LMFDB - Embedded newform 1792.2.e.e.895.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(895,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1792.895");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.e (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$14.3091920422$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12745506816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 23x^{4} + 1 $ x^8 + 23*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 896)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.5
Root			$-0.323042 + 0.323042i$ of defining polynomial
Character	$\chi$	$=$	1792.895
Dual form			1792.2.e.e.895.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.646084i q^{3} -0.646084 q^{5} +(2.44949 - 1.00000i) q^{7} +2.58258 q^{9} +O(q^{10})$	$q+0.646084i q^{3} -0.646084 q^{5} +(2.44949 - 1.00000i) q^{7} +2.58258 q^{9} -3.58258 q^{11} -5.54506 q^{13} -0.417424i q^{15} -6.19115i q^{17} -6.83723i q^{19} +(0.646084 + 1.58258i) q^{21} +5.58258i q^{23} -4.58258 q^{25} +3.60681i q^{27} -6.00000i q^{29} -6.19115 q^{31} -2.31464i q^{33} +(-1.58258 + 0.646084i) q^{35} +2.00000i q^{37} -3.58258i q^{39} +1.29217i q^{41} +3.58258 q^{43} -1.66856 q^{45} -6.19115 q^{47} +(5.00000 - 4.89898i) q^{49} +4.00000 q^{51} -9.16515i q^{53} +2.31464 q^{55} +4.41742 q^{57} +5.54506i q^{59} -10.4440 q^{61} +(6.32599 - 2.58258i) q^{63} +3.58258 q^{65} +4.41742 q^{67} -3.60681 q^{69} +9.16515i q^{71} -7.48331i q^{73} -2.96073i q^{75} +(-8.77548 + 3.58258i) q^{77} +2.00000i q^{79} +5.41742 q^{81} +0.646084i q^{83} +4.00000i q^{85} +3.87650 q^{87} -7.48331i q^{89} +(-13.5826 + 5.54506i) q^{91} -4.00000i q^{93} +4.41742i q^{95} -18.5734i q^{97} -9.25227 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{9}+O(q^{10}) $ 8 * q - 16 * q^9	$ 8 q - 16 q^{9} + 8 q^{11} + 24 q^{35} - 8 q^{43} + 40 q^{49} + 32 q^{51} + 72 q^{57} - 8 q^{65} + 72 q^{67} + 80 q^{81} - 72 q^{91} - 184 q^{99}+O(q^{100}) $ 8 * q - 16 * q^9 + 8 * q^11 + 24 * q^35 - 8 * q^43 + 40 * q^49 + 32 * q^51 + 72 * q^57 - 8 * q^65 + 72 * q^67 + 80 * q^81 - 72 * q^91 - 184 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$			0.646084i			0.373017i	0.982453	+	0.186508i	$0.0597171\pi$
$3$			0.646084i			0.373017i	−0.982453	+	0.186508i	$0.940283\pi$
$4$		0			0
$5$	−0.646084			−0.288937			−0.144469	−	0.989509i	$-0.546147\pi$
$5$	−0.646084			−0.288937			−0.144469	+	0.989509i	$0.546147\pi$
$6$		0			0
$7$	2.44949	−	1.00000i	0.925820	−	0.377964i
$7$	2.44949	−	1.00000i	0.925820	−	0.377964i
$8$		0			0
$9$	2.58258			0.860859
$10$		0			0
$11$	−3.58258			−1.08019			−0.540094	−	0.841605i	$-0.681611\pi$
$11$	−3.58258			−1.08019			−0.540094	+	0.841605i	$0.681611\pi$
$12$		0			0
$13$	−5.54506			−1.53792			−0.768962	−	0.639295i	$-0.779226\pi$
$13$	−5.54506			−1.53792			−0.768962	+	0.639295i	$0.779226\pi$
$14$		0			0
$15$		−	0.417424i		−	0.107778i
$16$		0			0
$17$		−	6.19115i		−	1.50157i	−0.660545	−	0.750787i	$-0.729675\pi$
$17$		−	6.19115i		−	1.50157i	0.660545	−	0.750787i	$-0.270325\pi$
$18$		0			0
$19$		−	6.83723i		−	1.56857i	−0.620402	−	0.784284i	$-0.713030\pi$
$19$		−	6.83723i		−	1.56857i	0.620402	−	0.784284i	$-0.286970\pi$
$20$		0			0
$21$	0.646084	+	1.58258i	0.140987	+	0.345346i
$22$		0			0
$23$			5.58258i			1.16405i	0.813172	+	0.582024i	$0.197739\pi$
$23$			5.58258i			1.16405i	−0.813172	+	0.582024i	$0.802261\pi$
$24$		0			0
$25$	−4.58258			−0.916515
$26$		0			0
$27$			3.60681i			0.694131i
$28$		0			0
$29$		−	6.00000i		−	1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$		−	6.00000i		−	1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$		0			0
$31$	−6.19115			−1.11196			−0.555981	−	0.831195i	$-0.687657\pi$
$31$	−6.19115			−1.11196			−0.555981	+	0.831195i	$0.687657\pi$
$32$		0			0
$33$		−	2.31464i		−	0.402928i
$34$		0			0
$35$	−1.58258	+	0.646084i	−0.267504	+	0.109208i
$36$		0			0
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$		−	3.58258i		−	0.573671i
$40$		0			0
$41$			1.29217i			0.201803i	0.994896	+	0.100901i	$0.0321726\pi$
$41$			1.29217i			0.201803i	−0.994896	+	0.100901i	$0.967827\pi$
$42$		0			0
$43$	3.58258			0.546338			0.273169	−	0.961966i	$-0.411928\pi$
$43$	3.58258			0.546338			0.273169	+	0.961966i	$0.411928\pi$
$44$		0			0
$45$	−1.66856			−0.248734
$46$		0			0
$47$	−6.19115			−0.903072			−0.451536	−	0.892253i	$-0.649124\pi$
$47$	−6.19115			−0.903072			−0.451536	+	0.892253i	$0.649124\pi$
$48$		0			0
$49$	5.00000	−	4.89898i	0.714286	−	0.699854i
$50$		0			0
$51$	4.00000			0.560112
$52$		0			0
$53$		−	9.16515i		−	1.25893i	−0.777029	−	0.629465i	$-0.783274\pi$
$53$		−	9.16515i		−	1.25893i	0.777029	−	0.629465i	$-0.216726\pi$
$54$		0			0
$55$	2.31464			0.312107
$56$		0			0
$57$	4.41742			0.585102
$58$		0			0
$59$			5.54506i			0.721906i	0.932584	+	0.360953i	$0.117548\pi$
$59$			5.54506i			0.721906i	−0.932584	+	0.360953i	$0.882452\pi$
$60$		0			0
$61$	−10.4440			−1.33722			−0.668611	−	0.743612i	$-0.733111\pi$
$61$	−10.4440			−1.33722			−0.668611	+	0.743612i	$0.733111\pi$
$62$		0			0
$63$	6.32599	−	2.58258i	0.797000	−	0.325374i
$64$		0			0
$65$	3.58258			0.444364
$66$		0			0
$67$	4.41742			0.539674			0.269837	−	0.962906i	$-0.413030\pi$
$67$	4.41742			0.539674			0.269837	+	0.962906i	$0.413030\pi$
$68$		0			0
$69$	−3.60681			−0.434209
$70$		0			0
$71$			9.16515i			1.08770i	0.839181	+	0.543852i	$0.183035\pi$
$71$			9.16515i			1.08770i	−0.839181	+	0.543852i	$0.816965\pi$
$72$		0			0
$73$		−	7.48331i		−	0.875856i	−0.899010	−	0.437928i	$-0.855713\pi$
