LMFDB - Embedded newform 1386.4.c.a.197.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,4,Mod(197,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1386.197");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1386.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:	$81.7766472680$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.5
Character	$\chi$	$=$	1386.197
Dual form			1386.4.c.a.197.32

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} -15.1163i q^{5} +7.00000i q^{7} -8.00000 q^{8} +O(q^{10})$	\(q-2.00000 q^{2} +4.00000 q^{4} -15.1163i q^{5} +7.00000i q^{7} -8.00000 q^{8} +30.2327i q^{10} +(-3.96615 + 36.2666i) q^{11} -14.6513i q^{13} -14.0000i q^{14} +16.0000 q^{16} +6.09184 q^{17} +107.758i q^{19} -60.4653i q^{20} +(7.93230 - 72.5333i) q^{22} -78.7051i q^{23} -103.504 q^{25} +29.3026i q^{26} +28.0000i q^{28} -282.559 q^{29} +232.869 q^{31} -32.0000 q^{32} -12.1837 q^{34} +105.814 q^{35} -69.3915 q^{37} -215.517i q^{38} +120.931i q^{40} +492.545 q^{41} +226.464i q^{43} +(-15.8646 + 145.067i) q^{44} +157.410i q^{46} -494.658i q^{47} -49.0000 q^{49} +207.007 q^{50} -58.6053i q^{52} -294.614i q^{53} +(548.219 + 59.9536i) q^{55} -56.0000i q^{56} +565.119 q^{58} -786.262i q^{59} -806.740i q^{61} -465.737 q^{62} +64.0000 q^{64} -221.474 q^{65} -455.014 q^{67} +24.3674 q^{68} -211.629 q^{70} +666.889i q^{71} +599.723i q^{73} +138.783 q^{74} +431.033i q^{76} +(-253.867 - 27.7630i) q^{77} -440.150i q^{79} -241.861i q^{80} -985.090 q^{82} +728.076 q^{83} -92.0863i q^{85} -452.927i q^{86} +(31.7292 - 290.133i) q^{88} -1091.39i q^{89} +102.559 q^{91} -314.821i q^{92} +989.316i q^{94} +1628.91 q^{95} -1803.73 q^{97} +98.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 72 q^{2} + 144 q^{4} - 288 q^{8}+O(q^{10}) $ 36 * q - 72 * q^2 + 144 * q^4 - 288 * q^8	$ 36 q - 72 q^{2} + 144 q^{4} - 288 q^{8} - 36 q^{11} + 576 q^{16} + 144 q^{17} + 72 q^{22} - 444 q^{25} + 432 q^{29} - 48 q^{31} - 1152 q^{32} - 288 q^{34} + 504 q^{35} + 24 q^{37} - 144 q^{41} - 144 q^{44} - 1764 q^{49} + 888 q^{50} + 2448 q^{55} - 864 q^{58} + 96 q^{62} + 2304 q^{64} - 2400 q^{65} + 624 q^{67} + 576 q^{68} - 1008 q^{70} - 48 q^{74} - 168 q^{77} + 288 q^{82} + 1296 q^{83} + 288 q^{88} - 6096 q^{95} + 768 q^{97} + 3528 q^{98}+O(q^{100}) $ 36 * q - 72 * q^2 + 144 * q^4 - 288 * q^8 - 36 * q^11 + 576 * q^16 + 144 * q^17 + 72 * q^22 - 444 * q^25 + 432 * q^29 - 48 * q^31 - 1152 * q^32 - 288 * q^34 + 504 * q^35 + 24 * q^37 - 144 * q^41 - 144 * q^44 - 1764 * q^49 + 888 * q^50 + 2448 * q^55 - 864 * q^58 + 96 * q^62 + 2304 * q^64 - 2400 * q^65 + 624 * q^67 + 576 * q^68 - 1008 * q^70 - 48 * q^74 - 168 * q^77 + 288 * q^82 + 1296 * q^83 + 288 * q^88 - 6096 * q^95 + 768 * q^97 + 3528 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$1135$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	−2.00000			−0.707107
$2$	−2.00000			−0.707107
$3$		0			0
$3$		0			0
$4$	4.00000			0.500000
$5$		−	15.1163i		−	1.35205i	−0.736881	−	0.676023i	$-0.763702\pi$
$5$		−	15.1163i		−	1.35205i	0.736881	−	0.676023i	$-0.236298\pi$
$6$		0			0
$7$			7.00000i			0.377964i
$7$			7.00000i			0.377964i
$8$	−8.00000			−0.353553
$9$		0			0
$10$			30.2327i			0.956041i
$11$	−3.96615	+	36.2666i	−0.108713	+	0.994073i
$11$	−3.96615	+	36.2666i	−0.108713	+	0.994073i
$12$		0			0
$13$		−	14.6513i		−	0.312580i	−0.987711	−	0.156290i	$-0.950047\pi$
$13$		−	14.6513i		−	0.312580i	0.987711	−	0.156290i	$-0.0499535\pi$
$14$		−	14.0000i		−	0.267261i
$15$		0			0
$16$	16.0000			0.250000
$17$	6.09184			0.0869111			0.0434555	−	0.999055i	$-0.486163\pi$
$17$	6.09184			0.0869111			0.0434555	+	0.999055i	$0.486163\pi$
$18$		0			0
$19$			107.758i			1.30113i	0.759451	+	0.650564i	$0.225468\pi$
$19$			107.758i			1.30113i	−0.759451	+	0.650564i	$0.774532\pi$
$20$		−	60.4653i		−	0.676023i
$21$		0			0
$22$	7.93230	−	72.5333i	0.0768714	−	0.702916i
$23$		−	78.7051i		−	0.713528i	−0.934194	−	0.356764i	$-0.883880\pi$
$23$		−	78.7051i		−	0.713528i	0.934194	−	0.356764i	$-0.116120\pi$
$24$		0			0
$25$	−103.504			−0.828028
$26$			29.3026i			0.221028i
$27$		0			0
$28$			28.0000i			0.188982i
$29$	−282.559			−1.80931			−0.904654	−	0.426146i	$-0.859871\pi$
$29$	−282.559			−1.80931			−0.904654	+	0.426146i	$0.859871\pi$
$30$		0			0
$31$	232.869			1.34918			0.674588	−	0.738195i	$-0.264321\pi$
$31$	232.869			1.34918			0.674588	+	0.738195i	$0.264321\pi$
$32$	−32.0000			−0.176777
$33$		0			0
$34$	−12.1837			−0.0614554
$35$	105.814			0.511025
$36$		0			0
$37$	−69.3915			−0.308321			−0.154161	−	0.988046i	$-0.549267\pi$
$37$	−69.3915			−0.308321			−0.154161	+	0.988046i	$0.549267\pi$
$38$		−	215.517i		−	0.920037i
$39$		0			0
$40$			120.931i			0.478020i
$41$	492.545			1.87616			0.938081	−	0.346417i	$-0.112602\pi$
$41$	492.545			1.87616			0.938081	+	0.346417i	$0.112602\pi$
$42$		0			0
$43$			226.464i			0.803148i	0.915827	+	0.401574i	$0.131537\pi$
$43$			226.464i			0.803148i	−0.915827	+	0.401574i	$0.868463\pi$
$44$	−15.8646	+	145.067i	−0.0543563	+	0.497037i
$45$		0			0
$46$			157.410i			0.504541i
$47$		−	494.658i		−	1.53518i	−0.640943	−	0.767588i	$-0.721457\pi$
$47$		−	494.658i		−	1.53518i	0.640943	−	0.767588i	$-0.278543\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$	207.007			0.585505
$51$		0			0
$52$		−	58.6053i		−	0.156290i
$53$		−	294.614i		−	0.763553i	−0.924255	−	0.381776i	$-0.875312\pi$
$53$		−	294.614i		−	0.763553i	0.924255	−	0.381776i	$-0.124688\pi$
$54$		0			0
$55$	548.219	+	59.9536i	1.34403	+	0.146984i
$56$		−	56.0000i		−	0.133631i
$57$		0			0
$58$	565.119			1.27937
$59$		−	786.262i		−	1.73496i	−0.497472	−	0.867480i	$-0.665738\pi$
$59$		−	786.262i		−	1.73496i	0.497472	−	0.867480i	$-0.334262\pi$
$60$		0			0
$61$		−	806.740i		−	1.69332i	−0.532134	−	0.846660i	$-0.678610\pi$
$61$		−	806.740i		−	1.69332i	0.532134	−	0.846660i	$-0.321390\pi$
$62$	−465.737			−0.954011
$63$		0			0
$64$	64.0000			0.125000
