LMFDB - Embedded newform 3380.2.a.r.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(1,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, 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("3380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.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:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 19x^{7} + 16x^{6} + 106x^{5} - 87x^{4} - 153x^{3} + 149x^{2} - 26x + 1 $ x^9 - x^8 - 19x^7 + 16x^6 + 106x^5 - 87x^4 - 153x^3 + 149x^2 - 26*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.715841$ of defining polynomial
Character	$\chi$	$=$	3380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.715841 q^{3} -1.00000 q^{5} -3.76323 q^{7} -2.48757 q^{9} +O(q^{10})$	$q-0.715841 q^{3} -1.00000 q^{5} -3.76323 q^{7} -2.48757 q^{9} +1.96386 q^{11} +0.715841 q^{15} -5.65737 q^{17} -0.336153 q^{19} +2.69387 q^{21} +1.38546 q^{23} +1.00000 q^{25} +3.92823 q^{27} -1.05504 q^{29} -9.97744 q^{31} -1.40581 q^{33} +3.76323 q^{35} +0.428134 q^{37} -3.99625 q^{41} +1.41775 q^{43} +2.48757 q^{45} -10.5023 q^{47} +7.16191 q^{49} +4.04977 q^{51} -0.428839 q^{53} -1.96386 q^{55} +0.240632 q^{57} -8.47905 q^{59} +12.5518 q^{61} +9.36131 q^{63} +8.25232 q^{67} -0.991769 q^{69} +11.6464 q^{71} -10.8433 q^{73} -0.715841 q^{75} -7.39046 q^{77} +9.33649 q^{79} +4.65073 q^{81} +10.2129 q^{83} +5.65737 q^{85} +0.755237 q^{87} +9.45621 q^{89} +7.14225 q^{93} +0.336153 q^{95} -15.8156 q^{97} -4.88525 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9	$ 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9} + 7 q^{11} + q^{15} + 13 q^{17} + 4 q^{19} + 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} - 13 q^{31} - 34 q^{33} + q^{35} - q^{37} + 6 q^{41} + q^{43} - 12 q^{45} + 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} - 38 q^{57} - 15 q^{59} + 21 q^{61} + 17 q^{63} + 7 q^{67} + 15 q^{69} + 7 q^{71} + 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} - 45 q^{83} - 13 q^{85} + 28 q^{87} + 41 q^{89} + 11 q^{93} - 4 q^{95} - 8 q^{97} + 81 q^{99}+O(q^{100}) $ 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9 + 7 * q^11 + q^15 + 13 * q^17 + 4 * q^19 + 3 * q^21 + 12 * q^23 + 9 * q^25 - 4 * q^27 + 16 * q^29 - 13 * q^31 - 34 * q^33 + q^35 - q^37 + 6 * q^41 + q^43 - 12 * q^45 + 2 * q^47 + 20 * q^49 + 11 * q^51 + 30 * q^53 - 7 * q^55 - 38 * q^57 - 15 * q^59 + 21 * q^61 + 17 * q^63 + 7 * q^67 + 15 * q^69 + 7 * q^71 + 28 * q^73 - q^75 + 46 * q^77 + 31 * q^79 + 41 * q^81 - 45 * q^83 - 13 * q^85 + 28 * q^87 + 41 * q^89 + 11 * q^93 - 4 * q^95 - 8 * q^97 + 81 * 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
$3$	−0.715841	−0.413291	−0.206645	−	0.978416i	$-0.566255\pi$
$3$	−0.715841	−0.413291	−0.206645	+	0.978416i	$0.566255\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.76323	−1.42237	−0.711184	−	0.703006i	$-0.751841\pi$
$7$	−3.76323	−1.42237	−0.711184	+	0.703006i	$0.751841\pi$
$8$	0	0
$9$	−2.48757	−0.829191
$10$	0	0
$11$	1.96386	0.592126	0.296063	−	0.955168i	$-0.404326\pi$
$11$	1.96386	0.592126	0.296063	+	0.955168i	$0.404326\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.715841	0.184829
$16$	0	0
$17$	−5.65737	−1.37211	−0.686057	−	0.727548i	$-0.740660\pi$
$17$	−5.65737	−1.37211	−0.686057	+	0.727548i	$0.740660\pi$
$18$	0	0
$19$	−0.336153	−0.0771189	−0.0385594	−	0.999256i	$-0.512277\pi$
$19$	−0.336153	−0.0771189	−0.0385594	+	0.999256i	$0.512277\pi$
$20$	0	0
$21$	2.69387	0.587851
$22$	0	0
$23$	1.38546	0.288889	0.144444	−	0.989513i	$-0.453861\pi$
$23$	1.38546	0.288889	0.144444	+	0.989513i	$0.453861\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.92823	0.755988
$28$	0	0
$29$	−1.05504	−0.195915	−0.0979576	−	0.995191i	$-0.531231\pi$
$29$	−1.05504	−0.195915	−0.0979576	+	0.995191i	$0.531231\pi$
$30$	0	0
$31$	−9.97744	−1.79200	−0.896000	−	0.444053i	$-0.853540\pi$
$31$	−9.97744	−1.79200	−0.896000	+	0.444053i	$0.853540\pi$
$32$	0	0
$33$	−1.40581	−0.244720
$34$	0	0
$35$	3.76323	0.636102
$36$	0	0
$37$	0.428134	0.0703848	0.0351924	−	0.999381i	$-0.488796\pi$
$37$	0.428134	0.0703848	0.0351924	+	0.999381i	$0.488796\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.99625	−0.624110	−0.312055	−	0.950064i	$-0.601017\pi$
$41$	−3.99625	−0.624110	−0.312055	+	0.950064i	$0.601017\pi$
$42$	0	0
$43$	1.41775	0.216205	0.108103	−	0.994140i	$-0.465523\pi$
$43$	1.41775	0.216205	0.108103	+	0.994140i	$0.465523\pi$
$44$	0	0
$45$	2.48757	0.370825
$46$	0	0
$47$	−10.5023	−1.53192	−0.765961	−	0.642887i	$-0.777736\pi$
$47$	−10.5023	−1.53192	−0.765961	+	0.642887i	$0.777736\pi$
$48$	0	0
$49$	7.16191	1.02313
$50$	0	0
$51$	4.04977	0.567082
$52$	0	0
$53$	−0.428839	−0.0589055	−0.0294528	−	0.999566i	$-0.509376\pi$
$53$	−0.428839	−0.0589055	−0.0294528	+	0.999566i	$0.509376\pi$
$54$	0	0
