.47	−0.431610
$267$	1669.66	0.382702
$268$	−1833.69	−0.417949
$269$	4603.64	1.04345	0.521726	−	0.853113i	$-0.325288\pi$
$269$	4603.64	1.04345	0.521726	+	0.853113i	$0.325288\pi$
$270$	0	0
$271$	−1521.31	−0.341008	−0.170504	−	0.985357i	$-0.554540\pi$
$271$	−1521.31	−0.341008	−0.170504	+	0.985357i	$0.554540\pi$
$272$	−367.471	−0.0819162
$273$	−426.812	−0.0946222
$274$	3542.18	0.780988
$275$	0	0
$276$	−166.285	−0.0362650
$277$	3293.74	0.714446	0.357223	−	0.934019i	$-0.383724\pi$
$277$	3293.74	0.714446	0.357223	+	0.934019i	$0.383724\pi$
$278$	5578.16	1.20344
$279$	−568.912	−0.122078
$280$	0	0
$281$	8857.86	1.88048	0.940242	−	0.340508i	$-0.110599\pi$
$281$	8857.86	1.88048	0.940242	+	0.340508i	$0.110599\pi$
$282$	2197.41	0.464020
$283$	5470.44	1.14906	0.574530	−	0.818483i	$-0.305185\pi$
$283$	5470.44	1.14906	0.574530	+	0.818483i	$0.305185\pi$
$284$	4051.30	0.846481
$285$	0	0
$286$	−2218.04	−0.458585
$287$	1175.29	0.241726
$288$	−759.461	−0.155388
$289$	−4385.52	−0.892636
$290$	0	0
$291$	304.066	0.0612531
$292$	3002.09	0.601657
$293$	−629.599	−0.125534	−0.0627672	−	0.998028i	$-0.519993\pi$
$293$	−629.599	−0.125534	−0.0627672	+	0.998028i	$0.519993\pi$
$294$	934.292	0.185337
$295$	0	0
$296$	−2356.57	−0.462747
$297$	3959.68	0.773616
$298$	−4380.10	−0.851452
$299$	−590.694	−0.114250
$300$	0	0
$301$	−4616.40	−0.884004
$302$	−3139.17	−0.598143
$303$	1144.85	0.217062
$304$	1629.17	0.307366
$305$	0	0
$306$	1090.16	0.203660
$307$	791.929	0.147224	0.0736120	−	0.997287i	$-0.476547\pi$
$307$	791.929	0.147224	0.0736120	+	0.997287i	$0.476547\pi$
$308$	−1588.19	−0.293816
$309$	2350.81	0.432793
$310$	0	0
$311$	2109.26	0.384582	0.192291	−	0.981338i	$-0.438408\pi$
$311$	2109.26	0.384582	0.192291	+	0.981338i	$0.438408\pi$
$312$	371.355	0.0673840
$313$	207.600	0.0374896	0.0187448	−	0.999824i	$-0.494033\pi$
$313$	207.600	0.0374896	0.0187448	+	0.999824i	$0.494033\pi$
$314$	−5267.72	−0.946734
$315$	0	0
$316$	−2566.59	−0.456905
$317$	4013.96	0.711188	0.355594	−	0.934640i	$-0.384279\pi$
$317$	4013.96	0.711188	0.355594	+	0.934640i	$0.384279\pi$
$318$	2437.35	0.429811
$319$	−6334.06	−1.11172
$320$	0	0
$321$	−895.803	−0.155760
$322$	−422.957	−0.0732002
$323$	−2338.56	−0.402852
$324$	1900.23	0.325829
$325$	0	0
$326$	575.313	0.0977413
$327$	358.252	0.0605853
$328$	−1022.58	−0.172142
$329$	5589.27	0.936615
$330$	0	0
$331$	1196.79	0.198736	0.0993678	−	0.995051i	$-0.468318\pi$
$331$	1196.79	0.198736	0.0993678	+	0.995051i	$0.468318\pi$
$332$	−3600.75	−0.595231
$333$	6991.12	1.15048
$334$	−2591.08	−0.424484
$335$	0	0
$336$	265.902	0.0431731
$337$	−3245.42	−0.524597	−0.262299	−	0.964987i	$-0.584481\pi$
$337$	−3245.42	−0.524597	−0.262299	+	0.964987i	$0.584481\pi$
$338$	−3074.83	−0.494819
$339$	1569.86	0.251514
$340$	0	0
$341$	1035.13	0.164385
$342$	−4833.16	−0.764174
$343$	5530.23	0.870566
$344$	4016.57	0.629532
$345$	0	0
$346$	2626.93	0.408163
$347$	−10510.0	−1.62596	−0.812981	−	0.582290i	$-0.802157\pi$
$347$	−10510.0	−1.62596	−0.812981	+	0.582290i	$0.802157\pi$
$348$	1060.48	0.163355
$349$	6150.02	0.943276	0.471638	−	0.881792i	$-0.343663\pi$
$349$	6150.02	0.943276	0.471638	+	0.881792i	$0.343663\pi$
$350$	0	0
$351$	−2355.00	−0.358121
$352$	1381.83	0.209238
$353$	−1123.35	−0.169376	−0.0846882	−	0.996408i	$-0.526989\pi$
$353$	−1123.35	−0.169376	−0.0846882	+	0.996408i	$0.526989\pi$
$354$	−53.5206	−0.00803556
$355$	0	0
$356$	−3695.08	−0.550109
$357$	−381.685	−0.0565852
$358$	3832.75	0.565829
$359$	2209.46	0.324821	0.162411	−	0.986723i	$-0.448073\pi$
$359$	2209.46	0.324821	0.162411	+	0.986723i	$0.448073\pi$
$360$	0	0
$361$	3508.93	0.511581
$362$	8886.63	1.29025
$363$	−964.624	−0.139476
$364$	944.568	0.136013
$365$	0	0
$366$	1729.90	0.247058