$73$		−	7.48331i		−	0.875856i	0.899010	−	0.437928i	$-0.144287\pi$
$74$		0			0
$75$		−	2.96073i		−	0.341875i
$76$		0			0
$77$	−8.77548	+	3.58258i	−1.00006	+	0.408272i
$78$		0			0
$79$			2.00000i			0.225018i	0.993651	+	0.112509i	$0.0358886\pi$
$79$			2.00000i			0.225018i	−0.993651	+	0.112509i	$0.964111\pi$
$80$		0			0
$81$	5.41742			0.601936
$82$		0			0
$83$			0.646084i			0.0709169i	0.999371	+	0.0354585i	$0.0112891\pi$
$83$			0.646084i			0.0709169i	−0.999371	+	0.0354585i	$0.988711\pi$
$84$		0			0
$85$			4.00000i			0.433861i
$86$		0			0
$87$	3.87650			0.415605
$88$		0			0
$89$		−	7.48331i		−	0.793230i	−0.917985	−	0.396615i	$-0.870185\pi$
$89$		−	7.48331i		−	0.793230i	0.917985	−	0.396615i	$-0.129815\pi$
$90$		0			0
$91$	−13.5826	+	5.54506i	−1.42384	+	0.581281i
$92$		0			0
$93$		−	4.00000i		−	0.414781i
$94$		0			0
$95$			4.41742i			0.453218i
$96$		0			0
$97$		−	18.5734i		−	1.88585i	−0.333009	−	0.942924i	$-0.608064\pi$
$97$		−	18.5734i		−	1.88585i	0.333009	−	0.942924i	$-0.391936\pi$
$98$		0			0
$99$	−9.25227			−0.929888
$100$		0			0
$101$	−2.96073			−0.294603			−0.147302	−	0.989092i	$-0.547059\pi$
$101$	−2.96073			−0.294603			−0.147302	+	0.989092i	$0.547059\pi$
$102$		0			0
$103$	1.29217			0.127321			0.0636605	−	0.997972i	$-0.479723\pi$
$103$	1.29217			0.127321			0.0636605	+	0.997972i	$0.479723\pi$
$104$		0			0
$105$	−0.417424	−	1.02248i	−0.0407364	−	0.0997835i
$106$		0			0
$107$	15.5826			1.50642			0.753212	−	0.657778i	$-0.228503\pi$
$107$	15.5826			1.50642			0.753212	+	0.657778i	$0.228503\pi$
$108$		0			0
$109$		−	5.16515i		−	0.494732i	−0.968922	−	0.247366i	$-0.920435\pi$
$109$		−	5.16515i		−	0.494732i	0.968922	−	0.247366i	$-0.0795650\pi$
$110$		0			0
$111$	−1.29217			−0.122647
$112$		0			0
$113$	−13.5826			−1.27774			−0.638871	−	0.769314i	$-0.720598\pi$
$113$	−13.5826			−1.27774			−0.638871	+	0.769314i	$0.720598\pi$
$114$		0			0
$115$		−	3.60681i		−	0.336337i
$116$		0			0
$117$	−14.3205			−1.32393
$118$		0			0
$119$	−6.19115	−	15.1652i	−0.567542	−	1.39019i
$120$		0			0
$121$	1.83485			0.166804
$122$		0			0
$123$	−0.834849			−0.0752758
$124$		0			0
$125$	6.19115			0.553753
$126$		0			0
$127$		−	17.5826i		−	1.56020i	−0.625654	−	0.780101i	$-0.715168\pi$
$127$		−	17.5826i		−	1.56020i	0.625654	−	0.780101i	$-0.284832\pi$
$128$		0			0
$129$			2.31464i			0.203793i
$130$		0			0
$131$			7.85971i			0.686706i	0.939206	+	0.343353i	$0.111563\pi$
$131$			7.85971i			0.686706i	−0.939206	+	0.343353i	$0.888437\pi$
$132$		0			0
$133$	−6.83723	−	16.7477i	−0.592863	−	1.45221i
$134$		0			0
$135$		−	2.33030i		−	0.200561i
$136$		0			0
$137$	1.16515			0.0995456			0.0497728	−	0.998761i	$-0.484150\pi$
$137$	1.16515			0.0995456			0.0497728	+	0.998761i	$0.484150\pi$
$138$		0			0
$139$		−	19.2195i		−	1.63018i	−0.579335	−	0.815089i	$-0.696688\pi$
$139$		−	19.2195i		−	1.63018i	0.579335	−	0.815089i	$-0.303312\pi$
$140$		0			0
$141$		−	4.00000i		−	0.336861i
$142$		0			0
$143$	19.8656			1.66125
$144$		0			0
$145$			3.87650i			0.321926i
$146$		0			0
$147$	3.16515	+	3.23042i	0.261057	+	0.266440i
$148$		0			0
$149$			1.16515i			0.0954529i	0.998860	+	0.0477265i	$0.0151976\pi$
$149$			1.16515i			0.0954529i	−0.998860	+	0.0477265i	$0.984802\pi$
$150$		0			0
$151$		−	2.41742i		−	0.196727i	−0.995151	−	0.0983636i	$-0.968639\pi$
$151$		−	2.41742i		−	0.196727i	0.995151	−	0.0983636i	$-0.0313608\pi$
$152$		0			0
$153$		−	15.9891i		−	1.29264i
$154$		0			0
$155$	4.00000			0.321288
$156$		0			0
$157$	−2.96073			−0.236292			−0.118146	−	0.992996i	$-0.537695\pi$
$157$	−2.96073			−0.236292			−0.118146	+	0.992996i	$0.537695\pi$
$158$		0			0
$159$	5.92146			0.469602
$160$		0			0
$161$	5.58258	+	13.6745i	0.439969	+	1.07770i
$162$		0			0
$163$	3.58258			0.280609			0.140304	−	0.990108i	$-0.455192\pi$
$163$	3.58258			0.280609			0.140304	+	0.990108i	$0.455192\pi$
$164$		0			0
$165$			1.49545i			0.116421i
$166$		0			0
$167$	−11.0901			−0.858180			−0.429090	−	0.903262i	$-0.641166\pi$
$167$	−11.0901			−0.858180			−0.429090	+	0.903262i	$0.641166\pi$
$168$		0			0
$169$	17.7477			1.36521
$170$		0			0
$171$		−	17.6577i		−	1.35032i
$172$		0			0
$173$	16.6352			1.26475			0.632375	−	0.774662i	$-0.282080\pi$
$173$	16.6352			1.26475			0.632375	+	0.774662i	$0.282080\pi$
$174$		0			0
$175$	−11.2250	+	4.58258i	−0.848528	+	0.346410i
$176$		0			0
$177$	−3.58258			−0.269283
$178$		0			0
$179$	7.58258			0.566748			0.283374	−	0.959009i	$-0.408546\pi$
$179$	7.58258			0.566748			0.283374	+	0.959009i	$0.408546\pi$
$180$		0			0
$181$	4.52259			0.336161			0.168081	−	0.985773i	$-0.446243\pi$
$181$	4.52259			0.336161			0.168081	+	0.985773i	$0.446243\pi$
$182$		0			0
$183$		−	6.74773i		−	0.498806i
$184$		0			0
$185$		−	1.29217i		−	0.0950021i
$186$		0			0
$187$			22.1803i			1.62198i
$188$		0			0
$189$	3.60681	+	8.83485i	0.262357	+	0.642641i
$190$		0			0
$191$			21.1652i			1.53146i	0.643164	+	0.765728i	$0.277621\pi$
$191$			21.1652i			1.53146i	−0.643164	+	0.765728i	$0.722379\pi$
$192$		0			0
$193$	9.58258			0.689769			0.344884	−	0.938645i	$-0.387918\pi$
$193$	9.58258			0.689769			0.344884	+	0.938645i	$0.387918\pi$
$194$		0			0
$195$			2.31464i			0.165755i
$196$		0			0
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$		0			0
$199$	−1.29217			−0.0915993			−0.0457997	−	0.998951i	$-0.514584\pi$
$199$	−1.29217			−0.0915993			−0.0457997	+	0.998951i	$0.514584\pi$
$200$		0			0
$201$			2.85403i			0.201307i
$202$		0			0
$203$	−6.00000	−	14.6969i	−0.421117	−	1.03152i
$204$		0			0
$205$		−	0.834849i		−	0.0583084i
$206$		0			0