$55$	−1.96386	−0.264807
$56$	0	0
$57$	0.240632	0.0318725
$58$	0	0
$59$	−8.47905	−1.10388	−0.551939	−	0.833884i	$-0.686112\pi$
$59$	−8.47905	−1.10388	−0.551939	+	0.833884i	$0.686112\pi$
$60$	0	0
$61$	12.5518	1.60709	0.803545	−	0.595244i	$-0.202945\pi$
$61$	12.5518	1.60709	0.803545	+	0.595244i	$0.202945\pi$
$62$	0	0
$63$	9.36131	1.17941
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.25232	1.00818	0.504091	−	0.863651i	$-0.331828\pi$
$67$	8.25232	1.00818	0.504091	+	0.863651i	$0.331828\pi$
$68$	0	0
$69$	−0.991769	−0.119395
$70$	0	0
$71$	11.6464	1.38218	0.691089	−	0.722770i	$-0.257131\pi$
$71$	11.6464	1.38218	0.691089	+	0.722770i	$0.257131\pi$
$72$	0	0
$73$	−10.8433	−1.26912	−0.634558	−	0.772875i	$-0.718818\pi$
$73$	−10.8433	−1.26912	−0.634558	+	0.772875i	$0.718818\pi$
$74$	0	0
$75$	−0.715841	−0.0826582
$76$	0	0
$77$	−7.39046	−0.842221
$78$	0	0
$79$	9.33649	1.05044	0.525219	−	0.850967i	$-0.323984\pi$
$79$	9.33649	1.05044	0.525219	+	0.850967i	$0.323984\pi$
$80$	0	0
$81$	4.65073	0.516748
$82$	0	0
$83$	10.2129	1.12102	0.560508	−	0.828149i	$-0.310606\pi$
$83$	10.2129	1.12102	0.560508	+	0.828149i	$0.310606\pi$
$84$	0	0
$85$	5.65737	0.613628
$86$	0	0
$87$	0.755237	0.0809699
$88$	0	0
$89$	9.45621	1.00236	0.501178	−	0.865344i	$-0.332900\pi$
$89$	9.45621	1.00236	0.501178	+	0.865344i	$0.332900\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	7.14225	0.740617
$94$	0	0
$95$	0.336153	0.0344886
$96$	0	0
$97$	−15.8156	−1.60583	−0.802915	−	0.596094i	$-0.796718\pi$
$97$	−15.8156	−1.60583	−0.802915	+	0.596094i	$0.796718\pi$
$98$	0	0
$99$	−4.88525	−0.490986
$100$	0	0
$101$	17.5424	1.74554	0.872769	−	0.488133i	$-0.162322\pi$
$101$	17.5424	1.74554	0.872769	+	0.488133i	$0.162322\pi$
$102$	0	0
$103$	−0.760790	−0.0749628	−0.0374814	−	0.999297i	$-0.511934\pi$
$103$	−0.760790	−0.0749628	−0.0374814	+	0.999297i	$0.511934\pi$
$104$	0	0
$105$	−2.69387	−0.262895
$106$	0	0
$107$	−6.38654	−0.617410	−0.308705	−	0.951158i	$-0.599896\pi$
$107$	−6.38654	−0.617410	−0.308705	+	0.951158i	$0.599896\pi$
$108$	0	0
$109$	12.1620	1.16491	0.582456	−	0.812862i	$-0.302092\pi$
$109$	12.1620	1.16491	0.582456	+	0.812862i	$0.302092\pi$
$110$	0	0
$111$	−0.306476	−0.0290894
$112$	0	0
$113$	20.2283	1.90292	0.951459	−	0.307775i	$-0.0995845\pi$
$113$	20.2283	1.90292	0.951459	+	0.307775i	$0.0995845\pi$
$114$	0	0
$115$	−1.38546	−0.129195
$116$	0	0
$117$	0	0
$118$	0	0
$119$	21.2900	1.95165
$120$	0	0
$121$	−7.14325	−0.649386
$122$	0	0
$123$	2.86068	0.257939
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−16.7363	−1.48511	−0.742555	−	0.669785i	$-0.766386\pi$
$127$	−16.7363	−1.48511	−0.742555	+	0.669785i	$0.766386\pi$
$128$	0	0
$129$	−1.01488	−0.0893556
$130$	0	0
$131$	−3.46690	−0.302904	−0.151452	−	0.988465i	$-0.548395\pi$
$131$	−3.46690	−0.302904	−0.151452	+	0.988465i	$0.548395\pi$
$132$	0	0
$133$	1.26502	0.109691
$134$	0	0
$135$	−3.92823	−0.338088
$136$	0	0
$137$	−2.22396	−0.190006	−0.0950029	−	0.995477i	$-0.530286\pi$
$137$	−2.22396	−0.190006	−0.0950029	+	0.995477i	$0.530286\pi$
$138$	0	0
$139$	8.90313	0.755154	0.377577	−	0.925978i	$-0.376757\pi$
$139$	8.90313	0.755154	0.377577	+	0.925978i	$0.376757\pi$
$140$	0	0
$141$	7.51799	0.633129
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1.05504	0.0876159
$146$	0	0
$147$	−5.12678	−0.422850
$148$	0	0
$149$	15.6217	1.27978	0.639892	−	0.768465i	$-0.278979\pi$
$149$	15.6217	1.27978	0.639892	+	0.768465i	$0.278979\pi$
$150$	0	0
$151$	−7.21151	−0.586864	−0.293432	−	0.955980i	$-0.594797\pi$
$151$	−7.21151	−0.586864	−0.293432	+	0.955980i	$0.594797\pi$
$152$	0	0
$153$	14.0731	1.13774
$154$	0	0
$155$	9.97744	0.801407
$156$	0	0
$157$	21.1160	1.68524	0.842622	−	0.538506i	$-0.181011\pi$
$157$	21.1160	1.68524	0.842622	+	0.538506i	$0.181011\pi$
$158$	0	0
$159$	0.306980	0.0243451
$160$	0	0
$161$	−5.21381	−0.410906
$162$	0	0
$163$	−11.2476	−0.880982	−0.440491	−	0.897757i	$-0.645196\pi$
$163$	−11.2476	−0.880982	−0.440491	+	0.897757i	$0.645196\pi$
$164$	0	0
$165$	1.40581	0.109442
$166$	0	0
$167$	−1.57105	−0.121572	−0.0607859	−	0.998151i	$-0.519361\pi$
$167$	−1.57105	−0.121572	−0.0607859	+	0.998151i	$0.519361\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0.836206	0.0639462
$172$	0	0
$173$	−2.92266	−0.222206	−0.111103	−	0.993809i	$-0.535438\pi$
$173$	−2.92266	−0.222206	−0.111103	+	0.993809i	$0.535438\pi$
$174$	0	0
$175$	−3.76323	−0.284474
$176$	0	0
$177$	6.06965	0.456223
$178$	0	0
$179$	10.0344	0.750010	0.375005	−	0.927023i	$-0.377641\pi$
$179$	10.0344	0.750010	0.375005	+	0.927023i	$0.377641\pi$
$180$	0	0
$181$	24.6169	1.82976	0.914879	−	0.403727i	$-0.132286\pi$
$181$	24.6169	1.82976	0.914879	+	0.403727i	$0.132286\pi$
$182$	0	0
$183$	−8.98507	−0.664196
$184$	0	0
$185$	−0.428134	−0.0314771
$186$	0	0
$187$	−11.1103	−0.812464
$188$	0	0
$189$	−14.7828	−1.07529
