LMFDB - Embedded newform 2736.3.o.r.721.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(721,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, 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("2736.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.o (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:	$74.5506003290$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} + 80 x^{18} - 152 x^{17} + 4326 x^{16} - 10096 x^{15} + 70116 x^{14} - 93436 x^{13} + \cdots + 36100 $ x^20 - 4x^19 + 80x^18 - 152x^17 + 4326x^16 - 10096x^15 + 70116x^14 - 93436x^13 + 597327x^12 - 837564x^11 + 2838686x^10 - 2898276x^9 + 7652658x^8 - 6988600x^7 + 12843126x^6 - 6054124x^5 + 7434909x^4 - 621764x^3 + 3099194x^2 - 329080*x + 36100
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{37} $
Twist minimal:	no (minimal twist has level 456)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.7
Root			$-3.49479 - 6.05315i$ of defining polynomial
Character	$\chi$	$=$	2736.721
Dual form			2736.3.o.r.721.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73712 q^{5} -11.2420 q^{7} +O(q^{10})$	$q-2.73712 q^{5} -11.2420 q^{7} -0.310403 q^{11} -13.9860i q^{13} -17.6502 q^{17} +(-17.8456 + 6.52193i) q^{19} -16.4750 q^{23} -17.5082 q^{25} -21.3495i q^{29} +28.1204i q^{31} +30.7708 q^{35} +42.1270i q^{37} -0.885710i q^{41} -53.5279 q^{43} +63.2951 q^{47} +77.3834 q^{49} -101.266i q^{53} +0.849611 q^{55} -41.6822i q^{59} -70.5688 q^{61} +38.2815i q^{65} -6.33916i q^{67} +18.0082i q^{71} +40.3728 q^{73} +3.48956 q^{77} -40.1903i q^{79} -114.546 q^{83} +48.3107 q^{85} -36.3308i q^{89} +157.232i q^{91} +(48.8455 - 17.8513i) q^{95} -36.4451i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 16 q^{7}+O(q^{10}) $ 20 * q + 16 * q^7	$ 20 q + 16 q^{7} + 16 q^{11} - 32 q^{17} - 40 q^{19} + 64 q^{23} + 68 q^{25} - 208 q^{35} - 64 q^{43} + 48 q^{47} + 20 q^{49} + 336 q^{55} + 184 q^{61} + 104 q^{73} - 88 q^{77} + 224 q^{83} - 136 q^{85} - 320 q^{95}+O(q^{100}) $ 20 * q + 16 * q^7 + 16 * q^11 - 32 * q^17 - 40 * q^19 + 64 * q^23 + 68 * q^25 - 208 * q^35 - 64 * q^43 + 48 * q^47 + 20 * q^49 + 336 * q^55 + 184 * q^61 + 104 * q^73 - 88 * q^77 + 224 * q^83 - 136 * q^85 - 320 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	−2.73712			−0.547424			−0.273712	−	0.961812i	$-0.588252\pi$
$5$	−2.73712			−0.547424			−0.273712	+	0.961812i	$0.588252\pi$
$6$		0			0
$7$	−11.2420			−1.60601			−0.803003	−	0.595975i	$-0.796766\pi$
$7$	−11.2420			−1.60601			−0.803003	+	0.595975i	$0.796766\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.310403			−0.0282185			−0.0141092	−	0.999900i	$-0.504491\pi$
$11$	−0.310403			−0.0282185			−0.0141092	+	0.999900i	$0.504491\pi$
$12$		0			0
$13$		−	13.9860i		−	1.07585i	−0.842993	−	0.537924i	$-0.819209\pi$
$13$		−	13.9860i		−	1.07585i	0.842993	−	0.537924i	$-0.180791\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−17.6502			−1.03825			−0.519123	−	0.854700i	$-0.673741\pi$
$17$	−17.6502			−1.03825			−0.519123	+	0.854700i	$0.673741\pi$
$18$		0			0
$19$	−17.8456	+	6.52193i	−0.939241	+	0.343260i
$19$	−17.8456	+	6.52193i	−0.939241	+	0.343260i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−16.4750			−0.716303			−0.358152	−	0.933663i	$-0.616593\pi$
$23$	−16.4750			−0.716303			−0.358152	+	0.933663i	$0.616593\pi$
$24$		0			0
$25$	−17.5082			−0.700327
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	21.3495i		−	0.736191i	−0.929788	−	0.368096i	$-0.880010\pi$
$29$		−	21.3495i		−	0.736191i	0.929788	−	0.368096i	$-0.119990\pi$
$30$		0			0
$31$			28.1204i			0.907111i	0.891228	+	0.453555i	$0.149845\pi$
$31$			28.1204i			0.907111i	−0.891228	+	0.453555i	$0.850155\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	30.7708			0.879166
$36$		0			0
$37$			42.1270i			1.13857i	0.822141	+	0.569284i	$0.192780\pi$
$37$			42.1270i			1.13857i	−0.822141	+	0.569284i	$0.807220\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	0.885710i		−	0.0216027i	−0.999942	−	0.0108013i	$-0.996562\pi$
$41$		−	0.885710i		−	0.0216027i	0.999942	−	0.0108013i	$-0.00343824\pi$
$42$		0			0
$43$	−53.5279			−1.24483			−0.622417	−	0.782685i	$-0.713849\pi$
$43$	−53.5279			−1.24483			−0.622417	+	0.782685i	$0.713849\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	63.2951			1.34670			0.673352	−	0.739322i	$-0.264854\pi$
$47$	63.2951			1.34670			0.673352	+	0.739322i	$0.264854\pi$
$48$		0			0
$49$	77.3834			1.57925
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	101.266i		−	1.91067i	−0.295520	−	0.955337i	$-0.595493\pi$
$53$		−	101.266i		−	1.91067i	0.295520	−	0.955337i	$-0.404507\pi$
$54$		0			0
$55$	0.849611			0.0154475
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	41.6822i		−	0.706478i	−0.935533	−	0.353239i	$-0.885080\pi$
$59$		−	41.6822i		−	0.706478i	0.935533	−	0.353239i	$-0.114920\pi$
$60$		0			0
$61$	−70.5688			−1.15687			−0.578433	−	0.815730i	$-0.696336\pi$
$61$	−70.5688			−1.15687			−0.578433	+	0.815730i	$0.696336\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			38.2815i			0.588946i
$66$		0			0
