LMFDB - Embedded newform 1710.4.f.a.341.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1710,4,Mod(341,1710)] 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("1710.341"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1710, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1710.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [40,-80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$100.893266110$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			341.11
Character	$\chi$	$=$	1710.341
Dual form			1710.4.f.a.341.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} -5.00000i q^{5} +26.6820 q^{7} -8.00000 q^{8} +10.0000i q^{10} +4.03096i q^{11} +20.1697i q^{13} -53.3639 q^{14} +16.0000 q^{16} +61.6549i q^{17} +(78.1463 + 27.4256i) q^{19} -20.0000i q^{20} -8.06192i q^{22} +63.3084i q^{23} -25.0000 q^{25} -40.3393i q^{26} +106.728 q^{28} -175.684 q^{29} +282.447i q^{31} -32.0000 q^{32} -123.310i q^{34} -133.410i q^{35} -197.954i q^{37} +(-156.293 - 54.8512i) q^{38} +40.0000i q^{40} -205.876 q^{41} -203.226 q^{43} +16.1238i q^{44} -126.617i q^{46} +504.324i q^{47} +368.928 q^{49} +50.0000 q^{50} +80.6787i q^{52} -165.512 q^{53} +20.1548 q^{55} -213.456 q^{56} +351.368 q^{58} -577.107 q^{59} -495.177 q^{61} -564.893i q^{62} +64.0000 q^{64} +100.848 q^{65} +138.983i q^{67} +246.619i q^{68} +266.820i q^{70} -342.233 q^{71} -418.213 q^{73} +395.909i q^{74} +(312.585 + 109.702i) q^{76} +107.554i q^{77} +860.281i q^{79} -80.0000i q^{80} +411.752 q^{82} -595.093i q^{83} +308.274 q^{85} +406.453 q^{86} -32.2477i q^{88} +25.7141 q^{89} +538.166i q^{91} +253.234i q^{92} -1008.65i q^{94} +(137.128 - 390.731i) q^{95} +310.597i q^{97} -737.855 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 80 q^{2} + 160 q^{4} - 56 q^{7} - 320 q^{8} + 112 q^{14} + 640 q^{16} - 76 q^{19} - 1000 q^{25} - 224 q^{28} - 120 q^{29} - 1280 q^{32} + 152 q^{38} - 312 q^{41} + 56 q^{43} + 2112 q^{49} + 2000 q^{50}+ \cdots - 4224 q^{98}+O(q^{100}) $ 40 * q - 80 * q^2 + 160 * q^4 - 56 * q^7 - 320 * q^8 + 112 * q^14 + 640 * q^16 - 76 * q^19 - 1000 * q^25 - 224 * q^28 - 120 * q^29 - 1280 * q^32 + 152 * q^38 - 312 * q^41 + 56 * q^43 + 2112 * q^49 + 2000 * q^50 + 288 * q^53 + 448 * q^56 + 240 * q^58 + 912 * q^59 - 848 * q^61 + 2560 * q^64 + 240 * q^65 - 1248 * q^71 + 512 * q^73 - 304 * q^76 + 624 * q^82 - 112 * q^86 - 552 * q^89 - 60 * q^95 - 4224 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1710\mathbb{Z}\right)^\times$.

$n$	$191$	$1027$	$1351$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	−2.00000			−0.707107
$2$	−2.00000			−0.707107
$3$		0			0
$3$		0			0
$4$	4.00000			0.500000
$5$		−	5.00000i		−	0.447214i
$5$		−	5.00000i		−	0.447214i
$6$		0			0
$7$	26.6820			1.44069			0.720346	−	0.693615i	$-0.243983\pi$
$7$	26.6820			1.44069			0.720346	+	0.693615i	$0.243983\pi$
$8$	−8.00000			−0.353553
$9$		0			0
$10$			10.0000i			0.316228i
$11$			4.03096i			0.110489i	0.998473	+	0.0552446i	$0.0175939\pi$
$11$			4.03096i			0.110489i	−0.998473	+	0.0552446i	$0.982406\pi$
$12$		0			0
$13$			20.1697i			0.430312i	0.976580	+	0.215156i	$0.0690260\pi$
$13$			20.1697i			0.430312i	−0.976580	+	0.215156i	$0.930974\pi$
$14$	−53.3639			−1.01872
$15$		0			0
$16$	16.0000			0.250000
$17$			61.6549i			0.879618i	0.898091	+	0.439809i	$0.144954\pi$
$17$			61.6549i			0.879618i	−0.898091	+	0.439809i	$0.855046\pi$
$18$		0			0
$19$	78.1463	+	27.4256i	0.943578	+	0.331151i
$19$	78.1463	+	27.4256i	0.943578	+	0.331151i
$20$		−	20.0000i		−	0.223607i
$21$		0			0
$22$		−	8.06192i		−	0.0781276i
$23$			63.3084i			0.573944i	0.957939	+	0.286972i	$0.0926487\pi$
$23$			63.3084i			0.573944i	−0.957939	+	0.286972i	$0.907351\pi$
$24$		0			0
$25$	−25.0000			−0.200000
$26$		−	40.3393i		−	0.304277i
$27$		0			0
$28$	106.728			0.720346
$29$	−175.684			−1.12496			−0.562478	−	0.826813i	$-0.690152\pi$
$29$	−175.684			−1.12496			−0.562478	+	0.826813i	$0.690152\pi$
$30$		0			0
$31$			282.447i			1.63642i	0.574922	+	0.818208i	$0.305032\pi$
$31$			282.447i			1.63642i	−0.574922	+	0.818208i	$0.694968\pi$
$32$	−32.0000			−0.176777
$33$		0			0
$34$		−	123.310i		−	0.621984i
$35$		−	133.410i		−	0.644297i
$36$		0			0
$37$		−	197.954i		−	0.879554i	−0.898107	−	0.439777i	$-0.855058\pi$
$37$		−	197.954i		−	0.879554i	0.898107	−	0.439777i	$-0.144942\pi$
$38$	−156.293	−	54.8512i	−0.667210	−	0.234159i
$39$		0			0
$40$			40.0000i			0.158114i
$41$	−205.876			−0.784206			−0.392103	−	0.919921i	$-0.628252\pi$
$41$	−205.876			−0.784206			−0.392103	+	0.919921i	$0.628252\pi$
$42$		0			0
$43$	−203.226			−0.720738			−0.360369	−	0.932810i	$-0.617349\pi$
$43$	−203.226			−0.720738			−0.360369	+	0.932810i	$0.617349\pi$
$44$			16.1238i			0.0552446i
$45$		0			0
$46$		−	126.617i		−	0.405840i
$47$			504.324i			1.56517i	0.622541	+	0.782587i	$0.286100\pi$
$47$			504.324i			1.56517i	−0.622541	+	0.782587i	$0.713900\pi$
$48$		0			0
$49$	368.928			1.07559
$50$	50.0000			0.141421
$51$		0			0
$52$			80.6787i			0.215156i
$53$	−165.512			−0.428959			−0.214480	−	0.976728i	$-0.568806\pi$
$53$	−165.512			−0.428959			−0.214480	+	0.976728i	$0.568806\pi$
$54$		0			0
$55$	20.1548			0.0494123
$56$	−213.456			−0.509361
$57$		0			0
$58$	351.368			0.795463
$59$	−577.107			−1.27344			−0.636720	−	0.771095i	$-0.719709\pi$
$59$	−577.107			−1.27344			−0.636720	+	0.771095i	$0.719709\pi$
$60$		0			0
$61$	−495.177			−1.03936			−0.519680	−	0.854361i	$-0.673949\pi$
$61$	−495.177			−1.03936			−0.519680	+	0.854361i	$0.673949\pi$
$62$		−	564.893i		−	1.15712i
$63$		0			0
$64$	64.0000			0.125000
$65$	100.848			0.192441
$66$		0			0
$67$			138.983i			0.253425i	0.991939	+	0.126712i	$0.0404425\pi$
$67$			138.983i			0.253425i	−0.991939	+	0.126712i	$0.959557\pi$
$68$			246.619i			0.439809i
$69$		0			0
