LMFDB - Embedded newform 150.4.c.b.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,4,Mod(49,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("150.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	150.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.85028650086$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	150.49
Dual form			150.4.c.b.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000i q^{2} -3.00000i q^{3} -4.00000 q^{4} -6.00000 q^{6} -23.0000i q^{7} +8.00000i q^{8} -9.00000 q^{9} -30.0000 q^{11} +12.0000i q^{12} +29.0000i q^{13} -46.0000 q^{14} +16.0000 q^{16} -78.0000i q^{17} +18.0000i q^{18} -149.000 q^{19} -69.0000 q^{21} +60.0000i q^{22} +150.000i q^{23} +24.0000 q^{24} +58.0000 q^{26} +27.0000i q^{27} +92.0000i q^{28} +234.000 q^{29} -217.000 q^{31} -32.0000i q^{32} +90.0000i q^{33} -156.000 q^{34} +36.0000 q^{36} -146.000i q^{37} +298.000i q^{38} +87.0000 q^{39} -156.000 q^{41} +138.000i q^{42} -433.000i q^{43} +120.000 q^{44} +300.000 q^{46} -30.0000i q^{47} -48.0000i q^{48} -186.000 q^{49} -234.000 q^{51} -116.000i q^{52} -552.000i q^{53} +54.0000 q^{54} +184.000 q^{56} +447.000i q^{57} -468.000i q^{58} +270.000 q^{59} +275.000 q^{61} +434.000i q^{62} +207.000i q^{63} -64.0000 q^{64} +180.000 q^{66} -803.000i q^{67} +312.000i q^{68} +450.000 q^{69} +660.000 q^{71} -72.0000i q^{72} -646.000i q^{73} -292.000 q^{74} +596.000 q^{76} +690.000i q^{77} -174.000i q^{78} -992.000 q^{79} +81.0000 q^{81} +312.000i q^{82} -846.000i q^{83} +276.000 q^{84} -866.000 q^{86} -702.000i q^{87} -240.000i q^{88} +1488.00 q^{89} +667.000 q^{91} -600.000i q^{92} +651.000i q^{93} -60.0000 q^{94} -96.0000 q^{96} +319.000i q^{97} +372.000i q^{98} +270.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} - 12 q^{6} - 18 q^{9} - 60 q^{11} - 92 q^{14} + 32 q^{16} - 298 q^{19} - 138 q^{21} + 48 q^{24} + 116 q^{26} + 468 q^{29} - 434 q^{31} - 312 q^{34} + 72 q^{36} + 174 q^{39} - 312 q^{41} + 240 q^{44}+ \cdots + 540 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 - 12 * q^6 - 18 * q^9 - 60 * q^11 - 92 * q^14 + 32 * q^16 - 298 * q^19 - 138 * q^21 + 48 * q^24 + 116 * q^26 + 468 * q^29 - 434 * q^31 - 312 * q^34 + 72 * q^36 + 174 * q^39 - 312 * q^41 + 240 * q^44 + 600 * q^46 - 372 * q^49 - 468 * q^51 + 108 * q^54 + 368 * q^56 + 540 * q^59 + 550 * q^61 - 128 * q^64 + 360 * q^66 + 900 * q^69 + 1320 * q^71 - 584 * q^74 + 1192 * q^76 - 1984 * q^79 + 162 * q^81 + 552 * q^84 - 1732 * q^86 + 2976 * q^89 + 1334 * q^91 - 120 * q^94 - 192 * q^96 + 540 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/150\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−6.00000	−0.408248
$7$	− 23.0000i	− 1.24188i	−0.783857	−	0.620942i	$-0.786750\pi$
$7$	− 23.0000i	− 1.24188i	0.783857	−	0.620942i	$-0.213250\pi$
$8$	8.00000i	0.353553i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−30.0000	−0.822304	−0.411152	−	0.911567i	$-0.634873\pi$
$11$	−30.0000	−0.822304	−0.411152	+	0.911567i	$0.634873\pi$
$12$	12.0000i	0.288675i
$13$	29.0000i	0.618704i	0.950948	+	0.309352i	$0.100112\pi$
$13$	29.0000i	0.618704i	−0.950948	+	0.309352i	$0.899888\pi$
$14$	−46.0000	−0.878144
$15$	0	0
$16$	16.0000	0.250000
$17$	− 78.0000i	− 1.11281i	−0.830911	−	0.556405i	$-0.812180\pi$
$17$	− 78.0000i	− 1.11281i	0.830911	−	0.556405i	$-0.187820\pi$
$18$	18.0000i	0.235702i
$19$	−149.000	−1.79910	−0.899551	−	0.436815i	$-0.856106\pi$
$19$	−149.000	−1.79910	−0.899551	+	0.436815i	$0.856106\pi$
$20$	0	0
$21$	−69.0000	−0.717002
$22$	60.0000i	0.581456i
$23$	150.000i	1.35988i	0.733269	+	0.679938i	$0.237993\pi$
$23$	150.000i	1.35988i	−0.733269	+	0.679938i	$0.762007\pi$
$24$	24.0000	0.204124
$25$	0	0
$26$	58.0000	0.437490
$27$	27.0000i	0.192450i
$28$	92.0000i	0.620942i
$29$	234.000	1.49837	0.749185	−	0.662361i	$-0.230446\pi$
$29$	234.000	1.49837	0.749185	+	0.662361i	$0.230446\pi$
$30$	0	0
$31$	−217.000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−217.000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	− 32.0000i	− 0.176777i
$33$	90.0000i	0.474757i
$34$	−156.000	−0.786876
$35$	0	0
$36$	36.0000	0.166667
$37$	− 146.000i	− 0.648710i	−0.945936	−	0.324355i	$-0.894853\pi$
$37$	− 146.000i	− 0.648710i	0.945936	−	0.324355i	$-0.105147\pi$
$38$	298.000i	1.27216i
$39$	87.0000	0.357209
$40$	0	0
$41$	−156.000	−0.594222	−0.297111	−	0.954843i	$-0.596023\pi$
$41$	−156.000	−0.594222	−0.297111	+	0.954843i	$0.596023\pi$
$42$	138.000i	0.506997i
$43$	− 433.000i	− 1.53563i	−0.640675	−	0.767813i	$-0.721345\pi$
$43$	− 433.000i	− 1.53563i	0.640675	−	0.767813i	$-0.278655\pi$
$44$	120.000	0.411152
$45$	0	0
$46$	300.000	0.961578
$47$	− 30.0000i	− 0.0931053i	−0.998916	−	0.0465527i	$-0.985176\pi$
$47$	− 30.0000i	− 0.0931053i	0.998916	−	0.0465527i	$-0.0148235\pi$
$48$	− 48.0000i	− 0.144338i
$49$	−186.000	−0.542274
$50$	0	0
$51$	−234.000	−0.642481
$52$	− 116.000i	− 0.309352i
$53$	− 552.000i	− 1.43062i	−0.698806	−	0.715312i	$-0.746285\pi$
$53$	− 552.000i	− 1.43062i	0.698806	−	0.715312i	$-0.253715\pi$
$54$	54.0000	0.136083
$55$	0	0
$56$	184.000	0.439072
$57$	447.000i	1.03871i
$58$	− 468.000i	− 1.05951i
$59$	270.000	0.595780	0.297890	−	0.954600i	$-0.403717\pi$
$59$	270.000	0.595780	0.297890	+	0.954600i	$0.403717\pi$
$60$	0	0
$61$	275.000	0.577215	0.288608	−	0.957447i	$-0.406808\pi$
$61$	275.000	0.577215	0.288608	+	0.957447i	$0.406808\pi$
$62$	434.000i	0.889001i
$63$	207.000i	0.413961i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	180.000	0.335704
$67$	− 803.000i	− 1.46421i	−0.681192	−	0.732105i	$-0.738538\pi$
$67$	− 803.000i	− 1.46421i	0.681192	−	0.732105i	$-0.261462\pi$
