LMFDB - Embedded newform 476.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [476,2,Mod(1,476)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(476, 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("476.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 476 = 2^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	476.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:	$3.80087913621$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	476.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.302776 q^{3} -3.30278 q^{5} +1.00000 q^{7} -2.90833 q^{9} +O(q^{10})$	$q+0.302776 q^{3} -3.30278 q^{5} +1.00000 q^{7} -2.90833 q^{9} +4.60555 q^{11} -6.60555 q^{13} -1.00000 q^{15} -1.00000 q^{17} -6.00000 q^{19} +0.302776 q^{21} -2.60555 q^{23} +5.90833 q^{25} -1.78890 q^{27} -8.60555 q^{29} -6.69722 q^{31} +1.39445 q^{33} -3.30278 q^{35} +7.21110 q^{37} -2.00000 q^{39} +9.51388 q^{41} -4.30278 q^{43} +9.60555 q^{45} +10.0000 q^{47} +1.00000 q^{49} -0.302776 q^{51} +0.697224 q^{53} -15.2111 q^{55} -1.81665 q^{57} +5.21110 q^{59} -4.30278 q^{61} -2.90833 q^{63} +21.8167 q^{65} +2.69722 q^{67} -0.788897 q^{69} -2.00000 q^{71} -13.5139 q^{73} +1.78890 q^{75} +4.60555 q^{77} -6.00000 q^{79} +8.18335 q^{81} +9.21110 q^{83} +3.30278 q^{85} -2.60555 q^{87} -2.00000 q^{89} -6.60555 q^{91} -2.02776 q^{93} +19.8167 q^{95} -2.09167 q^{97} -13.3944 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9	$ 2 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9} + 2 q^{11} - 6 q^{13} - 2 q^{15} - 2 q^{17} - 12 q^{19} - 3 q^{21} + 2 q^{23} + q^{25} - 18 q^{27} - 10 q^{29} - 17 q^{31} + 10 q^{33} - 3 q^{35} - 4 q^{39} + q^{41} - 5 q^{43} + 12 q^{45} + 20 q^{47} + 2 q^{49} + 3 q^{51} + 5 q^{53} - 16 q^{55} + 18 q^{57} - 4 q^{59} - 5 q^{61} + 5 q^{63} + 22 q^{65} + 9 q^{67} - 16 q^{69} - 4 q^{71} - 9 q^{73} + 18 q^{75} + 2 q^{77} - 12 q^{79} + 38 q^{81} + 4 q^{83} + 3 q^{85} + 2 q^{87} - 4 q^{89} - 6 q^{91} + 32 q^{93} + 18 q^{95} - 15 q^{97} - 34 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9 + 2 * q^11 - 6 * q^13 - 2 * q^15 - 2 * q^17 - 12 * q^19 - 3 * q^21 + 2 * q^23 + q^25 - 18 * q^27 - 10 * q^29 - 17 * q^31 + 10 * q^33 - 3 * q^35 - 4 * q^39 + q^41 - 5 * q^43 + 12 * q^45 + 20 * q^47 + 2 * q^49 + 3 * q^51 + 5 * q^53 - 16 * q^55 + 18 * q^57 - 4 * q^59 - 5 * q^61 + 5 * q^63 + 22 * q^65 + 9 * q^67 - 16 * q^69 - 4 * q^71 - 9 * q^73 + 18 * q^75 + 2 * q^77 - 12 * q^79 + 38 * q^81 + 4 * q^83 + 3 * q^85 + 2 * q^87 - 4 * q^89 - 6 * q^91 + 32 * q^93 + 18 * q^95 - 15 * q^97 - 34 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.302776	0.174808	0.0874038	−	0.996173i	$-0.472143\pi$
$3$	0.302776	0.174808	0.0874038	+	0.996173i	$0.472143\pi$
$4$	0	0
$5$	−3.30278	−1.47705	−0.738523	−	0.674228i	$-0.764476\pi$
$5$	−3.30278	−1.47705	−0.738523	+	0.674228i	$0.764476\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.90833	−0.969442
$10$	0	0
$11$	4.60555	1.38863	0.694313	−	0.719673i	$-0.255708\pi$
$11$	4.60555	1.38863	0.694313	+	0.719673i	$0.255708\pi$
$12$	0	0
$13$	−6.60555	−1.83205	−0.916025	−	0.401121i	$-0.868621\pi$
$13$	−6.60555	−1.83205	−0.916025	+	0.401121i	$0.868621\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	0.302776	0.0660711
$22$	0	0
$23$	−2.60555	−0.543295	−0.271647	−	0.962397i	$-0.587568\pi$
$23$	−2.60555	−0.543295	−0.271647	+	0.962397i	$0.587568\pi$
$24$	0	0
$25$	5.90833	1.18167
$26$	0	0
$27$	−1.78890	−0.344273
$28$	0	0
$29$	−8.60555	−1.59801	−0.799005	−	0.601324i	$-0.794640\pi$
$29$	−8.60555	−1.59801	−0.799005	+	0.601324i	$0.794640\pi$
$30$	0	0
$31$	−6.69722	−1.20286	−0.601429	−	0.798927i	$-0.705402\pi$
$31$	−6.69722	−1.20286	−0.601429	+	0.798927i	$0.705402\pi$
$32$	0	0
$33$	1.39445	0.242742
$34$	0	0
$35$	−3.30278	−0.558271
$36$	0	0
$37$	7.21110	1.18550	0.592749	−	0.805387i	$-0.298043\pi$
$37$	7.21110	1.18550	0.592749	+	0.805387i	$0.298043\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	9.51388	1.48582	0.742909	−	0.669392i	$-0.233445\pi$
$41$	9.51388	1.48582	0.742909	+	0.669392i	$0.233445\pi$
$42$	0	0
$43$	−4.30278	−0.656167	−0.328084	−	0.944649i	$-0.606403\pi$
$43$	−4.30278	−0.656167	−0.328084	+	0.944649i	$0.606403\pi$
$44$	0	0
$45$	9.60555	1.43191
$46$	0	0
$47$	10.0000	1.45865	0.729325	−	0.684167i	$-0.239834\pi$
$47$	10.0000	1.45865	0.729325	+	0.684167i	$0.239834\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.302776	−0.0423971
$52$	0	0
$53$	0.697224	0.0957711	0.0478856	−	0.998853i	$-0.484752\pi$
$53$	0.697224	0.0957711	0.0478856	+	0.998853i	$0.484752\pi$
$54$	0	0
$55$	−15.2111	−2.05106
$56$	0	0
$57$	−1.81665	−0.240622
$58$	0	0
$59$	5.21110	0.678428	0.339214	−	0.940709i	$-0.389839\pi$
$59$	5.21110	0.678428	0.339214	+	0.940709i	$0.389839\pi$