$67$		−	6.33916i		−	0.0946144i	−0.998880	−	0.0473072i	$-0.984936\pi$
$67$		−	6.33916i		−	0.0946144i	0.998880	−	0.0473072i	$-0.0150640\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			18.0082i			0.253637i	0.991926	+	0.126818i	$0.0404765\pi$
$71$			18.0082i			0.253637i	−0.991926	+	0.126818i	$0.959523\pi$
$72$		0			0
$73$	40.3728			0.553052			0.276526	−	0.961006i	$-0.410817\pi$
$73$	40.3728			0.553052			0.276526	+	0.961006i	$0.410817\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.48956			0.0453190
$78$		0			0
$79$		−	40.1903i		−	0.508737i	−0.967107	−	0.254369i	$-0.918132\pi$
$79$		−	40.1903i		−	0.508737i	0.967107	−	0.254369i	$-0.0818677\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−114.546			−1.38008			−0.690038	−	0.723773i	$-0.742406\pi$
$83$	−114.546			−1.38008			−0.690038	+	0.723773i	$0.742406\pi$
$84$		0			0
$85$	48.3107			0.568361
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	36.3308i		−	0.408211i	−0.978949	−	0.204106i	$-0.934571\pi$
$89$		−	36.3308i		−	0.408211i	0.978949	−	0.204106i	$-0.0654286\pi$
$90$		0			0
$91$			157.232i			1.72782i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	48.8455	−	17.8513i	0.514163	−	0.187909i
$96$		0			0
$97$		−	36.4451i		−	0.375723i	−0.982196	−	0.187861i	$-0.939844\pi$
$97$		−	36.4451i		−	0.375723i	0.982196	−	0.187861i	$-0.0601556\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	154.340			1.52812			0.764058	−	0.645148i	$-0.223204\pi$
$101$	154.340			1.52812			0.764058	+	0.645148i	$0.223204\pi$
$102$		0			0
$103$			151.897i			1.47473i	0.675494	+	0.737365i	$0.263930\pi$
$103$			151.897i			1.47473i	−0.675494	+	0.737365i	$0.736070\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			110.097i			1.02894i	0.857507	+	0.514472i	$0.172012\pi$
$107$			110.097i			1.02894i	−0.857507	+	0.514472i	$0.827988\pi$
$108$		0			0
$109$		−	122.349i		−	1.12247i	−0.827658	−	0.561233i	$-0.810327\pi$
$109$		−	122.349i		−	1.12247i	0.827658	−	0.561233i	$-0.189673\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	101.324i		−	0.896674i	−0.893865	−	0.448337i	$-0.852016\pi$
$113$		−	101.324i		−	0.896674i	0.893865	−	0.448337i	$-0.147984\pi$
$114$		0			0
$115$	45.0940			0.392122
$116$		0			0
$117$		0			0
$118$		0			0
$119$	198.424			1.66743
$120$		0			0
$121$	−120.904			−0.999204
$122$		0			0
$123$		0			0
$124$		0			0
$125$	116.350			0.930800
$126$		0			0
$127$			171.079i			1.34708i	0.739152	+	0.673538i	$0.235226\pi$
$127$			171.079i			1.34708i	−0.739152	+	0.673538i	$0.764774\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−17.9514			−0.137033			−0.0685167	−	0.997650i	$-0.521827\pi$
$131$	−17.9514			−0.137033			−0.0685167	+	0.997650i	$0.521827\pi$
$132$		0			0
$133$	200.621	−	73.3198i	1.50843	−	0.551277i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	20.7429			0.151408			0.0757039	−	0.997130i	$-0.475880\pi$
$137$	20.7429			0.151408			0.0757039	+	0.997130i	$0.475880\pi$
$138$		0			0
$139$	89.4222			0.643325			0.321662	−	0.946854i	$-0.395758\pi$
$139$	89.4222			0.643325			0.321662	+	0.946854i	$0.395758\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			4.34131i			0.0303588i
$144$		0			0
$145$			58.4363i			0.403009i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	231.949			1.55670			0.778352	−	0.627828i	$-0.216056\pi$
$149$	231.949			1.55670			0.778352	+	0.627828i	$0.216056\pi$
$150$		0			0
$151$			278.606i			1.84507i	0.385913	+	0.922535i	$0.373886\pi$
$151$			278.606i			1.84507i	−0.385913	+	0.922535i	$0.626114\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	76.9690i		−	0.496574i
$156$		0			0
$157$	47.3989			0.301904			0.150952	−	0.988541i	$-0.451766\pi$
$157$	47.3989			0.301904			0.150952	+	0.988541i	$0.451766\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	185.212			1.15039
$162$		0			0
$163$	64.5557			0.396047			0.198024	−	0.980197i	$-0.436548\pi$
$163$	64.5557			0.396047			0.198024	+	0.980197i	$0.436548\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	38.2586i		−	0.229093i	−0.993418	−	0.114547i	$-0.963458\pi$
$167$		−	38.2586i		−	0.229093i	0.993418	−	0.114547i	$-0.0365416\pi$
$168$		0			0
$169$	−26.6092			−0.157451
$170$		0			0
$171$		0			0
$172$		0			0
$173$			322.248i			1.86270i	0.364120	+	0.931352i	$0.381370\pi$
$173$			322.248i			1.86270i	−0.364120	+	0.931352i	$0.618630\pi$
$174$		0			0
$175$	196.828			1.12473
$176$		0			0
$177$		0			0
$178$		0			0
$179$			291.809i			1.63022i	0.579309	+	0.815108i	$0.303322\pi$
$179$			291.809i			1.63022i	−0.579309	+	0.815108i	$0.696678\pi$
$180$		0			0
$181$			88.2061i			0.487327i	0.969860	+	0.243663i	$0.0783492\pi$
$181$			88.2061i			0.487327i	−0.969860	+	0.243663i	$0.921651\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	115.307i		−	0.623280i
$186$		0			0
$187$	5.47867			0.0292977
$188$		0			0
$189$		0			0
$190$		0			0
$191$	101.238			0.530042			0.265021	−	0.964243i	$-0.414621\pi$
$191$	101.238			0.530042			0.265021	+	0.964243i	$0.414621\pi$
$192$		0			0
$193$			130.303i			0.675145i	0.941300	+	0.337573i	$0.109606\pi$
