LMFDB - Embedded newform 1840.3.k.e.321.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,3,Mod(321,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1840.321");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.k (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:	$50.1363686423$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.2
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.e.321.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.39985 q^{3} +2.23607i q^{5} +8.04288i q^{7} +20.1584 q^{9} +O(q^{10})$	$q-5.39985 q^{3} +2.23607i q^{5} +8.04288i q^{7} +20.1584 q^{9} +13.7869i q^{11} -25.1879 q^{13} -12.0744i q^{15} -26.9746i q^{17} +15.1400i q^{19} -43.4304i q^{21} +(-21.4525 + 8.29404i) q^{23} -5.00000 q^{25} -60.2538 q^{27} -19.5567 q^{29} -10.6769 q^{31} -74.4471i q^{33} -17.9844 q^{35} +53.1540i q^{37} +136.011 q^{39} +3.97941 q^{41} -14.5235i q^{43} +45.0756i q^{45} -81.1768 q^{47} -15.6879 q^{49} +145.659i q^{51} +81.9833i q^{53} -30.8284 q^{55} -81.7536i q^{57} +31.6526 q^{59} +56.8359i q^{61} +162.132i q^{63} -56.3219i q^{65} +0.391452i q^{67} +(115.840 - 44.7866i) q^{69} -90.6432 q^{71} +60.1639 q^{73} +26.9993 q^{75} -110.886 q^{77} +17.5857i q^{79} +143.936 q^{81} -46.2826i q^{83} +60.3171 q^{85} +105.603 q^{87} +144.667i q^{89} -202.583i q^{91} +57.6537 q^{93} -33.8540 q^{95} -115.940i q^{97} +277.922i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 128 q^{9}+O(q^{10}) $ 48 * q + 128 * q^9	$ 48 q + 128 q^{9} - 8 q^{23} - 240 q^{25} + 72 q^{29} - 32 q^{31} + 40 q^{35} + 96 q^{39} - 104 q^{41} - 128 q^{47} - 344 q^{49} - 80 q^{55} - 248 q^{59} + 292 q^{69} - 208 q^{71} + 224 q^{73} - 288 q^{77} + 184 q^{81} + 48 q^{87} - 672 q^{93}+O(q^{100}) $ 48 * q + 128 * q^9 - 8 * q^23 - 240 * q^25 + 72 * q^29 - 32 * q^31 + 40 * q^35 + 96 * q^39 - 104 * q^41 - 128 * q^47 - 344 * q^49 - 80 * q^55 - 248 * q^59 + 292 * q^69 - 208 * q^71 + 224 * q^73 - 288 * q^77 + 184 * q^81 + 48 * q^87 - 672 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	−5.39985			−1.79995			−0.899976	−	0.435940i	$-0.856416\pi$
$3$	−5.39985			−1.79995			−0.899976	+	0.435940i	$0.856416\pi$
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$			8.04288i			1.14898i	0.818511	+	0.574491i	$0.194800\pi$
$7$			8.04288i			1.14898i	−0.818511	+	0.574491i	$0.805200\pi$
$8$		0			0
$9$	20.1584			2.23982
$10$		0			0
$11$			13.7869i			1.25335i	0.779280	+	0.626676i	$0.215585\pi$
$11$			13.7869i			1.25335i	−0.779280	+	0.626676i	$0.784415\pi$
$12$		0			0
$13$	−25.1879			−1.93753			−0.968765	−	0.247979i	$-0.920234\pi$
$13$	−25.1879			−1.93753			−0.968765	+	0.247979i	$0.920234\pi$
$14$		0			0
$15$		−	12.0744i		−	0.804963i
$16$		0			0
$17$		−	26.9746i		−	1.58674i	−0.608737	−	0.793372i	$-0.708324\pi$
$17$		−	26.9746i		−	1.58674i	0.608737	−	0.793372i	$-0.291676\pi$
$18$		0			0
$19$			15.1400i			0.796840i	0.917203	+	0.398420i	$0.130441\pi$
$19$			15.1400i			0.796840i	−0.917203	+	0.398420i	$0.869559\pi$
$20$		0			0
$21$		−	43.4304i		−	2.06811i
$22$		0			0
$23$	−21.4525	+	8.29404i	−0.932717	+	0.360610i
$23$	−21.4525	+	8.29404i	−0.932717	+	0.360610i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	−60.2538			−2.23162
$28$		0			0
$29$	−19.5567			−0.674369			−0.337184	−	0.941439i	$-0.609475\pi$
$29$	−19.5567			−0.674369			−0.337184	+	0.941439i	$0.609475\pi$
$30$		0			0
$31$	−10.6769			−0.344416			−0.172208	−	0.985061i	$-0.555090\pi$
$31$	−10.6769			−0.344416			−0.172208	+	0.985061i	$0.555090\pi$
$32$		0			0
$33$		−	74.4471i		−	2.25597i
$34$		0			0
$35$	−17.9844			−0.513840
$36$		0			0
$37$			53.1540i			1.43659i	0.695736	+	0.718297i	$0.255078\pi$
$37$			53.1540i			1.43659i	−0.695736	+	0.718297i	$0.744922\pi$
$38$		0			0
$39$	136.011			3.48746
$40$		0			0
$41$	3.97941			0.0970589			0.0485295	−	0.998822i	$-0.484547\pi$
$41$	3.97941			0.0970589			0.0485295	+	0.998822i	$0.484547\pi$
$42$		0			0
$43$		−	14.5235i		−	0.337756i	−0.985637	−	0.168878i	$-0.945986\pi$
$43$		−	14.5235i		−	0.337756i	0.985637	−	0.168878i	$-0.0540143\pi$
$44$		0			0
$45$			45.0756i			1.00168i
$46$		0			0
$47$	−81.1768			−1.72717			−0.863583	−	0.504208i	$-0.831785\pi$
$47$	−81.1768			−1.72717			−0.863583	+	0.504208i	$0.831785\pi$
$48$		0			0
$49$	−15.6879			−0.320160
$50$		0			0
$51$			145.659i			2.85606i
$52$		0			0
$53$			81.9833i			1.54685i	0.633885	+	0.773427i	$0.281459\pi$
$53$			81.9833i			1.54685i	−0.633885	+	0.773427i	$0.718541\pi$
$54$		0			0
$55$	−30.8284			−0.560516
$56$		0			0
$57$		−	81.7536i		−	1.43427i
$58$		0			0
$59$	31.6526			0.536485			0.268242	−	0.963351i	$-0.413557\pi$
$59$	31.6526			0.536485			0.268242	+	0.963351i	$0.413557\pi$
$60$		0			0
$61$			56.8359i			0.931736i	0.884854	+	0.465868i	$0.154258\pi$
$61$			56.8359i			0.931736i	−0.884854	+	0.465868i	$0.845742\pi$
$62$		0			0
$63$			162.132i			2.57352i
$64$		0			0
$65$		−	56.3219i		−	0.866490i
$66$		0			0
$67$			0.391452i			0.00584257i	0.999996	+	0.00292128i	$0.000929875\pi$
$67$			0.391452i			0.00584257i	−0.999996	+	0.00292128i	$0.999070\pi$
$68$		0			0
$69$	115.840	−	44.7866i	1.67884	−	0.649081i
$70$		0			0
$71$	−90.6432			−1.27666			−0.638332	−	0.769761i	$-0.720375\pi$
$71$	−90.6432			−1.27666			−0.638332	+	0.769761i	$0.720375\pi$
$72$		0			0