$65$	−221.474			−0.422623
$66$		0			0
$67$	−455.014			−0.829684			−0.414842	−	0.909893i	$-0.636163\pi$
$67$	−455.014			−0.829684			−0.414842	+	0.909893i	$0.636163\pi$
$68$	24.3674			0.0434555
$69$		0			0
$70$	−211.629			−0.361349
$71$			666.889i			1.11472i	0.830271	+	0.557360i	$0.188186\pi$
$71$			666.889i			1.11472i	−0.830271	+	0.557360i	$0.811814\pi$
$72$		0			0
$73$			599.723i			0.961539i	0.876847	+	0.480769i	$0.159643\pi$
$73$			599.723i			0.961539i	−0.876847	+	0.480769i	$0.840357\pi$
$74$	138.783			0.218016
$75$		0			0
$76$			431.033i			0.650564i
$77$	−253.867	−	27.7630i	−0.375724	−	0.0410895i
$78$		0			0
$79$		−	440.150i		−	0.626845i	−0.949614	−	0.313422i	$-0.898524\pi$
$79$		−	440.150i		−	0.626845i	0.949614	−	0.313422i	$-0.101476\pi$
$80$		−	241.861i		−	0.338012i
$81$		0			0
$82$	−985.090			−1.32665
$83$	728.076			0.962852			0.481426	−	0.876487i	$-0.340119\pi$
$83$	728.076			0.962852			0.481426	+	0.876487i	$0.340119\pi$
$84$		0			0
$85$		−	92.0863i		−	0.117508i
$86$		−	452.927i		−	0.567911i
$87$		0			0
$88$	31.7292	−	290.133i	0.0384357	−	0.351458i
$89$		−	1091.39i		−	1.29985i	−0.759997	−	0.649926i	$-0.774800\pi$
$89$		−	1091.39i		−	1.29985i	0.759997	−	0.649926i	$-0.225200\pi$
$90$		0			0
$91$	102.559			0.118144
$92$		−	314.821i		−	0.356764i
$93$		0			0
$94$			989.316i			1.08553i
$95$	1628.91			1.75919
$96$		0			0
$97$	−1803.73			−1.88805			−0.944024	−	0.329878i	$-0.892992\pi$
$97$	−1803.73			−1.88805			−0.944024	+	0.329878i	$0.892992\pi$
$98$	98.0000			0.101015
$99$		0			0
$100$	−414.014			−0.414014
$101$	755.399			0.744208			0.372104	−	0.928191i	$-0.378636\pi$
$101$	755.399			0.744208			0.372104	+	0.928191i	$0.378636\pi$
$102$		0			0
$103$	−265.206			−0.253704			−0.126852	−	0.991922i	$-0.540487\pi$
$103$	−265.206			−0.253704			−0.126852	+	0.991922i	$0.540487\pi$
$104$			117.211i			0.110514i
$105$		0			0
$106$			589.227i			0.539913i
$107$	667.326			0.602924			0.301462	−	0.953478i	$-0.402525\pi$
$107$	667.326			0.602924			0.301462	+	0.953478i	$0.402525\pi$
$108$		0			0
$109$			1397.57i			1.22810i	0.789266	+	0.614051i	$0.210461\pi$
$109$			1397.57i			1.22810i	−0.789266	+	0.614051i	$0.789539\pi$
$110$	−1096.44	−	119.907i	−0.950375	−	0.103934i
$111$		0			0
$112$			112.000i			0.0944911i
$113$			2236.57i			1.86193i	0.365104	+	0.930967i	$0.381034\pi$
$113$			2236.57i			1.86193i	−0.365104	+	0.930967i	$0.618966\pi$
$114$		0			0
$115$	−1189.73			−0.964723
$116$	−1130.24			−0.904654
$117$		0			0
$118$			1572.52i			1.22680i
$119$			42.6429i			0.0328493i
$120$		0			0
$121$	−1299.54	−	287.678i	−0.976363	−	0.216137i
$122$			1613.48i			1.19736i
$123$		0			0
$124$	931.474			0.674588
$125$		−	324.947i		−	0.232513i
$126$		0			0
$127$		−	2026.87i		−	1.41618i	−0.706120	−	0.708092i	$-0.749556\pi$
$127$		−	2026.87i		−	1.41618i	0.706120	−	0.708092i	$-0.250444\pi$
$128$	−128.000			−0.0883883
$129$		0			0
$130$	442.948			0.298840
$131$	−173.964			−0.116025			−0.0580127	−	0.998316i	$-0.518476\pi$
$131$	−173.964			−0.116025			−0.0580127	+	0.998316i	$0.518476\pi$
$132$		0			0
$133$	−754.308			−0.491780
$134$	910.028			0.586675
$135$		0			0
$136$	−48.7347			−0.0307277
$137$		−	1380.55i		−	0.860938i	−0.902605	−	0.430469i	$-0.858348\pi$
$137$		−	1380.55i		−	0.860938i	0.902605	−	0.430469i	$-0.141652\pi$
$138$		0			0
$139$		−	117.469i		−	0.0716806i	−0.999358	−	0.0358403i	$-0.988589\pi$
$139$		−	117.469i		−	0.0716806i	0.999358	−	0.0358403i	$-0.0114108\pi$
$140$	423.257			0.255513
$141$		0			0
$142$		−	1333.78i		−	0.788226i
$143$	531.354	+	58.1093i	0.310728	+	0.0339814i
$144$		0			0
$145$			4271.26i			2.44627i
$146$		−	1199.45i		−	0.679910i
$147$		0			0
$148$	−277.566			−0.154161
$149$	−3332.18			−1.83210			−0.916050	−	0.401063i	$-0.868641\pi$
$149$	−3332.18			−1.83210			−0.916050	+	0.401063i	$0.868641\pi$
$150$		0			0
$151$		−	72.2428i		−	0.0389340i	−0.999810	−	0.0194670i	$-0.993803\pi$
$151$		−	72.2428i		−	0.0389340i	0.999810	−	0.0194670i	$-0.00619693\pi$
$152$		−	862.066i		−	0.460018i
$153$		0			0
$154$	507.733	+	55.5261i	0.265677	+	0.0290547i
$155$		−	3520.12i		−	1.82415i
$156$		0			0
$157$	−1386.66			−0.704890			−0.352445	−	0.935832i	$-0.614650\pi$
$157$	−1386.66			−0.704890			−0.352445	+	0.935832i	$0.614650\pi$
$158$			880.300i			0.443246i
$159$		0			0
$160$			483.723i			0.239010i
$161$	550.936			0.269688
$162$		0			0
$163$	−2810.39			−1.35047			−0.675235	−	0.737603i	$-0.735958\pi$
$163$	−2810.39			−1.35047			−0.675235	+	0.737603i	$0.735958\pi$
$164$	1970.18			0.938081
$165$		0			0
$166$	−1456.15			−0.680839
$167$	−2738.69			−1.26902			−0.634509	−	0.772916i	$-0.718797\pi$
$167$	−2738.69			−1.26902			−0.634509	+	0.772916i	$0.718797\pi$
$168$		0			0
$169$	1982.34			0.902294
$170$			184.173i			0.0830906i
$171$		0			0
$172$			905.854i			0.401574i
$173$	602.715			0.264876			0.132438	−	0.991191i	$-0.457719\pi$
$173$	602.715			0.264876			0.132438	+	0.991191i	$0.457719\pi$
$174$		0			0
$175$		−	724.525i		−	0.312965i
$176$	−63.4584	+	580.266i	−0.0271782	+	0.248518i
$177$		0			0
$178$			2182.77i			0.919134i
$179$		−	1175.69i		−	0.490922i	−0.969406	−	0.245461i	$-0.921061\pi$
$179$		−	1175.69i		−	0.490922i	0.969406	−	0.245461i	$-0.0789393\pi$
$180$		0			0
$181$	1503.10			0.617263			0.308631	−	0.951182i	$-0.400129\pi$
$181$	1503.10			0.617263			0.308631	+	0.951182i	$0.400129\pi$
$182$	−205.118			−0.0835406
$183$		0			0
$184$			629.641i			0.252270i
$185$			1048.94i			0.416865i
$186$		0			0
$187$	−24.1611	+	220.931i	−0.00944833	+	0.0863960i
$188$		−	1978.63i		−	0.767588i
$189$		0			0
$190$	−3257.82			−1.24393
$191$		−	3042.06i		−	1.15244i	−0.817295	−	0.576220i	$-0.804527\pi$
$191$		−	3042.06i		−	1.15244i	0.817295	−	0.576220i	$-0.195473\pi$
$192$		0			0
$193$			2786.21i			1.03915i	0.854426	+	0.519574i	$0.173909\pi$
$193$			2786.21i			1.03915i	−0.854426	+	0.519574i	$0.826091\pi$
$194$	3607.45			1.33505
$195$		0			0
$196$	−196.000			−0.0714286