$193$			130.303i			0.675145i	−0.941300	+	0.337573i	$0.890394\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	137.621			0.698582			0.349291	−	0.937014i	$-0.386422\pi$
$197$	137.621			0.698582			0.349291	+	0.937014i	$0.386422\pi$
$198$		0			0
$199$	−13.0530			−0.0655931			−0.0327965	−	0.999462i	$-0.510441\pi$
$199$	−13.0530			−0.0655931			−0.0327965	+	0.999462i	$0.510441\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			240.012i			1.18233i
$204$		0			0
$205$			2.42429i			0.0118258i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.53932	−	2.02443i	0.0265039	−	0.00968626i
$210$		0			0
$211$		−	337.082i		−	1.59755i	−0.601632	−	0.798773i	$-0.705483\pi$
$211$		−	337.082i		−	1.59755i	0.601632	−	0.798773i	$-0.294517\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	146.512			0.681453
$216$		0			0
$217$		−	316.131i		−	1.45682i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			246.856i			1.11700i
$222$		0			0
$223$		−	318.785i		−	1.42953i	−0.699364	−	0.714765i	$-0.746533\pi$
$223$		−	318.785i		−	1.42953i	0.699364	−	0.714765i	$-0.253467\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	208.298i		−	0.917612i	−0.888537	−	0.458806i	$-0.848277\pi$
$227$		−	208.298i		−	0.917612i	0.888537	−	0.458806i	$-0.151723\pi$
$228$		0			0
$229$	222.774			0.972813			0.486407	−	0.873733i	$-0.338307\pi$
$229$	222.774			0.972813			0.486407	+	0.873733i	$0.338307\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−194.289			−0.833857			−0.416928	−	0.908939i	$-0.636893\pi$
$233$	−194.289			−0.833857			−0.416928	+	0.908939i	$0.636893\pi$
$234$		0			0
$235$	−173.246			−0.737218
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−28.6579			−0.119907			−0.0599537	−	0.998201i	$-0.519095\pi$
$239$	−28.6579			−0.119907			−0.0599537	+	0.998201i	$0.519095\pi$
$240$		0			0
$241$		−	406.028i		−	1.68476i	−0.538880	−	0.842382i	$-0.681152\pi$
$241$		−	406.028i		−	1.68476i	0.538880	−	0.842382i	$-0.318848\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−211.808			−0.864521
$246$		0			0
$247$	91.2160	+	249.589i	0.369296	+	1.01048i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−164.500			−0.655380			−0.327690	−	0.944785i	$-0.606270\pi$
$251$	−164.500			−0.655380			−0.327690	+	0.944785i	$0.606270\pi$
$252$		0			0
$253$	5.11388			0.0202130
$254$		0			0
$255$		0			0
$256$		0			0
$257$			102.591i			0.399187i	0.979879	+	0.199594i	$0.0639622\pi$
$257$			102.591i			0.399187i	−0.979879	+	0.199594i	$0.936038\pi$
$258$		0			0
$259$		−	473.594i		−	1.82855i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	150.632			0.572744			0.286372	−	0.958119i	$-0.407551\pi$
$263$	150.632			0.572744			0.286372	+	0.958119i	$0.407551\pi$
$264$		0			0
$265$			277.176i			1.04595i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	111.844i		−	0.415776i	−0.978153	−	0.207888i	$-0.933341\pi$
$269$		−	111.844i		−	0.415776i	0.978153	−	0.207888i	$-0.0666590\pi$
$270$		0			0
$271$	−19.5406			−0.0721055			−0.0360528	−	0.999350i	$-0.511478\pi$
$271$	−19.5406			−0.0721055			−0.0360528	+	0.999350i	$0.511478\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.43459			0.0197621
$276$		0			0
$277$	−340.777			−1.23024			−0.615121	−	0.788433i	$-0.710893\pi$
$277$	−340.777			−1.23024			−0.615121	+	0.788433i	$0.710893\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			21.3074i			0.0758269i	0.999281	+	0.0379135i	$0.0120711\pi$
$281$			21.3074i			0.0758269i	−0.999281	+	0.0379135i	$0.987929\pi$
$282$		0			0
$283$	137.592			0.486190			0.243095	−	0.970002i	$-0.421837\pi$
$283$	137.592			0.486190			0.243095	+	0.970002i	$0.421837\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			9.95718i			0.0346940i
$288$		0			0
$289$	22.5286			0.0779537
$290$		0			0
$291$		0			0
$292$		0			0
$293$			325.749i			1.11177i	0.831259	+	0.555886i	$0.187621\pi$
$293$			325.749i			1.11177i	−0.831259	+	0.555886i	$0.812379\pi$
$294$		0			0
$295$			114.089i			0.386743i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			230.420i			0.770634i
$300$		0			0
$301$	601.763			1.99921
$302$		0			0
$303$		0			0
$304$		0			0
$305$	193.155			0.633296
$306$		0			0
$307$			2.35033i			0.00765580i	0.999993	+	0.00382790i	$0.00121846\pi$
$307$			2.35033i			0.00765580i	−0.999993	+	0.00382790i	$0.998782\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	449.753			1.44615			0.723076	−	0.690768i	$-0.242728\pi$
$311$	449.753			1.44615			0.723076	+	0.690768i	$0.242728\pi$
$312$		0			0
$313$	−157.341			−0.502686			−0.251343	−	0.967898i	$-0.580872\pi$
$313$	−157.341			−0.502686			−0.251343	+	0.967898i	$0.580872\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	460.983i		−	1.45420i	−0.686529	−	0.727102i	$-0.740867\pi$
$317$		−	460.983i		−	1.45420i	0.686529	−	0.727102i	$-0.259133\pi$
$318$		0			0
$319$			6.62696i			0.0207742i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	314.977	−	115.113i	0.975162	−	0.356388i
$324$		0			0
$325$			244.870i			0.753446i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−711.566			−2.16281
$330$		0			0
$331$			109.605i			0.331134i	0.986199	+	0.165567i	$0.0529455\pi$
