LMFDB - Embedded newform 3042.2.b.k.1351.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1351,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3042.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$24.2904922949$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 25 $ x^4 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 234)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.2
Root			$-1.58114 + 1.58114i$ of defining polynomial
Character	$\chi$	$=$	3042.1351
Dual form			3042.2.b.k.1351.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +2.16228i q^{5} -3.16228i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +2.16228i q^{5} -3.16228i q^{7} +1.00000i q^{8} +2.16228 q^{10} +5.16228i q^{11} -3.16228 q^{14} +1.00000 q^{16} +3.00000 q^{17} -1.16228i q^{19} -2.16228i q^{20} +5.16228 q^{22} -0.837722 q^{23} +0.324555 q^{25} +3.16228i q^{28} -8.16228 q^{29} -6.32456i q^{31} -1.00000i q^{32} -3.00000i q^{34} +6.83772 q^{35} +10.1623i q^{37} -1.16228 q^{38} -2.16228 q^{40} -3.00000i q^{41} -2.83772 q^{43} -5.16228i q^{44} +0.837722i q^{46} +11.1623i q^{47} -3.00000 q^{49} -0.324555i q^{50} +6.48683 q^{53} -11.1623 q^{55} +3.16228 q^{56} +8.16228i q^{58} +10.3246i q^{59} -6.16228 q^{61} -6.32456 q^{62} -1.00000 q^{64} +9.16228i q^{67} -3.00000 q^{68} -6.83772i q^{70} +0.837722i q^{71} -1.00000i q^{73} +10.1623 q^{74} +1.16228i q^{76} +16.3246 q^{77} -4.00000 q^{79} +2.16228i q^{80} -3.00000 q^{82} -15.4868i q^{83} +6.48683i q^{85} +2.83772i q^{86} -5.16228 q^{88} +12.0000i q^{89} +0.837722 q^{92} +11.1623 q^{94} +2.51317 q^{95} +4.00000i q^{97} +3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} - 4 q^{10} + 4 q^{16} + 12 q^{17} + 8 q^{22} - 16 q^{23} - 24 q^{25} - 20 q^{29} + 40 q^{35} + 8 q^{38} + 4 q^{40} - 24 q^{43} - 12 q^{49} - 12 q^{53} - 32 q^{55} - 12 q^{61} - 4 q^{64} - 12 q^{68} + 28 q^{74} + 40 q^{77} - 16 q^{79} - 12 q^{82} - 8 q^{88} + 16 q^{92} + 32 q^{94} + 48 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^10 + 4 * q^16 + 12 * q^17 + 8 * q^22 - 16 * q^23 - 24 * q^25 - 20 * q^29 + 40 * q^35 + 8 * q^38 + 4 * q^40 - 24 * q^43 - 12 * q^49 - 12 * q^53 - 32 * q^55 - 12 * q^61 - 4 * q^64 - 12 * q^68 + 28 * q^74 + 40 * q^77 - 16 * q^79 - 12 * q^82 - 8 * q^88 + 16 * q^92 + 32 * q^94 + 48 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	2.16228i	0.967000i	0.875344	+	0.483500i	$0.160635\pi$
$5$	2.16228i	0.967000i	−0.875344	+	0.483500i	$0.839365\pi$
$6$	0	0
$7$	− 3.16228i	− 1.19523i	−0.801784	−	0.597614i	$-0.796115\pi$
$7$	− 3.16228i	− 1.19523i	0.801784	−	0.597614i	$-0.203885\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	2.16228	0.683772
$11$	5.16228i	1.55649i	0.627964	+	0.778243i	$0.283889\pi$
$11$	5.16228i	1.55649i	−0.627964	+	0.778243i	$0.716111\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−3.16228	−0.845154
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	− 1.16228i	− 0.266645i	−0.991073	−	0.133322i	$-0.957435\pi$
$19$	− 1.16228i	− 0.266645i	0.991073	−	0.133322i	$-0.0425646\pi$
$20$	− 2.16228i	− 0.483500i
$21$	0	0
$22$	5.16228	1.10060
$23$	−0.837722	−0.174677	−0.0873386	−	0.996179i	$-0.527836\pi$
$23$	−0.837722	−0.174677	−0.0873386	+	0.996179i	$0.527836\pi$
$24$	0	0
$25$	0.324555	0.0649111
$26$	0	0
$27$	0	0
$28$	3.16228i	0.597614i
$29$	−8.16228	−1.51570	−0.757848	−	0.652431i	$-0.773749\pi$
$29$	−8.16228	−1.51570	−0.757848	+	0.652431i	$0.773749\pi$
$30$	0	0
$31$	− 6.32456i	− 1.13592i	−0.823055	−	0.567962i	$-0.807732\pi$
$31$	− 6.32456i	− 1.13592i	0.823055	−	0.567962i	$-0.192268\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	− 3.00000i	− 0.514496i
$35$	6.83772	1.15579
$36$	0	0
$37$	10.1623i	1.67067i	0.549743	+	0.835334i	$0.314726\pi$
$37$	10.1623i	1.67067i	−0.549743	+	0.835334i	$0.685274\pi$
$38$	−1.16228	−0.188546
$39$	0	0
$40$	−2.16228	−0.341886
$41$	− 3.00000i	− 0.468521i	−0.972174	−	0.234261i	$-0.924733\pi$
$41$	− 3.00000i	− 0.468521i	0.972174	−	0.234261i	$-0.0752669\pi$
$42$	0	0
$43$	−2.83772	−0.432749	−0.216374	−	0.976310i	$-0.569423\pi$
$43$	−2.83772	−0.432749	−0.216374	+	0.976310i	$0.569423\pi$
$44$	− 5.16228i	− 0.778243i
$45$	0	0
$46$	0.837722i	0.123515i
$47$	11.1623i	1.62819i	0.580735	+	0.814093i	$0.302765\pi$
$47$	11.1623i	1.62819i	−0.580735	+	0.814093i	$0.697235\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	− 0.324555i	− 0.0458991i
$51$	0	0
$52$	0	0
$53$	6.48683	0.891035	0.445518	−	0.895273i	$-0.353020\pi$
$53$	6.48683	0.891035	0.445518	+	0.895273i	$0.353020\pi$
$54$	0	0
$55$	−11.1623	−1.50512
$56$	3.16228	0.422577
$57$	0	0
$58$	8.16228i	1.07176i
$59$	10.3246i	1.34414i	0.740486	+	0.672071i	$0.234595\pi$
$59$	10.3246i	1.34414i	−0.740486	+	0.672071i	$0.765405\pi$
$60$	0	0
$61$	−6.16228	−0.788999	−0.394499	−	0.918896i	$-0.629082\pi$
$61$	−6.16228	−0.788999	−0.394499	+	0.918896i	$0.629082\pi$
$62$	−6.32456	−0.803219
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	9.16228i	1.11935i	0.828712	+	0.559675i	$0.189074\pi$
$67$	9.16228i	1.11935i	−0.828712	+	0.559675i	$0.810926\pi$
$68$	−3.00000	−0.363803