$197$	161.116			0.0582691			0.0291346	−	0.999575i	$-0.490725\pi$
$197$	161.116			0.0582691			0.0291346	+	0.999575i	$0.490725\pi$
$198$		0			0
$199$	4385.83			1.56233			0.781164	−	0.624326i	$-0.214626\pi$
$199$	4385.83			1.56233			0.781164	+	0.624326i	$0.214626\pi$
$200$	828.028			0.292752
$201$		0			0
$202$	−1510.80			−0.526235
$203$		−	1977.92i		−	0.683854i
$204$		0			0
$205$		−	7445.47i		−	2.53666i
$206$	530.412			0.179396
$207$		0			0
$208$		−	234.421i		−	0.0781451i
$209$	−3908.03	−	427.385i	−1.29342	−	0.141449i
$210$		0			0
$211$		−	4960.80i		−	1.61856i	−0.587425	−	0.809279i	$-0.699858\pi$
$211$		−	4960.80i		−	1.61856i	0.587425	−	0.809279i	$-0.300142\pi$
$212$		−	1178.45i		−	0.381776i
$213$		0			0
$214$	−1334.65			−0.426332
$215$	3423.30			1.08589
$216$		0			0
$217$			1630.08i			0.509940i
$218$		−	2795.15i		−	0.868400i
$219$		0			0
$220$	2192.87	+	239.815i	0.672016	+	0.0734922i
$221$		−	89.2535i		−	0.0271667i
$222$		0			0
$223$	941.995			0.282873			0.141436	−	0.989947i	$-0.454828\pi$
$223$	941.995			0.282873			0.141436	+	0.989947i	$0.454828\pi$
$224$		−	224.000i		−	0.0668153i
$225$		0			0
$226$		−	4473.13i		−	1.31659i
$227$	−5050.18			−1.47662			−0.738308	−	0.674463i	$-0.764375\pi$
$227$	−5050.18			−1.47662			−0.738308	+	0.674463i	$0.764375\pi$
$228$		0			0
$229$	1839.98			0.530959			0.265479	−	0.964117i	$-0.414470\pi$
$229$	1839.98			0.530959			0.265479	+	0.964117i	$0.414470\pi$
$230$	2379.47			0.682162
$231$		0			0
$232$	2260.47			0.639687
$233$	−3951.54			−1.11105			−0.555523	−	0.831501i	$-0.687482\pi$
$233$	−3951.54			−1.11105			−0.555523	+	0.831501i	$0.687482\pi$
$234$		0			0
$235$	−7477.42			−2.07563
$236$		−	3145.05i		−	0.867480i
$237$		0			0
$238$		−	85.2858i		−	0.0232280i
$239$	1496.90			0.405132			0.202566	−	0.979269i	$-0.435072\pi$
$239$	1496.90			0.405132			0.202566	+	0.979269i	$0.435072\pi$
$240$		0			0
$241$			1106.91i			0.295859i	0.988998	+	0.147930i	$0.0472609\pi$
$241$			1106.91i			0.295859i	−0.988998	+	0.147930i	$0.952739\pi$
$242$	2599.08	+	575.356i	0.690393	+	0.152832i
$243$		0			0
$244$		−	3226.96i		−	0.846660i
$245$			740.700i			0.193149i
$246$		0			0
$247$	1578.80			0.406707
$248$	−1862.95			−0.477005
$249$		0			0
$250$			649.895i			0.164412i
$251$		−	5678.91i		−	1.42809i	−0.700101	−	0.714044i	$-0.746862\pi$
$251$		−	5678.91i		−	1.42809i	0.700101	−	0.714044i	$-0.253138\pi$
$252$		0			0
$253$	2854.37	+	312.156i	0.709300	+	0.0775695i
$254$			4053.73i			1.00139i
$255$		0			0
$256$	256.000			0.0625000
$257$		−	4747.63i		−	1.15233i	−0.817333	−	0.576166i	$-0.804548\pi$
$257$		−	4747.63i		−	1.15233i	0.817333	−	0.576166i	$-0.195452\pi$
$258$		0			0
$259$		−	485.740i		−	0.116535i
$260$	−885.897			−0.211311
$261$		0			0
$262$	347.928			0.0820423
$263$	113.485			0.0266075			0.0133038	−	0.999912i	$-0.495765\pi$
$263$	113.485			0.0266075			0.0133038	+	0.999912i	$0.495765\pi$
$264$		0			0
$265$	−4453.48			−1.03236
$266$	1508.62			0.347741
$267$		0			0
$268$	−1820.06			−0.414842
$269$		−	6731.99i		−	1.52586i	−0.646481	−	0.762930i	$-0.723760\pi$
$269$		−	6731.99i		−	1.52586i	0.646481	−	0.762930i	$-0.276240\pi$
$270$		0			0
$271$		−	4809.63i		−	1.07810i	−0.842275	−	0.539048i	$-0.818784\pi$
$271$		−	4809.63i		−	1.07810i	0.842275	−	0.539048i	$-0.181216\pi$
$272$	97.4695			0.0217278
$273$		0			0
$274$			2761.10i			0.608775i
$275$	410.510	−	3753.73i	0.0900171	−	0.823121i
$276$		0			0
$277$			5267.40i			1.14255i	0.820758	+	0.571276i	$0.193552\pi$
$277$			5267.40i			1.14255i	−0.820758	+	0.571276i	$0.806448\pi$
$278$			234.938i			0.0506858i
$279$		0			0
$280$	−846.515			−0.180675
$281$	−6726.22			−1.42794			−0.713972	−	0.700174i	$-0.753106\pi$
$281$	−6726.22			−1.42794			−0.713972	+	0.700174i	$0.753106\pi$
$282$		0			0
$283$			3049.51i			0.640547i	0.947325	+	0.320273i	$0.103775\pi$
$283$			3049.51i			0.640547i	−0.947325	+	0.320273i	$0.896225\pi$
$284$			2667.55i			0.557360i
$285$		0			0
$286$	−1062.71	−	116.219i	−0.219718	−	0.0240285i
$287$			3447.81i			0.709122i
$288$		0			0
$289$	−4875.89			−0.992446
$290$		−	8542.52i		−	1.72977i
$291$		0			0
$292$			2398.89i			0.480769i
$293$	−7413.59			−1.47818			−0.739090	−	0.673607i	$-0.764744\pi$
$293$	−7413.59			−1.47818			−0.739090	+	0.673607i	$0.764744\pi$
$294$		0			0
$295$	−11885.4			−2.34575
$296$	555.132			0.109008
$297$		0			0
$298$	6664.36			1.29549
$299$	−1153.13			−0.223035
$300$		0			0
$301$	−1585.24			−0.303561
$302$			144.486i			0.0275305i
$303$		0			0
$304$			1724.13i			0.325282i
$305$	−12195.0			−2.28945
$306$		0			0
$307$		−	4299.19i		−	0.799243i	−0.916680	−	0.399622i	$-0.869142\pi$
$307$		−	4299.19i		−	0.799243i	0.916680	−	0.399622i	$-0.130858\pi$
$308$	−1015.47	−	111.052i	−0.187862	−	0.0205448i
$309$		0			0
$310$			7040.24i			1.28987i
$311$			9629.27i			1.75571i	0.478927	+	0.877855i	$0.341026\pi$
$311$			9629.27i			1.75571i	−0.478927	+	0.877855i	$0.658974\pi$
$312$		0			0
$313$	982.302			0.177390			0.0886949	−	0.996059i	$-0.471730\pi$
$313$	982.302			0.177390			0.0886949	+	0.996059i	$0.471730\pi$
$314$	2773.33			0.498433
$315$		0			0
$316$		−	1760.60i		−	0.313422i
$317$		−	9721.75i		−	1.72248i	−0.508194	−	0.861242i	$-0.669687\pi$
$317$		−	9721.75i		−	1.72248i	0.508194	−	0.861242i	$-0.330313\pi$
$318$		0			0
$319$	1120.67	−	10247.5i	0.196695	−	1.79859i
$320$		−	967.445i		−	0.169006i
$321$		0			0
$322$	−1101.87			−0.190698
$323$			656.446i			0.113083i
$324$		0			0
$325$			1516.46i			0.258825i
$326$	5620.77			0.954926
$327$		0			0
$328$	−3940.36			−0.663323
$329$	3462.61			0.580242
$330$		0			0
$331$	998.864			0.165869			0.0829343	−	0.996555i	$-0.473571\pi$
$331$	998.864			0.165869			0.0829343	+	0.996555i	$0.473571\pi$
$332$	2912.30			0.481426
$333$		0			0
$334$	5477.37			0.897331
$335$			6878.15i			1.12177i
$336$		0			0
$337$		−	6063.53i		−	0.980123i	−0.871688	−	0.490061i	$-0.836974\pi$
$337$		−	6063.53i		−	0.980123i	0.871688	−	0.490061i	$-0.163026\pi$
