LMFDB - Embedded newform 2576.2.a.bc.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2576,2,Mod(1,2576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2576 = 2^{4} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2576.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:	$20.5694635607$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.8580816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 12x^{3} + 10x^{2} + 20x + 6 $ x^5 - x^4 - 12x^3 + 10x^2 + 20*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 644)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.465082$ of defining polynomial
Character	$\chi$	$=$	2576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.465082 q^{3} +3.21918 q^{5} -1.00000 q^{7} -2.78370 q^{9} +O(q^{10})$	$q+0.465082 q^{3} +3.21918 q^{5} -1.00000 q^{7} -2.78370 q^{9} -1.43298 q^{11} +4.35072 q^{13} +1.49719 q^{15} +3.88564 q^{17} +4.37416 q^{19} -0.465082 q^{21} -1.00000 q^{23} +5.36314 q^{25} -2.68990 q^{27} -7.29190 q^{29} -1.13154 q^{31} -0.666453 q^{33} -3.21918 q^{35} +11.3043 q^{37} +2.02344 q^{39} +0.579445 q^{41} +12.0058 q^{43} -8.96124 q^{45} -8.87423 q^{47} +1.00000 q^{49} +1.80714 q^{51} +14.0700 q^{53} -4.61302 q^{55} +2.03434 q^{57} -2.48852 q^{59} +2.08514 q^{61} +2.78370 q^{63} +14.0058 q^{65} +7.64262 q^{67} -0.465082 q^{69} +5.42055 q^{71} +0.982187 q^{73} +2.49430 q^{75} +1.43298 q^{77} -13.1757 q^{79} +7.10007 q^{81} +12.1703 q^{83} +12.5086 q^{85} -3.39133 q^{87} -12.6119 q^{89} -4.35072 q^{91} -0.526257 q^{93} +14.0812 q^{95} -9.22495 q^{97} +3.98898 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} + 4 q^{5} - 5 q^{7} + 10 q^{9}+O(q^{10}) $ 5 * q - q^3 + 4 * q^5 - 5 * q^7 + 10 * q^9	$ 5 q - q^{3} + 4 q^{5} - 5 q^{7} + 10 q^{9} - 2 q^{11} + 3 q^{13} + 6 q^{15} + 4 q^{17} - 6 q^{19} + q^{21} - 5 q^{23} + 15 q^{25} - q^{27} + 5 q^{29} + q^{31} - 4 q^{35} + 22 q^{37} + q^{39} + 15 q^{41} - 12 q^{43} + 36 q^{45} + 21 q^{47} + 5 q^{49} - 24 q^{51} + 2 q^{53} + 22 q^{55} - 34 q^{57} - 12 q^{61} - 10 q^{63} - 2 q^{65} - 22 q^{67} + q^{69} + 15 q^{71} + 17 q^{73} + 47 q^{75} + 2 q^{77} - 2 q^{79} + 37 q^{81} + 16 q^{83} - 8 q^{85} + 33 q^{87} - 24 q^{89} - 3 q^{91} + 5 q^{93} + 8 q^{95} + 38 q^{97} + 36 q^{99}+O(q^{100}) $ 5 * q - q^3 + 4 * q^5 - 5 * q^7 + 10 * q^9 - 2 * q^11 + 3 * q^13 + 6 * q^15 + 4 * q^17 - 6 * q^19 + q^21 - 5 * q^23 + 15 * q^25 - q^27 + 5 * q^29 + q^31 - 4 * q^35 + 22 * q^37 + q^39 + 15 * q^41 - 12 * q^43 + 36 * q^45 + 21 * q^47 + 5 * q^49 - 24 * q^51 + 2 * q^53 + 22 * q^55 - 34 * q^57 - 12 * q^61 - 10 * q^63 - 2 * q^65 - 22 * q^67 + q^69 + 15 * q^71 + 17 * q^73 + 47 * q^75 + 2 * q^77 - 2 * q^79 + 37 * q^81 + 16 * q^83 - 8 * q^85 + 33 * q^87 - 24 * q^89 - 3 * q^91 + 5 * q^93 + 8 * q^95 + 38 * q^97 + 36 * 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
$3$	0.465082	0.268515	0.134258	−	0.990946i	$-0.457135\pi$
$3$	0.465082	0.268515	0.134258	+	0.990946i	$0.457135\pi$
$4$	0	0
$5$	3.21918	1.43966	0.719831	−	0.694149i	$-0.244219\pi$
$5$	3.21918	1.43966	0.719831	+	0.694149i	$0.244219\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.78370	−0.927899
$10$	0	0
$11$	−1.43298	−0.432060	−0.216030	−	0.976387i	$-0.569311\pi$
$11$	−1.43298	−0.432060	−0.216030	+	0.976387i	$0.569311\pi$
$12$	0	0
$13$	4.35072	1.20667	0.603336	−	0.797487i	$-0.293838\pi$
$13$	4.35072	1.20667	0.603336	+	0.797487i	$0.293838\pi$
$14$	0	0
$15$	1.49719	0.386572
$16$	0	0
$17$	3.88564	0.942405	0.471203	−	0.882025i	$-0.343820\pi$
$17$	3.88564	0.942405	0.471203	+	0.882025i	$0.343820\pi$
$18$	0	0
$19$	4.37416	1.00350	0.501751	−	0.865012i	$-0.332689\pi$
$19$	4.37416	1.00350	0.501751	+	0.865012i	$0.332689\pi$
$20$	0	0
$21$	−0.465082	−0.101489
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	5.36314	1.07263
$26$	0	0
$27$	−2.68990	−0.517671
$28$	0	0
$29$	−7.29190	−1.35407	−0.677036	−	0.735950i	$-0.736736\pi$
$29$	−7.29190	−1.35407	−0.677036	+	0.735950i	$0.736736\pi$
$30$	0	0
$31$	−1.13154	−0.203230	−0.101615	−	0.994824i	$-0.532401\pi$
$31$	−1.13154	−0.203230	−0.101615	+	0.994824i	$0.532401\pi$
$32$	0	0
$33$	−0.666453	−0.116015
$34$	0	0
$35$	−3.21918	−0.544141
$36$	0	0
$37$	11.3043	1.85842	0.929210	−	0.369552i	$-0.120489\pi$
$37$	11.3043	1.85842	0.929210	+	0.369552i	$0.120489\pi$
$38$	0	0
$39$	2.02344	0.324010
$40$	0	0
$41$	0.579445	0.0904942	0.0452471	−	0.998976i	$-0.485592\pi$
$41$	0.579445	0.0904942	0.0452471	+	0.998976i	$0.485592\pi$
$42$	0	0
$43$	12.0058	1.83086	0.915431	−	0.402475i	$-0.131850\pi$
$43$	12.0058	1.83086	0.915431	+	0.402475i	$0.131850\pi$
$44$	0	0
$45$	−8.96124	−1.33586
$46$	0	0
$47$	−8.87423	−1.29444	−0.647220	−	0.762304i	$-0.724068\pi$
$47$	−8.87423	−1.29444	−0.647220	+	0.762304i	$0.724068\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.80714	0.253050
$52$	0	0
$53$	14.0700	1.93266	0.966330	−	0.257308i	$-0.0828354\pi$
