LMFDB - Embedded newform 1150.4.b.m.599.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1150,4,Mod(599,1150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1150.599"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

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:traces = [8,0,0,-32,0,16,0,0,-128,0,186] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$67.8521965066$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 168x^{6} + 9540x^{4} + 208777x^{2} + 1542564 $ x^8 + 168x^6 + 9540x^4 + 208777*x^2 + 1542564
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.7
Root			$4.26018i$ of defining polynomial
Character	$\chi$	$=$	1150.599
Dual form			1150.4.b.m.599.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000i q^{2} +3.26018i q^{3} -4.00000 q^{4} -6.52037 q^{6} +27.7921i q^{7} -8.00000i q^{8} +16.3712 q^{9} -10.8182 q^{11} -13.0407i q^{12} -36.9683i q^{13} -55.5842 q^{14} +16.0000 q^{16} +118.996i q^{17} +32.7424i q^{18} +19.3321 q^{19} -90.6073 q^{21} -21.6365i q^{22} -23.0000i q^{23} +26.0815 q^{24} +73.9366 q^{26} +141.398i q^{27} -111.168i q^{28} -234.499 q^{29} +165.319 q^{31} +32.0000i q^{32} -35.2694i q^{33} -237.992 q^{34} -65.4848 q^{36} +202.301i q^{37} +38.6643i q^{38} +120.523 q^{39} -295.846 q^{41} -181.215i q^{42} +65.9221i q^{43} +43.2729 q^{44} +46.0000 q^{46} -110.279i q^{47} +52.1630i q^{48} -429.400 q^{49} -387.949 q^{51} +147.873i q^{52} +688.135i q^{53} -282.796 q^{54} +222.337 q^{56} +63.0263i q^{57} -468.998i q^{58} -10.5847 q^{59} +110.579 q^{61} +330.639i q^{62} +454.990i q^{63} -64.0000 q^{64} +70.5388 q^{66} -643.290i q^{67} -475.984i q^{68} +74.9842 q^{69} +143.216 q^{71} -130.970i q^{72} +158.213i q^{73} -404.601 q^{74} -77.3285 q^{76} -300.661i q^{77} +241.047i q^{78} -1123.34 q^{79} -18.9616 q^{81} -591.692i q^{82} +824.600i q^{83} +362.429 q^{84} -131.844 q^{86} -764.510i q^{87} +86.5458i q^{88} +879.672 q^{89} +1027.43 q^{91} +92.0000i q^{92} +538.971i q^{93} +220.557 q^{94} -104.326 q^{96} -938.437i q^{97} -858.801i q^{98} -177.107 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 32 q^{4} + 16 q^{6} - 128 q^{9} + 186 q^{11} - 104 q^{14} + 128 q^{16} - 370 q^{19} + 604 q^{21} - 64 q^{24} + 128 q^{26} - 588 q^{29} - 422 q^{31} - 432 q^{34} + 512 q^{36} + 842 q^{39} - 738 q^{41}+ \cdots - 4026 q^{99}+O(q^{100}) $ 8 * q - 32 * q^4 + 16 * q^6 - 128 * q^9 + 186 * q^11 - 104 * q^14 + 128 * q^16 - 370 * q^19 + 604 * q^21 - 64 * q^24 + 128 * q^26 - 588 * q^29 - 422 * q^31 - 432 * q^34 + 512 * q^36 + 842 * q^39 - 738 * q^41 - 744 * q^44 + 368 * q^46 - 1200 * q^49 - 630 * q^51 - 1036 * q^54 + 416 * q^56 + 66 * q^59 - 614 * q^61 - 512 * q^64 - 948 * q^66 - 184 * q^69 - 2514 * q^71 - 20 * q^74 + 1480 * q^76 - 2404 * q^79 - 1984 * q^81 - 2416 * q^84 - 400 * q^86 + 1968 * q^89 - 1990 * q^91 + 1452 * q^94 + 256 * q^96 - 4026 * 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
$3$	3.26018i	0.627423i	0.949518	+	0.313711i	$0.101572\pi$
$3$	3.26018i	0.627423i	−0.949518	+	0.313711i	$0.898428\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−6.52037	−0.443655
$7$	27.7921i	1.50063i	0.661079	+	0.750316i	$0.270099\pi$
$7$	27.7921i	1.50063i	−0.661079	+	0.750316i	$0.729901\pi$
$8$	− 8.00000i	− 0.353553i
$9$	16.3712	0.606341
$10$	0	0
$11$	−10.8182	−0.296529	−0.148264	−	0.988948i	$-0.547369\pi$
$11$	−10.8182	−0.296529	−0.148264	+	0.988948i	$0.547369\pi$
$12$	− 13.0407i	− 0.313711i
$13$	− 36.9683i	− 0.788705i	−0.918959	−	0.394352i	$-0.870969\pi$
$13$	− 36.9683i	− 0.788705i	0.918959	−	0.394352i	$-0.129031\pi$
$14$	−55.5842	−1.06111
$15$	0	0
$16$	16.0000	0.250000
$17$	118.996i	1.69769i	0.528639	+	0.848847i	$0.322703\pi$
$17$	118.996i	1.69769i	−0.528639	+	0.848847i	$0.677297\pi$
$18$	32.7424i	0.428748i
$19$	19.3321	0.233426	0.116713	−	0.993166i	$-0.462764\pi$
$19$	19.3321	0.233426	0.116713	+	0.993166i	$0.462764\pi$
$20$	0	0
$21$	−90.6073	−0.941531
$22$	− 21.6365i	− 0.209678i
$23$	− 23.0000i	− 0.208514i
$23$	− 23.0000i	− 0.208514i
$24$	26.0815	0.221827
$25$	0	0
$26$	73.9366	0.557699
$27$	141.398i	1.00785i
$28$	− 111.168i	− 0.750316i
$29$	−234.499	−1.50156	−0.750782	−	0.660550i	$-0.770323\pi$
$29$	−234.499	−1.50156	−0.750782	+	0.660550i	$0.770323\pi$
$30$	0	0
$31$	165.319	0.957814	0.478907	−	0.877866i	$-0.341033\pi$
$31$	165.319	0.957814	0.478907	+	0.877866i	$0.341033\pi$
$32$	32.0000i	0.176777i
$33$	− 35.2694i	− 0.186049i
$34$	−237.992	−1.20045
$35$	0	0
$36$	−65.4848	−0.303170
$37$	202.301i	0.898865i	0.893314	+	0.449432i	$0.148374\pi$
$37$	202.301i	0.898865i	−0.893314	+	0.449432i	$0.851626\pi$
$38$	38.6643i	0.165057i
$39$	120.523	0.494851
$40$	0	0
$41$	−295.846	−1.12691	−0.563456	−	0.826146i	$-0.690529\pi$
$41$	−295.846	−1.12691	−0.563456	+	0.826146i	$0.690529\pi$
$42$	− 181.215i	− 0.665763i
$43$	65.9221i	0.233791i	0.993144	+	0.116896i	$0.0372943\pi$
$43$	65.9221i	0.233791i	−0.993144	+	0.116896i	$0.962706\pi$
$44$	43.2729	0.148264
$45$	0	0
$46$	46.0000	0.147442
$47$	− 110.279i	− 0.342251i	−0.985249	−	0.171126i	$-0.945260\pi$
$47$	− 110.279i	− 0.342251i	0.985249	−	0.171126i	$-0.0547404\pi$
$48$	52.1630i	0.156856i
$49$	−429.400	−1.25190
$50$	0	0
$51$	−387.949	−1.06517
$52$	147.873i	0.394352i
$53$	688.135i	1.78345i	0.452582	+	0.891723i	$0.350503\pi$
$53$	688.135i	1.78345i	−0.452582	+	0.891723i	$0.649497\pi$
$54$	−282.796	−0.712661
$55$	0	0
$56$	222.337	0.530553
$57$	63.0263i	0.146457i
$58$	− 468.998i	− 1.06177i
$59$	−10.5847	−0.0233561	−0.0116781	−	0.999932i	$-0.503717\pi$
$59$	−10.5847	−0.0233561	−0.0116781	+	0.999932i	$0.503717\pi$
$60$	0	0
