LMFDB - Embedded newform 1710.4.f.b.341.1

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)

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")

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);

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			341.1
Character	$\chi$	$=$	1710.341
Dual form			1710.4.f.b.341.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} +4.00000 q^{4} -5.00000i q^{5} -27.0790 q^{7} +8.00000 q^{8} -10.0000i q^{10} +56.8780i q^{11} -83.3882i q^{13} -54.1581 q^{14} +16.0000 q^{16} -16.3398i q^{17} +(-80.2268 - 20.5587i) q^{19} -20.0000i q^{20} +113.756i q^{22} +200.235i q^{23} -25.0000 q^{25} -166.776i q^{26} -108.316 q^{28} +200.451 q^{29} -102.512i q^{31} +32.0000 q^{32} -32.6795i q^{34} +135.395i q^{35} -54.1715i q^{37} +(-160.454 - 41.1173i) q^{38} -40.0000i q^{40} -31.9092 q^{41} +254.378 q^{43} +227.512i q^{44} +400.470i q^{46} +110.831i q^{47} +390.274 q^{49} -50.0000 q^{50} -333.553i q^{52} -191.250 q^{53} +284.390 q^{55} -216.632 q^{56} +400.902 q^{58} +42.7800 q^{59} +84.0413 q^{61} -205.024i q^{62} +64.0000 q^{64} -416.941 q^{65} +923.661i q^{67} -65.3591i q^{68} +270.790i q^{70} +985.746 q^{71} +928.660 q^{73} -108.343i q^{74} +(-320.907 - 82.2346i) q^{76} -1540.20i q^{77} -1000.19i q^{79} -80.0000i q^{80} -63.8184 q^{82} +652.200i q^{83} -81.6988 q^{85} +508.756 q^{86} +455.024i q^{88} +56.9916 q^{89} +2258.07i q^{91} +800.940i q^{92} +221.662i q^{94} +(-102.793 + 401.134i) q^{95} +803.288i q^{97} +780.548 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−27.0790			−1.46213			−0.731065	−	0.682308i	$-0.760976\pi$
$7$	−27.0790			−1.46213			−0.731065	+	0.682308i	$0.760976\pi$
$8$	8.00000			0.353553
$9$		0			0
$10$		−	10.0000i		−	0.316228i
$11$			56.8780i			1.55903i	0.626382	+	0.779516i	$0.284535\pi$
$11$			56.8780i			1.55903i	−0.626382	+	0.779516i	$0.715465\pi$
$12$		0			0
$13$		−	83.3882i		−	1.77906i	−0.456881	−	0.889528i	$-0.651033\pi$
$13$		−	83.3882i		−	1.77906i	0.456881	−	0.889528i	$-0.348967\pi$
$14$	−54.1581			−1.03388
$15$		0			0
$16$	16.0000			0.250000
$17$		−	16.3398i		−	0.233116i	−0.993184	−	0.116558i	$-0.962814\pi$
$17$		−	16.3398i		−	0.233116i	0.993184	−	0.116558i	$-0.0371861\pi$
$18$		0			0
$19$	−80.2268	−	20.5587i	−0.968700	−	0.248236i
$19$	−80.2268	−	20.5587i	−0.968700	−	0.248236i
$20$		−	20.0000i		−	0.223607i
$21$		0			0
$22$			113.756i			1.10240i
$23$			200.235i			1.81530i	0.419728	+	0.907650i	$0.362125\pi$
$23$			200.235i			1.81530i	−0.419728	+	0.907650i	$0.637875\pi$
$24$		0			0
$25$	−25.0000			−0.200000
$26$		−	166.776i		−	1.25798i
$27$		0			0
$28$	−108.316			−0.731065
$29$	200.451			1.28355			0.641773	−	0.766895i	$-0.278199\pi$
$29$	200.451			1.28355			0.641773	+	0.766895i	$0.278199\pi$
$30$		0			0
$31$		−	102.512i		−	0.593925i	−0.954889	−	0.296963i	$-0.904026\pi$
$31$		−	102.512i		−	0.593925i	0.954889	−	0.296963i	$-0.0959737\pi$
$32$	32.0000			0.176777
$33$		0			0
$34$		−	32.6795i		−	0.164838i
$35$			135.395i			0.653884i
$36$		0			0
$37$		−	54.1715i		−	0.240696i	−0.992732	−	0.120348i	$-0.961599\pi$
$37$		−	54.1715i		−	0.240696i	0.992732	−	0.120348i	$-0.0384010\pi$
$38$	−160.454	−	41.1173i	−0.684974	−	0.175529i
$39$		0			0
$40$		−	40.0000i		−	0.158114i
$41$	−31.9092			−0.121546			−0.0607729	−	0.998152i	$-0.519357\pi$
$41$	−31.9092			−0.121546			−0.0607729	+	0.998152i	$0.519357\pi$
$42$		0			0
$43$	254.378			0.902147			0.451073	−	0.892487i	$-0.351041\pi$
$43$	254.378			0.902147			0.451073	+	0.892487i	$0.351041\pi$
$44$			227.512i			0.779516i
$45$		0			0
$46$			400.470i			1.28361i
$47$			110.831i			0.343966i	0.985100	+	0.171983i	$0.0550174\pi$
$47$			110.831i			0.343966i	−0.985100	+	0.171983i	$0.944983\pi$
$48$		0			0
$49$	390.274			1.13782
$50$	−50.0000			−0.141421
$51$		0			0
$52$		−	333.553i		−	0.889528i
$53$	−191.250			−0.495664			−0.247832	−	0.968803i	$-0.579718\pi$
$53$	−191.250			−0.495664			−0.247832	+	0.968803i	$0.579718\pi$
$54$		0			0
$55$	284.390			0.697220
$56$	−216.632			−0.516941
$57$		0			0
$58$	400.902			0.907604
$59$	42.7800			0.0943981			0.0471990	−	0.998886i	$-0.484970\pi$
$59$	42.7800			0.0943981			0.0471990	+	0.998886i	$0.484970\pi$
$60$		0			0
$61$	84.0413			0.176400			0.0881999	−	0.996103i	$-0.471889\pi$
$61$	84.0413			0.176400			0.0881999	+	0.996103i	$0.471889\pi$
$62$		−	205.024i		−	0.419969i
$63$		0			0
$64$	64.0000			0.125000
$65$	−416.941			−0.795618
$66$		0			0
$67$			923.661i			1.68423i	0.539302	+	0.842113i	$0.318688\pi$
$67$			923.661i			1.68423i	−0.539302	+	0.842113i	$0.681312\pi$
$68$		−	65.3591i		−	0.116558i
$69$		0			0
$70$			270.790i			0.462366i
$71$	985.746			1.64770			0.823849	−	0.566810i	$-0.191823\pi$
$71$	985.746			1.64770			0.823849	+	0.566810i	$0.191823\pi$
$72$		0			0
$73$	928.660			1.48892			0.744462	−	0.667665i	$-0.232706\pi$
$73$	928.660			1.48892			0.744462	+	0.667665i	$0.232706\pi$
$74$		−	108.343i		−	0.170198i
$75$		0			0
$76$	−320.907	−	82.2346i	−0.484350	−	0.124118i
$77$		−	1540.20i		−	2.27951i
$78$		0			0
$79$		−	1000.19i		−	1.42443i	−0.701959	−	0.712217i	$-0.747691\pi$
$79$		−	1000.19i		−	1.42443i	0.701959	−	0.712217i	$-0.252309\pi$
$80$		−	80.0000i		−	0.111803i
$81$		0			0
$82$	−63.8184			−0.0859459
$83$			652.200i			0.862509i	0.902230	+	0.431254i	$0.141929\pi$
$83$			652.200i			0.862509i	−0.902230	+	0.431254i	$0.858071\pi$
$84$		0			0
$85$	−81.6988			−0.104253
$86$	508.756			0.637914
$87$		0			0
$88$			455.024i			0.551201i
$89$	56.9916			0.0678775			0.0339388	−	0.999424i	$-0.489195\pi$
