LMFDB - Embedded newform 1150.4.b.s.599.8

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 = [12,0,0,-48,0,-20,0,0,-122,0,-98] 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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 219x^{10} + 17685x^{8} + 640366x^{6} + 10000368x^{4} + 54897345x^{2} + 95531076 $ x^12 + 219x^10 + 17685x^8 + 640366x^6 + 10000368x^4 + 54897345*x^2 + 95531076
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.8
Root			$-6.96868i$ of defining polynomial
Character	$\chi$	$=$	1150.599
Dual form			1150.4.b.s.599.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000i q^{2} -5.96868i q^{3} -4.00000 q^{4} +11.9374 q^{6} -15.2484i q^{7} -8.00000i q^{8} -8.62516 q^{9} +12.0268 q^{11} +23.8747i q^{12} +90.1983i q^{13} +30.4968 q^{14} +16.0000 q^{16} -83.9831i q^{17} -17.2503i q^{18} -74.8503 q^{19} -91.0130 q^{21} +24.0537i q^{22} +23.0000i q^{23} -47.7495 q^{24} -180.397 q^{26} -109.674i q^{27} +60.9937i q^{28} -94.5321 q^{29} -178.393 q^{31} +32.0000i q^{32} -71.7844i q^{33} +167.966 q^{34} +34.5007 q^{36} -296.487i q^{37} -149.701i q^{38} +538.365 q^{39} +193.074 q^{41} -182.026i q^{42} +15.4576i q^{43} -48.1074 q^{44} -46.0000 q^{46} -128.899i q^{47} -95.4989i q^{48} +110.486 q^{49} -501.269 q^{51} -360.793i q^{52} +342.961i q^{53} +219.347 q^{54} -121.987 q^{56} +446.758i q^{57} -189.064i q^{58} -725.012 q^{59} -784.288 q^{61} -356.786i q^{62} +131.520i q^{63} -64.0000 q^{64} +143.569 q^{66} +782.822i q^{67} +335.932i q^{68} +137.280 q^{69} -141.768 q^{71} +69.0013i q^{72} +404.340i q^{73} +592.973 q^{74} +299.401 q^{76} -183.390i q^{77} +1076.73i q^{78} +721.038 q^{79} -887.486 q^{81} +386.147i q^{82} -985.391i q^{83} +364.052 q^{84} -30.9153 q^{86} +564.232i q^{87} -96.2148i q^{88} -463.054 q^{89} +1375.38 q^{91} -92.0000i q^{92} +1064.77i q^{93} +257.798 q^{94} +190.998 q^{96} +1680.05i q^{97} +220.971i q^{98} -103.734 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 48 q^{4} - 20 q^{6} - 122 q^{9} - 98 q^{11} + 168 q^{14} + 192 q^{16} + 458 q^{19} + 184 q^{21} + 80 q^{24} - 64 q^{26} + 364 q^{29} + 228 q^{31} + 700 q^{34} + 488 q^{36} + 286 q^{39} + 486 q^{41}+ \cdots + 284 q^{99}+O(q^{100}) $ 12 * q - 48 * q^4 - 20 * q^6 - 122 * q^9 - 98 * q^11 + 168 * q^14 + 192 * q^16 + 458 * q^19 + 184 * q^21 + 80 * q^24 - 64 * q^26 + 364 * q^29 + 228 * q^31 + 700 * q^34 + 488 * q^36 + 286 * q^39 + 486 * q^41 + 392 * q^44 - 552 * q^46 - 1296 * q^49 - 2062 * q^51 + 1376 * q^54 - 672 * q^56 + 1118 * q^59 - 1376 * q^61 - 768 * q^64 - 2564 * q^66 - 230 * q^69 + 1168 * q^71 + 256 * q^74 - 1832 * q^76 + 2864 * q^79 + 68 * q^81 - 736 * q^84 - 728 * q^86 + 1182 * q^89 - 5728 * q^91 + 1992 * q^94 - 320 * q^96 + 284 * 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
$3$	− 5.96868i	− 1.14867i	−0.818619	−	0.574337i	$-0.805260\pi$
$3$	− 5.96868i	− 1.14867i	0.818619	−	0.574337i	$-0.194740\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	11.9374	0.812235
$7$	− 15.2484i	− 0.823337i	−0.911334	−	0.411669i	$-0.864946\pi$
$7$	− 15.2484i	− 0.823337i	0.911334	−	0.411669i	$-0.135054\pi$
$8$	− 8.00000i	− 0.353553i
$9$	−8.62516	−0.319451
$10$	0	0
$11$	12.0268	0.329657	0.164829	−	0.986322i	$-0.447293\pi$
$11$	12.0268	0.329657	0.164829	+	0.986322i	$0.447293\pi$
$12$	23.8747i	0.574337i
$13$	90.1983i	1.92435i	0.272438	+	0.962173i	$0.412170\pi$
$13$	90.1983i	1.92435i	−0.272438	+	0.962173i	$0.587830\pi$
$14$	30.4968	0.582187
$15$	0	0
$16$	16.0000	0.250000
$17$	− 83.9831i	− 1.19817i	−0.800685	−	0.599085i	$-0.795531\pi$
$17$	− 83.9831i	− 1.19817i	0.800685	−	0.599085i	$-0.204469\pi$
$18$	− 17.2503i	− 0.225886i
$19$	−74.8503	−0.903781	−0.451890	−	0.892073i	$-0.649250\pi$
$19$	−74.8503	−0.903781	−0.451890	+	0.892073i	$0.649250\pi$
$20$	0	0
$21$	−91.0130	−0.945746
$22$	24.0537i	0.233103i
$23$	23.0000i	0.208514i
$23$	23.0000i	0.208514i
$24$	−47.7495	−0.406117
$25$	0	0
$26$	−180.397	−1.36072
$27$	− 109.674i	− 0.781729i
$28$	60.9937i	0.411669i
$29$	−94.5321	−0.605316	−0.302658	−	0.953099i	$-0.597874\pi$
$29$	−94.5321	−0.605316	−0.302658	+	0.953099i	$0.597874\pi$
$30$	0	0
$31$	−178.393	−1.03356	−0.516779	−	0.856119i	$-0.672869\pi$
$31$	−178.393	−1.03356	−0.516779	+	0.856119i	$0.672869\pi$
$32$	32.0000i	0.176777i
$33$	− 71.7844i	− 0.378669i
$34$	167.966	0.847234
$35$	0	0
$36$	34.5007	0.159725
$37$	− 296.487i	− 1.31735i	−0.752426	−	0.658677i	$-0.771116\pi$
$37$	− 296.487i	− 1.31735i	0.752426	−	0.658677i	$-0.228884\pi$
$38$	− 149.701i	− 0.639069i
$39$	538.365	2.21045
$40$	0	0
$41$	193.074	0.735440	0.367720	−	0.929937i	$-0.380139\pi$
$41$	193.074	0.735440	0.367720	+	0.929937i	$0.380139\pi$
$42$	− 182.026i	− 0.668743i
$43$	15.4576i	0.0548202i	0.999624	+	0.0274101i	$0.00872600\pi$
$43$	15.4576i	0.0548202i	−0.999624	+	0.0274101i	$0.991274\pi$
$44$	−48.1074	−0.164829
$45$	0	0
$46$	−46.0000	−0.147442
$47$	− 128.899i	− 0.400040i	−0.979792	−	0.200020i	$-0.935899\pi$
$47$	− 128.899i	− 0.400040i	0.979792	−	0.200020i	$-0.0641006\pi$
$48$	− 95.4989i	− 0.287168i
$49$	110.486	0.322116
$50$	0	0
$51$	−501.269	−1.37631
$52$	− 360.793i	− 0.962173i
$53$	342.961i	0.888855i	0.895815	+	0.444428i	$0.146593\pi$
$53$	342.961i	0.888855i	−0.895815	+	0.444428i	$0.853407\pi$
$54$	219.347	0.552766
$55$	0	0
$56$	−121.987	−0.291094
$57$	446.758i	1.03815i
$58$	− 189.064i	− 0.428023i
$59$	−725.012	−1.59980	−0.799902	−	0.600130i	$-0.795115\pi$
$59$	−725.012	−1.59980	−0.799902	+	0.600130i	$0.795115\pi$
$60$	0	0
$61$	−784.288	−1.64619	−0.823097	−	0.567901i	$-0.807756\pi$
$61$	−784.288	−1.64619	−0.823097	+	0.567901i	$0.807756\pi$
