LMFDB - Embedded newform 9968.2.a.z.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9968,2,Mod(1,9968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9968, 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("9968.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9968 = 2^{4} \cdot 7 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9968.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:	$79.5948807348$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.135076.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 2 $ x^5 - x^4 - 5x^3 + 4x^2 + 4*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1246)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.26848$ of defining polynomial
Character	$\chi$	$=$	9968.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.45619 q^{3} -3.65944 q^{5} -1.00000 q^{7} +8.94527 q^{9} +O(q^{10})$	$q+3.45619 q^{3} -3.65944 q^{5} -1.00000 q^{7} +8.94527 q^{9} -0.165497 q^{11} +0.358607 q^{13} -12.6477 q^{15} +5.94527 q^{17} -1.21124 q^{19} -3.45619 q^{21} +5.56984 q^{23} +8.39149 q^{25} +20.5480 q^{27} -9.81809 q^{29} +1.53565 q^{31} -0.571988 q^{33} +3.65944 q^{35} +3.48709 q^{37} +1.23942 q^{39} -9.70716 q^{41} +2.90910 q^{43} -32.7347 q^{45} -4.06577 q^{47} +1.00000 q^{49} +20.5480 q^{51} +1.46905 q^{53} +0.605625 q^{55} -4.18626 q^{57} +7.87266 q^{59} -0.0478841 q^{61} -8.94527 q^{63} -1.31230 q^{65} +2.00396 q^{67} +19.2505 q^{69} +4.79874 q^{71} +7.03685 q^{73} +29.0026 q^{75} +0.165497 q^{77} +3.53226 q^{79} +44.1820 q^{81} -15.1778 q^{83} -21.7563 q^{85} -33.9332 q^{87} -1.00000 q^{89} -0.358607 q^{91} +5.30751 q^{93} +4.43244 q^{95} -17.8521 q^{97} -1.48041 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 6 q^{3} - 10 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) $ 5 * q + 6 * q^3 - 10 * q^5 - 5 * q^7 + 7 * q^9	$ 5 q + 6 q^{3} - 10 q^{5} - 5 q^{7} + 7 q^{9} + 8 q^{11} - 8 q^{13} - 6 q^{15} - 8 q^{17} + 10 q^{19} - 6 q^{21} + 2 q^{23} + 17 q^{25} + 24 q^{27} - 10 q^{29} + 10 q^{31} + 10 q^{35} - 4 q^{37} - 24 q^{39} - 28 q^{41} + 10 q^{43} - 12 q^{45} + 10 q^{47} + 5 q^{49} + 24 q^{51} + 4 q^{53} - 4 q^{55} + 2 q^{57} + 10 q^{59} - 16 q^{61} - 7 q^{63} - 26 q^{65} - 2 q^{69} + 16 q^{71} + 10 q^{73} + 10 q^{75} - 8 q^{77} + 8 q^{79} + 41 q^{81} + 14 q^{83} + 18 q^{85} - 22 q^{87} - 5 q^{89} + 8 q^{91} + 22 q^{93} - 34 q^{95} + 6 q^{97} + 8 q^{99}+O(q^{100}) $ 5 * q + 6 * q^3 - 10 * q^5 - 5 * q^7 + 7 * q^9 + 8 * q^11 - 8 * q^13 - 6 * q^15 - 8 * q^17 + 10 * q^19 - 6 * q^21 + 2 * q^23 + 17 * q^25 + 24 * q^27 - 10 * q^29 + 10 * q^31 + 10 * q^35 - 4 * q^37 - 24 * q^39 - 28 * q^41 + 10 * q^43 - 12 * q^45 + 10 * q^47 + 5 * q^49 + 24 * q^51 + 4 * q^53 - 4 * q^55 + 2 * q^57 + 10 * q^59 - 16 * q^61 - 7 * q^63 - 26 * q^65 - 2 * q^69 + 16 * q^71 + 10 * q^73 + 10 * q^75 - 8 * q^77 + 8 * q^79 + 41 * q^81 + 14 * q^83 + 18 * q^85 - 22 * q^87 - 5 * q^89 + 8 * q^91 + 22 * q^93 - 34 * q^95 + 6 * q^97 + 8 * 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
$3$	3.45619	1.99543	0.997717	−	0.0675348i	$-0.0215134\pi$
$3$	3.45619	1.99543	0.997717	+	0.0675348i	$0.0215134\pi$
$4$	0	0
$5$	−3.65944	−1.63655	−0.818275	−	0.574826i	$-0.805070\pi$
$5$	−3.65944	−1.63655	−0.818275	+	0.574826i	$0.805070\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	8.94527	2.98176
$10$	0	0
$11$	−0.165497	−0.0498991	−0.0249495	−	0.999689i	$-0.507943\pi$
$11$	−0.165497	−0.0498991	−0.0249495	+	0.999689i	$0.507943\pi$
$12$	0	0
$13$	0.358607	0.0994598	0.0497299	−	0.998763i	$-0.484164\pi$
$13$	0.358607	0.0994598	0.0497299	+	0.998763i	$0.484164\pi$
$14$	0	0
$15$	−12.6477	−3.26563
$16$	0	0
$17$	5.94527	1.44194	0.720970	−	0.692967i	$-0.243697\pi$
$17$	5.94527	1.44194	0.720970	+	0.692967i	$0.243697\pi$
$18$	0	0
$19$	−1.21124	−0.277876	−0.138938	−	0.990301i	$-0.544369\pi$
$19$	−1.21124	−0.277876	−0.138938	+	0.990301i	$0.544369\pi$
$20$	0	0
$21$	−3.45619	−0.754203
$22$	0	0
$23$	5.56984	1.16139	0.580696	−	0.814120i	$-0.302780\pi$
$23$	5.56984	1.16139	0.580696	+	0.814120i	$0.302780\pi$
$24$	0	0
$25$	8.39149	1.67830
$26$	0	0
$27$	20.5480	3.95446
$28$	0	0
$29$	−9.81809	−1.82317	−0.911587	−	0.411108i	$-0.865142\pi$
$29$	−9.81809	−1.82317	−0.911587	+	0.411108i	$0.865142\pi$
$30$	0	0
$31$	1.53565	0.275812	0.137906	−	0.990445i	$-0.455963\pi$
$31$	1.53565	0.275812	0.137906	+	0.990445i	$0.455963\pi$
$32$	0	0
$33$	−0.571988	−0.0995703
$34$	0	0
$35$	3.65944	0.618558
$36$	0	0
$37$	3.48709	0.573275	0.286637	−	0.958039i	$-0.407463\pi$
$37$	3.48709	0.573275	0.286637	+	0.958039i	$0.407463\pi$
$38$	0	0
$39$	1.23942	0.198465
$40$	0	0
$41$	−9.70716	−1.51600	−0.758002	−	0.652253i	$-0.773824\pi$
$41$	−9.70716	−1.51600	−0.758002	+	0.652253i	$0.773824\pi$
$42$	0	0
$43$	2.90910	0.443633	0.221817	−	0.975088i	$-0.428801\pi$
$43$	2.90910	0.443633	0.221817	+	0.975088i	$0.428801\pi$
$44$	0	0
