LMFDB - Embedded newform 1444.3.c.d.721.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1444,3,Mod(721,1444)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1444, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1444.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1444 = 2^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1444.c (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:	$39.3461501736$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.9
Character	$\chi$	$=$	1444.721
Dual form			1444.3.c.d.721.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.77829i q^{3} +5.00808 q^{5} +3.79291 q^{7} +5.83769 q^{9} +O(q^{10})$	$q-1.77829i q^{3} +5.00808 q^{5} +3.79291 q^{7} +5.83769 q^{9} -11.3816 q^{11} -8.41246i q^{13} -8.90582i q^{15} -22.6179 q^{17} -6.74489i q^{21} -17.7553 q^{23} +0.0808977 q^{25} -26.3857i q^{27} -21.3023i q^{29} -48.7049i q^{31} +20.2398i q^{33} +18.9952 q^{35} -32.1316i q^{37} -14.9598 q^{39} -27.7510i q^{41} -45.9531 q^{43} +29.2356 q^{45} +35.7165 q^{47} -34.6138 q^{49} +40.2212i q^{51} +65.2916i q^{53} -57.0001 q^{55} +19.4040i q^{59} +88.8005 q^{61} +22.1418 q^{63} -42.1303i q^{65} -54.8912i q^{67} +31.5741i q^{69} -80.1083i q^{71} +108.838 q^{73} -0.143859i q^{75} -43.1695 q^{77} -101.257i q^{79} +5.61780 q^{81} +75.6113 q^{83} -113.272 q^{85} -37.8816 q^{87} +164.434i q^{89} -31.9077i q^{91} -86.6114 q^{93} +75.3270i q^{97} -66.4424 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9}+O(q^{10}) $ 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9	$ 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9} - 64 q^{11} - 60 q^{17} + 100 q^{23} + 24 q^{25} - 144 q^{35} + 68 q^{39} - 16 q^{43} + 248 q^{45} + 180 q^{47} - 208 q^{49} - 532 q^{55} + 104 q^{61} - 1216 q^{63} - 484 q^{73} + 136 q^{77} + 460 q^{81} + 744 q^{83} + 32 q^{85} - 1296 q^{87} - 256 q^{93} + 804 q^{99}+O(q^{100}) $ 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9 - 64 * q^11 - 60 * q^17 + 100 * q^23 + 24 * q^25 - 144 * q^35 + 68 * q^39 - 16 * q^43 + 248 * q^45 + 180 * q^47 - 208 * q^49 - 532 * q^55 + 104 * q^61 - 1216 * q^63 - 484 * q^73 + 136 * q^77 + 460 * q^81 + 744 * q^83 + 32 * q^85 - 1296 * q^87 - 256 * q^93 + 804 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times$.

$n$	$723$	$1085$
$\chi(n)$	$1$	$-1$

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
$3$	− 1.77829i	− 0.592763i	−0.955070	−	0.296382i	$-0.904220\pi$
$3$	− 1.77829i	− 0.592763i	0.955070	−	0.296382i	$-0.0957800\pi$
$4$	0	0
$5$	5.00808	1.00162	0.500808	−	0.865558i	$-0.333036\pi$
$5$	5.00808	1.00162	0.500808	+	0.865558i	$0.333036\pi$
$6$	0	0
$7$	3.79291	0.541844	0.270922	−	0.962601i	$-0.412671\pi$
$7$	3.79291	0.541844	0.270922	+	0.962601i	$0.412671\pi$
$8$	0	0
$9$	5.83769	0.648632
$10$	0	0
$11$	−11.3816	−1.03469	−0.517346	−	0.855776i	$-0.673080\pi$
$11$	−11.3816	−1.03469	−0.517346	+	0.855776i	$0.673080\pi$
$12$	0	0
$13$	− 8.41246i	− 0.647113i	−0.946209	−	0.323556i	$-0.895122\pi$
$13$	− 8.41246i	− 0.647113i	0.946209	−	0.323556i	$-0.104878\pi$
$14$	0	0
$15$	− 8.90582i	− 0.593721i
$16$	0	0
$17$	−22.6179	−1.33047	−0.665233	−	0.746636i	$-0.731668\pi$
$17$	−22.6179	−1.33047	−0.665233	+	0.746636i	$0.731668\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	− 6.74489i	− 0.321185i
$22$	0	0
$23$	−17.7553	−0.771971	−0.385986	−	0.922505i	$-0.626139\pi$
$23$	−17.7553	−0.771971	−0.385986	+	0.922505i	$0.626139\pi$
$24$	0	0
$25$	0.0808977	0.00323591
$26$	0	0
$27$	− 26.3857i	− 0.977248i
$28$	0	0
$29$	− 21.3023i	− 0.734561i	−0.930110	−	0.367280i	$-0.880289\pi$
$29$	− 21.3023i	− 0.734561i	0.930110	−	0.367280i	$-0.119711\pi$
$30$	0	0
$31$	− 48.7049i	− 1.57113i	−0.618782	−	0.785563i	$-0.712374\pi$
$31$	− 48.7049i	− 1.57113i	0.618782	−	0.785563i	$-0.287626\pi$
$32$	0	0
$33$	20.2398i	0.613328i
$34$	0	0
$35$	18.9952	0.542720
$36$	0	0
$37$	− 32.1316i	− 0.868422i	−0.900811	−	0.434211i	$-0.857027\pi$
$37$	− 32.1316i	− 0.868422i	0.900811	−	0.434211i	$-0.142973\pi$
$38$	0	0
$39$	−14.9598	−0.383584
$40$	0	0
$41$	− 27.7510i	− 0.676853i	−0.940993	−	0.338426i	$-0.890105\pi$
$41$	− 27.7510i	− 0.676853i	0.940993	−	0.338426i	$-0.109895\pi$
$42$	0	0
$43$	−45.9531	−1.06868	−0.534339	−	0.845271i	$-0.679439\pi$
$43$	−45.9531	−1.06868	−0.534339	+	0.845271i	$0.679439\pi$
$44$	0	0
$45$	29.2356	0.649681
$46$	0	0
$47$	35.7165	0.759925	0.379962	−	0.925002i	$-0.375937\pi$
$47$	35.7165	0.759925	0.379962	+	0.925002i	$0.375937\pi$
$48$	0	0
$49$	−34.6138	−0.706405
$50$	0	0
$51$	40.2212i	0.788651i
$52$	0	0
$53$	65.2916i	1.23192i	0.787778	+	0.615959i	$0.211231\pi$
$53$	65.2916i	1.23192i	−0.787778	+	0.615959i	$0.788769\pi$
$54$	0	0
$55$	−57.0001	−1.03637
$56$	0	0
$57$	0	0
$58$	0	0
$59$	19.4040i	0.328881i	0.986387	+	0.164440i	$0.0525818\pi$
$59$	19.4040i	0.328881i	−0.986387	+	0.164440i	$0.947418\pi$
$60$	0	0
