LMFDB - Embedded newform 1710.4.f.a.341.8

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			341.8
Character	$\chi$	$=$	1710.341
Dual form			1710.4.f.a.341.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} +5.00000i q^{5} +22.4759 q^{7} -8.00000 q^{8} -10.0000i q^{10} -42.4536i q^{11} -2.38543i q^{13} -44.9517 q^{14} +16.0000 q^{16} +29.9073i q^{17} +(-17.5831 - 80.9310i) q^{19} +20.0000i q^{20} +84.9071i q^{22} +200.012i q^{23} -25.0000 q^{25} +4.77086i q^{26} +89.9034 q^{28} -228.522 q^{29} -94.1519i q^{31} -32.0000 q^{32} -59.8147i q^{34} +112.379i q^{35} +325.706i q^{37} +(35.1662 + 161.862i) q^{38} -40.0000i q^{40} +241.016 q^{41} +59.1163 q^{43} -169.814i q^{44} -400.023i q^{46} +341.405i q^{47} +162.164 q^{49} +50.0000 q^{50} -9.54173i q^{52} -327.424 q^{53} +212.268 q^{55} -179.807 q^{56} +457.044 q^{58} +20.0811 q^{59} +571.115 q^{61} +188.304i q^{62} +64.0000 q^{64} +11.9272 q^{65} +230.632i q^{67} +119.629i q^{68} -224.759i q^{70} +97.3627 q^{71} +889.552 q^{73} -651.411i q^{74} +(-70.3324 - 323.724i) q^{76} -954.180i q^{77} +932.003i q^{79} +80.0000i q^{80} -482.032 q^{82} +518.948i q^{83} -149.537 q^{85} -118.233 q^{86} +339.629i q^{88} +304.991 q^{89} -53.6146i q^{91} +800.047i q^{92} -682.810i q^{94} +(404.655 - 87.9155i) q^{95} -451.834i q^{97} -324.328 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	22.4759			1.21358			0.606791	−	0.794861i	$-0.292456\pi$
$7$	22.4759			1.21358			0.606791	+	0.794861i	$0.292456\pi$
$8$	−8.00000			−0.353553
$9$		0			0
$10$		−	10.0000i		−	0.316228i
$11$		−	42.4536i		−	1.16366i	−0.813311	−	0.581829i	$-0.802337\pi$
$11$		−	42.4536i		−	1.16366i	0.813311	−	0.581829i	$-0.197663\pi$
$12$		0			0
$13$		−	2.38543i		−	0.0508923i	−0.999676	−	0.0254461i	$-0.991899\pi$
$13$		−	2.38543i		−	0.0508923i	0.999676	−	0.0254461i	$-0.00810064\pi$
$14$	−44.9517			−0.858132
$15$		0			0
$16$	16.0000			0.250000
$17$			29.9073i			0.426682i	0.976978	+	0.213341i	$0.0684345\pi$
$17$			29.9073i			0.426682i	−0.976978	+	0.213341i	$0.931565\pi$
$18$		0			0
$19$	−17.5831	−	80.9310i	−0.212307	−	0.977203i
$19$	−17.5831	−	80.9310i	−0.212307	−	0.977203i
$20$			20.0000i			0.223607i
$21$		0			0
$22$			84.9071i			0.822830i
$23$			200.012i			1.81327i	0.421911	+	0.906637i	$0.361359\pi$
$23$			200.012i			1.81327i	−0.421911	+	0.906637i	$0.638641\pi$
$24$		0			0
$25$	−25.0000			−0.200000
$26$			4.77086i			0.0359863i
$27$		0			0
$28$	89.9034			0.606791
$29$	−228.522			−1.46329			−0.731646	−	0.681685i	$-0.761247\pi$
$29$	−228.522			−1.46329			−0.731646	+	0.681685i	$0.761247\pi$
$30$		0			0
$31$		−	94.1519i		−	0.545490i	−0.962086	−	0.272745i	$-0.912069\pi$
$31$		−	94.1519i		−	0.545490i	0.962086	−	0.272745i	$-0.0879315\pi$
$32$	−32.0000			−0.176777
$33$		0			0
$34$		−	59.8147i		−	0.301710i
$35$			112.379i			0.542730i
$36$		0			0
$37$			325.706i			1.44718i	0.690230	+	0.723590i	$0.257509\pi$
$37$			325.706i			1.44718i	−0.690230	+	0.723590i	$0.742491\pi$
$38$	35.1662	+	161.862i	0.150124	+	0.690987i
$39$		0			0
$40$		−	40.0000i		−	0.158114i
$41$	241.016			0.918059			0.459029	−	0.888421i	$-0.348197\pi$
$41$	241.016			0.918059			0.459029	+	0.888421i	$0.348197\pi$
$42$		0			0
$43$	59.1163			0.209655			0.104827	−	0.994490i	$-0.466571\pi$
$43$	59.1163			0.209655			0.104827	+	0.994490i	$0.466571\pi$
$44$		−	169.814i		−	0.581829i
$45$		0			0
$46$		−	400.023i		−	1.28218i
$47$			341.405i			1.05955i	0.848137	+	0.529777i	$0.177724\pi$
$47$			341.405i			1.05955i	−0.848137	+	0.529777i	$0.822276\pi$
$48$		0			0
$49$	162.164			0.472782
$50$	50.0000			0.141421
$51$		0			0
$52$		−	9.54173i		−	0.0254461i
$53$	−327.424			−0.848587			−0.424294	−	0.905525i	$-0.639478\pi$
$53$	−327.424			−0.848587			−0.424294	+	0.905525i	$0.639478\pi$
$54$		0			0
$55$	212.268			0.520403
$56$	−179.807			−0.429066
$57$		0			0
$58$	457.044			1.03470
$59$	20.0811			0.0443109			0.0221554	−	0.999755i	$-0.492947\pi$
$59$	20.0811			0.0443109			0.0221554	+	0.999755i	$0.492947\pi$
$60$		0			0
$61$	571.115			1.19875			0.599376	−	0.800468i	$-0.295416\pi$
$61$	571.115			1.19875			0.599376	+	0.800468i	$0.295416\pi$
$62$			188.304i			0.385720i
$63$		0			0
$64$	64.0000			0.125000
$65$	11.9272			0.0227597
$66$		0			0
$67$			230.632i			0.420539i	0.977643	+	0.210270i	$0.0674342\pi$
$67$			230.632i			0.420539i	−0.977643	+	0.210270i	$0.932566\pi$
$68$			119.629i			0.213341i
$69$		0			0
$70$		−	224.759i		−	0.383768i
$71$	97.3627			0.162744			0.0813720	−	0.996684i	$-0.474070\pi$
$71$	97.3627			0.162744			0.0813720	+	0.996684i	$0.474070\pi$
$72$		0			0
$73$	889.552			1.42622			0.713111	−	0.701052i	$-0.247286\pi$
$73$	889.552			1.42622			0.713111	+	0.701052i	$0.247286\pi$
$74$		−	651.411i		−	1.02331i
$75$		0			0
$76$	−70.3324	−	323.724i	−0.106154	−	0.488601i
$77$		−	954.180i		−	1.41219i
$78$		0			0
$79$			932.003i			1.32732i	0.748033	+	0.663661i	$0.230998\pi$
$79$			932.003i			1.32732i	−0.748033	+	0.663661i	$0.769002\pi$
$80$			80.0000i			0.111803i
$81$		0			0
$82$	−482.032			−0.649166
$83$			518.948i			0.686288i	0.939283	+	0.343144i	$0.111492\pi$
$83$			518.948i			0.686288i	−0.939283	+	0.343144i	$0.888508\pi$
$84$		0			0
$85$	−149.537			−0.190818
$86$	−118.233			−0.148248
$87$		0			0
$88$			339.629i			0.411415i
$89$	304.991			0.363247			0.181624	−	0.983368i	$-0.441865\pi$
$89$	304.991			0.363247			0.181624	+	0.983368i	$0.441865\pi$
$90$		0			0
$91$		−	53.6146i		−	0.0617620i
$92$			800.047i			0.906637i
$93$		0			0
$94$		−	682.810i		−	0.749218i
$95$	404.655	−	87.9155i	0.437018	−	0.0949467i
