LMFDB - Embedded newform 4140.3.d.c.2161.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,3,Mod(2161,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4140.2161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	4140.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:	$112.806829445$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 1380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2161.3
Character	$\chi$	$=$	4140.2161
Dual form			4140.3.d.c.2161.30

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} -10.0406i q^{7} +O(q^{10})$	$q-2.23607i q^{5} -10.0406i q^{7} +9.12315i q^{11} -5.62799 q^{13} -14.7225i q^{17} -9.77817i q^{19} +(15.4411 + 17.0462i) q^{23} -5.00000 q^{25} +22.5112 q^{29} +9.14318 q^{31} -22.4514 q^{35} +0.327438i q^{37} +64.1288 q^{41} -42.8185i q^{43} +11.5474 q^{47} -51.8135 q^{49} -63.3474i q^{53} +20.4000 q^{55} +51.5822 q^{59} +46.2561i q^{61} +12.5846i q^{65} -59.6241i q^{67} -11.3134 q^{71} -67.7398 q^{73} +91.6019 q^{77} -84.5653i q^{79} -40.8352i q^{83} -32.9205 q^{85} +4.77000i q^{89} +56.5083i q^{91} -21.8647 q^{95} -41.8758i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q + 24 q^{13} - 64 q^{23} - 160 q^{25} + 60 q^{29} - 4 q^{31} + 60 q^{35} + 108 q^{41} - 136 q^{47} - 428 q^{49} + 120 q^{55} + 84 q^{59} - 188 q^{71} + 472 q^{73} + 120 q^{77} + 60 q^{85} - 80 q^{95}+O(q^{100}) $ 32 * q + 24 * q^13 - 64 * q^23 - 160 * q^25 + 60 * q^29 - 4 * q^31 + 60 * q^35 + 108 * q^41 - 136 * q^47 - 428 * q^49 + 120 * q^55 + 84 * q^59 - 188 * q^71 + 472 * q^73 + 120 * q^77 + 60 * q^85 - 80 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$		−	10.0406i		−	1.43437i	−0.696883	−	0.717185i	$-0.745430\pi$
$7$		−	10.0406i		−	1.43437i	0.696883	−	0.717185i	$-0.254570\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			9.12315i			0.829377i	0.909963	+	0.414689i	$0.136110\pi$
$11$			9.12315i			0.829377i	−0.909963	+	0.414689i	$0.863890\pi$
$12$		0			0
$13$	−5.62799			−0.432922			−0.216461	−	0.976291i	$-0.569451\pi$
$13$	−5.62799			−0.432922			−0.216461	+	0.976291i	$0.569451\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	14.7225i		−	0.866028i	−0.901387	−	0.433014i	$-0.857450\pi$
$17$		−	14.7225i		−	0.866028i	0.901387	−	0.433014i	$-0.142550\pi$
$18$		0			0
$19$		−	9.77817i		−	0.514641i	−0.966326	−	0.257320i	$-0.917160\pi$
$19$		−	9.77817i		−	0.514641i	0.966326	−	0.257320i	$-0.0828395\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	15.4411	+	17.0462i	0.671354	+	0.741137i
$23$	15.4411	+	17.0462i	0.671354	+	0.741137i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	22.5112			0.776249			0.388124	−	0.921607i	$-0.373123\pi$
$29$	22.5112			0.776249			0.388124	+	0.921607i	$0.373123\pi$
$30$		0			0
$31$	9.14318			0.294941			0.147471	−	0.989066i	$-0.452887\pi$
$31$	9.14318			0.294941			0.147471	+	0.989066i	$0.452887\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−22.4514			−0.641470
$36$		0			0
$37$			0.327438i			0.00884968i	0.999990	+	0.00442484i	$0.00140847\pi$
$37$			0.327438i			0.00884968i	−0.999990	+	0.00442484i	$0.998592\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	64.1288			1.56412			0.782059	−	0.623205i	$-0.214170\pi$
$41$	64.1288			1.56412			0.782059	+	0.623205i	$0.214170\pi$
$42$		0			0
$43$		−	42.8185i		−	0.995780i	−0.867240	−	0.497890i	$-0.834108\pi$
$43$		−	42.8185i		−	0.995780i	0.867240	−	0.497890i	$-0.165892\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	11.5474			0.245688			0.122844	−	0.992426i	$-0.460798\pi$
$47$	11.5474			0.245688			0.122844	+	0.992426i	$0.460798\pi$
$48$		0			0
$49$	−51.8135			−1.05742
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	63.3474i		−	1.19523i	−0.801782	−	0.597617i	$-0.796114\pi$
$53$		−	63.3474i		−	1.19523i	0.801782	−	0.597617i	$-0.203886\pi$
$54$		0			0
$55$	20.4000			0.370909
$56$		0			0
$57$		0			0
$58$		0			0
$59$	51.5822			0.874274			0.437137	−	0.899395i	$-0.355993\pi$
$59$	51.5822			0.874274			0.437137	+	0.899395i	$0.355993\pi$
$60$		0			0
$61$			46.2561i			0.758297i	0.925336	+	0.379148i	$0.123783\pi$
$61$			46.2561i			0.758297i	−0.925336	+	0.379148i	$0.876217\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			12.5846i			0.193609i
$66$		0			0
$67$		−	59.6241i		−	0.889912i	−0.895552	−	0.444956i	$-0.853219\pi$
$67$		−	59.6241i		−	0.889912i	0.895552	−	0.444956i	$-0.146781\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−11.3134			−0.159344			−0.0796722	−	0.996821i	$-0.525387\pi$
$71$	−11.3134			−0.159344			−0.0796722	+	0.996821i	$0.525387\pi$
$72$		0			0
$73$	−67.7398			−0.927943			−0.463971	−	0.885850i	$-0.653576\pi$
$73$	−67.7398			−0.927943			−0.463971	+	0.885850i	$0.653576\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	91.6019			1.18963
$78$		0			0
$79$		−	84.5653i		−	1.07045i	−0.844710	−	0.535224i	$-0.820227\pi$
$79$		−	84.5653i		−	1.07045i	0.844710	−	0.535224i	$-0.179773\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	40.8352i		−	0.491991i	−0.969271	−	0.245995i	$-0.920885\pi$