$62$	− 356.786i	− 0.730837i
$63$	131.520i	0.263016i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	143.569	0.267759
$67$	782.822i	1.42742i	0.700443	+	0.713708i	$0.252986\pi$
$67$	782.822i	1.42742i	−0.700443	+	0.713708i	$0.747014\pi$
$68$	335.932i	0.599085i
$69$	137.280	0.239515
$70$	0	0
$71$	−141.768	−0.236968	−0.118484	−	0.992956i	$-0.537803\pi$
$71$	−141.768	−0.236968	−0.118484	+	0.992956i	$0.537803\pi$
$72$	69.0013i	0.112943i
$73$	404.340i	0.648280i	0.946009	+	0.324140i	$0.105075\pi$
$73$	404.340i	0.648280i	−0.946009	+	0.324140i	$0.894925\pi$
$74$	592.973	0.931510
$75$	0	0
$76$	299.401	0.451890
$77$	− 183.390i	− 0.271419i
$78$	1076.73i	1.56302i
$79$	721.038	1.02687	0.513437	−	0.858127i	$-0.328372\pi$
$79$	721.038	1.02687	0.513437	+	0.858127i	$0.328372\pi$
$80$	0	0
$81$	−887.486	−1.21740
$82$	386.147i	0.520035i
$83$	− 985.391i	− 1.30314i	−0.758588	−	0.651570i	$-0.774110\pi$
$83$	− 985.391i	− 1.30314i	0.758588	−	0.651570i	$-0.225890\pi$
$84$	364.052	0.472873
$85$	0	0
$86$	−30.9153	−0.0387637
$87$	564.232i	0.695311i
$88$	− 96.2148i	− 0.116551i
$89$	−463.054	−0.551502	−0.275751	−	0.961229i	$-0.588926\pi$
$89$	−463.054	−0.551502	−0.275751	+	0.961229i	$0.588926\pi$
$90$	0	0
$91$	1375.38	1.58439
$92$	− 92.0000i	− 0.104257i
$93$	1064.77i	1.18722i
$94$	257.798	0.282871
$95$	0	0
$96$	190.998	0.203059
$97$	1680.05i	1.75859i	0.476277	+	0.879295i	$0.341986\pi$
$97$	1680.05i	1.75859i	−0.476277	+	0.879295i	$0.658014\pi$
$98$	220.971i	0.227770i
$99$	−103.734	−0.105309
$100$	0	0
$101$	−1142.69	−1.12576	−0.562882	−	0.826537i	$-0.690308\pi$
$101$	−1142.69	−1.12576	−0.562882	+	0.826537i	$0.690308\pi$
$102$	− 1002.54i	− 0.973196i
$103$	716.090i	0.685033i	0.939512	+	0.342517i	$0.111279\pi$
$103$	716.090i	0.685033i	−0.939512	+	0.342517i	$0.888721\pi$
$104$	721.586	0.680359
$105$	0	0
$106$	−685.922	−0.628515
$107$	− 710.395i	− 0.641836i	−0.947107	−	0.320918i	$-0.896009\pi$
$107$	− 710.395i	− 0.641836i	0.947107	−	0.320918i	$-0.103991\pi$
$108$	438.694i	0.390865i
$109$	827.349	0.727024	0.363512	−	0.931589i	$-0.381578\pi$
$109$	827.349	0.727024	0.363512	+	0.931589i	$0.381578\pi$
$110$	0	0
$111$	−1769.63	−1.51321
$112$	− 243.975i	− 0.205834i
$113$	1425.51i	1.18673i	0.804934	+	0.593364i	$0.202201\pi$
$113$	1425.51i	1.18673i	−0.804934	+	0.593364i	$0.797799\pi$
$114$	−893.515	−0.734082
$115$	0	0
$116$	378.128	0.302658
$117$	− 777.975i	− 0.614734i
$118$	− 1450.02i	− 1.13123i
$119$	−1280.61	−0.986498
$120$	0	0
$121$	−1186.35	−0.891326
$122$	− 1568.58i	− 1.16403i
$123$	− 1152.40i	− 0.844780i
$124$	713.572	0.516779
$125$	0	0
$126$	−263.040	−0.185980
$127$	1410.48i	0.985514i	0.870167	+	0.492757i	$0.164011\pi$
$127$	1410.48i	0.985514i	−0.870167	+	0.492757i	$0.835989\pi$
$128$	− 128.000i	− 0.0883883i
$129$	92.2617	0.0629705
$130$	0	0
$131$	1187.59	0.792060	0.396030	−	0.918238i	$-0.370388\pi$
$131$	1187.59	0.792060	0.396030	+	0.918238i	$0.370388\pi$
$132$	287.138i	0.189334i
$133$	1141.35i	0.744116i
$134$	−1565.64	−1.00934
$135$	0	0
$136$	−671.865	−0.423617
$137$	− 925.039i	− 0.576872i	−0.957499	−	0.288436i	$-0.906865\pi$
$137$	− 925.039i	− 0.576872i	0.957499	−	0.288436i	$-0.0931352\pi$
$138$	274.559i	0.169363i
$139$	−1600.12	−0.976409	−0.488204	−	0.872729i	$-0.662348\pi$
$139$	−1600.12	−0.976409	−0.488204	+	0.872729i	$0.662348\pi$
$140$	0	0
$141$	−769.357	−0.459515
$142$	− 283.535i	− 0.167562i
$143$	1084.80i	0.634375i
$144$	−138.003	−0.0798626
$145$	0	0
$146$	−808.680	−0.458403
$147$	− 659.454i	− 0.370006i
$148$	1185.95i	0.658677i
$149$	−3494.49	−1.92134	−0.960671	−	0.277688i	$-0.910432\pi$
$149$	−3494.49	−1.92134	−0.960671	+	0.277688i	$0.910432\pi$
$150$	0	0
$151$	2981.78	1.60698	0.803488	−	0.595321i	$-0.202975\pi$
$151$	2981.78	1.60698	0.803488	+	0.595321i	$0.202975\pi$
$152$	598.802i	0.319535i
$153$	724.368i	0.382756i
$154$	366.781	0.191922
$155$	0	0
$156$	−2153.46	−1.10522
$157$	− 2061.78i	− 1.04808i	−0.851695	−	0.524038i	$-0.824425\pi$
$157$	− 2061.78i	− 1.04808i	0.851695	−	0.524038i	$-0.175575\pi$
$158$	1442.08i	0.726110i
$159$	2047.03	1.02100
$160$	0	0
$161$	350.714	0.171678
$162$	− 1774.97i	− 0.860833i
$163$	1017.82i	0.489092i	0.969638	+	0.244546i	$0.0786389\pi$
$163$	1017.82i	0.489092i	−0.969638	+	0.244546i	$0.921361\pi$
$164$	−772.294	−0.367720
$165$	0	0
$166$	1970.78	0.921460
$167$	2501.83i	1.15927i	0.814877	+	0.579633i	$0.196804\pi$
$167$	2501.83i	1.15927i	−0.814877	+	0.579633i	$0.803196\pi$
$168$	728.104i	0.334372i
$169$	−5938.73	−2.70311
$170$	0	0
$171$	645.596	0.288713
$172$	− 61.8306i	− 0.0274101i
$173$	248.051i	0.109011i	0.998513	+	0.0545057i	$0.0173583\pi$
$173$	248.051i	0.109011i	−0.998513	+	0.0545057i	$0.982642\pi$
$174$	−1128.46	−0.491659
$175$	0	0
$176$	192.430	0.0824144
$177$	4327.36i	1.83765i
$178$	− 926.109i	− 0.389971i
$179$	−1394.61	−0.582336	−0.291168	−	0.956672i	$-0.594044\pi$
$179$	−1394.61	−0.582336	−0.291168	+	0.956672i	$0.594044\pi$
$180$	0	0
$181$	−546.627	−0.224478	−0.112239	−	0.993681i	$-0.535802\pi$
$181$	−546.627	−0.224478	−0.112239	+	0.993681i	$0.535802\pi$
$182$	2750.76i	1.12033i
$183$	4681.17i	1.89094i
$184$	184.000	0.0737210
$185$	0	0
$186$	−2129.54	−0.839492
$187$	− 1010.05i	− 0.394986i
$188$	515.596i	0.200020i
$189$	−1672.35	−0.643627
$190$	0	0
$191$	992.925	0.376154	0.188077	−	0.982154i	$-0.439774\pi$
$191$	992.925	0.376154	0.188077	+	0.982154i	$0.439774\pi$
$192$	381.996i	0.143584i
$193$	4855.73i	1.81100i	0.424347	+	0.905500i	$0.360504\pi$
$193$	4855.73i	1.81100i	−0.424347	+	0.905500i	$0.639496\pi$
$194$	−3360.10	−1.24351