$45$	−32.7347	−4.87980
$46$	0	0
$47$	−4.06577	−0.593053	−0.296526	−	0.955025i	$-0.595828\pi$
$47$	−4.06577	−0.593053	−0.296526	+	0.955025i	$0.595828\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	20.5480	2.87729
$52$	0	0
$53$	1.46905	0.201789	0.100895	−	0.994897i	$-0.467829\pi$
$53$	1.46905	0.201789	0.100895	+	0.994897i	$0.467829\pi$
$54$	0	0
$55$	0.605625	0.0816624
$56$	0	0
$57$	−4.18626	−0.554484
$58$	0	0
$59$	7.87266	1.02493	0.512466	−	0.858707i	$-0.328732\pi$
$59$	7.87266	1.02493	0.512466	+	0.858707i	$0.328732\pi$
$60$	0	0
$61$	−0.0478841	−0.00613093	−0.00306546	−	0.999995i	$-0.500976\pi$
$61$	−0.0478841	−0.00613093	−0.00306546	+	0.999995i	$0.500976\pi$
$62$	0	0
$63$	−8.94527	−1.12700
$64$	0	0
$65$	−1.31230	−0.162771
$66$	0	0
$67$	2.00396	0.244823	0.122411	−	0.992479i	$-0.460937\pi$
$67$	2.00396	0.244823	0.122411	+	0.992479i	$0.460937\pi$
$68$	0	0
$69$	19.2505	2.31748
$70$	0	0
$71$	4.79874	0.569505	0.284753	−	0.958601i	$-0.408089\pi$
$71$	4.79874	0.569505	0.284753	+	0.958601i	$0.408089\pi$
$72$	0	0
$73$	7.03685	0.823600	0.411800	−	0.911274i	$-0.364900\pi$
$73$	7.03685	0.823600	0.411800	+	0.911274i	$0.364900\pi$
$74$	0	0
$75$	29.0026	3.34893
$76$	0	0
$77$	0.165497	0.0188601
$78$	0	0
$79$	3.53226	0.397410	0.198705	−	0.980059i	$-0.436326\pi$
$79$	3.53226	0.397410	0.198705	+	0.980059i	$0.436326\pi$
$80$	0	0
$81$	44.1820	4.90911
$82$	0	0
$83$	−15.1778	−1.66598	−0.832992	−	0.553285i	$-0.813374\pi$
$83$	−15.1778	−1.66598	−0.832992	+	0.553285i	$0.813374\pi$
$84$	0	0
$85$	−21.7563	−2.35981
$86$	0	0
$87$	−33.9332	−3.63802
$88$	0	0
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−0.358607	−0.0375923
$92$	0	0
$93$	5.30751	0.550364
$94$	0	0
$95$	4.43244	0.454759
$96$	0	0
$97$	−17.8521	−1.81261	−0.906304	−	0.422627i	$-0.861108\pi$
$97$	−17.8521	−1.81261	−0.906304	+	0.422627i	$0.861108\pi$
$98$	0	0
$99$	−1.48041	−0.148787
$100$	0	0
$101$	−19.0918	−1.89971	−0.949853	−	0.312698i	$-0.898767\pi$
$101$	−19.0918	−1.89971	−0.949853	+	0.312698i	$0.898767\pi$
$102$	0	0
$103$	12.1552	1.19769	0.598844	−	0.800866i	$-0.295627\pi$
$103$	12.1552	1.19769	0.598844	+	0.800866i	$0.295627\pi$
$104$	0	0
$105$	12.6477	1.23429
$106$	0	0
$107$	13.5759	1.31243	0.656214	−	0.754575i	$-0.272157\pi$
$107$	13.5759	1.31243	0.656214	+	0.754575i	$0.272157\pi$
$108$	0	0
$109$	8.92771	0.855120	0.427560	−	0.903987i	$-0.359373\pi$
$109$	8.92771	0.855120	0.427560	+	0.903987i	$0.359373\pi$
$110$	0	0
$111$	12.0521	1.14393
$112$	0	0
$113$	4.90842	0.461746	0.230873	−	0.972984i	$-0.425842\pi$
$113$	4.90842	0.461746	0.230873	+	0.972984i	$0.425842\pi$
$114$	0	0
$115$	−20.3825	−1.90068
$116$	0	0
$117$	3.20784	0.296565
$118$	0	0
$119$	−5.94527	−0.545002
$120$	0	0
$121$	−10.9726	−0.997510
$122$	0	0
$123$	−33.5498	−3.02508
$124$	0	0
$125$	−12.4110	−1.11007
$126$	0	0
$127$	17.5425	1.55664	0.778321	−	0.627867i	$-0.216072\pi$
$127$	17.5425	1.55664	0.778321	+	0.627867i	$0.216072\pi$
$128$	0	0
$129$	10.0544	0.885241
$130$	0	0
$131$	8.04087	0.702534	0.351267	−	0.936275i	$-0.385751\pi$
$131$	8.04087	0.702534	0.351267	+	0.936275i	$0.385751\pi$
$132$	0	0
$133$	1.21124	0.105027
$134$	0	0
$135$	−75.1941	−6.47168
$136$	0	0
$137$	13.4898	1.15251	0.576256	−	0.817269i	$-0.304513\pi$
$137$	13.4898	1.15251	0.576256	+	0.817269i	$0.304513\pi$
$138$	0	0
$139$	9.29398	0.788305	0.394153	−	0.919045i	$-0.371038\pi$
$139$	9.29398	0.788305	0.394153	+	0.919045i	$0.371038\pi$
$140$	0	0
$141$	−14.0521	−1.18340
$142$	0	0
$143$	−0.0593483	−0.00496295
$144$	0	0
$145$	35.9287	2.98372
$146$	0	0
$147$	3.45619	0.285062
$148$	0	0
$149$	2.22741	0.182476	0.0912382	−	0.995829i	$-0.470918\pi$
$149$	2.22741	0.182476	0.0912382	+	0.995829i	$0.470918\pi$
$150$	0	0
$151$	20.9145	1.70200	0.850999	−	0.525167i	$-0.175997\pi$
$151$	20.9145	1.70200	0.850999	+	0.525167i	$0.175997\pi$
$152$	0	0
$153$	53.1820	4.29951
$154$	0	0
$155$	−5.61963	−0.451380
$156$	0	0
$157$	17.3637	1.38578	0.692888	−	0.721046i	$-0.256338\pi$
$157$	17.3637	1.38578	0.692888	+	0.721046i	$0.256338\pi$
$158$	0	0
$159$	5.07732	0.402657
$160$	0	0
$161$	−5.56984	−0.438965
$162$	0	0
$163$	−15.5831	−1.22056	−0.610281	−	0.792185i	$-0.708944\pi$
$163$	−15.5831	−1.22056	−0.610281	+	0.792185i	$0.708944\pi$
$164$	0	0
$165$	2.09316	0.162952
$166$	0	0
$167$	22.8527	1.76840	0.884198	−	0.467113i	$-0.154706\pi$
$167$	22.8527	1.76840	0.884198	+	0.467113i	$0.154706\pi$
$168$	0	0
$169$	−12.8714	−0.990108
$170$	0	0
$171$	−10.8348	−0.828560
$172$	0	0
$173$	9.89466	0.752277	0.376139	−	0.926563i	$-0.377252\pi$
$173$	9.89466	0.752277	0.376139	+	0.926563i	$0.377252\pi$
$174$	0	0
$175$	−8.39149	−0.634337
$176$	0	0
$177$	27.2094	2.04518
$178$	0	0
$179$	15.5595	1.16297	0.581487	−	0.813555i	$-0.302471\pi$
$179$	15.5595	1.16297	0.581487	+	0.813555i	$0.302471\pi$
$180$	0	0
$181$	17.2533	1.28243	0.641215	−	0.767361i	$-0.278431\pi$
