LMFDB - Embedded newform 8008.2.a.t.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 17x^{8} + 13x^{7} + 96x^{6} - 49x^{5} - 207x^{4} + 58x^{3} + 169x^{2} - 12x - 32 $ x^10 - x^9 - 17x^8 + 13x^7 + 96x^6 - 49x^5 - 207x^4 + 58x^3 + 169x^2 - 12x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.514400$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.514400 q^{3} -2.27351 q^{5} +1.00000 q^{7} -2.73539 q^{9} +O(q^{10})$	$q+0.514400 q^{3} -2.27351 q^{5} +1.00000 q^{7} -2.73539 q^{9} -1.00000 q^{11} +1.00000 q^{13} -1.16950 q^{15} +5.22587 q^{17} -3.46490 q^{19} +0.514400 q^{21} +6.62511 q^{23} +0.168869 q^{25} -2.95028 q^{27} +1.19747 q^{29} +6.39771 q^{31} -0.514400 q^{33} -2.27351 q^{35} -3.23056 q^{37} +0.514400 q^{39} -9.16231 q^{41} -4.90728 q^{43} +6.21896 q^{45} -7.40956 q^{47} +1.00000 q^{49} +2.68819 q^{51} +4.70297 q^{53} +2.27351 q^{55} -1.78234 q^{57} +5.05287 q^{59} +2.00856 q^{61} -2.73539 q^{63} -2.27351 q^{65} +4.52843 q^{67} +3.40795 q^{69} -12.9191 q^{71} +2.01524 q^{73} +0.0868664 q^{75} -1.00000 q^{77} -7.96114 q^{79} +6.68855 q^{81} -3.59050 q^{83} -11.8811 q^{85} +0.615977 q^{87} -11.2018 q^{89} +1.00000 q^{91} +3.29098 q^{93} +7.87750 q^{95} -4.52497 q^{97} +2.73539 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9}+O(q^{10}) $ 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9	$ 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9} - 10 q^{11} + 10 q^{13} + 9 q^{15} + 7 q^{17} + 17 q^{19} + q^{21} + 15 q^{23} + 13 q^{25} + 7 q^{27} - q^{29} - 6 q^{31} - q^{33} + 3 q^{35} - 12 q^{37} + q^{39} + 4 q^{41} + 17 q^{43} - 15 q^{45} + 21 q^{47} + 10 q^{49} + 3 q^{51} + 20 q^{53} - 3 q^{55} + 12 q^{57} + 9 q^{59} + 6 q^{61} + 5 q^{63} + 3 q^{65} + 26 q^{67} + 30 q^{69} + 18 q^{71} - 4 q^{73} + 48 q^{75} - 10 q^{77} + 19 q^{79} - 14 q^{81} + 40 q^{83} - 25 q^{85} - 25 q^{87} - 19 q^{89} + 10 q^{91} + q^{93} + 11 q^{95} - 22 q^{97} - 5 q^{99}+O(q^{100}) $ 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9 - 10 * q^11 + 10 * q^13 + 9 * q^15 + 7 * q^17 + 17 * q^19 + q^21 + 15 * q^23 + 13 * q^25 + 7 * q^27 - q^29 - 6 * q^31 - q^33 + 3 * q^35 - 12 * q^37 + q^39 + 4 * q^41 + 17 * q^43 - 15 * q^45 + 21 * q^47 + 10 * q^49 + 3 * q^51 + 20 * q^53 - 3 * q^55 + 12 * q^57 + 9 * q^59 + 6 * q^61 + 5 * q^63 + 3 * q^65 + 26 * q^67 + 30 * q^69 + 18 * q^71 - 4 * q^73 + 48 * q^75 - 10 * q^77 + 19 * q^79 - 14 * q^81 + 40 * q^83 - 25 * q^85 - 25 * q^87 - 19 * q^89 + 10 * q^91 + q^93 + 11 * q^95 - 22 * q^97 - 5 * 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
$3$	0.514400	0.296989	0.148494	−	0.988913i	$-0.452557\pi$
$3$	0.514400	0.296989	0.148494	+	0.988913i	$0.452557\pi$
$4$	0	0
$5$	−2.27351	−1.01675	−0.508373	−	0.861137i	$-0.669753\pi$
$5$	−2.27351	−1.01675	−0.508373	+	0.861137i	$0.669753\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.73539	−0.911798
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.16950	−0.301962
$16$	0	0
$17$	5.22587	1.26746	0.633730	−	0.773554i	$-0.281523\pi$
$17$	5.22587	1.26746	0.633730	+	0.773554i	$0.281523\pi$
$18$	0	0
$19$	−3.46490	−0.794902	−0.397451	−	0.917623i	$-0.630105\pi$
$19$	−3.46490	−0.794902	−0.397451	+	0.917623i	$0.630105\pi$
$20$	0	0
$21$	0.514400	0.112251
$22$	0	0
$23$	6.62511	1.38143	0.690715	−	0.723127i	$-0.257296\pi$
$23$	6.62511	1.38143	0.690715	+	0.723127i	$0.257296\pi$
$24$	0	0
$25$	0.168869	0.0337739
$26$	0	0
$27$	−2.95028	−0.567783
$28$	0	0
$29$	1.19747	0.222364	0.111182	−	0.993800i	$-0.464536\pi$
$29$	1.19747	0.222364	0.111182	+	0.993800i	$0.464536\pi$
$30$	0	0
$31$	6.39771	1.14906	0.574531	−	0.818483i	$-0.305184\pi$
$31$	6.39771	1.14906	0.574531	+	0.818483i	$0.305184\pi$
$32$	0	0
$33$	−0.514400	−0.0895455
$34$	0	0
$35$	−2.27351	−0.384294
$36$	0	0
$37$	−3.23056	−0.531101	−0.265550	−	0.964097i	$-0.585554\pi$
$37$	−3.23056	−0.531101	−0.265550	+	0.964097i	$0.585554\pi$
$38$	0	0
$39$	0.514400	0.0823699
$40$	0	0
$41$	−9.16231	−1.43091	−0.715456	−	0.698657i	$-0.753781\pi$
$41$	−9.16231	−1.43091	−0.715456	+	0.698657i	$0.753781\pi$
$42$	0	0
$43$	−4.90728	−0.748354	−0.374177	−	0.927357i	$-0.622075\pi$
$43$	−4.90728	−0.748354	−0.374177	+	0.927357i	$0.622075\pi$
$44$	0	0
$45$	6.21896	0.927067
$46$	0	0
$47$	−7.40956	−1.08080	−0.540398	−	0.841410i	$-0.681726\pi$
$47$	−7.40956	−1.08080	−0.540398	+	0.841410i	$0.681726\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.68819	0.376422
$52$	0	0
$53$	4.70297	0.646002	0.323001	−	0.946399i	$-0.395308\pi$
$53$	4.70297	0.646002	0.323001	+	0.946399i	$0.395308\pi$
$54$	0	0
$55$	2.27351	0.306561
$56$	0	0
$57$	−1.78234	−0.236077
$58$	0	0
$59$	5.05287	0.657828	0.328914	−	0.944360i	$-0.393317\pi$
$59$	5.05287	0.657828	0.328914	+	0.944360i	$0.393317\pi$
$60$	0	0
$61$	2.00856	0.257169	0.128585	−	0.991699i	$-0.458957\pi$
