LMFDB - Embedded newform 1344.4.b.g.895.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(895,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1344.895");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 2x^{10} - 6x^{9} + 56x^{7} - 448x^{6} + 448x^{5} - 3072x^{3} - 8192x^{2} - 32768x + 262144 $ x^12 - x^11 - 2x^10 - 6x^9 + 56x^7 - 448x^6 + 448x^5 - 3072x^3 - 8192x^2 - 32768x + 262144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{30} $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.7
Root			$2.16644 + 1.81839i$ of defining polynomial
Character	$\chi$	$=$	1344.895
Dual form			1344.4.b.g.895.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +4.47531i q^{5} +(14.9825 + 10.8869i) q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +4.47531i q^{5} +(14.9825 + 10.8869i) q^{7} +9.00000 q^{9} -7.61561i q^{11} +13.3620i q^{13} -13.4259i q^{15} -55.3574i q^{17} +73.9099 q^{19} +(-44.9475 - 32.6607i) q^{21} -133.446i q^{23} +104.972 q^{25} -27.0000 q^{27} -23.3760 q^{29} -241.005 q^{31} +22.8468i q^{33} +(-48.7222 + 67.0514i) q^{35} -178.979 q^{37} -40.0861i q^{39} -494.101i q^{41} +72.6137i q^{43} +40.2778i q^{45} +59.7992 q^{47} +(105.951 + 326.226i) q^{49} +166.072i q^{51} +569.477 q^{53} +34.0822 q^{55} -221.730 q^{57} -59.5343 q^{59} -629.889i q^{61} +(134.843 + 97.9820i) q^{63} -59.7992 q^{65} -599.636i q^{67} +400.338i q^{69} +407.701i q^{71} +680.155i q^{73} -314.915 q^{75} +(82.9103 - 114.101i) q^{77} +1084.52i q^{79} +81.0000 q^{81} -935.224 q^{83} +247.741 q^{85} +70.1281 q^{87} +12.1437i q^{89} +(-145.471 + 200.197i) q^{91} +723.016 q^{93} +330.770i q^{95} -1433.48i q^{97} -68.5405i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 36 q^{3} - 10 q^{7} + 108 q^{9}+O(q^{10}) $ 12 * q - 36 * q^3 - 10 * q^7 + 108 * q^9	$ 12 q - 36 q^{3} - 10 q^{7} + 108 q^{9} - 84 q^{19} + 30 q^{21} - 216 q^{25} - 324 q^{27} - 200 q^{29} + 384 q^{31} + 84 q^{35} + 244 q^{37} - 280 q^{47} - 424 q^{49} + 16 q^{53} - 212 q^{55} + 252 q^{57} + 1168 q^{59} - 90 q^{63} + 280 q^{65} + 648 q^{75} - 968 q^{77} + 972 q^{81} - 968 q^{83} + 852 q^{85} + 600 q^{87} + 1648 q^{91} - 1152 q^{93}+O(q^{100}) $ 12 * q - 36 * q^3 - 10 * q^7 + 108 * q^9 - 84 * q^19 + 30 * q^21 - 216 * q^25 - 324 * q^27 - 200 * q^29 + 384 * q^31 + 84 * q^35 + 244 * q^37 - 280 * q^47 - 424 * q^49 + 16 * q^53 - 212 * q^55 + 252 * q^57 + 1168 * q^59 - 90 * q^63 + 280 * q^65 + 648 * q^75 - 968 * q^77 + 972 * q^81 - 968 * q^83 + 852 * q^85 + 600 * q^87 + 1648 * q^91 - 1152 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	−3.00000			−0.577350
$3$	−3.00000			−0.577350
$4$		0			0
$5$			4.47531i			0.400284i	0.979767	+	0.200142i	$0.0641403\pi$
$5$			4.47531i			0.400284i	−0.979767	+	0.200142i	$0.935860\pi$
$6$		0			0
$7$	14.9825	+	10.8869i	0.808979	+	0.587837i
$7$	14.9825	+	10.8869i	0.808979	+	0.587837i
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$		−	7.61561i		−	0.208745i	−0.994538	−	0.104372i	$-0.966717\pi$
$11$		−	7.61561i		−	0.208745i	0.994538	−	0.104372i	$-0.0332834\pi$
$12$		0			0
$13$			13.3620i			0.285074i	0.989790	+	0.142537i	$0.0455259\pi$
$13$			13.3620i			0.285074i	−0.989790	+	0.142537i	$0.954474\pi$
$14$		0			0
$15$		−	13.4259i		−	0.231104i
$16$		0			0
$17$		−	55.3574i		−	0.789773i	−0.918730	−	0.394886i	$-0.870784\pi$
$17$		−	55.3574i		−	0.789773i	0.918730	−	0.394886i	$-0.129216\pi$
$18$		0			0
$19$	73.9099			0.892426			0.446213	−	0.894927i	$-0.352772\pi$
$19$	73.9099			0.892426			0.446213	+	0.894927i	$0.352772\pi$
$20$		0			0
$21$	−44.9475	−	32.6607i	−0.467064	−	0.339388i
$22$		0			0
$23$		−	133.446i		−	1.20980i	−0.796301	−	0.604901i	$-0.793213\pi$
$23$		−	133.446i		−	1.20980i	0.796301	−	0.604901i	$-0.206787\pi$
$24$		0			0
$25$	104.972			0.839773
$26$		0			0
$27$	−27.0000			−0.192450
$28$		0			0
$29$	−23.3760			−0.149684			−0.0748418	−	0.997195i	$-0.523845\pi$
$29$	−23.3760			−0.149684			−0.0748418	+	0.997195i	$0.523845\pi$
$30$		0			0
$31$	−241.005			−1.39632			−0.698159	−	0.715943i	$-0.745997\pi$
$31$	−241.005			−1.39632			−0.698159	+	0.715943i	$0.745997\pi$
$32$		0			0
$33$			22.8468i			0.120519i
$34$		0			0
$35$	−48.7222	+	67.0514i	−0.235302	+	0.323821i
$36$		0			0
$37$	−178.979			−0.795240			−0.397620	−	0.917550i	$-0.630164\pi$
$37$	−178.979			−0.795240			−0.397620	+	0.917550i	$0.630164\pi$
$38$		0			0
$39$		−	40.0861i		−	0.164587i
$40$		0			0
$41$		−	494.101i		−	1.88209i	−0.338282	−	0.941045i	$-0.609846\pi$
$41$		−	494.101i		−	1.88209i	0.338282	−	0.941045i	$-0.390154\pi$
$42$		0			0
$43$			72.6137i			0.257523i	0.991676	+	0.128761i	$0.0411001\pi$
$43$			72.6137i			0.257523i	−0.991676	+	0.128761i	$0.958900\pi$
$44$		0			0
$45$			40.2778i			0.133428i
$46$		0			0
$47$	59.7992			0.185587			0.0927937	−	0.995685i	$-0.470420\pi$
$47$	59.7992			0.185587			0.0927937	+	0.995685i	$0.470420\pi$
$48$		0			0
$49$	105.951	+	326.226i	0.308895	+	0.951096i
$50$		0			0
$51$			166.072i			0.455976i
$52$		0			0
$53$	569.477			1.47592			0.737960	−	0.674845i	$-0.235789\pi$
$53$	569.477			1.47592			0.737960	+	0.674845i	$0.235789\pi$
$54$		0			0
$55$	34.0822			0.0835572
$56$		0			0
$57$	−221.730			−0.515242
$58$		0			0
$59$	−59.5343			−0.131368			−0.0656839	−	0.997840i	$-0.520923\pi$
$59$	−59.5343			−0.131368			−0.0656839	+	0.997840i	$0.520923\pi$
$60$		0			0
$61$		−	629.889i		−	1.32212i	−0.750335	−	0.661058i	$-0.770108\pi$
$61$		−	629.889i		−	1.32212i	0.750335	−	0.661058i	$-0.229892\pi$
$62$		0			0
$63$	134.843	+	97.9820i	0.269660	+	0.195946i
$64$		0			0
$65$	−59.7992			−0.114110
$66$		0			0
$67$		−	599.636i		−	1.09339i	−0.837332	−	0.546695i	$-0.815886\pi$
