LMFDB - Embedded newform 804.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [804,2,Mod(1,804)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(804, 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("804.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 804 = 2^{2} \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	804.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:	$6.41997232251$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1076.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 8x - 6 $ x^3 - 8*x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.14743$ of defining polynomial
Character	$\chi$	$=$	804.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -4.14743 q^{5} -3.38854 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -4.14743 q^{5} -3.38854 q^{7} +1.00000 q^{9} +5.90632 q^{11} -3.90632 q^{13} +4.14743 q^{15} -3.90632 q^{17} +0.388543 q^{19} +3.38854 q^{21} -0.147430 q^{23} +12.2012 q^{25} -1.00000 q^{27} +8.00000 q^{29} -2.51777 q^{31} -5.90632 q^{33} +14.0537 q^{35} +4.61146 q^{37} +3.90632 q^{39} +12.0537 q^{41} -3.77709 q^{43} -4.14743 q^{45} +8.68340 q^{47} +4.48223 q^{49} +3.90632 q^{51} +5.27666 q^{53} -24.4960 q^{55} -0.388543 q^{57} -1.27666 q^{59} +7.42409 q^{61} -3.38854 q^{63} +16.2012 q^{65} -1.00000 q^{67} +0.147430 q^{69} -12.4960 q^{71} -16.8126 q^{73} -12.2012 q^{75} -20.0138 q^{77} -9.51777 q^{79} +1.00000 q^{81} +0.723339 q^{83} +16.2012 q^{85} -8.00000 q^{87} +11.4241 q^{89} +13.2367 q^{91} +2.51777 q^{93} -1.61146 q^{95} +6.38854 q^{97} +5.90632 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9} + 4 q^{11} + 2 q^{13} + 3 q^{15} + 2 q^{17} - 4 q^{19} + 5 q^{21} + 9 q^{23} + 4 q^{25} - 3 q^{27} + 24 q^{29} + q^{31} - 4 q^{33} + 19 q^{35} + 19 q^{37} - 2 q^{39} + 13 q^{41} - q^{43} - 3 q^{45} + 2 q^{47} + 22 q^{49} - 2 q^{51} + 3 q^{53} - 22 q^{55} + 4 q^{57} + 9 q^{59} - 5 q^{63} + 16 q^{65} - 3 q^{67} - 9 q^{69} + 14 q^{71} - 23 q^{73} - 4 q^{75} - 20 q^{79} + 3 q^{81} + 15 q^{83} + 16 q^{85} - 24 q^{87} + 12 q^{89} - 10 q^{91} - q^{93} - 10 q^{95} + 14 q^{97} + 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9 + 4 * q^11 + 2 * q^13 + 3 * q^15 + 2 * q^17 - 4 * q^19 + 5 * q^21 + 9 * q^23 + 4 * q^25 - 3 * q^27 + 24 * q^29 + q^31 - 4 * q^33 + 19 * q^35 + 19 * q^37 - 2 * q^39 + 13 * q^41 - q^43 - 3 * q^45 + 2 * q^47 + 22 * q^49 - 2 * q^51 + 3 * q^53 - 22 * q^55 + 4 * q^57 + 9 * q^59 - 5 * q^63 + 16 * q^65 - 3 * q^67 - 9 * q^69 + 14 * q^71 - 23 * q^73 - 4 * q^75 - 20 * q^79 + 3 * q^81 + 15 * q^83 + 16 * q^85 - 24 * q^87 + 12 * q^89 - 10 * q^91 - q^93 - 10 * q^95 + 14 * q^97 + 4 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−4.14743	−1.85479	−0.927394	−	0.374087i	$-0.877956\pi$
$5$	−4.14743	−1.85479	−0.927394	+	0.374087i	$0.877956\pi$
$6$	0	0
$7$	−3.38854	−1.28075	−0.640375	−	0.768063i	$-0.721221\pi$
$7$	−3.38854	−1.28075	−0.640375	+	0.768063i	$0.721221\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.90632	1.78082	0.890411	−	0.455158i	$-0.150417\pi$
$11$	5.90632	1.78082	0.890411	+	0.455158i	$0.150417\pi$
$12$	0	0
$13$	−3.90632	−1.08342	−0.541709	−	0.840566i	$-0.682222\pi$
$13$	−3.90632	−1.08342	−0.541709	+	0.840566i	$0.682222\pi$
$14$	0	0
$15$	4.14743	1.07086
$16$	0	0
$17$	−3.90632	−0.947421	−0.473711	−	0.880681i	$-0.657086\pi$
$17$	−3.90632	−0.947421	−0.473711	+	0.880681i	$0.657086\pi$
$18$	0	0
$19$	0.388543	0.0891380	0.0445690	−	0.999006i	$-0.485809\pi$
$19$	0.388543	0.0891380	0.0445690	+	0.999006i	$0.485809\pi$
$20$	0	0
$21$	3.38854	0.739441
$22$	0	0
$23$	−0.147430	−0.0307413	−0.0153707	−	0.999882i	$-0.504893\pi$
$23$	−0.147430	−0.0307413	−0.0153707	+	0.999882i	$0.504893\pi$
$24$	0	0
$25$	12.2012	2.44024
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	−2.51777	−0.452206	−0.226103	−	0.974103i	$-0.572599\pi$
$31$	−2.51777	−0.452206	−0.226103	+	0.974103i	$0.572599\pi$
$32$	0	0
$33$	−5.90632	−1.02816
$34$	0	0
$35$	14.0537	2.37552
$36$	0	0
$37$	4.61146	0.758119	0.379059	−	0.925372i	$-0.376248\pi$
$37$	4.61146	0.758119	0.379059	+	0.925372i	$0.376248\pi$
$38$	0	0
$39$	3.90632	0.625511
$40$	0	0
$41$	12.0537	1.88248	0.941240	−	0.337740i	$-0.109662\pi$
$41$	12.0537	1.88248	0.941240	+	0.337740i	$0.109662\pi$
$42$	0	0
$43$	−3.77709	−0.576000	−0.288000	−	0.957630i	$-0.592990\pi$
$43$	−3.77709	−0.576000	−0.288000	+	0.957630i	$0.592990\pi$
$44$	0	0
$45$	−4.14743	−0.618262
$46$	0	0
$47$	8.68340	1.26660	0.633302	−	0.773905i	$-0.281699\pi$
$47$	8.68340	1.26660	0.633302	+	0.773905i	$0.281699\pi$
$48$	0	0
$49$	4.48223	0.640318
$50$	0	0
$51$	3.90632	0.546994
$52$	0	0
$53$	5.27666	0.724805	0.362403	−	0.932022i	$-0.381957\pi$
$53$	5.27666	0.724805	0.362403	+	0.932022i	$0.381957\pi$
$54$	0	0
$55$	−24.4960	−3.30305
$56$	0	0
