LMFDB - Embedded newform 1352.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1352,4,Mod(1,1352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1352 = 2^{3} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1352.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:	$79.7705823278$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{321}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 80 $ x^2 - x - 80
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$9.45824$ of defining polynomial
Character	$\chi$	$=$	1352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.45824 q^{3} -3.45824 q^{5} +26.3747 q^{7} +62.4582 q^{9} +O(q^{10})$	$q-9.45824 q^{3} -3.45824 q^{5} +26.3747 q^{7} +62.4582 q^{9} -58.0000 q^{11} +32.7088 q^{15} +88.2077 q^{17} -99.8329 q^{19} -249.458 q^{21} +9.49884 q^{23} -113.041 q^{25} -335.372 q^{27} -6.00000 q^{29} -23.4176 q^{31} +548.578 q^{33} -91.2100 q^{35} -44.9571 q^{37} -58.0812 q^{41} +457.955 q^{43} -215.995 q^{45} +496.208 q^{47} +352.625 q^{49} -834.289 q^{51} +601.747 q^{53} +200.578 q^{55} +944.244 q^{57} -211.336 q^{59} +414.912 q^{61} +1647.32 q^{63} -744.167 q^{67} -89.8422 q^{69} -882.537 q^{71} +319.499 q^{73} +1069.16 q^{75} -1529.73 q^{77} -490.664 q^{79} +1485.66 q^{81} +224.420 q^{83} -305.043 q^{85} +56.7494 q^{87} +580.501 q^{89} +221.490 q^{93} +345.246 q^{95} -403.499 q^{97} -3622.58 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 11 q^{5} - q^{7} + 107 q^{9}+O(q^{10}) $ 2 * q - q^3 + 11 * q^5 - q^7 + 107 * q^9	$ 2 q - q^{3} + 11 q^{5} - q^{7} + 107 q^{9} - 116 q^{11} + 155 q^{15} + 51 q^{17} - 128 q^{19} - 481 q^{21} - 196 q^{23} - 29 q^{25} - 187 q^{27} - 12 q^{29} - 226 q^{31} + 58 q^{33} - 487 q^{35} + 143 q^{37} + 278 q^{41} + 253 q^{43} + 428 q^{45} + 867 q^{47} + 759 q^{49} - 1149 q^{51} + 666 q^{53} - 638 q^{55} + 706 q^{57} - 996 q^{59} - 66 q^{61} + 428 q^{63} - 1560 q^{67} - 1828 q^{69} - 923 q^{71} + 424 q^{73} + 1780 q^{75} + 58 q^{77} - 408 q^{79} + 1538 q^{81} + 1058 q^{83} - 843 q^{85} + 6 q^{87} + 1376 q^{89} - 1492 q^{93} - 62 q^{95} - 592 q^{97} - 6206 q^{99}+O(q^{100}) $ 2 * q - q^3 + 11 * q^5 - q^7 + 107 * q^9 - 116 * q^11 + 155 * q^15 + 51 * q^17 - 128 * q^19 - 481 * q^21 - 196 * q^23 - 29 * q^25 - 187 * q^27 - 12 * q^29 - 226 * q^31 + 58 * q^33 - 487 * q^35 + 143 * q^37 + 278 * q^41 + 253 * q^43 + 428 * q^45 + 867 * q^47 + 759 * q^49 - 1149 * q^51 + 666 * q^53 - 638 * q^55 + 706 * q^57 - 996 * q^59 - 66 * q^61 + 428 * q^63 - 1560 * q^67 - 1828 * q^69 - 923 * q^71 + 424 * q^73 + 1780 * q^75 + 58 * q^77 - 408 * q^79 + 1538 * q^81 + 1058 * q^83 - 843 * q^85 + 6 * q^87 + 1376 * q^89 - 1492 * q^93 - 62 * q^95 - 592 * q^97 - 6206 * 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
$3$	−9.45824	−1.82024	−0.910119	−	0.414346i	$-0.864010\pi$
$3$	−9.45824	−1.82024	−0.910119	+	0.414346i	$0.864010\pi$
$4$	0	0
$5$	−3.45824	−0.309314	−0.154657	−	0.987968i	$-0.549427\pi$
$5$	−3.45824	−0.309314	−0.154657	+	0.987968i	$0.549427\pi$
$6$	0	0
$7$	26.3747	1.42410	0.712050	−	0.702129i	$-0.247767\pi$
$7$	26.3747	1.42410	0.712050	+	0.702129i	$0.247767\pi$
$8$	0	0
$9$	62.4582	2.31327
$10$	0	0
$11$	−58.0000	−1.58979	−0.794894	−	0.606749i	$-0.792473\pi$
$11$	−58.0000	−1.58979	−0.794894	+	0.606749i	$0.792473\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	32.7088	0.563025
$16$	0	0
$17$	88.2077	1.25844	0.629221	−	0.777227i	$-0.283374\pi$
$17$	88.2077	1.25844	0.629221	+	0.777227i	$0.283374\pi$
$18$	0	0
$19$	−99.8329	−1.20543	−0.602717	−	0.797955i	$-0.705915\pi$
$19$	−99.8329	−1.20543	−0.602717	+	0.797955i	$0.705915\pi$
$20$	0	0
$21$	−249.458	−2.59220
$22$	0	0
$23$	9.49884	0.0861150	0.0430575	−	0.999073i	$-0.486290\pi$
$23$	9.49884	0.0861150	0.0430575	+	0.999073i	$0.486290\pi$
$24$	0	0
$25$	−113.041	−0.904325
$26$	0	0
$27$	−335.372	−2.39046
$28$	0	0
$29$	−6.00000	−0.0384197	−0.0192099	−	0.999815i	$-0.506115\pi$
$29$	−6.00000	−0.0384197	−0.0192099	+	0.999815i	$0.506115\pi$
$30$	0	0
$31$	−23.4176	−0.135675	−0.0678376	−	0.997696i	$-0.521610\pi$
$31$	−23.4176	−0.135675	−0.0678376	+	0.997696i	$0.521610\pi$
$32$	0	0
$33$	548.578	2.89379
$34$	0	0
$35$	−91.2100	−0.440494
$36$	0	0
$37$	−44.9571	−0.199754	−0.0998770	−	0.995000i	$-0.531845\pi$
$37$	−44.9571	−0.199754	−0.0998770	+	0.995000i	$0.531845\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−58.0812	−0.221238	−0.110619	−	0.993863i	$-0.535283\pi$
$41$	−58.0812	−0.221238	−0.110619	+	0.993863i	$0.535283\pi$
$42$	0	0
$43$	457.955	1.62413	0.812063	−	0.583569i	$-0.198344\pi$
$43$	457.955	1.62413	0.812063	+	0.583569i	$0.198344\pi$
$44$	0	0
$45$	−215.995	−0.715526
$46$	0	0
$47$	496.208	1.53999	0.769993	−	0.638053i	$-0.220260\pi$
$47$	496.208	1.53999	0.769993	+	0.638053i	$0.220260\pi$
$48$	0	0
$49$	352.625	1.02806
$50$	0	0
$51$	−834.289	−2.29066
$52$	0	0
