LMFDB - Embedded newform 1710.4.f.a.341.36

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			341.36
Character	$\chi$	$=$	1710.341
Dual form			1710.4.f.a.341.35

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} +5.00000i q^{5} -27.7896 q^{7} -8.00000 q^{8} -10.0000i q^{10} +36.7025i q^{11} +30.3210i q^{13} +55.5793 q^{14} +16.0000 q^{16} +91.9154i q^{17} +(-38.7500 + 73.1945i) q^{19} +20.0000i q^{20} -73.4051i q^{22} +100.326i q^{23} -25.0000 q^{25} -60.6421i q^{26} -111.159 q^{28} +14.9075 q^{29} +114.008i q^{31} -32.0000 q^{32} -183.831i q^{34} -138.948i q^{35} +292.082i q^{37} +(77.5000 - 146.389i) q^{38} -40.0000i q^{40} +159.698 q^{41} +400.192 q^{43} +146.810i q^{44} -200.652i q^{46} -35.0953i q^{47} +429.264 q^{49} +50.0000 q^{50} +121.284i q^{52} +19.3413 q^{53} -183.513 q^{55} +222.317 q^{56} -29.8149 q^{58} -26.1972 q^{59} -25.1304 q^{61} -228.016i q^{62} +64.0000 q^{64} -151.605 q^{65} +617.868i q^{67} +367.662i q^{68} +277.896i q^{70} +831.421 q^{71} -768.355 q^{73} -584.165i q^{74} +(-155.000 + 292.778i) q^{76} -1019.95i q^{77} +270.654i q^{79} +80.0000i q^{80} -319.396 q^{82} -348.171i q^{83} -459.577 q^{85} -800.383 q^{86} -293.620i q^{88} -1274.76 q^{89} -842.610i q^{91} +401.304i q^{92} +70.1905i q^{94} +(-365.973 - 193.750i) q^{95} +1490.22i q^{97} -858.527 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−27.7896			−1.50050			−0.750250	−	0.661155i	$-0.770067\pi$
$7$	−27.7896			−1.50050			−0.750250	+	0.661155i	$0.770067\pi$
$8$	−8.00000			−0.353553
$9$		0			0
$10$		−	10.0000i		−	0.316228i
$11$			36.7025i			1.00602i	0.864280	+	0.503010i	$0.167774\pi$
$11$			36.7025i			1.00602i	−0.864280	+	0.503010i	$0.832226\pi$
$12$		0			0
$13$			30.3210i			0.646888i	0.946247	+	0.323444i	$0.104841\pi$
$13$			30.3210i			0.646888i	−0.946247	+	0.323444i	$0.895159\pi$
$14$	55.5793			1.06101
$15$		0			0
$16$	16.0000			0.250000
$17$			91.9154i			1.31134i	0.755048	+	0.655670i	$0.227614\pi$
$17$			91.9154i			1.31134i	−0.755048	+	0.655670i	$0.772386\pi$
$18$		0			0
$19$	−38.7500	+	73.1945i	−0.467887	+	0.883788i
$19$	−38.7500	+	73.1945i	−0.467887	+	0.883788i
$20$			20.0000i			0.223607i
$21$		0			0
$22$		−	73.4051i		−	0.711364i
$23$			100.326i			0.909539i	0.890609	+	0.454769i	$0.150278\pi$
$23$			100.326i			0.909539i	−0.890609	+	0.454769i	$0.849722\pi$
$24$		0			0
$25$	−25.0000			−0.200000
$26$		−	60.6421i		−	0.457419i
$27$		0			0
$28$	−111.159			−0.750250
$29$	14.9075			0.0954568			0.0477284	−	0.998860i	$-0.484802\pi$
$29$	14.9075			0.0954568			0.0477284	+	0.998860i	$0.484802\pi$
$30$		0			0
$31$			114.008i			0.660530i	0.943888	+	0.330265i	$0.107138\pi$
$31$			114.008i			0.660530i	−0.943888	+	0.330265i	$0.892862\pi$
$32$	−32.0000			−0.176777
$33$		0			0
$34$		−	183.831i		−	0.927257i
$35$		−	138.948i		−	0.671044i
$36$		0			0
$37$			292.082i			1.29779i	0.760880	+	0.648893i	$0.224768\pi$
$37$			292.082i			1.29779i	−0.760880	+	0.648893i	$0.775232\pi$
$38$	77.5000	−	146.389i	0.330846	−	0.624933i
$39$		0			0
$40$		−	40.0000i		−	0.158114i
$41$	159.698			0.608308			0.304154	−	0.952623i	$-0.401626\pi$
$41$	159.698			0.608308			0.304154	+	0.952623i	$0.401626\pi$
$42$		0			0
$43$	400.192			1.41927			0.709636	−	0.704569i	$-0.248860\pi$
$43$	400.192			1.41927			0.709636	+	0.704569i	$0.248860\pi$
$44$			146.810i			0.503010i
$45$		0			0
$46$		−	200.652i		−	0.643141i
$47$		−	35.0953i		−	0.108919i	−0.998516	−	0.0544593i	$-0.982656\pi$
$47$		−	35.0953i		−	0.108919i	0.998516	−	0.0544593i	$-0.0173435\pi$
$48$		0			0
$49$	429.264			1.25150
$50$	50.0000			0.141421
$51$		0			0
$52$			121.284i			0.323444i
$53$	19.3413			0.0501270			0.0250635	−	0.999686i	$-0.492021\pi$
$53$	19.3413			0.0501270			0.0250635	+	0.999686i	$0.492021\pi$
$54$		0			0
$55$	−183.513			−0.449906
$56$	222.317			0.530507
$57$		0			0
$58$	−29.8149			−0.0674981
$59$	−26.1972			−0.0578066			−0.0289033	−	0.999582i	$-0.509201\pi$
$59$	−26.1972			−0.0578066			−0.0289033	+	0.999582i	$0.509201\pi$
$60$		0			0
$61$	−25.1304			−0.0527478			−0.0263739	−	0.999652i	$-0.508396\pi$
$61$	−25.1304			−0.0527478			−0.0263739	+	0.999652i	$0.508396\pi$
$62$		−	228.016i		−	0.467065i
$63$		0			0
$64$	64.0000			0.125000
$65$	−151.605			−0.289297
$66$		0			0
$67$			617.868i			1.12664i	0.826240	+	0.563318i	$0.190475\pi$
$67$			617.868i			1.12664i	−0.826240	+	0.563318i	$0.809525\pi$
$68$			367.662i			0.655670i
$69$		0			0
$70$			277.896i			0.474499i
$71$	831.421			1.38974			0.694870	−	0.719136i	$-0.255462\pi$
$71$	831.421			1.38974			0.694870	+	0.719136i	$0.255462\pi$
$72$		0			0
$73$	−768.355			−1.23191			−0.615953	−	0.787783i	$-0.711229\pi$
$73$	−768.355			−1.23191			−0.615953	+	0.787783i	$0.711229\pi$
$74$		−	584.165i		−	0.917673i
$75$		0			0
$76$	−155.000	+	292.778i	−0.233944	+	0.441894i
$77$		−	1019.95i		−	1.50953i
$78$		0			0
$79$			270.654i			0.385455i	0.981252	+	0.192728i	$0.0617334\pi$
$79$			270.654i			0.385455i	−0.981252	+	0.192728i	$0.938267\pi$
$80$			80.0000i			0.111803i
$81$		0			0
$82$	−319.396			−0.430138
$83$		−	348.171i		−	0.460442i	−0.973138	−	0.230221i	$-0.926055\pi$
$83$		−	348.171i		−	0.460442i	0.973138	−	0.230221i	$-0.0739449\pi$
$84$		0			0
$85$	−459.577			−0.586449
$86$	−800.383			−1.00358
$87$		0			0
$88$		−	293.620i		−	0.355682i
$89$	−1274.76			−1.51825			−0.759125	−	0.650944i	$-0.774373\pi$
$89$	−1274.76			−1.51825			−0.759125	+	0.650944i	$0.774373\pi$
$90$		0			0
$91$		−	842.610i		−	0.970654i
$92$			401.304i			0.454769i
$93$		0			0
$94$			70.1905i			0.0770170i
