LMFDB - Embedded newform 8016.2.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
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} - 3x^{8} - 16x^{7} + 45x^{6} + 67x^{5} - 166x^{4} - 83x^{3} + 152x^{2} + 51x - 10 $ x^9 - 3x^8 - 16x^7 + 45x^6 + 67x^5 - 166x^4 - 83x^3 + 152x^2 + 51x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.985991$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.98599 q^{5} +0.0909950 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.98599 q^{5} +0.0909950 q^{7} +1.00000 q^{9} +2.35607 q^{11} +0.834094 q^{13} +1.98599 q^{15} +7.08503 q^{17} +1.66738 q^{19} -0.0909950 q^{21} +2.49969 q^{23} -1.05584 q^{25} -1.00000 q^{27} -3.96700 q^{29} +3.52738 q^{31} -2.35607 q^{33} -0.180715 q^{35} -1.64278 q^{37} -0.834094 q^{39} +0.142888 q^{41} +5.36216 q^{43} -1.98599 q^{45} +6.35086 q^{47} -6.99172 q^{49} -7.08503 q^{51} +13.4619 q^{53} -4.67913 q^{55} -1.66738 q^{57} -4.39116 q^{59} +0.0766485 q^{61} +0.0909950 q^{63} -1.65650 q^{65} -4.51021 q^{67} -2.49969 q^{69} -3.17932 q^{71} +9.30247 q^{73} +1.05584 q^{75} +0.214390 q^{77} +17.3241 q^{79} +1.00000 q^{81} -0.0141153 q^{83} -14.0708 q^{85} +3.96700 q^{87} -7.96099 q^{89} +0.0758984 q^{91} -3.52738 q^{93} -3.31139 q^{95} -12.8801 q^{97} +2.35607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{3} - 6 q^{5} + 11 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q - 9 * q^3 - 6 * q^5 + 11 * q^7 + 9 * q^9	$ 9 q - 9 q^{3} - 6 q^{5} + 11 q^{7} + 9 q^{9} - q^{11} - 4 q^{13} + 6 q^{15} - 9 q^{17} + 8 q^{19} - 11 q^{21} + 7 q^{23} - q^{25} - 9 q^{27} - 9 q^{29} + 25 q^{31} + q^{33} - 5 q^{35} - 6 q^{37} + 4 q^{39} - 4 q^{41} + 24 q^{43} - 6 q^{45} + 16 q^{47} + 4 q^{49} + 9 q^{51} - 26 q^{53} + 29 q^{55} - 8 q^{57} + 4 q^{59} - 20 q^{61} + 11 q^{63} - 8 q^{65} + 25 q^{67} - 7 q^{69} + 15 q^{71} - 10 q^{73} + q^{75} - 20 q^{77} + 34 q^{79} + 9 q^{81} + 4 q^{83} - 13 q^{85} + 9 q^{87} + 13 q^{89} + 21 q^{91} - 25 q^{93} + 7 q^{95} - 4 q^{97} - q^{99}+O(q^{100}) $ 9 * q - 9 * q^3 - 6 * q^5 + 11 * q^7 + 9 * q^9 - q^11 - 4 * q^13 + 6 * q^15 - 9 * q^17 + 8 * q^19 - 11 * q^21 + 7 * q^23 - q^25 - 9 * q^27 - 9 * q^29 + 25 * q^31 + q^33 - 5 * q^35 - 6 * q^37 + 4 * q^39 - 4 * q^41 + 24 * q^43 - 6 * q^45 + 16 * q^47 + 4 * q^49 + 9 * q^51 - 26 * q^53 + 29 * q^55 - 8 * q^57 + 4 * q^59 - 20 * q^61 + 11 * q^63 - 8 * q^65 + 25 * q^67 - 7 * q^69 + 15 * q^71 - 10 * q^73 + q^75 - 20 * q^77 + 34 * q^79 + 9 * q^81 + 4 * q^83 - 13 * q^85 + 9 * q^87 + 13 * q^89 + 21 * q^91 - 25 * q^93 + 7 * q^95 - 4 * q^97 - 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.98599	−0.888162	−0.444081	−	0.895987i	$-0.646470\pi$
$5$	−1.98599	−0.888162	−0.444081	+	0.895987i	$0.646470\pi$
$6$	0	0
$7$	0.0909950	0.0343929	0.0171964	−	0.999852i	$-0.494526\pi$
$7$	0.0909950	0.0343929	0.0171964	+	0.999852i	$0.494526\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.35607	0.710380	0.355190	−	0.934794i	$-0.384416\pi$
$11$	2.35607	0.710380	0.355190	+	0.934794i	$0.384416\pi$
$12$	0	0
$13$	0.834094	0.231336	0.115668	−	0.993288i	$-0.463099\pi$
$13$	0.834094	0.231336	0.115668	+	0.993288i	$0.463099\pi$
$14$	0	0
$15$	1.98599	0.512781
$16$	0	0
$17$	7.08503	1.71837	0.859186	−	0.511663i	$-0.170970\pi$
$17$	7.08503	1.71837	0.859186	+	0.511663i	$0.170970\pi$
$18$	0	0
$19$	1.66738	0.382522	0.191261	−	0.981539i	$-0.438742\pi$
$19$	1.66738	0.382522	0.191261	+	0.981539i	$0.438742\pi$
$20$	0	0
$21$	−0.0909950	−0.0198567
$22$	0	0
$23$	2.49969	0.521221	0.260611	−	0.965444i	$-0.416076\pi$
$23$	2.49969	0.521221	0.260611	+	0.965444i	$0.416076\pi$
$24$	0	0
$25$	−1.05584	−0.211168
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.96700	−0.736652	−0.368326	−	0.929697i	$-0.620069\pi$
$29$	−3.96700	−0.736652	−0.368326	+	0.929697i	$0.620069\pi$
$30$	0	0
$31$	3.52738	0.633535	0.316768	−	0.948503i	$-0.397402\pi$
$31$	3.52738	0.633535	0.316768	+	0.948503i	$0.397402\pi$
$32$	0	0
$33$	−2.35607	−0.410138
$34$	0	0
$35$	−0.180715	−0.0305465
$36$	0	0
$37$	−1.64278	−0.270071	−0.135036	−	0.990841i	$-0.543115\pi$
$37$	−1.64278	−0.270071	−0.135036	+	0.990841i	$0.543115\pi$
$38$	0	0
$39$	−0.834094	−0.133562
$40$	0	0
$41$	0.142888	0.0223154	0.0111577	−	0.999938i	$-0.496448\pi$
$41$	0.142888	0.0223154	0.0111577	+	0.999938i	$0.496448\pi$
$42$	0	0
$43$	5.36216	0.817721	0.408861	−	0.912597i	$-0.365926\pi$
$43$	5.36216	0.817721	0.408861	+	0.912597i	$0.365926\pi$
$44$	0	0
$45$	−1.98599	−0.296054
$46$	0	0
$47$	6.35086	0.926369	0.463184	−	0.886262i	$-0.346707\pi$
$47$	6.35086	0.926369	0.463184	+	0.886262i	$0.346707\pi$
$48$	0	0
$49$	−6.99172	−0.998817
$50$	0	0
$51$	−7.08503	−0.992103
$52$	0	0
$53$	13.4619	1.84914	0.924569	−	0.381015i	$-0.124425\pi$
$53$	13.4619	1.84914	0.924569	+	0.381015i	$0.124425\pi$
$54$	0	0
$55$	−4.67913	−0.630933
$56$	0	0
