LMFDB - Embedded newform 756.4.f.e.377.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [756,4,Mod(377,756)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("756.377"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	756.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$44.6054439643$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 36 x^{14} - 780 x^{13} + 17052 x^{12} - 100012 x^{11} + 336938 x^{10} + \cdots + 1976222698452 $ x^16 - 4x^15 - 36x^14 - 780x^13 + 17052x^12 - 100012x^11 + 336938x^10 - 6730840x^9 + 90353572x^8 - 474561804x^7 + 689078700x^6 + 9372890772x^5 - 37121562243x^4 - 109129608396x^3 + 718624177776x^2 + 62521834824*x + 1976222698452
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{26} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			377.6
Root			$-8.90572 + 7.58540i$ of defining polynomial
Character	$\chi$	$=$	756.377
Dual form			756.4.f.e.377.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-6.67001 q^{5} +(-18.2446 + 3.18348i) q^{7} -34.9270i q^{11} +63.3080i q^{13} -32.0860 q^{17} -125.448i q^{19} -63.6979i q^{23} -80.5110 q^{25} +303.036i q^{29} +5.46382i q^{31} +(121.692 - 21.2338i) q^{35} +45.0218 q^{37} +185.881 q^{41} -224.975 q^{43} +534.796 q^{47} +(322.731 - 116.163i) q^{49} -373.269i q^{53} +232.963i q^{55} +701.931 q^{59} -74.5006i q^{61} -422.265i q^{65} +425.353 q^{67} +936.115i q^{71} +957.209i q^{73} +(111.189 + 637.229i) q^{77} -129.399 q^{79} +719.682 q^{83} +214.014 q^{85} +274.227 q^{89} +(-201.540 - 1155.03i) q^{91} +836.739i q^{95} -1350.11i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 14 q^{7} + 388 q^{25} - 344 q^{37} - 404 q^{43} - 14 q^{49} + 1772 q^{67} - 1456 q^{79} - 4704 q^{85} + 2436 q^{91}+O(q^{100}) $ 16 * q + 14 * q^7 + 388 * q^25 - 344 * q^37 - 404 * q^43 - 14 * q^49 + 1772 * q^67 - 1456 * q^79 - 4704 * q^85 + 2436 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$-1$	$-1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	−6.67001			−0.596583			−0.298292	−	0.954475i	$-0.596417\pi$
$5$	−6.67001			−0.596583			−0.298292	+	0.954475i	$0.596417\pi$
$6$		0			0
$7$	−18.2446	+	3.18348i	−0.985116	+	0.171892i
$7$	−18.2446	+	3.18348i	−0.985116	+	0.171892i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	34.9270i		−	0.957353i	−0.877991	−	0.478677i	$-0.841117\pi$
$11$		−	34.9270i		−	0.957353i	0.877991	−	0.478677i	$-0.158883\pi$
$12$		0			0
$13$			63.3080i			1.35065i	0.737519	+	0.675327i	$0.235997\pi$
$13$			63.3080i			1.35065i	−0.737519	+	0.675327i	$0.764003\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−32.0860			−0.457765			−0.228882	−	0.973454i	$-0.573507\pi$
$17$	−32.0860			−0.457765			−0.228882	+	0.973454i	$0.573507\pi$
$18$		0			0
$19$		−	125.448i		−	1.51472i	−0.652995	−	0.757362i	$-0.726488\pi$
$19$		−	125.448i		−	1.51472i	0.652995	−	0.757362i	$-0.273512\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	63.6979i		−	0.577475i	−0.957408	−	0.288737i	$-0.906765\pi$
$23$		−	63.6979i		−	0.577475i	0.957408	−	0.288737i	$-0.0932355\pi$
$24$		0			0
$25$	−80.5110			−0.644088
$26$		0			0
$27$		0			0
$28$		0			0
$29$			303.036i			1.94043i	0.242252	+	0.970213i	$0.422114\pi$
$29$			303.036i			1.94043i	−0.242252	+	0.970213i	$0.577886\pi$
$30$		0			0
$31$			5.46382i			0.0316559i	0.999875	+	0.0158279i	$0.00503840\pi$
$31$			5.46382i			0.0316559i	−0.999875	+	0.0158279i	$0.994962\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	121.692	−	21.2338i	0.587704	−	0.102548i
$36$		0			0
$37$	45.0218			0.200042			0.100021	−	0.994985i	$-0.468109\pi$
$37$	45.0218			0.200042			0.100021	+	0.994985i	$0.468109\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	185.881			0.708043			0.354021	−	0.935237i	$-0.384814\pi$
$41$	185.881			0.708043			0.354021	+	0.935237i	$0.384814\pi$
$42$		0			0
$43$	−224.975			−0.797870			−0.398935	−	0.916979i	$-0.630620\pi$
$43$	−224.975			−0.797870			−0.398935	+	0.916979i	$0.630620\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	534.796			1.65974			0.829872	−	0.557954i	$-0.188413\pi$
$47$	534.796			1.65974			0.829872	+	0.557954i	$0.188413\pi$
$48$		0			0
$49$	322.731	−	116.163i	0.940907	−	0.338666i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	373.269i		−	0.967405i	−0.875232	−	0.483703i	$-0.839292\pi$
$53$		−	373.269i		−	0.967405i	0.875232	−	0.483703i	$-0.160708\pi$
$54$		0			0
$55$			232.963i			0.571141i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	701.931			1.54887			0.774437	−	0.632650i	$-0.218033\pi$
$59$	701.931			1.54887			0.774437	+	0.632650i	$0.218033\pi$
$60$		0			0
$61$		−	74.5006i		−	0.156374i	−0.996939	−	0.0781871i	$-0.975087\pi$
$61$		−	74.5006i		−	0.156374i	0.996939	−	0.0781871i	$-0.0249132\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	422.265i		−	0.805777i
$66$		0			0
$67$	425.353			0.775599			0.387799	−	0.921744i	$-0.373235\pi$
$67$	425.353			0.775599			0.387799	+	0.921744i	$0.373235\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			936.115i			1.56474i	0.622815	+	0.782369i	$0.285989\pi$
