LMFDB - Embedded newform 6030.2.d.k.2411.7

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			2411.7
Character	$\chi$	$=$	6030.2411
Dual form			6030.2.d.k.2411.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.65115i q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.65115i q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.37646 q^{11} +3.20813i q^{13} +2.65115i q^{14} +1.00000 q^{16} -7.87143i q^{17} -0.480302 q^{19} -1.00000 q^{20} +2.37646 q^{22} -9.18285i q^{23} +1.00000 q^{25} -3.20813i q^{26} -2.65115i q^{28} -0.463977i q^{29} -0.385676i q^{31} -1.00000 q^{32} +7.87143i q^{34} +2.65115i q^{35} -1.46251 q^{37} +0.480302 q^{38} +1.00000 q^{40} +3.83598 q^{41} -2.60950i q^{43} -2.37646 q^{44} +9.18285i q^{46} -1.91486i q^{47} -0.0286042 q^{49} -1.00000 q^{50} +3.20813i q^{52} +2.63457 q^{53} +2.37646 q^{55} +2.65115i q^{56} +0.463977i q^{58} -3.35047i q^{59} +14.7145i q^{61} +0.385676i q^{62} +1.00000 q^{64} -3.20813i q^{65} +(2.65259 + 7.74363i) q^{67} -7.87143i q^{68} -2.65115i q^{70} +6.09202i q^{71} -9.36227 q^{73} +1.46251 q^{74} -0.480302 q^{76} +6.30036i q^{77} -6.73632i q^{79} -1.00000 q^{80} -3.83598 q^{82} -5.22557i q^{83} +7.87143i q^{85} +2.60950i q^{86} +2.37646 q^{88} -7.05666i q^{89} +8.50523 q^{91} -9.18285i q^{92} +1.91486i q^{94} +0.480302 q^{95} -1.67744i q^{97} +0.0286042 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	2.65115i		−	1.00204i	−0.865435	−	0.501021i	$-0.832958\pi$
$7$		−	2.65115i		−	1.00204i	0.865435	−	0.501021i	$-0.167042\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$	1.00000			0.316228
$11$	−2.37646			−0.716530			−0.358265	−	0.933620i	$-0.616632\pi$
$11$	−2.37646			−0.716530			−0.358265	+	0.933620i	$0.616632\pi$
$12$		0			0
$13$			3.20813i			0.889774i	0.895587	+	0.444887i	$0.146756\pi$
$13$			3.20813i			0.889774i	−0.895587	+	0.444887i	$0.853244\pi$
$14$			2.65115i			0.708550i
$15$		0			0
$16$	1.00000			0.250000
$17$		−	7.87143i		−	1.90910i	−0.298047	−	0.954551i	$-0.596335\pi$
$17$		−	7.87143i		−	1.90910i	0.298047	−	0.954551i	$-0.403665\pi$
$18$		0			0
$19$	−0.480302			−0.110189			−0.0550944	−	0.998481i	$-0.517546\pi$
$19$	−0.480302			−0.110189			−0.0550944	+	0.998481i	$0.517546\pi$
$20$	−1.00000			−0.223607
$21$		0			0
$22$	2.37646			0.506664
$23$		−	9.18285i		−	1.91476i	−0.288836	−	0.957379i	$-0.593268\pi$
$23$		−	9.18285i		−	1.91476i	0.288836	−	0.957379i	$-0.406732\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		−	3.20813i		−	0.629165i
$27$		0			0
$28$		−	2.65115i		−	0.501021i
$29$		−	0.463977i		−	0.0861583i	−0.999072	−	0.0430792i	$-0.986283\pi$
$29$		−	0.463977i		−	0.0861583i	0.999072	−	0.0430792i	$-0.0137168\pi$
$30$		0			0
$31$		−	0.385676i		−	0.0692695i	−0.999400	−	0.0346347i	$-0.988973\pi$
$31$		−	0.385676i		−	0.0692695i	0.999400	−	0.0346347i	$-0.0110268\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$			7.87143i			1.34994i
$35$			2.65115i			0.448126i
$36$		0			0
$37$	−1.46251			−0.240435			−0.120218	−	0.992748i	$-0.538359\pi$
$37$	−1.46251			−0.240435			−0.120218	+	0.992748i	$0.538359\pi$
$38$	0.480302			0.0779152
$39$		0			0
$40$	1.00000			0.158114
$41$	3.83598			0.599079			0.299540	−	0.954084i	$-0.403167\pi$
$41$	3.83598			0.599079			0.299540	+	0.954084i	$0.403167\pi$
$42$		0			0
$43$		−	2.60950i		−	0.397946i	−0.980005	−	0.198973i	$-0.936239\pi$
$43$		−	2.60950i		−	0.397946i	0.980005	−	0.198973i	$-0.0637605\pi$
$44$	−2.37646			−0.358265
$45$		0			0
$46$			9.18285i			1.35394i
$47$		−	1.91486i		−	0.279312i	−0.990200	−	0.139656i	$-0.955400\pi$
$47$		−	1.91486i		−	0.279312i	0.990200	−	0.139656i	$-0.0445996\pi$
$48$		0			0
$49$	−0.0286042			−0.00408631
$50$	−1.00000			−0.141421
$51$		0			0
$52$			3.20813i			0.444887i
$53$	2.63457			0.361887			0.180943	−	0.983494i	$-0.442085\pi$
$53$	2.63457			0.361887			0.180943	+	0.983494i	$0.442085\pi$
$54$		0			0
$55$	2.37646			0.320442
$56$			2.65115i			0.354275i
$57$		0			0
$58$			0.463977i			0.0609232i
$59$		−	3.35047i		−	0.436194i	−0.975927	−	0.218097i	$-0.930015\pi$
$59$		−	3.35047i		−	0.436194i	0.975927	−	0.218097i	$-0.0699849\pi$
$60$		0			0
$61$			14.7145i			1.88400i	0.335613	+	0.942000i	$0.391057\pi$
$61$			14.7145i			1.88400i	−0.335613	+	0.942000i	$0.608943\pi$
$62$			0.385676i			0.0489809i
$63$		0			0
$64$	1.00000			0.125000
$65$		−	3.20813i		−	0.397919i
$66$		0			0
$67$	2.65259	+	7.74363i	0.324065	+	0.946035i
$67$	2.65259	+	7.74363i	0.324065	+	0.946035i
$68$		−	7.87143i		−	0.954551i
$69$		0			0
$70$		−	2.65115i		−	0.316873i
$71$			6.09202i			0.722990i	0.932374	+	0.361495i	$0.117734\pi$
$71$			6.09202i			0.722990i	−0.932374	+	0.361495i	$0.882266\pi$
$72$		0			0
$73$	−9.36227			−1.09577			−0.547886	−	0.836553i	$-0.684567\pi$
$73$	−9.36227			−1.09577			−0.547886	+	0.836553i	$0.684567\pi$
$74$	1.46251			0.170013
$75$		0			0
$76$	−0.480302			−0.0550944
$77$			6.30036i			0.717993i
$78$		0			0
$79$		−	6.73632i		−	0.757895i	−0.925418	−	0.378947i	$-0.876286\pi$
$79$		−	6.73632i		−	0.757895i	0.925418	−	0.378947i	$-0.123714\pi$
$80$	−1.00000			−0.111803
$81$		0			0
$82$	−3.83598			−0.423613
$83$		−	5.22557i		−	0.573581i	−0.957993	−	0.286790i	$-0.907412\pi$
$83$		−	5.22557i		−	0.573581i	0.957993	−	0.286790i	$-0.0925884\pi$
$84$		0			0
$85$			7.87143i			0.853777i
$86$			2.60950i			0.281390i
$87$		0			0
$88$	2.37646			0.253332
$89$		−	7.05666i		−	0.748005i	−0.927428	−	0.374002i	$-0.877985\pi$
$89$		−	7.05666i		−	0.748005i	0.927428	−	0.374002i	$-0.122015\pi$