$57$	−1.66738	−0.220849
$58$	0	0
$59$	−4.39116	−0.571680	−0.285840	−	0.958277i	$-0.592273\pi$
$59$	−4.39116	−0.571680	−0.285840	+	0.958277i	$0.592273\pi$
$60$	0	0
$61$	0.0766485	0.00981383	0.00490692	−	0.999988i	$-0.498438\pi$
$61$	0.0766485	0.00981383	0.00490692	+	0.999988i	$0.498438\pi$
$62$	0	0
$63$	0.0909950	0.0114643
$64$	0	0
$65$	−1.65650	−0.205464
$66$	0	0
$67$	−4.51021	−0.551010	−0.275505	−	0.961300i	$-0.588845\pi$
$67$	−4.51021	−0.551010	−0.275505	+	0.961300i	$0.588845\pi$
$68$	0	0
$69$	−2.49969	−0.300927
$70$	0	0
$71$	−3.17932	−0.377316	−0.188658	−	0.982043i	$-0.560414\pi$
$71$	−3.17932	−0.377316	−0.188658	+	0.982043i	$0.560414\pi$
$72$	0	0
$73$	9.30247	1.08877	0.544386	−	0.838835i	$-0.316763\pi$
$73$	9.30247	1.08877	0.544386	+	0.838835i	$0.316763\pi$
$74$	0	0
$75$	1.05584	0.121918
$76$	0	0
$77$	0.214390	0.0244320
$78$	0	0
$79$	17.3241	1.94912	0.974558	−	0.224137i	$-0.0719564\pi$
$79$	17.3241	1.94912	0.974558	+	0.224137i	$0.0719564\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.0141153	−0.00154936	−0.000774679	−	1.00000i	$-0.500247\pi$
$83$	−0.0141153	−0.00154936	−0.000774679		1.00000i	$0.500247\pi$
$84$	0	0
$85$	−14.0708	−1.52619
$86$	0	0
$87$	3.96700	0.425307
$88$	0	0
$89$	−7.96099	−0.843863	−0.421932	−	0.906628i	$-0.638648\pi$
$89$	−7.96099	−0.843863	−0.421932	+	0.906628i	$0.638648\pi$
$90$	0	0
$91$	0.0758984	0.00795631
$92$	0	0
$93$	−3.52738	−0.365772
$94$	0	0
$95$	−3.31139	−0.339742
$96$	0	0
$97$	−12.8801	−1.30778	−0.653888	−	0.756591i	$-0.726863\pi$
$97$	−12.8801	−1.30778	−0.653888	+	0.756591i	$0.726863\pi$
$98$	0	0
$99$	2.35607	0.236793
$100$	0	0
$101$	−14.6926	−1.46197	−0.730983	−	0.682396i	$-0.760938\pi$
$101$	−14.6926	−1.46197	−0.730983	+	0.682396i	$0.760938\pi$
$102$	0	0
$103$	4.05118	0.399174	0.199587	−	0.979880i	$-0.436040\pi$
$103$	4.05118	0.399174	0.199587	+	0.979880i	$0.436040\pi$
$104$	0	0
$105$	0.180715	0.0176360
$106$	0	0
$107$	−10.1492	−0.981161	−0.490581	−	0.871396i	$-0.663215\pi$
$107$	−10.1492	−0.981161	−0.490581	+	0.871396i	$0.663215\pi$
$108$	0	0
$109$	−4.51988	−0.432926	−0.216463	−	0.976291i	$-0.569452\pi$
$109$	−4.51988	−0.432926	−0.216463	+	0.976291i	$0.569452\pi$
$110$	0	0
$111$	1.64278	0.155926
$112$	0	0
$113$	−14.4542	−1.35974	−0.679869	−	0.733333i	$-0.737963\pi$
$113$	−14.4542	−1.35974	−0.679869	+	0.733333i	$0.737963\pi$
$114$	0	0
$115$	−4.96436	−0.462929
$116$	0	0
$117$	0.834094	0.0771120
$118$	0	0
$119$	0.644702	0.0590998
$120$	0	0
$121$	−5.44896	−0.495360
$122$	0	0
$123$	−0.142888	−0.0128838
$124$	0	0
$125$	12.0268	1.07571
$126$	0	0
$127$	18.7384	1.66277	0.831383	−	0.555700i	$-0.187550\pi$
$127$	18.7384	1.66277	0.831383	+	0.555700i	$0.187550\pi$
$128$	0	0
$129$	−5.36216	−0.472112
$130$	0	0
$131$	−6.50104	−0.567998	−0.283999	−	0.958825i	$-0.591661\pi$
$131$	−6.50104	−0.567998	−0.283999	+	0.958825i	$0.591661\pi$
$132$	0	0
$133$	0.151723	0.0131560
$134$	0	0
$135$	1.98599	0.170927
$136$	0	0
$137$	−4.41640	−0.377319	−0.188659	−	0.982043i	$-0.560414\pi$
$137$	−4.41640	−0.377319	−0.188659	+	0.982043i	$0.560414\pi$
$138$	0	0
$139$	−3.57252	−0.303017	−0.151508	−	0.988456i	$-0.548413\pi$
$139$	−3.57252	−0.303017	−0.151508	+	0.988456i	$0.548413\pi$
$140$	0	0
$141$	−6.35086	−0.534839
$142$	0	0
$143$	1.96518	0.164337
$144$	0	0
$145$	7.87842	0.654267
$146$	0	0
$147$	6.99172	0.576667
$148$	0	0
$149$	−3.73856	−0.306275	−0.153137	−	0.988205i	$-0.548938\pi$
$149$	−3.73856	−0.306275	−0.153137	+	0.988205i	$0.548938\pi$
$150$	0	0
$151$	9.88374	0.804327	0.402164	−	0.915568i	$-0.368258\pi$
$151$	9.88374	0.804327	0.402164	+	0.915568i	$0.368258\pi$
$152$	0	0
$153$	7.08503	0.572791
$154$	0	0
$155$	−7.00534	−0.562682
$156$	0	0
$157$	−0.704086	−0.0561922	−0.0280961	−	0.999605i	$-0.508944\pi$
$157$	−0.704086	−0.0561922	−0.0280961	+	0.999605i	$0.508944\pi$
$158$	0	0
$159$	−13.4619	−1.06760
$160$	0	0
$161$	0.227459	0.0179263
$162$	0	0
$163$	3.79586	0.297315	0.148658	−	0.988889i	$-0.452505\pi$
$163$	3.79586	0.297315	0.148658	+	0.988889i	$0.452505\pi$
$164$	0	0
$165$	4.67913	0.364269
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−12.3043	−0.946484
$170$	0	0
$171$	1.66738	0.127507
$172$	0	0
$173$	−5.35209	−0.406912	−0.203456	−	0.979084i	$-0.565217\pi$
$173$	−5.35209	−0.406912	−0.203456	+	0.979084i	$0.565217\pi$
$174$	0	0
$175$	−0.0960759	−0.00726266
$176$	0	0
$177$	4.39116	0.330060
$178$	0	0
$179$	2.68131	0.200410	0.100205	−	0.994967i	$-0.468050\pi$
$179$	2.68131	0.200410	0.100205	+	0.994967i	$0.468050\pi$
$180$	0	0
$181$	−21.8000	−1.62038	−0.810189	−	0.586169i	$-0.800635\pi$
$181$	−21.8000	−1.62038	−0.810189	+	0.586169i	$0.800635\pi$
$182$	0	0
$183$	−0.0766485	−0.00566602
$184$	0	0
$185$	3.26255	0.239867
$186$	0	0
$187$	16.6928	1.22070
$188$	0	0
$189$	−0.0909950	−0.00661891
$190$	0	0
$191$	26.2790	1.90148	0.950740	−	0.309989i	$-0.100325\pi$
