LMFDB - Embedded newform 8048.2.a.t.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.a (trivial)

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:	yes
Analytic conductor:	$64.2636035467$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	no (minimal twist has level 2012)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0551142 q^{3} +0.393007 q^{5} -3.87485 q^{7} -2.99696 q^{9} +O(q^{10})$	$q-0.0551142 q^{3} +0.393007 q^{5} -3.87485 q^{7} -2.99696 q^{9} -1.57160 q^{11} -5.66419 q^{13} -0.0216603 q^{15} -7.20439 q^{17} +1.16936 q^{19} +0.213559 q^{21} +6.02345 q^{23} -4.84555 q^{25} +0.330518 q^{27} -2.81854 q^{29} -7.23219 q^{31} +0.0866174 q^{33} -1.52284 q^{35} -8.36968 q^{37} +0.312177 q^{39} +2.39990 q^{41} +11.9873 q^{43} -1.17783 q^{45} -3.86551 q^{47} +8.01447 q^{49} +0.397064 q^{51} +5.56941 q^{53} -0.617650 q^{55} -0.0644483 q^{57} -11.9043 q^{59} -9.68030 q^{61} +11.6128 q^{63} -2.22607 q^{65} -6.38639 q^{67} -0.331978 q^{69} -7.68406 q^{71} -1.31232 q^{73} +0.267058 q^{75} +6.08971 q^{77} -8.47642 q^{79} +8.97267 q^{81} +14.2293 q^{83} -2.83138 q^{85} +0.155342 q^{87} +0.290799 q^{89} +21.9479 q^{91} +0.398596 q^{93} +0.459567 q^{95} -13.1242 q^{97} +4.71003 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9}+O(q^{10}) $ 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9	$ 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9} + 9 q^{11} - 16 q^{13} + 22 q^{15} + q^{17} + 12 q^{19} - 2 q^{21} + 22 q^{23} + 18 q^{25} + 43 q^{27} - 13 q^{29} + 28 q^{31} + 3 q^{33} + 12 q^{35} - 39 q^{37} + 11 q^{39} + 4 q^{41} + 50 q^{43} - 6 q^{45} + 27 q^{47} + 16 q^{49} + 37 q^{51} - 24 q^{53} + 49 q^{55} - q^{57} + 22 q^{59} - 22 q^{61} + 49 q^{63} - 14 q^{65} + 62 q^{67} - 17 q^{69} + 21 q^{71} - 6 q^{73} + 52 q^{75} - 24 q^{77} + 65 q^{79} + 29 q^{81} + 18 q^{83} - 54 q^{85} + 31 q^{87} + q^{89} + 45 q^{91} - 26 q^{93} + 53 q^{95} - 2 q^{97} + 58 q^{99}+O(q^{100}) $ 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9 + 9 * q^11 - 16 * q^13 + 22 * q^15 + q^17 + 12 * q^19 - 2 * q^21 + 22 * q^23 + 18 * q^25 + 43 * q^27 - 13 * q^29 + 28 * q^31 + 3 * q^33 + 12 * q^35 - 39 * q^37 + 11 * q^39 + 4 * q^41 + 50 * q^43 - 6 * q^45 + 27 * q^47 + 16 * q^49 + 37 * q^51 - 24 * q^53 + 49 * q^55 - q^57 + 22 * q^59 - 22 * q^61 + 49 * q^63 - 14 * q^65 + 62 * q^67 - 17 * q^69 + 21 * q^71 - 6 * q^73 + 52 * q^75 - 24 * q^77 + 65 * q^79 + 29 * q^81 + 18 * q^83 - 54 * q^85 + 31 * q^87 + q^89 + 45 * q^91 - 26 * q^93 + 53 * q^95 - 2 * q^97 + 58 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−0.0551142	−0.0318202	−0.0159101	−	0.999873i	$-0.505065\pi$
$3$	−0.0551142	−0.0318202	−0.0159101	+	0.999873i	$0.505065\pi$
$4$	0	0
$5$	0.393007	0.175758	0.0878790	−	0.996131i	$-0.471991\pi$
$5$	0.393007	0.175758	0.0878790	+	0.996131i	$0.471991\pi$
$6$	0	0
$7$	−3.87485	−1.46456	−0.732278	−	0.681006i	$-0.761543\pi$
$7$	−3.87485	−1.46456	−0.732278	+	0.681006i	$0.761543\pi$
$8$	0	0
$9$	−2.99696	−0.998987
$10$	0	0
$11$	−1.57160	−0.473855	−0.236928	−	0.971527i	$-0.576140\pi$
$11$	−1.57160	−0.473855	−0.236928	+	0.971527i	$0.576140\pi$
$12$	0	0
$13$	−5.66419	−1.57096	−0.785482	−	0.618884i	$-0.787585\pi$
$13$	−5.66419	−1.57096	−0.785482	+	0.618884i	$0.787585\pi$
$14$	0	0
$15$	−0.0216603	−0.00559265
$16$	0	0
$17$	−7.20439	−1.74732	−0.873660	−	0.486536i	$-0.838260\pi$
$17$	−7.20439	−1.74732	−0.873660	+	0.486536i	$0.838260\pi$
$18$	0	0
$19$	1.16936	0.268270	0.134135	−	0.990963i	$-0.457174\pi$
$19$	1.16936	0.268270	0.134135	+	0.990963i	$0.457174\pi$
$20$	0	0
$21$	0.213559	0.0466024
$22$	0	0
$23$	6.02345	1.25598	0.627989	−	0.778223i	$-0.283878\pi$
$23$	6.02345	1.25598	0.627989	+	0.778223i	$0.283878\pi$
$24$	0	0
$25$	−4.84555	−0.969109
$26$	0	0
$27$	0.330518	0.0636082
$28$	0	0
$29$	−2.81854	−0.523390	−0.261695	−	0.965151i	$-0.584281\pi$
$29$	−2.81854	−0.523390	−0.261695	+	0.965151i	$0.584281\pi$
$30$	0	0
$31$	−7.23219	−1.29894	−0.649470	−	0.760388i	$-0.725009\pi$
$31$	−7.23219	−1.29894	−0.649470	+	0.760388i	$0.725009\pi$
$32$	0	0
$33$	0.0866174	0.0150782
$34$	0	0
$35$	−1.52284	−0.257408
$36$	0	0
$37$	−8.36968	−1.37597	−0.687983	−	0.725727i	$-0.741504\pi$
$37$	−8.36968	−1.37597	−0.687983	+	0.725727i	$0.741504\pi$
$38$	0	0
$39$	0.312177	0.0499884
$40$	0	0
$41$	2.39990	0.374801	0.187401	−	0.982284i	$-0.439994\pi$
$41$	2.39990	0.374801	0.187401	+	0.982284i	$0.439994\pi$
$42$	0	0
$43$	11.9873	1.82804	0.914021	−	0.405667i	$-0.132961\pi$
$43$	11.9873	1.82804	0.914021	+	0.405667i	$0.132961\pi$
$44$	0	0
$45$	−1.17783	−0.175580
$46$	0	0
$47$	−3.86551	−0.563843	−0.281921	−	0.959438i	$-0.590972\pi$
$47$	−3.86551	−0.563843	−0.281921	+	0.959438i	$0.590972\pi$
$48$	0	0
$49$	8.01447	1.14492
$50$	0	0
$51$	0.397064	0.0556001
$52$	0	0
$53$	5.56941	0.765017	0.382509	−	0.923952i	$-0.375060\pi$
