LMFDB - Embedded newform 1150.4.b.p.599.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1150,4,Mod(599,1150)]

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([1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1150.599");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$67.8521965066$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 77x^{8} + 1924x^{6} + 16594x^{4} + 33128x^{2} + 10201 $ x^10 + 77x^8 + 1924x^6 + 16594x^4 + 33128x^2 + 10201
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.8
Root			$-5.76842i$ of defining polynomial
Character	$\chi$	$=$	1150.599
Dual form			1150.4.b.p.599.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000i q^{2} +1.80744i q^{3} -4.00000 q^{4} -3.61488 q^{6} -9.19471i q^{7} -8.00000i q^{8} +23.7332 q^{9} +43.1821 q^{11} -7.22976i q^{12} +25.6824i q^{13} +18.3894 q^{14} +16.0000 q^{16} -22.9669i q^{17} +47.4663i q^{18} -101.823 q^{19} +16.6189 q^{21} +86.3643i q^{22} -23.0000i q^{23} +14.4595 q^{24} -51.3647 q^{26} +91.6972i q^{27} +36.7788i q^{28} +146.682 q^{29} +23.9712 q^{31} +32.0000i q^{32} +78.0491i q^{33} +45.9339 q^{34} -94.9326 q^{36} -294.572i q^{37} -203.646i q^{38} -46.4193 q^{39} -127.823 q^{41} +33.2378i q^{42} -502.072i q^{43} -172.729 q^{44} +46.0000 q^{46} -607.879i q^{47} +28.9190i q^{48} +258.457 q^{49} +41.5114 q^{51} -102.729i q^{52} +674.254i q^{53} -183.394 q^{54} -73.5577 q^{56} -184.039i q^{57} +293.365i q^{58} -14.8056 q^{59} -478.549 q^{61} +47.9424i q^{62} -218.220i q^{63} -64.0000 q^{64} -156.098 q^{66} -458.422i q^{67} +91.8678i q^{68} +41.5711 q^{69} +1012.83 q^{71} -189.865i q^{72} -750.521i q^{73} +589.144 q^{74} +407.292 q^{76} -397.047i q^{77} -92.8387i q^{78} +641.648 q^{79} +475.058 q^{81} -255.645i q^{82} +900.187i q^{83} -66.4756 q^{84} +1004.14 q^{86} +265.120i q^{87} -345.457i q^{88} +923.770 q^{89} +236.142 q^{91} +92.0000i q^{92} +43.3265i q^{93} +1215.76 q^{94} -57.8381 q^{96} -168.230i q^{97} +516.915i q^{98} +1024.85 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 40 q^{4} - 48 q^{6} - 22 q^{9} - 108 q^{11} - 96 q^{14} + 160 q^{16} + 100 q^{19} - 316 q^{21} + 192 q^{24} - 144 q^{26} + 208 q^{29} - 684 q^{31} + 528 q^{34} + 88 q^{36} - 386 q^{39} + 4 q^{41}+ \cdots + 3480 q^{99}+O(q^{100}) $ 10 * q - 40 * q^4 - 48 * q^6 - 22 * q^9 - 108 * q^11 - 96 * q^14 + 160 * q^16 + 100 * q^19 - 316 * q^21 + 192 * q^24 - 144 * q^26 + 208 * q^29 - 684 * q^31 + 528 * q^34 + 88 * q^36 - 386 * q^39 + 4 * q^41 + 432 * q^44 + 460 * q^46 + 882 * q^49 + 2400 * q^51 + 1956 * q^54 + 384 * q^56 + 554 * q^59 + 964 * q^61 - 640 * q^64 + 1392 * q^66 + 552 * q^69 + 416 * q^71 + 1520 * q^74 - 400 * q^76 + 6888 * q^79 + 5866 * q^81 + 1264 * q^84 + 456 * q^86 - 280 * q^89 + 3952 * q^91 + 3864 * q^94 - 768 * q^96 + 3480 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.00000i	0.707107i
$2$	2.00000i	0.707107i
$3$	1.80744i	0.347842i	0.984760	+	0.173921i	$0.0556437\pi$
$3$	1.80744i	0.347842i	−0.984760	+	0.173921i	$0.944356\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−3.61488	−0.245961
$7$	− 9.19471i	− 0.496468i	−0.968700	−	0.248234i	$-0.920150\pi$
$7$	− 9.19471i	− 0.496468i	0.968700	−	0.248234i	$-0.0798501\pi$
$8$	− 8.00000i	− 0.353553i
$9$	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	$0.298412\pi$
$12$	− 7.22976i	− 0.173921i
$13$	25.6824i	0.547924i	0.961741	+	0.273962i	$0.0883342\pi$
$13$	25.6824i	0.547924i	−0.961741	+	0.273962i	$0.911666\pi$
$14$	18.3894	0.351056
$15$	0	0
$16$	16.0000	0.250000
$17$	− 22.9669i	− 0.327665i	−0.986488	−	0.163832i	$-0.947614\pi$
$17$	− 22.9669i	− 0.327665i	0.986488	−	0.163832i	$-0.0523856\pi$
$18$	47.4663i	0.621551i
$19$	−101.823	−1.22946	−0.614732	−	0.788736i	$-0.710736\pi$
$19$	−101.823	−1.22946	−0.614732	+	0.788736i	$0.710736\pi$
$20$	0	0
$21$	16.6189	0.172692
$22$	86.3643i	0.836951i
$23$	− 23.0000i	− 0.208514i
$23$	− 23.0000i	− 0.208514i
$24$	14.4595	0.122981
$25$	0	0
$26$	−51.3647	−0.387440
$27$	91.6972i	0.653597i
$28$	36.7788i	0.248234i
$29$	146.682	0.939250	0.469625	−	0.882866i	$-0.344389\pi$
$29$	146.682	0.939250	0.469625	+	0.882866i	$0.344389\pi$
$30$	0	0
$31$	23.9712	0.138882	0.0694412	−	0.997586i	$-0.477878\pi$
$31$	23.9712	0.138882	0.0694412	+	0.997586i	$0.477878\pi$
$32$	32.0000i	0.176777i
$33$	78.0491i	0.411715i
$34$	45.9339	0.231694
$35$	0	0
$36$	−94.9326	−0.439503
$37$	− 294.572i	− 1.30885i	−0.756129	−	0.654423i	$-0.772912\pi$
$37$	− 294.572i	− 1.30885i	0.756129	−	0.654423i	$-0.227088\pi$
$38$	− 203.646i	− 0.869362i
$39$	−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	$0.578278\pi$
$42$	33.2378i	0.122112i
$43$	− 502.072i	− 1.78059i	−0.455388	−	0.890293i	$-0.650499\pi$
$43$	− 502.072i	− 1.78059i	0.455388	−	0.890293i	$-0.349501\pi$
$44$	−172.729	−0.591814
$45$	0	0
$46$	46.0000	0.147442
$47$	− 607.879i	− 1.88656i	−0.332002	−	0.943279i	$-0.607724\pi$
$47$	− 607.879i	− 1.88656i	0.332002	−	0.943279i	$-0.392276\pi$
$48$	28.9190i	0.0869605i
$49$	258.457	0.753520
$50$	0	0
$51$	41.5114	0.113976
$52$	− 102.729i	− 0.273962i
$53$	674.254i	1.74747i	0.486401	+	0.873735i	$0.338309\pi$
$53$	674.254i	1.74747i	−0.486401	+	0.873735i	$0.661691\pi$
$54$	−183.394	−0.462163
$55$	0	0
$56$	−73.5577	−0.175528
$57$	− 184.039i	− 0.427659i