$90$		0			0
$91$	8.50523			0.891590
$92$		−	9.18285i		−	0.957379i
$93$		0			0
$94$			1.91486i			0.197503i
$95$	0.480302			0.0492779
$96$		0			0
$97$		−	1.67744i		−	0.170318i	−0.996367	−	0.0851591i	$-0.972860\pi$
$97$		−	1.67744i		−	0.170318i	0.996367	−	0.0851591i	$-0.0271399\pi$
$98$	0.0286042			0.00288946
$99$		0			0
$100$	1.00000			0.100000
$101$	−7.89090			−0.785174			−0.392587	−	0.919715i	$-0.628420\pi$
$101$	−7.89090			−0.785174			−0.392587	+	0.919715i	$0.628420\pi$
$102$		0			0
$103$	−3.53634			−0.348446			−0.174223	−	0.984706i	$-0.555741\pi$
$103$	−3.53634			−0.348446			−0.174223	+	0.984706i	$0.555741\pi$
$104$		−	3.20813i		−	0.314583i
$105$		0			0
$106$	−2.63457			−0.255892
$107$			1.89580i			0.183274i	0.995792	+	0.0916371i	$0.0292100\pi$
$107$			1.89580i			0.183274i	−0.995792	+	0.0916371i	$0.970790\pi$
$108$		0			0
$109$		−	9.65659i		−	0.924933i	−0.886637	−	0.462467i	$-0.846965\pi$
$109$		−	9.65659i		−	0.924933i	0.886637	−	0.462467i	$-0.153035\pi$
$110$	−2.37646			−0.226587
$111$		0			0
$112$		−	2.65115i		−	0.250510i
$113$	3.44947			0.324499			0.162250	−	0.986750i	$-0.448125\pi$
$113$	3.44947			0.324499			0.162250	+	0.986750i	$0.448125\pi$
$114$		0			0
$115$			9.18285i			0.856305i
$116$		−	0.463977i		−	0.0430792i
$117$		0			0
$118$			3.35047i			0.308436i
$119$	−20.8684			−1.91300
$120$		0			0
$121$	−5.35243			−0.486584
$122$		−	14.7145i		−	1.33219i
$123$		0			0
$124$		−	0.385676i		−	0.0346347i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−1.91263			−0.169719			−0.0848593	−	0.996393i	$-0.527044\pi$
$127$	−1.91263			−0.169719			−0.0848593	+	0.996393i	$0.527044\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$			3.20813i			0.281371i
$131$			8.11155i			0.708709i	0.935111	+	0.354355i	$0.115299\pi$
$131$			8.11155i			0.708709i	−0.935111	+	0.354355i	$0.884701\pi$
$132$		0			0
$133$			1.27335i			0.110414i
$134$	−2.65259	−	7.74363i	−0.229149	−	0.668948i
$135$		0			0
$136$			7.87143i			0.674970i
$137$	−16.9399			−1.44727			−0.723637	−	0.690181i	$-0.757531\pi$
$137$	−16.9399			−1.44727			−0.723637	+	0.690181i	$0.757531\pi$
$138$		0			0
$139$		−	21.8155i		−	1.85037i	−0.379521	−	0.925183i	$-0.623911\pi$
$139$		−	21.8155i		−	1.85037i	0.379521	−	0.925183i	$-0.376089\pi$
$140$			2.65115i			0.224063i
$141$		0			0
$142$		−	6.09202i		−	0.511231i
$143$		−	7.62399i		−	0.637550i
$144$		0			0
$145$			0.463977i			0.0385312i
$146$	9.36227			0.774827
$147$		0			0
$148$	−1.46251			−0.120218
$149$			16.0388i			1.31395i	0.753911	+	0.656976i	$0.228165\pi$
$149$			16.0388i			1.31395i	−0.753911	+	0.656976i	$0.771835\pi$
$150$		0			0
$151$	15.9855			1.30088			0.650441	−	0.759557i	$-0.274584\pi$
$151$	15.9855			1.30088			0.650441	+	0.759557i	$0.274584\pi$
$152$	0.480302			0.0389576
$153$		0			0
$154$		−	6.30036i		−	0.507698i
$155$			0.385676i			0.0309783i
$156$		0			0
$157$	14.5645			1.16237			0.581187	−	0.813770i	$-0.302588\pi$
$157$	14.5645			1.16237			0.581187	+	0.813770i	$0.302588\pi$
$158$			6.73632i			0.535913i
$159$		0			0
$160$	1.00000			0.0790569
$161$	−24.3451			−1.91867
$162$		0			0
$163$	12.0911			0.947051			0.473526	−	0.880780i	$-0.342981\pi$
$163$	12.0911			0.947051			0.473526	+	0.880780i	$0.342981\pi$
$164$	3.83598			0.299540
$165$		0			0
$166$			5.22557i			0.405583i
$167$		−	10.4319i		−	0.807241i	−0.914926	−	0.403621i	$-0.867752\pi$
$167$		−	10.4319i		−	0.807241i	0.914926	−	0.403621i	$-0.132248\pi$
$168$		0			0
$169$	2.70792			0.208302
$170$		−	7.87143i		−	0.603711i
$171$		0			0
$172$		−	2.60950i		−	0.198973i
$173$			9.38289i			0.713368i	0.934225	+	0.356684i	$0.116093\pi$
$173$			9.38289i			0.713368i	−0.934225	+	0.356684i	$0.883907\pi$
$174$		0			0
$175$		−	2.65115i		−	0.200408i
$176$	−2.37646			−0.179133
$177$		0			0
$178$			7.05666i			0.528919i
$179$	−21.2531			−1.58853			−0.794265	−	0.607571i	$-0.792144\pi$
$179$	−21.2531			−1.58853			−0.794265	+	0.607571i	$0.792144\pi$
$180$		0			0
$181$	−14.9159			−1.10869			−0.554347	−	0.832286i	$-0.687032\pi$
$181$	−14.9159			−1.10869			−0.554347	+	0.832286i	$0.687032\pi$
$182$	−8.50523			−0.630450
$183$		0			0
$184$			9.18285i			0.676969i
$185$	1.46251			0.107526
$186$		0			0
$187$			18.7062i			1.36793i
$188$		−	1.91486i		−	0.139656i
$189$		0			0
$190$	−0.480302			−0.0348448
$191$	−11.6633			−0.843923			−0.421962	−	0.906614i	$-0.638658\pi$
$191$	−11.6633			−0.843923			−0.421962	+	0.906614i	$0.638658\pi$
$192$		0			0
$193$	−6.38356			−0.459499			−0.229749	−	0.973250i	$-0.573791\pi$
$193$	−6.38356			−0.459499			−0.229749	+	0.973250i	$0.573791\pi$
$194$			1.67744i			0.120433i
$195$		0			0
$196$	−0.0286042			−0.00204315
$197$	−4.17857			−0.297711			−0.148855	−	0.988859i	$-0.547559\pi$
$197$	−4.17857			−0.297711			−0.148855	+	0.988859i	$0.547559\pi$
$198$		0			0
$199$	−3.96063			−0.280761			−0.140381	−	0.990098i	$-0.544833\pi$
$199$	−3.96063			−0.280761			−0.140381	+	0.990098i	$0.544833\pi$
$200$	−1.00000			−0.0707107
$201$		0			0
$202$	7.89090			0.555202
$203$	−1.23007			−0.0863342
$204$		0			0
$205$	−3.83598			−0.267916
$206$	3.53634			0.246388
$207$		0			0
$208$			3.20813i			0.222444i
$209$	1.14142			0.0789536
$210$		0			0
$211$	−18.2205			−1.25435			−0.627176	−	0.778877i	$-0.715790\pi$
$211$	−18.2205			−1.25435			−0.627176	+	0.778877i	$0.715790\pi$
$212$	2.63457			0.180943
$213$		0			0
$214$		−	1.89580i		−	0.129594i
$215$			2.60950i			0.177967i
$216$		0			0
$217$	−1.02249			−0.0694109
$218$			9.65659i			0.654026i
$219$		0			0
$220$	2.37646			0.160221
