LMFDB - Embedded newform 804.2.a.f.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:	$5$
Coefficient field:	5.5.24571284.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 15x^{3} + 10x^{2} + 64x + 38 $ x^5 - 2x^4 - 15x^3 + 10x^2 + 64x + 38
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.96721$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.96721 q^{5} -0.804360 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.96721 q^{5} -0.804360 q^{7} +1.00000 q^{9} +2.20700 q^{11} -0.207001 q^{13} -2.96721 q^{15} +7.18218 q^{17} +6.36097 q^{19} -0.804360 q^{21} -0.813248 q^{23} +3.80436 q^{25} +1.00000 q^{27} -3.78046 q^{29} +10.1583 q^{31} +2.20700 q^{33} +2.38671 q^{35} +8.58482 q^{37} -0.207001 q^{39} -3.58143 q^{41} +8.53179 q^{43} -2.96721 q^{45} -11.5962 q^{47} -6.35300 q^{49} +7.18218 q^{51} +0.760214 q^{53} -6.54864 q^{55} +6.36097 q^{57} +4.54068 q^{59} +1.41950 q^{61} -0.804360 q^{63} +0.614215 q^{65} +1.00000 q^{67} -0.813248 q^{69} -3.79300 q^{71} +0.443394 q^{73} +3.80436 q^{75} -1.77522 q^{77} -4.32571 q^{79} +1.00000 q^{81} -17.1086 q^{83} -21.3111 q^{85} -3.78046 q^{87} +13.1166 q^{89} +0.166503 q^{91} +10.1583 q^{93} -18.8744 q^{95} -4.94671 q^{97} +2.20700 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 + 3 * q^5 + 5 * q^7 + 5 * q^9	$ 5 q + 5 q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9} + 6 q^{11} + 4 q^{13} + 3 q^{15} + q^{17} + 13 q^{19} + 5 q^{21} + 10 q^{25} + 5 q^{27} + 3 q^{29} + 3 q^{31} + 6 q^{33} - 9 q^{35} + 12 q^{37} + 4 q^{39} + 11 q^{41} + 3 q^{43} + 3 q^{45} - 13 q^{47} + 24 q^{49} + q^{51} - 9 q^{53} + 14 q^{55} + 13 q^{57} - 12 q^{59} + 4 q^{61} + 5 q^{63} - 8 q^{65} + 5 q^{67} - 24 q^{71} + 12 q^{73} + 10 q^{75} - 4 q^{77} - 4 q^{79} + 5 q^{81} - 27 q^{83} - 4 q^{85} + 3 q^{87} - 5 q^{89} + 14 q^{91} + 3 q^{93} - 30 q^{95} + 8 q^{97} + 6 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + 3 * q^5 + 5 * q^7 + 5 * q^9 + 6 * q^11 + 4 * q^13 + 3 * q^15 + q^17 + 13 * q^19 + 5 * q^21 + 10 * q^25 + 5 * q^27 + 3 * q^29 + 3 * q^31 + 6 * q^33 - 9 * q^35 + 12 * q^37 + 4 * q^39 + 11 * q^41 + 3 * q^43 + 3 * q^45 - 13 * q^47 + 24 * q^49 + q^51 - 9 * q^53 + 14 * q^55 + 13 * q^57 - 12 * q^59 + 4 * q^61 + 5 * q^63 - 8 * q^65 + 5 * q^67 - 24 * q^71 + 12 * q^73 + 10 * q^75 - 4 * q^77 - 4 * q^79 + 5 * q^81 - 27 * q^83 - 4 * q^85 + 3 * q^87 - 5 * q^89 + 14 * q^91 + 3 * q^93 - 30 * q^95 + 8 * q^97 + 6 * 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
$5$	−2.96721	−1.32698	−0.663489	−	0.748186i	$-0.730925\pi$
$5$	−2.96721	−1.32698	−0.663489	+	0.748186i	$0.730925\pi$
$6$	0	0
$7$	−0.804360	−0.304020	−0.152010	−	0.988379i	$-0.548575\pi$
$7$	−0.804360	−0.304020	−0.152010	+	0.988379i	$0.548575\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.20700	0.665436	0.332718	−	0.943026i	$-0.392034\pi$
$11$	2.20700	0.665436	0.332718	+	0.943026i	$0.392034\pi$
$12$	0	0
$13$	−0.207001	−0.0574116	−0.0287058	−	0.999588i	$-0.509139\pi$
$13$	−0.207001	−0.0574116	−0.0287058	+	0.999588i	$0.509139\pi$
$14$	0	0
$15$	−2.96721	−0.766131
$16$	0	0
$17$	7.18218	1.74194	0.870968	−	0.491340i	$-0.163493\pi$
$17$	7.18218	1.74194	0.870968	+	0.491340i	$0.163493\pi$
$18$	0	0
$19$	6.36097	1.45931	0.729653	−	0.683818i	$-0.239682\pi$
$19$	6.36097	1.45931	0.729653	+	0.683818i	$0.239682\pi$
$20$	0	0
$21$	−0.804360	−0.175526
$22$	0	0
$23$	−0.813248	−0.169574	−0.0847870	−	0.996399i	$-0.527021\pi$
$23$	−0.813248	−0.169574	−0.0847870	+	0.996399i	$0.527021\pi$
$24$	0	0
$25$	3.80436	0.760872
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.78046	−0.702014	−0.351007	−	0.936373i	$-0.614161\pi$
$29$	−3.78046	−0.702014	−0.351007	+	0.936373i	$0.614161\pi$
$30$	0	0
$31$	10.1583	1.82448	0.912241	−	0.409654i	$-0.134351\pi$
$31$	10.1583	1.82448	0.912241	+	0.409654i	$0.134351\pi$
$32$	0	0
$33$	2.20700	0.384189
$34$	0	0
$35$	2.38671	0.403428
$36$	0	0
$37$	8.58482	1.41134	0.705668	−	0.708543i	$-0.250647\pi$
$37$	8.58482	1.41134	0.705668	+	0.708543i	$0.250647\pi$
$38$	0	0
$39$	−0.207001	−0.0331466
$40$	0	0
$41$	−3.58143	−0.559325	−0.279663	−	0.960098i	$-0.590223\pi$
$41$	−3.58143	−0.559325	−0.279663	+	0.960098i	$0.590223\pi$
$42$	0	0
$43$	8.53179	1.30109	0.650543	−	0.759470i	$-0.274541\pi$
$43$	8.53179	1.30109	0.650543	+	0.759470i	$0.274541\pi$
$44$	0	0
$45$	−2.96721	−0.442326
$46$	0	0
$47$	−11.5962	−1.69148	−0.845739	−	0.533597i	$-0.820840\pi$
$47$	−11.5962	−1.69148	−0.845739	+	0.533597i	$0.820840\pi$
$48$	0	0
$49$	−6.35300	−0.907572
$50$	0	0
$51$	7.18218	1.00571
$52$	0	0
$53$	0.760214	0.104423	0.0522117	−	0.998636i	$-0.483373\pi$
$53$	0.760214	0.104423	0.0522117	+	0.998636i	$0.483373\pi$
$54$	0	0
$55$	−6.54864	−0.883019
$56$	0	0
$57$	6.36097	0.842531
$58$	0	0
$59$	4.54068	0.591146	0.295573	−	0.955320i	$-0.404489\pi$
