LMFDB - Embedded newform 2760.2.k.f.2209.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2760,2,Mod(2209,2760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2760, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2760.2209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2760.k (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:	$22.0387109579$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2209.2
Character	$\chi$	$=$	2760.2209
Dual form			2760.2.k.f.2209.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-2.06524 + 0.857190i) q^{5} -3.22845i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(-2.06524 + 0.857190i) q^{5} -3.22845i q^{7} -1.00000 q^{9} +0.0657386 q^{11} -6.97643i q^{13} +(0.857190 + 2.06524i) q^{15} +5.91838i q^{17} -8.45018 q^{19} -3.22845 q^{21} -1.00000i q^{23} +(3.53045 - 3.54061i) q^{25} +1.00000i q^{27} -1.27134 q^{29} +6.03405 q^{31} -0.0657386i q^{33} +(2.76739 + 6.66752i) q^{35} +7.79905i q^{37} -6.97643 q^{39} -9.18897 q^{41} -6.44751i q^{43} +(2.06524 - 0.857190i) q^{45} -8.47051i q^{47} -3.42286 q^{49} +5.91838 q^{51} +12.9057i q^{53} +(-0.135766 + 0.0563505i) q^{55} +8.45018i q^{57} -0.969284 q^{59} +4.13599 q^{61} +3.22845i q^{63} +(5.98012 + 14.4080i) q^{65} -1.43654i q^{67} -1.00000 q^{69} +5.89720 q^{71} -0.325102i q^{73} +(-3.54061 - 3.53045i) q^{75} -0.212234i q^{77} -0.193553 q^{79} +1.00000 q^{81} +7.81029i q^{83} +(-5.07318 - 12.2229i) q^{85} +1.27134i q^{87} +7.43659 q^{89} -22.5230 q^{91} -6.03405i q^{93} +(17.4517 - 7.24341i) q^{95} +13.9149i q^{97} -0.0657386 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 2 q^{5} - 22 q^{9}+O(q^{10}) $ 22 * q - 2 * q^5 - 22 * q^9	$ 22 q - 2 q^{5} - 22 q^{9} - 16 q^{19} + 6 q^{21} + 2 q^{25} - 22 q^{29} + 6 q^{31} + 14 q^{35} - 12 q^{39} + 6 q^{41} + 2 q^{45} - 68 q^{49} + 10 q^{51} + 12 q^{55} - 18 q^{59} + 16 q^{61} + 44 q^{65} - 22 q^{69} - 30 q^{71} + 4 q^{75} + 36 q^{79} + 22 q^{81} + 34 q^{85} - 16 q^{89} + 28 q^{91} + 20 q^{95}+O(q^{100}) $ 22 * q - 2 * q^5 - 22 * q^9 - 16 * q^19 + 6 * q^21 + 2 * q^25 - 22 * q^29 + 6 * q^31 + 14 * q^35 - 12 * q^39 + 6 * q^41 + 2 * q^45 - 68 * q^49 + 10 * q^51 + 12 * q^55 - 18 * q^59 + 16 * q^61 + 44 * q^65 - 22 * q^69 - 30 * q^71 + 4 * q^75 + 36 * q^79 + 22 * q^81 + 34 * q^85 - 16 * q^89 + 28 * q^91 + 20 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2760\mathbb{Z}\right)^\times$.

$n$	$1201$	$1381$	$1657$	$1841$	$2071$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	−2.06524	+	0.857190i	−0.923604	+	0.383347i
$5$	−2.06524	+	0.857190i	−0.923604	+	0.383347i
$6$		0			0
$7$		−	3.22845i		−	1.22024i	−0.792310	−	0.610119i	$-0.791122\pi$
$7$		−	3.22845i		−	1.22024i	0.792310	−	0.610119i	$-0.208878\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	0.0657386			0.0198209			0.00991047	−	0.999951i	$-0.496845\pi$
$11$	0.0657386			0.0198209			0.00991047	+	0.999951i	$0.496845\pi$
$12$		0			0
$13$		−	6.97643i		−	1.93491i	−0.253037	−	0.967457i	$-0.581430\pi$
$13$		−	6.97643i		−	1.93491i	0.253037	−	0.967457i	$-0.418570\pi$
$14$		0			0
$15$	0.857190	+	2.06524i	0.221326	+	0.533243i
$16$		0			0
$17$			5.91838i			1.43542i	0.696343	+	0.717709i	$0.254809\pi$
$17$			5.91838i			1.43542i	−0.696343	+	0.717709i	$0.745191\pi$
$18$		0			0
$19$	−8.45018			−1.93860			−0.969302	−	0.245873i	$-0.920925\pi$
$19$	−8.45018			−1.93860			−0.969302	+	0.245873i	$0.920925\pi$
$20$		0			0
$21$	−3.22845			−0.704505
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	3.53045	−	3.54061i	0.706090	−	0.708122i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−1.27134			−0.236083			−0.118041	−	0.993009i	$-0.537662\pi$
$29$	−1.27134			−0.236083			−0.118041	+	0.993009i	$0.537662\pi$
$30$		0			0
$31$	6.03405			1.08375			0.541873	−	0.840460i	$-0.317715\pi$
$31$	6.03405			1.08375			0.541873	+	0.840460i	$0.317715\pi$
$32$		0			0
$33$		−	0.0657386i		−	0.0114436i
$34$		0			0
$35$	2.76739	+	6.66752i	0.467775	+	1.12702i
$36$		0			0
$37$			7.79905i			1.28216i	0.767476	+	0.641078i	$0.221513\pi$
$37$			7.79905i			1.28216i	−0.767476	+	0.641078i	$0.778487\pi$
$38$		0			0
$39$	−6.97643			−1.11712
$40$		0			0
$41$	−9.18897			−1.43508			−0.717538	−	0.696519i	$-0.754731\pi$
$41$	−9.18897			−1.43508			−0.717538	+	0.696519i	$0.754731\pi$
$42$		0			0
$43$		−	6.44751i		−	0.983236i	−0.870811	−	0.491618i	$-0.836406\pi$
$43$		−	6.44751i		−	0.983236i	0.870811	−	0.491618i	$-0.163594\pi$
$44$		0			0
$45$	2.06524	−	0.857190i	0.307868	−	0.127782i
$46$		0			0
$47$		−	8.47051i		−	1.23555i	−0.786355	−	0.617775i	$-0.788034\pi$
$47$		−	8.47051i		−	1.23555i	0.786355	−	0.617775i	$-0.211966\pi$
$48$		0			0
$49$	−3.42286			−0.488981
$50$		0			0
$51$	5.91838			0.828739
$52$		0			0
$53$			12.9057i			1.77274i	0.462978	+	0.886370i	$0.346781\pi$
$53$			12.9057i			1.77274i	−0.462978	+	0.886370i	$0.653219\pi$
$54$		0			0
$55$	−0.135766	+	0.0563505i	−0.0183067	+	0.00759830i
$56$		0			0
$57$			8.45018i			1.11925i
$58$		0			0
$59$	−0.969284			−0.126190			−0.0630950	−	0.998008i	$-0.520097\pi$
$59$	−0.969284			−0.126190			−0.0630950	+	0.998008i	$0.520097\pi$
$60$		0			0
$61$	4.13599			0.529560			0.264780	−	0.964309i	$-0.414701\pi$
$61$	4.13599			0.529560			0.264780	+	0.964309i	$0.414701\pi$
$62$		0			0
$63$			3.22845i			0.406746i
$64$		0			0
$65$	5.98012	+	14.4080i	0.741743	+	1.78709i
$66$		0			0
$67$		−	1.43654i		−	0.175501i	−0.996142	−	0.0877506i	$-0.972032\pi$
$67$		−	1.43654i		−	0.175501i	0.996142	−	0.0877506i	$-0.0279679\pi$
$68$		0			0
$69$	−1.00000			−0.120386
$70$		0			0
$71$	5.89720			0.699869			0.349934	−	0.936774i	$-0.386204\pi$
