LMFDB - Embedded newform 1352.2.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1352,2,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, 2, names="a")

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

chi := DirichletCharacter("1352.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1352 = 2^{3} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
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:	$10.7957743533$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.3728753.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 11x^{3} + 22x^{2} - 18x - 13 $ x^6 - x^5 - 10x^4 + 11x^3 + 22x^2 - 18x - 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.516708$ 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)

$f(q)$	$=$	$q-0.0716659 q^{3} -3.69476 q^{5} -1.67432 q^{7} -2.99486 q^{9} +O(q^{10})$	$q-0.0716659 q^{3} -3.69476 q^{5} -1.67432 q^{7} -2.99486 q^{9} -5.28797 q^{11} +0.264789 q^{15} -0.533731 q^{17} +4.65621 q^{19} +0.119992 q^{21} +8.51440 q^{23} +8.65127 q^{25} +0.429628 q^{27} +4.48486 q^{29} +0.899181 q^{31} +0.378967 q^{33} +6.18622 q^{35} +3.17170 q^{37} +10.2036 q^{41} -5.24467 q^{43} +11.0653 q^{45} -4.09333 q^{47} -4.19665 q^{49} +0.0382503 q^{51} -9.76260 q^{53} +19.5378 q^{55} -0.333692 q^{57} -10.8913 q^{59} +0.994321 q^{61} +5.01437 q^{63} +11.7805 q^{67} -0.610193 q^{69} -12.3134 q^{71} -4.70617 q^{73} -0.620002 q^{75} +8.85376 q^{77} +8.31096 q^{79} +8.95380 q^{81} -12.5820 q^{83} +1.97201 q^{85} -0.321412 q^{87} +2.12572 q^{89} -0.0644407 q^{93} -17.2036 q^{95} +13.2660 q^{97} +15.8368 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{3} + 2 q^{5} - 3 q^{7} + 9 q^{9}+O(q^{10}) $ 6 * q + 3 * q^3 + 2 * q^5 - 3 * q^7 + 9 * q^9	$ 6 q + 3 q^{3} + 2 q^{5} - 3 q^{7} + 9 q^{9} - 13 q^{11} + 14 q^{15} + 11 q^{17} - 15 q^{19} + 22 q^{21} + 15 q^{23} + 16 q^{25} + 21 q^{27} + 9 q^{29} + 3 q^{31} + 2 q^{33} + 14 q^{35} - 14 q^{37} + 20 q^{41} + 10 q^{43} + 9 q^{45} + 10 q^{47} + 27 q^{49} + 5 q^{51} + q^{53} + 14 q^{55} - 9 q^{57} - 50 q^{59} - 2 q^{63} - 6 q^{67} + 32 q^{69} - 9 q^{71} + 6 q^{73} + 40 q^{75} - 2 q^{77} + 13 q^{79} + 42 q^{81} - 36 q^{83} + 31 q^{85} + 22 q^{87} + 18 q^{89} + 12 q^{93} - 21 q^{95} + 14 q^{97} - 18 q^{99}+O(q^{100}) $ 6 * q + 3 * q^3 + 2 * q^5 - 3 * q^7 + 9 * q^9 - 13 * q^11 + 14 * q^15 + 11 * q^17 - 15 * q^19 + 22 * q^21 + 15 * q^23 + 16 * q^25 + 21 * q^27 + 9 * q^29 + 3 * q^31 + 2 * q^33 + 14 * q^35 - 14 * q^37 + 20 * q^41 + 10 * q^43 + 9 * q^45 + 10 * q^47 + 27 * q^49 + 5 * q^51 + q^53 + 14 * q^55 - 9 * q^57 - 50 * q^59 - 2 * q^63 - 6 * q^67 + 32 * q^69 - 9 * q^71 + 6 * q^73 + 40 * q^75 - 2 * q^77 + 13 * q^79 + 42 * q^81 - 36 * q^83 + 31 * q^85 + 22 * q^87 + 18 * q^89 + 12 * q^93 - 21 * q^95 + 14 * q^97 - 18 * 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
$3$	−0.0716659	−0.0413764	−0.0206882	−	0.999786i	$-0.506586\pi$
$3$	−0.0716659	−0.0413764	−0.0206882	+	0.999786i	$0.506586\pi$
$4$	0	0
$5$	−3.69476	−1.65235	−0.826174	−	0.563415i	$-0.809487\pi$
$5$	−3.69476	−1.65235	−0.826174	+	0.563415i	$0.809487\pi$
$6$	0	0
$7$	−1.67432	−0.632834	−0.316417	−	0.948620i	$-0.602480\pi$
$7$	−1.67432	−0.632834	−0.316417	+	0.948620i	$0.602480\pi$
$8$	0	0
$9$	−2.99486	−0.998288
$10$	0	0
$11$	−5.28797	−1.59438	−0.797192	−	0.603726i	$-0.793682\pi$
$11$	−5.28797	−1.59438	−0.797192	+	0.603726i	$0.793682\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.264789	0.0683681
$16$	0	0
$17$	−0.533731	−0.129449	−0.0647244	−	0.997903i	$-0.520617\pi$
$17$	−0.533731	−0.129449	−0.0647244	+	0.997903i	$0.520617\pi$
$18$	0	0
$19$	4.65621	1.06821	0.534104	−	0.845419i	$-0.320649\pi$
$19$	4.65621	1.06821	0.534104	+	0.845419i	$0.320649\pi$
$20$	0	0
$21$	0.119992	0.0261844
$22$	0	0
$23$	8.51440	1.77538	0.887688	−	0.460446i	$-0.152311\pi$
$23$	8.51440	1.77538	0.887688	+	0.460446i	$0.152311\pi$
$24$	0	0
$25$	8.65127	1.73025
$26$	0	0
$27$	0.429628	0.0826819
$28$	0	0
$29$	4.48486	0.832818	0.416409	−	0.909177i	$-0.363288\pi$
$29$	4.48486	0.832818	0.416409	+	0.909177i	$0.363288\pi$
$30$	0	0
$31$	0.899181	0.161498	0.0807488	−	0.996734i	$-0.474269\pi$
$31$	0.899181	0.161498	0.0807488	+	0.996734i	$0.474269\pi$
$32$	0	0
$33$	0.378967	0.0659698
$34$	0	0
$35$	6.18622	1.04566
$36$	0	0
$37$	3.17170	0.521424	0.260712	−	0.965417i	$-0.416043\pi$
$37$	3.17170	0.521424	0.260712	+	0.965417i	$0.416043\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.2036	1.59353	0.796767	−	0.604287i	$-0.206542\pi$
$41$	10.2036	1.59353	0.796767	+	0.604287i	$0.206542\pi$
$42$	0	0
$43$	−5.24467	−0.799805	−0.399903	−	0.916558i	$-0.630956\pi$
$43$	−5.24467	−0.799805	−0.399903	+	0.916558i	$0.630956\pi$
$44$	0	0
$45$	11.0653	1.64952
$46$	0	0
$47$	−4.09333	−0.597073	−0.298537	−	0.954398i	$-0.596498\pi$
$47$	−4.09333	−0.597073	−0.298537	+	0.954398i	$0.596498\pi$
$48$	0	0
$49$	−4.19665	−0.599521
$50$	0	0
$51$	0.0382503	0.00535612
$52$	0	0