$71$			936.115i			1.56474i	−0.622815	+	0.782369i	$0.714011\pi$
$72$		0			0
$73$			957.209i			1.53470i	0.641230	+	0.767349i	$0.278424\pi$
$73$			957.209i			1.53470i	−0.641230	+	0.767349i	$0.721576\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	111.189	+	637.229i	0.164561	+	0.943104i
$78$		0			0
$79$	−129.399			−0.184285			−0.0921426	−	0.995746i	$-0.529372\pi$
$79$	−129.399			−0.184285			−0.0921426	+	0.995746i	$0.529372\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	719.682			0.951752			0.475876	−	0.879512i	$-0.342131\pi$
$83$	719.682			0.951752			0.475876	+	0.879512i	$0.342131\pi$
$84$		0			0
$85$	214.014			0.273095
$86$		0			0
$87$		0			0
$88$		0			0
$89$	274.227			0.326606			0.163303	−	0.986576i	$-0.447785\pi$
$89$	274.227			0.326606			0.163303	+	0.986576i	$0.447785\pi$
$90$		0			0
$91$	−201.540	−	1155.03i	−0.232166	−	1.33055i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			836.739i			0.903659i
$96$		0			0
$97$		−	1350.11i		−	1.41323i	−0.707600	−	0.706613i	$-0.750222\pi$
$97$		−	1350.11i		−	1.41323i	0.707600	−	0.706613i	$-0.249778\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	851.130			0.838521			0.419261	−	0.907866i	$-0.362289\pi$
$101$	851.130			0.838521			0.419261	+	0.907866i	$0.362289\pi$
$102$		0			0
$103$			403.968i			0.386448i	0.981155	+	0.193224i	$0.0618944\pi$
$103$			403.968i			0.386448i	−0.981155	+	0.193224i	$0.938106\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	542.506i		−	0.490149i	−0.969504	−	0.245075i	$-0.921187\pi$
$107$		−	542.506i		−	0.490149i	0.969504	−	0.245075i	$-0.0788125\pi$
$108$		0			0
$109$	−5.88298			−0.00516961			−0.00258480	−	0.999997i	$-0.500823\pi$
$109$	−5.88298			−0.00516961			−0.00258480	+	0.999997i	$0.500823\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	2218.62i		−	1.84700i	−0.383603	−	0.923498i	$-0.625317\pi$
$113$		−	2218.62i		−	1.84700i	0.383603	−	0.923498i	$-0.374683\pi$
$114$		0			0
$115$			424.865i			0.344512i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	585.397	−	102.145i	0.450951	−	0.0786859i
$120$		0			0
$121$	111.105			0.0834747
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1370.76			0.980836
$126$		0			0
$127$	−2033.56			−1.42086			−0.710430	−	0.703767i	$-0.751500\pi$
$127$	−2033.56			−1.42086			−0.710430	+	0.703767i	$0.751500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2512.81			1.67592			0.837960	−	0.545732i	$-0.183748\pi$
$131$	2512.81			1.67592			0.837960	+	0.545732i	$0.183748\pi$
$132$		0			0
$133$	399.361	+	2288.75i	0.260368	+	1.49218i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			1062.51i			0.662599i	0.943526	+	0.331299i	$0.107487\pi$
$137$			1062.51i			0.662599i	−0.943526	+	0.331299i	$0.892513\pi$
$138$		0			0
$139$			1688.63i			1.03042i	0.857065	+	0.515208i	$0.172285\pi$
$139$			1688.63i			1.03042i	−0.857065	+	0.515208i	$0.827715\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2211.16			1.29305
$144$		0			0
$145$		−	2021.25i		−	1.15763i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			3134.13i			1.72321i	0.507583	+	0.861603i	$0.330539\pi$
$149$			3134.13i			1.72321i	−0.507583	+	0.861603i	$0.669461\pi$
$150$		0			0
$151$	1234.07			0.665083			0.332541	−	0.943089i	$-0.392094\pi$
$151$	1234.07			0.665083			0.332541	+	0.943089i	$0.392094\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	36.4437i		−	0.0188854i
$156$		0			0
$157$		−	2453.89i		−	1.24740i	−0.781664	−	0.623699i	$-0.785629\pi$
$157$		−	2453.89i		−	1.24740i	0.781664	−	0.623699i	$-0.214371\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	202.781	+	1162.14i	0.0992631	+	0.568880i
$162$		0			0
$163$	−1952.40			−0.938183			−0.469091	−	0.883150i	$-0.655419\pi$
$163$	−1952.40			−0.938183			−0.469091	+	0.883150i	$0.655419\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	548.691			0.254246			0.127123	−	0.991887i	$-0.459426\pi$
$167$	548.691			0.254246			0.127123	+	0.991887i	$0.459426\pi$
$168$		0			0
$169$	−1810.91			−0.824264
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2455.13			1.07896			0.539479	−	0.841999i	$-0.318621\pi$
$173$	2455.13			1.07896			0.539479	+	0.841999i	$0.318621\pi$
$174$		0			0
$175$	1468.89	−	256.305i	0.634502	−	0.110713i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			716.885i			0.299344i	0.988736	+	0.149672i	$0.0478217\pi$
$179$			716.885i			0.299344i	−0.988736	+	0.149672i	$0.952178\pi$
$180$		0			0
$181$			2029.15i			0.833290i	0.909069	+	0.416645i	$0.136794\pi$
$181$			2029.15i			0.833290i	−0.909069	+	0.416645i	$0.863206\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−300.296			−0.119342
$186$		0			0
$187$			1120.67i			0.438243i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			1097.05i			0.415600i	0.978171	+	0.207800i	$0.0666303\pi$
$191$			1097.05i			0.415600i	−0.978171	+	0.207800i	$0.933370\pi$
$192$		0			0
$193$	3636.04			1.35610			0.678051	−	0.735015i	$-0.262825\pi$
$193$	3636.04			1.35610			0.678051	+	0.735015i	$0.262825\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	1745.96i		−	0.631444i	−0.948852	−	0.315722i	$-0.897753\pi$