$338$	−3964.68			−0.638018
$339$		0			0
$340$		−	368.345i		−	0.0587539i
$341$	−923.591	+	8445.36i	−0.146672	+	1.34118i
$342$		0			0
$343$		−	343.000i		−	0.0539949i
$344$		−	1811.71i		−	0.283956i
$345$		0			0
$346$	−1205.43			−0.187296
$347$	7035.81			1.08848			0.544239	−	0.838930i	$-0.316818\pi$
$347$	7035.81			1.08848			0.544239	+	0.838930i	$0.316818\pi$
$348$		0			0
$349$		−	6976.68i		−	1.07007i	−0.844831	−	0.535033i	$-0.820299\pi$
$349$		−	6976.68i		−	1.07007i	0.844831	−	0.535033i	$-0.179701\pi$
$350$			1449.05i			0.221300i
$351$		0			0
$352$	126.917	−	1160.53i	0.0192179	−	0.175729i
$353$		−	1993.74i		−	0.300613i	−0.988639	−	0.150306i	$-0.951974\pi$
$353$		−	1993.74i		−	0.300613i	0.988639	−	0.150306i	$-0.0480260\pi$
$354$		0			0
$355$	10080.9			1.50715
$356$		−	4365.55i		−	0.649926i
$357$		0			0
$358$			2351.37i			0.347134i
$359$	−7102.06			−1.04410			−0.522051	−	0.852914i	$-0.674833\pi$
$359$	−7102.06			−1.04410			−0.522051	+	0.852914i	$0.674833\pi$
$360$		0			0
$361$	−4752.85			−0.692936
$362$	−3006.20			−0.436471
$363$		0			0
$364$	410.237			0.0590721
$365$	9065.62			1.30004
$366$		0			0
$367$	−8530.67			−1.21334			−0.606672	−	0.794952i	$-0.707496\pi$
$367$	−8530.67			−1.21334			−0.606672	+	0.794952i	$0.707496\pi$
$368$		−	1259.28i		−	0.178382i
$369$		0			0
$370$		−	2097.89i		−	0.294768i
$371$	2062.30			0.288596
$372$		0			0
$373$		−	9349.27i		−	1.29782i	−0.760865	−	0.648910i	$-0.775225\pi$
$373$		−	9349.27i		−	1.29782i	0.760865	−	0.648910i	$-0.224775\pi$
$374$	48.3223	−	441.861i	0.00668098	−	0.0610912i
$375$		0			0
$376$			3957.27i			0.542767i
$377$			4139.87i			0.565554i
$378$		0			0
$379$	−22.5634			−0.00305806			−0.00152903	−	0.999999i	$-0.500487\pi$
$379$	−22.5634			−0.00305806			−0.00152903	+	0.999999i	$0.500487\pi$
$380$	6515.64			0.879593
$381$		0			0
$382$			6084.12i			0.814898i
$383$		−	13059.8i		−	1.74236i	−0.490967	−	0.871178i	$-0.663356\pi$
$383$		−	13059.8i		−	1.74236i	0.490967	−	0.871178i	$-0.336644\pi$
$384$		0			0
$385$	−419.675	+	3837.53i	−0.0555549	+	0.507997i
$386$		−	5572.41i		−	0.734788i
$387$		0			0
$388$	−7214.90			−0.944024
$389$		−	13092.8i		−	1.70650i	−0.521499	−	0.853252i	$-0.674627\pi$
$389$		−	13092.8i		−	1.70650i	0.521499	−	0.853252i	$-0.325373\pi$
$390$		0			0
$391$		−	479.459i		−	0.0620135i
$392$	392.000			0.0505076
$393$		0			0
$394$	−322.232			−0.0412025
$395$	−6653.45			−0.847523
$396$		0			0
$397$	−10525.1			−1.33058			−0.665289	−	0.746586i	$-0.731692\pi$
$397$	−10525.1			−1.33058			−0.665289	+	0.746586i	$0.731692\pi$
$398$	−8771.66			−1.10473
$399$		0			0
$400$	−1656.06			−0.207007
$401$			13914.3i			1.73279i	0.499358	+	0.866396i	$0.333569\pi$
$401$			13914.3i			1.73279i	−0.499358	+	0.866396i	$0.666431\pi$
$402$		0			0
$403$		−	3411.83i		−	0.421726i
$404$	3021.60			0.372104
$405$		0			0
$406$			3955.83i			0.483558i
$407$	275.217	−	2516.60i	0.0335184	−	0.306494i
$408$		0			0
$409$		−	410.754i		−	0.0496589i	−0.999692	−	0.0248294i	$-0.992096\pi$
$409$		−	410.754i		−	0.0496589i	0.999692	−	0.0248294i	$-0.00790427\pi$
$410$			14890.9i			1.79369i
$411$		0			0
$412$	−1060.82			−0.126852
$413$	5503.84			0.655753
$414$		0			0
$415$		−	11005.8i		−	1.30182i
$416$			468.842i			0.0552569i
$417$		0			0
$418$	7816.06	+	854.771i	0.914584	+	0.100020i
$419$			13406.2i			1.56310i	0.623845	+	0.781548i	$0.285570\pi$
$419$			13406.2i			1.56310i	−0.623845	+	0.781548i	$0.714430\pi$
$420$		0			0
$421$	7917.47			0.916566			0.458283	−	0.888806i	$-0.348465\pi$
$421$	7917.47			0.916566			0.458283	+	0.888806i	$0.348465\pi$
$422$			9921.60i			1.14449i
$423$		0			0
$424$			2356.91i			0.269957i
$425$	−630.527			−0.0719649
$426$		0			0
$427$	5647.18			0.640015
$428$	2669.31			0.301462
$429$		0			0
$430$	−6846.60			−0.767842
$431$	3052.48			0.341143			0.170572	−	0.985345i	$-0.445439\pi$
$431$	3052.48			0.341143			0.170572	+	0.985345i	$0.445439\pi$
$432$		0			0
$433$	8283.39			0.919340			0.459670	−	0.888090i	$-0.347968\pi$
$433$	8283.39			0.919340			0.459670	+	0.888090i	$0.347968\pi$
$434$		−	3260.16i		−	0.360582i
$435$		0			0
$436$			5590.29i			0.614051i
$437$	8481.13			0.928392
$438$		0			0
$439$			2325.67i			0.252843i	0.991977	+	0.126421i	$0.0403492\pi$
$439$			2325.67i			0.252843i	−0.991977	+	0.126421i	$0.959651\pi$
$440$	−4385.75	−	479.629i	−0.475187	−	0.0519669i
$441$		0			0
$442$			178.507i			0.0192098i
$443$			4907.56i			0.526333i	0.964750	+	0.263166i	$0.0847668\pi$
$443$			4907.56i			0.526333i	−0.964750	+	0.263166i	$0.915233\pi$
$444$		0			0
$445$	−16497.8			−1.75746
$446$	−1883.99			−0.200021
$447$		0			0
$448$			448.000i			0.0472456i
$449$		−	4286.42i		−	0.450532i	−0.974297	−	0.225266i	$-0.927675\pi$
$449$		−	4286.42i		−	0.450532i	0.974297	−	0.225266i	$-0.0723251\pi$
$450$		0			0
$451$	−1953.51	+	17863.0i	−0.203962	+	1.86504i
$452$			8946.27i			0.930967i
$453$		0			0
$454$	10100.4			1.04413
$455$		−	1550.32i		−	0.159736i
$456$		0			0
$457$		−	1860.47i		−	0.190436i	−0.995456	−	0.0952179i	$-0.969645\pi$
$457$		−	1860.47i		−	0.190436i	0.995456	−	0.0952179i	$-0.0303548\pi$
$458$	−3679.97			−0.375445
$459$		0			0
$460$	−4758.93			−0.482362
$461$	−6958.02			−0.702966			−0.351483	−	0.936194i	$-0.614322\pi$
$461$	−6958.02			−0.702966			−0.351483	+	0.936194i	$0.614322\pi$
$462$		0			0
$463$	16832.1			1.68953			0.844766	−	0.535137i	$-0.179740\pi$
$463$	16832.1			1.68953			0.844766	+	0.535137i	$0.179740\pi$
$464$	−4520.95			−0.452327
$465$		0			0
$466$	7903.07			0.785628
$467$			9356.95i			0.927169i	0.886053	+	0.463584i	$0.153437\pi$
$467$			9356.95i			0.927169i	−0.886053	+	0.463584i	$0.846563\pi$
$468$		0			0
$469$		−	3185.10i		−	0.313591i
$470$	14954.8			1.46769
$471$		0			0
$472$			6290.10i			0.613401i
$473$	−8213.07	−	898.188i	−0.798388	−	0.0873123i
$474$		0			0
$475$		−	11153.4i		−	1.07737i
$476$			170.572i			0.0164247i
$477$		0			0
$478$	−2993.80			−0.286472