$71$	5.89720			0.699869			0.349934	+	0.936774i	$0.386204\pi$
$72$		0			0
$73$		−	0.325102i		−	0.0380503i	−0.999819	−	0.0190251i	$-0.993944\pi$
$73$		−	0.325102i		−	0.0380503i	0.999819	−	0.0190251i	$-0.00605625\pi$
$74$		0			0
$75$	−3.54061	−	3.53045i	−0.408834	−	0.407661i
$76$		0			0
$77$		−	0.212234i		−	0.0241863i
$78$		0			0
$79$	−0.193553			−0.0217764			−0.0108882	−	0.999941i	$-0.503466\pi$
$79$	−0.193553			−0.0217764			−0.0108882	+	0.999941i	$0.503466\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			7.81029i			0.857290i	0.903473	+	0.428645i	$0.141009\pi$
$83$			7.81029i			0.857290i	−0.903473	+	0.428645i	$0.858991\pi$
$84$		0			0
$85$	−5.07318	−	12.2229i	−0.550263	−	1.32576i
$86$		0			0
$87$			1.27134i			0.136302i
$88$		0			0
$89$	7.43659			0.788277			0.394138	−	0.919051i	$-0.371043\pi$
$89$	7.43659			0.788277			0.394138	+	0.919051i	$0.371043\pi$
$90$		0			0
$91$	−22.5230			−2.36105
$92$		0			0
$93$		−	6.03405i		−	0.625701i
$94$		0			0
$95$	17.4517	−	7.24341i	1.79050	−	0.743158i
$96$		0			0
$97$			13.9149i			1.41285i	0.707789	+	0.706424i	$0.249693\pi$
$97$			13.9149i			1.41285i	−0.707789	+	0.706424i	$0.750307\pi$
$98$		0			0
$99$	−0.0657386			−0.00660698
$100$		0			0
$101$	−10.8534			−1.07996			−0.539978	−	0.841679i	$-0.681568\pi$
$101$	−10.8534			−1.07996			−0.539978	+	0.841679i	$0.681568\pi$
$102$		0			0
$103$			16.1224i			1.58859i	0.607534	+	0.794293i	$0.292159\pi$
$103$			16.1224i			1.58859i	−0.607534	+	0.794293i	$0.707841\pi$
$104$		0			0
$105$	6.66752	−	2.76739i	0.650684	−	0.270070i
$106$		0			0
$107$			9.90935i			0.957973i	0.877822	+	0.478987i	$0.158996\pi$
$107$			9.90935i			0.957973i	−0.877822	+	0.478987i	$0.841004\pi$
$108$		0			0
$109$	−20.7118			−1.98383			−0.991915	−	0.126904i	$-0.959496\pi$
$109$	−20.7118			−1.98383			−0.991915	+	0.126904i	$0.959496\pi$
$110$		0			0
$111$	7.79905			0.740253
$112$		0			0
$113$		−	14.3137i		−	1.34652i	−0.739407	−	0.673259i	$-0.764894\pi$
$113$		−	14.3137i		−	1.34652i	0.739407	−	0.673259i	$-0.235106\pi$
$114$		0			0
$115$	0.857190	+	2.06524i	0.0799334	+	0.192585i
$116$		0			0
$117$			6.97643i			0.644971i
$118$		0			0
$119$	19.1072			1.75155
$120$		0			0
$121$	−10.9957			−0.999607
$122$		0			0
$123$			9.18897i			0.828542i
$124$		0			0
$125$	−4.25626	+	10.3385i	−0.380691	+	0.924702i
$126$		0			0
$127$			10.7551i			0.954362i	0.878805	+	0.477181i	$0.158341\pi$
$127$			10.7551i			0.954362i	−0.878805	+	0.477181i	$0.841659\pi$
$128$		0			0
$129$	−6.44751			−0.567672
$130$		0			0
$131$	−10.5052			−0.917845			−0.458923	−	0.888476i	$-0.651765\pi$
$131$	−10.5052			−0.917845			−0.458923	+	0.888476i	$0.651765\pi$
$132$		0			0
$133$			27.2810i			2.36556i
$134$		0			0
$135$	−0.857190	−	2.06524i	−0.0737752	−	0.177748i
$136$		0			0
$137$			19.4160i			1.65882i	0.558641	+	0.829410i	$0.311323\pi$
$137$			19.4160i			1.65882i	−0.558641	+	0.829410i	$0.688677\pi$
$138$		0			0
$139$	3.87016			0.328262			0.164131	−	0.986439i	$-0.447518\pi$
$139$	3.87016			0.328262			0.164131	+	0.986439i	$0.447518\pi$
$140$		0			0
$141$	−8.47051			−0.713346
$142$		0			0
$143$		−	0.458621i		−	0.0383518i
$144$		0			0
$145$	2.62563	−	1.08978i	0.218047	−	0.0905016i
$146$		0			0
$147$			3.42286i			0.282313i
$148$		0			0
$149$	2.18138			0.178705			0.0893527	−	0.996000i	$-0.471520\pi$
$149$	2.18138			0.178705			0.0893527	+	0.996000i	$0.471520\pi$
$150$		0			0
$151$	−11.6114			−0.944926			−0.472463	−	0.881351i	$-0.656635\pi$
$151$	−11.6114			−0.944926			−0.472463	+	0.881351i	$0.656635\pi$
$152$		0			0
$153$		−	5.91838i		−	0.478473i
$154$		0			0
$155$	−12.4618	+	5.17232i	−1.00095	+	0.415451i
$156$		0			0
$157$		−	19.7684i		−	1.57769i	−0.614591	−	0.788846i	$-0.710679\pi$
$157$		−	19.7684i		−	1.57769i	0.614591	−	0.788846i	$-0.289321\pi$
$158$		0			0
$159$	12.9057			1.02349
$160$		0			0
$161$	−3.22845			−0.254437
$162$		0			0
$163$			18.5086i			1.44971i	0.688902	+	0.724855i	$0.258093\pi$
$163$			18.5086i			1.44971i	−0.688902	+	0.724855i	$0.741907\pi$
$164$		0			0
$165$	0.0563505	+	0.135766i	0.00438688	+	0.0105694i
$166$		0			0
$167$			7.70765i			0.596436i	0.954498	+	0.298218i	$0.0963922\pi$
$167$			7.70765i			0.596436i	−0.954498	+	0.298218i	$0.903608\pi$
$168$		0			0
$169$	−35.6706			−2.74389
$170$		0			0
$171$	8.45018			0.646201
$172$		0			0
$173$		−	19.1212i		−	1.45376i	−0.686765	−	0.726880i	$-0.740970\pi$
$173$		−	19.1212i		−	1.45376i	0.686765	−	0.726880i	$-0.259030\pi$
$174$		0			0
$175$	−11.4307	−	11.3979i	−0.864077	−	0.861598i
$176$		0			0
$177$			0.969284i			0.0728559i
$178$		0			0
$179$	3.49767			0.261428			0.130714	−	0.991420i	$-0.458273\pi$
$179$	3.49767			0.261428			0.130714	+	0.991420i	$0.458273\pi$
$180$		0			0
$181$	−13.1785			−0.979553			−0.489776	−	0.871848i	$-0.662922\pi$
$181$	−13.1785			−0.979553			−0.489776	+	0.871848i	$0.662922\pi$
$182$		0			0
$183$		−	4.13599i		−	0.305741i
$184$		0			0
$185$	−6.68527	−	16.1069i	−0.491511	−	1.18420i
$186$		0			0
$187$			0.389066i			0.0284514i
$188$		0			0
$189$	3.22845			0.234835
$190$		0			0
$191$	−4.03471			−0.291941			−0.145971	−	0.989289i	$-0.546631\pi$
$191$	−4.03471			−0.291941			−0.145971	+	0.989289i	$0.546631\pi$
$192$		0			0
$193$		−	7.67518i		−	0.552471i	−0.961090	−	0.276236i	$-0.910913\pi$
$193$		−	7.67518i		−	0.552471i	0.961090	−	0.276236i	$-0.0890870\pi$
$194$		0			0
$195$	14.4080	−	5.98012i	1.03178	−	0.428246i
$196$		0			0
$197$			16.5915i			1.18209i	0.806638	+	0.591046i	$0.201285\pi$
$197$			16.5915i			1.18209i	−0.806638	+	0.591046i	$0.798715\pi$
$198$		0			0