$197$		−	1745.96i		−	0.631444i	0.948852	−	0.315722i	$-0.102247\pi$
$198$		0			0
$199$		−	1614.59i		−	0.575150i	−0.957758	−	0.287575i	$-0.907151\pi$
$199$		−	1614.59i		−	0.575150i	0.957758	−	0.287575i	$-0.0928491\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−964.708	−	5528.77i	−0.333543	−	1.91155i
$204$		0			0
$205$	−1239.83			−0.422407
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4381.52			−1.45013
$210$		0			0
$211$	3228.35			1.05331			0.526656	−	0.850079i	$-0.323446\pi$
$211$	3228.35			1.05331			0.526656	+	0.850079i	$0.323446\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1500.59			0.475996
$216$		0			0
$217$	−17.3940	−	99.6853i	−0.00544138	−	0.0311847i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	2031.30i		−	0.618282i
$222$		0			0
$223$		−	5196.27i		−	1.56039i	−0.625534	−	0.780197i	$-0.715119\pi$
$223$		−	5196.27i		−	1.56039i	0.625534	−	0.780197i	$-0.284881\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2366.27			−0.691873			−0.345937	−	0.938258i	$-0.612439\pi$
$227$	−2366.27			−0.691873			−0.345937	+	0.938258i	$0.612439\pi$
$228$		0			0
$229$			3576.13i			1.03195i	0.856602	+	0.515977i	$0.172571\pi$
$229$			3576.13i			1.03195i	−0.856602	+	0.515977i	$0.827429\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	500.638i		−	0.140763i	−0.997520	−	0.0703817i	$-0.977578\pi$
$233$		−	500.638i		−	0.140763i	0.997520	−	0.0703817i	$-0.0224217\pi$
$234$		0			0
$235$	−3567.09			−0.990176
$236$		0			0
$237$		0			0
$238$		0			0
$239$			4519.92i			1.22330i	0.791128	+	0.611651i	$0.209494\pi$
$239$			4519.92i			1.22330i	−0.791128	+	0.611651i	$0.790506\pi$
$240$		0			0
$241$		−	284.612i		−	0.0760726i	−0.999276	−	0.0380363i	$-0.987890\pi$
$241$		−	284.612i		−	0.0760726i	0.999276	−	0.0380363i	$-0.0121102\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2152.62	+	774.805i	−0.561329	+	0.202043i
$246$		0			0
$247$	7941.87			2.04587
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1625.38			−0.408739			−0.204369	−	0.978894i	$-0.565514\pi$
$251$	−1625.38			−0.408739			−0.204369	+	0.978894i	$0.565514\pi$
$252$		0			0
$253$	−2224.78			−0.552847
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4911.97			−1.19222			−0.596110	−	0.802903i	$-0.703288\pi$
$257$	−4911.97			−1.19222			−0.596110	+	0.802903i	$0.703288\pi$
$258$		0			0
$259$	−821.405	+	143.326i	−0.197064	+	0.0343855i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	3816.72i		−	0.894864i	−0.894318	−	0.447432i	$-0.852339\pi$
$263$		−	3816.72i		−	0.894864i	0.894318	−	0.447432i	$-0.147661\pi$
$264$		0			0
$265$			2489.71i			0.577138i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2705.82			−0.613296			−0.306648	−	0.951823i	$-0.599208\pi$
$269$	−2705.82			−0.613296			−0.306648	+	0.951823i	$0.599208\pi$
$270$		0			0
$271$			6302.28i			1.41268i	0.707872	+	0.706340i	$0.249655\pi$
$271$			6302.28i			1.41268i	−0.707872	+	0.706340i	$0.750345\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			2812.01i			0.616620i
$276$		0			0
$277$	−5432.50			−1.17837			−0.589183	−	0.808000i	$-0.700550\pi$
$277$	−5432.50			−1.17837			−0.589183	+	0.808000i	$0.700550\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	67.3177i		−	0.0142912i	−0.999974	−	0.00714562i	$-0.997725\pi$
$281$		−	67.3177i		−	0.0142912i	0.999974	−	0.00714562i	$-0.00227454\pi$
$282$		0			0
$283$			1322.44i			0.277776i	0.990308	+	0.138888i	$0.0443528\pi$
$283$			1322.44i			0.277776i	−0.990308	+	0.138888i	$0.955647\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3391.33	+	591.748i	−0.697504	+	0.121707i
$288$		0			0
$289$	−3883.49			−0.790451
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6300.41			−1.25623			−0.628113	−	0.778122i	$-0.716172\pi$
$293$	−6300.41			−1.25623			−0.628113	+	0.778122i	$0.716172\pi$
$294$		0			0
$295$	−4681.88			−0.924033
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4032.59			0.779968
$300$		0			0
$301$	4104.59	−	716.204i	0.785995	−	0.137147i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			496.919i			0.0932902i
$306$		0			0
$307$		−	6071.64i		−	1.12875i	−0.825518	−	0.564376i	$-0.809117\pi$
$307$		−	6071.64i		−	1.12875i	0.825518	−	0.564376i	$-0.190883\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	6945.30			1.26634			0.633170	−	0.774013i	$-0.281753\pi$
$311$	6945.30			1.26634			0.633170	+	0.774013i	$0.281753\pi$
$312$		0			0
$313$		−	5951.98i		−	1.07484i	−0.843314	−	0.537421i	$-0.819399\pi$
$313$		−	5951.98i		−	1.07484i	0.843314	−	0.537421i	$-0.180601\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	7117.54i		−	1.26108i	−0.776159	−	0.630538i	$-0.782834\pi$
$317$		−	7117.54i		−	1.26108i	0.776159	−	0.630538i	$-0.217166\pi$
$318$		0			0
$319$	10584.1			1.85767
$320$		0			0
$321$		0			0
$322$		0			0
$323$			4025.13i			0.693387i
$324$		0			0
$325$		−	5097.00i		−	0.869940i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−9757.14	+	1702.51i	−1.63504	+	0.285296i
$330$		0			0
$331$	−7385.35			−1.22639			−0.613195	−	0.789931i	$-0.710116\pi$