$479$	−3095.66			−0.295291			−0.147645	−	0.989040i	$-0.547169\pi$
$479$	−3095.66			−0.295291			−0.147645	+	0.989040i	$0.547169\pi$
$480$		0			0
$481$			1016.68i			0.0963752i
$482$		−	2213.81i		−	0.209204i
$483$		0			0
$484$	−5198.16	−	1150.71i	−0.488182	−	0.108068i
$485$			27265.7i			2.55273i
$486$		0			0
$487$	−5313.14			−0.494377			−0.247188	−	0.968967i	$-0.579507\pi$
$487$	−5313.14			−0.494377			−0.247188	+	0.968967i	$0.579507\pi$
$488$			6453.92i			0.598679i
$489$		0			0
$490$		−	1481.40i		−	0.136577i
$491$	5905.46			0.542789			0.271395	−	0.962468i	$-0.412515\pi$
$491$	5905.46			0.542789			0.271395	+	0.962468i	$0.412515\pi$
$492$		0			0
$493$	−1721.31			−0.157249
$494$	−3157.60			−0.287585
$495$		0			0
$496$	3725.90			0.337294
$497$	−4668.22			−0.421324
$498$		0			0
$499$	−3680.75			−0.330206			−0.165103	−	0.986276i	$-0.552796\pi$
$499$	−3680.75			−0.330206			−0.165103	+	0.986276i	$0.552796\pi$
$500$		−	1299.79i		−	0.116257i
$501$		0			0
$502$			11357.8i			1.00981i
$503$	−5925.23			−0.525235			−0.262617	−	0.964900i	$-0.584586\pi$
$503$	−5925.23			−0.525235			−0.262617	+	0.964900i	$0.584586\pi$
$504$		0			0
$505$		−	11418.9i		−	1.00620i
$506$	−5708.74	−	624.312i	−0.501550	−	0.0548500i
$507$		0			0
$508$		−	8107.47i		−	0.708092i
$509$			7531.22i			0.655826i	0.944708	+	0.327913i	$0.106345\pi$
$509$			7531.22i			0.655826i	−0.944708	+	0.327913i	$0.893655\pi$
$510$		0			0
$511$	−4198.06			−0.363427
$512$	−512.000			−0.0441942
$513$		0			0
$514$			9495.27i			0.814822i
$515$			4008.94i			0.343019i
$516$		0			0
$517$	17939.6	+	1961.89i	1.52608	+	0.166893i
$518$			971.481i			0.0824024i
$519$		0			0
$520$	1771.79			0.149420
$521$			12324.2i			1.03634i	0.855279	+	0.518168i	$0.173386\pi$
$521$			12324.2i			1.03634i	−0.855279	+	0.518168i	$0.826614\pi$
$522$		0			0
$523$		−	9652.00i		−	0.806984i	−0.914983	−	0.403492i	$-0.867796\pi$
$523$		−	9652.00i		−	0.806984i	0.914983	−	0.403492i	$-0.132204\pi$
$524$	−695.857			−0.0580127
$525$		0			0
$526$	−226.970			−0.0188144
$527$	1418.60			0.117258
$528$		0			0
$529$	5972.50			0.490877
$530$	8906.96			0.729988
$531$		0			0
$532$	−3017.23			−0.245890
$533$		−	7216.43i		−	0.586451i
$534$		0			0
$535$		−	10087.5i		−	0.815181i
$536$	3640.11			0.293338
$537$		0			0
$538$			13464.0i			1.07895i
$539$	194.341	−	1777.07i	0.0155304	−	0.142010i
$540$		0			0
$541$			17267.2i			1.37223i	0.727493	+	0.686115i	$0.240685\pi$
$541$			17267.2i			1.37223i	−0.727493	+	0.686115i	$0.759315\pi$
$542$			9619.25i			0.762329i
$543$		0			0
$544$	−194.939			−0.0153639
$545$	21126.2			1.66045
$546$		0			0
$547$		−	10969.2i		−	0.857422i	−0.903442	−	0.428711i	$-0.858968\pi$
$547$		−	10969.2i		−	0.857422i	0.903442	−	0.428711i	$-0.141032\pi$
$548$		−	5522.21i		−	0.430469i
$549$		0			0
$550$	−821.021	+	7507.45i	−0.0636517	+	0.582034i
$551$		−	30448.1i		−	2.35414i
$552$		0			0
$553$	3081.05			0.236925
$554$		−	10534.8i		−	0.807907i
$555$		0			0
$556$		−	469.877i		−	0.0358403i
$557$	9182.25			0.698500			0.349250	−	0.937030i	$-0.386436\pi$
$557$	9182.25			0.698500			0.349250	+	0.937030i	$0.386436\pi$
$558$		0			0
$559$	3317.99			0.251048
$560$	1693.03			0.127756
$561$		0			0
$562$	13452.4			1.00971
$563$	−21425.1			−1.60384			−0.801919	−	0.597432i	$-0.796188\pi$
$563$	−21425.1			−1.60384			−0.801919	+	0.597432i	$0.796188\pi$
$564$		0			0
$565$	33808.7			2.51742
$566$		−	6099.03i		−	0.452935i
$567$		0			0
$568$		−	5335.11i		−	0.394113i
$569$	7145.36			0.526448			0.263224	−	0.964735i	$-0.415214\pi$
$569$	7145.36			0.526448			0.263224	+	0.964735i	$0.415214\pi$
$570$		0			0
$571$		−	15554.8i		−	1.14001i	−0.821641	−	0.570006i	$-0.806941\pi$
$571$		−	15554.8i		−	1.14001i	0.821641	−	0.570006i	$-0.193059\pi$
$572$	2125.42	+	232.437i	0.155364	+	0.0169907i
$573$		0			0
$574$		−	6895.63i		−	0.501425i
$575$			8146.26i			0.590822i
$576$		0			0
$577$	12594.6			0.908699			0.454349	−	0.890824i	$-0.349872\pi$
$577$	12594.6			0.908699			0.454349	+	0.890824i	$0.349872\pi$
$578$	9751.78			0.701766
$579$		0			0
$580$			17085.0i			1.22313i
$581$			5096.53i			0.363924i
$582$		0			0
$583$	10684.6	+	1168.48i	0.759027	+	0.0830078i
$584$		−	4797.79i		−	0.339955i
$585$		0			0
$586$	14827.2			1.04523
$587$			12554.9i			0.882787i	0.897314	+	0.441394i	$0.145516\pi$
$587$			12554.9i			0.882787i	−0.897314	+	0.441394i	$0.854484\pi$
$588$		0			0
$589$			25093.5i			1.75545i
$590$	23770.8			1.65869
$591$		0			0
$592$	−1110.26			−0.0770803
$593$	12541.4			0.868490			0.434245	−	0.900795i	$-0.357015\pi$
$593$	12541.4			0.868490			0.434245	+	0.900795i	$0.357015\pi$
$594$		0			0
$595$	644.604			0.0444138
$596$	−13328.7			−0.916050
$597$		0			0
$598$	2306.27			0.157710
$599$		−	7912.60i		−	0.539733i	−0.962898	−	0.269866i	$-0.913020\pi$
$599$		−	7912.60i		−	0.539733i	0.962898	−	0.269866i	$-0.0869795\pi$
$600$		0			0
$601$			15400.7i			1.04527i	0.852555	+	0.522637i	$0.175052\pi$
$601$			15400.7i			1.04527i	−0.852555	+	0.522637i	$0.824948\pi$
$602$	3170.49			0.214650
$603$		0			0
$604$		−	288.971i		−	0.0194670i
$605$	−4348.63	+	19644.3i	−0.292227	+	1.32009i
$606$		0			0
$607$			3866.08i			0.258516i	0.991611	+	0.129258i	$0.0412596\pi$
$607$			3866.08i			0.258516i	−0.991611	+	0.129258i	$0.958740\pi$
$608$		−	3448.26i		−	0.230009i
$609$		0			0
$610$	24389.9			1.61888
$611$	−7247.39			−0.479866
$612$		0			0
$613$		−	11395.6i		−	0.750837i	−0.926855	−	0.375419i	$-0.877499\pi$
$613$		−	11395.6i		−	0.750837i	0.926855	−	0.375419i	$-0.122501\pi$
$614$			8598.38i			0.565150i
$615$		0			0
$616$	2030.93	+	222.104i	0.132839	+	0.0145273i
$617$		−	2813.59i		−	0.183583i	−0.995778	−	0.0917916i	$-0.970741\pi$
$617$		−	2813.59i		−	0.183583i	0.995778	−	0.0917916i	$-0.0292594\pi$
$618$		0			0
$619$	3608.59			0.234316			0.117158	−	0.993113i	$-0.462622\pi$
$619$	3608.59			0.234316			0.117158	+	0.993113i	$0.462622\pi$
$620$		−	14080.5i		−	0.912074i
$621$		0			0