$53$	−9.76260	−1.34100	−0.670498	−	0.741911i	$-0.733920\pi$
$53$	−9.76260	−1.34100	−0.670498	+	0.741911i	$0.733920\pi$
$54$	0	0
$55$	19.5378	2.63448
$56$	0	0
$57$	−0.333692	−0.0441986
$58$	0	0
$59$	−10.8913	−1.41793	−0.708963	−	0.705245i	$-0.750837\pi$
$59$	−10.8913	−1.41793	−0.708963	+	0.705245i	$0.750837\pi$
$60$	0	0
$61$	0.994321	0.127310	0.0636549	−	0.997972i	$-0.479724\pi$
$61$	0.994321	0.127310	0.0636549	+	0.997972i	$0.479724\pi$
$62$	0	0
$63$	5.01437	0.631751
$64$	0	0
$65$	0	0
$66$	0	0
$67$	11.7805	1.43921	0.719607	−	0.694382i	$-0.244322\pi$
$67$	11.7805	1.43921	0.719607	+	0.694382i	$0.244322\pi$
$68$	0	0
$69$	−0.610193	−0.0734585
$70$	0	0
$71$	−12.3134	−1.46133	−0.730666	−	0.682735i	$-0.760790\pi$
$71$	−12.3134	−1.46133	−0.730666	+	0.682735i	$0.760790\pi$
$72$	0	0
$73$	−4.70617	−0.550815	−0.275408	−	0.961328i	$-0.588813\pi$
$73$	−4.70617	−0.550815	−0.275408	+	0.961328i	$0.588813\pi$
$74$	0	0
$75$	−0.620002	−0.0715916
$76$	0	0
$77$	8.85376	1.00898
$78$	0	0
$79$	8.31096	0.935056	0.467528	−	0.883978i	$-0.345145\pi$
$79$	8.31096	0.935056	0.467528	+	0.883978i	$0.345145\pi$
$80$	0	0
$81$	8.95380	0.994867
$82$	0	0
$83$	−12.5820	−1.38105	−0.690525	−	0.723308i	$-0.742621\pi$
$83$	−12.5820	−1.38105	−0.690525	+	0.723308i	$0.742621\pi$
$84$	0	0
$85$	1.97201	0.213894
$86$	0	0
$87$	−0.321412	−0.0344590
$88$	0	0
$89$	2.12572	0.225326	0.112663	−	0.993633i	$-0.464062\pi$
$89$	2.12572	0.225326	0.112663	+	0.993633i	$0.464062\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.0644407	−0.00668218
$94$	0	0
$95$	−17.2036	−1.76505
$96$	0	0
$97$	13.2660	1.34695	0.673477	−	0.739208i	$-0.264800\pi$
$97$	13.2660	1.34695	0.673477	+	0.739208i	$0.264800\pi$
$98$	0	0
$99$	15.8368	1.59165
$100$	0	0
$101$	12.9918	1.29273	0.646366	−	0.763028i	$-0.276288\pi$
$101$	12.9918	1.29273	0.646366	+	0.763028i	$0.276288\pi$
$102$	0	0
$103$	3.10136	0.305586	0.152793	−	0.988258i	$-0.451173\pi$
$103$	3.10136	0.305586	0.152793	+	0.988258i	$0.451173\pi$
$104$	0	0
$105$	−0.443341	−0.0432657
$106$	0	0
$107$	−13.0054	−1.25728	−0.628638	−	0.777698i	$-0.716387\pi$
$107$	−13.0054	−1.25728	−0.628638	+	0.777698i	$0.716387\pi$
$108$	0	0
$109$	8.11738	0.777504	0.388752	−	0.921342i	$-0.372906\pi$
$109$	8.11738	0.777504	0.388752	+	0.921342i	$0.372906\pi$
$110$	0	0
$111$	−0.227303	−0.0215746
$112$	0	0
$113$	−0.518801	−0.0488047	−0.0244024	−	0.999702i	$-0.507768\pi$
$113$	−0.518801	−0.0488047	−0.0244024	+	0.999702i	$0.507768\pi$
$114$	0	0
$115$	−31.4587	−2.93354
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.893637	0.0819196
$120$	0	0
$121$	16.9626	1.54206
$122$	0	0
$123$	−0.731250	−0.0659346
$124$	0	0
$125$	−13.4906	−1.20663
$126$	0	0
$127$	6.76662	0.600441	0.300220	−	0.953870i	$-0.402940\pi$
$127$	6.76662	0.600441	0.300220	+	0.953870i	$0.402940\pi$
$128$	0	0
$129$	0.375864	0.0330930
$130$	0	0
$131$	−2.38674	−0.208530	−0.104265	−	0.994550i	$-0.533249\pi$
$131$	−2.38674	−0.208530	−0.104265	+	0.994550i	$0.533249\pi$
$132$	0	0
$133$	−7.79599	−0.675998
$134$	0	0
$135$	−1.58737	−0.136619
$136$	0	0
$137$	3.78629	0.323485	0.161742	−	0.986833i	$-0.448289\pi$
$137$	3.78629	0.323485	0.161742	+	0.986833i	$0.448289\pi$
$138$	0	0
$139$	−12.6724	−1.07486	−0.537432	−	0.843307i	$-0.680605\pi$
$139$	−12.6724	−1.07486	−0.537432	+	0.843307i	$0.680605\pi$
$140$	0	0
$141$	0.293352	0.0247047
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−16.5705	−1.37611
$146$	0	0
$147$	0.300757	0.0248060
$148$	0	0
$149$	14.0701	1.15267	0.576333	−	0.817215i	$-0.304483\pi$
$149$	14.0701	1.15267	0.576333	+	0.817215i	$0.304483\pi$
$150$	0	0
$151$	2.15428	0.175313	0.0876564	−	0.996151i	$-0.472062\pi$
$151$	2.15428	0.175313	0.0876564	+	0.996151i	$0.472062\pi$
$152$	0	0
$153$	1.59845	0.129227
$154$	0	0
$155$	−3.32226	−0.266850
$156$	0	0
$157$	16.3197	1.30245	0.651227	−	0.758883i	$-0.274254\pi$
$157$	16.3197	1.30245	0.651227	+	0.758883i	$0.274254\pi$
$158$	0	0
$159$	0.699646	0.0554855
$160$	0	0
$161$	−14.2558	−1.12352
$162$	0	0
$163$	−7.47419	−0.585424	−0.292712	−	0.956201i	$-0.594558\pi$
$163$	−7.47419	−0.585424	−0.292712	+	0.956201i	$0.594558\pi$
$164$	0	0
$165$	−1.40019	−0.109005
$166$	0	0
$167$	10.9398	0.846548	0.423274	−	0.906002i	$-0.360881\pi$
$167$	10.9398	0.846548	0.423274	+	0.906002i	$0.360881\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−13.9447	−1.06638
$172$	0	0
$173$	19.7868	1.50437	0.752183	−	0.658955i	$-0.229001\pi$
$173$	19.7868	1.50437	0.752183	+	0.658955i	$0.229001\pi$
$174$	0	0
$175$	−14.4850	−1.09496
$176$	0	0
$177$	0.780536	0.0586686
$178$	0	0
$179$	−5.60501	−0.418938	−0.209469	−	0.977815i	$-0.567174\pi$
$179$	−5.60501	−0.418938	−0.209469	+	0.977815i	$0.567174\pi$
$180$	0	0
$181$	1.48417	0.110317	0.0551586	−	0.998478i	$-0.482434\pi$
$181$	1.48417	0.110317	0.0551586	+	0.998478i	$0.482434\pi$
$182$	0	0
$183$	−0.0712590	−0.00526761
$184$	0	0
$185$	−11.7187	−0.861574
$186$	0	0
$187$	2.82235	0.206391
$188$	0	0
$189$	−0.719335	−0.0523239