$57$	−0.388543	−0.0514638
$58$	0	0
$59$	−1.27666	−0.166207	−0.0831035	−	0.996541i	$-0.526483\pi$
$59$	−1.27666	−0.166207	−0.0831035	+	0.996541i	$0.526483\pi$
$60$	0	0
$61$	7.42409	0.950557	0.475279	−	0.879835i	$-0.342347\pi$
$61$	7.42409	0.950557	0.475279	+	0.879835i	$0.342347\pi$
$62$	0	0
$63$	−3.38854	−0.426916
$64$	0	0
$65$	16.2012	2.00951
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	0.147430	0.0177485
$70$	0	0
$71$	−12.4960	−1.48301	−0.741503	−	0.670949i	$-0.765887\pi$
$71$	−12.4960	−1.48301	−0.741503	+	0.670949i	$0.765887\pi$
$72$	0	0
$73$	−16.8126	−1.96777	−0.983885	−	0.178802i	$-0.942778\pi$
$73$	−16.8126	−1.96777	−0.983885	+	0.178802i	$0.942778\pi$
$74$	0	0
$75$	−12.2012	−1.40887
$76$	0	0
$77$	−20.0138	−2.28079
$78$	0	0
$79$	−9.51777	−1.07083	−0.535417	−	0.844588i	$-0.679845\pi$
$79$	−9.51777	−1.07083	−0.535417	+	0.844588i	$0.679845\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.723339	0.0793968	0.0396984	−	0.999212i	$-0.487360\pi$
$83$	0.723339	0.0793968	0.0396984	+	0.999212i	$0.487360\pi$
$84$	0	0
$85$	16.2012	1.75726
$86$	0	0
$87$	−8.00000	−0.857690
$88$	0	0
$89$	11.4241	1.21095	0.605476	−	0.795864i	$-0.292983\pi$
$89$	11.4241	1.21095	0.605476	+	0.795864i	$0.292983\pi$
$90$	0	0
$91$	13.2367	1.38759
$92$	0	0
$93$	2.51777	0.261081
$94$	0	0
$95$	−1.61146	−0.165332
$96$	0	0
$97$	6.38854	0.648658	0.324329	−	0.945944i	$-0.394861\pi$
$97$	6.38854	0.648658	0.324329	+	0.945944i	$0.394861\pi$
$98$	0	0
$99$	5.90632	0.593607
$100$	0	0
$101$	−8.97826	−0.893371	−0.446685	−	0.894691i	$-0.647396\pi$
$101$	−8.97826	−0.893371	−0.446685	+	0.894691i	$0.647396\pi$
$102$	0	0
$103$	2.20118	0.216888	0.108444	−	0.994103i	$-0.465413\pi$
$103$	2.20118	0.216888	0.108444	+	0.994103i	$0.465413\pi$
$104$	0	0
$105$	−14.0537	−1.37151
$106$	0	0
$107$	11.1656	1.07942	0.539711	−	0.841850i	$-0.318533\pi$
$107$	11.1656	1.07942	0.539711	+	0.841850i	$0.318533\pi$
$108$	0	0
$109$	4.38854	0.420346	0.210173	−	0.977664i	$-0.432597\pi$
$109$	4.38854	0.420346	0.210173	+	0.977664i	$0.432597\pi$
$110$	0	0
$111$	−4.61146	−0.437700
$112$	0	0
$113$	−17.7190	−1.66686	−0.833429	−	0.552626i	$-0.813626\pi$
$113$	−17.7190	−1.66686	−0.833429	+	0.552626i	$0.813626\pi$
$114$	0	0
$115$	0.611457	0.0570186
$116$	0	0
$117$	−3.90632	−0.361139
$118$	0	0
$119$	13.2367	1.21341
$120$	0	0
$121$	23.8846	2.17133
$122$	0	0
$123$	−12.0537	−1.08685
$124$	0	0
$125$	−29.8664	−2.67133
$126$	0	0
$127$	6.20118	0.550266	0.275133	−	0.961406i	$-0.411278\pi$
$127$	6.20118	0.550266	0.275133	+	0.961406i	$0.411278\pi$
$128$	0	0
$129$	3.77709	0.332554
$130$	0	0
$131$	16.9245	1.47870	0.739351	−	0.673320i	$-0.235132\pi$
$131$	16.9245	1.47870	0.739351	+	0.673320i	$0.235132\pi$
$132$	0	0
$133$	−1.31660	−0.114163
$134$	0	0
$135$	4.14743	0.356954
$136$	0	0
$137$	18.3486	1.56763	0.783814	−	0.620996i	$-0.213272\pi$
$137$	18.3486	1.56763	0.783814	+	0.620996i	$0.213272\pi$
$138$	0	0
$139$	−2.12923	−0.180599	−0.0902995	−	0.995915i	$-0.528782\pi$
$139$	−2.12923	−0.180599	−0.0902995	+	0.995915i	$0.528782\pi$
$140$	0	0
$141$	−8.68340	−0.731275
$142$	0	0
$143$	−23.0719	−1.92937
$144$	0	0
$145$	−33.1794	−2.75540
$146$	0	0
$147$	−4.48223	−0.369688
$148$	0	0
$149$	7.81263	0.640036	0.320018	−	0.947411i	$-0.396311\pi$
$149$	7.81263	0.640036	0.320018	+	0.947411i	$0.396311\pi$
$150$	0	0
$151$	−23.4605	−1.90919	−0.954594	−	0.297910i	$-0.903710\pi$
$151$	−23.4605	−1.90919	−0.954594	+	0.297910i	$0.903710\pi$
$152$	0	0
$153$	−3.90632	−0.315807
$154$	0	0
$155$	10.4423	0.838745
$156$	0	0
$157$	12.3304	0.984074	0.492037	−	0.870574i	$-0.336253\pi$
$157$	12.3304	0.984074	0.492037	+	0.870574i	$0.336253\pi$
$158$	0	0
$159$	−5.27666	−0.418466
$160$	0	0
$161$	0.499574	0.0393719
$162$	0	0
$163$	18.8846	1.47915	0.739577	−	0.673072i	$-0.235025\pi$
$163$	18.8846	1.47915	0.739577	+	0.673072i	$0.235025\pi$
$164$	0	0
$165$	24.4960	1.90701
$166$	0	0
$167$	−3.88897	−0.300938	−0.150469	−	0.988615i	$-0.548078\pi$
$167$	−3.88897	−0.300938	−0.150469	+	0.988615i	$0.548078\pi$
$168$	0	0
$169$	2.25931	0.173793
$170$	0	0
$171$	0.388543	0.0297127
$172$	0	0
$173$	17.2731	1.31325	0.656626	−	0.754217i	$-0.271983\pi$
$173$	17.2731	1.31325	0.656626	+	0.754217i	$0.271983\pi$
$174$	0	0
$175$	−41.3442	−3.12533
$176$	0	0
$177$	1.27666	0.0959597
$178$	0	0
$179$	−18.2012	−1.36042	−0.680210	−	0.733017i	$-0.738111\pi$
$179$	−18.2012	−1.36042	−0.680210	+	0.733017i	$0.738111\pi$
$180$	0	0
$181$	−2.90632	−0.216025	−0.108012	−	0.994150i	$-0.534449\pi$
$181$	−2.90632	−0.216025	−0.108012	+	0.994150i	$0.534449\pi$
$182$	0	0
$183$	−7.42409	−0.548805
$184$	0	0
$185$	−19.1257	−1.40615
$186$	0	0
$187$	−23.0719	−1.68719
$188$	0	0
$189$	3.38854	0.246480
$190$	0	0
$191$	−4.77709	−0.345658	−0.172829	−	0.984952i	$-0.555291\pi$
