LMFDB - Embedded newform 378.4.d.b.377.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(377,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("378.377");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.d (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:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			377.4
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	378.377
Dual form			378.4.d.b.377.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -4.00000 q^{4} +3.46410 q^{5} +(14.0000 - 12.1244i) q^{7} -8.00000i q^{8} +O(q^{10})$	\(q+2.00000i q^{2} -4.00000 q^{4} +3.46410 q^{5} +(14.0000 - 12.1244i) q^{7} -8.00000i q^{8} +6.92820i q^{10} -30.0000i q^{11} +43.3013i q^{13} +(24.2487 + 28.0000i) q^{14} +16.0000 q^{16} -105.655 q^{17} -90.0666i q^{19} -13.8564 q^{20} +60.0000 q^{22} -201.000i q^{23} -113.000 q^{25} -86.6025 q^{26} +(-56.0000 + 48.4974i) q^{28} +111.000i q^{29} -202.650i q^{31} +32.0000i q^{32} -211.310i q^{34} +(48.4974 - 42.0000i) q^{35} +430.000 q^{37} +180.133 q^{38} -27.7128i q^{40} -117.779 q^{41} +355.000 q^{43} +120.000i q^{44} +402.000 q^{46} -588.897 q^{47} +(49.0000 - 339.482i) q^{49} -226.000i q^{50} -173.205i q^{52} -33.0000i q^{53} -103.923i q^{55} +(-96.9948 - 112.000i) q^{56} -222.000 q^{58} +583.701 q^{59} -692.820i q^{61} +405.300 q^{62} -64.0000 q^{64} +150.000i q^{65} +575.000 q^{67} +422.620 q^{68} +(84.0000 + 96.9948i) q^{70} -159.000i q^{71} -48.4974i q^{73} +860.000i q^{74} +360.267i q^{76} +(-363.731 - 420.000i) q^{77} -376.000 q^{79} +55.4256 q^{80} -235.559i q^{82} -232.095 q^{83} -366.000 q^{85} +710.000i q^{86} -240.000 q^{88} +566.381 q^{89} +(525.000 + 606.218i) q^{91} +804.000i q^{92} -1177.79i q^{94} -312.000i q^{95} +630.466i q^{97} +(678.964 + 98.0000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4} + 56 q^{7}+O(q^{10}) $ 4 * q - 16 * q^4 + 56 * q^7	$ 4 q - 16 q^{4} + 56 q^{7} + 64 q^{16} + 240 q^{22} - 452 q^{25} - 224 q^{28} + 1720 q^{37} + 1420 q^{43} + 1608 q^{46} + 196 q^{49} - 888 q^{58} - 256 q^{64} + 2300 q^{67} + 336 q^{70} - 1504 q^{79} - 1464 q^{85} - 960 q^{88} + 2100 q^{91}+O(q^{100}) $ 4 * q - 16 * q^4 + 56 * q^7 + 64 * q^16 + 240 * q^22 - 452 * q^25 - 224 * q^28 + 1720 * q^37 + 1420 * q^43 + 1608 * q^46 + 196 * q^49 - 888 * q^58 - 256 * q^64 + 2300 * q^67 + 336 * q^70 - 1504 * q^79 - 1464 * q^85 - 960 * q^88 + 2100 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			2.00000i			0.707107i
$2$			2.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−4.00000			−0.500000
$5$	3.46410			0.309839			0.154919	−	0.987927i	$-0.450488\pi$
$5$	3.46410			0.309839			0.154919	+	0.987927i	$0.450488\pi$
$6$		0			0
$7$	14.0000	−	12.1244i	0.755929	−	0.654654i
$7$	14.0000	−	12.1244i	0.755929	−	0.654654i
$8$		−	8.00000i		−	0.353553i
$9$		0			0
$10$			6.92820i			0.219089i
$11$		−	30.0000i		−	0.822304i	−0.911567	−	0.411152i	$-0.865127\pi$
$11$		−	30.0000i		−	0.822304i	0.911567	−	0.411152i	$-0.134873\pi$
$12$		0			0
$13$			43.3013i			0.923816i	0.886928	+	0.461908i	$0.152835\pi$
$13$			43.3013i			0.923816i	−0.886928	+	0.461908i	$0.847165\pi$
$14$	24.2487	+	28.0000i	0.462910	+	0.534522i
$15$		0			0
$16$	16.0000			0.250000
$17$	−105.655			−1.50736			−0.753680	−	0.657241i	$-0.771723\pi$
$17$	−105.655			−1.50736			−0.753680	+	0.657241i	$0.771723\pi$
$18$		0			0
$19$		−	90.0666i		−	1.08751i	−0.839244	−	0.543755i	$-0.817002\pi$
$19$		−	90.0666i		−	1.08751i	0.839244	−	0.543755i	$-0.182998\pi$
$20$	−13.8564			−0.154919
$21$		0			0
$22$	60.0000			0.581456
$23$		−	201.000i		−	1.82223i	−0.412147	−	0.911117i	$-0.635221\pi$
$23$		−	201.000i		−	1.82223i	0.412147	−	0.911117i	$-0.364779\pi$
$24$		0			0
$25$	−113.000			−0.904000
$26$	−86.6025			−0.653237
$27$		0			0
$28$	−56.0000	+	48.4974i	−0.377964	+	0.327327i
$29$			111.000i			0.710765i	0.934721	+	0.355382i	$0.115649\pi$
$29$			111.000i			0.710765i	−0.934721	+	0.355382i	$0.884351\pi$
$30$		0			0
$31$		−	202.650i		−	1.17410i	−0.809552	−	0.587048i	$-0.800290\pi$
$31$		−	202.650i		−	1.17410i	0.809552	−	0.587048i	$-0.199710\pi$
$32$			32.0000i			0.176777i
$33$		0			0
$34$		−	211.310i		−	1.06586i
$35$	48.4974	−	42.0000i	0.234216	−	0.202837i
$36$		0			0
$37$	430.000			1.91058			0.955291	−	0.295666i	$-0.0955415\pi$
$37$	430.000			1.91058			0.955291	+	0.295666i	$0.0955415\pi$
$38$	180.133			0.768986
$39$		0			0
$40$		−	27.7128i		−	0.109545i
$41$	−117.779			−0.448636			−0.224318	−	0.974516i	$-0.572015\pi$
$41$	−117.779			−0.448636			−0.224318	+	0.974516i	$0.572015\pi$
$42$		0			0
$43$	355.000			1.25900			0.629500	−	0.777001i	$-0.283260\pi$
$43$	355.000			1.25900			0.629500	+	0.777001i	$0.283260\pi$
$44$			120.000i			0.411152i
$45$		0			0
$46$	402.000			1.28851
$47$	−588.897			−1.82765			−0.913824	−	0.406109i	$-0.866885\pi$
$47$	−588.897			−1.82765			−0.913824	+	0.406109i	$0.866885\pi$
$48$		0			0
$49$	49.0000	−	339.482i	0.142857	−	0.989743i
$50$		−	226.000i		−	0.639225i
$51$		0			0
$52$		−	173.205i		−	0.461908i
$53$		−	33.0000i		−	0.0855264i	−0.999085	−	0.0427632i	$-0.986384\pi$
$53$		−	33.0000i		−	0.0855264i	0.999085	−	0.0427632i	$-0.0136161\pi$
$54$		0			0
$55$		−	103.923i		−	0.254781i
$56$	−96.9948	−	112.000i	−0.231455	−	0.267261i
$57$		0			0
$58$	−222.000			−0.502587
$59$	583.701			1.28799			0.643995	−	0.765030i	$-0.277276\pi$
$59$	583.701			1.28799			0.643995	+	0.765030i	$0.277276\pi$
$60$		0			0
$61$		−	692.820i		−	1.45421i	−0.686529	−	0.727103i	$-0.740866\pi$
$61$		−	692.820i		−	1.45421i	0.686529	−	0.727103i	$-0.259134\pi$
$62$	405.300			0.830212
$63$		0			0
$64$	−64.0000			−0.125000
$65$			150.000i			0.286234i
$66$		0			0
$67$	575.000			1.04847			0.524235	−	0.851574i	$-0.324352\pi$
$67$	575.000			1.04847			0.524235	+	0.851574i	$0.324352\pi$
$68$	422.620			0.753680
$69$		0			0
$70$	84.0000	+	96.9948i	0.143427	+	0.165616i
$71$		−	159.000i		−	0.265772i	−0.991131	−	0.132886i	$-0.957576\pi$