$53$	5.56941	0.765017	0.382509	+	0.923952i	$0.375060\pi$
$54$	0	0
$55$	−0.617650	−0.0832839
$56$	0	0
$57$	−0.0644483	−0.00853639
$58$	0	0
$59$	−11.9043	−1.54981	−0.774905	−	0.632078i	$-0.782202\pi$
$59$	−11.9043	−1.54981	−0.774905	+	0.632078i	$0.782202\pi$
$60$	0	0
$61$	−9.68030	−1.23944	−0.619718	−	0.784825i	$-0.712753\pi$
$61$	−9.68030	−1.23944	−0.619718	+	0.784825i	$0.712753\pi$
$62$	0	0
$63$	11.6128	1.46307
$64$	0	0
$65$	−2.22607	−0.276110
$66$	0	0
$67$	−6.38639	−0.780222	−0.390111	−	0.920768i	$-0.627563\pi$
$67$	−6.38639	−0.780222	−0.390111	+	0.920768i	$0.627563\pi$
$68$	0	0
$69$	−0.331978	−0.0399654
$70$	0	0
$71$	−7.68406	−0.911931	−0.455965	−	0.889998i	$-0.650706\pi$
$71$	−7.68406	−0.911931	−0.455965	+	0.889998i	$0.650706\pi$
$72$	0	0
$73$	−1.31232	−0.153595	−0.0767976	−	0.997047i	$-0.524470\pi$
$73$	−1.31232	−0.153595	−0.0767976	+	0.997047i	$0.524470\pi$
$74$	0	0
$75$	0.267058	0.0308372
$76$	0	0
$77$	6.08971	0.693987
$78$	0	0
$79$	−8.47642	−0.953672	−0.476836	−	0.878992i	$-0.658216\pi$
$79$	−8.47642	−0.953672	−0.476836	+	0.878992i	$0.658216\pi$
$80$	0	0
$81$	8.97267	0.996963
$82$	0	0
$83$	14.2293	1.56187	0.780934	−	0.624613i	$-0.214743\pi$
$83$	14.2293	1.56187	0.780934	+	0.624613i	$0.214743\pi$
$84$	0	0
$85$	−2.83138	−0.307106
$86$	0	0
$87$	0.155342	0.0166544
$88$	0	0
$89$	0.290799	0.0308247	0.0154123	−	0.999881i	$-0.495094\pi$
$89$	0.290799	0.0308247	0.0154123	+	0.999881i	$0.495094\pi$
$90$	0	0
$91$	21.9479	2.30077
$92$	0	0
$93$	0.398596	0.0413325
$94$	0	0
$95$	0.459567	0.0471506
$96$	0	0
$97$	−13.1242	−1.33256	−0.666282	−	0.745700i	$-0.732115\pi$
$97$	−13.1242	−1.33256	−0.666282	+	0.745700i	$0.732115\pi$
$98$	0	0
$99$	4.71003	0.473375
$100$	0	0
$101$	1.67272	0.166442	0.0832211	−	0.996531i	$-0.473479\pi$
$101$	1.67272	0.166442	0.0832211	+	0.996531i	$0.473479\pi$
$102$	0	0
$103$	9.98384	0.983737	0.491868	−	0.870670i	$-0.336314\pi$
$103$	9.98384	0.983737	0.491868	+	0.870670i	$0.336314\pi$
$104$	0	0
$105$	0.0839303	0.00819076
$106$	0	0
$107$	−12.1390	−1.17352	−0.586759	−	0.809761i	$-0.699597\pi$
$107$	−12.1390	−1.17352	−0.586759	+	0.809761i	$0.699597\pi$
$108$	0	0
$109$	17.6179	1.68749	0.843745	−	0.536744i	$-0.180346\pi$
$109$	17.6179	1.68749	0.843745	+	0.536744i	$0.180346\pi$
$110$	0	0
$111$	0.461288	0.0437835
$112$	0	0
$113$	−0.175796	−0.0165375	−0.00826875	−	0.999966i	$-0.502632\pi$
$113$	−0.175796	−0.0165375	−0.00826875	+	0.999966i	$0.502632\pi$
$114$	0	0
$115$	2.36726	0.220748
$116$	0	0
$117$	16.9754	1.56937
$118$	0	0
$119$	27.9159	2.55905
$120$	0	0
$121$	−8.53007	−0.775461
$122$	0	0
$123$	−0.132269	−0.0119263
$124$	0	0
$125$	−3.86937	−0.346087
$126$	0	0
$127$	12.4370	1.10360	0.551802	−	0.833975i	$-0.313941\pi$
$127$	12.4370	1.10360	0.551802	+	0.833975i	$0.313941\pi$
$128$	0	0
$129$	−0.660669	−0.0581686
$130$	0	0
$131$	19.0424	1.66374	0.831869	−	0.554971i	$-0.187271\pi$
$131$	19.0424	1.66374	0.831869	+	0.554971i	$0.187271\pi$
$132$	0	0
$133$	−4.53110	−0.392896
$134$	0	0
$135$	0.129896	0.0111796
$136$	0	0
$137$	−11.7979	−1.00796	−0.503980	−	0.863715i	$-0.668132\pi$
$137$	−11.7979	−1.00796	−0.503980	+	0.863715i	$0.668132\pi$
$138$	0	0
$139$	−15.4218	−1.30806	−0.654029	−	0.756470i	$-0.726923\pi$
$139$	−15.4218	−1.30806	−0.654029	+	0.756470i	$0.726923\pi$
$140$	0	0
$141$	0.213044	0.0179416
$142$	0	0
$143$	8.90184	0.744410
$144$	0	0
$145$	−1.10771	−0.0919900
$146$	0	0
$147$	−0.441711	−0.0364317
$148$	0	0
$149$	17.2425	1.41256	0.706279	−	0.707933i	$-0.250372\pi$
$149$	17.2425	1.41256	0.706279	+	0.707933i	$0.250372\pi$
$150$	0	0
$151$	−18.8690	−1.53554	−0.767770	−	0.640726i	$-0.778634\pi$
$151$	−18.8690	−1.53554	−0.767770	+	0.640726i	$0.778634\pi$
$152$	0	0
$153$	21.5913	1.74555
$154$	0	0
$155$	−2.84230	−0.228299
$156$	0	0
$157$	−12.7485	−1.01744	−0.508720	−	0.860932i	$-0.669881\pi$
$157$	−12.7485	−1.01744	−0.508720	+	0.860932i	$0.669881\pi$
$158$	0	0
$159$	−0.306953	−0.0243430
$160$	0	0
$161$	−23.3400	−1.83945
$162$	0	0
$163$	19.1512	1.50004	0.750019	−	0.661416i	$-0.230044\pi$
$163$	19.1512	1.50004	0.750019	+	0.661416i	$0.230044\pi$
$164$	0	0
$165$	0.0340413	0.00265011
$166$	0	0
$167$	−12.7878	−0.989546	−0.494773	−	0.869022i	$-0.664749\pi$
$167$	−12.7878	−0.989546	−0.494773	+	0.869022i	$0.664749\pi$
$168$	0	0
$169$	19.0831	1.46793
$170$	0	0
$171$	−3.50453	−0.267998
$172$	0	0
$173$	20.9991	1.59653	0.798267	−	0.602304i	$-0.205751\pi$
$173$	20.9991	1.59653	0.798267	+	0.602304i	$0.205751\pi$
$174$	0	0
$175$	18.7758	1.41931
$176$	0	0
$177$	0.656096	0.0493152
$178$	0	0
$179$	12.1735	0.909889	0.454945	−	0.890520i	$-0.349659\pi$
$179$	12.1735	0.909889	0.454945	+	0.890520i	$0.349659\pi$
$180$	0	0
$181$	−18.2165	−1.35402	−0.677009	−	0.735974i	$-0.736724\pi$
$181$	−18.2165	−1.35402	−0.677009	+	0.735974i	$0.736724\pi$
$182$	0	0
$183$	0.533522	0.0394391
$184$	0	0