$53$	14.0700	1.93266	0.966330	+	0.257308i	$0.0828354\pi$
$54$	0	0
$55$	−4.61302	−0.622020
$56$	0	0
$57$	2.03434	0.269456
$58$	0	0
$59$	−2.48852	−0.323978	−0.161989	−	0.986793i	$-0.551791\pi$
$59$	−2.48852	−0.323978	−0.161989	+	0.986793i	$0.551791\pi$
$60$	0	0
$61$	2.08514	0.266975	0.133488	−	0.991050i	$-0.457382\pi$
$61$	2.08514	0.266975	0.133488	+	0.991050i	$0.457382\pi$
$62$	0	0
$63$	2.78370	0.350713
$64$	0	0
$65$	14.0058	1.73720
$66$	0	0
$67$	7.64262	0.933695	0.466847	−	0.884338i	$-0.345390\pi$
$67$	7.64262	0.933695	0.466847	+	0.884338i	$0.345390\pi$
$68$	0	0
$69$	−0.465082	−0.0559893
$70$	0	0
$71$	5.42055	0.643301	0.321651	−	0.946858i	$-0.395762\pi$
$71$	5.42055	0.643301	0.321651	+	0.946858i	$0.395762\pi$
$72$	0	0
$73$	0.982187	0.114956	0.0574782	−	0.998347i	$-0.481694\pi$
$73$	0.982187	0.114956	0.0574782	+	0.998347i	$0.481694\pi$
$74$	0	0
$75$	2.49430	0.288017
$76$	0	0
$77$	1.43298	0.163303
$78$	0	0
$79$	−13.1757	−1.48238	−0.741189	−	0.671296i	$-0.765738\pi$
$79$	−13.1757	−1.48238	−0.741189	+	0.671296i	$0.765738\pi$
$80$	0	0
$81$	7.10007	0.788897
$82$	0	0
$83$	12.1703	1.33586	0.667931	−	0.744223i	$-0.267180\pi$
$83$	12.1703	1.33586	0.667931	+	0.744223i	$0.267180\pi$
$84$	0	0
$85$	12.5086	1.35675
$86$	0	0
$87$	−3.39133	−0.363589
$88$	0	0
$89$	−12.6119	−1.33686	−0.668431	−	0.743774i	$-0.733034\pi$
$89$	−12.6119	−1.33686	−0.668431	+	0.743774i	$0.733034\pi$
$90$	0	0
$91$	−4.35072	−0.456079
$92$	0	0
$93$	−0.526257	−0.0545703
$94$	0	0
$95$	14.0812	1.44470
$96$	0	0
$97$	−9.22495	−0.936652	−0.468326	−	0.883556i	$-0.655143\pi$
$97$	−9.22495	−0.936652	−0.468326	+	0.883556i	$0.655143\pi$
$98$	0	0
$99$	3.98898	0.400908
$100$	0	0
$101$	−10.4976	−1.04455	−0.522273	−	0.852778i	$-0.674916\pi$
$101$	−10.4976	−1.04455	−0.522273	+	0.852778i	$0.674916\pi$
$102$	0	0
$103$	18.2096	1.79425	0.897125	−	0.441778i	$-0.145652\pi$
$103$	18.2096	1.79425	0.897125	+	0.441778i	$0.145652\pi$
$104$	0	0
$105$	−1.49719	−0.146110
$106$	0	0
$107$	16.1811	1.56428	0.782141	−	0.623101i	$-0.214128\pi$
$107$	16.1811	1.56428	0.782141	+	0.623101i	$0.214128\pi$
$108$	0	0
$109$	5.06984	0.485602	0.242801	−	0.970076i	$-0.421934\pi$
$109$	5.06984	0.485602	0.242801	+	0.970076i	$0.421934\pi$
$110$	0	0
$111$	5.25744	0.499014
$112$	0	0
$113$	−16.5618	−1.55800	−0.779000	−	0.627024i	$-0.784273\pi$
$113$	−16.5618	−1.55800	−0.779000	+	0.627024i	$0.784273\pi$
$114$	0	0
$115$	−3.21918	−0.300190
$116$	0	0
$117$	−12.1111	−1.11967
$118$	0	0
$119$	−3.88564	−0.356196
$120$	0	0
$121$	−8.94657	−0.813325
$122$	0	0
$123$	0.269490	0.0242991
$124$	0	0
$125$	1.16903	0.104561
$126$	0	0
$127$	−10.8533	−0.963074	−0.481537	−	0.876426i	$-0.659921\pi$
$127$	−10.8533	−0.963074	−0.481537	+	0.876426i	$0.659921\pi$
$128$	0	0
$129$	5.58367	0.491615
$130$	0	0
$131$	−1.85782	−0.162319	−0.0811594	−	0.996701i	$-0.525862\pi$
$131$	−1.85782	−0.162319	−0.0811594	+	0.996701i	$0.525862\pi$
$132$	0	0
$133$	−4.37416	−0.379288
$134$	0	0
$135$	−8.65927	−0.745271
$136$	0	0
$137$	14.1454	1.20853	0.604263	−	0.796785i	$-0.293468\pi$
$137$	14.1454	1.20853	0.604263	+	0.796785i	$0.293468\pi$
$138$	0	0
$139$	4.44023	0.376616	0.188308	−	0.982110i	$-0.439700\pi$
$139$	4.44023	0.376616	0.188308	+	0.982110i	$0.439700\pi$
$140$	0	0
$141$	−4.12725	−0.347577
$142$	0	0
$143$	−6.23449	−0.521354
$144$	0	0
$145$	−23.4740	−1.94941
$146$	0	0
$147$	0.465082	0.0383593
$148$	0	0
$149$	−8.20964	−0.672560	−0.336280	−	0.941762i	$-0.609169\pi$
$149$	−8.20964	−0.672560	−0.336280	+	0.941762i	$0.609169\pi$
$150$	0	0
$151$	−1.34509	−0.109462	−0.0547309	−	0.998501i	$-0.517430\pi$
$151$	−1.34509	−0.109462	−0.0547309	+	0.998501i	$0.517430\pi$
$152$	0	0
$153$	−10.8164	−0.874457
$154$	0	0
$155$	−3.64262	−0.292582
$156$	0	0
$157$	−8.76237	−0.699313	−0.349657	−	0.936878i	$-0.613702\pi$
$157$	−8.76237	−0.699313	−0.349657	+	0.936878i	$0.613702\pi$
$158$	0	0
$159$	6.54369	0.518949
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−17.1632	−1.34433	−0.672165	−	0.740402i	$-0.734635\pi$
$163$	−17.1632	−1.34433	−0.672165	+	0.740402i	$0.734635\pi$
$164$	0	0
$165$	−2.14544	−0.167022
$166$	0	0
$167$	21.3895	1.65517	0.827583	−	0.561343i	$-0.189715\pi$
$167$	21.3895	1.65517	0.827583	+	0.561343i	$0.189715\pi$
$168$	0	0
$169$	5.92876	0.456058
$170$	0	0
$171$	−12.1763	−0.931149
$172$	0	0
$173$	4.93593	0.375272	0.187636	−	0.982239i	$-0.439918\pi$
$173$	4.93593	0.375272	0.187636	+	0.982239i	$0.439918\pi$
$174$	0	0
$175$	−5.36314	−0.405416
$176$	0	0
$177$	−1.15737	−0.0869932
$178$	0	0
$179$	−0.515239	−0.0385108	−0.0192554	−	0.999815i	$-0.506130\pi$
$179$	−0.515239	−0.0385108	−0.0192554	+	0.999815i	$0.506130\pi$
$180$	0	0
$181$	11.6818	0.868298	0.434149	−	0.900841i	$-0.357049\pi$
$181$	11.6818	0.868298	0.434149	+	0.900841i	$0.357049\pi$
$182$	0	0
$183$	0.969763	0.0716869
$184$	0	0
$185$	36.3907	2.67550