$73$	60.1639			0.824162			0.412081	−	0.911147i	$-0.364802\pi$
$73$	60.1639			0.824162			0.412081	+	0.911147i	$0.364802\pi$
$74$		0			0
$75$	26.9993			0.359990
$76$		0			0
$77$	−110.886			−1.44008
$78$		0			0
$79$			17.5857i			0.222604i	0.993787	+	0.111302i	$0.0355021\pi$
$79$			17.5857i			0.222604i	−0.993787	+	0.111302i	$0.964498\pi$
$80$		0			0
$81$	143.936			1.77699
$82$		0			0
$83$		−	46.2826i		−	0.557621i	−0.960346	−	0.278811i	$-0.910060\pi$
$83$		−	46.2826i		−	0.557621i	0.960346	−	0.278811i	$-0.0899402\pi$
$84$		0			0
$85$	60.3171			0.709613
$86$		0			0
$87$	105.603			1.21383
$88$		0			0
$89$			144.667i			1.62547i	0.582631	+	0.812737i	$0.302023\pi$
$89$			144.667i			1.62547i	−0.582631	+	0.812737i	$0.697977\pi$
$90$		0			0
$91$		−	202.583i		−	2.22619i
$92$		0			0
$93$	57.6537			0.619932
$94$		0			0
$95$	−33.8540			−0.356358
$96$		0			0
$97$		−	115.940i		−	1.19526i	−0.801772	−	0.597630i	$-0.796109\pi$
$97$		−	115.940i		−	1.19526i	0.801772	−	0.597630i	$-0.203891\pi$
$98$		0			0
$99$			277.922i			2.80729i
$100$		0			0
$101$	−62.4990			−0.618802			−0.309401	−	0.950932i	$-0.600128\pi$
$101$	−62.4990			−0.618802			−0.309401	+	0.950932i	$0.600128\pi$
$102$		0			0
$103$		−	50.6667i		−	0.491910i	−0.969281	−	0.245955i	$-0.920899\pi$
$103$		−	50.6667i		−	0.491910i	0.969281	−	0.245955i	$-0.0791015\pi$
$104$		0			0
$105$	97.1132			0.924888
$106$		0			0
$107$			77.8036i			0.727136i	0.931568	+	0.363568i	$0.118442\pi$
$107$			77.8036i			0.727136i	−0.931568	+	0.363568i	$0.881558\pi$
$108$		0			0
$109$			92.6231i			0.849753i	0.905251	+	0.424876i	$0.139682\pi$
$109$			92.6231i			0.849753i	−0.905251	+	0.424876i	$0.860318\pi$
$110$		0			0
$111$		−	287.024i		−	2.58580i
$112$		0			0
$113$		−	51.6680i		−	0.457239i	−0.973516	−	0.228619i	$-0.926579\pi$
$113$		−	51.6680i		−	0.457239i	0.973516	−	0.228619i	$-0.0734212\pi$
$114$		0			0
$115$	−18.5460	−	47.9692i	−0.161270	−	0.417124i
$116$		0			0
$117$	−507.748			−4.33973
$118$		0			0
$119$	216.954			1.82314
$120$		0			0
$121$	−69.0779			−0.570892
$122$		0			0
$123$	−21.4883			−0.174701
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	13.0929			0.103094			0.0515470	−	0.998671i	$-0.483585\pi$
$127$	13.0929			0.103094			0.0515470	+	0.998671i	$0.483585\pi$
$128$		0			0
$129$			78.4248i			0.607944i
$130$		0			0
$131$	7.13213			0.0544437			0.0272219	−	0.999629i	$-0.491334\pi$
$131$	7.13213			0.0544437			0.0272219	+	0.999629i	$0.491334\pi$
$132$		0			0
$133$	−121.769			−0.915555
$134$		0			0
$135$		−	134.732i		−	0.998012i
$136$		0			0
$137$			101.464i			0.740612i	0.928910	+	0.370306i	$0.120747\pi$
$137$			101.464i			0.740612i	−0.928910	+	0.370306i	$0.879253\pi$
$138$		0			0
$139$	16.1321			0.116058			0.0580290	−	0.998315i	$-0.481518\pi$
$139$	16.1321			0.116058			0.0580290	+	0.998315i	$0.481518\pi$
$140$		0			0
$141$	438.343			3.10881
$142$		0			0
$143$		−	347.262i		−	2.42841i
$144$		0			0
$145$		−	43.7301i		−	0.301587i
$146$		0			0
$147$	84.7121			0.576273
$148$		0			0
$149$		−	254.415i		−	1.70748i	−0.520697	−	0.853742i	$-0.674328\pi$
$149$		−	254.415i		−	1.70748i	0.520697	−	0.853742i	$-0.325672\pi$
$150$		0			0
$151$	−226.064			−1.49711			−0.748555	−	0.663072i	$-0.769252\pi$
$151$	−226.064			−1.49711			−0.748555	+	0.663072i	$0.769252\pi$
$152$		0			0
$153$		−	543.766i		−	3.55403i
$154$		0			0
$155$		−	23.8743i		−	0.154028i
$156$		0			0
$157$		−	281.723i		−	1.79441i	−0.441611	−	0.897206i	$-0.645593\pi$
$157$		−	281.723i		−	1.79441i	0.441611	−	0.897206i	$-0.354407\pi$
$158$		0			0
$159$		−	442.698i		−	2.78426i
$160$		0			0
$161$	−66.7079	−	172.540i	−0.414335	−	1.07167i
$162$		0			0
$163$	−29.8237			−0.182968			−0.0914838	−	0.995807i	$-0.529161\pi$
$163$	−29.8237			−0.182968			−0.0914838	+	0.995807i	$0.529161\pi$
$164$		0			0
$165$	166.469			1.00890
$166$		0			0
$167$	82.5024			0.494026			0.247013	−	0.969012i	$-0.420551\pi$
$167$	82.5024			0.494026			0.247013	+	0.969012i	$0.420551\pi$
$168$		0			0
$169$	465.430			2.75403
$170$		0			0
$171$			305.198i			1.78478i
$172$		0			0
$173$	302.299			1.74739			0.873696	−	0.486472i	$-0.161716\pi$
$173$	302.299			1.74739			0.873696	+	0.486472i	$0.161716\pi$
$174$		0			0
$175$		−	40.2144i		−	0.229796i
$176$		0			0
$177$	−170.919			−0.965647
$178$		0			0
$179$	201.373			1.12499			0.562495	−	0.826801i	$-0.309842\pi$
$179$	201.373			1.12499			0.562495	+	0.826801i	$0.309842\pi$
$180$		0			0
$181$		−	268.324i		−	1.48245i	−0.671256	−	0.741226i	$-0.734245\pi$
$181$		−	268.324i		−	1.48245i	0.671256	−	0.741226i	$-0.265755\pi$
$182$		0			0
$183$		−	306.906i		−	1.67708i
$184$		0			0
$185$	−118.856			−0.642465
$186$		0			0
$187$	371.896			1.98875
$188$		0			0
$189$		−	484.614i		−	2.56410i
$190$		0			0
$191$			102.219i			0.535179i	0.963533	+	0.267590i	$0.0862271\pi$
$191$			102.219i			0.535179i	−0.963533	+	0.267590i	$0.913773\pi$
$192$		0			0
$193$	131.590			0.681815			0.340907	−	0.940097i	$-0.389266\pi$
$193$	131.590			0.681815			0.340907	+	0.940097i	$0.389266\pi$
$194$		0			0
$195$			304.130i			1.55964i
$196$		0			0
$197$	−316.194			−1.60504			−0.802522	−	0.596622i	$-0.796509\pi$
$197$	−316.194			−1.60504			−0.802522	+	0.596622i	$0.796509\pi$
$198$		0			0
$199$			197.949i			0.994719i	0.867545	+	0.497359i	$0.165697\pi$
$199$			197.949i			0.994719i	−0.867545	+	0.497359i	$0.834303\pi$