$67$		−	599.636i		−	1.09339i	0.837332	−	0.546695i	$-0.184114\pi$
$68$		0			0
$69$			400.338i			0.698479i
$70$		0			0
$71$			407.701i			0.681482i	0.940157	+	0.340741i	$0.110678\pi$
$71$			407.701i			0.681482i	−0.940157	+	0.340741i	$0.889322\pi$
$72$		0			0
$73$			680.155i			1.09050i	0.838275	+	0.545248i	$0.183564\pi$
$73$			680.155i			1.09050i	−0.838275	+	0.545248i	$0.816436\pi$
$74$		0			0
$75$	−314.915			−0.484843
$76$		0			0
$77$	82.9103	−	114.101i	0.122708	−	0.168870i
$78$		0			0
$79$			1084.52i			1.54454i	0.635296	+	0.772269i	$0.280878\pi$
$79$			1084.52i			1.54454i	−0.635296	+	0.772269i	$0.719122\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	−935.224			−1.23680			−0.618398	−	0.785865i	$-0.712218\pi$
$83$	−935.224			−1.23680			−0.618398	+	0.785865i	$0.712218\pi$
$84$		0			0
$85$	247.741			0.316133
$86$		0			0
$87$	70.1281			0.0864198
$88$		0			0
$89$			12.1437i			0.0144633i	0.999974	+	0.00723163i	$0.00230192\pi$
$89$			12.1437i			0.0144633i	−0.999974	+	0.00723163i	$0.997698\pi$
$90$		0			0
$91$	−145.471	+	200.197i	−0.167577	+	0.230619i
$92$		0			0
$93$	723.016			0.806164
$94$		0			0
$95$			330.770i			0.357224i
$96$		0			0
$97$		−	1433.48i		−	1.50049i	−0.661160	−	0.750245i	$-0.729936\pi$
$97$		−	1433.48i		−	1.50049i	0.661160	−	0.750245i	$-0.270064\pi$
$98$		0			0
$99$		−	68.5405i		−	0.0695816i
$100$		0			0
$101$		−	1771.45i		−	1.74521i	−0.488427	−	0.872605i	$-0.662429\pi$
$101$		−	1771.45i		−	1.74521i	0.488427	−	0.872605i	$-0.337571\pi$
$102$		0			0
$103$	−1667.56			−1.59524			−0.797618	−	0.603163i	$-0.793907\pi$
$103$	−1667.56			−1.59524			−0.797618	+	0.603163i	$0.793907\pi$
$104$		0			0
$105$	146.167	−	201.154i	0.135851	−	0.186958i
$106$		0			0
$107$			1105.21i			0.998551i	0.866443	+	0.499276i	$0.166400\pi$
$107$			1105.21i			0.998551i	−0.866443	+	0.499276i	$0.833600\pi$
$108$		0			0
$109$	353.412			0.310557			0.155279	−	0.987871i	$-0.450372\pi$
$109$	353.412			0.310557			0.155279	+	0.987871i	$0.450372\pi$
$110$		0			0
$111$	536.936			0.459132
$112$		0			0
$113$	796.483			0.663070			0.331535	−	0.943443i	$-0.392434\pi$
$113$	796.483			0.663070			0.331535	+	0.943443i	$0.392434\pi$
$114$		0			0
$115$	597.213			0.484264
$116$		0			0
$117$			120.258i			0.0950245i
$118$		0			0
$119$	602.670	−	829.392i	0.464258	−	0.638910i
$120$		0			0
$121$	1273.00			0.956426
$122$		0			0
$123$			1482.30i			1.08662i
$124$		0			0
$125$			1029.19i			0.736431i
$126$		0			0
$127$		−	1382.89i		−	0.966230i	−0.875557	−	0.483115i	$-0.839505\pi$
$127$		−	1382.89i		−	0.966230i	0.875557	−	0.483115i	$-0.160495\pi$
$128$		0			0
$129$		−	217.841i		−	0.148681i
$130$		0			0
$131$	2930.78			1.95468			0.977342	−	0.211668i	$-0.0678895\pi$
$131$	2930.78			1.95468			0.977342	+	0.211668i	$0.0678895\pi$
$132$		0			0
$133$	1107.36	+	804.649i	0.721954	+	0.524601i
$134$		0			0
$135$		−	120.833i		−	0.0770347i
$136$		0			0
$137$	1514.84			0.944685			0.472343	−	0.881415i	$-0.343409\pi$
$137$	1514.84			0.944685			0.472343	+	0.881415i	$0.343409\pi$
$138$		0			0
$139$	−1121.88			−0.684582			−0.342291	−	0.939594i	$-0.611203\pi$
$139$	−1121.88			−0.684582			−0.342291	+	0.939594i	$0.611203\pi$
$140$		0			0
$141$	−179.398			−0.107149
$142$		0			0
$143$	101.760			0.0595076
$144$		0			0
$145$		−	104.615i		−	0.0599159i
$146$		0			0
$147$	−317.853	−	978.678i	−0.178341	−	0.549116i
$148$		0			0
$149$	201.082			0.110559			0.0552796	−	0.998471i	$-0.482395\pi$
$149$	201.082			0.110559			0.0552796	+	0.998471i	$0.482395\pi$
$150$		0			0
$151$		−	1803.92i		−	0.972192i	−0.873906	−	0.486096i	$-0.838421\pi$
$151$		−	1803.92i		−	0.972192i	0.873906	−	0.486096i	$-0.161579\pi$
$152$		0			0
$153$		−	498.216i		−	0.263258i
$154$		0			0
$155$		−	1078.57i		−	0.558923i
$156$		0			0
$157$		−	1163.09i		−	0.591241i	−0.955305	−	0.295621i	$-0.904474\pi$
$157$		−	1163.09i		−	0.591241i	0.955305	−	0.295621i	$-0.0955265\pi$
$158$		0			0
$159$	−1708.43			−0.852122
$160$		0			0
$161$	1452.81	−	1999.36i	0.711166	−	0.978705i
$162$		0			0
$163$		−	1394.86i		−	0.670271i	−0.942170	−	0.335135i	$-0.891218\pi$
$163$		−	1394.86i		−	0.670271i	0.942170	−	0.335135i	$-0.108782\pi$
$164$		0			0
$165$	−102.247			−0.0482417
$166$		0			0
$167$	1161.55			0.538222			0.269111	−	0.963109i	$-0.413270\pi$
$167$	1161.55			0.538222			0.269111	+	0.963109i	$0.413270\pi$
$168$		0			0
$169$	2018.46			0.918733
$170$		0			0
$171$	665.189			0.297475
$172$		0			0
$173$		−	2133.98i		−	0.937825i	−0.883245	−	0.468913i	$-0.844646\pi$
$173$		−	2133.98i		−	0.937825i	0.883245	−	0.468913i	$-0.155354\pi$
$174$		0			0
$175$	1572.74	+	1142.81i	0.679359	+	0.493650i
$176$		0			0
$177$	178.603			0.0758452
$178$		0			0
$179$		−	2551.23i		−	1.06530i	−0.846337	−	0.532648i	$-0.821197\pi$
$179$		−	2551.23i		−	1.06530i	0.846337	−	0.532648i	$-0.178803\pi$
$180$		0			0
$181$			715.321i			0.293754i	0.989155	+	0.146877i	$0.0469221\pi$
$181$			715.321i			0.293754i	−0.989155	+	0.146877i	$0.953078\pi$
$182$		0			0
$183$			1889.67i			0.763324i
$184$		0			0
$185$		−	800.985i		−	0.318322i
$186$		0			0
$187$	−421.580			−0.164861
$188$		0			0
$189$	−404.528	−	293.946i	−0.155688	−	0.113129i
$190$		0			0
$191$		−	2766.30i		−	1.04797i	−0.851727	−	0.523986i	$-0.824444\pi$
$191$		−	2766.30i		−	1.04797i	0.851727	−	0.523986i	$-0.175556\pi$
$192$		0			0
$193$	1182.88			0.441169			0.220584	−	0.975368i	$-0.429204\pi$
$193$	1182.88			0.441169			0.220584	+	0.975368i	$0.429204\pi$
$194$		0			0
$195$	179.398			0.0658817
$196$		0			0
$197$	4748.28			1.71726			0.858631	−	0.512593i	$-0.171315\pi$