$60$	0	0
$61$	−4.30278	−0.550914	−0.275457	−	0.961313i	$-0.588829\pi$
$61$	−4.30278	−0.550914	−0.275457	+	0.961313i	$0.588829\pi$
$62$	0	0
$63$	−2.90833	−0.366415
$64$	0	0
$65$	21.8167	2.70602
$66$	0	0
$67$	2.69722	0.329518	0.164759	−	0.986334i	$-0.447315\pi$
$67$	2.69722	0.329518	0.164759	+	0.986334i	$0.447315\pi$
$68$	0	0
$69$	−0.788897	−0.0949721
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	−13.5139	−1.58168	−0.790840	−	0.612023i	$-0.790356\pi$
$73$	−13.5139	−1.58168	−0.790840	+	0.612023i	$0.790356\pi$
$74$	0	0
$75$	1.78890	0.206564
$76$	0	0
$77$	4.60555	0.524851
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	8.18335	0.909261
$82$	0	0
$83$	9.21110	1.01105	0.505525	−	0.862812i	$-0.331299\pi$
$83$	9.21110	1.01105	0.505525	+	0.862812i	$0.331299\pi$
$84$	0	0
$85$	3.30278	0.358236
$86$	0	0
$87$	−2.60555	−0.279344
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−6.60555	−0.692450
$92$	0	0
$93$	−2.02776	−0.210269
$94$	0	0
$95$	19.8167	2.03315
$96$	0	0
$97$	−2.09167	−0.212377	−0.106189	−	0.994346i	$-0.533865\pi$
$97$	−2.09167	−0.212377	−0.106189	+	0.994346i	$0.533865\pi$
$98$	0	0
$99$	−13.3944	−1.34619
$100$	0	0
$101$	−12.4222	−1.23606	−0.618028	−	0.786156i	$-0.712068\pi$
$101$	−12.4222	−1.23606	−0.618028	+	0.786156i	$0.712068\pi$
$102$	0	0
$103$	13.2111	1.30173	0.650864	−	0.759194i	$-0.274407\pi$
$103$	13.2111	1.30173	0.650864	+	0.759194i	$0.274407\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	−9.39445	−0.908196	−0.454098	−	0.890952i	$-0.650038\pi$
$107$	−9.39445	−0.908196	−0.454098	+	0.890952i	$0.650038\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	2.18335	0.207234
$112$	0	0
$113$	19.2111	1.80723	0.903614	−	0.428347i	$-0.140904\pi$
$113$	19.2111	1.80723	0.903614	+	0.428347i	$0.140904\pi$
$114$	0	0
$115$	8.60555	0.802472
$116$	0	0
$117$	19.2111	1.77607
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	10.2111	0.928282
$122$	0	0
$123$	2.88057	0.259732
$124$	0	0
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−5.11943	−0.454276	−0.227138	−	0.973863i	$-0.572937\pi$
$127$	−5.11943	−0.454276	−0.227138	+	0.973863i	$0.572937\pi$
$128$	0	0
$129$	−1.30278	−0.114703
$130$	0	0
$131$	−10.4222	−0.910592	−0.455296	−	0.890340i	$-0.650467\pi$
$131$	−10.4222	−0.910592	−0.455296	+	0.890340i	$0.650467\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	0	0
$135$	5.90833	0.508508
$136$	0	0
$137$	−3.09167	−0.264139	−0.132070	−	0.991240i	$-0.542162\pi$
$137$	−3.09167	−0.264139	−0.132070	+	0.991240i	$0.542162\pi$
$138$	0	0
$139$	−1.48612	−0.126051	−0.0630256	−	0.998012i	$-0.520075\pi$
$139$	−1.48612	−0.126051	−0.0630256	+	0.998012i	$0.520075\pi$
$140$	0	0
$141$	3.02776	0.254983
$142$	0	0
$143$	−30.4222	−2.54403
$144$	0	0
$145$	28.4222	2.36034
$146$	0	0
$147$	0.302776	0.0249725
$148$	0	0
$149$	4.90833	0.402106	0.201053	−	0.979580i	$-0.435564\pi$
$149$	4.90833	0.402106	0.201053	+	0.979580i	$0.435564\pi$
$150$	0	0
$151$	14.1194	1.14902	0.574511	−	0.818497i	$-0.305192\pi$
$151$	14.1194	1.14902	0.574511	+	0.818497i	$0.305192\pi$
$152$	0	0
$153$	2.90833	0.235124
$154$	0	0
$155$	22.1194	1.77668
$156$	0	0
$157$	−0.183346	−0.0146326	−0.00731631	−	0.999973i	$-0.502329\pi$
$157$	−0.183346	−0.0146326	−0.00731631	+	0.999973i	$0.502329\pi$
$158$	0	0
$159$	0.211103	0.0167415
$160$	0	0
$161$	−2.60555	−0.205346
$162$	0	0
$163$	−21.8167	−1.70881	−0.854406	−	0.519606i	$-0.826079\pi$
$163$	−21.8167	−1.70881	−0.854406	+	0.519606i	$0.826079\pi$
$164$	0	0
$165$	−4.60555	−0.358542
$166$	0	0
$167$	−2.11943	−0.164006	−0.0820032	−	0.996632i	$-0.526132\pi$
$167$	−2.11943	−0.164006	−0.0820032	+	0.996632i	$0.526132\pi$
$168$	0	0
$169$	30.6333	2.35641
$170$	0	0
$171$	17.4500	1.33443
$172$	0	0
$173$	4.90833	0.373173	0.186587	−	0.982439i	$-0.440258\pi$
$173$	4.90833	0.373173	0.186587	+	0.982439i	$0.440258\pi$
$174$	0	0
$175$	5.90833	0.446628
$176$	0	0
$177$	1.57779	0.118594
$178$	0	0
$179$	11.0917	0.829031	0.414515	−	0.910042i	$-0.363951\pi$
$179$	11.0917	0.829031	0.414515	+	0.910042i	$0.363951\pi$
$180$	0	0
$181$	−21.6333	−1.60799	−0.803996	−	0.594635i	$-0.797296\pi$
$181$	−21.6333	−1.60799	−0.803996	+	0.594635i	$0.797296\pi$
$182$	0	0
$183$	−1.30278	−0.0963039
$184$	0	0
$185$	−23.8167	−1.75104
$186$	0	0
$187$	−4.60555	−0.336791
$188$	0	0
$189$	−1.78890	−0.130123
$190$	0	0
$191$	−23.7250	−1.71668	−0.858340	−	0.513082i	$-0.828504\pi$
$191$	−23.7250	−1.71668	−0.858340	+	0.513082i	$0.828504\pi$
$192$	0	0
$193$	−5.39445	−0.388301	−0.194150	−	0.980972i	$-0.562195\pi$
$193$	−5.39445	−0.388301	−0.194150	+	0.980972i	$0.562195\pi$
$194$	0	0
$195$	6.60555	0.473033
$196$	0	0
$197$	−23.0278	−1.64066	−0.820330	−	0.571891i	$-0.806210\pi$