$69$	0	0
$70$	− 6.83772i	− 0.817264i
$71$	0.837722i	0.0994194i	0.998764	+	0.0497097i	$0.0158296\pi$
$71$	0.837722i	0.0994194i	−0.998764	+	0.0497097i	$0.984170\pi$
$72$	0	0
$73$	− 1.00000i	− 0.117041i	−0.998286	−	0.0585206i	$-0.981362\pi$
$73$	− 1.00000i	− 0.117041i	0.998286	−	0.0585206i	$-0.0186383\pi$
$74$	10.1623	1.18134
$75$	0	0
$76$	1.16228i	0.133322i
$77$	16.3246	1.86036
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	2.16228i	0.241750i
$81$	0	0
$82$	−3.00000	−0.331295
$83$	− 15.4868i	− 1.69990i	−0.526863	−	0.849950i	$-0.676632\pi$
$83$	− 15.4868i	− 1.69990i	0.526863	−	0.849950i	$-0.323368\pi$
$84$	0	0
$85$	6.48683i	0.703596i
$86$	2.83772i	0.305999i
$87$	0	0
$88$	−5.16228	−0.550301
$89$	12.0000i	1.27200i	0.771690	+	0.635999i	$0.219412\pi$
$89$	12.0000i	1.27200i	−0.771690	+	0.635999i	$0.780588\pi$
$90$	0	0
$91$	0	0
$92$	0.837722	0.0873386
$93$	0	0
$94$	11.1623	1.15130
$95$	2.51317	0.257845
$96$	0	0
$97$	4.00000i	0.406138i	0.979164	+	0.203069i	$0.0650917\pi$
$97$	4.00000i	0.406138i	−0.979164	+	0.203069i	$0.934908\pi$
$98$	3.00000i	0.303046i
$99$	0	0
$100$	−0.324555	−0.0324555
$101$	−14.1623	−1.40920	−0.704600	−	0.709605i	$-0.748873\pi$
$101$	−14.1623	−1.40920	−0.704600	+	0.709605i	$0.748873\pi$
$102$	0	0
$103$	−15.8114	−1.55794	−0.778971	−	0.627060i	$-0.784258\pi$
$103$	−15.8114	−1.55794	−0.778971	+	0.627060i	$0.784258\pi$
$104$	0	0
$105$	0	0
$106$	− 6.48683i	− 0.630057i
$107$	13.8114	1.33520	0.667599	−	0.744521i	$-0.267322\pi$
$107$	13.8114	1.33520	0.667599	+	0.744521i	$0.267322\pi$
$108$	0	0
$109$	18.6491i	1.78626i	0.449798	+	0.893130i	$0.351496\pi$
$109$	18.6491i	1.78626i	−0.449798	+	0.893130i	$0.648504\pi$
$110$	11.1623i	1.06428i
$111$	0	0
$112$	− 3.16228i	− 0.298807i
$113$	19.3246	1.81790	0.908951	−	0.416904i	$-0.136885\pi$
$113$	19.3246	1.81790	0.908951	+	0.416904i	$0.136885\pi$
$114$	0	0
$115$	− 1.81139i	− 0.168913i
$116$	8.16228	0.757848
$117$	0	0
$118$	10.3246	0.950452
$119$	− 9.48683i	− 0.869657i
$120$	0	0
$121$	−15.6491	−1.42265
$122$	6.16228i	0.557906i
$123$	0	0
$124$	6.32456i	0.567962i
$125$	11.5132i	1.02977i
$126$	0	0
$127$	5.67544	0.503614	0.251807	−	0.967777i	$-0.418975\pi$
$127$	5.67544	0.503614	0.251807	+	0.967777i	$0.418975\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−8.64911	−0.755676	−0.377838	−	0.925872i	$-0.623332\pi$
$131$	−8.64911	−0.755676	−0.377838	+	0.925872i	$0.623332\pi$
$132$	0	0
$133$	−3.67544	−0.318701
$134$	9.16228	0.791500
$135$	0	0
$136$	3.00000i	0.257248i
$137$	− 3.00000i	− 0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	− 3.00000i	− 0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$	0	0
$139$	6.32456	0.536442	0.268221	−	0.963357i	$-0.413564\pi$
$139$	6.32456	0.536442	0.268221	+	0.963357i	$0.413564\pi$
$140$	−6.83772	−0.577893
$141$	0	0
$142$	0.837722	0.0703001
$143$	0	0
$144$	0	0
$145$	− 17.6491i	− 1.46568i
$146$	−1.00000	−0.0827606
$147$	0	0
$148$	− 10.1623i	− 0.835334i
$149$	18.4868i	1.51450i	0.653125	+	0.757250i	$0.273458\pi$
$149$	18.4868i	1.51450i	−0.653125	+	0.757250i	$0.726542\pi$
$150$	0	0
$151$	7.16228i	0.582858i	0.956592	+	0.291429i	$0.0941307\pi$
$151$	7.16228i	0.582858i	−0.956592	+	0.291429i	$0.905869\pi$
$152$	1.16228	0.0942732
$153$	0	0
$154$	− 16.3246i	− 1.31547i
$155$	13.6754	1.09844
$156$	0	0
$157$	8.48683	0.677323	0.338662	−	0.940908i	$-0.390026\pi$
$157$	8.48683	0.677323	0.338662	+	0.940908i	$0.390026\pi$
$158$	4.00000i	0.318223i
$159$	0	0
$160$	2.16228	0.170943
$161$	2.64911i	0.208779i
$162$	0	0
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	3.00000i	0.234261i
$165$	0	0
$166$	−15.4868	−1.20201
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	0	0
$170$	6.48683	0.497517
$171$	0	0
$172$	2.83772	0.216374
$173$	2.64911	0.201408	0.100704	−	0.994916i	$-0.467890\pi$
$173$	2.64911	0.201408	0.100704	+	0.994916i	$0.467890\pi$
$174$	0	0
$175$	− 1.02633i	− 0.0775836i
$176$	5.16228i	0.389121i
$177$	0	0
$178$	12.0000	0.899438
$179$	3.48683	0.260618	0.130309	−	0.991473i	$-0.458403\pi$
$179$	3.48683	0.260618	0.130309	+	0.991473i	$0.458403\pi$
$180$	0	0
$181$	−10.1623	−0.755356	−0.377678	−	0.925937i	$-0.623277\pi$
$181$	−10.1623	−0.755356	−0.377678	+	0.925937i	$0.623277\pi$
$182$	0	0
$183$	0	0
$184$	− 0.837722i	− 0.0617577i
$185$	−21.9737	−1.61554
$186$	0	0
$187$	15.4868i	1.13251i
$188$	− 11.1623i	− 0.814093i
$189$	0	0
$190$	− 2.51317i	− 0.182324i
$191$	1.67544	0.121231	0.0606155	−	0.998161i	$-0.480694\pi$
$191$	1.67544	0.121231	0.0606155	+	0.998161i	$0.480694\pi$
$192$	0	0
$193$	17.9737i	1.29377i	0.762586	+	0.646886i	$0.223929\pi$
$193$	17.9737i	1.29377i	−0.762586	+	0.646886i	$0.776071\pi$
$194$	4.00000	0.287183
$195$	0	0
$196$	3.00000	0.214286
$197$	18.9737i	1.35182i	0.736985	+	0.675909i	$0.236249\pi$
$197$	18.9737i	1.35182i	−0.736985	+	0.675909i	$0.763751\pi$
$198$	0	0
$199$	25.4868	1.80671	0.903357	−	0.428890i	$-0.141095\pi$
$199$	25.4868	1.80671	0.903357	+	0.428890i	$0.141095\pi$
$200$	0.324555i	0.0229495i
$201$	0	0
$202$	14.1623i	0.996454i
$203$	25.8114i	1.81160i
$204$	0	0
$205$	6.48683	0.453060