$191$	26.2790	1.90148	0.950740	+	0.309989i	$0.100325\pi$
$192$	0	0
$193$	18.3316	1.31953	0.659767	−	0.751470i	$-0.270655\pi$
$193$	18.3316	1.31953	0.659767	+	0.751470i	$0.270655\pi$
$194$	0	0
$195$	1.65650	0.118625
$196$	0	0
$197$	14.1522	1.00830	0.504150	−	0.863616i	$-0.331806\pi$
$197$	14.1522	1.00830	0.504150	+	0.863616i	$0.331806\pi$
$198$	0	0
$199$	17.7862	1.26083	0.630414	−	0.776259i	$-0.282885\pi$
$199$	17.7862	1.26083	0.630414	+	0.776259i	$0.282885\pi$
$200$	0	0
$201$	4.51021	0.318126
$202$	0	0
$203$	−0.360977	−0.0253356
$204$	0	0
$205$	−0.283775	−0.0198197
$206$	0	0
$207$	2.49969	0.173740
$208$	0	0
$209$	3.92845	0.271736
$210$	0	0
$211$	−2.74876	−0.189232	−0.0946161	−	0.995514i	$-0.530162\pi$
$211$	−2.74876	−0.189232	−0.0946161	+	0.995514i	$0.530162\pi$
$212$	0	0
$213$	3.17932	0.217843
$214$	0	0
$215$	−10.6492	−0.726269
$216$	0	0
$217$	0.320973	0.0217891
$218$	0	0
$219$	−9.30247	−0.628602
$220$	0	0
$221$	5.90958	0.397522
$222$	0	0
$223$	5.97542	0.400143	0.200072	−	0.979781i	$-0.435882\pi$
$223$	5.97542	0.400143	0.200072	+	0.979781i	$0.435882\pi$
$224$	0	0
$225$	−1.05584	−0.0703892
$226$	0	0
$227$	−0.438914	−0.0291318	−0.0145659	−	0.999894i	$-0.504637\pi$
$227$	−0.438914	−0.0291318	−0.0145659	+	0.999894i	$0.504637\pi$
$228$	0	0
$229$	1.49354	0.0986960	0.0493480	−	0.998782i	$-0.484286\pi$
$229$	1.49354	0.0986960	0.0493480	+	0.998782i	$0.484286\pi$
$230$	0	0
$231$	−0.214390	−0.0141058
$232$	0	0
$233$	−17.5018	−1.14658	−0.573291	−	0.819352i	$-0.694333\pi$
$233$	−17.5018	−1.14658	−0.573291	+	0.819352i	$0.694333\pi$
$234$	0	0
$235$	−12.6128	−0.822766
$236$	0	0
$237$	−17.3241	−1.12532
$238$	0	0
$239$	3.14251	0.203272	0.101636	−	0.994822i	$-0.467592\pi$
$239$	3.14251	0.203272	0.101636	+	0.994822i	$0.467592\pi$
$240$	0	0
$241$	18.6616	1.20210	0.601050	−	0.799211i	$-0.294749\pi$
$241$	18.6616	1.20210	0.601050	+	0.799211i	$0.294749\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	13.8855	0.887112
$246$	0	0
$247$	1.39075	0.0884912
$248$	0	0
$249$	0.0141153	0.000894522 0
$250$	0	0
$251$	−3.34821	−0.211337	−0.105668	−	0.994401i	$-0.533698\pi$
$251$	−3.34821	−0.211337	−0.105668	+	0.994401i	$0.533698\pi$
$252$	0	0
$253$	5.88943	0.370265
$254$	0	0
$255$	14.0708	0.881148
$256$	0	0
$257$	7.22045	0.450399	0.225200	−	0.974313i	$-0.427697\pi$
$257$	7.22045	0.450399	0.225200	+	0.974313i	$0.427697\pi$
$258$	0	0
$259$	−0.149485	−0.00928853
$260$	0	0
$261$	−3.96700	−0.245551
$262$	0	0
$263$	−7.01628	−0.432642	−0.216321	−	0.976322i	$-0.569406\pi$
$263$	−7.01628	−0.432642	−0.216321	+	0.976322i	$0.569406\pi$
$264$	0	0
$265$	−26.7353	−1.64233
$266$	0	0
$267$	7.96099	0.487205
$268$	0	0
$269$	17.2316	1.05063	0.525316	−	0.850907i	$-0.323947\pi$
$269$	17.2316	1.05063	0.525316	+	0.850907i	$0.323947\pi$
$270$	0	0
$271$	27.6572	1.68005	0.840027	−	0.542545i	$-0.182539\pi$
$271$	27.6572	1.68005	0.840027	+	0.542545i	$0.182539\pi$
$272$	0	0
$273$	−0.0758984	−0.00459358
$274$	0	0
$275$	−2.48762	−0.150009
$276$	0	0
$277$	24.7248	1.48557	0.742785	−	0.669529i	$-0.233504\pi$
$277$	24.7248	1.48557	0.742785	+	0.669529i	$0.233504\pi$
$278$	0	0
$279$	3.52738	0.211178
$280$	0	0
$281$	−1.24892	−0.0745041	−0.0372520	−	0.999306i	$-0.511860\pi$
$281$	−1.24892	−0.0745041	−0.0372520	+	0.999306i	$0.511860\pi$
$282$	0	0
$283$	26.7744	1.59158	0.795788	−	0.605576i	$-0.207057\pi$
$283$	26.7744	1.59158	0.795788	+	0.605576i	$0.207057\pi$
$284$	0	0
$285$	3.31139	0.196150
$286$	0	0
$287$	0.0130021	0.000767491 0
$288$	0	0
$289$	33.1977	1.95280
$290$	0	0
$291$	12.8801	0.755045
$292$	0	0
$293$	25.2333	1.47414	0.737072	−	0.675814i	$-0.236208\pi$
$293$	25.2333	1.47414	0.737072	+	0.675814i	$0.236208\pi$
$294$	0	0
$295$	8.72080	0.507745
$296$	0	0
$297$	−2.35607	−0.136713
$298$	0	0
$299$	2.08498	0.120577
$300$	0	0
$301$	0.487929	0.0281238
$302$	0	0
$303$	14.6926	0.844066
$304$	0	0
$305$	−0.152223	−0.00871628
$306$	0	0
$307$	−3.20268	−0.182786	−0.0913932	−	0.995815i	$-0.529132\pi$
$307$	−3.20268	−0.182786	−0.0913932	+	0.995815i	$0.529132\pi$
$308$	0	0
$309$	−4.05118	−0.230463
$310$	0	0
$311$	3.45554	0.195946	0.0979729	−	0.995189i	$-0.468764\pi$
$311$	3.45554	0.195946	0.0979729	+	0.995189i	$0.468764\pi$
$312$	0	0
$313$	31.7617	1.79528	0.897638	−	0.440734i	$-0.145282\pi$
$313$	31.7617	1.79528	0.897638	+	0.440734i	$0.145282\pi$
$314$	0	0
$315$	−0.180715	−0.0101822
$316$	0	0
$317$	−25.1696	−1.41367	−0.706834	−	0.707380i	$-0.749877\pi$
$317$	−25.1696	−1.41367	−0.706834	+	0.707380i	$0.749877\pi$
$318$	0	0
$319$	−9.34650	−0.523304
$320$	0	0
$321$	10.1492	0.566474
$322$	0	0
$323$	11.8134	0.657316
$324$	0	0
$325$	−0.880668	−0.0488507
$326$	0	0
$327$	4.51988	0.249950
$328$	0	0
$329$	0.577897	0.0318605
$330$	0	0
$331$	8.94004	0.491389	0.245694	−	0.969347i	$-0.420984\pi$