$190$	0	0
$191$	10.2535	0.741918	0.370959	−	0.928649i	$-0.379029\pi$
$191$	10.2535	0.741918	0.370959	+	0.928649i	$0.379029\pi$
$192$	0	0
$193$	14.3016	1.02945	0.514727	−	0.857354i	$-0.327893\pi$
$193$	14.3016	1.02945	0.514727	+	0.857354i	$0.327893\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.01800	0.143777	0.0718883	−	0.997413i	$-0.477097\pi$
$197$	2.01800	0.143777	0.0718883	+	0.997413i	$0.477097\pi$
$198$	0	0
$199$	3.02262	0.214268	0.107134	−	0.994245i	$-0.465833\pi$
$199$	3.02262	0.214268	0.107134	+	0.994245i	$0.465833\pi$
$200$	0	0
$201$	−0.844258	−0.0595494
$202$	0	0
$203$	−7.50910	−0.527036
$204$	0	0
$205$	−37.6999	−2.63307
$206$	0	0
$207$	−25.4995	−1.77234
$208$	0	0
$209$	−24.6219	−1.70313
$210$	0	0
$211$	−5.51742	−0.379835	−0.189917	−	0.981800i	$-0.560822\pi$
$211$	−5.51742	−0.379835	−0.189917	+	0.981800i	$0.560822\pi$
$212$	0	0
$213$	0.882451	0.0604646
$214$	0	0
$215$	19.3778	1.32156
$216$	0	0
$217$	−1.50552	−0.102201
$218$	0	0
$219$	0.337272	0.0227907
$220$	0	0
$221$	0	0
$222$	0	0
$223$	18.6506	1.24894	0.624469	−	0.781050i	$-0.285316\pi$
$223$	18.6506	1.24894	0.624469	+	0.781050i	$0.285316\pi$
$224$	0	0
$225$	−25.9094	−1.72729
$226$	0	0
$227$	−8.32529	−0.552569	−0.276284	−	0.961076i	$-0.589103\pi$
$227$	−8.32529	−0.552569	−0.276284	+	0.961076i	$0.589103\pi$
$228$	0	0
$229$	13.1782	0.870843	0.435422	−	0.900227i	$-0.356599\pi$
$229$	13.1782	0.870843	0.435422	+	0.900227i	$0.356599\pi$
$230$	0	0
$231$	−0.634513	−0.0417479
$232$	0	0
$233$	−3.46818	−0.227208	−0.113604	−	0.993526i	$-0.536240\pi$
$233$	−3.46818	−0.227208	−0.113604	+	0.993526i	$0.536240\pi$
$234$	0	0
$235$	15.1239	0.986573
$236$	0	0
$237$	−0.595613	−0.0386892
$238$	0	0
$239$	28.8087	1.86348	0.931741	−	0.363123i	$-0.118290\pi$
$239$	28.8087	1.86348	0.931741	+	0.363123i	$0.118290\pi$
$240$	0	0
$241$	11.6764	0.752140	0.376070	−	0.926591i	$-0.377275\pi$
$241$	11.6764	0.752140	0.376070	+	0.926591i	$0.377275\pi$
$242$	0	0
$243$	−1.93057	−0.123846
$244$	0	0
$245$	15.5056	0.990618
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0.901699	0.0571428
$250$	0	0
$251$	5.51270	0.347958	0.173979	−	0.984749i	$-0.444337\pi$
$251$	5.51270	0.347958	0.173979	+	0.984749i	$0.444337\pi$
$252$	0	0
$253$	−45.0239	−2.83063
$254$	0	0
$255$	−0.141326	−0.00885017
$256$	0	0
$257$	−5.58883	−0.348622	−0.174311	−	0.984691i	$-0.555770\pi$
$257$	−5.58883	−0.348622	−0.174311	+	0.984691i	$0.555770\pi$
$258$	0	0
$259$	−5.31044	−0.329975
$260$	0	0
$261$	−13.4316	−0.831392
$262$	0	0
$263$	3.68238	0.227065	0.113533	−	0.993534i	$-0.463783\pi$
$263$	3.68238	0.227065	0.113533	+	0.993534i	$0.463783\pi$
$264$	0	0
$265$	36.0705	2.21579
$266$	0	0
$267$	−0.152342	−0.00932316
$268$	0	0
$269$	−4.52398	−0.275832	−0.137916	−	0.990444i	$-0.544040\pi$
$269$	−4.52398	−0.275832	−0.137916	+	0.990444i	$0.544040\pi$
$270$	0	0
$271$	20.8679	1.26764	0.633819	−	0.773482i	$-0.281487\pi$
$271$	20.8679	1.26764	0.633819	+	0.773482i	$0.281487\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−45.7477	−2.75869
$276$	0	0
$277$	−3.15734	−0.189706	−0.0948532	−	0.995491i	$-0.530238\pi$
$277$	−3.15734	−0.189706	−0.0948532	+	0.995491i	$0.530238\pi$
$278$	0	0
$279$	−2.69292	−0.161221
$280$	0	0
$281$	20.0961	1.19883	0.599416	−	0.800437i	$-0.295399\pi$
$281$	20.0961	1.19883	0.599416	+	0.800437i	$0.295399\pi$
$282$	0	0
$283$	19.1802	1.14015	0.570074	−	0.821594i	$-0.306915\pi$
$283$	19.1802	1.14015	0.570074	+	0.821594i	$0.306915\pi$
$284$	0	0
$285$	1.23291	0.0730314
$286$	0	0
$287$	−17.0841	−1.00844
$288$	0	0
$289$	−16.7151	−0.983243
$290$	0	0
$291$	−0.950717	−0.0557320
$292$	0	0
$293$	15.0724	0.880539	0.440270	−	0.897866i	$-0.354883\pi$
$293$	15.0724	0.880539	0.440270	+	0.897866i	$0.354883\pi$
$294$	0	0
$295$	40.2408	2.34291
$296$	0	0
$297$	−2.27186	−0.131827
$298$	0	0
$299$	0	0
$300$	0	0
$301$	8.78127	0.506144
$302$	0	0
$303$	−0.931069	−0.0534885
$304$	0	0
$305$	−3.67378	−0.210360
$306$	0	0
$307$	−6.74275	−0.384829	−0.192414	−	0.981314i	$-0.561632\pi$
$307$	−6.74275	−0.384829	−0.192414	+	0.981314i	$0.561632\pi$
$308$	0	0
$309$	−0.222262	−0.0126440
$310$	0	0
$311$	−21.7596	−1.23387	−0.616937	−	0.787012i	$-0.711627\pi$
$311$	−21.7596	−1.23387	−0.616937	+	0.787012i	$0.711627\pi$
$312$	0	0
$313$	23.1707	1.30969	0.654844	−	0.755764i	$-0.272734\pi$
$313$	23.1707	1.30969	0.654844	+	0.755764i	$0.272734\pi$
$314$	0	0
$315$	−18.5269	−1.04387
$316$	0	0
$317$	−5.31095	−0.298293	−0.149146	−	0.988815i	$-0.547653\pi$
$317$	−5.31095	−0.298293	−0.149146	+	0.988815i	$0.547653\pi$
$318$	0	0
$319$	−23.7158	−1.32783
$320$	0	0
$321$	0.932042	0.0520215
$322$	0	0
$323$	−2.48516	−0.138278
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−0.581740	−0.0321703
$328$	0	0
$329$	6.85355	0.377848
$330$	0	0
$331$	2.58872	0.142289	0.0711445	−	0.997466i	$-0.477335\pi$
$331$	2.58872	0.142289	0.0711445	+	0.997466i	$0.477335\pi$
$332$	0	0
$333$	−9.49881	−0.520531
$334$	0	0