$367$	9017.63	1.28261	0.641303	−	0.767287i	$-0.278394\pi$
$367$	9017.63	1.28261	0.641303	+	0.767287i	$0.278394\pi$
$368$	368.000	0.0521286
$369$	3033.63	0.427980
$370$	0	0
$371$	6199.57	0.867563
$372$	−173.306	−0.0241546
$373$	−2000.52	−0.277702	−0.138851	−	0.990313i	$-0.544341\pi$
$373$	−2000.52	−0.277702	−0.138851	+	0.990313i	$0.544341\pi$
$374$	−1983.52	−0.274239
$375$	0	0
$376$	−4863.03	−0.666999
$377$	3767.15	0.514637
$378$	−1686.26	−0.229449
$379$	5404.84	0.732528	0.366264	−	0.930511i	$-0.380637\pi$
$379$	5404.84	0.732528	0.366264	+	0.930511i	$0.380637\pi$
$380$	0	0
$381$	3890.18	0.523097
$382$	3807.09	0.509915
$383$	9355.83	1.24820	0.624100	−	0.781345i	$-0.285466\pi$
$383$	9355.83	1.24820	0.624100	+	0.781345i	$0.285466\pi$
$384$	−231.352	−0.0307452
$385$	0	0
$386$	−6835.20	−0.901302
$387$	−11915.7	−1.56515
$388$	−672.920	−0.0880473
$389$	7771.25	1.01290	0.506450	−	0.862269i	$-0.330958\pi$
$389$	7771.25	1.01290	0.506450	+	0.862269i	$0.330958\pi$
$390$	0	0
$391$	−528.240	−0.0683229
$392$	−2067.66	−0.266409
$393$	−3120.16	−0.400487
$394$	−3219.14	−0.411619
$395$	0	0
$396$	−4099.39	−0.520208
$397$	2509.29	0.317223	0.158612	−	0.987341i	$-0.449298\pi$
$397$	2509.29	0.317223	0.158612	+	0.987341i	$0.449298\pi$
$398$	−9037.81	−1.13825
$399$	1692.19	0.212319
$400$	0	0
$401$	1502.71	0.187136	0.0935681	−	0.995613i	$-0.470173\pi$
$401$	1502.71	0.187136	0.0935681	+	0.995613i	$0.470173\pi$
$402$	1657.14	0.205599
$403$	−615.637	−0.0760969
$404$	−2533.64	−0.312013
$405$	0	0
$406$	2697.41	0.329729
$407$	−12720.2	−1.54919
$408$	332.091	0.0402965
$409$	6281.54	0.759419	0.379709	−	0.925106i	$-0.376024\pi$
$409$	6281.54	0.759419	0.379709	+	0.925106i	$0.376024\pi$
$410$	0	0
$411$	−3201.14	−0.384186
$412$	−5202.52	−0.622111
$413$	−136.134	−0.0162196
$414$	−1091.73	−0.129602
$415$	0	0
$416$	−821.836	−0.0968601
$417$	−5041.09	−0.591999
$418$	8793.87	1.02900
$419$	−5437.48	−0.633982	−0.316991	−	0.948429i	$-0.602672\pi$
$419$	−5437.48	−0.633982	−0.316991	+	0.948429i	$0.602672\pi$
$420$	0	0
$421$	8424.12	0.975217	0.487608	−	0.873062i	$-0.337869\pi$
$421$	8424.12	0.975217	0.487608	+	0.873062i	$0.337869\pi$
$422$	10817.3	1.24781
$423$	14426.9	1.65830
$424$	−5394.03	−0.617824
$425$	0	0
$426$	−3661.24	−0.416403
$427$	4400.12	0.498681
$428$	1982.48	0.223894
$429$	2004.49	0.225589
$430$	0	0
$431$	−2280.19	−0.254832	−0.127416	−	0.991849i	$-0.540668\pi$
$431$	−2280.19	−0.254832	−0.127416	+	0.991849i	$0.540668\pi$
$432$	1467.15	0.163399
$433$	9676.90	1.07400	0.537000	−	0.843582i	$-0.319557\pi$
$433$	9676.90	1.07400	0.537000	+	0.843582i	$0.319557\pi$
$434$	−440.816	−0.0487554
$435$	0	0
$436$	−792.839	−0.0870874
$437$	2341.93	0.256361
$438$	−2713.04	−0.295969
$439$	−10267.6	−1.11628	−0.558138	−	0.829748i	$-0.688484\pi$
$439$	−10267.6	−1.11628	−0.558138	+	0.829748i	$0.688484\pi$
$440$	0	0
$441$	6134.01	0.662348
$442$	1179.69	0.126951
$443$	−2947.70	−0.316139	−0.158069	−	0.987428i	$-0.550527\pi$
$443$	−2947.70	−0.316139	−0.158069	+	0.987428i	$0.550527\pi$
$444$	2129.68	0.227636
$445$	0	0
$446$	−11625.5	−1.23427
$447$	3958.39	0.418849
$448$	−588.462	−0.0620585
$449$	−14322.2	−1.50536	−0.752682	−	0.658385i	$-0.771240\pi$
$449$	−14322.2	−1.50536	−0.752682	+	0.658385i	$0.771240\pi$
$450$	0	0
$451$	−5519.65	−0.576297
$452$	−3474.22	−0.361534
$453$	2836.94	0.294240
$454$	−4066.59	−0.420385
$455$	0	0
$456$	−1472.31	−0.151200
$457$	13404.4	1.37206	0.686029	−	0.727574i	$-0.259352\pi$
$457$	13404.4	1.37206	0.686029	+	0.727574i	$0.259352\pi$
$458$	−871.805	−0.0889449
$459$	−2106.00	−0.214161
$460$	0	0
$461$	−16928.6	−1.71029	−0.855144	−	0.518390i	$-0.826532\pi$
$461$	−16928.6	−1.71029	−0.855144	+	0.518390i	$0.826532\pi$
$462$	1435.28	0.144535