$331$	8.94004	0.491389	0.245694	+	0.969347i	$0.420984\pi$
$332$	0	0
$333$	−1.64278	−0.0900238
$334$	0	0
$335$	8.95724	0.489387
$336$	0	0
$337$	−2.25049	−0.122592	−0.0612959	−	0.998120i	$-0.519523\pi$
$337$	−2.25049	−0.122592	−0.0612959	+	0.998120i	$0.519523\pi$
$338$	0	0
$339$	14.4542	0.785045
$340$	0	0
$341$	8.31073	0.450051
$342$	0	0
$343$	−1.27318	−0.0687451
$344$	0	0
$345$	4.96436	0.267272
$346$	0	0
$347$	−4.88275	−0.262119	−0.131060	−	0.991374i	$-0.541838\pi$
$347$	−4.88275	−0.262119	−0.131060	+	0.991374i	$0.541838\pi$
$348$	0	0
$349$	6.94280	0.371639	0.185820	−	0.982584i	$-0.440506\pi$
$349$	6.94280	0.371639	0.185820	+	0.982584i	$0.440506\pi$
$350$	0	0
$351$	−0.834094	−0.0445207
$352$	0	0
$353$	−12.9864	−0.691199	−0.345599	−	0.938382i	$-0.612324\pi$
$353$	−12.9864	−0.691199	−0.345599	+	0.938382i	$0.612324\pi$
$354$	0	0
$355$	6.31410	0.335118
$356$	0	0
$357$	−0.644702	−0.0341213
$358$	0	0
$359$	2.00276	0.105702	0.0528509	−	0.998602i	$-0.483169\pi$
$359$	2.00276	0.105702	0.0528509	+	0.998602i	$0.483169\pi$
$360$	0	0
$361$	−16.2199	−0.853677
$362$	0	0
$363$	5.44896	0.285996
$364$	0	0
$365$	−18.4746	−0.967006
$366$	0	0
$367$	−28.1645	−1.47017	−0.735087	−	0.677973i	$-0.762859\pi$
$367$	−28.1645	−1.47017	−0.735087	+	0.677973i	$0.762859\pi$
$368$	0	0
$369$	0.142888	0.00743848
$370$	0	0
$371$	1.22497	0.0635971
$372$	0	0
$373$	−0.470865	−0.0243804	−0.0121902	−	0.999926i	$-0.503880\pi$
$373$	−0.470865	−0.0243804	−0.0121902	+	0.999926i	$0.503880\pi$
$374$	0	0
$375$	−12.0268	−0.621063
$376$	0	0
$377$	−3.30885	−0.170414
$378$	0	0
$379$	16.1101	0.827519	0.413760	−	0.910386i	$-0.364215\pi$
$379$	16.1101	0.827519	0.413760	+	0.910386i	$0.364215\pi$
$380$	0	0
$381$	−18.7384	−0.959999
$382$	0	0
$383$	−37.3788	−1.90997	−0.954984	−	0.296656i	$-0.904128\pi$
$383$	−37.3788	−1.90997	−0.954984	+	0.296656i	$0.904128\pi$
$384$	0	0
$385$	−0.425777	−0.0216996
$386$	0	0
$387$	5.36216	0.272574
$388$	0	0
$389$	8.93497	0.453021	0.226511	−	0.974009i	$-0.427268\pi$
$389$	8.93497	0.453021	0.226511	+	0.974009i	$0.427268\pi$
$390$	0	0
$391$	17.7104	0.895652
$392$	0	0
$393$	6.50104	0.327934
$394$	0	0
$395$	−34.4055	−1.73113
$396$	0	0
$397$	0.164468	0.00825442	0.00412721	−	0.999991i	$-0.498686\pi$
$397$	0.164468	0.00825442	0.00412721	+	0.999991i	$0.498686\pi$
$398$	0	0
$399$	−0.151723	−0.00759564
$400$	0	0
$401$	5.07799	0.253583	0.126791	−	0.991929i	$-0.459532\pi$
$401$	5.07799	0.253583	0.126791	+	0.991929i	$0.459532\pi$
$402$	0	0
$403$	2.94216	0.146560
$404$	0	0
$405$	−1.98599	−0.0986847
$406$	0	0
$407$	−3.87050	−0.191853
$408$	0	0
$409$	31.1261	1.53909	0.769544	−	0.638594i	$-0.220484\pi$
$409$	31.1261	1.53909	0.769544	+	0.638594i	$0.220484\pi$
$410$	0	0
$411$	4.41640	0.217845
$412$	0	0
$413$	−0.399573	−0.0196617
$414$	0	0
$415$	0.0280329	0.00137608
$416$	0	0
$417$	3.57252	0.174947
$418$	0	0
$419$	23.9570	1.17037	0.585187	−	0.810898i	$-0.301021\pi$
$419$	23.9570	1.17037	0.585187	+	0.810898i	$0.301021\pi$
$420$	0	0
$421$	19.3408	0.942612	0.471306	−	0.881970i	$-0.343783\pi$
$421$	19.3408	0.942612	0.471306	+	0.881970i	$0.343783\pi$
$422$	0	0
$423$	6.35086	0.308790
$424$	0	0
$425$	−7.48064	−0.362865
$426$	0	0
$427$	0.00697463	0.000337526 0
$428$	0	0
$429$	−1.96518	−0.0948798
$430$	0	0
$431$	7.52144	0.362295	0.181147	−	0.983456i	$-0.442019\pi$
$431$	7.52144	0.362295	0.181147	+	0.983456i	$0.442019\pi$
$432$	0	0
$433$	35.3240	1.69756	0.848781	−	0.528745i	$-0.177337\pi$
$433$	35.3240	1.69756	0.848781	+	0.528745i	$0.177337\pi$
$434$	0	0
$435$	−7.87842	−0.377741
$436$	0	0
$437$	4.16792	0.199379
$438$	0	0
$439$	3.59035	0.171358	0.0856790	−	0.996323i	$-0.472694\pi$
$439$	3.59035	0.171358	0.0856790	+	0.996323i	$0.472694\pi$
$440$	0	0
$441$	−6.99172	−0.332939
$442$	0	0
$443$	−11.2209	−0.533121	−0.266560	−	0.963818i	$-0.585887\pi$
$443$	−11.2209	−0.533121	−0.266560	+	0.963818i	$0.585887\pi$
$444$	0	0
$445$	15.8105	0.749488
$446$	0	0
$447$	3.73856	0.176828
$448$	0	0
$449$	−14.4857	−0.683622	−0.341811	−	0.939769i	$-0.611040\pi$
$449$	−14.4857	−0.683622	−0.341811	+	0.939769i	$0.611040\pi$
$450$	0	0
$451$	0.336655	0.0158524
$452$	0	0
$453$	−9.88374	−0.464379
$454$	0	0
$455$	−0.150734	−0.00706650
$456$	0	0
$457$	18.0839	0.845929	0.422965	−	0.906146i	$-0.360989\pi$
$457$	18.0839	0.845929	0.422965	+	0.906146i	$0.360989\pi$
$458$	0	0
$459$	−7.08503	−0.330701
$460$	0	0
$461$	−11.4205	−0.531904	−0.265952	−	0.963986i	$-0.585686\pi$
$461$	−11.4205	−0.531904	−0.265952	+	0.963986i	$0.585686\pi$
$462$	0	0
$463$	−11.4961	−0.534268	−0.267134	−	0.963659i	$-0.586077\pi$
$463$	−11.4961	−0.534268	−0.267134	+	0.963659i	$0.586077\pi$
$464$	0	0
$465$	7.00534	0.324865
$466$	0	0
$467$	−28.3176	−1.31038	−0.655191	−	0.755463i	$-0.727412\pi$
$467$	−28.3176	−1.31038	−0.655191	+	0.755463i	$0.727412\pi$
$468$	0	0
$469$	−0.410407	−0.0189508