$68$	312.000i	0.556405i
$69$	450.000	0.785125
$70$	0	0
$71$	660.000	1.10321	0.551603	−	0.834107i	$-0.314016\pi$
$71$	660.000	1.10321	0.551603	+	0.834107i	$0.314016\pi$
$72$	− 72.0000i	− 0.117851i
$73$	− 646.000i	− 1.03573i	−0.855461	−	0.517867i	$-0.826726\pi$
$73$	− 646.000i	− 1.03573i	0.855461	−	0.517867i	$-0.173274\pi$
$74$	−292.000	−0.458707
$75$	0	0
$76$	596.000	0.899551
$77$	690.000i	1.02121i
$78$	− 174.000i	− 0.252585i
$79$	−992.000	−1.41277	−0.706384	−	0.707829i	$-0.749675\pi$
$79$	−992.000	−1.41277	−0.706384	+	0.707829i	$0.749675\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	312.000i	0.420178i
$83$	− 846.000i	− 1.11880i	−0.828897	−	0.559401i	$-0.811031\pi$
$83$	− 846.000i	− 1.11880i	0.828897	−	0.559401i	$-0.188969\pi$
$84$	276.000	0.358501
$85$	0	0
$86$	−866.000	−1.08585
$87$	− 702.000i	− 0.865084i
$88$	− 240.000i	− 0.290728i
$89$	1488.00	1.77222	0.886111	−	0.463474i	$-0.153397\pi$
$89$	1488.00	1.77222	0.886111	+	0.463474i	$0.153397\pi$
$90$	0	0
$91$	667.000	0.768358
$92$	− 600.000i	− 0.679938i
$93$	651.000i	0.725866i
$94$	−60.0000	−0.0658354
$95$	0	0
$96$	−96.0000	−0.102062
$97$	319.000i	0.333913i	0.985964	+	0.166956i	$0.0533939\pi$
$97$	319.000i	0.333913i	−0.985964	+	0.166956i	$0.946606\pi$
$98$	372.000i	0.383446i
$99$	270.000	0.274101
$100$	0	0
$101$	−792.000	−0.780267	−0.390133	−	0.920758i	$-0.627571\pi$
$101$	−792.000	−0.780267	−0.390133	+	0.920758i	$0.627571\pi$
$102$	468.000i	0.454303i
$103$	812.000i	0.776784i	0.921494	+	0.388392i	$0.126969\pi$
$103$	812.000i	0.776784i	−0.921494	+	0.388392i	$0.873031\pi$
$104$	−232.000	−0.218745
$105$	0	0
$106$	−1104.00	−1.01160
$107$	1416.00i	1.27934i	0.768648	+	0.639672i	$0.220930\pi$
$107$	1416.00i	1.27934i	−0.768648	+	0.639672i	$0.779070\pi$
$108$	− 108.000i	− 0.0962250i
$109$	55.0000	0.0483307	0.0241653	−	0.999708i	$-0.492307\pi$
$109$	55.0000	0.0483307	0.0241653	+	0.999708i	$0.492307\pi$
$110$	0	0
$111$	−438.000	−0.374533
$112$	− 368.000i	− 0.310471i
$113$	1404.00i	1.16882i	0.811457	+	0.584412i	$0.198675\pi$
$113$	1404.00i	1.16882i	−0.811457	+	0.584412i	$0.801325\pi$
$114$	894.000	0.734480
$115$	0	0
$116$	−936.000	−0.749185
$117$	− 261.000i	− 0.206235i
$118$	− 540.000i	− 0.421280i
$119$	−1794.00	−1.38198
$120$	0	0
$121$	−431.000	−0.323817
$122$	− 550.000i	− 0.408153i
$123$	468.000i	0.343074i
$124$	868.000	0.628619
$125$	0	0
$126$	414.000	0.292715
$127$	− 1280.00i	− 0.894344i	−0.894448	−	0.447172i	$-0.852431\pi$
$127$	− 1280.00i	− 0.894344i	0.894448	−	0.447172i	$-0.147569\pi$
$128$	128.000i	0.0883883i
$129$	−1299.00	−0.886594
$130$	0	0
$131$	480.000	0.320136	0.160068	−	0.987106i	$-0.448829\pi$
$131$	480.000	0.320136	0.160068	+	0.987106i	$0.448829\pi$
$132$	− 360.000i	− 0.237379i
$133$	3427.00i	2.23428i
$134$	−1606.00	−1.03535
$135$	0	0
$136$	624.000	0.393438
$137$	282.000i	0.175860i	0.996127	+	0.0879302i	$0.0280253\pi$
$137$	282.000i	0.175860i	−0.996127	+	0.0879302i	$0.971975\pi$
$138$	− 900.000i	− 0.555167i
$139$	−1604.00	−0.978773	−0.489387	−	0.872067i	$-0.662779\pi$
$139$	−1604.00	−0.978773	−0.489387	+	0.872067i	$0.662779\pi$
$140$	0	0
$141$	−90.0000	−0.0537544
$142$	− 1320.00i	− 0.780084i
$143$	− 870.000i	− 0.508763i
$144$	−144.000	−0.0833333
$145$	0	0
$146$	−1292.00	−0.732375
$147$	558.000i	0.313082i
$148$	584.000i	0.324355i
$149$	774.000	0.425561	0.212780	−	0.977100i	$-0.431748\pi$
$149$	774.000	0.425561	0.212780	+	0.977100i	$0.431748\pi$
$150$	0	0
$151$	293.000	0.157907	0.0789536	−	0.996878i	$-0.474842\pi$
$151$	293.000	0.157907	0.0789536	+	0.996878i	$0.474842\pi$
$152$	− 1192.00i	− 0.636079i
$153$	702.000i	0.370937i
$154$	1380.00	0.722101
$155$	0	0
$156$	−348.000	−0.178604
$157$	1729.00i	0.878912i	0.898264	+	0.439456i	$0.144829\pi$
$157$	1729.00i	0.878912i	−0.898264	+	0.439456i	$0.855171\pi$
$158$	1984.00i	0.998978i
$159$	−1656.00	−0.825971
$160$	0	0
$161$	3450.00	1.68881
$162$	− 162.000i	− 0.0785674i
$163$	− 1123.00i	− 0.539633i	−0.962912	−	0.269816i	$-0.913037\pi$
$163$	− 1123.00i	− 0.539633i	0.962912	−	0.269816i	$-0.0869630\pi$
$164$	624.000	0.297111
$165$	0	0
$166$	−1692.00	−0.791112
$167$	− 1200.00i	− 0.556041i	−0.960575	−	0.278020i	$-0.910322\pi$
$167$	− 1200.00i	− 0.556041i	0.960575	−	0.278020i	$-0.0896783\pi$
$168$	− 552.000i	− 0.253498i
$169$	1356.00	0.617205
$170$	0	0
$171$	1341.00	0.599701
$172$	1732.00i	0.767813i
$173$	− 1734.00i	− 0.762044i	−0.924566	−	0.381022i	$-0.875572\pi$
$173$	− 1734.00i	− 0.762044i	0.924566	−	0.381022i	$-0.124428\pi$
$174$	−1404.00	−0.611707
$175$	0	0
$176$	−480.000	−0.205576
$177$	− 810.000i	− 0.343974i
$178$	− 2976.00i	− 1.25315i
$179$	−2586.00	−1.07981	−0.539907	−	0.841725i	$-0.681541\pi$
$179$	−2586.00	−1.07981	−0.539907	+	0.841725i	$0.681541\pi$
$180$	0	0
$181$	−3931.00	−1.61430	−0.807152	−	0.590344i	$-0.798992\pi$
$181$	−3931.00	−1.61430	−0.807152	+	0.590344i	$0.798992\pi$
$182$	− 1334.00i	− 0.543311i
$183$	− 825.000i	− 0.333255i
$184$	−1200.00	−0.480789
$185$	0	0
$186$	1302.00	0.513265
$187$	2340.00i	0.915068i
$188$	120.000i	0.0465527i
$189$	621.000	0.239001
$190$	0	0
$191$	1566.00	0.593255	0.296628	−	0.954993i	$-0.404138\pi$
$191$	1566.00	0.593255	0.296628	+	0.954993i	$0.404138\pi$
$192$	192.000i	0.0721688i
$193$	2291.00i	0.854455i	0.904144	+	0.427227i	$0.140510\pi$
$193$	2291.00i	0.854455i	−0.904144	+	0.427227i	$0.859490\pi$
$194$	638.000	0.236112
$195$	0	0
$196$	744.000	0.271137
$197$	− 2142.00i	− 0.774676i	−0.921938	−	0.387338i	$-0.873395\pi$
$197$	− 2142.00i	− 0.774676i	0.921938	−	0.387338i	$-0.126605\pi$
$198$	− 540.000i	− 0.193819i
$199$	4903.00	1.74656	0.873278	−	0.487223i	$-0.161990\pi$