$53$	601.747	1.55955	0.779777	−	0.626058i	$-0.215333\pi$
$53$	601.747	1.55955	0.779777	+	0.626058i	$0.215333\pi$
$54$	0	0
$55$	200.578	0.491744
$56$	0	0
$57$	944.244	2.19418
$58$	0	0
$59$	−211.336	−0.466333	−0.233167	−	0.972437i	$-0.574909\pi$
$59$	−211.336	−0.466333	−0.233167	+	0.972437i	$0.574909\pi$
$60$	0	0
$61$	414.912	0.870885	0.435443	−	0.900216i	$-0.356592\pi$
$61$	414.912	0.870885	0.435443	+	0.900216i	$0.356592\pi$
$62$	0	0
$63$	1647.32	3.29433
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−744.167	−1.35693	−0.678466	−	0.734632i	$-0.737355\pi$
$67$	−744.167	−1.35693	−0.678466	+	0.734632i	$0.737355\pi$
$68$	0	0
$69$	−89.8422	−0.156750
$70$	0	0
$71$	−882.537	−1.47518	−0.737591	−	0.675248i	$-0.764037\pi$
$71$	−882.537	−1.47518	−0.737591	+	0.675248i	$0.764037\pi$
$72$	0	0
$73$	319.499	0.512254	0.256127	−	0.966643i	$-0.417554\pi$
$73$	319.499	0.512254	0.256127	+	0.966643i	$0.417554\pi$
$74$	0	0
$75$	1069.16	1.64609
$76$	0	0
$77$	−1529.73	−2.26402
$78$	0	0
$79$	−490.664	−0.698784	−0.349392	−	0.936977i	$-0.613612\pi$
$79$	−490.664	−0.698784	−0.349392	+	0.936977i	$0.613612\pi$
$80$	0	0
$81$	1485.66	2.03794
$82$	0	0
$83$	224.420	0.296787	0.148393	−	0.988928i	$-0.452590\pi$
$83$	224.420	0.296787	0.148393	+	0.988928i	$0.452590\pi$
$84$	0	0
$85$	−305.043	−0.389254
$86$	0	0
$87$	56.7494	0.0699331
$88$	0	0
$89$	580.501	0.691382	0.345691	−	0.938348i	$-0.387645\pi$
$89$	580.501	0.691382	0.345691	+	0.938348i	$0.387645\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	221.490	0.246961
$94$	0	0
$95$	345.246	0.372858
$96$	0	0
$97$	−403.499	−0.422362	−0.211181	−	0.977447i	$-0.567731\pi$
$97$	−403.499	−0.422362	−0.211181	+	0.977447i	$0.567731\pi$
$98$	0	0
$99$	−3622.58	−3.67760
$100$	0	0
$101$	−698.244	−0.687899	−0.343950	−	0.938988i	$-0.611765\pi$
$101$	−698.244	−0.687899	−0.343950	+	0.938988i	$0.611765\pi$
$102$	0	0
$103$	1285.33	1.22958	0.614791	−	0.788690i	$-0.289240\pi$
$103$	1285.33	1.22958	0.614791	+	0.788690i	$0.289240\pi$
$104$	0	0
$105$	862.686	0.801805
$106$	0	0
$107$	1329.82	1.20149	0.600743	−	0.799443i	$-0.294872\pi$
$107$	1329.82	1.20149	0.600743	+	0.799443i	$0.294872\pi$
$108$	0	0
$109$	951.034	0.835711	0.417856	−	0.908513i	$-0.362782\pi$
$109$	951.034	0.835711	0.417856	+	0.908513i	$0.362782\pi$
$110$	0	0
$111$	425.215	0.363600
$112$	0	0
$113$	89.8283	0.0747817	0.0373909	−	0.999301i	$-0.488095\pi$
$113$	89.8283	0.0747817	0.0373909	+	0.999301i	$0.488095\pi$
$114$	0	0
$115$	−32.8492	−0.0266366
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2326.45	1.79215
$120$	0	0
$121$	2033.00	1.52742
$122$	0	0
$123$	549.346	0.402706
$124$	0	0
$125$	823.201	0.589034
$126$	0	0
$127$	345.012	0.241062	0.120531	−	0.992710i	$-0.461540\pi$
$127$	345.012	0.241062	0.120531	+	0.992710i	$0.461540\pi$
$128$	0	0
$129$	−4331.44	−2.95630
$130$	0	0
$131$	−2052.03	−1.36860	−0.684301	−	0.729200i	$-0.739892\pi$
$131$	−2052.03	−1.36860	−0.684301	+	0.729200i	$0.739892\pi$
$132$	0	0
$133$	−2633.06	−1.71666
$134$	0	0
$135$	1159.80	0.739403
$136$	0	0
$137$	−1952.58	−1.21767	−0.608833	−	0.793298i	$-0.708362\pi$
$137$	−1952.58	−1.21767	−0.608833	+	0.793298i	$0.708362\pi$
$138$	0	0
$139$	−2238.69	−1.36606	−0.683032	−	0.730389i	$-0.739339\pi$
$139$	−2238.69	−1.36606	−0.683032	+	0.730389i	$0.739339\pi$
$140$	0	0
$141$	−4693.25	−2.80314
$142$	0	0
$143$	0	0
$144$	0	0
$145$	20.7494	0.0118838
$146$	0	0
$147$	−3335.21	−1.87132
$148$	0	0
$149$	1097.81	0.603598	0.301799	−	0.953372i	$-0.402413\pi$
$149$	1097.81	0.603598	0.301799	+	0.953372i	$0.402413\pi$
$150$	0	0
$151$	−3593.17	−1.93648	−0.968239	−	0.250026i	$-0.919561\pi$
$151$	−3593.17	−1.93648	−0.968239	+	0.250026i	$0.919561\pi$
$152$	0	0
$153$	5509.29	2.91111
$154$	0	0
$155$	80.9837	0.0419663
$156$	0	0
$157$	−2517.48	−1.27972	−0.639861	−	0.768490i	$-0.721008\pi$
$157$	−2517.48	−1.27972	−0.639861	+	0.768490i	$0.721008\pi$
$158$	0	0
$159$	−5691.47	−2.83876
$160$	0	0
$161$	250.529	0.122636
$162$	0	0
$163$	251.084	0.120653	0.0603263	−	0.998179i	$-0.480786\pi$
$163$	251.084	0.120653	0.0603263	+	0.998179i	$0.480786\pi$
$164$	0	0
$165$	−1897.11	−0.895090
$166$	0	0
$167$	450.596	0.208792	0.104396	−	0.994536i	$-0.466709\pi$
$167$	450.596	0.208792	0.104396	+	0.994536i	$0.466709\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−6235.39	−2.78849
$172$	0	0
$173$	−247.914	−0.108951	−0.0544756	−	0.998515i	$-0.517349\pi$
$173$	−247.914	−0.108951	−0.0544756	+	0.998515i	$0.517349\pi$
$174$	0	0
$175$	−2981.41	−1.28785
$176$	0	0
$177$	1998.87	0.848838
$178$	0	0
$179$	−1266.87	−0.528995	−0.264497	−	0.964386i	$-0.585206\pi$
$179$	−1266.87	−0.528995	−0.264497	+	0.964386i	$0.585206\pi$
$180$	0	0
$181$	4048.38	1.66251	0.831254	−	0.555893i	$-0.187624\pi$
$181$	4048.38	1.66251	0.831254	+	0.555893i	$0.187624\pi$
$182$	0	0
$183$	−3924.33	−1.58522
$184$	0	0
$185$	155.472	0.0617867