$335$	−43.5260	−2.37808
$336$	0	0
$337$	−24.9245	−1.35773	−0.678863	−	0.734265i	$-0.737527\pi$
$337$	−24.9245	−1.35773	−0.678863	+	0.734265i	$0.737527\pi$
$338$	0	0
$339$	0.0371804	0.00201936
$340$	0	0
$341$	−4.75484	−0.257489
$342$	0	0
$343$	18.7468	1.01223
$344$	0	0
$345$	2.25452	0.121379
$346$	0	0
$347$	−19.2255	−1.03208	−0.516039	−	0.856565i	$-0.672594\pi$
$347$	−19.2255	−1.03208	−0.516039	+	0.856565i	$0.672594\pi$
$348$	0	0
$349$	−21.1276	−1.13093	−0.565466	−	0.824772i	$-0.691304\pi$
$349$	−21.1276	−1.13093	−0.565466	+	0.824772i	$0.691304\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−20.7644	−1.10518	−0.552588	−	0.833454i	$-0.686360\pi$
$353$	−20.7644	−1.10518	−0.552588	+	0.833454i	$0.686360\pi$
$354$	0	0
$355$	45.4951	2.41463
$356$	0	0
$357$	−0.0640433	−0.00338953
$358$	0	0
$359$	7.78925	0.411101	0.205550	−	0.978647i	$-0.434102\pi$
$359$	7.78925	0.411101	0.205550	+	0.978647i	$0.434102\pi$
$360$	0	0
$361$	2.68030	0.141069
$362$	0	0
$363$	−1.21564	−0.0638047
$364$	0	0
$365$	17.3882	0.910139
$366$	0	0
$367$	−18.4871	−0.965016	−0.482508	−	0.875891i	$-0.660274\pi$
$367$	−18.4871	−0.965016	−0.482508	+	0.875891i	$0.660274\pi$
$368$	0	0
$369$	−30.5584	−1.59081
$370$	0	0
$371$	16.3457	0.848628
$372$	0	0
$373$	−20.8723	−1.08073	−0.540365	−	0.841431i	$-0.681714\pi$
$373$	−20.8723	−1.08073	−0.540365	+	0.841431i	$0.681714\pi$
$374$	0	0
$375$	0.966815	0.0499261
$376$	0	0
$377$	0	0
$378$	0	0
$379$	3.55386	0.182550	0.0912748	−	0.995826i	$-0.470906\pi$
$379$	3.55386	0.182550	0.0912748	+	0.995826i	$0.470906\pi$
$380$	0	0
$381$	−0.484936	−0.0248440
$382$	0	0
$383$	22.3629	1.14269	0.571345	−	0.820710i	$-0.306422\pi$
$383$	22.3629	1.14269	0.571345	+	0.820710i	$0.306422\pi$
$384$	0	0
$385$	−32.7126	−1.66719
$386$	0	0
$387$	15.7071	0.798436
$388$	0	0
$389$	−25.4960	−1.29270	−0.646349	−	0.763042i	$-0.723705\pi$
$389$	−25.4960	−1.29270	−0.646349	+	0.763042i	$0.723705\pi$
$390$	0	0
$391$	−4.54440	−0.229820
$392$	0	0
$393$	0.171048	0.00862821
$394$	0	0
$395$	−30.7070	−1.54504
$396$	0	0
$397$	−0.278830	−0.0139941	−0.00699703	−	0.999976i	$-0.502227\pi$
$397$	−0.278830	−0.0139941	−0.00699703	+	0.999976i	$0.502227\pi$
$398$	0	0
$399$	0.558707	0.0279703
$400$	0	0
$401$	2.67605	0.133636	0.0668178	−	0.997765i	$-0.478715\pi$
$401$	2.67605	0.133636	0.0668178	+	0.997765i	$0.478715\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−33.0822	−1.64387
$406$	0	0
$407$	−16.7719	−0.831350
$408$	0	0
$409$	−20.2214	−0.999886	−0.499943	−	0.866058i	$-0.666646\pi$
$409$	−20.2214	−0.999886	−0.499943	+	0.866058i	$0.666646\pi$
$410$	0	0
$411$	−0.271348	−0.0133846
$412$	0	0
$413$	18.2355	0.897312
$414$	0	0
$415$	46.4874	2.28198
$416$	0	0
$417$	0.908183	0.0444739
$418$	0	0
$419$	−19.9554	−0.974887	−0.487443	−	0.873155i	$-0.662070\pi$
$419$	−19.9554	−0.974887	−0.487443	+	0.873155i	$0.662070\pi$
$420$	0	0
$421$	−0.710675	−0.0346362	−0.0173181	−	0.999850i	$-0.505513\pi$
$421$	−0.710675	−0.0346362	−0.0173181	+	0.999850i	$0.505513\pi$
$422$	0	0
$423$	12.2590	0.596051
$424$	0	0
$425$	−4.61745	−0.223979
$426$	0	0
$427$	−1.66481	−0.0805660
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7.66813	−0.369361	−0.184680	−	0.982799i	$-0.559125\pi$
$431$	−7.66813	−0.369361	−0.184680	+	0.982799i	$0.559125\pi$
$432$	0	0
$433$	28.4412	1.36680	0.683398	−	0.730046i	$-0.260501\pi$
$433$	28.4412	1.36680	0.683398	+	0.730046i	$0.260501\pi$
$434$	0	0
$435$	1.18754	0.0569382
$436$	0	0
$437$	39.6448	1.89647
$438$	0	0
$439$	−34.7701	−1.65949	−0.829743	−	0.558146i	$-0.811513\pi$
$439$	−34.7701	−1.65949	−0.829743	+	0.558146i	$0.811513\pi$
$440$	0	0
$441$	12.5684	0.598495
$442$	0	0
$443$	31.9654	1.51872	0.759360	−	0.650670i	$-0.225512\pi$
$443$	31.9654	1.51872	0.759360	+	0.650670i	$0.225512\pi$
$444$	0	0
$445$	−7.85403	−0.372317
$446$	0	0
$447$	−1.00835	−0.0476931
$448$	0	0
$449$	−35.4247	−1.67180	−0.835898	−	0.548885i	$-0.815052\pi$
$449$	−35.4247	−1.67180	−0.835898	+	0.548885i	$0.815052\pi$
$450$	0	0
$451$	−53.9563	−2.54070
$452$	0	0
$453$	−0.154388	−0.00725380
$454$	0	0
$455$	0	0
$456$	0	0
$457$	23.9105	1.11849	0.559244	−	0.829003i	$-0.311092\pi$
$457$	23.9105	1.11849	0.559244	+	0.829003i	$0.311092\pi$
$458$	0	0
$459$	−0.229305	−0.0107031
$460$	0	0
$461$	14.9473	0.696167	0.348084	−	0.937463i	$-0.386832\pi$
$461$	14.9473	0.696167	0.348084	+	0.937463i	$0.386832\pi$
$462$	0	0
$463$	11.9227	0.554094	0.277047	−	0.960856i	$-0.410644\pi$
$463$	11.9227	0.554094	0.277047	+	0.960856i	$0.410644\pi$
$464$	0	0
$465$	0.238093	0.0110413
$466$	0	0
$467$	22.6973	1.05031	0.525154	−	0.851007i	$-0.324008\pi$
$467$	22.6973	1.05031	0.525154	+	0.851007i	$0.324008\pi$
$468$	0	0
$469$	−19.7243	−0.910783
$470$	0	0
$471$	−1.16957	−0.0538908
$472$	0	0
$473$	27.7337	1.27520
$474$	0	0
$475$	40.2821	1.84827
$476$	0	0
$477$	29.2377	1.33870
$478$	0	0
$479$	−5.06174	−0.231277	−0.115638	−	0.993291i	$-0.536891\pi$