$207$			14.4174i			1.00208i
$208$		0			0
$209$			24.4949i			1.69435i
$210$		0			0
$211$	−25.9129			−1.78392			−0.891958	−	0.452118i	$-0.850669\pi$
$211$	−25.9129			−1.78392			−0.891958	+	0.452118i	$0.850669\pi$
$212$		0			0
$213$	−5.92146			−0.405731
$214$		0			0
$215$	−2.31464			−0.157857
$216$		0			0
$217$	−15.1652	+	6.19115i	−1.02948	+	0.420282i
$218$		0			0
$219$	4.83485			0.326709
$220$		0			0
$221$			34.3303i			2.30931i
$222$		0			0
$223$	2.58434			0.173060			0.0865299	−	0.996249i	$-0.472422\pi$
$223$	2.58434			0.173060			0.0865299	+	0.996249i	$0.472422\pi$
$224$		0			0
$225$	−11.8348			−0.788990
$226$		0			0
$227$			12.7587i			0.846824i	0.905937	+	0.423412i	$0.139168\pi$
$227$			12.7587i			0.846824i	−0.905937	+	0.423412i	$0.860832\pi$
$228$		0			0
$229$	1.66856			0.110262			0.0551308	−	0.998479i	$-0.482442\pi$
$229$	1.66856			0.110262			0.0551308	+	0.998479i	$0.482442\pi$
$230$		0			0
$231$	−2.31464	−	5.66970i	−0.152292	−	0.373039i
$232$		0			0
$233$	−1.16515			−0.0763316			−0.0381658	−	0.999271i	$-0.512152\pi$
$233$	−1.16515			−0.0763316			−0.0381658	+	0.999271i	$0.512152\pi$
$234$		0			0
$235$	4.00000			0.260931
$236$		0			0
$237$	−1.29217			−0.0839353
$238$		0			0
$239$		−	25.5826i		−	1.65480i	−0.561614	−	0.827400i	$-0.689819\pi$
$239$		−	25.5826i		−	1.65480i	0.561614	−	0.827400i	$-0.310181\pi$
$240$		0			0
$241$		−	6.19115i		−	0.398807i	−0.979917	−	0.199403i	$-0.936100\pi$
$241$		−	6.19115i		−	0.398807i	0.979917	−	0.199403i	$-0.0639004\pi$
$242$		0			0
$243$			14.3205i			0.918663i
$244$		0			0
$245$	−3.23042	+	3.16515i	−0.206384	+	0.202214i
$246$		0			0
$247$			37.9129i			2.41234i
$248$		0			0
$249$	−0.417424			−0.0264532
$250$		0			0
$251$			10.4440i			0.659222i	0.944117	+	0.329611i	$0.106918\pi$
$251$			10.4440i			0.659222i	−0.944117	+	0.329611i	$0.893082\pi$
$252$		0			0
$253$		−	20.0000i		−	1.25739i
$254$		0			0
$255$	−2.58434			−0.161837
$256$		0			0
$257$			9.79796i			0.611180i	0.952163	+	0.305590i	$0.0988537\pi$
$257$			9.79796i			0.611180i	−0.952163	+	0.305590i	$0.901146\pi$
$258$		0			0
$259$	2.00000	+	4.89898i	0.124274	+	0.304408i
$260$		0			0
$261$		−	15.4955i		−	0.959145i
$262$		0			0
$263$		−	6.83485i		−	0.421455i	−0.977545	−	0.210727i	$-0.932417\pi$
$263$		−	6.83485i		−	0.421455i	0.977545	−	0.210727i	$-0.0675832\pi$
$264$		0			0
$265$			5.92146i			0.363752i
$266$		0			0
$267$	4.83485			0.295888
$268$		0			0
$269$	23.8488			1.45409			0.727044	−	0.686591i	$-0.240894\pi$
$269$	23.8488			1.45409			0.727044	+	0.686591i	$0.240894\pi$
$270$		0			0
$271$	17.5510			1.06615			0.533073	−	0.846070i	$-0.321037\pi$
$271$	17.5510			1.06615			0.533073	+	0.846070i	$0.321037\pi$
$272$		0			0
$273$	−3.58258	−	8.77548i	−0.216827	−	0.531116i
$274$		0			0
$275$	16.4174			0.990008
$276$		0			0
$277$		−	8.33030i		−	0.500519i	−0.968179	−	0.250260i	$-0.919484\pi$
$277$		−	8.33030i		−	0.500519i	0.968179	−	0.250260i	$-0.0805160\pi$
$278$		0			0
$279$	−15.9891			−0.957243
$280$		0			0
$281$	10.0000			0.596550			0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000			0.596550			0.298275	+	0.954480i	$0.403589\pi$
$282$		0			0
$283$			20.2420i			1.20326i	0.798774	+	0.601631i	$0.205482\pi$
$283$			20.2420i			1.20326i	−0.798774	+	0.601631i	$0.794518\pi$
$284$		0			0
$285$	−2.85403			−0.169058
$286$		0			0
$287$	1.29217	+	3.16515i	0.0762742	+	0.186833i
$288$		0			0
$289$	−21.3303			−1.25472
$290$		0			0
$291$	12.0000			0.703452
$292$		0			0
$293$	1.66856			0.0974783			0.0487392	−	0.998812i	$-0.484480\pi$
$293$	1.66856			0.0974783			0.0487392	+	0.998812i	$0.484480\pi$
$294$		0			0
$295$		−	3.58258i		−	0.208586i
$296$		0			0
$297$		−	12.9217i		−	0.749792i
$298$		0			0
$299$		−	30.9557i		−	1.79022i
$300$		0			0
$301$	8.77548	−	3.58258i	0.505810	−	0.206496i
$302$		0			0
$303$		−	1.91288i		−	0.109892i
$304$		0			0
$305$	6.74773			0.386374
$306$		0			0
$307$		−	24.1185i		−	1.37652i	−0.725466	−	0.688258i	$-0.758376\pi$
$307$		−	24.1185i		−	1.37652i	0.725466	−	0.688258i	$-0.241624\pi$
$308$		0			0
$309$			0.834849i			0.0474929i
$310$		0			0
$311$	−0.269691			−0.0152928			−0.00764639	−	0.999971i	$-0.502434\pi$
$311$	−0.269691			−0.0152928			−0.00764639	+	0.999971i	$0.502434\pi$
$312$		0			0
$313$			11.0901i			0.626851i	0.949613	+	0.313426i	$0.101477\pi$
$313$			11.0901i			0.626851i	−0.949613	+	0.313426i	$0.898523\pi$
$314$		0			0
$315$	−4.08712	+	1.66856i	−0.230283	+	0.0940127i
$316$		0			0
$317$		−	8.33030i		−	0.467876i	−0.972251	−	0.233938i	$-0.924839\pi$
$317$		−	8.33030i		−	0.467876i	0.972251	−	0.233938i	$-0.0751613\pi$
$318$		0			0
$319$			21.4955i			1.20351i
$320$		0			0
$321$			10.0677i			0.561921i
$322$		0			0
$323$	−42.3303			−2.35532
$324$		0			0
$325$	25.4107			1.40953
$326$		0			0
$327$	3.33712			0.184543
$328$		0			0
$329$	−15.1652	+	6.19115i	−0.836082	+	0.341329i
$330$		0			0
$331$	24.4174			1.34210			0.671052	−	0.741411i	$-0.265843\pi$
$331$	24.4174			1.34210			0.671052	+	0.741411i	$0.265843\pi$
$332$		0			0
$333$			5.16515i			0.283049i
$334$		0			0
$335$	−2.85403			−0.155932
$336$		0			0
$337$	8.74773			0.476519			0.238260	−	0.971202i	$-0.423423\pi$
$337$	8.74773			0.476519			0.238260	+	0.971202i	$0.423423\pi$
$338$		0			0
$339$		−	8.77548i		−	0.476619i
$340$		0			0
$341$	22.1803			1.20113
$342$		0			0
$343$	7.34847	−	17.0000i	0.396780	−	0.917914i
$344$		0			0
$345$	2.33030			0.125459
$346$		0			0
$347$	−26.7477			−1.43589			−0.717947	−	0.696098i	$-0.754918\pi$
$347$	−26.7477			−1.43589			−0.717947	+	0.696098i	$0.754918\pi$
$348$		0			0