$58$	293.365i	0.664150i
$59$	−14.8056	−0.0326700	−0.0163350	−	0.999867i	$-0.505200\pi$
$59$	−14.8056	−0.0326700	−0.0163350	+	0.999867i	$0.505200\pi$
$60$	0	0
$61$	−478.549	−1.00446	−0.502229	−	0.864735i	$-0.667487\pi$
$61$	−478.549	−1.00446	−0.502229	+	0.864735i	$0.667487\pi$
$62$	47.9424i	0.0982047i
$63$	− 218.220i	− 0.436398i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−156.098	−0.291127
$67$	− 458.422i	− 0.835898i	−0.908471	−	0.417949i	$-0.862749\pi$
$67$	− 458.422i	− 0.835898i	0.908471	−	0.417949i	$-0.137251\pi$
$68$	91.8678i	0.163832i
$69$	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	$0.178717\pi$
$72$	− 189.865i	− 0.310776i
$73$	− 750.521i	− 1.20331i	−0.798755	−	0.601657i	$-0.794508\pi$
$73$	− 750.521i	− 1.20331i	0.798755	−	0.601657i	$-0.205492\pi$
$74$	589.144	0.925494
$75$	0	0
$76$	407.292	0.614732
$77$	− 397.047i	− 0.587633i
$78$	− 92.8387i	− 0.134768i
$79$	641.648	0.913811	0.456905	−	0.889515i	$-0.348958\pi$
$79$	641.648	0.913811	0.456905	+	0.889515i	$0.348958\pi$
$80$	0	0
$81$	475.058	0.651657
$82$	− 255.645i	− 0.344284i
$83$	900.187i	1.19046i	0.803554	+	0.595231i	$0.202940\pi$
$83$	900.187i	1.19046i	−0.803554	+	0.595231i	$0.797060\pi$
$84$	−66.4756	−0.0863462
$85$	0	0
$86$	1004.14	1.25906
$87$	265.120i	0.326711i
$88$	− 345.457i	− 0.418476i
$89$	923.770	1.10022	0.550109	−	0.835093i	$-0.314586\pi$
$89$	923.770	1.10022	0.550109	+	0.835093i	$0.314586\pi$
$90$	0	0
$91$	236.142	0.272026
$92$	92.0000i	0.104257i
$93$	43.3265i	0.0483091i
$94$	1215.76	1.33400
$95$	0	0
$96$	−57.8381	−0.0614904
$97$	− 168.230i	− 0.176095i	−0.996116	−	0.0880473i	$-0.971937\pi$
$97$	− 168.230i	− 0.176095i	0.996116	−	0.0880473i	$-0.0280627\pi$
$98$	516.915i	0.532819i
$99$	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	$0.601003\pi$
$102$	83.0228i	0.0805929i
$103$	1300.63i	1.24422i	0.782929	+	0.622111i	$0.213725\pi$
$103$	1300.63i	1.24422i	−0.782929	+	0.622111i	$0.786275\pi$
$104$	205.459	0.193720
$105$	0	0
$106$	−1348.51	−1.23565
$107$	495.620i	0.447789i	0.974613	+	0.223894i	$0.0718770\pi$
$107$	495.620i	0.447789i	−0.974613	+	0.223894i	$0.928123\pi$
$108$	− 366.789i	− 0.326799i
$109$	198.210	0.174175	0.0870874	−	0.996201i	$-0.472244\pi$
$109$	198.210	0.174175	0.0870874	+	0.996201i	$0.472244\pi$
$110$	0	0
$111$	532.421	0.455272
$112$	− 147.115i	− 0.124117i
$113$	868.555i	0.723069i	0.932359	+	0.361534i	$0.117747\pi$
$113$	868.555i	0.723069i	−0.932359	+	0.361534i	$0.882253\pi$
$114$	368.078	0.302401
$115$	0	0
$116$	−586.730	−0.469625
$117$	609.524i	0.481628i
$118$	− 29.6113i	− 0.0231012i
$119$	−211.174	−0.162675
$120$	0	0
$121$	533.696	0.400974
$122$	− 957.098i	− 0.710259i
$123$	− 231.032i	− 0.169361i
$124$	−95.8848	−0.0694412
$125$	0	0
$126$	436.439	0.308580
$127$	− 2152.32i	− 1.50384i	−0.659257	−	0.751918i	$-0.729129\pi$
$127$	− 2152.32i	− 1.50384i	0.659257	−	0.751918i	$-0.270871\pi$
$128$	− 128.000i	− 0.0883883i
$129$	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	$0.304740\pi$
$132$	− 312.196i	− 0.205858i
$133$	936.233i	0.610389i
$134$	916.844	0.591069
$135$	0	0
$136$	−183.736	−0.115847
$137$	1771.09i	1.10448i	0.833684	+	0.552242i	$0.186227\pi$
$137$	1771.09i	1.10448i	−0.833684	+	0.552242i	$0.813773\pi$
$138$	83.1423i	0.0512865i
$139$	−2789.08	−1.70192	−0.850959	−	0.525232i	$-0.823979\pi$
$139$	−2789.08	−1.70192	−0.850959	+	0.525232i	$0.823979\pi$
$140$	0	0
$141$	1098.70	0.656224
$142$	2025.65i	1.19710i
$143$	1109.02i	0.648537i
$144$	379.731	0.219751
$145$	0	0
$146$	1501.04	0.850871
$147$	467.146i	0.262106i
$148$	1178.29i	0.654423i
$149$	2190.05	1.20413	0.602067	−	0.798445i	$-0.294344\pi$
$149$	2190.05	1.20413	0.602067	+	0.798445i	$0.294344\pi$
$150$	0	0
$151$	−1569.59	−0.845902	−0.422951	−	0.906153i	$-0.639006\pi$
$151$	−1569.59	−0.845902	−0.422951	+	0.906153i	$0.639006\pi$
$152$	814.584i	0.434681i
$153$	− 545.078i	− 0.288019i
$154$	794.094	0.415519
$155$	0	0
$156$	185.677	0.0952954
$157$	− 2633.86i	− 1.33888i	−0.742864	−	0.669442i	$-0.766533\pi$
$157$	− 2633.86i	− 1.33888i	0.742864	−	0.669442i	$-0.233467\pi$
$158$	1283.30i	0.646162i
$159$	−1218.67	−0.607844
$160$	0	0
$161$	−211.478	−0.103521
$162$	950.116i	0.460791i
$163$	− 287.657i	− 0.138227i	−0.997609	−	0.0691135i	$-0.977983\pi$
$163$	− 287.657i	− 0.138227i	0.997609	−	0.0691135i	$-0.0220171\pi$
$164$	511.290	0.243445
$165$	0	0
$166$	−1800.37	−0.841784
$167$	− 1295.54i	− 0.600310i	−0.953890	−	0.300155i	$-0.902962\pi$
$167$	− 1295.54i	− 0.600310i	0.953890	−	0.300155i	$-0.0970385\pi$
$168$	− 132.951i	− 0.0610560i
$169$	1537.42	0.699780
$170$	0	0
$171$	−2416.58	−1.08071
$172$	2008.29i	0.890293i
$173$	− 1313.46i	− 0.577230i	−0.957445	−	0.288615i	$-0.906805\pi$
$173$	− 1313.46i	− 0.577230i	0.957445	−	0.288615i	$-0.0931947\pi$
$174$	−530.240	−0.231019
$175$	0	0
$176$	690.914	0.295907
$177$	− 26.7603i	− 0.0113640i
$178$	1847.54i	0.777972i
$179$	−1916.37	−0.800204	−0.400102	−	0.916471i	$-0.631025\pi$
$179$	−1916.37	−0.800204	−0.400102	+	0.916471i	$0.631025\pi$
$180$	0	0
$181$	4443.32	1.82469	0.912345	−	0.409421i	$-0.134269\pi$
$181$	4443.32	1.82469	0.912345	+	0.409421i	$0.134269\pi$
$182$	472.284i	0.192352i
$183$	− 864.949i	− 0.349393i
$184$	−184.000	−0.0737210
$185$	0	0
$186$	−86.6530	−0.0341597