$185$	−3.28934	−0.241837
$186$	0	0
$187$	11.3224	0.827977
$188$	0	0
$189$	−1.28071	−0.0931577
$190$	0	0
$191$	15.1276	1.09460	0.547298	−	0.836938i	$-0.315656\pi$
$191$	15.1276	1.09460	0.547298	+	0.836938i	$0.315656\pi$
$192$	0	0
$193$	−14.7814	−1.06399	−0.531993	−	0.846749i	$-0.678557\pi$
$193$	−14.7814	−1.06399	−0.531993	+	0.846749i	$0.678557\pi$
$194$	0	0
$195$	0.122688	0.00878586
$196$	0	0
$197$	−15.3051	−1.09045	−0.545223	−	0.838291i	$-0.683555\pi$
$197$	−15.3051	−1.09045	−0.545223	+	0.838291i	$0.683555\pi$
$198$	0	0
$199$	15.0401	1.06616	0.533082	−	0.846063i	$-0.321034\pi$
$199$	15.0401	1.06616	0.533082	+	0.846063i	$0.321034\pi$
$200$	0	0
$201$	0.351981	0.0248268
$202$	0	0
$203$	10.9214	0.766534
$204$	0	0
$205$	0.943178	0.0658744
$206$	0	0
$207$	−18.0521	−1.25471
$208$	0	0
$209$	−1.83777	−0.127121
$210$	0	0
$211$	4.33711	0.298579	0.149290	−	0.988794i	$-0.452301\pi$
$211$	4.33711	0.298579	0.149290	+	0.988794i	$0.452301\pi$
$212$	0	0
$213$	0.423501	0.0290178
$214$	0	0
$215$	4.71108	0.321293
$216$	0	0
$217$	28.0236	1.90237
$218$	0	0
$219$	0.0723274	0.00488743
$220$	0	0
$221$	40.8070	2.74498
$222$	0	0
$223$	−17.2237	−1.15338	−0.576691	−	0.816963i	$-0.695656\pi$
$223$	−17.2237	−1.15338	−0.576691	+	0.816963i	$0.695656\pi$
$224$	0	0
$225$	14.5219	0.968128
$226$	0	0
$227$	−5.27466	−0.350091	−0.175046	−	0.984560i	$-0.556007\pi$
$227$	−5.27466	−0.350091	−0.175046	+	0.984560i	$0.556007\pi$
$228$	0	0
$229$	18.1676	1.20055	0.600275	−	0.799794i	$-0.295058\pi$
$229$	18.1676	1.20055	0.600275	+	0.799794i	$0.295058\pi$
$230$	0	0
$231$	−0.335630	−0.0220828
$232$	0	0
$233$	6.99180	0.458048	0.229024	−	0.973421i	$-0.426447\pi$
$233$	6.99180	0.458048	0.229024	+	0.973421i	$0.426447\pi$
$234$	0	0
$235$	−1.51917	−0.0990999
$236$	0	0
$237$	0.467171	0.0303460
$238$	0	0
$239$	7.07366	0.457557	0.228778	−	0.973479i	$-0.426527\pi$
$239$	7.07366	0.457557	0.228778	+	0.973479i	$0.426527\pi$
$240$	0	0
$241$	−8.50819	−0.548061	−0.274030	−	0.961721i	$-0.588357\pi$
$241$	−8.50819	−0.548061	−0.274030	+	0.961721i	$0.588357\pi$
$242$	0	0
$243$	−1.48607	−0.0953317
$244$	0	0
$245$	3.14974	0.201230
$246$	0	0
$247$	−6.62348	−0.421442
$248$	0	0
$249$	−0.784236	−0.0496990
$250$	0	0
$251$	2.03147	0.128225	0.0641127	−	0.997943i	$-0.479578\pi$
$251$	2.03147	0.128225	0.0641127	+	0.997943i	$0.479578\pi$
$252$	0	0
$253$	−9.46646	−0.595151
$254$	0	0
$255$	0.156049	0.00977216
$256$	0	0
$257$	30.9030	1.92768	0.963839	−	0.266485i	$-0.0858624\pi$
$257$	30.9030	1.92768	0.963839	+	0.266485i	$0.0858624\pi$
$258$	0	0
$259$	32.4313	2.01518
$260$	0	0
$261$	8.44706	0.522860
$262$	0	0
$263$	−8.24786	−0.508585	−0.254292	−	0.967127i	$-0.581843\pi$
$263$	−8.24786	−0.508585	−0.254292	+	0.967127i	$0.581843\pi$
$264$	0	0
$265$	2.18882	0.134458
$266$	0	0
$267$	−0.0160272	−0.000980847 0
$268$	0	0
$269$	−12.0666	−0.735716	−0.367858	−	0.929882i	$-0.619909\pi$
$269$	−12.0666	−0.735716	−0.367858	+	0.929882i	$0.619909\pi$
$270$	0	0
$271$	−21.2675	−1.29191	−0.645953	−	0.763377i	$-0.723540\pi$
$271$	−21.2675	−1.29191	−0.645953	+	0.763377i	$0.723540\pi$
$272$	0	0
$273$	−1.20964	−0.0732108
$274$	0	0
$275$	7.61526	0.459217
$276$	0	0
$277$	6.21148	0.373211	0.186606	−	0.982435i	$-0.440251\pi$
$277$	6.21148	0.373211	0.186606	+	0.982435i	$0.440251\pi$
$278$	0	0
$279$	21.6746	1.29762
$280$	0	0
$281$	21.9293	1.30819	0.654097	−	0.756411i	$-0.273049\pi$
$281$	21.9293	1.30819	0.654097	+	0.756411i	$0.273049\pi$
$282$	0	0
$283$	9.05101	0.538027	0.269013	−	0.963136i	$-0.413302\pi$
$283$	9.05101	0.538027	0.269013	+	0.963136i	$0.413302\pi$
$284$	0	0
$285$	−0.0253287	−0.00150034
$286$	0	0
$287$	−9.29926	−0.548918
$288$	0	0
$289$	34.9032	2.05313
$290$	0	0
$291$	0.723331	0.0424024
$292$	0	0
$293$	−23.6183	−1.37980	−0.689898	−	0.723906i	$-0.742345\pi$
$293$	−23.6183	−1.37980	−0.689898	+	0.723906i	$0.742345\pi$
$294$	0	0
$295$	−4.67848	−0.272392
$296$	0	0
$297$	−0.519441	−0.0301410
$298$	0	0
$299$	−34.1180	−1.97310
$300$	0	0
$301$	−46.4489	−2.67727
$302$	0	0
$303$	−0.0921907	−0.00529622
$304$	0	0
$305$	−3.80442	−0.217841
$306$	0	0
$307$	−26.0257	−1.48536	−0.742682	−	0.669644i	$-0.766447\pi$
$307$	−26.0257	−1.48536	−0.742682	+	0.669644i	$0.766447\pi$
$308$	0	0
$309$	−0.550251	−0.0313027
$310$	0	0
$311$	29.6716	1.68252	0.841261	−	0.540630i	$-0.181814\pi$
$311$	29.6716	1.68252	0.841261	+	0.540630i	$0.181814\pi$
$312$	0	0
$313$	−21.2878	−1.20326	−0.601628	−	0.798776i	$-0.705481\pi$
$313$	−21.2878	−1.20326	−0.601628	+	0.798776i	$0.705481\pi$
$314$	0	0
$315$	4.56391	0.257147
$316$	0	0
$317$	−3.12593	−0.175570	−0.0877849	−	0.996139i	$-0.527979\pi$
$317$	−3.12593	−0.175570	−0.0877849	+	0.996139i	$0.527979\pi$
$318$	0	0
$319$	4.42962	0.248011
$320$	0	0
$321$	0.669029	0.0373416
$322$	0	0
$323$	−8.42453	−0.468753
$324$	0	0
$325$	27.4461	1.52244
$326$	0	0
$327$	−0.970997	−0.0536962
