LMFDB - Embedded newform 6030.2.d.l.2411.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6030,2,Mod(2411,6030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6030, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6030.2411");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6030.d (of order $2$, degree $1$, 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:	$48.1497924188$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2411.2
Character	$\chi$	$=$	6030.2411
Dual form			6030.2.d.l.2411.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.69493i q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.69493i q^{7} +1.00000 q^{8} +1.00000 q^{10} +5.10258 q^{11} -6.05495i q^{13} -3.69493i q^{14} +1.00000 q^{16} -0.252749i q^{17} +4.17889 q^{19} +1.00000 q^{20} +5.10258 q^{22} +5.65818i q^{23} +1.00000 q^{25} -6.05495i q^{26} -3.69493i q^{28} -9.12065i q^{29} -7.22881i q^{31} +1.00000 q^{32} -0.252749i q^{34} -3.69493i q^{35} -5.83907 q^{37} +4.17889 q^{38} +1.00000 q^{40} -9.99657 q^{41} -5.80931i q^{43} +5.10258 q^{44} +5.65818i q^{46} +13.0372i q^{47} -6.65251 q^{49} +1.00000 q^{50} -6.05495i q^{52} -3.38087 q^{53} +5.10258 q^{55} -3.69493i q^{56} -9.12065i q^{58} +11.2319i q^{59} +2.77473i q^{61} -7.22881i q^{62} +1.00000 q^{64} -6.05495i q^{65} +(6.06213 - 5.50005i) q^{67} -0.252749i q^{68} -3.69493i q^{70} +8.07879i q^{71} +2.89962 q^{73} -5.83907 q^{74} +4.17889 q^{76} -18.8537i q^{77} +8.22636i q^{79} +1.00000 q^{80} -9.99657 q^{82} +12.6131i q^{83} -0.252749i q^{85} -5.80931i q^{86} +5.10258 q^{88} -2.69800i q^{89} -22.3726 q^{91} +5.65818i q^{92} +13.0372i q^{94} +4.17889 q^{95} -3.99115i q^{97} -6.65251 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{2} + 24 q^{4} + 24 q^{5} + 24 q^{8}+O(q^{10}) $ 24 * q + 24 * q^2 + 24 * q^4 + 24 * q^5 + 24 * q^8	$ 24 q + 24 q^{2} + 24 q^{4} + 24 q^{5} + 24 q^{8} + 24 q^{10} + 12 q^{11} + 24 q^{16} + 4 q^{19} + 24 q^{20} + 12 q^{22} + 24 q^{25} + 24 q^{32} - 16 q^{37} + 4 q^{38} + 24 q^{40} + 8 q^{41} + 12 q^{44} - 20 q^{49} + 24 q^{50} + 24 q^{53} + 12 q^{55} + 24 q^{64} - 32 q^{67} - 4 q^{73} - 16 q^{74} + 4 q^{76} + 24 q^{80} + 8 q^{82} + 12 q^{88} + 4 q^{95} - 20 q^{98}+O(q^{100}) $ 24 * q + 24 * q^2 + 24 * q^4 + 24 * q^5 + 24 * q^8 + 24 * q^10 + 12 * q^11 + 24 * q^16 + 4 * q^19 + 24 * q^20 + 12 * q^22 + 24 * q^25 + 24 * q^32 - 16 * q^37 + 4 * q^38 + 24 * q^40 + 8 * q^41 + 12 * q^44 - 20 * q^49 + 24 * q^50 + 24 * q^53 + 12 * q^55 + 24 * q^64 - 32 * q^67 - 4 * q^73 - 16 * q^74 + 4 * q^76 + 24 * q^80 + 8 * q^82 + 12 * q^88 + 4 * q^95 - 20 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6030\mathbb{Z}\right)^\times$.

$n$	$1207$	$3151$	$4691$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$		−	3.69493i		−	1.39655i	−0.715829	−	0.698276i	$-0.753951\pi$
$7$		−	3.69493i		−	1.39655i	0.715829	−	0.698276i	$-0.246049\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$	1.00000			0.316228
$11$	5.10258			1.53849			0.769243	−	0.638957i	$-0.220634\pi$
$11$	5.10258			1.53849			0.769243	+	0.638957i	$0.220634\pi$
$12$		0			0
$13$		−	6.05495i		−	1.67934i	−0.543096	−	0.839671i	$-0.682748\pi$
$13$		−	6.05495i		−	1.67934i	0.543096	−	0.839671i	$-0.317252\pi$
$14$		−	3.69493i		−	0.987512i
$15$		0			0
$16$	1.00000			0.250000
$17$		−	0.252749i		−	0.0613006i	−0.999530	−	0.0306503i	$-0.990242\pi$
$17$		−	0.252749i		−	0.0613006i	0.999530	−	0.0306503i	$-0.00975782\pi$
$18$		0			0
$19$	4.17889			0.958703			0.479352	−	0.877623i	$-0.340872\pi$
$19$	4.17889			0.958703			0.479352	+	0.877623i	$0.340872\pi$
$20$	1.00000			0.223607
$21$		0			0
$22$	5.10258			1.08787
$23$			5.65818i			1.17981i	0.807472	+	0.589907i	$0.200835\pi$
$23$			5.65818i			1.17981i	−0.807472	+	0.589907i	$0.799165\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		−	6.05495i		−	1.18747i
$27$		0			0
$28$		−	3.69493i		−	0.698276i
$29$		−	9.12065i		−	1.69366i	−0.531862	−	0.846831i	$-0.678508\pi$
$29$		−	9.12065i		−	1.69366i	0.531862	−	0.846831i	$-0.321492\pi$
$30$		0			0
$31$		−	7.22881i		−	1.29833i	−0.760647	−	0.649166i	$-0.775118\pi$
$31$		−	7.22881i		−	1.29833i	0.760647	−	0.649166i	$-0.224882\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$		−	0.252749i		−	0.0433461i
$35$		−	3.69493i		−	0.624557i
$36$		0			0
$37$	−5.83907			−0.959937			−0.479969	−	0.877286i	$-0.659352\pi$
$37$	−5.83907			−0.959937			−0.479969	+	0.877286i	$0.659352\pi$
$38$	4.17889			0.677906
$39$		0			0
$40$	1.00000			0.158114
$41$	−9.99657			−1.56120			−0.780601	−	0.625029i	$-0.785087\pi$
$41$	−9.99657			−1.56120			−0.780601	+	0.625029i	$0.785087\pi$
$42$		0			0
$43$		−	5.80931i		−	0.885911i	−0.896544	−	0.442955i	$-0.853930\pi$
$43$		−	5.80931i		−	0.885911i	0.896544	−	0.442955i	$-0.146070\pi$
$44$	5.10258			0.769243
$45$		0			0
$46$			5.65818i			0.834254i
$47$			13.0372i			1.90166i	0.309706	+	0.950832i	$0.399769\pi$
$47$			13.0372i			1.90166i	−0.309706	+	0.950832i	$0.600231\pi$
$48$		0			0
$49$	−6.65251			−0.950358
$50$	1.00000			0.141421
$51$		0			0
$52$		−	6.05495i		−	0.839671i
$53$	−3.38087			−0.464398			−0.232199	−	0.972668i	$-0.574592\pi$
$53$	−3.38087			−0.464398			−0.232199	+	0.972668i	$0.574592\pi$
$54$		0			0
$55$	5.10258			0.688032
$56$		−	3.69493i		−	0.493756i
$57$		0			0
$58$		−	9.12065i		−	1.19760i
$59$			11.2319i			1.46226i	0.682237	+	0.731131i	$0.261007\pi$
$59$			11.2319i			1.46226i	−0.682237	+	0.731131i	$0.738993\pi$
$60$		0			0
$61$			2.77473i			0.355268i	0.984097	+	0.177634i	$0.0568444\pi$
$61$			2.77473i			0.355268i	−0.984097	+	0.177634i	$0.943156\pi$
$62$		−	7.22881i		−	0.918059i
$63$		0			0
$64$	1.00000			0.125000
$65$		−	6.05495i		−	0.751024i
$66$		0			0
$67$	6.06213	−	5.50005i	0.740607	−	0.671938i