$187$	− 991.762i	− 0.387833i
$188$	2431.51i	0.943279i
$189$	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	$0.382584\pi$
$192$	− 115.676i	− 0.0434803i
$193$	3417.60i	1.27463i	0.770602	+	0.637317i	$0.219956\pi$
$193$	3417.60i	1.27463i	−0.770602	+	0.637317i	$0.780044\pi$
$194$	336.460	0.124518
$195$	0	0
$196$	−1033.83	−0.376760
$197$	− 1609.57i	− 0.582117i	−0.956705	−	0.291059i	$-0.905993\pi$
$197$	− 1609.57i	− 0.582117i	0.956705	−	0.291059i	$-0.0940075\pi$
$198$	2049.70i	0.735685i
$199$	4518.91	1.60973	0.804866	−	0.593456i	$-0.202237\pi$
$199$	4518.91	1.60973	0.804866	+	0.593456i	$0.202237\pi$
$200$	0	0
$201$	828.570	0.290760
$202$	− 1266.82i	− 0.441253i
$203$	− 1348.70i	− 0.466307i
$204$	−166.046	−0.0569878
$205$	0	0
$206$	−2601.26	−0.879798
$207$	− 545.863i	− 0.183285i
$208$	410.918i	0.136981i
$209$	−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	$0.155969\pi$
$212$	− 2697.02i	− 0.873735i
$213$	1830.62i	0.588883i
$214$	−991.240	−0.316634
$215$	0	0
$216$	733.577	0.231082
$217$	− 220.408i	− 0.0689506i
$218$	396.420i	0.123160i
$219$	1356.52	0.418563
$220$	0	0
$221$	589.846	0.179535
$222$	1064.84i	0.321926i
$223$	5812.74i	1.74552i	0.488154	+	0.872758i	$0.337671\pi$
$223$	5812.74i	1.74552i	−0.488154	+	0.872758i	$0.662329\pi$
$224$	294.231	0.0877639
$225$	0	0
$226$	−1737.11	−0.511287
$227$	− 2033.30i	− 0.594514i	−0.954798	−	0.297257i	$-0.903928\pi$
$227$	− 2033.30i	− 0.594514i	0.954798	−	0.297257i	$-0.0960717\pi$
$228$	736.156i	0.213830i
$229$	435.903	0.125787	0.0628936	−	0.998020i	$-0.479967\pi$
$229$	435.903	0.125787	0.0628936	+	0.998020i	$0.479967\pi$
$230$	0	0
$231$	717.639	0.204403
$232$	− 1173.46i	− 0.332075i
$233$	− 1582.05i	− 0.444822i	−0.974953	−	0.222411i	$-0.928607\pi$
$233$	− 1582.05i	− 0.444822i	0.974953	−	0.222411i	$-0.0713927\pi$
$234$	−1219.05	−0.340562
$235$	0	0
$236$	59.2226	0.0163350
$237$	1159.74i	0.317862i
$238$	− 422.349i	− 0.115029i
$239$	1732.60	0.468923	0.234461	−	0.972125i	$-0.424667\pi$
$239$	1732.60	0.468923	0.234461	+	0.972125i	$0.424667\pi$
$240$	0	0
$241$	−4441.73	−1.18721	−0.593604	−	0.804757i	$-0.702296\pi$
$241$	−4441.73	−1.18721	−0.593604	+	0.804757i	$0.702296\pi$
$242$	1067.39i	0.283531i
$243$	3334.46i	0.880271i
$244$	1914.20	0.502229
$245$	0	0
$246$	462.063	0.119756
$247$	− 2615.06i	− 0.673652i
$248$	− 191.770i	− 0.0491023i
$249$	−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	$0.298028\pi$
$252$	872.878i	0.218199i
$253$	− 993.189i	− 0.246803i
$254$	4304.63	1.06337
$255$	0	0
$256$	256.000	0.0625000
$257$	− 4229.41i	− 1.02655i	−0.858224	−	0.513275i	$-0.828432\pi$
$257$	− 4229.41i	− 1.02655i	0.858224	−	0.513275i	$-0.171568\pi$
$258$	1814.93i	0.437956i
$259$	−2708.50	−0.649800
$260$	0	0
$261$	3481.24	0.825606
$262$	3452.57i	0.814125i
$263$	4848.80i	1.13684i	0.822737	+	0.568422i	$0.192446\pi$
$263$	4848.80i	1.13684i	−0.822737	+	0.568422i	$0.807554\pi$
$264$	624.393	0.145563
$265$	0	0
$266$	−1872.47	−0.431610
$267$	1669.66i	0.382702i
$268$	1833.69i	0.417949i
$269$	−4603.64	−1.04345	−0.521726	−	0.853113i	$-0.674712\pi$
$269$	−4603.64	−1.04345	−0.521726	+	0.853113i	$0.674712\pi$
$270$	0	0
$271$	−1521.31	−0.341008	−0.170504	−	0.985357i	$-0.554540\pi$
$271$	−1521.31	−0.341008	−0.170504	+	0.985357i	$0.554540\pi$
$272$	− 367.471i	− 0.0819162i
$273$	426.812i	0.0946222i
$274$	−3542.18	−0.780988
$275$	0	0
$276$	−166.285	−0.0362650
$277$	3293.74i	0.714446i	0.934019	+	0.357223i	$0.116276\pi$
$277$	3293.74i	0.714446i	−0.934019	+	0.357223i	$0.883724\pi$
$278$	− 5578.16i	− 1.20344i
$279$	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	$0.110599\pi$
$282$	2197.41i	0.464020i
$283$	− 5470.44i	− 1.14906i	−0.818483	−	0.574530i	$-0.805185\pi$
$283$	− 5470.44i	− 1.14906i	0.818483	−	0.574530i	$-0.194815\pi$
$284$	−4051.30	−0.846481
$285$	0	0
$286$	−2218.04	−0.458585
$287$	1175.29i	0.241726i
$288$	759.461i	0.155388i
$289$	4385.52	0.892636
$290$	0	0
$291$	304.066	0.0612531
$292$	3002.09i	0.601657i
$293$	629.599i	0.125534i	0.998028	+	0.0627672i	$0.0199926\pi$
$293$	629.599i	0.125534i	−0.998028	+	0.0627672i	$0.980007\pi$
$294$	−934.292	−0.185337
$295$	0	0
$296$	−2356.57	−0.462747
$297$	3959.68i	0.773616i
$298$	4380.10i	0.851452i
$299$	590.694	0.114250
$300$	0	0
$301$	−4616.40	−0.884004
$302$	− 3139.17i	− 0.598143i
$303$	− 1144.85i	− 0.217062i
$304$	−1629.17	−0.307366
$305$	0	0
$306$	1090.16	0.203660
$307$	791.929i	0.147224i	0.997287	+	0.0736120i	$0.0234526\pi$
$307$	791.929i	0.147224i	−0.997287	+	0.0736120i	$0.976547\pi$
$308$	1588.19i	0.293816i
$309$	−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	$0.438408\pi$
$312$	371.355i	0.0673840i
$313$	− 207.600i	− 0.0374896i	−0.999824	−	0.0187448i	$-0.994033\pi$
$313$	− 207.600i	− 0.0374896i	0.999824	−	0.0187448i	$-0.00596701\pi$
$314$	5267.72	0.946734
$315$	0	0
$316$	−2566.59	−0.456905
$317$	4013.96i	0.711188i	0.934640	+	0.355594i	$0.115721\pi$
$317$	4013.96i	0.711188i	−0.934640	+	0.355594i	$0.884279\pi$
$318$	− 2437.35i	− 0.429811i
$319$	6334.06	1.11172
$320$	0	0
$321$	−895.803	−0.155760
$322$	− 422.957i	− 0.0732002i
$323$	2338.56i	0.402852i
$324$	−1900.23	−0.325829
$325$	0	0
$326$	575.313	0.0977413
$327$	358.252i	0.0605853i
$328$	1022.58i	0.172142i
$329$	−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	$0.468318\pi$