$206$	15.8114i	1.10163i
$207$	0	0
$208$	0	0
$209$	6.00000	0.415029
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−6.48683	−0.445518
$213$	0	0
$214$	− 13.8114i	− 0.944127i
$215$	− 6.13594i	− 0.418468i
$216$	0	0
$217$	−20.0000	−1.35769
$218$	18.6491	1.26308
$219$	0	0
$220$	11.1623	0.752561
$221$	0	0
$222$	0	0
$223$	14.3246i	0.959243i	0.877475	+	0.479622i	$0.159226\pi$
$223$	14.3246i	0.959243i	−0.877475	+	0.479622i	$0.840774\pi$
$224$	−3.16228	−0.211289
$225$	0	0
$226$	− 19.3246i	− 1.28545i
$227$	− 3.48683i	− 0.231429i	−0.993283	−	0.115715i	$-0.963084\pi$
$227$	− 3.48683i	− 0.231429i	0.993283	−	0.115715i	$-0.0369158\pi$
$228$	0	0
$229$	9.67544i	0.639371i	0.947524	+	0.319686i	$0.103577\pi$
$229$	9.67544i	0.639371i	−0.947524	+	0.319686i	$0.896423\pi$
$230$	−1.81139	−0.119439
$231$	0	0
$232$	− 8.16228i	− 0.535880i
$233$	−8.64911	−0.566622	−0.283311	−	0.959028i	$-0.591433\pi$
$233$	−8.64911	−0.566622	−0.283311	+	0.959028i	$0.591433\pi$
$234$	0	0
$235$	−24.1359	−1.57446
$236$	− 10.3246i	− 0.672071i
$237$	0	0
$238$	−9.48683	−0.614940
$239$	2.51317i	0.162563i	0.996691	+	0.0812816i	$0.0259013\pi$
$239$	2.51317i	0.162563i	−0.996691	+	0.0812816i	$0.974099\pi$
$240$	0	0
$241$	0.675445i	0.0435092i	0.999763	+	0.0217546i	$0.00692525\pi$
$241$	0.675445i	0.0435092i	−0.999763	+	0.0217546i	$0.993075\pi$
$242$	15.6491i	1.00596i
$243$	0	0
$244$	6.16228	0.394499
$245$	− 6.48683i	− 0.414429i
$246$	0	0
$247$	0	0
$248$	6.32456	0.401610
$249$	0	0
$250$	11.5132	0.728157
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	− 4.32456i	− 0.271882i
$254$	− 5.67544i	− 0.356109i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−15.9737	−0.996410	−0.498205	−	0.867059i	$-0.666007\pi$
$257$	−15.9737	−0.996410	−0.498205	+	0.867059i	$0.666007\pi$
$258$	0	0
$259$	32.1359	1.99683
$260$	0	0
$261$	0	0
$262$	8.64911i	0.534344i
$263$	−21.4868	−1.32493	−0.662467	−	0.749091i	$-0.730491\pi$
$263$	−21.4868	−1.32493	−0.662467	+	0.749091i	$0.730491\pi$
$264$	0	0
$265$	14.0263i	0.861631i
$266$	3.67544i	0.225356i
$267$	0	0
$268$	− 9.16228i	− 0.559675i
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	6.32456i	0.384189i	0.981376	+	0.192095i	$0.0615281\pi$
$271$	6.32456i	0.384189i	−0.981376	+	0.192095i	$0.938472\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	−3.00000	−0.181237
$275$	1.67544i	0.101033i
$276$	0	0
$277$	20.8114	1.25044	0.625218	−	0.780451i	$-0.285010\pi$
$277$	20.8114	1.25044	0.625218	+	0.780451i	$0.285010\pi$
$278$	− 6.32456i	− 0.379322i
$279$	0	0
$280$	6.83772i	0.408632i
$281$	− 13.3246i	− 0.794876i	−0.917629	−	0.397438i	$-0.869899\pi$
$281$	− 13.3246i	− 0.794876i	0.917629	−	0.397438i	$-0.130101\pi$
$282$	0	0
$283$	9.16228	0.544641	0.272320	−	0.962207i	$-0.412209\pi$
$283$	9.16228	0.544641	0.272320	+	0.962207i	$0.412209\pi$
$284$	− 0.837722i	− 0.0497097i
$285$	0	0
$286$	0	0
$287$	−9.48683	−0.559990
$288$	0	0
$289$	−8.00000	−0.470588
$290$	−17.6491	−1.03639
$291$	0	0
$292$	1.00000i	0.0585206i
$293$	8.16228i	0.476845i	0.971161	+	0.238423i	$0.0766304\pi$
$293$	8.16228i	0.476845i	−0.971161	+	0.238423i	$0.923370\pi$
$294$	0	0
$295$	−22.3246	−1.29979
$296$	−10.1623	−0.590670
$297$	0	0
$298$	18.4868	1.07091
$299$	0	0
$300$	0	0
$301$	8.97367i	0.517234i
$302$	7.16228	0.412143
$303$	0	0
$304$	− 1.16228i	− 0.0666612i
$305$	− 13.3246i	− 0.762962i
$306$	0	0
$307$	− 7.48683i	− 0.427296i	−0.976911	−	0.213648i	$-0.931465\pi$
$307$	− 7.48683i	− 0.427296i	0.976911	−	0.213648i	$-0.0685346\pi$
$308$	−16.3246	−0.930178
$309$	0	0
$310$	− 13.6754i	− 0.776713i
$311$	−2.51317	−0.142509	−0.0712543	−	0.997458i	$-0.522700\pi$
$311$	−2.51317	−0.142509	−0.0712543	+	0.997458i	$0.522700\pi$
$312$	0	0
$313$	−4.00000	−0.226093	−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000	−0.226093	−0.113047	+	0.993590i	$0.536061\pi$
$314$	− 8.48683i	− 0.478940i
$315$	0	0
$316$	4.00000	0.225018
$317$	6.48683i	0.364337i	0.983267	+	0.182168i	$0.0583116\pi$
$317$	6.48683i	0.364337i	−0.983267	+	0.182168i	$0.941688\pi$
$318$	0	0
$319$	− 42.1359i	− 2.35916i
$320$	− 2.16228i	− 0.120875i
$321$	0	0
$322$	2.64911	0.147629
$323$	− 3.48683i	− 0.194013i
$324$	0	0
$325$	0	0
$326$	−16.0000	−0.886158
$327$	0	0
$328$	3.00000	0.165647
$329$	35.2982	1.94605
$330$	0	0
$331$	− 26.9737i	− 1.48261i	−0.671170	−	0.741303i	$-0.734208\pi$
$331$	− 26.9737i	− 1.48261i	0.671170	−	0.741303i	$-0.265792\pi$
$332$	15.4868i	0.849950i
$333$	0	0
$334$	12.0000	0.656611
$335$	−19.8114	−1.08241
$336$	0	0
$337$	−11.0000	−0.599208	−0.299604	−	0.954064i	$-0.596855\pi$
$337$	−11.0000	−0.599208	−0.299604	+	0.954064i	$0.596855\pi$
$338$	0	0
$339$	0	0
$340$	− 6.48683i	− 0.351798i
$341$	32.6491	1.76805
$342$	0	0
$343$	− 12.6491i	− 0.682988i
$344$	− 2.83772i	− 0.153000i
$345$	0	0
$346$	− 2.64911i	− 0.142417i
$347$	3.48683	0.187183	0.0935915	−	0.995611i	$-0.470165\pi$
$347$	3.48683	0.187183	0.0935915	+	0.995611i	$0.470165\pi$
$348$	0	0
$349$	− 30.6491i	− 1.64061i	−0.571927	−	0.820305i	$-0.693804\pi$
$349$	− 30.6491i	− 1.64061i	0.571927	−	0.820305i	$-0.306196\pi$
$350$	−1.02633	−0.0548599
$351$	0	0