$70$			266.820i			0.455587i
$71$	−342.233			−0.572051			−0.286025	−	0.958222i	$-0.592334\pi$
$71$	−342.233			−0.572051			−0.286025	+	0.958222i	$0.592334\pi$
$72$		0			0
$73$	−418.213			−0.670522			−0.335261	−	0.942125i	$-0.608824\pi$
$73$	−418.213			−0.670522			−0.335261	+	0.942125i	$0.608824\pi$
$74$			395.909i			0.621938i
$75$		0			0
$76$	312.585	+	109.702i	0.471789	+	0.165575i
$77$			107.554i			0.159181i
$78$		0			0
$79$			860.281i			1.22518i	0.790401	+	0.612590i	$0.209872\pi$
$79$			860.281i			1.22518i	−0.790401	+	0.612590i	$0.790128\pi$
$80$		−	80.0000i		−	0.111803i
$81$		0			0
$82$	411.752			0.554517
$83$		−	595.093i		−	0.786987i	−0.919327	−	0.393494i	$-0.871266\pi$
$83$		−	595.093i		−	0.786987i	0.919327	−	0.393494i	$-0.128734\pi$
$84$		0			0
$85$	308.274			0.393377
$86$	406.453			0.509638
$87$		0			0
$88$		−	32.2477i		−	0.0390638i
$89$	25.7141			0.0306257			0.0153129	−	0.999883i	$-0.495126\pi$
$89$	25.7141			0.0306257			0.0153129	+	0.999883i	$0.495126\pi$
$90$		0			0
$91$			538.166i			0.619947i
$92$			253.234i			0.286972i
$93$		0			0
$94$		−	1008.65i		−	1.10675i
$95$	137.128	−	390.731i	0.148095	−	0.421981i
$96$		0			0
$97$			310.597i			0.325117i	0.986699	+	0.162558i	$0.0519745\pi$
$97$			310.597i			0.325117i	−0.986699	+	0.162558i	$0.948025\pi$
$98$	−737.855			−0.760558
$99$		0			0
$100$	−100.000			−0.100000
$101$		−	1413.46i		−	1.39252i	−0.717790	−	0.696259i	$-0.754846\pi$
$101$		−	1413.46i		−	1.39252i	0.717790	−	0.696259i	$-0.245154\pi$
$102$		0			0
$103$		−	1039.28i		−	0.994206i	−0.867691	−	0.497103i	$-0.834397\pi$
$103$		−	1039.28i		−	0.994206i	0.867691	−	0.497103i	$-0.165603\pi$
$104$		−	161.357i		−	0.152138i
$105$		0			0
$106$	331.024			0.303320
$107$	15.7183			0.0142014			0.00710069	−	0.999975i	$-0.497740\pi$
$107$	15.7183			0.0142014			0.00710069	+	0.999975i	$0.497740\pi$
$108$		0			0
$109$		−	1811.26i		−	1.59163i	−0.605541	−	0.795814i	$-0.707043\pi$
$109$		−	1811.26i		−	1.59163i	0.605541	−	0.795814i	$-0.292957\pi$
$110$	−40.3096			−0.0349397
$111$		0			0
$112$	426.912			0.360173
$113$	−253.224			−0.210808			−0.105404	−	0.994429i	$-0.533614\pi$
$113$	−253.224			−0.210808			−0.105404	+	0.994429i	$0.533614\pi$
$114$		0			0
$115$	316.542			0.256676
$116$	−702.736			−0.562478
$117$		0			0
$118$	1154.21			0.900458
$119$			1645.07i			1.26726i
$120$		0			0
$121$	1314.75			0.987792
$122$	990.354			0.734938
$123$		0			0
$124$			1129.79i			0.818208i
$125$			125.000i			0.0894427i
$126$		0			0
$127$			1241.87i			0.867700i	0.900985	+	0.433850i	$0.142845\pi$
$127$			1241.87i			0.867700i	−0.900985	+	0.433850i	$0.857155\pi$
$128$	−128.000			−0.0883883
$129$		0			0
$130$	−201.697			−0.136077
$131$			174.213i			0.116191i	0.998311	+	0.0580956i	$0.0185028\pi$
$131$			174.213i			0.116191i	−0.998311	+	0.0580956i	$0.981497\pi$
$132$		0			0
$133$	2085.10	+	731.769i	1.35940	+	0.477086i
$134$		−	277.966i		−	0.179198i
$135$		0			0
$136$		−	493.239i		−	0.310992i
$137$		−	744.656i		−	0.464382i	−0.972670	−	0.232191i	$-0.925411\pi$
$137$		−	744.656i		−	0.464382i	0.972670	−	0.232191i	$-0.0745894\pi$
$138$		0			0
$139$	−2687.71			−1.64006			−0.820030	−	0.572321i	$-0.806043\pi$
$139$	−2687.71			−1.64006			−0.820030	+	0.572321i	$0.806043\pi$
$140$		−	533.639i		−	0.322148i
$141$		0			0
$142$	684.466			0.404501
$143$	−81.3032			−0.0475448
$144$		0			0
$145$			878.420i			0.503095i
$146$	836.425			0.474130
$147$		0			0
$148$		−	791.817i		−	0.439777i
$149$		−	208.033i		−	0.114381i	−0.998363	−	0.0571903i	$-0.981786\pi$
$149$		−	208.033i		−	0.114381i	0.998363	−	0.0571903i	$-0.0182142\pi$
$150$		0			0
$151$		−	642.170i		−	0.346087i	−0.984914	−	0.173043i	$-0.944640\pi$
$151$		−	642.170i		−	0.346087i	0.984914	−	0.173043i	$-0.0553600\pi$
$152$	−625.170	−	219.405i	−0.333605	−	0.117079i
$153$		0			0
$154$		−	215.108i		−	0.112558i
$155$	1412.23			0.731828
$156$		0			0
$157$	1889.60			0.960551			0.480275	−	0.877118i	$-0.340537\pi$
$157$	1889.60			0.960551			0.480275	+	0.877118i	$0.340537\pi$
$158$		−	1720.56i		−	0.866333i
$159$		0			0
$160$			160.000i			0.0790569i
$161$			1689.19i			0.826876i
$162$		0			0
$163$	−2279.93			−1.09557			−0.547784	−	0.836619i	$-0.684529\pi$
$163$	−2279.93			−1.09557			−0.547784	+	0.836619i	$0.684529\pi$
$164$	−823.504			−0.392103
$165$		0			0
$166$			1190.19i			0.556484i
$167$	653.334			0.302734			0.151367	−	0.988478i	$-0.451633\pi$
$167$	653.334			0.302734			0.151367	+	0.988478i	$0.451633\pi$
$168$		0			0
$169$	1790.18			0.814831
$170$	−616.549			−0.278159
$171$		0			0
$172$	−812.905			−0.360369
$173$	−3562.25			−1.56551			−0.782754	−	0.622331i	$-0.786186\pi$
$173$	−3562.25			−1.56551			−0.782754	+	0.622331i	$0.786186\pi$
$174$		0			0
$175$	−667.049			−0.288138
$176$			64.4954i			0.0276223i
$177$		0			0
$178$	−51.4282			−0.0216557
$179$	−3680.25			−1.53673			−0.768365	−	0.640011i	$-0.778930\pi$
$179$	−3680.25			−1.53673			−0.768365	+	0.640011i	$0.778930\pi$
$180$		0			0
$181$			3018.01i			1.23938i	0.784848	+	0.619688i	$0.212741\pi$
$181$			3018.01i			1.23938i	−0.784848	+	0.619688i	$0.787259\pi$
$182$		−	1076.33i		−	0.438369i
$183$		0			0
$184$		−	506.467i		−	0.202920i
$185$	−989.772			−0.393348
$186$		0			0
$187$	−248.528			−0.0971882
$188$			2017.30i			0.782587i
$189$		0			0
$190$	−274.256	+	781.463i	−0.104719	+	0.298386i
$191$			1060.96i			0.401929i	0.979599	+	0.200965i	$0.0644076\pi$
$191$			1060.96i			0.401929i	−0.979599	+	0.200965i	$0.935592\pi$
$192$		0			0
$193$		−	343.242i		−	0.128016i	−0.997949	−	0.0640081i	$-0.979612\pi$
$193$		−	343.242i		−	0.128016i	0.997949	−	0.0640081i	$-0.0203883\pi$
$194$		−	621.193i		−	0.229892i
$195$		0			0
$196$	1475.71			0.537796
$197$		−	1413.46i		−	0.511194i	−0.966783	−	0.255597i	$-0.917728\pi$
$197$		−	1413.46i		−	0.511194i	0.966783	−	0.255597i	$-0.0822720\pi$
