LMFDB - Embedded newform 1344.4.b.j.895.10

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:	$24$
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.10
Character	$\chi$	$=$	1344.895
Dual form			1344.4.b.j.895.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -4.15525i q^{5} +(10.5863 + 15.1964i) q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -4.15525i q^{5} +(10.5863 + 15.1964i) q^{7} +9.00000 q^{9} +26.7188i q^{11} -36.8834i q^{13} -12.4658i q^{15} -41.0070i q^{17} -110.241 q^{19} +(31.7588 + 45.5892i) q^{21} +69.5113i q^{23} +107.734 q^{25} +27.0000 q^{27} +155.035 q^{29} +98.2305 q^{31} +80.1563i q^{33} +(63.1449 - 43.9886i) q^{35} +210.639 q^{37} -110.650i q^{39} +0.600048i q^{41} +354.960i q^{43} -37.3973i q^{45} +258.195 q^{47} +(-118.861 + 321.747i) q^{49} -123.021i q^{51} +274.991 q^{53} +111.023 q^{55} -330.722 q^{57} -301.151 q^{59} -469.215i q^{61} +(95.2765 + 136.768i) q^{63} -153.260 q^{65} -605.217i q^{67} +208.534i q^{69} +497.779i q^{71} +429.333i q^{73} +323.202 q^{75} +(-406.029 + 282.852i) q^{77} +1043.89i q^{79} +81.0000 q^{81} -1020.65 q^{83} -170.394 q^{85} +465.106 q^{87} +936.707i q^{89} +(560.495 - 390.458i) q^{91} +294.691 q^{93} +458.077i q^{95} +407.850i q^{97} +240.469i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9}+O(q^{10}) $ 24 * q + 72 * q^3 - 20 * q^7 + 216 * q^9	$ 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9} - 56 q^{19} - 60 q^{21} - 432 q^{25} + 648 q^{27} + 464 q^{31} + 568 q^{35} - 504 q^{37} + 560 q^{47} - 128 q^{49} + 784 q^{53} + 424 q^{55} - 168 q^{57} + 800 q^{59} - 180 q^{63} + 560 q^{65} - 1296 q^{75} + 1568 q^{77} + 1944 q^{81} + 1936 q^{83} - 3000 q^{85} - 496 q^{91} + 1392 q^{93}+O(q^{100}) $ 24 * q + 72 * q^3 - 20 * q^7 + 216 * q^9 - 56 * q^19 - 60 * q^21 - 432 * q^25 + 648 * q^27 + 464 * q^31 + 568 * q^35 - 504 * q^37 + 560 * q^47 - 128 * q^49 + 784 * q^53 + 424 * q^55 - 168 * q^57 + 800 * q^59 - 180 * q^63 + 560 * q^65 - 1296 * q^75 + 1568 * q^77 + 1944 * q^81 + 1936 * q^83 - 3000 * q^85 - 496 * q^91 + 1392 * 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
$5$		−	4.15525i		−	0.371657i	−0.982582	−	0.185828i	$-0.940503\pi$
$5$		−	4.15525i		−	0.371657i	0.982582	−	0.185828i	$-0.0594969\pi$
$6$		0			0
$7$	10.5863	+	15.1964i	0.571605	+	0.820529i
$7$	10.5863	+	15.1964i	0.571605	+	0.820529i
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$			26.7188i			0.732364i	0.930543	+	0.366182i	$0.119335\pi$
$11$			26.7188i			0.732364i	−0.930543	+	0.366182i	$0.880665\pi$
$12$		0			0
$13$		−	36.8834i		−	0.786894i	−0.919347	−	0.393447i	$-0.871283\pi$
$13$		−	36.8834i		−	0.786894i	0.919347	−	0.393447i	$-0.128717\pi$
$14$		0			0
$15$		−	12.4658i		−	0.214576i
$16$		0			0
$17$		−	41.0070i		−	0.585039i	−0.956260	−	0.292519i	$-0.905506\pi$
$17$		−	41.0070i		−	0.585039i	0.956260	−	0.292519i	$-0.0944936\pi$
$18$		0			0
$19$	−110.241			−1.33110			−0.665551	−	0.746353i	$-0.731803\pi$
$19$	−110.241			−1.33110			−0.665551	+	0.746353i	$0.731803\pi$
$20$		0			0
$21$	31.7588	+	45.5892i	0.330017	+	0.473732i
$22$		0			0
$23$			69.5113i			0.630178i	0.949062	+	0.315089i	$0.102034\pi$
$23$			69.5113i			0.630178i	−0.949062	+	0.315089i	$0.897966\pi$
$24$		0			0
$25$	107.734			0.861871
$26$		0			0
$27$	27.0000			0.192450
$28$		0			0
$29$	155.035			0.992736			0.496368	−	0.868112i	$-0.334667\pi$
$29$	155.035			0.992736			0.496368	+	0.868112i	$0.334667\pi$
$30$		0			0
$31$	98.2305			0.569120			0.284560	−	0.958658i	$-0.408153\pi$
$31$	98.2305			0.569120			0.284560	+	0.958658i	$0.408153\pi$
$32$		0			0
$33$			80.1563i			0.422831i
$34$		0			0
$35$	63.1449	−	43.9886i	0.304955	−	0.212441i
$36$		0			0
$37$	210.639			0.935916			0.467958	−	0.883751i	$-0.344990\pi$
$37$	210.639			0.935916			0.467958	+	0.883751i	$0.344990\pi$
$38$		0			0
$39$		−	110.650i		−	0.454313i
$40$		0			0
$41$			0.600048i			0.00228565i	0.999999	+	0.00114283i	$0.000363773\pi$
$41$			0.600048i			0.00228565i	−0.999999	+	0.00114283i	$0.999636\pi$
$42$		0			0
$43$			354.960i			1.25886i	0.777058	+	0.629429i	$0.216711\pi$
$43$			354.960i			1.25886i	−0.777058	+	0.629429i	$0.783289\pi$
$44$		0			0
$45$		−	37.3973i		−	0.123886i
$46$		0			0
$47$	258.195			0.801311			0.400655	−	0.916229i	$-0.368782\pi$
$47$	258.195			0.801311			0.400655	+	0.916229i	$0.368782\pi$
$48$		0			0
$49$	−118.861	+	321.747i	−0.346534	+	0.938037i
$50$		0			0
$51$		−	123.021i		−	0.337772i
$52$		0			0
$53$	274.991			0.712697			0.356349	−	0.934353i	$-0.384022\pi$
$53$	274.991			0.712697			0.356349	+	0.934353i	$0.384022\pi$
$54$		0			0
$55$	111.023			0.272188
$56$		0			0
$57$	−330.722			−0.768512
$58$		0			0
$59$	−301.151			−0.664516			−0.332258	−	0.943188i	$-0.607811\pi$
$59$	−301.151			−0.664516			−0.332258	+	0.943188i	$0.607811\pi$
$60$		0			0
$61$		−	469.215i		−	0.984867i	−0.870350	−	0.492433i	$-0.836108\pi$
$61$		−	469.215i		−	0.984867i	0.870350	−	0.492433i	$-0.163892\pi$
$62$		0			0
$63$	95.2765	+	136.768i	0.190535	+	0.273510i
$64$		0			0
$65$	−153.260			−0.292454
$66$		0			0
$67$		−	605.217i		−	1.10357i	−0.833987	−	0.551784i	$-0.813947\pi$
$67$		−	605.217i		−	1.10357i	0.833987	−	0.551784i	$-0.186053\pi$
$68$		0			0
$69$			208.534i			0.363834i
$70$		0			0
$71$			497.779i			0.832050i	0.909353	+	0.416025i	$0.136577\pi$
$71$			497.779i			0.832050i	−0.909353	+	0.416025i	$0.863423\pi$
$72$		0			0
$73$			429.333i			0.688351i	0.938905	+	0.344175i	$0.111841\pi$
$73$			429.333i			0.688351i	−0.938905	+	0.344175i	$0.888159\pi$
$74$		0			0