$200$		0			0
$201$		−	2.11378i		−	0.0105163i
$202$		0			0
$203$		−	157.292i		−	0.774838i
$204$		0			0
$205$			8.89824i			0.0434061i
$206$		0			0
$207$	−432.448	+	167.195i	−2.08912	+	0.807704i
$208$		0			0
$209$	−208.733			−0.998722
$210$		0			0
$211$	−0.317225			−0.00150344			−0.000751719	−	1.00000i	$-0.500239\pi$
$211$	−0.317225			−0.00150344			−0.000751719		1.00000i	$0.500239\pi$
$212$		0			0
$213$	489.460			2.29793
$214$		0			0
$215$	32.4755			0.151049
$216$		0			0
$217$		−	85.8730i		−	0.395728i
$218$		0			0
$219$	−324.876			−1.48345
$220$		0			0
$221$			679.435i			3.07436i
$222$		0			0
$223$	−77.1949			−0.346165			−0.173083	−	0.984907i	$-0.555373\pi$
$223$	−77.1949			−0.346165			−0.173083	+	0.984907i	$0.555373\pi$
$224$		0			0
$225$	−100.792			−0.447965
$226$		0			0
$227$			146.979i			0.647485i	0.946145	+	0.323743i	$0.104941\pi$
$227$			146.979i			0.647485i	−0.946145	+	0.323743i	$0.895059\pi$
$228$		0			0
$229$			9.23275i			0.0403177i	0.999797	+	0.0201588i	$0.00641719\pi$
$229$			9.23275i			0.0403177i	−0.999797	+	0.0201588i	$0.993583\pi$
$230$		0			0
$231$	598.769			2.59207
$232$		0			0
$233$	−61.1352			−0.262383			−0.131191	−	0.991357i	$-0.541880\pi$
$233$	−61.1352			−0.262383			−0.131191	+	0.991357i	$0.541880\pi$
$234$		0			0
$235$		−	181.517i		−	0.772412i
$236$		0			0
$237$		−	94.9602i		−	0.400676i
$238$		0			0
$239$	200.384			0.838428			0.419214	−	0.907887i	$-0.362306\pi$
$239$	200.384			0.838428			0.419214	+	0.907887i	$0.362306\pi$
$240$		0			0
$241$			292.211i			1.21249i	0.795277	+	0.606246i	$0.207325\pi$
$241$			292.211i			1.21249i	−0.795277	+	0.606246i	$0.792675\pi$
$242$		0			0
$243$	−234.950			−0.966870
$244$		0			0
$245$		−	35.0791i		−	0.143180i
$246$		0			0
$247$		−	381.344i		−	1.54390i
$248$		0			0
$249$			249.919i			1.00369i
$250$		0			0
$251$		−	348.599i		−	1.38884i	−0.719570	−	0.694419i	$-0.755661\pi$
$251$		−	348.599i		−	1.38884i	0.719570	−	0.694419i	$-0.244339\pi$
$252$		0			0
$253$	−114.349	−	295.763i	−0.451972	−	1.16902i
$254$		0			0
$255$	−325.704			−1.27727
$256$		0			0
$257$	260.468			1.01349			0.506747	−	0.862095i	$-0.330848\pi$
$257$	260.468			1.01349			0.506747	+	0.862095i	$0.330848\pi$
$258$		0			0
$259$	−427.511			−1.65062
$260$		0			0
$261$	−394.232			−1.51047
$262$		0			0
$263$		−	118.680i		−	0.451256i	−0.974214	−	0.225628i	$-0.927557\pi$
$263$		−	118.680i		−	0.451256i	0.974214	−	0.225628i	$-0.0724433\pi$
$264$		0			0
$265$	−183.320			−0.691774
$266$		0			0
$267$		−	781.181i		−	2.92577i
$268$		0			0
$269$	86.9509			0.323238			0.161619	−	0.986853i	$-0.448329\pi$
$269$	86.9509			0.323238			0.161619	+	0.986853i	$0.448329\pi$
$270$		0			0
$271$	447.049			1.64963			0.824814	−	0.565404i	$-0.191280\pi$
$271$	447.049			1.64963			0.824814	+	0.565404i	$0.191280\pi$
$272$		0			0
$273$			1093.92i			4.00703i
$274$		0			0
$275$		−	68.9344i		−	0.250670i
$276$		0			0
$277$	152.840			0.551770			0.275885	−	0.961191i	$-0.411029\pi$
$277$	152.840			0.551770			0.275885	+	0.961191i	$0.411029\pi$
$278$		0			0
$279$	−215.229			−0.771432
$280$		0			0
$281$		−	167.202i		−	0.595027i	−0.954718	−	0.297513i	$-0.903843\pi$
$281$		−	167.202i		−	0.595027i	0.954718	−	0.297513i	$-0.0961573\pi$
$282$		0			0
$283$			315.576i			1.11511i	0.830140	+	0.557555i	$0.188260\pi$
$283$			315.576i			1.11511i	−0.830140	+	0.557555i	$0.811740\pi$
$284$		0			0
$285$	182.807			0.641427
$286$		0			0
$287$			32.0059i			0.111519i
$288$		0			0
$289$	−438.631			−1.51776
$290$		0			0
$291$			626.060i			2.15141i
$292$		0			0
$293$			111.448i			0.380368i	0.981748	+	0.190184i	$0.0609085\pi$
$293$			111.448i			0.380368i	−0.981748	+	0.190184i	$0.939092\pi$
$294$		0			0
$295$			70.7774i			0.239923i
$296$		0			0
$297$		−	830.712i		−	2.79701i
$298$		0			0
$299$	540.343	−	208.909i	1.80717	−	0.698694i
$300$		0			0
$301$	116.811			0.388075
$302$		0			0
$303$	337.485			1.11381
$304$		0			0
$305$	−127.089			−0.416685
$306$		0			0
$307$	−479.807			−1.56289			−0.781445	−	0.623974i	$-0.785517\pi$
$307$	−479.807			−1.56289			−0.781445	+	0.623974i	$0.785517\pi$
$308$		0			0
$309$			273.593i			0.885413i
$310$		0			0
$311$	−54.4076			−0.174944			−0.0874720	−	0.996167i	$-0.527879\pi$
$311$	−54.4076			−0.174944			−0.0874720	+	0.996167i	$0.527879\pi$
$312$		0			0
$313$			150.487i			0.480788i	0.970675	+	0.240394i	$0.0772766\pi$
$313$			150.487i			0.480788i	−0.970675	+	0.240394i	$0.922723\pi$
$314$		0			0
$315$	−362.537			−1.15091
$316$		0			0
$317$	47.9849			0.151372			0.0756859	−	0.997132i	$-0.475885\pi$
$317$	47.9849			0.151372			0.0756859	+	0.997132i	$0.475885\pi$
$318$		0			0
$319$		−	269.626i		−	0.845221i
$320$		0			0
$321$		−	420.128i		−	1.30881i
$322$		0			0
$323$	408.395			1.26438
$324$		0			0
$325$	125.939			0.387506
$326$		0			0
$327$		−	500.151i		−	1.52951i
$328$		0			0
$329$		−	652.895i		−	1.98448i
$330$		0			0
$331$	239.686			0.724128			0.362064	−	0.932153i	$-0.382072\pi$
$331$	239.686			0.724128			0.362064	+	0.932153i	$0.382072\pi$
$332$		0			0
$333$			1071.50i			3.21772i
$334$		0			0
$335$	−0.875314			−0.00261288
$336$		0			0
$337$			433.814i			1.28728i	0.765327	+	0.643642i	$0.222577\pi$
$337$			433.814i			1.28728i	−0.765327	+	0.643642i	$0.777423\pi$
$338$		0			0
$339$			279.000i			0.823008i
$340$		0			0