$198$		0			0
$199$	4623.97			1.64716			0.823580	−	0.567201i	$-0.191974\pi$
$199$	4623.97			1.64716			0.823580	+	0.567201i	$0.191974\pi$
$200$	200.000			0.0707107
$201$		0			0
$202$			2826.92i			0.984659i
$203$	−4687.60			−1.62071
$204$		0			0
$205$			1029.38i			0.350707i
$206$			2078.56i			0.703010i
$207$		0			0
$208$			322.715i			0.107578i
$209$	−110.551	+	315.005i	−0.0365885	+	0.104255i
$210$		0			0
$211$		−	1734.58i		−	0.565939i	−0.959129	−	0.282970i	$-0.908680\pi$
$211$		−	1734.58i		−	0.565939i	0.959129	−	0.282970i	$-0.0913196\pi$
$212$	−662.048			−0.214480
$213$		0			0
$214$	−31.4367			−0.0100419
$215$			1016.13i			0.322324i
$216$		0			0
$217$			7536.23i			2.35757i
$218$			3622.52i			1.12545i
$219$		0			0
$220$	80.6192			0.0247061
$221$	−1243.56			−0.378510
$222$		0			0
$223$			3856.25i			1.15800i	0.815329	+	0.578999i	$0.196556\pi$
$223$			3856.25i			1.15800i	−0.815329	+	0.578999i	$0.803444\pi$
$224$	−853.823			−0.254681
$225$		0			0
$226$	506.448			0.149064
$227$	2197.61			0.642556			0.321278	−	0.946985i	$-0.395888\pi$
$227$	2197.61			0.642556			0.321278	+	0.946985i	$0.395888\pi$
$228$		0			0
$229$	1376.60			0.397242			0.198621	−	0.980076i	$-0.436354\pi$
$229$	1376.60			0.397242			0.198621	+	0.980076i	$0.436354\pi$
$230$	−633.084			−0.181497
$231$		0			0
$232$	1405.47			0.397732
$233$		−	2531.43i		−	0.711758i	−0.934532	−	0.355879i	$-0.884181\pi$
$233$		−	2531.43i		−	0.711758i	0.934532	−	0.355879i	$-0.115819\pi$
$234$		0			0
$235$	2521.62			0.699967
$236$	−2308.43			−0.636720
$237$		0			0
$238$		−	3290.15i		−	0.896086i
$239$			1936.19i			0.524025i	0.965065	+	0.262012i	$0.0843861\pi$
$239$			1936.19i			0.524025i	−0.965065	+	0.262012i	$0.915614\pi$
$240$		0			0
$241$		−	1723.16i		−	0.460574i	−0.973123	−	0.230287i	$-0.926033\pi$
$241$		−	1723.16i		−	0.460574i	0.973123	−	0.230287i	$-0.0739666\pi$
$242$	−2629.50			−0.698475
$243$		0			0
$244$	−1980.71			−0.519680
$245$		−	1844.64i		−	0.481019i
$246$		0			0
$247$	−553.165	+	1576.18i	−0.142498	+	0.406033i
$248$		−	2259.57i		−	0.578561i
$249$		0			0
$250$		−	250.000i		−	0.0632456i
$251$			281.572i			0.0708074i	0.999373	+	0.0354037i	$0.0112717\pi$
$251$			281.572i			0.0708074i	−0.999373	+	0.0354037i	$0.988728\pi$
$252$		0			0
$253$	−255.194			−0.0634146
$254$		−	2483.73i		−	0.613556i
$255$		0			0
$256$	256.000			0.0625000
$257$	−7841.48			−1.90326			−0.951631	−	0.307244i	$-0.900593\pi$
$257$	−7841.48			−1.90326			−0.951631	+	0.307244i	$0.900593\pi$
$258$		0			0
$259$		−	5281.81i		−	1.26717i
$260$	403.393			0.0962207
$261$		0			0
$262$		−	348.425i		−	0.0821595i
$263$			4722.12i			1.10714i	0.832802	+	0.553572i	$0.186735\pi$
$263$			4722.12i			1.10714i	−0.832802	+	0.553572i	$0.813265\pi$
$264$		0			0
$265$			827.560i			0.191836i
$266$	−4170.19	−	1463.54i	−0.961244	−	0.337351i
$267$		0			0
$268$			555.931i			0.126712i
$269$	1607.37			0.364323			0.182161	−	0.983269i	$-0.441691\pi$
$269$	1607.37			0.364323			0.182161	+	0.983269i	$0.441691\pi$
$270$		0			0
$271$	−924.365			−0.207200			−0.103600	−	0.994619i	$-0.533036\pi$
$271$	−924.365			−0.207200			−0.103600	+	0.994619i	$0.533036\pi$
$272$			986.478i			0.219904i
$273$		0			0
$274$			1489.31i			0.328367i
$275$		−	100.774i		−	0.0220978i
$276$		0			0
$277$	−116.985			−0.0253753			−0.0126876	−	0.999920i	$-0.504039\pi$
$277$	−116.985			−0.0253753			−0.0126876	+	0.999920i	$0.504039\pi$
$278$	5375.41			1.15970
$279$		0			0
$280$			1067.28i			0.227793i
$281$	−62.8583			−0.0133445			−0.00667226	−	0.999978i	$-0.502124\pi$
$281$	−62.8583			−0.0133445			−0.00667226	+	0.999978i	$0.502124\pi$
$282$		0			0
$283$	−1626.02			−0.341544			−0.170772	−	0.985311i	$-0.554626\pi$
$283$	−1626.02			−0.341544			−0.170772	+	0.985311i	$0.554626\pi$
$284$	−1368.93			−0.286025
$285$		0			0
$286$	162.606			0.0336193
$287$	−5493.18			−1.12980
$288$		0			0
$289$	1111.68			0.226273
$290$		−	1756.84i		−	0.355742i
$291$		0			0
$292$	−1672.85			−0.335261
$293$	3273.01			0.652599			0.326300	−	0.945266i	$-0.394198\pi$
$293$	3273.01			0.652599			0.326300	+	0.945266i	$0.394198\pi$
$294$		0			0
$295$			2885.54i			0.569500i
$296$			1583.63i			0.310969i
$297$		0			0
$298$			416.065i			0.0808793i
$299$	−1276.91			−0.246975
$300$		0			0
$301$	−5422.48			−1.03836
$302$			1284.34i			0.244720i
$303$		0			0
$304$	1250.34	+	438.809i	0.235894	+	0.0827876i
$305$			2475.89i			0.464816i
$306$		0			0
$307$			3005.62i			0.558762i	0.960180	+	0.279381i	$0.0901292\pi$
$307$			3005.62i			0.558762i	−0.960180	+	0.279381i	$0.909871\pi$
$308$			430.216i			0.0795904i
$309$		0			0
$310$	−2824.47			−0.517480
$311$			3280.50i			0.598135i	0.954232	+	0.299067i	$0.0966755\pi$
$311$			3280.50i			0.598135i	−0.954232	+	0.299067i	$0.903324\pi$
$312$		0			0
$313$	−8508.09			−1.53644			−0.768220	−	0.640186i	$-0.778857\pi$
$313$	−8508.09			−1.53644			−0.768220	+	0.640186i	$0.778857\pi$
$314$	−3779.20			−0.679212
$315$		0			0
$316$			3441.12i			0.612590i
$317$	494.452			0.0876063			0.0438032	−	0.999040i	$-0.486053\pi$
$317$	494.452			0.0876063			0.0438032	+	0.999040i	$0.486053\pi$
$318$		0			0
$319$		−	708.176i		−	0.124295i
$320$		−	320.000i		−	0.0559017i
$321$		0			0
$322$		−	3378.39i		−	0.584690i
$323$	−1690.92	+	4818.10i	−0.291286	+	0.829988i
$324$		0			0
$325$		−	504.242i		−	0.0860624i
$326$	4559.86			0.774684
$327$		0			0
$328$	1647.01			0.277259
$329$			13456.4i			2.25493i
$330$		0			0
$331$			5575.70i			0.925886i	0.886388	+	0.462943i	$0.153207\pi$
$331$			5575.70i			0.925886i	−0.886388	+	0.462943i	$0.846793\pi$
$332$		−	2380.37i		−	0.393494i
$333$		0			0
$334$	−1306.67			−0.214065
$335$	694.914			0.113335
$336$		0			0
$337$		−	7583.38i		−	1.22580i	−0.790162	−	0.612898i	$-0.790004\pi$