$89$	56.9916			0.0678775			0.0339388	+	0.999424i	$0.489195\pi$
$90$		0			0
$91$			2258.07i			2.60121i
$92$			800.940i			0.907650i
$93$		0			0
$94$			221.662i			0.243221i
$95$	−102.793	+	401.134i	−0.111014	+	0.433216i
$96$		0			0
$97$			803.288i			0.840840i	0.907330	+	0.420420i	$0.138117\pi$
$97$			803.288i			0.840840i	−0.907330	+	0.420420i	$0.861883\pi$
$98$	780.548			0.804564
$99$		0			0
$100$	−100.000			−0.100000
$101$			430.479i			0.424101i	0.977259	+	0.212051i	$0.0680142\pi$
$101$			430.479i			0.424101i	−0.977259	+	0.212051i	$0.931986\pi$
$102$		0			0
$103$		−	186.426i		−	0.178341i	−0.996016	−	0.0891705i	$-0.971578\pi$
$103$		−	186.426i		−	0.178341i	0.996016	−	0.0891705i	$-0.0284216\pi$
$104$		−	667.106i		−	0.628991i
$105$		0			0
$106$	−382.500			−0.350488
$107$	894.683			0.808339			0.404170	−	0.914684i	$-0.367561\pi$
$107$	894.683			0.808339			0.404170	+	0.914684i	$0.367561\pi$
$108$		0			0
$109$		−	1022.64i		−	0.898635i	−0.893372	−	0.449317i	$-0.851667\pi$
$109$		−	1022.64i		−	0.898635i	0.893372	−	0.449317i	$-0.148333\pi$
$110$	568.780			0.493009
$111$		0			0
$112$	−433.264			−0.365533
$113$	1924.24			1.60192			0.800960	−	0.598719i	$-0.204323\pi$
$113$	1924.24			1.60192			0.800960	+	0.598719i	$0.204323\pi$
$114$		0			0
$115$	1001.18			0.811827
$116$	801.805			0.641773
$117$		0			0
$118$	85.5601			0.0667495
$119$			442.465i			0.340846i
$120$		0			0
$121$	−1904.10			−1.43058
$122$	168.083			0.124733
$123$		0			0
$124$		−	410.048i		−	0.296963i
$125$			125.000i			0.0894427i
$126$		0			0
$127$			2611.90i			1.82495i	0.409134	+	0.912474i	$0.365831\pi$
$127$			2611.90i			1.82495i	−0.409134	+	0.912474i	$0.634169\pi$
$128$	128.000			0.0883883
$129$		0			0
$130$	−833.882			−0.562587
$131$			2605.88i			1.73799i	0.494817	+	0.868997i	$0.335235\pi$
$131$			2605.88i			1.73799i	−0.494817	+	0.868997i	$0.664765\pi$
$132$		0			0
$133$	2172.46	+	556.709i	1.41636	+	0.362953i
$134$			1847.32i			1.19093i
$135$		0			0
$136$		−	130.718i		−	0.0824190i
$137$		−	803.705i		−	0.501205i	−0.968090	−	0.250603i	$-0.919371\pi$
$137$		−	803.705i		−	0.501205i	0.968090	−	0.250603i	$-0.0806288\pi$
$138$		0			0
$139$	121.716			0.0742721			0.0371361	−	0.999310i	$-0.488177\pi$
$139$	121.716			0.0742721			0.0371361	+	0.999310i	$0.488177\pi$
$140$			541.581i			0.326942i
$141$		0			0
$142$	1971.49			1.16510
$143$	4742.95			2.77361
$144$		0			0
$145$		−	1002.26i		−	0.574019i
$146$	1857.32			1.05283
$147$		0			0
$148$		−	216.686i		−	0.120348i
$149$		−	2055.65i		−	1.13024i	−0.825009	−	0.565119i	$-0.808830\pi$
$149$		−	2055.65i		−	1.13024i	0.825009	−	0.565119i	$-0.191170\pi$
$150$		0			0
$151$		−	2421.41i		−	1.30498i	−0.757798	−	0.652490i	$-0.773725\pi$
$151$		−	2421.41i		−	1.30498i	0.757798	−	0.652490i	$-0.226275\pi$
$152$	−641.815	−	164.469i	−0.342487	−	0.0877646i
$153$		0			0
$154$		−	3080.40i		−	1.61186i
$155$	−512.560			−0.265611
$156$		0			0
$157$	3511.18			1.78486			0.892428	−	0.451190i	$-0.149000\pi$
$157$	3511.18			1.78486			0.892428	+	0.451190i	$0.149000\pi$
$158$		−	2000.38i		−	1.00723i
$159$		0			0
$160$		−	160.000i		−	0.0790569i
$161$		−	5422.17i		−	2.65420i
$162$		0			0
$163$	1447.68			0.695651			0.347825	−	0.937559i	$-0.386920\pi$
$163$	1447.68			0.695651			0.347825	+	0.937559i	$0.386920\pi$
$164$	−127.637			−0.0607729
$165$		0			0
$166$			1304.40i			0.609886i
$167$	−4273.75			−1.98032			−0.990159	−	0.139950i	$-0.955306\pi$
$167$	−4273.75			−1.98032			−0.990159	+	0.139950i	$0.955306\pi$
$168$		0			0
$169$	−4756.59			−2.16504
$170$	−163.398			−0.0737178
$171$		0			0
$172$	1017.51			0.451073
$173$	3913.88			1.72004			0.860019	−	0.510261i	$-0.170451\pi$
$173$	3913.88			1.72004			0.860019	+	0.510261i	$0.170451\pi$
$174$		0			0
$175$	676.976			0.292426
$176$			910.047i			0.389758i
$177$		0			0
$178$	113.983			0.0479966
$179$	1410.39			0.588924			0.294462	−	0.955663i	$-0.404860\pi$
$179$	1410.39			0.588924			0.294462	+	0.955663i	$0.404860\pi$
$180$		0			0
$181$		−	4189.31i		−	1.72038i	−0.509973	−	0.860191i	$-0.670345\pi$
$181$		−	4189.31i		−	1.72038i	0.509973	−	0.860191i	$-0.329655\pi$
$182$			4516.14i			1.83933i
$183$		0			0
$184$			1601.88i			0.641805i
$185$	−270.858			−0.107642
$186$		0			0
$187$	929.373			0.363436
$188$			443.325i			0.171983i
$189$		0			0
$190$	−205.587	+	802.268i	−0.0784991	+	0.306330i
$191$			3846.69i			1.45726i	0.684907	+	0.728630i	$0.259843\pi$
$191$			3846.69i			1.45726i	−0.684907	+	0.728630i	$0.740157\pi$
$192$		0			0
$193$			3875.90i			1.44556i	0.691078	+	0.722780i	$0.257136\pi$
$193$			3875.90i			1.44556i	−0.691078	+	0.722780i	$0.742864\pi$
$194$			1606.58i			0.594564i
$195$		0			0
$196$	1561.10			0.568912
$197$		−	219.930i		−	0.0795401i	−0.999209	−	0.0397700i	$-0.987337\pi$
$197$		−	219.930i		−	0.0795401i	0.999209	−	0.0397700i	$-0.0126625\pi$
$198$		0			0
$199$	607.112			0.216267			0.108133	−	0.994136i	$-0.465513\pi$
$199$	607.112			0.216267			0.108133	+	0.994136i	$0.465513\pi$
$200$	−200.000			−0.0707107
$201$		0			0
$202$			860.958i			0.299885i
$203$	−5428.02			−1.87671
$204$		0			0
$205$			159.546i			0.0543569i
$206$		−	372.853i		−	0.126106i
$207$		0			0
$208$		−	1334.21i		−	0.444764i
$209$	1169.33	−	4563.14i	0.387008	−	1.51023i
$210$		0			0
$211$			657.718i			0.214593i	0.994227	+	0.107297i	$0.0342194\pi$
$211$			657.718i			0.214593i	−0.994227	+	0.107297i	$0.965781\pi$
$212$	−765.000			−0.247832
$213$		0			0
$214$	1789.37			0.571582
$215$		−	1271.89i		−	0.403452i
$216$		0			0
$217$			2775.92i			0.868396i
$218$		−	2045.28i		−	0.635431i
$219$		0			0
$220$	1137.56			0.348610
