LMFDB - Embedded newform 1386.4.c.a.197.12

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			197.12
Character	$\chi$	$=$	1386.197
Dual form			1386.4.c.a.197.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} -6.55239i q^{5} -7.00000i q^{7} -8.00000 q^{8} +O(q^{10})$	\(q-2.00000 q^{2} +4.00000 q^{4} -6.55239i q^{5} -7.00000i q^{7} -8.00000 q^{8} +13.1048i q^{10} +(-6.90013 - 35.8244i) q^{11} -55.6316i q^{13} +14.0000i q^{14} +16.0000 q^{16} +91.6068 q^{17} +61.7006i q^{19} -26.2095i q^{20} +(13.8003 + 71.6488i) q^{22} -111.403i q^{23} +82.0663 q^{25} +111.263i q^{26} -28.0000i q^{28} +197.239 q^{29} +24.6978 q^{31} -32.0000 q^{32} -183.214 q^{34} -45.8667 q^{35} +38.7754 q^{37} -123.401i q^{38} +52.4191i q^{40} +246.872 q^{41} +127.759i q^{43} +(-27.6005 - 143.298i) q^{44} +222.805i q^{46} -277.313i q^{47} -49.0000 q^{49} -164.133 q^{50} -222.527i q^{52} +388.273i q^{53} +(-234.735 + 45.2123i) q^{55} +56.0000i q^{56} -394.478 q^{58} -657.239i q^{59} +257.509i q^{61} -49.3956 q^{62} +64.0000 q^{64} -364.520 q^{65} +80.1573 q^{67} +366.427 q^{68} +91.7334 q^{70} -802.499i q^{71} +159.670i q^{73} -77.5508 q^{74} +246.802i q^{76} +(-250.771 + 48.3009i) q^{77} -589.098i q^{79} -104.838i q^{80} -493.744 q^{82} +989.356 q^{83} -600.243i q^{85} -255.517i q^{86} +(55.2010 + 286.595i) q^{88} +1483.15i q^{89} -389.421 q^{91} -445.610i q^{92} +554.626i q^{94} +404.286 q^{95} -1092.23 q^{97} +98.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$		−	6.55239i		−	0.586063i	−0.956103	−	0.293032i	$-0.905336\pi$
$5$		−	6.55239i		−	0.586063i	0.956103	−	0.293032i	$-0.0946641\pi$
$6$		0			0
$7$		−	7.00000i		−	0.377964i
$7$		−	7.00000i		−	0.377964i
$8$	−8.00000			−0.353553
$9$		0			0
$10$			13.1048i			0.414409i
$11$	−6.90013	−	35.8244i	−0.189133	−	0.981951i
$11$	−6.90013	−	35.8244i	−0.189133	−	0.981951i
$12$		0			0
$13$		−	55.6316i		−	1.18688i	−0.804878	−	0.593440i	$-0.797769\pi$
$13$		−	55.6316i		−	1.18688i	0.804878	−	0.593440i	$-0.202231\pi$
$14$			14.0000i			0.267261i
$15$		0			0
$16$	16.0000			0.250000
$17$	91.6068			1.30694			0.653468	−	0.756954i	$-0.273314\pi$
$17$	91.6068			1.30694			0.653468	+	0.756954i	$0.273314\pi$
$18$		0			0
$19$			61.7006i			0.745004i	0.928031	+	0.372502i	$0.121500\pi$
$19$			61.7006i			0.745004i	−0.928031	+	0.372502i	$0.878500\pi$
$20$		−	26.2095i		−	0.293032i
$21$		0			0
$22$	13.8003	+	71.6488i	0.133737	+	0.694344i
$23$		−	111.403i		−	1.00996i	−0.863132	−	0.504979i	$-0.831500\pi$
$23$		−	111.403i		−	1.00996i	0.863132	−	0.504979i	$-0.168500\pi$
$24$		0			0
$25$	82.0663			0.656530
$26$			111.263i			0.839251i
$27$		0			0
$28$		−	28.0000i		−	0.188982i
$29$	197.239			1.26298			0.631488	−	0.775385i	$-0.282444\pi$
$29$	197.239			1.26298			0.631488	+	0.775385i	$0.282444\pi$
$30$		0			0
$31$	24.6978			0.143092			0.0715461	−	0.997437i	$-0.477207\pi$
$31$	24.6978			0.143092			0.0715461	+	0.997437i	$0.477207\pi$
$32$	−32.0000			−0.176777
$33$		0			0
$34$	−183.214			−0.924143
$35$	−45.8667			−0.221511
$36$		0			0
$37$	38.7754			0.172288			0.0861438	−	0.996283i	$-0.472546\pi$
$37$	38.7754			0.172288			0.0861438	+	0.996283i	$0.472546\pi$
$38$		−	123.401i		−	0.526798i
$39$		0			0
$40$			52.4191i			0.207205i
$41$	246.872			0.940363			0.470182	−	0.882570i	$-0.344188\pi$
$41$	246.872			0.940363			0.470182	+	0.882570i	$0.344188\pi$
$42$		0			0
$43$			127.759i			0.453093i	0.974000	+	0.226546i	$0.0727435\pi$
$43$			127.759i			0.453093i	−0.974000	+	0.226546i	$0.927256\pi$
$44$	−27.6005	−	143.298i	−0.0945667	−	0.490976i
$45$		0			0
$46$			222.805i			0.714148i
$47$		−	277.313i		−	0.860644i	−0.902676	−	0.430322i	$-0.858400\pi$
$47$		−	277.313i		−	0.860644i	0.902676	−	0.430322i	$-0.141600\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$	−164.133			−0.464237
$51$		0			0
$52$		−	222.527i		−	0.593440i
$53$			388.273i			1.00629i	0.864202	+	0.503145i	$0.167824\pi$
$53$			388.273i			1.00629i	−0.864202	+	0.503145i	$0.832176\pi$
$54$		0			0
$55$	−234.735	+	45.2123i	−0.575486	+	0.110844i
$56$			56.0000i			0.133631i
$57$		0			0
$58$	−394.478			−0.893059
$59$		−	657.239i		−	1.45026i	−0.688613	−	0.725129i	$-0.741780\pi$
$59$		−	657.239i		−	1.45026i	0.688613	−	0.725129i	$-0.258220\pi$
$60$		0			0
$61$			257.509i			0.540502i	0.962790	+	0.270251i	$0.0871066\pi$
$61$			257.509i			0.540502i	−0.962790	+	0.270251i	$0.912893\pi$
$62$	−49.3956			−0.101181
$63$		0			0
$64$	64.0000			0.125000
$65$	−364.520			−0.695587
$66$		0			0
$67$	80.1573			0.146161			0.0730804	−	0.997326i	$-0.476717\pi$
$67$	80.1573			0.146161			0.0730804	+	0.997326i	$0.476717\pi$
$68$	366.427			0.653468
$69$		0			0
$70$	91.7334			0.156632
$71$		−	802.499i		−	1.34140i	−0.741731	−	0.670698i	$-0.765995\pi$
$71$		−	802.499i		−	1.34140i	0.741731	−	0.670698i	$-0.234005\pi$
$72$		0			0
$73$			159.670i			0.256000i	0.991774	+	0.128000i	$0.0408557\pi$
$73$			159.670i			0.256000i	−0.991774	+	0.128000i	$0.959144\pi$
$74$	−77.5508			−0.121826
$75$		0			0
$76$			246.802i			0.372502i
$77$	−250.771	+	48.3009i	−0.371143	+	0.0714857i
$78$		0			0
$79$		−	589.098i		−	0.838970i	−0.907762	−	0.419485i	$-0.862211\pi$
$79$		−	589.098i		−	0.838970i	0.907762	−	0.419485i	$-0.137789\pi$
$80$		−	104.838i		−	0.146516i
$81$		0			0
$82$	−493.744			−0.664937
$83$	989.356			1.30839			0.654193	−	0.756328i	$-0.273009\pi$
$83$	989.356			1.30839			0.654193	+	0.756328i	$0.273009\pi$
$84$		0			0
$85$		−	600.243i		−	0.765947i
$86$		−	255.517i		−	0.320385i
$87$		0			0
$88$	55.2010	+	286.595i	0.0668687	+	0.347172i
$89$			1483.15i			1.76645i	0.468954	+	0.883223i	$0.344631\pi$