$199$	17.3031			1.22659			0.613294	−	0.789855i	$-0.289844\pi$
$199$	17.3031			1.22659			0.613294	+	0.789855i	$0.289844\pi$
$200$		0			0
$201$	−1.43654			−0.101326
$202$		0			0
$203$			4.10447i			0.288077i
$204$		0			0
$205$	18.9775	−	7.87670i	1.32544	−	0.550132i
$206$		0			0
$207$			1.00000i			0.0695048i
$208$		0			0
$209$	−0.555503			−0.0384250
$210$		0			0
$211$	18.4560			1.27057			0.635283	−	0.772280i	$-0.280884\pi$
$211$	18.4560			1.27057			0.635283	+	0.772280i	$0.280884\pi$
$212$		0			0
$213$		−	5.89720i		−	0.404070i
$214$		0			0
$215$	5.52674	+	13.3157i	0.376921	+	0.908121i
$216$		0			0
$217$		−	19.4806i		−	1.32243i
$218$		0			0
$219$	−0.325102			−0.0219683
$220$		0			0
$221$	41.2892			2.77741
$222$		0			0
$223$			9.09648i			0.609146i	0.952489	+	0.304573i	$0.0985137\pi$
$223$			9.09648i			0.609146i	−0.952489	+	0.304573i	$0.901486\pi$
$224$		0			0
$225$	−3.53045	+	3.54061i	−0.235363	+	0.236041i
$226$		0			0
$227$		−	25.2495i		−	1.67587i	−0.545772	−	0.837934i	$-0.683764\pi$
$227$		−	25.2495i		−	1.67587i	0.545772	−	0.837934i	$-0.316236\pi$
$228$		0			0
$229$	10.0618			0.664902			0.332451	−	0.943121i	$-0.392124\pi$
$229$	10.0618			0.664902			0.332451	+	0.943121i	$0.392124\pi$
$230$		0			0
$231$	−0.212234			−0.0139640
$232$		0			0
$233$		−	2.82475i		−	0.185056i	−0.995710	−	0.0925278i	$-0.970505\pi$
$233$		−	2.82475i		−	0.185056i	0.995710	−	0.0925278i	$-0.0294947\pi$
$234$		0			0
$235$	7.26084	+	17.4937i	0.473645	+	1.14116i
$236$		0			0
$237$			0.193553i			0.0125726i
$238$		0			0
$239$	−0.364657			−0.0235877			−0.0117938	−	0.999930i	$-0.503754\pi$
$239$	−0.364657			−0.0235877			−0.0117938	+	0.999930i	$0.503754\pi$
$240$		0			0
$241$	9.57384			0.616705			0.308352	−	0.951272i	$-0.400222\pi$
$241$	9.57384			0.616705			0.308352	+	0.951272i	$0.400222\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	7.06905	−	2.93405i	0.451625	−	0.187449i
$246$		0			0
$247$			58.9521i			3.75103i
$248$		0			0
$249$	7.81029			0.494957
$250$		0			0
$251$	−13.7873			−0.870249			−0.435124	−	0.900370i	$-0.643296\pi$
$251$	−13.7873			−0.870249			−0.435124	+	0.900370i	$0.643296\pi$
$252$		0			0
$253$		−	0.0657386i		−	0.00413295i
$254$		0			0
$255$	−12.2229	+	5.07318i	−0.765427	+	0.317695i
$256$		0			0
$257$		−	6.64434i		−	0.414462i	−0.978292	−	0.207231i	$-0.933555\pi$
$257$		−	6.64434i		−	0.414462i	0.978292	−	0.207231i	$-0.0664452\pi$
$258$		0			0
$259$	25.1788			1.56454
$260$		0			0
$261$	1.27134			0.0786942
$262$		0			0
$263$		−	2.43357i		−	0.150060i	−0.997181	−	0.0750302i	$-0.976095\pi$
$263$		−	2.43357i		−	0.150060i	0.997181	−	0.0750302i	$-0.0239053\pi$
$264$		0			0
$265$	−11.0627	−	26.6535i	−0.679574	−	1.63731i
$266$		0			0
$267$		−	7.43659i		−	0.455112i
$268$		0			0
$269$	−5.77031			−0.351822			−0.175911	−	0.984406i	$-0.556287\pi$
$269$	−5.77031			−0.351822			−0.175911	+	0.984406i	$0.556287\pi$
$270$		0			0
$271$	−7.33639			−0.445654			−0.222827	−	0.974858i	$-0.571528\pi$
$271$	−7.33639			−0.445654			−0.222827	+	0.974858i	$0.571528\pi$
$272$		0			0
$273$			22.5230i			1.36316i
$274$		0			0
$275$	0.232087	−	0.232755i	0.0139954	−	0.0140356i
$276$		0			0
$277$			5.04718i			0.303256i	0.988438	+	0.151628i	$0.0484515\pi$
$277$			5.04718i			0.303256i	−0.988438	+	0.151628i	$0.951548\pi$
$278$		0			0
$279$	−6.03405			−0.361249
$280$		0			0
$281$	−5.26255			−0.313938			−0.156969	−	0.987604i	$-0.550172\pi$
$281$	−5.26255			−0.313938			−0.156969	+	0.987604i	$0.550172\pi$
$282$		0			0
$283$		−	21.9216i		−	1.30310i	−0.758604	−	0.651552i	$-0.774118\pi$
$283$		−	21.9216i		−	1.30310i	0.758604	−	0.651552i	$-0.225882\pi$
$284$		0			0
$285$	−7.24341	−	17.4517i	−0.429063	−	1.03375i
$286$		0			0
$287$			29.6661i			1.75113i
$288$		0			0
$289$	−18.0273			−1.06043
$290$		0			0
$291$	13.9149			0.815708
$292$		0			0
$293$			22.8901i			1.33725i	0.743598	+	0.668627i	$0.233118\pi$
$293$			22.8901i			1.33725i	−0.743598	+	0.668627i	$0.766882\pi$
$294$		0			0
$295$	2.00181	−	0.830861i	0.116550	−	0.0483746i
$296$		0			0
$297$			0.0657386i			0.00381454i
$298$		0			0
$299$	−6.97643			−0.403457
$300$		0			0
$301$	−20.8154			−1.19978
$302$		0			0
$303$			10.8534i			0.623513i
$304$		0			0
$305$	−8.54183	+	3.54533i	−0.489104	+	0.203005i
$306$		0			0
$307$		−	6.34198i		−	0.361956i	−0.983487	−	0.180978i	$-0.942074\pi$
$307$		−	6.34198i		−	0.361956i	0.983487	−	0.180978i	$-0.0579263\pi$
$308$		0			0
$309$	16.1224			0.917171
$310$		0			0
$311$	1.15390			0.0654317			0.0327159	−	0.999465i	$-0.489584\pi$
$311$	1.15390			0.0654317			0.0327159	+	0.999465i	$0.489584\pi$
$312$		0			0
$313$		−	8.87225i		−	0.501489i	−0.968053	−	0.250745i	$-0.919325\pi$
$313$		−	8.87225i		−	0.501489i	0.968053	−	0.250745i	$-0.0806755\pi$
$314$		0			0
$315$	−2.76739	−	6.66752i	−0.155925	−	0.375672i
$316$		0			0
$317$			4.78940i			0.269000i	0.990914	+	0.134500i	$0.0429428\pi$
$317$			4.78940i			0.269000i	−0.990914	+	0.134500i	$0.957057\pi$
$318$		0			0
$319$	−0.0835765			−0.00467938
$320$		0			0
$321$	9.90935			0.553086
$322$		0			0
$323$		−	50.0114i		−	2.78271i
$324$		0			0
$325$	−24.7008	−	24.6299i	−1.37015	−	1.36622i
$326$		0			0
$327$			20.7118i			1.14536i
$328$		0			0
$329$	−27.3466			−1.50767
$330$		0			0
$331$	−27.6627			−1.52048			−0.760239	−	0.649644i	$-0.774918\pi$
$331$	−27.6627			−1.52048			−0.760239	+	0.649644i	$0.774918\pi$
$332$		0			0
$333$		−	7.79905i		−	0.427385i
$334$		0			0
$335$	1.23139	+	2.96680i	0.0672779	+	0.162094i
$336$		0			0
$337$			16.7742i			0.913750i	0.889531	+	0.456875i	$0.151031\pi$
$337$			16.7742i			0.913750i	−0.889531	+	0.456875i	$0.848969\pi$