$337$		−	7583.38i		−	1.22580i	0.790162	−	0.612898i	$-0.209996\pi$
$338$	−3580.37			−0.576173
$339$		0			0
$340$	1233.10			0.196688
$341$	−1138.53			−0.180806
$342$		0			0
$343$	691.803			0.108903
$344$	1625.81			0.254819
$345$		0			0
$346$	7124.51			1.10698
$347$			1079.04i			0.166934i	0.996511	+	0.0834671i	$0.0265993\pi$
$347$			1079.04i			0.166934i	−0.996511	+	0.0834671i	$0.973401\pi$
$348$		0			0
$349$	−5777.05			−0.886071			−0.443035	−	0.896504i	$-0.646098\pi$
$349$	−5777.05			−0.886071			−0.443035	+	0.896504i	$0.646098\pi$
$350$	1334.10			0.203744
$351$		0			0
$352$		−	128.991i		−	0.0195319i
$353$			632.906i			0.0954283i	0.998861	+	0.0477141i	$0.0151936\pi$
$353$			632.906i			0.0954283i	−0.998861	+	0.0477141i	$0.984806\pi$
$354$		0			0
$355$			1711.17i			0.255829i
$356$	102.856			0.0153129
$357$		0			0
$358$	7360.50			1.08663
$359$			4168.76i			0.612866i	0.951892	+	0.306433i	$0.0991355\pi$
$359$			4168.76i			0.612866i	−0.951892	+	0.306433i	$0.900864\pi$
$360$		0			0
$361$	5354.67	+	4286.41i	0.780679	+	0.624933i
$362$		−	6036.03i		−	0.876372i
$363$		0			0
$364$			2152.67i			0.309973i
$365$			2091.06i			0.299866i
$366$		0			0
$367$	−8361.04			−1.18922			−0.594609	−	0.804015i	$-0.702693\pi$
$367$	−8361.04			−1.18922			−0.594609	+	0.804015i	$0.702693\pi$
$368$			1012.93i			0.143486i
$369$		0			0
$370$	1979.54			0.278139
$371$	−4416.19			−0.617997
$372$		0			0
$373$		−	3633.75i		−	0.504419i	−0.967673	−	0.252210i	$-0.918843\pi$
$373$		−	3633.75i		−	0.504419i	0.967673	−	0.252210i	$-0.0811572\pi$
$374$	497.057			0.0687224
$375$		0			0
$376$		−	4034.59i		−	0.553373i
$377$		−	3543.49i		−	0.484082i
$378$		0			0
$379$			12979.0i			1.75907i	0.475836	+	0.879534i	$0.342146\pi$
$379$			12979.0i			1.75907i	−0.475836	+	0.879534i	$0.657854\pi$
$380$	548.512	−	1562.93i	0.0740475	−	0.210990i
$381$		0			0
$382$		−	2121.92i		−	0.284207i
$383$	5083.40			0.678197			0.339098	−	0.940751i	$-0.389878\pi$
$383$	5083.40			0.678197			0.339098	+	0.940751i	$0.389878\pi$
$384$		0			0
$385$	537.770			0.0711878
$386$			686.485i			0.0905211i
$387$		0			0
$388$			1242.39i			0.162558i
$389$			5187.19i			0.676095i	0.941129	+	0.338048i	$0.109766\pi$
$389$			5187.19i			0.676095i	−0.941129	+	0.338048i	$0.890234\pi$
$390$		0			0
$391$	−3903.27			−0.504851
$392$	−2951.42			−0.380279
$393$		0			0
$394$			2826.93i			0.361469i
$395$	4301.41			0.547917
$396$		0			0
$397$	−14120.3			−1.78508			−0.892538	−	0.450972i	$-0.851078\pi$
$397$	−14120.3			−1.78508			−0.892538	+	0.450972i	$0.851078\pi$
$398$	−9247.94			−1.16472
$399$		0			0
$400$	−400.000			−0.0500000
$401$	956.546			0.119121			0.0595607	−	0.998225i	$-0.481030\pi$
$401$	956.546			0.119121			0.0595607	+	0.998225i	$0.481030\pi$
$402$		0			0
$403$	−5696.85			−0.704170
$404$		−	5653.83i		−	0.696259i
$405$		0			0
$406$	9375.19			1.14602
$407$	797.946			0.0971812
$408$		0			0
$409$			4640.46i			0.561017i	0.959852	+	0.280508i	$0.0905031\pi$
$409$			4640.46i			0.561017i	−0.959852	+	0.280508i	$0.909497\pi$
$410$		−	2058.76i		−	0.247988i
$411$		0			0
$412$		−	4157.12i		−	0.497103i
$413$	−15398.4			−1.83463
$414$		0			0
$415$	−2975.47			−0.351951
$416$		−	645.429i		−	0.0760692i
$417$		0			0
$418$	221.103	−	630.009i	0.0258720	−	0.0737195i
$419$			10115.3i			1.17939i	0.807627	+	0.589694i	$0.200752\pi$
$419$			10115.3i			1.17939i	−0.807627	+	0.589694i	$0.799248\pi$
$420$		0			0
$421$			15180.7i			1.75739i	0.477387	+	0.878693i	$0.341584\pi$
$421$			15180.7i			1.75739i	−0.477387	+	0.878693i	$0.658416\pi$
$422$			3469.15i			0.400180i
$423$		0			0
$424$	1324.10			0.151660
$425$		−	1541.37i		−	0.175924i
$426$		0			0
$427$	−13212.3			−1.49740
$428$	62.8733			0.00710069
$429$		0			0
$430$		−	2032.26i		−	0.227917i
$431$	14107.0			1.57659			0.788295	−	0.615297i	$-0.210964\pi$
$431$	14107.0			1.57659			0.788295	+	0.615297i	$0.210964\pi$
$432$		0			0
$433$		−	7215.50i		−	0.800820i	−0.916336	−	0.400410i	$-0.868868\pi$
$433$		−	7215.50i		−	0.800820i	0.916336	−	0.400410i	$-0.131132\pi$
$434$		−	15072.5i		−	1.66705i
$435$		0			0
$436$		−	7245.05i		−	0.795814i
$437$	−1736.27	+	4947.31i	−0.190062	+	0.541561i
$438$		0			0
$439$			13614.9i			1.48019i	0.672500	+	0.740097i	$0.265220\pi$
$439$			13614.9i			1.48019i	−0.672500	+	0.740097i	$0.734780\pi$
$440$	−161.238			−0.0174699
$441$		0			0
$442$	2487.12			0.267647
$443$			12246.5i			1.31342i	0.754142	+	0.656712i	$0.228053\pi$
$443$			12246.5i			1.31342i	−0.754142	+	0.656712i	$0.771947\pi$
$444$		0			0
$445$		−	128.571i		−	0.0136962i
$446$		−	7712.49i		−	0.818828i
$447$		0			0
$448$	1707.65			0.180086
$449$	11066.6			1.16318			0.581589	−	0.813483i	$-0.302431\pi$
$449$	11066.6			1.16318			0.581589	+	0.813483i	$0.302431\pi$
$450$		0			0
$451$		−	829.879i		−	0.0866462i
$452$	−1012.90			−0.105404
$453$		0			0
$454$	−4395.21			−0.454356
$455$	2690.83			0.277249
$456$		0			0
$457$	1096.54			0.112241			0.0561206	−	0.998424i	$-0.482127\pi$
$457$	1096.54			0.112241			0.0561206	+	0.998424i	$0.482127\pi$
$458$	−2753.20			−0.280892
$459$		0			0
$460$	1266.17			0.128338
$461$		−	10189.5i		−	1.02944i	−0.857359	−	0.514719i	$-0.827896\pi$
$461$		−	10189.5i		−	1.02944i	0.857359	−	0.514719i	$-0.172104\pi$
$462$		0			0
$463$	11492.8			1.15359			0.576797	−	0.816888i	$-0.304303\pi$
$463$	11492.8			1.15359			0.576797	+	0.816888i	$0.304303\pi$
$464$	−2810.94			−0.281239
$465$		0			0
$466$			5062.87i			0.503289i
$467$		−	14018.0i		−	1.38903i	−0.719477	−	0.694516i	$-0.755619\pi$
$467$		−	14018.0i		−	1.38903i	0.719477	−	0.694516i	$-0.244381\pi$
$468$		0			0
$469$			3708.34i			0.365107i
$470$	−5043.24			−0.494952
$471$		0			0
$472$	4616.86			0.450229
$473$		−	819.197i		−	0.0796337i
$474$		0			0
$475$	−1953.66	−	685.640i	−0.188716	−	0.0662301i
$476$			6580.29i			0.633629i
$477$		0			0