$83$		−	40.8352i		−	0.491991i	0.969271	−	0.245995i	$-0.0791148\pi$
$84$		0			0
$85$	−32.9205			−0.387300
$86$		0			0
$87$		0			0
$88$		0			0
$89$			4.77000i			0.0535955i	0.999641	+	0.0267978i	$0.00853101\pi$
$89$			4.77000i			0.0535955i	−0.999641	+	0.0267978i	$0.991469\pi$
$90$		0			0
$91$			56.5083i			0.620971i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−21.8647			−0.230154
$96$		0			0
$97$		−	41.8758i		−	0.431709i	−0.976426	−	0.215854i	$-0.930746\pi$
$97$		−	41.8758i		−	0.431709i	0.976426	−	0.215854i	$-0.0692537\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−138.361			−1.36991			−0.684957	−	0.728583i	$-0.740179\pi$
$101$	−138.361			−1.36991			−0.684957	+	0.728583i	$0.740179\pi$
$102$		0			0
$103$		−	128.686i		−	1.24938i	−0.780872	−	0.624691i	$-0.785225\pi$
$103$		−	128.686i		−	1.24938i	0.780872	−	0.624691i	$-0.214775\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			136.527i			1.27595i	0.770056	+	0.637976i	$0.220228\pi$
$107$			136.527i			1.27595i	−0.770056	+	0.637976i	$0.779772\pi$
$108$		0			0
$109$		−	106.933i		−	0.981037i	−0.871431	−	0.490519i	$-0.836807\pi$
$109$		−	106.933i		−	0.981037i	0.871431	−	0.490519i	$-0.163193\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	8.44021i		−	0.0746921i	−0.999302	−	0.0373460i	$-0.988110\pi$
$113$		−	8.44021i		−	0.0746921i	0.999302	−	0.0373460i	$-0.0118904\pi$
$114$		0			0
$115$	38.1164	−	34.5274i	0.331447	−	0.300239i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−147.822			−1.24221
$120$		0			0
$121$	37.7681			0.312133
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	−66.2807			−0.521895			−0.260948	−	0.965353i	$-0.584035\pi$
$127$	−66.2807			−0.521895			−0.260948	+	0.965353i	$0.584035\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	146.278			1.11663			0.558313	−	0.829630i	$-0.311449\pi$
$131$	146.278			1.11663			0.558313	+	0.829630i	$0.311449\pi$
$132$		0			0
$133$	−98.1786			−0.738185
$134$		0			0
$135$		0			0
$136$		0			0
$137$			89.3470i			0.652168i	0.945341	+	0.326084i	$0.105729\pi$
$137$			89.3470i			0.652168i	−0.945341	+	0.326084i	$0.894271\pi$
$138$		0			0
$139$	−76.3745			−0.549457			−0.274728	−	0.961522i	$-0.588588\pi$
$139$	−76.3745			−0.549457			−0.274728	+	0.961522i	$0.588588\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	51.3450i		−	0.359056i
$144$		0			0
$145$		−	50.3366i		−	0.347149i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	39.0326i		−	0.261964i	−0.991385	−	0.130982i	$-0.958187\pi$
$149$		−	39.0326i		−	0.261964i	0.991385	−	0.130982i	$-0.0418130\pi$
$150$		0			0
$151$	189.166			1.25276			0.626378	−	0.779519i	$-0.284537\pi$
$151$	189.166			1.25276			0.626378	+	0.779519i	$0.284537\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	20.4448i		−	0.131902i
$156$		0			0
$157$		−	131.225i		−	0.835825i	−0.908487	−	0.417913i	$-0.862762\pi$
$157$		−	131.225i		−	0.835825i	0.908487	−	0.417913i	$-0.137238\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	171.153	−	155.038i	1.06307	−	0.962970i
$162$		0			0
$163$	−164.832			−1.01124			−0.505619	−	0.862757i	$-0.668736\pi$
$163$	−164.832			−1.01124			−0.505619	+	0.862757i	$0.668736\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−122.831			−0.735514			−0.367757	−	0.929922i	$-0.619874\pi$
$167$	−122.831			−0.735514			−0.367757	+	0.929922i	$0.619874\pi$
$168$		0			0
$169$	−137.326			−0.812578
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−118.439			−0.684620			−0.342310	−	0.939587i	$-0.611209\pi$
$173$	−118.439			−0.684620			−0.342310	+	0.939587i	$0.611209\pi$
$174$		0			0
$175$			50.2030i			0.286874i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−92.8346			−0.518629			−0.259315	−	0.965793i	$-0.583497\pi$
$179$	−92.8346			−0.518629			−0.259315	+	0.965793i	$0.583497\pi$
$180$		0			0
$181$			122.806i			0.678488i	0.940698	+	0.339244i	$0.110171\pi$
$181$			122.806i			0.678488i	−0.940698	+	0.339244i	$0.889829\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.732174			0.00395770
$186$		0			0
$187$	134.315			0.718264
$188$		0			0
$189$		0			0
$190$		0			0
$191$			169.982i			0.889960i	0.895541	+	0.444980i	$0.146789\pi$
$191$			169.982i			0.889960i	−0.895541	+	0.444980i	$0.853211\pi$
$192$		0			0
$193$	−31.3026			−0.162190			−0.0810948	−	0.996706i	$-0.525842\pi$
$193$	−31.3026			−0.162190			−0.0810948	+	0.996706i	$0.525842\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−83.5245			−0.423982			−0.211991	−	0.977272i	$-0.567995\pi$
$197$	−83.5245			−0.423982			−0.211991	+	0.977272i	$0.567995\pi$
$198$		0			0
$199$			236.332i			1.18760i	0.804614	+	0.593798i	$0.202372\pi$
$199$			236.332i			1.18760i	−0.804614	+	0.593798i	$0.797628\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	226.026i		−	1.11343i
$204$		0			0
$205$		−	143.396i		−	0.699495i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	89.2078			0.426831