$61$	88.8005	1.45575	0.727873	−	0.685712i	$-0.240509\pi$
$61$	88.8005	1.45575	0.727873	+	0.685712i	$0.240509\pi$
$62$	0	0
$63$	22.1418	0.351457
$64$	0	0
$65$	− 42.1303i	− 0.648159i
$66$	0	0
$67$	− 54.8912i	− 0.819272i	−0.912249	−	0.409636i	$-0.865656\pi$
$67$	− 54.8912i	− 0.819272i	0.912249	−	0.409636i	$-0.134344\pi$
$68$	0	0
$69$	31.5741i	0.457596i
$70$	0	0
$71$	− 80.1083i	− 1.12829i	−0.825677	−	0.564143i	$-0.809207\pi$
$71$	− 80.1083i	− 1.12829i	0.825677	−	0.564143i	$-0.190793\pi$
$72$	0	0
$73$	108.838	1.49093	0.745465	−	0.666545i	$-0.232227\pi$
$73$	108.838	1.49093	0.745465	+	0.666545i	$0.232227\pi$
$74$	0	0
$75$	− 0.143859i	− 0.00191813i
$76$	0	0
$77$	−43.1695	−0.560642
$78$	0	0
$79$	− 101.257i	− 1.28174i	−0.767651	−	0.640869i	$-0.778574\pi$
$79$	− 101.257i	− 1.28174i	0.767651	−	0.640869i	$-0.221426\pi$
$80$	0	0
$81$	5.61780	0.0693555
$82$	0	0
$83$	75.6113	0.910979	0.455490	−	0.890241i	$-0.349464\pi$
$83$	75.6113	0.910979	0.455490	+	0.890241i	$0.349464\pi$
$84$	0	0
$85$	−113.272	−1.33262
$86$	0	0
$87$	−37.8816	−0.435420
$88$	0	0
$89$	164.434i	1.84757i	0.382909	+	0.923786i	$0.374922\pi$
$89$	164.434i	1.84757i	−0.382909	+	0.923786i	$0.625078\pi$
$90$	0	0
$91$	− 31.9077i	− 0.350634i
$92$	0	0
$93$	−86.6114	−0.931305
$94$	0	0
$95$	0	0
$96$	0	0
$97$	75.3270i	0.776567i	0.921540	+	0.388283i	$0.126932\pi$
$97$	75.3270i	0.776567i	−0.921540	+	0.388283i	$0.873068\pi$
$98$	0	0
$99$	−66.4424	−0.671135
$100$	0	0
$101$	163.001	1.61388	0.806938	−	0.590636i	$-0.201123\pi$
$101$	163.001	1.61388	0.806938	+	0.590636i	$0.201123\pi$
$102$	0	0
$103$	− 141.143i	− 1.37032i	−0.728393	−	0.685159i	$-0.759733\pi$
$103$	− 141.143i	− 1.37032i	0.728393	−	0.685159i	$-0.240267\pi$
$104$	0	0
$105$	− 33.7790i	− 0.321704i
$106$	0	0
$107$	117.557i	1.09866i	0.835606	+	0.549330i	$0.185117\pi$
$107$	117.557i	1.09866i	−0.835606	+	0.549330i	$0.814883\pi$
$108$	0	0
$109$	− 140.689i	− 1.29073i	−0.763875	−	0.645364i	$-0.776706\pi$
$109$	− 140.689i	− 1.29073i	0.763875	−	0.645364i	$-0.223294\pi$
$110$	0	0
$111$	−57.1393	−0.514769
$112$	0	0
$113$	− 148.143i	− 1.31100i	−0.755194	−	0.655501i	$-0.772457\pi$
$113$	− 148.143i	− 1.31100i	0.755194	−	0.655501i	$-0.227543\pi$
$114$	0	0
$115$	−88.9202	−0.773219
$116$	0	0
$117$	− 49.1093i	− 0.419738i
$118$	0	0
$119$	−85.7877	−0.720905
$120$	0	0
$121$	8.54133	0.0705895
$122$	0	0
$123$	−49.3492	−0.401213
$124$	0	0
$125$	−124.797	−0.998376
$126$	0	0
$127$	192.688i	1.51723i	0.651540	+	0.758615i	$0.274123\pi$
$127$	192.688i	1.51723i	−0.651540	+	0.758615i	$0.725877\pi$
$128$	0	0
$129$	81.7179i	0.633472i
$130$	0	0
$131$	−122.973	−0.938724	−0.469362	−	0.883006i	$-0.655516\pi$
$131$	−122.973	−0.938724	−0.469362	+	0.883006i	$0.655516\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 132.142i	− 0.978828i
$136$	0	0
$137$	104.029	0.759334	0.379667	−	0.925123i	$-0.376039\pi$
$137$	104.029	0.759334	0.379667	+	0.925123i	$0.376039\pi$
$138$	0	0
$139$	−207.392	−1.49203	−0.746015	−	0.665929i	$-0.768035\pi$
$139$	−207.392	−1.49203	−0.746015	+	0.665929i	$0.768035\pi$
$140$	0	0
$141$	− 63.5142i	− 0.450455i
$142$	0	0
$143$	95.7475i	0.669563i
$144$	0	0
$145$	− 106.684i	− 0.735748i
$146$	0	0
$147$	61.5534i	0.418731i
$148$	0	0
$149$	5.73209	0.0384704	0.0192352	−	0.999815i	$-0.493877\pi$
$149$	5.73209	0.0384704	0.0192352	+	0.999815i	$0.493877\pi$
$150$	0	0
$151$	− 195.812i	− 1.29677i	−0.761313	−	0.648385i	$-0.775445\pi$
$151$	− 195.812i	− 1.29677i	0.761313	−	0.648385i	$-0.224555\pi$
$152$	0	0
$153$	−132.036	−0.862983
$154$	0	0
$155$	− 243.918i	− 1.57367i
$156$	0	0
$157$	−9.30741	−0.0592829	−0.0296414	−	0.999561i	$-0.509437\pi$
$157$	−9.30741	−0.0592829	−0.0296414	+	0.999561i	$0.509437\pi$
$158$	0	0
$159$	116.107	0.730235
$160$	0	0
$161$	−67.3444	−0.418288
$162$	0	0
$163$	−289.900	−1.77853	−0.889264	−	0.457394i	$-0.848783\pi$
$163$	−289.900	−1.77853	−0.889264	+	0.457394i	$0.848783\pi$
$164$	0	0
$165$	101.363i	0.614319i
$166$	0	0
$167$	− 63.3848i	− 0.379550i	−0.981828	−	0.189775i	$-0.939224\pi$
$167$	− 63.3848i	− 0.379550i	0.981828	−	0.189775i	$-0.0607758\pi$
$168$	0	0
$169$	98.2305	0.581245
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 94.4125i	− 0.545737i	−0.962051	−	0.272869i	$-0.912028\pi$
$173$	− 94.4125i	− 0.545737i	0.962051	−	0.272869i	$-0.0879724\pi$
$174$	0	0
$175$	0.306838	0.00175336
$176$	0	0
$177$	34.5059	0.194948
$178$	0	0
$179$	− 44.0009i	− 0.245815i	−0.992418	−	0.122908i	$-0.960778\pi$
$179$	− 44.0009i	− 0.245815i	0.992418	−	0.122908i	$-0.0392218\pi$
$180$	0	0
$181$	27.9096i	0.154197i	0.997023	+	0.0770985i	$0.0245656\pi$
$181$	27.9096i	0.154197i	−0.997023	+	0.0770985i	$0.975434\pi$
$182$	0	0
$183$	− 157.913i	− 0.862912i
$184$	0	0
$185$	− 160.918i	− 0.869826i
$186$	0	0
$187$	257.429	1.37662
$188$	0	0
$189$	− 100.079i	− 0.529516i
$190$	0	0
$191$	−213.556	−1.11809	−0.559047	−	0.829136i	$-0.688833\pi$