$349$	1.93825			0.103752			0.0518761	−	0.998654i	$-0.483480\pi$
$349$	1.93825			0.103752			0.0518761	+	0.998654i	$0.483480\pi$
$350$		0			0
$351$		−	20.0000i		−	1.06752i
$352$		0			0
$353$			17.0116i			0.905435i	0.891654	+	0.452718i	$0.149545\pi$
$353$			17.0116i			0.905435i	−0.891654	+	0.452718i	$0.850455\pi$
$354$		0			0
$355$		−	5.92146i		−	0.314278i
$356$		0			0
$357$	9.79796	−	4.00000i	0.518563	−	0.211702i
$358$		0			0
$359$			13.5826i			0.716861i	0.933556	+	0.358430i	$0.116688\pi$
$359$			13.5826i			0.716861i	−0.933556	+	0.358430i	$0.883312\pi$
$360$		0			0
$361$	−27.7477			−1.46041
$362$		0			0
$363$			1.18547i			0.0622208i
$364$		0			0
$365$			4.83485i			0.253068i
$366$		0			0
$367$	−29.3939			−1.53435			−0.767174	−	0.641439i	$-0.778338\pi$
$367$	−29.3939			−1.53435			−0.767174	+	0.641439i	$0.778338\pi$
$368$		0			0
$369$			3.33712i			0.173724i
$370$		0			0
$371$	−9.16515	−	22.4499i	−0.475831	−	1.16554i
$372$		0			0
$373$			28.3303i			1.46689i	0.679750	+	0.733444i	$0.262088\pi$
$373$			28.3303i			1.46689i	−0.679750	+	0.733444i	$0.737912\pi$
$374$		0			0
$375$			4.00000i			0.206559i
$376$		0			0
$377$			33.2704i			1.71351i
$378$		0			0
$379$	−29.9129			−1.53652			−0.768261	−	0.640137i	$-0.778878\pi$
$379$	−29.9129			−1.53652			−0.768261	+	0.640137i	$0.778878\pi$
$380$		0			0
$381$	11.3598			0.581981
$382$		0			0
$383$	−23.7421			−1.21317			−0.606583	−	0.795020i	$-0.707460\pi$
$383$	−23.7421			−1.21317			−0.606583	+	0.795020i	$0.707460\pi$
$384$		0			0
$385$	5.66970	−	2.31464i	0.288955	−	0.117965i
$386$		0			0
$387$	9.25227			0.470319
$388$		0			0
$389$			2.00000i			0.101404i	0.998714	+	0.0507020i	$0.0161459\pi$
$389$			2.00000i			0.101404i	−0.998714	+	0.0507020i	$0.983854\pi$
$390$		0			0
$391$	34.5625			1.74790
$392$		0			0
$393$	−5.07803			−0.256153
$394$		0			0
$395$		−	1.29217i		−	0.0650160i
$396$		0			0
$397$	−22.8263			−1.14562			−0.572811	−	0.819688i	$-0.694147\pi$
$397$	−22.8263			−1.14562			−0.572811	+	0.819688i	$0.694147\pi$
$398$		0			0
$399$	10.8204	−	4.41742i	0.541699	−	0.221148i
$400$		0			0
$401$	−6.41742			−0.320471			−0.160235	−	0.987079i	$-0.551225\pi$
$401$	−6.41742			−0.320471			−0.160235	+	0.987079i	$0.551225\pi$
$402$		0			0
$403$	34.3303			1.71011
$404$		0			0
$405$	−3.50011			−0.173922
$406$		0			0
$407$		−	7.16515i		−	0.355163i
$408$		0			0
$409$			31.2254i			1.54400i	0.635624	+	0.771999i	$0.280743\pi$
$409$			31.2254i			1.54400i	−0.635624	+	0.771999i	$0.719257\pi$
$410$		0			0
$411$			0.752785i			0.0371322i
$412$		0			0
$413$	5.54506	+	13.5826i	0.272855	+	0.668355i
$414$		0			0
$415$		−	0.417424i		−	0.0204906i
$416$		0			0
$417$	12.4174			0.608084
$418$		0			0
$419$			27.9950i			1.36765i	0.729648	+	0.683823i	$0.239684\pi$
$419$			27.9950i			1.36765i	−0.729648	+	0.683823i	$0.760316\pi$
$420$		0			0
$421$			31.4955i			1.53499i	0.641052	+	0.767497i	$0.278498\pi$
$421$			31.4955i			1.53499i	−0.641052	+	0.767497i	$0.721502\pi$
$422$		0			0
$423$	−15.9891			−0.777417
$424$		0			0
$425$			28.3714i			1.37622i
$426$		0			0
$427$	−25.5826	+	10.4440i	−1.23803	+	0.505423i
$428$		0			0
$429$			12.8348i			0.619672i
$430$		0			0
$431$			4.74773i			0.228690i	0.993441	+	0.114345i	$0.0364769\pi$
$431$			4.74773i			0.228690i	−0.993441	+	0.114345i	$0.963523\pi$
$432$		0			0
$433$		−	10.8204i		−	0.519997i	−0.965609	−	0.259998i	$-0.916278\pi$
$433$		−	10.8204i		−	0.519997i	0.965609	−	0.259998i	$-0.0837221\pi$
$434$		0			0
$435$	−2.50455			−0.120084
$436$		0			0
$437$	38.1694			1.82589
$438$		0			0
$439$	19.8656			0.948134			0.474067	−	0.880489i	$-0.342785\pi$
$439$	19.8656			0.948134			0.474067	+	0.880489i	$0.342785\pi$
$440$		0			0
$441$	12.9129	−	12.6520i	0.614899	−	0.602475i
$442$		0			0
$443$	22.7477			1.08078			0.540389	−	0.841416i	$-0.318277\pi$
$443$	22.7477			1.08078			0.540389	+	0.841416i	$0.318277\pi$
$444$		0			0
$445$			4.83485i			0.229194i
$446$		0			0
$447$	−0.752785			−0.0356055
$448$		0			0
$449$	−35.4955			−1.67513			−0.837567	−	0.546335i	$-0.816023\pi$
$449$	−35.4955			−1.67513			−0.837567	+	0.546335i	$0.816023\pi$
$450$		0			0
$451$		−	4.62929i		−	0.217985i
$452$		0			0
$453$	1.56186			0.0733825
$454$		0			0
$455$	8.77548	−	3.58258i	0.411401	−	0.167954i
$456$		0			0
$457$	4.74773			0.222089			0.111045	−	0.993815i	$-0.464580\pi$
$457$	4.74773			0.222089			0.111045	+	0.993815i	$0.464580\pi$
$458$		0			0
$459$	22.3303			1.04229
$460$		0			0
$461$	24.1185			1.12331			0.561655	−	0.827371i	$-0.310165\pi$
$461$	24.1185			1.12331			0.561655	+	0.827371i	$0.310165\pi$
$462$		0			0
$463$		−	17.1652i		−	0.797732i	−0.917009	−	0.398866i	$-0.869404\pi$
$463$		−	17.1652i		−	0.797732i	0.917009	−	0.398866i	$-0.130596\pi$
$464$		0			0
$465$			2.58434i			0.119846i
$466$		0			0
$467$		−	36.2311i		−	1.67658i	−0.545228	−	0.838288i	$-0.683557\pi$
$467$		−	36.2311i		−	1.67658i	0.545228	−	0.838288i	$-0.316443\pi$
$468$		0			0
$469$	10.8204	−	4.41742i	0.499641	−	0.203978i
$470$		0			0
$471$		−	1.91288i		−	0.0881408i
$472$		0			0
$473$	−12.8348			−0.590147
$474$		0			0
$475$			31.3321i			1.43762i
$476$		0			0
$477$		−	23.6697i		−	1.08376i
$478$		0			0
$479$	15.9891			0.730561			0.365280	−	0.930898i	$-0.380973\pi$
$479$	15.9891			0.730561			0.365280	+	0.930898i	$0.380973\pi$
$480$		0			0
$481$		−	11.0901i		−	0.505666i
$482$		0			0
$483$	−8.83485	+	3.60681i	−0.402000	+	0.164116i
$484$		0			0
$485$			12.0000i			0.544892i
$486$		0			0
$487$			7.25227i			0.328632i	0.986408	+	0.164316i	$0.0525417\pi$