$622$		−	19258.5i		−	1.24147i
$623$	7639.71			0.491298
$624$		0			0
$625$	−17850.0			−1.14240
$626$	−1964.60			−0.125434
$627$		0			0
$628$	−5546.65			−0.352445
$629$	−422.722			−0.0267965
$630$		0			0
$631$	16214.0			1.02293			0.511466	−	0.859303i	$-0.329103\pi$
$631$	16214.0			1.02293			0.511466	+	0.859303i	$0.329103\pi$
$632$			3521.20i			0.221623i
$633$		0			0
$634$			19443.5i			1.21798i
$635$	−30638.8			−1.91475
$636$		0			0
$637$			717.914i			0.0446543i
$638$	−2241.34	+	20495.0i	−0.139084	+	1.27179i
$639$		0			0
$640$			1934.89i			0.119505i
$641$			11452.6i			0.705693i	0.935681	+	0.352846i	$0.114786\pi$
$641$			11452.6i			0.705693i	−0.935681	+	0.352846i	$0.885214\pi$
$642$		0			0
$643$	2909.16			0.178423			0.0892114	−	0.996013i	$-0.471565\pi$
$643$	2909.16			0.178423			0.0892114	+	0.996013i	$0.471565\pi$
$644$	2203.74			0.134844
$645$		0			0
$646$		−	1312.89i		−	0.0799614i
$647$			10984.7i			0.667472i	0.942667	+	0.333736i	$0.108309\pi$
$647$			10984.7i			0.667472i	−0.942667	+	0.333736i	$0.891691\pi$
$648$		0			0
$649$	28515.1	+	3118.43i	1.72468	+	0.188612i
$650$		−	3032.93i		−	0.183017i
$651$		0			0
$652$	−11241.5			−0.675235
$653$		−	29424.6i		−	1.76336i	−0.471849	−	0.881679i	$-0.656413\pi$
$653$		−	29424.6i		−	1.76336i	0.471849	−	0.881679i	$-0.343587\pi$
$654$		0			0
$655$			2629.70i			0.156872i
$656$	7880.72			0.469040
$657$		0			0
$658$	−6925.21			−0.410293
$659$	20469.0			1.20995			0.604977	−	0.796243i	$-0.293182\pi$
$659$	20469.0			1.20995			0.604977	+	0.796243i	$0.293182\pi$
$660$		0			0
$661$	5493.08			0.323231			0.161616	−	0.986854i	$-0.448330\pi$
$661$	5493.08			0.323231			0.161616	+	0.986854i	$0.448330\pi$
$662$	−1997.73			−0.117287
$663$		0			0
$664$	−5824.61			−0.340420
$665$			11402.4i			0.664910i
$666$		0			0
$667$			22238.9i			1.29099i
$668$	−10954.7			−0.634509
$669$		0			0
$670$		−	13756.3i		−	0.793212i
$671$	29257.8	+	3199.65i	1.68328	+	0.184085i
$672$		0			0
$673$		−	9655.08i		−	0.553010i	−0.961012	−	0.276505i	$-0.910824\pi$
$673$		−	9655.08i		−	0.553010i	0.961012	−	0.276505i	$-0.0891763\pi$
$674$			12127.1i			0.693051i
$675$		0			0
$676$	7929.36			0.451147
$677$	2787.58			0.158250			0.0791251	−	0.996865i	$-0.474787\pi$
$677$	2787.58			0.158250			0.0791251	+	0.996865i	$0.474787\pi$
$678$		0			0
$679$		−	12626.1i		−	0.713615i
$680$			736.691i			0.0415453i
$681$		0			0
$682$	1847.18	−	16890.7i	0.103713	−	0.948357i
$683$			19476.8i			1.09116i	0.838060	+	0.545578i	$0.183690\pi$
$683$			19476.8i			1.09116i	−0.838060	+	0.545578i	$0.816310\pi$
$684$		0			0
$685$	−20868.9			−1.16403
$686$			686.000i			0.0381802i
$687$		0			0
$688$			3623.42i			0.200787i
$689$	−4316.48			−0.238671
$690$		0			0
$691$	3447.11			0.189775			0.0948873	−	0.995488i	$-0.469751\pi$
$691$	3447.11			0.189775			0.0948873	+	0.995488i	$0.469751\pi$
$692$	2410.86			0.132438
$693$		0			0
$694$	−14071.6			−0.769671
$695$	−1775.70			−0.0969155
$696$		0			0
$697$	3000.51			0.163059
$698$			13953.4i			0.756651i
$699$		0			0
$700$		−	2898.10i		−	0.156483i
$701$	−11319.4			−0.609883			−0.304942	−	0.952371i	$-0.598637\pi$
$701$	−11319.4			−0.609883			−0.304942	+	0.952371i	$0.598637\pi$
$702$		0			0
$703$		−	7477.51i		−	0.401166i
$704$	−253.834	+	2321.07i	−0.0135891	+	0.124259i
$705$		0			0
$706$			3987.49i			0.212565i
$707$			5287.79i			0.281284i
$708$		0			0
$709$	−6330.06			−0.335304			−0.167652	−	0.985846i	$-0.553618\pi$
$709$	−6330.06			−0.335304			−0.167652	+	0.985846i	$0.553618\pi$
$710$	−20161.8			−1.06572
$711$		0			0
$712$			8731.10i			0.459567i
$713$		−	18328.0i		−	0.962675i
$714$		0			0
$715$	878.399	−	8032.13i	0.0459444	−	0.420118i
$716$		−	4702.75i		−	0.245461i
$717$		0			0
$718$	14204.1			0.738291
$719$		−	18750.3i		−	0.972558i	−0.873804	−	0.486279i	$-0.838354\pi$
$719$		−	18750.3i		−	0.972558i	0.873804	−	0.486279i	$-0.161646\pi$
$720$		0			0
$721$		−	1856.44i		−	0.0958911i
$722$	9505.69			0.489980
$723$		0			0
$724$	6012.40			0.308631
$725$	29245.9			1.49816
$726$		0			0
$727$	8030.42			0.409673			0.204836	−	0.978796i	$-0.434334\pi$
$727$	8030.42			0.409673			0.204836	+	0.978796i	$0.434334\pi$
$728$	−820.474			−0.0417703
$729$		0			0
$730$	−18131.2			−0.919270
$731$			1379.58i			0.0698025i
$732$		0			0
$733$		−	6486.50i		−	0.326854i	−0.986555	−	0.163427i	$-0.947745\pi$
$733$		−	6486.50i		−	0.326854i	0.986555	−	0.163427i	$-0.0522549\pi$
$734$	17061.3			0.857964
$735$		0			0
$736$			2518.56i			0.126135i
$737$	1804.65	−	16501.8i	0.0901971	−	0.824767i
$738$		0			0
$739$			21884.8i			1.08937i	0.838641	+	0.544684i	$0.183351\pi$
$739$			21884.8i			1.08937i	−0.838641	+	0.544684i	$0.816649\pi$
$740$			4195.78i			0.208432i
$741$		0			0
$742$	−4124.59			−0.204068
$743$	36948.3			1.82436			0.912180	−	0.409789i	$-0.134398\pi$
$743$	36948.3			1.82436			0.912180	+	0.409789i	$0.134398\pi$
$744$		0			0
$745$			50370.4i			2.47708i
$746$			18698.5i			0.917698i
$747$		0			0
$748$	−96.6446	+	883.723i	−0.00472417	+	0.0431980i
$749$			4671.28i			0.227884i
$750$		0			0
$751$	4858.30			0.236061			0.118031	−	0.993010i	$-0.462342\pi$
$751$	4858.30			0.236061			0.118031	+	0.993010i	$0.462342\pi$
$752$		−	7914.53i		−	0.383794i
$753$		0			0
$754$		−	8279.73i		−	0.399907i
$755$	−1092.05			−0.0526406
$756$		0			0
$757$	−17350.6			−0.833049			−0.416524	−	0.909125i	$-0.636752\pi$
$757$	−17350.6			−0.833049			−0.416524	+	0.909125i	$0.636752\pi$
$758$	45.1269			0.00216238
$759$		0			0
$760$	−13031.3			−0.621966
$761$	−26097.2			−1.24313			−0.621565	−	0.783362i	$-0.713503\pi$
$761$	−26097.2			−1.24313			−0.621565	+	0.783362i	$0.713503\pi$
$762$		0			0
$763$	−9783.01			−0.464179
$764$		−	12168.2i		−	0.576220i
$765$		0			0
$766$			26119.5i			1.23203i
$767$	−11519.8			−0.542314
$768$		0			0
$769$		−	18376.6i		−	0.861738i	−0.902414	−	0.430869i	$-0.858207\pi$
$769$		−	18376.6i		−	0.861738i	0.902414	−	0.430869i	$-0.141793\pi$