$59$	4.54068	0.591146	0.295573	+	0.955320i	$0.404489\pi$
$60$	0	0
$61$	1.41950	0.181748	0.0908739	−	0.995862i	$-0.471034\pi$
$61$	1.41950	0.181748	0.0908739	+	0.995862i	$0.471034\pi$
$62$	0	0
$63$	−0.804360	−0.101340
$64$	0	0
$65$	0.614215	0.0761840
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	−0.813248	−0.0979036
$70$	0	0
$71$	−3.79300	−0.450146	−0.225073	−	0.974342i	$-0.572262\pi$
$71$	−3.79300	−0.450146	−0.225073	+	0.974342i	$0.572262\pi$
$72$	0	0
$73$	0.443394	0.0518953	0.0259477	−	0.999663i	$-0.491740\pi$
$73$	0.443394	0.0518953	0.0259477	+	0.999663i	$0.491740\pi$
$74$	0	0
$75$	3.80436	0.439290
$76$	0	0
$77$	−1.77522	−0.202306
$78$	0	0
$79$	−4.32571	−0.486680	−0.243340	−	0.969941i	$-0.578243\pi$
$79$	−4.32571	−0.486680	−0.243340	+	0.969941i	$0.578243\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−17.1086	−1.87792	−0.938959	−	0.344030i	$-0.888208\pi$
$83$	−17.1086	−1.87792	−0.938959	+	0.344030i	$0.888208\pi$
$84$	0	0
$85$	−21.3111	−2.31151
$86$	0	0
$87$	−3.78046	−0.405308
$88$	0	0
$89$	13.1166	1.39036	0.695179	−	0.718837i	$-0.255325\pi$
$89$	13.1166	1.39036	0.695179	+	0.718837i	$0.255325\pi$
$90$	0	0
$91$	0.166503	0.0174543
$92$	0	0
$93$	10.1583	1.05337
$94$	0	0
$95$	−18.8744	−1.93647
$96$	0	0
$97$	−4.94671	−0.502262	−0.251131	−	0.967953i	$-0.580803\pi$
$97$	−4.94671	−0.502262	−0.251131	+	0.967953i	$0.580803\pi$
$98$	0	0
$99$	2.20700	0.221812
$100$	0	0
$101$	−0.614215	−0.0611167	−0.0305584	−	0.999533i	$-0.509729\pi$
$101$	−0.614215	−0.0611167	−0.0305584	+	0.999533i	$0.509729\pi$
$102$	0	0
$103$	−4.58050	−0.451331	−0.225665	−	0.974205i	$-0.572456\pi$
$103$	−4.58050	−0.451331	−0.225665	+	0.974205i	$0.572456\pi$
$104$	0	0
$105$	2.38671	0.232919
$106$	0	0
$107$	−10.0034	−0.967065	−0.483533	−	0.875326i	$-0.660647\pi$
$107$	−10.0034	−0.967065	−0.483533	+	0.875326i	$0.660647\pi$
$108$	0	0
$109$	14.5554	1.39416	0.697079	−	0.716995i	$-0.254483\pi$
$109$	14.5554	1.39416	0.697079	+	0.716995i	$0.254483\pi$
$110$	0	0
$111$	8.58482	0.814835
$112$	0	0
$113$	14.0759	1.32415	0.662073	−	0.749440i	$-0.269677\pi$
$113$	14.0759	1.32415	0.662073	+	0.749440i	$0.269677\pi$
$114$	0	0
$115$	2.41308	0.225021
$116$	0	0
$117$	−0.207001	−0.0191372
$118$	0	0
$119$	−5.77706	−0.529583
$120$	0	0
$121$	−6.12915	−0.557195
$122$	0	0
$123$	−3.58143	−0.322927
$124$	0	0
$125$	3.54772	0.317318
$126$	0	0
$127$	6.39468	0.567436	0.283718	−	0.958908i	$-0.408432\pi$
$127$	6.39468	0.567436	0.283718	+	0.958908i	$0.408432\pi$
$128$	0	0
$129$	8.53179	0.751182
$130$	0	0
$131$	16.0713	1.40415	0.702077	−	0.712101i	$-0.252256\pi$
$131$	16.0713	1.40415	0.702077	+	0.712101i	$0.252256\pi$
$132$	0	0
$133$	−5.11651	−0.443658
$134$	0	0
$135$	−2.96721	−0.255377
$136$	0	0
$137$	−0.760214	−0.0649494	−0.0324747	−	0.999473i	$-0.510339\pi$
$137$	−0.760214	−0.0649494	−0.0324747	+	0.999473i	$0.510339\pi$
$138$	0	0
$139$	0.822136	0.0697326	0.0348663	−	0.999392i	$-0.488899\pi$
$139$	0.822136	0.0697326	0.0348663	+	0.999392i	$0.488899\pi$
$140$	0	0
$141$	−11.5962	−0.976575
$142$	0	0
$143$	−0.456851	−0.0382038
$144$	0	0
$145$	11.2174	0.931558
$146$	0	0
$147$	−6.35300	−0.523987
$148$	0	0
$149$	10.5839	0.867067	0.433534	−	0.901137i	$-0.357267\pi$
$149$	10.5839	0.867067	0.433534	+	0.901137i	$0.357267\pi$
$150$	0	0
$151$	−1.55074	−0.126197	−0.0630987	−	0.998007i	$-0.520098\pi$
$151$	−1.55074	−0.126197	−0.0630987	+	0.998007i	$0.520098\pi$
$152$	0	0
$153$	7.18218	0.580645
$154$	0	0
$155$	−30.1418	−2.42105
$156$	0	0
$157$	23.8554	1.90387	0.951934	−	0.306303i	$-0.0990920\pi$
$157$	23.8554	1.90387	0.951934	+	0.306303i	$0.0990920\pi$
$158$	0	0
$159$	0.760214	0.0602889
$160$	0	0
$161$	0.654145	0.0515538
$162$	0	0
$163$	−11.3714	−0.890677	−0.445339	−	0.895362i	$-0.646917\pi$
$163$	−11.3714	−0.890677	−0.445339	+	0.895362i	$0.646917\pi$
$164$	0	0
$165$	−6.54864	−0.509811
$166$	0	0
$167$	−11.7547	−0.909607	−0.454804	−	0.890592i	$-0.650291\pi$
$167$	−11.7547	−0.909607	−0.454804	+	0.890592i	$0.650291\pi$
$168$	0	0
$169$	−12.9572	−0.996704
$170$	0	0
$171$	6.36097	0.486435
$172$	0	0
$173$	−15.9041	−1.20917	−0.604584	−	0.796542i	$-0.706661\pi$
$173$	−15.9041	−1.20917	−0.604584	+	0.796542i	$0.706661\pi$
$174$	0	0
$175$	−3.06008	−0.231320
$176$	0	0
$177$	4.54068	0.341298
$178$	0	0
$179$	−11.3134	−0.845605	−0.422803	−	0.906222i	$-0.638954\pi$
$179$	−11.3134	−0.845605	−0.422803	+	0.906222i	$0.638954\pi$
$180$	0	0
$181$	−12.9810	−0.964874	−0.482437	−	0.875931i	$-0.660248\pi$
$181$	−12.9810	−0.964874	−0.482437	+	0.875931i	$0.660248\pi$
$182$	0	0
$183$	1.41950	0.104932
$184$	0	0
$185$	−25.4730	−1.87281
$186$	0	0
$187$	15.8511	1.15915
$188$	0	0
$189$	−0.804360	−0.0585086
$190$	0	0
$191$	3.63144	0.262762	0.131381	−	0.991332i	$-0.458059\pi$
$191$	3.63144	0.262762	0.131381	+	0.991332i	$0.458059\pi$
$192$	0	0