$181$	17.2533	1.28243	0.641215	+	0.767361i	$0.278431\pi$
$182$	0	0
$183$	−0.165497	−0.0122339
$184$	0	0
$185$	−12.7608	−0.938193
$186$	0	0
$187$	−0.983922	−0.0719515
$188$	0	0
$189$	−20.5480	−1.49465
$190$	0	0
$191$	9.02209	0.652815	0.326408	−	0.945229i	$-0.394162\pi$
$191$	9.02209	0.652815	0.326408	+	0.945229i	$0.394162\pi$
$192$	0	0
$193$	12.0564	0.867836	0.433918	−	0.900952i	$-0.357131\pi$
$193$	12.0564	0.867836	0.433918	+	0.900952i	$0.357131\pi$
$194$	0	0
$195$	−4.53557	−0.324799
$196$	0	0
$197$	−14.8920	−1.06101	−0.530506	−	0.847681i	$-0.677998\pi$
$197$	−14.8920	−1.06101	−0.530506	+	0.847681i	$0.677998\pi$
$198$	0	0
$199$	−18.4802	−1.31003	−0.655013	−	0.755618i	$-0.727337\pi$
$199$	−18.4802	−1.31003	−0.655013	+	0.755618i	$0.727337\pi$
$200$	0	0
$201$	6.92608	0.488528
$202$	0	0
$203$	9.81809	0.689095
$204$	0	0
$205$	35.5228	2.48102
$206$	0	0
$207$	49.8237	3.46299
$208$	0	0
$209$	0.200455	0.0138658
$210$	0	0
$211$	−1.84119	−0.126753	−0.0633763	−	0.997990i	$-0.520187\pi$
$211$	−1.84119	−0.126753	−0.0633763	+	0.997990i	$0.520187\pi$
$212$	0	0
$213$	16.5854	1.13641
$214$	0	0
$215$	−10.6457	−0.726028
$216$	0	0
$217$	−1.53565	−0.104247
$218$	0	0
$219$	24.3207	1.64344
$220$	0	0
$221$	2.13202	0.143415
$222$	0	0
$223$	11.9566	0.800675	0.400338	−	0.916368i	$-0.368893\pi$
$223$	11.9566	0.800675	0.400338	+	0.916368i	$0.368893\pi$
$224$	0	0
$225$	75.0641	5.00428
$226$	0	0
$227$	−3.55096	−0.235685	−0.117843	−	0.993032i	$-0.537598\pi$
$227$	−3.55096	−0.235685	−0.117843	+	0.993032i	$0.537598\pi$
$228$	0	0
$229$	−3.55139	−0.234683	−0.117341	−	0.993092i	$-0.537437\pi$
$229$	−3.55139	−0.234683	−0.117341	+	0.993092i	$0.537437\pi$
$230$	0	0
$231$	0.571988	0.0376341
$232$	0	0
$233$	−3.67352	−0.240660	−0.120330	−	0.992734i	$-0.538395\pi$
$233$	−3.67352	−0.240660	−0.120330	+	0.992734i	$0.538395\pi$
$234$	0	0
$235$	14.8784	0.970561
$236$	0	0
$237$	12.2082	0.793005
$238$	0	0
$239$	8.73692	0.565144	0.282572	−	0.959246i	$-0.408812\pi$
$239$	8.73692	0.565144	0.282572	+	0.959246i	$0.408812\pi$
$240$	0	0
$241$	12.4290	0.800625	0.400313	−	0.916379i	$-0.368902\pi$
$241$	12.4290	0.800625	0.400313	+	0.916379i	$0.368902\pi$
$242$	0	0
$243$	91.0576	5.84135
$244$	0	0
$245$	−3.65944	−0.233793
$246$	0	0
$247$	−0.434358	−0.0276375
$248$	0	0
$249$	−52.4575	−3.32436
$250$	0	0
$251$	−1.49840	−0.0945783	−0.0472891	−	0.998881i	$-0.515058\pi$
$251$	−1.49840	−0.0945783	−0.0472891	+	0.998881i	$0.515058\pi$
$252$	0	0
$253$	−0.921790	−0.0579524
$254$	0	0
$255$	−75.1941	−4.70884
$256$	0	0
$257$	0.612697	0.0382190	0.0191095	−	0.999817i	$-0.493917\pi$
$257$	0.612697	0.0382190	0.0191095	+	0.999817i	$0.493917\pi$
$258$	0	0
$259$	−3.48709	−0.216678
$260$	0	0
$261$	−87.8254	−5.43626
$262$	0	0
$263$	−4.32252	−0.266538	−0.133269	−	0.991080i	$-0.542547\pi$
$263$	−4.32252	−0.266538	−0.133269	+	0.991080i	$0.542547\pi$
$264$	0	0
$265$	−5.37589	−0.330239
$266$	0	0
$267$	−3.45619	−0.211516
$268$	0	0
$269$	1.16952	0.0713071	0.0356536	−	0.999364i	$-0.488649\pi$
$269$	1.16952	0.0713071	0.0356536	+	0.999364i	$0.488649\pi$
$270$	0	0
$271$	0.0899982	0.00546700	0.00273350	−	0.999996i	$-0.499130\pi$
$271$	0.0899982	0.00546700	0.00273350	+	0.999996i	$0.499130\pi$
$272$	0	0
$273$	−1.23942	−0.0750129
$274$	0	0
$275$	−1.38876	−0.0837456
$276$	0	0
$277$	−18.6948	−1.12326	−0.561630	−	0.827389i	$-0.689826\pi$
$277$	−18.6948	−1.12326	−0.561630	+	0.827389i	$0.689826\pi$
$278$	0	0
$279$	13.7368	0.822403
$280$	0	0
$281$	−5.92994	−0.353751	−0.176875	−	0.984233i	$-0.556599\pi$
$281$	−5.92994	−0.353751	−0.176875	+	0.984233i	$0.556599\pi$
$282$	0	0
$283$	−24.6866	−1.46746	−0.733732	−	0.679439i	$-0.762223\pi$
$283$	−24.6866	−1.46746	−0.733732	+	0.679439i	$0.762223\pi$
$284$	0	0
$285$	15.3194	0.907441
$286$	0	0
$287$	9.70716	0.572995
$288$	0	0
$289$	18.3462	1.07919
$290$	0	0
$291$	−61.7003	−3.61694
$292$	0	0
$293$	−8.27518	−0.483441	−0.241721	−	0.970346i	$-0.577712\pi$
$293$	−8.27518	−0.483441	−0.241721	+	0.970346i	$0.577712\pi$
$294$	0	0
$295$	−28.8095	−1.67735
$296$	0	0
$297$	−3.40062	−0.197324
$298$	0	0
$299$	1.99739	0.115512
$300$	0	0
$301$	−2.90910	−0.167678
$302$	0	0
$303$	−65.9849	−3.79074
$304$	0	0
$305$	0.175229	0.0100336
$306$	0	0
$307$	−0.0448352	−0.00255888	−0.00127944	−	0.999999i	$-0.500407\pi$
$307$	−0.0448352	−0.00255888	−0.00127944	+	0.999999i	$0.500407\pi$
$308$	0	0
$309$	42.0107	2.38991
$310$	0	0
$311$	14.4948	0.821925	0.410963	−	0.911652i	$-0.365193\pi$
$311$	14.4948	0.821925	0.410963	+	0.911652i	$0.365193\pi$
$312$	0	0
$313$	31.0681	1.75608	0.878038	−	0.478591i	$-0.158852\pi$
$313$	31.0681	1.75608	0.878038	+	0.478591i	$0.158852\pi$
$314$	0	0
$315$	32.7347	1.84439
$316$	0	0
$317$	4.16996	0.234208	0.117104	−	0.993120i	$-0.462639\pi$
$317$	4.16996	0.234208	0.117104	+	0.993120i	$0.462639\pi$
$318$	0	0
$319$	1.62486	0.0909747
$320$	0	0
$321$	46.9208	2.61886
$322$	0	0