$61$	2.00856	0.257169	0.128585	+	0.991699i	$0.458957\pi$
$62$	0	0
$63$	−2.73539	−0.344627
$64$	0	0
$65$	−2.27351	−0.281995
$66$	0	0
$67$	4.52843	0.553236	0.276618	−	0.960980i	$-0.410786\pi$
$67$	4.52843	0.553236	0.276618	+	0.960980i	$0.410786\pi$
$68$	0	0
$69$	3.40795	0.410269
$70$	0	0
$71$	−12.9191	−1.53322	−0.766610	−	0.642113i	$-0.778058\pi$
$71$	−12.9191	−1.53322	−0.766610	+	0.642113i	$0.778058\pi$
$72$	0	0
$73$	2.01524	0.235866	0.117933	−	0.993022i	$-0.462373\pi$
$73$	2.01524	0.235866	0.117933	+	0.993022i	$0.462373\pi$
$74$	0	0
$75$	0.0868664	0.0100305
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−7.96114	−0.895698	−0.447849	−	0.894109i	$-0.647810\pi$
$79$	−7.96114	−0.895698	−0.447849	+	0.894109i	$0.647810\pi$
$80$	0	0
$81$	6.68855	0.743173
$82$	0	0
$83$	−3.59050	−0.394108	−0.197054	−	0.980393i	$-0.563137\pi$
$83$	−3.59050	−0.394108	−0.197054	+	0.980393i	$0.563137\pi$
$84$	0	0
$85$	−11.8811	−1.28869
$86$	0	0
$87$	0.615977	0.0660397
$88$	0	0
$89$	−11.2018	−1.18739	−0.593693	−	0.804692i	$-0.702331\pi$
$89$	−11.2018	−1.18739	−0.593693	+	0.804692i	$0.702331\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	3.29098	0.341259
$94$	0	0
$95$	7.87750	0.808214
$96$	0	0
$97$	−4.52497	−0.459441	−0.229720	−	0.973257i	$-0.573781\pi$
$97$	−4.52497	−0.459441	−0.229720	+	0.973257i	$0.573781\pi$
$98$	0	0
$99$	2.73539	0.274917
$100$	0	0
$101$	6.07670	0.604655	0.302327	−	0.953204i	$-0.402236\pi$
$101$	6.07670	0.604655	0.302327	+	0.953204i	$0.402236\pi$
$102$	0	0
$103$	3.42886	0.337856	0.168928	−	0.985628i	$-0.445969\pi$
$103$	3.42886	0.337856	0.168928	+	0.985628i	$0.445969\pi$
$104$	0	0
$105$	−1.16950	−0.114131
$106$	0	0
$107$	13.3520	1.29078	0.645392	−	0.763852i	$-0.276694\pi$
$107$	13.3520	1.29078	0.645392	+	0.763852i	$0.276694\pi$
$108$	0	0
$109$	8.99068	0.861151	0.430575	−	0.902555i	$-0.358311\pi$
$109$	8.99068	0.861151	0.430575	+	0.902555i	$0.358311\pi$
$110$	0	0
$111$	−1.66180	−0.157731
$112$	0	0
$113$	17.1674	1.61498	0.807488	−	0.589884i	$-0.200827\pi$
$113$	17.1674	1.61498	0.807488	+	0.589884i	$0.200827\pi$
$114$	0	0
$115$	−15.0623	−1.40456
$116$	0	0
$117$	−2.73539	−0.252887
$118$	0	0
$119$	5.22587	0.479055
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.71309	−0.424965
$124$	0	0
$125$	10.9836	0.982407
$126$	0	0
$127$	−10.5129	−0.932869	−0.466434	−	0.884556i	$-0.654462\pi$
$127$	−10.5129	−0.932869	−0.466434	+	0.884556i	$0.654462\pi$
$128$	0	0
$129$	−2.52431	−0.222253
$130$	0	0
$131$	0.134174	0.0117229	0.00586144	−	0.999983i	$-0.498134\pi$
$131$	0.134174	0.0117229	0.00586144	+	0.999983i	$0.498134\pi$
$132$	0	0
$133$	−3.46490	−0.300445
$134$	0	0
$135$	6.70752	0.577291
$136$	0	0
$137$	−15.0170	−1.28299	−0.641496	−	0.767126i	$-0.721686\pi$
$137$	−15.0170	−1.28299	−0.641496	+	0.767126i	$0.721686\pi$
$138$	0	0
$139$	17.8791	1.51648	0.758242	−	0.651973i	$-0.226059\pi$
$139$	17.8791	1.51648	0.758242	+	0.651973i	$0.226059\pi$
$140$	0	0
$141$	−3.81148	−0.320984
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−2.72246	−0.226088
$146$	0	0
$147$	0.514400	0.0424270
$148$	0	0
$149$	19.2096	1.57371	0.786854	−	0.617139i	$-0.211708\pi$
$149$	19.2096	1.57371	0.786854	+	0.617139i	$0.211708\pi$
$150$	0	0
$151$	−9.45026	−0.769051	−0.384525	−	0.923114i	$-0.625635\pi$
$151$	−9.45026	−0.769051	−0.384525	+	0.923114i	$0.625635\pi$
$152$	0	0
$153$	−14.2948	−1.15567
$154$	0	0
$155$	−14.5453	−1.16830
$156$	0	0
$157$	17.0577	1.36135	0.680675	−	0.732585i	$-0.261687\pi$
$157$	17.0577	1.36135	0.680675	+	0.732585i	$0.261687\pi$
$158$	0	0
$159$	2.41920	0.191855
$160$	0	0
$161$	6.62511	0.522132
$162$	0	0
$163$	15.3651	1.20349	0.601744	−	0.798689i	$-0.294473\pi$
$163$	15.3651	1.20349	0.601744	+	0.798689i	$0.294473\pi$
$164$	0	0
$165$	1.16950	0.0910451
$166$	0	0
$167$	11.7558	0.909692	0.454846	−	0.890570i	$-0.349694\pi$
$167$	11.7558	0.909692	0.454846	+	0.890570i	$0.349694\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	9.47786	0.724790
$172$	0	0
$173$	−1.32173	−0.100489	−0.0502447	−	0.998737i	$-0.516000\pi$
$173$	−1.32173	−0.100489	−0.0502447	+	0.998737i	$0.516000\pi$
$174$	0	0
$175$	0.168869	0.0127653
$176$	0	0
$177$	2.59920	0.195368
$178$	0	0
$179$	−0.734423	−0.0548933	−0.0274467	−	0.999623i	$-0.508738\pi$
$179$	−0.734423	−0.0548933	−0.0274467	+	0.999623i	$0.508738\pi$
$180$	0	0
$181$	9.05987	0.673415	0.336707	−	0.941609i	$-0.390687\pi$
$181$	9.05987	0.673415	0.336707	+	0.941609i	$0.390687\pi$
$182$	0	0
$183$	1.03320	0.0763765
$184$	0	0
$185$	7.34473	0.539995
$186$	0	0
$187$	−5.22587	−0.382154
$188$	0	0
$189$	−2.95028	−0.214602
$190$	0	0
$191$	9.73155	0.704150	0.352075	−	0.935972i	$-0.385476\pi$
$191$	9.73155	0.704150	0.352075	+	0.935972i	$0.385476\pi$
$192$	0	0