$197$	−23.0278	−1.64066	−0.820330	+	0.571891i	$0.806210\pi$
$198$	0	0
$199$	13.5139	0.957973	0.478987	−	0.877822i	$-0.341004\pi$
$199$	13.5139	0.957973	0.478987	+	0.877822i	$0.341004\pi$
$200$	0	0
$201$	0.816654	0.0576023
$202$	0	0
$203$	−8.60555	−0.603991
$204$	0	0
$205$	−31.4222	−2.19462
$206$	0	0
$207$	7.57779	0.526693
$208$	0	0
$209$	−27.6333	−1.91144
$210$	0	0
$211$	−27.0278	−1.86067	−0.930334	−	0.366714i	$-0.880483\pi$
$211$	−27.0278	−1.86067	−0.930334	+	0.366714i	$0.880483\pi$
$212$	0	0
$213$	−0.605551	−0.0414917
$214$	0	0
$215$	14.2111	0.969189
$216$	0	0
$217$	−6.69722	−0.454637
$218$	0	0
$219$	−4.09167	−0.276490
$220$	0	0
$221$	6.60555	0.444337
$222$	0	0
$223$	−21.6333	−1.44867	−0.724337	−	0.689446i	$-0.757854\pi$
$223$	−21.6333	−1.44867	−0.724337	+	0.689446i	$0.757854\pi$
$224$	0	0
$225$	−17.1833	−1.14556
$226$	0	0
$227$	0.0916731	0.00608456	0.00304228	−	0.999995i	$-0.499032\pi$
$227$	0.0916731	0.00608456	0.00304228	+	0.999995i	$0.499032\pi$
$228$	0	0
$229$	−5.81665	−0.384375	−0.192188	−	0.981358i	$-0.561558\pi$
$229$	−5.81665	−0.384375	−0.192188	+	0.981358i	$0.561558\pi$
$230$	0	0
$231$	1.39445	0.0917480
$232$	0	0
$233$	13.2111	0.865488	0.432744	−	0.901517i	$-0.357545\pi$
$233$	13.2111	0.865488	0.432744	+	0.901517i	$0.357545\pi$
$234$	0	0
$235$	−33.0278	−2.15449
$236$	0	0
$237$	−1.81665	−0.118004
$238$	0	0
$239$	−12.3028	−0.795800	−0.397900	−	0.917429i	$-0.630261\pi$
$239$	−12.3028	−0.795800	−0.397900	+	0.917429i	$0.630261\pi$
$240$	0	0
$241$	13.9083	0.895914	0.447957	−	0.894055i	$-0.352152\pi$
$241$	13.9083	0.895914	0.447957	+	0.894055i	$0.352152\pi$
$242$	0	0
$243$	7.84441	0.503219
$244$	0	0
$245$	−3.30278	−0.211007
$246$	0	0
$247$	39.6333	2.52181
$248$	0	0
$249$	2.78890	0.176739
$250$	0	0
$251$	8.78890	0.554750	0.277375	−	0.960762i	$-0.410536\pi$
$251$	8.78890	0.554750	0.277375	+	0.960762i	$0.410536\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	0	0
$255$	1.00000	0.0626224
$256$	0	0
$257$	−13.8167	−0.861859	−0.430930	−	0.902386i	$-0.641814\pi$
$257$	−13.8167	−0.861859	−0.430930	+	0.902386i	$0.641814\pi$
$258$	0	0
$259$	7.21110	0.448076
$260$	0	0
$261$	25.0278	1.54918
$262$	0	0
$263$	22.4222	1.38261	0.691306	−	0.722562i	$-0.257036\pi$
$263$	22.4222	1.38261	0.691306	+	0.722562i	$0.257036\pi$
$264$	0	0
$265$	−2.30278	−0.141458
$266$	0	0
$267$	−0.605551	−0.0370591
$268$	0	0
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	−31.8167	−1.93272	−0.966362	−	0.257186i	$-0.917205\pi$
$271$	−31.8167	−1.93272	−0.966362	+	0.257186i	$0.917205\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	27.2111	1.64089
$276$	0	0
$277$	25.0278	1.50377	0.751886	−	0.659293i	$-0.229144\pi$
$277$	25.0278	1.50377	0.751886	+	0.659293i	$0.229144\pi$
$278$	0	0
$279$	19.4777	1.16610
$280$	0	0
$281$	−16.6972	−0.996073	−0.498036	−	0.867156i	$-0.665945\pi$
$281$	−16.6972	−0.996073	−0.498036	+	0.867156i	$0.665945\pi$
$282$	0	0
$283$	6.69722	0.398109	0.199054	−	0.979988i	$-0.436213\pi$
$283$	6.69722	0.398109	0.199054	+	0.979988i	$0.436213\pi$
$284$	0	0
$285$	6.00000	0.355409
$286$	0	0
$287$	9.51388	0.561586
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−0.633308	−0.0371252
$292$	0	0
$293$	1.81665	0.106130	0.0530650	−	0.998591i	$-0.483101\pi$
$293$	1.81665	0.106130	0.0530650	+	0.998591i	$0.483101\pi$
$294$	0	0
$295$	−17.2111	−1.00207
$296$	0	0
$297$	−8.23886	−0.478067
$298$	0	0
$299$	17.2111	0.995344
$300$	0	0
$301$	−4.30278	−0.248008
$302$	0	0
$303$	−3.76114	−0.216072
$304$	0	0
$305$	14.2111	0.813725
$306$	0	0
$307$	−26.0000	−1.48390	−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000	−1.48390	−0.741949	+	0.670456i	$0.766098\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	−15.7250	−0.891682	−0.445841	−	0.895112i	$-0.647095\pi$
$311$	−15.7250	−0.891682	−0.445841	+	0.895112i	$0.647095\pi$
$312$	0	0
$313$	−5.72498	−0.323595	−0.161798	−	0.986824i	$-0.551729\pi$
$313$	−5.72498	−0.323595	−0.161798	+	0.986824i	$0.551729\pi$
$314$	0	0
$315$	9.60555	0.541212
$316$	0	0
$317$	−12.6056	−0.707998	−0.353999	−	0.935246i	$-0.615178\pi$
$317$	−12.6056	−0.707998	−0.353999	+	0.935246i	$0.615178\pi$
$318$	0	0
$319$	−39.6333	−2.21904
$320$	0	0
$321$	−2.84441	−0.158759
$322$	0	0
$323$	6.00000	0.333849
$324$	0	0
$325$	−39.0278	−2.16487
$326$	0	0
$327$	−0.605551	−0.0334871
$328$	0	0
$329$	10.0000	0.551318
$330$	0	0
$331$	−0.275019	−0.0151164	−0.00755821	−	0.999971i	$-0.502406\pi$
$331$	−0.275019	−0.0151164	−0.00755821	+	0.999971i	$0.502406\pi$
$332$	0	0
$333$	−20.9722	−1.14927
$334$	0	0
$335$	−8.90833	−0.486714
$336$	0	0
$337$	10.7889	0.587709	0.293854	−	0.955850i	$-0.405062\pi$
$337$	10.7889	0.587709	0.293854	+	0.955850i	$0.405062\pi$
$338$	0	0
$339$	5.81665	0.315917
$340$	0	0
$341$	−30.8444	−1.67032