$61$	110.579	0.232101	0.116050	−	0.993243i	$-0.462977\pi$
$61$	110.579	0.232101	0.116050	+	0.993243i	$0.462977\pi$
$62$	330.639i	0.677276i
$63$	454.990i	0.909894i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	70.5388	0.131557
$67$	− 643.290i	− 1.17299i	−0.809952	−	0.586496i	$-0.800507\pi$
$67$	− 643.290i	− 1.17299i	0.809952	−	0.586496i	$-0.199493\pi$
$68$	− 475.984i	− 0.848847i
$69$	74.9842	0.130827
$70$	0	0
$71$	143.216	0.239389	0.119695	−	0.992811i	$-0.461808\pi$
$71$	143.216	0.239389	0.119695	+	0.992811i	$0.461808\pi$
$72$	− 130.970i	− 0.214374i
$73$	158.213i	0.253664i	0.991924	+	0.126832i	$0.0404809\pi$
$73$	158.213i	0.253664i	−0.991924	+	0.126832i	$0.959519\pi$
$74$	−404.601	−0.635593
$75$	0	0
$76$	−77.3285	−0.116713
$77$	− 300.661i	− 0.444981i
$78$	241.047i	0.349913i
$79$	−1123.34	−1.59982	−0.799912	−	0.600118i	$-0.795120\pi$
$79$	−1123.34	−1.59982	−0.799912	+	0.600118i	$0.795120\pi$
$80$	0	0
$81$	−18.9616	−0.0260104
$82$	− 591.692i	− 0.796847i
$83$	824.600i	1.09050i	0.838273	+	0.545251i	$0.183566\pi$
$83$	824.600i	1.09050i	−0.838273	+	0.545251i	$0.816434\pi$
$84$	362.429	0.470765
$85$	0	0
$86$	−131.844	−0.165316
$87$	− 764.510i	− 0.942115i
$88$	86.5458i	0.104839i
$89$	879.672	1.04770	0.523849	−	0.851811i	$-0.324496\pi$
$89$	879.672	1.04770	0.523849	+	0.851811i	$0.324496\pi$
$90$	0	0
$91$	1027.43	1.18356
$92$	92.0000i	0.104257i
$93$	538.971i	0.600954i
$94$	220.557	0.242008
$95$	0	0
$96$	−104.326	−0.110914
$97$	− 938.437i	− 0.982308i	−0.871073	−	0.491154i	$-0.836575\pi$
$97$	− 938.437i	− 0.982308i	0.871073	−	0.491154i	$-0.163425\pi$
$98$	− 858.801i	− 0.885224i
$99$	−177.107	−0.179798
$100$	0	0
$101$	−688.428	−0.678229	−0.339114	−	0.940745i	$-0.610127\pi$
$101$	−688.428	−0.678229	−0.339114	+	0.940745i	$0.610127\pi$
$102$	− 775.899i	− 0.753190i
$103$	− 2041.55i	− 1.95301i	−0.215500	−	0.976504i	$-0.569138\pi$
$103$	− 2041.55i	− 1.95301i	0.215500	−	0.976504i	$-0.430862\pi$
$104$	−295.746	−0.278849
$105$	0	0
$106$	−1376.27	−1.26109
$107$	287.420i	0.259682i	0.991535	+	0.129841i	$0.0414466\pi$
$107$	287.420i	0.259682i	−0.991535	+	0.129841i	$0.958553\pi$
$108$	− 565.592i	− 0.503927i
$109$	1211.29	1.06441	0.532204	−	0.846616i	$-0.321364\pi$
$109$	1211.29	1.06441	0.532204	+	0.846616i	$0.321364\pi$
$110$	0	0
$111$	−659.537	−0.563968
$112$	444.673i	0.375158i
$113$	− 741.316i	− 0.617143i	−0.951201	−	0.308571i	$-0.900149\pi$
$113$	− 741.316i	− 0.617143i	0.951201	−	0.308571i	$-0.0998509\pi$
$114$	−126.053	−0.103561
$115$	0	0
$116$	937.996	0.750782
$117$	− 605.215i	− 0.478224i
$118$	− 21.1694i	− 0.0165153i
$119$	−3307.15	−2.54761
$120$	0	0
$121$	−1213.97	−0.912071
$122$	221.157i	0.164120i
$123$	− 964.513i	− 0.707050i
$124$	−661.277	−0.478907
$125$	0	0
$126$	−909.980	−0.643392
$127$	1579.99i	1.10395i	0.833860	+	0.551976i	$0.186126\pi$
$127$	1579.99i	1.10395i	−0.833860	+	0.551976i	$0.813874\pi$
$128$	− 128.000i	− 0.0883883i
$129$	−214.918	−0.146686
$130$	0	0
$131$	−2348.86	−1.56657	−0.783285	−	0.621662i	$-0.786458\pi$
$131$	−2348.86	−1.56657	−0.783285	+	0.621662i	$0.786458\pi$
$132$	141.078i	0.0930245i
$133$	537.280i	0.350287i
$134$	1286.58	0.829430
$135$	0	0
$136$	951.969	0.600225
$137$	− 617.567i	− 0.385127i	−0.981285	−	0.192563i	$-0.938320\pi$
$137$	− 617.567i	− 0.385127i	0.981285	−	0.192563i	$-0.0616801\pi$
$138$	149.968i	0.0925084i
$139$	−1509.75	−0.921260	−0.460630	−	0.887592i	$-0.652377\pi$
$139$	−1509.75	−0.921260	−0.460630	+	0.887592i	$0.652377\pi$
$140$	0	0
$141$	359.529	0.214736
$142$	286.432i	0.169274i
$143$	399.932i	0.233874i
$144$	261.939	0.151585
$145$	0	0
$146$	−316.427	−0.179368
$147$	− 1399.92i	− 0.785468i
$148$	− 809.202i	− 0.449432i
$149$	−1025.46	−0.563818	−0.281909	−	0.959441i	$-0.590968\pi$
$149$	−1025.46	−0.563818	−0.281909	+	0.959441i	$0.590968\pi$
$150$	0	0
$151$	−1516.67	−0.817383	−0.408692	−	0.912672i	$-0.634015\pi$
$151$	−1516.67	−0.817383	−0.408692	+	0.912672i	$0.634015\pi$
$152$	− 154.657i	− 0.0825286i
$153$	1948.11i	1.02938i
$154$	601.322	0.314649
$155$	0	0
$156$	−482.094	−0.247426
$157$	− 2757.31i	− 1.40164i	−0.713339	−	0.700819i	$-0.752818\pi$
$157$	− 2757.31i	− 1.40164i	0.713339	−	0.700819i	$-0.247182\pi$
$158$	− 2246.69i	− 1.13125i
$159$	−2243.45	−1.11897
$160$	0	0
$161$	639.218	0.312903
$162$	− 37.9231i	− 0.0183921i
$163$	− 1263.29i	− 0.607047i	−0.952824	−	0.303524i	$-0.901837\pi$
$163$	− 1263.29i	− 0.607047i	0.952824	−	0.303524i	$-0.0981631\pi$
$164$	1183.38	0.563456
$165$	0	0
$166$	−1649.20	−0.771101
$167$	− 349.646i	− 0.162014i	−0.996714	−	0.0810072i	$-0.974186\pi$
$167$	− 349.646i	− 0.162014i	0.996714	−	0.0810072i	$-0.0258137\pi$
$168$	724.859i	0.332881i
$169$	830.344	0.377945
$170$	0	0
$171$	316.490	0.141536
$172$	− 263.689i	− 0.116896i
$173$	− 2313.83i	− 1.01686i	−0.861102	−	0.508432i	$-0.830225\pi$
$173$	− 2313.83i	− 1.01686i	0.861102	−	0.508432i	$-0.169775\pi$
$174$	1529.02	0.666176
$175$	0	0
$176$	−173.092	−0.0741322
$177$	− 34.5081i	− 0.0146542i
$178$	1759.34i	0.740834i
$179$	2347.69	0.980306	0.490153	−	0.871636i	$-0.336941\pi$
$179$	2347.69	0.980306	0.490153	+	0.871636i	$0.336941\pi$
$180$	0	0
$181$	4396.31	1.80539	0.902695	−	0.430282i	$-0.141586\pi$
$181$	4396.31	1.80539	0.902695	+	0.430282i	$0.141586\pi$
$182$	2054.85i	0.836900i
$183$	360.507i	0.145625i
$184$	−184.000	−0.0737210
$185$	0	0
$186$	−1077.94	−0.424939
$187$	− 1287.33i	− 0.503415i
$188$	441.115i	0.171126i
$189$	−3929.75	−1.51242
$190$	0	0
$191$	4153.14	1.57335	0.786677	−	0.617365i	$-0.211800\pi$
$191$	4153.14	1.57335	0.786677	+	0.617365i	$0.211800\pi$
$192$	− 208.652i	− 0.0784278i