$221$	−1362.54			−0.414727
$222$		0			0
$223$		−	692.879i		−	0.208066i	−0.994574	−	0.104033i	$-0.966825\pi$
$223$		−	692.879i		−	0.208066i	0.994574	−	0.104033i	$-0.0331747\pi$
$224$	−866.529			−0.258471
$225$		0			0
$226$	3848.47			1.13273
$227$	920.861			0.269250			0.134625	−	0.990897i	$-0.457017\pi$
$227$	920.861			0.269250			0.134625	+	0.990897i	$0.457017\pi$
$228$		0			0
$229$	843.230			0.243328			0.121664	−	0.992571i	$-0.461177\pi$
$229$	843.230			0.243328			0.121664	+	0.992571i	$0.461177\pi$
$230$	2002.35			0.574048
$231$		0			0
$232$	1603.61			0.453802
$233$			4162.66i			1.17041i	0.810887	+	0.585203i	$0.198985\pi$
$233$			4162.66i			1.17041i	−0.810887	+	0.585203i	$0.801015\pi$
$234$		0			0
$235$	554.156			0.153826
$236$	171.120			0.0471990
$237$		0			0
$238$			884.930i			0.241015i
$239$			4444.70i			1.20294i	0.798894	+	0.601472i	$0.205419\pi$
$239$			4444.70i			1.20294i	−0.798894	+	0.601472i	$0.794581\pi$
$240$		0			0
$241$		−	169.996i		−	0.0454373i	−0.999742	−	0.0227187i	$-0.992768\pi$
$241$		−	169.996i		−	0.0454373i	0.999742	−	0.0227187i	$-0.00723219\pi$
$242$	−3808.20			−1.01157
$243$		0			0
$244$	336.165			0.0881999
$245$		−	1951.37i		−	0.508851i
$246$		0			0
$247$	−1714.35	+	6689.97i	−0.441625	+	1.72337i
$248$		−	820.095i		−	0.209984i
$249$		0			0
$250$			250.000i			0.0632456i
$251$		−	5866.46i		−	1.47525i	−0.675211	−	0.737624i	$-0.735948\pi$
$251$		−	5866.46i		−	1.47525i	0.675211	−	0.737624i	$-0.264052\pi$
$252$		0			0
$253$	−11389.0			−2.83011
$254$			5223.79i			1.29043i
$255$		0			0
$256$	256.000			0.0625000
$257$	3825.17			0.928433			0.464216	−	0.885722i	$-0.346336\pi$
$257$	3825.17			0.928433			0.464216	+	0.885722i	$0.346336\pi$
$258$		0			0
$259$			1466.91i			0.351928i
$260$	−1667.76			−0.397809
$261$		0			0
$262$			5211.77i			1.22895i
$263$		−	2873.38i		−	0.673689i	−0.941560	−	0.336844i	$-0.890640\pi$
$263$		−	2873.38i		−	0.673689i	0.941560	−	0.336844i	$-0.109360\pi$
$264$		0			0
$265$			956.250i			0.221668i
$266$	4344.93	+	1113.42i	1.00152	+	0.256647i
$267$		0			0
$268$			3694.64i			0.842113i
$269$	1348.54			0.305657			0.152828	−	0.988253i	$-0.451162\pi$
$269$	1348.54			0.305657			0.152828	+	0.988253i	$0.451162\pi$
$270$		0			0
$271$	−4961.24			−1.11208			−0.556040	−	0.831156i	$-0.687680\pi$
$271$	−4961.24			−1.11208			−0.556040	+	0.831156i	$0.687680\pi$
$272$		−	261.436i		−	0.0582791i
$273$		0			0
$274$		−	1607.41i		−	0.354406i
$275$		−	1421.95i		−	0.311806i
$276$		0			0
$277$	−8357.14			−1.81275			−0.906376	−	0.422473i	$-0.861162\pi$
$277$	−8357.14			−1.81275			−0.906376	+	0.422473i	$0.861162\pi$
$278$	243.432			0.0525183
$279$		0			0
$280$			1083.16i			0.231183i
$281$	3470.02			0.736669			0.368334	−	0.929693i	$-0.379928\pi$
$281$	3470.02			0.736669			0.368334	+	0.929693i	$0.379928\pi$
$282$		0			0
$283$	−5646.98			−1.18614			−0.593071	−	0.805150i	$-0.702085\pi$
$283$	−5646.98			−1.18614			−0.593071	+	0.805150i	$0.702085\pi$
$284$	3942.98			0.823849
$285$		0			0
$286$	9485.90			1.96123
$287$	864.070			0.177716
$288$		0			0
$289$	4646.01			0.945657
$290$		−	2004.51i		−	0.405893i
$291$		0			0
$292$	3714.64			0.744462
$293$	−5662.14			−1.12896			−0.564481	−	0.825446i	$-0.690924\pi$
$293$	−5662.14			−1.12896			−0.564481	+	0.825446i	$0.690924\pi$
$294$		0			0
$295$		−	213.900i		−	0.0422161i
$296$		−	433.372i		−	0.0850988i
$297$		0			0
$298$		−	4111.30i		−	0.799199i
$299$	16697.2			3.22952
$300$		0			0
$301$	−6888.31			−1.31906
$302$		−	4842.83i		−	0.922760i
$303$		0			0
$304$	−1283.63	−	328.939i	−0.242175	−	0.0620590i
$305$		−	420.206i		−	0.0788884i
$306$		0			0
$307$		−	3219.65i		−	0.598550i	−0.954167	−	0.299275i	$-0.903255\pi$
$307$		−	3219.65i		−	0.598550i	0.954167	−	0.299275i	$-0.0967449\pi$
$308$		−	6160.80i		−	1.13975i
$309$		0			0
$310$	−1025.12			−0.187816
$311$		−	3117.56i		−	0.568426i	−0.958761	−	0.284213i	$-0.908268\pi$
$311$		−	3117.56i		−	0.568426i	0.958761	−	0.284213i	$-0.0917323\pi$
$312$		0			0
$313$	4974.77			0.898373			0.449187	−	0.893438i	$-0.351714\pi$
$313$	4974.77			0.898373			0.449187	+	0.893438i	$0.351714\pi$
$314$	7022.35			1.26208
$315$		0			0
$316$		−	4000.77i		−	0.712217i
$317$	2624.18			0.464948			0.232474	−	0.972603i	$-0.425318\pi$
$317$	2624.18			0.464948			0.232474	+	0.972603i	$0.425318\pi$
$318$		0			0
$319$			11401.3i			2.00109i
$320$		−	320.000i		−	0.0559017i
$321$		0			0
$322$		−	10844.3i		−	1.87681i
$323$	−335.924	+	1310.89i	−0.0578678	+	0.225820i
$324$		0			0
$325$			2084.71i			0.355811i
$326$	2895.36			0.491900
$327$		0			0
$328$	−255.273			−0.0429729
$329$		−	3001.20i		−	0.502923i
$330$		0			0
$331$			4553.13i			0.756081i	0.925789	+	0.378041i	$0.123402\pi$
$331$			4553.13i			0.756081i	−0.925789	+	0.378041i	$0.876598\pi$
$332$			2608.80i			0.431254i
$333$		0			0
$334$	−8547.51			−1.40030
$335$	4618.30			0.753208
$336$		0			0
$337$			9507.76i			1.53686i	0.639936	+	0.768428i	$0.278961\pi$
$337$			9507.76i			1.53686i	−0.639936	+	0.768428i	$0.721039\pi$
$338$	−9513.19			−1.53091
$339$		0			0
$340$	−326.795			−0.0521264
$341$	5830.67			0.925948
$342$		0			0
$343$	−1280.13			−0.201518
$344$	2035.03			0.318957
$345$		0			0
$346$	7827.76			1.21625
$347$			11673.7i			1.80599i	0.429649	+	0.902996i	$0.358637\pi$
$347$			11673.7i			1.80599i	−0.429649	+	0.902996i	$0.641363\pi$
$348$		0			0
$349$	−2788.79			−0.427738			−0.213869	−	0.976862i	$-0.568607\pi$
$349$	−2788.79			−0.427738			−0.213869	+	0.976862i	$0.568607\pi$
$350$	1353.95			0.206776
$351$		0			0
$352$			1820.09i			0.275600i