$186$	0	0
$187$	−5.56804	−0.407175
$188$	0	0
$189$	2.68990	0.195661
$190$	0	0
$191$	9.57241	0.692635	0.346318	−	0.938117i	$-0.387432\pi$
$191$	9.57241	0.692635	0.346318	+	0.938117i	$0.387432\pi$
$192$	0	0
$193$	−2.70285	−0.194555	−0.0972775	−	0.995257i	$-0.531013\pi$
$193$	−2.70285	−0.194555	−0.0972775	+	0.995257i	$0.531013\pi$
$194$	0	0
$195$	6.51383	0.466465
$196$	0	0
$197$	−1.91235	−0.136249	−0.0681247	−	0.997677i	$-0.521702\pi$
$197$	−1.91235	−0.136249	−0.0681247	+	0.997677i	$0.521702\pi$
$198$	0	0
$199$	14.4334	1.02315	0.511577	−	0.859238i	$-0.329062\pi$
$199$	14.4334	1.02315	0.511577	+	0.859238i	$0.329062\pi$
$200$	0	0
$201$	3.55445	0.250711
$202$	0	0
$203$	7.29190	0.511791
$204$	0	0
$205$	1.86534	0.130281
$206$	0	0
$207$	2.78370	0.193480
$208$	0	0
$209$	−6.26808	−0.433572
$210$	0	0
$211$	−18.7656	−1.29188	−0.645940	−	0.763388i	$-0.723534\pi$
$211$	−18.7656	−1.29188	−0.645940	+	0.763388i	$0.723534\pi$
$212$	0	0
$213$	2.52100	0.172736
$214$	0	0
$215$	38.6488	2.63582
$216$	0	0
$217$	1.13154	0.0768136
$218$	0	0
$219$	0.456798	0.0308675
$220$	0	0
$221$	16.9053	1.13717
$222$	0	0
$223$	−1.48289	−0.0993020	−0.0496510	−	0.998767i	$-0.515811\pi$
$223$	−1.48289	−0.0993020	−0.0496510	+	0.998767i	$0.515811\pi$
$224$	0	0
$225$	−14.9294	−0.995292
$226$	0	0
$227$	−1.33291	−0.0884681	−0.0442341	−	0.999021i	$-0.514085\pi$
$227$	−1.33291	−0.0884681	−0.0442341	+	0.999021i	$0.514085\pi$
$228$	0	0
$229$	6.02531	0.398164	0.199082	−	0.979983i	$-0.436204\pi$
$229$	6.02531	0.398164	0.199082	+	0.979983i	$0.436204\pi$
$230$	0	0
$231$	0.666453	0.0438494
$232$	0	0
$233$	−9.60687	−0.629367	−0.314683	−	0.949197i	$-0.601898\pi$
$233$	−9.60687	−0.629367	−0.314683	+	0.949197i	$0.601898\pi$
$234$	0	0
$235$	−28.5678	−1.86356
$236$	0	0
$237$	−6.12777	−0.398042
$238$	0	0
$239$	6.78332	0.438777	0.219388	−	0.975638i	$-0.429594\pi$
$239$	6.78332	0.438777	0.219388	+	0.975638i	$0.429594\pi$
$240$	0	0
$241$	16.5871	1.06847	0.534234	−	0.845337i	$-0.320600\pi$
$241$	16.5871	1.06847	0.534234	+	0.845337i	$0.320600\pi$
$242$	0	0
$243$	11.3718	0.729502
$244$	0	0
$245$	3.21918	0.205666
$246$	0	0
$247$	19.0307	1.21090
$248$	0	0
$249$	5.66018	0.358700
$250$	0	0
$251$	−18.0008	−1.13620	−0.568099	−	0.822960i	$-0.692321\pi$
$251$	−18.0008	−1.13620	−0.568099	+	0.822960i	$0.692321\pi$
$252$	0	0
$253$	1.43298	0.0900906
$254$	0	0
$255$	5.81752	0.364307
$256$	0	0
$257$	−19.0828	−1.19035	−0.595175	−	0.803596i	$-0.702917\pi$
$257$	−19.0828	−1.19035	−0.595175	+	0.803596i	$0.702917\pi$
$258$	0	0
$259$	−11.3043	−0.702417
$260$	0	0
$261$	20.2985	1.25644
$262$	0	0
$263$	2.89430	0.178470	0.0892350	−	0.996011i	$-0.471558\pi$
$263$	2.89430	0.178470	0.0892350	+	0.996011i	$0.471558\pi$
$264$	0	0
$265$	45.2938	2.78238
$266$	0	0
$267$	−5.86558	−0.358968
$268$	0	0
$269$	−24.2637	−1.47938	−0.739692	−	0.672946i	$-0.765029\pi$
$269$	−24.2637	−1.47938	−0.739692	+	0.672946i	$0.765029\pi$
$270$	0	0
$271$	9.48289	0.576045	0.288022	−	0.957624i	$-0.407002\pi$
$271$	9.48289	0.576045	0.288022	+	0.957624i	$0.407002\pi$
$272$	0	0
$273$	−2.02344	−0.122464
$274$	0	0
$275$	−7.68527	−0.463439
$276$	0	0
$277$	27.6661	1.66229	0.831146	−	0.556054i	$-0.187685\pi$
$277$	27.6661	1.66229	0.831146	+	0.556054i	$0.187685\pi$
$278$	0	0
$279$	3.14985	0.188577
$280$	0	0
$281$	−13.4891	−0.804693	−0.402347	−	0.915487i	$-0.631805\pi$
$281$	−13.4891	−0.804693	−0.402347	+	0.915487i	$0.631805\pi$
$282$	0	0
$283$	0.759880	0.0451702	0.0225851	−	0.999745i	$-0.492810\pi$
$283$	0.759880	0.0451702	0.0225851	+	0.999745i	$0.492810\pi$
$284$	0	0
$285$	6.54893	0.387925
$286$	0	0
$287$	−0.579445	−0.0342036
$288$	0	0
$289$	−1.90183	−0.111872
$290$	0	0
$291$	−4.29036	−0.251505
$292$	0	0
$293$	−17.9314	−1.04756	−0.523782	−	0.851853i	$-0.675479\pi$
$293$	−17.9314	−1.04756	−0.523782	+	0.851853i	$0.675479\pi$
$294$	0	0
$295$	−8.01102	−0.466419
$296$	0	0
$297$	3.85456	0.223665
$298$	0	0
$299$	−4.35072	−0.251609
$300$	0	0
$301$	−12.0058	−0.692001
$302$	0	0
$303$	−4.88223	−0.280477
$304$	0	0
$305$	6.71246	0.384354
$306$	0	0
$307$	0.641145	0.0365921	0.0182960	−	0.999833i	$-0.494176\pi$
$307$	0.641145	0.0365921	0.0182960	+	0.999833i	$0.494176\pi$
$308$	0	0
$309$	8.46898	0.481783
$310$	0	0
$311$	8.13604	0.461353	0.230676	−	0.973031i	$-0.425906\pi$
$311$	8.13604	0.461353	0.230676	+	0.973031i	$0.425906\pi$
$312$	0	0
$313$	1.11999	0.0633057	0.0316529	−	0.999499i	$-0.489923\pi$
$313$	1.11999	0.0633057	0.0316529	+	0.999499i	$0.489923\pi$
$314$	0	0
$315$	8.96124	0.504908
$316$	0	0
$317$	10.5948	0.595064	0.297532	−	0.954712i	$-0.403836\pi$
$317$	10.5948	0.595064	0.297532	+	0.954712i	$0.403836\pi$
$318$	0	0
$319$	10.4491	0.585040
$320$	0	0
$321$	7.52552	0.420034
$322$	0	0
$323$	16.9964	0.945705
$324$	0	0
$325$	23.3335	1.29431
$326$	0	0
$327$	2.35789	0.130392
$328$	0	0
$329$	8.87423	0.489252
$330$	0	0