$193$	4.69721	0.338112	0.169056	−	0.985606i	$-0.445928\pi$
$193$	4.69721	0.338112	0.169056	+	0.985606i	$0.445928\pi$
$194$	0	0
$195$	−1.16950	−0.0837493
$196$	0	0
$197$	21.1935	1.50997	0.754986	−	0.655741i	$-0.227644\pi$
$197$	21.1935	1.50997	0.754986	+	0.655741i	$0.227644\pi$
$198$	0	0
$199$	1.91009	0.135403	0.0677013	−	0.997706i	$-0.478434\pi$
$199$	1.91009	0.135403	0.0677013	+	0.997706i	$0.478434\pi$
$200$	0	0
$201$	2.32942	0.164305
$202$	0	0
$203$	1.19747	0.0840458
$204$	0	0
$205$	20.8307	1.45488
$206$	0	0
$207$	−18.1223	−1.25959
$208$	0	0
$209$	3.46490	0.239672
$210$	0	0
$211$	4.66930	0.321448	0.160724	−	0.986999i	$-0.448617\pi$
$211$	4.66930	0.321448	0.160724	+	0.986999i	$0.448617\pi$
$212$	0	0
$213$	−6.64560	−0.455349
$214$	0	0
$215$	11.1568	0.760886
$216$	0	0
$217$	6.39771	0.434305
$218$	0	0
$219$	1.03664	0.0700494
$220$	0	0
$221$	5.22587	0.351530
$222$	0	0
$223$	11.7238	0.785087	0.392544	−	0.919733i	$-0.371595\pi$
$223$	11.7238	0.785087	0.392544	+	0.919733i	$0.371595\pi$
$224$	0	0
$225$	−0.461924	−0.0307949
$226$	0	0
$227$	5.07288	0.336699	0.168350	−	0.985727i	$-0.446156\pi$
$227$	5.07288	0.336699	0.168350	+	0.985727i	$0.446156\pi$
$228$	0	0
$229$	28.5584	1.88719	0.943596	−	0.331098i	$-0.107419\pi$
$229$	28.5584	1.88719	0.943596	+	0.331098i	$0.107419\pi$
$230$	0	0
$231$	−0.514400	−0.0338450
$232$	0	0
$233$	3.72582	0.244086	0.122043	−	0.992525i	$-0.461055\pi$
$233$	3.72582	0.244086	0.122043	+	0.992525i	$0.461055\pi$
$234$	0	0
$235$	16.8457	1.09890
$236$	0	0
$237$	−4.09521	−0.266012
$238$	0	0
$239$	10.1919	0.659256	0.329628	−	0.944111i	$-0.393077\pi$
$239$	10.1919	0.659256	0.329628	+	0.944111i	$0.393077\pi$
$240$	0	0
$241$	2.14869	0.138409	0.0692045	−	0.997602i	$-0.477954\pi$
$241$	2.14869	0.138409	0.0692045	+	0.997602i	$0.477954\pi$
$242$	0	0
$243$	12.2914	0.788496
$244$	0	0
$245$	−2.27351	−0.145250
$246$	0	0
$247$	−3.46490	−0.220466
$248$	0	0
$249$	−1.84695	−0.117046
$250$	0	0
$251$	−2.13601	−0.134824	−0.0674120	−	0.997725i	$-0.521474\pi$
$251$	−2.13601	−0.134824	−0.0674120	+	0.997725i	$0.521474\pi$
$252$	0	0
$253$	−6.62511	−0.416517
$254$	0	0
$255$	−6.11163	−0.382725
$256$	0	0
$257$	1.36190	0.0849528	0.0424764	−	0.999097i	$-0.486475\pi$
$257$	1.36190	0.0849528	0.0424764	+	0.999097i	$0.486475\pi$
$258$	0	0
$259$	−3.23056	−0.200737
$260$	0	0
$261$	−3.27555	−0.202751
$262$	0	0
$263$	−5.90764	−0.364281	−0.182140	−	0.983273i	$-0.558303\pi$
$263$	−5.90764	−0.364281	−0.182140	+	0.983273i	$0.558303\pi$
$264$	0	0
$265$	−10.6923	−0.656820
$266$	0	0
$267$	−5.76219	−0.352640
$268$	0	0
$269$	4.43032	0.270122	0.135061	−	0.990837i	$-0.456877\pi$
$269$	4.43032	0.270122	0.135061	+	0.990837i	$0.456877\pi$
$270$	0	0
$271$	5.74175	0.348786	0.174393	−	0.984676i	$-0.444204\pi$
$271$	5.74175	0.348786	0.174393	+	0.984676i	$0.444204\pi$
$272$	0	0
$273$	0.514400	0.0311329
$274$	0	0
$275$	−0.168869	−0.0101832
$276$	0	0
$277$	−1.27349	−0.0765166	−0.0382583	−	0.999268i	$-0.512181\pi$
$277$	−1.27349	−0.0765166	−0.0382583	+	0.999268i	$0.512181\pi$
$278$	0	0
$279$	−17.5002	−1.04771
$280$	0	0
$281$	−3.38208	−0.201758	−0.100879	−	0.994899i	$-0.532165\pi$
$281$	−3.38208	−0.201758	−0.100879	+	0.994899i	$0.532165\pi$
$282$	0	0
$283$	2.27265	0.135095	0.0675476	−	0.997716i	$-0.478483\pi$
$283$	2.27265	0.135095	0.0675476	+	0.997716i	$0.478483\pi$
$284$	0	0
$285$	4.05218	0.240031
$286$	0	0
$287$	−9.16231	−0.540834
$288$	0	0
$289$	10.3097	0.606456
$290$	0	0
$291$	−2.32764	−0.136449
$292$	0	0
$293$	−0.664307	−0.0388092	−0.0194046	−	0.999812i	$-0.506177\pi$
$293$	−0.664307	−0.0388092	−0.0194046	+	0.999812i	$0.506177\pi$
$294$	0	0
$295$	−11.4878	−0.668844
$296$	0	0
$297$	2.95028	0.171193
$298$	0	0
$299$	6.62511	0.383140
$300$	0	0
$301$	−4.90728	−0.282851
$302$	0	0
$303$	3.12585	0.179576
$304$	0	0
$305$	−4.56649	−0.261476
$306$	0	0
$307$	1.68321	0.0960660	0.0480330	−	0.998846i	$-0.484705\pi$
$307$	1.68321	0.0960660	0.0480330	+	0.998846i	$0.484705\pi$
$308$	0	0
$309$	1.76381	0.100339
$310$	0	0
$311$	−20.1249	−1.14118	−0.570589	−	0.821236i	$-0.693285\pi$
$311$	−20.1249	−1.14118	−0.570589	+	0.821236i	$0.693285\pi$
$312$	0	0
$313$	−8.43043	−0.476516	−0.238258	−	0.971202i	$-0.576576\pi$
$313$	−8.43043	−0.476516	−0.238258	+	0.971202i	$0.576576\pi$
$314$	0	0
$315$	6.21896	0.350398
$316$	0	0
$317$	−20.1541	−1.13197	−0.565984	−	0.824416i	$-0.691504\pi$
$317$	−20.1541	−1.13197	−0.565984	+	0.824416i	$0.691504\pi$
$318$	0	0
$319$	−1.19747	−0.0670454
$320$	0	0
$321$	6.86825	0.383348
$322$	0	0
$323$	−18.1071	−1.00751
$324$	0	0
$325$	0.168869	0.00936719
$326$	0	0
$327$	4.62480	0.255752
$328$	0	0
$329$	−7.40956	−0.408502
$330$	0	0
$331$	8.76069	0.481531	0.240765	−	0.970583i	$-0.422602\pi$
$331$	8.76069	0.481531	0.240765	+	0.970583i	$0.422602\pi$
$332$	0	0