$75$	323.202			0.497602
$76$		0			0
$77$	−406.029	+	282.852i	−0.600926	+	0.418623i
$78$		0			0
$79$			1043.89i			1.48667i	0.668922	+	0.743333i	$0.266756\pi$
$79$			1043.89i			1.48667i	−0.668922	+	0.743333i	$0.733244\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	−1020.65			−1.34976			−0.674882	−	0.737926i	$-0.735805\pi$
$83$	−1020.65			−1.34976			−0.674882	+	0.737926i	$0.735805\pi$
$84$		0			0
$85$	−170.394			−0.217434
$86$		0			0
$87$	465.106			0.573156
$88$		0			0
$89$			936.707i			1.11563i	0.829967	+	0.557813i	$0.188359\pi$
$89$			936.707i			1.11563i	−0.829967	+	0.557813i	$0.811641\pi$
$90$		0			0
$91$	560.495	−	390.458i	0.645669	−	0.449793i
$92$		0			0
$93$	294.691			0.328582
$94$		0			0
$95$			458.077i			0.494713i
$96$		0			0
$97$			407.850i			0.426917i	0.976952	+	0.213458i	$0.0684727\pi$
$97$			407.850i			0.426917i	−0.976952	+	0.213458i	$0.931527\pi$
$98$		0			0
$99$			240.469i			0.244121i
$100$		0			0
$101$		−	831.061i		−	0.818749i	−0.912366	−	0.409375i	$-0.865747\pi$
$101$		−	831.061i		−	0.818749i	0.912366	−	0.409375i	$-0.134253\pi$
$102$		0			0
$103$	1652.29			1.58063			0.790316	−	0.612700i	$-0.209917\pi$
$103$	1652.29			1.58063			0.790316	+	0.612700i	$0.209917\pi$
$104$		0			0
$105$	189.435	−	131.966i	0.176066	−	0.122653i
$106$		0			0
$107$			303.007i			0.273765i	0.990587	+	0.136882i	$0.0437082\pi$
$107$			303.007i			0.273765i	−0.990587	+	0.136882i	$0.956292\pi$
$108$		0			0
$109$	1020.52			0.896768			0.448384	−	0.893841i	$-0.352000\pi$
$109$	1020.52			0.896768			0.448384	+	0.893841i	$0.352000\pi$
$110$		0			0
$111$	631.918			0.540351
$112$		0			0
$113$	417.458			0.347532			0.173766	−	0.984787i	$-0.444406\pi$
$113$	417.458			0.347532			0.173766	+	0.984787i	$0.444406\pi$
$114$		0			0
$115$	288.837			0.234210
$116$		0			0
$117$		−	331.951i		−	0.262298i
$118$		0			0
$119$	623.159	−	434.112i	0.480041	−	0.334411i
$120$		0			0
$121$	617.108			0.463642
$122$		0			0
$123$			1.80014i			0.00131962i
$124$		0			0
$125$		−	967.068i		−	0.691977i
$126$		0			0
$127$			1193.68i			0.834034i	0.908899	+	0.417017i	$0.136925\pi$
$127$			1193.68i			0.834034i	−0.908899	+	0.417017i	$0.863075\pi$
$128$		0			0
$129$			1064.88i			0.726802i
$130$		0			0
$131$	962.451			0.641906			0.320953	−	0.947095i	$-0.395997\pi$
$131$	962.451			0.641906			0.320953	+	0.947095i	$0.395997\pi$
$132$		0			0
$133$	−1167.04	−	1675.26i	−0.760865	−	1.09221i
$134$		0			0
$135$		−	112.192i		−	0.0715254i
$136$		0			0
$137$	−2842.24			−1.77247			−0.886237	−	0.463232i	$-0.846690\pi$
$137$	−2842.24			−1.77247			−0.886237	+	0.463232i	$0.846690\pi$
$138$		0			0
$139$	2012.76			1.22820			0.614101	−	0.789227i	$-0.289519\pi$
$139$	2012.76			1.22820			0.614101	+	0.789227i	$0.289519\pi$
$140$		0			0
$141$	774.585			0.462637
$142$		0			0
$143$	985.479			0.576293
$144$		0			0
$145$		−	644.211i		−	0.368957i
$146$		0			0
$147$	−356.584	+	965.240i	−0.200072	+	0.541576i
$148$		0			0
$149$	1487.06			0.817616			0.408808	−	0.912620i	$-0.365945\pi$
$149$	1487.06			0.817616			0.408808	+	0.912620i	$0.365945\pi$
$150$		0			0
$151$			267.813i			0.144333i	0.997393	+	0.0721666i	$0.0229913\pi$
$151$			267.813i			0.144333i	−0.997393	+	0.0721666i	$0.977009\pi$
$152$		0			0
$153$		−	369.063i		−	0.195013i
$154$		0			0
$155$		−	408.172i		−	0.211517i
$156$		0			0
$157$			569.423i			0.289458i	0.989471	+	0.144729i	$0.0462310\pi$
$157$			569.423i			0.289458i	−0.989471	+	0.144729i	$0.953769\pi$
$158$		0			0
$159$	824.974			0.411476
$160$		0			0
$161$	−1056.32	+	735.866i	−0.517079	+	0.360213i
$162$		0			0
$163$		−	1377.89i		−	0.662113i	−0.943611	−	0.331057i	$-0.892595\pi$
$163$		−	1377.89i		−	0.662113i	0.943611	−	0.331057i	$-0.107405\pi$
$164$		0			0
$165$	333.069			0.157148
$166$		0			0
$167$	2791.51			1.29349			0.646747	−	0.762705i	$-0.276129\pi$
$167$	2791.51			1.29349			0.646747	+	0.762705i	$0.276129\pi$
$168$		0			0
$169$	836.614			0.380798
$170$		0			0
$171$	−992.165			−0.443700
$172$		0			0
$173$			1433.66i			0.630053i	0.949083	+	0.315027i	$0.102013\pi$
$173$			1433.66i			0.630053i	−0.949083	+	0.315027i	$0.897987\pi$
$174$		0			0
$175$	1140.50	+	1637.17i	0.492650	+	0.707190i
$176$		0			0
$177$	−903.452			−0.383659
$178$		0			0
$179$			3472.47i			1.44997i	0.688765	+	0.724985i	$0.258153\pi$
$179$			3472.47i			1.44997i	−0.688765	+	0.724985i	$0.741847\pi$
$180$		0			0
$181$		−	407.085i		−	0.167174i	−0.996501	−	0.0835868i	$-0.973362\pi$
$181$		−	407.085i		−	0.167174i	0.996501	−	0.0835868i	$-0.0266376\pi$
$182$		0			0
$183$		−	1407.65i		−	0.568613i
$184$		0			0
$185$		−	875.259i		−	0.347840i
$186$		0			0
$187$	1095.66			0.428462
$188$		0			0
$189$	285.830	+	410.303i	0.110006	+	0.157911i
$190$		0			0
$191$			3898.48i			1.47688i	0.674318	+	0.738441i	$0.264438\pi$
$191$			3898.48i			1.47688i	−0.674318	+	0.738441i	$0.735562\pi$
$192$		0			0
$193$	2366.15			0.882483			0.441241	−	0.897389i	$-0.354538\pi$
$193$	2366.15			0.882483			0.441241	+	0.897389i	$0.354538\pi$
$194$		0			0
$195$	−459.779			−0.168849
$196$		0			0
$197$	−2173.29			−0.785991			−0.392995	−	0.919540i	$-0.628561\pi$
$197$	−2173.29			−0.785991			−0.392995	+	0.919540i	$0.628561\pi$
$198$		0			0
$199$	−2789.79			−0.993784			−0.496892	−	0.867812i	$-0.665525\pi$
$199$	−2789.79			−0.993784			−0.496892	+	0.867812i	$0.665525\pi$
$200$		0			0
$201$		−	1815.65i		−	0.637145i
$202$		0			0
$203$	1641.25	+	2355.98i	0.567453	+	0.814568i
$204$		0			0
$205$	2.49335			0.000849479
$206$		0			0
$207$			625.601i			0.210059i
$208$		0			0