$332$	− 3600.75i	− 0.595231i
$333$	− 6991.12i	− 1.15048i
$334$	2591.08	0.424484
$335$	0	0
$336$	265.902	0.0431731
$337$	− 3245.42i	− 0.524597i	−0.964987	−	0.262299i	$-0.915519\pi$
$337$	− 3245.42i	− 0.524597i	0.964987	−	0.262299i	$-0.0844806\pi$
$338$	3074.83i	0.494819i
$339$	−1569.86	−0.251514
$340$	0	0
$341$	1035.13	0.164385
$342$	− 4833.16i	− 0.764174i
$343$	− 5530.23i	− 0.870566i
$344$	−4016.57	−0.629532
$345$	0	0
$346$	2626.93	0.408163
$347$	− 10510.0i	− 1.62596i	−0.582290	−	0.812981i	$-0.697843\pi$
$347$	− 10510.0i	− 1.62596i	0.582290	−	0.812981i	$-0.302157\pi$
$348$	− 1060.48i	− 0.163355i
$349$	−6150.02	−0.943276	−0.471638	−	0.881792i	$-0.656337\pi$
$349$	−6150.02	−0.943276	−0.471638	+	0.881792i	$0.656337\pi$
$350$	0	0
$351$	−2355.00	−0.358121
$352$	1381.83i	0.209238i
$353$	1123.35i	0.169376i	0.996408	+	0.0846882i	$0.0269894\pi$
$353$	1123.35i	0.169376i	−0.996408	+	0.0846882i	$0.973011\pi$
$354$	53.5206	0.00803556
$355$	0	0
$356$	−3695.08	−0.550109
$357$	− 381.685i	− 0.0565852i
$358$	− 3832.75i	− 0.565829i
$359$	−2209.46	−0.324821	−0.162411	−	0.986723i	$-0.551927\pi$
$359$	−2209.46	−0.324821	−0.162411	+	0.986723i	$0.551927\pi$
$360$	0	0
$361$	3508.93	0.511581
$362$	8886.63i	1.29025i
$363$	964.624i	0.139476i
$364$	−944.568	−0.136013
$365$	0	0
$366$	1729.90	0.247058
$367$	9017.63i	1.28261i	0.767287	+	0.641303i	$0.221606\pi$
$367$	9017.63i	1.28261i	−0.767287	+	0.641303i	$0.778394\pi$
$368$	− 368.000i	− 0.0521286i
$369$	−3033.63	−0.427980
$370$	0	0
$371$	6199.57	0.867563
$372$	− 173.306i	− 0.0241546i
$373$	2000.52i	0.277702i	0.990313	+	0.138851i	$0.0443410\pi$
$373$	2000.52i	0.277702i	−0.990313	+	0.138851i	$0.955659\pi$
$374$	1983.52	0.274239
$375$	0	0
$376$	−4863.03	−0.666999
$377$	3767.15i	0.514637i
$378$	1686.26i	0.229449i
$379$	−5404.84	−0.732528	−0.366264	−	0.930511i	$-0.619363\pi$
$379$	−5404.84	−0.732528	−0.366264	+	0.930511i	$0.619363\pi$
$380$	0	0
$381$	3890.18	0.523097
$382$	3807.09i	0.509915i
$383$	− 9355.83i	− 1.24820i	−0.781345	−	0.624100i	$-0.785466\pi$
$383$	− 9355.83i	− 1.24820i	0.781345	−	0.624100i	$-0.214534\pi$
$384$	231.352	0.0307452
$385$	0	0
$386$	−6835.20	−0.901302
$387$	− 11915.7i	− 1.56515i
$388$	672.920i	0.0880473i
$389$	−7771.25	−1.01290	−0.506450	−	0.862269i	$-0.669042\pi$
$389$	−7771.25	−1.01290	−0.506450	+	0.862269i	$0.669042\pi$
$390$	0	0
$391$	−528.240	−0.0683229
$392$	− 2067.66i	− 0.266409i
$393$	3120.16i	0.400487i
$394$	3219.14	0.411619
$395$	0	0
$396$	−4099.39	−0.520208
$397$	2509.29i	0.317223i	0.987341	+	0.158612i	$0.0507018\pi$
$397$	2509.29i	0.317223i	−0.987341	+	0.158612i	$0.949298\pi$
$398$	9037.81i	1.13825i
$399$	−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	$0.470173\pi$
$402$	1657.14i	0.205599i
$403$	615.637i	0.0760969i
$404$	2533.64	0.312013
$405$	0	0
$406$	2697.41	0.329729
$407$	− 12720.2i	− 1.54919i
$408$	− 332.091i	− 0.0402965i
$409$	−6281.54	−0.759419	−0.379709	−	0.925106i	$-0.623976\pi$
$409$	−6281.54	−0.759419	−0.379709	+	0.925106i	$0.623976\pi$
$410$	0	0
$411$	−3201.14	−0.384186
$412$	− 5202.52i	− 0.622111i
$413$	136.134i	0.0162196i
$414$	1091.73	0.129602
$415$	0	0
$416$	−821.836	−0.0968601
$417$	− 5041.09i	− 0.591999i
$418$	− 8793.87i	− 1.02900i
$419$	5437.48	0.633982	0.316991	−	0.948429i	$-0.397328\pi$
$419$	5437.48	0.633982	0.316991	+	0.948429i	$0.397328\pi$
$420$	0	0
$421$	8424.12	0.975217	0.487608	−	0.873062i	$-0.337869\pi$
$421$	8424.12	0.975217	0.487608	+	0.873062i	$0.337869\pi$
$422$	10817.3i	1.24781i
$423$	− 14426.9i	− 1.65830i
$424$	5394.03	0.617824
$425$	0	0
$426$	−3661.24	−0.416403
$427$	4400.12i	0.498681i
$428$	− 1982.48i	− 0.223894i
$429$	−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	$0.540668\pi$
$432$	1467.15i	0.163399i
$433$	− 9676.90i	− 1.07400i	−0.843582	−	0.537000i	$-0.819557\pi$
$433$	− 9676.90i	− 1.07400i	0.843582	−	0.537000i	$-0.180443\pi$
$434$	440.816	0.0487554
$435$	0	0
$436$	−792.839	−0.0870874
$437$	2341.93i	0.256361i
$438$	2713.04i	0.295969i
$439$	10267.6	1.11628	0.558138	−	0.829748i	$-0.311516\pi$
$439$	10267.6	1.11628	0.558138	+	0.829748i	$0.311516\pi$
$440$	0	0
$441$	6134.01	0.662348
$442$	1179.69i	0.126951i
$443$	2947.70i	0.316139i	0.987428	+	0.158069i	$0.0505269\pi$
$443$	2947.70i	0.316139i	−0.987428	+	0.158069i	$0.949473\pi$
$444$	−2129.68	−0.227636
$445$	0	0
$446$	−11625.5	−1.23427
$447$	3958.39i	0.418849i
$448$	588.462i	0.0620585i
$449$	14322.2	1.50536	0.752682	−	0.658385i	$-0.228760\pi$
$449$	14322.2	1.50536	0.752682	+	0.658385i	$0.228760\pi$
$450$	0	0
$451$	−5519.65	−0.576297
$452$	− 3474.22i	− 0.361534i
$453$	− 2836.94i	− 0.294240i
$454$	4066.59	0.420385
$455$	0	0
$456$	−1472.31	−0.151200
$457$	13404.4i	1.37206i	0.727574	+	0.686029i	$0.240648\pi$
$457$	13404.4i	1.37206i	−0.727574	+	0.686029i	$0.759352\pi$
$458$	871.805i	0.0889449i
$459$	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	$0.826532\pi$
$462$	1435.28i	0.144535i
$463$	− 13109.3i	− 1.31585i	−0.753083	−	0.657925i	$-0.771434\pi$
$463$	− 13109.3i	− 1.31585i	0.753083	−	0.657925i	$-0.228566\pi$
$464$	2346.92	0.234812
$465$	0	0
$466$	3164.10	0.314537
$467$	− 4676.02i	− 0.463341i	−0.972794	−	0.231671i	$-0.925581\pi$
$467$	− 4676.02i	− 0.463341i	0.972794	−	0.231671i	$-0.0744191\pi$