$331$	−7385.35			−1.22639			−0.613195	+	0.789931i	$0.710116\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2837.11			−0.462709
$336$		0			0
$337$	3458.19			0.558991			0.279495	−	0.960147i	$-0.409833\pi$
$337$	3458.19			0.558991			0.279495	+	0.960147i	$0.409833\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	190.835			0.0303058
$342$		0			0
$343$	−5518.30	+	3146.75i	−0.868688	+	0.495359i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			1344.60i			0.208017i	0.994576	+	0.104008i	$0.0331669\pi$
$347$			1344.60i			0.208017i	−0.994576	+	0.104008i	$0.966833\pi$
$348$		0			0
$349$			5462.13i			0.837768i	0.908040	+	0.418884i	$0.137579\pi$
$349$			5462.13i			0.837768i	−0.908040	+	0.418884i	$0.862421\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−12042.9			−1.81580			−0.907902	−	0.419183i	$-0.862316\pi$
$353$	−12042.9			−1.81580			−0.907902	+	0.419183i	$0.862316\pi$
$354$		0			0
$355$		−	6243.89i		−	0.933497i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	9506.58i		−	1.39760i	−0.715318	−	0.698799i	$-0.753718\pi$
$359$		−	9506.58i		−	1.39760i	0.715318	−	0.698799i	$-0.246282\pi$
$360$		0			0
$361$	−8878.21			−1.29439
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	6384.59i		−	0.915575i
$366$		0			0
$367$			9381.08i			1.33430i	0.744923	+	0.667150i	$0.232486\pi$
$367$			9381.08i			1.33430i	−0.744923	+	0.667150i	$0.767514\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1188.29	+	6810.15i	0.166289	+	0.953006i
$372$		0			0
$373$	8404.46			1.16667			0.583333	−	0.812233i	$-0.301748\pi$
$373$	8404.46			1.16667			0.583333	+	0.812233i	$0.301748\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−19184.6			−2.62084
$378$		0			0
$379$	3578.24			0.484965			0.242482	−	0.970156i	$-0.422038\pi$
$379$	3578.24			0.484965			0.242482	+	0.970156i	$0.422038\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5774.08			0.770344			0.385172	−	0.922845i	$-0.374142\pi$
$383$	5774.08			0.770344			0.385172	+	0.922845i	$0.374142\pi$
$384$		0			0
$385$	−741.633	−	4250.32i	−0.0981744	−	0.562640i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			2464.93i			0.321277i	0.987013	+	0.160639i	$0.0513554\pi$
$389$			2464.93i			0.321277i	−0.987013	+	0.160639i	$0.948645\pi$
$390$		0			0
$391$			2043.81i			0.264348i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	863.092			0.109942
$396$		0			0
$397$		−	8175.01i		−	1.03348i	−0.856142	−	0.516740i	$-0.827145\pi$
$397$		−	8175.01i		−	1.03348i	0.856142	−	0.516740i	$-0.172855\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	9511.16i		−	1.18445i	−0.805772	−	0.592226i	$-0.798249\pi$
$401$		−	9511.16i		−	1.18445i	0.805772	−	0.592226i	$-0.201751\pi$
$402$		0			0
$403$	−345.904			−0.0427561
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	1572.48i		−	0.191511i
$408$		0			0
$409$			11798.2i			1.42636i	0.700981	+	0.713180i	$0.252746\pi$
$409$			11798.2i			1.42636i	−0.700981	+	0.713180i	$0.747254\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−12806.4	+	2234.58i	−1.52582	+	0.266239i
$414$		0			0
$415$	−4800.29			−0.567799
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6473.98			−0.754832			−0.377416	−	0.926044i	$-0.623187\pi$
$419$	−6473.98			−0.754832			−0.377416	+	0.926044i	$0.623187\pi$
$420$		0			0
$421$	6721.14			0.778072			0.389036	−	0.921222i	$-0.372808\pi$
$421$	6721.14			0.778072			0.389036	+	0.921222i	$0.372808\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2583.28			0.294841
$426$		0			0
$427$	237.171	+	1359.23i	0.0268794	+	0.154047i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	4295.83i		−	0.480099i	−0.970761	−	0.240049i	$-0.922836\pi$
$431$		−	4295.83i		−	0.480099i	0.970761	−	0.240049i	$-0.0771637\pi$
$432$		0			0
$433$			5369.78i			0.595970i	0.954571	+	0.297985i	$0.0963145\pi$
$433$			5369.78i			0.595970i	−0.954571	+	0.297985i	$0.903685\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−7990.77			−0.874715
$438$		0			0
$439$		−	1919.29i		−	0.208662i	−0.994543	−	0.104331i	$-0.966730\pi$
$439$		−	1919.29i		−	0.208662i	0.994543	−	0.104331i	$-0.0332702\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			9984.13i			1.07079i	0.844602	+	0.535395i	$0.179837\pi$
$443$			9984.13i			1.07079i	−0.844602	+	0.535395i	$0.820163\pi$
$444$		0			0
$445$	−1829.09			−0.194848
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	2542.74i		−	0.267259i	−0.991031	−	0.133629i	$-0.957337\pi$
$449$		−	2542.74i		−	0.267259i	0.991031	−	0.133629i	$-0.0426632\pi$
$450$		0			0
$451$		−	6492.27i		−	0.677847i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1344.27	+	7704.06i	0.138506	+	0.793784i
$456$		0			0
$457$	10949.3			1.12075			0.560377	−	0.828238i	$-0.310656\pi$
$457$	10949.3			1.12075			0.560377	+	0.828238i	$0.310656\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8412.41			−0.849902			−0.424951	−	0.905216i	$-0.639709\pi$
$461$	−8412.41			−0.849902			−0.424951	+	0.905216i	$0.639709\pi$
$462$		0			0
$463$	16374.8			1.64363			0.821815	−	0.569754i	$-0.192961\pi$
$463$	16374.8			1.64363			0.821815	+	0.569754i	$0.192961\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	10195.0			1.01021			0.505104	−	0.863058i	$-0.331454\pi$