$323$	−7.20112	−0.400681
$324$	0	0
$325$	3.00925	0.166923
$326$	0	0
$327$	30.8559	1.70633
$328$	0	0
$329$	4.06577	0.224153
$330$	0	0
$331$	30.6935	1.68707	0.843533	−	0.537078i	$-0.180472\pi$
$331$	30.6935	1.68707	0.843533	+	0.537078i	$0.180472\pi$
$332$	0	0
$333$	31.1930	1.70937
$334$	0	0
$335$	−7.33338	−0.400665
$336$	0	0
$337$	−26.6869	−1.45373	−0.726864	−	0.686782i	$-0.759023\pi$
$337$	−26.6869	−1.45373	−0.726864	+	0.686782i	$0.759023\pi$
$338$	0	0
$339$	16.9645	0.921383
$340$	0	0
$341$	−0.254145	−0.0137627
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−70.4458	−3.79268
$346$	0	0
$347$	−24.1124	−1.29442	−0.647211	−	0.762311i	$-0.724065\pi$
$347$	−24.1124	−1.29442	−0.647211	+	0.762311i	$0.724065\pi$
$348$	0	0
$349$	2.19540	0.117517	0.0587584	−	0.998272i	$-0.481286\pi$
$349$	2.19540	0.117517	0.0587584	+	0.998272i	$0.481286\pi$
$350$	0	0
$351$	7.36866	0.393310
$352$	0	0
$353$	−15.5498	−0.827633	−0.413816	−	0.910360i	$-0.635804\pi$
$353$	−15.5498	−0.827633	−0.413816	+	0.910360i	$0.635804\pi$
$354$	0	0
$355$	−17.5607	−0.932024
$356$	0	0
$357$	−20.5480	−1.08752
$358$	0	0
$359$	20.7963	1.09759	0.548795	−	0.835957i	$-0.315087\pi$
$359$	20.7963	1.09759	0.548795	+	0.835957i	$0.315087\pi$
$360$	0	0
$361$	−17.5329	−0.922785
$362$	0	0
$363$	−37.9235	−1.99047
$364$	0	0
$365$	−25.7509	−1.34786
$366$	0	0
$367$	20.7678	1.08407	0.542035	−	0.840356i	$-0.317654\pi$
$367$	20.7678	1.08407	0.542035	+	0.840356i	$0.317654\pi$
$368$	0	0
$369$	−86.8331	−4.52035
$370$	0	0
$371$	−1.46905	−0.0762692
$372$	0	0
$373$	−27.5709	−1.42757	−0.713784	−	0.700366i	$-0.753020\pi$
$373$	−27.5709	−1.42757	−0.713784	+	0.700366i	$0.753020\pi$
$374$	0	0
$375$	−42.8946	−2.21507
$376$	0	0
$377$	−3.52084	−0.181332
$378$	0	0
$379$	−22.2921	−1.14507	−0.572533	−	0.819882i	$-0.694039\pi$
$379$	−22.2921	−1.14507	−0.572533	+	0.819882i	$0.694039\pi$
$380$	0	0
$381$	60.6301	3.10617
$382$	0	0
$383$	−17.6672	−0.902751	−0.451376	−	0.892334i	$-0.649066\pi$
$383$	−17.6672	−0.902751	−0.451376	+	0.892334i	$0.649066\pi$
$384$	0	0
$385$	−0.605625	−0.0308655
$386$	0	0
$387$	26.0227	1.32281
$388$	0	0
$389$	10.7023	0.542627	0.271314	−	0.962491i	$-0.412542\pi$
$389$	10.7023	0.542627	0.271314	+	0.962491i	$0.412542\pi$
$390$	0	0
$391$	33.1142	1.67466
$392$	0	0
$393$	27.7908	1.40186
$394$	0	0
$395$	−12.9261	−0.650382
$396$	0	0
$397$	9.63166	0.483399	0.241700	−	0.970351i	$-0.422295\pi$
$397$	9.63166	0.483399	0.241700	+	0.970351i	$0.422295\pi$
$398$	0	0
$399$	4.18626	0.209575
$400$	0	0
$401$	17.5800	0.877905	0.438953	−	0.898510i	$-0.355350\pi$
$401$	17.5800	0.877905	0.438953	+	0.898510i	$0.355350\pi$
$402$	0	0
$403$	0.550697	0.0274322
$404$	0	0
$405$	−161.681	−8.03401
$406$	0	0
$407$	−0.577102	−0.0286059
$408$	0	0
$409$	−6.17751	−0.305458	−0.152729	−	0.988268i	$-0.548806\pi$
$409$	−6.17751	−0.305458	−0.152729	+	0.988268i	$0.548806\pi$
$410$	0	0
$411$	46.6234	2.29976
$412$	0	0
$413$	−7.87266	−0.387388
$414$	0	0
$415$	55.5424	2.72647
$416$	0	0
$417$	32.1218	1.57301
$418$	0	0
$419$	14.2031	0.693866	0.346933	−	0.937890i	$-0.387223\pi$
$419$	14.2031	0.693866	0.346933	+	0.937890i	$0.387223\pi$
$420$	0	0
$421$	−3.45742	−0.168504	−0.0842522	−	0.996444i	$-0.526850\pi$
$421$	−3.45742	−0.168504	−0.0842522	+	0.996444i	$0.526850\pi$
$422$	0	0
$423$	−36.3694	−1.76834
$424$	0	0
$425$	49.8897	2.42000
$426$	0	0
$427$	0.0478841	0.00231727
$428$	0	0
$429$	−0.205119	−0.00990325
$430$	0	0
$431$	6.59565	0.317701	0.158851	−	0.987303i	$-0.449221\pi$
$431$	6.59565	0.317701	0.158851	+	0.987303i	$0.449221\pi$
$432$	0	0
$433$	14.7485	0.708766	0.354383	−	0.935100i	$-0.384691\pi$
$433$	14.7485	0.708766	0.354383	+	0.935100i	$0.384691\pi$
$434$	0	0
$435$	124.176	5.95381
$436$	0	0
$437$	−6.74639	−0.322724
$438$	0	0
$439$	−32.1910	−1.53639	−0.768195	−	0.640215i	$-0.778845\pi$
$439$	−32.1910	−1.53639	−0.768195	+	0.640215i	$0.778845\pi$
$440$	0	0
$441$	8.94527	0.425965
$442$	0	0
$443$	6.93810	0.329639	0.164819	−	0.986324i	$-0.447296\pi$
$443$	6.93810	0.329639	0.164819	+	0.986324i	$0.447296\pi$
$444$	0	0
$445$	3.65944	0.173474
$446$	0	0
$447$	7.69835	0.364120
$448$	0	0
$449$	34.8687	1.64556	0.822779	−	0.568362i	$-0.192423\pi$
$449$	34.8687	1.64556	0.822779	+	0.568362i	$0.192423\pi$
$450$	0	0
$451$	1.60650	0.0756472
$452$	0	0
$453$	72.2846	3.39623
$454$	0	0
$455$	1.31230	0.0615217
$456$	0	0
$457$	29.3507	1.37297	0.686483	−	0.727145i	$-0.259153\pi$
$457$	29.3507	1.37297	0.686483	+	0.727145i	$0.259153\pi$
$458$	0	0
$459$	122.163	5.70210
$460$	0	0
$461$	−20.7149	−0.964788	−0.482394	−	0.875954i	$-0.660233\pi$
$461$	−20.7149	−0.964788	−0.482394	+	0.875954i	$0.660233\pi$
$462$	0	0
$463$	−13.5023	−0.627503	−0.313752	−	0.949505i	$-0.601586\pi$
$463$	−13.5023	−0.627503	−0.313752	+	0.949505i	$0.601586\pi$
$464$	0	0
$465$	−19.4225	−0.900698
$466$	0	0
$467$	−0.567365	−0.0262545	−0.0131273	−	0.999914i	$-0.504179\pi$