$468$	− 2438.09i	− 0.240814i
$469$	−4215.06	−0.414996
$470$	0	0
$471$	4760.54	0.465720
$472$	118.445i	0.0115506i
$473$	− 21680.5i	− 2.10755i
$474$	−2319.48	−0.224762
$475$	0	0
$476$	844.698	0.0813375
$477$	16002.2i	1.53604i
$478$	3465.20i	0.331579i
$479$	13518.0	1.28947	0.644734	−	0.764407i	$-0.276968\pi$
$479$	13518.0	1.28947	0.644734	+	0.764407i	$0.276968\pi$
$480$	0	0
$481$	7565.30	0.717148
$482$	− 8883.47i	− 0.839483i
$483$	− 382.234i	− 0.0360088i
$484$	−2134.79	−0.200487
$485$	0	0
$486$	−6668.92	−0.622446
$487$	14914.7i	1.38778i	0.720082	+	0.693889i	$0.244104\pi$
$487$	14914.7i	1.38778i	−0.720082	+	0.693889i	$0.755896\pi$
$488$	3828.39i	0.355129i
$489$	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	$0.554493\pi$
$492$	924.126i	0.0846806i
$493$	− 3368.85i	− 0.307759i
$494$	5230.11	0.476344
$495$	0	0
$496$	383.539	0.0347206
$497$	− 9312.64i	− 0.840501i
$498$	− 3254.07i	− 0.292808i
$499$	−13037.0	−1.16957	−0.584784	−	0.811189i	$-0.698821\pi$
$499$	−13037.0	−1.16957	−0.584784	+	0.811189i	$0.698821\pi$
$500$	0	0
$501$	2341.61	0.208813
$502$	9429.08i	0.838328i
$503$	− 2976.34i	− 0.263834i	−0.991261	−	0.131917i	$-0.957887\pi$
$503$	− 2976.34i	− 0.263834i	0.991261	−	0.131917i	$-0.0421132\pi$
$504$	−1745.76	−0.154290
$505$	0	0
$506$	1986.38	0.174516
$507$	2778.79i	0.243413i
$508$	8609.26i	0.751918i
$509$	−20818.7	−1.81292	−0.906458	−	0.422297i	$-0.861224\pi$
$509$	−20818.7	−1.81292	−0.906458	+	0.422297i	$0.861224\pi$
$510$	0	0
$511$	−6900.83	−0.597406
$512$	512.000i	0.0441942i
$513$	− 9336.88i	− 0.803574i
$514$	8458.82	0.725881
$515$	0	0
$516$	−3629.86	−0.309681
$517$	− 26249.5i	− 2.23298i
$518$	− 5417.00i	− 0.459478i
$519$	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	$0.644641\pi$
$522$	6962.48i	0.583792i
$523$	− 22364.9i	− 1.86988i	−0.354801	−	0.934942i	$-0.615451\pi$
$523$	− 22364.9i	− 1.86988i	0.354801	−	0.934942i	$-0.384549\pi$
$524$	−6905.15	−0.575673
$525$	0	0
$526$	−9697.60	−0.803870
$527$	− 550.545i	− 0.0455069i
$528$	1248.79i	0.102929i
$529$	−529.000	−0.0434783
$530$	0	0
$531$	−351.385	−0.0287171
$532$	− 3744.93i	− 0.305194i
$533$	− 3282.78i	− 0.266779i
$534$	−3339.32	−0.270611
$535$	0	0
$536$	−3667.38	−0.295535
$537$	− 3463.73i	− 0.278344i
$538$	− 9207.27i	− 0.737832i
$539$	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	$0.851086\pi$
$542$	− 3042.63i	− 0.241129i
$543$	8031.03i	0.634704i
$544$	734.942	0.0579235
$545$	0	0
$546$	−853.625	−0.0669080
$547$	4367.35i	0.341379i	0.985325	+	0.170690i	$0.0545996\pi$
$547$	4367.35i	0.341379i	−0.985325	+	0.170690i	$0.945400\pi$
$548$	− 7084.35i	− 0.552242i
$549$	−11357.5	−0.882924
$550$	0	0
$551$	−14935.7	−1.15477
$552$	− 332.569i	− 0.0256433i
$553$	− 5899.77i	− 0.453677i
$554$	−6587.47	−0.505189
$555$	0	0
$556$	11156.3	0.850959
$557$	− 13805.9i	− 1.05022i	−0.851034	−	0.525110i	$-0.824024\pi$
$557$	− 13805.9i	− 1.05022i	0.851034	−	0.525110i	$-0.175976\pi$
$558$	1137.82i	0.0863225i
$559$	12894.4	0.975625
$560$	0	0
$561$	1792.55	0.134905
$562$	17715.7i	1.32970i
$563$	− 8149.77i	− 0.610075i	−0.952340	−	0.305037i	$-0.901331\pi$
$563$	− 8149.77i	− 0.610075i	0.952340	−	0.305037i	$-0.0986689\pi$
$564$	−4394.82	−0.328112
$565$	0	0
$566$	10940.9	0.812508
$567$	− 4368.02i	− 0.323527i
$568$	− 8102.60i	− 0.598552i
$569$	11539.9	0.850226	0.425113	−	0.905140i	$-0.360234\pi$
$569$	11539.9	0.850226	0.425113	+	0.905140i	$0.360234\pi$
$570$	0	0
$571$	−2436.53	−0.178573	−0.0892867	−	0.996006i	$-0.528459\pi$
$571$	−2436.53	−0.178573	−0.0892867	+	0.996006i	$0.528459\pi$
$572$	− 4436.08i	− 0.324269i
$573$	3440.54i	0.250839i
$574$	−2350.58	−0.170926
$575$	0	0
$576$	−1518.92	−0.109876
$577$	− 6965.59i	− 0.502567i	−0.967913	−	0.251284i	$-0.919147\pi$
$577$	− 6965.59i	− 0.502567i	0.967913	−	0.251284i	$-0.0808527\pi$
$578$	8771.04i	0.631189i
$579$	−6177.11	−0.443371
$580$	0	0
$581$	8276.96	0.591026
$582$	608.132i	0.0433125i
$583$	29115.7i	2.06835i
$584$	−6004.17	−0.425435
$585$	0	0
$586$	−1259.20	−0.0887662
$587$	3080.79i	0.216623i	0.994117	+	0.108311i	$0.0345444\pi$
$587$	3080.79i	0.216623i	−0.994117	+	0.108311i	$0.965456\pi$
$588$	− 1868.58i	− 0.131053i
$589$	−2440.82	−0.170751
$590$	0	0
$591$	2909.20	0.202485
$592$	− 4713.15i	− 0.327211i
$593$	13741.0i	0.951563i	0.879564	+	0.475782i	$0.157835\pi$
$593$	13741.0i	0.951563i	−0.879564	+	0.475782i	$0.842165\pi$
$594$	−7919.36	−0.547029
$595$	0	0
$596$	−8760.21	−0.602067
$597$	8167.65i	0.559933i
$598$	1181.39i	0.0807869i
$599$	−24131.0	−1.64602	−0.823011	−	0.568026i	$-0.807707\pi$
$599$	−24131.0	−1.64602	−0.823011	+	0.568026i	$0.807707\pi$
$600$	0	0
$601$	−18108.3	−1.22904	−0.614520	−	0.788901i	$-0.710650\pi$
$601$	−18108.3	−1.22904	−0.614520	+	0.788901i	$0.710650\pi$
$602$	− 9232.81i	− 0.625085i
$603$	− 10879.8i	− 0.734759i
$604$	6278.35	0.422951
$605$	0	0
$606$	2289.70	0.153486
$607$	10257.1i	0.685873i	0.939359	+	0.342936i	$0.111422\pi$
$607$	10257.1i	0.685873i	−0.939359	+	0.342936i	$0.888578\pi$
$608$	− 3258.34i	− 0.217340i
$609$	2437.70	0.162201
$610$	0	0
$611$	15611.8	1.03369
$612$	2180.31i	0.144010i
$613$	2710.49i	0.178590i	0.996005	+	0.0892951i	$0.0284614\pi$
$613$	2710.49i	0.178590i	−0.996005	+	0.0892951i	$0.971539\pi$
$614$	−1583.86	−0.104103
$615$	0	0