$352$	5.16228	0.275150
$353$	− 16.6754i	− 0.887544i	−0.896140	−	0.443772i	$-0.853640\pi$
$353$	− 16.6754i	− 0.887544i	0.896140	−	0.443772i	$-0.146360\pi$
$354$	0	0
$355$	−1.81139	−0.0961385
$356$	− 12.0000i	− 0.635999i
$357$	0	0
$358$	− 3.48683i	− 0.184285i
$359$	− 28.4605i	− 1.50209i	−0.660252	−	0.751044i	$-0.729551\pi$
$359$	− 28.4605i	− 1.50209i	0.660252	−	0.751044i	$-0.270449\pi$
$360$	0	0
$361$	17.6491	0.928901
$362$	10.1623i	0.534117i
$363$	0	0
$364$	0	0
$365$	2.16228	0.113179
$366$	0	0
$367$	−6.51317	−0.339985	−0.169992	−	0.985445i	$-0.554374\pi$
$367$	−6.51317	−0.339985	−0.169992	+	0.985445i	$0.554374\pi$
$368$	−0.837722	−0.0436693
$369$	0	0
$370$	21.9737i	1.14236i
$371$	− 20.5132i	− 1.06499i
$372$	0	0
$373$	2.48683	0.128763	0.0643817	−	0.997925i	$-0.479492\pi$
$373$	2.48683	0.128763	0.0643817	+	0.997925i	$0.479492\pi$
$374$	15.4868	0.800805
$375$	0	0
$376$	−11.1623	−0.575651
$377$	0	0
$378$	0	0
$379$	− 30.3246i	− 1.55767i	−0.627230	−	0.778834i	$-0.715811\pi$
$379$	− 30.3246i	− 1.55767i	0.627230	−	0.778834i	$-0.284189\pi$
$380$	−2.51317	−0.128923
$381$	0	0
$382$	− 1.67544i	− 0.0857232i
$383$	6.97367i	0.356338i	0.984000	+	0.178169i	$0.0570173\pi$
$383$	6.97367i	0.356338i	−0.984000	+	0.178169i	$0.942983\pi$
$384$	0	0
$385$	35.2982i	1.79896i
$386$	17.9737	0.914836
$387$	0	0
$388$	− 4.00000i	− 0.203069i
$389$	6.48683	0.328895	0.164448	−	0.986386i	$-0.447416\pi$
$389$	6.48683	0.328895	0.164448	+	0.986386i	$0.447416\pi$
$390$	0	0
$391$	−2.51317	−0.127096
$392$	− 3.00000i	− 0.151523i
$393$	0	0
$394$	18.9737	0.955879
$395$	− 8.64911i	− 0.435184i
$396$	0	0
$397$	26.0000i	1.30490i	0.757831	+	0.652451i	$0.226259\pi$
$397$	26.0000i	1.30490i	−0.757831	+	0.652451i	$0.773741\pi$
$398$	− 25.4868i	− 1.27754i
$399$	0	0
$400$	0.324555	0.0162278
$401$	− 15.0000i	− 0.749064i	−0.927214	−	0.374532i	$-0.877803\pi$
$401$	− 15.0000i	− 0.749064i	0.927214	−	0.374532i	$-0.122197\pi$
$402$	0	0
$403$	0	0
$404$	14.1623	0.704600
$405$	0	0
$406$	25.8114	1.28100
$407$	−52.4605	−2.60037
$408$	0	0
$409$	− 27.3246i	− 1.35111i	−0.737309	−	0.675556i	$-0.763904\pi$
$409$	− 27.3246i	− 1.35111i	0.737309	−	0.675556i	$-0.236096\pi$
$410$	− 6.48683i	− 0.320362i
$411$	0	0
$412$	15.8114	0.778971
$413$	32.6491	1.60656
$414$	0	0
$415$	33.4868	1.64380
$416$	0	0
$417$	0	0
$418$	− 6.00000i	− 0.293470i
$419$	6.97367	0.340686	0.170343	−	0.985385i	$-0.445512\pi$
$419$	6.97367	0.340686	0.170343	+	0.985385i	$0.445512\pi$
$420$	0	0
$421$	30.1623i	1.47002i	0.678057	+	0.735010i	$0.262822\pi$
$421$	30.1623i	1.47002i	−0.678057	+	0.735010i	$0.737178\pi$
$422$	4.00000i	0.194717i
$423$	0	0
$424$	6.48683i	0.315028i
$425$	0.973666	0.0472297
$426$	0	0
$427$	19.4868i	0.943034i
$428$	−13.8114	−0.667599
$429$	0	0
$430$	−6.13594	−0.295901
$431$	− 9.48683i	− 0.456965i	−0.973548	−	0.228482i	$-0.926624\pi$
$431$	− 9.48683i	− 0.456965i	0.973548	−	0.228482i	$-0.0733763\pi$
$432$	0	0
$433$	−3.32456	−0.159768	−0.0798840	−	0.996804i	$-0.525455\pi$
$433$	−3.32456	−0.159768	−0.0798840	+	0.996804i	$0.525455\pi$
$434$	20.0000i	0.960031i
$435$	0	0
$436$	− 18.6491i	− 0.893130i
$437$	0.973666i	0.0465768i
$438$	0	0
$439$	6.51317	0.310857	0.155428	−	0.987847i	$-0.450324\pi$
$439$	6.51317	0.310857	0.155428	+	0.987847i	$0.450324\pi$
$440$	− 11.1623i	− 0.532141i
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−25.9473	−1.23002
$446$	14.3246	0.678287
$447$	0	0
$448$	3.16228i	0.149404i
$449$	32.6491i	1.54081i	0.637557	+	0.770403i	$0.279945\pi$
$449$	32.6491i	1.54081i	−0.637557	+	0.770403i	$0.720055\pi$
$450$	0	0
$451$	15.4868	0.729246
$452$	−19.3246	−0.908951
$453$	0	0
$454$	−3.48683	−0.163645
$455$	0	0
$456$	0	0
$457$	− 9.32456i	− 0.436184i	−0.975928	−	0.218092i	$-0.930017\pi$
$457$	− 9.32456i	− 0.436184i	0.975928	−	0.218092i	$-0.0699833\pi$
$458$	9.67544	0.452104
$459$	0	0
$460$	1.81139i	0.0844564i
$461$	0.486833i	0.0226741i	0.999936	+	0.0113370i	$0.00360877\pi$
$461$	0.486833i	0.0226741i	−0.999936	+	0.0113370i	$0.996391\pi$
$462$	0	0
$463$	8.83772i	0.410724i	0.978686	+	0.205362i	$0.0658371\pi$
$463$	8.83772i	0.410724i	−0.978686	+	0.205362i	$0.934163\pi$
$464$	−8.16228	−0.378924
$465$	0	0
$466$	8.64911i	0.400662i
$467$	−37.8114	−1.74970	−0.874851	−	0.484392i	$-0.839041\pi$
$467$	−37.8114	−1.74970	−0.874851	+	0.484392i	$0.839041\pi$
$468$	0	0
$469$	28.9737	1.33788
$470$	24.1359i	1.11331i
$471$	0	0
$472$	−10.3246	−0.475226
$473$	− 14.6491i	− 0.673567i
$474$	0	0
$475$	− 0.377223i	− 0.0173082i
$476$	9.48683i	0.434828i
$477$	0	0
$478$	2.51317	0.114950
$479$	− 24.0000i	− 1.09659i	−0.836286	−	0.548294i	$-0.815277\pi$
$479$	− 24.0000i	− 1.09659i	0.836286	−	0.548294i	$-0.184723\pi$
$480$	0	0
$481$	0	0
$482$	0.675445	0.0307657
$483$	0	0
$484$	15.6491	0.711323
$485$	−8.64911	−0.392736
$486$	0	0
$487$	− 5.48683i	− 0.248632i	−0.992243	−	0.124316i	$-0.960326\pi$
$487$	− 5.48683i	− 0.248632i	0.992243	−	0.124316i	$-0.0396737\pi$
$488$	− 6.16228i	− 0.278953i
$489$	0	0
$490$	−6.48683	−0.293045
$491$	25.8114	1.16485	0.582426	−	0.812884i	$-0.302104\pi$
$491$	25.8114	1.16485	0.582426	+	0.812884i	$0.302104\pi$