$470$	0	0
$471$	0.704086	0.0324426
$472$	0	0
$473$	12.6336	0.580893
$474$	0	0
$475$	−1.76048	−0.0807763
$476$	0	0
$477$	13.4619	0.616379
$478$	0	0
$479$	−0.851972	−0.0389276	−0.0194638	−	0.999811i	$-0.506196\pi$
$479$	−0.851972	−0.0389276	−0.0194638	+	0.999811i	$0.506196\pi$
$480$	0	0
$481$	−1.37023	−0.0624772
$482$	0	0
$483$	−0.227459	−0.0103498
$484$	0	0
$485$	25.5798	1.16152
$486$	0	0
$487$	16.8841	0.765091	0.382546	−	0.923937i	$-0.375047\pi$
$487$	16.8841	0.765091	0.382546	+	0.923937i	$0.375047\pi$
$488$	0	0
$489$	−3.79586	−0.171655
$490$	0	0
$491$	−40.5909	−1.83184	−0.915921	−	0.401359i	$-0.868538\pi$
$491$	−40.5909	−1.83184	−0.915921	+	0.401359i	$0.868538\pi$
$492$	0	0
$493$	−28.1063	−1.26584
$494$	0	0
$495$	−4.67913	−0.210311
$496$	0	0
$497$	−0.289302	−0.0129770
$498$	0	0
$499$	−1.31828	−0.0590143	−0.0295071	−	0.999565i	$-0.509394\pi$
$499$	−1.31828	−0.0590143	−0.0295071	+	0.999565i	$0.509394\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	20.3191	0.905983	0.452991	−	0.891515i	$-0.350357\pi$
$503$	20.3191	0.905983	0.452991	+	0.891515i	$0.350357\pi$
$504$	0	0
$505$	29.1793	1.29846
$506$	0	0
$507$	12.3043	0.546453
$508$	0	0
$509$	39.7984	1.76403	0.882016	−	0.471219i	$-0.156186\pi$
$509$	39.7984	1.76403	0.882016	+	0.471219i	$0.156186\pi$
$510$	0	0
$511$	0.846478	0.0374460
$512$	0	0
$513$	−1.66738	−0.0736164
$514$	0	0
$515$	−8.04560	−0.354532
$516$	0	0
$517$	14.9631	0.658074
$518$	0	0
$519$	5.35209	0.234931
$520$	0	0
$521$	8.30626	0.363904	0.181952	−	0.983307i	$-0.441758\pi$
$521$	8.30626	0.363904	0.181952	+	0.983307i	$0.441758\pi$
$522$	0	0
$523$	26.3330	1.15146	0.575731	−	0.817639i	$-0.304717\pi$
$523$	26.3330	1.15146	0.575731	+	0.817639i	$0.304717\pi$
$524$	0	0
$525$	0.0960759	0.00419310
$526$	0	0
$527$	24.9916	1.08865
$528$	0	0
$529$	−16.7516	−0.728328
$530$	0	0
$531$	−4.39116	−0.190560
$532$	0	0
$533$	0.119182	0.00516236
$534$	0	0
$535$	20.1562	0.871431
$536$	0	0
$537$	−2.68131	−0.115707
$538$	0	0
$539$	−16.4729	−0.709540
$540$	0	0
$541$	−14.1502	−0.608365	−0.304182	−	0.952614i	$-0.598383\pi$
$541$	−14.1502	−0.608365	−0.304182	+	0.952614i	$0.598383\pi$
$542$	0	0
$543$	21.8000	0.935525
$544$	0	0
$545$	8.97645	0.384509
$546$	0	0
$547$	24.6028	1.05194	0.525971	−	0.850503i	$-0.323702\pi$
$547$	24.6028	1.05194	0.525971	+	0.850503i	$0.323702\pi$
$548$	0	0
$549$	0.0766485	0.00327128
$550$	0	0
$551$	−6.61447	−0.281786
$552$	0	0
$553$	1.57641	0.0670357
$554$	0	0
$555$	−3.26255	−0.138487
$556$	0	0
$557$	−23.1405	−0.980493	−0.490246	−	0.871584i	$-0.663093\pi$
$557$	−23.1405	−0.980493	−0.490246	+	0.871584i	$0.663093\pi$
$558$	0	0
$559$	4.47254	0.189168
$560$	0	0
$561$	−16.6928	−0.704770
$562$	0	0
$563$	−18.8347	−0.793787	−0.396894	−	0.917865i	$-0.629912\pi$
$563$	−18.8347	−0.793787	−0.396894	+	0.917865i	$0.629912\pi$
$564$	0	0
$565$	28.7060	1.20767
$566$	0	0
$567$	0.0909950	0.00382143
$568$	0	0
$569$	38.2495	1.60350	0.801752	−	0.597657i	$-0.203901\pi$
$569$	38.2495	1.60350	0.801752	+	0.597657i	$0.203901\pi$
$570$	0	0
$571$	−28.5374	−1.19425	−0.597126	−	0.802148i	$-0.703691\pi$
$571$	−28.5374	−1.19425	−0.597126	+	0.802148i	$0.703691\pi$
$572$	0	0
$573$	−26.2790	−1.09782
$574$	0	0
$575$	−2.63927	−0.110065
$576$	0	0
$577$	−11.7985	−0.491178	−0.245589	−	0.969374i	$-0.578981\pi$
$577$	−11.7985	−0.491178	−0.245589	+	0.969374i	$0.578981\pi$
$578$	0	0
$579$	−18.3316	−0.761833
$580$	0	0
$581$	−0.00128442	−5.32868e−5 0
$582$	0	0
$583$	31.7172	1.31359
$584$	0	0
$585$	−1.65650	−0.0684880
$586$	0	0
$587$	1.16607	0.0481287	0.0240644	−	0.999710i	$-0.492339\pi$
$587$	1.16607	0.0481287	0.0240644	+	0.999710i	$0.492339\pi$
$588$	0	0
$589$	5.88146	0.242341
$590$	0	0
$591$	−14.1522	−0.582142
$592$	0	0
$593$	20.2574	0.831870	0.415935	−	0.909394i	$-0.363454\pi$
$593$	20.2574	0.831870	0.415935	+	0.909394i	$0.363454\pi$
$594$	0	0
$595$	−1.28037	−0.0524902
$596$	0	0
$597$	−17.7862	−0.727940
$598$	0	0
$599$	35.8580	1.46512	0.732559	−	0.680703i	$-0.238326\pi$
$599$	35.8580	1.46512	0.732559	+	0.680703i	$0.238326\pi$
$600$	0	0
$601$	−29.0525	−1.18507	−0.592537	−	0.805543i	$-0.701874\pi$
$601$	−29.0525	−1.18507	−0.592537	+	0.805543i	$0.701874\pi$
$602$	0	0
$603$	−4.51021	−0.183670
$604$	0	0
$605$	10.8216	0.439960
$606$	0	0
$607$	18.0950	0.734455	0.367227	−	0.930131i	$-0.380307\pi$
$607$	18.0950	0.734455	0.367227	+	0.930131i	$0.380307\pi$
$608$	0	0
$609$	0.360977	0.0146275
$610$	0	0
$611$	5.29722	0.214303
$612$	0	0
$613$	36.0458	1.45588	0.727939	−	0.685642i	$-0.240478\pi$
$613$	36.0458	1.45588	0.727939	+	0.685642i	$0.240478\pi$
$614$	0	0
$615$	0.283775	0.0114429
$616$	0	0
$617$	24.3584	0.980632	0.490316	−	0.871545i	$-0.336881\pi$
$617$	24.3584	0.980632	0.490316	+	0.871545i	$0.336881\pi$
$618$	0	0
$619$	26.2196	1.05386	0.526928	−	0.849910i	$-0.323344\pi$