$331$	−20.7357	−1.13973	−0.569867	−	0.821737i	$-0.693005\pi$
$331$	−20.7357	−1.13973	−0.569867	+	0.821737i	$0.693005\pi$
$332$	0	0
$333$	−31.4678	−1.72443
$334$	0	0
$335$	24.6030	1.34421
$336$	0	0
$337$	−6.65082	−0.362293	−0.181147	−	0.983456i	$-0.557981\pi$
$337$	−6.65082	−0.362293	−0.181147	+	0.983456i	$0.557981\pi$
$338$	0	0
$339$	−7.70258	−0.418347
$340$	0	0
$341$	1.62147	0.0878074
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.49719	−0.0806057
$346$	0	0
$347$	7.84301	0.421035	0.210517	−	0.977590i	$-0.432485\pi$
$347$	7.84301	0.421035	0.210517	+	0.977590i	$0.432485\pi$
$348$	0	0
$349$	19.4206	1.03956	0.519779	−	0.854301i	$-0.326014\pi$
$349$	19.4206	1.03956	0.519779	+	0.854301i	$0.326014\pi$
$350$	0	0
$351$	−11.7030	−0.624659
$352$	0	0
$353$	1.83689	0.0977676	0.0488838	−	0.998804i	$-0.484434\pi$
$353$	1.83689	0.0977676	0.0488838	+	0.998804i	$0.484434\pi$
$354$	0	0
$355$	17.4498	0.926137
$356$	0	0
$357$	−1.80714	−0.0956440
$358$	0	0
$359$	−17.6318	−0.930573	−0.465287	−	0.885160i	$-0.654049\pi$
$359$	−17.6318	−0.930573	−0.465287	+	0.885160i	$0.654049\pi$
$360$	0	0
$361$	0.133289	0.00701520
$362$	0	0
$363$	−4.16089	−0.218390
$364$	0	0
$365$	3.16184	0.165498
$366$	0	0
$367$	−34.7772	−1.81535	−0.907677	−	0.419670i	$-0.862146\pi$
$367$	−34.7772	−1.81535	−0.907677	+	0.419670i	$0.862146\pi$
$368$	0	0
$369$	−1.61300	−0.0839695
$370$	0	0
$371$	−14.0700	−0.730476
$372$	0	0
$373$	37.2048	1.92639	0.963195	−	0.268804i	$-0.0866285\pi$
$373$	37.2048	1.92639	0.963195	+	0.268804i	$0.0866285\pi$
$374$	0	0
$375$	0.543694	0.0280762
$376$	0	0
$377$	−31.7250	−1.63392
$378$	0	0
$379$	−6.26459	−0.321791	−0.160895	−	0.986971i	$-0.551438\pi$
$379$	−6.26459	−0.321791	−0.160895	+	0.986971i	$0.551438\pi$
$380$	0	0
$381$	−5.04767	−0.258600
$382$	0	0
$383$	−11.6424	−0.594898	−0.297449	−	0.954738i	$-0.596136\pi$
$383$	−11.6424	−0.594898	−0.297449	+	0.954738i	$0.596136\pi$
$384$	0	0
$385$	4.61302	0.235101
$386$	0	0
$387$	−33.4204	−1.69886
$388$	0	0
$389$	−1.77409	−0.0899498	−0.0449749	−	0.998988i	$-0.514321\pi$
$389$	−1.77409	−0.0899498	−0.0449749	+	0.998988i	$0.514321\pi$
$390$	0	0
$391$	−3.88564	−0.196505
$392$	0	0
$393$	−0.864041	−0.0435851
$394$	0	0
$395$	−42.4149	−2.13413
$396$	0	0
$397$	−24.9937	−1.25440	−0.627199	−	0.778859i	$-0.715799\pi$
$397$	−24.9937	−1.25440	−0.627199	+	0.778859i	$0.715799\pi$
$398$	0	0
$399$	−2.03434	−0.101845
$400$	0	0
$401$	−6.70645	−0.334904	−0.167452	−	0.985880i	$-0.553554\pi$
$401$	−6.70645	−0.334904	−0.167452	+	0.985880i	$0.553554\pi$
$402$	0	0
$403$	−4.92299	−0.245232
$404$	0	0
$405$	22.8564	1.13575
$406$	0	0
$407$	−16.1989	−0.802948
$408$	0	0
$409$	−31.7644	−1.57065	−0.785324	−	0.619086i	$-0.787503\pi$
$409$	−31.7644	−1.57065	−0.785324	+	0.619086i	$0.787503\pi$
$410$	0	0
$411$	6.57879	0.324508
$412$	0	0
$413$	2.48852	0.122452
$414$	0	0
$415$	39.1784	1.92319
$416$	0	0
$417$	2.06507	0.101127
$418$	0	0
$419$	−37.8939	−1.85124	−0.925619	−	0.378457i	$-0.876455\pi$
$419$	−37.8939	−1.85124	−0.925619	+	0.378457i	$0.876455\pi$
$420$	0	0
$421$	−29.2516	−1.42564	−0.712819	−	0.701348i	$-0.752582\pi$
$421$	−29.2516	−1.42564	−0.712819	+	0.701348i	$0.752582\pi$
$422$	0	0
$423$	24.7032	1.20111
$424$	0	0
$425$	20.8392	1.01085
$426$	0	0
$427$	−2.08514	−0.100907
$428$	0	0
$429$	−2.89955	−0.139992
$430$	0	0
$431$	−13.6251	−0.656296	−0.328148	−	0.944626i	$-0.606424\pi$
$431$	−13.6251	−0.656296	−0.328148	+	0.944626i	$0.606424\pi$
$432$	0	0
$433$	24.3946	1.17233	0.586165	−	0.810192i	$-0.300637\pi$
$433$	24.3946	1.17233	0.586165	+	0.810192i	$0.300637\pi$
$434$	0	0
$435$	−10.9173	−0.523446
$436$	0	0
$437$	−4.37416	−0.209245
$438$	0	0
$439$	−32.4986	−1.55108	−0.775538	−	0.631301i	$-0.782521\pi$
$439$	−32.4986	−1.55108	−0.775538	+	0.631301i	$0.782521\pi$
$440$	0	0
$441$	−2.78370	−0.132557
$442$	0	0
$443$	20.2865	0.963841	0.481921	−	0.876215i	$-0.339939\pi$
$443$	20.2865	0.963841	0.481921	+	0.876215i	$0.339939\pi$
$444$	0	0
$445$	−40.6001	−1.92463
$446$	0	0
$447$	−3.81816	−0.180593
$448$	0	0
$449$	12.8893	0.608283	0.304142	−	0.952627i	$-0.401630\pi$
$449$	12.8893	0.608283	0.304142	+	0.952627i	$0.401630\pi$
$450$	0	0
$451$	−0.830333	−0.0390989
$452$	0	0
$453$	−0.625577	−0.0293922
$454$	0	0
$455$	−14.0058	−0.656600
$456$	0	0
$457$	3.42773	0.160342	0.0801711	−	0.996781i	$-0.474453\pi$
$457$	3.42773	0.160342	0.0801711	+	0.996781i	$0.474453\pi$
$458$	0	0
$459$	−10.4520	−0.487856
$460$	0	0
$461$	−19.5594	−0.910974	−0.455487	−	0.890242i	$-0.650535\pi$
$461$	−19.5594	−0.910974	−0.455487	+	0.890242i	$0.650535\pi$
$462$	0	0
$463$	23.1867	1.07758	0.538788	−	0.842441i	$-0.318882\pi$
$463$	23.1867	1.07758	0.538788	+	0.842441i	$0.318882\pi$
$464$	0	0
$465$	−1.69412	−0.0785629
$466$	0	0
$467$	−25.3793	−1.17441	−0.587207	−	0.809437i	$-0.699773\pi$
$467$	−25.3793	−1.17441	−0.587207	+	0.809437i	$0.699773\pi$