$467$	−0.567365	−0.0262545	−0.0131273	+	0.999914i	$0.504179\pi$
$468$	0	0
$469$	−2.00396	−0.0925344
$470$	0	0
$471$	60.0123	2.76522
$472$	0	0
$473$	−0.481446	−0.0221369
$474$	0	0
$475$	−10.1641	−0.466360
$476$	0	0
$477$	13.1410	0.601687
$478$	0	0
$479$	17.1083	0.781700	0.390850	−	0.920454i	$-0.372181\pi$
$479$	17.1083	0.781700	0.390850	+	0.920454i	$0.372181\pi$
$480$	0	0
$481$	1.25050	0.0570178
$482$	0	0
$483$	−19.2505	−0.875926
$484$	0	0
$485$	65.3287	2.96642
$486$	0	0
$487$	−20.5358	−0.930566	−0.465283	−	0.885162i	$-0.654047\pi$
$487$	−20.5358	−0.930566	−0.465283	+	0.885162i	$0.654047\pi$
$488$	0	0
$489$	−53.8582	−2.43555
$490$	0	0
$491$	−27.3716	−1.23526	−0.617632	−	0.786467i	$-0.711908\pi$
$491$	−27.3716	−1.23526	−0.617632	+	0.786467i	$0.711908\pi$
$492$	0	0
$493$	−58.3712	−2.62891
$494$	0	0
$495$	5.41747	0.243497
$496$	0	0
$497$	−4.79874	−0.215253
$498$	0	0
$499$	−12.2846	−0.549933	−0.274967	−	0.961454i	$-0.588667\pi$
$499$	−12.2846	−0.549933	−0.274967	+	0.961454i	$0.588667\pi$
$500$	0	0
$501$	78.9834	3.52872
$502$	0	0
$503$	3.93581	0.175489	0.0877445	−	0.996143i	$-0.472034\pi$
$503$	3.93581	0.175489	0.0877445	+	0.996143i	$0.472034\pi$
$504$	0	0
$505$	69.8653	3.10896
$506$	0	0
$507$	−44.4860	−1.97569
$508$	0	0
$509$	−2.82255	−0.125107	−0.0625537	−	0.998042i	$-0.519924\pi$
$509$	−2.82255	−0.125107	−0.0625537	+	0.998042i	$0.519924\pi$
$510$	0	0
$511$	−7.03685	−0.311292
$512$	0	0
$513$	−24.8885	−1.09885
$514$	0	0
$515$	−44.4812	−1.96008
$516$	0	0
$517$	0.672870	0.0295928
$518$	0	0
$519$	34.1979	1.50112
$520$	0	0
$521$	12.2876	0.538331	0.269165	−	0.963094i	$-0.413252\pi$
$521$	12.2876	0.538331	0.269165	+	0.963094i	$0.413252\pi$
$522$	0	0
$523$	3.05299	0.133498	0.0667489	−	0.997770i	$-0.478737\pi$
$523$	3.05299	0.133498	0.0667489	+	0.997770i	$0.478737\pi$
$524$	0	0
$525$	−29.0026	−1.26578
$526$	0	0
$527$	9.12987	0.397703
$528$	0	0
$529$	8.02315	0.348833
$530$	0	0
$531$	70.4230	3.05610
$532$	0	0
$533$	−3.48106	−0.150781
$534$	0	0
$535$	−49.6800	−2.14785
$536$	0	0
$537$	53.7768	2.32064
$538$	0	0
$539$	−0.165497	−0.00712844
$540$	0	0
$541$	22.8562	0.982666	0.491333	−	0.870972i	$-0.336510\pi$
$541$	22.8562	0.982666	0.491333	+	0.870972i	$0.336510\pi$
$542$	0	0
$543$	59.6309	2.55900
$544$	0	0
$545$	−32.6704	−1.39945
$546$	0	0
$547$	−20.8457	−0.891298	−0.445649	−	0.895208i	$-0.647027\pi$
$547$	−20.8457	−0.891298	−0.445649	+	0.895208i	$0.647027\pi$
$548$	0	0
$549$	−0.428336	−0.0182809
$550$	0	0
$551$	11.8920	0.506617
$552$	0	0
$553$	−3.53226	−0.150207
$554$	0	0
$555$	−44.1038	−1.87210
$556$	0	0
$557$	−44.9382	−1.90409	−0.952045	−	0.305957i	$-0.901024\pi$
$557$	−44.9382	−1.90409	−0.952045	+	0.305957i	$0.901024\pi$
$558$	0	0
$559$	1.04322	0.0441237
$560$	0	0
$561$	−3.40062	−0.143574
$562$	0	0
$563$	27.6217	1.16412	0.582058	−	0.813147i	$-0.302248\pi$
$563$	27.6217	1.16412	0.582058	+	0.813147i	$0.302248\pi$
$564$	0	0
$565$	−17.9621	−0.755670
$566$	0	0
$567$	−44.1820	−1.85547
$568$	0	0
$569$	−21.7222	−0.910642	−0.455321	−	0.890327i	$-0.650476\pi$
$569$	−21.7222	−0.910642	−0.455321	+	0.890327i	$0.650476\pi$
$570$	0	0
$571$	9.15009	0.382919	0.191460	−	0.981500i	$-0.438678\pi$
$571$	9.15009	0.382919	0.191460	+	0.981500i	$0.438678\pi$
$572$	0	0
$573$	31.1821	1.30265
$574$	0	0
$575$	46.7393	1.94916
$576$	0	0
$577$	−21.9081	−0.912046	−0.456023	−	0.889968i	$-0.650726\pi$
$577$	−21.9081	−0.912046	−0.456023	+	0.889968i	$0.650726\pi$
$578$	0	0
$579$	41.6691	1.73171
$580$	0	0
$581$	15.1778	0.629683
$582$	0	0
$583$	−0.243123	−0.0100691
$584$	0	0
$585$	−11.7389	−0.485343
$586$	0	0
$587$	−44.5255	−1.83776	−0.918881	−	0.394534i	$-0.870906\pi$
$587$	−44.5255	−1.83776	−0.918881	+	0.394534i	$0.870906\pi$
$588$	0	0
$589$	−1.86004	−0.0766415
$590$	0	0
$591$	−51.4696	−2.11718
$592$	0	0
$593$	−27.6879	−1.13701	−0.568503	−	0.822681i	$-0.692477\pi$
$593$	−27.6879	−1.13701	−0.568503	+	0.822681i	$0.692477\pi$
$594$	0	0
$595$	21.7563	0.891923
$596$	0	0
$597$	−63.8711	−2.61407
$598$	0	0
$599$	−1.14704	−0.0468669	−0.0234335	−	0.999725i	$-0.507460\pi$
$599$	−1.14704	−0.0468669	−0.0234335	+	0.999725i	$0.507460\pi$
$600$	0	0
$601$	−16.1645	−0.659364	−0.329682	−	0.944092i	$-0.606942\pi$
$601$	−16.1645	−0.659364	−0.329682	+	0.944092i	$0.606942\pi$
$602$	0	0
$603$	17.9260	0.730002
$604$	0	0
$605$	40.1536	1.63248
$606$	0	0
$607$	17.1463	0.695945	0.347973	−	0.937505i	$-0.386870\pi$
$607$	17.1463	0.695945	0.347973	+	0.937505i	$0.386870\pi$
$608$	0	0
$609$	33.9332	1.37504
$610$	0	0
$611$	−1.45801	−0.0589849
$612$	0	0
$613$	24.9361	1.00716	0.503580	−	0.863949i	$-0.332016\pi$
$613$	24.9361	1.00716	0.503580	+	0.863949i	$0.332016\pi$
$614$	0	0
$615$	122.773	4.95070
$616$	0	0
$617$	27.7399	1.11677	0.558383	−	0.829584i	$-0.311422\pi$
$617$	27.7399	1.11677	0.558383	+	0.829584i	$0.311422\pi$
$618$	0	0
$619$	−22.5631	−0.906887	−0.453443	−	0.891285i	$-0.649805\pi$