$479$	−5.06174	−0.231277	−0.115638	+	0.993291i	$0.536891\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	1.02166	0.0464871
$484$	0	0
$485$	−49.0146	−2.22564
$486$	0	0
$487$	−12.6163	−0.571701	−0.285851	−	0.958274i	$-0.592276\pi$
$487$	−12.6163	−0.571701	−0.285851	+	0.958274i	$0.592276\pi$
$488$	0	0
$489$	0.535645	0.0242227
$490$	0	0
$491$	15.9685	0.720647	0.360323	−	0.932827i	$-0.382666\pi$
$491$	15.9685	0.720647	0.360323	+	0.932827i	$0.382666\pi$
$492$	0	0
$493$	−2.39371	−0.107807
$494$	0	0
$495$	−58.5130	−2.62997
$496$	0	0
$497$	20.6166	0.924780
$498$	0	0
$499$	−8.09104	−0.362205	−0.181102	−	0.983464i	$-0.557967\pi$
$499$	−8.09104	−0.362205	−0.181102	+	0.983464i	$0.557967\pi$
$500$	0	0
$501$	−0.784012	−0.0350271
$502$	0	0
$503$	−7.96763	−0.355259	−0.177630	−	0.984097i	$-0.556843\pi$
$503$	−7.96763	−0.355259	−0.177630	+	0.984097i	$0.556843\pi$
$504$	0	0
$505$	−48.0016	−2.13604
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.5236	0.732395	0.366197	−	0.930537i	$-0.380660\pi$
$509$	16.5236	0.732395	0.366197	+	0.930537i	$0.380660\pi$
$510$	0	0
$511$	7.87964	0.348575
$512$	0	0
$513$	2.00044	0.0883214
$514$	0	0
$515$	−11.4588	−0.504935
$516$	0	0
$517$	21.6454	0.951963
$518$	0	0
$519$	−1.41804	−0.0622451
$520$	0	0
$521$	26.4714	1.15973	0.579867	−	0.814711i	$-0.303105\pi$
$521$	26.4714	1.15973	0.579867	+	0.814711i	$0.303105\pi$
$522$	0	0
$523$	21.3659	0.934266	0.467133	−	0.884187i	$-0.345287\pi$
$523$	21.3659	0.934266	0.467133	+	0.884187i	$0.345287\pi$
$524$	0	0
$525$	1.03808	0.0453056
$526$	0	0
$527$	−0.479921	−0.0209057
$528$	0	0
$529$	49.4950	2.15196
$530$	0	0
$531$	32.6180	1.41550
$532$	0	0
$533$	0	0
$534$	0	0
$535$	48.0518	2.07746
$536$	0	0
$537$	0.401688	0.0173341
$538$	0	0
$539$	22.1918	0.955866
$540$	0	0
$541$	−29.1482	−1.25318	−0.626590	−	0.779349i	$-0.715550\pi$
$541$	−29.1482	−1.25318	−0.626590	+	0.779349i	$0.715550\pi$
$542$	0	0
$543$	−0.106364	−0.00456452
$544$	0	0
$545$	−29.9918	−1.28471
$546$	0	0
$547$	28.1998	1.20574	0.602869	−	0.797840i	$-0.294024\pi$
$547$	28.1998	1.20574	0.602869	+	0.797840i	$0.294024\pi$
$548$	0	0
$549$	−2.97786	−0.127092
$550$	0	0
$551$	20.8825	0.889623
$552$	0	0
$553$	−13.9152	−0.591735
$554$	0	0
$555$	0.839830	0.0356488
$556$	0	0
$557$	7.93293	0.336129	0.168065	−	0.985776i	$-0.446248\pi$
$557$	7.93293	0.336129	0.168065	+	0.985776i	$0.446248\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−0.202267	−0.00853970
$562$	0	0
$563$	−30.0142	−1.26495	−0.632473	−	0.774582i	$-0.717960\pi$
$563$	−30.0142	−1.26495	−0.632473	+	0.774582i	$0.717960\pi$
$564$	0	0
$565$	1.91685	0.0806424
$566$	0	0
$567$	−14.9915	−0.629586
$568$	0	0
$569$	16.5520	0.693894	0.346947	−	0.937885i	$-0.387218\pi$
$569$	16.5520	0.693894	0.346947	+	0.937885i	$0.387218\pi$
$570$	0	0
$571$	13.3677	0.559420	0.279710	−	0.960084i	$-0.409762\pi$
$571$	13.3677	0.559420	0.279710	+	0.960084i	$0.409762\pi$
$572$	0	0
$573$	−0.734828	−0.0306979
$574$	0	0
$575$	73.6604	3.07185
$576$	0	0
$577$	3.39112	0.141174	0.0705872	−	0.997506i	$-0.477513\pi$
$577$	3.39112	0.141174	0.0705872	+	0.997506i	$0.477513\pi$
$578$	0	0
$579$	−1.02494	−0.0425950
$580$	0	0
$581$	21.0663	0.873976
$582$	0	0
$583$	51.6243	2.13806
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15.7995	−0.652117	−0.326058	−	0.945350i	$-0.605721\pi$
$587$	−15.7995	−0.652117	−0.326058	+	0.945350i	$0.605721\pi$
$588$	0	0
$589$	4.18678	0.172513
$590$	0	0
$591$	−0.144622	−0.00594895
$592$	0	0
$593$	35.5013	1.45787	0.728933	−	0.684586i	$-0.240017\pi$
$593$	35.5013	1.45787	0.728933	+	0.684586i	$0.240017\pi$
$594$	0	0
$595$	−3.30178	−0.135360
$596$	0	0
$597$	−0.216619	−0.00886561
$598$	0	0
$599$	−37.2878	−1.52354	−0.761770	−	0.647848i	$-0.775669\pi$
$599$	−37.2878	−1.52354	−0.761770	+	0.647848i	$0.775669\pi$
$600$	0	0
$601$	26.8092	1.09357	0.546785	−	0.837273i	$-0.315852\pi$
$601$	26.8092	1.09357	0.546785	+	0.837273i	$0.315852\pi$
$602$	0	0
$603$	−35.2809	−1.43675
$604$	0	0
$605$	−62.6729	−2.54802
$606$	0	0
$607$	21.0571	0.854682	0.427341	−	0.904091i	$-0.359450\pi$
$607$	21.0571	0.854682	0.427341	+	0.904091i	$0.359450\pi$
$608$	0	0
$609$	0.538147	0.0218068
$610$	0	0
$611$	0	0
$612$	0	0
$613$	3.59143	0.145057	0.0725283	−	0.997366i	$-0.476893\pi$
$613$	3.59143	0.145057	0.0725283	+	0.997366i	$0.476893\pi$
$614$	0	0
$615$	2.70180	0.108947
$616$	0	0
$617$	−13.9080	−0.559917	−0.279958	−	0.960012i	$-0.590321\pi$
$617$	−13.9080	−0.559917	−0.279958	+	0.960012i	$0.590321\pi$
$618$	0	0
$619$	−23.0010	−0.924487	−0.462243	−	0.886753i	$-0.652955\pi$
$619$	−23.0010	−0.924487	−0.462243	+	0.886753i	$0.652955\pi$
$620$	0	0
$621$	3.65802	0.146791
$622$	0	0
$623$	−3.55914	−0.142594
$624$	0	0
$625$	6.58814	0.263526
$626$	0	0
$627$	1.76455	0.0704694
$628$	0	0
$629$	−1.69283	−0.0674977
$630$	0	0
$631$	−42.0774	−1.67507	−0.837537	−	0.546381i	$-0.816005\pi$
$631$	−42.0774	−1.67507	−0.837537	+	0.546381i	$0.816005\pi$
$632$	0	0
$633$	0.395411	0.0157162
$634$	0	0
$635$	−25.0011	−0.992137