$331$			109.605i			0.331134i	−0.986199	+	0.165567i	$0.947055\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			17.3511i			0.0517942i
$336$		0			0
$337$			176.867i			0.524829i	0.964955	+	0.262414i	$0.0845187\pi$
$337$			176.867i			0.524829i	−0.964955	+	0.262414i	$0.915481\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	8.72867i		−	0.0255973i
$342$		0			0
$343$	−319.087			−0.930283
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−302.378			−0.871406			−0.435703	−	0.900091i	$-0.643500\pi$
$347$	−302.378			−0.871406			−0.435703	+	0.900091i	$0.643500\pi$
$348$		0			0
$349$	461.182			1.32144			0.660720	−	0.750633i	$-0.270251\pi$
$349$	461.182			1.32144			0.660720	+	0.750633i	$0.270251\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	301.447			0.853959			0.426979	−	0.904261i	$-0.359578\pi$
$353$	301.447			0.853959			0.426979	+	0.904261i	$0.359578\pi$
$354$		0			0
$355$		−	49.2906i		−	0.138847i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−376.519			−1.04880			−0.524400	−	0.851472i	$-0.675710\pi$
$359$	−376.519			−1.04880			−0.524400	+	0.851472i	$0.675710\pi$
$360$		0			0
$361$	275.929	−	232.775i	0.764346	−	0.644807i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−110.505			−0.302754
$366$		0			0
$367$	−726.293			−1.97900			−0.989500	−	0.144534i	$-0.953832\pi$
$367$	−726.293			−1.97900			−0.989500	+	0.144534i	$0.953832\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			1138.43i			3.06855i
$372$		0			0
$373$			30.9287i			0.0829188i	0.999140	+	0.0414594i	$0.0132007\pi$
$373$			30.9287i			0.0829188i	−0.999140	+	0.0414594i	$0.986799\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−298.595			−0.792030
$378$		0			0
$379$		−	557.441i		−	1.47082i	−0.677622	−	0.735411i	$-0.736989\pi$
$379$		−	557.441i		−	1.47082i	0.677622	−	0.735411i	$-0.263011\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	193.524i		−	0.505285i	−0.967560	−	0.252643i	$-0.918700\pi$
$383$		−	193.524i		−	0.505285i	0.967560	−	0.252643i	$-0.0812997\pi$
$384$		0			0
$385$	−9.55135			−0.0248087
$386$		0			0
$387$		0			0
$388$		0			0
$389$	524.738			1.34894			0.674471	−	0.738302i	$-0.264372\pi$
$389$	524.738			1.34894			0.674471	+	0.738302i	$0.264372\pi$
$390$		0			0
$391$	290.786			0.743699
$392$		0			0
$393$		0			0
$394$		0			0
$395$			110.006i			0.278495i
$396$		0			0
$397$	−488.227			−1.22979			−0.614896	−	0.788608i	$-0.710802\pi$
$397$	−488.227			−1.22979			−0.614896	+	0.788608i	$0.710802\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			524.946i			1.30909i	0.756022	+	0.654546i	$0.227140\pi$
$401$			524.946i			1.30909i	−0.756022	+	0.654546i	$0.772860\pi$
$402$		0			0
$403$	393.293			0.975914
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	13.0764i		−	0.0321286i
$408$		0			0
$409$		−	58.6328i		−	0.143357i	−0.997428	−	0.0716783i	$-0.977165\pi$
$409$		−	58.6328i		−	0.143357i	0.997428	−	0.0716783i	$-0.0228355\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			468.593i			1.13461i
$414$		0			0
$415$	313.527			0.755487
$416$		0			0
$417$		0			0
$418$		0			0
$419$	603.597			1.44057			0.720283	−	0.693680i	$-0.244012\pi$
$419$	603.597			1.44057			0.720283	+	0.693680i	$0.244012\pi$
$420$		0			0
$421$			501.791i			1.19190i	0.803020	+	0.595951i	$0.203225\pi$
$421$			501.791i			1.19190i	−0.803020	+	0.595951i	$0.796775\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	309.022			0.727111
$426$		0			0
$427$	793.337			1.85793
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	735.726i		−	1.70702i	−0.521076	−	0.853510i	$-0.674469\pi$
$431$		−	735.726i		−	1.70702i	0.521076	−	0.853510i	$-0.325531\pi$
$432$		0			0
$433$			208.030i			0.480439i	0.970719	+	0.240219i	$0.0772194\pi$
$433$			208.030i			0.480439i	−0.970719	+	0.240219i	$0.922781\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	294.005	−	107.449i	0.672781	−	0.245878i
$438$		0			0
$439$		−	384.593i		−	0.876066i	−0.898959	−	0.438033i	$-0.855675\pi$
$439$		−	384.593i		−	0.876066i	0.898959	−	0.438033i	$-0.144325\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	513.920			1.16009			0.580045	−	0.814584i	$-0.303035\pi$
$443$	513.920			1.16009			0.580045	+	0.814584i	$0.303035\pi$
$444$		0			0
$445$			99.4417i			0.223465i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			604.857i			1.34712i	0.739132	+	0.673560i	$0.235236\pi$
$449$			604.857i			1.34712i	−0.739132	+	0.673560i	$0.764764\pi$
$450$		0			0
$451$			0.274927i			0.000609594i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	430.362i		−	0.945850i
$456$		0			0
$457$	−694.667			−1.52006			−0.760029	−	0.649889i	$-0.774815\pi$
$457$	−694.667			−1.52006			−0.760029	+	0.649889i	$0.774815\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−51.7190			−0.112189			−0.0560944	−	0.998425i	$-0.517865\pi$
$461$	−51.7190			−0.112189			−0.0560944	+	0.998425i	$0.517865\pi$
$462$		0			0
$463$	541.836			1.17027			0.585136	−	0.810935i	$-0.301041\pi$
$463$	541.836			1.17027			0.585136	+	0.810935i	$0.301041\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−104.192			−0.223110			−0.111555	−	0.993758i	$-0.535583\pi$