$353$			4315.08i			0.650619i	0.945608	+	0.325310i	$0.105468\pi$
$353$			4315.08i			0.650619i	−0.945608	+	0.325310i	$0.894532\pi$
$354$		0			0
$355$		−	4928.73i		−	0.736873i
$356$	227.966			0.0339388
$357$		0			0
$358$	2820.78			0.416432
$359$		−	55.5904i		−	0.00817257i	−0.999992	−	0.00408628i	$-0.998699\pi$
$359$		−	55.5904i		−	0.00817257i	0.999992	−	0.00408628i	$-0.00130071\pi$
$360$		0			0
$361$	6013.68	+	3298.71i	0.876758	+	0.480932i
$362$		−	8378.62i		−	1.21649i
$363$		0			0
$364$			9032.29i			1.30061i
$365$		−	4643.30i		−	0.665867i
$366$		0			0
$367$	11390.7			1.62013			0.810065	−	0.586340i	$-0.199432\pi$
$367$	11390.7			1.62013			0.810065	+	0.586340i	$0.199432\pi$
$368$			3203.76i			0.453825i
$369$		0			0
$370$	−541.715			−0.0761147
$371$	5178.86			0.724726
$372$		0			0
$373$			9200.18i			1.27712i	0.769570	+	0.638562i	$0.220471\pi$
$373$			9200.18i			1.27712i	−0.769570	+	0.638562i	$0.779529\pi$
$374$	1858.75			0.256988
$375$		0			0
$376$			886.650i			0.121610i
$377$		−	16715.3i		−	2.28350i
$378$		0			0
$379$		−	4476.53i		−	0.606713i	−0.952877	−	0.303356i	$-0.901893\pi$
$379$		−	4476.53i		−	0.606713i	0.952877	−	0.303356i	$-0.0981072\pi$
$380$	−411.173	+	1604.54i	−0.0555072	+	0.216608i
$381$		0			0
$382$			7693.38i			1.03044i
$383$	3244.22			0.432825			0.216413	−	0.976302i	$-0.430564\pi$
$383$	3244.22			0.432825			0.216413	+	0.976302i	$0.430564\pi$
$384$		0			0
$385$	−7701.00			−1.01943
$386$			7751.79i			1.02217i
$387$		0			0
$388$			3213.15i			0.420420i
$389$		−	5641.19i		−	0.735270i	−0.929970	−	0.367635i	$-0.880168\pi$
$389$		−	5641.19i		−	0.735270i	0.929970	−	0.367635i	$-0.119832\pi$
$390$		0			0
$391$	3271.79			0.423176
$392$	3122.19			0.402282
$393$		0			0
$394$		−	439.861i		−	0.0562433i
$395$	−5000.96			−0.637027
$396$		0			0
$397$	3463.04			0.437796			0.218898	−	0.975748i	$-0.429754\pi$
$397$	3463.04			0.437796			0.218898	+	0.975748i	$0.429754\pi$
$398$	1214.22			0.152924
$399$		0			0
$400$	−400.000			−0.0500000
$401$	−8233.62			−1.02536			−0.512678	−	0.858581i	$-0.671346\pi$
$401$	−8233.62			−1.02536			−0.512678	+	0.858581i	$0.671346\pi$
$402$		0			0
$403$	−8548.29			−1.05663
$404$			1721.92i			0.212051i
$405$		0			0
$406$	−10856.0			−1.32704
$407$	3081.16			0.375252
$408$		0			0
$409$			3217.85i			0.389028i	0.980900	+	0.194514i	$0.0623130\pi$
$409$			3217.85i			0.389028i	−0.980900	+	0.194514i	$0.937687\pi$
$410$			319.092i			0.0384362i
$411$		0			0
$412$		−	745.705i		−	0.0891705i
$413$	−1158.44			−0.138022
$414$		0			0
$415$	3261.00			0.385726
$416$		−	2668.42i		−	0.314496i
$417$		0			0
$418$	2338.67	−	9126.27i	0.273656	−	1.06790i
$419$		−	9258.09i		−	1.07944i	−0.841843	−	0.539722i	$-0.818529\pi$
$419$		−	9258.09i		−	1.07944i	0.841843	−	0.539722i	$-0.181471\pi$
$420$		0			0
$421$			11343.0i			1.31312i	0.754273	+	0.656561i	$0.227990\pi$
$421$			11343.0i			1.31312i	−0.754273	+	0.656561i	$0.772010\pi$
$422$			1315.44i			0.151740i
$423$		0			0
$424$	−1530.00			−0.175244
$425$			408.494i			0.0466232i
$426$		0			0
$427$	−2275.76			−0.257919
$428$	3578.73			0.404170
$429$		0			0
$430$		−	2543.78i		−	0.285284i
$431$	−8713.64			−0.973831			−0.486916	−	0.873449i	$-0.661878\pi$
$431$	−8713.64			−0.973831			−0.486916	+	0.873449i	$0.661878\pi$
$432$		0			0
$433$			1889.61i			0.209720i	0.994487	+	0.104860i	$0.0334394\pi$
$433$			1889.61i			0.209720i	−0.994487	+	0.104860i	$0.966561\pi$
$434$			5551.85i			0.614049i
$435$		0			0
$436$		−	4090.56i		−	0.449317i
$437$	4116.56	−	16064.2i	0.450622	−	1.75848i
$438$		0			0
$439$			10212.2i			1.11025i	0.831766	+	0.555126i	$0.187330\pi$
$439$			10212.2i			1.11025i	−0.831766	+	0.555126i	$0.812670\pi$
$440$	2275.12			0.246505
$441$		0			0
$442$	−2725.09			−0.293256
$443$		−	5486.33i		−	0.588405i	−0.955743	−	0.294202i	$-0.904946\pi$
$443$		−	5486.33i		−	0.588405i	0.955743	−	0.294202i	$-0.0950540\pi$
$444$		0			0
$445$		−	284.958i		−	0.0303557i
$446$		−	1385.76i		−	0.147125i
$447$		0			0
$448$	−1733.06			−0.182766
$449$	−7105.19			−0.746803			−0.373402	−	0.927670i	$-0.621809\pi$
$449$	−7105.19			−0.746803			−0.373402	+	0.927670i	$0.621809\pi$
$450$		0			0
$451$		−	1814.93i		−	0.189494i
$452$	7696.94			0.800960
$453$		0			0
$454$	1841.72			0.190388
$455$	11290.4			1.16330
$456$		0			0
$457$	−3358.09			−0.343731			−0.171865	−	0.985120i	$-0.554979\pi$
$457$	−3358.09			−0.343731			−0.171865	+	0.985120i	$0.554979\pi$
$458$	1686.46			0.172059
$459$		0			0
$460$	4004.70			0.405913
$461$			8590.73i			0.867918i	0.900933	+	0.433959i	$0.142884\pi$
$461$			8590.73i			0.867918i	−0.900933	+	0.433959i	$0.857116\pi$
$462$		0			0
$463$	1904.75			0.191191			0.0955953	−	0.995420i	$-0.469525\pi$
$463$	1904.75			0.191191			0.0955953	+	0.995420i	$0.469525\pi$
$464$	3207.22			0.320887
$465$		0			0
$466$			8325.31i			0.827602i
$467$		−	14715.2i		−	1.45812i	−0.684452	−	0.729058i	$-0.739958\pi$
$467$		−	14715.2i		−	1.45812i	0.684452	−	0.729058i	$-0.260042\pi$
$468$		0			0
$469$		−	25011.8i		−	2.46256i
$470$	1108.31			0.108772
$471$		0			0
$472$	342.240			0.0333748
$473$			14468.5i			1.40648i
$474$		0			0
$475$	2005.67	+	513.967i	0.193740	+	0.0496472i
$476$			1769.86i			0.170423i
$477$		0			0
$478$			8889.40i			0.850610i
$479$		−	10323.7i		−	0.984765i	−0.870379	−	0.492382i	$-0.836126\pi$
$479$		−	10323.7i		−	0.984765i	0.870379	−	0.492382i	$-0.163874\pi$
$480$		0			0
$481$	−4517.27			−0.428211
$482$		−	339.992i		−	0.0321290i
$483$		0			0
$484$	−7616.41			−0.715290
$485$	4016.44			0.376035
$486$		0			0
$487$		−	19238.4i		−	1.79010i	−0.445969	−	0.895048i	$-0.647141\pi$