$210$		0			0
$211$	5.63314			0.0266974			0.0133487	−	0.999911i	$-0.495751\pi$
$211$	5.63314			0.0266974			0.0133487	+	0.999911i	$0.495751\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−95.7452			−0.445326
$216$		0			0
$217$		−	91.8030i		−	0.423055i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			82.8579i			0.374923i
$222$		0			0
$223$	−424.750			−1.90471			−0.952353	−	0.304997i	$-0.901345\pi$
$223$	−424.750			−1.90471			−0.952353	+	0.304997i	$0.901345\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			33.3874i			0.147081i	0.997292	+	0.0735405i	$0.0234298\pi$
$227$			33.3874i			0.147081i	−0.997292	+	0.0735405i	$0.976570\pi$
$228$		0			0
$229$		−	13.6077i		−	0.0594221i	−0.999559	−	0.0297110i	$-0.990541\pi$
$229$		−	13.6077i		−	0.0594221i	0.999559	−	0.0297110i	$-0.00945871\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−185.350			−0.795494			−0.397747	−	0.917495i	$-0.630208\pi$
$233$	−185.350			−0.795494			−0.397747	+	0.917495i	$0.630208\pi$
$234$		0			0
$235$		−	25.8207i		−	0.109875i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−309.490			−1.29494			−0.647468	−	0.762093i	$-0.724172\pi$
$239$	−309.490			−1.29494			−0.647468	+	0.762093i	$0.724172\pi$
$240$		0			0
$241$			193.873i			0.804451i	0.915541	+	0.402226i	$0.131763\pi$
$241$			193.873i			0.804451i	−0.915541	+	0.402226i	$0.868237\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			115.859i			0.472892i
$246$		0			0
$247$			55.0314i			0.222799i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			353.939i			1.41012i	0.709149	+	0.705058i	$0.249079\pi$
$251$			353.939i			1.41012i	−0.709149	+	0.705058i	$0.750921\pi$
$252$		0			0
$253$	−155.515	+	140.872i	−0.614682	+	0.556806i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−294.098			−1.14435			−0.572174	−	0.820132i	$-0.693900\pi$
$257$	−294.098			−1.14435			−0.572174	+	0.820132i	$0.693900\pi$
$258$		0			0
$259$	3.28767			0.0126937
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	128.714i		−	0.489407i	−0.969598	−	0.244703i	$-0.921309\pi$
$263$		−	128.714i		−	0.489407i	0.969598	−	0.244703i	$-0.0786906\pi$
$264$		0			0
$265$	−141.649			−0.534525
$266$		0			0
$267$		0			0
$268$		0			0
$269$	312.826			1.16292			0.581462	−	0.813574i	$-0.302481\pi$
$269$	312.826			1.16292			0.581462	+	0.813574i	$0.302481\pi$
$270$		0			0
$271$	−477.115			−1.76057			−0.880286	−	0.474444i	$-0.842649\pi$
$271$	−477.115			−1.76057			−0.880286	+	0.474444i	$0.842649\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	45.6158i		−	0.165875i
$276$		0			0
$277$	249.074			0.899183			0.449591	−	0.893234i	$-0.351570\pi$
$277$	249.074			0.899183			0.449591	+	0.893234i	$0.351570\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	31.5385i		−	0.112236i	−0.998424	−	0.0561182i	$-0.982128\pi$
$281$		−	31.5385i		−	0.112236i	0.998424	−	0.0561182i	$-0.0178724\pi$
$282$		0			0
$283$			71.4582i			0.252503i	0.991998	+	0.126251i	$0.0402946\pi$
$283$			71.4582i			0.252503i	−0.991998	+	0.126251i	$0.959705\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	643.891i		−	2.24352i
$288$		0			0
$289$	72.2487			0.249995
$290$		0			0
$291$		0			0
$292$		0			0
$293$			77.4804i			0.264438i	0.991221	+	0.132219i	$0.0422102\pi$
$293$			77.4804i			0.264438i	−0.991221	+	0.132219i	$0.957790\pi$
$294$		0			0
$295$		−	115.341i		−	0.390987i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−86.9025	−	95.9355i	−0.290644	−	0.320855i
$300$		0			0
$301$	−429.924			−1.42832
$302$		0			0
$303$		0			0
$304$		0			0
$305$	103.432			0.339121
$306$		0			0
$307$	−118.471			−0.385901			−0.192950	−	0.981209i	$-0.561806\pi$
$307$	−118.471			−0.385901			−0.192950	+	0.981209i	$0.561806\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	70.7478			0.227485			0.113742	−	0.993510i	$-0.463716\pi$
$311$	70.7478			0.227485			0.113742	+	0.993510i	$0.463716\pi$
$312$		0			0
$313$		−	247.750i		−	0.791533i	−0.918351	−	0.395766i	$-0.870479\pi$
$313$		−	247.750i		−	0.791533i	0.918351	−	0.395766i	$-0.129521\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−283.470			−0.894228			−0.447114	−	0.894477i	$-0.647548\pi$
$317$	−283.470			−0.894228			−0.447114	+	0.894477i	$0.647548\pi$
$318$		0			0
$319$			205.373i			0.643803i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−143.959			−0.445693
$324$		0			0
$325$	28.1399			0.0865844
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	115.942i		−	0.352408i
$330$		0			0
$331$	7.28169			0.0219991			0.0109995	−	0.999940i	$-0.496499\pi$
$331$	7.28169			0.0219991			0.0109995	+	0.999940i	$0.496499\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−133.324			−0.397981
$336$		0			0
$337$		−	256.166i		−	0.760136i	−0.924959	−	0.380068i	$-0.875901\pi$
$337$		−	256.166i		−	0.760136i	0.924959	−	0.380068i	$-0.124099\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			83.4146i			0.244618i
$342$		0			0