$199$	4903.00	1.74656	0.873278	+	0.487223i	$0.161990\pi$
$200$	0	0
$201$	−2409.00	−0.845362
$202$	1584.00i	0.551732i
$203$	− 5382.00i	− 1.86080i
$204$	936.000	0.321241
$205$	0	0
$206$	1624.00	0.549269
$207$	− 1350.00i	− 0.453292i
$208$	464.000i	0.154676i
$209$	4470.00	1.47941
$210$	0	0
$211$	605.000	0.197393	0.0986965	−	0.995118i	$-0.468533\pi$
$211$	605.000	0.197393	0.0986965	+	0.995118i	$0.468533\pi$
$212$	2208.00i	0.715312i
$213$	− 1980.00i	− 0.636936i
$214$	2832.00	0.904633
$215$	0	0
$216$	−216.000	−0.0680414
$217$	4991.00i	1.56134i
$218$	− 110.000i	− 0.0341750i
$219$	−1938.00	−0.597981
$220$	0	0
$221$	2262.00	0.688500
$222$	876.000i	0.264835i
$223$	− 145.000i	− 0.0435422i	−0.999763	−	0.0217711i	$-0.993069\pi$
$223$	− 145.000i	− 0.0435422i	0.999763	−	0.0217711i	$-0.00693051\pi$
$224$	−736.000	−0.219536
$225$	0	0
$226$	2808.00	0.826484
$227$	− 2964.00i	− 0.866641i	−0.901240	−	0.433321i	$-0.857342\pi$
$227$	− 2964.00i	− 0.866641i	0.901240	−	0.433321i	$-0.142658\pi$
$228$	− 1788.00i	− 0.519356i
$229$	5635.00	1.62608	0.813038	−	0.582211i	$-0.197812\pi$
$229$	5635.00	1.62608	0.813038	+	0.582211i	$0.197812\pi$
$230$	0	0
$231$	2070.00	0.589593
$232$	1872.00i	0.529754i
$233$	4164.00i	1.17078i	0.810750	+	0.585392i	$0.199059\pi$
$233$	4164.00i	1.17078i	−0.810750	+	0.585392i	$0.800941\pi$
$234$	−522.000	−0.145830
$235$	0	0
$236$	−1080.00	−0.297890
$237$	2976.00i	0.815662i
$238$	3588.00i	0.977208i
$239$	1944.00	0.526138	0.263069	−	0.964777i	$-0.415265\pi$
$239$	1944.00	0.526138	0.263069	+	0.964777i	$0.415265\pi$
$240$	0	0
$241$	857.000	0.229063	0.114532	−	0.993420i	$-0.463463\pi$
$241$	857.000	0.229063	0.114532	+	0.993420i	$0.463463\pi$
$242$	862.000i	0.228973i
$243$	− 243.000i	− 0.0641500i
$244$	−1100.00	−0.288608
$245$	0	0
$246$	936.000	0.242590
$247$	− 4321.00i	− 1.11311i
$248$	− 1736.00i	− 0.444500i
$249$	−2538.00	−0.645941
$250$	0	0
$251$	−3924.00	−0.986776	−0.493388	−	0.869809i	$-0.664242\pi$
$251$	−3924.00	−0.986776	−0.493388	+	0.869809i	$0.664242\pi$
$252$	− 828.000i	− 0.206981i
$253$	− 4500.00i	− 1.11823i
$254$	−2560.00	−0.632396
$255$	0	0
$256$	256.000	0.0625000
$257$	2844.00i	0.690287i	0.938550	+	0.345144i	$0.112170\pi$
$257$	2844.00i	0.690287i	−0.938550	+	0.345144i	$0.887830\pi$
$258$	2598.00i	0.626916i
$259$	−3358.00	−0.805621
$260$	0	0
$261$	−2106.00	−0.499456
$262$	− 960.000i	− 0.226370i
$263$	− 6060.00i	− 1.42082i	−0.703788	−	0.710410i	$-0.748510\pi$
$263$	− 6060.00i	− 1.42082i	0.703788	−	0.710410i	$-0.251490\pi$
$264$	−720.000	−0.167852
$265$	0	0
$266$	6854.00	1.57987
$267$	− 4464.00i	− 1.02319i
$268$	3212.00i	0.732105i
$269$	−3906.00	−0.885327	−0.442664	−	0.896688i	$-0.645966\pi$
$269$	−3906.00	−0.885327	−0.442664	+	0.896688i	$0.645966\pi$
$270$	0	0
$271$	2144.00	0.480586	0.240293	−	0.970700i	$-0.422757\pi$
$271$	2144.00	0.480586	0.240293	+	0.970700i	$0.422757\pi$
$272$	− 1248.00i	− 0.278203i
$273$	− 2001.00i	− 0.443612i
$274$	564.000	0.124352
$275$	0	0
$276$	−1800.00	−0.392563
$277$	− 2321.00i	− 0.503449i	−0.967799	−	0.251725i	$-0.919002\pi$
$277$	− 2321.00i	− 0.503449i	0.967799	−	0.251725i	$-0.0809977\pi$
$278$	3208.00i	0.692097i
$279$	1953.00	0.419079
$280$	0	0
$281$	−6822.00	−1.44828	−0.724140	−	0.689654i	$-0.757763\pi$
$281$	−6822.00	−1.44828	−0.724140	+	0.689654i	$0.757763\pi$
$282$	180.000i	0.0380101i
$283$	4049.00i	0.850488i	0.905079	+	0.425244i	$0.139812\pi$
$283$	4049.00i	0.850488i	−0.905079	+	0.425244i	$0.860188\pi$
$284$	−2640.00	−0.551603
$285$	0	0
$286$	−1740.00	−0.359749
$287$	3588.00i	0.737955i
$288$	288.000i	0.0589256i
$289$	−1171.00	−0.238347
$290$	0	0
$291$	957.000	0.192785
$292$	2584.00i	0.517867i
$293$	2238.00i	0.446230i	0.974792	+	0.223115i	$0.0716225\pi$
$293$	2238.00i	0.446230i	−0.974792	+	0.223115i	$0.928377\pi$
$294$	1116.00	0.221382
$295$	0	0
$296$	1168.00	0.229353
$297$	− 810.000i	− 0.158252i
$298$	− 1548.00i	− 0.300917i
$299$	−4350.00	−0.841361
$300$	0	0
$301$	−9959.00	−1.90707
$302$	− 586.000i	− 0.111657i
$303$	2376.00i	0.450487i
$304$	−2384.00	−0.449776
$305$	0	0
$306$	1404.00	0.262292
$307$	− 1385.00i	− 0.257479i	−0.991678	−	0.128740i	$-0.958907\pi$
$307$	− 1385.00i	− 0.257479i	0.991678	−	0.128740i	$-0.0410931\pi$
$308$	− 2760.00i	− 0.510603i
$309$	2436.00	0.448476
$310$	0	0
$311$	−5670.00	−1.03381	−0.516907	−	0.856042i	$-0.672917\pi$
$311$	−5670.00	−1.03381	−0.516907	+	0.856042i	$0.672917\pi$
$312$	696.000i	0.126292i
$313$	− 421.000i	− 0.0760266i	−0.999277	−	0.0380133i	$-0.987897\pi$
$313$	− 421.000i	− 0.0760266i	0.999277	−	0.0380133i	$-0.0121029\pi$
$314$	3458.00	0.621485
$315$	0	0
$316$	3968.00	0.706384
$317$	− 9984.00i	− 1.76895i	−0.466587	−	0.884475i	$-0.654517\pi$
$317$	− 9984.00i	− 1.76895i	0.466587	−	0.884475i	$-0.345483\pi$
$318$	3312.00i	0.584049i
$319$	−7020.00	−1.23211
$320$	0	0
$321$	4248.00	0.738630
$322$	− 6900.00i	− 1.19417i
$323$	11622.0i	2.00206i
$324$	−324.000	−0.0555556
$325$	0	0
$326$	−2246.00	−0.381578
$327$	− 165.000i	− 0.0279037i
$328$	− 1248.00i	− 0.210089i
$329$	−690.000	−0.115626
$330$	0	0
$331$	−4228.00	−0.702090	−0.351045	−	0.936359i	$-0.614174\pi$
$331$	−4228.00	−0.702090	−0.351045	+	0.936359i	$0.614174\pi$
$332$	3384.00i	0.559401i
$333$	1314.00i	0.216237i
$334$	−2400.00	−0.393180
$335$	0	0
$336$	−1104.00	−0.179250
$337$	− 5393.00i	− 0.871737i	−0.900010	−	0.435869i	$-0.856441\pi$
$337$	− 5393.00i	− 0.871737i	0.900010	−	0.435869i	$-0.143559\pi$
$338$	− 2712.00i	− 0.436430i
$339$	4212.00	0.674821
$340$	0	0
$341$	6510.00	1.03383
$342$	− 2682.00i	− 0.424052i
$343$	− 3611.00i	− 0.568442i
$344$	3464.00	0.542925
$345$	0	0