$209$		−	2945.49i		−	0.974851i
$210$		0			0
$211$			894.802i			0.291947i	0.989289	+	0.145973i	$0.0466313\pi$
$211$			894.802i			0.291947i	−0.989289	+	0.145973i	$0.953369\pi$
$212$		0			0
$213$			1493.34i			0.480384i
$214$		0			0
$215$	1474.95			0.467864
$216$		0			0
$217$	1039.90	+	1492.75i	0.325312	+	0.466979i
$218$		0			0
$219$			1288.00i			0.397420i
$220$		0			0
$221$	−1512.48			−0.460363
$222$		0			0
$223$	2804.30			0.842108			0.421054	−	0.907036i	$-0.361660\pi$
$223$	2804.30			0.842108			0.421054	+	0.907036i	$0.361660\pi$
$224$		0			0
$225$	969.605			0.287290
$226$		0			0
$227$	5559.46			1.62553			0.812763	−	0.582595i	$-0.197963\pi$
$227$	5559.46			1.62553			0.812763	+	0.582595i	$0.197963\pi$
$228$		0			0
$229$		−	2366.25i		−	0.682820i	−0.939914	−	0.341410i	$-0.889095\pi$
$229$		−	2366.25i		−	0.682820i	0.939914	−	0.341410i	$-0.110905\pi$
$230$		0			0
$231$	−1218.09	+	848.557i	−0.346945	+	0.241692i
$232$		0			0
$233$	−3111.46			−0.874843			−0.437422	−	0.899257i	$-0.644108\pi$
$233$	−3111.46			−0.874843			−0.437422	+	0.899257i	$0.644108\pi$
$234$		0			0
$235$		−	1072.86i		−	0.297813i
$236$		0			0
$237$			3131.66i			0.858326i
$238$		0			0
$239$		−	4494.98i		−	1.21655i	−0.793725	−	0.608277i	$-0.791861\pi$
$239$		−	4494.98i		−	1.21655i	0.793725	−	0.608277i	$-0.208139\pi$
$240$		0			0
$241$			2555.56i			0.683061i	0.939871	+	0.341531i	$0.110945\pi$
$241$			2555.56i			0.683061i	−0.939871	+	0.341531i	$0.889055\pi$
$242$		0			0
$243$	243.000			0.0641500
$244$		0			0
$245$	1336.94	+	493.899i	0.348628	+	0.128792i
$246$		0			0
$247$			4066.05i			1.04743i
$248$		0			0
$249$	−3061.94			−0.779286
$250$		0			0
$251$	−803.874			−0.202152			−0.101076	−	0.994879i	$-0.532228\pi$
$251$	−803.874			−0.202152			−0.101076	+	0.994879i	$0.532228\pi$
$252$		0			0
$253$	−1857.25			−0.461520
$254$		0			0
$255$	−511.183			−0.125535
$256$		0			0
$257$		−	7577.19i		−	1.83911i	−0.392956	−	0.919557i	$-0.628548\pi$
$257$		−	7577.19i		−	1.83911i	0.392956	−	0.919557i	$-0.371452\pi$
$258$		0			0
$259$	2229.89	+	3200.96i	0.534975	+	0.767946i
$260$		0			0
$261$	1395.32			0.330912
$262$		0			0
$263$		−	3561.82i		−	0.835099i	−0.908654	−	0.417550i	$-0.862889\pi$
$263$		−	3561.82i		−	0.835099i	0.908654	−	0.417550i	$-0.137111\pi$
$264$		0			0
$265$		−	1142.66i		−	0.264879i
$266$		0			0
$267$			2810.12i			0.644107i
$268$		0			0
$269$		−	3944.71i		−	0.894102i	−0.894509	−	0.447051i	$-0.852474\pi$
$269$		−	3944.71i		−	0.894102i	0.894509	−	0.447051i	$-0.147526\pi$
$270$		0			0
$271$	−3972.84			−0.890526			−0.445263	−	0.895400i	$-0.646890\pi$
$271$	−3972.84			−0.890526			−0.445263	+	0.895400i	$0.646890\pi$
$272$		0			0
$273$	1681.49	−	1171.37i	0.372777	−	0.259688i
$274$		0			0
$275$			2878.52i			0.631204i
$276$		0			0
$277$	−7036.30			−1.52625			−0.763123	−	0.646253i	$-0.776335\pi$
$277$	−7036.30			−1.52625			−0.763123	+	0.646253i	$0.776335\pi$
$278$		0			0
$279$	884.074			0.189707
$280$		0			0
$281$	−652.815			−0.138590			−0.0692948	−	0.997596i	$-0.522075\pi$
$281$	−652.815			−0.138590			−0.0692948	+	0.997596i	$0.522075\pi$
$282$		0			0
$283$	6289.21			1.32104			0.660520	−	0.750808i	$-0.270336\pi$
$283$	6289.21			1.32104			0.660520	+	0.750808i	$0.270336\pi$
$284$		0			0
$285$			1374.23i			0.285623i
$286$		0			0
$287$	−9.11857	+	6.35228i	−0.00187544	+	0.00130649i
$288$		0			0
$289$	3231.43			0.657730
$290$		0			0
$291$			1223.55i			0.246480i
$292$		0			0
$293$			3451.83i			0.688254i	0.938923	+	0.344127i	$0.111825\pi$
$293$			3451.83i			0.688254i	−0.938923	+	0.344127i	$0.888175\pi$
$294$		0			0
$295$			1251.36i			0.246972i
$296$		0			0
$297$			721.406i			0.140944i
$298$		0			0
$299$	2563.81			0.495883
$300$		0			0
$301$	−5394.12	+	3757.71i	−1.03293	+	0.719570i
$302$		0			0
$303$		−	2493.18i		−	0.472705i
$304$		0			0
$305$	−1949.71			−0.366033
$306$		0			0
$307$	1879.04			0.349324			0.174662	−	0.984628i	$-0.444117\pi$
$307$	1879.04			0.349324			0.174662	+	0.984628i	$0.444117\pi$
$308$		0			0
$309$	4956.87			0.912578
$310$		0			0
$311$	4029.94			0.734780			0.367390	−	0.930067i	$-0.380251\pi$
$311$	4029.94			0.734780			0.367390	+	0.930067i	$0.380251\pi$
$312$		0			0
$313$			4724.35i			0.853151i	0.904452	+	0.426575i	$0.140280\pi$
$313$			4724.35i			0.853151i	−0.904452	+	0.426575i	$0.859720\pi$
$314$		0			0
$315$	568.304	−	395.898i	0.101652	−	0.0708137i
$316$		0			0
$317$	−8399.49			−1.48821			−0.744104	−	0.668063i	$-0.767123\pi$
$317$	−8399.49			−1.48821			−0.744104	+	0.668063i	$0.767123\pi$
$318$		0			0
$319$			4142.35i			0.727044i
$320$		0			0
$321$			909.022i			0.158058i
$322$		0			0
$323$			4520.63i			0.778746i
$324$		0			0
$325$		−	3973.59i		−	0.678201i
$326$		0			0
$327$	3061.55			0.517749
$328$		0			0
$329$	2733.32	+	3923.64i	0.458034	+	0.657498i
$330$		0			0
$331$			175.263i			0.0291037i	0.999894	+	0.0145518i	$0.00463216\pi$
$331$			175.263i			0.0291037i	−0.999894	+	0.0145518i	$0.995368\pi$
$332$		0			0
$333$	1895.75			0.311972
$334$		0			0
$335$	−2514.83			−0.410149
$336$		0			0
$337$	9534.83			1.54123			0.770616	−	0.637299i	$-0.219949\pi$
$337$	9534.83			1.54123			0.770616	+	0.637299i	$0.219949\pi$
$338$		0			0
$339$	1252.37			0.200648
$340$		0			0
$341$			2624.60i			0.416803i
$342$		0			0
$343$	−6147.69	+	1599.84i	−0.967767	+	0.251846i
$344$		0			0
$345$	866.510			0.135221
$346$		0			0
$347$		−	10295.8i		−	1.59282i	−0.604758	−	0.796409i	$-0.706730\pi$
$347$		−	10295.8i		−	1.59282i	0.604758	−	0.796409i	$-0.293270\pi$
$348$		0			0