$343$			28.2492i			0.0823592i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−539.358			−1.55435			−0.777173	−	0.629287i	$-0.783347\pi$
$347$	−539.358			−1.55435			−0.777173	+	0.629287i	$0.783347\pi$
$348$		0			0
$349$	522.961			1.49846			0.749228	−	0.662313i	$-0.230425\pi$
$349$	522.961			1.49846			0.749228	+	0.662313i	$0.230425\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	114.179			0.323452			0.161726	−	0.986836i	$-0.448294\pi$
$353$	114.179			0.323452			0.161726	+	0.986836i	$0.448294\pi$
$354$		0			0
$355$			25.2976i			0.0712610i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	402.611i		−	1.12148i	−0.827992	−	0.560739i	$-0.810517\pi$
$359$		−	402.611i		−	1.12148i	0.827992	−	0.560739i	$-0.189483\pi$
$360$		0			0
$361$	265.387			0.735145
$362$		0			0
$363$		0			0
$364$		0			0
$365$			151.471i			0.414989i
$366$		0			0
$367$		−	281.669i		−	0.767491i	−0.923439	−	0.383746i	$-0.874634\pi$
$367$		−	281.669i		−	0.767491i	0.923439	−	0.383746i	$-0.125366\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−636.045			−1.71441
$372$		0			0
$373$		−	60.6525i		−	0.162607i	−0.996689	−	0.0813036i	$-0.974092\pi$
$373$		−	60.6525i		−	0.162607i	0.996689	−	0.0813036i	$-0.0259083\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−126.693			−0.336055
$378$		0			0
$379$			24.3760i			0.0643167i	0.999483	+	0.0321584i	$0.0102381\pi$
$379$			24.3760i			0.0643167i	−0.999483	+	0.0321584i	$0.989762\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	433.729i		−	1.13245i	−0.824250	−	0.566226i	$-0.808403\pi$
$383$		−	433.729i		−	1.13245i	0.824250	−	0.566226i	$-0.191597\pi$
$384$		0			0
$385$		−	204.828i		−	0.532021i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			68.2154i			0.175361i	0.996149	+	0.0876805i	$0.0279455\pi$
$389$			68.2154i			0.175361i	−0.996149	+	0.0876805i	$0.972055\pi$
$390$		0			0
$391$	250.962	−	227.332i	0.641845	−	0.581411i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−189.094			−0.478719
$396$		0			0
$397$	682.441			1.71899			0.859497	−	0.511140i	$-0.170777\pi$
$397$	682.441			1.71899			0.859497	+	0.511140i	$0.170777\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	602.054i		−	1.50138i	−0.660653	−	0.750691i	$-0.729721\pi$
$401$		−	602.054i		−	1.50138i	0.660653	−	0.750691i	$-0.270279\pi$
$402$		0			0
$403$	−51.4577			−0.127687
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.98727			−0.00733972
$408$		0			0
$409$	−433.312			−1.05944			−0.529722	−	0.848172i	$-0.677704\pi$
$409$	−433.312			−1.05944			−0.529722	+	0.848172i	$0.677704\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	517.915i		−	1.25403i
$414$		0			0
$415$	−91.3104			−0.220025
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	478.597i		−	1.14224i	−0.820868	−	0.571118i	$-0.806510\pi$
$419$		−	478.597i		−	1.14224i	0.820868	−	0.571118i	$-0.193490\pi$
$420$		0			0
$421$		−	202.181i		−	0.480241i	−0.970743	−	0.240120i	$-0.922813\pi$
$421$		−	202.181i		−	0.480241i	0.970743	−	0.240120i	$-0.0771869\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			73.6124i			0.173206i
$426$		0			0
$427$	464.439			1.08768
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	322.802i		−	0.748961i	−0.927235	−	0.374481i	$-0.877821\pi$
$431$		−	322.802i		−	0.748961i	0.927235	−	0.374481i	$-0.122179\pi$
$432$		0			0
$433$			117.649i			0.271706i	0.990729	+	0.135853i	$0.0433774\pi$
$433$			117.649i			0.271706i	−0.990729	+	0.135853i	$0.956623\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	166.680	−	150.986i	0.381419	−	0.345506i
$438$		0			0
$439$	565.496			1.28814			0.644072	−	0.764964i	$-0.277244\pi$
$439$	565.496			1.28814			0.644072	+	0.764964i	$0.277244\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−138.667			−0.313019			−0.156509	−	0.987676i	$-0.550024\pi$
$443$	−138.667			−0.313019			−0.156509	+	0.987676i	$0.550024\pi$
$444$		0			0
$445$	10.6660			0.0239686
$446$		0			0
$447$		0			0
$448$		0			0
$449$	531.355			1.18342			0.591709	−	0.806152i	$-0.298453\pi$
$449$	531.355			1.18342			0.591709	+	0.806152i	$0.298453\pi$
$450$		0			0
$451$			585.057i			1.29724i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	126.356			0.277707
$456$		0			0
$457$		−	432.106i		−	0.945527i	−0.881189	−	0.472764i	$-0.843256\pi$
$457$		−	432.106i		−	0.945527i	0.881189	−	0.472764i	$-0.156744\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	319.935			0.694003			0.347002	−	0.937865i	$-0.387200\pi$
$461$	319.935			0.694003			0.347002	+	0.937865i	$0.387200\pi$
$462$		0			0
$463$	399.580			0.863023			0.431512	−	0.902107i	$-0.357980\pi$
$463$	399.580			0.863023			0.431512	+	0.902107i	$0.357980\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	43.2841i		−	0.0926855i	−0.998926	−	0.0463428i	$-0.985243\pi$
$467$		−	43.2841i		−	0.0926855i	0.998926	−	0.0463428i	$-0.0147566\pi$
$468$		0			0
$469$	−598.661			−1.27646
$470$		0			0
$471$		0			0
$472$		0			0
$473$	390.640			0.825878
$474$		0			0