$67$	6.06213	−	5.50005i	0.740607	−	0.671938i
$68$		−	0.252749i		−	0.0306503i
$69$		0			0
$70$		−	3.69493i		−	0.441629i
$71$			8.07879i			0.958776i	0.877603	+	0.479388i	$0.159141\pi$
$71$			8.07879i			0.958776i	−0.877603	+	0.479388i	$0.840859\pi$
$72$		0			0
$73$	2.89962			0.339374			0.169687	−	0.985498i	$-0.445724\pi$
$73$	2.89962			0.339374			0.169687	+	0.985498i	$0.445724\pi$
$74$	−5.83907			−0.678778
$75$		0			0
$76$	4.17889			0.479352
$77$		−	18.8537i		−	2.14858i
$78$		0			0
$79$			8.22636i			0.925537i	0.886479	+	0.462769i	$0.153144\pi$
$79$			8.22636i			0.925537i	−0.886479	+	0.462769i	$0.846856\pi$
$80$	1.00000			0.111803
$81$		0			0
$82$	−9.99657			−1.10394
$83$			12.6131i			1.38447i	0.721671	+	0.692236i	$0.243374\pi$
$83$			12.6131i			1.38447i	−0.721671	+	0.692236i	$0.756626\pi$
$84$		0			0
$85$		−	0.252749i		−	0.0274145i
$86$		−	5.80931i		−	0.626433i
$87$		0			0
$88$	5.10258			0.543937
$89$		−	2.69800i		−	0.285988i	−0.989724	−	0.142994i	$-0.954327\pi$
$89$		−	2.69800i		−	0.285988i	0.989724	−	0.142994i	$-0.0456730\pi$
$90$		0			0
$91$	−22.3726			−2.34529
$92$			5.65818i			0.589907i
$93$		0			0
$94$			13.0372i			1.34468i
$95$	4.17889			0.428745
$96$		0			0
$97$		−	3.99115i		−	0.405240i	−0.979257	−	0.202620i	$-0.935054\pi$
$97$		−	3.99115i		−	0.405240i	0.979257	−	0.202620i	$-0.0649456\pi$
$98$	−6.65251			−0.672005
$99$		0			0
$100$	1.00000			0.100000
$101$	−1.22506			−0.121898			−0.0609490	−	0.998141i	$-0.519413\pi$
$101$	−1.22506			−0.121898			−0.0609490	+	0.998141i	$0.519413\pi$
$102$		0			0
$103$	5.92251			0.583562			0.291781	−	0.956485i	$-0.405752\pi$
$103$	5.92251			0.583562			0.291781	+	0.956485i	$0.405752\pi$
$104$		−	6.05495i		−	0.593737i
$105$		0			0
$106$	−3.38087			−0.328379
$107$		−	7.10549i		−	0.686914i	−0.939168	−	0.343457i	$-0.888402\pi$
$107$		−	7.10549i		−	0.686914i	0.939168	−	0.343457i	$-0.111598\pi$
$108$		0			0
$109$			3.14887i			0.301607i	0.988564	+	0.150803i	$0.0481861\pi$
$109$			3.14887i			0.301607i	−0.988564	+	0.150803i	$0.951814\pi$
$110$	5.10258			0.486512
$111$		0			0
$112$		−	3.69493i		−	0.349138i
$113$	2.07969			0.195641			0.0978203	−	0.995204i	$-0.468813\pi$
$113$	2.07969			0.195641			0.0978203	+	0.995204i	$0.468813\pi$
$114$		0			0
$115$			5.65818i			0.527628i
$116$		−	9.12065i		−	0.846831i
$117$		0			0
$118$			11.2319i			1.03398i
$119$	−0.933890			−0.0856095
$120$		0			0
$121$	15.0363			1.36694
$122$			2.77473i			0.251213i
$123$		0			0
$124$		−	7.22881i		−	0.649166i
$125$	1.00000			0.0894427
$126$		0			0
$127$	−10.2156			−0.906486			−0.453243	−	0.891387i	$-0.649733\pi$
$127$	−10.2156			−0.906486			−0.453243	+	0.891387i	$0.649733\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$		−	6.05495i		−	0.531054i
$131$		−	17.2121i		−	1.50383i	−0.659262	−	0.751913i	$-0.729131\pi$
$131$		−	17.2121i		−	1.50383i	0.659262	−	0.751913i	$-0.270869\pi$
$132$		0			0
$133$		−	15.4407i		−	1.33888i
$134$	6.06213	−	5.50005i	0.523688	−	0.475132i
$135$		0			0
$136$		−	0.252749i		−	0.0216730i
$137$	19.1870			1.63926			0.819628	−	0.572895i	$-0.194180\pi$
$137$	19.1870			1.63926			0.819628	+	0.572895i	$0.194180\pi$
$138$		0			0
$139$			6.43228i			0.545579i	0.962074	+	0.272790i	$0.0879463\pi$
$139$			6.43228i			0.545579i	−0.962074	+	0.272790i	$0.912054\pi$
$140$		−	3.69493i		−	0.312279i
$141$		0			0
$142$			8.07879i			0.677957i
$143$		−	30.8959i		−	2.58364i
$144$		0			0
$145$		−	9.12065i		−	0.757429i
$146$	2.89962			0.239974
$147$		0			0
$148$	−5.83907			−0.479969
$149$			14.0925i			1.15450i	0.816566	+	0.577252i	$0.195875\pi$
$149$			14.0925i			1.15450i	−0.816566	+	0.577252i	$0.804125\pi$
$150$		0			0
$151$	−22.8467			−1.85924			−0.929621	−	0.368517i	$-0.879866\pi$
$151$	−22.8467			−1.85924			−0.929621	+	0.368517i	$0.879866\pi$
$152$	4.17889			0.338953
$153$		0			0
$154$		−	18.8537i		−	1.51927i
$155$		−	7.22881i		−	0.580632i
$156$		0			0
$157$	−19.3930			−1.54773			−0.773865	−	0.633351i	$-0.781679\pi$
$157$	−19.3930			−1.54773			−0.773865	+	0.633351i	$0.781679\pi$
$158$			8.22636i			0.654454i
$159$		0			0
$160$	1.00000			0.0790569
$161$	20.9066			1.64767
$162$		0			0
$163$	−9.65798			−0.756471			−0.378236	−	0.925709i	$-0.623469\pi$
$163$	−9.65798			−0.756471			−0.378236	+	0.925709i	$0.623469\pi$
$164$	−9.99657			−0.780601
$165$		0			0
$166$			12.6131i			0.978970i
$167$		−	17.3264i		−	1.34075i	−0.742020	−	0.670377i	$-0.766132\pi$
$167$		−	17.3264i		−	1.34075i	0.742020	−	0.670377i	$-0.233868\pi$
$168$		0			0
$169$	−23.6624			−1.82019
$170$		−	0.252749i		−	0.0193850i
$171$		0			0
$172$		−	5.80931i		−	0.442955i
$173$			5.05361i			0.384219i	0.981373	+	0.192110i	$0.0615329\pi$
$173$			5.05361i			0.384219i	−0.981373	+	0.192110i	$0.938467\pi$
$174$		0			0
$175$		−	3.69493i		−	0.279310i
$176$	5.10258			0.384621
$177$		0			0
$178$		−	2.69800i		−	0.202224i
$179$	−22.7443			−1.69999			−0.849995	−	0.526792i	$-0.823395\pi$
$179$	−22.7443			−1.69999			−0.849995	+	0.526792i	$0.823395\pi$
$180$		0			0
$181$	18.4253			1.36954			0.684770	−	0.728759i	$-0.259903\pi$
$181$	18.4253			1.36954			0.684770	+	0.728759i	$0.259903\pi$
$182$	−22.3726			−1.65837
$183$		0			0
$184$			5.65818i			0.417127i
$185$	−5.83907			−0.429297
$186$		0			0
$187$		−	1.28967i		−	0.0943101i
$188$			13.0372i			0.950832i
$189$		0			0
$190$	4.17889			0.303169
$191$	20.9633			1.51685			0.758424	−	0.651761i	$-0.225969\pi$
$191$	20.9633			1.51685			0.758424	+	0.651761i	$0.225969\pi$
$192$		0			0
$193$	−7.26799			−0.523162			−0.261581	−	0.965182i	$-0.584244\pi$
$193$	−7.26799			−0.523162			−0.261581	+	0.965182i	$0.584244\pi$
$194$		−	3.99115i		−	0.286548i
$195$		0			0
$196$	−6.65251			−0.475179
$197$	−24.5976			−1.75251			−0.876253	−	0.481851i	$-0.839965\pi$