$328$	0	0
$329$	14.9783	0.825779
$330$	0	0
$331$	7.51569	0.413099	0.206550	−	0.978436i	$-0.433776\pi$
$331$	7.51569	0.413099	0.206550	+	0.978436i	$0.433776\pi$
$332$	0	0
$333$	25.0836	1.37457
$334$	0	0
$335$	−2.50990	−0.137130
$336$	0	0
$337$	19.6404	1.06988	0.534940	−	0.844890i	$-0.320334\pi$
$337$	19.6404	1.06988	0.534940	+	0.844890i	$0.320334\pi$
$338$	0	0
$339$	0.00968885	0.000526226 0
$340$	0	0
$341$	11.3661	0.615509
$342$	0	0
$343$	−3.93093	−0.212250
$344$	0	0
$345$	−0.130470	−0.00702425
$346$	0	0
$347$	−0.975774	−0.0523823	−0.0261911	−	0.999657i	$-0.508338\pi$
$347$	−0.975774	−0.0523823	−0.0261911	+	0.999657i	$0.508338\pi$
$348$	0	0
$349$	−23.0700	−1.23491	−0.617455	−	0.786606i	$-0.711836\pi$
$349$	−23.0700	−1.23491	−0.617455	+	0.786606i	$0.711836\pi$
$350$	0	0
$351$	−1.87212	−0.0999261
$352$	0	0
$353$	11.7297	0.624311	0.312156	−	0.950031i	$-0.398949\pi$
$353$	11.7297	0.624311	0.312156	+	0.950031i	$0.398949\pi$
$354$	0	0
$355$	−3.01989	−0.160279
$356$	0	0
$357$	−1.53856	−0.0814294
$358$	0	0
$359$	−30.7685	−1.62390	−0.811950	−	0.583727i	$-0.801594\pi$
$359$	−30.7685	−1.62390	−0.811950	+	0.583727i	$0.801594\pi$
$360$	0	0
$361$	−17.6326	−0.928031
$362$	0	0
$363$	0.470128	0.0246753
$364$	0	0
$365$	−0.515750	−0.0269956
$366$	0	0
$367$	−12.9929	−0.678226	−0.339113	−	0.940746i	$-0.610127\pi$
$367$	−12.9929	−0.678226	−0.339113	+	0.940746i	$0.610127\pi$
$368$	0	0
$369$	−7.19241	−0.374422
$370$	0	0
$371$	−21.5806	−1.12041
$372$	0	0
$373$	−32.6383	−1.68995	−0.844974	−	0.534807i	$-0.820384\pi$
$373$	−32.6383	−1.68995	−0.844974	+	0.534807i	$0.820384\pi$
$374$	0	0
$375$	0.213257	0.0110125
$376$	0	0
$377$	15.9648	0.822227
$378$	0	0
$379$	−2.79558	−0.143599	−0.0717997	−	0.997419i	$-0.522874\pi$
$379$	−2.79558	−0.143599	−0.0717997	+	0.997419i	$0.522874\pi$
$380$	0	0
$381$	−0.685454	−0.0351169
$382$	0	0
$383$	−10.9543	−0.559738	−0.279869	−	0.960038i	$-0.590291\pi$
$383$	−10.9543	−0.559738	−0.279869	+	0.960038i	$0.590291\pi$
$384$	0	0
$385$	2.39330	0.121974
$386$	0	0
$387$	−35.9254	−1.82619
$388$	0	0
$389$	−23.4931	−1.19115	−0.595573	−	0.803301i	$-0.703075\pi$
$389$	−23.4931	−1.19115	−0.595573	+	0.803301i	$0.703075\pi$
$390$	0	0
$391$	−43.3953	−2.19460
$392$	0	0
$393$	−1.04950	−0.0529405
$394$	0	0
$395$	−3.33129	−0.167616
$396$	0	0
$397$	−6.07213	−0.304751	−0.152376	−	0.988323i	$-0.548692\pi$
$397$	−6.07213	−0.304751	−0.152376	+	0.988323i	$0.548692\pi$
$398$	0	0
$399$	0.249728	0.0125020
$400$	0	0
$401$	3.01333	0.150478	0.0752392	−	0.997166i	$-0.476028\pi$
$401$	3.01333	0.150478	0.0752392	+	0.997166i	$0.476028\pi$
$402$	0	0
$403$	40.9645	2.04059
$404$	0	0
$405$	3.52632	0.175224
$406$	0	0
$407$	13.1538	0.652009
$408$	0	0
$409$	−36.8401	−1.82163	−0.910814	−	0.412818i	$-0.864545\pi$
$409$	−36.8401	−1.82163	−0.910814	+	0.412818i	$0.864545\pi$
$410$	0	0
$411$	0.650230	0.0320735
$412$	0	0
$413$	46.1274	2.26978
$414$	0	0
$415$	5.59222	0.274511
$416$	0	0
$417$	0.849958	0.0416226
$418$	0	0
$419$	−26.5701	−1.29804	−0.649018	−	0.760773i	$-0.724820\pi$
$419$	−26.5701	−1.29804	−0.649018	+	0.760773i	$0.724820\pi$
$420$	0	0
$421$	−8.55428	−0.416910	−0.208455	−	0.978032i	$-0.566844\pi$
$421$	−8.55428	−0.416910	−0.208455	+	0.978032i	$0.566844\pi$
$422$	0	0
$423$	11.5848	0.563272
$424$	0	0
$425$	34.9092	1.69334
$426$	0	0
$427$	37.5097	1.81522
$428$	0	0
$429$	−0.490618	−0.0236872
$430$	0	0
$431$	−33.6376	−1.62027	−0.810133	−	0.586246i	$-0.800605\pi$
$431$	−33.6376	−1.62027	−0.810133	+	0.586246i	$0.800605\pi$
$432$	0	0
$433$	35.1631	1.68983	0.844916	−	0.534899i	$-0.179650\pi$
$433$	35.1631	1.68983	0.844916	+	0.534899i	$0.179650\pi$
$434$	0	0
$435$	0.0610503	0.00292714
$436$	0	0
$437$	7.04359	0.336941
$438$	0	0
$439$	16.1849	0.772462	0.386231	−	0.922402i	$-0.373777\pi$
$439$	16.1849	0.772462	0.386231	+	0.922402i	$0.373777\pi$
$440$	0	0
$441$	−24.0191	−1.14377
$442$	0	0
$443$	6.63691	0.315329	0.157665	−	0.987493i	$-0.449604\pi$
$443$	6.63691	0.315329	0.157665	+	0.987493i	$0.449604\pi$
$444$	0	0
$445$	0.114286	0.00541769
$446$	0	0
$447$	−0.950305	−0.0449479
$448$	0	0
$449$	−34.5801	−1.63194	−0.815968	−	0.578097i	$-0.803796\pi$
$449$	−34.5801	−1.63194	−0.815968	+	0.578097i	$0.803796\pi$
$450$	0	0
$451$	−3.77168	−0.177602
$452$	0	0
$453$	1.03995	0.0488612
$454$	0	0
$455$	8.62568	0.404378
$456$	0	0
$457$	−4.35985	−0.203945	−0.101973	−	0.994787i	$-0.532515\pi$
$457$	−4.35985	−0.203945	−0.101973	+	0.994787i	$0.532515\pi$
$458$	0	0
$459$	−2.38118	−0.111144
$460$	0	0
$461$	1.01621	0.0473298	0.0236649	−	0.999720i	$-0.492467\pi$
$461$	1.01621	0.0473298	0.0236649	+	0.999720i	$0.492467\pi$
$462$	0	0
$463$	−5.61029	−0.260732	−0.130366	−	0.991466i	$-0.541615\pi$
$463$	−5.61029	−0.260732	−0.130366	+	0.991466i	$0.541615\pi$
$464$	0	0
$465$	0.156651	0.00726452
$466$	0	0
$467$	7.22975	0.334553	0.167276	−	0.985910i	$-0.446503\pi$