$478$		−	3872.39i		−	0.370541i
$479$			7395.19i			0.705417i	0.935733	+	0.352709i	$0.114739\pi$
$479$			7395.19i			0.705417i	−0.935733	+	0.352709i	$0.885261\pi$
$480$		0			0
$481$	3992.67			0.378483
$482$			3446.32i			0.325675i
$483$		0			0
$484$	5259.01			0.493896
$485$	1552.98			0.145397
$486$		0			0
$487$		−	7380.20i		−	0.686712i	−0.939205	−	0.343356i	$-0.888436\pi$
$487$		−	7380.20i		−	0.686712i	0.939205	−	0.343356i	$-0.111564\pi$
$488$	3961.42			0.367469
$489$		0			0
$490$			3689.28i			0.340132i
$491$			16979.1i			1.56061i	0.625402	+	0.780303i	$0.284935\pi$
$491$			16979.1i			1.56061i	−0.625402	+	0.780303i	$0.715065\pi$
$492$		0			0
$493$		−	10831.8i		−	0.989530i
$494$	1106.33	−	3152.37i	0.100761	−	0.287109i
$495$		0			0
$496$			4519.15i			0.409104i
$497$	−9131.46			−0.824148
$498$		0			0
$499$	14389.7			1.29092			0.645461	−	0.763793i	$-0.276665\pi$
$499$	14389.7			1.29092			0.645461	+	0.763793i	$0.276665\pi$
$500$			500.000i			0.0447214i
$501$		0			0
$502$		−	563.143i		−	0.0500684i
$503$		−	1467.64i		−	0.130097i	−0.997882	−	0.0650486i	$-0.979280\pi$
$503$		−	1467.64i		−	0.130097i	0.997882	−	0.0650486i	$-0.0207202\pi$
$504$		0			0
$505$	−7067.29			−0.622753
$506$	510.387			0.0448409
$507$		0			0
$508$			4967.47i			0.433850i
$509$	13699.0			1.19293			0.596463	−	0.802640i	$-0.296572\pi$
$509$	13699.0			1.19293			0.596463	+	0.802640i	$0.296572\pi$
$510$		0			0
$511$	−11158.7			−0.966015
$512$	−512.000			−0.0441942
$513$		0			0
$514$	15683.0			1.34581
$515$	−5196.40			−0.444623
$516$		0			0
$517$	−2032.91			−0.172935
$518$			10563.6i			0.896021i
$519$		0			0
$520$	−806.787			−0.0680383
$521$	−12407.9			−1.04338			−0.521690	−	0.853135i	$-0.674698\pi$
$521$	−12407.9			−1.04338			−0.521690	+	0.853135i	$0.674698\pi$
$522$		0			0
$523$			983.734i			0.0822479i	0.999154	+	0.0411240i	$0.0130939\pi$
$523$			983.734i			0.0822479i	−0.999154	+	0.0411240i	$0.986906\pi$
$524$			696.851i			0.0580956i
$525$		0			0
$526$		−	9444.25i		−	0.782868i
$527$	−17414.2			−1.43942
$528$		0			0
$529$	8159.05			0.670588
$530$		−	1655.12i		−	0.135649i
$531$		0			0
$532$	8340.39	+	2927.07i	0.679702	+	0.238543i
$533$		−	4152.45i		−	0.337453i
$534$		0			0
$535$		−	78.5916i		−	0.00635105i
$536$		−	1111.86i		−	0.0895992i
$537$		0			0
$538$	−3214.73			−0.257615
$539$			1487.13i			0.118841i
$540$		0			0
$541$	−9038.28			−0.718273			−0.359137	−	0.933285i	$-0.616929\pi$
$541$	−9038.28			−0.718273			−0.359137	+	0.933285i	$0.616929\pi$
$542$	1848.73			0.146513
$543$		0			0
$544$		−	1972.96i		−	0.155496i
$545$	−9056.31			−0.711798
$546$		0			0
$547$		−	5904.92i		−	0.461565i	−0.973005	−	0.230782i	$-0.925871\pi$
$547$		−	5904.92i		−	0.461565i	0.973005	−	0.230782i	$-0.0741286\pi$
$548$		−	2978.62i		−	0.232191i
$549$		0			0
$550$			201.548i			0.0156255i
$551$	−13729.0	−	4818.24i	−1.06148	−	0.372530i
$552$		0			0
$553$			22954.0i			1.76511i
$554$	233.970			0.0179430
$555$		0			0
$556$	−10750.8			−0.820030
$557$			7436.51i			0.565701i	0.959164	+	0.282850i	$0.0912800\pi$
$557$			7436.51i			0.565701i	−0.959164	+	0.282850i	$0.908720\pi$
$558$		0			0
$559$		−	4099.01i		−	0.310142i
$560$		−	2134.56i		−	0.161074i
$561$		0			0
$562$	125.717			0.00943601
$563$	24313.8			1.82008			0.910041	−	0.414518i	$-0.136050\pi$
$563$	24313.8			1.82008			0.910041	+	0.414518i	$0.136050\pi$
$564$		0			0
$565$			1266.12i			0.0942762i
$566$	3252.05			0.241508
$567$		0			0
$568$	2737.87			0.202250
$569$	19127.1			1.40922			0.704612	−	0.709593i	$-0.251121\pi$
$569$	19127.1			1.40922			0.704612	+	0.709593i	$0.251121\pi$
$570$		0			0
$571$	−20402.5			−1.49530			−0.747650	−	0.664093i	$-0.768818\pi$
$571$	−20402.5			−1.49530			−0.747650	+	0.664093i	$0.768818\pi$
$572$	−325.213			−0.0237724
$573$		0			0
$574$	10986.4			0.798888
$575$		−	1582.71i		−	0.114789i
$576$		0			0
$577$	−17950.0			−1.29509			−0.647547	−	0.762025i	$-0.724205\pi$
$577$	−17950.0			−1.29509			−0.647547	+	0.762025i	$0.724205\pi$
$578$	−2223.36			−0.159999
$579$		0			0
$580$			3513.68i			0.251548i
$581$		−	15878.3i		−	1.13381i
$582$		0			0
$583$		−	667.173i		−	0.0473953i
$584$	3345.70			0.237065
$585$		0			0
$586$	−6546.03			−0.461457
$587$		−	6463.47i		−	0.454474i	−0.973840	−	0.227237i	$-0.927031\pi$
$587$		−	6463.47i		−	0.454474i	0.973840	−	0.227237i	$-0.0729692\pi$
$588$		0			0
$589$	−7746.26	+	22072.1i	−0.541900	+	1.54409i
$590$		−	5771.07i		−	0.402697i
$591$		0			0
$592$		−	3167.27i		−	0.219888i
$593$			4805.94i			0.332810i	0.986057	+	0.166405i	$0.0532159\pi$
$593$			4805.94i			0.332810i	−0.986057	+	0.166405i	$0.946784\pi$
$594$		0			0
$595$	8225.37			0.566735
$596$		−	832.131i		−	0.0571903i
$597$		0			0
$598$	2553.82			0.174638
$599$	14593.4			0.995441			0.497721	−	0.867337i	$-0.334170\pi$
$599$	14593.4			0.995441			0.497721	+	0.867337i	$0.334170\pi$
$600$		0			0
$601$		−	15730.3i		−	1.06764i	−0.845598	−	0.533821i	$-0.820756\pi$
$601$		−	15730.3i		−	1.06764i	0.845598	−	0.533821i	$-0.179244\pi$
$602$	10845.0			0.734232
$603$		0			0
$604$		−	2568.68i		−	0.173043i
$605$		−	6573.76i		−	0.441754i
$606$		0			0
$607$		−	18717.6i		−	1.25161i	−0.779981	−	0.625803i	$-0.784771\pi$
$607$		−	18717.6i		−	1.25161i	0.779981	−	0.625803i	$-0.215229\pi$
$608$	−2500.68	−	877.619i	−0.166803	−	0.0585397i
$609$		0			0
$610$		−	4951.77i		−	0.328674i
$611$	−10172.0			−0.673514
$612$		0			0
$613$	−21059.5			−1.38758			−0.693788	−	0.720180i	$-0.744059\pi$
$613$	−21059.5			−1.38758			−0.693788	+	0.720180i	$0.744059\pi$
$614$		−	6011.24i		−	0.395104i
$615$		0			0
$616$		−	860.432i		−	0.0562789i
$617$			23163.4i			1.51138i	0.654928	+	0.755691i	$0.272699\pi$
$617$			23163.4i			1.51138i	−0.654928	+	0.755691i	$0.727301\pi$
$618$		0			0
$619$	15131.7			0.982545			0.491273	−	0.871006i	$-0.336532\pi$