$475$			48.8909i			0.102928i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	749.848i		−	1.56545i	−0.622371	−	0.782723i	$-0.713830\pi$
$479$		−	749.848i		−	1.56545i	0.622371	−	0.782723i	$-0.286170\pi$
$480$		0			0
$481$		−	1.84282i		−	0.00383122i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−93.6370			−0.193066
$486$		0			0
$487$	950.201			1.95113			0.975566	−	0.219708i	$-0.0705105\pi$
$487$	950.201			1.95113			0.975566	+	0.219708i	$0.0705105\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−795.564			−1.62029			−0.810147	−	0.586227i	$-0.800613\pi$
$491$	−795.564			−1.62029			−0.810147	+	0.586227i	$0.800613\pi$
$492$		0			0
$493$		−	331.421i		−	0.672253i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			113.594i			0.228559i
$498$		0			0
$499$	−623.584			−1.24967			−0.624833	−	0.780758i	$-0.714833\pi$
$499$	−623.584			−1.24967			−0.624833	+	0.780758i	$0.714833\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	626.481i		−	1.24549i	−0.782425	−	0.622745i	$-0.786018\pi$
$503$		−	626.481i		−	1.24549i	0.782425	−	0.622745i	$-0.213982\pi$
$504$		0			0
$505$			309.385i			0.612644i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−251.571			−0.494245			−0.247122	−	0.968984i	$-0.579485\pi$
$509$	−251.571			−0.494245			−0.247122	+	0.968984i	$0.579485\pi$
$510$		0			0
$511$			680.148i			1.33101i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−287.752			−0.558741
$516$		0			0
$517$			105.348i			0.203768i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	739.126i		−	1.41867i	−0.704873	−	0.709334i	$-0.748996\pi$
$521$		−	739.126i		−	1.41867i	0.704873	−	0.709334i	$-0.251004\pi$
$522$		0			0
$523$		−	319.540i		−	0.610976i	−0.952196	−	0.305488i	$-0.901180\pi$
$523$		−	319.540i		−	0.610976i	0.952196	−	0.305488i	$-0.0988195\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	134.610i		−	0.255427i
$528$		0			0
$529$	−52.1426	+	526.424i	−0.0985682	+	0.995130i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−360.916			−0.677141
$534$		0			0
$535$	305.283			0.570623
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	472.702i		−	0.876999i
$540$		0			0
$541$	−577.121			−1.06677			−0.533383	−	0.845874i	$-0.679080\pi$
$541$	−577.121			−1.06677			−0.533383	+	0.845874i	$0.679080\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−239.110			−0.438733
$546$		0			0
$547$	246.044			0.449806			0.224903	−	0.974381i	$-0.427793\pi$
$547$	246.044			0.449806			0.224903	+	0.974381i	$0.427793\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	220.118i		−	0.399489i
$552$		0			0
$553$	−849.086			−1.53542
$554$		0			0
$555$		0			0
$556$		0			0
$557$			509.963i			0.915553i	0.889067	+	0.457777i	$0.151354\pi$
$557$			509.963i			0.915553i	−0.889067	+	0.457777i	$0.848646\pi$
$558$		0			0
$559$			240.982i			0.431095i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			529.080i			0.939750i	0.882733	+	0.469875i	$0.155701\pi$
$563$			529.080i			0.939750i	−0.882733	+	0.469875i	$0.844299\pi$
$564$		0			0
$565$	−18.8729			−0.0334033
$566$		0			0
$567$		0			0
$568$		0			0
$569$			87.5647i			0.153892i	0.997035	+	0.0769461i	$0.0245169\pi$
$569$			87.5647i			0.153892i	−0.997035	+	0.0769461i	$0.975483\pi$
$570$		0			0
$571$		−	664.465i		−	1.16369i	−0.813301	−	0.581844i	$-0.802332\pi$
$571$		−	664.465i		−	1.16369i	0.813301	−	0.581844i	$-0.197668\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−77.2057	−	85.2308i	−0.134271	−	0.148227i
$576$		0			0
$577$	−768.980			−1.33272			−0.666360	−	0.745630i	$-0.732149\pi$
$577$	−768.980			−1.33272			−0.666360	+	0.745630i	$0.732149\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−410.010			−0.705697
$582$		0			0
$583$	577.928			0.991300
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−172.409			−0.293712			−0.146856	−	0.989158i	$-0.546915\pi$
$587$	−172.409			−0.293712			−0.146856	+	0.989158i	$0.546915\pi$
$588$		0			0
$589$		−	89.4036i		−	0.151789i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−17.5748			−0.0296371			−0.0148185	−	0.999890i	$-0.504717\pi$
$593$	−17.5748			−0.0296371			−0.0148185	+	0.999890i	$0.504717\pi$
$594$		0			0
$595$			330.541i			0.555531i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−168.233			−0.280857			−0.140428	−	0.990091i	$-0.544848\pi$
$599$	−168.233			−0.280857			−0.140428	+	0.990091i	$0.544848\pi$
$600$		0			0
$601$	409.508			0.681378			0.340689	−	0.940176i	$-0.389340\pi$
$601$	409.508			0.681378			0.340689	+	0.940176i	$0.389340\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	84.4520i		−	0.139590i
$606$		0			0
$607$	−287.691			−0.473956			−0.236978	−	0.971515i	$-0.576157\pi$
$607$	−287.691			−0.473956			−0.236978	+	0.971515i	$0.576157\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−64.9884			−0.106364
$612$		0			0
$613$		−	92.4723i		−	0.150852i	−0.997151	−	0.0754260i	$-0.975968\pi$
$613$		−	92.4723i		−	0.150852i	0.997151	−	0.0754260i	$-0.0240317\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1130.37i			1.83203i	0.401139	+	0.916017i	$0.368614\pi$