$197$	−24.5976			−1.75251			−0.876253	+	0.481851i	$0.839965\pi$
$198$		0			0
$199$	16.2070			1.14888			0.574441	−	0.818546i	$-0.305220\pi$
$199$	16.2070			1.14888			0.574441	+	0.818546i	$0.305220\pi$
$200$	1.00000			0.0707107
$201$		0			0
$202$	−1.22506			−0.0861949
$203$	−33.7002			−2.36529
$204$		0			0
$205$	−9.99657			−0.698191
$206$	5.92251			0.412641
$207$		0			0
$208$		−	6.05495i		−	0.419835i
$209$	21.3231			1.47495
$210$		0			0
$211$	23.5763			1.62306			0.811529	−	0.584313i	$-0.198636\pi$
$211$	23.5763			1.62306			0.811529	+	0.584313i	$0.198636\pi$
$212$	−3.38087			−0.232199
$213$		0			0
$214$		−	7.10549i		−	0.485722i
$215$		−	5.80931i		−	0.396191i
$216$		0			0
$217$	−26.7099			−1.81319
$218$			3.14887i			0.213268i
$219$		0			0
$220$	5.10258			0.344016
$221$	−1.53038			−0.102945
$222$		0			0
$223$	1.31407			0.0879965			0.0439983	−	0.999032i	$-0.485990\pi$
$223$	1.31407			0.0879965			0.0439983	+	0.999032i	$0.485990\pi$
$224$		−	3.69493i		−	0.246878i
$225$		0			0
$226$	2.07969			0.138339
$227$		−	18.0703i		−	1.19937i	−0.800238	−	0.599683i	$-0.795293\pi$
$227$		−	18.0703i		−	1.19937i	0.800238	−	0.599683i	$-0.204707\pi$
$228$		0			0
$229$			9.73041i			0.643004i	0.946909	+	0.321502i	$0.104188\pi$
$229$			9.73041i			0.643004i	−0.946909	+	0.321502i	$0.895812\pi$
$230$			5.65818i			0.373090i
$231$		0			0
$232$		−	9.12065i		−	0.598800i
$233$	24.9008			1.63130			0.815652	−	0.578542i	$-0.196378\pi$
$233$	24.9008			1.63130			0.815652	+	0.578542i	$0.196378\pi$
$234$		0			0
$235$			13.0372i			0.850450i
$236$			11.2319i			0.731131i
$237$		0			0
$238$	−0.933890			−0.0605351
$239$	21.1041			1.36511			0.682555	−	0.730834i	$-0.260869\pi$
$239$	21.1041			1.36511			0.682555	+	0.730834i	$0.260869\pi$
$240$		0			0
$241$	15.7911			1.01719			0.508597	−	0.861005i	$-0.330164\pi$
$241$	15.7911			1.01719			0.508597	+	0.861005i	$0.330164\pi$
$242$	15.0363			0.966571
$243$		0			0
$244$			2.77473i			0.177634i
$245$	−6.65251			−0.425013
$246$		0			0
$247$		−	25.3030i		−	1.60999i
$248$		−	7.22881i		−	0.459030i
$249$		0			0
$250$	1.00000			0.0632456
$251$	−2.85903			−0.180461			−0.0902303	−	0.995921i	$-0.528760\pi$
$251$	−2.85903			−0.180461			−0.0902303	+	0.995921i	$0.528760\pi$
$252$		0			0
$253$			28.8713i			1.81513i
$254$	−10.2156			−0.640982
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	21.0214i		−	1.31128i	−0.755074	−	0.655640i	$-0.772399\pi$
$257$		−	21.0214i		−	1.31128i	0.755074	−	0.655640i	$-0.227601\pi$
$258$		0			0
$259$			21.5750i			1.34060i
$260$		−	6.05495i		−	0.375512i
$261$		0			0
$262$		−	17.2121i		−	1.06337i
$263$		−	20.6299i		−	1.27209i	−0.771650	−	0.636047i	$-0.780568\pi$
$263$		−	20.6299i		−	1.27209i	0.771650	−	0.636047i	$-0.219432\pi$
$264$		0			0
$265$	−3.38087			−0.207685
$266$		−	15.4407i		−	0.946731i
$267$		0			0
$268$	6.06213	−	5.50005i	0.370304	−	0.335969i
$269$			24.7520i			1.50915i	0.656212	+	0.754577i	$0.272158\pi$
$269$			24.7520i			1.50915i	−0.656212	+	0.754577i	$0.727842\pi$
$270$		0			0
$271$		−	6.60242i		−	0.401068i	−0.979687	−	0.200534i	$-0.935732\pi$
$271$		−	6.60242i		−	0.401068i	0.979687	−	0.200534i	$-0.0642678\pi$
$272$		−	0.252749i		−	0.0153252i
$273$		0			0
$274$	19.1870			1.15913
$275$	5.10258			0.307697
$276$		0			0
$277$	18.3675			1.10360			0.551798	−	0.833978i	$-0.313942\pi$
$277$	18.3675			1.10360			0.551798	+	0.833978i	$0.313942\pi$
$278$			6.43228i			0.385783i
$279$		0			0
$280$		−	3.69493i		−	0.220814i
$281$	25.3203			1.51048			0.755240	−	0.655448i	$-0.227520\pi$
$281$	25.3203			1.51048			0.755240	+	0.655448i	$0.227520\pi$
$282$		0			0
$283$	11.0733			0.658237			0.329118	−	0.944289i	$-0.393248\pi$
$283$	11.0733			0.658237			0.329118	+	0.944289i	$0.393248\pi$
$284$			8.07879i			0.479388i
$285$		0			0
$286$		−	30.8959i		−	1.82691i
$287$			36.9366i			2.18030i
$288$		0			0
$289$	16.9361			0.996242
$290$		−	9.12065i		−	0.535583i
$291$		0			0
$292$	2.89962			0.169687
$293$		−	18.9742i		−	1.10848i	−0.832356	−	0.554242i	$-0.813008\pi$
$293$		−	18.9742i		−	1.10848i	0.832356	−	0.554242i	$-0.186992\pi$
$294$		0			0
$295$			11.2319i			0.653944i
$296$	−5.83907			−0.339389
$297$		0			0
$298$			14.0925i			0.816358i
$299$	34.2600			1.98131
$300$		0			0
$301$	−21.4650			−1.23722
$302$	−22.8467			−1.31468
$303$		0			0
$304$	4.17889			0.239676
$305$			2.77473i			0.158881i
$306$		0			0
$307$	17.4969			0.998599			0.499299	−	0.866430i	$-0.333591\pi$
$307$	17.4969			0.998599			0.499299	+	0.866430i	$0.333591\pi$
$308$		−	18.8537i		−	1.07429i
$309$		0			0
$310$		−	7.22881i		−	0.410569i
$311$	11.7799			0.667978			0.333989	−	0.942577i	$-0.391605\pi$
$311$	11.7799			0.667978			0.333989	+	0.942577i	$0.391605\pi$
$312$		0			0
$313$			23.4886i			1.32766i	0.747886	+	0.663828i	$0.231069\pi$
$313$			23.4886i			1.32766i	−0.747886	+	0.663828i	$0.768931\pi$
$314$	−19.3930			−1.09441
$315$		0			0
$316$			8.22636i			0.462769i
$317$			29.0150i			1.62964i	0.579711	+	0.814822i	$0.303165\pi$
$317$			29.0150i			1.62964i	−0.579711	+	0.814822i	$0.696835\pi$
$318$		0			0
$319$		−	46.5388i		−	2.60567i
$320$	1.00000			0.0559017
$321$		0			0
$322$	20.9066			1.16508
$323$		−	1.05621i		−	0.0587691i
$324$		0			0
$325$		−	6.05495i		−	0.335868i
$326$	−9.65798			−0.534906
$327$		0			0
$328$	−9.99657			−0.551968
$329$	48.1714			2.65577
$330$		0			0
$331$			19.2208i			1.05647i	0.849099	+	0.528234i	$0.177146\pi$
$331$			19.2208i			1.05647i	−0.849099	+	0.528234i	$0.822854\pi$
$332$			12.6131i			0.692236i
$333$		0			0
$334$		−	17.3264i		−	0.948057i
$335$	6.06213	−	5.50005i	0.331210	−	0.300500i
$336$		0			0
$337$			1.60819i			0.0876037i	0.999040	+	0.0438018i	$0.0139470\pi$
$337$			1.60819i			0.0876037i	−0.999040	+	0.0438018i	$0.986053\pi$
$338$	−23.6624			−1.28707