$349$			12476.8i			1.91367i	0.290638	+	0.956833i	$0.406132\pi$
$349$			12476.8i			1.91367i	−0.290638	+	0.956833i	$0.593868\pi$
$350$		0			0
$351$		−	995.852i		−	0.151438i
$352$		0			0
$353$		−	10018.3i		−	1.51054i	−0.655415	−	0.755269i	$-0.727506\pi$
$353$		−	10018.3i		−	1.51054i	0.655415	−	0.755269i	$-0.272494\pi$
$354$		0			0
$355$	2068.40			0.309237
$356$		0			0
$357$	1869.48	−	1302.33i	0.277152	−	0.193072i
$358$		0			0
$359$		−	3440.45i		−	0.505793i	−0.967493	−	0.252897i	$-0.918617\pi$
$359$		−	3440.45i		−	0.505793i	0.967493	−	0.252897i	$-0.0813833\pi$
$360$		0			0
$361$	5293.98			0.771830
$362$		0			0
$363$	1851.32			0.267684
$364$		0			0
$365$	1783.99			0.255830
$366$		0			0
$367$	−7372.02			−1.04855			−0.524273	−	0.851550i	$-0.675663\pi$
$367$	−7372.02			−1.04855			−0.524273	+	0.851550i	$0.675663\pi$
$368$		0			0
$369$			5.40043i			0.000761884i
$370$		0			0
$371$	2911.13	+	4178.88i	0.407382	+	0.584788i
$372$		0			0
$373$	7872.77			1.09286			0.546430	−	0.837505i	$-0.315986\pi$
$373$	7872.77			1.09286			0.546430	+	0.837505i	$0.315986\pi$
$374$		0			0
$375$		−	2901.20i		−	0.399513i
$376$		0			0
$377$		−	5718.23i		−	0.781177i
$378$		0			0
$379$		−	3906.79i		−	0.529495i	−0.964318	−	0.264747i	$-0.914711\pi$
$379$		−	3906.79i		−	0.529495i	0.964318	−	0.264747i	$-0.0852885\pi$
$380$		0			0
$381$			3581.05i			0.481530i
$382$		0			0
$383$	823.361			0.109848			0.0549240	−	0.998491i	$-0.482508\pi$
$383$	823.361			0.109848			0.0549240	+	0.998491i	$0.482508\pi$
$384$		0			0
$385$	1175.32	+	1687.15i	0.155584	+	0.223338i
$386$		0			0
$387$			3194.64i			0.419620i
$388$		0			0
$389$	440.522			0.0574174			0.0287087	−	0.999588i	$-0.490860\pi$
$389$	440.522			0.0574174			0.0287087	+	0.999588i	$0.490860\pi$
$390$		0			0
$391$	2850.45			0.368679
$392$		0			0
$393$	2887.35			0.370605
$394$		0			0
$395$	4337.62			0.552529
$396$		0			0
$397$		−	1031.17i		−	0.130360i	−0.997874	−	0.0651801i	$-0.979238\pi$
$397$		−	1031.17i		−	0.130360i	0.997874	−	0.0651801i	$-0.0207622\pi$
$398$		0			0
$399$	−3501.11	−	5025.78i	−0.439285	−	0.630586i
$400$		0			0
$401$	−8879.49			−1.10579			−0.552893	−	0.833252i	$-0.686476\pi$
$401$	−8879.49			−1.10579			−0.552893	+	0.833252i	$0.686476\pi$
$402$		0			0
$403$		−	3623.08i		−	0.447837i
$404$		0			0
$405$		−	336.575i		−	0.0412952i
$406$		0			0
$407$			5628.02i			0.685432i
$408$		0			0
$409$			9256.56i			1.11909i	0.828800	+	0.559545i	$0.189024\pi$
$409$			9256.56i			1.11909i	−0.828800	+	0.559545i	$0.810976\pi$
$410$		0			0
$411$	−8526.72			−1.02334
$412$		0			0
$413$	−3188.06	−	4576.41i	−0.379841	−	0.545255i
$414$		0			0
$415$			4241.04i			0.501649i
$416$		0			0
$417$	6038.28			0.709103
$418$		0			0
$419$	−11247.8			−1.31143			−0.655716	−	0.755008i	$-0.727633\pi$
$419$	−11247.8			−1.31143			−0.655716	+	0.755008i	$0.727633\pi$
$420$		0			0
$421$	−368.940			−0.0427103			−0.0213552	−	0.999772i	$-0.506798\pi$
$421$	−368.940			−0.0427103			−0.0213552	+	0.999772i	$0.506798\pi$
$422$		0			0
$423$	2323.75			0.267104
$424$		0			0
$425$		−	4417.84i		−	0.504228i
$426$		0			0
$427$	7130.39	−	4967.25i	0.808111	−	0.562955i
$428$		0			0
$429$	2956.44			0.332723
$430$		0			0
$431$		−	5179.71i		−	0.578881i	−0.957196	−	0.289441i	$-0.906531\pi$
$431$		−	5179.71i		−	0.578881i	0.957196	−	0.289441i	$-0.0934693\pi$
$432$		0			0
$433$		−	1421.33i		−	0.157748i	−0.996885	−	0.0788738i	$-0.974868\pi$
$433$		−	1421.33i		−	0.157748i	0.996885	−	0.0788738i	$-0.0251324\pi$
$434$		0			0
$435$		−	1932.63i		−	0.213017i
$436$		0			0
$437$		−	7662.96i		−	0.838831i
$438$		0			0
$439$	−2696.52			−0.293161			−0.146581	−	0.989199i	$-0.546827\pi$
$439$	−2696.52			−0.293161			−0.146581	+	0.989199i	$0.546827\pi$
$440$		0			0
$441$	−1069.75	+	2895.72i	−0.115511	+	0.312679i
$442$		0			0
$443$			6901.80i			0.740213i	0.928989	+	0.370106i	$0.120679\pi$
$443$			6901.80i			0.740213i	−0.928989	+	0.370106i	$0.879321\pi$
$444$		0			0
$445$	3892.25			0.414630
$446$		0			0
$447$	4461.18			0.472051
$448$		0			0
$449$	−1515.38			−0.159276			−0.0796381	−	0.996824i	$-0.525376\pi$
$449$	−1515.38			−0.159276			−0.0796381	+	0.996824i	$0.525376\pi$
$450$		0			0
$451$	−16.0325			−0.00167393
$452$		0			0
$453$			803.439i			0.0833308i
$454$		0			0
$455$	−1622.45	−	2329.00i	−0.167169	−	0.239967i
$456$		0			0
$457$	6514.32			0.666799			0.333399	−	0.942786i	$-0.391804\pi$
$457$	6514.32			0.666799			0.333399	+	0.942786i	$0.391804\pi$
$458$		0			0
$459$		−	1107.19i		−	0.112591i
$460$		0			0
$461$			12310.7i			1.24375i	0.783118	+	0.621874i	$0.213628\pi$
$461$			12310.7i			1.24375i	−0.783118	+	0.621874i	$0.786372\pi$
$462$		0			0
$463$		−	7610.16i		−	0.763875i	−0.924188	−	0.381938i	$-0.875257\pi$
$463$		−	7610.16i		−	0.763875i	0.924188	−	0.381938i	$-0.124743\pi$
$464$		0			0
$465$		−	1224.52i		−	0.122120i
$466$		0			0
$467$	−16451.3			−1.63014			−0.815069	−	0.579364i	$-0.803301\pi$
$467$	−16451.3			−1.63014			−0.815069	+	0.579364i	$0.803301\pi$
$468$		0			0
$469$	9197.13	−	6407.00i	0.905509	−	0.630806i
$470$		0			0
$471$			1708.27i			0.167119i
$472$		0			0
$473$	−9484.10			−0.921943
$474$		0			0
$475$	−11876.6			−1.14724
$476$		0			0
$477$	2474.92			0.237566
$478$		0			0
$479$	−8221.60			−0.784247			−0.392124	−	0.919912i	$-0.628259\pi$
$479$	−8221.60			−0.784247			−0.392124	+	0.919912i	$0.628259\pi$
$480$		0			0
$481$		−	7769.10i		−	0.736467i
$482$		0			0
$483$	−3168.96	+	2207.60i	−0.298536	+	0.207969i
$484$		0			0
$485$	1694.72			0.158666
$486$		0			0
$487$		−	14062.9i		−	1.30853i	−0.756267	−	0.654263i	$-0.772979\pi$