$619$	26.2196	1.05386	0.526928	+	0.849910i	$0.323344\pi$
$620$	0	0
$621$	−2.49969	−0.100309
$622$	0	0
$623$	−0.724410	−0.0290229
$624$	0	0
$625$	−18.6060	−0.744241
$626$	0	0
$627$	−3.92845	−0.156887
$628$	0	0
$629$	−11.6391	−0.464083
$630$	0	0
$631$	−39.0723	−1.55544	−0.777722	−	0.628609i	$-0.783625\pi$
$631$	−39.0723	−1.55544	−0.777722	+	0.628609i	$0.783625\pi$
$632$	0	0
$633$	2.74876	0.109253
$634$	0	0
$635$	−37.2144	−1.47681
$636$	0	0
$637$	−5.83175	−0.231062
$638$	0	0
$639$	−3.17932	−0.125772
$640$	0	0
$641$	13.3708	0.528116	0.264058	−	0.964507i	$-0.414939\pi$
$641$	13.3708	0.528116	0.264058	+	0.964507i	$0.414939\pi$
$642$	0	0
$643$	−28.3204	−1.11685	−0.558424	−	0.829555i	$-0.688594\pi$
$643$	−28.3204	−1.11685	−0.558424	+	0.829555i	$0.688594\pi$
$644$	0	0
$645$	10.6492	0.419312
$646$	0	0
$647$	−13.7492	−0.540539	−0.270269	−	0.962785i	$-0.587113\pi$
$647$	−13.7492	−0.540539	−0.270269	+	0.962785i	$0.587113\pi$
$648$	0	0
$649$	−10.3459	−0.406110
$650$	0	0
$651$	−0.320973	−0.0125799
$652$	0	0
$653$	−23.5266	−0.920667	−0.460334	−	0.887746i	$-0.652270\pi$
$653$	−23.5266	−0.920667	−0.460334	+	0.887746i	$0.652270\pi$
$654$	0	0
$655$	12.9110	0.504475
$656$	0	0
$657$	9.30247	0.362924
$658$	0	0
$659$	37.7144	1.46914	0.734571	−	0.678531i	$-0.237383\pi$
$659$	37.7144	1.46914	0.734571	+	0.678531i	$0.237383\pi$
$660$	0	0
$661$	−22.3286	−0.868483	−0.434241	−	0.900797i	$-0.642984\pi$
$661$	−22.3286	−0.868483	−0.434241	+	0.900797i	$0.642984\pi$
$662$	0	0
$663$	−5.90958	−0.229509
$664$	0	0
$665$	−0.301320	−0.0116847
$666$	0	0
$667$	−9.91626	−0.383959
$668$	0	0
$669$	−5.97542	−0.231023
$670$	0	0
$671$	0.180589	0.00697155
$672$	0	0
$673$	−1.89185	−0.0729256	−0.0364628	−	0.999335i	$-0.511609\pi$
$673$	−1.89185	−0.0729256	−0.0364628	+	0.999335i	$0.511609\pi$
$674$	0	0
$675$	1.05584	0.0406392
$676$	0	0
$677$	31.6117	1.21494	0.607468	−	0.794344i	$-0.292185\pi$
$677$	31.6117	1.21494	0.607468	+	0.794344i	$0.292185\pi$
$678$	0	0
$679$	−1.17202	−0.0449782
$680$	0	0
$681$	0.438914	0.0168192
$682$	0	0
$683$	−19.1816	−0.733962	−0.366981	−	0.930228i	$-0.619609\pi$
$683$	−19.1816	−0.733962	−0.366981	+	0.930228i	$0.619609\pi$
$684$	0	0
$685$	8.77094	0.335120
$686$	0	0
$687$	−1.49354	−0.0569822
$688$	0	0
$689$	11.2285	0.427772
$690$	0	0
$691$	−19.4533	−0.740037	−0.370019	−	0.929024i	$-0.620649\pi$
$691$	−19.4533	−0.740037	−0.370019	+	0.929024i	$0.620649\pi$
$692$	0	0
$693$	0.214390	0.00814401
$694$	0	0
$695$	7.09499	0.269128
$696$	0	0
$697$	1.01237	0.0383462
$698$	0	0
$699$	17.5018	0.661979
$700$	0	0
$701$	−2.81290	−0.106242	−0.0531208	−	0.998588i	$-0.516917\pi$
$701$	−2.81290	−0.106242	−0.0531208	+	0.998588i	$0.516917\pi$
$702$	0	0
$703$	−2.73913	−0.103308
$704$	0	0
$705$	12.6128	0.475024
$706$	0	0
$707$	−1.33695	−0.0502812
$708$	0	0
$709$	−33.0623	−1.24168	−0.620841	−	0.783936i	$-0.713209\pi$
$709$	−33.0623	−1.24168	−0.620841	+	0.783936i	$0.713209\pi$
$710$	0	0
$711$	17.3241	0.649705
$712$	0	0
$713$	8.81734	0.330212
$714$	0	0
$715$	−3.90283	−0.145958
$716$	0	0
$717$	−3.14251	−0.117359
$718$	0	0
$719$	46.3189	1.72740	0.863701	−	0.504004i	$-0.168140\pi$
$719$	46.3189	1.72740	0.863701	+	0.504004i	$0.168140\pi$
$720$	0	0
$721$	0.368637	0.0137287
$722$	0	0
$723$	−18.6616	−0.694033
$724$	0	0
$725$	4.18850	0.155557
$726$	0	0
$727$	−22.5142	−0.835004	−0.417502	−	0.908676i	$-0.637094\pi$
$727$	−22.5142	−0.835004	−0.417502	+	0.908676i	$0.637094\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	37.9910	1.40515
$732$	0	0
$733$	−22.7476	−0.840202	−0.420101	−	0.907477i	$-0.638005\pi$
$733$	−22.7476	−0.840202	−0.420101	+	0.907477i	$0.638005\pi$
$734$	0	0
$735$	−13.8855	−0.512174
$736$	0	0
$737$	−10.6264	−0.391427
$738$	0	0
$739$	50.1147	1.84350	0.921750	−	0.387785i	$-0.126760\pi$
$739$	50.1147	1.84350	0.921750	+	0.387785i	$0.126760\pi$
$740$	0	0
$741$	−1.39075	−0.0510904
$742$	0	0
$743$	29.3798	1.07784	0.538920	−	0.842357i	$-0.318833\pi$
$743$	29.3798	1.07784	0.538920	+	0.842357i	$0.318833\pi$
$744$	0	0
$745$	7.42474	0.272022
$746$	0	0
$747$	−0.0141153	−0.000516452 0
$748$	0	0
$749$	−0.923527	−0.0337449
$750$	0	0
$751$	45.8288	1.67232	0.836159	−	0.548487i	$-0.184796\pi$
$751$	45.8288	1.67232	0.836159	+	0.548487i	$0.184796\pi$
$752$	0	0
$753$	3.34821	0.122015
$754$	0	0
$755$	−19.6290	−0.714373
$756$	0	0
$757$	−1.36638	−0.0496621	−0.0248310	−	0.999692i	$-0.507905\pi$
$757$	−1.36638	−0.0496621	−0.0248310	+	0.999692i	$0.507905\pi$
$758$	0	0
$759$	−5.88943	−0.213773
$760$	0	0
$761$	41.1549	1.49187	0.745933	−	0.666021i	$-0.232004\pi$
$761$	41.1549	1.49187	0.745933	+	0.666021i	$0.232004\pi$
$762$	0	0
$763$	−0.411287	−0.0148896
$764$	0	0
$765$	−14.0708	−0.508731
$766$	0	0
$767$	−3.66264	−0.132250
$768$	0	0
$769$	−6.65538	−0.239999	−0.120000	−	0.992774i	$-0.538289\pi$
$769$	−6.65538	−0.239999	−0.120000	+	0.992774i	$0.538289\pi$