$467$	10195.0			1.01021			0.505104	+	0.863058i	$0.331454\pi$
$468$		0			0
$469$	−7760.39	+	1354.10i	−0.764055	+	0.133319i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			7857.71i			0.763844i
$474$		0			0
$475$			10100.0i			0.975616i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7439.08			0.709604			0.354802	−	0.934942i	$-0.384548\pi$
$479$	7439.08			0.709604			0.354802	+	0.934942i	$0.384548\pi$
$480$		0			0
$481$			2850.24i			0.270187i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			9005.25i			0.843108i
$486$		0			0
$487$	−4081.97			−0.379819			−0.189909	−	0.981802i	$-0.560819\pi$
$487$	−4081.97			−0.379819			−0.189909	+	0.981802i	$0.560819\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	8145.58i		−	0.748686i	−0.927290	−	0.374343i	$-0.877868\pi$
$491$		−	8145.58i		−	0.748686i	0.927290	−	0.374343i	$-0.122132\pi$
$492$		0			0
$493$		−	9723.22i		−	0.888259i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2980.10	−	17079.0i	−0.268965	−	1.54145i
$498$		0			0
$499$	−4200.98			−0.376878			−0.188439	−	0.982085i	$-0.560343\pi$
$499$	−4200.98			−0.376878			−0.188439	+	0.982085i	$0.560343\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	6954.26			0.616451			0.308226	−	0.951313i	$-0.400265\pi$
$503$	6954.26			0.616451			0.308226	+	0.951313i	$0.400265\pi$
$504$		0			0
$505$	−5677.04			−0.500248
$506$		0			0
$507$		0			0
$508$		0			0
$509$	12719.0			1.10759			0.553793	−	0.832654i	$-0.313180\pi$
$509$	12719.0			1.10759			0.553793	+	0.832654i	$0.313180\pi$
$510$		0			0
$511$	−3047.25	−	17463.9i	−0.263802	−	1.51185i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	2694.47i		−	0.230548i
$516$		0			0
$517$		−	18678.8i		−	1.58896i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12516.3			1.05249			0.526246	−	0.850332i	$-0.323599\pi$
$521$	12516.3			1.05249			0.526246	+	0.850332i	$0.323599\pi$
$522$		0			0
$523$			14004.2i			1.17086i	0.810722	+	0.585431i	$0.199075\pi$
$523$			14004.2i			1.17086i	−0.810722	+	0.585431i	$0.800925\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	175.312i		−	0.0144909i
$528$		0			0
$529$	8109.58			0.666523
$530$		0			0
$531$		0			0
$532$		0			0
$533$			11767.8i			0.956320i
$534$		0			0
$535$			3618.52i			0.292415i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4057.21	−	11272.0i	−0.324223	−	0.900780i
$540$		0			0
$541$	1507.92			0.119835			0.0599173	−	0.998203i	$-0.480916\pi$
$541$	1507.92			0.119835			0.0599173	+	0.998203i	$0.480916\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	39.2395			0.00308410
$546$		0			0
$547$	−15198.8			−1.18803			−0.594015	−	0.804454i	$-0.702458\pi$
$547$	−15198.8			−1.18803			−0.594015	+	0.804454i	$0.702458\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	38015.3			2.93921
$552$		0			0
$553$	2360.83	−	411.939i	0.181542	−	0.0316771i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	1630.11i		−	0.124004i	−0.998076	−	0.0620018i	$-0.980252\pi$
$557$		−	1630.11i		−	0.124004i	0.998076	−	0.0620018i	$-0.0197484\pi$
$558$		0			0
$559$		−	14242.7i		−	1.07765i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	20439.9			1.53009			0.765043	−	0.643980i	$-0.222718\pi$
$563$	20439.9			1.53009			0.765043	+	0.643980i	$0.222718\pi$
$564$		0			0
$565$			14798.2i			1.10189i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			3587.85i			0.264342i	0.991227	+	0.132171i	$0.0421948\pi$
$569$			3587.85i			0.264342i	−0.991227	+	0.132171i	$0.957805\pi$
$570$		0			0
$571$	2733.33			0.200326			0.100163	−	0.994971i	$-0.468064\pi$
$571$	2733.33			0.200326			0.100163	+	0.994971i	$0.468064\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			5128.38i			0.371945i
$576$		0			0
$577$			22496.0i			1.62309i	0.584293	+	0.811543i	$0.301372\pi$
$577$			22496.0i			1.62309i	−0.584293	+	0.811543i	$0.698628\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−13130.3	+	2291.09i	−0.937586	+	0.163598i
$582$		0			0
$583$	−13037.2			−0.926148
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−15892.0			−1.11743			−0.558716	−	0.829359i	$-0.688706\pi$
$587$	−15892.0			−1.11743			−0.558716	+	0.829359i	$0.688706\pi$
$588$		0			0
$589$	685.426			0.0479499
$590$		0			0
$591$		0			0
$592$		0			0
$593$	20915.4			1.44839			0.724193	−	0.689597i	$-0.242212\pi$
$593$	20915.4			1.44839			0.724193	+	0.689597i	$0.242212\pi$
$594$		0			0
$595$	−3904.60	+	681.308i	−0.269030	+	0.0469427i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	5303.11i		−	0.361735i	−0.983507	−	0.180868i	$-0.942109\pi$
$599$		−	5303.11i		−	0.361735i	0.983507	−	0.180868i	$-0.0578905\pi$
$600$		0			0
$601$			10622.7i			0.720977i	0.932764	+	0.360488i	$0.117390\pi$
$601$			10622.7i			0.720977i	−0.932764	+	0.360488i	$0.882610\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−741.070			−0.0497996
$606$		0			0
$607$		−	4772.74i		−	0.319142i	−0.987186	−	0.159571i	$-0.948989\pi$
$607$		−	4772.74i		−	0.319142i	0.987186	−	0.159571i	$-0.0510112\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			33856.9i			2.24174i