$96$		0			0
$97$		−	451.834i		−	0.472956i	−0.971637	−	0.236478i	$-0.924007\pi$
$97$		−	451.834i		−	0.472956i	0.971637	−	0.236478i	$-0.0759932\pi$
$98$	−324.328			−0.334307
$99$		0			0
$100$	−100.000			−0.100000
$101$			265.567i			0.261632i	0.991407	+	0.130816i	$0.0417598\pi$
$101$			265.567i			0.261632i	−0.991407	+	0.130816i	$0.958240\pi$
$102$		0			0
$103$		−	572.517i		−	0.547687i	−0.961774	−	0.273844i	$-0.911705\pi$
$103$		−	572.517i		−	0.547687i	0.961774	−	0.273844i	$-0.0882951\pi$
$104$			19.0835i			0.0179931i
$105$		0			0
$106$	654.848			0.600042
$107$	−878.043			−0.793305			−0.396653	−	0.917969i	$-0.629828\pi$
$107$	−878.043			−0.793305			−0.396653	+	0.917969i	$0.629828\pi$
$108$		0			0
$109$		−	450.105i		−	0.395525i	−0.980250	−	0.197763i	$-0.936632\pi$
$109$		−	450.105i		−	0.395525i	0.980250	−	0.197763i	$-0.0633675\pi$
$110$	−424.536			−0.367981
$111$		0			0
$112$	359.614			0.303396
$113$	−98.2942			−0.0818295			−0.0409148	−	0.999163i	$-0.513027\pi$
$113$	−98.2942			−0.0818295			−0.0409148	+	0.999163i	$0.513027\pi$
$114$		0			0
$115$	−1000.06			−0.810921
$116$	−914.087			−0.731646
$117$		0			0
$118$	−40.1623			−0.0313325
$119$			672.193i			0.517814i
$120$		0			0
$121$	−471.306			−0.354099
$122$	−1142.23			−0.847645
$123$		0			0
$124$		−	376.608i		−	0.272745i
$125$		−	125.000i		−	0.0894427i
$126$		0			0
$127$		−	1945.78i		−	1.35953i	−0.733432	−	0.679763i	$-0.762083\pi$
$127$		−	1945.78i		−	1.35953i	0.733432	−	0.679763i	$-0.237917\pi$
$128$	−128.000			−0.0883883
$129$		0			0
$130$	−23.8543			−0.0160936
$131$			675.534i			0.450547i	0.974296	+	0.225274i	$0.0723276\pi$
$131$			675.534i			0.450547i	−0.974296	+	0.225274i	$0.927672\pi$
$132$		0			0
$133$	−395.195	−	1818.99i	−0.257652	−	1.18592i
$134$		−	461.263i		−	0.297366i
$135$		0			0
$136$		−	239.259i		−	0.150855i
$137$			2939.09i			1.83287i	0.400181	+	0.916436i	$0.368947\pi$
$137$			2939.09i			1.83287i	−0.400181	+	0.916436i	$0.631053\pi$
$138$		0			0
$139$	−1142.63			−0.697240			−0.348620	−	0.937264i	$-0.613350\pi$
$139$	−1142.63			−0.697240			−0.348620	+	0.937264i	$0.613350\pi$
$140$			449.517i			0.271365i
$141$		0			0
$142$	−194.725			−0.115077
$143$	−101.270			−0.0592212
$144$		0			0
$145$		−	1142.61i		−	0.654404i
$146$	−1779.10			−1.00849
$147$		0			0
$148$			1302.82i			0.723590i
$149$		−	2696.61i		−	1.48265i	−0.671146	−	0.741325i	$-0.734198\pi$
$149$		−	2696.61i		−	1.48265i	0.671146	−	0.741325i	$-0.265802\pi$
$150$		0			0
$151$			2865.80i			1.54447i	0.635336	+	0.772236i	$0.280862\pi$
$151$			2865.80i			1.54447i	−0.635336	+	0.772236i	$0.719138\pi$
$152$	140.665	+	647.448i	0.0750620	+	0.345493i
$153$		0			0
$154$			1908.36i			0.998572i
$155$	470.760			0.243950
$156$		0			0
$157$	−689.498			−0.350497			−0.175248	−	0.984524i	$-0.556073\pi$
$157$	−689.498			−0.350497			−0.175248	+	0.984524i	$0.556073\pi$
$158$		−	1864.01i		−	0.938559i
$159$		0			0
$160$		−	160.000i		−	0.0790569i
$161$			4495.43i			2.20056i
$162$		0			0
$163$	3641.25			1.74972			0.874861	−	0.484373i	$-0.160952\pi$
$163$	3641.25			1.74972			0.874861	+	0.484373i	$0.160952\pi$
$164$	964.065			0.459029
$165$		0			0
$166$		−	1037.90i		−	0.485279i
$167$	1039.34			0.481595			0.240797	−	0.970575i	$-0.422591\pi$
$167$	1039.34			0.481595			0.240797	+	0.970575i	$0.422591\pi$
$168$		0			0
$169$	2191.31			0.997410
$170$	299.073			0.134929
$171$		0			0
$172$	236.465			0.104827
$173$	4156.02			1.82645			0.913227	−	0.407451i	$-0.133582\pi$
$173$	4156.02			1.82645			0.913227	+	0.407451i	$0.133582\pi$
$174$		0			0
$175$	−561.896			−0.242716
$176$		−	679.257i		−	0.290914i
$177$		0			0
$178$	−609.982			−0.256855
$179$	−2417.06			−1.00927			−0.504635	−	0.863333i	$-0.668373\pi$
$179$	−2417.06			−1.00927			−0.504635	+	0.863333i	$0.668373\pi$
$180$		0			0
$181$		−	1971.67i		−	0.809684i	−0.914387	−	0.404842i	$-0.867326\pi$
$181$		−	1971.67i		−	0.809684i	0.914387	−	0.404842i	$-0.132674\pi$
$182$			107.229i			0.0436723i
$183$		0			0
$184$		−	1600.09i		−	0.641089i
$185$	−1628.53			−0.647199
$186$		0			0
$187$	1269.67			0.496512
$188$			1365.62i			0.529777i
$189$		0			0
$190$	−809.310	+	175.831i	−0.309019	+	0.0671375i
$191$			533.346i			0.202050i	0.994884	+	0.101025i	$0.0322122\pi$
$191$			533.346i			0.202050i	−0.994884	+	0.101025i	$0.967788\pi$
$192$		0			0
$193$			824.873i			0.307646i	0.988098	+	0.153823i	$0.0491586\pi$
$193$			824.873i			0.307646i	−0.988098	+	0.153823i	$0.950841\pi$
$194$			903.668i			0.334431i
$195$		0			0
$196$	648.657			0.236391
$197$			2475.48i			0.895283i	0.894213	+	0.447641i	$0.147736\pi$
$197$			2475.48i			0.895283i	−0.894213	+	0.447641i	$0.852264\pi$
$198$		0			0
$199$	2963.14			1.05553			0.527767	−	0.849389i	$-0.323029\pi$
$199$	2963.14			1.05553			0.527767	+	0.849389i	$0.323029\pi$
$200$	200.000			0.0707107
$201$		0			0
$202$		−	531.133i		−	0.185002i
$203$	−5136.22			−1.77582
$204$		0			0
$205$			1205.08i			0.410568i
$206$			1145.03i			0.387273i
$207$		0			0
$208$		−	38.1669i		−	0.0127231i
$209$	−3435.81	+	746.465i	−1.13713	+	0.247053i
$210$		0			0
$211$			4285.91i			1.39836i	0.714944	+	0.699181i	$0.246452\pi$
$211$			4285.91i			1.39836i	−0.714944	+	0.699181i	$0.753548\pi$
$212$	−1309.70			−0.424294
$213$		0			0
$214$	1756.09			0.560951
$215$			295.581i			0.0937604i
$216$		0			0
$217$		−	2116.14i		−	0.661997i
$218$			900.210i			0.279679i
$219$		0			0
$220$	849.071			0.260202
$221$	71.3419			0.0217148
$222$		0			0
$223$		−	128.779i		−	0.0386711i	−0.999813	−	0.0193356i	$-0.993845\pi$