$467$	7.22975	0.334553	0.167276	+	0.985910i	$0.446503\pi$
$468$	0	0
$469$	24.7463	1.14268
$470$	0	0
$471$	0.702623	0.0323751
$472$	0	0
$473$	−18.8392	−0.866227
$474$	0	0
$475$	−5.66619	−0.259983
$476$	0	0
$477$	−16.6913	−0.764243
$478$	0	0
$479$	14.0320	0.641140	0.320570	−	0.947225i	$-0.396126\pi$
$479$	14.0320	0.641140	0.320570	+	0.947225i	$0.396126\pi$
$480$	0	0
$481$	47.4075	2.16159
$482$	0	0
$483$	1.28636	0.0585316
$484$	0	0
$485$	−5.15792	−0.234209
$486$	0	0
$487$	14.2176	0.644263	0.322131	−	0.946695i	$-0.395601\pi$
$487$	14.2176	0.644263	0.322131	+	0.946695i	$0.395601\pi$
$488$	0	0
$489$	−1.05550	−0.0477315
$490$	0	0
$491$	20.5040	0.925333	0.462666	−	0.886533i	$-0.346893\pi$
$491$	20.5040	0.925333	0.462666	+	0.886533i	$0.346893\pi$
$492$	0	0
$493$	20.3059	0.914530
$494$	0	0
$495$	1.85107	0.0831995
$496$	0	0
$497$	29.7746	1.33557
$498$	0	0
$499$	37.1763	1.66424	0.832121	−	0.554594i	$-0.187127\pi$
$499$	37.1763	1.66424	0.832121	+	0.554594i	$0.187127\pi$
$500$	0	0
$501$	0.704786	0.0314875
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	0.657392	0.0292536
$506$	0	0
$507$	−1.05175	−0.0467098
$508$	0	0
$509$	−22.4908	−0.996886	−0.498443	−	0.866923i	$-0.666095\pi$
$509$	−22.4908	−0.996886	−0.498443	+	0.866923i	$0.666095\pi$
$510$	0	0
$511$	5.08504	0.224949
$512$	0	0
$513$	0.386494	0.0170641
$514$	0	0
$515$	3.92372	0.172900
$516$	0	0
$517$	6.07504	0.267180
$518$	0	0
$519$	−1.15735	−0.0508020
$520$	0	0
$521$	40.9522	1.79415	0.897073	−	0.441882i	$-0.145689\pi$
$521$	40.9522	1.79415	0.897073	+	0.441882i	$0.145689\pi$
$522$	0	0
$523$	−7.00014	−0.306095	−0.153047	−	0.988219i	$-0.548909\pi$
$523$	−7.00014	−0.306095	−0.153047	+	0.988219i	$0.548909\pi$
$524$	0	0
$525$	−1.03481	−0.0451629
$526$	0	0
$527$	52.1035	2.26966
$528$	0	0
$529$	13.2820	0.577479
$530$	0	0
$531$	35.6768	1.54824
$532$	0	0
$533$	−13.5935	−0.588800
$534$	0	0
$535$	−4.77070	−0.206255
$536$	0	0
$537$	−0.670932	−0.0289528
$538$	0	0
$539$	−12.5955	−0.542528
$540$	0	0
$541$	−18.3096	−0.787192	−0.393596	−	0.919284i	$-0.628769\pi$
$541$	−18.3096	−0.787192	−0.393596	+	0.919284i	$0.628769\pi$
$542$	0	0
$543$	1.00399	0.0430851
$544$	0	0
$545$	6.92396	0.296590
$546$	0	0
$547$	21.4683	0.917920	0.458960	−	0.888457i	$-0.348222\pi$
$547$	21.4683	0.917920	0.458960	+	0.888457i	$0.348222\pi$
$548$	0	0
$549$	29.0115	1.23818
$550$	0	0
$551$	−3.29589	−0.140410
$552$	0	0
$553$	32.8449	1.39671
$554$	0	0
$555$	0.181289	0.00769531
$556$	0	0
$557$	42.8763	1.81673	0.908364	−	0.418180i	$-0.137332\pi$
$557$	42.8763	1.81673	0.908364	+	0.418180i	$0.137332\pi$
$558$	0	0
$559$	−67.8982	−2.87179
$560$	0	0
$561$	−0.624026	−0.0263464
$562$	0	0
$563$	10.6367	0.448285	0.224142	−	0.974556i	$-0.428042\pi$
$563$	10.6367	0.448285	0.224142	+	0.974556i	$0.428042\pi$
$564$	0	0
$565$	−0.0690891	−0.00290660
$566$	0	0
$567$	−34.7678	−1.46011
$568$	0	0
$569$	−27.0739	−1.13500	−0.567499	−	0.823374i	$-0.692089\pi$
$569$	−27.0739	−1.13500	−0.567499	+	0.823374i	$0.692089\pi$
$570$	0	0
$571$	−23.2054	−0.971116	−0.485558	−	0.874204i	$-0.661384\pi$
$571$	−23.2054	−0.971116	−0.485558	+	0.874204i	$0.661384\pi$
$572$	0	0
$573$	−0.833747	−0.0348303
$574$	0	0
$575$	−29.1869	−1.21718
$576$	0	0
$577$	−2.52331	−0.105047	−0.0525234	−	0.998620i	$-0.516726\pi$
$577$	−2.52331	−0.105047	−0.0525234	+	0.998620i	$0.516726\pi$
$578$	0	0
$579$	0.814662	0.0338562
$580$	0	0
$581$	−55.1364	−2.28744
$582$	0	0
$583$	−8.75288	−0.362507
$584$	0	0
$585$	6.67144	0.275830
$586$	0	0
$587$	−38.5577	−1.59145	−0.795724	−	0.605660i	$-0.792909\pi$
$587$	−38.5577	−1.59145	−0.795724	+	0.605660i	$0.792909\pi$
$588$	0	0
$589$	−8.45703	−0.348466
$590$	0	0
$591$	0.843529	0.0346982
$592$	0	0
$593$	−13.5081	−0.554711	−0.277356	−	0.960767i	$-0.589458\pi$
$593$	−13.5081	−0.554711	−0.277356	+	0.960767i	$0.589458\pi$
$594$	0	0
$595$	10.9712	0.449774
$596$	0	0
$597$	−0.828923	−0.0339255
$598$	0	0
$599$	−3.95474	−0.161586	−0.0807932	−	0.996731i	$-0.525745\pi$
$599$	−3.95474	−0.161586	−0.0807932	+	0.996731i	$0.525745\pi$
$600$	0	0
$601$	−3.84696	−0.156921	−0.0784603	−	0.996917i	$-0.525000\pi$
$601$	−3.84696	−0.156921	−0.0784603	+	0.996917i	$0.525000\pi$
$602$	0	0
$603$	19.1398	0.779432
$604$	0	0
$605$	−3.35238	−0.136294
$606$	0	0
$607$	−24.1353	−0.979622	−0.489811	−	0.871829i	$-0.662934\pi$
$607$	−24.1353	−0.979622	−0.489811	+	0.871829i	$0.662934\pi$
$608$	0	0
$609$	−0.601926	−0.0243913
$610$	0	0
$611$	21.8950	0.885777
$612$	0	0
$613$	−0.865350	−0.0349512	−0.0174756	−	0.999847i	$-0.505563\pi$
$613$	−0.865350	−0.0349512	−0.0174756	+	0.999847i	$0.505563\pi$
$614$	0	0
$615$	−0.0519825	−0.00209614
$616$	0	0
$617$	0.798540	0.0321480	0.0160740	−	0.999871i	$-0.494883\pi$
$617$	0.798540	0.0321480	0.0160740	+	0.999871i	$0.494883\pi$
$618$	0	0
$619$	−9.56838	−0.384586	−0.192293	−	0.981338i	$-0.561592\pi$