$195$	0	0
$196$	−441.943	−0.161058
$197$	3461.50i	1.25189i	0.779868	+	0.625944i	$0.215286\pi$
$197$	3461.50i	1.25189i	−0.779868	+	0.625944i	$0.784714\pi$
$198$	− 207.467i	− 0.0744649i
$199$	2461.35	0.876785	0.438393	−	0.898784i	$-0.355548\pi$
$199$	2461.35	0.876785	0.438393	+	0.898784i	$0.355548\pi$
$200$	0	0
$201$	4672.42	1.63964
$202$	− 2285.39i	− 0.796036i
$203$	1441.46i	0.498379i
$204$	2005.07	0.688153
$205$	0	0
$206$	−1432.18	−0.484391
$207$	− 198.379i	− 0.0666100i
$208$	1443.17i	0.481087i
$209$	−900.213	−0.297938
$210$	0	0
$211$	−3338.23	−1.08916	−0.544581	−	0.838708i	$-0.683311\pi$
$211$	−3338.23	−1.08916	−0.544581	+	0.838708i	$0.683311\pi$
$212$	− 1371.84i	− 0.444428i
$213$	846.166i	0.272199i
$214$	1420.79	0.453846
$215$	0	0
$216$	−877.388	−0.276383
$217$	2720.21i	0.850968i
$218$	1654.70i	0.514084i
$219$	2413.38	0.744662
$220$	0	0
$221$	7575.13	2.30570
$222$	− 3539.27i	− 1.07000i
$223$	− 3805.53i	− 1.14277i	−0.820683	−	0.571384i	$-0.806407\pi$
$223$	− 3805.53i	− 1.14277i	0.820683	−	0.571384i	$-0.193593\pi$
$224$	487.949	0.145547
$225$	0	0
$226$	−2851.01	−0.839144
$227$	− 2552.09i	− 0.746204i	−0.927790	−	0.373102i	$-0.878294\pi$
$227$	− 2552.09i	− 0.746204i	0.927790	−	0.373102i	$-0.121706\pi$
$228$	− 1787.03i	− 0.519074i
$229$	281.217	0.0811500	0.0405750	−	0.999176i	$-0.487081\pi$
$229$	281.217	0.0811500	0.0405750	+	0.999176i	$0.487081\pi$
$230$	0	0
$231$	−1094.60	−0.311772
$232$	756.257i	0.214012i
$233$	5732.97i	1.61193i	0.591965	+	0.805964i	$0.298352\pi$
$233$	5732.97i	1.61193i	−0.591965	+	0.805964i	$0.701648\pi$
$234$	1555.95	0.434682
$235$	0	0
$236$	2900.05	0.799902
$237$	− 4303.64i	− 1.17954i
$238$	− 2561.22i	− 0.697560i
$239$	−170.096	−0.0460360	−0.0230180	−	0.999735i	$-0.507328\pi$
$239$	−170.096	−0.0460360	−0.0230180	+	0.999735i	$0.507328\pi$
$240$	0	0
$241$	−4733.72	−1.26525	−0.632626	−	0.774458i	$-0.718023\pi$
$241$	−4733.72	−1.26525	−0.632626	+	0.774458i	$0.718023\pi$
$242$	− 2372.71i	− 0.630263i
$243$	2335.94i	0.616668i
$244$	3137.15	0.823097
$245$	0	0
$246$	2304.79	0.597350
$247$	− 6751.37i	− 1.73919i
$248$	1427.14i	0.365418i
$249$	−5881.48	−1.49688
$250$	0	0
$251$	−222.109	−0.0558543	−0.0279271	−	0.999610i	$-0.508891\pi$
$251$	−222.109	−0.0558543	−0.0279271	+	0.999610i	$0.508891\pi$
$252$	− 526.080i	− 0.131508i
$253$	276.618i	0.0687383i
$254$	−2820.97	−0.696864
$255$	0	0
$256$	256.000	0.0625000
$257$	− 818.457i	− 0.198654i	−0.995055	−	0.0993268i	$-0.968331\pi$
$257$	− 818.457i	− 0.198654i	0.995055	−	0.0993268i	$-0.0316689\pi$
$258$	184.523i	0.0445269i
$259$	−4520.95	−1.08463
$260$	0	0
$261$	815.355	0.193369
$262$	2375.17i	0.560071i
$263$	− 6242.92i	− 1.46371i	−0.681462	−	0.731853i	$-0.738656\pi$
$263$	− 6242.92i	− 1.46371i	0.681462	−	0.731853i	$-0.261344\pi$
$264$	−574.275	−0.133880
$265$	0	0
$266$	−2282.70	−0.526170
$267$	2763.82i	0.633496i
$268$	− 3131.29i	− 0.713708i
$269$	−1088.75	−0.246774	−0.123387	−	0.992359i	$-0.539376\pi$
$269$	−1088.75	−0.246774	−0.123387	+	0.992359i	$0.539376\pi$
$270$	0	0
$271$	−6897.04	−1.54600	−0.772999	−	0.634408i	$-0.781244\pi$
$271$	−6897.04	−1.54600	−0.772999	+	0.634408i	$0.781244\pi$
$272$	− 1343.73i	− 0.299543i
$273$	− 8209.21i	− 1.81994i
$274$	1850.08	0.407910
$275$	0	0
$276$	−549.119	−0.119757
$277$	− 6455.61i	− 1.40029i	−0.714001	−	0.700145i	$-0.753119\pi$
$277$	− 6455.61i	− 1.40029i	0.714001	−	0.700145i	$-0.246881\pi$
$278$	− 3200.25i	− 0.690425i
$279$	1538.67	0.330171
$280$	0	0
$281$	−1521.46	−0.322999	−0.161499	−	0.986873i	$-0.551633\pi$
$281$	−1521.46	−0.322999	−0.161499	+	0.986873i	$0.551633\pi$
$282$	− 1538.71i	− 0.324926i
$283$	3613.92i	0.759101i	0.925171	+	0.379550i	$0.123921\pi$
$283$	3613.92i	0.759101i	−0.925171	+	0.379550i	$0.876079\pi$
$284$	567.071	0.118484
$285$	0	0
$286$	−2169.60	−0.448571
$287$	− 2944.07i	− 0.605515i
$288$	− 276.005i	− 0.0564714i
$289$	−2140.16	−0.435613
$290$	0	0
$291$	10027.7	2.02005
$292$	− 1617.36i	− 0.324140i
$293$	− 4848.54i	− 0.966740i	−0.875416	−	0.483370i	$-0.839413\pi$
$293$	− 4848.54i	− 0.966740i	0.875416	−	0.483370i	$-0.160587\pi$
$294$	1318.91	0.261634
$295$	0	0
$296$	−2371.89	−0.465755
$297$	− 1319.03i	− 0.257703i
$298$	− 6988.99i	− 1.35859i
$299$	−2074.56	−0.401254
$300$	0	0
$301$	235.705	0.0451355
$302$	5963.55i	1.13630i
$303$	6820.37i	1.29314i
$304$	−1197.60	−0.225945
$305$	0	0
$306$	−1448.74	−0.270650
$307$	1976.04i	0.367356i	0.982986	+	0.183678i	$0.0588004\pi$
$307$	1976.04i	0.367356i	−0.982986	+	0.183678i	$0.941200\pi$
$308$	733.562i	0.135710i
$309$	4274.11	0.786879
$310$	0	0
$311$	3020.57	0.550742	0.275371	−	0.961338i	$-0.411199\pi$
$311$	3020.57	0.550742	0.275371	+	0.961338i	$0.411199\pi$
$312$	− 4306.92i	− 0.781511i
$313$	− 775.315i	− 0.140011i	−0.997547	−	0.0700054i	$-0.977698\pi$
$313$	− 775.315i	− 0.140011i	0.997547	−	0.0700054i	$-0.0223017\pi$
$314$	4123.56	0.741101
$315$	0	0
$316$	−2884.15	−0.513437
$317$	4878.60i	0.864384i	0.901782	+	0.432192i	$0.142260\pi$
$317$	4878.60i	0.864384i	−0.901782	+	0.432192i	$0.857740\pi$
$318$	4094.05i	0.721959i
$319$	−1136.92	−0.199547
$320$	0	0
$321$	−4240.12	−0.737260
$322$	701.427i	0.121394i
$323$	6286.16i	1.08288i
$324$	3549.94	0.608701
$325$	0	0
$326$	−2035.64	−0.345840
$327$	− 4938.18i	− 0.835113i
$328$	− 1544.59i	− 0.260017i
$329$	−1965.51	−0.329367
$330$	0	0
$331$	3250.81	0.539822	0.269911	−	0.962885i	$-0.413006\pi$
$331$	3250.81	0.539822	0.269911	+	0.962885i	$0.413006\pi$
$332$	3941.56i	0.651570i
$333$	2557.25i	0.420830i
$334$	−5003.66	−0.819725
$335$	0	0
$336$	−1456.21	−0.236436