$71$		−	159.000i		−	0.265772i	0.991131	−	0.132886i	$-0.0424245\pi$
$72$		0			0
$73$		−	48.4974i		−	0.0777561i	−0.999244	−	0.0388780i	$-0.987622\pi$
$73$		−	48.4974i		−	0.0777561i	0.999244	−	0.0388780i	$-0.0123784\pi$
$74$			860.000i			1.35099i
$75$		0			0
$76$			360.267i			0.543755i
$77$	−363.731	−	420.000i	−0.538324	−	0.621603i
$78$		0			0
$79$	−376.000			−0.535485			−0.267742	−	0.963491i	$-0.586278\pi$
$79$	−376.000			−0.535485			−0.267742	+	0.963491i	$0.586278\pi$
$80$	55.4256			0.0774597
$81$		0			0
$82$		−	235.559i		−	0.317233i
$83$	−232.095			−0.306936			−0.153468	−	0.988154i	$-0.549044\pi$
$83$	−232.095			−0.306936			−0.153468	+	0.988154i	$0.549044\pi$
$84$		0			0
$85$	−366.000			−0.467039
$86$			710.000i			0.890247i
$87$		0			0
$88$	−240.000			−0.290728
$89$	566.381			0.674564			0.337282	−	0.941404i	$-0.390492\pi$
$89$	566.381			0.674564			0.337282	+	0.941404i	$0.390492\pi$
$90$		0			0
$91$	525.000	+	606.218i	0.604780	+	0.698339i
$92$			804.000i			0.911117i
$93$		0			0
$94$		−	1177.79i		−	1.29234i
$95$		−	312.000i		−	0.336953i
$96$		0			0
$97$			630.466i			0.659940i	0.943991	+	0.329970i	$0.107039\pi$
$97$			630.466i			0.659940i	−0.943991	+	0.329970i	$0.892961\pi$
$98$	678.964	+	98.0000i	0.699854	+	0.101015i
$99$		0			0
$100$	452.000			0.452000
$101$	−374.123			−0.368580			−0.184290	−	0.982872i	$-0.558999\pi$
$101$	−374.123			−0.368580			−0.184290	+	0.982872i	$0.558999\pi$
$102$		0			0
$103$		−	271.932i		−	0.260138i	−0.991505	−	0.130069i	$-0.958480\pi$
$103$		−	271.932i		−	0.260138i	0.991505	−	0.130069i	$-0.0415199\pi$
$104$	346.410			0.326618
$105$		0			0
$106$	66.0000			0.0604763
$107$		−	864.000i		−	0.780617i	−0.920684	−	0.390309i	$-0.872368\pi$
$107$		−	864.000i		−	0.780617i	0.920684	−	0.390309i	$-0.127632\pi$
$108$		0			0
$109$	−140.000			−0.123024			−0.0615118	−	0.998106i	$-0.519592\pi$
$109$	−140.000			−0.123024			−0.0615118	+	0.998106i	$0.519592\pi$
$110$	207.846			0.180158
$111$		0			0
$112$	224.000	−	193.990i	0.188982	−	0.163663i
$113$		−	582.000i		−	0.484513i	−0.970212	−	0.242256i	$-0.922112\pi$
$113$		−	582.000i		−	0.484513i	0.970212	−	0.242256i	$-0.0778875\pi$
$114$		0			0
$115$		−	696.284i		−	0.564599i
$116$		−	444.000i		−	0.355382i
$117$		0			0
$118$			1167.40i			0.910747i
$119$	−1479.17	+	1281.00i	−1.13946	+	0.986799i
$120$		0			0
$121$	431.000			0.323817
$122$	1385.64			1.02828
$123$		0			0
$124$			810.600i			0.587048i
$125$	−824.456			−0.589933
$126$		0			0
$127$	194.000			0.135549			0.0677745	−	0.997701i	$-0.478410\pi$
$127$	194.000			0.135549			0.0677745	+	0.997701i	$0.478410\pi$
$128$		−	128.000i		−	0.0883883i
$129$		0			0
$130$	−300.000			−0.202398
$131$	−971.681			−0.648062			−0.324031	−	0.946046i	$-0.605038\pi$
$131$	−971.681			−0.648062			−0.324031	+	0.946046i	$0.605038\pi$
$132$		0			0
$133$	−1092.00	−	1260.93i	−0.711943	−	0.822081i
$134$			1150.00i			0.741380i
$135$		0			0
$136$			845.241i			0.532932i
$137$			936.000i			0.583707i	0.956463	+	0.291854i	$0.0942720\pi$
$137$			936.000i			0.583707i	−0.956463	+	0.291854i	$0.905728\pi$
$138$		0			0
$139$			1978.00i			1.20699i	0.797366	+	0.603496i	$0.206226\pi$
$139$			1978.00i			1.20699i	−0.797366	+	0.603496i	$0.793774\pi$
$140$	−193.990	+	168.000i	−0.117108	+	0.101419i
$141$		0			0
$142$	318.000			0.187929
$143$	1299.04			0.759658
$144$		0			0
$145$			384.515i			0.220222i
$146$	96.9948			0.0549819
$147$		0			0
$148$	−1720.00			−0.955291
$149$		−	2613.00i		−	1.43668i	−0.695692	−	0.718340i	$-0.744902\pi$
$149$		−	2613.00i		−	1.43668i	0.695692	−	0.718340i	$-0.255098\pi$
$150$		0			0
$151$	−1856.00			−1.00026			−0.500130	−	0.865951i	$-0.666714\pi$
$151$	−1856.00			−1.00026			−0.500130	+	0.865951i	$0.666714\pi$
$152$	−720.533			−0.384493
$153$		0			0
$154$	840.000	−	727.461i	0.439540	−	0.380653i
$155$		−	702.000i		−	0.363781i
$156$		0			0
$157$			1564.04i			0.795058i	0.917590	+	0.397529i	$0.130132\pi$
$157$			1564.04i			0.795058i	−0.917590	+	0.397529i	$0.869868\pi$
$158$		−	752.000i		−	0.378645i
$159$		0			0
$160$			110.851i			0.0547723i
$161$	−2437.00	−	2814.00i	−1.19293	−	1.37748i
$162$		0			0
$163$	−2923.00			−1.40458			−0.702292	−	0.711889i	$-0.747840\pi$
$163$	−2923.00			−1.40458			−0.702292	+	0.711889i	$0.747840\pi$
$164$	471.118			0.224318
$165$		0			0
$166$		−	464.190i		−	0.217037i
$167$	1915.65			0.887649			0.443824	−	0.896114i	$-0.353621\pi$
$167$	1915.65			0.887649			0.443824	+	0.896114i	$0.353621\pi$
$168$		0			0
$169$	322.000			0.146563
$170$		−	732.000i		−	0.330246i
$171$		0			0
$172$	−1420.00			−0.629500
$173$	−4136.14			−1.81772			−0.908858	−	0.417107i	$-0.863044\pi$
$173$	−4136.14			−1.81772			−0.908858	+	0.417107i	$0.863044\pi$
$174$		0			0
$175$	−1582.00	+	1370.05i	−0.683360	+	0.591807i
$176$		−	480.000i		−	0.205576i
$177$		0			0
$178$			1132.76i			0.476989i
$179$		−	1110.00i		−	0.463493i	−0.972776	−	0.231747i	$-0.925556\pi$
$179$		−	1110.00i		−	0.463493i	0.972776	−	0.231747i	$-0.0744440\pi$
$180$		0			0
$181$			4117.08i			1.69072i	0.534196	+	0.845360i	$0.320614\pi$
$181$			4117.08i			1.69072i	−0.534196	+	0.845360i	$0.679386\pi$
$182$	−1212.44	+	1050.00i	−0.493801	+	0.427644i
$183$		0			0
$184$	−1608.00			−0.644257
$185$	1489.56			0.591972
$186$		0			0
$187$			3169.65i			1.23951i
$188$	2355.59			0.913824
$189$		0			0
$190$	624.000			0.238262
$191$			4512.00i			1.70930i	0.519202	+	0.854651i	$0.326229\pi$
$191$			4512.00i			1.70930i	−0.519202	+	0.854651i	$0.673771\pi$
$192$		0			0
$193$	997.000			0.371843			0.185921	−	0.982565i	$-0.440473\pi$
$193$	997.000			0.371843			0.185921	+	0.982565i	$0.440473\pi$
$194$	−1260.93			−0.466648
$195$		0			0
$196$	−196.000	+	1357.93i	−0.0714286	+	0.494872i
$197$		−	354.000i		−	0.128028i	−0.997949	−	0.0640138i	$-0.979610\pi$
$197$		−	354.000i		−	0.128028i	0.997949	−	0.0640138i	$-0.0203902\pi$
$198$		0			0
$199$		−	5007.36i		−	1.78373i	−0.452302	−	0.891865i	$-0.649397\pi$
