LMFDB - Embedded newform 8048.2.a.w.1.20

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:	$29$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q+1.15550 q^{3} -3.26045 q^{5} +1.92982 q^{7} -1.66482 q^{9} +O(q^{10})$	$q+1.15550 q^{3} -3.26045 q^{5} +1.92982 q^{7} -1.66482 q^{9} -1.97192 q^{11} -0.983643 q^{13} -3.76746 q^{15} -4.76368 q^{17} -4.17055 q^{19} +2.22991 q^{21} -8.45293 q^{23} +5.63055 q^{25} -5.39020 q^{27} -0.913684 q^{29} -7.19000 q^{31} -2.27856 q^{33} -6.29208 q^{35} +11.2113 q^{37} -1.13660 q^{39} +5.68259 q^{41} +9.63950 q^{43} +5.42805 q^{45} +4.21503 q^{47} -3.27580 q^{49} -5.50444 q^{51} -1.59953 q^{53} +6.42936 q^{55} -4.81908 q^{57} -0.532957 q^{59} -8.66711 q^{61} -3.21279 q^{63} +3.20712 q^{65} +10.3799 q^{67} -9.76737 q^{69} +10.1043 q^{71} +13.1435 q^{73} +6.50611 q^{75} -3.80545 q^{77} +17.5418 q^{79} -1.23394 q^{81} -2.72424 q^{83} +15.5317 q^{85} -1.05576 q^{87} +0.00198390 q^{89} -1.89825 q^{91} -8.30806 q^{93} +13.5979 q^{95} +4.11333 q^{97} +3.28289 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9	$ 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9} + 27 q^{11} + 16 q^{13} + 14 q^{15} - 15 q^{17} + 14 q^{19} + q^{21} + 25 q^{23} + 21 q^{25} + 25 q^{27} - 13 q^{29} + 27 q^{31} - 9 q^{33} + 29 q^{35} + 35 q^{37} + 38 q^{39} - 30 q^{41} + 38 q^{43} + q^{45} + 35 q^{47} + 14 q^{49} + 21 q^{51} + 2 q^{53} + 25 q^{55} - 25 q^{57} + 40 q^{59} + 10 q^{61} + 56 q^{63} - 50 q^{65} + 31 q^{67} + 11 q^{69} + 65 q^{71} - 23 q^{73} + 32 q^{75} + 13 q^{77} + 44 q^{79} - 7 q^{81} + 41 q^{83} + 26 q^{85} + 25 q^{87} - 48 q^{89} + 44 q^{91} + 25 q^{93} + 75 q^{95} - 18 q^{97} + 80 q^{99}+O(q^{100}) $ 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9 + 27 * q^11 + 16 * q^13 + 14 * q^15 - 15 * q^17 + 14 * q^19 + q^21 + 25 * q^23 + 21 * q^25 + 25 * q^27 - 13 * q^29 + 27 * q^31 - 9 * q^33 + 29 * q^35 + 35 * q^37 + 38 * q^39 - 30 * q^41 + 38 * q^43 + q^45 + 35 * q^47 + 14 * q^49 + 21 * q^51 + 2 * q^53 + 25 * q^55 - 25 * q^57 + 40 * q^59 + 10 * q^61 + 56 * q^63 - 50 * q^65 + 31 * q^67 + 11 * q^69 + 65 * q^71 - 23 * q^73 + 32 * q^75 + 13 * q^77 + 44 * q^79 - 7 * q^81 + 41 * q^83 + 26 * q^85 + 25 * q^87 - 48 * q^89 + 44 * q^91 + 25 * q^93 + 75 * q^95 - 18 * q^97 + 80 * 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
$3$	1.15550	0.667129	0.333565	−	0.942727i	$-0.391748\pi$
$3$	1.15550	0.667129	0.333565	+	0.942727i	$0.391748\pi$
$4$	0	0
$5$	−3.26045	−1.45812	−0.729059	−	0.684451i	$-0.760042\pi$
$5$	−3.26045	−1.45812	−0.729059	+	0.684451i	$0.760042\pi$
$6$	0	0
$7$	1.92982	0.729403	0.364701	−	0.931125i	$-0.381171\pi$
$7$	1.92982	0.729403	0.364701	+	0.931125i	$0.381171\pi$
$8$	0	0
$9$	−1.66482	−0.554939
$10$	0	0
$11$	−1.97192	−0.594557	−0.297278	−	0.954791i	$-0.596079\pi$
$11$	−1.97192	−0.594557	−0.297278	+	0.954791i	$0.596079\pi$
$12$	0	0
$13$	−0.983643	−0.272814	−0.136407	−	0.990653i	$-0.543555\pi$
$13$	−0.983643	−0.272814	−0.136407	+	0.990653i	$0.543555\pi$
$14$	0	0
$15$	−3.76746	−0.972754
$16$	0	0
$17$	−4.76368	−1.15536	−0.577681	−	0.816263i	$-0.696042\pi$
$17$	−4.76368	−1.15536	−0.577681	+	0.816263i	$0.696042\pi$
$18$	0	0
$19$	−4.17055	−0.956791	−0.478395	−	0.878145i	$-0.658781\pi$
$19$	−4.17055	−0.956791	−0.478395	+	0.878145i	$0.658781\pi$
$20$	0	0
$21$	2.22991	0.486606
$22$	0	0
$23$	−8.45293	−1.76256	−0.881278	−	0.472598i	$-0.843316\pi$
$23$	−8.45293	−1.76256	−0.881278	+	0.472598i	$0.843316\pi$
$24$	0	0
$25$	5.63055	1.12611
$26$	0	0
$27$	−5.39020	−1.03735
$28$	0	0
$29$	−0.913684	−0.169667	−0.0848335	−	0.996395i	$-0.527036\pi$
$29$	−0.913684	−0.169667	−0.0848335	+	0.996395i	$0.527036\pi$
$30$	0	0
$31$	−7.19000	−1.29136	−0.645681	−	0.763607i	$-0.723426\pi$
$31$	−7.19000	−1.29136	−0.645681	+	0.763607i	$0.723426\pi$
$32$	0	0
$33$	−2.27856	−0.396646
$34$	0	0
$35$	−6.29208	−1.06356
$36$	0	0
$37$	11.2113	1.84313	0.921564	−	0.388226i	$-0.126912\pi$
$37$	11.2113	1.84313	0.921564	+	0.388226i	$0.126912\pi$
$38$	0	0
$39$	−1.13660	−0.182002
$40$	0	0
$41$	5.68259	0.887472	0.443736	−	0.896158i	$-0.353653\pi$
$41$	5.68259	0.887472	0.443736	+	0.896158i	$0.353653\pi$
$42$	0	0
$43$	9.63950	1.47001	0.735005	−	0.678061i	$-0.237180\pi$
$43$	9.63950	1.47001	0.735005	+	0.678061i	$0.237180\pi$
$44$	0	0
$45$	5.42805	0.809166
$46$	0	0
$47$	4.21503	0.614825	0.307412	−	0.951576i	$-0.400537\pi$
$47$	4.21503	0.614825	0.307412	+	0.951576i	$0.400537\pi$
$48$	0	0
$49$	−3.27580	−0.467972
$50$	0	0
$51$	−5.50444	−0.770775
$52$	0	0
$53$	−1.59953	−0.219713	−0.109856	−	0.993947i	$-0.535039\pi$
$53$	−1.59953	−0.219713	−0.109856	+	0.993947i	$0.535039\pi$
$54$	0	0
$55$	6.42936	0.866934
$56$	0	0
$57$	−4.81908	−0.638303
$58$	0	0
$59$	−0.532957	−0.0693852	−0.0346926	−	0.999398i	$-0.511045\pi$