$193$	1020.52i	0.380616i	0.981724	+	0.190308i	$0.0609487\pi$
$193$	1020.52i	0.380616i	−0.981724	+	0.190308i	$0.939051\pi$
$194$	1876.87	0.694596
$195$	0	0
$196$	1717.60	0.625948
$197$	− 398.643i	− 0.144173i	−0.997398	−	0.0720866i	$-0.977034\pi$
$197$	− 398.643i	− 0.144173i	0.997398	−	0.0720866i	$-0.0229658\pi$
$198$	− 354.215i	− 0.127136i
$199$	−4131.40	−1.47169	−0.735847	−	0.677148i	$-0.763216\pi$
$199$	−4131.40	−1.47169	−0.735847	+	0.677148i	$0.763216\pi$
$200$	0	0
$201$	2097.24	0.735961
$202$	− 1376.86i	− 0.479580i
$203$	− 6517.22i	− 2.25329i
$204$	1551.80	0.532586
$205$	0	0
$206$	4083.10	1.38098
$207$	− 376.538i	− 0.126431i
$208$	− 591.493i	− 0.197176i
$209$	−209.139	−0.0692176
$210$	0	0
$211$	826.315	0.269601	0.134801	−	0.990873i	$-0.456961\pi$
$211$	826.315	0.269601	0.134801	+	0.990873i	$0.456961\pi$
$212$	− 2752.54i	− 0.891723i
$213$	466.911i	0.150198i
$214$	−574.840	−0.183623
$215$	0	0
$216$	1131.18	0.356330
$217$	4594.57i	1.43733i
$218$	2422.58i	0.752650i
$219$	−515.805	−0.159155
$220$	0	0
$221$	4399.08	1.33898
$222$	− 1319.07i	− 0.398786i
$223$	− 1110.09i	− 0.333351i	−0.986012	−	0.166676i	$-0.946697\pi$
$223$	− 1110.09i	− 0.333351i	0.986012	−	0.166676i	$-0.0533033\pi$
$224$	−889.347	−0.265277
$225$	0	0
$226$	1482.63	0.436386
$227$	− 4917.90i	− 1.43794i	−0.695040	−	0.718971i	$-0.744614\pi$
$227$	− 4917.90i	− 1.43794i	0.695040	−	0.718971i	$-0.255386\pi$
$228$	− 252.105i	− 0.0732284i
$229$	1390.62	0.401288	0.200644	−	0.979664i	$-0.435696\pi$
$229$	1390.62	0.401288	0.200644	+	0.979664i	$0.435696\pi$
$230$	0	0
$231$	980.211	0.279191
$232$	1875.99i	0.530883i
$233$	3409.59i	0.958668i	0.877633	+	0.479334i	$0.159122\pi$
$233$	3409.59i	0.958668i	−0.877633	+	0.479334i	$0.840878\pi$
$234$	1210.43	0.338155
$235$	0	0
$236$	42.3388	0.0116781
$237$	− 3662.31i	− 1.00377i
$238$	− 6614.30i	− 1.80143i
$239$	−2056.35	−0.556544	−0.278272	−	0.960502i	$-0.589762\pi$
$239$	−2056.35	−0.556544	−0.278272	+	0.960502i	$0.589762\pi$
$240$	0	0
$241$	−1957.03	−0.523085	−0.261543	−	0.965192i	$-0.584231\pi$
$241$	−1957.03	−0.523085	−0.261543	+	0.965192i	$0.584231\pi$
$242$	− 2427.93i	− 0.644931i
$243$	3755.93i	0.991535i
$244$	−442.315	−0.116050
$245$	0	0
$246$	1929.03	0.499960
$247$	− 714.676i	− 0.184104i
$248$	− 1322.55i	− 0.338638i
$249$	−2688.35	−0.684205
$250$	0	0
$251$	−1954.13	−0.491410	−0.245705	−	0.969345i	$-0.579019\pi$
$251$	−1954.13	−0.491410	−0.245705	+	0.969345i	$0.579019\pi$
$252$	− 1819.96i	− 0.454947i
$253$	248.819i	0.0618306i
$254$	−3159.99	−0.780612
$255$	0	0
$256$	256.000	0.0625000
$257$	1958.40i	0.475338i	0.971346	+	0.237669i	$0.0763834\pi$
$257$	1958.40i	0.475338i	−0.971346	+	0.237669i	$0.923617\pi$
$258$	− 429.837i	− 0.103723i
$259$	−5622.35	−1.34887
$260$	0	0
$261$	−3839.03	−0.910459
$262$	− 4697.72i	− 1.10773i
$263$	− 7205.23i	− 1.68933i	−0.535295	−	0.844665i	$-0.679800\pi$
$263$	− 7205.23i	− 1.68933i	0.535295	−	0.844665i	$-0.320200\pi$
$264$	−282.155	−0.0657783
$265$	0	0
$266$	−1074.56	−0.247690
$267$	2867.89i	0.657349i
$268$	2573.16i	0.586496i
$269$	3764.30	0.853210	0.426605	−	0.904438i	$-0.359709\pi$
$269$	3764.30	0.853210	0.426605	+	0.904438i	$0.359709\pi$
$270$	0	0
$271$	−5208.88	−1.16759	−0.583795	−	0.811901i	$-0.698433\pi$
$271$	−5208.88	−1.16759	−0.583795	+	0.811901i	$0.698433\pi$
$272$	1903.94i	0.424423i
$273$	3349.60i	0.742590i
$274$	1235.13	0.272326
$275$	0	0
$276$	−299.937	−0.0654133
$277$	1550.21i	0.336256i	0.985765	+	0.168128i	$0.0537723\pi$
$277$	1550.21i	0.336256i	−0.985765	+	0.168128i	$0.946228\pi$
$278$	− 3019.50i	− 0.651430i
$279$	2706.47	0.580761
$280$	0	0
$281$	7997.53	1.69784	0.848920	−	0.528522i	$-0.177254\pi$
$281$	7997.53	1.69784	0.848920	+	0.528522i	$0.177254\pi$
$282$	719.058i	0.151841i
$283$	5808.41i	1.22005i	0.792382	+	0.610025i	$0.208841\pi$
$283$	5808.41i	1.22005i	−0.792382	+	0.610025i	$0.791159\pi$
$284$	−572.864	−0.119695
$285$	0	0
$286$	−799.863	−0.165374
$287$	− 8222.18i	− 1.69108i
$288$	523.878i	0.107187i
$289$	−9247.08	−1.88216
$290$	0	0
$291$	3059.48	0.616322
$292$	− 632.854i	− 0.126832i
$293$	8854.66i	1.76551i	0.469832	+	0.882756i	$0.344314\pi$
$293$	8854.66i	1.76551i	−0.469832	+	0.882756i	$0.655686\pi$
$294$	2799.85	0.555410
$295$	0	0
$296$	1618.40	0.317797
$297$	− 1529.68i	− 0.298858i
$298$	− 2050.92i	− 0.398680i
$299$	−850.271	−0.164456
$300$	0	0
$301$	−1832.11	−0.350835
$302$	− 3033.34i	− 0.577977i
$303$	− 2244.40i	− 0.425536i
$304$	309.314	0.0583565
$305$	0	0
$306$	−3896.22	−0.727882
$307$	− 1501.03i	− 0.279050i	−0.990219	−	0.139525i	$-0.955443\pi$
$307$	− 1501.03i	− 0.279050i	0.990219	−	0.139525i	$-0.0445575\pi$
$308$	1202.64i	0.222490i
$309$	6655.83	1.22536
$310$	0	0
$311$	−7440.48	−1.35663	−0.678313	−	0.734773i	$-0.737289\pi$
$311$	−7440.48	−1.35663	−0.678313	+	0.734773i	$0.737289\pi$
$312$	− 964.188i	− 0.174956i
$313$	2528.63i	0.456634i	0.973587	+	0.228317i	$0.0733222\pi$
$313$	2528.63i	0.456634i	−0.973587	+	0.228317i	$0.926678\pi$
$314$	5514.61	0.991107
$315$	0	0
$316$	4493.38	0.799912
$317$	− 9636.72i	− 1.70742i	−0.520749	−	0.853710i	$-0.674347\pi$
$317$	− 9636.72i	− 1.70742i	0.520749	−	0.853710i	$-0.325653\pi$
$318$	− 4486.89i	− 0.791234i
$319$	2536.86	0.445257
$320$	0	0
$321$	−937.042	−0.162930
$322$	1278.44i	0.221256i
$323$	2300.45i	0.396286i
$324$	75.8462	0.0130052
$325$	0	0
$326$	2526.58	0.429247
$327$	3949.03i	0.667834i
$328$	2366.77i	0.398424i
$329$	3064.88	0.513593
$330$	0	0
$331$	2875.82	0.477551	0.238776	−	0.971075i	$-0.423254\pi$
$331$	2875.82	0.477551	0.238776	+	0.971075i	$0.423254\pi$
$332$	− 3298.40i	− 0.545251i
$333$	3311.90i	0.545018i