$612$		0			0
$613$	−17746.6			−1.16929			−0.584647	−	0.811288i	$-0.698767\pi$
$613$	−17746.6			−1.16929			−0.584647	+	0.811288i	$0.698767\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			17932.4i			1.17007i	0.811009	+	0.585034i	$0.198919\pi$
$617$			17932.4i			1.17007i	−0.811009	+	0.585034i	$0.801081\pi$
$618$		0			0
$619$		−	21183.7i		−	1.37551i	−0.725941	−	0.687757i	$-0.758595\pi$
$619$		−	21183.7i		−	1.37551i	0.725941	−	0.687757i	$-0.241405\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−5003.16	+	872.994i	−0.321745	+	0.0561409i
$624$		0			0
$625$	920.903			0.0589378
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1444.57			−0.0915720
$630$		0			0
$631$	21849.4			1.37846			0.689231	−	0.724542i	$-0.257949\pi$
$631$	21849.4			1.37846			0.689231	+	0.724542i	$0.257949\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	13563.9			0.847662
$636$		0			0
$637$	7354.02	+	20431.5i	0.457421	+	1.27084i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			1183.05i			0.0728980i	0.999336	+	0.0364490i	$0.0116047\pi$
$641$			1183.05i			0.0728980i	−0.999336	+	0.0364490i	$0.988395\pi$
$642$		0			0
$643$			11781.0i			0.722545i	0.932460	+	0.361273i	$0.117658\pi$
$643$			11781.0i			0.722545i	−0.932460	+	0.361273i	$0.882342\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	19929.5			1.21099			0.605494	−	0.795850i	$-0.292976\pi$
$647$	19929.5			1.21099			0.605494	+	0.795850i	$0.292976\pi$
$648$		0			0
$649$		−	24516.3i		−	1.48282i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			3112.47i			0.186524i	0.995642	+	0.0932622i	$0.0297295\pi$
$653$			3112.47i			0.186524i	−0.995642	+	0.0932622i	$0.970271\pi$
$654$		0			0
$655$	−16760.5			−0.999826
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	26019.6i		−	1.53806i	−0.639213	−	0.769030i	$-0.720740\pi$
$659$		−	26019.6i		−	1.53806i	0.639213	−	0.769030i	$-0.279260\pi$
$660$		0			0
$661$		−	14030.0i		−	0.825574i	−0.910828	−	0.412787i	$-0.864555\pi$
$661$		−	14030.0i		−	0.825574i	0.910828	−	0.412787i	$-0.135445\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2663.74	−	15266.0i	−0.155331	−	0.890209i
$666$		0			0
$667$	19302.7			1.12055
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−2602.08			−0.149705
$672$		0			0
$673$	1528.46			0.0875452			0.0437726	−	0.999042i	$-0.486062\pi$
$673$	1528.46			0.0875452			0.0437726	+	0.999042i	$0.486062\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	17195.5			0.976183			0.488092	−	0.872792i	$-0.337693\pi$
$677$	17195.5			0.976183			0.488092	+	0.872792i	$0.337693\pi$
$678$		0			0
$679$	4298.05	+	24632.2i	0.242922	+	1.39219i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			7106.05i			0.398105i	0.979989	+	0.199052i	$0.0637864\pi$
$683$			7106.05i			0.398105i	−0.979989	+	0.199052i	$0.936214\pi$
$684$		0			0
$685$		−	7086.92i		−	0.395296i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	23630.9			1.30663
$690$		0			0
$691$		−	6292.69i		−	0.346433i	−0.984884	−	0.173217i	$-0.944584\pi$
$691$		−	6292.69i		−	0.346433i	0.984884	−	0.173217i	$-0.0554161\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	11263.2i		−	0.614729i
$696$		0			0
$697$	−5964.19			−0.324117
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	16206.3i		−	0.873188i	−0.899659	−	0.436594i	$-0.856185\pi$
$701$		−	16206.3i		−	0.873188i	0.899659	−	0.436594i	$-0.143815\pi$
$702$		0			0
$703$		−	5647.90i		−	0.303008i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−15528.5	+	2709.55i	−0.826040	+	0.144135i
$708$		0			0
$709$	2700.59			0.143050			0.0715251	−	0.997439i	$-0.477213\pi$
$709$	2700.59			0.143050			0.0715251	+	0.997439i	$0.477213\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	348.034			0.0182805
$714$		0			0
$715$	−14748.4			−0.771414
$716$		0			0
$717$		0			0
$718$		0			0
$719$	37780.1			1.95961			0.979804	−	0.199960i	$-0.0640811\pi$
$719$	37780.1			1.95961			0.979804	+	0.199960i	$0.0640811\pi$
$720$		0			0
$721$	−1286.02	−	7370.23i	−0.0664272	−	0.380696i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	24397.7i		−	1.24981i
$726$		0			0
$727$			1290.88i			0.0658541i	0.999458	+	0.0329271i	$0.0104829\pi$
$727$			1290.88i			0.0658541i	−0.999458	+	0.0329271i	$0.989517\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	7218.56			0.365237
$732$		0			0
$733$			20976.9i			1.05702i	0.848926	+	0.528512i	$0.177250\pi$
$733$			20976.9i			1.05702i	−0.848926	+	0.528512i	$0.822750\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	14856.3i		−	0.742522i
$738$		0			0
$739$	19887.3			0.989939			0.494969	−	0.868910i	$-0.335179\pi$
$739$	19887.3			0.989939			0.494969	+	0.868910i	$0.335179\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			24419.0i			1.20572i	0.797848	+	0.602858i	$0.205971\pi$
$743$			24419.0i			1.20572i	−0.797848	+	0.602858i	$0.794029\pi$
$744$		0			0
$745$		−	20904.6i		−	1.02804i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1727.05	+	9897.80i	0.0842526	+	0.482854i
$750$		0			0
$751$	−4166.80			−0.202462			−0.101231	−	0.994863i	$-0.532278\pi$