$223$		−	128.779i		−	0.0386711i	0.999813	−	0.0193356i	$-0.00615508\pi$
$224$	−719.227			−0.214533
$225$		0			0
$226$	196.588			0.0578622
$227$	−2591.65			−0.757769			−0.378884	−	0.925444i	$-0.623692\pi$
$227$	−2591.65			−0.757769			−0.378884	+	0.925444i	$0.623692\pi$
$228$		0			0
$229$	2209.90			0.637705			0.318853	−	0.947804i	$-0.396703\pi$
$229$	2209.90			0.637705			0.318853	+	0.947804i	$0.396703\pi$
$230$	2000.12			0.573408
$231$		0			0
$232$	1828.17			0.517352
$233$			2329.46i			0.654970i	0.944856	+	0.327485i	$0.106201\pi$
$233$			2329.46i			0.654970i	−0.944856	+	0.327485i	$0.893799\pi$
$234$		0			0
$235$	−1707.03			−0.473847
$236$	80.3246			0.0221554
$237$		0			0
$238$		−	1344.39i		−	0.366150i
$239$			1301.18i			0.352161i	0.984376	+	0.176080i	$0.0563418\pi$
$239$			1301.18i			0.352161i	−0.984376	+	0.176080i	$0.943658\pi$
$240$		0			0
$241$		−	1772.29i		−	0.473708i	−0.971545	−	0.236854i	$-0.923884\pi$
$241$		−	1772.29i		−	0.473708i	0.971545	−	0.236854i	$-0.0761162\pi$
$242$	942.611			0.250386
$243$		0			0
$244$	2284.46			0.599376
$245$			810.821i			0.211434i
$246$		0			0
$247$	−193.056	+	41.9433i	−0.0497321	+	0.0108048i
$248$			753.215i			0.192860i
$249$		0			0
$250$			250.000i			0.0632456i
$251$		−	830.188i		−	0.208769i	−0.994537	−	0.104385i	$-0.966713\pi$
$251$		−	830.188i		−	0.208769i	0.994537	−	0.104385i	$-0.0332873\pi$
$252$		0			0
$253$	8491.21			2.11003
$254$			3891.55i			0.961329i
$255$		0			0
$256$	256.000			0.0625000
$257$	3792.51			0.920508			0.460254	−	0.887787i	$-0.347758\pi$
$257$	3792.51			0.920508			0.460254	+	0.887787i	$0.347758\pi$
$258$		0			0
$259$			7320.51i			1.75627i
$260$	47.7086			0.0113799
$261$		0			0
$262$		−	1351.07i		−	0.318585i
$263$			1337.84i			0.313669i	0.987625	+	0.156835i	$0.0501290\pi$
$263$			1337.84i			0.313669i	−0.987625	+	0.156835i	$0.949871\pi$
$264$		0			0
$265$		−	1637.12i		−	0.379500i
$266$	790.390	+	3637.99i	0.182188	+	0.838569i
$267$		0			0
$268$			922.526i			0.210270i
$269$	584.117			0.132395			0.0661975	−	0.997807i	$-0.478913\pi$
$269$	584.117			0.132395			0.0661975	+	0.997807i	$0.478913\pi$
$270$		0			0
$271$	2899.39			0.649909			0.324955	−	0.945730i	$-0.394651\pi$
$271$	2899.39			0.649909			0.324955	+	0.945730i	$0.394651\pi$
$272$			478.517i			0.106671i
$273$		0			0
$274$		−	5878.18i		−	1.29604i
$275$			1061.34i			0.232732i
$276$		0			0
$277$	−3606.86			−0.782366			−0.391183	−	0.920313i	$-0.627934\pi$
$277$	−3606.86			−0.782366			−0.391183	+	0.920313i	$0.627934\pi$
$278$	2285.25			0.493023
$279$		0			0
$280$		−	899.034i		−	0.191884i
$281$	3157.73			0.670370			0.335185	−	0.942152i	$-0.391201\pi$
$281$	3157.73			0.670370			0.335185	+	0.942152i	$0.391201\pi$
$282$		0			0
$283$	−3538.83			−0.743328			−0.371664	−	0.928367i	$-0.621213\pi$
$283$	−3538.83			−0.743328			−0.371664	+	0.928367i	$0.621213\pi$
$284$	389.451			0.0813720
$285$		0			0
$286$	202.540			0.0418757
$287$	5417.05			1.11414
$288$		0			0
$289$	4018.55			0.817942
$290$			2285.22i			0.462733i
$291$		0			0
$292$	3558.21			0.713111
$293$	−3466.47			−0.691172			−0.345586	−	0.938387i	$-0.612320\pi$
$293$	−3466.47			−0.691172			−0.345586	+	0.938387i	$0.612320\pi$
$294$		0			0
$295$			100.406i			0.0198164i
$296$		−	2605.64i		−	0.511655i
$297$		0			0
$298$			5393.22i			1.04839i
$299$	477.114			0.0922817
$300$		0			0
$301$	1328.69			0.254433
$302$		−	5731.59i		−	1.09211i
$303$		0			0
$304$	−281.330	−	1294.90i	−0.0530768	−	0.244301i
$305$			2855.58i			0.536098i
$306$		0			0
$307$			6671.05i			1.24018i	0.784529	+	0.620092i	$0.212905\pi$
$307$			6671.05i			1.24018i	−0.784529	+	0.620092i	$0.787095\pi$
$308$		−	3816.72i		−	0.706097i
$309$		0			0
$310$	−941.519			−0.172499
$311$		−	861.554i		−	0.157088i	−0.996911	−	0.0785438i	$-0.974973\pi$
$311$		−	861.554i		−	0.157088i	0.996911	−	0.0785438i	$-0.0250271\pi$
$312$		0			0
$313$	3303.29			0.596528			0.298264	−	0.954483i	$-0.403592\pi$
$313$	3303.29			0.596528			0.298264	+	0.954483i	$0.403592\pi$
$314$	1379.00			0.247838
$315$		0			0
$316$			3728.01i			0.663661i
$317$	1675.68			0.296895			0.148448	−	0.988920i	$-0.452572\pi$
$317$	1675.68			0.296895			0.148448	+	0.988920i	$0.452572\pi$
$318$		0			0
$319$			9701.57i			1.70277i
$320$			320.000i			0.0559017i
$321$		0			0
$322$		−	8990.87i		−	1.55603i
$323$	2420.43	−	525.864i	0.416955	−	0.0905877i
$324$		0			0
$325$			59.6358i			0.0101785i
$326$	−7282.50			−1.23724
$327$		0			0
$328$	−1928.13			−0.324583
$329$			7673.37i			1.28586i
$330$		0			0
$331$		−	373.893i		−	0.0620876i	−0.999518	−	0.0310438i	$-0.990117\pi$
$331$		−	373.893i		−	0.0620876i	0.999518	−	0.0310438i	$-0.00988314\pi$
$332$			2075.79i			0.343144i
$333$		0			0
$334$	−2078.67			−0.340539
$335$	−1153.16			−0.188071
$336$		0			0
$337$			7012.92i			1.13359i	0.823861	+	0.566793i	$0.191816\pi$
$337$			7012.92i			1.13359i	−0.823861	+	0.566793i	$0.808184\pi$
$338$	−4382.62			−0.705275
$339$		0			0
$340$	−598.147			−0.0954090
$341$	−3997.08			−0.634763
$342$		0			0
$343$	−4064.44			−0.639823
$344$	−472.930			−0.0741241
$345$		0			0
$346$	−8312.05			−1.29150
$347$			7327.96i			1.13368i	0.823829	+	0.566838i	$0.191833\pi$
$347$			7327.96i			1.13368i	−0.823829	+	0.566838i	$0.808167\pi$
$348$		0			0
$349$	−8323.44			−1.27663			−0.638315	−	0.769776i	$-0.720368\pi$
$349$	−8323.44			−1.27663			−0.638315	+	0.769776i	$0.720368\pi$
$350$	1123.79			0.171626
$351$		0			0
$352$			1358.51i			0.205708i
$353$		−	2261.55i		−	0.340992i	−0.985358	−	0.170496i	$-0.945463\pi$
$353$		−	2261.55i		−	0.340992i	0.985358	−	0.170496i	$-0.0545370\pi$