$89$			1483.15i			1.76645i	−0.468954	+	0.883223i	$0.655369\pi$
$90$		0			0
$91$	−389.421			−0.448598
$92$		−	445.610i		−	0.504979i
$93$		0			0
$94$			554.626i			0.608567i
$95$	404.286			0.436620
$96$		0			0
$97$	−1092.23			−1.14329			−0.571647	−	0.820500i	$-0.693695\pi$
$97$	−1092.23			−1.14329			−0.571647	+	0.820500i	$0.693695\pi$
$98$	98.0000			0.101015
$99$		0			0
$100$	328.265			0.328265
$101$	1187.05			1.16947			0.584733	−	0.811226i	$-0.301199\pi$
$101$	1187.05			1.16947			0.584733	+	0.811226i	$0.301199\pi$
$102$		0			0
$103$	36.7459			0.0351522			0.0175761	−	0.999846i	$-0.494405\pi$
$103$	36.7459			0.0351522			0.0175761	+	0.999846i	$0.494405\pi$
$104$			445.053i			0.419625i
$105$		0			0
$106$		−	776.546i		−	0.711555i
$107$	−376.827			−0.340460			−0.170230	−	0.985404i	$-0.554451\pi$
$107$	−376.827			−0.340460			−0.170230	+	0.985404i	$0.554451\pi$
$108$		0			0
$109$			398.475i			0.350156i	0.984555	+	0.175078i	$0.0560177\pi$
$109$			398.475i			0.350156i	−0.984555	+	0.175078i	$0.943982\pi$
$110$	469.471	−	90.4246i	0.406930	−	0.0783786i
$111$		0			0
$112$		−	112.000i		−	0.0944911i
$113$		−	1675.30i		−	1.39468i	−0.716738	−	0.697342i	$-0.754366\pi$
$113$		−	1675.30i		−	1.39468i	0.716738	−	0.697342i	$-0.245634\pi$
$114$		0			0
$115$	−729.952			−0.591899
$116$	788.955			0.631488
$117$		0			0
$118$			1314.48i			1.02549i
$119$		−	641.247i		−	0.493975i
$120$		0			0
$121$	−1235.78	+	494.386i	−0.928457	+	0.371440i
$122$		−	515.017i		−	0.382192i
$123$		0			0
$124$	98.7912			0.0715461
$125$		−	1356.78i		−	0.970831i
$126$		0			0
$127$			2369.88i			1.65585i	0.560840	+	0.827924i	$0.310478\pi$
$127$			2369.88i			1.65585i	−0.560840	+	0.827924i	$0.689522\pi$
$128$	−128.000			−0.0883883
$129$		0			0
$130$	729.040			0.491854
$131$	−933.527			−0.622616			−0.311308	−	0.950309i	$-0.600767\pi$
$131$	−933.527			−0.622616			−0.311308	+	0.950309i	$0.600767\pi$
$132$		0			0
$133$	431.904			0.281585
$134$	−160.315			−0.103351
$135$		0			0
$136$	−732.854			−0.462071
$137$			282.408i			0.176115i	0.996115	+	0.0880574i	$0.0280659\pi$
$137$			282.408i			0.176115i	−0.996115	+	0.0880574i	$0.971934\pi$
$138$		0			0
$139$			600.462i			0.366407i	0.983075	+	0.183203i	$0.0586467\pi$
$139$			600.462i			0.366407i	−0.983075	+	0.183203i	$0.941353\pi$
$140$	−183.467			−0.110756
$141$		0			0
$142$			1605.00i			0.948510i
$143$	−1992.97	+	383.865i	−1.16546	+	0.224479i
$144$		0			0
$145$		−	1292.38i		−	0.740184i
$146$		−	319.341i		−	0.181019i
$147$		0			0
$148$	155.102			0.0861438
$149$	3144.80			1.72907			0.864537	−	0.502569i	$-0.167612\pi$
$149$	3144.80			1.72907			0.864537	+	0.502569i	$0.167612\pi$
$150$		0			0
$151$		−	186.035i		−	0.100260i	−0.998743	−	0.0501301i	$-0.984036\pi$
$151$		−	186.035i		−	0.100260i	0.998743	−	0.0501301i	$-0.0159636\pi$
$152$		−	493.605i		−	0.263399i
$153$		0			0
$154$	501.542	−	96.6018i	0.262438	−	0.0505480i
$155$		−	161.830i		−	0.0838610i
$156$		0			0
$157$	−1775.26			−0.902426			−0.451213	−	0.892416i	$-0.649009\pi$
$157$	−1775.26			−0.902426			−0.451213	+	0.892416i	$0.649009\pi$
$158$			1178.20i			0.593242i
$159$		0			0
$160$			209.676i			0.103602i
$161$	−779.818			−0.381728
$162$		0			0
$163$	−464.647			−0.223276			−0.111638	−	0.993749i	$-0.535610\pi$
$163$	−464.647			−0.223276			−0.111638	+	0.993749i	$0.535610\pi$
$164$	987.487			0.470182
$165$		0			0
$166$	−1978.71			−0.925168
$167$	1565.01			0.725172			0.362586	−	0.931950i	$-0.381894\pi$
$167$	1565.01			0.725172			0.362586	+	0.931950i	$0.381894\pi$
$168$		0			0
$169$	−897.879			−0.408684
$170$			1200.49i			0.541606i
$171$		0			0
$172$			511.034i			0.226546i
$173$	−4263.73			−1.87379			−0.936895	−	0.349612i	$-0.886313\pi$
$173$	−4263.73			−1.87379			−0.936895	+	0.349612i	$0.886313\pi$
$174$		0			0
$175$		−	574.464i		−	0.248145i
$176$	−110.402	−	573.191i	−0.0472833	−	0.245488i
$177$		0			0
$178$		−	2966.30i		−	1.24907i
$179$		−	2763.50i		−	1.15393i	−0.816768	−	0.576966i	$-0.804236\pi$
$179$		−	2763.50i		−	1.15393i	0.816768	−	0.576966i	$-0.195764\pi$
$180$		0			0
$181$	−327.290			−0.134405			−0.0672024	−	0.997739i	$-0.521407\pi$
$181$	−327.290			−0.134405			−0.0672024	+	0.997739i	$0.521407\pi$
$182$	778.843			0.317207
$183$		0			0
$184$			891.220i			0.357074i
$185$		−	254.071i		−	0.100971i
$186$		0			0
$187$	−632.098	−	3281.76i	−0.247185	−	1.28335i
$188$		−	1109.25i		−	0.430322i
$189$		0			0
$190$	−808.572			−0.308737
$191$		−	456.383i		−	0.172894i	−0.996256	−	0.0864469i	$-0.972449\pi$
$191$		−	456.383i		−	0.172894i	0.996256	−	0.0864469i	$-0.0275513\pi$
$192$		0			0
$193$		−	894.750i		−	0.333707i	−0.985982	−	0.166854i	$-0.946639\pi$
$193$		−	894.750i		−	0.333707i	0.985982	−	0.166854i	$-0.0533608\pi$
$194$	2184.47			0.808431
$195$		0			0
$196$	−196.000			−0.0714286
$197$	1186.17			0.428990			0.214495	−	0.976725i	$-0.431189\pi$
$197$	1186.17			0.428990			0.214495	+	0.976725i	$0.431189\pi$
$198$		0			0
$199$	−957.549			−0.341100			−0.170550	−	0.985349i	$-0.554554\pi$
$199$	−957.549			−0.341100			−0.170550	+	0.985349i	$0.554554\pi$
$200$	−656.530			−0.232118
$201$		0			0
$202$	−2374.10			−0.826938
$203$		−	1380.67i		−	0.477360i
$204$		0			0
$205$		−	1617.60i		−	0.551112i
$206$	−73.4918			−0.0248564
$207$		0			0
$208$		−	890.106i		−	0.296720i
$209$	2210.39	−	425.742i	0.731558	−	0.140905i
$210$		0			0
$211$		−	2777.83i		−	0.906322i	−0.891429	−	0.453161i	$-0.850296\pi$
$211$		−	2777.83i		−	0.906322i	0.891429	−	0.453161i	$-0.149704\pi$
$212$			1553.09i			0.503145i
$213$		0			0
$214$	753.653			0.240741
$215$	837.123			0.265541
$216$		0			0
$217$		−	172.885i		−	0.0540838i