$193$	−6.06805	−0.436788	−0.218394	−	0.975861i	$-0.570082\pi$
$193$	−6.06805	−0.436788	−0.218394	+	0.975861i	$0.570082\pi$
$194$	0	0
$195$	0.614215	0.0439849
$196$	0	0
$197$	22.6518	1.61387	0.806937	−	0.590638i	$-0.201124\pi$
$197$	22.6518	1.61387	0.806937	+	0.590638i	$0.201124\pi$
$198$	0	0
$199$	−10.2104	−0.723796	−0.361898	−	0.932218i	$-0.617871\pi$
$199$	−10.2104	−0.723796	−0.361898	+	0.932218i	$0.617871\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	3.04085	0.213426
$204$	0	0
$205$	10.6269	0.742213
$206$	0	0
$207$	−0.813248	−0.0565247
$208$	0	0
$209$	14.0387	0.971074
$210$	0	0
$211$	−19.9909	−1.37623	−0.688114	−	0.725603i	$-0.741561\pi$
$211$	−19.9909	−1.37623	−0.688114	+	0.725603i	$0.741561\pi$
$212$	0	0
$213$	−3.79300	−0.259892
$214$	0	0
$215$	−25.3156	−1.72651
$216$	0	0
$217$	−8.17092	−0.554678
$218$	0	0
$219$	0.443394	0.0299618
$220$	0	0
$221$	−1.48672	−0.100007
$222$	0	0
$223$	−12.5134	−0.837958	−0.418979	−	0.907996i	$-0.637612\pi$
$223$	−12.5134	−0.837958	−0.418979	+	0.907996i	$0.637612\pi$
$224$	0	0
$225$	3.80436	0.253624
$226$	0	0
$227$	−3.34596	−0.222079	−0.111039	−	0.993816i	$-0.535418\pi$
$227$	−3.34596	−0.222079	−0.111039	+	0.993816i	$0.535418\pi$
$228$	0	0
$229$	19.0544	1.25915	0.629576	−	0.776939i	$-0.283228\pi$
$229$	19.0544	1.25915	0.629576	+	0.776939i	$0.283228\pi$
$230$	0	0
$231$	−1.77522	−0.116801
$232$	0	0
$233$	−20.1191	−1.31805	−0.659023	−	0.752123i	$-0.729030\pi$
$233$	−20.1191	−1.31805	−0.659023	+	0.752123i	$0.729030\pi$
$234$	0	0
$235$	34.4084	2.24455
$236$	0	0
$237$	−4.32571	−0.280985
$238$	0	0
$239$	9.56587	0.618765	0.309382	−	0.950938i	$-0.399878\pi$
$239$	9.56587	0.618765	0.309382	+	0.950938i	$0.399878\pi$
$240$	0	0
$241$	−6.07154	−0.391102	−0.195551	−	0.980694i	$-0.562650\pi$
$241$	−6.07154	−0.391102	−0.195551	+	0.980694i	$0.562650\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	18.8507	1.20433
$246$	0	0
$247$	−1.31672	−0.0837812
$248$	0	0
$249$	−17.1086	−1.08422
$250$	0	0
$251$	−7.56092	−0.477241	−0.238621	−	0.971113i	$-0.576695\pi$
$251$	−7.56092	−0.477241	−0.238621	+	0.971113i	$0.576695\pi$
$252$	0	0
$253$	−1.79484	−0.112841
$254$	0	0
$255$	−21.3111	−1.33455
$256$	0	0
$257$	−3.78046	−0.235819	−0.117909	−	0.993024i	$-0.537619\pi$
$257$	−3.78046	−0.235819	−0.117909	+	0.993024i	$0.537619\pi$
$258$	0	0
$259$	−6.90529	−0.429074
$260$	0	0
$261$	−3.78046	−0.234005
$262$	0	0
$263$	−30.3034	−1.86859	−0.934293	−	0.356507i	$-0.883968\pi$
$263$	−30.3034	−1.86859	−0.934293	+	0.356507i	$0.883968\pi$
$264$	0	0
$265$	−2.25572	−0.138568
$266$	0	0
$267$	13.1166	0.802724
$268$	0	0
$269$	−10.8971	−0.664406	−0.332203	−	0.943208i	$-0.607792\pi$
$269$	−10.8971	−0.664406	−0.332203	+	0.943208i	$0.607792\pi$
$270$	0	0
$271$	−19.9321	−1.21079	−0.605394	−	0.795926i	$-0.706984\pi$
$271$	−19.9321	−1.21079	−0.605394	+	0.795926i	$0.706984\pi$
$272$	0	0
$273$	0.166503	0.0100772
$274$	0	0
$275$	8.39623	0.506311
$276$	0	0
$277$	26.0499	1.56518	0.782592	−	0.622535i	$-0.213897\pi$
$277$	26.0499	1.56518	0.782592	+	0.622535i	$0.213897\pi$
$278$	0	0
$279$	10.1583	0.608161
$280$	0	0
$281$	2.66385	0.158912	0.0794560	−	0.996838i	$-0.474682\pi$
$281$	2.66385	0.158912	0.0794560	+	0.996838i	$0.474682\pi$
$282$	0	0
$283$	−18.6438	−1.10826	−0.554130	−	0.832430i	$-0.686949\pi$
$283$	−18.6438	−1.10826	−0.554130	+	0.832430i	$0.686949\pi$
$284$	0	0
$285$	−18.8744	−1.11802
$286$	0	0
$287$	2.88076	0.170046
$288$	0	0
$289$	34.5837	2.03434
$290$	0	0
$291$	−4.94671	−0.289981
$292$	0	0
$293$	−9.95715	−0.581703	−0.290851	−	0.956768i	$-0.593939\pi$
$293$	−9.95715	−0.581703	−0.290851	+	0.956768i	$0.593939\pi$
$294$	0	0
$295$	−13.4732	−0.784438
$296$	0	0
$297$	2.20700	0.128063
$298$	0	0
$299$	0.168343	0.00973552
$300$	0	0
$301$	−6.86263	−0.395556
$302$	0	0
$303$	−0.614215	−0.0352857
$304$	0	0
$305$	−4.21195	−0.241175
$306$	0	0
$307$	7.10632	0.405579	0.202790	−	0.979222i	$-0.434999\pi$
$307$	7.10632	0.405579	0.202790	+	0.979222i	$0.434999\pi$
$308$	0	0
$309$	−4.58050	−0.260576
$310$	0	0
$311$	27.1394	1.53894	0.769468	−	0.638685i	$-0.220521\pi$
$311$	27.1394	1.53894	0.769468	+	0.638685i	$0.220521\pi$
$312$	0	0
$313$	−9.41950	−0.532421	−0.266211	−	0.963915i	$-0.585772\pi$
$313$	−9.41950	−0.532421	−0.266211	+	0.963915i	$0.585772\pi$
$314$	0	0
$315$	2.38671	0.134476
$316$	0	0
$317$	1.42863	0.0802401	0.0401200	−	0.999195i	$-0.487226\pi$
$317$	1.42863	0.0802401	0.0401200	+	0.999195i	$0.487226\pi$
$318$	0	0
$319$	−8.34348	−0.467145
$320$	0	0
$321$	−10.0034	−0.558335
$322$	0	0
$323$	45.6856	2.54202
$324$	0	0
$325$	−0.787505	−0.0436829
$326$	0	0
$327$	14.5554	0.804917
$328$	0	0
$329$	9.32751	0.514242
$330$	0	0
$331$	29.6131	1.62769	0.813843	−	0.581085i	$-0.197372\pi$
$331$	29.6131	1.62769	0.813843	+	0.581085i	$0.197372\pi$
$332$	0	0
$333$	8.58482	0.470445