$221$	25.2526			1.69867
$222$		0			0
$223$	12.8448			0.860148			0.430074	−	0.902794i	$-0.358487\pi$
$223$	12.8448			0.860148			0.430074	+	0.902794i	$0.358487\pi$
$224$			2.65115i			0.177138i
$225$		0			0
$226$	−3.44947			−0.229456
$227$			20.6700i			1.37191i	0.727642	+	0.685957i	$0.240616\pi$
$227$			20.6700i			1.37191i	−0.727642	+	0.685957i	$0.759384\pi$
$228$		0			0
$229$			6.40386i			0.423179i	0.977359	+	0.211590i	$0.0678640\pi$
$229$			6.40386i			0.423179i	−0.977359	+	0.211590i	$0.932136\pi$
$230$		−	9.18285i		−	0.605499i
$231$		0			0
$232$			0.463977i			0.0304616i
$233$	8.18874			0.536462			0.268231	−	0.963355i	$-0.413561\pi$
$233$	8.18874			0.536462			0.268231	+	0.963355i	$0.413561\pi$
$234$		0			0
$235$			1.91486i			0.124912i
$236$		−	3.35047i		−	0.218097i
$237$		0			0
$238$	20.8684			1.35269
$239$	5.93875			0.384146			0.192073	−	0.981381i	$-0.438479\pi$
$239$	5.93875			0.384146			0.192073	+	0.981381i	$0.438479\pi$
$240$		0			0
$241$	8.24641			0.531198			0.265599	−	0.964084i	$-0.414430\pi$
$241$	8.24641			0.531198			0.265599	+	0.964084i	$0.414430\pi$
$242$	5.35243			0.344067
$243$		0			0
$244$			14.7145i			0.942000i
$245$	0.0286042			0.00182745
$246$		0			0
$247$		−	1.54087i		−	0.0980432i
$248$			0.385676i			0.0244905i
$249$		0			0
$250$	1.00000			0.0632456
$251$	−19.4380			−1.22692			−0.613458	−	0.789727i	$-0.710222\pi$
$251$	−19.4380			−1.22692			−0.613458	+	0.789727i	$0.710222\pi$
$252$		0			0
$253$			21.8227i			1.37198i
$254$	1.91263			0.120009
$255$		0			0
$256$	1.00000			0.0625000
$257$			1.41274i			0.0881240i	0.999029	+	0.0440620i	$0.0140299\pi$
$257$			1.41274i			0.0881240i	−0.999029	+	0.0440620i	$0.985970\pi$
$258$		0			0
$259$			3.87734i			0.240926i
$260$		−	3.20813i		−	0.198960i
$261$		0			0
$262$		−	8.11155i		−	0.501133i
$263$			11.1047i			0.684745i	0.939564	+	0.342373i	$0.111231\pi$
$263$			11.1047i			0.684745i	−0.939564	+	0.342373i	$0.888769\pi$
$264$		0			0
$265$	−2.63457			−0.161841
$266$		−	1.27335i		−	0.0780743i
$267$		0			0
$268$	2.65259	+	7.74363i	0.162033	+	0.473017i
$269$			13.5863i			0.828373i	0.910192	+	0.414187i	$0.135934\pi$
$269$			13.5863i			0.828373i	−0.910192	+	0.414187i	$0.864066\pi$
$270$		0			0
$271$			22.0533i			1.33964i	0.742523	+	0.669820i	$0.233629\pi$
$271$			22.0533i			1.33964i	−0.742523	+	0.669820i	$0.766371\pi$
$272$		−	7.87143i		−	0.477276i
$273$		0			0
$274$	16.9399			1.02338
$275$	−2.37646			−0.143306
$276$		0			0
$277$	5.10726			0.306866			0.153433	−	0.988159i	$-0.450967\pi$
$277$	5.10726			0.306866			0.153433	+	0.988159i	$0.450967\pi$
$278$			21.8155i			1.30841i
$279$		0			0
$280$		−	2.65115i		−	0.158437i
$281$	−5.35384			−0.319383			−0.159692	−	0.987167i	$-0.551050\pi$
$281$	−5.35384			−0.319383			−0.159692	+	0.987167i	$0.551050\pi$
$282$		0			0
$283$	−17.6403			−1.04861			−0.524303	−	0.851532i	$-0.675674\pi$
$283$	−17.6403			−1.04861			−0.524303	+	0.851532i	$0.675674\pi$
$284$			6.09202i			0.361495i
$285$		0			0
$286$			7.62399i			0.450816i
$287$		−	10.1698i		−	0.600302i
$288$		0			0
$289$	−44.9594			−2.64467
$290$		−	0.463977i		−	0.0272457i
$291$		0			0
$292$	−9.36227			−0.547886
$293$		−	26.6654i		−	1.55781i	−0.627142	−	0.778905i	$-0.715776\pi$
$293$		−	26.6654i		−	1.55781i	0.627142	−	0.778905i	$-0.284224\pi$
$294$		0			0
$295$			3.35047i			0.195072i
$296$	1.46251			0.0850067
$297$		0			0
$298$		−	16.0388i		−	0.929104i
$299$	29.4598			1.70370
$300$		0			0
$301$	−6.91819			−0.398758
$302$	−15.9855			−0.919862
$303$		0			0
$304$	−0.480302			−0.0275472
$305$		−	14.7145i		−	0.842550i
$306$		0			0
$307$	23.4194			1.33661			0.668307	−	0.743886i	$-0.267019\pi$
$307$	23.4194			1.33661			0.668307	+	0.743886i	$0.267019\pi$
$308$			6.30036i			0.358996i
$309$		0			0
$310$		−	0.385676i		−	0.0219049i
$311$	−8.30709			−0.471052			−0.235526	−	0.971868i	$-0.575681\pi$
$311$	−8.30709			−0.471052			−0.235526	+	0.971868i	$0.575681\pi$
$312$		0			0
$313$			29.8306i			1.68613i	0.537814	+	0.843064i	$0.319250\pi$
$313$			29.8306i			1.68613i	−0.537814	+	0.843064i	$0.680750\pi$
$314$	−14.5645			−0.821923
$315$		0			0
$316$		−	6.73632i		−	0.378947i
$317$			3.82342i			0.214745i	0.994219	+	0.107372i	$0.0342437\pi$
$317$			3.82342i			0.214745i	−0.994219	+	0.107372i	$0.965756\pi$
$318$		0			0
$319$			1.10262i			0.0617351i
$320$	−1.00000			−0.0559017
$321$		0			0
$322$	24.3451			1.35670
$323$			3.78066i			0.210362i
$324$		0			0
$325$			3.20813i			0.177955i
$326$	−12.0911			−0.669666
$327$		0			0
$328$	−3.83598			−0.211807
$329$	−5.07660			−0.279882
$330$		0			0
$331$			13.5228i			0.743282i	0.928377	+	0.371641i	$0.121205\pi$
$331$			13.5228i			0.743282i	−0.928377	+	0.371641i	$0.878795\pi$
$332$		−	5.22557i		−	0.286790i
$333$		0			0
$334$			10.4319i			0.570806i
$335$	−2.65259	−	7.74363i	−0.144926	−	0.423080i
$336$		0			0
$337$		−	17.3834i		−	0.946933i	−0.880812	−	0.473466i	$-0.843003\pi$
$337$		−	17.3834i		−	0.946933i	0.880812	−	0.473466i	$-0.156997\pi$
$338$	−2.70792			−0.147292
$339$		0			0
$340$			7.87143i			0.426888i
$341$			0.916545i			0.0496337i
$342$		0			0
$343$		−	18.4822i		−	0.997946i
$344$			2.60950i			0.140695i
$345$		0			0
$346$		−	9.38289i		−	0.504427i
$347$	4.16658			0.223674			0.111837	−	0.993727i	$-0.464327\pi$
$347$	4.16658			0.223674			0.111837	+	0.993727i	$0.464327\pi$
$348$		0			0
$349$	−15.4686			−0.828017			−0.414008	−	0.910273i	$-0.635872\pi$
$349$	−15.4686			−0.828017			−0.414008	+	0.910273i	$0.635872\pi$
$350$			2.65115i			0.141710i