$619$	15131.7			0.982545			0.491273	+	0.871006i	$0.336532\pi$
$620$	5648.93			0.365914
$621$		0			0
$622$		−	6560.99i		−	0.422945i
$623$	686.103			0.0441222
$624$		0			0
$625$	625.000			0.0400000
$626$	17016.2			1.08643
$627$		0			0
$628$	7558.40			0.480275
$629$	12204.8			0.773671
$630$		0			0
$631$	−12268.2			−0.773989			−0.386995	−	0.922082i	$-0.626487\pi$
$631$	−12268.2			−0.773989			−0.386995	+	0.922082i	$0.626487\pi$
$632$		−	6882.25i		−	0.433166i
$633$		0			0
$634$	−988.904			−0.0619470
$635$	6209.34			0.388047
$636$		0			0
$637$			7441.15i			0.462840i
$638$			1416.35i			0.0878901i
$639$		0			0
$640$			640.000i			0.0395285i
$641$	24506.9			1.51009			0.755043	−	0.655675i	$-0.227616\pi$
$641$	24506.9			1.51009			0.755043	+	0.655675i	$0.227616\pi$
$642$		0			0
$643$	11464.3			0.703123			0.351562	−	0.936165i	$-0.385651\pi$
$643$	11464.3			0.703123			0.351562	+	0.936165i	$0.385651\pi$
$644$			6756.77i			0.413438i
$645$		0			0
$646$	3381.84	−	9636.19i	0.205970	−	0.586890i
$647$		−	17774.5i		−	1.08005i	−0.841650	−	0.540023i	$-0.818416\pi$
$647$		−	17774.5i		−	1.08005i	0.841650	−	0.540023i	$-0.181584\pi$
$648$		0			0
$649$		−	2326.30i		−	0.140701i
$650$			1008.48i			0.0608553i
$651$		0			0
$652$	−9119.71			−0.547784
$653$			6125.96i			0.367117i	0.983009	+	0.183559i	$0.0587617\pi$
$653$			6125.96i			0.367117i	−0.983009	+	0.183559i	$0.941238\pi$
$654$		0			0
$655$	871.064			0.0519622
$656$	−3294.02			−0.196051
$657$		0			0
$658$		−	26912.7i		−	1.59448i
$659$	32207.3			1.90382			0.951912	−	0.306372i	$-0.0991152\pi$
$659$	32207.3			1.90382			0.951912	+	0.306372i	$0.0991152\pi$
$660$		0			0
$661$		−	12144.7i		−	0.714635i	−0.933983	−	0.357318i	$-0.883691\pi$
$661$		−	12144.7i		−	0.714635i	0.933983	−	0.357318i	$-0.116309\pi$
$662$		−	11151.4i		−	0.654700i
$663$		0			0
$664$			4760.74i			0.278242i
$665$	3658.84	−	10425.5i	0.213359	−	0.607944i
$666$		0			0
$667$		−	11122.3i		−	0.645661i
$668$	2613.34			0.151367
$669$		0			0
$670$	−1389.83			−0.0801400
$671$		−	1996.04i		−	0.114838i
$672$		0			0
$673$		−	20816.8i		−	1.19231i	−0.802868	−	0.596157i	$-0.796694\pi$
$673$		−	20816.8i		−	1.19231i	0.802868	−	0.596157i	$-0.203306\pi$
$674$			15166.8i			0.866769i
$675$		0			0
$676$	7160.74			0.407416
$677$	31366.0			1.78064			0.890320	−	0.455336i	$-0.150481\pi$
$677$	31366.0			1.78064			0.890320	+	0.455336i	$0.150481\pi$
$678$		0			0
$679$			8287.33i			0.468393i
$680$	−2466.19			−0.139080
$681$		0			0
$682$	2277.06			0.127849
$683$	3882.25			0.217497			0.108748	−	0.994069i	$-0.465316\pi$
$683$	3882.25			0.217497			0.108748	+	0.994069i	$0.465316\pi$
$684$		0			0
$685$	−3723.28			−0.207678
$686$	−1383.61			−0.0770063
$687$		0			0
$688$	−3251.62			−0.180184
$689$		−	3338.32i		−	0.184586i
$690$		0			0
$691$	25246.9			1.38993			0.694963	−	0.719045i	$-0.255421\pi$
$691$	25246.9			1.38993			0.694963	+	0.719045i	$0.255421\pi$
$692$	−14249.0			−0.782754
$693$		0			0
$694$		−	2158.09i		−	0.118040i
$695$			13438.5i			0.733457i
$696$		0			0
$697$		−	12693.3i		−	0.689801i
$698$	11554.1			0.626547
$699$		0			0
$700$	−2668.20			−0.144069
$701$		−	11434.8i		−	0.616102i	−0.951370	−	0.308051i	$-0.900323\pi$
$701$		−	11434.8i		−	0.616102i	0.951370	−	0.308051i	$-0.0996767\pi$
$702$		0			0
$703$	5429.01	−	15469.4i	0.291265	−	0.829928i
$704$			257.982i			0.0138111i
$705$		0			0
$706$		−	1265.81i		−	0.0674780i
$707$		−	37713.9i		−	2.00619i
$708$		0			0
$709$	11523.8			0.610419			0.305209	−	0.952285i	$-0.401274\pi$
$709$	11523.8			0.610419			0.305209	+	0.952285i	$0.401274\pi$
$710$		−	3422.33i		−	0.180898i
$711$		0			0
$712$	−205.713			−0.0108278
$713$	−17881.2			−0.939211
$714$		0			0
$715$			406.516i			0.0212627i
$716$	−14721.0			−0.768365
$717$		0			0
$718$		−	8337.53i		−	0.433362i
$719$		−	27869.1i		−	1.44554i	−0.691089	−	0.722770i	$-0.742869\pi$
$719$		−	27869.1i		−	1.44554i	0.691089	−	0.722770i	$-0.257131\pi$
$720$		0			0
$721$		−	27730.0i		−	1.43234i
$722$	−10709.3	−	8572.83i	−0.552023	−	0.441894i
$723$		0			0
$724$			12072.1i			0.619688i
$725$	4392.10			0.224991
$726$		0			0
$727$	−9227.10			−0.470721			−0.235360	−	0.971908i	$-0.575627\pi$
$727$	−9227.10			−0.470721			−0.235360	+	0.971908i	$0.575627\pi$
$728$		−	4305.33i		−	0.219184i
$729$		0			0
$730$		−	4182.13i		−	0.212038i
$731$		−	12529.9i		−	0.633973i
$732$		0			0
$733$	1154.07			0.0581538			0.0290769	−	0.999577i	$-0.490743\pi$
$733$	1154.07			0.0581538			0.0290769	+	0.999577i	$0.490743\pi$
$734$	16722.1			0.840904
$735$		0			0
$736$		−	2025.87i		−	0.101460i
$737$	−560.235			−0.0280007
$738$		0			0
$739$	14139.6			0.703834			0.351917	−	0.936031i	$-0.385530\pi$
$739$	14139.6			0.703834			0.351917	+	0.936031i	$0.385530\pi$
$740$	−3959.09			−0.196674
$741$		0			0
$742$	8832.38			0.436990
$743$	13812.8			0.682020			0.341010	−	0.940060i	$-0.389231\pi$
$743$	13812.8			0.682020			0.341010	+	0.940060i	$0.389231\pi$
$744$		0			0
$745$	−1040.16			−0.0511525
$746$			7267.50i			0.356678i
$747$		0			0
$748$	−994.113			−0.0485941
$749$	419.396			0.0204598
$750$		0			0
$751$		−	13002.8i		−	0.631794i	−0.948793	−	0.315897i	$-0.897695\pi$
$751$		−	13002.8i		−	0.631794i	0.948793	−	0.315897i	$-0.102305\pi$
$752$			8069.18i			0.391294i
$753$		0			0
$754$			7086.98i			0.342298i
$755$	−3210.85			−0.154775
$756$		0			0
$757$	2210.89			0.106151			0.0530754	−	0.998591i	$-0.483098\pi$
$757$	2210.89			0.106151			0.0530754	+	0.998591i	$0.483098\pi$
$758$		−	25958.0i		−	1.24385i
$759$		0			0
$760$	−1097.02	+	3125.85i	−0.0523595	+	0.149193i
$761$			28674.6i			1.36590i	0.730463	+	0.682952i	$0.239304\pi$
$761$			28674.6i			1.36590i	−0.730463	+	0.682952i	$0.760696\pi$
$762$		0			0
$763$		−	48328.1i		−	2.29304i
$764$			4243.84i			0.200965i
$765$		0			0
$766$	−10166.8			−0.479558