$334$	699.291	0.114561
$335$	0	0
$336$	−1449.72	−0.235383
$337$	9911.19i	1.60207i	0.598619	+	0.801034i	$0.295717\pi$
$337$	9911.19i	1.60207i	−0.598619	+	0.801034i	$0.704283\pi$
$338$	1660.69i	0.267247i
$339$	2416.83	0.387209
$340$	0	0
$341$	−1788.46	−0.284019
$342$	632.980i	0.100081i
$343$	− 2401.24i	− 0.378003i
$344$	527.377	0.0826578
$345$	0	0
$346$	4627.67	0.719031
$347$	4497.70i	0.695820i	0.937528	+	0.347910i	$0.113108\pi$
$347$	4497.70i	0.695820i	−0.937528	+	0.347910i	$0.886892\pi$
$348$	3058.04i	0.471058i
$349$	−888.112	−0.136216	−0.0681082	−	0.997678i	$-0.521696\pi$
$349$	−888.112	−0.136216	−0.0681082	+	0.997678i	$0.521696\pi$
$350$	0	0
$351$	5227.25	0.794900
$352$	− 346.183i	− 0.0524194i
$353$	11722.2i	1.76745i	0.468002	+	0.883727i	$0.344974\pi$
$353$	11722.2i	1.76745i	−0.468002	+	0.883727i	$0.655026\pi$
$354$	69.0161	0.0103621
$355$	0	0
$356$	−3518.69	−0.523849
$357$	− 10781.9i	− 1.59843i
$358$	4695.39i	0.693181i
$359$	1257.49	0.184869	0.0924343	−	0.995719i	$-0.470535\pi$
$359$	1257.49	0.184869	0.0924343	+	0.995719i	$0.470535\pi$
$360$	0	0
$361$	−6485.27	−0.945512
$362$	8792.63i	1.27660i
$363$	− 3957.75i	− 0.572254i
$364$	−4109.71	−0.591778
$365$	0	0
$366$	−721.014	−0.102973
$367$	10122.2i	1.43972i	0.694120	+	0.719859i	$0.255794\pi$
$367$	10122.2i	1.43972i	−0.694120	+	0.719859i	$0.744206\pi$
$368$	− 368.000i	− 0.0521286i
$369$	−4843.36	−0.683293
$370$	0	0
$371$	−19124.7	−2.67629
$372$	− 2155.89i	− 0.300477i
$373$	− 6530.08i	− 0.906473i	−0.891390	−	0.453237i	$-0.850269\pi$
$373$	− 6530.08i	− 0.906473i	0.891390	−	0.453237i	$-0.149731\pi$
$374$	2574.65	0.355968
$375$	0	0
$376$	−882.230	−0.121004
$377$	8669.03i	1.18429i
$378$	− 7859.50i	− 1.06944i
$379$	−277.850	−0.0376575	−0.0188288	−	0.999823i	$-0.505994\pi$
$379$	−277.850	−0.0376575	−0.0188288	+	0.999823i	$0.505994\pi$
$380$	0	0
$381$	−5151.07	−0.692644
$382$	8306.28i	1.11253i
$383$	10679.6i	1.42482i	0.701766	+	0.712408i	$0.252395\pi$
$383$	10679.6i	1.42482i	−0.701766	+	0.712408i	$0.747605\pi$
$384$	417.304	0.0554569
$385$	0	0
$386$	−2041.05	−0.269136
$387$	1079.22i	0.141757i
$388$	3753.75i	0.491154i
$389$	−1546.05	−0.201511	−0.100756	−	0.994911i	$-0.532126\pi$
$389$	−1546.05	−0.201511	−0.100756	+	0.994911i	$0.532126\pi$
$390$	0	0
$391$	2736.91	0.353994
$392$	3435.20i	0.442612i
$393$	− 7657.71i	− 0.982902i
$394$	797.285	0.101946
$395$	0	0
$396$	708.429	0.0898988
$397$	− 12901.8i	− 1.63103i	−0.578733	−	0.815517i	$-0.696453\pi$
$397$	− 12901.8i	− 1.63103i	0.578733	−	0.815517i	$-0.303547\pi$
$398$	− 8262.80i	− 1.04064i
$399$	−1751.63	−0.219778
$400$	0	0
$401$	−8534.31	−1.06280	−0.531400	−	0.847121i	$-0.678334\pi$
$401$	−8534.31	−1.06280	−0.531400	+	0.847121i	$0.678334\pi$
$402$	4194.49i	0.520403i
$403$	− 6111.57i	− 0.755432i
$404$	2753.71	0.339114
$405$	0	0
$406$	13034.4	1.59332
$407$	− 2188.53i	− 0.266539i
$408$	3103.59i	0.376595i
$409$	−650.968	−0.0787000	−0.0393500	−	0.999225i	$-0.512529\pi$
$409$	−650.968	−0.0787000	−0.0393500	+	0.999225i	$0.512529\pi$
$410$	0	0
$411$	2013.38	0.241637
$412$	8166.20i	0.976504i
$413$	− 294.171i	− 0.0350489i
$414$	753.075	0.0894001
$415$	0	0
$416$	1182.99	0.139425
$417$	− 4922.06i	− 0.578020i
$418$	− 418.279i	− 0.0489442i
$419$	−13266.2	−1.54677	−0.773384	−	0.633938i	$-0.781437\pi$
$419$	−13266.2	−1.54677	−0.773384	+	0.633938i	$0.781437\pi$
$420$	0	0
$421$	−8274.69	−0.957918	−0.478959	−	0.877837i	$-0.658986\pi$
$421$	−8274.69	−0.957918	−0.478959	+	0.877837i	$0.658986\pi$
$422$	1652.63i	0.190637i
$423$	− 1805.39i	− 0.207521i
$424$	5505.08	0.630543
$425$	0	0
$426$	−933.822	−0.106206
$427$	3073.21i	0.348298i
$428$	− 1149.68i	− 0.129841i
$429$	−1303.85	−0.146738
$430$	0	0
$431$	724.084	0.0809232	0.0404616	−	0.999181i	$-0.487117\pi$
$431$	724.084	0.0809232	0.0404616	+	0.999181i	$0.487117\pi$
$432$	2262.37i	0.251964i
$433$	11996.5i	1.33145i	0.746198	+	0.665724i	$0.231877\pi$
$433$	11996.5i	1.33145i	−0.746198	+	0.665724i	$0.768123\pi$
$434$	−9189.14	−1.01634
$435$	0	0
$436$	−4845.16	−0.532204
$437$	− 444.639i	− 0.0486727i
$438$	− 1031.61i	− 0.112539i
$439$	7765.40	0.844242	0.422121	−	0.906539i	$-0.361286\pi$
$439$	7765.40	0.844242	0.422121	+	0.906539i	$0.361286\pi$
$440$	0	0
$441$	−7029.80	−0.759075
$442$	8798.17i	0.946801i
$443$	9061.54i	0.971844i	0.874002	+	0.485922i	$0.161516\pi$
$443$	9061.54i	0.971844i	−0.874002	+	0.485922i	$0.838484\pi$
$444$	2638.15	0.281984
$445$	0	0
$446$	2220.19	0.235715
$447$	− 3343.19i	− 0.353752i
$448$	− 1778.69i	− 0.187579i
$449$	10944.3	1.15032	0.575159	−	0.818041i	$-0.304940\pi$
$449$	10944.3	1.15032	0.575159	+	0.818041i	$0.304940\pi$
$450$	0	0
$451$	3200.53	0.334162
$452$	2965.26i	0.308571i
$453$	− 4944.63i	− 0.512845i
$454$	9835.81	1.01678
$455$	0	0
$456$	504.211	0.0517803
$457$	10934.9i	1.11928i	0.828735	+	0.559641i	$0.189061\pi$
$457$	10934.9i	1.11928i	−0.828735	+	0.559641i	$0.810939\pi$
$458$	2781.25i	0.283754i
$459$	−16825.8	−1.71103
$460$	0	0
$461$	2652.25	0.267956	0.133978	−	0.990984i	$-0.457225\pi$
$461$	2652.25	0.267956	0.133978	+	0.990984i	$0.457225\pi$
$462$	1960.42i	0.197418i
$463$	− 707.699i	− 0.0710358i	−0.999369	−	0.0355179i	$-0.988692\pi$
$463$	− 707.699i	− 0.0710358i	0.999369	−	0.0355179i	$-0.0113081\pi$
$464$	−3751.98	−0.375391
$465$	0	0
$466$	−6819.18	−0.677880
$467$	− 10671.1i	− 1.05738i	−0.848814	−	0.528691i	$-0.822683\pi$
$467$	− 10671.1i	− 1.05738i	0.848814	−	0.528691i	$-0.177317\pi$
$468$	2420.86i	0.239112i
$469$	17878.4	1.76023
$470$	0	0
$471$	8989.33	0.879419
$472$	84.6776i	0.00825763i
$473$	− 713.161i	− 0.0693259i
$474$	7324.62	0.709770
$475$	0	0