$199$		−	5007.36i		−	1.78373i	0.452302	−	0.891865i	$-0.350603\pi$
$200$			904.000i			0.319612i
$201$		0			0
$202$		−	748.246i		−	0.260626i
$203$	1345.80	+	1554.00i	0.465305	+	0.537288i
$204$		0			0
$205$	−408.000			−0.139005
$206$	543.864			0.183946
$207$		0			0
$208$			692.820i			0.230954i
$209$	−2702.00			−0.894264
$210$		0			0
$211$	1657.00			0.540628			0.270314	−	0.962772i	$-0.412872\pi$
$211$	1657.00			0.540628			0.270314	+	0.962772i	$0.412872\pi$
$212$			132.000i			0.0427632i
$213$		0			0
$214$	1728.00			0.551980
$215$	1229.76			0.390087
$216$		0			0
$217$	−2457.00	−	2837.10i	−0.768627	−	0.887534i
$218$		−	280.000i		−	0.0869908i
$219$		0			0
$220$			415.692i			0.127391i
$221$		−	4575.00i		−	1.39252i
$222$		0			0
$223$			5975.58i			1.79441i	0.441611	+	0.897207i	$0.354407\pi$
$223$			5975.58i			1.79441i	−0.441611	+	0.897207i	$0.645593\pi$
$224$	387.979	+	448.000i	0.115728	+	0.133631i
$225$		0			0
$226$	1164.00			0.342602
$227$	−5.19615			−0.00151930			−0.000759649	−	1.00000i	$-0.500242\pi$
$227$	−5.19615			−0.00151930			−0.000759649		1.00000i	$0.500242\pi$
$228$		0			0
$229$			5258.51i			1.51743i	0.651422	+	0.758716i	$0.274173\pi$
$229$			5258.51i			1.51743i	−0.651422	+	0.758716i	$0.725827\pi$
$230$	1392.57			0.399232
$231$		0			0
$232$	888.000			0.251293
$233$			2784.00i			0.782772i	0.920227	+	0.391386i	$0.128004\pi$
$233$			2784.00i			0.782772i	−0.920227	+	0.391386i	$0.871996\pi$
$234$		0			0
$235$	−2040.00			−0.566276
$236$	−2334.80			−0.643995
$237$		0			0
$238$	−2562.00	−	2958.34i	−0.697772	−	0.805718i
$239$		−	2376.00i		−	0.643057i	−0.946900	−	0.321529i	$-0.895803\pi$
$239$		−	2376.00i		−	0.643057i	0.946900	−	0.321529i	$-0.104197\pi$
$240$		0			0
$241$		−	4392.48i		−	1.17404i	−0.809571	−	0.587022i	$-0.800300\pi$
$241$		−	4392.48i		−	1.17404i	0.809571	−	0.587022i	$-0.199700\pi$
$242$			862.000i			0.228973i
$243$		0			0
$244$			2771.28i			0.727103i
$245$	169.741	−	1176.00i	0.0442627	−	0.306661i
$246$		0			0
$247$	3900.00			1.00466
$248$	−1621.20			−0.415106
$249$		0			0
$250$		−	1648.91i		−	0.417145i
$251$	2393.69			0.601947			0.300973	−	0.953633i	$-0.402688\pi$
$251$	2393.69			0.601947			0.300973	+	0.953633i	$0.402688\pi$
$252$		0			0
$253$	−6030.00			−1.49843
$254$			388.000i			0.0958476i
$255$		0			0
$256$	256.000			0.0625000
$257$	1351.00			0.327911			0.163955	−	0.986468i	$-0.447575\pi$
$257$	1351.00			0.327911			0.163955	+	0.986468i	$0.447575\pi$
$258$		0			0
$259$	6020.00	−	5213.47i	1.44426	−	1.25077i
$260$		−	600.000i		−	0.143117i
$261$		0			0
$262$		−	1943.36i		−	0.458249i
$263$			6795.00i			1.59315i	0.604542	+	0.796573i	$0.293356\pi$
$263$			6795.00i			1.59315i	−0.604542	+	0.796573i	$0.706644\pi$
$264$		0			0
$265$		−	114.315i		−	0.0264994i
$266$	2521.87	−	2184.00i	0.581299	−	0.503420i
$267$		0			0
$268$	−2300.00			−0.524235
$269$	6772.32			1.53500			0.767501	−	0.641048i	$-0.221500\pi$
$269$	6772.32			1.53500			0.767501	+	0.641048i	$0.221500\pi$
$270$		0			0
$271$		−	3781.07i		−	0.847540i	−0.905770	−	0.423770i	$-0.860706\pi$
$271$		−	3781.07i		−	0.847540i	0.905770	−	0.423770i	$-0.139294\pi$
$272$	−1690.48			−0.376840
$273$		0			0
$274$	−1872.00			−0.412743
$275$			3390.00i			0.743363i
$276$		0			0
$277$	−148.000			−0.0321027			−0.0160514	−	0.999871i	$-0.505110\pi$
$277$	−148.000			−0.0321027			−0.0160514	+	0.999871i	$0.505110\pi$
$278$	−3956.00			−0.853473
$279$		0			0
$280$	−336.000	−	387.979i	−0.0717137	−	0.0828079i
$281$			5070.00i			1.07634i	0.842837	+	0.538169i	$0.180884\pi$
$281$			5070.00i			1.07634i	−0.842837	+	0.538169i	$0.819116\pi$
$282$		0			0
$283$		−	2514.94i		−	0.528260i	−0.964487	−	0.264130i	$-0.914915\pi$
$283$		−	2514.94i		−	0.528260i	0.964487	−	0.264130i	$-0.0850848\pi$
$284$			636.000i			0.132886i
$285$		0			0
$286$			2598.08i			0.537159i
$287$	−1648.91	+	1428.00i	−0.339137	+	0.293701i
$288$		0			0
$289$	6250.00			1.27214
$290$	−769.031			−0.155721
$291$		0			0
$292$			193.990i			0.0388780i
$293$	17.3205			0.00345350			0.00172675	−	0.999999i	$-0.499450\pi$
$293$	17.3205			0.00345350			0.00172675	+	0.999999i	$0.499450\pi$
$294$		0			0
$295$	2022.00			0.399069
$296$		−	3440.00i		−	0.675493i
$297$		0			0
$298$	5226.00			1.01589
$299$	8703.56			1.68341
$300$		0			0
$301$	4970.00	−	4304.15i	0.951714	−	0.824209i
$302$		−	3712.00i		−	0.707290i
$303$		0			0
$304$		−	1441.07i		−	0.271878i
$305$		−	2400.00i		−	0.450569i
$306$		0			0
$307$		−	1964.15i		−	0.365146i	−0.983192	−	0.182573i	$-0.941557\pi$
$307$		−	1964.15i		−	0.365146i	0.983192	−	0.182573i	$-0.0584425\pi$
$308$	1454.92	+	1680.00i	0.269162	+	0.310802i
$309$		0			0
$310$	1404.00			0.257232
$311$	9117.52			1.66240			0.831201	−	0.555972i	$-0.187654\pi$
$311$	9117.52			1.66240			0.831201	+	0.555972i	$0.187654\pi$
$312$		0			0
$313$			4517.19i			0.815740i	0.913040	+	0.407870i	$0.133728\pi$
$313$			4517.19i			0.815740i	−0.913040	+	0.407870i	$0.866272\pi$
$314$	−3128.08			−0.562191
$315$		0			0
$316$	1504.00			0.267742
$317$		−	8586.00i		−	1.52125i	−0.649189	−	0.760627i	$-0.724891\pi$
$317$		−	8586.00i		−	1.52125i	0.649189	−	0.760627i	$-0.275109\pi$
$318$		0			0
$319$	3330.00			0.584465
$320$	−221.703			−0.0387298
$321$		0			0
$322$	5628.00	−	4873.99i	0.974025	−	0.843531i
$323$			9516.00i			1.63927i
$324$		0			0
$325$		−	4893.04i		−	0.835130i
$326$		−	5846.00i		−	0.993190i
$327$		0			0
$328$			942.236i			0.158617i
$329$	−8244.56	+	7140.00i	−1.38157	+	1.19648i
$330$		0			0
$331$	−7967.00			−1.32298			−0.661489	−	0.749955i	$-0.730075\pi$
$331$	−7967.00			−1.32298			−0.661489	+	0.749955i	$0.730075\pi$
$332$	928.379			0.153468
$333$		0			0
$334$			3831.30i			0.627662i
$335$	1991.86			0.324856
$336$		0			0
$337$	6181.00			0.999111			0.499556	−	0.866282i	$-0.333497\pi$
$337$	6181.00			0.999111			0.499556	+	0.866282i	$0.333497\pi$
$338$			644.000i			0.103636i
$339$		0			0