$333$	8.83685	0.484256
$334$	0	0
$335$	−10.2955	−0.562501
$336$	0	0
$337$	12.7033	0.691995	0.345997	−	0.938236i	$-0.387541\pi$
$337$	12.7033	0.691995	0.345997	+	0.938236i	$0.387541\pi$
$338$	0	0
$339$	8.83092	0.479630
$340$	0	0
$341$	−6.39771	−0.346455
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−7.74803	−0.417140
$346$	0	0
$347$	34.3620	1.84465	0.922323	−	0.386419i	$-0.126288\pi$
$347$	34.3620	1.84465	0.922323	+	0.386419i	$0.126288\pi$
$348$	0	0
$349$	−2.03569	−0.108968	−0.0544839	−	0.998515i	$-0.517351\pi$
$349$	−2.03569	−0.108968	−0.0544839	+	0.998515i	$0.517351\pi$
$350$	0	0
$351$	−2.95028	−0.157475
$352$	0	0
$353$	19.5624	1.04120	0.520602	−	0.853800i	$-0.325708\pi$
$353$	19.5624	1.04120	0.520602	+	0.853800i	$0.325708\pi$
$354$	0	0
$355$	29.3719	1.55890
$356$	0	0
$357$	2.68819	0.142274
$358$	0	0
$359$	20.7999	1.09778	0.548888	−	0.835896i	$-0.315051\pi$
$359$	20.7999	1.09778	0.548888	+	0.835896i	$0.315051\pi$
$360$	0	0
$361$	−6.99447	−0.368130
$362$	0	0
$363$	0.514400	0.0269990
$364$	0	0
$365$	−4.58167	−0.239816
$366$	0	0
$367$	−19.4336	−1.01442	−0.507212	−	0.861821i	$-0.669324\pi$
$367$	−19.4336	−1.01442	−0.507212	+	0.861821i	$0.669324\pi$
$368$	0	0
$369$	25.0625	1.30470
$370$	0	0
$371$	4.70297	0.244166
$372$	0	0
$373$	−31.4033	−1.62600	−0.813002	−	0.582261i	$-0.802168\pi$
$373$	−31.4033	−1.62600	−0.813002	+	0.582261i	$0.802168\pi$
$374$	0	0
$375$	5.64998	0.291764
$376$	0	0
$377$	1.19747	0.0616728
$378$	0	0
$379$	25.2474	1.29687	0.648435	−	0.761270i	$-0.275424\pi$
$379$	25.2474	1.29687	0.648435	+	0.761270i	$0.275424\pi$
$380$	0	0
$381$	−5.40783	−0.277052
$382$	0	0
$383$	11.3953	0.582274	0.291137	−	0.956681i	$-0.405966\pi$
$383$	11.3953	0.582274	0.291137	+	0.956681i	$0.405966\pi$
$384$	0	0
$385$	2.27351	0.115869
$386$	0	0
$387$	13.4233	0.682347
$388$	0	0
$389$	0.303702	0.0153983	0.00769916	−	0.999970i	$-0.497549\pi$
$389$	0.303702	0.0153983	0.00769916	+	0.999970i	$0.497549\pi$
$390$	0	0
$391$	34.6220	1.75091
$392$	0	0
$393$	0.0690193	0.00348156
$394$	0	0
$395$	18.0998	0.910698
$396$	0	0
$397$	−0.469177	−0.0235473	−0.0117737	−	0.999931i	$-0.503748\pi$
$397$	−0.469177	−0.0235473	−0.0117737	+	0.999931i	$0.503748\pi$
$398$	0	0
$399$	−1.78234	−0.0892288
$400$	0	0
$401$	12.8204	0.640219	0.320109	−	0.947381i	$-0.396280\pi$
$401$	12.8204	0.640219	0.320109	+	0.947381i	$0.396280\pi$
$402$	0	0
$403$	6.39771	0.318692
$404$	0	0
$405$	−15.2065	−0.755618
$406$	0	0
$407$	3.23056	0.160133
$408$	0	0
$409$	−10.2861	−0.508615	−0.254308	−	0.967123i	$-0.581848\pi$
$409$	−10.2861	−0.508615	−0.254308	+	0.967123i	$0.581848\pi$
$410$	0	0
$411$	−7.72476	−0.381034
$412$	0	0
$413$	5.05287	0.248636
$414$	0	0
$415$	8.16305	0.400708
$416$	0	0
$417$	9.19699	0.450379
$418$	0	0
$419$	−27.1545	−1.32658	−0.663292	−	0.748361i	$-0.730841\pi$
$419$	−27.1545	−1.32658	−0.663292	+	0.748361i	$0.730841\pi$
$420$	0	0
$421$	−31.1889	−1.52005	−0.760027	−	0.649892i	$-0.774814\pi$
$421$	−31.1889	−1.52005	−0.760027	+	0.649892i	$0.774814\pi$
$422$	0	0
$423$	20.2681	0.985467
$424$	0	0
$425$	0.882490	0.0428071
$426$	0	0
$427$	2.00856	0.0972009
$428$	0	0
$429$	−0.514400	−0.0248355
$430$	0	0
$431$	16.4184	0.790845	0.395422	−	0.918499i	$-0.370598\pi$
$431$	16.4184	0.790845	0.395422	+	0.918499i	$0.370598\pi$
$432$	0	0
$433$	−34.7064	−1.66788	−0.833941	−	0.551853i	$-0.813921\pi$
$433$	−34.7064	−1.66788	−0.833941	+	0.551853i	$0.813921\pi$
$434$	0	0
$435$	−1.40043	−0.0671457
$436$	0	0
$437$	−22.9553	−1.09810
$438$	0	0
$439$	22.7569	1.08613	0.543064	−	0.839691i	$-0.317264\pi$
$439$	22.7569	1.08613	0.543064	+	0.839691i	$0.317264\pi$
$440$	0	0
$441$	−2.73539	−0.130257
$442$	0	0
$443$	8.35355	0.396889	0.198445	−	0.980112i	$-0.436411\pi$
$443$	8.35355	0.396889	0.198445	+	0.980112i	$0.436411\pi$
$444$	0	0
$445$	25.4674	1.20727
$446$	0	0
$447$	9.88139	0.467374
$448$	0	0
$449$	22.2566	1.05036	0.525178	−	0.850993i	$-0.323999\pi$
$449$	22.2566	1.05036	0.525178	+	0.850993i	$0.323999\pi$
$450$	0	0
$451$	9.16231	0.431436
$452$	0	0
$453$	−4.86121	−0.228400
$454$	0	0
$455$	−2.27351	−0.106584
$456$	0	0
$457$	−19.8474	−0.928422	−0.464211	−	0.885725i	$-0.653662\pi$
$457$	−19.8474	−0.928422	−0.464211	+	0.885725i	$0.653662\pi$
$458$	0	0
$459$	−15.4178	−0.719642
$460$	0	0
$461$	0.657405	0.0306184	0.0153092	−	0.999883i	$-0.495127\pi$
$461$	0.657405	0.0306184	0.0153092	+	0.999883i	$0.495127\pi$
$462$	0	0
$463$	−30.7146	−1.42743	−0.713715	−	0.700436i	$-0.752989\pi$
$463$	−30.7146	−1.42743	−0.713715	+	0.700436i	$0.752989\pi$
$464$	0	0
$465$	−7.48209	−0.346974
$466$	0	0
$467$	15.0371	0.695833	0.347916	−	0.937526i	$-0.386889\pi$
$467$	15.0371	0.695833	0.347916	+	0.937526i	$0.386889\pi$
$468$	0	0
$469$	4.52843	0.209103
$470$	0	0
$471$	8.77446	0.404306
$472$	0	0
$473$	4.90728	0.225637