$751$	−4166.80			−0.202462			−0.101231	+	0.994863i	$0.532278\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8231.28			−0.396777
$756$		0			0
$757$	18265.8			0.876993			0.438496	−	0.898733i	$-0.355511\pi$
$757$	18265.8			0.876993			0.438496	+	0.898733i	$0.355511\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	31339.2			1.49283			0.746416	−	0.665480i	$-0.231773\pi$
$761$	31339.2			1.49283			0.746416	+	0.665480i	$0.231773\pi$
$762$		0			0
$763$	107.333	−	18.7283i	0.00509266	−	0.000888612i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			44437.9i			2.09199i
$768$		0			0
$769$		−	19632.9i		−	0.920653i	−0.887750	−	0.460326i	$-0.847732\pi$
$769$		−	19632.9i		−	0.920653i	0.887750	−	0.460326i	$-0.152268\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−2575.55			−0.119840			−0.0599199	−	0.998203i	$-0.519085\pi$
$773$	−2575.55			−0.119840			−0.0599199	+	0.998203i	$0.519085\pi$
$774$		0			0
$775$		−	439.898i		−	0.0203892i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	23318.4i		−	1.07249i
$780$		0			0
$781$	32695.7			1.49801
$782$		0			0
$783$		0			0
$784$		0			0
$785$			16367.4i			0.744177i
$786$		0			0
$787$		−	733.646i		−	0.0332295i	−0.999862	−	0.0166148i	$-0.994711\pi$
$787$		−	733.646i		−	0.0332295i	0.999862	−	0.0166148i	$-0.00528889\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	7062.94	+	40477.9i	0.317483	+	1.81951i
$792$		0			0
$793$	4716.49			0.211207
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−12408.4			−0.551479			−0.275739	−	0.961232i	$-0.588923\pi$
$797$	−12408.4			−0.551479			−0.275739	+	0.961232i	$0.588923\pi$
$798$		0			0
$799$	−17159.5			−0.759773
$800$		0			0
$801$		0			0
$802$		0			0
$803$	33432.5			1.46925
$804$		0			0
$805$	−1352.55	−	7751.49i	−0.0592187	−	0.339384i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			23954.1i			1.04101i	0.853858	+	0.520507i	$0.174257\pi$
$809$			23954.1i			1.04101i	−0.853858	+	0.520507i	$0.825743\pi$
$810$		0			0
$811$			42759.3i			1.85140i	0.378261	+	0.925699i	$0.376522\pi$
$811$			42759.3i			1.85140i	−0.378261	+	0.925699i	$0.623478\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	13022.5			0.559704
$816$		0			0
$817$			28222.7i			1.20855i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			22180.7i			0.942887i	0.881896	+	0.471443i	$0.156267\pi$
$821$			22180.7i			0.942887i	−0.881896	+	0.471443i	$0.843733\pi$
$822$		0			0
$823$	17415.1			0.737610			0.368805	−	0.929507i	$-0.379767\pi$
$823$	17415.1			0.737610			0.368805	+	0.929507i	$0.379767\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	29147.9i		−	1.22560i	−0.790237	−	0.612801i	$-0.790043\pi$
$827$		−	29147.9i		−	1.22560i	0.790237	−	0.612801i	$-0.209957\pi$
$828$		0			0
$829$			27289.2i			1.14330i	0.820499	+	0.571648i	$0.193696\pi$
$829$			27289.2i			1.14330i	−0.820499	+	0.571648i	$0.806304\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−10355.2	+	3727.19i	−0.430714	+	0.155030i
$834$		0			0
$835$	−3659.77			−0.151679
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−9144.35			−0.376279			−0.188139	−	0.982142i	$-0.560246\pi$
$839$	−9144.35			−0.376279			−0.188139	+	0.982142i	$0.560246\pi$
$840$		0			0
$841$	−67441.8			−2.76526
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12078.8			0.491742
$846$		0			0
$847$	−2027.06	+	353.700i	−0.0822323	+	0.0143486i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	2867.79i		−	0.115519i
$852$		0			0
$853$		−	31690.4i		−	1.27205i	−0.771668	−	0.636025i	$-0.780577\pi$
$853$		−	31690.4i		−	1.27205i	0.771668	−	0.636025i	$-0.219423\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−10928.0			−0.435583			−0.217792	−	0.975995i	$-0.569885\pi$
$857$	−10928.0			−0.435583			−0.217792	+	0.975995i	$0.569885\pi$
$858$		0			0
$859$		−	22475.0i		−	0.892709i	−0.894856	−	0.446355i	$-0.852722\pi$
$859$		−	22475.0i		−	0.892709i	0.894856	−	0.446355i	$-0.147278\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	38790.6i		−	1.53006i	−0.643992	−	0.765032i	$-0.722723\pi$
$863$		−	38790.6i		−	1.53006i	0.643992	−	0.765032i	$-0.277277\pi$
$864$		0			0
$865$	−16375.7			−0.643689
$866$		0			0
$867$		0			0
$868$		0			0
$869$			4519.52i			0.176426i
$870$		0			0
$871$			26928.3i			1.04756i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−25009.0	+	4363.78i	−0.966237	+	0.168597i
$876$		0			0
$877$	−42382.0			−1.63186			−0.815928	−	0.578153i	$-0.803774\pi$
$877$	−42382.0			−1.63186			−0.815928	+	0.578153i	$0.803774\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−22017.1			−0.841968			−0.420984	−	0.907068i	$-0.638315\pi$
$881$	−22017.1			−0.841968			−0.420984	+	0.907068i	$0.638315\pi$
$882$		0			0
$883$	32785.0			1.24949			0.624746	−	0.780828i	$-0.285203\pi$
$883$	32785.0			1.24949			0.624746	+	0.780828i	$0.285203\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	27596.5			1.04464			0.522322	−	0.852748i	$-0.325066\pi$
$887$	27596.5			1.04464			0.522322	+	0.852748i	$0.325066\pi$
$888$		0			0
$889$	37101.5	−	6473.79i	1.39971	−	0.244234i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	67089.1i		−	2.51405i