$191$	−213.556	−1.11809	−0.559047	+	0.829136i	$0.688833\pi$
$192$	0	0
$193$	173.537i	0.899155i	0.893241	+	0.449577i	$0.148425\pi$
$193$	173.537i	0.899155i	−0.893241	+	0.449577i	$0.851575\pi$
$194$	0	0
$195$	−74.9199	−0.384204
$196$	0	0
$197$	98.3826	0.499404	0.249702	−	0.968323i	$-0.419667\pi$
$197$	98.3826	0.499404	0.249702	+	0.968323i	$0.419667\pi$
$198$	0	0
$199$	202.844	1.01932	0.509659	−	0.860376i	$-0.329772\pi$
$199$	202.844	1.01932	0.509659	+	0.860376i	$0.329772\pi$
$200$	0	0
$201$	−97.6125	−0.485634
$202$	0	0
$203$	− 80.7975i	− 0.398017i
$204$	0	0
$205$	− 138.979i	− 0.677947i
$206$	0	0
$207$	−103.650	−0.500725
$208$	0	0
$209$	0	0
$210$	0	0
$211$	318.388i	1.50895i	0.656330	+	0.754474i	$0.272108\pi$
$211$	318.388i	1.50895i	−0.656330	+	0.754474i	$0.727892\pi$
$212$	0	0
$213$	−142.456	−0.668806
$214$	0	0
$215$	−230.137	−1.07040
$216$	0	0
$217$	− 184.733i	− 0.851305i
$218$	0	0
$219$	− 193.545i	− 0.883768i
$220$	0	0
$221$	190.272i	0.860961i
$222$	0	0
$223$	341.990i	1.53359i	0.641893	+	0.766794i	$0.278149\pi$
$223$	341.990i	1.53359i	−0.641893	+	0.766794i	$0.721851\pi$
$224$	0	0
$225$	0.472255	0.00209891
$226$	0	0
$227$	− 21.0303i	− 0.0926443i	−0.998927	−	0.0463222i	$-0.985250\pi$
$227$	− 21.0303i	− 0.0926443i	0.998927	−	0.0463222i	$-0.0147501\pi$
$228$	0	0
$229$	−77.1274	−0.336801	−0.168401	−	0.985719i	$-0.553860\pi$
$229$	−77.1274	−0.336801	−0.168401	+	0.985719i	$0.553860\pi$
$230$	0	0
$231$	76.7678i	0.332328i
$232$	0	0
$233$	136.555	0.586074	0.293037	−	0.956101i	$-0.405334\pi$
$233$	136.555	0.586074	0.293037	+	0.956101i	$0.405334\pi$
$234$	0	0
$235$	178.871	0.761153
$236$	0	0
$237$	−180.065	−0.759766
$238$	0	0
$239$	−2.88902	−0.0120879	−0.00604397	−	0.999982i	$-0.501924\pi$
$239$	−2.88902	−0.0120879	−0.00604397	+	0.999982i	$0.501924\pi$
$240$	0	0
$241$	169.246i	0.702265i	0.936326	+	0.351132i	$0.114203\pi$
$241$	169.246i	0.702265i	−0.936326	+	0.351132i	$0.885797\pi$
$242$	0	0
$243$	− 247.461i	− 1.01836i
$244$	0	0
$245$	−173.349	−0.707547
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 134.459i	− 0.539995i
$250$	0	0
$251$	−24.5710	−0.0978922	−0.0489461	−	0.998801i	$-0.515586\pi$
$251$	−24.5710	−0.0978922	−0.0489461	+	0.998801i	$0.515586\pi$
$252$	0	0
$253$	202.085	0.798753
$254$	0	0
$255$	201.431i	0.789926i
$256$	0	0
$257$	− 65.6494i	− 0.255445i	−0.991810	−	0.127723i	$-0.959233\pi$
$257$	− 65.6494i	− 0.255445i	0.991810	−	0.127723i	$-0.0407667\pi$
$258$	0	0
$259$	− 121.872i	− 0.470550i
$260$	0	0
$261$	− 124.356i	− 0.476460i
$262$	0	0
$263$	3.99567	0.0151927	0.00759633	−	0.999971i	$-0.497582\pi$
$263$	3.99567	0.0151927	0.00759633	+	0.999971i	$0.497582\pi$
$264$	0	0
$265$	326.986i	1.23391i
$266$	0	0
$267$	292.411	1.09517
$268$	0	0
$269$	− 403.633i	− 1.50049i	−0.661158	−	0.750247i	$-0.729935\pi$
$269$	− 403.633i	− 1.50049i	0.661158	−	0.750247i	$-0.270065\pi$
$270$	0	0
$271$	−51.6943	−0.190754	−0.0953769	−	0.995441i	$-0.530406\pi$
$271$	−51.6943	−0.190754	−0.0953769	+	0.995441i	$0.530406\pi$
$272$	0	0
$273$	−56.7411	−0.207843
$274$	0	0
$275$	−0.920747	−0.00334817
$276$	0	0
$277$	259.434	0.936587	0.468293	−	0.883573i	$-0.344869\pi$
$277$	259.434	0.936587	0.468293	+	0.883573i	$0.344869\pi$
$278$	0	0
$279$	− 284.324i	− 1.01908i
$280$	0	0
$281$	517.299i	1.84092i	0.390835	+	0.920461i	$0.372186\pi$
$281$	517.299i	1.84092i	−0.390835	+	0.920461i	$0.627814\pi$
$282$	0	0
$283$	8.31819	0.0293929	0.0146965	−	0.999892i	$-0.495322\pi$
$283$	8.31819	0.0293929	0.0146965	+	0.999892i	$0.495322\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 105.257i	− 0.366749i
$288$	0	0
$289$	222.570	0.770140
$290$	0	0
$291$	133.953	0.460320
$292$	0	0
$293$	231.460i	0.789965i	0.918689	+	0.394983i	$0.129249\pi$
$293$	231.460i	0.789965i	−0.918689	+	0.394983i	$0.870751\pi$
$294$	0	0
$295$	97.1767i	0.329412i
$296$	0	0
$297$	300.312i	1.01115i
$298$	0	0
$299$	149.366i	0.499552i
$300$	0	0
$301$	−174.296	−0.579057
$302$	0	0
$303$	− 289.864i	− 0.956646i
$304$	0	0
$305$	444.720	1.45810
$306$	0	0
$307$	287.395i	0.936140i	0.883691	+	0.468070i	$0.155051\pi$
$307$	287.395i	0.936140i	−0.883691	+	0.468070i	$0.844949\pi$
$308$	0	0
$309$	−250.993	−0.812274
$310$	0	0
$311$	−226.974	−0.729821	−0.364911	−	0.931043i	$-0.618900\pi$
$311$	−226.974	−0.729821	−0.364911	+	0.931043i	$0.618900\pi$
$312$	0	0
$313$	294.297	0.940247	0.470124	−	0.882601i	$-0.344209\pi$
$313$	294.297	0.940247	0.470124	+	0.882601i	$0.344209\pi$
$314$	0	0
$315$	110.888	0.352026
$316$	0	0
$317$	− 79.7382i	− 0.251540i	−0.992059	−	0.125770i	$-0.959860\pi$
$317$	− 79.7382i	− 0.251540i	0.992059	−	0.125770i	$-0.0401402\pi$
$318$	0	0
$319$	242.454i	0.760045i
$320$	0	0
$321$	209.050	0.651245
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 0.680549i	− 0.00209400i
$326$	0	0
$327$	−250.186	−0.765096
$328$	0	0
$329$	135.469	0.411761
$330$	0	0
