LMFDB - Embedded newform 1520.3.h.a.721.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,3,Mod(721,1520)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1520.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1520.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$41.4170001828$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 30x^{10} + 329x^{8} + 1620x^{6} + 3479x^{4} + 2470x^{2} + 55 $ x^12 + 30x^10 + 329x^8 + 1620x^6 + 3479x^4 + 2470*x^2 + 55
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.12
Root			$-0.151673i$ of defining polynomial
Character	$\chi$	$=$	1520.721
Dual form			1520.3.h.a.721.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.10387i q^{3} +2.23607 q^{5} +6.35478 q^{7} -17.0495 q^{9} +O(q^{10})$	$q+5.10387i q^{3} +2.23607 q^{5} +6.35478 q^{7} -17.0495 q^{9} -10.0445 q^{11} -3.79583i q^{13} +11.4126i q^{15} -13.4309 q^{17} +(-2.99005 - 18.7633i) q^{19} +32.4340i q^{21} -16.7800 q^{23} +5.00000 q^{25} -41.0834i q^{27} -24.9494i q^{29} -46.3492i q^{31} -51.2658i q^{33} +14.2097 q^{35} +68.4543i q^{37} +19.3734 q^{39} -47.5852i q^{41} -43.4186 q^{43} -38.1238 q^{45} -37.4512 q^{47} -8.61675 q^{49} -68.5494i q^{51} -9.48017i q^{53} -22.4602 q^{55} +(95.7651 - 15.2608i) q^{57} -87.7096i q^{59} -95.9399 q^{61} -108.346 q^{63} -8.48774i q^{65} +75.1263i q^{67} -85.6428i q^{69} +77.6831i q^{71} -120.344 q^{73} +25.5193i q^{75} -63.8306 q^{77} +12.8248i q^{79} +56.2390 q^{81} +108.761 q^{83} -30.0324 q^{85} +127.338 q^{87} -156.732i q^{89} -24.1217i q^{91} +236.560 q^{93} +(-6.68596 - 41.9559i) q^{95} +56.1983i q^{97} +171.253 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 20 q^{7} - 48 q^{9}+O(q^{10}) $ 12 * q - 20 * q^7 - 48 * q^9	$ 12 q - 20 q^{7} - 48 q^{9} - 32 q^{11} - 44 q^{17} - 8 q^{19} - 36 q^{23} + 60 q^{25} + 40 q^{35} - 76 q^{39} - 320 q^{43} - 40 q^{45} + 56 q^{47} + 72 q^{49} + 60 q^{57} - 296 q^{61} + 96 q^{63} - 244 q^{73} - 200 q^{77} - 372 q^{81} + 160 q^{83} + 160 q^{85} - 444 q^{87} + 296 q^{93} + 80 q^{95} + 312 q^{99}+O(q^{100}) $ 12 * q - 20 * q^7 - 48 * q^9 - 32 * q^11 - 44 * q^17 - 8 * q^19 - 36 * q^23 + 60 * q^25 + 40 * q^35 - 76 * q^39 - 320 * q^43 - 40 * q^45 + 56 * q^47 + 72 * q^49 + 60 * q^57 - 296 * q^61 + 96 * q^63 - 244 * q^73 - 200 * q^77 - 372 * q^81 + 160 * q^83 + 160 * q^85 - 444 * q^87 + 296 * q^93 + 80 * q^95 + 312 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$			5.10387i			1.70129i	0.525741	+	0.850645i	$0.323788\pi$
$3$			5.10387i			1.70129i	−0.525741	+	0.850645i	$0.676212\pi$
$4$		0			0
$5$	2.23607			0.447214
$5$	2.23607			0.447214
$6$		0			0
$7$	6.35478			0.907826			0.453913	−	0.891046i	$-0.350028\pi$
$7$	6.35478			0.907826			0.453913	+	0.891046i	$0.350028\pi$
$8$		0			0
$9$	−17.0495			−1.89438
$10$		0			0
$11$	−10.0445			−0.913136			−0.456568	−	0.889688i	$-0.650922\pi$
$11$	−10.0445			−0.913136			−0.456568	+	0.889688i	$0.650922\pi$
$12$		0			0
$13$		−	3.79583i		−	0.291987i	−0.989286	−	0.145994i	$-0.953362\pi$
$13$		−	3.79583i		−	0.291987i	0.989286	−	0.145994i	$-0.0466379\pi$
$14$		0			0
$15$			11.4126i			0.760840i
$16$		0			0
$17$	−13.4309			−0.790052			−0.395026	−	0.918670i	$-0.629264\pi$
$17$	−13.4309			−0.790052			−0.395026	+	0.918670i	$0.629264\pi$
$18$		0			0
$19$	−2.99005	−	18.7633i	−0.157371	−	0.987540i
$19$	−2.99005	−	18.7633i	−0.157371	−	0.987540i
$20$		0			0
$21$			32.4340i			1.54447i
$22$		0			0
$23$	−16.7800			−0.729564			−0.364782	−	0.931093i	$-0.618856\pi$
$23$	−16.7800			−0.729564			−0.364782	+	0.931093i	$0.618856\pi$
$24$		0			0
$25$	5.00000			0.200000
$26$		0			0
$27$		−	41.0834i		−	1.52161i
$28$		0			0
$29$		−	24.9494i		−	0.860323i	−0.902752	−	0.430161i	$-0.858457\pi$
$29$		−	24.9494i		−	0.860323i	0.902752	−	0.430161i	$-0.141543\pi$
$30$		0			0
$31$		−	46.3492i		−	1.49513i	−0.664186	−	0.747567i	$-0.731222\pi$
$31$		−	46.3492i		−	1.49513i	0.664186	−	0.747567i	$-0.268778\pi$
$32$		0			0
$33$		−	51.2658i		−	1.55351i
$34$		0			0
$35$	14.2097			0.405992
$36$		0			0
$37$			68.4543i			1.85012i	0.379825	+	0.925058i	$0.375984\pi$
$37$			68.4543i			1.85012i	−0.379825	+	0.925058i	$0.624016\pi$
$38$		0			0
$39$	19.3734			0.496755
$40$		0			0
$41$		−	47.5852i		−	1.16061i	−0.814398	−	0.580307i	$-0.802932\pi$
$41$		−	47.5852i		−	1.16061i	0.814398	−	0.580307i	$-0.197068\pi$
$42$		0			0
$43$	−43.4186			−1.00974			−0.504868	−	0.863197i	$-0.668459\pi$
$43$	−43.4186			−1.00974			−0.504868	+	0.863197i	$0.668459\pi$
$44$		0			0
$45$	−38.1238			−0.847194
$46$		0			0
$47$	−37.4512			−0.796834			−0.398417	−	0.917204i	$-0.630440\pi$
$47$	−37.4512			−0.796834			−0.398417	+	0.917204i	$0.630440\pi$
$48$		0			0
$49$	−8.61675			−0.175852
$50$		0			0
$51$		−	68.5494i		−	1.34411i
$52$		0			0
$53$		−	9.48017i		−	0.178871i	−0.995993	−	0.0894356i	$-0.971494\pi$
$53$		−	9.48017i		−	0.178871i	0.995993	−	0.0894356i	$-0.0285063\pi$
$54$		0			0
$55$	−22.4602			−0.408367
$56$		0			0
$57$	95.7651	−	15.2608i	1.68009	−	0.267734i
$58$		0			0
$59$		−	87.7096i		−	1.48660i	−0.668956	−	0.743302i	$-0.733259\pi$
$59$		−	87.7096i		−	1.48660i	0.668956	−	0.743302i	$-0.266741\pi$
$60$		0			0
$61$	−95.9399			−1.57279			−0.786393	−	0.617727i	$-0.788054\pi$
$61$	−95.9399			−1.57279			−0.786393	+	0.617727i	$0.788054\pi$
$62$		0			0
$63$	−108.346			−1.71977
$64$		0			0
$65$		−	8.48774i		−	0.130581i
$66$		0			0
$67$			75.1263i			1.12129i	0.828057	+	0.560644i	$0.189446\pi$