$340$	1464.00			0.233519
$341$	−6079.50			−0.965464
$342$		0			0
$343$	−3430.00	−	5346.84i	−0.539949	−	0.841698i
$344$		−	2840.00i		−	0.445124i
$345$		0			0
$346$		−	8272.27i		−	1.28532i
$347$			1746.00i			0.270116i	0.990838	+	0.135058i	$0.0431220\pi$
$347$			1746.00i			0.270116i	−0.990838	+	0.135058i	$0.956878\pi$
$348$		0			0
$349$		−	2890.79i		−	0.443383i	−0.975117	−	0.221691i	$-0.928842\pi$
$349$		−	2890.79i		−	0.443383i	0.975117	−	0.221691i	$-0.0711578\pi$
$350$	−2740.10	−	3164.00i	−0.418471	−	0.483208i
$351$		0			0
$352$	960.000			0.145364
$353$	−510.955			−0.0770408			−0.0385204	−	0.999258i	$-0.512264\pi$
$353$	−510.955			−0.0770408			−0.0385204	+	0.999258i	$0.512264\pi$
$354$		0			0
$355$		−	550.792i		−	0.0823465i
$356$	−2265.52			−0.337282
$357$		0			0
$358$	2220.00			0.327739
$359$		−	2673.00i		−	0.392968i	−0.980507	−	0.196484i	$-0.937048\pi$
$359$		−	2673.00i		−	0.392968i	0.980507	−	0.196484i	$-0.0629524\pi$
$360$		0			0
$361$	−1253.00			−0.182680
$362$	−8234.17			−1.19552
$363$		0			0
$364$	−2100.00	−	2424.87i	−0.302390	−	0.349170i
$365$		−	168.000i		−	0.0240918i
$366$		0			0
$367$		−	3957.74i		−	0.562921i	−0.959573	−	0.281461i	$-0.909181\pi$
$367$		−	3957.74i		−	0.562921i	0.959573	−	0.281461i	$-0.0908189\pi$
$368$		−	3216.00i		−	0.455559i
$369$		0			0
$370$			2979.13i			0.418588i
$371$	−400.104	−	462.000i	−0.0559902	−	0.0646519i
$372$		0			0
$373$	6572.00			0.912293			0.456146	−	0.889905i	$-0.349229\pi$
$373$	6572.00			0.912293			0.456146	+	0.889905i	$0.349229\pi$
$374$	−6339.31			−0.876464
$375$		0			0
$376$			4711.18i			0.646171i
$377$	−4806.44			−0.656616
$378$		0			0
$379$	520.000			0.0704765			0.0352383	−	0.999379i	$-0.488781\pi$
$379$	520.000			0.0704765			0.0352383	+	0.999379i	$0.488781\pi$
$380$			1248.00i			0.168476i
$381$		0			0
$382$	−9024.00			−1.20866
$383$	9411.96			1.25569			0.627845	−	0.778339i	$-0.283937\pi$
$383$	9411.96			1.25569			0.627845	+	0.778339i	$0.283937\pi$
$384$		0			0
$385$	−1260.00	−	1454.92i	−0.166794	−	0.192597i
$386$			1994.00i			0.262932i
$387$		0			0
$388$		−	2521.87i		−	0.329970i
$389$			354.000i			0.0461401i	0.999734	+	0.0230701i	$0.00734408\pi$
$389$			354.000i			0.0461401i	−0.999734	+	0.0230701i	$0.992656\pi$
$390$		0			0
$391$			21236.7i			2.74676i
$392$	−2715.86	−	392.000i	−0.349927	−	0.0505076i
$393$		0			0
$394$	708.000			0.0905293
$395$	−1302.50			−0.165914
$396$		0			0
$397$		−	3200.83i		−	0.404647i	−0.979319	−	0.202324i	$-0.935151\pi$
$397$		−	3200.83i		−	0.404647i	0.979319	−	0.202324i	$-0.0648493\pi$
$398$	10014.7			1.26129
$399$		0			0
$400$	−1808.00			−0.226000
$401$			4914.00i			0.611954i	0.952039	+	0.305977i	$0.0989830\pi$
$401$			4914.00i			0.611954i	−0.952039	+	0.305977i	$0.901017\pi$
$402$		0			0
$403$	8775.00			1.08465
$404$	1496.49			0.184290
$405$		0			0
$406$	−3108.00	+	2691.61i	−0.379920	+	0.329020i
$407$		−	12900.0i		−	1.57108i
$408$		0			0
$409$		−	8684.50i		−	1.04993i	−0.851124	−	0.524965i	$-0.824079\pi$
$409$		−	8684.50i		−	1.04993i	0.851124	−	0.524965i	$-0.175921\pi$
$410$		−	816.000i		−	0.0982911i
$411$		0			0
$412$			1087.73i			0.130069i
$413$	8171.82	−	7077.00i	0.973629	−	0.843187i
$414$		0			0
$415$	−804.000			−0.0951007
$416$	−1385.64			−0.163309
$417$		0			0
$418$		−	5404.00i		−	0.632340i
$419$	−10788.9			−1.25793			−0.628967	−	0.777432i	$-0.716522\pi$
$419$	−10788.9			−1.25793			−0.628967	+	0.777432i	$0.716522\pi$
$420$		0			0
$421$	4112.00			0.476025			0.238013	−	0.971262i	$-0.423504\pi$
$421$	4112.00			0.476025			0.238013	+	0.971262i	$0.423504\pi$
$422$			3314.00i			0.382282i
$423$		0			0
$424$	−264.000			−0.0302381
$425$	11939.0			1.36265
$426$		0			0
$427$	−8400.00	−	9699.48i	−0.952001	−	1.09928i
$428$			3456.00i			0.390309i
$429$		0			0
$430$			2459.51i			0.275833i
$431$			12396.0i			1.38537i	0.721240	+	0.692685i	$0.243572\pi$
$431$			12396.0i			1.38537i	−0.721240	+	0.692685i	$0.756428\pi$
$432$		0			0
$433$			12034.3i			1.33564i	0.744324	+	0.667819i	$0.232772\pi$
$433$			12034.3i			1.33564i	−0.744324	+	0.667819i	$0.767228\pi$
$434$	5674.20	−	4914.00i	0.627581	−	0.543501i
$435$		0			0
$436$	560.000			0.0615118
$437$	−18103.4			−1.98170
$438$		0			0
$439$		−	10013.0i		−	1.08860i	−0.838892	−	0.544298i	$-0.816796\pi$
$439$		−	10013.0i		−	1.08860i	0.838892	−	0.544298i	$-0.183204\pi$
$440$	−831.384			−0.0900789
$441$		0			0
$442$	9150.00			0.984663
$443$		−	3834.00i		−	0.411194i	−0.978637	−	0.205597i	$-0.934086\pi$
$443$		−	3834.00i		−	0.411194i	0.978637	−	0.205597i	$-0.0659136\pi$
$444$		0			0
$445$	1962.00			0.209006
$446$	−11951.2			−1.26884
$447$		0			0
$448$	−896.000	+	775.959i	−0.0944911	+	0.0818317i
$449$		−	5910.00i		−	0.621181i	−0.950544	−	0.310590i	$-0.899473\pi$
$449$		−	5910.00i		−	0.621181i	0.950544	−	0.310590i	$-0.100527\pi$
$450$		0			0
$451$			3533.38i			0.368915i
$452$			2328.00i			0.242256i
$453$		0			0
$454$		−	10.3923i		−	0.00107431i
$455$	1818.65	+	2100.00i	0.187384	+	0.216373i
$456$		0			0
$457$	−15329.0			−1.56906			−0.784530	−	0.620091i	$-0.787096\pi$
$457$	−15329.0			−1.56906			−0.784530	+	0.620091i	$0.787096\pi$
$458$	−10517.0			−1.07299
$459$		0			0
$460$			2785.14i			0.282299i
$461$	−5771.19			−0.583061			−0.291531	−	0.956561i	$-0.594165\pi$
$461$	−5771.19			−0.583061			−0.291531	+	0.956561i	$0.594165\pi$
$462$		0			0
$463$	−9124.00			−0.915828			−0.457914	−	0.888997i	$-0.651403\pi$
$463$	−9124.00			−0.915828			−0.457914	+	0.888997i	$0.651403\pi$
$464$			1776.00i			0.177691i
$465$		0			0
$466$	−5568.00			−0.553503
$467$	11379.6			1.12759			0.563794	−	0.825915i	$-0.309341\pi$
$467$	11379.6			1.12759			0.563794	+	0.825915i	$0.309341\pi$
$468$		0			0
$469$	8050.00	−	6971.50i	0.792568	−	0.686384i
$470$		−	4080.00i		−	0.400418i
$471$		0			0
$472$		−	4669.61i		−	0.455373i
$473$		−	10650.0i		−	1.03528i
$474$		0			0
$475$			10177.5i			0.983110i
$476$	5916.69	−	5124.00i	0.569729	−	0.493399i