$474$	0	0
$475$	−0.585116	−0.0268469
$476$	0	0
$477$	−12.8645	−0.589023
$478$	0	0
$479$	11.0002	0.502611	0.251306	−	0.967908i	$-0.419140\pi$
$479$	11.0002	0.502611	0.251306	+	0.967908i	$0.419140\pi$
$480$	0	0
$481$	−3.23056	−0.147301
$482$	0	0
$483$	3.40795	0.155067
$484$	0	0
$485$	10.2876	0.467135
$486$	0	0
$487$	−30.3228	−1.37406	−0.687028	−	0.726631i	$-0.741085\pi$
$487$	−30.3228	−1.37406	−0.687028	+	0.726631i	$0.741085\pi$
$488$	0	0
$489$	7.90381	0.357423
$490$	0	0
$491$	37.9198	1.71130	0.855649	−	0.517556i	$-0.173158\pi$
$491$	37.9198	1.71130	0.855649	+	0.517556i	$0.173158\pi$
$492$	0	0
$493$	6.25782	0.281838
$494$	0	0
$495$	−6.21896	−0.279521
$496$	0	0
$497$	−12.9191	−0.579503
$498$	0	0
$499$	28.0365	1.25509	0.627543	−	0.778582i	$-0.284061\pi$
$499$	28.0365	1.25509	0.627543	+	0.778582i	$0.284061\pi$
$500$	0	0
$501$	6.04718	0.270168
$502$	0	0
$503$	−19.7739	−0.881673	−0.440837	−	0.897587i	$-0.645318\pi$
$503$	−19.7739	−0.881673	−0.440837	+	0.897587i	$0.645318\pi$
$504$	0	0
$505$	−13.8155	−0.614781
$506$	0	0
$507$	0.514400	0.0228453
$508$	0	0
$509$	−34.9819	−1.55055	−0.775273	−	0.631626i	$-0.782388\pi$
$509$	−34.9819	−1.55055	−0.775273	+	0.631626i	$0.782388\pi$
$510$	0	0
$511$	2.01524	0.0891488
$512$	0	0
$513$	10.2224	0.451332
$514$	0	0
$515$	−7.79557	−0.343514
$516$	0	0
$517$	7.40956	0.325872
$518$	0	0
$519$	−0.679898	−0.0298442
$520$	0	0
$521$	24.7254	1.08324	0.541621	−	0.840623i	$-0.317811\pi$
$521$	24.7254	1.08324	0.541621	+	0.840623i	$0.317811\pi$
$522$	0	0
$523$	20.9727	0.917073	0.458537	−	0.888675i	$-0.348374\pi$
$523$	20.9727	0.917073	0.458537	+	0.888675i	$0.348374\pi$
$524$	0	0
$525$	0.0868664	0.00379116
$526$	0	0
$527$	33.4336	1.45639
$528$	0	0
$529$	20.8921	0.908350
$530$	0	0
$531$	−13.8216	−0.599806
$532$	0	0
$533$	−9.16231	−0.396864
$534$	0	0
$535$	−30.3559	−1.31240
$536$	0	0
$537$	−0.377787	−0.0163027
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−33.4341	−1.43745	−0.718723	−	0.695296i	$-0.755273\pi$
$541$	−33.4341	−1.43745	−0.718723	+	0.695296i	$0.755273\pi$
$542$	0	0
$543$	4.66039	0.199997
$544$	0	0
$545$	−20.4404	−0.875572
$546$	0	0
$547$	15.6497	0.669134	0.334567	−	0.942372i	$-0.391410\pi$
$547$	15.6497	0.669134	0.334567	+	0.942372i	$0.391410\pi$
$548$	0	0
$549$	−5.49419	−0.234487
$550$	0	0
$551$	−4.14911	−0.176758
$552$	0	0
$553$	−7.96114	−0.338542
$554$	0	0
$555$	3.77813	0.160372
$556$	0	0
$557$	−21.1565	−0.896429	−0.448215	−	0.893926i	$-0.647940\pi$
$557$	−21.1565	−0.896429	−0.448215	+	0.893926i	$0.647940\pi$
$558$	0	0
$559$	−4.90728	−0.207556
$560$	0	0
$561$	−2.68819	−0.113495
$562$	0	0
$563$	30.6597	1.29215	0.646076	−	0.763273i	$-0.276409\pi$
$563$	30.6597	1.29215	0.646076	+	0.763273i	$0.276409\pi$
$564$	0	0
$565$	−39.0304	−1.64202
$566$	0	0
$567$	6.68855	0.280893
$568$	0	0
$569$	22.9637	0.962690	0.481345	−	0.876531i	$-0.340148\pi$
$569$	22.9637	0.962690	0.481345	+	0.876531i	$0.340148\pi$
$570$	0	0
$571$	−38.6555	−1.61768	−0.808842	−	0.588027i	$-0.799905\pi$
$571$	−38.6555	−1.61768	−0.808842	+	0.588027i	$0.799905\pi$
$572$	0	0
$573$	5.00591	0.209125
$574$	0	0
$575$	1.11878	0.0466563
$576$	0	0
$577$	−11.1502	−0.464189	−0.232094	−	0.972693i	$-0.574558\pi$
$577$	−11.1502	−0.464189	−0.232094	+	0.972693i	$0.574558\pi$
$578$	0	0
$579$	2.41624	0.100416
$580$	0	0
$581$	−3.59050	−0.148959
$582$	0	0
$583$	−4.70297	−0.194777
$584$	0	0
$585$	6.21896	0.257122
$586$	0	0
$587$	37.3249	1.54057	0.770283	−	0.637703i	$-0.220115\pi$
$587$	37.3249	1.54057	0.770283	+	0.637703i	$0.220115\pi$
$588$	0	0
$589$	−22.1674	−0.913392
$590$	0	0
$591$	10.9019	0.448445
$592$	0	0
$593$	9.87605	0.405561	0.202780	−	0.979224i	$-0.435002\pi$
$593$	9.87605	0.405561	0.202780	+	0.979224i	$0.435002\pi$
$594$	0	0
$595$	−11.8811	−0.487078
$596$	0	0
$597$	0.982549	0.0402131
$598$	0	0
$599$	37.7728	1.54335	0.771677	−	0.636014i	$-0.219418\pi$
$599$	37.7728	1.54335	0.771677	+	0.636014i	$0.219418\pi$
$600$	0	0
$601$	31.0180	1.26525	0.632625	−	0.774458i	$-0.281978\pi$
$601$	31.0180	1.26525	0.632625	+	0.774458i	$0.281978\pi$
$602$	0	0
$603$	−12.3870	−0.504439
$604$	0	0
$605$	−2.27351	−0.0924315
$606$	0	0
$607$	36.7748	1.49264	0.746322	−	0.665585i	$-0.231818\pi$
$607$	36.7748	1.49264	0.746322	+	0.665585i	$0.231818\pi$
$608$	0	0
$609$	0.615977	0.0249607
$610$	0	0
$611$	−7.40956	−0.299759
$612$	0	0
$613$	−32.3145	−1.30517	−0.652585	−	0.757716i	$-0.726315\pi$
$613$	−32.3145	−1.30517	−0.652585	+	0.757716i	$0.726315\pi$
$614$	0	0
$615$	10.7153	0.432082
$616$	0	0
$617$	−0.580699	−0.0233781	−0.0116890	−	0.999932i	$-0.503721\pi$
$617$	−0.580699	−0.0233781	−0.0116890	+	0.999932i	$0.503721\pi$
$618$	0	0
$619$	−14.8254	−0.595882	−0.297941	−	0.954584i	$-0.596300\pi$
$619$	−14.8254	−0.595882	−0.297941	+	0.954584i	$0.596300\pi$