$487$		−	14062.9i		−	1.30853i	0.756267	−	0.654263i	$-0.227021\pi$
$488$		0			0
$489$		−	4133.66i		−	0.382271i
$490$		0			0
$491$			3844.14i			0.353327i	0.984271	+	0.176664i	$0.0565305\pi$
$491$			3844.14i			0.353327i	−0.984271	+	0.176664i	$0.943470\pi$
$492$		0			0
$493$		−	6357.53i		−	0.580789i
$494$		0			0
$495$	999.208			0.0907294
$496$		0			0
$497$	−7564.46	+	5269.63i	−0.682721	+	0.475604i
$498$		0			0
$499$		−	5609.32i		−	0.503221i	−0.967829	−	0.251611i	$-0.919040\pi$
$499$		−	5609.32i		−	0.503221i	0.967829	−	0.251611i	$-0.0809602\pi$
$500$		0			0
$501$	8374.52			0.746799
$502$		0			0
$503$	9310.28			0.825298			0.412649	−	0.910890i	$-0.364604\pi$
$503$	9310.28			0.825298			0.412649	+	0.910890i	$0.364604\pi$
$504$		0			0
$505$	−3453.27			−0.304294
$506$		0			0
$507$	2509.84			0.219854
$508$		0			0
$509$		−	16749.0i		−	1.45852i	−0.684238	−	0.729259i	$-0.739865\pi$
$509$		−	16749.0i		−	1.45852i	0.684238	−	0.729259i	$-0.260135\pi$
$510$		0			0
$511$	−6524.31	+	4545.04i	−0.564812	+	0.393465i
$512$		0			0
$513$	−2976.50			−0.256171
$514$		0			0
$515$		−	6865.68i		−	0.587453i
$516$		0			0
$517$			6898.65i			0.586852i
$518$		0			0
$519$			4300.98i			0.363761i
$520$		0			0
$521$			18811.8i			1.58188i	0.611891	+	0.790942i	$0.290409\pi$
$521$			18811.8i			1.58188i	−0.611891	+	0.790942i	$0.709591\pi$
$522$		0			0
$523$	−3116.84			−0.260593			−0.130296	−	0.991475i	$-0.541593\pi$
$523$	−3116.84			−0.260593			−0.130296	+	0.991475i	$0.541593\pi$
$524$		0			0
$525$	3421.50	+	4911.50i	0.284432	+	0.408296i
$526$		0			0
$527$		−	4028.14i		−	0.332957i
$528$		0			0
$529$	7335.18			0.602875
$530$		0			0
$531$	−2710.35			−0.221505
$532$		0			0
$533$	22.1318			0.00179857
$534$		0			0
$535$	1259.07			0.101747
$536$		0			0
$537$			10417.4i			0.837141i
$538$		0			0
$539$	−8596.67	−	3175.83i	−0.686985	−	0.253790i
$540$		0			0
$541$	−2604.72			−0.206997			−0.103499	−	0.994630i	$-0.533004\pi$
$541$	−2604.72			−0.206997			−0.103499	+	0.994630i	$0.533004\pi$
$542$		0			0
$543$		−	1221.26i		−	0.0965177i
$544$		0			0
$545$		−	4240.50i		−	0.333290i
$546$		0			0
$547$		−	15543.1i		−	1.21495i	−0.794340	−	0.607473i	$-0.792183\pi$
$547$		−	15543.1i		−	1.21495i	0.794340	−	0.607473i	$-0.207817\pi$
$548$		0			0
$549$		−	4222.94i		−	0.328289i
$550$		0			0
$551$	−17091.2			−1.32143
$552$		0			0
$553$	−15863.3	+	11050.9i	−1.21985	+	0.849786i
$554$		0			0
$555$		−	2625.78i		−	0.200825i
$556$		0			0
$557$	−18480.0			−1.40579			−0.702894	−	0.711295i	$-0.748109\pi$
$557$	−18480.0			−1.40579			−0.702894	+	0.711295i	$0.748109\pi$
$558$		0			0
$559$	13092.1			0.990588
$560$		0			0
$561$	3286.97			0.247372
$562$		0			0
$563$	−3343.28			−0.250271			−0.125135	−	0.992140i	$-0.539937\pi$
$563$	−3343.28			−0.250271			−0.125135	+	0.992140i	$0.539937\pi$
$564$		0			0
$565$		−	1734.64i		−	0.129163i
$566$		0			0
$567$	857.489	+	1230.91i	0.0635117	+	0.0911698i
$568$		0			0
$569$	−7915.57			−0.583195			−0.291598	−	0.956541i	$-0.594187\pi$
$569$	−7915.57			−0.583195			−0.291598	+	0.956541i	$0.594187\pi$
$570$		0			0
$571$		−	1800.43i		−	0.131954i	−0.997821	−	0.0659768i	$-0.978984\pi$
$571$		−	1800.43i		−	0.131954i	0.997821	−	0.0659768i	$-0.0210163\pi$
$572$		0			0
$573$			11695.5i			0.852678i
$574$		0			0
$575$			7488.72i			0.543132i
$576$		0			0
$577$			8357.84i			0.603018i	0.953463	+	0.301509i	$0.0974904\pi$
$577$			8357.84i			0.603018i	−0.953463	+	0.301509i	$0.902510\pi$
$578$		0			0
$579$	7098.45			0.509502
$580$		0			0
$581$	−10804.8	−	15510.1i	−0.771532	−	1.10752i
$582$		0			0
$583$			7347.42i			0.521954i
$584$		0			0
$585$	−1379.34			−0.0974848
$586$		0			0
$587$	−26414.1			−1.85729			−0.928643	−	0.370976i	$-0.879023\pi$
$587$	−26414.1			−1.85729			−0.928643	+	0.370976i	$0.879023\pi$
$588$		0			0
$589$	−10829.0			−0.757556
$590$		0			0
$591$	−6519.86			−0.453792
$592$		0			0
$593$		−	27293.9i		−	1.89009i	−0.326938	−	0.945046i	$-0.606017\pi$
$593$		−	27293.9i		−	1.89009i	0.326938	−	0.945046i	$-0.393983\pi$
$594$		0			0
$595$	−1803.84	−	2589.38i	−0.124286	−	0.178411i
$596$		0			0
$597$	−8369.37			−0.573761
$598$		0			0
$599$		−	1254.96i		−	0.0856034i	−0.999084	−	0.0428017i	$-0.986372\pi$
$599$		−	1254.96i		−	0.0856034i	0.999084	−	0.0428017i	$-0.0136284\pi$
$600$		0			0
$601$			13532.3i			0.918460i	0.888317	+	0.459230i	$0.151875\pi$
$601$			13532.3i			0.918460i	−0.888317	+	0.459230i	$0.848125\pi$
$602$		0			0
$603$		−	5446.96i		−	0.367856i
$604$		0			0
$605$		−	2564.24i		−	0.172316i
$606$		0			0
$607$	21041.4			1.40699			0.703497	−	0.710699i	$-0.251621\pi$
$607$	21041.4			1.40699			0.703497	+	0.710699i	$0.251621\pi$
$608$		0			0
$609$	4923.74	+	7067.94i	0.327619	+	0.470291i
$610$		0			0
$611$		−	9523.11i		−	0.630546i
$612$		0			0
$613$	6166.85			0.406324			0.203162	−	0.979145i	$-0.434878\pi$
$613$	6166.85			0.406324			0.203162	+	0.979145i	$0.434878\pi$
$614$		0			0
$615$	7.48005			0.000490447
$616$		0			0
$617$	−28680.9			−1.87139			−0.935697	−	0.352805i	$-0.885228\pi$
$617$	−28680.9			−1.87139			−0.935697	+	0.352805i	$0.885228\pi$
$618$		0			0
$619$	−21715.8			−1.41007			−0.705035	−	0.709173i	$-0.749069\pi$
$619$	−21715.8			−1.41007			−0.705035	+	0.709173i	$0.749069\pi$
$620$		0			0
$621$			1876.80i			0.121278i
$622$		0			0
$623$	−14234.6	+	9916.24i	−0.915403	+	0.637698i
$624$		0			0
$625$	9448.33			0.604693
$626$		0			0
$627$		−	8836.47i		−	0.562831i
$628$		0			0
$629$		−	8637.69i		−	0.547547i
$630$		0			0