$334$	0	0
$335$	−2.96721	−0.162116
$336$	0	0
$337$	35.7232	1.94597	0.972984	−	0.230871i	$-0.0741576\pi$
$337$	35.7232	1.94597	0.972984	+	0.230871i	$0.0741576\pi$
$338$	0	0
$339$	14.0759	0.764496
$340$	0	0
$341$	22.4193	1.21408
$342$	0	0
$343$	10.7406	0.579939
$344$	0	0
$345$	2.41308	0.129916
$346$	0	0
$347$	−7.93938	−0.426208	−0.213104	−	0.977030i	$-0.568357\pi$
$347$	−7.93938	−0.426208	−0.213104	+	0.977030i	$0.568357\pi$
$348$	0	0
$349$	−20.1936	−1.08094	−0.540471	−	0.841363i	$-0.681754\pi$
$349$	−20.1936	−1.08094	−0.540471	+	0.841363i	$0.681754\pi$
$350$	0	0
$351$	−0.207001	−0.0110489
$352$	0	0
$353$	11.8289	0.629590	0.314795	−	0.949160i	$-0.398064\pi$
$353$	11.8289	0.629590	0.314795	+	0.949160i	$0.398064\pi$
$354$	0	0
$355$	11.2546	0.597334
$356$	0	0
$357$	−5.77706	−0.305755
$358$	0	0
$359$	31.8450	1.68071	0.840356	−	0.542034i	$-0.182346\pi$
$359$	31.8450	1.68071	0.840356	+	0.542034i	$0.182346\pi$
$360$	0	0
$361$	21.4619	1.12957
$362$	0	0
$363$	−6.12915	−0.321697
$364$	0	0
$365$	−1.31564	−0.0688640
$366$	0	0
$367$	0.0882937	0.00460889	0.00230445	−	0.999997i	$-0.499266\pi$
$367$	0.0882937	0.00460889	0.00230445	+	0.999997i	$0.499266\pi$
$368$	0	0
$369$	−3.58143	−0.186442
$370$	0	0
$371$	−0.611486	−0.0317468
$372$	0	0
$373$	−19.3699	−1.00293	−0.501467	−	0.865177i	$-0.667206\pi$
$373$	−19.3699	−1.00293	−0.501467	+	0.865177i	$0.667206\pi$
$374$	0	0
$375$	3.54772	0.183203
$376$	0	0
$377$	0.782558	0.0403038
$378$	0	0
$379$	−13.8429	−0.711060	−0.355530	−	0.934665i	$-0.615700\pi$
$379$	−13.8429	−0.711060	−0.355530	+	0.934665i	$0.615700\pi$
$380$	0	0
$381$	6.39468	0.327609
$382$	0	0
$383$	−7.91171	−0.404269	−0.202135	−	0.979358i	$-0.564788\pi$
$383$	−7.91171	−0.404269	−0.202135	+	0.979358i	$0.564788\pi$
$384$	0	0
$385$	5.26747	0.268455
$386$	0	0
$387$	8.53179	0.433695
$388$	0	0
$389$	−11.1891	−0.567308	−0.283654	−	0.958927i	$-0.591547\pi$
$389$	−11.1891	−0.567308	−0.283654	+	0.958927i	$0.591547\pi$
$390$	0	0
$391$	−5.84090	−0.295387
$392$	0	0
$393$	16.0713	0.810689
$394$	0	0
$395$	12.8353	0.645814
$396$	0	0
$397$	−30.9005	−1.55085	−0.775425	−	0.631440i	$-0.782464\pi$
$397$	−30.9005	−1.55085	−0.775425	+	0.631440i	$0.782464\pi$
$398$	0	0
$399$	−5.11651	−0.256146
$400$	0	0
$401$	19.5109	0.974328	0.487164	−	0.873310i	$-0.338031\pi$
$401$	19.5109	0.974328	0.487164	+	0.873310i	$0.338031\pi$
$402$	0	0
$403$	−2.10277	−0.104747
$404$	0	0
$405$	−2.96721	−0.147442
$406$	0	0
$407$	18.9467	0.939154
$408$	0	0
$409$	−29.3961	−1.45354	−0.726771	−	0.686880i	$-0.758980\pi$
$409$	−29.3961	−1.45354	−0.726771	+	0.686880i	$0.758980\pi$
$410$	0	0
$411$	−0.760214	−0.0374986
$412$	0	0
$413$	−3.65234	−0.179720
$414$	0	0
$415$	50.7650	2.49196
$416$	0	0
$417$	0.822136	0.0402602
$418$	0	0
$419$	3.95153	0.193045	0.0965226	−	0.995331i	$-0.469228\pi$
$419$	3.95153	0.193045	0.0965226	+	0.995331i	$0.469228\pi$
$420$	0	0
$421$	31.7420	1.54701	0.773506	−	0.633789i	$-0.218501\pi$
$421$	31.7420	1.54701	0.773506	+	0.633789i	$0.218501\pi$
$422$	0	0
$423$	−11.5962	−0.563826
$424$	0	0
$425$	27.3236	1.32539
$426$	0	0
$427$	−1.14179	−0.0552549
$428$	0	0
$429$	−0.456851	−0.0220570
$430$	0	0
$431$	−37.2038	−1.79205	−0.896023	−	0.444008i	$-0.853556\pi$
$431$	−37.2038	−1.79205	−0.896023	+	0.444008i	$0.853556\pi$
$432$	0	0
$433$	−16.2351	−0.780207	−0.390104	−	0.920771i	$-0.627561\pi$
$433$	−16.2351	−0.780207	−0.390104	+	0.920771i	$0.627561\pi$
$434$	0	0
$435$	11.2174	0.537835
$436$	0	0
$437$	−5.17304	−0.247460
$438$	0	0
$439$	8.46190	0.403864	0.201932	−	0.979400i	$-0.435278\pi$
$439$	8.46190	0.403864	0.201932	+	0.979400i	$0.435278\pi$
$440$	0	0
$441$	−6.35300	−0.302524
$442$	0	0
$443$	33.2774	1.58106	0.790528	−	0.612426i	$-0.209806\pi$
$443$	33.2774	1.58106	0.790528	+	0.612426i	$0.209806\pi$
$444$	0	0
$445$	−38.9198	−1.84498
$446$	0	0
$447$	10.5839	0.500601
$448$	0	0
$449$	40.4967	1.91116	0.955580	−	0.294732i	$-0.0952305\pi$
$449$	40.4967	1.91116	0.955580	+	0.294732i	$0.0952305\pi$
$450$	0	0
$451$	−7.90422	−0.372195
$452$	0	0
$453$	−1.55074	−0.0728601
$454$	0	0
$455$	−0.494050	−0.0231614
$456$	0	0
$457$	−26.5000	−1.23962	−0.619810	−	0.784752i	$-0.712790\pi$
$457$	−26.5000	−1.23962	−0.619810	+	0.784752i	$0.712790\pi$
$458$	0	0
$459$	7.18218	0.335236
$460$	0	0
$461$	24.1380	1.12422	0.562110	−	0.827062i	$-0.309990\pi$
$461$	24.1380	1.12422	0.562110	+	0.827062i	$0.309990\pi$
$462$	0	0
$463$	−1.16378	−0.0540854	−0.0270427	−	0.999634i	$-0.508609\pi$
$463$	−1.16378	−0.0540854	−0.0270427	+	0.999634i	$0.508609\pi$
$464$	0	0
$465$	−30.1418	−1.39779
$466$	0	0
$467$	8.20700	0.379775	0.189887	−	0.981806i	$-0.439188\pi$
$467$	8.20700	0.379775	0.189887	+	0.981806i	$0.439188\pi$
$468$	0	0
$469$	−0.804360	−0.0371419
$470$	0	0
$471$	23.8554	1.09920
$472$	0	0
$473$	18.8297	0.865789
$474$	0	0