$190$	0	0
$191$	2.01914	0.146100	0.0730498	−	0.997328i	$-0.476727\pi$
$191$	2.01914	0.146100	0.0730498	+	0.997328i	$0.476727\pi$
$192$	0	0
$193$	8.31501	0.598527	0.299264	−	0.954170i	$-0.403259\pi$
$193$	8.31501	0.598527	0.299264	+	0.954170i	$0.403259\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.49617	0.605327	0.302664	−	0.953097i	$-0.402124\pi$
$197$	8.49617	0.605327	0.302664	+	0.953097i	$0.402124\pi$
$198$	0	0
$199$	−19.0657	−1.35153	−0.675767	−	0.737115i	$-0.736188\pi$
$199$	−19.0657	−1.35153	−0.675767	+	0.737115i	$0.736188\pi$
$200$	0	0
$201$	−5.90734	−0.416672
$202$	0	0
$203$	3.97034	0.278663
$204$	0	0
$205$	3.99625	0.279110
$206$	0	0
$207$	−3.44643	−0.239544
$208$	0	0
$209$	−0.660158	−0.0456641
$210$	0	0
$211$	17.2368	1.18663	0.593316	−	0.804970i	$-0.297818\pi$
$211$	17.2368	1.18663	0.593316	+	0.804970i	$0.297818\pi$
$212$	0	0
$213$	−8.33699	−0.571241
$214$	0	0
$215$	−1.41775	−0.0966899
$216$	0	0
$217$	37.5474	2.54888
$218$	0	0
$219$	7.76210	0.524514
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−0.633842	−0.0424452	−0.0212226	−	0.999775i	$-0.506756\pi$
$223$	−0.633842	−0.0424452	−0.0212226	+	0.999775i	$0.506756\pi$
$224$	0	0
$225$	−2.48757	−0.165838
$226$	0	0
$227$	−13.9133	−0.923459	−0.461730	−	0.887021i	$-0.652771\pi$
$227$	−13.9133	−0.923459	−0.461730	+	0.887021i	$0.652771\pi$
$228$	0	0
$229$	21.7562	1.43769	0.718846	−	0.695169i	$-0.244671\pi$
$229$	21.7562	1.43769	0.718846	+	0.695169i	$0.244671\pi$
$230$	0	0
$231$	5.29039	0.348082
$232$	0	0
$233$	−9.45256	−0.619258	−0.309629	−	0.950857i	$-0.600205\pi$
$233$	−9.45256	−0.619258	−0.309629	+	0.950857i	$0.600205\pi$
$234$	0	0
$235$	10.5023	0.685096
$236$	0	0
$237$	−6.68344	−0.434136
$238$	0	0
$239$	−3.50513	−0.226728	−0.113364	−	0.993554i	$-0.536163\pi$
$239$	−3.50513	−0.226728	−0.113364	+	0.993554i	$0.536163\pi$
$240$	0	0
$241$	−12.2931	−0.791870	−0.395935	−	0.918279i	$-0.629579\pi$
$241$	−12.2931	−0.791870	−0.395935	+	0.918279i	$0.629579\pi$
$242$	0	0
$243$	−15.1139	−0.969555
$244$	0	0
$245$	−7.16191	−0.457557
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−7.31083	−0.463305
$250$	0	0
$251$	−7.89241	−0.498164	−0.249082	−	0.968482i	$-0.580129\pi$
$251$	−7.89241	−0.498164	−0.249082	+	0.968482i	$0.580129\pi$
$252$	0	0
$253$	2.72085	0.171059
$254$	0	0
$255$	−4.04977	−0.253607
$256$	0	0
$257$	4.33042	0.270124	0.135062	−	0.990837i	$-0.456877\pi$
$257$	4.33042	0.270124	0.135062	+	0.990837i	$0.456877\pi$
$258$	0	0
$259$	−1.61117	−0.100113
$260$	0	0
$261$	2.62448	0.162451
$262$	0	0
$263$	25.1235	1.54918	0.774592	−	0.632462i	$-0.217955\pi$
$263$	25.1235	1.54918	0.774592	+	0.632462i	$0.217955\pi$
$264$	0	0
$265$	0.428839	0.0263434
$266$	0	0
$267$	−6.76914	−0.414265
$268$	0	0
$269$	−32.0183	−1.95219	−0.976095	−	0.217345i	$-0.930260\pi$
$269$	−32.0183	−1.95219	−0.976095	+	0.217345i	$0.930260\pi$
$270$	0	0
$271$	−12.1300	−0.736846	−0.368423	−	0.929658i	$-0.620102\pi$
$271$	−12.1300	−0.736846	−0.368423	+	0.929658i	$0.620102\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.96386	0.118425
$276$	0	0
$277$	3.24396	0.194911	0.0974554	−	0.995240i	$-0.468930\pi$
$277$	3.24396	0.194911	0.0974554	+	0.995240i	$0.468930\pi$
$278$	0	0
$279$	24.8196	1.48591
$280$	0	0
$281$	−14.5156	−0.865930	−0.432965	−	0.901411i	$-0.642533\pi$
$281$	−14.5156	−0.865930	−0.432965	+	0.901411i	$0.642533\pi$
$282$	0	0
$283$	−17.7093	−1.05271	−0.526353	−	0.850266i	$-0.676441\pi$
$283$	−17.7093	−1.05271	−0.526353	+	0.850266i	$0.676441\pi$
$284$	0	0
$285$	−0.240632	−0.0142538
$286$	0	0
$287$	15.0388	0.887714
$288$	0	0
$289$	15.0058	0.882694
$290$	0	0
$291$	11.3214	0.663674
$292$	0	0
$293$	−21.7599	−1.27123	−0.635614	−	0.772007i	$-0.719253\pi$
$293$	−21.7599	−1.27123	−0.635614	+	0.772007i	$0.719253\pi$
$294$	0	0
$295$	8.47905	0.493670
$296$	0	0
$297$	7.71449	0.447640
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−5.33533	−0.307523
$302$	0	0
$303$	−12.5576	−0.721415
$304$	0	0
$305$	−12.5518	−0.718713
$306$	0	0
$307$	−10.6875	−0.609971	−0.304985	−	0.952357i	$-0.598652\pi$
$307$	−10.6875	−0.609971	−0.304985	+	0.952357i	$0.598652\pi$
$308$	0	0
$309$	0.544604	0.0309814
$310$	0	0
$311$	−26.4611	−1.50047	−0.750236	−	0.661170i	$-0.770060\pi$
$311$	−26.4611	−1.50047	−0.750236	+	0.661170i	$0.770060\pi$
$312$	0	0
$313$	19.7056	1.11383	0.556914	−	0.830570i	$-0.311985\pi$
$313$	19.7056	1.11383	0.556914	+	0.830570i	$0.311985\pi$
$314$	0	0
$315$	−9.36131	−0.527450
$316$	0	0
$317$	−10.7487	−0.603706	−0.301853	−	0.953355i	$-0.597605\pi$
$317$	−10.7487	−0.603706	−0.301853	+	0.953355i	$0.597605\pi$
$318$	0	0
$319$	−2.07194	−0.116007
$320$	0	0
$321$	4.57174	0.255170
$322$	0	0
$323$	1.90174	0.105816
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−8.70608	−0.481447
$328$	0	0
$329$	39.5227	2.17896