$59$	−0.532957	−0.0693852	−0.0346926	+	0.999398i	$0.511045\pi$
$60$	0	0
$61$	−8.66711	−1.10971	−0.554855	−	0.831947i	$-0.687226\pi$
$61$	−8.66711	−1.10971	−0.554855	+	0.831947i	$0.687226\pi$
$62$	0	0
$63$	−3.21279	−0.404774
$64$	0	0
$65$	3.20712	0.397794
$66$	0	0
$67$	10.3799	1.26811	0.634055	−	0.773288i	$-0.281389\pi$
$67$	10.3799	1.26811	0.634055	+	0.773288i	$0.281389\pi$
$68$	0	0
$69$	−9.76737	−1.17585
$70$	0	0
$71$	10.1043	1.19916	0.599580	−	0.800315i	$-0.295334\pi$
$71$	10.1043	1.19916	0.599580	+	0.800315i	$0.295334\pi$
$72$	0	0
$73$	13.1435	1.53833	0.769163	−	0.639052i	$-0.220673\pi$
$73$	13.1435	1.53833	0.769163	+	0.639052i	$0.220673\pi$
$74$	0	0
$75$	6.50611	0.751261
$76$	0	0
$77$	−3.80545	−0.433671
$78$	0	0
$79$	17.5418	1.97361	0.986804	−	0.161918i	$-0.0517680\pi$
$79$	17.5418	1.97361	0.986804	+	0.161918i	$0.0517680\pi$
$80$	0	0
$81$	−1.23394	−0.137105
$82$	0	0
$83$	−2.72424	−0.299024	−0.149512	−	0.988760i	$-0.547770\pi$
$83$	−2.72424	−0.299024	−0.149512	+	0.988760i	$0.547770\pi$
$84$	0	0
$85$	15.5317	1.68465
$86$	0	0
$87$	−1.05576	−0.113190
$88$	0	0
$89$	0.00198390	0.000210293 0	0.000105147	−	1.00000i	$-0.499967\pi$
$89$	0.00198390	0.000210293 0	0.000105147		1.00000i	$0.499967\pi$
$90$	0	0
$91$	−1.89825	−0.198991
$92$	0	0
$93$	−8.30806	−0.861506
$94$	0	0
$95$	13.5979	1.39511
$96$	0	0
$97$	4.11333	0.417645	0.208822	−	0.977954i	$-0.433037\pi$
$97$	4.11333	0.417645	0.208822	+	0.977954i	$0.433037\pi$
$98$	0	0
$99$	3.28289	0.329942
$100$	0	0
$101$	−6.50261	−0.647034	−0.323517	−	0.946222i	$-0.604865\pi$
$101$	−6.50261	−0.647034	−0.323517	+	0.946222i	$0.604865\pi$
$102$	0	0
$103$	−12.6003	−1.24155	−0.620774	−	0.783990i	$-0.713182\pi$
$103$	−12.6003	−1.24155	−0.620774	+	0.783990i	$0.713182\pi$
$104$	0	0
$105$	−7.27051	−0.709529
$106$	0	0
$107$	−7.55504	−0.730373	−0.365187	−	0.930934i	$-0.618995\pi$
$107$	−7.55504	−0.730373	−0.365187	+	0.930934i	$0.618995\pi$
$108$	0	0
$109$	20.6907	1.98181	0.990904	−	0.134568i	$-0.0429646\pi$
$109$	20.6907	1.98181	0.990904	+	0.134568i	$0.0429646\pi$
$110$	0	0
$111$	12.9547	1.22961
$112$	0	0
$113$	−0.0580032	−0.00545648	−0.00272824	−	0.999996i	$-0.500868\pi$
$113$	−0.0580032	−0.00545648	−0.00272824	+	0.999996i	$0.500868\pi$
$114$	0	0
$115$	27.5604	2.57002
$116$	0	0
$117$	1.63758	0.151395
$118$	0	0
$119$	−9.19303	−0.842724
$120$	0	0
$121$	−7.11152	−0.646502
$122$	0	0
$123$	6.56625	0.592059
$124$	0	0
$125$	−2.05587	−0.183882
$126$	0	0
$127$	5.95882	0.528760	0.264380	−	0.964419i	$-0.414833\pi$
$127$	5.95882	0.528760	0.264380	+	0.964419i	$0.414833\pi$
$128$	0	0
$129$	11.1385	0.980687
$130$	0	0
$131$	17.4107	1.52118	0.760591	−	0.649231i	$-0.224909\pi$
$131$	17.4107	1.52118	0.760591	+	0.649231i	$0.224909\pi$
$132$	0	0
$133$	−8.04841	−0.697886
$134$	0	0
$135$	17.5745	1.51257
$136$	0	0
$137$	−17.6965	−1.51191	−0.755957	−	0.654621i	$-0.772828\pi$
$137$	−17.6965	−1.51191	−0.755957	+	0.654621i	$0.772828\pi$
$138$	0	0
$139$	−21.1638	−1.79509	−0.897546	−	0.440920i	$-0.854652\pi$
$139$	−21.1638	−1.79509	−0.897546	+	0.440920i	$0.854652\pi$
$140$	0	0
$141$	4.87047	0.410168
$142$	0	0
$143$	1.93967	0.162203
$144$	0	0
$145$	2.97902	0.247394
$146$	0	0
$147$	−3.78519	−0.312198
$148$	0	0
$149$	−0.261784	−0.0214462	−0.0107231	−	0.999943i	$-0.503413\pi$
$149$	−0.261784	−0.0214462	−0.0107231	+	0.999943i	$0.503413\pi$
$150$	0	0
$151$	3.74445	0.304719	0.152360	−	0.988325i	$-0.451313\pi$
$151$	3.74445	0.304719	0.152360	+	0.988325i	$0.451313\pi$
$152$	0	0
$153$	7.93064	0.641154
$154$	0	0
$155$	23.4426	1.88296
$156$	0	0
$157$	−1.95939	−0.156376	−0.0781880	−	0.996939i	$-0.524913\pi$
$157$	−1.95939	−0.156376	−0.0781880	+	0.996939i	$0.524913\pi$
$158$	0	0
$159$	−1.84826	−0.146577
$160$	0	0
$161$	−16.3126	−1.28561
$162$	0	0
$163$	−19.2234	−1.50570	−0.752848	−	0.658194i	$-0.771321\pi$
$163$	−19.2234	−1.50570	−0.752848	+	0.658194i	$0.771321\pi$
$164$	0	0
$165$	7.42913	0.578357
$166$	0	0
$167$	−8.05915	−0.623636	−0.311818	−	0.950142i	$-0.600938\pi$
$167$	−8.05915	−0.623636	−0.311818	+	0.950142i	$0.600938\pi$
$168$	0	0
$169$	−12.0324	−0.925573
$170$	0	0
$171$	6.94320	0.530960
$172$	0	0
$173$	13.6123	1.03493	0.517463	−	0.855705i	$-0.326876\pi$
$173$	13.6123	1.03493	0.517463	+	0.855705i	$0.326876\pi$
$174$	0	0
$175$	10.8659	0.821387
$176$	0	0
$177$	−0.615833	−0.0462889
$178$	0	0
$179$	9.57059	0.715339	0.357670	−	0.933848i	$-0.383571\pi$
$179$	9.57059	0.715339	0.357670	+	0.933848i	$0.383571\pi$
$180$	0	0
$181$	−2.56071	−0.190336	−0.0951682	−	0.995461i	$-0.530339\pi$
$181$	−2.56071	−0.190336	−0.0951682	+	0.995461i	$0.530339\pi$
$182$	0	0
$183$	−10.0149	−0.740320
$184$	0	0
$185$	−36.5540	−2.68750
$186$	0	0
$187$	9.39360	0.686928
$188$	0	0
$189$	−10.4021	−0.756642
$190$	0	0