$477$		0			0
$478$	4752.00			0.454710
$479$	10222.6			0.975117			0.487558	−	0.873090i	$-0.337888\pi$
$479$	10222.6			0.975117			0.487558	+	0.873090i	$0.337888\pi$
$480$		0			0
$481$			18619.5i			1.76503i
$482$	8784.96			0.830174
$483$		0			0
$484$	−1724.00			−0.161908
$485$			2184.00i			0.204475i
$486$		0			0
$487$	5932.00			0.551960			0.275980	−	0.961163i	$-0.410998\pi$
$487$	5932.00			0.551960			0.275980	+	0.961163i	$0.410998\pi$
$488$	−5542.56			−0.514139
$489$		0			0
$490$	2352.00	+	339.482i	0.216842	+	0.0312984i
$491$			13302.0i			1.22263i	0.791388	+	0.611315i	$0.209359\pi$
$491$			13302.0i			1.22263i	−0.791388	+	0.611315i	$0.790641\pi$
$492$		0			0
$493$		−	11727.7i		−	1.07138i
$494$			7800.00i			0.710402i
$495$		0			0
$496$		−	3242.40i		−	0.293524i
$497$	−1927.77	−	2226.00i	−0.173989	−	0.200905i
$498$		0			0
$499$	13180.0			1.18240			0.591200	−	0.806525i	$-0.298654\pi$
$499$	13180.0			1.18240			0.591200	+	0.806525i	$0.298654\pi$
$500$	3297.82			0.294966
$501$		0			0
$502$			4787.39i			0.425641i
$503$	10596.7			0.939330			0.469665	−	0.882845i	$-0.344375\pi$
$503$	10596.7			0.939330			0.469665	+	0.882845i	$0.344375\pi$
$504$		0			0
$505$	−1296.00			−0.114200
$506$		−	12060.0i		−	1.05955i
$507$		0			0
$508$	−776.000			−0.0677745
$509$	−1375.25			−0.119758			−0.0598790	−	0.998206i	$-0.519071\pi$
$509$	−1375.25			−0.119758			−0.0598790	+	0.998206i	$0.519071\pi$
$510$		0			0
$511$	−588.000	−	678.964i	−0.0509033	−	0.0587781i
$512$			512.000i			0.0441942i
$513$		0			0
$514$			2702.00i			0.231868i
$515$		−	942.000i		−	0.0806009i
$516$		0			0
$517$			17666.9i			1.50288i
$518$	10426.9	+	12040.0i	0.884428	+	1.02125i
$519$		0			0
$520$	1200.00			0.101199
$521$	−18226.4			−1.53265			−0.766326	−	0.642452i	$-0.777917\pi$
$521$	−18226.4			−1.53265			−0.766326	+	0.642452i	$0.777917\pi$
$522$		0			0
$523$			19825.1i			1.65753i	0.559596	+	0.828766i	$0.310956\pi$
$523$			19825.1i			1.65753i	−0.559596	+	0.828766i	$0.689044\pi$
$524$	3886.72			0.324031
$525$		0			0
$526$	−13590.0			−1.12653
$527$			21411.0i			1.76979i
$528$		0			0
$529$	−28234.0			−2.32054
$530$	228.631			0.0187379
$531$		0			0
$532$	4368.00	+	5043.73i	0.355971	+	0.411040i
$533$		−	5100.00i		−	0.414457i
$534$		0			0
$535$		−	2992.98i		−	0.241865i
$536$		−	4600.00i		−	0.370690i
$537$		0			0
$538$			13544.6i			1.08541i
$539$	−10184.5	−	1470.00i	−0.813870	−	0.117472i
$540$		0			0
$541$	−18610.0			−1.47894			−0.739470	−	0.673190i	$-0.764924\pi$
$541$	−18610.0			−1.47894			−0.739470	+	0.673190i	$0.764924\pi$
$542$	7562.13			0.599302
$543$		0			0
$544$		−	3380.96i		−	0.266466i
$545$	−484.974			−0.0381175
$546$		0			0
$547$	−160.000			−0.0125066			−0.00625330	−	0.999980i	$-0.501990\pi$
$547$	−160.000			−0.0125066			−0.00625330	+	0.999980i	$0.501990\pi$
$548$		−	3744.00i		−	0.291854i
$549$		0			0
$550$	−6780.00			−0.525637
$551$	9997.40			0.772965
$552$		0			0
$553$	−5264.00	+	4558.76i	−0.404789	+	0.350557i
$554$		−	296.000i		−	0.0227001i
$555$		0			0
$556$		−	7912.01i		−	0.603496i
$557$			14757.0i			1.12257i	0.827621	+	0.561287i	$0.189694\pi$
$557$			14757.0i			1.12257i	−0.827621	+	0.561287i	$0.810306\pi$
$558$		0			0
$559$			15372.0i			1.16308i
$560$	775.959	−	672.000i	0.0585540	−	0.0507093i
$561$		0			0
$562$	−10140.0			−0.761086
$563$	3559.36			0.266446			0.133223	−	0.991086i	$-0.457467\pi$
$563$	3559.36			0.266446			0.133223	+	0.991086i	$0.457467\pi$
$564$		0			0
$565$		−	2016.11i		−	0.150121i
$566$	5029.88			0.373536
$567$		0			0
$568$	−1272.00			−0.0939647
$569$		−	13734.0i		−	1.01188i	−0.862569	−	0.505940i	$-0.831146\pi$
$569$		−	13734.0i		−	1.01188i	0.862569	−	0.505940i	$-0.168854\pi$
$570$		0			0
$571$	−6145.00			−0.450368			−0.225184	−	0.974316i	$-0.572298\pi$
$571$	−6145.00			−0.450368			−0.225184	+	0.974316i	$0.572298\pi$
$572$	−5196.15			−0.379829
$573$		0			0
$574$	−2856.00	−	3297.82i	−0.207678	−	0.239806i
$575$			22713.0i			1.64730i
$576$		0			0
$577$		−	2698.54i		−	0.194699i	−0.995250	−	0.0973496i	$-0.968963\pi$
$577$		−	2698.54i		−	0.194699i	0.995250	−	0.0973496i	$-0.0310365\pi$
$578$			12500.0i			0.899535i
$579$		0			0
$580$		−	1538.06i		−	0.110111i
$581$	−3249.33	+	2814.00i	−0.232022	+	0.200937i
$582$		0			0
$583$	−990.000			−0.0703287
$584$	−387.979			−0.0274909
$585$		0			0
$586$			34.6410i			0.00244199i
$587$	10355.9			0.728169			0.364084	−	0.931366i	$-0.381382\pi$
$587$	10355.9			0.728169			0.364084	+	0.931366i	$0.381382\pi$
$588$		0			0
$589$	−18252.0			−1.27684
$590$			4044.00i			0.282184i
$591$		0			0
$592$	6880.00			0.477646
$593$	7281.54			0.504245			0.252122	−	0.967695i	$-0.418871\pi$
$593$	7281.54			0.504245			0.252122	+	0.967695i	$0.418871\pi$
$594$		0			0
$595$	−5124.00	+	4437.51i	−0.353048	+	0.305748i
$596$			10452.0i			0.718340i
$597$		0			0
$598$			17407.1i			1.19035i
$599$		−	6291.00i		−	0.429121i	−0.976711	−	0.214560i	$-0.931168\pi$
$599$		−	6291.00i		−	0.429121i	0.976711	−	0.214560i	$-0.0688319\pi$
$600$		0			0
$601$			10901.5i			0.739904i	0.929051	+	0.369952i	$0.120626\pi$
$601$			10901.5i			0.739904i	−0.929051	+	0.369952i	$0.879374\pi$
$602$	8608.29	+	9940.00i	0.582804	+	0.672964i
$603$		0			0
$604$	7424.00			0.500130
$605$	1493.03			0.100331
$606$		0			0
$607$		−	20523.1i		−	1.37233i	−0.727445	−	0.686166i	$-0.759292\pi$
$607$		−	20523.1i		−	1.37233i	0.727445	−	0.686166i	$-0.240708\pi$
$608$	2882.13			0.192247
$609$		0			0
$610$	4800.00			0.318601
$611$		−	25500.0i		−	1.68841i
$612$		0			0
$613$	17036.0			1.12248			0.561238	−	0.827655i	$-0.310325\pi$
$613$	17036.0			1.12248			0.561238	+	0.827655i	$0.310325\pi$
$614$	3928.29			0.258197
$615$		0			0
$616$	−3360.00	+	2909.85i	−0.219770	+	0.190326i
$617$		−	27450.0i		−	1.79108i	−0.444983	−	0.895539i	$-0.646790\pi$
$617$		−	27450.0i		−	1.79108i	0.444983	−	0.895539i	$-0.353210\pi$
$618$		0			0
$619$			13277.9i			0.862171i	0.902311	+	0.431086i	$0.141869\pi$