$330$	0	0
$331$	−3.78709	−0.208157	−0.104079	−	0.994569i	$-0.533189\pi$
$331$	−3.78709	−0.208157	−0.104079	+	0.994569i	$0.533189\pi$
$332$	0	0
$333$	−1.06502	−0.0583625
$334$	0	0
$335$	−8.25232	−0.450872
$336$	0	0
$337$	20.9235	1.13978	0.569888	−	0.821722i	$-0.306987\pi$
$337$	20.9235	1.13978	0.569888	+	0.821722i	$0.306987\pi$
$338$	0	0
$339$	−14.4802	−0.786458
$340$	0	0
$341$	−19.5943	−1.06109
$342$	0	0
$343$	−0.609288	−0.0328985
$344$	0	0
$345$	0.991769	0.0533951
$346$	0	0
$347$	21.8741	1.17426	0.587132	−	0.809491i	$-0.300257\pi$
$347$	21.8741	1.17426	0.587132	+	0.809491i	$0.300257\pi$
$348$	0	0
$349$	−7.94720	−0.425404	−0.212702	−	0.977117i	$-0.568226\pi$
$349$	−7.94720	−0.425404	−0.212702	+	0.977117i	$0.568226\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.3059	−1.13400	−0.567000	−	0.823718i	$-0.691896\pi$
$353$	−21.3059	−1.13400	−0.567000	+	0.823718i	$0.691896\pi$
$354$	0	0
$355$	−11.6464	−0.618128
$356$	0	0
$357$	−15.2402	−0.806599
$358$	0	0
$359$	−15.0831	−0.796054	−0.398027	−	0.917374i	$-0.630305\pi$
$359$	−15.0831	−0.796054	−0.398027	+	0.917374i	$0.630305\pi$
$360$	0	0
$361$	−18.8870	−0.994053
$362$	0	0
$363$	5.11343	0.268385
$364$	0	0
$365$	10.8433	0.567566
$366$	0	0
$367$	−13.7040	−0.715344	−0.357672	−	0.933847i	$-0.616429\pi$
$367$	−13.7040	−0.715344	−0.357672	+	0.933847i	$0.616429\pi$
$368$	0	0
$369$	9.94097	0.517506
$370$	0	0
$371$	1.61382	0.0837853
$372$	0	0
$373$	−3.30720	−0.171240	−0.0856202	−	0.996328i	$-0.527287\pi$
$373$	−3.30720	−0.171240	−0.0856202	+	0.996328i	$0.527287\pi$
$374$	0	0
$375$	0.715841	0.0369658
$376$	0	0
$377$	0	0
$378$	0	0
$379$	30.6788	1.57586	0.787931	−	0.615763i	$-0.211152\pi$
$379$	30.6788	1.57586	0.787931	+	0.615763i	$0.211152\pi$
$380$	0	0
$381$	11.9806	0.613783
$382$	0	0
$383$	18.6469	0.952812	0.476406	−	0.879226i	$-0.341939\pi$
$383$	18.6469	0.952812	0.476406	+	0.879226i	$0.341939\pi$
$384$	0	0
$385$	7.39046	0.376653
$386$	0	0
$387$	−3.52676	−0.179275
$388$	0	0
$389$	17.7486	0.899889	0.449944	−	0.893057i	$-0.351444\pi$
$389$	17.7486	0.899889	0.449944	+	0.893057i	$0.351444\pi$
$390$	0	0
$391$	−7.83806	−0.396388
$392$	0	0
$393$	2.48174	0.125187
$394$	0	0
$395$	−9.33649	−0.469770
$396$	0	0
$397$	36.9431	1.85412	0.927061	−	0.374911i	$-0.122327\pi$
$397$	36.9431	1.85412	0.927061	+	0.374911i	$0.122327\pi$
$398$	0	0
$399$	−0.905554	−0.0453344
$400$	0	0
$401$	17.6654	0.882167	0.441083	−	0.897466i	$-0.354594\pi$
$401$	17.6654	0.882167	0.441083	+	0.897466i	$0.354594\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−4.65073	−0.231097
$406$	0	0
$407$	0.840796	0.0416767
$408$	0	0
$409$	29.2194	1.44481	0.722404	−	0.691471i	$-0.243037\pi$
$409$	29.2194	1.44481	0.722404	+	0.691471i	$0.243037\pi$
$410$	0	0
$411$	1.59200	0.0785277
$412$	0	0
$413$	31.9086	1.57012
$414$	0	0
$415$	−10.2129	−0.501333
$416$	0	0
$417$	−6.37322	−0.312098
$418$	0	0
$419$	22.7266	1.11027	0.555133	−	0.831762i	$-0.312667\pi$
$419$	22.7266	1.11027	0.555133	+	0.831762i	$0.312667\pi$
$420$	0	0
$421$	34.8180	1.69692	0.848462	−	0.529256i	$-0.177529\pi$
$421$	34.8180	1.69692	0.848462	+	0.529256i	$0.177529\pi$
$422$	0	0
$423$	26.1253	1.27026
$424$	0	0
$425$	−5.65737	−0.274423
$426$	0	0
$427$	−47.2352	−2.28587
$428$	0	0
$429$	0	0
$430$	0	0
$431$	30.0595	1.44791	0.723957	−	0.689845i	$-0.242321\pi$
$431$	30.0595	1.44791	0.723957	+	0.689845i	$0.242321\pi$
$432$	0	0
$433$	27.8940	1.34050	0.670251	−	0.742135i	$-0.266187\pi$
$433$	27.8940	1.34050	0.670251	+	0.742135i	$0.266187\pi$
$434$	0	0
$435$	−0.755237	−0.0362108
$436$	0	0
$437$	−0.465727	−0.0222788
$438$	0	0
$439$	20.2627	0.967087	0.483544	−	0.875320i	$-0.339349\pi$
$439$	20.2627	0.967087	0.483544	+	0.875320i	$0.339349\pi$
$440$	0	0
$441$	−17.8158	−0.848369
$442$	0	0
$443$	−34.5143	−1.63983	−0.819913	−	0.572489i	$-0.805978\pi$
$443$	−34.5143	−1.63983	−0.819913	+	0.572489i	$0.805978\pi$
$444$	0	0
$445$	−9.45621	−0.448268
$446$	0	0
$447$	−11.1827	−0.528923
$448$	0	0
$449$	2.64277	0.124720	0.0623600	−	0.998054i	$-0.480137\pi$
$449$	2.64277	0.124720	0.0623600	+	0.998054i	$0.480137\pi$
$450$	0	0
$451$	−7.84809	−0.369552
$452$	0	0
$453$	5.16229	0.242546
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−35.0036	−1.63740	−0.818699	−	0.574223i	$-0.805304\pi$
$457$	−35.0036	−1.63740	−0.818699	+	0.574223i	$0.805304\pi$
$458$	0	0
$459$	−22.2234	−1.03730
$460$	0	0
$461$	−2.16356	−0.100767	−0.0503836	−	0.998730i	$-0.516044\pi$
$461$	−2.16356	−0.100767	−0.0503836	+	0.998730i	$0.516044\pi$
$462$	0	0
$463$	5.59470	0.260008	0.130004	−	0.991513i	$-0.458501\pi$
$463$	5.59470	0.260008	0.130004	+	0.991513i	$0.458501\pi$
$464$	0	0
$465$	−7.14225	−0.331214
$466$	0	0
$467$	22.4990	1.04113	0.520565	−	0.853822i	$-0.325721\pi$
$467$	22.4990	1.04113	0.520565	+	0.853822i	$0.325721\pi$
$468$	0	0