$341$		−	147.201i		−	0.431675i
$342$		0			0
$343$			267.925i			0.781124i
$344$		0			0
$345$	100.146	+	259.027i	0.290278	+	0.750802i
$346$		0			0
$347$	−141.901			−0.408938			−0.204469	−	0.978873i	$-0.565547\pi$
$347$	−141.901			−0.408938			−0.204469	+	0.978873i	$0.565547\pi$
$348$		0			0
$349$	−602.412			−1.72611			−0.863054	−	0.505111i	$-0.831451\pi$
$349$	−602.412			−1.72611			−0.863054	+	0.505111i	$0.831451\pi$
$350$		0			0
$351$	1517.67			4.32384
$352$		0			0
$353$	424.341			1.20210			0.601050	−	0.799212i	$-0.294749\pi$
$353$	424.341			1.20210			0.601050	+	0.799212i	$0.294749\pi$
$354$		0			0
$355$		−	202.684i		−	0.570942i
$356$		0			0
$357$	−1171.52			−3.28156
$358$		0			0
$359$		−	703.596i		−	1.95988i	−0.199297	−	0.979939i	$-0.563866\pi$
$359$		−	703.596i		−	1.95988i	0.199297	−	0.979939i	$-0.436134\pi$
$360$		0			0
$361$	131.781			0.365045
$362$		0			0
$363$	373.010			1.02758
$364$		0			0
$365$			134.530i			0.368577i
$366$		0			0
$367$		−	313.213i		−	0.853440i	−0.904384	−	0.426720i	$-0.859669\pi$
$367$		−	313.213i		−	0.853440i	0.904384	−	0.426720i	$-0.140331\pi$
$368$		0			0
$369$	80.2187			0.217395
$370$		0			0
$371$	−659.382			−1.77731
$372$		0			0
$373$		−	244.870i		−	0.656488i	−0.944593	−	0.328244i	$-0.893543\pi$
$373$		−	244.870i		−	0.656488i	0.944593	−	0.328244i	$-0.106457\pi$
$374$		0			0
$375$			60.3722i			0.160993i
$376$		0			0
$377$	492.592			1.30661
$378$		0			0
$379$			518.150i			1.36715i	0.729880	+	0.683576i	$0.239576\pi$
$379$			518.150i			1.36715i	−0.729880	+	0.683576i	$0.760424\pi$
$380$		0			0
$381$	−70.7000			−0.185564
$382$		0			0
$383$		−	149.868i		−	0.391300i	−0.980674	−	0.195650i	$-0.937318\pi$
$383$		−	149.868i		−	0.391300i	0.980674	−	0.195650i	$-0.0626816\pi$
$384$		0			0
$385$		−	247.949i		−	0.644023i
$386$		0			0
$387$		−	292.771i		−	0.756514i
$388$		0			0
$389$		−	302.997i		−	0.778912i	−0.921045	−	0.389456i	$-0.872663\pi$
$389$		−	302.997i		−	0.778912i	0.921045	−	0.389456i	$-0.127337\pi$
$390$		0			0
$391$	223.729	+	578.673i	0.572196	+	1.47998i
$392$		0			0
$393$	−38.5125			−0.0979961
$394$		0			0
$395$	−39.3228			−0.0995515
$396$		0			0
$397$	−393.412			−0.990962			−0.495481	−	0.868619i	$-0.665008\pi$
$397$	−393.412			−0.990962			−0.495481	+	0.868619i	$0.665008\pi$
$398$		0			0
$399$	657.534			1.64796
$400$		0			0
$401$			671.483i			1.67452i	0.546805	+	0.837260i	$0.315844\pi$
$401$			671.483i			1.67452i	−0.546805	+	0.837260i	$0.684156\pi$
$402$		0			0
$403$	268.929			0.667317
$404$		0			0
$405$			321.851i			0.794694i
$406$		0			0
$407$	−732.827			−1.80056
$408$		0			0
$409$	461.171			1.12756			0.563779	−	0.825926i	$-0.309347\pi$
$409$	461.171			1.12756			0.563779	+	0.825926i	$0.309347\pi$
$410$		0			0
$411$		−	547.890i		−	1.33307i
$412$		0			0
$413$			254.578i			0.616412i
$414$		0			0
$415$	103.491			0.249376
$416$		0			0
$417$	−87.1108			−0.208899
$418$		0			0
$419$		−	466.232i		−	1.11273i	−0.830940	−	0.556363i	$-0.812197\pi$
$419$		−	466.232i		−	1.11273i	0.830940	−	0.556363i	$-0.187803\pi$
$420$		0			0
$421$			496.007i			1.17816i	0.808073	+	0.589082i	$0.200511\pi$
$421$			496.007i			1.17816i	−0.808073	+	0.589082i	$0.799489\pi$
$422$		0			0
$423$	−1636.40			−3.86855
$424$		0			0
$425$			134.873i			0.317349i
$426$		0			0
$427$	−457.124			−1.07055
$428$		0			0
$429$			1875.17i			4.37102i
$430$		0			0
$431$		−	749.497i		−	1.73897i	−0.493957	−	0.869486i	$-0.664450\pi$
$431$		−	749.497i		−	1.73897i	0.493957	−	0.869486i	$-0.335550\pi$
$432$		0			0
$433$			75.7220i			0.174878i	0.996170	+	0.0874388i	$0.0278682\pi$
$433$			75.7220i			0.174878i	−0.996170	+	0.0874388i	$0.972132\pi$
$434$		0			0
$435$			236.136i			0.542842i
$436$		0			0
$437$	−125.571	−	324.790i	−0.287349	−	0.743226i
$438$		0			0
$439$	643.908			1.46676			0.733380	−	0.679819i	$-0.237942\pi$
$439$	643.908			1.46676			0.733380	+	0.679819i	$0.237942\pi$
$440$		0			0
$441$	−316.242			−0.717103
$442$		0			0
$443$	−659.518			−1.48875			−0.744377	−	0.667760i	$-0.767253\pi$
$443$	−659.518			−1.48875			−0.744377	+	0.667760i	$0.767253\pi$
$444$		0			0
$445$	−323.486			−0.726934
$446$		0			0
$447$			1373.80i			3.07339i
$448$		0			0
$449$	72.4823			0.161430			0.0807152	−	0.996737i	$-0.474280\pi$
$449$	72.4823			0.161430			0.0807152	+	0.996737i	$0.474280\pi$
$450$		0			0
$451$			54.8637i			0.121649i
$452$		0			0
$453$	1220.71			2.69473
$454$		0			0
$455$	452.990			0.995582
$456$		0			0
$457$			772.339i			1.69002i	0.534750	+	0.845010i	$0.320406\pi$
$457$			772.339i			1.69002i	−0.534750	+	0.845010i	$0.679594\pi$
$458$		0			0
$459$			1625.33i			3.54101i
$460$		0			0
$461$	−111.433			−0.241720			−0.120860	−	0.992670i	$-0.538565\pi$
$461$	−111.433			−0.241720			−0.120860	+	0.992670i	$0.538565\pi$
$462$		0			0
$463$	−317.712			−0.686203			−0.343102	−	0.939298i	$-0.611478\pi$
$463$	−317.712			−0.686203			−0.343102	+	0.939298i	$0.611478\pi$
$464$		0			0
$465$			128.918i			0.277242i
$466$		0			0
$467$		−	288.070i		−	0.616852i	−0.951248	−	0.308426i	$-0.900198\pi$
$467$		−	288.070i		−	0.616852i	0.951248	−	0.308426i	$-0.0998022\pi$
$468$		0			0
$469$	−3.14840			−0.00671301
$470$		0			0
$471$			1521.26i			3.22986i
$472$		0			0
$473$	200.234			0.423327
$474$		0			0
$475$		−	75.6998i		−	0.159368i
$476$		0			0