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	2.60555	0.140278
$346$	0	0
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	0	0
$349$	19.2111	1.02835	0.514173	−	0.857686i	$-0.328099\pi$
$349$	19.2111	1.02835	0.514173	+	0.857686i	$0.328099\pi$
$350$	0	0
$351$	11.8167	0.630726
$352$	0	0
$353$	4.60555	0.245129	0.122564	−	0.992461i	$-0.460888\pi$
$353$	4.60555	0.245129	0.122564	+	0.992461i	$0.460888\pi$
$354$	0	0
$355$	6.60555	0.350586
$356$	0	0
$357$	−0.302776	−0.0160246
$358$	0	0
$359$	−17.4861	−0.922882	−0.461441	−	0.887171i	$-0.652667\pi$
$359$	−17.4861	−0.922882	−0.461441	+	0.887171i	$0.652667\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	3.09167	0.162271
$364$	0	0
$365$	44.6333	2.33621
$366$	0	0
$367$	26.7250	1.39503	0.697516	−	0.716569i	$-0.254288\pi$
$367$	26.7250	1.39503	0.697516	+	0.716569i	$0.254288\pi$
$368$	0	0
$369$	−27.6695	−1.44041
$370$	0	0
$371$	0.697224	0.0361981
$372$	0	0
$373$	12.3028	0.637014	0.318507	−	0.947921i	$-0.396819\pi$
$373$	12.3028	0.637014	0.318507	+	0.947921i	$0.396819\pi$
$374$	0	0
$375$	−0.908327	−0.0469058
$376$	0	0
$377$	56.8444	2.92764
$378$	0	0
$379$	−28.2389	−1.45053	−0.725266	−	0.688468i	$-0.758283\pi$
$379$	−28.2389	−1.45053	−0.725266	+	0.688468i	$0.758283\pi$
$380$	0	0
$381$	−1.55004	−0.0794109
$382$	0	0
$383$	−3.02776	−0.154711	−0.0773556	−	0.997004i	$-0.524648\pi$
$383$	−3.02776	−0.154711	−0.0773556	+	0.997004i	$0.524648\pi$
$384$	0	0
$385$	−15.2111	−0.775230
$386$	0	0
$387$	12.5139	0.636116
$388$	0	0
$389$	−14.0917	−0.714476	−0.357238	−	0.934013i	$-0.616282\pi$
$389$	−14.0917	−0.714476	−0.357238	+	0.934013i	$0.616282\pi$
$390$	0	0
$391$	2.60555	0.131768
$392$	0	0
$393$	−3.15559	−0.159178
$394$	0	0
$395$	19.8167	0.997084
$396$	0	0
$397$	−6.09167	−0.305732	−0.152866	−	0.988247i	$-0.548850\pi$
$397$	−6.09167	−0.305732	−0.152866	+	0.988247i	$0.548850\pi$
$398$	0	0
$399$	−1.81665	−0.0909464
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	44.2389	2.20369
$404$	0	0
$405$	−27.0278	−1.34302
$406$	0	0
$407$	33.2111	1.64621
$408$	0	0
$409$	−3.02776	−0.149713	−0.0748565	−	0.997194i	$-0.523850\pi$
$409$	−3.02776	−0.149713	−0.0748565	+	0.997194i	$0.523850\pi$
$410$	0	0
$411$	−0.936083	−0.0461736
$412$	0	0
$413$	5.21110	0.256422
$414$	0	0
$415$	−30.4222	−1.49337
$416$	0	0
$417$	−0.449961	−0.0220347
$418$	0	0
$419$	−1.30278	−0.0636448	−0.0318224	−	0.999494i	$-0.510131\pi$
$419$	−1.30278	−0.0636448	−0.0318224	+	0.999494i	$0.510131\pi$
$420$	0	0
$421$	−0.908327	−0.0442691	−0.0221346	−	0.999755i	$-0.507046\pi$
$421$	−0.908327	−0.0442691	−0.0221346	+	0.999755i	$0.507046\pi$
$422$	0	0
$423$	−29.0833	−1.41408
$424$	0	0
$425$	−5.90833	−0.286596
$426$	0	0
$427$	−4.30278	−0.208226
$428$	0	0
$429$	−9.21110	−0.444716
$430$	0	0
$431$	18.8444	0.907703	0.453852	−	0.891077i	$-0.350050\pi$
$431$	18.8444	0.907703	0.453852	+	0.891077i	$0.350050\pi$
$432$	0	0
$433$	11.6333	0.559061	0.279531	−	0.960137i	$-0.409821\pi$
$433$	11.6333	0.559061	0.279531	+	0.960137i	$0.409821\pi$
$434$	0	0
$435$	8.60555	0.412605
$436$	0	0
$437$	15.6333	0.747843
$438$	0	0
$439$	−0.302776	−0.0144507	−0.00722535	−	0.999974i	$-0.502300\pi$
$439$	−0.302776	−0.0144507	−0.00722535	+	0.999974i	$0.502300\pi$
$440$	0	0
$441$	−2.90833	−0.138492
$442$	0	0
$443$	−10.4222	−0.495174	−0.247587	−	0.968866i	$-0.579638\pi$
$443$	−10.4222	−0.495174	−0.247587	+	0.968866i	$0.579638\pi$
$444$	0	0
$445$	6.60555	0.313133
$446$	0	0
$447$	1.48612	0.0702911
$448$	0	0
$449$	−32.2389	−1.52145	−0.760723	−	0.649077i	$-0.775155\pi$
$449$	−32.2389	−1.52145	−0.760723	+	0.649077i	$0.775155\pi$
$450$	0	0
$451$	43.8167	2.06325
$452$	0	0
$453$	4.27502	0.200858
$454$	0	0
$455$	21.8167	1.02278
$456$	0	0
$457$	6.09167	0.284956	0.142478	−	0.989798i	$-0.454493\pi$
$457$	6.09167	0.284956	0.142478	+	0.989798i	$0.454493\pi$
$458$	0	0
$459$	1.78890	0.0834986
$460$	0	0
$461$	9.21110	0.429004	0.214502	−	0.976724i	$-0.431187\pi$
$461$	9.21110	0.429004	0.214502	+	0.976724i	$0.431187\pi$
$462$	0	0
$463$	36.5416	1.69823	0.849117	−	0.528205i	$-0.177135\pi$
$463$	36.5416	1.69823	0.849117	+	0.528205i	$0.177135\pi$
$464$	0	0
$465$	6.69722	0.310576
$466$	0	0
$467$	17.2111	0.796435	0.398217	−	0.917291i	$-0.369629\pi$
$467$	17.2111	0.796435	0.398217	+	0.917291i	$0.369629\pi$
$468$	0	0
$469$	2.69722	0.124546
$470$	0	0
$471$	−0.0555128	−0.00255789
$472$	0	0
$473$	−19.8167	−0.911171
$474$	0	0
$475$	−35.4500	−1.62656
$476$	0	0
$477$	−2.02776	−0.0928446
$478$	0	0
$479$	−24.3028	−1.11042	−0.555211	−	0.831709i	$-0.687363\pi$
$479$	−24.3028	−1.11042	−0.555211	+	0.831709i	$0.687363\pi$
$480$	0	0
$481$	−47.6333	−2.17189
$482$	0	0
$483$	−0.788897	−0.0358961
$484$	0	0
$485$	6.90833	0.313691
$486$	0	0