$631$		−	691.303i		−	0.0436139i	−0.999762	−	0.0218069i	$-0.993058\pi$
$631$		−	691.303i		−	0.0436139i	0.999762	−	0.0218069i	$-0.00694191\pi$
$632$		0			0
$633$			2684.41i			0.168555i
$634$		0			0
$635$	4960.06			0.309975
$636$		0			0
$637$	11867.1	+	4384.01i	0.738136	+	0.272686i
$638$		0			0
$639$			4480.01i			0.277350i
$640$		0			0
$641$	−5076.71			−0.312821			−0.156410	−	0.987692i	$-0.549992\pi$
$641$	−5076.71			−0.312821			−0.156410	+	0.987692i	$0.549992\pi$
$642$		0			0
$643$	−23620.0			−1.44865			−0.724325	−	0.689458i	$-0.757849\pi$
$643$	−23620.0			−1.44865			−0.724325	+	0.689458i	$0.757849\pi$
$644$		0			0
$645$	4424.85			0.270121
$646$		0			0
$647$	−22871.5			−1.38976			−0.694878	−	0.719128i	$-0.744542\pi$
$647$	−22871.5			−1.38976			−0.694878	+	0.719128i	$0.744542\pi$
$648$		0			0
$649$		−	8046.37i		−	0.486668i
$650$		0			0
$651$	3119.69	+	4478.25i	0.187819	+	0.269611i
$652$		0			0
$653$	−11952.1			−0.716265			−0.358133	−	0.933671i	$-0.616586\pi$
$653$	−11952.1			−0.716265			−0.358133	+	0.933671i	$0.616586\pi$
$654$		0			0
$655$		−	3999.22i		−	0.238569i
$656$		0			0
$657$			3864.00i			0.229450i
$658$		0			0
$659$		−	15849.4i		−	0.936879i	−0.883496	−	0.468440i	$-0.844816\pi$
$659$		−	15849.4i		−	0.936879i	0.883496	−	0.468440i	$-0.155184\pi$
$660$		0			0
$661$		−	23354.3i		−	1.37425i	−0.726540	−	0.687124i	$-0.758873\pi$
$661$		−	23354.3i		−	1.37425i	0.726540	−	0.687124i	$-0.241127\pi$
$662$		0			0
$663$	−4537.43			−0.265791
$664$		0			0
$665$	−6961.13	+	4849.33i	−0.405926	+	0.282781i
$666$		0			0
$667$			10776.7i			0.625600i
$668$		0			0
$669$	8412.91			0.486191
$670$		0			0
$671$	12536.9			0.721281
$672$		0			0
$673$	−13707.0			−0.785093			−0.392546	−	0.919732i	$-0.628406\pi$
$673$	−13707.0			−0.785093			−0.392546	+	0.919732i	$0.628406\pi$
$674$		0			0
$675$	2908.82			0.165867
$676$		0			0
$677$		−	26246.5i		−	1.49001i	−0.667060	−	0.745004i	$-0.732448\pi$
$677$		−	26246.5i		−	1.49001i	0.667060	−	0.745004i	$-0.267552\pi$
$678$		0			0
$679$	−6197.85	+	4317.62i	−0.350297	+	0.244028i
$680$		0			0
$681$	16678.4			0.938497
$682$		0			0
$683$		−	15624.1i		−	0.875317i	−0.899141	−	0.437658i	$-0.855808\pi$
$683$		−	15624.1i		−	0.875317i	0.899141	−	0.437658i	$-0.144192\pi$
$684$		0			0
$685$			11810.2i			0.658752i
$686$		0			0
$687$		−	7098.74i		−	0.394227i
$688$		0			0
$689$		−	10142.6i		−	0.560817i
$690$		0			0
$691$	−13845.8			−0.762254			−0.381127	−	0.924523i	$-0.624464\pi$
$691$	−13845.8			−0.762254			−0.381127	+	0.924523i	$0.624464\pi$
$692$		0			0
$693$	−3654.26	+	2545.67i	−0.200309	+	0.139541i
$694$		0			0
$695$		−	8363.52i		−	0.456470i
$696$		0			0
$697$	24.6062			0.00133720
$698$		0			0
$699$	−9334.38			−0.505091
$700$		0			0
$701$	−5302.85			−0.285714			−0.142857	−	0.989743i	$-0.545629\pi$
$701$	−5302.85			−0.285714			−0.142857	+	0.989743i	$0.545629\pi$
$702$		0			0
$703$	−23221.0			−1.24580
$704$		0			0
$705$		−	3218.59i		−	0.171942i
$706$		0			0
$707$	12629.1	−	8797.85i	0.671807	−	0.468002i
$708$		0			0
$709$	30171.8			1.59820			0.799102	−	0.601196i	$-0.205309\pi$
$709$	30171.8			1.59820			0.799102	+	0.601196i	$0.205309\pi$
$710$		0			0
$711$			9394.99i			0.495555i
$712$		0			0
$713$			6828.12i			0.358647i
$714$		0			0
$715$		−	4094.91i		−	0.214183i
$716$		0			0
$717$		−	13484.9i		−	0.702377i
$718$		0			0
$719$	14915.6			0.773655			0.386828	−	0.922152i	$-0.373571\pi$
$719$	14915.6			0.773655			0.386828	+	0.922152i	$0.373571\pi$
$720$		0			0
$721$	17491.6	+	25108.9i	0.903498	+	1.29695i
$722$		0			0
$723$			7666.67i			0.394366i
$724$		0			0
$725$	16702.6			0.855610
$726$		0			0
$727$	−17914.4			−0.913905			−0.456952	−	0.889491i	$-0.651059\pi$
$727$	−17914.4			−0.913905			−0.456952	+	0.889491i	$0.651059\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	14555.9			0.736481
$732$		0			0
$733$			7442.95i			0.375050i	0.982260	+	0.187525i	$0.0600465\pi$
$733$			7442.95i			0.375050i	−0.982260	+	0.187525i	$0.939953\pi$
$734$		0			0
$735$	4010.82	+	1481.70i	0.201280	+	0.0743581i
$736$		0			0
$737$	16170.7			0.808214
$738$		0			0
$739$			6072.67i			0.302283i	0.988512	+	0.151141i	$0.0482949\pi$
$739$			6072.67i			0.302283i	−0.988512	+	0.151141i	$0.951705\pi$
$740$		0			0
$741$			12198.1i			0.604737i
$742$		0			0
$743$		−	25677.6i		−	1.26786i	−0.773392	−	0.633929i	$-0.781441\pi$
$743$		−	25677.6i		−	1.26786i	0.773392	−	0.633929i	$-0.218559\pi$
$744$		0			0
$745$		−	6179.11i		−	0.303873i
$746$		0			0
$747$	−9185.81			−0.449921
$748$		0			0
$749$	−4604.62	+	3207.72i	−0.224632	+	0.156485i
$750$		0			0
$751$		−	28754.7i		−	1.39717i	−0.715528	−	0.698584i	$-0.753814\pi$
$751$		−	28754.7i		−	1.39717i	0.715528	−	0.698584i	$-0.246186\pi$
$752$		0			0
$753$	−2411.62			−0.116712
$754$		0			0
$755$	1112.83			0.0536424
$756$		0			0
$757$	28046.8			1.34660			0.673301	−	0.739368i	$-0.264876\pi$
$757$	28046.8			1.34660			0.673301	+	0.739368i	$0.264876\pi$
$758$		0			0
$759$	−5571.76			−0.266459
$760$		0			0
$761$		−	14783.1i		−	0.704186i	−0.935965	−	0.352093i	$-0.885470\pi$
$761$		−	14783.1i		−	0.704186i	0.935965	−	0.352093i	$-0.114530\pi$
$762$		0			0
$763$	10803.5	+	15508.2i	0.512598	+	0.735824i
$764$		0			0
$765$	−1533.55			−0.0724779
$766$		0			0
$767$			11107.5i			0.522904i
$768$		0			0
$769$		−	31012.1i		−	1.45426i	−0.686500	−	0.727130i	$-0.740854\pi$
$769$		−	31012.1i		−	1.45426i	0.686500	−	0.727130i	$-0.259146\pi$
$770$		0			0
$771$		−	22731.6i		−	1.06181i
$772$		0			0
$773$		−	34993.2i		−	1.62823i	−0.580706	−	0.814113i	$-0.697223\pi$