$186$	0	0
$187$	−5116.04	−2.00065
$188$	0	0
$189$	−8845.35	−3.40426
$190$	0	0
$191$	−4021.38	−1.52344	−0.761720	−	0.647907i	$-0.775645\pi$
$191$	−4021.38	−1.52344	−0.761720	+	0.647907i	$0.775645\pi$
$192$	0	0
$193$	1529.58	0.570476	0.285238	−	0.958457i	$-0.407927\pi$
$193$	1529.58	0.570476	0.285238	+	0.958457i	$0.407927\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1146.04	−0.414477	−0.207239	−	0.978290i	$-0.566448\pi$
$197$	−1146.04	−0.414477	−0.207239	+	0.978290i	$0.566448\pi$
$198$	0	0
$199$	−5190.08	−1.84882	−0.924409	−	0.381403i	$-0.875441\pi$
$199$	−5190.08	−1.84882	−0.924409	+	0.381403i	$0.875441\pi$
$200$	0	0
$201$	7038.51	2.46994
$202$	0	0
$203$	−158.248	−0.0547135
$204$	0	0
$205$	200.859	0.0684320
$206$	0	0
$207$	593.281	0.199207
$208$	0	0
$209$	5790.31	1.91638
$210$	0	0
$211$	1272.63	0.415219	0.207610	−	0.978212i	$-0.433432\pi$
$211$	1272.63	0.415219	0.207610	+	0.978212i	$0.433432\pi$
$212$	0	0
$213$	8347.24	2.68518
$214$	0	0
$215$	−1583.72	−0.502365
$216$	0	0
$217$	−617.633	−0.193215
$218$	0	0
$219$	−3021.90	−0.932424
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−3340.87	−1.00323	−0.501617	−	0.865090i	$-0.667261\pi$
$223$	−3340.87	−1.00323	−0.501617	+	0.865090i	$0.667261\pi$
$224$	0	0
$225$	−7060.32	−2.09195
$226$	0	0
$227$	3577.07	1.04590	0.522948	−	0.852365i	$-0.324832\pi$
$227$	3577.07	1.04590	0.522948	+	0.852365i	$0.324832\pi$
$228$	0	0
$229$	−6215.35	−1.79355	−0.896773	−	0.442492i	$-0.854095\pi$
$229$	−6215.35	−1.79355	−0.896773	+	0.442492i	$0.854095\pi$
$230$	0	0
$231$	14468.6	4.12105
$232$	0	0
$233$	2163.02	0.608171	0.304085	−	0.952645i	$-0.401649\pi$
$233$	2163.02	0.608171	0.304085	+	0.952645i	$0.401649\pi$
$234$	0	0
$235$	−1716.00	−0.476339
$236$	0	0
$237$	4640.81	1.27195
$238$	0	0
$239$	2908.51	0.787180	0.393590	−	0.919286i	$-0.371233\pi$
$239$	2908.51	0.787180	0.393590	+	0.919286i	$0.371233\pi$
$240$	0	0
$241$	−1391.63	−0.371963	−0.185981	−	0.982553i	$-0.559546\pi$
$241$	−1391.63	−0.371963	−0.185981	+	0.982553i	$0.559546\pi$
$242$	0	0
$243$	−4996.66	−1.31908
$244$	0	0
$245$	−1219.46	−0.317994
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−2122.62	−0.540222
$250$	0	0
$251$	2634.12	0.662407	0.331204	−	0.943559i	$-0.392545\pi$
$251$	2634.12	0.662407	0.331204	+	0.943559i	$0.392545\pi$
$252$	0	0
$253$	−550.933	−0.136904
$254$	0	0
$255$	2885.17	0.708534
$256$	0	0
$257$	441.841	0.107242	0.0536212	−	0.998561i	$-0.482924\pi$
$257$	441.841	0.107242	0.0536212	+	0.998561i	$0.482924\pi$
$258$	0	0
$259$	−1185.73	−0.284470
$260$	0	0
$261$	−374.749	−0.0888751
$262$	0	0
$263$	−2100.32	−0.492439	−0.246220	−	0.969214i	$-0.579188\pi$
$263$	−2100.32	−0.492439	−0.246220	+	0.969214i	$0.579188\pi$
$264$	0	0
$265$	−2080.98	−0.482392
$266$	0	0
$267$	−5490.52	−1.25848
$268$	0	0
$269$	6432.40	1.45796	0.728978	−	0.684537i	$-0.239995\pi$
$269$	6432.40	1.45796	0.728978	+	0.684537i	$0.239995\pi$
$270$	0	0
$271$	−3093.14	−0.693339	−0.346669	−	0.937987i	$-0.612687\pi$
$271$	−3093.14	−0.693339	−0.346669	+	0.937987i	$0.612687\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6556.35	1.43768
$276$	0	0
$277$	−1082.57	−0.234820	−0.117410	−	0.993084i	$-0.537459\pi$
$277$	−1082.57	−0.234820	−0.117410	+	0.993084i	$0.537459\pi$
$278$	0	0
$279$	−1462.62	−0.313853
$280$	0	0
$281$	−4885.85	−1.03724	−0.518621	−	0.855004i	$-0.673555\pi$
$281$	−4885.85	−1.03724	−0.518621	+	0.855004i	$0.673555\pi$
$282$	0	0
$283$	−8190.58	−1.72042	−0.860211	−	0.509938i	$-0.829668\pi$
$283$	−8190.58	−1.72042	−0.860211	+	0.509938i	$0.829668\pi$
$284$	0	0
$285$	−3265.42	−0.678690
$286$	0	0
$287$	−1531.87	−0.315065
$288$	0	0
$289$	2867.59	0.583674
$290$	0	0
$291$	3816.39	0.768799
$292$	0	0
$293$	1167.54	0.232793	0.116397	−	0.993203i	$-0.462866\pi$
$293$	1167.54	0.232793	0.116397	+	0.993203i	$0.462866\pi$
$294$	0	0
$295$	730.851	0.144243
$296$	0	0
$297$	19451.6	3.80032
$298$	0	0
$299$	0	0
$300$	0	0
$301$	12078.4	2.31292
$302$	0	0
$303$	6604.15	1.25214
$304$	0	0
$305$	−1434.86	−0.269377
$306$	0	0
$307$	−3928.01	−0.730240	−0.365120	−	0.930961i	$-0.618972\pi$
$307$	−3928.01	−0.730240	−0.365120	+	0.930961i	$0.618972\pi$
$308$	0	0
$309$	−12156.9	−2.23813
$310$	0	0
$311$	−2848.86	−0.519433	−0.259717	−	0.965685i	$-0.583629\pi$
$311$	−2848.86	−0.519433	−0.259717	+	0.965685i	$0.583629\pi$
$312$	0	0
$313$	−5990.54	−1.08181	−0.540903	−	0.841085i	$-0.681917\pi$
$313$	−5990.54	−1.08181	−0.540903	+	0.841085i	$0.681917\pi$
$314$	0	0
$315$	−5696.81	−1.01898
$316$	0	0
$317$	3521.83	0.623992	0.311996	−	0.950083i	$-0.399002\pi$
$317$	3521.83	0.623992	0.311996	+	0.950083i	$0.399002\pi$
$318$	0	0
$319$	348.000	0.0610792
$320$	0	0
$321$	−12577.8	−2.18699
$322$	0	0
$323$	−8806.03	−1.51697
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−8995.10	−1.52119
$328$	0	0
$329$	13087.3	2.19309
$330$	0	0