$339$		0			0
$340$		−	0.252749i		−	0.0137072i
$341$		−	36.8856i		−	1.99747i
$342$		0			0
$343$		−	1.28396i		−	0.0693275i
$344$		−	5.80931i		−	0.313217i
$345$		0			0
$346$			5.05361i			0.271684i
$347$	3.77771			0.202798			0.101399	−	0.994846i	$-0.467668\pi$
$347$	3.77771			0.202798			0.101399	+	0.994846i	$0.467668\pi$
$348$		0			0
$349$	−21.3534			−1.14302			−0.571510	−	0.820595i	$-0.693642\pi$
$349$	−21.3534			−1.14302			−0.571510	+	0.820595i	$0.693642\pi$
$350$		−	3.69493i		−	0.197502i
$351$		0			0
$352$	5.10258			0.271968
$353$	−21.9172			−1.16653			−0.583267	−	0.812280i	$-0.698226\pi$
$353$	−21.9172			−1.16653			−0.583267	+	0.812280i	$0.698226\pi$
$354$		0			0
$355$			8.07879i			0.428778i
$356$		−	2.69800i		−	0.142994i
$357$		0			0
$358$	−22.7443			−1.20207
$359$			29.2681i			1.54471i	0.635189	+	0.772357i	$0.280922\pi$
$359$			29.2681i			1.54471i	−0.635189	+	0.772357i	$0.719078\pi$
$360$		0			0
$361$	−1.53687			−0.0808878
$362$	18.4253			0.968411
$363$		0			0
$364$	−22.3726			−1.17264
$365$	2.89962			0.151773
$366$		0			0
$367$		−	25.5770i		−	1.33511i	−0.744560	−	0.667555i	$-0.767341\pi$
$367$		−	25.5770i		−	1.33511i	0.744560	−	0.667555i	$-0.232659\pi$
$368$			5.65818i			0.294953i
$369$		0			0
$370$	−5.83907			−0.303559
$371$			12.4921i			0.648556i
$372$		0			0
$373$		−	17.3266i		−	0.897138i	−0.893748	−	0.448569i	$-0.851934\pi$
$373$		−	17.3266i		−	0.897138i	0.893748	−	0.448569i	$-0.148066\pi$
$374$		−	1.28967i		−	0.0666873i
$375$		0			0
$376$			13.0372i			0.672340i
$377$	−55.2251			−2.84424
$378$		0			0
$379$		−	7.93843i		−	0.407770i	−0.978995	−	0.203885i	$-0.934643\pi$
$379$		−	7.93843i		−	0.407770i	0.978995	−	0.203885i	$-0.0653568\pi$
$380$	4.17889			0.214373
$381$		0			0
$382$	20.9633			1.07257
$383$	17.9184			0.915587			0.457793	−	0.889059i	$-0.348640\pi$
$383$	17.9184			0.915587			0.457793	+	0.889059i	$0.348640\pi$
$384$		0			0
$385$		−	18.8537i		−	0.960872i
$386$	−7.26799			−0.369931
$387$		0			0
$388$		−	3.99115i		−	0.202620i
$389$		−	22.9267i		−	1.16243i	−0.813750	−	0.581215i	$-0.802577\pi$
$389$		−	22.9267i		−	1.16243i	0.813750	−	0.581215i	$-0.197423\pi$
$390$		0			0
$391$	1.43010			0.0723233
$392$	−6.65251			−0.336002
$393$		0			0
$394$	−24.5976			−1.23921
$395$			8.22636i			0.413913i
$396$		0			0
$397$	8.23403			0.413254			0.206627	−	0.978420i	$-0.433751\pi$
$397$	8.23403			0.413254			0.206627	+	0.978420i	$0.433751\pi$
$398$	16.2070			0.812382
$399$		0			0
$400$	1.00000			0.0500000
$401$	4.68848			0.234132			0.117066	−	0.993124i	$-0.462651\pi$
$401$	4.68848			0.234132			0.117066	+	0.993124i	$0.462651\pi$
$402$		0			0
$403$	−43.7701			−2.18034
$404$	−1.22506			−0.0609490
$405$		0			0
$406$	−33.7002			−1.67251
$407$	−29.7943			−1.47685
$408$		0			0
$409$			24.6532i			1.21902i	0.792778	+	0.609511i	$0.208634\pi$
$409$			24.6532i			1.21902i	−0.792778	+	0.609511i	$0.791366\pi$
$410$	−9.99657			−0.493696
$411$		0			0
$412$	5.92251			0.291781
$413$	41.5009			2.04213
$414$		0			0
$415$			12.6131i			0.619155i
$416$		−	6.05495i		−	0.296868i
$417$		0			0
$418$	21.3231			1.04295
$419$		−	24.8106i		−	1.21208i	−0.795436	−	0.606038i	$-0.792758\pi$
$419$		−	24.8106i		−	1.21208i	0.795436	−	0.606038i	$-0.207242\pi$
$420$		0			0
$421$	−8.25483			−0.402316			−0.201158	−	0.979559i	$-0.564470\pi$
$421$	−8.25483			−0.402316			−0.201158	+	0.979559i	$0.564470\pi$
$422$	23.5763			1.14767
$423$		0			0
$424$	−3.38087			−0.164189
$425$		−	0.252749i		−	0.0122601i
$426$		0			0
$427$	10.2525			0.496151
$428$		−	7.10549i		−	0.343457i
$429$		0			0
$430$		−	5.80931i		−	0.280150i
$431$		−	3.92206i		−	0.188919i	−0.995529	−	0.0944596i	$-0.969888\pi$
$431$		−	3.92206i		−	0.188919i	0.995529	−	0.0944596i	$-0.0301123\pi$
$432$		0			0
$433$		−	12.5714i		−	0.604141i	−0.953286	−	0.302070i	$-0.902322\pi$
$433$		−	12.5714i		−	0.604141i	0.953286	−	0.302070i	$-0.0976778\pi$
$434$	−26.7099			−1.28212
$435$		0			0
$436$			3.14887i			0.150803i
$437$			23.6449i			1.13109i
$438$		0			0
$439$	−38.7852			−1.85112			−0.925558	−	0.378606i	$-0.876403\pi$
$439$	−38.7852			−1.85112			−0.925558	+	0.378606i	$0.876403\pi$
$440$	5.10258			0.243256
$441$		0			0
$442$	−1.53038			−0.0727929
$443$	34.5708			1.64251			0.821255	−	0.570562i	$-0.193274\pi$
$443$	34.5708			1.64251			0.821255	+	0.570562i	$0.193274\pi$
$444$		0			0
$445$		−	2.69800i		−	0.127898i
$446$	1.31407			0.0622229
$447$		0			0
$448$		−	3.69493i		−	0.174569i
$449$			34.8770i			1.64595i	0.568079	+	0.822974i	$0.307687\pi$
$449$			34.8770i			1.64595i	−0.568079	+	0.822974i	$0.692313\pi$
$450$		0			0
$451$	−51.0083			−2.40189
$452$	2.07969			0.0978203
$453$		0			0
$454$		−	18.0703i		−	0.848080i
$455$	−22.3726			−1.04884
$456$		0			0
$457$	−1.01076			−0.0472816			−0.0236408	−	0.999721i	$-0.507526\pi$
$457$	−1.01076			−0.0472816			−0.0236408	+	0.999721i	$0.507526\pi$
$458$			9.73041i			0.454672i
$459$		0			0
$460$			5.65818i			0.263814i
$461$		−	19.6304i		−	0.914279i	−0.889395	−	0.457139i	$-0.848874\pi$
$461$		−	19.6304i		−	0.914279i	0.889395	−	0.457139i	$-0.151126\pi$
$462$		0			0
$463$		−	8.70958i		−	0.404769i	−0.979306	−	0.202384i	$-0.935131\pi$
$463$		−	8.70958i		−	0.404769i	0.979306	−	0.202384i	$-0.0648690\pi$
$464$		−	9.12065i		−	0.423415i
$465$		0			0
$466$	24.9008			1.15351
$467$		−	23.3816i		−	1.08197i	−0.841032	−	0.540985i	$-0.818051\pi$
$467$		−	23.3816i		−	1.08197i	0.841032	−	0.540985i	$-0.181949\pi$
$468$		0			0
$469$	−20.3223	−	22.3992i	−0.938397	−	1.03430i
$470$			13.0372i			0.601359i
$471$		0			0
$472$			11.2319i			0.516988i
$473$		−	29.6424i		−	1.36296i
$474$		0			0
$475$	4.17889			0.191741
$476$	−0.933890			−0.0428048
$477$		0			0
$478$	21.1041			0.965279
$479$			14.5193i			0.663406i	0.943384	+	0.331703i	$0.107623\pi$