$617$			1130.37i			1.83203i	−0.401139	+	0.916017i	$0.631386\pi$
$618$		0			0
$619$		−	421.743i		−	0.681329i	−0.940185	−	0.340665i	$-0.889348\pi$
$619$		−	421.743i		−	0.681329i	0.940185	−	0.340665i	$-0.110652\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	47.8936			0.0768758
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	4.82070			0.00766407
$630$		0			0
$631$			548.612i			0.869433i	0.900567	+	0.434716i	$0.143151\pi$
$631$			548.612i			0.869433i	−0.900567	+	0.434716i	$0.856849\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			148.208i			0.233399i
$636$		0			0
$637$	291.606			0.457780
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	196.679i		−	0.306832i	−0.988162	−	0.153416i	$-0.950973\pi$
$641$		−	196.679i		−	0.306832i	0.988162	−	0.153416i	$-0.0490275\pi$
$642$		0			0
$643$			526.582i			0.818946i	0.912322	+	0.409473i	$0.134287\pi$
$643$			526.582i			0.818946i	−0.912322	+	0.409473i	$0.865713\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1040.63			−1.60840			−0.804199	−	0.594361i	$-0.797405\pi$
$647$	−1040.63			−1.60840			−0.804199	+	0.594361i	$0.797405\pi$
$648$		0			0
$649$			470.592i			0.725103i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−425.108			−0.651007			−0.325504	−	0.945541i	$-0.605534\pi$
$653$	−425.108			−0.651007			−0.325504	+	0.945541i	$0.605534\pi$
$654$		0			0
$655$		−	327.088i		−	0.499371i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	109.808i		−	0.166628i	−0.996523	−	0.0833139i	$-0.973450\pi$
$659$		−	109.808i		−	0.166628i	0.996523	−	0.0833139i	$-0.0265504\pi$
$660$		0			0
$661$			836.648i			1.26573i	0.774262	+	0.632866i	$0.218121\pi$
$661$			836.648i			1.26573i	−0.774262	+	0.632866i	$0.781879\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			219.534i			0.330126i
$666$		0			0
$667$	347.599	+	383.729i	0.521137	+	0.575307i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−422.001			−0.628914
$672$		0			0
$673$	1205.42			1.79112			0.895558	−	0.444946i	$-0.146777\pi$
$673$	1205.42			1.79112			0.895558	+	0.444946i	$0.146777\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			470.370i			0.694786i	0.937720	+	0.347393i	$0.112933\pi$
$677$			470.370i			0.694786i	−0.937720	+	0.347393i	$0.887067\pi$
$678$		0			0
$679$	−420.457			−0.619230
$680$		0			0
$681$		0			0
$682$		0			0
$683$	106.607			0.156086			0.0780430	−	0.996950i	$-0.475133\pi$
$683$	106.607			0.156086			0.0780430	+	0.996950i	$0.475133\pi$
$684$		0			0
$685$	199.786			0.291658
$686$		0			0
$687$		0			0
$688$		0			0
$689$			356.518i			0.517443i
$690$		0			0
$691$	112.497			0.162803			0.0814017	−	0.996681i	$-0.474060\pi$
$691$	112.497			0.162803			0.0814017	+	0.996681i	$0.474060\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			170.779i			0.245725i
$696$		0			0
$697$		−	944.135i		−	1.35457i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			694.162i			0.990246i	0.868823	+	0.495123i	$0.164877\pi$
$701$			694.162i			0.990246i	−0.868823	+	0.495123i	$0.835123\pi$
$702$		0			0
$703$	3.20175			0.00455440
$704$		0			0
$705$		0			0
$706$		0			0
$707$			1389.23i			1.96496i
$708$		0			0
$709$		−	477.427i		−	0.673380i	−0.941615	−	0.336690i	$-0.890692\pi$
$709$		−	477.427i		−	0.673380i	0.941615	−	0.336690i	$-0.109308\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	141.181	+	155.856i	0.198010	+	0.218592i
$714$		0			0
$715$	−114.811			−0.160575
$716$		0			0
$717$		0			0
$718$		0			0
$719$	283.942			0.394912			0.197456	−	0.980312i	$-0.436732\pi$
$719$	283.942			0.394912			0.197456	+	0.980312i	$0.436732\pi$
$720$		0			0
$721$	−1292.09			−1.79208
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−112.556			−0.155250
$726$		0			0
$727$		−	1190.53i		−	1.63759i	−0.574085	−	0.818796i	$-0.694642\pi$
$727$		−	1190.53i		−	1.63759i	0.574085	−	0.818796i	$-0.305358\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−630.395			−0.862373
$732$		0			0
$733$			1268.28i			1.73025i	0.501553	+	0.865127i	$0.332762\pi$
$733$			1268.28i			1.73025i	−0.501553	+	0.865127i	$0.667238\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	543.960			0.738073
$738$		0			0
$739$	−267.970			−0.362612			−0.181306	−	0.983427i	$-0.558032\pi$
$739$	−267.970			−0.362612			−0.181306	+	0.983427i	$0.558032\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	446.286i		−	0.600654i	−0.953836	−	0.300327i	$-0.902904\pi$
$743$		−	446.286i		−	0.600654i	0.953836	−	0.300327i	$-0.0970958\pi$
$744$		0			0
$745$	−87.2795			−0.117154
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1370.81			1.83019
$750$		0			0
$751$		−	704.234i		−	0.937728i	−0.883270	−	0.468864i	$-0.844663\pi$
$751$		−	704.234i		−	0.937728i	0.883270	−	0.468864i	$-0.155337\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	422.988i		−	0.560250i
$756$		0			0
$757$		−	1004.71i		−	1.32722i	−0.748077	−	0.663612i	$-0.769022\pi$