$218$		−	796.949i		−	0.247597i
$219$		0			0
$220$	−938.941	+	180.849i	−0.287743	+	0.0554221i
$221$		−	5096.23i		−	1.55118i
$222$		0			0
$223$	398.568			0.119686			0.0598432	−	0.998208i	$-0.480940\pi$
$223$	398.568			0.119686			0.0598432	+	0.998208i	$0.480940\pi$
$224$			224.000i			0.0668153i
$225$		0			0
$226$			3350.61i			0.986191i
$227$	1985.51			0.580541			0.290270	−	0.956945i	$-0.406255\pi$
$227$	1985.51			0.580541			0.290270	+	0.956945i	$0.406255\pi$
$228$		0			0
$229$	4188.02			1.20853			0.604263	−	0.796785i	$-0.293468\pi$
$229$	4188.02			1.20853			0.604263	+	0.796785i	$0.293468\pi$
$230$	1459.90			0.418536
$231$		0			0
$232$	−1577.91			−0.446530
$233$	−1419.86			−0.399220			−0.199610	−	0.979875i	$-0.563968\pi$
$233$	−1419.86			−0.399220			−0.199610	+	0.979875i	$0.563968\pi$
$234$		0			0
$235$	−1817.06			−0.504392
$236$		−	2628.96i		−	0.725129i
$237$		0			0
$238$			1282.49i			0.349293i
$239$	1368.54			0.370391			0.185195	−	0.982702i	$-0.440708\pi$
$239$	1368.54			0.370391			0.185195	+	0.982702i	$0.440708\pi$
$240$		0			0
$241$		−	1562.62i		−	0.417664i	−0.977952	−	0.208832i	$-0.933034\pi$
$241$		−	1562.62i		−	0.417664i	0.977952	−	0.208832i	$-0.0669662\pi$
$242$	2471.55	−	988.772i	0.656518	−	0.262647i
$243$		0			0
$244$			1030.03i			0.270251i
$245$			321.067i			0.0837233i
$246$		0			0
$247$	3432.50			0.884231
$248$	−197.582			−0.0505907
$249$		0			0
$250$			2713.56i			0.686481i
$251$			1196.62i			0.300917i	0.988616	+	0.150459i	$0.0480751\pi$
$251$			1196.62i			0.300917i	−0.988616	+	0.150459i	$0.951925\pi$
$252$		0			0
$253$	−3990.93	+	768.692i	−0.991730	+	0.191017i
$254$		−	4739.76i		−	1.17086i
$255$		0			0
$256$	256.000			0.0625000
$257$		−	3964.28i		−	0.962199i	−0.876666	−	0.481100i	$-0.840238\pi$
$257$		−	3964.28i		−	0.962199i	0.876666	−	0.481100i	$-0.159762\pi$
$258$		0			0
$259$		−	271.428i		−	0.0651186i
$260$	−1458.08			−0.347793
$261$		0			0
$262$	1867.05			0.440256
$263$	−3488.64			−0.817943			−0.408971	−	0.912547i	$-0.634112\pi$
$263$	−3488.64			−0.817943			−0.408971	+	0.912547i	$0.634112\pi$
$264$		0			0
$265$	2544.11			0.589750
$266$	−863.808			−0.199111
$267$		0			0
$268$	320.629			0.0730804
$269$		−	4435.23i		−	1.00528i	−0.864495	−	0.502641i	$-0.832362\pi$
$269$		−	4435.23i		−	1.00528i	0.864495	−	0.502641i	$-0.167638\pi$
$270$		0			0
$271$		−	2049.09i		−	0.459312i	−0.973272	−	0.229656i	$-0.926240\pi$
$271$		−	2049.09i		−	0.459312i	0.973272	−	0.229656i	$-0.0737600\pi$
$272$	1465.71			0.326734
$273$		0			0
$274$		−	564.815i		−	0.124532i
$275$	−566.268	−	2939.97i	−0.124172	−	0.644681i
$276$		0			0
$277$		−	632.392i		−	0.137172i	−0.997645	−	0.0685862i	$-0.978151\pi$
$277$		−	632.392i		−	0.137172i	0.997645	−	0.0685862i	$-0.0218488\pi$
$278$		−	1200.92i		−	0.259089i
$279$		0			0
$280$	366.934			0.0783160
$281$	−8601.62			−1.82609			−0.913043	−	0.407864i	$-0.866274\pi$
$281$	−8601.62			−1.82609			−0.913043	+	0.407864i	$0.866274\pi$
$282$		0			0
$283$		−	1082.51i		−	0.227380i	−0.993516	−	0.113690i	$-0.963733\pi$
$283$		−	1082.51i		−	0.227380i	0.993516	−	0.113690i	$-0.0362671\pi$
$284$		−	3209.99i		−	0.670698i
$285$		0			0
$286$	3985.94	−	767.731i	0.824104	−	0.158730i
$287$		−	1728.10i		−	0.355424i
$288$		0			0
$289$	3478.80			0.708080
$290$			2584.77i			0.523389i
$291$		0			0
$292$			638.681i			0.128000i
$293$	−7293.56			−1.45425			−0.727124	−	0.686506i	$-0.759143\pi$
$293$	−7293.56			−1.45425			−0.727124	+	0.686506i	$0.759143\pi$
$294$		0			0
$295$	−4306.49			−0.849943
$296$	−310.203			−0.0609129
$297$		0			0
$298$	−6289.60			−1.22264
$299$	−6197.51			−1.19870
$300$		0			0
$301$	894.310			0.171253
$302$			372.070i			0.0708947i
$303$		0			0
$304$			987.209i			0.186251i
$305$	1687.30			0.316768
$306$		0			0
$307$		−	7483.28i		−	1.39118i	−0.718438	−	0.695591i	$-0.755142\pi$
$307$		−	7483.28i		−	1.39118i	0.718438	−	0.695591i	$-0.244858\pi$
$308$	−1003.08	+	193.204i	−0.185571	+	0.0357428i
$309$		0			0
$310$			323.659i			0.0592987i
$311$		−	3297.65i		−	0.601263i	−0.953740	−	0.300632i	$-0.902803\pi$
$311$		−	3297.65i		−	0.601263i	0.953740	−	0.300632i	$-0.0971975\pi$
$312$		0			0
$313$	−10454.9			−1.88800			−0.944002	−	0.329938i	$-0.892972\pi$
$313$	−10454.9			−1.88800			−0.944002	+	0.329938i	$0.892972\pi$
$314$	3550.51			0.638112
$315$		0			0
$316$		−	2356.39i		−	0.419485i
$317$		−	6260.50i		−	1.10923i	−0.832108	−	0.554614i	$-0.812866\pi$
$317$		−	6260.50i		−	1.10923i	0.832108	−	0.554614i	$-0.187134\pi$
$318$		0			0
$319$	−1360.97	−	7065.96i	−0.238871	−	1.24018i
$320$		−	419.353i		−	0.0732579i
$321$		0			0
$322$	1559.64			0.269923
$323$			5652.19i			0.973673i
$324$		0			0
$325$		−	4565.48i		−	0.779222i
$326$	929.293			0.157880
$327$		0			0
$328$	−1974.97			−0.332469
$329$	−1941.19			−0.325293
$330$		0			0
$331$	−7940.75			−1.31862			−0.659310	−	0.751871i	$-0.729151\pi$
$331$	−7940.75			−1.31862			−0.659310	+	0.751871i	$0.729151\pi$
$332$	3957.43			0.654193
$333$		0			0
$334$	−3130.01			−0.512774
$335$		−	525.221i		−	0.0856594i
$336$		0			0
$337$			7685.43i			1.24229i	0.783695	+	0.621146i	$0.213332\pi$
$337$			7685.43i			1.24229i	−0.783695	+	0.621146i	$0.786668\pi$
$338$	1795.76			0.288983
$339$		0			0
$340$		−	2400.97i		−	0.382973i
$341$	−170.418	−	884.784i	−0.0270635	−	0.140510i
$342$		0			0
$343$			343.000i			0.0539949i
$344$		−	1022.07i		−	0.160193i
$345$		0			0
$346$	8527.46			1.32497
$347$	−4420.72			−0.683910			−0.341955	−	0.939716i	$-0.611089\pi$
$347$	−4420.72			−0.683910			−0.341955	+	0.939716i	$0.611089\pi$
$348$		0			0
$349$		−	566.787i		−	0.0869324i	−0.999055	−	0.0434662i	$-0.986160\pi$