$351$		0			0
$352$	2.37646			0.126666
$353$	−14.7586			−0.785519			−0.392760	−	0.919641i	$-0.628480\pi$
$353$	−14.7586			−0.785519			−0.392760	+	0.919641i	$0.628480\pi$
$354$		0			0
$355$		−	6.09202i		−	0.323331i
$356$		−	7.05666i		−	0.374002i
$357$		0			0
$358$	21.2531			1.12326
$359$		−	22.1470i		−	1.16887i	−0.811439	−	0.584437i	$-0.801315\pi$
$359$		−	22.1470i		−	1.16887i	0.811439	−	0.584437i	$-0.198685\pi$
$360$		0			0
$361$	−18.7693			−0.987858
$362$	14.9159			0.783964
$363$		0			0
$364$	8.50523			0.445795
$365$	9.36227			0.490044
$366$		0			0
$367$		−	18.2022i		−	0.950149i	−0.879945	−	0.475075i	$-0.842421\pi$
$367$		−	18.2022i		−	0.950149i	0.879945	−	0.475075i	$-0.157579\pi$
$368$		−	9.18285i		−	0.478689i
$369$		0			0
$370$	−1.46251			−0.0760323
$371$		−	6.98465i		−	0.362625i
$372$		0			0
$373$			7.24041i			0.374894i	0.982275	+	0.187447i	$0.0600213\pi$
$373$			7.24041i			0.374894i	−0.982275	+	0.187447i	$0.939979\pi$
$374$		−	18.7062i		−	0.967273i
$375$		0			0
$376$			1.91486i			0.0987516i
$377$	1.48850			0.0766615
$378$		0			0
$379$		−	13.7524i		−	0.706414i	−0.935545	−	0.353207i	$-0.885091\pi$
$379$		−	13.7524i		−	0.706414i	0.935545	−	0.353207i	$-0.114909\pi$
$380$	0.480302			0.0246390
$381$		0			0
$382$	11.6633			0.596744
$383$	−37.5336			−1.91788			−0.958938	−	0.283617i	$-0.908465\pi$
$383$	−37.5336			−1.91788			−0.958938	+	0.283617i	$0.908465\pi$
$384$		0			0
$385$		−	6.30036i		−	0.321096i
$386$	6.38356			0.324915
$387$		0			0
$388$		−	1.67744i		−	0.0851591i
$389$			0.288667i			0.0146360i	0.999973	+	0.00731800i	$0.00232941\pi$
$389$			0.288667i			0.0146360i	−0.999973	+	0.00731800i	$0.997671\pi$
$390$		0			0
$391$	−72.2822			−3.65547
$392$	0.0286042			0.00144473
$393$		0			0
$394$	4.17857			0.210513
$395$			6.73632i			0.338941i
$396$		0			0
$397$	−28.4458			−1.42765			−0.713826	−	0.700323i	$-0.753039\pi$
$397$	−28.4458			−1.42765			−0.713826	+	0.700323i	$0.753039\pi$
$398$	3.96063			0.198528
$399$		0			0
$400$	1.00000			0.0500000
$401$	17.2500			0.861423			0.430712	−	0.902490i	$-0.358263\pi$
$401$	17.2500			0.861423			0.430712	+	0.902490i	$0.358263\pi$
$402$		0			0
$403$	1.23730			0.0616342
$404$	−7.89090			−0.392587
$405$		0			0
$406$	1.23007			0.0610475
$407$	3.47560			0.172279
$408$		0			0
$409$			20.4706i			1.01221i	0.862472	+	0.506104i	$0.168915\pi$
$409$			20.4706i			1.01221i	−0.862472	+	0.506104i	$0.831085\pi$
$410$	3.83598			0.189446
$411$		0			0
$412$	−3.53634			−0.174223
$413$	−8.88261			−0.437084
$414$		0			0
$415$			5.22557i			0.256513i
$416$		−	3.20813i		−	0.157291i
$417$		0			0
$418$	−1.14142			−0.0558286
$419$		−	25.2025i		−	1.23122i	−0.788049	−	0.615612i	$-0.788909\pi$
$419$		−	25.2025i		−	1.23122i	0.788049	−	0.615612i	$-0.211091\pi$
$420$		0			0
$421$	24.5302			1.19553			0.597765	−	0.801671i	$-0.296055\pi$
$421$	24.5302			1.19553			0.597765	+	0.801671i	$0.296055\pi$
$422$	18.2205			0.886961
$423$		0			0
$424$	−2.63457			−0.127946
$425$		−	7.87143i		−	0.381821i
$426$		0			0
$427$	39.0104			1.88785
$428$			1.89580i			0.0916371i
$429$		0			0
$430$		−	2.60950i		−	0.125841i
$431$		−	24.0897i		−	1.16036i	−0.814488	−	0.580180i	$-0.802982\pi$
$431$		−	24.0897i		−	1.16036i	0.814488	−	0.580180i	$-0.197018\pi$
$432$		0			0
$433$		−	25.6231i		−	1.23137i	−0.787993	−	0.615684i	$-0.788880\pi$
$433$		−	25.6231i		−	1.23137i	0.787993	−	0.615684i	$-0.211120\pi$
$434$	1.02249			0.0490809
$435$		0			0
$436$		−	9.65659i		−	0.462467i
$437$			4.41054i			0.210985i
$438$		0			0
$439$	−33.9366			−1.61971			−0.809853	−	0.586633i	$-0.800453\pi$
$439$	−33.9366			−1.61971			−0.809853	+	0.586633i	$0.800453\pi$
$440$	−2.37646			−0.113293
$441$		0			0
$442$	−25.2526			−1.20114
$443$	−21.1506			−1.00490			−0.502449	−	0.864607i	$-0.667567\pi$
$443$	−21.1506			−1.00490			−0.502449	+	0.864607i	$0.667567\pi$
$444$		0			0
$445$			7.05666i			0.334518i
$446$	−12.8448			−0.608217
$447$		0			0
$448$		−	2.65115i		−	0.125255i
$449$			16.8866i			0.796926i	0.917184	+	0.398463i	$0.130456\pi$
$449$			16.8866i			0.796926i	−0.917184	+	0.398463i	$0.869544\pi$
$450$		0			0
$451$	−9.11606			−0.429259
$452$	3.44947			0.162250
$453$		0			0
$454$		−	20.6700i		−	0.970090i
$455$	−8.50523			−0.398731
$456$		0			0
$457$	5.97624			0.279557			0.139778	−	0.990183i	$-0.455361\pi$
$457$	5.97624			0.279557			0.139778	+	0.990183i	$0.455361\pi$
$458$		−	6.40386i		−	0.299233i
$459$		0			0
$460$			9.18285i			0.428153i
$461$		−	10.4601i		−	0.487175i	−0.969879	−	0.243587i	$-0.921676\pi$
$461$		−	10.4601i		−	0.487175i	0.969879	−	0.243587i	$-0.0783243\pi$
$462$		0			0
$463$		−	32.9643i		−	1.53198i	−0.642852	−	0.765990i	$-0.722249\pi$
$463$		−	32.9643i		−	1.53198i	0.642852	−	0.765990i	$-0.277751\pi$
$464$		−	0.463977i		−	0.0215396i
$465$		0			0
$466$	−8.18874			−0.379336
$467$			14.0031i			0.647988i	0.946059	+	0.323994i	$0.105026\pi$
$467$			14.0031i			0.647988i	−0.946059	+	0.323994i	$0.894974\pi$
$468$		0			0
$469$	20.5295	−	7.03241i	0.947966	−	0.324727i
$470$		−	1.91486i		−	0.0883261i
$471$		0			0
$472$			3.35047i			0.154218i
$473$			6.20139i			0.285140i
$474$		0			0
$475$	−0.480302			−0.0220378
$476$	−20.8684			−0.956500
$477$		0			0
$478$	−5.93875			−0.271632
$479$			17.8841i			0.817145i	0.912726	+	0.408572i	$0.133973\pi$
$479$			17.8841i			0.817145i	−0.912726	+	0.408572i	$0.866027\pi$
$480$		0			0
$481$		−	4.69192i		−	0.213933i
$482$	−8.24641			−0.375614
$483$		0			0
$484$	−5.35243			−0.243292
$485$			1.67744i			0.0761687i