$475$	24.1994	1.11035
$476$	0	0
$477$	0.760214	0.0348078
$478$	0	0
$479$	−1.17411	−0.0536466	−0.0268233	−	0.999640i	$-0.508539\pi$
$479$	−1.17411	−0.0536466	−0.0268233	+	0.999640i	$0.508539\pi$
$480$	0	0
$481$	−1.77706	−0.0810271
$482$	0	0
$483$	0.654145	0.0297646
$484$	0	0
$485$	14.6779	0.666491
$486$	0	0
$487$	−10.0817	−0.456847	−0.228423	−	0.973562i	$-0.573357\pi$
$487$	−10.0817	−0.456847	−0.228423	+	0.973562i	$0.573357\pi$
$488$	0	0
$489$	−11.3714	−0.514233
$490$	0	0
$491$	5.97261	0.269540	0.134770	−	0.990877i	$-0.456970\pi$
$491$	5.97261	0.269540	0.134770	+	0.990877i	$0.456970\pi$
$492$	0	0
$493$	−27.1520	−1.22286
$494$	0	0
$495$	−6.54864	−0.294340
$496$	0	0
$497$	3.05094	0.136853
$498$	0	0
$499$	−18.6901	−0.836683	−0.418341	−	0.908290i	$-0.637388\pi$
$499$	−18.6901	−0.836683	−0.418341	+	0.908290i	$0.637388\pi$
$500$	0	0
$501$	−11.7547	−0.525162
$502$	0	0
$503$	−12.9399	−0.576963	−0.288481	−	0.957486i	$-0.593150\pi$
$503$	−12.9399	−0.576963	−0.288481	+	0.957486i	$0.593150\pi$
$504$	0	0
$505$	1.82251	0.0811006
$506$	0	0
$507$	−12.9572	−0.575447
$508$	0	0
$509$	−1.08290	−0.0479987	−0.0239994	−	0.999712i	$-0.507640\pi$
$509$	−1.08290	−0.0479987	−0.0239994	+	0.999712i	$0.507640\pi$
$510$	0	0
$511$	−0.356648	−0.0157772
$512$	0	0
$513$	6.36097	0.280844
$514$	0	0
$515$	13.5913	0.598906
$516$	0	0
$517$	−25.5928	−1.12557
$518$	0	0
$519$	−15.9041	−0.698113
$520$	0	0
$521$	−30.8020	−1.34946	−0.674730	−	0.738065i	$-0.735740\pi$
$521$	−30.8020	−1.34946	−0.674730	+	0.738065i	$0.735740\pi$
$522$	0	0
$523$	19.5306	0.854014	0.427007	−	0.904248i	$-0.359568\pi$
$523$	19.5306	0.854014	0.427007	+	0.904248i	$0.359568\pi$
$524$	0	0
$525$	−3.06008	−0.133553
$526$	0	0
$527$	72.9587	3.17813
$528$	0	0
$529$	−22.3386	−0.971245
$530$	0	0
$531$	4.54068	0.197049
$532$	0	0
$533$	0.741358	0.0321118
$534$	0	0
$535$	29.6822	1.28327
$536$	0	0
$537$	−11.3134	−0.488210
$538$	0	0
$539$	−14.0211	−0.603931
$540$	0	0
$541$	39.4616	1.69659	0.848294	−	0.529525i	$-0.177630\pi$
$541$	39.4616	1.69659	0.848294	+	0.529525i	$0.177630\pi$
$542$	0	0
$543$	−12.9810	−0.557070
$544$	0	0
$545$	−43.1891	−1.85002
$546$	0	0
$547$	−19.7773	−0.845616	−0.422808	−	0.906219i	$-0.638956\pi$
$547$	−19.7773	−0.845616	−0.422808	+	0.906219i	$0.638956\pi$
$548$	0	0
$549$	1.41950	0.0605826
$550$	0	0
$551$	−24.0474	−1.02445
$552$	0	0
$553$	3.47943	0.147960
$554$	0	0
$555$	−25.4730	−1.08127
$556$	0	0
$557$	12.1892	0.516474	0.258237	−	0.966082i	$-0.416858\pi$
$557$	12.1892	0.516474	0.258237	+	0.966082i	$0.416858\pi$
$558$	0	0
$559$	−1.76609	−0.0746975
$560$	0	0
$561$	15.8511	0.669233
$562$	0	0
$563$	−20.5787	−0.867287	−0.433644	−	0.901084i	$-0.642772\pi$
$563$	−20.5787	−0.867287	−0.433644	+	0.901084i	$0.642772\pi$
$564$	0	0
$565$	−41.7661	−1.75711
$566$	0	0
$567$	−0.804360	−0.0337800
$568$	0	0
$569$	8.03710	0.336933	0.168466	−	0.985707i	$-0.446119\pi$
$569$	8.03710	0.336933	0.168466	+	0.985707i	$0.446119\pi$
$570$	0	0
$571$	−21.8968	−0.916353	−0.458177	−	0.888861i	$-0.651497\pi$
$571$	−21.8968	−0.916353	−0.458177	+	0.888861i	$0.651497\pi$
$572$	0	0
$573$	3.63144	0.151706
$574$	0	0
$575$	−3.09389	−0.129024
$576$	0	0
$577$	−9.64922	−0.401702	−0.200851	−	0.979622i	$-0.564371\pi$
$577$	−9.64922	−0.401702	−0.200851	+	0.979622i	$0.564371\pi$
$578$	0	0
$579$	−6.06805	−0.252180
$580$	0	0
$581$	13.7615	0.570924
$582$	0	0
$583$	1.67779	0.0694870
$584$	0	0
$585$	0.614215	0.0253947
$586$	0	0
$587$	14.3257	0.591286	0.295643	−	0.955299i	$-0.404466\pi$
$587$	14.3257	0.591286	0.295643	+	0.955299i	$0.404466\pi$
$588$	0	0
$589$	64.6165	2.66248
$590$	0	0
$591$	22.6518	0.931770
$592$	0	0
$593$	−4.43635	−0.182179	−0.0910894	−	0.995843i	$-0.529035\pi$
$593$	−4.43635	−0.182179	−0.0910894	+	0.995843i	$0.529035\pi$
$594$	0	0
$595$	17.1418	0.702745
$596$	0	0
$597$	−10.2104	−0.417884
$598$	0	0
$599$	−22.5646	−0.921964	−0.460982	−	0.887410i	$-0.652503\pi$
$599$	−22.5646	−0.921964	−0.460982	+	0.887410i	$0.652503\pi$
$600$	0	0
$601$	8.88689	0.362504	0.181252	−	0.983437i	$-0.441985\pi$
$601$	8.88689	0.362504	0.181252	+	0.983437i	$0.441985\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	18.1865	0.739386
$606$	0	0
$607$	−6.57136	−0.266723	−0.133362	−	0.991067i	$-0.542577\pi$
$607$	−6.57136	−0.266723	−0.133362	+	0.991067i	$0.542577\pi$
$608$	0	0
$609$	3.04085	0.123222
$610$	0	0
$611$	2.40042	0.0971105
$612$	0	0
$613$	29.3079	1.18374	0.591868	−	0.806035i	$-0.298391\pi$
$613$	29.3079	1.18374	0.591868	+	0.806035i	$0.298391\pi$
$614$	0	0
$615$	10.6269	0.428517
$616$	0	0
$617$	−7.53296	−0.303266	−0.151633	−	0.988437i	$-0.548453\pi$
$617$	−7.53296	−0.303266	−0.151633	+	0.988437i	$0.548453\pi$
$618$	0	0
$619$	6.51677	0.261931	0.130966	−	0.991387i	$-0.458192\pi$
$619$	6.51677	0.261931	0.130966	+	0.991387i	$0.458192\pi$
$620$	0	0