$67$			75.1263i			1.12129i	−0.828057	+	0.560644i	$0.810554\pi$
$68$		0			0
$69$		−	85.6428i		−	1.24120i
$70$		0			0
$71$			77.6831i			1.09413i	0.837091	+	0.547064i	$0.184255\pi$
$71$			77.6831i			1.09413i	−0.837091	+	0.547064i	$0.815745\pi$
$72$		0			0
$73$	−120.344			−1.64854			−0.824272	−	0.566193i	$-0.808416\pi$
$73$	−120.344			−1.64854			−0.824272	+	0.566193i	$0.808416\pi$
$74$		0			0
$75$			25.5193i			0.340258i
$76$		0			0
$77$	−63.8306			−0.828969
$78$		0			0
$79$			12.8248i			0.162339i	0.996700	+	0.0811693i	$0.0258655\pi$
$79$			12.8248i			0.162339i	−0.996700	+	0.0811693i	$0.974135\pi$
$80$		0			0
$81$	56.2390			0.694308
$82$		0			0
$83$	108.761			1.31037			0.655186	−	0.755467i	$-0.272590\pi$
$83$	108.761			1.31037			0.655186	+	0.755467i	$0.272590\pi$
$84$		0			0
$85$	−30.0324			−0.353322
$86$		0			0
$87$	127.338			1.46366
$88$		0			0
$89$		−	156.732i		−	1.76103i	−0.474016	−	0.880516i	$-0.657196\pi$
$89$		−	156.732i		−	1.76103i	0.474016	−	0.880516i	$-0.342804\pi$
$90$		0			0
$91$		−	24.1217i		−	0.265074i
$92$		0			0
$93$	236.560			2.54366
$94$		0			0
$95$	−6.68596	−	41.9559i	−0.0703786	−	0.441641i
$96$		0			0
$97$			56.1983i			0.579364i	0.957123	+	0.289682i	$0.0935496\pi$
$97$			56.1983i			0.579364i	−0.957123	+	0.289682i	$0.906450\pi$
$98$		0			0
$99$	171.253			1.72983
$100$		0			0
$101$	−27.4569			−0.271850			−0.135925	−	0.990719i	$-0.543401\pi$
$101$	−27.4569			−0.271850			−0.135925	+	0.990719i	$0.543401\pi$
$102$		0			0
$103$			20.2246i			0.196355i	0.995169	+	0.0981777i	$0.0313013\pi$
$103$			20.2246i			0.196355i	−0.995169	+	0.0981777i	$0.968699\pi$
$104$		0			0
$105$			72.5245i			0.690710i
$106$		0			0
$107$		−	29.0356i		−	0.271361i	−0.990753	−	0.135680i	$-0.956678\pi$
$107$		−	29.0356i		−	0.271361i	0.990753	−	0.135680i	$-0.0433220\pi$
$108$		0			0
$109$			103.073i			0.945627i	0.881162	+	0.472814i	$0.156762\pi$
$109$			103.073i			0.945627i	−0.881162	+	0.472814i	$0.843238\pi$
$110$		0			0
$111$	−349.382			−3.14758
$112$		0			0
$113$			130.480i			1.15469i	0.816500	+	0.577346i	$0.195912\pi$
$113$			130.480i			1.15469i	−0.816500	+	0.577346i	$0.804088\pi$
$114$		0			0
$115$	−37.5212			−0.326271
$116$		0			0
$117$			64.7169i			0.553136i
$118$		0			0
$119$	−85.3503			−0.717230
$120$		0			0
$121$	−20.1081			−0.166182
$122$		0			0
$123$	242.868			1.97454
$124$		0			0
$125$	11.1803			0.0894427
$126$		0			0
$127$			51.2735i			0.403728i	0.979414	+	0.201864i	$0.0646999\pi$
$127$			51.2735i			0.403728i	−0.979414	+	0.201864i	$0.935300\pi$
$128$		0			0
$129$		−	221.603i		−	1.71785i
$130$		0			0
$131$	111.464			0.850867			0.425434	−	0.904990i	$-0.360122\pi$
$131$	111.464			0.850867			0.425434	+	0.904990i	$0.360122\pi$
$132$		0			0
$133$	−19.0011	−	119.236i	−0.142866	−	0.896514i
$134$		0			0
$135$		−	91.8652i		−	0.680483i
$136$		0			0
$137$	−103.250			−0.753646			−0.376823	−	0.926285i	$-0.622984\pi$
$137$	−103.250			−0.753646			−0.376823	+	0.926285i	$0.622984\pi$
$138$		0			0
$139$	−12.1509			−0.0874164			−0.0437082	−	0.999044i	$-0.513917\pi$
$139$	−12.1509			−0.0874164			−0.0437082	+	0.999044i	$0.513917\pi$
$140$		0			0
$141$		−	191.146i		−	1.35564i
$142$		0			0
$143$			38.1272i			0.266624i
$144$		0			0
$145$		−	55.7885i		−	0.384748i
$146$		0			0
$147$		−	43.9787i		−	0.299175i
$148$		0			0
$149$	247.081			1.65826			0.829130	−	0.559056i	$-0.188836\pi$
$149$	247.081			1.65826			0.829130	+	0.559056i	$0.188836\pi$
$150$		0			0
$151$		−	61.9073i		−	0.409982i	−0.978764	−	0.204991i	$-0.934283\pi$
$151$		−	61.9073i		−	0.409982i	0.978764	−	0.204991i	$-0.0657165\pi$
$152$		0			0
$153$	228.989			1.49666
$154$		0			0
$155$		−	103.640i		−	0.668644i
$156$		0			0
$157$	74.5130			0.474605			0.237303	−	0.971436i	$-0.423737\pi$
$157$	74.5130			0.474605			0.237303	+	0.971436i	$0.423737\pi$
$158$		0			0
$159$	48.3855			0.304311
$160$		0			0
$161$	−106.633			−0.662317
$162$		0			0
$163$	67.2648			0.412667			0.206334	−	0.978482i	$-0.433847\pi$
$163$	67.2648			0.412667			0.206334	+	0.978482i	$0.433847\pi$
$164$		0			0
$165$		−	114.634i		−	0.694750i
$166$		0			0
$167$		−	165.664i		−	0.992001i	−0.868322	−	0.496001i	$-0.834801\pi$
$167$		−	165.664i		−	0.992001i	0.868322	−	0.496001i	$-0.165199\pi$
$168$		0			0
$169$	154.592			0.914743
$170$		0			0
$171$	50.9788	+	319.903i	0.298122	+	1.87078i
$172$		0			0
$173$		−	309.445i		−	1.78870i	−0.447370	−	0.894349i	$-0.647639\pi$
$173$		−	309.445i		−	1.78870i	0.447370	−	0.894349i	$-0.352361\pi$
$174$		0			0
$175$	31.7739			0.181565
$176$		0			0
$177$	447.658			2.52914
$178$		0			0
$179$			211.904i			1.18382i	0.806005	+	0.591909i	$0.201626\pi$
$179$			211.904i			1.18382i	−0.806005	+	0.591909i	$0.798374\pi$
$180$		0			0
$181$		−	101.114i		−	0.558639i	−0.960198	−	0.279319i	$-0.909891\pi$
$181$		−	101.114i		−	0.558639i	0.960198	−	0.279319i	$-0.0901089\pi$
$182$		0			0
$183$		−	489.665i		−	2.67576i
$184$		0			0
$185$			153.068i			0.827397i
$186$		0			0
$187$	134.906			0.721425
$188$		0			0
$189$		−	261.076i		−	1.38135i
$190$		0			0
$191$	57.2008			0.299481			0.149740	−	0.988725i	$-0.452156\pi$
$191$	57.2008			0.299481			0.149740	+	0.988725i	$0.452156\pi$
$192$		0			0
$193$			156.431i			0.810524i	0.914201	+	0.405262i	$0.132820\pi$
$193$			156.431i			0.810524i	−0.914201	+	0.405262i	$0.867180\pi$
$194$		0			0
$195$	43.3203			0.222155
$196$		0			0
$197$	168.534			0.855503			0.427751	−	0.903896i	$-0.359306\pi$
$197$	168.534			0.855503			0.427751	+	0.903896i	$0.359306\pi$