$619$	−22.5631	−0.906887	−0.453443	+	0.891285i	$0.649805\pi$
$620$	0	0
$621$	114.449	4.59268
$622$	0	0
$623$	1.00000	0.0400642
$624$	0	0
$625$	3.45966	0.138386
$626$	0	0
$627$	0.692812	0.0276683
$628$	0	0
$629$	20.7317	0.826628
$630$	0	0
$631$	17.2943	0.688476	0.344238	−	0.938882i	$-0.388137\pi$
$631$	17.2943	0.688476	0.344238	+	0.938882i	$0.388137\pi$
$632$	0	0
$633$	−6.36350	−0.252926
$634$	0	0
$635$	−64.1955	−2.54752
$636$	0	0
$637$	0.358607	0.0142085
$638$	0	0
$639$	42.9260	1.69813
$640$	0	0
$641$	12.7228	0.502521	0.251260	−	0.967920i	$-0.419155\pi$
$641$	12.7228	0.502521	0.251260	+	0.967920i	$0.419155\pi$
$642$	0	0
$643$	−5.92880	−0.233809	−0.116904	−	0.993143i	$-0.537297\pi$
$643$	−5.92880	−0.233809	−0.116904	+	0.993143i	$0.537297\pi$
$644$	0	0
$645$	−36.7935	−1.44874
$646$	0	0
$647$	−43.1398	−1.69600	−0.848000	−	0.529996i	$-0.822193\pi$
$647$	−43.1398	−1.69600	−0.848000	+	0.529996i	$0.822193\pi$
$648$	0	0
$649$	−1.30290	−0.0511432
$650$	0	0
$651$	−5.30751	−0.208018
$652$	0	0
$653$	−26.9025	−1.05278	−0.526389	−	0.850244i	$-0.676454\pi$
$653$	−26.9025	−1.05278	−0.526389	+	0.850244i	$0.676454\pi$
$654$	0	0
$655$	−29.4251	−1.14973
$656$	0	0
$657$	62.9465	2.45578
$658$	0	0
$659$	−1.62318	−0.0632302	−0.0316151	−	0.999500i	$-0.510065\pi$
$659$	−1.62318	−0.0632302	−0.0316151	+	0.999500i	$0.510065\pi$
$660$	0	0
$661$	17.6491	0.686471	0.343236	−	0.939249i	$-0.388477\pi$
$661$	17.6491	0.686471	0.343236	+	0.939249i	$0.388477\pi$
$662$	0	0
$663$	7.36866	0.286175
$664$	0	0
$665$	−4.43244	−0.171883
$666$	0	0
$667$	−54.6852	−2.11742
$668$	0	0
$669$	41.3244	1.59769
$670$	0	0
$671$	0.00792465	0.000305928 0
$672$	0	0
$673$	−7.55743	−0.291317	−0.145659	−	0.989335i	$-0.546530\pi$
$673$	−7.55743	−0.291317	−0.145659	+	0.989335i	$0.546530\pi$
$674$	0	0
$675$	172.428	6.63677
$676$	0	0
$677$	−45.5082	−1.74902	−0.874511	−	0.485006i	$-0.838817\pi$
$677$	−45.5082	−1.74902	−0.874511	+	0.485006i	$0.838817\pi$
$678$	0	0
$679$	17.8521	0.685101
$680$	0	0
$681$	−12.2728	−0.470295
$682$	0	0
$683$	26.4469	1.01196	0.505981	−	0.862545i	$-0.331131\pi$
$683$	26.4469	1.01196	0.505981	+	0.862545i	$0.331131\pi$
$684$	0	0
$685$	−49.3652	−1.88615
$686$	0	0
$687$	−12.2743	−0.468294
$688$	0	0
$689$	0.526812	0.0200699
$690$	0	0
$691$	23.0912	0.878431	0.439215	−	0.898382i	$-0.355256\pi$
$691$	23.0912	0.878431	0.439215	+	0.898382i	$0.355256\pi$
$692$	0	0
$693$	1.48041	0.0562362
$694$	0	0
$695$	−34.0108	−1.29010
$696$	0	0
$697$	−57.7117	−2.18599
$698$	0	0
$699$	−12.6964	−0.480222
$700$	0	0
$701$	10.2571	0.387406	0.193703	−	0.981060i	$-0.437950\pi$
$701$	10.2571	0.387406	0.193703	+	0.981060i	$0.437950\pi$
$702$	0	0
$703$	−4.22369	−0.159300
$704$	0	0
$705$	51.4227	1.93669
$706$	0	0
$707$	19.0918	0.718021
$708$	0	0
$709$	−8.29366	−0.311475	−0.155737	−	0.987798i	$-0.549775\pi$
$709$	−8.29366	−0.311475	−0.155737	+	0.987798i	$0.549775\pi$
$710$	0	0
$711$	31.5970	1.18498
$712$	0	0
$713$	8.55335	0.320325
$714$	0	0
$715$	0.217181	0.00812213
$716$	0	0
$717$	30.1965	1.12771
$718$	0	0
$719$	−23.8072	−0.887859	−0.443929	−	0.896062i	$-0.646416\pi$
$719$	−23.8072	−0.887859	−0.443929	+	0.896062i	$0.646416\pi$
$720$	0	0
$721$	−12.1552	−0.452683
$722$	0	0
$723$	42.9572	1.59759
$724$	0	0
$725$	−82.3884	−3.05983
$726$	0	0
$727$	18.0102	0.667961	0.333981	−	0.942580i	$-0.391608\pi$
$727$	18.0102	0.667961	0.333981	+	0.942580i	$0.391608\pi$
$728$	0	0
$729$	182.167	6.74691
$730$	0	0
$731$	17.2954	0.639692
$732$	0	0
$733$	−29.8727	−1.10337	−0.551687	−	0.834051i	$-0.686016\pi$
$733$	−29.8727	−1.10337	−0.551687	+	0.834051i	$0.686016\pi$
$734$	0	0
$735$	−12.6477	−0.466518
$736$	0	0
$737$	−0.331649	−0.0122164
$738$	0	0
$739$	25.4458	0.936039	0.468019	−	0.883718i	$-0.344968\pi$
$739$	25.4458	0.936039	0.468019	+	0.883718i	$0.344968\pi$
$740$	0	0
$741$	−1.50123	−0.0551489
$742$	0	0
$743$	−21.1223	−0.774901	−0.387450	−	0.921891i	$-0.626644\pi$
$743$	−21.1223	−0.774901	−0.387450	+	0.921891i	$0.626644\pi$
$744$	0	0
$745$	−8.15106	−0.298632
$746$	0	0
$747$	−135.770	−4.96756
$748$	0	0
$749$	−13.5759	−0.496051
$750$	0	0
$751$	−25.4708	−0.929444	−0.464722	−	0.885457i	$-0.653846\pi$
$751$	−25.4708	−0.929444	−0.464722	+	0.885457i	$0.653846\pi$
$752$	0	0
$753$	−5.17876	−0.188725
$754$	0	0
$755$	−76.5354	−2.78541
$756$	0	0
$757$	46.7062	1.69756	0.848782	−	0.528742i	$-0.177336\pi$
$757$	46.7062	1.69756	0.848782	+	0.528742i	$0.177336\pi$
$758$	0	0
$759$	−3.18588	−0.115640
$760$	0	0
$761$	−6.49203	−0.235336	−0.117668	−	0.993053i	$-0.537542\pi$
$761$	−6.49203	−0.235336	−0.117668	+	0.993053i	$0.537542\pi$
$762$	0	0
$763$	−8.92771	−0.323205
$764$	0	0
$765$	−194.616	−7.03637
$766$	0	0
$767$	2.82319	0.101940
$768$	0	0
$769$	−30.9529	−1.11619	−0.558096	−	0.829777i	$-0.688468\pi$
$769$	−30.9529	−1.11619	−0.558096	+	0.829777i	$0.688468\pi$
$770$	0	0
$771$	2.11760	0.0762635