$487$		−	19238.4i		−	1.79010i	0.445969	−	0.895048i	$-0.352859\pi$
$488$	672.330			0.0623667
$489$		0			0
$490$		−	3902.74i		−	0.359812i
$491$		−	734.667i		−	0.0675256i	−0.999430	−	0.0337628i	$-0.989251\pi$
$491$		−	734.667i		−	0.0675256i	0.999430	−	0.0337628i	$-0.0107491\pi$
$492$		0			0
$493$		−	3275.33i		−	0.299215i
$494$	−3428.70	+	13379.9i	−0.312276	+	1.21861i
$495$		0			0
$496$		−	1640.19i		−	0.148481i
$497$	−26693.0			−2.40915
$498$		0			0
$499$	10038.6			0.900579			0.450290	−	0.892883i	$-0.351321\pi$
$499$	10038.6			0.900579			0.450290	+	0.892883i	$0.351321\pi$
$500$			500.000i			0.0447214i
$501$		0			0
$502$		−	11732.9i		−	1.04316i
$503$			8304.30i			0.736124i	0.929801	+	0.368062i	$0.119979\pi$
$503$			8304.30i			0.736124i	−0.929801	+	0.368062i	$0.880021\pi$
$504$		0			0
$505$	2152.39			0.189664
$506$	−22777.9			−2.00119
$507$		0			0
$508$			10447.6i			0.912474i
$509$	−8973.94			−0.781459			−0.390730	−	0.920506i	$-0.627777\pi$
$509$	−8973.94			−0.781459			−0.390730	+	0.920506i	$0.627777\pi$
$510$		0			0
$511$	−25147.2			−2.17700
$512$	512.000			0.0441942
$513$		0			0
$514$	7650.33			0.656501
$515$	−932.131			−0.0797565
$516$		0			0
$517$	−6303.85			−0.536254
$518$			2933.82i			0.248851i
$519$		0			0
$520$	−3335.53			−0.281293
$521$	15105.6			1.27023			0.635113	−	0.772419i	$-0.280953\pi$
$521$	15105.6			1.27023			0.635113	+	0.772419i	$0.280953\pi$
$522$		0			0
$523$			10254.9i			0.857390i	0.903449	+	0.428695i	$0.141027\pi$
$523$			10254.9i			0.857390i	−0.903449	+	0.428695i	$0.858973\pi$
$524$			10423.5i			0.868997i
$525$		0			0
$526$		−	5746.76i		−	0.476370i
$527$	−1675.02			−0.138454
$528$		0			0
$529$	−27927.1			−2.29531
$530$			1912.50i			0.156743i
$531$		0			0
$532$	8689.86	+	2226.83i	0.708182	+	0.181477i
$533$			2660.85i			0.216237i
$534$		0			0
$535$		−	4473.42i		−	0.361500i
$536$			7389.28i			0.595464i
$537$		0			0
$538$	2697.07			0.216132
$539$			22198.0i			1.77390i
$540$		0			0
$541$	−4979.41			−0.395715			−0.197857	−	0.980231i	$-0.563398\pi$
$541$	−4979.41			−0.395715			−0.197857	+	0.980231i	$0.563398\pi$
$542$	−9922.47			−0.786359
$543$		0			0
$544$		−	522.873i		−	0.0412095i
$545$	−5113.20			−0.401882
$546$		0			0
$547$		−	20579.8i		−	1.60864i	−0.594194	−	0.804322i	$-0.702529\pi$
$547$		−	20579.8i		−	1.60864i	0.594194	−	0.804322i	$-0.297471\pi$
$548$		−	3214.82i		−	0.250603i
$549$		0			0
$550$		−	2843.90i		−	0.220480i
$551$	−16081.6	−	4121.01i	−1.24337	−	0.318622i
$552$		0			0
$553$			27084.2i			2.08271i
$554$	−16714.3			−1.28181
$555$		0			0
$556$	486.864			0.0371361
$557$		−	650.447i		−	0.0494799i	−0.999694	−	0.0247400i	$-0.992124\pi$
$557$		−	650.447i		−	0.0494799i	0.999694	−	0.0247400i	$-0.00787578\pi$
$558$		0			0
$559$		−	21212.1i		−	1.60497i
$560$			2166.32i			0.163471i
$561$		0			0
$562$	6940.04			0.520904
$563$	−17206.2			−1.28802			−0.644008	−	0.765019i	$-0.722730\pi$
$563$	−17206.2			−1.28802			−0.644008	+	0.765019i	$0.722730\pi$
$564$		0			0
$565$		−	9621.18i		−	0.716400i
$566$	−11294.0			−0.838729
$567$		0			0
$568$	7885.97			0.582549
$569$	−15474.3			−1.14010			−0.570050	−	0.821610i	$-0.693076\pi$
$569$	−15474.3			−1.14010			−0.570050	+	0.821610i	$0.693076\pi$
$570$		0			0
$571$	6229.57			0.456566			0.228283	−	0.973595i	$-0.426689\pi$
$571$	6229.57			0.456566			0.228283	+	0.973595i	$0.426689\pi$
$572$	18971.8			1.38680
$573$		0			0
$574$	1728.14			0.125664
$575$		−	5005.88i		−	0.363060i
$576$		0			0
$577$	9682.04			0.698559			0.349280	−	0.937019i	$-0.386426\pi$
$577$	9682.04			0.698559			0.349280	+	0.937019i	$0.386426\pi$
$578$	9292.02			0.668680
$579$		0			0
$580$		−	4009.02i		−	0.287010i
$581$		−	17660.9i		−	1.26110i
$582$		0			0
$583$		−	10877.9i		−	0.772756i
$584$	7429.28			0.526414
$585$		0			0
$586$	−11324.3			−0.798296
$587$			16716.3i			1.17540i	0.809080	+	0.587698i	$0.199966\pi$
$587$			16716.3i			1.17540i	−0.809080	+	0.587698i	$0.800034\pi$
$588$		0			0
$589$	−2107.51	+	8224.20i	−0.147434	+	0.575335i
$590$		−	427.800i		−	0.0298513i
$591$		0			0
$592$		−	866.744i		−	0.0601739i
$593$			6332.50i			0.438524i	0.975666	+	0.219262i	$0.0703649\pi$
$593$			6332.50i			0.438524i	−0.975666	+	0.219262i	$0.929635\pi$
$594$		0			0
$595$	2212.33			0.152431
$596$		−	8222.60i		−	0.565119i
$597$		0			0
$598$	33394.5			2.28362
$599$	21933.9			1.49615			0.748077	−	0.663612i	$-0.230977\pi$
$599$	21933.9			1.49615			0.748077	+	0.663612i	$0.230977\pi$
$600$		0			0
$601$		−	5773.08i		−	0.391828i	−0.980621	−	0.195914i	$-0.937233\pi$
$601$		−	5773.08i		−	0.391828i	0.980621	−	0.195914i	$-0.0627674\pi$
$602$	−13776.6			−0.932713
$603$		0			0
$604$		−	9685.65i		−	0.652490i
$605$			9520.51i			0.639775i
$606$		0			0
$607$		−	4900.06i		−	0.327656i	−0.986489	−	0.163828i	$-0.947616\pi$
$607$		−	4900.06i		−	0.327656i	0.986489	−	0.163828i	$-0.0523842\pi$
$608$	−2567.26	−	657.877i	−0.171244	−	0.0438823i
$609$		0			0
$610$		−	840.413i		−	0.0557825i
$611$	9242.02			0.611935
$612$		0			0
$613$	−16217.1			−1.06852			−0.534261	−	0.845319i	$-0.679410\pi$
$613$	−16217.1			−1.06852			−0.534261	+	0.845319i	$0.679410\pi$
$614$		−	6439.29i		−	0.423239i
$615$		0			0
$616$		−	12321.6i		−	0.805928i
$617$		−	14489.8i		−	0.945438i	−0.881213	−	0.472719i	$-0.843273\pi$
$617$		−	14489.8i		−	0.945438i	0.881213	−	0.472719i	$-0.156727\pi$
$618$		0			0
$619$	−28205.2			−1.83144			−0.915720	−	0.401816i	$-0.868379\pi$
$619$	−28205.2			−1.83144			−0.915720	+	0.401816i	$0.868379\pi$
$620$	−2050.24			−0.132806
$621$		0			0
$622$		−	6235.12i		−	0.401938i
$623$	−1543.28			−0.0992457