$467$	−104.192			−0.223110			−0.111555	+	0.993758i	$0.535583\pi$
$468$		0			0
$469$			71.2651i			0.151951i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	16.6152			0.0351273
$474$		0			0
$475$	312.443	−	114.187i	0.657775	−	0.240394i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−633.822			−1.32322			−0.661609	−	0.749849i	$-0.730126\pi$
$479$	−633.822			−1.32322			−0.661609	+	0.749849i	$0.730126\pi$
$480$		0			0
$481$	589.190			1.22493
$482$		0			0
$483$		0			0
$484$		0			0
$485$			99.7546i			0.205680i
$486$		0			0
$487$		−	505.544i		−	1.03808i	−0.854750	−	0.519039i	$-0.826290\pi$
$487$		−	505.544i		−	1.03808i	0.854750	−	0.519039i	$-0.173710\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	730.747			1.48828			0.744141	−	0.668022i	$-0.232859\pi$
$491$	730.747			1.48828			0.744141	+	0.668022i	$0.232859\pi$
$492$		0			0
$493$			376.823i			0.764347i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	202.449i		−	0.407342i
$498$		0			0
$499$	−371.043			−0.743574			−0.371787	−	0.928318i	$-0.621255\pi$
$499$	−371.043			−0.743574			−0.371787	+	0.928318i	$0.621255\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−293.762			−0.584020			−0.292010	−	0.956415i	$-0.594324\pi$
$503$	−293.762			−0.584020			−0.292010	+	0.956415i	$0.594324\pi$
$504$		0			0
$505$	−422.446			−0.836527
$506$		0			0
$507$		0			0
$508$		0			0
$509$			512.759i			1.00739i	0.863883	+	0.503693i	$0.168026\pi$
$509$			512.759i			1.00739i	−0.863883	+	0.503693i	$0.831974\pi$
$510$		0			0
$511$	−453.873			−0.888205
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	415.761i		−	0.807303i
$516$		0			0
$517$	−19.6470			−0.0380019
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	681.610i		−	1.30827i	−0.756377	−	0.654136i	$-0.773032\pi$
$521$		−	681.610i		−	1.30827i	0.756377	−	0.654136i	$-0.226968\pi$
$522$		0			0
$523$		−	582.249i		−	1.11329i	−0.830752	−	0.556643i	$-0.812089\pi$
$523$		−	582.249i		−	1.11329i	0.830752	−	0.556643i	$-0.187911\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	496.331i		−	0.941804i
$528$		0			0
$529$	−257.575			−0.486910
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−12.3876			−0.0232412
$534$		0			0
$535$		−	301.349i		−	0.563269i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−24.0200			−0.0445641
$540$		0			0
$541$	−503.596			−0.930861			−0.465431	−	0.885084i	$-0.654100\pi$
$541$	−503.596			−0.930861			−0.465431	+	0.885084i	$0.654100\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			334.884i			0.614465i
$546$		0			0
$547$			773.633i			1.41432i	0.707053	+	0.707160i	$0.250024\pi$
$547$			773.633i			1.41432i	−0.707053	+	0.707160i	$0.749976\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	139.240	+	380.995i	0.252705	+	0.691460i
$552$		0			0
$553$			451.820i			0.817035i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−155.918			−0.279925			−0.139962	−	0.990157i	$-0.544698\pi$
$557$	−155.918			−0.279925			−0.139962	+	0.990157i	$0.544698\pi$
$558$		0			0
$559$			748.643i			1.33925i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	848.915i		−	1.50784i	−0.656965	−	0.753921i	$-0.728160\pi$
$563$		−	848.915i		−	1.50784i	0.656965	−	0.753921i	$-0.271840\pi$
$564$		0			0
$565$			277.336i			0.490861i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	578.479i		−	1.01666i	−0.861163	−	0.508329i	$-0.830263\pi$
$569$		−	578.479i		−	1.01666i	0.861163	−	0.508329i	$-0.169737\pi$
$570$		0			0
$571$	−68.8352			−0.120552			−0.0602760	−	0.998182i	$-0.519198\pi$
$571$	−68.8352			−0.120552			−0.0602760	+	0.998182i	$0.519198\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	288.447			0.501646
$576$		0			0
$577$	−789.827			−1.36885			−0.684425	−	0.729083i	$-0.739947\pi$
$577$	−789.827			−1.36885			−0.684425	+	0.729083i	$0.739947\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1287.73			2.21641
$582$		0			0
$583$			31.4332i			0.0539163i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−795.144			−1.35459			−0.677294	−	0.735712i	$-0.736848\pi$
$587$	−795.144			−1.35459			−0.677294	+	0.735712i	$0.736848\pi$
$588$		0			0
$589$	−183.400	−	501.825i	−0.311375	−	0.851995i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−392.928			−0.662610			−0.331305	−	0.943524i	$-0.607489\pi$
$593$	−392.928			−0.662610			−0.331305	+	0.943524i	$0.607489\pi$
$594$		0			0
$595$	−543.110			−0.912790
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	466.332i		−	0.778518i	−0.921128	−	0.389259i	$-0.872731\pi$
$599$		−	466.332i		−	0.778518i	0.921128	−	0.389259i	$-0.127269\pi$
$600$		0			0
$601$			992.020i			1.65062i	0.564683	+	0.825308i	$0.308998\pi$
$601$			992.020i			1.65062i	−0.564683	+	0.825308i	$0.691002\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	330.928			0.546988
$606$		0			0
$607$			879.411i			1.44878i	0.689389	+	0.724391i	$0.257879\pi$
$607$			879.411i			1.44878i	−0.689389	+	0.724391i	$0.742121\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	885.247i		−	1.44885i
$612$		0			0
$613$	309.791			0.505368			0.252684	−	0.967549i	$-0.418687\pi$