$619$	−9.56838	−0.384586	−0.192293	+	0.981338i	$0.561592\pi$
$620$	0	0
$621$	1.99086	0.0798904
$622$	0	0
$623$	−1.12680	−0.0451445
$624$	0	0
$625$	22.7070	0.908282
$626$	0	0
$627$	0.101287	0.00404501
$628$	0	0
$629$	60.2984	2.40426
$630$	0	0
$631$	8.84224	0.352004	0.176002	−	0.984390i	$-0.443683\pi$
$631$	8.84224	0.352004	0.176002	+	0.984390i	$0.443683\pi$
$632$	0	0
$633$	−0.239036	−0.00950084
$634$	0	0
$635$	4.88782	0.193967
$636$	0	0
$637$	−45.3955	−1.79864
$638$	0	0
$639$	23.0289	0.911007
$640$	0	0
$641$	32.6896	1.29116	0.645581	−	0.763692i	$-0.276615\pi$
$641$	32.6896	1.29116	0.645581	+	0.763692i	$0.276615\pi$
$642$	0	0
$643$	−2.01428	−0.0794356	−0.0397178	−	0.999211i	$-0.512646\pi$
$643$	−2.01428	−0.0794356	−0.0397178	+	0.999211i	$0.512646\pi$
$644$	0	0
$645$	−0.259647	−0.0102236
$646$	0	0
$647$	−26.4318	−1.03914	−0.519571	−	0.854428i	$-0.673908\pi$
$647$	−26.4318	−1.03914	−0.519571	+	0.854428i	$0.673908\pi$
$648$	0	0
$649$	18.7088	0.734385
$650$	0	0
$651$	−1.54450	−0.0605337
$652$	0	0
$653$	10.7224	0.419598	0.209799	−	0.977745i	$-0.432719\pi$
$653$	10.7224	0.419598	0.209799	+	0.977745i	$0.432719\pi$
$654$	0	0
$655$	7.48378	0.292416
$656$	0	0
$657$	3.93297	0.153440
$658$	0	0
$659$	−32.4661	−1.26470	−0.632350	−	0.774683i	$-0.717909\pi$
$659$	−32.4661	−1.26470	−0.632350	+	0.774683i	$0.717909\pi$
$660$	0	0
$661$	−7.61064	−0.296020	−0.148010	−	0.988986i	$-0.547287\pi$
$661$	−7.61064	−0.296020	−0.148010	+	0.988986i	$0.547287\pi$
$662$	0	0
$663$	−2.24905	−0.0873457
$664$	0	0
$665$	−1.78075	−0.0690547
$666$	0	0
$667$	−16.9774	−0.657366
$668$	0	0
$669$	0.949267	0.0367008
$670$	0	0
$671$	15.2136	0.587313
$672$	0	0
$673$	45.8894	1.76891	0.884453	−	0.466629i	$-0.154532\pi$
$673$	45.8894	1.76891	0.884453	+	0.466629i	$0.154532\pi$
$674$	0	0
$675$	−1.60154	−0.0616432
$676$	0	0
$677$	11.9372	0.458783	0.229391	−	0.973334i	$-0.426326\pi$
$677$	11.9372	0.458783	0.229391	+	0.973334i	$0.426326\pi$
$678$	0	0
$679$	50.8545	1.95162
$680$	0	0
$681$	0.290709	0.0111400
$682$	0	0
$683$	12.2314	0.468021	0.234010	−	0.972234i	$-0.424815\pi$
$683$	12.2314	0.468021	0.234010	+	0.972234i	$0.424815\pi$
$684$	0	0
$685$	−4.63665	−0.177157
$686$	0	0
$687$	−1.00129	−0.0382017
$688$	0	0
$689$	−31.5462	−1.20182
$690$	0	0
$691$	0.978061	0.0372072	0.0186036	−	0.999827i	$-0.494078\pi$
$691$	0.978061	0.0372072	0.0186036	+	0.999827i	$0.494078\pi$
$692$	0	0
$693$	−18.2506	−0.693285
$694$	0	0
$695$	−6.06087	−0.229902
$696$	0	0
$697$	−17.2898	−0.654899
$698$	0	0
$699$	−0.385347	−0.0145752
$700$	0	0
$701$	−26.3979	−0.997034	−0.498517	−	0.866880i	$-0.666122\pi$
$701$	−26.3979	−0.997034	−0.498517	+	0.866880i	$0.666122\pi$
$702$	0	0
$703$	−9.78717	−0.369130
$704$	0	0
$705$	0.0837280	0.00315338
$706$	0	0
$707$	−6.48155	−0.243764
$708$	0	0
$709$	8.71411	0.327265	0.163633	−	0.986521i	$-0.447679\pi$
$709$	8.71411	0.327265	0.163633	+	0.986521i	$0.447679\pi$
$710$	0	0
$711$	25.4035	0.952706
$712$	0	0
$713$	−43.5627	−1.63144
$714$	0	0
$715$	3.49849	0.130836
$716$	0	0
$717$	−0.389859	−0.0145595
$718$	0	0
$719$	−8.38209	−0.312599	−0.156300	−	0.987710i	$-0.549957\pi$
$719$	−8.38209	−0.312599	−0.156300	+	0.987710i	$0.549957\pi$
$720$	0	0
$721$	−38.6859	−1.44074
$722$	0	0
$723$	0.468922	0.0174394
$724$	0	0
$725$	13.6574	0.507222
$726$	0	0
$727$	−20.3729	−0.755590	−0.377795	−	0.925889i	$-0.623318\pi$
$727$	−20.3729	−0.755590	−0.377795	+	0.925889i	$0.623318\pi$
$728$	0	0
$729$	−26.8361	−0.993930
$730$	0	0
$731$	−86.3610	−3.19418
$732$	0	0
$733$	9.55353	0.352868	0.176434	−	0.984312i	$-0.443544\pi$
$733$	9.55353	0.352868	0.176434	+	0.984312i	$0.443544\pi$
$734$	0	0
$735$	−0.173596	−0.00640317
$736$	0	0
$737$	10.0369	0.369712
$738$	0	0
$739$	−5.39782	−0.198562	−0.0992809	−	0.995059i	$-0.531654\pi$
$739$	−5.39782	−0.198562	−0.0992809	+	0.995059i	$0.531654\pi$
$740$	0	0
$741$	0.365048	0.0134104
$742$	0	0
$743$	48.5369	1.78065	0.890323	−	0.455329i	$-0.150478\pi$
$743$	48.5369	1.78065	0.890323	+	0.455329i	$0.150478\pi$
$744$	0	0
$745$	6.77642	0.248269
$746$	0	0
$747$	−42.6447	−1.56029
$748$	0	0
$749$	47.0367	1.71868
$750$	0	0
$751$	34.1862	1.24747	0.623736	−	0.781635i	$-0.285614\pi$
$751$	34.1862	1.24747	0.623736	+	0.781635i	$0.285614\pi$
$752$	0	0
$753$	−0.111963	−0.00408015
$754$	0	0
$755$	−7.41566	−0.269884
$756$	0	0
$757$	0.787643	0.0286274	0.0143137	−	0.999898i	$-0.495444\pi$
$757$	0.787643	0.0286274	0.0143137	+	0.999898i	$0.495444\pi$
$758$	0	0
$759$	0.521736	0.0189378
$760$	0	0
$761$	−46.2864	−1.67788	−0.838941	−	0.544222i	$-0.816825\pi$
$761$	−46.2864	−1.67788	−0.838941	+	0.544222i	$0.816825\pi$
$762$	0	0
$763$	−68.2668	−2.47142
$764$	0	0
$765$	8.48553	0.306795
$766$	0	0
$767$	67.4283	2.43470
$768$	0	0
$769$	13.4886	0.486413	0.243207	−	0.969975i	$-0.421801\pi$
$769$	13.4886	0.486413	0.243207	+	0.969975i	$0.421801\pi$