$331$	−1487.38	−0.246990	−0.123495	−	0.992345i	$-0.539410\pi$
$331$	−1487.38	−0.246990	−0.123495	+	0.992345i	$0.539410\pi$
$332$	0	0
$333$	−2807.94	−0.462085
$334$	0	0
$335$	2573.51	0.419718
$336$	0	0
$337$	−2559.21	−0.413676	−0.206838	−	0.978375i	$-0.566317\pi$
$337$	−2559.21	−0.413676	−0.206838	+	0.978375i	$0.566317\pi$
$338$	0	0
$339$	−849.617	−0.136121
$340$	0	0
$341$	1358.22	0.215695
$342$	0	0
$343$	253.864	0.0399632
$344$	0	0
$345$	310.696	0.0484849
$346$	0	0
$347$	11229.6	1.73728	0.868639	−	0.495446i	$-0.164996\pi$
$347$	11229.6	1.73728	0.868639	+	0.495446i	$0.164996\pi$
$348$	0	0
$349$	−2570.45	−0.394249	−0.197124	−	0.980378i	$-0.563160\pi$
$349$	−2570.45	−0.394249	−0.197124	+	0.980378i	$0.563160\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5401.58	−0.814439	−0.407219	−	0.913330i	$-0.633502\pi$
$353$	−5401.58	−0.814439	−0.407219	+	0.913330i	$0.633502\pi$
$354$	0	0
$355$	3052.02	0.456294
$356$	0	0
$357$	−22004.1	−3.26213
$358$	0	0
$359$	−2650.71	−0.389691	−0.194846	−	0.980834i	$-0.562421\pi$
$359$	−2650.71	−0.389691	−0.194846	+	0.980834i	$0.562421\pi$
$360$	0	0
$361$	3107.62	0.453071
$362$	0	0
$363$	−19228.6	−2.78027
$364$	0	0
$365$	−1104.90	−0.158447
$366$	0	0
$367$	−7298.53	−1.03809	−0.519046	−	0.854746i	$-0.673713\pi$
$367$	−7298.53	−1.03809	−0.519046	+	0.854746i	$0.673713\pi$
$368$	0	0
$369$	−3627.65	−0.511783
$370$	0	0
$371$	15870.9	2.22096
$372$	0	0
$373$	−8960.03	−1.24379	−0.621894	−	0.783102i	$-0.713636\pi$
$373$	−8960.03	−1.24379	−0.621894	+	0.783102i	$0.713636\pi$
$374$	0	0
$375$	−7786.03	−1.07218
$376$	0	0
$377$	0	0
$378$	0	0
$379$	3470.36	0.470344	0.235172	−	0.971954i	$-0.424435\pi$
$379$	3470.36	0.470344	0.235172	+	0.971954i	$0.424435\pi$
$380$	0	0
$381$	−3263.20	−0.438790
$382$	0	0
$383$	7611.42	1.01547	0.507736	−	0.861513i	$-0.330483\pi$
$383$	7611.42	1.01547	0.507736	+	0.861513i	$0.330483\pi$
$384$	0	0
$385$	5290.18	0.700292
$386$	0	0
$387$	28603.0	3.75704
$388$	0	0
$389$	−9394.26	−1.22444	−0.612221	−	0.790687i	$-0.709724\pi$
$389$	−9394.26	−1.22444	−0.612221	+	0.790687i	$0.709724\pi$
$390$	0	0
$391$	837.870	0.108371
$392$	0	0
$393$	19408.6	2.49118
$394$	0	0
$395$	1696.83	0.216144
$396$	0	0
$397$	−8377.68	−1.05910	−0.529551	−	0.848278i	$-0.677640\pi$
$397$	−8377.68	−1.05910	−0.529551	+	0.848278i	$0.677640\pi$
$398$	0	0
$399$	24904.2	3.12473
$400$	0	0
$401$	1171.50	0.145890	0.0729448	−	0.997336i	$-0.476760\pi$
$401$	1171.50	0.145890	0.0729448	+	0.997336i	$0.476760\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−5137.76	−0.630364
$406$	0	0
$407$	2607.51	0.317566
$408$	0	0
$409$	−1320.85	−0.159687	−0.0798434	−	0.996807i	$-0.525442\pi$
$409$	−1320.85	−0.159687	−0.0798434	+	0.996807i	$0.525442\pi$
$410$	0	0
$411$	18468.0	2.21644
$412$	0	0
$413$	−5573.94	−0.664105
$414$	0	0
$415$	−776.097	−0.0918003
$416$	0	0
$417$	21174.0	2.48656
$418$	0	0
$419$	−1553.50	−0.181129	−0.0905647	−	0.995891i	$-0.528867\pi$
$419$	−1553.50	−0.181129	−0.0905647	+	0.995891i	$0.528867\pi$
$420$	0	0
$421$	6746.63	0.781022	0.390511	−	0.920598i	$-0.372298\pi$
$421$	6746.63	0.781022	0.390511	+	0.920598i	$0.372298\pi$
$422$	0	0
$423$	30992.3	3.56240
$424$	0	0
$425$	−9971.05	−1.13804
$426$	0	0
$427$	10943.2	1.24023
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2646.81	0.295805	0.147903	−	0.989002i	$-0.452748\pi$
$431$	2646.81	0.295805	0.147903	+	0.989002i	$0.452748\pi$
$432$	0	0
$433$	9197.82	1.02083	0.510415	−	0.859928i	$-0.329492\pi$
$433$	9197.82	1.02083	0.510415	+	0.859928i	$0.329492\pi$
$434$	0	0
$435$	−196.253	−0.0216313
$436$	0	0
$437$	−948.297	−0.103806
$438$	0	0
$439$	−9360.53	−1.01766	−0.508831	−	0.860866i	$-0.669922\pi$
$439$	−9360.53	−1.01766	−0.508831	+	0.860866i	$0.669922\pi$
$440$	0	0
$441$	22024.4	2.37818
$442$	0	0
$443$	3196.32	0.342803	0.171401	−	0.985201i	$-0.445170\pi$
$443$	3196.32	0.342803	0.171401	+	0.985201i	$0.445170\pi$
$444$	0	0
$445$	−2007.51	−0.213854
$446$	0	0
$447$	−10383.3	−1.09869
$448$	0	0
$449$	2844.45	0.298970	0.149485	−	0.988764i	$-0.452238\pi$
$449$	2844.45	0.298970	0.149485	+	0.988764i	$0.452238\pi$
$450$	0	0
$451$	3368.71	0.351721
$452$	0	0
$453$	33985.1	3.52485
$454$	0	0
$455$	0	0
$456$	0	0
$457$	4468.90	0.457431	0.228716	−	0.973493i	$-0.426547\pi$
$457$	4468.90	0.457431	0.228716	+	0.973493i	$0.426547\pi$
$458$	0	0
$459$	−29582.4	−3.00825
$460$	0	0
$461$	−15006.8	−1.51613	−0.758067	−	0.652177i	$-0.773856\pi$
$461$	−15006.8	−1.51613	−0.758067	+	0.652177i	$0.773856\pi$
$462$	0	0
$463$	−6247.37	−0.627084	−0.313542	−	0.949574i	$-0.601516\pi$
$463$	−6247.37	−0.627084	−0.313542	+	0.949574i	$0.601516\pi$
$464$	0	0
$465$	−765.963	−0.0763886
$466$	0	0
$467$	−977.327	−0.0968422	−0.0484211	−	0.998827i	$-0.515419\pi$
$467$	−977.327	−0.0968422	−0.0484211	+	0.998827i	$0.515419\pi$
$468$	0	0
$469$	−19627.2	−1.93241
$470$	0	0
$471$	23810.9	2.32940
$472$	0	0