$619$			13277.9i			0.862171i	−0.902311	+	0.431086i	$0.858131\pi$
$620$			2808.00i			0.181890i
$621$		0			0
$622$			18235.0i			1.17550i
$623$	7929.33	−	6867.00i	0.509923	−	0.441606i
$624$		0			0
$625$	11269.0			0.721216
$626$	−9034.38			−0.576815
$627$		0			0
$628$		−	6256.17i		−	0.397529i
$629$	−45431.7			−2.87994
$630$		0			0
$631$	−24730.0			−1.56020			−0.780100	−	0.625655i	$-0.784832\pi$
$631$	−24730.0			−1.56020			−0.780100	+	0.625655i	$0.784832\pi$
$632$			3008.00i			0.189322i
$633$		0			0
$634$	17172.0			1.07569
$635$	672.036			0.0419983
$636$		0			0
$637$	14700.0	+	2121.76i	0.914341	+	0.131974i
$638$			6660.00i			0.413279i
$639$		0			0
$640$		−	443.405i		−	0.0273861i
$641$		−	31320.0i		−	1.92990i	−0.262435	−	0.964950i	$-0.584525\pi$
$641$		−	31320.0i		−	1.92990i	0.262435	−	0.964950i	$-0.415475\pi$
$642$		0			0
$643$		−	21148.3i		−	1.29706i	−0.761189	−	0.648530i	$-0.775384\pi$
$643$		−	21148.3i		−	1.29706i	0.761189	−	0.648530i	$-0.224616\pi$
$644$	9747.98	+	11256.0i	0.596466	+	0.688740i
$645$		0			0
$646$	−19032.0			−1.15914
$647$	−14414.1			−0.875854			−0.437927	−	0.899010i	$-0.644287\pi$
$647$	−14414.1			−0.875854			−0.437927	+	0.899010i	$0.644287\pi$
$648$		0			0
$649$		−	17511.0i		−	1.05912i
$650$	9786.09			0.590526
$651$		0			0
$652$	11692.0			0.702292
$653$			12321.0i			0.738374i	0.929355	+	0.369187i	$0.120364\pi$
$653$			12321.0i			0.738374i	−0.929355	+	0.369187i	$0.879636\pi$
$654$		0			0
$655$	−3366.00			−0.200795
$656$	−1884.47			−0.112159
$657$		0			0
$658$	−14280.0	−	16489.1i	−0.846037	−	0.976919i
$659$			8544.00i			0.505049i	0.967591	+	0.252524i	$0.0812608\pi$
$659$			8544.00i			0.505049i	−0.967591	+	0.252524i	$0.918739\pi$
$660$		0			0
$661$			9650.99i			0.567897i	0.958840	+	0.283948i	$0.0916445\pi$
$661$			9650.99i			0.567897i	−0.958840	+	0.283948i	$0.908356\pi$
$662$		−	15934.0i		−	0.935487i
$663$		0			0
$664$			1856.76i			0.108518i
$665$	−3782.80	−	4368.00i	−0.220587	−	0.254712i
$666$		0			0
$667$	22311.0			1.29518
$668$	−7662.59			−0.443824
$669$		0			0
$670$			3983.72i			0.229708i
$671$	−20784.6			−1.19580
$672$		0			0
$673$	27631.0			1.58261			0.791305	−	0.611421i	$-0.209402\pi$
$673$	27631.0			1.58261			0.791305	+	0.611421i	$0.209402\pi$
$674$			12362.0i			0.706478i
$675$		0			0
$676$	−1288.00			−0.0732817
$677$	22831.9			1.29616			0.648080	−	0.761572i	$-0.275572\pi$
$677$	22831.9			1.29616			0.648080	+	0.761572i	$0.275572\pi$
$678$		0			0
$679$	7644.00	+	8826.53i	0.432032	+	0.498868i
$680$			2928.00i			0.165123i
$681$		0			0
$682$		−	12159.0i		−	0.682686i
$683$			5046.00i			0.282694i	0.989960	+	0.141347i	$0.0451433\pi$
$683$			5046.00i			0.282694i	−0.989960	+	0.141347i	$0.954857\pi$
$684$		0			0
$685$			3242.40i			0.180855i
$686$	10693.7	−	6860.00i	0.595170	−	0.381802i
$687$		0			0
$688$	5680.00			0.314750
$689$	1428.94			0.0790107
$690$		0			0
$691$			18363.2i			1.01095i	0.862840	+	0.505477i	$0.168683\pi$
$691$			18363.2i			1.01095i	−0.862840	+	0.505477i	$0.831317\pi$
$692$	16544.5			0.908858
$693$		0			0
$694$	−3492.00			−0.191001
$695$			6852.00i			0.373973i
$696$		0			0
$697$	12444.0			0.676256
$698$	5781.59			0.313519
$699$		0			0
$700$	6328.00	−	5480.21i	0.341680	−	0.295903i
$701$		−	20262.0i		−	1.09170i	−0.837881	−	0.545852i	$-0.816206\pi$
$701$		−	20262.0i		−	1.09170i	0.837881	−	0.545852i	$-0.183794\pi$
$702$		0			0
$703$		−	38728.7i		−	2.07778i
$704$			1920.00i			0.102788i
$705$		0			0
$706$		−	1021.91i		−	0.0544760i
$707$	−5237.72	+	4536.00i	−0.278621	+	0.241293i
$708$		0			0
$709$	−15446.0			−0.818176			−0.409088	−	0.912495i	$-0.634153\pi$
$709$	−15446.0			−0.818176			−0.409088	+	0.912495i	$0.634153\pi$
$710$	1101.58			0.0582278
$711$		0			0
$712$		−	4531.04i		−	0.238495i
$713$	−40732.6			−2.13948
$714$		0			0
$715$	4500.00			0.235371
$716$			4440.00i			0.231747i
$717$		0			0
$718$	5346.00			0.277870
$719$	28606.6			1.48379			0.741895	−	0.670517i	$-0.233927\pi$
$719$	28606.6			1.48379			0.741895	+	0.670517i	$0.233927\pi$
$720$		0			0
$721$	−3297.00	−	3807.05i	−0.170301	−	0.196646i
$722$		−	2506.00i		−	0.129174i
$723$		0			0
$724$		−	16468.3i		−	0.845360i
$725$		−	12543.0i		−	0.642531i
$726$		0			0
$727$		−	5918.42i		−	0.301928i	−0.988539	−	0.150964i	$-0.951762\pi$
$727$		−	5918.42i		−	0.301928i	0.988539	−	0.150964i	$-0.0482378\pi$
$728$	4849.74	−	4200.00i	0.246900	−	0.213822i
$729$		0			0
$730$	336.000			0.0170355
$731$	−37507.6			−1.89777
$732$		0			0
$733$		−	9150.42i		−	0.461090i	−0.973062	−	0.230545i	$-0.925949\pi$
$733$		−	9150.42i		−	0.461090i	0.973062	−	0.230545i	$-0.0740508\pi$
$734$	7915.47			0.398046
$735$		0			0
$736$	6432.00			0.322129
$737$		−	17250.0i		−	0.862160i
$738$		0			0
$739$	−4876.00			−0.242715			−0.121358	−	0.992609i	$-0.538725\pi$
$739$	−4876.00			−0.242715			−0.121358	+	0.992609i	$0.538725\pi$
$740$	−5958.25			−0.295986
$741$		0			0
$742$	924.000	−	800.207i	0.0457158	−	0.0395910i
$743$		−	9171.00i		−	0.452828i	−0.974031	−	0.226414i	$-0.927300\pi$
$743$		−	9171.00i		−	0.452828i	0.974031	−	0.226414i	$-0.0727003\pi$
$744$		0			0
$745$		−	9051.70i		−	0.445139i
$746$			13144.0i			0.645089i
$747$		0			0
$748$		−	12678.6i		−	0.619754i
$749$	−10475.4	−	12096.0i	−0.511034	−	0.590091i
$750$		0			0
$751$	−20942.0			−1.01756			−0.508778	−	0.860898i	$-0.669903\pi$
$751$	−20942.0			−1.01756			−0.508778	+	0.860898i	$0.669903\pi$
$752$	−9422.36			−0.456912
$753$		0			0
$754$		−	9612.88i		−	0.464298i
$755$	−6429.37			−0.309919
$756$		0			0
$757$	3022.00			0.145094			0.0725472	−	0.997365i	$-0.476887\pi$
$757$	3022.00			0.145094			0.0725472	+	0.997365i	$0.476887\pi$
$758$			1040.00i			0.0498344i
$759$		0			0
$760$	−2496.00			−0.119131
$761$	5031.61			0.239679			0.119839	−	0.992793i	$-0.461762\pi$
$761$	5031.61			0.239679			0.119839	+	0.992793i	$0.461762\pi$
$762$		0			0
$763$	−1960.00	+	1697.41i	−0.0929971	+	0.0805378i