$354$		0			0
$355$			486.814i			0.0727813i
$356$	1219.96			0.181624
$357$		0			0
$358$	4834.12			0.713662
$359$			4750.84i			0.698439i	0.937041	+	0.349220i	$0.113553\pi$
$359$			4750.84i			0.698439i	−0.937041	+	0.349220i	$0.886447\pi$
$360$		0			0
$361$	−6240.67	+	2846.04i	−0.909851	+	0.414935i
$362$			3943.33i			0.572533i
$363$		0			0
$364$		−	214.459i		−	0.0308810i
$365$			4447.76i			0.637825i
$366$		0			0
$367$	9673.76			1.37593			0.687965	−	0.725744i	$-0.258504\pi$
$367$	9673.76			1.37593			0.687965	+	0.725744i	$0.258504\pi$
$368$			3200.19i			0.453319i
$369$		0			0
$370$	3257.06			0.457638
$371$	−7359.13			−1.02983
$372$		0			0
$373$			6848.65i			0.950697i	0.879798	+	0.475348i	$0.157678\pi$
$373$			6848.65i			0.950697i	−0.879798	+	0.475348i	$0.842322\pi$
$374$	−2539.35			−0.351087
$375$		0			0
$376$		−	2731.24i		−	0.374609i
$377$			545.123i			0.0744702i
$378$		0			0
$379$			9697.05i			1.31426i	0.753778	+	0.657129i	$0.228229\pi$
$379$			9697.05i			1.31426i	−0.753778	+	0.657129i	$0.771771\pi$
$380$	1618.62	−	351.662i	0.218509	−	0.0474734i
$381$		0			0
$382$		−	1066.69i		−	0.142871i
$383$	13738.5			1.83292			0.916458	−	0.400132i	$-0.131036\pi$
$383$	13738.5			1.83292			0.916458	+	0.400132i	$0.131036\pi$
$384$		0			0
$385$	4770.90			0.631552
$386$		−	1649.75i		−	0.217539i
$387$		0			0
$388$		−	1807.34i		−	0.236478i
$389$		−	5890.69i		−	0.767789i	−0.923377	−	0.383894i	$-0.874583\pi$
$389$		−	5890.69i		−	0.767789i	0.923377	−	0.383894i	$-0.125417\pi$
$390$		0			0
$391$	−5981.82			−0.773692
$392$	−1297.31			−0.167154
$393$		0			0
$394$		−	4950.96i		−	0.633060i
$395$	−4660.01			−0.593597
$396$		0			0
$397$	−3831.67			−0.484398			−0.242199	−	0.970227i	$-0.577869\pi$
$397$	−3831.67			−0.484398			−0.242199	+	0.970227i	$0.577869\pi$
$398$	−5926.28			−0.746376
$399$		0			0
$400$	−400.000			−0.0500000
$401$	11464.7			1.42773			0.713867	−	0.700282i	$-0.246942\pi$
$401$	11464.7			1.42773			0.713867	+	0.700282i	$0.246942\pi$
$402$		0			0
$403$	−224.593			−0.0277612
$404$			1062.27i			0.130816i
$405$		0			0
$406$	10272.4			1.25570
$407$	13827.4			1.68402
$408$		0			0
$409$		−	8708.80i		−	1.05287i	−0.850216	−	0.526433i	$-0.823529\pi$
$409$		−	8708.80i		−	1.05287i	0.850216	−	0.526433i	$-0.176471\pi$
$410$		−	2410.16i		−	0.290316i
$411$		0			0
$412$		−	2290.07i		−	0.273844i
$413$	451.341			0.0537749
$414$		0			0
$415$	−2594.74			−0.306917
$416$			76.3338i			0.00899657i
$417$		0			0
$418$	6871.62	−	1492.93i	0.804072	−	0.174693i
$419$			5333.99i			0.621915i	0.950424	+	0.310957i	$0.100650\pi$
$419$			5333.99i			0.621915i	−0.950424	+	0.310957i	$0.899350\pi$
$420$		0			0
$421$			9410.70i			1.08943i	0.838622	+	0.544715i	$0.183362\pi$
$421$			9410.70i			1.08943i	−0.838622	+	0.544715i	$0.816638\pi$
$422$		−	8571.83i		−	0.988792i
$423$		0			0
$424$	2619.39			0.300021
$425$		−	747.683i		−	0.0853364i
$426$		0			0
$427$	12836.3			1.45478
$428$	−3512.17			−0.396653
$429$		0			0
$430$		−	591.163i		−	0.0662986i
$431$	5216.40			0.582982			0.291491	−	0.956574i	$-0.405849\pi$
$431$	5216.40			0.582982			0.291491	+	0.956574i	$0.405849\pi$
$432$		0			0
$433$			12148.2i			1.34828i	0.738603	+	0.674141i	$0.235486\pi$
$433$			12148.2i			1.34828i	−0.738603	+	0.674141i	$0.764514\pi$
$434$			4232.29i			0.468102i
$435$		0			0
$436$		−	1800.42i		−	0.197763i
$437$	16187.2	−	3516.82i	1.77194	−	0.384971i
$438$		0			0
$439$			5737.96i			0.623822i	0.950111	+	0.311911i	$0.100969\pi$
$439$			5737.96i			0.623822i	−0.950111	+	0.311911i	$0.899031\pi$
$440$	−1698.14			−0.183990
$441$		0			0
$442$	−142.684			−0.0153547
$443$		−	1622.04i		−	0.173963i	−0.996210	−	0.0869816i	$-0.972278\pi$
$443$		−	1622.04i		−	0.173963i	0.996210	−	0.0869816i	$-0.0277221\pi$
$444$		0			0
$445$			1524.96i			0.162449i
$446$			257.557i			0.0273446i
$447$		0			0
$448$	1438.45			0.151698
$449$	−17823.7			−1.87339			−0.936693	−	0.350152i	$-0.886130\pi$
$449$	−17823.7			−1.87339			−0.936693	+	0.350152i	$0.886130\pi$
$450$		0			0
$451$		−	10232.0i		−	1.06831i
$452$	−393.177			−0.0409148
$453$		0			0
$454$	5183.29			0.535824
$455$	268.073			0.0276208
$456$		0			0
$457$	4786.20			0.489911			0.244955	−	0.969534i	$-0.421227\pi$
$457$	4786.20			0.489911			0.244955	+	0.969534i	$0.421227\pi$
$458$	−4419.81			−0.450926
$459$		0			0
$460$	−4000.23			−0.405460
$461$			5467.18i			0.552347i	0.961108	+	0.276173i	$0.0890664\pi$
$461$			5467.18i			0.552347i	−0.961108	+	0.276173i	$0.910934\pi$
$462$		0			0
$463$	−1702.99			−0.170939			−0.0854695	−	0.996341i	$-0.527239\pi$
$463$	−1702.99			−0.170939			−0.0854695	+	0.996341i	$0.527239\pi$
$464$	−3656.35			−0.365823
$465$		0			0
$466$		−	4658.92i		−	0.463134i
$467$			4063.14i			0.402612i	0.979528	+	0.201306i	$0.0645186\pi$
$467$			4063.14i			0.402612i	−0.979528	+	0.201306i	$0.935481\pi$
$468$		0			0
$469$			5183.64i			0.510359i
$470$	3414.05			0.335060
$471$		0			0
$472$	−160.649			−0.0156663
$473$		−	2509.70i		−	0.243966i
$474$		0			0
$475$	439.577	+	2023.28i	0.0424615	+	0.195441i
$476$			2688.77i			0.258907i
$477$		0			0
$478$		−	2602.36i		−	0.249015i
$479$			19517.8i			1.86177i	0.365307	+	0.930887i	$0.380964\pi$
$479$			19517.8i			1.86177i	−0.365307	+	0.930887i	$0.619036\pi$
$480$		0			0
$481$	776.948			0.0736503
$482$			3544.59i			0.334962i
$483$		0			0
$484$	−1885.22			−0.177049
$485$	2259.17			0.211513
$486$		0			0
$487$			3339.95i			0.310775i	0.987854	+	0.155388i	$0.0496626\pi$
$487$			3339.95i			0.310775i	−0.987854	+	0.155388i	$0.950337\pi$