$492$	0	0
$493$	−24.4868	−1.10283
$494$	0	0
$495$	0	0
$496$	− 6.32456i	− 0.283981i
$497$	2.64911	0.118829
$498$	0	0
$499$	12.6491i	0.566252i	0.959083	+	0.283126i	$0.0913715\pi$
$499$	12.6491i	0.566252i	−0.959083	+	0.283126i	$0.908629\pi$
$500$	− 11.5132i	− 0.514884i
$501$	0	0
$502$	− 12.0000i	− 0.535586i
$503$	−31.8114	−1.41840	−0.709200	−	0.705007i	$-0.750944\pi$
$503$	−31.8114	−1.41840	−0.709200	+	0.705007i	$0.750944\pi$
$504$	0	0
$505$	− 30.6228i	− 1.36270i
$506$	−4.32456	−0.192250
$507$	0	0
$508$	−5.67544	−0.251807
$509$	21.8377i	0.967940i	0.875085	+	0.483970i	$0.160806\pi$
$509$	21.8377i	0.967940i	−0.875085	+	0.483970i	$0.839194\pi$
$510$	0	0
$511$	−3.16228	−0.139891
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	15.9737i	0.704568i
$515$	− 34.1886i	− 1.50653i
$516$	0	0
$517$	−57.6228	−2.53425
$518$	− 32.1359i	− 1.41197i
$519$	0	0
$520$	0	0
$521$	−27.0000	−1.18289	−0.591446	−	0.806345i	$-0.701443\pi$
$521$	−27.0000	−1.18289	−0.591446	+	0.806345i	$0.701443\pi$
$522$	0	0
$523$	1.16228	0.0508229	0.0254114	−	0.999677i	$-0.491910\pi$
$523$	1.16228	0.0508229	0.0254114	+	0.999677i	$0.491910\pi$
$524$	8.64911	0.377838
$525$	0	0
$526$	21.4868i	0.936870i
$527$	− 18.9737i	− 0.826506i
$528$	0	0
$529$	−22.2982	−0.969488
$530$	14.0263	0.609265
$531$	0	0
$532$	3.67544	0.159351
$533$	0	0
$534$	0	0
$535$	29.8641i	1.29114i
$536$	−9.16228	−0.395750
$537$	0	0
$538$	− 6.00000i	− 0.258678i
$539$	− 15.4868i	− 0.667065i
$540$	0	0
$541$	− 34.4868i	− 1.48270i	−0.671116	−	0.741352i	$-0.734185\pi$
$541$	− 34.4868i	− 1.48270i	0.671116	−	0.741352i	$-0.265815\pi$
$542$	6.32456	0.271663
$543$	0	0
$544$	− 3.00000i	− 0.128624i
$545$	−40.3246	−1.72731
$546$	0	0
$547$	−2.18861	−0.0935783	−0.0467891	−	0.998905i	$-0.514899\pi$
$547$	−2.18861	−0.0935783	−0.0467891	+	0.998905i	$0.514899\pi$
$548$	3.00000i	0.128154i
$549$	0	0
$550$	1.67544	0.0714412
$551$	9.48683i	0.404153i
$552$	0	0
$553$	12.6491i	0.537895i
$554$	− 20.8114i	− 0.884191i
$555$	0	0
$556$	−6.32456	−0.268221
$557$	0.486833i	0.0206278i	0.999947	+	0.0103139i	$0.00328307\pi$
$557$	0.486833i	0.0206278i	−0.999947	+	0.0103139i	$0.996717\pi$
$558$	0	0
$559$	0	0
$560$	6.83772	0.288947
$561$	0	0
$562$	−13.3246	−0.562062
$563$	−15.3509	−0.646963	−0.323481	−	0.946235i	$-0.604853\pi$
$563$	−15.3509	−0.646963	−0.323481	+	0.946235i	$0.604853\pi$
$564$	0	0
$565$	41.7851i	1.75791i
$566$	− 9.16228i	− 0.385119i
$567$	0	0
$568$	−0.837722	−0.0351500
$569$	23.2982	0.976712	0.488356	−	0.872644i	$-0.337597\pi$
$569$	23.2982	0.976712	0.488356	+	0.872644i	$0.337597\pi$
$570$	0	0
$571$	−8.13594	−0.340479	−0.170239	−	0.985403i	$-0.554454\pi$
$571$	−8.13594	−0.340479	−0.170239	+	0.985403i	$0.554454\pi$
$572$	0	0
$573$	0	0
$574$	9.48683i	0.395973i
$575$	−0.271887	−0.0113385
$576$	0	0
$577$	− 25.6491i	− 1.06779i	−0.845552	−	0.533893i	$-0.820728\pi$
$577$	− 25.6491i	− 1.06779i	0.845552	−	0.533893i	$-0.179272\pi$
$578$	8.00000i	0.332756i
$579$	0	0
$580$	17.6491i	0.732839i
$581$	−48.9737	−2.03177
$582$	0	0
$583$	33.4868i	1.38688i
$584$	1.00000	0.0413803
$585$	0	0
$586$	8.16228	0.337181
$587$	− 3.35089i	− 0.138306i	−0.997606	−	0.0691530i	$-0.977970\pi$
$587$	− 3.35089i	− 0.138306i	0.997606	−	0.0691530i	$-0.0220297\pi$
$588$	0	0
$589$	−7.35089	−0.302888
$590$	22.3246i	0.919087i
$591$	0	0
$592$	10.1623i	0.417667i
$593$	13.3246i	0.547174i	0.961847	+	0.273587i	$0.0882101\pi$
$593$	13.3246i	0.547174i	−0.961847	+	0.273587i	$0.911790\pi$
$594$	0	0
$595$	20.5132	0.840958
$596$	− 18.4868i	− 0.757250i
$597$	0	0
$598$	0	0
$599$	39.6228	1.61894	0.809471	−	0.587159i	$-0.199754\pi$
$599$	39.6228	1.61894	0.809471	+	0.587159i	$0.199754\pi$
$600$	0	0
$601$	16.2982	0.664818	0.332409	−	0.943135i	$-0.392139\pi$
$601$	16.2982	0.664818	0.332409	+	0.943135i	$0.392139\pi$
$602$	8.97367	0.365739
$603$	0	0
$604$	− 7.16228i	− 0.291429i
$605$	− 33.8377i	− 1.37570i
$606$	0	0
$607$	−33.2982	−1.35153	−0.675767	−	0.737116i	$-0.736187\pi$
$607$	−33.2982	−1.35153	−0.675767	+	0.737116i	$0.736187\pi$
$608$	−1.16228	−0.0471366
$609$	0	0
$610$	−13.3246	−0.539495
$611$	0	0
$612$	0	0
$613$	− 1.51317i	− 0.0611162i	−0.999533	−	0.0305581i	$-0.990272\pi$
$613$	− 1.51317i	− 0.0611162i	0.999533	−	0.0305581i	$-0.00972847\pi$
$614$	−7.48683	−0.302144
$615$	0	0
$616$	16.3246i	0.657735i
$617$	− 36.6228i	− 1.47438i	−0.675687	−	0.737189i	$-0.736153\pi$
$617$	− 36.6228i	− 1.47438i	0.675687	−	0.737189i	$-0.263847\pi$
$618$	0	0
$619$	− 0.649111i	− 0.0260900i	−0.999915	−	0.0130450i	$-0.995848\pi$
$619$	− 0.649111i	− 0.0260900i	0.999915	−	0.0130450i	$-0.00415246\pi$
$620$	−13.6754	−0.549219
$621$	0	0
$622$	2.51317i	0.100769i
$623$	37.9473	1.52033
$624$	0	0
$625$	−23.2719	−0.930875
$626$	4.00000i	0.159872i
$627$	0	0
$628$	−8.48683	−0.338662
$629$	30.4868i	1.21559i
$630$	0	0
$631$	25.2982i	1.00711i	0.863964	+	0.503553i	$0.167974\pi$
$631$	25.2982i	1.00711i	−0.863964	+	0.503553i	$0.832026\pi$
$632$	− 4.00000i	− 0.159111i
$633$	0	0
$634$	6.48683	0.257625
$635$	12.2719i	0.486995i
$636$	0	0
$637$	0	0
$638$	−42.1359	−1.66818