$486$		0			0
$487$			28.1809i			1.27700i	0.769622	+	0.638500i	$0.220445\pi$
$487$			28.1809i			1.27700i	−0.769622	+	0.638500i	$0.779555\pi$
$488$		−	14.7145i		−	0.666095i
$489$		0			0
$490$	−0.0286042			−0.00129220
$491$			18.2542i			0.823801i	0.911229	+	0.411900i	$0.135135\pi$
$491$			18.2542i			0.823801i	−0.911229	+	0.411900i	$0.864865\pi$
$492$		0			0
$493$	−3.65216			−0.164485
$494$			1.54087i			0.0693270i
$495$		0			0
$496$		−	0.385676i		−	0.0173174i
$497$	16.1509			0.724466
$498$		0			0
$499$		−	6.71500i		−	0.300605i	−0.988640	−	0.150302i	$-0.951975\pi$
$499$		−	6.71500i		−	0.300605i	0.988640	−	0.150302i	$-0.0480247\pi$
$500$	−1.00000			−0.0447214
$501$		0			0
$502$	19.4380			0.867561
$503$	−8.01567			−0.357401			−0.178701	−	0.983904i	$-0.557189\pi$
$503$	−8.01567			−0.357401			−0.178701	+	0.983904i	$0.557189\pi$
$504$		0			0
$505$	7.89090			0.351141
$506$		−	21.8227i		−	0.970138i
$507$		0			0
$508$	−1.91263			−0.0848593
$509$			23.5516i			1.04391i	0.852974	+	0.521954i	$0.174797\pi$
$509$			23.5516i			1.04391i	−0.852974	+	0.521954i	$0.825203\pi$
$510$		0			0
$511$			24.8208i			1.09801i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$		−	1.41274i		−	0.0623131i
$515$	3.53634			0.155830
$516$		0			0
$517$			4.55060i			0.200135i
$518$		−	3.87734i		−	0.170360i
$519$		0			0
$520$			3.20813i			0.140686i
$521$	−16.3825			−0.717731			−0.358865	−	0.933389i	$-0.616836\pi$
$521$	−16.3825			−0.717731			−0.358865	+	0.933389i	$0.616836\pi$
$522$		0			0
$523$	29.5918			1.29396			0.646980	−	0.762507i	$-0.276032\pi$
$523$	29.5918			1.29396			0.646980	+	0.762507i	$0.276032\pi$
$524$			8.11155i			0.354355i
$525$		0			0
$526$		−	11.1047i		−	0.484188i
$527$	−3.03582			−0.132243
$528$		0			0
$529$	−61.3248			−2.66630
$530$	2.63457			0.114439
$531$		0			0
$532$			1.27335i			0.0552068i
$533$			12.3063i			0.533045i
$534$		0			0
$535$		−	1.89580i		−	0.0819627i
$536$	−2.65259	−	7.74363i	−0.114574	−	0.334474i
$537$		0			0
$538$		−	13.5863i		−	0.585748i
$539$	0.0679767			0.00292796
$540$		0			0
$541$			13.4662i			0.578955i	0.957185	+	0.289478i	$0.0934816\pi$
$541$			13.4662i			0.578955i	−0.957185	+	0.289478i	$0.906518\pi$
$542$		−	22.0533i		−	0.947269i
$543$		0			0
$544$			7.87143i			0.337485i
$545$			9.65659i			0.413643i
$546$		0			0
$547$			14.8725i			0.635900i	0.948107	+	0.317950i	$0.102994\pi$
$547$			14.8725i			0.635900i	−0.948107	+	0.317950i	$0.897006\pi$
$548$	−16.9399			−0.723637
$549$		0			0
$550$	2.37646			0.101333
$551$			0.222849i			0.00949368i
$552$		0			0
$553$	−17.8590			−0.759442
$554$	−5.10726			−0.216987
$555$		0			0
$556$		−	21.8155i		−	0.925183i
$557$			13.8312i			0.586048i	0.956105	+	0.293024i	$0.0946616\pi$
$557$			13.8312i			0.586048i	−0.956105	+	0.293024i	$0.905338\pi$
$558$		0			0
$559$	8.37162			0.354082
$560$			2.65115i			0.112032i
$561$		0			0
$562$	5.35384			0.225838
$563$	2.26916			0.0956338			0.0478169	−	0.998856i	$-0.484774\pi$
$563$	2.26916			0.0956338			0.0478169	+	0.998856i	$0.484774\pi$
$564$		0			0
$565$	−3.44947			−0.145120
$566$	17.6403			0.741477
$567$		0			0
$568$		−	6.09202i		−	0.255616i
$569$			24.9292i			1.04508i	0.852613	+	0.522542i	$0.175016\pi$
$569$			24.9292i			1.04508i	−0.852613	+	0.522542i	$0.824984\pi$
$570$		0			0
$571$	−34.8020			−1.45642			−0.728209	−	0.685356i	$-0.759647\pi$
$571$	−34.8020			−1.45642			−0.728209	+	0.685356i	$0.759647\pi$
$572$		−	7.62399i		−	0.318775i
$573$		0			0
$574$			10.1698i			0.424478i
$575$		−	9.18285i		−	0.382951i
$576$		0			0
$577$		−	10.9263i		−	0.454866i	−0.973794	−	0.227433i	$-0.926967\pi$
$577$		−	10.9263i		−	0.454866i	0.973794	−	0.227433i	$-0.0730333\pi$
$578$	44.9594			1.87007
$579$		0			0
$580$			0.463977i			0.0192656i
$581$	−13.8538			−0.574752
$582$		0			0
$583$	−6.26097			−0.259303
$584$	9.36227			0.387414
$585$		0			0
$586$			26.6654i			1.10154i
$587$	34.7114			1.43269			0.716347	−	0.697744i	$-0.245813\pi$
$587$	34.7114			1.43269			0.716347	+	0.697744i	$0.245813\pi$
$588$		0			0
$589$			0.185241i			0.00763272i
$590$		−	3.35047i		−	0.137937i
$591$		0			0
$592$	−1.46251			−0.0601088
$593$	31.1174			1.27784			0.638920	−	0.769273i	$-0.279381\pi$
$593$	31.1174			1.27784			0.638920	+	0.769273i	$0.279381\pi$
$594$		0			0
$595$	20.8684			0.855519
$596$			16.0388i			0.656976i
$597$		0			0
$598$	−29.4598			−1.20470
$599$	15.1681			0.619754			0.309877	−	0.950777i	$-0.399712\pi$
$599$	15.1681			0.619754			0.309877	+	0.950777i	$0.399712\pi$
$600$		0			0
$601$	−14.7425			−0.601358			−0.300679	−	0.953725i	$-0.597213\pi$
$601$	−14.7425			−0.601358			−0.300679	+	0.953725i	$0.597213\pi$
$602$	6.91819			0.281964
$603$		0			0
$604$	15.9855			0.650441
$605$	5.35243			0.217607
$606$		0			0
$607$	9.38573			0.380955			0.190477	−	0.981692i	$-0.438996\pi$
$607$	9.38573			0.380955			0.190477	+	0.981692i	$0.438996\pi$
$608$	0.480302			0.0194788
$609$		0			0
$610$			14.7145i			0.595773i
$611$	6.14313			0.248524
$612$		0			0
$613$	−23.8173			−0.961971			−0.480986	−	0.876728i	$-0.659721\pi$
$613$	−23.8173			−0.961971			−0.480986	+	0.876728i	$0.659721\pi$
$614$	−23.4194			−0.945129
$615$		0			0
$616$		−	6.30036i		−	0.253849i
$617$			9.20247i			0.370477i	0.982694	+	0.185239i	$0.0593058\pi$
$617$			9.20247i			0.370477i	−0.982694	+	0.185239i	$0.940694\pi$
$618$		0			0
$619$	−1.14435			−0.0459954			−0.0229977	−	0.999736i	$-0.507321\pi$
$619$	−1.14435			−0.0459954			−0.0229977	+	0.999736i	$0.507321\pi$
$620$			0.385676i			0.0154891i