$488$	−4568.92			−0.423822
$489$		0			0
$490$		−	1621.64i		−	0.149507i
$491$		−	4937.72i		−	0.453842i	−0.973913	−	0.226921i	$-0.927134\pi$
$491$		−	4937.72i		−	0.453842i	0.973913	−	0.226921i	$-0.0728659\pi$
$492$		0			0
$493$		−	6834.48i		−	0.624360i
$494$	386.111	−	83.8866i	0.0351659	−	0.00764015i
$495$		0			0
$496$		−	1506.43i		−	0.136372i
$497$	2188.31			0.197503
$498$		0			0
$499$	−17209.5			−1.54389			−0.771946	−	0.635688i	$-0.780716\pi$
$499$	−17209.5			−1.54389			−0.771946	+	0.635688i	$0.780716\pi$
$500$		−	500.000i		−	0.0447214i
$501$		0			0
$502$			1660.38i			0.147622i
$503$		−	10473.6i		−	0.928418i	−0.885726	−	0.464209i	$-0.846339\pi$
$503$		−	10473.6i		−	0.928418i	0.885726	−	0.464209i	$-0.153661\pi$
$504$		0			0
$505$	−1327.83			−0.117006
$506$	−16982.4			−1.49202
$507$		0			0
$508$		−	7783.10i		−	0.679763i
$509$	2031.80			0.176931			0.0884657	−	0.996079i	$-0.471804\pi$
$509$	2031.80			0.176931			0.0884657	+	0.996079i	$0.471804\pi$
$510$		0			0
$511$	19993.4			1.73084
$512$	−512.000			−0.0441942
$513$		0			0
$514$	−7585.03			−0.650897
$515$	2862.59			0.244933
$516$		0			0
$517$	14493.9			1.23296
$518$		−	14641.0i		−	1.24187i
$519$		0			0
$520$	−95.4173			−0.00804678
$521$	12633.1			1.06232			0.531159	−	0.847272i	$-0.321757\pi$
$521$	12633.1			1.06232			0.531159	+	0.847272i	$0.321757\pi$
$522$		0			0
$523$		−	3942.96i		−	0.329663i	−0.986322	−	0.164831i	$-0.947292\pi$
$523$		−	3942.96i		−	0.329663i	0.986322	−	0.164831i	$-0.0527080\pi$
$524$			2702.14i			0.225274i
$525$		0			0
$526$		−	2675.69i		−	0.221798i
$527$	2815.83			0.232751
$528$		0			0
$529$	−27837.7			−2.28796
$530$			3274.24i			0.268347i
$531$		0			0
$532$	−1580.78	−	7275.98i	−0.128826	−	0.592958i
$533$		−	574.928i		−	0.0467221i
$534$		0			0
$535$		−	4390.22i		−	0.354777i
$536$		−	1845.05i		−	0.148683i
$537$		0			0
$538$	−1168.23			−0.0936174
$539$		−	6884.45i		−	0.550156i
$540$		0			0
$541$	−9920.35			−0.788372			−0.394186	−	0.919031i	$-0.628973\pi$
$541$	−9920.35			−0.788372			−0.394186	+	0.919031i	$0.628973\pi$
$542$	−5798.78			−0.459555
$543$		0			0
$544$		−	957.035i		−	0.0754274i
$545$	2250.53			0.176884
$546$		0			0
$547$			6562.70i			0.512982i	0.966547	+	0.256491i	$0.0825664\pi$
$547$			6562.70i			0.512982i	−0.966547	+	0.256491i	$0.917434\pi$
$548$			11756.4i			0.916436i
$549$		0			0
$550$		−	2122.68i		−	0.164566i
$551$	4018.12	+	18494.5i	0.310667	+	1.42993i
$552$		0			0
$553$			20947.6i			1.61082i
$554$	7213.73			0.553217
$555$		0			0
$556$	−4570.51			−0.348620
$557$			4222.81i			0.321232i	0.987017	+	0.160616i	$0.0513480\pi$
$557$			4222.81i			0.321232i	−0.987017	+	0.160616i	$0.948652\pi$
$558$		0			0
$559$		−	141.018i		−	0.0106698i
$560$			1798.07i			0.135683i
$561$		0			0
$562$	−6315.45			−0.474023
$563$	−3323.03			−0.248755			−0.124377	−	0.992235i	$-0.539693\pi$
$563$	−3323.03			−0.248755			−0.124377	+	0.992235i	$0.539693\pi$
$564$		0			0
$565$		−	491.471i		−	0.0365953i
$566$	7077.66			0.525612
$567$		0			0
$568$	−778.902			−0.0575387
$569$	−20572.3			−1.51570			−0.757852	−	0.652427i	$-0.773751\pi$
$569$	−20572.3			−1.51570			−0.757852	+	0.652427i	$0.773751\pi$
$570$		0			0
$571$	6592.85			0.483191			0.241596	−	0.970377i	$-0.422329\pi$
$571$	6592.85			0.483191			0.241596	+	0.970377i	$0.422329\pi$
$572$	−405.080			−0.0296106
$573$		0			0
$574$	−10834.1			−0.787816
$575$		−	5000.29i		−	0.362655i
$576$		0			0
$577$	4473.99			0.322799			0.161399	−	0.986889i	$-0.448399\pi$
$577$	4473.99			0.322799			0.161399	+	0.986889i	$0.448399\pi$
$578$	−8037.10			−0.578373
$579$		0			0
$580$		−	4570.44i		−	0.327202i
$581$			11663.8i			0.832867i
$582$		0			0
$583$			13900.3i			0.987465i
$584$	−7116.41			−0.504245
$585$		0			0
$586$	6932.94			0.488732
$587$		−	8353.93i		−	0.587399i	−0.955898	−	0.293700i	$-0.905113\pi$
$587$		−	8353.93i		−	0.587399i	0.955898	−	0.293700i	$-0.0948866\pi$
$588$		0			0
$589$	−7619.81	+	1655.48i	−0.533054	+	0.115811i
$590$		−	200.811i		−	0.0140123i
$591$		0			0
$592$			5211.29i			0.361795i
$593$		−	18308.0i		−	1.26782i	−0.773406	−	0.633911i	$-0.781449\pi$
$593$		−	18308.0i		−	1.26782i	0.773406	−	0.633911i	$-0.218551\pi$
$594$		0			0
$595$	−3360.97			−0.231573
$596$		−	10786.4i		−	0.741325i
$597$		0			0
$598$	−954.228			−0.0652530
$599$	21546.2			1.46971			0.734853	−	0.678226i	$-0.237251\pi$
$599$	21546.2			1.46971			0.734853	+	0.678226i	$0.237251\pi$
$600$		0			0
$601$		−	316.494i		−	0.0214810i	−0.999942	−	0.0107405i	$-0.996581\pi$
$601$		−	316.494i		−	0.0214810i	0.999942	−	0.0107405i	$-0.00341887\pi$
$602$	−2657.38			−0.179911
$603$		0			0
$604$			11463.2i			0.772236i
$605$		−	2356.53i		−	0.158358i
$606$		0			0
$607$		−	20223.3i		−	1.35229i	−0.736769	−	0.676144i	$-0.763650\pi$
$607$		−	20223.3i		−	1.35229i	0.736769	−	0.676144i	$-0.236350\pi$
$608$	562.659	+	2589.79i	0.0375310	+	0.172747i
$609$		0			0
$610$		−	5711.15i		−	0.379078i
$611$	814.399			0.0539231
$612$		0			0
$613$	−3531.01			−0.232653			−0.116327	−	0.993211i	$-0.537112\pi$
$613$	−3531.01			−0.232653			−0.116327	+	0.993211i	$0.537112\pi$
$614$		−	13342.1i		−	0.876943i
$615$		0			0
$616$			7633.44i			0.499286i
$617$		−	23322.1i		−	1.52174i	−0.648906	−	0.760868i	$-0.724773\pi$
$617$		−	23322.1i		−	1.52174i	0.648906	−	0.760868i	$-0.275227\pi$
$618$		0			0
$619$	−22505.5			−1.46134			−0.730672	−	0.682728i	$-0.760793\pi$
$619$	−22505.5			−1.46134			−0.730672	+	0.682728i	$0.760793\pi$
$620$	1883.04			0.121975
$621$		0			0
$622$			1723.11i			0.111078i
$623$	6854.94			0.440830
$624$		0			0