$613$	309.791			0.505368			0.252684	+	0.967549i	$0.418687\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	477.444			0.773816			0.386908	−	0.922118i	$-0.373543\pi$
$617$	477.444			0.773816			0.386908	+	0.922118i	$0.373543\pi$
$618$		0			0
$619$	519.736			0.839638			0.419819	−	0.907608i	$-0.362094\pi$
$619$	519.736			0.839638			0.419819	+	0.907608i	$0.362094\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			408.432i			0.655589i
$624$		0			0
$625$	119.240			0.190785
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	743.549i		−	1.18211i
$630$		0			0
$631$	158.408			0.251043			0.125522	−	0.992091i	$-0.459940\pi$
$631$	158.408			0.251043			0.125522	+	0.992091i	$0.459940\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	468.263i		−	0.737422i
$636$		0			0
$637$		−	1082.29i		−	1.69904i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	997.278i		−	1.55582i	−0.628378	−	0.777908i	$-0.716281\pi$
$641$		−	997.278i		−	1.55582i	0.628378	−	0.777908i	$-0.283719\pi$
$642$		0			0
$643$	−317.649			−0.494010			−0.247005	−	0.969014i	$-0.579446\pi$
$643$	−317.649			−0.494010			−0.247005	+	0.969014i	$0.579446\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−715.136			−1.10531			−0.552655	−	0.833410i	$-0.686385\pi$
$647$	−715.136			−1.10531			−0.552655	+	0.833410i	$0.686385\pi$
$648$		0			0
$649$			12.9383i			0.0199357i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	92.8522			0.142193			0.0710966	−	0.997469i	$-0.477350\pi$
$653$	92.8522			0.142193			0.0710966	+	0.997469i	$0.477350\pi$
$654$		0			0
$655$	49.1351			0.0750154
$656$		0			0
$657$		0			0
$658$		0			0
$659$			1192.93i			1.81021i	0.425190	+	0.905104i	$0.360207\pi$
$659$			1192.93i			1.81021i	−0.425190	+	0.905104i	$0.639793\pi$
$660$		0			0
$661$			106.919i			0.161754i	0.996724	+	0.0808770i	$0.0257721\pi$
$661$			106.919i			0.161754i	−0.996724	+	0.0808770i	$0.974228\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−549.123	+	200.685i	−0.825748	+	0.301782i
$666$		0			0
$667$			351.733i			0.527336i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	21.9048			0.0326450
$672$		0			0
$673$			916.053i			1.36115i	0.732679	+	0.680574i	$0.238270\pi$
$673$			916.053i			1.36115i	−0.732679	+	0.680574i	$0.761730\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			88.0723i			0.130092i	0.997882	+	0.0650460i	$0.0207194\pi$
$677$			88.0723i			0.130092i	−0.997882	+	0.0650460i	$0.979281\pi$
$678$		0			0
$679$			409.717i			0.603413i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			795.552i			1.16479i	0.812906	+	0.582395i	$0.197884\pi$
$683$			795.552i			1.16479i	−0.812906	+	0.582395i	$0.802116\pi$
$684$		0			0
$685$	−56.7758			−0.0828843
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1416.31			−2.05560
$690$		0			0
$691$	822.997			1.19102			0.595511	−	0.803347i	$-0.296950\pi$
$691$	822.997			1.19102			0.595511	+	0.803347i	$0.296950\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−244.759			−0.352172
$696$		0			0
$697$			15.6329i			0.0224289i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1122.62			−1.60146			−0.800728	−	0.599028i	$-0.795554\pi$
$701$	−1122.62			−1.60146			−0.800728	+	0.599028i	$0.795554\pi$
$702$		0			0
$703$	−274.750	−	751.781i	−0.390825	−	1.06939i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1735.09			−2.45416
$708$		0			0
$709$	622.483			0.877973			0.438986	−	0.898494i	$-0.355338\pi$
$709$	622.483			0.877973			0.438986	+	0.898494i	$0.355338\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	463.284i		−	0.649767i
$714$		0			0
$715$		−	11.8827i		−	0.0166191i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	565.183			0.786068			0.393034	−	0.919524i	$-0.371426\pi$
$719$	565.183			0.786068			0.393034	+	0.919524i	$0.371426\pi$
$720$		0			0
$721$		−	1707.63i		−	2.36842i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			373.791i			0.515574i
$726$		0			0
$727$	−1088.05			−1.49663			−0.748316	−	0.663342i	$-0.769137\pi$
$727$	−1088.05			−1.49663			−0.748316	+	0.663342i	$0.769137\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	944.777			1.29244
$732$		0			0
$733$	48.2593			0.0658380			0.0329190	−	0.999458i	$-0.489520\pi$
$733$	48.2593			0.0658380			0.0329190	+	0.999458i	$0.489520\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1.96770i			0.00266987i
$738$		0			0
$739$	−1222.53			−1.65430			−0.827150	−	0.561981i	$-0.810039\pi$
$739$	−1222.53			−1.65430			−0.827150	+	0.561981i	$0.810039\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			208.838i			0.281074i	0.990075	+	0.140537i	$0.0448829\pi$
$743$			208.838i			0.281074i	−0.990075	+	0.140537i	$0.955117\pi$
$744$		0			0
$745$	−634.872			−0.852177
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	1237.71i		−	1.65249i
$750$		0			0
$751$			570.887i			0.760169i	0.924952	+	0.380084i	$0.124105\pi$
$751$			570.887i			0.760169i	−0.924952	+	0.380084i	$0.875895\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	762.577i		−	1.01004i
$756$		0			0
$757$	870.524			1.14997			0.574983	−	0.818165i	$-0.305009\pi$