$469$	−31.0554	−1.43400
$470$	0	0
$471$	−15.1157	−0.696495
$472$	0	0
$473$	2.78427	0.128021
$474$	0	0
$475$	−0.336153	−0.0154238
$476$	0	0
$477$	1.06677	0.0488439
$478$	0	0
$479$	27.8803	1.27388	0.636941	−	0.770912i	$-0.280199\pi$
$479$	27.8803	1.27388	0.636941	+	0.770912i	$0.280199\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	3.73226	0.169824
$484$	0	0
$485$	15.8156	0.718149
$486$	0	0
$487$	−20.0548	−0.908771	−0.454386	−	0.890805i	$-0.650141\pi$
$487$	−20.0548	−0.908771	−0.454386	+	0.890805i	$0.650141\pi$
$488$	0	0
$489$	8.05151	0.364102
$490$	0	0
$491$	−30.3986	−1.37187	−0.685935	−	0.727663i	$-0.740607\pi$
$491$	−30.3986	−1.37187	−0.685935	+	0.727663i	$0.740607\pi$
$492$	0	0
$493$	5.96872	0.268818
$494$	0	0
$495$	4.88525	0.219575
$496$	0	0
$497$	−43.8282	−1.96596
$498$	0	0
$499$	21.0685	0.943154	0.471577	−	0.881825i	$-0.343685\pi$
$499$	21.0685	0.943154	0.471577	+	0.881825i	$0.343685\pi$
$500$	0	0
$501$	1.12462	0.0502445
$502$	0	0
$503$	−25.8008	−1.15040	−0.575201	−	0.818012i	$-0.695076\pi$
$503$	−25.8008	−1.15040	−0.575201	+	0.818012i	$0.695076\pi$
$504$	0	0
$505$	−17.5424	−0.780629
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−15.2047	−0.673936	−0.336968	−	0.941516i	$-0.609401\pi$
$509$	−15.2047	−0.673936	−0.336968	+	0.941516i	$0.609401\pi$
$510$	0	0
$511$	40.8060	1.80515
$512$	0	0
$513$	−1.32049	−0.0583009
$514$	0	0
$515$	0.760790	0.0335244
$516$	0	0
$517$	−20.6251	−0.907091
$518$	0	0
$519$	2.09216	0.0918355
$520$	0	0
$521$	12.0690	0.528754	0.264377	−	0.964419i	$-0.414834\pi$
$521$	12.0690	0.528754	0.264377	+	0.964419i	$0.414834\pi$
$522$	0	0
$523$	15.6516	0.684397	0.342199	−	0.939628i	$-0.388828\pi$
$523$	15.6516	0.684397	0.342199	+	0.939628i	$0.388828\pi$
$524$	0	0
$525$	2.69387	0.117570
$526$	0	0
$527$	56.4460	2.45883
$528$	0	0
$529$	−21.0805	−0.916543
$530$	0	0
$531$	21.0923	0.915326
$532$	0	0
$533$	0	0
$534$	0	0
$535$	6.38654	0.276114
$536$	0	0
$537$	−7.18307	−0.309972
$538$	0	0
$539$	14.0650	0.605822
$540$	0	0
$541$	20.5902	0.885242	0.442621	−	0.896709i	$-0.354049\pi$
$541$	20.5902	0.885242	0.442621	+	0.896709i	$0.354049\pi$
$542$	0	0
$543$	−17.6218	−0.756222
$544$	0	0
$545$	−12.1620	−0.520965
$546$	0	0
$547$	−28.8896	−1.23523	−0.617616	−	0.786480i	$-0.711901\pi$
$547$	−28.8896	−1.23523	−0.617616	+	0.786480i	$0.711901\pi$
$548$	0	0
$549$	−31.2235	−1.33258
$550$	0	0
$551$	0.354654	0.0151088
$552$	0	0
$553$	−35.1354	−1.49411
$554$	0	0
$555$	0.306476	0.0130092
$556$	0	0
$557$	−33.2861	−1.41038	−0.705189	−	0.709020i	$-0.749138\pi$
$557$	−33.2861	−1.41038	−0.705189	+	0.709020i	$0.749138\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	7.95319	0.335784
$562$	0	0
$563$	−29.2175	−1.23137	−0.615686	−	0.787991i	$-0.711121\pi$
$563$	−29.2175	−1.23137	−0.615686	+	0.787991i	$0.711121\pi$
$564$	0	0
$565$	−20.2283	−0.851011
$566$	0	0
$567$	−17.5018	−0.735006
$568$	0	0
$569$	29.3646	1.23103	0.615514	−	0.788126i	$-0.288948\pi$
$569$	29.3646	1.23103	0.615514	+	0.788126i	$0.288948\pi$
$570$	0	0
$571$	−34.0977	−1.42694	−0.713472	−	0.700683i	$-0.752879\pi$
$571$	−34.0977	−1.42694	−0.713472	+	0.700683i	$0.752879\pi$
$572$	0	0
$573$	−1.44538	−0.0603816
$574$	0	0
$575$	1.38546	0.0577777
$576$	0	0
$577$	−6.46819	−0.269274	−0.134637	−	0.990895i	$-0.542987\pi$
$577$	−6.46819	−0.269274	−0.134637	+	0.990895i	$0.542987\pi$
$578$	0	0
$579$	−5.95222	−0.247366
$580$	0	0
$581$	−38.4336	−1.59450
$582$	0	0
$583$	−0.842180	−0.0348795
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−17.0018	−0.701738	−0.350869	−	0.936425i	$-0.614114\pi$
$587$	−17.0018	−0.701738	−0.350869	+	0.936425i	$0.614114\pi$
$588$	0	0
$589$	3.35395	0.138197
$590$	0	0
$591$	−6.08191	−0.250176
$592$	0	0
$593$	10.2611	0.421375	0.210687	−	0.977553i	$-0.432430\pi$
$593$	10.2611	0.421375	0.210687	+	0.977553i	$0.432430\pi$
$594$	0	0
$595$	−21.2900	−0.872804
$596$	0	0
$597$	13.6480	0.558577
$598$	0	0
$599$	−10.9528	−0.447520	−0.223760	−	0.974644i	$-0.571833\pi$
$599$	−10.9528	−0.447520	−0.223760	+	0.974644i	$0.571833\pi$
$600$	0	0
$601$	17.9210	0.731011	0.365506	−	0.930809i	$-0.380896\pi$
$601$	17.9210	0.731011	0.365506	+	0.930809i	$0.380896\pi$
$602$	0	0
$603$	−20.5282	−0.835974
$604$	0	0
$605$	7.14325	0.290414
$606$	0	0
$607$	10.9922	0.446161	0.223081	−	0.974800i	$-0.428389\pi$
$607$	10.9922	0.446161	0.223081	+	0.974800i	$0.428389\pi$
$608$	0	0
$609$	−2.84213	−0.115169
$610$	0	0
$611$	0	0
$612$	0	0
$613$	2.06699	0.0834850	0.0417425	−	0.999128i	$-0.486709\pi$
$613$	2.06699	0.0834850	0.0417425	+	0.999128i	$0.486709\pi$
$614$	0	0
$615$	−2.86068	−0.115354
$616$	0	0
$617$	−39.0188	−1.57084	−0.785420	−	0.618963i	$-0.787553\pi$
$617$	−39.0188	−1.57084	−0.785420	+	0.618963i	$0.787553\pi$
$618$	0	0
$619$	15.8716	0.637935	0.318967	−	0.947766i	$-0.396664\pi$