$639$	0	0
$640$	−2.16228	−0.0854715
$641$	−41.6491	−1.64504	−0.822520	−	0.568735i	$-0.807433\pi$
$641$	−41.6491	−1.64504	−0.822520	+	0.568735i	$0.807433\pi$
$642$	0	0
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	− 2.64911i	− 0.104390i
$645$	0	0
$646$	−3.48683	−0.137188
$647$	15.3509	0.603506	0.301753	−	0.953386i	$-0.402428\pi$
$647$	15.3509	0.603506	0.301753	+	0.953386i	$0.402428\pi$
$648$	0	0
$649$	−53.2982	−2.09214
$650$	0	0
$651$	0	0
$652$	16.0000i	0.626608i
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	0	0
$655$	− 18.7018i	− 0.730739i
$656$	− 3.00000i	− 0.117130i
$657$	0	0
$658$	− 35.2982i	− 1.37607i
$659$	−42.9737	−1.67402	−0.837008	−	0.547190i	$-0.815697\pi$
$659$	−42.9737	−1.67402	−0.837008	+	0.547190i	$0.815697\pi$
$660$	0	0
$661$	7.51317i	0.292228i	0.989268	+	0.146114i	$0.0466767\pi$
$661$	7.51317i	0.292228i	−0.989268	+	0.146114i	$0.953323\pi$
$662$	−26.9737	−1.04836
$663$	0	0
$664$	15.4868	0.601006
$665$	− 7.94733i	− 0.308184i
$666$	0	0
$667$	6.83772	0.264758
$668$	− 12.0000i	− 0.464294i
$669$	0	0
$670$	19.8114i	0.765381i
$671$	− 31.8114i	− 1.22807i
$672$	0	0
$673$	−27.3246	−1.05328	−0.526642	−	0.850087i	$-0.676549\pi$
$673$	−27.3246	−1.05328	−0.526642	+	0.850087i	$0.676549\pi$
$674$	11.0000i	0.423704i
$675$	0	0
$676$	0	0
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	12.6491	0.485428
$680$	−6.48683	−0.248759
$681$	0	0
$682$	− 32.6491i	− 1.25020i
$683$	27.6228i	1.05696i	0.848947	+	0.528478i	$0.177237\pi$
$683$	27.6228i	1.05696i	−0.848947	+	0.528478i	$0.822763\pi$
$684$	0	0
$685$	6.48683	0.247849
$686$	−12.6491	−0.482945
$687$	0	0
$688$	−2.83772	−0.108187
$689$	0	0
$690$	0	0
$691$	− 2.83772i	− 0.107952i	−0.998542	−	0.0539760i	$-0.982811\pi$
$691$	− 2.83772i	− 0.107952i	0.998542	−	0.0539760i	$-0.0171895\pi$
$692$	−2.64911	−0.100704
$693$	0	0
$694$	− 3.48683i	− 0.132358i
$695$	13.6754i	0.518739i
$696$	0	0
$697$	− 9.00000i	− 0.340899i
$698$	−30.6491	−1.16009
$699$	0	0
$700$	1.02633i	0.0387918i
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	11.8114	0.445475
$704$	− 5.16228i	− 0.194561i
$705$	0	0
$706$	−16.6754	−0.627589
$707$	44.7851i	1.68432i
$708$	0	0
$709$	− 11.4605i	− 0.430408i	−0.976569	−	0.215204i	$-0.930958\pi$
$709$	− 11.4605i	− 0.430408i	0.976569	−	0.215204i	$-0.0690417\pi$
$710$	1.81139i	0.0679802i
$711$	0	0
$712$	−12.0000	−0.449719
$713$	5.29822i	0.198420i
$714$	0	0
$715$	0	0
$716$	−3.48683	−0.130309
$717$	0	0
$718$	−28.4605	−1.06214
$719$	25.6754	0.957533	0.478766	−	0.877942i	$-0.341084\pi$
$719$	25.6754	0.957533	0.478766	+	0.877942i	$0.341084\pi$
$720$	0	0
$721$	50.0000i	1.86210i
$722$	− 17.6491i	− 0.656832i
$723$	0	0
$724$	10.1623	0.377678
$725$	−2.64911	−0.0983855
$726$	0	0
$727$	15.1623	0.562338	0.281169	−	0.959658i	$-0.409278\pi$
$727$	15.1623	0.562338	0.281169	+	0.959658i	$0.409278\pi$
$728$	0	0
$729$	0	0
$730$	− 2.16228i	− 0.0800295i
$731$	−8.51317	−0.314871
$732$	0	0
$733$	35.4605i	1.30976i	0.755731	+	0.654882i	$0.227282\pi$
$733$	35.4605i	1.30976i	−0.755731	+	0.654882i	$0.772718\pi$
$734$	6.51317i	0.240405i
$735$	0	0
$736$	0.837722i	0.0308789i
$737$	−47.2982	−1.74225
$738$	0	0
$739$	23.6228i	0.868978i	0.900677	+	0.434489i	$0.143071\pi$
$739$	23.6228i	0.868978i	−0.900677	+	0.434489i	$0.856929\pi$
$740$	21.9737	0.807768
$741$	0	0
$742$	−20.5132	−0.753062
$743$	10.3246i	0.378771i	0.981903	+	0.189386i	$0.0606496\pi$
$743$	10.3246i	0.378771i	−0.981903	+	0.189386i	$0.939350\pi$
$744$	0	0
$745$	−39.9737	−1.46452
$746$	− 2.48683i	− 0.0910494i
$747$	0	0
$748$	− 15.4868i	− 0.566255i
$749$	− 43.6754i	− 1.59587i
$750$	0	0
$751$	30.7851	1.12336	0.561681	−	0.827354i	$-0.310155\pi$
$751$	30.7851	1.12336	0.561681	+	0.827354i	$0.310155\pi$
$752$	11.1623i	0.407046i
$753$	0	0
$754$	0	0
$755$	−15.4868	−0.563624
$756$	0	0
$757$	−1.35089	−0.0490989	−0.0245495	−	0.999699i	$-0.507815\pi$
$757$	−1.35089	−0.0490989	−0.0245495	+	0.999699i	$0.507815\pi$
$758$	−30.3246	−1.10144
$759$	0	0
$760$	2.51317i	0.0911621i
$761$	− 12.0000i	− 0.435000i	−0.976060	−	0.217500i	$-0.930210\pi$
$761$	− 12.0000i	− 0.435000i	0.976060	−	0.217500i	$-0.0697902\pi$
$762$	0	0
$763$	58.9737	2.13499
$764$	−1.67544	−0.0606155
$765$	0	0
$766$	6.97367	0.251969
$767$	0	0
$768$	0	0
$769$	− 28.6491i	− 1.03311i	−0.856253	−	0.516557i	$-0.827214\pi$
$769$	− 28.6491i	− 1.03311i	0.856253	−	0.516557i	$-0.172786\pi$
$770$	35.2982	1.27206
$771$	0	0
$772$	− 17.9737i	− 0.646886i
$773$	2.64911i	0.0952819i	0.998865	+	0.0476409i	$0.0151703\pi$
$773$	2.64911i	0.0952819i	−0.998865	+	0.0476409i	$0.984830\pi$
$774$	0	0
$775$	− 2.05267i	− 0.0737340i
$776$	−4.00000	−0.143592
$777$	0	0
$778$	− 6.48683i	− 0.232564i
$779$	−3.48683	−0.124929
$780$	0	0
$781$	−4.32456	−0.154745
$782$	2.51317i	0.0898707i
$783$	0	0
$784$	−3.00000	−0.107143
$785$	18.3509i	0.654971i
$786$	0	0
$787$	− 46.9737i	− 1.67443i	−0.546874	−	0.837215i	$-0.684182\pi$
$787$	− 46.9737i	− 1.67443i	0.546874	−	0.837215i	$-0.315818\pi$
$788$	− 18.9737i	− 0.675909i
$789$	0	0
$790$	−8.64911	−0.307722
$791$	− 61.1096i	− 2.17281i
$792$	0	0
$793$	0	0