$197$	4748.28			1.71726			0.858631	+	0.512593i	$0.171315\pi$
$198$		0			0
$199$	918.553			0.327209			0.163604	−	0.986526i	$-0.447688\pi$
$199$	918.553			0.327209			0.163604	+	0.986526i	$0.447688\pi$
$200$		0			0
$201$			1798.91i			0.631269i
$202$		0			0
$203$	−350.232	−	254.493i	−0.121091	−	0.0879895i
$204$		0			0
$205$	2211.26			0.753370
$206$		0			0
$207$		−	1201.02i		−	0.403267i
$208$		0			0
$209$		−	562.869i		−	0.186289i
$210$		0			0
$211$			4595.79i			1.49946i	0.661742	+	0.749732i	$0.269818\pi$
$211$			4595.79i			1.49946i	−0.661742	+	0.749732i	$0.730182\pi$
$212$		0			0
$213$		−	1223.10i		−	0.393454i
$214$		0			0
$215$	−324.969			−0.103082
$216$		0			0
$217$	−3610.86	−	2623.80i	−1.12959	−	0.820807i
$218$		0			0
$219$		−	2040.47i		−	0.629598i
$220$		0			0
$221$	739.686			0.225143
$222$		0			0
$223$	2989.01			0.897575			0.448787	−	0.893639i	$-0.351856\pi$
$223$	2989.01			0.897575			0.448787	+	0.893639i	$0.351856\pi$
$224$		0			0
$225$	944.744			0.279924
$226$		0			0
$227$	1938.17			0.566698			0.283349	−	0.959017i	$-0.408554\pi$
$227$	1938.17			0.566698			0.283349	+	0.959017i	$0.408554\pi$
$228$		0			0
$229$		−	5281.13i		−	1.52396i	−0.647600	−	0.761980i	$-0.724227\pi$
$229$		−	5281.13i		−	1.52396i	0.647600	−	0.761980i	$-0.275773\pi$
$230$		0			0
$231$	−248.731	+	342.303i	−0.0708454	+	0.0974972i
$232$		0			0
$233$	−1490.57			−0.419101			−0.209550	−	0.977798i	$-0.567200\pi$
$233$	−1490.57			−0.419101			−0.209550	+	0.977798i	$0.567200\pi$
$234$		0			0
$235$			267.620i			0.0742876i
$236$		0			0
$237$		−	3253.57i		−	0.891739i
$238$		0			0
$239$			2036.18i			0.551085i	0.961289	+	0.275543i	$0.0888575\pi$
$239$			2036.18i			0.551085i	−0.961289	+	0.275543i	$0.911142\pi$
$240$		0			0
$241$		−	6102.51i		−	1.63111i	−0.578681	−	0.815554i	$-0.696432\pi$
$241$		−	6102.51i		−	1.63111i	0.578681	−	0.815554i	$-0.303568\pi$
$242$		0			0
$243$	−243.000			−0.0641500
$244$		0			0
$245$	−1459.96	+	474.164i	−0.380708	+	0.123646i
$246$		0			0
$247$			987.586i			0.254407i
$248$		0			0
$249$	2805.67			0.714065
$250$		0			0
$251$	1319.61			0.331846			0.165923	−	0.986139i	$-0.446940\pi$
$251$	1319.61			0.331846			0.165923	+	0.986139i	$0.446940\pi$
$252$		0			0
$253$	−1016.27			−0.252540
$254$		0			0
$255$	−743.224			−0.182520
$256$		0			0
$257$			4469.92i			1.08492i	0.840080	+	0.542462i	$0.182508\pi$
$257$			4469.92i			1.08492i	−0.840080	+	0.542462i	$0.817492\pi$
$258$		0			0
$259$	−2681.55	−	1948.52i	−0.643333	−	0.467472i
$260$		0			0
$261$	−210.384			−0.0498945
$262$		0			0
$263$			4673.64i			1.09577i	0.836552	+	0.547887i	$0.184568\pi$
$263$			4673.64i			1.09577i	−0.836552	+	0.547887i	$0.815432\pi$
$264$		0			0
$265$			2548.59i			0.590787i
$266$		0			0
$267$		−	36.4311i		−	0.00835036i
$268$		0			0
$269$			4633.12i			1.05013i	0.851061	+	0.525067i	$0.175960\pi$
$269$			4633.12i			1.05013i	−0.851061	+	0.525067i	$0.824040\pi$
$270$		0			0
$271$	−3004.01			−0.673359			−0.336680	−	0.941619i	$-0.609304\pi$
$271$	−3004.01			−0.673359			−0.336680	+	0.941619i	$0.609304\pi$
$272$		0			0
$273$	436.413	−	600.590i	0.0967505	−	0.133148i
$274$		0			0
$275$		−	799.423i		−	0.175298i
$276$		0			0
$277$	8852.78			1.92026			0.960130	−	0.279552i	$-0.0901861\pi$
$277$	8852.78			1.92026			0.960130	+	0.279552i	$0.0901861\pi$
$278$		0			0
$279$	−2169.05			−0.465439
$280$		0			0
$281$	378.664			0.0803885			0.0401943	−	0.999192i	$-0.487202\pi$
$281$	378.664			0.0803885			0.0401943	+	0.999192i	$0.487202\pi$
$282$		0			0
$283$	−1515.73			−0.318378			−0.159189	−	0.987248i	$-0.550888\pi$
$283$	−1515.73			−0.318378			−0.159189	+	0.987248i	$0.550888\pi$
$284$		0			0
$285$		−	992.309i		−	0.206243i
$286$		0			0
$287$	5379.23	−	7402.88i	1.10636	−	1.52257i
$288$		0			0
$289$	1848.56			0.376259
$290$		0			0
$291$			4300.43i			0.866308i
$292$		0			0
$293$		−	5745.66i		−	1.14561i	−0.819690	−	0.572807i	$-0.805854\pi$
$293$		−	5745.66i		−	1.14561i	0.819690	−	0.572807i	$-0.194146\pi$
$294$		0			0
$295$		−	266.434i		−	0.0525844i
$296$		0			0
$297$			205.621i			0.0401729i
$298$		0			0
$299$	1783.11			0.344883
$300$		0			0
$301$	−790.537	+	1087.93i	−0.151381	+	0.208331i
$302$		0			0
$303$			5314.36i			1.00760i
$304$		0			0
$305$	2818.95			0.529222
$306$		0			0
$307$	−591.984			−0.110053			−0.0550265	−	0.998485i	$-0.517524\pi$
$307$	−591.984			−0.110053			−0.0550265	+	0.998485i	$0.517524\pi$
$308$		0			0
$309$	5002.67			0.921010
$310$		0			0
$311$	−2628.66			−0.479284			−0.239642	−	0.970861i	$-0.577030\pi$
$311$	−2628.66			−0.479284			−0.239642	+	0.970861i	$0.577030\pi$
$312$		0			0
$313$		−	3781.59i		−	0.682902i	−0.939900	−	0.341451i	$-0.889082\pi$
$313$		−	3781.59i		−	0.682902i	0.939900	−	0.341451i	$-0.110918\pi$
$314$		0			0
$315$	−438.500	+	603.462i	−0.0784339	+	0.107940i
$316$		0			0
$317$	−3073.59			−0.544575			−0.272288	−	0.962216i	$-0.587780\pi$
$317$	−3073.59			−0.544575			−0.272288	+	0.962216i	$0.587780\pi$
$318$		0			0
$319$			178.023i			0.0312457i
$320$		0			0
$321$		−	3315.64i		−	0.576514i
$322$		0			0
$323$		−	4091.46i		−	0.704814i
$324$		0			0
$325$			1402.63i			0.239397i
$326$		0			0
$327$	−1060.24			−0.179300
$328$		0			0
$329$	895.942	+	651.027i	0.150136	+	0.109095i
$330$		0			0
$331$			6921.62i			1.14939i	0.818369	+	0.574693i	$0.194878\pi$
$331$			6921.62i			1.14939i	−0.818369	+	0.574693i	$0.805122\pi$
$332$		0			0
$333$	−1610.81			−0.265080
$334$		0			0
$335$	2683.56			0.437667
$336$		0			0
$337$	8596.94			1.38963			0.694815	−	0.719189i	$-0.255486\pi$
$337$	8596.94			1.38963			0.694815	+	0.719189i	$0.255486\pi$
$338$		0			0
$339$	−2389.45			−0.382823
$340$		0			0
$341$			1835.40i			0.291474i