$191$	−4.77709	−0.345658	−0.172829	+	0.984952i	$0.555291\pi$
$192$	0	0
$193$	11.2949	0.813022	0.406511	−	0.913646i	$-0.366745\pi$
$193$	11.2949	0.813022	0.406511	+	0.913646i	$0.366745\pi$
$194$	0	0
$195$	−16.2012	−1.16019
$196$	0	0
$197$	22.3850	1.59487	0.797433	−	0.603408i	$-0.206191\pi$
$197$	22.3850	1.59487	0.797433	+	0.603408i	$0.206191\pi$
$198$	0	0
$199$	−9.51777	−0.674697	−0.337349	−	0.941380i	$-0.609530\pi$
$199$	−9.51777	−0.674697	−0.337349	+	0.941380i	$0.609530\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	−27.1083	−1.90263
$204$	0	0
$205$	−49.9921	−3.49160
$206$	0	0
$207$	−0.147430	−0.0102471
$208$	0	0
$209$	2.29486	0.158739
$210$	0	0
$211$	21.0719	1.45065	0.725326	−	0.688405i	$-0.241689\pi$
$211$	21.0719	1.45065	0.725326	+	0.688405i	$0.241689\pi$
$212$	0	0
$213$	12.4960	0.856214
$214$	0	0
$215$	15.6652	1.06836
$216$	0	0
$217$	8.53159	0.579162
$218$	0	0
$219$	16.8126	1.13609
$220$	0	0
$221$	15.2593	1.02645
$222$	0	0
$223$	13.0355	0.872925	0.436462	−	0.899722i	$-0.356231\pi$
$223$	13.0355	0.872925	0.436462	+	0.899722i	$0.356231\pi$
$224$	0	0
$225$	12.2012	0.813412
$226$	0	0
$227$	−3.75889	−0.249486	−0.124743	−	0.992189i	$-0.539811\pi$
$227$	−3.75889	−0.249486	−0.124743	+	0.992189i	$0.539811\pi$
$228$	0	0
$229$	−0.107495	−0.00710344	−0.00355172	−	0.999994i	$-0.501131\pi$
$229$	−0.107495	−0.00710344	−0.00355172	+	0.999994i	$0.501131\pi$
$230$	0	0
$231$	20.0138	1.31681
$232$	0	0
$233$	−6.88812	−0.451256	−0.225628	−	0.974214i	$-0.572443\pi$
$233$	−6.88812	−0.451256	−0.225628	+	0.974214i	$0.572443\pi$
$234$	0	0
$235$	−36.0138	−2.34928
$236$	0	0
$237$	9.51777	0.618246
$238$	0	0
$239$	−11.5178	−0.745023	−0.372511	−	0.928028i	$-0.621503\pi$
$239$	−11.5178	−0.745023	−0.372511	+	0.928028i	$0.621503\pi$
$240$	0	0
$241$	17.6834	1.13909	0.569544	−	0.821961i	$-0.307120\pi$
$241$	17.6834	1.13909	0.569544	+	0.821961i	$0.307120\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−18.5897	−1.18765
$246$	0	0
$247$	−1.51777	−0.0965736
$248$	0	0
$249$	−0.723339	−0.0458398
$250$	0	0
$251$	8.96445	0.565831	0.282916	−	0.959145i	$-0.408698\pi$
$251$	8.96445	0.565831	0.282916	+	0.959145i	$0.408698\pi$
$252$	0	0
$253$	−0.870770	−0.0547448
$254$	0	0
$255$	−16.2012	−1.01456
$256$	0	0
$257$	1.55417	0.0969467	0.0484733	−	0.998824i	$-0.484564\pi$
$257$	1.55417	0.0969467	0.0484733	+	0.998824i	$0.484564\pi$
$258$	0	0
$259$	−15.6261	−0.970960
$260$	0	0
$261$	8.00000	0.495188
$262$	0	0
$263$	9.75889	0.601759	0.300879	−	0.953662i	$-0.402720\pi$
$263$	9.75889	0.601759	0.300879	+	0.953662i	$0.402720\pi$
$264$	0	0
$265$	−21.8846	−1.34436
$266$	0	0
$267$	−11.4241	−0.699143
$268$	0	0
$269$	0.647004	0.0394485	0.0197243	−	0.999805i	$-0.493721\pi$
$269$	0.647004	0.0394485	0.0197243	+	0.999805i	$0.493721\pi$
$270$	0	0
$271$	8.03640	0.488177	0.244088	−	0.969753i	$-0.421511\pi$
$271$	8.03640	0.488177	0.244088	+	0.969753i	$0.421511\pi$
$272$	0	0
$273$	−13.2367	−0.801123
$274$	0	0
$275$	72.0640	4.34562
$276$	0	0
$277$	3.48223	0.209227	0.104613	−	0.994513i	$-0.466639\pi$
$277$	3.48223	0.209227	0.104613	+	0.994513i	$0.466639\pi$
$278$	0	0
$279$	−2.51777	−0.150735
$280$	0	0
$281$	−8.20118	−0.489241	−0.244621	−	0.969619i	$-0.578663\pi$
$281$	−8.20118	−0.489241	−0.244621	+	0.969619i	$0.578663\pi$
$282$	0	0
$283$	12.0138	0.714147	0.357073	−	0.934076i	$-0.383775\pi$
$283$	12.0138	0.714147	0.357073	+	0.934076i	$0.383775\pi$
$284$	0	0
$285$	1.61146	0.0954545
$286$	0	0
$287$	−40.8446	−2.41098
$288$	0	0
$289$	−1.74069	−0.102393
$290$	0	0
$291$	−6.38854	−0.374503
$292$	0	0
$293$	26.3660	1.54032	0.770158	−	0.637853i	$-0.220177\pi$
$293$	26.3660	1.54032	0.770158	+	0.637853i	$0.220177\pi$
$294$	0	0
$295$	5.29486	0.308279
$296$	0	0
$297$	−5.90632	−0.342719
$298$	0	0
$299$	0.575909	0.0333057
$300$	0	0
$301$	12.7988	0.737712
$302$	0	0
$303$	8.97826	0.515788
$304$	0	0
$305$	−30.7909	−1.76308
$306$	0	0
$307$	16.5897	0.946825	0.473413	−	0.880841i	$-0.343022\pi$
$307$	16.5897	0.946825	0.473413	+	0.880841i	$0.343022\pi$
$308$	0	0
$309$	−2.20118	−0.125221
$310$	0	0
$311$	5.46049	0.309636	0.154818	−	0.987943i	$-0.450521\pi$
$311$	5.46049	0.309636	0.154818	+	0.987943i	$0.450521\pi$
$312$	0	0
$313$	−16.9783	−0.959668	−0.479834	−	0.877359i	$-0.659303\pi$
$313$	−16.9783	−0.959668	−0.479834	+	0.877359i	$0.659303\pi$
$314$	0	0
$315$	14.0537	0.791839
$316$	0	0
$317$	18.2376	1.02432	0.512162	−	0.858889i	$-0.328845\pi$
$317$	18.2376	1.02432	0.512162	+	0.858889i	$0.328845\pi$
$318$	0	0
$319$	47.2505	2.64552
$320$	0	0
$321$	−11.1656	−0.623205
$322$	0	0
$323$	−1.51777	−0.0844512
$324$	0	0
$325$	−47.6617	−2.64379
$326$	0	0
$327$	−4.38854	−0.242687
$328$	0	0
$329$	−29.4241	−1.62220
$330$	0	0
$331$	−20.4743	−1.12537	−0.562685	−	0.826672i	$-0.690231\pi$