$757$		−	1004.71i		−	1.32722i	0.748077	−	0.663612i	$-0.230978\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−388.433			−0.510425			−0.255213	−	0.966885i	$-0.582145\pi$
$761$	−388.433			−0.510425			−0.255213	+	0.966885i	$0.582145\pi$
$762$		0			0
$763$	−1073.67			−1.40717
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−290.304			−0.378493
$768$		0			0
$769$			278.185i			0.361749i	0.983506	+	0.180874i	$0.0578928\pi$
$769$			278.185i			0.361749i	−0.983506	+	0.180874i	$0.942107\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			624.739i			0.808200i	0.914715	+	0.404100i	$0.132415\pi$
$773$			624.739i			0.808200i	−0.914715	+	0.404100i	$0.867585\pi$
$774$		0			0
$775$	−45.7159			−0.0589883
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	627.063i		−	0.804959i
$780$		0			0
$781$		−	103.214i		−	0.132157i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−293.427			−0.373792
$786$		0			0
$787$		−	497.966i		−	0.632740i	−0.948636	−	0.316370i	$-0.897536\pi$
$787$		−	497.966i		−	0.632740i	0.948636	−	0.316370i	$-0.102464\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−84.7447			−0.107136
$792$		0			0
$793$		−	260.329i		−	0.328283i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	245.617i		−	0.308177i	−0.988057	−	0.154089i	$-0.950756\pi$
$797$		−	245.617i		−	0.308177i	0.988057	−	0.154089i	$-0.0492441\pi$
$798$		0			0
$799$		−	170.006i		−	0.212773i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	618.001i		−	0.769615i
$804$		0			0
$805$	−346.676	−	382.711i	−0.430653	−	0.475417i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	402.926			0.498054			0.249027	−	0.968497i	$-0.419889\pi$
$809$	402.926			0.498054			0.249027	+	0.968497i	$0.419889\pi$
$810$		0			0
$811$	−457.449			−0.564055			−0.282027	−	0.959406i	$-0.591007\pi$
$811$	−457.449			−0.564055			−0.282027	+	0.959406i	$0.591007\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			368.575i			0.452239i
$816$		0			0
$817$	−418.687			−0.512469
$818$		0			0
$819$		0			0
$820$		0			0
$821$	579.812			0.706226			0.353113	−	0.935581i	$-0.385123\pi$
$821$	579.812			0.706226			0.353113	+	0.935581i	$0.385123\pi$
$822$		0			0
$823$	210.842			0.256187			0.128094	−	0.991762i	$-0.459114\pi$
$823$	210.842			0.256187			0.128094	+	0.991762i	$0.459114\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			253.146i			0.306102i	0.988218	+	0.153051i	$0.0489098\pi$
$827$			253.146i			0.306102i	−0.988218	+	0.153051i	$0.951090\pi$
$828$		0			0
$829$	604.073			0.728677			0.364339	−	0.931267i	$-0.381295\pi$
$829$	604.073			0.728677			0.364339	+	0.931267i	$0.381295\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			762.823i			0.915754i
$834$		0			0
$835$			274.658i			0.328932i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1239.51i		−	1.47736i	−0.674054	−	0.738682i	$-0.735448\pi$
$839$		−	1239.51i		−	1.47736i	0.674054	−	0.738682i	$-0.264552\pi$
$840$		0			0
$841$	−334.245			−0.397438
$842$		0			0
$843$		0			0
$844$		0			0
$845$			307.070i			0.363396i
$846$		0			0
$847$		−	379.214i		−	0.447714i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−5.58156	+	5.05602i	−0.00655882	+	0.00594127i
$852$		0			0
$853$	753.543			0.883403			0.441701	−	0.897162i	$-0.354375\pi$
$853$	753.543			0.883403			0.441701	+	0.897162i	$0.354375\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	828.344			0.966562			0.483281	−	0.875465i	$-0.339445\pi$
$857$	828.344			0.966562			0.483281	+	0.875465i	$0.339445\pi$
$858$		0			0
$859$	−1086.41			−1.26473			−0.632367	−	0.774669i	$-0.717916\pi$
$859$	−1086.41			−1.26473			−0.632367	+	0.774669i	$0.717916\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1379.92			1.59898			0.799490	−	0.600679i	$-0.205103\pi$
$863$	1379.92			1.59898			0.799490	+	0.600679i	$0.205103\pi$
$864$		0			0
$865$			264.838i			0.306171i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	771.502			0.887805
$870$		0			0
$871$			335.564i			0.385262i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	112.257			0.128294
$876$		0			0
$877$	139.181			0.158701			0.0793507	−	0.996847i	$-0.474715\pi$
$877$	139.181			0.158701			0.0793507	+	0.996847i	$0.474715\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1576.93i			1.78993i	0.446137	+	0.894965i	$0.352799\pi$
$881$			1576.93i			1.78993i	−0.446137	+	0.894965i	$0.647201\pi$
$882$		0			0
$883$	−1387.20			−1.57101			−0.785506	−	0.618854i	$-0.787597\pi$
$883$	−1387.20			−1.57101			−0.785506	+	0.618854i	$0.787597\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1060.90			1.19605			0.598027	−	0.801476i	$-0.295952\pi$
$887$	1060.90			1.19605			0.598027	+	0.801476i	$0.295952\pi$
$888$		0			0
$889$			665.498i			0.748591i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	112.912i		−	0.126441i
$894$		0			0
$895$			207.584i			0.231938i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	205.824			0.228948
$900$		0			0
$901$	−932.630			−1.03511
$902$		0			0
$903$		0			0