$473$	−26561.4	−2.58202
$474$	0	0
$475$	11285.2	1.09010
$476$	0	0
$477$	37584.1	3.60766
$478$	0	0
$479$	−8832.44	−0.842515	−0.421257	−	0.906941i	$-0.638411\pi$
$479$	−8832.44	−0.842515	−0.421257	+	0.906941i	$0.638411\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−2369.56	−0.223227
$484$	0	0
$485$	1395.39	0.130642
$486$	0	0
$487$	9906.44	0.921774	0.460887	−	0.887459i	$-0.347531\pi$
$487$	9906.44	0.921774	0.460887	+	0.887459i	$0.347531\pi$
$488$	0	0
$489$	−2374.81	−0.219617
$490$	0	0
$491$	11073.6	1.01781	0.508906	−	0.860822i	$-0.330050\pi$
$491$	11073.6	1.01781	0.508906	+	0.860822i	$0.330050\pi$
$492$	0	0
$493$	−529.246	−0.0483490
$494$	0	0
$495$	12527.7	1.13753
$496$	0	0
$497$	−23276.7	−2.10081
$498$	0	0
$499$	3268.09	0.293186	0.146593	−	0.989197i	$-0.453169\pi$
$499$	3268.09	0.293186	0.146593	+	0.989197i	$0.453169\pi$
$500$	0	0
$501$	−4261.85	−0.380050
$502$	0	0
$503$	−18758.6	−1.66283	−0.831416	−	0.555651i	$-0.812469\pi$
$503$	−18758.6	−1.66283	−0.831416	+	0.555651i	$0.812469\pi$
$504$	0	0
$505$	2414.69	0.212777
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−6121.28	−0.533047	−0.266523	−	0.963828i	$-0.585875\pi$
$509$	−6121.28	−0.533047	−0.266523	+	0.963828i	$0.585875\pi$
$510$	0	0
$511$	8426.69	0.729501
$512$	0	0
$513$	33481.2	2.88154
$514$	0	0
$515$	−4444.97	−0.380327
$516$	0	0
$517$	−28780.0	−2.44825
$518$	0	0
$519$	2344.83	0.198317
$520$	0	0
$521$	−12976.1	−1.09116	−0.545578	−	0.838060i	$-0.683690\pi$
$521$	−12976.1	−1.09116	−0.545578	+	0.838060i	$0.683690\pi$
$522$	0	0
$523$	−11563.9	−0.966832	−0.483416	−	0.875391i	$-0.660604\pi$
$523$	−11563.9	−0.966832	−0.483416	+	0.875391i	$0.660604\pi$
$524$	0	0
$525$	28198.9	2.34419
$526$	0	0
$527$	−2065.61	−0.170739
$528$	0	0
$529$	−12076.8	−0.992584
$530$	0	0
$531$	−13199.7	−1.07875
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−4598.84	−0.371636
$536$	0	0
$537$	11982.3	0.962897
$538$	0	0
$539$	−20452.3	−1.63440
$540$	0	0
$541$	11231.9	0.892603	0.446302	−	0.894883i	$-0.352741\pi$
$541$	11231.9	0.892603	0.446302	+	0.894883i	$0.352741\pi$
$542$	0	0
$543$	−38290.6	−3.02616
$544$	0	0
$545$	−3288.90	−0.258497
$546$	0	0
$547$	5604.16	0.438056	0.219028	−	0.975719i	$-0.429711\pi$
$547$	5604.16	0.438056	0.219028	+	0.975719i	$0.429711\pi$
$548$	0	0
$549$	25914.7	2.01459
$550$	0	0
$551$	598.998	0.0463124
$552$	0	0
$553$	−12941.1	−0.995139
$554$	0	0
$555$	−1470.49	−0.112467
$556$	0	0
$557$	−13863.4	−1.05460	−0.527299	−	0.849680i	$-0.676795\pi$
$557$	−13863.4	−1.05460	−0.527299	+	0.849680i	$0.676795\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	48388.8	3.64167
$562$	0	0
$563$	9602.64	0.718833	0.359417	−	0.933177i	$-0.382976\pi$
$563$	9602.64	0.718833	0.359417	+	0.933177i	$0.382976\pi$
$564$	0	0
$565$	−310.647	−0.0231310
$566$	0	0
$567$	39183.8	2.90223
$568$	0	0
$569$	−21102.9	−1.55480	−0.777399	−	0.629008i	$-0.783461\pi$
$569$	−21102.9	−1.55480	−0.777399	+	0.629008i	$0.783461\pi$
$570$	0	0
$571$	20570.0	1.50758	0.753790	−	0.657115i	$-0.228224\pi$
$571$	20570.0	1.50758	0.753790	+	0.657115i	$0.228224\pi$
$572$	0	0
$573$	38035.2	2.77302
$574$	0	0
$575$	−1073.75	−0.0778759
$576$	0	0
$577$	−7896.95	−0.569765	−0.284882	−	0.958562i	$-0.591955\pi$
$577$	−7896.95	−0.569765	−0.284882	+	0.958562i	$0.591955\pi$
$578$	0	0
$579$	−14467.2	−1.03840
$580$	0	0
$581$	5919.01	0.422654
$582$	0	0
$583$	−34901.3	−2.47936
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15872.7	1.11608	0.558039	−	0.829815i	$-0.311554\pi$
$587$	15872.7	1.11608	0.558039	+	0.829815i	$0.311554\pi$
$588$	0	0
$589$	2337.85	0.163548
$590$	0	0
$591$	10839.5	0.754447
$592$	0	0
$593$	4037.35	0.279585	0.139793	−	0.990181i	$-0.455356\pi$
$593$	4037.35	0.279585	0.139793	+	0.990181i	$0.455356\pi$
$594$	0	0
$595$	−8045.42	−0.554336
$596$	0	0
$597$	49089.0	3.36529
$598$	0	0
$599$	−18225.7	−1.24321	−0.621606	−	0.783330i	$-0.713519\pi$
$599$	−18225.7	−1.24321	−0.621606	+	0.783330i	$0.713519\pi$
$600$	0	0
$601$	5555.22	0.377042	0.188521	−	0.982069i	$-0.439631\pi$
$601$	5555.22	0.377042	0.188521	+	0.982069i	$0.439631\pi$
$602$	0	0
$603$	−46479.4	−3.13895
$604$	0	0
$605$	−7030.59	−0.472453
$606$	0	0
$607$	4083.66	0.273066	0.136533	−	0.990636i	$-0.456404\pi$
$607$	4083.66	0.273066	0.136533	+	0.990636i	$0.456404\pi$
$608$	0	0
$609$	1496.75	0.0995917
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1065.87	−0.0702285	−0.0351143	−	0.999383i	$-0.511180\pi$
$613$	−1065.87	−0.0702285	−0.0351143	+	0.999383i	$0.511180\pi$
$614$	0	0
$615$	−1899.77	−0.124563
$616$	0	0
$617$	3183.51	0.207720	0.103860	−	0.994592i	$-0.466881\pi$
$617$	3183.51	0.207720	0.103860	+	0.994592i	$0.466881\pi$
$618$	0	0
$619$	3361.17	0.218250	0.109125	−	0.994028i	$-0.465195\pi$
$619$	3361.17	0.218250	0.109125	+	0.994028i	$0.465195\pi$
$620$	0	0
$621$	−3185.65	−0.205854
$622$	0	0
$623$	15310.5	0.984597