$770$	0	0
$771$	−7.22045	−0.260038
$772$	0	0
$773$	−30.8303	−1.10889	−0.554444	−	0.832221i	$-0.687069\pi$
$773$	−30.8303	−1.10889	−0.554444	+	0.832221i	$0.687069\pi$
$774$	0	0
$775$	−3.72434	−0.133782
$776$	0	0
$777$	0.149485	0.00536273
$778$	0	0
$779$	0.238249	0.00853615
$780$	0	0
$781$	−7.49068	−0.268038
$782$	0	0
$783$	3.96700	0.141769
$784$	0	0
$785$	1.39831	0.0499078
$786$	0	0
$787$	−9.92617	−0.353830	−0.176915	−	0.984226i	$-0.556612\pi$
$787$	−9.92617	−0.353830	−0.176915	+	0.984226i	$0.556612\pi$
$788$	0	0
$789$	7.01628	0.249786
$790$	0	0
$791$	−1.31526	−0.0467653
$792$	0	0
$793$	0.0639321	0.00227029
$794$	0	0
$795$	26.7353	0.948202
$796$	0	0
$797$	−20.4507	−0.724402	−0.362201	−	0.932100i	$-0.617975\pi$
$797$	−20.4507	−0.724402	−0.362201	+	0.932100i	$0.617975\pi$
$798$	0	0
$799$	44.9961	1.59185
$800$	0	0
$801$	−7.96099	−0.281288
$802$	0	0
$803$	21.9172	0.773442
$804$	0	0
$805$	−0.451732	−0.0159215
$806$	0	0
$807$	−17.2316	−0.606582
$808$	0	0
$809$	−11.0254	−0.387631	−0.193815	−	0.981038i	$-0.562086\pi$
$809$	−11.0254	−0.387631	−0.193815	+	0.981038i	$0.562086\pi$
$810$	0	0
$811$	−39.7185	−1.39471	−0.697353	−	0.716728i	$-0.745639\pi$
$811$	−39.7185	−1.39471	−0.697353	+	0.716728i	$0.745639\pi$
$812$	0	0
$813$	−27.6572	−0.969979
$814$	0	0
$815$	−7.53855	−0.264064
$816$	0	0
$817$	8.94073	0.312797
$818$	0	0
$819$	0.0758984	0.00265210
$820$	0	0
$821$	4.78084	0.166853	0.0834263	−	0.996514i	$-0.473414\pi$
$821$	4.78084	0.166853	0.0834263	+	0.996514i	$0.473414\pi$
$822$	0	0
$823$	−11.8312	−0.412408	−0.206204	−	0.978509i	$-0.566111\pi$
$823$	−11.8312	−0.412408	−0.206204	+	0.978509i	$0.566111\pi$
$824$	0	0
$825$	2.48762	0.0866079
$826$	0	0
$827$	−25.7394	−0.895045	−0.447523	−	0.894273i	$-0.647694\pi$
$827$	−25.7394	−0.895045	−0.447523	+	0.894273i	$0.647694\pi$
$828$	0	0
$829$	6.37091	0.221271	0.110636	−	0.993861i	$-0.464711\pi$
$829$	6.37091	0.221271	0.110636	+	0.993861i	$0.464711\pi$
$830$	0	0
$831$	−24.7248	−0.857695
$832$	0	0
$833$	−49.5366	−1.71634
$834$	0	0
$835$	−1.98599	−0.0687281
$836$	0	0
$837$	−3.52738	−0.121924
$838$	0	0
$839$	−35.3021	−1.21876	−0.609381	−	0.792877i	$-0.708582\pi$
$839$	−35.3021	−1.21876	−0.609381	+	0.792877i	$0.708582\pi$
$840$	0	0
$841$	−13.2630	−0.457343
$842$	0	0
$843$	1.24892	0.0430150
$844$	0	0
$845$	24.4362	0.840631
$846$	0	0
$847$	−0.495828	−0.0170368
$848$	0	0
$849$	−26.7744	−0.918897
$850$	0	0
$851$	−4.10644	−0.140767
$852$	0	0
$853$	6.70883	0.229706	0.114853	−	0.993383i	$-0.463360\pi$
$853$	6.70883	0.229706	0.114853	+	0.993383i	$0.463360\pi$
$854$	0	0
$855$	−3.31139	−0.113247
$856$	0	0
$857$	25.0043	0.854130	0.427065	−	0.904221i	$-0.359548\pi$
$857$	25.0043	0.854130	0.427065	+	0.904221i	$0.359548\pi$
$858$	0	0
$859$	21.4079	0.730429	0.365214	−	0.930923i	$-0.380996\pi$
$859$	21.4079	0.730429	0.365214	+	0.930923i	$0.380996\pi$
$860$	0	0
$861$	−0.0130021	−0.000443111 0
$862$	0	0
$863$	−23.1799	−0.789053	−0.394527	−	0.918885i	$-0.629091\pi$
$863$	−23.1799	−0.789053	−0.394527	+	0.918885i	$0.629091\pi$
$864$	0	0
$865$	10.6292	0.361404
$866$	0	0
$867$	−33.1977	−1.12745
$868$	0	0
$869$	40.8167	1.38461
$870$	0	0
$871$	−3.76194	−0.127469
$872$	0	0
$873$	−12.8801	−0.435926
$874$	0	0
$875$	1.09438	0.0369969
$876$	0	0
$877$	−19.9817	−0.674734	−0.337367	−	0.941373i	$-0.609536\pi$
$877$	−19.9817	−0.674734	−0.337367	+	0.941373i	$0.609536\pi$
$878$	0	0
$879$	−25.2333	−0.851098
$880$	0	0
$881$	31.0252	1.04527	0.522633	−	0.852557i	$-0.324950\pi$
$881$	31.0252	1.04527	0.522633	+	0.852557i	$0.324950\pi$
$882$	0	0
$883$	6.97713	0.234799	0.117400	−	0.993085i	$-0.462544\pi$
$883$	6.97713	0.234799	0.117400	+	0.993085i	$0.462544\pi$
$884$	0	0
$885$	−8.72080	−0.293147
$886$	0	0
$887$	−31.9000	−1.07110	−0.535549	−	0.844504i	$-0.679895\pi$
$887$	−31.9000	−1.07110	−0.535549	+	0.844504i	$0.679895\pi$
$888$	0	0
$889$	1.70510	0.0571873
$890$	0	0
$891$	2.35607	0.0789312
$892$	0	0
$893$	10.5893	0.354357
$894$	0	0
$895$	−5.32506	−0.177997
$896$	0	0
$897$	−2.08498	−0.0696153
$898$	0	0
$899$	−13.9931	−0.466695
$900$	0	0
$901$	95.3782	3.17751
$902$	0	0
$903$	−0.487929	−0.0162373
$904$	0	0
$905$	43.2945	1.43916
$906$	0	0
$907$	9.59465	0.318585	0.159293	−	0.987231i	$-0.449079\pi$
$907$	9.59465	0.318585	0.159293	+	0.987231i	$0.449079\pi$
$908$	0	0
$909$	−14.6926	−0.487322
$910$	0	0
$911$	−26.1540	−0.866520	−0.433260	−	0.901269i	$-0.642637\pi$
$911$	−26.1540	−0.866520	−0.433260	+	0.901269i	$0.642637\pi$
$912$	0	0
$913$	−0.0332566	−0.00110063
$914$	0	0
$915$	0.152223	0.00503234
$916$	0	0
$917$	−0.591562	−0.0195351
$918$	0	0
$919$	−1.18144	−0.0389722	−0.0194861	−	0.999810i	$-0.506203\pi$
$919$	−1.18144	−0.0389722	−0.0194861	+	0.999810i	$0.506203\pi$
$920$	0	0
$921$	3.20268	0.105532
$922$	0	0
$923$	−2.65185	−0.0872867
$924$	0	0