$624$		0			0
$625$	625.000			0.0400000
$626$	9949.54			0.635246
$627$		0			0
$628$	14044.7			0.892428
$629$	−885.150			−0.0561101
$630$		0			0
$631$	16999.2			1.07247			0.536233	−	0.844070i	$-0.319847\pi$
$631$	16999.2			1.07247			0.536233	+	0.844070i	$0.319847\pi$
$632$		−	8001.53i		−	0.503614i
$633$		0			0
$634$	5248.35			0.328768
$635$	13059.5			0.816142
$636$		0			0
$637$		−	32544.2i		−	2.02425i
$638$			22802.5i			1.41498i
$639$		0			0
$640$		−	640.000i		−	0.0395285i
$641$	4806.18			0.296151			0.148075	−	0.988976i	$-0.452692\pi$
$641$	4806.18			0.296151			0.148075	+	0.988976i	$0.452692\pi$
$642$		0			0
$643$	−21763.2			−1.33477			−0.667386	−	0.744712i	$-0.732587\pi$
$643$	−21763.2			−1.33477			−0.667386	+	0.744712i	$0.732587\pi$
$644$		−	21688.7i		−	1.32710i
$645$		0			0
$646$	−671.848	+	2621.78i	−0.0409187	+	0.159679i
$647$		−	13185.6i		−	0.801204i	−0.916252	−	0.400602i	$-0.868801\pi$
$647$		−	13185.6i		−	0.801204i	0.916252	−	0.400602i	$-0.131199\pi$
$648$		0			0
$649$			2433.24i			0.147170i
$650$			4169.41i			0.251597i
$651$		0			0
$652$	5790.72			0.347825
$653$		−	7286.29i		−	0.436653i	−0.975876	−	0.218327i	$-0.929940\pi$
$653$		−	7286.29i		−	0.436653i	0.975876	−	0.218327i	$-0.0700598\pi$
$654$		0			0
$655$	13029.4			0.777254
$656$	−510.547			−0.0303864
$657$		0			0
$658$		−	6002.40i		−	0.355620i
$659$	−23207.0			−1.37180			−0.685900	−	0.727696i	$-0.740591\pi$
$659$	−23207.0			−1.37180			−0.685900	+	0.727696i	$0.740591\pi$
$660$		0			0
$661$		−	14215.2i		−	0.836472i	−0.908338	−	0.418236i	$-0.862648\pi$
$661$		−	14215.2i		−	0.836472i	0.908338	−	0.418236i	$-0.137352\pi$
$662$			9106.27i			0.534630i
$663$		0			0
$664$			5217.60i			0.304943i
$665$	2783.54	−	10862.3i	0.162318	−	0.633418i
$666$		0			0
$667$			40137.3i			2.33002i
$668$	−17095.0			−0.990159
$669$		0			0
$670$	9236.61			0.532599
$671$			4780.10i			0.275013i
$672$		0			0
$673$			18676.1i			1.06970i	0.844946	+	0.534852i	$0.179633\pi$
$673$			18676.1i			1.06970i	−0.844946	+	0.534852i	$0.820367\pi$
$674$			19015.5i			1.08672i
$675$		0			0
$676$	−19026.4			−1.08252
$677$	8198.69			0.465438			0.232719	−	0.972544i	$-0.425238\pi$
$677$	8198.69			0.465438			0.232719	+	0.972544i	$0.425238\pi$
$678$		0			0
$679$		−	21752.3i		−	1.22942i
$680$	−653.591			−0.0368589
$681$		0			0
$682$	11661.3			0.654744
$683$	−18171.9			−1.01805			−0.509026	−	0.860751i	$-0.669994\pi$
$683$	−18171.9			−1.01805			−0.509026	+	0.860751i	$0.669994\pi$
$684$		0			0
$685$	−4018.52			−0.224146
$686$	−2560.26			−0.142494
$687$		0			0
$688$	4070.05			0.225537
$689$			15948.0i			0.881814i
$690$		0			0
$691$	12977.8			0.714472			0.357236	−	0.934014i	$-0.383719\pi$
$691$	12977.8			0.714472			0.357236	+	0.934014i	$0.383719\pi$
$692$	15655.5			0.860019
$693$		0			0
$694$			23347.5i			1.27703i
$695$		−	608.580i		−	0.0332155i
$696$		0			0
$697$			521.389i			0.0283343i
$698$	−5577.58			−0.302456
$699$		0			0
$700$	2707.90			0.146213
$701$		−	4673.25i		−	0.251792i	−0.992043	−	0.125896i	$-0.959819\pi$
$701$		−	4673.25i		−	0.251792i	0.992043	−	0.125896i	$-0.0401805\pi$
$702$		0			0
$703$	−1113.69	+	4346.01i	−0.0597493	+	0.233162i
$704$			3640.19i			0.194879i
$705$		0			0
$706$			8630.16i			0.460057i
$707$		−	11656.9i		−	0.620091i
$708$		0			0
$709$	16493.4			0.873658			0.436829	−	0.899544i	$-0.356101\pi$
$709$	16493.4			0.873658			0.436829	+	0.899544i	$0.356101\pi$
$710$		−	9857.46i		−	0.521048i
$711$		0			0
$712$	455.933			0.0239983
$713$	20526.5			1.07815
$714$		0			0
$715$		−	23714.8i		−	1.24039i
$716$	5641.56			0.294462
$717$		0			0
$718$		−	111.181i		−	0.00577888i
$719$		−	1898.96i		−	0.0984970i	−0.998787	−	0.0492485i	$-0.984317\pi$
$719$		−	1898.96i		−	0.0984970i	0.998787	−	0.0492485i	$-0.0156826\pi$
$720$		0			0
$721$			5048.24i			0.260758i
$722$	12027.4	+	6597.42i	0.619962	+	0.340070i
$723$		0			0
$724$		−	16757.2i		−	0.860191i
$725$	−5011.28			−0.256709
$726$		0			0
$727$	−5044.44			−0.257342			−0.128671	−	0.991687i	$-0.541071\pi$
$727$	−5044.44			−0.257342			−0.128671	+	0.991687i	$0.541071\pi$
$728$			18064.6i			0.919667i
$729$		0			0
$730$		−	9286.60i		−	0.470839i
$731$		−	4156.48i		−	0.210305i
$732$		0			0
$733$	31602.7			1.59246			0.796230	−	0.604994i	$-0.206825\pi$
$733$	31602.7			1.59246			0.796230	+	0.604994i	$0.206825\pi$
$734$	22781.3			1.14560
$735$		0			0
$736$			6407.52i			0.320903i
$737$	−52535.9			−2.62576
$738$		0			0
$739$	4635.15			0.230727			0.115363	−	0.993323i	$-0.463197\pi$
$739$	4635.15			0.230727			0.115363	+	0.993323i	$0.463197\pi$
$740$	−1083.43			−0.0538212
$741$		0			0
$742$	10357.7			0.512458
$743$	−32892.0			−1.62408			−0.812040	−	0.583602i	$-0.801643\pi$
$743$	−32892.0			−1.62408			−0.812040	+	0.583602i	$0.801643\pi$
$744$		0			0
$745$	−10278.3			−0.505458
$746$			18400.4i			0.903063i
$747$		0			0
$748$	3717.49			0.181718
$749$	−24227.1			−1.18190
$750$		0			0
$751$			23364.6i			1.13527i	0.823280	+	0.567635i	$0.192141\pi$
$751$			23364.6i			1.13527i	−0.823280	+	0.567635i	$0.807859\pi$
$752$			1773.30i			0.0859915i
$753$		0			0
$754$		−	33430.5i		−	1.61468i
$755$	−12107.1			−0.583604
$756$		0			0
$757$	−14191.6			−0.681379			−0.340689	−	0.940176i	$-0.610660\pi$
$757$	−14191.6			−0.681379			−0.340689	+	0.940176i	$0.610660\pi$
$758$		−	8953.07i		−	0.429011i
$759$		0			0
$760$	−822.346	+	3209.07i	−0.0392495	+	0.153165i
$761$			14367.5i			0.684390i	0.939629	+	0.342195i	$0.111170\pi$
$761$			14367.5i			0.684390i	−0.939629	+	0.342195i	$0.888830\pi$
$762$		0			0
$763$			27692.1i			1.31392i