$621$	−0.813248	−0.0326345
$622$	0	0
$623$	−10.5505	−0.422696
$624$	0	0
$625$	−29.5486	−1.18195
$626$	0	0
$627$	14.0387	0.560650
$628$	0	0
$629$	61.6578	2.45846
$630$	0	0
$631$	−21.6174	−0.860573	−0.430287	−	0.902692i	$-0.641588\pi$
$631$	−21.6174	−0.860573	−0.430287	+	0.902692i	$0.641588\pi$
$632$	0	0
$633$	−19.9909	−0.794565
$634$	0	0
$635$	−18.9744	−0.752975
$636$	0	0
$637$	1.31508	0.0521052
$638$	0	0
$639$	−3.79300	−0.150049
$640$	0	0
$641$	−8.19748	−0.323781	−0.161891	−	0.986809i	$-0.551759\pi$
$641$	−8.19748	−0.323781	−0.161891	+	0.986809i	$0.551759\pi$
$642$	0	0
$643$	−33.0579	−1.30368	−0.651839	−	0.758358i	$-0.726002\pi$
$643$	−33.0579	−1.30368	−0.651839	+	0.758358i	$0.726002\pi$
$644$	0	0
$645$	−25.3156	−0.996803
$646$	0	0
$647$	−15.5727	−0.612224	−0.306112	−	0.951995i	$-0.599028\pi$
$647$	−15.5727	−0.612224	−0.306112	+	0.951995i	$0.599028\pi$
$648$	0	0
$649$	10.0213	0.393369
$650$	0	0
$651$	−8.17092	−0.320244
$652$	0	0
$653$	−14.5827	−0.570666	−0.285333	−	0.958428i	$-0.592104\pi$
$653$	−14.5827	−0.570666	−0.285333	+	0.958428i	$0.592104\pi$
$654$	0	0
$655$	−47.6870	−1.86328
$656$	0	0
$657$	0.443394	0.0172984
$658$	0	0
$659$	−20.6662	−0.805040	−0.402520	−	0.915411i	$-0.631866\pi$
$659$	−20.6662	−0.805040	−0.402520	+	0.915411i	$0.631866\pi$
$660$	0	0
$661$	−7.24621	−0.281845	−0.140922	−	0.990021i	$-0.545007\pi$
$661$	−7.24621	−0.281845	−0.140922	+	0.990021i	$0.545007\pi$
$662$	0	0
$663$	−1.48672	−0.0577393
$664$	0	0
$665$	15.1818	0.588724
$666$	0	0
$667$	3.07445	0.119043
$668$	0	0
$669$	−12.5134	−0.483795
$670$	0	0
$671$	3.13283	0.120941
$672$	0	0
$673$	26.1323	1.00733	0.503663	−	0.863900i	$-0.331985\pi$
$673$	26.1323	1.00733	0.503663	+	0.863900i	$0.331985\pi$
$674$	0	0
$675$	3.80436	0.146430
$676$	0	0
$677$	14.9949	0.576300	0.288150	−	0.957585i	$-0.406960\pi$
$677$	14.9949	0.576300	0.288150	+	0.957585i	$0.406960\pi$
$678$	0	0
$679$	3.97894	0.152698
$680$	0	0
$681$	−3.34596	−0.128217
$682$	0	0
$683$	15.8476	0.606391	0.303196	−	0.952928i	$-0.401946\pi$
$683$	15.8476	0.606391	0.303196	+	0.952928i	$0.401946\pi$
$684$	0	0
$685$	2.25572	0.0861865
$686$	0	0
$687$	19.0544	0.726972
$688$	0	0
$689$	−0.157365	−0.00599512
$690$	0	0
$691$	32.4982	1.23629	0.618145	−	0.786064i	$-0.287884\pi$
$691$	32.4982	1.23629	0.618145	+	0.786064i	$0.287884\pi$
$692$	0	0
$693$	−1.77522	−0.0674352
$694$	0	0
$695$	−2.43945	−0.0925337
$696$	0	0
$697$	−25.7225	−0.974308
$698$	0	0
$699$	−20.1191	−0.760974
$700$	0	0
$701$	−10.8729	−0.410663	−0.205331	−	0.978692i	$-0.565827\pi$
$701$	−10.8729	−0.410663	−0.205331	+	0.978692i	$0.565827\pi$
$702$	0	0
$703$	54.6078	2.05957
$704$	0	0
$705$	34.4084	1.29589
$706$	0	0
$707$	0.494050	0.0185807
$708$	0	0
$709$	−16.1154	−0.605228	−0.302614	−	0.953113i	$-0.597859\pi$
$709$	−16.1154	−0.605228	−0.302614	+	0.953113i	$0.597859\pi$
$710$	0	0
$711$	−4.32571	−0.162227
$712$	0	0
$713$	−8.26121	−0.309385
$714$	0	0
$715$	1.35557	0.0506956
$716$	0	0
$717$	9.56587	0.357244
$718$	0	0
$719$	−24.3936	−0.909728	−0.454864	−	0.890561i	$-0.650312\pi$
$719$	−24.3936	−0.909728	−0.454864	+	0.890561i	$0.650312\pi$
$720$	0	0
$721$	3.68438	0.137213
$722$	0	0
$723$	−6.07154	−0.225803
$724$	0	0
$725$	−14.3822	−0.534143
$726$	0	0
$727$	−48.2622	−1.78995	−0.894973	−	0.446120i	$-0.852805\pi$
$727$	−48.2622	−1.78995	−0.894973	+	0.446120i	$0.852805\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	61.2769	2.26641
$732$	0	0
$733$	−47.4757	−1.75356	−0.876778	−	0.480895i	$-0.840312\pi$
$733$	−47.4757	−1.75356	−0.876778	+	0.480895i	$0.840312\pi$
$734$	0	0
$735$	18.8507	0.695319
$736$	0	0
$737$	2.20700	0.0812959
$738$	0	0
$739$	36.5613	1.34493	0.672465	−	0.740129i	$-0.265236\pi$
$739$	36.5613	1.34493	0.672465	+	0.740129i	$0.265236\pi$
$740$	0	0
$741$	−1.31672	−0.0483711
$742$	0	0
$743$	−14.3900	−0.527918	−0.263959	−	0.964534i	$-0.585028\pi$
$743$	−14.3900	−0.527918	−0.263959	+	0.964534i	$0.585028\pi$
$744$	0	0
$745$	−31.4047	−1.15058
$746$	0	0
$747$	−17.1086	−0.625973
$748$	0	0
$749$	8.04634	0.294007
$750$	0	0
$751$	−2.91380	−0.106326	−0.0531631	−	0.998586i	$-0.516930\pi$
$751$	−2.91380	−0.106326	−0.0531631	+	0.998586i	$0.516930\pi$
$752$	0	0
$753$	−7.56092	−0.275535
$754$	0	0
$755$	4.60138	0.167461
$756$	0	0
$757$	−19.2739	−0.700521	−0.350260	−	0.936652i	$-0.613907\pi$
$757$	−19.2739	−0.700521	−0.350260	+	0.936652i	$0.613907\pi$
$758$	0	0
$759$	−1.79484	−0.0651485
$760$	0	0
$761$	−49.8530	−1.80717	−0.903586	−	0.428408i	$-0.859075\pi$
$761$	−49.8530	−1.80717	−0.903586	+	0.428408i	$0.859075\pi$
$762$	0	0
$763$	−11.7078	−0.423851
$764$	0	0
$765$	−21.3111	−0.770504
$766$	0	0
$767$	−0.939923	−0.0339386
$768$	0	0
$769$	−26.4266	−0.952969	−0.476485	−	0.879183i	$-0.658089\pi$
$769$	−26.4266	−0.952969	−0.476485	+	0.879183i	$0.658089\pi$
$770$	0	0