$764$		−	18048.0i		−	0.854651i
$765$		0			0
$766$			18823.9i			0.887906i
$767$			25275.0i			1.18987i
$768$		0			0
$769$		−	8767.64i		−	0.411143i	−0.978642	−	0.205572i	$-0.934095\pi$
$769$		−	8767.64i		−	0.411143i	0.978642	−	0.205572i	$-0.0659054\pi$
$770$	2909.85	−	2520.00i	0.136186	−	0.117941i
$771$		0			0
$772$	−3988.00			−0.185921
$773$	28603.1			1.33089			0.665447	−	0.746445i	$-0.268241\pi$
$773$	28603.1			1.33089			0.665447	+	0.746445i	$0.268241\pi$
$774$		0			0
$775$			22899.4i			1.06138i
$776$	5043.73			0.233324
$777$		0			0
$778$	−708.000			−0.0326260
$779$			10608.0i			0.487896i
$780$		0			0
$781$	−4770.00			−0.218545
$782$	−42473.3			−1.94226
$783$		0			0
$784$	784.000	−	5431.71i	0.0357143	−	0.247436i
$785$			5418.00i			0.246340i
$786$		0			0
$787$		−	16277.8i		−	0.737283i	−0.929572	−	0.368641i	$-0.879823\pi$
$787$		−	16277.8i		−	0.737283i	0.929572	−	0.368641i	$-0.120177\pi$
$788$			1416.00i			0.0640138i
$789$		0			0
$790$		−	2605.00i		−	0.117319i
$791$	−7056.37	−	8148.00i	−0.317188	−	0.366257i
$792$		0			0
$793$	30000.0			1.34342
$794$	6401.66			0.286129
$795$		0			0
$796$			20029.4i			0.891865i
$797$	−4084.18			−0.181517			−0.0907584	−	0.995873i	$-0.528929\pi$
$797$	−4084.18			−0.181517			−0.0907584	+	0.995873i	$0.528929\pi$
$798$		0			0
$799$	62220.0			2.75493
$800$		−	3616.00i		−	0.159806i
$801$		0			0
$802$	−9828.00			−0.432717
$803$	−1454.92			−0.0639391
$804$		0			0
$805$	−8442.00	−	9747.98i	−0.369617	−	0.426797i
$806$			17550.0i			0.766963i
$807$		0			0
$808$			2992.98i			0.130313i
$809$			22128.0i			0.961655i	0.876815	+	0.480828i	$0.159664\pi$
$809$			22128.0i			0.961655i	−0.876815	+	0.480828i	$0.840336\pi$
$810$		0			0
$811$		−	39023.1i		−	1.68963i	−0.535062	−	0.844813i	$-0.679712\pi$
$811$		−	39023.1i		−	1.68963i	0.535062	−	0.844813i	$-0.320288\pi$
$812$	−5383.21	−	6216.00i	−0.232652	−	0.268644i
$813$		0			0
$814$	25800.0			1.11092
$815$	−10125.6			−0.435194
$816$		0			0
$817$		−	31973.7i		−	1.36918i
$818$	17369.0			0.742412
$819$		0			0
$820$	1632.00			0.0695023
$821$			4107.00i			0.174586i	0.996183	+	0.0872931i	$0.0278217\pi$
$821$			4107.00i			0.174586i	−0.996183	+	0.0872931i	$0.972178\pi$
$822$		0			0
$823$	−3958.00			−0.167639			−0.0838197	−	0.996481i	$-0.526712\pi$
$823$	−3958.00			−0.167639			−0.0838197	+	0.996481i	$0.526712\pi$
$824$	−2175.46			−0.0919728
$825$		0			0
$826$	14154.0	+	16343.6i	0.596224	+	0.688460i
$827$		−	21048.0i		−	0.885019i	−0.896764	−	0.442509i	$-0.854088\pi$
$827$		−	21048.0i		−	0.885019i	0.896764	−	0.442509i	$-0.145912\pi$
$828$		0			0
$829$			25675.9i			1.07571i	0.843038	+	0.537854i	$0.180765\pi$
$829$			25675.9i			1.07571i	−0.843038	+	0.537854i	$0.819235\pi$
$830$		−	1608.00i		−	0.0672464i
$831$		0			0
$832$		−	2771.28i		−	0.115477i
$833$	−5177.10	+	35868.0i	−0.215337	+	1.49190i
$834$		0			0
$835$	6636.00			0.275028
$836$	10808.0			0.447132
$837$		0			0
$838$		−	21577.9i		−	0.889494i
$839$	18169.2			0.747641			0.373821	−	0.927501i	$-0.378048\pi$
$839$	18169.2			0.747641			0.373821	+	0.927501i	$0.378048\pi$
$840$		0			0
$841$	12068.0			0.494813
$842$			8224.00i			0.336601i
$843$		0			0
$844$	−6628.00			−0.270314
$845$	1115.44			0.0454110
$846$		0			0
$847$	6034.00	−	5225.60i	0.244782	−	0.211988i
$848$		−	528.000i		−	0.0213816i
$849$		0			0
$850$			23878.1i			0.963542i
$851$		−	86430.0i		−	3.48153i
$852$		0			0
$853$		−	11876.7i		−	0.476729i	−0.971176	−	0.238364i	$-0.923389\pi$
$853$		−	11876.7i		−	0.476729i	0.971176	−	0.238364i	$-0.0766112\pi$
$854$	19399.0	−	16800.0i	0.777306	−	0.673166i
$855$		0			0
$856$	−6912.00			−0.275990
$857$	5693.25			0.226929			0.113464	−	0.993542i	$-0.463805\pi$
$857$	5693.25			0.226929			0.113464	+	0.993542i	$0.463805\pi$
$858$		0			0
$859$		−	4887.85i		−	0.194146i	−0.995277	−	0.0970729i	$-0.969052\pi$
$859$		−	4887.85i		−	0.194146i	0.995277	−	0.0970729i	$-0.0309480\pi$
$860$	−4919.02			−0.195043
$861$		0			0
$862$	−24792.0			−0.979604
$863$			10647.0i			0.419963i	0.977705	+	0.209982i	$0.0673404\pi$
$863$			10647.0i			0.419963i	−0.977705	+	0.209982i	$0.932660\pi$
$864$		0			0
$865$	−14328.0			−0.563198
$866$	−24068.6			−0.944438
$867$		0			0
$868$	9828.00	+	11348.4i	0.384313	+	0.443767i
$869$			11280.0i			0.440331i
$870$		0			0
$871$			24898.2i			0.968593i
$872$			1120.00i			0.0434954i
$873$		0			0
$874$		−	36206.8i		−	1.40127i
$875$	−11542.4	+	9996.00i	−0.445947	+	0.386202i
$876$		0			0
$877$	32744.0			1.26076			0.630380	−	0.776287i	$-0.282899\pi$
$877$	32744.0			1.26076			0.630380	+	0.776287i	$0.282899\pi$
$878$	20026.0			0.769754
$879$		0			0
$880$		−	1662.77i		−	0.0636954i
$881$	31483.5			1.20398			0.601990	−	0.798504i	$-0.294375\pi$
$881$	31483.5			1.20398			0.601990	+	0.798504i	$0.294375\pi$
$882$		0			0
$883$	−17201.0			−0.655560			−0.327780	−	0.944754i	$-0.606301\pi$
$883$	−17201.0			−0.655560			−0.327780	+	0.944754i	$0.606301\pi$
$884$			18300.0i			0.696262i
$885$		0			0
$886$	7668.00			0.290758
$887$	34495.5			1.30580			0.652901	−	0.757443i	$-0.273552\pi$
$887$	34495.5			1.30580			0.652901	+	0.757443i	$0.273552\pi$
$888$		0			0
$889$	2716.00	−	2352.12i	0.102465	−	0.0887376i
$890$			3924.00i			0.147790i
$891$		0			0
$892$		−	23902.3i		−	0.897207i
$893$			53040.0i			1.98759i
$894$		0			0
$895$		−	3845.15i		−	0.143608i
$896$	−1551.92	−	1792.00i	−0.0578638	−	0.0668153i
$897$		0			0
$898$	11820.0			0.439241
$899$	22494.1			0.834507
$900$		0			0
$901$			3486.62i			0.128919i
$902$	−7066.77			−0.260862
$903$		0			0
$904$	−4656.00			−0.171301
$905$			14262.0i			0.523851i
$906$		0			0
$907$	38284.0			1.40154			0.700771	−	0.713386i	$-0.252839\pi$
$907$	38284.0			1.40154			0.700771	+	0.713386i	$0.252839\pi$
$908$	20.7846			0.000759649
$909$		0			0
$910$	−4200.00	+	3637.31i	−0.152999	+	0.132501i
$911$			23232.0i			0.844907i	0.906385	+	0.422454i	$0.138831\pi$