$477$			1652.65i			3.46468i
$478$		0			0
$479$		−	733.811i		−	1.53196i	−0.642862	−	0.765982i	$-0.722253\pi$
$479$		−	733.811i		−	1.53196i	0.642862	−	0.765982i	$-0.277747\pi$
$480$		0			0
$481$		−	1338.84i		−	2.78345i
$482$		0			0
$483$	360.213	+	931.689i	0.745782	+	1.92896i
$484$		0			0
$485$	259.250			0.534537
$486$		0			0
$487$	703.638			1.44484			0.722421	−	0.691454i	$-0.243029\pi$
$487$	703.638			1.44484			0.722421	+	0.691454i	$0.243029\pi$
$488$		0			0
$489$	161.044			0.329333
$490$		0			0
$491$	647.347			1.31843			0.659213	−	0.751956i	$-0.270890\pi$
$491$	647.347			1.31843			0.659213	+	0.751956i	$0.270890\pi$
$492$		0			0
$493$			527.535i			1.07005i
$494$		0			0
$495$	−621.452			−1.25546
$496$		0			0
$497$		−	729.032i		−	1.46686i
$498$		0			0
$499$	−177.883			−0.356480			−0.178240	−	0.983987i	$-0.557040\pi$
$499$	−177.883			−0.356480			−0.178240	+	0.983987i	$0.557040\pi$
$500$		0			0
$501$	−445.501			−0.889223
$502$		0			0
$503$			714.277i			1.42003i	0.704184	+	0.710017i	$0.251313\pi$
$503$			714.277i			1.42003i	−0.704184	+	0.710017i	$0.748687\pi$
$504$		0			0
$505$		−	139.752i		−	0.276737i
$506$		0			0
$507$	−2513.26			−4.95711
$508$		0			0
$509$	−557.237			−1.09477			−0.547385	−	0.836881i	$-0.684376\pi$
$509$	−557.237			−1.09477			−0.547385	+	0.836881i	$0.684376\pi$
$510$		0			0
$511$			483.890i			0.946948i
$512$		0			0
$513$		−	912.241i		−	1.77825i
$514$		0			0
$515$	113.294			0.219989
$516$		0			0
$517$		−	1119.17i		−	2.16475i
$518$		0			0
$519$	−1632.37			−3.14522
$520$		0			0
$521$		−	682.440i		−	1.30987i	−0.755687	−	0.654933i	$-0.772697\pi$
$521$		−	682.440i		−	1.30987i	0.755687	−	0.654933i	$-0.227303\pi$
$522$		0			0
$523$		−	181.095i		−	0.346262i	−0.984899	−	0.173131i	$-0.944612\pi$
$523$		−	181.095i		−	0.346262i	0.984899	−	0.173131i	$-0.0553885\pi$
$524$		0			0
$525$			217.152i			0.413622i
$526$		0			0
$527$			288.006i			0.546500i
$528$		0			0
$529$	391.418	−	355.855i	0.739920	−	0.672694i
$530$		0			0
$531$	638.067			1.20163
$532$		0			0
$533$	−100.233			−0.188055
$534$		0			0
$535$	−173.974			−0.325185
$536$		0			0
$537$	−1087.39			−2.02493
$538$		0			0
$539$		−	216.286i		−	0.401273i
$540$		0			0
$541$	70.2247			0.129805			0.0649026	−	0.997892i	$-0.479326\pi$
$541$	70.2247			0.129805			0.0649026	+	0.997892i	$0.479326\pi$
$542$		0			0
$543$			1448.91i			2.66834i
$544$		0			0
$545$	−207.111			−0.380021
$546$		0			0
$547$	−390.468			−0.713836			−0.356918	−	0.934136i	$-0.616172\pi$
$547$	−390.468			−0.713836			−0.356918	+	0.934136i	$0.616172\pi$
$548$		0			0
$549$			1145.72i			2.08693i
$550$		0			0
$551$		−	296.088i		−	0.537364i
$552$		0			0
$553$	−141.440			−0.255768
$554$		0			0
$555$	641.805			1.15640
$556$		0			0
$557$			1084.94i			1.94783i	0.226913	+	0.973915i	$0.427137\pi$
$557$			1084.94i			1.94783i	−0.226913	+	0.973915i	$0.572863\pi$
$558$		0			0
$559$			365.816i			0.654412i
$560$		0			0
$561$	−2008.18			−3.57965
$562$		0			0
$563$			350.964i			0.623383i	0.950183	+	0.311691i	$0.100895\pi$
$563$			350.964i			0.623383i	−0.950183	+	0.311691i	$0.899105\pi$
$564$		0			0
$565$	115.533			0.204483
$566$		0			0
$567$			1157.66i			2.04173i
$568$		0			0
$569$		−	735.062i		−	1.29185i	−0.763401	−	0.645924i	$-0.776472\pi$
$569$		−	735.062i		−	1.29185i	0.763401	−	0.645924i	$-0.223528\pi$
$570$		0			0
$571$		−	271.220i		−	0.474992i	−0.971389	−	0.237496i	$-0.923673\pi$
$571$		−	271.220i		−	0.474992i	0.971389	−	0.237496i	$-0.0763266\pi$
$572$		0			0
$573$		−	551.969i		−	0.963296i
$574$		0			0
$575$	107.262	−	41.4702i	0.186543	−	0.0721221i
$576$		0			0
$577$	−759.911			−1.31700			−0.658501	−	0.752579i	$-0.728809\pi$
$577$	−759.911			−1.31700			−0.658501	+	0.752579i	$0.728809\pi$
$578$		0			0
$579$	−710.568			−1.22723
$580$		0			0
$581$	372.245			0.640697
$582$		0			0
$583$	−1130.29			−1.93875
$584$		0			0
$585$		−	1135.36i		−	1.94079i
$586$		0			0
$587$	−292.685			−0.498611			−0.249305	−	0.968425i	$-0.580202\pi$
$587$	−292.685			−0.498611			−0.249305	+	0.968425i	$0.580202\pi$
$588$		0			0
$589$		−	161.648i		−	0.274445i
$590$		0			0
$591$	1707.40			2.88900
$592$		0			0
$593$	497.470			0.838903			0.419452	−	0.907778i	$-0.362222\pi$
$593$	497.470			0.838903			0.419452	+	0.907778i	$0.362222\pi$
$594$		0			0
$595$			485.123i			0.815333i
$596$		0			0
$597$		−	1068.90i		−	1.79045i
$598$		0			0
$599$	−5.08743			−0.00849321			−0.00424660	−	0.999991i	$-0.501352\pi$
$599$	−5.08743			−0.00849321			−0.00424660	+	0.999991i	$0.501352\pi$
$600$		0			0
$601$	1020.13			1.69738			0.848692	−	0.528888i	$-0.177391\pi$
$601$	1020.13			1.69738			0.848692	+	0.528888i	$0.177391\pi$
$602$		0			0
$603$			7.89106i			0.0130863i
$604$		0			0
$605$		−	154.463i		−	0.255310i
$606$		0			0
$607$	319.519			0.526390			0.263195	−	0.964743i	$-0.415224\pi$
$607$	319.519			0.526390			0.263195	+	0.964743i	$0.415224\pi$
$608$		0			0
$609$			849.354i			1.39467i
$610$		0			0
$611$	2044.67			3.34644
$612$		0			0
$613$		−	417.181i		−	0.680557i	−0.940325	−	0.340279i	$-0.889479\pi$
$613$		−	417.181i		−	0.680557i	0.940325	−	0.340279i	$-0.110521\pi$
$614$		0			0
$615$		−	48.0492i		−	0.0781288i
$616$		0			0
$617$		−	164.906i		−	0.267271i	−0.991031	−	0.133636i	$-0.957335\pi$
$617$		−	164.906i		−	0.267271i	0.991031	−	0.133636i	$-0.0426652\pi$