$624$	0	0
$625$	11283.3	0.722128
$626$	0	0
$627$	−54766.1	−3.48828
$628$	0	0
$629$	−3965.56	−0.251379
$630$	0	0
$631$	−17306.6	−1.09186	−0.545932	−	0.837829i	$-0.683824\pi$
$631$	−17306.6	−1.09186	−0.545932	+	0.837829i	$0.683824\pi$
$632$	0	0
$633$	−12036.8	−0.755798
$634$	0	0
$635$	−1193.13	−0.0745638
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−55121.7	−3.41249
$640$	0	0
$641$	20526.1	1.26479	0.632397	−	0.774644i	$-0.282071\pi$
$641$	20526.1	1.26479	0.632397	+	0.774644i	$0.282071\pi$
$642$	0	0
$643$	−17395.8	−1.06691	−0.533456	−	0.845828i	$-0.679107\pi$
$643$	−17395.8	−1.06691	−0.533456	+	0.845828i	$0.679107\pi$
$644$	0	0
$645$	14979.2	0.914424
$646$	0	0
$647$	1475.37	0.0896486	0.0448243	−	0.998995i	$-0.485727\pi$
$647$	1475.37	0.0896486	0.0448243	+	0.998995i	$0.485727\pi$
$648$	0	0
$649$	12257.5	0.741371
$650$	0	0
$651$	5841.72	0.351698
$652$	0	0
$653$	−4918.78	−0.294773	−0.147387	−	0.989079i	$-0.547086\pi$
$653$	−4918.78	−0.294773	−0.147387	+	0.989079i	$0.547086\pi$
$654$	0	0
$655$	7096.41	0.423328
$656$	0	0
$657$	19955.3	1.18498
$658$	0	0
$659$	−3938.97	−0.232839	−0.116419	−	0.993200i	$-0.537142\pi$
$659$	−3938.97	−0.232839	−0.116419	+	0.993200i	$0.537142\pi$
$660$	0	0
$661$	−31950.2	−1.88006	−0.940028	−	0.341097i	$-0.889202\pi$
$661$	−31950.2	−1.88006	−0.940028	+	0.341097i	$0.889202\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9105.76	0.530987
$666$	0	0
$667$	−56.9930	−0.00330851
$668$	0	0
$669$	31598.7	1.82612
$670$	0	0
$671$	−24064.9	−1.38452
$672$	0	0
$673$	14912.6	0.854141	0.427070	−	0.904218i	$-0.359546\pi$
$673$	14912.6	0.854141	0.427070	+	0.904218i	$0.359546\pi$
$674$	0	0
$675$	37910.7	2.16175
$676$	0	0
$677$	−24876.2	−1.41221	−0.706107	−	0.708105i	$-0.749550\pi$
$677$	−24876.2	−1.41221	−0.706107	+	0.708105i	$0.749550\pi$
$678$	0	0
$679$	−10642.2	−0.601486
$680$	0	0
$681$	−33832.8	−1.90378
$682$	0	0
$683$	5819.10	0.326005	0.163003	−	0.986626i	$-0.447882\pi$
$683$	5819.10	0.326005	0.163003	+	0.986626i	$0.447882\pi$
$684$	0	0
$685$	6752.49	0.376642
$686$	0	0
$687$	58786.2	3.26468
$688$	0	0
$689$	0	0
$690$	0	0
$691$	23926.4	1.31722	0.658612	−	0.752483i	$-0.271144\pi$
$691$	23926.4	1.31722	0.658612	+	0.752483i	$0.271144\pi$
$692$	0	0
$693$	−95544.4	−5.23728
$694$	0	0
$695$	7741.90	0.422543
$696$	0	0
$697$	−5123.21	−0.278415
$698$	0	0
$699$	−20458.3	−1.10702
$700$	0	0
$701$	−11493.3	−0.619251	−0.309626	−	0.950859i	$-0.600204\pi$
$701$	−11493.3	−0.619251	−0.309626	+	0.950859i	$0.600204\pi$
$702$	0	0
$703$	4488.20	0.240790
$704$	0	0
$705$	16230.4	0.867051
$706$	0	0
$707$	−18416.0	−0.979638
$708$	0	0
$709$	−11011.4	−0.583276	−0.291638	−	0.956529i	$-0.594200\pi$
$709$	−11011.4	−0.583276	−0.291638	+	0.956529i	$0.594200\pi$
$710$	0	0
$711$	−30646.0	−1.61648
$712$	0	0
$713$	−222.440	−0.0116837
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−27509.4	−1.43286
$718$	0	0
$719$	−14546.9	−0.754529	−0.377265	−	0.926106i	$-0.623135\pi$
$719$	−14546.9	−0.754529	−0.377265	+	0.926106i	$0.623135\pi$
$720$	0	0
$721$	33900.1	1.75105
$722$	0	0
$723$	13162.4	0.677061
$724$	0	0
$725$	678.244	0.0347439
$726$	0	0
$727$	7971.12	0.406647	0.203323	−	0.979112i	$-0.434826\pi$
$727$	7971.12	0.406647	0.203323	+	0.979112i	$0.434826\pi$
$728$	0	0
$729$	7146.79	0.363095
$730$	0	0
$731$	40395.1	2.04387
$732$	0	0
$733$	4191.29	0.211199	0.105599	−	0.994409i	$-0.466324\pi$
$733$	4191.29	0.211199	0.105599	+	0.994409i	$0.466324\pi$
$734$	0	0
$735$	11534.0	0.578825
$736$	0	0
$737$	43161.7	2.15723
$738$	0	0
$739$	−15062.1	−0.749756	−0.374878	−	0.927074i	$-0.622315\pi$
$739$	−15062.1	−0.749756	−0.374878	+	0.927074i	$0.622315\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8785.14	0.433776	0.216888	−	0.976197i	$-0.430409\pi$
$743$	8785.14	0.433776	0.216888	+	0.976197i	$0.430409\pi$
$744$	0	0
$745$	−3796.49	−0.186701
$746$	0	0
$747$	14016.9	0.686547
$748$	0	0
$749$	35073.7	1.71104
$750$	0	0
$751$	18437.0	0.895841	0.447920	−	0.894073i	$-0.352165\pi$
$751$	18437.0	0.895841	0.447920	+	0.894073i	$0.352165\pi$
$752$	0	0
$753$	−24914.1	−1.20574
$754$	0	0
$755$	12426.0	0.598980
$756$	0	0
$757$	−26856.4	−1.28945	−0.644725	−	0.764415i	$-0.723028\pi$
$757$	−26856.4	−1.28945	−0.644725	+	0.764415i	$0.723028\pi$
$758$	0	0
$759$	5210.85	0.249199
$760$	0	0
$761$	21550.1	1.02653	0.513265	−	0.858230i	$-0.328436\pi$
$761$	21550.1	1.02653	0.513265	+	0.858230i	$0.328436\pi$
$762$	0	0
$763$	25083.2	1.19014
$764$	0	0
$765$	−19052.4	−0.900448
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−35567.3	−1.66787	−0.833934	−	0.551864i	$-0.813917\pi$
$769$	−35567.3	−1.66787	−0.833934	+	0.551864i	$0.813917\pi$
$770$	0	0
$771$	−4179.04	−0.195207
$772$	0	0
$773$	−20762.0	−0.966053	−0.483026	−	0.875606i	$-0.660462\pi$
$773$	−20762.0	−0.966053	−0.483026	+	0.875606i	$0.660462\pi$
$774$	0	0
$775$	2647.14	0.122694