$770$	0	0
$771$	−1.70319	−0.0613391
$772$	0	0
$773$	−31.3281	−1.12679	−0.563396	−	0.826187i	$-0.690505\pi$
$773$	−31.3281	−1.12679	−0.563396	+	0.826187i	$0.690505\pi$
$774$	0	0
$775$	35.0439	1.25881
$776$	0	0
$777$	−1.78742	−0.0641234
$778$	0	0
$779$	2.80635	0.100548
$780$	0	0
$781$	12.0763	0.432123
$782$	0	0
$783$	−0.931578	−0.0332919
$784$	0	0
$785$	−5.01025	−0.178823
$786$	0	0
$787$	34.6855	1.23640	0.618202	−	0.786019i	$-0.287861\pi$
$787$	34.6855	1.23640	0.618202	+	0.786019i	$0.287861\pi$
$788$	0	0
$789$	0.454574	0.0161833
$790$	0	0
$791$	0.681183	0.0242201
$792$	0	0
$793$	54.8311	1.94711
$794$	0	0
$795$	−0.120635	−0.00427848
$796$	0	0
$797$	−31.5707	−1.11829	−0.559146	−	0.829069i	$-0.688871\pi$
$797$	−31.5707	−1.11829	−0.559146	+	0.829069i	$0.688871\pi$
$798$	0	0
$799$	27.8486	0.985214
$800$	0	0
$801$	−0.871515	−0.0307935
$802$	0	0
$803$	2.06244	0.0727819
$804$	0	0
$805$	−9.17278	−0.323298
$806$	0	0
$807$	0.665043	0.0234106
$808$	0	0
$809$	−20.9171	−0.735406	−0.367703	−	0.929943i	$-0.619856\pi$
$809$	−20.9171	−0.735406	−0.367703	+	0.929943i	$0.619856\pi$
$810$	0	0
$811$	32.0531	1.12554	0.562768	−	0.826615i	$-0.309736\pi$
$811$	32.0531	1.12554	0.562768	+	0.826615i	$0.309736\pi$
$812$	0	0
$813$	1.17214	0.0411087
$814$	0	0
$815$	7.52656	0.263644
$816$	0	0
$817$	14.0174	0.490408
$818$	0	0
$819$	−65.7770	−2.29844
$820$	0	0
$821$	23.5009	0.820186	0.410093	−	0.912044i	$-0.365496\pi$
$821$	23.5009	0.820186	0.410093	+	0.912044i	$0.365496\pi$
$822$	0	0
$823$	−18.2470	−0.636049	−0.318025	−	0.948082i	$-0.603019\pi$
$823$	−18.2470	−0.636049	−0.318025	+	0.948082i	$0.603019\pi$
$824$	0	0
$825$	−0.419709	−0.0146124
$826$	0	0
$827$	24.3506	0.846752	0.423376	−	0.905954i	$-0.360845\pi$
$827$	24.3506	0.846752	0.423376	+	0.905954i	$0.360845\pi$
$828$	0	0
$829$	−44.6365	−1.55029	−0.775145	−	0.631784i	$-0.782323\pi$
$829$	−44.6365	−1.55029	−0.775145	+	0.631784i	$0.782323\pi$
$830$	0	0
$831$	−0.342340	−0.0118757
$832$	0	0
$833$	−57.7394	−2.00055
$834$	0	0
$835$	−5.02568	−0.173921
$836$	0	0
$837$	−2.39037	−0.0826231
$838$	0	0
$839$	−3.13565	−0.108255	−0.0541273	−	0.998534i	$-0.517238\pi$
$839$	−3.13565	−0.108255	−0.0541273	+	0.998534i	$0.517238\pi$
$840$	0	0
$841$	−21.0558	−0.726063
$842$	0	0
$843$	−1.20862	−0.0416270
$844$	0	0
$845$	7.49979	0.258000
$846$	0	0
$847$	33.0528	1.13571
$848$	0	0
$849$	−0.498839	−0.0171201
$850$	0	0
$851$	−50.4144	−1.72818
$852$	0	0
$853$	−38.5399	−1.31958	−0.659790	−	0.751450i	$-0.729355\pi$
$853$	−38.5399	−1.31958	−0.659790	+	0.751450i	$0.729355\pi$
$854$	0	0
$855$	−1.37730	−0.0471028
$856$	0	0
$857$	47.1571	1.61086	0.805428	−	0.592694i	$-0.201936\pi$
$857$	47.1571	1.61086	0.805428	+	0.592694i	$0.201936\pi$
$858$	0	0
$859$	−39.1445	−1.33559	−0.667796	−	0.744344i	$-0.732762\pi$
$859$	−39.1445	−1.33559	−0.667796	+	0.744344i	$0.732762\pi$
$860$	0	0
$861$	0.512521	0.0174667
$862$	0	0
$863$	29.2074	0.994231	0.497116	−	0.867684i	$-0.334392\pi$
$863$	29.2074	0.994231	0.497116	+	0.867684i	$0.334392\pi$
$864$	0	0
$865$	8.25280	0.280604
$866$	0	0
$867$	−1.92366	−0.0653310
$868$	0	0
$869$	13.3215	0.451902
$870$	0	0
$871$	36.1738	1.22570
$872$	0	0
$873$	39.3328	1.33122
$874$	0	0
$875$	14.9932	0.506864
$876$	0	0
$877$	−18.0825	−0.610601	−0.305301	−	0.952256i	$-0.598757\pi$
$877$	−18.0825	−0.610601	−0.305301	+	0.952256i	$0.598757\pi$
$878$	0	0
$879$	1.30170	0.0439054
$880$	0	0
$881$	23.5166	0.792294	0.396147	−	0.918187i	$-0.370347\pi$
$881$	23.5166	0.792294	0.396147	+	0.918187i	$0.370347\pi$
$882$	0	0
$883$	26.9537	0.907065	0.453532	−	0.891240i	$-0.350164\pi$
$883$	26.9537	0.907065	0.453532	+	0.891240i	$0.350164\pi$
$884$	0	0
$885$	0.257850	0.00866755
$886$	0	0
$887$	16.7349	0.561904	0.280952	−	0.959722i	$-0.409350\pi$
$887$	16.7349	0.561904	0.280952	+	0.959722i	$0.409350\pi$
$888$	0	0
$889$	−48.1914	−1.61629
$890$	0	0
$891$	−14.1014	−0.472416
$892$	0	0
$893$	−4.52018	−0.151262
$894$	0	0
$895$	4.78427	0.159920
$896$	0	0
$897$	1.88039	0.0627843
$898$	0	0
$899$	20.3842	0.679852
$900$	0	0
$901$	−40.1242	−1.33673
$902$	0	0
$903$	2.55999	0.0851912
$904$	0	0
$905$	−7.15920	−0.237980
$906$	0	0
$907$	14.3040	0.474957	0.237479	−	0.971393i	$-0.423679\pi$
$907$	14.3040	0.474957	0.237479	+	0.971393i	$0.423679\pi$
$908$	0	0
$909$	−5.01309	−0.166274
$910$	0	0
$911$	30.4572	1.00909	0.504547	−	0.863384i	$-0.331660\pi$
$911$	30.4572	1.00909	0.504547	+	0.863384i	$0.331660\pi$
$912$	0	0
$913$	−22.3628	−0.740100
$914$	0	0
$915$	0.209678	0.00693173
$916$	0	0
$917$	−73.7863	−2.43664
$918$	0	0
$919$	−36.9524	−1.21895	−0.609473	−	0.792807i	$-0.708619\pi$
$919$	−36.9524	−1.21895	−0.609473	+	0.792807i	$0.708619\pi$
$920$	0	0
$921$	1.43438	0.0472646
$922$	0	0
$923$	43.5240	1.43261
$924$	0	0
$925$	40.5557	1.33346
$926$	0	0
$927$	−29.9212	−0.982741
$928$	0	0