$767$		−	11640.1i		−	0.547977i
$768$		0			0
$769$	−14992.4			−0.703045			−0.351522	−	0.936179i	$-0.614336\pi$
$769$	−14992.4			−0.703045			−0.351522	+	0.936179i	$0.614336\pi$
$770$	−1075.54			−0.0503374
$771$		0			0
$772$		−	1372.97i		−	0.0640081i
$773$	10268.3			0.477780			0.238890	−	0.971047i	$-0.423217\pi$
$773$	10268.3			0.477780			0.238890	+	0.971047i	$0.423217\pi$
$774$		0			0
$775$		−	7061.17i		−	0.327283i
$776$		−	2484.77i		−	0.114946i
$777$		0			0
$778$		−	10374.4i		−	0.478071i
$779$	−16088.4	−	5646.27i	−0.739959	−	0.259690i
$780$		0			0
$781$		−	1379.53i		−	0.0632054i
$782$	7806.54			0.356984
$783$		0			0
$784$	5902.84			0.268898
$785$		−	9448.00i		−	0.429571i
$786$		0			0
$787$			12071.0i			0.546741i	0.961909	+	0.273371i	$0.0881385\pi$
$787$			12071.0i			0.546741i	−0.961909	+	0.273371i	$0.911861\pi$
$788$		−	5653.86i		−	0.255597i
$789$		0			0
$790$	−8602.81			−0.387436
$791$	−6756.51			−0.303709
$792$		0			0
$793$		−	9987.56i		−	0.447249i
$794$	28240.5			1.26224
$795$		0			0
$796$	18495.9			0.823580
$797$	−1910.48			−0.0849091			−0.0424546	−	0.999098i	$-0.513518\pi$
$797$	−1910.48			−0.0849091			−0.0424546	+	0.999098i	$0.513518\pi$
$798$		0			0
$799$	−31094.0			−1.37675
$800$	800.000			0.0353553
$801$		0			0
$802$	−1913.09			−0.0842315
$803$		−	1685.80i		−	0.0740854i
$804$		0			0
$805$	8445.96			0.369790
$806$	11393.7			0.497923
$807$		0			0
$808$			11307.7i			0.492330i
$809$		−	1901.61i		−	0.0826415i	−0.999146	−	0.0413207i	$-0.986843\pi$
$809$		−	1901.61i		−	0.0826415i	0.999146	−	0.0413207i	$-0.0131565\pi$
$810$		0			0
$811$		−	11256.8i		−	0.487399i	−0.969851	−	0.243699i	$-0.921639\pi$
$811$		−	11256.8i		−	0.487399i	0.969851	−	0.243699i	$-0.0783611\pi$
$812$	−18750.4			−0.810357
$813$		0			0
$814$	−1595.89			−0.0687175
$815$			11399.6i			0.489953i
$816$		0			0
$817$	−15881.4	−	5573.60i	−0.680072	−	0.238673i
$818$		−	9280.92i		−	0.396699i
$819$		0			0
$820$			4117.52i			0.175354i
$821$			21475.0i			0.912888i	0.889752	+	0.456444i	$0.150877\pi$
$821$			21475.0i			0.912888i	−0.889752	+	0.456444i	$0.849123\pi$
$822$		0			0
$823$	−19604.0			−0.830320			−0.415160	−	0.909748i	$-0.636274\pi$
$823$	−19604.0			−0.830320			−0.415160	+	0.909748i	$0.636274\pi$
$824$			8314.24i			0.351505i
$825$		0			0
$826$	30796.7			1.29728
$827$	−11420.9			−0.480224			−0.240112	−	0.970745i	$-0.577184\pi$
$827$	−11420.9			−0.480224			−0.240112	+	0.970745i	$0.577184\pi$
$828$		0			0
$829$		−	18934.9i		−	0.793290i	−0.917972	−	0.396645i	$-0.870174\pi$
$829$		−	18934.9i		−	0.793290i	0.917972	−	0.396645i	$-0.129826\pi$
$830$	5950.93			0.248867
$831$		0			0
$832$			1290.86i			0.0537890i
$833$			22746.2i			0.946109i
$834$		0			0
$835$		−	3266.67i		−	0.135387i
$836$	−442.206	+	1260.02i	−0.0182943	+	0.0521276i
$837$		0			0
$838$		−	20230.5i		−	0.833953i
$839$	29742.8			1.22388			0.611940	−	0.790904i	$-0.290390\pi$
$839$	29742.8			1.22388			0.611940	+	0.790904i	$0.290390\pi$
$840$		0			0
$841$	6475.87			0.265524
$842$		−	30361.3i		−	1.24266i
$843$		0			0
$844$		−	6938.31i		−	0.282970i
$845$		−	8950.92i		−	0.364404i
$846$		0			0
$847$	35080.2			1.42310
$848$	−2648.19			−0.107240
$849$		0			0
$850$			3082.74i			0.124397i
$851$	12532.2			0.504815
$852$		0			0
$853$	32182.7			1.29181			0.645906	−	0.763417i	$-0.276480\pi$
$853$	32182.7			1.29181			0.645906	+	0.763417i	$0.276480\pi$
$854$	26424.6			1.05882
$855$		0			0
$856$	−125.747			−0.00502095
$857$	16396.0			0.653530			0.326765	−	0.945106i	$-0.394041\pi$
$857$	16396.0			0.653530			0.326765	+	0.945106i	$0.394041\pi$
$858$		0			0
$859$	31217.2			1.23995			0.619976	−	0.784621i	$-0.287142\pi$
$859$	31217.2			1.23995			0.619976	+	0.784621i	$0.287142\pi$
$860$			4064.53i			0.161162i
$861$		0			0
$862$	−28214.0			−1.11482
$863$	−42965.3			−1.69474			−0.847368	−	0.531006i	$-0.821814\pi$
$863$	−42965.3			−1.69474			−0.847368	+	0.531006i	$0.821814\pi$
$864$		0			0
$865$			17811.3i			0.700117i
$866$			14431.0i			0.566265i
$867$		0			0
$868$			30144.9i			1.17879i
$869$	−3467.76			−0.135369
$870$		0			0
$871$	−2803.24			−0.109052
$872$			14490.1i			0.562725i
$873$		0			0
$874$	3472.54	−	9894.63i	0.134394	−	0.382941i
$875$			3335.25i			0.128859i
$876$		0			0
$877$			36973.1i			1.42359i	0.702386	+	0.711797i	$0.252118\pi$
$877$			36973.1i			1.42359i	−0.702386	+	0.711797i	$0.747882\pi$
$878$		−	27229.9i		−	1.04665i
$879$		0			0
$880$	322.477			0.0123531
$881$		−	16080.5i		−	0.614943i	−0.951557	−	0.307472i	$-0.900517\pi$
$881$		−	16080.5i		−	0.614943i	0.951557	−	0.307472i	$-0.0994829\pi$
$882$		0			0
$883$	3465.26			0.132067			0.0660335	−	0.997817i	$-0.478966\pi$
$883$	3465.26			0.132067			0.0660335	+	0.997817i	$0.478966\pi$
$884$	−4974.23			−0.189255
$885$		0			0
$886$		−	24492.9i		−	0.928731i
$887$	9025.97			0.341671			0.170836	−	0.985300i	$-0.445353\pi$
$887$	9025.97			0.341671			0.170836	+	0.985300i	$0.445353\pi$
$888$		0			0
$889$			33135.5i			1.25009i
$890$			257.141i			0.00968471i
$891$		0			0
$892$			15425.0i			0.578999i
$893$	−13831.4	+	39411.0i	−0.518308	+	1.47686i
$894$		0			0
$895$			18401.3i			0.687247i
$896$	−3415.29			−0.127340
$897$		0			0
$898$	−22133.3			−0.822491
$899$		−	49621.4i		−	1.84090i
$900$		0			0
$901$		−	10204.6i		−	0.377320i
$902$			1659.76i			0.0612681i
$903$		0			0
$904$	2025.79			0.0745319
$905$	15090.1			0.554266
$906$		0			0
$907$			42483.2i			1.55527i	0.628714	+	0.777636i	$0.283582\pi$
$907$			42483.2i			1.55527i	−0.628714	+	0.777636i	$0.716418\pi$
$908$	8790.42			0.321278
$909$		0			0
$910$	−5381.66			−0.196044
$911$	25475.8			0.926509			0.463254	−	0.886225i	$-0.346682\pi$
$911$	25475.8			0.926509			0.463254	+	0.886225i	$0.346682\pi$
$912$		0			0
$913$	2398.80			0.0869536
$914$	−2193.09			−0.0793664
$915$		0			0