$619$	15.8716	0.637935	0.318967	+	0.947766i	$0.396664\pi$
$620$	0	0
$621$	5.44240	0.218396
$622$	0	0
$623$	−35.5859	−1.42572
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.472568	0.0188726
$628$	0	0
$629$	−2.42211	−0.0965760
$630$	0	0
$631$	−6.54118	−0.260400	−0.130200	−	0.991488i	$-0.541562\pi$
$631$	−6.54118	−0.260400	−0.130200	+	0.991488i	$0.541562\pi$
$632$	0	0
$633$	−12.3388	−0.490424
$634$	0	0
$635$	16.7363	0.664162
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−28.9713	−1.14609
$640$	0	0
$641$	−27.9790	−1.10510	−0.552552	−	0.833478i	$-0.686346\pi$
$641$	−27.9790	−1.10510	−0.552552	+	0.833478i	$0.686346\pi$
$642$	0	0
$643$	−0.570593	−0.0225020	−0.0112510	−	0.999937i	$-0.503581\pi$
$643$	−0.570593	−0.0225020	−0.0112510	+	0.999937i	$0.503581\pi$
$644$	0	0
$645$	1.01488	0.0399610
$646$	0	0
$647$	−35.4222	−1.39259	−0.696296	−	0.717755i	$-0.745170\pi$
$647$	−35.4222	−1.39259	−0.696296	+	0.717755i	$0.745170\pi$
$648$	0	0
$649$	−16.6517	−0.653636
$650$	0	0
$651$	−26.8780	−1.05343
$652$	0	0
$653$	−12.4879	−0.488688	−0.244344	−	0.969689i	$-0.578573\pi$
$653$	−12.4879	−0.488688	−0.244344	+	0.969689i	$0.578573\pi$
$654$	0	0
$655$	3.46690	0.135463
$656$	0	0
$657$	26.9736	1.05234
$658$	0	0
$659$	−48.6745	−1.89609	−0.948045	−	0.318135i	$-0.896944\pi$
$659$	−48.6745	−1.89609	−0.948045	+	0.318135i	$0.896944\pi$
$660$	0	0
$661$	3.30190	0.128429	0.0642145	−	0.997936i	$-0.479546\pi$
$661$	3.30190	0.128429	0.0642145	+	0.997936i	$0.479546\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.26502	−0.0490555
$666$	0	0
$667$	−1.46171	−0.0565976
$668$	0	0
$669$	0.453730	0.0175422
$670$	0	0
$671$	24.6499	0.951600
$672$	0	0
$673$	24.2607	0.935181	0.467590	−	0.883945i	$-0.345122\pi$
$673$	24.2607	0.935181	0.467590	+	0.883945i	$0.345122\pi$
$674$	0	0
$675$	3.92823	0.151198
$676$	0	0
$677$	25.5400	0.981583	0.490791	−	0.871277i	$-0.336708\pi$
$677$	25.5400	0.981583	0.490791	+	0.871277i	$0.336708\pi$
$678$	0	0
$679$	59.5177	2.28408
$680$	0	0
$681$	9.95972	0.381657
$682$	0	0
$683$	42.5536	1.62827	0.814133	−	0.580678i	$-0.197212\pi$
$683$	42.5536	1.62827	0.814133	+	0.580678i	$0.197212\pi$
$684$	0	0
$685$	2.22396	0.0849732
$686$	0	0
$687$	−15.5740	−0.594185
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−16.6314	−0.632686	−0.316343	−	0.948645i	$-0.602455\pi$
$691$	−16.6314	−0.632686	−0.316343	+	0.948645i	$0.602455\pi$
$692$	0	0
$693$	18.3843	0.698362
$694$	0	0
$695$	−8.90313	−0.337715
$696$	0	0
$697$	22.6083	0.856350
$698$	0	0
$699$	6.76653	0.255934
$700$	0	0
$701$	−17.5058	−0.661185	−0.330593	−	0.943774i	$-0.607249\pi$
$701$	−17.5058	−0.661185	−0.330593	+	0.943774i	$0.607249\pi$
$702$	0	0
$703$	−0.143919	−0.00542800
$704$	0	0
$705$	−7.51799	−0.283144
$706$	0	0
$707$	−66.0163	−2.48280
$708$	0	0
$709$	16.8149	0.631496	0.315748	−	0.948843i	$-0.397745\pi$
$709$	16.8149	0.631496	0.315748	+	0.948843i	$0.397745\pi$
$710$	0	0
$711$	−23.2252	−0.871013
$712$	0	0
$713$	−13.8233	−0.517688
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.50911	0.0937046
$718$	0	0
$719$	30.3694	1.13259	0.566294	−	0.824203i	$-0.308377\pi$
$719$	30.3694	1.13259	0.566294	+	0.824203i	$0.308377\pi$
$720$	0	0
$721$	2.86303	0.106625
$722$	0	0
$723$	8.79992	0.327273
$724$	0	0
$725$	−1.05504	−0.0391830
$726$	0	0
$727$	14.3641	0.532733	0.266367	−	0.963872i	$-0.414177\pi$
$727$	14.3641	0.532733	0.266367	+	0.963872i	$0.414177\pi$
$728$	0	0
$729$	−3.13308	−0.116040
$730$	0	0
$731$	−8.02074	−0.296658
$732$	0	0
$733$	45.8191	1.69237	0.846183	−	0.532892i	$-0.178895\pi$
$733$	45.8191	1.69237	0.846183	+	0.532892i	$0.178895\pi$
$734$	0	0
$735$	5.12678	0.189104
$736$	0	0
$737$	16.2064	0.596971
$738$	0	0
$739$	−5.48314	−0.201701	−0.100850	−	0.994902i	$-0.532156\pi$
$739$	−5.48314	−0.201701	−0.100850	+	0.994902i	$0.532156\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−53.0367	−1.94573	−0.972864	−	0.231378i	$-0.925677\pi$
$743$	−53.0367	−1.94573	−0.972864	+	0.231378i	$0.925677\pi$
$744$	0	0
$745$	−15.6217	−0.572337
$746$	0	0
$747$	−25.4054	−0.929535
$748$	0	0
$749$	24.0340	0.878184
$750$	0	0
$751$	0.975080	0.0355812	0.0177906	−	0.999842i	$-0.494337\pi$
$751$	0.975080	0.0355812	0.0177906	+	0.999842i	$0.494337\pi$
$752$	0	0
$753$	5.64971	0.205887
$754$	0	0
$755$	7.21151	0.262454
$756$	0	0
$757$	24.4113	0.887244	0.443622	−	0.896214i	$-0.353693\pi$
$757$	24.4113	0.887244	0.443622	+	0.896214i	$0.353693\pi$
$758$	0	0
$759$	−1.94770	−0.0706969
$760$	0	0
$761$	3.66922	0.133009	0.0665046	−	0.997786i	$-0.478815\pi$
$761$	3.66922	0.133009	0.0665046	+	0.997786i	$0.478815\pi$
$762$	0	0
$763$	−45.7686	−1.65693
$764$	0	0
$765$	−14.0731	−0.508814
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−40.4300	−1.45794	−0.728972	−	0.684544i	$-0.760001\pi$
$769$	−40.4300	−1.45794	−0.728972	+	0.684544i	$0.760001\pi$