$625$	625.000			0.0400000
$626$	−6606.59			−0.421809
$627$		0			0
$628$	−2757.99			−0.175248
$629$	−9740.98			−0.617486
$630$		0			0
$631$	−10839.7			−0.683868			−0.341934	−	0.939724i	$-0.611082\pi$
$631$	−10839.7			−0.683868			−0.341934	+	0.939724i	$0.611082\pi$
$632$		−	7456.02i		−	0.469279i
$633$		0			0
$634$	−3351.37			−0.209937
$635$	9728.88			0.607998
$636$		0			0
$637$		−	386.832i		−	0.0240610i
$638$		−	19403.1i		−	1.20404i
$639$		0			0
$640$		−	640.000i		−	0.0395285i
$641$	23380.1			1.44065			0.720325	−	0.693636i	$-0.243992\pi$
$641$	23380.1			1.44065			0.720325	+	0.693636i	$0.243992\pi$
$642$		0			0
$643$	−16998.5			−1.04255			−0.521273	−	0.853390i	$-0.674543\pi$
$643$	−16998.5			−1.04255			−0.521273	+	0.853390i	$0.674543\pi$
$644$			17981.7i			1.10028i
$645$		0			0
$646$	−4840.86	+	1051.73i	−0.294832	+	0.0640552i
$647$		−	30776.4i		−	1.87009i	−0.354536	−	0.935043i	$-0.615361\pi$
$647$		−	30776.4i		−	1.87009i	0.354536	−	0.935043i	$-0.384639\pi$
$648$		0			0
$649$		−	852.516i		−	0.0515627i
$650$		−	119.272i		−	0.00719726i
$651$		0			0
$652$	14565.0			0.874861
$653$			17565.5i			1.05267i	0.850279	+	0.526333i	$0.176433\pi$
$653$			17565.5i			1.05267i	−0.850279	+	0.526333i	$0.823567\pi$
$654$		0			0
$655$	−3377.67			−0.201491
$656$	3856.26			0.229515
$657$		0			0
$658$		−	15346.7i		−	0.909237i
$659$	−11056.4			−0.653560			−0.326780	−	0.945100i	$-0.605964\pi$
$659$	−11056.4			−0.653560			−0.326780	+	0.945100i	$0.605964\pi$
$660$		0			0
$661$			17060.3i			1.00389i	0.864900	+	0.501944i	$0.167382\pi$
$661$			17060.3i			1.00389i	−0.864900	+	0.501944i	$0.832618\pi$
$662$			747.785i			0.0439026i
$663$		0			0
$664$		−	4151.58i		−	0.242640i
$665$	9094.97	−	1975.98i	0.530358	−	0.115226i
$666$		0			0
$667$		−	45707.0i		−	2.65335i
$668$	4157.35			0.240797
$669$		0			0
$670$	2306.32			0.132986
$671$		−	24245.9i		−	1.39494i
$672$		0			0
$673$			16848.9i			0.965046i	0.875883	+	0.482523i	$0.160280\pi$
$673$			16848.9i			0.965046i	−0.875883	+	0.482523i	$0.839720\pi$
$674$		−	14025.8i		−	0.801566i
$675$		0			0
$676$	8765.24			0.498705
$677$	−26123.6			−1.48303			−0.741514	−	0.670937i	$-0.765892\pi$
$677$	−26123.6			−1.48303			−0.741514	+	0.670937i	$0.765892\pi$
$678$		0			0
$679$		−	10155.4i		−	0.573972i
$680$	1196.29			0.0674644
$681$		0			0
$682$	7994.17			0.448845
$683$	−24701.4			−1.38385			−0.691927	−	0.721967i	$-0.743238\pi$
$683$	−24701.4			−1.38385			−0.691927	+	0.721967i	$0.743238\pi$
$684$		0			0
$685$	−14695.5			−0.819685
$686$	8128.88			0.452423
$687$		0			0
$688$	945.861			0.0524137
$689$			781.048i			0.0431866i
$690$		0			0
$691$	3254.24			0.179156			0.0895781	−	0.995980i	$-0.471448\pi$
$691$	3254.24			0.179156			0.0895781	+	0.995980i	$0.471448\pi$
$692$	16624.1			0.913227
$693$		0			0
$694$		−	14655.9i		−	0.801630i
$695$		−	5713.13i		−	0.311815i
$696$		0			0
$697$			7208.15i			0.391719i
$698$	16646.9			0.902713
$699$		0			0
$700$	−2247.59			−0.121358
$701$		−	23601.1i		−	1.27161i	−0.771849	−	0.635806i	$-0.780668\pi$
$701$		−	23601.1i		−	1.27161i	0.771849	−	0.635806i	$-0.219332\pi$
$702$		0			0
$703$	26359.7	−	5726.91i	1.41419	−	0.307247i
$704$		−	2717.03i		−	0.145457i
$705$		0			0
$706$			4523.10i			0.241118i
$707$			5968.84i			0.317512i
$708$		0			0
$709$	−3542.29			−0.187635			−0.0938176	−	0.995589i	$-0.529907\pi$
$709$	−3542.29			−0.187635			−0.0938176	+	0.995589i	$0.529907\pi$
$710$		−	973.627i		−	0.0514642i
$711$		0			0
$712$	−2439.93			−0.128427
$713$	18831.5			0.989123
$714$		0			0
$715$		−	506.351i		−	0.0264845i
$716$	−9668.24			−0.504635
$717$		0			0
$718$		−	9501.68i		−	0.493871i
$719$		−	34404.2i		−	1.78451i	−0.451536	−	0.892253i	$-0.649124\pi$
$719$		−	34404.2i		−	1.78451i	0.451536	−	0.892253i	$-0.350876\pi$
$720$		0			0
$721$		−	12867.8i		−	0.664664i
$722$	12481.3	−	5692.07i	0.643362	−	0.293403i
$723$		0			0
$724$		−	7886.67i		−	0.404842i
$725$	5713.05			0.292658
$726$		0			0
$727$	−26173.7			−1.33525			−0.667626	−	0.744497i	$-0.732689\pi$
$727$	−26173.7			−1.33525			−0.667626	+	0.744497i	$0.732689\pi$
$728$			428.917i			0.0218362i
$729$		0			0
$730$		−	8895.52i		−	0.451011i
$731$			1768.01i			0.0894559i
$732$		0			0
$733$	30180.2			1.52078			0.760391	−	0.649466i	$-0.225007\pi$
$733$	30180.2			1.52078			0.760391	+	0.649466i	$0.225007\pi$
$734$	−19347.5			−0.972929
$735$		0			0
$736$		−	6400.37i		−	0.320545i
$737$	9791.13			0.489364
$738$		0			0
$739$	−28512.5			−1.41928			−0.709641	−	0.704563i	$-0.751143\pi$
$739$	−28512.5			−1.41928			−0.709641	+	0.704563i	$0.751143\pi$
$740$	−6514.11			−0.323599
$741$		0			0
$742$	14718.3			0.728200
$743$	−27376.4			−1.35174			−0.675871	−	0.737020i	$-0.736232\pi$
$743$	−27376.4			−1.35174			−0.675871	+	0.737020i	$0.736232\pi$
$744$		0			0
$745$	13483.0			0.663061
$746$		−	13697.3i		−	0.672244i
$747$		0			0
$748$	5078.69			0.248256
$749$	−19734.8			−0.962741
$750$		0			0
$751$		−	17264.9i		−	0.838889i	−0.907781	−	0.419445i	$-0.862225\pi$
$751$		−	17264.9i		−	0.838889i	0.907781	−	0.419445i	$-0.137775\pi$
$752$			5462.48i			0.264889i
$753$		0			0
$754$		−	1090.25i		−	0.0526584i
$755$	−14329.0			−0.690709
$756$		0			0
$757$	612.504			0.0294080			0.0147040	−	0.999892i	$-0.495319\pi$
$757$	612.504			0.0294080			0.0147040	+	0.999892i	$0.495319\pi$
$758$		−	19394.1i		−	0.929321i
$759$		0			0
$760$	−3237.24	+	703.324i	−0.154509	+	0.0335687i
$761$		−	38634.7i		−	1.84035i	−0.391505	−	0.920176i	$-0.628046\pi$
$761$		−	38634.7i		−	1.84035i	0.391505	−	0.920176i	$-0.371954\pi$
$762$		0			0