$916$	5506.40			0.198621
$917$			4648.34i			0.167395i
$918$		0			0
$919$	15286.2			0.548688			0.274344	−	0.961632i	$-0.411539\pi$
$919$	15286.2			0.548688			0.274344	+	0.961632i	$0.411539\pi$
$920$	−2532.34			−0.0907485
$921$		0			0
$922$			20379.0i			0.727923i
$923$		−	6902.73i		−	0.246160i
$924$		0			0
$925$			4948.86i			0.175911i
$926$	−22985.5			−0.815713
$927$		0			0
$928$	5621.89			0.198866
$929$			37261.0i			1.31592i	0.753051	+	0.657962i	$0.228581\pi$
$929$			37261.0i			1.31592i	−0.753051	+	0.657962i	$0.771419\pi$
$930$		0			0
$931$	28830.3	+	10118.1i	1.01490	+	0.356183i
$932$		−	10125.7i		−	0.355879i
$933$		0			0
$934$			28036.1i			0.982194i
$935$			1242.64i			0.0434639i
$936$		0			0
$937$	−32537.4			−1.13442			−0.567210	−	0.823573i	$-0.691977\pi$
$937$	−32537.4			−1.13442			−0.567210	+	0.823573i	$0.691977\pi$
$938$		−	7416.67i		−	0.258170i
$939$		0			0
$940$	10086.5			0.349984
$941$	−26041.2			−0.902145			−0.451072	−	0.892487i	$-0.648958\pi$
$941$	−26041.2			−0.902145			−0.451072	+	0.892487i	$0.648958\pi$
$942$		0			0
$943$		−	13033.7i		−	0.450090i
$944$	−9233.71			−0.318360
$945$		0			0
$946$			1638.39i			0.0563095i
$947$		−	24656.9i		−	0.846083i	−0.906110	−	0.423041i	$-0.860963\pi$
$947$		−	24656.9i		−	0.846083i	0.906110	−	0.423041i	$-0.139037\pi$
$948$		0			0
$949$		−	8435.21i		−	0.288534i
$950$	3907.31	+	1371.28i	0.133442	+	0.0468318i
$951$		0			0
$952$		−	13160.6i		−	0.448043i
$953$	31283.4			1.06335			0.531673	−	0.846949i	$-0.321563\pi$
$953$	31283.4			1.06335			0.531673	+	0.846949i	$0.321563\pi$
$954$		0			0
$955$	5304.81			0.179748
$956$			7744.77i			0.262012i
$957$		0			0
$958$		−	14790.4i		−	0.498805i
$959$		−	19868.9i		−	0.669030i
$960$		0			0
$961$	−49985.1			−1.67786
$962$	−7985.35			−0.267628
$963$		0			0
$964$		−	6892.64i		−	0.230287i
$965$	−1716.21			−0.0572506
$966$		0			0
$967$	1529.17			0.0508528			0.0254264	−	0.999677i	$-0.491906\pi$
$967$	1529.17			0.0508528			0.0254264	+	0.999677i	$0.491906\pi$
$968$	−10518.0			−0.349237
$969$		0			0
$970$	−3105.97			−0.102811
$971$	−10432.8			−0.344802			−0.172401	−	0.985027i	$-0.555153\pi$
$971$	−10432.8			−0.344802			−0.172401	+	0.985027i	$0.555153\pi$
$972$		0			0
$973$	−71713.3			−2.36282
$974$			14760.4i			0.485579i
$975$		0			0
$976$	−7922.84			−0.259840
$977$	40750.3			1.33441			0.667204	−	0.744875i	$-0.267491\pi$
$977$	40750.3			1.33441			0.667204	+	0.744875i	$0.267491\pi$
$978$		0			0
$979$			103.653i			0.00338381i
$980$		−	7378.55i		−	0.240509i
$981$		0			0
$982$		−	33958.2i		−	1.10351i
$983$	−40513.4			−1.31452			−0.657262	−	0.753662i	$-0.728286\pi$
$983$	−40513.4			−1.31452			−0.657262	+	0.753662i	$0.728286\pi$
$984$		0			0
$985$	−7067.32			−0.228613
$986$			21663.5i			0.699704i
$987$		0			0
$988$	−2212.66	+	6304.74i	−0.0712491	+	0.203017i
$989$		−	12865.9i		−	0.413663i
$990$		0			0
$991$		−	27563.7i		−	0.883541i	−0.897128	−	0.441771i	$-0.854350\pi$
$991$		−	27563.7i		−	0.883541i	0.897128	−	0.441771i	$-0.145650\pi$
$992$		−	9038.29i		−	0.289280i
$993$		0			0
$994$	18262.9			0.582761
$995$		−	23119.9i		−	0.736632i
$996$		0			0
$997$	15712.0			0.499100			0.249550	−	0.968362i	$-0.419717\pi$
$997$	15712.0			0.499100			0.249550	+	0.968362i	$0.419717\pi$
$998$	−28779.3			−0.912820
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1710.4.f.a.341.11	✓	40
3.2	odd	2		1710.4.f.b.341.12	yes	40
19.18	odd	2		1710.4.f.b.341.11	yes	40
57.56	even	2	inner	1710.4.f.a.341.12	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1710.4.f.a.341.11	✓	40	1.1	even	1	trivial
1710.4.f.a.341.12	yes	40	57.56	even	2	inner
1710.4.f.b.341.11	yes	40	19.18	odd	2
1710.4.f.b.341.12	yes	40	3.2	odd	2

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:	\( N \)	\(=\)	\( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1710.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -5.00000i q^{5} +26.6820 q^{7} -8.00000 q^{8} +10.0000i q^{10} +4.03096i q^{11} +20.1697i q^{13} -53.3639 q^{14} +16.0000 q^{16} +61.6549i q^{17} +(78.1463 + 27.4256i) q^{19} -20.0000i q^{20} -8.06192i q^{22} +63.3084i q^{23} -25.0000 q^{25} -40.3393i q^{26} +106.728 q^{28} -175.684 q^{29} +282.447i q^{31} -32.0000 q^{32} -123.310i q^{34} -133.410i q^{35} -197.954i q^{37} +(-156.293 - 54.8512i) q^{38} +40.0000i q^{40} -205.876 q^{41} -203.226 q^{43} +16.1238i q^{44} -126.617i q^{46} +504.324i q^{47} +368.928 q^{49} +50.0000 q^{50} +80.6787i q^{52} -165.512 q^{53} +20.1548 q^{55} -213.456 q^{56} +351.368 q^{58} -577.107 q^{59} -495.177 q^{61} -564.893i q^{62} +64.0000 q^{64} +100.848 q^{65} +138.983i q^{67} +246.619i q^{68} +266.820i q^{70} -342.233 q^{71} -418.213 q^{73} +395.909i q^{74} +(312.585 + 109.702i) q^{76} +107.554i q^{77} +860.281i q^{79} -80.0000i q^{80} +411.752 q^{82} -595.093i q^{83} +308.274 q^{85} +406.453 q^{86} -32.2477i q^{88} +25.7141 q^{89} +538.166i q^{91} +253.234i q^{92} -1008.65i q^{94} +(137.128 - 390.731i) q^{95} +310.597i q^{97} -737.855 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 80 q^{2} + 160 q^{4} - 56 q^{7} - 320 q^{8} + 112 q^{14} + 640 q^{16} - 76 q^{19} - 1000 q^{25} - 224 q^{28} - 120 q^{29} - 1280 q^{32} + 152 q^{38} - 312 q^{41} + 56 q^{43} + 2112 q^{49} + 2000 q^{50}+ \cdots - 4224 q^{98}+O(q^{100}) \) 40 * q - 80 * q^2 + 160 * q^4 - 56 * q^7 - 320 * q^8 + 112 * q^14 + 640 * q^16 - 76 * q^19 - 1000 * q^25 - 224 * q^28 - 120 * q^29 - 1280 * q^32 + 152 * q^38 - 312 * q^41 + 56 * q^43 + 2112 * q^49 + 2000 * q^50 + 288 * q^53 + 448 * q^56 + 240 * q^58 + 912 * q^59 - 848 * q^61 + 2560 * q^64 + 240 * q^65 - 1248 * q^71 + 512 * q^73 - 304 * q^76 + 624 * q^82 - 112 * q^86 - 552 * q^89 - 60 * q^95 - 4224 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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