$198$		0			0
$199$	−40.9710			−0.205885			−0.102942	−	0.994687i	$-0.532826\pi$
$199$	−40.9710			−0.205885			−0.102942	+	0.994687i	$0.532826\pi$
$200$		0			0
$201$	−383.435			−1.90764
$202$		0			0
$203$		−	158.548i		−	0.781023i
$204$		0			0
$205$		−	106.404i		−	0.519043i
$206$		0			0
$207$	286.089			1.38207
$208$		0			0
$209$	30.0336	+	188.467i	0.143701	+	0.901758i
$210$		0			0
$211$			264.352i			1.25285i	0.779481	+	0.626426i	$0.215483\pi$
$211$			264.352i			1.25285i	−0.779481	+	0.626426i	$0.784517\pi$
$212$		0			0
$213$	−396.484			−1.86143
$214$		0			0
$215$	−97.0870			−0.451568
$216$		0			0
$217$		−	294.539i		−	1.35732i
$218$		0			0
$219$		−	614.219i		−	2.80465i
$220$		0			0
$221$			50.9814i			0.230685i
$222$		0			0
$223$			220.658i			0.989498i	0.869036	+	0.494749i	$0.164740\pi$
$223$			220.658i			0.989498i	−0.869036	+	0.494749i	$0.835260\pi$
$224$		0			0
$225$	−85.2473			−0.378877
$226$		0			0
$227$		−	156.019i		−	0.687308i	−0.939096	−	0.343654i	$-0.888335\pi$
$227$		−	156.019i		−	0.687308i	0.939096	−	0.343654i	$-0.111665\pi$
$228$		0			0
$229$	259.477			1.13309			0.566545	−	0.824031i	$-0.308280\pi$
$229$	259.477			1.13309			0.566545	+	0.824031i	$0.308280\pi$
$230$		0			0
$231$		−	325.783i		−	1.41032i
$232$		0			0
$233$	−353.499			−1.51716			−0.758582	−	0.651578i	$-0.774107\pi$
$233$	−353.499			−1.51716			−0.758582	+	0.651578i	$0.774107\pi$
$234$		0			0
$235$	−83.7434			−0.356355
$236$		0			0
$237$	−65.4558			−0.276185
$238$		0			0
$239$	176.322			0.737750			0.368875	−	0.929479i	$-0.379743\pi$
$239$	176.322			0.737750			0.368875	+	0.929479i	$0.379743\pi$
$240$		0			0
$241$		−	35.4360i		−	0.147037i	−0.997294	−	0.0735187i	$-0.976577\pi$
$241$		−	35.4360i		−	0.147037i	0.997294	−	0.0735187i	$-0.0234229\pi$
$242$		0			0
$243$		−	82.7142i		−	0.340388i
$244$		0			0
$245$	−19.2676			−0.0786434
$246$		0			0
$247$	−71.2222	+	11.3497i	−0.288349	+	0.0459504i
$248$		0			0
$249$			555.101i			2.22932i
$250$		0			0
$251$	−305.098			−1.21553			−0.607764	−	0.794118i	$-0.707933\pi$
$251$	−305.098			−1.21553			−0.607764	+	0.794118i	$0.707933\pi$
$252$		0			0
$253$	168.546			0.666191
$254$		0			0
$255$		−	153.281i		−	0.601103i
$256$		0			0
$257$			294.709i			1.14673i	0.819300	+	0.573364i	$0.194362\pi$
$257$			294.709i			1.14673i	−0.819300	+	0.573364i	$0.805638\pi$
$258$		0			0
$259$			435.012i			1.67958i
$260$		0			0
$261$			425.373i			1.62978i
$262$		0			0
$263$	−339.127			−1.28946			−0.644728	−	0.764412i	$-0.723029\pi$
$263$	−339.127			−1.28946			−0.644728	+	0.764412i	$0.723029\pi$
$264$		0			0
$265$		−	21.1983i		−	0.0799936i
$266$		0			0
$267$	799.939			2.99603
$268$		0			0
$269$			33.2611i			0.123647i	0.998087	+	0.0618235i	$0.0196916\pi$
$269$			33.2611i			0.123647i	−0.998087	+	0.0618235i	$0.980308\pi$
$270$		0			0
$271$	−381.340			−1.40716			−0.703579	−	0.710618i	$-0.748416\pi$
$271$	−381.340			−1.40716			−0.703579	+	0.710618i	$0.748416\pi$
$272$		0			0
$273$	123.114			0.450967
$274$		0			0
$275$	−50.2225			−0.182627
$276$		0			0
$277$	33.7216			0.121739			0.0608694	−	0.998146i	$-0.480613\pi$
$277$	33.7216			0.121739			0.0608694	+	0.998146i	$0.480613\pi$
$278$		0			0
$279$			790.228i			2.83236i
$280$		0			0
$281$			209.260i			0.744699i	0.928093	+	0.372349i	$0.121448\pi$
$281$			209.260i			0.744699i	−0.928093	+	0.372349i	$0.878552\pi$
$282$		0			0
$283$	341.019			1.20501			0.602506	−	0.798114i	$-0.294169\pi$
$283$	341.019			1.20501			0.602506	+	0.798114i	$0.294169\pi$
$284$		0			0
$285$	214.137	−	34.1243i	0.751359	−	0.119734i
$286$		0			0
$287$		−	302.394i		−	1.05364i
$288$		0			0
$289$	−108.611			−0.375818
$290$		0			0
$291$	−286.829			−0.985666
$292$		0			0
$293$		−	409.506i		−	1.39763i	−0.715303	−	0.698815i	$-0.753711\pi$
$293$		−	409.506i		−	1.39763i	0.715303	−	0.698815i	$-0.246289\pi$
$294$		0			0
$295$		−	196.125i		−	0.664829i
$296$		0			0
$297$			412.662i			1.38943i
$298$		0			0
$299$			63.6940i			0.213023i
$300$		0			0
$301$	−275.916			−0.916665
$302$		0			0
$303$		−	140.136i		−	0.462496i
$304$		0			0
$305$	−214.528			−0.703371
$306$		0			0
$307$			235.337i			0.766571i	0.923630	+	0.383286i	$0.125207\pi$
$307$			235.337i			0.766571i	−0.923630	+	0.383286i	$0.874793\pi$
$308$		0			0
$309$	−103.224			−0.334057
$310$		0			0
$311$	197.036			0.633558			0.316779	−	0.948499i	$-0.397399\pi$
$311$	197.036			0.633558			0.316779	+	0.948499i	$0.397399\pi$
$312$		0			0
$313$	−27.8992			−0.0891348			−0.0445674	−	0.999006i	$-0.514191\pi$
$313$	−27.8992			−0.0891348			−0.0445674	+	0.999006i	$0.514191\pi$
$314$		0			0
$315$	−242.268			−0.769105
$316$		0			0
$317$			372.696i			1.17570i	0.808971	+	0.587848i	$0.200025\pi$
$317$			372.696i			1.17570i	−0.808971	+	0.587848i	$0.799975\pi$
$318$		0			0
$319$			250.604i			0.785592i
$320$		0			0
$321$	148.194			0.461663
$322$		0			0
$323$	40.1591	+	252.007i	0.124331	+	0.780207i
$324$		0			0
$325$		−	18.9792i		−	0.0583974i
$326$		0			0
$327$	−526.073			−1.60879
$328$		0			0
$329$	−237.994			−0.723386
$330$		0			0
$331$		−	3.74805i		−	0.0113234i	−0.999984	−	0.00566170i	$-0.998198\pi$
$331$		−	3.74805i		−	0.0113234i	0.999984	−	0.00566170i	$-0.00180219\pi$
$332$		0			0
$333$		−	1167.11i		−	3.50483i
$334$		0			0
$335$			167.988i			0.501455i
$336$		0			0
$337$		−	410.554i		−	1.21826i	−0.793070	−	0.609131i	$-0.791518\pi$
$337$		−	410.554i		−	1.21826i	0.793070	−	0.609131i	$-0.208482\pi$
$338$		0			0
$339$	−665.953			−1.96446
$340$		0			0