$342$		0			0
$343$	−1964.17	+	6041.16i	−0.309200	+	0.950997i
$344$		0			0
$345$	−1791.64			−0.279590
$346$		0			0
$347$		−	4939.05i		−	0.764098i	−0.924142	−	0.382049i	$-0.875219\pi$
$347$		−	4939.05i		−	0.764098i	0.924142	−	0.382049i	$-0.124781\pi$
$348$		0			0
$349$		−	195.755i		−	0.0300244i	−0.999887	−	0.0150122i	$-0.995221\pi$
$349$		−	195.755i		−	0.0300244i	0.999887	−	0.0150122i	$-0.00477872\pi$
$350$		0			0
$351$		−	360.775i		−	0.0548624i
$352$		0			0
$353$		−	837.552i		−	0.126284i	−0.998005	−	0.0631422i	$-0.979888\pi$
$353$		−	837.552i		−	0.126284i	0.998005	−	0.0631422i	$-0.0201122\pi$
$354$		0			0
$355$	−1824.59			−0.272786
$356$		0			0
$357$	−1808.01	+	2488.18i	−0.268039	+	0.368875i
$358$		0			0
$359$		−	5984.33i		−	0.879779i	−0.898052	−	0.439890i	$-0.855018\pi$
$359$		−	5984.33i		−	0.879779i	0.898052	−	0.439890i	$-0.144982\pi$
$360$		0			0
$361$	−1396.32			−0.203576
$362$		0			0
$363$	−3819.01			−0.552193
$364$		0			0
$365$	−3043.91			−0.436508
$366$		0			0
$367$	−3550.58			−0.505010			−0.252505	−	0.967596i	$-0.581254\pi$
$367$	−3550.58			−0.505010			−0.252505	+	0.967596i	$0.581254\pi$
$368$		0			0
$369$		−	4446.91i		−	0.627363i
$370$		0			0
$371$	8532.20	+	6199.84i	1.19399	+	0.867600i
$372$		0			0
$373$	−4863.68			−0.675152			−0.337576	−	0.941298i	$-0.609607\pi$
$373$	−4863.68			−0.675152			−0.337576	+	0.941298i	$0.609607\pi$
$374$		0			0
$375$		−	3087.58i		−	0.425179i
$376$		0			0
$377$		−	312.351i		−	0.0426708i
$378$		0			0
$379$			6050.73i			0.820066i	0.912071	+	0.410033i	$0.134483\pi$
$379$			6050.73i			0.820066i	−0.912071	+	0.410033i	$0.865517\pi$
$380$		0			0
$381$			4148.66i			0.557853i
$382$		0			0
$383$	3136.38			0.418437			0.209219	−	0.977869i	$-0.432908\pi$
$383$	3136.38			0.418437			0.209219	+	0.977869i	$0.432908\pi$
$384$		0			0
$385$	510.637	+	371.049i	0.0675960	+	0.0491180i
$386$		0			0
$387$			653.523i			0.0858409i
$388$		0			0
$389$	4086.79			0.532669			0.266335	−	0.963881i	$-0.414187\pi$
$389$	4086.79			0.532669			0.266335	+	0.963881i	$0.414187\pi$
$390$		0			0
$391$	−7387.23			−0.955469
$392$		0			0
$393$	−8792.34			−1.12854
$394$		0			0
$395$	−4853.58			−0.618254
$396$		0			0
$397$			4953.21i			0.626182i	0.949723	+	0.313091i	$0.101365\pi$
$397$			4953.21i			0.626182i	−0.949723	+	0.313091i	$0.898635\pi$
$398$		0			0
$399$	−3322.07	−	2413.95i	−0.416821	−	0.302879i
$400$		0			0
$401$	6920.41			0.861818			0.430909	−	0.902396i	$-0.358193\pi$
$401$	6920.41			0.861818			0.430909	+	0.902396i	$0.358193\pi$
$402$		0			0
$403$		−	3220.32i		−	0.398053i
$404$		0			0
$405$			362.500i			0.0444760i
$406$		0			0
$407$			1363.03i			0.166002i
$408$		0			0
$409$			15351.1i			1.85590i	0.372710	+	0.927948i	$0.378429\pi$
$409$			15351.1i			1.85590i	−0.372710	+	0.927948i	$0.621571\pi$
$410$		0			0
$411$	−4544.53			−0.545414
$412$		0			0
$413$	−891.973	−	648.143i	−0.106274	−	0.0772229i
$414$		0			0
$415$		−	4185.42i		−	0.495070i
$416$		0			0
$417$	3365.65			0.395244
$418$		0			0
$419$	12665.4			1.47672			0.738358	−	0.674409i	$-0.235601\pi$
$419$	12665.4			1.47672			0.738358	+	0.674409i	$0.235601\pi$
$420$		0			0
$421$	−11040.6			−1.27812			−0.639059	−	0.769158i	$-0.720676\pi$
$421$	−11040.6			−1.27812			−0.639059	+	0.769158i	$0.720676\pi$
$422$		0			0
$423$	538.193			0.0618625
$424$		0			0
$425$		−	5810.95i		−	0.663230i
$426$		0			0
$427$	6857.54	−	9437.32i	0.777189	−	1.06956i
$428$		0			0
$429$	−305.280			−0.0343567
$430$		0			0
$431$		−	11223.6i		−	1.25434i	−0.778881	−	0.627171i	$-0.784213\pi$
$431$		−	11223.6i		−	1.25434i	0.778881	−	0.627171i	$-0.215787\pi$
$432$		0			0
$433$			3804.05i			0.422196i	0.977465	+	0.211098i	$0.0677040\pi$
$433$			3804.05i			0.422196i	−0.977465	+	0.211098i	$0.932296\pi$
$434$		0			0
$435$			313.845i			0.0345925i
$436$		0			0
$437$		−	9862.99i		−	1.07966i
$438$		0			0
$439$	−729.294			−0.0792877			−0.0396438	−	0.999214i	$-0.512622\pi$
$439$	−729.294			−0.0792877			−0.0396438	+	0.999214i	$0.512622\pi$
$440$		0			0
$441$	953.560	+	2936.03i	0.102965	+	0.317032i
$442$		0			0
$443$		−	3801.56i		−	0.407714i	−0.979001	−	0.203857i	$-0.934652\pi$
$443$		−	3801.56i		−	0.407714i	0.979001	−	0.203857i	$-0.0653478\pi$
$444$		0			0
$445$	−54.3468			−0.00578941
$446$		0			0
$447$	−603.247			−0.0638313
$448$		0			0
$449$	−14008.3			−1.47237			−0.736185	−	0.676780i	$-0.763375\pi$
$449$	−14008.3			−1.47237			−0.736185	+	0.676780i	$0.763375\pi$
$450$		0			0
$451$	−3762.88			−0.392876
$452$		0			0
$453$			5411.76i			0.561295i
$454$		0			0
$455$	−895.942	−	651.027i	−0.0923130	−	0.0670783i
$456$		0			0
$457$	−15859.0			−1.62331			−0.811656	−	0.584136i	$-0.801434\pi$
$457$	−15859.0			−1.62331			−0.811656	+	0.584136i	$0.801434\pi$
$458$		0			0
$459$			1494.65i			0.151992i
$460$		0			0
$461$			7619.91i			0.769836i	0.922951	+	0.384918i	$0.125770\pi$
$461$			7619.91i			0.769836i	−0.922951	+	0.384918i	$0.874230\pi$
$462$		0			0
$463$			5312.60i			0.533255i	0.963800	+	0.266628i	$0.0859094\pi$
$463$			5312.60i			0.533255i	−0.963800	+	0.266628i	$0.914091\pi$
$464$		0			0
$465$			3235.72i			0.322694i
$466$		0			0
$467$	−17971.9			−1.78082			−0.890408	−	0.455163i	$-0.849581\pi$
$467$	−17971.9			−1.78082			−0.890408	+	0.455163i	$0.849581\pi$
$468$		0			0
$469$	6528.17	−	8984.05i	0.642735	−	0.884530i
$470$		0			0
$471$			3489.28i			0.341353i
$472$		0			0
$473$	552.997			0.0537565
$474$		0			0
$475$	7758.44			0.749435
$476$		0			0
$477$	5125.30			0.491973
$478$		0			0
$479$	11971.4			1.14194			0.570970	−	0.820971i	$-0.306567\pi$
$479$	11971.4			1.14194			0.570970	+	0.820971i	$0.306567\pi$