$479$			14.5193i			0.663406i	−0.943384	+	0.331703i	$0.892377\pi$
$480$		0			0
$481$			35.3553i			1.61206i
$482$	15.7911			0.719264
$483$		0			0
$484$	15.0363			0.683469
$485$		−	3.99115i		−	0.181229i
$486$		0			0
$487$		−	13.5245i		−	0.612853i	−0.951894	−	0.306426i	$-0.900867\pi$
$487$		−	13.5245i		−	0.612853i	0.951894	−	0.306426i	$-0.0991333\pi$
$488$			2.77473i			0.125606i
$489$		0			0
$490$	−6.65251			−0.300530
$491$			17.6500i			0.796532i	0.917270	+	0.398266i	$0.130388\pi$
$491$			17.6500i			0.796532i	−0.917270	+	0.398266i	$0.869612\pi$
$492$		0			0
$493$	−2.30523			−0.103823
$494$		−	25.3030i		−	1.13844i
$495$		0			0
$496$		−	7.22881i		−	0.324583i
$497$	29.8506			1.33898
$498$		0			0
$499$			32.0141i			1.43315i	0.697511	+	0.716574i	$0.254291\pi$
$499$			32.0141i			1.43315i	−0.697511	+	0.716574i	$0.745709\pi$
$500$	1.00000			0.0447214
$501$		0			0
$502$	−2.85903			−0.127605
$503$	−17.0489			−0.760173			−0.380086	−	0.924951i	$-0.624106\pi$
$503$	−17.0489			−0.760173			−0.380086	+	0.924951i	$0.624106\pi$
$504$		0			0
$505$	−1.22506			−0.0545144
$506$			28.8713i			1.28349i
$507$		0			0
$508$	−10.2156			−0.453243
$509$			23.2788i			1.03182i	0.856644	+	0.515908i	$0.172545\pi$
$509$			23.2788i			1.03182i	−0.856644	+	0.515908i	$0.827455\pi$
$510$		0			0
$511$		−	10.7139i		−	0.473954i
$512$	1.00000			0.0441942
$513$		0			0
$514$		−	21.0214i		−	0.927215i
$515$	5.92251			0.260977
$516$		0			0
$517$			66.5231i			2.92568i
$518$			21.5750i			0.947949i
$519$		0			0
$520$		−	6.05495i		−	0.265527i
$521$	17.5382			0.768365			0.384182	−	0.923257i	$-0.374483\pi$
$521$	17.5382			0.768365			0.384182	+	0.923257i	$0.374483\pi$
$522$		0			0
$523$	16.4608			0.719782			0.359891	−	0.932994i	$-0.382814\pi$
$523$	16.4608			0.719782			0.359891	+	0.932994i	$0.382814\pi$
$524$		−	17.2121i		−	0.751913i
$525$		0			0
$526$		−	20.6299i		−	0.899506i
$527$	−1.82707			−0.0795886
$528$		0			0
$529$	−9.01505			−0.391959
$530$	−3.38087			−0.146855
$531$		0			0
$532$		−	15.4407i		−	0.669440i
$533$			60.5288i			2.62179i
$534$		0			0
$535$		−	7.10549i		−	0.307197i
$536$	6.06213	−	5.50005i	0.261844	−	0.237566i
$537$		0			0
$538$			24.7520i			1.06713i
$539$	−33.9449			−1.46211
$540$		0			0
$541$		−	4.12941i		−	0.177537i	−0.996052	−	0.0887686i	$-0.971707\pi$
$541$		−	4.12941i		−	0.177537i	0.996052	−	0.0887686i	$-0.0282932\pi$
$542$		−	6.60242i		−	0.283598i
$543$		0			0
$544$		−	0.252749i		−	0.0108365i
$545$			3.14887i			0.134883i
$546$		0			0
$547$			26.5064i			1.13333i	0.823947	+	0.566666i	$0.191767\pi$
$547$			26.5064i			1.13333i	−0.823947	+	0.566666i	$0.808233\pi$
$548$	19.1870			0.819628
$549$		0			0
$550$	5.10258			0.217575
$551$		−	38.1142i		−	1.62372i
$552$		0			0
$553$	30.3958			1.29256
$554$	18.3675			0.780360
$555$		0			0
$556$			6.43228i			0.272790i
$557$			2.63483i			0.111641i	0.998441	+	0.0558207i	$0.0177775\pi$
$557$			2.63483i			0.111641i	−0.998441	+	0.0558207i	$0.982222\pi$
$558$		0			0
$559$	−35.1751			−1.48775
$560$		−	3.69493i		−	0.156139i
$561$		0			0
$562$	25.3203			1.06807
$563$	−38.2567			−1.61233			−0.806164	−	0.591692i	$-0.798460\pi$
$563$	−38.2567			−1.61233			−0.806164	+	0.591692i	$0.798460\pi$
$564$		0			0
$565$	2.07969			0.0874932
$566$	11.0733			0.465444
$567$		0			0
$568$			8.07879i			0.338979i
$569$		−	5.20352i		−	0.218143i	−0.994034	−	0.109071i	$-0.965212\pi$
$569$		−	5.20352i		−	0.218143i	0.994034	−	0.109071i	$-0.0347877\pi$
$570$		0			0
$571$	10.3913			0.434862			0.217431	−	0.976076i	$-0.430232\pi$
$571$	10.3913			0.434862			0.217431	+	0.976076i	$0.430232\pi$
$572$		−	30.8959i		−	1.29182i
$573$		0			0
$574$			36.9366i			1.54171i
$575$			5.65818i			0.235963i
$576$		0			0
$577$			19.2770i			0.802512i	0.915966	+	0.401256i	$0.131426\pi$
$577$			19.2770i			0.802512i	−0.915966	+	0.401256i	$0.868574\pi$
$578$	16.9361			0.704450
$579$		0			0
$580$		−	9.12065i		−	0.378714i
$581$	46.6047			1.93349
$582$		0			0
$583$	−17.2511			−0.714469
$584$	2.89962			0.119987
$585$		0			0
$586$		−	18.9742i		−	0.783817i
$587$	−2.17246			−0.0896671			−0.0448336	−	0.998994i	$-0.514276\pi$
$587$	−2.17246			−0.0896671			−0.0448336	+	0.998994i	$0.514276\pi$
$588$		0			0
$589$		−	30.2084i		−	1.24472i
$590$			11.2319i			0.462408i
$591$		0			0
$592$	−5.83907			−0.239984
$593$	5.74134			0.235769			0.117884	−	0.993027i	$-0.462389\pi$
$593$	5.74134			0.235769			0.117884	+	0.993027i	$0.462389\pi$
$594$		0			0
$595$	−0.933890			−0.0382857
$596$			14.0925i			0.577252i
$597$		0			0
$598$	34.2600			1.40100
$599$	−8.81011			−0.359971			−0.179986	−	0.983669i	$-0.557605\pi$
$599$	−8.81011			−0.359971			−0.179986	+	0.983669i	$0.557605\pi$
$600$		0			0
$601$	11.7296			0.478460			0.239230	−	0.970963i	$-0.423105\pi$
$601$	11.7296			0.478460			0.239230	+	0.970963i	$0.423105\pi$
$602$	−21.4650			−0.874847
$603$		0			0
$604$	−22.8467			−0.929621
$605$	15.0363			0.611313
$606$		0			0
$607$	34.7653			1.41108			0.705540	−	0.708670i	$-0.250705\pi$
$607$	34.7653			1.41108			0.705540	+	0.708670i	$0.250705\pi$
$608$	4.17889			0.169476
$609$		0			0
$610$			2.77473i			0.112346i
$611$	78.9394			3.19354
$612$		0			0
$613$	−0.566756			−0.0228911			−0.0114455	−	0.999934i	$-0.503643\pi$
$613$	−0.566756			−0.0228911			−0.0114455	+	0.999934i	$0.503643\pi$
$614$	17.4969			0.706116
$615$		0			0
$616$		−	18.8537i		−	0.759636i
$617$			18.1445i			0.730471i	0.930915	+	0.365235i	$0.119011\pi$
$617$			18.1445i			0.730471i	−0.930915	+	0.365235i	$0.880989\pi$
$618$		0			0
$619$	17.2328			0.692645			0.346323	−	0.938115i	$-0.387430\pi$
$619$	17.2328			0.692645			0.346323	+	0.938115i	$0.387430\pi$
$620$		−	7.22881i		−	0.290316i
$621$		0			0
$622$	11.7799			0.472331
$623$	−9.96894			−0.399397
$624$		0			0