$331$	− 124.905i	− 0.377355i	−0.982039	−	0.188678i	$-0.939580\pi$
$331$	− 124.905i	− 0.377355i	0.982039	−	0.188678i	$-0.0604201\pi$
$332$	0	0
$333$	− 187.574i	− 0.563287i
$334$	0	0
$335$	− 274.900i	− 0.820597i
$336$	0	0
$337$	− 505.126i	− 1.49889i	−0.662066	−	0.749446i	$-0.730320\pi$
$337$	− 505.126i	− 1.49889i	0.662066	−	0.749446i	$-0.269680\pi$
$338$	0	0
$339$	−263.441	−0.777114
$340$	0	0
$341$	554.341i	1.62563i
$342$	0	0
$343$	−317.140	−0.924606
$344$	0	0
$345$	158.126i	0.458336i
$346$	0	0
$347$	516.500	1.48847	0.744236	−	0.667917i	$-0.232814\pi$
$347$	516.500	1.48847	0.744236	+	0.667917i	$0.232814\pi$
$348$	0	0
$349$	−556.163	−1.59359	−0.796795	−	0.604249i	$-0.793473\pi$
$349$	−556.163	−1.59359	−0.796795	+	0.604249i	$0.793473\pi$
$350$	0	0
$351$	−221.969	−0.632389
$352$	0	0
$353$	610.107	1.72835	0.864174	−	0.503193i	$-0.167841\pi$
$353$	610.107	1.72835	0.864174	+	0.503193i	$0.167841\pi$
$354$	0	0
$355$	− 401.189i	− 1.13011i
$356$	0	0
$357$	152.555i	0.427326i
$358$	0	0
$359$	110.697	0.308348	0.154174	−	0.988044i	$-0.450728\pi$
$359$	110.697	0.308348	0.154174	+	0.988044i	$0.450728\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	− 15.1890i	− 0.0418428i
$364$	0	0
$365$	545.069	1.49334
$366$	0	0
$367$	266.521	0.726217	0.363108	−	0.931747i	$-0.381715\pi$
$367$	266.521	0.726217	0.363108	+	0.931747i	$0.381715\pi$
$368$	0	0
$369$	− 162.002i	− 0.439028i
$370$	0	0
$371$	247.645i	0.667507i
$372$	0	0
$373$	− 158.234i	− 0.424221i	−0.977246	−	0.212111i	$-0.931966\pi$
$373$	− 158.234i	− 0.424221i	0.977246	−	0.212111i	$-0.0680337\pi$
$374$	0	0
$375$	221.925i	0.591800i
$376$	0	0
$377$	−179.204	−0.475343
$378$	0	0
$379$	331.503i	0.874677i	0.899297	+	0.437339i	$0.144079\pi$
$379$	331.503i	0.874677i	−0.899297	+	0.437339i	$0.855921\pi$
$380$	0	0
$381$	342.655	0.899357
$382$	0	0
$383$	207.713i	0.542332i	0.962533	+	0.271166i	$0.0874092\pi$
$383$	207.713i	0.542332i	−0.962533	+	0.271166i	$0.912591\pi$
$384$	0	0
$385$	−216.196	−0.561549
$386$	0	0
$387$	−268.260	−0.693178
$388$	0	0
$389$	229.550	0.590102	0.295051	−	0.955482i	$-0.404663\pi$
$389$	229.550	0.590102	0.295051	+	0.955482i	$0.404663\pi$
$390$	0	0
$391$	401.589	1.02708
$392$	0	0
$393$	218.681i	0.556441i
$394$	0	0
$395$	− 507.105i	− 1.28381i
$396$	0	0
$397$	466.037	1.17390	0.586949	−	0.809624i	$-0.300329\pi$
$397$	466.037	1.17390	0.586949	+	0.809624i	$0.300329\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	455.007i	1.13468i	0.823483	+	0.567340i	$0.192028\pi$
$401$	455.007i	1.13468i	−0.823483	+	0.567340i	$0.807972\pi$
$402$	0	0
$403$	−409.728	−1.01670
$404$	0	0
$405$	28.1344	0.0694676
$406$	0	0
$407$	365.710i	0.898551i
$408$	0	0
$409$	313.971i	0.767655i	0.923405	+	0.383827i	$0.125394\pi$
$409$	313.971i	0.767655i	−0.923405	+	0.383827i	$0.874606\pi$
$410$	0	0
$411$	− 184.993i	− 0.450105i
$412$	0	0
$413$	73.5975i	0.178202i
$414$	0	0
$415$	378.668	0.912452
$416$	0	0
$417$	368.803i	0.884420i
$418$	0	0
$419$	627.213	1.49693	0.748464	−	0.663176i	$-0.230792\pi$
$419$	627.213	1.49693	0.748464	+	0.663176i	$0.230792\pi$
$420$	0	0
$421$	333.666i	0.792556i	0.918131	+	0.396278i	$0.129698\pi$
$421$	333.666i	0.792556i	−0.918131	+	0.396278i	$0.870302\pi$
$422$	0	0
$423$	208.502	0.492911
$424$	0	0
$425$	−1.82974	−0.00430526
$426$	0	0
$427$	336.812	0.788787
$428$	0	0
$429$	170.267	0.396892
$430$	0	0
$431$	− 793.629i	− 1.84137i	−0.390312	−	0.920683i	$-0.627633\pi$
$431$	− 793.629i	− 1.84137i	0.390312	−	0.920683i	$-0.372367\pi$
$432$	0	0
$433$	− 269.413i	− 0.622201i	−0.950377	−	0.311100i	$-0.899302\pi$
$433$	− 269.413i	− 0.622201i	0.950377	−	0.311100i	$-0.100698\pi$
$434$	0	0
$435$	−189.714	−0.436124
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 342.030i	− 0.779111i	−0.921003	−	0.389556i	$-0.872629\pi$
$439$	− 342.030i	− 0.779111i	0.921003	−	0.389556i	$-0.127371\pi$
$440$	0	0
$441$	−202.065	−0.458197
$442$	0	0
$443$	−697.166	−1.57374	−0.786869	−	0.617120i	$-0.788299\pi$
$443$	−697.166	−1.57374	−0.786869	+	0.617120i	$0.788299\pi$
$444$	0	0
$445$	823.499i	1.85056i
$446$	0	0
$447$	− 10.1933i	− 0.0228038i
$448$	0	0
$449$	669.410i	1.49089i	0.666567	+	0.745445i	$0.267763\pi$
$449$	669.410i	1.49089i	−0.666567	+	0.745445i	$0.732237\pi$
$450$	0	0
$451$	315.851i	0.700335i
$452$	0	0
$453$	−348.211	−0.768677
$454$	0	0
$455$	− 159.796i	− 0.351201i
$456$	0	0
$457$	432.366	0.946096	0.473048	−	0.881037i	$-0.343154\pi$
$457$	432.366	0.946096	0.473048	+	0.881037i	$0.343154\pi$
$458$	0	0
$459$	596.790i	1.30020i
$460$	0	0
$461$	109.368	0.237240	0.118620	−	0.992940i	$-0.462153\pi$
$461$	109.368	0.237240	0.118620	+	0.992940i	$0.462153\pi$
$462$	0	0
$463$	130.406	0.281655	0.140828	−	0.990034i	$-0.455024\pi$
$463$	130.406	0.281655	0.140828	+	0.990034i	$0.455024\pi$
$464$	0	0
$465$	−433.757	−0.932811
$466$	0	0
$467$	−766.386	−1.64108	−0.820541	−	0.571587i	$-0.806328\pi$