$337$	836.952i	0.135287i	0.997710	+	0.0676435i	$0.0215480\pi$
$337$	836.952i	0.135287i	−0.997710	+	0.0676435i	$0.978452\pi$
$338$	− 11877.5i	− 1.91139i
$339$	8508.39	1.36316
$340$	0	0
$341$	−2145.51	−0.340720
$342$	1291.19i	0.204151i
$343$	− 6914.94i	− 1.08855i
$344$	123.661	0.0193819
$345$	0	0
$346$	−496.102	−0.0770827
$347$	2965.99i	0.458856i	0.973326	+	0.229428i	$0.0736855\pi$
$347$	2965.99i	0.458856i	−0.973326	+	0.229428i	$0.926315\pi$
$348$	− 2256.93i	− 0.347655i
$349$	869.965	0.133433	0.0667165	−	0.997772i	$-0.478748\pi$
$349$	869.965	0.133433	0.0667165	+	0.997772i	$0.478748\pi$
$350$	0	0
$351$	9892.37	1.50432
$352$	384.859i	0.0582757i
$353$	1984.81i	0.299265i	0.988742	+	0.149633i	$0.0478091\pi$
$353$	1984.81i	0.299265i	−0.988742	+	0.149633i	$0.952191\pi$
$354$	−8654.73	−1.29942
$355$	0	0
$356$	1852.22	0.275751
$357$	7643.55i	1.13316i
$358$	− 2789.22i	− 0.411774i
$359$	778.464	0.114445	0.0572225	−	0.998361i	$-0.481776\pi$
$359$	778.464	0.114445	0.0572225	+	0.998361i	$0.481776\pi$
$360$	0	0
$361$	−1256.43	−0.183180
$362$	− 1093.25i	− 0.158730i
$363$	7080.98i	1.02384i
$364$	−5501.53	−0.792193
$365$	0	0
$366$	−9362.33	−1.33710
$367$	− 3782.98i	− 0.538066i	−0.963131	−	0.269033i	$-0.913296\pi$
$367$	− 3782.98i	− 0.538066i	0.963131	−	0.269033i	$-0.0867040\pi$
$368$	368.000i	0.0521286i
$369$	−1665.29	−0.234937
$370$	0	0
$371$	5229.61	0.731827
$372$	− 4259.08i	− 0.593611i
$373$	− 4252.72i	− 0.590342i	−0.955444	−	0.295171i	$-0.904623\pi$
$373$	− 4252.72i	− 0.590342i	0.955444	−	0.295171i	$-0.0953766\pi$
$374$	2020.10	0.279297
$375$	0	0
$376$	−1031.19	−0.141435
$377$	− 8526.63i	− 1.16484i
$378$	− 3344.70i	− 0.455113i
$379$	−4743.71	−0.642923	−0.321462	−	0.946923i	$-0.604174\pi$
$379$	−4743.71	−0.642923	−0.321462	+	0.946923i	$0.604174\pi$
$380$	0	0
$381$	8418.74	1.13203
$382$	1985.85i	0.265981i
$383$	6966.42i	0.929419i	0.885463	+	0.464710i	$0.153841\pi$
$383$	6966.42i	0.929419i	−0.885463	+	0.464710i	$0.846159\pi$
$384$	−763.991	−0.101529
$385$	0	0
$386$	−9711.45	−1.28057
$387$	− 133.325i	− 0.0175123i
$388$	− 6720.20i	− 0.879295i
$389$	8686.79	1.13223	0.566115	−	0.824326i	$-0.308446\pi$
$389$	8686.79	1.13223	0.566115	+	0.824326i	$0.308446\pi$
$390$	0	0
$391$	1931.61	0.249836
$392$	− 883.886i	− 0.113885i
$393$	− 7088.32i	− 0.909818i
$394$	−6923.01	−0.885219
$395$	0	0
$396$	414.934	0.0526546
$397$	− 13526.4i	− 1.71000i	−0.518631	−	0.854998i	$-0.673558\pi$
$397$	− 13526.4i	− 1.71000i	0.518631	−	0.854998i	$-0.326442\pi$
$398$	4922.69i	0.619981i
$399$	6812.35	0.854747
$400$	0	0
$401$	−6804.61	−0.847396	−0.423698	−	0.905803i	$-0.639268\pi$
$401$	−6804.61	−0.847396	−0.423698	+	0.905803i	$0.639268\pi$
$402$	9344.83i	1.15940i
$403$	− 16090.7i	− 1.98893i
$404$	4570.77	0.562882
$405$	0	0
$406$	−2882.93	−0.352407
$407$	− 3565.80i	− 0.434276i
$408$	4010.15i	0.486598i
$409$	−3156.92	−0.381662	−0.190831	−	0.981623i	$-0.561118\pi$
$409$	−3156.92	−0.381662	−0.190831	+	0.981623i	$0.561118\pi$
$410$	0	0
$411$	−5521.27	−0.662637
$412$	− 2864.36i	− 0.342517i
$413$	11055.3i	1.31718i
$414$	396.758	0.0471004
$415$	0	0
$416$	−2886.35	−0.340180
$417$	9550.64i	1.12157i
$418$	− 1800.43i	− 0.210674i
$419$	13696.5	1.59694	0.798468	−	0.602038i	$-0.205644\pi$
$419$	13696.5	1.59694	0.798468	+	0.602038i	$0.205644\pi$
$420$	0	0
$421$	−3910.11	−0.452653	−0.226326	−	0.974052i	$-0.572672\pi$
$421$	−3910.11	−0.452653	−0.226326	+	0.974052i	$0.572672\pi$
$422$	− 6676.46i	− 0.770154i
$423$	1111.78i	0.127793i
$424$	2743.69	0.314258
$425$	0	0
$426$	−1692.33	−0.192474
$427$	11959.2i	1.35537i
$428$	2841.58i	0.320918i
$429$	6474.83	0.728690
$430$	0	0
$431$	5415.70	0.605256	0.302628	−	0.953109i	$-0.402136\pi$
$431$	5415.70	0.605256	0.302628	+	0.953109i	$0.402136\pi$
$432$	− 1754.78i	− 0.195432i
$433$	15906.0i	1.76534i	0.469989	+	0.882672i	$0.344258\pi$
$433$	15906.0i	1.76534i	−0.469989	+	0.882672i	$0.655742\pi$
$434$	−5440.42	−0.601725
$435$	0	0
$436$	−3309.39	−0.363512
$437$	− 1721.56i	− 0.188451i
$438$	4826.75i	0.526555i
$439$	−6939.90	−0.754495	−0.377247	−	0.926112i	$-0.623129\pi$
$439$	−6939.90	−0.754495	−0.377247	+	0.926112i	$0.623129\pi$
$440$	0	0
$441$	−952.958	−0.102900
$442$	15150.3i	1.63037i
$443$	− 16617.0i	− 1.78216i	−0.453844	−	0.891081i	$-0.649948\pi$
$443$	− 16617.0i	− 1.78216i	0.453844	−	0.891081i	$-0.350052\pi$
$444$	7078.54	0.756605
$445$	0	0
$446$	7611.07	0.808059
$447$	20857.5i	2.20700i
$448$	975.899i	0.102917i
$449$	−12669.8	−1.33168	−0.665841	−	0.746093i	$-0.731927\pi$
$449$	−12669.8	−1.33168	−0.665841	+	0.746093i	$0.731927\pi$
$450$	0	0
$451$	2322.07	0.242443
$452$	− 5702.02i	− 0.593364i
$453$	− 17797.3i	− 1.84589i
$454$	5104.19	0.527646
$455$	0	0
$456$	3574.06	0.367041
$457$	− 7479.70i	− 0.765614i	−0.923828	−	0.382807i	$-0.874957\pi$
$457$	− 7479.70i	− 0.765614i	0.923828	−	0.382807i	$-0.125043\pi$
$458$	562.435i	0.0573818i
$459$	−9210.73	−0.936645
$460$	0	0
$461$	3260.42	0.329399	0.164699	−	0.986344i	$-0.447335\pi$
$461$	3260.42	0.329399	0.164699	+	0.986344i	$0.447335\pi$
$462$	− 2189.20i	− 0.220456i
$463$	− 10666.2i	− 1.07062i	−0.844654	−	0.535312i	$-0.820194\pi$
$463$	− 10666.2i	− 1.07062i	0.844654	−	0.535312i	$-0.179806\pi$
$464$	−1512.51	−0.151329
$465$	0	0
$466$	−11465.9	−1.13980
$467$	6202.48i	0.614597i	0.951613	+	0.307298i	$0.0994250\pi$
$467$	6202.48i	0.614597i	−0.951613	+	0.307298i	$0.900575\pi$
$468$	3111.90i	0.307367i
$469$	11936.8	1.17525
$470$	0	0
$471$	−12306.1	−1.20390
$472$	5800.09i	0.565616i
$473$	185.907i	0.0180719i
$474$	8607.29	0.834063
$475$	0	0
$476$	5122.44	0.493249
$477$	− 2958.10i	− 0.283945i