$331$	−20.4743	−1.12537	−0.562685	+	0.826672i	$0.690231\pi$
$332$	0	0
$333$	4.61146	0.252706
$334$	0	0
$335$	4.14743	0.226598
$336$	0	0
$337$	14.4596	0.787667	0.393833	−	0.919182i	$-0.371149\pi$
$337$	14.4596	0.787667	0.393833	+	0.919182i	$0.371149\pi$
$338$	0	0
$339$	17.7190	0.962361
$340$	0	0
$341$	−14.8708	−0.805297
$342$	0	0
$343$	8.53159	0.460662
$344$	0	0
$345$	−0.611457	−0.0329197
$346$	0	0
$347$	3.22291	0.173015	0.0865075	−	0.996251i	$-0.472429\pi$
$347$	3.22291	0.173015	0.0865075	+	0.996251i	$0.472429\pi$
$348$	0	0
$349$	29.4960	1.57889	0.789443	−	0.613823i	$-0.210369\pi$
$349$	29.4960	1.57889	0.789443	+	0.613823i	$0.210369\pi$
$350$	0	0
$351$	3.90632	0.208504
$352$	0	0
$353$	23.5142	1.25154	0.625768	−	0.780009i	$-0.284786\pi$
$353$	23.5142	1.25154	0.625768	+	0.780009i	$0.284786\pi$
$354$	0	0
$355$	51.8264	2.75066
$356$	0	0
$357$	−13.2367	−0.700562
$358$	0	0
$359$	−21.5142	−1.13548	−0.567739	−	0.823209i	$-0.692182\pi$
$359$	−21.5142	−1.13548	−0.567739	+	0.823209i	$0.692182\pi$
$360$	0	0
$361$	−18.8490	−0.992054
$362$	0	0
$363$	−23.8846	−1.25362
$364$	0	0
$365$	69.7292	3.64979
$366$	0	0
$367$	11.8490	0.618515	0.309257	−	0.950978i	$-0.399920\pi$
$367$	11.8490	0.618515	0.309257	+	0.950978i	$0.399920\pi$
$368$	0	0
$369$	12.0537	0.627493
$370$	0	0
$371$	−17.8802	−0.928293
$372$	0	0
$373$	−4.97826	−0.257765	−0.128882	−	0.991660i	$-0.541139\pi$
$373$	−4.97826	−0.257765	−0.128882	+	0.991660i	$0.541139\pi$
$374$	0	0
$375$	29.8664	1.54229
$376$	0	0
$377$	−31.2505	−1.60948
$378$	0	0
$379$	−7.64700	−0.392800	−0.196400	−	0.980524i	$-0.562925\pi$
$379$	−7.64700	−0.392800	−0.196400	+	0.980524i	$0.562925\pi$
$380$	0	0
$381$	−6.20118	−0.317696
$382$	0	0
$383$	−6.66959	−0.340800	−0.170400	−	0.985375i	$-0.554506\pi$
$383$	−6.66959	−0.340800	−0.170400	+	0.985375i	$0.554506\pi$
$384$	0	0
$385$	83.0059	4.23037
$386$	0	0
$387$	−3.77709	−0.192000
$388$	0	0
$389$	−24.1648	−1.22520	−0.612601	−	0.790392i	$-0.709877\pi$
$389$	−24.1648	−1.22520	−0.612601	+	0.790392i	$0.709877\pi$
$390$	0	0
$391$	0.575909	0.0291250
$392$	0	0
$393$	−16.9245	−0.853729
$394$	0	0
$395$	39.4743	1.98617
$396$	0	0
$397$	9.81263	0.492482	0.246241	−	0.969209i	$-0.420805\pi$
$397$	9.81263	0.492482	0.246241	+	0.969209i	$0.420805\pi$
$398$	0	0
$399$	1.31660	0.0659122
$400$	0	0
$401$	9.82998	0.490886	0.245443	−	0.969411i	$-0.421067\pi$
$401$	9.82998	0.490886	0.245443	+	0.969411i	$0.421067\pi$
$402$	0	0
$403$	9.83522	0.489927
$404$	0	0
$405$	−4.14743	−0.206087
$406$	0	0
$407$	27.2367	1.35007
$408$	0	0
$409$	−0.777087	−0.0384245	−0.0192122	−	0.999815i	$-0.506116\pi$
$409$	−0.777087	−0.0384245	−0.0192122	+	0.999815i	$0.506116\pi$
$410$	0	0
$411$	−18.3486	−0.905070
$412$	0	0
$413$	4.32602	0.212870
$414$	0	0
$415$	−3.00000	−0.147264
$416$	0	0
$417$	2.12923	0.104269
$418$	0	0
$419$	−28.7736	−1.40568	−0.702840	−	0.711348i	$-0.748085\pi$
$419$	−28.7736	−1.40568	−0.702840	+	0.711348i	$0.748085\pi$
$420$	0	0
$421$	8.74154	0.426037	0.213018	−	0.977048i	$-0.431671\pi$
$421$	8.74154	0.426037	0.213018	+	0.977048i	$0.431671\pi$
$422$	0	0
$423$	8.68340	0.422202
$424$	0	0
$425$	−47.6617	−2.31193
$426$	0	0
$427$	−25.1569	−1.21743
$428$	0	0
$429$	23.0719	1.11392
$430$	0	0
$431$	31.1621	1.50103	0.750513	−	0.660856i	$-0.229807\pi$
$431$	31.1621	1.50103	0.750513	+	0.660856i	$0.229807\pi$
$432$	0	0
$433$	−14.0711	−0.676214	−0.338107	−	0.941108i	$-0.609787\pi$
$433$	−14.0711	−0.676214	−0.338107	+	0.941108i	$0.609787\pi$
$434$	0	0
$435$	33.1794	1.59083
$436$	0	0
$437$	−0.0572830	−0.00274022
$438$	0	0
$439$	8.81349	0.420645	0.210322	−	0.977632i	$-0.432549\pi$
$439$	8.81349	0.420645	0.210322	+	0.977632i	$0.432549\pi$
$440$	0	0
$441$	4.48223	0.213439
$442$	0	0
$443$	9.23672	0.438850	0.219425	−	0.975629i	$-0.429582\pi$
$443$	9.23672	0.438850	0.219425	+	0.975629i	$0.429582\pi$
$444$	0	0
$445$	−47.3806	−2.24606
$446$	0	0
$447$	−7.81263	−0.369525
$448$	0	0
$449$	−9.71895	−0.458666	−0.229333	−	0.973348i	$-0.573654\pi$
$449$	−9.71895	−0.458666	−0.229333	+	0.973348i	$0.573654\pi$
$450$	0	0
$451$	71.1933	3.35236
$452$	0	0
$453$	23.4605	1.10227
$454$	0	0
$455$	−54.8984	−2.57368
$456$	0	0
$457$	−26.5897	−1.24381	−0.621907	−	0.783091i	$-0.713642\pi$
$457$	−26.5897	−1.24381	−0.621907	+	0.783091i	$0.713642\pi$
$458$	0	0
$459$	3.90632	0.182331
$460$	0	0
$461$	−17.1430	−0.798431	−0.399216	−	0.916857i	$-0.630717\pi$
$461$	−17.1430	−0.798431	−0.399216	+	0.916857i	$0.630717\pi$
$462$	0	0
$463$	−3.09368	−0.143776	−0.0718878	−	0.997413i	$-0.522902\pi$
$463$	−3.09368	−0.143776	−0.0718878	+	0.997413i	$0.522902\pi$
$464$	0	0
$465$	−10.4423	−0.484250
$466$	0	0
$467$	−0.496038	−0.0229539	−0.0114770	−	0.999934i	$-0.503653\pi$
$467$	−0.496038	−0.0229539	−0.0114770	+	0.999934i	$0.503653\pi$
$468$	0	0
$469$	3.38854	0.156468