$338$		0			0
$339$	−14.3137			−0.777412
$340$		0			0
$341$	0.396670			0.0214809
$342$		0			0
$343$		−	11.5486i		−	0.623565i
$344$		0			0
$345$	2.06524	−	0.857190i	0.111189	−	0.0461496i
$346$		0			0
$347$		−	5.96963i		−	0.320467i	−0.987079	−	0.160233i	$-0.948775\pi$
$347$		−	5.96963i		−	0.320467i	0.987079	−	0.160233i	$-0.0512247\pi$
$348$		0			0
$349$	1.13668			0.0608450			0.0304225	−	0.999537i	$-0.490315\pi$
$349$	1.13668			0.0608450			0.0304225	+	0.999537i	$0.490315\pi$
$350$		0			0
$351$	6.97643			0.372374
$352$		0			0
$353$		−	21.5211i		−	1.14545i	−0.819747	−	0.572725i	$-0.805886\pi$
$353$		−	21.5211i		−	1.14545i	0.819747	−	0.572725i	$-0.194114\pi$
$354$		0			0
$355$	−12.1791	+	5.05502i	−0.646402	+	0.268293i
$356$		0			0
$357$		−	19.1072i		−	1.01126i
$358$		0			0
$359$	37.0522			1.95554			0.977770	−	0.209680i	$-0.0672423\pi$
$359$	37.0522			1.95554			0.977770	+	0.209680i	$0.0672423\pi$
$360$		0			0
$361$	52.4055			2.75819
$362$		0			0
$363$			10.9957i			0.577123i
$364$		0			0
$365$	0.278674	+	0.671413i	0.0145865	+	0.0351434i
$366$		0			0
$367$		−	17.6408i		−	0.920843i	−0.887700	−	0.460421i	$-0.847698\pi$
$367$		−	17.6408i		−	0.920843i	0.887700	−	0.460421i	$-0.152302\pi$
$368$		0			0
$369$	9.18897			0.478359
$370$		0			0
$371$	41.6655			2.16316
$372$		0			0
$373$			4.23283i			0.219168i	0.993978	+	0.109584i	$0.0349518\pi$
$373$			4.23283i			0.219168i	−0.993978	+	0.109584i	$0.965048\pi$
$374$		0			0
$375$	10.3385	+	4.25626i	0.533877	+	0.219792i
$376$		0			0
$377$			8.86944i			0.456800i
$378$		0			0
$379$	−10.0989			−0.518746			−0.259373	−	0.965777i	$-0.583516\pi$
$379$	−10.0989			−0.518746			−0.259373	+	0.965777i	$0.583516\pi$
$380$		0			0
$381$	10.7551			0.551001
$382$		0			0
$383$		−	9.37015i		−	0.478792i	−0.970922	−	0.239396i	$-0.923051\pi$
$383$		−	9.37015i		−	0.478792i	0.970922	−	0.239396i	$-0.0769494\pi$
$384$		0			0
$385$	0.181925	+	0.438314i	0.00927174	+	0.0223385i
$386$		0			0
$387$			6.44751i			0.327745i
$388$		0			0
$389$	−8.26196			−0.418898			−0.209449	−	0.977820i	$-0.567167\pi$
$389$	−8.26196			−0.418898			−0.209449	+	0.977820i	$0.567167\pi$
$390$		0			0
$391$	5.91838			0.299305
$392$		0			0
$393$			10.5052i			0.529918i
$394$		0			0
$395$	0.399734	−	0.165912i	0.0201128	−	0.00834793i
$396$		0			0
$397$			16.6821i			0.837251i	0.908159	+	0.418626i	$0.137488\pi$
$397$			16.6821i			0.837251i	−0.908159	+	0.418626i	$0.862512\pi$
$398$		0			0
$399$	27.2810			1.36576
$400$		0			0
$401$	−32.5600			−1.62597			−0.812985	−	0.582285i	$-0.802159\pi$
$401$	−32.5600			−1.62597			−0.812985	+	0.582285i	$0.802159\pi$
$402$		0			0
$403$		−	42.0961i		−	2.09696i
$404$		0			0
$405$	−2.06524	+	0.857190i	−0.102623	+	0.0425941i
$406$		0			0
$407$			0.512699i			0.0254135i
$408$		0			0
$409$	−36.9534			−1.82723			−0.913614	−	0.406583i	$-0.866720\pi$
$409$	−36.9534			−1.82723			−0.913614	+	0.406583i	$0.866720\pi$
$410$		0			0
$411$	19.4160			0.957720
$412$		0			0
$413$			3.12928i			0.153982i
$414$		0			0
$415$	−6.69490	−	16.1301i	−0.328640	−	0.791797i
$416$		0			0
$417$		−	3.87016i		−	0.189522i
$418$		0			0
$419$	−18.9211			−0.924357			−0.462178	−	0.886787i	$-0.652932\pi$
$419$	−18.9211			−0.924357			−0.462178	+	0.886787i	$0.652932\pi$
$420$		0			0
$421$	−15.0194			−0.732001			−0.366001	−	0.930615i	$-0.619273\pi$
$421$	−15.0194			−0.732001			−0.366001	+	0.930615i	$0.619273\pi$
$422$		0			0
$423$			8.47051i			0.411850i
$424$		0			0
$425$	20.9547	+	20.8946i	1.01645	+	1.01353i
$426$		0			0
$427$		−	13.3528i		−	0.646189i
$428$		0			0
$429$	−0.458621			−0.0221424
$430$		0			0
$431$	29.6509			1.42823			0.714117	−	0.700026i	$-0.246828\pi$
$431$	29.6509			1.42823			0.714117	+	0.700026i	$0.246828\pi$
$432$		0			0
$433$		−	18.8586i		−	0.906286i	−0.891438	−	0.453143i	$-0.850303\pi$
$433$		−	18.8586i		−	0.906286i	0.891438	−	0.453143i	$-0.149697\pi$
$434$		0			0
$435$	−1.08978	−	2.62563i	−0.0522511	−	0.125890i
$436$		0			0
$437$			8.45018i			0.404227i
$438$		0			0
$439$	6.38449			0.304715			0.152358	−	0.988325i	$-0.451313\pi$
$439$	6.38449			0.304715			0.152358	+	0.988325i	$0.451313\pi$
$440$		0			0
$441$	3.42286			0.162994
$442$		0			0
$443$		−	12.7718i		−	0.606806i	−0.952862	−	0.303403i	$-0.901877\pi$
$443$		−	12.7718i		−	0.606806i	0.952862	−	0.303403i	$-0.0981228\pi$
$444$		0			0
$445$	−15.3584	+	6.37457i	−0.728056	+	0.302184i
$446$		0			0
$447$		−	2.18138i		−	0.103176i
$448$		0			0
$449$	−10.0794			−0.475676			−0.237838	−	0.971305i	$-0.576439\pi$
$449$	−10.0794			−0.475676			−0.237838	+	0.971305i	$0.576439\pi$
$450$		0			0
$451$	−0.604071			−0.0284446
$452$		0			0
$453$			11.6114i			0.545553i
$454$		0			0
$455$	46.5155	−	19.3065i	2.18068	−	0.905103i
$456$		0			0
$457$		−	12.4330i		−	0.581593i	−0.956785	−	0.290796i	$-0.906080\pi$
$457$		−	12.4330i		−	0.581593i	0.956785	−	0.290796i	$-0.0939202\pi$
$458$		0			0
$459$	−5.91838			−0.276246
$460$		0			0
$461$	−4.09734			−0.190832			−0.0954160	−	0.995437i	$-0.530418\pi$
$461$	−4.09734			−0.190832			−0.0954160	+	0.995437i	$0.530418\pi$
$462$		0			0
$463$		−	2.93672i		−	0.136481i	−0.997669	−	0.0682405i	$-0.978261\pi$
$463$		−	2.93672i		−	0.136481i	0.997669	−	0.0682405i	$-0.0217385\pi$
$464$		0			0
$465$	5.17232	+	12.4618i	0.239861	+	0.577901i
$466$		0			0
$467$		−	25.7239i		−	1.19036i	−0.803593	−	0.595179i	$-0.797081\pi$
$467$		−	25.7239i		−	1.19036i	0.803593	−	0.595179i	$-0.202919\pi$
$468$		0			0
$469$	−4.63779			−0.214153
$470$		0			0
$471$	−19.7684			−0.910880
$472$		0			0
$473$		−	0.423851i		−	0.0194887i
$474$		0			0
$475$	−29.8329	+	29.9188i	−1.36883	+	1.37277i