$770$	839.351	−	7675.06i	0.0392832	−	0.359208i
$771$		0			0
$772$			11144.8i			0.519574i
$773$			16769.9i			0.780299i	0.920752	+	0.390150i	$0.127577\pi$
$773$			16769.9i			0.780299i	−0.920752	+	0.390150i	$0.872423\pi$
$774$		0			0
$775$	−24102.7			−1.11716
$776$	14429.8			0.667526
$777$		0			0
$778$			26185.5i			1.20668i
$779$			53075.8i			2.44113i
$780$		0			0
$781$	−24185.8	−	2644.98i	−1.10811	−	0.121184i
$782$			958.918i			0.0438502i
$783$		0			0
$784$	−784.000			−0.0357143
$785$			20961.3i			0.953044i
$786$		0			0
$787$		−	34925.2i		−	1.58189i	−0.611886	−	0.790946i	$-0.709589\pi$
$787$		−	34925.2i		−	1.58189i	0.611886	−	0.790946i	$-0.290411\pi$
$788$	644.463			0.0291346
$789$		0			0
$790$	13306.9			0.599289
$791$	−15656.0			−0.703745
$792$		0			0
$793$	−11819.8			−0.529298
$794$	21050.2			0.940861
$795$		0			0
$796$	17543.3			0.781164
$797$			1971.23i			0.0876093i	0.999040	+	0.0438047i	$0.0139479\pi$
$797$			1971.23i			0.0876093i	−0.999040	+	0.0438047i	$0.986052\pi$
$798$		0			0
$799$		−	3013.38i		−	0.133424i
$800$	3312.11			0.146376
$801$		0			0
$802$		−	27828.7i		−	1.22527i
$803$	−21750.0	−	2378.59i	−0.955840	−	0.104531i
$804$		0			0
$805$		−	8328.13i		−	0.364631i
$806$			6823.66i			0.298205i
$807$		0			0
$808$	−6043.19			−0.263117
$809$	11922.7			0.518147			0.259074	−	0.965858i	$-0.416583\pi$
$809$	11922.7			0.518147			0.259074	+	0.965858i	$0.416583\pi$
$810$		0			0
$811$			12222.3i			0.529201i	0.964358	+	0.264600i	$0.0852400\pi$
$811$			12222.3i			0.529201i	−0.964358	+	0.264600i	$0.914760\pi$
$812$		−	7911.66i		−	0.341927i
$813$		0			0
$814$	−550.434	+	5033.19i	−0.0237011	+	0.216724i
$815$			42482.7i			1.82590i
$816$		0			0
$817$	−24403.3			−1.04500
$818$			821.508i			0.0351141i
$819$		0			0
$820$		−	29781.9i		−	1.26833i
$821$	−34115.1			−1.45021			−0.725106	−	0.688637i	$-0.758209\pi$
$821$	−34115.1			−1.45021			−0.725106	+	0.688637i	$0.758209\pi$
$822$		0			0
$823$	−35922.7			−1.52149			−0.760745	−	0.649051i	$-0.775166\pi$
$823$	−35922.7			−1.52149			−0.760745	+	0.649051i	$0.775166\pi$
$824$	2121.65			0.0896979
$825$		0			0
$826$	−11007.7			−0.463688
$827$	−44657.0			−1.87772			−0.938861	−	0.344295i	$-0.888118\pi$
$827$	−44657.0			−1.87772			−0.938861	+	0.344295i	$0.888118\pi$
$828$		0			0
$829$	7492.36			0.313897			0.156948	−	0.987607i	$-0.449834\pi$
$829$	7492.36			0.313897			0.156948	+	0.987607i	$0.449834\pi$
$830$			22011.7i			0.920526i
$831$		0			0
$832$		−	937.684i		−	0.0390725i
$833$	−298.500			−0.0124159
$834$		0			0
$835$			41398.9i			1.71577i
$836$	−15632.1	−	1709.54i	−0.646709	−	0.0707245i
$837$		0			0
$838$		−	26812.5i		−	1.10528i
$839$			21384.5i			0.879947i	0.898011	+	0.439973i	$0.145012\pi$
$839$			21384.5i			0.879947i	−0.898011	+	0.439973i	$0.854988\pi$
$840$		0			0
$841$	55450.8			2.27360
$842$	−15834.9			−0.648110
$843$		0			0
$844$		−	19843.2i		−	0.809279i
$845$		−	29965.7i		−	1.21994i
$846$		0			0
$847$	2013.74	−	9096.78i	0.0816920	−	0.369031i
$848$		−	4713.82i		−	0.190888i
$849$		0			0
$850$	1261.05			0.0508868
$851$			5461.47i			0.219996i
$852$		0			0
$853$			1962.32i			0.0787673i	0.999224	+	0.0393837i	$0.0125395\pi$
$853$			1962.32i			0.0787673i	−0.999224	+	0.0393837i	$0.987461\pi$
$854$	−11294.4			−0.452559
$855$		0			0
$856$	−5338.61			−0.213166
$857$	−45782.4			−1.82485			−0.912425	−	0.409244i	$-0.865793\pi$
$857$	−45782.4			−1.82485			−0.912425	+	0.409244i	$0.865793\pi$
$858$		0			0
$859$	16194.6			0.643252			0.321626	−	0.946867i	$-0.395771\pi$
$859$	16194.6			0.643252			0.321626	+	0.946867i	$0.395771\pi$
$860$	13693.2			0.542947
$861$		0			0
$862$	−6104.96			−0.241225
$863$		−	22177.8i		−	0.874785i	−0.899271	−	0.437393i	$-0.855902\pi$
$863$		−	22177.8i		−	0.874785i	0.899271	−	0.437393i	$-0.144098\pi$
$864$		0			0
$865$		−	9110.85i		−	0.358125i
$866$	−16566.8			−0.650072
$867$		0			0
$868$			6520.32i			0.254970i
$869$	15962.8	+	1745.70i	0.623129	+	0.0681459i
$870$		0			0
$871$			6666.56i			0.259343i
$872$		−	11180.6i		−	0.434200i
$873$		0			0
$874$	−16962.3			−0.656472
$875$	2274.63			0.0878818
$876$		0			0
$877$			11873.7i			0.457179i	0.973523	+	0.228590i	$0.0734114\pi$
$877$			11873.7i			0.457179i	−0.973523	+	0.228590i	$0.926589\pi$
$878$		−	4651.33i		−	0.178787i
$879$		0			0
$880$	8771.50	+	959.258i	0.336008	+	0.0367461i
$881$		−	9169.28i		−	0.350648i	−0.984511	−	0.175324i	$-0.943903\pi$
$881$		−	9169.28i		−	0.350648i	0.984511	−	0.175324i	$-0.0560973\pi$
$882$		0			0
$883$	9452.31			0.360244			0.180122	−	0.983644i	$-0.442351\pi$
$883$	9452.31			0.360244			0.180122	+	0.983644i	$0.442351\pi$
$884$		−	357.014i		−	0.0135833i
$885$		0			0
$886$		−	9815.13i		−	0.372173i
$887$	−2188.97			−0.0828620			−0.0414310	−	0.999141i	$-0.513192\pi$
$887$	−2188.97			−0.0828620			−0.0414310	+	0.999141i	$0.513192\pi$
$888$		0			0
$889$	14188.1			0.535267
$890$	32995.6			1.24271
$891$		0			0
$892$	3767.98			0.141436
$893$	53303.5			1.99746
$894$		0			0
$895$	−17772.1			−0.663749
$896$		−	896.000i		−	0.0334077i
$897$		0			0
$898$			8572.85i			0.318574i
$899$	−65799.2			−2.44107
$900$		0			0
$901$		−	1794.74i		−	0.0663612i
$902$	3907.01	−	35725.9i	0.144223	−	1.31878i
$903$		0			0
$904$		−	17892.5i		−	0.658293i
$905$		−	22721.4i		−	0.834568i
$906$		0			0
$907$	11275.0			0.412766			0.206383	−	0.978471i	$-0.433831\pi$
$907$	11275.0			0.412766			0.206383	+	0.978471i	$0.433831\pi$
$908$	−20200.7			−0.738308
$909$		0			0
$910$			3100.64i			0.112951i
$911$			28805.5i			1.04761i	0.851840	+	0.523803i	$0.175487\pi$
$911$			28805.5i			1.04761i	−0.851840	+	0.523803i	$0.824513\pi$
$912$		0			0
$913$	−2887.66	+	26404.9i	−0.104674	+	0.957145i
$914$			3720.94i			0.134658i
$915$		0			0
$916$	7359.94			0.265479
$917$		−	1217.75i		−	0.0438535i
$918$		0			0
$919$			24979.4i			0.896621i	0.893878	+	0.448311i	$0.147974\pi$
$919$			24979.4i			0.896621i	−0.893878	+	0.448311i	$0.852026\pi$
$920$	9517.86			0.341081
$921$		0			0