$757$	870.524			1.14997			0.574983	+	0.818165i	$0.305009\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1334.91			−1.75415			−0.877073	−	0.480357i	$-0.840507\pi$
$761$	−1334.91			−1.75415			−0.877073	+	0.480357i	$0.840507\pi$
$762$		0			0
$763$			1375.45i			1.80269i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−582.968			−0.760063
$768$		0			0
$769$	−44.4989			−0.0578660			−0.0289330	−	0.999581i	$-0.509211\pi$
$769$	−44.4989			−0.0578660			−0.0289330	+	0.999581i	$0.509211\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			207.598i			0.268561i	0.990943	+	0.134281i	$0.0428724\pi$
$773$			207.598i			0.268561i	−0.990943	+	0.134281i	$0.957128\pi$
$774$		0			0
$775$		−	492.337i		−	0.635274i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.77654	+	15.8060i	0.00741533	+	0.0202901i
$780$		0			0
$781$		−	5.58980i		−	0.00715723i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−129.737			−0.165270
$786$		0			0
$787$		−	608.480i		−	0.773164i	−0.922255	−	0.386582i	$-0.873656\pi$
$787$		−	608.480i		−	0.773164i	0.922255	−	0.386582i	$-0.126344\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1139.09i			1.44006i
$792$		0			0
$793$			986.978i			1.24461i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	280.571i		−	0.352034i	−0.984387	−	0.176017i	$-0.943679\pi$
$797$		−	280.571i		−	0.352034i	0.984387	−	0.176017i	$-0.0563214\pi$
$798$		0			0
$799$	−1117.17			−1.39821
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−12.5318			−0.0156063
$804$		0			0
$805$	−506.948			−0.629750
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−593.982			−0.734218			−0.367109	−	0.930178i	$-0.619652\pi$
$809$	−593.982			−0.734218			−0.367109	+	0.930178i	$0.619652\pi$
$810$		0			0
$811$		−	265.930i		−	0.327904i	−0.986468	−	0.163952i	$-0.947576\pi$
$811$		−	265.930i		−	0.327904i	0.986468	−	0.163952i	$-0.0524243\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−176.697			−0.216806
$816$		0			0
$817$	955.236	−	349.105i	1.16920	−	0.427302i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1313.39			1.59975			0.799873	−	0.600169i	$-0.204900\pi$
$821$	1313.39			1.59975			0.799873	+	0.600169i	$0.204900\pi$
$822$		0			0
$823$	−131.486			−0.159764			−0.0798821	−	0.996804i	$-0.525454\pi$
$823$	−131.486			−0.159764			−0.0798821	+	0.996804i	$0.525454\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	6.29770i		−	0.00761511i	−0.999993	−	0.00380756i	$-0.998788\pi$
$827$		−	6.29770i		−	0.00761511i	0.999993	−	0.00380756i	$-0.00121199\pi$
$828$		0			0
$829$			27.4439i			0.0331048i	0.999863	+	0.0165524i	$0.00526903\pi$
$829$			27.4439i			0.0331048i	−0.999863	+	0.0165524i	$0.994731\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1365.83			−1.63965
$834$		0			0
$835$			104.718i			0.125411i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	252.647i		−	0.301129i	−0.988600	−	0.150565i	$-0.951891\pi$
$839$		−	252.647i		−	0.301129i	0.988600	−	0.150565i	$-0.0481091\pi$
$840$		0			0
$841$	385.197			0.458023
$842$		0			0
$843$		0			0
$844$		0			0
$845$	72.8325			0.0861923
$846$		0			0
$847$	1359.20			1.60473
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	694.042i		−	0.815560i
$852$		0			0
$853$	−222.513			−0.260859			−0.130429	−	0.991458i	$-0.541636\pi$
$853$	−222.513			−0.260859			−0.130429	+	0.991458i	$0.541636\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	61.3955i		−	0.0716401i	−0.999358	−	0.0358200i	$-0.988596\pi$
$857$		−	61.3955i		−	0.0716401i	0.999358	−	0.0358200i	$-0.0114043\pi$
$858$		0			0
$859$	−217.863			−0.253624			−0.126812	−	0.991927i	$-0.540474\pi$
$859$	−217.863			−0.253624			−0.126812	+	0.991927i	$0.540474\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1687.15i		−	1.95498i	−0.210974	−	0.977492i	$-0.567663\pi$
$863$		−	1687.15i		−	1.95498i	0.210974	−	0.977492i	$-0.432337\pi$
$864$		0			0
$865$		−	882.031i		−	1.01969i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			12.4752i			0.0143558i
$870$		0			0
$871$	−88.6598			−0.101791
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1308.01			−1.49487
$876$		0			0
$877$			170.030i			0.193877i	0.995290	+	0.0969383i	$0.0309049\pi$
$877$			170.030i			0.193877i	−0.995290	+	0.0969383i	$0.969095\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1242.99			1.41089			0.705445	−	0.708765i	$-0.250747\pi$
$881$	1242.99			1.41089			0.705445	+	0.708765i	$0.250747\pi$
$882$		0			0
$883$	−1056.62			−1.19663			−0.598314	−	0.801262i	$-0.704163\pi$
$883$	−1056.62			−1.19663			−0.598314	+	0.801262i	$0.704163\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	266.107i		−	0.300008i	−0.988685	−	0.150004i	$-0.952071\pi$
$887$		−	266.107i		−	0.300008i	0.988685	−	0.150004i	$-0.0479287\pi$
$888$		0			0
$889$		−	1923.27i		−	2.16341i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1129.54	+	412.806i	−1.26488	+	0.462269i
$894$		0			0
$895$		−	798.716i		−	0.892420i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	600.358			0.667807
$900$		0			0
$901$			1787.36i			1.98375i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	241.431i		−	0.266774i
$906$		0			0