$487$			7.25227i			0.328632i	−0.986408	+	0.164316i	$0.947458\pi$
$488$		0			0
$489$			2.31464i			0.104672i
$490$		0			0
$491$	3.58258			0.161679			0.0808397	−	0.996727i	$-0.474240\pi$
$491$	3.58258			0.161679			0.0808397	+	0.996727i	$0.474240\pi$
$492$		0			0
$493$	−37.1469			−1.67301
$494$		0			0
$495$	5.97774			0.268680
$496$		0			0
$497$	9.16515	+	22.4499i	0.411113	+	1.00702i
$498$		0			0
$499$	11.5826			0.518507			0.259254	−	0.965809i	$-0.416523\pi$
$499$	11.5826			0.518507			0.259254	+	0.965809i	$0.416523\pi$
$500$		0			0
$501$		−	7.16515i		−	0.320115i
$502$		0			0
$503$	25.0343			1.11622			0.558112	−	0.829766i	$-0.311526\pi$
$503$	25.0343			1.11622			0.558112	+	0.829766i	$0.311526\pi$
$504$		0			0
$505$	1.91288			0.0851220
$506$		0			0
$507$			11.4665i			0.509246i
$508$		0			0
$509$	36.5008			1.61787			0.808935	−	0.587898i	$-0.200045\pi$
$509$	36.5008			1.61787			0.808935	+	0.587898i	$0.200045\pi$
$510$		0			0
$511$	−7.48331	−	18.3303i	−0.331042	−	0.810885i
$512$		0			0
$513$	24.6606			1.08879
$514$		0			0
$515$	−0.834849			−0.0367878
$516$		0			0
$517$	22.1803			0.975486
$518$		0			0
$519$			10.7477i			0.471773i
$520$		0			0
$521$			20.8881i			0.915124i	0.889178	+	0.457562i	$0.151277\pi$
$521$			20.8881i			0.915124i	−0.889178	+	0.457562i	$0.848723\pi$
$522$		0			0
$523$			0.915775i			0.0400440i	0.999800	+	0.0200220i	$0.00637363\pi$
$523$			0.915775i			0.0400440i	−0.999800	+	0.0200220i	$0.993626\pi$
$524$		0			0
$525$	−2.96073	−	7.25227i	−0.129217	−	0.316515i
$526$		0			0
$527$			38.3303i			1.66969i
$528$		0			0
$529$	−8.16515			−0.355007
$530$		0			0
$531$			14.3205i			0.621459i
$532$		0			0
$533$		−	7.16515i		−	0.310357i
$534$		0			0
$535$	−10.0677			−0.435262
$536$		0			0
$537$			4.89898i			0.211407i
$538$		0			0
$539$	−17.9129	+	17.5510i	−0.771562	+	0.755974i
$540$		0			0
$541$		−	18.8348i		−	0.809773i	−0.914367	−	0.404887i	$-0.867311\pi$
$541$		−	18.8348i		−	0.809773i	0.914367	−	0.404887i	$-0.132689\pi$
$542$		0			0
$543$			2.92197i			0.125394i
$544$		0			0
$545$			3.33712i			0.142947i
$546$		0			0
$547$	11.5826			0.495235			0.247618	−	0.968858i	$-0.420352\pi$
$547$	11.5826			0.495235			0.247618	+	0.968858i	$0.420352\pi$
$548$		0			0
$549$	−26.9725			−1.15116
$550$		0			0
$551$	−41.0234			−1.74765
$552$		0			0
$553$	2.00000	+	4.89898i	0.0850487	+	0.208326i
$554$		0			0
$555$	0.834849			0.0354373
$556$		0			0
$557$		−	27.4955i		−	1.16502i	−0.812824	−	0.582510i	$-0.802071\pi$
$557$		−	27.4955i		−	1.16502i	0.812824	−	0.582510i	$-0.197929\pi$
$558$		0			0
$559$	−19.8656			−0.840226
$560$		0			0
$561$	−14.3303			−0.605026
$562$		0			0
$563$		−	1.66856i		−	0.0703214i	−0.999382	−	0.0351607i	$-0.988806\pi$
$563$		−	1.66856i		−	0.0703214i	0.999382	−	0.0351607i	$-0.0111943\pi$
$564$		0			0
$565$	8.77548			0.369187
$566$		0			0
$567$	13.2699	−	5.41742i	0.557284	−	0.227510i
$568$		0			0
$569$	−17.5826			−0.737100			−0.368550	−	0.929608i	$-0.620146\pi$
$569$	−17.5826			−0.737100			−0.368550	+	0.929608i	$0.620146\pi$
$570$		0			0
$571$	3.58258			0.149926			0.0749631	−	0.997186i	$-0.476116\pi$
$571$	3.58258			0.149926			0.0749631	+	0.997186i	$0.476116\pi$
$572$		0			0
$573$	−13.6745			−0.571259
$574$		0			0
$575$		−	25.5826i		−	1.06687i
$576$		0			0
$577$		−	34.5625i		−	1.43886i	−0.694566	−	0.719429i	$-0.744404\pi$
$577$		−	34.5625i		−	1.43886i	0.694566	−	0.719429i	$-0.255596\pi$
$578$		0			0
$579$			6.19115i			0.257295i
$580$		0			0
$581$	0.646084	+	1.58258i	0.0268041	+	0.0656563i
$582$		0			0
$583$			32.8348i			1.35988i
$584$		0			0
$585$	9.25227			0.382534
$586$		0			0
$587$		−	6.83723i		−	0.282203i	−0.989995	−	0.141101i	$-0.954936\pi$
$587$		−	6.83723i		−	0.282203i	0.989995	−	0.141101i	$-0.0450643\pi$
$588$		0			0
$589$			42.3303i			1.74419i
$590$		0			0
$591$	−11.6295			−0.478374
$592$		0			0
$593$		−	5.16867i		−	0.212252i	−0.994353	−	0.106126i	$-0.966155\pi$
$593$		−	5.16867i		−	0.212252i	0.994353	−	0.106126i	$-0.0338447\pi$
$594$		0			0
$595$	4.00000	+	9.79796i	0.163984	+	0.401677i
$596$		0			0
$597$		−	0.834849i		−	0.0341681i
$598$		0			0
$599$		−	10.0000i		−	0.408589i	−0.978909	−	0.204294i	$-0.934510\pi$
$599$		−	10.0000i		−	0.408589i	0.978909	−	0.204294i	$-0.0654900\pi$
$600$		0			0
$601$		−	19.8656i		−	0.810335i	−0.914242	−	0.405168i	$-0.867213\pi$
$601$		−	19.8656i		−	0.810335i	0.914242	−	0.405168i	$-0.132787\pi$
$602$		0			0
$603$	11.4083			0.464583
$604$		0			0
$605$	−1.18547			−0.0481960
$606$		0			0
$607$	−27.3489			−1.11006			−0.555029	−	0.831831i	$-0.687293\pi$
$607$	−27.3489			−1.11006			−0.555029	+	0.831831i	$0.687293\pi$
$608$		0			0
$609$	9.49545	−	3.87650i	0.384775	−	0.157084i
$610$		0			0
$611$	34.3303			1.38886
$612$		0			0
$613$		−	21.1652i		−	0.854852i	−0.904050	−	0.427426i	$-0.859420\pi$
$613$		−	21.1652i		−	0.854852i	0.904050	−	0.427426i	$-0.140580\pi$
$614$		0			0
$615$	0.539382			0.0217500
$616$		0			0
$617$	−2.41742			−0.0973218			−0.0486609	−	0.998815i	$-0.515495\pi$
$617$	−2.41742			−0.0973218			−0.0486609	+	0.998815i	$0.515495\pi$
$618$		0			0
$619$		−	29.2872i		−	1.17715i	−0.808442	−	0.588575i	$-0.799689\pi$
$619$		−	29.2872i		−	1.17715i	0.808442	−	0.588575i	$-0.200311\pi$
$620$		0			0
$621$	−20.1353			−0.808002
$622$		0			0
$623$	−7.48331	−	18.3303i	−0.299813	−	0.734388i
$624$		0			0
$625$	18.9129			0.756515
$626$		0			0
$627$	−15.8258			−0.632020
$628$		0			0
$629$	12.3823			0.493714
$630$		0			0
$631$		−	3.66970i		−	0.146088i	−0.997329	−	0.0730442i	$-0.976729\pi$
$631$		−	3.66970i		−	0.146088i	0.997329	−	0.0730442i	$-0.0232714\pi$