$925$	1.73451	0.0570303
$926$	0	0
$927$	4.05118	0.133058
$928$	0	0
$929$	43.5999	1.43046	0.715232	−	0.698887i	$-0.246321\pi$
$929$	43.5999	1.43046	0.715232	+	0.698887i	$0.246321\pi$
$930$	0	0
$931$	−11.6578	−0.382070
$932$	0	0
$933$	−3.45554	−0.113129
$934$	0	0
$935$	−33.1518	−1.08418
$936$	0	0
$937$	15.5235	0.507130	0.253565	−	0.967318i	$-0.418397\pi$
$937$	15.5235	0.507130	0.253565	+	0.967318i	$0.418397\pi$
$938$	0	0
$939$	−31.7617	−1.03650
$940$	0	0
$941$	−15.1806	−0.494874	−0.247437	−	0.968904i	$-0.579588\pi$
$941$	−15.1806	−0.494874	−0.247437	+	0.968904i	$0.579588\pi$
$942$	0	0
$943$	0.357177	0.0116313
$944$	0	0
$945$	0.180715	0.00587867
$946$	0	0
$947$	22.6669	0.736574	0.368287	−	0.929712i	$-0.379944\pi$
$947$	22.6669	0.736574	0.368287	+	0.929712i	$0.379944\pi$
$948$	0	0
$949$	7.75913	0.251872
$950$	0	0
$951$	25.1696	0.816181
$952$	0	0
$953$	−56.4316	−1.82800	−0.914000	−	0.405715i	$-0.867022\pi$
$953$	−56.4316	−1.82800	−0.914000	+	0.405715i	$0.867022\pi$
$954$	0	0
$955$	−52.1898	−1.68882
$956$	0	0
$957$	9.34650	0.302129
$958$	0	0
$959$	−0.401871	−0.0129771
$960$	0	0
$961$	−18.5576	−0.598633
$962$	0	0
$963$	−10.1492	−0.327054
$964$	0	0
$965$	−36.4063	−1.17196
$966$	0	0
$967$	−14.1287	−0.454349	−0.227174	−	0.973854i	$-0.572949\pi$
$967$	−14.1287	−0.454349	−0.227174	+	0.973854i	$0.572949\pi$
$968$	0	0
$969$	−11.8134	−0.379501
$970$	0	0
$971$	16.4175	0.526863	0.263431	−	0.964678i	$-0.415146\pi$
$971$	16.4175	0.526863	0.263431	+	0.964678i	$0.415146\pi$
$972$	0	0
$973$	−0.325081	−0.0104216
$974$	0	0
$975$	0.880668	0.0282040
$976$	0	0
$977$	−15.1986	−0.486248	−0.243124	−	0.969995i	$-0.578172\pi$
$977$	−15.1986	−0.486248	−0.243124	+	0.969995i	$0.578172\pi$
$978$	0	0
$979$	−18.7566	−0.599464
$980$	0	0
$981$	−4.51988	−0.144309
$982$	0	0
$983$	11.0518	0.352499	0.176249	−	0.984346i	$-0.443603\pi$
$983$	11.0518	0.352499	0.176249	+	0.984346i	$0.443603\pi$
$984$	0	0
$985$	−28.1061	−0.895534
$986$	0	0
$987$	−0.577897	−0.0183947
$988$	0	0
$989$	13.4037	0.426214
$990$	0	0
$991$	−13.0573	−0.414780	−0.207390	−	0.978258i	$-0.566497\pi$
$991$	−13.0573	−0.414780	−0.207390	+	0.978258i	$0.566497\pi$
$992$	0	0
$993$	−8.94004	−0.283703
$994$	0	0
$995$	−35.3232	−1.11982
$996$	0	0
$997$	40.9089	1.29560	0.647799	−	0.761811i	$-0.275690\pi$
$997$	40.9089	1.29560	0.647799	+	0.761811i	$0.275690\pi$
$998$	0	0
$999$	1.64278	0.0519752

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.ba.1.3		9
4.3	odd	2		4008.2.a.i.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.i.1.3	✓	9	4.3	odd	2
8016.2.a.ba.1.3		9	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.98599 q^{5} +0.0909950 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.98599 q^{5} +0.0909950 q^{7} +1.00000 q^{9} +2.35607 q^{11} +0.834094 q^{13} +1.98599 q^{15} +7.08503 q^{17} +1.66738 q^{19} -0.0909950 q^{21} +2.49969 q^{23} -1.05584 q^{25} -1.00000 q^{27} -3.96700 q^{29} +3.52738 q^{31} -2.35607 q^{33} -0.180715 q^{35} -1.64278 q^{37} -0.834094 q^{39} +0.142888 q^{41} +5.36216 q^{43} -1.98599 q^{45} +6.35086 q^{47} -6.99172 q^{49} -7.08503 q^{51} +13.4619 q^{53} -4.67913 q^{55} -1.66738 q^{57} -4.39116 q^{59} +0.0766485 q^{61} +0.0909950 q^{63} -1.65650 q^{65} -4.51021 q^{67} -2.49969 q^{69} -3.17932 q^{71} +9.30247 q^{73} +1.05584 q^{75} +0.214390 q^{77} +17.3241 q^{79} +1.00000 q^{81} -0.0141153 q^{83} -14.0708 q^{85} +3.96700 q^{87} -7.96099 q^{89} +0.0758984 q^{91} -3.52738 q^{93} -3.31139 q^{95} -12.8801 q^{97} +2.35607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{3} - 6 q^{5} + 11 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q - 9 * q^3 - 6 * q^5 + 11 * q^7 + 9 * q^9	\( 9 q - 9 q^{3} - 6 q^{5} + 11 q^{7} + 9 q^{9} - q^{11} - 4 q^{13} + 6 q^{15} - 9 q^{17} + 8 q^{19} - 11 q^{21} + 7 q^{23} - q^{25} - 9 q^{27} - 9 q^{29} + 25 q^{31} + q^{33} - 5 q^{35} - 6 q^{37} + 4 q^{39} - 4 q^{41} + 24 q^{43} - 6 q^{45} + 16 q^{47} + 4 q^{49} + 9 q^{51} - 26 q^{53} + 29 q^{55} - 8 q^{57} + 4 q^{59} - 20 q^{61} + 11 q^{63} - 8 q^{65} + 25 q^{67} - 7 q^{69} + 15 q^{71} - 10 q^{73} + q^{75} - 20 q^{77} + 34 q^{79} + 9 q^{81} + 4 q^{83} - 13 q^{85} + 9 q^{87} + 13 q^{89} + 21 q^{91} - 25 q^{93} + 7 q^{95} - 4 q^{97} - q^{99}+O(q^{100}) \) 9 * q - 9 * q^3 - 6 * q^5 + 11 * q^7 + 9 * q^9 - q^11 - 4 * q^13 + 6 * q^15 - 9 * q^17 + 8 * q^19 - 11 * q^21 + 7 * q^23 - q^25 - 9 * q^27 - 9 * q^29 + 25 * q^31 + q^33 - 5 * q^35 - 6 * q^37 + 4 * q^39 - 4 * q^41 + 24 * q^43 - 6 * q^45 + 16 * q^47 + 4 * q^49 + 9 * q^51 - 26 * q^53 + 29 * q^55 - 8 * q^57 + 4 * q^59 - 20 * q^61 + 11 * q^63 - 8 * q^65 + 25 * q^67 - 7 * q^69 + 15 * q^71 - 10 * q^73 + q^75 - 20 * q^77 + 34 * q^79 + 9 * q^81 + 4 * q^83 - 13 * q^85 + 9 * q^87 + 13 * q^89 + 21 * q^91 - 25 * q^93 + 7 * q^95 - 4 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more