$95$	−365.973	−	193.750i	−0.395242	−	0.209246i
$96$		0			0
$97$			1490.22i			1.55989i	0.625850	+	0.779944i	$0.284752\pi$
$97$			1490.22i			1.55989i	−0.625850	+	0.779944i	$0.715248\pi$
$98$	−858.527			−0.884942
$99$		0			0
$100$	−100.000			−0.100000
$101$		−	808.241i		−	0.796267i	−0.917327	−	0.398134i	$-0.869658\pi$
$101$		−	808.241i		−	0.796267i	0.917327	−	0.398134i	$-0.130342\pi$
$102$		0			0
$103$			1236.46i			1.18284i	0.806364	+	0.591420i	$0.201432\pi$
$103$			1236.46i			1.18284i	−0.806364	+	0.591420i	$0.798568\pi$
$104$		−	242.568i		−	0.228709i
$105$		0			0
$106$	−38.6826			−0.0354452
$107$	11.9236			0.0107729			0.00538646	−	0.999985i	$-0.498285\pi$
$107$	11.9236			0.0107729			0.00538646	+	0.999985i	$0.498285\pi$
$108$		0			0
$109$		−	347.569i		−	0.305423i	−0.988271	−	0.152711i	$-0.951200\pi$
$109$		−	347.569i		−	0.305423i	0.988271	−	0.152711i	$-0.0488005\pi$
$110$	367.025			0.318132
$111$		0			0
$112$	−444.634			−0.375125
$113$	48.5045			0.0403798			0.0201899	−	0.999796i	$-0.493573\pi$
$113$	48.5045			0.0403798			0.0201899	+	0.999796i	$0.493573\pi$
$114$		0			0
$115$	−501.629			−0.406758
$116$	59.6298			0.0477284
$117$		0			0
$118$	52.3944			0.0408754
$119$		−	2554.30i		−	1.96766i
$120$		0			0
$121$	−16.0761			−0.0120782
$122$	50.2608			0.0372984
$123$		0			0
$124$			456.032i			0.330265i
$125$		−	125.000i		−	0.0894427i
$126$		0			0
$127$			1783.37i			1.24605i	0.782201	+	0.623027i	$0.214097\pi$
$127$			1783.37i			1.24605i	−0.782201	+	0.623027i	$0.785903\pi$
$128$	−128.000			−0.0883883
$129$		0			0
$130$	303.210			0.204564
$131$		−	499.428i		−	0.333093i	−0.986034	−	0.166547i	$-0.946738\pi$
$131$		−	499.428i		−	0.333093i	0.986034	−	0.166547i	$-0.0532616\pi$
$132$		0			0
$133$	1076.85	−	2034.05i	0.702065	−	1.32612i
$134$		−	1235.74i		−	0.796651i
$135$		0			0
$136$		−	735.323i		−	0.463628i
$137$			1206.85i			0.752613i	0.926495	+	0.376306i	$0.122806\pi$
$137$			1206.85i			0.752613i	−0.926495	+	0.376306i	$0.877194\pi$
$138$		0			0
$139$	−341.515			−0.208395			−0.104198	−	0.994557i	$-0.533227\pi$
$139$	−341.515			−0.208395			−0.104198	+	0.994557i	$0.533227\pi$
$140$		−	555.793i		−	0.335522i
$141$		0			0
$142$	−1662.84			−0.982694
$143$	−1112.86			−0.650783
$144$		0			0
$145$			74.5373i			0.0426896i
$146$	1536.71			0.871089
$147$		0			0
$148$			1168.33i			0.648893i
$149$		−	506.899i		−	0.278703i	−0.990243	−	0.139352i	$-0.955498\pi$
$149$		−	506.899i		−	0.278703i	0.990243	−	0.139352i	$-0.0445019\pi$
$150$		0			0
$151$			884.637i			0.476760i	0.971172	+	0.238380i	$0.0766164\pi$
$151$			884.637i			0.476760i	−0.971172	+	0.238380i	$0.923384\pi$
$152$	310.000	−	585.556i	0.165423	−	0.312466i
$153$		0			0
$154$			2039.90i			1.06740i
$155$	−570.039			−0.295398
$156$		0			0
$157$	−3372.54			−1.71438			−0.857191	−	0.514999i	$-0.827792\pi$
$157$	−3372.54			−1.71438			−0.857191	+	0.514999i	$0.827792\pi$
$158$		−	541.308i		−	0.272558i
$159$		0			0
$160$		−	160.000i		−	0.0790569i
$161$		−	2788.02i		−	1.36476i
$162$		0			0
$163$	1245.55			0.598520			0.299260	−	0.954172i	$-0.403260\pi$
$163$	1245.55			0.598520			0.299260	+	0.954172i	$0.403260\pi$
$164$	638.791			0.304154
$165$		0			0
$166$			696.341i			0.325582i
$167$	141.300			0.0654737			0.0327369	−	0.999464i	$-0.489578\pi$
$167$	141.300			0.0654737			0.0327369	+	0.999464i	$0.489578\pi$
$168$		0			0
$169$	1277.64			0.581536
$170$	919.154			0.414682
$171$		0			0
$172$	1600.77			0.709636
$173$	2539.60			1.11608			0.558040	−	0.829814i	$-0.311553\pi$
$173$	2539.60			1.11608			0.558040	+	0.829814i	$0.311553\pi$
$174$		0			0
$175$	694.741			0.300100
$176$			587.241i			0.251505i
$177$		0			0
$178$	2549.52			1.07357
$179$	1201.07			0.501521			0.250761	−	0.968049i	$-0.419319\pi$
$179$	1201.07			0.501521			0.250761	+	0.968049i	$0.419319\pi$
$180$		0			0
$181$			1781.99i			0.731793i	0.930656	+	0.365897i	$0.119238\pi$
$181$			1781.99i			0.731793i	−0.930656	+	0.365897i	$0.880762\pi$
$182$			1685.22i			0.686356i
$183$		0			0
$184$		−	802.607i		−	0.321571i
$185$	−1460.41			−0.580387
$186$		0			0
$187$	−3373.53			−1.31923
$188$		−	140.381i		−	0.0544593i
$189$		0			0
$190$	731.945	+	387.500i	0.279478	+	0.147959i
$191$		−	1327.99i		−	0.503088i	−0.967846	−	0.251544i	$-0.919062\pi$
$191$		−	1327.99i		−	0.503088i	0.967846	−	0.251544i	$-0.0809383\pi$
$192$		0			0
$193$		−	3316.75i		−	1.23702i	−0.785777	−	0.618510i	$-0.787737\pi$
$193$		−	3316.75i		−	1.23702i	0.785777	−	0.618510i	$-0.212263\pi$
$194$		−	2980.44i		−	1.10301i
$195$		0			0
$196$	1717.05			0.625749
$197$			4932.01i			1.78371i	0.452318	+	0.891857i	$0.350597\pi$
$197$			4932.01i			1.78371i	−0.452318	+	0.891857i	$0.649403\pi$
$198$		0			0
$199$	2089.34			0.744268			0.372134	−	0.928179i	$-0.378626\pi$
$199$	2089.34			0.744268			0.372134	+	0.928179i	$0.378626\pi$
$200$	200.000			0.0707107
$201$		0			0
$202$			1616.48i			0.563046i
$203$	−414.273			−0.143233
$204$		0			0
$205$			798.489i			0.272043i
$206$		−	2472.93i		−	0.836394i
$207$		0			0
$208$			485.137i			0.161722i
$209$	−2686.42	−	1422.22i	−0.889109	−	0.470705i
$210$		0			0
$211$			3693.55i			1.20509i	0.798084	+	0.602546i	$0.205847\pi$
$211$			3693.55i			1.20509i	−0.798084	+	0.602546i	$0.794153\pi$
$212$	77.3652			0.0250635
$213$		0			0
$214$	−23.8473			−0.00761760
$215$			2000.96i			0.634717i
$216$		0			0
$217$		−	3168.24i		−	0.991124i
$218$			695.138i			0.215967i
$219$		0			0
$220$	−734.051			−0.224953
$221$	−2786.97			−0.848289
$222$		0			0
$223$			1742.01i			0.523112i	0.965188	+	0.261556i	$0.0842355\pi$