$620$	0	0
$621$	−19.5460	−0.784352
$622$	0	0
$623$	−11.2018	−0.448789
$624$	0	0
$625$	−25.8158	−1.03263
$626$	0	0
$627$	1.78234	0.0711799
$628$	0	0
$629$	−16.8825	−0.673149
$630$	0	0
$631$	8.04466	0.320253	0.160126	−	0.987097i	$-0.448810\pi$
$631$	8.04466	0.320253	0.160126	+	0.987097i	$0.448810\pi$
$632$	0	0
$633$	2.40189	0.0954664
$634$	0	0
$635$	23.9012	0.948491
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	35.3389	1.39799
$640$	0	0
$641$	34.3729	1.35765	0.678824	−	0.734301i	$-0.262490\pi$
$641$	34.3729	1.35765	0.678824	+	0.734301i	$0.262490\pi$
$642$	0	0
$643$	−18.4974	−0.729467	−0.364734	−	0.931112i	$-0.618840\pi$
$643$	−18.4974	−0.729467	−0.364734	+	0.931112i	$0.618840\pi$
$644$	0	0
$645$	5.73905	0.225975
$646$	0	0
$647$	−3.08731	−0.121375	−0.0606874	−	0.998157i	$-0.519329\pi$
$647$	−3.08731	−0.121375	−0.0606874	+	0.998157i	$0.519329\pi$
$648$	0	0
$649$	−5.05287	−0.198343
$650$	0	0
$651$	3.29098	0.128984
$652$	0	0
$653$	−18.7682	−0.734458	−0.367229	−	0.930131i	$-0.619694\pi$
$653$	−18.7682	−0.734458	−0.367229	+	0.930131i	$0.619694\pi$
$654$	0	0
$655$	−0.305048	−0.0119192
$656$	0	0
$657$	−5.51246	−0.215062
$658$	0	0
$659$	−1.59326	−0.0620648	−0.0310324	−	0.999518i	$-0.509880\pi$
$659$	−1.59326	−0.0620648	−0.0310324	+	0.999518i	$0.509880\pi$
$660$	0	0
$661$	−40.7370	−1.58449	−0.792243	−	0.610206i	$-0.791087\pi$
$661$	−40.7370	−1.58449	−0.792243	+	0.610206i	$0.791087\pi$
$662$	0	0
$663$	2.68819	0.104401
$664$	0	0
$665$	7.87750	0.305476
$666$	0	0
$667$	7.93336	0.307181
$668$	0	0
$669$	6.03074	0.233162
$670$	0	0
$671$	−2.00856	−0.0775395
$672$	0	0
$673$	−39.8431	−1.53584	−0.767919	−	0.640547i	$-0.778708\pi$
$673$	−39.8431	−1.53584	−0.767919	+	0.640547i	$0.778708\pi$
$674$	0	0
$675$	−0.498213	−0.0191762
$676$	0	0
$677$	25.7827	0.990910	0.495455	−	0.868634i	$-0.335001\pi$
$677$	25.7827	0.990910	0.495455	+	0.868634i	$0.335001\pi$
$678$	0	0
$679$	−4.52497	−0.173652
$680$	0	0
$681$	2.60949	0.0999959
$682$	0	0
$683$	−3.12203	−0.119461	−0.0597305	−	0.998215i	$-0.519024\pi$
$683$	−3.12203	−0.119461	−0.0597305	+	0.998215i	$0.519024\pi$
$684$	0	0
$685$	34.1415	1.30448
$686$	0	0
$687$	14.6904	0.560475
$688$	0	0
$689$	4.70297	0.179169
$690$	0	0
$691$	−20.2463	−0.770207	−0.385103	−	0.922873i	$-0.625834\pi$
$691$	−20.2463	−0.770207	−0.385103	+	0.922873i	$0.625834\pi$
$692$	0	0
$693$	2.73539	0.103909
$694$	0	0
$695$	−40.6484	−1.54188
$696$	0	0
$697$	−47.8811	−1.81362
$698$	0	0
$699$	1.91656	0.0724909
$700$	0	0
$701$	−28.5877	−1.07974	−0.539871	−	0.841748i	$-0.681527\pi$
$701$	−28.5877	−1.07974	−0.539871	+	0.841748i	$0.681527\pi$
$702$	0	0
$703$	11.1936	0.422173
$704$	0	0
$705$	8.66545	0.326360
$706$	0	0
$707$	6.07670	0.228538
$708$	0	0
$709$	11.5746	0.434694	0.217347	−	0.976094i	$-0.430260\pi$
$709$	11.5746	0.434694	0.217347	+	0.976094i	$0.430260\pi$
$710$	0	0
$711$	21.7768	0.816696
$712$	0	0
$713$	42.3855	1.58735
$714$	0	0
$715$	2.27351	0.0850246
$716$	0	0
$717$	5.24269	0.195792
$718$	0	0
$719$	37.8987	1.41338	0.706691	−	0.707522i	$-0.250187\pi$
$719$	37.8987	1.41338	0.706691	+	0.707522i	$0.250187\pi$
$720$	0	0
$721$	3.42886	0.127697
$722$	0	0
$723$	1.10528	0.0411059
$724$	0	0
$725$	0.202216	0.00751011
$726$	0	0
$727$	−9.05035	−0.335659	−0.167829	−	0.985816i	$-0.553676\pi$
$727$	−9.05035	−0.335659	−0.167829	+	0.985816i	$0.553676\pi$
$728$	0	0
$729$	−13.7429	−0.508998
$730$	0	0
$731$	−25.6448	−0.948509
$732$	0	0
$733$	−3.52650	−0.130254	−0.0651271	−	0.997877i	$-0.520745\pi$
$733$	−3.52650	−0.130254	−0.0651271	+	0.997877i	$0.520745\pi$
$734$	0	0
$735$	−1.16950	−0.0431375
$736$	0	0
$737$	−4.52843	−0.166807
$738$	0	0
$739$	−4.04998	−0.148981	−0.0744905	−	0.997222i	$-0.523733\pi$
$739$	−4.04998	−0.148981	−0.0744905	+	0.997222i	$0.523733\pi$
$740$	0	0
$741$	−1.78234	−0.0654760
$742$	0	0
$743$	16.0518	0.588885	0.294443	−	0.955669i	$-0.404866\pi$
$743$	16.0518	0.588885	0.294443	+	0.955669i	$0.404866\pi$
$744$	0	0
$745$	−43.6732	−1.60006
$746$	0	0
$747$	9.82142	0.359347
$748$	0	0
$749$	13.3520	0.487870
$750$	0	0
$751$	−19.5746	−0.714286	−0.357143	−	0.934050i	$-0.616249\pi$
$751$	−19.5746	−0.714286	−0.357143	+	0.934050i	$0.616249\pi$
$752$	0	0
$753$	−1.09876	−0.0400412
$754$	0	0
$755$	21.4853	0.781930
$756$	0	0
$757$	−14.1224	−0.513288	−0.256644	−	0.966506i	$-0.582617\pi$
$757$	−14.1224	−0.513288	−0.256644	+	0.966506i	$0.582617\pi$
$758$	0	0
$759$	−3.40795	−0.123701
$760$	0	0
$761$	33.4786	1.21360	0.606799	−	0.794855i	$-0.292453\pi$
$761$	33.4786	1.21360	0.606799	+	0.794855i	$0.292453\pi$
$762$	0	0
$763$	8.99068	0.325484
$764$	0	0
$765$	32.4995	1.17502
$766$	0	0
$767$	5.05287	0.182449
$768$	0	0
$769$	13.7035	0.494159	0.247080	−	0.968995i	$-0.420529\pi$
$769$	13.7035	0.494159	0.247080	+	0.968995i	$0.420529\pi$