$772$	0	0
$773$	−14.9641	−0.538220	−0.269110	−	0.963109i	$-0.586730\pi$
$773$	−14.9641	−0.538220	−0.269110	+	0.963109i	$0.586730\pi$
$774$	0	0
$775$	12.8864	0.462894
$776$	0	0
$777$	−12.0521	−0.432366
$778$	0	0
$779$	11.7577	0.421262
$780$	0	0
$781$	−0.794174	−0.0284178
$782$	0	0
$783$	−201.742	−7.20967
$784$	0	0
$785$	−63.5414	−2.26789
$786$	0	0
$787$	27.2792	0.972397	0.486198	−	0.873848i	$-0.338383\pi$
$787$	27.2792	0.972397	0.486198	+	0.873848i	$0.338383\pi$
$788$	0	0
$789$	−14.9394	−0.531858
$790$	0	0
$791$	−4.90842	−0.174523
$792$	0	0
$793$	−0.0171716	−0.000609781 0
$794$	0	0
$795$	−18.5801	−0.658969
$796$	0	0
$797$	−5.48188	−0.194178	−0.0970890	−	0.995276i	$-0.530953\pi$
$797$	−5.48188	−0.194178	−0.0970890	+	0.995276i	$0.530953\pi$
$798$	0	0
$799$	−24.1721	−0.855146
$800$	0	0
$801$	−8.94527	−0.316066
$802$	0	0
$803$	−1.16457	−0.0410969
$804$	0	0
$805$	20.3825	0.718389
$806$	0	0
$807$	4.04210	0.142289
$808$	0	0
$809$	−29.0158	−1.02014	−0.510071	−	0.860132i	$-0.670381\pi$
$809$	−29.0158	−1.02014	−0.510071	+	0.860132i	$0.670381\pi$
$810$	0	0
$811$	37.5071	1.31705	0.658527	−	0.752557i	$-0.271180\pi$
$811$	37.5071	1.31705	0.658527	+	0.752557i	$0.271180\pi$
$812$	0	0
$813$	0.311051	0.0109090
$814$	0	0
$815$	57.0254	1.99751
$816$	0	0
$817$	−3.52360	−0.123275
$818$	0	0
$819$	−3.20784	−0.112091
$820$	0	0
$821$	13.2699	0.463122	0.231561	−	0.972820i	$-0.425617\pi$
$821$	13.2699	0.463122	0.231561	+	0.972820i	$0.425617\pi$
$822$	0	0
$823$	−48.2863	−1.68315	−0.841577	−	0.540138i	$-0.818372\pi$
$823$	−48.2863	−1.68315	−0.841577	+	0.540138i	$0.818372\pi$
$824$	0	0
$825$	−4.79983	−0.167109
$826$	0	0
$827$	−47.9777	−1.66835	−0.834174	−	0.551501i	$-0.814055\pi$
$827$	−47.9777	−1.66835	−0.834174	+	0.551501i	$0.814055\pi$
$828$	0	0
$829$	36.8486	1.27981	0.639903	−	0.768456i	$-0.278974\pi$
$829$	36.8486	1.27981	0.639903	+	0.768456i	$0.278974\pi$
$830$	0	0
$831$	−64.6127	−2.24139
$832$	0	0
$833$	5.94527	0.205991
$834$	0	0
$835$	−83.6281	−2.89407
$836$	0	0
$837$	31.5546	1.09069
$838$	0	0
$839$	−23.0505	−0.795792	−0.397896	−	0.917431i	$-0.630259\pi$
$839$	−23.0505	−0.795792	−0.397896	+	0.917431i	$0.630259\pi$
$840$	0	0
$841$	67.3948	2.32396
$842$	0	0
$843$	−20.4950	−0.705886
$844$	0	0
$845$	47.1021	1.62036
$846$	0	0
$847$	10.9726	0.377023
$848$	0	0
$849$	−85.3215	−2.92823
$850$	0	0
$851$	19.4226	0.665797
$852$	0	0
$853$	−32.9697	−1.12886	−0.564430	−	0.825481i	$-0.690904\pi$
$853$	−32.9697	−1.12886	−0.564430	+	0.825481i	$0.690904\pi$
$854$	0	0
$855$	39.6494	1.35598
$856$	0	0
$857$	−11.1896	−0.382230	−0.191115	−	0.981568i	$-0.561210\pi$
$857$	−11.1896	−0.382230	−0.191115	+	0.981568i	$0.561210\pi$
$858$	0	0
$859$	7.60460	0.259466	0.129733	−	0.991549i	$-0.458588\pi$
$859$	7.60460	0.259466	0.129733	+	0.991549i	$0.458588\pi$
$860$	0	0
$861$	33.5498	1.14337
$862$	0	0
$863$	−18.5722	−0.632206	−0.316103	−	0.948725i	$-0.602375\pi$
$863$	−18.5722	−0.632206	−0.316103	+	0.948725i	$0.602375\pi$
$864$	0	0
$865$	−36.2089	−1.23114
$866$	0	0
$867$	63.4081	2.15345
$868$	0	0
$869$	−0.584577	−0.0198304
$870$	0	0
$871$	0.718636	0.0243500
$872$	0	0
$873$	−159.692	−5.40475
$874$	0	0
$875$	12.4110	0.419567
$876$	0	0
$877$	56.0632	1.89312	0.946559	−	0.322530i	$-0.104533\pi$
$877$	56.0632	1.89312	0.946559	+	0.322530i	$0.104533\pi$
$878$	0	0
$879$	−28.6006	−0.964675
$880$	0	0
$881$	26.7642	0.901708	0.450854	−	0.892598i	$-0.351120\pi$
$881$	26.7642	0.901708	0.450854	+	0.892598i	$0.351120\pi$
$882$	0	0
$883$	19.3810	0.652224	0.326112	−	0.945331i	$-0.394261\pi$
$883$	19.3810	0.652224	0.326112	+	0.945331i	$0.394261\pi$
$884$	0	0
$885$	−99.5712	−3.34705
$886$	0	0
$887$	19.5756	0.657285	0.328642	−	0.944454i	$-0.393409\pi$
$887$	19.5756	0.657285	0.328642	+	0.944454i	$0.393409\pi$
$888$	0	0
$889$	−17.5425	−0.588355
$890$	0	0
$891$	−7.31197	−0.244960
$892$	0	0
$893$	4.92460	0.164795
$894$	0	0
$895$	−56.9392	−1.90327
$896$	0	0
$897$	6.90335	0.230496
$898$	0	0
$899$	−15.0772	−0.502852
$900$	0	0
$901$	8.73389	0.290968
$902$	0	0
$903$	−10.0544	−0.334590
$904$	0	0
$905$	−63.1375	−2.09876
$906$	0	0
$907$	44.2704	1.46998	0.734988	−	0.678080i	$-0.237188\pi$
$907$	44.2704	1.46998	0.734988	+	0.678080i	$0.237188\pi$
$908$	0	0
$909$	−170.781	−5.66446
$910$	0	0
$911$	−32.6588	−1.08204	−0.541018	−	0.841011i	$-0.681961\pi$
$911$	−32.6588	−1.08204	−0.541018	+	0.841011i	$0.681961\pi$
$912$	0	0
$913$	2.51188	0.0831311
$914$	0	0
$915$	0.605625	0.0200213
$916$	0	0
$917$	−8.04087	−0.265533
$918$	0	0
$919$	31.0244	1.02340	0.511700	−	0.859164i	$-0.329016\pi$
$919$	31.0244	1.02340	0.511700	+	0.859164i	$0.329016\pi$
$920$	0	0
$921$	−0.154959	−0.00510607
$922$	0	0
$923$	1.72086	0.0566429
$924$	0	0
$925$	29.2619	0.962126
$926$	0	0
$927$	108.732	3.57121
$928$	0	0
$929$	17.7544	0.582503	0.291251	−	0.956647i	$-0.405928\pi$
$929$	17.7544	0.582503	0.291251	+	0.956647i	$0.405928\pi$