$346$	−3468.00	−0.538846
$347$	7914.00i	1.22434i	0.790726	+	0.612170i	$0.209703\pi$
$347$	7914.00i	1.22434i	−0.790726	+	0.612170i	$0.790297\pi$
$348$	2808.00i	0.432542i
$349$	−1010.00	−0.154911	−0.0774557	−	0.996996i	$-0.524680\pi$
$349$	−1010.00	−0.154911	−0.0774557	+	0.996996i	$0.524680\pi$
$350$	0	0
$351$	−783.000	−0.119070
$352$	960.000i	0.145364i
$353$	4722.00i	0.711974i	0.934491	+	0.355987i	$0.115855\pi$
$353$	4722.00i	0.711974i	−0.934491	+	0.355987i	$0.884145\pi$
$354$	−1620.00	−0.243226
$355$	0	0
$356$	−5952.00	−0.886111
$357$	5382.00i	0.797887i
$358$	5172.00i	0.763544i
$359$	−6204.00	−0.912074	−0.456037	−	0.889961i	$-0.650732\pi$
$359$	−6204.00	−0.912074	−0.456037	+	0.889961i	$0.650732\pi$
$360$	0	0
$361$	15342.0	2.23677
$362$	7862.00i	1.14148i
$363$	1293.00i	0.186956i
$364$	−2668.00	−0.384179
$365$	0	0
$366$	−1650.00	−0.235647
$367$	− 1361.00i	− 0.193579i	−0.995305	−	0.0967897i	$-0.969143\pi$
$367$	− 1361.00i	− 0.193579i	0.995305	−	0.0967897i	$-0.0308574\pi$
$368$	2400.00i	0.339969i
$369$	1404.00	0.198074
$370$	0	0
$371$	−12696.0	−1.77667
$372$	− 2604.00i	− 0.362933i
$373$	− 913.000i	− 0.126738i	−0.997990	−	0.0633691i	$-0.979815\pi$
$373$	− 913.000i	− 0.126738i	0.997990	−	0.0633691i	$-0.0201845\pi$
$374$	4680.00	0.647051
$375$	0	0
$376$	240.000	0.0329177
$377$	6786.00i	0.927047i
$378$	− 1242.00i	− 0.168999i
$379$	8881.00	1.20366	0.601829	−	0.798625i	$-0.294439\pi$
$379$	8881.00	1.20366	0.601829	+	0.798625i	$0.294439\pi$
$380$	0	0
$381$	−3840.00	−0.516350
$382$	− 3132.00i	− 0.419495i
$383$	5460.00i	0.728441i	0.931313	+	0.364221i	$0.118665\pi$
$383$	5460.00i	0.728441i	−0.931313	+	0.364221i	$0.881335\pi$
$384$	384.000	0.0510310
$385$	0	0
$386$	4582.00	0.604191
$387$	3897.00i	0.511875i
$388$	− 1276.00i	− 0.166956i
$389$	13884.0	1.80963	0.904816	−	0.425803i	$-0.140008\pi$
$389$	13884.0	1.80963	0.904816	+	0.425803i	$0.140008\pi$
$390$	0	0
$391$	11700.0	1.51328
$392$	− 1488.00i	− 0.191723i
$393$	− 1440.00i	− 0.184831i
$394$	−4284.00	−0.547779
$395$	0	0
$396$	−1080.00	−0.137051
$397$	3781.00i	0.477992i	0.971021	+	0.238996i	$0.0768183\pi$
$397$	3781.00i	0.477992i	−0.971021	+	0.238996i	$0.923182\pi$
$398$	− 9806.00i	− 1.23500i
$399$	10281.0	1.28996
$400$	0	0
$401$	9024.00	1.12378	0.561892	−	0.827211i	$-0.310074\pi$
$401$	9024.00	1.12378	0.561892	+	0.827211i	$0.310074\pi$
$402$	4818.00i	0.597761i
$403$	− 6293.00i	− 0.777858i
$404$	3168.00	0.390133
$405$	0	0
$406$	−10764.0	−1.31578
$407$	4380.00i	0.533436i
$408$	− 1872.00i	− 0.227151i
$409$	−14789.0	−1.78794	−0.893972	−	0.448123i	$-0.852093\pi$
$409$	−14789.0	−1.78794	−0.893972	+	0.448123i	$0.852093\pi$
$410$	0	0
$411$	846.000	0.101533
$412$	− 3248.00i	− 0.388392i
$413$	− 6210.00i	− 0.739889i
$414$	−2700.00	−0.320526
$415$	0	0
$416$	928.000	0.109372
$417$	4812.00i	0.565095i
$418$	− 8940.00i	− 1.04610i
$419$	−9840.00	−1.14729	−0.573646	−	0.819103i	$-0.694472\pi$
$419$	−9840.00	−1.14729	−0.573646	+	0.819103i	$0.694472\pi$
$420$	0	0
$421$	5510.00	0.637865	0.318932	−	0.947778i	$-0.396676\pi$
$421$	5510.00	0.637865	0.318932	+	0.947778i	$0.396676\pi$
$422$	− 1210.00i	− 0.139578i
$423$	270.000i	0.0310351i
$424$	4416.00	0.505802
$425$	0	0
$426$	−3960.00	−0.450382
$427$	− 6325.00i	− 0.716834i
$428$	− 5664.00i	− 0.639672i
$429$	−2610.00	−0.293734
$430$	0	0
$431$	11070.0	1.23718	0.618588	−	0.785715i	$-0.287705\pi$
$431$	11070.0	1.23718	0.618588	+	0.785715i	$0.287705\pi$
$432$	432.000i	0.0481125i
$433$	− 12133.0i	− 1.34659i	−0.739373	−	0.673297i	$-0.764878\pi$
$433$	− 12133.0i	− 1.34659i	0.739373	−	0.673297i	$-0.235122\pi$
$434$	9982.00	1.10404
$435$	0	0
$436$	−220.000	−0.0241653
$437$	− 22350.0i	− 2.44656i
$438$	3876.00i	0.422837i
$439$	1873.00	0.203630	0.101815	−	0.994803i	$-0.467535\pi$
$439$	1873.00	0.203630	0.101815	+	0.994803i	$0.467535\pi$
$440$	0	0
$441$	1674.00	0.180758
$442$	− 4524.00i	− 0.486843i
$443$	− 576.000i	− 0.0617756i	−0.999523	−	0.0308878i	$-0.990167\pi$
$443$	− 576.000i	− 0.0617756i	0.999523	−	0.0308878i	$-0.00983345\pi$
$444$	1752.00	0.187266
$445$	0	0
$446$	−290.000	−0.0307890
$447$	− 2322.00i	− 0.245698i
$448$	1472.00i	0.155235i
$449$	4884.00	0.513341	0.256671	−	0.966499i	$-0.417374\pi$
$449$	4884.00	0.513341	0.256671	+	0.966499i	$0.417374\pi$
$450$	0	0
$451$	4680.00	0.488631
$452$	− 5616.00i	− 0.584412i
$453$	− 879.000i	− 0.0911678i
$454$	−5928.00	−0.612808
$455$	0	0
$456$	−3576.00	−0.367240
$457$	15802.0i	1.61748i	0.588169	+	0.808738i	$0.299849\pi$
$457$	15802.0i	1.61748i	−0.588169	+	0.808738i	$0.700151\pi$
$458$	− 11270.0i	− 1.14981i
$459$	2106.00	0.214160
$460$	0	0
$461$	−15360.0	−1.55181	−0.775907	−	0.630847i	$-0.782708\pi$
$461$	−15360.0	−1.55181	−0.775907	+	0.630847i	$0.782708\pi$
$462$	− 4140.00i	− 0.416905i
$463$	1712.00i	0.171843i	0.996302	+	0.0859216i	$0.0273835\pi$
$463$	1712.00i	0.171843i	−0.996302	+	0.0859216i	$0.972617\pi$
$464$	3744.00	0.374592
$465$	0	0
$466$	8328.00	0.827869
$467$	16278.0i	1.61297i	0.591256	+	0.806484i	$0.298632\pi$
$467$	16278.0i	1.61297i	−0.591256	+	0.806484i	$0.701368\pi$
$468$	1044.00i	0.103117i
$469$	−18469.0	−1.81838
$470$	0	0
$471$	5187.00	0.507440
$472$	2160.00i	0.210640i
$473$	12990.0i	1.26275i
$474$	5952.00	0.576760
$475$	0	0
$476$	7176.00	0.690990
$477$	4968.00i	0.476874i
$478$	− 3888.00i	− 0.372036i
$479$	14766.0	1.40851	0.704254	−	0.709948i	$-0.251281\pi$
$479$	14766.0	1.40851	0.704254	+	0.709948i	$0.251281\pi$
$480$	0	0
$481$	4234.00	0.401359
$482$	− 1714.00i	− 0.161972i
$483$	− 10350.0i	− 0.975034i
$484$	1724.00	0.161908
$485$	0	0
$486$	−486.000	−0.0453609
$487$	3319.00i	0.308826i	0.988006	+	0.154413i	$0.0493486\pi$