$636$	0	0
$637$	0	0
$638$	0	0
$639$	36.8770	1.45883
$640$	0	0
$641$	−31.8923	−1.25967	−0.629835	−	0.776729i	$-0.716877\pi$
$641$	−31.8923	−1.25967	−0.629835	+	0.776729i	$0.716877\pi$
$642$	0	0
$643$	28.6374	1.12935	0.564674	−	0.825314i	$-0.309002\pi$
$643$	28.6374	1.12935	0.564674	+	0.825314i	$0.309002\pi$
$644$	0	0
$645$	−1.38873	−0.0546812
$646$	0	0
$647$	−20.9639	−0.824175	−0.412088	−	0.911144i	$-0.635200\pi$
$647$	−20.9639	−0.824175	−0.412088	+	0.911144i	$0.635200\pi$
$648$	0	0
$649$	57.5929	2.26072
$650$	0	0
$651$	0.107894	0.00422871
$652$	0	0
$653$	41.5312	1.62524	0.812621	−	0.582792i	$-0.198040\pi$
$653$	41.5312	1.62524	0.812621	+	0.582792i	$0.198040\pi$
$654$	0	0
$655$	8.81842	0.344564
$656$	0	0
$657$	14.0943	0.549872
$658$	0	0
$659$	16.6864	0.650009	0.325004	−	0.945713i	$-0.394634\pi$
$659$	16.6864	0.650009	0.325004	+	0.945713i	$0.394634\pi$
$660$	0	0
$661$	−20.9744	−0.815810	−0.407905	−	0.913024i	$-0.633740\pi$
$661$	−20.9744	−0.815810	−0.407905	+	0.913024i	$0.633740\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	28.8043	1.11698
$666$	0	0
$667$	38.1859	1.47856
$668$	0	0
$669$	−1.33661	−0.0516765
$670$	0	0
$671$	−5.25794	−0.202981
$672$	0	0
$673$	19.1510	0.738217	0.369108	−	0.929386i	$-0.379663\pi$
$673$	19.1510	0.738217	0.369108	+	0.929386i	$0.379663\pi$
$674$	0	0
$675$	3.71682	0.143061
$676$	0	0
$677$	−9.27886	−0.356616	−0.178308	−	0.983975i	$-0.557062\pi$
$677$	−9.27886	−0.356616	−0.178308	+	0.983975i	$0.557062\pi$
$678$	0	0
$679$	−22.2115	−0.852398
$680$	0	0
$681$	0.596640	0.0228633
$682$	0	0
$683$	23.6964	0.906718	0.453359	−	0.891328i	$-0.350226\pi$
$683$	23.6964	0.906718	0.453359	+	0.891328i	$0.350226\pi$
$684$	0	0
$685$	−13.9895	−0.534509
$686$	0	0
$687$	−0.944432	−0.0360323
$688$	0	0
$689$	0	0
$690$	0	0
$691$	15.9767	0.607781	0.303891	−	0.952707i	$-0.401714\pi$
$691$	15.9767	0.607781	0.303891	+	0.952707i	$0.401714\pi$
$692$	0	0
$693$	−26.5158	−1.00725
$694$	0	0
$695$	46.8217	1.77605
$696$	0	0
$697$	−5.44597	−0.206281
$698$	0	0
$699$	0.248550	0.00940104
$700$	0	0
$701$	−18.8538	−0.712099	−0.356050	−	0.934467i	$-0.615877\pi$
$701$	−18.8538	−0.712099	−0.356050	+	0.934467i	$0.615877\pi$
$702$	0	0
$703$	14.7681	0.556989
$704$	0	0
$705$	−1.08387	−0.0408208
$706$	0	0
$707$	−21.7524	−0.818085
$708$	0	0
$709$	18.6647	0.700969	0.350485	−	0.936568i	$-0.386017\pi$
$709$	18.6647	0.700969	0.350485	+	0.936568i	$0.386017\pi$
$710$	0	0
$711$	−24.8902	−0.933455
$712$	0	0
$713$	7.65599	0.286719
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−2.06461	−0.0771041
$718$	0	0
$719$	7.68075	0.286444	0.143222	−	0.989691i	$-0.454254\pi$
$719$	7.68075	0.286444	0.143222	+	0.989691i	$0.454254\pi$
$720$	0	0
$721$	−5.19267	−0.193385
$722$	0	0
$723$	−0.836797	−0.0311208
$724$	0	0
$725$	38.7998	1.44099
$726$	0	0
$727$	43.9048	1.62834	0.814169	−	0.580628i	$-0.197193\pi$
$727$	43.9048	1.62834	0.814169	+	0.580628i	$0.197193\pi$
$728$	0	0
$729$	−26.7231	−0.989743
$730$	0	0
$731$	2.79924	0.103534
$732$	0	0
$733$	−29.8392	−1.10214	−0.551068	−	0.834460i	$-0.685779\pi$
$733$	−29.8392	−1.10214	−0.551068	+	0.834460i	$0.685779\pi$
$734$	0	0
$735$	−1.11122	−0.0409881
$736$	0	0
$737$	−62.2948	−2.29466
$738$	0	0
$739$	−46.1813	−1.69881	−0.849404	−	0.527744i	$-0.823038\pi$
$739$	−46.1813	−1.69881	−0.849404	+	0.527744i	$0.823038\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−23.3668	−0.857245	−0.428622	−	0.903484i	$-0.641001\pi$
$743$	−23.3668	−0.857245	−0.428622	+	0.903484i	$0.641001\pi$
$744$	0	0
$745$	−51.9856	−1.90461
$746$	0	0
$747$	37.6813	1.37869
$748$	0	0
$749$	21.7752	0.795648
$750$	0	0
$751$	19.2991	0.704235	0.352118	−	0.935956i	$-0.385462\pi$
$751$	19.2991	0.704235	0.352118	+	0.935956i	$0.385462\pi$
$752$	0	0
$753$	−0.395073	−0.0143972
$754$	0	0
$755$	−7.95955	−0.289678
$756$	0	0
$757$	26.4677	0.961985	0.480993	−	0.876725i	$-0.340276\pi$
$757$	26.4677	0.961985	0.480993	+	0.876725i	$0.340276\pi$
$758$	0	0
$759$	3.22668	0.117121
$760$	0	0
$761$	44.3291	1.60693	0.803464	−	0.595354i	$-0.202988\pi$
$761$	44.3291	1.60693	0.803464	+	0.595354i	$0.202988\pi$
$762$	0	0
$763$	−13.5911	−0.492031
$764$	0	0
$765$	−5.90590	−0.213528
$766$	0	0
$767$	0	0
$768$	0	0
$769$	23.3110	0.840616	0.420308	−	0.907381i	$-0.361922\pi$
$769$	23.3110	0.840616	0.420308	+	0.907381i	$0.361922\pi$
$770$	0	0
$771$	0.400529	0.0144247
$772$	0	0
$773$	−6.80051	−0.244597	−0.122299	−	0.992493i	$-0.539027\pi$
$773$	−6.80051	−0.244597	−0.122299	+	0.992493i	$0.539027\pi$
$774$	0	0
$775$	7.77906	0.279432
$776$	0	0
$777$	0.380578	0.0136532
$778$	0	0
$779$	47.5101	1.70223
$780$	0	0
$781$	65.1129	2.32992
$782$	0	0
$783$	1.92682	0.0688589
$784$	0	0
$785$	−60.2975	−2.15211
$786$	0	0
$787$	−18.6193	−0.663706	−0.331853	−	0.943331i	$-0.607674\pi$
$787$	−18.6193	−0.663706	−0.331853	+	0.943331i	$0.607674\pi$
$788$	0	0
$789$	−0.263901	−0.00939514
$790$	0	0
$791$	0.868640	0.0308853