$773$		−	34993.2i		−	1.62823i	0.580706	−	0.814113i	$-0.302777\pi$
$774$		0			0
$775$	10582.8			0.490508
$776$		0			0
$777$	6689.66	+	9602.88i	0.308868	+	0.443374i
$778$		0			0
$779$		−	66.1497i		−	0.00304244i
$780$		0			0
$781$	−13300.0			−0.609364
$782$		0			0
$783$	4185.95			0.191052
$784$		0			0
$785$	2366.10			0.107579
$786$		0			0
$787$	12762.8			0.578073			0.289037	−	0.957318i	$-0.406665\pi$
$787$	12762.8			0.578073			0.289037	+	0.957318i	$0.406665\pi$
$788$		0			0
$789$		−	10685.5i		−	0.482145i
$790$		0			0
$791$	4419.33	+	6343.86i	0.198651	+	0.285160i
$792$		0			0
$793$	−17306.3			−0.774985
$794$		0			0
$795$		−	3427.97i		−	0.152928i
$796$		0			0
$797$			43638.6i			1.93947i	0.244155	+	0.969736i	$0.421490\pi$
$797$			43638.6i			1.93947i	−0.244155	+	0.969736i	$0.578510\pi$
$798$		0			0
$799$		−	10587.8i		−	0.468798i
$800$		0			0
$801$			8430.36i			0.371875i
$802$		0			0
$803$	−11471.2			−0.504124
$804$		0			0
$805$	3057.71	+	4389.28i	0.133876	+	0.192176i
$806$		0			0
$807$		−	11834.1i		−	0.516210i
$808$		0			0
$809$	344.856			0.0149870			0.00749350	−	0.999972i	$-0.497615\pi$
$809$	344.856			0.0149870			0.00749350	+	0.999972i	$0.497615\pi$
$810$		0			0
$811$	−18390.9			−0.796291			−0.398145	−	0.917322i	$-0.630346\pi$
$811$	−18390.9			−0.796291			−0.398145	+	0.917322i	$0.630346\pi$
$812$		0			0
$813$	−11918.5			−0.514146
$814$		0			0
$815$	−5725.47			−0.246079
$816$		0			0
$817$		−	39131.0i		−	1.67567i
$818$		0			0
$819$	5044.46	−	3514.12i	0.215223	−	0.149931i
$820$		0			0
$821$	−34681.7			−1.47430			−0.737149	−	0.675730i	$-0.763828\pi$
$821$	−34681.7			−1.47430			−0.737149	+	0.675730i	$0.763828\pi$
$822$		0			0
$823$		−	33952.8i		−	1.43806i	−0.694981	−	0.719028i	$-0.744587\pi$
$823$		−	33952.8i		−	1.43806i	0.694981	−	0.719028i	$-0.255413\pi$
$824$		0			0
$825$			8635.55i			0.364426i
$826$		0			0
$827$		−	1625.51i		−	0.0683487i	−0.999416	−	0.0341744i	$-0.989120\pi$
$827$		−	1625.51i		−	0.0683487i	0.999416	−	0.0341744i	$-0.0108802\pi$
$828$		0			0
$829$			13544.9i			0.567470i	0.958903	+	0.283735i	$0.0915735\pi$
$829$			13544.9i			0.567470i	−0.958903	+	0.283735i	$0.908426\pi$
$830$		0			0
$831$	−21108.9			−0.881179
$832$		0			0
$833$	13193.9	+	4874.15i	0.548788	+	0.202736i
$834$		0			0
$835$		−	11599.4i		−	0.480736i
$836$		0			0
$837$	2652.22			0.109527
$838$		0			0
$839$	37264.6			1.53339			0.766697	−	0.642009i	$-0.221899\pi$
$839$	37264.6			1.53339			0.766697	+	0.642009i	$0.221899\pi$
$840$		0			0
$841$	−353.057			−0.0144761
$842$		0			0
$843$	−1958.44			−0.0800147
$844$		0			0
$845$		−	3476.34i		−	0.141526i
$846$		0			0
$847$	6532.88	+	9377.82i	0.265020	+	0.380432i
$848$		0			0
$849$	18867.6			0.762703
$850$		0			0
$851$			14641.8i			0.589794i
$852$		0			0
$853$			48255.7i			1.93698i	0.249056	+	0.968489i	$0.419880\pi$
$853$			48255.7i			1.93698i	−0.249056	+	0.968489i	$0.580120\pi$
$854$		0			0
$855$			4122.69i			0.164904i
$856$		0			0
$857$			29668.8i			1.18257i	0.806461	+	0.591287i	$0.201380\pi$
$857$			29668.8i			1.18257i	−0.806461	+	0.591287i	$0.798620\pi$
$858$		0			0
$859$	−3516.98			−0.139695			−0.0698474	−	0.997558i	$-0.522251\pi$
$859$	−3516.98			−0.139695			−0.0698474	+	0.997558i	$0.522251\pi$
$860$		0			0
$861$	−27.3557	+	19.0568i	−0.00108279	+	0.000754303i
$862$		0			0
$863$		−	419.237i		−	0.0165365i	−0.999966	−	0.00826825i	$-0.997368\pi$
$863$		−	419.237i		−	0.0165365i	0.999966	−	0.00826825i	$-0.00263190\pi$
$864$		0			0
$865$	5957.22			0.234164
$866$		0			0
$867$	9694.28			0.379740
$868$		0			0
$869$	−27891.4			−1.08878
$870$		0			0
$871$	−22322.5			−0.868391
$872$		0			0
$873$			3670.65i			0.142306i
$874$		0			0
$875$	14695.9	−	10237.6i	0.567787	−	0.395538i
$876$		0			0
$877$	2715.11			0.104541			0.0522707	−	0.998633i	$-0.483354\pi$
$877$	2715.11			0.104541			0.0522707	+	0.998633i	$0.483354\pi$
$878$		0			0
$879$			10355.5i			0.397363i
$880$		0			0
$881$		−	29828.1i		−	1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$881$		−	29828.1i		−	1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$882$		0			0
$883$			46118.5i			1.75766i	0.477137	+	0.878829i	$0.341674\pi$
$883$			46118.5i			1.75766i	−0.477137	+	0.878829i	$0.658326\pi$
$884$		0			0
$885$			3754.07i			0.142589i
$886$		0			0
$887$	−3840.97			−0.145397			−0.0726985	−	0.997354i	$-0.523161\pi$
$887$	−3840.97			−0.145397			−0.0726985	+	0.997354i	$0.523161\pi$
$888$		0			0
$889$	−18139.7	+	12636.7i	−0.684349	+	0.476739i
$890$		0			0
$891$			2164.22i			0.0813738i
$892$		0			0
$893$	−28463.6			−1.06663
$894$		0			0
$895$	14429.0			0.538891
$896$		0			0
$897$	7691.44			0.286298
$898$		0			0
$899$	15229.2			0.564986
$900$		0			0
$901$		−	11276.6i		−	0.416955i
$902$		0			0
$903$	−16182.4	+	11273.1i	−0.596362	+	0.415444i
$904$		0			0
$905$	−1691.54			−0.0621312
$906$		0			0
$907$			52814.3i			1.93348i	0.255754	+	0.966742i	$0.417676\pi$
$907$			52814.3i			1.93348i	−0.255754	+	0.966742i	$0.582324\pi$
$908$		0			0
$909$		−	7479.55i		−	0.272916i
$910$		0			0
$911$			29690.1i			1.07978i	0.841737	+	0.539888i	$0.181533\pi$
$911$			29690.1i			1.07978i	−0.841737	+	0.539888i	$0.818467\pi$
$912$		0			0
$913$		−	27270.4i		−	0.988519i
$914$		0			0
$915$	−5849.12			−0.211329
$916$		0			0
$917$	10188.8	+	14625.8i	0.366917	+	0.526702i
$918$		0			0
$919$			28193.7i			1.01200i	0.862534	+	0.505999i	$0.168876\pi$
$919$			28193.7i			1.01200i	−0.862534	+	0.505999i	$0.831124\pi$
$920$		0			0
$921$	5637.11			0.201682
$922$		0			0
$923$	18359.8			0.654735