$632$		0			0
$633$		−	16.7419i		−	0.665430i
$634$		0			0
$635$			11.3598i			0.450801i
$636$		0			0
$637$	−27.7253	+	27.1652i	−1.09852	+	1.07632i
$638$		0			0
$639$			23.6697i			0.936359i
$640$		0			0
$641$	−20.7477			−0.819486			−0.409743	−	0.912201i	$-0.634382\pi$
$641$	−20.7477			−0.819486			−0.409743	+	0.912201i	$0.634382\pi$
$642$		0			0
$643$		−	24.3882i		−	0.961777i	−0.876782	−	0.480888i	$-0.840314\pi$
$643$		−	24.3882i		−	0.961777i	0.876782	−	0.480888i	$-0.159686\pi$
$644$		0			0
$645$		−	1.49545i		−	0.0588835i
$646$		0			0
$647$	43.0683			1.69319			0.846596	−	0.532237i	$-0.178648\pi$
$647$	43.0683			1.69319			0.846596	+	0.532237i	$0.178648\pi$
$648$		0			0
$649$		−	19.8656i		−	0.779793i
$650$		0			0
$651$	−4.00000	−	9.79796i	−0.156772	−	0.384012i
$652$		0			0
$653$			27.4955i			1.07598i	0.842951	+	0.537990i	$0.180816\pi$
$653$			27.4955i			1.07598i	−0.842951	+	0.537990i	$0.819184\pi$
$654$		0			0
$655$		−	5.07803i		−	0.198415i
$656$		0			0
$657$		−	19.3262i		−	0.753988i
$658$		0			0
$659$	−17.2523			−0.672053			−0.336027	−	0.941853i	$-0.609083\pi$
$659$	−17.2523			−0.672053			−0.336027	+	0.941853i	$0.609083\pi$
$660$		0			0
$661$	−47.8606			−1.86156			−0.930781	−	0.365577i	$-0.880872\pi$
$661$	−47.8606			−1.86156			−0.930781	+	0.365577i	$0.880872\pi$
$662$		0			0
$663$	−22.1803			−0.861410
$664$		0			0
$665$	4.41742	+	10.8204i	0.171300	+	0.419598i
$666$		0			0
$667$	33.4955			1.29695
$668$		0			0
$669$			1.66970i			0.0645542i
$670$		0			0
$671$	37.4166			1.44445
$672$		0			0
$673$	36.3303			1.40043			0.700215	−	0.713932i	$-0.253087\pi$
$673$	36.3303			1.40043			0.700215	+	0.713932i	$0.253087\pi$
$674$		0			0
$675$		−	16.5285i		−	0.636182i
$676$		0			0
$677$	11.7362			0.451059			0.225530	−	0.974236i	$-0.427589\pi$
$677$	11.7362			0.451059			0.225530	+	0.974236i	$0.427589\pi$
$678$		0			0
$679$	−18.5734	−	45.4955i	−0.712783	−	1.74596i
$680$		0			0
$681$	−8.24318			−0.315879
$682$		0			0
$683$	−0.417424			−0.0159723			−0.00798615	−	0.999968i	$-0.502542\pi$
$683$	−0.417424			−0.0159723			−0.00798615	+	0.999968i	$0.502542\pi$
$684$		0			0
$685$	−0.752785			−0.0287625
$686$		0			0
$687$			1.07803i			0.0411294i
$688$		0			0
$689$			50.8213i			1.93614i
$690$		0			0
$691$			25.4107i			0.966668i	0.875436	+	0.483334i	$0.160574\pi$
$691$			25.4107i			0.966668i	−0.875436	+	0.483334i	$0.839426\pi$
$692$		0			0
$693$	−22.6633	+	9.25227i	−0.860909	+	0.351465i
$694$		0			0
$695$			12.4174i			0.471020i
$696$		0			0
$697$	8.00000			0.303022
$698$		0			0
$699$		−	0.752785i		−	0.0284730i
$700$		0			0
$701$		−	39.4955i		−	1.49172i	−0.666101	−	0.745861i	$-0.732038\pi$
$701$		−	39.4955i		−	1.49172i	0.666101	−	0.745861i	$-0.267962\pi$
$702$		0			0
$703$	13.6745			0.515742
$704$		0			0
$705$			2.58434i			0.0973317i
$706$		0			0
$707$	−7.25227	+	2.96073i	−0.272750	+	0.111350i
$708$		0			0
$709$			5.16515i			0.193981i	0.995285	+	0.0969907i	$0.0309217\pi$
$709$			5.16515i			0.193981i	−0.995285	+	0.0969907i	$0.969078\pi$
$710$		0			0
$711$			5.16515i			0.193708i
$712$		0			0
$713$		−	34.5625i		−	1.29438i
$714$		0			0
$715$	−12.8348			−0.479996
$716$		0			0
$717$	16.5285			0.617268
$718$		0			0
$719$	40.7537			1.51986			0.759928	−	0.650007i	$-0.225234\pi$
$719$	40.7537			1.51986			0.759928	+	0.650007i	$0.225234\pi$
$720$		0			0
$721$	3.16515	−	1.29217i	0.117876	−	0.0481228i
$722$		0			0
$723$	4.00000			0.148762
$724$		0			0
$725$			27.4955i			1.02116i
$726$		0			0
$727$	−31.2254			−1.15809			−0.579043	−	0.815297i	$-0.696574\pi$
$727$	−31.2254			−1.15809			−0.579043	+	0.815297i	$0.696574\pi$
$728$		0			0
$729$	7.00000			0.259259
$730$		0			0
$731$		−	22.1803i		−	0.820366i
$732$		0			0
$733$	7.10692			0.262500			0.131250	−	0.991349i	$-0.458101\pi$
$733$	7.10692			0.262500			0.131250	+	0.991349i	$0.458101\pi$
$734$		0			0
$735$	−2.04495	−	2.08712i	−0.0754292	−	0.0769846i
$736$		0			0
$737$	−15.8258			−0.582949
$738$		0			0
$739$	−26.7477			−0.983931			−0.491966	−	0.870615i	$-0.663721\pi$
$739$	−26.7477			−0.983931			−0.491966	+	0.870615i	$0.663721\pi$
$740$		0			0
$741$	−24.4949			−0.899843
$742$		0			0
$743$		−	40.7477i		−	1.49489i	−0.664324	−	0.747445i	$-0.731281\pi$
$743$		−	40.7477i		−	1.49489i	0.664324	−	0.747445i	$-0.268719\pi$
$744$		0			0
$745$		−	0.752785i		−	0.0275799i
$746$		0			0
$747$			1.66856i			0.0610494i
$748$		0			0
$749$	38.1694	−	15.5826i	1.39468	−	0.569375i
$750$		0			0
$751$			11.0780i			0.404243i	0.979360	+	0.202122i	$0.0647836\pi$
$751$			11.0780i			0.404243i	−0.979360	+	0.202122i	$0.935216\pi$
$752$		0			0
$753$	−6.74773			−0.245901
$754$		0			0
$755$			1.56186i			0.0568419i
$756$		0			0
$757$		−	30.8348i		−	1.12071i	−0.828252	−	0.560356i	$-0.810664\pi$
$757$		−	30.8348i		−	1.12071i	0.828252	−	0.560356i	$-0.189336\pi$
$758$		0			0
$759$	12.9217			0.469027
$760$		0			0
$761$		−	28.6411i		−	1.03824i	−0.854702	−	0.519119i	$-0.826260\pi$
$761$		−	28.6411i		−	1.03824i	0.854702	−	0.519119i	$-0.173740\pi$
$762$		0			0
$763$	−5.16515	−	12.6520i	−0.186991	−	0.458033i
$764$		0			0
$765$			10.3303i			0.373493i
$766$		0			0
$767$		−	30.7477i		−	1.11024i
$768$		0			0
$769$			35.5850i			1.28323i	0.767027	+	0.641614i	$0.221735\pi$
$769$			35.5850i			1.28323i	−0.767027	+	0.641614i	$0.778265\pi$
$770$		0			0
$771$	−6.33030			−0.227980
$772$		0			0
$773$	−10.4440			−0.375646			−0.187823	−	0.982203i	$-0.560143\pi$
$773$	−10.4440			−0.375646			−0.187823	+	0.982203i	$0.560143\pi$
$774$		0			0
$775$	28.3714			1.01913
$776$		0			0