$223$			1742.01i			0.523112i	−0.965188	+	0.261556i	$0.915764\pi$
$224$	889.268			0.265253
$225$		0			0
$226$	−97.0089			−0.0285528
$227$	3048.20			0.891259			0.445630	−	0.895217i	$-0.352980\pi$
$227$	3048.20			0.891259			0.445630	+	0.895217i	$0.352980\pi$
$228$		0			0
$229$	−2754.53			−0.794865			−0.397433	−	0.917631i	$-0.630099\pi$
$229$	−2754.53			−0.794865			−0.397433	+	0.917631i	$0.630099\pi$
$230$	1003.26			0.287621
$231$		0			0
$232$	−119.260			−0.0337491
$233$		−	1913.43i		−	0.537995i	−0.963141	−	0.268997i	$-0.913308\pi$
$233$		−	1913.43i		−	0.537995i	0.963141	−	0.268997i	$-0.0866923\pi$
$234$		0			0
$235$	175.476			0.0487099
$236$	−104.789			−0.0289033
$237$		0			0
$238$			5108.59i			1.39135i
$239$		−	569.205i		−	0.154054i	−0.997029	−	0.0770269i	$-0.975457\pi$
$239$		−	569.205i		−	0.154054i	0.997029	−	0.0770269i	$-0.0245427\pi$
$240$		0			0
$241$		−	5326.93i		−	1.42381i	−0.702277	−	0.711904i	$-0.747833\pi$
$241$		−	5326.93i		−	1.42381i	0.702277	−	0.711904i	$-0.252167\pi$
$242$	32.1521			0.00854056
$243$		0			0
$244$	−100.522			−0.0263739
$245$			2146.32i			0.559687i
$246$		0			0
$247$	−2219.33	−	1174.94i	−0.571712	−	0.302671i
$248$		−	912.063i		−	0.233533i
$249$		0			0
$250$			250.000i			0.0632456i
$251$		−	975.035i		−	0.245194i	−0.992457	−	0.122597i	$-0.960878\pi$
$251$		−	975.035i		−	0.245194i	0.992457	−	0.122597i	$-0.0391223\pi$
$252$		0			0
$253$	−3682.21			−0.915015
$254$		−	3566.75i		−	0.881093i
$255$		0			0
$256$	256.000			0.0625000
$257$	−6200.98			−1.50508			−0.752542	−	0.658544i	$-0.771173\pi$
$257$	−6200.98			−1.50508			−0.752542	+	0.658544i	$0.771173\pi$
$258$		0			0
$259$		−	8116.86i		−	1.94733i
$260$	−606.421			−0.144649
$261$		0			0
$262$			998.856i			0.235533i
$263$		−	3640.55i		−	0.853560i	−0.904356	−	0.426780i	$-0.859648\pi$
$263$		−	3640.55i		−	0.853560i	0.904356	−	0.426780i	$-0.140352\pi$
$264$		0			0
$265$			96.7065i			0.0224175i
$266$	−2153.70	+	4068.10i	−0.496435	+	0.937711i
$267$		0			0
$268$			2471.47i			0.563318i
$269$	−6322.32			−1.43301			−0.716503	−	0.697584i	$-0.754258\pi$
$269$	−6322.32			−1.43301			−0.716503	+	0.697584i	$0.754258\pi$
$270$		0			0
$271$	8259.44			1.85139			0.925693	−	0.378277i	$-0.123483\pi$
$271$	8259.44			1.85139			0.925693	+	0.378277i	$0.123483\pi$
$272$			1470.65i			0.327835i
$273$		0			0
$274$		−	2413.69i		−	0.532178i
$275$		−	917.563i		−	0.201204i
$276$		0			0
$277$	−364.578			−0.0790807			−0.0395403	−	0.999218i	$-0.512589\pi$
$277$	−364.578			−0.0790807			−0.0395403	+	0.999218i	$0.512589\pi$
$278$	683.030			0.147358
$279$		0			0
$280$			1111.59i			0.237250i
$281$	−1410.01			−0.299339			−0.149669	−	0.988736i	$-0.547821\pi$
$281$	−1410.01			−0.299339			−0.149669	+	0.988736i	$0.547821\pi$
$282$		0			0
$283$	−6322.36			−1.32801			−0.664003	−	0.747730i	$-0.731144\pi$
$283$	−6322.36			−1.32801			−0.664003	+	0.747730i	$0.731144\pi$
$284$	3325.68			0.694870
$285$		0			0
$286$	2225.72			0.460173
$287$	−4437.94			−0.912765
$288$		0			0
$289$	−3535.45			−0.719611
$290$		−	149.075i		−	0.0301861i
$291$		0			0
$292$	−3073.42			−0.615953
$293$	−1239.75			−0.247191			−0.123595	−	0.992333i	$-0.539442\pi$
$293$	−1239.75			−0.247191			−0.123595	+	0.992333i	$0.539442\pi$
$294$		0			0
$295$		−	130.986i		−	0.0258519i
$296$		−	2336.66i		−	0.458836i
$297$		0			0
$298$			1013.80i			0.197073i
$299$	−3041.98			−0.588370
$300$		0			0
$301$	−11121.2			−2.12962
$302$		−	1769.27i		−	0.337120i
$303$		0			0
$304$	−620.000	+	1171.11i	−0.116972	+	0.220947i
$305$		−	125.652i		−	0.0235896i
$306$		0			0
$307$			736.196i			0.136863i	0.997656	+	0.0684315i	$0.0217994\pi$
$307$			736.196i			0.136863i	−0.997656	+	0.0684315i	$0.978201\pi$
$308$		−	4079.80i		−	0.754767i
$309$		0			0
$310$	1140.08			0.208878
$311$		−	6618.80i		−	1.20681i	−0.797435	−	0.603405i	$-0.793810\pi$
$311$		−	6618.80i		−	1.20681i	0.797435	−	0.603405i	$-0.206190\pi$
$312$		0			0
$313$	−10467.4			−1.89027			−0.945136	−	0.326679i	$-0.894071\pi$
$313$	−10467.4			−1.89027			−0.945136	+	0.326679i	$0.894071\pi$
$314$	6745.08			1.21225
$315$		0			0
$316$			1082.62i			0.192728i
$317$	9361.20			1.65860			0.829302	−	0.558801i	$-0.188738\pi$
$317$	9361.20			1.65860			0.829302	+	0.558801i	$0.188738\pi$
$318$		0			0
$319$			547.142i			0.0960315i
$320$			320.000i			0.0559017i
$321$		0			0
$322$			5576.04i			0.965032i
$323$	−6727.71	−	3561.72i	−1.15895	−	0.613559i
$324$		0			0
$325$		−	758.026i		−	0.129378i
$326$	−2491.09			−0.423217
$327$		0			0
$328$	−1277.58			−0.215069
$329$			975.285i			0.163432i
$330$		0			0
$331$		−	920.881i		−	0.152919i	−0.997073	−	0.0764595i	$-0.975638\pi$
$331$		−	920.881i		−	0.152919i	0.997073	−	0.0764595i	$-0.0243616\pi$
$332$		−	1392.68i		−	0.230221i
$333$		0			0
$334$	−282.600			−0.0462969
$335$	−3089.34			−0.503847
$336$		0			0
$337$			8436.13i			1.36364i	0.731522	+	0.681818i	$0.238810\pi$
$337$			8436.13i			1.36364i	−0.731522	+	0.681818i	$0.761190\pi$
$338$	−2555.27			−0.411208
$339$		0			0
$340$	−1838.31			−0.293224
$341$	−4184.38			−0.664507
$342$		0			0
$343$	−2397.23			−0.377371
$344$	−3201.53			−0.501788
$345$		0			0
$346$	−5079.19			−0.789188
$347$		−	2614.02i		−	0.404404i	−0.979344	−	0.202202i	$-0.935190\pi$
$347$		−	2614.02i		−	0.404404i	0.979344	−	0.202202i	$-0.0648097\pi$
$348$		0			0
$349$	3290.23			0.504647			0.252324	−	0.967643i	$-0.418805\pi$
$349$	3290.23			0.504647			0.252324	+	0.967643i	$0.418805\pi$
$350$	−1389.48			−0.212203
$351$		0			0
$352$		−	1174.48i		−	0.177841i
$353$		−	4219.58i		−	0.636220i	−0.948054	−	0.318110i	$-0.896952\pi$