$468$	0	0
$469$	−7.64262	−0.352903
$470$	0	0
$471$	−4.07522	−0.187776
$472$	0	0
$473$	−17.2040	−0.791041
$474$	0	0
$475$	23.4593	1.07638
$476$	0	0
$477$	−39.1666	−1.79331
$478$	0	0
$479$	11.6843	0.533867	0.266934	−	0.963715i	$-0.413990\pi$
$479$	11.6843	0.533867	0.266934	+	0.963715i	$0.413990\pi$
$480$	0	0
$481$	49.1819	2.24250
$482$	0	0
$483$	0.465082	0.0211620
$484$	0	0
$485$	−29.6968	−1.34846
$486$	0	0
$487$	−14.1689	−0.642053	−0.321027	−	0.947070i	$-0.604028\pi$
$487$	−14.1689	−0.642053	−0.321027	+	0.947070i	$0.604028\pi$
$488$	0	0
$489$	−7.98232	−0.360973
$490$	0	0
$491$	21.4349	0.967343	0.483672	−	0.875250i	$-0.339303\pi$
$491$	21.4349	0.967343	0.483672	+	0.875250i	$0.339303\pi$
$492$	0	0
$493$	−28.3337	−1.27608
$494$	0	0
$495$	12.8413	0.577172
$496$	0	0
$497$	−5.42055	−0.243145
$498$	0	0
$499$	−31.2974	−1.40107	−0.700533	−	0.713620i	$-0.747054\pi$
$499$	−31.2974	−1.40107	−0.700533	+	0.713620i	$0.747054\pi$
$500$	0	0
$501$	9.94786	0.444438
$502$	0	0
$503$	6.81907	0.304047	0.152024	−	0.988377i	$-0.451421\pi$
$503$	6.81907	0.304047	0.152024	+	0.988377i	$0.451421\pi$
$504$	0	0
$505$	−33.7936	−1.50379
$506$	0	0
$507$	2.75736	0.122459
$508$	0	0
$509$	3.09151	0.137029	0.0685145	−	0.997650i	$-0.478174\pi$
$509$	3.09151	0.137029	0.0685145	+	0.997650i	$0.478174\pi$
$510$	0	0
$511$	−0.982187	−0.0434494
$512$	0	0
$513$	−11.7660	−0.519483
$514$	0	0
$515$	58.6202	2.58311
$516$	0	0
$517$	12.7166	0.559275
$518$	0	0
$519$	2.29561	0.100766
$520$	0	0
$521$	36.5401	1.60085	0.800424	−	0.599434i	$-0.204608\pi$
$521$	36.5401	1.60085	0.800424	+	0.599434i	$0.204608\pi$
$522$	0	0
$523$	−33.9506	−1.48455	−0.742277	−	0.670093i	$-0.766254\pi$
$523$	−33.9506	−1.48455	−0.742277	+	0.670093i	$0.766254\pi$
$524$	0	0
$525$	−2.49430	−0.108860
$526$	0	0
$527$	−4.39674	−0.191525
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	6.92730	0.300619
$532$	0	0
$533$	2.52100	0.109197
$534$	0	0
$535$	52.0898	2.25204
$536$	0	0
$537$	−0.239629	−0.0103407
$538$	0	0
$539$	−1.43298	−0.0617228
$540$	0	0
$541$	−14.1345	−0.607692	−0.303846	−	0.952721i	$-0.598271\pi$
$541$	−14.1345	−0.607692	−0.303846	+	0.952721i	$0.598271\pi$
$542$	0	0
$543$	5.43298	0.233151
$544$	0	0
$545$	16.3207	0.699103
$546$	0	0
$547$	−22.5965	−0.966155	−0.483078	−	0.875578i	$-0.660481\pi$
$547$	−22.5965	−0.966155	−0.483078	+	0.875578i	$0.660481\pi$
$548$	0	0
$549$	−5.80441	−0.247726
$550$	0	0
$551$	−31.8960	−1.35881
$552$	0	0
$553$	13.1757	0.560287
$554$	0	0
$555$	16.9247	0.718412
$556$	0	0
$557$	24.0343	1.01837	0.509184	−	0.860658i	$-0.329947\pi$
$557$	24.0343	1.01837	0.509184	+	0.860658i	$0.329947\pi$
$558$	0	0
$559$	52.2337	2.20925
$560$	0	0
$561$	−2.58960	−0.109333
$562$	0	0
$563$	5.73192	0.241572	0.120786	−	0.992679i	$-0.461459\pi$
$563$	5.73192	0.241572	0.120786	+	0.992679i	$0.461459\pi$
$564$	0	0
$565$	−53.3154	−2.24299
$566$	0	0
$567$	−7.10007	−0.298175
$568$	0	0
$569$	−43.1390	−1.80848	−0.904241	−	0.427022i	$-0.859563\pi$
$569$	−43.1390	−1.80848	−0.904241	+	0.427022i	$0.859563\pi$
$570$	0	0
$571$	−30.3922	−1.27188	−0.635938	−	0.771740i	$-0.719386\pi$
$571$	−30.3922	−1.27188	−0.635938	+	0.771740i	$0.719386\pi$
$572$	0	0
$573$	4.45196	0.185983
$574$	0	0
$575$	−5.36314	−0.223659
$576$	0	0
$577$	−8.43120	−0.350995	−0.175498	−	0.984480i	$-0.556153\pi$
$577$	−8.43120	−0.350995	−0.175498	+	0.984480i	$0.556153\pi$
$578$	0	0
$579$	−1.25705	−0.0522410
$580$	0	0
$581$	−12.1703	−0.504908
$582$	0	0
$583$	−20.1620	−0.835024
$584$	0	0
$585$	−38.9878	−1.61195
$586$	0	0
$587$	14.4759	0.597483	0.298741	−	0.954334i	$-0.403433\pi$
$587$	14.4759	0.597483	0.298741	+	0.954334i	$0.403433\pi$
$588$	0	0
$589$	−4.94952	−0.203941
$590$	0	0
$591$	−0.889401	−0.0365851
$592$	0	0
$593$	−30.2688	−1.24299	−0.621496	−	0.783417i	$-0.713475\pi$
$593$	−30.2688	−1.24299	−0.621496	+	0.783417i	$0.713475\pi$
$594$	0	0
$595$	−12.5086	−0.512802
$596$	0	0
$597$	6.71270	0.274732
$598$	0	0
$599$	−0.846874	−0.0346023	−0.0173012	−	0.999850i	$-0.505507\pi$
$599$	−0.846874	−0.0346023	−0.0173012	+	0.999850i	$0.505507\pi$
$600$	0	0
$601$	−12.4427	−0.507549	−0.253775	−	0.967263i	$-0.581672\pi$
$601$	−12.4427	−0.507549	−0.253775	+	0.967263i	$0.581672\pi$
$602$	0	0
$603$	−21.2748	−0.866375
$604$	0	0
$605$	−28.8007	−1.17091
$606$	0	0
$607$	17.0709	0.692888	0.346444	−	0.938071i	$-0.387389\pi$
$607$	17.0709	0.692888	0.346444	+	0.938071i	$0.387389\pi$
$608$	0	0
$609$	3.39133	0.137424
$610$	0	0
$611$	−38.6093	−1.56196
$612$	0	0
$613$	25.7372	1.03952	0.519758	−	0.854314i	$-0.326022\pi$
$613$	25.7372	1.03952	0.519758	+	0.854314i	$0.326022\pi$
$614$	0	0
$615$	0.867537	0.0349825
$616$	0	0
$617$	−9.62083	−0.387320	−0.193660	−	0.981069i	$-0.562036\pi$
$617$	−9.62083	−0.387320	−0.193660	+	0.981069i	$0.562036\pi$
$618$	0	0
$619$	32.3586	1.30060	0.650302	−	0.759676i	$-0.274642\pi$