$776$	0	0
$777$	11214.9	0.517803
$778$	0	0
$779$	5798.42	0.266688
$780$	0	0
$781$	51187.2	2.34522
$782$	0	0
$783$	2012.23	0.0918409
$784$	0	0
$785$	8706.03	0.395836
$786$	0	0
$787$	19483.4	0.882475	0.441237	−	0.897390i	$-0.354540\pi$
$787$	19483.4	0.882475	0.441237	+	0.897390i	$0.354540\pi$
$788$	0	0
$789$	19865.4	0.896357
$790$	0	0
$791$	2369.20	0.106497
$792$	0	0
$793$	0	0
$794$	0	0
$795$	19682.4	0.878068
$796$	0	0
$797$	−11672.8	−0.518783	−0.259392	−	0.965772i	$-0.583522\pi$
$797$	−11672.8	−0.518783	−0.259392	+	0.965772i	$0.583522\pi$
$798$	0	0
$799$	43769.3	1.93798
$800$	0	0
$801$	36257.1	1.59935
$802$	0	0
$803$	−18530.9	−0.814374
$804$	0	0
$805$	−866.389	−0.0379332
$806$	0	0
$807$	−60839.2	−2.65383
$808$	0	0
$809$	−18423.5	−0.800663	−0.400331	−	0.916370i	$-0.631105\pi$
$809$	−18423.5	−0.800663	−0.400331	+	0.916370i	$0.631105\pi$
$810$	0	0
$811$	−16287.6	−0.705222	−0.352611	−	0.935770i	$-0.614706\pi$
$811$	−16287.6	−0.705222	−0.352611	+	0.935770i	$0.614706\pi$
$812$	0	0
$813$	29255.6	1.26204
$814$	0	0
$815$	−868.306	−0.0373196
$816$	0	0
$817$	−45719.0	−1.95778
$818$	0	0
$819$	0	0
$820$	0	0
$821$	26118.8	1.11030	0.555148	−	0.831751i	$-0.312661\pi$
$821$	26118.8	1.11030	0.555148	+	0.831751i	$0.312661\pi$
$822$	0	0
$823$	−13571.1	−0.574798	−0.287399	−	0.957811i	$-0.592791\pi$
$823$	−13571.1	−0.574798	−0.287399	+	0.957811i	$0.592791\pi$
$824$	0	0
$825$	−62011.6	−2.61693
$826$	0	0
$827$	−45842.8	−1.92758	−0.963790	−	0.266662i	$-0.914079\pi$
$827$	−45842.8	−1.92758	−0.963790	+	0.266662i	$0.914079\pi$
$828$	0	0
$829$	2847.06	0.119279	0.0596397	−	0.998220i	$-0.481005\pi$
$829$	2847.06	0.119279	0.0596397	+	0.998220i	$0.481005\pi$
$830$	0	0
$831$	10239.2	0.427429
$832$	0	0
$833$	31104.2	1.29376
$834$	0	0
$835$	−1558.27	−0.0645822
$836$	0	0
$837$	7853.63	0.324326
$838$	0	0
$839$	28995.5	1.19313	0.596564	−	0.802565i	$-0.296532\pi$
$839$	28995.5	1.19313	0.596564	+	0.802565i	$0.296532\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	0	0
$843$	46211.5	1.88803
$844$	0	0
$845$	0	0
$846$	0	0
$847$	53619.8	2.17520
$848$	0	0
$849$	77468.4	3.13158
$850$	0	0
$851$	−427.040	−0.0172018
$852$	0	0
$853$	−14179.1	−0.569148	−0.284574	−	0.958654i	$-0.591852\pi$
$853$	−14179.1	−0.569148	−0.284574	+	0.958654i	$0.591852\pi$
$854$	0	0
$855$	21563.5	0.862520
$856$	0	0
$857$	−28026.0	−1.11709	−0.558547	−	0.829473i	$-0.688641\pi$
$857$	−28026.0	−1.11709	−0.558547	+	0.829473i	$0.688641\pi$
$858$	0	0
$859$	−20995.6	−0.833946	−0.416973	−	0.908919i	$-0.636909\pi$
$859$	−20995.6	−0.833946	−0.416973	+	0.908919i	$0.636909\pi$
$860$	0	0
$861$	14488.8	0.573494
$862$	0	0
$863$	25139.2	0.991596	0.495798	−	0.868438i	$-0.334876\pi$
$863$	25139.2	0.991596	0.495798	+	0.868438i	$0.334876\pi$
$864$	0	0
$865$	857.346	0.0337001
$866$	0	0
$867$	−27122.3	−1.06243
$868$	0	0
$869$	28458.5	1.11092
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−25201.8	−0.977036
$874$	0	0
$875$	21711.7	0.838844
$876$	0	0
$877$	−34676.4	−1.33516	−0.667582	−	0.744537i	$-0.732670\pi$
$877$	−34676.4	−1.33516	−0.667582	+	0.744537i	$0.732670\pi$
$878$	0	0
$879$	−11042.9	−0.423739
$880$	0	0
$881$	−25864.8	−0.989110	−0.494555	−	0.869146i	$-0.664669\pi$
$881$	−25864.8	−0.989110	−0.494555	+	0.869146i	$0.664669\pi$
$882$	0	0
$883$	−29347.0	−1.11847	−0.559233	−	0.829011i	$-0.688904\pi$
$883$	−29347.0	−1.11847	−0.559233	+	0.829011i	$0.688904\pi$
$884$	0	0
$885$	−6912.56	−0.262557
$886$	0	0
$887$	14292.1	0.541015	0.270508	−	0.962718i	$-0.412808\pi$
$887$	14292.1	0.541015	0.270508	+	0.962718i	$0.412808\pi$
$888$	0	0
$889$	9099.58	0.343296
$890$	0	0
$891$	−86168.2	−3.23989
$892$	0	0
$893$	−49537.9	−1.85635
$894$	0	0
$895$	4381.12	0.163626
$896$	0	0
$897$	0	0
$898$	0	0
$899$	140.506	0.00521260
$900$	0	0
$901$	53078.7	1.96261
$902$	0	0
$903$	−114241.	−4.21006
$904$	0	0
$905$	−14000.3	−0.514237
$906$	0	0
$907$	−19319.4	−0.707264	−0.353632	−	0.935385i	$-0.615054\pi$
$907$	−19319.4	−0.707264	−0.353632	+	0.935385i	$0.615054\pi$
$908$	0	0
$909$	−43611.1	−1.59130
$910$	0	0
$911$	−47558.1	−1.72961	−0.864803	−	0.502112i	$-0.832557\pi$
$911$	−47558.1	−1.72961	−0.864803	+	0.502112i	$0.832557\pi$
$912$	0	0
$913$	−13016.4	−0.471828
$914$	0	0
$915$	13571.3	0.490331
$916$	0	0
$917$	−54121.7	−1.94903
$918$	0	0
$919$	49795.2	1.78737	0.893684	−	0.448696i	$-0.148111\pi$
$919$	49795.2	1.78737	0.893684	+	0.448696i	$0.148111\pi$
$920$	0	0
$921$	37152.1	1.32921
$922$	0	0
$923$	0	0
$924$	0	0
$925$	5081.97	0.180642
$926$	0	0
$927$	80279.3	2.84435
$928$	0	0
$929$	18163.8	0.641479	0.320739	−	0.947167i	$-0.396069\pi$
$929$	18163.8	0.641479	0.320739	+	0.947167i	$0.396069\pi$
$930$	0	0
$931$	−35203.6	−1.23926
$932$	0	0
$933$	26945.2	0.945493
$934$	0	0
$935$	17692.5	0.618830
$936$	0	0
$937$	3915.72	0.136522	0.0682610	−	0.997667i	$-0.478255\pi$