$794$	26.0000	0.922705
$795$	0	0
$796$	−25.4868	−0.903357
$797$	−18.9737	−0.672082	−0.336041	−	0.941847i	$-0.609088\pi$
$797$	−18.9737	−0.672082	−0.336041	+	0.941847i	$0.609088\pi$
$798$	0	0
$799$	33.4868i	1.18468i
$800$	− 0.324555i	− 0.0114748i
$801$	0	0
$802$	−15.0000	−0.529668
$803$	5.16228	0.182173
$804$	0	0
$805$	−5.72811	−0.201889
$806$	0	0
$807$	0	0
$808$	− 14.1623i	− 0.498227i
$809$	3.00000	0.105474	0.0527372	−	0.998608i	$-0.483205\pi$
$809$	3.00000	0.105474	0.0527372	+	0.998608i	$0.483205\pi$
$810$	0	0
$811$	− 38.9737i	− 1.36855i	−0.729224	−	0.684275i	$-0.760119\pi$
$811$	− 38.9737i	− 1.36855i	0.729224	−	0.684275i	$-0.239881\pi$
$812$	− 25.8114i	− 0.905802i
$813$	0	0
$814$	52.4605i	1.83874i
$815$	34.5964	1.21186
$816$	0	0
$817$	3.29822i	0.115390i
$818$	−27.3246	−0.955381
$819$	0	0
$820$	−6.48683	−0.226530
$821$	14.6491i	0.511257i	0.966775	+	0.255629i	$0.0822825\pi$
$821$	14.6491i	0.511257i	−0.966775	+	0.255629i	$0.917718\pi$
$822$	0	0
$823$	−2.70178	−0.0941781	−0.0470890	−	0.998891i	$-0.514994\pi$
$823$	−2.70178	−0.0941781	−0.0470890	+	0.998891i	$0.514994\pi$
$824$	− 15.8114i	− 0.550816i
$825$	0	0
$826$	− 32.6491i	− 1.13601i
$827$	30.9737i	1.07706i	0.842606	+	0.538530i	$0.181020\pi$
$827$	30.9737i	1.07706i	−0.842606	+	0.538530i	$0.818980\pi$
$828$	0	0
$829$	−5.83772	−0.202752	−0.101376	−	0.994848i	$-0.532325\pi$
$829$	−5.83772	−0.202752	−0.101376	+	0.994848i	$0.532325\pi$
$830$	− 33.4868i	− 1.16234i
$831$	0	0
$832$	0	0
$833$	−9.00000	−0.311832
$834$	0	0
$835$	−25.9473	−0.897944
$836$	−6.00000	−0.207514
$837$	0	0
$838$	− 6.97367i	− 0.240901i
$839$	− 20.6491i	− 0.712886i	−0.934317	−	0.356443i	$-0.883989\pi$
$839$	− 20.6491i	− 0.712886i	0.934317	−	0.356443i	$-0.116011\pi$
$840$	0	0
$841$	37.6228	1.29734
$842$	30.1623	1.03946
$843$	0	0
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	49.4868i	1.70039i
$848$	6.48683	0.222759
$849$	0	0
$850$	− 0.973666i	− 0.0333965i
$851$	− 8.51317i	− 0.291828i
$852$	0	0
$853$	10.8641i	0.371978i	0.982552	+	0.185989i	$0.0595489\pi$
$853$	10.8641i	0.371978i	−0.982552	+	0.185989i	$0.940451\pi$
$854$	19.4868	0.666826
$855$	0	0
$856$	13.8114i	0.472064i
$857$	0.350889	0.0119862	0.00599308	−	0.999982i	$-0.498092\pi$
$857$	0.350889	0.0119862	0.00599308	+	0.999982i	$0.498092\pi$
$858$	0	0
$859$	47.4868	1.62023	0.810115	−	0.586271i	$-0.199405\pi$
$859$	47.4868	1.62023	0.810115	+	0.586271i	$0.199405\pi$
$860$	6.13594i	0.209234i
$861$	0	0
$862$	−9.48683	−0.323123
$863$	− 16.1886i	− 0.551067i	−0.961292	−	0.275533i	$-0.911146\pi$
$863$	− 16.1886i	− 0.551067i	0.961292	−	0.275533i	$-0.0888545\pi$
$864$	0	0
$865$	5.72811i	0.194762i
$866$	3.32456i	0.112973i
$867$	0	0
$868$	20.0000	0.678844
$869$	− 20.6491i	− 0.700473i
$870$	0	0
$871$	0	0
$872$	−18.6491	−0.631539
$873$	0	0
$874$	0.973666	0.0329347
$875$	36.4078	1.23081
$876$	0	0
$877$	0.864056i	0.0291771i	0.999894	+	0.0145886i	$0.00464385\pi$
$877$	0.864056i	0.0291771i	−0.999894	+	0.0145886i	$0.995356\pi$
$878$	− 6.51317i	− 0.219809i
$879$	0	0
$880$	−11.1623	−0.376280
$881$	27.9737	0.942457	0.471228	−	0.882011i	$-0.343811\pi$
$881$	27.9737	0.942457	0.471228	+	0.882011i	$0.343811\pi$
$882$	0	0
$883$	36.6491	1.23334	0.616670	−	0.787221i	$-0.288481\pi$
$883$	36.6491	1.23334	0.616670	+	0.787221i	$0.288481\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.6754	0.862097	0.431049	−	0.902329i	$-0.358144\pi$
$887$	25.6754	0.862097	0.431049	+	0.902329i	$0.358144\pi$
$888$	0	0
$889$	− 17.9473i	− 0.601934i
$890$	25.9473i	0.869757i
$891$	0	0
$892$	− 14.3246i	− 0.479622i
$893$	12.9737	0.434147
$894$	0	0
$895$	7.53950i	0.252018i
$896$	3.16228	0.105644
$897$	0	0
$898$	32.6491	1.08951
$899$	51.6228i	1.72172i
$900$	0	0
$901$	19.4605	0.648323
$902$	− 15.4868i	− 0.515655i
$903$	0	0
$904$	19.3246i	0.642725i
$905$	− 21.9737i	− 0.730429i
$906$	0	0
$907$	52.2719	1.73566	0.867830	−	0.496862i	$-0.165514\pi$
$907$	52.2719	1.73566	0.867830	+	0.496862i	$0.165514\pi$
$908$	3.48683i	0.115715i
$909$	0	0
$910$	0	0
$911$	25.9473	0.859673	0.429837	−	0.902907i	$-0.358571\pi$
$911$	25.9473	0.859673	0.429837	+	0.902907i	$0.358571\pi$
$912$	0	0
$913$	79.9473	2.64587
$914$	−9.32456	−0.308429
$915$	0	0
$916$	− 9.67544i	− 0.319686i
$917$	27.3509i	0.903206i
$918$	0	0
$919$	28.6491	0.945047	0.472523	−	0.881318i	$-0.343343\pi$
$919$	28.6491	0.945047	0.472523	+	0.881318i	$0.343343\pi$
$920$	1.81139	0.0597197
$921$	0	0
$922$	0.486833	0.0160330
$923$	0	0
$924$	0	0
$925$	3.29822i	0.108445i
$926$	8.83772	0.290426
$927$	0	0
$928$	8.16228i	0.267940i
$929$	46.9473i	1.54029i	0.637868	+	0.770146i	$0.279817\pi$
$929$	46.9473i	1.54029i	−0.637868	+	0.770146i	$0.720183\pi$
$930$	0	0
$931$	3.48683i	0.114276i
$932$	8.64911	0.283311
$933$	0	0
$934$	37.8114i	1.23723i
$935$	−33.4868	−1.09514
$936$	0	0
$937$	−36.2982	−1.18581	−0.592906	−	0.805272i	$-0.702019\pi$
$937$	−36.2982	−1.18581	−0.592906	+	0.805272i	$0.702019\pi$
$938$	− 28.9737i	− 0.946024i
$939$	0	0
$940$	24.1359	0.787228
$941$	− 52.5964i	− 1.71460i	−0.514821	−	0.857298i	$-0.672142\pi$