$476$	13228.6	1.27381
$477$	11265.6i	1.08138i
$478$	− 4112.69i	− 0.393536i
$479$	−1951.17	−0.186119	−0.0930597	−	0.995661i	$-0.529665\pi$
$479$	−1951.17	−0.186119	−0.0930597	+	0.995661i	$0.529665\pi$
$480$	0	0
$481$	7478.71	0.708939
$482$	− 3914.06i	− 0.369877i
$483$	2083.97i	0.196323i
$484$	4855.86	0.456035
$485$	0	0
$486$	−7511.86	−0.701121
$487$	17071.2i	1.58844i	0.607630	+	0.794220i	$0.292120\pi$
$487$	17071.2i	1.58844i	−0.607630	+	0.794220i	$0.707880\pi$
$488$	− 884.629i	− 0.0820600i
$489$	4118.57	0.380875
$490$	0	0
$491$	16979.9	1.56067	0.780337	−	0.625359i	$-0.215047\pi$
$491$	16979.9	1.56067	0.780337	+	0.625359i	$0.215047\pi$
$492$	3858.05i	0.353525i
$493$	− 27904.5i	− 2.54920i
$494$	1429.35	0.130181
$495$	0	0
$496$	2645.11	0.239453
$497$	3980.28i	0.359235i
$498$	− 5376.70i	− 0.483806i
$499$	2788.83	0.250191	0.125095	−	0.992145i	$-0.460076\pi$
$499$	2788.83	0.250191	0.125095	+	0.992145i	$0.460076\pi$
$500$	0	0
$501$	1139.91	0.101651
$502$	− 3908.27i	− 0.347479i
$503$	15525.1i	1.37620i	0.725614	+	0.688102i	$0.241556\pi$
$503$	15525.1i	1.37620i	−0.725614	+	0.688102i	$0.758444\pi$
$504$	3639.92	0.321696
$505$	0	0
$506$	−497.638	−0.0437208
$507$	2707.08i	0.237131i
$508$	− 6319.98i	− 0.551976i
$509$	−9039.20	−0.787143	−0.393571	−	0.919294i	$-0.628761\pi$
$509$	−9039.20	−0.787143	−0.393571	+	0.919294i	$0.628761\pi$
$510$	0	0
$511$	−4397.08	−0.380657
$512$	512.000i	0.0441942i
$513$	2733.53i	0.235260i
$514$	−3916.81	−0.336115
$515$	0	0
$516$	859.673	0.0733430
$517$	1193.02i	0.101487i
$518$	− 11244.7i	− 0.953792i
$519$	7543.52	0.638004
$520$	0	0
$521$	1093.03	0.0919128	0.0459564	−	0.998943i	$-0.485366\pi$
$521$	1093.03	0.0919128	0.0459564	+	0.998943i	$0.485366\pi$
$522$	− 7678.06i	− 0.643792i
$523$	4660.98i	0.389695i	0.980834	+	0.194848i	$0.0624212\pi$
$523$	4660.98i	0.389695i	−0.980834	+	0.194848i	$0.937579\pi$
$524$	9395.44	0.783285
$525$	0	0
$526$	14410.5	1.19454
$527$	19672.4i	1.62607i
$528$	− 564.311i	− 0.0465123i
$529$	−529.000	−0.0434783
$530$	0	0
$531$	−173.284	−0.0141618
$532$	− 2149.12i	− 0.175143i
$533$	10936.9i	0.888801i
$534$	−5735.79	−0.464816
$535$	0	0
$536$	−5146.32	−0.414715
$537$	7653.91i	0.615066i
$538$	7528.60i	0.603311i
$539$	4645.35	0.371223
$540$	0	0
$541$	−8072.40	−0.641515	−0.320757	−	0.947161i	$-0.603937\pi$
$541$	−8072.40	−0.641515	−0.320757	+	0.947161i	$0.603937\pi$
$542$	− 10417.8i	− 0.825611i
$543$	14332.8i	1.13274i
$544$	−3807.88	−0.300113
$545$	0	0
$546$	−6699.20	−0.525090
$547$	− 2877.27i	− 0.224905i	−0.993657	−	0.112452i	$-0.964129\pi$
$547$	− 2877.27i	− 0.224905i	0.993657	−	0.112452i	$-0.0358706\pi$
$548$	2470.27i	0.192563i
$549$	1810.31	0.140732
$550$	0	0
$551$	−4533.36	−0.350504
$552$	− 599.874i	− 0.0462542i
$553$	− 31220.1i	− 2.40075i
$554$	−3100.42	−0.237769
$555$	0	0
$556$	6038.99	0.460630
$557$	15928.7i	1.21171i	0.795575	+	0.605855i	$0.207169\pi$
$557$	15928.7i	1.21171i	−0.795575	+	0.605855i	$0.792831\pi$
$558$	5412.95i	0.410660i
$559$	2437.03	0.184392
$560$	0	0
$561$	4196.92	0.315854
$562$	15995.1i	1.20055i
$563$	− 7988.51i	− 0.598003i	−0.954253	−	0.299001i	$-0.903346\pi$
$563$	− 7988.51i	− 0.598003i	0.954253	−	0.299001i	$-0.0966535\pi$
$564$	−1438.12	−0.107368
$565$	0	0
$566$	−11616.8	−0.862706
$567$	− 526.981i	− 0.0390320i
$568$	− 1145.73i	− 0.0846368i
$569$	6429.49	0.473705	0.236853	−	0.971546i	$-0.423884\pi$
$569$	6429.49	0.473705	0.236853	+	0.971546i	$0.423884\pi$
$570$	0	0
$571$	2886.51	0.211553	0.105776	−	0.994390i	$-0.466267\pi$
$571$	2886.51	0.211553	0.105776	+	0.994390i	$0.466267\pi$
$572$	− 1599.73i	− 0.116937i
$573$	13540.0i	0.987158i
$574$	16444.4	1.19577
$575$	0	0
$576$	−1047.76	−0.0757926
$577$	− 9966.59i	− 0.719089i	−0.933128	−	0.359545i	$-0.882932\pi$
$577$	− 9966.59i	− 0.719089i	0.933128	−	0.359545i	$-0.117068\pi$
$578$	− 18494.2i	− 1.33089i
$579$	−3327.10	−0.238807
$580$	0	0
$581$	−22917.4	−1.63644
$582$	6118.95i	0.435806i
$583$	− 7444.40i	− 0.528843i
$584$	1265.71	0.0896838
$585$	0	0
$586$	−17709.3	−1.24841
$587$	− 3522.91i	− 0.247710i	−0.992300	−	0.123855i	$-0.960474\pi$
$587$	− 3522.91i	− 0.247710i	0.992300	−	0.123855i	$-0.0395258\pi$
$588$	5599.70i	0.392734i
$589$	3195.97	0.223579
$590$	0	0
$591$	1299.65	0.0904575
$592$	3236.81i	0.224716i
$593$	28713.0i	1.98837i	0.107695	+	0.994184i	$0.465653\pi$
$593$	28713.0i	1.98837i	−0.107695	+	0.994184i	$0.534347\pi$
$594$	3059.35	0.211325
$595$	0	0
$596$	4101.84	0.281909
$597$	− 13469.1i	− 0.923374i
$598$	− 1700.54i	− 0.116288i
$599$	−7736.44	−0.527717	−0.263858	−	0.964561i	$-0.584995\pi$
$599$	−7736.44	−0.527717	−0.263858	+	0.964561i	$0.584995\pi$
$600$	0	0
$601$	−3863.89	−0.262249	−0.131124	−	0.991366i	$-0.541859\pi$
$601$	−3863.89	−0.262249	−0.131124	+	0.991366i	$0.541859\pi$
$602$	− 3664.23i	− 0.248078i
$603$	− 10531.4i	− 0.711232i
$604$	6066.68	0.408692
$605$	0	0
$606$	4488.80	0.300900
$607$	24954.6i	1.66866i	0.551265	+	0.834330i	$0.314145\pi$
$607$	24954.6i	1.66866i	−0.551265	+	0.834330i	$0.685855\pi$
$608$	618.628i	0.0412643i
$609$	21247.3	1.41377
$610$	0	0
$611$	−4076.82	−0.269935
$612$	− 7792.44i	− 0.514690i
$613$	− 12712.0i	− 0.837574i	−0.908085	−	0.418787i	$-0.862455\pi$
$613$	− 12712.0i	− 0.837574i	0.908085	−	0.418787i	$-0.137545\pi$
$614$	3002.06	0.197318
$615$	0	0
$616$	−2405.29	−0.157324
$617$	11017.6i	0.718884i	0.933167	+	0.359442i	$0.117033\pi$
$617$	11017.6i	0.718884i	−0.933167	+	0.359442i	$0.882967\pi$
$618$	13311.7i	0.866461i
$619$	−6859.65	−0.445416	−0.222708	−	0.974885i	$-0.571490\pi$
$619$	−6859.65	−0.445416	−0.222708	+	0.974885i	$0.571490\pi$
$620$	0	0