$476$		0			0
$477$		−	12.9057i		−	0.590913i
$478$		0			0
$479$	16.0276			0.732319			0.366159	−	0.930552i	$-0.380673\pi$
$479$	16.0276			0.732319			0.366159	+	0.930552i	$0.380673\pi$
$480$		0			0
$481$	54.4095			2.48086
$482$		0			0
$483$			3.22845i			0.146899i
$484$		0			0
$485$	−11.9277	−	28.7377i	−0.541611	−	1.30491i
$486$		0			0
$487$			9.25118i			0.419211i	0.977786	+	0.209605i	$0.0672179\pi$
$487$			9.25118i			0.419211i	−0.977786	+	0.209605i	$0.932782\pi$
$488$		0			0
$489$	18.5086			0.836990
$490$		0			0
$491$	8.29900			0.374529			0.187264	−	0.982310i	$-0.440038\pi$
$491$	8.29900			0.374529			0.187264	+	0.982310i	$0.440038\pi$
$492$		0			0
$493$		−	7.52430i		−	0.338878i
$494$		0			0
$495$	0.135766	−	0.0563505i	0.00610224	−	0.00253277i
$496$		0			0
$497$		−	19.0388i		−	0.854007i
$498$		0			0
$499$	−10.8879			−0.487410			−0.243705	−	0.969849i	$-0.578363\pi$
$499$	−10.8879			−0.487410			−0.243705	+	0.969849i	$0.578363\pi$
$500$		0			0
$501$	7.70765			0.344352
$502$		0			0
$503$			1.63379i			0.0728472i	0.999336	+	0.0364236i	$0.0115966\pi$
$503$			1.63379i			0.0728472i	−0.999336	+	0.0364236i	$0.988403\pi$
$504$		0			0
$505$	22.4150	−	9.30345i	0.997453	−	0.413998i
$506$		0			0
$507$			35.6706i			1.58418i
$508$		0			0
$509$	−17.8511			−0.791237			−0.395618	−	0.918415i	$-0.629470\pi$
$509$	−17.8511			−0.791237			−0.395618	+	0.918415i	$0.629470\pi$
$510$		0			0
$511$	−1.04957			−0.0464304
$512$		0			0
$513$		−	8.45018i		−	0.373085i
$514$		0			0
$515$	−13.8200	−	33.2966i	−0.608980	−	1.46723i
$516$		0			0
$517$		−	0.556840i		−	0.0244898i
$518$		0			0
$519$	−19.1212			−0.839329
$520$		0			0
$521$	−30.3584			−1.33003			−0.665013	−	0.746832i	$-0.731574\pi$
$521$	−30.3584			−1.33003			−0.665013	+	0.746832i	$0.731574\pi$
$522$		0			0
$523$		−	30.8925i		−	1.35083i	−0.737437	−	0.675416i	$-0.763964\pi$
$523$		−	30.8925i		−	1.35083i	0.737437	−	0.675416i	$-0.236036\pi$
$524$		0			0
$525$	−11.3979	+	11.4307i	−0.497444	+	0.498875i
$526$		0			0
$527$			35.7118i			1.55563i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	0.969284			0.0420634
$532$		0			0
$533$			64.1062i			2.77675i
$534$		0			0
$535$	−8.49420	−	20.4652i	−0.367236	−	0.884788i
$536$		0			0
$537$		−	3.49767i		−	0.150936i
$538$		0			0
$539$	−0.225015			−0.00969206
$540$		0			0
$541$	14.9325			0.641997			0.320999	−	0.947080i	$-0.395982\pi$
$541$	14.9325			0.641997			0.320999	+	0.947080i	$0.395982\pi$
$542$		0			0
$543$			13.1785i			0.565545i
$544$		0			0
$545$	42.7749	−	17.7539i	1.83227	−	0.760495i
$546$		0			0
$547$		−	9.02414i		−	0.385844i	−0.981214	−	0.192922i	$-0.938204\pi$
$547$		−	9.02414i		−	0.385844i	0.981214	−	0.192922i	$-0.0617965\pi$
$548$		0			0
$549$	−4.13599			−0.176520
$550$		0			0
$551$	10.7431			0.457671
$552$		0			0
$553$			0.624876i			0.0265724i
$554$		0			0
$555$	−16.1069	+	6.68527i	−0.683701	+	0.283774i
$556$		0			0
$557$		−	7.83975i		−	0.332181i	−0.986111	−	0.166090i	$-0.946886\pi$
$557$		−	7.83975i		−	0.332181i	0.986111	−	0.166090i	$-0.0531144\pi$
$558$		0			0
$559$	−44.9806			−1.90248
$560$		0			0
$561$	0.389066			0.0164264
$562$		0			0
$563$		−	42.8111i		−	1.80427i	−0.431449	−	0.902137i	$-0.641998\pi$
$563$		−	42.8111i		−	1.80427i	0.431449	−	0.902137i	$-0.358002\pi$
$564$		0			0
$565$	12.2695	+	29.5612i	0.516183	+	1.24365i
$566$		0			0
$567$		−	3.22845i		−	0.135582i
$568$		0			0
$569$	−19.6657			−0.824429			−0.412214	−	0.911087i	$-0.635244\pi$
$569$	−19.6657			−0.824429			−0.412214	+	0.911087i	$0.635244\pi$
$570$		0			0
$571$	4.50381			0.188479			0.0942393	−	0.995550i	$-0.469958\pi$
$571$	4.50381			0.188479			0.0942393	+	0.995550i	$0.469958\pi$
$572$		0			0
$573$			4.03471i			0.168552i
$574$		0			0
$575$	−3.54061	−	3.53045i	−0.147654	−	0.147230i
$576$		0			0
$577$			7.11875i			0.296357i	0.988961	+	0.148179i	$0.0473411\pi$
$577$			7.11875i			0.296357i	−0.988961	+	0.148179i	$0.952659\pi$
$578$		0			0
$579$	−7.67518			−0.318970
$580$		0			0
$581$	25.2151			1.04610
$582$		0			0
$583$			0.848406i			0.0351374i
$584$		0			0
$585$	−5.98012	−	14.4080i	−0.247248	−	0.595698i
$586$		0			0
$587$			32.6995i			1.34965i	0.737976	+	0.674826i	$0.235782\pi$
$587$			32.6995i			1.34965i	−0.737976	+	0.674826i	$0.764218\pi$
$588$		0			0
$589$	−50.9888			−2.10096
$590$		0			0
$591$	16.5915			0.682481
$592$		0			0
$593$		−	4.20534i		−	0.172693i	−0.996265	−	0.0863463i	$-0.972481\pi$
$593$		−	4.20534i		−	0.172693i	0.996265	−	0.0863463i	$-0.0275192\pi$
$594$		0			0
$595$	−39.4610	+	16.3785i	−1.61774	+	0.671452i
$596$		0			0
$597$		−	17.3031i		−	0.708171i
$598$		0			0
$599$	14.7313			0.601905			0.300953	−	0.953639i	$-0.402695\pi$
$599$	14.7313			0.601905			0.300953	+	0.953639i	$0.402695\pi$
$600$		0			0
$601$	0.146957			0.00599451			0.00299725	−	0.999996i	$-0.499046\pi$
$601$	0.146957			0.00599451			0.00299725	+	0.999996i	$0.499046\pi$
$602$		0			0
$603$			1.43654i			0.0585004i
$604$		0			0
$605$	22.7087	−	9.42539i	0.923242	−	0.383196i
$606$		0			0
$607$			24.4044i			0.990545i	0.868738	+	0.495272i	$0.164932\pi$
$607$			24.4044i			0.990545i	−0.868738	+	0.495272i	$0.835068\pi$
$608$		0			0
$609$	4.10447			0.166321
$610$		0			0
$611$	−59.0939			−2.39068
$612$		0			0
$613$		−	46.8776i		−	1.89337i	−0.322160	−	0.946685i	$-0.604409\pi$
$613$		−	46.8776i		−	1.89337i	0.322160	−	0.946685i	$-0.395591\pi$
$614$		0			0
$615$	−7.87670	−	18.9775i	−0.317619	−	0.765245i
$616$		0			0
$617$			36.0685i			1.45206i	0.687662	+	0.726031i	$0.258637\pi$
$617$			36.0685i			1.45206i	−0.687662	+	0.726031i	$0.741363\pi$
$618$		0			0