$349$		−	566.787i		−	0.0869324i	0.999055	−	0.0434662i	$-0.0138401\pi$
$350$			1148.93i			0.175465i
$351$		0			0
$352$	220.804	+	1146.38i	0.0334344	+	0.173586i
$353$		−	1252.37i		−	0.188830i	−0.995533	−	0.0944148i	$-0.969902\pi$
$353$		−	1252.37i		−	0.188830i	0.995533	−	0.0944148i	$-0.0300980\pi$
$354$		0			0
$355$	−5258.28			−0.786142
$356$			5932.60i			0.883223i
$357$		0			0
$358$			5527.01i			0.815954i
$359$	2269.45			0.333640			0.166820	−	0.985987i	$-0.446650\pi$
$359$	2269.45			0.333640			0.166820	+	0.985987i	$0.446650\pi$
$360$		0			0
$361$	3052.04			0.444969
$362$	654.580			0.0950386
$363$		0			0
$364$	−1557.69			−0.224299
$365$	1046.22			0.150032
$366$		0			0
$367$	5324.93			0.757382			0.378691	−	0.925523i	$-0.376374\pi$
$367$	5324.93			0.757382			0.378691	+	0.925523i	$0.376374\pi$
$368$		−	1782.44i		−	0.252490i
$369$		0			0
$370$			508.143i			0.0713976i
$371$	2717.91			0.380342
$372$		0			0
$373$			1393.44i			0.193430i	0.995312	+	0.0967150i	$0.0308335\pi$
$373$			1393.44i			0.193430i	−0.995312	+	0.0967150i	$0.969166\pi$
$374$	1264.20	+	6563.52i	0.174786	+	0.907463i
$375$		0			0
$376$			2218.50i			0.304284i
$377$		−	10972.7i		−	1.49900i
$378$		0			0
$379$	−846.685			−0.114753			−0.0573764	−	0.998353i	$-0.518273\pi$
$379$	−846.685			−0.114753			−0.0573764	+	0.998353i	$0.518273\pi$
$380$	1617.14			0.218310
$381$		0			0
$382$			912.766i			0.122254i
$383$			7939.82i			1.05928i	0.848221	+	0.529642i	$0.177674\pi$
$383$			7939.82i			1.05928i	−0.848221	+	0.529642i	$0.822326\pi$
$384$		0			0
$385$	316.486	+	1643.15i	0.0418951	+	0.217513i
$386$			1789.50i			0.235967i
$387$		0			0
$388$	−4368.94			−0.571647
$389$		−	1584.30i		−	0.206497i	−0.994656	−	0.103249i	$-0.967076\pi$
$389$		−	1584.30i		−	0.206497i	0.994656	−	0.103249i	$-0.0329237\pi$
$390$		0			0
$391$		−	10205.2i		−	1.31995i
$392$	392.000			0.0505076
$393$		0			0
$394$	−2372.34			−0.303342
$395$	−3859.99			−0.491690
$396$		0			0
$397$	1990.55			0.251644			0.125822	−	0.992053i	$-0.459843\pi$
$397$	1990.55			0.251644			0.125822	+	0.992053i	$0.459843\pi$
$398$	1915.10			0.241194
$399$		0			0
$400$	1313.06			0.164133
$401$			6728.41i			0.837907i	0.908008	+	0.418953i	$0.137603\pi$
$401$			6728.41i			0.837907i	−0.908008	+	0.418953i	$0.862397\pi$
$402$		0			0
$403$		−	1373.98i		−	0.169833i
$404$	4748.21			0.584733
$405$		0			0
$406$			2761.34i			0.337545i
$407$	−267.555	−	1389.11i	−0.0325853	−	0.169178i
$408$		0			0
$409$		−	16033.3i		−	1.93837i	−0.246334	−	0.969185i	$-0.579226\pi$
$409$		−	16033.3i		−	1.93837i	0.246334	−	0.969185i	$-0.420774\pi$
$410$			3235.20i			0.389695i
$411$		0			0
$412$	146.984			0.0175761
$413$	−4600.68			−0.548146
$414$		0			0
$415$		−	6482.64i		−	0.766796i
$416$			1780.21i			0.209813i
$417$		0			0
$418$	−4420.77	+	851.484i	−0.517290	+	0.0996350i
$419$			11191.5i			1.30487i	0.757844	+	0.652436i	$0.226253\pi$
$419$			11191.5i			1.30487i	−0.757844	+	0.652436i	$0.773747\pi$
$420$		0			0
$421$	1222.85			0.141563			0.0707816	−	0.997492i	$-0.477451\pi$
$421$	1222.85			0.141563			0.0707816	+	0.997492i	$0.477451\pi$
$422$			5555.67i			0.640867i
$423$		0			0
$424$		−	3106.18i		−	0.355777i
$425$	7517.82			0.858042
$426$		0			0
$427$	1802.56			0.204290
$428$	−1507.31			−0.170230
$429$		0			0
$430$	−1674.25			−0.187766
$431$	1558.89			0.174221			0.0871103	−	0.996199i	$-0.472237\pi$
$431$	1558.89			0.174221			0.0871103	+	0.996199i	$0.472237\pi$
$432$		0			0
$433$	824.465			0.0915040			0.0457520	−	0.998953i	$-0.485432\pi$
$433$	824.465			0.0915040			0.0457520	+	0.998953i	$0.485432\pi$
$434$			345.769i			0.0382430i
$435$		0			0
$436$			1593.90i			0.175078i
$437$	6873.60			0.752423
$438$		0			0
$439$			2798.79i			0.304280i	0.988359	+	0.152140i	$0.0486165\pi$
$439$			2798.79i			0.304280i	−0.988359	+	0.152140i	$0.951384\pi$
$440$	1877.88	−	361.698i	0.203465	−	0.0391893i
$441$		0			0
$442$			10192.5i			1.09685i
$443$		−	14770.6i		−	1.58414i	−0.610430	−	0.792070i	$-0.709003\pi$
$443$		−	14770.6i		−	1.58414i	0.610430	−	0.792070i	$-0.290997\pi$
$444$		0			0
$445$	9718.17			1.03525
$446$	−797.135			−0.0846310
$447$		0			0
$448$		−	448.000i		−	0.0472456i
$449$			17127.8i			1.80025i	0.435634	+	0.900124i	$0.356524\pi$
$449$			17127.8i			1.80025i	−0.435634	+	0.900124i	$0.643476\pi$
$450$		0			0
$451$	−1703.45	−	8844.04i	−0.177854	−	0.923391i
$452$		−	6701.22i		−	0.697342i
$453$		0			0
$454$	−3971.01			−0.410504
$455$			2551.64i			0.262907i
$456$		0			0
$457$			12809.8i			1.31120i	0.755110	+	0.655599i	$0.227584\pi$
$457$			12809.8i			1.31120i	−0.755110	+	0.655599i	$0.772416\pi$
$458$	−8376.05			−0.854557
$459$		0			0
$460$	−2919.81			−0.295950
$461$	−8218.30			−0.830291			−0.415146	−	0.909755i	$-0.636269\pi$
$461$	−8218.30			−0.830291			−0.415146	+	0.909755i	$0.636269\pi$
$462$		0			0
$463$	14333.5			1.43873			0.719366	−	0.694631i	$-0.244432\pi$
$463$	14333.5			1.43873			0.719366	+	0.694631i	$0.244432\pi$
$464$	3155.82			0.315744
$465$		0			0
$466$	2839.72			0.282291
$467$			8195.08i			0.812041i	0.913864	+	0.406021i	$0.133084\pi$
$467$			8195.08i			0.812041i	−0.913864	+	0.406021i	$0.866916\pi$
$468$		0			0
$469$		−	561.101i		−	0.0552436i
$470$	3634.12			0.356659
$471$		0			0
$472$			5257.91i			0.512744i
$473$	4576.87	−	881.550i	0.444915	−	0.0856950i
$474$		0			0
$475$			5063.53i			0.489118i
$476$		−	2564.99i		−	0.246988i
$477$		0			0
$478$	−2737.08			−0.261906
$479$	2525.03			0.240859			0.120430	−	0.992722i	$-0.461573\pi$
$479$	2525.03			0.240859			0.120430	+	0.992722i	$0.461573\pi$
$480$		0			0
$481$		−	2157.14i		−	0.204485i
$482$			3125.23i			0.295333i