$770$	0	0
$771$	−3.09989	−0.111640
$772$	0	0
$773$	31.6144	1.13709	0.568546	−	0.822651i	$-0.307506\pi$
$773$	31.6144	1.13709	0.568546	+	0.822651i	$0.307506\pi$
$774$	0	0
$775$	−9.97744	−0.358400
$776$	0	0
$777$	1.15334	0.0413758
$778$	0	0
$779$	1.34335	0.0481306
$780$	0	0
$781$	22.8720	0.818424
$782$	0	0
$783$	−4.14442	−0.148109
$784$	0	0
$785$	−21.1160	−0.753664
$786$	0	0
$787$	−7.11487	−0.253618	−0.126809	−	0.991927i	$-0.540473\pi$
$787$	−7.11487	−0.253618	−0.126809	+	0.991927i	$0.540473\pi$
$788$	0	0
$789$	−17.9844	−0.640263
$790$	0	0
$791$	−76.1237	−2.70665
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−0.306980	−0.0108875
$796$	0	0
$797$	−0.224720	−0.00795999	−0.00398000	−	0.999992i	$-0.501267\pi$
$797$	−0.224720	−0.00795999	−0.00398000	+	0.999992i	$0.501267\pi$
$798$	0	0
$799$	59.4155	2.10197
$800$	0	0
$801$	−23.5230	−0.831145
$802$	0	0
$803$	−21.2948	−0.751477
$804$	0	0
$805$	5.21381	0.183763
$806$	0	0
$807$	22.9200	0.806822
$808$	0	0
$809$	6.22088	0.218714	0.109357	−	0.994003i	$-0.465121\pi$
$809$	6.22088	0.218714	0.109357	+	0.994003i	$0.465121\pi$
$810$	0	0
$811$	−25.6039	−0.899074	−0.449537	−	0.893262i	$-0.648411\pi$
$811$	−25.6039	−0.899074	−0.449537	+	0.893262i	$0.648411\pi$
$812$	0	0
$813$	8.68316	0.304532
$814$	0	0
$815$	11.2476	0.393987
$816$	0	0
$817$	−0.476582	−0.0166735
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42.9424	−1.49870	−0.749350	−	0.662174i	$-0.769634\pi$
$821$	−42.9424	−1.49870	−0.749350	+	0.662174i	$0.769634\pi$
$822$	0	0
$823$	45.1061	1.57230	0.786150	−	0.618036i	$-0.212071\pi$
$823$	45.1061	1.57230	0.786150	+	0.618036i	$0.212071\pi$
$824$	0	0
$825$	−1.40581	−0.0489441
$826$	0	0
$827$	−10.5868	−0.368139	−0.184070	−	0.982913i	$-0.558927\pi$
$827$	−10.5868	−0.368139	−0.184070	+	0.982913i	$0.558927\pi$
$828$	0	0
$829$	−17.8655	−0.620495	−0.310247	−	0.950656i	$-0.600412\pi$
$829$	−17.8655	−0.620495	−0.310247	+	0.950656i	$0.600412\pi$
$830$	0	0
$831$	−2.32216	−0.0805548
$832$	0	0
$833$	−40.5175	−1.40385
$834$	0	0
$835$	1.57105	0.0543685
$836$	0	0
$837$	−39.1936	−1.35473
$838$	0	0
$839$	−10.2341	−0.353320	−0.176660	−	0.984272i	$-0.556529\pi$
$839$	−10.2341	−0.353320	−0.176660	+	0.984272i	$0.556529\pi$
$840$	0	0
$841$	−27.8869	−0.961617
$842$	0	0
$843$	10.3909	0.357881
$844$	0	0
$845$	0	0
$846$	0	0
$847$	26.8817	0.923666
$848$	0	0
$849$	12.6770	0.435074
$850$	0	0
$851$	0.593163	0.0203334
$852$	0	0
$853$	47.9341	1.64123	0.820616	−	0.571480i	$-0.193631\pi$
$853$	47.9341	1.64123	0.820616	+	0.571480i	$0.193631\pi$
$854$	0	0
$855$	−0.836206	−0.0285976
$856$	0	0
$857$	−30.6992	−1.04866	−0.524332	−	0.851514i	$-0.675685\pi$
$857$	−30.6992	−1.04866	−0.524332	+	0.851514i	$0.675685\pi$
$858$	0	0
$859$	−55.0682	−1.87890	−0.939451	−	0.342685i	$-0.888664\pi$
$859$	−55.0682	−1.87890	−0.939451	+	0.342685i	$0.888664\pi$
$860$	0	0
$861$	−10.7654	−0.366884
$862$	0	0
$863$	21.9551	0.747360	0.373680	−	0.927558i	$-0.378096\pi$
$863$	21.9551	0.747360	0.373680	+	0.927558i	$0.378096\pi$
$864$	0	0
$865$	2.92266	0.0993734
$866$	0	0
$867$	−10.7418	−0.364809
$868$	0	0
$869$	18.3356	0.621991
$870$	0	0
$871$	0	0
$872$	0	0
$873$	39.3424	1.33154
$874$	0	0
$875$	3.76323	0.127220
$876$	0	0
$877$	−11.0497	−0.373120	−0.186560	−	0.982444i	$-0.559734\pi$
$877$	−11.0497	−0.373120	−0.186560	+	0.982444i	$0.559734\pi$
$878$	0	0
$879$	15.5766	0.525387
$880$	0	0
$881$	11.3150	0.381211	0.190606	−	0.981667i	$-0.438955\pi$
$881$	11.3150	0.381211	0.190606	+	0.981667i	$0.438955\pi$
$882$	0	0
$883$	47.0917	1.58476	0.792381	−	0.610026i	$-0.208841\pi$
$883$	47.0917	1.58476	0.792381	+	0.610026i	$0.208841\pi$
$884$	0	0
$885$	−6.06965	−0.204029
$886$	0	0
$887$	−28.0464	−0.941707	−0.470853	−	0.882211i	$-0.656054\pi$
$887$	−28.0464	−0.941707	−0.470853	+	0.882211i	$0.656054\pi$
$888$	0	0
$889$	62.9827	2.11237
$890$	0	0
$891$	9.13339	0.305980
$892$	0	0
$893$	3.53039	0.118140
$894$	0	0
$895$	−10.0344	−0.335415
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.5265	0.351080
$900$	0	0
$901$	2.42610	0.0808250
$902$	0	0
$903$	3.81924	0.127096
$904$	0	0
$905$	−24.6169	−0.818293
$906$	0	0
$907$	−40.2492	−1.33645	−0.668226	−	0.743958i	$-0.732946\pi$
$907$	−40.2492	−1.33645	−0.668226	+	0.743958i	$0.732946\pi$
$908$	0	0
$909$	−43.6381	−1.44738
$910$	0	0
$911$	1.23171	0.0408085	0.0204042	−	0.999792i	$-0.493505\pi$
$911$	1.23171	0.0408085	0.0204042	+	0.999792i	$0.493505\pi$
$912$	0	0
$913$	20.0568	0.663783
$914$	0	0
$915$	8.98507	0.297037
$916$	0	0
$917$	13.0467	0.430841
$918$	0	0
$919$	−14.0110	−0.462180	−0.231090	−	0.972932i	$-0.574229\pi$
$919$	−14.0110	−0.462180	−0.231090	+	0.972932i	$0.574229\pi$
$920$	0	0
$921$	7.65058	0.252095
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0.428134	0.0140770
$926$	0	0
$927$	1.89252	0.0621585
$928$	0	0