$478$	− 340.193i	− 0.0325524i
$479$	13749.4	1.31154	0.655768	−	0.754962i	$-0.272345\pi$
$479$	13749.4	1.31154	0.655768	+	0.754962i	$0.272345\pi$
$480$	0	0
$481$	26742.6	2.53505
$482$	− 9467.43i	− 0.894668i
$483$	− 2093.30i	− 0.197202i
$484$	4745.42	0.445663
$485$	0	0
$486$	−4671.87	−0.436050
$487$	9673.72i	0.900119i	0.892999	+	0.450060i	$0.148597\pi$
$487$	9673.72i	0.900119i	−0.892999	+	0.450060i	$0.851403\pi$
$488$	6274.31i	0.582017i
$489$	6075.05	0.561807
$490$	0	0
$491$	5908.70	0.543087	0.271544	−	0.962426i	$-0.412466\pi$
$491$	5908.70	0.543087	0.271544	+	0.962426i	$0.412466\pi$
$492$	4609.58i	0.422390i
$493$	7939.10i	0.725272i
$494$	13502.7	1.22979
$495$	0	0
$496$	−2854.29	−0.258390
$497$	2161.73i	0.195105i
$498$	− 11763.0i	− 1.05846i
$499$	12119.8	1.08729	0.543644	−	0.839316i	$-0.317044\pi$
$499$	12119.8	1.08729	0.543644	+	0.839316i	$0.317044\pi$
$500$	0	0
$501$	14932.6	1.33162
$502$	− 444.219i	− 0.0394949i
$503$	17552.8i	1.55595i	0.628298	+	0.777973i	$0.283752\pi$
$503$	17552.8i	1.55595i	−0.628298	+	0.777973i	$0.716248\pi$
$504$	1052.16	0.0929900
$505$	0	0
$506$	−553.235	−0.0486053
$507$	35446.4i	3.10499i
$508$	− 5641.94i	− 0.492757i
$509$	−7377.43	−0.642434	−0.321217	−	0.947006i	$-0.604092\pi$
$509$	−7377.43	−0.642434	−0.321217	+	0.947006i	$0.604092\pi$
$510$	0	0
$511$	6165.55	0.533753
$512$	512.000i	0.0441942i
$513$	8209.10i	0.706512i
$514$	1636.91	0.140469
$515$	0	0
$516$	−369.047	−0.0314852
$517$	− 1550.25i	− 0.131876i
$518$	− 9041.91i	− 0.766947i
$519$	1480.54	0.125219
$520$	0	0
$521$	−1512.93	−0.127222	−0.0636111	−	0.997975i	$-0.520262\pi$
$521$	−1512.93	−0.127222	−0.0636111	+	0.997975i	$0.520262\pi$
$522$	1630.71i	0.136732i
$523$	− 13736.6i	− 1.14849i	−0.818684	−	0.574244i	$-0.805296\pi$
$523$	− 13736.6i	− 1.14849i	0.818684	−	0.574244i	$-0.194704\pi$
$524$	−4750.34	−0.396030
$525$	0	0
$526$	12485.8	1.03500
$527$	14982.0i	1.23838i
$528$	− 1148.55i	− 0.0946672i
$529$	−529.000	−0.0434783
$530$	0	0
$531$	6253.34	0.511058
$532$	− 4565.39i	− 0.372058i
$533$	17414.9i	1.41524i
$534$	−5527.65	−0.447949
$535$	0	0
$536$	6262.58	0.504668
$537$	8323.99i	0.668914i
$538$	− 2177.50i	− 0.174496i
$539$	1328.80	0.106188
$540$	0	0
$541$	24827.7	1.97306	0.986529	−	0.163587i	$-0.0523063\pi$
$541$	24827.7	1.97306	0.986529	+	0.163587i	$0.0523063\pi$
$542$	− 13794.1i	− 1.09319i
$543$	3262.64i	0.257852i
$544$	2687.46	0.211809
$545$	0	0
$546$	16418.4	1.28689
$547$	− 4902.67i	− 0.383223i	−0.981471	−	0.191612i	$-0.938629\pi$
$547$	− 4902.67i	− 0.383223i	0.981471	−	0.191612i	$-0.0613714\pi$
$548$	3700.16i	0.288436i
$549$	6764.62	0.525877
$550$	0	0
$551$	7075.75	0.547073
$552$	− 1098.24i	− 0.0846813i
$553$	− 10994.7i	− 0.845464i
$554$	12911.2	0.990154
$555$	0	0
$556$	6400.50	0.488204
$557$	− 2422.01i	− 0.184244i	−0.995748	−	0.0921220i	$-0.970635\pi$
$557$	− 2422.01i	− 0.184244i	0.995748	−	0.0921220i	$-0.0293650\pi$
$558$	3077.34i	0.233466i
$559$	−1394.25	−0.105493
$560$	0	0
$561$	−6028.68	−0.453710
$562$	− 3042.92i	− 0.228395i
$563$	− 18122.4i	− 1.35660i	−0.734785	−	0.678300i	$-0.762717\pi$
$563$	− 18122.4i	− 1.35660i	0.734785	−	0.678300i	$-0.237283\pi$
$564$	3077.43	0.229757
$565$	0	0
$566$	−7227.85	−0.536765
$567$	13532.8i	1.00233i
$568$	1134.14i	0.0837809i
$569$	−15845.3	−1.16744	−0.583718	−	0.811956i	$-0.698403\pi$
$569$	−15845.3	−1.16744	−0.583718	+	0.811956i	$0.698403\pi$
$570$	0	0
$571$	−6699.07	−0.490976	−0.245488	−	0.969400i	$-0.578948\pi$
$571$	−6699.07	−0.490976	−0.245488	+	0.969400i	$0.578948\pi$
$572$	− 4339.21i	− 0.317188i
$573$	− 5926.45i	− 0.432079i
$574$	5888.13	0.428164
$575$	0	0
$576$	552.011	0.0399313
$577$	19546.6i	1.41029i	0.709065	+	0.705143i	$0.249117\pi$
$577$	19546.6i	1.41029i	−0.709065	+	0.705143i	$0.750883\pi$
$578$	− 4280.33i	− 0.308025i
$579$	28982.3	2.08025
$580$	0	0
$581$	−15025.7	−1.07292
$582$	20055.4i	1.42839i
$583$	4124.74i	0.293018i
$584$	3234.72	0.229201
$585$	0	0
$586$	9697.08	0.683588
$587$	− 1488.14i	− 0.104638i	−0.998630	−	0.0523188i	$-0.983339\pi$
$587$	− 1488.14i	− 0.104638i	0.998630	−	0.0523188i	$-0.0166612\pi$
$588$	2637.82i	0.185003i
$589$	13352.8	0.934111
$590$	0	0
$591$	20660.6	1.43801
$592$	− 4743.79i	− 0.329339i
$593$	− 8930.13i	− 0.618409i	−0.950996	−	0.309205i	$-0.899937\pi$
$593$	− 8930.13i	− 0.618409i	0.950996	−	0.309205i	$-0.100063\pi$
$594$	2638.05	0.182223
$595$	0	0
$596$	13978.0	0.960671
$597$	− 14691.0i	− 1.00714i
$598$	− 4149.12i	− 0.283729i
$599$	−18051.8	−1.23135	−0.615674	−	0.788001i	$-0.711116\pi$
$599$	−18051.8	−1.23135	−0.615674	+	0.788001i	$0.711116\pi$
$600$	0	0
$601$	−22172.1	−1.50486	−0.752428	−	0.658674i	$-0.771118\pi$
$601$	−22172.1	−1.50486	−0.752428	+	0.658674i	$0.771118\pi$
$602$	471.409i	0.0319156i
$603$	− 6751.97i	− 0.455989i
$604$	−11927.1	−0.803488
$605$	0	0
$606$	−13640.7	−0.914385
$607$	− 11705.4i	− 0.782713i	−0.920239	−	0.391357i	$-0.872006\pi$
$607$	− 11705.4i	− 0.782713i	0.920239	−	0.391357i	$-0.127994\pi$
$608$	− 2395.21i	− 0.159767i
$609$	8603.64	0.572475
$610$	0	0
$611$	11626.5	0.769815
$612$	− 2897.47i	− 0.191378i
$613$	10911.8i	0.718962i	0.933152	+	0.359481i	$0.117046\pi$
$613$	10911.8i	0.718962i	−0.933152	+	0.359481i	$0.882954\pi$
$614$	−3952.07	−0.259760
$615$	0	0
$616$	−1467.12	−0.0959612
$617$	− 23999.1i	− 1.56591i	−0.622079	−	0.782955i	$-0.713712\pi$
$617$	− 23999.1i	− 1.56591i	0.622079	−	0.782955i	$-0.286288\pi$
$618$	8548.22i	0.556408i
$619$	−9204.68	−0.597685	−0.298843	−	0.954302i	$-0.596601\pi$
$619$	−9204.68	−0.597685	−0.298843	+	0.954302i	$0.596601\pi$
$620$	0	0
$621$	2522.49	0.163002