$483$		0			0
$484$	−4943.11	+	1977.54i	−0.464229	+	0.185720i
$485$			7156.74i			0.670043i
$486$		0			0
$487$	21460.7			1.99688			0.998438	−	0.0558720i	$-0.0177939\pi$
$487$	21460.7			1.99688			0.998438	+	0.0558720i	$0.0177939\pi$
$488$		−	2060.07i		−	0.191096i
$489$		0			0
$490$		−	642.134i		−	0.0592013i
$491$	−18033.6			−1.65753			−0.828763	−	0.559600i	$-0.810955\pi$
$491$	−18033.6			−1.65753			−0.828763	+	0.559600i	$0.810955\pi$
$492$		0			0
$493$	18068.4			1.65063
$494$	−6865.01			−0.625246
$495$		0			0
$496$	395.165			0.0357730
$497$	−5617.49			−0.507000
$498$		0			0
$499$	−743.132			−0.0666677			−0.0333338	−	0.999444i	$-0.510612\pi$
$499$	−743.132			−0.0666677			−0.0333338	+	0.999444i	$0.510612\pi$
$500$		−	5427.11i		−	0.485416i
$501$		0			0
$502$		−	2393.25i		−	0.212781i
$503$	437.842			0.0388120			0.0194060	−	0.999812i	$-0.493822\pi$
$503$	437.842			0.0388120			0.0194060	+	0.999812i	$0.493822\pi$
$504$		0			0
$505$		−	7778.02i		−	0.685381i
$506$	7981.86	−	1537.38i	0.701259	−	0.135069i
$507$		0			0
$508$			9479.51i			0.827924i
$509$			16840.6i			1.46650i	0.679961	+	0.733249i	$0.261997\pi$
$509$			16840.6i			1.46650i	−0.679961	+	0.733249i	$0.738003\pi$
$510$		0			0
$511$	1117.69			0.0967589
$512$	−512.000			−0.0441942
$513$		0			0
$514$			7928.57i			0.680378i
$515$		−	240.773i		−	0.0206014i
$516$		0			0
$517$	−9934.57	+	1913.50i	−0.845110	+	0.162776i
$518$			542.856i			0.0460458i
$519$		0			0
$520$	2916.16			0.245927
$521$			6723.58i			0.565384i	0.959211	+	0.282692i	$0.0912275\pi$
$521$			6723.58i			0.565384i	−0.959211	+	0.282692i	$0.908773\pi$
$522$		0			0
$523$		−	13936.3i		−	1.16518i	−0.812765	−	0.582591i	$-0.802039\pi$
$523$		−	13936.3i		−	1.16518i	0.812765	−	0.582591i	$-0.197961\pi$
$524$	−3734.11			−0.311308
$525$		0			0
$526$	6977.29			0.578373
$527$	2262.49			0.187012
$528$		0			0
$529$	−243.530			−0.0200156
$530$	−5088.23			−0.417016
$531$		0			0
$532$	1727.62			0.140793
$533$		−	13733.9i		−	1.11610i
$534$		0			0
$535$			2469.11i			0.199531i
$536$	−641.258			−0.0516756
$537$		0			0
$538$			8870.47i			0.710842i
$539$	338.106	+	1755.40i	0.0270191	+	0.140279i
$540$		0			0
$541$			15574.9i			1.23774i	0.785493	+	0.618870i	$0.212409\pi$
$541$			15574.9i			1.23774i	−0.785493	+	0.618870i	$0.787591\pi$
$542$			4098.18i			0.324782i
$543$		0			0
$544$	−2931.42			−0.231036
$545$	2610.96			0.205213
$546$		0			0
$547$			5721.54i			0.447231i	0.974677	+	0.223615i	$0.0717860\pi$
$547$			5721.54i			0.447231i	−0.974677	+	0.223615i	$0.928214\pi$
$548$			1129.63i			0.0880574i
$549$		0			0
$550$	1132.54	+	5879.95i	0.0878027	+	0.455858i
$551$			12169.7i			0.940923i
$552$		0			0
$553$	−4123.68			−0.317101
$554$			1264.78i			0.0969955i
$555$		0			0
$556$			2401.85i			0.183203i
$557$	−15418.8			−1.17292			−0.586458	−	0.809980i	$-0.699478\pi$
$557$	−15418.8			−1.17292			−0.586458	+	0.809980i	$0.699478\pi$
$558$		0			0
$559$	7107.42			0.537767
$560$	−733.867			−0.0553778
$561$		0			0
$562$	17203.2			1.29124
$563$	18422.2			1.37904			0.689522	−	0.724265i	$-0.257821\pi$
$563$	18422.2			1.37904			0.689522	+	0.724265i	$0.257821\pi$
$564$		0			0
$565$	−10977.2			−0.817373
$566$			2165.02i			0.160782i
$567$		0			0
$568$			6419.99i			0.474255i
$569$	2695.46			0.198594			0.0992968	−	0.995058i	$-0.468341\pi$
$569$	2695.46			0.198594			0.0992968	+	0.995058i	$0.468341\pi$
$570$		0			0
$571$			818.760i			0.0600071i	0.999550	+	0.0300035i	$0.00955185\pi$
$571$			818.760i			0.0600071i	−0.999550	+	0.0300035i	$0.990448\pi$
$572$	−7971.88	+	1535.46i	−0.582729	+	0.112239i
$573$		0			0
$574$			3456.21i			0.251323i
$575$		−	9142.39i		−	0.663068i
$576$		0			0
$577$	5754.42			0.415182			0.207591	−	0.978216i	$-0.433438\pi$
$577$	5754.42			0.415182			0.207591	+	0.978216i	$0.433438\pi$
$578$	−6957.59			−0.500688
$579$		0			0
$580$		−	5169.54i		−	0.370092i
$581$		−	6925.49i		−	0.494523i
$582$		0			0
$583$	13909.7	−	2679.13i	0.988128	−	0.190323i
$584$		−	1277.36i		−	0.0905096i
$585$		0			0
$586$	14587.1			1.02831
$587$			6869.11i			0.482996i	0.970401	+	0.241498i	$0.0776387\pi$
$587$			6869.11i			0.482996i	−0.970401	+	0.241498i	$0.922361\pi$
$588$		0			0
$589$			1523.87i			0.106604i
$590$	8612.97			0.601001
$591$		0			0
$592$	620.407			0.0430719
$593$	−9493.02			−0.657389			−0.328695	−	0.944436i	$-0.606609\pi$
$593$	−9493.02			−0.657389			−0.328695	+	0.944436i	$0.606609\pi$
$594$		0			0
$595$	−4201.70			−0.289501
$596$	12579.2			0.864537
$597$		0			0
$598$	12395.0			0.847608
$599$			20741.9i			1.41484i	0.706791	+	0.707422i	$0.250142\pi$
$599$			20741.9i			1.41484i	−0.706791	+	0.707422i	$0.749858\pi$
$600$		0			0
$601$		−	2514.11i		−	0.170637i	−0.996354	−	0.0853183i	$-0.972809\pi$
$601$		−	2514.11i		−	0.170637i	0.996354	−	0.0853183i	$-0.0271907\pi$
$602$	−1788.62			−0.121094
$603$		0			0
$604$		−	744.139i		−	0.0501301i
$605$	3239.41	+	8097.28i	0.217687	+	0.544134i
$606$		0			0
$607$		−	2793.73i		−	0.186810i	−0.995628	−	0.0934052i	$-0.970225\pi$
$607$		−	2793.73i		−	0.186810i	0.995628	−	0.0934052i	$-0.0297752\pi$
$608$		−	1974.42i		−	0.131699i
$609$		0			0
$610$	−3374.59			−0.223989
$611$	−15427.4			−1.02148
$612$		0			0
$613$			28317.5i			1.86580i	0.360141	+	0.932898i	$0.382729\pi$
$613$			28317.5i			1.86580i	−0.360141	+	0.932898i	$0.617271\pi$
$614$			14966.6i			0.983715i
$615$		0			0
$616$	2006.17	−	386.407i	0.131219	−	0.0252740i
$617$			8068.47i			0.526458i	0.964733	+	0.263229i	$0.0847874\pi$
$617$			8068.47i			0.526458i	−0.964733	+	0.263229i	$0.915213\pi$
$618$		0			0
$619$	3983.89			0.258685			0.129342	−	0.991600i	$-0.458713\pi$