$467$	−766.386	−1.64108	−0.820541	+	0.571587i	$0.806328\pi$
$468$	0	0
$469$	− 208.197i	− 0.443918i
$470$	0	0
$471$	16.5513i	0.0351407i
$472$	0	0
$473$	523.021	1.10575
$474$	0	0
$475$	0	0
$476$	0	0
$477$	381.152i	0.799061i
$478$	0	0
$479$	123.982	0.258835	0.129417	−	0.991590i	$-0.458689\pi$
$479$	123.982	0.258835	0.129417	+	0.991590i	$0.458689\pi$
$480$	0	0
$481$	−270.306	−0.561967
$482$	0	0
$483$	119.758i	0.247946i
$484$	0	0
$485$	377.244i	0.777822i
$486$	0	0
$487$	− 138.109i	− 0.283592i	−0.989896	−	0.141796i	$-0.954712\pi$
$487$	− 138.109i	− 0.283592i	0.989896	−	0.141796i	$-0.0452877\pi$
$488$	0	0
$489$	515.526i	1.05425i
$490$	0	0
$491$	−463.340	−0.943667	−0.471833	−	0.881688i	$-0.656408\pi$
$491$	−463.340	−0.943667	−0.471833	+	0.881688i	$0.656408\pi$
$492$	0	0
$493$	481.813i	0.977308i
$494$	0	0
$495$	−332.749	−0.672220
$496$	0	0
$497$	− 303.844i	− 0.611355i
$498$	0	0
$499$	500.582	1.00317	0.501585	−	0.865108i	$-0.332750\pi$
$499$	500.582	1.00317	0.501585	+	0.865108i	$0.332750\pi$
$500$	0	0
$501$	−112.717	−0.224983
$502$	0	0
$503$	748.669	1.48841	0.744204	−	0.667952i	$-0.232829\pi$
$503$	748.669	1.48841	0.744204	+	0.667952i	$0.232829\pi$
$504$	0	0
$505$	816.325	1.61649
$506$	0	0
$507$	− 174.682i	− 0.344541i
$508$	0	0
$509$	270.477i	0.531388i	0.964057	+	0.265694i	$0.0856011\pi$
$509$	270.477i	0.531388i	−0.964057	+	0.265694i	$0.914399\pi$
$510$	0	0
$511$	412.812	0.807852
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 706.855i	− 1.37253i
$516$	0	0
$517$	−406.511	−0.786289
$518$	0	0
$519$	−167.893	−0.323493
$520$	0	0
$521$	337.810i	0.648388i	0.945991	+	0.324194i	$0.105093\pi$
$521$	337.810i	0.648388i	−0.945991	+	0.324194i	$0.894907\pi$
$522$	0	0
$523$	− 235.293i	− 0.449891i	−0.974371	−	0.224945i	$-0.927780\pi$
$523$	− 235.293i	− 0.449891i	0.974371	−	0.224945i	$-0.0722203\pi$
$524$	0	0
$525$	− 0.545646i	− 0.00103933i
$526$	0	0
$527$	1101.60i	2.09033i
$528$	0	0
$529$	−213.748	−0.404061
$530$	0	0
$531$	113.274i	0.213323i
$532$	0	0
$533$	−233.454	−0.438000
$534$	0	0
$535$	588.733i	1.10044i
$536$	0	0
$537$	−78.2463	−0.145710
$538$	0	0
$539$	393.962	0.730912
$540$	0	0
$541$	−843.668	−1.55946	−0.779730	−	0.626116i	$-0.784644\pi$
$541$	−843.668	−1.55946	−0.779730	+	0.626116i	$0.784644\pi$
$542$	0	0
$543$	49.6314	0.0914022
$544$	0	0
$545$	− 704.584i	− 1.29281i
$546$	0	0
$547$	− 924.535i	− 1.69019i	−0.534615	−	0.845096i	$-0.679543\pi$
$547$	− 924.535i	− 1.69019i	0.534615	−	0.845096i	$-0.320457\pi$
$548$	0	0
$549$	518.390	0.944243
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 384.060i	− 0.694502i
$554$	0	0
$555$	−286.159	−0.515601
$556$	0	0
$557$	−66.8445	−0.120008	−0.0600040	−	0.998198i	$-0.519111\pi$
$557$	−66.8445	−0.120008	−0.0600040	+	0.998198i	$0.519111\pi$
$558$	0	0
$559$	386.579i	0.691554i
$560$	0	0
$561$	− 457.783i	− 0.816012i
$562$	0	0
$563$	− 88.0324i	− 0.156363i	−0.996939	−	0.0781816i	$-0.975089\pi$
$563$	− 88.0324i	− 0.156363i	0.996939	−	0.0781816i	$-0.0249114\pi$
$564$	0	0
$565$	− 741.914i	− 1.31312i
$566$	0	0
$567$	21.3078	0.0375799
$568$	0	0
$569$	− 716.409i	− 1.25907i	−0.776973	−	0.629534i	$-0.783246\pi$
$569$	− 716.409i	− 1.25907i	0.776973	−	0.629534i	$-0.216754\pi$
$570$	0	0
$571$	−573.651	−1.00464	−0.502321	−	0.864681i	$-0.667520\pi$
$571$	−573.651	−1.00464	−0.502321	+	0.864681i	$0.667520\pi$
$572$	0	0
$573$	379.764i	0.662764i
$574$	0	0
$575$	−1.43637	−0.00249803
$576$	0	0
$577$	−730.417	−1.26589	−0.632944	−	0.774198i	$-0.718154\pi$
$577$	−730.417	−1.26589	−0.632944	+	0.774198i	$0.718154\pi$
$578$	0	0
$579$	308.599	0.532986
$580$	0	0
$581$	286.787	0.493609
$582$	0	0
$583$	− 743.125i	− 1.27466i
$584$	0	0
$585$	− 245.944i	− 0.420416i
$586$	0	0
$587$	688.851	1.17351	0.586755	−	0.809764i	$-0.300405\pi$
$587$	688.851	1.17351	0.586755	+	0.809764i	$0.300405\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	− 174.953i	− 0.296028i
$592$	0	0
$593$	−213.988	−0.360857	−0.180428	−	0.983588i	$-0.557748\pi$
$593$	−213.988	−0.360857	−0.180428	+	0.983588i	$0.557748\pi$
$594$	0	0
$595$	−429.632	−0.722071
$596$	0	0
$597$	− 360.716i	− 0.604214i
$598$	0	0
$599$	997.685i	1.66558i	0.553586	+	0.832792i	$0.313259\pi$
$599$	997.685i	1.66558i	−0.553586	+	0.832792i	$0.686741\pi$
$600$	0	0
$601$	551.397i	0.917465i	0.888574	+	0.458733i	$0.151696\pi$
$601$	551.397i	0.917465i	−0.888574	+	0.458733i	$0.848304\pi$
$602$	0	0
$603$	− 320.438i	− 0.531406i
$604$	0	0
$605$	42.7757	0.0707036
$606$	0	0
$607$	− 232.503i	− 0.383036i	−0.981489	−	0.191518i	$-0.938659\pi$
$607$	− 232.503i	− 0.383036i	0.981489	−	0.191518i	$-0.0613410\pi$
$608$	0	0
$609$	−143.681	−0.235930
$610$	0	0
$611$	− 300.463i	− 0.491757i
$612$	0	0
$613$	−16.8744	−0.0275275	−0.0137637	−	0.999905i	$-0.504381\pi$
$613$	−16.8744	−0.0275275	−0.0137637	+	0.999905i	$0.504381\pi$
$614$	0	0
$615$	−247.145	−0.401862
$616$	0	0