$191$	10.5004	0.759781	0.379891	−	0.925031i	$-0.375962\pi$
$191$	10.5004	0.759781	0.379891	+	0.925031i	$0.375962\pi$
$192$	0	0
$193$	21.1558	1.52282	0.761412	−	0.648268i	$-0.224506\pi$
$193$	21.1558	1.52282	0.761412	+	0.648268i	$0.224506\pi$
$194$	0	0
$195$	3.70583	0.265380
$196$	0	0
$197$	−8.01788	−0.571251	−0.285625	−	0.958341i	$-0.592201\pi$
$197$	−8.01788	−0.571251	−0.285625	+	0.958341i	$0.592201\pi$
$198$	0	0
$199$	16.8675	1.19571	0.597854	−	0.801605i	$-0.296020\pi$
$199$	16.8675	1.19571	0.597854	+	0.801605i	$0.296020\pi$
$200$	0	0
$201$	11.9940	0.845993
$202$	0	0
$203$	−1.76324	−0.123756
$204$	0	0
$205$	−18.5278	−1.29404
$206$	0	0
$207$	14.0726	0.978111
$208$	0	0
$209$	8.22401	0.568867
$210$	0	0
$211$	3.78113	0.260304	0.130152	−	0.991494i	$-0.458453\pi$
$211$	3.78113	0.260304	0.130152	+	0.991494i	$0.458453\pi$
$212$	0	0
$213$	11.6755	0.799995
$214$	0	0
$215$	−31.4291	−2.14345
$216$	0	0
$217$	−13.8754	−0.941923
$218$	0	0
$219$	15.1873	1.02626
$220$	0	0
$221$	4.68576	0.315198
$222$	0	0
$223$	−7.81962	−0.523641	−0.261820	−	0.965117i	$-0.584323\pi$
$223$	−7.81962	−0.523641	−0.261820	+	0.965117i	$0.584323\pi$
$224$	0	0
$225$	−9.37382	−0.624921
$226$	0	0
$227$	17.6507	1.17152	0.585758	−	0.810486i	$-0.300797\pi$
$227$	17.6507	1.17152	0.585758	+	0.810486i	$0.300797\pi$
$228$	0	0
$229$	−2.54980	−0.168496	−0.0842478	−	0.996445i	$-0.526849\pi$
$229$	−2.54980	−0.168496	−0.0842478	+	0.996445i	$0.526849\pi$
$230$	0	0
$231$	−4.39721	−0.289315
$232$	0	0
$233$	−24.0102	−1.57296	−0.786479	−	0.617617i	$-0.788098\pi$
$233$	−24.0102	−1.57296	−0.786479	+	0.617617i	$0.788098\pi$
$234$	0	0
$235$	−13.7429	−0.896487
$236$	0	0
$237$	20.2696	1.31665
$238$	0	0
$239$	−10.2539	−0.663272	−0.331636	−	0.943407i	$-0.607601\pi$
$239$	−10.2539	−0.663272	−0.331636	+	0.943407i	$0.607601\pi$
$240$	0	0
$241$	−20.4625	−1.31811	−0.659053	−	0.752097i	$-0.729043\pi$
$241$	−20.4625	−1.31811	−0.659053	+	0.752097i	$0.729043\pi$
$242$	0	0
$243$	14.7448	0.945878
$244$	0	0
$245$	10.6806	0.682358
$246$	0	0
$247$	4.10234	0.261026
$248$	0	0
$249$	−3.14786	−0.199488
$250$	0	0
$251$	−16.4873	−1.04067	−0.520334	−	0.853963i	$-0.674192\pi$
$251$	−16.4873	−1.04067	−0.520334	+	0.853963i	$0.674192\pi$
$252$	0	0
$253$	16.6685	1.04794
$254$	0	0
$255$	17.9469	1.12388
$256$	0	0
$257$	−16.7982	−1.04784	−0.523922	−	0.851766i	$-0.675532\pi$
$257$	−16.7982	−1.04784	−0.523922	+	0.851766i	$0.675532\pi$
$258$	0	0
$259$	21.6358	1.34438
$260$	0	0
$261$	1.52112	0.0941547
$262$	0	0
$263$	5.48596	0.338279	0.169140	−	0.985592i	$-0.445901\pi$
$263$	5.48596	0.338279	0.169140	+	0.985592i	$0.445901\pi$
$264$	0	0
$265$	5.21520	0.320367
$266$	0	0
$267$	0.00229240	0.000140293 0
$268$	0	0
$269$	−5.32940	−0.324939	−0.162470	−	0.986714i	$-0.551946\pi$
$269$	−5.32940	−0.324939	−0.162470	+	0.986714i	$0.551946\pi$
$270$	0	0
$271$	15.4702	0.939750	0.469875	−	0.882733i	$-0.344299\pi$
$271$	15.4702	0.939750	0.469875	+	0.882733i	$0.344299\pi$
$272$	0	0
$273$	−2.19343	−0.132753
$274$	0	0
$275$	−11.1030	−0.669536
$276$	0	0
$277$	20.8189	1.25089	0.625443	−	0.780270i	$-0.284918\pi$
$277$	20.8189	1.25089	0.625443	+	0.780270i	$0.284918\pi$
$278$	0	0
$279$	11.9700	0.716627
$280$	0	0
$281$	−11.9951	−0.715568	−0.357784	−	0.933804i	$-0.616468\pi$
$281$	−11.9951	−0.715568	−0.357784	+	0.933804i	$0.616468\pi$
$282$	0	0
$283$	−3.47789	−0.206739	−0.103369	−	0.994643i	$-0.532962\pi$
$283$	−3.47789	−0.206739	−0.103369	+	0.994643i	$0.532962\pi$
$284$	0	0
$285$	15.7124	0.930722
$286$	0	0
$287$	10.9664	0.647325
$288$	0	0
$289$	5.69261	0.334859
$290$	0	0
$291$	4.75296	0.278623
$292$	0	0
$293$	6.80153	0.397350	0.198675	−	0.980065i	$-0.436336\pi$
$293$	6.80153	0.397350	0.198675	+	0.980065i	$0.436336\pi$
$294$	0	0
$295$	1.73768	0.101172
$296$	0	0
$297$	10.6291	0.616761
$298$	0	0
$299$	8.31466	0.480849
$300$	0	0
$301$	18.6025	1.07223
$302$	0	0
$303$	−7.51377	−0.431655
$304$	0	0
$305$	28.2587	1.61809
$306$	0	0
$307$	4.90709	0.280063	0.140031	−	0.990147i	$-0.455280\pi$
$307$	4.90709	0.280063	0.140031	+	0.990147i	$0.455280\pi$
$308$	0	0
$309$	−14.5597	−0.828273
$310$	0	0
$311$	4.40793	0.249951	0.124975	−	0.992160i	$-0.460115\pi$
$311$	4.40793	0.249951	0.124975	+	0.992160i	$0.460115\pi$
$312$	0	0
$313$	−19.0681	−1.07779	−0.538896	−	0.842372i	$-0.681158\pi$
$313$	−19.0681	−1.07779	−0.538896	+	0.842372i	$0.681158\pi$
$314$	0	0
$315$	10.4752	0.590208
$316$	0	0
$317$	35.3612	1.98608	0.993042	−	0.117757i	$-0.0375704\pi$
$317$	35.3612	1.98608	0.993042	+	0.117757i	$0.0375704\pi$
$318$	0	0
$319$	1.80171	0.100877
$320$	0	0
$321$	−8.72986	−0.487253
$322$	0	0
$323$	19.8672	1.10544
$324$	0	0
$325$	−5.53845	−0.307218
$326$	0	0
$327$	23.9081	1.32212
$328$	0	0
$329$	8.13423	0.448455
$330$	0	0
$331$	13.5365	0.744036	0.372018	−	0.928226i	$-0.378666\pi$