$764$			15386.8i			0.728630i
$765$		0			0
$766$	6488.44			0.306054
$767$		−	3567.35i		−	0.167939i
$768$		0			0
$769$	−20326.6			−0.953181			−0.476590	−	0.879126i	$-0.658127\pi$
$769$	−20326.6			−0.953181			−0.476590	+	0.879126i	$0.658127\pi$
$770$	−15402.0			−0.720844
$771$		0			0
$772$			15503.6i			0.722780i
$773$	10857.4			0.505191			0.252595	−	0.967572i	$-0.418716\pi$
$773$	10857.4			0.505191			0.252595	+	0.967572i	$0.418716\pi$
$774$		0			0
$775$			2562.80i			0.118785i
$776$			6426.30i			0.297282i
$777$		0			0
$778$		−	11282.4i		−	0.519914i
$779$	2559.97	+	656.010i	0.117741	+	0.0301720i
$780$		0			0
$781$			56067.2i			2.56881i
$782$	6543.59			0.299230
$783$		0			0
$784$	6244.38			0.284456
$785$		−	17555.9i		−	0.798212i
$786$		0			0
$787$		−	1366.48i		−	0.0618931i	−0.999521	−	0.0309465i	$-0.990148\pi$
$787$		−	1366.48i		−	0.0618931i	0.999521	−	0.0309465i	$-0.00985216\pi$
$788$		−	879.722i		−	0.0397700i
$789$		0			0
$790$	−10001.9			−0.450446
$791$	−52106.4			−2.34221
$792$		0			0
$793$		−	7008.05i		−	0.313825i
$794$	6926.09			0.309569
$795$		0			0
$796$	2428.45			0.108133
$797$	−15168.3			−0.674137			−0.337068	−	0.941480i	$-0.609435\pi$
$797$	−15168.3			−0.674137			−0.337068	+	0.941480i	$0.609435\pi$
$798$		0			0
$799$	1810.96			0.0801840
$800$	−800.000			−0.0353553
$801$		0			0
$802$	−16467.2			−0.725036
$803$			52820.3i			2.32128i
$804$		0			0
$805$	−27110.9			−1.18700
$806$	−17096.6			−0.747148
$807$		0			0
$808$			3443.83i			0.149942i
$809$			33431.8i			1.45290i	0.687217	+	0.726452i	$0.258832\pi$
$809$			33431.8i			1.45290i	−0.687217	+	0.726452i	$0.741168\pi$
$810$		0			0
$811$		−	21252.7i		−	0.920201i	−0.887867	−	0.460101i	$-0.847813\pi$
$811$		−	21252.7i		−	0.920201i	0.887867	−	0.460101i	$-0.152187\pi$
$812$	−21712.1			−0.938356
$813$		0			0
$814$	6162.33			0.265343
$815$		−	7238.40i		−	0.311105i
$816$		0			0
$817$	−20407.9	−	5229.67i	−0.873909	−	0.223945i
$818$			6435.70i			0.275084i
$819$		0			0
$820$			638.184i			0.0271785i
$821$			32993.3i			1.40253i	0.712902	+	0.701264i	$0.247380\pi$
$821$			32993.3i			1.40253i	−0.712902	+	0.701264i	$0.752620\pi$
$822$		0			0
$823$	−13462.5			−0.570196			−0.285098	−	0.958498i	$-0.592026\pi$
$823$	−13462.5			−0.570196			−0.285098	+	0.958498i	$0.592026\pi$
$824$		−	1491.41i		−	0.0630531i
$825$		0			0
$826$	−2316.88			−0.0975965
$827$	11252.2			0.473127			0.236563	−	0.971616i	$-0.423979\pi$
$827$	11252.2			0.473127			0.236563	+	0.971616i	$0.423979\pi$
$828$		0			0
$829$		−	3441.74i		−	0.144194i	−0.997398	−	0.0720968i	$-0.977031\pi$
$829$		−	3441.74i		−	0.144194i	0.997398	−	0.0720968i	$-0.0229691\pi$
$830$	6522.00			0.272749
$831$		0			0
$832$		−	5336.85i		−	0.222382i
$833$		−	6376.98i		−	0.265245i
$834$		0			0
$835$			21368.8i			0.885625i
$836$	4677.34	−	18252.5i	0.193504	−	0.755117i
$837$		0			0
$838$		−	18516.2i		−	0.763282i
$839$	23999.6			0.987555			0.493777	−	0.869588i	$-0.335616\pi$
$839$	23999.6			0.987555			0.493777	+	0.869588i	$0.335616\pi$
$840$		0			0
$841$	15791.7			0.647491
$842$			22686.0i			0.928517i
$843$		0			0
$844$			2630.87i			0.107297i
$845$			23783.0i			0.968236i
$846$		0			0
$847$	51561.2			2.09169
$848$	−3060.00			−0.123916
$849$		0			0
$850$			816.988i			0.0329676i
$851$	10847.0			0.436935
$852$		0			0
$853$	27198.9			1.09176			0.545881	−	0.837862i	$-0.316195\pi$
$853$	27198.9			1.09176			0.545881	+	0.837862i	$0.316195\pi$
$854$	−4551.51			−0.182377
$855$		0			0
$856$	7157.46			0.285791
$857$	44102.3			1.75788			0.878942	−	0.476928i	$-0.158250\pi$
$857$	44102.3			1.75788			0.878942	+	0.476928i	$0.158250\pi$
$858$		0			0
$859$	−33712.9			−1.33908			−0.669540	−	0.742776i	$-0.733509\pi$
$859$	−33712.9			−1.33908			−0.669540	+	0.742776i	$0.733509\pi$
$860$		−	5087.56i		−	0.201726i
$861$		0			0
$862$	−17427.3			−0.688603
$863$	−46554.1			−1.83629			−0.918146	−	0.396242i	$-0.870314\pi$
$863$	−46554.1			−1.83629			−0.918146	+	0.396242i	$0.870314\pi$
$864$		0			0
$865$		−	19569.4i		−	0.769225i
$866$			3779.21i			0.148294i
$867$		0			0
$868$			11103.7i			0.434198i
$869$	56888.9			2.22074
$870$		0			0
$871$	77022.4			2.99633
$872$		−	8181.12i		−	0.317715i
$873$		0			0
$874$	8233.13	−	32128.4i	0.318638	−	1.24343i
$875$		−	3384.88i		−	0.130777i
$876$		0			0
$877$		−	32258.8i		−	1.24208i	−0.783780	−	0.621039i	$-0.786711\pi$
$877$		−	32258.8i		−	1.24208i	0.783780	−	0.621039i	$-0.213289\pi$
$878$			20424.3i			0.785066i
$879$		0			0
$880$	4550.24			0.174305
$881$			13413.9i			0.512969i	0.966548	+	0.256485i	$0.0825643\pi$
$881$			13413.9i			0.512969i	−0.966548	+	0.256485i	$0.917436\pi$
$882$		0			0
$883$	−37251.7			−1.41973			−0.709864	−	0.704339i	$-0.751244\pi$
$883$	−37251.7			−1.41973			−0.709864	+	0.704339i	$0.751244\pi$
$884$	−5450.18			−0.207363
$885$		0			0
$886$		−	10972.7i		−	0.416065i
$887$	−37868.6			−1.43349			−0.716743	−	0.697337i	$-0.754368\pi$
$887$	−37868.6			−1.43349			−0.716743	+	0.697337i	$0.754368\pi$
$888$		0			0
$889$		−	70727.6i		−	2.66831i
$890$		−	569.916i		−	0.0214648i
$891$		0			0
$892$		−	2771.52i		−	0.104033i
$893$	2278.54	−	8891.64i	0.0853847	−	0.333200i
$894$		0			0
$895$		−	7051.95i		−	0.263375i
$896$	−3466.12			−0.129235
$897$		0			0
$898$	−14210.4			−0.528070
$899$		−	20548.6i		−	0.762331i
$900$		0			0
$901$			3124.98i			0.115547i
$902$		−	3629.86i		−	0.133992i
$903$		0			0
$904$	15393.9			0.566364
$905$	−20946.6			−0.769378
$906$		0			0
$907$		−	29578.0i		−	1.08282i	−0.840757	−	0.541412i	$-0.817890\pi$
$907$		−	29578.0i		−	1.08282i	0.840757	−	0.541412i	$-0.182110\pi$