$616$	−3176.38	−0.207760
$617$	20517.7i	1.33875i	0.742923	+	0.669377i	$0.233439\pi$
$617$	20517.7i	1.33875i	−0.742923	+	0.669377i	$0.766561\pi$
$618$	− 4701.62i	− 0.306031i
$619$	5220.67	0.338992	0.169496	−	0.985531i	$-0.445786\pi$
$619$	5220.67	0.338992	0.169496	+	0.985531i	$0.445786\pi$
$620$	0	0
$621$	2109.03	0.136284
$622$	4218.52i	0.271941i
$623$	− 8493.80i	− 0.546223i
$624$	−742.710	−0.0476477
$625$	0	0
$626$	415.200	0.0265092
$627$	− 7947.20i	− 0.506189i
$628$	10535.4i	0.669442i
$629$	−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	$0.705837\pi$
$632$	− 5133.18i	− 0.323081i
$633$	9775.80i	0.613828i
$634$	−8027.93	−0.502886
$635$	0	0
$636$	4874.70	0.303922
$637$	6637.79i	0.412871i
$638$	12668.1i	0.786106i
$639$	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	$0.346108\pi$
$642$	− 1791.61i	− 0.110139i
$643$	− 1568.01i	− 0.0961687i	−0.998843	−	0.0480843i	$-0.984688\pi$
$643$	− 1568.01i	− 0.0961687i	0.998843	−	0.0480843i	$-0.0153116\pi$
$644$	845.913	0.0517603
$645$	0	0
$646$	−4677.13	−0.284859
$647$	− 15035.4i	− 0.913603i	−0.889569	−	0.456801i	$-0.848995\pi$
$647$	− 15035.4i	− 0.913603i	0.889569	−	0.456801i	$-0.151005\pi$
$648$	− 3800.47i	− 0.230396i
$649$	−639.339	−0.0386691
$650$	0	0
$651$	398.375	0.0239839
$652$	1150.63i	0.0691135i
$653$	22403.7i	1.34261i	0.741180	+	0.671307i	$0.234267\pi$
$653$	22403.7i	1.34261i	−0.741180	+	0.671307i	$0.765733\pi$
$654$	−716.505	−0.0428403
$655$	0	0
$656$	−2045.16	−0.121723
$657$	− 17812.2i	− 1.05772i
$658$	− 11178.5i	− 0.662287i
$659$	−890.918	−0.0526635	−0.0263317	−	0.999653i	$-0.508383\pi$
$659$	−890.918	−0.0526635	−0.0263317	+	0.999653i	$0.508383\pi$
$660$	0	0
$661$	−28636.0	−1.68504	−0.842519	−	0.538667i	$-0.818928\pi$
$661$	−28636.0	−1.68504	−0.842519	+	0.538667i	$0.818928\pi$
$662$	2393.58i	0.140527i
$663$	1066.11i	0.0624499i
$664$	7201.50	0.420892
$665$	0	0
$666$	13982.2	0.813515
$667$	− 3373.70i	− 0.195847i
$668$	5182.16i	0.300155i
$669$	−10506.2	−0.607164
$670$	0	0
$671$	−20664.8	−1.18890
$672$	531.805i	0.0305280i
$673$	− 15056.8i	− 0.862405i	−0.902255	−	0.431203i	$-0.858089\pi$
$673$	− 15056.8i	− 0.862405i	0.902255	−	0.431203i	$-0.141911\pi$
$674$	6490.84	0.370946
$675$	0	0
$676$	−6149.66	−0.349890
$677$	− 1612.72i	− 0.0915539i	−0.998952	−	0.0457770i	$-0.985424\pi$
$677$	− 1612.72i	− 0.0915539i	0.998952	−	0.0457770i	$-0.0145764\pi$
$678$	− 3139.72i	− 0.177847i
$679$	−1546.83	−0.0874253
$680$	0	0
$681$	3675.06	0.206797
$682$	2070.25i	0.116238i
$683$	− 20264.3i	− 1.13527i	−0.823279	−	0.567637i	$-0.807858\pi$
$683$	− 20264.3i	− 1.13527i	0.823279	−	0.567637i	$-0.192142\pi$
$684$	9666.33	0.540353
$685$	0	0
$686$	11060.5	0.615583
$687$	787.868i	0.0437540i
$688$	− 8033.15i	− 0.445147i
$689$	−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	$0.567343\pi$
$692$	5253.85i	0.288615i
$693$	− 9423.19i	− 0.516533i
$694$	21020.1	1.14973
$695$	0	0
$696$	2120.96	0.115510
$697$	2935.69i	0.159537i
$698$	− 12300.0i	− 0.666997i
$699$	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	$0.336823\pi$
$702$	− 4710.00i	− 0.253230i
$703$	29994.2i	1.60918i
$704$	−2763.66	−0.147953
$705$	0	0
$706$	−2246.70	−0.119767
$707$	5824.02i	0.309809i
$708$	107.041i	0.00568200i
$709$	−14793.4	−0.783607	−0.391804	−	0.920049i	$-0.628149\pi$
$709$	−14793.4	−0.783607	−0.391804	+	0.920049i	$0.628149\pi$
$710$	0	0
$711$	15228.3	0.803245
$712$	− 7390.16i	− 0.388986i
$713$	− 551.337i	− 0.0289590i
$714$	763.370	0.0400118
$715$	0	0
$716$	7665.49	0.400102
$717$	3131.57i	0.163111i
$718$	− 4418.92i	− 0.229683i
$719$	−32215.9	−1.67100	−0.835500	−	0.549490i	$-0.814822\pi$
$719$	−32215.9	−1.67100	−0.835500	+	0.549490i	$0.814822\pi$
$720$	0	0
$721$	11958.9	0.617716
$722$	7017.86i	0.361742i
$723$	− 8028.17i	− 0.412961i
$724$	−17773.3	−0.912345
$725$	0	0
$726$	−1929.25	−0.0986242
$727$	− 1853.37i	− 0.0945497i	−0.998882	−	0.0472748i	$-0.984946\pi$
$727$	− 1853.37i	− 0.0945497i	0.998882	−	0.0472748i	$-0.0150537\pi$
$728$	− 1889.14i	− 0.0961759i
$729$	6799.73	0.345462
$730$	0	0
$731$	−11531.1	−0.583436
$732$	3459.80i	0.174696i
$733$	6958.47i	0.350637i	0.984512	+	0.175319i	$0.0560955\pi$
$733$	6958.47i	0.350637i	−0.984512	+	0.175319i	$0.943904\pi$
$734$	−18035.3	−0.906940
$735$	0	0
$736$	736.000	0.0368605
$737$	− 19795.6i	− 0.989392i
$738$	− 6067.26i	− 0.302628i
$739$	−6469.54	−0.322038	−0.161019	−	0.986951i	$-0.551478\pi$
$739$	−6469.54	−0.322038	−0.161019	+	0.986951i	$0.551478\pi$
$740$	0	0
$741$	4726.56	0.234325
$742$	12399.1i	0.613460i
$743$	6992.55i	0.345265i	0.984986	+	0.172632i	$0.0552273\pi$
$743$	6992.55i	0.345265i	−0.984986	+	0.172632i	$0.944773\pi$
$744$	346.612	0.0170799
$745$	0	0
$746$	−4001.04	−0.196365
$747$	21364.3i	1.04642i
$748$	3967.05i	0.193917i
$749$	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	$0.296148\pi$
$752$	− 9726.06i	− 0.471639i
$753$	8521.25i	0.412393i
$754$	−7534.30	−0.363903
$755$	0	0
$756$	−3372.52	−0.162245
$757$	35778.4i	1.71782i	0.512128	+	0.858909i	$0.328857\pi$
$757$	35778.4i	1.71782i	−0.512128	+	0.858909i	$0.671143\pi$
$758$	− 10809.7i	− 0.517975i
$759$	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	$0.882988\pi$
$762$	7780.37i	0.369886i
$763$	− 1822.48i	− 0.0864722i
$764$	−7614.18	−0.360564
$765$	0	0
$766$	18711.7	0.882610