$331$	13.5365	0.744036	0.372018	+	0.928226i	$0.378666\pi$
$332$	0	0
$333$	−18.6648	−1.02282
$334$	0	0
$335$	−33.8432	−1.84905
$336$	0	0
$337$	4.46560	0.243257	0.121628	−	0.992576i	$-0.461188\pi$
$337$	4.46560	0.243257	0.121628	+	0.992576i	$0.461188\pi$
$338$	0	0
$339$	−0.0670228	−0.00364018
$340$	0	0
$341$	14.1781	0.767788
$342$	0	0
$343$	−19.8304	−1.07074
$344$	0	0
$345$	31.8460	1.71453
$346$	0	0
$347$	17.3752	0.932748	0.466374	−	0.884588i	$-0.345560\pi$
$347$	17.3752	0.932748	0.466374	+	0.884588i	$0.345560\pi$
$348$	0	0
$349$	29.7959	1.59494	0.797469	−	0.603360i	$-0.206172\pi$
$349$	29.7959	1.59494	0.797469	+	0.603360i	$0.206172\pi$
$350$	0	0
$351$	5.30204	0.283002
$352$	0	0
$353$	−2.46145	−0.131010	−0.0655048	−	0.997852i	$-0.520866\pi$
$353$	−2.46145	−0.131010	−0.0655048	+	0.997852i	$0.520866\pi$
$354$	0	0
$355$	−32.9446	−1.74852
$356$	0	0
$357$	−10.6226	−0.562206
$358$	0	0
$359$	19.5034	1.02935	0.514674	−	0.857386i	$-0.327913\pi$
$359$	19.5034	1.02935	0.514674	+	0.857386i	$0.327913\pi$
$360$	0	0
$361$	−1.60647	−0.0845512
$362$	0	0
$363$	−8.21738	−0.431301
$364$	0	0
$365$	−42.8536	−2.24306
$366$	0	0
$367$	−30.3322	−1.58333	−0.791663	−	0.610958i	$-0.790784\pi$
$367$	−30.3322	−1.58333	−0.791663	+	0.610958i	$0.790784\pi$
$368$	0	0
$369$	−9.46047	−0.492492
$370$	0	0
$371$	−3.08681	−0.160259
$372$	0	0
$373$	−5.42971	−0.281140	−0.140570	−	0.990071i	$-0.544893\pi$
$373$	−5.42971	−0.281140	−0.140570	+	0.990071i	$0.544893\pi$
$374$	0	0
$375$	−2.37556	−0.122673
$376$	0	0
$377$	0.898739	0.0462874
$378$	0	0
$379$	34.4737	1.77080	0.885398	−	0.464835i	$-0.153886\pi$
$379$	34.4737	1.77080	0.885398	+	0.464835i	$0.153886\pi$
$380$	0	0
$381$	6.88543	0.352751
$382$	0	0
$383$	−9.85691	−0.503665	−0.251832	−	0.967771i	$-0.581033\pi$
$383$	−9.85691	−0.503665	−0.251832	+	0.967771i	$0.581033\pi$
$384$	0	0
$385$	12.4075	0.632344
$386$	0	0
$387$	−16.0480	−0.815765
$388$	0	0
$389$	−23.5924	−1.19618	−0.598091	−	0.801428i	$-0.704074\pi$
$389$	−23.5924	−1.19618	−0.598091	+	0.801428i	$0.704074\pi$
$390$	0	0
$391$	40.2670	2.03639
$392$	0	0
$393$	20.1181	1.01483
$394$	0	0
$395$	−57.1942	−2.87775
$396$	0	0
$397$	−6.42961	−0.322693	−0.161347	−	0.986898i	$-0.551584\pi$
$397$	−6.42961	−0.322693	−0.161347	+	0.986898i	$0.551584\pi$
$398$	0	0
$399$	−9.29996	−0.465580
$400$	0	0
$401$	14.5137	0.724777	0.362389	−	0.932027i	$-0.381961\pi$
$401$	14.5137	0.724777	0.362389	+	0.932027i	$0.381961\pi$
$402$	0	0
$403$	7.07240	0.352301
$404$	0	0
$405$	4.02321	0.199915
$406$	0	0
$407$	−22.1078	−1.09584
$408$	0	0
$409$	20.5504	1.01615	0.508076	−	0.861312i	$-0.330357\pi$
$409$	20.5504	1.01615	0.508076	+	0.861312i	$0.330357\pi$
$410$	0	0
$411$	−20.4483	−1.00864
$412$	0	0
$413$	−1.02851	−0.0506097
$414$	0	0
$415$	8.88225	0.436013
$416$	0	0
$417$	−24.4548	−1.19756
$418$	0	0
$419$	−22.2567	−1.08731	−0.543657	−	0.839308i	$-0.682961\pi$
$419$	−22.2567	−1.08731	−0.543657	+	0.839308i	$0.682961\pi$
$420$	0	0
$421$	25.4817	1.24190	0.620952	−	0.783849i	$-0.286746\pi$
$421$	25.4817	1.24190	0.620952	+	0.783849i	$0.286746\pi$
$422$	0	0
$423$	−7.01724	−0.341190
$424$	0	0
$425$	−26.8221	−1.30106
$426$	0	0
$427$	−16.7260	−0.809426
$428$	0	0
$429$	2.24129	0.108210
$430$	0	0
$431$	12.0335	0.579635	0.289817	−	0.957082i	$-0.406405\pi$
$431$	12.0335	0.579635	0.289817	+	0.957082i	$0.406405\pi$
$432$	0	0
$433$	−12.9773	−0.623648	−0.311824	−	0.950140i	$-0.600940\pi$
$433$	−12.9773	−0.623648	−0.311824	+	0.950140i	$0.600940\pi$
$434$	0	0
$435$	3.44227	0.165044
$436$	0	0
$437$	35.2534	1.68640
$438$	0	0
$439$	6.96631	0.332484	0.166242	−	0.986085i	$-0.446837\pi$
$439$	6.96631	0.332484	0.166242	+	0.986085i	$0.446837\pi$
$440$	0	0
$441$	5.45360	0.259695
$442$	0	0
$443$	22.3604	1.06238	0.531188	−	0.847254i	$-0.321746\pi$
$443$	22.3604	1.06238	0.531188	+	0.847254i	$0.321746\pi$
$444$	0	0
$445$	−0.00646842	−0.000306632 0
$446$	0	0
$447$	−0.302492	−0.0143074
$448$	0	0
$449$	−41.3012	−1.94912	−0.974562	−	0.224117i	$-0.928050\pi$
$449$	−41.3012	−1.94912	−0.974562	+	0.224117i	$0.928050\pi$
$450$	0	0
$451$	−11.2056	−0.527653
$452$	0	0
$453$	4.32672	0.203287
$454$	0	0
$455$	6.18916	0.290152
$456$	0	0
$457$	−7.19418	−0.336530	−0.168265	−	0.985742i	$-0.553816\pi$
$457$	−7.19418	−0.336530	−0.168265	+	0.985742i	$0.553816\pi$
$458$	0	0
$459$	25.6772	1.19851
$460$	0	0
$461$	30.4170	1.41666	0.708329	−	0.705882i	$-0.249449\pi$
$461$	30.4170	1.41666	0.708329	+	0.705882i	$0.249449\pi$
$462$	0	0
$463$	17.5190	0.814175	0.407088	−	0.913389i	$-0.366544\pi$
$463$	17.5190	0.814175	0.407088	+	0.913389i	$0.366544\pi$
$464$	0	0
$465$	27.0880	1.25618
$466$	0	0
$467$	−22.0087	−1.01844	−0.509220	−	0.860636i	$-0.670066\pi$
$467$	−22.0087	−1.01844	−0.509220	+	0.860636i	$0.670066\pi$
$468$	0	0