$621$		0			0
$622$	8.30709			0.333084
$623$	−18.7083			−0.749532
$624$		0			0
$625$	1.00000			0.0400000
$626$		−	29.8306i		−	1.19227i
$627$		0			0
$628$	14.5645			0.581187
$629$			11.5121i			0.459016i
$630$		0			0
$631$		−	1.84923i		−	0.0736169i	−0.999322	−	0.0368084i	$-0.988281\pi$
$631$		−	1.84923i		−	0.0736169i	0.999322	−	0.0368084i	$-0.0117191\pi$
$632$			6.73632i			0.267956i
$633$		0			0
$634$		−	3.82342i		−	0.151847i
$635$	1.91263			0.0759005
$636$		0			0
$637$		−	0.0917657i		−	0.00363589i
$638$		−	1.10262i		−	0.0436533i
$639$		0			0
$640$	1.00000			0.0395285
$641$	−50.1811			−1.98204			−0.991018	−	0.133731i	$-0.957304\pi$
$641$	−50.1811			−1.98204			−0.991018	+	0.133731i	$0.957304\pi$
$642$		0			0
$643$	−12.4053			−0.489218			−0.244609	−	0.969622i	$-0.578660\pi$
$643$	−12.4053			−0.489218			−0.244609	+	0.969622i	$0.578660\pi$
$644$	−24.3451			−0.959333
$645$		0			0
$646$		−	3.78066i		−	0.148748i
$647$	32.4968			1.27758			0.638790	−	0.769381i	$-0.279435\pi$
$647$	32.4968			1.27758			0.638790	+	0.769381i	$0.279435\pi$
$648$		0			0
$649$			7.96227i			0.312546i
$650$		−	3.20813i		−	0.125833i
$651$		0			0
$652$	12.0911			0.473526
$653$	−13.5854			−0.531637			−0.265818	−	0.964023i	$-0.585642\pi$
$653$	−13.5854			−0.531637			−0.265818	+	0.964023i	$0.585642\pi$
$654$		0			0
$655$		−	8.11155i		−	0.316944i
$656$	3.83598			0.149770
$657$		0			0
$658$	5.07660			0.197906
$659$			46.8596i			1.82539i	0.408641	+	0.912695i	$0.366003\pi$
$659$			46.8596i			1.82539i	−0.408641	+	0.912695i	$0.633997\pi$
$660$		0			0
$661$		−	34.2464i		−	1.33203i	−0.745939	−	0.666015i	$-0.767999\pi$
$661$		−	34.2464i		−	1.33203i	0.745939	−	0.666015i	$-0.232001\pi$
$662$		−	13.5228i		−	0.525580i
$663$		0			0
$664$			5.22557i			0.202791i
$665$		−	1.27335i		−	0.0493785i
$666$		0			0
$667$	−4.26063			−0.164972
$668$		−	10.4319i		−	0.403621i
$669$		0			0
$670$	2.65259	+	7.74363i	0.102478	+	0.299162i
$671$		−	34.9685i		−	1.34994i
$672$		0			0
$673$			13.4148i			0.517102i	0.965998	+	0.258551i	$0.0832451\pi$
$673$			13.4148i			0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$674$			17.3834i			0.669583i
$675$		0			0
$676$	2.70792			0.104151
$677$	−25.8703			−0.994277			−0.497139	−	0.867671i	$-0.665616\pi$
$677$	−25.8703			−0.994277			−0.497139	+	0.867671i	$0.665616\pi$
$678$		0			0
$679$	−4.44715			−0.170666
$680$		−	7.87143i		−	0.301856i
$681$		0			0
$682$		−	0.916545i		−	0.0350963i
$683$	−19.8947			−0.761251			−0.380626	−	0.924729i	$-0.624291\pi$
$683$	−19.8947			−0.761251			−0.380626	+	0.924729i	$0.624291\pi$
$684$		0			0
$685$	16.9399			0.647240
$686$			18.4822i			0.705655i
$687$		0			0
$688$		−	2.60950i		−	0.0994864i
$689$			8.45205i			0.321997i
$690$		0			0
$691$	19.4984			0.741756			0.370878	−	0.928682i	$-0.379057\pi$
$691$	19.4984			0.741756			0.370878	+	0.928682i	$0.379057\pi$
$692$			9.38289i			0.356684i
$693$		0			0
$694$	−4.16658			−0.158161
$695$			21.8155i			0.827509i
$696$		0			0
$697$		−	30.1947i		−	1.14370i
$698$	15.4686			0.585496
$699$		0			0
$700$		−	2.65115i		−	0.100204i
$701$	−12.6504			−0.477801			−0.238900	−	0.971044i	$-0.576787\pi$
$701$	−12.6504			−0.477801			−0.238900	+	0.971044i	$0.576787\pi$
$702$		0			0
$703$	0.702446			0.0264933
$704$	−2.37646			−0.0895663
$705$		0			0
$706$	14.7586			0.555446
$707$			20.9200i			0.786777i
$708$		0			0
$709$	−3.83312			−0.143956			−0.0719779	−	0.997406i	$-0.522931\pi$
$709$	−3.83312			−0.143956			−0.0719779	+	0.997406i	$0.522931\pi$
$710$			6.09202i			0.228629i
$711$		0			0
$712$			7.05666i			0.264460i
$713$	−3.54161			−0.132634
$714$		0			0
$715$			7.62399i			0.285121i
$716$	−21.2531			−0.794265
$717$		0			0
$718$			22.1470i			0.826519i
$719$			11.6258i			0.433569i	0.976219	+	0.216785i	$0.0695570\pi$
$719$			11.6258i			0.433569i	−0.976219	+	0.216785i	$0.930443\pi$
$720$		0			0
$721$			9.37537i			0.349157i
$722$	18.7693			0.698521
$723$		0			0
$724$	−14.9159			−0.554347
$725$		−	0.463977i		−	0.0172317i
$726$		0			0
$727$			10.5085i			0.389739i	0.980829	+	0.194870i	$0.0624284\pi$
$727$			10.5085i			0.389739i	−0.980829	+	0.194870i	$0.937572\pi$
$728$	−8.50523			−0.315225
$729$		0			0
$730$	−9.36227			−0.346513
$731$	−20.5405			−0.759719
$732$		0			0
$733$		−	1.92140i		−	0.0709684i	−0.999370	−	0.0354842i	$-0.988703\pi$
$733$		−	1.92140i		−	0.0709684i	0.999370	−	0.0354842i	$-0.0112973\pi$
$734$			18.2022i			0.671857i
$735$		0			0
$736$			9.18285i			0.338484i
$737$	−6.30378	−	18.4024i	−0.232203	−	0.677863i
$738$		0			0
$739$			28.9587i			1.06526i	0.846347	+	0.532632i	$0.178797\pi$
$739$			28.9587i			1.06526i	−0.846347	+	0.532632i	$0.821203\pi$
$740$	1.46251			0.0537630
$741$		0			0
$742$			6.98465i			0.256415i
$743$		−	23.0996i		−	0.847442i	−0.905793	−	0.423721i	$-0.860724\pi$
$743$		−	23.0996i		−	0.847442i	0.905793	−	0.423721i	$-0.139276\pi$
$744$		0			0
$745$		−	16.0388i		−	0.587617i
$746$		−	7.24041i		−	0.265090i
$747$		0			0
$748$			18.7062i			0.683965i
$749$	5.02606			0.183648
$750$		0			0
$751$	2.45486			0.0895793			0.0447896	−	0.998996i	$-0.485738\pi$
$751$	2.45486			0.0895793			0.0447896	+	0.998996i	$0.485738\pi$
$752$		−	1.91486i		−	0.0698279i
$753$		0			0
$754$	−1.48850			−0.0542079
$755$	−15.9855			−0.581772
$756$		0			0
$757$		−	16.8241i		−	0.611482i	−0.952115	−	0.305741i	$-0.901096\pi$
$757$		−	16.8241i		−	0.611482i	0.952115	−	0.305741i	$-0.0989041\pi$
$758$			13.7524i			0.499510i
$759$		0			0
$760$	−0.480302			−0.0174224