$622$	6041.14i	0.389433i
$623$	7060.85i	0.454072i
$624$	8613.84	0.552611
$625$	0	0
$626$	1550.63	0.0990026
$627$	5373.09i	0.342233i
$628$	8247.11i	0.524038i
$629$	−24899.9	−1.57842
$630$	0	0
$631$	−2047.31	−0.129163	−0.0645817	−	0.997912i	$-0.520571\pi$
$631$	−2047.31	−0.129163	−0.0645817	+	0.997912i	$0.520571\pi$
$632$	− 5768.30i	− 0.363055i
$633$	19924.8i	1.25109i
$634$	−9757.21	−0.611212
$635$	0	0
$636$	−8188.10	−0.510502
$637$	9965.62i	0.619863i
$638$	− 2273.85i	− 0.141101i
$639$	1222.77	0.0756996
$640$	0	0
$641$	16728.4	1.03078	0.515390	−	0.856956i	$-0.327647\pi$
$641$	16728.4	1.03078	0.515390	+	0.856956i	$0.327647\pi$
$642$	− 8480.24i	− 0.521321i
$643$	− 23471.6i	− 1.43955i	−0.694209	−	0.719774i	$-0.744245\pi$
$643$	− 23471.6i	− 1.43955i	0.694209	−	0.719774i	$-0.255755\pi$
$644$	−1402.85	−0.0858388
$645$	0	0
$646$	−12572.3	−0.765714
$647$	8626.14i	0.524156i	0.965047	+	0.262078i	$0.0844077\pi$
$647$	8626.14i	0.524156i	−0.965047	+	0.262078i	$0.915592\pi$
$648$	7099.89i	0.430417i
$649$	−8719.60	−0.527387
$650$	0	0
$651$	16236.1	0.977484
$652$	− 4071.29i	− 0.244546i
$653$	− 22752.2i	− 1.36350i	−0.731587	−	0.681748i	$-0.761220\pi$
$653$	− 22752.2i	− 1.36350i	0.731587	−	0.681748i	$-0.238780\pi$
$654$	9876.36	0.590514
$655$	0	0
$656$	3089.18	0.183860
$657$	− 3487.50i	− 0.207093i
$658$	− 3931.01i	− 0.232898i
$659$	31487.4	1.86127	0.930633	−	0.365954i	$-0.119257\pi$
$659$	31487.4	1.86127	0.930633	+	0.365954i	$0.119257\pi$
$660$	0	0
$661$	2034.20	0.119699	0.0598496	−	0.998207i	$-0.480938\pi$
$661$	2034.20	0.119699	0.0598496	+	0.998207i	$0.480938\pi$
$662$	6501.63i	0.381712i
$663$	− 45213.6i	− 2.64849i
$664$	−7883.13	−0.460730
$665$	0	0
$666$	−5114.49	−0.297571
$667$	− 2174.24i	− 0.126217i
$668$	− 10007.3i	− 0.579633i
$669$	−22714.0	−1.31267
$670$	0	0
$671$	−9432.52	−0.542680
$672$	− 2912.41i	− 0.167186i
$673$	− 2759.40i	− 0.158049i	−0.996873	−	0.0790246i	$-0.974819\pi$
$673$	− 2759.40i	− 0.158049i	0.996873	−	0.0790246i	$-0.0251806\pi$
$674$	−1673.90	−0.0956623
$675$	0	0
$676$	23754.9	1.35156
$677$	− 19232.2i	− 1.09181i	−0.837849	−	0.545903i	$-0.816187\pi$
$677$	− 19232.2i	− 1.09181i	0.837849	−	0.545903i	$-0.183813\pi$
$678$	17016.8i	0.963902i
$679$	25618.1	1.44791
$680$	0	0
$681$	−15232.6	−0.857145
$682$	− 4291.01i	− 0.240926i
$683$	− 1381.53i	− 0.0773980i	−0.999251	−	0.0386990i	$-0.987679\pi$
$683$	− 1381.53i	− 0.0773980i	0.999251	−	0.0386990i	$-0.0123213\pi$
$684$	−2582.38	−0.144357
$685$	0	0
$686$	13829.9	0.769719
$687$	− 1678.50i	− 0.0932149i
$688$	247.322i	0.0137050i
$689$	−30934.5	−1.71047
$690$	0	0
$691$	−31009.3	−1.70716	−0.853581	−	0.520960i	$-0.825574\pi$
$691$	−31009.3	−1.70716	−0.853581	+	0.520960i	$0.825574\pi$
$692$	− 992.204i	− 0.0545057i
$693$	1581.77i	0.0867050i
$694$	−5931.99	−0.324460
$695$	0	0
$696$	4513.86	0.245829
$697$	− 16214.9i	− 0.881182i
$698$	1739.93i	0.0943514i
$699$	34218.2	1.85158
$700$	0	0
$701$	−14937.6	−0.804830	−0.402415	−	0.915457i	$-0.631829\pi$
$701$	−14937.6	−0.804830	−0.402415	+	0.915457i	$0.631829\pi$
$702$	19784.7i	1.06371i
$703$	22192.1i	1.19060i
$704$	−769.718	−0.0412072
$705$	0	0
$706$	−3969.61	−0.211612
$707$	17424.3i	0.926884i
$708$	− 17309.5i	− 0.918826i
$709$	7067.82	0.374383	0.187191	−	0.982323i	$-0.440062\pi$
$709$	7067.82	0.374383	0.187191	+	0.982323i	$0.440062\pi$
$710$	0	0
$711$	−6219.07	−0.328036
$712$	3704.43i	0.194985i
$713$	− 4103.04i	− 0.215512i
$714$	−15287.1	−0.801268
$715$	0	0
$716$	5578.45	0.291168
$717$	1015.25i	0.0528804i
$718$	1556.93i	0.0809249i
$719$	13129.8	0.681028	0.340514	−	0.940239i	$-0.389399\pi$
$719$	13129.8	0.681028	0.340514	+	0.940239i	$0.389399\pi$
$720$	0	0
$721$	10919.2	0.564013
$722$	− 2512.87i	− 0.129528i
$723$	28254.1i	1.45336i
$724$	2186.51	0.112239
$725$	0	0
$726$	−14162.0	−0.723966
$727$	− 3432.63i	− 0.175116i	−0.996159	−	0.0875580i	$-0.972094\pi$
$727$	− 3432.63i	− 0.175116i	0.996159	−	0.0875580i	$-0.0279063\pi$
$728$	− 11003.1i	− 0.560165i
$729$	−10019.7	−0.509052
$730$	0	0
$731$	1298.18	0.0656839
$732$	− 18724.7i	− 0.945469i
$733$	3319.64i	0.167277i	0.996496	+	0.0836383i	$0.0266540\pi$
$733$	3319.64i	0.167277i	−0.996496	+	0.0836383i	$0.973346\pi$
$734$	7565.97	0.380470
$735$	0	0
$736$	−736.000	−0.0368605
$737$	9414.88i	0.470559i
$738$	− 3330.58i	− 0.166125i
$739$	−29135.2	−1.45028	−0.725139	−	0.688602i	$-0.758225\pi$
$739$	−29135.2	−1.45028	−0.725139	+	0.688602i	$0.758225\pi$
$740$	0	0
$741$	−40296.8	−1.99776
$742$	10459.2i	0.517480i
$743$	11376.3i	0.561718i	0.959749	+	0.280859i	$0.0906194\pi$
$743$	11376.3i	0.561718i	−0.959749	+	0.280859i	$0.909381\pi$
$744$	8518.17	0.419746
$745$	0	0
$746$	8505.44	0.417435
$747$	8499.16i	0.416289i
$748$	4040.21i	0.197493i
$749$	−10832.4	−0.528447
$750$	0	0
$751$	−6225.31	−0.302483	−0.151241	−	0.988497i	$-0.548327\pi$
$751$	−6225.31	−0.302483	−0.151241	+	0.988497i	$0.548327\pi$
$752$	− 2062.38i	− 0.100010i
$753$	1325.70i	0.0641583i
$754$	17053.3	0.823665
$755$	0	0
$756$	6689.39	0.321813
$757$	− 1885.26i	− 0.0905163i	−0.998975	−	0.0452582i	$-0.985589\pi$
$757$	− 1885.26i	− 0.0905163i	0.998975	−	0.0452582i	$-0.0144110\pi$
$758$	− 9487.42i	− 0.454615i
$759$	1651.04	0.0789579
$760$	0	0
$761$	15635.5	0.744791	0.372395	−	0.928074i	$-0.378537\pi$
$761$	15635.5	0.744791	0.372395	+	0.928074i	$0.378537\pi$
$762$	16837.5i	0.800469i
$763$	− 12615.8i	− 0.598586i
$764$	−3971.70	−0.188077
$765$	0	0
$766$	−13932.8	−0.657199
$767$	− 65394.8i	− 3.07858i
$768$	− 1527.98i	− 0.0717921i
$769$	−18918.5	−0.887151	−0.443576	−	0.896237i	$-0.646290\pi$
$769$	−18918.5	−0.887151	−0.443576	+	0.896237i	$0.646290\pi$