$625$	1.00000			0.0400000
$626$			23.4886i			0.938794i
$627$		0			0
$628$	−19.3930			−0.773865
$629$			1.47582i			0.0588448i
$630$		0			0
$631$			16.6732i			0.663749i	0.943324	+	0.331874i	$0.107681\pi$
$631$			16.6732i			0.663749i	−0.943324	+	0.331874i	$0.892319\pi$
$632$			8.22636i			0.327227i
$633$		0			0
$634$			29.0150i			1.15233i
$635$	−10.2156			−0.405393
$636$		0			0
$637$			40.2806i			1.59598i
$638$		−	46.5388i		−	1.84249i
$639$		0			0
$640$	1.00000			0.0395285
$641$	−11.4261			−0.451304			−0.225652	−	0.974208i	$-0.572451\pi$
$641$	−11.4261			−0.451304			−0.225652	+	0.974208i	$0.572451\pi$
$642$		0			0
$643$	−38.0283			−1.49969			−0.749845	−	0.661614i	$-0.769872\pi$
$643$	−38.0283			−1.49969			−0.749845	+	0.661614i	$0.769872\pi$
$644$	20.9066			0.823835
$645$		0			0
$646$		−	1.05621i		−	0.0415560i
$647$	−31.0064			−1.21899			−0.609493	−	0.792792i	$-0.708627\pi$
$647$	−31.0064			−1.21899			−0.609493	+	0.792792i	$0.708627\pi$
$648$		0			0
$649$			57.3114i			2.24967i
$650$		−	6.05495i		−	0.237495i
$651$		0			0
$652$	−9.65798			−0.378236
$653$	27.1561			1.06270			0.531351	−	0.847152i	$-0.321685\pi$
$653$	27.1561			1.06270			0.531351	+	0.847152i	$0.321685\pi$
$654$		0			0
$655$		−	17.2121i		−	0.672531i
$656$	−9.99657			−0.390301
$657$		0			0
$658$	48.1714			1.87792
$659$			38.9300i			1.51650i	0.651964	+	0.758249i	$0.273945\pi$
$659$			38.9300i			1.51650i	−0.651964	+	0.758249i	$0.726055\pi$
$660$		0			0
$661$		−	45.9057i		−	1.78553i	−0.450527	−	0.892763i	$-0.648764\pi$
$661$		−	45.9057i		−	1.78553i	0.450527	−	0.892763i	$-0.351236\pi$
$662$			19.2208i			0.747036i
$663$		0			0
$664$			12.6131i			0.489485i
$665$		−	15.4407i		−	0.598765i
$666$		0			0
$667$	51.6063			1.99820
$668$		−	17.3264i		−	0.670377i
$669$		0			0
$670$	6.06213	−	5.50005i	0.234201	−	0.212486i
$671$			14.1583i			0.546575i
$672$		0			0
$673$			19.3170i			0.744616i	0.928109	+	0.372308i	$0.121434\pi$
$673$			19.3170i			0.744616i	−0.928109	+	0.372308i	$0.878566\pi$
$674$			1.60819i			0.0619451i
$675$		0			0
$676$	−23.6624			−0.910094
$677$	−36.3806			−1.39822			−0.699110	−	0.715014i	$-0.746420\pi$
$677$	−36.3806			−1.39822			−0.699110	+	0.715014i	$0.746420\pi$
$678$		0			0
$679$	−14.7470			−0.565939
$680$		−	0.252749i		−	0.00969248i
$681$		0			0
$682$		−	36.8856i		−	1.41242i
$683$	13.5354			0.517918			0.258959	−	0.965888i	$-0.416620\pi$
$683$	13.5354			0.517918			0.258959	+	0.965888i	$0.416620\pi$
$684$		0			0
$685$	19.1870			0.733098
$686$		−	1.28396i		−	0.0490219i
$687$		0			0
$688$		−	5.80931i		−	0.221478i
$689$			20.4710i			0.779882i
$690$		0			0
$691$	−35.8320			−1.36312			−0.681558	−	0.731764i	$-0.738697\pi$
$691$	−35.8320			−1.36312			−0.681558	+	0.731764i	$0.738697\pi$
$692$			5.05361i			0.192110i
$693$		0			0
$694$	3.77771			0.143400
$695$			6.43228i			0.243990i
$696$		0			0
$697$			2.52662i			0.0957027i
$698$	−21.3534			−0.808237
$699$		0			0
$700$		−	3.69493i		−	0.139655i
$701$	−31.6230			−1.19438			−0.597192	−	0.802099i	$-0.703717\pi$
$701$	−31.6230			−1.19438			−0.597192	+	0.802099i	$0.703717\pi$
$702$		0			0
$703$	−24.4008			−0.920295
$704$	5.10258			0.192311
$705$		0			0
$706$	−21.9172			−0.824864
$707$			4.52651i			0.170237i
$708$		0			0
$709$	−5.65099			−0.212228			−0.106114	−	0.994354i	$-0.533841\pi$
$709$	−5.65099			−0.212228			−0.106114	+	0.994354i	$0.533841\pi$
$710$			8.07879i			0.303192i
$711$		0			0
$712$		−	2.69800i		−	0.101112i
$713$	40.9019			1.53179
$714$		0			0
$715$		−	30.8959i		−	1.15544i
$716$	−22.7443			−0.849995
$717$		0			0
$718$			29.2681i			1.09228i
$719$			15.4960i			0.577904i	0.957344	+	0.288952i	$0.0933067\pi$
$719$			15.4960i			0.577904i	−0.957344	+	0.288952i	$0.906693\pi$
$720$		0			0
$721$		−	21.8832i		−	0.814975i
$722$	−1.53687			−0.0571963
$723$		0			0
$724$	18.4253			0.684770
$725$		−	9.12065i		−	0.338732i
$726$		0			0
$727$		−	30.3663i		−	1.12622i	−0.826381	−	0.563111i	$-0.809604\pi$
$727$		−	30.3663i		−	1.12622i	0.826381	−	0.563111i	$-0.190396\pi$
$728$	−22.3726			−0.829185
$729$		0			0
$730$	2.89962			0.107320
$731$	−1.46830			−0.0543069
$732$		0			0
$733$		−	1.88141i		−	0.0694914i	−0.999396	−	0.0347457i	$-0.988938\pi$
$733$		−	1.88141i		−	0.0694914i	0.999396	−	0.0347457i	$-0.0110621\pi$
$734$		−	25.5770i		−	0.944066i
$735$		0			0
$736$			5.65818i			0.208563i
$737$	30.9325	−	28.0644i	1.13941	−	1.03377i
$738$		0			0
$739$			30.4545i			1.12029i	0.828395	+	0.560144i	$0.189254\pi$
$739$			30.4545i			1.12029i	−0.828395	+	0.560144i	$0.810746\pi$
$740$	−5.83907			−0.214649
$741$		0			0
$742$			12.4921i			0.458598i
$743$		−	8.21651i		−	0.301434i	−0.988577	−	0.150717i	$-0.951842\pi$
$743$		−	8.21651i		−	0.301434i	0.988577	−	0.150717i	$-0.0481583\pi$
$744$		0			0
$745$			14.0925i			0.516310i
$746$		−	17.3266i		−	0.634373i
$747$		0			0
$748$		−	1.28967i		−	0.0471551i
$749$	−26.2543			−0.959311
$750$		0			0
$751$	6.45700			0.235619			0.117810	−	0.993036i	$-0.462413\pi$
$751$	6.45700			0.235619			0.117810	+	0.993036i	$0.462413\pi$
$752$			13.0372i			0.475416i
$753$		0			0
$754$	−55.2251			−2.01118
$755$	−22.8467			−0.831478
$756$		0			0
$757$			5.20766i			0.189276i	0.995512	+	0.0946379i	$0.0301693\pi$
$757$			5.20766i			0.189276i	−0.995512	+	0.0946379i	$0.969831\pi$
$758$		−	7.93843i		−	0.288337i
$759$		0			0
$760$	4.17889			0.151584
$761$			15.7464i			0.570805i	0.958408	+	0.285403i	$0.0921273\pi$
$761$			15.7464i			0.570805i	−0.958408	+	0.285403i	$0.907873\pi$
$762$		0			0
$763$	11.6349			0.421210
$764$	20.9633			0.758424
$765$		0			0
$766$	17.9184			0.647418
$767$	68.0083			2.45564
$768$		0			0
$769$			38.6659i			1.39433i	0.716912	+	0.697164i	$0.245555\pi$
$769$			38.6659i			1.39433i	−0.716912	+	0.697164i	$0.754445\pi$