$929$	−24.7198	−0.811029	−0.405515	−	0.914089i	$-0.632908\pi$
$929$	−24.7198	−0.811029	−0.405515	+	0.914089i	$0.632908\pi$
$930$	0	0
$931$	−2.40750	−0.0789026
$932$	0	0
$933$	18.9419	0.620131
$934$	0	0
$935$	11.1103	0.363345
$936$	0	0
$937$	42.9404	1.40280	0.701401	−	0.712767i	$-0.252558\pi$
$937$	42.9404	1.40280	0.701401	+	0.712767i	$0.252558\pi$
$938$	0	0
$939$	−14.1061	−0.460335
$940$	0	0
$941$	39.1437	1.27605	0.638024	−	0.770016i	$-0.279752\pi$
$941$	39.1437	1.27605	0.638024	+	0.770016i	$0.279752\pi$
$942$	0	0
$943$	−5.53665	−0.180298
$944$	0	0
$945$	14.7828	0.480885
$946$	0	0
$947$	−17.0370	−0.553627	−0.276813	−	0.960924i	$-0.589278\pi$
$947$	−17.0370	−0.553627	−0.276813	+	0.960924i	$0.589278\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	7.69434	0.249506
$952$	0	0
$953$	−5.06939	−0.164214	−0.0821068	−	0.996624i	$-0.526165\pi$
$953$	−5.06939	−0.164214	−0.0821068	+	0.996624i	$0.526165\pi$
$954$	0	0
$955$	−2.01914	−0.0653378
$956$	0	0
$957$	1.48318	0.0479444
$958$	0	0
$959$	8.36928	0.270258
$960$	0	0
$961$	68.5493	2.21127
$962$	0	0
$963$	15.8870	0.511951
$964$	0	0
$965$	−8.31501	−0.267670
$966$	0	0
$967$	21.3805	0.687550	0.343775	−	0.939052i	$-0.388294\pi$
$967$	21.3805	0.687550	0.343775	+	0.939052i	$0.388294\pi$
$968$	0	0
$969$	−1.36134	−0.0437327
$970$	0	0
$971$	0.427629	0.0137233	0.00686164	−	0.999976i	$-0.497816\pi$
$971$	0.427629	0.0137233	0.00686164	+	0.999976i	$0.497816\pi$
$972$	0	0
$973$	−33.5045	−1.07411
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−6.58761	−0.210756	−0.105378	−	0.994432i	$-0.533605\pi$
$977$	−6.58761	−0.210756	−0.105378	+	0.994432i	$0.533605\pi$
$978$	0	0
$979$	18.5707	0.593522
$980$	0	0
$981$	−30.2540	−0.965934
$982$	0	0
$983$	−45.4129	−1.44844	−0.724222	−	0.689566i	$-0.757801\pi$
$983$	−45.4129	−1.44844	−0.724222	+	0.689566i	$0.757801\pi$
$984$	0	0
$985$	−8.49617	−0.270711
$986$	0	0
$987$	−28.2919	−0.900542
$988$	0	0
$989$	1.96424	0.0624592
$990$	0	0
$991$	45.1183	1.43323	0.716614	−	0.697470i	$-0.245691\pi$
$991$	45.1183	1.43323	0.716614	+	0.697470i	$0.245691\pi$
$992$	0	0
$993$	2.71095	0.0860295
$994$	0	0
$995$	19.0657	0.604425
$996$	0	0
$997$	24.3712	0.771844	0.385922	−	0.922531i	$-0.373883\pi$
$997$	24.3712	0.771844	0.385922	+	0.922531i	$0.373883\pi$
$998$	0	0
$999$	1.68181	0.0532101

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.a.r.1.4	✓	9
13.5	odd	4		3380.2.f.j.3041.8		18
13.8	odd	4		3380.2.f.j.3041.7		18
13.12	even	2		3380.2.a.s.1.4	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.2.a.r.1.4	✓	9	1.1	even	1	trivial
3380.2.a.s.1.4	yes	9	13.12	even	2
3380.2.f.j.3041.7		18	13.8	odd	4
3380.2.f.j.3041.8		18	13.5	odd	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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.a (trivial)

\(f(q)\)	\(=\)	\(q-0.715841 q^{3} -1.00000 q^{5} -3.76323 q^{7} -2.48757 q^{9} +O(q^{10})\)	\(q-0.715841 q^{3} -1.00000 q^{5} -3.76323 q^{7} -2.48757 q^{9} +1.96386 q^{11} +0.715841 q^{15} -5.65737 q^{17} -0.336153 q^{19} +2.69387 q^{21} +1.38546 q^{23} +1.00000 q^{25} +3.92823 q^{27} -1.05504 q^{29} -9.97744 q^{31} -1.40581 q^{33} +3.76323 q^{35} +0.428134 q^{37} -3.99625 q^{41} +1.41775 q^{43} +2.48757 q^{45} -10.5023 q^{47} +7.16191 q^{49} +4.04977 q^{51} -0.428839 q^{53} -1.96386 q^{55} +0.240632 q^{57} -8.47905 q^{59} +12.5518 q^{61} +9.36131 q^{63} +8.25232 q^{67} -0.991769 q^{69} +11.6464 q^{71} -10.8433 q^{73} -0.715841 q^{75} -7.39046 q^{77} +9.33649 q^{79} +4.65073 q^{81} +10.2129 q^{83} +5.65737 q^{85} +0.755237 q^{87} +9.45621 q^{89} +7.14225 q^{93} +0.336153 q^{95} -15.8156 q^{97} -4.88525 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9	\( 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9} + 7 q^{11} + q^{15} + 13 q^{17} + 4 q^{19} + 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} - 13 q^{31} - 34 q^{33} + q^{35} - q^{37} + 6 q^{41} + q^{43} - 12 q^{45} + 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} - 38 q^{57} - 15 q^{59} + 21 q^{61} + 17 q^{63} + 7 q^{67} + 15 q^{69} + 7 q^{71} + 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} - 45 q^{83} - 13 q^{85} + 28 q^{87} + 41 q^{89} + 11 q^{93} - 4 q^{95} - 8 q^{97} + 81 q^{99}+O(q^{100}) \) 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9 + 7 * q^11 + q^15 + 13 * q^17 + 4 * q^19 + 3 * q^21 + 12 * q^23 + 9 * q^25 - 4 * q^27 + 16 * q^29 - 13 * q^31 - 34 * q^33 + q^35 - q^37 + 6 * q^41 + q^43 - 12 * q^45 + 2 * q^47 + 20 * q^49 + 11 * q^51 + 30 * q^53 - 7 * q^55 - 38 * q^57 - 15 * q^59 + 21 * q^61 + 17 * q^63 + 7 * q^67 + 15 * q^69 + 7 * q^71 + 28 * q^73 - q^75 + 46 * q^77 + 31 * q^79 + 41 * q^81 - 45 * q^83 - 13 * q^85 + 28 * q^87 + 41 * q^89 + 11 * q^93 - 4 * q^95 - 8 * q^97 + 81 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more