$619$	3983.89			0.258685			0.129342	+	0.991600i	$0.458713\pi$
$620$		−	647.318i		−	0.0419305i
$621$		0			0
$622$			6595.31i			0.425157i
$623$	10382.1			0.667654
$624$		0			0
$625$	1368.15			0.0875617
$626$	20909.8			1.33502
$627$		0			0
$628$	−7101.03			−0.451213
$629$	3552.09			0.225169
$630$		0			0
$631$	−26978.1			−1.70203			−0.851015	−	0.525141i	$-0.824012\pi$
$631$	−26978.1			−1.70203			−0.851015	+	0.525141i	$0.824012\pi$
$632$			4712.78i			0.296621i
$633$		0			0
$634$			12521.0i			0.784342i
$635$	15528.4			0.970431
$636$		0			0
$637$			2725.95i			0.169554i
$638$	2721.95	+	14131.9i	0.168907	+	0.876941i
$639$		0			0
$640$			838.705i			0.0518012i
$641$			29063.6i			1.79086i	0.445197	+	0.895432i	$0.353133\pi$
$641$			29063.6i			1.79086i	−0.445197	+	0.895432i	$0.646867\pi$
$642$		0			0
$643$	−13056.7			−0.800787			−0.400394	−	0.916343i	$-0.631127\pi$
$643$	−13056.7			−0.800787			−0.400394	+	0.916343i	$0.631127\pi$
$644$	−3119.27			−0.190864
$645$		0			0
$646$		−	11304.4i		−	0.688490i
$647$		−	14420.4i		−	0.876236i	−0.898918	−	0.438118i	$-0.855645\pi$
$647$		−	14420.4i		−	0.876236i	0.898918	−	0.438118i	$-0.144355\pi$
$648$		0			0
$649$	−23545.2	+	4535.04i	−1.42408	+	0.274292i
$650$			9130.96i			0.550993i
$651$		0			0
$652$	−1858.59			−0.111638
$653$			31174.5i			1.86823i	0.356977	+	0.934113i	$0.383807\pi$
$653$			31174.5i			1.86823i	−0.356977	+	0.934113i	$0.616193\pi$
$654$		0			0
$655$			6116.83i			0.364892i
$656$	3949.95			0.235091
$657$		0			0
$658$	3882.38			0.230017
$659$	−8920.73			−0.527317			−0.263659	−	0.964616i	$-0.584929\pi$
$659$	−8920.73			−0.527317			−0.263659	+	0.964616i	$0.584929\pi$
$660$		0			0
$661$	13146.8			0.773603			0.386801	−	0.922163i	$-0.373580\pi$
$661$	13146.8			0.773603			0.386801	+	0.922163i	$0.373580\pi$
$662$	15881.5			0.932405
$663$		0			0
$664$	−7914.85			−0.462584
$665$		−	2830.00i		−	0.165027i
$666$		0			0
$667$		−	21972.9i		−	1.27555i
$668$	6260.02			0.362586
$669$		0			0
$670$			1050.44i			0.0605704i
$671$	9225.09	−	1776.84i	0.530746	−	0.102227i
$672$		0			0
$673$		−	12496.5i		−	0.715759i	−0.933768	−	0.357879i	$-0.883500\pi$
$673$		−	12496.5i		−	0.715759i	0.933768	−	0.357879i	$-0.116500\pi$
$674$		−	15370.9i		−	0.878432i
$675$		0			0
$676$	−3591.51			−0.204342
$677$	−21017.3			−1.19315			−0.596573	−	0.802559i	$-0.703471\pi$
$677$	−21017.3			−1.19315			−0.596573	+	0.802559i	$0.703471\pi$
$678$		0			0
$679$			7645.64i			0.432125i
$680$			4801.94i			0.270803i
$681$		0			0
$682$	340.836	+	1769.57i	0.0191368	+	0.0993553i
$683$		−	22271.2i		−	1.24771i	−0.781541	−	0.623854i	$-0.785566\pi$
$683$		−	22271.2i		−	1.24771i	0.781541	−	0.623854i	$-0.214434\pi$
$684$		0			0
$685$	1850.44			0.103214
$686$		−	686.000i		−	0.0381802i
$687$		0			0
$688$			2044.14i			0.113273i
$689$	21600.3			1.19435
$690$		0			0
$691$	32044.2			1.76414			0.882069	−	0.471120i	$-0.156150\pi$
$691$	32044.2			1.76414			0.882069	+	0.471120i	$0.156150\pi$
$692$	−17054.9			−0.936895
$693$		0			0
$694$	8841.44			0.483597
$695$	3934.46			0.214738
$696$		0			0
$697$	22615.1			1.22899
$698$			1133.57i			0.0614705i
$699$		0			0
$700$		−	2297.86i		−	0.124073i
$701$	−7142.39			−0.384828			−0.192414	−	0.981314i	$-0.561632\pi$
$701$	−7142.39			−0.384828			−0.192414	+	0.981314i	$0.561632\pi$
$702$		0			0
$703$			2392.47i			0.128355i
$704$	−441.608	−	2292.76i	−0.0236417	−	0.122744i
$705$		0			0
$706$			2504.74i			0.133523i
$707$		−	8309.37i		−	0.442017i
$708$		0			0
$709$	25495.1			1.35048			0.675239	−	0.737599i	$-0.264040\pi$
$709$	25495.1			1.35048			0.675239	+	0.737599i	$0.264040\pi$
$710$	10516.6			0.555886
$711$		0			0
$712$		−	11865.2i		−	0.624533i
$713$		−	2751.40i		−	0.144517i
$714$		0			0
$715$	2515.23	+	13058.7i	0.131559	+	0.683032i
$716$		−	11054.0i		−	0.576966i
$717$		0			0
$718$	−4538.89			−0.235919
$719$			1018.61i			0.0528341i	0.999651	+	0.0264171i	$0.00840979\pi$
$719$			1018.61i			0.0528341i	−0.999651	+	0.0264171i	$0.991590\pi$
$720$		0			0
$721$		−	257.221i		−	0.0132863i
$722$	−6104.08			−0.314640
$723$		0			0
$724$	−1309.16			−0.0672024
$725$	16186.6			0.829182
$726$		0			0
$727$	8321.66			0.424530			0.212265	−	0.977212i	$-0.431916\pi$
$727$	8321.66			0.424530			0.212265	+	0.977212i	$0.431916\pi$
$728$	3115.37			0.158604
$729$		0			0
$730$	−2092.44			−0.106089
$731$			11703.5i			0.592163i
$732$		0			0
$733$			1793.13i			0.0903557i	0.998979	+	0.0451779i	$0.0143855\pi$
$733$			1793.13i			0.0903557i	−0.998979	+	0.0451779i	$0.985615\pi$
$734$	−10649.9			−0.535550
$735$		0			0
$736$			3564.88i			0.178537i
$737$	−553.096	−	2871.59i	−0.0276439	−	0.143523i
$738$		0			0
$739$		−	13301.3i		−	0.662104i	−0.943613	−	0.331052i	$-0.892597\pi$
$739$		−	13301.3i		−	0.662104i	0.943613	−	0.331052i	$-0.107403\pi$
$740$		−	1016.29i		−	0.0504857i
$741$		0			0
$742$	−5435.82			−0.268942
$743$	−32784.8			−1.61879			−0.809394	−	0.587267i	$-0.800204\pi$
$743$	−32784.8			−1.61879			−0.809394	+	0.587267i	$0.800204\pi$
$744$		0			0
$745$		−	20605.9i		−	1.01335i
$746$		−	2786.87i		−	0.136776i
$747$		0			0
$748$	−2528.39	−	13127.0i	−0.123593	−	0.641674i
$749$			2637.79i			0.128682i
$750$		0			0
$751$	29432.2			1.43009			0.715043	−	0.699080i	$-0.246407\pi$
$751$	29432.2			1.43009			0.715043	+	0.699080i	$0.246407\pi$
$752$		−	4437.01i		−	0.215161i
$753$		0			0
$754$			21945.4i			1.05995i
$755$	−1218.97			−0.0587589
$756$		0			0
$757$	−36000.8			−1.72849			−0.864247	−	0.503068i	$-0.832205\pi$
$757$	−36000.8			−1.72849			−0.864247	+	0.503068i	$0.832205\pi$
$758$	1693.37			0.0811424
$759$		0			0
$760$	−3234.29			−0.154368