$480$		0			0
$481$		−	2391.52i		−	0.226702i
$482$		0			0
$483$	−4358.44	+	5998.07i	−0.410592	+	0.565055i
$484$		0			0
$485$	6415.25			0.600622
$486$		0			0
$487$			16208.1i			1.50813i	0.656802	+	0.754063i	$0.271909\pi$
$487$			16208.1i			1.50813i	−0.656802	+	0.754063i	$0.728091\pi$
$488$		0			0
$489$			4184.59i			0.386981i
$490$		0			0
$491$		−	4844.19i		−	0.445245i	−0.974905	−	0.222622i	$-0.928538\pi$
$491$		−	4844.19i		−	0.445245i	0.974905	−	0.222622i	$-0.0714617\pi$
$492$		0			0
$493$			1294.04i			0.118216i
$494$		0			0
$495$	306.740			0.0278524
$496$		0			0
$497$	−4438.60	+	6108.39i	−0.400600	+	0.551305i
$498$		0			0
$499$		−	3372.90i		−	0.302589i	−0.988489	−	0.151294i	$-0.951656\pi$
$499$		−	3372.90i		−	0.302589i	0.988489	−	0.151294i	$-0.0483441\pi$
$500$		0			0
$501$	−3484.64			−0.310743
$502$		0			0
$503$	18916.6			1.67684			0.838420	−	0.545024i	$-0.183480\pi$
$503$	18916.6			1.67684			0.838420	+	0.545024i	$0.183480\pi$
$504$		0			0
$505$	7927.80			0.698579
$506$		0			0
$507$	−6055.37			−0.530431
$508$		0			0
$509$			14473.0i			1.26033i	0.776463	+	0.630163i	$0.217012\pi$
$509$			14473.0i			1.26033i	−0.776463	+	0.630163i	$0.782988\pi$
$510$		0			0
$511$	−7404.78	+	10190.4i	−0.641033	+	0.882188i
$512$		0			0
$513$	−1995.57			−0.171747
$514$		0			0
$515$		−	7462.83i		−	0.638547i
$516$		0			0
$517$		−	455.407i		−	0.0387404i
$518$		0			0
$519$			6401.95i			0.541454i
$520$		0			0
$521$		−	3641.31i		−	0.306197i	−0.988211	−	0.153098i	$-0.951075\pi$
$521$		−	3641.31i		−	0.306197i	0.988211	−	0.153098i	$-0.0489251\pi$
$522$		0			0
$523$	−16089.9			−1.34525			−0.672623	−	0.739985i	$-0.734832\pi$
$523$	−16089.9			−1.34525			−0.672623	+	0.739985i	$0.734832\pi$
$524$		0			0
$525$	−4718.21	−	3428.44i	−0.392228	−	0.285009i
$526$		0			0
$527$			13341.4i			1.10277i
$528$		0			0
$529$	−5640.87			−0.463620
$530$		0			0
$531$	−535.808			−0.0437893
$532$		0			0
$533$	6602.19			0.536534
$534$		0			0
$535$	−4946.17			−0.399704
$536$		0			0
$537$			7653.69i			0.615049i
$538$		0			0
$539$	2484.41	−	806.882i	0.198536	−	0.0644803i
$540$		0			0
$541$	−18144.0			−1.44190			−0.720952	−	0.692985i	$-0.756295\pi$
$541$	−18144.0			−1.44190			−0.720952	+	0.692985i	$0.756295\pi$
$542$		0			0
$543$		−	2145.96i		−	0.169599i
$544$		0			0
$545$			1581.63i			0.124311i
$546$		0			0
$547$			12520.2i			0.978653i	0.872101	+	0.489326i	$0.162757\pi$
$547$			12520.2i			0.978653i	−0.872101	+	0.489326i	$0.837243\pi$
$548$		0			0
$549$		−	5669.00i		−	0.440705i
$550$		0			0
$551$	−1727.72			−0.133582
$552$		0			0
$553$	−11807.1	+	16248.9i	−0.907936	+	1.24950i
$554$		0			0
$555$			2402.95i			0.183783i
$556$		0			0
$557$	−8252.03			−0.627737			−0.313869	−	0.949466i	$-0.601625\pi$
$557$	−8252.03			−0.627737			−0.313869	+	0.949466i	$0.601625\pi$
$558$		0			0
$559$	−970.265			−0.0734130
$560$		0			0
$561$	1264.74			0.0951825
$562$		0			0
$563$	18486.4			1.38386			0.691928	−	0.721967i	$-0.256762\pi$
$563$	18486.4			1.38386			0.691928	+	0.721967i	$0.256762\pi$
$564$		0			0
$565$			3564.51i			0.265416i
$566$		0			0
$567$	1213.58	+	881.838i	0.0898866	+	0.0653152i
$568$		0			0
$569$	−22842.2			−1.68294			−0.841471	−	0.540302i	$-0.818310\pi$
$569$	−22842.2			−1.68294			−0.841471	+	0.540302i	$0.818310\pi$
$570$		0			0
$571$			7404.29i			0.542662i	0.962486	+	0.271331i	$0.0874638\pi$
$571$			7404.29i			0.542662i	−0.962486	+	0.271331i	$0.912536\pi$
$572$		0			0
$573$			8298.91i			0.605047i
$574$		0			0
$575$		−	14008.1i		−	1.01596i
$576$		0			0
$577$		−	680.722i		−	0.0491141i	−0.999698	−	0.0245570i	$-0.992182\pi$
$577$		−	680.722i		−	0.0491141i	0.999698	−	0.0245570i	$-0.00781753\pi$
$578$		0			0
$579$	−3548.64			−0.254709
$580$		0			0
$581$	−14012.0	−	10181.7i	−1.00054	−	0.727035i
$582$		0			0
$583$		−	4336.92i		−	0.308090i
$584$		0			0
$585$	−538.193			−0.0380368
$586$		0			0
$587$	21647.2			1.52210			0.761052	−	0.648691i	$-0.224683\pi$
$587$	21647.2			1.52210			0.761052	+	0.648691i	$0.224683\pi$
$588$		0			0
$589$	−17812.7			−1.24611
$590$		0			0
$591$	−14244.8			−0.991462
$592$		0			0
$593$		−	14571.9i		−	1.00910i	−0.863383	−	0.504549i	$-0.831659\pi$
$593$		−	14571.9i		−	1.00910i	0.863383	−	0.504549i	$-0.168341\pi$
$594$		0			0
$595$	3711.79	+	2697.13i	0.255745	+	0.185835i
$596$		0			0
$597$	−2755.66			−0.188914
$598$		0			0
$599$		−	23843.1i		−	1.62638i	−0.581997	−	0.813191i	$-0.697729\pi$
$599$		−	23843.1i		−	1.62638i	0.581997	−	0.813191i	$-0.302271\pi$
$600$		0			0
$601$		−	3664.55i		−	0.248719i	−0.992237	−	0.124360i	$-0.960312\pi$
$601$		−	3664.55i		−	0.248719i	0.992237	−	0.124360i	$-0.0396876\pi$
$602$		0			0
$603$		−	5396.72i		−	0.364463i
$604$		0			0
$605$			5697.08i			0.382842i
$606$		0			0
$607$	1982.67			0.132577			0.0662885	−	0.997801i	$-0.478884\pi$
$607$	1982.67			0.132577			0.0662885	+	0.997801i	$0.478884\pi$
$608$		0			0
$609$	1050.70	+	763.478i	0.0699119	+	0.0508008i
$610$		0			0
$611$			799.038i			0.0529061i
$612$		0			0
$613$	10215.4			0.673077			0.336538	−	0.941670i	$-0.390744\pi$
$613$	10215.4			0.673077			0.336538	+	0.941670i	$0.390744\pi$
$614$		0			0
$615$	−6633.77			−0.434958
$616$		0			0
$617$	−10912.4			−0.712022			−0.356011	−	0.934482i	$-0.615863\pi$
$617$	−10912.4			−0.712022			−0.356011	+	0.934482i	$0.615863\pi$
$618$		0			0
$619$	−6816.95			−0.442644			−0.221322	−	0.975201i	$-0.571037\pi$
$619$	−6816.95			−0.442644			−0.221322	+	0.975201i	$0.571037\pi$
$620$		0			0
$621$			3603.05i			0.232826i
$622$		0			0
$623$	−132.207	+	181.943i	−0.00850204	+	0.0117005i
$624$		0			0
$625$	8515.49			0.544991
$626$		0			0
$627$			1688.61i			0.107554i