$761$			20.9442i			0.759225i	0.925146	+	0.379612i	$0.123943\pi$
$761$			20.9442i			0.759225i	−0.925146	+	0.379612i	$0.876057\pi$
$762$		0			0
$763$	−25.6011			−0.926821
$764$	−11.6633			−0.421962
$765$		0			0
$766$	37.5336			1.35614
$767$	10.7487			0.388114
$768$		0			0
$769$		−	31.4567i		−	1.13436i	−0.823595	−	0.567179i	$-0.808035\pi$
$769$		−	31.4567i		−	1.13436i	0.823595	−	0.567179i	$-0.191965\pi$
$770$			6.30036i			0.227049i
$771$		0			0
$772$	−6.38356			−0.229749
$773$			32.9803i			1.18622i	0.805121	+	0.593110i	$0.202100\pi$
$773$			32.9803i			1.18622i	−0.805121	+	0.593110i	$0.797900\pi$
$774$		0			0
$775$		−	0.385676i		−	0.0138539i
$776$			1.67744i			0.0602166i
$777$		0			0
$778$		−	0.288667i		−	0.0103492i
$779$	−1.84243			−0.0660118
$780$		0			0
$781$		−	14.4775i		−	0.518044i
$782$	72.2822			2.58481
$783$		0			0
$784$	−0.0286042			−0.00102158
$785$	−14.5645			−0.519829
$786$		0			0
$787$		−	7.10846i		−	0.253389i	−0.991942	−	0.126695i	$-0.959563\pi$
$787$		−	7.10846i		−	0.253389i	0.991942	−	0.126695i	$-0.0404368\pi$
$788$	−4.17857			−0.148855
$789$		0			0
$790$		−	6.73632i		−	0.239667i
$791$		−	9.14508i		−	0.325161i
$792$		0			0
$793$	−47.2060			−1.67633
$794$	28.4458			1.00950
$795$		0			0
$796$	−3.96063			−0.140381
$797$		−	16.2740i		−	0.576456i	−0.957562	−	0.288228i	$-0.906934\pi$
$797$		−	16.2740i		−	0.576456i	0.957562	−	0.288228i	$-0.0930660\pi$
$798$		0			0
$799$	−15.0727			−0.533235
$800$	−1.00000			−0.0353553
$801$		0			0
$802$	−17.2500			−0.609118
$803$	22.2491			0.785153
$804$		0			0
$805$	24.3451			0.858053
$806$	−1.23730			−0.0435820
$807$		0			0
$808$	7.89090			0.277601
$809$	54.0083			1.89883			0.949416	−	0.314023i	$-0.101677\pi$
$809$	54.0083			1.89883			0.949416	+	0.314023i	$0.101677\pi$
$810$		0			0
$811$			0.701210i			0.0246228i	0.999924	+	0.0123114i	$0.00391894\pi$
$811$			0.701210i			0.0246228i	−0.999924	+	0.0123114i	$0.996081\pi$
$812$	−1.23007			−0.0431671
$813$		0			0
$814$	−3.47560			−0.121820
$815$	−12.0911			−0.423534
$816$		0			0
$817$			1.25335i			0.0438491i
$818$		−	20.4706i		−	0.715739i
$819$		0			0
$820$	−3.83598			−0.133958
$821$			41.9831i			1.46522i	0.680648	+	0.732610i	$0.261698\pi$
$821$			41.9831i			1.46522i	−0.680648	+	0.732610i	$0.738302\pi$
$822$		0			0
$823$	43.5730			1.51886			0.759429	−	0.650590i	$-0.225478\pi$
$823$	43.5730			1.51886			0.759429	+	0.650590i	$0.225478\pi$
$824$	3.53634			0.123194
$825$		0			0
$826$	8.88261			0.309065
$827$		−	18.6914i		−	0.649964i	−0.945720	−	0.324982i	$-0.894642\pi$
$827$		−	18.6914i		−	0.649964i	0.945720	−	0.324982i	$-0.105358\pi$
$828$		0			0
$829$	37.8614			1.31498			0.657491	−	0.753462i	$-0.271618\pi$
$829$	37.8614			1.31498			0.657491	+	0.753462i	$0.271618\pi$
$830$		−	5.22557i		−	0.181382i
$831$		0			0
$832$			3.20813i			0.111222i
$833$			0.225156i			0.00780118i
$834$		0			0
$835$			10.4319i			0.361009i
$836$	1.14142			0.0394768
$837$		0			0
$838$			25.2025i			0.870607i
$839$		−	22.3073i		−	0.770134i	−0.922889	−	0.385067i	$-0.874178\pi$
$839$		−	22.3073i		−	0.770134i	0.922889	−	0.385067i	$-0.125822\pi$
$840$		0			0
$841$	28.7847			0.992577
$842$	−24.5302			−0.845368
$843$		0			0
$844$	−18.2205			−0.627176
$845$	−2.70792			−0.0931554
$846$		0			0
$847$			14.1901i			0.487577i
$848$	2.63457			0.0904716
$849$		0			0
$850$			7.87143i			0.269988i
$851$			13.4300i			0.460375i
$852$		0			0
$853$	−20.5513			−0.703662			−0.351831	−	0.936064i	$-0.614441\pi$
$853$	−20.5513			−0.703662			−0.351831	+	0.936064i	$0.614441\pi$
$854$	−39.0104			−1.33491
$855$		0			0
$856$		−	1.89580i		−	0.0647972i
$857$	−52.0384			−1.77760			−0.888799	−	0.458298i	$-0.848459\pi$
$857$	−52.0384			−1.77760			−0.888799	+	0.458298i	$0.848459\pi$
$858$		0			0
$859$	−40.8224			−1.39284			−0.696421	−	0.717633i	$-0.745225\pi$
$859$	−40.8224			−1.39284			−0.696421	+	0.717633i	$0.745225\pi$
$860$			2.60950i			0.0889833i
$861$		0			0
$862$			24.0897i			0.820499i
$863$			21.3140i			0.725537i	0.931879	+	0.362768i	$0.118168\pi$
$863$			21.3140i			0.725537i	−0.931879	+	0.362768i	$0.881832\pi$
$864$		0			0
$865$		−	9.38289i		−	0.319028i
$866$			25.6231i			0.870709i
$867$		0			0
$868$	−1.02249			−0.0347054
$869$			16.0086i			0.543055i
$870$		0			0
$871$	−24.8425	+	8.50984i	−0.841757	+	0.288345i
$872$			9.65659i			0.327013i
$873$		0			0
$874$		−	4.41054i		−	0.149189i
$875$			2.65115i			0.0896253i
$876$		0			0
$877$	31.9109			1.07755			0.538777	−	0.842448i	$-0.318886\pi$
$877$	31.9109			1.07755			0.538777	+	0.842448i	$0.318886\pi$
$878$	33.9366			1.14530
$879$		0			0
$880$	2.37646			0.0801105
$881$			24.7203i			0.832848i	0.909171	+	0.416424i	$0.136717\pi$
$881$			24.7203i			0.832848i	−0.909171	+	0.416424i	$0.863283\pi$
$882$		0			0
$883$			13.8008i			0.464434i	0.972664	+	0.232217i	$0.0745980\pi$
$883$			13.8008i			0.464434i	−0.972664	+	0.232217i	$0.925402\pi$
$884$	25.2526			0.849335
$885$		0			0
$886$	21.1506			0.710570
$887$		−	37.5267i		−	1.26002i	−0.776586	−	0.630011i	$-0.783050\pi$
$887$		−	37.5267i		−	1.26002i	0.776586	−	0.630011i	$-0.216950\pi$
$888$		0			0
$889$			5.07068i			0.170065i
$890$		−	7.05666i		−	0.236540i
$891$		0			0
$892$	12.8448			0.430074
$893$			0.919713i			0.0307770i
$894$		0			0
$895$	21.2531			0.710412
$896$			2.65115i			0.0885688i
$897$		0			0
$898$		−	16.8866i		−	0.563512i
$899$	−0.178945			−0.00596814
$900$		0			0
$901$		−	20.7379i		−	0.690879i
$902$	9.11606			0.303532
$903$		0			0
$904$	−3.44947			−0.114728
$905$	14.9159			0.495823
$906$		0			0