$470$	0	0
$471$	−12.3304	−0.568155
$472$	0	0
$473$	−22.3087	−1.02575
$474$	0	0
$475$	4.74069	0.217518
$476$	0	0
$477$	5.27666	0.241602
$478$	0	0
$479$	2.79443	0.127681	0.0638405	−	0.997960i	$-0.479665\pi$
$479$	2.79443	0.127681	0.0638405	+	0.997960i	$0.479665\pi$
$480$	0	0
$481$	−18.0138	−0.821359
$482$	0	0
$483$	−0.499574	−0.0227314
$484$	0	0
$485$	−26.4960	−1.20312
$486$	0	0
$487$	−18.1794	−0.823789	−0.411895	−	0.911232i	$-0.635133\pi$
$487$	−18.1794	−0.823789	−0.411895	+	0.911232i	$0.635133\pi$
$488$	0	0
$489$	−18.8846	−0.853990
$490$	0	0
$491$	−25.2332	−1.13876	−0.569379	−	0.822075i	$-0.692816\pi$
$491$	−25.2332	−1.13876	−0.569379	+	0.822075i	$0.692816\pi$
$492$	0	0
$493$	−31.2505	−1.40745
$494$	0	0
$495$	−24.4960	−1.10102
$496$	0	0
$497$	42.3434	1.89936
$498$	0	0
$499$	5.11542	0.228998	0.114499	−	0.993423i	$-0.463474\pi$
$499$	5.11542	0.228998	0.114499	+	0.993423i	$0.463474\pi$
$500$	0	0
$501$	3.88897	0.173746
$502$	0	0
$503$	20.7545	0.925397	0.462699	−	0.886516i	$-0.346881\pi$
$503$	20.7545	0.925397	0.462699	+	0.886516i	$0.346881\pi$
$504$	0	0
$505$	37.2367	1.65701
$506$	0	0
$507$	−2.25931	−0.100340
$508$	0	0
$509$	6.01381	0.266558	0.133279	−	0.991079i	$-0.457449\pi$
$509$	6.01381	0.266558	0.133279	+	0.991079i	$0.457449\pi$
$510$	0	0
$511$	56.9703	2.52022
$512$	0	0
$513$	−0.388543	−0.0171546
$514$	0	0
$515$	−9.12923	−0.402282
$516$	0	0
$517$	51.2869	2.25560
$518$	0	0
$519$	−17.2731	−0.758206
$520$	0	0
$521$	−5.75535	−0.252147	−0.126073	−	0.992021i	$-0.540237\pi$
$521$	−5.75535	−0.252147	−0.126073	+	0.992021i	$0.540237\pi$
$522$	0	0
$523$	−2.76328	−0.120830	−0.0604148	−	0.998173i	$-0.519242\pi$
$523$	−2.76328	−0.120830	−0.0604148	+	0.998173i	$0.519242\pi$
$524$	0	0
$525$	41.3442	1.80441
$526$	0	0
$527$	9.83522	0.428429
$528$	0	0
$529$	−22.9783	−0.999055
$530$	0	0
$531$	−1.27666	−0.0554023
$532$	0	0
$533$	−47.0858	−2.03951
$534$	0	0
$535$	−46.3087	−2.00210
$536$	0	0
$537$	18.2012	0.785439
$538$	0	0
$539$	26.4734	1.14029
$540$	0	0
$541$	20.1075	0.864489	0.432244	−	0.901757i	$-0.357722\pi$
$541$	20.1075	0.864489	0.432244	+	0.901757i	$0.357722\pi$
$542$	0	0
$543$	2.90632	0.124722
$544$	0	0
$545$	−18.2012	−0.779653
$546$	0	0
$547$	−0.316596	−0.0135367	−0.00676834	−	0.999977i	$-0.502154\pi$
$547$	−0.316596	−0.0135367	−0.00676834	+	0.999977i	$0.502154\pi$
$548$	0	0
$549$	7.42409	0.316852
$550$	0	0
$551$	3.10835	0.132420
$552$	0	0
$553$	32.2514	1.37147
$554$	0	0
$555$	19.1257	0.811841
$556$	0	0
$557$	−18.9783	−0.804135	−0.402067	−	0.915610i	$-0.631708\pi$
$557$	−18.9783	−0.804135	−0.402067	+	0.915610i	$0.631708\pi$
$558$	0	0
$559$	14.7545	0.624049
$560$	0	0
$561$	23.0719	0.974098
$562$	0	0
$563$	−14.4596	−0.609401	−0.304701	−	0.952448i	$-0.598556\pi$
$563$	−14.4596	−0.609401	−0.304701	+	0.952448i	$0.598556\pi$
$564$	0	0
$565$	73.4881	3.09167
$566$	0	0
$567$	−3.38854	−0.142305
$568$	0	0
$569$	−22.8708	−0.958793	−0.479396	−	0.877599i	$-0.659144\pi$
$569$	−22.8708	−0.958793	−0.479396	+	0.877599i	$0.659144\pi$
$570$	0	0
$571$	−41.3806	−1.73173	−0.865863	−	0.500282i	$-0.833230\pi$
$571$	−41.3806	−1.73173	−0.865863	+	0.500282i	$0.833230\pi$
$572$	0	0
$573$	4.77709	0.199566
$574$	0	0
$575$	−1.79882	−0.0750161
$576$	0	0
$577$	−0.669592	−0.0278755	−0.0139377	−	0.999903i	$-0.504437\pi$
$577$	−0.669592	−0.0278755	−0.0139377	+	0.999903i	$0.504437\pi$
$578$	0	0
$579$	−11.2949	−0.469398
$580$	0	0
$581$	−2.45107	−0.101687
$582$	0	0
$583$	31.1656	1.29075
$584$	0	0
$585$	16.2012	0.669836
$586$	0	0
$587$	14.0711	0.580776	0.290388	−	0.956909i	$-0.406216\pi$
$587$	14.0711	0.580776	0.290388	+	0.956909i	$0.406216\pi$
$588$	0	0
$589$	−0.978264	−0.0403087
$590$	0	0
$591$	−22.3850	−0.920796
$592$	0	0
$593$	36.1976	1.48646	0.743229	−	0.669037i	$-0.233293\pi$
$593$	36.1976	1.48646	0.743229	+	0.669037i	$0.233293\pi$
$594$	0	0
$595$	−54.8984	−2.25061
$596$	0	0
$597$	9.51777	0.389537
$598$	0	0
$599$	−24.0138	−0.981178	−0.490589	−	0.871391i	$-0.663218\pi$
$599$	−24.0138	−0.981178	−0.490589	+	0.871391i	$0.663218\pi$
$600$	0	0
$601$	38.3660	1.56498	0.782490	−	0.622663i	$-0.213949\pi$
$601$	38.3660	1.56498	0.782490	+	0.622663i	$0.213949\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	−99.0596	−4.02735
$606$	0	0
$607$	24.7181	1.00328	0.501638	−	0.865077i	$-0.332731\pi$
$607$	24.7181	1.00328	0.501638	+	0.865077i	$0.332731\pi$
$608$	0	0
$609$	27.1083	1.09849
$610$	0	0
$611$	−33.9201	−1.37226
$612$	0	0
$613$	−7.74069	−0.312643	−0.156322	−	0.987706i	$-0.549964\pi$
$613$	−7.74069	−0.312643	−0.156322	+	0.987706i	$0.549964\pi$
$614$	0	0
$615$	49.9921	2.01588
$616$	0	0
$617$	43.7554	1.76152	0.880762	−	0.473559i	$-0.157031\pi$
$617$	43.7554	1.76152	0.880762	+	0.473559i	$0.157031\pi$
$618$	0	0
$619$	23.5316	0.945814	0.472907	−	0.881112i	$-0.343205\pi$