$619$	32.3586	1.30060	0.650302	+	0.759676i	$0.274642\pi$
$620$	0	0
$621$	2.68990	0.107942
$622$	0	0
$623$	12.6119	0.505286
$624$	0	0
$625$	−23.0524	−0.922096
$626$	0	0
$627$	−2.91517	−0.116421
$628$	0	0
$629$	43.9245	1.75138
$630$	0	0
$631$	18.8123	0.748905	0.374453	−	0.927246i	$-0.377831\pi$
$631$	18.8123	0.748905	0.374453	+	0.927246i	$0.377831\pi$
$632$	0	0
$633$	−8.72757	−0.346890
$634$	0	0
$635$	−34.9387	−1.38650
$636$	0	0
$637$	4.35072	0.172382
$638$	0	0
$639$	−15.0892	−0.596919
$640$	0	0
$641$	7.54959	0.298191	0.149095	−	0.988823i	$-0.452364\pi$
$641$	7.54959	0.298191	0.149095	+	0.988823i	$0.452364\pi$
$642$	0	0
$643$	34.2240	1.34966	0.674831	−	0.737972i	$-0.264216\pi$
$643$	34.2240	1.34966	0.674831	+	0.737972i	$0.264216\pi$
$644$	0	0
$645$	17.9749	0.707759
$646$	0	0
$647$	−27.3126	−1.07377	−0.536884	−	0.843656i	$-0.680399\pi$
$647$	−27.3126	−1.07377	−0.536884	+	0.843656i	$0.680399\pi$
$648$	0	0
$649$	3.56600	0.139978
$650$	0	0
$651$	0.526257	0.0206256
$652$	0	0
$653$	−24.3457	−0.952721	−0.476361	−	0.879250i	$-0.658044\pi$
$653$	−24.3457	−0.952721	−0.476361	+	0.879250i	$0.658044\pi$
$654$	0	0
$655$	−5.98067	−0.233684
$656$	0	0
$657$	−2.73411	−0.106668
$658$	0	0
$659$	4.35059	0.169475	0.0847375	−	0.996403i	$-0.472995\pi$
$659$	4.35059	0.169475	0.0847375	+	0.996403i	$0.472995\pi$
$660$	0	0
$661$	42.1265	1.63853	0.819266	−	0.573414i	$-0.194381\pi$
$661$	42.1265	1.63853	0.819266	+	0.573414i	$0.194381\pi$
$662$	0	0
$663$	7.86236	0.305349
$664$	0	0
$665$	−14.0812	−0.546047
$666$	0	0
$667$	7.29190	0.282344
$668$	0	0
$669$	−0.689668	−0.0266641
$670$	0	0
$671$	−2.98797	−0.115349
$672$	0	0
$673$	31.1266	1.19984	0.599921	−	0.800059i	$-0.295199\pi$
$673$	31.1266	1.19984	0.599921	+	0.800059i	$0.295199\pi$
$674$	0	0
$675$	−14.4263	−0.555268
$676$	0	0
$677$	−33.5186	−1.28823	−0.644113	−	0.764930i	$-0.722773\pi$
$677$	−33.5186	−1.28823	−0.644113	+	0.764930i	$0.722773\pi$
$678$	0	0
$679$	9.22495	0.354021
$680$	0	0
$681$	−0.619911	−0.0237550
$682$	0	0
$683$	−3.41106	−0.130521	−0.0652603	−	0.997868i	$-0.520788\pi$
$683$	−3.41106	−0.130521	−0.0652603	+	0.997868i	$0.520788\pi$
$684$	0	0
$685$	45.5368	1.73987
$686$	0	0
$687$	2.80226	0.106913
$688$	0	0
$689$	61.2145	2.33209
$690$	0	0
$691$	−26.2143	−0.997239	−0.498620	−	0.866821i	$-0.666160\pi$
$691$	−26.2143	−0.997239	−0.498620	+	0.866821i	$0.666160\pi$
$692$	0	0
$693$	−3.98898	−0.151529
$694$	0	0
$695$	14.2939	0.542200
$696$	0	0
$697$	2.25151	0.0852822
$698$	0	0
$699$	−4.46798	−0.168995
$700$	0	0
$701$	22.4947	0.849615	0.424807	−	0.905284i	$-0.360342\pi$
$701$	22.4947	0.849615	0.424807	+	0.905284i	$0.360342\pi$
$702$	0	0
$703$	49.4469	1.86493
$704$	0	0
$705$	−13.2864	−0.500393
$706$	0	0
$707$	10.4976	0.394801
$708$	0	0
$709$	32.7536	1.23009	0.615044	−	0.788493i	$-0.289138\pi$
$709$	32.7536	1.23009	0.615044	+	0.788493i	$0.289138\pi$
$710$	0	0
$711$	36.6771	1.37550
$712$	0	0
$713$	1.13154	0.0423763
$714$	0	0
$715$	−20.0700	−0.750574
$716$	0	0
$717$	3.15480	0.117818
$718$	0	0
$719$	−33.3046	−1.24205	−0.621027	−	0.783789i	$-0.713284\pi$
$719$	−33.3046	−1.24205	−0.621027	+	0.783789i	$0.713284\pi$
$720$	0	0
$721$	−18.2096	−0.678162
$722$	0	0
$723$	7.71435	0.286900
$724$	0	0
$725$	−39.1075	−1.45242
$726$	0	0
$727$	−3.94437	−0.146289	−0.0731443	−	0.997321i	$-0.523303\pi$
$727$	−3.94437	−0.146289	−0.0731443	+	0.997321i	$0.523303\pi$
$728$	0	0
$729$	−16.0114	−0.593015
$730$	0	0
$731$	46.6500	1.72541
$732$	0	0
$733$	−46.6499	−1.72305	−0.861526	−	0.507714i	$-0.830491\pi$
$733$	−46.6499	−1.72305	−0.861526	+	0.507714i	$0.830491\pi$
$734$	0	0
$735$	1.49719	0.0552245
$736$	0	0
$737$	−10.9517	−0.403412
$738$	0	0
$739$	0.0891814	0.00328059	0.00164030	−	0.999999i	$-0.499478\pi$
$739$	0.0891814	0.00328059	0.00164030	+	0.999999i	$0.499478\pi$
$740$	0	0
$741$	8.85086	0.325145
$742$	0	0
$743$	32.7413	1.20116	0.600581	−	0.799564i	$-0.294936\pi$
$743$	32.7413	1.20116	0.600581	+	0.799564i	$0.294936\pi$
$744$	0	0
$745$	−26.4283	−0.968260
$746$	0	0
$747$	−33.8784	−1.23955
$748$	0	0
$749$	−16.1811	−0.591243
$750$	0	0
$751$	33.9576	1.23913	0.619565	−	0.784945i	$-0.287309\pi$
$751$	33.9576	1.23913	0.619565	+	0.784945i	$0.287309\pi$
$752$	0	0
$753$	−8.37183	−0.305086
$754$	0	0
$755$	−4.33009	−0.157588
$756$	0	0
$757$	32.9381	1.19716	0.598578	−	0.801064i	$-0.295733\pi$
$757$	32.9381	1.19716	0.598578	+	0.801064i	$0.295733\pi$
$758$	0	0
$759$	0.666453	0.0241907
$760$	0	0
$761$	−16.2439	−0.588839	−0.294420	−	0.955676i	$-0.595126\pi$
$761$	−16.2439	−0.588839	−0.294420	+	0.955676i	$0.595126\pi$
$762$	0	0
$763$	−5.06984	−0.183540
$764$	0	0
$765$	−34.8201	−1.25892
$766$	0	0
$767$	−10.8269	−0.390936
$768$	0	0
$769$	−47.5358	−1.71419	−0.857093	−	0.515162i	$-0.827732\pi$
$769$	−47.5358	−1.71419	−0.857093	+	0.515162i	$0.827732\pi$
$770$	0	0
$771$	−8.87506	−0.319627
$772$	0	0