$353$		−	4219.58i		−	0.636220i	0.948054	−	0.318110i	$-0.103048\pi$
$354$		0			0
$355$			4157.10i			0.621510i
$356$	−5099.04			−0.759125
$357$		0			0
$358$	−2402.14			−0.354629
$359$			5533.74i			0.813537i	0.913531	+	0.406769i	$0.133344\pi$
$359$			5533.74i			0.813537i	−0.913531	+	0.406769i	$0.866656\pi$
$360$		0			0
$361$	−3855.87	−	5672.58i	−0.562163	−	0.827027i
$362$		−	3563.99i		−	0.517456i
$363$		0			0
$364$		−	3370.44i		−	0.485327i
$365$		−	3841.77i		−	0.550925i
$366$		0			0
$367$	−8991.17			−1.27884			−0.639422	−	0.768856i	$-0.720826\pi$
$367$	−8991.17			−1.27884			−0.639422	+	0.768856i	$0.720826\pi$
$368$			1605.21i			0.227385i
$369$		0			0
$370$	2920.82			0.410396
$371$	−537.488			−0.0752156
$372$		0			0
$373$		−	5559.77i		−	0.771780i	−0.922545	−	0.385890i	$-0.873894\pi$
$373$		−	5559.77i		−	0.771780i	0.922545	−	0.385890i	$-0.126106\pi$
$374$	6747.06			0.932840
$375$		0			0
$376$			280.762i			0.0385085i
$377$			452.010i			0.0617498i
$378$		0			0
$379$			1282.64i			0.173839i	0.996215	+	0.0869194i	$0.0277023\pi$
$379$			1282.64i			0.173839i	−0.996215	+	0.0869194i	$0.972298\pi$
$380$	−1463.89	−	775.000i	−0.197621	−	0.104623i
$381$		0			0
$382$			2655.98i			0.355737i
$383$	10263.6			1.36930			0.684652	−	0.728871i	$-0.259954\pi$
$383$	10263.6			1.36930			0.684652	+	0.728871i	$0.259954\pi$
$384$		0			0
$385$	5099.75			0.675084
$386$			6633.50i			0.874705i
$387$		0			0
$388$			5960.89i			0.779944i
$389$		−	4082.19i		−	0.532070i	−0.963963	−	0.266035i	$-0.914286\pi$
$389$		−	4082.19i		−	0.532070i	0.963963	−	0.266035i	$-0.0857136\pi$
$390$		0			0
$391$	−9221.50			−1.19271
$392$	−3434.11			−0.442471
$393$		0			0
$394$		−	9864.03i		−	1.26128i
$395$	−1353.27			−0.172381
$396$		0			0
$397$	−1208.88			−0.152826			−0.0764128	−	0.997076i	$-0.524347\pi$
$397$	−1208.88			−0.152826			−0.0764128	+	0.997076i	$0.524347\pi$
$398$	−4178.68			−0.526277
$399$		0			0
$400$	−400.000			−0.0500000
$401$	−12794.7			−1.59336			−0.796681	−	0.604400i	$-0.793413\pi$
$401$	−12794.7			−1.59336			−0.796681	+	0.604400i	$0.793413\pi$
$402$		0			0
$403$	−3456.84			−0.427289
$404$		−	3232.96i		−	0.398134i
$405$		0			0
$406$	828.546			0.101281
$407$	−10720.2			−1.30560
$408$		0			0
$409$			13677.8i			1.65361i	0.562492	+	0.826803i	$0.309843\pi$
$409$			13677.8i			1.65361i	−0.562492	+	0.826803i	$0.690157\pi$
$410$		−	1596.98i		−	0.192364i
$411$		0			0
$412$			4945.86i			0.591420i
$413$	728.011			0.0867387
$414$		0			0
$415$	1740.85			0.205916
$416$		−	970.273i		−	0.114355i
$417$		0			0
$418$	5372.85	+	2844.45i	0.628695	+	0.332838i
$419$		−	7137.88i		−	0.832240i	−0.909310	−	0.416120i	$-0.863390\pi$
$419$		−	7137.88i		−	0.832240i	0.909310	−	0.416120i	$-0.136610\pi$
$420$		0			0
$421$		−	8825.01i		−	1.02163i	−0.859692	−	0.510813i	$-0.829344\pi$
$421$		−	8825.01i		−	1.02163i	0.859692	−	0.510813i	$-0.170656\pi$
$422$		−	7387.10i		−	0.852129i
$423$		0			0
$424$	−154.730			−0.0177226
$425$		−	2297.89i		−	0.262268i
$426$		0			0
$427$	698.365			0.0791481
$428$	47.6946			0.00538646
$429$		0			0
$430$		−	4001.92i		−	0.448813i
$431$	14136.8			1.57992			0.789961	−	0.613157i	$-0.210101\pi$
$431$	14136.8			1.57992			0.789961	+	0.613157i	$0.210101\pi$
$432$		0			0
$433$		−	17162.9i		−	1.90484i	−0.304790	−	0.952420i	$-0.598586\pi$
$433$		−	17162.9i		−	1.90484i	0.304790	−	0.952420i	$-0.401414\pi$
$434$			6336.47i			0.700831i
$435$		0			0
$436$		−	1390.28i		−	0.152711i
$437$	−7343.30	−	3887.63i	−0.803839	−	0.425562i
$438$		0			0
$439$		−	2630.03i		−	0.285933i	−0.989728	−	0.142966i	$-0.954336\pi$
$439$		−	2630.03i		−	0.285933i	0.989728	−	0.142966i	$-0.0456641\pi$
$440$	1468.10			0.159066
$441$		0			0
$442$	5573.94			0.599831
$443$		−	10271.8i		−	1.10164i	−0.834624	−	0.550821i	$-0.814315\pi$
$443$		−	10271.8i		−	1.10164i	0.834624	−	0.550821i	$-0.185685\pi$
$444$		0			0
$445$		−	6373.80i		−	0.678982i
$446$		−	3484.03i		−	0.369896i
$447$		0			0
$448$	−1778.54			−0.187562
$449$	3111.53			0.327043			0.163521	−	0.986540i	$-0.447715\pi$
$449$	3111.53			0.327043			0.163521	+	0.986540i	$0.447715\pi$
$450$		0			0
$451$			5861.32i			0.611970i
$452$	194.018			0.0201899
$453$		0			0
$454$	−6096.39			−0.630215
$455$	4213.05			0.434090
$456$		0			0
$457$	13897.2			1.42250			0.711251	−	0.702938i	$-0.248129\pi$
$457$	13897.2			1.42250			0.711251	+	0.702938i	$0.248129\pi$
$458$	5509.05			0.562055
$459$		0			0
$460$	−2006.52			−0.203379
$461$			1623.62i			0.164034i	0.996631	+	0.0820171i	$0.0261362\pi$
$461$			1623.62i			0.164034i	−0.996631	+	0.0820171i	$0.973864\pi$
$462$		0			0
$463$	−5800.06			−0.582185			−0.291092	−	0.956695i	$-0.594019\pi$
$463$	−5800.06			−0.582185			−0.291092	+	0.956695i	$0.594019\pi$
$464$	238.519			0.0238642
$465$		0			0
$466$			3826.85i			0.380420i
$467$			1945.62i			0.192789i	0.995343	+	0.0963945i	$0.0307310\pi$
$467$			1945.62i			0.192789i	−0.995343	+	0.0963945i	$0.969269\pi$
$468$		0			0
$469$		−	17170.3i		−	1.69051i
$470$	−350.953			−0.0344431
$471$		0			0
$472$	209.578			0.0204377
$473$			14688.1i			1.42782i
$474$		0			0
$475$	968.750	−	1829.86i	0.0935775	−	0.176758i
$476$		−	10217.2i		−	0.983832i
$477$		0			0
$478$			1138.41i			0.108932i
$479$		−	4187.24i		−	0.399415i	−0.979856	−	0.199708i	$-0.936001\pi$
$479$		−	4187.24i		−	0.399415i	0.979856	−	0.199708i	$-0.0639992\pi$
$480$		0			0
$481$	−8856.24			−0.839521
$482$			10653.9i			1.00678i
$483$		0			0
$484$	−64.3042			−0.00603909
$485$	−7451.11			−0.697603
$486$		0			0