$771$	−3.78046	−0.136150
$772$	0	0
$773$	−18.9911	−0.683063	−0.341531	−	0.939870i	$-0.610946\pi$
$773$	−18.9911	−0.683063	−0.341531	+	0.939870i	$0.610946\pi$
$774$	0	0
$775$	38.6458	1.38820
$776$	0	0
$777$	−6.90529	−0.247726
$778$	0	0
$779$	−22.7814	−0.816227
$780$	0	0
$781$	−8.37115	−0.299543
$782$	0	0
$783$	−3.78046	−0.135103
$784$	0	0
$785$	−70.7841	−2.52639
$786$	0	0
$787$	30.8214	1.09867	0.549333	−	0.835604i	$-0.314882\pi$
$787$	30.8214	1.09867	0.549333	+	0.835604i	$0.314882\pi$
$788$	0	0
$789$	−30.3034	−1.07883
$790$	0	0
$791$	−11.3221	−0.402566
$792$	0	0
$793$	−0.293836	−0.0104344
$794$	0	0
$795$	−2.25572	−0.0800020
$796$	0	0
$797$	19.8899	0.704537	0.352269	−	0.935899i	$-0.385410\pi$
$797$	19.8899	0.704537	0.352269	+	0.935899i	$0.385410\pi$
$798$	0	0
$799$	−83.2859	−2.94644
$800$	0	0
$801$	13.1166	0.463453
$802$	0	0
$803$	0.978570	0.0345330
$804$	0	0
$805$	−1.94099	−0.0684108
$806$	0	0
$807$	−10.8971	−0.383595
$808$	0	0
$809$	−4.46401	−0.156946	−0.0784732	−	0.996916i	$-0.525005\pi$
$809$	−4.46401	−0.156946	−0.0784732	+	0.996916i	$0.525005\pi$
$810$	0	0
$811$	45.3844	1.59366	0.796830	−	0.604203i	$-0.206508\pi$
$811$	45.3844	1.59366	0.796830	+	0.604203i	$0.206508\pi$
$812$	0	0
$813$	−19.9321	−0.699048
$814$	0	0
$815$	33.7414	1.18191
$816$	0	0
$817$	54.2704	1.89868
$818$	0	0
$819$	0.166503	0.00581809
$820$	0	0
$821$	−6.62935	−0.231366	−0.115683	−	0.993286i	$-0.536906\pi$
$821$	−6.62935	−0.231366	−0.115683	+	0.993286i	$0.536906\pi$
$822$	0	0
$823$	38.4563	1.34050	0.670250	−	0.742135i	$-0.266187\pi$
$823$	38.4563	1.34050	0.670250	+	0.742135i	$0.266187\pi$
$824$	0	0
$825$	8.39623	0.292319
$826$	0	0
$827$	3.03047	0.105380	0.0526898	−	0.998611i	$-0.483221\pi$
$827$	3.03047	0.105380	0.0526898	+	0.998611i	$0.483221\pi$
$828$	0	0
$829$	−23.1943	−0.805572	−0.402786	−	0.915294i	$-0.631958\pi$
$829$	−23.1943	−0.805572	−0.402786	+	0.915294i	$0.631958\pi$
$830$	0	0
$831$	26.0499	0.903660
$832$	0	0
$833$	−45.6284	−1.58093
$834$	0	0
$835$	34.8788	1.20703
$836$	0	0
$837$	10.1583	0.351122
$838$	0	0
$839$	43.0009	1.48456	0.742278	−	0.670092i	$-0.233745\pi$
$839$	43.0009	1.48456	0.742278	+	0.670092i	$0.233745\pi$
$840$	0	0
$841$	−14.7081	−0.507176
$842$	0	0
$843$	2.66385	0.0917479
$844$	0	0
$845$	38.4466	1.32260
$846$	0	0
$847$	4.93004	0.169398
$848$	0	0
$849$	−18.6438	−0.639854
$850$	0	0
$851$	−6.98159	−0.239326
$852$	0	0
$853$	11.7081	0.400878	0.200439	−	0.979706i	$-0.435763\pi$
$853$	11.7081	0.400878	0.200439	+	0.979706i	$0.435763\pi$
$854$	0	0
$855$	−18.8744	−0.645489
$856$	0	0
$857$	−18.2729	−0.624192	−0.312096	−	0.950051i	$-0.601031\pi$
$857$	−18.2729	−0.624192	−0.312096	+	0.950051i	$0.601031\pi$
$858$	0	0
$859$	−7.47408	−0.255012	−0.127506	−	0.991838i	$-0.540697\pi$
$859$	−7.47408	−0.255012	−0.127506	+	0.991838i	$0.540697\pi$
$860$	0	0
$861$	2.88076	0.0981760
$862$	0	0
$863$	−23.5975	−0.803267	−0.401634	−	0.915800i	$-0.631557\pi$
$863$	−23.5975	−0.803267	−0.401634	+	0.915800i	$0.631557\pi$
$864$	0	0
$865$	47.1909	1.60454
$866$	0	0
$867$	34.5837	1.17453
$868$	0	0
$869$	−9.54684	−0.323854
$870$	0	0
$871$	−0.207001	−0.00701395
$872$	0	0
$873$	−4.94671	−0.167421
$874$	0	0
$875$	−2.85364	−0.0964708
$876$	0	0
$877$	−23.8849	−0.806536	−0.403268	−	0.915082i	$-0.632126\pi$
$877$	−23.8849	−0.806536	−0.403268	+	0.915082i	$0.632126\pi$
$878$	0	0
$879$	−9.95715	−0.335846
$880$	0	0
$881$	−27.3236	−0.920556	−0.460278	−	0.887775i	$-0.652250\pi$
$881$	−27.3236	−0.920556	−0.460278	+	0.887775i	$0.652250\pi$
$882$	0	0
$883$	−49.0724	−1.65142	−0.825709	−	0.564096i	$-0.809225\pi$
$883$	−49.0724	−1.65142	−0.825709	+	0.564096i	$0.809225\pi$
$884$	0	0
$885$	−13.4732	−0.452895
$886$	0	0
$887$	−7.68760	−0.258124	−0.129062	−	0.991637i	$-0.541197\pi$
$887$	−7.68760	−0.258124	−0.129062	+	0.991637i	$0.541197\pi$
$888$	0	0
$889$	−5.14363	−0.172512
$890$	0	0
$891$	2.20700	0.0739373
$892$	0	0
$893$	−73.7629	−2.46838
$894$	0	0
$895$	33.5694	1.12210
$896$	0	0
$897$	0.168343	0.00562081
$898$	0	0
$899$	−38.4030	−1.28081
$900$	0	0
$901$	5.45999	0.181899
$902$	0	0
$903$	−6.86263	−0.228374
$904$	0	0
$905$	38.5176	1.28037
$906$	0	0
$907$	−10.2355	−0.339863	−0.169932	−	0.985456i	$-0.554355\pi$
$907$	−10.2355	−0.339863	−0.169932	+	0.985456i	$0.554355\pi$
$908$	0	0
$909$	−0.614215	−0.0203722
$910$	0	0
$911$	36.5623	1.21136	0.605682	−	0.795707i	$-0.292900\pi$
$911$	36.5623	1.21136	0.605682	+	0.795707i	$0.292900\pi$
$912$	0	0
$913$	−37.7588	−1.24963
$914$	0	0
$915$	−4.21195	−0.139243
$916$	0	0
$917$	−12.9271	−0.426891
$918$	0	0
$919$	25.1404	0.829304	0.414652	−	0.909980i	$-0.363903\pi$
$919$	25.1404	0.829304	0.414652	+	0.909980i	$0.363903\pi$
$920$	0	0
$921$	7.10632	0.234161
$922$	0	0
$923$	0.785153	0.0258436
$924$	0	0
$925$	32.6598	1.07385
$926$	0	0
$927$	−4.58050	−0.150444
$928$	0	0