$469$	20.0314	0.924963
$470$	0	0
$471$	−2.26407	−0.104323
$472$	0	0
$473$	−19.0083	−0.874005
$474$	0	0
$475$	−23.4825	−1.07745
$476$	0	0
$477$	2.66293	0.121927
$478$	0	0
$479$	0.140520	0.00642050	0.00321025	−	0.999995i	$-0.498978\pi$
$479$	0.140520	0.00642050	0.00321025	+	0.999995i	$0.498978\pi$
$480$	0	0
$481$	−11.0279	−0.502831
$482$	0	0
$483$	−18.8493	−0.857671
$484$	0	0
$485$	−13.4113	−0.608976
$486$	0	0
$487$	22.9064	1.03799	0.518995	−	0.854777i	$-0.326306\pi$
$487$	22.9064	1.03799	0.518995	+	0.854777i	$0.326306\pi$
$488$	0	0
$489$	−22.2127	−1.00449
$490$	0	0
$491$	29.7589	1.34300	0.671500	−	0.741005i	$-0.265650\pi$
$491$	29.7589	1.34300	0.671500	+	0.741005i	$0.265650\pi$
$492$	0	0
$493$	4.35250	0.196027
$494$	0	0
$495$	−10.7037	−0.481095
$496$	0	0
$497$	19.4995	0.874671
$498$	0	0
$499$	19.9972	0.895198	0.447599	−	0.894234i	$-0.352279\pi$
$499$	19.9972	0.895198	0.447599	+	0.894234i	$0.352279\pi$
$500$	0	0
$501$	−9.31236	−0.416046
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	21.2014	0.943452
$506$	0	0
$507$	−13.9035	−0.617477
$508$	0	0
$509$	2.29798	0.101856	0.0509282	−	0.998702i	$-0.483782\pi$
$509$	2.29798	0.101856	0.0509282	+	0.998702i	$0.483782\pi$
$510$	0	0
$511$	25.3645	1.12206
$512$	0	0
$513$	22.4801	0.992522
$514$	0	0
$515$	41.0828	1.81032
$516$	0	0
$517$	−8.31170	−0.365548
$518$	0	0
$519$	15.7291	0.690430
$520$	0	0
$521$	−13.3648	−0.585522	−0.292761	−	0.956186i	$-0.594574\pi$
$521$	−13.3648	−0.585522	−0.292761	+	0.956186i	$0.594574\pi$
$522$	0	0
$523$	36.5939	1.60014	0.800071	−	0.599906i	$-0.204795\pi$
$523$	36.5939	1.60014	0.800071	+	0.599906i	$0.204795\pi$
$524$	0	0
$525$	12.5556	0.547972
$526$	0	0
$527$	34.2508	1.49199
$528$	0	0
$529$	48.4519	2.10661
$530$	0	0
$531$	0.887276	0.0385045
$532$	0	0
$533$	−5.58964	−0.242114
$534$	0	0
$535$	24.6328	1.06497
$536$	0	0
$537$	11.0588	0.477224
$538$	0	0
$539$	6.45962	0.278236
$540$	0	0
$541$	−26.5138	−1.13992	−0.569959	−	0.821673i	$-0.693041\pi$
$541$	−26.5138	−1.13992	−0.569959	+	0.821673i	$0.693041\pi$
$542$	0	0
$543$	−2.95891	−0.126979
$544$	0	0
$545$	−67.4610	−2.88971
$546$	0	0
$547$	6.04597	0.258507	0.129254	−	0.991612i	$-0.458742\pi$
$547$	6.04597	0.258507	0.129254	+	0.991612i	$0.458742\pi$
$548$	0	0
$549$	14.4291	0.615821
$550$	0	0
$551$	3.81057	0.162336
$552$	0	0
$553$	33.8525	1.43956
$554$	0	0
$555$	−42.2382	−1.79291
$556$	0	0
$557$	31.6034	1.33908	0.669540	−	0.742776i	$-0.266491\pi$
$557$	31.6034	1.33908	0.669540	+	0.742776i	$0.266491\pi$
$558$	0	0
$559$	−9.48183	−0.401039
$560$	0	0
$561$	10.8543	0.458270
$562$	0	0
$563$	6.88343	0.290102	0.145051	−	0.989424i	$-0.453665\pi$
$563$	6.88343	0.290102	0.145051	+	0.989424i	$0.453665\pi$
$564$	0	0
$565$	0.189117	0.00795620
$566$	0	0
$567$	−2.38129	−0.100005
$568$	0	0
$569$	30.3332	1.27163	0.635817	−	0.771840i	$-0.280663\pi$
$569$	30.3332	1.27163	0.635817	+	0.771840i	$0.280663\pi$
$570$	0	0
$571$	−35.3930	−1.48115	−0.740576	−	0.671973i	$-0.765447\pi$
$571$	−35.3930	−1.48115	−0.740576	+	0.671973i	$0.765447\pi$
$572$	0	0
$573$	12.1332	0.506872
$574$	0	0
$575$	−47.5946	−1.98483
$576$	0	0
$577$	13.9961	0.582667	0.291333	−	0.956622i	$-0.405901\pi$
$577$	13.9961	0.582667	0.291333	+	0.956622i	$0.405901\pi$
$578$	0	0
$579$	24.4455	1.01592
$580$	0	0
$581$	−5.25729	−0.218109
$582$	0	0
$583$	3.15415	0.130632
$584$	0	0
$585$	−5.33927	−0.220751
$586$	0	0
$587$	44.8955	1.85303	0.926517	−	0.376253i	$-0.122788\pi$
$587$	44.8955	1.85303	0.926517	+	0.376253i	$0.122788\pi$
$588$	0	0
$589$	29.9863	1.23556
$590$	0	0
$591$	−9.26468	−0.381098
$592$	0	0
$593$	−20.7630	−0.852636	−0.426318	−	0.904573i	$-0.640189\pi$
$593$	−20.7630	−0.852636	−0.426318	+	0.904573i	$0.640189\pi$
$594$	0	0
$595$	29.9734	1.22879
$596$	0	0
$597$	19.4905	0.797692
$598$	0	0
$599$	30.1464	1.23175	0.615874	−	0.787845i	$-0.288803\pi$
$599$	30.1464	1.23175	0.615874	+	0.787845i	$0.288803\pi$
$600$	0	0
$601$	−33.4859	−1.36592	−0.682959	−	0.730456i	$-0.739307\pi$
$601$	−33.4859	−1.36592	−0.682959	+	0.730456i	$0.739307\pi$
$602$	0	0
$603$	−17.2807	−0.703723
$604$	0	0
$605$	23.1868	0.942677
$606$	0	0
$607$	−38.7080	−1.57111	−0.785554	−	0.618793i	$-0.787622\pi$
$607$	−38.7080	−1.57111	−0.785554	+	0.618793i	$0.787622\pi$
$608$	0	0
$609$	−2.03743	−0.0825609
$610$	0	0
$611$	−4.14608	−0.167733
$612$	0	0
$613$	−0.233177	−0.00941792	−0.00470896	−	0.999989i	$-0.501499\pi$
$613$	−0.233177	−0.00941792	−0.00470896	+	0.999989i	$0.501499\pi$
$614$	0	0
$615$	−21.4089	−0.863292
$616$	0	0
$617$	−21.9488	−0.883624	−0.441812	−	0.897108i	$-0.645664\pi$
$617$	−21.9488	−0.883624	−0.441812	+	0.897108i	$0.645664\pi$
$618$	0	0
$619$	−25.7059	−1.03321	−0.516603	−	0.856225i	$-0.672804\pi$
$619$	−25.7059	−1.03321	−0.516603	+	0.856225i	$0.672804\pi$
$620$	0	0