$628$		0			0
$629$			9907.79i			0.628059i
$630$		0			0
$631$		−	27480.9i		−	1.73375i	−0.498527	−	0.866874i	$-0.666125\pi$
$631$		−	27480.9i		−	1.73375i	0.498527	−	0.866874i	$-0.333875\pi$
$632$		0			0
$633$		−	13787.4i		−	0.865716i
$634$		0			0
$635$	6188.84			0.386766
$636$		0			0
$637$	−4359.04	+	1415.72i	−0.271132	+	0.0880579i
$638$		0			0
$639$			3669.31i			0.227161i
$640$		0			0
$641$	31255.0			1.92589			0.962946	−	0.269694i	$-0.0869225\pi$
$641$	31255.0			1.92589			0.962946	+	0.269694i	$0.0869225\pi$
$642$		0			0
$643$	3994.76			0.245004			0.122502	−	0.992468i	$-0.460908\pi$
$643$	3994.76			0.245004			0.122502	+	0.992468i	$0.460908\pi$
$644$		0			0
$645$	974.906			0.0595145
$646$		0			0
$647$	12996.2			0.789697			0.394848	−	0.918746i	$-0.370797\pi$
$647$	12996.2			0.789697			0.394848	+	0.918746i	$0.370797\pi$
$648$		0			0
$649$			453.390i			0.0274223i
$650$		0			0
$651$	10832.6	+	7871.40i	0.652170	+	0.473893i
$652$		0			0
$653$	−18282.4			−1.09563			−0.547814	−	0.836600i	$-0.684540\pi$
$653$	−18282.4			−1.09563			−0.547814	+	0.836600i	$0.684540\pi$
$654$		0			0
$655$			13116.2i			0.782428i
$656$		0			0
$657$			6121.40i			0.363498i
$658$		0			0
$659$		−	19470.9i		−	1.15095i	−0.817818	−	0.575477i	$-0.804816\pi$
$659$		−	19470.9i		−	1.15095i	0.817818	−	0.575477i	$-0.195184\pi$
$660$		0			0
$661$			25139.3i			1.47928i	0.673000	+	0.739642i	$0.265005\pi$
$661$			25139.3i			1.47928i	−0.673000	+	0.739642i	$0.734995\pi$
$662$		0			0
$663$	−2219.06			−0.129987
$664$		0			0
$665$	−3601.06	+	4955.76i	−0.209989	+	0.288987i
$666$		0			0
$667$			3119.44i			0.181087i
$668$		0			0
$669$	−8967.04			−0.518215
$670$		0			0
$671$	−4796.99			−0.275985
$672$		0			0
$673$	12277.8			0.703232			0.351616	−	0.936144i	$-0.385632\pi$
$673$	12277.8			0.703232			0.351616	+	0.936144i	$0.385632\pi$
$674$		0			0
$675$	−2834.23			−0.161614
$676$		0			0
$677$		−	8737.50i		−	0.496026i	−0.968757	−	0.248013i	$-0.920222\pi$
$677$		−	8737.50i		−	0.496026i	0.968757	−	0.248013i	$-0.0797776\pi$
$678$		0			0
$679$	15606.1	−	21477.1i	0.882044	−	1.21387i
$680$		0			0
$681$	−5814.50			−0.327183
$682$		0			0
$683$			21358.4i			1.19657i	0.801284	+	0.598284i	$0.204151\pi$
$683$			21358.4i			1.19657i	−0.801284	+	0.598284i	$0.795849\pi$
$684$		0			0
$685$			6779.40i			0.378142i
$686$		0			0
$687$			15843.4i			0.879859i
$688$		0			0
$689$			7609.37i			0.420746i
$690$		0			0
$691$	−27244.9			−1.49992			−0.749961	−	0.661483i	$-0.769928\pi$
$691$	−27244.9			−1.49992			−0.749961	+	0.661483i	$0.769928\pi$
$692$		0			0
$693$	746.193	−	1026.91i	0.0409026	−	0.0562901i
$694$		0			0
$695$		−	5020.78i		−	0.274027i
$696$		0			0
$697$	−27352.2			−1.48642
$698$		0			0
$699$	4471.71			0.241968
$700$		0			0
$701$	−9913.36			−0.534126			−0.267063	−	0.963679i	$-0.586053\pi$
$701$	−9913.36			−0.534126			−0.267063	+	0.963679i	$0.586053\pi$
$702$		0			0
$703$	−13228.3			−0.709693
$704$		0			0
$705$		−	802.860i		−	0.0428900i
$706$		0			0
$707$	19285.6	−	26540.8i	1.02590	−	1.41184i
$708$		0			0
$709$	4521.15			0.239486			0.119743	−	0.992805i	$-0.461793\pi$
$709$	4521.15			0.239486			0.119743	+	0.992805i	$0.461793\pi$
$710$		0			0
$711$			9760.71i			0.514846i
$712$		0			0
$713$			32161.2i			1.68927i
$714$		0			0
$715$			455.407i			0.0238199i
$716$		0			0
$717$		−	6108.53i		−	0.318169i
$718$		0			0
$719$	14246.5			0.738948			0.369474	−	0.929241i	$-0.379538\pi$
$719$	14246.5			0.738948			0.369474	+	0.929241i	$0.379538\pi$
$720$		0			0
$721$	−24984.2	−	18154.5i	−1.29051	−	0.937738i
$722$		0			0
$723$			18307.5i			0.941720i
$724$		0			0
$725$	−2453.82			−0.125700
$726$		0			0
$727$	13979.5			0.713164			0.356582	−	0.934264i	$-0.383942\pi$
$727$	13979.5			0.713164			0.356582	+	0.934264i	$0.383942\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	4019.70			0.203384
$732$		0			0
$733$			22386.3i			1.12805i	0.825759	+	0.564023i	$0.190747\pi$
$733$			22386.3i			1.12805i	−0.825759	+	0.564023i	$0.809253\pi$
$734$		0			0
$735$	4379.89	−	1422.49i	0.219802	−	0.0713869i
$736$		0			0
$737$	−4566.59			−0.228240
$738$		0			0
$739$		−	19233.7i		−	0.957406i	−0.877977	−	0.478703i	$-0.841107\pi$
$739$		−	19233.7i		−	0.957406i	0.877977	−	0.478703i	$-0.158893\pi$
$740$		0			0
$741$		−	2962.76i		−	0.146882i
$742$		0			0
$743$			10980.6i			0.542181i	0.962554	+	0.271090i	$0.0873842\pi$
$743$			10980.6i			0.542181i	−0.962554	+	0.271090i	$0.912616\pi$
$744$		0			0
$745$			899.906i			0.0442550i
$746$		0			0
$747$	−8417.01			−0.412266
$748$		0			0
$749$	−12032.3	+	16558.9i	−0.586985	+	0.807807i
$750$		0			0
$751$		−	3799.25i		−	0.184603i	−0.995731	−	0.0923014i	$-0.970578\pi$
$751$		−	3799.25i		−	0.184603i	0.995731	−	0.0923014i	$-0.0294223\pi$
$752$		0			0
$753$	−3958.84			−0.191591
$754$		0			0
$755$	8073.10			0.389153
$756$		0			0
$757$	16774.7			0.805400			0.402700	−	0.915332i	$-0.368072\pi$
$757$	16774.7			0.805400			0.402700	+	0.915332i	$0.368072\pi$
$758$		0			0
$759$	3048.82			0.145804
$760$		0			0
$761$			37256.0i			1.77468i	0.461119	+	0.887338i	$0.347448\pi$
$761$			37256.0i			1.77468i	−0.461119	+	0.887338i	$0.652552\pi$
$762$		0			0
$763$	5295.00	+	3847.56i	0.251235	+	0.182557i
$764$		0			0
$765$	2229.67			0.105378
$766$		0			0
$767$		−	795.498i		−	0.0374495i
$768$		0			0
$769$		−	3677.87i		−	0.172468i	−0.996275	−	0.0862338i	$-0.972517\pi$
$769$		−	3677.87i		−	0.172468i	0.996275	−	0.0862338i	$-0.0274832\pi$
$770$		0			0
$771$		−	13409.7i		−	0.626382i
$772$		0			0
$773$		−	1000.27i		−	0.0465421i	−0.999729	−	0.0232711i	$-0.992592\pi$
$773$		−	1000.27i		−	0.0465421i	0.999729	−	0.0232711i	$-0.00740808\pi$