$929$	47.5694	1.56070	0.780350	−	0.625343i	$-0.215041\pi$
$929$	47.5694	1.56070	0.780350	+	0.625343i	$0.215041\pi$
$930$	0	0
$931$	−40.4112	−1.32443
$932$	0	0
$933$	27.1394	0.888505
$934$	0	0
$935$	−47.0336	−1.53816
$936$	0	0
$937$	−8.93757	−0.291978	−0.145989	−	0.989286i	$-0.546636\pi$
$937$	−8.93757	−0.291978	−0.145989	+	0.989286i	$0.546636\pi$
$938$	0	0
$939$	−9.41950	−0.307394
$940$	0	0
$941$	51.2259	1.66992	0.834958	−	0.550313i	$-0.185492\pi$
$941$	51.2259	1.66992	0.834958	+	0.550313i	$0.185492\pi$
$942$	0	0
$943$	2.91259	0.0948470
$944$	0	0
$945$	2.38671	0.0776397
$946$	0	0
$947$	42.7116	1.38794	0.693971	−	0.720003i	$-0.255860\pi$
$947$	42.7116	1.38794	0.693971	+	0.720003i	$0.255860\pi$
$948$	0	0
$949$	−0.0917828	−0.00297939
$950$	0	0
$951$	1.42863	0.0463266
$952$	0	0
$953$	−2.10668	−0.0682421	−0.0341211	−	0.999418i	$-0.510863\pi$
$953$	−2.10668	−0.0682421	−0.0341211	+	0.999418i	$0.510863\pi$
$954$	0	0
$955$	−10.7753	−0.348680
$956$	0	0
$957$	−8.34348	−0.269706
$958$	0	0
$959$	0.611486	0.0197459
$960$	0	0
$961$	72.1907	2.32873
$962$	0	0
$963$	−10.0034	−0.322355
$964$	0	0
$965$	18.0052	0.579608
$966$	0	0
$967$	−6.62520	−0.213052	−0.106526	−	0.994310i	$-0.533973\pi$
$967$	−6.62520	−0.213052	−0.106526	+	0.994310i	$0.533973\pi$
$968$	0	0
$969$	45.6856	1.46763
$970$	0	0
$971$	−41.4608	−1.33054	−0.665271	−	0.746602i	$-0.731684\pi$
$971$	−41.4608	−1.33054	−0.665271	+	0.746602i	$0.731684\pi$
$972$	0	0
$973$	−0.661294	−0.0212001
$974$	0	0
$975$	−0.787505	−0.0252203
$976$	0	0
$977$	10.5251	0.336728	0.168364	−	0.985725i	$-0.446152\pi$
$977$	10.5251	0.336728	0.168364	+	0.985725i	$0.446152\pi$
$978$	0	0
$979$	28.9484	0.925194
$980$	0	0
$981$	14.5554	0.464719
$982$	0	0
$983$	−54.6723	−1.74378	−0.871888	−	0.489706i	$-0.837104\pi$
$983$	−54.6723	−1.74378	−0.871888	+	0.489706i	$0.837104\pi$
$984$	0	0
$985$	−67.2127	−2.14158
$986$	0	0
$987$	9.32751	0.296898
$988$	0	0
$989$	−6.93846	−0.220630
$990$	0	0
$991$	−7.24345	−0.230096	−0.115048	−	0.993360i	$-0.536702\pi$
$991$	−7.24345	−0.230096	−0.115048	+	0.993360i	$0.536702\pi$
$992$	0	0
$993$	29.6131	0.939745
$994$	0	0
$995$	30.2964	0.960462
$996$	0	0
$997$	−17.8569	−0.565535	−0.282768	−	0.959188i	$-0.591253\pi$
$997$	−17.8569	−0.565535	−0.282768	+	0.959188i	$0.591253\pi$
$998$	0	0
$999$	8.58482	0.271612

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.a.f.1.1	✓	5
3.2	odd	2		2412.2.a.j.1.5		5
4.3	odd	2		3216.2.a.ba.1.1		5
12.11	even	2		9648.2.a.ca.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.a.f.1.1	✓	5	1.1	even	1	trivial
2412.2.a.j.1.5		5	3.2	odd	2
3216.2.a.ba.1.1		5	4.3	odd	2
9648.2.a.ca.1.5		5	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.96721 q^{5} -0.804360 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.96721 q^{5} -0.804360 q^{7} +1.00000 q^{9} +2.20700 q^{11} -0.207001 q^{13} -2.96721 q^{15} +7.18218 q^{17} +6.36097 q^{19} -0.804360 q^{21} -0.813248 q^{23} +3.80436 q^{25} +1.00000 q^{27} -3.78046 q^{29} +10.1583 q^{31} +2.20700 q^{33} +2.38671 q^{35} +8.58482 q^{37} -0.207001 q^{39} -3.58143 q^{41} +8.53179 q^{43} -2.96721 q^{45} -11.5962 q^{47} -6.35300 q^{49} +7.18218 q^{51} +0.760214 q^{53} -6.54864 q^{55} +6.36097 q^{57} +4.54068 q^{59} +1.41950 q^{61} -0.804360 q^{63} +0.614215 q^{65} +1.00000 q^{67} -0.813248 q^{69} -3.79300 q^{71} +0.443394 q^{73} +3.80436 q^{75} -1.77522 q^{77} -4.32571 q^{79} +1.00000 q^{81} -17.1086 q^{83} -21.3111 q^{85} -3.78046 q^{87} +13.1166 q^{89} +0.166503 q^{91} +10.1583 q^{93} -18.8744 q^{95} -4.94671 q^{97} +2.20700 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 + 3 * q^5 + 5 * q^7 + 5 * q^9	\( 5 q + 5 q^{3} + 3 q^{5} + 5 q^{7} + 5 q^{9} + 6 q^{11} + 4 q^{13} + 3 q^{15} + q^{17} + 13 q^{19} + 5 q^{21} + 10 q^{25} + 5 q^{27} + 3 q^{29} + 3 q^{31} + 6 q^{33} - 9 q^{35} + 12 q^{37} + 4 q^{39} + 11 q^{41} + 3 q^{43} + 3 q^{45} - 13 q^{47} + 24 q^{49} + q^{51} - 9 q^{53} + 14 q^{55} + 13 q^{57} - 12 q^{59} + 4 q^{61} + 5 q^{63} - 8 q^{65} + 5 q^{67} - 24 q^{71} + 12 q^{73} + 10 q^{75} - 4 q^{77} - 4 q^{79} + 5 q^{81} - 27 q^{83} - 4 q^{85} + 3 q^{87} - 5 q^{89} + 14 q^{91} + 3 q^{93} - 30 q^{95} + 8 q^{97} + 6 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + 3 * q^5 + 5 * q^7 + 5 * q^9 + 6 * q^11 + 4 * q^13 + 3 * q^15 + q^17 + 13 * q^19 + 5 * q^21 + 10 * q^25 + 5 * q^27 + 3 * q^29 + 3 * q^31 + 6 * q^33 - 9 * q^35 + 12 * q^37 + 4 * q^39 + 11 * q^41 + 3 * q^43 + 3 * q^45 - 13 * q^47 + 24 * q^49 + q^51 - 9 * q^53 + 14 * q^55 + 13 * q^57 - 12 * q^59 + 4 * q^61 + 5 * q^63 - 8 * q^65 + 5 * q^67 - 24 * q^71 + 12 * q^73 + 10 * q^75 - 4 * q^77 - 4 * q^79 + 5 * q^81 - 27 * q^83 - 4 * q^85 + 3 * q^87 - 5 * q^89 + 14 * q^91 + 3 * q^93 - 30 * q^95 + 8 * q^97 + 6 * q^99

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