$911$			23232.0i			0.844907i	−0.906385	+	0.422454i	$0.861169\pi$
$912$		0			0
$913$			6962.84i			0.252395i
$914$		−	30658.0i		−	1.10949i
$915$		0			0
$916$		−	21034.0i		−	0.758716i
$917$	−13603.5	+	11781.0i	−0.489889	+	0.424256i
$918$		0			0
$919$	−22208.0			−0.797143			−0.398571	−	0.917137i	$-0.630494\pi$
$919$	−22208.0			−0.797143			−0.398571	+	0.917137i	$0.630494\pi$
$920$	−5570.28			−0.199616
$921$		0			0
$922$		−	11542.4i		−	0.412287i
$923$	6884.90			0.245525
$924$		0			0
$925$	−48590.0			−1.72717
$926$		−	18248.0i		−	0.647588i
$927$		0			0
$928$	−3552.00			−0.125647
$929$	−35881.2			−1.26719			−0.633597	−	0.773663i	$-0.718422\pi$
$929$	−35881.2			−1.26719			−0.633597	+	0.773663i	$0.718422\pi$
$930$		0			0
$931$	−30576.0	−	4413.27i	−1.07636	−	0.155359i
$932$		−	11136.0i		−	0.391386i
$933$		0			0
$934$			22759.1i			0.797326i
$935$			10980.0i			0.384047i
$936$		0			0
$937$		−	25042.0i		−	0.873091i	−0.899682	−	0.436545i	$-0.856202\pi$
$937$		−	25042.0i		−	0.873091i	0.899682	−	0.436545i	$-0.143798\pi$
$938$	13943.0	+	16100.0i	0.485347	+	0.560430i
$939$		0			0
$940$	8160.00			0.283138
$941$	27162.0			0.940974			0.470487	−	0.882407i	$-0.344078\pi$
$941$	27162.0			0.940974			0.470487	+	0.882407i	$0.344078\pi$
$942$		0			0
$943$			23673.7i			0.817519i
$944$	9339.22			0.321998
$945$		0			0
$946$	21300.0			0.732054
$947$		−	40044.0i		−	1.37408i	−0.726619	−	0.687041i	$-0.758909\pi$
$947$		−	40044.0i		−	1.37408i	0.726619	−	0.687041i	$-0.241091\pi$
$948$		0			0
$949$	2100.00			0.0718323
$950$	−20355.1			−0.695164
$951$		0			0
$952$	10248.0	+	11833.4i	0.348886	+	0.402859i
$953$			31914.0i			1.08478i	0.840127	+	0.542390i	$0.182481\pi$
$953$			31914.0i			1.08478i	−0.840127	+	0.542390i	$0.817519\pi$
$954$		0			0
$955$			15630.0i			0.529608i
$956$			9504.00i			0.321529i
$957$		0			0
$958$			20445.1i			0.689512i
$959$	11348.4	+	13104.0i	0.382126	+	0.441241i
$960$		0			0
$961$	−11276.0			−0.378504
$962$	−37239.1			−1.24806
$963$		0			0
$964$			17569.9i			0.587022i
$965$	3453.71			0.115211
$966$		0			0
$967$	−13114.0			−0.436109			−0.218055	−	0.975937i	$-0.569971\pi$
$967$	−13114.0			−0.436109			−0.218055	+	0.975937i	$0.569971\pi$
$968$		−	3448.00i		−	0.114486i
$969$		0			0
$970$	−4368.00			−0.144586
$971$	−2523.60			−0.0834049			−0.0417024	−	0.999130i	$-0.513278\pi$
$971$	−2523.60			−0.0834049			−0.0417024	+	0.999130i	$0.513278\pi$
$972$		0			0
$973$	23982.0	+	27692.0i	0.790162	+	0.912400i
$974$			11864.0i			0.390295i
$975$		0			0
$976$		−	11085.1i		−	0.363551i
$977$		−	23376.0i		−	0.765470i	−0.923858	−	0.382735i	$-0.874982\pi$
$977$		−	23376.0i		−	0.765470i	0.923858	−	0.382735i	$-0.125018\pi$
$978$		0			0
$979$		−	16991.4i		−	0.554697i
$980$	−678.964	+	4704.00i	−0.0221313	+	0.153330i
$981$		0			0
$982$	−26604.0			−0.864529
$983$	7167.23			0.232552			0.116276	−	0.993217i	$-0.462904\pi$
$983$	7167.23			0.232552			0.116276	+	0.993217i	$0.462904\pi$
$984$		0			0
$985$		−	1226.29i		−	0.0396679i
$986$	23455.4			0.757579
$987$		0			0
$988$	−15600.0			−0.502330
$989$		−	71355.0i		−	2.29419i
$990$		0			0
$991$	34954.0			1.12043			0.560217	−	0.828346i	$-0.310718\pi$
$991$	34954.0			1.12043			0.560217	+	0.828346i	$0.310718\pi$
$992$	6484.80			0.207553
$993$		0			0
$994$	4452.00	−	3855.55i	0.142061	−	0.123029i
$995$		−	17346.0i		−	0.552669i
$996$		0			0
$997$			1823.85i			0.0579357i	0.999580	+	0.0289679i	$0.00922204\pi$
$997$			1823.85i			0.0579357i	−0.999580	+	0.0289679i	$0.990778\pi$
$998$			26360.0i			0.836083i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.d.b.377.4	yes	4
3.2	odd	2	inner	378.4.d.b.377.1	✓	4
7.6	odd	2	inner	378.4.d.b.377.3	yes	4
21.20	even	2	inner	378.4.d.b.377.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.d.b.377.1	✓	4	3.2	odd	2	inner
378.4.d.b.377.2	yes	4	21.20	even	2	inner
378.4.d.b.377.3	yes	4	7.6	odd	2	inner
378.4.d.b.377.4	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -4.00000 q^{4} +3.46410 q^{5} +(14.0000 - 12.1244i) q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+2.00000i q^{2} -4.00000 q^{4} +3.46410 q^{5} +(14.0000 - 12.1244i) q^{7} -8.00000i q^{8} +6.92820i q^{10} -30.0000i q^{11} +43.3013i q^{13} +(24.2487 + 28.0000i) q^{14} +16.0000 q^{16} -105.655 q^{17} -90.0666i q^{19} -13.8564 q^{20} +60.0000 q^{22} -201.000i q^{23} -113.000 q^{25} -86.6025 q^{26} +(-56.0000 + 48.4974i) q^{28} +111.000i q^{29} -202.650i q^{31} +32.0000i q^{32} -211.310i q^{34} +(48.4974 - 42.0000i) q^{35} +430.000 q^{37} +180.133 q^{38} -27.7128i q^{40} -117.779 q^{41} +355.000 q^{43} +120.000i q^{44} +402.000 q^{46} -588.897 q^{47} +(49.0000 - 339.482i) q^{49} -226.000i q^{50} -173.205i q^{52} -33.0000i q^{53} -103.923i q^{55} +(-96.9948 - 112.000i) q^{56} -222.000 q^{58} +583.701 q^{59} -692.820i q^{61} +405.300 q^{62} -64.0000 q^{64} +150.000i q^{65} +575.000 q^{67} +422.620 q^{68} +(84.0000 + 96.9948i) q^{70} -159.000i q^{71} -48.4974i q^{73} +860.000i q^{74} +360.267i q^{76} +(-363.731 - 420.000i) q^{77} -376.000 q^{79} +55.4256 q^{80} -235.559i q^{82} -232.095 q^{83} -366.000 q^{85} +710.000i q^{86} -240.000 q^{88} +566.381 q^{89} +(525.000 + 606.218i) q^{91} +804.000i q^{92} -1177.79i q^{94} -312.000i q^{95} +630.466i q^{97} +(678.964 + 98.0000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4} + 56 q^{7}+O(q^{10}) \) 4 * q - 16 * q^4 + 56 * q^7	\( 4 q - 16 q^{4} + 56 q^{7} + 64 q^{16} + 240 q^{22} - 452 q^{25} - 224 q^{28} + 1720 q^{37} + 1420 q^{43} + 1608 q^{46} + 196 q^{49} - 888 q^{58} - 256 q^{64} + 2300 q^{67} + 336 q^{70} - 1504 q^{79} - 1464 q^{85} - 960 q^{88} + 2100 q^{91}+O(q^{100}) \) 4 * q - 16 * q^4 + 56 * q^7 + 64 * q^16 + 240 * q^22 - 452 * q^25 - 224 * q^28 + 1720 * q^37 + 1420 * q^43 + 1608 * q^46 + 196 * q^49 - 888 * q^58 - 256 * q^64 + 2300 * q^67 + 336 * q^70 - 1504 * q^79 - 1464 * q^85 - 960 * q^88 + 2100 * q^91

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