$770$		−	18.8537i		−	0.679439i
$771$		0			0
$772$	−7.26799			−0.261581
$773$		−	1.53017i		−	0.0550362i	−0.999621	−	0.0275181i	$-0.991240\pi$
$773$		−	1.53017i		−	0.0550362i	0.999621	−	0.0275181i	$-0.00876039\pi$
$774$		0			0
$775$		−	7.22881i		−	0.259666i
$776$		−	3.99115i		−	0.143274i
$777$		0			0
$778$		−	22.9267i		−	0.821963i
$779$	−41.7746			−1.49673
$780$		0			0
$781$			41.2227i			1.47506i
$782$	1.43010			0.0511403
$783$		0			0
$784$	−6.65251			−0.237590
$785$	−19.3930			−0.692166
$786$		0			0
$787$			29.0785i			1.03654i	0.855218	+	0.518268i	$0.173423\pi$
$787$			29.0785i			1.03654i	−0.855218	+	0.518268i	$0.826577\pi$
$788$	−24.5976			−0.876253
$789$		0			0
$790$			8.22636i			0.292681i
$791$		−	7.68430i		−	0.273222i
$792$		0			0
$793$	16.8009			0.596617
$794$	8.23403			0.292215
$795$		0			0
$796$	16.2070			0.574441
$797$		−	40.4456i		−	1.43266i	−0.697763	−	0.716329i	$-0.745821\pi$
$797$		−	40.4456i		−	1.43266i	0.697763	−	0.716329i	$-0.254179\pi$
$798$		0			0
$799$	3.29513			0.116573
$800$	1.00000			0.0353553
$801$		0			0
$802$	4.68848			0.165556
$803$	14.7955			0.522123
$804$		0			0
$805$	20.9066			0.736861
$806$	−43.7701			−1.54174
$807$		0			0
$808$	−1.22506			−0.0430974
$809$	41.0065			1.44171			0.720857	−	0.693084i	$-0.243749\pi$
$809$	41.0065			1.44171			0.720857	+	0.693084i	$0.243749\pi$
$810$		0			0
$811$		−	43.6942i		−	1.53431i	−0.641462	−	0.767155i	$-0.721672\pi$
$811$		−	43.6942i		−	1.53431i	0.641462	−	0.767155i	$-0.278328\pi$
$812$	−33.7002			−1.18264
$813$		0			0
$814$	−29.7943			−1.04429
$815$	−9.65798			−0.338304
$816$		0			0
$817$		−	24.2765i		−	0.849326i
$818$			24.6532i			0.861978i
$819$		0			0
$820$	−9.99657			−0.349095
$821$			1.40258i			0.0489503i	0.999700	+	0.0244752i	$0.00779146\pi$
$821$			1.40258i			0.0489503i	−0.999700	+	0.0244752i	$0.992209\pi$
$822$		0			0
$823$	−35.9384			−1.25273			−0.626367	−	0.779528i	$-0.715459\pi$
$823$	−35.9384			−1.25273			−0.626367	+	0.779528i	$0.715459\pi$
$824$	5.92251			0.206320
$825$		0			0
$826$	41.5009			1.44400
$827$			10.6722i			0.371110i	0.982634	+	0.185555i	$0.0594084\pi$
$827$			10.6722i			0.371110i	−0.982634	+	0.185555i	$0.940592\pi$
$828$		0			0
$829$	−19.9736			−0.693711			−0.346855	−	0.937919i	$-0.612751\pi$
$829$	−19.9736			−0.693711			−0.346855	+	0.937919i	$0.612751\pi$
$830$			12.6131i			0.437808i
$831$		0			0
$832$		−	6.05495i		−	0.209918i
$833$			1.68141i			0.0582575i
$834$		0			0
$835$		−	17.3264i		−	0.599604i
$836$	21.3231			0.737476
$837$		0			0
$838$		−	24.8106i		−	0.857067i
$839$		−	8.04265i		−	0.277663i	−0.990316	−	0.138832i	$-0.955665\pi$
$839$		−	8.04265i		−	0.277663i	0.990316	−	0.138832i	$-0.0443347\pi$
$840$		0			0
$841$	−54.1862			−1.86849
$842$	−8.25483			−0.284480
$843$		0			0
$844$	23.5763			0.811529
$845$	−23.6624			−0.814013
$846$		0			0
$847$		−	55.5581i		−	1.90900i
$848$	−3.38087			−0.116099
$849$		0			0
$850$		−	0.252749i		−	0.00866922i
$851$		−	33.0385i		−	1.13255i
$852$		0			0
$853$	−32.8493			−1.12474			−0.562369	−	0.826886i	$-0.690110\pi$
$853$	−32.8493			−1.12474			−0.562369	+	0.826886i	$0.690110\pi$
$854$	10.2525			0.350832
$855$		0			0
$856$		−	7.10549i		−	0.242861i
$857$	−8.10834			−0.276975			−0.138488	−	0.990364i	$-0.544224\pi$
$857$	−8.10834			−0.276975			−0.138488	+	0.990364i	$0.544224\pi$
$858$		0			0
$859$	51.2358			1.74814			0.874072	−	0.485796i	$-0.161470\pi$
$859$	51.2358			1.74814			0.874072	+	0.485796i	$0.161470\pi$
$860$		−	5.80931i		−	0.198096i
$861$		0			0
$862$		−	3.92206i		−	0.133586i
$863$		−	17.6393i		−	0.600450i	−0.953868	−	0.300225i	$-0.902938\pi$
$863$		−	17.6393i		−	0.600450i	0.953868	−	0.300225i	$-0.0970618\pi$
$864$		0			0
$865$			5.05361i			0.171828i
$866$		−	12.5714i		−	0.427192i
$867$		0			0
$868$	−26.7099			−0.906594
$869$			41.9756i			1.42393i
$870$		0			0
$871$	−33.3025	−	36.7059i	−1.12841	−	1.24373i
$872$			3.14887i			0.106634i
$873$		0			0
$874$			23.6449i			0.799802i
$875$		−	3.69493i		−	0.124911i
$876$		0			0
$877$	−51.7958			−1.74902			−0.874511	−	0.485006i	$-0.838817\pi$
$877$	−51.7958			−1.74902			−0.874511	+	0.485006i	$0.838817\pi$
$878$	−38.7852			−1.30894
$879$		0			0
$880$	5.10258			0.172008
$881$			26.1742i			0.881830i	0.897549	+	0.440915i	$0.145346\pi$
$881$			26.1742i			0.881830i	−0.897549	+	0.440915i	$0.854654\pi$
$882$		0			0
$883$			13.4097i			0.451273i	0.974212	+	0.225636i	$0.0724461\pi$
$883$			13.4097i			0.451273i	−0.974212	+	0.225636i	$0.927554\pi$
$884$	−1.53038			−0.0514723
$885$		0			0
$886$	34.5708			1.16143
$887$			15.8830i			0.533298i	0.963794	+	0.266649i	$0.0859164\pi$
$887$			15.8830i			0.533298i	−0.963794	+	0.266649i	$0.914084\pi$
$888$		0			0
$889$			37.7458i			1.26595i
$890$		−	2.69800i		−	0.0904373i
$891$		0			0
$892$	1.31407			0.0439983
$893$			54.4809i			1.82313i
$894$		0			0
$895$	−22.7443			−0.760258
$896$		−	3.69493i		−	0.123439i
$897$		0			0
$898$			34.8770i			1.16386i
$899$	−65.9314			−2.19894
$900$		0			0
$901$			0.854510i			0.0284679i
$902$	−51.0083			−1.69839
$903$		0			0
$904$	2.07969			0.0691694
$905$	18.4253			0.612477
$906$		0			0
$907$	32.8398			1.09043			0.545213	−	0.838298i	$-0.316449\pi$
$907$	32.8398			1.09043			0.545213	+	0.838298i	$0.316449\pi$
$908$		−	18.0703i		−	0.599683i
$909$		0			0
$910$	−22.3726			−0.741645
$911$		−	38.8299i		−	1.28649i	−0.765659	−	0.643246i	$-0.777587\pi$
$911$		−	38.8299i		−	1.28649i	0.765659	−	0.643246i	$-0.222413\pi$
$912$		0			0
$913$			64.3595i			2.12999i
$914$	−1.01076			−0.0334331
$915$		0			0
$916$			9.73041i			0.321502i
$917$	−63.5974			−2.10017
$918$		0			0
$919$			17.0194i			0.561418i	0.959793	+	0.280709i	$0.0905696\pi$
$919$			17.0194i			0.561418i	−0.959793	+	0.280709i	$0.909430\pi$
$920$			5.65818i			0.186545i