$487$	3319.00i	0.308826i	−0.988006	+	0.154413i	$0.950651\pi$
$488$	2200.00i	0.204076i
$489$	−3369.00	−0.311557
$490$	0	0
$491$	11064.0	1.01693	0.508464	−	0.861083i	$-0.330214\pi$
$491$	11064.0	1.01693	0.508464	+	0.861083i	$0.330214\pi$
$492$	− 1872.00i	− 0.171537i
$493$	− 18252.0i	− 1.66740i
$494$	−8642.00	−0.787089
$495$	0	0
$496$	−3472.00	−0.314309
$497$	− 15180.0i	− 1.37005i
$498$	5076.00i	0.456749i
$499$	14131.0	1.26772	0.633858	−	0.773449i	$-0.281470\pi$
$499$	14131.0	1.26772	0.633858	+	0.773449i	$0.281470\pi$
$500$	0	0
$501$	−3600.00	−0.321030
$502$	7848.00i	0.697756i
$503$	11988.0i	1.06266i	0.847165	+	0.531331i	$0.178308\pi$
$503$	11988.0i	1.06266i	−0.847165	+	0.531331i	$0.821692\pi$
$504$	−1656.00	−0.146357
$505$	0	0
$506$	−9000.00	−0.790709
$507$	− 4068.00i	− 0.356344i
$508$	5120.00i	0.447172i
$509$	−10806.0	−0.940997	−0.470499	−	0.882401i	$-0.655926\pi$
$509$	−10806.0	−0.940997	−0.470499	+	0.882401i	$0.655926\pi$
$510$	0	0
$511$	−14858.0	−1.28626
$512$	− 512.000i	− 0.0441942i
$513$	− 4023.00i	− 0.346237i
$514$	5688.00	0.488107
$515$	0	0
$516$	5196.00	0.443297
$517$	900.000i	0.0765608i
$518$	6716.00i	0.569660i
$519$	−5202.00	−0.439966
$520$	0	0
$521$	22578.0	1.89858	0.949290	−	0.314402i	$-0.101804\pi$
$521$	22578.0	1.89858	0.949290	+	0.314402i	$0.101804\pi$
$522$	4212.00i	0.353169i
$523$	12065.0i	1.00873i	0.863491	+	0.504365i	$0.168273\pi$
$523$	12065.0i	1.00873i	−0.863491	+	0.504365i	$0.831727\pi$
$524$	−1920.00	−0.160068
$525$	0	0
$526$	−12120.0	−1.00467
$527$	16926.0i	1.39907i
$528$	1440.00i	0.118689i
$529$	−10333.0	−0.849264
$530$	0	0
$531$	−2430.00	−0.198593
$532$	− 13708.0i	− 1.11714i
$533$	− 4524.00i	− 0.367648i
$534$	−8928.00	−0.723506
$535$	0	0
$536$	6424.00	0.517676
$537$	7758.00i	0.623431i
$538$	7812.00i	0.626021i
$539$	5580.00	0.445914
$540$	0	0
$541$	−12055.0	−0.958013	−0.479006	−	0.877811i	$-0.659003\pi$
$541$	−12055.0	−0.958013	−0.479006	+	0.877811i	$0.659003\pi$
$542$	− 4288.00i	− 0.339825i
$543$	11793.0i	0.932019i
$544$	−2496.00	−0.196719
$545$	0	0
$546$	−4002.00	−0.313681
$547$	− 6176.00i	− 0.482754i	−0.970431	−	0.241377i	$-0.922401\pi$
$547$	− 6176.00i	− 0.482754i	0.970431	−	0.241377i	$-0.0775991\pi$
$548$	− 1128.00i	− 0.0879302i
$549$	−2475.00	−0.192405
$550$	0	0
$551$	−34866.0	−2.69572
$552$	3600.00i	0.277584i
$553$	22816.0i	1.75449i
$554$	−4642.00	−0.355992
$555$	0	0
$556$	6416.00	0.489387
$557$	− 8274.00i	− 0.629409i	−0.949190	−	0.314704i	$-0.898095\pi$
$557$	− 8274.00i	− 0.629409i	0.949190	−	0.314704i	$-0.101905\pi$
$558$	− 3906.00i	− 0.296334i
$559$	12557.0	0.950098
$560$	0	0
$561$	7020.00	0.528315
$562$	13644.0i	1.02409i
$563$	966.000i	0.0723127i	0.999346	+	0.0361563i	$0.0115114\pi$
$563$	966.000i	0.0723127i	−0.999346	+	0.0361563i	$0.988489\pi$
$564$	360.000	0.0268772
$565$	0	0
$566$	8098.00	0.601386
$567$	− 1863.00i	− 0.137987i
$568$	5280.00i	0.390042i
$569$	−19002.0	−1.40001	−0.700005	−	0.714138i	$-0.746819\pi$
$569$	−19002.0	−1.40001	−0.700005	+	0.714138i	$0.746819\pi$
$570$	0	0
$571$	8645.00	0.633594	0.316797	−	0.948493i	$-0.397393\pi$
$571$	8645.00	0.633594	0.316797	+	0.948493i	$0.397393\pi$
$572$	3480.00i	0.254381i
$573$	− 4698.00i	− 0.342516i
$574$	7176.00	0.521813
$575$	0	0
$576$	576.000	0.0416667
$577$	− 10931.0i	− 0.788672i	−0.918966	−	0.394336i	$-0.870975\pi$
$577$	− 10931.0i	− 0.788672i	0.918966	−	0.394336i	$-0.129025\pi$
$578$	2342.00i	0.168537i
$579$	6873.00	0.493320
$580$	0	0
$581$	−19458.0	−1.38942
$582$	− 1914.00i	− 0.136319i
$583$	16560.0i	1.17641i
$584$	5168.00	0.366187
$585$	0	0
$586$	4476.00	0.315532
$587$	− 8904.00i	− 0.626077i	−0.949740	−	0.313039i	$-0.898653\pi$
$587$	− 8904.00i	− 0.626077i	0.949740	−	0.313039i	$-0.101347\pi$
$588$	− 2232.00i	− 0.156541i
$589$	32333.0	2.26190
$590$	0	0
$591$	−6426.00	−0.447259
$592$	− 2336.00i	− 0.162177i
$593$	8820.00i	0.610782i	0.952227	+	0.305391i	$0.0987872\pi$
$593$	8820.00i	0.610782i	−0.952227	+	0.305391i	$0.901213\pi$
$594$	−1620.00	−0.111901
$595$	0	0
$596$	−3096.00	−0.212780
$597$	− 14709.0i	− 1.00837i
$598$	8700.00i	0.594932i
$599$	−9804.00	−0.668749	−0.334374	−	0.942440i	$-0.608525\pi$
$599$	−9804.00	−0.668749	−0.334374	+	0.942440i	$0.608525\pi$
$600$	0	0
$601$	−23437.0	−1.59071	−0.795354	−	0.606146i	$-0.792715\pi$
$601$	−23437.0	−1.59071	−0.795354	+	0.606146i	$0.792715\pi$
$602$	19918.0i	1.34850i
$603$	7227.00i	0.488070i
$604$	−1172.00	−0.0789536
$605$	0	0
$606$	4752.00	0.318543
$607$	− 2648.00i	− 0.177066i	−0.996073	−	0.0885330i	$-0.971782\pi$
$607$	− 2648.00i	− 0.177066i	0.996073	−	0.0885330i	$-0.0282179\pi$
$608$	4768.00i	0.318039i
$609$	−16146.0	−1.07433
$610$	0	0
$611$	870.000	0.0576046
$612$	− 2808.00i	− 0.185468i
$613$	794.000i	0.0523154i	0.999658	+	0.0261577i	$0.00832721\pi$
$613$	794.000i	0.0523154i	−0.999658	+	0.0261577i	$0.991673\pi$
$614$	−2770.00	−0.182065
$615$	0	0
$616$	−5520.00	−0.361051
$617$	− 18720.0i	− 1.22146i	−0.791840	−	0.610728i	$-0.790877\pi$
$617$	− 18720.0i	− 1.22146i	0.791840	−	0.610728i	$-0.209123\pi$
$618$	− 4872.00i	− 0.317121i
$619$	8959.00	0.581733	0.290866	−	0.956764i	$-0.406056\pi$
$619$	8959.00	0.581733	0.290866	+	0.956764i	$0.406056\pi$
$620$	0	0
$621$	−4050.00	−0.261708
$622$	11340.0i	0.731017i
$623$	− 34224.0i	− 2.20089i
$624$	1392.00	0.0893022
$625$	0	0
$626$	−842.000	−0.0537589
$627$	− 13410.0i	− 0.854137i
$628$	− 6916.00i	− 0.439456i
$629$	−11388.0	−0.721891
$630$	0	0
$631$	−12373.0	−0.780604	−0.390302	−	0.920687i	$-0.627629\pi$
$631$	−12373.0	−0.780604	−0.390302	+	0.920687i	$0.627629\pi$
$632$	− 7936.00i	− 0.499489i