$463$	13109.3	1.31585	0.657925	−	0.753083i	$-0.271434\pi$
$463$	13109.3	1.31585	0.657925	+	0.753083i	$0.271434\pi$
$464$	−2346.92	−0.234812
$465$	0	0
$466$	3164.10	0.314537
$467$	−4676.02	−0.463341	−0.231671	−	0.972794i	$-0.574419\pi$
$467$	−4676.02	−0.463341	−0.231671	+	0.972794i	$0.574419\pi$
$468$	2438.09	0.240814
$469$	4215.06	0.414996
$470$	0	0
$471$	4760.54	0.465720
$472$	118.445	0.0115506
$473$	21680.5	2.10755
$474$	2319.48	0.224762
$475$	0	0
$476$	844.698	0.0813375
$477$	16002.2	1.53604
$478$	−3465.20	−0.331579
$479$	−13518.0	−1.28947	−0.644734	−	0.764407i	$-0.723032\pi$
$479$	−13518.0	−1.28947	−0.644734	+	0.764407i	$0.723032\pi$
$480$	0	0
$481$	7565.30	0.717148
$482$	−8883.47	−0.839483
$483$	382.234	0.0360088
$484$	2134.79	0.200487
$485$	0	0
$486$	−6668.92	−0.622446
$487$	14914.7	1.38778	0.693889	−	0.720082i	$-0.255896\pi$
$487$	14914.7	1.38778	0.693889	+	0.720082i	$0.255896\pi$
$488$	−3828.39	−0.355129
$489$	−519.922	−0.0480812
$490$	0	0
$491$	−3706.99	−0.340722	−0.170361	−	0.985382i	$-0.554493\pi$
$491$	−3706.99	−0.340722	−0.170361	+	0.985382i	$0.554493\pi$
$492$	924.126	0.0846806
$493$	3368.85	0.307759
$494$	−5230.11	−0.476344
$495$	0	0
$496$	383.539	0.0347206
$497$	−9312.64	−0.840501
$498$	3254.07	0.292808
$499$	13037.0	1.16957	0.584784	−	0.811189i	$-0.301179\pi$
$499$	13037.0	1.16957	0.584784	+	0.811189i	$0.301179\pi$
$500$	0	0
$501$	2341.61	0.208813
$502$	9429.08	0.838328
$503$	2976.34	0.263834	0.131917	−	0.991261i	$-0.457887\pi$
$503$	2976.34	0.263834	0.131917	+	0.991261i	$0.457887\pi$
$504$	1745.76	0.154290
$505$	0	0
$506$	1986.38	0.174516
$507$	2778.79	0.243413
$508$	−8609.26	−0.751918
$509$	20818.7	1.81292	0.906458	−	0.422297i	$-0.138776\pi$
$509$	20818.7	1.81292	0.906458	+	0.422297i	$0.138776\pi$
$510$	0	0
$511$	−6900.83	−0.597406
$512$	512.000	0.0441942
$513$	9336.88	0.803574
$514$	−8458.82	−0.725881
$515$	0	0
$516$	−3629.86	−0.309681
$517$	−26249.5	−2.23298
$518$	5417.00	0.459478
$519$	−2374.01	−0.200785
$520$	0	0
$521$	−10439.5	−0.877855	−0.438927	−	0.898522i	$-0.644641\pi$
$521$	−10439.5	−0.877855	−0.438927	+	0.898522i	$0.644641\pi$
$522$	6962.48	0.583792
$523$	22364.9	1.86988	0.934942	−	0.354801i	$-0.115451\pi$
$523$	22364.9	1.86988	0.934942	+	0.354801i	$0.115451\pi$
$524$	6905.15	0.575673
$525$	0	0
$526$	−9697.60	−0.803870
$527$	−550.545	−0.0455069
$528$	−1248.79	−0.102929
$529$	529.000	0.0434783
$530$	0	0
$531$	−351.385	−0.0287171
$532$	−3744.93	−0.305194
$533$	3282.78	0.266779
$534$	3339.32	0.270611
$535$	0	0
$536$	−3667.38	−0.295535
$537$	−3463.73	−0.278344
$538$	9207.27	0.737832
$539$	−11160.7	−0.891887
$540$	0	0
$541$	−22462.5	−1.78510	−0.892550	−	0.450949i	$-0.851086\pi$
$541$	−22462.5	−1.78510	−0.892550	+	0.450949i	$0.851086\pi$
$542$	−3042.63	−0.241129
$543$	−8031.03	−0.634704
$544$	−734.942	−0.0579235
$545$	0	0
$546$	−853.625	−0.0669080
$547$	4367.35	0.341379	0.170690	−	0.985325i	$-0.445400\pi$
$547$	4367.35	0.341379	0.170690	+	0.985325i	$0.445400\pi$
$548$	7084.35	0.552242
$549$	11357.5	0.882924
$550$	0	0
$551$	−14935.7	−1.15477
$552$	−332.569	−0.0256433
$553$	5899.77	0.453677
$554$	6587.47	0.505189
$555$	0	0
$556$	11156.3	0.850959
$557$	−13805.9	−1.05022	−0.525110	−	0.851034i	$-0.675976\pi$
$557$	−13805.9	−1.05022	−0.525110	+	0.851034i	$0.675976\pi$
$558$	−1137.82	−0.0863225
$559$	−12894.4	−0.975625
$560$	0	0
$561$	1792.55	0.134905
$562$	17715.7	1.32970
$563$	8149.77	0.610075	0.305037	−	0.952340i	$-0.401331\pi$
$563$	8149.77	0.610075	0.305037	+	0.952340i	$0.401331\pi$
$564$	4394.82	0.328112
$565$	0	0
$566$	10940.9	0.812508
$567$	−4368.02	−0.323527
$568$	8102.60	0.598552
$569$	−11539.9	−0.850226	−0.425113	−	0.905140i	$-0.639766\pi$
$569$	−11539.9	−0.850226	−0.425113	+	0.905140i	$0.639766\pi$