$921$		0			0
$922$		−	19.6304i		−	0.646493i
$923$	48.9167			1.61011
$924$		0			0
$925$	−5.83907			−0.191987
$926$		−	8.70958i		−	0.286215i
$927$		0			0
$928$		−	9.12065i		−	0.299400i
$929$	26.1858			0.859128			0.429564	−	0.903036i	$-0.358667\pi$
$929$	26.1858			0.859128			0.429564	+	0.903036i	$0.358667\pi$
$930$		0			0
$931$	−27.8001			−0.911112
$932$	24.9008			0.815652
$933$		0			0
$934$		−	23.3816i		−	0.765069i
$935$		−	1.28967i		−	0.0421768i
$936$		0			0
$937$		−	1.58651i		−	0.0518290i	−0.999664	−	0.0259145i	$-0.991750\pi$
$937$		−	1.58651i		−	0.0518290i	0.999664	−	0.0259145i	$-0.00824977\pi$
$938$	−20.3223	−	22.3992i	−0.663547	−	0.731358i
$939$		0			0
$940$			13.0372i			0.425225i
$941$	−18.2168			−0.593852			−0.296926	−	0.954901i	$-0.595961\pi$
$941$	−18.2168			−0.593852			−0.296926	+	0.954901i	$0.595961\pi$
$942$		0			0
$943$		−	56.5625i		−	1.84193i
$944$			11.2319i			0.365566i
$945$		0			0
$946$		−	29.6424i		−	0.963759i
$947$			38.8832i			1.26353i	0.775158	+	0.631767i	$0.217670\pi$
$947$			38.8832i			1.26353i	−0.775158	+	0.631767i	$0.782330\pi$
$948$		0			0
$949$		−	17.5570i		−	0.569925i
$950$	4.17889			0.135581
$951$		0			0
$952$	−0.933890			−0.0302675
$953$		−	35.5052i		−	1.15013i	−0.818109	−	0.575063i	$-0.804978\pi$
$953$		−	35.5052i		−	1.15013i	0.818109	−	0.575063i	$-0.195022\pi$
$954$		0			0
$955$	20.9633			0.678355
$956$	21.1041			0.682555
$957$		0			0
$958$			14.5193i			0.469099i
$959$		−	70.8947i		−	2.28931i
$960$		0			0
$961$	−21.2557			−0.685666
$962$			35.3553i			1.13990i
$963$		0			0
$964$	15.7911			0.508597
$965$	−7.26799			−0.233965
$966$		0			0
$967$	6.46649			0.207948			0.103974	−	0.994580i	$-0.466844\pi$
$967$	6.46649			0.207948			0.103974	+	0.994580i	$0.466844\pi$
$968$	15.0363			0.483285
$969$		0			0
$970$		−	3.99115i		−	0.128148i
$971$			14.2357i			0.456845i	0.973562	+	0.228423i	$0.0733568\pi$
$971$			14.2357i			0.456845i	−0.973562	+	0.228423i	$0.926643\pi$
$972$		0			0
$973$	23.7668			0.761930
$974$		−	13.5245i		−	0.433352i
$975$		0			0
$976$			2.77473i			0.0888171i
$977$		−	2.57587i		−	0.0824093i	−0.999151	−	0.0412047i	$-0.986880\pi$
$977$		−	2.57587i		−	0.0824093i	0.999151	−	0.0412047i	$-0.0131196\pi$
$978$		0			0
$979$		−	13.7668i		−	0.439988i
$980$	−6.65251			−0.212507
$981$		0			0
$982$			17.6500i			0.563233i
$983$	−48.3618			−1.54250			−0.771251	−	0.636531i	$-0.780369\pi$
$983$	−48.3618			−1.54250			−0.771251	+	0.636531i	$0.780369\pi$
$984$		0			0
$985$	−24.5976			−0.783745
$986$	−2.30523			−0.0734136
$987$		0			0
$988$		−	25.3030i		−	0.804995i
$989$	32.8701			1.04521
$990$		0			0
$991$			31.8339i			1.01124i	0.862757	+	0.505619i	$0.168736\pi$
$991$			31.8339i			1.01124i	−0.862757	+	0.505619i	$0.831264\pi$
$992$		−	7.22881i		−	0.229515i
$993$		0			0
$994$	29.8506			0.946802
$995$	16.2070			0.513796
$996$		0			0
$997$	40.5571			1.28446			0.642228	−	0.766513i	$-0.278010\pi$
$997$	40.5571			1.28446			0.642228	+	0.766513i	$0.278010\pi$
$998$			32.0141i			1.01339i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6030.2.d.l.2411.2	yes	24
3.2	odd	2		6030.2.d.k.2411.2	✓	24
67.66	odd	2		6030.2.d.k.2411.23	yes	24
201.200	even	2	inner	6030.2.d.l.2411.23	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6030.2.d.k.2411.2	✓	24	3.2	odd	2
6030.2.d.k.2411.23	yes	24	67.66	odd	2
6030.2.d.l.2411.2	yes	24	1.1	even	1	trivial
6030.2.d.l.2411.23	yes	24	201.200	even	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:	\( N \)	\(=\)	\( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6030.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.69493i q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -3.69493i q^{7} +1.00000 q^{8} +1.00000 q^{10} +5.10258 q^{11} -6.05495i q^{13} -3.69493i q^{14} +1.00000 q^{16} -0.252749i q^{17} +4.17889 q^{19} +1.00000 q^{20} +5.10258 q^{22} +5.65818i q^{23} +1.00000 q^{25} -6.05495i q^{26} -3.69493i q^{28} -9.12065i q^{29} -7.22881i q^{31} +1.00000 q^{32} -0.252749i q^{34} -3.69493i q^{35} -5.83907 q^{37} +4.17889 q^{38} +1.00000 q^{40} -9.99657 q^{41} -5.80931i q^{43} +5.10258 q^{44} +5.65818i q^{46} +13.0372i q^{47} -6.65251 q^{49} +1.00000 q^{50} -6.05495i q^{52} -3.38087 q^{53} +5.10258 q^{55} -3.69493i q^{56} -9.12065i q^{58} +11.2319i q^{59} +2.77473i q^{61} -7.22881i q^{62} +1.00000 q^{64} -6.05495i q^{65} +(6.06213 - 5.50005i) q^{67} -0.252749i q^{68} -3.69493i q^{70} +8.07879i q^{71} +2.89962 q^{73} -5.83907 q^{74} +4.17889 q^{76} -18.8537i q^{77} +8.22636i q^{79} +1.00000 q^{80} -9.99657 q^{82} +12.6131i q^{83} -0.252749i q^{85} -5.80931i q^{86} +5.10258 q^{88} -2.69800i q^{89} -22.3726 q^{91} +5.65818i q^{92} +13.0372i q^{94} +4.17889 q^{95} -3.99115i q^{97} -6.65251 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{2} + 24 q^{4} + 24 q^{5} + 24 q^{8}+O(q^{10}) \) 24 * q + 24 * q^2 + 24 * q^4 + 24 * q^5 + 24 * q^8	\( 24 q + 24 q^{2} + 24 q^{4} + 24 q^{5} + 24 q^{8} + 24 q^{10} + 12 q^{11} + 24 q^{16} + 4 q^{19} + 24 q^{20} + 12 q^{22} + 24 q^{25} + 24 q^{32} - 16 q^{37} + 4 q^{38} + 24 q^{40} + 8 q^{41} + 12 q^{44} - 20 q^{49} + 24 q^{50} + 24 q^{53} + 12 q^{55} + 24 q^{64} - 32 q^{67} - 4 q^{73} - 16 q^{74} + 4 q^{76} + 24 q^{80} + 8 q^{82} + 12 q^{88} + 4 q^{95} - 20 q^{98}+O(q^{100}) \) 24 * q + 24 * q^2 + 24 * q^4 + 24 * q^5 + 24 * q^8 + 24 * q^10 + 12 * q^11 + 24 * q^16 + 4 * q^19 + 24 * q^20 + 12 * q^22 + 24 * q^25 + 24 * q^32 - 16 * q^37 + 4 * q^38 + 24 * q^40 + 8 * q^41 + 12 * q^44 - 20 * q^49 + 24 * q^50 + 24 * q^53 + 12 * q^55 + 24 * q^64 - 32 * q^67 - 4 * q^73 - 16 * q^74 + 4 * q^76 + 24 * q^80 + 8 * q^82 + 12 * q^88 + 4 * q^95 - 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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