$618$		0			0
$619$			703.432i			1.13640i	0.822890	+	0.568201i	$0.192360\pi$
$619$			703.432i			1.13640i	−0.822890	+	0.568201i	$0.807640\pi$
$620$		0			0
$621$	1292.59	−	499.748i	2.08147	−	0.804746i
$622$		0			0
$623$	−1163.54			−1.86764
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	1127.13			1.79765
$628$		0			0
$629$	1433.81			2.27951
$630$		0			0
$631$			580.844i			0.920514i	0.887786	+	0.460257i	$0.152243\pi$
$631$			580.844i			0.920514i	−0.887786	+	0.460257i	$0.847757\pi$
$632$		0			0
$633$	1.71297			0.00270612
$634$		0			0
$635$			29.2767i			0.0461051i
$636$		0			0
$637$	395.144			0.620320
$638$		0			0
$639$	−1827.22			−2.85950
$640$		0			0
$641$			91.7850i			0.143190i	0.997434	+	0.0715952i	$0.0228090\pi$
$641$			91.7850i			0.143190i	−0.997434	+	0.0715952i	$0.977191\pi$
$642$		0			0
$643$		−	361.493i		−	0.562198i	−0.959679	−	0.281099i	$-0.909301\pi$
$643$		−	361.493i		−	0.562198i	0.959679	−	0.281099i	$-0.0906989\pi$
$644$		0			0
$645$	−175.363			−0.271881
$646$		0			0
$647$	221.902			0.342970			0.171485	−	0.985187i	$-0.445143\pi$
$647$	221.902			0.342970			0.171485	+	0.985187i	$0.445143\pi$
$648$		0			0
$649$			436.391i			0.672404i
$650$		0			0
$651$			463.701i			0.712291i
$652$		0			0
$653$	−196.932			−0.301581			−0.150790	−	0.988566i	$-0.548182\pi$
$653$	−196.932			−0.301581			−0.150790	+	0.988566i	$0.548182\pi$
$654$		0			0
$655$			15.9479i			0.0243480i
$656$		0			0
$657$	1212.81			1.84598
$658$		0			0
$659$		−	237.033i		−	0.359685i	−0.983695	−	0.179843i	$-0.942441\pi$
$659$		−	237.033i		−	0.359685i	0.983695	−	0.179843i	$-0.0575589\pi$
$660$		0			0
$661$		−	187.404i		−	0.283515i	−0.989901	−	0.141758i	$-0.954725\pi$
$661$		−	187.404i		−	0.283515i	0.989901	−	0.141758i	$-0.0452754\pi$
$662$		0			0
$663$		−	3668.85i		−	5.53371i
$664$		0			0
$665$		−	272.283i		−	0.409449i
$666$		0			0
$667$	419.540	−	162.204i	0.628995	−	0.243184i
$668$		0			0
$669$	416.841			0.623081
$670$		0			0
$671$	−783.590			−1.16779
$672$		0			0
$673$	1023.47			1.52076			0.760378	−	0.649481i	$-0.225014\pi$
$673$	1023.47			1.52076			0.760378	+	0.649481i	$0.225014\pi$
$674$		0			0
$675$	301.269			0.446325
$676$		0			0
$677$			328.952i			0.485897i	0.970039	+	0.242948i	$0.0781146\pi$
$677$			328.952i			0.485897i	−0.970039	+	0.242948i	$0.921885\pi$
$678$		0			0
$679$	932.493			1.37333
$680$		0			0
$681$		−	793.666i		−	1.16544i
$682$		0			0
$683$	852.623			1.24835			0.624175	−	0.781285i	$-0.285435\pi$
$683$	852.623			1.24835			0.624175	+	0.781285i	$0.285435\pi$
$684$		0			0
$685$	−226.880			−0.331212
$686$		0			0
$687$		−	49.8555i		−	0.0725699i
$688$		0			0
$689$		−	2064.99i		−	2.99708i
$690$		0			0
$691$	152.451			0.220624			0.110312	−	0.993897i	$-0.464815\pi$
$691$	152.451			0.220624			0.110312	+	0.993897i	$0.464815\pi$
$692$		0			0
$693$	−2235.29			−3.22552
$694$		0			0
$695$			36.0724i			0.0519028i
$696$		0			0
$697$		−	107.343i		−	0.154008i
$698$		0			0
$699$	330.121			0.472276
$700$		0			0
$701$			347.450i			0.495649i	0.968805	+	0.247824i	$0.0797156\pi$
$701$			347.450i			0.495649i	−0.968805	+	0.247824i	$0.920284\pi$
$702$		0			0
$703$	−804.750			−1.14474
$704$		0			0
$705$			980.164i			1.39030i
$706$		0			0
$707$		−	502.672i		−	0.710992i
$708$		0			0
$709$		−	565.772i		−	0.797986i	−0.916954	−	0.398993i	$-0.869360\pi$
$709$		−	565.772i		−	0.797986i	0.916954	−	0.398993i	$-0.130640\pi$
$710$		0			0
$711$			354.500i			0.498593i
$712$		0			0
$713$	229.046	−	88.5546i	0.321243	−	0.124200i
$714$		0			0
$715$	776.502			1.08602
$716$		0			0
$717$	−1082.05			−1.50913
$718$		0			0
$719$	742.441			1.03260			0.516301	−	0.856407i	$-0.327309\pi$
$719$	742.441			1.03260			0.516301	+	0.856407i	$0.327309\pi$
$720$		0			0
$721$	407.506			0.565195
$722$		0			0
$723$		−	1577.90i		−	2.18243i
$724$		0			0
$725$	97.7835			0.134874
$726$		0			0
$727$			732.792i			1.00797i	0.863713	+	0.503984i	$0.168133\pi$
$727$			732.792i			1.00797i	−0.863713	+	0.503984i	$0.831867\pi$
$728$		0			0
$729$	−26.7322			−0.0366697
$730$		0			0
$731$	−391.766			−0.535932
$732$		0			0
$733$			238.339i			0.325156i	0.986696	+	0.162578i	$0.0519809\pi$
$733$			238.339i			0.325156i	−0.986696	+	0.162578i	$0.948019\pi$
$734$		0			0
$735$			189.422i			0.257717i
$736$		0			0
$737$	−5.39690			−0.00732280
$738$		0			0
$739$	−226.514			−0.306514			−0.153257	−	0.988186i	$-0.548976\pi$
$739$	−226.514			−0.306514			−0.153257	+	0.988186i	$0.548976\pi$
$740$		0			0
$741$			2059.20i			2.77895i
$742$		0			0
$743$		−	1075.55i		−	1.44758i	−0.690022	−	0.723788i	$-0.742399\pi$
$743$		−	1075.55i		−	1.44758i	0.690022	−	0.723788i	$-0.257601\pi$
$744$		0			0
$745$	568.889			0.763610
$746$		0			0
$747$		−	932.983i		−	1.24897i
$748$		0			0
$749$	−625.764			−0.835467
$750$		0			0
$751$			821.759i			1.09422i	0.837061	+	0.547110i	$0.184272\pi$
$751$			821.759i			1.09422i	−0.837061	+	0.547110i	$0.815728\pi$
$752$		0			0
$753$			1882.38i			2.49984i
$754$		0			0
$755$		−	505.494i		−	0.669528i
$756$		0			0
$757$		−	798.872i		−	1.05531i	−0.849458	−	0.527657i	$-0.823071\pi$
$757$		−	798.872i		−	1.05531i	0.849458	−	0.527657i	$-0.176929\pi$
$758$		0			0
$759$	617.467	+	1597.08i	0.813527	+	2.10418i
$760$		0			0
$761$	−965.013			−1.26809			−0.634043	−	0.773298i	$-0.718606\pi$