$929$	21.7736	0.714369	0.357184	−	0.934034i	$-0.383737\pi$
$929$	21.7736	0.714369	0.357184	+	0.934034i	$0.383737\pi$
$930$	0	0
$931$	9.37181	0.307149
$932$	0	0
$933$	−1.63532	−0.0535381
$934$	0	0
$935$	4.44979	0.145524
$936$	0	0
$937$	−11.6432	−0.380366	−0.190183	−	0.981749i	$-0.560908\pi$
$937$	−11.6432	−0.380366	−0.190183	+	0.981749i	$0.560908\pi$
$938$	0	0
$939$	1.17326	0.0382879
$940$	0	0
$941$	−7.70362	−0.251131	−0.125565	−	0.992085i	$-0.540075\pi$
$941$	−7.70362	−0.251131	−0.125565	+	0.992085i	$0.540075\pi$
$942$	0	0
$943$	14.4557	0.470742
$944$	0	0
$945$	−0.503327	−0.0163732
$946$	0	0
$947$	18.1730	0.590544	0.295272	−	0.955413i	$-0.404590\pi$
$947$	18.1730	0.590544	0.295272	+	0.955413i	$0.404590\pi$
$948$	0	0
$949$	7.43323	0.241293
$950$	0	0
$951$	0.172283	0.00558666
$952$	0	0
$953$	−6.04662	−0.195869	−0.0979346	−	0.995193i	$-0.531224\pi$
$953$	−6.04662	−0.195869	−0.0979346	+	0.995193i	$0.531224\pi$
$954$	0	0
$955$	5.94526	0.192384
$956$	0	0
$957$	−0.244135	−0.00789176
$958$	0	0
$959$	45.7150	1.47621
$960$	0	0
$961$	21.3045	0.687243
$962$	0	0
$963$	36.3800	1.17233
$964$	0	0
$965$	−5.80918	−0.187004
$966$	0	0
$967$	42.3294	1.36122	0.680611	−	0.732645i	$-0.261714\pi$
$967$	42.3294	1.36122	0.680611	+	0.732645i	$0.261714\pi$
$968$	0	0
$969$	0.464311	0.0149158
$970$	0	0
$971$	31.4448	1.00911	0.504556	−	0.863379i	$-0.331656\pi$
$971$	31.4448	1.00911	0.504556	+	0.863379i	$0.331656\pi$
$972$	0	0
$973$	59.7571	1.91572
$974$	0	0
$975$	−1.51267	−0.0484442
$976$	0	0
$977$	−2.64693	−0.0846828	−0.0423414	−	0.999103i	$-0.513482\pi$
$977$	−2.64693	−0.0846828	−0.0423414	+	0.999103i	$0.513482\pi$
$978$	0	0
$979$	−0.457020	−0.0146064
$980$	0	0
$981$	−52.8002	−1.68578
$982$	0	0
$983$	−35.7056	−1.13883	−0.569416	−	0.822050i	$-0.692830\pi$
$983$	−35.7056	−1.13883	−0.569416	+	0.822050i	$0.692830\pi$
$984$	0	0
$985$	−6.01502	−0.191655
$986$	0	0
$987$	−0.825516	−0.0262765
$988$	0	0
$989$	72.2048	2.29598
$990$	0	0
$991$	7.98107	0.253527	0.126763	−	0.991933i	$-0.459541\pi$
$991$	7.98107	0.253527	0.126763	+	0.991933i	$0.459541\pi$
$992$	0	0
$993$	−0.414221	−0.0131449
$994$	0	0
$995$	5.91087	0.187387
$996$	0	0
$997$	−23.9835	−0.759566	−0.379783	−	0.925076i	$-0.624001\pi$
$997$	−23.9835	−0.759566	−0.379783	+	0.925076i	$0.624001\pi$
$998$	0	0
$999$	−2.76633	−0.0875227

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.t.1.9		21
4.3	odd	2		2012.2.a.a.1.13	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2012.2.a.a.1.13	✓	21	4.3	odd	2
8048.2.a.t.1.9		21	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0551142 q^{3} +0.393007 q^{5} -3.87485 q^{7} -2.99696 q^{9} +O(q^{10})\)	\(q-0.0551142 q^{3} +0.393007 q^{5} -3.87485 q^{7} -2.99696 q^{9} -1.57160 q^{11} -5.66419 q^{13} -0.0216603 q^{15} -7.20439 q^{17} +1.16936 q^{19} +0.213559 q^{21} +6.02345 q^{23} -4.84555 q^{25} +0.330518 q^{27} -2.81854 q^{29} -7.23219 q^{31} +0.0866174 q^{33} -1.52284 q^{35} -8.36968 q^{37} +0.312177 q^{39} +2.39990 q^{41} +11.9873 q^{43} -1.17783 q^{45} -3.86551 q^{47} +8.01447 q^{49} +0.397064 q^{51} +5.56941 q^{53} -0.617650 q^{55} -0.0644483 q^{57} -11.9043 q^{59} -9.68030 q^{61} +11.6128 q^{63} -2.22607 q^{65} -6.38639 q^{67} -0.331978 q^{69} -7.68406 q^{71} -1.31232 q^{73} +0.267058 q^{75} +6.08971 q^{77} -8.47642 q^{79} +8.97267 q^{81} +14.2293 q^{83} -2.83138 q^{85} +0.155342 q^{87} +0.290799 q^{89} +21.9479 q^{91} +0.398596 q^{93} +0.459567 q^{95} -13.1242 q^{97} +4.71003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9}+O(q^{10}) \) 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9	\( 21 q + 10 q^{3} - 3 q^{5} + 15 q^{7} + 21 q^{9} + 9 q^{11} - 16 q^{13} + 22 q^{15} + q^{17} + 12 q^{19} - 2 q^{21} + 22 q^{23} + 18 q^{25} + 43 q^{27} - 13 q^{29} + 28 q^{31} + 3 q^{33} + 12 q^{35} - 39 q^{37} + 11 q^{39} + 4 q^{41} + 50 q^{43} - 6 q^{45} + 27 q^{47} + 16 q^{49} + 37 q^{51} - 24 q^{53} + 49 q^{55} - q^{57} + 22 q^{59} - 22 q^{61} + 49 q^{63} - 14 q^{65} + 62 q^{67} - 17 q^{69} + 21 q^{71} - 6 q^{73} + 52 q^{75} - 24 q^{77} + 65 q^{79} + 29 q^{81} + 18 q^{83} - 54 q^{85} + 31 q^{87} + q^{89} + 45 q^{91} - 26 q^{93} + 53 q^{95} - 2 q^{97} + 58 q^{99}+O(q^{100}) \) 21 * q + 10 * q^3 - 3 * q^5 + 15 * q^7 + 21 * q^9 + 9 * q^11 - 16 * q^13 + 22 * q^15 + q^17 + 12 * q^19 - 2 * q^21 + 22 * q^23 + 18 * q^25 + 43 * q^27 - 13 * q^29 + 28 * q^31 + 3 * q^33 + 12 * q^35 - 39 * q^37 + 11 * q^39 + 4 * q^41 + 50 * q^43 - 6 * q^45 + 27 * q^47 + 16 * q^49 + 37 * q^51 - 24 * q^53 + 49 * q^55 - q^57 + 22 * q^59 - 22 * q^61 + 49 * q^63 - 14 * q^65 + 62 * q^67 - 17 * q^69 + 21 * q^71 - 6 * q^73 + 52 * q^75 - 24 * q^77 + 65 * q^79 + 29 * q^81 + 18 * q^83 - 54 * q^85 + 31 * q^87 + q^89 + 45 * q^91 - 26 * q^93 + 53 * q^95 - 2 * q^97 + 58 * q^99

Introduction

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