$894$		0			0
$895$		−	4781.63i		−	0.178583i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1655.74			−0.0614259
$900$		0			0
$901$			11976.7i			0.442844i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	13534.4i		−	0.497127i
$906$		0			0
$907$	29047.5			1.06340			0.531701	−	0.846932i	$-0.321553\pi$
$907$	29047.5			1.06340			0.531701	+	0.846932i	$0.321553\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	12198.7i		−	0.443647i	−0.975087	−	0.221823i	$-0.928799\pi$
$911$		−	12198.7i		−	0.443647i	0.975087	−	0.221823i	$-0.0712008\pi$
$912$		0			0
$913$		−	25136.3i		−	0.911163i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−45845.3	+	7999.48i	−1.65097	+	0.288076i
$918$		0			0
$919$	−4615.21			−0.165660			−0.0828302	−	0.996564i	$-0.526396\pi$
$919$	−4615.21			−0.165660			−0.0828302	+	0.996564i	$0.526396\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−59263.6			−2.11342
$924$		0			0
$925$	−3624.75			−0.128844
$926$		0			0
$927$		0			0
$928$		0			0
$929$	19211.9			0.678495			0.339248	−	0.940697i	$-0.389828\pi$
$929$	19211.9			0.678495			0.339248	+	0.940697i	$0.389828\pi$
$930$		0			0
$931$	−14572.4	−	40486.0i	−0.512986	−	1.42521i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	7474.86i		−	0.261448i
$936$		0			0
$937$			631.925i			0.0220321i	0.999939	+	0.0110161i	$0.00350659\pi$
$937$			631.925i			0.0220321i	−0.999939	+	0.0110161i	$0.996493\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−25685.2			−0.889814			−0.444907	−	0.895577i	$-0.646763\pi$
$941$	−25685.2			−0.889814			−0.444907	+	0.895577i	$0.646763\pi$
$942$		0			0
$943$		−	11840.2i		−	0.408877i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	27811.0i		−	0.954316i	−0.878817	−	0.477158i	$-0.841667\pi$
$947$		−	27811.0i		−	0.954316i	0.878817	−	0.477158i	$-0.158333\pi$
$948$		0			0
$949$	−60599.1			−2.07284
$950$		0			0
$951$		0			0
$952$		0			0
$953$			30431.9i			1.03440i	0.855863	+	0.517202i	$0.173026\pi$
$953$			30431.9i			1.03440i	−0.855863	+	0.517202i	$0.826974\pi$
$954$		0			0
$955$		−	7317.32i		−	0.247940i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3382.46	−	19385.0i	−0.113895	−	0.652737i
$960$		0			0
$961$	29761.1			0.998998
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−24252.4			−0.809028
$966$		0			0
$967$	−23229.3			−0.772495			−0.386247	−	0.922395i	$-0.626229\pi$
$967$	−23229.3			−0.772495			−0.386247	+	0.922395i	$0.626229\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−44322.5			−1.46486			−0.732428	−	0.680845i	$-0.761613\pi$
$971$	−44322.5			−1.46486			−0.732428	+	0.680845i	$0.761613\pi$
$972$		0			0
$973$	−5375.72	−	30808.4i	−0.177120	−	1.01508i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			39956.4i			1.30841i	0.756317	+	0.654206i	$0.226997\pi$
$977$			39956.4i			1.30841i	−0.756317	+	0.654206i	$0.773003\pi$
$978$		0			0
$979$		−	9577.92i		−	0.312678i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	27872.5			0.904370			0.452185	−	0.891924i	$-0.350645\pi$
$983$	27872.5			0.904370			0.452185	+	0.891924i	$0.350645\pi$
$984$		0			0
$985$			11645.6i			0.376709i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			14330.4i			0.460750i
$990$		0			0
$991$	8699.00			0.278842			0.139421	−	0.990233i	$-0.455476\pi$
$991$	8699.00			0.278842			0.139421	+	0.990233i	$0.455476\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			10769.3i			0.343125i
$996$		0			0
$997$			41387.1i			1.31469i	0.753591	+	0.657344i	$0.228320\pi$
$997$			41387.1i			1.31469i	−0.753591	+	0.657344i	$0.771680\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.4.f.e.377.6	yes	16
3.2	odd	2	inner	756.4.f.e.377.12	yes	16
7.6	odd	2	inner	756.4.f.e.377.11	yes	16
21.20	even	2	inner	756.4.f.e.377.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.4.f.e.377.5	✓	16	21.20	even	2	inner
756.4.f.e.377.6	yes	16	1.1	even	1	trivial
756.4.f.e.377.11	yes	16	7.6	odd	2	inner
756.4.f.e.377.12	yes	16	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	756.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-6.67001 q^{5} +(-18.2446 + 3.18348i) q^{7} -34.9270i q^{11} +63.3080i q^{13} -32.0860 q^{17} -125.448i q^{19} -63.6979i q^{23} -80.5110 q^{25} +303.036i q^{29} +5.46382i q^{31} +(121.692 - 21.2338i) q^{35} +45.0218 q^{37} +185.881 q^{41} -224.975 q^{43} +534.796 q^{47} +(322.731 - 116.163i) q^{49} -373.269i q^{53} +232.963i q^{55} +701.931 q^{59} -74.5006i q^{61} -422.265i q^{65} +425.353 q^{67} +936.115i q^{71} +957.209i q^{73} +(111.189 + 637.229i) q^{77} -129.399 q^{79} +719.682 q^{83} +214.014 q^{85} +274.227 q^{89} +(-201.540 - 1155.03i) q^{91} +836.739i q^{95} -1350.11i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 14 q^{7} + 388 q^{25} - 344 q^{37} - 404 q^{43} - 14 q^{49} + 1772 q^{67} - 1456 q^{79} - 4704 q^{85} + 2436 q^{91}+O(q^{100}) \) 16 * q + 14 * q^7 + 388 * q^25 - 344 * q^37 - 404 * q^43 - 14 * q^49 + 1772 * q^67 - 1456 * q^79 - 4704 * q^85 + 2436 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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