$487$		−	8199.09i		−	0.762908i	−0.924388	−	0.381454i	$-0.875423\pi$
$487$		−	8199.09i		−	0.762908i	0.924388	−	0.381454i	$-0.124577\pi$
$488$	201.043			0.0186492
$489$		0			0
$490$		−	4292.64i		−	0.395758i
$491$			16187.9i			1.48788i	0.668246	+	0.743941i	$0.267046\pi$
$491$			16187.9i			1.48788i	−0.668246	+	0.743941i	$0.732954\pi$
$492$		0			0
$493$			1370.23i			0.125176i
$494$	4438.67	+	2349.88i	0.404261	+	0.214021i
$495$		0			0
$496$			1824.13i			0.165132i
$497$	−23104.9			−2.08530
$498$		0			0
$499$	10973.4			0.984447			0.492223	−	0.870469i	$-0.336184\pi$
$499$	10973.4			0.984447			0.492223	+	0.870469i	$0.336184\pi$
$500$		−	500.000i		−	0.0447214i
$501$		0			0
$502$			1950.07i			0.173378i
$503$			11195.2i			0.992380i	0.868214	+	0.496190i	$0.165268\pi$
$503$			11195.2i			0.992380i	−0.868214	+	0.496190i	$0.834732\pi$
$504$		0			0
$505$	4041.20			0.356102
$506$	7364.43			0.647013
$507$		0			0
$508$			7133.49i			0.623027i
$509$	6726.65			0.585763			0.292882	−	0.956149i	$-0.405386\pi$
$509$	6726.65			0.585763			0.292882	+	0.956149i	$0.405386\pi$
$510$		0			0
$511$	21352.3			1.84847
$512$	−512.000			−0.0441942
$513$		0			0
$514$	12402.0			1.06426
$515$	−6182.32			−0.528982
$516$		0			0
$517$	1288.09			0.109574
$518$			16233.7i			1.37697i
$519$		0			0
$520$	1212.84			0.102282
$521$	7521.63			0.632493			0.316246	−	0.948677i	$-0.397577\pi$
$521$	7521.63			0.632493			0.316246	+	0.948677i	$0.397577\pi$
$522$		0			0
$523$			843.506i			0.0705238i	0.999378	+	0.0352619i	$0.0112265\pi$
$523$			843.506i			0.0705238i	−0.999378	+	0.0352619i	$0.988773\pi$
$524$		−	1997.71i		−	0.166547i
$525$		0			0
$526$			7281.11i			0.603558i
$527$	−10479.1			−0.866179
$528$		0			0
$529$	2101.72			0.172739
$530$		−	193.413i		−	0.0158516i
$531$		0			0
$532$	4307.39	−	8136.19i	0.351032	−	0.663062i
$533$			4842.20i			0.393507i
$534$		0			0
$535$			59.6182i			0.00481779i
$536$		−	4942.94i		−	0.398326i
$537$		0			0
$538$	12644.6			1.01329
$539$			15755.1i			1.25903i
$540$		0			0
$541$	5585.32			0.443866			0.221933	−	0.975062i	$-0.428763\pi$
$541$	5585.32			0.443866			0.221933	+	0.975062i	$0.428763\pi$
$542$	−16518.9			−1.30913
$543$		0			0
$544$		−	2941.29i		−	0.231814i
$545$	1737.85			0.136589
$546$		0			0
$547$		−	5559.39i		−	0.434556i	−0.976110	−	0.217278i	$-0.930282\pi$
$547$		−	5559.39i		−	0.434556i	0.976110	−	0.217278i	$-0.0697178\pi$
$548$			4827.39i			0.376306i
$549$		0			0
$550$			1835.13i			0.142273i
$551$	−577.664	+	1091.14i	−0.0446630	+	0.0843635i
$552$		0			0
$553$		−	7521.38i		−	0.578375i
$554$	729.155			0.0559185
$555$		0			0
$556$	−1366.06			−0.104198
$557$			2447.26i			0.186165i	0.995658	+	0.0930825i	$0.0296720\pi$
$557$			2447.26i			0.186165i	−0.995658	+	0.0930825i	$0.970328\pi$
$558$		0			0
$559$			12134.2i			0.918109i
$560$		−	2223.17i		−	0.167761i
$561$		0			0
$562$	2820.02			0.211664
$563$	26016.1			1.94751			0.973756	−	0.227593i	$-0.0730855\pi$
$563$	26016.1			1.94751			0.973756	+	0.227593i	$0.0730855\pi$
$564$		0			0
$565$			242.522i			0.0180584i
$566$	12644.7			0.939041
$567$		0			0
$568$	−6651.37			−0.491347
$569$	2588.82			0.190736			0.0953682	−	0.995442i	$-0.469597\pi$
$569$	2588.82			0.190736			0.0953682	+	0.995442i	$0.469597\pi$
$570$		0			0
$571$	−9015.39			−0.660740			−0.330370	−	0.943852i	$-0.607173\pi$
$571$	−9015.39			−0.660740			−0.330370	+	0.943852i	$0.607173\pi$
$572$	−4451.43			−0.325391
$573$		0			0
$574$	8875.89			0.645422
$575$		−	2508.15i		−	0.181908i
$576$		0			0
$577$	21709.3			1.56633			0.783163	−	0.621817i	$-0.213605\pi$
$577$	21709.3			1.56633			0.783163	+	0.621817i	$0.213605\pi$
$578$	7070.89			0.508842
$579$		0			0
$580$			298.149i			0.0213448i
$581$			9675.53i			0.690893i
$582$		0			0
$583$			709.875i			0.0504289i
$584$	6146.84			0.435544
$585$		0			0
$586$	2479.50			0.174790
$587$			8448.39i			0.594042i	0.954871	+	0.297021i	$0.0959931\pi$
$587$			8448.39i			0.594042i	−0.954871	+	0.297021i	$0.904007\pi$
$588$		0			0
$589$	−8344.75	−	4417.81i	−0.583768	−	0.309054i
$590$			261.972i			0.0182800i
$591$		0			0
$592$			4673.32i			0.324446i
$593$			17978.4i			1.24500i	0.782620	+	0.622499i	$0.213883\pi$
$593$			17978.4i			1.24500i	−0.782620	+	0.622499i	$0.786117\pi$
$594$		0			0
$595$	12771.5			0.879966
$596$		−	2027.60i		−	0.139352i
$597$		0			0
$598$	6083.97			0.416040
$599$	−72.4662			−0.00494305			−0.00247153	−	0.999997i	$-0.500787\pi$
$599$	−72.4662			−0.00494305			−0.00247153	+	0.999997i	$0.500787\pi$
$600$		0			0
$601$		−	4061.23i		−	0.275642i	−0.990457	−	0.137821i	$-0.955990\pi$
$601$		−	4061.23i		−	0.275642i	0.990457	−	0.137821i	$-0.0440099\pi$
$602$	22242.4			1.50587
$603$		0			0
$604$			3538.55i			0.238380i
$605$		−	80.3803i		−	0.00540153i
$606$		0			0
$607$		−	8611.24i		−	0.575814i	−0.957658	−	0.287907i	$-0.907041\pi$
$607$		−	8611.24i		−	0.575814i	0.957658	−	0.287907i	$-0.0929595\pi$
$608$	1240.00	−	2342.22i	0.0827116	−	0.156233i
$609$		0			0
$610$			251.304i			0.0166803i
$611$	1064.12			0.0704581
$612$		0			0
$613$	−23161.2			−1.52606			−0.763029	−	0.646364i	$-0.776289\pi$
$613$	−23161.2			−1.52606			−0.763029	+	0.646364i	$0.776289\pi$
$614$		−	1472.39i		−	0.0967767i
$615$		0			0
$616$			8159.60i			0.533701i
$617$			13082.7i			0.853632i	0.904338	+	0.426816i	$0.140365\pi$
$617$			13082.7i			0.853632i	−0.904338	+	0.426816i	$0.859635\pi$
$618$		0			0
$619$	−15969.7			−1.03696			−0.518479	−	0.855090i	$-0.673502\pi$
$619$	−15969.7			−1.03696			−0.518479	+	0.855090i	$0.673502\pi$
$620$	−2280.16			−0.147699
$621$		0			0
$622$			13237.6i			0.853344i