$341$			465.554i			1.36526i
$342$		0			0
$343$	−366.142			−1.06747
$344$		0			0
$345$		−	191.503i		−	0.555081i
$346$		0			0
$347$	−210.073			−0.605399			−0.302700	−	0.953086i	$-0.597888\pi$
$347$	−210.073			−0.605399			−0.302700	+	0.953086i	$0.597888\pi$
$348$		0			0
$349$	−308.461			−0.883843			−0.441921	−	0.897054i	$-0.645703\pi$
$349$	−308.461			−0.883843			−0.441921	+	0.897054i	$0.645703\pi$
$350$		0			0
$351$	−155.946			−0.444290
$352$		0			0
$353$	556.817			1.57739			0.788693	−	0.614787i	$-0.210758\pi$
$353$	556.817			1.57739			0.788693	+	0.614787i	$0.210758\pi$
$354$		0			0
$355$			173.705i			0.489309i
$356$		0			0
$357$		−	435.617i		−	1.22021i
$358$		0			0
$359$	−118.795			−0.330904			−0.165452	−	0.986218i	$-0.552908\pi$
$359$	−118.795			−0.330904			−0.165452	+	0.986218i	$0.552908\pi$
$360$		0			0
$361$	−343.119	+	112.206i	−0.950469	+	0.310821i
$362$		0			0
$363$		−	102.629i		−	0.282724i
$364$		0			0
$365$	−269.097			−0.737252
$366$		0			0
$367$	−378.940			−1.03253			−0.516267	−	0.856428i	$-0.672679\pi$
$367$	−378.940			−1.03253			−0.516267	+	0.856428i	$0.672679\pi$
$368$		0			0
$369$			811.302i			2.19865i
$370$		0			0
$371$		−	60.2444i		−	0.162384i
$372$		0			0
$373$			201.901i			0.541290i	0.962679	+	0.270645i	$0.0872369\pi$
$373$			201.901i			0.541290i	−0.962679	+	0.270645i	$0.912763\pi$
$374$		0			0
$375$			57.0630i			0.152168i
$376$		0			0
$377$	−94.7036			−0.251203
$378$		0			0
$379$		−	267.286i		−	0.705241i	−0.935766	−	0.352620i	$-0.885291\pi$
$379$		−	267.286i		−	0.705241i	0.935766	−	0.352620i	$-0.114709\pi$
$380$		0			0
$381$	−261.693			−0.686858
$382$		0			0
$383$			146.990i			0.383785i	0.981416	+	0.191893i	$0.0614626\pi$
$383$			146.990i			0.383785i	−0.981416	+	0.191893i	$0.938537\pi$
$384$		0			0
$385$	−142.730			−0.370726
$386$		0			0
$387$	740.265			1.91283
$388$		0			0
$389$	−187.901			−0.483035			−0.241518	−	0.970396i	$-0.577645\pi$
$389$	−187.901			−0.483035			−0.241518	+	0.970396i	$0.577645\pi$
$390$		0			0
$391$	225.370			0.576393
$392$		0			0
$393$			568.896i			1.44757i
$394$		0			0
$395$			28.6770i			0.0726000i
$396$		0			0
$397$	26.5137			0.0667852			0.0333926	−	0.999442i	$-0.489369\pi$
$397$	26.5137			0.0667852			0.0333926	+	0.999442i	$0.489369\pi$
$398$		0			0
$399$	608.567	−	96.9793i	1.52523	−	0.243056i
$400$		0			0
$401$		−	38.9717i		−	0.0971864i	−0.998819	−	0.0485932i	$-0.984526\pi$
$401$		−	38.9717i		−	0.0971864i	0.998819	−	0.0485932i	$-0.0154738\pi$
$402$		0			0
$403$	−175.934			−0.436560
$404$		0			0
$405$	125.754			0.310504
$406$		0			0
$407$		−	687.589i		−	1.68941i
$408$		0			0
$409$		−	111.913i		−	0.273626i	−0.990597	−	0.136813i	$-0.956314\pi$
$409$		−	111.913i		−	0.273626i	0.990597	−	0.136813i	$-0.0436860\pi$
$410$		0			0
$411$		−	526.972i		−	1.28217i
$412$		0			0
$413$		−	557.375i		−	1.34958i
$414$		0			0
$415$	243.197			0.586016
$416$		0			0
$417$		−	62.0165i		−	0.148721i
$418$		0			0
$419$	−207.979			−0.496370			−0.248185	−	0.968713i	$-0.579834\pi$
$419$	−207.979			−0.496370			−0.248185	+	0.968713i	$0.579834\pi$
$420$		0			0
$421$		−	182.812i		−	0.434232i	−0.976146	−	0.217116i	$-0.930335\pi$
$421$		−	182.812i		−	0.434232i	0.976146	−	0.217116i	$-0.0696650\pi$
$422$		0			0
$423$	638.523			1.50951
$424$		0			0
$425$	−67.1544			−0.158010
$426$		0			0
$427$	−609.677			−1.42782
$428$		0			0
$429$	−194.596			−0.453605
$430$		0			0
$431$			175.672i			0.407592i	0.979013	+	0.203796i	$0.0653279\pi$
$431$			175.672i			0.407592i	−0.979013	+	0.203796i	$0.934672\pi$
$432$		0			0
$433$			343.766i			0.793918i	0.917836	+	0.396959i	$0.129934\pi$
$433$			343.766i			0.793918i	−0.917836	+	0.396959i	$0.870066\pi$
$434$		0			0
$435$	284.737			0.654568
$436$		0			0
$437$	50.1730	+	314.847i	0.114812	+	0.720473i
$438$		0			0
$439$		−	448.069i		−	1.02066i	−0.859979	−	0.510329i	$-0.829524\pi$
$439$		−	448.069i		−	1.02066i	0.859979	−	0.510329i	$-0.170476\pi$
$440$		0			0
$441$	146.911			0.333131
$442$		0			0
$443$	193.700			0.437246			0.218623	−	0.975809i	$-0.429843\pi$
$443$	193.700			0.437246			0.218623	+	0.975809i	$0.429843\pi$
$444$		0			0
$445$		−	350.463i		−	0.787558i
$446$		0			0
$447$			1261.07i			2.82118i
$448$		0			0
$449$		−	754.877i		−	1.68124i	−0.541625	−	0.840621i	$-0.682191\pi$
$449$		−	754.877i		−	1.68124i	0.541625	−	0.840621i	$-0.317809\pi$
$450$		0			0
$451$			477.969i			1.05980i
$452$		0			0
$453$	315.967			0.697498
$454$		0			0
$455$		−	53.9378i		−	0.118545i
$456$		0			0
$457$	−213.763			−0.467753			−0.233876	−	0.972266i	$-0.575141\pi$
$457$	−213.763			−0.467753			−0.233876	+	0.972266i	$0.575141\pi$
$458$		0			0
$459$			551.786i			1.20215i
$460$		0			0
$461$	−660.616			−1.43301			−0.716504	−	0.697583i	$-0.754259\pi$
$461$	−660.616			−1.43301			−0.716504	+	0.697583i	$0.754259\pi$
$462$		0			0
$463$	−481.283			−1.03949			−0.519744	−	0.854322i	$-0.673973\pi$
$463$	−481.283			−1.03949			−0.519744	+	0.854322i	$0.673973\pi$
$464$		0			0
$465$	528.964			1.13756
$466$		0			0
$467$	−420.429			−0.900277			−0.450139	−	0.892959i	$-0.648625\pi$
$467$	−420.429			−0.900277			−0.450139	+	0.892959i	$0.648625\pi$
$468$		0			0
$469$			477.411i			1.01793i
$470$		0			0
$471$			380.305i			0.807441i
$472$		0			0
$473$	436.118			0.922026
$474$		0			0
$475$	−14.9503	−	93.8163i	−0.0314742	−	0.197508i
$476$		0			0
$477$			161.632i			0.338851i
$478$		0			0
$479$	124.589			0.260101			0.130051	−	0.991507i	$-0.458486\pi$