$617$	288.254	0.467187	0.233593	−	0.972334i	$-0.424952\pi$
$617$	288.254	0.467187	0.233593	+	0.972334i	$0.424952\pi$
$618$	0	0
$619$	−38.8490	−0.0627608	−0.0313804	−	0.999508i	$-0.509990\pi$
$619$	−38.8490	−0.0627608	−0.0313804	+	0.999508i	$0.509990\pi$
$620$	0	0
$621$	468.487i	0.754407i
$622$	0	0
$623$	623.683i	1.00110i
$624$	0	0
$625$	−627.016	−1.00323
$626$	0	0
$627$	0	0
$628$	0	0
$629$	726.751i	1.15541i
$630$	0	0
$631$	76.1296	0.120649	0.0603245	−	0.998179i	$-0.480786\pi$
$631$	76.1296	0.120649	0.0603245	+	0.998179i	$0.480786\pi$
$632$	0	0
$633$	566.186	0.894448
$634$	0	0
$635$	964.998i	1.51968i
$636$	0	0
$637$	291.188i	0.457123i
$638$	0	0
$639$	− 467.647i	− 0.731842i
$640$	0	0
$641$	− 228.121i	− 0.355882i	−0.984041	−	0.177941i	$-0.943056\pi$
$641$	− 228.121i	− 0.355882i	0.984041	−	0.177941i	$-0.0569437\pi$
$642$	0	0
$643$	−314.234	−0.488699	−0.244350	−	0.969687i	$-0.578574\pi$
$643$	−314.234	−0.488699	−0.244350	+	0.969687i	$0.578574\pi$
$644$	0	0
$645$	409.250i	0.634496i
$646$	0	0
$647$	535.939	0.828344	0.414172	−	0.910199i	$-0.364071\pi$
$647$	535.939	0.828344	0.414172	+	0.910199i	$0.364071\pi$
$648$	0	0
$649$	− 220.849i	− 0.340291i
$650$	0	0
$651$	−328.509	−0.504622
$652$	0	0
$653$	−286.892	−0.439345	−0.219672	−	0.975574i	$-0.570499\pi$
$653$	−286.892	−0.439345	−0.219672	+	0.975574i	$0.570499\pi$
$654$	0	0
$655$	−615.858	−0.940242
$656$	0	0
$657$	635.362	0.967065
$658$	0	0
$659$	606.219i	0.919907i	0.887943	+	0.459954i	$0.152134\pi$
$659$	606.219i	0.919907i	−0.887943	+	0.459954i	$0.847866\pi$
$660$	0	0
$661$	44.9988i	0.0680768i	0.999421	+	0.0340384i	$0.0108369\pi$
$661$	44.9988i	0.0680768i	−0.999421	+	0.0340384i	$0.989163\pi$
$662$	0	0
$663$	338.359	0.510346
$664$	0	0
$665$	0	0
$666$	0	0
$667$	378.229i	0.567060i
$668$	0	0
$669$	608.157	0.909054
$670$	0	0
$671$	−1010.69	−1.50625
$672$	0	0
$673$	− 878.369i	− 1.30515i	−0.757722	−	0.652577i	$-0.773688\pi$
$673$	− 878.369i	− 1.30515i	0.757722	−	0.652577i	$-0.226312\pi$
$674$	0	0
$675$	− 2.13454i	− 0.00316228i
$676$	0	0
$677$	222.102i	0.328068i	0.986455	+	0.164034i	$0.0524507\pi$
$677$	222.102i	0.328068i	−0.986455	+	0.164034i	$0.947549\pi$
$678$	0	0
$679$	285.708i	0.420778i
$680$	0	0
$681$	−37.3979	−0.0549161
$682$	0	0
$683$	1261.07i	1.84637i	0.384361	+	0.923183i	$0.374422\pi$
$683$	1261.07i	1.84637i	−0.384361	+	0.923183i	$0.625578\pi$
$684$	0	0
$685$	520.984	0.760561
$686$	0	0
$687$	137.155i	0.199643i
$688$	0	0
$689$	549.263	0.797189
$690$	0	0
$691$	677.841	0.980956	0.490478	−	0.871453i	$-0.336822\pi$
$691$	677.841	0.980956	0.490478	+	0.871453i	$0.336822\pi$
$692$	0	0
$693$	−252.010	−0.363651
$694$	0	0
$695$	−1038.64	−1.49444
$696$	0	0
$697$	627.669i	0.900530i
$698$	0	0
$699$	− 242.835i	− 0.347403i
$700$	0	0
$701$	−505.112	−0.720559	−0.360280	−	0.932844i	$-0.617319\pi$
$701$	−505.112	−0.720559	−0.360280	+	0.932844i	$0.617319\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	− 318.084i	− 0.451183i
$706$	0	0
$707$	618.250	0.874469
$708$	0	0
$709$	1297.23	1.82967	0.914834	−	0.403831i	$-0.132321\pi$
$709$	1297.23	1.82967	0.914834	+	0.403831i	$0.132321\pi$
$710$	0	0
$711$	− 591.108i	− 0.831376i
$712$	0	0
$713$	864.772i	1.21286i
$714$	0	0
$715$	479.511i	0.670645i
$716$	0	0
$717$	5.13751i	0.00716529i
$718$	0	0
$719$	393.953	0.547918	0.273959	−	0.961741i	$-0.411667\pi$
$719$	393.953	0.547918	0.273959	+	0.961741i	$0.411667\pi$
$720$	0	0
$721$	− 535.342i	− 0.742499i
$722$	0	0
$723$	300.968	0.416277
$724$	0	0
$725$	− 1.72330i	− 0.00237697i
$726$	0	0
$727$	406.967	0.559790	0.279895	−	0.960031i	$-0.409700\pi$
$727$	406.967	0.559790	0.279895	+	0.960031i	$0.409700\pi$
$728$	0	0
$729$	−389.498	−0.534290
$730$	0	0
$731$	1039.36	1.42184
$732$	0	0
$733$	94.2759	0.128616	0.0643082	−	0.997930i	$-0.479516\pi$
$733$	94.2759	0.128616	0.0643082	+	0.997930i	$0.479516\pi$
$734$	0	0
$735$	308.265i	0.419408i
$736$	0	0
$737$	624.751i	0.847695i
$738$	0	0
$739$	115.195	0.155880	0.0779400	−	0.996958i	$-0.475166\pi$
$739$	115.195	0.155880	0.0779400	+	0.996958i	$0.475166\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 96.0157i	− 0.129227i	−0.997910	−	0.0646136i	$-0.979419\pi$
$743$	− 96.0157i	− 0.129227i	0.997910	−	0.0646136i	$-0.0205815\pi$
$744$	0	0
$745$	28.7068	0.0385326
$746$	0	0
$747$	441.395	0.590890
$748$	0	0
$749$	445.881i	0.595302i
$750$	0	0
$751$	− 1471.66i	− 1.95960i	−0.199987	−	0.979799i	$-0.564090\pi$
$751$	− 1471.66i	− 1.95960i	0.199987	−	0.979799i	$-0.435910\pi$
$752$	0	0
$753$	43.6943i	0.0580269i
$754$	0	0
$755$	− 980.644i	− 1.29887i
$756$	0	0
$757$	−298.108	−0.393802	−0.196901	−	0.980423i	$-0.563088\pi$
$757$	−298.108	−0.393802	−0.196901	+	0.980423i	$0.563088\pi$
$758$	0	0
$759$	− 359.365i	− 0.473471i
$760$	0	0
$761$	892.453	1.17274	0.586369	−	0.810044i	$-0.300557\pi$
$761$	892.453	1.17274	0.586369	+	0.810044i	$0.300557\pi$
$762$	0	0
$763$	− 533.622i	− 0.699374i