$621$	3252.16	0.210152
$622$	− 14881.0i	− 0.959280i
$623$	24447.9i	1.57221i
$624$	1928.38	0.123713
$625$	0	0
$626$	−5057.25	−0.322889
$627$	− 681.833i	− 0.0434287i
$628$	11029.2i	0.700819i
$629$	−24073.0	−1.52600
$630$	0	0
$631$	−14104.0	−0.889815	−0.444907	−	0.895577i	$-0.646763\pi$
$631$	−14104.0	−0.889815	−0.444907	+	0.895577i	$0.646763\pi$
$632$	8986.75i	0.565623i
$633$	2693.94i	0.169154i
$634$	19273.4	1.20733
$635$	0	0
$636$	8973.79	0.559487
$637$	15874.2i	0.987376i
$638$	5073.73i	0.314844i
$639$	2344.62	0.145151
$640$	0	0
$641$	−144.931	−0.00893047	−0.00446524	−	0.999990i	$-0.501421\pi$
$641$	−144.931	−0.00893047	−0.00446524	+	0.999990i	$0.501421\pi$
$642$	− 1874.08i	− 0.115209i
$643$	− 30891.4i	− 1.89461i	−0.320328	−	0.947307i	$-0.603793\pi$
$643$	− 30891.4i	− 1.89461i	0.320328	−	0.947307i	$-0.396207\pi$
$644$	−2556.87	−0.156452
$645$	0	0
$646$	−4600.90	−0.280217
$647$	22784.1i	1.38444i	0.721684	+	0.692222i	$0.243368\pi$
$647$	22784.1i	1.38444i	−0.721684	+	0.692222i	$0.756632\pi$
$648$	151.692i	0.00919605i
$649$	114.508	0.00692576
$650$	0	0
$651$	−14979.1	−0.901811
$652$	5053.17i	0.303524i
$653$	32125.7i	1.92523i	0.270868	+	0.962617i	$0.412689\pi$
$653$	32125.7i	1.92523i	−0.270868	+	0.962617i	$0.587311\pi$
$654$	−7898.05	−0.472230
$655$	0	0
$656$	−4733.54	−0.281728
$657$	2590.14i	0.153807i
$658$	6129.75i	0.363165i
$659$	−15521.7	−0.917508	−0.458754	−	0.888563i	$-0.651704\pi$
$659$	−15521.7	−0.917508	−0.458754	+	0.888563i	$0.651704\pi$
$660$	0	0
$661$	19428.4	1.14323	0.571617	−	0.820521i	$-0.306316\pi$
$661$	19428.4	1.14323	0.571617	+	0.820521i	$0.306316\pi$
$662$	5751.64i	0.337680i
$663$	14341.8i	0.840106i
$664$	6596.80	0.385550
$665$	0	0
$666$	−6623.80	−0.385386
$667$	5393.48i	0.313098i
$668$	1398.58i	0.0810072i
$669$	3619.11	0.209152
$670$	0	0
$671$	−1196.27	−0.0688246
$672$	− 2899.43i	− 0.166441i
$673$	− 18992.7i	− 1.08784i	−0.839138	−	0.543919i	$-0.816940\pi$
$673$	− 18992.7i	− 1.08784i	0.839138	−	0.543919i	$-0.183060\pi$
$674$	−19822.4	−1.13283
$675$	0	0
$676$	−3321.38	−0.188972
$677$	19104.3i	1.08455i	0.840202	+	0.542273i	$0.182436\pi$
$677$	19104.3i	1.08455i	−0.840202	+	0.542273i	$0.817564\pi$
$678$	4833.65i	0.273798i
$679$	26081.1	1.47408
$680$	0	0
$681$	16033.3	0.902197
$682$	− 3576.92i	− 0.200832i
$683$	24737.2i	1.38586i	0.721004	+	0.692931i	$0.243681\pi$
$683$	24737.2i	1.38586i	−0.721004	+	0.692931i	$0.756319\pi$
$684$	−1265.96	−0.0707679
$685$	0	0
$686$	4802.49	0.267288
$687$	4533.69i	0.251778i
$688$	1054.75i	0.0584479i
$689$	25439.2	1.40661
$690$	0	0
$691$	12957.9	0.713373	0.356687	−	0.934224i	$-0.383906\pi$
$691$	12957.9	0.713373	0.356687	+	0.934224i	$0.383906\pi$
$692$	9255.33i	0.508432i
$693$	− 4922.18i	− 0.269810i
$694$	−8995.41	−0.492019
$695$	0	0
$696$	−6116.08	−0.333088
$697$	− 35204.5i	− 1.91315i
$698$	− 1776.22i	− 0.0963195i
$699$	−11115.9	−0.601490
$700$	0	0
$701$	20730.7	1.11696	0.558478	−	0.829519i	$-0.311385\pi$
$701$	20730.7	1.11696	0.558478	+	0.829519i	$0.311385\pi$
$702$	10454.5i	0.562079i
$703$	3910.90i	0.209819i
$704$	692.367	0.0370661
$705$	0	0
$706$	−23444.5	−1.24978
$707$	− 19132.8i	− 1.01777i
$708$	138.032i	0.00732708i
$709$	22191.2	1.17547	0.587735	−	0.809053i	$-0.300020\pi$
$709$	22191.2	1.17547	0.587735	+	0.809053i	$0.300020\pi$
$710$	0	0
$711$	−18390.5	−0.970038
$712$	− 7037.38i	− 0.370417i
$713$	− 3802.34i	− 0.199718i
$714$	21563.8	1.13026
$715$	0	0
$716$	−9390.77	−0.490153
$717$	− 6704.07i	− 0.349188i
$718$	2514.98i	0.130722i
$719$	773.962	0.0401445	0.0200723	−	0.999799i	$-0.493610\pi$
$719$	773.962	0.0401445	0.0200723	+	0.999799i	$0.493610\pi$
$720$	0	0
$721$	56738.9	2.93075
$722$	− 12970.5i	− 0.668578i
$723$	− 6380.29i	− 0.328196i
$724$	−17585.3	−0.902695
$725$	0	0
$726$	7915.51	0.404645
$727$	− 32484.7i	− 1.65721i	−0.559835	−	0.828604i	$-0.689136\pi$
$727$	− 32484.7i	− 1.65721i	0.559835	−	0.828604i	$-0.310864\pi$
$728$	− 8219.41i	− 0.418450i
$729$	−12757.0	−0.648122
$730$	0	0
$731$	−7844.48	−0.396906
$732$	− 1442.03i	− 0.0728127i
$733$	13701.5i	0.690416i	0.938526	+	0.345208i	$0.112192\pi$
$733$	13701.5i	0.690416i	−0.938526	+	0.345208i	$0.887808\pi$
$734$	−20244.5	−1.01803
$735$	0	0
$736$	736.000	0.0368605
$737$	6959.26i	0.347826i
$738$	− 9686.71i	− 0.483161i
$739$	31474.2	1.56671	0.783354	−	0.621576i	$-0.213507\pi$
$739$	31474.2	1.56671	0.783354	+	0.621576i	$0.213507\pi$
$740$	0	0
$741$	2329.98	0.115511
$742$	− 38249.4i	− 1.89243i
$743$	− 28664.2i	− 1.41533i	−0.706549	−	0.707664i	$-0.749749\pi$
$743$	− 28664.2i	− 1.41533i	0.706549	−	0.707664i	$-0.250251\pi$
$744$	4311.77	0.212469
$745$	0	0
$746$	13060.2	0.640974
$747$	13499.7i	0.661215i
$748$	5149.31i	0.251708i
$749$	−7988.00	−0.389687
$750$	0	0
$751$	−25904.0	−1.25866	−0.629328	−	0.777140i	$-0.716670\pi$
$751$	−25904.0	−1.25866	−0.629328	+	0.777140i	$0.716670\pi$
$752$	− 1764.46i	− 0.0855628i
$753$	− 6370.84i	− 0.308322i
$754$	−17338.1	−0.837420
$755$	0	0
$756$	15719.0	0.756209
$757$	2569.70i	0.123378i	0.998095	+	0.0616892i	$0.0196488\pi$
$757$	2569.70i	0.123378i	−0.998095	+	0.0616892i	$0.980351\pi$
$758$	− 555.700i	− 0.0266279i
$759$	−811.197	−0.0387939
$760$	0	0
$761$	−23219.4	−1.10605	−0.553024	−	0.833165i	$-0.686526\pi$
$761$	−23219.4	−1.10605	−0.553024	+	0.833165i	$0.686526\pi$
$762$	− 10302.1i	− 0.489774i
$763$	33664.3i	1.59728i
$764$	−16612.6	−0.786677
$765$	0	0
$766$	−21359.3	−1.00750
$767$	391.298i	0.0184211i
$768$	834.607i	0.0392139i
$769$	−34771.1	−1.63053	−0.815265	−	0.579088i	$-0.803409\pi$
$769$	−34771.1	−1.63053	−0.815265	+	0.579088i	$0.803409\pi$
$770$	0	0
$771$	−6384.76	−0.298238