$907$	−0.620637			−0.0206079			−0.0103040	−	0.999947i	$-0.503280\pi$
$907$	−0.620637			−0.0206079			−0.0103040	+	0.999947i	$0.503280\pi$
$908$			20.6700i			0.685957i
$909$		0			0
$910$	8.50523			0.281946
$911$		−	7.96346i		−	0.263841i	−0.991260	−	0.131921i	$-0.957886\pi$
$911$		−	7.96346i		−	0.263841i	0.991260	−	0.131921i	$-0.0421144\pi$
$912$		0			0
$913$			12.4184i			0.410988i
$914$	−5.97624			−0.197676
$915$		0			0
$916$			6.40386i			0.211590i
$917$	21.5049			0.710156
$918$		0			0
$919$		−	52.8028i		−	1.74180i	−0.491457	−	0.870902i	$-0.663536\pi$
$919$		−	52.8028i		−	1.74180i	0.491457	−	0.870902i	$-0.336464\pi$
$920$		−	9.18285i		−	0.302750i
$921$		0			0
$922$			10.4601i			0.344485i
$923$	−19.5440			−0.643298
$924$		0			0
$925$	−1.46251			−0.0480871
$926$			32.9643i			1.08327i
$927$		0			0
$928$			0.463977i			0.0152308i
$929$	47.9449			1.57302			0.786510	−	0.617578i	$-0.211886\pi$
$929$	47.9449			1.57302			0.786510	+	0.617578i	$0.211886\pi$
$930$		0			0
$931$	0.0137386			0.000450265
$932$	8.18874			0.268231
$933$		0			0
$934$		−	14.0031i		−	0.458197i
$935$		−	18.7062i		−	0.611757i
$936$		0			0
$937$		−	8.77209i		−	0.286572i	−0.989681	−	0.143286i	$-0.954233\pi$
$937$		−	8.77209i		−	0.286572i	0.989681	−	0.143286i	$-0.0457668\pi$
$938$	−20.5295	+	7.03241i	−0.670313	+	0.229616i
$939$		0			0
$940$			1.91486i			0.0624560i
$941$	−42.5830			−1.38817			−0.694083	−	0.719895i	$-0.744190\pi$
$941$	−42.5830			−1.38817			−0.694083	+	0.719895i	$0.744190\pi$
$942$		0			0
$943$		−	35.2252i		−	1.14709i
$944$		−	3.35047i		−	0.109049i
$945$		0			0
$946$		−	6.20139i		−	0.201624i
$947$		−	43.2794i		−	1.40639i	−0.710997	−	0.703195i	$-0.751756\pi$
$947$		−	43.2794i		−	1.40639i	0.710997	−	0.703195i	$-0.248244\pi$
$948$		0			0
$949$		−	30.0354i		−	0.974989i
$950$	0.480302			0.0155830
$951$		0			0
$952$	20.8684			0.676347
$953$			26.5130i			0.858841i	0.903105	+	0.429420i	$0.141282\pi$
$953$			26.5130i			0.858841i	−0.903105	+	0.429420i	$0.858718\pi$
$954$		0			0
$955$	11.6633			0.377414
$956$	5.93875			0.192073
$957$		0			0
$958$		−	17.8841i		−	0.577809i
$959$			44.9102i			1.45023i
$960$		0			0
$961$	30.8513			0.995202
$962$			4.69192i			0.151274i
$963$		0			0
$964$	8.24641			0.265599
$965$	6.38356			0.205494
$966$		0			0
$967$	25.9357			0.834037			0.417019	−	0.908898i	$-0.363075\pi$
$967$	25.9357			0.834037			0.417019	+	0.908898i	$0.363075\pi$
$968$	5.35243			0.172033
$969$		0			0
$970$		−	1.67744i		−	0.0538594i
$971$		−	32.4691i		−	1.04198i	−0.853562	−	0.520992i	$-0.825562\pi$
$971$		−	32.4691i		−	1.04198i	0.853562	−	0.520992i	$-0.174438\pi$
$972$		0			0
$973$	−57.8362			−1.85414
$974$		−	28.1809i		−	0.902976i
$975$		0			0
$976$			14.7145i			0.471000i
$977$		−	2.19934i		−	0.0703632i	−0.999381	−	0.0351816i	$-0.988799\pi$
$977$		−	2.19934i		−	0.0703632i	0.999381	−	0.0351816i	$-0.0112010\pi$
$978$		0			0
$979$			16.7699i			0.535968i
$980$	0.0286042			0.000913726
$981$		0			0
$982$		−	18.2542i		−	0.582515i
$983$	−25.9976			−0.829196			−0.414598	−	0.910005i	$-0.636078\pi$
$983$	−25.9976			−0.829196			−0.414598	+	0.910005i	$0.636078\pi$
$984$		0			0
$985$	4.17857			0.133140
$986$	3.65216			0.116309
$987$		0			0
$988$		−	1.54087i		−	0.0490216i
$989$	−23.9627			−0.761969
$990$		0			0
$991$			26.0361i			0.827063i	0.910490	+	0.413532i	$0.135705\pi$
$991$			26.0361i			0.827063i	−0.910490	+	0.413532i	$0.864295\pi$
$992$			0.385676i			0.0122452i
$993$		0			0
$994$	−16.1509			−0.512275
$995$	3.96063			0.125560
$996$		0			0
$997$	22.9518			0.726892			0.363446	−	0.931615i	$-0.381600\pi$
$997$	22.9518			0.726892			0.363446	+	0.931615i	$0.381600\pi$
$998$			6.71500i			0.212560i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6030.2.d.k.2411.7	✓	24
3.2	odd	2		6030.2.d.l.2411.7	yes	24
67.66	odd	2		6030.2.d.l.2411.18	yes	24
201.200	even	2	inner	6030.2.d.k.2411.18	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6030.2.d.k.2411.7	✓	24	1.1	even	1	trivial
6030.2.d.k.2411.18	yes	24	201.200	even	2	inner
6030.2.d.l.2411.7	yes	24	3.2	odd	2
6030.2.d.l.2411.18	yes	24	67.66	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.65115i q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.65115i q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.37646 q^{11} +3.20813i q^{13} +2.65115i q^{14} +1.00000 q^{16} -7.87143i q^{17} -0.480302 q^{19} -1.00000 q^{20} +2.37646 q^{22} -9.18285i q^{23} +1.00000 q^{25} -3.20813i q^{26} -2.65115i q^{28} -0.463977i q^{29} -0.385676i q^{31} -1.00000 q^{32} +7.87143i q^{34} +2.65115i q^{35} -1.46251 q^{37} +0.480302 q^{38} +1.00000 q^{40} +3.83598 q^{41} -2.60950i q^{43} -2.37646 q^{44} +9.18285i q^{46} -1.91486i q^{47} -0.0286042 q^{49} -1.00000 q^{50} +3.20813i q^{52} +2.63457 q^{53} +2.37646 q^{55} +2.65115i q^{56} +0.463977i q^{58} -3.35047i q^{59} +14.7145i q^{61} +0.385676i q^{62} +1.00000 q^{64} -3.20813i q^{65} +(2.65259 + 7.74363i) q^{67} -7.87143i q^{68} -2.65115i q^{70} +6.09202i q^{71} -9.36227 q^{73} +1.46251 q^{74} -0.480302 q^{76} +6.30036i q^{77} -6.73632i q^{79} -1.00000 q^{80} -3.83598 q^{82} -5.22557i q^{83} +7.87143i q^{85} +2.60950i q^{86} +2.37646 q^{88} -7.05666i q^{89} +8.50523 q^{91} -9.18285i q^{92} +1.91486i q^{94} +0.480302 q^{95} -1.67744i q^{97} +0.0286042 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} + 24 q^{4} - 24 q^{5} - 24 q^{8}+O(q^{10}) \) 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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