$908$	3683.44			0.134625
$909$		0			0
$910$	22580.7			0.822575
$911$	8597.34			0.312670			0.156335	−	0.987704i	$-0.450032\pi$
$911$	8597.34			0.312670			0.156335	+	0.987704i	$0.450032\pi$
$912$		0			0
$913$	−37095.8			−1.34468
$914$	−6716.18			−0.243054
$915$		0			0
$916$	3372.92			0.121664
$917$		−	70564.8i		−	2.54117i
$918$		0			0
$919$	−23979.5			−0.860731			−0.430366	−	0.902655i	$-0.641615\pi$
$919$	−23979.5			−0.860731			−0.430366	+	0.902655i	$0.641615\pi$
$920$	8009.40			0.287024
$921$		0			0
$922$			17181.5i			0.613711i
$923$		−	82199.6i		−	2.93135i
$924$		0			0
$925$			1354.29i			0.0481391i
$926$	3809.50			0.135192
$927$		0			0
$928$	6414.44			0.226901
$929$			16310.3i			0.576022i	0.957627	+	0.288011i	$0.0929940\pi$
$929$			16310.3i			0.576022i	−0.957627	+	0.288011i	$0.907006\pi$
$930$		0			0
$931$	−31310.4	−	8023.51i	−1.10221	−	0.282449i
$932$			16650.6i			0.585203i
$933$		0			0
$934$		−	29430.5i		−	1.03104i
$935$		−	4646.86i		−	0.162533i
$936$		0			0
$937$	55020.5			1.91829			0.959147	−	0.282910i	$-0.0912996\pi$
$937$	55020.5			1.91829			0.959147	+	0.282910i	$0.0912996\pi$
$938$		−	50023.7i		−	1.74129i
$939$		0			0
$940$	2216.62			0.0769131
$941$	−10730.5			−0.371736			−0.185868	−	0.982575i	$-0.559510\pi$
$941$	−10730.5			−0.371736			−0.185868	+	0.982575i	$0.559510\pi$
$942$		0			0
$943$		−	6389.34i		−	0.220642i
$944$	684.481			0.0235995
$945$		0			0
$946$			28937.0i			0.994528i
$947$			55547.3i			1.90607i	0.302864	+	0.953034i	$0.402057\pi$
$947$			55547.3i			1.90607i	−0.302864	+	0.953034i	$0.597943\pi$
$948$		0			0
$949$		−	77439.3i		−	2.64888i
$950$	4011.34	+	1027.93i	0.136995	+	0.0351058i
$951$		0			0
$952$			3539.72i			0.120507i
$953$	−28184.3			−0.958006			−0.479003	−	0.877813i	$-0.659002\pi$
$953$	−28184.3			−0.958006			−0.479003	+	0.877813i	$0.659002\pi$
$954$		0			0
$955$	19233.5			0.651707
$956$			17778.8i			0.601472i
$957$		0			0
$958$		−	20647.4i		−	0.696334i
$959$			21763.5i			0.732828i
$960$		0			0
$961$	19282.3			0.647253
$962$	−9034.53			−0.302791
$963$		0			0
$964$		−	679.983i		−	0.0227187i
$965$	19379.5			0.646474
$966$		0			0
$967$	44740.7			1.48786			0.743931	−	0.668256i	$-0.232959\pi$
$967$	44740.7			1.48786			0.743931	+	0.668256i	$0.232959\pi$
$968$	−15232.8			−0.505786
$969$		0			0
$970$	8032.88			0.265897
$971$	38742.6			1.28044			0.640221	−	0.768191i	$-0.278843\pi$
$971$	38742.6			1.28044			0.640221	+	0.768191i	$0.278843\pi$
$972$		0			0
$973$	−3295.95			−0.108595
$974$		−	38476.9i		−	1.26579i
$975$		0			0
$976$	1344.66			0.0440999
$977$	28420.1			0.930646			0.465323	−	0.885141i	$-0.345938\pi$
$977$	28420.1			0.930646			0.465323	+	0.885141i	$0.345938\pi$
$978$		0			0
$979$			3241.57i			0.105823i
$980$		−	7805.48i		−	0.254425i
$981$		0			0
$982$		−	1469.33i		−	0.0477478i
$983$	8407.09			0.272782			0.136391	−	0.990655i	$-0.456450\pi$
$983$	8407.09			0.272782			0.136391	+	0.990655i	$0.456450\pi$
$984$		0			0
$985$	−1099.65			−0.0355714
$986$		−	6550.65i		−	0.211577i
$987$		0			0
$988$	−6857.40	+	26759.9i	−0.220813	+	0.861686i
$989$			50935.4i			1.63767i
$990$		0			0
$991$		−	502.063i		−	0.0160934i	−0.999968	−	0.00804670i	$-0.997439\pi$
$991$		−	502.063i		−	0.0160934i	0.999968	−	0.00804670i	$-0.00256137\pi$
$992$		−	3280.38i		−	0.104992i
$993$		0			0
$994$	−53386.1			−1.70352
$995$		−	3035.56i		−	0.0967174i
$996$		0			0
$997$	2194.17			0.0696992			0.0348496	−	0.999393i	$-0.488905\pi$
$997$	2194.17			0.0696992			0.0348496	+	0.999393i	$0.488905\pi$
$998$	20077.2			0.636806
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1710.4.f.b.341.1	yes	40
3.2	odd	2		1710.4.f.a.341.2	yes	40
19.18	odd	2		1710.4.f.a.341.1	✓	40
57.56	even	2	inner	1710.4.f.b.341.2	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1710.4.f.a.341.1	✓	40	19.18	odd	2
1710.4.f.a.341.2	yes	40	3.2	odd	2
1710.4.f.b.341.1	yes	40	1.1	even	1	trivial
1710.4.f.b.341.2	yes	40	57.56	even	2	inner

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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} -5.00000i q^{5} -27.0790 q^{7} +8.00000 q^{8} -10.0000i q^{10} +56.8780i q^{11} -83.3882i q^{13} -54.1581 q^{14} +16.0000 q^{16} -16.3398i q^{17} +(-80.2268 - 20.5587i) q^{19} -20.0000i q^{20} +113.756i q^{22} +200.235i q^{23} -25.0000 q^{25} -166.776i q^{26} -108.316 q^{28} +200.451 q^{29} -102.512i q^{31} +32.0000 q^{32} -32.6795i q^{34} +135.395i q^{35} -54.1715i q^{37} +(-160.454 - 41.1173i) q^{38} -40.0000i q^{40} -31.9092 q^{41} +254.378 q^{43} +227.512i q^{44} +400.470i q^{46} +110.831i q^{47} +390.274 q^{49} -50.0000 q^{50} -333.553i q^{52} -191.250 q^{53} +284.390 q^{55} -216.632 q^{56} +400.902 q^{58} +42.7800 q^{59} +84.0413 q^{61} -205.024i q^{62} +64.0000 q^{64} -416.941 q^{65} +923.661i q^{67} -65.3591i q^{68} +270.790i q^{70} +985.746 q^{71} +928.660 q^{73} -108.343i q^{74} +(-320.907 - 82.2346i) q^{76} -1540.20i q^{77} -1000.19i q^{79} -80.0000i q^{80} -63.8184 q^{82} +652.200i q^{83} -81.6988 q^{85} +508.756 q^{86} +455.024i q^{88} +56.9916 q^{89} +2258.07i q^{91} +800.940i q^{92} +221.662i q^{94} +(-102.793 + 401.134i) q^{95} +803.288i q^{97} +780.548 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 80 q^{2} + 160 q^{4} - 56 q^{7} + 320 q^{8} - 112 q^{14} + 640 q^{16} - 76 q^{19} - 1000 q^{25} - 224 q^{28} + 120 q^{29} + 1280 q^{32} - 152 q^{38} + 312 q^{41} + 56 q^{43} + 2112 q^{49} - 2000 q^{50}+ \cdots + 4224 q^{98}+O(q^{100}) \) 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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