$479$	124.589			0.260101			0.130051	+	0.991507i	$0.458486\pi$
$480$		0			0
$481$	259.841			0.540210
$482$		0			0
$483$		−	544.241i		−	1.12679i
$484$		0			0
$485$			125.663i			0.259099i
$486$		0			0
$487$			770.701i			1.58255i	0.611462	+	0.791274i	$0.290582\pi$
$487$			770.701i			1.58255i	−0.611462	+	0.791274i	$0.709418\pi$
$488$		0			0
$489$			343.311i			0.702067i
$490$		0			0
$491$	−54.3027			−0.110596			−0.0552981	−	0.998470i	$-0.517611\pi$
$491$	−54.3027			−0.110596			−0.0552981	+	0.998470i	$0.517611\pi$
$492$		0			0
$493$			335.092i			0.679700i
$494$		0			0
$495$	382.934			0.773604
$496$		0			0
$497$			493.659i			0.993278i
$498$		0			0
$499$	192.861			0.386495			0.193247	−	0.981150i	$-0.438098\pi$
$499$	192.861			0.386495			0.193247	+	0.981150i	$0.438098\pi$
$500$		0			0
$501$	845.528			1.68768
$502$		0			0
$503$	−544.862			−1.08322			−0.541612	−	0.840629i	$-0.682186\pi$
$503$	−544.862			−1.08322			−0.541612	+	0.840629i	$0.682186\pi$
$504$		0			0
$505$	−61.3954			−0.121575
$506$		0			0
$507$			789.015i			1.55624i
$508$		0			0
$509$		−	629.472i		−	1.23668i	−0.785909	−	0.618342i	$-0.787805\pi$
$509$		−	629.472i		−	1.23668i	0.785909	−	0.618342i	$-0.212195\pi$
$510$		0			0
$511$	−764.758			−1.49659
$512$		0			0
$513$	−770.858	+	122.841i	−1.50265	+	0.239457i
$514$		0			0
$515$			45.2236i			0.0878128i
$516$		0			0
$517$	376.178			0.727618
$518$		0			0
$519$	1579.36			3.04309
$520$		0			0
$521$		−	535.499i		−	1.02783i	−0.857841	−	0.513915i	$-0.828195\pi$
$521$		−	535.499i		−	1.02783i	0.857841	−	0.513915i	$-0.171805\pi$
$522$		0			0
$523$		−	693.075i		−	1.32519i	−0.748978	−	0.662595i	$-0.769455\pi$
$523$		−	693.075i		−	1.32519i	0.748978	−	0.662595i	$-0.230545\pi$
$524$		0			0
$525$			162.170i			0.308895i
$526$		0			0
$527$			622.510i			1.18123i
$528$		0			0
$529$	−247.432			−0.467736
$530$		0			0
$531$			1495.40i			2.81620i
$532$		0			0
$533$	−180.626			−0.338885
$534$		0			0
$535$		−	64.9256i		−	0.121356i
$536$		0			0
$537$	−1081.53			−2.01402
$538$		0			0
$539$	86.5509			0.160577
$540$		0			0
$541$	217.348			0.401752			0.200876	−	0.979617i	$-0.435621\pi$
$541$	217.348			0.401752			0.200876	+	0.979617i	$0.435621\pi$
$542$		0			0
$543$	516.071			0.950406
$544$		0			0
$545$			230.479i			0.422897i
$546$		0			0
$547$		−	288.990i		−	0.528318i	−0.964479	−	0.264159i	$-0.914906\pi$
$547$		−	288.990i		−	0.528318i	0.964479	−	0.264159i	$-0.0850944\pi$
$548$		0			0
$549$	1635.72			2.97946
$550$		0			0
$551$	−468.131	+	74.5999i	−0.849603	+	0.135390i
$552$		0			0
$553$			81.4985i			0.147375i
$554$		0			0
$555$	−781.241			−1.40764
$556$		0			0
$557$	216.252			0.388245			0.194122	−	0.980977i	$-0.437814\pi$
$557$	216.252			0.388245			0.194122	+	0.980977i	$0.437814\pi$
$558$		0			0
$559$			164.810i			0.294830i
$560$		0			0
$561$			688.545i			1.22735i
$562$		0			0
$563$			132.093i			0.234624i	0.993095	+	0.117312i	$0.0374277\pi$
$563$			132.093i			0.234624i	−0.993095	+	0.117312i	$0.962572\pi$
$564$		0			0
$565$			291.763i			0.516394i
$566$		0			0
$567$	357.386			0.630311
$568$		0			0
$569$			540.812i			0.950460i	0.879861	+	0.475230i	$0.157635\pi$
$569$			540.812i			0.950460i	−0.879861	+	0.475230i	$0.842365\pi$
$570$		0			0
$571$	173.764			0.304316			0.152158	−	0.988356i	$-0.451378\pi$
$571$	173.764			0.304316			0.152158	+	0.988356i	$0.451378\pi$
$572$		0			0
$573$			291.945i			0.509503i
$574$		0			0
$575$	−83.8999			−0.145913
$576$		0			0
$577$	−252.461			−0.437541			−0.218770	−	0.975776i	$-0.570205\pi$
$577$	−252.461			−0.437541			−0.218770	+	0.975776i	$0.570205\pi$
$578$		0			0
$579$	−798.404			−1.37894
$580$		0			0
$581$	691.152			1.18959
$582$		0			0
$583$			95.2235i			0.163334i
$584$		0			0
$585$			144.711i			0.247370i
$586$		0			0
$587$	378.687			0.645123			0.322562	−	0.946548i	$-0.395456\pi$
$587$	378.687			0.645123			0.322562	+	0.946548i	$0.395456\pi$
$588$		0			0
$589$	−869.661	+	138.586i	−1.47650	+	0.235291i
$590$		0			0
$591$			860.175i			1.45546i
$592$		0			0
$593$	501.502			0.845704			0.422852	−	0.906199i	$-0.361029\pi$
$593$	501.502			0.845704			0.422852	+	0.906199i	$0.361029\pi$
$594$		0			0
$595$	−190.849			−0.320755
$596$		0			0
$597$		−	209.111i		−	0.350269i
$598$		0			0
$599$			52.0491i			0.0868933i	0.999056	+	0.0434467i	$0.0138339\pi$
$599$			52.0491i			0.0868933i	−0.999056	+	0.0434467i	$0.986166\pi$
$600$		0			0
$601$		−	59.3311i		−	0.0987206i	−0.998781	−	0.0493603i	$-0.984282\pi$
$601$		−	59.3311i		−	0.0987206i	0.998781	−	0.0493603i	$-0.0157183\pi$
$602$		0			0
$603$		−	1280.86i		−	2.12415i
$604$		0			0
$605$	−44.9630			−0.0743191
$606$		0			0
$607$		−	95.8823i		−	0.157961i	−0.996876	−	0.0789805i	$-0.974834\pi$
$607$		−	95.8823i		−	0.157961i	0.996876	−	0.0789805i	$-0.0251665\pi$
$608$		0			0
$609$	809.207			1.32875
$610$		0			0
$611$			142.158i			0.232665i
$612$		0			0
$613$	740.997			1.20880			0.604402	−	0.796679i	$-0.293412\pi$
$613$	740.997			1.20880			0.604402	+	0.796679i	$0.293412\pi$
$614$		0			0
$615$	543.070			0.883041
$616$		0			0
$617$	−274.604			−0.445064			−0.222532	−	0.974925i	$-0.571432\pi$
$617$	−274.604			−0.445064			−0.222532	+	0.974925i	$0.571432\pi$
$618$		0			0
$619$	−610.524			−0.986307			−0.493153	−	0.869942i	$-0.664156\pi$
$619$	−610.524			−0.986307			−0.493153	+	0.869942i	$0.664156\pi$
$620$		0			0
$621$			689.378i			1.11011i
$622$		0			0
$623$		−	995.997i		−	1.59871i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−961.913	+	153.287i	−1.53415	+	0.244478i