$772$	− 4082.10i	− 0.190308i
$773$	36377.2i	1.69262i	0.532687	+	0.846312i	$0.321182\pi$
$773$	36377.2i	1.69262i	−0.532687	+	0.846312i	$0.678818\pi$
$774$	−2158.45	−0.100238
$775$	0	0
$776$	−7507.50	−0.347298
$777$	− 18329.9i	− 0.846309i
$778$	− 3092.10i	− 0.142490i
$779$	−5719.34	−0.263051
$780$	0	0
$781$	−1549.34	−0.0709858
$782$	5473.82i	0.250311i
$783$	− 33157.7i	− 1.51336i
$784$	−6870.40	−0.312974
$785$	0	0
$786$	15315.4	0.695017
$787$	− 1814.43i	− 0.0821824i	−0.999155	−	0.0410912i	$-0.986917\pi$
$787$	− 1814.43i	− 0.0821824i	0.999155	−	0.0410912i	$-0.0130834\pi$
$788$	1594.57i	0.0720866i
$789$	23490.4	1.05992
$790$	0	0
$791$	20602.7	0.926104
$792$	1416.86i	0.0635680i
$793$	− 4087.91i	− 0.183059i
$794$	25803.5	1.15332
$795$	0	0
$796$	16525.6	0.735847
$797$	− 7958.23i	− 0.353695i	−0.984238	−	0.176848i	$-0.943410\pi$
$797$	− 7958.23i	− 0.353695i	0.984238	−	0.176848i	$-0.0565900\pi$
$798$	− 3503.27i	− 0.155406i
$799$	13122.7	0.581038
$800$	0	0
$801$	14401.3	0.635261
$802$	− 17068.6i	− 0.751513i
$803$	− 1711.59i	− 0.0752188i
$804$	−8388.98	−0.367981
$805$	0	0
$806$	12223.1	0.534171
$807$	12272.3i	0.535323i
$808$	5507.42i	0.239790i
$809$	−20506.9	−0.891203	−0.445601	−	0.895232i	$-0.647010\pi$
$809$	−20506.9	−0.891203	−0.445601	+	0.895232i	$0.647010\pi$
$810$	0	0
$811$	1962.00	0.0849506	0.0424753	−	0.999098i	$-0.486476\pi$
$811$	1962.00	0.0849506	0.0424753	+	0.999098i	$0.486476\pi$
$812$	26068.9i	1.12665i
$813$	− 16981.9i	− 0.732573i
$814$	4377.07	0.188472
$815$	0	0
$816$	−6207.19	−0.266293
$817$	1274.42i	0.0545730i
$818$	− 1301.94i	− 0.0556493i
$819$	16820.2	0.717638
$820$	0	0
$821$	25168.1	1.06988	0.534941	−	0.844889i	$-0.320334\pi$
$821$	25168.1	1.06988	0.534941	+	0.844889i	$0.320334\pi$
$822$	4026.77i	0.170863i
$823$	− 3337.33i	− 0.141351i	−0.997499	−	0.0706756i	$-0.977484\pi$
$823$	− 3337.33i	− 0.141351i	0.997499	−	0.0706756i	$-0.0225155\pi$
$824$	−16332.4	−0.690492
$825$	0	0
$826$	588.342	0.0247833
$827$	− 13087.4i	− 0.550294i	−0.961402	−	0.275147i	$-0.911273\pi$
$827$	− 13087.4i	− 0.550294i	0.961402	−	0.275147i	$-0.0887265\pi$
$828$	1506.15i	0.0632154i
$829$	38486.8	1.61243	0.806213	−	0.591626i	$-0.201514\pi$
$829$	38486.8	1.61243	0.806213	+	0.591626i	$0.201514\pi$
$830$	0	0
$831$	−5053.97	−0.210975
$832$	2365.97i	0.0985881i
$833$	− 51097.0i	− 2.12534i
$834$	9844.12	0.408722
$835$	0	0
$836$	836.558	0.0346088
$837$	23375.8i	0.965337i
$838$	− 26532.4i	− 1.09373i
$839$	−8192.40	−0.337107	−0.168554	−	0.985692i	$-0.553910\pi$
$839$	−8192.40	−0.337107	−0.168554	+	0.985692i	$0.553910\pi$
$840$	0	0
$841$	30600.7	1.25469
$842$	− 16549.4i	− 0.677350i
$843$	26073.4i	1.06526i
$844$	−3305.26	−0.134801
$845$	0	0
$846$	3610.79	0.146739
$847$	− 33738.7i	− 1.36868i
$848$	11010.2i	0.445861i
$849$	−18936.5	−0.765487
$850$	0	0
$851$	4652.91	0.187426
$852$	− 1867.64i	− 0.0750991i
$853$	6774.83i	0.271941i	0.990713	+	0.135971i	$0.0434152\pi$
$853$	6774.83i	0.271941i	−0.990713	+	0.135971i	$0.956585\pi$
$854$	−6146.42	−0.246284
$855$	0	0
$856$	2299.36	0.0918113
$857$	19272.3i	0.768177i	0.923296	+	0.384089i	$0.125484\pi$
$857$	19272.3i	0.768177i	−0.923296	+	0.384089i	$0.874516\pi$
$858$	− 2607.70i	− 0.103759i
$859$	5991.11	0.237968	0.118984	−	0.992896i	$-0.462036\pi$
$859$	5991.11	0.237968	0.118984	+	0.992896i	$0.462036\pi$
$860$	0	0
$861$	26805.8	1.06102
$862$	1448.17i	0.0572214i
$863$	− 20711.4i	− 0.816945i	−0.912771	−	0.408472i	$-0.866062\pi$
$863$	− 20711.4i	− 0.816945i	0.912771	−	0.408472i	$-0.133938\pi$
$864$	−4524.74	−0.178165
$865$	0	0
$866$	−23993.1	−0.941476
$867$	− 30147.2i	− 1.18091i
$868$	− 18378.3i	− 0.718663i
$869$	12152.6	0.474394
$870$	0	0
$871$	−23781.4	−0.925144
$872$	− 9690.31i	− 0.376325i
$873$	− 15363.3i	− 0.595613i
$874$	889.278	0.0344168
$875$	0	0
$876$	2063.22	0.0795774
$877$	− 45757.4i	− 1.76182i	−0.473281	−	0.880912i	$-0.656930\pi$
$877$	− 45757.4i	− 1.76182i	0.473281	−	0.880912i	$-0.343070\pi$
$878$	15530.8i	0.596969i
$879$	−28867.8	−1.10772
$880$	0	0
$881$	37019.6	1.41569	0.707845	−	0.706367i	$-0.249667\pi$
$881$	37019.6	1.41569	0.707845	+	0.706367i	$0.249667\pi$
$882$	− 14059.6i	− 0.536747i
$883$	9311.25i	0.354868i	0.984133	+	0.177434i	$0.0567797\pi$
$883$	9311.25i	0.354868i	−0.984133	+	0.177434i	$0.943220\pi$
$884$	−17596.3	−0.669490
$885$	0	0
$886$	−18123.1	−0.687197
$887$	33041.8i	1.25077i	0.780315	+	0.625387i	$0.215059\pi$
$887$	33041.8i	1.25077i	−0.780315	+	0.625387i	$0.784941\pi$
$888$	5276.30i	0.199393i
$889$	−43911.4	−1.65662
$890$	0	0
$891$	205.130	0.00771283
$892$	4440.37i	0.166676i
$893$	− 2131.92i	− 0.0798903i
$894$	6686.37	0.250141
$895$	0	0
$896$	3557.39	0.132638
$897$	− 2772.04i	− 0.103184i
$898$	21888.6i	0.813398i
$899$	−38767.2	−1.43822
$900$	0	0
$901$	−81885.4	−3.02774
$902$	6401.06i	0.236288i
$903$	− 5973.03i	− 0.220122i
$904$	−5930.53	−0.218193
$905$	0	0
$906$	9889.25	0.362636
$907$	− 41359.1i	− 1.51412i	−0.653346	−	0.757060i	$-0.726635\pi$
$907$	− 41359.1i	− 1.51412i	0.653346	−	0.757060i	$-0.273365\pi$
$908$	19671.6i	0.718971i
$909$	−11270.4	−0.411238
$910$	0	0
$911$	−16960.3	−0.616818	−0.308409	−	0.951254i	$-0.599797\pi$
$911$	−16960.3	−0.616818	−0.308409	+	0.951254i	$0.599797\pi$
$912$	1008.42i	0.0366142i
$913$	− 8920.71i	− 0.323365i
$914$	−21869.8	−0.791452
$915$	0	0
$916$	−5562.50	−0.200644
$917$	− 65279.7i	− 2.35085i
$918$	− 33651.7i	− 1.20988i
$919$	−3085.38	−0.110748	−0.0553739	−	0.998466i	$-0.517635\pi$
$919$	−3085.38	−0.110748	−0.0553739	+	0.998466i	$0.517635\pi$
$920$	0	0
$921$	4893.63	0.175082
$922$	5304.51i	0.189474i
$923$	− 5294.46i	− 0.188807i