$907$		−	373.578i		−	0.411883i	−0.978564	−	0.205942i	$-0.933974\pi$
$907$		−	373.578i		−	0.411883i	0.978564	−	0.205942i	$-0.0660257\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1268.23i			1.39212i	0.717981	+	0.696062i	$0.245066\pi$
$911$			1268.23i			1.39212i	−0.717981	+	0.696062i	$0.754934\pi$
$912$		0			0
$913$	35.5555			0.0389436
$914$		0			0
$915$		0			0
$916$		0			0
$917$	201.810			0.220076
$918$		0			0
$919$	535.759			0.582980			0.291490	−	0.956574i	$-0.405849\pi$
$919$	535.759			0.582980			0.291490	+	0.956574i	$0.405849\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	251.863			0.272875
$924$		0			0
$925$		−	737.567i		−	0.797370i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1634.82			−1.75977			−0.879883	−	0.475190i	$-0.842379\pi$
$929$	−1634.82			−1.75977			−0.879883	+	0.475190i	$0.842379\pi$
$930$		0			0
$931$	−1380.95	+	504.689i	−1.48330	+	0.542094i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−14.9958			−0.0160383
$936$		0			0
$937$	−212.861			−0.227173			−0.113586	−	0.993528i	$-0.536234\pi$
$937$	−212.861			−0.227173			−0.113586	+	0.993528i	$0.536234\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1252.46i			1.33099i	0.746403	+	0.665495i	$0.231779\pi$
$941$			1252.46i			1.33099i	−0.746403	+	0.665495i	$0.768221\pi$
$942$		0			0
$943$			14.5920i			0.0154741i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	558.077			0.589310			0.294655	−	0.955604i	$-0.404795\pi$
$947$	558.077			0.589310			0.294655	+	0.955604i	$0.404795\pi$
$948$		0			0
$949$		−	564.655i		−	0.595000i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1888.29i		−	1.98142i	−0.135992	−	0.990710i	$-0.543422\pi$
$953$		−	1888.29i		−	1.98142i	0.135992	−	0.990710i	$-0.456578\pi$
$954$		0			0
$955$	−277.100			−0.290158
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−233.192			−0.243162
$960$		0			0
$961$	170.241			0.177150
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	356.655i		−	0.369591i
$966$		0			0
$967$	−156.093			−0.161420			−0.0807099	−	0.996738i	$-0.525719\pi$
$967$	−156.093			−0.161420			−0.0807099	+	0.996738i	$0.525719\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			438.742i			0.451846i	0.974145	+	0.225923i	$0.0725397\pi$
$971$			438.742i			0.451846i	−0.974145	+	0.225923i	$0.927460\pi$
$972$		0			0
$973$	−1005.29			−1.03318
$974$		0			0
$975$		0			0
$976$		0			0
$977$			493.815i			0.505440i	0.967539	+	0.252720i	$0.0813252\pi$
$977$			493.815i			0.505440i	−0.967539	+	0.252720i	$0.918675\pi$
$978$		0			0
$979$			11.2772i			0.0115191i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	185.597i		−	0.188807i	−0.995534	−	0.0944036i	$-0.969906\pi$
$983$		−	185.597i		−	0.188807i	0.995534	−	0.0944036i	$-0.0300944\pi$
$984$		0			0
$985$	−376.685			−0.382421
$986$		0			0
$987$		0			0
$988$		0			0
$989$	881.871			0.891679
$990$		0			0
$991$			573.094i			0.578298i	0.957284	+	0.289149i	$0.0933723\pi$
$991$			573.094i			0.578298i	−0.957284	+	0.289149i	$0.906628\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	35.7277			0.0359072
$996$		0			0
$997$	−402.997			−0.404210			−0.202105	−	0.979364i	$-0.564778\pi$
$997$	−402.997			−0.404210			−0.202105	+	0.979364i	$0.564778\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.o.r.721.7		20
3.2	odd	2		912.3.o.e.721.7		20
4.3	odd	2		1368.3.o.c.721.7		20
12.11	even	2		456.3.o.a.265.17	yes	20
19.18	odd	2	inner	2736.3.o.r.721.8		20
57.56	even	2		912.3.o.e.721.17		20
76.75	even	2		1368.3.o.c.721.8		20
228.227	odd	2		456.3.o.a.265.7	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.3.o.a.265.7	✓	20	228.227	odd	2
456.3.o.a.265.17	yes	20	12.11	even	2
912.3.o.e.721.7		20	3.2	odd	2
912.3.o.e.721.17		20	57.56	even	2
1368.3.o.c.721.7		20	4.3	odd	2
1368.3.o.c.721.8		20	76.75	even	2
2736.3.o.r.721.7		20	1.1	even	1	trivial
2736.3.o.r.721.8		20	19.18	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.73712 q^{5} -11.2420 q^{7} +O(q^{10})\)	\(q-2.73712 q^{5} -11.2420 q^{7} -0.310403 q^{11} -13.9860i q^{13} -17.6502 q^{17} +(-17.8456 + 6.52193i) q^{19} -16.4750 q^{23} -17.5082 q^{25} -21.3495i q^{29} +28.1204i q^{31} +30.7708 q^{35} +42.1270i q^{37} -0.885710i q^{41} -53.5279 q^{43} +63.2951 q^{47} +77.3834 q^{49} -101.266i q^{53} +0.849611 q^{55} -41.6822i q^{59} -70.5688 q^{61} +38.2815i q^{65} -6.33916i q^{67} +18.0082i q^{71} +40.3728 q^{73} +3.48956 q^{77} -40.1903i q^{79} -114.546 q^{83} +48.3107 q^{85} -36.3308i q^{89} +157.232i q^{91} +(48.8455 - 17.8513i) q^{95} -36.4451i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 16 q^{7}+O(q^{10}) \) 20 * q + 16 * q^7	\( 20 q + 16 q^{7} + 16 q^{11} - 32 q^{17} - 40 q^{19} + 64 q^{23} + 68 q^{25} - 208 q^{35} - 64 q^{43} + 48 q^{47} + 20 q^{49} + 336 q^{55} + 184 q^{61} + 104 q^{73} - 88 q^{77} + 224 q^{83} - 136 q^{85} - 320 q^{95}+O(q^{100}) \) 20 * q + 16 * q^7 + 16 * q^11 - 32 * q^17 - 40 * q^19 + 64 * q^23 + 68 * q^25 - 208 * q^35 - 64 * q^43 + 48 * q^47 + 20 * q^49 + 336 * q^55 + 184 * q^61 + 104 * q^73 - 88 * q^77 + 224 * q^83 - 136 * q^85 - 320 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more