$774$		0			0
$775$	−25298.7			−1.17259
$776$		0			0
$777$	8044.64	+	5845.56i	0.371429	+	0.269895i
$778$		0			0
$779$		−	36519.0i		−	1.67963i
$780$		0			0
$781$	3104.89			0.142256
$782$		0			0
$783$	631.153			0.0288066
$784$		0			0
$785$	5205.20			0.236664
$786$		0			0
$787$	−29106.1			−1.31832			−0.659161	−	0.752002i	$-0.729088\pi$
$787$	−29106.1			−1.31832			−0.659161	+	0.752002i	$0.729088\pi$
$788$		0			0
$789$		−	14020.9i		−	0.632646i
$790$		0			0
$791$	11933.3	+	8671.23i	0.536410	+	0.389777i
$792$		0			0
$793$	8416.59			0.376900
$794$		0			0
$795$		−	7645.76i		−	0.341091i
$796$		0			0
$797$		−	26054.3i		−	1.15796i	−0.815343	−	0.578978i	$-0.803452\pi$
$797$		−	26054.3i		−	1.15796i	0.815343	−	0.578978i	$-0.196548\pi$
$798$		0			0
$799$		−	3310.33i		−	0.146572i
$800$		0			0
$801$			109.293i			0.00482108i
$802$		0			0
$803$	5179.80			0.227635
$804$		0			0
$805$	8947.75	+	6501.79i	0.391760	+	0.284668i
$806$		0			0
$807$		−	13899.3i		−	0.606295i
$808$		0			0
$809$	−29749.3			−1.29287			−0.646433	−	0.762971i	$-0.723740\pi$
$809$	−29749.3			−1.29287			−0.646433	+	0.762971i	$0.723740\pi$
$810$		0			0
$811$	−2585.38			−0.111942			−0.0559709	−	0.998432i	$-0.517825\pi$
$811$	−2585.38			−0.111942			−0.0559709	+	0.998432i	$0.517825\pi$
$812$		0			0
$813$	9012.02			0.388764
$814$		0			0
$815$	6242.44			0.268299
$816$		0			0
$817$			5366.87i			0.229820i
$818$		0			0
$819$	−1309.24	+	1801.77i	−0.0558589	+	0.0768729i
$820$		0			0
$821$	3421.88			0.145462			0.0727311	−	0.997352i	$-0.476829\pi$
$821$	3421.88			0.145462			0.0727311	+	0.997352i	$0.476829\pi$
$822$		0			0
$823$			16719.6i			0.708150i	0.935217	+	0.354075i	$0.115204\pi$
$823$			16719.6i			0.708150i	−0.935217	+	0.354075i	$0.884796\pi$
$824$		0			0
$825$			2398.27i			0.101208i
$826$		0			0
$827$		−	2715.80i		−	0.114193i	−0.998369	−	0.0570966i	$-0.981816\pi$
$827$		−	2715.80i		−	0.114193i	0.998369	−	0.0570966i	$-0.0181843\pi$
$828$		0			0
$829$			2183.04i			0.0914599i	0.998954	+	0.0457299i	$0.0145614\pi$
$829$			2183.04i			0.0914599i	−0.998954	+	0.0457299i	$0.985439\pi$
$830$		0			0
$831$	−26558.4			−1.10866
$832$		0			0
$833$	18059.0	−	5865.17i	0.751150	−	0.243957i
$834$		0			0
$835$			5198.28i			0.215442i
$836$		0			0
$837$	6507.14			0.268721
$838$		0			0
$839$	18292.1			0.752696			0.376348	−	0.926478i	$-0.377180\pi$
$839$	18292.1			0.752696			0.376348	+	0.926478i	$0.377180\pi$
$840$		0			0
$841$	−23842.6			−0.977595
$842$		0			0
$843$	−1135.99			−0.0464123
$844$		0			0
$845$			9033.22i			0.367754i
$846$		0			0
$847$	19072.8	+	13859.0i	0.773729	+	0.562222i
$848$		0			0
$849$	4547.19			0.183815
$850$		0			0
$851$			23884.0i			0.962083i
$852$		0			0
$853$		−	39811.6i		−	1.59804i	−0.601307	−	0.799018i	$-0.705353\pi$
$853$		−	39811.6i		−	1.59804i	0.601307	−	0.799018i	$-0.294647\pi$
$854$		0			0
$855$			2976.93i			0.119075i
$856$		0			0
$857$			8630.80i			0.344017i	0.985095	+	0.172008i	$0.0550256\pi$
$857$			8630.80i			0.344017i	−0.985095	+	0.172008i	$0.944974\pi$
$858$		0			0
$859$	1012.27			0.0402073			0.0201036	−	0.999798i	$-0.493600\pi$
$859$	1012.27			0.0402073			0.0201036	+	0.999798i	$0.493600\pi$
$860$		0			0
$861$	−16137.7	+	22208.6i	−0.638758	+	0.879057i
$862$		0			0
$863$		−	18567.1i		−	0.732367i	−0.930543	−	0.366184i	$-0.880664\pi$
$863$		−	18567.1i		−	0.732367i	0.930543	−	0.366184i	$-0.119336\pi$
$864$		0			0
$865$	9550.24			0.375396
$866$		0			0
$867$	−5545.68			−0.217233
$868$		0			0
$869$	8259.31			0.322414
$870$		0			0
$871$	8012.34			0.311697
$872$		0			0
$873$		−	12901.3i		−	0.500163i
$874$		0			0
$875$	−11204.7	+	15419.9i	−0.432902	+	0.595758i
$876$		0			0
$877$	2228.75			0.0858149			0.0429075	−	0.999079i	$-0.486338\pi$
$877$	2228.75			0.0858149			0.0429075	+	0.999079i	$0.486338\pi$
$878$		0			0
$879$			17237.0i			0.661420i
$880$		0			0
$881$		−	28095.1i		−	1.07440i	−0.843454	−	0.537201i	$-0.819482\pi$
$881$		−	28095.1i		−	1.07440i	0.843454	−	0.537201i	$-0.180518\pi$
$882$		0			0
$883$			29038.8i			1.10672i	0.832942	+	0.553361i	$0.186655\pi$
$883$			29038.8i			1.10672i	−0.832942	+	0.553361i	$0.813345\pi$
$884$		0			0
$885$			799.303i			0.0303596i
$886$		0			0
$887$	−45085.5			−1.70668			−0.853338	−	0.521357i	$-0.825426\pi$
$887$	−45085.5			−1.70668			−0.853338	+	0.521357i	$0.825426\pi$
$888$		0			0
$889$	15055.3	−	20719.1i	0.567986	−	0.781660i
$890$		0			0
$891$		−	616.864i		−	0.0231939i
$892$		0			0
$893$	4419.75			0.165623
$894$		0			0
$895$	11417.5			0.426421
$896$		0			0
$897$	−5349.33			−0.199118
$898$		0			0
$899$	5633.75			0.209006
$900$		0			0
$901$		−	31524.8i		−	1.16564i
$902$		0			0
$903$	2371.61	−	3263.80i	0.0874001	−	0.120280i
$904$		0			0
$905$	−3201.28			−0.117585
$906$		0			0
$907$		−	9186.50i		−	0.336309i	−0.985761	−	0.168155i	$-0.946219\pi$
$907$		−	9186.50i		−	0.336309i	0.985761	−	0.168155i	$-0.0537808\pi$
$908$		0			0
$909$		−	15943.1i		−	0.581737i
$910$		0			0
$911$		−	13945.1i		−	0.507158i	−0.967315	−	0.253579i	$-0.918392\pi$
$911$		−	13945.1i		−	0.507158i	0.967315	−	0.253579i	$-0.0816077\pi$
$912$		0			0
$913$			7122.30i			0.258175i
$914$		0			0
$915$	−8456.85			−0.305546
$916$		0			0
$917$	43910.4	+	31907.1i	1.58130	+	1.14904i
$918$		0			0
$919$			16186.3i			0.580996i	0.956876	+	0.290498i	$0.0938209\pi$
$919$			16186.3i			0.580996i	−0.956876	+	0.290498i	$0.906179\pi$
$920$		0			0
$921$	1775.95			0.0635392
$922$		0			0
$923$	−5447.71			−0.194273
$924$		0			0
$925$	−18787.7			−0.667821
$926$		0			0
$927$	−15008.0			−0.531745
$928$		0			0
$929$		−	12983.8i		−	0.458541i	−0.973363	−	0.229271i	$-0.926366\pi$