$770$	0	0
$771$	−4885.11	−0.228188
$772$	− 19422.9i	− 0.905500i
$773$	32813.2i	1.52679i	0.645931	+	0.763395i	$0.276469\pi$
$773$	32813.2i	1.52679i	−0.645931	+	0.763395i	$0.723531\pi$
$774$	266.649	0.0123831
$775$	0	0
$776$	13440.4	0.621756
$777$	26984.1i	1.24588i
$778$	17373.6i	0.800608i
$779$	−14451.6	−0.664676
$780$	0	0
$781$	−1705.02	−0.0781183
$782$	3863.22i	0.176661i
$783$	10367.7i	0.473193i
$784$	1767.77	0.0805290
$785$	0	0
$786$	14176.6	0.643339
$787$	− 10855.2i	− 0.491672i	−0.969311	−	0.245836i	$-0.920938\pi$
$787$	− 10855.2i	− 0.491672i	0.969311	−	0.245836i	$-0.0790625\pi$
$788$	− 13846.0i	− 0.625944i
$789$	−37262.0	−1.68132
$790$	0	0
$791$	21736.7	0.977078
$792$	829.868i	0.0372324i
$793$	− 70741.5i	− 3.16785i
$794$	27052.7	1.20915
$795$	0	0
$796$	−9845.39	−0.438393
$797$	− 20171.5i	− 0.896499i	−0.893908	−	0.448250i	$-0.852048\pi$
$797$	− 20171.5i	− 0.896499i	0.893908	−	0.448250i	$-0.147952\pi$
$798$	13624.7i	0.604397i
$799$	−10825.3	−0.479316
$800$	0	0
$801$	3993.92	0.176178
$802$	− 13609.2i	− 0.599200i
$803$	4862.94i	0.213710i
$804$	−18689.7	−0.819818
$805$	0	0
$806$	32181.5	1.40638
$807$	6498.40i	0.283463i
$808$	9141.54i	0.398018i
$809$	−6849.64	−0.297677	−0.148838	−	0.988862i	$-0.547553\pi$
$809$	−6849.64	−0.297677	−0.148838	+	0.988862i	$0.547553\pi$
$810$	0	0
$811$	−21138.6	−0.915262	−0.457631	−	0.889142i	$-0.651302\pi$
$811$	−21138.6	−0.915262	−0.457631	+	0.889142i	$0.651302\pi$
$812$	− 5765.86i	− 0.249190i
$813$	41166.2i	1.77585i
$814$	7131.60	0.307079
$815$	0	0
$816$	−8020.30	−0.344077
$817$	− 1157.01i	− 0.0495454i
$818$	− 6313.84i	− 0.269876i
$819$	−11862.9	−0.506133
$820$	0	0
$821$	260.761	0.0110848	0.00554240	−	0.999985i	$-0.498236\pi$
$821$	260.761	0.0110848	0.00554240	+	0.999985i	$0.498236\pi$
$822$	− 11042.5i	− 0.468555i
$823$	20615.0i	0.873139i	0.899671	+	0.436569i	$0.143807\pi$
$823$	20615.0i	0.873139i	−0.899671	+	0.436569i	$0.856193\pi$
$824$	5728.72	0.242196
$825$	0	0
$826$	−22110.6	−0.931386
$827$	10281.8i	0.432327i	0.976357	+	0.216164i	$0.0693545\pi$
$827$	10281.8i	0.432327i	−0.976357	+	0.216164i	$0.930646\pi$
$828$	793.515i	0.0333050i
$829$	14144.4	0.592590	0.296295	−	0.955097i	$-0.404249\pi$
$829$	14144.4	0.592590	0.296295	+	0.955097i	$0.404249\pi$
$830$	0	0
$831$	−38531.5	−1.60847
$832$	− 5772.69i	− 0.240543i
$833$	− 9278.94i	− 0.385950i
$834$	−19101.3	−0.793073
$835$	0	0
$836$	3600.85	0.148969
$837$	19565.0i	0.807963i
$838$	27392.9i	1.12920i
$839$	38790.8	1.59619	0.798097	−	0.602529i	$-0.205840\pi$
$839$	38790.8	1.59619	0.798097	+	0.602529i	$0.205840\pi$
$840$	0	0
$841$	−15452.7	−0.633592
$842$	− 7820.21i	− 0.320074i
$843$	9081.11i	0.371020i
$844$	13352.9	0.544581
$845$	0	0
$846$	−2223.55	−0.0903632
$847$	18090.0i	0.733862i
$848$	5487.38i	0.222214i
$849$	21570.4	0.871959
$850$	0	0
$851$	6819.19	0.274687
$852$	− 3384.67i	− 0.136099i
$853$	− 38874.0i	− 1.56040i	−0.625530	−	0.780200i	$-0.715117\pi$
$853$	− 38874.0i	− 1.56040i	0.625530	−	0.780200i	$-0.284883\pi$
$854$	−23918.3	−0.958393
$855$	0	0
$856$	−5683.16	−0.226923
$857$	− 11643.1i	− 0.464086i	−0.972706	−	0.232043i	$-0.925459\pi$
$857$	− 11643.1i	− 0.464086i	0.972706	−	0.232043i	$-0.0745410\pi$
$858$	12949.7i	0.515262i
$859$	−32626.4	−1.29592	−0.647961	−	0.761673i	$-0.724378\pi$
$859$	−32626.4	−1.29592	−0.647961	+	0.761673i	$0.724378\pi$
$860$	0	0
$861$	−17572.2	−0.695539
$862$	10831.4i	0.427980i
$863$	− 30383.6i	− 1.19846i	−0.800578	−	0.599229i	$-0.795474\pi$
$863$	− 30383.6i	− 1.19846i	0.800578	−	0.599229i	$-0.204526\pi$
$864$	3509.55	0.138191
$865$	0	0
$866$	−31812.0	−1.24829
$867$	12774.0i	0.500377i
$868$	− 10880.8i	− 0.425484i
$869$	8671.81	0.338517
$870$	0	0
$871$	−70609.2	−2.74684
$872$	− 6618.79i	− 0.257042i
$873$	− 14490.7i	− 0.561783i
$874$	3443.11	0.133255
$875$	0	0
$876$	−9653.51	−0.372331
$877$	4652.82i	0.179150i	0.995980	+	0.0895750i	$0.0285509\pi$
$877$	4652.82i	0.179150i	−0.995980	+	0.0895750i	$0.971449\pi$
$878$	− 13879.8i	− 0.533509i
$879$	−28939.4	−1.11047
$880$	0	0
$881$	−33643.5	−1.28658	−0.643290	−	0.765622i	$-0.722431\pi$
$881$	−33643.5	−1.28658	−0.643290	+	0.765622i	$0.722431\pi$
$882$	− 1905.92i	− 0.0727613i
$883$	− 33311.2i	− 1.26955i	−0.772698	−	0.634774i	$-0.781093\pi$
$883$	− 33311.2i	− 1.26955i	0.772698	−	0.634774i	$-0.218907\pi$
$884$	−30300.5	−1.15285
$885$	0	0
$886$	33234.0	1.26018
$887$	− 25636.1i	− 0.970435i	−0.874394	−	0.485217i	$-0.838741\pi$
$887$	− 25636.1i	− 0.970435i	0.874394	−	0.485217i	$-0.161259\pi$
$888$	14157.1i	0.535000i
$889$	21507.7	0.811410
$890$	0	0
$891$	−10673.7	−0.401326
$892$	15222.1i	0.571384i
$893$	9648.13i	0.361548i
$894$	−41715.0	−1.56058
$895$	0	0
$896$	−1951.80	−0.0727734
$897$	12382.4i	0.460910i
$898$	− 25339.6i	− 0.941642i
$899$	16863.9	0.625630
$900$	0	0
$901$	28802.9	1.06500
$902$	4644.13i	0.171433i
$903$	− 1406.85i	− 0.0518459i
$904$	11404.0	0.419572
$905$	0	0
$906$	35594.5	1.30524
$907$	− 14681.1i	− 0.537460i	−0.963215	−	0.268730i	$-0.913396\pi$
$907$	− 14681.1i	− 0.537460i	0.963215	−	0.268730i	$-0.0866040\pi$
$908$	10208.4i	0.373102i
$909$	9855.91	0.359626
$910$	0	0
$911$	−13950.7	−0.507364	−0.253682	−	0.967288i	$-0.581642\pi$
$911$	−13950.7	−0.507364	−0.253682	+	0.967288i	$0.581642\pi$
$912$	7148.12i	0.259537i
$913$	− 11851.1i	− 0.429590i
$914$	14959.4	0.541371
$915$	0	0
$916$	−1124.87	−0.0405750
$917$	− 18108.8i	− 0.652133i
$918$	− 18421.5i	− 0.662308i
$919$	9495.79	0.340846	0.170423	−	0.985371i	$-0.445487\pi$
$919$	9495.79	0.340846	0.170423	+	0.985371i	$0.445487\pi$
$920$	0	0
$921$	11794.3	0.421972