$924$		0			0
$925$	22693.0			0.806639
$926$		0			0
$927$	14870.6			0.526877
$928$		0			0
$929$		−	5554.27i		−	0.196157i	−0.995179	−	0.0980784i	$-0.968730\pi$
$929$		−	5554.27i		−	0.196157i	0.995179	−	0.0980784i	$-0.0312696\pi$
$930$		0			0
$931$	13103.3	−	35469.5i	0.461272	−	1.24862i
$932$		0			0
$933$	12089.8			0.424226
$934$		0			0
$935$		−	4552.73i		−	0.159241i
$936$		0			0
$937$		−	6999.65i		−	0.244043i	−0.992527	−	0.122022i	$-0.961062\pi$
$937$		−	6999.65i		−	0.244043i	0.992527	−	0.122022i	$-0.0389377\pi$
$938$		0			0
$939$			14173.1i			0.492567i
$940$		0			0
$941$			31782.0i			1.10103i	0.834827	+	0.550513i	$0.185568\pi$
$941$			31782.0i			1.10103i	−0.834827	+	0.550513i	$0.814432\pi$
$942$		0			0
$943$	−41.7101			−0.00144037
$944$		0			0
$945$	1704.91	−	1187.69i	0.0586886	−	0.0408843i
$946$		0			0
$947$			34722.9i			1.19149i	0.803173	+	0.595746i	$0.203143\pi$
$947$			34722.9i			1.19149i	−0.803173	+	0.595746i	$0.796857\pi$
$948$		0			0
$949$	15835.3			0.541659
$950$		0			0
$951$	−25198.5			−0.859218
$952$		0			0
$953$	−16968.8			−0.576783			−0.288391	−	0.957513i	$-0.593120\pi$
$953$	−16968.8			−0.576783			−0.288391	+	0.957513i	$0.593120\pi$
$954$		0			0
$955$	16199.2			0.548893
$956$		0			0
$957$			12427.1i			0.419759i
$958$		0			0
$959$	−30088.8	−	43191.8i	−1.01316	−	1.45437i
$960$		0			0
$961$	−20141.8			−0.676103
$962$		0			0
$963$			2727.07i			0.0912549i
$964$		0			0
$965$		−	9831.94i		−	0.327981i
$966$		0			0
$967$			6717.59i			0.223395i	0.993742	+	0.111698i	$0.0356288\pi$
$967$			6717.59i			0.223395i	−0.993742	+	0.111698i	$0.964371\pi$
$968$		0			0
$969$			13561.9i			0.449609i
$970$		0			0
$971$	32476.5			1.07335			0.536674	−	0.843789i	$-0.319680\pi$
$971$	32476.5			1.07335			0.536674	+	0.843789i	$0.319680\pi$
$972$		0			0
$973$	21307.6	+	30586.7i	0.702047	+	1.00778i
$974$		0			0
$975$		−	11920.8i		−	0.391559i
$976$		0			0
$977$	−9656.90			−0.316225			−0.158112	−	0.987421i	$-0.550541\pi$
$977$	−9656.90			−0.316225			−0.158112	+	0.987421i	$0.550541\pi$
$978$		0			0
$979$	−25027.6			−0.817045
$980$		0			0
$981$	9184.65			0.298923
$982$		0			0
$983$	731.732			0.0237422			0.0118711	−	0.999930i	$-0.496221\pi$
$983$	731.732			0.0237422			0.0118711	+	0.999930i	$0.496221\pi$
$984$		0			0
$985$			9030.55i			0.292119i
$986$		0			0
$987$	8199.97	+	11770.9i	0.264446	+	0.379607i
$988$		0			0
$989$	−24673.7			−0.793305
$990$		0			0
$991$		−	50307.5i		−	1.61258i	−0.591519	−	0.806291i	$-0.701471\pi$
$991$		−	50307.5i		−	1.61258i	0.591519	−	0.806291i	$-0.298529\pi$
$992$		0			0
$993$			525.789i			0.0168030i
$994$		0			0
$995$			11592.3i			0.369347i
$996$		0			0
$997$		−	1821.15i		−	0.0578501i	−0.999582	−	0.0289250i	$-0.990792\pi$
$997$		−	1821.15i		−	0.0578501i	0.999582	−	0.0289250i	$-0.00920841\pi$
$998$		0			0
$999$	5687.26			0.180117

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.b.j.895.10		24
4.3	odd	2		1344.4.b.i.895.10		24
7.6	odd	2		1344.4.b.i.895.15		24
8.3	odd	2		672.4.b.b.223.15	yes	24
8.5	even	2		672.4.b.a.223.15	yes	24
28.27	even	2	inner	1344.4.b.j.895.15		24
56.13	odd	2		672.4.b.b.223.10	yes	24
56.27	even	2		672.4.b.a.223.10	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.b.a.223.10	✓	24	56.27	even	2
672.4.b.a.223.15	yes	24	8.5	even	2
672.4.b.b.223.10	yes	24	56.13	odd	2
672.4.b.b.223.15	yes	24	8.3	odd	2
1344.4.b.i.895.10		24	4.3	odd	2
1344.4.b.i.895.15		24	7.6	odd	2
1344.4.b.j.895.10		24	1.1	even	1	trivial
1344.4.b.j.895.15		24	28.27	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -4.15525i q^{5} +(10.5863 + 15.1964i) q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -4.15525i q^{5} +(10.5863 + 15.1964i) q^{7} +9.00000 q^{9} +26.7188i q^{11} -36.8834i q^{13} -12.4658i q^{15} -41.0070i q^{17} -110.241 q^{19} +(31.7588 + 45.5892i) q^{21} +69.5113i q^{23} +107.734 q^{25} +27.0000 q^{27} +155.035 q^{29} +98.2305 q^{31} +80.1563i q^{33} +(63.1449 - 43.9886i) q^{35} +210.639 q^{37} -110.650i q^{39} +0.600048i q^{41} +354.960i q^{43} -37.3973i q^{45} +258.195 q^{47} +(-118.861 + 321.747i) q^{49} -123.021i q^{51} +274.991 q^{53} +111.023 q^{55} -330.722 q^{57} -301.151 q^{59} -469.215i q^{61} +(95.2765 + 136.768i) q^{63} -153.260 q^{65} -605.217i q^{67} +208.534i q^{69} +497.779i q^{71} +429.333i q^{73} +323.202 q^{75} +(-406.029 + 282.852i) q^{77} +1043.89i q^{79} +81.0000 q^{81} -1020.65 q^{83} -170.394 q^{85} +465.106 q^{87} +936.707i q^{89} +(560.495 - 390.458i) q^{91} +294.691 q^{93} +458.077i q^{95} +407.850i q^{97} +240.469i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9}+O(q^{10}) \) 24 * q + 72 * q^3 - 20 * q^7 + 216 * q^9	\( 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9} - 56 q^{19} - 60 q^{21} - 432 q^{25} + 648 q^{27} + 464 q^{31} + 568 q^{35} - 504 q^{37} + 560 q^{47} - 128 q^{49} + 784 q^{53} + 424 q^{55} - 168 q^{57} + 800 q^{59} - 180 q^{63} + 560 q^{65} - 1296 q^{75} + 1568 q^{77} + 1944 q^{81} + 1936 q^{83} - 3000 q^{85} - 496 q^{91} + 1392 q^{93}+O(q^{100}) \) 24 * q + 72 * q^3 - 20 * q^7 + 216 * q^9 - 56 * q^19 - 60 * q^21 - 432 * q^25 + 648 * q^27 + 464 * q^31 + 568 * q^35 - 504 * q^37 + 560 * q^47 - 128 * q^49 + 784 * q^53 + 424 * q^55 - 168 * q^57 + 800 * q^59 - 180 * q^63 + 560 * q^65 - 1296 * q^75 + 1568 * q^77 + 1944 * q^81 + 1936 * q^83 - 3000 * q^85 - 496 * q^91 + 1392 * q^93

Introduction

L-functions

Modular forms

Varieties

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Database

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