$761$	−965.013			−1.26809			−0.634043	+	0.773298i	$0.718606\pi$
$762$		0			0
$763$	−744.956			−0.976351
$764$		0			0
$765$	1215.90			1.58941
$766$		0			0
$767$	−797.263			−1.03946
$768$		0			0
$769$		−	184.486i		−	0.239904i	−0.992780	−	0.119952i	$-0.961726\pi$
$769$		−	184.486i		−	0.239904i	0.992780	−	0.119952i	$-0.0382740\pi$
$770$		0			0
$771$	−1406.49			−1.82424
$772$		0			0
$773$			898.807i			1.16275i	0.813635	+	0.581376i	$0.197485\pi$
$773$			898.807i			1.16275i	−0.813635	+	0.581376i	$0.802515\pi$
$774$		0			0
$775$	53.3845			0.0688832
$776$		0			0
$777$	2308.50			2.97104
$778$		0			0
$779$			60.2482i			0.0773404i
$780$		0			0
$781$		−	1249.69i		−	1.60011i
$782$		0			0
$783$	1178.37			1.50494
$784$		0			0
$785$	629.951			0.802486
$786$		0			0
$787$		−	522.240i		−	0.663583i	−0.943353	−	0.331791i	$-0.892347\pi$
$787$		−	522.240i		−	0.663583i	0.943353	−	0.331791i	$-0.107653\pi$
$788$		0			0
$789$			640.856i			0.812238i
$790$		0			0
$791$	415.559			0.525359
$792$		0			0
$793$		−	1431.58i		−	1.80527i
$794$		0			0
$795$	989.902			1.24516
$796$		0			0
$797$		−	772.292i		−	0.968999i	−0.874792	−	0.484499i	$-0.839002\pi$
$797$		−	772.292i		−	0.968999i	0.874792	−	0.484499i	$-0.160998\pi$
$798$		0			0
$799$			2189.71i			2.74057i
$800$		0			0
$801$			2916.26i			3.64078i
$802$		0			0
$803$			829.471i			1.03297i
$804$		0			0
$805$	385.810	−	149.163i	0.479268	−	0.185296i
$806$		0			0
$807$	−469.522			−0.581812
$808$		0			0
$809$	−730.064			−0.902427			−0.451214	−	0.892416i	$-0.649009\pi$
$809$	−730.064			−0.902427			−0.451214	+	0.892416i	$0.649009\pi$
$810$		0			0
$811$	−1002.64			−1.23630			−0.618150	−	0.786060i	$-0.712118\pi$
$811$	−1002.64			−1.23630			−0.618150	+	0.786060i	$0.712118\pi$
$812$		0			0
$813$	−2414.00			−2.96925
$814$		0			0
$815$		−	66.6879i		−	0.0818256i
$816$		0			0
$817$	219.885			0.269137
$818$		0			0
$819$		−	4083.76i		−	4.98627i
$820$		0			0
$821$	−1481.16			−1.80409			−0.902046	−	0.431640i	$-0.857935\pi$
$821$	−1481.16			−1.80409			−0.902046	+	0.431640i	$0.857935\pi$
$822$		0			0
$823$	−823.606			−1.00074			−0.500368	−	0.865813i	$-0.666802\pi$
$823$	−823.606			−1.00074			−0.500368	+	0.865813i	$0.666802\pi$
$824$		0			0
$825$			372.235i			0.451195i
$826$		0			0
$827$			236.233i			0.285651i	0.989748	+	0.142825i	$0.0456187\pi$
$827$			236.233i			0.285651i	−0.989748	+	0.142825i	$0.954381\pi$
$828$		0			0
$829$	−269.757			−0.325400			−0.162700	−	0.986676i	$-0.552020\pi$
$829$	−269.757			−0.325400			−0.162700	+	0.986676i	$0.552020\pi$
$830$		0			0
$831$	−825.316			−0.993160
$832$		0			0
$833$			423.174i			0.508012i
$834$		0			0
$835$			184.481i			0.220935i
$836$		0			0
$837$	643.324			0.768607
$838$		0			0
$839$			869.577i			1.03644i	0.855246	+	0.518222i	$0.173406\pi$
$839$			869.577i			1.03644i	−0.855246	+	0.518222i	$0.826594\pi$
$840$		0			0
$841$	−458.536			−0.545227
$842$		0			0
$843$			902.869i			1.07102i
$844$		0			0
$845$			1040.73i			1.23164i
$846$		0			0
$847$		−	555.585i		−	0.655944i
$848$		0			0
$849$		−	1704.06i		−	2.00714i
$850$		0			0
$851$	−440.861	−	1140.29i	−0.518051	−	1.33994i
$852$		0			0
$853$	−188.879			−0.221429			−0.110715	−	0.993852i	$-0.535314\pi$
$853$	−188.879			−0.221429			−0.110715	+	0.993852i	$0.535314\pi$
$854$		0			0
$855$	−682.443			−0.798179
$856$		0			0
$857$	−929.185			−1.08423			−0.542115	−	0.840304i	$-0.682376\pi$
$857$	−929.185			−1.08423			−0.542115	+	0.840304i	$0.682376\pi$
$858$		0			0
$859$	−124.076			−0.144442			−0.0722210	−	0.997389i	$-0.523009\pi$
$859$	−124.076			−0.144442			−0.0722210	+	0.997389i	$0.523009\pi$
$860$		0			0
$861$		−	172.827i		−	0.200729i
$862$		0			0
$863$	−177.001			−0.205100			−0.102550	−	0.994728i	$-0.532700\pi$
$863$	−177.001			−0.205100			−0.102550	+	0.994728i	$0.532700\pi$
$864$		0			0
$865$			675.961i			0.781458i
$866$		0			0
$867$	2368.54			2.73189
$868$		0			0
$869$	−242.452			−0.279001
$870$		0			0
$871$		−	9.85986i		−	0.0113202i
$872$		0			0
$873$		−	2337.17i		−	2.67717i
$874$		0			0
$875$	89.9221			0.102768
$876$		0			0
$877$	376.880			0.429737			0.214869	−	0.976643i	$-0.431068\pi$
$877$	376.880			0.429737			0.214869	+	0.976643i	$0.431068\pi$
$878$		0			0
$879$		−	601.802i		−	0.684644i
$880$		0			0
$881$			109.423i			0.124203i	0.998070	+	0.0621015i	$0.0197802\pi$
$881$			109.423i			0.124203i	−0.998070	+	0.0621015i	$0.980220\pi$
$882$		0			0
$883$	−193.790			−0.219468			−0.109734	−	0.993961i	$-0.535000\pi$
$883$	−193.790			−0.219468			−0.109734	+	0.993961i	$0.535000\pi$
$884$		0			0
$885$		−	382.188i		−	0.431850i
$886$		0			0
$887$	462.603			0.521537			0.260768	−	0.965401i	$-0.416024\pi$
$887$	462.603			0.521537			0.260768	+	0.965401i	$0.416024\pi$
$888$		0			0
$889$			105.305i			0.118453i
$890$		0			0
$891$			1984.43i			2.22719i
$892$		0			0
$893$		−	1229.01i		−	1.37627i
$894$		0			0
$895$			450.284i			0.503111i
$896$		0			0
$897$	−2917.77	+	1128.08i	−3.25281	+	1.25761i
$898$		0			0
$899$	208.805			0.232263
$900$		0			0
$901$	2211.47			2.45446
$902$		0			0
$903$	−630.761			−0.698517
$904$		0			0
$905$	599.990			0.662972
$906$		0			0
$907$			410.320i			0.452393i	0.974082	+	0.226196i	$0.0726291\pi$
$907$			410.320i			0.452393i	−0.974082	+	0.226196i	$0.927371\pi$
$908$		0			0
$909$	−1259.88			−1.38601
$910$		0			0