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−2.58503	−0.0916814
$796$	0	0
$797$	47.5364	1.68383	0.841913	−	0.539613i	$-0.181429\pi$
$797$	47.5364	1.68383	0.841913	+	0.539613i	$0.181429\pi$
$798$	0	0
$799$	2.18473	0.0772904
$800$	0	0
$801$	−6.36624	−0.224940
$802$	0	0
$803$	24.8861	0.878211
$804$	0	0
$805$	52.6720	1.85644
$806$	0	0
$807$	0.324216	0.0114129
$808$	0	0
$809$	−48.8878	−1.71880	−0.859401	−	0.511302i	$-0.829163\pi$
$809$	−48.8878	−1.71880	−0.859401	+	0.511302i	$0.829163\pi$
$810$	0	0
$811$	16.4564	0.577863	0.288932	−	0.957350i	$-0.406700\pi$
$811$	16.4564	0.577863	0.288932	+	0.957350i	$0.406700\pi$
$812$	0	0
$813$	−1.49552	−0.0524502
$814$	0	0
$815$	27.6154	0.967324
$816$	0	0
$817$	−24.4203	−0.854358
$818$	0	0
$819$	0	0
$820$	0	0
$821$	18.0180	0.628834	0.314417	−	0.949285i	$-0.398191\pi$
$821$	18.0180	0.628834	0.314417	+	0.949285i	$0.398191\pi$
$822$	0	0
$823$	−0.198815	−0.00693024	−0.00346512	−	0.999994i	$-0.501103\pi$
$823$	−0.198815	−0.00693024	−0.00346512	+	0.999994i	$0.501103\pi$
$824$	0	0
$825$	3.27855	0.114144
$826$	0	0
$827$	9.51904	0.331009	0.165505	−	0.986209i	$-0.447075\pi$
$827$	9.51904	0.331009	0.165505	+	0.986209i	$0.447075\pi$
$828$	0	0
$829$	−47.8686	−1.66254	−0.831272	−	0.555866i	$-0.812387\pi$
$829$	−47.8686	−1.66254	−0.831272	+	0.555866i	$0.812387\pi$
$830$	0	0
$831$	0.226274	0.00784936
$832$	0	0
$833$	2.23988	0.0776072
$834$	0	0
$835$	−40.4200	−1.39879
$836$	0	0
$837$	0.386313	0.0133529
$838$	0	0
$839$	−8.70234	−0.300438	−0.150219	−	0.988653i	$-0.547998\pi$
$839$	−8.70234	−0.300438	−0.150219	+	0.988653i	$0.547998\pi$
$840$	0	0
$841$	−8.88601	−0.306414
$842$	0	0
$843$	−1.44021	−0.0496033
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−28.4009	−0.975867
$848$	0	0
$849$	−1.37457	−0.0471751
$850$	0	0
$851$	27.0051	0.925723
$852$	0	0
$853$	−20.7251	−0.709616	−0.354808	−	0.934939i	$-0.615454\pi$
$853$	−20.7251	−0.709616	−0.354808	+	0.934939i	$0.615454\pi$
$854$	0	0
$855$	51.5224	1.76203
$856$	0	0
$857$	−13.1607	−0.449563	−0.224781	−	0.974409i	$-0.572167\pi$
$857$	−13.1607	−0.449563	−0.224781	+	0.974409i	$0.572167\pi$
$858$	0	0
$859$	−28.7604	−0.981293	−0.490647	−	0.871359i	$-0.663239\pi$
$859$	−28.7604	−0.981293	−0.490647	+	0.871359i	$0.663239\pi$
$860$	0	0
$861$	1.22435	0.0417257
$862$	0	0
$863$	−12.3842	−0.421562	−0.210781	−	0.977533i	$-0.567601\pi$
$863$	−12.3842	−0.421562	−0.210781	+	0.977533i	$0.567601\pi$
$864$	0	0
$865$	−73.1077	−2.48573
$866$	0	0
$867$	1.19791	0.0406830
$868$	0	0
$869$	−43.9481	−1.49084
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−39.7297	−1.34465
$874$	0	0
$875$	22.5876	0.763599
$876$	0	0
$877$	6.61104	0.223239	0.111619	−	0.993751i	$-0.464396\pi$
$877$	6.61104	0.223239	0.111619	+	0.993751i	$0.464396\pi$
$878$	0	0
$879$	−1.08018	−0.0364335
$880$	0	0
$881$	−12.4830	−0.420563	−0.210281	−	0.977641i	$-0.567438\pi$
$881$	−12.4830	−0.420563	−0.210281	+	0.977641i	$0.567438\pi$
$882$	0	0
$883$	−46.1900	−1.55442	−0.777209	−	0.629242i	$-0.783365\pi$
$883$	−46.1900	−1.55442	−0.777209	+	0.629242i	$0.783365\pi$
$884$	0	0
$885$	−2.88389	−0.0969410
$886$	0	0
$887$	50.7786	1.70498	0.852489	−	0.522746i	$-0.175092\pi$
$887$	50.7786	1.70498	0.852489	+	0.522746i	$0.175092\pi$
$888$	0	0
$889$	−11.3295	−0.379979
$890$	0	0
$891$	−47.3474	−1.58620
$892$	0	0
$893$	−19.0594	−0.637798
$894$	0	0
$895$	20.7092	0.692232
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.03270	0.134498
$900$	0	0
$901$	5.21060	0.173590
$902$	0	0
$903$	−0.629318	−0.0209424
$904$	0	0
$905$	−5.48364	−0.182282
$906$	0	0
$907$	−7.12962	−0.236735	−0.118368	−	0.992970i	$-0.537766\pi$
$907$	−7.12962	−0.236735	−0.118368	+	0.992970i	$0.537766\pi$
$908$	0	0
$909$	−38.9087	−1.29052
$910$	0	0
$911$	30.8696	1.02276	0.511378	−	0.859356i	$-0.329135\pi$
$911$	30.8696	1.02276	0.511378	+	0.859356i	$0.329135\pi$
$912$	0	0
$913$	66.5331	2.20192
$914$	0	0
$915$	0.263285	0.00870393
$916$	0	0
$917$	3.99616	0.131965
$918$	0	0
$919$	21.7514	0.717513	0.358756	−	0.933431i	$-0.383201\pi$
$919$	21.7514	0.717513	0.358756	+	0.933431i	$0.383201\pi$
$920$	0	0
$921$	0.483225	0.0159228
$922$	0	0
$923$	0	0
$924$	0	0
$925$	27.4392	0.902196
$926$	0	0
$927$	−9.28815	−0.305063
$928$	0	0
$929$	16.4586	0.539989	0.269995	−	0.962862i	$-0.412978\pi$
$929$	16.4586	0.539989	0.269995	+	0.962862i	$0.412978\pi$
$930$	0	0
$931$	−19.5405	−0.640413
$932$	0	0
$933$	1.55942	0.0510532
$934$	0	0
$935$	−10.4279	−0.341030
$936$	0	0
$937$	−15.7483	−0.514476	−0.257238	−	0.966348i	$-0.582812\pi$
$937$	−15.7483	−0.514476	−0.257238	+	0.966348i	$0.582812\pi$
$938$	0	0
$939$	−1.66055	−0.0541901
$940$	0	0
$941$	7.71671	0.251558	0.125779	−	0.992058i	$-0.459857\pi$
$941$	7.71671	0.251558	0.125779	+	0.992058i	$0.459857\pi$
$942$	0	0
$943$	86.8775	2.82912
$944$	0	0
$945$	2.65777	0.0864573
$946$	0	0
$947$	9.40299	0.305556	0.152778	−	0.988261i	$-0.451178\pi$
$947$	9.40299	0.305556	0.152778	+	0.988261i	$0.451178\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0.380614	0.0123423