$761$	35579.7			1.69483			0.847414	−	0.530932i	$-0.178158\pi$
$761$	35579.7			1.69483			0.847414	+	0.530932i	$0.178158\pi$
$762$		0			0
$763$	2789.32			0.132346
$764$		−	1825.53i		−	0.0864469i
$765$		0			0
$766$		−	15879.6i		−	0.749027i
$767$	−36563.3			−1.72128
$768$		0			0
$769$			31790.3i			1.49075i	0.666646	+	0.745375i	$0.267729\pi$
$769$			31790.3i			1.49075i	−0.666646	+	0.745375i	$0.732271\pi$
$770$	−632.972	−	3286.29i	−0.0296243	−	0.153805i
$771$		0			0
$772$		−	3579.00i		−	0.166854i
$773$		−	9126.70i		−	0.424663i	−0.977198	−	0.212332i	$-0.931894\pi$
$773$		−	9126.70i		−	0.424663i	0.977198	−	0.212332i	$-0.0681057\pi$
$774$		0			0
$775$	2026.86			0.0939443
$776$	8737.87			0.404216
$777$		0			0
$778$			3168.61i			0.146015i
$779$			15232.1i			0.700575i
$780$		0			0
$781$	−28749.0	+	5537.34i	−1.31718	+	0.253703i
$782$			20410.5i			0.933346i
$783$		0			0
$784$	−784.000			−0.0357143
$785$			11632.2i			0.528879i
$786$		0			0
$787$			20200.8i			0.914967i	0.889218	+	0.457484i	$0.151249\pi$
$787$			20200.8i			0.914967i	−0.889218	+	0.457484i	$0.848751\pi$
$788$	4744.68			0.214495
$789$		0			0
$790$	7719.99			0.347677
$791$	−11727.1			−0.527141
$792$		0			0
$793$	14325.6			0.641510
$794$	−3981.10			−0.177939
$795$		0			0
$796$	−3830.19			−0.170550
$797$		−	19361.7i		−	0.860509i	−0.902708	−	0.430255i	$-0.858424\pi$
$797$		−	19361.7i		−	0.860509i	0.902708	−	0.430255i	$-0.141576\pi$
$798$		0			0
$799$		−	25403.7i		−	1.12481i
$800$	−2626.12			−0.116059
$801$		0			0
$802$		−	13456.8i		−	0.592490i
$803$	5720.09	−	1101.75i	0.251379	−	0.0484181i
$804$		0			0
$805$			5109.67i			0.223717i
$806$			2747.96i			0.120090i
$807$		0			0
$808$	−9496.42			−0.413469
$809$	−17267.9			−0.750442			−0.375221	−	0.926935i	$-0.622433\pi$
$809$	−17267.9			−0.750442			−0.375221	+	0.926935i	$0.622433\pi$
$810$		0			0
$811$			32477.9i			1.40623i	0.711075	+	0.703116i	$0.248209\pi$
$811$			32477.9i			1.40623i	−0.711075	+	0.703116i	$0.751791\pi$
$812$		−	5522.69i		−	0.238680i
$813$		0			0
$814$	535.111	+	2778.21i	0.0230413	+	0.119627i
$815$			3044.54i			0.130854i
$816$		0			0
$817$	−7882.78			−0.337556
$818$			32066.5i			1.37063i
$819$		0			0
$820$		−	6470.40i		−	0.275556i
$821$	5800.29			0.246567			0.123283	−	0.992371i	$-0.460658\pi$
$821$	5800.29			0.246567			0.123283	+	0.992371i	$0.460658\pi$
$822$		0			0
$823$	22373.1			0.947601			0.473801	−	0.880632i	$-0.342882\pi$
$823$	22373.1			0.947601			0.473801	+	0.880632i	$0.342882\pi$
$824$	−293.967			−0.0124282
$825$		0			0
$826$	9201.35			0.387598
$827$	−16389.5			−0.689142			−0.344571	−	0.938760i	$-0.611976\pi$
$827$	−16389.5			−0.689142			−0.344571	+	0.938760i	$0.611976\pi$
$828$		0			0
$829$	−12472.6			−0.522549			−0.261274	−	0.965265i	$-0.584143\pi$
$829$	−12472.6			−0.522549			−0.261274	+	0.965265i	$0.584143\pi$
$830$			12965.3i			0.542207i
$831$		0			0
$832$		−	3560.42i		−	0.148360i
$833$	−4488.73			−0.186705
$834$		0			0
$835$		−	10254.5i		−	0.424997i
$836$	8841.55	−	1702.97i	0.365779	−	0.0704526i
$837$		0			0
$838$		−	22383.0i		−	0.922684i
$839$		−	10576.2i		−	0.435199i	−0.976038	−	0.217599i	$-0.930177\pi$
$839$		−	10576.2i		−	0.435199i	0.976038	−	0.217599i	$-0.0698226\pi$
$840$		0			0
$841$	14514.1			0.595110
$842$	−2445.70			−0.100100
$843$		0			0
$844$		−	11111.3i		−	0.453161i
$845$			5883.25i			0.239515i
$846$		0			0
$847$	3460.70	+	8650.44i	0.140391	+	0.350924i
$848$			6212.37i			0.251573i
$849$		0			0
$850$	−15035.6			−0.606728
$851$		−	4319.68i		−	0.174003i
$852$		0			0
$853$			18701.4i			0.750672i	0.926889	+	0.375336i	$0.122473\pi$
$853$			18701.4i			0.750672i	−0.926889	+	0.375336i	$0.877527\pi$
$854$	−3605.12			−0.144455
$855$		0			0
$856$	3014.61			0.120371
$857$	27299.6			1.08814			0.544069	−	0.839040i	$-0.316883\pi$
$857$	27299.6			1.08814			0.544069	+	0.839040i	$0.316883\pi$
$858$		0			0
$859$	−43416.5			−1.72451			−0.862254	−	0.506477i	$-0.830948\pi$
$859$	−43416.5			−1.72451			−0.862254	+	0.506477i	$0.830948\pi$
$860$	3348.49			0.132771
$861$		0			0
$862$	−3117.78			−0.123193
$863$			14537.3i			0.573414i	0.958018	+	0.286707i	$0.0925607\pi$
$863$			14537.3i			0.573414i	−0.958018	+	0.286707i	$0.907439\pi$
$864$		0			0
$865$			27937.6i			1.09816i
$866$	−1648.93			−0.0647031
$867$		0			0
$868$		−	691.539i		−	0.0270419i
$869$	−21104.1	+	4064.85i	−0.823828	+	0.158677i
$870$		0			0
$871$		−	4459.28i		−	0.173475i
$872$		−	3187.80i		−	0.123799i
$873$		0			0
$874$	−13747.2			−0.532044
$875$	−9497.44			−0.366940
$876$		0			0
$877$		−	36411.3i		−	1.40196i	−0.713180	−	0.700981i	$-0.752746\pi$
$877$		−	36411.3i		−	1.40196i	0.713180	−	0.700981i	$-0.247254\pi$
$878$		−	5597.58i		−	0.215159i
$879$		0			0
$880$	−3755.77	+	723.397i	−0.143871	+	0.0277110i
$881$			5941.19i			0.227201i	0.993527	+	0.113600i	$0.0362383\pi$
$881$			5941.19i			0.227201i	−0.993527	+	0.113600i	$0.963762\pi$
$882$		0			0
$883$	−38950.0			−1.48445			−0.742225	−	0.670150i	$-0.766230\pi$
$883$	−38950.0			−1.48445			−0.742225	+	0.670150i	$0.766230\pi$
$884$		−	20384.9i		−	0.775588i
$885$		0			0
$886$			29541.3i			1.12016i
$887$	3824.53			0.144775			0.0723873	−	0.997377i	$-0.476938\pi$
$887$	3824.53			0.144775			0.0723873	+	0.997377i	$0.476938\pi$
$888$		0			0
$889$	16589.1			0.625852
$890$	−19436.3			−0.732031
$891$		0			0
$892$	1594.27			0.0598432
$893$	17110.4			0.641183
$894$		0			0
$895$	−18107.5			−0.676277
$896$			896.000i			0.0334077i
$897$		0			0
$898$		−	34255.6i		−	1.27297i
$899$	4871.37			0.180722
$900$		0			0
$901$			35568.4i			1.31516i
$902$	3406.89	+	17688.1i	0.125762	+	0.652936i
$903$		0			0
$904$			13402.4i			0.493095i