$628$		0			0
$629$		−	919.402i		−	1.46169i
$630$		0			0
$631$	819.064			1.29804			0.649021	−	0.760771i	$-0.275179\pi$
$631$	819.064			1.29804			0.649021	+	0.760771i	$0.275179\pi$
$632$		0			0
$633$	−1349.22			−2.13146
$634$		0			0
$635$			114.651i			0.180553i
$636$		0			0
$637$			32.7078i			0.0513465i
$638$		0			0
$639$		−	1324.45i		−	2.07270i
$640$		0			0
$641$			1110.13i			1.73187i	0.500152	+	0.865937i	$0.333277\pi$
$641$			1110.13i			1.73187i	−0.500152	+	0.865937i	$0.666723\pi$
$642$		0			0
$643$	346.176			0.538376			0.269188	−	0.963088i	$-0.413245\pi$
$643$	346.176			0.538376			0.269188	+	0.963088i	$0.413245\pi$
$644$		0			0
$645$		−	495.519i		−	0.768247i
$646$		0			0
$647$	708.711			1.09538			0.547690	−	0.836681i	$-0.315507\pi$
$647$	708.711			1.09538			0.547690	+	0.836681i	$0.315507\pi$
$648$		0			0
$649$			880.999i			1.35747i
$650$		0			0
$651$	1503.29			2.30920
$652$		0			0
$653$	−628.097			−0.961864			−0.480932	−	0.876758i	$-0.659702\pi$
$653$	−628.097			−0.961864			−0.480932	+	0.876758i	$0.659702\pi$
$654$		0			0
$655$	249.240			0.380519
$656$		0			0
$657$	2051.80			3.12298
$658$		0			0
$659$		−	635.622i		−	0.964525i	−0.876027	−	0.482262i	$-0.839815\pi$
$659$		−	635.622i		−	0.964525i	0.876027	−	0.482262i	$-0.160185\pi$
$660$		0			0
$661$			435.287i			0.658528i	0.944238	+	0.329264i	$0.106801\pi$
$661$			435.287i			0.658528i	−0.944238	+	0.329264i	$0.893199\pi$
$662$		0			0
$663$	−260.202			−0.392462
$664$		0			0
$665$	−42.4878	−	266.621i	−0.0638915	−	0.400933i
$666$		0			0
$667$			418.650i			0.627661i
$668$		0			0
$669$	−1126.21			−1.68342
$670$		0			0
$671$	963.668			1.43617
$672$		0			0
$673$		−	428.742i		−	0.637061i	−0.947913	−	0.318530i	$-0.896811\pi$
$673$		−	428.742i		−	0.637061i	0.947913	−	0.318530i	$-0.103189\pi$
$674$		0			0
$675$		−	205.417i		−	0.304321i
$676$		0			0
$677$		−	162.012i		−	0.239309i	−0.992816	−	0.119654i	$-0.961821\pi$
$677$		−	162.012i		−	0.239309i	0.992816	−	0.119654i	$-0.0381786\pi$
$678$		0			0
$679$			357.128i			0.525962i
$680$		0			0
$681$	796.300			1.16931
$682$		0			0
$683$			884.714i			1.29533i	0.761923	+	0.647667i	$0.224255\pi$
$683$			884.714i			1.29533i	−0.761923	+	0.647667i	$0.775745\pi$
$684$		0			0
$685$	−230.873			−0.337041
$686$		0			0
$687$			1324.34i			1.92771i
$688$		0			0
$689$	−35.9852			−0.0522281
$690$		0			0
$691$	1042.25			1.50832			0.754160	−	0.656691i	$-0.228044\pi$
$691$	1042.25			1.50832			0.754160	+	0.656691i	$0.228044\pi$
$692$		0			0
$693$	1088.28			1.57039
$694$		0			0
$695$	−27.1702			−0.0390938
$696$		0			0
$697$			639.111i			0.916946i
$698$		0			0
$699$		−	1804.21i		−	2.58113i
$700$		0			0
$701$	−713.782			−1.01823			−0.509117	−	0.860697i	$-0.670028\pi$
$701$	−713.782			−1.01823			−0.509117	+	0.860697i	$0.670028\pi$
$702$		0			0
$703$	1284.43	−	204.682i	1.82706	−	0.291155i
$704$		0			0
$705$		−	427.415i		−	0.606263i
$706$		0			0
$707$	−174.482			−0.246793
$708$		0			0
$709$	−227.442			−0.320792			−0.160396	−	0.987053i	$-0.551277\pi$
$709$	−227.442			−0.320792			−0.160396	+	0.987053i	$0.551277\pi$
$710$		0			0
$711$		−	218.655i		−	0.307532i
$712$		0			0
$713$			777.738i			1.09080i
$714$		0			0
$715$			85.2551i			0.119238i
$716$		0			0
$717$			899.926i			1.25513i
$718$		0			0
$719$	−413.703			−0.575386			−0.287693	−	0.957723i	$-0.592888\pi$
$719$	−413.703			−0.575386			−0.287693	+	0.957723i	$0.592888\pi$
$720$		0			0
$721$			128.523i			0.178256i
$722$		0			0
$723$	180.861			0.250153
$724$		0			0
$725$		−	124.747i		−	0.172065i
$726$		0			0
$727$	297.592			0.409342			0.204671	−	0.978831i	$-0.434388\pi$
$727$	297.592			0.409342			0.204671	+	0.978831i	$0.434388\pi$
$728$		0			0
$729$	928.313			1.27341
$730$		0			0
$731$	583.151			0.797744
$732$		0			0
$733$	−772.246			−1.05354			−0.526771	−	0.850007i	$-0.676597\pi$
$733$	−772.246			−1.05354			−0.526771	+	0.850007i	$0.676597\pi$
$734$		0			0
$735$		−	98.3395i		−	0.133795i
$736$		0			0
$737$		−	754.606i		−	1.02389i
$738$		0			0
$739$	−1209.35			−1.63646			−0.818231	−	0.574889i	$-0.805045\pi$
$739$	−1209.35			−1.63646			−0.818231	+	0.574889i	$0.805045\pi$
$740$		0			0
$741$	−57.9276	−	363.509i	−0.0781749	−	0.490565i
$742$		0			0
$743$		−	62.2804i		−	0.0838229i	−0.999121	−	0.0419114i	$-0.986655\pi$
$743$		−	62.2804i		−	0.0838229i	0.999121	−	0.0419114i	$-0.0133447\pi$
$744$		0			0
$745$	552.489			0.741596
$746$		0			0
$747$	−1854.31			−2.48235
$748$		0			0
$749$		−	184.515i		−	0.246348i
$750$		0			0
$751$		−	1440.03i		−	1.91748i	−0.284285	−	0.958740i	$-0.591756\pi$
$751$		−	1440.03i		−	1.91748i	0.284285	−	0.958740i	$-0.408244\pi$
$752$		0			0
$753$		−	1557.18i		−	2.06796i
$754$		0			0
$755$		−	138.429i		−	0.183350i
$756$		0			0
$757$	−513.483			−0.678313			−0.339157	−	0.940730i	$-0.610142\pi$
$757$	−513.483			−0.678313			−0.339157	+	0.940730i	$0.610142\pi$
$758$		0			0
$759$			860.238i			1.13338i
$760$		0			0
$761$	134.340			0.176531			0.0882654	−	0.996097i	$-0.471868\pi$
$761$	134.340			0.176531			0.0882654	+	0.996097i	$0.471868\pi$
$762$		0			0
$763$			655.009i			0.858465i
$764$		0			0
$765$	512.036			0.669328
$766$		0			0
$767$	−332.931			−0.434069
$768$		0			0
$769$	−103.497			−0.134586			−0.0672932	−	0.997733i	$-0.521436\pi$
$769$	−103.497			−0.134586			−0.0672932	+	0.997733i	$0.521436\pi$
$770$		0			0
$771$	−1504.16			−1.95092
$772$		0			0
$773$		−	1361.14i		−	1.76086i	−0.474177	−	0.880429i	$-0.657254\pi$