$621$	45.5630	1.82838
$622$	0	0
$623$	0.00382857	0.000153388 0
$624$	0	0
$625$	−21.4497	−0.857987
$626$	0	0
$627$	9.50286	0.379508
$628$	0	0
$629$	−53.4071	−2.12948
$630$	0	0
$631$	−17.1563	−0.682981	−0.341490	−	0.939885i	$-0.610932\pi$
$631$	−17.1563	−0.682981	−0.341490	+	0.939885i	$0.610932\pi$
$632$	0	0
$633$	4.36911	0.173656
$634$	0	0
$635$	−19.4284	−0.770994
$636$	0	0
$637$	3.22222	0.127669
$638$	0	0
$639$	−16.8218	−0.665460
$640$	0	0
$641$	−21.7362	−0.858529	−0.429264	−	0.903179i	$-0.641227\pi$
$641$	−21.7362	−0.858529	−0.429264	+	0.903179i	$0.641227\pi$
$642$	0	0
$643$	−17.0424	−0.672085	−0.336042	−	0.941847i	$-0.609089\pi$
$643$	−17.0424	−0.672085	−0.336042	+	0.941847i	$0.609089\pi$
$644$	0	0
$645$	−36.3164	−1.42996
$646$	0	0
$647$	25.2034	0.990847	0.495424	−	0.868652i	$-0.335013\pi$
$647$	25.2034	0.990847	0.495424	+	0.868652i	$0.335013\pi$
$648$	0	0
$649$	1.05095	0.0412534
$650$	0	0
$651$	−16.0330	−0.628385
$652$	0	0
$653$	43.5540	1.70440	0.852200	−	0.523217i	$-0.175268\pi$
$653$	43.5540	1.70440	0.852200	+	0.523217i	$0.175268\pi$
$654$	0	0
$655$	−56.7668	−2.21806
$656$	0	0
$657$	−21.8814	−0.853677
$658$	0	0
$659$	6.34008	0.246975	0.123487	−	0.992346i	$-0.460592\pi$
$659$	6.34008	0.246975	0.123487	+	0.992346i	$0.460592\pi$
$660$	0	0
$661$	29.9696	1.16568	0.582841	−	0.812586i	$-0.301941\pi$
$661$	29.9696	1.16568	0.582841	+	0.812586i	$0.301941\pi$
$662$	0	0
$663$	5.41440	0.210278
$664$	0	0
$665$	26.2415	1.01760
$666$	0	0
$667$	7.72331	0.299048
$668$	0	0
$669$	−9.03559	−0.349336
$670$	0	0
$671$	17.0909	0.659786
$672$	0	0
$673$	−33.5701	−1.29403	−0.647015	−	0.762477i	$-0.723983\pi$
$673$	−33.5701	−1.29403	−0.647015	+	0.762477i	$0.723983\pi$
$674$	0	0
$675$	−30.3498	−1.16816
$676$	0	0
$677$	−22.2557	−0.855355	−0.427677	−	0.903931i	$-0.640668\pi$
$677$	−22.2557	−0.855355	−0.427677	+	0.903931i	$0.640668\pi$
$678$	0	0
$679$	7.93797	0.304631
$680$	0	0
$681$	20.3954	0.781553
$682$	0	0
$683$	−12.3966	−0.474344	−0.237172	−	0.971468i	$-0.576220\pi$
$683$	−12.3966	−0.474344	−0.237172	+	0.971468i	$0.576220\pi$
$684$	0	0
$685$	57.6986	2.20455
$686$	0	0
$687$	−2.94630	−0.112408
$688$	0	0
$689$	1.57337	0.0599406
$690$	0	0
$691$	−22.0680	−0.839506	−0.419753	−	0.907638i	$-0.637883\pi$
$691$	−22.0680	−0.839506	−0.419753	+	0.907638i	$0.637883\pi$
$692$	0	0
$693$	6.33537	0.240661
$694$	0	0
$695$	69.0037	2.61746
$696$	0	0
$697$	−27.0700	−1.02535
$698$	0	0
$699$	−27.7438	−1.04937
$700$	0	0
$701$	−15.9963	−0.604170	−0.302085	−	0.953281i	$-0.597683\pi$
$701$	−15.9963	−0.604170	−0.302085	+	0.953281i	$0.597683\pi$
$702$	0	0
$703$	−46.7574	−1.76349
$704$	0	0
$705$	−15.8799	−0.598073
$706$	0	0
$707$	−12.5489	−0.471948
$708$	0	0
$709$	27.1114	1.01819	0.509095	−	0.860711i	$-0.329980\pi$
$709$	27.1114	1.01819	0.509095	+	0.860711i	$0.329980\pi$
$710$	0	0
$711$	−29.2039	−1.09523
$712$	0	0
$713$	60.7765	2.27610
$714$	0	0
$715$	−6.32419	−0.236511
$716$	0	0
$717$	−11.8485	−0.442488
$718$	0	0
$719$	23.2408	0.866735	0.433367	−	0.901217i	$-0.357325\pi$
$719$	23.2408	0.866735	0.433367	+	0.901217i	$0.357325\pi$
$720$	0	0
$721$	−24.3164	−0.905589
$722$	0	0
$723$	−23.6444	−0.879346
$724$	0	0
$725$	−5.14454	−0.191063
$726$	0	0
$727$	33.1038	1.22775	0.613876	−	0.789402i	$-0.289609\pi$
$727$	33.1038	1.22775	0.613876	+	0.789402i	$0.289609\pi$
$728$	0	0
$729$	20.7395	0.768128
$730$	0	0
$731$	−45.9195	−1.69839
$732$	0	0
$733$	46.3254	1.71107	0.855533	−	0.517748i	$-0.173230\pi$
$733$	46.3254	1.71107	0.855533	+	0.517748i	$0.173230\pi$
$734$	0	0
$735$	12.3414	0.455221
$736$	0	0
$737$	−20.4684	−0.753963
$738$	0	0
$739$	−31.5923	−1.16214	−0.581070	−	0.813854i	$-0.697366\pi$
$739$	−31.5923	−1.16214	−0.581070	+	0.813854i	$0.697366\pi$
$740$	0	0
$741$	4.74026	0.174138
$742$	0	0
$743$	−18.9414	−0.694892	−0.347446	−	0.937700i	$-0.612951\pi$
$743$	−18.9414	−0.694892	−0.347446	+	0.937700i	$0.612951\pi$
$744$	0	0
$745$	0.853534	0.0312711
$746$	0	0
$747$	4.53536	0.165940
$748$	0	0
$749$	−14.5798	−0.532736
$750$	0	0
$751$	−18.6252	−0.679643	−0.339822	−	0.940490i	$-0.610367\pi$
$751$	−18.6252	−0.679643	−0.339822	+	0.940490i	$0.610367\pi$
$752$	0	0
$753$	−19.0511	−0.694260
$754$	0	0
$755$	−12.2086	−0.444317
$756$	0	0
$757$	4.42740	0.160917	0.0804583	−	0.996758i	$-0.474362\pi$
$757$	4.42740	0.160917	0.0804583	+	0.996758i	$0.474362\pi$
$758$	0	0
$759$	19.2605	0.699112
$760$	0	0
$761$	−48.8700	−1.77154	−0.885768	−	0.464129i	$-0.846368\pi$
$761$	−48.8700	−1.77154	−0.885768	+	0.464129i	$0.846368\pi$
$762$	0	0
$763$	39.9293	1.44554
$764$	0	0
$765$	−25.8575	−0.934879
$766$	0	0
$767$	0.524240	0.0189292
$768$	0	0
$769$	−30.1276	−1.08643	−0.543214	−	0.839594i	$-0.682793\pi$
$769$	−30.1276	−1.08643	−0.543214	+	0.839594i	$0.682793\pi$
$770$	0	0
$771$	−19.4104	−0.699047