$905$			2144.53i			0.0787697i
$906$		0			0
$907$	41597.2			1.52284			0.761419	−	0.648260i	$-0.224503\pi$
$907$	41597.2			1.52284			0.761419	+	0.648260i	$0.224503\pi$
$908$	7942.03			0.290270
$909$		0			0
$910$		−	5103.28i		−	0.185903i
$911$		−	14147.7i		−	0.514525i	−0.966342	−	0.257263i	$-0.917179\pi$
$911$		−	14147.7i		−	0.514525i	0.966342	−	0.257263i	$-0.0828206\pi$
$912$		0			0
$913$	−6826.69	−	35443.1i	−0.247459	−	1.28477i
$914$		−	25619.6i		−	0.927156i
$915$		0			0
$916$	16752.1			0.604263
$917$			6534.69i			0.235327i
$918$		0			0
$919$		−	28122.8i		−	1.00945i	−0.863280	−	0.504726i	$-0.831594\pi$
$919$		−	28122.8i		−	1.00945i	0.863280	−	0.504726i	$-0.168406\pi$
$920$	5839.62			0.209268
$921$		0			0
$922$	16436.6			0.587105
$923$	−44644.3			−1.59207
$924$		0			0
$925$	3182.15			0.113112
$926$	−28667.0			−1.01734
$927$		0			0
$928$	−6311.64			−0.223265
$929$			29235.7i			1.03250i	0.856439	+	0.516249i	$0.172672\pi$
$929$			29235.7i			1.03250i	−0.856439	+	0.516249i	$0.827328\pi$
$930$		0			0
$931$		−	3023.33i		−	0.106429i
$932$	−5679.45			−0.199610
$933$		0			0
$934$		−	16390.2i		−	0.574200i
$935$	−21503.3	+	4141.75i	−0.752122	+	0.144866i
$936$		0			0
$937$		−	34811.6i		−	1.21371i	−0.794813	−	0.606855i	$-0.792431\pi$
$937$		−	34811.6i		−	1.21371i	0.794813	−	0.606855i	$-0.207569\pi$
$938$			1122.20i			0.0390631i
$939$		0			0
$940$	−7268.25			−0.252196
$941$	32395.8			1.12229			0.561145	−	0.827718i	$-0.310361\pi$
$941$	32395.8			1.12229			0.561145	+	0.827718i	$0.310361\pi$
$942$		0			0
$943$		−	27502.2i		−	0.949728i
$944$		−	10515.8i		−	0.362565i
$945$		0			0
$946$	−9153.75	+	1763.10i	−0.314603	+	0.0605955i
$947$		−	56206.5i		−	1.92869i	−0.264654	−	0.964343i	$-0.585258\pi$
$947$		−	56206.5i		−	1.92869i	0.264654	−	0.964343i	$-0.414742\pi$
$948$		0			0
$949$	8882.72			0.303841
$950$		−	10127.1i		−	0.345858i
$951$		0			0
$952$			5129.98i			0.174647i
$953$	45721.9			1.55412			0.777060	−	0.629426i	$-0.216710\pi$
$953$	45721.9			1.55412			0.777060	+	0.629426i	$0.216710\pi$
$954$		0			0
$955$	−2990.40			−0.101327
$956$	5474.15			0.185195
$957$		0			0
$958$	−5050.06			−0.170313
$959$	1976.85			0.0665651
$960$		0			0
$961$	−29181.0			−0.979525
$962$			4314.28i			0.144592i
$963$		0			0
$964$		−	6250.47i		−	0.208832i
$965$	−5862.75			−0.195574
$966$		0			0
$967$			17778.4i			0.591225i	0.955308	+	0.295613i	$0.0955237\pi$
$967$			17778.4i			0.591225i	−0.955308	+	0.295613i	$0.904476\pi$
$968$	9886.21	−	3955.09i	0.328259	−	0.131324i
$969$		0			0
$970$		−	14313.5i		−	0.473792i
$971$			30070.2i			0.993819i	0.867802	+	0.496910i	$0.165532\pi$
$971$			30070.2i			0.993819i	−0.867802	+	0.496910i	$0.834468\pi$
$972$		0			0
$973$	4203.24			0.138489
$974$	−42921.5			−1.41200
$975$		0			0
$976$			4120.14i			0.135125i
$977$		−	39082.5i		−	1.27980i	−0.768460	−	0.639898i	$-0.778977\pi$
$977$		−	39082.5i		−	1.27980i	0.768460	−	0.639898i	$-0.221023\pi$
$978$		0			0
$979$	53133.0	−	10233.9i	1.73456	−	0.334094i
$980$			1284.27i			0.0418617i
$981$		0			0
$982$	36067.2			1.17205
$983$			28242.6i			0.916377i	0.888855	+	0.458189i	$0.151502\pi$
$983$			28242.6i			0.916377i	−0.888855	+	0.458189i	$0.848498\pi$
$984$		0			0
$985$		−	7772.24i		−	0.251415i
$986$	−36136.8			−1.16717
$987$		0			0
$988$	13730.0			0.442115
$989$	14232.6			0.457605
$990$		0			0
$991$	−6944.91			−0.222616			−0.111308	−	0.993786i	$-0.535504\pi$
$991$	−6944.91			−0.222616			−0.111308	+	0.993786i	$0.535504\pi$
$992$	−790.330			−0.0252954
$993$		0			0
$994$	11235.0			0.358503
$995$			6274.23i			0.199906i
$996$		0			0
$997$			1158.45i			0.0367989i	0.999831	+	0.0183995i	$0.00585706\pi$
$997$			1158.45i			0.0367989i	−0.999831	+	0.0183995i	$0.994143\pi$
$998$	1486.26			0.0471412
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.4.c.a.197.12	✓	36
3.2	odd	2		1386.4.c.b.197.25	yes	36
11.10	odd	2		1386.4.c.b.197.12	yes	36
33.32	even	2	inner	1386.4.c.a.197.25	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.4.c.a.197.12	✓	36	1.1	even	1	trivial
1386.4.c.a.197.25	yes	36	33.32	even	2	inner
1386.4.c.b.197.12	yes	36	11.10	odd	2
1386.4.c.b.197.25	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -6.55239i q^{5} -7.00000i q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} -6.55239i q^{5} -7.00000i q^{7} -8.00000 q^{8} +13.1048i q^{10} +(-6.90013 - 35.8244i) q^{11} -55.6316i q^{13} +14.0000i q^{14} +16.0000 q^{16} +91.6068 q^{17} +61.7006i q^{19} -26.2095i q^{20} +(13.8003 + 71.6488i) q^{22} -111.403i q^{23} +82.0663 q^{25} +111.263i q^{26} -28.0000i q^{28} +197.239 q^{29} +24.6978 q^{31} -32.0000 q^{32} -183.214 q^{34} -45.8667 q^{35} +38.7754 q^{37} -123.401i q^{38} +52.4191i q^{40} +246.872 q^{41} +127.759i q^{43} +(-27.6005 - 143.298i) q^{44} +222.805i q^{46} -277.313i q^{47} -49.0000 q^{49} -164.133 q^{50} -222.527i q^{52} +388.273i q^{53} +(-234.735 + 45.2123i) q^{55} +56.0000i q^{56} -394.478 q^{58} -657.239i q^{59} +257.509i q^{61} -49.3956 q^{62} +64.0000 q^{64} -364.520 q^{65} +80.1573 q^{67} +366.427 q^{68} +91.7334 q^{70} -802.499i q^{71} +159.670i q^{73} -77.5508 q^{74} +246.802i q^{76} +(-250.771 + 48.3009i) q^{77} -589.098i q^{79} -104.838i q^{80} -493.744 q^{82} +989.356 q^{83} -600.243i q^{85} -255.517i q^{86} +(55.2010 + 286.595i) q^{88} +1483.15i q^{89} -389.421 q^{91} -445.610i q^{92} +554.626i q^{94} +404.286 q^{95} -1092.23 q^{97} +98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 72 q^{2} + 144 q^{4} - 288 q^{8}+O(q^{10}) \) 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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