$911$		−	159.182i		−	0.174733i	−0.996176	−	0.0873666i	$-0.972155\pi$
$911$		−	159.182i		−	0.174733i	0.996176	−	0.0873666i	$-0.0278452\pi$
$912$		0			0
$913$	638.092			0.698896
$914$		0			0
$915$	686.262			0.750013
$916$		0			0
$917$			57.3628i			0.0625549i
$918$		0			0
$919$		−	592.465i		−	0.644685i	−0.946623	−	0.322342i	$-0.895530\pi$
$919$		−	592.465i		−	0.644685i	0.946623	−	0.322342i	$-0.104470\pi$
$920$		0			0
$921$	2590.89			2.81313
$922$		0			0
$923$	2283.11			2.47358
$924$		0			0
$925$		−	265.770i		−	0.287319i
$926$		0			0
$927$		−	1021.36i		−	1.10179i
$928$		0			0
$929$	−1048.70			−1.12885			−0.564424	−	0.825485i	$-0.690902\pi$
$929$	−1048.70			−1.12885			−0.564424	+	0.825485i	$0.690902\pi$
$930$		0			0
$931$		−	237.514i		−	0.255117i
$932$		0			0
$933$	293.793			0.314891
$934$		0			0
$935$			831.585i			0.889395i
$936$		0			0
$937$			322.200i			0.343863i	0.985109	+	0.171932i	$0.0550008\pi$
$937$			322.200i			0.343863i	−0.985109	+	0.171932i	$0.944999\pi$
$938$		0			0
$939$		−	812.606i		−	0.865395i
$940$		0			0
$941$		−	147.366i		−	0.156605i	−0.996930	−	0.0783026i	$-0.975050\pi$
$941$		−	147.366i		−	0.156605i	0.996930	−	0.0783026i	$-0.0249500\pi$
$942$		0			0
$943$	−85.3683	+	33.0054i	−0.0905284	+	0.0350004i
$944$		0			0
$945$	1083.63			1.14670
$946$		0			0
$947$	412.468			0.435552			0.217776	−	0.975999i	$-0.430120\pi$
$947$	412.468			0.435552			0.217776	+	0.975999i	$0.430120\pi$
$948$		0			0
$949$	−1515.40			−1.59684
$950$		0			0
$951$	−259.111			−0.272462
$952$		0			0
$953$			319.732i			0.335501i	0.985830	+	0.167750i	$0.0536502\pi$
$953$			319.732i			0.335501i	−0.985830	+	0.167750i	$0.946350\pi$
$954$		0			0
$955$	−228.569			−0.239339
$956$		0			0
$957$			1455.94i			1.52136i
$958$		0			0
$959$	−816.061			−0.850950
$960$		0			0
$961$	−847.004			−0.881378
$962$		0			0
$963$			1568.40i			1.62866i
$964$		0			0
$965$			294.245i			0.304917i
$966$		0			0
$967$	203.799			0.210754			0.105377	−	0.994432i	$-0.466395\pi$
$967$	203.799			0.210754			0.105377	+	0.994432i	$0.466395\pi$
$968$		0			0
$969$	−2205.27			−2.27582
$970$		0			0
$971$		−	102.425i		−	0.105484i	−0.998608	−	0.0527421i	$-0.983204\pi$
$971$		−	102.425i		−	0.105484i	0.998608	−	0.0527421i	$-0.0167961\pi$
$972$		0			0
$973$			129.748i			0.133349i
$974$		0			0
$975$	−680.055			−0.697492
$976$		0			0
$977$			1339.39i			1.37092i	0.728110	+	0.685461i	$0.240399\pi$
$977$			1339.39i			1.37092i	−0.728110	+	0.685461i	$0.759601\pi$
$978$		0			0
$979$	−1994.51			−2.03729
$980$		0			0
$981$			1867.13i			1.90330i
$982$		0			0
$983$			372.816i			0.379263i	0.981855	+	0.189632i	$0.0607294\pi$
$983$			372.816i			0.379263i	−0.981855	+	0.189632i	$0.939271\pi$
$984$		0			0
$985$		−	707.031i		−	0.717798i
$986$		0			0
$987$			3525.54i			3.57197i
$988$		0			0
$989$	120.458	+	311.565i	0.121798	+	0.315030i
$990$		0			0
$991$	−337.629			−0.340695			−0.170348	−	0.985384i	$-0.554489\pi$
$991$	−337.629			−0.340695			−0.170348	+	0.985384i	$0.554489\pi$
$992$		0			0
$993$	−1294.27			−1.30339
$994$		0			0
$995$	−442.628			−0.444852
$996$		0			0
$997$	1080.17			1.08342			0.541711	−	0.840565i	$-0.317777\pi$
$997$	1080.17			1.08342			0.541711	+	0.840565i	$0.317777\pi$
$998$		0			0
$999$		−	3202.73i		−	3.20594i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.e.321.2		48
4.3	odd	2		920.3.k.a.321.48	yes	48
23.22	odd	2	inner	1840.3.k.e.321.1		48
92.91	even	2		920.3.k.a.321.47	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.3.k.a.321.47	✓	48	92.91	even	2
920.3.k.a.321.48	yes	48	4.3	odd	2
1840.3.k.e.321.1		48	23.22	odd	2	inner
1840.3.k.e.321.2		48	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.39985 q^{3} +2.23607i q^{5} +8.04288i q^{7} +20.1584 q^{9} +O(q^{10})\)	\(q-5.39985 q^{3} +2.23607i q^{5} +8.04288i q^{7} +20.1584 q^{9} +13.7869i q^{11} -25.1879 q^{13} -12.0744i q^{15} -26.9746i q^{17} +15.1400i q^{19} -43.4304i q^{21} +(-21.4525 + 8.29404i) q^{23} -5.00000 q^{25} -60.2538 q^{27} -19.5567 q^{29} -10.6769 q^{31} -74.4471i q^{33} -17.9844 q^{35} +53.1540i q^{37} +136.011 q^{39} +3.97941 q^{41} -14.5235i q^{43} +45.0756i q^{45} -81.1768 q^{47} -15.6879 q^{49} +145.659i q^{51} +81.9833i q^{53} -30.8284 q^{55} -81.7536i q^{57} +31.6526 q^{59} +56.8359i q^{61} +162.132i q^{63} -56.3219i q^{65} +0.391452i q^{67} +(115.840 - 44.7866i) q^{69} -90.6432 q^{71} +60.1639 q^{73} +26.9993 q^{75} -110.886 q^{77} +17.5857i q^{79} +143.936 q^{81} -46.2826i q^{83} +60.3171 q^{85} +105.603 q^{87} +144.667i q^{89} -202.583i q^{91} +57.6537 q^{93} -33.8540 q^{95} -115.940i q^{97} +277.922i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 128 q^{9}+O(q^{10}) \) 48 * q + 128 * q^9	\( 48 q + 128 q^{9} - 8 q^{23} - 240 q^{25} + 72 q^{29} - 32 q^{31} + 40 q^{35} + 96 q^{39} - 104 q^{41} - 128 q^{47} - 344 q^{49} - 80 q^{55} - 248 q^{59} + 292 q^{69} - 208 q^{71} + 224 q^{73} - 288 q^{77} + 184 q^{81} + 48 q^{87} - 672 q^{93}+O(q^{100}) \) 48 * q + 128 * q^9 - 8 * q^23 - 240 * q^25 + 72 * q^29 - 32 * q^31 + 40 * q^35 + 96 * q^39 - 104 * q^41 - 128 * q^47 - 344 * q^49 - 80 * q^55 - 248 * q^59 + 292 * q^69 - 208 * q^71 + 224 * q^73 - 288 * q^77 + 184 * q^81 + 48 * q^87 - 672 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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