$619$	23.5316	0.945814	0.472907	+	0.881112i	$0.343205\pi$
$620$	0	0
$621$	0.147430	0.00591617
$622$	0	0
$623$	−38.7110	−1.55092
$624$	0	0
$625$	62.8628	2.51451
$626$	0	0
$627$	−2.29486	−0.0916479
$628$	0	0
$629$	−18.0138	−0.718258
$630$	0	0
$631$	11.2229	0.446777	0.223389	−	0.974729i	$-0.428288\pi$
$631$	11.2229	0.446777	0.223389	+	0.974729i	$0.428288\pi$
$632$	0	0
$633$	−21.0719	−0.837535
$634$	0	0
$635$	−25.7190	−1.02063
$636$	0	0
$637$	−17.5090	−0.693732
$638$	0	0
$639$	−12.4960	−0.494336
$640$	0	0
$641$	32.5862	1.28708	0.643538	−	0.765414i	$-0.277466\pi$
$641$	32.5862	1.28708	0.643538	+	0.765414i	$0.277466\pi$
$642$	0	0
$643$	−47.3442	−1.86707	−0.933537	−	0.358481i	$-0.883295\pi$
$643$	−47.3442	−1.86707	−0.933537	+	0.358481i	$0.883295\pi$
$644$	0	0
$645$	−15.6652	−0.616817
$646$	0	0
$647$	28.5462	1.12227	0.561134	−	0.827725i	$-0.310365\pi$
$647$	28.5462	1.12227	0.561134	+	0.827725i	$0.310365\pi$
$648$	0	0
$649$	−7.54036	−0.295985
$650$	0	0
$651$	−8.53159	−0.334379
$652$	0	0
$653$	18.4787	0.723127	0.361564	−	0.932347i	$-0.382243\pi$
$653$	18.4787	0.723127	0.361564	+	0.932347i	$0.382243\pi$
$654$	0	0
$655$	−70.1933	−2.74268
$656$	0	0
$657$	−16.8126	−0.655923
$658$	0	0
$659$	15.5577	0.606042	0.303021	−	0.952984i	$-0.402005\pi$
$659$	15.5577	0.606042	0.303021	+	0.952984i	$0.402005\pi$
$660$	0	0
$661$	−29.2593	−1.13806	−0.569028	−	0.822318i	$-0.692680\pi$
$661$	−29.2593	−1.13806	−0.569028	+	0.822318i	$0.692680\pi$
$662$	0	0
$663$	−15.2593	−0.592623
$664$	0	0
$665$	5.46049	0.211749
$666$	0	0
$667$	−1.17944	−0.0456682
$668$	0	0
$669$	−13.0355	−0.503983
$670$	0	0
$671$	43.8490	1.69277
$672$	0	0
$673$	3.71895	0.143355	0.0716775	−	0.997428i	$-0.477165\pi$
$673$	3.71895	0.143355	0.0716775	+	0.997428i	$0.477165\pi$
$674$	0	0
$675$	−12.2012	−0.469624
$676$	0	0
$677$	−16.7806	−0.644932	−0.322466	−	0.946581i	$-0.604512\pi$
$677$	−16.7806	−0.644932	−0.322466	+	0.946581i	$0.604512\pi$
$678$	0	0
$679$	−21.6479	−0.830768
$680$	0	0
$681$	3.75889	0.144041
$682$	0	0
$683$	2.23758	0.0856185	0.0428093	−	0.999083i	$-0.486369\pi$
$683$	2.23758	0.0856185	0.0428093	+	0.999083i	$0.486369\pi$
$684$	0	0
$685$	−76.0996	−2.90762
$686$	0	0
$687$	0.107495	0.00410118
$688$	0	0
$689$	−20.6123	−0.785266
$690$	0	0
$691$	48.5462	1.84679	0.923393	−	0.383855i	$-0.125404\pi$
$691$	48.5462	1.84679	0.923393	+	0.383855i	$0.125404\pi$
$692$	0	0
$693$	−20.0138	−0.760262
$694$	0	0
$695$	8.83083	0.334973
$696$	0	0
$697$	−47.0858	−1.78350
$698$	0	0
$699$	6.88812	0.260533
$700$	0	0
$701$	34.0676	1.28671	0.643357	−	0.765566i	$-0.277541\pi$
$701$	34.0676	1.28671	0.643357	+	0.765566i	$0.277541\pi$
$702$	0	0
$703$	1.79175	0.0675772
$704$	0	0
$705$	36.0138	1.35636
$706$	0	0
$707$	30.4232	1.14418
$708$	0	0
$709$	−20.2940	−0.762157	−0.381079	−	0.924543i	$-0.624447\pi$
$709$	−20.2940	−0.762157	−0.381079	+	0.924543i	$0.624447\pi$
$710$	0	0
$711$	−9.51777	−0.356944
$712$	0	0
$713$	0.371196	0.0139014
$714$	0	0
$715$	95.6893	3.57858
$716$	0	0
$717$	11.5178	0.430139
$718$	0	0
$719$	2.53597	0.0945759	0.0472879	−	0.998881i	$-0.484942\pi$
$719$	2.53597	0.0945759	0.0472879	+	0.998881i	$0.484942\pi$
$720$	0	0
$721$	−7.45879	−0.277780
$722$	0	0
$723$	−17.6834	−0.657653
$724$	0	0
$725$	97.6094	3.62512
$726$	0	0
$727$	−49.2514	−1.82663	−0.913316	−	0.407251i	$-0.866488\pi$
$727$	−49.2514	−1.82663	−0.913316	+	0.407251i	$0.866488\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	14.7545	0.545715
$732$	0	0
$733$	−6.58972	−0.243397	−0.121698	−	0.992567i	$-0.538834\pi$
$733$	−6.58972	−0.243397	−0.121698	+	0.992567i	$0.538834\pi$
$734$	0	0
$735$	18.5897	0.685692
$736$	0	0
$737$	−5.90632	−0.217562
$738$	0	0
$739$	−40.6617	−1.49576	−0.747882	−	0.663832i	$-0.768929\pi$
$739$	−40.6617	−1.49576	−0.747882	+	0.663832i	$0.768929\pi$
$740$	0	0
$741$	1.51777	0.0557568
$742$	0	0
$743$	−37.7016	−1.38314	−0.691569	−	0.722311i	$-0.743080\pi$
$743$	−37.7016	−1.38314	−0.691569	+	0.722311i	$0.743080\pi$
$744$	0	0
$745$	−32.4024	−1.18713
$746$	0	0
$747$	0.723339	0.0264656
$748$	0	0
$749$	−37.8352	−1.38247
$750$	0	0
$751$	24.8118	0.905395	0.452697	−	0.891664i	$-0.350462\pi$
$751$	24.8118	0.905395	0.452697	+	0.891664i	$0.350462\pi$
$752$	0	0
$753$	−8.96445	−0.326683
$754$	0	0
$755$	97.3007	3.54114
$756$	0	0
$757$	10.7771	0.391700	0.195850	−	0.980634i	$-0.437253\pi$
$757$	10.7771	0.391700	0.195850	+	0.980634i	$0.437253\pi$
$758$	0	0
$759$	0.870770	0.0316069
$760$	0	0
$761$	1.07195	0.0388581	0.0194290	−	0.999811i	$-0.493815\pi$
$761$	1.07195	0.0388581	0.0194290	+	0.999811i	$0.493815\pi$
$762$	0	0
$763$	−14.8708	−0.538358
$764$	0	0
$765$	16.2012	0.585755
$766$	0	0
$767$	4.98704	0.180072
$768$	0	0
$769$	51.1569	1.84476	0.922382	−	0.386280i	$-0.126240\pi$