$633$	− 1815.00i	− 0.113965i
$634$	−19968.0	−1.25084
$635$	0	0
$636$	6624.00	0.412985
$637$	− 5394.00i	− 0.335507i
$638$	14040.0i	0.871237i
$639$	−5940.00	−0.367735
$640$	0	0
$641$	24900.0	1.53431	0.767154	−	0.641463i	$-0.221672\pi$
$641$	24900.0	1.53431	0.767154	+	0.641463i	$0.221672\pi$
$642$	− 8496.00i	− 0.522290i
$643$	− 14668.0i	− 0.899610i	−0.893127	−	0.449805i	$-0.851493\pi$
$643$	− 14668.0i	− 0.899610i	0.893127	−	0.449805i	$-0.148507\pi$
$644$	−13800.0	−0.844404
$645$	0	0
$646$	23244.0	1.41567
$647$	10788.0i	0.655518i	0.944761	+	0.327759i	$0.106293\pi$
$647$	10788.0i	0.655518i	−0.944761	+	0.327759i	$0.893707\pi$
$648$	648.000i	0.0392837i
$649$	−8100.00	−0.489912
$650$	0	0
$651$	14973.0	0.901441
$652$	4492.00i	0.269816i
$653$	− 14214.0i	− 0.851817i	−0.904766	−	0.425909i	$-0.859954\pi$
$653$	− 14214.0i	− 0.851817i	0.904766	−	0.425909i	$-0.140046\pi$
$654$	−330.000	−0.0197309
$655$	0	0
$656$	−2496.00	−0.148556
$657$	5814.00i	0.345245i
$658$	1380.00i	0.0817599i
$659$	588.000	0.0347576	0.0173788	−	0.999849i	$-0.494468\pi$
$659$	588.000	0.0347576	0.0173788	+	0.999849i	$0.494468\pi$
$660$	0	0
$661$	−3166.00	−0.186298	−0.0931491	−	0.995652i	$-0.529693\pi$
$661$	−3166.00	−0.186298	−0.0931491	+	0.995652i	$0.529693\pi$
$662$	8456.00i	0.496453i
$663$	− 6786.00i	− 0.397506i
$664$	6768.00	0.395556
$665$	0	0
$666$	2628.00	0.152902
$667$	35100.0i	2.03760i
$668$	4800.00i	0.278020i
$669$	−435.000	−0.0251391
$670$	0	0
$671$	−8250.00	−0.474646
$672$	2208.00i	0.126749i
$673$	9182.00i	0.525914i	0.964808	+	0.262957i	$0.0846977\pi$
$673$	9182.00i	0.525914i	−0.964808	+	0.262957i	$0.915302\pi$
$674$	−10786.0	−0.616411
$675$	0	0
$676$	−5424.00	−0.308603
$677$	− 11742.0i	− 0.666590i	−0.942823	−	0.333295i	$-0.891839\pi$
$677$	− 11742.0i	− 0.666590i	0.942823	−	0.333295i	$-0.108161\pi$
$678$	− 8424.00i	− 0.477171i
$679$	7337.00	0.414681
$680$	0	0
$681$	−8892.00	−0.500356
$682$	− 13020.0i	− 0.731029i
$683$	− 6024.00i	− 0.337485i	−0.985660	−	0.168742i	$-0.946029\pi$
$683$	− 6024.00i	− 0.337485i	0.985660	−	0.168742i	$-0.0539706\pi$
$684$	−5364.00	−0.299850
$685$	0	0
$686$	−7222.00	−0.401949
$687$	− 16905.0i	− 0.938815i
$688$	− 6928.00i	− 0.383906i
$689$	16008.0	0.885132
$690$	0	0
$691$	9344.00	0.514418	0.257209	−	0.966356i	$-0.417197\pi$
$691$	9344.00	0.514418	0.257209	+	0.966356i	$0.417197\pi$
$692$	6936.00i	0.381022i
$693$	− 6210.00i	− 0.340402i
$694$	15828.0	0.865739
$695$	0	0
$696$	5616.00	0.305853
$697$	12168.0i	0.661257i
$698$	2020.00i	0.109539i
$699$	12492.0	0.675953
$700$	0	0
$701$	21234.0	1.14408	0.572038	−	0.820227i	$-0.306153\pi$
$701$	21234.0	1.14408	0.572038	+	0.820227i	$0.306153\pi$
$702$	1566.00i	0.0841950i
$703$	21754.0i	1.16709i
$704$	1920.00	0.102788
$705$	0	0
$706$	9444.00	0.503441
$707$	18216.0i	0.969000i
$708$	3240.00i	0.171987i
$709$	1723.00	0.0912675	0.0456337	−	0.998958i	$-0.485469\pi$
$709$	1723.00	0.0912675	0.0456337	+	0.998958i	$0.485469\pi$
$710$	0	0
$711$	8928.00	0.470923
$712$	11904.0i	0.626575i
$713$	− 32550.0i	− 1.70969i
$714$	10764.0	0.564191
$715$	0	0
$716$	10344.0	0.539907
$717$	− 5832.00i	− 0.303766i
$718$	12408.0i	0.644934i
$719$	−18510.0	−0.960093	−0.480046	−	0.877243i	$-0.659380\pi$
$719$	−18510.0	−0.960093	−0.480046	+	0.877243i	$0.659380\pi$
$720$	0	0
$721$	18676.0	0.964675
$722$	− 30684.0i	− 1.58163i
$723$	− 2571.00i	− 0.132250i
$724$	15724.0	0.807152
$725$	0	0
$726$	2586.00	0.132198
$727$	1009.00i	0.0514742i	0.999669	+	0.0257371i	$0.00819328\pi$
$727$	1009.00i	0.0514742i	−0.999669	+	0.0257371i	$0.991807\pi$
$728$	5336.00i	0.271656i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−33774.0	−1.70886
$732$	3300.00i	0.166628i
$733$	− 21994.0i	− 1.10828i	−0.832425	−	0.554138i	$-0.813048\pi$
$733$	− 21994.0i	− 1.10828i	0.832425	−	0.554138i	$-0.186952\pi$
$734$	−2722.00	−0.136881
$735$	0	0
$736$	4800.00	0.240394
$737$	24090.0i	1.20403i
$738$	− 2808.00i	− 0.140059i
$739$	13948.0	0.694297	0.347148	−	0.937810i	$-0.387150\pi$
$739$	13948.0	0.694297	0.347148	+	0.937810i	$0.387150\pi$
$740$	0	0
$741$	−12963.0	−0.642655
$742$	25392.0i	1.25629i
$743$	− 26508.0i	− 1.30886i	−0.756122	−	0.654431i	$-0.772908\pi$
$743$	− 26508.0i	− 1.30886i	0.756122	−	0.654431i	$-0.227092\pi$
$744$	−5208.00	−0.256632
$745$	0	0
$746$	−1826.00	−0.0896174
$747$	7614.00i	0.372934i
$748$	− 9360.00i	− 0.457534i
$749$	32568.0	1.58880
$750$	0	0
$751$	−1600.00	−0.0777428	−0.0388714	−	0.999244i	$-0.512376\pi$
$751$	−1600.00	−0.0777428	−0.0388714	+	0.999244i	$0.512376\pi$
$752$	− 480.000i	− 0.0232763i
$753$	11772.0i	0.569715i
$754$	13572.0	0.655521
$755$	0	0
$756$	−2484.00	−0.119500
$757$	− 30101.0i	− 1.44523i	−0.691250	−	0.722615i	$-0.742940\pi$
$757$	− 30101.0i	− 1.44523i	0.691250	−	0.722615i	$-0.257060\pi$
$758$	− 17762.0i	− 0.851115i
$759$	−13500.0	−0.645611
$760$	0	0
$761$	35628.0	1.69713	0.848564	−	0.529093i	$-0.177468\pi$
$761$	35628.0	1.69713	0.848564	+	0.529093i	$0.177468\pi$
$762$	7680.00i	0.365114i
$763$	− 1265.00i	− 0.0600211i
$764$	−6264.00	−0.296628
$765$	0	0
$766$	10920.0	0.515086
$767$	7830.00i	0.368611i
$768$	− 768.000i	− 0.0360844i
$769$	12517.0	0.586963	0.293482	−	0.955965i	$-0.405186\pi$
$769$	12517.0	0.586963	0.293482	+	0.955965i	$0.405186\pi$
$770$	0	0
$771$	8532.00	0.398538
$772$	− 9164.00i	− 0.427227i
$773$	14124.0i	0.657186i	0.944472	+	0.328593i	$0.106574\pi$
$773$	14124.0i	0.657186i	−0.944472	+	0.328593i	$0.893426\pi$
$774$	7794.00	0.361950
$775$	0	0
$776$	−2552.00	−0.118056
$777$	10074.0i	0.465126i
$778$	− 27768.0i	− 1.27960i
$779$	23244.0	1.06907
$780$	0	0
$781$	−19800.0	−0.907170
$782$	− 23400.0i	− 1.07005i