$764$	0	0
$765$	−661.249	−0.864378
$766$	0	0
$767$	163.235	0.212823
$768$	0	0
$769$	−921.103	−1.19779	−0.598896	−	0.800827i	$-0.704394\pi$
$769$	−921.103	−1.19779	−0.598896	+	0.800827i	$0.704394\pi$
$770$	0	0
$771$	−116.744	−0.151418
$772$	0	0
$773$	1428.19i	1.84759i	0.382884	+	0.923796i	$0.374931\pi$
$773$	1428.19i	1.84759i	−0.382884	+	0.923796i	$0.625069\pi$
$774$	0	0
$775$	− 3.94011i	− 0.00508402i
$776$	0	0
$777$	−216.724	−0.278924
$778$	0	0
$779$	0	0
$780$	0	0
$781$	911.762i	1.16743i
$782$	0	0
$783$	−562.075	−0.717848
$784$	0	0
$785$	−46.6123	−0.0593787
$786$	0	0
$787$	− 883.115i	− 1.12213i	−0.827772	−	0.561064i	$-0.810392\pi$
$787$	− 883.115i	− 1.12213i	0.827772	−	0.561064i	$-0.189608\pi$
$788$	0	0
$789$	− 7.10546i	− 0.00900565i
$790$	0	0
$791$	− 561.894i	− 0.710359i
$792$	0	0
$793$	− 747.031i	− 0.942031i
$794$	0	0
$795$	581.476	0.731416
$796$	0	0
$797$	− 247.046i	− 0.309970i	−0.987917	−	0.154985i	$-0.950467\pi$
$797$	− 247.046i	− 0.309970i	0.987917	−	0.154985i	$-0.0495329\pi$
$798$	0	0
$799$	−807.832	−1.01105
$800$	0	0
$801$	959.914i	1.19839i
$802$	0	0
$803$	−1238.75	−1.54265
$804$	0	0
$805$	−337.266	−0.418964
$806$	0	0
$807$	−717.776	−0.889437
$808$	0	0
$809$	−146.171	−0.180681	−0.0903404	−	0.995911i	$-0.528796\pi$
$809$	−146.171	−0.180681	−0.0903404	+	0.995911i	$0.528796\pi$
$810$	0	0
$811$	− 1090.41i	− 1.34452i	−0.740313	−	0.672262i	$-0.765323\pi$
$811$	− 1090.41i	− 1.34452i	0.740313	−	0.672262i	$-0.234677\pi$
$812$	0	0
$813$	91.9274i	0.113072i
$814$	0	0
$815$	−1451.84	−1.78140
$816$	0	0
$817$	0	0
$818$	0	0
$819$	− 186.267i	− 0.227433i
$820$	0	0
$821$	1088.34	1.32563	0.662813	−	0.748785i	$-0.269362\pi$
$821$	1088.34	1.32563	0.662813	+	0.748785i	$0.269362\pi$
$822$	0	0
$823$	1321.73	1.60599	0.802997	−	0.595983i	$-0.203237\pi$
$823$	1321.73	1.60599	0.802997	+	0.595983i	$0.203237\pi$
$824$	0	0
$825$	1.63735i	0.00198467i
$826$	0	0
$827$	− 1296.70i	− 1.56796i	−0.620786	−	0.783980i	$-0.713187\pi$
$827$	− 1296.70i	− 1.56796i	0.620786	−	0.783980i	$-0.286813\pi$
$828$	0	0
$829$	− 419.575i	− 0.506121i	−0.967450	−	0.253061i	$-0.918563\pi$
$829$	− 419.575i	− 0.506121i	0.967450	−	0.253061i	$-0.0814372\pi$
$830$	0	0
$831$	− 461.349i	− 0.555174i
$832$	0	0
$833$	782.893	0.939848
$834$	0	0
$835$	− 317.436i	− 0.380163i
$836$	0	0
$837$	−1285.11	−1.53538
$838$	0	0
$839$	− 1109.57i	− 1.32249i	−0.750171	−	0.661244i	$-0.770029\pi$
$839$	− 1109.57i	− 1.32249i	0.750171	−	0.661244i	$-0.229971\pi$
$840$	0	0
$841$	387.214	0.460420
$842$	0	0
$843$	919.907	1.09123
$844$	0	0
$845$	491.946	0.582185
$846$	0	0
$847$	32.3965	0.0382485
$848$	0	0
$849$	− 14.7922i	− 0.0174230i
$850$	0	0
$851$	570.508i	0.670397i
$852$	0	0
$853$	−437.306	−0.512668	−0.256334	−	0.966588i	$-0.582515\pi$
$853$	−437.306	−0.512668	−0.256334	+	0.966588i	$0.582515\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 824.968i	− 0.962623i	−0.876550	−	0.481311i	$-0.840161\pi$
$857$	− 824.968i	− 0.962623i	0.876550	−	0.481311i	$-0.159839\pi$
$858$	0	0
$859$	−1409.14	−1.64044	−0.820219	−	0.572050i	$-0.806148\pi$
$859$	−1409.14	−1.64044	−0.820219	+	0.572050i	$0.806148\pi$
$860$	0	0
$861$	−187.177	−0.217395
$862$	0	0
$863$	− 346.969i	− 0.402050i	−0.979586	−	0.201025i	$-0.935573\pi$
$863$	− 346.969i	− 0.402050i	0.979586	−	0.201025i	$-0.0644273\pi$
$864$	0	0
$865$	− 472.826i	− 0.546619i
$866$	0	0
$867$	− 395.795i	− 0.456510i
$868$	0	0
$869$	1152.47i	1.32620i
$870$	0	0
$871$	−461.770	−0.530161
$872$	0	0
$873$	439.735i	0.503706i
$874$	0	0
$875$	−473.343	−0.540964
$876$	0	0
$877$	− 1490.25i	− 1.69925i	−0.527385	−	0.849627i	$-0.676827\pi$
$877$	− 1490.25i	− 1.69925i	0.527385	−	0.849627i	$-0.323173\pi$
$878$	0	0
$879$	411.602	0.468262
$880$	0	0
$881$	1049.96	1.19178	0.595890	−	0.803066i	$-0.296799\pi$
$881$	1049.96	1.19178	0.595890	+	0.803066i	$0.296799\pi$
$882$	0	0
$883$	513.050	0.581031	0.290515	−	0.956870i	$-0.406173\pi$
$883$	513.050	0.581031	0.290515	+	0.956870i	$0.406173\pi$
$884$	0	0
$885$	172.808	0.195263
$886$	0	0
$887$	110.695i	0.124797i	0.998051	+	0.0623984i	$0.0198749\pi$
$887$	110.695i	0.124797i	−0.998051	+	0.0623984i	$0.980125\pi$
$888$	0	0
$889$	730.849i	0.822102i
$890$	0	0
$891$	−63.9396	−0.0717617
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 220.360i	− 0.246212i
$896$	0	0
$897$	265.616	0.296116
$898$	0	0
$899$	−1037.52	−1.15409
$900$	0	0
$901$	− 1476.76i	− 1.63902i
$902$	0	0
$903$	309.949i	0.343243i
$904$	0	0
$905$	139.774i	0.154446i
$906$	0	0
$907$	771.639i	0.850759i	0.905015	+	0.425380i	$0.139859\pi$
$907$	771.639i	0.850759i	−0.905015	+	0.425380i	$0.860141\pi$
$908$	0	0
$909$	951.552	1.04681
$910$	0	0
$911$	− 1700.40i	− 1.86652i	−0.359197	−	0.933262i	$-0.616949\pi$
$911$	− 1700.40i	− 1.86652i	0.359197	−	0.933262i	$-0.383051\pi$
$912$	0	0
$913$	−860.579	−0.942584
$914$	0	0
$915$	− 790.841i	− 0.864307i
$916$	0	0