$772$	0	0
$773$	11.4422	0.411547	0.205774	−	0.978600i	$-0.434029\pi$
$773$	11.4422	0.411547	0.205774	+	0.978600i	$0.434029\pi$
$774$	0	0
$775$	−40.4836	−1.45421
$776$	0	0
$777$	25.0002	0.896877
$778$	0	0
$779$	−23.6996	−0.849125
$780$	0	0
$781$	−19.9249	−0.712969
$782$	0	0
$783$	4.92494	0.176003
$784$	0	0
$785$	6.38848	0.228015
$786$	0	0
$787$	−4.32502	−0.154170	−0.0770851	−	0.997025i	$-0.524561\pi$
$787$	−4.32502	−0.154170	−0.0770851	+	0.997025i	$0.524561\pi$
$788$	0	0
$789$	6.33904	0.225676
$790$	0	0
$791$	−0.111936	−0.00397998
$792$	0	0
$793$	8.52535	0.302744
$794$	0	0
$795$	6.02618	0.213726
$796$	0	0
$797$	−40.4668	−1.43341	−0.716704	−	0.697378i	$-0.754350\pi$
$797$	−40.4668	−1.43341	−0.716704	+	0.697378i	$0.754350\pi$
$798$	0	0
$799$	−20.0790	−0.710345
$800$	0	0
$801$	−0.00330283	−0.000116700 0
$802$	0	0
$803$	−25.9179	−0.914623
$804$	0	0
$805$	53.1865	1.87458
$806$	0	0
$807$	−6.15813	−0.216777
$808$	0	0
$809$	−13.3192	−0.468277	−0.234139	−	0.972203i	$-0.575227\pi$
$809$	−13.3192	−0.468277	−0.234139	+	0.972203i	$0.575227\pi$
$810$	0	0
$811$	9.62900	0.338120	0.169060	−	0.985606i	$-0.445927\pi$
$811$	9.62900	0.338120	0.169060	+	0.985606i	$0.445927\pi$
$812$	0	0
$813$	17.8759	0.626935
$814$	0	0
$815$	62.6771	2.19548
$816$	0	0
$817$	−40.2021	−1.40649
$818$	0	0
$819$	3.16024	0.110428
$820$	0	0
$821$	−50.4726	−1.76151	−0.880754	−	0.473575i	$-0.842963\pi$
$821$	−50.4726	−1.76151	−0.880754	+	0.473575i	$0.842963\pi$
$822$	0	0
$823$	7.06420	0.246243	0.123121	−	0.992392i	$-0.460710\pi$
$823$	7.06420	0.246243	0.123121	+	0.992392i	$0.460710\pi$
$824$	0	0
$825$	−12.8295	−0.446667
$826$	0	0
$827$	41.9997	1.46047	0.730237	−	0.683194i	$-0.239410\pi$
$827$	41.9997	1.46047	0.730237	+	0.683194i	$0.239410\pi$
$828$	0	0
$829$	−1.91390	−0.0664723	−0.0332362	−	0.999448i	$-0.510581\pi$
$829$	−1.91390	−0.0664723	−0.0332362	+	0.999448i	$0.510581\pi$
$830$	0	0
$831$	24.0562	0.834502
$832$	0	0
$833$	15.6049	0.540676
$834$	0	0
$835$	26.2765	0.909335
$836$	0	0
$837$	38.7556	1.33959
$838$	0	0
$839$	−19.8138	−0.684047	−0.342023	−	0.939691i	$-0.611112\pi$
$839$	−19.8138	−0.684047	−0.342023	+	0.939691i	$0.611112\pi$
$840$	0	0
$841$	−28.1652	−0.971213
$842$	0	0
$843$	−13.8604	−0.477377
$844$	0	0
$845$	39.2312	1.34959
$846$	0	0
$847$	−13.7240	−0.471561
$848$	0	0
$849$	−4.01870	−0.137922
$850$	0	0
$851$	−94.7684	−3.24862
$852$	0	0
$853$	−9.43175	−0.322937	−0.161469	−	0.986878i	$-0.551623\pi$
$853$	−9.43175	−0.322937	−0.161469	+	0.986878i	$0.551623\pi$
$854$	0	0
$855$	−22.6380	−0.774203
$856$	0	0
$857$	0.0817322	0.00279192	0.00139596	−	0.999999i	$-0.499556\pi$
$857$	0.0817322	0.00279192	0.00139596	+	0.999999i	$0.499556\pi$
$858$	0	0
$859$	20.2568	0.691152	0.345576	−	0.938391i	$-0.387684\pi$
$859$	20.2568	0.691152	0.345576	+	0.938391i	$0.387684\pi$
$860$	0	0
$861$	12.6717	0.431849
$862$	0	0
$863$	−24.6759	−0.839978	−0.419989	−	0.907529i	$-0.637966\pi$
$863$	−24.6759	−0.839978	−0.419989	+	0.907529i	$0.637966\pi$
$864$	0	0
$865$	−44.3824	−1.50905
$866$	0	0
$867$	6.57782	0.223394
$868$	0	0
$869$	−34.5911	−1.17342
$870$	0	0
$871$	−10.2101	−0.345958
$872$	0	0
$873$	−6.84793	−0.231767
$874$	0	0
$875$	−3.96745	−0.134124
$876$	0	0
$877$	−49.2534	−1.66317	−0.831585	−	0.555397i	$-0.812566\pi$
$877$	−49.2534	−1.66317	−0.831585	+	0.555397i	$0.812566\pi$
$878$	0	0
$879$	7.85918	0.265084
$880$	0	0
$881$	8.18119	0.275631	0.137816	−	0.990458i	$-0.455992\pi$
$881$	8.18119	0.275631	0.137816	+	0.990458i	$0.455992\pi$
$882$	0	0
$883$	54.7020	1.84087	0.920434	−	0.390897i	$-0.127835\pi$
$883$	54.7020	1.84087	0.920434	+	0.390897i	$0.127835\pi$
$884$	0	0
$885$	2.00790	0.0674947
$886$	0	0
$887$	28.5302	0.957951	0.478976	−	0.877828i	$-0.341008\pi$
$887$	28.5302	0.957951	0.478976	+	0.877828i	$0.341008\pi$
$888$	0	0
$889$	11.4994	0.385679
$890$	0	0
$891$	2.43324	0.0815166
$892$	0	0
$893$	−17.5790	−0.588259
$894$	0	0
$895$	−31.2044	−1.04305
$896$	0	0
$897$	9.60761	0.320789
$898$	0	0
$899$	6.56939	0.219101
$900$	0	0
$901$	7.61966	0.253848
$902$	0	0
$903$	21.4952	0.715316
$904$	0	0
$905$	8.34908	0.277533
$906$	0	0
$907$	−25.5848	−0.849528	−0.424764	−	0.905304i	$-0.639643\pi$
$907$	−25.5848	−0.849528	−0.424764	+	0.905304i	$0.639643\pi$
$908$	0	0
$909$	10.8256	0.359064
$910$	0	0
$911$	4.38307	0.145218	0.0726088	−	0.997360i	$-0.476868\pi$
$911$	4.38307	0.145218	0.0726088	+	0.997360i	$0.476868\pi$
$912$	0	0
$913$	5.37199	0.177787
$914$	0	0
$915$	32.6530	1.07947
$916$	0	0
$917$	33.5995	1.10955
$918$	0	0
$919$	42.0803	1.38810	0.694051	−	0.719926i	$-0.255824\pi$
$919$	42.0803	1.38810	0.694051	+	0.719926i	$0.255824\pi$
$920$	0	0
$921$	5.67016	0.186838
$922$	0	0
$923$	−9.93903	−0.327147
$924$	0	0
$925$	63.1258	2.07556
$926$	0	0
$927$	20.9772	0.688983