$783$	6318.00i	0.288361i
$784$	−2976.00	−0.135569
$785$	0	0
$786$	−2880.00	−0.130695
$787$	− 40433.0i	− 1.83136i	−0.401907	−	0.915680i	$-0.631653\pi$
$787$	− 40433.0i	− 1.83136i	0.401907	−	0.915680i	$-0.368347\pi$
$788$	8568.00i	0.387338i
$789$	−18180.0	−0.820311
$790$	0	0
$791$	32292.0	1.45154
$792$	2160.00i	0.0969094i
$793$	7975.00i	0.357126i
$794$	7562.00	0.337992
$795$	0	0
$796$	−19612.0	−0.873278
$797$	27300.0i	1.21332i	0.794962	+	0.606660i	$0.207491\pi$
$797$	27300.0i	1.21332i	−0.794962	+	0.606660i	$0.792509\pi$
$798$	− 20562.0i	− 0.912139i
$799$	−2340.00	−0.103609
$800$	0	0
$801$	−13392.0	−0.590740
$802$	− 18048.0i	− 0.794635i
$803$	19380.0i	0.851688i
$804$	9636.00	0.422681
$805$	0	0
$806$	−12586.0	−0.550028
$807$	11718.0i	0.511144i
$808$	− 6336.00i	− 0.275866i
$809$	2856.00	0.124118	0.0620591	−	0.998072i	$-0.480233\pi$
$809$	2856.00	0.124118	0.0620591	+	0.998072i	$0.480233\pi$
$810$	0	0
$811$	−12619.0	−0.546379	−0.273189	−	0.961960i	$-0.588079\pi$
$811$	−12619.0	−0.546379	−0.273189	+	0.961960i	$0.588079\pi$
$812$	21528.0i	0.930400i
$813$	− 6432.00i	− 0.277466i
$814$	8760.00	0.377196
$815$	0	0
$816$	−3744.00	−0.160620
$817$	64517.0i	2.76275i
$818$	29578.0i	1.26427i
$819$	−6003.00	−0.256119
$820$	0	0
$821$	−29082.0	−1.23626	−0.618130	−	0.786076i	$-0.712109\pi$
$821$	−29082.0	−1.23626	−0.618130	+	0.786076i	$0.712109\pi$
$822$	− 1692.00i	− 0.0717947i
$823$	10235.0i	0.433499i	0.976227	+	0.216749i	$0.0695455\pi$
$823$	10235.0i	0.433499i	−0.976227	+	0.216749i	$0.930455\pi$
$824$	−6496.00	−0.274635
$825$	0	0
$826$	−12420.0	−0.523180
$827$	− 26976.0i	− 1.13428i	−0.823622	−	0.567139i	$-0.808050\pi$
$827$	− 26976.0i	− 1.13428i	0.823622	−	0.567139i	$-0.191950\pi$
$828$	5400.00i	0.226646i
$829$	−37802.0	−1.58374	−0.791868	−	0.610692i	$-0.790891\pi$
$829$	−37802.0	−1.58374	−0.791868	+	0.610692i	$0.790891\pi$
$830$	0	0
$831$	−6963.00	−0.290666
$832$	− 1856.00i	− 0.0773380i
$833$	14508.0i	0.603448i
$834$	9624.00	0.399583
$835$	0	0
$836$	−17880.0	−0.739704
$837$	− 5859.00i	− 0.241955i
$838$	19680.0i	0.811258i
$839$	16974.0	0.698460	0.349230	−	0.937037i	$-0.386443\pi$
$839$	16974.0	0.698460	0.349230	+	0.937037i	$0.386443\pi$
$840$	0	0
$841$	30367.0	1.24511
$842$	− 11020.0i	− 0.451038i
$843$	20466.0i	0.836164i
$844$	−2420.00	−0.0986965
$845$	0	0
$846$	540.000	0.0219451
$847$	9913.00i	0.402143i
$848$	− 8832.00i	− 0.357656i
$849$	12147.0	0.491029
$850$	0	0
$851$	21900.0	0.882165
$852$	7920.00i	0.318468i
$853$	− 24937.0i	− 1.00097i	−0.865745	−	0.500485i	$-0.833155\pi$
$853$	− 24937.0i	− 1.00097i	0.865745	−	0.500485i	$-0.166845\pi$
$854$	−12650.0	−0.506878
$855$	0	0
$856$	−11328.0	−0.452317
$857$	− 15756.0i	− 0.628022i	−0.949419	−	0.314011i	$-0.898327\pi$
$857$	− 15756.0i	− 0.628022i	0.949419	−	0.314011i	$-0.101673\pi$
$858$	5220.00i	0.207701i
$859$	−38144.0	−1.51508	−0.757542	−	0.652787i	$-0.773600\pi$
$859$	−38144.0	−1.51508	−0.757542	+	0.652787i	$0.773600\pi$
$860$	0	0
$861$	10764.0	0.426058
$862$	− 22140.0i	− 0.874816i
$863$	5448.00i	0.214892i	0.994211	+	0.107446i	$0.0342673\pi$
$863$	5448.00i	0.214892i	−0.994211	+	0.107446i	$0.965733\pi$
$864$	864.000	0.0340207
$865$	0	0
$866$	−24266.0	−0.952185
$867$	3513.00i	0.137610i
$868$	− 19964.0i	− 0.780671i
$869$	29760.0	1.16172
$870$	0	0
$871$	23287.0	0.905913
$872$	440.000i	0.0170875i
$873$	− 2871.00i	− 0.111304i
$874$	−44700.0	−1.72998
$875$	0	0
$876$	7752.00	0.298991
$877$	− 21191.0i	− 0.815928i	−0.912998	−	0.407964i	$-0.866239\pi$
$877$	− 21191.0i	− 0.815928i	0.912998	−	0.407964i	$-0.133761\pi$
$878$	− 3746.00i	− 0.143988i
$879$	6714.00	0.257631
$880$	0	0
$881$	18216.0	0.696609	0.348305	−	0.937381i	$-0.386758\pi$
$881$	18216.0	0.696609	0.348305	+	0.937381i	$0.386758\pi$
$882$	− 3348.00i	− 0.127815i
$883$	12767.0i	0.486573i	0.969955	+	0.243286i	$0.0782255\pi$
$883$	12767.0i	0.486573i	−0.969955	+	0.243286i	$0.921775\pi$
$884$	−9048.00	−0.344250
$885$	0	0
$886$	−1152.00	−0.0436819
$887$	− 11010.0i	− 0.416775i	−0.978046	−	0.208388i	$-0.933178\pi$
$887$	− 11010.0i	− 0.416775i	0.978046	−	0.208388i	$-0.0668215\pi$
$888$	− 3504.00i	− 0.132417i
$889$	−29440.0	−1.11067
$890$	0	0
$891$	−2430.00	−0.0913671
$892$	580.000i	0.0217711i
$893$	4470.00i	0.167506i
$894$	−4644.00	−0.173734
$895$	0	0
$896$	2944.00	0.109768
$897$	13050.0i	0.485760i
$898$	− 9768.00i	− 0.362987i
$899$	−50778.0	−1.88381
$900$	0	0
$901$	−43056.0	−1.59201
$902$	− 9360.00i	− 0.345514i
$903$	29877.0i	1.10105i
$904$	−11232.0	−0.413242
$905$	0	0
$906$	−1758.00	−0.0644654
$907$	− 22772.0i	− 0.833662i	−0.908984	−	0.416831i	$-0.863141\pi$
$907$	− 22772.0i	− 0.833662i	0.908984	−	0.416831i	$-0.136859\pi$
$908$	11856.0i	0.433321i
$909$	7128.00	0.260089
$910$	0	0
$911$	29802.0	1.08385	0.541923	−	0.840428i	$-0.317696\pi$
$911$	29802.0	1.08385	0.541923	+	0.840428i	$0.317696\pi$
$912$	7152.00i	0.259678i
$913$	25380.0i	0.919995i
$914$	31604.0	1.14373
$915$	0	0
$916$	−22540.0	−0.813038
$917$	− 11040.0i	− 0.397571i
$918$	− 4212.00i	− 0.151434i
$919$	−48941.0	−1.75671	−0.878354	−	0.478011i	$-0.841358\pi$
$919$	−48941.0	−1.75671	−0.878354	+	0.478011i	$0.841358\pi$
$920$	0	0
$921$	−4155.00	−0.148656
$922$	30720.0i	1.09730i
$923$	19140.0i	0.682558i
$924$	−8280.00	−0.294797
$925$	0	0
$926$	3424.00	0.121511
$927$	− 7308.00i	− 0.258928i
$928$	− 7488.00i	− 0.264877i
$929$	−31026.0	−1.09573	−0.547863	−	0.836568i	$-0.684559\pi$
$929$	−31026.0	−1.09573	−0.547863	+	0.836568i	$0.684559\pi$
$930$	0	0
$931$	27714.0	0.975607
$932$	− 16656.0i	− 0.585392i
$933$	17010.0i	0.596873i
$934$	32556.0	1.14054
$935$	0	0
$936$	2088.00	0.0729150