$770$	0	0
$771$	0.700559	0.0252300
$772$	0	0
$773$	33.3104	1.19809	0.599045	−	0.800715i	$-0.295547\pi$
$773$	33.3104	1.19809	0.599045	+	0.800715i	$0.295547\pi$
$774$	0	0
$775$	1.08038	0.0388083
$776$	0	0
$777$	−1.66180	−0.0596167
$778$	0	0
$779$	31.7465	1.13744
$780$	0	0
$781$	12.9191	0.462283
$782$	0	0
$783$	−3.53287	−0.126255
$784$	0	0
$785$	−38.7809	−1.38415
$786$	0	0
$787$	6.89306	0.245711	0.122856	−	0.992425i	$-0.460795\pi$
$787$	6.89306	0.245711	0.122856	+	0.992425i	$0.460795\pi$
$788$	0	0
$789$	−3.03889	−0.108187
$790$	0	0
$791$	17.1674	0.610403
$792$	0	0
$793$	2.00856	0.0713260
$794$	0	0
$795$	−5.50010	−0.195068
$796$	0	0
$797$	16.5987	0.587956	0.293978	−	0.955812i	$-0.405021\pi$
$797$	16.5987	0.587956	0.293978	+	0.955812i	$0.405021\pi$
$798$	0	0
$799$	−38.7214	−1.36987
$800$	0	0
$801$	30.6412	1.08266
$802$	0	0
$803$	−2.01524	−0.0711162
$804$	0	0
$805$	−15.0623	−0.530876
$806$	0	0
$807$	2.27896	0.0802231
$808$	0	0
$809$	−36.0087	−1.26600	−0.633000	−	0.774152i	$-0.718177\pi$
$809$	−36.0087	−1.26600	−0.633000	+	0.774152i	$0.718177\pi$
$810$	0	0
$811$	29.6830	1.04231	0.521155	−	0.853462i	$-0.325501\pi$
$811$	29.6830	1.04231	0.521155	+	0.853462i	$0.325501\pi$
$812$	0	0
$813$	2.95355	0.103586
$814$	0	0
$815$	−34.9328	−1.22364
$816$	0	0
$817$	17.0032	0.594868
$818$	0	0
$819$	−2.73539	−0.0955824
$820$	0	0
$821$	41.0342	1.43210	0.716052	−	0.698047i	$-0.245947\pi$
$821$	41.0342	1.43210	0.716052	+	0.698047i	$0.245947\pi$
$822$	0	0
$823$	−0.491778	−0.0171423	−0.00857116	−	0.999963i	$-0.502728\pi$
$823$	−0.491778	−0.0171423	−0.00857116	+	0.999963i	$0.502728\pi$
$824$	0	0
$825$	−0.0868664	−0.00302430
$826$	0	0
$827$	−16.5801	−0.576546	−0.288273	−	0.957548i	$-0.593081\pi$
$827$	−16.5801	−0.576546	−0.288273	+	0.957548i	$0.593081\pi$
$828$	0	0
$829$	31.2047	1.08378	0.541892	−	0.840448i	$-0.317708\pi$
$829$	31.2047	1.08378	0.541892	+	0.840448i	$0.317708\pi$
$830$	0	0
$831$	−0.655083	−0.0227246
$832$	0	0
$833$	5.22587	0.181066
$834$	0	0
$835$	−26.7270	−0.924926
$836$	0	0
$837$	−18.8751	−0.652417
$838$	0	0
$839$	13.6955	0.472822	0.236411	−	0.971653i	$-0.424029\pi$
$839$	13.6955	0.472822	0.236411	+	0.971653i	$0.424029\pi$
$840$	0	0
$841$	−27.5661	−0.950554
$842$	0	0
$843$	−1.73974	−0.0599198
$844$	0	0
$845$	−2.27351	−0.0782113
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	1.16905	0.0401218
$850$	0	0
$851$	−21.4028	−0.733679
$852$	0	0
$853$	−13.0674	−0.447421	−0.223710	−	0.974656i	$-0.571817\pi$
$853$	−13.0674	−0.447421	−0.223710	+	0.974656i	$0.571817\pi$
$854$	0	0
$855$	−21.5481	−0.736928
$856$	0	0
$857$	9.55664	0.326448	0.163224	−	0.986589i	$-0.447811\pi$
$857$	9.55664	0.326448	0.163224	+	0.986589i	$0.447811\pi$
$858$	0	0
$859$	24.1841	0.825151	0.412575	−	0.910923i	$-0.364629\pi$
$859$	24.1841	0.825151	0.412575	+	0.910923i	$0.364629\pi$
$860$	0	0
$861$	−4.71309	−0.160622
$862$	0	0
$863$	−54.9431	−1.87029	−0.935143	−	0.354271i	$-0.884729\pi$
$863$	−54.9431	−1.87029	−0.935143	+	0.354271i	$0.884729\pi$
$864$	0	0
$865$	3.00498	0.102172
$866$	0	0
$867$	5.30333	0.180111
$868$	0	0
$869$	7.96114	0.270063
$870$	0	0
$871$	4.52843	0.153440
$872$	0	0
$873$	12.3776	0.418917
$874$	0	0
$875$	10.9836	0.371315
$876$	0	0
$877$	−49.9397	−1.68634	−0.843172	−	0.537643i	$-0.819315\pi$
$877$	−49.9397	−1.68634	−0.843172	+	0.537643i	$0.819315\pi$
$878$	0	0
$879$	−0.341719	−0.0115259
$880$	0	0
$881$	28.7281	0.967875	0.483938	−	0.875103i	$-0.339206\pi$
$881$	28.7281	0.967875	0.483938	+	0.875103i	$0.339206\pi$
$882$	0	0
$883$	−25.7961	−0.868106	−0.434053	−	0.900887i	$-0.642917\pi$
$883$	−25.7961	−0.868106	−0.434053	+	0.900887i	$0.642917\pi$
$884$	0	0
$885$	−5.90931	−0.198639
$886$	0	0
$887$	−33.6267	−1.12907	−0.564537	−	0.825408i	$-0.690945\pi$
$887$	−33.6267	−1.12907	−0.564537	+	0.825408i	$0.690945\pi$
$888$	0	0
$889$	−10.5129	−0.352591
$890$	0	0
$891$	−6.68855	−0.224075
$892$	0	0
$893$	25.6734	0.859127
$894$	0	0
$895$	1.66972	0.0558126
$896$	0	0
$897$	3.40795	0.113788
$898$	0	0
$899$	7.66105	0.255510
$900$	0	0
$901$	24.5771	0.818782
$902$	0	0
$903$	−2.52431	−0.0840036
$904$	0	0
$905$	−20.5977	−0.684692
$906$	0	0
$907$	25.7148	0.853845	0.426923	−	0.904288i	$-0.359598\pi$
$907$	25.7148	0.853845	0.426923	+	0.904288i	$0.359598\pi$
$908$	0	0
$909$	−16.6222	−0.551323
$910$	0	0
$911$	0.000897645 0	2.97403e−5 0	1.48702e−5	−	1.00000i	$-0.499995\pi$
$911$	0.000897645 0	2.97403e−5 0	1.48702e−5		1.00000i	$0.499995\pi$
$912$	0	0
$913$	3.59050	0.118828
$914$	0	0
$915$	−2.34900	−0.0776555
$916$	0	0
$917$	0.134174	0.00443083
$918$	0	0
$919$	41.7614	1.37758	0.688791	−	0.724960i	$-0.258142\pi$
$919$	41.7614	1.37758	0.688791	+	0.724960i	$0.258142\pi$
$920$	0	0
$921$	0.865844	0.0285305
$922$	0	0
$923$	−12.9191	−0.425239
$924$	0	0