$773$		−	1361.14i		−	1.76086i	0.474177	−	0.880429i	$-0.342746\pi$
$774$		0			0
$775$		−	231.746i		−	0.299027i
$776$		0			0
$777$	−2220.24			−2.85746
$778$		0			0
$779$	−892.853	+	142.282i	−1.14615	+	0.182647i
$780$		0			0
$781$		−	780.288i		−	0.999088i
$782$		0			0
$783$	−1025.00			−1.30907
$784$		0			0
$785$	166.616			0.212250
$786$		0			0
$787$		−	945.437i		−	1.20132i	−0.799506	−	0.600659i	$-0.794905\pi$
$787$		−	945.437i		−	1.20132i	0.799506	−	0.600659i	$-0.205095\pi$
$788$		0			0
$789$		−	1730.86i		−	2.19374i
$790$		0			0
$791$			829.173i			1.04826i
$792$		0			0
$793$			364.172i			0.459233i
$794$		0			0
$795$	108.193			0.136092
$796$		0			0
$797$		−	642.975i		−	0.806745i	−0.915036	−	0.403372i	$-0.867838\pi$
$797$		−	642.975i		−	0.806745i	0.915036	−	0.403372i	$-0.132162\pi$
$798$		0			0
$799$	503.002			0.629540
$800$		0			0
$801$			2672.19i			3.33607i
$802$		0			0
$803$	1208.79			1.50535
$804$		0			0
$805$	−238.439			−0.296197
$806$		0			0
$807$	−169.760			−0.210359
$808$		0			0
$809$	−1206.61			−1.49148			−0.745740	−	0.666238i	$-0.767904\pi$
$809$	−1206.61			−1.49148			−0.745740	+	0.666238i	$0.767904\pi$
$810$		0			0
$811$			1322.86i			1.63115i	0.578651	+	0.815575i	$0.303579\pi$
$811$			1322.86i			1.63115i	−0.578651	+	0.815575i	$0.696421\pi$
$812$		0			0
$813$		−	1946.31i		−	2.39398i
$814$		0			0
$815$	150.409			0.184550
$816$		0			0
$817$	129.824	+	814.675i	0.158903	+	0.997154i
$818$		0			0
$819$			411.262i			0.502151i
$820$		0			0
$821$	−288.808			−0.351776			−0.175888	−	0.984410i	$-0.556280\pi$
$821$	−288.808			−0.351776			−0.175888	+	0.984410i	$0.556280\pi$
$822$		0			0
$823$	−1068.46			−1.29825			−0.649123	−	0.760684i	$-0.724864\pi$
$823$	−1068.46			−1.29825			−0.649123	+	0.760684i	$0.724864\pi$
$824$		0			0
$825$		−	256.329i		−	0.310702i
$826$		0			0
$827$		−	393.538i		−	0.475862i	−0.971282	−	0.237931i	$-0.923531\pi$
$827$		−	393.538i		−	0.475862i	0.971282	−	0.237931i	$-0.0764692\pi$
$828$		0			0
$829$			814.069i			0.981989i	0.871163	+	0.490995i	$0.163367\pi$
$829$			814.069i			0.981989i	−0.871163	+	0.490995i	$0.836633\pi$
$830$		0			0
$831$			172.111i			0.207113i
$832$		0			0
$833$	115.731			0.138932
$834$		0			0
$835$		−	370.436i		−	0.443636i
$836$		0			0
$837$	−1904.18			−2.27501
$838$		0			0
$839$			219.914i			0.262115i	0.991375	+	0.131057i	$0.0418372\pi$
$839$			219.914i			0.262115i	−0.991375	+	0.131057i	$0.958163\pi$
$840$		0			0
$841$	218.529			0.259845
$842$		0			0
$843$	−1068.04			−1.26695
$844$		0			0
$845$	345.677			0.409086
$846$		0			0
$847$	−127.782			−0.150865
$848$		0			0
$849$			1740.51i			2.05008i
$850$		0			0
$851$		−	1148.66i		−	1.34978i
$852$		0			0
$853$	719.082			0.843003			0.421502	−	0.906828i	$-0.361503\pi$
$853$	719.082			0.843003			0.421502	+	0.906828i	$0.361503\pi$
$854$		0			0
$855$	113.992	+	715.326i	0.133324	+	0.836638i
$856$		0			0
$857$			153.486i			0.179097i	0.995982	+	0.0895485i	$0.0285424\pi$
$857$			153.486i			0.179097i	−0.995982	+	0.0895485i	$0.971458\pi$
$858$		0			0
$859$	−914.298			−1.06438			−0.532188	−	0.846627i	$-0.678630\pi$
$859$	−914.298			−1.06438			−0.532188	+	0.846627i	$0.678630\pi$
$860$		0			0
$861$	1543.38			1.79254
$862$		0			0
$863$		−	1647.58i		−	1.90914i	−0.297992	−	0.954568i	$-0.596317\pi$
$863$		−	1647.58i		−	1.90914i	0.297992	−	0.954568i	$-0.403683\pi$
$864$		0			0
$865$		−	691.939i		−	0.799930i
$866$		0			0
$867$		−	554.338i		−	0.639375i
$868$		0			0
$869$		−	128.818i		−	0.148237i
$870$		0			0
$871$	285.167			0.327402
$872$		0			0
$873$		−	958.151i		−	1.09754i
$874$		0			0
$875$	71.0486			0.0811984
$876$		0			0
$877$		−	819.184i		−	0.934076i	−0.884237	−	0.467038i	$-0.845321\pi$
$877$		−	819.184i		−	0.934076i	0.884237	−	0.467038i	$-0.154679\pi$
$878$		0			0
$879$	2090.06			2.37777
$880$		0			0
$881$	899.647			1.02117			0.510583	−	0.859829i	$-0.329430\pi$
$881$	899.647			1.02117			0.510583	+	0.859829i	$0.329430\pi$
$882$		0			0
$883$	1007.40			1.14088			0.570442	−	0.821338i	$-0.306772\pi$
$883$	1007.40			1.14088			0.570442	+	0.821338i	$0.306772\pi$
$884$		0			0
$885$	1000.99			1.13107
$886$		0			0
$887$		−	668.366i		−	0.753513i	−0.926312	−	0.376756i	$-0.877039\pi$
$887$		−	668.366i		−	0.753513i	0.926312	−	0.376756i	$-0.122961\pi$
$888$		0			0
$889$			325.832i			0.366515i
$890$		0			0
$891$	−564.892			−0.633998
$892$		0			0
$893$	111.981	+	702.706i	0.125399	+	0.786905i
$894$		0			0
$895$			473.831i			0.529420i
$896$		0			0
$897$	−325.086			−0.362414
$898$		0			0
$899$	−1156.38			−1.28630
$900$		0			0
$901$			127.327i			0.141317i
$902$		0			0
$903$		−	1408.24i		−	1.55951i
$904$		0			0
$905$		−	226.097i		−	0.249831i
$906$		0			0
$907$			1186.27i			1.30791i	0.756534	+	0.653954i	$0.226891\pi$
$907$			1186.27i			1.30791i	−0.756534	+	0.653954i	$0.773109\pi$
$908$		0			0
$909$	468.125			0.514989
$910$		0			0
$911$		−	526.830i		−	0.578298i	−0.957284	−	0.289149i	$-0.906628\pi$
$911$		−	526.830i		−	0.578298i	0.957284	−	0.289149i	$-0.0933724\pi$
$912$		0			0
$913$	−1092.45			−1.19655
$914$		0			0
$915$		−	1094.92i		−	1.19664i
$916$		0			0
$917$	708.327			0.772439
$918$		0			0
$919$	−959.827			−1.04443			−0.522213	−	0.852815i	$-0.674893\pi$
$919$	−959.827			−1.04443			−0.522213	+	0.852815i	$0.674893\pi$
$920$		0			0
$921$	−1201.13			−1.30416
$922$		0			0
$923$	294.872			0.319471
$924$		0			0
$925$			342.272i			0.370023i
$926$		0			0
$927$		−	344.818i		−	0.371972i
$928$		0			0