$570$	0	0
$571$	−2436.53	−0.178573	−0.0892867	−	0.996006i	$-0.528459\pi$
$571$	−2436.53	−0.178573	−0.0892867	+	0.996006i	$0.528459\pi$
$572$	−4436.08	−0.324269
$573$	−3440.54	−0.250839
$574$	2350.58	0.170926
$575$	0	0
$576$	−1518.92	−0.109876
$577$	−6965.59	−0.502567	−0.251284	−	0.967913i	$-0.580853\pi$
$577$	−6965.59	−0.502567	−0.251284	+	0.967913i	$0.580853\pi$
$578$	−8771.04	−0.631189
$579$	6177.11	0.443371
$580$	0	0
$581$	8276.96	0.591026
$582$	608.132	0.0433125
$583$	−29115.7	−2.06835
$584$	6004.17	0.425435
$585$	0	0
$586$	−1259.20	−0.0887662
$587$	3080.79	0.216623	0.108311	−	0.994117i	$-0.465456\pi$
$587$	3080.79	0.216623	0.108311	+	0.994117i	$0.465456\pi$
$588$	1868.58	0.131053
$589$	2440.82	0.170751
$590$	0	0
$591$	2909.20	0.202485
$592$	−4713.15	−0.327211
$593$	−13741.0	−0.951563	−0.475782	−	0.879564i	$-0.657835\pi$
$593$	−13741.0	−0.951563	−0.475782	+	0.879564i	$0.657835\pi$
$594$	7919.36	0.547029
$595$	0	0
$596$	−8760.21	−0.602067
$597$	8167.65	0.559933
$598$	−1181.39	−0.0807869
$599$	24131.0	1.64602	0.823011	−	0.568026i	$-0.192293\pi$
$599$	24131.0	1.64602	0.823011	+	0.568026i	$0.192293\pi$
$600$	0	0
$601$	−18108.3	−1.22904	−0.614520	−	0.788901i	$-0.710650\pi$
$601$	−18108.3	−1.22904	−0.614520	+	0.788901i	$0.710650\pi$
$602$	−9232.81	−0.625085
$603$	10879.8	0.734759
$604$	−6278.35	−0.422951
$605$	0	0
$606$	2289.70	0.153486
$607$	10257.1	0.685873	0.342936	−	0.939359i	$-0.388578\pi$
$607$	10257.1	0.685873	0.342936	+	0.939359i	$0.388578\pi$
$608$	3258.34	0.217340
$609$	−2437.70	−0.162201
$610$	0	0
$611$	15611.8	1.03369
$612$	2180.31	0.144010
$613$	−2710.49	−0.178590	−0.0892951	−	0.996005i	$-0.528461\pi$
$613$	−2710.49	−0.178590	−0.0892951	+	0.996005i	$0.528461\pi$
$614$	1583.86	0.104103
$615$	0	0
$616$	−3176.38	−0.207760
$617$	20517.7	1.33875	0.669377	−	0.742923i	$-0.266561\pi$
$617$	20517.7	1.33875	0.669377	+	0.742923i	$0.266561\pi$
$618$	4701.62	0.306031
$619$	−5220.67	−0.338992	−0.169496	−	0.985531i	$-0.554214\pi$
$619$	−5220.67	−0.338992	−0.169496	+	0.985531i	$0.554214\pi$
$620$	0	0
$621$	2109.03	0.136284
$622$	4218.52	0.271941
$623$	8493.80	0.546223
$624$	742.710	0.0476477
$625$	0	0
$626$	415.200	0.0265092
$627$	−7947.20	−0.506189
$628$	−10535.4	−0.669442
$629$	6765.41	0.428863
$630$	0	0
$631$	−19100.5	−1.20504	−0.602520	−	0.798103i	$-0.705837\pi$
$631$	−19100.5	−1.20504	−0.602520	+	0.798103i	$0.705837\pi$
$632$	−5133.18	−0.323081
$633$	−9775.80	−0.613828
$634$	8027.93	0.502886
$635$	0	0
$636$	4874.70	0.303922
$637$	6637.79	0.412871
$638$	−12668.1	−0.786106
$639$	−24037.5	−1.48812
$640$	0	0
$641$	15088.0	0.929701	0.464851	−	0.885389i	$-0.346108\pi$
$641$	15088.0	0.929701	0.464851	+	0.885389i	$0.346108\pi$
$642$	−1791.61	−0.110139
$643$	1568.01	0.0961687	0.0480843	−	0.998843i	$-0.484688\pi$
$643$	1568.01	0.0961687	0.0480843	+	0.998843i	$0.484688\pi$
$644$	−845.913	−0.0517603
$645$	0	0
$646$	−4677.13	−0.284859
$647$	−15035.4	−0.913603	−0.456801	−	0.889569i	$-0.651005\pi$
$647$	−15035.4	−0.913603	−0.456801	+	0.889569i	$0.651005\pi$
$648$	3800.47	0.230396
$649$	639.339	0.0386691
$650$	0	0
$651$	398.375	0.0239839
$652$	1150.63	0.0691135
$653$	−22403.7	−1.34261	−0.671307	−	0.741180i	$-0.734267\pi$
$653$	−22403.7	−1.34261	−0.671307	+	0.741180i	$0.734267\pi$
$654$	716.505	0.0428403
$655$	0	0
$656$	−2045.16	−0.121723
$657$	−17812.2	−1.05772
$658$	11178.5	0.662287
$659$	890.918	0.0526635	0.0263317	−	0.999653i	$-0.491617\pi$
$659$	890.918	0.0526635	0.0263317	+	0.999653i	$0.491617\pi$
$660$	0	0
$661$	−28636.0	−1.68504	−0.842519	−	0.538667i	$-0.818928\pi$
$661$	−28636.0	−1.68504	−0.842519	+	0.538667i	$0.818928\pi$
$662$	2393.58	0.140527
$663$	−1066.11	−0.0624499
$664$	−7201.50	−0.420892