$487$	13.6333	0.617784	0.308892	−	0.951097i	$-0.400042\pi$
$487$	13.6333	0.617784	0.308892	+	0.951097i	$0.400042\pi$
$488$	0	0
$489$	−6.60555	−0.298713
$490$	0	0
$491$	30.6972	1.38535	0.692673	−	0.721252i	$-0.256433\pi$
$491$	30.6972	1.38535	0.692673	+	0.721252i	$0.256433\pi$
$492$	0	0
$493$	8.60555	0.387575
$494$	0	0
$495$	44.2389	1.98839
$496$	0	0
$497$	−2.00000	−0.0897123
$498$	0	0
$499$	−6.00000	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$500$	0	0
$501$	−0.641712	−0.0286696
$502$	0	0
$503$	10.8806	0.485141	0.242570	−	0.970134i	$-0.422009\pi$
$503$	10.8806	0.485141	0.242570	+	0.970134i	$0.422009\pi$
$504$	0	0
$505$	41.0278	1.82571
$506$	0	0
$507$	9.27502	0.411918
$508$	0	0
$509$	3.39445	0.150456	0.0752281	−	0.997166i	$-0.476031\pi$
$509$	3.39445	0.150456	0.0752281	+	0.997166i	$0.476031\pi$
$510$	0	0
$511$	−13.5139	−0.597819
$512$	0	0
$513$	10.7334	0.473891
$514$	0	0
$515$	−43.6333	−1.92271
$516$	0	0
$517$	46.0555	2.02552
$518$	0	0
$519$	1.48612	0.0652335
$520$	0	0
$521$	7.33053	0.321156	0.160578	−	0.987023i	$-0.448664\pi$
$521$	7.33053	0.321156	0.160578	+	0.987023i	$0.448664\pi$
$522$	0	0
$523$	17.0278	0.744572	0.372286	−	0.928118i	$-0.378574\pi$
$523$	17.0278	0.744572	0.372286	+	0.928118i	$0.378574\pi$
$524$	0	0
$525$	1.78890	0.0780739
$526$	0	0
$527$	6.69722	0.291736
$528$	0	0
$529$	−16.2111	−0.704831
$530$	0	0
$531$	−15.1556	−0.657697
$532$	0	0
$533$	−62.8444	−2.72209
$534$	0	0
$535$	31.0278	1.34145
$536$	0	0
$537$	3.35829	0.144921
$538$	0	0
$539$	4.60555	0.198375
$540$	0	0
$541$	3.81665	0.164091	0.0820454	−	0.996629i	$-0.473855\pi$
$541$	3.81665	0.164091	0.0820454	+	0.996629i	$0.473855\pi$
$542$	0	0
$543$	−6.55004	−0.281089
$544$	0	0
$545$	6.60555	0.282951
$546$	0	0
$547$	−39.4500	−1.68676	−0.843379	−	0.537319i	$-0.819437\pi$
$547$	−39.4500	−1.68676	−0.843379	+	0.537319i	$0.819437\pi$
$548$	0	0
$549$	12.5139	0.534079
$550$	0	0
$551$	51.6333	2.19965
$552$	0	0
$553$	−6.00000	−0.255146
$554$	0	0
$555$	−7.21110	−0.306094
$556$	0	0
$557$	11.2111	0.475030	0.237515	−	0.971384i	$-0.423667\pi$
$557$	11.2111	0.475030	0.237515	+	0.971384i	$0.423667\pi$
$558$	0	0
$559$	28.4222	1.20213
$560$	0	0
$561$	−1.39445	−0.0588737
$562$	0	0
$563$	32.2389	1.35871	0.679353	−	0.733812i	$-0.262261\pi$
$563$	32.2389	1.35871	0.679353	+	0.733812i	$0.262261\pi$
$564$	0	0
$565$	−63.4500	−2.66936
$566$	0	0
$567$	8.18335	0.343668
$568$	0	0
$569$	−41.5139	−1.74035	−0.870176	−	0.492741i	$-0.835995\pi$
$569$	−41.5139	−1.74035	−0.870176	+	0.492741i	$0.835995\pi$
$570$	0	0
$571$	−33.6333	−1.40751	−0.703755	−	0.710443i	$-0.748495\pi$
$571$	−33.6333	−1.40751	−0.703755	+	0.710443i	$0.748495\pi$
$572$	0	0
$573$	−7.18335	−0.300089
$574$	0	0
$575$	−15.3944	−0.641993
$576$	0	0
$577$	39.0278	1.62475	0.812373	−	0.583138i	$-0.198175\pi$
$577$	39.0278	1.62475	0.812373	+	0.583138i	$0.198175\pi$
$578$	0	0
$579$	−1.63331	−0.0678779
$580$	0	0
$581$	9.21110	0.382141
$582$	0	0
$583$	3.21110	0.132990
$584$	0	0
$585$	−63.4500	−2.62333
$586$	0	0
$587$	−17.2111	−0.710378	−0.355189	−	0.934794i	$-0.615584\pi$
$587$	−17.2111	−0.710378	−0.355189	+	0.934794i	$0.615584\pi$
$588$	0	0
$589$	40.1833	1.65573
$590$	0	0
$591$	−6.97224	−0.286800
$592$	0	0
$593$	9.63331	0.395593	0.197796	−	0.980243i	$-0.436622\pi$
$593$	9.63331	0.395593	0.197796	+	0.980243i	$0.436622\pi$
$594$	0	0
$595$	3.30278	0.135401
$596$	0	0
$597$	4.09167	0.167461
$598$	0	0
$599$	−32.5416	−1.32962	−0.664808	−	0.747015i	$-0.731486\pi$
$599$	−32.5416	−1.32962	−0.664808	+	0.747015i	$0.731486\pi$
$600$	0	0
$601$	19.2111	0.783637	0.391819	−	0.920042i	$-0.371846\pi$
$601$	19.2111	0.783637	0.391819	+	0.920042i	$0.371846\pi$
$602$	0	0
$603$	−7.84441	−0.319449
$604$	0	0
$605$	−33.7250	−1.37112
$606$	0	0
$607$	−16.0917	−0.653141	−0.326570	−	0.945173i	$-0.605893\pi$
$607$	−16.0917	−0.653141	−0.326570	+	0.945173i	$0.605893\pi$
$608$	0	0
$609$	−2.60555	−0.105582
$610$	0	0
$611$	−66.0555	−2.67232
$612$	0	0
$613$	−30.9361	−1.24950	−0.624748	−	0.780826i	$-0.714798\pi$
$613$	−30.9361	−1.24950	−0.624748	+	0.780826i	$0.714798\pi$
$614$	0	0
$615$	−9.51388	−0.383637
$616$	0	0
$617$	4.60555	0.185413	0.0927063	−	0.995694i	$-0.470448\pi$
$617$	4.60555	0.185413	0.0927063	+	0.995694i	$0.470448\pi$
$618$	0	0
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	0	0
$621$	4.66106	0.187042
$622$	0	0
$623$	−2.00000	−0.0801283
$624$	0	0
$625$	−19.6333	−0.785332
$626$	0	0
$627$	−8.36669	−0.334134
$628$	0	0
$629$	−7.21110	−0.287525
$630$	0	0
$631$	13.1194	0.522276	0.261138	−	0.965301i	$-0.415902\pi$
$631$	13.1194	0.522276	0.261138	+	0.965301i	$0.415902\pi$
$632$	0	0
$633$	−8.18335	−0.325259
$634$	0	0
$635$	16.9083	0.670986
$636$	0	0
$637$	−6.60555	−0.261721
$638$	0	0