$763$		−	10116.5i		−	0.480002i
$764$			2133.39i			0.101025i
$765$		0			0
$766$	−27477.1			−1.29607
$767$		−	47.9022i		−	0.00225508i
$768$		0			0
$769$	−25600.5			−1.20049			−0.600246	−	0.799815i	$-0.704931\pi$
$769$	−25600.5			−1.20049			−0.600246	+	0.799815i	$0.704931\pi$
$770$	−9541.80			−0.446575
$771$		0			0
$772$			3299.49i			0.153823i
$773$	−5903.06			−0.274668			−0.137334	−	0.990525i	$-0.543853\pi$
$773$	−5903.06			−0.274668			−0.137334	+	0.990525i	$0.543853\pi$
$774$		0			0
$775$			2353.80i			0.109098i
$776$			3614.67i			0.167215i
$777$		0			0
$778$			11781.4i			0.542909i
$779$	−4237.81	−	19505.7i	−0.194911	−	0.897130i
$780$		0			0
$781$		−	4133.39i		−	0.189378i
$782$	11963.6			0.547083
$783$		0			0
$784$	2594.63			0.118195
$785$		−	3447.49i		−	0.156747i
$786$		0			0
$787$		−	23987.9i		−	1.08650i	−0.839571	−	0.543250i	$-0.817194\pi$
$787$		−	23987.9i		−	1.08650i	0.839571	−	0.543250i	$-0.182806\pi$
$788$			9901.92i			0.447641i
$789$		0			0
$790$	9320.03			0.419736
$791$	−2209.25			−0.0993069
$792$		0			0
$793$		−	1362.36i		−	0.0610072i
$794$	7663.34			0.342521
$795$		0			0
$796$	11852.6			0.527767
$797$	35133.4			1.56147			0.780733	−	0.624865i	$-0.214846\pi$
$797$	35133.4			1.56147			0.780733	+	0.624865i	$0.214846\pi$
$798$		0			0
$799$	−10210.5			−0.452093
$800$	800.000			0.0353553
$801$		0			0
$802$	−22929.5			−1.00956
$803$		−	37764.6i		−	1.65963i
$804$		0			0
$805$	−22477.2			−0.984119
$806$	449.186			0.0196302
$807$		0			0
$808$		−	2124.53i		−	0.0925010i
$809$			20827.9i			0.905156i	0.891725	+	0.452578i	$0.149496\pi$
$809$			20827.9i			0.905156i	−0.891725	+	0.452578i	$0.850504\pi$
$810$		0			0
$811$			26513.2i			1.14797i	0.818865	+	0.573986i	$0.194604\pi$
$811$			26513.2i			1.14797i	−0.818865	+	0.573986i	$0.805396\pi$
$812$	−20544.9			−0.887912
$813$		0			0
$814$	−27654.7			−1.19078
$815$			18206.3i			0.782500i
$816$		0			0
$817$	−1039.45	−	4784.34i	−0.0445112	−	0.204875i
$818$			17417.6i			0.744489i
$819$		0			0
$820$			4820.32i			0.205284i
$821$		−	17644.1i		−	0.750039i	−0.927017	−	0.375019i	$-0.877636\pi$
$821$		−	17644.1i		−	0.750039i	0.927017	−	0.375019i	$-0.122364\pi$
$822$		0			0
$823$	11304.5			0.478797			0.239398	−	0.970921i	$-0.423050\pi$
$823$	11304.5			0.478797			0.239398	+	0.970921i	$0.423050\pi$
$824$			4580.14i			0.193637i
$825$		0			0
$826$	−902.682			−0.0380246
$827$	−11918.6			−0.501150			−0.250575	−	0.968097i	$-0.580620\pi$
$827$	−11918.6			−0.501150			−0.250575	+	0.968097i	$0.580620\pi$
$828$		0			0
$829$		−	1394.07i		−	0.0584056i	−0.999574	−	0.0292028i	$-0.990703\pi$
$829$		−	1394.07i		−	0.0584056i	0.999574	−	0.0292028i	$-0.00929686\pi$
$830$	5189.48			0.217023
$831$		0			0
$832$		−	152.668i		−	0.00636154i
$833$			4849.90i			0.201728i
$834$		0			0
$835$			5196.68i			0.215376i
$836$	−13743.2	+	2985.86i	−0.568565	+	0.123527i
$837$		0			0
$838$		−	10668.0i		−	0.439760i
$839$	17.1301			0.000704884			0.000352442	−	1.00000i	$-0.499888\pi$
$839$	17.1301			0.000704884			0.000352442		1.00000i	$0.499888\pi$
$840$		0			0
$841$	27833.2			1.14122
$842$		−	18821.4i		−	0.770343i
$843$		0			0
$844$			17143.7i			0.699181i
$845$			10956.5i			0.446055i
$846$		0			0
$847$	−10593.0			−0.429728
$848$	−5238.78			−0.212147
$849$		0			0
$850$			1495.37i			0.0603419i
$851$	−65144.9			−2.62413
$852$		0			0
$853$	3452.91			0.138599			0.0692997	−	0.997596i	$-0.477924\pi$
$853$	3452.91			0.138599			0.0692997	+	0.997596i	$0.477924\pi$
$854$	−25672.6			−1.02869
$855$		0			0
$856$	7024.35			0.280476
$857$	46709.0			1.86178			0.930892	−	0.365295i	$-0.119032\pi$
$857$	46709.0			1.86178			0.930892	+	0.365295i	$0.119032\pi$
$858$		0			0
$859$	38063.0			1.51187			0.755933	−	0.654649i	$-0.227183\pi$
$859$	38063.0			1.51187			0.755933	+	0.654649i	$0.227183\pi$
$860$			1182.33i			0.0468802i
$861$		0			0
$862$	−10432.8			−0.412230
$863$	7848.64			0.309584			0.154792	−	0.987947i	$-0.450529\pi$
$863$	7848.64			0.309584			0.154792	+	0.987947i	$0.450529\pi$
$864$		0			0
$865$			20780.1i			0.816815i
$866$		−	24296.4i		−	0.953380i
$867$		0			0
$868$		−	8464.58i		−	0.330998i
$869$	39566.8			1.54455
$870$		0			0
$871$	550.156			0.0214022
$872$			3600.84i			0.139839i
$873$		0			0
$874$	−32374.3	+	7033.65i	−1.25295	+	0.272216i
$875$		−	2809.48i		−	0.108546i
$876$		0			0
$877$			45942.0i			1.76893i	0.466606	+	0.884465i	$0.345477\pi$
$877$			45942.0i			1.76893i	−0.466606	+	0.884465i	$0.654523\pi$
$878$		−	11475.9i		−	0.441109i
$879$		0			0
$880$	3396.29			0.130101
$881$		−	9920.82i		−	0.379388i	−0.981843	−	0.189694i	$-0.939250\pi$
$881$		−	9920.82i		−	0.379388i	0.981843	−	0.189694i	$-0.0607496\pi$
$882$		0			0
$883$	34368.3			1.30984			0.654918	−	0.755700i	$-0.272703\pi$
$883$	34368.3			1.30984			0.654918	+	0.755700i	$0.272703\pi$
$884$	285.368			0.0108574
$885$		0			0
$886$			3244.09i			0.123010i
$887$	41880.5			1.58535			0.792677	−	0.609642i	$-0.208687\pi$
$887$	41880.5			1.58535			0.792677	+	0.609642i	$0.208687\pi$
$888$		0			0
$889$		−	43733.0i		−	1.64990i
$890$		−	3049.91i		−	0.114869i
$891$		0			0
$892$		−	515.115i		−	0.0193356i
$893$	27630.3	−	6002.96i	1.03540	−	0.224951i
$894$		0			0
$895$		−	12085.3i		−	0.451360i
$896$	−2876.91			−0.107267
$897$		0			0
$898$	35647.3			1.32468
$899$			21515.8i			0.798210i
$900$		0			0
$901$		−	9792.38i		−	0.362077i
$902$			20464.0i			0.755406i
$903$		0			0
$904$	786.353			0.0289311
$905$	9858.34			0.362102
$906$		0			0
$907$		−	49300.8i		−	1.80486i	−0.430840	−	0.902428i	$-0.641783\pi$