$777$	−3.16515	+	1.29217i	−0.113549	+	0.0463563i
$778$		0			0
$779$	8.83485			0.316541
$780$		0			0
$781$		−	32.8348i		−	1.17492i
$782$		0			0
$783$	21.6409			0.773382
$784$		0			0
$785$	1.91288			0.0682736
$786$		0			0
$787$			8.12940i			0.289782i	0.989448	+	0.144891i	$0.0462831\pi$
$787$			8.12940i			0.289782i	−0.989448	+	0.144891i	$0.953717\pi$
$788$		0			0
$789$	4.41589			0.157210
$790$		0			0
$791$	−33.2704	+	13.5826i	−1.18296	+	0.482941i
$792$		0			0
$793$	57.9129			2.05655
$794$		0			0
$795$	−3.82576			−0.135686
$796$		0			0
$797$	−5.54506			−0.196416			−0.0982081	−	0.995166i	$-0.531311\pi$
$797$	−5.54506			−0.196416			−0.0982081	+	0.995166i	$0.531311\pi$
$798$		0			0
$799$			38.3303i			1.35603i
$800$		0			0
$801$		−	19.3262i		−	0.682859i
$802$		0			0
$803$			26.8095i			0.946088i
$804$		0			0
$805$	−3.60681	−	8.83485i	−0.127123	−	0.311387i
$806$		0			0
$807$			15.4083i			0.542399i
$808$		0			0
$809$	−1.58258			−0.0556404			−0.0278202	−	0.999613i	$-0.508857\pi$
$809$	−1.58258			−0.0556404			−0.0278202	+	0.999613i	$0.508857\pi$
$810$		0			0
$811$		−	1.93825i		−	0.0680612i	−0.999421	−	0.0340306i	$-0.989166\pi$
$811$		−	1.93825i		−	0.0680612i	0.999421	−	0.0340306i	$-0.0108344\pi$
$812$		0			0
$813$			11.3394i			0.397690i
$814$		0			0
$815$	−2.31464			−0.0810784
$816$		0			0
$817$		−	24.4949i		−	0.856968i
$818$		0			0
$819$	−35.0780	+	14.3205i	−1.22573	+	0.500400i
$820$		0			0
$821$		−	54.6606i		−	1.90767i	−0.300334	−	0.953834i	$-0.597098\pi$
$821$		−	54.6606i		−	1.90767i	0.300334	−	0.953834i	$-0.402902\pi$
$822$		0			0
$823$			18.8348i			0.656542i	0.944584	+	0.328271i	$0.106466\pi$
$823$			18.8348i			0.656542i	−0.944584	+	0.328271i	$0.893534\pi$
$824$		0			0
$825$			10.6070i			0.369289i
$826$		0			0
$827$	−25.9129			−0.901079			−0.450540	−	0.892756i	$-0.648768\pi$
$827$	−25.9129			−0.901079			−0.450540	+	0.892756i	$0.648768\pi$
$828$		0			0
$829$	3.98320			0.138342			0.0691712	−	0.997605i	$-0.477965\pi$
$829$	3.98320			0.138342			0.0691712	+	0.997605i	$0.477965\pi$
$830$		0			0
$831$	5.38207			0.186702
$832$		0			0
$833$	−30.3303	−	30.9557i	−1.05088	−	1.07255i
$834$		0			0
$835$	7.16515			0.247960
$836$		0			0
$837$		−	22.3303i		−	0.771848i
$838$		0			0
$839$	−20.8881			−0.721137			−0.360568	−	0.932733i	$-0.617417\pi$
$839$	−20.8881			−0.721137			−0.360568	+	0.932733i	$0.617417\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$			6.46084i			0.222523i
$844$		0			0
$845$	−11.4665			−0.394460
$846$		0			0
$847$	4.49444	−	1.83485i	0.154431	−	0.0630461i
$848$		0			0
$849$	−13.0780			−0.448837
$850$		0			0
$851$	−11.1652			−0.382736
$852$		0			0
$853$	−47.3212			−1.62025			−0.810124	−	0.586258i	$-0.800600\pi$
$853$	−47.3212			−1.62025			−0.810124	+	0.586258i	$0.800600\pi$
$854$		0			0
$855$			11.4083i			0.390157i
$856$		0			0
$857$		−	3.33712i		−	0.113994i	−0.998374	−	0.0569969i	$-0.981847\pi$
$857$		−	3.33712i		−	0.113994i	0.998374	−	0.0569969i	$-0.0181525\pi$
$858$		0			0
$859$		−	13.7812i		−	0.470207i	−0.971970	−	0.235104i	$-0.924457\pi$
$859$		−	13.7812i		−	0.470207i	0.971970	−	0.235104i	$-0.0755429\pi$
$860$		0			0
$861$	−2.04495	+	0.834849i	−0.0696918	+	0.0284516i
$862$		0			0
$863$			14.8348i			0.504984i	0.967599	+	0.252492i	$0.0812502\pi$
$863$			14.8348i			0.504984i	−0.967599	+	0.252492i	$0.918750\pi$
$864$		0			0
$865$	−10.7477			−0.365434
$866$		0			0
$867$		−	13.7812i		−	0.468033i
$868$		0			0
$869$		−	7.16515i		−	0.243061i
$870$		0			0
$871$	−24.4949			−0.829978
$872$		0			0
$873$		−	47.9673i		−	1.62345i
$874$		0			0
$875$	15.1652	−	6.19115i	0.512676	−	0.209299i
$876$		0			0
$877$		−	32.3303i		−	1.09172i	−0.837877	−	0.545858i	$-0.816204\pi$
$877$		−	32.3303i		−	1.09172i	0.837877	−	0.545858i	$-0.183796\pi$
$878$		0			0
$879$			1.07803i			0.0363610i
$880$		0			0
$881$		−	35.1019i		−	1.18261i	−0.806447	−	0.591307i	$-0.798612\pi$
$881$		−	35.1019i		−	1.18261i	0.806447	−	0.591307i	$-0.201388\pi$
$882$		0			0
$883$	13.0780			0.440111			0.220055	−	0.975487i	$-0.429376\pi$
$883$	13.0780			0.440111			0.220055	+	0.975487i	$0.429376\pi$
$884$		0			0
$885$	2.31464			0.0778059
$886$		0			0
$887$	28.6411			0.961674			0.480837	−	0.876810i	$-0.340333\pi$
$887$	28.6411			0.961674			0.480837	+	0.876810i	$0.340333\pi$
$888$		0			0
$889$	−17.5826	−	43.0683i	−0.589701	−	1.44447i
$890$		0			0
$891$	−19.4083			−0.650204
$892$		0			0
$893$			42.3303i			1.41653i
$894$		0			0
$895$	−4.89898			−0.163755
$896$		0			0
$897$	20.0000			0.667781
$898$		0			0
$899$			37.1469i			1.23892i
$900$		0			0
$901$	−56.7428			−1.89038
$902$		0			0
$903$	2.31464	+	5.66970i	0.0770265	+	0.188676i
$904$		0			0
$905$	−2.92197			−0.0971296
$906$		0			0
$907$	−5.91288			−0.196334			−0.0981670	−	0.995170i	$-0.531298\pi$
$907$	−5.91288			−0.196334			−0.0981670	+	0.995170i	$0.531298\pi$
$908$		0			0
$909$	−7.64630			−0.253612
$910$		0			0
$911$			14.4174i			0.477671i	0.971060	+	0.238835i	$0.0767656\pi$
$911$			14.4174i			0.477671i	−0.971060	+	0.238835i	$0.923234\pi$
$912$		0			0
$913$		−	2.31464i		−	0.0766035i
$914$		0			0
$915$			4.35960i			0.144124i
$916$		0			0
$917$	7.85971	+	19.2523i	0.259550	+	0.635766i
$918$		0			0
$919$		−	13.1652i		−	0.434278i	−0.976141	−	0.217139i	$-0.930327\pi$
$919$		−	13.1652i		−	0.434278i	0.976141	−	0.217139i	$-0.0696725\pi$
$920$		0			0
$921$	15.5826			0.513463
$922$		0			0
$923$		−	50.8213i		−	1.67280i
$924$		0			0
$925$		−	9.16515i		−	0.301348i
$926$		0			0
$927$	3.33712			0.109605
$928$		0			0
$929$			33.0007i			1.08272i	0.840792	+	0.541359i	$0.182090\pi$