$917$	−466.425	−0.508642
$918$	0	0
$919$	−227.859	−0.247942	−0.123971	−	0.992286i	$-0.539563\pi$
$919$	−227.859	−0.247942	−0.123971	+	0.992286i	$0.539563\pi$
$920$	0	0
$921$	511.072	0.554909
$922$	0	0
$923$	−673.908	−0.730128
$924$	0	0
$925$	− 2.59937i	− 0.00281013i
$926$	0	0
$927$	− 823.948i	− 0.888833i
$928$	0	0
$929$	−1125.89	−1.21194	−0.605969	−	0.795488i	$-0.707214\pi$
$929$	−1125.89	−1.21194	−0.605969	+	0.795488i	$0.707214\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	403.626i	0.432611i
$934$	0	0
$935$	1289.22	1.37885
$936$	0	0
$937$	−65.0552	−0.0694292	−0.0347146	−	0.999397i	$-0.511052\pi$
$937$	−65.0552	−0.0694292	−0.0347146	+	0.999397i	$0.511052\pi$
$938$	0	0
$939$	− 523.346i	− 0.557344i
$940$	0	0
$941$	876.944i	0.931928i	0.884804	+	0.465964i	$0.154292\pi$
$941$	876.944i	0.931928i	−0.884804	+	0.465964i	$0.845708\pi$
$942$	0	0
$943$	492.728i	0.522511i
$944$	0	0
$945$	− 501.202i	− 0.530372i
$946$	0	0
$947$	261.819	0.276472	0.138236	−	0.990399i	$-0.455857\pi$
$947$	261.819	0.276472	0.138236	+	0.990399i	$0.455857\pi$
$948$	0	0
$949$	− 915.595i	− 0.964799i
$950$	0	0
$951$	−141.798	−0.149104
$952$	0	0
$953$	− 1429.67i	− 1.50018i	−0.661338	−	0.750088i	$-0.730011\pi$
$953$	− 1429.67i	− 1.50018i	0.661338	−	0.750088i	$-0.269989\pi$
$954$	0	0
$955$	−1069.51	−1.11990
$956$	0	0
$957$	431.154	0.450526
$958$	0	0
$959$	394.571	0.411441
$960$	0	0
$961$	−1411.17	−1.46844
$962$	0	0
$963$	686.259i	0.712626i
$964$	0	0
$965$	869.087i	0.900609i
$966$	0	0
$967$	−852.166	−0.881248	−0.440624	−	0.897692i	$-0.645243\pi$
$967$	−852.166	−0.881248	−0.440624	+	0.897692i	$0.645243\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	256.268i	0.263922i	0.991255	+	0.131961i	$0.0421273\pi$
$971$	256.268i	0.263922i	−0.991255	+	0.131961i	$0.957873\pi$
$972$	0	0
$973$	−786.620	−0.808448
$974$	0	0
$975$	−1.21021	−0.00124124
$976$	0	0
$977$	998.166i	1.02166i	0.859680	+	0.510832i	$0.170663\pi$
$977$	998.166i	1.02166i	−0.859680	+	0.510832i	$0.829337\pi$
$978$	0	0
$979$	− 1871.52i	− 1.91167i
$980$	0	0
$981$	− 821.301i	− 0.837208i
$982$	0	0
$983$	374.980i	0.381464i	0.981642	+	0.190732i	$0.0610862\pi$
$983$	374.980i	0.381464i	−0.981642	+	0.190732i	$0.938914\pi$
$984$	0	0
$985$	492.708	0.500212
$986$	0	0
$987$	− 240.904i	− 0.244077i
$988$	0	0
$989$	815.913	0.824988
$990$	0	0
$991$	933.796i	0.942277i	0.882059	+	0.471138i	$0.156157\pi$
$991$	933.796i	0.942277i	−0.882059	+	0.471138i	$0.843843\pi$
$992$	0	0
$993$	−222.117	−0.223682
$994$	0	0
$995$	1015.86	1.02097
$996$	0	0
$997$	350.932	0.351988	0.175994	−	0.984391i	$-0.443686\pi$
$997$	350.932	0.351988	0.175994	+	0.984391i	$0.443686\pi$
$998$	0	0
$999$	−847.816	−0.848664

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1444.3.c.d.721.9	✓	24
19.18	odd	2	inner	1444.3.c.d.721.16	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1444.3.c.d.721.9	✓	24	1.1	even	1	trivial
1444.3.c.d.721.16	yes	24	19.18	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1444 = 2^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1444.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.77829i q^{3} +5.00808 q^{5} +3.79291 q^{7} +5.83769 q^{9} +O(q^{10})\)	\(q-1.77829i q^{3} +5.00808 q^{5} +3.79291 q^{7} +5.83769 q^{9} -11.3816 q^{11} -8.41246i q^{13} -8.90582i q^{15} -22.6179 q^{17} -6.74489i q^{21} -17.7553 q^{23} +0.0808977 q^{25} -26.3857i q^{27} -21.3023i q^{29} -48.7049i q^{31} +20.2398i q^{33} +18.9952 q^{35} -32.1316i q^{37} -14.9598 q^{39} -27.7510i q^{41} -45.9531 q^{43} +29.2356 q^{45} +35.7165 q^{47} -34.6138 q^{49} +40.2212i q^{51} +65.2916i q^{53} -57.0001 q^{55} +19.4040i q^{59} +88.8005 q^{61} +22.1418 q^{63} -42.1303i q^{65} -54.8912i q^{67} +31.5741i q^{69} -80.1083i q^{71} +108.838 q^{73} -0.143859i q^{75} -43.1695 q^{77} -101.257i q^{79} +5.61780 q^{81} +75.6113 q^{83} -113.272 q^{85} -37.8816 q^{87} +164.434i q^{89} -31.9077i q^{91} -86.6114 q^{93} +75.3270i q^{97} -66.4424 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9}+O(q^{10}) \) 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9	\( 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9} - 64 q^{11} - 60 q^{17} + 100 q^{23} + 24 q^{25} - 144 q^{35} + 68 q^{39} - 16 q^{43} + 248 q^{45} + 180 q^{47} - 208 q^{49} - 532 q^{55} + 104 q^{61} - 1216 q^{63} - 484 q^{73} + 136 q^{77} + 460 q^{81} + 744 q^{83} + 32 q^{85} - 1296 q^{87} - 256 q^{93} + 804 q^{99}+O(q^{100}) \) 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9 - 64 * q^11 - 60 * q^17 + 100 * q^23 + 24 * q^25 - 144 * q^35 + 68 * q^39 - 16 * q^43 + 248 * q^45 + 180 * q^47 - 208 * q^49 - 532 * q^55 + 104 * q^61 - 1216 * q^63 - 484 * q^73 + 136 * q^77 + 460 * q^81 + 744 * q^83 + 32 * q^85 - 1296 * q^87 - 256 * q^93 + 804 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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