$623$	35425.1			2.27813
$624$		0			0
$625$	625.000			0.0400000
$626$	20934.9			1.33662
$627$		0			0
$628$	−13490.2			−0.857191
$629$	−26846.9			−1.70184
$630$		0			0
$631$	−3928.27			−0.247832			−0.123916	−	0.992293i	$-0.539545\pi$
$631$	−3928.27			−0.247832			−0.123916	+	0.992293i	$0.539545\pi$
$632$		−	2165.23i		−	0.136279i
$633$		0			0
$634$	−18722.4			−1.17281
$635$	−8916.87			−0.557252
$636$		0			0
$637$			13015.7i			0.809578i
$638$		−	1094.28i		−	0.0679045i
$639$		0			0
$640$		−	640.000i		−	0.0395285i
$641$	−1116.93			−0.0688235			−0.0344118	−	0.999408i	$-0.510956\pi$
$641$	−1116.93			−0.0688235			−0.0344118	+	0.999408i	$0.510956\pi$
$642$		0			0
$643$	−21477.2			−1.31723			−0.658616	−	0.752480i	$-0.728858\pi$
$643$	−21477.2			−1.31723			−0.658616	+	0.752480i	$0.728858\pi$
$644$		−	11152.1i		−	0.682381i
$645$		0			0
$646$	13455.4	+	7123.45i	0.819499	+	0.433852i
$647$		−	3654.77i		−	0.222077i	−0.993816	−	0.111039i	$-0.964582\pi$
$647$		−	3654.77i		−	0.222077i	0.993816	−	0.111039i	$-0.0354177\pi$
$648$		0			0
$649$		−	961.504i		−	0.0581546i
$650$			1516.05i			0.0914837i
$651$		0			0
$652$	4982.19			0.299260
$653$			11757.9i			0.704630i	0.935882	+	0.352315i	$0.114605\pi$
$653$			11757.9i			0.704630i	−0.935882	+	0.352315i	$0.885395\pi$
$654$		0			0
$655$	2497.14			0.148964
$656$	2555.17			0.152077
$657$		0			0
$658$		−	1950.57i		−	0.115564i
$659$	15340.2			0.906783			0.453392	−	0.891311i	$-0.350214\pi$
$659$	15340.2			0.906783			0.453392	+	0.891311i	$0.350214\pi$
$660$		0			0
$661$		−	11520.2i		−	0.677890i	−0.940806	−	0.338945i	$-0.889930\pi$
$661$		−	11520.2i		−	0.677890i	0.940806	−	0.338945i	$-0.110070\pi$
$662$			1841.76i			0.108130i
$663$		0			0
$664$			2785.36i			0.162791i
$665$	10170.2	+	5384.24i	0.593060	+	0.313973i
$666$		0			0
$667$			1495.60i			0.0868216i
$668$	565.199			0.0327369
$669$		0			0
$670$	6178.68			0.356273
$671$		−	922.349i		−	0.0530654i
$672$		0			0
$673$			25295.2i			1.44882i	0.689368	+	0.724412i	$0.257889\pi$
$673$			25295.2i			1.44882i	−0.689368	+	0.724412i	$0.742111\pi$
$674$		−	16872.3i		−	0.964236i
$675$		0			0
$676$	5110.54			0.290768
$677$	6669.71			0.378638			0.189319	−	0.981916i	$-0.439372\pi$
$677$	6669.71			0.378638			0.189319	+	0.981916i	$0.439372\pi$
$678$		0			0
$679$		−	41412.7i		−	2.34061i
$680$	3676.62			0.207341
$681$		0			0
$682$	8368.76			0.469877
$683$	−14015.5			−0.785192			−0.392596	−	0.919711i	$-0.628423\pi$
$683$	−14015.5			−0.785192			−0.392596	+	0.919711i	$0.628423\pi$
$684$		0			0
$685$	−6034.24			−0.336579
$686$	4794.47			0.266842
$687$		0			0
$688$	6403.07			0.354818
$689$			586.448i			0.0324266i
$690$		0			0
$691$	5813.29			0.320040			0.160020	−	0.987114i	$-0.448844\pi$
$691$	5813.29			0.320040			0.160020	+	0.987114i	$0.448844\pi$
$692$	10158.4			0.558040
$693$		0			0
$694$			5228.04i			0.285957i
$695$		−	1707.57i		−	0.0931971i
$696$		0			0
$697$			14678.7i			0.797698i
$698$	−6580.45			−0.356839
$699$		0			0
$700$	2778.96			0.150050
$701$		−	22840.9i		−	1.23066i	−0.788271	−	0.615328i	$-0.789023\pi$
$701$		−	22840.9i		−	1.23066i	0.788271	−	0.615328i	$-0.210977\pi$
$702$		0			0
$703$	−21378.8	−	11318.2i	−1.14697	−	0.607217i
$704$			2348.96i			0.125753i
$705$		0			0
$706$			8439.16i			0.449875i
$707$			22460.7i			1.19480i
$708$		0			0
$709$	10102.3			0.535118			0.267559	−	0.963542i	$-0.413783\pi$
$709$	10102.3			0.535118			0.267559	+	0.963542i	$0.413783\pi$
$710$		−	8314.21i		−	0.439474i
$711$		0			0
$712$	10198.1			0.536783
$713$	−11437.9			−0.600777
$714$		0			0
$715$		−	5564.29i		−	0.291039i
$716$	4804.29			0.250761
$717$		0			0
$718$		−	11067.5i		−	0.575258i
$719$		−	32602.7i		−	1.69106i	−0.533925	−	0.845532i	$-0.679284\pi$
$719$		−	32602.7i		−	1.69106i	0.533925	−	0.845532i	$-0.320716\pi$
$720$		0			0
$721$		−	34360.9i		−	1.77485i
$722$	7711.75	+	11345.2i	0.397509	+	0.584796i
$723$		0			0
$724$			7127.97i			0.365897i
$725$	−372.687			−0.0190914
$726$		0			0
$727$	25142.1			1.28262			0.641312	−	0.767280i	$-0.278391\pi$
$727$	25142.1			1.28262			0.641312	+	0.767280i	$0.278391\pi$
$728$			6740.88i			0.343178i
$729$		0			0
$730$			7683.55i			0.389563i
$731$			36783.8i			1.86115i
$732$		0			0
$733$	31303.5			1.57738			0.788691	−	0.614790i	$-0.210759\pi$
$733$	31303.5			1.57738			0.788691	+	0.614790i	$0.210759\pi$
$734$	17982.3			0.904279
$735$		0			0
$736$		−	3210.43i		−	0.160785i
$737$	−22677.3			−1.13342
$738$		0			0
$739$	4518.25			0.224907			0.112454	−	0.993657i	$-0.464129\pi$
$739$	4518.25			0.224907			0.112454	+	0.993657i	$0.464129\pi$
$740$	−5841.65			−0.290194
$741$		0			0
$742$	1074.98			0.0531854
$743$	−25941.7			−1.28090			−0.640451	−	0.767999i	$-0.721252\pi$
$743$	−25941.7			−1.28090			−0.640451	+	0.767999i	$0.721252\pi$
$744$		0			0
$745$	2534.50			0.124640
$746$			11119.5i			0.545731i
$747$		0			0
$748$	−13494.1			−0.659617
$749$	−331.353			−0.0161647
$750$		0			0
$751$			27387.3i			1.33073i	0.746520	+	0.665363i	$0.231723\pi$
$751$			27387.3i			1.33073i	−0.746520	+	0.665363i	$0.768277\pi$
$752$		−	561.524i		−	0.0272296i
$753$		0			0
$754$		−	904.019i		−	0.0436637i
$755$	−4423.19			−0.213214
$756$		0			0
$757$	7927.73			0.380632			0.190316	−	0.981723i	$-0.439049\pi$
$757$	7927.73			0.380632			0.190316	+	0.981723i	$0.439049\pi$
$758$		−	2565.29i		−	0.122923i
$759$		0			0
$760$	2927.78	+	1550.00i	0.139739	+	0.0739795i
$761$			16884.6i			0.804291i	0.915576	+	0.402146i	$0.131735\pi$
$761$			16884.6i			0.804291i	−0.915576	+	0.402146i	$0.868265\pi$
$762$		0			0