$941$	− 52.5964i	− 1.71460i	0.514821	−	0.857298i	$-0.327858\pi$
$942$	0	0
$943$	2.51317i	0.0818400i
$944$	10.3246i	0.336036i
$945$	0	0
$946$	−14.6491	−0.476284
$947$	− 18.9737i	− 0.616561i	−0.951295	−	0.308281i	$-0.900246\pi$
$947$	− 18.9737i	− 0.616561i	0.951295	−	0.308281i	$-0.0997536\pi$
$948$	0	0
$949$	0	0
$950$	−0.377223	−0.0122387
$951$	0	0
$952$	9.48683	0.307470
$953$	−23.2982	−0.754703	−0.377352	−	0.926070i	$-0.623165\pi$
$953$	−23.2982	−0.754703	−0.377352	+	0.926070i	$0.623165\pi$
$954$	0	0
$955$	3.62278i	0.117230i
$956$	− 2.51317i	− 0.0812816i
$957$	0	0
$958$	−24.0000	−0.775405
$959$	−9.48683	−0.306346
$960$	0	0
$961$	−9.00000	−0.290323
$962$	0	0
$963$	0	0
$964$	− 0.675445i	− 0.0217546i
$965$	−38.8641	−1.25108
$966$	0	0
$967$	− 53.4868i	− 1.72002i	−0.510277	−	0.860010i	$-0.670457\pi$
$967$	− 53.4868i	− 1.72002i	0.510277	−	0.860010i	$-0.329543\pi$
$968$	− 15.6491i	− 0.502981i
$969$	0	0
$970$	8.64911i	0.277706i
$971$	18.9737	0.608894	0.304447	−	0.952529i	$-0.401528\pi$
$971$	18.9737	0.608894	0.304447	+	0.952529i	$0.401528\pi$
$972$	0	0
$973$	− 20.0000i	− 0.641171i
$974$	−5.48683	−0.175809
$975$	0	0
$976$	−6.16228	−0.197250
$977$	− 21.0000i	− 0.671850i	−0.941889	−	0.335925i	$-0.890951\pi$
$977$	− 21.0000i	− 0.671850i	0.941889	−	0.335925i	$-0.109049\pi$
$978$	0	0
$979$	−61.9473	−1.97985
$980$	6.48683i	0.207214i
$981$	0	0
$982$	− 25.8114i	− 0.823674i
$983$	− 39.6228i	− 1.26377i	−0.775062	−	0.631885i	$-0.782281\pi$
$983$	− 39.6228i	− 1.26377i	0.775062	−	0.631885i	$-0.217719\pi$
$984$	0	0
$985$	−41.0263	−1.30721
$986$	24.4868i	0.779820i
$987$	0	0
$988$	0	0
$989$	2.37722	0.0755913
$990$	0	0
$991$	−42.7851	−1.35911	−0.679556	−	0.733624i	$-0.737828\pi$
$991$	−42.7851	−1.35911	−0.679556	+	0.733624i	$0.737828\pi$
$992$	−6.32456	−0.200805
$993$	0	0
$994$	− 2.64911i	− 0.0840247i
$995$	55.1096i	1.74709i
$996$	0	0
$997$	−2.53950	−0.0804268	−0.0402134	−	0.999191i	$-0.512804\pi$
$997$	−2.53950	−0.0804268	−0.0402134	+	0.999191i	$0.512804\pi$
$998$	12.6491	0.400401
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.b.k.1351.2		4
3.2	odd	2		3042.2.b.j.1351.3		4
13.5	odd	4		3042.2.a.q.1.2		2
13.7	odd	12		234.2.h.d.55.1	✓	4
13.8	odd	4		3042.2.a.w.1.1		2
13.11	odd	12		234.2.h.d.217.1	yes	4
13.12	even	2	inner	3042.2.b.k.1351.3		4
39.5	even	4		3042.2.a.x.1.1		2
39.8	even	4		3042.2.a.r.1.2		2
39.11	even	12		234.2.h.e.217.2	yes	4
39.20	even	12		234.2.h.e.55.2	yes	4
39.38	odd	2		3042.2.b.j.1351.2		4
52.7	even	12		1872.2.t.p.289.1		4
52.11	even	12		1872.2.t.p.1153.1		4
156.11	odd	12		1872.2.t.n.1153.2		4
156.59	odd	12		1872.2.t.n.289.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.h.d.55.1	✓	4	13.7	odd	12
234.2.h.d.217.1	yes	4	13.11	odd	12
234.2.h.e.55.2	yes	4	39.20	even	12
234.2.h.e.217.2	yes	4	39.11	even	12
1872.2.t.n.289.2		4	156.59	odd	12
1872.2.t.n.1153.2		4	156.11	odd	12
1872.2.t.p.289.1		4	52.7	even	12
1872.2.t.p.1153.1		4	52.11	even	12
3042.2.a.q.1.2		2	13.5	odd	4
3042.2.a.r.1.2		2	39.8	even	4
3042.2.a.w.1.1		2	13.8	odd	4
3042.2.a.x.1.1		2	39.5	even	4
3042.2.b.j.1351.2		4	39.38	odd	2
3042.2.b.j.1351.3		4	3.2	odd	2
3042.2.b.k.1351.2		4	1.1	even	1	trivial
3042.2.b.k.1351.3		4	13.12	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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.16228i q^{5} -3.16228i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.16228i q^{5} -3.16228i q^{7} +1.00000i q^{8} +2.16228 q^{10} +5.16228i q^{11} -3.16228 q^{14} +1.00000 q^{16} +3.00000 q^{17} -1.16228i q^{19} -2.16228i q^{20} +5.16228 q^{22} -0.837722 q^{23} +0.324555 q^{25} +3.16228i q^{28} -8.16228 q^{29} -6.32456i q^{31} -1.00000i q^{32} -3.00000i q^{34} +6.83772 q^{35} +10.1623i q^{37} -1.16228 q^{38} -2.16228 q^{40} -3.00000i q^{41} -2.83772 q^{43} -5.16228i q^{44} +0.837722i q^{46} +11.1623i q^{47} -3.00000 q^{49} -0.324555i q^{50} +6.48683 q^{53} -11.1623 q^{55} +3.16228 q^{56} +8.16228i q^{58} +10.3246i q^{59} -6.16228 q^{61} -6.32456 q^{62} -1.00000 q^{64} +9.16228i q^{67} -3.00000 q^{68} -6.83772i q^{70} +0.837722i q^{71} -1.00000i q^{73} +10.1623 q^{74} +1.16228i q^{76} +16.3246 q^{77} -4.00000 q^{79} +2.16228i q^{80} -3.00000 q^{82} -15.4868i q^{83} +6.48683i q^{85} +2.83772i q^{86} -5.16228 q^{88} +12.0000i q^{89} +0.837722 q^{92} +11.1623 q^{94} +2.51317 q^{95} +4.00000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} - 4 q^{10} + 4 q^{16} + 12 q^{17} + 8 q^{22} - 16 q^{23} - 24 q^{25} - 20 q^{29} + 40 q^{35} + 8 q^{38} + 4 q^{40} - 24 q^{43} - 12 q^{49} - 12 q^{53} - 32 q^{55} - 12 q^{61} - 4 q^{64} - 12 q^{68} + 28 q^{74} + 40 q^{77} - 16 q^{79} - 12 q^{82} - 8 q^{88} + 16 q^{92} + 32 q^{94} + 48 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^10 + 4 * q^16 + 12 * q^17 + 8 * q^22 - 16 * q^23 - 24 * q^25 - 20 * q^29 + 40 * q^35 + 8 * q^38 + 4 * q^40 - 24 * q^43 - 12 * q^49 - 12 * q^53 - 32 * q^55 - 12 * q^61 - 4 * q^64 - 12 * q^68 + 28 * q^74 + 40 * q^77 - 16 * q^79 - 12 * q^82 - 8 * q^88 + 16 * q^92 + 32 * q^94 + 48 * q^95

Introduction

L-functions

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