$639$	5.81665	0.230103
$640$	0	0
$641$	−7.57779	−0.299305	−0.149652	−	0.988739i	$-0.547815\pi$
$641$	−7.57779	−0.299305	−0.149652	+	0.988739i	$0.547815\pi$
$642$	0	0
$643$	27.3305	1.07781	0.538905	−	0.842366i	$-0.318838\pi$
$643$	27.3305	1.07781	0.538905	+	0.842366i	$0.318838\pi$
$644$	0	0
$645$	4.30278	0.169422
$646$	0	0
$647$	−27.3944	−1.07699	−0.538493	−	0.842630i	$-0.681006\pi$
$647$	−27.3944	−1.07699	−0.538493	+	0.842630i	$0.681006\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	0	0
$651$	−2.02776	−0.0794740
$652$	0	0
$653$	2.00000	0.0782660	0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000	0.0782660	0.0391330	+	0.999234i	$0.487540\pi$
$654$	0	0
$655$	34.4222	1.34499
$656$	0	0
$657$	39.3028	1.53335
$658$	0	0
$659$	7.54163	0.293780	0.146890	−	0.989153i	$-0.453074\pi$
$659$	7.54163	0.293780	0.146890	+	0.989153i	$0.453074\pi$
$660$	0	0
$661$	15.6333	0.608065	0.304033	−	0.952662i	$-0.401667\pi$
$661$	15.6333	0.608065	0.304033	+	0.952662i	$0.401667\pi$
$662$	0	0
$663$	2.00000	0.0776736
$664$	0	0
$665$	19.8167	0.768457
$666$	0	0
$667$	22.4222	0.868191
$668$	0	0
$669$	−6.55004	−0.253239
$670$	0	0
$671$	−19.8167	−0.765013
$672$	0	0
$673$	−20.8444	−0.803493	−0.401746	−	0.915751i	$-0.631597\pi$
$673$	−20.8444	−0.803493	−0.401746	+	0.915751i	$0.631597\pi$
$674$	0	0
$675$	−10.5694	−0.406816
$676$	0	0
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	0	0
$679$	−2.09167	−0.0802710
$680$	0	0
$681$	0.0277564	0.00106363
$682$	0	0
$683$	−11.0278	−0.421965	−0.210983	−	0.977490i	$-0.567666\pi$
$683$	−11.0278	−0.421965	−0.210983	+	0.977490i	$0.567666\pi$
$684$	0	0
$685$	10.2111	0.390146
$686$	0	0
$687$	−1.76114	−0.0671917
$688$	0	0
$689$	−4.60555	−0.175458
$690$	0	0
$691$	4.30278	0.163685	0.0818426	−	0.996645i	$-0.473920\pi$
$691$	4.30278	0.163685	0.0818426	+	0.996645i	$0.473920\pi$
$692$	0	0
$693$	−13.3944	−0.508813
$694$	0	0
$695$	4.90833	0.186183
$696$	0	0
$697$	−9.51388	−0.360364
$698$	0	0
$699$	4.00000	0.151294
$700$	0	0
$701$	−11.2111	−0.423437	−0.211719	−	0.977331i	$-0.567906\pi$
$701$	−11.2111	−0.423437	−0.211719	+	0.977331i	$0.567906\pi$
$702$	0	0
$703$	−43.2666	−1.63183
$704$	0	0
$705$	−10.0000	−0.376622
$706$	0	0
$707$	−12.4222	−0.467185
$708$	0	0
$709$	−24.1833	−0.908225	−0.454112	−	0.890944i	$-0.650044\pi$
$709$	−24.1833	−0.908225	−0.454112	+	0.890944i	$0.650044\pi$
$710$	0	0
$711$	17.4500	0.654425
$712$	0	0
$713$	17.4500	0.653506
$714$	0	0
$715$	100.478	3.75765
$716$	0	0
$717$	−3.72498	−0.139112
$718$	0	0
$719$	−2.48612	−0.0927167	−0.0463583	−	0.998925i	$-0.514762\pi$
$719$	−2.48612	−0.0927167	−0.0463583	+	0.998925i	$0.514762\pi$
$720$	0	0
$721$	13.2111	0.492007
$722$	0	0
$723$	4.21110	0.156613
$724$	0	0
$725$	−50.8444	−1.88831
$726$	0	0
$727$	49.2666	1.82720	0.913599	−	0.406617i	$-0.133292\pi$
$727$	49.2666	1.82720	0.913599	+	0.406617i	$0.133292\pi$
$728$	0	0
$729$	−22.1749	−0.821294
$730$	0	0
$731$	4.30278	0.159144
$732$	0	0
$733$	−15.6333	−0.577429	−0.288715	−	0.957415i	$-0.593228\pi$
$733$	−15.6333	−0.577429	−0.288715	+	0.957415i	$0.593228\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	12.4222	0.457578
$738$	0	0
$739$	−24.9083	−0.916268	−0.458134	−	0.888883i	$-0.651482\pi$
$739$	−24.9083	−0.916268	−0.458134	+	0.888883i	$0.651482\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	0	0
$743$	−33.3944	−1.22512	−0.612562	−	0.790423i	$-0.709861\pi$
$743$	−33.3944	−1.22512	−0.612562	+	0.790423i	$0.709861\pi$
$744$	0	0
$745$	−16.2111	−0.593929
$746$	0	0
$747$	−26.7889	−0.980155
$748$	0	0
$749$	−9.39445	−0.343266
$750$	0	0
$751$	−51.2111	−1.86872	−0.934360	−	0.356331i	$-0.884028\pi$
$751$	−51.2111	−1.86872	−0.934360	+	0.356331i	$0.884028\pi$
$752$	0	0
$753$	2.66106	0.0969746
$754$	0	0
$755$	−46.6333	−1.69716
$756$	0	0
$757$	23.0917	0.839281	0.419641	−	0.907690i	$-0.362156\pi$
$757$	23.0917	0.839281	0.419641	+	0.907690i	$0.362156\pi$
$758$	0	0
$759$	−3.63331	−0.131881
$760$	0	0
$761$	46.6056	1.68945	0.844725	−	0.535201i	$-0.179764\pi$
$761$	46.6056	1.68945	0.844725	+	0.535201i	$0.179764\pi$
$762$	0	0
$763$	−2.00000	−0.0724049
$764$	0	0
$765$	−9.60555	−0.347289
$766$	0	0
$767$	−34.4222	−1.24291
$768$	0	0
$769$	11.8167	0.426119	0.213060	−	0.977039i	$-0.431657\pi$
$769$	11.8167	0.426119	0.213060	+	0.977039i	$0.431657\pi$
$770$	0	0
$771$	−4.18335	−0.150660
$772$	0	0
$773$	−6.84441	−0.246176	−0.123088	−	0.992396i	$-0.539280\pi$
$773$	−6.84441	−0.246176	−0.123088	+	0.992396i	$0.539280\pi$
$774$	0	0
$775$	−39.5694	−1.42137
$776$	0	0
$777$	2.18335	0.0783271
$778$	0	0
$779$	−57.0833	−2.04522
$780$	0	0
$781$	−9.21110	−0.329599
$782$	0	0
$783$	15.3944	0.550153
$784$	0	0
$785$	0.605551	0.0216131
$786$	0	0
$787$	−14.4222	−0.514096	−0.257048	−	0.966399i	$-0.582750\pi$