$763$			9658.82i			0.458287i
$764$		−	5311.95i		−	0.251544i
$765$		0			0
$766$	−20527.1			−0.968244
$767$		−	794.327i		−	0.0373944i
$768$		0			0
$769$	23745.4			1.11350			0.556751	−	0.830680i	$-0.312048\pi$
$769$	23745.4			1.11350			0.556751	+	0.830680i	$0.312048\pi$
$770$	−10199.5			−0.477356
$771$		0			0
$772$		−	13267.0i		−	0.618510i
$773$	35255.4			1.64042			0.820212	−	0.572060i	$-0.193856\pi$
$773$	35255.4			1.64042			0.820212	+	0.572060i	$0.193856\pi$
$774$		0			0
$775$		−	2850.20i		−	0.132106i
$776$		−	11921.8i		−	0.551504i
$777$		0			0
$778$			8164.37i			0.376230i
$779$	−6188.29	+	11689.0i	−0.284620	+	0.537615i
$780$		0			0
$781$			30515.2i			1.39811i
$782$	18443.0			0.843376
$783$		0			0
$784$	6868.22			0.312874
$785$		−	16862.7i		−	0.766695i
$786$		0			0
$787$		−	38401.6i		−	1.73935i	−0.493625	−	0.869675i	$-0.664328\pi$
$787$		−	38401.6i		−	1.73935i	0.493625	−	0.869675i	$-0.335672\pi$
$788$			19728.1i			0.891857i
$789$		0			0
$790$	2706.54			0.121892
$791$	−1347.92			−0.0605898
$792$		0			0
$793$		−	761.980i		−	0.0341219i
$794$	2417.75			0.108064
$795$		0			0
$796$	8357.36			0.372134
$797$	−7919.12			−0.351957			−0.175979	−	0.984394i	$-0.556309\pi$
$797$	−7919.12			−0.351957			−0.175979	+	0.984394i	$0.556309\pi$
$798$		0			0
$799$	3225.80			0.142829
$800$	800.000			0.0353553
$801$		0			0
$802$	25589.5			1.12668
$803$		−	28200.6i		−	1.23932i
$804$		0			0
$805$	13940.1			0.610340
$806$	6913.67			0.302139
$807$		0			0
$808$			6465.93i			0.281523i
$809$		−	26165.6i		−	1.13713i	−0.822640	−	0.568563i	$-0.807500\pi$
$809$		−	26165.6i		−	1.13713i	0.822640	−	0.568563i	$-0.192500\pi$
$810$		0			0
$811$		−	9246.19i		−	0.400342i	−0.979761	−	0.200171i	$-0.935850\pi$
$811$		−	9246.19i		−	0.400342i	0.979761	−	0.200171i	$-0.0641498\pi$
$812$	−1657.09			−0.0716164
$813$		0			0
$814$	21440.3			0.923198
$815$			6227.73i			0.267666i
$816$		0			0
$817$	−15507.4	+	29291.8i	−0.664059	+	1.25434i
$818$		−	27355.6i		−	1.16928i
$819$		0			0
$820$			3193.96i			0.136022i
$821$			43130.6i			1.83346i	0.399509	+	0.916729i	$0.369180\pi$
$821$			43130.6i			1.83346i	−0.399509	+	0.916729i	$0.630820\pi$
$822$		0			0
$823$	32441.5			1.37405			0.687023	−	0.726636i	$-0.258917\pi$
$823$	32441.5			1.37405			0.687023	+	0.726636i	$0.258917\pi$
$824$		−	9891.71i		−	0.418197i
$825$		0			0
$826$	−1456.02			−0.0613335
$827$	−5433.35			−0.228459			−0.114230	−	0.993454i	$-0.536440\pi$
$827$	−5433.35			−0.228459			−0.114230	+	0.993454i	$0.536440\pi$
$828$		0			0
$829$		−	8293.44i		−	0.347459i	−0.984793	−	0.173729i	$-0.944418\pi$
$829$		−	8293.44i		−	0.347459i	0.984793	−	0.173729i	$-0.0555818\pi$
$830$	−3481.71			−0.145605
$831$		0			0
$832$			1940.55i			0.0808610i
$833$			39455.9i			1.64114i
$834$		0			0
$835$			706.499i			0.0292807i
$836$	−10745.7	−	5688.89i	−0.444555	−	0.235352i
$837$		0			0
$838$			14275.8i			0.588482i
$839$	45464.3			1.87080			0.935400	−	0.353592i	$-0.115040\pi$
$839$	45464.3			1.87080			0.935400	+	0.353592i	$0.115040\pi$
$840$		0			0
$841$	−24166.8			−0.990888
$842$			17650.0i			0.722399i
$843$		0			0
$844$			14774.2i			0.602546i
$845$			6388.18i			0.260071i
$846$		0			0
$847$	446.748			0.0181233
$848$	309.461			0.0125318
$849$		0			0
$850$			4595.77i			0.185451i
$851$	−29303.4			−1.18039
$852$		0			0
$853$	−8816.65			−0.353899			−0.176950	−	0.984220i	$-0.556623\pi$
$853$	−8816.65			−0.353899			−0.176950	+	0.984220i	$0.556623\pi$
$854$	−1396.73			−0.0559661
$855$		0			0
$856$	−95.3891			−0.00380880
$857$	11609.5			0.462747			0.231373	−	0.972865i	$-0.425678\pi$
$857$	11609.5			0.462747			0.231373	+	0.972865i	$0.425678\pi$
$858$		0			0
$859$	25640.6			1.01845			0.509223	−	0.860635i	$-0.329933\pi$
$859$	25640.6			1.01845			0.509223	+	0.860635i	$0.329933\pi$
$860$			8003.83i			0.317359i
$861$		0			0
$862$	−28273.6			−1.11717
$863$	31502.5			1.24259			0.621296	−	0.783576i	$-0.286606\pi$
$863$	31502.5			1.24259			0.621296	+	0.783576i	$0.286606\pi$
$864$		0			0
$865$			12698.0i			0.499127i
$866$			34325.8i			1.34692i
$867$		0			0
$868$		−	12672.9i		−	0.495562i
$869$	−9933.69			−0.387776
$870$		0			0
$871$	−18734.4			−0.728806
$872$			2780.55i			0.107983i
$873$		0			0
$874$	14686.6	+	7775.26i	0.568400	+	0.300918i
$875$			3473.70i			0.134209i
$876$		0			0
$877$			27873.1i			1.07321i	0.843833	+	0.536606i	$0.180294\pi$
$877$			27873.1i			1.07321i	−0.843833	+	0.536606i	$0.819706\pi$
$878$			5260.06i			0.202185i
$879$		0			0
$880$	−2936.20			−0.112477
$881$			24632.8i			0.941998i	0.882134	+	0.470999i	$0.156106\pi$
$881$			24632.8i			0.941998i	−0.882134	+	0.470999i	$0.843894\pi$
$882$		0			0
$883$	−11130.4			−0.424199			−0.212100	−	0.977248i	$-0.568030\pi$
$883$	−11130.4			−0.424199			−0.212100	+	0.977248i	$0.568030\pi$
$884$	−11147.9			−0.424145
$885$		0			0
$886$			20543.6i			0.778978i
$887$	36072.2			1.36549			0.682744	−	0.730658i	$-0.260787\pi$
$887$	36072.2			1.36549			0.682744	+	0.730658i	$0.260787\pi$
$888$		0			0
$889$		−	49559.3i		−	1.86970i
$890$			12747.6i			0.480113i
$891$		0			0
$892$			6968.06i			0.261556i
$893$	2568.78	+	1359.94i	0.0962609	+	0.0509616i
$894$		0			0
$895$			6005.36i			0.224287i
$896$	3557.07			0.132627
$897$		0			0
$898$	−6223.06			−0.231254
$899$			1699.57i			0.0630520i
$900$		0			0
$901$			1777.76i			0.0657336i
$902$		−	11722.6i		−	0.432728i
$903$		0			0
$904$	−388.036			−0.0142764
$905$	−8909.97			−0.327268
$906$		0			0
$907$		−	3019.26i		−	0.110532i	−0.998472	−	0.0552662i	$-0.982399\pi$