$773$	−3.12026	−0.112228	−0.0561140	−	0.998424i	$-0.517871\pi$
$773$	−3.12026	−0.112228	−0.0561140	+	0.998424i	$0.517871\pi$
$774$	0	0
$775$	−6.06859	−0.217990
$776$	0	0
$777$	−5.25744	−0.188610
$778$	0	0
$779$	2.53459	0.0908110
$780$	0	0
$781$	−7.76754	−0.277945
$782$	0	0
$783$	19.6145	0.700963
$784$	0	0
$785$	−28.2077	−1.00678
$786$	0	0
$787$	26.7656	0.954092	0.477046	−	0.878878i	$-0.341707\pi$
$787$	26.7656	0.954092	0.477046	+	0.878878i	$0.341707\pi$
$788$	0	0
$789$	1.34609	0.0479219
$790$	0	0
$791$	16.5618	0.588869
$792$	0	0
$793$	9.07187	0.322152
$794$	0	0
$795$	21.0654	0.747111
$796$	0	0
$797$	37.4252	1.32567	0.662834	−	0.748766i	$-0.269353\pi$
$797$	37.4252	1.32567	0.662834	+	0.748766i	$0.269353\pi$
$798$	0	0
$799$	−34.4820	−1.21989
$800$	0	0
$801$	35.1078	1.24047
$802$	0	0
$803$	−1.40745	−0.0496680
$804$	0	0
$805$	3.21918	0.113461
$806$	0	0
$807$	−11.2846	−0.397237
$808$	0	0
$809$	14.3827	0.505670	0.252835	−	0.967509i	$-0.418637\pi$
$809$	14.3827	0.505670	0.252835	+	0.967509i	$0.418637\pi$
$810$	0	0
$811$	−44.7573	−1.57164	−0.785821	−	0.618453i	$-0.787760\pi$
$811$	−44.7573	−1.57164	−0.785821	+	0.618453i	$0.787760\pi$
$812$	0	0
$813$	4.41033	0.154677
$814$	0	0
$815$	−55.2516	−1.93538
$816$	0	0
$817$	52.5152	1.83727
$818$	0	0
$819$	12.1111	0.423196
$820$	0	0
$821$	−23.1002	−0.806203	−0.403101	−	0.915155i	$-0.632068\pi$
$821$	−23.1002	−0.806203	−0.403101	+	0.915155i	$0.632068\pi$
$822$	0	0
$823$	18.1468	0.632559	0.316280	−	0.948666i	$-0.397566\pi$
$823$	18.1468	0.632559	0.316280	+	0.948666i	$0.397566\pi$
$824$	0	0
$825$	−3.57428	−0.124441
$826$	0	0
$827$	−55.5541	−1.93180	−0.965902	−	0.258907i	$-0.916638\pi$
$827$	−55.5541	−1.93180	−0.965902	+	0.258907i	$0.916638\pi$
$828$	0	0
$829$	−9.72066	−0.337612	−0.168806	−	0.985649i	$-0.553991\pi$
$829$	−9.72066	−0.337612	−0.168806	+	0.985649i	$0.553991\pi$
$830$	0	0
$831$	12.8670	0.446351
$832$	0	0
$833$	3.88564	0.134629
$834$	0	0
$835$	68.8566	2.38288
$836$	0	0
$837$	3.04371	0.105206
$838$	0	0
$839$	−12.5675	−0.433879	−0.216940	−	0.976185i	$-0.569607\pi$
$839$	−12.5675	−0.433879	−0.216940	+	0.976185i	$0.569607\pi$
$840$	0	0
$841$	24.1718	0.833511
$842$	0	0
$843$	−6.27355	−0.216073
$844$	0	0
$845$	19.0858	0.656570
$846$	0	0
$847$	8.94657	0.307408
$848$	0	0
$849$	0.353407	0.0121289
$850$	0	0
$851$	−11.3043	−0.387507
$852$	0	0
$853$	−7.50447	−0.256948	−0.128474	−	0.991713i	$-0.541008\pi$
$853$	−7.50447	−0.256948	−0.128474	+	0.991713i	$0.541008\pi$
$854$	0	0
$855$	−39.1979	−1.34054
$856$	0	0
$857$	31.7855	1.08577	0.542886	−	0.839807i	$-0.317332\pi$
$857$	31.7855	1.08577	0.542886	+	0.839807i	$0.317332\pi$
$858$	0	0
$859$	24.7494	0.844439	0.422219	−	0.906494i	$-0.361251\pi$
$859$	24.7494	0.844439	0.422219	+	0.906494i	$0.361251\pi$
$860$	0	0
$861$	−0.269490	−0.00918419
$862$	0	0
$863$	−41.5021	−1.41275	−0.706373	−	0.707840i	$-0.749670\pi$
$863$	−41.5021	−1.41275	−0.706373	+	0.707840i	$0.749670\pi$
$864$	0	0
$865$	15.8897	0.540265
$866$	0	0
$867$	−0.884505	−0.0300394
$868$	0	0
$869$	18.8805	0.640476
$870$	0	0
$871$	33.2509	1.12666
$872$	0	0
$873$	25.6795	0.869119
$874$	0	0
$875$	−1.16903	−0.0395203
$876$	0	0
$877$	−39.8485	−1.34559	−0.672795	−	0.739829i	$-0.734906\pi$
$877$	−39.8485	−1.34559	−0.672795	+	0.739829i	$0.734906\pi$
$878$	0	0
$879$	−8.33958	−0.281287
$880$	0	0
$881$	−13.0131	−0.438424	−0.219212	−	0.975677i	$-0.570349\pi$
$881$	−13.0131	−0.438424	−0.219212	+	0.975677i	$0.570349\pi$
$882$	0	0
$883$	26.9778	0.907875	0.453938	−	0.891033i	$-0.350019\pi$
$883$	26.9778	0.907875	0.453938	+	0.891033i	$0.350019\pi$
$884$	0	0
$885$	−3.72578	−0.125241
$886$	0	0
$887$	−32.6870	−1.09752	−0.548762	−	0.835979i	$-0.684901\pi$
$887$	−32.6870	−1.09752	−0.548762	+	0.835979i	$0.684901\pi$
$888$	0	0
$889$	10.8533	0.364008
$890$	0	0
$891$	−10.1743	−0.340850
$892$	0	0
$893$	−38.8173	−1.29897
$894$	0	0
$895$	−1.65865	−0.0554426
$896$	0	0
$897$	−2.02344	−0.0675608
$898$	0	0
$899$	8.25105	0.275188
$900$	0	0
$901$	54.6708	1.82135
$902$	0	0
$903$	−5.58367	−0.185813
$904$	0	0
$905$	37.6057	1.25006
$906$	0	0
$907$	−19.5669	−0.649709	−0.324854	−	0.945764i	$-0.605315\pi$
$907$	−19.5669	−0.649709	−0.324854	+	0.945764i	$0.605315\pi$
$908$	0	0
$909$	29.2220	0.969234
$910$	0	0
$911$	−39.7346	−1.31647	−0.658233	−	0.752814i	$-0.728696\pi$
$911$	−39.7346	−1.31647	−0.658233	+	0.752814i	$0.728696\pi$
$912$	0	0
$913$	−17.4398	−0.577172
$914$	0	0
$915$	3.12184	0.103205
$916$	0	0
$917$	1.85782	0.0613507
$918$	0	0
$919$	6.89148	0.227329	0.113665	−	0.993519i	$-0.463741\pi$
$919$	6.89148	0.227329	0.113665	+	0.993519i	$0.463741\pi$
$920$	0	0
$921$	0.298185	0.00982554
$922$	0	0
$923$	23.5833	0.776254
$924$	0	0
$925$	60.6267	1.99339
$926$	0	0
$927$	−50.6902	−1.66488
$928$	0	0
$929$	−29.3962	−0.964458	−0.482229	−	0.876045i	$-0.660173\pi$
$929$	−29.3962	−0.964458	−0.482229	+	0.876045i	$0.660173\pi$