$930$	0	0
$931$	−1.21124	−0.0396966
$932$	0	0
$933$	50.0969	1.64010
$934$	0	0
$935$	3.60060	0.117752
$936$	0	0
$937$	−31.8518	−1.04055	−0.520276	−	0.853998i	$-0.674171\pi$
$937$	−31.8518	−1.04055	−0.520276	+	0.853998i	$0.674171\pi$
$938$	0	0
$939$	107.378	3.50413
$940$	0	0
$941$	−10.1053	−0.329423	−0.164711	−	0.986342i	$-0.552669\pi$
$941$	−10.1053	−0.329423	−0.164711	+	0.986342i	$0.552669\pi$
$942$	0	0
$943$	−54.0673	−1.76068
$944$	0	0
$945$	75.1941	2.44607
$946$	0	0
$947$	−33.4845	−1.08810	−0.544051	−	0.839052i	$-0.683110\pi$
$947$	−33.4845	−1.08810	−0.544051	+	0.839052i	$0.683110\pi$
$948$	0	0
$949$	2.52347	0.0819151
$950$	0	0
$951$	14.4122	0.467347
$952$	0	0
$953$	25.5905	0.828958	0.414479	−	0.910059i	$-0.363964\pi$
$953$	25.5905	0.828958	0.414479	+	0.910059i	$0.363964\pi$
$954$	0	0
$955$	−33.0158	−1.06837
$956$	0	0
$957$	5.61583	0.181534
$958$	0	0
$959$	−13.4898	−0.435609
$960$	0	0
$961$	−28.6418	−0.923928
$962$	0	0
$963$	121.440	3.91334
$964$	0	0
$965$	−44.1195	−1.42026
$966$	0	0
$967$	−20.6751	−0.664867	−0.332433	−	0.943127i	$-0.607870\pi$
$967$	−20.6751	−0.664867	−0.332433	+	0.943127i	$0.607870\pi$
$968$	0	0
$969$	−24.8885	−0.799533
$970$	0	0
$971$	−39.8023	−1.27732	−0.638659	−	0.769490i	$-0.720510\pi$
$971$	−39.8023	−1.27732	−0.638659	+	0.769490i	$0.720510\pi$
$972$	0	0
$973$	−9.29398	−0.297951
$974$	0	0
$975$	10.4006	0.333084
$976$	0	0
$977$	−31.8179	−1.01795	−0.508973	−	0.860783i	$-0.669975\pi$
$977$	−31.8179	−1.01795	−0.508973	+	0.860783i	$0.669975\pi$
$978$	0	0
$979$	0.165497	0.00528929
$980$	0	0
$981$	79.8608	2.54976
$982$	0	0
$983$	−25.1904	−0.803449	−0.401725	−	0.915761i	$-0.631589\pi$
$983$	−25.1904	−0.803449	−0.401725	+	0.915761i	$0.631589\pi$
$984$	0	0
$985$	54.4964	1.73640
$986$	0	0
$987$	14.0521	0.447282
$988$	0	0
$989$	16.2032	0.515232
$990$	0	0
$991$	27.9417	0.887598	0.443799	−	0.896126i	$-0.353630\pi$
$991$	27.9417	0.887598	0.443799	+	0.896126i	$0.353630\pi$
$992$	0	0
$993$	106.083	3.36643
$994$	0	0
$995$	67.6271	2.14392
$996$	0	0
$997$	−12.2395	−0.387630	−0.193815	−	0.981038i	$-0.562086\pi$
$997$	−12.2395	−0.387630	−0.193815	+	0.981038i	$0.562086\pi$
$998$	0	0
$999$	71.6528	2.26699

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9968.2.a.z.1.5		5
4.3	odd	2		1246.2.a.n.1.1	✓	5
28.27	even	2		8722.2.a.x.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1246.2.a.n.1.1	✓	5	4.3	odd	2
8722.2.a.x.1.5		5	28.27	even	2
9968.2.a.z.1.5		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9968 = 2^{4} \cdot 7 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9968.a (trivial)

\(f(q)\)	\(=\)	\(q+3.45619 q^{3} -3.65944 q^{5} -1.00000 q^{7} +8.94527 q^{9} +O(q^{10})\)	\(q+3.45619 q^{3} -3.65944 q^{5} -1.00000 q^{7} +8.94527 q^{9} -0.165497 q^{11} +0.358607 q^{13} -12.6477 q^{15} +5.94527 q^{17} -1.21124 q^{19} -3.45619 q^{21} +5.56984 q^{23} +8.39149 q^{25} +20.5480 q^{27} -9.81809 q^{29} +1.53565 q^{31} -0.571988 q^{33} +3.65944 q^{35} +3.48709 q^{37} +1.23942 q^{39} -9.70716 q^{41} +2.90910 q^{43} -32.7347 q^{45} -4.06577 q^{47} +1.00000 q^{49} +20.5480 q^{51} +1.46905 q^{53} +0.605625 q^{55} -4.18626 q^{57} +7.87266 q^{59} -0.0478841 q^{61} -8.94527 q^{63} -1.31230 q^{65} +2.00396 q^{67} +19.2505 q^{69} +4.79874 q^{71} +7.03685 q^{73} +29.0026 q^{75} +0.165497 q^{77} +3.53226 q^{79} +44.1820 q^{81} -15.1778 q^{83} -21.7563 q^{85} -33.9332 q^{87} -1.00000 q^{89} -0.358607 q^{91} +5.30751 q^{93} +4.43244 q^{95} -17.8521 q^{97} -1.48041 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 6 q^{3} - 10 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) \) 5 * q + 6 * q^3 - 10 * q^5 - 5 * q^7 + 7 * q^9	\( 5 q + 6 q^{3} - 10 q^{5} - 5 q^{7} + 7 q^{9} + 8 q^{11} - 8 q^{13} - 6 q^{15} - 8 q^{17} + 10 q^{19} - 6 q^{21} + 2 q^{23} + 17 q^{25} + 24 q^{27} - 10 q^{29} + 10 q^{31} + 10 q^{35} - 4 q^{37} - 24 q^{39} - 28 q^{41} + 10 q^{43} - 12 q^{45} + 10 q^{47} + 5 q^{49} + 24 q^{51} + 4 q^{53} - 4 q^{55} + 2 q^{57} + 10 q^{59} - 16 q^{61} - 7 q^{63} - 26 q^{65} - 2 q^{69} + 16 q^{71} + 10 q^{73} + 10 q^{75} - 8 q^{77} + 8 q^{79} + 41 q^{81} + 14 q^{83} + 18 q^{85} - 22 q^{87} - 5 q^{89} + 8 q^{91} + 22 q^{93} - 34 q^{95} + 6 q^{97} + 8 q^{99}+O(q^{100}) \) 5 * q + 6 * q^3 - 10 * q^5 - 5 * q^7 + 7 * q^9 + 8 * q^11 - 8 * q^13 - 6 * q^15 - 8 * q^17 + 10 * q^19 - 6 * q^21 + 2 * q^23 + 17 * q^25 + 24 * q^27 - 10 * q^29 + 10 * q^31 + 10 * q^35 - 4 * q^37 - 24 * q^39 - 28 * q^41 + 10 * q^43 - 12 * q^45 + 10 * q^47 + 5 * q^49 + 24 * q^51 + 4 * q^53 - 4 * q^55 + 2 * q^57 + 10 * q^59 - 16 * q^61 - 7 * q^63 - 26 * q^65 - 2 * q^69 + 16 * q^71 + 10 * q^73 + 10 * q^75 - 8 * q^77 + 8 * q^79 + 41 * q^81 + 14 * q^83 + 18 * q^85 - 22 * q^87 - 5 * q^89 + 8 * q^91 + 22 * q^93 - 34 * q^95 + 6 * q^97 + 8 * q^99

Introduction

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