$787$	−14.4222	−0.514096	−0.257048	+	0.966399i	$0.582750\pi$
$788$	0	0
$789$	6.78890	0.241691
$790$	0	0
$791$	19.2111	0.683068
$792$	0	0
$793$	28.4222	1.00930
$794$	0	0
$795$	−0.697224	−0.0247280
$796$	0	0
$797$	14.7889	0.523850	0.261925	−	0.965088i	$-0.415643\pi$
$797$	14.7889	0.523850	0.261925	+	0.965088i	$0.415643\pi$
$798$	0	0
$799$	−10.0000	−0.353775
$800$	0	0
$801$	5.81665	0.205521
$802$	0	0
$803$	−62.2389	−2.19636
$804$	0	0
$805$	8.60555	0.303306
$806$	0	0
$807$	−1.81665	−0.0639492
$808$	0	0
$809$	45.4500	1.59794	0.798968	−	0.601374i	$-0.205380\pi$
$809$	45.4500	1.59794	0.798968	+	0.601374i	$0.205380\pi$
$810$	0	0
$811$	−38.3583	−1.34694	−0.673471	−	0.739214i	$-0.735197\pi$
$811$	−38.3583	−1.34694	−0.673471	+	0.739214i	$0.735197\pi$
$812$	0	0
$813$	−9.63331	−0.337855
$814$	0	0
$815$	72.0555	2.52399
$816$	0	0
$817$	25.8167	0.903210
$818$	0	0
$819$	19.2111	0.671290
$820$	0	0
$821$	40.4222	1.41074	0.705372	−	0.708837i	$-0.250780\pi$
$821$	40.4222	1.41074	0.705372	+	0.708837i	$0.250780\pi$
$822$	0	0
$823$	−5.02776	−0.175257	−0.0876283	−	0.996153i	$-0.527929\pi$
$823$	−5.02776	−0.175257	−0.0876283	+	0.996153i	$0.527929\pi$
$824$	0	0
$825$	8.23886	0.286840
$826$	0	0
$827$	18.8444	0.655284	0.327642	−	0.944802i	$-0.393746\pi$
$827$	18.8444	0.655284	0.327642	+	0.944802i	$0.393746\pi$
$828$	0	0
$829$	−29.3944	−1.02091	−0.510456	−	0.859904i	$-0.670523\pi$
$829$	−29.3944	−1.02091	−0.510456	+	0.859904i	$0.670523\pi$
$830$	0	0
$831$	7.57779	0.262871
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	7.00000	0.242245
$836$	0	0
$837$	11.9806	0.414112
$838$	0	0
$839$	21.2111	0.732289	0.366144	−	0.930558i	$-0.380678\pi$
$839$	21.2111	0.732289	0.366144	+	0.930558i	$0.380678\pi$
$840$	0	0
$841$	45.0555	1.55364
$842$	0	0
$843$	−5.05551	−0.174121
$844$	0	0
$845$	−101.175	−3.48052
$846$	0	0
$847$	10.2111	0.350858
$848$	0	0
$849$	2.02776	0.0695924
$850$	0	0
$851$	−18.7889	−0.644075
$852$	0	0
$853$	−19.2111	−0.657776	−0.328888	−	0.944369i	$-0.606674\pi$
$853$	−19.2111	−0.657776	−0.328888	+	0.944369i	$0.606674\pi$
$854$	0	0
$855$	−57.6333	−1.97102
$856$	0	0
$857$	12.5139	0.427466	0.213733	−	0.976892i	$-0.431438\pi$
$857$	12.5139	0.427466	0.213733	+	0.976892i	$0.431438\pi$
$858$	0	0
$859$	−45.6333	−1.55699	−0.778494	−	0.627652i	$-0.784016\pi$
$859$	−45.6333	−1.55699	−0.778494	+	0.627652i	$0.784016\pi$
$860$	0	0
$861$	2.88057	0.0981696
$862$	0	0
$863$	8.93608	0.304188	0.152094	−	0.988366i	$-0.451398\pi$
$863$	8.93608	0.304188	0.152094	+	0.988366i	$0.451398\pi$
$864$	0	0
$865$	−16.2111	−0.551194
$866$	0	0
$867$	0.302776	0.0102828
$868$	0	0
$869$	−27.6333	−0.937396
$870$	0	0
$871$	−17.8167	−0.603694
$872$	0	0
$873$	6.08327	0.205887
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	−6.18335	−0.208797	−0.104398	−	0.994536i	$-0.533292\pi$
$877$	−6.18335	−0.208797	−0.104398	+	0.994536i	$0.533292\pi$
$878$	0	0
$879$	0.550039	0.0185523
$880$	0	0
$881$	−7.11943	−0.239860	−0.119930	−	0.992782i	$-0.538267\pi$
$881$	−7.11943	−0.239860	−0.119930	+	0.992782i	$0.538267\pi$
$882$	0	0
$883$	55.7250	1.87529	0.937647	−	0.347588i	$-0.112999\pi$
$883$	55.7250	1.87529	0.937647	+	0.347588i	$0.112999\pi$
$884$	0	0
$885$	−5.21110	−0.175169
$886$	0	0
$887$	−12.2750	−0.412155	−0.206077	−	0.978536i	$-0.566070\pi$
$887$	−12.2750	−0.412155	−0.206077	+	0.978536i	$0.566070\pi$
$888$	0	0
$889$	−5.11943	−0.171700
$890$	0	0
$891$	37.6888	1.26262
$892$	0	0
$893$	−60.0000	−2.00782
$894$	0	0
$895$	−36.6333	−1.22452
$896$	0	0
$897$	5.21110	0.173994
$898$	0	0
$899$	57.6333	1.92218
$900$	0	0
$901$	−0.697224	−0.0232279
$902$	0	0
$903$	−1.30278	−0.0433537
$904$	0	0
$905$	71.4500	2.37508
$906$	0	0
$907$	−3.57779	−0.118799	−0.0593994	−	0.998234i	$-0.518919\pi$
$907$	−3.57779	−0.118799	−0.0593994	+	0.998234i	$0.518919\pi$
$908$	0	0
$909$	36.1278	1.19828
$910$	0	0
$911$	−10.4222	−0.345303	−0.172652	−	0.984983i	$-0.555233\pi$
$911$	−10.4222	−0.345303	−0.172652	+	0.984983i	$0.555233\pi$
$912$	0	0
$913$	42.4222	1.40397
$914$	0	0
$915$	4.30278	0.142245
$916$	0	0
$917$	−10.4222	−0.344172
$918$	0	0
$919$	−10.4861	−0.345905	−0.172953	−	0.984930i	$-0.555331\pi$
$919$	−10.4861	−0.345905	−0.172953	+	0.984930i	$0.555331\pi$
$920$	0	0
$921$	−7.87217	−0.259397
$922$	0	0
$923$	13.2111	0.434849
$924$	0	0
$925$	42.6056	1.40086
$926$	0	0
$927$	−38.4222	−1.26195
$928$	0	0
$929$	−24.6972	−0.810290	−0.405145	−	0.914253i	$-0.632779\pi$
$929$	−24.6972	−0.810290	−0.405145	+	0.914253i	$0.632779\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	−4.76114	−0.155873
$934$	0	0
$935$	15.2111	0.497456
$936$	0	0
$937$	−25.2111	−0.823611	−0.411805	−	0.911272i	$-0.635102\pi$
$937$	−25.2111	−0.823611	−0.411805	+	0.911272i	$0.635102\pi$