$907$		−	49300.8i		−	1.80486i	0.430840	−	0.902428i	$-0.358217\pi$
$908$	−10366.6			−0.378884
$909$		0			0
$910$	−536.146			−0.0195309
$911$	−29350.9			−1.06744			−0.533720	−	0.845661i	$-0.679207\pi$
$911$	−29350.9			−1.06744			−0.533720	+	0.845661i	$0.679207\pi$
$912$		0			0
$913$	22031.2			0.798605
$914$	−9572.41			−0.346419
$915$		0			0
$916$	8839.61			0.318853
$917$			15183.2i			0.546776i
$918$		0			0
$919$	−24760.2			−0.888751			−0.444376	−	0.895841i	$-0.646574\pi$
$919$	−24760.2			−0.888751			−0.444376	+	0.895841i	$0.646574\pi$
$920$	8000.47			0.286704
$921$		0			0
$922$		−	10934.4i		−	0.390568i
$923$		−	232.252i		−	0.00828242i
$924$		0			0
$925$		−	8142.64i		−	0.289436i
$926$	3405.98			0.120872
$927$		0			0
$928$	7312.70			0.258676
$929$			15223.8i			0.537651i	0.963189	+	0.268826i	$0.0866356\pi$
$929$			15223.8i			0.537651i	−0.963189	+	0.268826i	$0.913364\pi$
$930$		0			0
$931$	−2851.35	−	13124.1i	−0.100375	−	0.462004i
$932$			9317.85i			0.327485i
$933$		0			0
$934$		−	8126.28i		−	0.284690i
$935$			6348.37i			0.222047i
$936$		0			0
$937$	−45560.8			−1.58848			−0.794241	−	0.607603i	$-0.792131\pi$
$937$	−45560.8			−1.58848			−0.794241	+	0.607603i	$0.792131\pi$
$938$		−	10367.3i		−	0.360878i
$939$		0			0
$940$	−6828.10			−0.236923
$941$	47113.0			1.63214			0.816069	−	0.577955i	$-0.196149\pi$
$941$	47113.0			1.63214			0.816069	+	0.577955i	$0.196149\pi$
$942$		0			0
$943$			48206.0i			1.66469i
$944$	321.298			0.0110777
$945$		0			0
$946$			5019.40i			0.172510i
$947$			13466.3i			0.462086i	0.972944	+	0.231043i	$0.0742137\pi$
$947$			13466.3i			0.462086i	−0.972944	+	0.231043i	$0.925786\pi$
$948$		0			0
$949$		−	2121.96i		−	0.0725837i
$950$	−879.155	−	4046.55i	−0.0300248	−	0.138197i
$951$		0			0
$952$		−	5377.54i		−	0.183075i
$953$	−53686.4			−1.82484			−0.912421	−	0.409253i	$-0.865789\pi$
$953$	−53686.4			−1.82484			−0.912421	+	0.409253i	$0.865789\pi$
$954$		0			0
$955$	−2666.73			−0.0903596
$956$			5204.72i			0.176080i
$957$		0			0
$958$		−	39035.6i		−	1.31647i
$959$			66058.6i			2.22434i
$960$		0			0
$961$	20926.4			0.702441
$962$	−1553.90			−0.0520786
$963$		0			0
$964$		−	7089.18i		−	0.236854i
$965$	−4124.37			−0.137583
$966$		0			0
$967$	−14565.1			−0.484367			−0.242184	−	0.970230i	$-0.577864\pi$
$967$	−14565.1			−0.484367			−0.242184	+	0.970230i	$0.577864\pi$
$968$	3770.45			0.125193
$969$		0			0
$970$	−4518.34			−0.149562
$971$	−18361.1			−0.606834			−0.303417	−	0.952858i	$-0.598128\pi$
$971$	−18361.1			−0.606834			−0.303417	+	0.952858i	$0.598128\pi$
$972$		0			0
$973$	−25681.5			−0.846158
$974$		−	6679.90i		−	0.219751i
$975$		0			0
$976$	9137.84			0.299688
$977$	4120.05			0.134915			0.0674575	−	0.997722i	$-0.478511\pi$
$977$	4120.05			0.134915			0.0674575	+	0.997722i	$0.478511\pi$
$978$		0			0
$979$		−	12948.0i		−	0.422695i
$980$			3243.28i			0.105717i
$981$		0			0
$982$			9875.45i			0.320915i
$983$	11800.1			0.382874			0.191437	−	0.981505i	$-0.438685\pi$
$983$	11800.1			0.382874			0.191437	+	0.981505i	$0.438685\pi$
$984$		0			0
$985$	−12377.4			−0.400383
$986$			13669.0i			0.441489i
$987$		0			0
$988$	−772.222	+	167.773i	−0.0248661	+	0.00540240i
$989$			11823.9i			0.380161i
$990$		0			0
$991$			19341.0i			0.619965i	0.950742	+	0.309983i	$0.100323\pi$
$991$			19341.0i			0.619965i	−0.950742	+	0.309983i	$0.899677\pi$
$992$			3012.86i			0.0964299i
$993$		0			0
$994$	−4376.62			−0.139656
$995$			14815.7i			0.472049i
$996$		0			0
$997$	−30687.7			−0.974813			−0.487407	−	0.873175i	$-0.662057\pi$
$997$	−30687.7			−0.974813			−0.487407	+	0.873175i	$0.662057\pi$
$998$	34419.0			1.09170
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1710.4.f.a.341.8	yes	40
3.2	odd	2		1710.4.f.b.341.7	yes	40
19.18	odd	2		1710.4.f.b.341.8	yes	40
57.56	even	2	inner	1710.4.f.a.341.7	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1710.4.f.a.341.7	✓	40	57.56	even	2	inner
1710.4.f.a.341.8	yes	40	1.1	even	1	trivial
1710.4.f.b.341.7	yes	40	3.2	odd	2
1710.4.f.b.341.8	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +5.00000i q^{5} +22.4759 q^{7} -8.00000 q^{8} -10.0000i q^{10} -42.4536i q^{11} -2.38543i q^{13} -44.9517 q^{14} +16.0000 q^{16} +29.9073i q^{17} +(-17.5831 - 80.9310i) q^{19} +20.0000i q^{20} +84.9071i q^{22} +200.012i q^{23} -25.0000 q^{25} +4.77086i q^{26} +89.9034 q^{28} -228.522 q^{29} -94.1519i q^{31} -32.0000 q^{32} -59.8147i q^{34} +112.379i q^{35} +325.706i q^{37} +(35.1662 + 161.862i) q^{38} -40.0000i q^{40} +241.016 q^{41} +59.1163 q^{43} -169.814i q^{44} -400.023i q^{46} +341.405i q^{47} +162.164 q^{49} +50.0000 q^{50} -9.54173i q^{52} -327.424 q^{53} +212.268 q^{55} -179.807 q^{56} +457.044 q^{58} +20.0811 q^{59} +571.115 q^{61} +188.304i q^{62} +64.0000 q^{64} +11.9272 q^{65} +230.632i q^{67} +119.629i q^{68} -224.759i q^{70} +97.3627 q^{71} +889.552 q^{73} -651.411i q^{74} +(-70.3324 - 323.724i) q^{76} -954.180i q^{77} +932.003i q^{79} +80.0000i q^{80} -482.032 q^{82} +518.948i q^{83} -149.537 q^{85} -118.233 q^{86} +339.629i q^{88} +304.991 q^{89} -53.6146i q^{91} +800.047i q^{92} -682.810i q^{94} +(404.655 - 87.9155i) q^{95} -451.834i q^{97} -324.328 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 80 q^{2} + 160 q^{4} - 56 q^{7} - 320 q^{8} + 112 q^{14} + 640 q^{16} - 76 q^{19} - 1000 q^{25} - 224 q^{28} - 120 q^{29} - 1280 q^{32} + 152 q^{38} - 312 q^{41} + 56 q^{43} + 2112 q^{49} + 2000 q^{50}+ \cdots - 4224 q^{98}+O(q^{100}) \) 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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