$665$	0	0
$666$	13982.2	0.813515
$667$	−3373.70	−0.195847
$668$	−5182.16	−0.300155
$669$	10506.2	0.607164
$670$	0	0
$671$	−20664.8	−1.18890
$672$	531.805	0.0305280
$673$	15056.8	0.862405	0.431203	−	0.902255i	$-0.358089\pi$
$673$	15056.8	0.862405	0.431203	+	0.902255i	$0.358089\pi$
$674$	−6490.84	−0.370946
$675$	0	0
$676$	−6149.66	−0.349890
$677$	−1612.72	−0.0915539	−0.0457770	−	0.998952i	$-0.514576\pi$
$677$	−1612.72	−0.0915539	−0.0457770	+	0.998952i	$0.514576\pi$
$678$	3139.72	0.177847
$679$	1546.83	0.0874253
$680$	0	0
$681$	3675.06	0.206797
$682$	2070.25	0.116238
$683$	20264.3	1.13527	0.567637	−	0.823279i	$-0.307858\pi$
$683$	20264.3	1.13527	0.567637	+	0.823279i	$0.307858\pi$
$684$	−9666.33	−0.540353
$685$	0	0
$686$	11060.5	0.615583
$687$	787.868	0.0437540
$688$	8033.15	0.445147
$689$	17316.4	0.957481
$690$	0	0
$691$	−7628.56	−0.419977	−0.209988	−	0.977704i	$-0.567343\pi$
$691$	−7628.56	−0.419977	−0.209988	+	0.977704i	$0.567343\pi$
$692$	5253.85	0.288615
$693$	9423.19	0.516533
$694$	−21020.1	−1.14973
$695$	0	0
$696$	2120.96	0.115510
$697$	2935.69	0.159537
$698$	12300.0	0.666997
$699$	−2859.46	−0.154728
$700$	0	0
$701$	18206.5	0.980953	0.490477	−	0.871454i	$-0.336823\pi$
$701$	18206.5	0.980953	0.490477	+	0.871454i	$0.336823\pi$
$702$	−4710.00	−0.253230
$703$	−29994.2	−1.60918
$704$	2763.66	0.147953
$705$	0	0
$706$	−2246.70	−0.119767
$707$	5824.02	0.309809
$708$	−107.041	−0.00568200
$709$	14793.4	0.783607	0.391804	−	0.920049i	$-0.371851\pi$
$709$	14793.4	0.783607	0.391804	+	0.920049i	$0.371851\pi$
$710$	0	0
$711$	15228.3	0.803245
$712$	−7390.16	−0.388986
$713$	551.337	0.0289590
$714$	−763.370	−0.0400118
$715$	0	0
$716$	7665.49	0.400102
$717$	3131.57	0.163111
$718$	4418.92	0.229683
$719$	32215.9	1.67100	0.835500	−	0.549490i	$-0.185178\pi$
$719$	32215.9	1.67100	0.835500	+	0.549490i	$0.185178\pi$
$720$	0	0
$721$	11958.9	0.617716
$722$	7017.86	0.361742
$723$	8028.17	0.412961
$724$	17773.3	0.912345
$725$	0	0
$726$	−1929.25	−0.0986242
$727$	−1853.37	−0.0945497	−0.0472748	−	0.998882i	$-0.515054\pi$
$727$	−1853.37	−0.0945497	−0.0472748	+	0.998882i	$0.515054\pi$
$728$	1889.14	0.0961759
$729$	−6799.73	−0.345462
$730$	0	0
$731$	−11531.1	−0.583436
$732$	3459.80	0.174696
$733$	−6958.47	−0.350637	−0.175319	−	0.984512i	$-0.556096\pi$
$733$	−6958.47	−0.350637	−0.175319	+	0.984512i	$0.556096\pi$
$734$	18035.3	0.906940
$735$	0	0
$736$	736.000	0.0368605
$737$	−19795.6	−0.989392
$738$	6067.26	0.302628
$739$	6469.54	0.322038	0.161019	−	0.986951i	$-0.448522\pi$
$739$	6469.54	0.322038	0.161019	+	0.986951i	$0.448522\pi$
$740$	0	0
$741$	4726.56	0.234325
$742$	12399.1	0.613460
$743$	−6992.55	−0.345265	−0.172632	−	0.984986i	$-0.555227\pi$
$743$	−6992.55	−0.345265	−0.172632	+	0.984986i	$0.555227\pi$
$744$	−346.612	−0.0170799
$745$	0	0
$746$	−4001.04	−0.196365
$747$	21364.3	1.04642
$748$	−3967.05	−0.193917
$749$	−4557.08	−0.222313
$750$	0	0
$751$	24595.2	1.19506	0.597531	−	0.801846i	$-0.296148\pi$
$751$	24595.2	1.19506	0.597531	+	0.801846i	$0.296148\pi$
$752$	−9726.06	−0.471639
$753$	−8521.25	−0.412393
$754$	7534.30	0.363903
$755$	0	0
$756$	−3372.52	−0.162245
$757$	35778.4	1.71782	0.858909	−	0.512128i	$-0.171143\pi$
$757$	35778.4	1.71782	0.858909	+	0.512128i	$0.171143\pi$
$758$	10809.7	0.517975
$759$	−1795.13	−0.0858486
$760$	0	0
$761$	−39181.2	−1.86638	−0.933191	−	0.359380i	$-0.882988\pi$
$761$	−39181.2	−1.86638	−0.933191	+	0.359380i	$0.882988\pi$
$762$	7780.37	0.369886
$763$	1822.48	0.0864722
$764$	7614.18	0.360564
$765$	0	0
$766$	18711.7	0.882610
$767$	−380.244	−0.0179007
$768$	−462.705	−0.0217401
$769$	9455.39	0.443394	0.221697	−	0.975116i	$-0.428840\pi$
$769$	9455.39	0.443394	0.221697	+	0.975116i	$0.428840\pi$
$770$	0	0