$907$		−	3019.26i		−	0.110532i	0.998472	−	0.0552662i	$-0.0176007\pi$
$908$	12192.8			0.445630
$909$		0			0
$910$	−8426.10			−0.306948
$911$	40590.3			1.47620			0.738099	−	0.674692i	$-0.235724\pi$
$911$	40590.3			1.47620			0.738099	+	0.674692i	$0.235724\pi$
$912$		0			0
$913$	12778.7			0.463214
$914$	−27794.4			−1.00586
$915$		0			0
$916$	−11018.1			−0.397433
$917$			13878.9i			0.499806i
$918$		0			0
$919$	40269.5			1.44545			0.722725	−	0.691135i	$-0.242889\pi$
$919$	40269.5			1.44545			0.722725	+	0.691135i	$0.242889\pi$
$920$	4013.04			0.143811
$921$		0			0
$922$		−	3247.25i		−	0.115990i
$923$			25209.5i			0.899005i
$924$		0			0
$925$		−	7302.06i		−	0.259557i
$926$	11600.1			0.411667
$927$		0			0
$928$	−477.039			−0.0168745
$929$		−	43791.3i		−	1.54655i	−0.634070	−	0.773275i	$-0.718617\pi$
$929$		−	43791.3i		−	1.54655i	0.634070	−	0.773275i	$-0.281383\pi$
$930$		0			0
$931$	−16634.0	+	31419.7i	−0.585560	+	1.10606i
$932$		−	7653.71i		−	0.268997i
$933$		0			0
$934$		−	3891.24i		−	0.136322i
$935$		−	16867.6i		−	0.589980i
$936$		0			0
$937$	−6593.17			−0.229871			−0.114936	−	0.993373i	$-0.536666\pi$
$937$	−6593.17			−0.229871			−0.114936	+	0.993373i	$0.536666\pi$
$938$			34340.6i			1.19537i
$939$		0			0
$940$	701.905			0.0243549
$941$	26947.0			0.933525			0.466762	−	0.884383i	$-0.345420\pi$
$941$	26947.0			0.933525			0.466762	+	0.884383i	$0.345420\pi$
$942$		0			0
$943$			16021.8i			0.553279i
$944$	−419.155			−0.0144516
$945$		0			0
$946$		−	29376.1i		−	1.00962i
$947$			12761.7i			0.437909i	0.975735	+	0.218955i	$0.0702647\pi$
$947$			12761.7i			0.437909i	−0.975735	+	0.218955i	$0.929735\pi$
$948$		0			0
$949$		−	23297.3i		−	0.796905i
$950$	−1937.50	+	3659.73i	−0.0661693	+	0.124987i
$951$		0			0
$952$			20434.4i			0.695674i
$953$	57227.2			1.94519			0.972597	−	0.232498i	$-0.0746899\pi$
$953$	57227.2			1.94519			0.972597	+	0.232498i	$0.0746899\pi$
$954$		0			0
$955$	6639.94			0.224988
$956$		−	2276.82i		−	0.0770269i
$957$		0			0
$958$			8374.48i			0.282429i
$959$		−	33537.8i		−	1.12929i
$960$		0			0
$961$	16793.2			0.563700
$962$	17712.5			0.593631
$963$		0			0
$964$		−	21307.7i		−	0.711904i
$965$	16583.7			0.553212
$966$		0			0
$967$	29661.3			0.986393			0.493197	−	0.869918i	$-0.335828\pi$
$967$	29661.3			0.986393			0.493197	+	0.869918i	$0.335828\pi$
$968$	128.608			0.00427028
$969$		0			0
$970$	14902.2			0.493280
$971$	−46586.2			−1.53967			−0.769837	−	0.638241i	$-0.779662\pi$
$971$	−46586.2			−1.53967			−0.769837	+	0.638241i	$0.779662\pi$
$972$		0			0
$973$	9490.57			0.312697
$974$			16398.2i			0.539457i
$975$		0			0
$976$	−402.086			−0.0131870
$977$	−6435.36			−0.210732			−0.105366	−	0.994433i	$-0.533601\pi$
$977$	−6435.36			−0.210732			−0.105366	+	0.994433i	$0.533601\pi$
$978$		0			0
$979$		−	46786.9i		−	1.52739i
$980$			8585.27i			0.279843i
$981$		0			0
$982$		−	32375.8i		−	1.05209i
$983$	10380.3			0.336805			0.168403	−	0.985718i	$-0.446139\pi$
$983$	10380.3			0.336805			0.168403	+	0.985718i	$0.446139\pi$
$984$		0			0
$985$	−24660.1			−0.797701
$986$		−	2740.45i		−	0.0885129i
$987$		0			0
$988$	−8877.33	−	4699.76i	−0.285856	−	0.151335i
$989$			40149.6i			1.29088i
$990$		0			0
$991$			18846.9i			0.604127i	0.953288	+	0.302064i	$0.0976755\pi$
$991$			18846.9i			0.604127i	−0.953288	+	0.302064i	$0.902324\pi$
$992$		−	3648.25i		−	0.116766i
$993$		0			0
$994$	46209.7			1.47453
$995$			10446.7i			0.332847i
$996$		0			0
$997$	−26913.9			−0.854936			−0.427468	−	0.904031i	$-0.640594\pi$
$997$	−26913.9			−0.854936			−0.427468	+	0.904031i	$0.640594\pi$
$998$	−21946.9			−0.696109
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1710.4.f.a.341.36	yes	40
3.2	odd	2		1710.4.f.b.341.35	yes	40
19.18	odd	2		1710.4.f.b.341.36	yes	40
57.56	even	2	inner	1710.4.f.a.341.35	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1710.4.f.a.341.35	✓	40	57.56	even	2	inner
1710.4.f.a.341.36	yes	40	1.1	even	1	trivial
1710.4.f.b.341.35	yes	40	3.2	odd	2
1710.4.f.b.341.36	yes	40	19.18	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +5.00000i q^{5} -27.7896 q^{7} -8.00000 q^{8} -10.0000i q^{10} +36.7025i q^{11} +30.3210i q^{13} +55.5793 q^{14} +16.0000 q^{16} +91.9154i q^{17} +(-38.7500 + 73.1945i) q^{19} +20.0000i q^{20} -73.4051i q^{22} +100.326i q^{23} -25.0000 q^{25} -60.6421i q^{26} -111.159 q^{28} +14.9075 q^{29} +114.008i q^{31} -32.0000 q^{32} -183.831i q^{34} -138.948i q^{35} +292.082i q^{37} +(77.5000 - 146.389i) q^{38} -40.0000i q^{40} +159.698 q^{41} +400.192 q^{43} +146.810i q^{44} -200.652i q^{46} -35.0953i q^{47} +429.264 q^{49} +50.0000 q^{50} +121.284i q^{52} +19.3413 q^{53} -183.513 q^{55} +222.317 q^{56} -29.8149 q^{58} -26.1972 q^{59} -25.1304 q^{61} -228.016i q^{62} +64.0000 q^{64} -151.605 q^{65} +617.868i q^{67} +367.662i q^{68} +277.896i q^{70} +831.421 q^{71} -768.355 q^{73} -584.165i q^{74} +(-155.000 + 292.778i) q^{76} -1019.95i q^{77} +270.654i q^{79} +80.0000i q^{80} -319.396 q^{82} -348.171i q^{83} -459.577 q^{85} -800.383 q^{86} -293.620i q^{88} -1274.76 q^{89} -842.610i q^{91} +401.304i q^{92} +70.1905i q^{94} +(-365.973 - 193.750i) q^{95} +1490.22i q^{97} -858.527 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 80 q^{2} + 160 q^{4} - 56 q^{7} - 320 q^{8} + 112 q^{14} + 640 q^{16} - 76 q^{19} - 1000 q^{25} - 224 q^{28} - 120 q^{29} - 1280 q^{32} + 152 q^{38} - 312 q^{41} + 56 q^{43} + 2112 q^{49} + 2000 q^{50}+ \cdots - 4224 q^{98}+O(q^{100}) \) 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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