$924$	−3920.84	−0.139596
$925$	0	0
$926$	1415.40	0.0502299
$927$	− 33422.6i	− 1.18419i
$928$	− 7503.97i	− 0.265442i
$929$	42292.0	1.49360	0.746801	−	0.665048i	$-0.231589\pi$
$929$	42292.0	1.49360	0.746801	+	0.665048i	$0.231589\pi$
$930$	0	0
$931$	−8301.22	−0.292225
$932$	− 13638.4i	− 0.479334i
$933$	− 24257.3i	− 0.851178i
$934$	21342.1	0.747682
$935$	0	0
$936$	−4841.72	−0.169078
$937$	15746.4i	0.549000i	0.961587	+	0.274500i	$0.0885123\pi$
$937$	15746.4i	0.549000i	−0.961587	+	0.274500i	$0.911488\pi$
$938$	35756.8i	1.24467i
$939$	−8243.79	−0.286502
$940$	0	0
$941$	52033.4	1.80259	0.901296	−	0.433204i	$-0.142617\pi$
$941$	52033.4	1.80259	0.901296	+	0.433204i	$0.142617\pi$
$942$	17978.7i	0.621843i
$943$	6804.46i	0.234977i
$944$	−169.355	−0.00583903
$945$	0	0
$946$	1426.32	0.0490208
$947$	− 17293.2i	− 0.593405i	−0.954970	−	0.296702i	$-0.904113\pi$
$947$	− 17293.2i	− 0.593405i	0.954970	−	0.296702i	$-0.0958869\pi$
$948$	14649.2i	0.501883i
$949$	5848.88	0.200066
$950$	0	0
$951$	31417.5	1.07127
$952$	26457.2i	0.900717i
$953$	− 8989.54i	− 0.305561i	−0.988260	−	0.152781i	$-0.951177\pi$
$953$	− 8989.54i	− 0.305561i	0.988260	−	0.152781i	$-0.0488228\pi$
$954$	−22531.2	−0.764648
$955$	0	0
$956$	8225.39	0.278272
$957$	8270.64i	0.279365i
$958$	− 3902.34i	− 0.131606i
$959$	17163.5	0.577933
$960$	0	0
$961$	−2460.53	−0.0825931
$962$	14957.4i	0.501296i
$963$	4705.41i	0.157456i
$964$	7828.13	0.261543
$965$	0	0
$966$	−4167.94	−0.138821
$967$	23285.3i	0.774358i	0.922005	+	0.387179i	$0.126550\pi$
$967$	23285.3i	0.774358i	−0.922005	+	0.387179i	$0.873450\pi$
$968$	9711.73i	0.322466i
$969$	−7499.89	−0.248639
$970$	0	0
$971$	44316.5	1.46466	0.732330	−	0.680950i	$-0.238433\pi$
$971$	44316.5	1.46466	0.732330	+	0.680950i	$0.238433\pi$
$972$	− 15023.7i	− 0.495768i
$973$	− 41959.1i	− 1.38247i
$974$	−34142.4	−1.12320
$975$	0	0
$976$	1769.26	0.0580252
$977$	761.665i	0.0249415i	0.999922	+	0.0124707i	$0.00396966\pi$
$977$	761.665i	0.0249415i	−0.999922	+	0.0124707i	$0.996030\pi$
$978$	8237.13i	0.269319i
$979$	−9516.49	−0.310673
$980$	0	0
$981$	19830.3	0.645394
$982$	33959.8i	1.10356i
$983$	17423.3i	0.565328i	0.959219	+	0.282664i	$0.0912182\pi$
$983$	17423.3i	0.565328i	−0.959219	+	0.282664i	$0.908782\pi$
$984$	−7716.10	−0.249980
$985$	0	0
$986$	55808.9	1.80255
$987$	9992.06i	0.322240i
$988$	2858.70i	0.0920521i
$989$	1516.21	0.0487489
$990$	0	0
$991$	−33873.1	−1.08579	−0.542894	−	0.839801i	$-0.682671\pi$
$991$	−33873.1	−1.08579	−0.542894	+	0.839801i	$0.682671\pi$
$992$	5290.22i	0.169319i
$993$	9375.71i	0.299626i
$994$	−7960.55	−0.254017
$995$	0	0
$996$	10753.4	0.342103
$997$	− 31163.3i	− 0.989921i	−0.868915	−	0.494960i	$-0.835183\pi$
$997$	− 31163.3i	− 0.989921i	0.868915	−	0.494960i	$-0.164817\pi$
$998$	5577.66i	0.176912i
$999$	−28604.9	−0.905925

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.b.m.599.7		8
5.2	odd	4		1150.4.a.o.1.3		4
5.3	odd	4		230.4.a.i.1.2	✓	4
5.4	even	2	inner	1150.4.b.m.599.2		8
15.8	even	4		2070.4.a.bi.1.4		4
20.3	even	4		1840.4.a.l.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.i.1.2	✓	4	5.3	odd	4
1150.4.a.o.1.3		4	5.2	odd	4
1150.4.b.m.599.2		8	5.4	even	2	inner
1150.4.b.m.599.7		8	1.1	even	1	trivial
1840.4.a.l.1.3		4	20.3	even	4
2070.4.a.bi.1.4		4	15.8	even	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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +3.26018i q^{3} -4.00000 q^{4} -6.52037 q^{6} +27.7921i q^{7} -8.00000i q^{8} +16.3712 q^{9} -10.8182 q^{11} -13.0407i q^{12} -36.9683i q^{13} -55.5842 q^{14} +16.0000 q^{16} +118.996i q^{17} +32.7424i q^{18} +19.3321 q^{19} -90.6073 q^{21} -21.6365i q^{22} -23.0000i q^{23} +26.0815 q^{24} +73.9366 q^{26} +141.398i q^{27} -111.168i q^{28} -234.499 q^{29} +165.319 q^{31} +32.0000i q^{32} -35.2694i q^{33} -237.992 q^{34} -65.4848 q^{36} +202.301i q^{37} +38.6643i q^{38} +120.523 q^{39} -295.846 q^{41} -181.215i q^{42} +65.9221i q^{43} +43.2729 q^{44} +46.0000 q^{46} -110.279i q^{47} +52.1630i q^{48} -429.400 q^{49} -387.949 q^{51} +147.873i q^{52} +688.135i q^{53} -282.796 q^{54} +222.337 q^{56} +63.0263i q^{57} -468.998i q^{58} -10.5847 q^{59} +110.579 q^{61} +330.639i q^{62} +454.990i q^{63} -64.0000 q^{64} +70.5388 q^{66} -643.290i q^{67} -475.984i q^{68} +74.9842 q^{69} +143.216 q^{71} -130.970i q^{72} +158.213i q^{73} -404.601 q^{74} -77.3285 q^{76} -300.661i q^{77} +241.047i q^{78} -1123.34 q^{79} -18.9616 q^{81} -591.692i q^{82} +824.600i q^{83} +362.429 q^{84} -131.844 q^{86} -764.510i q^{87} +86.5458i q^{88} +879.672 q^{89} +1027.43 q^{91} +92.0000i q^{92} +538.971i q^{93} +220.557 q^{94} -104.326 q^{96} -938.437i q^{97} -858.801i q^{98} -177.107 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{4} + 16 q^{6} - 128 q^{9} + 186 q^{11} - 104 q^{14} + 128 q^{16} - 370 q^{19} + 604 q^{21} - 64 q^{24} + 128 q^{26} - 588 q^{29} - 422 q^{31} - 432 q^{34} + 512 q^{36} + 842 q^{39} - 738 q^{41}+ \cdots - 4026 q^{99}+O(q^{100}) \) 8 * q - 32 * q^4 + 16 * q^6 - 128 * q^9 + 186 * q^11 - 104 * q^14 + 128 * q^16 - 370 * q^19 + 604 * q^21 - 64 * q^24 + 128 * q^26 - 588 * q^29 - 422 * q^31 - 432 * q^34 + 512 * q^36 + 842 * q^39 - 738 * q^41 - 744 * q^44 + 368 * q^46 - 1200 * q^49 - 630 * q^51 - 1036 * q^54 + 416 * q^56 + 66 * q^59 - 614 * q^61 - 512 * q^64 - 948 * q^66 - 184 * q^69 - 2514 * q^71 - 20 * q^74 + 1480 * q^76 - 2404 * q^79 - 1984 * q^81 - 2416 * q^84 - 400 * q^86 + 1968 * q^89 - 1990 * q^91 + 1452 * q^94 + 256 * q^96 - 4026 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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