$904$		0			0
$905$	274.603			0.303429
$906$		0			0
$907$		−	302.161i		−	0.333143i	−0.986029	−	0.166571i	$-0.946730\pi$
$907$		−	302.161i		−	0.333143i	0.986029	−	0.166571i	$-0.0532697\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	626.574i		−	0.687787i	−0.939009	−	0.343894i	$-0.888254\pi$
$911$		−	626.574i		−	0.687787i	0.939009	−	0.343894i	$-0.111746\pi$
$912$		0			0
$913$	372.546			0.408046
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	1468.72i		−	1.60166i
$918$		0			0
$919$		−	316.798i		−	0.344720i	−0.985034	−	0.172360i	$-0.944861\pi$
$919$		−	316.798i		−	0.344720i	0.985034	−	0.172360i	$-0.0551392\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	63.6720			0.0689837
$924$		0			0
$925$		−	1.63719i		−	0.00176994i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−97.4561			−0.104904			−0.0524522	−	0.998623i	$-0.516704\pi$
$929$	−97.4561			−0.104904			−0.0524522	+	0.998623i	$0.516704\pi$
$930$		0			0
$931$			506.641i			0.544190i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	300.338i		−	0.321218i
$936$		0			0
$937$		−	1083.82i		−	1.15669i	−0.815791	−	0.578347i	$-0.803698\pi$
$937$		−	1083.82i		−	1.15669i	0.815791	−	0.578347i	$-0.196302\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1624.07i			1.72590i	0.505290	+	0.862950i	$0.331386\pi$
$941$			1624.07i			1.72590i	−0.505290	+	0.862950i	$0.668614\pi$
$942$		0			0
$943$	990.222	+	1093.15i	1.05008	+	1.15923i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1141.91			1.20582			0.602909	−	0.797810i	$-0.294008\pi$
$947$	1141.91			1.20582			0.602909	+	0.797810i	$0.294008\pi$
$948$		0			0
$949$	381.239			0.401727
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1555.11i			1.63180i	0.578190	+	0.815902i	$0.303759\pi$
$953$			1555.11i			1.63180i	−0.578190	+	0.815902i	$0.696241\pi$
$954$		0			0
$955$	380.092			0.398002
$956$		0			0
$957$		0			0
$958$		0			0
$959$	897.097			0.935450
$960$		0			0
$961$	−877.402			−0.913010
$962$		0			0
$963$		0			0
$964$		0			0
$965$			69.9947i			0.0725334i
$966$		0			0
$967$	1622.71			1.67808			0.839041	−	0.544068i	$-0.183117\pi$
$967$	1622.71			1.67808			0.839041	+	0.544068i	$0.183117\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	84.1960i		−	0.0867106i	−0.999060	−	0.0433553i	$-0.986195\pi$
$971$		−	84.1960i		−	0.0867106i	0.999060	−	0.0433553i	$-0.0138047\pi$
$972$		0			0
$973$			766.845i			0.788125i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			372.252i			0.381016i	0.981686	+	0.190508i	$0.0610135\pi$
$977$			372.252i			0.381016i	−0.981686	+	0.190508i	$0.938987\pi$
$978$		0			0
$979$	−43.5174			−0.0444509
$980$		0			0
$981$		0			0
$982$		0			0
$983$			1379.58i			1.40344i	0.712451	+	0.701722i	$0.247585\pi$
$983$			1379.58i			1.40344i	−0.712451	+	0.701722i	$0.752415\pi$
$984$		0			0
$985$			186.766i			0.189611i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	729.891	−	661.167i	0.738009	−	0.668521i
$990$		0			0
$991$	894.727			0.902852			0.451426	−	0.892308i	$-0.350915\pi$
$991$	894.727			0.902852			0.451426	+	0.892308i	$0.350915\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	528.453			0.531109
$996$		0			0
$997$	283.922			0.284777			0.142388	−	0.989811i	$-0.454522\pi$
$997$	283.922			0.284777			0.142388	+	0.989811i	$0.454522\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.3.d.c.2161.3		32
3.2	odd	2		1380.3.d.a.781.25	yes	32
23.22	odd	2	inner	4140.3.d.c.2161.30		32
69.68	even	2		1380.3.d.a.781.24	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.3.d.a.781.24	✓	32	69.68	even	2
1380.3.d.a.781.25	yes	32	3.2	odd	2
4140.3.d.c.2161.3		32	1.1	even	1	trivial
4140.3.d.c.2161.30		32	23.22	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4140.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} -10.0406i q^{7} +O(q^{10})\)	\(q-2.23607i q^{5} -10.0406i q^{7} +9.12315i q^{11} -5.62799 q^{13} -14.7225i q^{17} -9.77817i q^{19} +(15.4411 + 17.0462i) q^{23} -5.00000 q^{25} +22.5112 q^{29} +9.14318 q^{31} -22.4514 q^{35} +0.327438i q^{37} +64.1288 q^{41} -42.8185i q^{43} +11.5474 q^{47} -51.8135 q^{49} -63.3474i q^{53} +20.4000 q^{55} +51.5822 q^{59} +46.2561i q^{61} +12.5846i q^{65} -59.6241i q^{67} -11.3134 q^{71} -67.7398 q^{73} +91.6019 q^{77} -84.5653i q^{79} -40.8352i q^{83} -32.9205 q^{85} +4.77000i q^{89} +56.5083i q^{91} -21.8647 q^{95} -41.8758i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q + 24 q^{13} - 64 q^{23} - 160 q^{25} + 60 q^{29} - 4 q^{31} + 60 q^{35} + 108 q^{41} - 136 q^{47} - 428 q^{49} + 120 q^{55} + 84 q^{59} - 188 q^{71} + 472 q^{73} + 120 q^{77} + 60 q^{85} - 80 q^{95}+O(q^{100}) \) 32 * q + 24 * q^13 - 64 * q^23 - 160 * q^25 + 60 * q^29 - 4 * q^31 + 60 * q^35 + 108 * q^41 - 136 * q^47 - 428 * q^49 + 120 * q^55 + 84 * q^59 - 188 * q^71 + 472 * q^73 + 120 * q^77 + 60 * q^85 - 80 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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