$771$	7644.41	0.357077
$772$	−13670.4	−0.637317
$773$	−5683.75	−0.264464	−0.132232	−	0.991219i	$-0.542214\pi$
$773$	−5683.75	−0.264464	−0.132232	+	0.991219i	$0.542214\pi$
$774$	−23831.5	−1.10673
$775$	0	0
$776$	−1345.84	−0.0622588
$777$	−4895.46	−0.226028
$778$	15542.5	0.716228
$779$	−13015.3	−0.598615
$780$	0	0
$781$	43736.0	2.00384
$782$	−1056.48	−0.0483116
$783$	−13450.4	−0.613891
$784$	−4135.32	−0.188380
$785$	0	0
$786$	−6240.32	−0.283187
$787$	−18500.7	−0.837966	−0.418983	−	0.907994i	$-0.637613\pi$
$787$	−18500.7	−0.837966	−0.418983	+	0.907994i	$0.637613\pi$
$788$	−6438.28	−0.291059
$789$	8763.91	0.395442
$790$	0	0
$791$	7986.11	0.358980
$792$	−8198.79	−0.367842
$793$	12290.3	0.550366
$794$	5018.58	0.224311
$795$	0	0
$796$	−18075.6	−0.804866
$797$	30464.4	1.35396	0.676979	−	0.736002i	$-0.263289\pi$
$797$	30464.4	1.35396	0.676979	+	0.736002i	$0.263289\pi$
$798$	3384.37	0.150132
$799$	13961.1	0.618159
$800$	0	0
$801$	21924.0	0.967099
$802$	3005.42	0.132325
$803$	32409.1	1.42427
$804$	3314.28	0.145380
$805$	0	0
$806$	−1231.27	−0.0538087
$807$	−8320.80	−0.362957
$808$	−5067.28	−0.220626
$809$	−20168.5	−0.876500	−0.438250	−	0.898853i	$-0.644401\pi$
$809$	−20168.5	−0.876500	−0.438250	+	0.898853i	$0.644401\pi$
$810$	0	0
$811$	3472.00	0.150331	0.0751655	−	0.997171i	$-0.476051\pi$
$811$	3472.00	0.150331	0.0751655	+	0.997171i	$0.476051\pi$
$812$	5394.81	0.233154
$813$	2749.68	0.118617
$814$	−25440.5	−1.09544
$815$	0	0
$816$	664.182	0.0284939
$817$	51122.5	2.18917
$818$	12563.1	0.536990
$819$	−5604.39	−0.239113
$820$	0	0
$821$	7377.27	0.313603	0.156802	−	0.987630i	$-0.449882\pi$
$821$	7377.27	0.313603	0.156802	+	0.987630i	$0.449882\pi$
$822$	−6402.27	−0.271660
$823$	−19960.3	−0.845411	−0.422705	−	0.906267i	$-0.638919\pi$
$823$	−19960.3	−0.845411	−0.422705	+	0.906267i	$0.638919\pi$
$824$	−10405.0	−0.439899
$825$	0	0
$826$	−272.267	−0.0114690
$827$	−3105.37	−0.130574	−0.0652868	−	0.997867i	$-0.520796\pi$
$827$	−3105.37	−0.130574	−0.0652868	+	0.997867i	$0.520796\pi$
$828$	−2183.45	−0.0916427
$829$	−10780.3	−0.451646	−0.225823	−	0.974168i	$-0.572507\pi$
$829$	−10780.3	−0.451646	−0.225823	+	0.974168i	$0.572507\pi$
$830$	0	0
$831$	−5953.23	−0.248514
$832$	−1643.67	−0.0684905
$833$	5935.98	0.246902
$834$	−10082.2	−0.418606
$835$	0	0
$836$	17587.7	0.727613
$837$	2198.09	0.0907731
$838$	−10875.0	−0.448293
$839$	2884.27	0.118684	0.0593422	−	0.998238i	$-0.481100\pi$
$839$	2884.27	0.118684	0.0593422	+	0.998238i	$0.481100\pi$
$840$	0	0
$841$	−2873.26	−0.117810
$842$	16848.2	0.689582
$843$	−16010.1	−0.654111
$844$	21634.6	0.882337
$845$	0	0
$846$	28853.8	1.17259
$847$	−4907.18	−0.199071
$848$	−10788.1	−0.436868
$849$	−9887.50	−0.399691
$850$	0	0
$851$	−6775.15	−0.272913
$852$	−7322.49	−0.294442
$853$	−13694.1	−0.549681	−0.274841	−	0.961490i	$-0.588625\pi$
$853$	−13694.1	−0.549681	−0.274841	+	0.961490i	$0.588625\pi$
$854$	8800.24	0.352621
$855$	0	0
$856$	3964.96	0.158317
$857$	−31610.2	−1.25996	−0.629978	−	0.776613i	$-0.716936\pi$
$857$	−31610.2	−1.25996	−0.629978	+	0.776613i	$0.716936\pi$
$858$	4008.97	0.159515
$859$	−20316.3	−0.806964	−0.403482	−	0.914988i	$-0.632200\pi$
$859$	−20316.3	−0.806964	−0.403482	+	0.914988i	$0.632200\pi$
$860$	0	0
$861$	−2124.27	−0.0840823
$862$	−4560.37	−0.180194
$863$	−9902.42	−0.390593	−0.195297	−	0.980744i	$-0.562567\pi$
$863$	−9902.42	−0.390593	−0.195297	+	0.980744i	$0.562567\pi$
$864$	2934.31	0.115541
$865$	0	0
$866$	19353.8	0.759433
$867$	7926.56	0.310496
$868$	−881.633	−0.0344753
$869$	−27707.7	−1.08161
$870$	0	0
$871$	11773.4	0.458008
$872$	−1585.68	−0.0615801
$873$	3992.63	0.154788
$874$	4683.86	0.181275
$875$	0	0
$876$	−5426.09	−0.209281
$877$	−11027.7	−0.424606	−0.212303	−	0.977204i	$-0.568096\pi$