$619$	3.28991			0.132233			0.0661164	−	0.997812i	$-0.478939\pi$
$619$	3.28991			0.132233			0.0661164	+	0.997812i	$0.478939\pi$
$620$		0			0
$621$	1.00000			0.0401286
$622$		0			0
$623$		−	24.0086i		−	0.961885i
$624$		0			0
$625$	−0.0718380	−	24.9999i	−0.00287352	−	0.999996i
$626$		0			0
$627$			0.555503i			0.0221847i
$628$		0			0
$629$	−46.1578			−1.84043
$630$		0			0
$631$	−22.3500			−0.889738			−0.444869	−	0.895596i	$-0.646750\pi$
$631$	−22.3500			−0.889738			−0.444869	+	0.895596i	$0.646750\pi$
$632$		0			0
$633$		−	18.4560i		−	0.733561i
$634$		0			0
$635$	−9.21917	−	22.2119i	−0.365852	−	0.881453i
$636$		0			0
$637$			23.8794i			0.946135i
$638$		0			0
$639$	−5.89720			−0.233290
$640$		0			0
$641$	−15.6172			−0.616842			−0.308421	−	0.951250i	$-0.599801\pi$
$641$	−15.6172			−0.616842			−0.308421	+	0.951250i	$0.599801\pi$
$642$		0			0
$643$			42.8496i			1.68982i	0.534906	+	0.844912i	$0.320347\pi$
$643$			42.8496i			1.68982i	−0.534906	+	0.844912i	$0.679653\pi$
$644$		0			0
$645$	13.3157	−	5.52674i	0.524304	−	0.217615i
$646$		0			0
$647$			19.5472i			0.768482i	0.923233	+	0.384241i	$0.125537\pi$
$647$			19.5472i			0.768482i	−0.923233	+	0.384241i	$0.874463\pi$
$648$		0			0
$649$	−0.0637194			−0.00250121
$650$		0			0
$651$	−19.4806			−0.763505
$652$		0			0
$653$		−	18.6349i		−	0.729239i	−0.931157	−	0.364620i	$-0.881199\pi$
$653$		−	18.6349i		−	0.729239i	0.931157	−	0.364620i	$-0.118801\pi$
$654$		0			0
$655$	21.6958	−	9.00497i	0.847726	−	0.351853i
$656$		0			0
$657$			0.325102i			0.0126834i
$658$		0			0
$659$	−37.6555			−1.46685			−0.733425	−	0.679770i	$-0.762080\pi$
$659$	−37.6555			−1.46685			−0.733425	+	0.679770i	$0.762080\pi$
$660$		0			0
$661$	−23.8210			−0.926530			−0.463265	−	0.886220i	$-0.653322\pi$
$661$	−23.8210			−0.926530			−0.463265	+	0.886220i	$0.653322\pi$
$662$		0			0
$663$		−	41.2892i		−	1.60354i
$664$		0			0
$665$	−23.3850	−	56.3418i	−0.906830	−	2.18484i
$666$		0			0
$667$			1.27134i			0.0492267i
$668$		0			0
$669$	9.09648			0.351690
$670$		0			0
$671$	0.271895			0.0104964
$672$		0			0
$673$		−	1.25728i		−	0.0484645i	−0.999706	−	0.0242323i	$-0.992286\pi$
$673$		−	1.25728i		−	0.0484645i	0.999706	−	0.0242323i	$-0.00771412\pi$
$674$		0			0
$675$	3.54061	+	3.53045i	0.136278	+	0.135887i
$676$		0			0
$677$		−	32.1066i		−	1.23396i	−0.786980	−	0.616978i	$-0.788357\pi$
$677$		−	32.1066i		−	1.23396i	0.786980	−	0.616978i	$-0.211643\pi$
$678$		0			0
$679$	44.9236			1.72401
$680$		0			0
$681$	−25.2495			−0.967562
$682$		0			0
$683$			16.4438i			0.629203i	0.949224	+	0.314602i	$0.101871\pi$
$683$			16.4438i			0.629203i	−0.949224	+	0.314602i	$0.898129\pi$
$684$		0			0
$685$	−16.6432	−	40.0987i	−0.635903	−	1.53209i
$686$		0			0
$687$		−	10.0618i		−	0.383881i
$688$		0			0
$689$	90.0359			3.43010
$690$		0			0
$691$	7.84704			0.298515			0.149258	−	0.988798i	$-0.452312\pi$
$691$	7.84704			0.298515			0.149258	+	0.988798i	$0.452312\pi$
$692$		0			0
$693$			0.212234i			0.00806209i
$694$		0			0
$695$	−7.99281	+	3.31746i	−0.303185	+	0.125838i
$696$		0			0
$697$		−	54.3839i		−	2.05994i
$698$		0			0
$699$	−2.82475			−0.106842
$700$		0			0
$701$	−32.0170			−1.20926			−0.604632	−	0.796505i	$-0.706680\pi$
$701$	−32.0170			−1.20926			−0.604632	+	0.796505i	$0.706680\pi$
$702$		0			0
$703$		−	65.9034i		−	2.48559i
$704$		0			0
$705$	17.4937	−	7.26084i	0.658849	−	0.273459i
$706$		0			0
$707$			35.0397i			1.31780i
$708$		0			0
$709$	48.6956			1.82880			0.914401	−	0.404809i	$-0.132662\pi$
$709$	48.6956			1.82880			0.914401	+	0.404809i	$0.132662\pi$
$710$		0			0
$711$	0.193553			0.00725881
$712$		0			0
$713$		−	6.03405i		−	0.225977i
$714$		0			0
$715$	0.393125	+	0.947163i	0.0147021	+	0.0354219i
$716$		0			0
$717$			0.364657i			0.0136184i
$718$		0			0
$719$	−19.0133			−0.709078			−0.354539	−	0.935041i	$-0.615362\pi$
$719$	−19.0133			−0.709078			−0.354539	+	0.935041i	$0.615362\pi$
$720$		0			0
$721$	52.0503			1.93845
$722$		0			0
$723$		−	9.57384i		−	0.356055i
$724$		0			0
$725$	−4.48842	+	4.50133i	−0.166696	+	0.167175i
$726$		0			0
$727$		−	39.1687i		−	1.45269i	−0.687332	−	0.726343i	$-0.741218\pi$
$727$		−	39.1687i		−	1.45269i	0.687332	−	0.726343i	$-0.258782\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	38.1588			1.41136
$732$		0			0
$733$			15.1029i			0.557838i	0.960315	+	0.278919i	$0.0899762\pi$
$733$			15.1029i			0.557838i	−0.960315	+	0.278919i	$0.910024\pi$
$734$		0			0
$735$	−2.93405	−	7.06905i	−0.108224	−	0.260746i
$736$		0			0
$737$		−	0.0944362i		−	0.00347860i
$738$		0			0
$739$	14.4962			0.533252			0.266626	−	0.963800i	$-0.414091\pi$
$739$	14.4962			0.533252			0.266626	+	0.963800i	$0.414091\pi$
$740$		0			0
$741$	58.9521			2.16566
$742$		0			0
$743$			35.5357i			1.30368i	0.758357	+	0.651840i	$0.226003\pi$
$743$			35.5357i			1.30368i	−0.758357	+	0.651840i	$0.773997\pi$
$744$		0			0
$745$	−4.50507	+	1.86986i	−0.165053	+	0.0685062i
$746$		0			0
$747$		−	7.81029i		−	0.285763i
$748$		0			0
$749$	31.9918			1.16896
$750$		0			0
$751$	21.1876			0.773146			0.386573	−	0.922259i	$-0.373659\pi$
$751$	21.1876			0.773146			0.386573	+	0.922259i	$0.373659\pi$
$752$		0			0
$753$			13.7873i			0.502438i
$754$		0			0
$755$	23.9804	−	9.95321i	0.872738	−	0.362235i
$756$		0			0
$757$			16.1950i			0.588619i	0.955710	+	0.294309i	$0.0950896\pi$
$757$			16.1950i			0.588619i	−0.955710	+	0.294309i	$0.904910\pi$
$758$		0			0
$759$	−0.0657386			−0.00238616
$760$		0			0
$761$	43.2049			1.56618			0.783089	−	0.621910i	$-0.213643\pi$
$761$	43.2049			1.56618			0.783089	+	0.621910i	$0.213643\pi$
$762$		0			0
$763$			66.8669i			2.42074i
$764$		0			0