$922$	13916.0			0.497072
$923$	9770.79			0.348439
$924$		0			0
$925$	7182.27			0.255299
$926$	−33664.2			−1.19468
$927$		0			0
$928$	9041.90			0.319844
$929$			628.452i			0.0221947i	0.999938	+	0.0110973i	$0.00353246\pi$
$929$			628.452i			0.0221947i	−0.999938	+	0.0110973i	$0.996468\pi$
$930$		0			0
$931$		−	5280.16i		−	0.185876i
$932$	−15806.1			−0.555523
$933$		0			0
$934$		−	18713.9i		−	0.655607i
$935$	3339.66	+	365.228i	0.116811	+	0.0127746i
$936$		0			0
$937$			27858.5i			0.971289i	0.874156	+	0.485644i	$0.161415\pi$
$937$			27858.5i			0.971289i	−0.874156	+	0.485644i	$0.838585\pi$
$938$			6370.20i			0.221742i
$939$		0			0
$940$	−29909.7			−1.03781
$941$	13091.8			0.453540			0.226770	−	0.973948i	$-0.427183\pi$
$941$	13091.8			0.453540			0.226770	+	0.973948i	$0.427183\pi$
$942$		0			0
$943$		−	38765.8i		−	1.33869i
$944$		−	12580.2i		−	0.433740i
$945$		0			0
$946$	16426.1	+	1796.38i	0.564546	+	0.0617391i
$947$			24037.9i			0.824845i	0.910993	+	0.412422i	$0.135317\pi$
$947$			24037.9i			0.824845i	−0.910993	+	0.412422i	$0.864683\pi$
$948$		0			0
$949$	8786.74			0.300558
$950$			22306.7i			0.761817i
$951$		0			0
$952$		−	341.143i		−	0.0116140i
$953$	36780.2			1.25019			0.625093	−	0.780550i	$-0.285061\pi$
$953$	36780.2			1.25019			0.625093	+	0.780550i	$0.285061\pi$
$954$		0			0
$955$	−45984.8			−1.55815
$956$	5987.61			0.202566
$957$		0			0
$958$	6191.32			0.208802
$959$	9663.87			0.325404
$960$		0			0
$961$	24436.8			0.820274
$962$		−	2033.35i		−	0.0681475i
$963$		0			0
$964$			4427.62i			0.147930i
$965$	42117.2			1.40498
$966$		0			0
$967$		−	44416.6i		−	1.47708i	−0.674207	−	0.738542i	$-0.735515\pi$
$967$		−	44416.6i		−	1.47708i	0.674207	−	0.738542i	$-0.264485\pi$
$968$	10396.3	+	2301.42i	0.345196	+	0.0764158i
$969$		0			0
$970$		−	54531.4i		−	1.80505i
$971$		−	52909.3i		−	1.74865i	−0.485341	−	0.874325i	$-0.661305\pi$
$971$		−	52909.3i		−	1.74865i	0.485341	−	0.874325i	$-0.338695\pi$
$972$		0			0
$973$	822.284			0.0270927
$974$	10626.3			0.349577
$975$		0			0
$976$		−	12907.8i		−	0.423330i
$977$		−	2759.07i		−	0.0903486i	−0.998979	−	0.0451743i	$-0.985616\pi$
$977$		−	2759.07i		−	0.0903486i	0.998979	−	0.0451743i	$-0.0143843\pi$
$978$		0			0
$979$	39581.0	+	4328.60i	1.29215	+	0.141310i
$980$			2962.80i			0.0965747i
$981$		0			0
$982$	−11810.9			−0.383810
$983$		−	10377.0i		−	0.336699i	−0.985727	−	0.168349i	$-0.946156\pi$
$983$		−	10377.0i		−	0.336699i	0.985727	−	0.168349i	$-0.0538437\pi$
$984$		0			0
$985$		−	2435.48i		−	0.0787826i
$986$	3442.61			0.111192
$987$		0			0
$988$	6315.20			0.203354
$989$	17823.8			0.573069
$990$		0			0
$991$	−41859.1			−1.34177			−0.670886	−	0.741560i	$-0.734086\pi$
$991$	−41859.1			−1.34177			−0.670886	+	0.741560i	$0.734086\pi$
$992$	−7451.79			−0.238503
$993$		0			0
$994$	9336.44			0.297921
$995$		−	66297.7i		−	2.11234i
$996$		0			0
$997$		−	25563.7i		−	0.812045i	−0.913863	−	0.406023i	$-0.866915\pi$
$997$		−	25563.7i		−	0.812045i	0.913863	−	0.406023i	$-0.133085\pi$
$998$	7361.50			0.233491
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.4.c.a.197.5	✓	36
3.2	odd	2		1386.4.c.b.197.32	yes	36
11.10	odd	2		1386.4.c.b.197.5	yes	36
33.32	even	2	inner	1386.4.c.a.197.32	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.4.c.a.197.5	✓	36	1.1	even	1	trivial
1386.4.c.a.197.32	yes	36	33.32	even	2	inner
1386.4.c.b.197.5	yes	36	11.10	odd	2
1386.4.c.b.197.32	yes	36	3.2	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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1386.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -15.1163i q^{5} +7.00000i q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} -15.1163i q^{5} +7.00000i q^{7} -8.00000 q^{8} +30.2327i q^{10} +(-3.96615 + 36.2666i) q^{11} -14.6513i q^{13} -14.0000i q^{14} +16.0000 q^{16} +6.09184 q^{17} +107.758i q^{19} -60.4653i q^{20} +(7.93230 - 72.5333i) q^{22} -78.7051i q^{23} -103.504 q^{25} +29.3026i q^{26} +28.0000i q^{28} -282.559 q^{29} +232.869 q^{31} -32.0000 q^{32} -12.1837 q^{34} +105.814 q^{35} -69.3915 q^{37} -215.517i q^{38} +120.931i q^{40} +492.545 q^{41} +226.464i q^{43} +(-15.8646 + 145.067i) q^{44} +157.410i q^{46} -494.658i q^{47} -49.0000 q^{49} +207.007 q^{50} -58.6053i q^{52} -294.614i q^{53} +(548.219 + 59.9536i) q^{55} -56.0000i q^{56} +565.119 q^{58} -786.262i q^{59} -806.740i q^{61} -465.737 q^{62} +64.0000 q^{64} -221.474 q^{65} -455.014 q^{67} +24.3674 q^{68} -211.629 q^{70} +666.889i q^{71} +599.723i q^{73} +138.783 q^{74} +431.033i q^{76} +(-253.867 - 27.7630i) q^{77} -440.150i q^{79} -241.861i q^{80} -985.090 q^{82} +728.076 q^{83} -92.0863i q^{85} -452.927i q^{86} +(31.7292 - 290.133i) q^{88} -1091.39i q^{89} +102.559 q^{91} -314.821i q^{92} +989.316i q^{94} +1628.91 q^{95} -1803.73 q^{97} +98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 72 q^{2} + 144 q^{4} - 288 q^{8}+O(q^{10}) \) 36 * q - 72 * q^2 + 144 * q^4 - 288 * q^8	\( 36 q - 72 q^{2} + 144 q^{4} - 288 q^{8} - 36 q^{11} + 576 q^{16} + 144 q^{17} + 72 q^{22} - 444 q^{25} + 432 q^{29} - 48 q^{31} - 1152 q^{32} - 288 q^{34} + 504 q^{35} + 24 q^{37} - 144 q^{41} - 144 q^{44} - 1764 q^{49} + 888 q^{50} + 2448 q^{55} - 864 q^{58} + 96 q^{62} + 2304 q^{64} - 2400 q^{65} + 624 q^{67} + 576 q^{68} - 1008 q^{70} - 48 q^{74} - 168 q^{77} + 288 q^{82} + 1296 q^{83} + 288 q^{88} - 6096 q^{95} + 768 q^{97} + 3528 q^{98}+O(q^{100}) \) 36 * q - 72 * q^2 + 144 * q^4 - 288 * q^8 - 36 * q^11 + 576 * q^16 + 144 * q^17 + 72 * q^22 - 444 * q^25 + 432 * q^29 - 48 * q^31 - 1152 * q^32 - 288 * q^34 + 504 * q^35 + 24 * q^37 - 144 * q^41 - 144 * q^44 - 1764 * q^49 + 888 * q^50 + 2448 * q^55 - 864 * q^58 + 96 * q^62 + 2304 * q^64 - 2400 * q^65 + 624 * q^67 + 576 * q^68 - 1008 * q^70 - 48 * q^74 - 168 * q^77 + 288 * q^82 + 1296 * q^83 + 288 * q^88 - 6096 * q^95 + 768 * q^97 + 3528 * q^98

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