$937$	− 11183.0i	− 0.389896i	−0.980814	−	0.194948i	$-0.937546\pi$
$937$	− 11183.0i	− 0.389896i	0.980814	−	0.194948i	$-0.0624538\pi$
$938$	36938.0i	1.28579i
$939$	−1263.00	−0.0438940
$940$	0	0
$941$	−2562.00	−0.0887554	−0.0443777	−	0.999015i	$-0.514130\pi$
$941$	−2562.00	−0.0887554	−0.0443777	+	0.999015i	$0.514130\pi$
$942$	− 10374.0i	− 0.358814i
$943$	− 23400.0i	− 0.808069i
$944$	4320.00	0.148945
$945$	0	0
$946$	25980.0	0.892899
$947$	− 7638.00i	− 0.262093i	−0.991376	−	0.131046i	$-0.958166\pi$
$947$	− 7638.00i	− 0.262093i	0.991376	−	0.131046i	$-0.0418336\pi$
$948$	− 11904.0i	− 0.407831i
$949$	18734.0	0.640813
$950$	0	0
$951$	−29952.0	−1.02130
$952$	− 14352.0i	− 0.488604i
$953$	51432.0i	1.74821i	0.485735	+	0.874106i	$0.338552\pi$
$953$	51432.0i	1.74821i	−0.485735	+	0.874106i	$0.661448\pi$
$954$	9936.00	0.337201
$955$	0	0
$956$	−7776.00	−0.263069
$957$	21060.0i	0.711362i
$958$	− 29532.0i	− 0.995966i
$959$	6486.00	0.218398
$960$	0	0
$961$	17298.0	0.580645
$962$	− 8468.00i	− 0.283804i
$963$	− 12744.0i	− 0.426448i
$964$	−3428.00	−0.114532
$965$	0	0
$966$	−20700.0	−0.689453
$967$	− 39728.0i	− 1.32116i	−0.750754	−	0.660582i	$-0.770309\pi$
$967$	− 39728.0i	− 1.32116i	0.750754	−	0.660582i	$-0.229691\pi$
$968$	− 3448.00i	− 0.114486i
$969$	34866.0	1.15589
$970$	0	0
$971$	−47946.0	−1.58461	−0.792307	−	0.610123i	$-0.791120\pi$
$971$	−47946.0	−1.58461	−0.792307	+	0.610123i	$0.791120\pi$
$972$	972.000i	0.0320750i
$973$	36892.0i	1.21552i
$974$	6638.00	0.218373
$975$	0	0
$976$	4400.00	0.144304
$977$	22326.0i	0.731087i	0.930794	+	0.365544i	$0.119117\pi$
$977$	22326.0i	0.731087i	−0.930794	+	0.365544i	$0.880883\pi$
$978$	6738.00i	0.220304i
$979$	−44640.0	−1.45730
$980$	0	0
$981$	−495.000	−0.0161102
$982$	− 22128.0i	− 0.719076i
$983$	− 48468.0i	− 1.57262i	−0.617830	−	0.786312i	$-0.711988\pi$
$983$	− 48468.0i	− 1.57262i	0.617830	−	0.786312i	$-0.288012\pi$
$984$	−3744.00	−0.121295
$985$	0	0
$986$	−36504.0	−1.17903
$987$	2070.00i	0.0667567i
$988$	17284.0i	0.556556i
$989$	64950.0	2.08826
$990$	0	0
$991$	−25141.0	−0.805883	−0.402942	−	0.915226i	$-0.632012\pi$
$991$	−25141.0	−0.805883	−0.402942	+	0.915226i	$0.632012\pi$
$992$	6944.00i	0.222250i
$993$	12684.0i	0.405352i
$994$	−30360.0	−0.968773
$995$	0	0
$996$	10152.0	0.322970
$997$	35422.0i	1.12520i	0.826729	+	0.562601i	$0.190199\pi$
$997$	35422.0i	1.12520i	−0.826729	+	0.562601i	$0.809801\pi$
$998$	− 28262.0i	− 0.896411i
$999$	3942.00	0.124844

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.4.c.b.49.1		2
3.2	odd	2		450.4.c.h.199.2		2
4.3	odd	2		1200.4.f.q.49.2		2
5.2	odd	4		150.4.a.g.1.1	yes	1
5.3	odd	4		150.4.a.c.1.1	✓	1
5.4	even	2	inner	150.4.c.b.49.2		2
15.2	even	4		450.4.a.i.1.1		1
15.8	even	4		450.4.a.l.1.1		1
15.14	odd	2		450.4.c.h.199.1		2
20.3	even	4		1200.4.a.r.1.1		1
20.7	even	4		1200.4.a.v.1.1		1
20.19	odd	2		1200.4.f.q.49.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.4.a.c.1.1	✓	1	5.3	odd	4
150.4.a.g.1.1	yes	1	5.2	odd	4
150.4.c.b.49.1		2	1.1	even	1	trivial
150.4.c.b.49.2		2	5.4	even	2	inner
450.4.a.i.1.1		1	15.2	even	4
450.4.a.l.1.1		1	15.8	even	4
450.4.c.h.199.1		2	15.14	odd	2
450.4.c.h.199.2		2	3.2	odd	2
1200.4.a.r.1.1		1	20.3	even	4
1200.4.a.v.1.1		1	20.7	even	4
1200.4.f.q.49.1		2	20.19	odd	2
1200.4.f.q.49.2		2	4.3	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:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -3.00000i q^{3} -4.00000 q^{4} -6.00000 q^{6} -23.0000i q^{7} +8.00000i q^{8} -9.00000 q^{9} -30.0000 q^{11} +12.0000i q^{12} +29.0000i q^{13} -46.0000 q^{14} +16.0000 q^{16} -78.0000i q^{17} +18.0000i q^{18} -149.000 q^{19} -69.0000 q^{21} +60.0000i q^{22} +150.000i q^{23} +24.0000 q^{24} +58.0000 q^{26} +27.0000i q^{27} +92.0000i q^{28} +234.000 q^{29} -217.000 q^{31} -32.0000i q^{32} +90.0000i q^{33} -156.000 q^{34} +36.0000 q^{36} -146.000i q^{37} +298.000i q^{38} +87.0000 q^{39} -156.000 q^{41} +138.000i q^{42} -433.000i q^{43} +120.000 q^{44} +300.000 q^{46} -30.0000i q^{47} -48.0000i q^{48} -186.000 q^{49} -234.000 q^{51} -116.000i q^{52} -552.000i q^{53} +54.0000 q^{54} +184.000 q^{56} +447.000i q^{57} -468.000i q^{58} +270.000 q^{59} +275.000 q^{61} +434.000i q^{62} +207.000i q^{63} -64.0000 q^{64} +180.000 q^{66} -803.000i q^{67} +312.000i q^{68} +450.000 q^{69} +660.000 q^{71} -72.0000i q^{72} -646.000i q^{73} -292.000 q^{74} +596.000 q^{76} +690.000i q^{77} -174.000i q^{78} -992.000 q^{79} +81.0000 q^{81} +312.000i q^{82} -846.000i q^{83} +276.000 q^{84} -866.000 q^{86} -702.000i q^{87} -240.000i q^{88} +1488.00 q^{89} +667.000 q^{91} -600.000i q^{92} +651.000i q^{93} -60.0000 q^{94} -96.0000 q^{96} +319.000i q^{97} +372.000i q^{98} +270.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 12 q^{6} - 18 q^{9} - 60 q^{11} - 92 q^{14} + 32 q^{16} - 298 q^{19} - 138 q^{21} + 48 q^{24} + 116 q^{26} + 468 q^{29} - 434 q^{31} - 312 q^{34} + 72 q^{36} + 174 q^{39} - 312 q^{41} + 240 q^{44}+ \cdots + 540 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 - 12 * q^6 - 18 * q^9 - 60 * q^11 - 92 * q^14 + 32 * q^16 - 298 * q^19 - 138 * q^21 + 48 * q^24 + 116 * q^26 + 468 * q^29 - 434 * q^31 - 312 * q^34 + 72 * q^36 + 174 * q^39 - 312 * q^41 + 240 * q^44 + 600 * q^46 - 372 * q^49 - 468 * q^51 + 108 * q^54 + 368 * q^56 + 540 * q^59 + 550 * q^61 - 128 * q^64 + 360 * q^66 + 900 * q^69 + 1320 * q^71 - 584 * q^74 + 1192 * q^76 - 1984 * q^79 + 162 * q^81 + 552 * q^84 - 1732 * q^86 + 2976 * q^89 + 1334 * q^91 - 120 * q^94 - 192 * q^96 + 540 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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