$765$	5.07318	+	12.2229i	0.183421	+	0.441920i
$766$		0			0
$767$			6.76214i			0.244167i
$768$		0			0
$769$	−21.7056			−0.782724			−0.391362	−	0.920237i	$-0.627996\pi$
$769$	−21.7056			−0.782724			−0.391362	+	0.920237i	$0.627996\pi$
$770$		0			0
$771$	−6.64434			−0.239290
$772$		0			0
$773$			29.4351i			1.05871i	0.848402	+	0.529353i	$0.177565\pi$
$773$			29.4351i			1.05871i	−0.848402	+	0.529353i	$0.822435\pi$
$774$		0			0
$775$	21.3029	−	21.3642i	0.765223	−	0.767425i
$776$		0			0
$777$		−	25.1788i		−	0.903285i
$778$		0			0
$779$	77.6485			2.78205
$780$		0			0
$781$	0.387674			0.0138721
$782$		0			0
$783$		−	1.27134i		−	0.0454341i
$784$		0			0
$785$	16.9453	+	40.8266i	0.604803	+	1.45716i
$786$		0			0
$787$		−	46.8169i		−	1.66884i	−0.551128	−	0.834421i	$-0.685802\pi$
$787$		−	46.8169i		−	1.66884i	0.551128	−	0.834421i	$-0.314198\pi$
$788$		0			0
$789$	−2.43357			−0.0866374
$790$		0			0
$791$	−46.2109			−1.64307
$792$		0			0
$793$		−	28.8545i		−	1.02465i
$794$		0			0
$795$	−26.6535	+	11.0627i	−0.945301	+	0.392352i
$796$		0			0
$797$			33.3666i			1.18191i	0.806706	+	0.590953i	$0.201248\pi$
$797$			33.3666i			1.18191i	−0.806706	+	0.590953i	$0.798752\pi$
$798$		0			0
$799$	50.1317			1.77353
$800$		0			0
$801$	−7.43659			−0.262759
$802$		0			0
$803$		−	0.0213717i		−	0.000754192i
$804$		0			0
$805$	6.66752	−	2.76739i	0.234999	−	0.0975377i
$806$		0			0
$807$			5.77031i			0.203125i
$808$		0			0
$809$	15.7442			0.553536			0.276768	−	0.960937i	$-0.410737\pi$
$809$	15.7442			0.553536			0.276768	+	0.960937i	$0.410737\pi$
$810$		0			0
$811$	−38.8986			−1.36592			−0.682958	−	0.730458i	$-0.739307\pi$
$811$	−38.8986			−1.36592			−0.682958	+	0.730458i	$0.739307\pi$
$812$		0			0
$813$			7.33639i			0.257298i
$814$		0			0
$815$	−15.8654	−	38.2248i	−0.555742	−	1.33896i
$816$		0			0
$817$			54.4826i			1.90611i
$818$		0			0
$819$	22.5230			0.787018
$820$		0			0
$821$	−51.0285			−1.78091			−0.890453	−	0.455075i	$-0.849612\pi$
$821$	−51.0285			−1.78091			−0.890453	+	0.455075i	$0.849612\pi$
$822$		0			0
$823$		−	10.2071i		−	0.355796i	−0.984049	−	0.177898i	$-0.943070\pi$
$823$		−	10.2071i		−	0.355796i	0.984049	−	0.177898i	$-0.0569297\pi$
$824$		0			0
$825$	−0.232755	−	0.232087i	−0.00810349	−	0.00808023i
$826$		0			0
$827$			23.1561i			0.805216i	0.915373	+	0.402608i	$0.131896\pi$
$827$			23.1561i			0.805216i	−0.915373	+	0.402608i	$0.868104\pi$
$828$		0			0
$829$	42.0270			1.45966			0.729829	−	0.683630i	$-0.239600\pi$
$829$	42.0270			1.45966			0.729829	+	0.683630i	$0.239600\pi$
$830$		0			0
$831$	5.04718			0.175085
$832$		0			0
$833$		−	20.2578i		−	0.701892i
$834$		0			0
$835$	−6.60692	−	15.9182i	−0.228642	−	0.550871i
$836$		0			0
$837$			6.03405i			0.208567i
$838$		0			0
$839$	52.8804			1.82563			0.912817	−	0.408368i	$-0.133902\pi$
$839$	52.8804			1.82563			0.912817	+	0.408368i	$0.133902\pi$
$840$		0			0
$841$	−27.3837			−0.944265
$842$		0			0
$843$			5.26255i			0.181252i
$844$		0			0
$845$	73.6683	−	30.5764i	2.53427	−	1.05186i
$846$		0			0
$847$			35.4990i			1.21976i
$848$		0			0
$849$	−21.9216			−0.752348
$850$		0			0
$851$	7.79905			0.267348
$852$		0			0
$853$		−	16.5381i		−	0.566255i	−0.959082	−	0.283128i	$-0.908628\pi$
$853$		−	16.5381i		−	0.566255i	0.959082	−	0.283128i	$-0.0913720\pi$
$854$		0			0
$855$	−17.4517	+	7.24341i	−0.596834	+	0.247719i
$856$		0			0
$857$			32.4506i			1.10849i	0.832353	+	0.554246i	$0.186993\pi$
$857$			32.4506i			1.10849i	−0.832353	+	0.554246i	$0.813007\pi$
$858$		0			0
$859$	2.16762			0.0739583			0.0369791	−	0.999316i	$-0.488226\pi$
$859$	2.16762			0.0739583			0.0369791	+	0.999316i	$0.488226\pi$
$860$		0			0
$861$	29.6661			1.01102
$862$		0			0
$863$		−	29.9774i		−	1.02044i	−0.860043	−	0.510222i	$-0.829563\pi$
$863$		−	29.9774i		−	1.02044i	0.860043	−	0.510222i	$-0.170437\pi$
$864$		0			0
$865$	16.3905	+	39.4900i	0.557294	+	1.34270i
$866$		0			0
$867$			18.0273i			0.612238i
$868$		0			0
$869$	−0.0127239			−0.000431630
$870$		0			0
$871$	−10.0219			−0.339580
$872$		0			0
$873$		−	13.9149i		−	0.470949i
$874$		0			0
$875$	33.3772	+	13.7411i	1.12836	+	0.464534i
$876$		0			0
$877$		−	3.20629i		−	0.108269i	−0.998534	−	0.0541344i	$-0.982760\pi$
$877$		−	3.20629i		−	0.108269i	0.998534	−	0.0541344i	$-0.0172399\pi$
$878$		0			0
$879$	22.8901			0.772064
$880$		0			0
$881$	−24.4815			−0.824803			−0.412401	−	0.911002i	$-0.635310\pi$
$881$	−24.4815			−0.824803			−0.412401	+	0.911002i	$0.635310\pi$
$882$		0			0
$883$		−	12.8213i		−	0.431470i	−0.976452	−	0.215735i	$-0.930785\pi$
$883$		−	12.8213i		−	0.431470i	0.976452	−	0.215735i	$-0.0692148\pi$
$884$		0			0
$885$	−0.830861	−	2.00181i	−0.0279291	−	0.0672900i
$886$		0			0
$887$			15.4201i			0.517756i	0.965910	+	0.258878i	$0.0833528\pi$
$887$			15.4201i			0.517756i	−0.965910	+	0.258878i	$0.916647\pi$
$888$		0			0
$889$	34.7223			1.16455
$890$		0			0
$891$	0.0657386			0.00220233
$892$		0			0
$893$			71.5773i			2.39524i
$894$		0			0
$895$	−7.22354	+	2.99817i	−0.241456	+	0.100218i
$896$		0			0
$897$			6.97643i			0.232936i
$898$		0			0
$899$	−7.67135			−0.255854
$900$		0			0
$901$	−76.3811			−2.54462
$902$		0			0
$903$			20.8154i			0.692694i
$904$		0			0
$905$	27.2169	−	11.2965i	0.904719	−	0.375509i
$906$		0			0
$907$		−	40.0907i		−	1.33119i	−0.746313	−	0.665595i	$-0.768178\pi$
$907$		−	40.0907i		−	1.33119i	0.746313	−	0.665595i	$-0.231822\pi$
$908$		0			0
$909$	10.8534			0.359986
$910$		0			0
$911$	−6.30556			−0.208913			−0.104456	−	0.994529i	$-0.533310\pi$
$911$	−6.30556			−0.208913			−0.104456	+	0.994529i	$0.533310\pi$
$912$		0			0