$769$	51.1569	1.84476	0.922382	+	0.386280i	$0.126240\pi$
$770$	0	0
$771$	−1.55417	−0.0559722
$772$	0	0
$773$	−39.8126	−1.43196	−0.715980	−	0.698120i	$-0.754020\pi$
$773$	−39.8126	−1.43196	−0.715980	+	0.698120i	$0.754020\pi$
$774$	0	0
$775$	−30.7198	−1.10349
$776$	0	0
$777$	15.6261	0.560584
$778$	0	0
$779$	4.68340	0.167800
$780$	0	0
$781$	−73.8056	−2.64097
$782$	0	0
$783$	−8.00000	−0.285897
$784$	0	0
$785$	−51.1395	−1.82525
$786$	0	0
$787$	−28.9201	−1.03089	−0.515446	−	0.856922i	$-0.672374\pi$
$787$	−28.9201	−1.03089	−0.515446	+	0.856922i	$0.672374\pi$
$788$	0	0
$789$	−9.75889	−0.347426
$790$	0	0
$791$	60.0414	2.13483
$792$	0	0
$793$	−29.0009	−1.02985
$794$	0	0
$795$	21.8846	0.776166
$796$	0	0
$797$	−48.2288	−1.70835	−0.854176	−	0.519984i	$-0.825938\pi$
$797$	−48.2288	−1.70835	−0.854176	+	0.519984i	$0.825938\pi$
$798$	0	0
$799$	−33.9201	−1.20001
$800$	0	0
$801$	11.4241	0.403650
$802$	0	0
$803$	−99.3007	−3.50425
$804$	0	0
$805$	−2.07195	−0.0730265
$806$	0	0
$807$	−0.647004	−0.0227756
$808$	0	0
$809$	−13.9827	−0.491604	−0.245802	−	0.969320i	$-0.579051\pi$
$809$	−13.9827	−0.491604	−0.245802	+	0.969320i	$0.579051\pi$
$810$	0	0
$811$	−28.1430	−0.988236	−0.494118	−	0.869395i	$-0.664509\pi$
$811$	−28.1430	−0.988236	−0.494118	+	0.869395i	$0.664509\pi$
$812$	0	0
$813$	−8.03640	−0.281849
$814$	0	0
$815$	−78.3225	−2.74352
$816$	0	0
$817$	−1.46756	−0.0513435
$818$	0	0
$819$	13.2367	0.462529
$820$	0	0
$821$	−11.8126	−0.412264	−0.206132	−	0.978524i	$-0.566088\pi$
$821$	−11.8126	−0.412264	−0.206132	+	0.978524i	$0.566088\pi$
$822$	0	0
$823$	−21.7190	−0.757075	−0.378538	−	0.925586i	$-0.623573\pi$
$823$	−21.7190	−0.757075	−0.378538	+	0.925586i	$0.623573\pi$
$824$	0	0
$825$	−72.0640	−2.50895
$826$	0	0
$827$	−16.3087	−0.567108	−0.283554	−	0.958956i	$-0.591514\pi$
$827$	−16.3087	−0.567108	−0.283554	+	0.958956i	$0.591514\pi$
$828$	0	0
$829$	15.9201	0.552929	0.276464	−	0.961024i	$-0.410837\pi$
$829$	15.9201	0.552929	0.276464	+	0.961024i	$0.410837\pi$
$830$	0	0
$831$	−3.48223	−0.120797
$832$	0	0
$833$	−17.5090	−0.606651
$834$	0	0
$835$	16.1292	0.558175
$836$	0	0
$837$	2.51777	0.0870270
$838$	0	0
$839$	−23.0147	−0.794554	−0.397277	−	0.917699i	$-0.630045\pi$
$839$	−23.0147	−0.794554	−0.397277	+	0.917699i	$0.630045\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	8.20118	0.282464
$844$	0	0
$845$	−9.37034	−0.322350
$846$	0	0
$847$	−80.9339	−2.78092
$848$	0	0
$849$	−12.0138	−0.412313
$850$	0	0
$851$	−0.679868	−0.0233056
$852$	0	0
$853$	24.0711	0.824179	0.412089	−	0.911143i	$-0.364799\pi$
$853$	24.0711	0.824179	0.412089	+	0.911143i	$0.364799\pi$
$854$	0	0
$855$	−1.61146	−0.0551107
$856$	0	0
$857$	−4.88812	−0.166975	−0.0834875	−	0.996509i	$-0.526606\pi$
$857$	−4.88812	−0.166975	−0.0834875	+	0.996509i	$0.526606\pi$
$858$	0	0
$859$	−23.6479	−0.806854	−0.403427	−	0.915012i	$-0.632181\pi$
$859$	−23.6479	−0.806854	−0.403427	+	0.915012i	$0.632181\pi$
$860$	0	0
$861$	40.8446	1.39198
$862$	0	0
$863$	17.5715	0.598142	0.299071	−	0.954231i	$-0.403323\pi$
$863$	17.5715	0.598142	0.299071	+	0.954231i	$0.403323\pi$
$864$	0	0
$865$	−71.6391	−2.43580
$866$	0	0
$867$	1.74069	0.0591168
$868$	0	0
$869$	−56.2150	−1.90696
$870$	0	0
$871$	3.90632	0.132361
$872$	0	0
$873$	6.38854	0.216219
$874$	0	0
$875$	101.204	3.42130
$876$	0	0
$877$	2.37473	0.0801890	0.0400945	−	0.999196i	$-0.487234\pi$
$877$	2.37473	0.0801890	0.0400945	+	0.999196i	$0.487234\pi$
$878$	0	0
$879$	−26.3660	−0.889302
$880$	0	0
$881$	3.37388	0.113669	0.0568344	−	0.998384i	$-0.481899\pi$
$881$	3.37388	0.113669	0.0568344	+	0.998384i	$0.481899\pi$
$882$	0	0
$883$	−44.0355	−1.48191	−0.740957	−	0.671552i	$-0.765628\pi$
$883$	−44.0355	−1.48191	−0.740957	+	0.671552i	$0.765628\pi$
$884$	0	0
$885$	−5.29486	−0.177985
$886$	0	0
$887$	33.7518	1.13328	0.566638	−	0.823967i	$-0.308244\pi$
$887$	33.7518	1.13328	0.566638	+	0.823967i	$0.308244\pi$
$888$	0	0
$889$	−21.0130	−0.704752
$890$	0	0
$891$	5.90632	0.197869
$892$	0	0
$893$	3.37388	0.112903
$894$	0	0
$895$	75.4881	2.52329
$896$	0	0
$897$	−0.575909	−0.0192290
$898$	0	0
$899$	−20.1422	−0.671780
$900$	0	0
$901$	−20.6123	−0.686696
$902$	0	0
$903$	−12.7988	−0.425918
$904$	0	0
$905$	12.0537	0.400680
$906$	0	0
$907$	52.3660	1.73878	0.869392	−	0.494124i	$-0.164511\pi$
$907$	52.3660	1.73878	0.869392	+	0.494124i	$0.164511\pi$
$908$	0	0
$909$	−8.97826	−0.297790
$910$	0	0
$911$	20.2359	0.670444	0.335222	−	0.942139i	$-0.391189\pi$
$911$	20.2359	0.670444	0.335222	+	0.942139i	$0.391189\pi$
$912$	0	0
$913$	4.27227	0.141392
$914$	0	0
$915$	30.7909	1.01792
$916$	0	0
$917$	−57.3495	−1.89385
$918$	0	0
$919$	51.1803	1.68828	0.844141	−	0.536121i	$-0.180111\pi$
$919$	51.1803	1.68828	0.844141	+	0.536121i	$0.180111\pi$
$920$	0	0
$921$	−16.5897	−0.546650
$922$	0	0