$877$	−11027.7	−0.424606	−0.212303	+	0.977204i	$0.568096\pi$
$878$	−20535.2	−0.789326
$879$	1137.96	0.0436661
$880$	0	0
$881$	−27892.8	−1.06667	−0.533333	−	0.845906i	$-0.679061\pi$
$881$	−27892.8	−1.06667	−0.533333	+	0.845906i	$0.679061\pi$
$882$	12268.0	0.468351
$883$	−1618.76	−0.0616937	−0.0308468	−	0.999524i	$-0.509820\pi$
$883$	−1618.76	−0.0616937	−0.0308468	+	0.999524i	$0.509820\pi$
$884$	2359.38	0.0897677
$885$	0	0
$886$	−5895.40	−0.223544
$887$	9518.69	0.360323	0.180161	−	0.983637i	$-0.442338\pi$
$887$	9518.69	0.360323	0.180161	+	0.983637i	$0.442338\pi$
$888$	4259.37	0.160963
$889$	19789.9	0.746606
$890$	0	0
$891$	20514.0	0.771319
$892$	−23251.0	−0.872758
$893$	−61896.0	−2.31945
$894$	7916.78	0.296171
$895$	0	0
$896$	−1176.92	−0.0438820
$897$	1067.64	0.0397409
$898$	−28644.5	−1.06445
$899$	−3516.15	−0.130445
$900$	0	0
$901$	15485.6	0.572585
$902$	−11039.3	−0.407504
$903$	8343.87	0.307494
$904$	−6948.44	−0.255643
$905$	0	0
$906$	5673.87	0.208059
$907$	−9580.94	−0.350750	−0.175375	−	0.984502i	$-0.556114\pi$
$907$	−9580.94	−0.350750	−0.175375	+	0.984502i	$0.556114\pi$
$908$	−8133.18	−0.297257
$909$	15032.8	0.548522
$910$	0	0
$911$	−18016.6	−0.655232	−0.327616	−	0.944811i	$-0.606245\pi$
$911$	−18016.6	−0.655232	−0.327616	+	0.944811i	$0.606245\pi$
$912$	−2944.62	−0.106915
$913$	−38872.0	−1.40906
$914$	26808.8	0.970192
$915$	0	0
$916$	−1743.61	−0.0628936
$917$	−15872.7	−0.571606
$918$	−4212.01	−0.151435
$919$	−36712.5	−1.31777	−0.658887	−	0.752242i	$-0.728973\pi$
$919$	−36712.5	−1.31777	−0.658887	+	0.752242i	$0.728973\pi$
$920$	0	0
$921$	−1431.36	−0.0512107
$922$	−33857.2	−1.20936
$923$	−26011.8	−0.927614
$924$	2870.56	0.102202
$925$	0	0
$926$	26218.5	0.930447
$927$	30868.1	1.09368
$928$	−4693.84	−0.166037
$929$	4117.08	0.145400	0.0727002	−	0.997354i	$-0.476838\pi$
$929$	4117.08	0.145400	0.0727002	+	0.997354i	$0.476838\pi$
$930$	0	0
$931$	−26316.9	−0.926425
$932$	6328.20	0.222411
$933$	−3812.36	−0.133774
$934$	−9352.04	−0.327632
$935$	0	0
$936$	4876.19	0.170281
$937$	11540.5	0.402362	0.201181	−	0.979554i	$-0.435522\pi$
$937$	11540.5	0.402362	0.201181	+	0.979554i	$0.435522\pi$
$938$	8430.12	0.293447
$939$	−375.225	−0.0130405
$940$	0	0
$941$	−8600.43	−0.297945	−0.148972	−	0.988841i	$-0.547597\pi$
$941$	−8600.43	−0.297945	−0.148972	+	0.988841i	$0.547597\pi$
$942$	9521.08	0.329314
$943$	−2939.92	−0.101524
$944$	236.890	0.00816750
$945$	0	0
$946$	43361.0	1.49026
$947$	−48782.8	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$947$	−48782.8	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$948$	4638.96	0.158931
$949$	−19275.2	−0.659324
$950$	0	0
$951$	−7255.00	−0.247381
$952$	1689.40	0.0575143
$953$	−42550.4	−1.44632	−0.723160	−	0.690681i	$-0.757311\pi$
$953$	−42550.4	−1.44632	−0.723160	+	0.690681i	$0.757311\pi$
$954$	32004.4	1.08614
$955$	0	0
$956$	−6930.40	−0.234461
$957$	11448.4	0.386704
$958$	−27036.1	−0.911791
$959$	−16284.6	−0.548341
$960$	0	0
$961$	−29216.4	−0.980712
$962$	15130.6	0.507100
$963$	−11762.6	−0.393609
$964$	−17766.9	−0.593604
$965$	0	0
$966$	764.469	0.0254621
$967$	25119.6	0.835360	0.417680	−	0.908594i	$-0.362843\pi$
$967$	25119.6	0.835360	0.417680	+	0.908594i	$0.362843\pi$
$968$	4269.57	0.141766
$969$	4226.82	0.140129
$970$	0	0
$971$	11624.9	0.384204	0.192102	−	0.981375i	$-0.438470\pi$
$971$	11624.9	0.384204	0.192102	+	0.981375i	$0.438470\pi$
$972$	−13337.8	−0.440136
$973$	−25644.8	−0.844947
$974$	29829.3	0.981308
$975$	0	0
$976$	−7656.78	−0.251114
$977$	15427.4	0.505187	0.252593	−	0.967573i	$-0.418717\pi$
$977$	15427.4	0.505187	0.252593	+	0.967573i	$0.418717\pi$
$978$	−1039.84	−0.0339985
$979$	−39890.4	−1.30225
$980$	0	0
$981$	4704.14	0.153101
$982$	−7413.99	−0.240927