$913$			0.513438i			0.0169923i
$914$		0			0
$915$	3.54533	+	8.54183i	0.117205	+	0.282384i
$916$		0			0
$917$			33.9155i			1.11999i
$918$		0			0
$919$	−23.4471			−0.773447			−0.386724	−	0.922196i	$-0.626393\pi$
$919$	−23.4471			−0.773447			−0.386724	+	0.922196i	$0.626393\pi$
$920$		0			0
$921$	−6.34198			−0.208975
$922$		0			0
$923$		−	41.1414i		−	1.35419i
$924$		0			0
$925$	27.6134	+	27.5342i	0.907923	+	0.905318i
$926$		0			0
$927$		−	16.1224i		−	0.529529i
$928$		0			0
$929$	8.42414			0.276387			0.138193	−	0.990405i	$-0.455870\pi$
$929$	8.42414			0.276387			0.138193	+	0.990405i	$0.455870\pi$
$930$		0			0
$931$	28.9238			0.947940
$932$		0			0
$933$		−	1.15390i		−	0.0377770i
$934$		0			0
$935$	−0.333504	−	0.803516i	−0.0109067	−	0.0262778i
$936$		0			0
$937$			7.52677i			0.245889i	0.992414	+	0.122944i	$0.0392337\pi$
$937$			7.52677i			0.245889i	−0.992414	+	0.122944i	$0.960766\pi$
$938$		0			0
$939$	−8.87225			−0.289535
$940$		0			0
$941$	−12.3725			−0.403332			−0.201666	−	0.979454i	$-0.564636\pi$
$941$	−12.3725			−0.403332			−0.201666	+	0.979454i	$0.564636\pi$
$942$		0			0
$943$			9.18897i			0.299234i
$944$		0			0
$945$	−6.66752	+	2.76739i	−0.216895	+	0.0900233i
$946$		0			0
$947$			31.3397i			1.01840i	0.860647	+	0.509202i	$0.170059\pi$
$947$			31.3397i			1.01840i	−0.860647	+	0.509202i	$0.829941\pi$
$948$		0			0
$949$	−2.26805			−0.0736239
$950$		0			0
$951$	4.78940			0.155307
$952$		0			0
$953$		−	56.3853i		−	1.82650i	−0.407400	−	0.913250i	$-0.633564\pi$
$953$		−	56.3853i		−	1.82650i	0.407400	−	0.913250i	$-0.366436\pi$
$954$		0			0
$955$	8.33265	−	3.45851i	0.269638	−	0.111915i
$956$		0			0
$957$			0.0835765i			0.00270164i
$958$		0			0
$959$	62.6834			2.02415
$960$		0			0
$961$	5.40970			0.174507
$962$		0			0
$963$		−	9.90935i		−	0.319324i
$964$		0			0
$965$	6.57909	+	15.8511i	0.211788	+	0.510265i
$966$		0			0
$967$		−	59.2814i		−	1.90636i	−0.302400	−	0.953181i	$-0.597788\pi$
$967$		−	59.2814i		−	1.90636i	0.302400	−	0.953181i	$-0.402212\pi$
$968$		0			0
$969$	−50.0114			−1.60660
$970$		0			0
$971$	−2.49186			−0.0799677			−0.0399839	−	0.999200i	$-0.512731\pi$
$971$	−2.49186			−0.0799677			−0.0399839	+	0.999200i	$0.512731\pi$
$972$		0			0
$973$		−	12.4946i		−	0.400558i
$974$		0			0
$975$	−24.6299	+	24.7008i	−0.788789	+	0.791059i
$976$		0			0
$977$		−	13.8501i		−	0.443104i	−0.975149	−	0.221552i	$-0.928888\pi$
$977$		−	13.8501i		−	0.443104i	0.975149	−	0.221552i	$-0.0711122\pi$
$978$		0			0
$979$	0.488871			0.0156244
$980$		0			0
$981$	20.7118			0.661277
$982$		0			0
$983$		−	15.5777i		−	0.496851i	−0.968651	−	0.248426i	$-0.920087\pi$
$983$		−	15.5777i		−	0.496851i	0.968651	−	0.248426i	$-0.0799131\pi$
$984$		0			0
$985$	−14.2220	−	34.2654i	−0.453152	−	1.09179i
$986$		0			0
$987$			27.3466i			0.870451i
$988$		0			0
$989$	−6.44751			−0.205019
$990$		0			0
$991$	6.44736			0.204807			0.102404	−	0.994743i	$-0.467347\pi$
$991$	6.44736			0.204807			0.102404	+	0.994743i	$0.467347\pi$
$992$		0			0
$993$			27.6627i			0.877848i
$994$		0			0
$995$	−35.7352	+	14.8321i	−1.13288	+	0.470209i
$996$		0			0
$997$		−	46.6181i		−	1.47641i	−0.674576	−	0.738206i	$-0.735673\pi$
$997$		−	46.6181i		−	1.47641i	0.674576	−	0.738206i	$-0.264327\pi$
$998$		0			0
$999$	−7.79905			−0.246751

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2760.2.k.f.2209.2	✓	22
5.4	even	2	inner	2760.2.k.f.2209.13	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.k.f.2209.2	✓	22	1.1	even	1	trivial
2760.2.k.f.2209.13	yes	22	5.4	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:	\( N \)	\(=\)	\( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2760.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-2.06524 + 0.857190i) q^{5} -3.22845i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(-2.06524 + 0.857190i) q^{5} -3.22845i q^{7} -1.00000 q^{9} +0.0657386 q^{11} -6.97643i q^{13} +(0.857190 + 2.06524i) q^{15} +5.91838i q^{17} -8.45018 q^{19} -3.22845 q^{21} -1.00000i q^{23} +(3.53045 - 3.54061i) q^{25} +1.00000i q^{27} -1.27134 q^{29} +6.03405 q^{31} -0.0657386i q^{33} +(2.76739 + 6.66752i) q^{35} +7.79905i q^{37} -6.97643 q^{39} -9.18897 q^{41} -6.44751i q^{43} +(2.06524 - 0.857190i) q^{45} -8.47051i q^{47} -3.42286 q^{49} +5.91838 q^{51} +12.9057i q^{53} +(-0.135766 + 0.0563505i) q^{55} +8.45018i q^{57} -0.969284 q^{59} +4.13599 q^{61} +3.22845i q^{63} +(5.98012 + 14.4080i) q^{65} -1.43654i q^{67} -1.00000 q^{69} +5.89720 q^{71} -0.325102i q^{73} +(-3.54061 - 3.53045i) q^{75} -0.212234i q^{77} -0.193553 q^{79} +1.00000 q^{81} +7.81029i q^{83} +(-5.07318 - 12.2229i) q^{85} +1.27134i q^{87} +7.43659 q^{89} -22.5230 q^{91} -6.03405i q^{93} +(17.4517 - 7.24341i) q^{95} +13.9149i q^{97} -0.0657386 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 2 q^{5} - 22 q^{9}+O(q^{10}) \) 22 * q - 2 * q^5 - 22 * q^9	\( 22 q - 2 q^{5} - 22 q^{9} - 16 q^{19} + 6 q^{21} + 2 q^{25} - 22 q^{29} + 6 q^{31} + 14 q^{35} - 12 q^{39} + 6 q^{41} + 2 q^{45} - 68 q^{49} + 10 q^{51} + 12 q^{55} - 18 q^{59} + 16 q^{61} + 44 q^{65} - 22 q^{69} - 30 q^{71} + 4 q^{75} + 36 q^{79} + 22 q^{81} + 34 q^{85} - 16 q^{89} + 28 q^{91} + 20 q^{95}+O(q^{100}) \) 22 * q - 2 * q^5 - 22 * q^9 - 16 * q^19 + 6 * q^21 + 2 * q^25 - 22 * q^29 + 6 * q^31 + 14 * q^35 - 12 * q^39 + 6 * q^41 + 2 * q^45 - 68 * q^49 + 10 * q^51 + 12 * q^55 - 18 * q^59 + 16 * q^61 + 44 * q^65 - 22 * q^69 - 30 * q^71 + 4 * q^75 + 36 * q^79 + 22 * q^81 + 34 * q^85 - 16 * q^89 + 28 * q^91 + 20 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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