$928$	0	0
$929$	38.6208	1.26711	0.633554	−	0.773699i	$-0.281595\pi$
$929$	38.6208	1.26711	0.633554	+	0.773699i	$0.281595\pi$
$930$	0	0
$931$	13.6619	0.447751
$932$	0	0
$933$	5.09337	0.166749
$934$	0	0
$935$	−30.6274	−1.00162
$936$	0	0
$937$	33.3945	1.09095	0.545476	−	0.838126i	$-0.316349\pi$
$937$	33.3945	1.09095	0.545476	+	0.838126i	$0.316349\pi$
$938$	0	0
$939$	−22.0332	−0.719027
$940$	0	0
$941$	16.9997	0.554174	0.277087	−	0.960845i	$-0.410631\pi$
$941$	16.9997	0.554174	0.277087	+	0.960845i	$0.410631\pi$
$942$	0	0
$943$	−48.0345	−1.56422
$944$	0	0
$945$	33.9156	1.10327
$946$	0	0
$947$	−54.7458	−1.77900	−0.889500	−	0.456935i	$-0.848947\pi$
$947$	−54.7458	−1.77900	−0.889500	+	0.456935i	$0.848947\pi$
$948$	0	0
$949$	−12.9285	−0.419676
$950$	0	0
$951$	40.8600	1.32498
$952$	0	0
$953$	−9.64381	−0.312394	−0.156197	−	0.987726i	$-0.549923\pi$
$953$	−9.64381	−0.312394	−0.156197	+	0.987726i	$0.549923\pi$
$954$	0	0
$955$	−34.2360	−1.10785
$956$	0	0
$957$	2.08188	0.0672977
$958$	0	0
$959$	−34.1510	−1.10279
$960$	0	0
$961$	20.6961	0.667616
$962$	0	0
$963$	12.5777	0.405312
$964$	0	0
$965$	−68.9773	−2.22046
$966$	0	0
$967$	32.5842	1.04784	0.523918	−	0.851769i	$-0.324470\pi$
$967$	32.5842	1.04784	0.523918	+	0.851769i	$0.324470\pi$
$968$	0	0
$969$	22.9566	0.737471
$970$	0	0
$971$	−24.4450	−0.784476	−0.392238	−	0.919864i	$-0.628299\pi$
$971$	−24.4450	−0.784476	−0.392238	+	0.919864i	$0.628299\pi$
$972$	0	0
$973$	−40.8424	−1.30935
$974$	0	0
$975$	−6.39969	−0.204954
$976$	0	0
$977$	43.8778	1.40377	0.701887	−	0.712288i	$-0.252341\pi$
$977$	43.8778	1.40377	0.701887	+	0.712288i	$0.252341\pi$
$978$	0	0
$979$	−0.00391210	−0.000125031 0
$980$	0	0
$981$	−34.4462	−1.09978
$982$	0	0
$983$	−23.2295	−0.740906	−0.370453	−	0.928851i	$-0.620798\pi$
$983$	−23.2295	−0.740906	−0.370453	+	0.928851i	$0.620798\pi$
$984$	0	0
$985$	26.1419	0.832951
$986$	0	0
$987$	9.39912	0.299177
$988$	0	0
$989$	−81.4820	−2.59098
$990$	0	0
$991$	29.4959	0.936970	0.468485	−	0.883472i	$-0.344800\pi$
$991$	29.4959	0.936970	0.468485	+	0.883472i	$0.344800\pi$
$992$	0	0
$993$	15.6415	0.496368
$994$	0	0
$995$	−54.9958	−1.74348
$996$	0	0
$997$	50.4456	1.59763	0.798814	−	0.601578i	$-0.205461\pi$
$997$	50.4456	1.59763	0.798814	+	0.601578i	$0.205461\pi$
$998$	0	0
$999$	−60.4313	−1.91196

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.w.1.20		29
4.3	odd	2		4024.2.a.e.1.10	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.10	✓	29	4.3	odd	2
8048.2.a.w.1.20		29	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)

\(f(q)\)	\(=\)	\(q+1.15550 q^{3} -3.26045 q^{5} +1.92982 q^{7} -1.66482 q^{9} +O(q^{10})\)	\(q+1.15550 q^{3} -3.26045 q^{5} +1.92982 q^{7} -1.66482 q^{9} -1.97192 q^{11} -0.983643 q^{13} -3.76746 q^{15} -4.76368 q^{17} -4.17055 q^{19} +2.22991 q^{21} -8.45293 q^{23} +5.63055 q^{25} -5.39020 q^{27} -0.913684 q^{29} -7.19000 q^{31} -2.27856 q^{33} -6.29208 q^{35} +11.2113 q^{37} -1.13660 q^{39} +5.68259 q^{41} +9.63950 q^{43} +5.42805 q^{45} +4.21503 q^{47} -3.27580 q^{49} -5.50444 q^{51} -1.59953 q^{53} +6.42936 q^{55} -4.81908 q^{57} -0.532957 q^{59} -8.66711 q^{61} -3.21279 q^{63} +3.20712 q^{65} +10.3799 q^{67} -9.76737 q^{69} +10.1043 q^{71} +13.1435 q^{73} +6.50611 q^{75} -3.80545 q^{77} +17.5418 q^{79} -1.23394 q^{81} -2.72424 q^{83} +15.5317 q^{85} -1.05576 q^{87} +0.00198390 q^{89} -1.89825 q^{91} -8.30806 q^{93} +13.5979 q^{95} +4.11333 q^{97} +3.28289 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9	\( 29 q + 7 q^{3} - 4 q^{5} + 13 q^{7} + 20 q^{9} + 27 q^{11} + 16 q^{13} + 14 q^{15} - 15 q^{17} + 14 q^{19} + q^{21} + 25 q^{23} + 21 q^{25} + 25 q^{27} - 13 q^{29} + 27 q^{31} - 9 q^{33} + 29 q^{35} + 35 q^{37} + 38 q^{39} - 30 q^{41} + 38 q^{43} + q^{45} + 35 q^{47} + 14 q^{49} + 21 q^{51} + 2 q^{53} + 25 q^{55} - 25 q^{57} + 40 q^{59} + 10 q^{61} + 56 q^{63} - 50 q^{65} + 31 q^{67} + 11 q^{69} + 65 q^{71} - 23 q^{73} + 32 q^{75} + 13 q^{77} + 44 q^{79} - 7 q^{81} + 41 q^{83} + 26 q^{85} + 25 q^{87} - 48 q^{89} + 44 q^{91} + 25 q^{93} + 75 q^{95} - 18 q^{97} + 80 q^{99}+O(q^{100}) \) 29 * q + 7 * q^3 - 4 * q^5 + 13 * q^7 + 20 * q^9 + 27 * q^11 + 16 * q^13 + 14 * q^15 - 15 * q^17 + 14 * q^19 + q^21 + 25 * q^23 + 21 * q^25 + 25 * q^27 - 13 * q^29 + 27 * q^31 - 9 * q^33 + 29 * q^35 + 35 * q^37 + 38 * q^39 - 30 * q^41 + 38 * q^43 + q^45 + 35 * q^47 + 14 * q^49 + 21 * q^51 + 2 * q^53 + 25 * q^55 - 25 * q^57 + 40 * q^59 + 10 * q^61 + 56 * q^63 - 50 * q^65 + 31 * q^67 + 11 * q^69 + 65 * q^71 - 23 * q^73 + 32 * q^75 + 13 * q^77 + 44 * q^79 - 7 * q^81 + 41 * q^83 + 26 * q^85 + 25 * q^87 - 48 * q^89 + 44 * q^91 + 25 * q^93 + 75 * q^95 - 18 * q^97 + 80 * q^99

Introduction

L-functions

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