$930$	0	0
$931$	4.37416	0.143357
$932$	0	0
$933$	3.78393	0.123880
$934$	0	0
$935$	−17.9245	−0.586195
$936$	0	0
$937$	0.289659	0.00946276	0.00473138	−	0.999989i	$-0.498494\pi$
$937$	0.289659	0.00946276	0.00473138	+	0.999989i	$0.498494\pi$
$938$	0	0
$939$	0.520889	0.0169986
$940$	0	0
$941$	19.2998	0.629155	0.314577	−	0.949232i	$-0.398137\pi$
$941$	19.2998	0.629155	0.314577	+	0.949232i	$0.398137\pi$
$942$	0	0
$943$	−0.579445	−0.0188693
$944$	0	0
$945$	8.65927	0.281686
$946$	0	0
$947$	−57.7764	−1.87748	−0.938740	−	0.344626i	$-0.888006\pi$
$947$	−57.7764	−1.87748	−0.938740	+	0.344626i	$0.888006\pi$
$948$	0	0
$949$	4.27322	0.138715
$950$	0	0
$951$	4.92746	0.159784
$952$	0	0
$953$	−14.0133	−0.453935	−0.226968	−	0.973902i	$-0.572881\pi$
$953$	−14.0133	−0.453935	−0.226968	+	0.973902i	$0.572881\pi$
$954$	0	0
$955$	30.8153	0.997161
$956$	0	0
$957$	4.85971	0.157092
$958$	0	0
$959$	−14.1454	−0.456780
$960$	0	0
$961$	−29.7196	−0.958698
$962$	0	0
$963$	−45.0432	−1.45150
$964$	0	0
$965$	−8.70096	−0.280094
$966$	0	0
$967$	23.4457	0.753962	0.376981	−	0.926221i	$-0.376962\pi$
$967$	23.4457	0.753962	0.376981	+	0.926221i	$0.376962\pi$
$968$	0	0
$969$	7.90473	0.253936
$970$	0	0
$971$	14.4747	0.464517	0.232258	−	0.972654i	$-0.425389\pi$
$971$	14.4747	0.464517	0.232258	+	0.972654i	$0.425389\pi$
$972$	0	0
$973$	−4.44023	−0.142347
$974$	0	0
$975$	10.8520	0.347543
$976$	0	0
$977$	−25.9350	−0.829735	−0.414867	−	0.909882i	$-0.636172\pi$
$977$	−25.9350	−0.829735	−0.414867	+	0.909882i	$0.636172\pi$
$978$	0	0
$979$	18.0726	0.577604
$980$	0	0
$981$	−14.1129	−0.450590
$982$	0	0
$983$	−19.1380	−0.610409	−0.305204	−	0.952287i	$-0.598725\pi$
$983$	−19.1380	−0.610409	−0.305204	+	0.952287i	$0.598725\pi$
$984$	0	0
$985$	−6.15621	−0.196153
$986$	0	0
$987$	4.12725	0.131372
$988$	0	0
$989$	−12.0058	−0.381761
$990$	0	0
$991$	3.19195	0.101396	0.0506979	−	0.998714i	$-0.483855\pi$
$991$	3.19195	0.101396	0.0506979	+	0.998714i	$0.483855\pi$
$992$	0	0
$993$	−9.64379	−0.306036
$994$	0	0
$995$	46.4636	1.47300
$996$	0	0
$997$	−59.8455	−1.89533	−0.947663	−	0.319274i	$-0.896561\pi$
$997$	−59.8455	−1.89533	−0.947663	+	0.319274i	$0.896561\pi$
$998$	0	0
$999$	−30.4075	−0.962049

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2576.2.a.bc.1.3		5
4.3	odd	2		644.2.a.c.1.3	✓	5
12.11	even	2		5796.2.a.s.1.2		5
28.27	even	2		4508.2.a.g.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
644.2.a.c.1.3	✓	5	4.3	odd	2
2576.2.a.bc.1.3		5	1.1	even	1	trivial
4508.2.a.g.1.3		5	28.27	even	2
5796.2.a.s.1.2		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:	\( N \)	\(=\)	\( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2576.a (trivial)

\(f(q)\)	\(=\)	\(q+0.465082 q^{3} +3.21918 q^{5} -1.00000 q^{7} -2.78370 q^{9} +O(q^{10})\)	\(q+0.465082 q^{3} +3.21918 q^{5} -1.00000 q^{7} -2.78370 q^{9} -1.43298 q^{11} +4.35072 q^{13} +1.49719 q^{15} +3.88564 q^{17} +4.37416 q^{19} -0.465082 q^{21} -1.00000 q^{23} +5.36314 q^{25} -2.68990 q^{27} -7.29190 q^{29} -1.13154 q^{31} -0.666453 q^{33} -3.21918 q^{35} +11.3043 q^{37} +2.02344 q^{39} +0.579445 q^{41} +12.0058 q^{43} -8.96124 q^{45} -8.87423 q^{47} +1.00000 q^{49} +1.80714 q^{51} +14.0700 q^{53} -4.61302 q^{55} +2.03434 q^{57} -2.48852 q^{59} +2.08514 q^{61} +2.78370 q^{63} +14.0058 q^{65} +7.64262 q^{67} -0.465082 q^{69} +5.42055 q^{71} +0.982187 q^{73} +2.49430 q^{75} +1.43298 q^{77} -13.1757 q^{79} +7.10007 q^{81} +12.1703 q^{83} +12.5086 q^{85} -3.39133 q^{87} -12.6119 q^{89} -4.35072 q^{91} -0.526257 q^{93} +14.0812 q^{95} -9.22495 q^{97} +3.98898 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} + 4 q^{5} - 5 q^{7} + 10 q^{9}+O(q^{10}) \) 5 * q - q^3 + 4 * q^5 - 5 * q^7 + 10 * q^9	\( 5 q - q^{3} + 4 q^{5} - 5 q^{7} + 10 q^{9} - 2 q^{11} + 3 q^{13} + 6 q^{15} + 4 q^{17} - 6 q^{19} + q^{21} - 5 q^{23} + 15 q^{25} - q^{27} + 5 q^{29} + q^{31} - 4 q^{35} + 22 q^{37} + q^{39} + 15 q^{41} - 12 q^{43} + 36 q^{45} + 21 q^{47} + 5 q^{49} - 24 q^{51} + 2 q^{53} + 22 q^{55} - 34 q^{57} - 12 q^{61} - 10 q^{63} - 2 q^{65} - 22 q^{67} + q^{69} + 15 q^{71} + 17 q^{73} + 47 q^{75} + 2 q^{77} - 2 q^{79} + 37 q^{81} + 16 q^{83} - 8 q^{85} + 33 q^{87} - 24 q^{89} - 3 q^{91} + 5 q^{93} + 8 q^{95} + 38 q^{97} + 36 q^{99}+O(q^{100}) \) 5 * q - q^3 + 4 * q^5 - 5 * q^7 + 10 * q^9 - 2 * q^11 + 3 * q^13 + 6 * q^15 + 4 * q^17 - 6 * q^19 + q^21 - 5 * q^23 + 15 * q^25 - q^27 + 5 * q^29 + q^31 - 4 * q^35 + 22 * q^37 + q^39 + 15 * q^41 - 12 * q^43 + 36 * q^45 + 21 * q^47 + 5 * q^49 - 24 * q^51 + 2 * q^53 + 22 * q^55 - 34 * q^57 - 12 * q^61 - 10 * q^63 - 2 * q^65 - 22 * q^67 + q^69 + 15 * q^71 + 17 * q^73 + 47 * q^75 + 2 * q^77 - 2 * q^79 + 37 * q^81 + 16 * q^83 - 8 * q^85 + 33 * q^87 - 24 * q^89 - 3 * q^91 + 5 * q^93 + 8 * q^95 + 38 * q^97 + 36 * q^99

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