$938$	0	0
$939$	−1.73338	−0.0565669
$940$	0	0
$941$	−0.669468	−0.0218240	−0.0109120	−	0.999940i	$-0.503473\pi$
$941$	−0.669468	−0.0218240	−0.0109120	+	0.999940i	$0.503473\pi$
$942$	0	0
$943$	−24.7889	−0.807238
$944$	0	0
$945$	5.90833	0.192198
$946$	0	0
$947$	20.2389	0.657675	0.328837	−	0.944387i	$-0.393343\pi$
$947$	20.2389	0.657675	0.328837	+	0.944387i	$0.393343\pi$
$948$	0	0
$949$	89.2666	2.89772
$950$	0	0
$951$	−3.81665	−0.123763
$952$	0	0
$953$	26.7527	0.866606	0.433303	−	0.901248i	$-0.357348\pi$
$953$	26.7527	0.866606	0.433303	+	0.901248i	$0.357348\pi$
$954$	0	0
$955$	78.3583	2.53561
$956$	0	0
$957$	−12.0000	−0.387905
$958$	0	0
$959$	−3.09167	−0.0998353
$960$	0	0
$961$	13.8528	0.446865
$962$	0	0
$963$	27.3221	0.880443
$964$	0	0
$965$	17.8167	0.573538
$966$	0	0
$967$	−20.3028	−0.652893	−0.326447	−	0.945216i	$-0.605851\pi$
$967$	−20.3028	−0.652893	−0.326447	+	0.945216i	$0.605851\pi$
$968$	0	0
$969$	1.81665	0.0583593
$970$	0	0
$971$	−20.2389	−0.649496	−0.324748	−	0.945801i	$-0.605279\pi$
$971$	−20.2389	−0.649496	−0.324748	+	0.945801i	$0.605279\pi$
$972$	0	0
$973$	−1.48612	−0.0476429
$974$	0	0
$975$	−11.8167	−0.378436
$976$	0	0
$977$	12.5139	0.400354	0.200177	−	0.979760i	$-0.435848\pi$
$977$	12.5139	0.400354	0.200177	+	0.979760i	$0.435848\pi$
$978$	0	0
$979$	−9.21110	−0.294388
$980$	0	0
$981$	5.81665	0.185711
$982$	0	0
$983$	14.5139	0.462921	0.231460	−	0.972844i	$-0.425650\pi$
$983$	14.5139	0.462921	0.231460	+	0.972844i	$0.425650\pi$
$984$	0	0
$985$	76.0555	2.42333
$986$	0	0
$987$	3.02776	0.0963745
$988$	0	0
$989$	11.2111	0.356492
$990$	0	0
$991$	6.00000	0.190596	0.0952981	−	0.995449i	$-0.469620\pi$
$991$	6.00000	0.190596	0.0952981	+	0.995449i	$0.469620\pi$
$992$	0	0
$993$	−0.0832691	−0.00264247
$994$	0	0
$995$	−44.6333	−1.41497
$996$	0	0
$997$	32.9083	1.04222	0.521109	−	0.853490i	$-0.325519\pi$
$997$	32.9083	1.04222	0.521109	+	0.853490i	$0.325519\pi$
$998$	0	0
$999$	−12.8999	−0.408136

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	476.2.a.a.1.2	✓	2
3.2	odd	2		4284.2.a.p.1.2		2
4.3	odd	2		1904.2.a.l.1.1		2
7.6	odd	2		3332.2.a.n.1.1		2
8.3	odd	2		7616.2.a.m.1.2		2
8.5	even	2		7616.2.a.z.1.1		2
17.16	even	2		8092.2.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.a.a.1.2	✓	2	1.1	even	1	trivial
1904.2.a.l.1.1		2	4.3	odd	2
3332.2.a.n.1.1		2	7.6	odd	2
4284.2.a.p.1.2		2	3.2	odd	2
7616.2.a.m.1.2		2	8.3	odd	2
7616.2.a.z.1.1		2	8.5	even	2
8092.2.a.n.1.1		2	17.16	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:	\( N \)	\(=\)	\( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	476.a (trivial)

\(f(q)\)	\(=\)	\(q+0.302776 q^{3} -3.30278 q^{5} +1.00000 q^{7} -2.90833 q^{9} +O(q^{10})\)	\(q+0.302776 q^{3} -3.30278 q^{5} +1.00000 q^{7} -2.90833 q^{9} +4.60555 q^{11} -6.60555 q^{13} -1.00000 q^{15} -1.00000 q^{17} -6.00000 q^{19} +0.302776 q^{21} -2.60555 q^{23} +5.90833 q^{25} -1.78890 q^{27} -8.60555 q^{29} -6.69722 q^{31} +1.39445 q^{33} -3.30278 q^{35} +7.21110 q^{37} -2.00000 q^{39} +9.51388 q^{41} -4.30278 q^{43} +9.60555 q^{45} +10.0000 q^{47} +1.00000 q^{49} -0.302776 q^{51} +0.697224 q^{53} -15.2111 q^{55} -1.81665 q^{57} +5.21110 q^{59} -4.30278 q^{61} -2.90833 q^{63} +21.8167 q^{65} +2.69722 q^{67} -0.788897 q^{69} -2.00000 q^{71} -13.5139 q^{73} +1.78890 q^{75} +4.60555 q^{77} -6.00000 q^{79} +8.18335 q^{81} +9.21110 q^{83} +3.30278 q^{85} -2.60555 q^{87} -2.00000 q^{89} -6.60555 q^{91} -2.02776 q^{93} +19.8167 q^{95} -2.09167 q^{97} -13.3944 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9	\( 2 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9} + 2 q^{11} - 6 q^{13} - 2 q^{15} - 2 q^{17} - 12 q^{19} - 3 q^{21} + 2 q^{23} + q^{25} - 18 q^{27} - 10 q^{29} - 17 q^{31} + 10 q^{33} - 3 q^{35} - 4 q^{39} + q^{41} - 5 q^{43} + 12 q^{45} + 20 q^{47} + 2 q^{49} + 3 q^{51} + 5 q^{53} - 16 q^{55} + 18 q^{57} - 4 q^{59} - 5 q^{61} + 5 q^{63} + 22 q^{65} + 9 q^{67} - 16 q^{69} - 4 q^{71} - 9 q^{73} + 18 q^{75} + 2 q^{77} - 12 q^{79} + 38 q^{81} + 4 q^{83} + 3 q^{85} + 2 q^{87} - 4 q^{89} - 6 q^{91} + 32 q^{93} + 18 q^{95} - 15 q^{97} - 34 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9 + 2 * q^11 - 6 * q^13 - 2 * q^15 - 2 * q^17 - 12 * q^19 - 3 * q^21 + 2 * q^23 + q^25 - 18 * q^27 - 10 * q^29 - 17 * q^31 + 10 * q^33 - 3 * q^35 - 4 * q^39 + q^41 - 5 * q^43 + 12 * q^45 + 20 * q^47 + 2 * q^49 + 3 * q^51 + 5 * q^53 - 16 * q^55 + 18 * q^57 - 4 * q^59 - 5 * q^61 + 5 * q^63 + 22 * q^65 + 9 * q^67 - 16 * q^69 - 4 * q^71 - 9 * q^73 + 18 * q^75 + 2 * q^77 - 12 * q^79 + 38 * q^81 + 4 * q^83 + 3 * q^85 + 2 * q^87 - 4 * q^89 - 6 * q^91 + 32 * q^93 + 18 * q^95 - 15 * q^97 - 34 * q^99

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