$925$	−0.545543	−0.0179373
$926$	0	0
$927$	−9.37928	−0.308056
$928$	0	0
$929$	5.30666	0.174106	0.0870529	−	0.996204i	$-0.472255\pi$
$929$	5.30666	0.174106	0.0870529	+	0.996204i	$0.472255\pi$
$930$	0	0
$931$	−3.46490	−0.113557
$932$	0	0
$933$	−10.3522	−0.338917
$934$	0	0
$935$	11.8811	0.388553
$936$	0	0
$937$	4.81996	0.157461	0.0787306	−	0.996896i	$-0.474913\pi$
$937$	4.81996	0.157461	0.0787306	+	0.996896i	$0.474913\pi$
$938$	0	0
$939$	−4.33661	−0.141520
$940$	0	0
$941$	8.28467	0.270072	0.135036	−	0.990841i	$-0.456885\pi$
$941$	8.28467	0.270072	0.135036	+	0.990841i	$0.456885\pi$
$942$	0	0
$943$	−60.7013	−1.97671
$944$	0	0
$945$	6.70752	0.218195
$946$	0	0
$947$	34.1651	1.11022	0.555109	−	0.831778i	$-0.312677\pi$
$947$	34.1651	1.11022	0.555109	+	0.831778i	$0.312677\pi$
$948$	0	0
$949$	2.01524	0.0654173
$950$	0	0
$951$	−10.3673	−0.336182
$952$	0	0
$953$	−46.5135	−1.50672	−0.753361	−	0.657607i	$-0.771569\pi$
$953$	−46.5135	−1.50672	−0.753361	+	0.657607i	$0.771569\pi$
$954$	0	0
$955$	−22.1248	−0.715942
$956$	0	0
$957$	−0.615977	−0.0199117
$958$	0	0
$959$	−15.0170	−0.484926
$960$	0	0
$961$	9.93065	0.320343
$962$	0	0
$963$	−36.5229	−1.17693
$964$	0	0
$965$	−10.6792	−0.343775
$966$	0	0
$967$	−0.490715	−0.0157803	−0.00789017	−	0.999969i	$-0.502512\pi$
$967$	−0.490715	−0.0157803	−0.00789017	+	0.999969i	$0.502512\pi$
$968$	0	0
$969$	−9.31430	−0.299218
$970$	0	0
$971$	−16.0865	−0.516242	−0.258121	−	0.966113i	$-0.583103\pi$
$971$	−16.0865	−0.516242	−0.258121	+	0.966113i	$0.583103\pi$
$972$	0	0
$973$	17.8791	0.573177
$974$	0	0
$975$	0.0868664	0.00278195
$976$	0	0
$977$	5.48022	0.175328	0.0876638	−	0.996150i	$-0.472060\pi$
$977$	5.48022	0.175328	0.0876638	+	0.996150i	$0.472060\pi$
$978$	0	0
$979$	11.2018	0.358010
$980$	0	0
$981$	−24.5930	−0.785195
$982$	0	0
$983$	39.3065	1.25368	0.626841	−	0.779147i	$-0.284347\pi$
$983$	39.3065	1.25368	0.626841	+	0.779147i	$0.284347\pi$
$984$	0	0
$985$	−48.1837	−1.53526
$986$	0	0
$987$	−3.81148	−0.121321
$988$	0	0
$989$	−32.5113	−1.03380
$990$	0	0
$991$	−30.7837	−0.977877	−0.488939	−	0.872318i	$-0.662616\pi$
$991$	−30.7837	−0.977877	−0.488939	+	0.872318i	$0.662616\pi$
$992$	0	0
$993$	4.50650	0.143009
$994$	0	0
$995$	−4.34262	−0.137670
$996$	0	0
$997$	17.2437	0.546113	0.273056	−	0.961998i	$-0.411965\pi$
$997$	17.2437	0.546113	0.273056	+	0.961998i	$0.411965\pi$
$998$	0	0
$999$	9.53107	0.301550

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.t.1.6	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.t.1.6	✓	10	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+0.514400 q^{3} -2.27351 q^{5} +1.00000 q^{7} -2.73539 q^{9} +O(q^{10})\)	\(q+0.514400 q^{3} -2.27351 q^{5} +1.00000 q^{7} -2.73539 q^{9} -1.00000 q^{11} +1.00000 q^{13} -1.16950 q^{15} +5.22587 q^{17} -3.46490 q^{19} +0.514400 q^{21} +6.62511 q^{23} +0.168869 q^{25} -2.95028 q^{27} +1.19747 q^{29} +6.39771 q^{31} -0.514400 q^{33} -2.27351 q^{35} -3.23056 q^{37} +0.514400 q^{39} -9.16231 q^{41} -4.90728 q^{43} +6.21896 q^{45} -7.40956 q^{47} +1.00000 q^{49} +2.68819 q^{51} +4.70297 q^{53} +2.27351 q^{55} -1.78234 q^{57} +5.05287 q^{59} +2.00856 q^{61} -2.73539 q^{63} -2.27351 q^{65} +4.52843 q^{67} +3.40795 q^{69} -12.9191 q^{71} +2.01524 q^{73} +0.0868664 q^{75} -1.00000 q^{77} -7.96114 q^{79} +6.68855 q^{81} -3.59050 q^{83} -11.8811 q^{85} +0.615977 q^{87} -11.2018 q^{89} +1.00000 q^{91} +3.29098 q^{93} +7.87750 q^{95} -4.52497 q^{97} +2.73539 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9}+O(q^{10}) \) 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9	\( 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9} - 10 q^{11} + 10 q^{13} + 9 q^{15} + 7 q^{17} + 17 q^{19} + q^{21} + 15 q^{23} + 13 q^{25} + 7 q^{27} - q^{29} - 6 q^{31} - q^{33} + 3 q^{35} - 12 q^{37} + q^{39} + 4 q^{41} + 17 q^{43} - 15 q^{45} + 21 q^{47} + 10 q^{49} + 3 q^{51} + 20 q^{53} - 3 q^{55} + 12 q^{57} + 9 q^{59} + 6 q^{61} + 5 q^{63} + 3 q^{65} + 26 q^{67} + 30 q^{69} + 18 q^{71} - 4 q^{73} + 48 q^{75} - 10 q^{77} + 19 q^{79} - 14 q^{81} + 40 q^{83} - 25 q^{85} - 25 q^{87} - 19 q^{89} + 10 q^{91} + q^{93} + 11 q^{95} - 22 q^{97} - 5 q^{99}+O(q^{100}) \) 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9 - 10 * q^11 + 10 * q^13 + 9 * q^15 + 7 * q^17 + 17 * q^19 + q^21 + 15 * q^23 + 13 * q^25 + 7 * q^27 - q^29 - 6 * q^31 - q^33 + 3 * q^35 - 12 * q^37 + q^39 + 4 * q^41 + 17 * q^43 - 15 * q^45 + 21 * q^47 + 10 * q^49 + 3 * q^51 + 20 * q^53 - 3 * q^55 + 12 * q^57 + 9 * q^59 + 6 * q^61 + 5 * q^63 + 3 * q^65 + 26 * q^67 + 30 * q^69 + 18 * q^71 - 4 * q^73 + 48 * q^75 - 10 * q^77 + 19 * q^79 - 14 * q^81 + 40 * q^83 - 25 * q^85 - 25 * q^87 - 19 * q^89 + 10 * q^91 + q^93 + 11 * q^95 - 22 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more