$929$			33.0007i			1.08272i	−0.840792	+	0.541359i	$0.817910\pi$
$930$		0			0
$931$	−33.4955	−	34.1862i	−1.09777	−	1.12041i
$932$		0			0
$933$		−	0.174243i		−	0.00570446i
$934$		0			0
$935$		−	14.3303i		−	0.468651i
$936$		0			0
$937$		−	6.94393i		−	0.226848i	−0.993547	−	0.113424i	$-0.963818\pi$
$937$		−	6.94393i		−	0.226848i	0.993547	−	0.113424i	$-0.0361819\pi$
$938$		0			0
$939$	−7.16515			−0.233826
$940$		0			0
$941$	−25.6804			−0.837156			−0.418578	−	0.908181i	$-0.637471\pi$
$941$	−25.6804			−0.837156			−0.418578	+	0.908181i	$0.637471\pi$
$942$		0			0
$943$	−7.21362			−0.234908
$944$		0			0
$945$	−2.33030	−	5.70805i	−0.0758048	−	0.185683i
$946$		0			0
$947$	46.7477			1.51910			0.759549	−	0.650451i	$-0.225420\pi$
$947$	46.7477			1.51910			0.759549	+	0.650451i	$0.225420\pi$
$948$		0			0
$949$			41.4955i			1.34700i
$950$		0			0
$951$	5.38207			0.174526
$952$		0			0
$953$	−8.33030			−0.269845			−0.134922	−	0.990856i	$-0.543079\pi$
$953$	−8.33030			−0.269845			−0.134922	+	0.990856i	$0.543079\pi$
$954$		0			0
$955$		−	13.6745i		−	0.442495i
$956$		0			0
$957$	−13.8879			−0.448931
$958$		0			0
$959$	2.85403	−	1.16515i	0.0921613	−	0.0376247i
$960$		0			0
$961$	7.33030			0.236461
$962$		0			0
$963$	40.2432			1.29682
$964$		0			0
$965$	−6.19115			−0.199300
$966$		0			0
$967$		−	47.0780i		−	1.51393i	−0.653457	−	0.756964i	$-0.726682\pi$
$967$		−	47.0780i		−	1.51393i	0.653457	−	0.756964i	$-0.273318\pi$
$968$		0			0
$969$		−	27.3489i		−	0.878574i
$970$		0			0
$971$			37.5233i			1.20418i	0.798429	+	0.602090i	$0.205665\pi$
$971$			37.5233i			1.20418i	−0.798429	+	0.602090i	$0.794335\pi$
$972$		0			0
$973$	−19.2195	−	47.0780i	−0.616150	−	1.50925i
$974$		0			0
$975$			16.4174i			0.525778i
$976$		0			0
$977$	31.4955			1.00763			0.503814	−	0.863812i	$-0.331930\pi$
$977$	31.4955			1.00763			0.503814	+	0.863812i	$0.331930\pi$
$978$		0			0
$979$			26.8095i			0.856837i
$980$		0			0
$981$		−	13.3394i		−	0.425894i
$982$		0			0
$983$	−48.2370			−1.53852			−0.769261	−	0.638935i	$-0.779375\pi$
$983$	−48.2370			−1.53852			−0.769261	+	0.638935i	$0.779375\pi$
$984$		0			0
$985$		−	11.6295i		−	0.370547i
$986$		0			0
$987$	−4.00000	−	9.79796i	−0.127321	−	0.311872i
$988$		0			0
$989$			20.0000i			0.635963i
$990$		0			0
$991$			16.3303i			0.518749i	0.965777	+	0.259375i	$0.0835164\pi$
$991$			16.3303i			0.518749i	−0.965777	+	0.259375i	$0.916484\pi$
$992$		0			0
$993$			15.7757i			0.500627i
$994$		0			0
$995$	0.834849			0.0264665
$996$		0			0
$997$	−7.85971			−0.248919			−0.124460	−	0.992225i	$-0.539720\pi$
$997$	−7.85971			−0.248919			−0.124460	+	0.992225i	$0.539720\pi$
$998$		0			0
$999$	−7.21362			−0.228229

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.e.e.895.5		8
4.3	odd	2		1792.2.e.d.895.3		8
7.6	odd	2	inner	1792.2.e.e.895.4		8
8.3	odd	2	inner	1792.2.e.e.895.6		8
8.5	even	2		1792.2.e.d.895.4		8
16.3	odd	4		896.2.f.c.895.6	yes	8
16.5	even	4		896.2.f.d.895.5	yes	8
16.11	odd	4		896.2.f.d.895.3	yes	8
16.13	even	4		896.2.f.c.895.4	yes	8
28.27	even	2		1792.2.e.d.895.6		8
56.13	odd	2		1792.2.e.d.895.5		8
56.27	even	2	inner	1792.2.e.e.895.3		8
112.13	odd	4		896.2.f.c.895.5	yes	8
112.27	even	4		896.2.f.d.895.6	yes	8
112.69	odd	4		896.2.f.d.895.4	yes	8
112.83	even	4		896.2.f.c.895.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.f.c.895.3	✓	8	112.83	even	4
896.2.f.c.895.4	yes	8	16.13	even	4
896.2.f.c.895.5	yes	8	112.13	odd	4
896.2.f.c.895.6	yes	8	16.3	odd	4
896.2.f.d.895.3	yes	8	16.11	odd	4
896.2.f.d.895.4	yes	8	112.69	odd	4
896.2.f.d.895.5	yes	8	16.5	even	4
896.2.f.d.895.6	yes	8	112.27	even	4
1792.2.e.d.895.3		8	4.3	odd	2
1792.2.e.d.895.4		8	8.5	even	2
1792.2.e.d.895.5		8	56.13	odd	2
1792.2.e.d.895.6		8	28.27	even	2
1792.2.e.e.895.3		8	56.27	even	2	inner
1792.2.e.e.895.4		8	7.6	odd	2	inner
1792.2.e.e.895.5		8	1.1	even	1	trivial
1792.2.e.e.895.6		8	8.3	odd	2	inner

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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.646084i q^{3} -0.646084 q^{5} +(2.44949 - 1.00000i) q^{7} +2.58258 q^{9} +O(q^{10})\)	\(q+0.646084i q^{3} -0.646084 q^{5} +(2.44949 - 1.00000i) q^{7} +2.58258 q^{9} -3.58258 q^{11} -5.54506 q^{13} -0.417424i q^{15} -6.19115i q^{17} -6.83723i q^{19} +(0.646084 + 1.58258i) q^{21} +5.58258i q^{23} -4.58258 q^{25} +3.60681i q^{27} -6.00000i q^{29} -6.19115 q^{31} -2.31464i q^{33} +(-1.58258 + 0.646084i) q^{35} +2.00000i q^{37} -3.58258i q^{39} +1.29217i q^{41} +3.58258 q^{43} -1.66856 q^{45} -6.19115 q^{47} +(5.00000 - 4.89898i) q^{49} +4.00000 q^{51} -9.16515i q^{53} +2.31464 q^{55} +4.41742 q^{57} +5.54506i q^{59} -10.4440 q^{61} +(6.32599 - 2.58258i) q^{63} +3.58258 q^{65} +4.41742 q^{67} -3.60681 q^{69} +9.16515i q^{71} -7.48331i q^{73} -2.96073i q^{75} +(-8.77548 + 3.58258i) q^{77} +2.00000i q^{79} +5.41742 q^{81} +0.646084i q^{83} +4.00000i q^{85} +3.87650 q^{87} -7.48331i q^{89} +(-13.5826 + 5.54506i) q^{91} -4.00000i q^{93} +4.41742i q^{95} -18.5734i q^{97} -9.25227 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{9}+O(q^{10}) \) 8 * q - 16 * q^9	\( 8 q - 16 q^{9} + 8 q^{11} + 24 q^{35} - 8 q^{43} + 40 q^{49} + 32 q^{51} + 72 q^{57} - 8 q^{65} + 72 q^{67} + 80 q^{81} - 72 q^{91} - 184 q^{99}+O(q^{100}) \) 8 * q - 16 * q^9 + 8 * q^11 + 24 * q^35 - 8 * q^43 + 40 * q^49 + 32 * q^51 + 72 * q^57 - 8 * q^65 + 72 * q^67 + 80 * q^81 - 72 * q^91 - 184 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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