$952$	0	0
$953$	6.34709	0.205602	0.102801	−	0.994702i	$-0.467219\pi$
$953$	6.34709	0.205602	0.102801	+	0.994702i	$0.467219\pi$
$954$	0	0
$955$	−37.8843	−1.22591
$956$	0	0
$957$	1.69962	0.0549408
$958$	0	0
$959$	−6.33947	−0.204712
$960$	0	0
$961$	−30.1915	−0.973919
$962$	0	0
$963$	38.9493	1.25512
$964$	0	0
$965$	−52.8411	−1.70102
$966$	0	0
$967$	26.1512	0.840967	0.420484	−	0.907300i	$-0.361860\pi$
$967$	26.1512	0.840967	0.420484	+	0.907300i	$0.361860\pi$
$968$	0	0
$969$	0.178102	0.00572145
$970$	0	0
$971$	23.6066	0.757571	0.378786	−	0.925484i	$-0.376342\pi$
$971$	23.6066	0.757571	0.378786	+	0.925484i	$0.376342\pi$
$972$	0	0
$973$	21.2178	0.680210
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.66271	0.245152	0.122576	−	0.992459i	$-0.460885\pi$
$977$	7.66271	0.245152	0.122576	+	0.992459i	$0.460885\pi$
$978$	0	0
$979$	−11.2407	−0.359256
$980$	0	0
$981$	−24.3105	−0.776173
$982$	0	0
$983$	−54.3954	−1.73494	−0.867472	−	0.497486i	$-0.834256\pi$
$983$	−54.3954	−1.73494	−0.867472	+	0.497486i	$0.834256\pi$
$984$	0	0
$985$	−7.45604	−0.237569
$986$	0	0
$987$	−0.491166	−0.0156340
$988$	0	0
$989$	−44.6552	−1.41995
$990$	0	0
$991$	11.2481	0.357308	0.178654	−	0.983912i	$-0.442826\pi$
$991$	11.2481	0.357308	0.178654	+	0.983912i	$0.442826\pi$
$992$	0	0
$993$	−0.185523	−0.00588740
$994$	0	0
$995$	−11.1679	−0.354045
$996$	0	0
$997$	48.6774	1.54163	0.770815	−	0.637059i	$-0.219849\pi$
$997$	48.6774	1.54163	0.770815	+	0.637059i	$0.219849\pi$
$998$	0	0
$999$	1.36265	0.0431123

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1352.2.a.n.1.3	yes	6
4.3	odd	2		2704.2.a.bg.1.4		6
13.2	odd	12		1352.2.o.h.1161.7		24
13.3	even	3		1352.2.i.n.529.4		12
13.4	even	6		1352.2.i.m.1329.4		12
13.5	odd	4		1352.2.f.g.337.6		12
13.6	odd	12		1352.2.o.h.361.8		24
13.7	odd	12		1352.2.o.h.361.7		24
13.8	odd	4		1352.2.f.g.337.5		12
13.9	even	3		1352.2.i.n.1329.4		12
13.10	even	6		1352.2.i.m.529.4		12
13.11	odd	12		1352.2.o.h.1161.8		24
13.12	even	2		1352.2.a.m.1.3	✓	6
52.31	even	4		2704.2.f.r.337.8		12
52.47	even	4		2704.2.f.r.337.7		12
52.51	odd	2		2704.2.a.bf.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1352.2.a.m.1.3	✓	6	13.12	even	2
1352.2.a.n.1.3	yes	6	1.1	even	1	trivial
1352.2.f.g.337.5		12	13.8	odd	4
1352.2.f.g.337.6		12	13.5	odd	4
1352.2.i.m.529.4		12	13.10	even	6
1352.2.i.m.1329.4		12	13.4	even	6
1352.2.i.n.529.4		12	13.3	even	3
1352.2.i.n.1329.4		12	13.9	even	3
1352.2.o.h.361.7		24	13.7	odd	12
1352.2.o.h.361.8		24	13.6	odd	12
1352.2.o.h.1161.7		24	13.2	odd	12
1352.2.o.h.1161.8		24	13.11	odd	12
2704.2.a.bf.1.4		6	52.51	odd	2
2704.2.a.bg.1.4		6	4.3	odd	2
2704.2.f.r.337.7		12	52.47	even	4
2704.2.f.r.337.8		12	52.31	even	4

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1352.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0716659 q^{3} -3.69476 q^{5} -1.67432 q^{7} -2.99486 q^{9} +O(q^{10})\)	\(q-0.0716659 q^{3} -3.69476 q^{5} -1.67432 q^{7} -2.99486 q^{9} -5.28797 q^{11} +0.264789 q^{15} -0.533731 q^{17} +4.65621 q^{19} +0.119992 q^{21} +8.51440 q^{23} +8.65127 q^{25} +0.429628 q^{27} +4.48486 q^{29} +0.899181 q^{31} +0.378967 q^{33} +6.18622 q^{35} +3.17170 q^{37} +10.2036 q^{41} -5.24467 q^{43} +11.0653 q^{45} -4.09333 q^{47} -4.19665 q^{49} +0.0382503 q^{51} -9.76260 q^{53} +19.5378 q^{55} -0.333692 q^{57} -10.8913 q^{59} +0.994321 q^{61} +5.01437 q^{63} +11.7805 q^{67} -0.610193 q^{69} -12.3134 q^{71} -4.70617 q^{73} -0.620002 q^{75} +8.85376 q^{77} +8.31096 q^{79} +8.95380 q^{81} -12.5820 q^{83} +1.97201 q^{85} -0.321412 q^{87} +2.12572 q^{89} -0.0644407 q^{93} -17.2036 q^{95} +13.2660 q^{97} +15.8368 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{3} + 2 q^{5} - 3 q^{7} + 9 q^{9}+O(q^{10}) \) 6 * q + 3 * q^3 + 2 * q^5 - 3 * q^7 + 9 * q^9	\( 6 q + 3 q^{3} + 2 q^{5} - 3 q^{7} + 9 q^{9} - 13 q^{11} + 14 q^{15} + 11 q^{17} - 15 q^{19} + 22 q^{21} + 15 q^{23} + 16 q^{25} + 21 q^{27} + 9 q^{29} + 3 q^{31} + 2 q^{33} + 14 q^{35} - 14 q^{37} + 20 q^{41} + 10 q^{43} + 9 q^{45} + 10 q^{47} + 27 q^{49} + 5 q^{51} + q^{53} + 14 q^{55} - 9 q^{57} - 50 q^{59} - 2 q^{63} - 6 q^{67} + 32 q^{69} - 9 q^{71} + 6 q^{73} + 40 q^{75} - 2 q^{77} + 13 q^{79} + 42 q^{81} - 36 q^{83} + 31 q^{85} + 22 q^{87} + 18 q^{89} + 12 q^{93} - 21 q^{95} + 14 q^{97} - 18 q^{99}+O(q^{100}) \) 6 * q + 3 * q^3 + 2 * q^5 - 3 * q^7 + 9 * q^9 - 13 * q^11 + 14 * q^15 + 11 * q^17 - 15 * q^19 + 22 * q^21 + 15 * q^23 + 16 * q^25 + 21 * q^27 + 9 * q^29 + 3 * q^31 + 2 * q^33 + 14 * q^35 - 14 * q^37 + 20 * q^41 + 10 * q^43 + 9 * q^45 + 10 * q^47 + 27 * q^49 + 5 * q^51 + q^53 + 14 * q^55 - 9 * q^57 - 50 * q^59 - 2 * q^63 - 6 * q^67 + 32 * q^69 - 9 * q^71 + 6 * q^73 + 40 * q^75 - 2 * q^77 + 13 * q^79 + 42 * q^81 - 36 * q^83 + 31 * q^85 + 22 * q^87 + 18 * q^89 + 12 * q^93 - 21 * q^95 + 14 * q^97 - 18 * q^99

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