$929$	−288.443			−0.310488			−0.155244	−	0.987876i	$-0.549616\pi$
$929$	−288.443			−0.310488			−0.155244	+	0.987876i	$0.549616\pi$
$930$		0			0
$931$	25.7645	+	161.678i	0.0276740	+	0.173661i
$932$		0			0
$933$			1005.65i			1.07786i
$934$		0			0
$935$	301.660			0.322631
$936$		0			0
$937$	1072.58			1.14470			0.572350	−	0.820009i	$-0.306032\pi$
$937$	1072.58			1.14470			0.572350	+	0.820009i	$0.306032\pi$
$938$		0			0
$939$		−	142.394i		−	0.151644i
$940$		0			0
$941$			250.856i			0.266585i	0.991077	+	0.133293i	$0.0425550\pi$
$941$			250.856i			0.266585i	−0.991077	+	0.133293i	$0.957445\pi$
$942$		0			0
$943$			798.478i			0.846743i
$944$		0			0
$945$		−	583.783i		−	0.617760i
$946$		0			0
$947$	346.018			0.365383			0.182692	−	0.983170i	$-0.441519\pi$
$947$	346.018			0.365383			0.182692	+	0.983170i	$0.441519\pi$
$948$		0			0
$949$			456.805i			0.481354i
$950$		0			0
$951$	−1902.19			−2.00020
$952$		0			0
$953$		−	1194.91i		−	1.25384i	−0.779084	−	0.626920i	$-0.784315\pi$
$953$		−	1194.91i		−	1.25384i	0.779084	−	0.626920i	$-0.215685\pi$
$954$		0			0
$955$	127.905			0.133932
$956$		0			0
$957$	−1279.05			−1.33652
$958$		0			0
$959$	−656.128			−0.684180
$960$		0			0
$961$	−1187.24			−1.23543
$962$		0			0
$963$			495.042i			0.514062i
$964$		0			0
$965$			349.791i			0.362477i
$966$		0			0
$967$	−1235.74			−1.27791			−0.638955	−	0.769244i	$-0.720633\pi$
$967$	−1235.74			−1.27791			−0.638955	+	0.769244i	$0.720633\pi$
$968$		0			0
$969$	−1286.21	+	204.966i	−1.32736	+	0.211524i
$970$		0			0
$971$			882.995i			0.909367i	0.890653	+	0.454683i	$0.150248\pi$
$971$			882.995i			0.909367i	−0.890653	+	0.454683i	$0.849752\pi$
$972$		0			0
$973$	−77.2162			−0.0793589
$974$		0			0
$975$	96.8672			0.0993509
$976$		0			0
$977$			254.773i			0.260771i	0.991463	+	0.130385i	$0.0416214\pi$
$977$			254.773i			0.260771i	−0.991463	+	0.130385i	$0.958379\pi$
$978$		0			0
$979$			1574.29i			1.60806i
$980$		0			0
$981$		−	1757.35i		−	1.79138i
$982$		0			0
$983$			966.862i			0.983583i	0.870713	+	0.491791i	$0.163658\pi$
$983$			966.862i			0.983583i	−0.870713	+	0.491791i	$0.836342\pi$
$984$		0			0
$985$	376.854			0.382592
$986$		0			0
$987$		−	1214.69i		−	1.23069i
$988$		0			0
$989$	728.564			0.736667
$990$		0			0
$991$			1440.50i			1.45358i	0.686857	+	0.726792i	$0.258990\pi$
$991$			1440.50i			1.45358i	−0.686857	+	0.726792i	$0.741010\pi$
$992$		0			0
$993$	19.1295			0.0192644
$994$		0			0
$995$	−91.6140			−0.0920744
$996$		0			0
$997$	−1340.38			−1.34441			−0.672207	−	0.740363i	$-0.734654\pi$
$997$	−1340.38			−1.34441			−0.672207	+	0.740363i	$0.734654\pi$
$998$		0			0
$999$	2812.33			2.81515

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.3.h.a.721.12		12
4.3	odd	2		95.3.c.a.56.7	yes	12
12.11	even	2		855.3.e.a.721.6		12
19.18	odd	2	inner	1520.3.h.a.721.1		12
20.3	even	4		475.3.d.c.474.14		24
20.7	even	4		475.3.d.c.474.11		24
20.19	odd	2		475.3.c.g.151.6		12
76.75	even	2		95.3.c.a.56.6	✓	12
228.227	odd	2		855.3.e.a.721.7		12
380.227	odd	4		475.3.d.c.474.13		24
380.303	odd	4		475.3.d.c.474.12		24
380.379	even	2		475.3.c.g.151.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.3.c.a.56.6	✓	12	76.75	even	2
95.3.c.a.56.7	yes	12	4.3	odd	2
475.3.c.g.151.6		12	20.19	odd	2
475.3.c.g.151.7		12	380.379	even	2
475.3.d.c.474.11		24	20.7	even	4
475.3.d.c.474.12		24	380.303	odd	4
475.3.d.c.474.13		24	380.227	odd	4
475.3.d.c.474.14		24	20.3	even	4
855.3.e.a.721.6		12	12.11	even	2
855.3.e.a.721.7		12	228.227	odd	2
1520.3.h.a.721.1		12	19.18	odd	2	inner
1520.3.h.a.721.12		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1520.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.10387i q^{3} +2.23607 q^{5} +6.35478 q^{7} -17.0495 q^{9} +O(q^{10})\)	\(q+5.10387i q^{3} +2.23607 q^{5} +6.35478 q^{7} -17.0495 q^{9} -10.0445 q^{11} -3.79583i q^{13} +11.4126i q^{15} -13.4309 q^{17} +(-2.99005 - 18.7633i) q^{19} +32.4340i q^{21} -16.7800 q^{23} +5.00000 q^{25} -41.0834i q^{27} -24.9494i q^{29} -46.3492i q^{31} -51.2658i q^{33} +14.2097 q^{35} +68.4543i q^{37} +19.3734 q^{39} -47.5852i q^{41} -43.4186 q^{43} -38.1238 q^{45} -37.4512 q^{47} -8.61675 q^{49} -68.5494i q^{51} -9.48017i q^{53} -22.4602 q^{55} +(95.7651 - 15.2608i) q^{57} -87.7096i q^{59} -95.9399 q^{61} -108.346 q^{63} -8.48774i q^{65} +75.1263i q^{67} -85.6428i q^{69} +77.6831i q^{71} -120.344 q^{73} +25.5193i q^{75} -63.8306 q^{77} +12.8248i q^{79} +56.2390 q^{81} +108.761 q^{83} -30.0324 q^{85} +127.338 q^{87} -156.732i q^{89} -24.1217i q^{91} +236.560 q^{93} +(-6.68596 - 41.9559i) q^{95} +56.1983i q^{97} +171.253 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 20 q^{7} - 48 q^{9}+O(q^{10}) \) 12 * q - 20 * q^7 - 48 * q^9	\( 12 q - 20 q^{7} - 48 q^{9} - 32 q^{11} - 44 q^{17} - 8 q^{19} - 36 q^{23} + 60 q^{25} + 40 q^{35} - 76 q^{39} - 320 q^{43} - 40 q^{45} + 56 q^{47} + 72 q^{49} + 60 q^{57} - 296 q^{61} + 96 q^{63} - 244 q^{73} - 200 q^{77} - 372 q^{81} + 160 q^{83} + 160 q^{85} - 444 q^{87} + 296 q^{93} + 80 q^{95} + 312 q^{99}+O(q^{100}) \) 12 * q - 20 * q^7 - 48 * q^9 - 32 * q^11 - 44 * q^17 - 8 * q^19 - 36 * q^23 + 60 * q^25 + 40 * q^35 - 76 * q^39 - 320 * q^43 - 40 * q^45 + 56 * q^47 + 72 * q^49 + 60 * q^57 - 296 * q^61 + 96 * q^63 - 244 * q^73 - 200 * q^77 - 372 * q^81 + 160 * q^83 + 160 * q^85 - 444 * q^87 + 296 * q^93 + 80 * q^95 + 312 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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