$767$	− 380.244i	− 0.0179007i
$768$	462.705i	0.0217401i
$769$	−9455.39	−0.443394	−0.221697	−	0.975116i	$-0.571160\pi$
$769$	−9455.39	−0.443394	−0.221697	+	0.975116i	$0.571160\pi$
$770$	0	0
$771$	7644.41	0.357077
$772$	− 13670.4i	− 0.637317i
$773$	5683.75i	0.264464i	0.991219	+	0.132232i	$0.0422143\pi$
$773$	5683.75i	0.264464i	−0.991219	+	0.132232i	$0.957786\pi$
$774$	23831.5	1.10673
$775$	0	0
$776$	−1345.84	−0.0622588
$777$	− 4895.46i	− 0.226028i
$778$	− 15542.5i	− 0.716228i
$779$	13015.3	0.598615
$780$	0	0
$781$	43736.0	2.00384
$782$	− 1056.48i	− 0.0483116i
$783$	13450.4i	0.613891i
$784$	4135.32	0.188380
$785$	0	0
$786$	−6240.32	−0.283187
$787$	− 18500.7i	− 0.837966i	−0.907994	−	0.418983i	$-0.862387\pi$
$787$	− 18500.7i	− 0.837966i	0.907994	−	0.418983i	$-0.137613\pi$
$788$	6438.28i	0.291059i
$789$	−8763.91	−0.395442
$790$	0	0
$791$	7986.11	0.358980
$792$	− 8198.79i	− 0.367842i
$793$	− 12290.3i	− 0.550366i
$794$	−5018.58	−0.224311
$795$	0	0
$796$	−18075.6	−0.804866
$797$	30464.4i	1.35396i	0.736002	+	0.676979i	$0.236711\pi$
$797$	30464.4i	1.35396i	−0.736002	+	0.676979i	$0.763289\pi$
$798$	− 3384.37i	− 0.150132i
$799$	−13961.1	−0.618159
$800$	0	0
$801$	21924.0	0.967099
$802$	3005.42i	0.132325i
$803$	− 32409.1i	− 1.42427i
$804$	−3314.28	−0.145380
$805$	0	0
$806$	−1231.27	−0.0538087
$807$	− 8320.80i	− 0.362957i
$808$	5067.28i	0.220626i
$809$	20168.5	0.876500	0.438250	−	0.898853i	$-0.355599\pi$
$809$	20168.5	0.876500	0.438250	+	0.898853i	$0.355599\pi$
$810$	0	0
$811$	3472.00	0.150331	0.0751655	−	0.997171i	$-0.476051\pi$
$811$	3472.00	0.150331	0.0751655	+	0.997171i	$0.476051\pi$
$812$	5394.81i	0.233154i
$813$	− 2749.68i	− 0.118617i
$814$	25440.5	1.09544
$815$	0	0
$816$	664.182	0.0284939
$817$	51122.5i	2.18917i
$818$	− 12563.1i	− 0.536990i
$819$	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	$0.449882\pi$
$822$	− 6402.27i	− 0.271660i
$823$	19960.3i	0.845411i	0.906267	+	0.422705i	$0.138919\pi$
$823$	19960.3i	0.845411i	−0.906267	+	0.422705i	$0.861081\pi$
$824$	10405.0	0.439899
$825$	0	0
$826$	−272.267	−0.0114690
$827$	− 3105.37i	− 0.130574i	−0.997867	−	0.0652868i	$-0.979204\pi$
$827$	− 3105.37i	− 0.130574i	0.997867	−	0.0652868i	$-0.0207962\pi$
$828$	2183.45i	0.0916427i
$829$	10780.3	0.451646	0.225823	−	0.974168i	$-0.427493\pi$
$829$	10780.3	0.451646	0.225823	+	0.974168i	$0.427493\pi$
$830$	0	0
$831$	−5953.23	−0.248514
$832$	− 1643.67i	− 0.0684905i
$833$	− 5935.98i	− 0.246902i
$834$	10082.2	0.418606
$835$	0	0
$836$	17587.7	0.727613
$837$	2198.09i	0.0907731i
$838$	10875.0i	0.448293i
$839$	−2884.27	−0.118684	−0.0593422	−	0.998238i	$-0.518900\pi$
$839$	−2884.27	−0.118684	−0.0593422	+	0.998238i	$0.518900\pi$
$840$	0	0
$841$	−2873.26	−0.117810
$842$	16848.2i	0.689582i
$843$	16010.1i	0.654111i
$844$	−21634.6	−0.882337
$845$	0	0
$846$	28853.8	1.17259
$847$	− 4907.18i	− 0.199071i
$848$	10788.1i	0.436868i
$849$	9887.50	0.399691
$850$	0	0
$851$	−6775.15	−0.272913
$852$	− 7322.49i	− 0.294442i
$853$	13694.1i	0.549681i	0.961490	+	0.274841i	$0.0886251\pi$
$853$	13694.1i	0.549681i	−0.961490	+	0.274841i	$0.911375\pi$
$854$	−8800.24	−0.352621
$855$	0	0
$856$	3964.96	0.158317
$857$	− 31610.2i	− 1.25996i	−0.776613	−	0.629978i	$-0.783064\pi$
$857$	− 31610.2i	− 1.25996i	0.776613	−	0.629978i	$-0.216936\pi$
$858$	− 4008.97i	− 0.159515i
$859$	20316.3	0.806964	0.403482	−	0.914988i	$-0.367800\pi$
$859$	20316.3	0.806964	0.403482	+	0.914988i	$0.367800\pi$
$860$	0	0
$861$	−2124.27	−0.0840823
$862$	− 4560.37i	− 0.180194i
$863$	9902.42i	0.390593i	0.980744	+	0.195297i	$0.0625670\pi$
$863$	9902.42i	0.390593i	−0.980744	+	0.195297i	$0.937433\pi$
$864$	−2934.31	−0.115541
$865$	0	0
$866$	19353.8	0.759433
$867$	7926.56i	0.310496i
$868$	881.633i	0.0344753i
$869$	27707.7	1.08161
$870$	0	0
$871$	11773.4	0.458008
$872$	− 1585.68i	− 0.0615801i
$873$	− 3992.63i	− 0.154788i
$874$	−4683.86	−0.181275
$875$	0	0
$876$	−5426.09	−0.209281
$877$	− 11027.7i	− 0.424606i	−0.977204	−	0.212303i	$-0.931904\pi$
$877$	− 11027.7i	− 0.424606i	0.977204	−	0.212303i	$-0.0680965\pi$
$878$	20535.2i	0.789326i
$879$	−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	$0.679061\pi$
$882$	12268.0i	0.468351i
$883$	1618.76i	0.0616937i	0.999524	+	0.0308468i	$0.00982041\pi$
$883$	1618.76i	0.0616937i	−0.999524	+	0.0308468i	$0.990180\pi$
$884$	−2359.38	−0.0897677
$885$	0	0
$886$	−5895.40	−0.223544
$887$	9518.69i	0.360323i	0.983637	+	0.180161i	$0.0576620\pi$
$887$	9518.69i	0.360323i	−0.983637	+	0.180161i	$0.942338\pi$
$888$	− 4259.37i	− 0.160963i
$889$	−19789.9	−0.746606
$890$	0	0
$891$	20514.0	0.771319
$892$	− 23251.0i	− 0.872758i
$893$	61896.0i	2.31945i
$894$	−7916.78	−0.296171
$895$	0	0
$896$	−1176.92	−0.0438820
$897$	1067.64i	0.0397409i
$898$	28644.5i	1.06445i
$899$	3516.15	0.130445
$900$	0	0
$901$	15485.6	0.572585
$902$	− 11039.3i	− 0.407504i
$903$	− 8343.87i	− 0.307494i
$904$	6948.44	0.255643
$905$	0	0
$906$	5673.87	0.208059
$907$	− 9580.94i	− 0.350750i	−0.984502	−	0.175375i	$-0.943886\pi$
$907$	− 9580.94i	− 0.350750i	0.984502	−	0.175375i	$-0.0561137\pi$
$908$	8133.18i	0.297257i
$909$	−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	$0.606245\pi$
$912$	− 2944.62i	− 0.106915i
$913$	38872.0i	1.40906i
$914$	−26808.8	−0.970192