$937$	3915.72	0.136522	0.0682610	+	0.997667i	$0.478255\pi$
$938$	0	0
$939$	56659.9	1.96914
$940$	0	0
$941$	38184.8	1.32284	0.661418	−	0.750017i	$-0.269955\pi$
$941$	38184.8	1.32284	0.661418	+	0.750017i	$0.269955\pi$
$942$	0	0
$943$	−551.704	−0.0190519
$944$	0	0
$945$	30589.3	1.05298
$946$	0	0
$947$	35496.7	1.21804	0.609022	−	0.793154i	$-0.291562\pi$
$947$	35496.7	1.21804	0.609022	+	0.793154i	$0.291562\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−33310.3	−1.13581
$952$	0	0
$953$	51103.0	1.73703	0.868515	−	0.495663i	$-0.165075\pi$
$953$	51103.0	1.73703	0.868515	+	0.495663i	$0.165075\pi$
$954$	0	0
$955$	13906.9	0.471221
$956$	0	0
$957$	−3291.47	−0.111179
$958$	0	0
$959$	−51498.8	−1.73408
$960$	0	0
$961$	−29242.6	−0.981592
$962$	0	0
$963$	83058.4	2.77936
$964$	0	0
$965$	−5289.67	−0.176456
$966$	0	0
$967$	−52681.0	−1.75192	−0.875961	−	0.482383i	$-0.839771\pi$
$967$	−52681.0	−1.75192	−0.875961	+	0.482383i	$0.839771\pi$
$968$	0	0
$969$	83289.5	2.76124
$970$	0	0
$971$	28878.6	0.954437	0.477219	−	0.878785i	$-0.341645\pi$
$971$	28878.6	0.954437	0.477219	+	0.878785i	$0.341645\pi$
$972$	0	0
$973$	−59044.7	−1.94541
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20043.8	0.656355	0.328178	−	0.944616i	$-0.393566\pi$
$977$	20043.8	0.656355	0.328178	+	0.944616i	$0.393566\pi$
$978$	0	0
$979$	−33669.1	−1.09915
$980$	0	0
$981$	59399.9	1.93322
$982$	0	0
$983$	14937.1	0.484659	0.242330	−	0.970194i	$-0.422088\pi$
$983$	14937.1	0.484659	0.242330	+	0.970194i	$0.422088\pi$
$984$	0	0
$985$	3963.28	0.128204
$986$	0	0
$987$	−123783.	−3.99195
$988$	0	0
$989$	4350.04	0.139862
$990$	0	0
$991$	24226.9	0.776582	0.388291	−	0.921537i	$-0.373066\pi$
$991$	24226.9	0.776582	0.388291	+	0.921537i	$0.373066\pi$
$992$	0	0
$993$	14068.0	0.449581
$994$	0	0
$995$	17948.5	0.571865
$996$	0	0
$997$	−38.6217	−0.00122684	−0.000613421	−	1.00000i	$-0.500195\pi$
$997$	−38.6217	−0.00122684	−0.000613421		1.00000i	$0.500195\pi$
$998$	0	0
$999$	15077.4	0.477504

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1352.4.a.g.1.1		2
13.12	even	2		104.4.a.d.1.1	✓	2
39.38	odd	2		936.4.a.i.1.1		2
52.51	odd	2		208.4.a.i.1.2		2
104.51	odd	2		832.4.a.v.1.1		2
104.77	even	2		832.4.a.w.1.2		2
156.155	even	2		1872.4.a.bh.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.4.a.d.1.1	✓	2	13.12	even	2
208.4.a.i.1.2		2	52.51	odd	2
832.4.a.v.1.1		2	104.51	odd	2
832.4.a.w.1.2		2	104.77	even	2
936.4.a.i.1.1		2	39.38	odd	2
1352.4.a.g.1.1		2	1.1	even	1	trivial
1872.4.a.bh.1.1		2	156.155	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:	\( N \)	\(=\)	\( 1352 = 2^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1352.a (trivial)

\(f(q)\)	\(=\)	\(q-9.45824 q^{3} -3.45824 q^{5} +26.3747 q^{7} +62.4582 q^{9} +O(q^{10})\)	\(q-9.45824 q^{3} -3.45824 q^{5} +26.3747 q^{7} +62.4582 q^{9} -58.0000 q^{11} +32.7088 q^{15} +88.2077 q^{17} -99.8329 q^{19} -249.458 q^{21} +9.49884 q^{23} -113.041 q^{25} -335.372 q^{27} -6.00000 q^{29} -23.4176 q^{31} +548.578 q^{33} -91.2100 q^{35} -44.9571 q^{37} -58.0812 q^{41} +457.955 q^{43} -215.995 q^{45} +496.208 q^{47} +352.625 q^{49} -834.289 q^{51} +601.747 q^{53} +200.578 q^{55} +944.244 q^{57} -211.336 q^{59} +414.912 q^{61} +1647.32 q^{63} -744.167 q^{67} -89.8422 q^{69} -882.537 q^{71} +319.499 q^{73} +1069.16 q^{75} -1529.73 q^{77} -490.664 q^{79} +1485.66 q^{81} +224.420 q^{83} -305.043 q^{85} +56.7494 q^{87} +580.501 q^{89} +221.490 q^{93} +345.246 q^{95} -403.499 q^{97} -3622.58 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 11 q^{5} - q^{7} + 107 q^{9}+O(q^{10}) \) 2 * q - q^3 + 11 * q^5 - q^7 + 107 * q^9	\( 2 q - q^{3} + 11 q^{5} - q^{7} + 107 q^{9} - 116 q^{11} + 155 q^{15} + 51 q^{17} - 128 q^{19} - 481 q^{21} - 196 q^{23} - 29 q^{25} - 187 q^{27} - 12 q^{29} - 226 q^{31} + 58 q^{33} - 487 q^{35} + 143 q^{37} + 278 q^{41} + 253 q^{43} + 428 q^{45} + 867 q^{47} + 759 q^{49} - 1149 q^{51} + 666 q^{53} - 638 q^{55} + 706 q^{57} - 996 q^{59} - 66 q^{61} + 428 q^{63} - 1560 q^{67} - 1828 q^{69} - 923 q^{71} + 424 q^{73} + 1780 q^{75} + 58 q^{77} - 408 q^{79} + 1538 q^{81} + 1058 q^{83} - 843 q^{85} + 6 q^{87} + 1376 q^{89} - 1492 q^{93} - 62 q^{95} - 592 q^{97} - 6206 q^{99}+O(q^{100}) \) 2 * q - q^3 + 11 * q^5 - q^7 + 107 * q^9 - 116 * q^11 + 155 * q^15 + 51 * q^17 - 128 * q^19 - 481 * q^21 - 196 * q^23 - 29 * q^25 - 187 * q^27 - 12 * q^29 - 226 * q^31 + 58 * q^33 - 487 * q^35 + 143 * q^37 + 278 * q^41 + 253 * q^43 + 428 * q^45 + 867 * q^47 + 759 * q^49 - 1149 * q^51 + 666 * q^53 - 638 * q^55 + 706 * q^57 - 996 * q^59 - 66 * q^61 + 428 * q^63 - 1560 * q^67 - 1828 * q^69 - 923 * q^71 + 424 * q^73 + 1780 * q^75 + 58 * q^77 - 408 * q^79 + 1538 * q^81 + 1058 * q^83 - 843 * q^85 + 6 * q^87 + 1376 * q^89 - 1492 * q^93 - 62 * q^95 - 592 * q^97 - 6206 * q^99

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