$923$	48.8135	1.60672
$924$	0	0
$925$	56.2652	1.84999
$926$	0	0
$927$	2.20118	0.0722962
$928$	0	0
$929$	21.8699	0.717529	0.358764	−	0.933428i	$-0.383198\pi$
$929$	21.8699	0.717529	0.358764	+	0.933428i	$0.383198\pi$
$930$	0	0
$931$	1.74154	0.0570766
$932$	0	0
$933$	−5.46049	−0.178768
$934$	0	0
$935$	95.6893	3.12937
$936$	0	0
$937$	−3.57506	−0.116792	−0.0583960	−	0.998293i	$-0.518599\pi$
$937$	−3.57506	−0.116792	−0.0583960	+	0.998293i	$0.518599\pi$
$938$	0	0
$939$	16.9783	0.554065
$940$	0	0
$941$	−27.8264	−0.907116	−0.453558	−	0.891227i	$-0.649846\pi$
$941$	−27.8264	−0.907116	−0.453558	+	0.891227i	$0.649846\pi$
$942$	0	0
$943$	−1.77709	−0.0578699
$944$	0	0
$945$	−14.0537	−0.457168
$946$	0	0
$947$	20.9956	0.682266	0.341133	−	0.940015i	$-0.389189\pi$
$947$	20.9956	0.682266	0.341133	+	0.940015i	$0.389189\pi$
$948$	0	0
$949$	65.6755	2.13192
$950$	0	0
$951$	−18.2376	−0.591394
$952$	0	0
$953$	−15.9636	−0.517112	−0.258556	−	0.965996i	$-0.583247\pi$
$953$	−15.9636	−0.517112	−0.258556	+	0.965996i	$0.583247\pi$
$954$	0	0
$955$	19.8126	0.641122
$956$	0	0
$957$	−47.2505	−1.52739
$958$	0	0
$959$	−62.1751	−2.00774
$960$	0	0
$961$	−24.6608	−0.795510
$962$	0	0
$963$	11.1656	0.359807
$964$	0	0
$965$	−46.8446	−1.50798
$966$	0	0
$967$	−13.5680	−0.436317	−0.218158	−	0.975913i	$-0.570005\pi$
$967$	−13.5680	−0.436317	−0.218158	+	0.975913i	$0.570005\pi$
$968$	0	0
$969$	1.51777	0.0487579
$970$	0	0
$971$	−48.1941	−1.54662	−0.773311	−	0.634027i	$-0.781401\pi$
$971$	−48.1941	−1.54662	−0.773311	+	0.634027i	$0.781401\pi$
$972$	0	0
$973$	7.21499	0.231302
$974$	0	0
$975$	47.6617	1.52639
$976$	0	0
$977$	−7.74154	−0.247674	−0.123837	−	0.992303i	$-0.539520\pi$
$977$	−7.74154	−0.247674	−0.123837	+	0.992303i	$0.539520\pi$
$978$	0	0
$979$	67.4743	2.15649
$980$	0	0
$981$	4.38854	0.140115
$982$	0	0
$983$	3.59764	0.114747	0.0573735	−	0.998353i	$-0.481727\pi$
$983$	3.59764	0.114747	0.0573735	+	0.998353i	$0.481727\pi$
$984$	0	0
$985$	−92.8403	−2.95814
$986$	0	0
$987$	29.4241	0.936579
$988$	0	0
$989$	0.556857	0.0177070
$990$	0	0
$991$	45.4526	1.44385	0.721924	−	0.691972i	$-0.243258\pi$
$991$	45.4526	1.44385	0.721924	+	0.691972i	$0.243258\pi$
$992$	0	0
$993$	20.4743	0.649732
$994$	0	0
$995$	39.4743	1.25142
$996$	0	0
$997$	38.4517	1.21778	0.608889	−	0.793255i	$-0.291615\pi$
$997$	38.4517	1.21778	0.608889	+	0.793255i	$0.291615\pi$
$998$	0	0
$999$	−4.61146	−0.145900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.a.e.1.1	✓	3
3.2	odd	2		2412.2.a.h.1.3		3
4.3	odd	2		3216.2.a.t.1.1		3
12.11	even	2		9648.2.a.br.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.a.e.1.1	✓	3	1.1	even	1	trivial
2412.2.a.h.1.3		3	3.2	odd	2
3216.2.a.t.1.1		3	4.3	odd	2
9648.2.a.br.1.3		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.14743 q^{5} -3.38854 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -4.14743 q^{5} -3.38854 q^{7} +1.00000 q^{9} +5.90632 q^{11} -3.90632 q^{13} +4.14743 q^{15} -3.90632 q^{17} +0.388543 q^{19} +3.38854 q^{21} -0.147430 q^{23} +12.2012 q^{25} -1.00000 q^{27} +8.00000 q^{29} -2.51777 q^{31} -5.90632 q^{33} +14.0537 q^{35} +4.61146 q^{37} +3.90632 q^{39} +12.0537 q^{41} -3.77709 q^{43} -4.14743 q^{45} +8.68340 q^{47} +4.48223 q^{49} +3.90632 q^{51} +5.27666 q^{53} -24.4960 q^{55} -0.388543 q^{57} -1.27666 q^{59} +7.42409 q^{61} -3.38854 q^{63} +16.2012 q^{65} -1.00000 q^{67} +0.147430 q^{69} -12.4960 q^{71} -16.8126 q^{73} -12.2012 q^{75} -20.0138 q^{77} -9.51777 q^{79} +1.00000 q^{81} +0.723339 q^{83} +16.2012 q^{85} -8.00000 q^{87} +11.4241 q^{89} +13.2367 q^{91} +2.51777 q^{93} -1.61146 q^{95} +6.38854 q^{97} +5.90632 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9} + 4 q^{11} + 2 q^{13} + 3 q^{15} + 2 q^{17} - 4 q^{19} + 5 q^{21} + 9 q^{23} + 4 q^{25} - 3 q^{27} + 24 q^{29} + q^{31} - 4 q^{33} + 19 q^{35} + 19 q^{37} - 2 q^{39} + 13 q^{41} - q^{43} - 3 q^{45} + 2 q^{47} + 22 q^{49} - 2 q^{51} + 3 q^{53} - 22 q^{55} + 4 q^{57} + 9 q^{59} - 5 q^{63} + 16 q^{65} - 3 q^{67} - 9 q^{69} + 14 q^{71} - 23 q^{73} - 4 q^{75} - 20 q^{79} + 3 q^{81} + 15 q^{83} + 16 q^{85} - 24 q^{87} + 12 q^{89} - 10 q^{91} - q^{93} - 10 q^{95} + 14 q^{97} + 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9 + 4 * q^11 + 2 * q^13 + 3 * q^15 + 2 * q^17 - 4 * q^19 + 5 * q^21 + 9 * q^23 + 4 * q^25 - 3 * q^27 + 24 * q^29 + q^31 - 4 * q^33 + 19 * q^35 + 19 * q^37 - 2 * q^39 + 13 * q^41 - q^43 - 3 * q^45 + 2 * q^47 + 22 * q^49 - 2 * q^51 + 3 * q^53 - 22 * q^55 + 4 * q^57 + 9 * q^59 - 5 * q^63 + 16 * q^65 - 3 * q^67 - 9 * q^69 + 14 * q^71 - 23 * q^73 - 4 * q^75 - 20 * q^79 + 3 * q^81 + 15 * q^83 + 16 * q^85 - 24 * q^87 + 12 * q^89 - 10 * q^91 - q^93 - 10 * q^95 + 14 * q^97 + 4 * q^99

Introduction

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