$983$	42012.4	1.36316	0.681580	−	0.731744i	$-0.261293\pi$
$983$	42012.4	1.36316	0.681580	+	0.731744i	$0.261293\pi$
$984$	1848.25	0.0598782
$985$	0	0
$986$	6737.70	0.217619
$987$	−10102.3	−0.325794
$988$	−10460.2	−0.336826
$989$	11547.6	0.371278
$990$	0	0
$991$	30889.0	0.990132	0.495066	−	0.868855i	$-0.335144\pi$
$991$	30889.0	0.990132	0.495066	+	0.868855i	$0.335144\pi$
$992$	767.078	0.0245512
$993$	−2163.13	−0.0691286
$994$	−18625.3	−0.594324
$995$	0	0
$996$	6508.14	0.207046
$997$	−10011.2	−0.318011	−0.159005	−	0.987278i	$-0.550829\pi$
$997$	−10011.2	−0.318011	−0.159005	+	0.987278i	$0.550829\pi$
$998$	26073.9	0.827009
$999$	−27011.4	−0.855458

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.a.t.1.3	yes	5
5.2	odd	4		1150.4.b.p.599.8		10
5.3	odd	4		1150.4.b.p.599.3		10
5.4	even	2		1150.4.a.s.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1150.4.a.s.1.3	✓	5	5.4	even	2
1150.4.a.t.1.3	yes	5	1.1	even	1	trivial
1150.4.b.p.599.3		10	5.3	odd	4
1150.4.b.p.599.8		10	5.2	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -1.80744 q^{3} +4.00000 q^{4} -3.61488 q^{6} -9.19471 q^{7} +8.00000 q^{8} -23.7332 q^{9} +43.1821 q^{11} -7.22976 q^{12} -25.6824 q^{13} -18.3894 q^{14} +16.0000 q^{16} -22.9669 q^{17} -47.4663 q^{18} +101.823 q^{19} +16.6189 q^{21} +86.3643 q^{22} +23.0000 q^{23} -14.4595 q^{24} -51.3647 q^{26} +91.6972 q^{27} -36.7788 q^{28} -146.682 q^{29} +23.9712 q^{31} +32.0000 q^{32} -78.0491 q^{33} -45.9339 q^{34} -94.9326 q^{36} -294.572 q^{37} +203.646 q^{38} +46.4193 q^{39} -127.823 q^{41} +33.2378 q^{42} +502.072 q^{43} +172.729 q^{44} +46.0000 q^{46} -607.879 q^{47} -28.9190 q^{48} -258.457 q^{49} +41.5114 q^{51} -102.729 q^{52} -674.254 q^{53} +183.394 q^{54} -73.5577 q^{56} -184.039 q^{57} -293.365 q^{58} +14.8056 q^{59} -478.549 q^{61} +47.9424 q^{62} +218.220 q^{63} +64.0000 q^{64} -156.098 q^{66} -458.422 q^{67} -91.8678 q^{68} -41.5711 q^{69} +1012.83 q^{71} -189.865 q^{72} +750.521 q^{73} -589.144 q^{74} +407.292 q^{76} -397.047 q^{77} +92.8387 q^{78} -641.648 q^{79} +475.058 q^{81} -255.645 q^{82} -900.187 q^{83} +66.4756 q^{84} +1004.14 q^{86} +265.120 q^{87} +345.457 q^{88} -923.770 q^{89} +236.142 q^{91} +92.0000 q^{92} -43.3265 q^{93} -1215.76 q^{94} -57.8381 q^{96} -168.230 q^{97} -516.915 q^{98} -1024.85 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 10 q^{2} - 12 q^{3} + 20 q^{4} - 24 q^{6} + 24 q^{7} + 40 q^{8} + 11 q^{9} - 54 q^{11} - 48 q^{12} - 36 q^{13} + 48 q^{14} + 80 q^{16} - 132 q^{17} + 22 q^{18} - 50 q^{19} - 158 q^{21} - 108 q^{22}+ \cdots - 1740 q^{99}+O(q^{100}) \) 5 * q + 10 * q^2 - 12 * q^3 + 20 * q^4 - 24 * q^6 + 24 * q^7 + 40 * q^8 + 11 * q^9 - 54 * q^11 - 48 * q^12 - 36 * q^13 + 48 * q^14 + 80 * q^16 - 132 * q^17 + 22 * q^18 - 50 * q^19 - 158 * q^21 - 108 * q^22 + 115 * q^23 - 96 * q^24 - 72 * q^26 - 489 * q^27 + 96 * q^28 - 104 * q^29 - 342 * q^31 + 160 * q^32 + 348 * q^33 - 264 * q^34 + 44 * q^36 - 380 * q^37 - 100 * q^38 + 193 * q^39 + 2 * q^41 - 316 * q^42 + 114 * q^43 - 216 * q^44 + 230 * q^46 - 966 * q^47 - 192 * q^48 - 441 * q^49 + 1200 * q^51 - 144 * q^52 - 308 * q^53 - 978 * q^54 + 192 * q^56 - 1410 * q^57 - 208 * q^58 - 277 * q^59 + 482 * q^61 - 684 * q^62 + 222 * q^63 + 320 * q^64 + 696 * q^66 - 2044 * q^67 - 528 * q^68 - 276 * q^69 + 208 * q^71 + 88 * q^72 - 646 * q^73 - 760 * q^74 - 200 * q^76 - 2124 * q^77 + 386 * q^78 - 3444 * q^79 + 2933 * q^81 + 4 * q^82 - 1218 * q^83 - 632 * q^84 + 228 * q^86 - 103 * q^87 - 432 * q^88 + 140 * q^89 + 1976 * q^91 + 460 * q^92 - 473 * q^93 - 1932 * q^94 - 384 * q^96 - 2816 * q^97 - 882 * q^98 - 1740 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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