$922$	6520.84i	0.232920i
$923$	− 12787.2i	− 0.456009i
$924$	4378.40	0.155886
$925$	0	0
$926$	21332.3	0.757046
$927$	− 6176.39i	− 0.218834i
$928$	− 3025.03i	− 0.107006i
$929$	28148.4	0.994101	0.497051	−	0.867722i	$-0.334416\pi$
$929$	28148.4	0.994101	0.497051	+	0.867722i	$0.334416\pi$
$930$	0	0
$931$	−8269.89	−0.291122
$932$	− 22931.9i	− 0.805964i
$933$	− 18028.8i	− 0.632623i
$934$	−12405.0	−0.434586
$935$	0	0
$936$	−6223.80	−0.217341
$937$	− 39623.7i	− 1.38148i	−0.723102	−	0.690741i	$-0.757285\pi$
$937$	− 39623.7i	− 1.38148i	0.723102	−	0.690741i	$-0.242715\pi$
$938$	23873.6i	0.831024i
$939$	−4627.61	−0.160827
$940$	0	0
$941$	−24909.7	−0.862948	−0.431474	−	0.902125i	$-0.642006\pi$
$941$	−24909.7	−0.862948	−0.431474	+	0.902125i	$0.642006\pi$
$942$	− 24612.2i	− 0.851283i
$943$	4440.69i	0.153350i
$944$	−11600.2	−0.399951
$945$	0	0
$946$	−371.813	−0.0127787
$947$	− 51892.6i	− 1.78066i	−0.455320	−	0.890328i	$-0.650475\pi$
$947$	− 51892.6i	− 1.78066i	0.455320	−	0.890328i	$-0.349525\pi$
$948$	17214.6i	0.589772i
$949$	−36470.8	−1.24751
$950$	0	0
$951$	29118.8	0.992895
$952$	10244.9i	0.348780i
$953$	24462.4i	0.831496i	0.909480	+	0.415748i	$0.136480\pi$
$953$	24462.4i	0.831496i	−0.909480	+	0.415748i	$0.863520\pi$
$954$	5916.19	0.200780
$955$	0	0
$956$	680.385	0.0230180
$957$	6785.93i	0.229214i
$958$	27498.8i	0.927396i
$959$	−14105.4	−0.474960
$960$	0	0
$961$	2033.06	0.0682440
$962$	53485.2i	1.79255i
$963$	6127.27i	0.205035i
$964$	18934.9	0.632626
$965$	0	0
$966$	4186.60	0.139443
$967$	− 7340.86i	− 0.244122i	−0.992523	−	0.122061i	$-0.961050\pi$
$967$	− 7340.86i	− 0.244122i	0.992523	−	0.122061i	$-0.0389504\pi$
$968$	9490.84i	0.315131i
$969$	37520.1	1.24388
$970$	0	0
$971$	−33474.8	−1.10634	−0.553171	−	0.833068i	$-0.686582\pi$
$971$	−33474.8	−1.10634	−0.553171	+	0.833068i	$0.686582\pi$
$972$	− 9343.74i	− 0.308334i
$973$	24399.4i	0.803914i
$974$	−19347.4	−0.636480
$975$	0	0
$976$	−12548.6	−0.411548
$977$	− 29496.6i	− 0.965895i	−0.875649	−	0.482947i	$-0.839566\pi$
$977$	− 29496.6i	− 0.965895i	0.875649	−	0.482947i	$-0.160434\pi$
$978$	12150.1i	0.397257i
$979$	−5569.09	−0.181807
$980$	0	0
$981$	−7136.02	−0.232248
$982$	11817.4i	0.384021i
$983$	33214.4i	1.07770i	0.842403	+	0.538848i	$0.181140\pi$
$983$	33214.4i	1.07770i	−0.842403	+	0.538848i	$0.818860\pi$
$984$	−9219.16	−0.298675
$985$	0	0
$986$	−15878.2	−0.512845
$987$	11731.5i	0.378336i
$988$	27005.5i	0.869594i
$989$	−355.526	−0.0114308
$990$	0	0
$991$	−21527.1	−0.690041	−0.345020	−	0.938595i	$-0.612128\pi$
$991$	−21527.1	−0.690041	−0.345020	+	0.938595i	$0.612128\pi$
$992$	− 5708.58i	− 0.182709i
$993$	− 19403.1i	− 0.620079i
$994$	−4323.47	−0.137960
$995$	0	0
$996$	23525.9	0.748442
$997$	− 39017.1i	− 1.23940i	−0.784838	−	0.619701i	$-0.787254\pi$
$997$	− 39017.1i	− 1.23940i	0.784838	−	0.619701i	$-0.212746\pi$
$998$	24239.6i	0.768829i
$999$	−32516.7	−1.02981

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.b.s.599.8		12
5.2	odd	4		1150.4.a.w.1.2	✓	6
5.3	odd	4		1150.4.a.x.1.5	yes	6
5.4	even	2	inner	1150.4.b.s.599.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1150.4.a.w.1.2	✓	6	5.2	odd	4
1150.4.a.x.1.5	yes	6	5.3	odd	4
1150.4.b.s.599.5		12	5.4	even	2	inner
1150.4.b.s.599.8		12	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)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -5.96868i q^{3} -4.00000 q^{4} +11.9374 q^{6} -15.2484i q^{7} -8.00000i q^{8} -8.62516 q^{9} +12.0268 q^{11} +23.8747i q^{12} +90.1983i q^{13} +30.4968 q^{14} +16.0000 q^{16} -83.9831i q^{17} -17.2503i q^{18} -74.8503 q^{19} -91.0130 q^{21} +24.0537i q^{22} +23.0000i q^{23} -47.7495 q^{24} -180.397 q^{26} -109.674i q^{27} +60.9937i q^{28} -94.5321 q^{29} -178.393 q^{31} +32.0000i q^{32} -71.7844i q^{33} +167.966 q^{34} +34.5007 q^{36} -296.487i q^{37} -149.701i q^{38} +538.365 q^{39} +193.074 q^{41} -182.026i q^{42} +15.4576i q^{43} -48.1074 q^{44} -46.0000 q^{46} -128.899i q^{47} -95.4989i q^{48} +110.486 q^{49} -501.269 q^{51} -360.793i q^{52} +342.961i q^{53} +219.347 q^{54} -121.987 q^{56} +446.758i q^{57} -189.064i q^{58} -725.012 q^{59} -784.288 q^{61} -356.786i q^{62} +131.520i q^{63} -64.0000 q^{64} +143.569 q^{66} +782.822i q^{67} +335.932i q^{68} +137.280 q^{69} -141.768 q^{71} +69.0013i q^{72} +404.340i q^{73} +592.973 q^{74} +299.401 q^{76} -183.390i q^{77} +1076.73i q^{78} +721.038 q^{79} -887.486 q^{81} +386.147i q^{82} -985.391i q^{83} +364.052 q^{84} -30.9153 q^{86} +564.232i q^{87} -96.2148i q^{88} -463.054 q^{89} +1375.38 q^{91} -92.0000i q^{92} +1064.77i q^{93} +257.798 q^{94} +190.998 q^{96} +1680.05i q^{97} +220.971i q^{98} -103.734 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 48 q^{4} - 20 q^{6} - 122 q^{9} - 98 q^{11} + 168 q^{14} + 192 q^{16} + 458 q^{19} + 184 q^{21} + 80 q^{24} - 64 q^{26} + 364 q^{29} + 228 q^{31} + 700 q^{34} + 488 q^{36} + 286 q^{39} + 486 q^{41}+ \cdots + 284 q^{99}+O(q^{100}) \) 12 * q - 48 * q^4 - 20 * q^6 - 122 * q^9 - 98 * q^11 + 168 * q^14 + 192 * q^16 + 458 * q^19 + 184 * q^21 + 80 * q^24 - 64 * q^26 + 364 * q^29 + 228 * q^31 + 700 * q^34 + 488 * q^36 + 286 * q^39 + 486 * q^41 + 392 * q^44 - 552 * q^46 - 1296 * q^49 - 2062 * q^51 + 1376 * q^54 - 672 * q^56 + 1118 * q^59 - 1376 * q^61 - 768 * q^64 - 2564 * q^66 - 230 * q^69 + 1168 * q^71 + 256 * q^74 - 1832 * q^76 + 2864 * q^79 + 68 * q^81 - 736 * q^84 - 728 * q^86 + 1182 * q^89 - 5728 * q^91 + 1992 * q^94 - 320 * q^96 + 284 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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