$929$		−	12983.8i		−	0.458541i	0.973363	−	0.229271i	$-0.0736341\pi$
$930$		0			0
$931$	7830.84	+	24111.3i	0.275666	+	0.848783i
$932$		0			0
$933$	7885.97			0.276715
$934$		0			0
$935$		−	1886.70i		−	0.0659912i
$936$		0			0
$937$			14063.8i			0.490335i	0.969481	+	0.245168i	$0.0788430\pi$
$937$			14063.8i			0.490335i	−0.969481	+	0.245168i	$0.921157\pi$
$938$		0			0
$939$			11344.8i			0.394274i
$940$		0			0
$941$			16064.3i			0.556517i	0.960506	+	0.278259i	$0.0897572\pi$
$941$			16064.3i			0.556517i	−0.960506	+	0.278259i	$0.910243\pi$
$942$		0			0
$943$	−65935.9			−2.27696
$944$		0			0
$945$	1315.50	−	1810.39i	0.0452838	−	0.0623195i
$946$		0			0
$947$		−	4913.13i		−	0.168591i	−0.996441	−	0.0842953i	$-0.973136\pi$
$947$		−	4913.13i		−	0.168591i	0.996441	−	0.0842953i	$-0.0268639\pi$
$948$		0			0
$949$	−9088.25			−0.310871
$950$		0			0
$951$	9220.78			0.314411
$952$		0			0
$953$	−1609.89			−0.0547215			−0.0273607	−	0.999626i	$-0.508710\pi$
$953$	−1609.89			−0.0547215			−0.0273607	+	0.999626i	$0.508710\pi$
$954$		0			0
$955$	12380.1			0.419487
$956$		0			0
$957$		−	534.068i		−	0.0180397i
$958$		0			0
$959$	22696.2	+	16491.9i	0.764231	+	0.555321i
$960$		0			0
$961$	28292.5			0.949701
$962$		0			0
$963$			9946.92i			0.332850i
$964$		0			0
$965$			5293.75i			0.176593i
$966$		0			0
$967$			9176.82i			0.305177i	0.988290	+	0.152589i	$0.0487610\pi$
$967$			9176.82i			0.305177i	−0.988290	+	0.152589i	$0.951239\pi$
$968$		0			0
$969$			12274.4i			0.406924i
$970$		0			0
$971$	−9484.07			−0.313448			−0.156724	−	0.987642i	$-0.550093\pi$
$971$	−9484.07			−0.313448			−0.156724	+	0.987642i	$0.550093\pi$
$972$		0			0
$973$	−16808.6	−	12213.8i	−0.553813	−	0.402423i
$974$		0			0
$975$		−	4207.90i		−	0.138216i
$976$		0			0
$977$	10671.9			0.349461			0.174731	−	0.984616i	$-0.444095\pi$
$977$	10671.9			0.349461			0.174731	+	0.984616i	$0.444095\pi$
$978$		0			0
$979$	92.4817			0.00301913
$980$		0			0
$981$	3180.71			0.103519
$982$		0			0
$983$	−18046.5			−0.585547			−0.292774	−	0.956182i	$-0.594578\pi$
$983$	−18046.5			−0.585547			−0.292774	+	0.956182i	$0.594578\pi$
$984$		0			0
$985$			21250.0i			0.687393i
$986$		0			0
$987$	−2687.83	−	1953.08i	−0.0866813	−	0.0629861i
$988$		0			0
$989$	9690.01			0.311552
$990$		0			0
$991$			13201.9i			0.423181i	0.977358	+	0.211590i	$0.0678643\pi$
$991$			13201.9i			0.423181i	−0.977358	+	0.211590i	$0.932136\pi$
$992$		0			0
$993$		−	20764.9i		−	0.663598i
$994$		0			0
$995$			4110.81i			0.130976i
$996$		0			0
$997$		−	30223.1i		−	0.960054i	−0.877254	−	0.480027i	$-0.840627\pi$
$997$		−	30223.1i		−	0.960054i	0.877254	−	0.480027i	$-0.159373\pi$
$998$		0			0
$999$	4832.42			0.153044

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.b.g.895.7		12
4.3	odd	2		1344.4.b.h.895.7		12
7.6	odd	2		1344.4.b.h.895.6		12
8.3	odd	2		84.4.b.a.55.9	✓	12
8.5	even	2		84.4.b.b.55.10	yes	12
24.5	odd	2		252.4.b.e.55.3		12
24.11	even	2		252.4.b.f.55.4		12
28.27	even	2	inner	1344.4.b.g.895.6		12
56.13	odd	2		84.4.b.a.55.10	yes	12
56.27	even	2		84.4.b.b.55.9	yes	12
168.83	odd	2		252.4.b.e.55.4		12
168.125	even	2		252.4.b.f.55.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.4.b.a.55.9	✓	12	8.3	odd	2
84.4.b.a.55.10	yes	12	56.13	odd	2
84.4.b.b.55.9	yes	12	56.27	even	2
84.4.b.b.55.10	yes	12	8.5	even	2
252.4.b.e.55.3		12	24.5	odd	2
252.4.b.e.55.4		12	168.83	odd	2
252.4.b.f.55.3		12	168.125	even	2
252.4.b.f.55.4		12	24.11	even	2
1344.4.b.g.895.6		12	28.27	even	2	inner
1344.4.b.g.895.7		12	1.1	even	1	trivial
1344.4.b.h.895.6		12	7.6	odd	2
1344.4.b.h.895.7		12	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +4.47531i q^{5} +(14.9825 + 10.8869i) q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +4.47531i q^{5} +(14.9825 + 10.8869i) q^{7} +9.00000 q^{9} -7.61561i q^{11} +13.3620i q^{13} -13.4259i q^{15} -55.3574i q^{17} +73.9099 q^{19} +(-44.9475 - 32.6607i) q^{21} -133.446i q^{23} +104.972 q^{25} -27.0000 q^{27} -23.3760 q^{29} -241.005 q^{31} +22.8468i q^{33} +(-48.7222 + 67.0514i) q^{35} -178.979 q^{37} -40.0861i q^{39} -494.101i q^{41} +72.6137i q^{43} +40.2778i q^{45} +59.7992 q^{47} +(105.951 + 326.226i) q^{49} +166.072i q^{51} +569.477 q^{53} +34.0822 q^{55} -221.730 q^{57} -59.5343 q^{59} -629.889i q^{61} +(134.843 + 97.9820i) q^{63} -59.7992 q^{65} -599.636i q^{67} +400.338i q^{69} +407.701i q^{71} +680.155i q^{73} -314.915 q^{75} +(82.9103 - 114.101i) q^{77} +1084.52i q^{79} +81.0000 q^{81} -935.224 q^{83} +247.741 q^{85} +70.1281 q^{87} +12.1437i q^{89} +(-145.471 + 200.197i) q^{91} +723.016 q^{93} +330.770i q^{95} -1433.48i q^{97} -68.5405i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 36 q^{3} - 10 q^{7} + 108 q^{9}+O(q^{10}) \) 12 * q - 36 * q^3 - 10 * q^7 + 108 * q^9	\( 12 q - 36 q^{3} - 10 q^{7} + 108 q^{9} - 84 q^{19} + 30 q^{21} - 216 q^{25} - 324 q^{27} - 200 q^{29} + 384 q^{31} + 84 q^{35} + 244 q^{37} - 280 q^{47} - 424 q^{49} + 16 q^{53} - 212 q^{55} + 252 q^{57} + 1168 q^{59} - 90 q^{63} + 280 q^{65} + 648 q^{75} - 968 q^{77} + 972 q^{81} - 968 q^{83} + 852 q^{85} + 600 q^{87} + 1648 q^{91} - 1152 q^{93}+O(q^{100}) \) 12 * q - 36 * q^3 - 10 * q^7 + 108 * q^9 - 84 * q^19 + 30 * q^21 - 216 * q^25 - 324 * q^27 - 200 * q^29 + 384 * q^31 + 84 * q^35 + 244 * q^37 - 280 * q^47 - 424 * q^49 + 16 * q^53 - 212 * q^55 + 252 * q^57 + 1168 * q^59 - 90 * q^63 + 280 * q^65 + 648 * q^75 - 968 * q^77 + 972 * q^81 - 968 * q^83 + 852 * q^85 + 600 * q^87 + 1648 * q^91 - 1152 * q^93

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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