$915$	0	0
$916$	−1743.61	−0.0628936
$917$	− 15872.7i	− 0.571606i
$918$	4212.01i	0.151435i
$919$	36712.5	1.31777	0.658887	−	0.752242i	$-0.271027\pi$
$919$	36712.5	1.31777	0.658887	+	0.752242i	$0.271027\pi$
$920$	0	0
$921$	−1431.36	−0.0512107
$922$	− 33857.2i	− 1.20936i
$923$	26011.8i	0.927614i
$924$	−2870.56	−0.102202
$925$	0	0
$926$	26218.5	0.930447
$927$	30868.1i	1.09368i
$928$	4693.84i	0.166037i
$929$	−4117.08	−0.145400	−0.0727002	−	0.997354i	$-0.523162\pi$
$929$	−4117.08	−0.145400	−0.0727002	+	0.997354i	$0.523162\pi$
$930$	0	0
$931$	−26316.9	−0.926425
$932$	6328.20i	0.222411i
$933$	3812.36i	0.133774i
$934$	9352.04	0.327632
$935$	0	0
$936$	4876.19	0.170281
$937$	11540.5i	0.402362i	0.979554	+	0.201181i	$0.0644779\pi$
$937$	11540.5i	0.402362i	−0.979554	+	0.201181i	$0.935522\pi$
$938$	− 8430.12i	− 0.293447i
$939$	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	$0.547597\pi$
$942$	9521.08i	0.329314i
$943$	2939.92i	0.101524i
$944$	−236.890	−0.00816750
$945$	0	0
$946$	43361.0	1.49026
$947$	− 48782.8i	− 1.67395i	−0.547243	−	0.836974i	$-0.684323\pi$
$947$	− 48782.8i	− 1.67395i	0.547243	−	0.836974i	$-0.315677\pi$
$948$	− 4638.96i	− 0.158931i
$949$	19275.2	0.659324
$950$	0	0
$951$	−7255.00	−0.247381
$952$	1689.40i	0.0575143i
$953$	42550.4i	1.44632i	0.690681	+	0.723160i	$0.257311\pi$
$953$	42550.4i	1.44632i	−0.690681	+	0.723160i	$0.742689\pi$
$954$	−32004.4	−1.08614
$955$	0	0
$956$	−6930.40	−0.234461
$957$	11448.4i	0.386704i
$958$	27036.1i	0.911791i
$959$	16284.6	0.548341
$960$	0	0
$961$	−29216.4	−0.980712
$962$	15130.6i	0.507100i
$963$	11762.6i	0.393609i
$964$	17766.9	0.593604
$965$	0	0
$966$	764.469	0.0254621
$967$	25119.6i	0.835360i	0.908594	+	0.417680i	$0.137157\pi$
$967$	25119.6i	0.835360i	−0.908594	+	0.417680i	$0.862843\pi$
$968$	− 4269.57i	− 0.141766i
$969$	−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	$0.438470\pi$
$972$	− 13337.8i	− 0.440136i
$973$	25644.8i	0.844947i
$974$	−29829.3	−0.981308
$975$	0	0
$976$	−7656.78	−0.251114
$977$	15427.4i	0.505187i	0.967573	+	0.252593i	$0.0812835\pi$
$977$	15427.4i	0.505187i	−0.967573	+	0.252593i	$0.918717\pi$
$978$	1039.84i	0.0339985i
$979$	39890.4	1.30225
$980$	0	0
$981$	4704.14	0.153101
$982$	− 7413.99i	− 0.240927i
$983$	− 42012.4i	− 1.36316i	−0.731744	−	0.681580i	$-0.761293\pi$
$983$	− 42012.4i	− 1.36316i	0.731744	−	0.681580i	$-0.238707\pi$
$984$	−1848.25	−0.0598782
$985$	0	0
$986$	6737.70	0.217619
$987$	− 10102.3i	− 0.325794i
$988$	10460.2i	0.336826i
$989$	−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	$0.335144\pi$
$992$	767.078i	0.0245512i
$993$	2163.13i	0.0691286i
$994$	18625.3	0.594324
$995$	0	0
$996$	6508.14	0.207046
$997$	− 10011.2i	− 0.318011i	−0.987278	−	0.159005i	$-0.949171\pi$
$997$	− 10011.2i	− 0.318011i	0.987278	−	0.159005i	$-0.0508287\pi$
$998$	− 26073.9i	− 0.827009i
$999$	27011.4	0.855458

Twists

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

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

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +1.80744i q^{3} -4.00000 q^{4} -3.61488 q^{6} -9.19471i q^{7} -8.00000i q^{8} +23.7332 q^{9} +43.1821 q^{11} -7.22976i q^{12} +25.6824i q^{13} +18.3894 q^{14} +16.0000 q^{16} -22.9669i q^{17} +47.4663i q^{18} -101.823 q^{19} +16.6189 q^{21} +86.3643i q^{22} -23.0000i q^{23} +14.4595 q^{24} -51.3647 q^{26} +91.6972i q^{27} +36.7788i q^{28} +146.682 q^{29} +23.9712 q^{31} +32.0000i q^{32} +78.0491i q^{33} +45.9339 q^{34} -94.9326 q^{36} -294.572i q^{37} -203.646i q^{38} -46.4193 q^{39} -127.823 q^{41} +33.2378i q^{42} -502.072i q^{43} -172.729 q^{44} +46.0000 q^{46} -607.879i q^{47} +28.9190i q^{48} +258.457 q^{49} +41.5114 q^{51} -102.729i q^{52} +674.254i q^{53} -183.394 q^{54} -73.5577 q^{56} -184.039i q^{57} +293.365i q^{58} -14.8056 q^{59} -478.549 q^{61} +47.9424i q^{62} -218.220i q^{63} -64.0000 q^{64} -156.098 q^{66} -458.422i q^{67} +91.8678i q^{68} +41.5711 q^{69} +1012.83 q^{71} -189.865i q^{72} -750.521i q^{73} +589.144 q^{74} +407.292 q^{76} -397.047i q^{77} -92.8387i q^{78} +641.648 q^{79} +475.058 q^{81} -255.645i q^{82} +900.187i q^{83} -66.4756 q^{84} +1004.14 q^{86} +265.120i q^{87} -345.457i q^{88} +923.770 q^{89} +236.142 q^{91} +92.0000i q^{92} +43.3265i q^{93} +1215.76 q^{94} -57.8381 q^{96} -168.230i q^{97} +516.915i q^{98} +1024.85 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 40 q^{4} - 48 q^{6} - 22 q^{9} - 108 q^{11} - 96 q^{14} + 160 q^{16} + 100 q^{19} - 316 q^{21} + 192 q^{24} - 144 q^{26} + 208 q^{29} - 684 q^{31} + 528 q^{34} + 88 q^{36} - 386 q^{39} + 4 q^{41}+ \cdots + 3480 q^{99}+O(q^{100}) \) 10 * q - 40 * q^4 - 48 * q^6 - 22 * q^9 - 108 * q^11 - 96 * q^14 + 160 * q^16 + 100 * q^19 - 316 * q^21 + 192 * q^24 - 144 * q^26 + 208 * q^29 - 684 * q^31 + 528 * q^34 + 88 * q^36 - 386 * q^39 + 4 * q^41 + 432 * q^44 + 460 * q^46 + 882 * q^49 + 2400 * q^51 + 1956 * q^54 + 384 * q^56 + 554 * q^59 + 964 * q^61 - 640 * q^64 + 1392 * q^66 + 552 * q^69 + 416 * q^71 + 1520 * q^74 - 400 * q^76 + 6888 * q^79 + 5866 * q^81 + 1264 * q^84 + 456 * q^86 - 280 * q^89 + 3952 * q^91 + 3864 * q^94 - 768 * q^96 + 3480 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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